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+{"text":"\\section{Introduction}\n\nThe nucleation of Abrikosov vortices \\cite{Abrikosov57} in the\nmixed state of type-II superconductors with periodic artificial\npinning centers attracted a great attention since 1970's. Recent\nprogress in the fabrication of nanostructures provides the\npossibility to realize superconducting thin films containing\nartificial defects as pinning sites with well-defined size,\ngeometry and spatial arrangement \\cite{Mosh98,Schuller00}. Vortex\npinning was extensively explored by many groups to develop a\nfundamental understanding of flux dynamics and for its relevance\nin applications which require enhancements of the critical current\ndensity. Thus, several types of artificial pinning centers, such\nas square, rectangular or triangular arrays, have been introduced\nin a controlled way in the superconducting films. In particular,\nthe use of regular array of pinning centers such as antidots\n\\cite{Mosh98,Fiory78,Lykov93,Castellanos97} or magnetic dots\n\\cite{Schuller00,Schuller97,Mosh99,Schuller08} brings to new\ncommensurability effects, which give additional insight into the\npinning properties of vortices. The most notable phenomenon for\nthese studies is the so-called matching effect which occurs when\nthe vortex lattice is commensurate with the periodic pinning\narray. This situation occurs, in particular, at fractional or\ninteger values of the so-called first matching field\n$H_1=\\Phi_0\/a_{0}^{2}$, i.e., when the applied field $H$\ncorresponds to one flux quantum, $\\Phi_0=h\/2e$, per unit cell\narea, $a_{0}^{2}$, of the pinning array. Here $a_{0}$ is the\nlattice constant of the pinning arrangement. As a result, at the\nmatching field, the critical current density, $J_c$, is\ndrastically enhanced \\cite{Schuller00,Fiory78,Silhanek05} and\nmoreover, as a consequence of the Little-Parks effect\n\\cite{Little-Parks}, the upper critical magnetic field is\nincreased at the matching values. Recently antidot arrangements\nwith a big variety of symmetries have been investigated. Matching\neffects have been reported in perforated Nb thin films for\nantidots lattices with short range order \\cite{Ziemann}, or\nquasiperiodic fivefold Penrose structures \\cite{Kemmler06}.\nMoreover asymmetric pinning arrays have been suggested as\nsuperconducting rectifiers \\cite{Mosh05}.\\\\\nIf the artificial structure of defects is created by lithographic\ntechnique, the matching fields are usually in the range of a few\noersteds. For this reason, matching effects are observed in a very\nnarrow temperature region, close to the critical temperature\n$T_c$, for a reduced value $t=T\/T_c\\geq0.95$. In order to both\nincrease the matching field and decrease the temperature where the\neffect is present, the period of the pinning structure should be\nreduced to less than 100 nm. This gives, in fact, the possibility\nto increase $H_1$ up to 1 tesla or even higher. A reasonable\nmethod to achieve this goal is to use self-assembled substrates,\nsuch as, for example, $Al_{2}O_{3}$ templates with characteristic\nfeatures in the nanometric scale \\cite{Mosh07}. The pore diameter\nin $Al_{2}O_{3}$ substrates could easily be varied in the range\n25-200 nm with porosity (i.e. interpore spacing) around 50\\%, and\nthis gives the possibility to achieve matching fields of thousands\nof oersteds \\cite{Mosh07}. To prepare $Al_{2}O_{3}$ substrates\nbulk Al \\cite{Prischepa-Cryogenics94}, Al foils \\cite{Welp-PRB},\nand deposited thick Al films were used \\cite{Mosh07}.\\\\\nVery recently, another very promising material for self-assembled\nsubstrates and an optimum candidate for the Nb growth was\nproposed, namely, porous silicon (PS) \\cite{Trezza-JAP08}. PS is\nconstituted by a network of pores immersed in a nanocrystalline\nmatrix \\cite{Pavesi} and it is a material which offers a\nconsiderable technological interest in different fields, as for\ninstance micro and optoelectronics \\cite{Collins} and gas sensing\n\\cite{Cheraga,Lysenko}. The diameter of pores, ${\\O}$, in PS can\neasily be varied from 200 nm down to 5 nm by using substrates with\nappropriate doping (n or p) and different regimes of anodization.\nThe porosity, in fact, can be varied in the range 30-90\\% by\nadjusting parameters such as the acid solution, the anodizing\ncurrent density and the illumination of the substrate during the\nanodization. The regularity of the pores arrangement, however, is\nof the order of 10\\% lower than the one observed in $Al_{2}O_{3}$\ntemplates obtained by electrochemical oxidation \\cite{Piraux}. It\nhas been demonstrated \\cite{Trezza-JAP08} that thin Nb films\ndeposited on PS substrates can inherit their structure. The\nresulting samples then consist of porous Nb thin films with in\nplane geometrical dimensions, $a_{0}$ and ${\\O}$, comparable with\nthe superconducting coherence length, $\\xi(T)$. In these samples,\nmatching fields of the order of 1 Tesla\nwere experimentally observed \\cite{Trezza-JAP08}.\\\\\nAim of this work is to deepen the study of the matching effect in\nsuperconducting Nb thin films deposited on PS. Superconducting\nproperties  were investigated by transport measurements in the\npresence of magnetic fields applied perpendicularly to the samples\nsurface, down to t = 0.52. As a consequence of the high density of\nthe pore network, the (H,T) phase diagram presents a deviation\nfrom the classic linear dependence. This effect appears at the\nmatching field $H_{1}\\approx$ 1 Tesla, a value larger than those\ntypical of periodic pinning arrays obtained both by lithographic\ntechniques and by using another kind of self-organized templates.\nMoreover a new effect related to the commensurability between the\nvortex lattice and the underlaying pinning structure was found. It\nconsists in the appearance of pronounced structures in the\nderivative of the $R(H)$ curves, $dR$\/$dH$, which can be observed\nin correspondence of the first matching field and its fractional\nvalues.\n\n\\section{Fabrication}\n\nPorous layers were fabricated by electrochemical anodic etching of\nn-type, 0.01 $\\Omega$cm, monocrystalline silicon wafers. The\nelectrochemical dissolution was performed in 48\\% water solution\nof HF, applying a current density of 20 mA\/cm$^{2}$. The\nanodization time was chosen in the range of 0.5 - 4 min in order\nto get porous layers with a thickness ranging from 0.5 to 4\n$\\mu$m. The pores extend on a surface of about 1 cm$^{2}$. The\nintegral porosity was estimated by gravimetry to be of about 50\\%\n\\cite{Lazarouk}. The resulting porous substrates have ${\\O}$=10 nm\nand $a_{0}$ = 40 nm. For this lattice, if the formula\n$H_1=\\Phi_0\/a_{0}^{2}$ for the square lattice is used, the\nexpected first matching field is $H_{1}$ = 1.3 Tesla.\\\\\nNb thin films were grown on top of the porous Si substrates in a\nUHV dc diode magnetron sputtering system with a base pressure in\nthe low $10^{-8}$ mbar regime and sputtering Argon pressure of\n$3.5\\times10^{-3}$ mbar. In order to reduce the possible\ncontamination of the porous templates, the substrates were heated\nat $120^{\\circ}$C for one hour in the UHV chamber. The deposition\nwas then realized at room temperature after the cool off of the\nsubstrates. Films were deposited at typical rates of 0.33 nm\/s,\ncontrolled by a quartz crystal monitor calibrated by low-angle\nreflectivity measurements. Since the effect of the periodic\ntemplate would be reduced when the film thickness, $d_{Nb}$,\nexceeds the pore diameter, ${\\O}$, \\cite{Trezza-JAP08} the Nb\nthickness was chosen to be 8.5 nm for the sample analyzed in this\npaper. A reference Nb thin film of the same thickness was grown on\na non-porous Si substrate in the same deposition run.\n\n\\section{Experimental results and discussion}\n\nThe superconducting properties were resistively measured in a\n$^{4}$He cryostat using a standard dc four-probe technique on\nunstructured samples. The critical temperature was defined at the\nmidpoint of the $R(T)$ transition curves. The value of the\ntransition temperatures of the film grown on the porous substrate\nand of the reference sample in the absence of the magnetic field\nwere $T_c$ = 3.83 K and $T_c$ = 4.53 K, respectively. The critical\ntemperature depression in the case of the porous sample is\nconsistent with what already reported in literature for  films\ngrown both on $Al_{2}O_{3}$ \\cite{Mosh07} and on PS\n\\cite{Trezza-JAP08}. The first step for the characterization of\nthe behavior of the porous Nb sample in the presence of\nperpendicular magnetic field is the determination of its ($H$,$T$)\nphase diagram. The temperature dependence of the perpendicular\nupper critical field, $H_{c2\\bot}$, was obtained performing\nresistance vs. field, $R(H)$, measurements at fixed values of the\ntemperature with a temperature stability of 1 mK.\n$H_{c2\\bot}$ was defined at the midpoint of each of the $R(H)$ curves.\\\\\nIn Fig. \\ref{Fig.1} the ($H$,$T$) phase diagrams of the Nb thin\nfilms are shown. In general, the perpendicular upper critical\nfield of superconducting films of thickness $\\textit{d}$ obeys a\nlinear temperature dependence, $H_{c2\\bot}(T)$ =\n($\\Phi_{0}$\/2$\\pi\\xi_{0\\parallel}^{2}$)(1-$T$\/$T_{c}$)\n\\cite{Tinkham}. $\\xi_{0\\parallel}$ is the Ginzburg-Landau\ncoherence length parallel to the sample surface at $T$ = 0. The\ntemperature dependence of $\\xi_{\\parallel}$ is\n$\\xi_{\\parallel}(T)$ = $\\xi_{0\\parallel}$\/$\\sqrt{1-T\/T_{c}}$.\nAnother superconducting parameter to be taken into account is the\nmagnetic field penetration depth, $\\lambda$, whose temperature\ndependence is $\\lambda(T)$ = $\\lambda_{0}$\/$\\sqrt{1-T\/T_{c}}$,\nwhere $\\lambda_{0}$ is the penetration depth at $T$ =0.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig1.eps} \\caption{Left scale: Perpendicular upper\ncritical field $H_{c2\\bot}$ vs. temperature of the Nb thin film\nwith $d_{Nb}$ = 8.5 nm grown on (a) porous template and (b)\nnon-porous reference substrate. The linear fits to the data close\nto $T_{c}$ are also shown. Right scale: $dH_{c2\\bot}^{2}$\/$dT^{2}$\nversus temperature. The inset shows the comparison between the\nsecond derivatives as functions of the reduced temperature of two\nsamples, grown on the porous template (full circles) and on the\nnon-porous template (open circles). (Color online).} \\label{Fig.1}\n\\end{figure}\n\nThe $H_{c2\\bot}(T)$ curve obtained for the Nb film deposited on\nporous Si template, reported in Fig. \\ref{Fig.1}(a), presents some\npeculiarities, which indicate that the superconducting properties\nare influenced by the introduction of the porous array. In fact,\nif the $H_{c2\\bot}$ second derivative versus the temperature is\nplotted we can see that it changes its sign from positive to\nnegative at $H \\approx$ 1.16 Tesla. This field value is very close\nto the nominal first matching field that we expect for the porous\nSi template, $H_{1}\\approx$ 1.30 Tesla, assuming a square porous\narray. This change in concavity was already reported in a previous\nstudy on the same kind of samples, and it was ascribed to the\nformation of a commensurate vortex structure \\cite{Trezza-JAP08}.\nFrom the measured value of $H_{1}$ it follows that the period of\nthe porous template is $a$ = 42 nm. In the following we will\nidentify $a_{0}$ $\\equiv$ 42 nm. In Fig. \\ref{Fig.1}(b) is\nreported the $H_{c2\\bot}(T)$ curve for the Nb reference film of\nthe same thickness deposited on the non-porous template. As\nexpected the $H_{c2\\bot}(T)$ behavior is linear over the all\ntemperature range and the $H_{c2\\bot}$ second derivative versus\ntemperature does not present any peculiarity except for a shallow\npeak near $T_c$. In the inset of Fig. \\ref{Fig.1}(b), for sake of\ncomparison, the $dH_{c2\\bot}^{2}$\/$dT^{2}$ versus the reduced\ntemperature is reported for both the Nb films, in order to point\nout the difference in their magnitude. A fit to the data close to\n$T_{c}$ with the expression for $H_{c2\\bot}(T)$ reported above,\nyields a value of the Ginzburg-Landau coherence length at $T$ = 0,\n$\\xi_{0\\parallel}$ = 9.1 nm and $\\xi_{0\\parallel}$ = 9.5 nm,\nresulting in a superconducting coherence length $\\xi_{S}$ = 5.8 nm\nand $\\xi_{S}$ = 6.0 nm, for the Nb porous sample and the Nb\nreference film, respectively. The values of $\\xi_{0\\parallel}$ are\nsignificantly smaller than the BCS coherence length of Nb,\n$\\xi_{0}$ = 39 nm \\cite{Buckel}, indicating that our films are in\ndirty limit regime with an electron mean free path of $\\textit{l}$\n= 1.38 $\\xi_{0\\parallel}^{2}$ \/ $\\xi_{0}$ $\\approx$ 3 nm\n\\cite{Schmidt}. Since the film dimensions in the $xy$ plane are\nlarger than $\\xi_{\\parallel}(T)$, the expression for\n$H_{c2\\bot}(T)$, reported above, is verified in the whole\ntemperature range. The Ginzburg-Landau parameter, $\\kappa$ =\n$\\lambda$(0)\/$\\xi_{0\\parallel}$, can be estimated using the\nexpression $\\kappa$ = 0.72$\\lambda_{L}$\/$\\textit{l}$ = 9.6, where\n$\\lambda_{L}$ = 39 nm is the London penetration depth of Nb\n\\cite{Buckel}. Ratios of $\\xi_{0\\parallel}$\/$\\textit{a} \\approx$\n0.2 and $\\lambda$(0)\/$\\textit{a} \\approx$ 2.1, measured for\n$a_{0}$ = 42 nm, are larger than in previous works\n\\cite{Welp-PRB,Mosh06} on perforated Nb samples, and indicate that\nwe are in presence of individual vortex pinning \\cite{Brandt}.\nMoreover, the pore diameter, ${\\O}$, in our PS template is\ncomparable with the vortex core dimension at $T$=0, ${\\O} \\approx\n\\xi_{0\\parallel}$. This means that the saturation number, $n_{S}$\n= $\\frac{\\O}{2\\xi_{S}(T)}$, defined as the maximum number of\nvortices that fits into a pore with diameter ${\\O}$, is less or\nequal to 1, so that each pore can trap only one fluxon\n\\cite{Mkrtchyan}. Subsequently multiquanta vortex lattice\n\\cite{Mosh98} cannot be observed in our system.\\\\\nNow we move to a more careful inspection of the $R(H)$ curves of\nthe Nb porous film. This will lead to the observation of a\npeculiar behavior of these transitions, whose analysis represents\nthe main subject of this work. In Figs. \\ref{Fig.2}(a) and 2(b)\n$R(H)$ curves obtained for two different values of the\ntemperature, $T$ = 3.490 K and $T$ = 3.531 K, respectively, are\npresented.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig2.eps} \\caption{Left scale: $R(H)$ measurement\nat (a) $T$ = 3.490 K and (b) $T$ = 3.531 K. Right scale: $dR$\/$dH$\nversus the applied magnetic field. In both panels the arrow\nindicates the field where the bump is present. (Color online)}\n\\label{Fig.2}\n\\end{figure}\n\n\\noindent At first glance both the curves are rather smooth and do\nnot present any structures or enlargements due, for example, to\nsample inhomogeneities. However if the dependence of the first\nderivative $dR$\/$dH$ versus the applied magnetic field is\nanalyzed, some distinct features can be observed. In particular in\nboth the curves a small local maximum is present at specific\nvalues of the magnetic field. Let's focus on the position where\nthe bumps, as indicated by an arrow in Fig. \\ref{Fig.2}, start to\ndevelop. The bumps in the first derivative reflects the presence\nof a small dip in the corresponding magnetic field dependence of\nthe resistance $R(H)$ at the same value of $H$. This effect was\nascribed to a pinning enhancement when the period of the vortex\nstructure is commensurate with the period of the antidots\n\\cite{Patel}. The bumps in the $dR$\/$dH$ appear indeed in our\ncurves at values of the magnetic fields $H_{n}$ when the magnetic\nflux threading each unit cell is equal to the flux quantum,\n$\\Phi_0$, or to fractional values of $\\Phi_0$. In Fig.\n\\ref{Fig.2}(a), where the $R(H)$ measurement at $T$ = 3.490 K is\nshown, the peculiarity in $dR$\/$dH$ is, in fact, observed at\n$H_{bump}\\approx$ 0.126 Tesla. The period of the vortex lattice at\nthis field value is $a'$ = 128 nm, i.e. about three times the\ninterpore spacing of this analyzed sample, $a_{0}$ = 42 nm.\nConsequently this field value corresponds to one-ninth of the\nmatching field $H_{1}$\/9 $\\approx$ 0.129 Tesla. Similarly, in Fig.\n\\ref{Fig.2}(b) where the $R(H)$ measurement at $T$ = 3.531 K is\nshown, the bump in $dR$\/$dH$ develops at $H_{bump}\\approx$ 0.065\nTesla. The period of the vortex array at this field is then $a''$\n= 178 nm, which is about four times the interpore spacing of this\nsample. Consequently this field value corresponds to one-sixteenth\nof the matching field $H_{1}$\/16 $\\approx$ 0.072 Tesla. An\nadditional bump structure is present at $H\\approx$ 0.2 Tesla.\nHowever, this field value does not correspond to any commensurate\nvortex configuration (see discussion below) and does not survive\nrepeating the measurement in the same temperature range. Many\n$R(H)$ measurements at different temperatures have been performed\nand the behavior of all the corresponding $dR$\/$dH$ curves has\nbeen analyzed. A selection of these curves is reported in Fig.\n\\ref{Fig.3}. Some of them have been obtained by sweeping the field\nupward and downward and no hysteresis has been detected.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3.eps} \\caption{First derivatives, $dR$\/$dH$, as a\nfunction of the applied magnetic field at different temperatures.\nThe arrows indicate the field where the bump is present for each\ntemperature. (Color online)} \\label{Fig.3}\n\\end{figure}\n\nFor instance, the curves at $T$ = 2.551 K and $T$ = 3.304 K\npresent a bump at $H_{bump}$ = $H_{1}$ and $H_{bump}$ = $H_{1}$\/4,\nrespectively. By comparison a curve with no bump, measured at\ntemperature $T$ = 2.805 K, is also shown. In all curves the fields\nat which the bumps are observed are related to the first matching\nfield through the relation: $H$ = $H_{1}$\/$n^{2}$ with $n$ =\n1,...,4. The temperatures at which bumps are observed, the\ncorresponding fields and their values normalized to $H_{1}$, the\n$\\xi_{S}$ values, the vortex-vortex distances, $a$, and their\nvalues normalized to $a_{0}$, are summarized in Table \\ref{table}.\n\n\n\\begin{table}\n\\caption{Temperatures at which the bumps are observed,\ncorresponding fields and their values normalized to $H_{1}$,\n$\\xi_{S}$ values at that temperature, vortex-vortex distances,\n$a_{k\/l}$, and their values normalized to $a_{0}$ = 42 nm.}\n\\label{table}\n\\begin{center}\n\\begin{tabular}{cccccc}\n$T$(K)  & $H_{bump}$(T) & $\\frac{H_{bump}}{H_{1}}$ & $\\xi_{S}$(nm) & $a_{k\/l}$(nm) & $\\frac{a_{k\/l}}{a_{0}}$ \\\\\n\\hline\n2.551 & 1.160  & 1 & 10.02  & 42.0  & 1.00  \\\\\n\n3.304  & 0.275  & 1\/4 & 15.66  & 87.0  &  2.07  \\\\\n\n3.490  & 0.126  &  1\/9 & 19.44  & 128.0  & 3.05  \\\\\n\n3.531  & 0.065  & 1\/16 & 20.76  & 178.0  & 4.24  \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe argue that the presence of the observed bumps in the $dR$\/$dH$\ncurves can be related to different vortex lattice arrangements\nmade possible by the lattice of holes. The specific vortex lattice\nconfigurations occurring at the first matching field and at its\nfractional values are shown in Fig. \\ref{Fig.4}.\n\n\\begin{figure}\n\\includegraphics[width=7cm]{fig4.eps} \\caption{Vortex lattice configurations\noccurring at the first matching field and its fractional values.\nIncreasing the temperature the vortices diameter and their\nreciprocal distance increase, as reported in Table \\ref{table}.\nBlue circles represent the holes, pink ones represent the\nvortices. (Color online)} \\label{Fig.4}\n\\end{figure}\n\n\\noindent In the case of $H_{bump}$\/$H_{1}$ = 1 a commensurate\nsquare vortex configuration is formed, where each pore is occupied\nby a fluxon and the side of this square array is just $a_{0}$ = 42\nnm. Increasing the temperature the vortices diameter\n($\\approx$2$\\xi_{S}$) and their reciprocal distance increase, as\nreported in Table \\ref{table}. When $H_{bump}$\/$H_{1}$ = 1\/4, 1\/9\nand 1\/16 a square vortex lattice is again obtained with $a$ = 87\nnm, 128 nm and 178 nm, respectively. This means that the pores act\nas an ordered template of strong pinning centers, which is able to\npreserve the long range positional order of the flux lattice also\nat low fields value, i.e. at higher vortex spacing. As already\npointed out the optimization of the vortex structures leads to the\nformation of larger square flux lattices with respect to the\nunderlying artificial pinning array with the lattice constant $a$\nexactly equal to $na_{0}$. The vortices tend to be placed as far\nfrom each other as possible due to the repulsive interaction\nbetween them and at the same time they want to follow the imposed\nsquare potential induced by the antidots. This constraint gives\n$a$ = $a_{0} \\sqrt{l^{2}+k^{2}}$, where $l$ and $k$ are integer\nnumbers. Therefore, we should expect the fractional matching\nfields at $H$ = $H_{k\/l}$ = $\\Phi_0\/a^{2}$ =\n$\\Phi_0\/[a_{0}^{2}(l^{2}+k^{2})]$ = $H_{1}\/(l^{2}+k^{2})$\n\\cite{Mosh95}. We observed bumps at fractional matching fields\n$H_{0\/2}$, $H_{0\/3}$ and $H_{0\/4}$. The other bumps expected from\nthe equation above at fractional fields $H_{k\/l}$ with $k \\neq$ 0\nhave not been observed. All the fields values at which the bumps\nin the $dR$\/$dH$ appear are shown as points of coordinates\n($H_{bump}$,$T$) in Fig. \\ref{Fig.5}.\n\n\\begin{figure}\n\\includegraphics[width=8.8cm]{fig5.eps} \\caption{The points of coordinates\n($H_{bump}$,$T$) identify the values of the fields and\ntemperatures at which bumps have been observed in the $dR$\/$dH$\ncurves at fixed temperatures. The solid lines correspond to the\ndifferent matching field orders achieved with the interpore\nspacing $a_{0}$ = 42 nm, while the dotted lines are obtained\nforasmuch as the regularity of the pore distance is achieved\nwithin the 10 percents of the average distance. (Color online)}\n\\label{Fig.5}\n\\end{figure}\n\n\\noindent In this figure the solid lines correspond to the\nmatching fields of different order, as calculated assuming an\ninterpore spacing $a_{0}$ = 42 nm, through the formula\n$H_1=\\Phi_0\/a_{0}^{2}$.  The dotted lines are obtained considering\na deviation from the corresponding mean interpore distance of the\norder of 10\\% \\cite{Trezza-JAP08}. It is worth noticing that all\nthe data fall into the range theoretically estimated, suggesting\nthat the observed peculiarities in the $R(H)$ curves can be indeed\nascribed to commensurability effect between the porous structure\nof the Nb film and the vortex lattice. The distribution of the\nexperimental points is consistent with the observation that a\ncertain temperature dependence of the matching effect can be found\nfor the case of short-range ordered templates \\cite{Ziemann}. We\nwould also point out that the effect is observable in our sample\nonly up to $H$ = $H_{1}$, due to the very high value of the first\nmatching field. The second matching field in fact is $H_{2}$ =\n2$H_{1}$ = 2.32 Tesla. From a linear extrapolation of the\n$H_{c2\\bot}$ curve, it follows that in order to see at this field\na bump in the $dR$\/$dH$ we should measure a $R(H)$ curve at $T$ =\n1.73 K, temperature which cannot be reached in our experimental\nsetup. All the field values reported above have been calculated\nassuming a square lattice. The measured field values do not match\nwith the ones calculated if a triangular array for the pores is\nconsidered. In fact, at $T$ = 3.490 K (see Fig. \\ref{Fig.2}(a))\nthe structure in the $dR$\/$dH$ curve for a triangular lattice\nwould have been observed at a field $2\/\\sqrt{3}$ times higher than\n$H_{0\/3}$ = 0.126 Tesla, where no peculiar feature has been\ndetected. This supports our assumption of considering a square\nlattice of holes in our system.\n\n\\section{Conclusions}\n\nMatching effects have been reported for Nb thin film grown on\nporous silicon. Due to the extremely reduced values of the\ninterpore distance the effect is present at fields values higher\nthan 1 Tesla and down to reduced temperatures as low as t $\\simeq$\n0.52. The commensurability manifests both in the ($H$,$T$) phase\ndiagram and in the $R(H)$ transitions. The latter in particular\nreveal the formation of fractional matching states. As it was\nargued in many works the vortex configuration at fractional\nmatching fields are characterized by striking domain structure and\nassociated grain boundaries \\cite{Field02,Mosh03}. The presence of\nmultiple degenerate states with domain formation at the fractional\nfield, directly observed with scanning Hall probe microscopy\n\\cite{Field02}, seems to be high probable in our films. The\nreduced regularity of our templates, in fact, could be compensated\nby the formation of domain walls of different complexity. The\nparticular domain configuration is of course a matter of energy\nbalance between the cost in energy for the wall formation and the\nenergy gain due to the vortex pinning.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nBosonic sigma models with the metric action functional \npossess rigid symmetries induced by isometries\nof their target space. Such rigid symmetries may be gauged\nby the minimal coupling to the gauge fields of the isometry\ngroup. The gauged action is then invariant under arbitrary \nlocal gauge transformations. The minimal coupling does not work, \nhowever, for the topological Wess-Zumino term in the action \nfunctional of the sigma model, if such is present. In particular,\nit was shown in \\cite{JJMO,HS} for the two-dimensional sigma model \nwith the Wess-Zumino term corresponding to a closed $3$-form $H$ \non the target space that the gauging of rigid symmetries \nrequires satisfying certain conditions. Such conditions assure\nthe absence of local gauge anomalies and guarantee the existence\nof a gauging procedure that results in an action functional invariant \nunder infinitesimal local gauge transformations.  \nThe infinitesimal gauge invariance of the gauged action implies \nits invariance under all ``small'' local gauge transformations, \ni.e. the ones that are homotopic to unity. As was observed in\n\\cite{GSW}, it is possible, however, that the gauged action\nexhibits global gauge anomalies that lead to its non-invariance \nunder some ``large'' local gauge transformations non-homotopic \nto unity. The phenomenon was analyzed in detail for sigma models \non closed worldsheets in \\cite{GSW} and on worldsheets\nwith boundaries and defects in \\cite{GSW1}. In the case of Wess-Zumino-Witten \n(WZW) models of conformal field theory with Lie group $G=\\tilde G\/Z$ as the\ntarget, where $\\tilde G$ is the universal covering group of $G$ and $Z$ \nis a subgroup of the center $\\tilde Z$ of $\\tilde G$, \nwith the Wess-Zumino term corresponding to the bi-invariant \nclosed 3-form $H_k=\\frac{k}{12\\pi}\\,{\\rm tr}\\hspace{0.03cm}(g^{-1}dg)^3$, \nthe local gauge anomalies are absent for a restricted class\nof rigid symmetries. These include the symmetries induced \nby the adjoint action $g\\mapsto hgh^{-1}$ on $G$ for \n$h\\in\\tilde G\/\\tilde Z$, or by its twisted versions \n$g\\mapsto hg\\hspace{0.03cm}\\omega(h)^{-1}$, for \n$h\\in\\tilde G\/Z^\\omega$, where $\\omega$ is an automorphism\nof $\\tilde G$ and $Z^\\omega=\\{z\\in\\tilde Z\\,|\\,z\\hspace{0.02cm}\n\\omega(z)^{-1}\\in Z\\}$\nis the subgroup of elements in $\\tilde Z$ that acts trivially. \nIn these cases, the global gauge anomalies may occur for the target \ngroups $G$ that are not simply connected (corresponding to the so called \nnon-diagonal WZW models). They are detected by a cohomology class \n$\\varphi\\in H^2(\\tilde G\/Z^\\omega\\times G,U(1))$ that can be easily \ncomputed. Class $\\varphi$ is invariant under the \naction of $\\gamma\\in\\tilde G\/Z^\\omega$ on $\\tilde G\/\\tilde Z^\\omega\n\\times G$ given by \n\\begin{equation}\n(h,g)\\ \\mapsto\\ (\\gamma h\\gamma^{-1},\\gamma g\\hspace{0.04cm}\n\\omega(\\gamma)^{-1})\\,.\n\\label{ac}\n\\end{equation} \nThe simplest case when the anomaly class is nontrivial corresponds \nto $G=SU(3)\/\\mathbb{Z}_3$ at level $k=1$ or to $G=SU(4)\/\\mathbb{Z}_4$ \nat level $k=2$, both with $\\omega={I\\hspace{-0.04cm}d}$. Some other \ncases with global gauge anomalies for $\\omega={I\\hspace{-0.04cm}d}$ \nwere cited in \\cite{GSW}. In Sec.\\,\\ref{sec:3} of the present paper, \nwe obtain the full list of connected compact simple target groups $G$ \nfor which the WZW model with the gauged (twisted) adjoint action of \n$\\tilde G\/\\tilde Z^\\omega$ exhibits global gauge anomalies. In the twisted \ncase, we consider only outer automorphisms $\\omega$ since for \ninner automorphisms the twisted adjoint action may be \nreduced to the untwisted one by conjugating it with a right \ntranslation on $G$ which is a rigid symmetry of the WZW theory.\nThe classes of outer automorphisms of $\\tilde G$ modulo inner automorphisms\nare generated by automorphisms of the Lie algebra $\\mathfrak{g}$ that\npreserve the set of simple roots inducing a symmetry of the Dynkin \ndiagram of $\\mathfrak{g}$.\nGlobal gauge anomalies occur only for (non-simply connected) \ngroups $G$ with Lie algebras $\\mathfrak{g}=A_r,\\,D_r,\\,\\mathfrak{e}_6$ \nin the Cartan classification of simple Lie algebras\\footnote{We consider \nthe compact real forms $\\mathfrak{g}$ of complex simple Lie algebras that \nare in one-to-one correspondence with their complexifications \n$\\mathfrak{g}^{\\mathbb C}$.}.  \n\\vskip 0.1cm\n\nGauged WZW models serve to construct coset $G\/H$ models \\cite{GKO,Godd} \nof the two-dimensional conformal field theory \\cite{BRS,GK0,GK,KPSY}. \nIn such models, \none restricts the gauging to the (possibly twisted) adjoint action on \nthe target group of the subgroup \n$\\Gamma=\\tilde H\/(Z^\\omega\\cap\\tilde H)\\subset\n\\tilde G\/Z^\\omega$, where $\\tilde H$ a closed \nconnected subgroup of $\\tilde G$ (simply-connected or not). Global gauge \nanomalies are now detected by the pullback cohomology class \nin $H^2(\\Gamma\\times G,U(1))$. \nSecs.\\,\\ref{sec:4} and \\ref{sec:5} are devoted to finding out when the latter \nis nontrivial for groups $\\,G\\,$ as before and for a wide class of \nsubgroups $\\tilde H\\subset\\tilde G$ (the nontriviality of the pullback \nclass depends only on the subgroup $\\tilde H$ modulo conjugation \nby elements of $\\tilde G$ and it may occur only if the original anomaly class \n$\\varphi$ is nontrivial, hence for Lie algebras $\\mathfrak{g}$ \nenumerated above). Closed connected subgroups $\\tilde H\\subset\\tilde G$\nare in one-to-one correspondence to Lie subalgebras $\\mathfrak{h}\\subset\n\\mathfrak{g}$. We obtain the complete list of cases with global \ngauge anomalies for subgroups $\\tilde H$ with the Lie algebra $\\mathfrak{h}$ \nwhich is a semisimple regular subalgebra of $\\mathfrak{g}$ (i.e. such that \nthe roots of $\\mathfrak{h}$ form a  subset of roots of $\\mathfrak{g}$). \nThe complete classification (modulo conjugation) of regular subalgebras \nof simple Lie algebras was obtained in the classical work \\cite{Dynkin}\nof Dynkin. The complete classification of all semisimple subalgebras \nof simple Lie algebras is not known explicitly, except for low ranks \nand may be complicated. We give the complete list of non-regular \nsemisimple subalgebras $\\mathfrak{h}$ of $\\mathfrak{g}=\\mathfrak{e}_6$\ncorresponding to subgroups $\\tilde H\\subset\\tilde G$ that lead to global \ngauge anomalies. For $\\mathfrak{g}=A_r$ \nand $\\mathfrak{g}=D_r$, we limit ourselves to few examples of anomalous \nsubgroups $\\tilde H\\subset\\tilde G$ for which $\\mathfrak{h}$ is a non-regular \nsemisimple subalgebra of $\\mathfrak{g}$.\n\\vskip 0.1cm\n\nAs discussed in \\cite{GSW} for the untwisted case, the presence of global \ngauge anomalies of the type studied here renders the $G\/H$ coset models \ninconsistent on the quantum level (barring accidental degeneracies \nof the affine characters). Hence the importance of the classification \nof the anomalous cases.  \n\n\n\\section{No-anomaly condition}\n\\label{sec:2}\n\nThe WZ contribution to the action of the WZW model corresponding \nto the closed $3$-form $H_k$ on a connected compact simple Lie group $G=\\tilde G\/Z$ with $Z\\subset\\tilde Z$ may be defined (modulo $2\\pi$) whenever \nthe periods of $H_k$ (i.e.\nits integrals over closed $3$-cycles) belong to $2\\pi\\mathbb Z$. \nFor the standard \nnormalization of the invariant negative-definite quadratic \nform ${\\rm tr}$ on the Lie algebra \n$\\mathfrak{g}$ in which long roots (viewed as elements of \n$\\ii\\mathfrak t_{\\mathfrak g}$, where $\\mathfrak t_{\\mathfrak g}$ is the\nCartan subalgebra of $\\mathfrak g$) have length squared $2$, this happens \nfor levels $k\\in K_G\\subset\\mathbb Z$. If $G=\\tilde G$ then\n$K_G=\\mathbb Z$ whereas $K_G$ may be a proper subset \nof $\\mathbb Z$ if $G=\\tilde G\/Z$ with $Z$ nontrivial (i.e. $\\not=\\{1\\}$). \nSets $K_G$ of admissible levels are explicitly known \n\\cite{FGK,Gawedzki}. Besides, for $G=SO(2r)\/{\\mathbb Z_2}$ with $r$ even \n(where $K_G=\\mathbb{Z}$ when $4|r$ and $K_G\n=2\\mathbb{Z}$ if $4\\hspace{-0.2cm}\\not{\\hspace{-0.05cm}|}r$), there are \ntwo different consistent choices of the WZ term of the action. \nThe details of the construction of the WZ contribution \n$\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\big]$ to the Feynman \namplitude of the sigma-model field $g:\\Sigma\\to G$ defined on a closed \noriented worldsheet $\\Sigma$, discussed e.g. in \\cite{Gtop,GR}, will not \ninterest us here beyond the fact that the result is invariant under \nthe composition of fields $g$ with the left or right action of (fixed) \nelements of group $G$. The action functional with the (twisted) adjoint \nsymmetry of the WZW model gauged is a functional of field $g$ and of \ngauge-field $A$, a $\\mathfrak{g}$-valued $1$-form on $\\Sigma$. \nIt has the form\n\\begin{equation}\nS^{W\\hspace{-0.05cm}Z}_\\Sigma(g,A)\\,=\\,S^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\,\n+\\,\\frac{_k}{^{4\\pi}}\\int{\\rm tr}\\,\\big((g^{-1}dg)\\hspace{0.02cm}\\omega(A)\n+(dg)g^{-1}\\hspace{-0.05cm}A+g^{-1}\\hspace{-0.05cm}\nAg\\hspace{0.05cm}\\omega(A)\\big)\n\\label{gauged_action}\n\\end{equation}\n(for the untwisted case, $\\omega=I\\hspace{-0.03cm}d$).\nThe local gauge transformations $h:\\Sigma\\to\\tilde G\/Z^\\omega$ act \non the sigma model and gauge fields by\n\\begin{equation}\n{}^h\\hspace{-0.05cm}g\\,=\\,hg\\hspace{0.04cm}\\omega(h)^{-1}\\,,\n\\qquad{}^h\\hspace{-0.06cm}A\\,\n=\\,hA\\hspace{0.01cm}h^{-1}+hdh^{-1}\\,.\n\\end{equation}\nNote that $Z^\\omega=\\tilde Z$ for $\\omega=I\\hspace{-0.03cm}d$.\nIt is easy to show that the invariance of the gauged Feynman amplitudes \nunder such transformations: \n\\begin{equation}\n\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma({}^h\\hspace{-0.05cm}g,\n{}^h\\hspace{-0.06cm}A)\n\\big]\\,=\\,\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g,A)\\big]\n\\label{gaugeinv}\n\\end{equation}\nis equivalent to the identity\n\\begin{eqnarray}\n\\frac{\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma({}^h\\hspace{-0.05cm}g)\\big]}\n{\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\,\n+\\,\\frac{_{\\ii k}}{^{4\\pi}}\\int_\\Sigma\n{\\rm tr}\\,\\big(g^{-1}dg\\hspace{0.05cm}\\omega(h^{-1}dh)+(dg)g^{-1}h^{-1}dh\n+g^{-1}(h^{-1}dh)g\\hspace{0.05cm}\\omega(h^{-1}dh)\\big)\\big]}\\,\n=\\,1\\,,\\  \n\\label{eval}\n\\end{eqnarray}\nsee Appendix \\ref{app:1}.\nThe ratio on the left hand side belongs always to $U(1)$. It coincides \nwith the evaluation of the anomaly class $\\varphi\\in \nH^2(\\tilde G\/Z^\\omega\\times G,U(1))$ on the 2-cocycle that is the image of the \nfundamental class of $\\Sigma$ under the map $(h,g):\\Sigma\\to\\tilde G\/\nZ^\\omega\\times G$.\n\nA simple analysis \\cite{GSW}\nof the structure of cohomology group $H^2(\\tilde G\/Z^\\omega\\times G,U(1))$ \nbased on the K\\\"unneth Theorem shows that class $\\varphi$ is trivial \nif and only if\nidentity (\\ref{eval}) holds for $\\Sigma=S^1\\times S^1$ and \n\\begin{equation}\nh({\\rm e}^{\\ii\\sigma_1},{\\rm e}^{\\ii\\sigma_2})={\\rm e}^{\\ii\\sigma_1\\tilde M}\\,,\n\\qquad g({\\rm e}^{\\ii\\sigma_1},\n{\\rm e}^{\\ii\\sigma_2})={\\rm e}^{\\ii\\sigma_2M}\n\\label{hg}\n\\end{equation}\nwhere $\\tilde M,M\\in\\ii\\mathfrak t_{\\mathfrak g}$ and are such that, \nin terms of the exponential map with values in $\\tilde G$,\n\\begin{equation} \n\\tilde z\\equiv e^{2\\ii\\pi\\tilde M}\\in Z^\\omega\\quad\\ {\\rm and}\n\\quad\\ z\\equiv e^{2\\ii\\pi M}\\in Z\\,.\n\\label{tzz}\n\\end{equation}\nBoth $\\tilde M$ and $M$ have to belong to the coweight lattice \n$P^\\vee(\\mathfrak{g})\\subset\\ii\\mathfrak{t}_{\\mathfrak g}$ dual to the\nweight lattice of $\\mathfrak{g}$ and composed of \n$M\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ s.t. $\\exp[2\\ii\\pi M]\\in\\tilde Z$.\nFor $(h,g)$ given by Eqs.\\,(\\ref{hg}), the left hand side of Eq.\\,(\\ref{eval})\nis easily computable giving rise to the identity\n\\begin{equation}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}\\,\\exp\\big[-2\\ii\\pi k\\,\n{\\rm tr}(M\\omega(\\tilde M))\\big]\\,=\\,1\n\\label{idtytw}\n\\end{equation}\nwhich holds for all $\\tilde M,M\\in P^\\vee(\\mathfrak{g})$ \nas above if and only if there\nare no global gauge anomalies for the WZW model with gauged (twisted) adjoint \naction of $\\tilde G\/\\tilde Z^\\omega$ on the target group $G$. \nIn Eq.\\,(\\ref{idtytw}),\n\\begin{equation}\nZ^2\\ni(z,z')\\,\\mapsto\\,c_{z, z'}\\in U(1) \n\\label{bihom}\n\\end{equation}\nis a $k$-dependent bihomomorphism in $Hom(Z\\otimes Z,U(1))$\nwhose explicit form may be extracted from Appendix 2 of \\cite{FGK}.\nFor cyclic $Z\\equiv\\mathbb Z_p$ generated by $z_0=e^{2\\ii\\pi\\theta}$\nfor $\\theta\\in P^\\vee(\\mathfrak{g})$,\n\\begin{equation}\nc_{z_0^m,z_0^n}=\\exp[-\\ii\\pi k\\hspace{0.02cm}mn\\,{\\rm tr}(\\theta^2)].\n\\label{cc}\n\\end{equation}\nFor the only case with non-cyclic $Z$, we shall explicit $c_{z,z'}$\nin Sec.\\,\\ref{sec:D_reven_tw}. In the untwisted case with \n$\\omega=I\\hspace{-0.03cm}d$, \ncondition (\\ref{idtytw}) reduces to the requirement that\n\\begin{equation}\n\\exp\\big[-2\\ii\\pi k\\,{\\rm tr}(M\\tilde M)\\big]\\,=\\,1\\,.\n\\label{idty}\n\\end{equation}\nIf we gauge only the adjoint action of $\\tilde H\/(\\tilde Z^\\omega\n\\cap\\tilde H)$ \nthen there are no global gauge anomalies if and only if identity (\\ref{idtytw}) \nholds under the additional restriction that, as an element of $\\tilde G$, \n$\\,\\exp[2\\ii\\pi\\tilde M]\\in\\tilde H$. \n\\vskip 0.1cm\n\nIt is enough to check the above conditions for $\\tilde M,M$ in different\nclasses modulo the coroot lattice $Q^\\vee(\\mathfrak{g})$ (composed\nof $\\tilde M\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ s.t. $\\exp[2\\ii\\pi\\tilde M]=1$\nin $\\tilde G$) since ${\\rm tr}\\,\\tilde M M\\in{\\mathbb Z}$ if \n$\\tilde M\\in P^\\vee(\\mathfrak{g})$ and $M\\in Q^\\vee(\\mathfrak{g})$ \nor {\\it vice versa}. In particular, if $Z=\\{1\\}$, i.e. if $G$ is simply\nconnected, then conditions (\\ref{idtytw}) and (\\ref{idty}) are \nalways satisfied so that there are no global gauge anomalies in that case. \nIn the sequel, we shall describe for each Lie algebra $\\mathfrak{g}$ \nthe center $\\tilde Z$ \nof the corresponding simply connected group $\\tilde G$ in terms \nof coweights of $\\mathfrak{g}$. Then choosing a Lie \nsubalgebra $\\mathfrak{h}\\subset\\mathfrak{g}$, we shall restrict elements \n$\\tilde{M}$ by requiring that $e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$. \nNote that $e^{2\\ii\\pi\\tilde M}\\in\\tilde H\\,$ \nif and only if $\\,e^{2\\ii\\pi\\tilde M}\\in g\\tilde H g^{-1}\\,$ \nfor $\\,g\\in\\tilde G$ and $\\,e^{2\\ii\\pi\\tilde M}\\in\\tilde Z$. \nHence the no-anomaly conditions coincide for conjugate subgroups \n$\\tilde H\\subset\\tilde G$. Thus it is \nenough to consider one Lie subalgebra $\\,\\mathfrak h\\subset\\mathfrak g\\,$ \nin each class of subalgebras related by inner automorphisms of $\\mathfrak g$. \nWe may also require that the Cartan subalgebra $\\mathfrak t_{\\mathfrak h}$\nof $\\mathfrak h$ be contained in the Cartan subalgebra \n$\\mathfrak t_{\\mathfrak g}$ of $\\mathfrak g$. Then \n$e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$ if and only if there is \n$q^\\vee\\in Q^\\vee(\\mathfrak g)$ such that $\\tilde M+q^\\vee\\in\\ii\\mathfrak \nt_{\\mathfrak h}$. This is the condition that we shall impose \non $\\tilde M$. \n\\vskip 0.1cm\n\nThe no-anomaly conditions for Lie subalgebras $\\mathfrak h\\subset\\mathfrak g$\nrelated by outer automorphisms $\\omega'$ of $\\mathfrak g$ are also related.\nIndeed, it is easy to see that the expression on the right hand side\nof Eq.\\,(\\ref{eval}) for gauge transformation $h$ and fields $g$ coincides \nwith the similar expression for gauge transformation $\\omega'(h)$ and \nfield $\\omega'(g)$ if in the latter case subgroup $Z\\subset\\tilde Z$ is \nreplaced by $\\omega'(Z)$ and the twist $\\omega$ by \n$\\omega'\\omega\\hspace{0.04cm}\\omega'^{-1}$. The only exception is \nthe case of $G=SO(2r)\/\\mathbb{Z}_2$ for even $r$ and odd $k$ where \none may also have to interchange the two different consistent choices \nof the theory, see Sec.\\,\\ref{sec:D_reven_tw}.\n\\vskip 0.2cm\n\n{\\bf Summarizing:} the necessary and sufficient condition for \nthe absence of global gauge anomalies requires that Eq.\\,(\\ref{idtytw})\nholds for all $\\tilde M,M\\in P^\\vee(\\mathfrak{g})$ such that\n\\begin{equation} \n\\tilde z\\equiv e^{2\\ii\\pi\\tilde M}\\in Z^\\omega\\cap\\tilde H\\quad\\ {\\rm and}\n\\quad\\ z\\equiv e^{2\\ii\\pi M}\\in Z\\,.\n\\label{tzzH}\n\\end{equation}\nIn the untwisted case, this reduces to the condition\\footnote{In the conformal\nfiled theory terminology \\cite{SchellYank}, condition\n(\\ref{coset_anomaly}) means that the monodromy charge $Q_J(\\tilde J)$ for the \nsimple currents $\\tilde J$ and $J$ corresponding to the central elements \n$\\tilde z$ and $z$ has to vanish modulo 1.}\n\\begin{equation}\\label{coset_anomaly}k~\\text{tr} \n(M \\tilde{M}) \\in \\mathbb{Z}\\ \\ \\text{for all}\\ \\ \n\\tilde M,M \\in P^\\vee(\\mathfrak{g})\\ \\ \ns.t.\\ \\ \\tilde z\\in\\tilde H, \n\\ \\ z\\in Z\\,. \n\\end{equation}\nThe no-anomaly conditions for subgroups $\\tilde H\\subset\\tilde G$\ncorresponding to Lie subalgebras $\\mathfrak h\\subset\\mathfrak g$ related \nby inner (outer) automorphisms of $\\mathfrak g$ coincide (are simply\nrelated).\n\n\n\\section{Cases with $\\mathfrak{h}=\\mathfrak{g}$}\n\\label{sec:3}\n\nAs the first step, we shall consider the cases with \n$\\mathfrak h=\\mathfrak g$ for all simple \nalgebras $\\mathfrak g$ according to the Cartan classification, and for \narbitrary nontrivial subgroups $Z\\subset\\tilde Z$. \nIf there are no global gauge anomalies in that case, then the anomalies are \nabsent \nalso for other $\\mathfrak h\\subset\\mathfrak g$. In other words,\nupon restricting $\\mathfrak h$ to a smaller subalgebra, the anomalies may \nonly disappear. In this way, a lot of trivial cases can be already treated\nwithout specifying the subalgebra $\\mathfrak{h}$. We shall then consider \nin the next section the classification of subalgebras $\\mathfrak{h}\\subset\n\\mathfrak{g}$ up to conjugation only for the remaining cases: those \nwith possible anomalies. \n\n\n\n\\subsection{Case $A_r = \\mathfrak{su}(r+1)$, $r\\geq1$}\n\nLie algebra $\\mathfrak{g} = A_r $, corresponding to group \n$\\tilde G=SU(r+1)$, is composed of traceless anti-hermitian\nmatrices of size $r+1$. Its Cartan subalgebra $\\mathfrak{t}_{\\mathfrak g}$ \nmay be taken \nas the subalgebra of diagonal traceless matrices with imaginary entries. \nWe define $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g},\\ i = 1, \\ldots, r+1$, \\,as \na diagonal matrix with the $j$'s diagonal entry equal to $\\delta_{ij}$, \nso that tr$(e_i e_j) \n= \\delta_{ij}$. Roots (viewed as elements of $\\ii\\mathfrak{t}_{\\mathfrak g}$) \nand coroots of $\\mathfrak{su}(r+1)$ have then the form\n$e_i - e_j$ for $i \\neq j$ and the standard choice of simple roots is \n$\\alpha_i = e_i-e_{i+1}, \\,i = 1 \\ldots r$. The center $\\tilde{Z} \\cong \n\\mathbb{Z}_{r+1}$ may be generated by $z = e^{2i\\pi \\theta}$ with \n$\\theta = \\lambda^\\vee_r = (1\/(r+1)) \\sum_{i=1}^{r+1}e_i-e_{r+1}$ where\n$\\lambda^\\vee_i$ denotes the $i$-th simple coweight satisfying\n$\\,{\\rm tr}(\\lambda_i^\\vee\\alpha_j)=\\delta_{ij}$. \nSubgroups $Z$ of $\\tilde{Z}$ are of the form $Z \\cong \\mathbb{Z}_p$ with $p | \n(r+1)$, and may be generated by $z^q= e^{2i\\pi q \\theta} $ \nfor $r+1=pq$. The admissible levels\nfor the WZW model based on group $G=\\tilde G\/\\mathbb Z_p$ are: \n\\begin{equation}\\label{Consistency_Ar}\n\\begin{array}{ll}\n k \\in 2\\mathbb{Z} & \\text{if } p \\text{ even and } q \\text{ odd,}\\\\\n k \\in \\mathbb{Z}  & \\text{otherwise,}\n\\end{array}\n\\end{equation}\nsee \\cite{FGK,Gawedzki}. If we now represent $M$ and $\\tilde{M}$ \nin the Euclidian space spanned by vectors $e_i$,\n\\begin{eqnarray}\\label{M_Ar}\n&&M = aq \\theta = \\left( \\dfrac{a}{p}, \\ldots , \\dfrac{a}{p},- \\dfrac{ar}{p}\n\\right), \\hspace{2.1cm} a \\in \\mathbb{Z}\\,, \\\\\n\\label{tildeM_Ar}\n&&\\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{r+1}, \\ldots ,\n\\dfrac{\\tilde{a}}{r+1},- \\dfrac{\\tilde{a}r}{r+1} \\right), \\qquad \\tilde{a} \\in\n\\mathbb{Z}\\,,\n\\end{eqnarray}\nthe condition for $M$ in (\\ref{tzzH}) is satisfied and\n$e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{Z}$. \n\\subsubsection{Untwisted case}\n\n\nIf $\\omega=I\\hspace{-0.03cm}d$, the global gauge invariance\nfor $\\mathfrak{h}=\\mathfrak{g}$ is assured if \n\\begin{equation}\\label{Quantity_Ar}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{r a\\tilde{a}}{p}\\,\\in\\mathbb Z\\,.\n\\end{equation}\nIn particular, $k\\in p\\mathbb{Z}$ is a sufficient condition\nfor the absence of global anomalies.\nRecall that $p$ divides $r+1$. This implies that $p$ and $r$ are relatively \nprime. Hence $k\\in p\\mathbb{Z}$ is also a necessary condition for the absence\nof the anomalies if there are no further restrictions on the values of \n$\\tilde a$, \\,i.e. if $\\mathfrak h=\\mathfrak g$. \nTaking into account restrictions (\\ref{Consistency_Ar}), this leads to the \nfirst result:\n\n\\begin{prop}\\label{prop_Ar_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{su}(r+1)$, subgroups $Z \\cong \\mathbb{Z}_p$, $r+1 = pq$, \nand arbitrary subalgebras $\\mathfrak{h}$ do not have global gauge anomalies \nif $k \\in p\\mathbb{Z}$. \nThe models with $\\mathfrak h=\\mathfrak g$ and with \n$k \\notin p\\mathbb{Z}$ for $p>1$ \nodd or $q$ even, or with $k \\in 2\\mathbb{Z}\\setminus p\\mathbb{Z}$ \nfor $p > 2$ even and $q$ odd \nare anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\n\nFor $r>1$, there is one nontrivial outer automorphism of \n$\\mathfrak{su}(r+1)$. It maps simple root \n$\\alpha_i$ to $\\alpha_{r+1-i}$ so that for $\\tilde M$ given by \nEq.\\,(\\ref{tildeM_Ar}),\n\\begin{equation}\n\\omega(\\tilde M)\n= \\omega(\\tilde{a} \\theta) = \\left( \\dfrac{\\tilde{a}r}{r+1},\n\\dfrac{-\\tilde{a}}{r+1}, \\ldots ,\\dfrac{-\\tilde{a}}{r+1} \\right), \\qquad \n\\tilde{a} \\in\\mathbb{Z}\\,.\n\\label{omegaM_Ar}\n\\end{equation}\nThe condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nreduces to the requirement\n\\begin{eqnarray}\nq|\\tilde a\\quad\\ {\\rm for}\\ \\quad q\\ \\quad {\\rm odd\\ \\quad and}\n\\quad\\ \\frac{q}{2}|\\tilde a\\quad\\ {\\rm for}\\ \\quad q\\ \\ {\\rm even}\\,.\n\\end{eqnarray}\nIt follows that $Z^\\omega\\cong\\mathbb Z_p$ for $\\,q\\,$ odd and $Z^\\omega\n\\cong\\mathbb Z_{2p}$ for $\\,q\\,$ even.\nFrom Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp\\hspace{-0.07cm}\n\\big[-2\\ii\\pi k\\frac{\\tilde a a r}{p}\\big]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{M_Ar}) and (\\ref{omegaM_Ar}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[-2\\ii\\pi k\\frac{a\\tilde a}{p}\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) reduces to the identity\n\\begin{equation}\n\\exp[-2\\ii\\pi k a\\tilde aq]=1\n\\end{equation}\nwhich always holds implying\n\n\\begin{prop}\\label{prop_Ar_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{su}(r+1)$, subgroups $Z \\cong \\mathbb{Z}_p$, $r+1 = pq$, \nand arbitrary subalgebras $\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n \n\n\n\\subsection{Case $B_r = \\mathfrak{so}(2r+1)$, $r \\geq 2$}\n\nLie algebra $\\mathfrak{g} = B_r $, corresponding to group\n$\\tilde G=Spin(2r+1)$, is composed of real antisymmetric\nmatrices of size $2r+1$. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$ \nmay be taken as composed of $r$ blocks\n\n\\begin{equation}\n \\begin{pmatrix}\n0 & - t_i \\\\\n t_i  & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally, with the last diagonal entry vanishing. Let \n$e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ denote the\nmatrix corresponding to $t_j = \\ii\\delta_{ij}$. With the normalization such \nthat\ntr$(e_ie_j) = \\delta_{ij}$, roots of $\\mathfrak{g}$ have the form $\\pm e_i \\pm\ne_j$ for $i \\neq j$ and $\\pm e_i$, and one may choose $\\alpha_i = e_i -e_{i+1}$\nfor $i = 1 \\ldots r-1$ and $\\alpha_r = e_r$ as the simple roots. The center\n$\\tilde{Z} \\cong \\mathbb{Z}_2$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with\n$\\theta = \\lambda^\\vee_1 = e_1$, and the only nontrivial subgroup of the\ncenter is $Z =\n\\tilde{Z}$. If we describe $M$ and $\\tilde{M}$ in the Euclidian space spanned\nby vectors $e_i$, it is enough to take\n\\begin{equation}\n M = a \\theta = \\left( a, 0, \\ldots, 0 \\right), \\qquad \n \\tilde{M} = \\tilde{a} \\theta = \\left( \\tilde{a}, 0, \\ldots, 0 \\right), \\qquad\na,\\tilde{a} \\in \\mathbb{Z}\\,.\n\\end{equation}\nLie algebra $\\mathfrak{so}_{r+1}$ does not have nontrivial\nouter automorphisms. For $\\omega=I\\hspace{-0.03cm}d$, the global gauge \ninvariance is assured if\n\\begin{equation}\n k~\\text{tr} (M \\tilde{M}) = k a\\tilde{a} \\in \\mathbb{Z}\n\\end{equation}\nwhich is always the case leading to\n\\begin{prop}\nThe coset models corresponding to Lie algebra \n$g = \\mathfrak{so}(2r+1)$ and any subalgebra $\\mathfrak{h}$ do not have \nglobal gauge anomalies.\n\\end{prop}\n\n\n\\subsection{Case $C_r = \\mathfrak{sp}(2r)$, $r \\geq 3$}\n\nLie algebra $\\mathfrak{g} = C_r$, corresponding to group \n$\\tilde G=Sp(2r)$, is composed of antihermitian matrices $X$ of\nsize $2r$ such that $\\Omega X$ is symmetric, with $\\Omega$ built \nof $r$ blocks \n\n\\begin{equation}\n\\omega = \\begin{pmatrix}\n0 & -1 \\\\\n1 & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$ may be taken \nas composed of r blocks $t_i \\omega$ placed diagonally. \nLet $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ \ndenote the matrix corresponding to\n$t_j = \\ii\\delta_{ij}$. With the normalization tr$(e_ie_j) = 2 \\delta_{ij}$, \nroots of $\\mathfrak{g}$ have the form $(1\/2) (\\pm e_i \\pm e_j)$ for \n$i \\neq j$ and\n$\\pm e_i$. The simple roots may be chosen as $\\alpha_i = (1\/2) (e_i - e_{i+1})$\nfor $i = 1, \\ldots r-1$ and $\\alpha_r = e_r$. The center $\\tilde{Z} \\cong \n\\mathbb{Z}_2$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta =\n\\lambda^\\vee_r = (1\/2) \\sum_{i=1}^{r} e_i$, and its only nontrivial subgroup is\n$Z = \\tilde{Z}$. We then take $M$ and $\\tilde{M}$ in the\nEuclidian space spanned by vectors $e_i$ of the form \n\\begin{equation}\n M = a \\theta = \\left( \\dfrac{a}{2}, \\ldots, \\dfrac{a}{2} \\right) \\qquad \n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{2},  \\ldots,\n\\dfrac{\\tilde{a}}{2} \\right) \\qquad a,\\tilde{a} \\in \\mathbb{Z}.\n\\end{equation}\n \n\\noindent Lie algebra $\\mathfrak{sp}(2r)$ does not have nontrivial\nouter automorphisms. For $\\omega=I\\hspace{-0.03cm}d$,\ntaking into account the normalization of $\\rm tr$, we obtain: \n\\begin{equation}\\label{Quantity_Cr}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{a\\tilde{a}r}{2} ,\n\\end{equation}\nensuring the global gauge invariance if it is an integer. The admissible \nlevels $k$ are \n\\begin{eqnarray}\n&&k \\in \\mathbb{Z} \\qquad\\ \\text{\\,if } r \\text{ is even,}\\\\\n&&k \\in 2\\mathbb{Z} \\qquad \\text{if } r \\text{ is odd,}\n\\end{eqnarray}\nsee \\cite{FGK,Gawedzki}, so that the above condition is always satisfied\nleading to  \n\n\\begin{prop}\nThe coset models corresponding to Lie algebra \n$\\mathfrak{g} = \\mathfrak{sp}(2r)$ \nand any subalgebra $\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\\subsection{Case $D_r = \\mathfrak{so}(2r)$, $r \\geq 4$}\n\nLie algebra $\\mathfrak{g} = D_r$, corresponding to group \n$\\tilde G=Spin(2r)$, is composed of real antisymmetric\nmatrices of size $2r$. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$\nmay be taken as composed of $r$ blocks\n\\begin{equation}\n\\omega = \\begin{pmatrix}\n0 & -t_i \\\\\nt_i & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally. Let us denote by $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ \nthe matrix corresponding to $t_j =\\ii\\delta_{ij}$. \nWith the normalization tr$(e_ie_j) =  \\delta_{ij}$, roots of\n$\\mathfrak{g}$ have the form $\\pm e_i \\pm e_j$ for $i \\neq j$, and the simple\nroots may be chosen as $\\alpha_i = e_i - e_{i+1}$ for $i = 1 \\ldots r-1$ and\n$\\alpha_r = e_{r-1} + e_{r}$. \n\n\\paragraph{Case of $r$ odd.}\n\nIf $r$ is odd, the center $\\tilde{Z} \\cong \\mathbb{Z}_4$ is generated by $z =\ne^{2 \\ii \\pi \\theta}$ with $\\theta = \\lambda^\\vee_r = (1\/2) \\sum_{i=1}^{r} e_i$.\nThe possible nontrivial subgroups are \n$Z = \\tilde{Z}$ and $Z \\cong \\mathbb{Z}_2$, \ngenerated by $z^2$. In particular, $Spin(2r)\/\\mathbb{Z}_2=SO(2r)$. \nTaking the general form of $M$ and $\\tilde{M}$ in the Euclidian\nspace spanned by vectors $e_i$, \n\\begin{equation}\\label{MMtilde_Dr_odd}\n\\begin{array} {ll}\n M = a \\theta = \\left( \\dfrac{a}{2}, \\ldots, \\dfrac{a}{2} \\right), & a \\in\n\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n & a \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,, \\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{2},  \\ldots,\n\\dfrac{\\tilde{a}}{2} \\right), & \\tilde{a} \\in \\mathbb{Z}\\,.\n\\end{array}\n\\end{equation}\nThe admissibility condition for the levels\nin the corresponding WZW models are \\cite{Gawedzki}:\n\\begin{eqnarray}\n\\label{comp_Dr_odd}\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n&&k \\in \\mathbb{Z}\\hspace{0.06cm}\\ \\text{ if } Z \\cong \\mathbb{Z}_2\\,.\n\\end{eqnarray}\n\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if the quantity \n\\begin{equation}\\label{Quantity_Dr_odd}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{a\\tilde{a}r}{4} ,\n\\end{equation}\nis an integer. The latter holds for \n\\begin{eqnarray}\n&&k \\in 4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,.\n\\end{eqnarray}\n\n\\noindent Comparing to to the admissibility conditions\n(\\ref{comp_Dr_odd}), we deduce the following \n\\begin{prop}\\label{prop_Drodd_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ odd, and any subalgebra $\\mathfrak{h}$ \ndo not have global gauge anomalies for\n\\begin{eqnarray}\n&&k \\in 4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2.\n\\end{eqnarray}\nThe models with $\\mathfrak h=\\mathfrak g$ and $k \\in 2\\mathbb{Z}$\nwith odd $k\/2$ for $Z \\cong \\mathbb{Z}_4$\nor with $k$ odd for $Z \\cong \\mathbb{Z}_2$ are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\nThere is only one nontrivial outer automorphism $\\omega$ of \n$\\mathfrak{so}(2r)$ with odd $\\,r$, \\,It exchanges the simple roots \n$\\alpha_{r-1}$ and $\\alpha_{r}$ and does not change the other ones. Thus, \ntaking $M$ and $\\tilde{M}$ given by \\eqref{MMtilde_Dr_odd}, we get\n\\begin{equation}\\label{omegaM_Dr_odd}\n  \\omega(\\tilde{M}) = \\tilde{a}\\,\\omega(\\lambda_r^\\vee) = \\tilde{a} \n\\lambda_{r-1}^\\vee = -\\tilde{a}\\lambda_r^\\vee +\\tilde a q^\\vee \n= - \\tilde M +\\tilde a q^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(D_r)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\n\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nis always satisfied whatever the subgroup $Z \\cong \\mathbb Z_4$ \nor $\\mathbb Z_2$ considered. From Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp\\hspace{-0.07cm}\\big[\n\\hspace{-0.05cm}-\\ii\\pi k\\frac{\\tilde a a r}{2}\\big]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{MMtilde_Dr_odd}) and (\\ref{omegaM_Dr_odd}),\n\\begin{eqnarray}\n\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-2\\ii\\pi k\n\\,{\\rm tr}(M\\omega(\\tilde M))\\big]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}+\\ii\\pi k\\frac{a\\tilde a r}{2}\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) always holds implying\n\n\\begin{prop}\\label{prop_Drodd_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(2r)$, $r$ odd, subgroups $Z \\cong \\mathbb{Z}_4$ or \n$\\mathbb Z_2$, and arbitrary subalgebras $\\mathfrak{h}$ do not have \nglobal gauge anomalies.\n\\end{prop}\n\n\n\\paragraph{Case of $r$ even.}\n\nIf $r$ is even, the center $\\tilde{Z} \\cong \n\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ is\ngenerated by $z_1 = e^{2 \\ii \\pi \\theta_1}$ with $\\theta_1 = \n\\lambda^\\vee_r = (1\/2)\\sum_{i=1}^{r} e_i$\nand $z_2 = e^{2 \\ii \\pi \\theta_2}$ with $\\theta_2=\\lambda^\\vee_1 =\ne_1$. The possible nontrivial subgroups are given in Table \\ref{Subgroups_Dr}.\n\\begin{table}[htb]\n\\centering\n \\begin{tabular}{|c|c|c|}\n\\hline\n Subgroup $Z$ & Type & Generator(s) $z_i$ \\\\\n\\hline \\hline\n$\\tilde{Z}$ & $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ & $z_1,\\,z_2$ \\\\\n \\hline\n$Z_1 := \\mathbb{Z}_2 \\times \\lbrace 1 \\rbrace$ & $\\mathbb{Z}_2$ & $z_1$ \\\\\n\\hline \n$ Z_2 := \\lbrace 1 \\rbrace \\times \\mathbb{Z}_2 $ & $\\mathbb{Z}_2$  & $z_2$\n\\\\\n\\hline \n$ Z_{\\rm diag}$ & $\\mathbb{Z}_2$ & $z_1z_2$ \\\\\n\\hline\n \\end{tabular}\n\\caption{Subgroups of $\\tilde Z(Spin(2r)) \\cong \\mathbb{Z}_2 \\times\n\\mathbb{Z}_2$, $r$ even, and their generators. }\n\\label{Subgroups_Dr}\n\\end{table}\n\n\\noindent Here, $SO(2r)=Spin(2r)\/Z_2$. The general form \nof $M$ and $\\tilde{M}$ in the Euclidian space spanned\nby vectors $e_i$ is\n\\begin{equation}\\label{MMtilde_Dr_even}\n\\begin{array} {ll}\n M = a_1 \\theta_1 + a_2 \\theta_2 = \\left(\\dfrac{a_1}{2} + a_2, \\dfrac{a_1}{2},\n\\ldots, \\dfrac{a_1}{2} \\right), & a_1,a_2 \\in \\mathbb{Z} \\text{ if } Z =\n\\tilde{Z}, \\\\\n & a_1 \\in \\mathbb{Z}, a_2 = 0 \\text{ if } Z = Z_1, \\\\\n & a_1= 0 , a_2 \\in \\mathbb{Z} \\text{ if } Z = Z_2, \\\\\n & a_1=a_2 \\in \\mathbb{Z} \\text{ if } Z = Z_{\\rm diag}, \\\\\n \\tilde{M} = \\tilde{a}_1 \\theta_1 + \\tilde{a}_2 \\theta_2 =\n\\left(\\dfrac{\\tilde{a}_1}{2} + \\tilde{a}_2, \\dfrac{\\tilde{a}_1}{2},  \\ldots,\n\\dfrac{\\tilde{a}_1}{2} \\right), & \\tilde{a}_1, \\tilde{a}_2 \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\nIn this case, the conditions for admissible levels of the WZW model are \n\\cite{Gawedzki}:\n\\begin{equation}\\label{Consistency_Dreven}\n\\begin{array}{lcl}\n k \\in \\mathbb{Z} & \\text{ if } & r\/2 \\text{ is even for any } Z, \\\\\n& & r\/2 \\text{ is odd for } Z = Z_2,\\\\\n k \\in 2\\mathbb{Z} & \\text{ if } & r\/2 \\text{ is odd and } Z = \\tilde{Z}, Z_1\n\\text{ or } Z_\\text{diag}. \\\\\n\\end{array}\n\\end{equation}\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if\n\\begin{equation}\\label{Quantity_Dr_even}\n k~\\text{tr} (M \\tilde{M}) = k \\left( \\dfrac{a_1 \\tilde{a}_1 r}{4} + \\dfrac{a_1\n\\tilde{a}_2}{2} + \\dfrac{a_2 \\tilde{a}_1}{2}+a_2\\tilde a_2 \\right) ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 2\\mathbb{Z}$, whatever the subgroup considered. Comparing  \nto the admissibility conditions (\\ref{Consistency_Dreven}), \nwe deduce the following \n\n\\begin{prop}\\label{prop_Dreven_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ even, and any subalgebra $\\mathfrak{h}$ \ndo not have global gauge anomalies if $k \\in 2\\mathbb{Z}$. The models\nwith $\\mathfrak h=\\mathfrak g$ and with $k$ odd for $r\/2$ even and any \nnontrivial $Z$, or with $k$ odd for \n$r\/2$ odd and $Z = Z_2$, are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\n\\label{sec:D_reven_tw}\n\nFor $r>4$, there is only one nontrivial outer automorphism $\\omega$ \nof $\\mathfrak{so}(2r)$, which is the same as the one described in \nthe case of $r$ odd: it interchanges the simple roots $\\alpha_{r-1}$ \nand $\\alpha_r$. Thus, taking $M$ and \n$\\tilde{M}$ given by \\eqref{MMtilde_Dr_even}, we get\n\\begin{eqnarray}\\label{omegaM_Dr_even}\n\\omega (\\tilde{M})&=&\\tilde{a}_1\\omega(\\theta_1) + \\tilde{a}_2\n\\hspace{0.02cm}\\omega(\\theta_2)\\,=\\,\n\\tilde{a}_1 \\lambda_{r-1}^\\vee + \\tilde{a}_2\\lambda_1^\\vee \\cr\n&=& \\tilde{a}_1\\lambda_r^\\vee + (\\tilde{a}_1+\\tilde{a}_2)\\lambda_1^\\vee + \n\\tilde a_1q^\\vee\\,=\\,\n\\tilde{M} + \\tilde{a}_1\\theta_2 + \\tilde a_1q^\\vee\n\\end{eqnarray}\nwhere $q^\\vee \\in Q^\\vee(D_r)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})\n=e^{-2\\ii\\pi\\tilde a_1\\theta_2}\\,\\in\\,Z\n\\end{equation}\nis satisfied for arbitrary $\\tilde a_1$ if $Z = \\tilde Z$ or \n$Z_2$, and for $\\tilde a_1=0\\,{mod}\\,2$ if $Z = Z_1$ or $Z_{diag}$. \nFor $Z=\\tilde Z$, the expression\nfor  bihomomorphism (\\ref{bihom})\nextracted from \\cite{FGK} reads:\n\\begin{eqnarray}\nc_{z_1^{m_1}z_2^{m_2},z_1^{n_1}z_2^{n_2}}=\\Big(\\hspace{-0.1cm}\n\\pm \\exp\\hspace{-0.07cm}\\big[\\dfrac{\\ii\\pi k}{2}\\big]\n\\Big)^{m_1n_2-m_2n_1}\\,\n\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\frac{\\ii\\pi k}{2}(m_1n_1 \n\\dfrac{r}{2}+m_1n_2+m_2n_1+2m_2n_2)\\big]\n\\label{pmbihol}\n\\end{eqnarray}\nfor $m_i,n_i\\in\\mathbb Z$, with the sign $\\pm$ corresponding to the two choices\nof WZ action functional. For the cyclic subgroups of $\\tilde Z$,\nthe above expression reduces to the one given by Eq.\\,(\\ref{cc}). \nWe have:\n\\begin{equation}\n  c_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\big(\\pm 1)^{a_1\\tilde\n    a_1}\\,\\exp[\\ii\\pi k( a_1 \\tilde a_1 +  a_2 \\tilde a_1)]\n\\end{equation}\n\n\\noindent and, from Eqs.\\,(\\ref{MMtilde_Dr_even}) and (\\ref{omegaM_Dr_even}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\ii\\pi k \n\\big( (\\dfrac{r}{2}+1) a_1 \\tilde a_1 \n+ a_1 \\tilde a_2 + a_2 \\tilde a_1\\big)\\big].\n\\end{eqnarray} \nHence the no-anomaly condition \\eqref{idtytw} requires that\n\\begin{equation}\n\\label{Dreven_tw_c}\n  (\\pm 1)^{a_1 \\tilde a_1}\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}\n- \\ii \\pi k \\big(\\dfrac{r}{2} a_1\n    \\tilde a_1 + a_1 \\tilde a_2 \\big)\\big] = 1\n\\end{equation}\nConsidering each subgroup $Z$ and the corresponding values of\n$a_1,\\,a_2,\\,\\tilde a_1,$ and $\\tilde a_2$, and recalling the\nconditions \\eqref{Consistency_Dreven} for the admissible levels \nof the corresponding WZW model, we deduce the\n\n\\begin{prop}\\label{prop_Dreven_h=g_tw}\nThe twisted coset model corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(2r)$, $r>4$ even and arbitrary subalgebra \ndo not have anomalies for $Z = \\tilde Z$ (+ theory), $Z_1$ and\n$Z_{diag}$ if $k$ is even, and for $Z = Z_2$ if $k \\in \\mathbb Z$. \nThe twisted models with $\\mathfrak h = \\mathfrak g$ for \n$Z=\\tilde Z$ (- theory) and $k$ even, and for $Z = \\tilde Z$, $Z_1$ or \n$Z_{diag}$ and $k$ odd, $r\/2>2$ even, are anomalous.\n\\end{prop}\n\nFor $r=4$, there are more nontrivial outer automorphisms, because\nthe symmetries of the diagram of $D_4$ form the permutation group $S_3$\n(the well known ``triality''). They belong to two conjugacy classes, \nthe one composed of cyclic permutations of order 2,\n\\begin{equation}\n\\omega_1:\\,\\alpha_3\\rightarrow\\alpha_4\\rightarrow\\alpha_3\\,,\n\\qquad \\omega_2:\\,\\alpha_1\\rightarrow\\alpha_3\\rightarrow\\alpha_1\\,,\n\\qquad \\omega_3:\\,\\alpha_1\\rightarrow\\alpha_4\\rightarrow\\alpha_1\\,,\n\\label{omegas}\n\\end{equation}\nand the one containing cyclic permutations of order 3,\n\\begin{equation}\n\\omega_4:\\,\\alpha_1 \\rightarrow \\alpha_4 \\rightarrow \n\\alpha_3\\rightarrow \\alpha_1\\,,\\qquad\\omega_4^{-1}:\\,\n\\alpha_1 \\rightarrow \\alpha_3 \\rightarrow \n\\alpha_4\\rightarrow \\alpha_1\\,.\n\\label{permutalpha}\n\\end{equation}\nThe no-anomaly conditions for twists $\\omega$ and \n$\\omega'\\omega\\hspace{0.04cm}\\omega'^{-1}$ in the same conjugacy class\nare related, as was discussed at the end of Sec.\\,\\ref{sec:2}:\nthey coincide if in the latter case subgroup $Z\\subset\\tilde Z$ is \nreplaced by $\\omega'(Z)$. The only exception is \nthe case $Z=\\tilde Z$ for odd $k$ where one has also to interchange  \nthe $\\pm$ theories if $\\omega'$ is cyclic of order 2. \nIt is straightforward to see that\n\\begin{equation}\n\\omega_4\\omega_1\\omega_4^{-1}=\\omega_2\\,,\\qquad\\omega_4^{-1}\\omega_1\\omega_4=\n\\omega_3\n\\end{equation}\nand\n\\begin{equation}\n\\omega_4(Z_1)=Z_{diag}\\,,\\qquad\\omega_4(Z_2)=Z_1\\,,\\qquad\\omega_4(Z_{diag})\n=Z_2\\,.\n\\label{permutZ}\n\\end{equation}\nThe results of Proposition \\ref{prop_Dreven_h=g_tw} still hold for $r=4$ \nand twist $\\omega_1$ and the ones for $r=4$ and twists $\\omega_2$ and \n$\\omega_3$ follow from the latter by using the above remark \n(or by a direct calculation) giving:  \n\n\\begin{prop}\\label{prop_D4_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$ with twist $\\omega_1$ and arbitrary subalgebra \ndo not have anomalies for $Z = \\tilde Z$ (+ theory), $Z_1$ and\n$Z_{diag}$ if $k$ is even, and for $Z = Z_2$ if $k \\in \\mathbb Z$. \nThe models with $\\mathfrak h = \\mathfrak g$ for $Z=\\tilde Z$ (- theory)\nand $k$ even, and for $Z = \\tilde Z$, $Z_1$ or $Z_{diag}$ and $k$ odd are \nanomalous. The results for twist $\\omega_2$ ($\\omega_3$) are as \nthe ones for twist $\\omega_1$ except for the permutation (\\ref{permutZ})\nof the subgroups $Z\\rightarrow\\omega_4(Z)$ ($Z\\rightarrow\\omega_4^{-1}(Z))$.\n\\end{prop}\n\n\n\\noindent For the cyclic outer automorphism $\\omega_4$ of order 3,\ntaking $M$ and $\\tilde{M}$ given \nby Eqs.\\,\\eqref{MMtilde_Dr_even}, we obtain:\n\\begin{equation}\\label{omega2M_Dr_even}\n  \\omega_4 (\\tilde{M}) =\n (\\tilde{a}_1 + \\tilde{a}_2) \\theta_1 + \\tilde{a}_1 \n\\theta_2 + \\tilde a_1 q^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(D_4)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega_4(e^{-2\\ii\\pi\\tilde M})=\\exp[2\\ii\\pi(-\\tilde\n  a_2\\theta_1+ (\\tilde a_2 - \\tilde a_1) \\theta_2)]\\,\\in\\,Z\n\\end{equation}\nis satisfied for arbitrary $\\tilde a_1, \\tilde a_2$ if $Z = \\tilde Z$,\nand for $\\tilde a_1 =\\tilde a_2$, $\\tilde a_2 = 0$, $\\tilde a_1 = 0$,\nall $mod\\,2$, if $Z = Z_1,\\, Z_2$ or $Z_{diag}$ respectively. \nExpression (\\ref{pmbihol}) for the bihomomorphism gives here:\n\\begin{equation}\n   c_{\\tilde z\\hspace{0.02cm}\\omega_4(\\tilde z)^{-1},z}\n= (\\pm 1)^{-a_2 \\tilde a_2 + a_1\n     \\tilde a_1 - a_1 \\tilde a_2}\\exp[\\ii \\pi k(a_1\\tilde a_1+ a_2 \n\\tilde a_1 - a_2 \\tilde a_2)]\n\\end{equation}\nFrom Eqs.\\,(\\ref{MMtilde_Dr_even}) and (\\ref{omegaM_Dr_even}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega_4(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\ii\\pi k \n\\big( a_1 \\tilde a_1 + a_2 \\tilde a_1 + a_2 \\tilde\n  a_2 \\big)\\big]\n\\end{eqnarray} \nso that the no-anomaly condition \\eqref{idtytw} becomes\n\\begin{equation}\n  (\\pm 1)^{-a_2 \\tilde a_2 - a_1\n     \\tilde a_2 + a_1 \\tilde a_1}= 1.\n\\end{equation}\nConsidering each subgroup $Z$ and the corresponding values of\n$a_1,\\,a_2,\\,\\tilde a_1,$ and $\\tilde a_2$, and recalling the\nadmissible values \\eqref{Consistency_Dreven} of the level, we deduce\n\n\\begin{prop}\\label{prop_D4_h=g_tw2}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$, outer automorphism $\\omega_4$ and arbitrary\nsubalgebra do not have anomalies for $Z=\\tilde Z$ (+ theory) and\n$Z=Z_1,\\,Z_2$ or $Z_{diag}$. The models with $\\mathfrak h = \\mathfrak g$ \nand $Z = \\tilde Z$ (- theory) is anomalous.  \n\\end{prop}\n\n\\noindent The results for the twist $\\omega_4^{-1}$ may be deduced from\nthe above proposition if we observe that $\\omega_4^{-1}$ may be obtained\nfrom $\\omega_4$ by the conjugation by any cyclic outer automorphism\n$\\omega'$ of order 2. Hence the conditions for the absence or the presence\nof anomalies for the theory twisted by $\\omega_4^{-1}$ are as for\nthe ones for the twist $\\omega_4$ except for the exchange of the $\\pm$\ntheories for $Z=\\tilde Z$ and $k$ odd leading to\n\n\\begin{prop}\\label{prop_D4_h=g_tw3}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$, outer automorphism $\\omega_4^{-1}$ and arbitrary\nsubalgebra do not have anomalies for $Z=\\tilde Z$ (($-$)${}^k$ theory) and\n$Z=Z_1,\\,Z_2$ or $Z_{diag}$. The models with $\\mathfrak h = \\mathfrak g$, \nand $Z = \\tilde Z$ (($-$)${}^{k+1}$ theory) is anomalous.  \n\\end{prop}\n\n\\noindent This may be confirmed by a direct calculation.\n\n\n\n\\subsection{Case $\\mathfrak{e}_6$}\n\nThe imaginary part $\\,\\ii\\mathfrak{t}_{\\mathfrak g}\\,$ of the \ncomplexification of the Cartan \nsubalgebra $\\mathfrak{t}_{\\mathfrak g}$ of $\\mathfrak{g} = \\mathfrak{e}_6$ \nmay be identified \nwith the subspace of $\\mathbb{R}^{7}$ orthogonal to the vector \n$(1, \\ldots, 1, 0)$, with the scalar product inherited from \n$\\mathbb{R}^{7}$. The simple roots may be\ntaken as $\\alpha_i = e_i - e_{i+1}$ for $i=1 \\ldots 5$ and $\\alpha_6 =\n(1\/2)(-e_1-e_2-e_3+e_4+e_5+e_6) + (1\/\\sqrt{2})e_7$, where $e_i$ are the vectors\nof the canonical basis of $\\mathbb{R}^{7}$. The center $\\tilde{Z} \\cong\n\\mathbb{Z}_3$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta =\n\\lambda^\\vee_5 = (1\/6)(e_1 +e_2+e_3+e_4+e_5-5e_6) + (1\/\\sqrt{2})e_7$. The only\nnontrivial subgroup is $Z = \\tilde{Z}$. The general form of $M$ and\n$\\tilde{M}$ in the Euclidian space spanned by vectors $e_i$ is\n\n\\begin{equation}\\label{MMtilde_e6}\n\\begin{array}{l}\n M = a \\theta = \\left( \\dfrac{a}{6},\\ldots ,\\dfrac{a}{6},\\dfrac{-5a}{6},\n\\dfrac{a}{\\sqrt{2}} \\right) \\qquad  a \\in \\mathbb{Z},\\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{6},\\ldots\n,\\dfrac{\\tilde{a}}{6},\\dfrac{-5\\tilde{a}}{6}, \\dfrac{\\tilde{a}}{\\sqrt{2}}\n\\right) \\qquad \\tilde{a} \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if\n\\begin{equation}\\label{Quantity_e6}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{4a\\tilde{a}}{3} ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 3\\mathbb{Z}$. Since all integer levels $k \\in \\mathbb{Z}$ \nare admissible \\cite{FGK,Gawedzki}, \nwe deduce \n\n\\begin{prop}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{e}_6$ and arbitrary subalgebra $\\mathfrak{h}$ do not have global \ngauge anomalies if $k \\in 3\\mathbb{Z}$. The models $Z=\\mathbb Z_3$, \n$\\mathfrak h=\\mathfrak g$ and $k \\in\n\\mathbb{Z} \\setminus 3\\mathbb{Z}$ are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\nThere is only one nontrivial outer automorphism $\\omega$ of \n$\\mathfrak{e}_6$, which exchanges the simple roots $\\alpha_1$ \nand $\\alpha_2$ with $\\alpha_5$ and $\\alpha_4$ and does not change \nthe other ones. Thus, taking $M$ and $\\tilde{M}$ given by \n\\eqref{MMtilde_e6}, we get\n\\begin{equation}\\label{omegaM_e6}\n  \\omega(\\tilde{M}) = \\tilde{a}\\hspace{0.02cm}\\omega(\\lambda_5^\\vee) \n= \\tilde{a} \\lambda_{1}^\\vee = -\\tilde{a}\\lambda_5^\\vee + \\tilde a q^\\vee \n= - \\tilde M + \\tilde aq^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(\\mathfrak e_6)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nis always satisfied for $Z = \\tilde{Z}$. From Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp[-2\\ii\\pi k\\frac{4 \\tilde a a}{3}]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{MMtilde_e6}) and (\\ref{omegaM_e6}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}+2\\ii\\pi k\\frac{4 a\\tilde a }{3}\n\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) always holds implying\n\n\\begin{prop}\\label{prop_e6_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak e_6$, subgroup $Z \\cong \\mathbb{Z}_3$ and arbitrary subalgebras \n$\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n\\subsection{Case $\\mathfrak{e}_7$}\n\nThe imaginary part $\\,\\ii\\mathfrak{t}_{\\mathfrak g}\\,$ of the \ncomplexification of the Cartan \nsubalgebra $\\mathfrak{t}_{\\mathfrak g}$ of $\\mathfrak{g} = \\mathfrak{e}_7$ \nmay be \nidentified with the\nsubspace of $\\mathbb{R}^{8}$ orthogonal to the vector $(1, \\ldots, 1)$ with the\nsimple roots $\\alpha_i = e_i - e_{i+1}$ for $i=1 \\ldots 6$ and $\\alpha_7 =\n(1\/2)(-e_1-e_2-e_3-e_4+e_5+e_6+e_7+e_8)$, where $e_i$ are the vectors of the\ncanonical basis of $\\mathbb{R}^{8}$. The center $\\tilde{Z} \\cong \\mathbb{Z}_2$\nis generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta = \\lambda^\\vee_1 =\n(1\/4)(3,-1,\\ldots, -1, 3)$. The only nontrivial subgroup is $Z = \\tilde{Z}$.\nThe general form of $M$ and $\\tilde{M}$ in the Euclidian space generated\nby $e_i$ is \n\\begin{equation}\n\\begin{array}{l}\n M = a \\theta = \\left( \\dfrac{3a}{4},\\dfrac{-a}{4}, \\ldots, \\dfrac{-a}{4},\n\\dfrac{3a}{4} \\right) \\qquad  a \\in \\mathbb{Z},\\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{3\\tilde{a}}{4},\n\\dfrac{-\\tilde{a}}{4}, \\ldots,\\dfrac{-\\tilde{a}}{4}, \\dfrac{3\\tilde{a}}{4}\n\\right) \\qquad \\tilde{a} \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\nLie algebra $\\mathfrak{e}_7$ does not have nontrivial outer automorphisms\nso that we may take $\\omega=I\\hspace{-0.03cm}d$.\nThe global gauge invariance is then assured if the quantity\n\\begin{equation}\\label{Quantity_e7}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{3a\\tilde{a}}{2} ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 2\\mathbb{Z}$. The condition for admissible levels \nalso requires in this case that $k \\in 2\\mathbb{Z}$ \\cite{FGK,Gawedzki} \nso that we deduce: \n\n\\begin{prop}\nThe coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{e}_7$ and any subalgebra $\\mathfrak{h}$ do not have global\ngauge anomalies.\n\\end{prop}\n\n\n\n\\subsection{Case $\\mathfrak{g}_2$, $\\mathfrak{f}_4$ and $\\mathfrak{e}_8$}\n\n\nThe center of the simply connected groups  corresponding to Lie\nalgebras  $\\mathfrak{g} =\n\\mathfrak{g}_2, \\mathfrak{f}_4$ or $\\mathfrak{e}_8$ is trivial~: \n$\\tilde{Z} \\cong \\lbrace 1\\rbrace$ so that there are no nontrivial\nsubgroups $Z$ in that case and we infer:\n\n\\begin{prop}\nThe coset models corresponding to Lie algebras $\\mathfrak{g} =\n\\mathfrak{g}_2, \\mathfrak{f}_4$ or $\\mathfrak{e}_8$ and any subalgebra\n$\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n\\section{Regular subalgebras}\n\\label{sec:4}\n\nLooking back at the previous section, the global gauge anomalies of the coset\nmodels may appear only for $\\mathfrak{g} = A_r$, $D_r$ \nand $\\mathfrak{e}_6$ in the untwisted case, and only for $\\mathfrak{g}=D_r$\nwith even $r$ in the twisted case (note that these are all simply laced \nLie algebras). Now we have to specify the Lie subalgebra $\\mathfrak{h}$ of \na simple algebra $\\mathfrak g $ to see in which cases the anomalies \nsurvive the restriction of the symmetry group. The first class of \nsemisimple subalgebras that we shall consider are the regular ones, \nintroduced by Dynkin in \\cite{Dynkin}. A Lie subalgebra $\\mathfrak{h}$ of \nan algebra $\\mathfrak{g}$ is called regular if, for a choice of the Cartan \nsubalgebra $\\,t_{\\mathfrak g}\\subset\\mathfrak g\\,$\n(defined up to conjugation), it's complexification is of the form\n\\begin{equation}\n\\mathfrak h^{\\mathbb C}\\,=\\,\\mathfrak t_{\\mathfrak\nh}^{\\mathbb C}\\oplus\\Big(\\mathop{\\oplus}\\limits_{\\alpha\\in\n\\Delta_{\\mathfrak h}\\subset\\Delta_{\\mathfrak g}}\\mathbb C e_\\alpha\\Big)\n\\end{equation}\nwhere $\\mathfrak t_{\\mathfrak h}\\subset\\mathfrak t_{\\mathfrak g}$ is\na Cartan subalgebra of $\\mathfrak{h}$. Subalgebra $\\,\\mathfrak h\\,$ \nis semisimple if $\\,\\alpha\\in\\Delta_{\\mathfrak h}\\,$ implies that \n$\\,-\\alpha\\in\\Delta_{\\mathfrak h}\\,$ and if \n$\\,\\alpha\\in\\Delta_{\\mathfrak h}\\,$  span $\\,\\mathfrak t_{\\mathfrak h}^{\\mathbb C}$.\n$\\,\\Delta_{\\mathfrak h}\\,$ is then the set of roots of $\\,\\mathfrak h$. \n\n\\paragraph{Construction of regular subalgebras.}\nThere is a nice diagrammatic method to obtain all the regular \nsemisimple subalgebras \nof a given semisimple algebra (up to conjugation), \nproposed by Dynkin in \\cite{Dynkin} and summarized in \\cite{Lorente}. \nWe briefly describe it here:\n\n\\begin{enumerate}\n \\item Take the Dynkin diagram of the ambient algebra $\\mathfrak{g}$, \nand adjoin to it a node corresponding to the lowest root $\\delta=-\\phi$ \n(negative of the highest root $\\phi$) of $\\mathfrak{g}$, obtaining the \nextended Dynkin diagram of $\\mathfrak{g}$.\n\n\\item Remove arbitrarily one root from this diagram, in order to obtain \nat most $r+1$ different diagrams, which may split into orthogonal subdiagrams.\n\\item Reapply the firsts two steps to each connected subdiagram obtained\nabove, until no new diagram appears. This way one gets all the regular \nsubalgebras $\\mathfrak{h} \\subset \\mathfrak{g}$ of maximal rank.\n\\item Remove again an arbitrarily root from each diagram, and apply the full \nprocedure to each connected subdiagram obtained this way (including the\nlast step).\n\\end{enumerate}\nThe algorithm stops when no root can be removed, hence one will obtain \nall the regular subalgebras of $\\mathfrak{g}$.\n\n\\subsection{Regular semisimple subalgebras of $A_r$}\n\nThe semisimple regular subalgebras of $A_r$ are \ngiven in \\cite{Dynkin} (Chapter II, Table 9) and have the form: \n\n\\begin{equation}\\label{regular_subalg_Ar}\n \\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m}, \\qquad r_1 + 1 + \\ldots +\nr_m + 1 \\leq r+1\n\\end{equation}\n\n\\noindent The embedding of $\\mathfrak{h}$ in $\\mathfrak{g}$ realizing\nthe ideals $A_{r_i}$ as diagonal blocks in the matrices of $A_{r}$\nis unique up to an inner automorphism of $A_r$.\nTaking $M$ and $\\tilde{M}$ as given in Eqs. \\eqref{M_Ar} and \\eqref{tildeM_Ar} \nwe must require that $\\tilde M+q^\\vee\\in\\ii\\mathfrak{t}_{\\mathfrak h}$, \nfor some $q^\\vee\\in Q^\\vee(A_r)$. Looking block by block, we obtain \nthe conditions \n\\begin{equation}\n\\label{condA}\n \\dfrac{\\tilde{a}(r_i+1)}{r+1} \\in \\mathbb{Z} \\qquad \\forall i = 1, \\ldots, m\n\\end{equation}\nand that\n\\begin{equation}\n\\dfrac{\\tilde{a}}{r+1}\\in\\mathbb Z\n\\end{equation}\nif the inequality in (\\ref{regular_subalg_Ar}) is strict. The latter condition\nimplies that (\\ref{Quantity_Ar}) holds eliminating possible global \ngauge anomalies. We may then limit ourselves to the case when the inequality \nin (\\ref{regular_subalg_Ar}) is saturated. this implies that \nFor $i = 1, \\ldots, m$, we may then rewrite conditions (\\ref{condA}) as \n\\begin{equation}\n \\tilde{a}(r_i+1) = q_i (r+1) \\qquad q_i \\in \\mathbb{Z}.\n\\label{lst}\n\\end{equation}\nIn what follows, we shall denote by, respectively, \n$u_1 \\wedge\\cdots\\wedge u_n$ and $u_1\\vee\\cdots\\vee u_n$ the greatest \ncommon divisor and the least common multiple of $u_1,\\dots,u_n$. \nDividing both sides of Eq.\\,(\\ref{lst}) by $(r+1) \\wedge (r_i+1)$, we get\n\\begin{equation}\n \\tilde{a}\\frac{r_i+1}{(r+1) \\wedge (r_i+1)}= q_i\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}\n\\end{equation}\nso that $\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}|\\frac{\\tilde{a}\\,(r_i+1)}{(r+1)\\wedge(r_i+1)}$. Using the \nfact that $\\frac{r+1}{(r+1)\\wedge (r_i+1)}$ and \n$\\frac{r_i+1}{(r+1)\\wedge(r_i+1)}$ are relatively prime, we infer \nthat $\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}|\\tilde a$, i.e. that\n\\begin{equation}\n \\tilde{a} \\in \\dfrac{r+1}{(r+1) \\wedge (r_i+1)}\\mathbb{Z} \\qquad \\forall i\n= 1, \\ldots, m\n\\end{equation}\nwhich leads, according to Proposition \\ref{lcm} of Appendix \\ref{app:2}, \nto the condition\n\\begin{equation}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r+1) \\wedge (r_1+1)} \\vee \\cdots \\vee\n\\dfrac{r+1}{(r+1) \\wedge (r_m+1)} \\right) \\mathbb{Z}\n\\end{equation}\nThis property can be reformulated, using Proposition\n\\ref{lcmfrak} of Appendix \\ref{app:2}, as\n\\begin{equation}\\label{Condition_regular_Ar}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r+1) \\wedge (r_1+1) \\wedge \\cdots \\wedge\n(r_m+1)} \\right) \\mathbb{Z}\n\\end{equation}\nSince we assumed that $r_1 + 1 + \\ldots + r_m + 1 = r+1$,\ncondition (\\ref{Condition_regular_Ar}) may be simplified to \n\\begin{equation}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r_1+1) \\wedge \\cdots \\wedge (r_m+1)} \\right)\n\\mathbb{Z}\n\\end{equation}\nIn order to guarantee that the quantity \\eqref{Quantity_Ar}\nis an integer for every $a$ and $\\tilde{a}$, ensuring the global \ngauge invariance, it is enough to compute it for $a = 1$ and\n\\begin{equation}\n \\tilde{a} = \\dfrac{r+1}{(r_1+1) \\wedge \\cdots \\wedge (r_m+1)}\\,.\n\\end{equation}\nDenoting $(r_1+1) \\wedge \\ldots \\wedge\n(r_m+1)=l$, and $r+1 = pq$, the quantity \\eqref{Quantity_Ar} becomes\n\\begin{equation}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{r q}{{l}} = k r\\dfrac{q\/(q\\wedge\n{l})}{{l}\/(q\\wedge {l})}.\n\\label{lst1}\n\\end{equation}\nFinally, recalling that ${l} | (r+1)$ and, consequently, \n$\\frac{l}{q\\wedge {l}}$ and $r$ are relatively prime, \nwe infer that the right hand side of Eq.\\,(\\ref{lst1})\nis be an integer if and only if \n\\begin{equation}\n k \\in \\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z}\\,.\n\\end{equation}\nTaking into account condition \\eqref{Consistency_Ar}  for admissible\nlevels, we are now able to state \n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} = A_r$, \nsubgroup $Z \\cong \\mathbb{Z}_p$ for $ (r+1)=pq$ and any regular subalgebra\n$\\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m}$ do not have global \ngauge anomalies for \n\\begin{itemize}\n \\item $r_1+1 + \\ldots r_m+1 < r+1 \\qquad k \\in \n\\left\\lbrace\\begin{array}{l}\n2\\mathbb{Z} \\text{ if } p \\text{ even and } q \\text{ odd}\\\\ \n\\mathbb{Z} \\text{ otherwise}\\end{array}\\right.$\n\n\\item $r_1+1 + \\ldots r_m+1 = r+1 \\qquad k \\in \n\\left\\lbrace\\begin{array}{l}\n\\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z} \\cap 2\\mathbb{Z} \\text{ if } p \\text{ even\nand } q \\text{ odd} \\\\\n\\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z} \\text{ otherwise}\n\\end{array}\\right.$\n\\end{itemize}\nwhere ${l} = (r_1+1) \\wedge \\ldots \\wedge (r_m+1)$.\nThe other untwisted models with admissible levels are anomalous.\n\\end{prop}\n\n\\paragraph{Example 1:  $\\mathfrak g = A_4 = \\mathfrak{su}(5)$.}\n\nThe center $\\tilde Z \\cong \\mathbb Z_5$ of the corresponding group has \nonly one nontrivial subgroup, $Z = \\tilde Z \\cong \\mathbb Z_5$, so with \n$p=5$ odd and $q =1$ odd with the previous notations. The admissible \nlevels are $k \\in \\mathbb Z$, according \nto \\eqref{Consistency_Ar}. \nFollowing Proposition \\ref{prop_Ar_h=g}, the regular subalgebra \n$\\mathfrak h = \\mathfrak g$ leads to the condition $k \\in 5\\mathbb Z$ \nfor non-anomalous models. Then, applying the last proposition above, \nthe cases $\\mathfrak h = A_1,\\,A_1 \\oplus A_1\\equiv 2A_1,\\,A_2$ and $A_3$ \nleads to non-anomalous models for every $k \\in \\mathbb Z$, because \nhere we have \n$r_1+1 + \\ldots r_m+1 < r+1 = 5$. For $\\mathfrak h = A_2 \\oplus A_1$, \nwe have an equality. However, $l = (r_1+1) \\wedge (r_2+1) = 3 \\wedge 2 = 1$, \nso $l \/ (l \\wedge q) = 1$ and the model has no anomalies \nfor every $k \\in \\mathbb Z$. Consequently, the only anomalous models \ncorresponding to $\\mathfrak g = A_4$ and $\\mathfrak h$ regular are those \nwith $\\mathfrak h = \\mathfrak g$, $Z = \\tilde Z$ and $k \\in \\mathbb Z \n\\setminus 5\\mathbb Z$.\n\n\\paragraph{Example 2:  $\\mathfrak g = A_5 = \\mathfrak{su}(6)$.}\n\nHere the center $\\tilde Z \\cong \\mathbb Z_6$ has three nontrivial \nsubgroups : $Z \\cong \\mathbb Z_6, \\mathbb Z_3$ and $\\mathbb Z_2$ with \nthe respective admissible levels $k \\in 2 \\mathbb Z,\\,\\mathbb Z$ and \n$2 \\mathbb Z$. The models corresponding to the case \n$\\mathfrak h = \\mathfrak g$ will be non-anomalous for \n\\begin{equation}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    6 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_6 \\\\\n    3 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_3 \\\\    \n    2 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_2.\n  \\end{array}\n\\right.\n\\end{equation}\nRegular subalgebras $\\mathfrak h = A_1,\\,2A_1,\\,A_2,\\,A_2 \\oplus A_1, \nA_3$ and $A_4$ correspond to the strict inequality for ranks in \nthe proposition above, so there will be no anomalies for these models with\n\\begin{equation}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    2 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_6 \\text{ or } \\mathbb Z_2 \\\\\n    \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_3. \\\\    \n  \\end{array}\n\\right.\n\\end{equation}\nComputation shows that $\\mathfrak h = 2A_2$ leads to non-anomalous \nmodels for the same $k$ as for $\\mathfrak h = \\mathfrak g$, and that \nthe models corresponding to $\\mathfrak h = A_3 \\oplus A_1$ and to $3A_1$ \nhave no anomalies for $k \\in 2 \\mathbb Z$ if \n$Z \\cong \\mathbb Z_6$ or $\\mathbb Z_2$ and for $k\\in\\mathbb Z$ \nif $Z\\cong\\mathbb Z_3$ . \nThus, the anomalous models corresponding to $\\mathfrak g = A_5$ have either \n$\\mathfrak h = \\mathfrak g$ or $\\mathfrak{h}=2A_2$, \nwhere $k \\in 2\\mathbb Z \\setminus 6 \\mathbb Z$ for $Z \\cong \\mathbb Z_6$ \nand $k \\in \\mathbb Z \\setminus 3 \\mathbb Z$ for $Z \\cong \\mathbb Z_3$. \n\n\\subsection{Regular semisimple subalgebras of $D_r$}\n\nThe semisimple regular subalgebras of $D_r$ are given \nin \\cite{Dynkin} (Chapter II, Table 9) and have the form:\n\n\\begin{equation}\\label{regular_subalg_Dr}\n \\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus\n\\ldots \\oplus D_{s_n}\n\\end{equation}\nwhere $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n \\leq r$.\\footnote{To \ntake into account all the possible cases with this formula, we may need \nto consider $D_2$ instead of $2A_1$ and $D_3$ instead of $A_3$ \nto respect the inequality. See examples below.} The embedding of\n$D_{s_i}$ subalgebras realizes them as diagonal \nblocks in $D_r$. Instead of giving an explicit embedding of \nsubalgebras $A_{r_i}$, it is enough to see that\n$A_l$ is trivially embedded in $D_{l+1}$, by sending the $l$ simple roots\n$\\alpha_i^{A_l}$ of $A_l$ to the $l$ first simple roots $\\alpha_i^{D_{l+1}}$ of\n$D_{l+1}$. Then, the Serre construction allows us to reconstruct the full\nstructure of $A_l$, embedded in $D_{l+1}$, which is then easily embedded in\n$D_r$ as a diagonal block. The embedding of $\\mathfrak h$ into $\\mathfrak g$ \ndescribed above is unique, up to inner automorphisms of $\\mathfrak g$, except\nfor even $r$ if there are no $D_{s_i}$ and $r_1+1+\\ldots+r_m+1=r$ with \nall $r_i$ odd. In the latter case there is a second \nindependent embedding of $A_{r_1}\\oplus\\ldots\\oplus A_{r_m}$ into $D_r$ that\nsends the simple roots of $A_{r_m}$ to the last $r_m+1$ simple roots of \n$D_r$ omitting $\\alpha_{r-1}$. That embedding is related to the previous \none by the outer automorphism $\\omega$ of $D_r$ that permutes roots \n$\\alpha_{r_1}$ and $\\alpha_r$, but not by an inner automorphism. \nRecall that the coroot lattice $Q^\\vee(D_r)$ is composed of vectors \n\\begin{equation}\nq^\\vee=\\sum\\limits_{i=1}^rq^\\vee_ie_i\\quad{\\rm with}\\quad \nq^\\vee_i\\in\\mathbb Z\\quad{\\rm and}\\quad\\sum\\limits_{i=1}^rq^\\vee_i\\in2\\mathbb Z\\,.\n\\label{coroot_lat_Dr}\n\\end{equation} \n\n\n\\paragraph{Case of $r$ odd.}\nTaking $M$ and $\\tilde{M}$ as given in \\eqref{MMtilde_Dr_odd}, \nwe shall impose the condition $e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$. On the \nLie-algebra level, we have to show that for some \n$q^\\vee\\in Q^\\vee(\\mathfrak{g})$, $\\tilde M+q^\\vee$ belongs to\n$\\ii\\mathfrak{t}_{\\mathfrak{h}}$. Looking block by block, we infer that \n\\begin{equation}\n\\label{D1}\n\\frac{\\tilde{a}\\hspace{0.01cm}(r_i+1)}{2}\\in\\mathbb Z\\,, \\qquad i=1, \\dots, m\\,,\n\\end{equation}\nand that\n\\begin{equation}\n\\label{D2}\n\\frac{\\tilde{a}}{2}\\in\\mathbb Z\n\\end{equation}\nif $r_1 + 1 + \\ldots + r_m + 1 + s_1 +\\ldots s_n < r$. The condition \nthat the sum of components of vectors in $Q^\\vee(\\mathfrak{so}(2r))$ is even\nimposes the additional requirement that   \n\\begin{equation}\n\\label{add_odd}\n\\frac{\\tilde{a}\\hspace{0.01cm}r}{2}\\in2\\mathbb Z\\,,\n\\end{equation}\ni.e. $\\tilde a\\in 4\\mathbb Z$, in the absence of $D_{s_i}$ components \nin $\\mathfrak{h}$,  (in that case conditions (\\ref{D1}) and (\\ref{D2}) imply \nalready that $\\tilde a\\in2\\mathbb Z$).\nRe-examining the quantity \\eqref{Quantity_Dr_odd} which \nhas to be an integer with the above restrictions in mind \nand taking into account the conditions for admissible levels, \nwe deduce \n\n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ odd, and a regular subalgebra  $\\mathfrak{h} = A_{r_1}\n\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus \\ldots \\oplus D_{s_n}$ \ndo not have global gauge anomalies for the following cases \n\\begin{itemize}\n\n\\item\\hspace{-0.1cm}$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd and }\\quad k \\in \\left\\lbrace\n\\begin{array}{l} \n4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,\n\\end{array}\\right.$\n\\item $\\hspace{-0.3cm}\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\ and\\ some\\  \nr_{i}\\ even\\end{array}\\right\\rbrace\n\\ k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n\\mathbb{Z}\\ \\hspace{0.04cm} \\text{ if } Z \\cong \\mathbb{Z}_2\n\\end{array}\\right.$\n\\end{itemize}\nThe other untwisted models with admissible levels \nare not globally gauge invariant.\n\\end{prop}\n\n\n\\noindent{\\bf Remark}\\ \\ \\ In particular, the global gauge anomalies present\nif $\\mathfrak h=\\mathfrak g$ for $Z=\\mathbb Z_4$ and $k\\in2\\mathbb Z$,\n$k\/2$ odd, or for $Z=\\mathbb Z_2$ and $k$ odd, disappear for $\\mathfrak h= \nA_{r_1}\\oplus \\ldots\\oplus A_{r_m}\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r$ or if\n$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r$ with some\n$r_i$ even. Note that if there no $D_{s_i}$ and $r_1 + 1 + \\ldots + r_m + 1=r$\nthen all $r_i$ cannot be odd. \n\n\n\\paragraph{Example: $\\mathfrak{g}=D_5=\\mathfrak{so}(10)$.} The admissible\nlevels are $k \\in 2\\mathbb Z$ for \n$Z = \\tilde Z \\cong \\mathbb Z_4$ and $k \\in \\mathbb Z$ for $Z \\cong \n\\mathbb Z_2$. According to Proposition \\ref{prop_Drodd_h=g}, there are \nno gauge anomalies in the case $\\mathfrak h = \\mathfrak g$ for \n\\begin{equation}\\label{ex_Drodd_1}\n  k \\in \\left\\lbrace\n    \\begin{array}{ll}\n      4 \\mathbb Z \\text{ if } Z \\cong \\mathbb Z_4\\\\\n      2 \\mathbb Z \\text{ if } Z \\cong \\mathbb Z_2.\\\\\n    \\end{array}\n\\right.\n\\end{equation}\n For regular subalgebra $\\mathfrak h = A_1, 2A_1\\cong D_2, \nA_2, A_3\\cong D_3, D_4$, \nthe inequality on the ranks is strict so there are no anomalies for  \n\\begin{equation}\\label{ex_Drodd_2}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    2 \\mathbb Z & \\text{if } Z = \\mathbb Z_4 \\\\\n    \\mathbb Z & \\text{if } Z = \\mathbb Z_2. \\\\    \n  \\end{array}\n\\right.\n\\end{equation}\nIn the case $\\mathfrak h = A_4$ and $A_2\\oplus A_1$, the rank inequality \nis saturated and there is one $r_i$ even, so \\eqref{ex_Drodd_2} \nstill gives the no-anomaly condition for $k$.\n$D_5$ admits also $D_3\\oplus D_2\\cong A_3 \\oplus 2 A_1$, \n$A_1\\oplus D_3\\cong A_3 \\oplus A_1$, $A_2\\oplus D_2\\cong \nA_2 \\oplus 2 A_1$, $2D_2\\cong 4A_1$ and $A_1\\oplus D_2\\cong 3 A_1$,\nsee \\cite{Lorente} or the method described above,\nwhere only the left hand sides respect the inequality for ranks\nand should be used to extract the no-anomaly conditions.\nFor $A_2\\oplus D_2$, $2D_2$ and $A_1\\oplus D_2$ either the inequality\nfor ranks is saturated and there is an even $r_i$ or the inequality\nfor ranks is strict, hence there are no anomalies for levels\nsatisfying \\eqref{ex_Drodd_2}. Finally, for $D_3\\oplus D_2$ and $A_1\\oplus D_3$ \nthe rank inequality is saturated by there is no even $r_i$ and the\ngauge anomalies persist for \n$Z \\cong \\mathbb Z_4$ if $k \\in 2\\mathbb Z \\setminus 4 \\mathbb Z$ \nand for $Z \\cong \\mathbb Z_2$ if $k$ odd.\n\n\n\\paragraph{Case of $r$ even.}\n\nTaking $M$ and $\\tilde{M}$ as given in \\eqref{MMtilde_Dr_even} and \nfollowing the same reasoning as for the case of $r$ odd, we get \nthe same conditions: \n\\begin{equation}\n\\frac{\\tilde{a}_1(r_i+1)}{2}\\in\\mathbb Z\\,, \\qquad i=1,\\dots,m\\,,\n\\label{=}\n\\end{equation}\nand, if $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \n\\ldots s_n < r$, \n\\begin{equation}\n\\frac{\\tilde{a}_1}{2}\\in\\mathbb Z\n\\label{<}\n\\end{equation}\nAdditionally, if there are no $D_{s_i}$ components in $\\mathfrak{h}$,\nthen\n\\begin{eqnarray}\n&&\\tilde a_1\\frac{r}{2}+\\tilde a_2\\in2\\mathbb Z\\hspace{1.6cm}\\text{for the }\n1^{\\rm st}\\text{ \\,embedding}\\,\\cr\n&&\\tilde a_1\\big(\\frac{r}{2}-1\\big)+\\tilde a_2\\in2\\mathbb Z\\qquad\\text{for \nthe }2^{\\rm nd}\\text{ embedding}\\,\n\\end{eqnarray}\n(the last two conditions differ only if all $r_i$ are odd and the\nrank inequality is saturated because in the other cases $\\tilde a_1$ has\nto be even). Examining the quantity \n\\eqref{Quantity_Dr_even} which has to be an integer with this information \nin mind and taking into account the admissibility conditions \nfor the levels, we deduce \n\n\\begin{prop}\nThe untwisted  coset models built with Lie algebra $\\mathfrak{g} \n=\\mathfrak{so}(2r)$, $r$ even, and a regular \nsubalgebra  $\\mathfrak{h} = A_{r_1}\n\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus \n\\ldots \\oplus D_{s_n}$ do not\nhave global gauge anomalies for the following cases \n\\begin{itemize}\n\\item$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd }  \\\\ \\hspace*{0.2cm}k \\in \\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ for any } Z \\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=Z_1\n\\ \\ \\ \n\\text{ for the } 1^{\\rm st}\\text{\\, embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=Z_{diag}\n\\text{ for the } 2^{\\rm nd}\\text{ embedding}\n\\end{array}\\right.$\n\\item\\hspace{-0.3cm} \n$\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\n\\text{ and some } r_{i}\\text{ even}\\end{array}\n\\right\\rbrace\\\\ \\hspace*{0.26cm}k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z = \\tilde{Z},\\, Z_1 \\text{ or } Z_\\text{diag} \\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z = Z_2 \\\\\n\\mathbb Z\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and any } Z\\\\\n\\end{array}\\right.$\n\\end{itemize}\nThe other untwisted models with admissible levels are not globally \ngauge invariant.\n\\end{prop}\n\n\n\n\\noindent{\\bf Remark}\\ \\ \\ In particular, the global gauge anomalies present\nif $\\mathfrak h=\\mathfrak g$ for $Z=Z_2$ and $k$ odd disappear \nfor $\\mathfrak h=A_{r_1}\\oplus \\ldots\\oplus A_{r_m}\n\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r\\,$ or if\n$\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\,$ with some\n$r_i$ even.\n\n\\paragraph{Example: $\\mathfrak{g}=D_4=\\mathfrak{so}(8)$.} Here $r$ \nand $r\/2$ are both even, so all levels \n$k \\in \\mathbb Z$ are admissible for all $Z$ and there are no anomalies \nin the case $\\mathfrak h = \\mathfrak g$ for $k$ even according\nto Proposition \\ref{prop_Dreven_h=g}, whereas \nthe cases with $k$ odd are anomalous. The possible (proper, nontrivial) \nsubalgebras $\\mathfrak h$ are: $A_1$, $A_2$, $2A_1$, $A_3$ (the latter \ntwo with 2 inequivalent embeddings), $D_2$, $D_3$, $2D_2$ and $A_1\\oplus D_2$.\nNote that the two embeddings of $2A_1$ and that of $D_2$ are \nrelated by the outer automorphisms of $D_4$ and similarly for \nthe two embeddings of $A_3$ and the one of $D_3$. For regular subalgebra \n$\\mathfrak h = A_1$ or $A_2$, \nthe inequality on ranks is strict and there are no $D_{s_i}$ so there\nare no anomalies for $k\\in\\mathbb Z$ for all $Z$. For $D_2$ or $D_3$,\nthe rank inequality is still strict and there are no anomalies for $k$\neven and all $Z$ and for $k$ odd and $Z=Z_2$. For $A_1\\oplus D_2$ or \n$2D_2$, the rank inequality is saturated and there are no anomalies \nfor even $k$ and any $Z$. Finally, for $2A_1$ or $A_3$ the rank inequality is \nsaturated and there are no $D_{s_i}$ so there are no anomalies for\n$k$ even and any $Z$ and for $k$ odd and $Z=Z_1$ for the 1$^{\\rm st}$ embedding\nand $Z=Z_{diag}$ for the 2$^{\\rm nd}$ one. \n\\vskip 0.4cm\n\nRecall from Sec.\\,\\ref{sec:D_reven_tw} that the twisted coset\nmodels for $\\mathfrak g=\\mathfrak{so}(2r)=\\mathfrak h$ \nwith $r>4$ even have gauge \nanomalies for  $Z=\\tilde Z$ (- theory) if $k$ is even and for \n$Z=\\tilde Z,\\,Z_1$ or $Z_{diag}$ if $k$ is odd for $r\/2$ even. \nThese are the cases where the no-anomaly condition (\\ref{Dreven_tw_c}) \nmay be violated. The restriction $e^{2\\ii\\pi\\tilde M}\\in\\tilde H$\nfor $\\mathfrak h=A_{r_1}\\oplus \\ldots\\oplus A_{r_m}\n\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r\\,$ or if\n$\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\,$ with some\n$r_i$ even imposes the condition $\\tilde a_1\\in 2\\mathbb Z$ removing \nthe anomalies in the case $Z=\\tilde Z$ (- theory) for $k$ even and, if,\nadditionally, there are no $D_{s_i}$ components in $\\mathfrak{h}$, \nalso for $Z\\not=Z_2$ and $k$ odd. If there are no $D_{s_i}$ and\n$r_1+1+\\ldots+r_m+1=r$ with all $r_i$ odd then for $k$ odd ($r\/2$ even)\nthe anomalies for $Z=\\tilde Z$ are removed for the $+$ theory in the\ncase of the 1$^{\\rm st}$ embedding and for the $-$ theory in the case of\nthe 2$^{\\rm nd}$ embedding, and for $Z=Z_1,\\ Z_{diag}$ in the case \nof both embeddings. We obtain this way \n\n\\begin{prop}\nThe twisted  coset models built with Lie algebra $\\mathfrak{g} \n=\\mathfrak{so}(2r)$, $r>4$ even, and a regular subalgebra  \n$\\mathfrak{h} = A_{r_1}\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \n\\oplus \\ldots \\oplus D_{s_n}$ do not have global gauge anomalies for \nthe following cases \n\\begin{itemize}\n\\item$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd }  \\\\ \\hspace*{0.2cm}k \\in \\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z=\\tilde Z \\text{ (+ theory) or } Z=Z_1,\\,Z_{diag} \n\\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z=Z_2\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=\\tilde Z\n\\text{ (+ theory) for the } 1^{\\rm st}\\text{\\, embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=\\tilde Z\n\\text{ \\,(- theory) \\hspace{0.07cm}for the } 2^{\\rm nd}\\text{ embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and }\nZ=Z_1,\\,Z_{diag}\n\\end{array}\\right.$\n\\item\\hspace{-0.3cm} \n$\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\n\\text{ and some } r_{i}\\text{ even}\\end{array}\n\\right\\rbrace\\\\ \\hspace*{0.26cm}k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z = \\tilde{Z},\\, Z_1 \\text{ or } Z_\\text{diag} \\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z = Z_2 \\\\\n\\mathbb Z\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and } \nZ=\\tilde Z,\\,Z_1,\\,Z_{diag}\n\\end{array}\\right.$\n\\end{itemize}\nThe other twisted models with admissible levels are not globally \ngauge invariant.\n\\end{prop}\n\n\nThe above results also hold for the coset model with \n$\\mathfrak g=\\mathfrak{so}(8)$ with twist $\\omega_1$,\nsee (\\ref{omegas}). Hence, for $\\mathfrak h=A_1$ or $A_2$\nthere are no gauge anomalies. For $D_2$ or $D_3$ there are\nno anomalies if $k$ is even for any $Z$ and if $k$ is odd\nfor $Z=Z_2$. For $A_1\\oplus D_2$ or $2D_2$ there are no anomalies\nif $k$ is even for $Z=\\tilde Z$ (+ theory) or $Z=Z_1,\\,Z_2,\\,Z_{diag}$\nor if $k$ is odd and $Z=Z_2$. Finally, for $2A_1$ or $A_3$ there are no\nanomalies for $Z=\\tilde Z$ (+ theory for the 1$^{\\rm st}$ embedding, \n- theory for the 2$^{\\rm nd}$ one) and for $Z=Z_1,Z_2,Z_{diag}$. In accordance\nwith the discussion of Sec.\\,\\ref{sec:D_reven_tw}, we may obtain\nthe result for twist $\\omega_2$ from the one for $\\omega_1$\nby applying the permutation $Z\\rightarrow\\omega_4(Z)$ induced \nby the outer automorphism $\\omega_4$ on the cyclic subgroups \nof $\\tilde Z$, see Eqs.\\,(\\ref{permutZ}), and on the one\n$\\mathfrak h\\rightarrow\\omega_4(\\mathfrak h)$  \non subalgebras (modulo inner automorphisms) induced by the\naction (\\ref{permutalpha}) of $\\omega_4$ on simple roots:\n\\begin{equation}\n\\begin{array}{ll}\\label{permuth}\n&\\hspace*{-0.6cm}\\omega_4(A_1)=A_1,\\ \\,\\omega_4(A_2)=A_2,\\ \\,\n\\omega_4((2A_1)^{(1)})\n=(2A_1)^{(2)},\\ \\,\\omega_4((2A_1)^{(2)})=D_2,\\ \\,\n\\omega_4(A_3^{(1)})=A_3^{(2)},\\\\\n&\\hspace*{-0.6cm}\\omega_4(A_3^{(2)})=D_3,\\ \\,\\omega_4(D_2)=(2A_1)^{(1)},\\ \\,\n\\omega_4(D_3)=A_3^{(1)},\\ \\,\\omega_4(2D_2)=2D_2,\\ \\,\n\\omega_4(A_1\\oplus D_2)=A_1\\oplus D_2.\n\\end{array}\n\\end{equation}\nwhere the superscript $(i),\\ i=1,2$, labels the independent embeddings.\nSimilarly, the result for twist $\\omega_3$ from the one for $\\omega_1$\nby applying the inverse permutations $Z\\rightarrow\\omega_4^{-1}(Z)$\nand $\\mathfrak h\\rightarrow\\omega_4^{-1}(\\mathfrak h)$.\n\\,For twists $\\omega_4,\\omega_4^{-1}$, the the remaining\ngauge anomalies are lifted if $\\mathfrak h=A_1$ or $A_2$ \nimposing the restrictions $\\tilde a_1,\\tilde a_2\\in 2\\mathbb Z$\nresulting in \n\\begin{prop}\nThe twisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(8)$ with twist $\\omega_4$ have global gauge anomalies \nfor regular subalgebras \n$\\mathfrak{h}=2A_1$,\\,$A_3$,\\,$D_2$,\\,$D_3$,\\,$2D_2$,\\,$A_1\\oplus D_2$ \nand $Z=\\tilde Z$ (- theory). The other cases of coset models with \nLie algebra $\\mathfrak{so}(8)$ and twist $\\omega_4$ are without anomalies.\n\\end{prop}\n\n\\noindent Similarly\n\n\n\\begin{prop}\nThe twisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(8)$ with twist $\\omega_4^{-1}$ have global gauge anomalies \nfor regular subalgebras $\\mathfrak{h}=2A_1$,\\,$A_3$,\\,$D_2$,\\,$D_3$,\\,$2D_2$\nand $A_1\\oplus D_2$ and $Z=\\tilde Z$ (($-$)$^k$ theory). The other cases \nof coset models with Lie algebra $\\mathfrak{so}(8)$ and twist $\\omega_4^{-1}$ \nare without anomalies.\n\\end{prop}\n\n\n\n\\subsection{Regular semisimple subalgebras of $\\mathfrak e_{6}$}\n\nIn this case with fixed rank $r=6$, one can establish a complete list \nof regular semisimple subalgebras, up to conjugation, with an embedding, \nhowever, that is not explicit \\cite{Dynkin,Lorente}. We shall only need \nthe embedding of simple roots\nin the ambient algebra which is enough to reconstruct the full embedding \nusing the Serre construction. The element $M$ and $\\tilde{M}$ will be\ndescribed employing the explicit\nrealization of the coweight and coroot lattices of $\\mathfrak{e}_6$,\n\\begin{equation}\\label{Coweight_Lattice_e6}\n P^\\vee(\\mathfrak{e}_6) = \\left\\lbrace \\left( \\dfrac{a}{6} + q_1, \\ldots,\n\\dfrac{a}{6} + q_6, \\dfrac{b}{\\sqrt{2}} \\right) \\left| \n\\begin{array}{l}a,b, q_1,\n\\ldots, q_6 \\in \\mathbb{Z}\\\\ a + q_1 + \\ldots + q_6 = 0\\\\ a + b \\in 2\\mathbb{Z}\n\\end{array} \\right. \\right\\rbrace\n\\end{equation}\nand the coroot lattice $Q^\\vee(\\mathfrak{e}_6)$ is defined the same way but\nadding the condition $a \\in 3 \\mathbb{Z}$. We shall consider only the\nuntwisted coset models because the twisted ones are non-anomalous,\nsee Proposition \\ref{prop_e6_h=g_tw}.\nTaking $M$ and $\\tilde{M}$ in\n$P^\\vee(\\mathfrak{e}_6)$ with the corresponding coefficients, the quantity\n\\eqref{Quantity_e6} becomes\n\\begin{equation}\\label{Quantity_e6_bis}\n k~\\text{tr}(M\\tilde{M}) = k \\dfrac{a \\tilde{a}}{3} + m, \\qquad \\text{ with }\nm \\in \\mathbb{Z}\n\\end{equation}\nNow, specifying a subalgebra $\\mathfrak{h}\\subset \\mathfrak e_{6}$ \nand requiring that $e^{2\\ii \\pi\n\\tilde{M}} \\in \\tilde{Z}\\cap\\tilde{H}$, two possibilities arise: if one can\nshow that $\\tilde{a} \\in 3\\mathbb{Z}$ then the previous quantity is an integer\nfor\nevery $k \\in \\mathbb{Z}$ and all the corresponding coset models are globally\ngauge invariant. Otherwise, if there exist an element $\\tilde{M}$ such that\n$\\tilde{a} \\notin 3\\mathbb{Z}$, then we have to require $k \\in 3\\mathbb{Z}$ to\nhave a globally gauge invariant coset model, and the other coset models \nare anomalous. Before examining the anomaly problem for every regular\nsubalgebra of $\\mathfrak{e}_6$, one can make four remarks:\n\n\\begin{itemize}\n\\item if there are no anomalies for a given subalgebra $\\mathfrak{h}$ of\n$\\mathfrak{e}_6$ ($\\tilde{a} \\in 3\\mathbb{Z}$), then the regular subalgebras \nthat are smaller (and will be obtained from the Dynkin diagram of \n$\\mathfrak{h}$ by the procedure described above) lead also\nto the condition $\\tilde{a} \\in 3\\mathbb{Z}$, inheriting \nit from $\\mathfrak{h}$. In other words, the regular subalgebra with \nno anomalies protects the cases of its regular subalgebras. Consequently, \nwe will look only at the cases where the anomalies are present and treat \nthe problem by decreasing rank.\n\n\n\\item Among the regular subalgebras generated by the algorithm described\nat the beginning of Sec.\\,\\ref{sec:4},\nmany can still be mapped into each other by the conjugations \nthat normalize $\\,\\mathfrak t_{\\mathfrak e_6}\\,$ (and induce on it Weyl group \ntransformations) and, as a result, they lead to the same condition \nfor the absence of anomalies. We may then consider only one regular \nsubalgebra in each class of subalgebras related by Weyl group transformations.\nIn particular, there are Weyl group transformations that permute the simple \nroots $\\,\\alpha_i\\,$ and $\\,\\delta=-\\phi\\,$ according to the symmetries of the \nextended Dynkin diagrams (see, e.g., Appendix B of \\cite{Gawedzki1})\nand they permit to restrict the count of regular subalgebras. \n\n\\item The subalgebras related by the outer automorphism of $\\mathfrak{e}_6$ \nlead to the same no-anomaly condition, see the remark at the end of\nSec.\\,\\ref{sec:2}.\n\n\\item Since $e^{2\\ii \\pi\\tilde{M}} \\in\\tilde Z\\cap \\tilde{H}$ if and only if\n$\\tilde M\\in P^\\vee(\\mathfrak g)$ and $\\tilde{M}+q^\\vee\\in\\ii\n\\mathfrak{t}_{\\mathfrak h}\\subset\\ii\\mathfrak{t}_\\mathfrak{g}$ for some\n$q^\\vee\\in Q^\\vee(\\mathfrak{e}_6)$, it is enough to check the no-anomaly\ncondition (\\ref{coset_anomaly}) only for $\\tilde{M}\\in P^\\vee(\\mathfrak g)$\nperpendicular to the orthogonal complement $\\ii\\mathfrak{t}^\\bot_{\\mathfrak h}$\nof $\\ii \\mathfrak{t}_{\\mathfrak h}$ in $\\ii\\mathfrak{t}_{\\mathfrak g}$.\n\\end{itemize}\n\n \nWe now consider the regular semisimple subalgebras, beginning \nby those of rank 6 and then decreasing the rank. Subspace \n$\\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$  (which is small for high\nranks) is computed for each subalgebra and we look at the consequences \nof the condition $\\tilde M\\perp \\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$ \non $\\tilde{M}$. Upon using the protection \nproperty and the Weyl transformations described above, as well as\nthe outer automorphism of $\\mathfrak{e}_6$, only a few cases have \nto be treated. The explicit computation is given in Table \n\\ref{Computation_e6_regular}. The subalgebras of rank 6 are not represented \nbecause we have $\\mathfrak{t}_\\mathfrak{h}^\\bot = \\emptyset$, so there is \nno supplementary condition for $\\tilde{M}$ and there are always anomalies if \n$k \\notin 3\\mathbb{Z}$. Only subalgebras of rank 5 and 4 have \npotential anomalies, the ones of lower ranks being protected by \na possible inclusion into non-anomalous subalgebras. \n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n   $\\mathfrak{h}$ & simple roots of $\\mathfrak{h}$ &\nbasis of $\\ii\\mathfrak{t}_{\\mathfrak{h}}^\\bot$ & $\\tilde{M}$ \\\\\n\\hline \\hline\n $D_5$ &  $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4,\n\\alpha_6$ & $\\left(1,1,1,1,1,\\text{-}5, 3\\sqrt{2} \\right)$ &\n$\\tilde{a} \\in 3\\mathbb{Z} $ \\\\\n\\hline\n $A_3 \\oplus 2A_1$ &  $\\alpha_1, \\alpha_2, \\alpha_3\n\\oplus \\delta \\oplus \\alpha_5$ & $\\left(1,1,1,1,\\text{-}2,\\text{-}2, 0\n\\right)$ & $\\tilde{a} \\in 3\\mathbb{Z} $ \\\\\n\\hline \n  $A_4 \\oplus A_1$ &  $\\alpha_1, \\alpha_2, \\alpha_3,\n\\alpha_4 \\oplus \\delta$ & $\\left(1,1,1,1,1,\\text{-}5, 0 \\right)$ &\n$\\tilde{a} \\in 3\\mathbb{Z} $\\\\\n\\hline\n  $A_5$ &  $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5$\n& $\\left(0,0,0,0,0,0, 1 \\right)$ & $\\tilde{a}\\in 2\\mathbb Z$ \\\\\n\\hline \n $2A_2\\oplus A_1$ &  $\\alpha_1, \\alpha_2 \\oplus\n\\alpha_4,\\alpha_5 \\oplus \\alpha_6$ & $\\left(1,1,1,-1,-1,-1,3\\sqrt{2}\n\\right)$ & $\\tilde{a}\\in\\mathbb Z$ \\\\\n\\hline \\hline\n $2A_2$ & $\\alpha_1, \\alpha_2 \\oplus \\alpha_4,\n\\alpha_5$ & $\\left(1,1,1,\\text{-}1,\\text{-}1,\\text{-}1, 0 \\right)$ &\n$\\tilde{a}\\in2\\mathbb Z$\\\\\n & & $\\left(0,0,0,0,0,0,1 \\right)$ & \\\\\n\\hline\n \\end{tabular}\n\\caption{$\\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$ for the regular\nsubalgebras of $\\mathfrak{e}_6$ of rank $5$ and $4$ and consequences \nfor $\\tilde a$; the simple roots $\\alpha_i$ of $\\mathfrak{e}_6$ and \nits lowest root $\\delta$ are used to generate the regular subalgebras\n\\cite{Lorente}.}\\label{Computation_e6_regular}\n\\end{table}\n\n\\noindent We are thus able to state\n\n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} \n= \\mathfrak{e}_6$ \nand any regular subalgebra $\\mathfrak{h}$ do not have global gauge anomalies \nfor every $k\\in\\mathbb{Z}$, except for the cases \n$\\mathfrak{h} = \\mathfrak{e}_6, A_5 \\oplus A_1, 3A_2$, \nof rank 6, $A_5, 2A_2\\oplus A_1$,\nof rank 5, and $2A_2$ of rank 4, where the only globally gauge\ninvariant models are those with $k \\in 3 \\mathbb{Z}$. \n\\end{prop}\n\n\\section{R-subalgebras and S-subalgebras}\n\\label{sec:5}\n\nThe regular subalgebras are not the only possible Lie subalgebras for a given\nambient Lie algebra. We can use them, however, to classify all the remaining \nones. Let $\\mathfrak{h}$ be a semisimple subalgebra of $\\mathfrak{g}$. \nLet $\\mathcal{R}(\\mathfrak{h})$ be a minimal regular\nsubalgebra of $\\mathfrak{g}$ containing $\\mathfrak h$ (up to conjugation). \nIf $\\mathcal{R}(\\mathfrak{h}) =\\mathfrak{g}$, then \n$\\mathfrak{h}$ is called an S-subalgebra. Otherwise, \nit is called an R-subalgebra. For the exceptional simple algebras, the \nclassification of R- and S-subalgebras has been achieved by Dynkin \nin \\cite{Dynkin}. \nThe case of other simple algebras was discussed in \\cite{Dynk2} with\nless explicit results. In this section, we first treat completely the \ncase of non-regular subalgebras of the exceptional Lie algebra \n$\\mathfrak{g} = \\mathfrak{e}_6$ which may have anomalies and then we consider \nsome examples of non-regular subalgebras of classical Lie algebras.\n\n\\paragraph{Dynkin index.} Consider a simple Lie subalgebra $\\mathfrak\nh \\subset \\mathfrak g$ of a semisimple Lie algebra $\\mathfrak g$ \nand the corresponding embedding $\\iota$. \\,The relation \n\\begin{equation}\n  {\\rm tr}_{\\mathfrak{g}}(\\iota(X))^2\\,=\\,j\\,{\\rm tr}_{\\mathfrak{h}}X^2 \\qquad {\\rm for}\\quad X \\in \\mathfrak h\\,\n\\end{equation}\n\\noindent where the invariant quadratic forms $\\,{\\rm tr}_{\\mathfrak g}\\,$ \nand $\\,{\\rm tr}_{\\mathfrak h}\\,$ have the normalizations described in\nthe beginning of Sec.\\,\\ref{sec:2}, \ndefines the scalar factor $\\,j\\,$ (independent of $X$), \n called Dynkin index, which\nis always an integer \\cite{Dynkin}. Moreover, $\\,j\\,$ is invariant under\ncomposition of $\\,\\iota\\,$ with inner (and outer) automorphisms of \n$\\mathfrak g$, so that it depends on the class of equivalent \nembeddings.\n\n\n\\subsection{Simple nonregular subalgebras of $\\mathfrak{e}_6$}\n\\label{sec:SimpleCase}\n\n\\paragraph{Subalgebras of rank 1}\n\nAccording to Dynkin, the subalgebra $\\mathfrak{h} = A_1$ can be embedded in\nseveral different ways in $\\mathfrak{e}_6$, as regular, R- and S-subalgebra and\nthe embedding $\\iota$ is fully characterized by the embedding \nof the simple coroot $\\alpha^\\vee$ of $A_1$. Recall the compatibility \ncondition for $\\tilde{M}$ in the anomaly problem\n\n\\begin{equation}\n e^{2\\ii \\pi \\tilde{M}} \\in \\tilde{H} \\cap \\tilde{Z} \\subseteq\n\\mathcal{Z}(\\tilde{H})\\,,\n\\end{equation}\nwhere  $\\mathcal{Z}(\\tilde{H})=\\{1,e^{2\\ii\\pi\\iota(\\lambda^\\vee)}\\}$ \nwith $\\lambda^\\vee =\\frac{1}{2} \\alpha^\\vee$ is the center of $\\tilde H$\nwhich is either trivial (if $1=e^{2\\ii\\pi\\iota(\\lambda^\\vee)}$ and \n$\\tilde H\\cong SO(3)$) or is isomorphic to $\\mathbb Z_2$\n(if $1\\not=e^{2\\ii\\pi\\iota(\\lambda^\\vee)}$ and $\\tilde H\\cong SU(2)$). \nLooking at the embedding of $\\lambda^\\vee$ in $\\mathfrak{e}_6$, three\npossibilities can occur \n\n\n\\begin{enumerate}\n \\item If $\\iota(\\lambda^\\vee) \\notin P^\\vee(\\mathfrak{e}_6)$ \nthen $\\tilde{Z}\n\\cap \\tilde{H} = \\lbrace 1 \\rbrace$ and $\\tilde{M}$ is a coroot of\n$\\mathfrak{e}_6$, so the quantity \\eqref{Quantity_e6} is always \nan integer and there are no anomalies for this model.\n\\item If $\\iota(\\lambda^\\vee) \\in Q^\\vee(\\mathfrak{e}_6)$ \nthen $\\tilde{M}$ is still\nonly a coroot of $\\mathfrak{e}_6$, and there are no anomalies too.\n\\item If $\\,\\iota(\\lambda^\\vee) \\in P^\\vee(\\mathfrak{e}_6)\\setminus \nQ^\\vee(\\mathfrak{e}_6)$ then anomalies are\npossible and we have to check that the quantity \\eqref{Quantity_e6}\nis an integer for $\\tilde{M} = \\iota(\\lambda^\\vee)$ looking at the\ncorresponding value for $\\tilde{a}$, see Eq.\\,(\\ref{Quantity_e6_bis}).\n\\end{enumerate}\nThe explicit embeddings are given in \\cite{Dynkin} (Chapter III, Table 18), and\nthe computation of the intersection with the roots of $\\mathfrak{e}_6$ is done\nin Table \\ref{Computation_e6_rank1} for each subalgebra of rank 1: the\npossibility 3 never occurs, so there are no anomalies for \nthe corresponding coset models for any $k \\in \\mathbb{Z}$.\n\n\\begin{table}[htb]\n\\centering \n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ & Compatibility\\\\\n\\hline \\hline\n$A_1$ & 1 & $ \\left(0,0,0,0,0,0,\\frac{1}{\\sqrt{2}}\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \\hline\n$2 A_1$ & 2 &$ \\left( \\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\tfrac{1}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$3 A_1$ & 3 & $ \\left(\n\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{\\text{-}1}{4},\\tfrac{\\text{-}1}{4}\n,\\tfrac{\\text{-}1}{4},\\tfrac{3}{2\\sqrt{2}} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2$ & 4 &$ \\left( 0,0,0,0,0,0,\\sqrt{2} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2 \\oplus A_1$ & 5 & $ \\left(\n\\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\sqrt{2} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2 \\oplus 2A_1$ & 6 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\sqrt{2}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ 2A_2$ & 8 &$ \\left(1,0,0,0,0,\\text{-}1,\\sqrt{2}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$2 A_2 \\oplus A_1$ & 9 & $ \\left(\n\\tfrac{3}{4},\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{\\text{-}1}{4},\\tfrac{\\text{-}1}{4}\n,\\tfrac{\\text{-}3}{4},\\tfrac{5}{2\\sqrt{2}} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 $ & 10 & $ \\left( \\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\tfrac{3}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 \\oplus A_1$ & 11 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{3\n}{\\sqrt{2}} \\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 \\oplus 2A_1$ & 12 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{3}{\\sqrt{2}} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_4$ & 20 & $ \\left( 1,0,0,0,0,\\text{-}1,2\\sqrt{2} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_4 \\oplus A_1$ & 21 & $\n\\left(1,\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\text{-}1,2\\sqrt{2}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ D_4$ & 28 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{5}{\\sqrt{2}} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ D_5 (a_1)$ & 30 & $\n\\left(1,\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\text{-}1,\\tfrac{5}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_5$ & 35 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}3}{2},\\tfrac{5\n}{\\sqrt{2}}\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_5 \\oplus A_1$ & 36 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}3}{2},\\tfrac{5}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$D_5 $ & 60 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}3}{2},\\tfrac{7}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \\hline\n$ \\mathfrak{e}_6 (a_1)$ & 84 &$\n\\left(2,1,0,0,\\text{-}1,\\text{-}2,4\\sqrt{2}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$\\mathfrak{e}_6 $ & 156 & $ \\left(\n\\tfrac{5}{2},\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}3}{2}\n,\\tfrac{\\text{-}5}{2},\\tfrac{11}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \n  \\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for rank 1\nsubalgebras and its intersection with the lattices of\n$\\mathfrak{e}_6$.}\n\\label{Computation_e6_rank1}\n\\end{table}\n\n\n\\paragraph{Simple S-subalgebras of rank > 1}\n\nFollowing \\cite{Dynkin} (Chapter IV, Table 24), there exist four S-subalgebras\nof $\\mathfrak{e}_6$ of rank $>1$: $\\mathfrak{h} = A_2, \\mathfrak{g}_2, C_4$ \nand $\\mathfrak{f}_4$. For the cases $\\mathfrak{g}_2$ and $\\mathfrak{f}_4$, \nthe center\nof the corresponding group is $\\mathcal{Z}(\\tilde{H}) \\cong \\lbrace 1 \\rbrace$.\nThen $\\tilde{M}$ can be only a coroot of $\\mathfrak{e}_6$ and the quantity\n\\eqref{Quantity_e6} is always an integer. For the two remaining cases, the\nexplicit\nembedding is still given in \\cite{Dynkin}, and the strategy is the same as for\nrank one: we look how the generating element $\\iota(\\lambda^\\vee)$ of\n$\\mathcal{Z}(\\tilde{H})$ intersects with the lattices of $\\mathfrak{e}_6$ and\ncheck which possibility occurs among those listed in the case of rank one\n(except that we would also have to check that for the low multiples \nof $\\lambda^\\vee$ if $\\iota(\\lambda^\\vee)$ were\nnot in $Q^\\vee(\\mathfrak{g})$). The results are described in Table \n\\ref{Computation_e6_S} from which we infer that there are no gauge anomalies \nfor all simple S-subalgebras of $\\mathfrak e_6$.\n\n\\begin{table}[htb]\n \\centering\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ &\nCompatibility \\\\\n\\hline \\hline\n$A_2$ & $\\mathfrak{e}_6$ & 9 & $ \\left(\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\n\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\n\\tfrac{3}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$C_4$ & $\\mathfrak{e}_6$ & 1 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{1}{\\sqrt{2}}\\right)$ &\n$\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n\\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for simple S-subalgebras of\n$\\mathfrak{e}_6$ and its intersection with the lattices.}\n\\label{Computation_e6_S}\n\\end{table}\n\n\\paragraph{Simple R-subalgebras of rank > 1}\\label{SimpleR}\n\nWe only need to look at the R-subalgebras $\\mathfrak{h}$ with potential \nanomalies. Indeed, the subalgebra $\\mathcal{R}(\\mathfrak{h})$ is \nregular, so has been already treated. If $\\mathcal{R}(\\mathfrak{h})$ \ncorresponds to a model without anomalies, then it protects also \nthe R-subalgebra $\\mathfrak{h}$ included in it and there will be no anomalies \nfor the model built with $\\mathfrak{h}$. The list of the R-subalgebras of \n$\\mathfrak{e}_6$ is\ngiven in \\cite{Dynkin} (Chapter IV, Table 25), but without explicit embedding.\nThere remain five cases with potential anomalies: $\\mathfrak{h} = A_2$, with\n$\\mathcal{R}(\\mathfrak{h}) = A_5$,\\,$2A_2$,\\,$3A_2$, and  $\\mathfrak{h} = A_3$ \nor $C_3$ with $\\mathcal{R}(\\mathfrak{h}) = A_5$. If\n$\\mathcal{R}(\\mathfrak{h})$ is simple, then the embedding of $\\mathfrak{h}$ in\n$\\mathcal{R}(\\mathfrak{h})$ is given in \\cite{Lorente} (Table XIII), \nconsidering\n$\\mathfrak{h}$ as an S-subalgebra of $\\mathcal{R}(\\mathfrak{h})$. \n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n  \\hline\n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ &\nCompatibility & $\\tilde{a}$ \\\\\n\\hline \\hline\n$A_2$ & $2A_2(\\iota_1)$ & 2 & $ \\left(\n\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\\text{-}2}{3},\\tfrac{2}{3},\\tfrac{\\text{-}1}{3}\n,\\tfrac{\\text{-}1}{3},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ &\\\\\n\\hline\n$A_2$  & $2A_2 (\\iota_2)$ & 2 & $ \\left(\n\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\\text{-}2}{3},\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\n\\text{-}2}{3},0\\right)$ &\n$\\in P^\\vee(\\mathfrak{e}_6)\\setminus Q^\\vee(\\mathfrak{e}_6)$ & 2\\\\\n\\hline\n$A_2$ & $3A_2(\\iota_1)$ & 3 & $ \\left(\n0,0,-1,1,0,0,0\\right)$ &\n$\\in Q^\\vee(\\mathfrak{e}_6)$ &\\\\\n\\hline\n$A_2$ & $3A_2(\\iota_2)$ & 3 & $ \\left(0,0,-1,\\tfrac{2}{3},\\tfrac{2}{3}\n,\\tfrac{\\text{-}1}{3},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n$A_2$ & $A_5$ & 5& $ \\left(\n\\tfrac{2}{3},\\tfrac{2}{3},\\tfrac{\\text{-}1}{3},\\tfrac{2}{3},\\tfrac{\\text\n{-}1}{3},\\tfrac{\\text{-}4}{3},0\\right)$ &\n$\\in P^\\vee(\\mathfrak{e}_6)\\setminus Q^\\vee(\\mathfrak{e}_6)$ & 4\\\\\n\\hline\n$A_3$ & $A_5$ & 2 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},0\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n$C_3$ & $A_5$ & 1 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n\\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for simple R-subalgebras of\n$\\mathfrak{e}_6$ and its intersection with the lattices. In case of potential\nanomalies, the explicit value of $\\tilde{a}$ that enters quantity\n\\eqref{Quantity_e6_bis} is given.}\n\\label{Computation_e6_R}\n\\end{table}\n\n\n\\noindent If $\\mathcal{R}(\\mathfrak{h})$ is only semisimple, the problem \nof the embedding is\ntreated in \\cite{Minchenko}, where several inequivalent embeddings of\n$\\mathfrak{h}$ in $\\mathfrak{e}_6$ appear. \nFor the $\\mathfrak{h} = A_2$ and $\\mathcal{R}(\\mathfrak{h}) = 3 A_2$, the two\ninequivalent embeddings are the following, denoting by $\\tilde\\alpha_1^\\vee$ \nand $\\tilde\\alpha_2^\\vee$ the simple coroots of $A_2$.\n\\begin{align}\n \\iota_1(\\tilde\\alpha_1^\\vee) = \\alpha_1^\\vee + \\alpha_5^\\vee \n+ \\delta^\\vee & \\qquad &\n\\iota_2(\\tilde\\alpha_1^\\vee) = \\alpha_1^\\vee + \\alpha_4^\\vee + \\delta^\\vee\\\\\n \\iota_1(\\tilde\\alpha_2^\\vee) = \\alpha_2^\\vee \n+ \\alpha_4^\\vee + \\alpha_6^\\vee & \\qquad &\n\\iota_2(\\tilde\\alpha_2^\\vee) = \\alpha_2^\\vee + \\alpha_5^\\vee + \\alpha_6^\\vee\n\\end{align}\nwhere we have exchanged $\\alpha^\\vee_4$ and $\\alpha^\\vee_5$. The other possible\nexchanges are equivalent to $\\iota_1$ or $\\iota_2$ \\cite{Minchenko}. For\n$\\mathcal{R}(\\mathfrak{h}) = 2A_2$, the two embeddings are given by\nsimilar formulas but with omission of $\\alpha^\\vee_6$ and $\\delta^\\vee$. \nAgain, in order to find $Z(\\tilde H)\\cap\\tilde Z$, we have to check how \nthe generating element $\\iota(\\lambda^\\vee)$ of \n$\\mathcal{Z}(\\tilde{H})$ intersects \nwith the lattices of $\\mathfrak{e}_6$. An explicit calculation is done in \nTable \\ref{Computation_e6_R}, and this time potential anomalies occur. \nThen, looking at the value of $\\tilde{a}$ for $\\tilde{M} =  \n\\iota(\\lambda^\\vee)$, we deduce an, eventually more restrictive, condition \non level $k$ required to avoid the anomalies (to exclude the anomalies in \nthe case of $A_3\\subset A_5$, we also have to observe that $\\iota(2\\lambda^\\vee)\n\\in Q^\\vee(\\mathfrak{e}_6)$). \n\\eject\n\n\\noindent This way, we obtain the general result for \nsimple nonregular subalgebras of $\\mathfrak{e}_6$\n\n\\begin{prop}\nThe untwisted coset models with $\\mathfrak{g} = \\mathfrak{e}_6$ and any\nsimple, nonregular subalgebra $\\mathfrak{h}$ do not have global  gauge\nanomalies for $k \\in \\mathbb{Z}$ except for the R-subalgebras \n$\\mathfrak{h} = A_2$ with $\\mathcal{R}(\\mathfrak{h}) = A_5$ and \n$\\mathfrak h=A_2$ with $\\mathcal{R}(\\mathfrak h)=2A_2$ embedded \nvia $\\iota_2$. For those subalgebras, the global gauge invariance requires \nthat $k \\in 3\\mathbb{Z}$.\n\\end{prop}\n\n \n\n\\subsection{Semisimple nonregular subalgebras of $\\mathfrak{e}_6$}\n\nLet $\\mathfrak h$ be a semisimple subalgebra of $\\mathfrak e_6$:\n\\begin{equation}\n \\mathfrak h = \\mathop{\\oplus}\\limits_{i=1}^{n} \\mathfrak h_i\n\\end{equation}\nwhere the $\\mathfrak h_i$ are simple, and the corresponding subgroups \nare denoted by $\\tilde H_i$. The case $n =1$ has been already treated \nabove, so we now deal with $n \\geq 2$. First, suppose that one of the \n$\\mathfrak h_i$ considered as a simple subalgebra leads to anomalies: \nthere exists $\\tilde M_i$ such that $e^{2 \\ii \\pi \\tilde{M}_i} \\in \n\\tilde H_i \\cap \\tilde Z$ which imposes $k \\in 3 \\mathbb Z$ to ensure \nthat the quantity $(\\ref{Quantity_e6})$ is integral. Then, taking \n$\\tilde M = \\tilde M_i$ but now embedded in $\\mathfrak h$,\nwe shall still have to impose $k \\in 3\\mathbb Z$ \nto have a globally gauge invariant model with semisimple Lie algebra \n$\\mathfrak h$. In other words, semisimple algebras composed of simple \nideals with at least one leading to anomalies are also anomalous. \nHowever, the inverse \nis not true: one can have a semisimple subalgebra corresponding to \nan anomalous model with all its simple ideals without any anomaly. For \nexample, the model with regular subalgebra $2A_2$ of \n$\\mathfrak e_6$ is anomalous for $k \\in \\mathbb{Z}\\setminus 3\\mathbb{Z}$ \nwhereas the one with $A_2$ (still regular) is globally gauge invariant \nfor every $k \\in \\mathbb{Z}$. Thus we need to check all the cases where \nall the simple ideals correspond to models without anomaly. To do that, \nwe need to consider the elements $\\sum_{i=1}^n \\alpha_i \\iota \n(\\lambda^\\vee_i)$ where $\\alpha_i \\in \\mathbb{Z}$ and $\\lambda_i$ are \nthe generating elements of the center of the $\\tilde H_i$, which have \nall been described above in the simple case (Tables \n\\ref{Computation_e6_rank1}, \\ref{Computation_e6_S} and \n\\ref{Computation_e6_R}), and $\\iota: \\mathfrak h \\rightarrow \n\\mathfrak e_6$ is the embedding. Comparing how these elements are \ncompatible with the coroot and coweight lattices of $\\mathfrak e_6$, \nthe anomaly problem is reduced to the three possibilities described in \nthe simple case \\ref{sec:SimpleCase}. \n\n\\paragraph{S-subalgebras}\n\nIn \\cite{Dynkin}\n(Chapter V, Table 39) one can find all the S-subalgebra of $\\mathfrak e_6$ \nand their including relations. It turns out that subalgebra\n $\\mathfrak h = \\mathfrak g_2 \\oplus A_2$ \n(with the explicit embedding given in \\cite{Dynkin}, Chapter V, Table 35) \nleads to an anomaly if $k \\in \\mathbb Z \\setminus 3 \\mathbb Z$, and that \nthe other semisimple nonsimple S-subalgebras of $\\mathfrak e_6$ are protected. \n\n\\begin{table}[ht]\n\\centering\n  \\begin{tabular}{|c|c|c|c|c|}\n  \\hline \n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & $\\left\\lbrace \n\\mathcal{R}(\\mathfrak{h}_i) \\right\\rbrace$ & Indices &\n No anomaly for \\\\\n\\hline \\hline\n\n$A_2 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 2,1 &\n${k \\in 3\\mathbb{Z}}$  \\\\\n$A_3 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 2,1 &$k \\in \\mathbb{Z}$  \\\\\n$C_3 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 1,1 &$k \\in \\mathbb{Z}$  \\\\\n$A_1 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 35,1&$k \\in \\mathbb{Z}$  \\\\\n$(A_2 (\\iota_1) \\oplus A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ \n& $A_5, A_1$ & 2,3,1 &$k \\in\n\\mathbb{Z}$  \\\\\n$(A_2 (\\iota_2) \\oplus A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ \n& 2,3,1 &${k \\in\n3\\mathbb{Z}}$  \\\\\n$(2A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ \n& 8,3,1&$k \\in \\mathbb{Z}$  \\\\\n\\hline\n$A_1 \\oplus (2A_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 4,1,1&${k \\in\n3\\mathbb{Z}}$  \\\\\n$A_2 \\oplus A_2 (\\iota_1) $ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ &1,2 &$k \\in\n\\mathbb{Z}$  \\\\\n$A_2 \\oplus A_2 (\\iota_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 1,2&${k \\in\n3\\mathbb{Z}}$  \\\\\n$A_1 \\oplus A_2 (\\iota_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 4,2&${k \\in\n3\\mathbb{Z}}$  \\\\\n\\hline\n$A_1 \\oplus A_1 \\oplus A_2$ & $A_2 \\oplus A_2 \\oplus A_2$ \n& $A_2, A_2, A_2$ & 4,4,1&\n$k\\in \\mathbb{Z}$  \\\\\n\\hline\n$A_2(\\iota_2) \\oplus A_1$ & $A_5$ & $A_5 $ \n& 2,3&${k \\in 3\\mathbb{Z}}$  \\\\\n\\hline\n$  A_1 \\oplus A_2 (\\iota_2)$ & $A_1 \\oplus (2A_2)\n$ & $A_1, 2A_2$ & 1,2&${k \\in 3\\mathbb{Z}}$  \\\\\n\\hline\n$A_1 \\oplus A_1 \\oplus A_2$ & $A_1 \\oplus A_2 \\oplus A_2$ \n& $A_1, A_2, A_2$  & 1,4,1&$k \\in \\mathbb{Z}$  \\\\\n\\hline\n\\end{tabular}\\small\n\\caption{Semisimple nonsimple R-subalgebras of\n$\\mathfrak{e}_6$ with possible anomalies and the conditions\non $k$ required for their absence}\n\\label{Computation_e6_ssimple}\n\\end{table}\n\n\n\n\\paragraph{R-subalgebras}\nThe end of \\cite{Lorente} proposes a method to construct all the\nsemisimple R-subalgebras: the idea is to take the semisimple S-subalgebras \nof the semisimple regular subalgebras of $\\mathfrak e_6$, treating each \nsemisimple ideal independently. The semisimple S-subalgebras are described \nfor the classical \nalgebras up to rank 6 in \\cite{Lorente}, which is enough to construct all \nthe semisimple R-subalgebras of $\\mathfrak e_6$. However, we only need to \ntreat the R-subalgebras $\\mathfrak h$ where the regular \nsubalgebras $\\mathcal R(\\mathfrak h)$ lead to an anomaly problem, because\nthe other cases are protected against anomalies. \nThe computation is given in Table \\ref{Computation_e6_ssimple}, using \nthe fact that one ideal leads to an anomaly or computing the elements \nof the center as described before. Note that for the nonsimple S-subalgebra \n$A_2 \\oplus A_1 \\subset A_5$, $A_2$ is actually embedded in \n$A_2 \\oplus A_2$ \\cite{Lorente}, so the question of the two inequivalent \nembeddings $\\iota_1$ and $\\iota_2$ arises also here, as in \\ref{SimpleR}. \nWorking by decreasing rank, we have excluded some algebras from this Table \nsince they are protected by the ones of higher rank that do not have anomalies.\n\n\\eject\n\n\n\n\n\n\n\n\n\n\n\\noindent Putting all that together, we obtain the following result:\n\n\\begin{prop}\nThe untwisted coset models with $\\mathfrak{g} = \\mathfrak{e}_6$ and any\nnonregular nonsimple semisimple subalgebra \n$\\mathfrak{h}$ do not have global  gauge anomaly\nfor $k \\in \\mathbb{Z}$, except for the S-subalgebra $\\mathfrak{h} =\n\\mathfrak{g}_2 \\oplus A_2$ and the R-subalgebras appearing in \nTable \\ref{Computation_e6_ssimple} with the condition $k\\in 3\\mathbb Z$\nwhich exhibit global gauge anomaly for $k\\in\\mathbb Z\\setminus3\\mathbb Z$.\n\\end{prop}\n\n\\subsection{Examples of nonregular subalgebras of classical Lie algebras}\n\nThe semisimple nonregular subalgebra of classical algebra have been \nclassified explicitly in \\cite{Lorente} only up to rank 6. The general \nclassification proposed by Dynkin in \\cite{Dynk2} is less explicit and \ndoes not allow us to treat the anomaly problem in a general form as for \nregular subalgebras. Here we only give some example of classical algebras, \nbut the method is always the same once the explicit embedding of a \nsubalgebra is known : as for $\\mathfrak{e}_6$, we need to look how \nthe embedding of the generating element of the center of the considered \nsubalgebra is compatible with the coroot lattice of the ambient algebra.\n\n\\paragraph{Nonregular semisimple subalgebras of $A_4$.}\n\nThe coroot lattice of $A_4$ is given by\n\\begin{equation}\n  P^\\vee(A_4) = \\left\\lbrace \\left. \\left( \\dfrac{a}{5} + q_1, \\dots , \n\\dfrac{a}{5} + q_4, -\\dfrac{4a}{5} - q_1 - \\dots - q_4\\right) \\right| a, \nq_1, \\dots q_4 \\in \\mathbb Z \\right\\rbrace\n\\end{equation}\nand the coweight lattice $Q^\\vee(A_4)$ is given by the same formula but \nwith $a = 0$. According to \\cite{Lorente}, $A_4$ admits two S-subalgebras \nwhich are simple : $A_1$ and $B_2$. For $\\mathfrak h = A_1$, the embedding \nof the generating element $\\lambda^\\vee$ of the center of the corresponding \ngroup is given by\n\\begin{equation}\n  \\iota(\\lambda^\\vee) = (2,1,0,-1,-2) \\in Q^\\vee(A_4)\n\\end{equation}\nso the quantity $k$ tr$(M\\tilde M)$ will be integral for every $k \\in \n\\mathbb Z$ and there will be no anomaly for this model. For $\\mathfrak{h} \n= B_2$, one have\n\\begin{equation}\n   \\iota(\\lambda^\\vee) = (1,0,0,0,-1) \\in Q^\\vee(A_4)\n\\end{equation}\nwhich leads to the same conclusion. As we have seen in the regular case, \nall regular subalgebras of $A_4$ (except $A_4$) leads to non-anomalous \nmodels. We immediately conclude that all the R-subalgebra of $A_4$ are \nprotected by their regular $\\mathcal R(\\mathfrak h)$, so there is also \nno anomaly for these models. Finally, the only anomalous models corresponding \nto $\\mathfrak g = A_4$ and an arbitrary semisimple subalgebra are those with \n$\\mathfrak h = \\mathfrak g$, $Z = \\tilde Z \\cong Z_5$ and $k \\in \\mathbb Z \n\\setminus 5\\mathbb Z$. \n\n\\paragraph{S-subalgebras of $A_5$.}\n\nThe coroot lattice of $A_5$ is given by\n\\begin{equation}\n  P^\\vee(A_5) = \\left\\lbrace \\left. \\left( \\dfrac{a}{6} + q_1, \\dots , \n\\dfrac{a}{6} + q_5, -\\dfrac{5a}{6} - q_1 - \\dots - q_5\\right) \\right| a, \nq_1, \\dots q_5 \\in \\mathbb Z \\right\\rbrace\n\\end{equation}\nand the coweight lattice $Q^\\vee(A_5)$ is given by the same formula \nbut with $a = 0$. According to \\cite{Lorente}, $A_5$ admits six \nS-subalgebras : $A_1$, $A_2$, $A_3$, $C_3$, $A_1 \\oplus A_1$ and \n$A_2 \\oplus A_1$. For $\\mathfrak{h}=A_1$, one has   \n\\begin{equation}\n\\iota(\\lambda^\\vee) = \\left( \\dfrac{5}{2},\\dfrac{3}{2},\\dfrac{1}{2},\n-\\dfrac{1}{2},-\\dfrac{3}{2},-\\dfrac{5}{2} \\right),\n\\end{equation}\nsee Table VI of \\cite{Lorente}, whereas for $\\mathfrak h = A_2,\\,A_3$ \nand $C_3$, one has \n\\begin{eqnarray}\n\\iota(\\lambda^\\vee) = \\left( \\dfrac{2}{3},\\dfrac{2}{3},-\\dfrac{1}{3},\n\\dfrac{2}{3},-\\dfrac{1}{3},-\\dfrac{1}{3} \\right),\\ \n\\left( \\dfrac{1}{2},\\dfrac{1}{2},-\\dfrac{1}{2},\n\\dfrac{1}{2},-\\dfrac{1}{2},-\\dfrac{1}{2} \\right),\\ \n\\left( \\dfrac{1}{2},\\dfrac{1}{2},\\dfrac{1}{2},\n-\\dfrac{1}{2},-\\dfrac{1}{2},-\\dfrac{1}{2} \\right),\n\\end{eqnarray}\nrespectively, see the last 3 entries of Table 5 above.\nIn all 4 cases, $\\iota(\\lambda^\\vee)\\in P^\\vee(A_5) \\setminus Q^\\vee(A_5)$.\nTaking $\\iota(\\lambda^\\vee) = \\tilde M$ with $\\tilde a = 3,4,3,3$, respectively,\nand appropriate $\\tilde q_i$, and $M\\in P^\\vee(A_5)$ such that \n$e^{2\\ii\\pi M}\\in Z\\cong\\mathbb{Z}_p$, we obtain\n  \\begin{equation}\n    \\text{tr}(M\\tilde M) = \\dfrac{5 a \\tilde{a}}{p} + n\\,,\n  \\end{equation}\nwhere $n \\in \\mathbb Z$. There will be no anomaly for $k$ such that \n$k\\,{\\rm tr}(M\\tilde M) \\in \\mathbb Z$. \nFor $\\tilde a=3$, this imposes on $k$ the same \nrestrictions that the admissibility conditions (\\ref{Consistency_Ar}), \nso that the untwisted coset theories corresponding to the S-subalgebras \n$\\mathfrak{h}=A_1,A_3,C_3\\subset A_5$ do not have anomalies.\n\\,For the S-subalgebra $\\mathfrak{h}=A_2$, we obtain\nthe non-anomalous models with admissible levels for \n\\begin{equation}\n  k \\in \\left\\lbrace \n    \\begin{array}{ll}\n\\mathbb Z \\cap 2 \\mathbb Z = 2 \\mathbb Z &\\text{if } Z\\cong\\mathbb Z_2\\\\      \n3 \\mathbb Z   &\\text{if } Z\\cong\\mathbb Z_3\\\\\n3 \\mathbb Z \\cap 2 \\mathbb Z = 6 \\mathbb Z &\\text{if } Z\\cong\\mathbb Z_6      \n    \\end{array}\\right.\n\\end{equation}\nThe other untwisted models corresponding to the S-subalgebra \n$\\mathfrak h = A_2\\subset A_5$ and non-trivial subgroups $Z$ are anomalous.\n\\vskip 0.1cm\n\nThere are no conceptual or technical difficulties to obtain the no-anomaly \nconditions on $k$ for other subalgebras of $A_5$, and also for other \nclassical algebra $\\mathfrak g$, once the embeddings are known, but there \nis no general result so each case has to be treated separately. The previous \nexamples show that different anomaly conditions could appear according to \nthe subalgebra considered.\n\n\\section{Conclusions}\n\nWe have studied above the conditions for the absence of global gauge\nanomaly in the coset models of conformal field theory derived from\nWZW models with connected simple compact groups $G=\\tilde G\/Z$ as the targets  \nby gauging a subgroup of the rigid adjoint or twisted-adjoint symmetries\n$\\,G\\ni g\\mapsto hg\\hspace{0.03cm}\\omega(h)^{-1}\\in G$, \\,where $\\omega$ is \na, possible trivial, automorphism of $G$. \\,The full group of such symmetries \nis equal to $\\tilde G\/Z^\\omega$, where $Z^\\omega$ is the maximal subgroup \nof the center $\\tilde Z$ of the universal covering group $\\tilde G$ \nof $G$ for which the (twisted) adjoint action \nis well defined. We considered both the coset models where the full group \n$\\tilde G\/Z^\\omega$ was gauged and the ones where the gauging concerned only \na closed connected subgroup of $\\tilde G\/Z^\\omega$. Global gauge \nanomalies obstructing the invariance of the Feynman amplitudes of the theory \nunder ``large'' gauge transformations non-homotopic to unity may appear \nonly for non-simply connected groups $G$ corresponding to Lie algebras \n$\\mathfrak{g}$ of types $A_r,\\,D_r$ and $\\mathfrak{e}_6\\,$ (that are all\nsimply-laced). Using the results\n\\cite{Dynkin,Lorente,Minchenko} on the classification of semisimple \nLie subalgebras of simple Lie algebras, we obtained a complete \nlist of non-anomalous coset models (without boundaries) for groups $G$ \nwith the Lie algebra $A_r,\\,D_r$ or $\\mathfrak{e}_6$ if the gauged symmetry \nsubgroup $\\subset\\tilde G\/Z^\\omega$ corresponds to a regular Lie subalgebra \n$\\mathfrak{h}\\subset\\mathfrak{g}$ or, for $\\mathfrak{g}=\\mathfrak{e}_6$,\nto any semisimple Lie subalgebra. The global gauge anomalies that appear\nin the other coset model should render them inconsistent on the quantum\nlevel, as was argued in \\cite{GSW}. \n\\eject\n\n\n\\noindent{\\bf\\Large Appendices}\n\\vskip -0.5cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\ \\ \\ \\ Previous research has shown that using deep CNN models can be outstanding performance. Image classifiers or object detectors usually use VGG\\cite{vgg}, ResNet\\cite{resnet}, and other high FLOPs models as the backbone. In object detection, although the one-stage model led by YOLO\\cite{yolov1} is much faster than the two-stage model led by Faster-RCNN\\cite{fasterrcnn}, the backbone DarkNet used by YOLO is still a VGG-like or ResNet-like deep CNN model. YOLO has a computing overhead of at least \\textasciitilde20k MFLOPs. For edge computing devices, such high computing resource requirements cannot be carried. In the past, the number of model parameters was regarded as a primary goal of model lightweight. However, people soon discovered that the number of parameters is not positively correlated with the running speed. In recent years, more attention has been paid to the number of FLOPs. Although the model's actual running speed is still affected by the framework and OS, and implementation details, FLOPs are still recognized by the mainstream as the most theoretical metrics. In this paper, we use FLOPs as the guide to design a novel lightweight convolution method CSL-Module. We have theoretically proved that CSL-Module is 5 to 7 times faster than convolution-3x3. In our experiments, we have also shown that CSL-Module is faster and performs better than other lightweight convolution methods. Furthermore, CSL-Module was used to construct two highly efficient components, which were finally combined into a new lightweight object detector CSL-YOLO. Compared with other similar lightweight YOLOs, CSL-YOLO has reached the state-of-the-art level.\n\nThe rest of the paper is organized as follows: Section 2 briefly reviews the related techniques for lightweight model design, followed by the proposed CSL-Module, CSL-Bone, CSL-FPN, and CSL-YOLO in section 3, the tricks of CSL-YOLO in section 4, the experiments and discussion are presented in section 5, and finally, the conclusion in section 6.\n\n\\begin{figure*}\n  \\centering\n  \\bmvaHangBox{\\fbox{\\includegraphics[width=10cm]{_imgs\/cslyolo.png}}}\\\\\n  \\caption{Overall architecture of CSL-YOLO. the convolution-1x1 is weights-sharing.}\n  \\label{cslyolo}\n\\end{figure*}\n\n\\section{Related Work}\n\\ \\ \\ \\ In recent years, many lightweight CNN models have been proposed. In this section, we first review the lightweight convolution methods and backbone, and then we review the lightweight model of the one-stage object detector, especially the SSD\\cite{ssd}\\cite{retinanet} series and YOLO\\cite{yolov1} series.\n\n\\subsection{Lightweight Convolution Methods}\n\\ \\ \\ \\ The previous works have shown that convolution-3x3 is a powerful feature extraction method, but this method is still too expensive for edge computing. Especially in the deep CNN model, many convolution layers are used, which causes unbearable large FLOPs. Depthwise Separable Convolution (DSC) proposed by Google\\cite{xception}\\cite{mobilenet}. They uses a depthwise convolution to extract feature in space, and then use a pointwise convolution to extract feature in depth. This decoupling convolution method has shown in their experiments that it can approximate the performance of convolution-3x3 with fewer FLOPs. The backbone MobileNet\\cite{mobilenet} uses DSC extensively reduces FLOPs but maintains good accuracy. The successor backbone MobileNetv2\\cite{mobilenetv2} found that doing nonlinear transformation when the feature dimension is small, it will lose too much useful information. This problem is significant in lightweight models where the feature dimension is strictly controlled. Therefore, they use a pointwise convolution to expand the feature dimension to avoid the information loss caused by nonlinear transformation before the feature-maps passes through the DSC. They call this method Inverted Residual Block (IRB).\n\nIn addition to the two methods of decoupling in space and depth to reduce calculations, DSC and IRB, there are also some methods to divide the feature-maps into multiple groups to reduce calculations. ShuffleNet\\cite{shufflenet} divides the feature-maps into $G$ groups and passes them through DSC. The formula for FLOPs of convolution is shown as \\myeqref{convflops}. $H'$ denotes the height of output. $W'$ denotes the width of output. $C$ denotes the channel of input. $N$ denotes the channel of output. $K$ denotes the kernel size of convolution. According to the it, the feature-maps are grouped so that the input channel $C$ is reduced to $C\/G$, the output channel $N$ is reduced to $N\/G$, and the FLOPs only have the original $1\/G$. CSPNet\\cite{cspnet} divides the feature-maps into two halves. The first half of which is generated by convolution, and the other half is directly output after concatenating the first half. GhostNet\\cite{ghostnet} discusses this problem more systematically. The proposed Ghost Module uses half of the feature-maps to produce half of the output feature-maps with more expensive transformations, and then uses these feature-maps to generate more redundant feature-maps through cheap linear transformations, and finally concatenates the two halves. These lightweight methods performed well in their experiments.\n\n\\begin{equation}\nFLOPs_{conv}=H'*W'*C*K^2*N\n\\label{convflops}\n\\end{equation}\n\n\\subsection{Lightweight Object Detection Method}\n\\ \\ \\ \\ As generally recognized, object detectors can be classified into two-stage and one-stage. The two-stage model has a stage to generate ROI(Region of Interest), so the accuracy is usually higher, but the speed is also slower than the one-stage model. Although the one-stage model represented by YOLO focuses on the characteristics of real-time detection. However, most FLOPs of the one-stage model are still unacceptable for the tight computing resources of edge computing devices. Therefore, we will review the representative lightweight models in the YOLO and SSD series with extremely low FLOPs.\n\n\\subsubsection{SSD Series}\n\\ \\ \\ \\ SSD is an essential branch of the one-stage detectors, and many designs have led to later detectors. The multi-scale prediction they recommended has indirectly affected the proposal and contribute to the popularization of Feature Pyramid Network(FPN)\\cite{fpn}, and they effectively integrated the anchor box proposed by Faster-RCNN\\cite{fasterrcnn} to improve the performance significantly. We next introduce a few representative lightweight models in the SSD series. MobileNet-SSD\\cite{mobilenet} achieves quite good results with the simple combination of lightweight backbone MobileNet + SSD head. MobileNet-SSDLite further modified the SSD head to make the entire model more lightweight, and the accuracy has improved under the same FLOPs. MobileNetv2-SSDLite\\cite{mobilenetv2} replaced the backbone with a lighter MobileNetv2\\cite{mobilenetv2} and achieved fantastic speed and accuracy. PeleeNet\\cite{peleenet} constructed a DenseNet-like\\cite{densenet} lightweight backbone, and at the same time, reduced the output scale of the SSD head to reduce the calculation. They also got impressive results in the experiments.\n\n\\subsubsection{YOLO Series}\n\\ \\ \\ \\ As the one-stage model pioneer, YOLO discards the stage that generates the prior ROI and directly predicts the bounding box. This solution dramatically improves the speed of inference and achieves real-time on the GPU. But YOLO is still too large for embedding devices. YOLO has updated four official versions so far. Although there are corresponding tiny versions from YOLOv1 to YOLOv4\\cite{yolov1}\\cite{yolov2}\\cite{yolov3}\\cite{yolov4}, the official updated version focuses more on improving accuracy than speed. Therefore, Tiny-YOLOv1 to Tiny-YOLOv4\\cite{yolov1}\\cite{yolov2}\\cite{yolov3}\\cite{yolov4} always follow a similar compression strategy. They remove some convolutional layers or remove some multi-scale output layers in FPN. This strategy achieves a good compression ratio, but it also causes a large loss of accuracy. YOLO-LITE\\cite{yololite} follows a more aggressive reduction strategy. YOLO-LITE even only has 482 MFLOPs, but it also loses more considerable accuracy.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\bmvaHangBox{\\fbox{\\includegraphics[width=5cm]{_imgs\/cslm.png}}}&\n\\bmvaHangBox{\\fbox{\\includegraphics[width=5cm]{_imgs\/cslm_types.png}}}\\\\\n(a)&(b)\n\\end{tabular}\n\\caption{(a) Overall architecture of CSL-Module. (b) The two variants of CSL-Module are with attention and downsample version.}\n\\label{cslm}\n\\end{figure}\n\n\\section{Approaches}\n\\ \\ \\ \\ In this section, we first introduce the Cross-Stage Lightweight (CSL) Module. CSL-Module generates feature-maps with fewer FLOPs, and then we build two lightweight components necessary for object detectors based on CSL-Module.\n\\subsection{CSL-Module}\n\\ \\ \\ \\ Previous research has shown that use less computation to generate redundant feature-maps can reduce the FLOPs considerably. CSPNet\\cite{cspnet} presents a cross-stage method for solving it, and GhostNet\\cite{ghostnet} systematically verifies the effectiveness of the cheap operation in this issue. However, the problem is that the main operation to generate valuable feature-maps is still too expensive for edge computing. We propose dividing the input feature-maps into two branches. The first branch generates the half redundant feature-maps by a cheap operation like GhostNet did; the other branch generates the other half necessary feature-maps by lightweight main operation, then concatenate the two outputs together. The overall architecture is shown in \\myfigref{cslm}. The hyperparameter $t$ represents the ratio of feature expanding. We set $t$ as 3 in CSL-Bone, and set $t$ as 2 in else. We insert the SE module\\cite{squeezenet} or adaptive average pooling when down-sampling or attention is needed after expansion block. Besides, we use Mish\\cite{mish} as the activation function, and in their experiments show that Mish performs better in deep CNN models than ReLU\\cite{relu} and Swish\\cite{swish}.\n\\subsubsection{Difference from Existing Methods}\n\\ \\ \\ \\ The proposed CSL-Module generates half redundant feature-maps by a cheap operation at the skip branch. On the main branch, it is different from CSP Module and Ghost Module. We propose a lightweight main operation to generate the other half necessary feature-maps. In this branch, we design an IRB-like expansion block, using the input feature-maps and output feature-maps of the skip branch to generate intermediate candidate feature-maps by depthwise convolution. One of the great advantages of this block is pointwise convolution-free, and We all know that depthwise convolution has far fewer FLOPs than pointwise convolution. It is different from IRB. IRB uses pointwise convolution to generate candidate feature-maps. The other advantages of this block is it thoroughly considered all currently available features, this can minimizes redundant calculations. Besides, because there is already skip branch, the main branch only needs to generate half of the feature-maps, reducing FLOPs significantly. In general, the proposed CSL-Module reduces FLOPs by cheap operation and cross-stage ideas. On the other hand, we especially lightweight design for the main branch. We replace the convolution layers in VGG-16\\cite{vgg} to verify the effectiveness of CSL-Module, and the new models are denoted as IRB-VGG-16, Ghost-VGG-16, and CSL-VGG-16, respectively. We evaluate them on CIFAR-10, the training setting and tricks are all of the same (e.g., flip, affine, mix-up\\cite{mixup}, and steps learning rate). From \\mytabref{cslmoncifar}, it can be seen that CSL-Module is faster than other advanced lightweight convolution methods, and CSL-Module can more approximate the performance of convolution-3x3. This experiment proves that CSL-Module is a very competitive lightweight convolution method.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& MFLOPs & Acc.(\\%)\\\\\n\\hline\nVGG-16 & 299 & 92.6\\\\\nIRB-VGG-16 & 226 & 92.6\\\\\nGhost-VGG-16 & 169 & 90.2\\\\\n\\textbf{CSL-VGG-16} & \\textbf{128} & 92.0\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison of CSL-Module with the other lightweight convolution methods on CIFAR-10.}\n\\label{cslmoncifar}\n\\end{table}\n\n\\subsubsection{Analysis on FLOPs}\n\\ \\ \\ \\ The analysis is made with the well-known formula \\myeqref{convflops}. We assume the input shape and output shape are equal for computational simplicity to compare the speed-up ratio of FLOPs between CSL-Module and convolution-3x3. CSL-Module has a hyperparameter $t$ as same as IRB\\cite{mobilenetv2}, representing the expansion ratio of the expansion block. When $t$=2, the speed-up ratio reaches 7.2 times; when $t$=3, it reaches 5.1 times. The whole process is shown in \\myeqref{analysissr}\n\\begin{equation}\n\\begin{aligned}\nFLOPs_{csl}&=(H'*W'*C*0.5N)+t(H'*W'*K^2*0.5N)\\\\\n            &+(H'*W'*C*K^2)\\\\\n            &+(H'*W'*(C+0.5tN)*K^2)\\\\\n            &+(H'*W'*(C+0.5tN)*0.5N)\\\\\nwhen\\ C=&N,\\ K=3:\\\\\n&sr=\\frac{FLOPs_{conv}}{FLOPs_{csl}}\\simeq\\frac{9}{1+0.25t}\n\\label{analysissr}\n\\end{aligned}\n\\end{equation}\n\\subsection{Building Lightweight Components}\n\\ \\ \\ \\ We propose two lightweight components CSL-Bone and CSL-FPN. These two components are necessary for object detectors. CSL-Bone extracts the features of input image with fewer FLOPs than other backbone models; CSL-FPN predicts the bounding boxes on more different scales efficiently.\n\n\\subsubsection{Lightweight Backbone}\n\\ \\ \\ \\ The proposed CSL-Bone consists of several CSL-Module groups. SE Module\\ \\cite{squeezenet} has integrated to first CSL-Module in a group to enhance the feature extraction capabilities of the entire group.\\ Also, we insert pooling layers for down-sampling at appropriate locations to obtain high-level semantic feature. Finally, CSL-Bone outputs three different scale feature-maps. The overall architecture is shown in \\mytabref{cslyolo}. We evaluate CSL-Bone, MobileNetv2, and GhostNet on CIFAR-10 and applied the same training setting too. It can be seen from \\mytabref{cslboneoncifar}. Although the CSL-Bone gets lower accuracy than MobileNetv2, but the FLOPs of CSL-Bone just 58.7\\% than MobileNetv2. On the other hand, CSL-Bone gets higher accuracy than GhostNet, but only slightly increased FLOPs.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& MFLOPs & Acc.(\\%)\\\\\n\\hline\nMobileNetv2 & 75 & 91.3\\\\\nGhostNet & 40 & 89.3\\\\\n\\textbf{CSL-Bone} & 44 & 90.7\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison of CSL-Bone with the other lightweight backbons on CIFAR-10.}\n\\label{cslboneoncifar}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& MFLOPs & $AP$ & $AP_{50}$\\\\\n\\hline\nVanilla-FPN (conv-3x3) & 416 & 19.0 & 35.8\\\\\nCSL-FPN ($R$=1) & 127 & 18.7 & 35.5\\\\\nCSL-FPN ($R$=2) & 198 & 18.8 & 35.8\\\\\nCSL-FPN ($R$=3) & 268 & 18.8 & \\textbf{37.2}\\\\\nCSL-FPN ($R$=4) & 339 & \\textbf{19.8} & 37.0\\\\\nCSL-FPN ($R$=5) & 409 & \\textbf{19.8} & \\textbf{37.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The performance of the proposed CSL-FPN with different $R$ on MS-COCO.}\n\\label{cslfpnonmscoco}\n\\end{table}\n\n\\begin{equation}\n\\begin{aligned}\n&Let\\ l\\ be\\ the\\ number\\ of\\ layers.\\\\ \n&Let\\ k\\ be\\ the\\ number\\ of\\ anchors\\ per\\ layer.\\\\\n&Let\\ B\\ be\\ bounding\\ boxes\\ of\\ img.\\\\\n&Let\\ A_i\\ be\\ anchors\\ of\\ layer_i,\\ 0\\leq i<l\\\\\n&S=[0,\\ \\frac{1}{2^{l-1}},\\ \\frac{1}{2^{l-2}},\\ \\frac{1}{2^{l-3}},...,\\ \\frac{1}{2^{l-l}}]\\\\\n&X_i=\\{S_i\\leq b<S_{i+1}|\\forall b \\in B\\}\\\\\n&C_i=Kmeans(X_i,\\ k)\\ with\\ IoU\\ dist\\ function.\\\\\n&A_i=\\{Center(C_i^0),Center(C_i^1),...,Center(C_i^k)\\}\n\\label{anchorconstraint}\n\\end{aligned}\n\\end{equation}\n\n\\subsubsection{Lightweight FPN}\n\\ \\ \\ \\ Previous studies have shown that large-scale feature-maps have more object details, such as edges, corners, or textures, while small feature-maps have comprehensive semantic understanding. Vanilla FPN\\cite{fpn} up-sampling the small feature-maps then fuse them with the large feature-maps. On the other hand, vanilla FPN output three scale feature-maps. It is helpful for the model to detect objects of different sizes. The proposed CSL-FPN first replaces all convolution-3x3 in FPN to CSL-Module. Second, in the expansion stage, a middle-scale layer is formed between two scale layers, and these middle-scale layers can enhance the model's ability to detect objects of different sizes. Third, in the repeat stage,  there are (k)th layer, (k-1)th, and (k+1)th layer for feature fusion at the same time, but each time only odd or even layers are used. For example, there are only the 2nd and 4th layers in the first fusion, and in the second fusion, there are the 1st, 3rd, and 5th layers. In other words, the proposed CSL-FPN has the same number of convolutions as vanilla-FPN but has more feature fusion. The overall architecture is shown in \\myfigref{cslfpn}. In our implementation of the proposed CSL-FPN, to make element-wise addition easier, we set the channels of the five output layers to be the same in the layer expansion stage. The repeat stage uses a hyperparameter $R$ to indicate that CSL-FPN has stacked several blocks in total. A larger $R$ can achieve higher AP, but FLOPs also increase. There is a trade-off between speed and performance. We test the best value of $R$ on MS-COCO based on 320x320 CSL-YOLO. \\mytabref{cslfpnonmscoco} shows the results. As $R$ gets larger and larger, $AP$ also improved from 18.6\\% to 19.8\\%, $AP_{50}$ improved from 35.5\\% to 37.2\\%, but MFLOPs also deteriorated from 127 to 409. After considering the trade-off, we decided to set $R$ to 3.\n\\begin{figure}\n  \\centering\n  \\bmvaHangBox{\\fbox{\\includegraphics[width=10cm]{_imgs\/cslfpn.png}}}\\\\\n  \\caption{Overall architecture of CSL-FPN. the layer expansion stage is used to expand the output scale and enhance the continuity between the scales. the repeat stage uses a hyperparameter $R$ to represent the number of times the block is stacked. the blue arrow represents downsampling, the orange arrow represents upsampling, and the white hollow circle represents identity mapping.}\n  \\label{cslfpn}\n\\end{figure}\n\n\\section{TRICKS OF CSL-YOLO}\n\\subsection{Anchors Constraint}\n\\ \\ \\ \\ The original YOLO series uses K-means with the IoU distance function to cluster the height and width of ground truth, then take the center point as an anchor box. These $k$ anchors generated from $k$ cluster and assign to the output layers of FPN based on their scale. When we expand the output layers from 3 to 5, $k$ also increased from 9 to 15. If we use the above method, then most of these anchors are small-scale due to many small objects in MS-COCO\\cite{mscoco}. There will be high-level output layers forced to use small-scale anchors. However, it is well known that high-level feature-maps are not conducive to detect small objects. Therefore, we add the scale limit such as \\myeqref{anchorconstraint} before K-means so that the distribution of anchors generated is more in line with each output layer's scale. We conduct experiments in \\mytabref{improvementac}, and we can see that the original 3 output layers has worsened after being expanded to 5 output layers. After adding our constraint method, it has improved a considerable AP.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& $AP$ & $AP_{50}$\\\\\n\\hline\nOutput Layers x 3 w\/o AC & 17.9 & 36.1\\\\\nOutput Layers x 5 w\/o AC & 17.2 & 35.1\\\\\n\\textbf{Output Layers x 5 w\/ AC} & \\textbf{18.8} & \\textbf{37.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Improvement by anchors constraint.}\n\\label{improvementac}\n\\end{table}\n\n\\subsection{Non-Exponential Prediction}\nYOLO series actually predicts the offset of $x$, $y$, $w$, $h$, as follows:\n\n\\begin{equation}\n\\begin{aligned}\n&w=w_{anchor}*\\mathrm{e}^{w_{pred}},\\\\\n&h=h_{anchor}*\\mathrm{e}^{h_{pred}},\n\\label{logprediction}\n\\end{aligned}\n\\end{equation}\nwhere $h_{pred}$ and $w_{pred}$ denotes the offset of the object height and width predicted by the model, $h_{anchor}$ and $w_{anchor}$ denotes the height and width of anchor. Although the log function can limit the prediction of the model range, but the sensitivity of the exponential function makes the width and height quite unstable. Therefore, we removed the log function and let the model directly predict the offset. Then, \\myeqref{nonogprediction} can be modified as:\n\\begin{equation}\n\\begin{aligned}\n&w=w_{anchor}+w_{pred},\\\\\n&h=h_{anchor}+h_{pred}.\n\\label{nonogprediction}\n\\end{aligned}\n\\end{equation}\nFrom \\mytabref{nonexppred}, clearly non-exponential prediction can improve 1\\textasciitilde2\\% AP even under various image sizes. As shown in \\myfigref{cslyolo}, all components are then integrated to construct CSL-YOLO.  During inference, the soft-nms\\cite{softnms} technique is adopted to penalize overlapping boxes.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& Input Size & $AP$ & $AP_{50}$\\\\\n\\hline\nw\/ Exp & 224x224 & 15.0 & 31.1\\\\\nw\/ Exp & 320x320 & 18.8 & 37.2\\\\\nw\/ Exp & 416x416 & 22.0 & 41.9\\\\\nw\/ Exp & 512x512 & 25.1 & 45.7\\\\\n\\hline\nw\/o Exp & 224x224 & \\textbf{16.5} & \\textbf{32.2}\\\\\nw\/o Exp & 320x320 & \\textbf{21.4} & \\textbf{39.9}\\\\\nw\/o Exp & 416x416 & \\textbf{24.5} & \\textbf{44.0}\\\\\nw\/o Exp & 512x512 & \\textbf{26.3} & \\textbf{46.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Improvement by non-exponential prediction.}\n\\label{nonexppred}\n\\end{table}\n\n\n\n\\section{EXPERIMENTS}\n\\ \\ \\ \\ The proposed CSL-YOLO is not pre-trained on Imagenet\\cite{imagenet}. We follow the conclusion of \\cite{rethinking}, and train from scratch on MS-COCO. We applied the basic augmentation methods such as flip, rotate, and mix-up\\cite{mixup} instead of the fancy methods. We set the batch size as 16, and choosing Adam as the optimizer, The learning rate starts from 1e-3, after 50 epochs, it drops to 1e-4, after 40 epochs, it drops to 1e-5, and finally it is completed after another 20 periods. We train the model on two GTX 1080ti. The entire training process has \\textasciitilde800K iterations. Finally, the result of the prediction is shown in \\mytabref{cslyoloonmscoco}. Under the input scale of 416x416, the proposed CSL-YOLO uses 3.2M parameters and 1470 MFLOPs to obtain 42.8\\% $AP_{50}$, and Tiny-YOLOv4 uses 6.1M parameters and 3450 MFLOPs to obtain 40.2\\% $AP_{50}$. It can be said that CSL-YOLO consumes much less time(FLOPs) and space(parameters) than the advanced Tiny-YOLOv4, and can achieve impressive AP performance. Besides, at the input scale of 224x224, compared with the lightest YOLO-LITE, CSL-YOLO still achieves higher AP performance with lower FLOPs.\n\n\\begin{table}\n\\begin{center}\n\\setlength{\\tabcolsep}{0.7mm}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\nModel & Backbone & Input Size & MFLOPs & Params & $AP$ & $AP_{50}$ &  $AP_{75}$\\\\\n\\hline\nYOLO-LITE & DarkNet-LITE & 224x224 & 482 & - & - & 12.3 & -\\\\\nTiny-YOLOv3 & Tiny-DarkNet & 320x320 & 1650 & 8.7M & 14.0 & 29.1 & -\\\\\nTiny-YOLOv3 & Tiny-DarkNet & 416x416 & 2780 & 8.7M & 16.0 & 33.1 & -\\\\\nTiny-YOLOv4 & Tiny-CSPDarkNet & 416x416 & 3450 & 6.1M & 21.7 & 40.2 & -\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 224x224 & \\textbf{425} & \\textbf{3.2M} & \\textbf{16.5} & \\textbf{32.2} & \\textbf{15.0}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 320x320 & \\textbf{869} & \\textbf{3.2M} & \\textbf{21.4} & \\textbf{39.9} & \\textbf{20.6}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 416x416 & \\textbf{1470} & \\textbf{3.2M} & \\textbf{24.5} & \\textbf{44.0} & \\textbf{24.2}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 512x512 & \\textbf{2223} & \\textbf{3.2M} & \\textbf{26.3} & \\textbf{46.2} & \\textbf{26.5}\\\\\n\\hline\nMobileNet-SSD & MobileNet & 300x300 & 1200 & 6.8M & 19.3 & - & -\\\\\nMobileNet-SSDLite & MobileNet & 320x320 & 1300 & - & 22.2 & - & -\\\\\nMobileNetv2-SSDLite & MobileNetv2 & 320x320 & 800 & 4.3M & 22.1 & - & -\\\\\nPeleeNet & PeleeNet & 304x304 & 1290 & 6.0M & 22.4 & 38.3 & 22.9\\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\caption{Comparison of CSL-YOLO with the other state-of-the-art lightweight object detection system on MS-COCO.}\n\\label{cslyoloonmscoco}\n\\end{table}\n\n\\section{CONCLUSION}\n\\ \\ \\ \\ We proposed a new lightweight convolution method CSL-Module, in the experiment proved that it is competitive compared to other similar methods. We further proposed two components CSL-Bone and CSL-FPN, both of which achieve better performance with fewer FLOPs. We build a new lightweight object detector CSL-YOLO with these components. Besides, we proposed two tricks, a new anchor generation method and a non-exponential prediction to improve AP. Finally, we trained CSL-YOLO from scratch on MS-COCO. From \\mytabref{cslyoloonmscoco}, we can see that CSL-YOLO has better AP and fewer FLOPs and parameters than other lightweight YOLOs. In summary, we have three contributions:\n\\begin{enumerate}\n\\item We proposed a novel lightweight convolution method CSL-Module that costs fewer FLOPs to approximate the fitting ability of convolution-3x3.\n\\item We proposed two lightweight components for object detector, CSL-Bone and CSL-FPN which showed excellent results.\n\\item We effectively integrated these components, and two tricks to improve AP, and finally proposed a state-of-the-art level lightweight detector CSL-YOLO.\n\\end{enumerate}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRedshift is one of well-known effects of gravity, it plays an essential\nrole in relativistic astrophysics. Its description entered many textbooks.\nIn particles, it concerns propagation of light in the black hole background.\nThus main attention is focused on properties of light outside the\nevent horizon. Less  these properties inside the horizon are discussed.\nAlso, strange as it may seem, the question of the redshift of a photon\nmoving along the horizon dropped out from consideration almost completely.\nThere is general discussion of this issue in \\cite{lake} for the\nReissner-Nordstr\\\"{o}m - de Sitter metric but the relation between redshift or\nblueshift and the nature of the horizon was not revealed there.\n\nMeanwhile, there are several points that can serve as a motivation for such a\nconsideration.\n\n(i) Recently, important methodic issues were discussed in \\cite{au}, \\cit\n{kas} concerning properties of the world visible by an observer falling into\nthe Schwarzschild black hole and communicating by radio signals with another\nfalling one. In doing so, some incorrect statements were made in \\cite{au}\nabout \"ghosts\" of the first observer supposedly waiting for the second one\non the horizon. As was argued in \\cite{kas}, there are no such ghosts at all\nalthough the second observer does receive a signal from the first one\nemitted at the moment of crossing the horizon. This required detailed\ncalculation of the frequency shift for a photon propagating along the\nhorizon with the result that a finite redshift occurs in this case.\n\nIn the present work, we generalize these observations, demonstrate that such\na redshift is present for any horizon of a spherically symmetric nonextremal\nblack hole and find its value.\n\n(ii) If the metric contains an inner horizon (say, for the Reissner-Nordst\n\\\"{o}m black hole), the calculation of the frequency\nshift for a photon emitted along such a horizon is of special interest. We demonstrate that now,\ninstead of a redshift, a blueshift occurs.  In the limit when a photon\nis received near the bifurcation point, the blueshift becomes infinite. This\nestablishes the connection of the issue under discussion with the analogue\nof the Ba\\~{n}ados-Silk-West (BSW) effect \\cite{ban}. It consists in the\ninfinite growth of the energy of colliding particles in the centre of mass\nframe. Originally, it was found near the event horizon but, later on, it\nturned out that another similar effect is valid also near the inner horizon\n(see \\cite{inner} and references therein). From another hand, the issue\nunder discussion can be considered as an effect supplemental to a well-known\ninstability of the inner horizon \\cite{fn}, \\cite{ham}.\n\n(iii) In addition to the propagation along the nonextremal horizon, there is\nalso a question what happens in the extremal case. We argue that, by\ncontrast to two previous ones, now the frequency shift is absent.\n\n(iv) In papers \\cite{au}, \\cite{kas} the Kruskal-Szekerez (KS) coordinate\nsystem was used. We also exploit it. In addition, it is of interest to\ncompare the results using another powerful system - the Lemaitre one. We\nconstruct such a system for a whole class of metrics that includes the\nSchwarzchild one as a particular case.\n\nIn the present work, we restrict ourselves by the simplest case of radially\nmoving observers.\n\n\\section{Motion outside the nonextremal event horizon}\n\nWe consider the metri\n\\begin{equation}\nds^{2}=-fdt^{2}+\\frac{dr^{2}}{f}+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi\n^{2})\\text{.}  \\label{met}\n\\end{equation}\n\nWe suppose that the metric has the event horizon at $r=r_{+}$, so \nf(r_{+})=0 $. (For simplicity, we assumed that $g_{00}g_{11}=-1$ but this\ncondition can be relaxed easily.) We consider now a nonextremal black hole.\nNear the horizon\n\\begin{equation}\nf\\approx \\kappa (r-r_{+})\\text{,}  \\label{kar}\n\\end{equation\nwhere $\\kappa =\\frac{f^{\\prime }(r_{+})}{2}$ is the surface gravity.\n\nLet an observer has the four-velocity $u^{\\mu }=\\frac{dx^{\\mu }}{d\\tau }$,\nwhere $\\tau $ is the proper time. We restrict ourselves to the radial motion\nof a massive particle (we call it \"observer\"). Then, the four-velocity \nu^{\\mu }=(\\dot{t}$, $\\dot{r})$. Here, a dot denotes derivative with respect to \n$\\tau $.\n\nThe geodesic equations of motion for such a particle read\n\\begin{equation}\nm\\dot{t}=\\frac{E}{f}\\text{,}  \\label{mt}\n\\end{equation\n\\begin{equation}\nm\\dot{r}=-Z\\text{, }Z=\\sqrt{E^{2}-fm^{2}}\\text{,}  \\label{mr}\n\\end{equation\n$E=-mu_t$ is the conserved energy  of a particle.\n\nFor a photon having the wave vector $k^{\\mu }$, the equations of motion ar\n\\begin{equation}\nk^{t}=\\frac{\\omega _{0}}{f}\\text{,}  \\label{yt}\n\\end{equation\n\\begin{equation}\nk^{\\phi }=\\frac{l}{g_{\\phi }}\\text{,}\n\\end{equation\n\\begin{equation}\nk^{r}=-Q\\text{, }Q=\\sqrt{\\omega _{0}^{2}-f\\frac{l^{2}}{r^{2}}}  \\label{r}\n\\end{equation\nwhere $\\omega _{0}=-k_{0}$ is the conserved frequency, $l=k_{\\phi }$ is the\nconserved angular momentum.\n\nLet a free falling observer emits or receives a photon. Its frequency measured\nby this observer is equal to \n\\begin{equation}\n\\omega =-k_{\\mu }u^{\\mu }.  \\label{om}\n\\end{equation}\n\nTaking into account (\\ref{mt}) - (\\ref{r}), we find after straightforward\ncalculation\n\\begin{equation}\n\\omega =\\frac{E\\omega _{0}-\\varepsilon ZQ}{mf}\\text{.}  \\label{wzq}\n\\end{equation}\n\nHere, $\\varepsilon =+1$ if both objects (the observer and the photon) move in\nthe same direction and $\\varepsilon =-1$ if they do this in opposite ones.\nIf $\\varepsilon =-1$, this corresponds to head-on collision between a\nmassive and a massless particles and this means that either an observer receives\na photon emitted from a smaller value of $r$ or a falling observer emits a\nphoton in the backward direction.\n\nNear the horizon, $f\\ll 1.$ Let $\\varepsilon =+1$ and the observer looks\nback. Near the horizon, he will see the frequenc\n\\begin{equation}\n\\omega \\approx \\frac{1}{2}(\\frac{El^{2}}{m\\omega _{0}r_{+}^{2}}+\\frac{\\omega\n_{0}m}{E})\\text{.}  \\label{n1}\n\\end{equation\nIf $\\varepsilon =-1$\n\\begin{equation}\n\\frac{\\omega }{\\omega _{0}}\\approx \\frac{2E}{mf_{em}}\\text{.}  \\label{n2}\n\\end{equation\nHere $f_{em}$ is the value of the metric function in the point of emission.\nLet the observer emits in his frame a photon having the frequency $\\omega $\nand travelling to infinity. Then, at infinity it will be received with the\nfrequency $\\omega _{0}$ having the order $f_{em}$. Eq. (\\ref{n2}) agrees\nwith the standard result for the Schwarzschild metric (see Sec. XII.102 of \n\\cite{LL}, especially eq. 102.10).\n\nIt is instructive to compare this to another situation usually discussed in\ntextbooks when the emitter is not free falling but is static. In the latter\ncase, the frequency at infinity $\\omega _{0}=\\omega \\sqrt{f}$ (see, e.g. eq.\n88.6 of \\cite{LL}) has the order $\\sqrt{f_{em}}\\,$. Obviously, the\ndifference can be attributed to the motion of an emitter (the Doppler effect\nmakes the redshift stronger).\n\nSome reservations are in order. Throughout the paper, we assume that the\ngeometric optics is a reasonable approximation for propagation of light\nwaves. As usual, this implies that the wavelength $\\lambda $ satisfies\nrelations $\\frac{\\lambda }{2\\pi }\\ll \\mathcal{L},$ $\\mathcal{R},$ where \n\\lambda $ is the wave length, $\\mathcal{L}$ is a typical scale\ncharacterizing a wave packet and $\\mathcal{R}^{-2}$ is the typical component\nof the Riemann tensor (see eq. 22.23c in \\cite{grav}). For the Schwarzschild\nmetric, this entails, in terms of the frequency, the condition $\\omega M\\gg\n1,M$ is the black hole mass. In the more general case of, say, the\nReissner-Nordstr\\\"{o}m metric, this gives $\\omega r_{+}\\gg 1$, where $r_{+}$\nis the radius of the horison.\n\nAlso, we assume that backreaction of the photon on the metric is negligible.\nThis implies that, in any case, its energy is much less than the ADM mass of\na black hole, so in natural geometric units $\\omega \\ll M$. Thus we have the\ndouble inequality $r_{+}^{-1}\\ll \\omega \\ll M$. As for the Reissner-Nordstr\\\"{o}m\n metric$\\ M\\leq r_{+}\\leq 2M$, the conditions of validity of this\napproximation are practically the same for the Reissner-Nordstr\\\"{o}m and\nSchwarzschild metrics.\n\n\\section{Photon emitted at the horizon}\n\nWe see from (\\ref{n2}) that, as the point of emission of a photon is\napproaching the horizon, the frequency measured at infinity becomes smaller\nand smaller. However, this formula does not describe what happens if the\nphoton is emitted exactly on the horizon. Then, $\\omega _{0}=0$ and $l=0$, \nf_{em}=0$ from the very beginning (see also below), the photon does not\nreach infinity at all and moves along the leg of the horizon. Here, original\ncoordinates (\\ref{met}) are not applicable since the metric becomes\ndegenerate on the horizon. This can be remedied with the use of standard KS\ncoordinates or its generalization.\n\nLet us introduce coordinates $U$ and $V$, wher\n\\begin{equation}\nU=-\\exp (-\\kappa u)\\text{, }V=\\exp (\\kappa v)\\text{,}  \\label{uv}\n\\end{equation\n\\begin{equation}\nu=t-r^{\\ast }\\text{, }v=t+r^{\\ast }\\text{.}  \\label{uU}\n\\end{equation\n\\begin{equation}\nr^{\\ast }=\\int^{r}\\frac{dr}{f}  \\label{tc}\n\\end{equation\nis the tortoise coordinate. It is seen from (\\ref{uv}) tha\n\\begin{equation}\nUV=-\\exp (2\\kappa r^{\\ast })\\text{.}  \\label{uvr}\n\\end{equation}\n\nThen the metric read\n\\begin{equation}\nds^{2}=FdUdV+r^{2}d\\omega ^{2}\\text{,}  \\label{mF}\n\\end{equation\nHere\n\\begin{equation}\nF=-f\\frac{du}{dU}\\frac{dv}{dV}\\text{.}  \\label{fg}\n\\end{equation\nFor the transformation (\\ref{uv})\n\\begin{equation}\nF=\\frac{f}{UV\\kappa ^{2}}\\text{,}\n\\end{equation\n$F\\neq 0$ on the horizon. It is clear from (\\ref{met}), (\\ref{uvr}) that \nF=F(r)$. For example, for the Schwarzschild metric\n\\begin{equation}\nF=-\\frac{4r_{+}^{3}}{r}e^{-\\frac{r}{r_{+}}}\\text{,}\n\\end{equation\nprovided the constant of integration is chosen in (\\ref{tc}) in such a way\nthat $r^{\\ast }=r+r_{+}\\ln \\frac{r-r_{+}}{r_{+}}$.\n\nWe consider the vicinity of the future horizon, on which $U=0$. Along this\nhorizon, $V$ takes finite values.\n\nNear the horizo\n\\begin{equation}\nr^{\\ast }\\approx \\frac{1}{2\\kappa }\\ln \\frac{r-r_{+}}{r_{+}}+const\\text{,}\n\\end{equation\n\\begin{equation}\nUV\\approx \\frac{(r-r_{+})}{r_{+}}C_{0}\\text{,}  \\label{uvc}\n\\end{equation\nwhere $C_{0}$ is a constant. In the Schwarzschild case, $C_{0}=e$. It is\ninstructive to rewrite equations of motion for massive particles outside the\nevent horizon (\\ref{mt}) - (\\ref{mr})\\ in terms of the KS coordinates and\ntake the horizon limit afterwards. For an observer moving inward they rea\n\\begin{equation}\nmu^{U}=\\frac{E+Z}{f}\\frac{dU}{du}\\text{,}  \\label{mu}\n\\end{equation\n\\begin{equation}\nmu^{V}=\\frac{E-Z}{f}\\frac{dV}{dv}\\text{.}  \\label{mV}\n\\end{equation\nTaking also into account (\\ref{uv}) and (17), we hav\n\\begin{equation}\nm\\dot{U}=-\\frac{(E+Z)}{FV\\kappa }\\text{,}  \\label{uu}\n\\end{equation\n\\begin{equation}\nm\\dot{V}=\\frac{(E-Z)}{UF\\kappa }.  \\label{vv}\n\\end{equation}\n\nIn a similar way, we have for a photon moving in the outward direction\n\\begin{equation}\nk^{U}=\\frac{\\kappa U}{f}(Q-\\omega _{0})=\\frac{Q-\\omega _{0}}{F\\kappa V}\\text\n,}  \\label{ku}\n\\end{equation\n\\begin{equation}\nk^{V}=\\frac{\\kappa V}{f}(\\omega _{0}+Q)=\\frac{\\omega _{0}+Q}{FU\\kappa }\\text\n.}\n\\end{equation}\n\nLet $\\lambda $ be the affine parameter. On the future horizon $U=0$, $k^{U}\n\\frac{dU}{d\\lambda }=0=k_{V}$ which agrees with (\\ref{ku}) if we put $\\omega\n_{0}=0=f=Q$ in the right hand side. Thus only $k^{V\\text{ }}$remains\nnonzero. It follows from the geodesic equations that on the horizo\n\\begin{equation}\n\\frac{dk^{V}}{d\\lambda }=-\\Gamma _{VV}^{V}\\left( k^{V}\\right) ^{2}\\text{.}\n\\end{equation\nHere the Christoffel symbol $\\Gamma _{VV}^{V}\\sim \\frac{\\partial F}{\\partial\nV}\\sim \\frac{\\partial F}{\\partial r}\\frac{\\partial r}{\\partial V}\\sim f=0$\non the horizon. Therefore, $k^{V\\text{ }}=const$ along the horizon\ngenerator. We have from (\\ref{om}), (\\ref{uu}) on the horizo\n\\begin{equation}\n\\omega =-\\frac{1}{2} Fk^{V}u^{U}\\text{,}  \\label{omf0}\n\\end{equation\nand we have\n\n\\begin{equation}\n\\omega = \\frac{E}{Vm\\kappa }k^{V}\\text{.}  \\label{omf}\n\\end{equation}\n\nLet observer 1 crosses the horizon at some $V=V_{1}$ and the same for observer 2 but\nlater, at $V=V_{2}$. It follows from (\\ref{uv}), (\\ref{uU}) that $V_{2}>V_{1}\n. Assuming that observers are identical in that they have the same values of \n$E$ and $m$, we obtain\n\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{V_{1}}{V_{2}}<1\\text{.}  \\label{12}\n\\end{equation}\n\nThis agrees with eq. (A16) of \\cite{kas} obtained for the Schwarzschild\nmetric by another method.\n\nIt is also instructive to check that indeed $\\omega _{0}=0$. By definition, \n\\omega _{0}$ is a constant Killing frequency \n\\begin{equation}\n\\omega _{0}=-k_{\\mu }\\xi ^{\\mu }\\text{,}  \\label{fr}\n\\end{equation\nwhere $\\xi ^{\\mu }$ is the Killing vector. In the original coordinates (\\re\n{met}), \n\\begin{equation}\n\\xi ^{\\mu }=(1,0,0,0)\\text{, }\\xi _{\\mu }=(-f,0,0,0)\\text{.}\n\\end{equation\nPassing to KS coordinates, one obtain\n\\begin{equation}\n\\xi ^{U}=-\\kappa U\\text{, }\\xi ^{V}=\\kappa V\\text{.}\n\\end{equation\nThen, we see from (\\ref{fr}) that\n\n\\begin{equation}\n\\omega _{0}=-F(k^{V}\\xi ^{U}+k^{U}\\xi ^{V})=F\\kappa (k^{V}U-k^{U}V)\\text{.}\n\\end{equation\nOn the future horizon, $k^{U}=0$ and $U=0$, so we see that indeed $\\omega\n_{0}=0$.\n\nAlso, it is easy to check that for a photon propagating along the horizon \nl=0$. Indeed, if we write down the condition $k_{\\mu }k^{\\mu }=0$ on the\nfuture horizon, we obtain that $k_{\\phi }=0$. This agrees with previous\nobservations concerning the properties of trajectories on the horizon \\cit\n{circkerr}, \\cite{nh15}.\n\n\\section{Generalized Lemaitre frame}\n\n\\subsection{Form of metric}\n\nIt is instructive to reformulate the redshift value in the Lemaitre-like\ncoordinates $\\rho ,\\tau $. In contrast to the Kruskal ones, this frame is\nbased on free falling particles. The Lemaitre frame is well known for the\nSchwarzschild metric. Now, we suggest its generalization valid for the\nmetric (\\ref{met}).\n\nThe general theory of transformations that make the metric of a spherically\nsymmetric black hole regular, was developed in \\cite{f}. For our goals, it\nis sufficient to find a particular class of transformations that (i) makes the\nmetric regular on the horizon, (ii) generalizes the Lemaitre metric (in\nparticular, the metric should have $g_{\\tau \\tau }=-1$). We make the\ntransformatio\n\\begin{equation}\n\\rho =t+\\int \\frac{dr^{\\ast }}{\\sqrt{1-f}}\\text{,}  \\label{rof}\n\\end{equation\n\\begin{equation}\n\\tau =t+\\int dr^{\\ast }\\sqrt{1-f}  \\label{tauf}\n\\end{equation\nwhere $r^{\\ast }$ is given by (\\ref{tc}). Eqs. (\\ref{rof}), (\\ref{tauf}) are\ndirect generalization of eqs. 102.1 of \\cite{LL}. Then, it is easy to check\nthat\n\n\\begin{equation}\nds^{2}=-d\\tau ^{2}+(1-f)d\\rho ^{2}+r^{2}(\\rho ,\\tau )(d\\theta ^{2}+\\sin\n^{2}\\theta d\\phi ^{2})\\text{.}  \\label{metL}\n\\end{equation}\n\nOn the horizon, $f=0$, the metric coefficient is regular, $g_{\\rho \\rho }=1\n. In the particular case of the Schwarzschild metric, $f=1-\\frac{r_{+}}{r}$\nand we return to the standard formula for the Lemaitre metric, when $r$ is\nexpressed in terms of $\\rho $ and $\\tau $. The coordinates (\\ref{rof}), (\\re\n{tauf}) are suitable for the description of a black hole including both the\nouter R region and the contracting T$^{\\_}$ one \\cite{nov}. In a similar\nway, one can use the expanding version that would result in a changing sign at \n$\\tau $.\n\nNow, we want to pay attention to some nice properties of the metric (\\re\n{metL}). The proper distance between points 1 and 2 calculated for a given \n\\tau $ is equal to $l=\\int d\\rho \\sqrt{1-f}$. Requiring $d\\tau =0$ in (\\re\n{tauf}) and substituting $dt$ into (\\ref{rof}), we obtain from (\\ref{tc}), \n\\ref{rof}) tha\n\\begin{equation}\nl=r_{2}-r_{1}\\text{.}  \\label{dist}\n\\end{equation}\n\nIt is also instructive to calculate the velocity. Let, say, point 1 be fixed\nand let us focus on the velocity of free fall $v=\\frac{dl}{d\\tau }$ of\na particle with $E=m$, where $r_{2}\\equiv r$ changes depending on time.\nThen, it is easy to  find from (\\ref{mr}), (\\ref{dist}) tha\n\\begin{equation}\nv=-\\sqrt{1-f}\\text{.}\n\\end{equation\nTaking the derivative ones more, we obtain $\\frac{dv}{dr}=\\frac{1}{2\\sqrt{1-\n}}\\frac{df}{dr}$. On the horizon, this gives u\n\\begin{equation}\n\\left( \\frac{dv}{dr}\\right) _{H}=\\kappa \\text{,}  \\label{ka}\n\\end{equation\nwhere we took into account that for our metric the surface gravity $\\kappa \n\\frac{1}{2}\\left( \\frac{df}{dr}\\right) _{H}$. The subscript \"H\" means that the\ncorresponding quantity is calculated on the horizon. Eq. (\\ref{ka}) will be\nused below. It is worth noting that for the extremal horizon ($\\kappa =0$)\nwe have also $\\left( \\frac{dv}{dr}\\right) _{H}=0$.\n\n\\subsection{Redshift: from Kruskal coordinates to Lemaitre ones}\n\nThe above frame is especially useful for the presentation of the redshift \n\\ref{12}). On the horizon, $f=0$. Then, in its vicinity, we obtain from (\\re\n{uU}), (\\ref{rof}), (\\ref{tauf}) that on the horizo\n\\begin{equation}\nv=\\tau +C_{1}=\\rho +C_{2}\\text{,}\n\\end{equation\nwhere $C_{1,2}$ are constants. As a result, we obtain from (\\ref{12}) tha\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\exp (\\kappa (\\tau _{1}-\\tau _{2}))=\\exp\n(\\kappa (\\rho _{1}-\\rho _{2}))\\text{.}  \\label{omk}\n\\end{equation}\n\nThus the Lemaitre frame allows us to present the resulting redshift along\nthe horizon in a simple and intuitively clear picture -- the redshift grows\n(and, consequently, the emitter looks dimmer) exponentially with respect to\nLemaitre time that passes from emitting to observation.\n\nIn the last paragraph of Sec. II, we listed the general condition for the\ngeometrical optic to be valid. Now, we can express it in another way. Since\na physical wave packet has a finite length, parts of it will move away from\nthe black hole horizon even if its center is located exactly on the horizon.\nSince the equation of light geodesics in the generalized Lemaitre frame\nreads $dr\/d\\tau =1-\\sqrt{1-f}$ for outward propagation, the Lemaitre time needed to leave the\nvicinity of the horizon $r=r_{+}$ diverges as $\\left\\vert \\ln ({r\/r_{+}-1)\n\\right\\vert $. Suppose, the emitter radiates light with the wavelength \n\\lambda $. Since in any case the wave packets cannot be smaller than \n\\lambda $, we can roughly estimate initial scale as $r-r_{+}\\ \\sim \\lambda \n. Then, we find that after the Lemaitre time $\\tau \/r_{+}\\sim \\ln \nr_{+}\/\\lambda }\\sim \\ln {\\omega _{0}r_{+}}$ the wave packet will reach the\nscale of black hole horizon, the geometric optic approximation fails and, in\nparticular, Eq. (\\ref{omk}) evidently breaks down.\n\n\\section{Photon emitted at the inner horizon}\n\nLet us consider the situation similar to that considered above. An observer\nmoves beyond the event horizon $r_{+}$and approaches the inner horizon \nr_{-}<r_{+}$. When he crosses it, he emits a photon. Another observer who\nalso crosses the inner horizon later, receives this very photon. What can be\nsaid about its frequency?\n\n\\subsection{The coordinates and metric}\n\nThe metric between the outer and inner horizons represents so-called $T$\nregion \\cite{nov}. For the definiteness, we consider $T^{-}$ region that\ncorresponds to a black hole but similar formulas are valid for the $T^{+}$\nregions (white holes). Now, the metric can be formally obtained from (\\re\n{met}) if one takes into account that for $r_{\\_}<r<r_{+}$the metric\nfunction $f<0$, so spacelike and timelike coordinates mutually interchange.\nWe can write $f=-g$, \n\\begin{equation}\ny=t,T=-r\\text{.}  \\label{T}\n\\end{equation\nThen, the metric can be rewritten in the for\n\\begin{equation}\nds^{2}=-\\frac{dT^{2}}{g}+gdy^{2}+T^{2}d\\omega ^{2}\\text{.}\n\\end{equation}\n\nThe equations of motion for a massive particle have the for\n\\begin{equation}\nm\\dot{y}=\\frac{P}{g}\\text{,}\n\\end{equation\n\\begin{equation}\nm\\dot{T}=z\\equiv \\sqrt{P^{2}+m^{2}g}\\text{,}  \\label{z}\n\\end{equation\nwhere $P=mu_{y}$ is the conserved momentum.\n\nNow, the KS transformation somewhat changes and reads\n\n\\begin{equation}\nU=\\exp (-\\kappa _{-}u)\\text{,}  \\label{ut}\n\\end{equation\n\\begin{equation}\nV=\\exp (\\kappa _{-}v)\\text{,}  \\label{vt}\n\\end{equation\nwhere $r^{\\ast }$ is given by the formula\n\n\\begin{equation}\nr^{\\ast }=\\int \\frac{dr}{g}\\text{.}  \\label{tg}\n\\end{equation\nThe metric takes the form (\\ref{mF}) wit\n\\begin{equation}\nF=-\\frac{g}{UV\\kappa _{-}^{2}}\\text{.}\n\\end{equation\nwhere $\\kappa _{-}$ is the surface gravity associated with the inner horizon\nand $F\\neq 0$ is finite there.\n\nRepeating the calculations step by step, we arrive to the same formula (\\ref{12}\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{V_{1}}{V_{2}}.  \\label{vin}\n\\end{equation\nNow, we would like to pay attention that according to (\\ref{uv}), $V$ is a\nmonotonically increasing function of $v$. It is seen from (\\ref{uU}), (\\re\n{tc}) that for a fixed $u$, $\\frac{\\partial v}{\\partial r}>0$. However, it\nis seen from (\\ref{T}) that event 2 that takes place after 1, has \nr_{2}<r_{1}$. As a result, $v_{2}<v_{1}$ and $V_{2}<V_{1}$. Therefore, \n\\omega _{2}>\\omega _{1}$ and now we have a blueshift. Thus this is related\nto the fact that $r$ and $t$ coordinates change their character in the\nregion under discussion.\n\nThe results (\\ref{omk}) and (\\ref{vin}) can be united in one formul\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\exp [\\left( \\frac{dv}{dr}\\right) _{H}(\\rho\n_{1}-\\rho _{2})]=\\exp [\\left( \\frac{dv}{dr}\\right) _{H}(\\tau _{1}-\\tau _{2})\n\\text{,}\n\\end{equation\nwhere $\\tau _{2}>\\tau _{1}$, $\\rho _{2}>\\rho _{1}$. For the outer horizon we\ncan use eq. (\\ref{ka}) that gives us (\\ref{omk}) and we have a redshift.\nFor the inner horizon, the counterpart of (\\ref{ka}) gives us $\\left( \\frac\ndv}{dr}\\right) _{H}=-\\kappa _{-}$, where now $\\kappa _{-}=\\frac{1}{2\n\\left\\vert \\frac{df}{dr}\\right\\vert _{H}$ is the surface gravity of the\ninner horizon (where $\\left( \\frac{df}{dr}\\right) _{H}<0$). As a result, we\nobtain here a blueshift.\n\nIn a similar way, the procedure under discussion gives the same result when\nan observer crosses the event horizon of a white hole moving outward from the \nT^{+}$ to $R$ region. Then, he will detect all photons propagating along\nthis horizon to be blueshifted. In particular, this holds for the Schwarzschild\nmetric. Analogously, an observer entering $T^{+}$ region from the inner $R$\none (say, like in the Reissner-Nordstr\\\"{o}m metric)\\ will see a redshift at\nthe inner horizon. In other words, in both situations (either black or white\nhole) an observer crossing a horizon from the $T$ to $R$ region will see\na blueshift, while from the $R$ to $T$ region he will see a redshift.\n\n\\subsection{Relation to other effects}\n\nIn the previous subsection we have shown that the blueshift at the inner\nhorizon (and, consequently, the energy absorbed by the observer) grows\nexponentially with the Lemaitre time between the moments of emission and\nobservation. Here we compare this interesting effect with others known in\nliterature.\n\nIf two particle collide, their energy $E_{c.m.}$ in the centre of mass frame\ncan be defined on the point of collision according t\n\\begin{equation}\nE_{c.m.}^{2}=-P_{\\mu }P^{\\mu }\\text{,}\n\\end{equation\n$P^{\\mu }=p_{1}^{\\mu }+p_{2}^{\\mu }$ being the total momentum of two\nparticles. If particle 1 is massive and particle 2 is massless, $p_{1}^{\\mu\n}=mu^{\\mu }$ and $p_{2}^{\\mu }=k^{\\mu }$, where we put the Planck constant\nto unity. As a result\n\\begin{equation}\nE_{c.m.}^{2}=m^{2}+2m\\omega \\text{.}\n\\end{equation}\n\nIn the example under discussion, if $V_{1}=O(1)$ and $V_{2}\\rightarrow 0$,\nthe frequency $\\omega _{2}\\rightarrow \\infty $ according to (\\ref{vin}).\nThen, $E_{c.m.}\\rightarrow \\infty $ as well and we encounter a counterpart\nof the BSW effect near the inner horizon. But $V=0$ on the future horizon \nU=0$ is nothing else than the bifurcation point \\cite{inner} (see also below\nfor more details). Thus the present results for the blueshift agree with the\nprevious ones in the limit when the bifurcation point is reached.\n\nThere is also another issue to which we can compare the present\nconsideration. As is well known, near the inner (Cauchy) horizon an\ninstability develops inside black  holes. This happens when a decaying flux\nof radiation coming from infinity crosses the event horizon and concentrates\nnear the inner one - see, e.g. Chapter 14.3.1 in \\cite{fn}. (For a modern\nreview of the subject see \\cite{ham}.) However, now we consider radiation\nwhich is not coming from infinity but is emitted by an observer who crosses\nthe inner horizon. The resulting energy flux from an emitter at the inner\nhorizon appears to be finite, though it is not restricted from above if \nV\\rightarrow 0$.\n\nThus as far as the radiation near the inner horizon is concerned, we have\nthree situations: (i) the analogue of the BSW effect (relevant near the\nbifurcation point), (ii) blueshift of a photon in the situation under\ndiscussion (relevant near any point of the inner horizon, the blueshift is\nin general finite), (iii) the instability of the inner horizon (infinite\nblueshift due to concentration of radiation along the horizon). Cases (i)\nand (ii) are closely related in the sense that in the limit when the point\nwhere a photon is absorbed approaches the bifurcation point, one obtains (i)\nfrom (ii). Meanwhile, in case (iii) the effect is unbounded and this \npoints to a potential pathology connected with the nature of the \\textit{inner}\n horizon.\n\n\\section{Special case: emission at the bifurcation point}\n\nIn the Sec. V, we discussed briefly such spacetimes that contain $T^{+}$\nregions (white holes). Then the intersection between the future and past\nhorizons forms the so-called bifurcation point (sphere, if the angle\nvariables are taken into account), where it is possible to pass from the\nwhite hole region to the black one. White holes and bifurcation points do\nnot arise in the situation when a black hole is formed due to gravitational\ncollapse and in this sense they are not feasible astrophysically. However, they\nare inevitably present in the full picture of an eternal black-white hole.\nTherefore, we consider such objects for theoretical reasons and for\ncompleteness. In particular, in Sec. V, we saw that accounting for the\nbifurcation point arises naturally in the connection between our problem and\nthe BSW effect. In doing so, it is a receiver that passes near the\nbifurcation point.\n\nIn the present Section, we consider another case, when it is an emitter that\npasses through this point at the moment of radiation. Consideration of the\nfrequency shift when a photon emitted from the bifurcation point is a\nseparate case that does not follow directly from the previous formulas. For\nthe Reissner-Nordstr\\\"{o}m-de Sitter metric, such a problem was considered\nin Sec. IV b of \\cite{lake}. It follows from the corresponding results that\ndifferent cases are possible here: $\\omega _{2}<\\omega _{1}$, $\\omega\n_{2}=\\omega _{1}$, $\\omega _{2}>\\omega _{1}.$ On the first glance, this\ndisagrees with our results described above since we obtained either redshift\n(for the event horizon of a black hole or inner horizon of a white hole) or\nblueshift (for the inner one in a black hole or event horizon of a white\nhole). Fortunately, this contradiction is illusory. Now, we will explain how\none can obtain the results for the bifurcation point from ours. To this end,\nwe compare (i) the generic situation and (ii) that with crossing the\nbifurcation point and trace how (ii) arises from (i) within the limiting\ntransition.\n\nFor our purposes, it is sufficient to discuss the simplest metric that\npossesses the bifurcation point, so we can imply it to be, say, the\nSchwarzschild one. We assume that the emitter 1 moves from the inner\nexpanding $T^{+}$ region (i.e. white hole) \\cite{nov}, crosses the past\nhorizon and enters the R region. Afterwards, it crosses the event horizon\nfalling into a black hole. Let, as before, the emitter and receiver have\nequal masses $m_{1}=m_{2}=m$. However, now we cannot put \\ $E_{1}=E_{2}.$\nThis is because a particle with $E=m$ would escape to infinity instead of\nfalling into a black hole. We remind the reader that up to now, in all our\nconsiderations an emitter and an observer are set to be at rest in infinity.\nHowever, the most gereral case can easily be obtained by adding\ncorresponding Lorentz boosts. In the present subsection we meet the\nsituation where this procedure is needed.\n\nTherefore, we must use more general formula based on (\\ref{omf}\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{E_{2}}{E_{1}}\\frac{V_{1}}{V_{2}}\\text{\n}  \\label{gen}\n\\end{equation\nThe first factor can be interpreted as a Lorentz boost responsible for the\nDoppler effect. For $E_{1}=E_{2}$ we return to the case considered by us\nabove but now the first factor is not equal to one and plays now a crucial\nrole.\n\nIf, by assumption, particle 1 falls into a black hole, this means that it\nmust bounce from the potential barrier in the turning point $r=r_{0}$.\nAccording to equations of motion (\\ref{mr}), this means tha\n\\begin{equation}\nE=m\\sqrt{f(r_{0})}\\text{.}  \\label{ef}\n\\end{equation}\n\nIf $r_{0}\\rightarrow r_{+}$, $f(r_{0})\\rightarrow 0$, so $E\\rightarrow 0$ as\nwell. More precisely, it is seen from (\\ref{kar}), (\\ref{uvc}) tha\n\\begin{equation}\nE\\sim \\sqrt{r_{0}-r_{+}}\\sim \\sqrt{\\left\\vert U\\right\\vert V}\\text{.}\n\\end{equation\nAs a result\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}\\sim \\frac{\\alpha E_{2}}{V_{2}}\\text{, \n\\alpha \\equiv \\sqrt{\\frac{\\left\\vert V\\right\\vert _{1}}{U_{1}}}\\text{.}\n\\end{equation}\n\nIn the limit when the trajectory of particle 1 passes closer and closer to\nthe bifurcaiton point $U=0=V$, $\\alpha $ remains finite. Using equations of\nmotion in the T region (see the previous section), it is easy to show that\nin the limit $V\\rightarrow 0$, $U\\rightarrow 0$, the component of the\nvelocity $u^{U}$ contains just this factor $\\alpha $.\n\nThus depending on relation between $\\alpha $ and $V_{2}$ one can obtain any\nresult for $\\omega _{2}$ (redshift, blueshift, the absence of the frequency\nshift). In this sense, the general formula (\\ref{gen}) reproduces both\n\"standard\" fall of the emitter in a black hole and the behavior of the\nemitter that passes through the bifurcation point.\n\n\\section{Extremal horizon}\n\nLet an observer crosses the (ultra) extremal horizon $r_{+}$. By definition,\nthis means that near it the metric function is\n\\begin{equation}\nf\\sim (r-r_{+})^{n}\n\\end{equation\nwhere $n=2$ in the extremal case and $n=3,4...$ in the ultraextremal one.\nThe difference with the nonextremal case consists in a different nature of\ntransformation making the metric regular. Let the two-dimensional part of\nthe  metric has the same form as in (\\ref{met}). The subsequent procedure\nis known  - see, e.g., \\cite{lib}, \\cite{bron} (Sec. 3.5.1). We use the\nsame coordinates $u$, $v$ and want to find appropriate coordinates $U,$ $V$\n\\begin{equation}\nV=V(v)\\text{, }U=U(u)\\text{.}\n\\end{equation}\n\nNow, we are interested in the situation with emission of a photon exactly\nalong the horizon.Then, near the horizon it follows for the tortoise\ncoordinate (\\ref{tc}) tha\n\\begin{equation}\nr-r_{+}\\sim \\left\\vert r^{\\ast }\\right\\vert ^{\\frac{1}{1-n}}\\text{,}\n\\end{equation\n\\begin{equation}\nf\\sim \\left\\vert r^{\\ast }\\right\\vert ^{\\frac{n}{1-n}}\\text{.}\n\\end{equation\nWe consider the metric near the future horizon where $v$ is finite, $r^{\\ast\n}\\rightarrow -\\infty $, $u=v-2r^{\\ast }\\rightarrow +\\infty $. We hav\n\\begin{equation}\nf\\sim u^{\\frac{n}{1-n}}\\text{.}  \\label{fu}\n\\end{equation}\n\nWe try a transformation that behaves like \n\\begin{equation}\nU\\sim u^{-\\frac{1}{n-1}}\\text{,}  \\label{uextr}\n\\end{equation\nso that $U\\rightarrow 0$. Then, it is easy to check that the metric has the\nform (\\ref{mF}) where $F\\neq 0$ is finite on the horizon. To find the\nfrequency, we must use the expression for $u^{U}$ (\\ref{mu}) in which now \n\\ref{uextr}) is valid, so $\\frac{dU}{du}\\sim u^{\\frac{n}{1-n}}$. \\ It is\nseen from (\\ref{fu}) that $u^{U}\\rightarrow const$ on the horizon and it does\nnot contain $V$. Taking into account that $k^{V}$ is a constant along the\nhorizon generator as before, we come to the conclusion that $V$ drops out and \n\\frac{\\omega _{2}}{\\omega _{1}}=const.$ We see that in the horizon limit the\nquantity $V$ does not enter the frequency. In this sense, $\\frac{\\omega _{2\n}{\\omega _{1}}$ does not change along the horizon, so redshift or blueshift\nis absent.\n\nIn a sense, it is quite natural. Indeed, the extremal horizon is the double\none. The inner and outer horizons merge. But for an inner horizon we had\na blueshift, for the outer one we had a redshift. Together, they mutually cancel\nand produce no effect.\n\nThe absence of the redshift or blueshift formally agrees with (\\ref{omk}) if\none puts $\\kappa =0$ there. However, for (ultra)extremal black holes the\nKruskal-like transformation looks very different, so we could not use eq. \n\\ref{omk}) directly. Therefore, it was not obvious in advance, whether or\nnot the redshift for the extremal horizon can be obtained as the extremal\nlimit of a nonextremal one. Now, we see that this is the case.\n\n\\section{Summary}\n\nThus we showed that for emission along the outer horizon redshift occurs and\nwe derived a simple formula that generalized the one previously found in\nliterature. We also showed that along the inner horizon blueshift occurs and\nfound its relation with the BSW effect. We also showed how the previously known\nresults for the emission at the bifurcation point are reproduced from a\ngeneral formula and lead to a diversity of situations (redshift, blueshift\nor the absence of frequency shift). For (ultra)extremal horizons the effect\nis absent.\n\nThese observations have a quite general character in agreement with the\nuniversality of black hole physics. We also generalized the Lemaitre frame\nand in this frame derived a simple and instructive formula for a redshift\nalong the horizon in terms of the Lemaitre time and the surface gravity.\n\n\\section*{Acknowledgements}\n\nThe work was supported by the Russian Government Program of Competitive\nGrowth of Kazan Federal University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction}\n\n\nHelicity density (hereafter simply denoted as helicity) is defined as the inner product of velocity and vorticity and is known to play a crucial role termed as the $\\alpha$ effect in the dynamo action in magnetohydrodynamics \\cite{moffattbook}. \nIn contrast, the role of helicity in neutral hydrodynamic turbulence is not clearly understood to-date. Studies on helicity can be divided into two categories, namely studies on the emergence of helicity and studies on the effects of helicity on the dynamics of turbulence. In the former, the rise of statistically significant helicity spectrum of homogeneous turbulence is never found in the absence of ad hoc initialization or forcing \\cite{cj1989}. Conversely, helicity is known to emerge in rotating inhomogeneous turbulence such as a convection zone in a rotating sphere \\cite{duarteetal2016,ssd2014} or a rotating inhomogeneous turbulence in which the rotation axis is parallel to the inhomogeneous direction \\cite{gl1999,rd2014,kapylaetal2017}. Therefore, the key in the emergence of helicity corresponds to the inhomogeneity of rotating turbulence. In the latter, most studies focus on homogeneous turbulence and effects on energy cascade. A few studies revealed that helicity does not crucially influence hydrodynamic flows in the context of the energy cascade.\nFor example, with the aid of the eddy-damped quasi-normalized Markovian (EDQNM) approximation, Andr\\'e and Lesieur \\cite{al1977} showed that helicity does not affect the energy cascade once the inertial range is established. Rogers and Moin \\cite{rm1987} numerically showed that the correlation between helicity and the dissipation rate of the turbulent kinetic energy is tenuous in homogeneous isotropic turbulence, homogeneous shear turbulence, and turbulent channel flow. Wallace \\textit{et al.} \\cite{wallaceetal1992} experimentally confirmed the correlation between helicity and the dissipation rate in a turbulent boundary-layer, a two-stream mixing-layer, and grid-flow turbulence. They concluded that there is a tenuous relationship between small dissipation rate and large helicity except in the shear flows.\n\nIn contrast, helicity is expected to be important in dynamics of mean flow generation in inhomogeneous turbulence. This point was first discussed by Yokoi and Yoshizawa \\cite{yy1993} in terms of the closure scheme for the Reynolds-averaged Navier--Stokes (RANS) formulation. They suggested that the spatial gradient of helicity coupled with the vortical motion of fluid affects the Reynolds stress (velocity--velocity correlation) and diminishes the turbulent momentum transfer.\nRecently, Yokoi and Brandenburg \\cite{yb2016} numerically revealed that the mean flow is generated in a system with both inhomogeneous helicity and system rotation. This phenomenon can be explained with a model expression for the Reynolds stress obtained by Yokoi and Yoshizawa \\cite{yy1993}.\nFlow generation in the context of the large-scale flow instability was also discussed by Frisch \\textit{et al.} \\cite{aka}, and it is termed as the anisotropic kinetic alpha (AKA) effect. However, Yokoi and Brandenbrug \\cite{yb2016} noted that the flow generation due to the inhomogeneous helicity is suitable for treating flows at high Reynolds number, such as astro\/geophysical flows, while the AKA effect is valid only for flows at low Reynolds number. Thus the model proposed by Yokoi and Yoshizawa \\cite{yy1993} involves general physics of fully developed inhomogeneous turbulence. However, the origin of the helicity effect on the Reynolds stress was not demonstrated based on the Reynolds-stress transport equation. In this sense, the manner in which helicity affects the Reynolds-stress evolution continues to be unclear. \n\nThe Reynolds stress is typically modeled by the eddy-viscosity representation, which is one of the simplest models for the Reynolds stress. The eddy-viscosity model represents the momentum transfer enhanced by turbulence, and the effective viscosity is augmented by turbulent motions. Pope \\cite{pope1975} obtained a nonlinear eddy-viscosity model for the Reynolds stress from the Reynolds-stress transport equation model of Launder \\textit{et al.} \\cite{lrr1975} by neglecting the diffusion effect. The fore-mentioned nonlinear eddy-viscosity models represented a considerable improvement relative to the conventional models. However, in some flows, the models continue to exhibit difficulties in terms of performance. A representative case in which the models do not work well is a swirling flow in a straight pipe \\cite{steenbergen,kito1991}. In the flow, the mean axial velocity exhibits a dent profile in the center axis region of the pipe, and the dent profile is significantly more persistent in the downstream region than those predicted by the eddy-viscosity type models. \n\nYokoi and Yoshizawa \\cite{yy1993} applied the turbulence model with inhomogeneous helicity effect on the Reynolds stress to a swirling pipe flow and successfully reproduced the sustainment of the dent mean velocity. Another description of the effect of helicity on turbulence was constructed by Yoshizawa \\textit{et al.} \\cite{yoshizawaswirl2011}. They introduced a timescale of helical motion into the model and obtained good results in a swirling pipe flow. The results suggest the importance of helicity effect in describing the properties of swirling flows.\nThis helicity effect is also discussed in the context of the sub-grid scale (SGS) modeling in relation to the over-estimation of dissipation rate in the use of eddy-viscosity-type SGS stress models \\cite{yy2017}.\nHowever, the terms obtained in Yokoi and Yoshizawa \\cite{yy1993} or Yoshizawa \\textit{et al.} \\cite{yoshizawaswirl2011} were not directly linked to the systematic modeling of Pope \\cite{pope1975}. This is because the mechanism by which helicity affects the Reynolds stress is not fully known, and thus the helicity effect in the Reynolds-stress evolution is not explicitly considered. In order to reveal the helicity effect on the Reynolds stress, we investigate the physical origin of the effect at the level of the Reynolds-stress transport equation.\n\nIn this study, we perform a numerical simulation of a rotating inhomogeneous turbulence driven by a helical external forcing.\nAlthough the mechanism of the helicity generation is important, this is not examined here. We impose the helicity by external forcing in the present study and focus on the effect of inhomogeneous helicity on the mean flow.\nThe flow configuration is similar to that used by Yokoi and Brandenburg \\cite{yb2016}. It has two homogeneous directions and one inhomogeneous direction, and the rotation axis is perpendicular to the inhomogeneous direction. In the configuration the mean flow is expected to emerge in the rotation-axis direction.\nThis flow configuration is similar to the low-latitude region of rotating sphere in which turbulence is radially inhomogeneous and its rotation axis is mostly perpendicular to the inhomogeneous direction \\cite{duarteetal2016,ssd2014}. We also conduct simulations in non-rotating and\/or non-helical forcing cases to identify the condition for the mean-flow generation. The helicity effect is tested in relation to the Reynolds-stress transport equation, and the origin of the mean-flow generation is explored.\n\nThe rest of this study is organized as follows. Section~\\ref{sec:level2} summarizes the relationship between the eddy-viscosity-type turbulence model and the transport equation for the Reynolds stress. The model for the Reynolds stress including the helicity effect derived by Yokoi and Yoshizawa \\cite{yy1993} is also presented. Section~\\ref{sec:level3} presents the numerical setup and the simulation results. We also discuss the origin of the helicity effect on the Reynolds stress. A comparison between our results and the model expression of the Reynolds stress with helicity is given in Sec.~\\ref{sec:level4}. The conclusions are discussed in Sec.~\\ref{sec:level5}.\n\n\\section{\\label{sec:level2}Model representations of the Reynolds stress and helicity effect}\n\nThe Navier--Stokes equation and the continuity equation for an incompressible fluid in a rotating system are given respectively as follows:\n\\begin{align}\n\\frac{\\partial u_i}{\\partial t} & =\n- \\frac{\\partial}{\\partial x_j} u_i u_j - \\frac{\\partial p}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 u_i}{\\partial x_j \\partial x_j} + 2 \\epsilon_{ij\\ell} u_j \\Omega^F_\\ell + f_i, \n\\label{eq:2.1} \\\\\n\\frac{\\partial u_i}{\\partial x_i} & = 0, \n\\label{eq:2.2}\n\\end{align}\nwhere $u_i$ denotes the $i$-th component of the velocity, $p$ the pressure divided by the fluid density with centrifugal force included, $\\nu$ the kinematic viscosity, $\\Omega^F_i$ the angular velocity of the system, $f_i$ the external force, and $\\epsilon_{ij\\ell}$ the alternating tensor. We decompose a physical quantity $q [= (u_i, p, f_i)]$ into mean and fluctuation parts as follows:\n\\begin{align}\nq = Q + q', \\ \\ & Q = \\left< q \\right>,\n\\label{eq:2.3}\n\\end{align}\nwhere $\\left< \\cdot \\right>$ denotes an ensemble average. Substituting Eq.~(\\ref{eq:2.3}) into Eqs.~(\\ref{eq:2.1}) and (\\ref{eq:2.2}), we obtain the mean field equations,\n\\begin{align}\n\\frac{\\partial U_i}{\\partial t} & =\n- \\frac{\\partial}{\\partial x_j} \\left( U_i U_j + R_{ij} \\right) - \\frac{\\partial P}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 U_i}{\\partial x_j \\partial x_j} + 2 \\epsilon_{ij\\ell} U_j \\Omega^F_\\ell + F_i, \n\\label{eq:2.4} \\\\\n\\frac{\\partial U_i}{\\partial x_i} & = 0,\n\\label{eq:2.5}\n\\end{align}\nwhere $R_{ij} (= \\left< u_i' u_j' \\right>)$ denotes the Reynolds stress. The only difference between Eqs.~(\\ref{eq:2.1}) and (\\ref{eq:2.4}) corresponds to the Reynolds stress. Thus, the Reynolds stress solely represents the effects of turbulent motion on the mean velocity. In order to close the system of Eqs.~(\\ref{eq:2.4}) and (\\ref{eq:2.5}), a model expression for the Reynolds stress is required.\n\n\\subsection{\\label{sec:level2a}Relationship between model and transport equation for the Reynolds stress}\n\nThe simplest model for the Reynolds stress is the eddy-viscosity model that is expressed as follows:\n\\begin{align}\nR_{ij} = \\frac{2}{3} K \\delta_{ij} - 2 \\nu_T S_{ij},\n\\label{eq:2.6}\n\\end{align}\nwhere $K (= \\left<u_i' u_i' \\right>\/2)$ denotes the turbulent kinetic energy, $\\nu_T$ the eddy viscosity, \\linebreak $S_{ij} [= \\left( \\partial U_i \/ \\partial x_j + \\partial U_j \/ \\partial x_i \\right) \/ 2]$ the strain rate of the mean velocity, and $\\delta_{ij}$ the Kronecker delta. The eddy-viscosity model is not just an empirical model but can be obtained from the fundamental equation, i.e., the Navier--Stokes equation. Specifically, the model expression for the Reynolds stress is closely related to the transport mechanism of the Reynolds stress. A systematic way to obtain the eddy-viscosity-type model from the Reynolds-stress transport equation may be summarized as follows \\cite{pope1975,yoshibook}. The exact transport equation for the Reynolds stress is expressed as follows:\n\\begin{align}\n\\frac{\\mathrm{D} R_{ij}}{\\mathrm{D} t} & =\nP_{ij} - \\varepsilon_{ij} + \\Phi_{ij} + \\Pi_{ij} \n + T_{ij} + D_{ij} + C_{ij} + F_{ij}, \n\\label{eq:2.7}\n\\end{align}\nwhere $\\mathrm{D}\/\\mathrm{D} t = \\partial \/ \\partial t + U_\\ell \\partial \/ \\partial x_\\ell$ denotes the Lagrange derivative. Here, $P_{ij}$ denotes the production rate, $\\varepsilon_{ij}$ the destruction rate, $\\Phi_{ij}$ the pressure--strain correlation, $\\Pi_{ij}$ the pressure diffusion, $T_{ij}$ the turbulent diffusion, and $D_{ij}$ the viscous diffusion, $C_{ij}$ the Coriolis effect, and $F_{ij}$ the external work. They are respectively defined as follows:\n\\begin{subequations}\n\\begin{align}\nP_{ij} & = - R_{i\\ell} \\frac{\\partial U_j}{\\partial x_\\ell} - R_{j\\ell} \\frac{\\partial U_i}{\\partial x_\\ell}, \n\\label{eq:2.8a} \\\\\n\\varepsilon_{ij} & = \\left< 2 \\nu s_{i\\ell} \\frac{\\partial u_j'}{\\partial x_\\ell}\n   + 2 \\nu s_{j\\ell} \\frac{\\partial u_i'}{\\partial x_\\ell} \\right>, \n\\label{eq:2.8b} \\\\\n\\Phi_{ij} & = 2 \\left< p' s_{ij} \\right>, \n\\label{eq:2.8c} \\\\\n\\Pi_{ij} & = - \\frac{\\partial}{\\partial x_j} \\left< p' u_i' \\right>\n   - \\frac{\\partial}{\\partial x_i} \\left< p' u_j' \\right>,\n\\label{eq:2.8d} \\\\\nT_{ij} & = - \\frac{\\partial}{\\partial x_\\ell} \\left< u_i' u_j' u_\\ell' \\right>, \n\\label{eq:2.8e} \\\\\nD_{ij} & = \\frac{\\partial}{\\partial x_\\ell} \\left< 2 \\nu s_{i\\ell} u_j' + 2 \\nu s_{j\\ell} u_i' \\right>, \n\\label{eq:2.8f} \\\\\nC_{ij} & = 2\\left( \\epsilon_{im \\ell} R_{jm} + \\epsilon_{jm \\ell} R_{im} \\right) \\Omega_\\ell^F ,\n\\label{eq:2.8g} \\\\\nF_{ij} & = \\left< u_i' f_j + u_j' f_i \\right>,\n\\label{eq:2.8h}\n\\end{align}\n\\end{subequations}\nwhere $s_{ij}  [= \\left( \\partial u_i \/ \\partial x_j + \\partial u_j \/ \\partial x_i \\right) \/ 2]$ denotes the strain rate of the velocity. Pope \\cite{pope1975} obtained a general expression of the Reynolds stress based on the following two assumptions. First, to the right-hand side of Eq.~(\\ref{eq:2.7}), the model by Launder \\textit{et al.} \\cite{lrr1975} (LRR model) is adopted; $\\varepsilon_{ij}$ and $\\Phi_{ij}$ are modeled as follows:\n\\begin{align}\n\\varepsilon_{ij} & = \\frac{2}{3} \\varepsilon \\delta_{ij}, \n\\label{eq:2.9} \\\\\n\\Phi_{ij} & = - C_{S1} \\frac{\\varepsilon}{K} B_{ij} + C_{R1} K S_{ij} \\nonumber \\\\\n& \\hspace{1.2em} + C_{R2} \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n + C_{R3} \\left( B_{i\\ell} \\Omega_{\\ell j} + B_{j\\ell} \\Omega_{\\ell i} \\right) ,\n\\label{eq:2.10}\n\\end{align}\nwhere $\\varepsilon (= \\varepsilon_{ii}\/2)$ denotes the dissipation rate of the turbulent energy $K$, $B_{ij} [= R_{ij} -(2\/3) K \\delta_{ij}]$ the deviatoric part of the Reynolds stress, $\\Omega_{ij} = \\left( \\partial U_j \/ \\partial x_i - \\partial U_i \/ \\partial x_j \\right)\/2$, $\\left[ A_{ij} \\right]_D = A_{ij} - A_{\\ell \\ell} \\delta_{ij} \/ 3$, and $C_{S1}$, $C_{R1}$, $C_{R2}$, and $C_{R3}$ denote the model constants.\nThe term with $C_{S1}$ describes the `return to isotropy' model while the terms with $C_{R1}$, $C_{R2}$, and $C_{R3}$ correspond to the `isotropization of production' model \\cite{lrr1975}. Although there are more elaborate models for the pressure--strain correlation, such as Craft and Launder \\cite{tcl}, we focus on simple models proportional to $B_{ij}$. Second, quasi-homogeneity of the flow field is assumed, and the diffusion terms are neglected as $\\Pi_{ij} = T_{ij} = D_{ij} = 0$. In addition to the two assumptions, it is necessary to handle the time derivative term, $\\mathrm{D} R_{ij} \/ \\mathrm{D} t$. In the algebraic stress models, the weak-equilibrium assumption, $\\mathrm{D} (R_{ij}\/K)\/\\mathrm{D} t = 0$, is applied. Here this assumption is not used; we introduce an appropriate time derivative instead of the Lagrange derivative in order to satisfy the frame invariance of the turbulence equation in a rotating system \\cite{hamba2006,ariki2015}. When the upper convected time derivative, $\\mathfrak{D} A_{ij}\/\\mathfrak{D} t = \\mathrm{D} A_{ij}\/\\mathrm{D} t - A_{i\\ell} \\partial U_j \/ \\partial x_\\ell - A_{j\\ell} \\partial U_i \/ \\partial x_\\ell$, is adopted, Eq.~(\\ref{eq:2.7}) is expressed as follows:\n\\begin{align}\n\\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t} & =\n - C_{S1} \\frac{\\varepsilon}{K} B_{ij}\n- \\left( \\frac{4}{3} - C_{R1} \\right) K S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\left( 1 - C_{R2} \\right) \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\left( 1 - C_{R3} \\right) \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right),\n\\label{eq:2.11}\n\\end{align}\nwhere $\\Omega_{ij}^* (= \\Omega_{ij} + \\epsilon_{ij\\ell} \\Omega_\\ell^F)$ denotes the mean absolute vorticity tensor. Here it is assumed that the external work does not affect the Reynolds stress directly. The model for $\\Phi_{ij}$ is extended to a rotating system. Thus, we replace $\\Omega_{ij}$ in Eq.~(\\ref{eq:2.10}) by $\\Omega_{ij}^*$. This frame invariant formulation is performed to ensure the consistency of the equations in a rotating frame. The effect of rotation may affect the transport equation for $\\varepsilon$ \\cite{bfr1985}, and this type of a modification is needed to predict turbulent flows under the solid body rotation with the Reynolds-stress models. However, this point is beyond the scope of the present study that focuses on the effects on the mean flow. The first term on the right-hand side of Eq.~(\\ref{eq:2.11}) represents the destruction of $B_{ij}$ or the relaxation to an isotropic state. The second term denotes the production of $B_{ij}$ by the isotropic part of turbulence, while the third and fourth terms  denote the production by the anisotropic part of turbulence. Equation~(\\ref{eq:2.11}) is re-expressed as follows:\n\\begin{align}\nB_{ij} & =\n- 2 \\frac{4-3C_{R1}}{6C_{S1}} \\frac{K^2}{\\varepsilon} S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\frac{1-C_{R2}}{C_{S1}} \\frac{K}{\\varepsilon} \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\frac{1-C_{R3}}{C_{S1}} \\frac{K}{\\varepsilon} \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right) \n\\nonumber \\\\\n& \\hspace{1.2em}\n- \\frac{1}{C_{S1}} \\frac{K}{\\varepsilon} \\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t}.\n\\label{eq:2.12}\n\\end{align}\nSubstituting this expression iteratively into $B_{ij}$ on the right-hand side, we obtain the following:\n\\begin{align}\nB_{ij} & =\n- 2 C_\\nu \\frac{K^2}{\\varepsilon} S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n+ C_{q1} \\frac{K^3}{\\varepsilon^2} \\left[ S_{i\\ell} S_{\\ell j} + S_{j\\ell} S_{\\ell i} \\right]_D \n+ C_{q2} \\frac{K^3}{\\varepsilon^2} \\left( S_{i\\ell} \\Omega_{\\ell j}^* + S_{j\\ell} \\Omega_{\\ell i}^* \\right)\n\\nonumber \\\\\n& \\hspace{1.2em}\n+ C_d \\frac{K}{\\varepsilon} \\frac{\\mathfrak{D}}{\\mathfrak{D} t} \\left( \\frac{K^2}{\\varepsilon} S_{ij} \\right) \n+ \\text{(higher order terms)} , \n\\label{eq:2.13}\n\\end{align}\nwhere $C_\\nu = (4-3C_{R1})\/(6C_{S1})$, $C_{q1} = 2 C_\\nu (1-C_{R2}) \/C_{S1}$, $C_{q2} = 2 C_\\nu (1-C_{R3})\/C_{S1}$, and $C_d = 2C_\\nu\/C_{S1}$. In contrast to the formulation obtained by Pope \\cite{pope1975}, the time derivative term is retained in the right-hand side of Eq.~(\\ref{eq:2.13}) as shown in Yoshizawa \\cite{yoshibook}. This corresponds to a more general formulation when compared with that obtained by Pope \\cite{pope1975} since the time derivative term does not always disappear. The first term of Eq.~(\\ref{eq:2.13}) represents the eddy-viscosity term which corresponds to the second term on the right-hand side of Eq.~(\\ref{eq:2.6}), and this term is derived from the isotropic part of the production term. This reflects the point that the eddy-viscosity model constitutes a good approximation when the turbulence is nearly isotropic, quasi-homogeneous, and steady. \n\n\n\\subsection{\\label{sec:level2b}The Reynolds-stress expression accompanied with the helicity effect}\n\nThe eddy-viscosity-type models provide good results for simple flows such as free shear layer flows and channel flows. However, they perform poorly for more complex flows. An example in which the usual eddy-viscosity models do not work well is a swirling flow in a straight pipe \\cite{steenbergen,kito1991}. In the swirling-flow experiments, it is observed that the mean axial velocity shows a dent in the center axis region, and this inhomogeneous velocity profile is very persistent to the well downstream region. However, this type of a dent profile that is imposed at the pipe inlet cannot be sustained and decays rapidly in the usual eddy-viscosity model simulation \\cite{ky1987,steenbergen}. This is because the eddy viscosity is so strong that it smears out any large-scale velocity gradient. Jakirli\\'c \\textit{et al.} \\cite{jht2000} pointed out that even with an elaborate explicit Reynolds-stress model such as Craft \\textit{et al.} \\cite{cls1996} or Shih \\textit{et al.} \\cite{shihetal1997} as well as the standard eddy-viscosity model, it is difficult to accurately reproduce the fore-mentioned rotational flows without performing a few modifications in the model constants. With the aid of the two-scale direct-interaction approximation (TSDIA) \\cite{tsdia} that is an analytical statistical theory of inhomogeneous turbulence, Yokoi and Yoshizawa \\cite{yy1993} suggested that eddy viscosity may be suppressed by symmetry breaking swirling motion. They analytically constructed a new turbulence model in which the helicity effect is incorporated. In the formulation, homogeneous isotropic non-mirror-symmetric turbulence is assumed as the basic field, and the effects of inhomogeneity, anisotropy, and system rotation are incorporated in a perturbational manner based on the Navier--Stokes equation. Brief descriptions of the formulation are given in Appendix~\\ref{sec:a}. According to the formulation, the deviatoric or traceless part of the Reynolds stress is expressed as follows:\n\\begin{align}\nB_{ij} = - 2 \\nu_T S_{ij}\n+ \\eta \\left[ \\frac{\\partial H}{\\partial x_j} \\Omega^*_i + \\frac{\\partial H}{\\partial x_i} \\Omega^*_j \\right]_D,\n\\label{eq:2.14}\n\\end{align}\nwhere $\\eta$ denotes the transport coefficient, $H (= \\left< u_i' \\omega_i' \\right>)$ the turbulent helicity, and $\\Omega^*_i (= \\epsilon_{ij\\ell} \\partial U_\\ell \/ \\partial x_j + 2 \\Omega^F_i)$ the mean absolute vorticity. In this study, we refer to the model of Eq.~(\\ref{eq:2.14}) as the helicity model. This model allowed the successful reproduction of the sustainment of the dent mean axial velocity in a swirling flow. The helicity model is similar to the AKA model \\cite{aka} in the sense that the AKA describes the effect of lack of parity invariance on the mean flow. The helicity model is developed for high-Reynolds number flows since the TSDIA corresponds to perturbational expansion from fully developed homogeneous turbulence, while the AKA is valid for low-Reynolds number flows \\cite{yb2016}. Hence, it is expected the helicity model can be applied to realistic high-Reynolds number turbulent flows.\n\nIt is interesting to note that as pointed out in \\cite{yy1993} and \\cite{yb2016}, the present model accounts for the mean flow generation from the no-mean-velocity initial condition. Even if system does not have the mean velocity gradient, Eq.~(\\ref{eq:2.14}) may include a non-zero value when both the helicity gradient and the system rotation exist. In such cases, the deviatoric part of the Reynolds stress is expressed as follows:\n\\begin{align}\nB_{ij} = 2 \\eta \\left[ \\frac{\\partial H}{\\partial x_j} \\Omega^F_i + \\frac{\\partial H}{\\partial x_i} \\Omega^F_j \\right]_D \n\\neq 0.\n\\label{eq:2.15}\n\\end{align}\nThis suggests that the mean flow is generated by this helicity effect when the inhomogeneous helicity is coupled with the rotation since the mean velocity equation is expressed as\n\\begin{align}\n\\frac{\\partial U_i}{\\partial t} = - \\frac{\\partial}{\\partial x_j} \n\\left[ \\eta \\left( \\frac{\\partial H}{\\partial x_j} 2 \\Omega^F_i \n+ \\frac{\\partial H}{\\partial x_i} 2 \\Omega^F_j - \\frac{\\partial H}{\\partial x_\\ell} 2 \\Omega^F_\\ell \\frac{2}{3} \\delta_{ij} \\right) \\right] - \\frac{\\partial P}{\\partial x_i}\n\\neq 0.\n\\label{eq:2.16}\n\\end{align}\nYokoi and Brandenburg \\cite{yb2016} performed direct numerical simulations (DNSs) of a rotating inhomogeneous turbulence with an imposed turbulent helicity. They commenced with a no-mean-velocity configuration and observed a mean-flow generation in a rotating turbulence. Additionally, they confirmed that in the early stage of the simulation in which the mean-velocity gradient is not significantly developed, the Reynolds stress is well correlated with the middle part of Eq.~(\\ref{eq:2.15}). It is not possible to predict this type of a flow generation phenomenon by using a conventional model of the Reynolds stress as given by Eq.~(\\ref{eq:2.13}) since each term contains the mean shear rate.\n\nThe results indicate that inhomogeneous helicity coupled with the vortical motion of fluid affects the Reynolds stress and reduces turbulent momentum transport represented by the eddy viscosity. The following points should be noted. The model representation of Eq.~(\\ref{eq:2.14}) was analytically obtained from the Navier--Stokes equation with the aid of TSDIA. However, the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) is not obtained in a direct manner from the systematic construction of the model shown in Sec.~\\ref{sec:level2a}. This is because the turbulent helicity is not explicitly included in the Reynolds-stress transport equation given in Eq.~(\\ref{eq:2.11}) on which the model constitution is based. Yokoi and Brandenburg \\cite{yb2016} compared the profile of the Reynolds stress with that of Eq.~(\\ref{eq:2.14}) to determine a very good correlation between them. However, the origin of the helicity effect on the Reynolds-stress equation was not shown. As shown in Sec.~\\ref{sec:level2a}, the model expression of the Reynolds stress is related to  its transport mechanism. The effect of helicity corresponding to the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) should exist on the right-hand side of Eq.~(\\ref{eq:2.7}) as well as the production term corresponding to the eddy-viscosity term. Hence, the physical origin of the second term of Eq.~(\\ref{eq:2.14}) is not clarified in the sense  of the Reynolds-stress evolution.\n\n\\section{\\label{sec:level3}Numerical simulations}\n\nIn order to investigate the mechanism of the mean-flow generation and its relationship to the turbulent helicity, we perform a series of numerical simulations of a rotating inhomogeneous turbulence driven by a helical external force. We examine the transport equation for the Reynolds stress to explore the manner in which the turbulent helicity affects the Reynolds-stress transport.\n\n\\subsection{\\label{sec:level3a}Governing equations and numerical setup}\n\nIn order to simulate a high-Reynolds-number turbulent flow, the large eddy simulation (LES) is adopted instead of the DNS. The governing equations of the LES in a rotating system are expressed as follows:\n\\begin{align}\n\\frac{\\partial \\overline{u}_i}{\\partial t} & =\n - \\frac{\\partial}{\\partial x_j} \\overline{u}_i \\overline{u}_j\n  - \\frac{\\partial \\overline{p}}{\\partial x_i}\n + \\frac{\\partial}{\\partial x_j} 2 \\nu_{sgs} \\overline{s}_{ij} \n+ 2 \\epsilon_{ij\\ell} \\overline{u}_j \\Omega_\\ell^F\n  + \\overline{f}_i , \n\\label{eq:3.1} \\\\ \n\\frac{\\partial \\overline{u}_i}{\\partial x_i} & = 0,\n\\label{eq:3.2}\n\\end{align}\nwhere the kinematic viscosity is neglected, and $\\overline{q}$ denotes the grid-scale (resolved) component of $q$. It should be noted that $\\overline{q}$ is different from the ensemble average, $\\left< q \\right>$, which is already introduced in Eq.~(\\ref{eq:2.3}). With respect to the model of the subgrid-scale (SGS) viscosity, $\\nu_{sgs}$, the Smagorinsky model \\cite{smagorinsky1963},\n\\begin{align}\n\\nu_{sgs} = \\left( C_S \\Delta \\right)^2 \\sqrt{ 2 \\overline{s}_{ij} \\overline{s}_{ij}}, \n\\label{eq:3.3}\n\\end{align}\nis applied with the Smagorinsky constant $C_S = 0.19$, which is the optimized value for homogeneous isotropic turbulence \\cite{yoshibook}, and $\\Delta = \\left( \\Delta x \\Delta y \\Delta z \\right)^{1\/3}$ where $\\Delta x_i$ denotes the grid size of the $i$-th direction.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=0.45]{fig1.eps}\n\\caption{Computational domain and schematic profiles of turbulent energy and helicity. $K^{GS} (= \\left< \\overline{u}_i' \\overline{u}_i' \\right>\/2)$ and $H^{GS} (= \\left< \\overline{u}_i' \\overline{\\omega}_i' \\right>)$ denote the turbulent energy and helicity of the grid scale motions, respectively. An external forcing is applied only around $y=0$ plane.}\n\\label{fig:1}\n\\end{figure}\n\nIn the simulation, the computational domain is a rectangular parallelepiped region as shown in Fig.~\\ref{fig:1}. An external force applied around the center plane at $y=0$ injects turbulent energy and helicity. In the calculation, the rotation axis is set perpendicular to the inhomogeneous direction of the turbulence to assess the helicity model Eq.~(\\ref{eq:2.14}). This set up is similar to that used by Yokoi and Brandenburg \\cite{yb2016}. The configuration corresponds to the low-latitude region of a rotating  spherical convection in which the inhomogeneous direction of helicity is mainly perpendicular to the rotation axis in a low-latitude region \\cite{duarteetal2016,ssd2014}. The objective involves elucidating the effect of inhomogeneous helicity on the mean flow in rotating turbulence and not clarifying the mechanism of helicity generation, and thus helicity is injected by an external forcing to achieve simplicity in contrast to the simulation in which helicity emerges spontaneously \\cite{gl1999,rd2014,kapylaetal2017}. The external force is defined by the vector potential $\\overline{\\psi}_i$ as follows:\n\\begin{align}\n\\overline{f}_i = C \\epsilon_{ij\\ell} \\frac{\\partial}{\\partial x_j} \\left[ g (y) \\overline{\\psi}_\\ell \\right],\n\\label{eq:3.4}\n\\end{align}\nwhere $g(y)$ denotes a weighting function introduced to confine the external force around the $y=0$ plane. The coefficient $C$ is determined to satisfy $\\left< \\overline{u}_i' \\overline{u}_i' \\right>_S (y=0)\/2=1$ at each time step, where $\\left< \\cdot \\right>_S$ denotes the $x$--$z$ plane average and $\\overline{q}'$ denotes the fluctuation of $\\overline{q}$ around $\\left< \\overline{q} \\right>_S$; \n\\begin{align}\n\\overline{q} = \\left< \\overline{q} \\right>_S + \\overline{q}'. \n\\label{eq:3.5}\n\\end{align}\nThe force is solenoidal, $\\partial \\overline{f}_i \/ \\partial x_i = 0$. With respect to the weighting function, $g(y) = \\mathrm{exp} \\left[ -y^2\/\\sigma^2 \\right]$ with $\\sigma = L_y\/32 = 0.393$ is applied, and this is a value comparable to the forcing scale $\\pi\/k_f$ where $k_f$ is given in the following [Eq.~(\\ref{eq:3.6a})]. The vector potential $\\overline{\\psi}_i$ obeys a stochastic process like the Ornstein--Uhlenbeck process \\cite{ouforcing}, and is determined from the power and helicity spectra of $\\overline{f}_i$, $E^{ex}(k)$ and $E_H^{ex}(k)$ given as follows:\n\\begin{subequations}\n\\begin{align}\nE^{ex} (k) & \\propto\n\\begin{cases}\nk^{-5\/3} & k=k_f, 10 \\le k_f \\le 14 \\\\\n0 & \\text{otherwise} ,\n\\end{cases} \n\\label{eq:3.6a} \\\\\nE_H^{ex} (k) & = 2 \\alpha k E^{ex} (k) ,\n\\label{eq:3.6b}\n\\end{align}\n\\end{subequations}\nwhere $\\alpha$ denotes the parameter that determines the intensity of helicity of the external force. The spectrum $E^{ex}(k)$ is selected corresponding to the typical inertial-range form of turbulence, and $E^{ex}_H (k)$ corresponds to the statistical property of inertial wave when $\\alpha = \\pm 1$ \\cite{moffatt1970}. The range of $\\alpha$ should be $-1 \\le \\alpha \\le 1$ since the helicity spectrum must satisfy $|E_H^{ex}(k)| \\le 2k E^{ex}(k)$ \\cite{al1977}; $\\alpha = 0$ corresponds to the non-helical case and $\\alpha = 1$ ($-1$) is the most positively (negatively) helical case. Details of forcing are given in Appendix~\\ref{sec:b}.\n\nThe size of the computational domain is $L_x \\times L_y \\times L_z = 2\\pi \\times 4\\pi \\times 2\\pi$ and the number of the grid point is $N_x \\times N_y \\times N_z = 128 \\times 256 \\times 128$. The periodic boundary conditions are used in all directions, we use the second-order finite-difference scheme in space, and the Adams--Bashforth method is used for time integral. A triply periodic box is used, and thus the pseudo-spectral scheme may be more appropriate for DNS with the linear viscosity term. However, with respect to the LES, a complex nonlinear form of the SGS viscosity decreases the numerical accuracy of the pseudo-spectral scheme. Moreover, we are going to apply the code to homogeneous turbulence with a non-uniform grid. Thus, we adopt finite-difference scheme. The pressure is directly solved in the wave number space by using FFT. Parameters of the simulation are shown in Table~\\ref{tb:1}; namely Run 1 is non-helical and non-rotating case, Run 2 is helical but non-rotating, Run 3 is rotating but non-helical, and Runs 4, 5, and 6 are helical and rotating. We observe the effect of helical forcing by comparing Runs 1 and 2 for non-rotating case, and Runs 3, 4, and 6 for the rotating case. We also observe the effect of the system rotation by comparing Runs 2, 5, and 6. In all the runs, the external force is applied in the wavenumber band $10 \\le k \\le 14$. With respect to the helical cases, $\\alpha = 0.5$ for Runs 2 and 4 and $\\alpha = 0.2$ for Run 5, which are not fully helical ones, are adopted since the relative helicity [$u_i \\omega_i\/(|u_i||\\omega_i|)$] in realistic turbulence is modulated from the maximally helical case of the inertial wave, $\\alpha = \\pm 1$, due to buoyancy and nonlinear interaction of turbulence \\cite{ranjan2017}. $L^{GS}_0$ denotes the characteristic length scale of the turbulence and $\\mathrm{Ro}^{GS}_0$ denotes the Rossby number respectively defined by\n\\begin{align}\nL^{GS}_0 = \\frac{(K^{GS}_0)^{3\/2}}{\\varepsilon^{SGS}_0} \\ , \\ \\ \n\\mathrm{Ro}^{GS}_0 = \\frac{{K^{GS}_0}^{1\/2}}{L^{GS}_0 2 \\Omega^F} ,\n\\label{eq:3.7}\n\\end{align}\nwhere $K^{GS} = \\left< \\overline{u}_i' \\overline{u}_i' \\right>\/2$, $K^{GS}_0 = K^{GS} (y=0)$, $\\varepsilon^{SGS} = 2\\left< \\nu_{sgs} \\overline{s}_{ij} \\overline{s}_{ij}' \\right>$, $\\varepsilon^{SGS}_0 = \\varepsilon^{SGS} (y=0)$, and $\\left< \\cdot \\right>$ denotes the average over the homogeneous plane and over time. The time average is taken over $20 \\le t \\le 30$ as mentioned below. In our calculation, the length scale of inhomogeneity of turbulence is estimated as $\\ell^\\nabla = 0.4$ for all runs, in which $\\ell^\\nabla$ is defined such that $K^{GS} (y= \\ell^\\nabla) = \\mathrm{e}^{-1} K^{GS}_0$. The validity of turbulence models requires the length scale of inhomogeneity of turbulence is much longer than the scale of energy containing eddy $L_0^{GS}$ \\cite{corrsin1974}. These two scales are comparable in the simulation. However, the fore-mentioned lack of scale separation is often observed in actual turbulence such as in an atmospheric boundary layer \\cite{stull1993}. It should be emphasized that the mean velocity is set to zero in the initial condition, and the plane average of the external force is also zero, $\\left< \\overline{f}_i \\right>_S = 0$, such that the external force does not directly excite the mean velocity.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Calculation parameters.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nRun & $\\alpha$ & $\\Omega_x^F$ & $L^{GS}_0$ & $\\mathrm{Ro}^{GS}_0$ \\\\ \\hline\n1 & $0$ & $0$ & $0.506$ & $\\infty$ \\\\\n2 & $0.5$ & $0$ & $0.547$ & $\\infty$ \\\\\n3 & $0$ & $5$ & $0.542$ & $0.185$ \\\\\n4 & $0.2$ & $5$ & $0.550$ & $0.182$ \\\\\n5 & $0.5$ & $2$ & $0.544$ & $0.459$ \\\\\n6 & $0.5$ & $5$ & $0.602$ & $0.166$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tb:1}\n\\end{table}\n\n\\subsection{\\label{sec:level3b}Numerical results}\n\n\\subsubsection{\\label{sec:level3b1}Mean-flow generation}\n\nFigure~\\ref{fig:2} shows the time evolution of the mean axial velocity, $\\left< \\overline{u}_x \\right>_S$, for Run 6. The mean flow is generated around $y=0$ as time elapses and is sustained in subsequent periods. This result is the same as that obtained by Yokoi and Brandenburg \\cite{yb2016} in which the positive mean velocity directed to the rotation axis was generated around the positively helical region.\nIn the simulation performed by Yokoi and Brandenburg \\cite{yb2016}, helicity is distributed as $H(y) \\propto \\sin (\\pi y\/y_0)$ (in the original study, the inhomogeneous direction is $z$), and thus the positive axial velocity emerges in $y>0$ and the negative axial velocity emerges in $y<0$. Conversely, in the present simulation, the positive helicity is driven only in a limited region around $y=0$. Hence, the positive axial mean velocity emerges only around $y=0$.\nIt should be noted again that the mean velocity cannot be directly generated from the external force since the direct contribution from the external force is excluded in the calculation. Hereafter, we take the time average over $20 \\le t \\le 30$ as well as the homogeneous plane average.\nThe mean axial velocity of each run is given in Fig.~\\ref{fig:3}. Evidently, the positive axial mean velocity emerges only for the cases with both helicity injection and system rotation, namely Runs 4, 5, and 6. The difference between Run 3 and Runs 4 and 6 only corresponds to the existence of the helicity injection, and thus the external force with $\\alpha = 0$ does not influence the induction of the axial mean velocity. This indicates that neither inhomogeneous helicity nor system rotation by themselves are sufficient to obtain the mean-flow generation.\nIt is interesting to note that the maximum values of the mean flows for Runs 4 and 5 are the same. This suggests that the product of the helicity and the angular velocity of system rotation determines the mean-flow generation. The mean flow profile is expected to be symmetric about $y=0$. The present result is slightly asymmetric due to the limitations of time or ensemble average. \n\n\\begin{figure}[htp]\n \\begin{tabular}{c}\n  \\begin{minipage}{0.49\\hsize}\n   \\centering\n   \\includegraphics[scale=0.74]{fig2.eps}\n   \\caption{Time evolution of the axial mean velocity for Run 6. The horizontal axis denotes the time, and the vertical axis denotes the inhomogeneous direction, $y$, and the color contour denotes the value of $\\left< \\overline{u}_x \\right>_S$.}\n  \\label{fig:2}\n  \\end{minipage}\n  \\begin{minipage}{0.04\\hsize}\n  \\end{minipage}\n  \\begin{minipage}{0.49\\hsize}\n   \\centering\n   \\includegraphics[scale=0.67]{fig3.eps}\n   \\caption{Mean axial velocity of each run.}\n   \\label{fig:3}\n  \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nWhen the turbulent field is statistically steady, the equation for the mean axial velocity is expressed as follows:\n\\begin{align}\n\\frac{\\partial U_x}{\\partial t} \n= -\\frac{\\partial R_{xy}}{\\partial y} = 0 ,\n\\label{eq:3.8}\n\\end{align}\nwhere $R_{ij}$ satisfies $R_{ij} = R_{ij}^{GS} - 2\\left<\\nu_{sgs} \\overline{s}_{ij} \\right>$ in the framework of the eddy-viscosity representation of the SGS stress, and $R_{ij}^{GS} = \\left< \\overline{u}_i' \\overline{u}_j' \\right>$ denotes the Reynolds stress of the grid scale.\nIt should be noted that $\\overline{u}_i'$ denotes the fluctuation of the GS velocity $\\overline{u}_i$ and is defined as in Eq.~(\\ref{eq:3.5}).\nEquation~(\\ref{eq:3.8}) gives the Reynolds stress constant in the $y$ direction. The turbulence is inactive at the upper and lower boundaries, and thus $y=\\pm L_y\/2$, $R_{xy}$ disappears at this point. Therefore, the solution of the mean velocity equation is $R_{xy} = 0$. The green line with squares in Fig.~\\ref{fig:4} shows the profile of $R_{xy}$ for Run 6. It is nearly equal to zero although a slight non-zero value is observed around $y=0$ because the time averaging is insufficient for the statistically steady state. Here, we consider the appropriateness of the eddy-viscosity model,\n\\begin{align}\nR_{xy} = - \\nu_T \\frac{\\partial U_x}{\\partial y} \\ , \\ \\ \n\\nu_T = C_\\nu \\frac{K^2}{\\varepsilon},\n\\label{eq:3.9}\n\\end{align}\nIn Fig.~\\ref{fig:4}, the profile of $R_{xy}$ estimated by Eq.~(\\ref{eq:3.9}) is also plotted in the red line with crosses. It should be noted that $\\nu_T$ is evaluated by using $K^{GS}$ and $\\varepsilon^{SGS}$ instead of $K$ and $\\varepsilon$ as $\\nu_T = C\\nu (K^{GS})^2\/\\varepsilon^{SGS}$ with $C_\\nu =0.09$. This clearly indicates excessively high non-zero values around $y=0$. Since $\\nu_T \\neq 0$ around $y=0$, the velocity gradient must vanish in order to satisfy $R_{xy} = 0$. Therefore, the eddy-viscosity model is unable to reproduce the present result in which the mean flow is sustained around $y=0$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.72]{fig4.eps}\n\\caption{The Reynolds stress $R_{xy}$ for Run 6. The green line with squares denotes the directly evaluated value, $R_{xy} = R_{xy}^{GS} - 2 \\left< \\nu_{sgs} \\overline{s}_{xy} \\right>$, and the red line with crosses denotes the value estimated by the eddy-viscosity model that is given by Eq.~(\\ref{eq:3.9}) with $C_\\nu = 0.09$.}\n\\label{fig:4}\n\\end{figure}\n\nIn order to rectify the inadequacy of the eddy-viscosity model, let us assume the following generic expression for the model,\n\\begin{align}\nR_{xy} = - \\nu_T \\frac{\\partial U_x}{\\partial y} + N_{xy},\n\\label{eq:3.10}\n\\end{align}\nwhere $N_{xy}$ denotes an additional term. As shown in Fig.~\\ref{fig:4}, the eddy-viscosity term, $-\\nu_T \\partial U_x \/ \\partial y$, has a large positive gradient around $y=0$. In order to satisfy $R_{xy} = 0$, $N_{xy}$ must involve a large negative gradient around $y=0$ to counterbalance the eddy-viscosity term. We expect that the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) is a good candidate for $N_{xy}$ because the mean flow is only sustained when both the helical force and the system rotation are present.\n\n\n\\subsubsection{\\label{sec:level3b2}Origin of the helicity effect}\n\nIn order to investigate the origin of the additional term $N_{xy}$, we examine the transport equation for the Reynolds stress. The transport equation for $R_{xy}^{GS}$ is expressed as follows:\n\\begin{align}\n\\frac{\\partial R_{xy}^{GS}}{\\partial t} =\nP_{xy}^{GS} + \\Phi_{xy}^{GS} + \\Pi_{xy}^{GS} + C_{xy}^{GS},\n\\label{eq:3.11}\n\\end{align}\nwhere only the terms that significantly contribute to the simulation for Run 6 are included. Here $P_{xy}^{GS}$ denotes the production, $\\Phi_{xy}^{GS}$ the pressure--strain correlation, $\\Pi_{xy}^{GS}$ the pressure diffusion, and $C_{xy}^{GS}$ the Coriolis effect. They are respectively defined as follows:\n\\begin{subequations}\n\\begin{align}\nP_{xy}^{GS} & = - \\frac{2}{3} K^{GS} \\frac{\\partial U_x}{\\partial y} \n- B_{yy}^{GS} \\frac{\\partial U_x}{\\partial y} - B_{xz}^{GS} \\frac{\\partial U_z}{\\partial y}, \n\\label{eq:3.12a} \\\\\n\\Phi_{xy}^{GS} &\n = 2 \\left< \\overline{p}' \\overline{s}_{xy}' \\right>, \n\\label{eq:3.12b} \\\\\n\\Pi_{xy}^{GS} &\n = -\\frac{\\partial}{\\partial y} \\left< \\overline{p}' \\overline{u}_x' \\right>, \n\\label{eq:3.12c} \\\\\nC_{xy}^{GS} & = 2 R_{xz}^{GS} \\Omega_x^F,\n\\label{eq:3.12d}\n\\end{align}\n\\end{subequations}\nwhere $B_{ij}^{GS} = R_{ij}^{GS} - (2\/3) K^{GS} \\delta_{ij}$. The budget of the transport equation for $R_{xy}^{GS}$ for Run 6 is shown in Fig.~\\ref{fig:5}. It should be noted that the balance of the above four terms are mostly the same for Runs 4 and 5 especially in the sense that the pressure--strain correlation and the pressure diffusion are predominant (figures are not shown here). The production term plotted in the red line with crosses exhibits a positive gradient around $y=0$. It should be noted that with respect to the production term $P_{xy}^{GS}$, the first term on the right-hand side of Eq.~(\\ref{eq:3.12a}) is dominant [detailed contribution from each term in Eq.~(\\ref{eq:3.12a}) is not shown here]. Thus, as shown in Sec.~\\ref{sec:level2a}, the production term corresponds to the eddy-viscosity term, and it also exhibits a positive gradient around $y=0$ in Fig.~\\ref{fig:4}. Based on the discussion in Sec.~\\ref{sec:level3b1}, any candidate of the term corresponding to $N_{xy}$ that accounts for the sustainment of the mean velocity should exhibit a negative gradient around $y=0$. In Fig.~\\ref{fig:5}, two candidates are observed, namely the pressure diffusion $\\Pi_{xy}^{GS}$ (the blue line with circles) and the Coriolis effect $C_{xy}^{GS}$ (the magenta line with triangles). If the Coriolis effect corresponds to the origin of $N_{xy}$, the mean flow would be sustained for Run 3 in which  system rotation exists as well as for Runs 4, 5, and 6. Therefore, we focus on the pressure diffusion term. However, this does not deny the importance of Coriolis force in the flow generation phenomenon. As shown in Fig. 5, the Coriolis effect also contributes to the Reynolds stress in the sense that it sustains the mean flow. Additionally, the effect of the Coriolis force appears not only in the Coriolis effect but also in the pressure through the Poisson equation as discussed in the following paragraph.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.72]{fig5.eps}\n\\caption{Budget of the transport equation for $R_{xy}^{GS}$ for Run 6. The $y$ coordinate is limited to the region at $-2\\le y \\le 2$ where a high mean velocity exists.}\n\\label{fig:5}\n\\end{figure}\n\nIn order to investigate the pressure diffusion [Eq.(\\ref{eq:3.12c})], we consider the Poisson equation for the pressure fluctuation,\n\\begin{align}\n\\nabla^2 \\overline{p}' & =\n - 2\\overline{s}_{ab}' S_{ab} + \\overline{\\omega}_a' \\Omega_a^*\n- \\overline{s}_{ab}' \\overline{s}_{ab}'\n+ \\frac{1}{2} \\overline{\\omega}_a' \\overline{\\omega}_a' \n+ \\frac{\\partial^2}{\\partial x_a \\partial x_b} \\left[ 2\\left( \\nu_{sgs} \\overline{s}_{ab} - \\left< \\nu_{sgs} \\overline{s}_{ab} \\right> \\right) \\right].\n\\label{eq:3.13}\n\\end{align}\nWe approximate the left-hand side as\n\\begin{align}\n\\nabla^2 \\overline{p}' = - \\frac{\\overline{p}'}{\\ell_p^2},\n\\label{eq:3.14}\n\\end{align}\nwhere $\\ell_p$ denotes the length scale associated with the pressure fluctuation. Thus, the pressure diffusion $\\Pi_{xy}^{GS}$ is estimated as follows:\n\\begin{align}\n\\Pi_{xy}^{GS}\/\\ell_p^2 & =\n\\frac{\\partial}{\\partial y} \\left[\n - 2 \\left< \\overline{u}_x' \\overline{s}_{ab}'\\right> S_{ab}\n + \\left< \\overline{u}_x' \\overline{\\omega}_a' \\right> \\Omega_a^* \n- \\left< \\overline{u}_x' \\overline{s}_{ab}' \\overline{s}_{ab}' \\right> \n+ \\frac{1}{2} \\left< \\overline{u}_x' \\overline{\\omega}_a' \\overline{\\omega}_a' \\right> \n+ \\frac{\\partial^2}{\\partial y^2} \\left( 2 \\left< \\overline{u}_x' \\nu_{sgs} \\overline{s}_{ab} \\right> \\right) \\right] ,\n\\label{eq:3.15}\n\\end{align}\nwhere $\\ell_p$ is approximated as a constant in space for simplicity. Figure~\\ref{fig:6} shows the pressure diffusion $\\Pi_{xy}^{GS}$ evaluated from Eq.~(\\ref{eq:3.15}) for Run 6. As shown in the figure, the second term related to the mean absolute vorticity is dominant. Thus, $\\Pi_{xy}^{GS}$ is approximated as follows:\n\\begin{align}\n\\Pi_{xy}^{GS} \/ \\ell_p^2\n= \\frac{\\partial}{\\partial y} \\left( 2 \\left< \\overline{u}_x' \\overline{\\omega}_x' \\right> \\Omega_x^F \\right)\n= \\frac{\\partial}{\\partial y} \\left( \\frac{2}{3} H^{GS} \\Omega_x^F \\right),\n\\label{eq:3.16}\n\\end{align}\nand this includes $|\\Omega_i| \\ll |2\\Omega_i^F|$ and $\\left<\\overline{u}_x' \\overline{\\omega}_x' \\right> = \\left<\\overline{u}_y' \\overline{\\omega}_y' \\right> = \\left<\\overline{u}_z' \\overline{\\omega}_z' \\right> = H^{GS} \/3$. This indicates that the helicity gradient and the system rotation may account for the pressure diffusion that contributes to the mean velocity sustainment. A model expression of the pressure diffusion that is similar to Eq.~(\\ref{eq:3.16}) is also analytically obtained with the aid of the TSDIA \\cite{tsdia}. A brief introduction of the theory and the detailed calculation are given in Appendix~\\ref{sec:a}. The result is\n\\begin{align}\n\\Pi_{ij} & = \\frac{1}{3} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2\\Omega_i^F \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right] \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3), \n\\label{eq:3.17}\n\\end{align}\nwhere $L$ denotes the length scale related to the energy containing eddy and $u^{(00)}$ is the lowest-order velocity corresponding to homogeneous isotropic turbulence defined in Eq.~(\\ref{eq:a4}). This model expression for the pressure diffusion is in good agreement with  Eq.~(\\ref{eq:3.16}).\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.72]{fig6.eps}\n\\caption{Approximate evaluation of $\\Pi_{xy}^{GS}$ for Run 6.}\n\\label{fig:6}\n\\end{figure}\n\nIn Fig.~\\ref{fig:5}, the pressure--strain correlation $\\Phi_{ij}^{GS}$ (the green line with squares) also significantly contributes to the Reynolds-stress transport. One might consider that the pressure diffusion and the pressure--strain correlation cancel each other. However, the sum of these two terms (as denoted by the cyan line with diamonds in Fig.~\\ref{fig:5}) contributes to exhibit a negative gradient around $y=0$ and plays the same role as the pressure diffusion itself. This tendency is also theoretically demonstrated as follows. The model expression of the pressure--strain correlation $\\Phi_{ij}$ is obtained with the aid of the TSDIA \\cite{tsdia}, and it is possible to analytically examine the balance of the two terms. The analytical result of the pressure--strain correlation is as follows:\n\\begin{align}\n\\Phi_{ij} & = - \\frac{3}{10} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2 \\Omega_i^F \\right)\n   + \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right]_D \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3).\n\\label{eq:3.18}\n\\end{align}\nAlthough the helicity effect of the pressure--strain correlation $\\Phi_{ij}$ has the sign opposite to that of the pressure diffusion $\\Pi_{ij}$, its magnitude is slightly smaller. Thus the sum of $\\Pi_{ij}$ and $\\Phi_{ij}$,\n\\begin{align}\n\\Pi_{ij} + \\Phi_{ij} & = \n\\frac{1}{30} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2\\Omega_i^F \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right] \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3),\n\\label{eq:3.19}\n\\end{align}\ncontributes in the same manner as the pressure diffusion $\\Pi_{ij}$ and sustains the mean flow.\n\n\n\\section{\\label{sec:level4}Correspondence of the pressure diffusion to the helicity model}\n\nThe results indicated that the pressure diffusion plays an important role in the sustainment of the mean velocity  in inhomogeneous helical turbulence. This fact contradicts the assumption for the derivation of the model for the Reynolds stress as given in Sec.~\\ref{sec:level2a}. In the current construction, the flow is assumed to be quasi-homogeneous for the diffusion to be neglected. However, the effect of the pressure diffusion is required to improve the Reynolds-stress model for inhomogeneous helical turbulence. As shown in Sec.~\\ref{sec:level3b2}, the effect of helicity is explicitly incorporated in the pressure diffusion term for the Reynolds-stress transport equation. Here, we add the helicity effect that originates from the pressure diffusion term to the LRR model \\cite{lrr1975} as follows:\n\\begin{align}\n\\begin{split}\n& \\Phi_{ij} + \\left[ \\Pi_{ij} \\right]_D\n= \\Phi_{ij}^{LRR} + C_{PH} \\Gamma_{ij},\n\\label{eq:4.1}\n\\end{split}\n\\end{align}\nwhere $\\Phi_{ij}^{LRR}$ denotes the LRR model given by Eq.~(\\ref{eq:2.10}), $C_{PH}$ is a positive constant, and\n\\begin{align}\n\\Gamma_{ij} = \n \\left[ \\frac{\\partial}{\\partial x_j} \\left( \\frac{K^3}{\\varepsilon^2} H \\Omega_i^* \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( \\frac{K^3}{\\varepsilon^2} H \\Omega_j^* \\right) \\right]_D .\n\\label{eq:4.2}\n\\end{align}\nHere, the length scale that corresponds to $\\ell_p$ in Eq.~(\\ref{eq:3.14}) or $L$ in Eq.~(\\ref{eq:3.17}) is expressed in terms of $K$ and $\\varepsilon$. Thus, the Reynolds-stress equation is re-expressed as follows:\n\\begin{align}\n\\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t} & =\n- C_{S1} \\frac{\\varepsilon}{K} B_{ij}\n- \\left( \\frac{4}{3} - C_{R1} \\right) K S_{ij} \n+ C_{PH} \\Gamma_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\left( 1 - C_{R2} \\right) \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\left( 1 - C_{R3} \\right) \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right),\n\\label{eq:4.3}\n\\end{align}\nThe third term on the right-hand side denotes the only difference between Eqs.~(\\ref{eq:2.11}) and (\\ref{eq:4.3}). Thus, the model expression corresponding to Eq.~(\\ref{eq:2.13}) is given as follows:\n\\begin{align}\nB_{ij} & = - 2 C_\\nu \\frac{K^2}{\\varepsilon} S_{ij}\n+ C_\\gamma \\frac{K}{\\varepsilon} \\Gamma_{ij}\n+ \\cdots,\n\\end{align}\nwhere $C_\\gamma = C_{PH} \/ C_{S1}$. The second term is significantly similar to Eq.~(\\ref{eq:2.14}) obtained by Yokoi and Yoshizawa \\cite{yy1993}. Hence, the helicity model given by Eq.~(\\ref{eq:2.14}) can trace part of its origin to the pressure diffusion in inhomogeneous helical turbulence in a rotating system.\n\n\n\\section{\\label{sec:level5}Conclusions}\n\nThe mechanism of the mean-flow generation and its relationship to the turbulent helicity were investigated by using the numerical simulation of a rotating inhomogeneous turbulence. In the simulation, an external forcing was applied to inject turbulent energy and helicity and the rotation axis was perpendicular to the inhomogeneous direction. The initial mean velocity and the mean part of the external force were set to zero, and this implies that it is not possible to directly excite the mean flow by the external forcing. The results showed that the mean flow is generated and sustained only when both helical forcing and system rotation exist. The flow-generation phenomenon originates from both the turbulent helicity and the rotational motion of fluid. \n\nThe usual eddy-viscosity model is unable to reproduce the mean-flow generation observed in the simulation, and therefore an additional term is needed to explain the phenomenon. In order to explore candidates for the additional term, the budget of the Reynolds-stress transport equation was investigated. The results suggested that the pressure diffusion significantly influences the sustainment of the mean flow. The approximation to the Poisson equation for the pressure fluctuation was used to obtain an expression for the pressure diffusion in terms of the turbulent helicity and the angular velocity of the system rotation. The effect of helicity in relation to the pressure diffusion term was considered to obtain a model for the Reynolds stress, and the obtained model is considerably similar to the one obtained by Yokoi and Yoshizawa \\cite{yy1993}. The model implies that the inhomogeneity of helicity plays a crucial role in rotating turbulence such as the momentum transport due to turbulence in the low-latitude region of a rotating sphere \\cite{duarteetal2016,ssd2014}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n      It is known that quenched disorder can strongly affect the\n      large-scale, long-time behaviour of nonequilibrium driven systems\n      with interacting constituents.  The interplay of disorder,\n      interactions and drive opens up the possibility of new regimes of\n      complex and interesting behaviour arising in these systems\n      \\cite{Fisher}.  In the theoretical effort to delineate and explore\n      regimes of new behaviour, an important role is played by simple\n      models which capture some features of more complex physical systems.\n      In this paper, we study disordered driven diffusive systems by\n      analysing stochastically evolving lattice gas models, with quenched\n      disordered hopping rates \\cite {prl97}. \n\n      Driven diffusive systems in the absence of disorder have been studied\n      extensively and are reviewed in \\cite{DDS}.  Also, systems with\n      disorder and drive but no interactions between particles are well\n      studied and understood \\cite{BouGeo}.  But there have been only\n      sporadic studies of disordered driven diffusive systems of\n      interacting particles.  It has been argued that strong enough random\n      site dilution can substantially affect the transport properties of\n      particles with hard-core interactions, and can make the system\n      respond nonmonotonically to the driving field \\cite{RB,BR}.  On the\n      other hand, a low concentration of blocked sites was found\n      numerically not to affect the critical behaviour of a driven lattice\n      gas with additional attractive inter-particle interactions\n      \\cite{Lauritsen}.  Finally, a driven lattice gas with a quenched\n      noise distribution was studied using field-theoretic techniques in\n      \\cite{Janssen}, but the connection of this study with\n      particle-conserving disordered lattice gas models is not clear.\n\n      In this paper, we study disordered lattice gas models with a view\n      towards identifying different sorts of generic behaviour that can\n      arise on large scales as a consequence of disorder.  The only\n      interaction included is the hard-core constraint which limits the\n      allowed occupancy of each site.  Our results pertain mostly, but not\n      exclusively, to one dimension.  In the remainder of this\n      Introduction, we discuss the different types of behaviour displayed\n      by the lattice gas models under study.\n\n      We find three distinct regimes in disordered driven diffusive systems\n      in one dimension:\n\n      In the {\\it Homogeneous} regime, the state of the system is\n      characterized by a single density and a nonzero current.  Quenched\n      disorder induces variations of the density on the microscopic scale,\n      of the order of a few lattice spacings. However, the system has a\n      macroscopically homogeneous density. In the thermodynamic\n      limit, the current approaches a finite value.\n\n      In the {\\it Segregated-Density} regime, the state of the system is\n      characterized by two distinct values of density, and a nonzero\n      current.  Besides microscopic-scale variations of the density, there\n      are  macroscopic regions with differing high and low\n      densities. The state is thus characterized by phase separation of\n      the density, and a spatially constant time-averaged current which\n      remains finite in the thermodynamic limit.  \n\n\n      In the {\\it Vanishing-Current} regime, the state of the system is\n      characterized by two distinct values of the density, and an\n      essentially zero current.  The hallmark of this regime is that the\n      current decreases as the system size increases, and vanishes in\n      the thermodynamic limit. This is a consequence of rare but\n      rate-limiting backbends, or stretches of bonds which disfavour the\n      forward flow of current.  The density is inhomogeneous on a\n      macroscopic scale.\n\n      The density profiles in typical states in each of the three regimes\n      are depicted in Figure 1, while Figure 2 shows the variation of the\n      current with system size in the three cases.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=1.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Representative steady-state density profiles \nfor the (a) Homogeneous (b) Segregated-Density and (c) Vanishing-Current\nregimes in the Disordered Asymmetric Exclusion Process (DASEP). }\n\\label{fig:regimes1}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=2.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\noindent\\caption{Variation of the steady-state current with the system size\nfor the three DASEP regimes of Fig. 1: (a) Homogeneous (circles), (b)\nSegregated-Density (triangles) and (c) Vanishing-Current (squares).  In (a)\nand (b), the current approaches a finite value in the thermodynamic limit\nwhereas in (c) the current vanishes as a power of the system size. The\ndashed line corresponds to $J=0.125$ which is the limiting value of the\ncurrent in the regime (b) for the chosen values of the parameters.}\n\\label{fig:regimes2}\n\\end{figure}\n\n      Examples of these behaviours are discussed in this paper for two\n      types of lattice-gas models, namely the disordered drop-push process\n      (DDPP) and the disordered asymmetric simple exclusion process\n      (DASEP). The models are defined in detail in Sections \\ref{sec:DDPP}\n      and \\ref{sec:DASEP} respectively, but for the purpose of discussion\n      here, it suffices to note that the models are similar in that there\n      is a maximum occupancy of each site in both, and are different in the\n      dynamical moves ---  attempted nearest neighbour jumps in\n      the DASEP, and slightly longer-ranged leapfrogging moves in the DDPP.\n\n\n      The absence of detailed balance, together with the breaking of\n      translational invariance, in disordered off-equilibrium systems makes\n      the characterization of even the stationary state difficult in\n      general.  It is shown that the steady state of the disordered\n      drop-push process can be found explicitly -- the first such instance\n      we are aware of, in a system with disorder, interactions and drive\n      \\cite{Krug_Evans}. This determination -- which is based on the\n      condition of pairwise balance \\cite{SRB} -- shows that a product\n      measure form is valid in all dimensions. The form reflects the\n      microscopic inhomogeneities coming from the underlying disorder, and\n      results in a macroscopically homogeneous state.\n\n      For the disordered asymmetric exclusion process, the steady state\n      measure is not analytically characterizable, and we study the problem\n      within a site-wise inhomogeneous mean-field theory and by numerical\n      simulation.  The result depends crucially on whether or not the\n      system has backbends, which are stretches of the lattice where the\n      local bias is against the particle flow.  In the no-backbend case,\n      when the average particle density is sufficiently away from 1\/2, the\n      spatial profile of the density has microscopic shocks, but is uniform\n      on macroscopic scales (Fig.~\\ref{fig:regimes1}a).  However, in a\n      finite region around half-filling, disorder induces phase separation\n      into macroscopic regions of high and low\n      density (Fig.~\\ref{fig:regimes1}b).  We give approximate arguments to\n      understand the origin and nature of this phase separation, and to\n      obtain the form of the phase diagram in the current-density plane.\n      This sort of behaviour has also been seen earlier in a model with a\n      single weak bond \\cite{Janowsky}. We argue that disorder-induced\n      phase separation is a generic feature of systems in which the current\n      $J$ versus density $\\rho$ shows a maximum at some intermediate\n      density, in the absence of disorder.\n\n      In the version of the DASEP in which the easy direction of hopping is\n      itself a quenched random variable, the model represents a system of\n      hard-core particles in a random potential with an overall downward\n      tilt, but with backbends of arbitrary length.  Long backbends\n      severely limit the maximum current that can flow through the system,\n      and in fact the current decreases to zero as the system size\n      increases (Fig.~\\ref{fig:regimes2}); the system is in the\n      vanishing-current regime.\n\n      Although our emphasis in this paper is on the analysis of lattice\n      models, we comment briefly on certain constraints that are important\n      in a continuum description.  Such a description is expected to be\n      valid for the large-scale, long-time behaviour, and is based on\n      stochastic differential equations involving appropriate\n      coarse-grained variables. It is argued that quenched randomness is\n      manifest in random multiplicative coefficients in a gradient\n      expansion.  Conservation of particle number -- which implies spatial\n      constancy of the current in the steady state -- imposes strong\n      constraints on these terms.\n\n      In one dimension, using a well known mapping \\cite{solids}, the\n      particle models are equivalent to stochastic growth models of a 1-$d$\n      interface moving in a 2-$d$ medium. The interface moves with a speed\n      proportional to the current in the particle model. The\n      disordered jump rates now become local growth rates which are\n      disordered in a columnar fashion \n      for the moving interface \\cite{columnar}. The three principal\n      regimes of behaviour discussed above for the particle models\n      translate into distinct regimes for interface motion, namely (i) a\n      moving interface with normal roughness, (ii) a moving interface with\n      large segments with different mean slopes, and (iii) an interface\n      with different-slope segments, which is stationary in the\n      thermodynamic limit.\n\n      The paper is organized as follows. In Section \\ref{sec:DDPP} we\n      define and discuss the steady state properties of the disordered\n      drop-push process in arbitrary dimensionality. The disordered\n      asymmetric exclusion process with only forward-easy-direction of\n      hopping, but quenched random rates, is discussed in Section\n      \\ref{sec:DASEP}; the case in which there are some\n      backward-easy-direction bonds is discussed in Section\n      \\ref{sec:Sinai}. In Section \\ref{sec:continuum} we discuss the\n      constraints on a continuum description, while Section\n      \\ref{sec:height} discusses the implications of our results for models\n      interface growth in the presence of columnar disorder. Section\n      \\ref{sec:conclusion} is the conclusion.\n\n\\section{Disordered Drop-Push Process : DDPP} \\label{sec:DDPP}\n\n    The drop-push process was initially introduced in \\cite{BR,SRB} as a\n    model of activated flow involving transport through a series of traps\n    of equal depths. The dynamics consists of activated hops together with\n    a cascade of overflows following each move.  The disordered version of\n    the model may be considered as a discrete model of activated fluid flow\n    down an inclined rugged slope with lakes of varying depths; see\n    Figs.~\\ref{fig:lakes},\\ref{fig:ddpp-1d}. This is similar to\n    above-threshold behaviour of the model considered in\n    \\cite{Narayan_Fisher}. In this section we show that the steady state\n    and current can be found exactly in all dimensions for the DDPP and its\n    generalizations.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=3.eps,width=5.5cm}\n\\end{center}\n\\narrowtext\n\\noindent\\caption{Schematic diagram of water flowing down a rugged\nhill-side.  Water from a lake higher up cascades downhill, under the action\nof gravity, until it finds a partially filled lake. The unequal capacities\nof the lakes are the quenched variables in the system.}\n\\label{fig:lakes}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=4.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{A Disordered Drop-Push Process (DDPP) configuration and move in \n$d=1$. }\n\\label{fig:ddpp-1d}\n\\end{figure}\n\n   \\subsection{The model}   \\label{sub:defDDPP}\n\n    The model in $d$-dimensions is defined on a hypercubic lattice with\n    periodic boundary conditions along all the $d$ axes (with unit vectors\n    \\{$\\e_\\nu |\\nu=1\\cdots d$\\}). At each site ${r}$ is a well which can hold\n    at most $l_\\r$ particles (Figs.~\\ref{fig:ddpp-1d},\\ref{fig:ddpp-2d}) with\n    $l_\\r$'s chosen independently from some probability distribution\n    $P(l)$. The configuration $\\cal{C}$ of the system is specified by\n    specifying the set of occupation numbers \\{$n_\\r$\\} with ($0\\len_\\r\\le\n    l_\\r~, \\forall{r}$). Further, with each site ${r}$ is assigned a set\n    \\{$\\epsilon(\\nr|\\lr); n_\\r=1,\\cdots,l_\\r$\\} of positive random numbers chosen from\n    some given distribution \\cite{well_distbn}.  The dynamics is\n    stochastic. In a time interval $dt$, with a probability\n    $p_{\\pm\\nu}\\epsilon(\\nr|\\lr) dt$, the topmost particle in the well ${r}$ hops out,\n    and drops into well ${r}\\pm\\e_\\nu$, i.e. into the adjacent well in the\n    $\\pm\\nu$th direction. Here $\\{p_{\\pm\\nu}; \\nu=1,..,d\\}$ are a set of\n    site-independent positive numbers satisfying $\\sum_{\\nu=1}^d\n    (p_\\nu+p_{-\\nu})=1$. Now, if well ${r}\\pm\\e_\\nu$ is already full, then the\n    particle gets pushed further {\\it preserving the direction of the\n    initial jump} to the next site and so on. The cascade of transfers\n    terminates once a partially full well is encountered. Note that here\n    the set of jump-rates \\{$\\epsilon(\\nr|\\lr)$\\} are site-dependent as well as\n    functions of the occupation numbers. These rates, together with the\n    well-depths \\{$l_\\r$\\}, constitute the quenched random variables in the\n    model. The set of probabilities \\{$p_{\\pm\\nu}$\\} determines the\n    direction of the global bias ${\\vec E} = \\sum_{\\nu=1}^d\n    (p_\\nu-p_{-\\nu})\\e_\\nu$ and, as will be shown in Section\n    \\ref{sub:imDDPP}, also the direction and magnitude of the steady state\n    current in the model.  However, they do not enter the expression for\n    the normalized invariant measure.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=5.eps,width=7cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{The DDPP model in $d=2$. The model can be generalized to $d>2$\n(see text). The rates $\\epsilon(\\nr|\\lr)$ depend on the well depth $l_{r}$ as well as\nthe occupation number $n_{r}$.}\n\\label{fig:ddpp-2d}\n\\end{figure}\n\n    Though all the results we will discuss holds for any arbitrary choice\n    of the $\\epsilon$'s, in a physical system they should be determined\n    from the details of the trapping mechanisms etc., e.g. they may be\n    taken to be of the Kramers form $\\epsilon(\\nr|\\lr)\\propto exp[-g(l_\\r-n_\\r)]$ for\n    situations where the jumps are activated \\cite{kehr87}.\n\n\n     \\subsection{Invariant measure} \n     \\label{sub:imDDPP} \n\n     The time evolution of the probability\n     $\\cal{P({\\cal C})}$ for the system to be in configuration ${\\cal C}$ is\n     given by the master equation \\cite{vanKamp}\n\\begin{equation} {d\\over dt}{\\cal{P({\\cal C})}}\n                    =\\sum_{{\\calC^{\\prime\\prime}}}\nW({\\cdprime \\rightarrow \\calC}){\\cal P}({{\\cal C}''})-\\sum_{{\\cal{C}^{\\prime}}} W(\\calC \\rightarrow \\cprime){\\cal P}({\\cal C}).\n\\label{eq:master}\n\\end{equation}\n      Here the $W$'s are the transition matrix elements identified with the\n      rates $\\epsilon$'s defined in the model. e.g. if the transition\n      $\\calC \\rightarrow \\cprime$ involves moving the topmost of the $n_{r}$ particles at ${r}$ to\n      ${r^\\prime}$ along the $\\nu$th direction, then\n      $W({\\calC \\rightarrow \\cprime})=p_\\nu\\epsilon_{r}(n_{r}|l_{r})$. The steady-state or the\n      invariant measure of the dynamics is the set of time independent\n      weights $\\{\\mu({\\cal C})\\}$ satisfying (\\ref{eq:master}) above.\n      Hence the problem of finding the invariant measure reduces to that of\n      finding a set of positive weights $\\{\\mu({\\cal C})\\}$ such that the\n      total incoming flux into any configuration ${\\cal C}$ (the first sum\n      in (\\ref{eq:master})) equals the total flux out of ${\\cal C}$ (the\n      second sum in (\\ref{eq:master})).  The uniqueness of the invariant\n      measure is ensured by the connectedness property of the $W$-matrix,\n      i.e. every configuration can be reached from any other by a sequence\n      of transitions \\cite{vanKamp}.\n\n      We claim that the (unnormalized) measure of configuration\n      ${\\cal C}(\\{n_\\r\\})$ in the steady state has the product form\n\\begin{equation} \\mu({\\cal C})=\\prod_{{r}} u_{r}(n_\\r).\n\\label{eq:imDDPP}\n\\end{equation}\nHere $u_{r}$ are the single-site weights defined as\n\\begin{equation}\n u_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n               1 &\\mbox{~~~if~$n_\\r=0$} \\\\\n               \\tau_{r}(1)\\cdots \\tau_{r}(n_\\r) &\\mbox{~~~if~$0<n_\\r\\lel_\\r$}\n               \\end{array} \n             \\right. \n\\end{equation}\nwhere\n$\\tau_{r}(n_\\r)=\\epsilon_0\/\\epsilon(\\nr|\\lr)$ with $\\epsilon_0$ being a microscopic\nrate.\n\n       To show that (\\ref{eq:imDDPP}) is indeed the invariant measure for\n       the DDPP we show that it is possible to associate  \n       configuration ${\\calC^{\\prime\\prime}}$ in one-to-one correspondence with every\n       ${\\cal{C}^{\\prime}}$ obtained from $\\cal {C}$ by an elementary transition such\n       that\n\\begin{equation}\nW({\\calC \\rightarrow \\cprime}) \\mu({\\cal C}) = W({\\cdprime \\rightarrow \\calC}) \\mu({\\calC^{\\prime\\prime}}).\n\\label{eq:pwb}\n\\end{equation}\n       The above is the condition of {\\it pairwise balance} \\cite{SRB}\n       which ensures that the terms in the two sums on the right hand side\n       of (\\ref{eq:master}) cancel in pairs. Pairwise balance has been\n       used earlier to find steady states of translationally invariant\n       systems \\cite{SRB,BPB}.  We now see that it can be used effectively\n       to deduce the steady state of a disordered system as well.\n\n       Suppose the transition $\\calC \\rightarrow \\cprime$ involves hopping a particle at site\n       ${r}$ to a site ${r^\\prime}={r}+\\Delta r'\\e_\\nu$ with all wells in between\n       along the $\\nu$th axis full (Fig.~\\ref{fig:pwb-2d}). Also suppose\n       the well ${r''}={r}-\\Delta r''\\e_\\nu$ is not full but all wells between\n       ${r''}$ and ${r}$ are full.  The configuration ${\\calC^{\\prime\\prime}}$ is\n       constructed such that it is identical to ${\\cal C}$ at all sites except\n       at the sites ${r''}$ and ${r}$, at which $n^{\\prime\\prime}_{r''}=n_{r''}+1,~ n^{\\prime\\prime}_{r}\n       = n_{r}-1$. With ${\\calC^{\\prime\\prime}}$ thus defined, the pairwise balance\n       condition (\\ref{eq:pwb})\n\\begin{equation}\n\\epsilon(\\nr|\\lr) \\prod_{r} u_{r}(n_{r}) = \\epsilon(n^{\\prime\\prime}_{r''}|l_{r''})\n                             \\prod_{r} u_{r}(n^{\\prime\\prime}_{r})\n\\end{equation}\nreduces to \n\\begin{equation}\n\\epsilon(\\nr|\\lr) u_{{r''}}(n_{{r''}})u_{r}(n_{r})\n       = \\epsilon(n^{\\prime\\prime}_{{r''}}|l_{{r''}})\n             u_{{r''}}(n^{\\prime\\prime}_{{r''}}) u_{r}(n^{\\prime\\prime}_{r}),\n\\label{eq:pwb-check}\n\\end{equation}\n       since ${\\cal C}$ and ${\\calC^{\\prime\\prime}}$ differ only at the sites ${r''}$ and\n       ${r}$.  As can be explicitly checked, condition (\\ref{eq:pwb-check})\n       is satisfied in view of the form of the weights $u_{r}(n_{r})$.  \n       The prescription above ensures one-to-one correspondence between\n       configurations ${\\calC^{\\prime\\prime}}$ and $\\calC \\rightarrow \\cprime$ transitions.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=6.eps,width=7cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{Construction of configurations satisfying the  pairwise balance\ncondition in $d=1$.}\n\\label{fig:pwb-2d}\n\\end{figure}\n\n       In the limit of large $L$ we use the grand\n       canonical description and write the normalized probability of\n       configuration $\\cal{C}$ in the steady state as\n\\begin{equation}\n{\\cal P}({\\cal C}) = {z^{N_P}  \\mu({\\cal C}) \\over {\\prod_{r} Z_{r}}}\n\\end{equation}\nwhere the site generating functions are given by \n\\begin{equation}\nZ_{r}=\\sum^{l_\\r}_{n_\\r=0}\n     u_{r}(n_\\r)z^{n_\\r}\n\\label{eq:sitegen-ddpp}\n\\end{equation}\n     and $N_p$ is the number of particles in the configuration. Here $z$ is\n     the fugacity and will be shown to be directly related to\n     the steady-state current.  The mean particle density at site ${r}$ is\n     given as $\\langlen_\\r\\rangle = z \\partial ln Z_{r}\/\\partial z$\n     \\cite{angle_avg}.  Thus the fugacity $z$ is related to the mean\n     particle density $\\rho$ of the system through\n\\begin{equation}\n\\rho={1\\over L^d}\\sum_{r} \\langlen_\\r\\rangle = {z\\over V}\\sum_{r} {\\partial\nln Z_{r} \\over \\partial z }\n\\label{eq:implicit}\n\\end{equation}\n     The steady state is characterized by a spatially uniform $z$ but\n     inhomogeneous site densities.\n\n     \\subsection{Steady-state current}\n     \\label{sub:J0-DDPP}\n\n      For simplicity of presentation, we first derive a closed form\n      expression for the steady-state current $J_0$ for the 1-$d$ fully\n      asymmetric model. Then we use these results to\n      write down the current for the general $d$-dimensional case.\n\n      (i) {\\it $d=1$, with forward jumps}: In this case, for notational\n            convenience, we replace the lattice site index ${r}$ by a single\n            integer index $i$. Further, we allow jumps only in one\n            direction ($\\nu=1: p_1=p, p_{-1}=0$). For an infinite system we\n            write the current $J_{i,i+1}$ in the bond $(i,i+1)$ as\n\\begin{eqnarray}\n   J_{i,i+1}& = &\\sum_{j<i}\\sum_{n_j=1}^{l_j}p\\epsilon(n_j|l_j)P_j(n_j)\n            \\prod_{j<k\\le i}P_k(l_k) \\nonumber \\\\\n           & & +\\sum_{n_i=1}^{l_i}p\\epsilon(n_i|l_i)P_i(n_i)\n\\label{eq:Ji-i1} \n\\end{eqnarray}\n            where \n\\begin{equation}\nP_i(n_i)=u_i(n_i) z^{n_i}\/Z_i\n\\label{eq:site-prob}\n\\end{equation}\n            is the probability that site $i$ has occupancy $n_i$. Note\n            that the product form of the measure (\\ref{eq:imDDPP}) has\n            been exploited to write the above expression with\n            decoupled joint probabilities.\n\n            It is cumbersome, and not very instructive, to perform the\n            sum in (\\ref{eq:Ji-i1}) to obtain the current in a\n            closed form.  Instead we use the spatial constancy of the\n            current in the steady state to arrive at the result. We\n            note that in the above expression the first term\n            represents contribution of jumps originating from site $j$\n            to the left of site $i$ with all sites in between\n            full. The second sum represents jumps from the site $i$\n            itself contributing to $J_{i,i+1}$. The first sum may be\n            rewritten as\n\n$ P_i(n_i)\\left[\\sum_{j<i-1}\\sum_{n_j=1}^{l_j}\\epsilon(n_j|l_j)P_j(n_j)\n \\prod_{j<k\\le i-1}P_k(l_k)\\right. \\\\ \\left.~~~~~~~~~~~~~\n +\\sum_{n_{i-1}=1}^{l_{i-1}}\\epsilon(n_{i-1}|l_{i-1})P_{i-1}(n_{i-1})\\right]\n $.\n\n\n \\noindent The quantity within square brackets [$\\cdot$] is immediately recognised to\n            be $J_{i-1,i}$ by comparing it with\n            (\\ref{eq:Ji-i1}). Physically, this means that all\n            jumps contributing to $J_{i-1,i}$, for which site $i$ is\n            completely full, also contribute to $J_{i,i+1}$. Hence we\n            have the recursion\n\\begin{equation}\nJ_{i,i+1} = P_i(l_i)J_{i-1,i}\n            +\\sum_{n_i=1}^{l_i}p\\epsilon(n_i|l_i)P_i(n_i)\n\\label{eq:Jrecur}\n\\end{equation}\n            relating $J_{i,i+1}$ to $J_{i-1,i}$.\n            Now, since $\\{P_i(n_i)\\}$ are the steady-state site\n            probabilities, and since in the steady state all the bond\n            currents must be equal (i.e. $J_{i,i+1}=J_{i-1,i}=\n            ..... =J_0$), from (\\ref{eq:Jrecur}) we obtain\n\\begin{eqnarray}\nJ_0 & = & {{\\sum_{n_i=1}^{l_i}\\epsilon(n_i|l_i)P_i(n_i)}\n             \\over{1-P_i(l_i)}} \n      =  p\\epsilon_0 z\n\\label{eq:J0-1d}\n\\end{eqnarray}\n      where (\\ref{eq:imDDPP}) and (\\ref{eq:site-prob}) have been used in\n      the second step above. \n\n      Note that the steady-state current does not depend upon the detailed\n      spatial arrangement of wells. It is only a function, through the fugacity\n      $z$, of the density and the  total number of wells of different\n      types in a particular realization of disorder.\n\n      (ii) {\\it $d=1$, with jumps in both directions: }\n      We can write the current $J_{i,i+1}$ in bond $(i,i+1)$ as the\n      difference between the current $J^r_{i,i+1}$ due to the rightward\n      jumps and the current $J^l_{i,i+1}$ due to the leftward jumps. This\n      can be done since in each cascade the direction of the initial jump\n      is preserved. Now, we use the result for the fully asymmetric case\n      above to each of these currents separately to obtain\n      $J_{i,i+1} = J_0 = (p_1-p_{-1}) \\epsilon_0 z$\n\n\n     (iii) {\\it $d>1$}: To generalize the above results to\n     $d>1$ we note that for DDPP in $d>1$, the invariant measure\n     (\\ref{eq:imDDPP}) is the same if we single out a particular direction,\n     say $\\nu$, and allow jumps only along that direction. Together with\n     the direction preservation of individual jumps, this allows us to\n     write the expression for the current in any dimension:\n\n\\begin{eqnarray}\n\\vec J_0 &=& \\left[\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu\\right] \\epsilon_0 z \n          = \\epsilon_0 z {\\vec E}\n\\label{eq:J0-dd}\n\\end{eqnarray}\n         where ${\\vec E}\\equiv\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu$ is the external\n         drive. As in the $d=1$ case, the magnitude and the direction of\n         the steady-state current does not depend upon the detailed\n         arrangements of the wells.\n\n\\subsection{Static two-point correlation functions}\n\n          Because of the product form of the measure, the connected part of\n          the equal-time density-density correlation function\n\\begin{equation}\nG_{r}(\\Delta{r}) = \\langle n_{r} n_{{r}+\\Delta{r}}\\rangle-\\langle\nn_{r}\\rangle\\langle n_{{r}+\\Delta{r}}\\rangle\n\\label{eq:partcorr}\n\\end{equation}\n          vanishes identically for $\\Delta{r} \\ne 0$.  Consequently, the\n          fluctuation of the number of particles in $r$ consecutive sites\n          along a straight line can be computed exactly:\n\\begin{eqnarray}\n\\Gamma_i^2(r)& = & \\left\\langle\\left[\\sum_{j=i+1}^{i+r}\\left(n_j-\\langle\n                  n_j\\rangle\\right)\\right]^2 \\right\\rangle \\nonumber\\\\ \n           & = & \\sum_{j=i+1}^{i+r}\n                 \\left( z{\\partial\\over \\partial z}\\right)^2 \\ln Z_j.\n\\label{eq:hhcor-DDPP}\n\\end{eqnarray}\n\\noindent The second step follows from the product form of the measure.\n\n        For $d = 1$, a standard mapping discussed in Section VI introduces\n        height variables defined by\n\\begin{equation}\nh_i=\\sum_{j\\le i} 2(\\langle n_j\\rangle-n_j).\n\\label{eq:heightdef}\n\\end{equation}\n\\noindent Evidently, $\\Gamma_i^2(r)$ is the equal-time height-height \n        correlation $\\langle (h_{i+r}-h_i)^2\\rangle$.  Averaging over the\n        disorder distribution gives $\\overline {\\Gamma^2}(r)\\sim r$\n        implying that the `roughness' exponent $\\alpha$ (defined by\n        $\\overline {\\Gamma^2}(r) \\sim r^{2\\alpha}$) is $1\/2$.\n      \n  \\subsection{Two-rate DDPP model: Explicit results}\n  \\label{subsec:2rate}\n    \n     Let us consider a drop-push model where the maximum occupancy of each site\nis restricted to one, i.e. $l_\\r=1, \\forall{r}$, but the hopping rates\n$\\epsilon_{r}(1)$  are\ndisordered, and chosen independently from the binary distribution\n\n\\begin{equation}\nProb(x=\\epsilon_a) = 1-f, ~~~~~~Prob(x=\\epsilon_b)=f.\n\\end{equation}\n\n\\noindent  This model has the essential ingredients of disorder present in the\n      original DDPP, yet it is simple enough that explicit, closed form\n      relations between the mean density $\\rho$ and fugacity $z$, and hence\n      the steady-state current $\\vec{J_0}$, can be written down.\n\n      Let us denote by $Z_a$ and $Z_b$ the site generating functions\n      for the $a$ and the $b$ sites respectively. Using\n      (\\ref{eq:sitegen-ddpp}) and (\\ref{eq:imDDPP}), these are given by\n      $Z_a=1+\\epsilon_0 z\/\\epsilon_a$ and $Z_b=1+\\epsilon_0 z\/\\epsilon_b$. Now, since the fractions of\n$a$ and $b$ sites are $1-f$ and $f$ respectively, (\\ref{eq:implicit})\nreduces to \n\\begin{equation}\n\\rho=(1-f){\\epsilon_0 z\\over \\epsilon_a+\\epsilon_0 z}+f{\\epsilon_0 z\\over \\epsilon_b+\\epsilon_0 z}.\n\\end{equation}\nThis can be easily\ninverted to obtain $z$ as a function of $\\rho$, e.g. for $f=1\/2$ and\n$\\epsilon_a=\\epsilon_0=\\epsilon_b\/q $ we obtain\n\n\\begin{equation}\nz(\\rho) = {\\sqrt{(1-q)^2(1\/2-\\rho)^2+q}-(1+q)(1\/2-\\rho)\\over 2(1-\\rho)}\n\\label{eq:fugacity}\n\\end{equation}\nSince $z(\\rho)$ is known, the steady-state current is trivially obtained\nfrom (\\ref{eq:J0-dd}).\n\nFinally, the correlation function $\\Gamma_i^2(r)$ of (\\ref{eq:hhcor-DDPP}),\nupon disorder averaging, may be written as\n\\begin{eqnarray}\n{\\overline{\\Gamma^2}(r)} & = & r\\left( z{\\partial\\over \\partial z}\\right)^2\n                              \\left[(1-f)\\ln Z_a + f \\ln Z_b\\right] \\nonumber \\\\\n                         & = &\n                      {1\\over 2}\\left[{z\\over(1+z)^2}+{qz\\over(q+z)^2}\\right] r\n\\end{eqnarray}\nwhere $z$ is given by (\\ref{eq:fugacity}) above.\n\n  \\subsection{Generalized Disordered Drop-push Process: GDDP}\n  \\label{subsec:GDDP}\n\n    We may consider a generalized version of the drop-push process in\n    which, in addition to the particle moves, independent hole moves are\n    also allowed. For simplicity we restrict ourselves to the generalized\n    version of the single occupancy DDPP introduced above. This generalized\n    model may be regarded as the disordered lattice gas analogue of the\n    Toom interface dynamics in the low-noise limit \\cite{Toom}; see Section\n    \\ref{sec:height}. The techniques developed for the DDPP may be used to\n    obtain the exact steady-state measure and other quantities such as\n    current and static correlations provided a certain condition\n    [(\\ref{eq:constr-GDDP}) below] is met.\n\n    The model in $d$ dimensions is defined on a hypercubic lattice with\n    periodic boundary conditions along all the $d$ axes (with unit vectors\n    \\{$\\e_\\nu |\\nu=1\\cdots d$\\}), similar to the DDPP.  Each site ${r}$ of the\n    lattice can hold either a particle ($n_\\r=1$) or a hole ($n_\\r=0$). The\n    configuration $\\cal{C}$ of the system is specified by specifying the\n    occupation number of each well \\{$n_\\r$\\} with ($n_\\r\\in\\{0,1\\},\n    \\forall{r}$). Further, to each site ${r}$ is assigned a pair of positive\n    random numbers $(\\alpha_{r},\\beta_{r})$ chosen from some\n    distribution. The dynamics is stochastic and is very similar to that\n    for the DDPP dynamics: in a time interval $dt$, a particle at site ${r}$\n    is exchanged with the closest hole in the $\\pm\\nu$th direction with a\n    probability $p_{\\pm\\nu}\\alpha_\\r dt$ (Fig.~\\ref{fig:gddp-model}). For\n    identical particles this move is equivalent to a cascade of particle\n    moves terminating at the first vacant site as in the drop-push\n    dynamics.  Likewise, in interval $dt$, a hole at site ${r}$ is exchanged\n    with the closest particle along the $\\nu$th direction with probability\n    $q_{\\nu}\\beta_\\r dt$. This can be looked upon of as a cascade of hole-moves\n    analogous to the cascade of particle moves.  Here, as in the DDPP, the\n    $p_\\nu$'s and $q_\\nu$'s are all non-negative and satisfy\n    $\\sum_{\\nu=1}^d (a_\\nu+a_{-\\nu})=1; a=p,q$. Further we chose $\\alpha$'s\n    and $\\beta$'s such that\n\\begin{equation}\n\\alpha_\\r\\beta_\\r= K,\n\\label{eq:constr-GDDP}\n\\end{equation} \n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=7.eps,width=5cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{Generalized Disordered Drop-push Process (GDDP) configuration and\nmoves in $d=2$.}\n\\label{fig:gddp-model}\n\\end{figure}\n    \\noindent where $K$ is a constant independent of ${r}$.  As we will see\n    below, this particular choice of the jump-rates allows the exact\n    determination of the invariant measure of the model.  Physically this\n    choice is quite reasonable since it implies that the sites which act as\n    traps for particles (low $\\alpha_\\r$) are more transparent to holes (high\n    $\\beta_\\r$) and vice versa. As pointed out earlier the non-disordered\n    version of this model, i.e.  \\{$\\alpha_\\r=\\alpha, \\beta_\\r=\\beta, \\forall {r}$\\} in\n    1-$d$, is the lattice gas equivalent of the low-noise driven Toom\n    interface dynamics \\cite{Toom}.\n\n    The master equation (\\ref{eq:master}) governing the time evolution of\n    the system now includes terms corresponding to hole moves as well.\n    Since each micro-step involves either only particle moves or hole\n    moves, we use the principle of pairwise balance for the particle moves\n    and hole moves separately.  We work in the grand canonical picture in\n    the thermodynamic limit.\n\n     If only particle moves were allowed the invariant measure would be\ngiven by\n\\begin{equation}\n\\mu_{pcle}({{\\cal C}})=\\prod_{r} u_{r}(n_r)\n\\label{eq:muC-p}\n\\end{equation}\n     The single site weights $u_{r}$ are given by\n\\begin{equation}\n u_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n               1 & \\mbox{~~~if~$n_\\r=0$} \\\\ \n               \\epsilon_0\/\\alpha_\\r & \\mbox{~~~if~$n_\\r=1$}\n               \\end{array} \\right. \n\\end{equation}\n\\noindent   Introducing the fugacity $z$ and site generating functions \n     $Z_{r}=1+z u_{r}(1)$ we can write the normalized single site\n     probabilities as $P^{pcle}_{r}(0)=1\/Z_{r}$ and\n     $P^{pcle}_{r}(1)=zu_{r}(1)\/Z_{r}$.\n \n     Similarly, with only the hole moves, the invariant measure has the product\nform \n\\begin{equation}\n\\mu_{hole}({{\\cal C}})=\\prod_{r} v_{r}(n_r),\n\\label{eq:muC-h}\n\\end{equation}\nwhere $n_{r}=1$ ($0$) refers to the presence (absence) of a\n     hole. The single site weights $v_{r}$ are given by\n\\begin{equation}\n v_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n               1 & \\mbox{~~~if~$n_\\r=0$} \\\\ \n               \\epsilon_0\/\\beta_\\r & \\mbox{~~~if~$n_\\r=1$}\n               \\end{array} \\right. .\n\\end{equation}\n\\noindent  Introducing the fugacity $y$ for holes and site generating \n     functions $Y_{r}=1+y v_{r}(1)$ we can write the normalized single-site\n     probabilities as: $P^{hole}_{r}(0)=1\/Y_{r}$ and\n     $P^{hole}_{r}(1)=y v_{r}(1)\/Y_{r}$.\n\n     Now, since each site is occupied either by a particle or a hole, we\n     must have $P^{pcle}_{r}(0)=P^{hole}_{r}(1)$ and\n     $P^{pcle}_{r}(1)=P^{hole}_{r}(0)$.  Using the detailed forms of\n     $P^{pcle}_{r}$'s and $P^{hole}_{r}$'s, we arrive at the condition\n     (\\ref{eq:constr-GDDP}) with $K\\equiv\\epsilon_0^2 y z$.  If this\n     condition is satisfied then the invariant measure for GDDP is given by\n     either (\\ref{eq:muC-p}) or (\\ref{eq:muC-h}), since both are\n     equivalent.\n\n     In a similar manner as for the DDPP the current due to the particle\n     moves and hole moves may be computed. The total particle\n     current due to both types of moves, in $d$ dimensions, is given by\n\\begin{equation}\n\\vec {J_0} = \\epsilon_0 z\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu \n             - \\epsilon_0 y\\sum_{\\nu=1}^d (q_\\nu-q_{-\\nu})\\e_\\nu\n\\end{equation}\n\n    As for the DDPP, static density-density correlations in the steady\n    state vanish identically on account of the product form of the\n    steady-state measure.  In $d=1$ the height-height correlation is given\n    by (\\ref{eq:hhcor-DDPP}). \n \n\\section{Disordered Asymmetric Simple Exclusion Process: DASEP \\hfill} \n\\label{sec:DASEP}\n      The asymmetric simple exclusion process (ASEP) is a prototype model\n      for studying nonequilibrium phenomena in the context of lattice gases\n      \\cite{ligget,spohn}. When discussing the effect of quenched disorder,\n      it is important to distinguish between cases in which (a) the easy\n      direction of hopping in each bond is the same but hopping rates are\n      random, and (b) the easy direction is itself a random variable. The\n      latter case is studied in Section \\ref{sec:Sinai}. In this section,\n      we consider a 1-$d$ system with disorder of type (a) and show that\n      quenched disorder can induce macroscopic phase separation.  Using a\n      variety of arguments we sketch the phase coexistence curve in the\n      current ($J_0$) - mean density ($\\rho$) plane. This agrees\n      qualitatively with the results obtained from the Monte Carlo (MC)\n      simulations.\n\n      \\subsection{The model} \n      \\label{sub:defDASEP}\n \n      In one dimension, we define the disordered asymmetric simple\n      exclusion process on a ring of $L$ sites. Each site can hold either\n      $1$ or $0$ particle. To each bond $(i,i+1)$ of the lattice is\n      assigned a quenched random rate $\\alpha_{i,i+1}$ chosen independently\n      from some probability distribution $Prob(\\alpha)$. The dynamics is\n      stochastic: in a time interval $dt$ a particle at site $i$ attempts\n      to hop, with probability $p\\alpha_{i,i+1} dt$, to site $i+1$. The\n      move is completed if and only if site $i+1$ is unoccupied (see\n      Fig.~\\ref{fig:dasep-1d}).\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=8.eps,width=7cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{The disordered fully asymmetric simple exclusion process \n(DASEP) and moves in $d=1$.}\n\\label{fig:dasep-1d}\n\\end{figure}\n      For the model defined above no analytically exact characterization of\n      the steady-state measure could be obtained. A simpler \n      model in which there is only one defect bond has been studied in\n      detail by Janowsky and Lebowitz \\cite{Janowsky}, but in this case too\n      the exact steady-state measure is not known.  We use Monte Carlo\n      simulations and a mean-field approximation to demonstrate some\n      striking effects of quenched disorder.\n\n     \\subsection{Current-density plot and density profile in steady\n     state\\hfill} \\label{sub:imDASEP}\n\n     Figure \\ref{fig:dasep-jrho} shows the steady-state current $J_0$ vs\n     mean density $\\rho$ plot, obtained from MC simulations, for\n     a system of size $L=8000$ and the rates $\\alpha$ chosen from the\n     binary distribution\n\\begin{equation}\n     Prob(\\alpha=r)=f, ~~~~~Prob(\\alpha=1)=1-f.\n\\label{eq:binary}\n\\end{equation}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=9.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{The current-density plot for the DASEP  for a given realization of\ndisorder for a system of size $L=8000$. The hopping rates are chosen from\nthe distribution ({\\protect\\ref{eq:binary}}) with $r=f=1\/2$. The filled\ncircles are the MC results and the dashed line is the mean field curve. The\nsolid line represents the $J-\\rho$ curve for the Fully Segregated Model\n(FSM) for the same values of the parameters. }\n\\label{fig:dasep-jrho}\n\\end{figure}\n\\noindent Here $f$ is the fraction of weak bonds, and $r$ is a measure\n     of the strength of the weak bonds. We used the values\n     $r=1\/2,f=1\/2$ in our numerical work. For a specified mean density\n     $\\rho$, a random initial distribution of $N_p=\\rho L$ particles\n     on $L$ sites is chosen and the system is allowed to settle into a\n     steady state by evolving it for a sufficiently large number of MC\n     steps. Then the current across each bond is obtained by counting the\n     total number of jumps across that bond, over a large number of MC\n     steps. An average of all the bond currents thus computed is taken\n     as $J_0$, as currents across all bonds are equal in the steady\n     state. $J_0$ is a symmetric function of density around $\\rho=1\/2$\n     as a result of a certain symmetry with respect to particle-hole\n     interchange (see Appendix \\ref{apx:PHsym}). As may be expected, the\n     current for the disordered system lies between the corresponding\n     values of the two pure reference systems with $r=1$ and $r=1\/2$\n     on all the bonds.  The more striking qualitative difference\n     between the disordered and pure systems is the appearance of a\n     plateau (Regime B in Fig.~\\ref{fig:dasep-jrho}) for a range of\n     densities $|\\rho-1\/2|\\le \\Delta$. In this regime, the current is\n     independent of the mean density and equals the maximum allowed\n     current in the system.  The approximate size $\\Delta$ of regime\n     B, which is a function of $r$ and $f$, is obtained in subsection\n     \\ref{sub:FSM} below.\n\n     We studied the steady state density profiles characterized by the set\n     of site densities $\\{\\rho_i\\equiv\\langle n_i \\rangle\\}$ in both\n     regimes A and B, using MC simulations.  We found that in regime A the\n     system is homogeneous on a macroscopic scale, while in regime B it\n     shows macroscopic density segregation.  Figure\n     \\ref{fig:dasep-profiles} shows the steady state density profiles for\n     three representative mean densities --- one from regime A and two from\n     regime B. Evidently there is a large qualitative difference between\n     the profiles in the two regimes.  In regime A\n     (Fig.~\\ref{fig:dasep-profiles}a), there are density variations\n     (shocks) only over microscopic scales; coarse-graining over a few\n     lattice spacings leads to a spatially uniform density.  In contrast to\n     this, in the profiles corresponding to regime B\n     (Fig.~\\ref{fig:dasep-profiles}b), there are density inhomogeneities\n     over length scales comparable to the system size $L$, in addition to\n     the shocks on a microscopic scale.  This segregation into high and low\n     density phases, with large shocks separating them, is reminiscent of\n     phase separation, and occurs over the full range of mean particle\n     densities corresponding to regime B.  A qualitatively similar\n     phenomenon has been found in a system with one defect bond, studied in\n     \\cite{Janowsky}.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=10.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Density profiles for the DASEP for a system of size $L=8000$ for a\ngiven realization of disorder at three fillings (a) $\\rho=0.8$, (b)\n$\\rho=0.6$, and (c) $\\rho=0.5$. In d,e and f are shown the blow-ups of the\nregions enclosed in the dashed boxes in a,b and c respectively. Circles are\nMC profiles and the continuous lines are mean-field results.}\n\\label{fig:dasep-profiles}\n\\end{figure}\n\n     The {\\it number} of different large-scale regions of high and low\n     density shows fluctuations from one realization of disorder to\n     another.  As the size of the system is increased, we monitored the\n     mean number of these regions, and found that it is nearly constant, or\n     perhaps increases very mildly -- certainly, much less fast than\n     linearly in the size of the system.  This implies that the\n     characteristic length scale of density segregation increases\n     indefinitely in the thermodynamic limit.\n\n\n     \\subsection{ Mean-field approximation \\hfill}\n     \\label{sub:MFA}\n      \n      We now turn to a mean-field approximation, which assumes no\n     correlations between site densities in the steady state.  We will see\n     that it captures most of the steady-state features found in the MC\n     simulations above, including the steady-state density profile.\n     \n     The time-averaged steady-state current $J_{i,i+1}$ in the bond\n     $(i,i+1)$ is given by $J_{i,i+1} = \\alpha_{i,i+1} \\langle\n     n_i~(1-n_{i+1})\\rangle $.  In view of the mean-field approximation\n     $\\langle n_i n_j\\rangle =\\langle n_i\\rangle\\langle n_j\\rangle$ this\n     reduces to\n\\begin{equation} J_{i,i+1} = \\alpha_{i,i+1}\\langle\n     n_i\\rangle~\\langle 1-n_{i+1}\\rangle \\label{eq:MFA} \n\\end{equation}\n\n     To compute the steady-state current $J_0$ as a function of the mean\n     particle density $\\rho$ for a given realization of disorder, we use\n     the two iteration schemes described below, which yield equivalent\n     results.\n\n     (i) {\\it Constant current iteration scheme.} For a given system\n     of size $L$ and for a fixed value $J_0$ for the current, we\n     iterate the following set of equations\n\\begin{equation}\n    \\rho_{i+1}=1-J_0\/(\\alpha_{i,i+1}\\rho_i) ~~~~i=1,..,L\n\\end{equation} \n     around the chain starting with, say, some value $\\rho_1$ (periodic\n     boundary conditions imply $\\rho_{i+L}=\\rho_i$). If $J_0$ is less than a\n     certain value $J^{MF}_{max}$, which is the maximum current supported\n     by the system within the mean-field approximation, then the iteration\n     scheme converges, i.e. we get all the site densities in the physically\n     acceptable range $[0,1]$. The average of these site densities gives\n     the mean density of the system corresponding to the stationary current\n     $J_0$. There are in general two values of the mean particle density\n     corresponding to an allowed value of $J_0$ and the\n     iteration scheme converges to one or the other depending on the\n     initial value of the density $\\rho_1$.\n  \n     However, in this scheme we find that the number of iterations required\n     for convergence increases without bound, as the trial value $J_0$ gets\n     closer to $J^{MF}_{max}$ from below.  This divergence of the iteration\n     scheme is presumably due to the existence of the plateau in the\n     $J~vs~\\rho$ curve, i.e. there exist many values of $\\rho$ for $J_0$\n     very close to $J^{MF}_{max}$. Hence to obtain the density profile for\n     $\\rho$ in the density segregated regime we resort to the {\\it constant\n     density} iteration scheme described below.\n\n     (ii) {\\it Constant density iteration scheme.}  In this scheme we begin\n     by assigning site densities $\\{0\\le\\rho_i(0)\\le 1\\}$ to the lattice\n     sites subjected to the constraint ${1\\over L}\\sum_i \\rho_i(0)\n     =\\rho$. The site densities are updated in parallel according to:\n\\begin{equation}\n     \\rho_i(t+1)=\\rho_i(t)+J_{i-1,i}(t)-J_{i,i+1}(t);~~~i=1,..,L \n\\end{equation}\n     where $J_{i,i+1}(t)=\\alpha_{i,i+1}\\rho_i(t)[1-\\rho_{i+1}(t)]$ in view of\n     (\\ref{eq:MFA}).\n\n     We refer to this as the constant density scheme, since in each\n     iteration the total density remains unchanged, i.e\n     $\\sum_i\\rho_i(t+1)=\\sum_i\\rho_i(t)$. After a sufficient number of\n     iterations, which depends upon the starting mean density $\\rho$,\n     the set of site densities converge to a set of numbers\n     \\{$\\rho_i$\\} and the current on each bond converges to the steady\n     state current $J_0$.\n\n     The steady-state density profiles and the $J_0~vs ~\\rho$\n     plot ($0\\le\\rho\\le1$) for a given configuration of disorder obtained\n     using these schemes is shown in Figs.~\\ref{fig:dasep-profiles} and\n     \\ref{fig:dasep-jrho} respectively.  It is evident that the mean-field\n     approximation (\\ref{eq:MFA}) reproduces quite well not only the\n     $J-\\rho$ relationship, but also the density profile, including the\n     locations of shocks, though not the shapes of individual shocks.\n\n     \\subsection{Qualitative explanation of phase separation}\n     \\label{sub:MF-gross}    \n\n     Although the mean-field approximation of the previous subsection\n     successfully reproduces many features in the steady state, it does not\n     yield a simple understanding of the phase separation\n     (Fig.~\\ref{fig:dasep-profiles}) or the plateau in the $J$--$\\rho$\n     curve (Fig.~\\ref{fig:dasep-jrho}) in terms of the macroscopic\n     parameters of the model.  We conjecture that underlying the behaviour\n     of the DASEP in different regimes is a Maximum Current principle: For\n     a given mean density, the system settles into a steady-state \n     which maximizes the stationary current. Such a maximum\n     current principle has been used to describe phase separation in the\n     asymmetric simple exclusion process with open boundary conditions in\n     \\cite{Krug91}.\n \n     To use the maximum-current principle to have a qualitative\n     understanding of the phase separation in DASEP, let us assume that the\n     density in each stretch of like bonds is uniform. This approximation\n     is in fact exact in the Fully Segregated Model (FSM) discussed in the\n     following subsection. Let us denote stretches of $\\alpha=1$ bonds by X\n     and stretches of $\\alpha=r<1$ bonds by Y. The two parabolas in\n     Fig.~\\ref{fig:phases-expl} are the steady-state $J~vs~\\rho$ curves for\n     the two pure reference systems all X and all Y. In the disordered\n     system, since the steady-state current has to be spatially constant,\n     the possible densities are given by the four intersections of the line\n     $J=J_0$ with the two parabolas. If the mean density is in the range\n     $\\rho\\le 1\/2-\\Delta$ (or $\\rho\\ge 1\/2+\\Delta$), then the allowed\n     densities for the X and Y stretches are $\\rho_1,\\rho_2$ (or\n     $\\rho_4,\\rho_3$) respectively. The current is in fact determined by\n     the density constraint\n     $(1-f)\\rho_{1,4}(J_0)+f\\rho_{2,3}(J_0)=\\rho$. The variation of density\n     between $\\rho_1$ and $\\rho_2$ (or $\\rho_3$ and $\\rho_4$) between X and\n     Y stretches corresponds to the `sub-bands' seen in\n     Fig.~(\\ref{fig:dasep-profiles}a). On a macroscopic scale, however, the\n     system has uniform density. Now consider what happens when the mean\n     density is brought closer to $1\/2$. From Fig.~\\ref{fig:phases-expl},\n     it is evident that the current would tend to increase, and would\n     eventually reach the maximum allowed value $J_{max}^Y$ (which equals\n     $1\/4$ in the thermodynamic limit, as argued below). As the density is\n     increased further, the current remains constant at $J_{max}^Y$, in\n     accordance with the maximum current principle, and the excess density\n     is taken care of by converting some of the X stretches of density\n     $\\rho_1$ into ones with $\\rho_4$ (or vice versa if $\\rho > 1\/2$). This\n     conversion takes place adjacent to the largest stretch of Y bonds,\n     leading to two macroscopic regions of different mean densities -- one\n     with densities $(\\rho_1, \\rho_2)$ for the X and Y stretches, and the\n     other with $(\\rho_4, \\rho_3)$. The position of the principal shock\n     separating these regions is at the location of the largest Y\n     stretch. In the DASEP, the assumption of uniform density in each\n     stretch is not really true, on account of the finite length of most of\n     the stretches. However, the above argument provides a qualitative\n     picture towards understanding the reason for phase separation in the\n     DASEP.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=11.eps,width=6.5cm}\n\\end{center}\n\\narrowtext\n\\caption{Origin of phase separation in DASEP. The two parabolas\n$J=r\\rho(1-\\rho);~r=1,1\/2$, represent the\n$J-\\rho$ plots for the two reference non-disordered systems.}\n\\label{fig:phases-expl}.\n\\end{figure}\n     In a certain respect, the reason for the phase separation in the DASEP\n     is similar to that in the single defect bond model studied in\n     \\cite{Janowsky} --- both have local segments in the system where the\n     maximum allowed current is less than that allowed everywhere else. In\n     the single defect bond model, phase separation takes place when the\n     current carried by the rest of the system, with presumed uniform\n     density, is larger than the maximum current allowed through the weak\n     bond. In the DASEP, with an extensive number of weak bonds, the {\\it\n     largest stretch} of weak bonds acts as the current-limiting\n     segment. The length of this stretch increases as $\\ln L$ with system\n     size, and in the thermodynamic limit the maximum allowed current in\n     this stretch is ${1\\over 4}r$ -- equal to the maximal current in a\n     pure system with only weak bonds.\n\n     The essential point leading to phase separation is thus the maximum\n     current principle, coupled with localized current-limiting regions in\n     the system. In the DASEP, this limit is determined by long stretches\n     of weak bonds, and similar considerations should apply in related\n     models.  Consequently, we would expect a density-segregated phase in\n     disordered versions of models which, in the absence of disorder,\n     display a maximum in the steady-state current as a function of\n     density.\n \n     \\subsection{The Fully Segregated Model} \\label{sub:FSM}\n\n     It is useful to define a model for which many of the approximations\n     made in the previous subsection are actually exact. To this end, we\n     study a Fully Segregated Model (FSM), which is obtained from the\n     binary random model above by arranging all like bonds together. Thus,\n     in this model, one has {\\it two} large stretches of X and Y, of\n     lengths $(1-f)L$ and $fL$ respectively. For the FSM in the\n     thermodynamic limit, the assumption of uniform density within each\n     stretch is justified, as correlations due to the junctions decay with\n     increasing separation, and may be neglected in the bulk \\cite{FSM}.\n\n     Steady-state MC density profiles for the FSM at three different\n     fillings $\\rho\\le 1\/2$ are shown in Fig.~\\ref{fig:dp-seg} ---\n     symmetry under particle-hole exchange implies analogous behaviour\n     for $\\rho\\ge 1\/2$ (Appendix \\ref{apx:PHsym}). For low densities\n     ($\\rho<\\rho_c^-$), the two stretches have uniform bulk\n     densities $\\rho_x$ and $\\rho_y$ related to each other by the\n     requirement of equality of the two bulk currents,\n\\begin{equation}\n\\rho_x(1-\\rho_x)=r\\rho_y(1-\\rho_y)=J_0,\n\\label{eq:J-xy}\n\\end{equation}\nand the density constraint \n\\begin{equation}\n(1-f)\\rho_x+f\\rho_y=\\rho. \n\\label{eq:contr-rho}\n\\end{equation}\nThese three equations determine $\\rho_x, \\rho_y$ and $J_0$ uniquely for a given\n$\\rho$. For $f=1\/2$ we obtain\n\\begin{eqnarray}\n\\rho_y &=& {4\\rho-1-r\\pm\\sqrt{(4\\rho-1-r)^2+8(1-r)\\rho(1-2\\rho)}\\over\n{2(1-r)}}\\nonumber \\\\ \n\\rho_x&=&2\\rho-\\rho_y, ~~J_0=\\rho_x(1-\\rho_x)\n\\end{eqnarray}\n     This is analogous to the macroscopically homogeneous state of the\n     fully random system.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=12.eps,width=7.5cm}\n\\end{center}\n\\narrowtext\n\\caption{Representative density profiles of the FSM with $r=f=1\/2$ at \nthree different fillings: (a) $\\rho=0.24$ (dotted) (b) $\\rho=\\rho_c^-\n=0.324$ (solid) and (c) $\\rho=0.4$ (dashed). The Y stretch ($r=1\/2$) is in\nthe range $i\\in[2000,6000]$. The inset shows the variation of the bulk\ndensities in the X and Y stretches as a function of the filling\n$\\rho$.}\n\\label{fig:dp-seg}\n\\end{figure}\n     As the mean density is increased, the density of each stretch\n     increases, until, at a critical density $\\rho=\\rho_c^-={1\\over\n     2}-{1\\over 4}\\sqrt{1-r}$ (the corresponding critical density on the\n     higher side is $\\rho_c^+=1-\\rho_c^-$), $\\rho_y$ equals $1\/2$\n     (Fig.~\\ref{fig:dp-seg}).  At this density, the current equals the\n     maximum possible current in Y, namely, $J_0=J_{max}=r\/4$.  As $\\rho$\n     is increased further, $\\rho_y$ and $J_0$ remain constant at $1\/2$ and\n     $r\/4$ respectively. The density change is adjusted by creating a\n     density inhomogeneity in stretch X. The two densities $\\rho_x^{high}$ and\n     $\\rho_x^{low}$ are related by $\\rho_x^{high}+\\rho_x^{low}=1$ so that the\n     currents in the two phases are equal,\n     i.e. $\\rho_x^{high}(1-\\rho_x^{high})=\\rho_x^{low}(1-\\rho_x^{low})=r\/4$. This\n     implies $\\rho_x^{high,low}=(1\\pm\\sqrt{1-r})\/2$ (see inset in\n     Fig.~\\ref{fig:dp-seg}).  The fraction of these phases can be determined\n     from a lever rule and are given by\n     $f^{high,low}=|\\rho-\\rho^\\mp_c|\/(\\rho_x^{high}-\\rho_x^{low})$. This locking\n     of the density in the Y stretch at $\\rho_y=1\/2$ is a direct consequence\n     of the maximum current principle introduced above: any change of $\\rho_y$\n     from $1\/2$ would reduce the current in Y, and hence in the full\n     system.  All the arguments above can be applied for $\\rho>1\/2$ because\n     of particle-hole symmetry. Thus for $\\rho^-_c<\\rho<\\rho^+_c$ the state\n     of the segregated model is analogous to the phase separated regime B\n     of the random model. The size of regime B in the FSM is given by\n     $2\\Delta=\\rho_c^+-\\rho_c^-=\\sqrt{1-r}\/2$. It closely approximates the\n     size of the B regime in the DASEP (Fig.~\\ref{fig:dasep-jrho}).\n\n\\subsection{Phase-coexistence curve}\n     \\label{sub:pcc}\n\n     For the FSM, as $r$ is varied we obtain\n     different $J_0~vs~\\rho$ curves. The phase-coexistence curve in the\n     current-density plane in the parametric form\n\\begin{equation}\nJ_c={r\\over 4}, ~~~~~\\rho_c={1\\over 2}\\pm{1\\over 4}\\sqrt{1-r}\n\\label{eq:ph-diag}\n\\end{equation}\n    which is the parabola $J_c=1\/4-(1-2\\rho_c)^2$ in\n    Fig.~\\ref{fig:phase-diag}.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=13.eps,width=7.6cm}\n\\end{center}\n\\narrowtext\n\\caption{The phase coexistence curve for the FSM and the DASEP for $f=1\/2$.\nThe solid parabola is the coexistence curve of the FSM. The circles and the\ntriangles are respectively the MC and mean-field phase-coexistence curves\nfor the DASEP. The dashed parabola $J=\\rho(1-\\rho)$ demarcates the allowed\nregion for the DASEP. }\n\\label{fig:phase-diag}\n\\end{figure}\n\n    The difference between the phase boundaries of the DASEP and FSM\n    Fig.~\\ref{fig:phase-diag} comes from the fact that the interspersed\n    weak-bond stretches in the DASEP have finite lengths, and the density\n    in these small stretches is not quite equal to $1\/2$.  Close to the\n    phase separation, we anticipate the mean density in a Y-stretch of\n    length $l$ in the DASEP to be of the form $\\rho_Y(l)=1\/2\\pm\n    A(r)\/l^\\alpha(r)$, with $\\alpha(r)>0$.  Using the distribution of the\n    $Y$ stretches, namely $P_Y(l)=2^{-l}$, we obtain the $r$ dependence of\n    the critical density\n\\begin{equation}\n \\rho_c^\\pm = {1\\over 4} + \\rho_X \\pm A(r) \\sum_l P_Y(l) \\rho_Y(l) $$\n\\end{equation}\n    \\noindent where $\\rho_X(r<<1)\\sim r$ is the density in the X stretches\n    near the phase transition.  Comparing with the phase diagram for the\n    FSM in Fig.~\\ref{fig:phase-diag}, the correction term $A(r)$ seems to\n    be positive for all $r$. Further, let us suppose that the current in\n    the FSM is a lower bound to that in the DASEP, as suggested by\n    Fig.~\\ref{fig:phase-diag}.  Then the coexistence curve for the random\n    system must be quadratic near the critical point ($J^0=1\/4,\n    \\rho^0=1\/2$) -- being bounded by two quadratics, namely, the $J-\\rho$\n    curve for the pure system $J=\\rho(1-\\rho)$, and the coexistence curve\n    of the FSM $J=1\/4-(1-2\\rho)^2$.\n\n     \\subsection{Correlations in the steady state}\n     \n     Figure \\ref{fig:dasep-corln} shows the Monte Carlo results for the\n     site-averaged density-density and height-height correlation functions\n     $\\overline{G}(\\Delta{r})~{\\rm and}~ \\overline{\\Gamma^2}(r)$ in both the\n     homogeneous and density-segregated regimes of the DASEP.\n     $\\overline{G}(\\Delta{r})$ is seen to decay rapidly over a few lattice\n     spacings, accounting for the success of the mean field approximation.\n     It is found that $\\overline{\\Gamma^2}(r)$ grows as $r$ --- implying a\n     roughness exponent $\\alpha=1\/2$.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=14.eps,width=7.6cm}\n\\end{center}\n\\narrowtext\n\\caption{ Height-height correlation function $\\overline{\\Gamma^2}(r)$\nfor DASEP: (a) in Homogeneous (circles), and (b) Segregated-Density\n(triangles) regimes. Inset shows the site averaged density-density\ncorrelation function $\\overline G(\\Delta{r})$ defined in\n({\\protect\\ref{eq:partcorr}}). The small negative values at large\n$|\\Delta{r}|$ arise due to the finite size of the system.}\n\\label{fig:dasep-corln}\n\\end{figure}\n\n\\section{DASEP with  backbends}\n\\label{sec:Sinai}\n\n     As discussed at the beginning of Section \\ref{sec:DASEP}, the\n     introduction of randomness in the easy direction of individual bonds\n     alters the properties of one-dimensional disordered exclusion process\n     in a crucial way.  We study this in this section.\n\n     The model is defined as follows: Assign quenched arrows (pointing\n     either right or left) independently to each bond of a periodic chain,\n     with probability $f < {1\\over 2}$ for left arrows, and $1-f$ for right\n     arrows.  An arrow defines the easy direction of hopping on each bond:\n     a particle-hole exchange across a bond occurs with rate $w (1+g)$ if\n     the particle moves along the direction of the arrow, and $w (1-g)$ if\n     it moves opposite to the arrow.  Since $f < {1\\over 2}$, there is an\n     overall tendency for particles to circulate rightward, and the\n     question is whether there is a nonzero current even in the\n     thermodynamic limit.\n\n     The model represents a system of hard-core particles in a random\n     potential with a downward tilt.  A conglomeration of left pointing\n     arrows constitutes a backbend, where the potential climbs up before\n     going down again.  Within mean-field theory it is possible to obtain\n     an upper bound $J_\\l$ on the current that can be carried by mutually\n     excluding particles through a backbend of length $\\l$ \\cite{RB}.  To\n     this end, consider biased diffusion of hard core particles in the\n     segment [$0, l$] of a 1-d lattice, with the `optimal' boundary\n     conditions $\\rho(0) = 1$ and $\\rho (l) = 0$; these boundary conditions\n     force the largest possible current through the segment, opposite to\n     the bias.  The master equation that describes transport is invariant\n     under interchange of particles and holes and simultaneous relabelling\n     of sites in reverse order, i.e. $n_j \\rightarrow \\bar n_{l - j}$ The\n     boundary conditions respect this symmetry, implying that the\n     steady-state density $\\rho(j)$ at site $j$ satisfies $\\rho(j) = 1 -\n     \\rho (l - j)$.  Thus in the steady state the number of particles in\n     the backbend is $l\/2$ irrespective of the strength of the bias $g$.\n     The principal effect of increasing $g$ is to sharpen the region which\n     marks the transition from the particle-rich half of the backbend to\n     the hole-rich half.  The steady-state profile approaches a step\n     function centered at $j = l\/2$ as $ g \\rightarrow 1$.\n\n     The current in the steady state is the number of particles crossing\n     site $l$ in unit time. Results of a Monte Carlo study \\cite{RB} are\n     consistent with the large-$l$ asymptotic behaviour \n\\begin{equation}\n J_l \\sim \\exp  \\left( - {1 \\over 2}~l\/\\Lambda(g) \\right).\n\\label{eq:s1}\n\\end{equation}\n     where $\\Lambda(g)$ is a bias-induced length given by $\\Lambda^{-1}(g)=\n     \\ln \\{(1+g)\/(1-g)\\}$.  This can be seen by writing the current within\n     a mean field approximation as $J = W(1 + g) \\rho_j (1 - \\rho_{j+1}) -\n     W(1 -g) \\rho_{j+1} (1 - \\rho_j)$, and finding the value of $J$ for\n     which the boundary conditions $\\rho_0 = 1$, $\\rho_l = 0$ hold.  For\n     $l>>\\Lambda(g)>>1$,this leads to $J \\approx 2g~e^{-lg}$ \\cite{RB}, in\n     agreement with (\\ref{eq:s1}) when $g$ is small.\n\n     The origin of the factor $1\\over2$ in the exponent in (\\ref{eq:s1})\n     has been discussed in \\cite{RB}, and we recount the argument in brief.\n     The transport of a single particle through the backbend involves two\n     (approximately) causally independent steps which occur in parallel:\n     (i) the topmost particle (located at site $k \\approx l\/2$ in large\n     fields) has to be activated a distance $l\/2$, which requires an\n     activation time $\\tau_{1\/2} \\sim \\exp \\left({1\\over 2}~l \/\\Lambda(g)\n     \\right)$ and, (ii) the consequent hole that remains in the\n     steady-state distribution moves to the bottom and is filled up, by\n     moving each of $l\/2$ particles up through a lattice spacing.  The time\n     required is $\\tau_{1\/2}$ again.  The current $J$ is thus proportional\n     to $\\tau^{-1}_{1\/2}$, and consequently follows (\\ref{eq:s1}).\n\n     Since, for fixed $g$, the largest current that can flow through a\n     long backbend ($l>>\\Lambda(g)$) is exponentially small in its\n     length, the largest current through the 1-d lattice of length $L$\n     is determined by the length $\\l^*(L)$ of the largest backbend\n     encountered.  Since the probability of occurrences of $\\l$\n     consecutive left-pointing arrows on bonds is proportional to\n     $f^\\l$, we may estimate $\\l^*$ from $Lf^{\\l^*} = C$ where $C$ is\n     a constant of order unity.  Substituting in (\\ref{eq:s1}), we\n     find that the current falls with increasing lattice size as\n\\begin{equation}\nJ(L) \\sim L^{-{1\\over2} \\theta} \n\\label{eq:s2}\n\\end{equation}\n     with $ \\theta^{-1} = \\Lambda(g)\\ln f$.\n     Thus the current is expected to decay as a power law in $L$, with a\n     bias-dependent power, and to vanish in the thermodynamic limit.\n     Figure \\ref{fig:JL_sinai} shows the result of Monte Carlo simulations.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=15.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Size dependence of the steady-state current in the backbend\nmodel (DASEP with some reversed bonds) for two sets of parameters: (a)\n$g=0.33, f=0.25$ (triangles) and (b) $r=0.54, f=0.3$ (circles). Each point\nrepresents an average over $40$ realizations of disorder in (a) and $100$\nrealizations in (b). The straight lines have slopes of $-\\theta\/2$ with\n$\\theta=0.5$ and $1.0$ respectively as predicted by\n({\\protect\\ref{eq:s2}}).}\n\\label{fig:JL_sinai}\n\\end{figure}\n\n     As with the milder sort of disorder discussed in Section III, the\n     state is strongly inhomogeneous and shows macroscopic regions of high\n     and low density. Figure \\ref{fig:regimes1}c shows the time-averaged\n     density profile for a typical configuration of bonds. There is a large\n     shock around the rate-limiting backbend, which separates the two\n     regions.\n\n     For fixed lattice size $L$ with an associated longest backbend\n     $l^*(L)$, the current is a nonmonotonic function of $g$. This can\n     be seen as follows. If $g$ is low enough that\n     $\\Lambda(g)>>l^*(L)$, linear response theory would imply that\n     current $J$ grows linearly with $g$. On the other hand, if $g$ is\n     large enough that $\\Lambda(g)<<l^*(L)$, the current falls with\n     increasing $g$ according to (\\ref{eq:s1}). In between, $J$\n     achieves a local maximum when $\\Lambda(g=g_{max})\\simeq l^*(L)$,\n     which implies $g_{max}\\sim 1\/\\ln L$.\n\n     The argument given above implies that the current carried by a system\n     of hard-core particles through the randomly backbending lattice\n     vanishes in the thermodynamic limit, no matter how small the bias\n     $g$. This is in contrast to the behaviour of {\\it noninteracting}\n     particles in the same random environment, where the drift velocity\n     vanishes only if the bias is strong enough \\cite{BD,BR}. The\n     difference can be traced to the possibility, in the noninteracting\n     case, of compensation by a large build-up of density at the bottom of\n     a backbend, which then succeeds in driving a finite current over the\n     backbend.  This option does not exist once repulsive hard-core\n     interactions come into play, and the current vanishes in the\n     thermodynamic limit.\n\n\n\\section{Continuum description}\n\\label{sec:continuum}\n\n     It is interesting to ask whether the behaviours found above in the\n     disordered lattice gases can be reproduced using a continuum\n     description of the problem. Though we have not pursued this question\n     to its logical end, we discuss in this section some general\n     constraints that a continuum description should satisfy.\n\n      The steady-state of the disordered system of interacting particles is\n     described by a spatially varying time-averaged density profile\n     $\\rho_0({r}) \\equiv \\langle n({r}) \\rangle$.  The time evolutions of\n     fluctuations around the mean density profile are governed by the\n     continuity equation\n\\begin{equation}\n{\\partial\\over\\partial t} \\rho({r},t) + \\nabla\\cdot{\\vec J}(\\rho,{r},t) = 0.\n\\label{eq:continuity}\n\\end{equation}\n     The coarse-grained current ${\\vec J}$ may be phenomenologically\n    \ndivided into three parts\n\\begin{equation}\n{\\vec J}({r},t)=\\vec{J}_{sys}(\\rho({r},t),{r})+ {\\vec J}_{diff}\n +\\vec{\\epsilon}({r},t).\n\\label{eq:J=j+j+j}\n\\end{equation}\n     A local hydrodynamic assumption has been made in writing the {\\it\n     systematic} part of the current, $\\vec{J}_{sys}$ as a function of the\n     local density, $\\rho({r},t)$. The explicit ${r}$ dependence of\n     $\\vec{J}_{sys}$ comes from the breaking of translational invariance by\n     quenched disorder, and its exact form of $\\vec{J}_{sys}$ depends on\n     the microscopic dynamics of the model. The {\\it diffusive} current,\n     ${\\vec J}_{diff}=D({r})\\cdot\\nabla [\\rho({r},t)-\\rho_0({r})]$, involves a\n     quenched diffusion tensor, $D({r})$. The {\\it noise},\n     $\\vec{\\epsilon}({r},t)$, is to mimic, on a mesoscopic scale, the\n     stochastic nature of the evolution.  It is usually assumed to be\n     Gaussian distributed and $\\delta$-correlated in space and time with\n     vanishing spatial and temporal averages.\n\n     To obtain the time evolution of the density fluctuations\n     $\\tilde\\rho\\equiv\\rho({r},t)-\\rho_0({r})$, we expand $\\vec{J}_{sys}$ in powers\n     of $\\tilde\\rho$ as\n     $\\vec{J}_{sys}(\\rho_0({r}),{r})+\\vec{c}({r})\\tilde\\rho+\\vec{\\lambda}({r})\\tilde\\rho^2\\cdots$\n     and put it in (\\ref{eq:continuity}). This results in\n\\begin{equation}\n\\partial_t\\tilde\\rho =\n\\nabla\\cdot[D({r})\\cdot\\nabla\\tilde\\rho-\\vec{c}({r})\\tilde\\rho-\\vec{\\lambda}({r})\\tilde\\rho^2\n\\cdots-\\vec{\\epsilon}({r},t)]\n\\label{eq:dynamics-ddim}\n\\end{equation}\n\n\n     In one spatial dimension, the above reduces to (with ${r}$ replaced\n     with $x$) the form\n\\begin{equation}\n\\partial_t \\tilde\\rho=\\partial_x[D(x)\\partial_x \\tilde\\rho -c(x)\\tilde\\rho-\\lambda(x)\\tilde\\rho^2\\cdots-\\eta(x,t)]\n\\label{eq:dynamics-1dim}\n\\end{equation}\n     which was considered in \\cite{prl97}.  In steady state, the\n     time-averaged current must be independent of $x$.  As both $J_{diff}$\n     and $\\epsilon(x,t)$ vanish under time averaging, the constraint to be\n     satisfied is\n\\begin{equation}\n\\partial_x\\langle J_{sys}(x)\\rangle=0\n\\label{eq:constraint}\n\\end{equation}\n      which is important to account for, as the coefficients in $J_{sys}$\n      are explicitly space-dependent, {\\it i.e.},\n      $\\partial_x[\\lambda(x)\\langle \\tilde\\rho^2 \\rangle +...]$ must vanish.\n\n\n      In their attempt to study a continuum model which describes the\n      DASEP, Becker and Janssen (BJ) \\cite{Janssen} write the current $J'$\n      in powers of $\\phi(x,t) \\equiv \\rho(x,t)-\\rho$, the density\n      fluctuation away from the {\\it overall} particle density $\\rho$ in\n      the system. In one dimension, the form quoted \\cite{DDS} for the\n      DASEP is\n\\begin {equation}\nJ'=(1-2\\rho)\\phi(x,t)-\\phi^2(x,t)+\\eta(x)\n\\label{eq:BJ}\n\\end{equation}\n      where $\\eta(x)$ is an additive quenched noise term. Since the\n      time-averaged density $\\rho_0(x)$ varies in space, it is evident that\n      $\\phi$ has a nonzero expectation value $\\langle\\phi(x,t)\\rangle =\n      \\rho_0(x)-\\rho$. As we have seen in Section III, $\\rho_0(x)$, and\n      thus $\\langle \\phi(x,t) \\rangle$, can show strong variations,\n      especially in the density-segregated regime. Spatial constancy of the\n      average current demands that the time average of the right hand side\n      of (\\ref{eq:BJ}) must satisfy\n\\begin{equation}\n\\partial_x[(1-2\\rho)\\langle\\phi(x,t)\\rangle-\\langle\\phi^2(x,t)\\rangle+\\eta(x)]\n=0.\n\\label{eq:BJconstr}\n\\end{equation}\n      Even when this condition is satisfied, it is not entirely clear that\n      this continuum theory actually represents the DASEP in one dimension.\n      BJ argue that if $\\partial J\/\\partial \\rho \\ne 0$, then disorder is\n      irrelevant on the large scale, a conclusion that is supported by\n      \\cite{prl97,unpubl}. However, their conclusion, that the condition\n      $\\partial J\/ \\partial \\rho =0$ holds only if $\\rho={1 \\over 2}$, does\n      not seem to be correct, at least for the DASEP in one dimension; we\n      have seen in Section IV, there is an entire range of densities $({1\n      \\over 2}-\\Delta \\le \\rho \\le {1 \\over 2}+\\Delta)$ where $\\partial\n      J\/\\partial \\rho$ vanishes, associated with segregation of\n      density on a macroscopic scale.\n\n\n \\section{Equivalent interface models in one dimension}\n  \\label{sec:height}\n      \n      In 1-$d$, both the DASEP (with or without backbends) and the GDDP --\n      for which the maximum occupancy per site is 1 -- are equivalent to\n      stochastic growth models of a 1-$d$ interface moving in a 2-$d$\n      medium.  Corresponding to each particle-hole configuration $\\{n_i\\}$\n      is assigned an interface profile $\\{H_i\\}$ through $H_i=\\sum_{j\\le i}\n      (1-2 n_i)$ \\cite{solids}. Pictorially this means that each particle\n      corresponds to a $-45^\\circ$ downward line segment, while a hole\n      corresponds to an upward one (Fig.~\\ref{fig:map}). Thus, clusters of\n      particles and holes translate to $\\pm 45^\\circ$ slope segments, and\n      the interface has a mean slope which vanishes when the particle\n      density is $1\/2$.  Away from half-filling, periodic boundary\n      condition for the lattice becomes a helical boundary condition for\n      the interface.  Junctions between adjacent particle and hole clusters\n      correspond to corners in the interface profile.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=16.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{ Mapping between driven particle systems in $d=1$  and \ngrowing interfaces. (a) Toom Interface dynamics corresponding to the\nGDPP, (b) Single-step model corresponding to the DASEP. }\n\\label{fig:map}\n\\end{figure}\n      \n      Evolution of the interface is dictated by the dynamics of the\n      corresponding particle system. The GDDP corresponds to the slice-wise\n      motion of segments of a Toom interface in the low noise limit\n      \\cite{Toom}, while the DASEP corresponds to the corner-flip `single\n      step' growth model \\cite{solids} (Fig.~\\ref{fig:map}).  In both cases\n      particle movement to the right (or hole move to left in GDDP)\n      corresponds to local forward growth (deposition) of the interface\n      while a leftward particle move (e.g. in DASEP with backbends)\n      corresponds to local backward growth (evaporation). The quenched jump\n      rates for the particle moves implies {\\it columnar} disordered\n      growth-rates for the interface: the normal growth rate of the\n      interface at a fixed $i$ is the same irrespective of the location of\n      the interface at successive times.  In the long time limit, the mean\n      local forward speed of the interface is the same at all points\n      along the interface -- being proportional to the spatially constant\n      steady-state current of the corresponding particle model.\n\n      We now discuss the various qualitatively different regimes which\n      arise in the interface growth models. Figure \\ref{fig:interface}\n      shows time averaged steady-state interface profiles $\\langle\n      H_i\\rangle$ corresponding to the three regimes of driven particle\n      systems illustrated in Fig.~\\ref{fig:regimes1} in Section\n      \\ref{sec:introduction}.  In all the three cases we started from an\n      initially uniform profile -- corresponding to a random\n      distribution of particles on the lattice.\n      \n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=17.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Interface morphology in the three regimes: (a) Uniformly moving\ninterface with a uniform slope, (b) interface with large\nsegments of different slopes moving with a non-vanishing speed, and (c)\ninterface with large segments of different slopes moving with a speed which\nvanishes in the thermodynamic limit. The insets show blow-ups of regions\nenclosed in the dashed boxes.}\n\\label{fig:interface}\n\\end{figure}\n\n      Figure \\ref{fig:interface}a is an interface with a non-zero mean\n      tilt, which has net forward growth rate at all points. The interface\n      has a uniform slope on a macroscopic scale and moves with a finite\n      non-zero speed preserving its mean tilt. This case corresponds to the\n      {\\it Homogeneous} regime depicted in Fig.~\\ref{fig:regimes1}a. On a\n      microscopic scale the interface has frozen-in roughness \n      (Fig.~\\ref{fig:interface}a, inset) corresponding to the microscopic\n      shocks in the steady-state density profile of\n      Fig.~\\ref{fig:regimes1}a.\n\n\n      If the mean tilt vanishes, but the interface still has a net forward\n      growth rate at all points, then the initial uniform profile at $t=0$\n      coarsens into large segments of different mean slopes at long times\n      (Fig.~\\ref{fig:interface}b). These segments have frozen roughness on\n      microscopic scales, similar to the non-zero tilt case\n      (Fig.~\\ref{fig:interface}b, inset). The interface moves with a finite\n      speed preserving its mean shape and mean vanishing tilt. This\n      corresponds to the {\\it Segregated-Density } regime of\n      Fig.~\\ref{fig:regimes1}b.\n\n      In addition to the frozen roughness on the microscopic scales, we can\n      define the kinetic roughness as the equal-time mean-square height\n      fluctuations around the steady-state profile. We consider the\n      zero-mean height variables $h_i(t)\\equiv H_i(t)-\\langle H_i\\rangle$\n      defined in (\\ref{eq:heightdef}), and define the roughness exponent\n      $\\alpha$ through ${\\overline{\\langle(h_{i+r}-h_i)^2\\rangle}}\\sim\n      r^{2\\alpha}$.  As discussed in Subsections \\ref{sec:DDPP}D,E and\n      \\ref{sec:DASEP}F, the roughness exponent $\\alpha=1\/2$, in both the\n      cases above.\n      \n\n       In Fig.~\\ref{fig:interface}c, which corresponds to the {\\it\n      Vanishing-Current} regime of Fig.~\\ref{fig:regimes1}c, the profile\n      resembles that corresponding to the Segregated-Density regime in that\n      it has large segments of different mean slopes. However, in this case\n      the interface is stationary in the thermodynamic limit -- reflecting\n      the vanishing of the steady-state current in the particle system.  In\n      interface language, this situation can be visualized as a case of\n      interface growth where there are local stretches of the interface\n      having a net backward growth (evaporation) rate. Though, on the\n      average, there are more forward-growth regions than backward-growth\n      ones, in the limit of large system size, arbitrarily long evaporation\n      stretches effectively pin the interface.\n            \n      Turning now to continuum description of the dynamics of these\n      interfaces, at least in the cases where the mean speed of growth is\n      non-zero, the sum in the definition of the height variable is\n      replaced by an integral of the coarse-grained particle density,\n      $H(x,t)=\\int^x[1-2\\rho(x',t)] dx'$. The growth equation for $H(x,t)$\n      is governed by (spatially) integrated one dimensional version of the\n      continuity equation (\\ref{eq:continuity}): $\\partial_t\n      H(x,t)=J(\\partial_x H,x,t)$. The fluctuations $h(x,t)\\equiv\n      H(x,t)-H_0(x)$in $H$ around the steady-state profile $H_0(x)\\equiv\n      \\langle H(x,t)\\rangle$, are governed by\n\\begin{equation}\n\\partial_t h= D(x)\\partial_{xx} h - c(x)\\partial_x h + {1\\over 2}\\lambda(x)\\partial_x h^2\\cdots + 2\\eta(x,t)\n\\label{eq:height}\n\\end{equation}\nobtained by integrating (\\ref{eq:dynamics-1dim}). The absence of any additive\nquenched spatial noise term in (\\ref{eq:height}) is due to the spatial\nconstancy of the growth speed of the interface dictated by the same\nconstraint on the steady-state current. \nIn this respect (\\ref{eq:height}) differs from the model discussed\nin \\cite{krug} where such a term arises naturally due to absence of any\nsuch constraints. As can be readily verified by power counting, an additive\nquenched columnar term is highly relevant in the RG sense and leads to much\nrougher interfaces than the $\\alpha=1\/2$ interfaces described by\n(\\ref{eq:height}).\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\n      In this paper, we have studied the stationary current-carrying states\n      of driven lattice gas models with quenched disordered hopping rates.\n      The principal results are of two types --- first the exact\n      determination of the steady states for a class of disordered models,\n      and second the demarcation of distinct regimes of behaviour on\n      macroscopic length scales, as a result of disorder.  In this section,\n      we briefly review these results, and discuss some related open\n      problems.\n \n      The steady states of a family of disordered models --- the disordered\n      drop-push process (DDPP), and related models --- have been found in\n      all dimensions by an application of the condition of pairwise\n      balance.  The result is a product measure state, with site-dependent\n      weights, reflecting the microscopic disorder in the model.  The\n      current has been computed as well.  The system is characterized\n      by a strictly uniform current density, and a coarse-gained particle\n      density that is approximately uniform.  On a macroscopic scale, the\n      state is homogeneous.\n\n      Disorder can lead to macroscopically non-homogeneous states, as\n      in the 1-$d$ disordered asymmetric simple exclusion process\n      (DASEP).  Our numerical and mean-field results show that a\n      macroscopically density-segregated state occurs in the DASEP\n      model with no backbends, for densities in a finite range around\n      half-filling.  The origin of density separation is traceable to\n      the existence of a largest current that can be carried by a\n      stretch of weak bonds.  This low current can be sustained in the\n      rest of the lattice only by separating the density into distinct\n      large and small values in macroscopic regions of the lattice.\n\n      Backbends introduce a third type of possible behaviour in one\n      dimension.  Like the stretch of weak forward bonds in the\n      density-segregated regime, a backbend rate-limits the current,\n      leading to density segregation.  However, there is an important\n      difference: the longer the stretch of weak forward bonds, the closer\n      is the current to a finite asymptotic value; by contrast, the longer\n      the stretch of reverse-biased bonds, the smaller the current --- it\n      decreases exponentially fast with backbend length.  Since the\n      probability of occurrence of a backbend decreases exponentially with\n      its length, the result is a current that decreases as a\n      bias-dependent power of the overall size of the lattice.\n\n\n      The crucial physical point which underlies the behaviour in each of\n      these regimes is the requirement that the steady-state current be\n      constant at all points in the system. Continuum field-theoretical\n      approaches must ensure that this local constraint is respected; while\n      this is automatically assured for translationally invariant systems,\n      it may need special care to guarantee in disordered systems.\n\n      It would be of interest to generalize our results to higher\n      dimensions and also to include interactions between particles at\n      different sites. A few scattered results along these lines are\n      available. \\hfill \\break \\noindent (i) For the drop-push class\n      of problems, we have seen in Section \\ref{sec:DDPP} that the\n      exact steady state even in higher dimensions is characterized by\n      inhomogeneous product measure. On large scales, this leads to a\n      homogeneous state. \\hfill \\break \\noindent (ii) The transport of\n      particles with hard-core interactions through the infinite\n      cluster of a randomly diluted lattice above the percolation\n      concentration has been studied \\cite{RB,BR}.  In a certain\n      regime of dilution, backbends act as local traps, but unlike the\n      one-dimensional case considered in Section \\ref{sec:Sinai},\n      there exist infinitely long paths on which the length of every\n      backbend is less than a fixed value \\cite{BR2,RB,BR}. The\n      sub-network of such paths is expected to carry a current which\n      then remains finite in the thermodynamic limit. There is thus no\n      vanishing-current regime in this system. \\hfill \\break \\noindent\n      (iii) With attractive interactions between particles, the driven\n      lattice gas system with nearest-neighbour hopping is known to\n      undergo phase separation below a certain temperature\n      \\cite{Katz,DDS}. A numerical simulation showed that the addition\n      of a low concentration of blocked sites did not alter the\n      critical behaviour of this system \\cite{Lauritsen}.\n\n      More systematic studies of higher-dimensional systems are called for.\n      In particular, it would be interesting to know whether\n      disorder-induced large-scale inhomogeneities, akin to the phase\n      separation found in one dimension, persist in higher dimensions as\n      well.\n\n      Acknowledgements: We thank R. E. Amritkar, D. Dhar, S.\nKrishnamurthy, R. Lahiri, G. I. Menon and N. Trivedi for useful\ndiscussions.\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSXP 1062 is a Be\/X-ray binary (BeXRB) system discovered in the eastern wing of the Small Magellanic Cloud (SMC) \\citep{henault2012}. The detection of strong X-ray pulsations with a period of 1062 s revealed that the compact object in this system, is a slowly rotating pulsar. Furthermore, SXP 1062 is associated with a supernova remnant (SNR) and its kinematic age is calculated to be as young as 10--40 kyr \\citep{henault2012,haberl2012}. The theoretical contradiction between its long period and young age puts SXP 1062 into the center of a remarkable attention.\n\nThe spectral type of the optical counterpart 2dFS 3831 \\citep{evans2004} is specified to be B0-0.5(III)e+ \\citep{henault2012}. Moreover, spectroscopic observations imply that the large circumstellar disc around the counterpart has been growing in size or density \\citep{sturm2013}. I-band photometry carried on by \\textit{Optical Gravitational Lensing Experiment} \\citep[OGLE;][]{udalskietal2008} revealed periodic magnitude variations which signify that the orbital period of the binary system is likely to be $\\sim \\! 656$ d \\citep{schmidtke2012}.\n\nThe X-ray spectrum of SXP 1062 is basically described by an absorbed powerlaw. Additionally, thermal components are used to describe a possible soft excess below 1 keV \\citep{henault2012,sturm2013}. A $3\\sigma$ evidence for \\textit{Fe} K$\\alpha$ emission line at 6.4 keV is also reported by \\cite{sturm2013}. Furthermore, a search for a proton cyclotron absorption line on the 0.2--10 keV continuum yields no significant evidence \\citep{sturm2013}.\n\nFirst four pulse period measurements of SXP 1062 in 2010, demonstrate a very high spin-down rate of $\\dot{\\nu}=-\\,2.6 \\times10^{-12}$ Hz s$^{-1}$ during a short observation interval of 18 days \\citep{haberl2012}. The following period is measured $\\sim \\! 2.5$ years later, which implies a 40 factor lower long term spin-down rate \\citep{sturm2013}. In view of the fact that most BeXRBs show spin-up during their outbursts \\citep{bildsten1997}, SXP 1062 is atypical with its strong spin-down.\n\nThe extraordinary observational properties of SXP 1062; such as its long pulse period, strong spin-down and young age; have led several authors to implement different theoretical models in order to explain the true nature of the source \\citep{oskinova2013}. First, \\cite{haberl2012} suggested that the initial pulse period of the neutron star at birth might be unusually long and calculated a lower limit of 0.5 s. Another explanation given by \\cite{popov2012} indicated that the initial magnetic field of the neutron star at birth might be as large as $10^{14}$ G which then experienced a field decay. On the other hand, \\cite{ikhsanov2012} proposed that the initial magnetic field could be $\\sim \\! 4 \\times 10^{13}$ G once the magnetization of the accretion flow is introduced within the scope of the magnetic accretion scenario. \\cite{fu2012} pointed out that SXP 1062 may currently have a magnetic field higher than $10^{14}$ G, although no cyclotron line is detected on the X-ray spectrum. The possibility of high magnetic field nominates SXP 1062 as an accreting magnetar however, \\cite{postnov2014} claim that quasi-spherical accretion theory \\citep{shakura2012,shakura2013} estimates a lower limit for the magnetic field which is consistent with the standard values for neutron stars.\n\nIn this paper, we present timing analysis of the accretion powered pulsar SXP 1062 with \\textit{Swift}, \\textit{XMM-Newton} and \\textit{Chandra} observations between 2012 Oct. 9 and 2014 Nov. 23. First, we find orbital parameters of the binary system. In addition, we resolve a glitch event on MJD 56834.5 which is the first observational evidence that accretion powered pulsars may glitch like isolated pulsars. Furthermore, the fractional size of the glitch ensures that we discover the largest glitch reported upto now. In Section 2, we describe the observations. Then, in Section 3, we explain the timing analysis and its results. Finally in Section 4, we discuss that the observed glitch should be associated with the internal structure of the neutron star.\n\n\\section{Observations}\n\n\\begin{figure*}\n  \\center{\\includegraphics[width=9cm,angle=270]{burst2.eps}} \n  \\caption{\\textit{Upper panel:} I-band optical light curve of the counterpart provided by OGLE. The dashed lines indicate the times of optical outbursts calculated according to $\\sim\\!656$ d orbital period \\citep{schmidtke2012}. \\textit{Middle panel:} X-ray light curve of SXP 1062 from \\textit{Swift}--XRT observations. Each point represents an observation. The dashed line indicates the time of the X-ray outburst, that is MJD 56809.5. \\textit{Lower panel:} X-ray luminosity evolution in 0.3--10 keV energy band (calculated by using the distance to SMC: 60 kpc \\citep{hilditch2005}). The maximum luminosity, $1.3 \\times 10^{37}$ erg s$^{-1}$, is indicated with a dashed line.}\n  \\label{burst}\n\\end{figure*}\n\n\\textit{Swift} monitoring campaign of SXP 1062 begins during an X-ray outburst on 2012 Oct. 9 and continues until 2014 Nov. 23. Throughout a time about 2 years, 87 pointing observations have a total \\textit{Swift}--XRT \\citep[X-Ray Telescope;][]{burrows2005} exposure of $\\sim \\! 164$ ks. XRT operates in 0.2--10 keV energy range, with an effective area of 110 cm$^{2}$ at 1.5 keV. Its spatial resolution is 18 arcsec and its field of view (FOV) is 23.6 arcmin $\\times$ 23.6 arcmin. This focusing instrument automatically switches between four operation modes depending on the source count rate. The operation mode of SXP 1062 observations is photon counting (PC) mode, which has a timing resolution of $\\sim \\! 2.5$ s. Clean event files are produced with the script \\verb\"XRTPIPELINE v.0.13.2\" by using a default screening criteria. Light curves and spectra are extracted with \\verb\"XSELECT v.2.4d\". Circular regions selected for source and background extraction have a radius of 35 and 141 arcsec, respectively. \n\nIn order to investigate the evolution of X-ray luminosity, consecutive 8--10 observations are combined and spectral files are extracted for each group of observations. Spectra are re-binned to have at least 5 counts per bin and Cash statistic \\citep{cash1979} is preferred during spectral fitting with \\verb\"XSPEC v.12.9.0\". We model the spectra with an absorbed powerlaw and measure the X-ray flux in 0.3--10 keV energy band. Then, the flux values are converted to luminosity by using the distance to SMC, that is 60 kpc \\citep{hilditch2005}. X-ray luminosity evolution is plotted on the lower panel of Figure \\ref{burst}. \n\nOGLE \\citep{udalskietal2008} monitoring of SXP 1062 provides I-band photometry measurements of the optical counterpart, which are received from the X-Ray variables OGLE Monitoring System \\citep[XROM\\footnote{http:\/\/ogle.astrouw.edu.pl\/ogle4\/xrom\/xrom.html};][]{udalski2008}. Periodic optical outbursts are evident on the OGLE light curve (see the upper panel of Fig. \\ref{burst}). This periodicity of $\\sim \\! 656$ d is associated with the binary orbit of the system \\citep{schmidtke2012}.\n\nSXP 1062 is also observed with \\textit{XMM-Newton} and \\textit{Chandra} observatories. The \\textit{XMM-Newton} observation has a duration of $\\sim \\! 86$ ks on 2013 Oct. 11. We use single and double-pixel events (PATTERN 0--4) of EPIC-PN camera \\citep{struder2001} which is sensitive to the photons in 0.15--15 keV energy range and possesses a FOV of 30 arcmin with a spatial resolution of 6 arcsec. During this observation, EPIC-PN camera operated on full frame mode which has a time resolution of 73 ms. The data reduction is carried out with \\verb\"SAS v.15.0.0\" software. Filtering of high energy background flare times yields a net exposure of $\\sim \\! 47$ ks. We also avoid bad pixels by rejecting events with \\verb\"FLAG\" $\\neq 0$. In addition, 3 pointing observations taken with \\textit{Chandra}-ACIS detectors \\citep{garmire2003} are between 2014 June 29 and July 18, with a total exposure of $\\sim \\! 87$ ks. ACIS detectors have 17 arcmin $\\times$ 17 arcmin FOV and operates in 0.2--10 keV energy band. All observations are conducted in imaging mode with a time resolution of $\\sim \\! 0.5$ s. The data is analysed with \\verb\"CIAO v.4.9\" software. Default screening criteria are applied while producing the clean events. We extract 0.2--12 keV \\textit{XMM-Newton} and 0.2--10 keV \\textit{Chandra} lightcurves with 1 s bin time and all time series are converted to Solar System barycentre. Circular source extraction regions have radii of 25 and 8 arcsec for \\textit{XMM-Newton} and \\textit{Chandra}, respectively. Background emissions are estimated from source-free circular regions on the same detector chip as the source. \n\n\\section{Timing Analysis}\n\n\\begin{figure*}\n  \\center{\\includegraphics[width=9.6cm,angle=270]{glitch.eps}} \n  \\caption{\\textit{Upper panel:} Cycle counts of pulse phase obtained by expanding the TOAs. The time denoted with an arrow mark corresponds to the X-ray outburst of SXP 1062, that is MJD 56809.5. \\textit{Middle panel:} Phase offset series after the removal of the secular spin-down trend and the orbital model given in Table \\ref{soln}. The glitch event is evident on MJD 56834.5 and it is 25 days after the X-ray outburst. \\textit{Lower panel:} Residuals after an additional removal of the glitch event. \\textbf{The final model has a reduced $\\chi^2$ of 1.0.}}\n  \\label{glitch}\n\\end{figure*}\n\n\\begin{table}\n\\caption{Binary orbit, timing solution and glitch parameters of SXP 1062. A number given in parentheses is the $1\\sigma$ uncertainty in the least significant digit of a stated value.}\n\\label{soln}\n \\center{\\renewcommand{\\arraystretch}{1.2}\\begin{tabular}{lc}\n  \\hline \n  \\textbf{Circular Orbital Model:} & \\\\\n  Orbital Epoch (MJD) & 56351(10) \\\\\n $P_{\\mathrm{orb}}$ (days) & 656(2) \\\\\n $\\frac{a_{\\mathrm{x}}}{c} \\sin i $ (lt-s) & 1636(16) \\\\ \n \\hline\n \\textbf{Timing Solution:} & \\\\\n  Folding Epoch (MJD) & 56576.0 \\\\\n  Validity Range (MJD) & 56209 -- 56830 \\\\\n $\\nu_{\\mathrm{o}}$ (mHz) & 0.931787(5) \\\\\n $\\dot{\\nu}_{\\mathrm{o}}$ (Hz s$^{-1}$) & $-\\,4.29(7) \\times 10^{-14}$ \\\\\n \\hline\n \\textbf{Glitch Parameters:} & \\\\\n $t_\\mathrm{g}$ (MJD) & $\\sim \\! 56834.5$ \\\\\n Validity Range (MJD) & 56834 -- 56978 \\\\\n $\\Delta\\nu$ (Hz) & $1.28(5) \\times 10^{-6}$ \\\\\n $\\Delta \\dot{\\nu}$ (Hz s$^{-1}$) & $- \\, 1.5(9) \\times 10^{-14}$ \\\\\n $\\Delta\\nu \/ \\nu_{\\mathrm{o}}$ & $1.37(6) \\times 10^{-3}$ \\\\\n $\\Delta \\dot{\\nu} \/ \\dot{\\nu}_{\\mathrm{o}}$ & 0.3(2) \\\\\n  \\hline\n \\end{tabular}}\n\\end{table}\n\nWe extract barycentric light curves from \\textit{Swift}, \\textit{Chandra} and \\textit{XMM-Newton} observations described in the previous section. First, we search for periodicity on the light curve of the \\textit{XMM-Newton} observation on MJD 56576.7 by folding it on trial periods \\citep{leahy1983}. The period that give the maximum $\\chi^2$ value is 1073.5 s. Then, all observations are folded with the same ephemeris and frequency. For each observation, pulse profiles with 10 phase bins are constructed. Observations which have an exposure less than the pulse period of SXP 1062 yield pulse profiles with zero count bins, therefore they are excluded during the timing analysis. The pulse profiles are described with harmonic representation as \\citep{deeter1985}\n\n\\begin{equation}\n f(\\phi) = F_{\\mathrm{o}} + \\sum_{k=1}^{n} F_{k} \\, \\cos k(\\phi-\\phi_{k}) \\, ,\n\\end{equation}\n\n\\noindent where $n$ is the number of harmonics used for the representation of a pulse. Similarly, the template pulse profile is obtained from the longest observation ($\\sim\\!47$ ks) which is held on MJD 56576.7 with \\textit{XMM-Newton}. The template pulse profile is represented as\n\n\\begin{equation}\n g(\\phi) = G_{\\mathrm{o}} + \\sum_{k=1}^{n} G_{k} \\, \\cos k(\\phi-\\phi_{k}) \\, .\n\\end{equation}\n\nThe time of arrival (TOA) of each pulse is estimated by searching the location of the maximum in the cross-correlation with template pulse \\citep{deeter1985}\\citep[see also][for applications]{icdem2012},\n\n\\begin{equation}\n \\Delta\\Phi = \\dfrac{\\sum\\limits_{k=1}^{n} k \\, G_{k} \\, F_{k} \\, \\sin k \\Delta\\phi_{k}}{\\sum\\limits_{k=1}^{n}k^{2} \\,\nG_{k} \\, F_{k} \\, \\cos k \\Delta\\phi_{k}} \\,.\n\\end{equation}\n\n\\noindent To check how pulse timing is affected by the changes in pulse profiles, we measure TOAs by using different number of harmonics. The obtained TOAs are consistent within $1\\sigma$ level for all harmonics therefore, we decide to perform pulse timing analysis by using five harmonics. In order to examine the timing behaviour of SXP 1062, we first focus on pre-outburst TOAs which are described as\n\n\\begin{equation}\n \\Phi (t) = \\Phi_{\\mathrm{o}} + \\nu_{\\mathrm{o}} \\, (t-t_{\\mathrm{o}}) + \\frac{1}{2} \\, {\\dot\\nu_{\\mathrm{o}}} \\, (t-t_{\\mathrm{o}})^{2} \\, ,\n\\end{equation}\n\n\\noindent where $t_{\\mathrm{o}}$ is the folding epoch, $\\nu_{\\mathrm{o}}$ is the spin frequency and ${\\dot\\nu_{\\mathrm{o}}}$ is the derivative of the spin frequency, respectively. We expand TOAs around the spin-down rate previously reported by \\cite{sturm2013} and obtain a phase coherent timing solution (see the upper panel of Fig. \\ref{glitch}). We find that the source is spinning down with a rate of ${\\dot\\nu_{\\mathrm{o}}}= -\\,4.29(7) \\times 10^{-14}$ Hz s$^{-1}$ between MJD 56209 and MJD 56830 (see Table \\ref{soln}).\n\n\\begin{figure}\n  \\center{\\includegraphics[width=5.1cm,angle=270]{orbit_res.eps}} \n  \\caption{\\textit{Upper panel:} Doppler delays on the residuals after the removal of the spin-down from TOAs prior to the glitch, are fitted with a circular orbital model (solid line) \\textbf{with a reduced $\\chi^2$ of 1.0}. The orbital parameters are given in Table \\ref{soln}. \\textit{Lower panel:} Residuals after the removal of orbital model.}\n  \\label{orbit}\n\\end{figure}\n\nThe residuals after the removal of the spin-down trend are given in Figure \\ref{orbit}. The fluctuation on the residuals is consistent with Doppler delays due to orbital motion. In general for an eccentric orbit, it can be represented as \\citep{deeter1981}\\citep[see also][for applications]{zand2001,baykal2000,baykal2010}\n\n\\begin{equation}\n\\begin{split}\n \\delta t_{\\mathrm{orbit}} = \\, & x \\, \\sin (l) - \\frac{3}{2} \\, x \\, e \\sin (w) \\\\\n &+ \\frac{1}{2} \\, x \\, e \\cos (w) \\sin (2l) - \\frac{1}{2} \\, x \\, e \\sin (w) \\cos (2l) \\, ,\n \\end{split}\n \\label{eqn:orbit}\n \\end{equation}\n\n\\noindent where $x \\!=\\! \\frac{a_{\\mathrm{x}}}{c}\\sin i$ is the light travel time for projected semi-major axis, $i$ is the inclination angle between the line of sight and the orbital angular momentum vector, $l \\!=\\! 2 \\pi \\, (t - T_{\\mathrm{\\frac{\\pi}{2}}}) \\, \/ \\, P_{\\mathrm{orb}} + \\frac{\\pi}{2}$ is the mean orbital longitude at $t$, $T_{\\mathrm{\\frac{\\pi}{2}}}$ is the epoch when the mean orbital longitude is equal to $90\\degree$, $P_{\\mathrm{orb}}$ is the orbital period, $w$ is the longitude of periastron and $e$ is the eccentricity. Since the time span of fitted TOAs covers less than one orbital cycle, we fix the orbital period to $P_{\\mathrm{orb}} \\!=\\! 656(2)$ d as reported by \\cite{schmidtke2012} and seek for other orbital parameters. \n\n\\begin{figure}\n  \\center{\\includegraphics[width=5.8cm,angle=270]{inclination.eps}} \n  \\caption{The relation between the binary system inclination angle and the donor mass in units of M$_{\\odot}$. The solid curve is constructed by using the mass function value of 10.9M$_{\\odot}$, whereas the dashed curves are constructed for 1$\\sigma$ error in the mass function ($\\pm 0.3$M$_{\\odot}$). For a donor mass of $ \\sim \\! 15$M$_{\\odot}$ \\citep{henault2012}, we find that the inclination angle would be $\\simeq\\! 73(2) \\degree$.}\n  \\label{inclination}\n\\end{figure}\n\nThe best fit with a reduced $\\chi^2$ of 1.0 (see Fig. \\ref{orbit} and Table \\ref{soln}) yields a circular orbit with orbital parameters $\\frac{a_{\\mathrm{x}}}{c} \\sin i \\!=\\! 1636(16)$ lt-s and $T_{\\mathrm{\\frac{\\pi}{2}}} \\!=\\! 56351(10)$ MJD. \nFurthermore, a search for eccentricity leaving all orbital fit parameters free including the orbital period, gives an estimate for the upper limit to the eccentricity as 0.2 at 1.6$\\sigma$ that is 90 per cent confidence level \\citep{lampton1976}. Using these results, the mass function $f(M)$ can be calculated via\n\n\\begin{equation}\n f(M) = \\frac{4\\pi^{2}}{G} \\, \\frac{(a_{\\mathrm{x}} \\sin i)^{3}}{P_{\\mathrm{orb}}^{2}} = \\frac{(M_{\\mathrm{c}} \\sin i)^{3}}{(M_{\\mathrm{x}} \\! + \\! M_{\\mathrm{c}})^{2}} \\, ,\n\\end{equation}\n\n\\noindent where $M_{\\mathrm{c}}$ is the mass of the counterpart and $M_{\\mathrm{x}}$ is the mass of the neutron star. The mass function calculated via orbital parameters is $f(M) \\!\\simeq\\! 10.9(3)$M$_{\\odot}$. Assuming a neutron star with a mass of 1.4M$_{\\odot}$, we plot the relation between the inclination angle and the donor mass in Figure \\ref{inclination}. Considering the typical evolutionary mass of the optical counterpart as $ \\sim \\! 15$M$_{\\odot}$ \\citep{henault2012}, the inclination angle would be $i \\!\\simeq\\! 73(2) \\degree$.\n\nIn the middle panel of Figure \\ref{glitch}, we show residuals after the removal of both the spin-down trend and the orbital model from all TOAs. The TOAs after MJD 56834.5 show a sudden change of slope which indicates a possible glitch event. We model TOAs after the glitch with a second order polynomial $ \\Delta\\phi = \\phi_{\\mathrm{o}} + \\delta\\nu \\, (t-t_{\\mathrm{g}}) +\\delta \\dot{\\nu} \\, (t-t_{\\mathrm{g}})^2$, where $t_{\\mathrm{g}}$ is the time of the glitch event. \nThus, pulse frequencies receive a correction of $\\nu_{\\mathrm{o}} +\\delta\\nu +\\delta\\dot{\\nu} \\, (t-t_{\\mathrm{g}})$. We find that the glitch event occurs on MJD 56834.5 and it causes a frequency shift of $\\Delta\\nu = 1.28(5) \\times 10^{-6} $ Hz along with a change of spin-down rate $\\Delta \\dot{\\nu} = - \\, 1.5(9) \\times 10^{-14}$ Hz s$^{-1}$ (see Table \\ref{soln}). In the lower panel of Figure \\ref{glitch}, we show residuals after the glitch correction.\n\n\\section{Discussion}\n\n\\subsection{The Orbit}\n\nThe accretion powered pulsar SXP 1062 is in a BeXRB system. In BeXRBs, the neutron star accretes matter from the stellar wind of a main sequence star, which is in the form of a circumstellar disk around the mass donor. Generally, the orbits of BeXRBs are relatively wide and moderately eccentric \\citep[$P_\\mathrm{orb} \\! \\geq \\! 20$ d, $ e \\! \\geq \\! 0.3$;][]{reig2011}. While the neutron star passes through the edge of circumstellar disk of the Be companion, it interacts with the material and accretion takes place. The X-ray emission of most BeXRBs is transient, however persistent BeXRBs with low luminosities ($L_\\mathrm{x} \\! \\sim \\! 10^{34-35}$ erg s$^{-1}$) also exist. Persistent systems are known to contain slower pulsars that have wider orbits \\citep[$P_\\mathrm{s} \\! > \\! 200$ s, $P_\\mathrm{orb} \\! > \\! 200$ d;][]{reig1999,reig2011}. Although most BeXRBs have eccentric orbits, there are several systems (i.e. X Per, GS 0834--430, KS 1947+300) with low eccentricity ($e \\! < \\! 0.2$). \n\nTransient BeXRBS show two types of X-ray outbursts \\citep{stella1986}. Type I outbursts ($L_\\mathrm{x} \\! \\sim \\! 10^{36-37}$ erg s$^{-1}$) occur once in a while the pulsar passes from the periastron of the orbit, where accretion enhances. On the other hand, Type II outbursts are major events ($L_\\mathrm{x} \\! \\geq \\! 10^{37}$ erg s$^{-1}$) that occur during mass ejection episodes of the counterpart. Transient BeXRBs may have very low quiescence luminosities ($L_\\mathrm{x} \\! \\leq \\! 10^{33}$ erg s$^{-1}$). The luminosity increase during a Type II outburst can be 3--4 order of magnitudes, while it is only about one order of magnitude for a Type I outburst \\citep{reig2011}. When an X-ray outburst occurs in a BeXRB, the neutron star generally enters a spin-up episode due to enhanced accretion. \n\nAs the population of BeXRBs has grown in the past decades, the pulse periods ($P_\\mathrm{s}$) of pulsars in BeXRBs are still strongly correlated with the orbital periods ($P_\\mathrm{orb}$), as it is firstly demonstrated by \\cite{corbet1984}. Although there is a large scatter in data, a positive correlation is evident with $P_\\mathrm{s} \\! \\propto \\! P_\\mathrm{orb}^2$. The large BeXRB population in SMC also obeys this relation \\citep{knigge2011,yang2017}. Therefore in BeXRBs, pulsars with longer pulse periods reside in binary systems with wider orbits and consequently lower accretion rates.\n\nThe optical light curve of SXP 1062 (see Fig. \\ref{burst}) shows periodic variations devoted to an orbital period of $\\sim \\! 656$ days \\citep{schmidtke2012}. X-ray observations of the source reveal the occurrence of a Type I outburst ($L_\\mathrm{x} \\! \\! \\simeq \\! 1.3 \\times 10^{37}$ erg s$^{-1}$) that happen together with the optical enhancement. However, the long term spin-down of SXP 1062 is not interrupted by the outburst. Prior to the X-ray outburst a minimum luminosity of $2.4 \\times 10^{35}$ erg s$^{-1}$ is measured, hence the luminosity increases by a factor of $\\sim \\! 50$ during the outburst. The outburst of SXP 1062 is observed only during an observation with an exposure of 2.2 ks, however the actual duration of the outburst might be longer since neighbouring observations are 14 days apart. The luminosity drops to $3.6 \\times 10^{36}$ erg s$^{-1}$ therefore, the outburst finishes in the following observation. These characteristics classify SXP 1062 as a persistent BeXRB.\n\nWe are able to resolve the orbital motion of SXP 1062 from its X-ray emission observed for $ \\sim \\! 2$ years (see Fig. \\ref{orbit}). We determine the orbital epoch as 56351(10) MJD and the light travel time for projected semi-major axis as 1636(16) lt-s by considering a circular orbit with a period of 656(2) days. (see Table \\ref{soln}). We also report an upper limit of 0.2 to the eccentricity at 90 per cent confidence level, therefore SXP 1062 is claimed to be in a low eccentric orbit despite the fact that denser observational coverage is needed for a better assessment. The orbital and pulse periods of the system, position the source on a place in line with BeXRBs on the Corbet diagram, the uppermost right end of the existing correlation for BeXRBs. Moreover, the mass function of the system is calculated to be $f(M) \\!\\simeq\\! 10.9(3)$M$_{\\odot}$, which seems appropriate bearing in mind that the Be companion is suggested to have a mass of $ \\sim \\! 15$M$_{\\odot}$ \\citep{henault2012}. Consequently, the orbital inclination can be evaluated as $i \\!\\simeq\\! 73(2) \\degree$. If we allow variation of the donor mass, the effect on the inclination angle is plotted in Figure \\ref{inclination}. Using this relation, the minimum donor mass is determined to be 13.3(3)M$_{\\odot}$ for $i \\!=\\! 90 \\degree$.\n\n\n\\subsection{Magnetic Field Estimation from Secular Spin-down Trend before the Glitch}\n\nIf we consider that the source is accreting via a prograde accretion disc that is formed before the glitch, standard accretion disc theory \\citep{pringle1972,lamb1973,ghosh1979,wang1987,ghosh1994,torkelsson1998} can be used to estimate the surface magnetic field of the neutron star. For this scenario, the inner radius of the accretion disc at which the magnetosphere disrupts the Keplerian rotation depends on the accretion rate ($\\dot{M}$) and the magnetic dipole moment of the neutron star ($\\mu \\!=\\! BR^3$ where $B$ is the surface magnetic field and $R$ is the radius of the neutron star) as\n\n\\begin{equation}\nr_\\mathrm{o} = K \\, \\mu^{4\/7} \\, (GM)^{-1\/7} \\, \\dot{M}^{-2\/7} \\, ,\n\\label{eqn:r0}\n\\end{equation}\n\n\\noindent where $K$ is a dimensionless parameter of about 0.91 and $M$ is the mass of the neutron star \\citep{pringle1972,lamb1973}. The torque can then be estimated as\n\n\\begin{equation}\n2\\pi \\, I \\, \\dot{\\nu }= n(\\omega_\\mathrm{s}) \\, \\dot{M} \\, l_\\mathrm{K} \\, ,\n\\label{eqn:torque}\n\\end{equation} \n\n\\noindent where $I$ is the moment of inertia of the neutron star, $\\dot{\\nu}$ is the spin rate of the neutron star, $n(\\omega_\\mathrm{s})$ is the dimensionless torque which is a factor parametrising the material torque and magnetic torque contributions to the total torque, and $l_\\mathrm{K }= (GMr_\\mathrm{o})^{1\/2}$ is the angular momentum per mass added by the Keplerian disc at $r_\\mathrm{o}$. The dimensionless torque can be approximated as\n\n\\begin{equation}\nn(\\omega_\\mathrm{s}) \\approx 1.4 \\, (1-\\omega_\\mathrm{s}\/\\omega_\\mathrm{c}) \\, \/ \\, (1-\\omega_\\mathrm{s}) \\, ,\n\\label{eqn:dimtorque}\n\\end{equation}\n\n\\noindent where $\\omega_\\mathrm{s}$, being equal to the ratio of the neutron star's rotational frequency to the Keplerian frequency at the inner radius of the accretion disc, is known as the fastness parameter and can be expressed as\n\n\\begin{equation}\n\\omega_\\mathrm{s} = 2\\pi \\, K^{3\/2} \\, P^{-1} \\, (GM)^{-5\/7} \\, \\mu^{6\/7} \\, \\dot{M}^{-3\/7} \\, ,\n\\label{eqn:fastness}\n\\end{equation}\n\n\\noindent where $P$ is the pulse period of the neutron star. In Eqn. \\ref{eqn:dimtorque}, $\\omega_\\mathrm{c}$ is the critical fastness parameter which has been estimated to be $\\sim \\!0.35$ \\citep{ghosh1979,wang1987,ghosh1994,torkelsson1998}. For $\\omega_\\mathrm{s} \\!=\\! \\omega_\\mathrm{c}$, the total torque on the neutron star becomes zero (i.e. $n(\\omega_\\mathrm{s}) \\!=\\! 0$) due to the negative torque contribution coming from the magnetic torque exerted outside the co-rotation radius at which the neutron star's rotational frequency equals to the Keplerian frequency.\n\nFor $\\omega_\\mathrm{s} \\!>\\! \\omega_\\mathrm{c}$, spin-down contribution coming from the outer disc outside the co-rotation radius is greater in magnitude than the total spin-up contributions coming from the material torque at the inner radius and the magnetic torque inside the co-rotation radius. This leads to a net spin-down of the neutron star (i.e. $n(\\omega_\\mathrm{s}) \\!<\\! 0$). On the contrary, for $\\omega_\\mathrm{s} \\!<\\! \\omega_\\mathrm{c}$, spin-up contribution coming from the material and magnetic torques is greater in magnitude than the spin-down contribution coming from the magnetic torques from the outer disc (i.e. $n(\\omega_\\mathrm{s}) \\!>\\! 0$). \n\nFrom a quadratic fit to the arrival times prior to the glitch, SXP 1062 is found to show a secular spin-down with a rate of $-\\,4.29(7) \\times 10^{-14}$ Hz s$^{-1}$ when a maximum luminosity of $L_\\mathrm{x} \\!\\sim \\! 3.3 \\times 10^{36}$ erg s$^{-1}$ is observed. Considering this luminosity value to be nearly the total accretion luminosity (i.e. $L \\!=\\! GM \\dot{M} \/ R$) and assuming a typical neutron star with $I \\!=\\! 10^{45}$ g cm$^2$, $M \\!=\\! 1.4$M$_{\\odot}$ and $R \\!=\\! 10^6$ cm; Eqn.s \\ref{eqn:r0} - \\ref{eqn:fastness} are solved numerically to obtain $\\mu$ of about $1.5 \\times 10^{32}$ G cm$^{3}$ leading to a magnetic field estimate of about $1.5 \\times 10^{14}$ G with $n(\\omega_\\mathrm{s}) \\!\\approx \\! -\\,0.0123$ and $r_\\mathrm{o} \\!=\\! 8.78 \\times 10^9$ cm. \n\nSXP 1062 can be considered to be a member of a class of accretion powered pulsars in high-mass X-ray binaries with very slow pulsations and persistent spin-down states \\citep{reig2012,fu2012}. Long spin periods together with the spin-down behaviour of these pulsars are argued as an indication of their magnetar-like magnetic fields. Thus, this class is sometimes classified as ``accreting magnetars''. Alternatively by using a theoretical model based on quasi-spherical subsonic accretion, long spin periods of these systems have also been considered not to be necessarily related to magnetar fields \\citep{shakura2013}. \n\n\\cite{fu2012} previously made use of three different theoretical approaches to obtain an estimate of the magnetic field of SXP 1062: Firstly, they estimated the magnetic field strength by considering the time scale for the ejector phase being comparable to the estimated age of the pulsar. Secondly, they estimated the magnetic field strength assuming the short-term spin-down rate of $-2.6\\times 10^{-12}$ Hz s$^{-1}$ as being near the maximum spin-down rate in disk or spherical accretion \\citep{lynden1974,lipunov1982,bisnovatyi1991}. Their final approach was to make use of the spin-down mechanism proposed by \\cite{illarionov1990}. All these three approaches lead to a surface magnetic field of SXP 1062 as $\\gtrsim 10^{14}$ Gauss.\n\nOur timing analysis shows that the source has a long-term secular steady spin-down trend with a rate of $-4.29(1)\\times 10^{-14}$ Hz s$^{-1}$ which could be as a result of a steady disc accretion. Thus, using standard accretion theory, our magnetic field estimate for SXP 1062 follows consideration of accretion via prograde accretion disc with a small negative dimensionless torque. According to this theoretical framework, observed spin-down rate and luminosity of the source leads to a magnetar-like surface magnetic field estimation which is consistent with the previous estimations by \\cite{fu2012}. \n\n\\subsection{The Glitch}\n\n\\begin{figure}\n  \\center{\\includegraphics[width=4.1cm,angle=270]{freqs.eps}} \n  \\caption{Pulse frequency evolution of SXP 1062. Frequencies are calculated from the slopes of linear fits to the TOAs shown in Fig. \\ref{glitch}. The time intervals of linear fits are represented as x-axis error bars. The frequency jump on MJD 56834.5 is identified as a spin-up glitch event with $\\Delta \\nu = 1.28(5) \\times 10^{-6}$ Hz. The source continues to spin-down after the glitch with a change of frequency derivative $\\Delta \\dot \\nu = 1.5(9) \\times 10^{-14}$ Hz s$^{-1}$.}\n  \\label{freqs}\n\\end{figure}\n\nA glitch in the pulse frequency is observed 25 days after the X-ray outburst of SXP 1062. The source has not shown any spin-up trend during the outburst which may be due to a very short duration of the outburst. Actually, the outburst is displayed only in one of the observations, which has an exposure of about 2.2 ks. As seen from Figure \\ref{glitch} and Table \\ref{soln}, the glitch occurred on MJD 56834.5 with a change of pulse frequency $\\Delta \\nu = 1.28(5) \\times 10^{-6}$ Hz and a change of pulse frequency derivative $\\Delta \\dot \\nu = - \\,1.5(9) \\times 10^{-14}$ Hz s$^{-1}$. In Figure \\ref{freqs}, we also show the pulse frequency evolution which is constructed by measuring the slopes of the TOAs (see Fig. \\ref{glitch}) for time intervals of approximately 30--70 days. Since the occurrence of the glitch does not coincide with the time of the X-ray outburst, it should be associated with the internal structure of the neutron star. SXP 1062 continues to spin-down with a constant rate after the glitch event.\n\nA glitch is a sudden fractional change in frequency which is mostly pursued by a change of spin-down rate of a previously rather stable rotating pulsar. Almost 10 per cent of pulsars are observed to glitch and pulsars of all ages seem to have glitches \\citep{haskell2015} with fractional change of frequency ($\\Delta \\nu\/\\nu$) ranging from $10^{-11}$ to $10^{-5}$ and fractional change of frequency derivative ($\\Delta\\dot{\\nu}\/\\dot{\\nu}$) varying between $10^{-4}$ and $10^{-1}$ \\citep{espinoza2011,yu2013,dib2014}. The core of a neutron star contains a significant amount of neutron superfluid \\citep{lamb1978a,lamb1978b,sauls1989,lamb1991,datta1993,lattimer2007} therefore, the moment of inertia of the star resides mainly in the core. Moreover, the inner part of the crust lattice also contains an amount of neutron superfluid which carries $10^{-2}$ of the star's moment of inertia. Coupling time scales between crustal neutron superfluid and the rest of the crust is typically very long extending from months to years \\citep{alpar1981,alpar1993,akbal2015}. For radio pulsars which spin-down due to electromagnetic dipole radiation, it is possible to resolve moment of inertia of the crustal superfluid during the post glitch \\citep{espinoza2011,yu2013}. Like canonical pulsars, magnetars also exhibit glitches however, there are some distinguishing characteristics between these two groups. While almost all pulsar are radiatively quiet i.e. they are not accompanied by any burst or pulse profile changes after the glitch \\citep{espinoza2011,yu2013} \\citep[see][for exceptions]{archibald2016,manchester2011,livingstone2010}, magnetar glitches can either be radiatively loud i.e. they can be accompanied by flares, bursts and\/or pulse profile changes; or radiatively quiet \\citep{dib2014}.  \n\nIn magnetars, glitches are resolved with high fractional frequency changes at the order of $\\Delta \\nu \/ \\nu \\! \\sim \\! + \\, 10^{-5}$  and $- \\, 10^{-4}$ \\citep{kaspi2017}. Largest spin-down glitches observed are, the glitch of 1E 2259+586 with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-6}$ \\citep{archibald2013} and the glitch of SGR 1900+14 with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-4}$ within 80 days after a large outburst \\citep{woods1999,thompson2000}. There are a few net spin-down glitches \\citep{icdem2012,sasmaz2013,archibald2017} together with a large number of spin-up glitches \\citep{dib2014}. Large spin-down glitches can be explained by particle outflow from magnetic multipoles during an outburst, while this process induces vortex inflow from the crust. The density of vortex lines are proportional to the superfluid velocity therefore the angular momentum taken from the crust \\citep{thompson2000,duncan2013}. The spin-up glitches can be caused by sudden fractures of the crust and consequently vortex outflow in the crust superfluid \\citep{thompson2000}. For both cases of spin-down and spin-up glitches, vortex unpinning from the crust occurs and then the vortices creep and re-pin to the crustal nuclei, therefore the post glitch relaxation should be observed in both cases \\citep{gugercinoglu2014}.\n\nDue to the presence of dominant external torque noise, it is not easy to detect these types of glitches for accretion powered pulsars in X-ray binaries \\citep{baykal1997}. However, for KS 1947+300 \\cite{galloway2004} have discovered a spin-up glitch. KS 1947+300 was spinning up during this glitch, therefore the influence of the external torques is not clear yet; whether the glitch event is associated with internal or external torques. Recently, \\cite{ducci2015} have suggested that both glitches and anti-glitches are possible for accretion powered X-ray pulsars, furthermore glitches of binary pulsars should have longer rise and recovery time scales compared to isolated pulsars since they have pulse periods longer than those of isolated ones. \n\nSXP 1062 is found to be spinning down secularly until MJD 56834, that is 25 days subsequent to the X-ray outburst. Then, the source showed a spin-up glitch with a fractional frequency change of $\\Delta \\nu \/ \\nu \\! \\sim \\! 1.37(6) \\times 10^{-3}$ and a fractional change of frequency derivative $\\Delta \\dot \\nu \/ \\dot \\nu \\! \\sim \\! 0.3(2)$.\n\nDuring the secular spin-down of SXP 1062, the spin-down rate is measured to be $- \\, 4.29(7) \\times 10^{-14}$ Hz s$^{-1}$. If we consider that the observed glitch is due to a torque reversal (i.e. consider it as a frequency jump due to accretion torque) with a similar magnitude of spin-up rate, upper limit for ${{\\Delta \\nu} \/ {\\nu}}$ can be estimated to be about $\\sim \\! 1.5 \\times 10^{-5}$ for a maximum of $\\Delta t \\! \\sim \\! 4$ days (the time interval between two neighboring observations around the frequency jump) which is two orders of magnitude smaller than the observed ${{\\Delta \\nu} \/ {\\nu}}$ of the glitch. So, it is unlikely that the glitch is as a result of the accretion torques. Furthermore, the ratio of the core superfluid moment of inertia to the crust moment of inertia ($I_{\\mathrm{s}} \/ I_{\\mathrm{c}}$) should be at the order of $10^2$ \\citep{baykal1991,baykal1997}. Therefore the glitch event in SXP 1062 should be associated with the internal structure of the neutron star.\n\nRecently, \\cite{ducci2015} discussed observability of glitches in accretion powered pulsars by using the ``snowplow'' model of \\cite{pizzochero2011}. In the two component neutron star model \\citep{baym1969}, a neutron star consists of two components: the normal component where charged particles (protons and electrons) co-rotate with the neutron star's magnetic field with moment of inertia $I_{\\mathrm{c}}$ and the neutron superfluid with moment of inertia $I_{\\mathrm{s}}$. The rotating superfluid (both in the core and inner crust) is considered to be an array of vortices which are pinned to the crustal lattice of ions. When the neutron star slows down, a rotational lag is developed between the vortices and the normal component. Eventually, vortices are unpinned and suddenly move out after a certain critical value of rotational lag, leading to a glitch. The time required to build a glitch is inversely proportional to the spin-down rate therefore, pulsars with higher spin-down rates are expected to glitch more often. Moreover, the coupling time scales between the crust and core are proportional to the pulse period as $\\tau = 10-100 \\, P_{\\mathrm{s}}$ \\citep{alpar1984b,alpar1988,sidery2009}. Since SXP 1062 has a long pulse period along with a strong spin-down rate, it is a good candidate for observing such glitches. In accretion powered pulsars, the time scales for both glitch rise and decay are suggested to be long therefore, a glitch would appear as a single jump in frequency leaving the spin down-rate almost unchanged \\citep{ducci2015}. The jump in pulse frequency can be estimated via \\citep{ducci2015}\n\n\\begin{equation}\n \\Delta\\nu \\simeq 2\\times 10^{-5} \\, \\frac{Q_{0.95} \\, R_{6}^{2} \\, f_{15}}{M_{1.4} \\, [ \\, 1-Q_{0.95} \\, (1-Y_{0.05}) \\, ]} \\qquad \\mathrm{Hz} \\, \\mathrm{s^{-1}} \\, ,\n\\end{equation}\n\n\\noindent where $Q$ ($= I_{\\mathrm{s}} \/ (I_{\\mathrm{c}} + I_{\\mathrm{s}})$) is the fraction of superfluid in the neutron star ($Q_{0.95}$ in units of 0.95), $Y$ is the fraction of vortices coupled to normal crust ($Y_{0.05}$ in units of 0.05) and $f$ is the pinning force ($f_{15}$ in units of $10^{15}$ dyn cm$^{-1}$). The parameter $Y$ represents short time dynamics and approaches to 1 for long time scales (steady state). Assuming a neutron star with a mass of 1.4M$_{\\odot}$, a radius of 10 km, $f_{15} \\!\\simeq\\! 1$ dyn cm$^{-1}$ and by using the $\\Delta\\nu$ value observed for SXP 1062; we find that for a superfluid fraction around 95 per cent the fraction of coupled vortices is around 78 per cent. \n\nBoth glitch rise and decay times for SXP 1062 should be at the order of a day or less ($\\tau = 10-100 \\, P_{\\mathrm{s}} \\simeq 10^{4}-10^{5}$ s) however, the sampling of TOAs around the glitch is about 3--4 days. Therefore; we observe neither the rise nor the decay of the glitch, since the glitch rise and decay should have already finished within the observational gaps. Thus for SXP 1062, the observed step-like change in pulse frequency and its magnitude can be qualitatively explained by the model of \\cite{ducci2015}.\n\nSXP 1062 has a strong and steady spin-down rate among accretion powered X-ray pulsars. Moreover, SXP 1062 is associated with a young supernova remnant with an age of 10--40 kyr \\citep{henault2012,haberl2012}, therefore it is a young pulsar spinning down very fast in the remnant. The detection rate of glitches are observed to be higher for younger pulsars \\citep{espinoza2011} and long intervals of steady spin rates are expected to increase glitch possibility \\citep{ducci2015}. Therefore, these unique properties of SXP 1062 allows the vortices to creep and pin to the crustal nuclei \\citep{alpar1984a,alpar1984b}. Sudden unpinning of vortices may cause a large glitch event, which is observed in this case with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-3}$ being the largest value of fractional frequency jump reported as far. The fractional size of the glitch suggests that $I_{\\mathrm{s}}\/I_{\\mathrm{c}}$ is around $10^2$ which corresponds to soft equation of state \\citep{datta1993,delsate2016}. It is possible to observe a glitch in this source again. In addition, the long pulse period of SXP 1062 makes it possible to reveal glitch rise and crust core coupling time if future observations are sampled closely \\citep{newton2015}. Future monitoring of this source with \\textit{LOFT} and \\textit{NICER} can reveal more information about the interior of the neutron star. \n\n\\section*{Acknowledgements}\n\nWe acknowledge support from T\\\"{U}B\\.{I}TAK, the Scientific and Technological Research Council of Turkey through the research project MFAG 114F345. We thank M. Ali Alpar for helpful comments. We also thank the anonymous referee for the valuable comments that helped to improve the manuscript.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe rank of a finitely generated group is the minimum cardinality of a generating set. There are very few families of groups for which one knows how to compute the rank (see \\cite{kw} and references therein),  and there exists no  algorithm computing the rank of a word-hyperbolic group \\cite{sh}. \n\nBy Grushko's theorem, rank is additive under free product. It does not behave as nicely under direct product, even when one of the factors is $\\Z$: the solvable Baumslag-Solitar group $BS(1,2)=\\langle a,t\\mid tat\\m=a^2\\rangle$ and the product $BS(1,2)\\times \\Z$ both  have rank 2.\n\nIn this paper we consider semi-direct products $G=A\\rtimes_\\varphi\\Z$ (also known as mapping tori), with the generator of the cyclic group $\\Z$ acting on $A$ by some automorphism $\\varphi\\in Aut(A)$. This was motivated by the remark that, when $A$ is a free group $F_d$ and $\\varphi$ has finite order in $Out(F_d)$, then $G$ is a generalized Baumslag-Solitar group and its rank may be computed \\cite{le}. But we do not know how to compute the rank when $\\varphi$ has infinite order. Abelianizing does not help much, so we ask:\n\n\\begin{qst*} Given $\\varphi\\in GL(d,\\Z)$, can one compute the rank of $G=\\Z^d\\rtimes_\\varphi\\Z$?\n\\end{qst*}\n\nWe can prove:\n\n\\begin{thm} \\label{decid} Given $\\varphi\\in GL(d,\\Z)$,   one can decide whether $G=\\Z^d\\rtimes_\\varphi\\Z$ has rank 2 or not. \n\\end{thm}\n\n\nIt turns out that the rank of $G$  is 1 plus the minimum number $k$ such that $\\Z^d$ may be generated by $k$ orbits of $\\varphi$ (i.e.\\ there exist $g_1,\\dots,g_k\\in\\Z^d$ such that the elements $\\varphi^n(g_i)$, for $n\\in\\Z$ and $i=1,\\dots, k$, generate $\\Z^d$). \nIn particular, $G$\nhas rank 2 if and only if $\\Z^d$ may be generated by a single $\\varphi$-orbit. This happens precisely when\n$\\varphi$ is conjugate to the companion matrix having the same characteristic polynomial. This may be decided since the conjugacy problem is solvable in $GL(d,\\Z)$ \\cite{grunewald}. \n\nTheorem \\ref{decid}  extends to the case when $\\varphi$ is an automorphism of an arbitrary finitely generated nilpotent group $A$.\n\nWhen $G$ has rank 2, one can classify generating pairs up to Nielsen equivalence. In particular: \n\n\\begin{thm}Suppose that $G=\\Z^d\\rtimes_\\varphi\\Z$ has rank 2.\n There are infinitely many Nielsen classes of generating pairs  if and only if the cyclic subgroup of $GL(d,\\Z)$ generated by $\\varphi$ has infinite index in its centralizer. \n\\end{thm}\n\n\nOur next result is motivated by the following theorem due to J.\\ Souto:\n\\begin{thm}[\\cite{souto}] Let $A$ be the fundamental group of a closed orientable surface of genus $g\\ge 2$. Let $\\varphi$ be an automorphism of $A$ representing a pseudo-Anosov mapping class. Then there exists $n_0$ such that  the rank of $G_n=A\\rtimes_{\\varphi^n}\\Z$ is $2g+1$ for all $n\\ge n_0$.\n\\end{thm}\n\nWe prove:\n\\begin{thm} \\label{powe} Given $\\varphi$ of infinite order in $ GL(d,\\Z)$, with $d\\ge2$,  there exists $n_0$ such that  the rank of $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ is $\\ge3$ for all $n\\ge n_0$.\n\\end{thm}\n\nThe   theorem   becomes false if the hypothesis that $\\varphi$ has infinite order is dropped, or if 3 is replaced by 4. We do not know hypotheses that would guarantee that the rank is $d+1$ for $n$ large. \n\n\nAn equivalent formulation of Theorem \\ref{powe} is:\n\\begin{thm} \\label{pow2} Given a matrix $M$  of infinite order in $ GL(d,\\Z)$, with $d\\ge2$,  there exists $n_0$ such that  $M^n$ is not conjugate to a companion matrix  if $n\\ge n_0$.\n\\end{thm}\n\nOur proof is based on the Skolem-Mahler-Lech theorem on linear recurrent sequences \\cite{rs}. There are alternative approaches based on equations in $S$-units and Baker's theory on linear forms in logarithms. They are due to Amoroso-Zannier \\cite  {AZ} and yield uniformity: \\emph{one may take $n_0=[C d^6(\\log d)^6]$ where $C$ is a universal constant (independent of $M$).}\n\n\nWe conclude with a few   open questions. \n\nOur analysis on $\\Z^d$ uses the Cayley-Hamilton theorem. This is not available in a non-abelian free group $F_d$. Given $\\varphi\\in Aut(F_d)$, can one decide whether $F_d$ may be generated by a single  $\\varphi$-orbit? More basically: given $\\varphi\\in Aut(F_d)$ and $g\\in F_d$, can one decide whether the $\\varphi$-orbit of $g$ generates $F_d$?\n\nWhat about ascending HNN extensions? For instance, let $\\varphi$ be an injective endomorphism of $\\Z^d$ (a matrix with integral entries and non-zero determinant). Let $G=\\Z^d*_\\varphi=\\langle \\Z^d,t\\mid tgt\\m=\\varphi(g)\\rangle$. Can one decide whether $G$ has rank 2?\n\\medskip\n\n{\\it \\small\nAcknowledgements. We wish to thank J.-L.\\  Colliot-Th\\'el\\`ene, F.\\  Grunewald, P.\\ de la Harpe, G.\\ Henniart,  and number theorists in Caen, in particular F.\\  Amoroso, J.-P.\\  Bezivin, D.\\  Simon, for helpful conversations related to this work. The second author would also like to thank LMNO of Universit\\'e de Caen for their hospitality during the\npreparation of the present work.}\n\n \n   \\section{Generalities}\n  \nLet $A$ be a finitely generated group. The letters $a,b,v$ will always denote elements of $A$. We denote by $i_a$ the inner automorphism $v\\mapsto ava\\m$. \n\n\nGiven $\\varphi\n\\in Aut(A)$, we let $G$  \nbe the mapping torus $G=A\\rtimes_\\varphi \\Z=  \\langle A,t\\mid tat\\m=\\varphi(a)\\rangle$. There is an exact sequence $1\\to A\\to G\\to \\Z\\to 1$. Up to isomorphism, $G$ only depends on the image of $\\varphi$ in $Out(A)$. Any $g\\in G$ has   unique forms $at^n, t^na'$ with $n\\in\\Z$. \n\nIf $N$ is a characteristic subgroup of $A$, we denote by $\\bar \\varphi$ the automorphism induced on $A\/N$. There is an exact sequence $1\\to N\\to A\\rtimes_\\varphi\\Z\\to A\/N\\rtimes_{\\bar\\varphi}\\Z\\to 1$.\n\n \n The rank $rk(G)$ is the minimum cardinality of a generating set. We let $vrk(G)$ be the   minimum number of elements needed to generate a finite index subgroup: $vrk(G)=\\inf _H rk(H)$ with the infimum taken over all subgroups of finite index.\n \n Two generating sets are Nielsen equivalent if one can pass from one to the other by Nielsen operations:  permuting the generators, replacing $g_i$ by $g_i\\m$ or $g_ig_j$. For instance, any generating set of $\\Z$ is Nielsen equivalent to $\\{0,\\dots,0,1\\}$ by the Euclidean algorithm.\n \n The $\\varphi$-orbit of $a\\in A$ is $\\{\\varphi^n(a)\\mid n\\in\\Z\\}$. We denote by\n$\\Or(\\varphi)$ the minimum number of $\\varphi$-orbits needed to generate $A$. Clearly $\\Or(\\varphi)\\le rk(A)$.\n We also denote by \n $\\vor(\\varphi)$ the minimum number of $\\varphi$-orbits needed to generate a finite index subgroup of $A$, so $\\vor(\\varphi)\\le vrk(A)$.\n\n\n  \\begin{lem} Given $a,a_1,\\dots,a_k\\in A$, the intersection   $A'=  \\langle a_1,\\dots,a_k, at\\rangle\\cap A$   is generated by the $(i_a\\circ \\varphi)$-orbits of $a_1,\\dots,a_k$.  \n  \n\n The $(i_a\\circ \\varphi)$-orbits of $a_1,\\dots,a_k$ generate $A$ if and only if  $a_1,\\dots,a_k, at$ generate $G$.\n \\end{lem}\n \n \\begin{proof} One has $(i_a\\circ \\varphi)^n(v)=(at)^nv(at)^{-n}$ for $v\\in A$ and $n\\in\\Z$. This shows that the $(i_a\\circ \\varphi)$-orbit of $a_i$ is contained in $A'$. \n Conversely, if  $v\\in A'$, write it  in terms of $a_1,\\dots,a_k, at$. The exponent sum of $t$ is 0, so $v$ is a product of elements of the form $(at)^na_i(at)^{-n}$.\n \n If $A'=A$, then $\\langle a_1,\\dots,a_k, at \\rangle$ contains $A$ and $at$, so equals $G$. \n \\end{proof}\n \n \\begin{cor} $rk(G)=1+\\min_{a\\in A}\\Or(i_a\\circ \\varphi)$.\n\\end{cor}\n\n\\begin{proof}  $\\le$  is clear. For the converse, use that  any finite generating set of $G$ is Nielsen equivalent to a set $\\{a_1,\\dots,a_k, at\\}$ (Euclid's algorithm).\n\\end{proof}\n\n \\begin{cor}   $vrk(G)=1+\\min_{a\\in A, n\\ne0}\\vor(i_a\\circ \\varphi^n)$.\n\\end{cor}\n\n\\begin{proof}  If $n\\ne0$ and the $(i_a\\circ \\varphi^n)$-orbits of $a_1,\\dots,a_k$ generate a finite index subgroup of  $A$, the subgroup of $G$ generated by $a_1,\\dots,a_k, at^n$ has finite index because it maps  onto $n\\Z$ and it meets $A$ in a subgroup of  finite index.\n\nAny finite subset of $G$ generating a finite index subgroup is Nielsen equivalent to $\\{a_1,\\dots,a_k, at^n\\}$ with $n\\ne0$, and the $(i_a\\circ\\varphi^n)$-orbits of $a_1,\\dots,a_k$ generate a finite index subgroup of $A$. \\end{proof}\n\n \n \\begin{cor} Suppose that  \n    $A$ is abelian.\n     \\begin{enumerate}\n  \n\\item \n $rk(G)=1+ \\Or(  \\varphi)$ and $vrk(G)=1+ \\vor(  \\varphi )$. \n\\item  $G$ has rank $\\le2$ if and only if $A$  \nis generated by a single $\\varphi$-orbit. A pair $(a_1,at)$ generates $G$ if and only if the $\\varphi$-orbit of $a_1$ generates $A$. \n\\item $vrk(G)$ is computable.\n \\end{enumerate}\n\\end{cor}\n\n\\begin{proof}  $i_a$ is the identity and $\\vor( \\varphi )\\le \\vor( \\varphi^n)$, so 1 follows from previous results. 2 is clear. \n\nFor 3, first suppose $A=\\Z^d$. View $\\varphi$ as an automorphism of the vector space $\\Q^d$. Then $\\vor(  \\varphi )$ is    the minimum number of $\\varphi$-orbits needed to generate $\\Q^d$. This is computable (it is the number of blocks in the rational canonical form of $\\varphi$). If $A$ has a torsion subgroup $T$, then $A\/T\\simeq \\Z^d$ for some $d$. Let   $\\bar \\varphi$ be the automorphism  induced on  $\\Z^d$. Then    $\\vor(  \\varphi )=\\vor(\\bar  \\varphi )$ is computable.\n\\end{proof}\n\n\\section{Computability}\n\nSuppose $A=\\Z^d$ with $d\\ge1$. We view $\\varphi\\in Aut(A)$ as  an automorphism of $\\Z^d$ or as a matrix in $ GL(d,\\Z)$. Its companion  matrix $M_\\phi$ is the unique matrix of the form $$ \\left(\\begin{array}{ccccc}\n0&&&&*\\\\\n1&0&&&*\\\\\n&\\ddots&\\ddots&&*\\\\\n&&1&0&*\\\\\n&&&1&*\n \\end{array}\\right)$$ having the same characteristic polynomial as $\\varphi$ (the empty triangles are filled with 0's, and $*$ denotes an arbitrary integer).\n \n \n \n  \\begin{lem}  \\label{comp} \n  Let $\\varphi\\in GL(d,\\Z)$, with $d\\ge1$. \n  \\begin{enumerate}\n  \\item\n  The following are equivalent:\n  \\begin{enumerate}\n\\item  $G=\\Z^d\\rtimes_\\varphi \\Z$  has rank 2;\n\\item  $\\Z^d$ may be generated by a single $\\varphi$-orbit;\n\\item There exists $a\\in \\Z^d$ such that $\\{a,\\varphi(a),\\dots,\\varphi^{p-1}(a)\\}$ is a basis of $\\Z^d$. \n\\item  $\\phi$ is conjugate  to its companion matrix $M_\\varphi$ in $GL(d,\\Z)$.\n  \\end{enumerate}\n  \\item Suppose that the $\\varphi$-orbit of $a$ generates $\\Z^d$. Then the $\\varphi$-orbit of $b$ generates $\\Z^d$ if and only if $b=h(a)$ where $h\\in GL(d,\\Z)$ commutes with $\\varphi$. \n  \\end{enumerate}\n \\end{lem}\n \n\n\n\\begin{proof}   We already know that (a) is equivalent to (b).  If  $a$ is the first element of a basis of $\\Z^d$ in which $\\varphi$ is represented by the matrix $M_\\phi$, then the basis is $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ and \nthe $\\varphi$-orbit of $a$ generates $\\Z^d$, so $(d)\\Rightarrow(c)\\Rightarrow(b)$.\n\nConversely, suppose that the $\\varphi$-orbit of $a$ generates $\\Z^d$. By the Cayley-Hamilton theorem, $\\Z^d$ is generated by $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$. This set is a basis of $\\Z^d$ in which $\\varphi$ is represented by $M_\\phi$. This proves 1.\n\nTo prove 2, suppose that $h$ commutes with $\\varphi$, and define $b=h(a)$. The image of the basis $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ by $h$ is $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$, so the orbit of $b$ generates. Conversely, if the orbit of $b$ generates, define $h$ as the automorphism taking $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ to $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$. It commutes with $\\varphi$ because $M_\\varphi$ represents $\\varphi$ in both bases.\n\\end{proof}\n\n\n\n\n \n\n\\begin{prop}\\label{compu}\n If  \n $A$ is nilpotent, one can decide whether $G=A\\rtimes_\\varphi \\Z$ has rank 2 or not.\n\\end{prop}\n\n\\begin{proof}  If $A=\\Z^d$, one has to decide whether $\\phi$ is conjugate to its companion matrix $M_\\varphi$ in $GL(d,\\Z)$. This is possible because the conjugacy problem is solvable in $GL(d,\\Z)$ by \\cite{grunewald}.\n\nWe now assume   that $A$ is abelian. It fits in an exact sequence $0\\to T\\to A\\to \\Z^d\\to 0$ with $T$ finite. We denote by $a\\mapsto \\bar a$ the map $A\\to \\Z^d$, and by  $h\\mapsto \\bar h$ the natural epimorphism $Aut(A)\\to Aut(\\Z^d)$. They each have finite kernel. \n\nWe have to decide whether $A$ may be generated by a single $\\varphi$-orbit. We first check whether the matrix of $\\bar\\varphi$ is conjugate to its companion matrix.\nIf not, the answer to our question is no. If yes,  \\cite{grunewald} yields  a conjugator and therefore an explicit $u\\in\\Z^d$ whose $\\bar\\varphi$-orbit  generates $\\Z^d$. \n\nWe claim that $A$ may be generated by a single $\\varphi$-orbit if and only if there exist $a\\in A$ mapping onto $u$, and $\\psi\\in Aut(A)$ of the form $h\\varphi h\\m$ with $h\\in Aut(A)$ and $[\\bar h,\\bar\\varphi]=1$, such that the $\\psi$-orbit of $a$ generates $A$. \n\nThe ``if'' direction is clear. Conversely, suppose that the $\\varphi$-orbit of $b$ generates $A$. Then the $\\bar\\varphi$-orbit of $\\bar b$ generates $\\Z^d$, so by Lemma \\ref{comp} there exists $\\theta\\in Aut(\\Z^d)$ commuting with $\\bar \\varphi$ and mapping $\\bar b$ to $u$.\nLet $h$ be any lift of $\\theta$ to $Aut(A)$. Defining    $a=h(b)$ and $\\psi=h\\varphi h\\m$, it is easy to check that the $\\psi$-orbit of $a$ generates $A$. This proves the claim.\n\nWe now explain how to decide whether $a$ and $\\psi$ as above exist. Note that $a$ and $\\psi$ must belong to   explicit finite sets: $a$   belongs to  the preimage $A_u$ of $u$, and \n$\\psi$     belongs to     the preimage $X_\\varphi$ of $\\bar\\varphi$ in $Aut(A)$. \n\nBy Theorem C of \\cite{grunewald}, the centralizer of $\\bar\\varphi$  in $Aut(\\Z^d)$ is a finitely generated subgroup and one can compute a finite generating set. The same is true of \n  $D=\\{h\\in Aut(A)\\mid [\\bar h,\\bar\\varphi]=1\\}$, so we can list the elements $\\psi$ in the orbit $D\\varphi$ of $\\varphi$ for the action of $D$ on $X_\\varphi$ by conjugation.  \n  \nBy the claim proved above,    $A$ may be generated by a single $\\varphi$-orbit if and only if there exist \n    $a \\in A_u$ and $\\psi\\in D\\varphi$  such that the $\\psi$-orbit of $a$ generates $A$. To decide this, we enumerate the pairs $(a,\\psi)$ with $a\\in A_u$ and $\\psi\\in D\\varphi$. For each pair,     \n    we consider the increasing sequence of subgroups $A_N=\\langle \\psi^{-N}(a), \\dots, \\psi^{-1}(a), a,\\psi(a), \\dots\\psi^{N}(a)\\rangle$. It stabilizes and we check whether $A_N=A$ for $N$ large. \n    \n    This completes the proof for $A$ abelian. If $A$ is nilpotent, let $B$ be its abelianization and let $\\rho:B\\to B$ be the automorphism induced by $\\varphi$. If $G_\\varphi=A\\rtimes_\\varphi\\Z$ has rank 2, so does its quotient $G_\\rho=B\\rtimes_\\rho\\Z$. Conversely, if $G_\\rho$ has rank 2, it is generated by $t$ and some $b\\in B$ whose $\\rho$-orbit generates $B$. Let $a$ be any lift of $b$ to $A$. The subgroup of $A$ generated by the $\\varphi$-orbit of $a$ maps surjectively to $B$, so equals $A$ by a classical fact about nilpotent groups (see e.g.\\  Theorem 2.2.3(d) of \\cite{k}). Thus $G_\\varphi$ has rank 2.\n    \\end{proof}\n \n\\begin{cor} If $A=\\Z^2$ or $A=F_2$, one can compute the rank of $G$. \n\\end{cor}\n\n\\begin{proof}  The rank is 2 or 3, so this is clear from the proposition  if $A=\\Z^2$. \n\nRecall that the natural map $ Out(F_2)\\to Out(\\Z^2)=Aut(\\Z^2)$ is an isomorphism (both groups are isomorphic to $GL(2,\\Z)$).\nGiven $G=F_2\\rtimes_\\varphi \\Z$,   let $\\rho$ be the image of $\\varphi$ in $Aut(\\Z^2)$. \nConsider $G_\\rho=\\Z^2\\rtimes_{\\rho} \\Z$. We prove that $G$ and $G_\\rho$ have the same rank. \n\nClearly $2\\le rk(G_\\rho)\\le rk(G)\\le3$. If $G_\\rho$ has rank 2, Lemma \\ref{comp} lets us assume that $\\rho$ is of the form $ \\left(\\begin{array}{rr} 0&\\pm1\\\\ 1&n  \\end{array}\\right)$. Since $G$ only depends on the class of $\\varphi$ in $Out(F_2)$, it is isomorphic to $$\\langle a,b,t\\mid tat\\m=b,tbt\\m=a^{\\pm1}b^n\\rangle,$$ so has rank 2.\n\\end{proof}  \n\n\\section{Nielsen equivalence}\n\n  \\begin{prop} \\label {gp} Suppose  \nthat  $A$ is abelian and $G=A\\rtimes_\\varphi\\Z$ has rank 2. \n  \\begin{enumerate}\n  \n\\item  Any  generating pair of $G$  is Nielsen equivalent to a pair $(a,t)$ with $a\\in A$.\n  \n\\item  Two generating pairs  $(a,t)$ and $(b,t)$, with $a,b\\in A$, are Nielsen equivalent if and only if $b$ belongs to the $\\varphi$-orbit of $a$ or $a\\m$.\n\\end{enumerate}\n \\end{prop}\n\n\\begin{proof}   Given $x,y\\in A$, and $n$, write\n$$\n(x,ty)\\sim((ty)^nx(ty)^{-n},ty)=(\\varphi^n(x),ty)\n$$\nand\n$$\n(x,ty)\\sim (\\varphi^n(x),ty)\\sim(\\varphi^n(x),ty\\varphi^n(x))\\sim( x,ty\\varphi^n(x)).\n$$\n\nEvery generating pair is equivalent to some $(a,ty)$, with the $\\varphi$-orbit of $a$ generating $A$. But $(a,ty)\\sim ( a,ty\\varphi^n(a))$ so by an easy induction $(a,ty)\\sim (a,t)$. This proves 1.\n\nIf $b=\\varphi^n(a^\\varepsilon)$ with $\\varepsilon=\\pm1$, then $(b,t)=(\\varphi^n(a^\\varepsilon),t)=(t^na^\\varepsilon t^{-n},t)\\sim(a,t)$. The converse follows from Theorem 2.1 of \\cite{hw}.\nWe give a proof for completeness. If $(b,t)\\sim(a,t)$, we can write $b=w(a,t)$ with $w$   a primitive word with exponent sum 0 in $t$. Such a word is conjugate to $a^{\\pm1}$ in the free group $F(a,t)$, so $b$ is conjugate to $a^{\\pm1}$ in $G$. Since $A$ is abelian, $b$ belongs to the $\\varphi$-orbit of $a^{\\pm1}$. \n\\end{proof}\n\n\\begin{rem} More generally, if $A$ is abelian, any generating set of $G$  is Nielsen equivalent to a set of the form $\\{a_1,\\dots,a_k, t\\}$.\n\\end {rem}\n\n\\begin{rem} \\label{heis} The proposition does not extend to nilpotent groups. Let $A$ be the Heisenberg group $\\langle a,b,c\\mid [a,b]=c, [a,c]=[b,c]=1\\rangle$. Let $\\varphi$ map $a$ to $ab$ and $b$ to $b$. \nThe generating pairs $(a,t)$ and $(ac\\m,t)$ are Nielsen equivalent  (even conjugate) but $ac\\m$ does not belong to the $\\varphi$-orbit of $a^{\\pm1}$.\nMoreover, $(a, tc)$ is a generating pair which is not Nielsen equivalent to a pair $(x,t)$  with $x\\in A$. \nIndeed, if it were, then $t$ would be conjugate to $tca^k$ for some $k\\in\\Z$ by \\cite{hw}. Counting  exponent sum in $a$ yields $k=0$. But $t$ and $tc$ are not conjugate. \n\\end {rem}\n\n\n\\begin{cor} Let $A=\\Z^d$. If $G$  \nhas rank 2, the number of Nielsen classes of generating pairs is equal to the index of the  group generated by $\\varphi$ and $-Id$ in the centralizer of $\\varphi$ in $GL(d,\\Z)$.\n\\end{cor}\n\n\\begin{proof}   By Proposition \\ref{gp} we need only  consider generating pairs of the form $(a,t)$. Fix one. To any $b\\in \\Z^d$ such that  $(b,t)$ generates $G$ we associate the automorphism $\\psi_b$ of $\\Z^d$ taking the basis $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ to the basis $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$. By Lemma \\ref{comp}, the image of this map $b\\mapsto \\psi_b$ is the centralizer of $\\phi$ in $GL(d,\\Z)$. By Proposition \\ref{gp}, $(b,t)\\sim(a,t)$ if and only if $\\psi_b$ is $\\pm \\varphi^n$ for some $n\\in \\Z$. \n\\end{proof}\n\n  \\begin{example*}    The number of Nielsen classes of generating pairs is always finite if $d=2$. If $\\phi=\n  \\left(\\begin{array}{rrrr} 0&1&0&0\\\\ 1&1&0&0\\\\ 0 &0&0&1\\\\  0&0&1&0  \\end{array}\\right)$, this number is infinite.\n  \\end{example*}\n  \n\\section{Powers}\n\nFix $\\varphi\\in GL(d,\\Z)$.\nSay that $v\\in \\Z^d$ is \\emph{$\\varphi$-cyclic} if \n  its $\\varphi$-orbit generates $\\Z^d$, or equivalently if $\\{v,\\phi(v),\\dots,\\phi^{d-1}(v)\\}$ is a basis of $\\Z^d$. \n  The existence of such a $v$   is equivalent to \n  $\\phi$ being conjugate to its companion matrix, and also to $G$ having  rank 2.  If  $v$   is $\\varphi^n$-cyclic  for some $n\\ge2$, it  is   $\\varphi$-cyclic since its $\\varphi^n$-orbit is contained in its $\\varphi$-orbit. \n  \n  If $v$ is $\\varphi$-cyclic, we denote by $\\delta_n$ \n   the index of the subgroup of $\\Z^d$ generated by the $\\varphi^n$-orbit of $v$.\n    It   does not depend on the choice of $v$ since $\\varphi$ always has matrix $M_\\phi$ in the basis $\\{v,\\varphi(v),\\dots,\\varphi^{d-1}(v)\\}$. Also note that  $\\delta_1=1$. The group $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ has rank 2 (equivalently, $\\varphi^n$ is conjugate to its companion matrix) if and only if $\\delta_n=1$. \n\n\n\\begin{thm} \\label{po2}\nIf $\\varphi\\in GL(2,\\Z)$ has infinite order, the rank of  $G_n=\\Z^2\\rtimes_{\\varphi^n}\\Z$   is 3 for all $n\\ge3$. \n\\end{thm}\n\n\\begin{proof}\n  If $G_n$ has rank 2 for some $n$, there exists a $\\varphi^n$-cyclic element $v$. Such a $v$ is also $\\varphi$-cyclic.  In the basis $\\{v,\\varphi(v)\\}$, the matrix of $\\varphi$ has the form \n $ M=\\left(\\begin{array}{rr} 0&\\varepsilon \\\\ 1&\\tau  \\end{array}\\right)$ with $\\varepsilon=\\pm1$. If finite, the index \n$\\delta_n$  is the absolute value of the determinant $c_n$ of the matrix expressing the family $\\{v,\\varphi^n(v)\\}$ in the basis $\\{v,\\varphi(v)\\}$. We prove the theorem by showing   $ | c_n | >1$ for $n\\ge3$.\n \n The number $c_n$ is determined by the equation $M^n=c_n M+ d_n I$. It follows from the Cayley-Hamilton theorem that the sequence $c_n$ satisfies the recurrence relation $c_{n+2}-\\tau c_{n+1}-\\varepsilon c_n=0$.  \n \n  If $\\varepsilon=-1$ one has $$c_n=\\prod_{k=1}^{n-1}(\\tau-2\\cos\\frac{k\\pi}n) $$\n because $c_n$ is a monic polynomial  of degree $n-1$ in $\\tau$ which vanishes for $\\tau=2\\cos\\frac{k\\pi}n$ \n  (one also has $c_n=U_{n-1}(\\tau\/2)$, with $U_{n-1}$ a Chebyshev polynomial of the second kind). \n  \n   \n If  $\\varepsilon=1$ one has \n $$c_n=\\prod_{k=1}^{n-1}(\\tau-2i\\cos\\frac{k\\pi}n) .$$\n \n \n Since $\\varphi$ is assumed to have infinite order, one has $\\tau\\ne0$ if $\\varepsilon=1$, and $ | \\tau | \\ge2$ if $\\varepsilon=-1$. One checks that $ | c_n | >1$ for $n\\ge3$ (for $n\\ge 2$ if $\\varepsilon=-1$).\n \n \\end{proof}\n\n\n\n\\begin{thm} \\label{pow}\nSuppose that $\\varphi\\in GL(d,\\Z)$ has infinite order. \n\\begin{enumerate}\n\\item There exists $n_0$ such that $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ has rank $\\ge3$ for every $n\\ge n_0$. Equivalently: $\\varphi^n$ is not conjugate to its companion matrix for $n\\ge n_0$.\n\\item \nMore precisely, the minimum index of 2-generated subgroups of $G_n$ goes to infinity with $n$.\n\\end{enumerate}\n\\end{thm}\n\n\n Note that there are arbitrarily large values of $n$ for which the rank of $G_n$ is $d+1$ (whenever $\\varphi^n$ is the identity modulo some prime number). As already mentioned, it is proved in \\cite{AZ} that  $n_0$ may be chosen to depend only on $d$.\n \n The key step in the proof of Theorem \\ref{pow} \n is the following result.\n  \n  \\begin{prop} \\label{key}\n  If $\\varphi$ has infinite order and $v $ is $\\varphi$-cyclic, then the index $\\delta_n$ of the subgroup  of $\\Z^d$ generated by the $\\varphi^n$-orbit of $v$ goes to infinity with $n$.\n  \\end{prop} \n  \n   \\begin{proof}[Proof of the theorem from the proposition]\n   As above, if $G_n$ has rank 2 for some $n$, there exists a $\\varphi $-cyclic element $v$. \n  For $n$ large one has $\\delta_n>1$, so $G_n$ has rank $>2$. Assertion 1 is proved.\n   \n For Assertion 2, suppose that there are arbitrarily large values of $n$ such that $G_n$ contains a 2-generated subgroup $H_n$ of index $\\le C$, for some fixed $C$. This subgroup has a generating pair of the form $(a_n,t_n)$ with $a_n \\in\\Z^d$, and the  intersection of $H_n$ with $\\Z^d$ is generated by the $\\varphi^{nm_n}$-orbit of $a_n$ for some $m_n\\ge1$. It has index $\\le C$ in $\\Z^d$. \n \n \n The subgroup  of $\\Z^d$ generated by the $\\varphi$-orbit of $a_n$  has index $\\le C$, so we can assume that it does not depend on $n$. Call it $J$. It is $\\varphi$-invariant so we can apply the proposition to the action of $\\varphi$ on $J$, with $v=a_n$. This gives the required contradiction.\n \\end{proof}\n \n \n\n  \n    \\begin{proof}  [Proof of Proposition \\ref{key}]  When $d=2$, one  easily   checks that $  c_n  $, as computed above,  goes to infinity with $n$. The proof in the general case is more involved. \n    \n    Define numbers $u_k(i)$, \n    for $k=0,\\dots,d-1$ and $i\\ge0$,   by $\\varphi^i(v)=\\sum_{k=0}^{d-1}u_k(i)\\varphi^k(v)$. The sequences $u_0,\\dots, u_{d-1}$ \n     form a basis for the \n    space $\\cals$ of sequences satisfying the linear recurrence associated to the characteristic polynomial of $\\varphi$ (the recurrence is $\\sum_{j=0}^d a_ju_k(i+j)=0$ if the characteristic polynomial is $\\sum _{j=0}^d  a_jX^j$). \n    \n    The index $\\delta_n$  is the absolute value of the determinant $c_n$ of the matrix $(u_k(ni))_{0\\le i,k\\le d-1}$ (it is infinite if the determinant is $0$). We have to prove that, given $c\\ne0$, the set of $n$'s such that $c_n=c\n   $ is finite. We assume it is not and we work towards a contradiction. \n   \n   A sequence satisfies a linear recurrence if and only if it is a finite sum of polynomials times exponentials, so     $c_n$ also is a recurrent sequence. The Skolem-Mahler-Lech theorem \\cite{rs} then implies that $c_n=c\n   $ for all $n$ in an  arithmetic progression $\\N_0\\subset \\N$.\n    \n    \n    We shall now replace the basis $u_k$ of $\\cals$ by another basis $w_k$ depending on the eigenvalues of $\\varphi$. We  then assume  that    $D_n:=\\det (w_k(ni))_{0\\le i,k\\le d-1}=c'\\ne0$ for $n\\in\\N_0$. \n\nWe order  the eigenvalues $\\lambda_k$ of $\\varphi$  so that $0<| \\lambda_1 |\\le | \\lambda_2 |\\le\\dots\\le | \\lambda_{d} | $.\nFirst suppose that the eigenvalues  are all distinct. We then choose $w_k(i)= (\\lambda_{k+1})^i$. In this case $D_n$ is a Vandermonde determinant, for instance $$D_n=\n\\left |   \\begin{matrix} \n1 & 1&1   \\cr\n(\\lambda_1)^{n}& (\\lambda_2)^{n} & (\\lambda_3)^{n}  \\cr  \n(\\lambda_1)^{2n} &(\\lambda_2)^{2n} &(\\lambda_3)^{2n}  \n   \\end{matrix}\\right | $$ for $d=3$, \n   so $\\displaystyle D_n=\\prod_{1\\le k< m\\le d}\\bigl((\\lambda_m)^{n} -(\\lambda_k)^{n} \\bigr)$.\n   \nIf all moduli $ | \\lambda_k | $ are distinct, \nthen $ | D_n | $ goes to infinity with $n$ because  its diagonal  term $$ (\\lambda_2)^{n}      (\\lambda_3)^{2n}  \\dots(\\lambda_{d})^{(d-1)n}=\\biggl( \\lambda_2      (\\lambda_3)^{2}  \\dots(\\lambda_{d})^{(d-1)}\\biggr)^n$$ has modulus bigger than all others.  \n\n   If the $\\lambda_k$'s are distinct but their moduli are not, \n    expand      $D_n$  as   a sum $\\sum_j \\varepsilon_j\\mu_j{}^ n$ (with $\\varepsilon_j=\\pm1$). \n  Now  \n there may be several (possibly cancelling) terms for which $ | \\mu_j | $ takes its maximal value $K= |  \\lambda_2      (\\lambda_3)^{2}  \\dots(\\lambda_{d})^{(d-1)} | $. Note that $K>1$ because otherwise all $\\lambda_k$'s have modulus 1, hence are roots of unity by a classical result, and $\\varphi$   has  finite order. \n   \n Since $D_n=c'$ for $n\\in \\N_0$ and $K>1$, one has    $\\sum_{| \\mu_j | =K} \\varepsilon_j\\mu_j{}^ n=0$ for $n\\in\\N_0$. Call this sum $D_{n,K}$. Recall that $\\displaystyle D_n=\\prod_{1\\le k< m\\le d}\\bigl((\\lambda_m)^{n} -(\\lambda_k)^{n} \\bigr)$. To expand  this product, one  chooses one of   $(\\lambda_m)^{n}$ or \n$(\\lambda_k)^{n} $ for each couple  $k,m$. The corresponding term contributes to $D_{n,K}$ if and only if one always chooses a term of maximal modulus. In other words, $\\displaystyle D_{n,K}=\\prod_{1\\le k< m\\le p} E_{k,m}$ with $E_{k,m}=(\\lambda_m)^{n} -(\\lambda_k)^{n}$ if $ | \\lambda_m | = | \\lambda_k | $ and\n$E_{k,m}=(\\lambda_m)^{n} $ if $ | \\lambda_m | > | \\lambda_k | $. Since the $\\lambda_k$'s are non-zero, $D_{n,K}=0$ implies \n $(\\lambda_k)^n=(\\lambda_m)^n$ for some $k,m$ with $k\\ne m$, so that $D_n=0$, a contradiction.  \n\n   \n    This completes the proof when the eigenvalues of $\\varphi$ are distinct. In the remaining case, the basis $w_k$ must have a different form: if $\\lambda$ is an eigenvalue of multiplicity $r$, we use the sequences $\\lambda^i, i\\lambda^i, \\dots, i^{r-1}\\lambda^i$. For instance,\n    $$D_n=\n\\left |   \\begin{matrix} \n1 & 0&0&1   \\cr\n(\\lambda_1)^{n}& n(\\lambda_1)^{n} & n^2(\\lambda_1)^{n} & (\\lambda_4)^{n}  \\cr  \n(\\lambda_1)^{2n} &2n(\\lambda_1)^{2n} &(2n)^2(\\lambda_1)^{2n} &(\\lambda_4)^{2n} \\cr\n(\\lambda_1)^{3n} &3n(\\lambda_1)^{3n} &(3n)^2(\\lambda_1)^{3n} &(\\lambda_4)^{3n}  \n \n   \\end{matrix}\\right | $$\n   when $d=4$ and $\\lambda_1=\\lambda_2= \\lambda_3\\ne \\lambda_4$.\n    \n    Calling $\\nu_1,\\dots,\\nu_q$ the distinct eigenvalues of $\\varphi$, there exist   integers $a,b,c_k,d_{mk}$ (depending only on the multiplicities of the eigenvalues) such that $$D_n=an^b\\prod _{k=1}^q(\\nu_k)^{nc_k}\\prod _{1\\le k< m\\le q}\\bigl((\\nu_m)^{n} -(\\nu_k)^{n}\\bigr)^{d_{mk}}$$ (see \\cite{fh} or Theorem 21 in \\cite{kr}). For instance, $D_n$ as displayed above equals $2n^3(\\lambda_1)^{3n}((\\lambda_4)^{n} -(\\lambda_1)^{n})^3$.\n    \n    If $K>1$, we conclude as in the previous case. \n    If $K=1$,\n  all eigenvalues are roots of unity and $D_n=n^bE_n$ where $E_n$ only takes finitely many values and $b>0$ (an eigenvalue $\\nu_j$ of multiplicity $r\\ge 2$ contributes   $1+\\dots+(r-1)$ to $b$).  \nSuch a product cannot take a non-zero value infinitely often.\n     \\end{proof} \n     \n     \n     \\begin{cor} If $A$ is abelian, and $\\varphi\\in Aut(A)$ has infinite order, then \n     $G_n=A\\rtimes_{\\varphi^n}\\Z$ has rank $\\ge3$ for $n$ large. The minimum index of 2-generated subgroups of $G_n$ goes to infinity with $n$.\n     \\end{cor}\n     \n     This follows readily from Theorem \\ref{pow}, writing $A\/T\\sim\\Z^d$ with $T$ finite.\n The analogous result for nilpotent groups is false, as the following example shows.      \n Let $A$ be the Heisenberg group as in Remark \\ref{heis}. If $\\varphi$ maps $a$ to $bc$, $b$ to $ac^2$, and $c$ to $c\\m$, then $\\varphi^{2n+1}(a)=bc^{1-n}$, so  $G_{2n+1}$ has rank 2 since $a$ and $\\varphi^{2n+1}(a)$ generate $A$. The automorphism induced by $\\varphi$ on the abelianization  of $A$ has order 2.\n \n  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIntegrable quantum models are gaining an increasing role in modern physics. The access to an exact solution of a many-body interacting system gives an unprecedented opportunity to explore strongly correlated quantum systems beyond perturbative or numerical methods. Circumstances are especially encouraging in one-dimensional systems where some integrable models naturally appear and the theoretical predictions are of huge experimental value. For example the recent progresses in cold atoms experiments in optical lattices allow the manufacturing of different models in an almost ideally isolated environment. Many physical properties of some paradigmatic integrable models as the XXZ spin chain and the Lieb-Liniger model were indeed observed in such experiments\\cite{2004_Paredes_NATURE_429,2008_Amerongen_PRL_100,2014_Fabbri_arXiv} and similar observations were also possible in highly spatially anisotropic crystals \\cite{2013_Mourigal_NatPhys_9,2013_Lake_PRL_111}.\n\nThe standard technique to solve interacting integrable models is the Bethe Ansatz\\cite{KorepinBOOK}. This technique provides us a complete characterization of eigenstates and, at least in some cases, the matrix elements of physical operators between two eigenstates ({\\em form factors}). However these expressions are usually cumbersome since they depend on all the $N$ variables (\\emph{rapidities}) describing the eigenstates, where $N$ is the number of constituents in the system. Moreover in order to address the two-point correlation functions one has to perform a summation of form factors over the whole Hilbert space. Performing such summations is so far beyond our analytical abilities but they can be evaluated numerically \\cite{2005_Caux_PRL_95,2006_Caux_PRA_74,PhysRevA.89.033605}. In the conformal limit the summations simplifies allowing for an exact treatment \\cite{1742-5468-2012-09-P09001,1742-5468-2011-12-P12010}. This led to a derivation of the Luttinger liquid \\cite{1981_Haldane_PRL_47,GiamarchiBOOK} (an effective theory of gapless 1D models) results for correlation functions directly from a microscopic (integrable) theory. Similar results were also worked out in this direction: the phenomenological quantities entering the Luttinger liquid description of correlation functions  were connected with microscopic data \\cite{2011_Shashi_PRB_84,2012_Shashi_PRB_85}. However the full determination of correlation functions resisted so far the best efforts.\n\nOne of the main difficulty arises from a complicated structure of form factors as they are highly non-trivial functions of rapidities of two eigenstates. The thermodynamic limit at fixed density however allows for some simplifications. On the physical grounds we can reason that form factors of local operators are non-zero only when evaluated between two very similar eigenstates. Indeed a local operator is not expected to modify a macroscopic number of degrees of freedom and consequently its form factors are functions of parameters specifying one of the states as an \\emph{excited state} over the other which we call an \\emph{averaging state}. The number of such excitations is a sub-extensive number $n$  such that in the thermodynamic limit $n\/N \\to 0$. We can then distinguish two different situations depending on a type of the averaging state. When the averaging state is the ground state of a gapless theory each excitation with a finite momentum and energy gets \"dressed\" by an infinite number of zero-energy excitations. This sign of the criticality in the system can be seen in a non-integer scaling behavior of the form factors with the size of the system (here the length $L$) as $L^{- \\alpha}$, with $\\alpha$ a rational number \\cite{1990_Slavnov_TMP_82}. This makes the evaluation of the spectral sum, required for computation of dynamical correlation functions, a daunting task. Still some progress was achieved in the aforementioned conformal limit \\cite{1742-5468-2012-09-P09001,1742-5468-2011-12-P12010}. Here we focus on another regime, when the averaging state is a finite entropy state. Similar averaging states were already considered, for example, in \\cite{1742-5468-2011-01-P01011}.\n\nThe excited states of a finite entropy averaging state contribute indeed \\emph{individually} to the whole sum over the Hilbert space. This can be seen again at the level of the form factors as their scaling with the system size is determined by number of excitations $n$ as $L^{-n}$. In particular as we restrict here to the (repulsive) Lieb-Liniger model and to the density operator form factors the only relevant excitations are \\emph{particle-hole} excitations.\n\nOur main result is the thermodynamic limit of the form factors of the density operator between the finite entropy averaging state and its excited states with a number $n$ of particle-hole excitations. Such form factors are the building blocks of the correlation functions for a general, non-critical situation. The results are applicable to compute correlation functions at finite temperatures (when the averaging state is the thermal state) and also for systems out of equilibrium with the averaging state being the steady state of the unitary time evolution \\cite{2013_Caux_PRL_110,PhysRevA.89.033601}.\n\n\\subsection{Structure of the article}\n\nIn section \\ref{1DBoseGas} we recall the Bethe Ansatz solution of the Lieb-Liniger model and we collect all the necessary ingredients to compute the thermodynamic limit of the density form factors. The bulk of the computation is shown in section \\ref{thermodynamiclimit}. In section \\ref{regularization} we show how to regularize divergences present in form factors. In section \\ref{expansion} we compute the dynamical structure factor (density-density correlation function) in the $1\/c$ expansion. In section~\\ref{numerics} we numerically evaluate the dynamical structure factor by including only the simplest (1 particle-hole) excitations.\n\n\n\\section{1D Bose gas} \\label{1DBoseGas}\n\nThe Hamiltonian of $N$ bosonic particles confined in a one spatial dimension is  \\cite{1963_Lieb_PR_130_1,1998_Olshanii_PRL_81}\n\\begin{align}\\label{H}\n  H = -\\sum_{j=1}^N \\partial_{x_j}^2 + 2c\\sum_{j<k}^N \\delta\\left(x_j - x_k\\right) ,\n\\end{align}\nwhere $c$ is the strength of the two-body, repulsive ($c>0$) interactions and we set $\\hbar=1$ and $2m=1$. The wavefunctions are superpositions of plane waves \\cite{1963_Lieb_PR_130_1},\n\\begin{align} \\label{wv_fnc}\n  \\langle {\\boldsymbol{x}}|{\\boldsymbol{\\lambda}}\\rangle = \\Psi({\\boldsymbol{x}}|{\\boldsymbol{\\lambda}}) = \\prod_{j>k}^N \\textrm{sgn}\\left(x_j-x_k\\right)\\sum_{P_N} \\mathcal{A}_P e^{i\\sum_{j=1}^N \\lambda_{P_j} x_j},\n\\end{align}\nwhere the summation extends over all permutations of $N$ particles. We also adopt a shorthand notation in which ${\\boldsymbol{x}} = \\{x_j\\}_{j=1}^N$ and ${\\boldsymbol{\\lambda}} = \\{\\lambda_j\\}_{j=1}^N$. The effect of interaction is encapsulated in the coefficients\n\\begin{align}\n  \\mathcal{A}_P = (-1)^{[P]} e^{i\/2\\sum_{j>k} \\textrm{sgn}(x_j-x_k)\\theta(\\lambda_{P_j} - \\lambda_{P_k})},\n\\end{align}\nwith two-particle phase shift\n\\begin{align}\\label{phase_shift}\n  \\theta(\\lambda) = 2\\arctan\\left(\\lambda\/c\\right).\n\\end{align}\nThe energy of an eigenstate $|{\\boldsymbol{\\lambda}}\\rangle$ is\n\\begin{align}  \\label{energy_discrete}\n  E({\\boldsymbol{\\lambda}}) = \\sum_{j=1}^N \\lambda_j^2.\n\\end{align}\nThe operator of the total momentum, $\\hat{P} = -i\\sum_{j=1}^N \\partial_{x_j}$, commutes with the Hamiltonian \\eqref{H} and its eigenvalues are simply\n\\begin{align} \\label{momentum_discrete}\n  P({\\boldsymbol{\\lambda}}) = \\sum_{j=1}^N \\lambda_j.\n\\end{align}\nImposing the periodic boundary conditions constrains the set of rapidities ${\\boldsymbol{\\lambda}}$ to solutions of the Bethe equations\n\\begin{align}  \\label{bethe}\n  \\lambda_j = \\frac{2\\pi}{L}I_j - \\frac{1}{L} \\sum_{k=1}^N \\theta \\left(\\lambda_j - \\lambda_k\\right),\n\\end{align}\nwhere $L$ is the length of the system. Quantum numbers $I_j$ are integers (half-odd integers) for $N$ odd (even) and follow the Pauli principle - wave function vanishes identically if any two of them coincide. The Hilbert space is spanned by allowed choices of quantum numbers. It is customary to name the eigenstates of \\eqref{H} in the finite system the Bethe states. We follow this tradition.\n\nThe norm of the Bethe states admits a neat representation in the form of a determinant \\cite{1971_Gaudin_JMP_12_I,1982_Korepin_CMP_86}\n\\begin{align} \\label{norm}\n  |{\\boldsymbol{\\lambda}}|^2 \\equiv \\langle{\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle = c^N \\prod_{j\\neq k}^N \\frac{\\lambda_j - \\lambda_k + ic}{\\lambda_j - \\lambda_k} \\det_N \\mathcal{G},\n\\end{align}\nwhere the Gaudin matrix is\n\\begin{align} \\label{gaudin}\n  \\mathcal{G}_{jk} &= \\delta_{jk}\\left(L + \\sum_{m=1}^N K\\left(\\lambda_j - \\lambda_m\\right) \\right) - K\\left(\\lambda_j - \\lambda_k\\right),\\\\\n  K(\\lambda) &= \\frac{2c}{\\lambda^2 + c^2}.\\label{K}\n\\end{align}\nNote also that the kernel $K(\\lambda)$ is a derivative of the two-particle phase shift $\\theta(\\lambda)$.\n\nWe define the density operator as\n\\begin{equation}\n\\hat{\\rho}(x) = \\sum_{j=1}^N \\delta( x  - x_j)\n\\end{equation}\nwhere $\\{ x_j\\}_{j=1}^N$ are the positions of all the particles in the gas. The two-point correlation function of this operator is particularly relevant for both theory and experiment.\nThe form factors of the density operator are given by the Algebraic Bethe Ansatz approach \\cite{1990_Slavnov_TMP_82}\n\\begin{align} \\label{ff}\n\\langle {\\boldsymbol{\\mu}} | \\hat{\\rho}(0) | {\\boldsymbol{\\lambda}}\\rangle = \\left( \\sum_{j=1}^N (\\mu_j - \\lambda_j) \\right)\\prod_{j=1}^N \\left(V_j^+ - V_j^-\\right) \\prod_{j,k}^N \\left(\\frac{\\lambda_j - \\lambda_k + ic}{\\mu_j - \\lambda_k}\\right)\\frac{\\det_N \\left(\\delta_{jk} + U_{jk}\\right)}{V_p^+ - V_p^-},\n\\end{align}\nwhere both $|{\\boldsymbol{\\lambda}}\\rangle$ and $|{\\boldsymbol{\\mu}}\\rangle$ are Bethe states. Different factors appearing in \\eqref{ff} are\n\\begin{align} \\label{matrix_U}\n  V_j^{\\pm} &= \\prod_{k=1}^N \\frac{\\mu_k - \\lambda_j \\pm ic}{\\lambda_k - \\lambda_j \\pm ic},\\\\\n  U_{jk} &= i\\frac{\\mu_j - \\lambda_j}{V_j^+ - V_j^-} \\prod_{m\\neq j}^N \\left(\\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j}\\right) \\biggl(K\\left(\\lambda_j - \\lambda_k\\right) - K\\left(\\lambda_p - \\lambda_k\\right) \\biggl),\n\\end{align}\nand $\\lambda_p$ is an arbitrary number, not necessarily from the set ${\\boldsymbol{\\lambda}}$.\n\n\n\\subsection{Thermodynamic limit}\n\nWe consider now the thermodynamic limit $N\\rightarrow\\infty$ with fixed density $D = N\/L $. We denote such limit with $\\lim_{\\text{th}}$. The Bethe states can be then characterized by a filling function $\\vartheta(\\lambda)$ defined as a number of rapidities in an interval $(\\lambda,\\lambda+d\\lambda)$ divided by a maximal number of rapidities (in this interval). Due to an interacting nature of the gas the maximal number of the particles is not constant and the density of particles is connected with the filling function through an integral equation \\cite{1969_Yang_JMP_10}\n\\begin{align} \\label{rho_p}\n  2\\pi \\rho(\\lambda) &= \\vartheta(\\lambda)\\left(1 + \\int_{-\\infty}^{\\infty} d\\mu K(\\lambda -\\mu) \\rho(\\mu)\\right).\n\\end{align}\nThe filling function obeys\n\\begin{align}\\label{filling_condition}\n  0 \\leq \\vartheta(\\lambda) \\leq 1,\n\\end{align}\nwhich guarantees the existence of $\\rho(\\lambda)$ through \\eqref{rho_p}.\nThe filling function provides a complete macroscopic characterization of the Bethe states in the thermodynamic limit. For example the extensive part of the momentum and the energy is (c.f. with eqs. \\eqref{energy_discrete} and \\eqref{momentum_discrete})\n\\begin{align}\\label{energy_cont}\n  P[\\vartheta] &= L\\int_{-\\infty}^{\\infty} d\\lambda\\, \\rho(\\lambda)\\, \\lambda.\\\\\n  E[\\vartheta] &= L\\int_{-\\infty}^{\\infty} d\\lambda\\, \\rho(\\lambda)\\, \\lambda^2\\,.\n\\end{align}\nIn this work we focus on the regions of the Hilbert space that are characterized by a smooth (differentiable) filling function and are of the finite energy density: $E[\\vartheta]\/L < \\infty$.\n\nA given smooth filling function corresponds to many different microscopic eigenstates. The number of them is equal to the logarithm of the entropy $S[\\vartheta]$, the later is given by \\cite{1969_Yang_JMP_10}\n\\begin{align}\n  S[\\vartheta] &= L\\int_{-\\infty}^{\\infty} {\\rm d}\\lambda\\ s[\\vartheta; \\lambda],\\\\\n  s[\\vartheta;\\lambda] &= \\rho_t(\\lambda)\\log \\rho_t(\\lambda) - \\rho(\\lambda)\\log \\rho(\\lambda) - \\rho_h(\\lambda)\\log \\rho_h(\\lambda),\n\\end{align}\nwhere we introduced a shorthand notation\n\\begin{align}\n \\rho_t(\\lambda) &\\equiv \\rho(\\lambda)\/\\vartheta(\\lambda),\\\\\n \\rho_h(\\lambda) &\\equiv \\rho_t(\\lambda)\\left(1-\\vartheta(\\lambda)\\right),\n\\end{align}\nwhere $\\rho_t(\\lambda)$ has a meaning of the maximal density of the rapidities and $\\rho_h(\\lambda)$ denotes the density of holes.\n\n\n\nThe density operator is diagonal in the functional space of the filling functions. Its form factors are nonzero only when the two eigenstates are characterized by the same filling function $\\vartheta(\\lambda)$ and differ only by a number $n$ of excitations such that $\\lim_{\\text{th}} \\frac{n}{N} = 0$. The density form factors are zero for states containing different number of particles and therefore these excitations occur only as particle-hole pairs. We choose a set $\\{ \\lambda_j^-\\}_{j=1}^n$ of rapidities in the averaging state and we change their values to a new set $\\{ \\mu_j^+\\}_{j=1}^n$. We denote the pair of particle-hole as $\\{\\mu_j^+, \\mu_j^-\\}_{j=1}^n$, where $\\{ \\mu_j^+ \\}_{j=1}^n$ are particles: they are not present in the averaging state, while $\\{\\mu_j^-\\}_{j=1}^n$ are holes: they are related to the rapidities we have changed in the averaging state $\\{ \\lambda^-\\}_{j=1}^n$ by $1\/L$ corrections as $\\lambda_j^- = \\mu_j^- + \\left(\\frac{F(\\mu^-)}{L} \\right)$ for each $j=1, \\ldots , n$ and with the function $F(\\lambda)$  defined below. The rapidities $\\{ \\mu_j^- \\}_{j=1}^n$ are absent in the excited state. A single excitation will be denoted here with $\\mu^- \\to \\mu^+$.\n\nDue to the correlated nature of the gas, particle-hole excitations modify the density of particles not only in the vicinity of $\\mu_j^+$ and $\\mu_j^-$. In fact the density $\\rho(\\lambda)$ acquires a change of order of $1\/L$, as can be seen from studying the difference $\\mu_j - \\lambda_j$. This difference can be conveniently expressed as \\cite{KorepinBOOK}\n\\begin{align} \\label{back flow_def}\n  F(\\lambda_j) = -L\\rho_t(\\lambda_j)\\left(\\mu_j - \\lambda_j\\right) ,\n\\end{align}\nwhere $F(\\lambda)$ is the back-flow function that fulfills the following linear integral equation (for a single particle-hole excitation)\n\\begin{align} \\label{backflow}\n  2\\pi F\\left(\\lambda\\,|\\, \\mu^+, \\mu^-\\right) =&\\; \\theta(\\lambda-\\mu^+) - \\theta(\\lambda-\\mu^-) \\nonumber\\\\\n&+ \\int_{-\\infty}^{\\infty} d\\mu K(\\lambda-\\mu) \\vartheta(\\mu) F\\left(\\mu\\,|\\, \\mu^+, \\mu^-\\right) .\n\\end{align}\nThe linearity of the back-flow implies that for multiple particle-hole excitations the total back-flow is the sum of individual contributions\n\\begin{equation} \\label{back-sum-flow}\n  F\\left(\\lambda\\,|\\, \\{(\\mu_j^+, \\mu_j^-)\\}_{j=1}^n\\right) = \\sum_{j=1}^n F\\left(\\lambda\\,|\\, \\mu_j^+, \\mu_j^-\\right) .\n\\end{equation}\nIt also implies the back-flow can be further factorized in the particle and hole contributions. We define the back-flow for a single excitation as\n\\begin{align} \\label{back-single-flow}\n  2\\pi F\\left(\\lambda\\,|\\, \\mu \\right) &= \\theta(\\lambda-\\mu)\n+ \\int_{-\\infty}^{\\infty} d\\gamma K(\\lambda-\\gamma) \\vartheta(\\gamma) F\\left(\\gamma|\\, \\mu \\right) .\n\\end{align}\nThis allows to write the momentum and the energy of a single excitation as\n\\begin{align}\n & k[\\vartheta; \\mu] = \\mu  - \\int_{-\\infty}^{\\infty} \\rmd\\lambda \\vartheta(\\lambda) F(\\lambda|\\mu)\\label{exc_momentum} ,\\\\\n  & \\omega[\\vartheta; \\mu] = \\mu^2 - 2\\int_{-\\infty}^{\\infty} \\rmd \\lambda \\vartheta(\\lambda) \\lambda F(\\lambda|\\mu)\\label{exc_energy} ,\n\\end{align}\nwhich are the fundamental building blocks for the energy and momentum of a thermodynamic state with $n$ particle-holes\n\\begin{align}\n& \\Delta \\omega = \\sum_{j=1}^n \\omega[\\vartheta; \\mu^+_j] - \\omega[\\vartheta; \\mu^-_j] , \\nonumber \\\\&\n \\Delta k = \\sum_{j=1}^n k[\\vartheta; \\mu^+_j] - k[\\vartheta; \\mu^-_j] .\n\\end{align}\nThe excited states, due to the back-flow of their rapidities, have different entropy respect to the averaging state. The difference equals \\cite{1990_Korepin_NPB_340}\n\\begin{align}\n \\delta S[\\vartheta, \\mu^+, \\mu^-] = \\int_{-\\infty}^{\\infty} {\\rm d}\\lambda\\, s[\\vartheta;\\lambda]\\frac{\\partial}{\\partial \\lambda}\\left(\\frac{F(\\lambda| \\mu^+, \\mu^-)}{\\rho_t(\\lambda)}\\right)  \\equiv \\delta S[\\vartheta, \\mu^+] -  \\delta S[\\vartheta, \\mu^-] ,\\label{S_diff}\n\\end{align}\nwhere in the last step we used the back-flow function for a single excitation \\eqref{back-single-flow}. The differential entropy \\eqref{S_diff} corresponds to the number of microstates that share $n$ particle-hole excitations with the same thermodynamic energy and momentum (given by \\eqref{exc_energy}  and \\eqref{exc_momentum}) but with different sub-leading corrections to them.\n\nFinally, let us introduce the form factors and relate them to the correlation functions. We consider an ensemble average, denoted by $<\\cdot>$ of the density-density correlation function. We assume that the ensemble has a saddle-point configuration uniquely specifying the filling function $\\vartheta(\\lambda)$ \\cite{1969_Yang_JMP_10}. In order to compute the form factors, it is useful to directly refer to a specific microscopic configuration that has $\\vartheta(\\lambda)$ as its thermodynamic limit. We will choose one such finite size configuration and call it a averaging state $|{\\boldsymbol{\\lambda}}\\rangle$. Any other state, with the same filling function but with microscopic differences can be viewed as an excitation (with a positive or negative energy) over the averaging state. The choice of the averaging state is not unique, indeed there is a number $e^{S[\\vartheta]}$ of possible choices, but the correlation functions in the thermodynamic limit are independent of this choice for most of the relevant operators \\cite{KorepinBOOK}. For simplicity we choose here the averaging state with rapidities distributed such that for each interval $[\\lambda, \\lambda + d\\lambda]$ there are $\\rho(\\lambda) d\\lambda$ uniformly distributed rapidities: $ (\\lambda_j- \\lambda_k)= \\left(\\frac{j-k}{L \\rho(\\lambda_j) } \\right)+ \\mathcal{O}(L^{-2})$ when $\\lambda_j \\sim \\lambda_k$.\n\nWe define then the density density correlation function in the thermodynamic limit~as\n\\begin{align}\n  \\langle \\hat{\\rho}(x,t) \\hat{\\rho}(0,0)\\rangle = \\frac{\\langle \\vartheta|\\hat{\\rho}(x,t) \\hat{\\rho}(0,0)|\\vartheta\\rangle }{\\langle \\vartheta|\\vartheta\\rangle} = \\lim_{\\text{th}} \\frac{\\langle {\\boldsymbol{\\lambda}}|\\rho(x,t) \\rho(0,0)|{\\boldsymbol{\\lambda}}\\rangle }{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle} .\n\\end{align}\nThe correlation function in the finite system can be expanded using the complete basis of Bethe states\n\\begin{align}  \\label{corr_func_finite_N}\n\\frac{\\langle {\\boldsymbol{\\lambda}}|\\hat{\\rho}(x,t) \\hat{\\rho}(0,0)|{\\boldsymbol{\\lambda}}\\rangle}{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle} = \\sum_{\\{\\mu_j\\}_{j=1}^N} e^{ix(P_{\\mu} - P_{\\lambda}) -it(E_{\\mu} -E_{\\lambda})} |\\mathcal{F}_N\\left( {\\boldsymbol{\\mu}}, {\\boldsymbol{\\lambda}}\\right)|^2.\n\\end{align}\nwhere we defined the microscopic form factors as\n\\begin{align} \\label{micro_ff}\n  \\mathcal{F}_N\\left( {\\boldsymbol{\\mu}}, {\\boldsymbol{\\lambda}}\\right) = \\frac{\\langle{\\boldsymbol{\\mu}}| \\hat{\\rho}(0,0)|{\\boldsymbol{\\lambda}}\\rangle}{\\sqrt{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle \\langle {\\boldsymbol{\\mu}}|{\\boldsymbol{\\mu}}\\rangle}} ,\n\\end{align}\nand we used that\n\\begin{align}\n   \\hat{\\rho}(x,t) = e^{i \\left(Ht - Px\\right)} \\hat{\\rho}(0,0)e^{-i \\left(Ht - Px\\right)} .\n\\end{align}\nTo proceed further in taking the thermodynamic limit it is important to note two things. First the summation in eq.~\\eqref{corr_func_finite_N}  is constrained since the set of rapidities must be a solution to the Bethe equations \\eqref{bethe}. On the other hand the form factors become in the thermodynamic limit rather smooth functions of the filling $\\vartheta(\\lambda)$ and of the particles-holes momenta $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$. The only poles that appear are kinematic poles, when $\\mu_j^+\\rightarrow \\mu_k^-$, and they can be easily regularized (see Section~\\ref{regularization}). Therefore we do not need to evaluate the form factors precisely at the $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$ that follows from the solutions of the Bethe equations. In fact we can take now $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$ to be independent free parameters (macroscopic excitations) that we denote $\\{p_j, h_j\\}_{j=1}^n$. For each choice of $\\{p_j, h_j\\}_{j=1}^n$ there is a number $\\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right)$ (with $\\delta S$ defined in~\\eqref{S_diff}) of microscopic states which share the same form factor up to finite size corrections. In order then to use macroscopic variables we need to multiply the form factors at fixed $\\{p_j, h_j\\}_{j=1}^n$ times the number of microscopic states that are characterized by the same macroscopic excitations. This allows to define the thermodynamic limit of the form factors for smooth filling functions $\\vartheta$\n\\begin{equation}\n|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |= \\lim_{\\text{th}} \\Big( L^{n}  |\\mathcal{F}(\\vartheta; \\{\\mu^-\\}, \\{\\mu^+\\})|\\Big) \\times \\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right)  .\\label{FF_TL}\n\\end{equation}\nMoreover we can recast the sum over the macroscopic rapidites of the excitations into integrals by taking a special care of the divergences encountered (as is done in section~\\ref{regularization})\n\\begin{equation}\n\\sum_{\\mu_1^+< \\ldots < \\mu_n^+}  \\sum_{\\mu_1^-< \\ldots < \\mu_n^-}=  \\frac{1}{n!^2}\\sum_{\\{\\mu_j^+,\\mu_j^-\\}_{j=1}^n} =  L^{2n} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} dp_j  \\rho_h(p_j)   \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right).\n\\end{equation}\nwhere $ \\fint$ denotes the finite part of the integral, defined for a generic function $f(h)$ with a pole in $h=p$ as\n\\begin{equation}\n  \\fint_{-\\infty}^{\\infty} d h f(h) = \\lim_{\\epsilon \\to 0^+}  \\int_{-\\infty}^\\infty dh f(h + i \\epsilon) - \\pi i \\underset{h=p}{\\rm res} f(h).\n\\end{equation}\n\nWith this notation we can then write the correlation functions as a sum over all the possible excitations on the thermodynamic state~\\cite{1990_Korepin_NPB_340}\n\\begin{align}\n  & \\langle\\hat{\\rho}(x,t)  \\hat{\\rho}(0,0) \\rangle = \\sum_{n=0}^{\\infty} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j)\\:   \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right)|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j \\}_{j=1}^n \\rangle |^2 \\nonumber \\\\&\n\\times  \\prod_{j=1}^n  \\exp \\Big( {ix   (k(p_j) -k(h_j) ) -it (\\omega(p_j) - \\omega(h_j))}  \\Big) , \\label{corr_func_TL}\n\\end{align}\nwhere the momentum and energy of the excitation follows \\eqref{exc_momentum} and \\eqref{exc_energy} respectively.\n\nEq.~\\eqref{corr_func_TL} reminds the LeClair-Mussardo formula that appears in the context of integrable field theories~\\cite{1999_LeClair_NPB_552}\n\\begin{align}\n  & \\langle\\hat{\\rho}(x,t)  \\hat{\\rho}(0,0) \\rangle = \\sum_{n=0}^{\\infty} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j)    \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right)|\\langle 0 | \\hat{\\rho} | 0, \\{ h_j \\to p_j \\}_{j=1}^n \\rangle_{FT} |^2 \\nonumber \\\\&\n\\times  \\prod_{j=1}^n  \\exp \\Big( {ix   (k(p_j) -k(h_j) ) -it (\\omega(p_j) - \\omega(h_j))} \\Big)  \\label{corr_func_LM}.\n\\end{align}\nThe main difference between eqs.~\\eqref{corr_func_TL} and~\\eqref{corr_func_LM} are the form factors used. The field theoretic form factors simply come from excitations over a structure-less vacuum.  Here, eq.~\\eqref{corr_func_TL} suggests that the concept of vacuum is not appropriate for the strongly correlated systems. The form factors still depend \\emph{explicitly} on the properties of the averaging state (through the filling function $\\vartheta$). Conceptually this difference is responsible for the insufficiency of the field theoretical approach to the two-point correlation function (contrary to the one-point functions where the field theory approach is correct)~\\cite{2002_Saleur_NPB_602, 2002_Alvaredo_NPB_636}. On the computational level this was explicitly shown in~\\cite{1742-5468-2010-11-P11012}.\n\n\n\\subsection{Finite size corrections}\n\nIn order to compute the thermodynamic limit of the form factors we need to characterize the density of the particles and the back-flow function up to order $1\/L$. This is due to existence of products in eq.~\\eqref{ff} which are of order $N$ and in the thermodynamic limit can yield finite contributions from order $1\/L$ terms. The derivation is similar to the one presented in~\\cite{2012_Shashi_PRB_85} for the ground state distribution. Therefore here we simply state the result highlighting few differences between the two cases and for the details we refer to the section IV.B of \\cite{2012_Shashi_PRB_85}.\n\nThe filling function $\\vartheta(\\lambda)$, as well as the particle density $\\rho(\\lambda)$, comes from the thermodynamic limit of a certain class of Bethe states. Let $\\{\\lambda\\}_{j=1}^N$ be the Bethe roots of one of these states. Let us consider Bethe equations \\eqref{bethe} and define a variable $x$ that satisfies\n\\begin{align}\n  \\lambda(x) = 2\\pi x - \\frac{1}{L}\\sum_{k=1}^N \\theta(\\lambda(x) - \\lambda_k).\n\\end{align}\nClearly $\\lambda(I_j\/L) = \\lambda_j$ but we allow here $x$ to take any real value. Therefore $dx\/d\\lambda$ has a meaning of a number of possible quantum numbers in the range $d\\lambda$. Thus\n\\begin{align}\n  \\frac{dx}{d\\lambda} = \\rho_t(\\lambda).\n\\end{align}\nand from the Euler-Maclaurin formula we have\n\\begin{align}\n  \\rho_t(\\lambda) = \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{\\lambda_1}^{\\lambda_N} K(\\lambda-\\mu) \\rho(\\mu) d\\mu + \\frac{1}{2L}\\left(K(\\lambda-\\lambda_N) - K(\\lambda - \\lambda_1)\\right),\n\\end{align}\nwhere $\\lambda_{1,N}$ are the smallest and largest rapidities respectively. Incorporating the $1\/L$ in the boundaries of the integral yields\n\\begin{align}\n   \\rho_t(\\lambda) = \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-q}^{q} K(\\lambda-\\mu) \\rho(\\mu) d\\mu,\n\\end{align}\nwith $q_L = \\lambda_1 + 1\/(2L\\rho_p(\\lambda_1))$ and $q_R = \\lambda_N + 1\/(2L\\rho_p(\\lambda_N))$. In the thermodynamic limit we have $q_{R,L}\\rightarrow\\pm\\infty$ and it is convenient to separate the thermodynamic part from the finite size corrections. Before doing so, let us note that we can bound the finite-size corrections from above by choosing $q = \\textrm{min}(|q_L|, q_R)$. We have\n\\begin{align}\n \\rho_t(\\lambda) =& \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu - \\frac{1}{2\\pi}\\int_{q}^{\\infty} \\nonumber \\left(K(\\lambda-\\mu)+K(\\lambda+\\mu)\\right) \\rho(\\mu) d\\mu.\n\\\\ =& \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu\\nonumber\\\\ &- \\frac{1}{2\\pi}\\int_{0}^{\\infty} \\left(K(\\lambda-\\mu-q)+K(\\lambda+\\mu+q)\\right) \\rho(\\mu+q) d\\mu.\n\\end{align}\nThe last term that controls the finite-size corrections can be easily bounded (where $M$ is a positive constant)\n\\begin{align}\n  \\frac{1}{2\\pi}\\int_{q}^{\\infty} \\left(K(\\lambda-\\mu)+K(\\lambda+\\mu)\\right) \\rho(\\mu) d\\mu \\leq M\\frac{\\rho(q)}{2\\pi},\n\\end{align}\nand the finite size corrections are proportional to $\\rho(q)$. For the energy of the state to be finite we require $\\rho(\\lambda) \\sim \\lambda^{-3-\\epsilon}$ for large $\\lambda$ (c.f. \\eqref{energy_cont}). The boundary $q$ itself is a monotonically increasing function of $N$. For the particle density to spread over the whole real line, rather than to accumulate in the final interval of it, we should have $q \\sim N^{1+\\delta}$ with $\\delta > 0$. Therefore $\\rho(q) \\sim N^{-3-\\gamma}$ with $\\gamma>0$ and the finite-size corrections are at least of order $1\/L^3$ and thus are negligible in the further analysis.\nWe have\n\\begin{align}\n  \\rho_t(\\lambda) &= \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu + \\mathcal{O}(1\/L^3).\n\\end{align}\n\nThe finite-size corrections to the particle and hole rapidities follow from analogous computations as presented in \\cite{2012_Shashi_PRB_85} and are given by\n\\begin{align}\\label{finite_size_excitations}\n  \\mu_0^+ &= 2\\pi \\frac{I^+}{L} - \\int_{-\\infty}^{\\infty} \\theta(\\mu_0^+ - \\lambda) \\rho(\\lambda) d\\lambda, \\;\\;\\;\\;\\;\\;\n  \\mu_{1\/L}^+ = -\\frac{F(\\mu_0^+)}{L\\rho_t(\\mu_0^+)}, \\\\\n  \\lambda_0^- &= 2\\pi \\frac{I^-}{L} - \\int_{-\\infty}^{\\infty} \\theta(\\lambda_0^- - \\lambda) \\rho(\\lambda) d\\lambda, \\;\\;\\;\\;\\;\\;\n  \\lambda_{1\/L}^- = 0.\n\\end{align}\nFor the auxiliary rapidity $\\mu^-$ we have\n\\begin{align}\n  \\mu_0^- = \\lambda_0^-, \\;\\;\\;\\;\\;\\;\n  \\mu_{1\/L}^- = -\\frac{F(\\lambda_0^-)}{L\\rho_t(\\mu_0^-)},\n\\end{align}\nWe can consider now finite-size corrections to the back-flow function. The thermodynamic limit is given by eq. \\eqref{backflow}. The leading finite size corrections are\n\\begin{align}\n  2\\pi F_{1\/L}(\\lambda) &= \\left(\\int_{q_R}^{\\infty} + \\int_{-\\infty}^{q_L}\\right)d\\mu\\, K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu) \\nonumber\\\\\n  &- \\frac{1}{L}\\left[ K(\\lambda,\\mu_0^+)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^+)}{\\rho_t(\\mu_0^+)}\\right) - K(\\lambda,\\mu_0^-)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^-)}{\\rho_t(\\mu_0^-)}\\right)\\right] \\nonumber\\\\\n & + \\frac{1}{2L} \\int_{-\\infty}^{\\infty} d\\mu\\, \\rho(\\mu)K'(\\lambda-\\mu) \\left(\\frac{F(\\lambda)}{\\rho_t(\\lambda)} - \\frac{F(\\mu)}{\\rho_t(\\mu)}\\right)^2.\n\\end{align}\nThe first integral can be estimated in the following way\n\\begin{align}\n\\left|\\int_{q_R}^{\\infty} d\\mu\\,K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu)\\right| &\\leq \\int_{q_R}^{\\infty} d\\mu \\left|\\,K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu)\\right| \\nonumber\\\\ &\\leq |\\vartheta(q_R)| \\int_{q_R}^{\\infty} d\\mu \\left|K(\\lambda,\\mu) F(\\mu)\\right|.\n\\end{align}\nBut $\\vartheta(q_R)$ is proportional to $\\rho(q_R)$ which is of order $1\/L^3$. The same holds for the other integral between $-\\infty$ and $q_L$. Therefore both integrals can be neglected in the leading order and\n\\begin{align}\n  2\\pi L F_{1\/L}(\\lambda) =& - \\left[ K(\\lambda,\\mu_0^+)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^+)}{\\rho_t(\\mu_0^+)}\\right) - K(\\lambda,\\mu_0^-)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^-)}{\\rho_t(\\mu_0^-)}\\right)\\right] \\nonumber\\\\\n&+ \\frac{1}{2} \\int_{-\\infty}^{\\infty} d\\mu\\, \\rho(\\mu)K'(\\lambda-\\mu) \\left(\\frac{F(\\lambda)}{\\rho_t(\\lambda)} - \\frac{F(\\mu)}{\\rho_t(\\mu)}\\right)^2.\n\\end{align}\nFinally the density of the excited state is\n\\begin{align}\n  \\rho_{t,ex}(\\mu_j) = \\rho_t(\\lambda_j) + \\frac{1}{L}\\left(F'(\\lambda_j) - F(\\lambda_j)\\frac{\\rho_t'(\\lambda_j)}{\\rho_t(\\lambda_j)} \\right).\n\\end{align}\nThis completes the list of formulas required to take properly the thermodynamic limit of the form factors. This is achieved in the next section.\n\n\n\\section{Thermodynamic limit for smooth distribution of rapidities} \\label{thermodynamiclimit}\n\nIn this section we calculate the thermodynamic limit of the finite size (normalized) form factors~\\eqref{micro_ff} (see also~\\eqref{ff} and~\\eqref{norm} for explicit formulas). We are interested in the leading term in $1\/L$ so with the equivalence $\\sim$ we denote that we are neglecting extra sub-leading correction in $1\/L$.  We proceed as in~\\cite{2012_Shashi_PRB_85} since the two calculations share many common steps. Introducing the short-hand notation\n\\begin{equation}\n\\mu^{}_{ij} = \\mu^{}_i - \\mu^{}_j ,\n\\end{equation}\nwe start from an intermediate expression for the thermodynamic limit of the form factors given by (c.f. eq. 88 in \\cite{2012_Shashi_PRB_85})\n\\begin{align}\\label{starting}\n|\\mathcal{F}_N| \\sim& \\prod_{i=1}^n\\left( \\frac{F_L(\\lambda^-_i)}{L(\\rho_L(\\lambda^-_i)\\rho_{ex}(\\mu^+_i))^{1\/2}(\\lambda_i^- - \\mu_i^+)}\\right)\\frac{i \\Delta k \\: {\\det_N}(\\delta_{jk} + U_{jk})}{(V_p^+ - V_p^-){\\rm Det}\\left(1-\\frac{\\hat{K \\vartheta}}{2\\pi} \\right)}\\nonumber\\\\\n&\\times \\left[\\prod_{j,k} \\left(\\frac{(\\lambda_j - \\mu_k +ic)(\\lambda_j - \\mu_k + ic)}{(\\lambda_{jk} + ic)(\\mu_{jk} + ic)}\\right)^{1\/2}\\right] \\nonumber\\\\ &\n\\times \\left\\{\\prod_j\\frac{{\\rm sin}(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j\\neq k}\\left(\\frac{\\lambda_{jk} \\mu_{jk}}{(\\mu_j - \\lambda_k)^2}\\right)^{1\/2}\\right\\}\\nonumber\\\\ \\times\n&{\\rm exp}\\left[\\int_{-\\infty}^\\infty d\\lambda \\vartheta(\\lambda) \\left(\\frac{\\pi F_L(\\lambda) {\\rm cos}(\\pi F_L(\\lambda))}{ {\\rm sin}(\\pi F_L(\\lambda))} - 1\\right) \\right] \\nonumber \\\\ & \\times \\exp \\left[ \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\left[ \\left( F_L'(\\lambda) - \\frac{F_L(\\lambda)\\rho_t'(\\lambda)}{2\\rho_t(\\lambda)}\\right) + \\frac{1}{2}F_L'(\\lambda) \\right]\\right] .\n\\end{align}\nwhere ${\\rm Det}( 1- \\frac{K \\vartheta }{2 \\pi})$ is the Fredholm determinant of the kernel\n\\begin{align}\n -\\left[\\frac{K \\vartheta }{2 \\pi}\\right](\\mu,\\nu) = -\\frac{1}{2 \\pi}\\frac{2c}{(\\mu - \\nu)^2 + c^2} \\vartheta(\\nu) ,\n\\end{align}\nIn eq. \\eqref{starting} there are three groups of elements which are still written for a finite system. We denote them as\n\\begin{align}\n  M_1 &= \\prod_{j,k} \\left(\\frac{(\\lambda_j - \\mu_k +ic)(\\lambda_j - \\mu_k + ic)}{(\\lambda_{jk} + ic)(\\mu_{jk} + ic)}\\right)^{1\/2}, \\\\\n  M_2 &= \\prod_j\\frac{{\\rm sin}(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j\\neq k}\\left(\\frac{\\lambda_{jk} \\mu_{jk}}{(\\mu_j - \\lambda_k)^2}\\right)^{1\/2}, \\\\\n  \\Theta &= \\frac{\\det_N (\\delta_{jk} + U_{ij})}{V_p^+ - V_p^- }.\n\\end{align}\nThe thermodynamic limit of them requires some work. Calculation of $M_1$ is exactly the same as for the ground state form factors and thus we do not reproduce it here. For the details we refer again to \\cite{2012_Shashi_PRB_85}. On the other hand the term $M_2$ has a manifestly different thermodynamic limit and is responsible for different size dependence of the ground state (critical) form factors and the finite entropy state form factors. Computations are presented in the next section. The thermodynamic limit of $\\Theta$ was computed in \\cite{2012_Shashi_PRB_85} only for a specific type of excitations. As we require here the form factors for a generic particle-hole excitations we have to generalize the previous calculations. This is done in the subsequent section.\n\n\\subsection{\\texorpdfstring{Evalutation of $M_2$}{Evaluation of M(2)}}\nWe focus here on the evaluation of the double products given by\n\\begin{equation}\nM_2 = \\prod_j \\frac{\\sin(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j \\neq k =1}^N \\left(\\frac{\\lambda_{jk} \\mu_{kj}}{(\\mu_k - \\lambda_j)^2} \\right)^{1\/2} ,\n\\end{equation}\nwhich present formal differences in the thermodynamic limit when the states is described by a smooth distribution or when the distribution is discontinuous as for the ground state. Differently from the ground state situation this term is not expected to produce power law divergences in the system size as $1\/L^\\alpha$.\n\nFollowing \\cite{2012_Shashi_PRB_85} we decompose the product in three pieces\n\\begin{equation}\nM= T'' \\times T_{holes} \\times T_{particles} ,\n\\end{equation}\ndepending on which rapidities we let the sum run over.\n\\begin{align}\nT'' &=\\prod_{j \\neq k}'' \\left(\\frac{\\lambda_{jk} \\mu_{kj}}{(\\mu_k - \\lambda_j)^2} \\right)^{1\/2} \\nonumber \\\\\n&\\sim \\prod_{j \\neq k}'' \\left(1 + \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} \\left(1 - \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} \\nonumber\\\\\n&\\times \\left(1 + \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)}  -\\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_j)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} ,\n\\end{align}\nwhere by $\\prod_{j \\neq k}''$ we denoted the product where we excluded the particles excitations $\\{ \\mu^+_j \\}_{j=1}^n$ but we included the holes $\\{ \\mu^-_j \\}_{j=1}^n$. When the two rapidities get closer, i.e.\\\nwhen $j \\in [k-n^*, k+ n^*]$ where $n$ is a given sub-extensive cut-off such that $ n \\propto L^{1- \\alpha}$ with $\\alpha<1\/2$, then we substitute for the difference between the two\n\\begin{equation}\\label{Expanded_rap}\nL \\rho_t(\\lambda_j )(\\lambda_j- \\lambda_k)= \\frac{j-k}{\\vartheta(\\lambda_j)}  +  \\frac{ (j-k)^2 \\partial_\\lambda \\rho_t(\\lambda_j)}{2 L \\rho(\\lambda_j)^2} ,\n\\end{equation}\nwhile for all the other $j$ we can just exchange the sum for an integral over the rapidities (see figure~\\ref{M2_regions}).\nThe cut-off $n^*$ delimits the region where the approximation \\eqref{Expanded_rap} start to break down, which corresponds to a distance in rapidities\n\\begin{equation}\n|\\lambda_j - \\lambda_k | \\equiv \\nu^*(\\lambda) = \\frac{n^*}{L \\rho(\\lambda)} + \\mathcal{O}(n^*\/L^2) .\n\\end{equation}\nWe denote the two regions in $\\lambda_j - \\lambda_k$ separated by the cut-off as the region $I$ (smooth part) and $II$ (discrete part) where the approximation \\eqref{Expanded_rap} is valid (IIa where $j < k$ and IIb where $k< j$ ). $T''$ is then given by the product of these three terms\n\\begin{equation}\nT'' = T_I \\times T_{IIa} \\times T_{IIb} ,\n\\end{equation}\nFor the fist term we have the following\n\\begin{align}\n  \\log T_1 =& \\left[\\frac{1}{2} \\sum_{j \\neq [k - n^* , k+n^*]} \\frac{F_L(\\lambda_j) F_L(\\lambda_k)}{L^2 \\rho_t(\\lambda_j)\\rho_t(\\lambda_k)(\\lambda_j- \\lambda_j)} \\right] \\nonumber\\\\\n  \\sim& \\frac{1}{2} \\left( \\int d\\lambda \\vartheta(\\lambda) \\int_{-\\infty}^{\\lambda - \\nu^*(\\lambda)} \\!\\!d \\mu \\vartheta(\\mu) \\frac{F(\\lambda) F(\\mu)}{(\\lambda - \\mu)^2} +\\int d\\lambda \\vartheta(\\lambda) \\int_{\\lambda + \\nu^*(\\lambda)}^{\\infty} \\!\\!d\\mu \\vartheta(\\mu) \\frac{F(\\lambda) F(\\mu)}{(\\lambda - \\mu)^2} \\right) \\nonumber \\\\\n  =& -\\frac{1}{4} \\int d\\lambda  \\int d \\mu  \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2}  \\nonumber \\\\\n  &+ \\frac{1}{2} \\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2\\left( \\int_{-\\infty}^{\\lambda - \\nu^*(\\lambda)} d \\mu \\frac{1}{(\\lambda - \\mu)^2} + \\int_{\\lambda + \\nu^*(\\lambda)}^{\\infty} d \\mu  \\frac{1}{(\\lambda - \\mu)^2} \\right) \\nonumber \\\\\n  =& -\\frac{1}{4} \\int d\\lambda  \\int d \\mu  \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2}\n  + \\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\frac{1}{\\nu^*(\\lambda)}  \\nonumber \\\\\n  =& -\\frac{1}{4} \\int d\\lambda  \\int d \\mu  \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2} \\nonumber \\\\\n  &+ \\frac{L}{n^*}\\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\rho(\\lambda)  + \\mathcal{O}\\left(\\frac{L}{(n^*)^2} \\right).\n\\end{align}\nThe computation in the sector II can be done analogously as  in \\cite{2012_Shashi_PRB_85}  leading to\n\\begin{align}\n\\log T_{II}\n= & -  \\int d\\lambda  \\frac{\\rho'_t(\\lambda)}{\\rho_t(\\lambda)}\\frac{ F(\\lambda)} {2\\rho(\\lambda) \\partial_\\lambda(F(\\lambda)\\vartheta(\\lambda))}  \\frac{\\partial }{\\partial \\lambda} \\log \\left( \\frac{\\pi F(\\lambda) \\vartheta(\\lambda)}{\\sin \\pi F(\\lambda) \\vartheta(\\lambda)} \\right) \\\\& - \\frac{L}{ n^*}\\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\rho(\\lambda)  + \\mathcal{O}(L\/(n^*)^2) ,\n\\end{align}\nwhere the cut-off depended part cancels exactly the one in $T_I$ leading to a cut-off independent result.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{M2_regions.pdf}\\label{M2_regions}\n\\caption{Schematic of range of $\\lambda$'s. Cutoff parameter n* is used to separate regions I, II. Figure is taken from \\cite{2012_Shashi_PRB_85} and there are two extra regions (IIIa and IIIb) depicted. These are the regions in which the two rapidities are close two each other and simultaneously they are close to the edge of the distribution. For the smooth distribution of rapidities there is no edge and consequently the region III does not appear in the present calculations. Ultimately the existence or lack of region III determines the scaling of the form factors with the system size $L$. It is the region III that leads to the fractional power of the system size for the ground state form factors and consequently to the criticality of zero temperature 1D Bose gas. This shows an intimate connection between the distribution of rapidities and the form of the correlation function. (Figure reproduced from~\\cite{2012_Shashi_PRB_85}, courtesy of A. Shashi).}\n\\end{figure}\n\n\nNow we are left only with the computation of $T_{particle}$ and $T_{hole}$ defined as\n\\begin{equation}\nT_{hole} \\sim \\prod_{j}' \\prod_{k=1}^n \\left( 1 - \\frac{F_L(\\lambda_j)}{L \\rho(\\lambda_j)(\\lambda_j - \\lambda_k^-) } \\right)^{-1} ,\n\\end{equation}\n\\begin{equation}\nT_{particle} \\sim \\prod_{j}' \\prod_{k=1}^n \\left( 1 - \\frac{F_L(\\lambda_j)}{L \\rho(\\lambda_j)(\\lambda_j - \\lambda_k^-) } \\right) ,\n\\end{equation}\nwhere the product $\\prod_{j}'$ runs respectively over all the rapidities in the state but not over the holes(particle).\nProceeding as in \\cite{2012_Shashi_PRB_85} and using the fact that for a smooth distribution particle and holes are always arbitrarily close to an extensive number of rapidities of the state, we obtain\n\\begin{equation}\nT_{hole} =  \\prod_{k=1}^n \\frac{\\sin \\pi F(\\mu_k^-) \\vartheta(\\mu_k^-)}{\\pi F(\\mu_k^-)  \\vartheta(\\mu_k^-) } \\exp\\left( \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^-) }{(\\lambda - \\mu_k^-)  } \\right) .\n\\end{equation}\nFor $T_{particle}$ we have the same result\n\\begin{equation}\nT_{particle} = \\prod_{k=1}^n  \\frac{\\pi F(\\mu_k^+)   \\vartheta(\\mu_k^+) }{  \\sin \\pi F(\\mu_k^+)  \\vartheta(\\mu_k^+)} \\exp\\left( - \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^+) }{(\\lambda - \\mu_k^+)  } \\right) .\n\\end{equation}\n\nFinally we can then write for the whole factor $M_2$ in the thermodynamic limit\n\\begin{align}\n  M_2 & = \\exp\\left[ -  \\int d\\lambda  \\frac{\\rho'_t(\\lambda)}{\\rho_t(\\lambda)}\\frac{ F(\\lambda)} {2\\rho(\\lambda) \\partial_\\lambda(F(\\lambda)\\vartheta(\\lambda))}  \\frac{\\partial }{\\partial \\lambda} \\log\\left( \\frac{\\pi F(\\lambda) \\vartheta(\\lambda)}{\\sin \\pi F(\\lambda) \\vartheta(\\lambda)} \\right) \\right] \\nonumber \\\\&\n  \\times \\prod_{k=1}^n \\frac{\\sin \\pi F(\\mu_k^-) \\vartheta(\\mu_k^-)}{\\pi F(\\mu_k^-)  \\vartheta(\\mu_k^-) }  \\frac{\\pi F(\\mu_k^+)   \\vartheta(\\mu_k^+) }{  \\sin \\pi F(\\mu_k^+)  \\vartheta(\\mu_k^+)}  \\nonumber \\\\& \\times\n  \\exp\\left[ \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^-) }{(\\lambda - \\mu_k^-)  } )  - \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^+) }{(\\lambda - \\mu_k^+)  } \\right] \\nonumber \\\\& \\times\n  \\exp\\left[ \\frac{-1}{4} \\int d\\lambda  \\int d \\mu  \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2} \\right] ,\\label{M2_final}\n\\end{align}\n\n\n\\subsection{Fredholm determinant}\nWe are left with the problem of computing the thermodynamic limit of the determinant\n\\begin{equation}\\label{theta}\n\\Theta = \\frac{\\det_N (\\delta_{jk} + U_{jk})}{V_p^+ - V_p^- }\n\\end{equation}\nwhere the matrix $U$ is given in \\eqref{matrix_U} and $\\lambda_p$ is an arbitrary number. Analogously as is done in \\cite{2012_Shashi_PRB_85} we can take the limit $\\lambda_p \\to \\infty$ leading to\n\\begin{equation}\n\\Theta = \\frac{i c}{2 \\Delta k } \\det \\left(\\delta_{jk} + \\frac{i (\\mu_j - \\lambda_j)}{V_j^+ - V_j^- } \\prod_{m \\neq j} \\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j} \\left( \\frac{2 c}{(\\lambda_j - \\lambda_k)^2 + c^2} - \\frac{2}{c} \\right) \\right)\n\\end{equation}\nIt is useful to consider a vector\n\\begin{equation}\na_j = \\frac{i (\\mu_j - \\lambda_j)}{V_j^+ - V_j^-} \\prod_{m \\neq j} \\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j}.\n\\end{equation}\nWith this notation the determinant in \\eqref{theta} is expressed as\n\\begin{align}\n  \\det_N \\left(  \\delta_{ij}+ A_{ij}\\right),\n\\end{align}\nwith the matrix $A$ given by\n\\begin{align}\n  A_{jk} = a_j \\left(K(\\lambda_j-\\lambda_k) - \\frac{2}{c} \\right)\n\\end{align}\nDepending on $j$ the vector $a_j$ has a different scaling behaviors with the system size\n \\begin{align}\n   a_j \\sim \\begin{cases}\n     \\mathcal{O}(1\/L), &\\lambda_j \\notin \\{\\lambda_j^-\\}_{j=1}^n\\\\\n     \\mathcal{O}(1), &\\lambda_j \\in \\{\\lambda_j^-\\}_{j=1}^n\n   \\end{cases}.\n\\end{align}\nWe can use this property to simplify the computation of the determinant.\nWe denote by $\\tilde{a}_j$ a vector $a_j$ in which we substitute $\\mu_j$ by $\\mu_j^-$. Correspondingly we define a matrix $\\tilde{A}_{jk}$ with $\\tilde{a}_j$ as a prefactor instead of $a_j$. Thus the matrix elements of $\\tilde{A}_{jk}$ are all of $\\mathcal{O}(1\/L)$. By $B_{jk}$ we denote the difference of the two matrices: $B_{jk} = A_{jk} - \\tilde{A}_{jk}$. Note that the matrix $B_{jk}$ has only $n$ non-zero rows corresponding to the excited rapidites $\\{ \\lambda_j^-\\}_{j=1}^n$. Using standard proprieties of the determinant and assuming the matrix $\\delta_{ij} +\\tilde{A}_{ij} $ is invertible we can recast the determinant in a product of the determinant of an $N\\times N$ matrix and the determinant of an $n\\times n$ one\n\\begin{equation}\\label{two_det}\n  \\det_N (\\delta_{ij}+ \\tilde{A}_{ij}+ B_{ij} ) =  \\det_N( \\delta_{ij} +\\tilde{A}_{ij} ) \\times \\det_n \\left(   \\delta_{ij} + \\sum_{k=1}^N B_{ik} \\left(1+ \\tilde{A}\\right)^{-1}_{kj}\\right),\n\\end{equation}\nwhere the indices $i,j$ in the second determinant run only over the $n$ excited rapidities\nand $1$ denotes here the identity matrix. We can now take the thermodynamic limit. The first determinant becomes a Fredholm determinant ${\\rm Det}(1 + \\hat{A})$ with the kernel\n\\begin{equation} \\label{kernelA}\n  \\hat{A}(\\lambda, \\mu) = \\tilde{a}(\\lambda)\\left( K(\\lambda-\\mu) - \\frac{2}{c}\\right),\n\\end{equation}\nwhere $\\tilde{a}(\\lambda_j)$ is the thermodynamic limit of $\\rho(\\lambda_j){a}_j$. Standard computations give \\cite{2012_Shashi_PRB_85}\n\\begin{align}\n  \\tilde{a}(\\lambda ) = &  \\frac{\\vartheta(\\lambda)F(\\lambda)}{\\Gamma[1 + \\vartheta(\\lambda)F(\\lambda)]  \\Gamma[1 - \\vartheta(\\lambda)F(\\lambda)]} \\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n} \\frac{ \\mu_k^+ - \\lambda}{\\mu_k^- - \\lambda}\\exp \\left[ - {\\rm PV}\\! \\int d\\mu  \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda} \\right]\n    \\nonumber \\\\& \\times\n\\left[2 {\\rm Im} \\left( \\prod_{k=1}^n \\frac{\\mu_k^+ - \\lambda + i c}{\\mu_k^- - \\lambda+ ic}   \\exp\\left[- \\int d\\mu \\frac{\\vartheta(\\mu)F(\\mu)}{\\mu - \\lambda + ic} \\right] \\right)\\right]^{-1}. \\label{a_tilde_derivation}\n\\end{align}\nwhere the product $\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n} $ runs over all the holes (particles) except the $j-$th one when $\\lambda = \\mu_j^{-}$ for any $j=1,\\ldots, n$. With ${\\rm PV} \\int$ we denoted the principal value of the integral\n\\begin{equation}\n{\\rm PV} \\int dx \\frac{f(x)}{x} = \\lim_{\\epsilon \\to 0} \\left( \\int_{-\\infty}^{-\\epsilon}dx \\frac{f(x)}{x} + \\int_{\\epsilon}^{\\infty} dx \\frac{f(x)}{x} \\right)\n\\end{equation}\n The second line of eq.~\\eqref{a_tilde_derivation} can be still simplified. We separate the exponential term into real and imaginary parts and for the imaginary part use the integral equation for the backflow \\eqref{backflow} to obtain\n\\begin{align}\n  &\\exp \\left(-\\int d\\mu \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda +ic}\\right) = \\exp\\left(-\\frac{1}{2c} \\int d\\mu (\\mu-\\lambda)\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)\\right) \\nonumber\\\\ &\\times \\exp\\left(i\\pi F(\\lambda) -\\frac{i}{2}\\sum_{k=1}^n \\left(\\theta(\\lambda-\\mu_k^+) - \\theta(\\lambda - \\mu_k^-) \\right) \\right).\n\\end{align}\nFrom the definition of $\\theta(\\lambda) = 2{\\rm atan}(\\lambda\/c)$~\\eqref{phase_shift} and an identity \\\n\\begin{align}\n  \\exp\\left(2i{\\rm atan}(x)\\right) = \\frac{1 + ix}{1 - ix},\n\\end{align}\nit follows that\n\\begin{align}\n  &{\\rm Im}\\left[ \\prod_k^n\\frac{\\mu_k^+ - \\lambda +ic}{\\mu_k^- - \\lambda + ic}\\exp\\left(-\\int d\\mu \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda +ic}\\right) \\right] = \\sin \\pi F(\\lambda) \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^- - \\lambda)}{K(\\mu_k^+ -\\lambda)} \\right)^{1\/2}\n\\nonumber\\\\\n&\\times\n\\exp\\left(-\\frac{1}{2c} \\int d\\mu (\\mu-\\lambda)\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)\\right) .\n\\end{align}\nUsing Euler's reflection formula for $\\Gamma$ functions\n\\begin{align}\n  \\Gamma(1-z)\\Gamma(1+z) = \\frac{\\pi z}{\\sin \\pi z},\n\\end{align}\nwe obtain\n\\begin{align}\n  &\\tilde{a}(\\lambda) = \\frac{\\sin[ \\pi \\vartheta(\\lambda) F(\\lambda)]}{2\\pi \\sin [ \\pi F(\\lambda)]} \\nonumber\\\\& \\times  \\frac{\\prod_{k=1}^n \\mu_k^+ - \\lambda}{\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n}\\mu_k^- - \\lambda} \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^+ -\\lambda)}{K(\\mu_k^- - \\lambda)} \\right)^{1\/2}\n   \\exp \\left[ \\frac{c}{2}{\\rm PV}\\! \\int d\\mu  \\frac{\\vartheta(\\mu)F(\\mu) K(\\mu - \\lambda)}{\\mu - \\lambda} \\right] .\n\\end{align}\n\n\nIn the $n \\times n$ determinant in \\eqref{two_det} we note that we can neglect the $1\/L$ corrections to $B$ since $\\lim_{\\text{th}} \\frac{n}{L}=0$. We introduce the matrix $W = A (1+\\tilde{A})^{-1}$ as the solution of the following equation\n\\begin{equation}\nW_{ij} + \\sum_{k=1}^N W_{ik} \\tilde{A}_{kj}= A_{ij},\\;\\;\\;i,j=1,\\dots,n,\n\\end{equation}\nwhich in the thermodynamic limit becomes a linear integral equations for the function $W(\\lambda , \\mu)$\n\\begin{equation}\\label{kernelW}\n  W(\\lambda, \\mu) + \\int_{-\\infty}^\\infty d\\alpha W(\\lambda, \\alpha) \\tilde{a}(\\alpha) \\left(K(\\alpha - \\mu) - \\frac{2}{c} \\right)    =    b( \\lambda)\\left(K(\\lambda - \\mu) - \\frac{2}{c} \\right)   ,\n\\end{equation}\nwith the vector $b(\\lambda)$ given by\n\\begin{align}\n  b(\\lambda) & =- \\frac{\\sin [\\pi \\vartheta(\\lambda) F(\\lambda)]}{2\\pi \\vartheta(\\lambda) F(\\lambda) \\sin [\\pi F(\\lambda)]} \\nonumber\\\\\n  &\\times \\frac{\\prod_{k=1}^n \\mu_k^+ - \\lambda}{\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n}\\mu_k^- - \\lambda} \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^+ -\\lambda)}{K(\\mu_k^- - \\lambda)} \\right)^{1\/2}\n\\exp \\left[ \\frac{c}{2}{ \\rm PV} \\int d\\mu\\frac{\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)}{\\lambda - \\mu}\\right],\n\\end{align}\n\nPutting everything together we have then\n\\begin{equation}\n  \\lim\\nolimits_\\text{th} \\det_N (\\delta_{ij}+ \\tilde{A}_{ij}+ B_{ij} ) =  {\\rm Det}( 1 +\\hat{A} ) \\det_n \\left(  \\delta_{ij} + W(\\mu_i^-,\\mu_j^-) \\right),\n\\end{equation}\nand the determinant part of the form factors is then expressed as\n\\begin{equation}\n  \\Theta = \\frac{i c}{2 \\Delta k} {\\rm Det}( 1+ \\hat{A})   {\\rm det}_n \\left(  \\delta_{ij} + W(\\mu_i^-,\\mu_j^-) \\right). \\label{theta_final}\n\\end{equation}\nNote that the Fredholm determinant is still a function of the excitations and not only of the the averaging state. This unfortunately poses still serious problems to the computation of correlation functions. However from the numerical point of view its evaluation can be effectively approximated with the very first terms of its expansion in powers of the trace.\n\n\\subsection{Final result}\nWe report here the final expression for the thermodynamic limit of the form factors of the density operator between the representative state given by a smooth distribution $\\rho(\\lambda)$ and a number $n$ of particle-hole excitations. Most of the remaining computations can be carried out exactly as is done in \\cite{2012_Shashi_PRB_85} by simply rescalling the shift function as $\\tilde{F}(\\lambda) = \\vartheta (\\lambda)F(\\lambda)$.\nCombining the partial thermodynamic limit of the form factors from eq. \\eqref{starting} with the results for $M_1$, $M_2$ (\\eqref{M2_final} and $\\Theta$ (\\eqref{theta_final} we find the form factors between a thermodynamic state $|\\vartheta\\rangle$ and one of its excited states with $n$ particle-holes, as defined in \\eqref{FF_TL}, to be\n\\begin{align}\\label{FFd_final_expression}\n  & |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |= \\nonumber \\\\&\n  \\frac{c}{2} \\left[\\prod_{k=1}^n \\frac{F(h_k)}{ (\\rho_t(p_k) \\rho_t(h_k))^{1\/2} } \\frac{\\pi \\tilde{F}(p_k)    }{  \\sin \\pi \\tilde{F}(p_k)  } \\: \\frac{\\sin \\pi \\tilde{F}(h_k)}{\\pi \\tilde{F}(h_k)   } \\right] \\nonumber \\\\&  \\times\n  \\prod_{i,j=1}^n \\left[\\frac{(p_i - h_j + i c)^2}{(h_{i,j} + ic)(p_{i,j} + ic)} \\right]^{1\/2} \\frac{\\prod_{i< j =1}^n  h_{ij} p_{ij}}{\\prod_{i , j} (p_i - h_j)}\n  \\det_n \\left(  \\delta_{ij} + W(h_i,h_j) \\right) \\nonumber \\\\& \\times \\exp\\left(- \\frac{1}{4} \\int d\\lambda \\int d\\mu \\left( \\frac{\\tilde{F}(\\lambda) - \\tilde{F}(\\mu)}{\\lambda - \\mu}\\right)^2   - \\frac{1}{2} \\int d\\mu d \\lambda \\left( \\frac{\\tilde{F}(\\lambda)\\tilde{F}(\\mu)}{(\\lambda - \\mu + i c)^2}\\right) \\right) \\nonumber \\\\& \\times\n  \\exp\\left( \\sum_{k=1}^n  {\\rm PV} \\int_{-\\infty}^{\\infty} d\\lambda  \\: \\frac{\\tilde{F}(\\lambda)  (h_k - p_k)  }{(\\lambda - h_k) ( \\lambda - p_k)   }+  \\int d\\lambda \\frac{\\tilde{F}(\\lambda) (p_k - h_k)}{(\\lambda - h_k + i c) (\\lambda - p_k + i c)}  \\right)\n  \\nonumber \\\\& \\times \\frac{{\\rm Det}\\left(1 + \\hat{A} \\right)}{{\\rm Det}\\left(1  - \\frac{K \\vartheta}{2 \\pi }\\right)} \\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right),\n\\end{align}\nwith the kernels $\\hat{A}$ and $W$ given respectively in \\eqref{kernelA} and \\eqref{kernelW}. The form factors are now completely characterized by thermodynamic data. Knowing the $\\vartheta(\\lambda)$ function we can find the density $\\rho_t(\\lambda)$. Specifying the rapidities of the excitations $\\{h_j\\rightarrow p_j\\}_{j=1}^n$ the back-flow function $F(\\lambda|\\{h_j\\rightarrow p_j\\}_{j=1}^n)$ and the form factor itself follows. Note that in order to have a complete resolution of identity we need to include also the diagonal form factor with $n=0$\n\\begin{equation} \\label{diagonal}\n|\\langle \\vartheta | \\hat{\\rho} | \\vartheta \\rangle |= D\n\\end{equation}\nwhere the density of particles can be chosen to be unitary $D=1$.\n\nThe expression \\eqref{FFd_final_expression} is complicated and the meaning of many terms is rather obscure. The main difficulty is hidden in the Fredholm determinant which depends on the excitations and a factorization of it is still not possible. In order to have some insight on the structure of the form factors it is interesting to consider the small density limit. That is we let $\\vartheta(\\lambda) \\approx 0$ and obtain\n\\begin{align}\n  & |\\langle 0 | \\hat{\\rho} | 0, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |=\n\\frac{c}{2} \\left[\\prod_{k=1}^n  \\sum_{l=1}^n (\\theta(p_l - h_k)  - \\theta(h_l - h_k)  )\\right] \\nonumber \\\\&  \\times\n\\prod_{i,j=1}^n \\left[\\frac{(p_i - h_j + i c)^2}{(h_{i,j} + ic)(p_{i,j} + ic)} \\right]^{1\/2} \\frac{\\prod_{i< j =1}^n  h_{ij} p_{ij}}{\\prod_{i , j} (p_i - h_j)}\n \\det_n \\left(  \\delta_{ij} + W(h_i,h_j) \\right)\\label{FFd_zero_density},\n\\end{align}\nwhere we used that $\\rho_t(\\lambda) = 1\/(2\\pi) + \\mathcal{O}(\\vartheta(\\lambda))$ and\n\\begin{align}\n  F(\\lambda) = \\frac{1}{2\\pi} \\sum_{k=1}^n \\left(\\theta(p_k - \\lambda) - \\theta(h_k - \\lambda)\\right) + \\mathcal{O}(\\vartheta(\\lambda)).\n\\end{align}\nThe matrix $W(h,p)$ \\eqref{kernelW} also simplifies. The kernel $\\hat{A}$ becomes small and we obtain an explicit expression for $W(\\mu, \\lambda)$\n\\begin{align}\n \n  W(\\lambda, \\mu) =  b(\\lambda)\\left(K(\\lambda - \\mu) - \\frac{2}{c} \\right) + \\mathcal{O}(\\vartheta(\\lambda))\n\\end{align}\nwith\n\\begin{align}\n  b(\\lambda) = \\frac{-1}{2\\sin\\left[\\frac{1}{2} \\sum_{k=1}^n\\left(\\theta(p_k -\\lambda)-\\theta(h_k - \\lambda) \\right) \\right]} \\prod_{k=1}^n \\frac{K^{1\/2}(h_k-\\lambda)}{K^{1\/2}(p_k-\\lambda)}  \\frac{\\prod_{k=1}^n (p_k - \\lambda)}{\\underset{h_k \\neq \\lambda}{\\prod_{k=1}^n} (h_k - \\lambda)}.\n\\end{align}\nIn the case of 1 particle-hole excitation the form factor simplifies to\n\\begin{equation} \\label{1ph_small_density}\n |\\langle 0 | \\hat{\\rho} | 0, h \\to p \\rangle |  = \\frac{1}{2}\\frac{\\theta(p-h)}{ (p-h)} \\left((p-h)^2 + c^2\\right)^{1\/2} ,\n\\end{equation}\nNote that the form factor \\eqref{1ph_small_density} describes a process of creating a particle-hole excitation in a low density state. Therefore is very different from the form factor \\eqref{ff} for $N=1$. The later equals\n\\begin{equation}\n |\\langle \\mu| \\hat{\\rho} |\\lambda\\rangle| = c,\n\\end{equation}\nand describes the process of exciting a single particle state with momentum $\\lambda$ to momentum $\\mu$ (Since these are single particle states the momentum is equal to the rapidity.). This shows that particle-hole excitations over the averaging state cannot be identified with particle creation over the vacuum in the field theory. Note that contrary to the relativistic field theory \\cite{MussardoBOOK} there is no crossing symmetry that would allow to transform the hole in the ket state into a particle in the bra state in eq.~\\eqref{1ph_small_density}.\n\n\n\n\\section{Regularization of the divergences} \\label{regularization}\nTo compute correlation functions we need to perform an integration over all possible values of the rapidites of the excitations.\nThe form factors \\eqref{FFd_final_expression} have however a singularity whenever $h_j = p_k$ and they are finite only when we consider only one single particle-hole  $n=1$ with $p \\to h$, when the form factor becomes indeed diagonal. Therefore we need to be careful while rewriting the sums as integrals. The aim of this section is to show how this can be done. Let us start with the finite size form of the correlation function where we already neglect sub-leading corrections \\eqref{corr_func_TL}\n\\begin{align}\n& \\langle \\hat{\\rho}(x,t) \\hat{\\rho}(0) \\rangle = \\sum_{n=0}^\\infty \\frac{1}{n!^2} \\prod_{j=1}^n \\left[   \\frac{1}{L} \\sum_{p_j} \\frac{1}{L}\\sum_{h_j} \\right] \\\\&\n\\times    |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |^2 e^{\\sum_{j=1}^n  \\left[ - i x  (k(p_j) - k(h_j)) - i t ( \\omega(p_j)  - \\omega(h_j))\\right]}.\n\\end{align}\nThe sum over particle and holes rapidites transforms into a product of integrals under a proper regularization. The idea, already introduced to regularize the field theory form factors in \\cite{1742-5468-2010-11-P11012}, is to write the sum over the holes as a complex integral over all the values that the holes rapidites can take for a finite (but large) $L$ using \\eqref{finite_size_excitations}\n\\begin{equation}\nL Q(h) =L \\left( h + \\int d\\lambda \\theta(h - \\lambda) \\rho(\\lambda) \\right) = 2 \\pi I_j,\n\\end{equation}\nwhere $\\{ I_j \\}$ are all the quantum numbers of the averaging state at some large fixed system size $L$.\nWith a help of $Q(h)$  we can write the sum of a function $f(z)$ over all the values of hole rapidity $h$ as\n\\begin{align}\n \\frac{1}{L}\\sum_{h} f(h) =&   \\sum_{I_j}  \\oint_{I_j} \\frac{dz}{2 \\pi} \\frac{f(z) Q'(z)}{ e^{i L Q(z)} -1} \\nonumber\\\\\n=&  \\left(\\int_{\\mathbb{R} - i \\epsilon} -\\int_{\\mathbb{R} + i \\epsilon} \\right)\\frac{f(z) Q'(z)}{ e^{i L Q(z)} -1} \\frac{dz}{2 \\pi} -   \\sum_{\\text{poles(f)} \\in \\Gamma_\\epsilon} \\oint dz \\frac{f(z) Q'(z)}{ e^{i L Q(z)}-1} -  \\sum_{r_j \\not \\in \\{ I_j \\}  } f(z_j),\n\\end{align}\nwhere the first integrals are taken on a single contour including the poles in $Q(z) = 2 \\pi I_j$ where $I_j$ are all the possible quantum numbers of the hole. In the second step we modified the sum over all these contours in the integral over the line above and below the real axes. In order to do that we need to subtract extra poles that we do not want to include. One type of them are the poles of $f(z)$ in the stripe $\\Gamma_\\epsilon$ delimited by the two imaginary lines. Other poles are located at the values $z$ such that $Q(z)= 2 \\pi r_j$ with $r_j$ not a quantum number of the averaging state (where holes cannot be created). When $L \\to \\infty$ only the integral above the real line survives the limit (since $Q(z)$ is monotonic in $z$) leading to\n\\begin{equation}\n\\frac{1}{L}\\sum_{h} f(h)   =  \\int_{\\mathbb{R} + i \\epsilon} {f(z) \\rho(z)}{} dz -  \\pi i  \\sum_{\\text{res(f)} \\in \\Gamma_\\epsilon} {f(z) \\rho(z)}{ }.\n\\end{equation}\n If now we impose that $f(z)$ has only a double pole in $z=p$ we can then rewrite the sum in terms of the finite part of the integral over $h$\n \\begin{equation}\n\\frac{1}{L}\\sum_{h} f(h)  = \\lim_{\\epsilon \\to 0^+}  \\int_{-\\infty}^\\infty dh f(h + i \\epsilon) - \\pi i \\underset{h=p}{\\rm res} f(h) = \\: \\fint_{-\\infty}^{\\infty} d h f(h).\n\\end{equation}\nIn order to compute the finite part is then useful to compute the limit  $p_j \\to h_j$ of the form factors \\eqref{FFd_final_expression}\n\\begin{align} \\label{FF_recursion}\n  \\frac{|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |}{|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^{n-1} \\rangle |} {=} \\frac{F(h_n)}{\\rho_t(h_n)(p_n - h_n)}  + \\mathcal{O}(p_n - h_n).\n\\end{align}\nwhere the back-flow is now computed as the sum of the other back-flows for the residual excitation \\eqref{back-sum-flow}\n\\begin{equation}\n  F\\left(\\lambda\\,|\\, \\{(\\mu_j^+, \\mu_j^-)\\}_{j=1}^n\\right) = \\sum_{j=1}^{n-1} F\\left(\\lambda\\,|\\, \\mu_j^+, \\mu_j^-\\right) .\n\\end{equation}\n\n\n\\section{\\texorpdfstring{Dynamical structure factor in $1\/c$ expansion}{Dynamical structure factor in 1\/c expansion}} \\label{expansion}\nWe consider here the expansion in $1\/c$ of the dynamical structure factor, defined as the Fourier transform of  the density-density correlation\n\\begin{align}\\label{dsf}\n&S(q, \\omega) =\\int dx dt \\: e^{i q x - i \\omega t} \\langle \\rho(\\lambda) |  \\hat{\\rho}(x,t)  \\hat{\\rho}(0,0) | \\rho(\\lambda) \\rangle\n\\nonumber \\\\&\n=(2 \\pi)^2 \\sum_{n=0}^\\infty \\frac{1}{n!^2}\\left[ \\prod_{j=1}^n \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j)   \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j)    \\right] \\delta\\left(q-\\sum_{j=1}^n (k(p_j) - k(h_j))\\right)  \\nonumber\\\\\n& \\times \\delta\\left(\\omega - \\sum_{j=1}^n (\\omega(p_j) - \\omega(h_j) \\right)   |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |^2  , \\nonumber \\\\\n\\end{align}\nfor a generic thermal state at temperature $T=\\beta^{-1}$ and density $D=1$.\nExpanding at the first order in $1\/c$ the only relevant form factors are the ones with only 1 particle-hole excitation $p,h$\n\\begin{align}\n & |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, h \\to p \\rangle | \\nonumber \\\\&\n =\n  \\frac{1}{2 \\pi } \\frac{1 + \\frac{2}{ c} }{(\\rho_t(h) \\rho_t(p))^{1\/2}}\n \\left[1 - \\frac{(p - h)^2}{\\pi c} {\\rm PV} \\int d\\lambda \\frac{\\vartheta(\\lambda)}{(\\lambda - p)(\\lambda - h)} \\right] + \\mathcal{O}(1\/c^2),\n\\end{align}\nsince the ones with two or more particle-hole excitations contribute at the order $1\/c^2$ or higher. The filling fraction for a thermal state at temperature $\\beta$, including the $1\/c$ correction, is given by\n\\begin{align}\n \\vartheta(\\lambda)  = \\frac{1 + \\frac{2}{c}}{1 + e^{\\beta(\\lambda^2 - h)}} ,\n\\end{align}\nwith $h$ the chemical potential fixing the density $D=1$ of the gas.\nIn the 1 particle-hole spectrum dynamical structure factor at $S(q,\\omega)$ is given in terms of a single form factor with energy $\\omega$ and momentum $q$ times the density of states, which is simply the Jacobian of the transformation from the rapidities of the excitations to the energy and momentum variable\n\\begin{align}\np^2   - h^2   & = \\omega,   \\\\\np  - h  & = q \\Big( 1 + \\frac{2}{c} \\Big)^{-1},\n\\end{align}\nwhich gives a Jacobian factor $\\Big|\\det \\frac{\\partial ( \\omega, q)}{\\partial( p, h ) }  \\Big|= 2 q (1+2\/c)^{-1}$ with the rapidities of the excitations given by\n\\begin{align}\n& p =  \\frac{q}{2 (1 + 2\/c)} + \\frac{\\omega (1 + 2\/c)}{2 q} , \\\\\n& h = - \\frac{q}{2 (1 + 2\/c)}+  \\frac{\\omega (1 + 2\/c)}{2 q}.\n\\end{align}\n\nWe obtain then an expression for the thermal dynamical structure factor up to $1\/c^2$ corrections\n\\begin{align}\\label{dynamical_final}\nS(q, \\omega) &=  (2\\pi)^2 \\frac{1 + \\frac{2}{c}}{2 q }   \\Big[\\rho_h( p) \\rho( h)|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, h \\to p \\rangle |\\Big]\\nonumber\\\\\n&=\\frac{2 }{\\pi} \\left( \\pi \\frac{1 + \\frac{6}{c}}{4 q} +  \\frac{1}{2 c} {\\rm PV}\\int \\frac{\\vartheta(\\lambda + p)  - \\vartheta(\\lambda + h)}{\\lambda  }  \\right) {\\vartheta(h)\\left( 1 - \\vartheta(p)\\right)} + \\mathcal{O}(1\/c^2).\n\\end{align}\nUsing\n\\begin{align}\n  1-e^{-\\beta\\omega}=\\frac{\\vartheta(h) - \\vartheta(p)}{\\vartheta(h)(1- \\vartheta(p))}.\n\\end{align}\nwe obtain the same result as in \\cite{2005_Brand_PRA_72} (where here we have chosen unitary density $D=1$). Note that the limit $T \\to 0$ can be easily recovered from \\eqref{dynamical_final}.  The same is believed to be true for all the orders in $1\/c$ of the correlation functions. This is a non-trivial statement since the procedure to obtain the form factors when the averaging state is the ground state and when is a thermal state are manifestly different.\n\n\nThe example here is carried on for a thermal state, however this result can be extended to any filling fraction $\\vartheta(\\lambda)$ including for example the saddle point state after a quench in the Lieb-Liniger model \\cite{2014_DeNardis_PRA_89}\n\n\n\\section{\\texorpdfstring{Numerical evaluation of the dynamical structure factor}{Numerical evaluation}} \\label{numerics}\n\nThe dynamical structure factor \\eqref{dsf} can be computed through numerical evaluations of the exact formula \\eqref{FFd_final_expression}. The sum over all the possible number of excitations $n = 1, 2 , \\ldots$ requires a great numerical effort, mainly due to the complicated structure of the form factors. To simplify the problem we focus here only on the simplest excitations ($n=1$) consisting of a single particle-hole pair. This leads to an approximate expression for the correlation function which is shown in figures~\\ref{fig2} and~\\ref{fig3}. As in the $1\/c$ section, for concreteness we limit ourselves to thermal equilibrium correlations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.52]{fig2_new.pdf}\n\\caption{Correlation function $S(k,\\omega)$ at $T=1$, unitary density $D=1$ and with $c=16$ (\\emph{on the left}) and with $c=4$ (\\emph{on the right}). We plot it as a function of energy $\\omega$ for fixed value of momentum $k=k_F$. The 1 particle-hole approximation $S_{1ph}(q, \\omega)$ (blue) fits well the ABACUS results (green) and misses only on the correlation weight which is expected to come from multiple particle-hole excitations. For $c=16$ we plot also the results of the $1\/c$ expansion (red). Inset in the right figure shows the percentage of the f-sum rule \\eqref{fsumrule} saturation for some values of the interaction parameter $c$.}\n\\label{fig2}\n\\end{figure}\n\n\nThe approximation of the correlation function to a single particle-hole pair becomes exact in the limit of large interactions (c.f. previous section) and in the limit of very small momentum $k$.\\footnote{This is the same idea as presented in study of the dynamic structure factor of the XXZ spin chain at small momentum~\\cite{2007_Pereira_JSTAT_8}.} As the interaction is decreased and momentum is increased we expect the approximation to become worse. To quantify how far the resulting correlation function is from the true one we compare our results with an exact numerical evaluation of the correlation function in a finite system~\\cite{PhysRevA.89.033605} via the ABACUS algorithm~\\cite{2009_Caux_JMP_50}. This shows that even for values of $c\\sim 1$, which go well beyond the $1\/c$ expansion and at finite momentum $k=k_F$, the 1 particle-hole contribution captures the essential features of the dynamic structure factor. For $c\\approx 4$ the kinetic and potential energy \\eqref{H} of the ground state of the system are equal \\cite{JCS_Comment} and thus the correlation function is the most difficult to compute.\n\nAdditionally we consider the f-sum rule~\\cite{LL_StatPhys2_BOOK}, an exact equality obeyed by the dynamic structure factor for any fixed momentum $q$\n\\begin{equation} \\label{fsumrule}\n\\int_{-\\infty}^\\infty \\frac{d\\omega}{2 \\pi} \\omega S(q,\\omega) = D q^2.\n\\end{equation}\nIn the limit $c\\to \\infty$ or $k\\rightarrow 0$ the 1 particle-hole spectrum is the full excitation spectrum for the density operator and consequently the f-sum rule is completely saturated by including only these types of excitations in the sum \\eqref{dsf}. However as $c$ decreases with $k$ finite we observe that the f-sum rules is saturated only up to a certain precision and more excitations have to be taken into account in order to obtain the full correlation function. Again, even at values of $c \\sim 1$  and $k\\sim k_F$ the contribution of the 1 particle-hole excitations remains very significant (see insets of figures~\\ref{fig2} and~\\ref{fig3}).\n\nThe results of this section confirm that the form factors~\\eqref{FFd_final_expression} can be directly used to compute the dynamic structure factor or in general the density-density correlation on a generic state with non-zero entropy. Moreover it asserts that the expansion in particle-hole excitation numbers is an effective method to compute the correlation function.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.52]{fig3_new.pdf}\n\\caption{Correlation function $S(k,\\omega)$ at $T=1$, unitary density $D=1$ and with $c=16$ (\\emph{on the left}) and with $c=4$ (\\emph{on the right}) as a function of energy $\\omega$ for fixed value of momentum $k=k_F\/10$. For  $c=16$ we plot also the result of the $1\/c$ expansion. The f-sum rule (as plotted inside the inset) is saturated exactly (up to a numerical precision of $\\sim 1\\%$) since the 1 particle-hole approximation becomes exact at small momenta. Lower value of momentum allows for development of a second peak, at negative energies, due to the detailed balance relation $S(k,-\\omega) = S(k, \\omega) \\exp(-\\omega\/T)$~\\cite{LL_StatPhys2_BOOK}. Note that at higher momenta the correlation function is shifted towards higher energies and the negative energy peak becomes practically invisible (See figure~\\ref{fig2}).}\n\\label{fig3}\n\\end{figure}\n\n\n\n\\section{Conclusions} \\label{discussion}\n\nIn this work we studied the thermodynamic limit of the particle-hole form factors for the density operator of the 1D Bose gas. The computations presented here can be generalized to different operators (like the bosonic field operator $\\Psi$) but also to other Bethe Ansatz solvable models for which the microscopic matrix elements are known such as the XXZ spin chain. These problems will be addressed in the future.\nThese form factors constitute the building blocks to compute thermal or post-quench equilibrium correlation functions in the thermodynamic limit at fixed density of particles. They also provide a first step towards the post-quench time evolution as recently done in \\cite{me} for the Tonks-Girardeau ($c=\\infty$) regime.\n\nThe final formula \\eqref{FFd_final_expression} is valid in the thermodynamic limit and it is considerably simpler than its finite size version but still it is not suitable to obtain close-form expressions of correlation functions. The Fredholm determinant of the kernel $\\hat{A}$ is a non-trivial functions of the excitations parameters $\\{ p_j, h_j\\}_{j=1}^n$ and we were not able to obtain further simplifications. A fully factorized expression of the form factors involving a simple almost factorized part depending only on the excitation parameters is still under research.\n\nWe computed the exact thermodynamic dynamical structure factor including only the 1 particle-hole excitations over a thermal state. This approximation is qualitatively different from the usual perturbative one or the low energy limit. For example the perturbation theory in $1\/c$ breaks at $c\\sim 10$ yielding unphysical, negative values of the correlation~\\cite{2005_Brand_PRA_72}, while the (non-linear) Luttinger liquid theory is not able to reproduce the exact shape of the correlation function~\\cite{2012_Imambekov_RMP_84}. We showed that a thermodynamic Bethe Ansatz approach with only single particle-hole excitations produces a good estimate of the density correlations of the system for a wide range of values of the interaction parameter and momentum. Therefore the effect of the extra particle-hole excitations is mainly to increase  the weight of the correlation at large momentum.\n\n\nAnother interesting point is to compare our result with similar ones for the thermodynamic limit of one-point functions of the Lieb-Liniger model obtained from the non-relativistic limit of the sinh-Gordon model \\cite{2009_Kormos_PRA_81}. As shown in \\cite{1742-5468-2011-11-P11017} the large volume limit of the diagonal form factor obtained by Bethe Ansatz \\eqref{diagonal} can be expressed as a LeClair-Mussardo series \\cite{1999_LeClair_NPB_552} of the elementary form factors obtained via the bootstrap program \\cite{MussardoBOOK,2009_Kormos_PRA_81}. How to extend this relation to two-point functions remains to be clarified. The Bethe Ansatz approach, presented in this work, might shed a light on this important problem of the Quantum Integrable Field Theories.\n\nFinally, following \\cite{2011_Shashi_PRB_84} where a relation between the form factors and the prefactors of the Lutinger liquid correlation functions at zero temperature was established, it would be interesting to see whether such simple relations also exist at finite temperature or even out-of-equilibrium. The result of \\cite{2011_Kozlowski_JSTAT_P03019} where low temperature correlation functions were studied seem to suggest that such relations might exists. This will be also a subject of a further research.\n\n\n\n\\ack\nWe are very grateful to J.-S. Caux for his support and critical comments and to R. Konik for a stimulating and encouraging discussion.\nWe acknowledge useful and inspiring discussions with S. Eli\\\"{e}ns, G. Mussardo and H. Saleur.\n\\noindent J. De Nardis acknowledges support from the Netherlands Organisation for Scientific Research (NWO). M. Panfil acknowledges support from the Foundation for Fundamental Research on Matter (FOM) at the early stage of this work.\n\n\\noindent This work was supported by ERC under the Starting Grant n. 279391 EDEQS.\n\n\\section*{References}\n\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Missing Details of Section~\\ref{sec:flow}}~\\label{sec:appx_flow}\n\\section{Recap: Flow for Additive Valuations}\\label{sec:flow_additive}\nWhen the valuations are additive,   we simply view $t_{ij}$ as bidder $i$'s value for receiving item $j$. Although there are many possible ways to define a flow, we focus on a class of simple ones. Every flow in this class $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}\\in\\R^{nm}$. Based on $\\beta_i=\\{\\beta_{ij}\\}_{j\\in[m]}$, we first partition the type space $T_i$ for each bidder $i$ into $m+1$ regions:\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}<\\beta_{ij}$ for all $j\\in[m]$.\n\t\\item $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{t_{ik}-\\beta_{ik}\\}$.\n\\end{itemize}\n\nWe use essentially the same flow as in~\\cite{CaiDW16}. Here we provide a partial specification and state some desirable properties of the flow. See Figure~\\ref{fig:multiflow} for an example with $2$ items and~\\cite{CaiDW16} for a complete description of the flow.\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth}\n{\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\\begin{enumerate}[leftmargin=0.7cm]\n \\item For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$,  the flow goes directly to $\\varnothing$ (the super sink).\n \\item  For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from  $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n \\item For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\\end{enumerate}\n\\end{minipage}}\n\\caption{Partial Specification of the flow $\\lambda^{(\\beta)}$.}\n\\label{fig:flow specification}\n\\end{figure}\n\n\\begin{figure}\n  \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n  \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n  \\label{fig:multiflow}\n\\end{figure}\n\n\n\\begin{lemma}[\\cite{CaiDW16}\\footnote{Note that this Lemma is a special case of Lemma 3 in~\\cite{CaiDW16} when the valuations are additive. }]\\label{lem:additive flow properties}\n\tFor any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_{i}$ such that the corresponding virtual value function $\\Phi_{i}(t_{i}, \\cdot)$ satisfies the following properties:\n\t\\begin{itemize}[leftmargin=0.5cm]\n\t\t\\item For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi_{i}(t_{i},S) = \\sum_{k\\in S} t_{ik}$.\n\t\t\\item For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$,  $$\\Phi_{i}(t_{i},S)\\leq \\sum_{k\\in S \\land  k\\neq j} t_{ik}+{\\tilde{\\varphi}}_{ij}(t_{ij})\\cdot\\mathds{1}[j\\in S],$$ where ${\\tilde{\\varphi}}_{ij}(\\cdot)$ is Myerson's ironed virtual value function for  $D_{ij}$.\n\t\\end{itemize}\n\\end{lemma}\n\nThe properties above are crucial for showing the approximation results for simple mechanisms in~\\cite{CaiDW16}. One of the key challenges in approximating the optimal revenue is how to provide a tight upper bound. A trivial upper bound is the social welfare, which may be arbitrarily bad in the worst case. By plugging the virtual value functions in Lemma~\\ref{lem:additive flow properties} into the partial Lagrangian, we obtain a new upper bound that replaces the value of the buyer's favorite item with the corresponding Myerson's ironed virtual value. As demonstrated in~\\cite{CaiDW16}, this new upper bound is at most $8$ times larger than the optimal revenue when the buyers are additive, and its appealing structure allows the authors to compare the revenue of simple mechanisms to it. In \\notshow{the next section} {Section~\\ref{sec:flow}}, we identify some difficulties in directly applying this flow to subadditive valuations. Then we show how to overcome these difficulties by relaxing the subadditive valuations and obtain a similar upper bound.\n\n\\section{Proof of Lemma~\\ref{lem:subadditive flow properties}}\\label{sec:proof_virtual_relaxation}\n\\begin{lemma}\\label{lem:separate the favorite out in virtual value}\n\tFor any flow $\\lambda^{(\\beta)}_i$ that respects the partial specification in Figure~\\ref{fig:flow specification}, the corresponding virtual valuation function $\\Phi_i^{(\\beta_i)}$ of $v_i^{(\\beta_i)}$ for any buyer $i$ is:\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\n    \\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$, if $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$.\n\\item $v_i(t_i,S)$, otherwise.\n\\end{itemize}\n\n\\begin{comment}\n\t \\begin{equation*}\n\\begin{aligned}\n\\Phi_i^{(\\beta_i)}(t_i, S)=\n\\begin{cases}\nv_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)  &\\text{if }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:separate the favorite out in virtual value}\n\tThe proof follows the definitions of the virtual valuation function (Definition~\\ref{def:virtual value}) and relaxed valuation (Definition~\\ref{def:relaxed valuation}). We use $t_{i,-j}=\\langle t_{i{j'}}\\rangle_{j'\\not=j}$ to denote bidder $i$'s information for all items except item $j$. If $t_i\\in R_j^{(\\beta_i)}$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i,S) = v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})$. Since $\\lambda(t_i,t_i')>0$ only when $t_{i,-j}=t_{i,-j}'$ and $t_i'\\in R_j^{(\\beta_i)}$,  $v_i^{(\\beta_i)}(t'_i,S) =  v_i(t'_i, S\\backslash \\{j\\})+V_i(t'_{ij})= v_i(t_i, S\\backslash \\{j\\})+V_i(t'_{ij})$. Therefore,\n\\begin{align*}\n    \\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})\n    -\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)\n\\end{align*}\n\t\n\tIf $t_i\\in R_j^{(\\beta_i)}$ and $j\\notin S$ or $t_i\\in R_0^{(\\beta_i)}$, then $v_i^{(\\beta_i)}(t_i,S) =v_i(t_i, S)$. If $t_i\\in R_0^{(\\beta_i)}$, there is no flow entering $t_i$ except from the source, so clearly  $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$. If $t_i\\in R_j^{(\\beta_i)}$, then for any $t'_i$ that only differs from $t_i$ in the $j$-th coordinate, we have $v_i(t'_i, S)=v_i(t_i,S)$, because {$j\\not\\in S$}. Hence,  $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$.\n\\end{prevproof}\n\n\n\\begin{prevproof}{Lemma}{lem:subadditive flow properties}\n\nLet $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$. According to Lemma~\\ref{lem:separate the favorite out in virtual value}, it suffices to prove that for any $j>0$, any $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\n\\begin{claim}\nFor any type $t_{i}\\in R^{(\\beta_i)}_{j}$, if we only allow flow from type $t'_{i}$ to $t_{i}$, where $t_{ik}=t'_{ik}$ for all $k\\neq j$ and $t'_{ij}\\in \\argmin_{s\\in T_{ij} \\land V_i(s)> V_i(t_{ij})} V_i(s)$, and the flow $\\lambda(t_i',t_i)$ equals $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ fraction of the total in flow to $t_i'$, then there exists a flow $\\lambda$ such that\n\\begin{align*}\n\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))\n=V_i(t_{ij})-\\frac{\\left(V_i({t'_{ij}})-V_i(t_{ij})\\right)\\cdot\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]},\n\\end{align*} where $\\varphi_{ij}(V_i(t_{ij}))$ is the Myerson virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$. \\end{claim}\n\\begin{proof}\n{As the flow only goes from $t_i'$ and $t_i$, where $t_i'$ and $t_i$ only differs in the $j$-th coordinate, and \\\\\n\\noindent$t_{ij}\\in \\argmax_{s\\in T_{ij} \\land V_i(s)< V_i(t_{ij}')} V_i(s)$. If $t_{ij}$ is a type with the largest $V_i(t_{ij})$ value in $T_{ij}$, then there is no flow coming into it except the one from the source, so $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})$. For every other value of $t_{ij}$, the in flow is exactly\n\\begin{align*} \\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot  \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) \n=\\prod_{k} f_{ik}(t_{ik})\\cdot \\frac{\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v=V_i(t_{ij})]}.\\end{align*}\n {This is because each type of the form $(x,t_{i,-j})$ with $V_i(x) > V_i(t_{ij})$ is also in $R^{(\\beta_i)}_{j}$. So $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ of all flow that enters these types will be passed down to $t_{i}$ (and possibly further, before going to the sink), and the total amount of flow entering all of these types from the source is exactly {$\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) $}.} Therefore, $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))$. Whenever there is no more type $t_i\\in R_j^{(\\beta_i)}$ with smaller $V_i(t_{ij})$ value, we push all the flow to the sink.}\n\\end{proof}\n\n\nIf $F_{ij}$ is regular, this completes our proof. When $F_{ij}$ is not regular, we can iron the virtual value function in the same way as in \\cite{CaiDW16}. Basically, for two types $t_i,t_i'\\in R^{(\\beta_i)}_{j}$ that only differ in the $j$-th coordinate, if $\\Psi_{ij}^{(\\beta_i)}(t_i)>\\Psi_{ij}^{(\\beta_i)}(t_i')$ but $V_i(t_{ij})<V_i(t_{ij}')$, add a loop between $t_i$ and $t_i'$ with a proper weight to make $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\Psi_{ij}^{(\\beta_i)}(t_i')$.\n\\begin{lemma}\\cite{CaiDW16}\nFor any $\\beta$ and $i$, there exists a flow $\\lambda_i(\\beta)$ such that for any $t_i\\in R_j^{(\\beta_i)}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\\end{lemma}\n\\end{prevproof}\n\n\n\n\n\n\\section{Missing Details of Section~\\ref{sec:duality}}\\label{sec:appx_duality}\n\n\n\\section{Efficiently Finding the Mechanism for Identical Bidders}\nIn this section, we are going to prove that such a mechanism which approximately obtains the optimal revenue can be computed in polynomial time when bidders are identical. To say \\textbf{identical}, we mean all bidders share same private information set $\\tilde{T}$ and distribution $\\tilde{D}$ on this support. Besides, given a same private information, all bidders have the same item valuation, i.e., for any $t\\in \\tilde{T}$ and $S\\subseteq [m]$, $v_i(t,S)=v_j(t,S)$ holds for all $i,j\\in [n]$.\n\nWhen bidders are identical, it's reasonable to only consider symmetric mechanisms. In a symmetric mechanism, the ex-ante probability to win an item $j$ is the same for all bidder $i$, and thus no more than $\\frac{1}{n}$ due to no overallocation. The following lemma formally proves these observations.\n\n\\begin{lemma}\\label{lem:comp:symmetric}\nIf bidders are identical, then for any BIC mechanism $M$, there exists a symmetric BIC mechanism $M'$ that satisfies\n\\begin{itemize}\n\\item $\\textsc{Rev}(M,v,D)=\\textsc{Rev}(M',v,D)$\n\\item $\\forall i\\in [n], \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq \\frac{1}{n}$\n\\end{itemize}\nwhere $\\pi_{ij}'(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}'(t_i)$ and $\\sigma_{iS}'(t_i)$ is the probability of buyer i receiving exactly bundle S when her reported type is $t_i$ in mechanism $M'$.\n\\end{lemma}\n\\begin{proof}\nDefine mechanism $M'$ as follows: after receiving type profile $t=\\langle t_i\\rangle_{i=1}^n$, draw a random permutation $\\rho$ of size $n$. Then follow mechanism $M$ treating bidder $i$ as $\\rho(i)$ for all $i$. Formally, we denote by $M_i(t)$, $h_i(t)$ the (probably random) received item set and charged price to bidder $i$ by mechanism $M$ when type profile is $t$, and $M_i'(t)$, $h_i'(t)$ the same meaning for mechanism $M'$. After the permutation $\\rho$ is drawn, define type profile $t^{(\\rho)}=\\langle t_{\\rho(i)}\\rangle_{i=1}^n$. The allocation rule and price for mechanism $M'$ is then defined as: $M_i'(t)=M_{\\rho(i)}(t^{(\\rho)})$, $h_i'(t)=h_{\\rho(i)}(t^{(\\rho)})$. Since bidders are identical, with the random permutation, $M'$ will treat all bidders as a same role, and thus the ex-ante probability getting an item $j$ is the same for all bidders. Clearly $\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq 1$, thus each ex-ante probability $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq \\frac{1}{n}$.\n\nTo prove that $M'$ has the same expected revenue as $M$, it's enough to show $\\sum_t f(t)\\cdot \\sum_i h_i(t)=\\sum_t f(t)\\cdot \\sum_i h_{\\rho(i)}(t^{(\\rho)})$. Notice for all $t\\in T$, $f(t)=f(t^{(\\rho)})$ as all bidders' types are independent, we have\n\\begin{equation}\n\\sum_t f(t)\\cdot \\sum_i h_{\\rho(i)}(t^{(\\rho)})=\\sum_t f(t^{(\\rho)})\\cdot \\sum_i h_i(t^{(\\rho)})=\\sum_t f(t)\\cdot \\sum_i h_i(t)\n\\end{equation}\n\n\n\\end{proof}\n\nNow without loss of generality, we can assume $M$ is a symmetric mechanism. For every $i\\in [n], j\\in [m]$, select $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. With Lemma~\\ref{lem:comp:symmetric}, it's easy to check that $\\beta$ satisfies the two conditions in Lemma~\\ref{lem:requirement for beta} and thus Theorem~\\ref{thm:multi} works. We have to note here that in a discrete distribution $D_{ij}$, there might not exist such a $\\beta_{ij}$. Thus we have to assume that $\\forall i,j, \\exists \\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$, so that it's able to get the exact $\\beta_{ij}$. $\\beta_{ij}$ is sensitive to the proof and the mechanism used in the proof, and thus it's necessary to use the exact value.\n\nConsider this specific $\\beta$ for proofs in Section~\\ref{sec:multi}. Now we'll show all the posted price and entry fee for the mechanisms that we use in the proof can be computed in polynomial time.\n\nGiven bidder $i$'s type $t_i$, it'll need exponential size to explicitly described $i$'s value $v_i(t_i,\\cdot):2^{[m]}\\to R^{*}$. Thus it's usually assumed that there exist the following oracles access to $v_i(\\cdot,\\cdot)$.\n\n\\begin{itemize}\n\\item Value oracle: Given type $t_i\\in T_i$ and set $S$ as input, returns a non-negative real number that represents bidder's value $v_i(t_i,S)$.\n\\item XOS oracle(for XOS valuations): Given type $t_i\\in T_i$ and set $S$ as input, returns prices $\\{p_j\\}_{j\\in S}$ as the 1-supporting prices for $v_i(t_i,S)$.\n\\end{itemize}\n\nWe now state the main theorem for this section:\n\\begin{theorem}\nConsider the case that bidders are identical, and have XOS valuations. Suppose $\\forall i,j, \\exists \\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Besides, we are given\n\\begin{itemize}\n\\item $\\forall i,j$, the distribution for $v_i(t_{ij})$.\n\\item $\\forall i\\in [n]$, a value oracle for $v_i(\\cdot,\\cdot)$.\n\\item $\\forall i\\in [n]$, an XOS oracle for $v_i(\\cdot,\\cdot)$.\n\\item $\\forall t\\in T$, a black box algorithm $\\mathcal{A}$ that maximizes the social welfare for the combinatorial auction with bidder's valuation $v_i'(t_i,\\cdot)$.\n\\end{itemize}\n\nThen with $O()$ type samples from $T$, with probability at least ---, we can compute a DSIC mechanism in $POLY(n,m,\\sum_i\\sum_j |T_{ij}|, )$ time such that the revenue of this mechanism obtains at least $1\/$ the optimal revenue.\n\\end{theorem}\n\n\\begin{proof}\nFirstly for Lemma~\\ref{lem:multi_single}, as showed in \\cite{ChawlaHMS10, KleinbergW12}, the posted-price mechanism that used can be efficiently computed. We now consider the Sequential Posted Price Mechanism stated in Section~\\ref{subsection:tail} and Section~\\ref{subsection:core}.\n\nThe mechanism used in Corollary~\\ref{cor:bound tail } is determined by the posted price $\\xi_{ij}=\\beta_{ij}+P_{ij}$. Having distribution $D_{ij}$, $\\beta_{ij}$ and $P_{ij}$ can both be computed in $O(|T_{ij}|)$ time by brute force. Thus this mechanism can be computed in linear time.\n\nNow consider the ``CORE\" part. Recall that $\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S)$, which can be described as the expected social welfare of $M^{(\\beta)}$ in a new combinatorial auction. In this auction, when type profile $t\\in T$, bidder $i$ has valuation $v_i'(t_i,\\cdot)$. Let $M^{(\\beta),*}$ be the optimal mechanism among above and $\\sigma_{iS}^{(\\beta),*}$, $Q_j^{*}$ be the corresponding definition. Then according to Lemma~\\ref{lem:core and q_j}, $2\\alpha\\cdot\\sum_{j\\in [m]}Q_j^{*}\\geq \\textsc{Core}(\\beta)$ holds for all BIC mechanism $M^{(\\beta)}$. It implies that the mechanism with specific parameter $\\beta$ and $\\sigma_{iS}^{(\\beta)}=\\sigma_{iS}^{(\\beta),*}$ can guarantee a constant factor of the optimal revenue.\n\nThe Sequential Posted Price Mechanism with Entry Fee stated in Section~\\ref{subsection:core} use $Q_j^{*}$ as the posted price for item $j$. Recall that $Q_j^{*}=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta),*}(t_i)\\cdot \\gamma_j^{S}(t_i)$. Even exactly calculating one $\\sigma_{iS}^{(\\beta),*}(t_i)$ requires running mechanism $M^{(\\beta),*}$ under every type profile $t$, which takes exponential time. $Q_j^{*}$ can thus not be able to compute exactly in polynomial time.\n\n\n\n\n\nRecall that $Q_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)$. And even exactly calculating one $\\sigma_{iS}^{(\\beta)}(t_i)$ requires running mechanism $M$ under every type profile $t$, which takes exponential time. $Q_j$ can thus not be able to compute exactly in polynomial time. What we do is to generate enough samples of type profile $t\\in T$, run $M$ with those samples, and calculate the empirical approximation for $Q_j$.\n\n\n\n\n\n\n\n\\end{proof}\n\n\nCorollary~\\ref{cor:bound tail }, Lemma~\\ref{lem:beta_ij}, Lemma~\\ref{lem:tau_i}, and Definition~\\ref{def:entry fee}.\n\n\n\n\n\n\\section*{\\huge{Appendix}}\n\n\\section{Details from Preliminaries}\n\nWe first formally define various valuation classes.\n\\begin{definition}\\label{def:valuation classes}\nWe define several classes of valuations formally. Let $t$ be the type and $v(t,S)$ be the value for bundle $S\\in[m]$.\n\n\n\t\n\\begin{itemize}[leftmargin=0.5cm]\n\t\\item  \\textbf{Constrained Additive} \\\\\n$v(t,S) =\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} v(t, \\{j\\})$, where $\\mathcal{I}\\subseteq 2^{[m]}$ is a downward closed set system over the items specifying the feasible bundles. In particular, when $\\mathcal{I}=2^{[m]}$, the valuation is an \\textbf{additive function}; when $\\mathcal{I}=\\{\\{j\\}\\ |\\ j\\in[m] \\}$, the valuation is a \\textbf{unit-demand function}; when $\\mathcal{I}$ is a matroid, the valuation is a \\textbf{matroid-rank function}. An equivalent way to represent any constrained additive valuations is to view the function as additive but the bidder is only allowed to receive bundles that are feasible, i.e., bundles in $\\mathcal{I}$. To ease notations, we interpret $t$ as an $m$-dimensional vector $(t_1, t_2,\\cdots, t_m)$ such that $t_j =  v(t, \\{j\\})$.\n\n\\item \\textbf{XOS\/Fractionally Subadditive}\\\\\n $v(t,S) = \\max_{i\\in [{K}]} v^{(i)}(t, S)$, where ${K}$ is some finite number and $v^{(i)}(t,\\cdot)$ is an additive function for any $i\\in[{K}]$.\n\n\\item \\textbf{Subadditive}\\\\ \n$v(t,S_1\\cup S_2)\\leq v(t,S_1)+v(t,S_2)$ for any subset $S_1, S_2\\subseteq [m]$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{The following are a few examples of various valuation distributions which are over independent items (Definition~\\ref{def:subadditive independent}):\n\n\\begin{example}\\label{eg:valuation}\\cite{RubinsteinW15}\n$t=\\{t_j\\}_{j\\in[m]}$ where $t$ is drawn from $\\prod_j D_j$,\n\\begin{itemize}[leftmargin=0.5cm]\n\\item Additive: $t_j$ is the value of item $j$. $v(t,S)=\\sum_{j\\in S}t_j$.\n\\item Unit-demand: $t_j$ is the value of item $j$. $v(t,S)=\\max_{j\\in S}t_j$.\n\\item Constrained Additive: $t_j$ is the value of item $j$.\n $v(t,S)=\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} t_j$.\n\\item XOS\/Fractionally Subadditive: $t_j=\\{t_{j}^{(k)}\\}_{k\\in[K]}$ encodes all the possible values associated with item $j$, and $v(t,S)=\\max_{k\\in[K]}\\sum_{j\\in S}t_{j}^{(k)}$.\n\\end{itemize}\n\\end{example}\n\n\n}\n\n\\section{Improved Analysis for Constrained Additive Valuation}\n\nIn this section, we show that for constrained additive bidders, we do not need to relax the valuations, as applying directly the flow in Section~\\ref{sec:flow} already gives an upper bound with the right format. So we can take $M^{(\\beta)}$ to simply be $M$. In particular, we can derive the following improved upper bound for $\\textsc{Rev}(M,v,D)$ using essentially the same proof as in Section~\\ref{sec:flow}.\n\\begin{theorem}\\label{thm:revenue upperbound for constrained additive}\nIf for any bidder $i$ any type $t_i\\in T$, $v_i(t_i, \\cdot)$ is a constrained additive valuation, then for any mechanism $M$ and any $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}$,\n$$\\textsc{Rev}{(M,v,D)}\\leq \\textsc{Non-Favorite}(M,\\beta)+\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\nCombining the same upper bounds we obtained for \\\\\n\\noindent$\\textsc{Non-Favorite}(M,\\beta)$ and $\\textsc{Single}(M,\\beta)$ and the improved upper bound in Theorem~\\ref{thm:revenue upperbound for constrained additive}, we can improve the approximation ratio when the bidder(s) have constrained additive valuations.\n\\begin{theorem}\n\tFor a single buyer whose valuation is constrained additive,\n$$\\textsc{Rev}(M,v,D)\\leq 7\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev},$$\nfor any BIC mechanism $M$.\n\\end{theorem}\n\n\\begin{theorem}\n\tFor multiple buyers whose valuations are constrained additive,\n\\begin{equation}\n\\begin{aligned}\n\\textsc{Rev}(M,v,D)&\\leq 8\\cdot\\textsc{APostEnRev}\\\\\n&+\\left(6+\\frac{22}{1-b}+\\frac{4(b+1)}{(1-b)b}\\right)\\cdot \\textsc{PostRev}\n\\end{aligned}\n\\end{equation}\nfor any BIC mechanism $M$. In particular, if we set $b$ to be $\\frac{1}{4}$, then $$\\textsc{Rev}(M,v,D)\\leq 8\\cdot\\textsc{APostEnRev}+62\\cdot\\textsc{PostRev}.$$\n\\end{theorem}\n\n\\section*{\\homeworkProblemName\n   \\enterProblemHeader{\\homeworkProblemName}\n  {\\exitProblemHeader{\\homeworkProblemName}\n\n\\newcommand{\\homeworkSectionName}{\n\\newlength{\\homeworkSectionLabelLength}{\n\\newenvironment{homeworkSection}[1\n  \n\n   \\renewcommand{\\homeworkSectionName}{#1\n   \\settowidth{\\homeworkSectionLabelLength}{\\homeworkSectionName\n   \\addtolength{\\homeworkSectionLabelLength}{0.25in\n   \\changetext{}{-\\homeworkSectionLabelLength}{}{}{\n   \\subsection*{\\homeworkSectionName\n   \\enterProblemHeader{\\homeworkProblemName\\ [\\homeworkSectionName]}\n  {\\enterProblemHeader{\\homeworkProblemName\n\n  \n  \n   \\changetext{}{+\\homeworkSectionLabelLength}{}{}{}\n\n\\newcommand{\\Answer}{\\ \\\\\\textbf{Answer:} }\n\\newcommand{\\Intui}{\\ \\\\\\textbf{Intuition:} }\n\\newcommand{\\Proof}{\\ \\\\\\textbf{Proof:} }\n\\newcommand{\\Acknowledgement}[1]{\\ \\\\{\\bf Acknowledgement:} #1}\n\n\n\n\n\n\\title{\\textmd{\\bf \\Title}\n\\author{\\textbf{\\StudentName}}\\\\{\\large Instructed by \\textit{\\ClassInstructor}}\\\\\\normalsize\\vspace{0.1in}\\small{\\DueDate}}\n\\date{}\n\n\\begin{document}\n\\begin{spacing}{1.1}\n\\maketitle \\thispagestyle{empty}\n\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{definition}{Definition}\n\\newtheorem{assignment}{Homework Problem}\n\\newtheorem{notation}{Notation}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{conjecture}{Conjecture}\n\n\\section{Multiple Bidder, Additive With Matroid Constraint}\n\\subsection{Flow}\nFor a feasible $\\pi\\in P(\\mathcal{F},D)$, let M be the mechanism that induces $\\pi(\\cdot)$. For all $i\\in [n], j\\in [m]$, define\n\\begin{equation}\nq_{ij}^{\\pi}=b\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\pi_{ij}(t_i)\n\\end{equation}\nwhere $b\\in (0,1)$ is a constant parameter that we will determine later. Choose $\\beta_{ij}^{\\pi}$ such that $Pr[t_{ij}\\geq \\beta_{ij}^{\\pi}]=q_{ij}^{\\pi}$ (Assume all the distributions are continuous so $q_{ij}^{\\pi}$ exists).\n\nNotice that we have the following for all $j$ since $\\pi$ is feasible, i.e., without over-allocation:\n\\begin{equation}\n\\sum_i q_{ij}=b\\cdot\\sum_i \\sum_{t\\in T}f(t)\\pi_{ij}(t_i)=b\\cdot\\sum_{t\\in T}f(t)\\sum_i\\pi_{ij}(t_i)\\leq b\n\\end{equation}\n\n~\\\\\n\nLet $R_0^{\\pi}=\\{t_i\\in T_i|t_{ik}<\\beta_{ik},\\forall k\\in [m]\\}$, and $R_j^{\\pi}=\\{t_i\\in T_i|(\\forall k\\not=j,t_{ij}-\\beta_{ij}^{\\pi}\\geq t_{ik}-\\beta_{ik}^{\\pi})\\cap (t_{ij}\\geq \\beta_{ij}^{\\pi})\\}$.\n\nWe will define the same flow as before: $\\lambda^{\\pi}(t_i',t_i)>0$ if and only if\n\\begin{itemize}\n\\item $t_i\\in R_j^{\\pi}$\n\\item $t_{ik}'=t_{ik},\\forall k\\not=j$\n\\item $t_{ij}'>t_{ij}$\n\\end{itemize}\n\nThen after ironing, the Language function becomes:\n\\begin{equation}\nL(\\lambda^{\\pi},\\pi,p)=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\bigg(t_{ij}\\cdot \\mathds{1}\\big[t_i\\not\\in R_j^{\\pi}\\big]+\\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big]\\bigg)\n\\end{equation}\n\nBy the Strong Duality Theorem,\n\\begin{equation}\nREV=\\min_{\\lambda}\\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda,\\pi,p)\\leq \\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda^{\\pi},\\pi,p)\n\\end{equation}\n\nIt's enough to bound $L(\\lambda^{\\pi},\\pi,p)$ for all feasible $\\pi$ to obtain an upper bound for $REV$.\n\n\\subsection{Sequential Mechanism and c-Selectability}\nWe will use a new Sequential mechanism. The mechanism posts a price $\\theta_{ij}$ of each item $j$ for each bidder $i$ and determine an order $\\sigma$ for bidders. Each bidder comes in order $\\sigma$. When bidder $i$ with type $t_i$ comes, suppose the set of items left is $S_i$. The mechanism will let bidder $i$ know $S_i$ and charge him an entry fee $\\delta_i$. If he chooses to join the auction, he pays the entry fee and then takes his favorite bundle $S_i^{*}=\\arg\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})$ and pay $\\sum_{j\\in S_i^{*}}\\theta_{ij}$. Let the optimal revenue that such a mechanism can get is SEQ.\n\nWhen bidder $i$ is facing the remaining item set $S_i$, we have no idea on which item he will take. c-Selectability from [\\ref{feldman}] actually take control of it. The following lemma directly comes from the definition of c-Selectability:\n\\begin{lemma}\\label{feldman2015}\n(Feldman 2015) For a downward close $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS, then in the Sequential mechanism, when it's bidder $i$'s turn, as long as $j\\in S_i$ and $t_{ij}>\\theta_{ij}$, item $j$ is in $S_i^{*}$ with at least probability $c$. The probability is taken over the randomness of $S_i$.\n\\end{lemma}\n\nIf $\\mathcal{F}$ is a matroid, the paper shows that $c=1-b$.\n\n\n\\subsection{Separating into Pieces}\n\\begin{equation}\n\\begin{aligned}\n&L(\\lambda^{\\pi},\\pi,p) \\\\\n=&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big] \\text{(SINGLE)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\text{(SURPLUS)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE + SURPLUS }+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\text{(PROPHET)}\n\\end{aligned}\n\\end{equation}\n\n~\\\\\n\n\\begin{equation}\n\\begin{aligned}\nSURPLUS&=\\sum_i\\sum_{t_i: \\exists k,t_{ik}>\\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\text{ (TAIL)}\\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]\\text{ (CORE)}\\\\\n\\end{aligned}\n\\end{equation}\n\nHere, $\\tau_i^{\\pi}$ is chosen such that\n\\begin{equation}\n\\sum_j \\Pr_{t_{ij}}[t_{ij}\\geq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]=\\frac{1}{2}\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding SINGLE}\n\\begin{lemma}\nFor any downward close $\\mathcal{F}$, $\\text{SINGLE}\\leq \\text{OPT}^{\\text{COPIES}}$.\n\\end{lemma}\n\\begin{proof}\nWe build a new mechanism $M'$ in the Copies setting based on $M$. Whenever $M$ allocates item $j$ to bidder $i$ and $t_i\\in R_j^{\\pi}$, $M'$ serves the agent $(i,j)$. Then $M'$ is feasible in the Copies setting and SINGLE is the ironed virtual welfare of $M'$ with respect to $\\tilde{\\phi}(\\cdot)$. Since the maximum revenue in the Copies setting is achieved by the maximum virtual welfare, thus $\\text{OPT}^{\\text{COPIES}}$ is no less than SINGLE.\n\\end{proof}\n\nThe $\\text{OPT}^{\\text{COPIES}}$ can be achieved $\\frac{1}{6}$-approximately by a post-price mechanism.\n\n~\\\\\n\n\\subsection{Bounding PROPHET}\n\\begin{equation}\n\\begin{aligned}\n\\text{PROPHET }&=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi}\\\\\n&=\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}(t_i)\\\\\n&=\\frac{1}{b}\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\\label{prophet}\nFor $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS,\n\\[\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{(1-b)\\cdot c}\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider a Sequential mechanism without entry fee and post price $\\theta_{ij}=\\beta_{ij}^{\\pi}$. Then according to Lemma \\ref{feldman2015},\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>\\beta_{ij}^{\\pi}]\\cdot c\\\\\n&\\geq \\sum_i\\sum_j \\beta_{ij}\\cdot (\\sum_{l=1}^{i-1}q_{lj}^{\\pi})\\cdot q_{ij}^{\\pi}\\cdot c\\\\\n&\\geq (1-b)\\cdot c\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nSpecifically, if $\\mathcal{F}$ is a matroid,\n\\begin{equation}\n\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{\\cdot (1-b)^2}\\cdot SEQ\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding TAIL}\nWe will bound TAIL for any downward close $\\mathcal{F}$. Let $P_{ij}=\\arg\\max_{x\\geq \\tau_i^{\\pi}}x\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq x]$, and $r_{ij}=P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]$, $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. We have the following relationship between $r_i$ and $\\tau_i^{\\pi}$:\n\\begin{lemma}\\label{tail0}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot \\tau_i^{\\pi}$.\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n\\begin{aligned}\nr_i&= \\sum_j P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]\\\\\n&\\geq \\sum_j \\tau_i^{\\pi}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq \\tau_i^{\\pi}]\\\\\n&=\\frac{1}{2}\\cdot \\tau_i^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\nWe notice by the definition of $r_{ij}$,\n\\begin{equation}\\label{tail1}\n\\begin{aligned}\n\\text{TAIL }&=\\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\sum_{k\\not=j}\\Pr_{t_{ik}\\sim T_{ik}}\\big[t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}\\\\\n&\\leq \\sum_i r_i\\cdot \\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\\\\n&=\\frac{1}{2}\\cdot r\n\\end{aligned}\n\\end{equation}\n\nThe following lemma shows that $r$ can be approximately achieved by the Sequential mechanism.\n\\begin{lemma}\\label{tail2}\n\\[r\\leq \\frac{2}{(1-b)}\\text{ SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the Sequential mechanism with item price $\\theta_{ij}=P_{ij}+\\beta_{ij}^{\\pi}$, $\\sigma=(1,2,...,n)$ and without entry fee. When bidder $i$ comes, he will definitely take only item $j$ and pay $P_{ij}+\\beta_{ij}^{\\pi}$ if:\n\\begin{itemize}\n\\item $j\\in S_i$\n\\item $t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}$\n\\item $\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}$\n\\end{itemize}\n\nNotice that due to the second condition, every bidder will take item $j$ with at most $q_{ij}$ probability. Thus we have\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j (P_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}]\\\\\n&\\geq \\sum_i\\sum_j P_{ij}\\cdot \\big(\\sum_{l=1}^{i-1}q_{lj}^{\\pi}\\big)\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]\\\\\n&\\geq (1-b)\\cdot \\sum_i\\sum_j P_{ij}\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\big(1-\\sum_{k\\not=j}\\Pr_{t_{ik}}[t_{ik}\\leq P_{ik}+\\tau_i^{\\pi}]\\big)\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot \\sum_i\\sum_j r_{ij}\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot r\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nCombining Equation (\\ref{tail1}) and Lemma \\ref{tail2}, we have\n\\begin{equation}\nTAIL\\leq \\frac{1}{1-b}\\cdot SEQ\n\\end{equation}\n\n\\subsection{Bounding CORE}\nWe will bound CORE for matroid $\\mathcal{F}$. Define $t_{ij}'=(t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[\\beta_{ij}^{\\pi}\\leq t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}\\big]\\in [0,\\tau_i^{\\pi}]$. Then since $\\pi(\\cdot)$ is feasible,\n\\begin{equation}\nCORE=\\sum_i\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\n\\end{equation}\n\nConsider the Sequential mechanism with item price $\\theta_{ij}=\\beta_{ij}^{\\pi}$ and order $\\sigma=(1,2,...,n)$. When it's bidder $i$'s turn, suppose the set of items left is $S_i$. Define the entry fee $\\delta_i$ for bidder $i$ as:\n\\[\\Pr_{t_i'}[\\max_{S \\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\leq \\delta_i]=\\frac{2}{3}\\]\n\nNotice that if bidder $i$ enters the auction, the profit he gets is\n\\[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})\\geq \\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\]\nThus with probability at least $\\frac{1}{3}$, bidder $i$ will pay the entry fee $\\delta_i$. Besides, with the same argument in Lemma \\ref{prophet},\n\\begin{equation}\n\\text{SEQ}\\geq \\sum_i\\frac{1}{3}\\big(\\delta_i+(1-b)^2\\cdot \\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\big)\n\\end{equation}\n\n~\\\\\n\nThe following result from [Schechtman 1999] can be applied to bound $\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]$ for all $i$:\n\n\\begin{lemma}\\label{core2}\n(Schechtman, 1999)Let $f(t,S)$ be a value function that is additive under a downward close constaint, drawn from a distribution D over support $[0,\\tau]$ for some $\\tau\\geq 0$. Let $\\Delta$ be a value such that $\\Pr_{t\\sim D}\\big[f(t,[m])\\leq \\Delta\\big]=\\frac{2}{3}$. Then for all $k>0$,\n\\begin{equation}\n\\Pr_{t\\sim D}\\big[f(t,[m])\\geq 3\\Delta+k\\cdot \\tau\\big]\\leq \\frac{9}{4}\\cdot 2^{-k}\n\\end{equation}\n\\end{lemma}\n\n\\begin{corollary}\\label{core3}\nFor each $i$,\n\\[\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\\]\n\\end{corollary}\n\\begin{proof}\nLet $g(t_i')=\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Directly from Lemma \\ref{core2},\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{t_i'}\\big[g(t')\\big]&= \\mathbf{E}_{t_i'}\\big[g(t_i');g(t_i')<3\\delta_i\\big]+ \\mathbf{E}_{t_i'}\\big[g(t');g(t_i')\\geq 3\\delta_i\\big]\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}Pr[g(t_i')>3\\delta_i+y]dy\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}\\frac{9}{4}\\cdot 2^{-y\/\\tau}dy\\\\\n&\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nNow we can finish bounding the CORE with the following lemma.\n\n\\begin{lemma}\n\\[\\text{CORE }+3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the mechanism above. Assume $S^{*}=\\arg\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Notice for all $i,j$, $\\Pr_{t_{-i}}[j\\in S_i]\\geq 1-\\sum_i q_{ij}=1-b$. With Lemma \\ref{tail0}, \\ref{tail2}, and \\ref{core2}, we have\n\\begin{equation}\n\\begin{aligned}\nCORE&=\\sum_i\\mathbf{E}_{t_i'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\cdot \\mathds{1}[j\\in S_i]\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i (3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4ln(2)})\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\bigg(9\\cdot \\text{SEQ }-3(1-b)^2\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+\\frac{9r}{2\\cdot ln(2)}\\bigg)\\\\\n&\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}-3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\n\\subsection{Optimizing the Constant}\nPut all pieces together:\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda^{\\pi},\\pi,p)&\\leq \\text{SINGLE}+\\text{TAIL}+\\text{PROPHET}+\\text{CORE} \\\\\n&\\leq 6\\cdot \\text{SREV}+\\frac{1}{1-b}\\cdot \\text{SEQ}+\\big(\\frac{1}{b}-3(1-b)\\big)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+(\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\\\\n&\\leq 6\\cdot \\text{SREV}+\\bigg(\\frac{7}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2}+\\frac{1}{b(1-b)^2}\\bigg)\\cdot \\text{SEQ}\n\\end{aligned}\n\\end{equation}\n\nWhen $b=\\frac{1}{7}$. The larger one of SREV and SEQ gets a $1\/42$-approximately mechanism.\n\n~\\\\\n\n\n\\section{Single Bidder, Subadditive Valuation}\n\\subsection{Problem Setting\\textsuperscript{[\\ref{subadditive}]}}\nLet $m$ be the number of items, $I=[m]$ be the set of items. For each item $j$, let $\\Omega_j$ be a compact space. Each type $t_j\\in \\Omega_j$ represent the information for item $j$. For all $S\\subseteq I$, denote $\\Omega_S=\\times_{j\\in S}\\Omega_j$. $\\Omega=\\Omega_{I}$. The bidder's type $t=\\langle t_j \\rangle_{j\\in I}$ is drawn from a distribution $D$ on $\\Omega$. After $t$ is drawn, the bidder's valuation for a set of items S is only determined by the vector $\\langle t_j \\rangle_{j\\in S}$. Formally, denote $\\Omega^{*}=\\bigcup_{S\\subseteq I}\\Omega_S$. There exists a valuation function $V:\\Omega^{*}\\to \\mathcal{R}$ such that $\\forall t\\in \\Omega, S\\subseteq I$,\n\\begin{equation}\nv(t,S)=V(\\langle t_j \\rangle_{j\\in S},S)\n\\end{equation}\n\nIn this problem, we assume that the bidder's valuation is subadditive, i.e., $\\forall t\\in \\Omega$, $\\forall P,Q\\subseteq I$,\n\\begin{equation}\nv(t,P\\cup Q)\\leq v(t,P)+v(t,Q)\n\\end{equation}\n\n~\\\\\n\n\\subsection{Duality}\nWe use the following functions(variables) to describe the mechanism:\n\\begin{itemize}\n\\item $p(t)$, $t\\in \\Omega^{+}$: the prize that the bidder should pay when his type is $t$.\n\\item $\\phi(t,t')$, $t\\in \\Omega, t'\\in \\Omega^{+}$: the expect valuation if the bidder pretends to be $t'$ when his type is $t$.\n\\end{itemize}\n\nwhere $\\Omega^{+}=\\Omega\\cup \\{\\emptyset\\}$ which allows the bidder not to participate in the auction. When $t=\\emptyset$, $p(t)=\\phi(\\cdot,t)=0$.\n\n\\textbf{Remark:} For a mechanism, let $\\pi_S(t)$ be the probability for bidder with type $t$ to obtain a set $S$ of item. Then $\\phi(t,t')$ can be written as\n\\begin{equation}\n\\phi(t,t')=\\sum_{S\\subseteq I}\\pi_S(t)v_{t'}(S)\n\\end{equation}\n\nHere we use $\\phi(t,t')$ to replace the original variables $\\pi_j(t)=\\sum_{S:j\\in S}\\pi_S(t)$. The function in fact includes both the probability and the valuation. Like $\\pi$, all the $\\phi(t,t')$'s should stay in some feasible region to avoid over-allocation. We use $\\phi\\in \\mathcal{O}$ to represent it.\n\n~\\\\\n\nOur primal is:\n\\begin{itemize}\n\\item \\textbf{Variables}: $p(t),\\phi(t,t'),\\quad t\\in \\Omega, t'\\in \\Omega^{+}$\n\\item \\textbf{Constraint}:\n\n$\\quad(1)\\phi(t,t)-p(t)\\geq \\phi(t,t')-p(t'),\\quad \\forall t\\in \\Omega, t'\\in \\Omega^{+}$\n\n$\\quad(2)\\phi\\in \\mathcal{O}$\n\n\\item \\textbf{Objective}: \\text{min} $\\sum_{t\\in \\Omega}f(t)p(t)$\n\\end{itemize}\n\n~\\\\\n\nThe Language dual function $L(\\lambda,\\phi,p)$ is\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi,p)&=\\sum_{t\\in T}f(t)p(t)+\\sum_{t\\in \\Omega,t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg((\\phi(t,t)-p(t))-(\\phi(t,t')-p(t'))\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}\\bigg(f(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg)+\\sum_{t\\in \\Omega}\\bigg(\\phi(t,t)\\cdot\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')-\\sum_{t'\\in \\Omega}\\lambda(t',t)\\cdot \\phi(t',t)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\bigg(\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Phi(t)=\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\n\\end{equation}\n\nHere we need $\\forall t\\in \\Omega$,\n\\begin{equation}\nf(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in T^{+}}\\lambda(t,t')=0\n\\end{equation}\nto avoid the unlimited optimal value.\n\n\\subsection{Change the Valuation}\n\nWe will change the valuation and apply the duality to the new valuation. Let\n\n\\begin{equation}\nR_j=\\bigg\\{t\\in\\Omega: j=\\arg\\max_k V(t_k,\\{k\\})\\bigg\\}\n\\end{equation}\n\nWe define a new valuation $\\hat{v}(\\cdot)$. For a type $t\\in \\Omega$, if $t\\in R_j$, then\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t,S)=\n\\begin{cases}\nv(t,S\\backslash\\{j\\})+v(t,\\{j\\}), &j\\in S\\\\\nv(t,S), &j\\not\\in S\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\nSince $v(t,\\cdot)$ is subadditive, $\\hat{v}(t,S)\\geq v(t,S)$ for all subset S.\n\nLet $REV(v)$ and $REV(\\hat{v})$ be the optimal revenue for the two valuations. We would like to see that the two values are not far so that we can bound $REV(v)$ by bounding $REV(\\hat{v})$. The following two lemmas give the proof of this fact:\n\n\\begin{lemma}\n\\textsuperscript{[\\ref{subadditive}]}Consider a distributions D with two different valuations $v$ and $\\hat{v}$. Let $\\delta(t,S)=\\hat{v}(t,S)-v(t,S)$. If $\\delta(t,S)\\geq 0$ for all $t\\in T, S\\subseteq I$, then for any $\\epsilon\\in (0,1)$,\n\\begin{equation}\nREV(\\hat{v})\\geq (1-\\epsilon)(REV(v)-\\frac{\\bar\\delta}{\\epsilon})\n\\end{equation}\nwhere $\\bar\\delta=\\mathbb{E}_{t\\sim D}[\\max_{S\\subseteq I}\\delta(t,S)]$.\n\nIf we choose $\\epsilon=\\frac{1}{2}$,\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta\n\\end{equation}\n\n\\end{lemma}\n\n~\\\\\n\n\\begin{lemma}\nLet $\\bar\\delta'=\\mathbb{E}_{t\\sim D}\\bigg[v(t,I\\backslash\\{j\\}), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]$, then\n\\begin{equation}\n\\bar\\delta\\leq \\bar\\delta'\\leq 6BREV+9.2SREV\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nFor a fixed type $t$, we make a partition $I=C_t\\cup T_t$ based on some cutoff $r$, where $C_t=\\{j\\in I: v(t,\\{j\\})<r\\}$, $T_t=\\{j\\in I: v(t,\\{j\\})\\geq r\\}$. Where $r$ is chosen such that\n\\begin{equation}\n\\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=1\n\\end{equation}\n\nNotice by monotonicity of $v(t,\\cdot)$,\n\n\\begin{equation}\n\\begin{aligned}\n\\bar\\delta&=\\mathbb{E}_{t\\sim D}\\bigg[\\max_{S\\subseteq I}\\big(\\hat{v}(t,S)-v(t,S)\\big)\\bigg]\\\\\n&\\leq \\mathbb{E}_{t\\sim D}\\bigg[\\max_{S\\subseteq I}\\big(v(t,S\\backslash\\{j\\})\\big), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]\\\\\n&\\leq \\bar\\delta'\\\\\n&\\leq \\mathbb{E}_{t\\sim D}\\bigg[v(t,C_t)+\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\\\\n&= \\mathbb{E}_{t\\sim D}[v(t,C_t)]\\text{(CORE)}+\\mathbb{E}_{t\\sim D}\\bigg[\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\text{(TAIL)}\\\\\n\\end{aligned}\n\\end{equation}\n\n\nWe will bound the two parts separately.\n\n~\\\\\n\n\\subsubsection{Core}\n\nWe define function $F:\\Omega^{*}\\to R^{+}$ as follows. For a vector $\\langle t_j\\rangle_{j\\in S}$, let $C=\\{j\\in S: V(t_j,\\{j\\})\\leq r\\}$, then define\n\\begin{equation}\nF(\\langle t_j\\rangle_{j\\in S})=V(\\langle t_j\\rangle_{j\\in C},C)\n\\end{equation}\n\nIt's not hard to justify that F is monotone, subadditive and $r$-Lipschitz. We have the following lemma for functions with this property:\n\\begin{lemma}\\label{core2}\n(Rubinstein and Weinberg, 2015)Let $v$ be a subadditive valuation over independent items and $c$-Lipschitz. Let $a$ be the median of the value of the grand bundle $v([m])$. Then\n\\begin{equation}\\\n\\mathds{E}[v([m])]\\leq 3a+\\frac{4c}{ln(2)}\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nLet $\\delta$ be the median of $F(t)$ over distribution $D$. Now consider the mechanism that sell the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $V(t,I)\\geq F(t)$. Thus with probability at least $\\frac{1}{2}$, the bidder will buy the bundle. Thus,\n\\begin{equation}\nBREV\\geq \\frac{1}{2}\\delta\n\\end{equation}\n\nAccording to Lemma \\ref{core2},\n\\begin{equation}\\label{e1}\n\\text{(CORE)}=\\mathbb{E}_{t\\sim D}[F(t)]\\leq 3\\delta+\\frac{4r}{ln(2)}\n\\leq 6BREV+\\frac{4r}{ln(2)}\n\\end{equation}\n\n~\\\\\n\n\\subsubsection{Tail}\n\\begin{equation}\n\\begin{aligned}\n(TAIL)&=\\mathbb{E}_{t\\sim D}\\bigg[\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\\\\n&\\leq \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)\\cdot V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\n\\end{aligned}\n\\end{equation}\n\nFor all $t_j$, consider the auction that we sell each item at price $V(t_j,\\{j\\})$. We notice if there exists $k\\not= j$ such that $V(t_k,\\{k\\})>V(t_j,\\{j\\})$, there is at least one bundle with positive profit. The mechanism will definitely sell some item, obtaining expected revenue at least $V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]$. Thus,\n\\begin{equation}\nV(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\\leq SREV\n\\end{equation}\n\n\\begin{equation}\\label{e2}\n(TAIL)\\leq SREV\\cdot \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=SREV\n\\end{equation}\n\n~\\\\\n\nNow we can finish the prove of Lemma 2.\n\n\\begin{equation}\n\\begin{aligned}\nPr_t[\\exists j, V(t_j,\\{j\\})\\geq r]&\\geq 1-\\prod_{j}F_j(r)\\\\\n&\\geq 1-(\\frac{\\sum_j F_j(r)}{n})^n\\\\\n&\\geq 1-(1-1\/n)^n\\\\\n&=1-\\frac{1}{e}\n\\end{aligned}\n\\end{equation}\n\nConsider the auction that sells every item with price $r$. Then with probability at least $1-\\frac{1}{e}$, the mechanism will sell at least one item at price at least $r$, obtaining revenue at least $(1-\\frac{1}{e})r$. Thus,\n\\begin{equation}\\label{e3}\nr\\leq \\frac{1}{1-1\/e}\\cdot SREV\n\\end{equation}\n\nCombining Equation (\\ref{e1})(\\ref{e2})(\\ref{e3}), we have\n\\begin{equation}\n\\bar\\delta\\leq (CORE)+(TAIL)\n\\leq 6BREV+\\frac{4r}{ln(2)}+SREV\n\\leq 6BREV+9.2SREV\n\\end{equation}\nwhich finishes Lemma 2.\n\n~\\\\\n\n\\subsection{Construction of the Flow}\nWe apply duality on the new valuation $\\hat{v}$. For $t,t'\\in \\Omega$, $\\lambda(t',t)>0$ if and only if\n\\begin{itemize}\n\\item $t$ and $t'$ only differ on the $j$-th coordinate.\n\\item $t\\in R_j$.\n\\item $V(t_j',\\{j\\})>V(t_j,\\{j\\})$.\n\\end{itemize}\n\nWe now consider $\\hat{v}(t',S)-\\hat{v}(t,S)$ when $\\lambda(t',t)>0$. There are two conditions for subset S:\n\n(1)$j\\not\\in S$, notice that only the $j-$th coordinate is different,\n\\begin{equation}\n\\hat{v}(t',S)=V'(\\langle t_k' \\rangle_{k\\in S},S)=V'(\\langle t_k \\rangle_{k\\in S},S)=\\hat{v}(t,S)\n\\end{equation}\n\n(2)$j\\in S$, we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t',S)-\\hat{v}(t,S)&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j',\\{j\\})\\bigg)-\n\\bigg(V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j,\\{j\\})\\bigg)\\\\\n&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})-V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})\\bigg)+\\bigg(V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\\bigg)\\\\\n&=V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Bound the Language Function}\nFor all type $t$, the valuation when bidder tells the truth is\n\n\\begin{equation}\n\\begin{aligned}\n\\phi'(t,t)&=\\sum_{S\\subseteq I}\\pi_S(t)\\hat{v}(t,S)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\sum_{S:j\\in S}\\pi_S(t)\\big(\\hat{v}(t,S\\backslash\\{j\\})+\\hat{v}(t,\\{j\\})\\big)+\\sum_{S:j\\not\\in S}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\sum_{S\\subseteq I\\backslash\\{j\\}}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&\\leq \\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\hat{v}(t,I\\backslash\\{j\\})\\bigg)\n\\end{aligned}\n\\end{equation}\n\nThe virtual valuation $\\Phi(t)$:\n\n\\begin{equation}\n\\begin{aligned}\n\\Phi(t)&=\\phi'(t,t)-\\frac{1}{f(t)}\\sum_{S\\subseteq I}\\pi_S(t)\\sum_{t'\\in \\Omega}\\lambda(t',t)(\\hat{v}(t',S)-\\hat{v}(t,S))\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\big(\\hat{v}(t,\\{j\\})-\\sum_{t_j':V'(t_j',\\{j\\})>V'(t_j,\\{j\\})}\\frac{f_j(t_j')}{f_j(t_j)}\\big)\\bigg)\\\\\n&\\leq\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\tilde{\\phi}_j(t_j)\\bigg)\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\phi}_j(t_j)$ is the Myerson Virtual Value for item j with the marginal distribution on $\\Omega_j$ and valuation $V'(t_j,\\{j\\})$.\n\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi',p)&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\\\\\n&=\\bar\\delta'+\\sum_t\\sum_jf(t)\\pi_j(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\\\\n&\\leq \\bar\\delta'+\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\n\\begin{equation}\n\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\leq 6SREV\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nConsider the optimal revenue $OneRev$ when the mechanism is only allowed to sell one item. Assume a mechanism M allocate item j only if $t\\in R_j$. Then LHS is the virtual welfare of this mechanism, which is less than $OneRev$. Besides, this allocation rule is same as assuming bidder's valuation is unit-demand. Thus selling separately can reach a $\\frac{1}{6}$-approximation of this revenue.\n\\end{proof}\n\n~\\\\\n\n\\subsection{Conclusion}\nPut everything together we have\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta \\leq 2L(\\lambda,\\phi',p)+2\\bar\\delta \\leq 4\\bar\\delta'+12SREV \\leq 24BREV+49SREV\n\\end{equation}\n\n\n\n\n\\section{Duality}\\label{sec:duality}\n\n\n The focus of \\cite{CaiDW16} was on additive and unit-demand valuations and their respective dual was derived from an LP that is only meaningful for constrained additive valuations. In order to tackle general valuations, we need to apply the duality framework to an LP that is meaningful for general valuations. Instead of using the ``implicit forms'' LP from~\\cite{CaiDW13b, CaiDW16}, we choose a slightly different and more intuitive LP formulation (see Figure~\\ref{fig:LPRevenue}). For all bidders $i$ and types $t_i \\in T_i$, we use $p_i(t_i)$ as the interim price paid by bidder $i$ and $\\sigma_{iS}(t_i)$ as the interim probability of receiving the exact bundle $S$. To ease the notation, we use a special type $\\varnothing$ to represent the choice of not participating in the mechanism. More specifically, ${\\sigma}_{iS}(\\varnothing)=0$ for any $S$ and $p_{i}(\\varnothing)=0$. Now a Bayesian IR (BIR) constraint is simply another BIC constraint: for any type $t_{i}$, bidder $i$ will not want to lie to type $\\varnothing$.  We let $T_{i}^{+}=T_{i}\\cup \\{\\varnothing\\}$.\n\nFollowing the recipe provided by~\\cite{CaiDW16}, we take the partial Lagrangian dual of the LP in Figure~\\ref{fig:LPRevenue} by lagrangifying the BIC constraints. Let $\\lambda_i(t_i,t_i')$ be the Lagrange multiplier associated with the BIC constraint that if bidder $i$'s true type is $t_i$ she will not prefer to lie to type $t_i'$\n (see Figure~\\ref{fig:Lagrangian} and Definition~\\ref{def:Lagrangian}). As shown in~\\cite{CaiDW16}, the dual solution has finite value if and only if the dual variables $\\lambda_i$ form a valid flow for every bidder $i$. The reason is that the payments $p_i(t_i)$ are unconstrained variables, therefore the corresponding coefficients must be $0$ in order for the dual solution to have finite value. It turns out when all these coefficients are $0$, the dual variables $\\lambda$ can be interpreted as a flow described in Lemma~\\ref{lem:useful dual}. We refer the readers to~\\cite{CaiDW16} for a complete proof. From now on, we only consider $\\lambda$ that corresponds to a flow.\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $p_i(t_i)$, for all bidders $i$ and types $t_i \\in T_i$, denoting the expected price paid by bidder $i$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\item $\\sigma_{iS}(t_i)$, for all bidders $i$, all bundles of items $S\\subseteq[m]$, and types $t_i \\in T_i$, denoting the probability that bidder $i$ receives \\textbf{exactly} the bundle $S$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\sum_{S\\subseteq[m]} {\\sigma}_{iS}(t_i) \\cdot v_i(t_i,S) - p_i(t_i) \\geq\\sum_{S\\subseteq[m]}{\\sigma}_{iS}(t'_i) \\cdot v_i(t_i, S) - p_i(t'_i) $, for all bidders $i$, and types $t_i \\in T_i, t'_i \\in T_i^+$, guaranteeing that the reduced form mechanism $({\\bf{\\sigma}},{p})$ is BIC and Bayesian IR.\n\\item ${\\bf{\\sigma}} \\in {P(D)}$, guaranteeing ${\\sigma}$ is feasible.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\max \\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)$, the expected revenue.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{A Linear Program (LP) for Revenue Optimization.}\n\\label{fig:LPRevenue}\n\\end{figure}\n\n\\begin{definition}\\label{def:Lagrangian}\nLet ${\\mathcal{L}}(\\lambda, \\sigma, p)$ be the partial Lagrangian defined as follows:\n\\begin{align*}\n& {\\mathcal{L}}(\\lambda, \\sigma, p)\\\\\\stepcounter{equation}\\tag{\\theequation} \\label{eq:primal lagrangian}\n=&\\sum_{i=1}^{n} \\left(\\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)+\\sum_{t_{i}\\in T_{i},t_{i}'\\in T_i^{+}} \\lambda_{i}(t_{i},t_{i}')\\cdot \\left(\\sum_{S\\subseteq[m]} v_i(t_{i},S)\\cdot\\left(\\sigma_{iS}(t_{i})-\\sigma_{iS}({t_{i}'})\\right)-\\left((p_{i}(t_{i})-p_{i}(t_{i}')\\right)\\right)\\right)\\\\\n=& \\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}} p_{i}(t_{i})\\cdot\\left(f_{i}(t_{i})+\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})-\\sum_{t_{i}'\\in T_{i}^{+}} \\lambda_{i}(t_{i},t_{i}')\\right)\\right)\\\\\n&+\\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}}\\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot \\left(v_i(t_{i},S)\\cdot \\sum_{t_{i}'\\in T_{i}^{+}}\\lambda_{i}(t_{i},t_{i}')-\\sum_{t'_i\\in T_{i}}\\left(v_i(t'_{i},S)\\cdot \\lambda_{i}(t_{i}',t_{i})\\right)\\right)\\right)~~ ({\\sigma}_i(\\varnothing)=\\textbf{0},\\ p_{i}(\\varnothing)=0)\\stepcounter{equation}\\tag{\\theequation} \\label{eq:dual lagrangian}\n\\end{align*}\n\\end{definition}\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, the Lagrangian multipliers for Bayesian IC and IR constraints.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')\\geq 0$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, guaranteeing that the Lagrangian multipliers are non-negative.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\min_{\\lambda}\\max_{\\sigma\\in {P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)$.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{Partial Lagrangian of the Revenue Maximization LP.}\n\\label{fig:Lagrangian}\n\\end{figure}\n\n\n\\notshow{ \\begin{definition}[Useful Dual Variables~\\cite{CaiDW16}]\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$.\n\\end{definition}}\n\n\n\\begin{lemma}[Useful Dual Variables~\\cite{CaiDW16}]\\label{lem:useful dual}\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$. $\\lambda$ is useful iff for each bidder $i$, $\\lambda_{i}$ forms a valid flow, i.e., iff the following satisfies flow conservation at all nodes except the source and the sink:\n\n \\noindent\\textbf{\\emph{1.}} Nodes: A super source $s$ and a super sink $\\varnothing$, along with a node $t_{i}$ for every type $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{2.}} An edge from $s$ to $t_{i}$ with flow $f_i(t_{i})$, for all $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{3.}} An edge from $t_i$ to $t_i'$ with flow $\\lambda_i(t_i,t_i')$ for all $t_i\\in T_i$, and $t_i'\\in T_{i}^{+}$ (including the sink).\n\n\\end{lemma}\n\n\\begin{definition}[Virtual Value Function]\\label{def:virtual value}\nFor each flow $\\lambda$, we define a corresponding virtual value function $\\Phi(\\cdot)$, such that for every bidder $i$, every type $t_{i}\\in T_{i}$ and every set $S\\subseteq[m]$,\n$$\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right)$$.\n\\notshow{\n\\vspace{.1in}\\noindent$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-$\n\n$\\hspace{1.3cm}{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right).$\t\n}\n\\end{definition}\nThe proof of Theorem~\\ref{thm:revenue less than virtual welfare} is essentially the same as in~\\cite{CaiDW16}. We include it in Appendix~\\ref{sec:proof_duality} for completeness.\n\\begin{theorem}[Virtual Welfare $\\geq$ Revenue~\\cite{CaiDW16}]\\label{thm:revenue less than virtual welfare}\nFor any flow $\\lambda$ and any BIC mechanism $M=(\\sigma,p)$, the revenue of $M$ is $\\leq$  the virtual welfare of {$\\sigma$} w.r.t. the virtual valuation $\\Phi(\\cdot)$ corresponding to $\\lambda$.\n$$\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq \\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$$\n\n\\notshow{\n\\vspace{.05in} $\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq $\n\n$\\hspace{1cm}\\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$\n}\nLet $\\lambda^{*}$ be the optimal dual variables and $M^{*}=(\\sigma^{*},p^{*})$ be the revenue optimal BIC mechanism, then the expected virtual welfare with respect to $\\Phi^{*}$ (induced by $\\lambda^{*}$) under  $\\sigma^{*}$ equals to the expected revenue of $M^{*}$.\n\\end{theorem}\n\n\\section{Canonical Flow and Properties of the Virtual Valuations}\\label{sec:flow}\n\n\nIn this section, we present a canonical way of setting the dual variables\/flow that induces our benchmarks. A recap of the flow for additive valuations and the appealing properties of the corresponding virtual valuation functions can be found in Appendix~\\ref{sec:flow_additive}. We refer readers to that Section for more intuition about the flow.\n\nAlthough any flow can provide a finite upper bound of the optimal revenue, we focus on a particular class of flows, in which every flow $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n],j\\in[m]}\\in\\R^{nm}_{\\geq 0}$. Based on $\\beta$, we partition the type set $T_i$ of each buyer $i$ into $m+1$ regions: \\textbf{(i)} $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})<\\beta_{ij}$ for all $j\\in[m]$. \\textbf{(ii)} $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{V_i(t_{ik})-\\beta_{ik}\\}$. Intuitively, if we view $\\beta_{ij}$ as the price of item $j$ for bidder $i$, then $R^{(\\beta_i)}_0$ contains all types in $T_i$ that cannot afford any item, and any $R^{(\\beta_i)}_j$ with $j>0$ contains all types in $T_i$ whose ``favorite'' item is $j$. We first provide a {\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\n\\noindent\\textbf{1.} For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$,  the flow goes directly to $\\varnothing$ (the super sink).\n\n\\noindent \\textbf{2.}  For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from  $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n\n\\noindent \\textbf{3.} For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\n\\notshow{\n\\begin{figure}\n  \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n  \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n  \\label{fig:multiflow}\n\\end{figure}\n}\n\nFor additive valuations and any type $t_i \\in R_j^{(\\beta_i)}$ , the contribution to the virtual value function $\\Phi(t_i,S)$  from any type $t_i'\\in R_j^{(\\beta_i)}$ is either $0$ if $j\\notin S$, or {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))=\\lambda_i^{(\\beta)}(t_i', t_i)(t_{ij}'-t_{ij})$} if $t_i$, $t_i'$ only differs on the $j$-th coordinate and $j\\in S$. In either case, the contribution does not depend on $t_{ik}$ for any $k\\neq j$. This is the key property that allows~\\cite{CaiDW16} to choose a flow such that the value of the favorite item is replaced by the corresponding Myerson's ironed virtual value in the virtual value function $\\Phi_i(t_i,\\cdot)$.\nUnfortunately, this property no longer holds for subadditive valuations. When $j\\in S$ and $\\lambda_i^{(\\beta)}(t_i',t_i)>0$, the contribution {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))$} heavily depends on $t_{ik}$ of all the other item $k\\in S$. All we can conclude is that the contribution lies in the range {$[-\\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}), \\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}')]$}\\footnote{$v_i(t,\\cdot)$ is subadditive and monotone for every type $t\\in T_i$, therefore $v_i(t_i,S)\\in[v_i(t_i, S\\backslash\\{j\\}),v_i(t_i, S\\backslash\\{j\\})+V_{i}(t_{ij})]$ and $v_i(t'_i,S)\\in[v_i(t'_i, S\\backslash\\{j\\}),v_i(t'_i, S\\backslash\\{j\\})+V_{i}(t'_{ij})]$.}, but this is not sufficient for us to convert the value of item $j$ into the corresponding Myerson's ironed virtual value.\n\n\\subsection{Valuation Relaxation}\\label{sec:valuation relaxation}\nThis is the first major barrier for extending the duality framework to accommodate subadditive valuations. We overcome it by considering a relaxation of the valuation functions. More specifically, for any $\\beta$, we construct another function $v_i^{(\\beta_i)}(\\cdot,\\cdot): T_i\\times 2^{[m]}\\mapsto {\\mathbb{R}_{\\geq 0}}$ for every buyer $i$ such that: (i) for any $t_i$,  $v_i^{(\\beta_i)}(t_i,\\cdot)$ is subadditive and monotone, and for every bundle $S$ the new value $v_i^{(\\beta_i)}(t_i,S)$ is no smaller than the original value $v_i(t_i,S)$; (ii) for any BIC mechanism $M$ with respect to the original valuations, there exists another mechanism $M^{(\\beta)}$ that is BIC with respect to the new valuations and its revenue is comparable to the revenue of $M$; (iii) for the new valuations $v^{(\\beta)}$, there exists a flow whose induced virtual value functions have properties similar to those in the additive case.\nProperty (ii) implies that the optimal revenue with respect to $v^{(\\beta)}$ can serve as a proxy for the original optimal revenue. Moreover, due to Theorem~\\ref{thm:revenue less than virtual welfare}, the optimal revenue for $v^{(\\beta)}$ is upper bounded by the partial Lagrangian dual with respect to $v^{(\\beta)}$, which has an appealing format similar to the additive case by property (iii). Thus, we obtain a benchmark for subadditive bidders that resembles the benchmark for additive bidders in~\\cite{CaiDW16}\n\n\\begin{definition}[Relaxed Valuation]\\label{def:relaxed valuation}\n\tGiven $\\beta$, for any buyer $i$, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})$, if the ``favorite'' item is in $S$, i.e., $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$. Otherwise, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S)$.\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\nv_i^{(\\beta_i)}(t_i,S)=\n\\begin{cases}\nv_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})  &\\text{if the ``favorite'' item is in $S$, i.e., }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\\end{definition}\n\nIn the next Lemma, we show that for any BIC mechanism $M$ for $v$, there exists a BIC mechanism $M^{(\\beta)}$ for $v^{(\\beta)}$ such that its revenue is comparable to the revenue of $M$ (property (ii)). Moreover, the ex-ante probability for any buyer $i$ to receive any item $j$ in $M^{(\\beta)}$ is no greater than in $M$ (property (i)). We will see later that this is an important property for our analysis. The proof of Lemma~\\ref{lem:relaxed valuation} is similar to the $\\epsilon$-BIC to BIC reduction in~\\cite{HartlineKM11, BeiH11,DaskalakisW12} and can be found in Appendix~\\ref{sec:proof_relaxed_valuation}.\n\n\n\\begin{lemma}\\label{lem:relaxed valuation}\n\n\tFor any $\\beta$ and any BIC mechanism $M$ for subadditive valuation  $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\n \\vspace{.1in}\n  \\noindent \\emph{\\textbf{(i)}} $\\displaystyle\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\n  \\vspace{.1in}\n \\noindent \\emph{\\textbf{(ii)}} $\\displaystyle\\textsc{Rev}(M, v, D)\\leq2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\displaystyle+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).$\n\n\\vspace{0.05in}\n\\noindent$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$)\n\\end{lemma}\n\\notshow{\nFrom now on, we fix $M^{(\\beta)}$ to be the mechanism that is constructed by setting $\\eta$ to be $1\/2$ and $\\epsilon$ be a extremely tiny positive constant $\\epsilon_o$ in Lemma~\\ref{lem:relaxed valuation}.\n\\begin{corollary}\n\tFor any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that\n\t$$\\textsc{Rev}(M, v, D)\\leq 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\epsilon_0.$$\n\\end{corollary}}\n\\subsection{Virtual Valuation for the Relaxed Valuation}\\label{sec:virtual for relaxed}\nFor any $\\beta$, based on the same partition of the type sets as in the beginning of Section~\\ref{sec:flow}, we construct a flow $\\lambda^{(\\beta)}$ that respects the partial specification, such that the corresponding virtual valuation function for $v^{(\\beta)}$ has the same appealing properties as in the additive case.\nFor the relaxed valuation, as {$\\lambda_i^{(\\beta)}(t_i, t_i')$} is only positive for types $t_i$, $t_i'\\in R_j^{(\\beta_i)}$ that only differ in the $j$-th coordinate, the contribution from item $j$ to the virtual valuation solely depends on $t_{ij}$ and $t'_{ij}$ but not $t_{ik}$ for any other item $k\\in S$\n. Notice that this property does not hold for the original valuation, and it is the main reason why we choose the relaxed valuation as in Definition~\\ref{def:relaxed valuation}. Moreover, we can choose $\\lambda_i^{(\\beta)}$ carefully so that the virtual valuation of $v^{(\\beta)}$ has the following format:\n\n\n\n\n\\begin{lemma}\\label{lem:subadditive flow properties}\n\tLet $F_{ij}$ be the distribution of $V_i(t_{ij})$ when $t_{ij}$ is drawn from $D_{ij}$. For any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_i$ such that the corresponding virtual value function $\\Phi^{(\\beta_i)}_{i}(t_{i}, \\cdot)$ of valuation $v_i^{(\\beta_i)}(t_i,\\cdot)$ satisfies the following properties:\n\n\\vspace{.05in}\t\n\\noindent 1. For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi^{(\\beta_i)}_{i}(t_{i},S) = v_i(t_i, S)$.\n\n\\vspace{.05in}\n\\noindent 2. For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$,  $\\Phi_{i}^{(\\beta_i)}(t_{i},S)\\leq  v_i (t_{i}, S)\\cdot\\mathds{1}[j\\notin S]+\\left(v_i (t_{i}, S\\backslash\\{j\\})+{\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))\\right)\\cdot\\mathds{1}[j\\in S],$ where ${\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$ is the Myerson's ironed virtual value for  $V_i(t_{ij})$ with respect to $F_{ij}$.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem:subadditive flow properties} is postponed to Appendix~\\ref{sec:proof_virtual_relaxation}.\nNext, we use the virtual welfare of the allocation $\\sigma^{(\\beta)}$ to bound the revenue of $M^{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:upper bound the revenue of the relaxed mechanism}\n\tFor any $\\beta$, \\begin{align*} &\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot\\Phi^{(\\beta_i)}_i(t_i,S)\\\\\n \\leq &\t\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\backslash\\{j\\})+\\sum_{S:j\\notin S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\right)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\pi^{(\\beta)}_{ij}(t_i)\\cdot {\\tilde{\\varphi}}_{ij}(t_{ij}),\\end{align*}\n where $ \\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$. {\\bf \\textsc{Non-Favorite}$(M, \\beta)$} denotes the sum of the first two terms. {\\bf \\textsc{Single}$(M, \\beta)$} denotes the last term.  \\end{lemma}\n\n\\begin{proof}\nThe Lemma follows easily from the properties in Lemma~\\ref{lem:subadditive flow properties} and Theorem~\\ref{thm:revenue less than virtual welfare}.\n\\end{proof}\n\nWe obtain Theorem~\\ref{thm:revenue upperbound for subadditive} by combining Lemma~\\ref{lem:relaxed valuation} and~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n \\begin{theorem}\\label{thm:revenue upperbound for subadditive}\nFor any mechanism $M$ and any $\\beta$,\n$$\\textsc{Rev}{(M,v,D)}\\leq 4\\cdot\\textsc{Non-Favorite}(M, \\beta)+2\\cdot\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\n\\begin{prevproof}{Theorem}{thm:revenue upperbound for subadditive}\nFirst, let's look at the value of $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)$. When $t_i\\in R_j^{(\\beta_i)}$ for some $j>0$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)= v_i(t_i, S\\backslash\\{j\\})+V_i(t_{ij})-v_i(t_i,S)\\leq v_i(t_i, S\\backslash\\{j\\}),$ because $V_i(t_{ij})\\leq v_i(t_i,S)$. For the other cases, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)=0$. Therefore,\n\\begin{align*}\\label{eq:bounding delta}\n\t&\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\nonumber\\\\\n\\leq &\\sum_i\\sum_{t_i} f_i(t_i)\\sum_{j} \\mathds{1}[t_i\\in R_j^{(\\beta_i)}]\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\cdot v_i(t_i, S\\backslash\\{j\\})\\nonumber\\\\\n\\leq &\\textsc{Non-Favorite}(M,\\beta)~~~~~~~~~\\text{(Definition of $\\textsc{Non-Favorite}(M,\\beta)$)}\n\\end{align*}\n\nOur statement follows from combining Lemma~\\ref{lem:relaxed valuation}, Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism} with the inequality above.\n\\end{prevproof}\n\\begin{comment}\nNow we bound $\\textsc{Rev}(M, v, D)$. By Lemma~\\ref{lem:relaxed valuation},\n\t\\begin{align*}&\\textsc{Rev}(M, v, D)\\\\\n\t\\leq& 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\\\\n+&2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\\\\t\n\\leq &2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\textsc{Non-Favorite}(M,\\beta)~~~~\\text{(Equation~(\\ref{eq:bounding delta}))}\\\\\n\\leq & 4\\cdot\\textsc{Non-Favorite}(M,\\beta)+2\\cdot\\textsc{Single}(M,\\beta)~~~~\\text{(Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism})}.\n\t\\end{align*}\n\\end{comment}\n\n\n\n\\subsection{Upper Bound for the Revenue of Subadditive Buyers}~\\label{sec:choice of beta}\n In Section~\\ref{sec:valuation relaxation}, we have argued that for any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that its revenue with respect to the relaxed valuation $v^{(\\beta)}$ is comparable to the revenue of $M$ with respect to the original valuation. In Section~\\ref{sec:virtual for relaxed}, we have shown for any $\\beta$ how to choose a flow to obtain an upper bound for $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and also an upper bound for $\\textsc{Rev}(M,v,D)$. Now we specify our choice of $\\beta$.\n\nIn~\\cite{CaiDW16}, the authors fixed a particular $\\beta$, and shown that under any allocation rule, the corresponding benchmark can be bounded by the sum of the revenue of a few simple mechanisms. However, for valuations beyond additive and unit-demand, the benchmark becomes much more challenging to analyze\\footnote{Indeed, the difficulties already arise for valuations as simple as $k$-demand. A bidder's valuation is $k$-demand if her valuation is additive subject to a uniform matroid with rank $k$.}. We adopt an alternative and more flexible approach to obtain a new upper bound. Instead of fixing a single $\\beta$ for all mechanisms, we customize a different $\\beta$ for every different mechanism $M$. Next, we relax the valuation and design the flow based on the chosen $\\beta$ as specified in Section~\\ref{sec:valuation relaxation} and \\ref{sec:virtual for relaxed}.\n Then we upper bound the revenue of $M$ with the benchmark in Theorem~\\ref{thm:revenue upperbound for subadditive} and argue that for any mechanism $M$, the corresponding benchmark can be upper bounded by the sum of the revenue of a few simple mechanisms.  As we allow $\\beta$, in other words the flow $\\lambda^{(\\beta)}$, to depend on the mechanism, our new approach may provide a better upper bound. As it turns out, our new upper bound is indeed easier to analyze.\n\n Lemma~\\ref{lem:requirement for beta} specifies the two properties of our $\\beta$ that play the most crucial roles in our analysis. We construct such a $\\beta$ in the proof of Lemma~\\ref{lem:requirement for beta}, however the construction is not necessarily unique and any $\\beta$ satisfying these two properties suffices. Note that our construction heavily relies on property \\textbf{(i)} of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:requirement for beta}\n\tFor any constant $b\\in (0,1)$ and any mechanism $M$, there exists a $\\beta$ such that: for the mechanism $M^{(\\beta)} $ constructed in Lemma~\\ref{lem:relaxed valuation} according to $\\beta$, any $i\\in[n]$ and $j\\in[m]$,\n\n\\noindent\\emph{\\textbf{(i)}} $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]\\leq b$;\n\n\\noindent\\emph{\\textbf{(ii)}} $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\/ b$, where $\\pi_{ij}^{(\\beta)}(t_i) = \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)$.\n\\end{lemma}\n\nBefore proving Lemma~\\ref{lem:requirement for beta}, we provide some intuition behind the two required properties.\nProperty \\textbf{(i)} is used to guarantee that if item $j$'s price for bidder $i$ is higher than $\\beta_{ij}$ for all $i$ and $j$ in an RSPM, for any item $j'$ and any bidder $i'$, $j'$ is still available with probability at least $(1-b)$ when $i'$ is visited. As for any bidder $k\\neq i'$ to purchase item $j'$,  $V_k(t_{kj'})$ must be greater than her price for item $j'$. By the union bound, the probability that there exists such a bidder is upper bounded by the LHS of property (i), and therefore is at most $b$. With this guarantee, we can easily show that the RSPM achieves good revenue (Lemma~\\ref{lem:neprev}). Property \\textbf{(ii)} states that the ex-ante probability for bidder $i$ to receive an item $j$ in $M^{(\\beta)}$ is not much bigger than the probability that bidder $i$'s value is larger than item $j$. This is crucial for proving our key Lemma~\\ref{lem:hat Q}, in which we argue that two different valuations provide comparable welfare under the same allocation rule $\\sigma^{(\\beta)}$. With Lemma~\\ref{lem:hat Q}, we can show that the ASPE obtains good revenue.\n\n\\begin{prevproof}{Lemma}{lem:requirement for beta}\n\tWhen there is only one buyer, we can simply set every $\\beta_j$ to be $0$ and both conditions are satisfied.\n\tWhen there are multiple players, we let $$\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x\\right] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\},$$ where $ \\pi_{ij}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}(t_i)$. Clearly, when the distribution of $V_i(t_{ij})$ is continuous, then\n\\begin{equation}\\label{equ:beta_second_condition}\n\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i),\n\\end{equation}\nand therefore for any $j$, $$\\sum_i\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\leq b.$$\n\nSo the first condition is satisfied. The second condition holds because by the first property in Lemma~\\ref{lem:relaxed valuation}, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$.\n\nWhen the distribution for $V_i(t_{ij})$ is discrete, it is possible that Equation~\\ref{equ:beta_second_condition} does not hold, but this is essentially a tie breaking issue and not hard to fix. Let $\\epsilon>0$ be an extremely small constant that is smaller than $\\left|V_i(t_{ij})-V_i(t'_{ij})\\right|$ for any $t_{ij},  t'_{ij}\\in T_{ij}$, any $i$ and any $j$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[0,\\epsilon]$, and think of it as a random rebate that the seller gives to bidder $i$ when she purchases item $j$. Now we modify the definition of $\\beta_{ij}$ as $\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij},\\zeta_{ij}}[V_i(t_{ij})+\\zeta_{ij}\\geq x] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\}.$\n\n\\notshow{\\begin{equation}\n\\epsilon_1=\\epsilon\\cdot \\frac{\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)}{\\Pr_{t_{ij}}\\left[V_i(t_{ij})= \\beta_{ij}\\right]}\n\\end{equation}\n\nBy the definition of $\\beta_{ij}$, $\\epsilon_1\\in [0,\\epsilon)$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[\\epsilon_1-\\epsilon,\\epsilon_1]$.  It is not hard to verify that $\\Pr_{t_{ij},\\zeta_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+\\zeta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$. For those probabilities related to $\\beta_{ij}$ shown in the proofs below, which is written for simplification, we refer to this definition. With regard to this definition, we make essentially small changes for the mechanism described below. Instead of using fixed price, add a small disturbance $\\zeta_{ij}$ on item $j$'s price for bidder $i$. Since $\\epsilon$ can be chosen as small as possible, the revenue will only be affected by a small constant. All the argument maintain to be true.}\n\nBoth of the two properties in Lemma~\\ref{lem:requirement for beta} hold if we replace $V_i(t_{ij})$ with $V_i(t_{ij})+\\zeta_{ij}$. The only change we need to make in the mechanism is to actually give the bidders $\\zeta_{ij}$ as the corresponding rebate. Since we can choose $\\epsilon$ to be arbitrarily small, the sum of the rebate is also arbitrarily small.  For the simplicity of the presentation, we will omit $\\zeta_{ij}$ and $\\epsilon$ in the rest of the paper.\nThe random rebate indeed makes our mechanism randomized(according to the random variable $\\zeta_{ij}\\sim [0,\\epsilon]$). However, the randomized mechanism is a uniform distribution of deterministic DSIC mechanisms (after determining all $\\zeta_{ij}$), and the expected revenue of the randomized mechanism is simply the average revenue of all these deterministic mechanisms. Therefore, there must be one realization of the rebates such that the corresponding deterministic mechanism has revenue above the expectation, i.e., the expected revenue of the randomized one. Thus if the randomized mechanism is proved to achieve some approximation ratio, there must exist a deterministic one that achieves the same ratio. The deterministic mechanism will use a fixed value $z_{ij}\\in [0,\\epsilon]$ as the rebate.\n\nSimilarly, the same issue about discrete distributions arises when we define some other crucial parameters later, e.g., in the Definition of $c$, $c_i$ and $\\tau_i$. We can resolve all of them together using the trick (adding a random rebate) described above, and we will not include a detailed proof for those cases.\n\\end{prevproof}\n\n\n\n\n\\subsubsection{Bad Example for Chawla and Miller's Approach}\\label{sec:cs_ocrs}\nLet bidders be constrained additive and $\\mathcal{F}_i$ be bidder $i$ feasibility constraint. We use $P_{\\mathcal{F}_i}=conv(\\{1^S | S\\in \\mathcal{F}_i\\})$ to denote the feasibility polytope of bidder $i$. Let $\\{q_{ij}\\}_{i\\in[n],j\\in[m]}$ be a collection of probabilities that satisfy $\\sum_i q_{ij}\\leq 1\/2$ for all item $j$ and $\\boldsymbol{q}_i = (q_{i1},\\ldots, q_{im})\\in b\\cdot P_{\\mathcal{F}_i}$. Let $\\beta_{ij}=F_{ij}^{-1}(q_{ij})$. The analysis by Chawla and Miller~\\cite{ChawlaM16} needs to upper bound $\\sum_{i, j}\\beta_{ij}\\cdot q_{ij}$ using the revenue of some BIC mechanism. When $\\mathcal{F}_i$ is a matroid for every bidder $i$, this expression can be upper bounded by the revenue of a sequential posted price mechanism constructed using OCRS from~\\cite{FeldmanSZ16}. Here we show that if the bidders have general downward closed feasibility constraints, this expression is gigantic. More specifically, we prove that even when there is only one bidder, the expression could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare.\n\nConsider the following example.\n\\begin{example}\\label{ex:counterexample ocrs}\n\tThe seller is selling $m=k^2$ items to a single bidder. The bidder's value for each item is drawn i.i.d. from distribution $F$, which is the equal revenue distribution truncated at $k$, i.e.,\n\t\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x<k\\\\\n1,&\\text{if}~~x=k\n\\end{cases}\n\\]\n Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The bidder is additive subject to feasibility constraint $\\mathcal{F}=\\left\\{S\\subseteq [m]|\\exists i\\in[k], S\\subseteq A_i\\right\\}$.\n\\end{example}\n\n\\begin{lemma}\\label{lem:counter example}\nLet $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope for the bidder in Example~\\ref{ex:counterexample ocrs}. Let $SW$ be the optimal social welfare. Then for any constant $b>0$, there exists $q\\in b\\cdot P_{\\mathcal{F}}$ such that for sufficiently large $k$, $$\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{\\log k })\\cdot SW$$.\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule: w.p. $(1-b)$, don't allocate anything, and w.p. $b$, give the buyer one of the sets $A_i$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in b\\cdot P_{\\mathcal{F}}$. Since $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have $\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2$. We use $V_i$ to denote the random variable of the bidder's value for set $A_i$. It is not hard to see that $SW={\\mathbb{E}}[\\max_{i\\in[k]} V_i]$. \n\n\\begin{lemma}\nFor any $i\\in [k]$,\n\\[\\Pr\\left[V_i>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=2k$, and $|X|\\leq k$.\nFor every $i$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_i-k\\log(k)>t\\right]\\leq \\exp\\left(-\\frac{\\frac{1}{2}t^2}{2k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_i>3k\\log(k)\\right]\\leq \\exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nBy the union bound, $\\Pr[\\max_{i\\in[k]}V_i>3\\cdot k\\log(k)]\\leq k^{-2}$. Therefore, ${\\mathbb{E}}[\\max_{i\\in[k]} V_i]\\leq 3 k\\log k +k^2\\cdot k^{-2}\\leq 4 k\\log k$.\n\\end{proof}\n\n\\notshow{In the analysis of the paper by Chawla and Miller~\\cite{ChawlaM16}, they rely on the following lemma in single buyer auction.\n\\begin{lemma}\\label{lem:shuchi}\n~\\cite{ChawlaM16}~\\cite{FeldmanSZ16}Suppose the buyer is additive within a matroid constraint $\\mathcal{F}$ and let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. For any constant $b\\in (0,1)$ and ex-ante vector $q\\in bP_{\\mathcal{F}}$, let $\\beta$ be the corresponding ex-ante prices. In other words, $\\beta_j$ is chosen such that the probability that the value for item $j$\nexceeds this price is precisely $q_j$. Then the value $\\beta\\cdot q$ can be bounded within some constant factor by the revenue of a posted price mechanism with a more strict constraint, which guarantees that the ex-ante probability of the buyer getting item $j$ is at most $q_j$.\n\\end{lemma}\n\nThe result can be generated for $\\mathcal{F}$ beyond a matroid~\\cite{FeldmanSZ16}. However, Lemma~\\ref{lem:shuchi} does not hold for general downward-close $\\mathcal{F}$. In this section we provide a counterexample with some general downward-close constraint $\\mathcal{F}$, such that the term $\\beta\\cdot q$ cannot be upper bounded by any single buyer mechanism, within in a constant factor.\n\n\\begin{lemma}\nConsider the following single buyer auction with $m=k^2$ i.i.d. items. Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The value distribution $F$ for a single item is defined as the equal-revenue distribution truncated at value $k$, i.e.,\n\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x<k\\\\\n1,&\\text{if}~~x=k\n\\end{cases}\n\\]\nDefine a downward-close feasibility constraint $\\mathcal{F}$ for the buyer as $\\mathcal{F}=\\{S\\subseteq [m]|\\exists p, S\\subseteq A_p\\}$. Let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. Let $SW$ be the optimal social welfare among all BIC mechanisms in this auction. Then for any $b>0$, there exists $q\\in bP_{\\mathcal{F}}$ such that for sufficiently large $k$,\n\\begin{equation}\\label{equ:q times beta}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{log(k)})\\cdot SW\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule. With probability $(1-b)$, don't allocate anything. With probability $b$, give the buyer one of the sets $A_p$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in bP_{\\mathcal{F}}$.\n\nSince $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have\n\\begin{equation}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2\n\\end{equation}\n\nConsider the optimal social welfare. For every bundle $p$, denote $V_p$ the random variable of the buyer's value for bundle $p$. Notice $A_p\\in \\mathcal{F}$, $V_p$ is the sum of $k$ independent random variables with cdf $F$. With Bernstein Inequality, $V_p=O(klog(k))$ with high probability.\n\\notshow{\\begin{lemma}\\label{lem:bernstein}\n(Bernstein Inequality)~\\cite{bernstein1924modification}: Suppose $X_1,...,X_n$ are independent random variables with zero mean, and $|X_i|\\leq M$ almost surely for all $i$. Then for any $t>0$,\n\\[\\Pr\\left[\\sum_{i=1}X_i>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{\\sum_{i=1}^nE[X_i^2]+\\frac{1}{3}Mt}\\right)\\]\n\\end{lemma}}\n\\begin{lemma}\nFor any $p\\in [k]$,\n\\[\\Pr\\left[V_p>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=k-1$, and $|X|\\leq k$.\nFor every $p$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_p-k\\log(k)>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_p>3k\\log(k)\\right]\\leq exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nWith union bound,\n\\[\\Pr[\\max_{p\\in [k]}V_p\\geq 3k\\log(k)]\\leq \\sum_{p\\in [k]}\\Pr\\left[V_p>3k\\log(k)\\right]\\leq k^{-2}\\]\n\nNotice the social welfare for the mechanism is at most \\\\\n\\noindent$\\max_{p\\in [k]}V_p$ due to the feasible constraint $\\mathcal{F}$. Also notice that $\\max_{p\\in [k]}V_p\\leq k^2$, we have\n\\begin{align*}\nSW&\\leq E[\\max_{p\\in [k]}V_p]\\leq \\bigg(3k\\log(k)\\cdot \\Pr[\\max_{p\\in [k]}V_p\\leq 3k\\log(k)]\\\\\n&+k^2\\cdot\\Pr[\\max_{p\\in [k]}V_p> 3k\\log(k)]\\bigg)\\\\\n&\\leq 3k\\log(k)+k^2\\cdot k^{-2}=O(k\\log(k))\n\\end{align*}\n\n\\noindent When $k$ is sufficiently large, Equation~\\ref{equ:q times beta} holds.\n\n\\begin{comment}\nNow consider the optimal revenue in this auction. Let $REV_k$ be the optimal revenue selling the items in set $A_k$ to a single buyer. \\ref{HartN12} has shown that $REV_k=\\Theta(n\\log(n))$ for the equal-revenue distribution $F(x)$. Since people are only interested one set of items,\n\\begin{equation}\nREV\\leq \\sum_{k\\in [n]}REV_k=\\Theta(n^2\\log(n))\n\\end{equation}\n\\end{comment}\n\n\\end{proof}}\n\n\n\n\\section{Introduction}\nIn Mechanism Design, we aim to design a mechanism\/system such that a group of strategic participants, who are only interested in optimizing their own utilities, are incentivized to choose actions that also help achieve the designer's objective. Clearly, the quality of the solution with respect to the designer's objective is crucial. However, perhaps one should also pay equal attention to another criterion of a mechanism, that is, its simplicity. When facing a complicated mechanism, participants may be confused by the rules and thus unable to optimize their actions and react in unpredictable ways instead. This may lead to undesirable outcomes and poor performance of the mechanism. An ideal mechanism would be optimal and simple. However, such cases of simple mechanisms being optimal only exist in single-item auctions, with the seminal examples of auctions by Vickrey~\\cite{Vickrey61} and Myerson~\\cite{Myerson81}, while none has been discovered in broader settings. Indeed, we now know that even in fairly simple settings the optimal mechanisms suffer many undesirable properties including randomization, non-monotonicity, and others~\\cite{RochetC98, Tha04, Pavlov11a, HartN13, HartR12, BriestCKW10, DaskalakisDT13, DaskalakisDT14}.\nTo move forward, one has to compromise -- either settle with optimal but somewhat complex mechanisms or turn to simple but approximately optimal solutions.\n\nRecently, there has been extensive research effort focusing on the latter approach, that is, studying the performance of simple mechanisms through the lens of approximation. In particular, a central problem on this front is how to design simple and approximately revenue-optimal mechanisms in multi-item settings. For instance, when bidders have unit-demand valuations, we know sequential posted price mechanisms approximates the optimal revenue due to a line of work initiated by Chawla et al.~\\cite{ChawlaHK07, ChawlaHMS10, ChawlaMS15, CaiDW16}. When buyers have additive valuations, we know that either selling the items separately or running a VCG mechanism with per bidder entry fee approximates the optimal revenue due to a series of work initiated by Hart and Nisan~\\cite{HartN12, CaiH13, LiY13, BabaioffILW14, Yao15, CaiDW16}. Recently, Chawla and Miller~\\cite{ChawlaM16} generalized the two lines of work described above to matroid rank functions\\footnote{{Here is a hierarchy of the valuation functions. additive \\& unit-demand $\\subseteq$ matroid rank $\\subseteq$ constrained additive \\&  submodular\n $\\subseteq$ XOS $\\subseteq$ subadditive. A function is constrained additive if it is additive up to some downward closed feasibility constraints. The class of submodular functions is neither a superset nor a subset of the class of constrained additive functions.} See Definition~\\ref{def:valuation classes} for the formal definition. }. They show that a simple mechanism, the sequential two-part tariff mechanism, suffices to extract a constant fraction of the optimal revenue. For subadditive valuations beyond matroid rank functions, we only know how to handle a single buyer~\\cite{RubinsteinW15}\\footnote{All results mentioned above assume that the buyers' valuation distributions are over independent items. For additive and unit-demand valuations, this means a bidder's values for the items are independent. The definition is generalized to subadditive valuations by Rubinstein and Weinberg~\\cite{RubinsteinW15}. See Definition~\\ref{def:subadditive independent}.}. It is a major open problem to extend this result to multiple subadditive buyers.\n\nIn this paper, we unify and strengthen all the results mentioned above via an extension of the duality framework proposed by Cai et al.~\\cite{CaiDW16}. Moreover, we show that even when there are multiple buyers with XOS valuation functions, there exists a simple, deterministic and Dominant Strategy Incentive Compatible (DSIC) mechanism that achieves a constant fraction of the optimal Bayesian Incentive Compatible (BIC) revenue\\footnote{A mechanism is Bayesian Incentive Compatible (BIC) if it is in every bidder's interest to tell the truth, assuming that all other bidders' reported their values. A mechanism is Dominant Strategy Incentive Compatible (DSIC) if it is in every bidder's interest to tell the truth no matter what reports the other bidders make.}. For subadditive valuations, our approximation ratio degrades to $O(\\log m)$.\n\n\\begin{informaltheorem}\n\tThere exists a simple, deterministic and DSIC mechanism that achieves a constant fraction of the optimal BIC revenue in multi-item settings, when the buyers' valuation distributions are XOS over independent items. When the buyers' valuation distributions are subadditive over independent items, our mechanism achieves at least $\\Omega(\\frac{1}{\\log m})$ of the optimal BIC revenue, where $m$ is the number of items.\n\\end{informaltheorem}\n\nThe original paper by Cai et al.~\\cite{CaiDW16} provided a unified treatment  for additive and unit-demand valuations. However, it is inadequate to provide an analyzable benchmark for even a single subadditive bidder. In this paper, we show how to extend their duality framework to accommodate general subadditive valuations. Using this extended framework, we substantially improve the approximation ratios for many of the settings discussed above, and in the meantime generalize the results to broader cases. See Table~\\ref{table:comp} for the comparison between the best ratios reported in the literature and the new ratios obtained in this work.\n\n\\begin{table*}\n\n\\centering\n\\begin{tabular}{|c|l|p{2.1cm}|c|p{2cm}|c|c|}\n\\hline\n\t& &\\centering Additive or Unit-demand& \\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2.2cm}{\\centering Matroid-Rank}}}& \\centering Constrained Additive&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1cm}{\\centering XOS}}}&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2cm}{\\centering Subadditive}}} \\\\\n\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Single Buyer}}} & Previous&      6~\\cite{BabaioffILW14} or  4~\\cite{ChawlaMS15}& 31.1* &\\centering 31.1~\\cite{ChawlaM16} & 338* & 338~\\cite{RubinsteinW15} \\\\\\cline{2-7}\n\t\t&This Paper & \\centering - &{11*} &\\centering{11}&{40*}&  {40} \\\\\\cline{2-7}\n\n\\hline\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Multiple Buyer}}} & Previous&     8~\\cite{CaiDW16}  or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}&\\centering ? & ? &? \\\\ \\cline{2-7}\n\t\t& This Paper& \\centering - & 70*  &\\centering 70 & 268 &$O(\\log m)$  \\\\\n\\hline\n\\end{tabular}\\\\\n* The result is implied by another result for a more general setting.\n\\caption{Comparison of approximation ratios between previous and current work.}\n\t\t  \\label{table:comp}\n\\end{table*}\n\n\\notshow{\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\\hline\n\t\t                  & Additive or Unit-demand& Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer&      6~\\cite{BabaioffILW14} or  4~\\cite{ChawlaMS15}& 33.1~\\cite{ChawlaM16} &$\\rightarrow$ &$\\rightarrow$ & 338~\\cite{RubinsteinW15} \\\\\\hline\n\t\tMultiple Buyers&     8~\\cite{CaiDW16}  or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}& ? & ? &? \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t  \\caption{The best approximation ratios known prior to this work.}\n\t\t  \\label{table:old}\n\\end{table}\n\n\\begin{table}\\centering\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\\hline\n\t\t                   &Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer&  $\\rightarrow$ &14& $\\rightarrow$& 48 \\\\\\hline\n\t\tMultiple Buyers& $\\rightarrow$  & 70 & 268 &$O(\\log m)$ \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t  \\caption{New approximation ratios obtained in this work.}\n\t\t  \\label{table:new}\n\\end{table}\n\n}\nOur mechanism is either a \\emph{rationed sequential posted price mechanism} (\\textbf{RSPM}) or an \\emph{anonymous sequential posted price with entry fee mechanism} (\\textbf{ASPE}). In an RSPM, there is a price $p_{ij}$ for buyer $i$ if she wants to buy item $j$, and she is allowed to purchase at most one item. We visit the buyers in some arbitrary order and the buyer takes her favorite item among the available items given the item prices for her. Here we allow personalized prices, that is, $p_{ij}$ could be different from $p_{kj}$ if $i\\neq k$. In an ASPE, every buyer faces the same collection of item prices $\\{p_j\\}_{j\\in[m]}$. Again, we visit the buyers in some arbitrary order. For each buyer, we show her the available items and the associated price for each item. Then we ask her to pay the entry fee to enter the mechanism, which may depend on what items are still available and the identity of the buyer. If the buyer accepts the entry fee, she can proceed to purchase any item at the given prices; if she rejects the entry fee, then she will leave the mechanism without receiving anything. Given the entry fee and item prices, the decision making for the buyer is straightforward, as she only accepts the entry fee when the surplus for winning her favorite bundle is larger than the entry fee. Therefore, both RSPM and ASPE are DSIC and ex-post Individually Rational (ex-post IR).\n\n\n\\subsection{Our Contributions}\nTo obtain the new generalizations, we provide important extensions to the duality framework in~\\cite{CaiDW16}, as well as novel analytic techniques and new simple mechanisms.\n\n\\vspace{.05in}\n\\noindent \\textbf{1. Accommodating subadditive valuations:} the original duality framework in~\\cite{CaiDW16} already unified the additive case and unit-demand case by providing an approximately tight upper bound for the optimal revenue using a single dual solution. A trivial upper bound for the revenue is the social welfare, which may be arbitrarily bad in the worst case. The duality based upper bound in~\\cite{CaiDW16} improves this trivial upper bound, the social welfare, by substituting the value of each buyer's favorite item with the corresponding Myerson's virtual value. However, the substitution is viable only when the following condition holds -- the buyer's marginal gain for adding an item solely depends on her value for that item (assuming it's feasible to add that item\\footnote{WLOG, we can reduce any constrained additive valuation to an additive valuation with a  feasibility constraint (see Definition~\\ref{def:valuation classes})}), but not the set of items she has already received. This applies to valuations that are additive, unit-demand and more generally constrained additive, but breaks under more general valuation functions, e.g., submodular, XOS or subadditive valuations. As a consequence, the original dual solution from~\\cite{CaiDW16} fails to provide a nice upper bound for more general valuations. To overcome this difficulty, we take a different approach. Instead of directly studying the dual of the original problem, we first relax the valuations and argue that the optimal revenue of the relaxed valuation is comparable to the original one. Then, since we choose the relaxation in a particular way, by applying a dual solution similar to the one in~\\cite{CaiDW16} to the relaxed valuation, we recover an upper bound of the optimal revenue for the relaxed valuation resembling the appealing format of the one in~\\cite{CaiDW16}. Combining these two steps, we obtain an upper bound for subadditive valuations that is easy to analyze. Indeed, we use our new upper bound to improve the approximation ratio for a single subadditive buyer from $338$~\\cite{RubinsteinW15}  to $40$. See Section~\\ref{sec:valuation relaxation} for more details.\n\n\\vspace{.05in}\n\\noindent\\textbf{2. An adaptive dual:}  our second major change to the framework is that we choose the dual in an adaptive manner. In~\\cite{CaiDW16}, a dual solution $\\lambda$ is chosen up front inducing a virtual value function $\\Phi(\\cdot)$, then the corresponding optimal virtual welfare is used as a benchmark for the optimal revenue. Finally, it is shown that the revenue of some simple mechanism is within a constant factor of the optimal virtual welfare. Unfortunately, when the valuations are beyond additive and unit-demand, the optimal virtual welfare for this particular choice of virtual value function becomes extremely complex and hard to analyze. Indeed, it is already challenging to bound when the buyers' valuations are $k$-demand. In this paper, we take a more flexible approach. For any particular allocation rule $\\sigma$, we tailor a special dual $\\lambda^{(\\sigma)}$ based on $\\sigma$ in a fashion that is inspired by Chawla and Miller's ex-ante relaxation~\\cite{ChawlaM16}. Therefore, the induced virtual valuation $\\Phi^{(\\sigma)}$ also depends on $\\sigma$. By duality, we can show that the optimal revenue obtainable by $\\sigma$ is still upper bounded by the virtual welfare with respect to $\\Phi^{(\\sigma)}$ under allocation rule $\\sigma$. Since the virtual valuation is designed specifically for allocation $\\sigma$, the induced virtual welfare is much easier to analyze. Indeed, we manage to prove that for any allocation $\\sigma$ the induced virtual welfare is within a constant factor of the revenue of some simple mechanism, when bidders have XOS valuations. See Section~\\ref{sec:virtual for relaxed} and~\\ref{sec:choice of beta} for more details.\n\n\n\\vspace{.05in}\n\n\\noindent\\textbf{3. A novel analysis and new mechanism:} with the two contributions above, we manage to derive an upper bound of the optimal revenue similar to the one in \\cite{CaiDW16} but for subadditive bidders. The third major contribution of this paper is a novel approach to analyzing this upper bound. The analysis in~\\cite{CaiDW16} essentially breaks the upper bound into three different terms-- \\textsc{Single}, \\textsc{Tail}~ and \\textsc{Core}, and bound them separately. All three terms are relatively simple to bound for additive and unit-demand buyers, but for more general settings the $\\textsc{Core}$ becomes much more challenging to handle. Indeed, the analysis in~\\cite{CaiDW16} was  insufficient to tackle the $\\textsc{Core}$ even when the buyers have $k$-demand valuations\\footnote{The class of $k$-demand valuations is a generalization of unit-demand valuations, where the buyer's value is additive up to $k$ items.}-- a very special case of matroid rank valuations, which itself is a special case of XOS or subadditive valuations. Rubinstein and Weinberg~\\cite{RubinsteinW15} showed how to approximate the $\\textsc{Core}$ for a single subadditive bidder using grand bundling, but their approach does not apply to multiple bidders. Yao~\\cite{Yao15} showed how to approximate the $\\textsc{Core}$ for multiple additive bidders using a VCG with per bidder entry fee mechanism, but again it is unclear how his approach can be extended to multiple k-demand bidders. A recent paper by Chawla and Miller~\\cite{ChawlaM16} finally broke the barrier of analyzing the $\\textsc{Core}$ for multiple $k$-demand buyers. They showed how to bound the $\\textsc{Core}$ for matroid rank valuations  using a sequential posted price mechanism by applying the \\emph{online contention resolution scheme (OCRS)} developed by Feldman et al.~\\cite{FeldmanSZ16}. The connection with OCRS is an elegant observation, and one might hope the same technique applies to more general valuations. Unfortunately, OCRS is only known to exist for special cases of downward closed constraints, and as we show in Section~\\ref{sec:core comparison}, the approach by Chawla and Miller cannot yield any constant factor approximation for general constrained additive valuations.\n\nWe take an entirely different approach to bound the $\\textsc{Core}$. Here we provide some intuition behind our mechanism and analysis. The $\\textsc{Core}$ is essentially the optimal social welfare induced by some truncated valuation $v'$, and our goal is to design a mechanism that extracts a constant fraction of the welfare as revenue. Let $M$ be any sequential posted price mechanism. A key observation is that when bidder $i$'s valuation is subadditive over independent items, her utility in $M$, which is the largest surplus she can achieve from the unsold items, is also subadditive over independent items. If we can argue that her utility function is $a$-Lipschitz (Definition~\\ref{def:Lipschitz}) with some small $a$,  Talagrand's concentration inequality~\\cite{Talagrand1995concentration,Schechtman2003concentration} allows us to set an entry fee for the bidder so that we can extract a constant fraction of her utility just through the entry fee.\nIf we modify $M$ by introducing an entry fee for every bidder, according to Talagrand's concentration inequality, the new mechanism $M'$ should intuitively have revenue that is a constant fraction of the social welfare obtained by $M$~\\footnote{$M$'s welfare is simply its revenue plus the sum of utilities of the bidders, and $M'$ can extract some extra revenue from the entry fee, which is a constant fraction of the total utility from the bidders.}. Therefore, if there exists a sequential posted price mechanism $M$ that achieves a constant fraction of the optimal social welfare under the truncated valuation $v'$, the modified mechanism $M'$ can obtain a constant fraction of $\\textsc{Core}$ as revenue. Surprisingly, when the bidders have XOS valuations, Feldman et al.~\\cite{FeldmanGL15} showed that there exists an anonymous sequential posted price mechanism that always obtains at least half of the optimal social welfare. Hence, an anonymous sequential posted price with per bidder entry fee mechanism should approximate the $\\textsc{Core}$ well, and this is exactly the intuition behind our ASPE mechanism.\n\n To turn the intuition into a theorem, there are two technical difficulties that we need to address: (i) the Lipschitz constants of the bidders' utility functions turn out to be too large (ii) we deliberately neglected the difference in bidders' behavior under $M$ and $M'$ in hope to keep our discussion in the previous paragraph intuitive. However, due to the entry fee, bidders may end up purchasing completely different items under $M$ and $M'$, so it is not straightforward to see how one can relate the revenue of $M'$ to the welfare obtained by $M$.\n  See Section~\\ref{sec:core comparison} for a more detailed discussion on how we overcome these two difficulties.\n \n\n\n\n\n\n\n\n\n\n\\subsection{Related Work}\nIn recent years, we have witnessed several breakthroughs in designing (approximately) optimal mechanisms in multi-dimensional settings. The black-box reduction by Cai et al.~\\cite{CaiDW12a,CaiDW12b,CaiDW13a,CaiDW13b} shows that we can reduce any Bayesian mechanism design problem to a similar algorithm design problem via convex optimization. Through their reduction, it is proved that all optimal mechanisms can be characterized as a distribution of virtual welfare maximizers, where the virtual valuations are computed by an LP. Although this characterization provides important insights about the structure of the optimal mechanism, the optimal allocation rule is unavoidably randomized and might still be complex as the virtual valuations are only a solution of an LP.\n\nAnother line of work considers the ``Simple vs. Optimal'' auction design problem. For instance, a sequence of results~\\cite{ChawlaHK07,ChawlaHMS10,ChawlaMS10,ChawlaMS15} show that sequential posted price mechanism can achieve $\\frac{1}{33.75}$ of the optimal revenue, whenever the buyers have unit-demand valuations over independent items. Another series of results~\\cite{HartN12,CaiH13,LiY13,BabaioffILW14,Yao15} show that the better of selling the items separately and running the VCG mechanism with per bidder entry fee achieves $\\frac{1}{69}$ of the optimal revenue, whenever the buyers' valuations are additive over independent items. Cai et al.~\\cite{CaiDW16} unified the two lines of results and improved the approximation ratios to $\\frac{1}{8}$ for the additive case and $\\frac{1}{24}$ for the unit-demand case using their duality framework.\n\nSome recent works have shown that simple mechanisms can approximate the optimal revenue even when buyers have more sophisticated valuations. For instance, Chawla and Miller~\\cite{ChawlaM16} showed that the sequential two-part tariff mechanism can approximate the optimal revenue when buyers have matroid rank valuation functions over independent items. Their mechanism requires every buyer to pay an entry fee up front, and then run a sequential posted price mechanism on buyers who have accepted the entry fee. Our ASPE is similar to their mechanism, but with two major differences: (i) since buyers are asked to pay the entry fee before the seller visits them, the buyers have to make their decisions based on the expected utility (assuming every other buyer behaves truthfully) they can receive. Hence, the mechanism is only guaranteed to be BIC and interim IR. While in our mechanism, the buyers can see what items are still available before paying the entry fee, therefore the decision making is straightforward and the ASPE is DSIC and ex-post IR; (ii) the item prices in the ASPE are anonymous, while in the sequential two-part tariff mechanism, personalized prices are allowed. For valuations beyond matroid rank functions, Rubinstein and Weinberg~\\cite{RubinsteinW15} showed that for a single buyer whose valuation is subadditive over independent items, either grand bundling or selling the items separately achieves at least $\\frac{1}{338}$ of the optimal revenue.\n\nThe Cai-Devanur-Weinberg duality framework~\\cite{CaiDW16} has been applied to other intriguing Mechanism Design problems. For example, Eden et al. showed that the better of selling separately and bundling together gets an $O(d)$-approximation for a single bidder with ``complementarity-$d$ valuations over independent items''~\\cite{EdenFFTW16a}. The same authors also proved a Bulow-Klemperer result for regular i.i.d. and constrained additive bidders~\\cite{EdenFFTW16b}. Liu and Psomas provided a Bulow-Klemperer result for {dynamic auctions}~\\cite{LiuP16}. Finally, Brustle et al.~\\cite{BrustleCWZ17} extended the duality framework to two-sided markets and used it to design simple mechanisms for approximating the Gains from Trade.\n\nStrong duality frameworks have recently been developed for one additive buyer~\\cite{DaskalakisDT13,DaskalakisDT15,Giannakopoulos14a,GiannakopoulosK14,GiannakopoulosK15}. These frameworks show that the dual problem of revenue maximization can be viewed as an optimal transport\/bipartite matching problem. Hartline and Haghpanah provided an alternative duality framework in~\\cite{HartlineH15}. They showed that if certain paths exist, these paths provide a witness of the optimality of a certain Myerson-type mechanism, but these paths are not guaranteed to exist in general. Similar to the Cai-Devanur-Weinberg framework, Carroll~\\cite{Carroll15} independently made use of a partial Lagrangian over incentive constraints. These duality frameworks have been successfully provide conditions under which a certain type of mechanism is optimal when there is a single unit-demand or additive bidder. However, none of these frameworks succeeds in yielding any approximately optimal results in multi-buyer settings.\n\n\\input{prelim}\n\\input{roadmap}\n\\input{duality}\n\\input{single_subadditive}\n\\input{multi_XOS}\n\\input{example_ocrs}\n\\newpage\n\n\\section{Multiple Bidders}\\label{sec:multi}\n\nIn this section, we prove our main result -- simple mechanisms can approximate the optimal BIC revenue even when there are multiple XOS\/subadditive bidders.\nFirst, we need the definition of supporting prices.\n\\begin{definition}[Supporting Prices~\\cite{DobzinskiNS05}]\\label{def:supporting price}\nFor any $\\alpha\\geq 1$, a type $t$ and a subset $S\\subseteq[m]$, prices $\\{p_j\\}_{j\\in S}$\nare $\\alpha$-supporting prices for $v(t,S)$ if \\textbf{(i)}\t$v(t,S') \\geq \\sum_{j\\in S'} p_j$ for all $S'\\subseteq S$ and \\textbf{(ii)} $\\sum_{j\\in S}p_j\\geq \\frac{v(t,S)}{\\alpha}$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:multi}\nIf for any buyer $i$, any type $t_i\\in T_i$ and any bundle $S\\in [m]$, $v_i(t_i,S)$ has a set of $\\alpha$-supporting prices $\\{\\theta_j^{S}(t_i)\\}_{j\\in S}$, then for any BIC mechanism $M$ and any constant $b\\in (0, 1)$,\n\\begin{align*}\n\\textsc{Rev}(M,v,D)\\leq 32\\alpha \\cdot \\textsc{APostEnRev}\n+\\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev}\n\\end{align*}\n\n\\vspace{0.05in}\nIf $v_i(t_i,\\cdot)$ is an XOS valuation for all $i$ and $t_i\\in T_i$, then $\\alpha=1$. By setting $b$ to $\\frac{1}{4}$, we have $$\\textsc{Rev}(M,v,D)\\leq 236\\cdot\\textsc{PostRev}+32\\cdot\\textsc{APostEnRev}.$$ For general subadditive valuations, $\\alpha=O(\\log(m))$ by~\\cite{BhawalkarR11}, hence $$\\textsc{Rev}(M,v,D)\\leq O(\\log(m))\\cdot \\max\\{\\textsc{PostRev},\\textsc{APostEnRev}\\}.$$\n\\end{theorem}\nHere is a sketch of the proof for Theorem~\\ref{thm:multi}. We show how to upper bound $\\textsc{Single}(M,\\beta)$ in Lemma~\\ref{lem:multi_single}. Then, we decompose $\\textsc{Non-Favorite}(M,\\beta)$ into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ in Lemma~\\ref{lem:multi decomposition}. We show how to construct a simple mechanism to approximate $\\textsc{Tail}(M,\\beta)$ in Section~\\ref{subsection:tail} and how to approximate $\\textsc{Core}(M,\\beta)$ in Section~\\ref{subsection:core}.\n\n\\vspace{.1in}\n \\noindent\\textbf{Analysis of $\\textsc{Single}(M,\\beta)$:} \n\n\n\\begin{lemma}\\label{lem:multi_single}\nFor any mechanism $M$, $$\\textsc{Single}(M, \\beta)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one agent $(i,j)$ for each of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(M,\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M,\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$.\n\\end{proof}\n\n\n\\vspace{.05in}\n \\noindent\\textbf{Core-Tail Decomposition of $\\textsc{Non-Favorite}(M,\\beta)$:} we decompose $\\textsc{Non-Favorite}(M, \\beta)$ into two terms $\\textsc{Tail}(M, \\beta)$ and $\\textsc{Core}(M, \\beta)$\\footnote{In~\\cite{CaiDW16}, $\\textsc{Non-Favorite}$ is decomposed into four different terms $\\textsc{Under}$, $\\textsc{Over}$, $\\textsc{Core}$ and $\\textsc{Tail}$. We essentially merge the first three terms into $\\textsc{Core}(M, \\beta)$ in our decomposition.}. First, we need the following definition.\n\\begin{definition}\\label{def:c_i}\nFor every buyer $i$, let $c_i :=\\inf\\big\\{x\\geq 0:\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+x\\right]\\leq \\frac{1}{2}\\big\\}.$ For every $t_i \\in T_i$, let $\\mathcal{T}_i(t_i)=\\{j\\ |\\ V_i(t_{ij})\\geq \\beta_{ij}+c_i\\}$ and $\\mathcal{C}_i(t_i) = [m]\\backslash\\mathcal{T}_i(t_i)$.\n\\end{definition}\n Since $v_i(t_i,\\cdot)$ is subadditive for all $i$ and $t_i\\in T_i$, we have $v_i(t_i,S)\\leq v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. The term $\\textsc{Non-Favorite}(M,\\beta)$ can be decomposed into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ based on the inequality above. The complete proof of Lemma~\\ref{lem:multi decomposition} can be found in Appendix~\\ref{appx:multi}.\n\n \\begin{lemma}~\\label{lem:multi decomposition}\n\t\t\\begin{align*} &\\textsc{Non-Favorite}(M,\\beta)\\\\\n\t\t\t\\leq&  \\sum_i\\sum_{t_i}f_i(t_i) \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))~~~~~~~~~~(\\textsc{Core}(M,\\beta))\\\\\n\t\t\t+&\\sum_i\\sum_j  \\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot\\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]~~~~~~~(\\textsc{Tail}(M,\\beta))\n\t\t\\end{align*}\n\t\t\\end{lemma}\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\nIn this section we show how to bound $\\textsc{Tail}(M,\\beta)$ with the revenue of an RSPM.\n\\begin{lemma}\\label{lem:multi-tail}\n\tFor any BIC mechanism $M$, $\\textsc{Tail}(M, \\beta)\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{lemma}\n\nWe first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$\n\\begin{align*}\nr_{ij}&=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],\n\\end{align*}\n$r_i=\\sum_j r_{ij}$, and $r=\\sum_i r_i$. We show in the following Lemma that $r$ is an upper bound of $\\textsc{Tail}(M,\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M,\\beta)\\leq r.$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{proof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful for analyzing $\\textsc{Core}(M,\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be  a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by property (i) of Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$.\n\\end{proof}\n\n\\notshow{\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\n\\textsc{Tail}(M,\\beta)\\leq r\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n$$r= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{proof}\n}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n\\begin{equation}\\label{r and prev}\nr= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{prevproof}\n\n\n\nWe have done the analysis for $\\textsc{Tail}{(M,\\beta)}$. Before starting the analysis for $\\textsc{Core}{(M,\\beta)}$, we show that $r_i$ is within a constant factor of $c_i$. This Lemma is useful for bounding $\\textsc{Core}{(M,\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\n\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because when $c_i>0$,\n$\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]$ is at least $\\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Inequality~(\\ref{r and prev}), \n$\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\n\nIn this section we upper bound $\\textsc{Core}(M,\\beta)$. Recall that\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$$\nWe can view it as the welfare of another valuation function $v'$ under allocation $\\sigma^{(\\beta)}$ where $v'_i(t_i, S) = v_i(t_i,S\\cap\\mathcal{C}_i(t_i))$. In other words, we ``truncate'' the function at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold. The new function still satisfies monotonicity, subadditivity and no externalities.\n\nWe first compare existing methods for analyzing the $\\textsc{Core}$ with our approach before jumping into the proofs.\n\n\\subsubsection{Comparison between the Existing Methods and Our Approach}\\label{sec:core comparison}\nAs all results in the literature~\\cite{ChawlaHMS10, Yao15, CaiDW16,ChawlaM16} only study special cases of constrained additive valuations, we restrict our attention to constrained additive valuations in the comparison, but our approach also applies to XOS and subadditive valuations.\n\nWe compare our approach to the state of the art result by Chawla and Miller~\\cite{ChawlaM16}. They separate $\\textsc{Core}(M,\\beta) $ into two parts: (i) the welfare obtained from values below $\\beta$, and (ii) the welfare obtained from values between $\\beta$ and $\\beta+c$\\footnote{In particular, if bidder $i$ is awarded a bundle $S$ that is feasible for her, the contribution for the first part is $\\sum_{j\\in S} \\min\\left\\{\\beta_{ij},t_{ij}\\right\\}\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ and the contribution to the second part is $\\sum_{j\\in S} \\left(t_{ij}-\\beta_{ij}\\right)^+\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ }.\n It is not hard to show that the latter can be upper bounded by the revenue of a sequential posted price with per bidder entry fee mechanism.\n  Due to their choice of $\\beta$ (similar to the second property of Lemma~\\ref{lem:requirement for beta}), the former is upper bounded by $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$.\n   It turns out when every bidder's feasibility constraint is a matroid, one can use the OCRS from~\\cite{FeldmanSZ16} to design a sequential posted price mechanism to approximate this expression.\n    However, as we show in Example~\\ref{ex:counterexample ocrs}, $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$ could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare when the bidders have general downward closed feasibility constraints.\n     Hence, such approach cannot yield any constant factor approximation for general constrained additive valuations.\n\nAs explained in the intro, we take an entirely different approach. We first construct the posted prices $\\{Q_j\\}_{j\\in[m]}$ for our ASPE (Definition~\\ref{def:posted prices}), Feldman et al.~\\cite{FeldmanGL15} showed that the anonymous posted price mechanism with these prices achieves welfare $\\Omega\\left(\\textsc{Core}(M,\\beta)\\right)$. If all bidders have valuations that are subadditive over independent items, for any bidder $i$ and any set of available items $S$, $i$'s surplus for $S$ under valuation $v'_i(t_i, \\cdot)$ ($max_{S'\\subseteq S}~v'_i(t_i,S') -\\sum_{j\\in S'} Q_j$) is also subadditive over independent items. According to Talagrand's concentration inequality, the surplus concentrates and its expectation is upper bounded by its median and its Lipschitz constant $a$. One can extract at least half of the median by setting the median of the surplus as the entry fee. How about the Lipschitz constant $a$? Unfortunately, $a$ could be as large as $\\frac{1}{2}\\max_{j\\in[m]}\\{\\beta_{ij}+c_i\\}$, which is too large to be bounded.\n\nHere is how we overcome this difficulty. Instead of considering $v'$, we construct a new valuation $\\hat{v}$ that is always dominated by the true valuation $v$. We consider the social welfare induced by $\\sigma^{(\\beta)}$ under $\\hat{v}$ and define it as $\\widehat{\\textsc{Core}}(M,\\beta)$. In Section~\\ref{sec:proxy core}, we show that $\\widehat{\\textsc{Core}}(M,\\beta)$ is not too far away from $\\textsc{Core}(M,\\beta)$, so it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$ (Lemma~\\ref{lem:hat Q}). But why is $\\widehat{\\textsc{Core}}(M,\\beta)$ easier to approximate? The reason is two-fold. \\textbf{(i)} For any bidder $i$ and any set of available items $S$,  bidder $i$'s surplus for $S$ under $\\hat{v}_i(t_i,\\cdot)$ (defined as $\\mu_i(t_i,S)$ in Definition~\\ref{def:entry fee}, which is $max_{S'\\subseteq S}~ \\hat{v}_i(t_i,S') -\\sum_{j\\in S'} Q_j$), is not only subadditive over independent items, but also has a small Lipschitz constant $\\tau_i$ (Lemma~\\ref{lem:property of mu}). Indeed, these Lipschitz constants are so small that $\\sum_i \\tau_i$  and can be upper bounded by $\\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}). \\textbf{(ii)} If we set the entry fee of our ASPE to be the median of $\\mu_i(t_i,S)$ when $t_i$ is drawn from $D_i$, using a proof inspired by Feldman et al.~\\cite{FeldmanGL15}, we can show that our ASPE's revenue collected from the posted prices plus the expected surplus of the bidders (over the randomness of all bidders' types) approximates $\\widehat{\\textsc{Core}}(M,\\beta)$ (implied by Lemma~\\ref{lem:lower bounding mu}). Again by Talagrand's concentration inequality, we can bound bidder $i$'s expected surplus by our entry fee and $\\tau_i$ (Lemma~\\ref{lem:concentration entry fee}). As $\\hat{v}$ is always smaller than the true valuation $v$, thus for any type $t_i$ of bidder $i$ and any available items $S$, the surplus for $S$ under $v_i(t_i,\\cdot)$ must be larger than $\\mu_i(t_i,S)$, and the entry fee is accepted with probability at least $1\/2$. Putting everything together, we demonstrate that we can approximate $\\textsc{Core}(M,\\beta)$ with an ASPE or an RSPM (Lemma~\\ref{lem:upper bounding Q}).\n\\subsubsection{Construction of $\\widehat{\\textsc{Core}}(M,\\beta)$}\\label{sec:proxy core}\n\nWe first show that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{proof}\n\n\n\nNext, we define the prices of our ASPE.\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n{ $\\textsc{Core}(M,\\beta)$ can be upper bounded by $\\sum_{j\\in [m]}Q_j$. The proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).}\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta)$.\n\\end{lemma}\n\\begin{proof}\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\vspace{0.05in}\nIn the following definitions, we define $\\widehat{\\textsc{Core}}(M,\\beta)$ which is the welfare of another function $\\hat{v}$ under the same allocation $\\sigma^{(\\beta)}$.\n\n \n\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\}.$$\n\\end{definition}\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$. Moreover, let $$\\widehat{\\textsc{Core}}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{v}_i(t_i,S).$$\n\\end{definition}\n\nIn the next two Lemmas, we prove some useful properties of $\\tau_i$. In particular, we argue that $\\sum_{i\\in[n]} \\tau_i$ can be upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}).\n \\begin{lemma}\\label{lem:beta_ij}\n\\begin{align*}\n\\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\right]\n \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAccording to the definition of $\\tau_i$, for every buyer $i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$,\n and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. Our statement follows directly from Lemma~\\ref{lem:neprev}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n$$\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative,  \\begin{align*}\n  \\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\n  \\geq \\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right].\n  \\end{align*}\nAccording to the definition of $\\tau_i$, when $\\tau_i>0$, $$\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}.$\nTherefore, $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:beta_ij}.\n\\end{proof}\n\n\nIn the following two Lemmas, we compare $\\widehat{\\textsc{Core}}(M,\\beta)$ with $\\textsc{Core}(M,\\beta)$. The proof of Lemma~\\ref{lem:hat gamma} is postponed to Appendix~\\ref{appx:multi}.\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lem:hat gamma}, $\\sum_{j\\in[m]}\\hat{Q}_j\\leq \\widehat{\\textsc{Core}}(M,\\beta)\/2$. By Lemma~\\ref{lem:core and q_j}, $\\sum_{j\\in[m]}{Q}_j\\leq {\\textsc{Core}}(M,\\beta)\/2\\alpha$. Hence, Lemma~\\ref{lem:hat Q} shows that to approximate $\\textsc{Core}(M,\\beta)$, it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$. Indeed, we will use $\\sum_{j\\in[m]} \\hat{Q}_j$ as an proxy for $\\textsc{Core}(M,\\beta)$ in our analysis of the ASPE.\n\\subsubsection{Design and Analysis of Our ASPE}\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$ \\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\big(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\big).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n$\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\leq a]\\geq\\frac{1}{2}\\}$.\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nIn Lemma~\\ref{lem:property of mu}, we show that $\\tau_i$ is the Lipschitz constant for $\\mu_i(\\cdot,\\cdot)$ and the proof is postponed to Appendix~\\ref{appx:multi}. Moreover, $\\sum_i \\tau_i$ is upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:tau_i}.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\n\n\n\n\nThe following Lemma is crucial for our proof. We show that in expectation over all type profiles, we can lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i}))$ for all bidders. In particular, this lower bound plus our ASPE's revenue from the posted prices already approximates $\\widehat{\\textsc{Core}}(M,\\beta)$. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. For every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE. Then\n\n\\begin{align*}\n{\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]&\\geq \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot(2\\hat{Q}_j-Q_j)\\\\\n&\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\n\\end{align*}\n\n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq &\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\n\n\n\\vspace{0.05in}\nSince entry fee in the ASPE for every bidder as the median of her utility over the available items under $\\hat{v}$. Clearly, bidders accept the entry fee with probability at least $1\/2$, as their true utilities (under $v$) are always higher than their utilities under $\\hat{v}$. Combining the concentration property of the utility under $\\hat{v}$ and Lemma~\\ref{lem:lower bounding mu}, we can argue that the total revenue from our ASPE is comparable to $\\widehat{\\textsc{Core}}(M,\\beta)$, and therefore is comparable to $\\textsc{Core}(M,\\beta)$.\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j- \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $u_i(t_i,S) = \\max_{S'\\subseteq S} \\big( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\big).$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$\n\nFor any $i$ and $t_{<i}$, by Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}, we are able to derive a lower bound for $\\delta_i\\left(S_i(t_{<i})\\right)$, as shown in Lemma~\\ref{lem:concentration entry fee}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n\n\\begin{proof}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{proof}\n\n\\vspace{0.05in}\n\nBack to the proof of Lemma~\\ref{lem:entry fee acceptance probability}. According to Lemma~\\ref{lem:concentration entry fee}, the revenue from the entry fee is at least\n$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i$, which is equal to\n$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i$.\nCombining Lemma~\\ref{lem:tau_i} and Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $\\frac{1}{4}\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - (\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b(1-b)})\\textsc{PostRev}$. Since the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$, the total revenue of the ASPE is at least $\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$\n\\end{proof}\n\nCombining everything together, we have the main result of Section~\\ref{subsection:core}.\n\n\\begin{lemma}\\label{lem:upper bounding Q}\n\tFor any BIC mechanism $M$, {$\\textsc{Core}(M,\\beta)\\leq8\\alpha\\cdot\\textsc{APostEnRev}+4\\alpha\\left(\\frac{6}{1-b}+\\frac{1}{b(1-b)}\\right)\\textsc{PostRev}$.}\n\\end{lemma}\n\n\\begin{proof}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{proof}\n\nNow, we have upper bounded $\\textsc{Single}(M,\\beta)$, $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tThe proof is done by combining {Theorem~\\ref{thm:revenue upperbound for subadditive}}, Lemma~\\ref{lem:multi_single},~\\ref{lem:multi decomposition},~\\ref{lem:multi-tail} and~\\ref{lem:upper bounding Q}.\n\\end{prevproof}\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nWe focus on revenue maximization in the combinatorial auction with $n$ independent bidders and $m$ heterogenous items. Each bidder has a valuation that is \\textbf{subadditive over independent items} (see Definition~\\ref{def:subadditive independent}). We denote bidder $i$'s type $t_i$ as $\\langle t_{ij}\\rangle_{j=1}^m$, where $t_{ij}$ is bidder $i$'s private information about item $j$. For each $i$, $j$, we assume $t_{ij}$ is drawn independently from the distribution $D_{ij}$. Let $D_i=\\times_{j=1}^m D_{ij}$ be the distribution of bidder $i$'s type and $D=\\times_{i=1}^n D_i$ be the distribution of the type profile. We use $T_{ij}$ (or $T_i, T$) and $f_{ij}$ (or $f_i, f$) to denote the support and density function of $D_{ij}$ (or $D_i, D$). For notational convenience, we let  $t_{-i}$ to be the types of all bidders except $i$ and $t_{<i}$ (or $t_{\\leq i})$ to be the types of the first $i-1$ (or $i$) bidders. Similarly, we define $D_{-i}$, $T_{-i}$\n  and $f_{-i}$ for the corresponding  distributions, support sets and density functions. When bidder $i$'s type is $t_i$, her valuation for a set of items $S$ is denoted by $v_i(t_i,S)$.\n\n\\begin{definition}~\\cite{RubinsteinW15}\\label{def:subadditive independent}\nFor every bidder $i$, whose type is drawn from a product distribution $F_i=\\prod_j F_{ij}$, her distribution $\\mathcal{V}_i$ of valuation function $v_i(t_i,\\cdot)$ is \\textbf{subadditive over independent items} if:\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ has no externalities}, i.e., for each $t_i\\in T_i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ only depends on $\\langle t_{ij}\\rangle_{j\\in S}$, formally, for any $t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$, $v_i(t_i',S)=v_i(t_i,S)$.\n\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ is monotone}, i.e., for all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, $v_i(t_i,U)\\leq v_i(t_i,V)$.\n\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ is subadditive}, i.e., for all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, $v_i(t_i,U\\cup V)\\leq v_i(t_i,U)+ v_i(t_i,V)$.\n    \n\\end{itemize}\n\n We use $V_i(t_{ij})$ to denote $v_i(t_i,\\{j\\})$, as it only depends on $t_{ij}$. When $v_i(t_i,\\cdot)$ is XOS (or constrained additive) for all $i$ and $t_i\\in T_i$, we say $\\mathcal{V}_i$ is XOS (or constrained additive) over independent items.\n\\end{definition}\n\nWe first formally define various valuation classes.\n\\begin{definition}\\label{def:valuation classes}\nWe define several classes of valuations formally. Let $t$ be the type and $v(t,S)$ be the value for bundle $S\\in[m]$.\n\n\n\t\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item  \\textbf{Constrained Additive:}\n$v(t,S) =\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} v(t, \\{j\\})$, where $\\mathcal{I}\\subseteq 2^{[m]}$ is a downward closed set system over the items specifying the feasible bundles. In particular, when $\\mathcal{I}=2^{[m]}$, the valuation is an \\textbf{additive function}; when $\\mathcal{I}=\\{\\{j\\}\\ |\\ j\\in[m] \\}$, the valuation is a \\textbf{unit-demand function}; when $\\mathcal{I}$ is a matroid, the valuation is a \\textbf{matroid-rank function}. An equivalent way to represent any constrained additive valuations is to view the function as additive but the bidder is only allowed to receive bundles that are feasible, i.e., bundles in $\\mathcal{I}$. To ease notations, we interpret $t$ as an $m$-dimensional vector $(t_1, t_2,\\cdots, t_m)$ such that $t_j =  v(t, \\{j\\})$.\n\n\\item \\textbf{XOS\/Fractionally Subadditive:}\n $v(t,S) = \\max_{i\\in [{K}]} v^{(i)}(t, S)$, where ${K}$ is some finite number and $v^{(i)}(t,\\cdot)$ is an additive function for any $i\\in[{K}]$.\n\n\\item \\textbf{Subadditive:}\n$v(t,S_1\\cup S_2)\\leq v(t,S_1)+v(t,S_2)$ for any $S_1, S_2\\subseteq [m]$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{The following are a few examples of various valuation distributions which are over independent items (Definition~\\ref{def:subadditive independent}):\n\n\\begin{example}\\label{eg:valuation}\\cite{RubinsteinW15}\n$t=\\{t_j\\}_{j\\in[m]}$ where $t$ is drawn from $\\prod_j D_j$,\n\\begin{itemize}[leftmargin=0.7cm]\n\\item Additive: $t_j$ is the value of item $j$. $v(t,S)=\\sum_{j\\in S}t_j$.\n\\item Unit-demand: $t_j$ is the value of item $j$. $v(t,S)=\\max_{j\\in S}t_j$.\n\\item Constrained Additive: $t_j$ is the value of item $j$.\n $v(t,S)=$\\\\\n \\noindent$\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} t_j$.\n\\item XOS\/Fractionally Subadditive: $t_j=\\{t_{j}^{(k)}\\}_{k\\in[K]}$ encodes all the possible values associated with item $j$, and $v(t,S)=$\\\\\n    \\noindent$\\max_{k\\in[K]}\\sum_{j\\in S}t_{j}^{(k)}$.\n\\end{itemize}\n\\end{example}\n\n\n}\n\n\nGiven  $D$ and $v=\\{v_i(\\cdot,\\cdot)\\}_{i\\in [n]}$, we use \\textbf{$\\textsc{Rev}(M,v,D)$} to denote the expected revenue of a BIC mechanism $M$. Throughout the paper, we use the following notations for the simple mechanisms we consider.\n\n\\vspace{.05in}\n\\noindent\\textbf{Single-Bidder Mechanisms:}\n\\vspace{.03in}\n\n\\noindent \\textbf{- $\\textsc{SRev}(v,D)$} denotes the optimal expected revenue achievable by any posted price mechanism that only allows the buyer to purchase at most one item, and we use $\\textsc{SRev}$ for short if there is no confusion\\footnote{The mechanism is slightly different from selling separately, as we only allow the buyer to purchase at most one item.}.\n\n\\noindent \\textbf{- $\\textsc{BRev}(v,D)$} denotes the optimal expected revenue achievable by selling a grand bundle and we use $\\textsc{BRev}$ for short if there is no confusion.\n\n\\vspace{.05in}\n\\noindent\\textbf{Multi-Bidder Mechanisms:}\n\\vspace{.03in}\n\n\\noindent\\textbf{- $\\textsc{PostRev}(v,D)$} denotes the optimal expected revenue achievable by selling the items via an RSPM to the bidders, and we use $\\textsc{PostRev}$ for short when there is no confusion.\n\n\\noindent\\textbf{- $\\textsc{APostEnRev}(v,D)$} denotes the optimal expected revenue achievable by selling the items via an ASPE to the bidders, and we use $\\textsc{APostEnRev}$ for short when there is no confusion.\n\n\\vspace{.05in}\n\\noindent\\textbf{Single-Dimensional Copies Setting:} In the analysis for unit-demand bidders in~\\cite{ChawlaHMS10, CaiDW16}, the optimal revenue is upper bounded by the optimal revenue in the single-dimensional copies setting defined in~\\cite{ChawlaHMS10}. We use the same technique. We construct $nm$ agents, where agent $(i,j)$ has value $V_i(t_{ij})$ of being served with $t_{ij}\\sim D_{ij}$, and we are only allow to use matchings, that is, for each $i$ at most one agent $(i,k)$ is served and for each $j$ at most one agent $(k,j)$ is served\\footnote{This is exactly the copies setting used in~\\cite{ChawlaHMS10}, if every bidder $i$ is unit-demand and has value $V_i(t_{ij})$ with type $t_i$. Notice that this unit-demand multi-dimensional setting is equivalent as adding an extra constraint, each buyer can purchase at most one item, to the original setting with subadditive bidders.}.\nNotice that this is a single-dimensional setting, as each agent's type is specified by a single number. Let $\\textsc{OPT}^{\\textsc{Copies-UD}}$ be the optimal BIC revenue in this copies setting.\n\n\\vspace{.05in}\n\\noindent\\textbf{Continuous vs. Discrete Distributions:} We explicitly assume that the input distributions are discrete. Nevertheless, it is known that every $D$ can be discretized into $D^{+}$ such that the optimal revenue for $D$ and $D^{+}$ are within $(1\\pm\\epsilon)$ of each other~\\cite{CaiDW16}. So our results also apply to  continuous distributions. \n\\subsection{Our Mechanisms}\nIn this section, we introduce a class of mechanisms called \\emph{Sequential Posted Price with Entry Fee}. For each bidder $i$, the mechanism first determines a posted price $\\xi_{ij}$ for each item $j$ and an entry fee function ${\\delta_i}(\\cdot):2^{[m]}\\to {\\mathbb{R}_{\\geq 0}}$ for each bidder $i$ that maps the set of available items to a real value entry fee. The seller visits the bidders sequentially in some arbitrary order. For simplicity, we assume the bidders are visited in the lexicographical order. When bidder $i$ is visited, let $S_i(t_{<i})$ be the set of items that are still available. Clearly, this set only depends on the types of bidders who are visited before $i$. The mechanism shows the set $S_i(t_{<i})$ to bidder $i$ and asks her for an entry fee ${\\delta_i}(S_i(t_{<i}))$. If she accepts the entry fee, she can enter the mechanism and take her favorite bundle $S_i^{*}$ by paying $\\sum_{j\\in S_i^{*}} \\xi_{ij}$.\n\n If there exist multiple bundles with the same maximum surplus, the bidder can break ties arbitrarily. Sometimes, there is a feasibility constraint $\\mathcal{F}$ on what items a buyer can purchase. In particular, if we say the mechanism is rationed, then $\\mathcal{F}=\\{\\emptyset\\} \\cup \\{\\{j\\}\\ |\\ j\\in[m] \\}$, i.e., a buyer can purchase at most one item. Formally, the favorite bundle $S_i^{*}$ is defined as follows:\n$S_i^{*}={\\argmax}_{S\\subseteq S_i(t_{<i})\\land S\\in \\mathcal{F}} v_i(t_i,S)-\\sum_{j\\in S}\\xi_{ij}$.\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n    \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n        \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n        \\STATE $S\\gets S\\backslash S_i^{*}$.\n    \\ELSE\n        \\STATE $i$ gets nothing and pays $0$.\n    \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n\nSee Algorithm \\ref{alg:seq-mech} for the formal specification of the above mechanism. Notice that before the bidder decides whether to pay the entry fee, she is aware of the set $S_i(t_{<i})$ which contains all available items. Thus, she can compute her favorite bundle $S_i^{*}$ and the corresponding utility if she chooses to enter the mechanism. She can then compare that utility with the entry fee and accept the entry fee if the former is greater than the latter. The mechanism described above is therefore deterministic and DSIC. Throughout this paper, we  focus on the following two special cases of this class of mechanisms:\n\n\n\\vspace{.05in}\n \\noindent\\textbf{-Rationed Sequential Posted Price Mechanism (RSPM):}  Every buyer can purchase at most one item and the mechanism always charges $0$ entry fee, i.e., $\\mathcal{F}=\\{\\emptyset\\} \\cup \\{\\{j\\}\\ |\\ j\\in[m] \\}$ and ${\\delta_i}(S)=0$ for all $i$ and $S$.\n\n \\vspace{.05in}\n\\noindent\\textbf{-Anonymous Sequential Posted Price with Entry Fee Mechanism (ASPE):}  The mechanism uses anonymous posted prices, i.e., ${\\xi}_{ij} = \\xi_{kj}$ for any item $j$ and bidders $i\\neq k$, but may charge positive and personalized entry fee. Also, any buyer can purchase any bundle available once she has paid the entry fee, i.e., $\\mathcal{F} = 2^{[m]}$.\n\n\n\n\\section{Proof of Corollary~\\ref{corollary:concentrate}}\\label{sec:concentration_proof}\n\n\n\\section{Proof of Theorem~\\ref{thm:revenue less than virtual welfare}}\\label{sec:proof_duality}\n\\begin{prevproof}{Theorem}{thm:revenue less than virtual welfare}\nWhen $\\lambda$ is useful, we can simplify function ${\\mathcal{L}}(\\lambda, \\sigma, p)$ by removing the term associated with $p$ and replacing $\\sum_{t_{i}'\\in T_{i}^{+}}\\lambda(t_{i},t_{i}')$ with $f_{i}(t_{i})+\\sum_{t_{i}'\\in T_{i}} \\lambda(t_{i}',t_{i})$. After the simplification, we have\n\\begin{align*}\n&{\\mathcal{L}}(\\lambda, \\sigma, p) = \\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i})\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot \\\\\n&\\left( v_i(t_{i},S)-{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right)\\right)\\\\\n&\\hspace{1.3cm}=\\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i})\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S),\t\n\\end{align*}\n which is exactly the virtual welfare of $\\sigma$ with respect to $\\lambda$. Now, we only need to prove that ${\\mathcal{L}}(\\lambda, \\sigma, p)$ is greater than the revenue of $M$. Let us think of ${\\mathcal{L}}(\\lambda, \\sigma, p)$ using Expression~(\\ref{eq:primal lagrangian}). Since $M$ is a BIC mechanism, $$\\sum_{S\\subseteq[m]}v_i(t_i,S)\\cdot\\left(\\sigma_{iS}(t_{i})-\\sigma_{iS}({t_{i}'})\\right)-\\left((p_{i}(t_{i})-p_{i}(t_{i}')\\right)\\geq 0$$ for any $i$, $t_{i}\\in T_{i}$ and $t_i'\\in T_i^+$. Also, all the dual variables $\\lambda$ are nonnegative. Therefore, it is clear that ${\\mathcal{L}}(\\lambda, \\sigma, p)$ is at least as large as the revenue of $M$.\n\nWhen $\\lambda^{*}$ is the optimal dual variable, by strong duality, we know $\\max_{\\sigma\\in{P(D)}, p}{\\mathcal{L}}(\\lambda^{*}, \\sigma, p)$ equals to the revenue of $M^{*}=(\\sigma^*,p^*)$. But we also know that ${\\mathcal{L}}(\\lambda^{*}, \\sigma^{*}, p^{*})$ is at least as large as the revenue of $M^{*}$, therefore $\\sigma^{*}$ maximizes the virtual welfare.\n\\end{prevproof}\n\n\n\\section{Analysis for the Multi-Bidder Case}\\label{sec:multi_proof}\n\n\\subsection{Analyzing $\\textsc{Single}(\\beta)$ in the Multi-Bidder Case}\n\n\\begin{prevproof}{Lemma}{lem:multi_single}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one copy of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$\n\\end{prevproof}\n\n\\subsection{\\textsc{Core}--\\textsc{Tail}~Decomposition}\\label{sec:multi_nf}\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\notshow{=}\\mingfeinote{\\leq}&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(\\beta))\n\\end{align*}\n\\mingfeinote{[Mingfei: $t_i$ in region k if $V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}$ and also other $j'\\not=k,j$ is no larger than k]}\n\\end{prevproof}\n\n\\subsection{Analyzing $\\textsc{Tail}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\nIn this section, we show how to upper bound $\\textsc{Tail}(\\beta)$ with the revenue of an RSPM.\n\tLet us first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$ $$r_{ij}=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$ $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. The next Lemma shows that $r$ is an upper bound of $\\textsc{Tail}(\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\n$$\\textsc{Tail}(\\beta)\\leq r.$$\t\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]=1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful when bounding $\\textsc{Core}(\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nIf $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ is  a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $$\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right].$$\n\\end{proof}\n\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\nr\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$ (Definition~\\ref{def:c_i})). Therefore, $$r= \\sum_i\\sum_j(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\n\tIt follows from Lemma~\\ref{lem:tail and r} and Corollary~\\ref{cor:bound tail }.\n\\end{prevproof}\nFinally, we show that $r_i$ is within a constant factor of $c_i$. This Lemma will be useful for bounding $\\textsc{Core}{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\\\\\n&\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because if $c_i>0$, then $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]\\geq \\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Corollary~\\ref{cor:bound tail }, $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\subsection{Analyzing $\\textsc{Core}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$. Hence, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$.\\mingfeinote{Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$ for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then or any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nNext, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$  and any set $S$, then for any $t_i$ there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$ and any set $S$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. In particular, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\nAlso, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})>\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})\\leq\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)=\\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_j \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that   $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap X_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat},  $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\t\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n\\begin{equation*}\n\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\geq a]\\leq\\frac{1}{2}\\}.\n\\end{equation*}\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nWe separate the revenue of the ASPE into two different parts -- the entry fee part and the item pricing part. We first provide a lower bound of the entry fee part.\n We define another set of prices $\\{\\hat{Q}_j\\}_{j\\in[m]}$ similar to the way we define $\\{{Q}_j\\}_{j\\in[m]}$, and argue that $\\sum_j Q_j$ is not much larger than $\\sum_j \\hat{Q}_j$.\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:hat Q}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}). The second inequality is due to the choice of $\\beta_{ij}$ (Lemma~\\ref{lem:requirement for beta}). The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i})$ for all bidders. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}.\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. Then for every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE, then $${\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n=&\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\\notshow{\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i(t_i,t_{< i}) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\max_{S\\subseteq \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} \\hat{v}_i(t_i, S) - \\sum_{j\\in S} Q_j\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j}\\cdot \\mathds{1}[j\\in S_i(t_{<i})]\\cdot\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i, t'_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)~~~~~\\text{(Definition of $\\hat{Q}_j$ and $\\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq 1$ for all $j$)}\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{($\\forall j$, $Q_j-\\hat{Q}_j\\geq 0$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n{We apply the concentration bound in Corollary~\\ref{corollary:concentrate} to prove Lemma~\\ref{lem:concentration entry fee}. First, we argue that for any bidder $i$, $\\mu_i(\\cdot,\\cdot)$ is Lipschitz and for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{j\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{j\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all $U\\subseteq V\\subseteq S_i(t_{<i})$, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{prevproof}\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\n{\\mathbb{E}}_{t}\\left[\\mu_i(t_i,t_{< i})\\right]&= 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\int_{y=0}^{\\infty}\\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+y]dy\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot\\sum_{k=0}^m \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+k\\cdot \\tau_i] \\quad(\\mu_i(t_i,t_{< i})\\leq m\\cdot \\tau_i)\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot \\sum_{k=0}^2 \\Pr[\\mu_i(t_i,t_{< i})\\geq \\delta_i(t_{<i})]+ \\tau_i\\cdot\\sum_{k=3}^{m} \\frac{9}{4}\\cdot 2^{-k}\\\\\n&\\leq 3\\delta_i(t_{<i})+\\tau_i+\\frac{9}{16}\\tau_i\\\\\n&=3\\delta_i(t_{<i})+\\frac{25}{16}\\tau_i\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $$u_i(t_i,S) = \\max_{S'\\subseteq S} \\left( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\right).$$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $$\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$$ Due to Lemma~\\ref{lem:concentration entry fee}, this is no smaller than $$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i= \\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i.$$ Combining Lemma~\\ref{lem:tau_i} and by Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $$\\frac{1}{4}\\cdot\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\nOn the other hand, the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$. Therefore, the total revenue of the ASPE is at least $$\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\nFinally, we upper bound the $\\textsc{Core}(\\beta)$ by combining Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\n\n\\vspace{.1in}\n\\begin{prevproof}{Lemma}{lem:upper bounding Q}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{prevproof}\n\nNow, we have upper bounded $\\textsc{Single}(\\beta)$, $\\textsc{Tail}(\\beta)$ and $\\textsc{Core}(\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tCombining Lemma~\\ref{lem:multi_single}, Corollary~\\ref{cor:bound tail } and Lemma~\\ref{lem:upper bounding Q}, we have\n\t\\begin{equation*}\n\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b\\cdot(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev} + 32\\alpha \\cdot \\textsc{APostEnRev}.\n\\end{equation*}\n\\end{prevproof}\n\n\\notshow{\\yangnote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(\\beta)\\leq 120\\textsc{PostRev}+12\\textsc{APostEnRev}$ and $\\textsc{Tail}(\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 508\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf I optimize over $b$, I can get $\\textsc{Core}(\\beta)+\\textsc{Tail}(\\beta)\\leq 94\\textsc{PostRev}+12\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 388\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf we take $b=1\/4$, $\\textsc{Core}(\\beta)\\leq 18\\textsc{APostEnRev}+152\/3\\textsc{PostRev}$. $\\textsc{Tail}(\\beta)\\leq 8\/3\\textsc{PostRev}$. $\\textsc{Rev}\\leq (640\/3+12)\\textsc{PostRev}+72\\textsc{APostEnRev}\\leq 226\\textsc{PostRev}+72\\textsc{APostEnRev}.$\n}\\mingfeinote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(\\beta)\\leq 49\\textsc{PostRev}+18\\textsc{APostEnRev}$ and $\\textsc{Tail}(\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 224\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\nAfter optimizing over $b$($b=0.3$), I can get $\\textsc{Core}(\\beta)+\\textsc{Tail}(\\beta)\\leq 46\\textsc{PostRev}+18\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 196\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\n}}\n\n\n\n\\notshow{To bound $\\textsc{Core}(\\beta)$, notice the social welfare of new valuation$(\\textsc{Core})$ can be viewed as the sum of revenue and bidder's utility. With no entry fee, the revenue comes only from the posted price. We are going to show that with proper entry fee, the expected revenue from entry fee can approximately obtain the utility part.}\n\n\\notshow{For all $t_i$ and $S$, define $\\theta_j'(t_i)=\\gamma_j^{S}(t_i)(t_i)\\cdot \\mathds{1}\\left[V_i(t_{ij})\\leq Q_j+\\tau_i\\right]$. When each bidder is additive with item value $\\theta_j'(t_i)$, given t, the maximum utility can be written as\n\n\\begin{definition}\n\t$\\mu_i(t)=\\mathds{E}_{t_i\\sim T_i}\\left[\\max_{S\\subseteq S_i(t_{<i})}\\sum_{j\\in S}\\left(\\theta_j'(t_i)-Q_j\\right)\\right]$\n\\end{definition}\n\nDefine $\\textsc{Utility}=\\mathds{E}_{t\\in T}\\left[\\sum_i \\mu_i(t)\\right]$.\n\n\\begin{lemma}\n(citation)For every $i$ and every $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, for any $p\\in (0,1)$,\n\\begin{equation}\n\\Pr_{t_i}\\left[\\mu_i(t_i,t_{-i})\\leq (1-\\delta)\\cdot \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\right]\\leq e^{-\\frac{p^2}{2}}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIt comes directly from (citation) with the fact that $\\theta_j'(t_i)-Q_j\\leq \\tau_i$.\n\\end{proof}\n\n\nTo bound \\textsc{Utility}, for each $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, $\\mu_i(t)$ concentrates around its expectation $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})$. Define the entry fee as $\\delta_i(t_{\\geq i})=(1-p)\\cdot c\\cdot \\mathds{E}_{t_i}\\mu_i(t)$, where $p\\in (0,1)$ is a parameter to be decided. By the definition of supporting price, $v_i(t_i,S)\\geq c\\cdot \\sum_{j\\in S}\\theta_j'(t_i)$. Hence for any $t_{-i}$, if bidder $i$ with type $t_i$ accepts the entry fee, the maximum utility he can get is no less than $c\\cdot \\mu_i(t_i,t_{-i})$. Then with probability at least $(1-e^{-\\frac{p^2}{2}})$, bidder $i$ will accept the entry fee. When $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\leq 2\\tau_i$, we can directly use $2\\tau_i$ as the upper bound.\n\nGiven $t$, let $\\textsc{SOLD(t)}$ be the set of items sold during the auction. Then the expected revenue from posted price is $\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j$. Hence by analyzing the revenue of this sequential posted price mechanism, we have\n\\begin{equation}\n\\begin{aligned}\n\\textsc{APostEnRev}&\\geq \\mathds{E}_{t_{-i}}\\left[(1-e^{-\\frac{p^2}{2}})\\cdot \\sum_i\\delta_i(t_{\\geq i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&=\\mathds{E}_{t_{-i}}\\left[(1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot \\sum_i \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]\\quad(\\text{Definiton of $\\delta_i(t_{\\geq i})$})\\\\\n&+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&\\geq (1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot(\\textsc{Utility}-2\\sum_i\\tau_i)+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n\\end{aligned}\n\\end{equation}\n\t\nAfter picking a proper parameter $p=0.62$, by Lemma \\ref{lem:tau_i},\n\\begin{equation}\\label{equ:utility_upper_bound}\n\\textsc{Utility}+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq 2\\sum_i\\tau_i+8.25\\textsc{APostEnRev}\\leq (\\frac{8}{1-b}+8.25)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\n\nNotice that the term $\\textsc{Utility}$ is smaller than the real expect utility since item values $\\gamma_j^{S}(t_i)(t_i)$ are replaced by $\\theta_j'(t_i)$. We still need to make sure that these two parts are not far from each other to complete the proof. Lemma \\ref{lem:utility} describes this fact.\n\n\\begin{lemma}\\label{lem:utility}\n\\begin{equation}\n\\sum_{j\\in [m]}\\Pr_t\\left[j\\not\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq \\textsc{Utility}+\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nGiven a BIC mechanism $M$, let $M_i(t)$ be the set of items allocated to bidder $i$ when type profile is $t$. Notice that in the term $\\textsc{Utility}$, the favorite bundle just contains all items in $S_i(t_{<i})$ with positive surplus, we have\n\\begin{equation}\\label{equ:utility}\n\\begin{aligned}\n\\textsc{Utility}&=\\sum_i\\mathds{E}_{t\\sim T}\\left[\\sum_{j\\in S_i(t_{<i})}\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\right]\\\\\n&\\geq \\sum_i\\mathds{E}_{t_i,t_{-i},t_{-i}'}\\left[\\sum_{j\\in [m]}\\mathds{1}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t_{-i}'}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\mathds{E}_{t_i,t_{-i}}\\left[\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\\\\n\\end{aligned}\n\\end{equation}\n\nThe last inequality is because for each $t_i'$, the probability of $j\\in S_i(t_{<i}')$ can not exceed the probability of item $j$ is not sold after the sequential mechanism. Notice that for $j\\in A_i$, $\\theta_j'(t_i)$ and $\\gamma_j^{S}(t_i)(t_i)$ are different only if $V_i(t_{ij})\\geq Q_j+\\tau_i$. Using the fact that $\\gamma_j^{S}(t_i)(t_i)\\leq V_i(t_{ij})$, we have\n\n\\begin{equation}\n\\begin{aligned}\n&\\text{RHS of inequality \\ref{equ:utility}}\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\theta_j^{S\\cap\\mathcal{C}_i(t_i)}(t_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\left(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\gamma_j^{S}(t_i)(t_i)-Q_j\\right) \\quad(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)=1)\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot (\\beta_{ij}+c_i-Q_j) \\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\beta_{ij}+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\right)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\frac{1}{b}\\sum_i\\sum_{j\\not\\in A_i}\\beta_{ij}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-\\frac{1}{2}\\sum_i (c_i+\\tau_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev} \\quad (\\text{Lemma \\ref{lem:c_i},\\ref{lem:tau_i},\\ref{lem:beta_ij}})\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nThe inequality stated in Lemma \\ref{lem:core} comes directly from Equation \\ref{equ:core and q_j}, Inequality \\ref{equ:utility_upper_bound} and Lemma \\ref{lem:utility}.\n\nThe following lemma gives an upper bound for $\\textsc{Core}(\\beta)$.\n\\begin{lemma}\\label{lem:core}\nIf for all $i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ has a $c$-supporting price $\\{\\theta_j^{S}(t_i)\\}_{j=1}^m$, then\n\\[\\text{CORE }\\leq \\left(\\frac{24}{1-b}+\\frac{4}{b(1-b)}+16.5\\right)\\cdot c\\cdot \\textsc{APostEnRev}\\]\n\\end{lemma}\n}\n\n\\subsection{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nFor any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We describe the construction of the RSPM and the ASPE separately. Unfortunately, in order to approximate the optimal revenue, we need to know an (approximately) revenue-maximizing mechanism $M^*$. However, for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}, we show that we can directly construct the desired RSPM and ASPE that approximates the optimal revenue without knowing $M^*$.\n\n  Indeed, we can always directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue one can obtain from the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know the optimal revenue in the unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction does not depend on $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$'s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the computation requires sampling from the distribution, but it is not hard to argue that a polynomial number of samples suffices.\n\n  For symmetric bidders, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, select $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies the first property in Lemma~\\ref{lem:requirement for beta}. Notice that the optimal mechanism is symmetric, so for any item $j$ the ex-ante probability for any bidder to win it is the same, and therefore no more than $1\/n$. Hence, the second property is also satisfied. Given this $\\beta$, we can essentially follow the procedure mentioned above to construct the ASPE. The only difference is that when we construct the $Q_j$s, we no longer know the $\\sigma$. This can be resolved by using the welfare maximizing allocation with respect to $v'$.\n\n\n\n\n\\section{Analysis for the Multi-Bidder Case}\\label{sec:multi_proof}\n\n\\subsection{Analyzing $\\textsc{Single}(M,\\beta)$ in the Multi-Bidder Case}\n\n\n\\subsection{\\textsc{Core}--\\textsc{Tail}~Decomposition}\\label{sec:multi_nf}\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(M,\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(M,\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(M,\\beta))\n\\end{align*}\n\\end{prevproof}\n\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$. {Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$, for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then for any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nThen, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(M,\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that   $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat},  $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n\\begin{equation*}\n\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\leq a]\\geq\\frac{1}{2}\\}.\n\\end{equation*}\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nWe separate the revenue of the ASPE into two different parts -- the entry fee part and the item pricing part. We first provide a lower bound of the entry fee part.\n We define another set of prices $\\{\\hat{Q}_j\\}_{j\\in[m]}$ similar to the way we define $\\{{Q}_j\\}_{j\\in[m]}$, and argue that $\\sum_j Q_j$ is not much larger than $\\sum_j \\hat{Q}_j$.\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:hat Q}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i}))$ for all bidders. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}.\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. For every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE. Then $${\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq &\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\\notshow{\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i(t_i,t_{< i}) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\max_{S\\subseteq \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} \\hat{v}_i(t_i, S) - \\sum_{j\\in S} Q_j\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j}\\cdot \\mathds{1}[j\\in S_i(t_{<i})]\\cdot\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i, t'_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)~~~~~\\text{(Definition of $\\hat{Q}_j$ and $\\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq 1$ for all $j$)}\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{($\\forall j$, $Q_j-\\hat{Q}_j\\geq 0$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n{We apply the concentration bound in Corollary~\\ref{corollary:concentrate} to prove Lemma~\\ref{lem:concentration entry fee}. First, we argue that for any bidder $i$, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{k\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{k\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{prevproof}\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\n{\\mathbb{E}}_{t}\\left[\\mu_i(t_i,t_{< i})\\right]&= 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\int_{y=0}^{\\infty}\\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+y]dy\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot\\sum_{k=0}^m \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+k\\cdot \\tau_i] \\quad(\\mu_i(t_i,t_{< i})\\leq m\\cdot \\tau_i)\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot \\sum_{k=0}^2 \\Pr[\\mu_i(t_i,t_{< i})\\geq \\delta_i(t_{<i})]+ \\tau_i\\cdot\\sum_{k=3}^{m} \\frac{9}{4}\\cdot 2^{-k}\\\\\n&\\leq 3\\delta_i(t_{<i})+\\tau_i+\\frac{9}{16}\\tau_i\\\\\n&=3\\delta_i(t_{<i})+\\frac{25}{16}\\tau_i\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $$u_i(t_i,S) = \\max_{S'\\subseteq S} \\left( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\right).$$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $$\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$$ Due to Lemma~\\ref{lem:concentration entry fee}, this is no smaller than $$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i= \\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i.$$ Combining Lemma~\\ref{lem:tau_i} and by Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $$\\frac{1}{4}\\cdot\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\nOn the other hand, the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$. Therefore, the total revenue of the ASPE is at least $$\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\nFinally, we upper bound the $\\textsc{Core}(M,\\beta)$ by combining Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\n\n\\vspace{.1in}\n\\begin{prevproof}{Lemma}{lem:upper bounding Q}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{prevproof}\n\nNow, we have upper bounded $\\textsc{Single}(M,\\beta)$, $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tCombining {Theorem~\\ref{thm:revenue upperbound for subadditive}}, Lemma~\\ref{lem:multi_single},~\\ref{lem:multi decomposition},~\\ref{lem:multi-tail} and~\\ref{lem:upper bounding Q}, we have\n\t\\begin{equation*}\n\\textsc{Rev}{(M,v,D)}\\leq \\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b\\cdot(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev} + 32\\alpha \\cdot \\textsc{APostEnRev}.\n\\end{equation*}\n\\end{prevproof}\n\n\\notshow{\\yangnote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(M,\\beta)\\leq 120\\textsc{PostRev}+12\\textsc{APostEnRev}$ and $\\textsc{Tail}(M,\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(M,\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 508\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf I optimize over $b$, I can get $\\textsc{Core}(M,\\beta)+\\textsc{Tail}(M,\\beta)\\leq 94\\textsc{PostRev}+12\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 388\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf we take $b=1\/4$, $\\textsc{Core}(M,\\beta)\\leq 18\\textsc{APostEnRev}+152\/3\\textsc{PostRev}$. $\\textsc{Tail}(M,\\beta)\\leq 8\/3\\textsc{PostRev}$. $\\textsc{Rev}\\leq (640\/3+12)\\textsc{PostRev}+72\\textsc{APostEnRev}\\leq 226\\textsc{PostRev}+72\\textsc{APostEnRev}.$\n}\\mingfeinote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(M,\\beta)\\leq 49\\textsc{PostRev}+18\\textsc{APostEnRev}$ and $\\textsc{Tail}(M,\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(M,\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 224\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\nAfter optimizing over $b$($b=0.3$), I can get $\\textsc{Core}(M,\\beta)+\\textsc{Tail}(M,\\beta)\\leq 46\\textsc{PostRev}+18\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 196\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\n}}\n\n\n\n\\notshow{To bound $\\textsc{Core}(\\beta)$, notice the social welfare of new valuation$(\\textsc{Core})$ can be viewed as the sum of revenue and bidder's utility. With no entry fee, the revenue comes only from the posted price. We are going to show that with proper entry fee, the expected revenue from entry fee can approximately obtain the utility part.}\n\n\\notshow{For all $t_i$ and $S$, define $\\theta_j'(t_i)=\\gamma_j^{S}(t_i)(t_i)\\cdot \\mathds{1}\\left[V_i(t_{ij})\\leq Q_j+\\tau_i\\right]$. When each bidder is additive with item value $\\theta_j'(t_i)$, given t, the maximum utility can be written as\n\n\\begin{definition}\n\t$\\mu_i(t)=\\mathds{E}_{t_i\\sim T_i}\\left[\\max_{S\\subseteq S_i(t_{<i})}\\sum_{j\\in S}\\left(\\theta_j'(t_i)-Q_j\\right)\\right]$\n\\end{definition}\n\nDefine $\\textsc{Utility}=\\mathds{E}_{t\\in T}\\left[\\sum_i \\mu_i(t)\\right]$.\n\n\\begin{lemma}\n(citation)For every $i$ and every $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, for any $p\\in (0,1)$,\n\\begin{equation}\n\\Pr_{t_i}\\left[\\mu_i(t_i,t_{-i})\\leq (1-\\delta)\\cdot \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\right]\\leq e^{-\\frac{p^2}{2}}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIt comes directly from (citation) with the fact that $\\theta_j'(t_i)-Q_j\\leq \\tau_i$.\n\\end{proof}\n\n\nTo bound \\textsc{Utility}, for each $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, $\\mu_i(t)$ concentrates around its expectation $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})$. Define the entry fee as $\\delta_i(t_{\\geq i})=(1-p)\\cdot c\\cdot \\mathds{E}_{t_i}\\mu_i(t)$, where $p\\in (0,1)$ is a parameter to be decided. By the definition of supporting price, $v_i(t_i,S)\\geq c\\cdot \\sum_{j\\in S}\\theta_j'(t_i)$. Hence for any $t_{-i}$, if bidder $i$ with type $t_i$ accepts the entry fee, the maximum utility he can get is no less than $c\\cdot \\mu_i(t_i,t_{-i})$. Then with probability at least $(1-e^{-\\frac{p^2}{2}})$, bidder $i$ will accept the entry fee. When $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\leq 2\\tau_i$, we can directly use $2\\tau_i$ as the upper bound.\n\nGiven $t$, let $\\textsc{SOLD(t)}$ be the set of items sold during the auction. Then the expected revenue from posted price is $\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j$. Hence by analyzing the revenue of this sequential posted price mechanism, we have\n\\begin{equation}\n\\begin{aligned}\n\\textsc{APostEnRev}&\\geq \\mathds{E}_{t_{-i}}\\left[(1-e^{-\\frac{p^2}{2}})\\cdot \\sum_i\\delta_i(t_{\\geq i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&=\\mathds{E}_{t_{-i}}\\left[(1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot \\sum_i \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]\\quad(\\text{Definiton of $\\delta_i(t_{\\geq i})$})\\\\\n&+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&\\geq (1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot(\\textsc{Utility}-2\\sum_i\\tau_i)+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n\\end{aligned}\n\\end{equation}\n\t\nAfter picking a proper parameter $p=0.62$, by Lemma \\ref{lem:tau_i},\n\\begin{equation}\\label{equ:utility_upper_bound}\n\\textsc{Utility}+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq 2\\sum_i\\tau_i+8.25\\textsc{APostEnRev}\\leq (\\frac{8}{1-b}+8.25)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\n\nNotice that the term $\\textsc{Utility}$ is smaller than the real expect utility since item values $\\gamma_j^{S}(t_i)(t_i)$ are replaced by $\\theta_j'(t_i)$. We still need to make sure that these two parts are not far from each other to complete the proof. Lemma \\ref{lem:utility} describes this fact.\n\n\\begin{lemma}\\label{lem:utility}\n\\begin{equation}\n\\sum_{j\\in [m]}\\Pr_t\\left[j\\not\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq \\textsc{Utility}+\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nGiven a BIC mechanism $M$, let $M_i(t)$ be the set of items allocated to bidder $i$ when type profile is $t$. Notice that in the term $\\textsc{Utility}$, the favorite bundle just contains all items in $S_i(t_{<i})$ with positive surplus, we have\n\\begin{equation}\\label{equ:utility}\n\\begin{aligned}\n\\textsc{Utility}&=\\sum_i\\mathds{E}_{t\\sim T}\\left[\\sum_{j\\in S_i(t_{<i})}\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\right]\\\\\n&\\geq \\sum_i\\mathds{E}_{t_i,t_{-i},t_{-i}'}\\left[\\sum_{j\\in [m]}\\mathds{1}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t_{-i}'}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\mathds{E}_{t_i,t_{-i}}\\left[\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\\\\n\\end{aligned}\n\\end{equation}\n\nThe last inequality is because for each $t_i'$, the probability of $j\\in S_i(t_{<i}')$ can not exceed the probability of item $j$ is not sold after the sequential mechanism. Notice that for $j\\in A_i$, $\\theta_j'(t_i)$ and $\\gamma_j^{S}(t_i)(t_i)$ are different only if $V_i(t_{ij})\\geq Q_j+\\tau_i$. Using the fact that $\\gamma_j^{S}(t_i)(t_i)\\leq V_i(t_{ij})$, we have\n\n\\begin{equation}\n\\begin{aligned}\n&\\text{RHS of inequality \\ref{equ:utility}}\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\theta_j^{S\\cap\\mathcal{C}_i(t_i)}(t_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\left(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\gamma_j^{S}(t_i)(t_i)-Q_j\\right) \\quad(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)=1)\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot (\\beta_{ij}+c_i-Q_j) \\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\beta_{ij}+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\right)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\frac{1}{b}\\sum_i\\sum_{j\\not\\in A_i}\\beta_{ij}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-\\frac{1}{2}\\sum_i (c_i+\\tau_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev} \\quad (\\text{Lemma \\ref{lem:c_i},\\ref{lem:tau_i},\\ref{lem:beta_ij}})\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nThe inequality stated in Lemma \\ref{lem:core} comes directly from Equation \\ref{equ:core and q_j}, Inequality \\ref{equ:utility_upper_bound} and Lemma \\ref{lem:utility}.\n\nThe following lemma gives an upper bound for $\\textsc{Core}(M,\\beta)$.\n\\begin{lemma}\\label{lem:core}\nIf for all $i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ has a $c$-supporting price $\\{\\theta_j^{S}(t_i)\\}_{j=1}^m$, then\n\\[\\text{CORE }\\leq \\left(\\frac{24}{1-b}+\\frac{4}{b(1-b)}+16.5\\right)\\cdot c\\cdot \\textsc{APostEnRev}\\]\n\\end{lemma}\n}\n\n\\subsection{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n  Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n  When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\\section{Missing Proofs for the Multi-Bidder Case}\\label{appx:multi}\n\n\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(M,\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+$\\\\\n\\noindent$\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(M,\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(M,\\beta))\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{Lemma}{lem:hat gamma}\n\tSince for every set $S\\subseteq[m]$ and every subset $S'$ of $S$,  $v_i(t_i, S')\\geq \\sum_{j\\in S'}\\theta_j^S(t_i)$. Therefore,\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\tfor all $j\\in S$, we have $\\min\\{t_{ij}^{(k)}, Q_j+\\tau_i\\}\\geq \\hat{\\gamma}^S_j(t_i)$. Therefore, $\\hat{v}_i(t_i,S)\\geq \\sum_{j\\in S}\\hat{\\gamma}^S_j(t_i)$.\n\t\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{k\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{k\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\\end{prevproof}\n\n\\section{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n  Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n  When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\n\\section{Analysis for the Single-Bidder Case}\\label{sec:single_appx}\n\n\n\\begin{comment}\n\\begin{prevproof}[lemma]\\ref{lem:single-single}\nRecall that $\\textsc{Single}(M)=\\sum_{t\\in T}f(t)\\cdot$ \\\\\n\\noindent$\\sum_{j\\in[m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\pi^{(\\beta)}_{j}(t)\\cdot {\\tilde{\\varphi}}_{j}(V(t_{j}))$.\n\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to the buyer and $t\\in R_j^{(\\beta)}$, $M'$ serves the agent $j$. $M'$ is feasible in the copies setting as there is at most one agent being served, and $\\textsc{Single}(M)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M)$.\n\nAs shown in~\\cite{ChawlaHMS10}, when there is a single buyer, a simple posted-price mechanism with the constraint that the buyer can only purchase one item achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/2$ in the original setting. Therefore, by the definition of $\\textsc{SRev}$ we have $2\\textsc{SRev}\\geq\\textsc{OPT}^{\\textsc{Copies-UD}}$.\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{lemma}{lem:single decomposition}\n{Recall that for all $t\\in T$ and $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$.} We replace every $v(t,S)$ in $\\textsc{Non-Favorite}(M)$ with $v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. Also since $R^{\\beta}_0=\\emptyset$, the corresponding term is simply $0$. First, the contribution from $v\\left(t,S\\cap \\mathcal{C}(t)\\right)$ is upper bounded by the \\textsc{Core}(M).\n\n\\begin{align*}\n& \\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,(S\\backslash\\{j\\})\\cap \\mathcal{C}(t)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,S\\cap \\mathcal{C}(t)\\right)\\right)\\\\\n\\leq& \\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\\quad(\\textsc{Core}(M))\n\\end{align*}\n\nThe inequality comes from the monotonicity of $v(t,\\cdot)$.\n\nNext, we upper bound the contribution from $\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$ by the $\\textsc{Tail}(M)$.\n\\begin{align*}\n&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}(t)}V(t_k)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}(t)}V(t_k)\\right) \\\\\n=&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_j)\\cdot \\mathds{1}\\left[t\\not\\in R_j^{(\\beta)}\\right]\\cdot \\pi_j^{(\\beta)}(t)~~~~~~~\\text{{(Recall $\\pi_{j}^{(\\beta)}(t)=\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)$)}}\\\\\n\\leq &\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot\n{\\mathds{1}\\left[t \\not\\in R_j^{(\\beta)}\\right]}~~~\\text{($\\pi_j^{(\\beta)}(t)\\leq 1$)}\\\\\n\\leq &{\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot \\mathds{1}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~\\text{(Definition of $R_j^{(\\beta)}$)}}\\\\\n=&{\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]\\quad(\\textsc{Tail}(M))}\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsubsection{Analyzing $\\textsc{Tail}(M)$~in the Single-Bidder Case}\n\\begin{prevproof}[lemma]{lem:single-tail}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$, \\\\\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{prevproof}\n\\end{comment}\n\n\\begin{prevproof}{Lemma}{lem:single subadditive}\nWe argue the three properties one by one.\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\emph{Monotonicity:} For all $t\\in T$ and $U\\subseteq V\\subseteq [m]$, $U\\cap \\mathcal{C}(t)\\subseteq V\\cap \\mathcal{C}(t)$. Since $v(t,\\cdot)$ is monotone,\n$$v'(t,U)=v\\left(t,U\\cap \\mathcal{C}(t)\\right)\\leq v\\left(t,V\\cap \\mathcal{C}(t)\\right)=v'(t,V).$$ Thus, $v'(t,\\cdot)$ is monotone.\n\\item \\emph{Subadditivity:} For all $t\\in T$ and $U,V\\subseteq [m]$, notice $(U\\cup V)\\cap \\mathcal{C}(t)=\\left(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)$, we have\n$$v'(t,U\\cup V)=v\\left(\\left(t,(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)\\right)\\leq v\\left(t,U\\cap\\mathcal{C}(t)\\right)+v\\left(t,V\\cap\\mathcal{C}(t)\\right)=v'(t,U)+v'(t,V).$$\n\\item \\emph{No externalities:} For any $t\\in T$, $S\\subseteq [m]$, and any $t'\\in T$ such that $t_{j}=t_{j}'$ for all $j\\in S$, to prove $v'(t,S)=v'(t',S)$, it is enough to show $S\\cap \\mathcal{C}(t)=S\\cap \\mathcal{C}(t')$. Since $V(t_j)=V(t_j')$ for any $j\\in S$, $j\\in S\\cap \\mathcal{C}(t)$ if and only if $j\\in S\\cap \\mathcal{C}(t')$.\n\\end{itemize}\n\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:single Lipschitz}\nFor any $t,t'\\in T$, and set $X,Y\\subseteq [m]$, define set $H=\\left\\{j\\in X\\cap Y:t_j=t_j'\\right\\}$. Since $v'(\\cdot,\\cdot)$ has no externalities, $v'(t',H)=v'(t,H)$. Therefore,\n\\begin{align*}\n|v'(t,X)-v'(t',Y)|&=\\max\\left\\{v'(t,X)-v'(t',Y),v'(t',Y)-v'(t,X)\\right\\}\\\\\n&\\leq \\max\\left\\{v'(t,X)-v'(t',H),v'(t',Y)-v'(t,H)\\right\\}\\quad\\text{(Monotonicity)}\\\\\n&\\leq \\max\\left\\{v'(t,X\\backslash H),v'(t',Y\\backslash H)\\right\\}\\quad\\text{(Subadditivity)}\\\\\n& = \\max\\left\\{v\\left(t,(X\\backslash H)\\cap \\mathcal{C}(t)\\right),v\\left(t',(Y\\backslash H)\\cap\\mathcal{C}(t)\\right)\\right\\}\\quad\\text{(Definition of $v'(\\cdot,\\cdot)$)}\\\\\n&\\leq c\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\\\\n&\\leq c\\cdot (|X\\Delta Y|+|X\\cap Y|-|H|)\n\\end{align*}\nThe second last inequality is because both $v(t,\\cdot)$ and $v(t',\\cdot)$ are subadditive and for any item $j\\in \\mathcal{C}(t)$ ($\\mathcal{C}(t')$) the single-item valuation $V(t_j)$ ($V(t'_j)$) is less than $c$.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\n\tIn a $n$-player $m$-item combinatorial auction, for any absolute constant $\\eta\\in(0,1)$ and $\\epsilon>0$, any two type profile distributions $D, D'$ on type profile set $T$ and $T'$ accordingly($T$ and $T'$ might be different), any two valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$, $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$,  assume for every $i$ there exists a coupling $\\hat{D_i}$ for $D_i$ and $D_i'$ such that $\\forall t_i\\in T_i,t_i'\\in T_i', \\hat{D_i}(t_i,t_i')>0$, $v_i'(t_i',S)\\geq v_i(t_i,S)$ holds for subset $S$. Here $\\hat{D_i}(t_i,t_i')$ is the coupling probability. Then for any BIC mechanism $M$ for valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to $D$, there exists a BIC mechanism $M'$ for valuation functions $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to distribution $D'$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S}\\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M',v', D')}+\\frac{1}{\\eta}\\cdot \\sum_i\\sum_{t_i'\\in T_i'}\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]}\\hat{D_i}(t_i,t_i')\\cdot\\sigma'_{iS}(t_i')\\cdot \\left(v_i'(t_i', S)-v_i(t_i, S)\\right)+\\epsilon$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ is the revenue of the mechanism $M$ while the buyers' types are drawn from $D$ and their valuations are $v$ (similarly defined for $\\textsc{Rev}(M', v', D')$). $\\sigma'_{iS}(t_i')$ is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i'$ in mechanism $M'$ with respect to $D'$ and  $\\sigma_{iS}(t_i)$ is the probability for the same event in mechanism $M$ with respect to $D$\n\\end{lemma}\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M'$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} sampled i.i.d. from $D_i'$ and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i'$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the bidder $i$ on the left and the surrogates on the right. The edge weight between a replica (bidder $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i'(r_i,\\cdot)$ to receive bidder $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the expected payment of bidder $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(r_i,S) - (1-\\eta)p_i(s_i).$\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M'$ truthfully, then the distribution of types of the surrogate chosen for buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type and $\\ell-1$ replicas are sampled i.i.d. from the distribution $D_i'$, while $\\ell$ surrogates are sampled i.i.d. from the distribution $D_i$. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M'$ is a BIC mechanism with respect to valuation $v'$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate.  If buyer $i$ has type $t_i'$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is  $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(t_i',S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S} \\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M'$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, this surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for $i$ to win the item in $M'$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M',v', D')$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}\\in T_i$ and $r_i^{k}\\in T_i'$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. \t\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M'$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M'$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, $i$ pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M'$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider a different type of matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et. al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from  $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles.  If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or (b)$\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{r_i^{\\theta(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P$ and $V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P$ and $X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$,  we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i'(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means  whenever mechanism $M'$ awards buyer $i$ (with type $t_i$)  bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i'(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i E_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq  &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M',v',D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\cal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon\/(1-\\eta)$. Therefore, we finished the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\n\n\n\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\\label{sec:proof_relaxed_valuation}\nWe first prove some properties of $v^{(\\beta)}$, which will be useful for proving Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed larger}\n\tFor any $\\beta_i$, $t_i\\in T_i$ and $S\\in[m]$, $v_i^{(\\beta_i)}(t_i,S)\\geq v_i(t_i,S)$.\n\\end{lemma}\n\\begin{proof}\n\tThis follows from the fact that $v_i(t_i,\\cdot)$ is a subadditive function over bundles of items for all $t_i$.\n\\end{proof}\n\n\\begin{lemma}\n\tFor any $\\beta_i$ and $t_i\\in T_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is a monotone, subadditive function over the items.\n\\end{lemma}\n\\begin{proof}\nMonotonicity follows directly from the monotonicity of $v_i(t_i,\\cdot)$. We only argue subadditivity here. If $t_i$ belongs to $R_0^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t_i,\\cdot)=v_i(t_i,\\cdot)$. So it is clearly a subadditive function. If $t_i$ belongs to $R_j^{(\\beta_i)}$ for some $j>0$ and $j$ is not in either $U$ or $V$, then clearly $v_i^{(\\beta_i)}(t_i,U\\cup V)\\leq v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)$. If $j$ is in one of the two sets, without loss of generality let's assume it is in $U$. Then $v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)=v_i(t_i,U\\backslash\\{j\\})+V_i(t_{ij})+v_i(t_i,V)\\geq v_i(t_i,V\\cup (U\\backslash\\{j\\}))+V_i(t_{ij})= v_i^{(\\beta_i)}(t_i,U\\cup V)$.\n\\end{proof}\n\n\nHere we prove a stronger version of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\nFor any $\\beta$, any absolute constant $\\eta\\in(0,1)$ and any BIC mechanism $M$ for subadditive valuations $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+\\frac{1}{\\eta}\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$).\n\\end{lemma}\n\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M^{(\\beta)}$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the buyer $i$ on the left and the surrogates on the right. The edge weight between a replica (or buyer $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i^{(\\beta_i)}(r_i,\\cdot)$ to receive buyer $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the interim payment of buyer $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(r_i,S) - (1-\\eta)p_i(s_i)$, where $p_i(s_i)$ is the interim payment for buyer $i$ if she reported $s_i$.\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M^{(\\beta)}$ truthfully, then the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type is sampled from the distribution, then we sampled $\\ell-1$ replicas and $\\ell$ surrogates i.i.d. from the same distribution. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M^{(\\beta)}$ is a BIC mechanism with respect to valuation $v^{(\\beta)}$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate.  If buyer $i$ has type $t_i$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is  $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(t_i,S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful. It is critical that the other buyers are reporting truthfully, otherwise we cannot invoke Lemma~\\ref{lem:same distribution} and buyer $i$'s value for winning a surrogate with type $s_i$ may be different from the weight on the corresponding edge.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M^{(\\beta)}$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, the chosen surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for  buyer $i$ to win the item in $M^{(\\beta)}$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}$ and $r_i^{k}$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. We will slightly abuse notation by using $s_i^k$ (or $r_i^j$) $\\in V(\\bold{s_i},\\bold{r_i})$ to denote that $s_i^k$  (or $r_i^j$) is matched in the VCG matching $V(\\bold{s_i},\\bold{r_i})$.\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M^{(\\beta)}$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M^{(\\beta)}$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, she pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M^{(\\beta)}$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider another matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from  $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles.  If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}[Adapted from~\\cite{DaskalakisW12}]\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is matched in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or\\\\ (b) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ in $V(\\bold{s_i},\\bold{r_i})$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P \\cap V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P\\cap X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\nFor any $\\beta$,\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$,  we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M^{(\\beta)}$ awards buyer $i$ (with type $t_i$)  bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq  &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\mathcal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon$ for any $\\epsilon$. Let $P^{(\\beta)}$ be the set of all BIC mechanisms that satisfy the first condition in Lemma~\\ref{lem:relaxed valuation stronger}. Clearly, $P^{(\\beta)}$ is a compact set and contains all $M^{(\\beta)}$ we constructed (by choosing different values for $\\ell$). Notice that both $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and $\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ are linear functions over the allocation\/price rules of mechanism $M^{(\\beta)}$. Therefore, \\begin{align*}\n \t&\\textsc{Rev}(M, v, D)\\\\\n \t\\leq &\\max_{M^{(\\beta)}\\in P^{(\\beta)}} \\left(\\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).\n \\end{align*}\nThis completes the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\\section{Paper Organization}\\label{sec:roadmap}\nIn this section, we provide the roadmap to our paper. In Section~\\ref{sec:duality}, we review the Duality framework of~\\cite{CaiDW16}.\n\nIn Section~\\ref{sec:flow}, we derive an upper bound of the optimal revenue for subadditive bidders by combining the duality framework with our new techniques, i.e. valuation relaxation and adaptive dual variables. Our main result in this section,  Theorem~\\ref{thm:revenue upperbound for subadditive}, shows that the revenue can be upper bounded by two terms -- $\\textsc{Non-Favorite}$ and $\\textsc{Single}$ defined in Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n\nIn Section~\\ref{sec:single}, we use the single bidder case to familiarize the readers with some basic ideas and techniques used to bound $\\textsc{Single}$ and $\\textsc{Non-Favorite}$. The main result of this section, Theorem~\\ref{thm:single}, shows that the optimal revenue for a single subadditive bidder is upper bounded by $24\\textsc{SRev}$ and $16\\textsc{BRev}$.\n\nSection~\\ref{sec:multi} contains the main result of this paper. We show how to upper bound the optimal revenue for XOS (or subadditive) bidders with a constant number of (or $O(\\log m)$) $\\textsc{PostRev}$ (the optimal revenue obtainable by an RSPM) and $\\textsc{APostEnRev}$ ((the optimal revenue obtainable by an ASPE). In particular, $\\textsc{Single}$ can be upper bounded by the optimal revenue  $\\textsc{OPT}^{\\textsc{Copies-UD}}$ in the copies setting which is again upper bounded by $6\\textsc{PostRev}$. We further decompose $\\textsc{Non-Favorite}$ into two terms $\\textsc{Tail}$ and $\\textsc{Core}$, and show how to bound $\\textsc{Tail}$ in Section~\\ref{subsection:tail} and how to bound $\\textsc{Core}$ in Section~\\ref{subsection:core}.\n\n\n\n\\section{Warm Up: Single Bidder}\\label{sec:single}\nTo warm up, we first study the case where there is a single subadditive buyer and show how to improve the approximation ratio from $338$ to $40$. Since there is only one buyer, we will drop the subscript $i$ in the notations. As specified in Section~\\ref{sec:choice of beta}, we use a $\\beta$ that satisfies both properties in Lemma~\\ref{lem:requirement for beta}. For a single buyer, we can simply set $\\beta_{j}$ to be $0$ for all $j$. We use $\\textsc{Single}(M), \\textsc{Non-Favorite}(M)$ in the following proof to denote the corresponding terms in Theorem~\\ref{thm:revenue upperbound for subadditive} for $\\beta=\\textbf{0}$. Notice $R_0^{(\\textbf{0})}=\\emptyset$. Theorem~\\ref{thm:single} shows that the optimal revenue is within a constant factor of the better of selling separately and grand bundling.\n\n\\begin{theorem}\\label{thm:single}\nFor a single buyer whose valuation distribution is subadditive over independent items, \n\\[\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}\\]\nfor any BIC mechanism $M$.\n\\end{theorem}\n\nRecall that the revenue for mechanism $M$ is upper bounded by $4\\cdot \\textsc{Non-Favorite}(M)+2\\cdot\\textsc{Single}(M)$ (Theorem~\\ref{thm:revenue upperbound for subadditive}). We first upper bound $\\textsc{Single}(M)$ by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. Since $\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the original setting, $ \\mathds{1}[t\\in R_j^{(\\beta)}]\\cdot\\pi^{(\\beta)}_{j}(t)$ with $\\pi^{(\\beta)}_j(t)=\\sum_{S:j\\in S}\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the copies setting, and therefore $\\textsc{Single}(M)$ is the Myerson Virtual Welfare of a certain allocation in the copies setting, which is upper bounded by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. By~\\cite{ChawlaHMS10}, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is at most $2\\cdot\\textsc{SRev}$.\n\\begin{lemma}\\label{lem:single-single}\nFor any BIC mechanism $M$, $\\textsc{Single} (M)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2\\cdot\\textsc{SRev}.$\n\\end{lemma}\n\nWe decompose $\\textsc{Non-Favorite}(M)$ into two terms $\\textsc{Core}(M)$ and $\\textsc{Tail}(M)$, and bound them separately. For every $t\\in T$, define $\\mathcal{C}(t)=\\{j:V(t_j)< c\\}$, $\\mathcal{T}(t)=[m]\\backslash \\mathcal{C}(t)$. Here the threshold $c$ is chosen as\n\\begin{equation}\\label{equ:single-def of c}\nc:=\\inf\\left\\{x\\geq 0:\\  \\sum_j \\Pr_{t_j}\\left[V(t_j)\\geq x\\right]\\leq 2\\right\\}.\n\\end{equation}\nSince $v(t,\\cdot)$ is subadditive for all $t\\in T$ , we have for every $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. {We decompose $\\textsc{Non-Favorite}(M)$ based on the inequality above.} Proof of Lemma~\\ref{lem:single decomposition} can be found in Appendix~\\ref{sec:single_appx}.\n\n\n\n\n\\begin{lemma}\\label{lem:single decomposition}\n\\begin{align*}\t\\textsc{Non-Favorite}(M)\n\\leq &\t\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))~~~~~~~~~\\quad(\\textsc{Core}(M))\\\\\n+&\\sum_j\\sum_{t_{j}:V(t_{j})\\geq c}f_{j}(t_{j})\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~~\\quad(\\textsc{Tail}(M))\n\\end{align*}\n\\end{lemma}\n\n\nUsing the definition of $c$ and $\\textsc{SRev}$, we can upper bound $\\textsc{Tail}(M)$ with a similar argument as in~\\cite{CaiDW16}.  \n\\begin{lemma}\\label{lem:single-tail}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M)\\leq 2\\cdot\\textsc{SRev}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$,\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{proof}\n\n\nThe $\\textsc{Core}(M)$ is upper bounded by ${\\mathbb{E}}_{t}[v'(t,[m])]$ where $v'(t,S)$\n$= v(t,S\\cap \\mathcal{C}(t))$. We argue that $v'(t,\\cdot)$ is drawn from a distribution that is subadditive over independent items and $v'(\\cdot,\\cdot)$ is $c$-Lipschitz (see Definition~\\ref{def:Lipschitz}). Using a concentration bound by Schechtman~\\cite{Schechtman2003concentration}, we show ${\\mathbb{E}}_{t}[v'(t,[m])]$ is upper bounded by the median of random variable $v'(t,[m])$ and $c$, which are upper bounded by $\\textsc{BRev}$ and $\\textsc{SRev}$ respectively.\n\\begin{lemma}\\label{lem:single-core}\nFor any BIC mechanism $M$, $\\textsc{Core}(M) \\leq 3\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev}$.\n\\end{lemma}\n\nRecall that\n\\begin{equation}\n\\textsc{Core}(M)=\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq [m]}\\sigma_S^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\n\\end{equation}\n\nWe will bound $\\textsc{Core}(M)$ with a concentration inequality from~\\cite{Schechtman2003concentration}. It requires the following definition:\n\n\\begin{definition}\\label{def:Lipschitz}\nA function $v(\\cdot,\\cdot)$ is \\textbf{$a$-Lipschitz} if for any type $t,t'\\in T$, and set $X,Y\\subseteq [m]$,\n$$\\left|v(t,X)-v(t',Y)\\right|\\leq a\\cdot \\left(\\left|X\\Delta Y\\right|+\\left|\\{j\\in X\\cap Y:t_j\\not=t_j'\\}\\right|\\right),$$ where $X\\Delta Y=\\left(X\\backslash Y\\right)\\cup \\left(Y\\backslash X\\right)$ is the symmetric difference between $X$ and $Y$.\n\\end{definition}\n\nDefine a new valuation function for the bidder as $v'(t,S)=v(t,S\\cap \\mathcal{C}(t))$, for all $t\\in T$ and $S\\subseteq [m]$. Then $v'(\\cdot,\\cdot)$ is $c-$ Lipschitz, and when $t$ is drawn from the product distribution $D=\\prod_j D_j$, $v'(t,\\cdot)$ remains to be a valuation drawn from a distribution that is subadditive over independent items. See Appendix~\\ref{sec:single_appx} for the proof of Lemma~\\ref{lem:single subadditive} and Lemma~\\ref{lem:single Lipschitz}.\n\n\\begin{lemma}\\label{lem:single subadditive}\nFor all $t\\in T$, $v'(t,\\cdot)$ satisfies monotonicity, subadditivity and no externalities defined in Definition~\\ref{def:subadditive independent}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:single Lipschitz}\n$v'(\\cdot,\\cdot)$ is $c-$Lipschitz.\n\\end{lemma}\n\nNext, we apply the following concentration inequality to derive Corollary~\\ref{corollary:concentrate}, which is useful to analyze the $\\textsc{Core}(M)$. \n\n\\begin{lemma}~\\cite{Schechtman2003concentration}\\label{lem:schechtman}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is  subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then for all $a>0, k\\in \\{1,2,...,|I|\\}, q\\in \\mathbb{N}$,\n$$\\Pr_t[g(t,I)\\geq (q+1)a+k\\cdot c]\\leq \\Pr_t[g(t,I)\\leq a]^{-q}q^{-k}.$$\n\\end{lemma}\n\n\\begin{corollary}\\label{corollary:concentrate}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is  subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then if we let $a$ be the median of the value of the grand bundle $g(t,I)$, i.e. $a=\\inf\\left\\{x\\geq 0: \\Pr_t[g(t,I)\\leq x]\\geq \\frac{1}{2}\\right\\}$,\n$$\\mathds{E}_t[g(t,I)]\\leq 2a+\\frac{5c}{2}.$$\n\\end{corollary}\n\n\\begin{proof}\nLet $Y$ be $g(t,I)$. If we apply Lemma~\\ref{lem:schechtman} to the case where $a$ is the median and $q=2$, we have\n\n\\begin{align*}\n\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]&= 3a\\cdot \\Pr_t[Y\\geq 3a]+\\int_{y=0}^{\\infty}\\Pr_t[Y\\geq 3a+y]dy\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot\\sum_{k=0}^{|I|} \\Pr_t[Y\\geq 3a+k\\cdot c] \\quad(Y\\leq |I|\\cdot c)\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot \\sum_{k=0}^2 \\Pr_t[Y > a]+ c\\cdot\\sum_{k=3}^{|I|} 4\\cdot 2^{-k}\\quad(\\text{Lemma~\\ref{lem:schechtman}})\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n\\end{align*}\n\nWith the inequality above, we can upper bound the expected value of $Y$.\n\\begin{align*}\n{\\mathbb{E}}_{t}[Y]&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]\\\\\n&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n&= a+2a\\cdot \\Pr_{t}[Y>a]+\\frac{5}{2}c\\\\\n&\\leq 2a+\\frac{5}{2}c\n\\end{align*}\n\\end{proof}\n\n\nNow, we are ready to prove Lemma~\\ref{lem:single-core}.\n\n\\begin{prevproof}{Lemma}{lem:single-core}\nLet $\\delta$ be the median of $v'(t,[m])$ when $t$ is sampled from distribution $D$. Now consider the mechanism that sells the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $v(t,[m])\\geq v'(t,[m])$. Thus with probability at least $\\frac{1}{2}$,\n  the bidder purchases the bundle. Thus, $\\textsc{BRev}\\geq \\frac{1}{2}\\delta$.\n\nAccording to Corollary~\\ref{corollary:concentrate},\n\n\\begin{comment}\n\\begin{equation}\\label{equ:single-core}\n\\begin{aligned}\n\\textsc{Core}(M)&\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\\\\\n&< 4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev} \\text{(Lemma~\\ref{lem:single-bound for c}, Inequality~\\ref{equ:bound for delta})}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\\begin{equation}\\label{equ:single-core-prev}\n\\textsc{Core}(M)\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\n\\end{equation}\n\nIt remains to argue that the Lipchitz constant $c$ can be upper bounded using $\\textsc{SRev}$. Notice that by AM-GM Inequality,\n\\begin{align*}\n&\\Pr_t\\left[\\exists j\\in [m], V(t_j)\\geq c\\right]= 1-\\prod_{j}\\Pr_{t_j}[V(t_j)< c]\\\\\n\\geq& 1-(\\frac{\\sum_j \\Pr_{t_j}[V(t_j)< c]}{m})^m\n= 1-(1-\\frac{2}{m})^m\n\\geq 1-e^{-2}\n\\end{align*}\n\n\n\nConsider the mechanism that posts price $c$ for each item but only allow the buyer to purchase one item. Then with probability at least $1-e^{-2}$, the mechanism sells one item obtaining expected revenue $(1-e^{-2})\\cdot c$. Thus $c\\leq \\frac{1}{1-e^{-2}}\\cdot\\textsc{SRev}$. Inequality~\\eqref{equ:single-core-prev} becomes\n\n\\begin{equation}\\label{equ:single-core}\n\\textsc{Core}(M)\\leq 2\\delta+\\frac{5c}{2}<4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev}\n\\end{equation}\n\n\\end{prevproof}\n\n\\begin{prevproof}{Theorem}{thm:single}\nSince $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2 \\textsc{SRev}$ (Lemma~\\ref{lem:single-single}) and $\\textsc{Non-Favorite}(M)\\leq 5\\textsc{SRev}+4\\textsc{BRev}$ (Lemma~\\ref{lem:single-tail} and~\\ref{lem:single-core}), $\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}$ according to Theorem~\\ref{thm:revenue upperbound for subadditive}.\n\\end{prevproof}\n\n\n\n\n\n\n\n\n\n\\section{Sequentially Posted-Price Mechanism with Entry Fee}\\label{sec:spm}\nHere is the formal specification of the Sequential Posted Price with Entry Fee Mechanism.\\\\\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n    \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n        \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n        \\STATE $S\\gets S\\backslash S_i^{*}$.\n    \\ELSE\n        \\STATE $i$ gets nothing and pays $0$.\n    \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n\n\\section*{\\homeworkProblemName\n   \\enterProblemHeader{\\homeworkProblemName}\n  {\\exitProblemHeader{\\homeworkProblemName}\n\n\\newcommand{\\homeworkSectionName}{\n\\newlength{\\homeworkSectionLabelLength}{\n\\newenvironment{homeworkSection}[1\n  \n\n   \\renewcommand{\\homeworkSectionName}{#1\n   \\settowidth{\\homeworkSectionLabelLength}{\\homeworkSectionName\n   \\addtolength{\\homeworkSectionLabelLength}{0.25in\n   \\changetext{}{-\\homeworkSectionLabelLength}{}{}{\n   \\subsection*{\\homeworkSectionName\n   \\enterProblemHeader{\\homeworkProblemName\\ [\\homeworkSectionName]}\n  {\\enterProblemHeader{\\homeworkProblemName\n\n  \n  \n   \\changetext{}{+\\homeworkSectionLabelLength}{}{}{}\n\n\\newcommand{\\Answer}{\\ \\\\\\textbf{Answer:} }\n\\newcommand{\\Intui}{\\ \\\\\\textbf{Intuition:} }\n\\newcommand{\\Proof}{\\ \\\\\\textbf{Proof:} }\n\\newcommand{\\Acknowledgement}[1]{\\ \\\\{\\bf Acknowledgement:} #1}\n\n\n\n\n\n\\title{\\textmd{\\bf \\Title}\n\\author{\\textbf{\\StudentName}}\\\\{\\large Instructed by \\textit{\\ClassInstructor}}\\\\\\normalsize\\vspace{0.1in}\\small{\\DueDate}}\n\\date{}\n\n\\begin{document}\n\\begin{spacing}{1.1}\n\\maketitle \\thispagestyle{empty}\n\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{definition}{Definition}\n\\newtheorem{assignment}{Homework Problem}\n\\newtheorem{notation}{Notation}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{conjecture}{Conjecture}\n\n\\section{Multiple Bidder, Additive With Matroid Constraint}\n\\subsection{Flow}\nFor a feasible $\\pi\\in P(\\mathcal{F},D)$, let M be the mechanism that induces $\\pi(\\cdot)$. For all $i\\in [n], j\\in [m]$, define\n\\begin{equation}\nq_{ij}^{\\pi}=b\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\pi_{ij}(t_i)\n\\end{equation}\nwhere $b\\in (0,1)$ is a constant parameter that we will determine later. Choose $\\beta_{ij}^{\\pi}$ such that $Pr[t_{ij}\\geq \\beta_{ij}^{\\pi}]=q_{ij}^{\\pi}$ (Assume all the distributions are continuous so $q_{ij}^{\\pi}$ exists).\n\nNotice that we have the following for all $j$ since $\\pi$ is feasible, i.e., without over-allocation:\n\\begin{equation}\n\\sum_i q_{ij}=b\\cdot\\sum_i \\sum_{t\\in T}f(t)\\pi_{ij}(t_i)=b\\cdot\\sum_{t\\in T}f(t)\\sum_i\\pi_{ij}(t_i)\\leq b\n\\end{equation}\n\n~\\\\\n\nLet $R_0^{\\pi}=\\{t_i\\in T_i|t_{ik}<\\beta_{ik},\\forall k\\in [m]\\}$, and $R_j^{\\pi}=\\{t_i\\in T_i|(\\forall k\\not=j,t_{ij}-\\beta_{ij}^{\\pi}\\geq t_{ik}-\\beta_{ik}^{\\pi})\\cap (t_{ij}\\geq \\beta_{ij}^{\\pi})\\}$.\n\nWe will define the same flow as before: $\\lambda^{\\pi}(t_i',t_i)>0$ if and only if\n\\begin{itemize}\n\\item $t_i\\in R_j^{\\pi}$\n\\item $t_{ik}'=t_{ik},\\forall k\\not=j$\n\\item $t_{ij}'>t_{ij}$\n\\end{itemize}\n\nThen after ironing, the Language function becomes:\n\\begin{equation}\nL(\\lambda^{\\pi},\\pi,p)=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\bigg(t_{ij}\\cdot \\mathds{1}\\big[t_i\\not\\in R_j^{\\pi}\\big]+\\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big]\\bigg)\n\\end{equation}\n\nBy the Strong Duality Theorem,\n\\begin{equation}\nREV=\\min_{\\lambda}\\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda,\\pi,p)\\leq \\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda^{\\pi},\\pi,p)\n\\end{equation}\n\nIt's enough to bound $L(\\lambda^{\\pi},\\pi,p)$ for all feasible $\\pi$ to obtain an upper bound for $REV$.\n\n\\subsection{Sequential Mechanism and c-Selectability}\nWe will use a new Sequential mechanism. The mechanism posts a price $\\theta_{ij}$ of each item $j$ for each bidder $i$ and determine an order $\\sigma$ for bidders. Each bidder comes in order $\\sigma$. When bidder $i$ with type $t_i$ comes, suppose the set of items left is $S_i$. The mechanism will let bidder $i$ know $S_i$ and charge him an entry fee $\\delta_i$. If he chooses to join the auction, he pays the entry fee and then takes his favorite bundle $S_i^{*}=\\arg\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})$ and pay $\\sum_{j\\in S_i^{*}}\\theta_{ij}$. Let the optimal revenue that such a mechanism can get is SEQ.\n\nWhen bidder $i$ is facing the remaining item set $S_i$, we have no idea on which item he will take. c-Selectability from [\\ref{feldman}] actually take control of it. The following lemma directly comes from the definition of c-Selectability:\n\\begin{lemma}\\label{feldman2015}\n(Feldman 2015) For a downward close $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS, then in the Sequential mechanism, when it's bidder $i$'s turn, as long as $j\\in S_i$ and $t_{ij}>\\theta_{ij}$, item $j$ is in $S_i^{*}$ with at least probability $c$. The probability is taken over the randomness of $S_i$.\n\\end{lemma}\n\nIf $\\mathcal{F}$ is a matroid, the paper shows that $c=1-b$.\n\n\n\\subsection{Separating into Pieces}\n\\begin{equation}\n\\begin{aligned}\n&L(\\lambda^{\\pi},\\pi,p) \\\\\n=&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big] \\text{(SINGLE)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\text{(SURPLUS)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE + SURPLUS }+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\text{(PROPHET)}\n\\end{aligned}\n\\end{equation}\n\n~\\\\\n\n\\begin{equation}\n\\begin{aligned}\nSURPLUS&=\\sum_i\\sum_{t_i: \\exists k,t_{ik}>\\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\text{ (TAIL)}\\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]\\text{ (CORE)}\\\\\n\\end{aligned}\n\\end{equation}\n\nHere, $\\tau_i^{\\pi}$ is chosen such that\n\\begin{equation}\n\\sum_j \\Pr_{t_{ij}}[t_{ij}\\geq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]=\\frac{1}{2}\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding SINGLE}\n\\begin{lemma}\nFor any downward close $\\mathcal{F}$, $\\text{SINGLE}\\leq \\text{OPT}^{\\text{COPIES}}$.\n\\end{lemma}\n\\begin{proof}\nWe build a new mechanism $M'$ in the Copies setting based on $M$. Whenever $M$ allocates item $j$ to bidder $i$ and $t_i\\in R_j^{\\pi}$, $M'$ serves the agent $(i,j)$. Then $M'$ is feasible in the Copies setting and SINGLE is the ironed virtual welfare of $M'$ with respect to $\\tilde{\\phi}(\\cdot)$. Since the maximum revenue in the Copies setting is achieved by the maximum virtual welfare, thus $\\text{OPT}^{\\text{COPIES}}$ is no less than SINGLE.\n\\end{proof}\n\nThe $\\text{OPT}^{\\text{COPIES}}$ can be achieved $\\frac{1}{6}$-approximately by a post-price mechanism.\n\n~\\\\\n\n\\subsection{Bounding PROPHET}\n\\begin{equation}\n\\begin{aligned}\n\\text{PROPHET }&=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi}\\\\\n&=\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}(t_i)\\\\\n&=\\frac{1}{b}\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\\label{prophet}\nFor $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS,\n\\[\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{(1-b)\\cdot c}\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider a Sequential mechanism without entry fee and post price $\\theta_{ij}=\\beta_{ij}^{\\pi}$. Then according to Lemma \\ref{feldman2015},\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>\\beta_{ij}^{\\pi}]\\cdot c\\\\\n&\\geq \\sum_i\\sum_j \\beta_{ij}\\cdot (\\sum_{l=1}^{i-1}q_{lj}^{\\pi})\\cdot q_{ij}^{\\pi}\\cdot c\\\\\n&\\geq (1-b)\\cdot c\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nSpecifically, if $\\mathcal{F}$ is a matroid,\n\\begin{equation}\n\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{\\cdot (1-b)^2}\\cdot SEQ\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding TAIL}\nWe will bound TAIL for any downward close $\\mathcal{F}$. Let $P_{ij}=\\arg\\max_{x\\geq \\tau_i^{\\pi}}x\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq x]$, and $r_{ij}=P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]$, $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. We have the following relationship between $r_i$ and $\\tau_i^{\\pi}$:\n\\begin{lemma}\\label{tail0}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot \\tau_i^{\\pi}$.\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n\\begin{aligned}\nr_i&= \\sum_j P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]\\\\\n&\\geq \\sum_j \\tau_i^{\\pi}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq \\tau_i^{\\pi}]\\\\\n&=\\frac{1}{2}\\cdot \\tau_i^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\nWe notice by the definition of $r_{ij}$,\n\\begin{equation}\\label{tail1}\n\\begin{aligned}\n\\text{TAIL }&=\\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\sum_{k\\not=j}\\Pr_{t_{ik}\\sim T_{ik}}\\big[t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}\\\\\n&\\leq \\sum_i r_i\\cdot \\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\\\\n&=\\frac{1}{2}\\cdot r\n\\end{aligned}\n\\end{equation}\n\nThe following lemma shows that $r$ can be approximately achieved by the Sequential mechanism.\n\\begin{lemma}\\label{tail2}\n\\[r\\leq \\frac{2}{(1-b)}\\text{ SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the Sequential mechanism with item price $\\theta_{ij}=P_{ij}+\\beta_{ij}^{\\pi}$, $\\sigma=(1,2,...,n)$ and without entry fee. When bidder $i$ comes, he will definitely take only item $j$ and pay $P_{ij}+\\beta_{ij}^{\\pi}$ if:\n\\begin{itemize}\n\\item $j\\in S_i$\n\\item $t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}$\n\\item $\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}$\n\\end{itemize}\n\nNotice that due to the second condition, every bidder will take item $j$ with at most $q_{ij}$ probability. Thus we have\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j (P_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}]\\\\\n&\\geq \\sum_i\\sum_j P_{ij}\\cdot \\big(\\sum_{l=1}^{i-1}q_{lj}^{\\pi}\\big)\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]\\\\\n&\\geq (1-b)\\cdot \\sum_i\\sum_j P_{ij}\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\big(1-\\sum_{k\\not=j}\\Pr_{t_{ik}}[t_{ik}\\leq P_{ik}+\\tau_i^{\\pi}]\\big)\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot \\sum_i\\sum_j r_{ij}\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot r\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nCombining Equation (\\ref{tail1}) and Lemma \\ref{tail2}, we have\n\\begin{equation}\nTAIL\\leq \\frac{1}{1-b}\\cdot SEQ\n\\end{equation}\n\n\\subsection{Bounding CORE}\nWe will bound CORE for matroid $\\mathcal{F}$. Define $t_{ij}'=(t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[\\beta_{ij}^{\\pi}\\leq t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}\\big]\\in [0,\\tau_i^{\\pi}]$. Then since $\\pi(\\cdot)$ is feasible,\n\\begin{equation}\nCORE=\\sum_i\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\n\\end{equation}\n\nConsider the Sequential mechanism with item price $\\theta_{ij}=\\beta_{ij}^{\\pi}$ and order $\\sigma=(1,2,...,n)$. When it's bidder $i$'s turn, suppose the set of items left is $S_i$. Define the entry fee $\\delta_i$ for bidder $i$ as:\n\\[\\Pr_{t_i'}[\\max_{S \\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\leq \\delta_i]=\\frac{2}{3}\\]\n\nNotice that if bidder $i$ enters the auction, the profit he gets is\n\\[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})\\geq \\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\]\nThus with probability at least $\\frac{1}{3}$, bidder $i$ will pay the entry fee $\\delta_i$. Besides, with the same argument in Lemma \\ref{prophet},\n\\begin{equation}\n\\text{SEQ}\\geq \\sum_i\\frac{1}{3}\\big(\\delta_i+(1-b)^2\\cdot \\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\big)\n\\end{equation}\n\n~\\\\\n\nThe following result from [Schechtman 1999] can be applied to bound $\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]$ for all $i$:\n\n\\begin{lemma}\\label{core2}\n(Schechtman, 1999)Let $f(t,S)$ be a value function that is additive under a downward close constaint, drawn from a distribution D over support $[0,\\tau]$ for some $\\tau\\geq 0$. Let $\\Delta$ be a value such that $\\Pr_{t\\sim D}\\big[f(t,[m])\\leq \\Delta\\big]=\\frac{2}{3}$. Then for all $k>0$,\n\\begin{equation}\n\\Pr_{t\\sim D}\\big[f(t,[m])\\geq 3\\Delta+k\\cdot \\tau\\big]\\leq \\frac{9}{4}\\cdot 2^{-k}\n\\end{equation}\n\\end{lemma}\n\n\\begin{corollary}\\label{core3}\nFor each $i$,\n\\[\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\\]\n\\end{corollary}\n\\begin{proof}\nLet $g(t_i')=\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Directly from Lemma \\ref{core2},\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{t_i'}\\big[g(t')\\big]&= \\mathbf{E}_{t_i'}\\big[g(t_i');g(t_i')<3\\delta_i\\big]+ \\mathbf{E}_{t_i'}\\big[g(t');g(t_i')\\geq 3\\delta_i\\big]\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}Pr[g(t_i')>3\\delta_i+y]dy\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}\\frac{9}{4}\\cdot 2^{-y\/\\tau}dy\\\\\n&\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nNow we can finish bounding the CORE with the following lemma.\n\n\\begin{lemma}\n\\[\\text{CORE }+3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the mechanism above. Assume $S^{*}=\\arg\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Notice for all $i,j$, $\\Pr_{t_{-i}}[j\\in S_i]\\geq 1-\\sum_i q_{ij}=1-b$. With Lemma \\ref{tail0}, \\ref{tail2}, and \\ref{core2}, we have\n\\begin{equation}\n\\begin{aligned}\nCORE&=\\sum_i\\mathbf{E}_{t_i'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\cdot \\mathds{1}[j\\in S_i]\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i (3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4ln(2)})\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\bigg(9\\cdot \\text{SEQ }-3(1-b)^2\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+\\frac{9r}{2\\cdot ln(2)}\\bigg)\\\\\n&\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}-3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\n\\subsection{Optimizing the Constant}\nPut all pieces together:\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda^{\\pi},\\pi,p)&\\leq \\text{SINGLE}+\\text{TAIL}+\\text{PROPHET}+\\text{CORE} \\\\\n&\\leq 6\\cdot \\text{SREV}+\\frac{1}{1-b}\\cdot \\text{SEQ}+\\big(\\frac{1}{b}-3(1-b)\\big)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+(\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\\\\n&\\leq 6\\cdot \\text{SREV}+\\bigg(\\frac{7}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2}+\\frac{1}{b(1-b)^2}\\bigg)\\cdot \\text{SEQ}\n\\end{aligned}\n\\end{equation}\n\nWhen $b=\\frac{1}{7}$. The larger one of SREV and SEQ gets a $1\/42$-approximately mechanism.\n\n~\\\\\n\n\n\\section{Single Bidder, Subadditive Valuation}\n\\subsection{Problem Setting\\textsuperscript{[\\ref{subadditive}]}}\nLet $m$ be the number of items, $I=[m]$ be the set of items. For each item $j$, let $\\Omega_j$ be a compact space. Each type $t_j\\in \\Omega_j$ represent the information for item $j$. For all $S\\subseteq I$, denote $\\Omega_S=\\times_{j\\in S}\\Omega_j$. $\\Omega=\\Omega_{I}$. The bidder's type $t=\\langle t_j \\rangle_{j\\in I}$ is drawn from a distribution $D$ on $\\Omega$. After $t$ is drawn, the bidder's valuation for a set of items S is only determined by the vector $\\langle t_j \\rangle_{j\\in S}$. Formally, denote $\\Omega^{*}=\\bigcup_{S\\subseteq I}\\Omega_S$. There exists a valuation function $V:\\Omega^{*}\\to \\mathcal{R}$ such that $\\forall t\\in \\Omega, S\\subseteq I$,\n\\begin{equation}\nv(t,S)=V(\\langle t_j \\rangle_{j\\in S},S)\n\\end{equation}\n\nIn this problem, we assume that the bidder's valuation is subadditive, i.e., $\\forall t\\in \\Omega$, $\\forall P,Q\\subseteq I$,\n\\begin{equation}\nv(t,P\\cup Q)\\leq v(t,P)+v(t,Q)\n\\end{equation}\n\n~\\\\\n\n\\subsection{Duality}\nWe use the following functions(variables) to describe the mechanism:\n\\begin{itemize}\n\\item $p(t)$, $t\\in \\Omega^{+}$: the prize that the bidder should pay when his type is $t$.\n\\item $\\phi(t,t')$, $t\\in \\Omega, t'\\in \\Omega^{+}$: the expect valuation if the bidder pretends to be $t'$ when his type is $t$.\n\\end{itemize}\n\nwhere $\\Omega^{+}=\\Omega\\cup \\{\\emptyset\\}$ which allows the bidder not to participate in the auction. When $t=\\emptyset$, $p(t)=\\phi(\\cdot,t)=0$.\n\n\\textbf{Remark:} For a mechanism, let $\\pi_S(t)$ be the probability for bidder with type $t$ to obtain a set $S$ of item. Then $\\phi(t,t')$ can be written as\n\\begin{equation}\n\\phi(t,t')=\\sum_{S\\subseteq I}\\pi_S(t)v_{t'}(S)\n\\end{equation}\n\nHere we use $\\phi(t,t')$ to replace the original variables $\\pi_j(t)=\\sum_{S:j\\in S}\\pi_S(t)$. The function in fact includes both the probability and the valuation. Like $\\pi$, all the $\\phi(t,t')$'s should stay in some feasible region to avoid over-allocation. We use $\\phi\\in \\mathcal{O}$ to represent it.\n\n~\\\\\n\nOur primal is:\n\\begin{itemize}\n\\item \\textbf{Variables}: $p(t),\\phi(t,t'),\\quad t\\in \\Omega, t'\\in \\Omega^{+}$\n\\item \\textbf{Constraint}:\n\n$\\quad(1)\\phi(t,t)-p(t)\\geq \\phi(t,t')-p(t'),\\quad \\forall t\\in \\Omega, t'\\in \\Omega^{+}$\n\n$\\quad(2)\\phi\\in \\mathcal{O}$\n\n\\item \\textbf{Objective}: \\text{min} $\\sum_{t\\in \\Omega}f(t)p(t)$\n\\end{itemize}\n\n~\\\\\n\nThe Language dual function $L(\\lambda,\\phi,p)$ is\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi,p)&=\\sum_{t\\in T}f(t)p(t)+\\sum_{t\\in \\Omega,t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg((\\phi(t,t)-p(t))-(\\phi(t,t')-p(t'))\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}\\bigg(f(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg)+\\sum_{t\\in \\Omega}\\bigg(\\phi(t,t)\\cdot\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')-\\sum_{t'\\in \\Omega}\\lambda(t',t)\\cdot \\phi(t',t)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\bigg(\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Phi(t)=\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\n\\end{equation}\n\nHere we need $\\forall t\\in \\Omega$,\n\\begin{equation}\nf(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in T^{+}}\\lambda(t,t')=0\n\\end{equation}\nto avoid the unlimited optimal value.\n\n\\subsection{Change the Valuation}\n\nWe will change the valuation and apply the duality to the new valuation. Let\n\n\\begin{equation}\nR_j=\\bigg\\{t\\in\\Omega: j=\\arg\\max_k V(t_k,\\{k\\})\\bigg\\}\n\\end{equation}\n\nWe define a new valuation $\\hat{v}(\\cdot)$. For a type $t\\in \\Omega$, if $t\\in R_j$, then\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t,S)=\n\\begin{cases}\nv(t,S\\backslash\\{j\\})+v(t,\\{j\\}), &j\\in S\\\\\nv(t,S), &j\\not\\in S\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\nSince $v(t,\\cdot)$ is subadditive, $\\hat{v}(t,S)\\geq v(t,S)$ for all subset S.\n\nLet $REV(v)$ and $REV(\\hat{v})$ be the optimal revenue for the two valuations. We would like to see that the two values are not far so that we can bound $REV(v)$ by bounding $REV(\\hat{v})$. The following two lemmas give the proof of this fact:\n\n\\begin{lemma}\n\\textsuperscript{[\\ref{subadditive}]}Consider a distributions D with two different valuations $v$ and $\\hat{v}$. Let $\\delta(t,S)=\\hat{v}(t,S)-v(t,S)$. If $\\delta(t,S)\\geq 0$ for all $t\\in T, S\\subseteq I$, then for any $\\epsilon\\in (0,1)$,\n\\begin{equation}\nREV(\\hat{v})\\geq (1-\\epsilon)(REV(v)-\\frac{\\bar\\delta}{\\epsilon})\n\\end{equation}\nwhere $\\bar\\delta=\\mathbb{E}_{t\\sim D}[\\max_{S\\subseteq I}\\delta(t,S)]$.\n\nIf we choose $\\epsilon=\\frac{1}{2}$,\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta\n\\end{equation}\n\n\\end{lemma}\n\n~\\\\\n\n\\begin{lemma}\nLet $\\bar\\delta'=\\mathbb{E}_{t\\sim D}\\bigg[v(t,I\\backslash\\{j\\}), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]$, then\n\\begin{equation}\n\\bar\\delta\\leq \\bar\\delta'\\leq 6BREV+9.2SREV\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nFor a fixed type $t$, we make a partition $I=C_t\\cup T_t$ based on some cutoff $r$, where $C_t=\\{j\\in I: v(t,\\{j\\})<r\\}$, $T_t=\\{j\\in I: v(t,\\{j\\})\\geq r\\}$. Where $r$ is chosen such that\n\\begin{equation}\n\\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=1\n\\end{equation}\n\nNotice by monotonicity of $v(t,\\cdot)$,\n\n\\begin{equation}\n\\begin{aligned}\n\\bar\\delta&=\\mathbb{E}_{t\\sim D}\\bigg[\\max_{S\\subseteq I}\\big(\\hat{v}(t,S)-v(t,S)\\big)\\bigg]\\\\\n&\\leq \\mathbb{E}_{t\\sim D}\\bigg[\\max_{S\\subseteq I}\\big(v(t,S\\backslash\\{j\\})\\big), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]\\\\\n&\\leq \\bar\\delta'\\\\\n&\\leq \\mathbb{E}_{t\\sim D}\\bigg[v(t,C_t)+\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\\\\n&= \\mathbb{E}_{t\\sim D}[v(t,C_t)]\\text{(CORE)}+\\mathbb{E}_{t\\sim D}\\bigg[\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\text{(TAIL)}\\\\\n\\end{aligned}\n\\end{equation}\n\n\nWe will bound the two parts separately.\n\n~\\\\\n\n\\subsubsection{Core}\n\nWe define function $F:\\Omega^{*}\\to R^{+}$ as follows. For a vector $\\langle t_j\\rangle_{j\\in S}$, let $C=\\{j\\in S: V(t_j,\\{j\\})\\leq r\\}$, then define\n\\begin{equation}\nF(\\langle t_j\\rangle_{j\\in S})=V(\\langle t_j\\rangle_{j\\in C},C)\n\\end{equation}\n\nIt's not hard to justify that F is monotone, subadditive and $r$-Lipschitz. We have the following lemma for functions with this property:\n\\begin{lemma}\\label{core2}\n(Rubinstein and Weinberg, 2015)Let $v$ be a subadditive valuation over independent items and $c$-Lipschitz. Let $a$ be the median of the value of the grand bundle $v([m])$. Then\n\\begin{equation}\\\n\\mathds{E}[v([m])]\\leq 3a+\\frac{4c}{ln(2)}\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nLet $\\delta$ be the median of $F(t)$ over distribution $D$. Now consider the mechanism that sell the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $V(t,I)\\geq F(t)$. Thus with probability at least $\\frac{1}{2}$, the bidder will buy the bundle. Thus,\n\\begin{equation}\nBREV\\geq \\frac{1}{2}\\delta\n\\end{equation}\n\nAccording to Lemma \\ref{core2},\n\\begin{equation}\\label{e1}\n\\text{(CORE)}=\\mathbb{E}_{t\\sim D}[F(t)]\\leq 3\\delta+\\frac{4r}{ln(2)}\n\\leq 6BREV+\\frac{4r}{ln(2)}\n\\end{equation}\n\n~\\\\\n\n\\subsubsection{Tail}\n\\begin{equation}\n\\begin{aligned}\n(TAIL)&=\\mathbb{E}_{t\\sim D}\\bigg[\\sum_{j\\in T_t}v(t,\\{j\\})\\cdot \\mathds{1}_{[t\\not\\in R_j]}\\bigg]\\\\\n&\\leq \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)\\cdot V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\n\\end{aligned}\n\\end{equation}\n\nFor all $t_j$, consider the auction that we sell each item at price $V(t_j,\\{j\\})$. We notice if there exists $k\\not= j$ such that $V(t_k,\\{k\\})>V(t_j,\\{j\\})$, there is at least one bundle with positive profit. The mechanism will definitely sell some item, obtaining expected revenue at least $V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]$. Thus,\n\\begin{equation}\nV(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\\leq SREV\n\\end{equation}\n\n\\begin{equation}\\label{e2}\n(TAIL)\\leq SREV\\cdot \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=SREV\n\\end{equation}\n\n~\\\\\n\nNow we can finish the prove of Lemma 2.\n\n\\begin{equation}\n\\begin{aligned}\nPr_t[\\exists j, V(t_j,\\{j\\})\\geq r]&\\geq 1-\\prod_{j}F_j(r)\\\\\n&\\geq 1-(\\frac{\\sum_j F_j(r)}{n})^n\\\\\n&\\geq 1-(1-1\/n)^n\\\\\n&=1-\\frac{1}{e}\n\\end{aligned}\n\\end{equation}\n\nConsider the auction that sells every item with price $r$. Then with probability at least $1-\\frac{1}{e}$, the mechanism will sell at least one item at price at least $r$, obtaining revenue at least $(1-\\frac{1}{e})r$. Thus,\n\\begin{equation}\\label{e3}\nr\\leq \\frac{1}{1-1\/e}\\cdot SREV\n\\end{equation}\n\nCombining Equation (\\ref{e1})(\\ref{e2})(\\ref{e3}), we have\n\\begin{equation}\n\\bar\\delta\\leq (CORE)+(TAIL)\n\\leq 6BREV+\\frac{4r}{ln(2)}+SREV\n\\leq 6BREV+9.2SREV\n\\end{equation}\nwhich finishes Lemma 2.\n\n~\\\\\n\n\\subsection{Construction of the Flow}\nWe apply duality on the new valuation $\\hat{v}$. For $t,t'\\in \\Omega$, $\\lambda(t',t)>0$ if and only if\n\\begin{itemize}\n\\item $t$ and $t'$ only differ on the $j$-th coordinate.\n\\item $t\\in R_j$.\n\\item $V(t_j',\\{j\\})>V(t_j,\\{j\\})$.\n\\end{itemize}\n\nWe now consider $\\hat{v}(t',S)-\\hat{v}(t,S)$ when $\\lambda(t',t)>0$. There are two conditions for subset S:\n\n(1)$j\\not\\in S$, notice that only the $j-$th coordinate is different,\n\\begin{equation}\n\\hat{v}(t',S)=V'(\\langle t_k' \\rangle_{k\\in S},S)=V'(\\langle t_k \\rangle_{k\\in S},S)=\\hat{v}(t,S)\n\\end{equation}\n\n(2)$j\\in S$, we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t',S)-\\hat{v}(t,S)&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j',\\{j\\})\\bigg)-\n\\bigg(V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j,\\{j\\})\\bigg)\\\\\n&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})-V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})\\bigg)+\\bigg(V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\\bigg)\\\\\n&=V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Bound the Language Function}\nFor all type $t$, the valuation when bidder tells the truth is\n\n\\begin{equation}\n\\begin{aligned}\n\\phi'(t,t)&=\\sum_{S\\subseteq I}\\pi_S(t)\\hat{v}(t,S)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\sum_{S:j\\in S}\\pi_S(t)\\big(\\hat{v}(t,S\\backslash\\{j\\})+\\hat{v}(t,\\{j\\})\\big)+\\sum_{S:j\\not\\in S}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\sum_{S\\subseteq I\\backslash\\{j\\}}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&\\leq \\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\hat{v}(t,I\\backslash\\{j\\})\\bigg)\n\\end{aligned}\n\\end{equation}\n\nThe virtual valuation $\\Phi(t)$:\n\n\\begin{equation}\n\\begin{aligned}\n\\Phi(t)&=\\phi'(t,t)-\\frac{1}{f(t)}\\sum_{S\\subseteq I}\\pi_S(t)\\sum_{t'\\in \\Omega}\\lambda(t',t)(\\hat{v}(t',S)-\\hat{v}(t,S))\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\big(\\hat{v}(t,\\{j\\})-\\sum_{t_j':V'(t_j',\\{j\\})>V'(t_j,\\{j\\})}\\frac{f_j(t_j')}{f_j(t_j)}\\big)\\bigg)\\\\\n&\\leq\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\tilde{\\phi}_j(t_j)\\bigg)\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\phi}_j(t_j)$ is the Myerson Virtual Value for item j with the marginal distribution on $\\Omega_j$ and valuation $V'(t_j,\\{j\\})$.\n\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi',p)&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\\\\\n&=\\bar\\delta'+\\sum_t\\sum_jf(t)\\pi_j(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\\\\n&\\leq \\bar\\delta'+\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\n\\begin{equation}\n\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\leq 6SREV\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nConsider the optimal revenue $OneRev$ when the mechanism is only allowed to sell one item. Assume a mechanism M allocate item j only if $t\\in R_j$. Then LHS is the virtual welfare of this mechanism, which is less than $OneRev$. Besides, this allocation rule is same as assuming bidder's valuation is unit-demand. Thus selling separately can reach a $\\frac{1}{6}$-approximation of this revenue.\n\\end{proof}\n\n~\\\\\n\n\\subsection{Conclusion}\nPut everything together we have\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta \\leq 2L(\\lambda,\\phi',p)+2\\bar\\delta \\leq 4\\bar\\delta'+12SREV \\leq 24BREV+49SREV\n\\end{equation}\n\n\n\n\n\\section{Missing Details of Section~\\ref{sec:flow}}~\\label{sec:appx_flow}\n\\section{Recap: Flow for Additive Valuations}\\label{sec:flow_additive}\nWhen the valuations are additive,   we simply view $t_{ij}$ as bidder $i$'s value for receiving item $j$. Although there are many possible ways to define a flow, we focus on a class of simple ones. Every flow in this class $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}\\in\\R^{nm}$. Based on $\\beta_i=\\{\\beta_{ij}\\}_{j\\in[m]}$, we first partition the type space $T_i$ for each bidder $i$ into $m+1$ regions:\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}<\\beta_{ij}$ for all $j\\in[m]$.\n\t\\item $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{t_{ik}-\\beta_{ik}\\}$.\n\\end{itemize}\n\nWe use essentially the same flow as in~\\cite{CaiDW16}. Here we provide a partial specification and state some desirable properties of the flow. See Figure~\\ref{fig:multiflow} for an example with $2$ items and~\\cite{CaiDW16} for a complete description of the flow.\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth}\n{\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\\begin{enumerate}[leftmargin=0.7cm]\n \\item For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$,  the flow goes directly to $\\varnothing$ (the super sink).\n \\item  For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from  $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n \\item For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\\end{enumerate}\n\\end{minipage}}\n\\caption{Partial Specification of the flow $\\lambda^{(\\beta)}$.}\n\\label{fig:flow specification}\n\\end{figure}\n\n\\begin{figure}\n  \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n  \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n  \\label{fig:multiflow}\n\\end{figure}\n\n\n\\begin{lemma}[\\cite{CaiDW16}\\footnote{Note that this Lemma is a special case of Lemma 3 in~\\cite{CaiDW16} when the valuations are additive. }]\\label{lem:additive flow properties}\n\tFor any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_{i}$ such that the corresponding virtual value function $\\Phi_{i}(t_{i}, \\cdot)$ satisfies the following properties:\n\t\\begin{itemize}[leftmargin=0.5cm]\n\t\t\\item For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi_{i}(t_{i},S) = \\sum_{k\\in S} t_{ik}$.\n\t\t\\item For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$,  $$\\Phi_{i}(t_{i},S)\\leq \\sum_{k\\in S \\land  k\\neq j} t_{ik}+{\\tilde{\\varphi}}_{ij}(t_{ij})\\cdot\\mathds{1}[j\\in S],$$ where ${\\tilde{\\varphi}}_{ij}(\\cdot)$ is Myerson's ironed virtual value function for  $D_{ij}$.\n\t\\end{itemize}\n\\end{lemma}\n\nThe properties above are crucial for showing the approximation results for simple mechanisms in~\\cite{CaiDW16}. One of the key challenges in approximating the optimal revenue is how to provide a tight upper bound. A trivial upper bound is the social welfare, which may be arbitrarily bad in the worst case. By plugging the virtual value functions in Lemma~\\ref{lem:additive flow properties} into the partial Lagrangian, we obtain a new upper bound that replaces the value of the buyer's favorite item with the corresponding Myerson's ironed virtual value. As demonstrated in~\\cite{CaiDW16}, this new upper bound is at most $8$ times larger than the optimal revenue when the buyers are additive, and its appealing structure allows the authors to compare the revenue of simple mechanisms to it. In \\notshow{the next section} {Section~\\ref{sec:flow}}, we identify some difficulties in directly applying this flow to subadditive valuations. Then we show how to overcome these difficulties by relaxing the subadditive valuations and obtain a similar upper bound.\n\n\\section{Proof of Lemma~\\ref{lem:subadditive flow properties}}\\label{sec:proof_virtual_relaxation}\n\\begin{lemma}\\label{lem:separate the favorite out in virtual value}\n\tFor any flow $\\lambda^{(\\beta)}_i$ that respects the partial specification in Figure~\\ref{fig:flow specification}, the corresponding virtual valuation function $\\Phi_i^{(\\beta_i)}$ of $v_i^{(\\beta_i)}$ for any buyer $i$ is:\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\n    \\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$, if $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$.\n\\item $v_i(t_i,S)$, otherwise.\n\\end{itemize}\n\n\\begin{comment}\n\t \\begin{equation*}\n\\begin{aligned}\n\\Phi_i^{(\\beta_i)}(t_i, S)=\n\\begin{cases}\nv_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)  &\\text{if }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:separate the favorite out in virtual value}\n\tThe proof follows the definitions of the virtual valuation function (Definition~\\ref{def:virtual value}) and relaxed valuation (Definition~\\ref{def:relaxed valuation}). We use $t_{i,-j}=\\langle t_{i{j'}}\\rangle_{j'\\not=j}$ to denote bidder $i$'s information for all items except item $j$. If $t_i\\in R_j^{(\\beta_i)}$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i,S) = v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})$. Since $\\lambda(t_i,t_i')>0$ only when $t_{i,-j}=t_{i,-j}'$ and $t_i'\\in R_j^{(\\beta_i)}$,  $v_i^{(\\beta_i)}(t'_i,S) =  v_i(t'_i, S\\backslash \\{j\\})+V_i(t'_{ij})= v_i(t_i, S\\backslash \\{j\\})+V_i(t'_{ij})$. Therefore,\n\\begin{align*}\n    \\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})\n    -\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)\n\\end{align*}\n\t\n\tIf $t_i\\in R_j^{(\\beta_i)}$ and $j\\notin S$ or $t_i\\in R_0^{(\\beta_i)}$, then $v_i^{(\\beta_i)}(t_i,S) =v_i(t_i, S)$. If $t_i\\in R_0^{(\\beta_i)}$, there is no flow entering $t_i$ except from the source, so clearly  $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$. If $t_i\\in R_j^{(\\beta_i)}$, then for any $t'_i$ that only differs from $t_i$ in the $j$-th coordinate, we have $v_i(t'_i, S)=v_i(t_i,S)$, because {$j\\not\\in S$}. Hence,  $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$.\n\\end{prevproof}\n\n\n\\begin{prevproof}{Lemma}{lem:subadditive flow properties}\n\nLet $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$. According to Lemma~\\ref{lem:separate the favorite out in virtual value}, it suffices to prove that for any $j>0$, any $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\n\\begin{claim}\nFor any type $t_{i}\\in R^{(\\beta_i)}_{j}$, if we only allow flow from type $t'_{i}$ to $t_{i}$, where $t_{ik}=t'_{ik}$ for all $k\\neq j$ and $t'_{ij}\\in \\argmin_{s\\in T_{ij} \\land V_i(s)> V_i(t_{ij})} V_i(s)$, and the flow $\\lambda(t_i',t_i)$ equals $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ fraction of the total in flow to $t_i'$, then there exists a flow $\\lambda$ such that\n\\begin{align*}\n\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))\n=V_i(t_{ij})-\\frac{\\left(V_i({t'_{ij}})-V_i(t_{ij})\\right)\\cdot\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]},\n\\end{align*} where $\\varphi_{ij}(V_i(t_{ij}))$ is the Myerson virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$. \\end{claim}\n\\begin{proof}\n{As the flow only goes from $t_i'$ and $t_i$, where $t_i'$ and $t_i$ only differs in the $j$-th coordinate, and \\\\\n\\noindent$t_{ij}\\in \\argmax_{s\\in T_{ij} \\land V_i(s)< V_i(t_{ij}')} V_i(s)$. If $t_{ij}$ is a type with the largest $V_i(t_{ij})$ value in $T_{ij}$, then there is no flow coming into it except the one from the source, so $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})$. For every other value of $t_{ij}$, the in flow is exactly\n\\begin{align*} \\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot  \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) \n=\\prod_{k} f_{ik}(t_{ik})\\cdot \\frac{\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v=V_i(t_{ij})]}.\\end{align*}\n {This is because each type of the form $(x,t_{i,-j})$ with $V_i(x) > V_i(t_{ij})$ is also in $R^{(\\beta_i)}_{j}$. So $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ of all flow that enters these types will be passed down to $t_{i}$ (and possibly further, before going to the sink), and the total amount of flow entering all of these types from the source is exactly {$\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) $}.} Therefore, $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))$. Whenever there is no more type $t_i\\in R_j^{(\\beta_i)}$ with smaller $V_i(t_{ij})$ value, we push all the flow to the sink.}\n\\end{proof}\n\n\nIf $F_{ij}$ is regular, this completes our proof. When $F_{ij}$ is not regular, we can iron the virtual value function in the same way as in \\cite{CaiDW16}. Basically, for two types $t_i,t_i'\\in R^{(\\beta_i)}_{j}$ that only differ in the $j$-th coordinate, if $\\Psi_{ij}^{(\\beta_i)}(t_i)>\\Psi_{ij}^{(\\beta_i)}(t_i')$ but $V_i(t_{ij})<V_i(t_{ij}')$, add a loop between $t_i$ and $t_i'$ with a proper weight to make $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\Psi_{ij}^{(\\beta_i)}(t_i')$.\n\\begin{lemma}\\cite{CaiDW16}\nFor any $\\beta$ and $i$, there exists a flow $\\lambda_i(\\beta)$ such that for any $t_i\\in R_j^{(\\beta_i)}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\\end{lemma}\n\\end{prevproof}\n\n\n\n\n\n\\section{Missing Details of Section~\\ref{sec:duality}}\\label{sec:appx_duality}\n\n\n\\section{Efficiently Finding the Mechanism for Identical Bidders}\nIn this section, we are going to prove that such a mechanism which approximately obtains the optimal revenue can be computed in polynomial time when bidders are identical. To say \\textbf{identical}, we mean all bidders share same private information set $\\tilde{T}$ and distribution $\\tilde{D}$ on this support. Besides, given a same private information, all bidders have the same item valuation, i.e., for any $t\\in \\tilde{T}$ and $S\\subseteq [m]$, $v_i(t,S)=v_j(t,S)$ holds for all $i,j\\in [n]$.\n\nWhen bidders are identical, it's reasonable to only consider symmetric mechanisms. In a symmetric mechanism, the ex-ante probability to win an item $j$ is the same for all bidder $i$, and thus no more than $\\frac{1}{n}$ due to no overallocation. The following lemma formally proves these observations.\n\n\\begin{lemma}\\label{lem:comp:symmetric}\nIf bidders are identical, then for any BIC mechanism $M$, there exists a symmetric BIC mechanism $M'$ that satisfies\n\\begin{itemize}\n\\item $\\textsc{Rev}(M,v,D)=\\textsc{Rev}(M',v,D)$\n\\item $\\forall i\\in [n], \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq \\frac{1}{n}$\n\\end{itemize}\nwhere $\\pi_{ij}'(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}'(t_i)$ and $\\sigma_{iS}'(t_i)$ is the probability of buyer i receiving exactly bundle S when her reported type is $t_i$ in mechanism $M'$.\n\\end{lemma}\n\\begin{proof}\nDefine mechanism $M'$ as follows: after receiving type profile $t=\\langle t_i\\rangle_{i=1}^n$, draw a random permutation $\\rho$ of size $n$. Then follow mechanism $M$ treating bidder $i$ as $\\rho(i)$ for all $i$. Formally, we denote by $M_i(t)$, $h_i(t)$ the (probably random) received item set and charged price to bidder $i$ by mechanism $M$ when type profile is $t$, and $M_i'(t)$, $h_i'(t)$ the same meaning for mechanism $M'$. After the permutation $\\rho$ is drawn, define type profile $t^{(\\rho)}=\\langle t_{\\rho(i)}\\rangle_{i=1}^n$. The allocation rule and price for mechanism $M'$ is then defined as: $M_i'(t)=M_{\\rho(i)}(t^{(\\rho)})$, $h_i'(t)=h_{\\rho(i)}(t^{(\\rho)})$. Since bidders are identical, with the random permutation, $M'$ will treat all bidders as a same role, and thus the ex-ante probability getting an item $j$ is the same for all bidders. Clearly $\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq 1$, thus each ex-ante probability $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}'(t_i)\\leq \\frac{1}{n}$.\n\nTo prove that $M'$ has the same expected revenue as $M$, it's enough to show $\\sum_t f(t)\\cdot \\sum_i h_i(t)=\\sum_t f(t)\\cdot \\sum_i h_{\\rho(i)}(t^{(\\rho)})$. Notice for all $t\\in T$, $f(t)=f(t^{(\\rho)})$ as all bidders' types are independent, we have\n\\begin{equation}\n\\sum_t f(t)\\cdot \\sum_i h_{\\rho(i)}(t^{(\\rho)})=\\sum_t f(t^{(\\rho)})\\cdot \\sum_i h_i(t^{(\\rho)})=\\sum_t f(t)\\cdot \\sum_i h_i(t)\n\\end{equation}\n\n\n\\end{proof}\n\nNow without loss of generality, we can assume $M$ is a symmetric mechanism. For every $i\\in [n], j\\in [m]$, select $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. With Lemma~\\ref{lem:comp:symmetric}, it's easy to check that $\\beta$ satisfies the two conditions in Lemma~\\ref{lem:requirement for beta} and thus Theorem~\\ref{thm:multi} works. We have to note here that in a discrete distribution $D_{ij}$, there might not exist such a $\\beta_{ij}$. Thus we have to assume that $\\forall i,j, \\exists \\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$, so that it's able to get the exact $\\beta_{ij}$. $\\beta_{ij}$ is sensitive to the proof and the mechanism used in the proof, and thus it's necessary to use the exact value.\n\nConsider this specific $\\beta$ for proofs in Section~\\ref{sec:multi}. Now we'll show all the posted price and entry fee for the mechanisms that we use in the proof can be computed in polynomial time.\n\nGiven bidder $i$'s type $t_i$, it'll need exponential size to explicitly described $i$'s value $v_i(t_i,\\cdot):2^{[m]}\\to R^{*}$. Thus it's usually assumed that there exist the following oracles access to $v_i(\\cdot,\\cdot)$.\n\n\\begin{itemize}\n\\item Value oracle: Given type $t_i\\in T_i$ and set $S$ as input, returns a non-negative real number that represents bidder's value $v_i(t_i,S)$.\n\\item XOS oracle(for XOS valuations): Given type $t_i\\in T_i$ and set $S$ as input, returns prices $\\{p_j\\}_{j\\in S}$ as the 1-supporting prices for $v_i(t_i,S)$.\n\\end{itemize}\n\nWe now state the main theorem for this section:\n\\begin{theorem}\nConsider the case that bidders are identical, and have XOS valuations. Suppose $\\forall i,j, \\exists \\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Besides, we are given\n\\begin{itemize}\n\\item $\\forall i,j$, the distribution for $v_i(t_{ij})$.\n\\item $\\forall i\\in [n]$, a value oracle for $v_i(\\cdot,\\cdot)$.\n\\item $\\forall i\\in [n]$, an XOS oracle for $v_i(\\cdot,\\cdot)$.\n\\item $\\forall t\\in T$, a black box algorithm $\\mathcal{A}$ that maximizes the social welfare for the combinatorial auction with bidder's valuation $v_i'(t_i,\\cdot)$.\n\\end{itemize}\n\nThen with $O()$ type samples from $T$, with probability at least ---, we can compute a DSIC mechanism in $POLY(n,m,\\sum_i\\sum_j |T_{ij}|, )$ time such that the revenue of this mechanism obtains at least $1\/$ the optimal revenue.\n\\end{theorem}\n\n\\begin{proof}\nFirstly for Lemma~\\ref{lem:multi_single}, as showed in \\cite{ChawlaHMS10, KleinbergW12}, the posted-price mechanism that used can be efficiently computed. We now consider the Sequential Posted Price Mechanism stated in Section~\\ref{subsection:tail} and Section~\\ref{subsection:core}.\n\nThe mechanism used in Corollary~\\ref{cor:bound tail } is determined by the posted price $\\xi_{ij}=\\beta_{ij}+P_{ij}$. Having distribution $D_{ij}$, $\\beta_{ij}$ and $P_{ij}$ can both be computed in $O(|T_{ij}|)$ time by brute force. Thus this mechanism can be computed in linear time.\n\nNow consider the ``CORE\" part. Recall that $\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S)$, which can be described as the expected social welfare of $M^{(\\beta)}$ in a new combinatorial auction. In this auction, when type profile $t\\in T$, bidder $i$ has valuation $v_i'(t_i,\\cdot)$. Let $M^{(\\beta),*}$ be the optimal mechanism among above and $\\sigma_{iS}^{(\\beta),*}$, $Q_j^{*}$ be the corresponding definition. Then according to Lemma~\\ref{lem:core and q_j}, $2\\alpha\\cdot\\sum_{j\\in [m]}Q_j^{*}\\geq \\textsc{Core}(\\beta)$ holds for all BIC mechanism $M^{(\\beta)}$. It implies that the mechanism with specific parameter $\\beta$ and $\\sigma_{iS}^{(\\beta)}=\\sigma_{iS}^{(\\beta),*}$ can guarantee a constant factor of the optimal revenue.\n\nThe Sequential Posted Price Mechanism with Entry Fee stated in Section~\\ref{subsection:core} use $Q_j^{*}$ as the posted price for item $j$. Recall that $Q_j^{*}=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta),*}(t_i)\\cdot \\gamma_j^{S}(t_i)$. Even exactly calculating one $\\sigma_{iS}^{(\\beta),*}(t_i)$ requires running mechanism $M^{(\\beta),*}$ under every type profile $t$, which takes exponential time. $Q_j^{*}$ can thus not be able to compute exactly in polynomial time.\n\n\n\n\n\nRecall that $Q_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)$. And even exactly calculating one $\\sigma_{iS}^{(\\beta)}(t_i)$ requires running mechanism $M$ under every type profile $t$, which takes exponential time. $Q_j$ can thus not be able to compute exactly in polynomial time. What we do is to generate enough samples of type profile $t\\in T$, run $M$ with those samples, and calculate the empirical approximation for $Q_j$.\n\n\n\n\n\n\n\n\\end{proof}\n\n\nCorollary~\\ref{cor:bound tail }, Lemma~\\ref{lem:beta_ij}, Lemma~\\ref{lem:tau_i}, and Definition~\\ref{def:entry fee}.\n\n\n\n\n\n\\section*{\\huge{Appendix}}\n\n\\section{Details from Preliminaries}\n\nWe first formally define various valuation classes.\n\\begin{definition}\\label{def:valuation classes}\nWe define several classes of valuations formally. Let $t$ be the type and $v(t,S)$ be the value for bundle $S\\in[m]$.\n\n\n\t\n\\begin{itemize}[leftmargin=0.5cm]\n\t\\item  \\textbf{Constrained Additive} \\\\\n$v(t,S) =\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} v(t, \\{j\\})$, where $\\mathcal{I}\\subseteq 2^{[m]}$ is a downward closed set system over the items specifying the feasible bundles. In particular, when $\\mathcal{I}=2^{[m]}$, the valuation is an \\textbf{additive function}; when $\\mathcal{I}=\\{\\{j\\}\\ |\\ j\\in[m] \\}$, the valuation is a \\textbf{unit-demand function}; when $\\mathcal{I}$ is a matroid, the valuation is a \\textbf{matroid-rank function}. An equivalent way to represent any constrained additive valuations is to view the function as additive but the bidder is only allowed to receive bundles that are feasible, i.e., bundles in $\\mathcal{I}$. To ease notations, we interpret $t$ as an $m$-dimensional vector $(t_1, t_2,\\cdots, t_m)$ such that $t_j =  v(t, \\{j\\})$.\n\n\\item \\textbf{XOS\/Fractionally Subadditive}\\\\\n $v(t,S) = \\max_{i\\in [{K}]} v^{(i)}(t, S)$, where ${K}$ is some finite number and $v^{(i)}(t,\\cdot)$ is an additive function for any $i\\in[{K}]$.\n\n\\item \\textbf{Subadditive}\\\\ \n$v(t,S_1\\cup S_2)\\leq v(t,S_1)+v(t,S_2)$ for any subset $S_1, S_2\\subseteq [m]$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{The following are a few examples of various valuation distributions which are over independent items (Definition~\\ref{def:subadditive independent}):\n\n\\begin{example}\\label{eg:valuation}\\cite{RubinsteinW15}\n$t=\\{t_j\\}_{j\\in[m]}$ where $t$ is drawn from $\\prod_j D_j$,\n\\begin{itemize}[leftmargin=0.5cm]\n\\item Additive: $t_j$ is the value of item $j$. $v(t,S)=\\sum_{j\\in S}t_j$.\n\\item Unit-demand: $t_j$ is the value of item $j$. $v(t,S)=\\max_{j\\in S}t_j$.\n\\item Constrained Additive: $t_j$ is the value of item $j$.\n $v(t,S)=\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} t_j$.\n\\item XOS\/Fractionally Subadditive: $t_j=\\{t_{j}^{(k)}\\}_{k\\in[K]}$ encodes all the possible values associated with item $j$, and $v(t,S)=\\max_{k\\in[K]}\\sum_{j\\in S}t_{j}^{(k)}$.\n\\end{itemize}\n\\end{example}\n\n\n}\n\n\\section{Improved Analysis for Constrained Additive Valuation}\n\nIn this section, we show that for constrained additive bidders, we do not need to relax the valuations, as applying directly the flow in Section~\\ref{sec:flow} already gives an upper bound with the right format. So we can take $M^{(\\beta)}$ to simply be $M$. In particular, we can derive the following improved upper bound for $\\textsc{Rev}(M,v,D)$ using essentially the same proof as in Section~\\ref{sec:flow}.\n\\begin{theorem}\\label{thm:revenue upperbound for constrained additive}\nIf for any bidder $i$ any type $t_i\\in T$, $v_i(t_i, \\cdot)$ is a constrained additive valuation, then for any mechanism $M$ and any $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}$,\n$$\\textsc{Rev}{(M,v,D)}\\leq \\textsc{Non-Favorite}(M,\\beta)+\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\nCombining the same upper bounds we obtained for \\\\\n\\noindent$\\textsc{Non-Favorite}(M,\\beta)$ and $\\textsc{Single}(M,\\beta)$ and the improved upper bound in Theorem~\\ref{thm:revenue upperbound for constrained additive}, we can improve the approximation ratio when the bidder(s) have constrained additive valuations.\n\\begin{theorem}\n\tFor a single buyer whose valuation is constrained additive,\n$$\\textsc{Rev}(M,v,D)\\leq 7\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev},$$\nfor any BIC mechanism $M$.\n\\end{theorem}\n\n\\begin{theorem}\n\tFor multiple buyers whose valuations are constrained additive,\n\\begin{equation}\n\\begin{aligned}\n\\textsc{Rev}(M,v,D)&\\leq 8\\cdot\\textsc{APostEnRev}\\\\\n&+\\left(6+\\frac{22}{1-b}+\\frac{4(b+1)}{(1-b)b}\\right)\\cdot \\textsc{PostRev}\n\\end{aligned}\n\\end{equation}\nfor any BIC mechanism $M$. In particular, if we set $b$ to be $\\frac{1}{4}$, then $$\\textsc{Rev}(M,v,D)\\leq 8\\cdot\\textsc{APostEnRev}+62\\cdot\\textsc{PostRev}.$$\n\\end{theorem}\n\n\\section{Duality}\\label{sec:duality}\n\n\n The focus of \\cite{CaiDW16} was on additive and unit-demand valuations and their respective dual was derived from an LP that is only meaningful for constrained additive valuations. In order to tackle general valuations, we need to apply the duality framework to an LP that is meaningful for general valuations. Instead of using the ``implicit forms'' LP from~\\cite{CaiDW13b, CaiDW16}, we choose a slightly different and more intuitive LP formulation (see Figure~\\ref{fig:LPRevenue}). For all bidders $i$ and types $t_i \\in T_i$, we use $p_i(t_i)$ as the interim price paid by bidder $i$ and $\\sigma_{iS}(t_i)$ as the interim probability of receiving the exact bundle $S$. To ease the notation, we use a special type $\\varnothing$ to represent the choice of not participating in the mechanism. More specifically, ${\\sigma}_{iS}(\\varnothing)=0$ for any $S$ and $p_{i}(\\varnothing)=0$. Now a Bayesian IR (BIR) constraint is simply another BIC constraint: for any type $t_{i}$, bidder $i$ will not want to lie to type $\\varnothing$.  We let $T_{i}^{+}=T_{i}\\cup \\{\\varnothing\\}$.\n\nFollowing the recipe provided by~\\cite{CaiDW16}, we take the partial Lagrangian dual of the LP in Figure~\\ref{fig:LPRevenue} by lagrangifying the BIC constraints. Let $\\lambda_i(t_i,t_i')$ be the Lagrange multiplier associated with the BIC constraint that if bidder $i$'s true type is $t_i$ she will not prefer to lie to type $t_i'$\n (see Figure~\\ref{fig:Lagrangian} and Definition~\\ref{def:Lagrangian}). As shown in~\\cite{CaiDW16}, the dual solution has finite value if and only if the dual variables $\\lambda_i$ form a valid flow for every bidder $i$. The reason is that the payments $p_i(t_i)$ are unconstrained variables, therefore the corresponding coefficients must be $0$ in order for the dual solution to have finite value. It turns out when all these coefficients are $0$, the dual variables $\\lambda$ can be interpreted as a flow described in Lemma~\\ref{lem:useful dual}. We refer the readers to~\\cite{CaiDW16} for a complete proof. From now on, we only consider $\\lambda$ that corresponds to a flow.\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $p_i(t_i)$, for all bidders $i$ and types $t_i \\in T_i$, denoting the expected price paid by bidder $i$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\item $\\sigma_{iS}(t_i)$, for all bidders $i$, all bundles of items $S\\subseteq[m]$, and types $t_i \\in T_i$, denoting the probability that bidder $i$ receives \\textbf{exactly} the bundle $S$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\sum_{S\\subseteq[m]} {\\sigma}_{iS}(t_i) \\cdot v_i(t_i,S) - p_i(t_i) \\geq\\sum_{S\\subseteq[m]}{\\sigma}_{iS}(t'_i) \\cdot v_i(t_i, S) - p_i(t'_i) $, for all bidders $i$, and types $t_i \\in T_i, t'_i \\in T_i^+$, guaranteeing that the reduced form mechanism $({\\bf{\\sigma}},{p})$ is BIC and Bayesian IR.\n\\item ${\\bf{\\sigma}} \\in {P(D)}$, guaranteeing ${\\sigma}$ is feasible.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\max \\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)$, the expected revenue.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{A Linear Program (LP) for Revenue Optimization.}\n\\label{fig:LPRevenue}\n\\end{figure}\n\n\\begin{definition}\\label{def:Lagrangian}\nLet ${\\mathcal{L}}(\\lambda, \\sigma, p)$ be the partial Lagrangian defined as follows:\n\\begin{align*}\n& {\\mathcal{L}}(\\lambda, \\sigma, p)\\\\\\stepcounter{equation}\\tag{\\theequation} \\label{eq:primal lagrangian}\n=&\\sum_{i=1}^{n} \\left(\\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)+\\sum_{t_{i}\\in T_{i},t_{i}'\\in T_i^{+}} \\lambda_{i}(t_{i},t_{i}')\\cdot \\left(\\sum_{S\\subseteq[m]} v_i(t_{i},S)\\cdot\\left(\\sigma_{iS}(t_{i})-\\sigma_{iS}({t_{i}'})\\right)-\\left((p_{i}(t_{i})-p_{i}(t_{i}')\\right)\\right)\\right)\\\\\n=& \\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}} p_{i}(t_{i})\\cdot\\left(f_{i}(t_{i})+\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})-\\sum_{t_{i}'\\in T_{i}^{+}} \\lambda_{i}(t_{i},t_{i}')\\right)\\right)\\\\\n&+\\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}}\\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot \\left(v_i(t_{i},S)\\cdot \\sum_{t_{i}'\\in T_{i}^{+}}\\lambda_{i}(t_{i},t_{i}')-\\sum_{t'_i\\in T_{i}}\\left(v_i(t'_{i},S)\\cdot \\lambda_{i}(t_{i}',t_{i})\\right)\\right)\\right)~~ ({\\sigma}_i(\\varnothing)=\\textbf{0},\\ p_{i}(\\varnothing)=0)\\stepcounter{equation}\\tag{\\theequation} \\label{eq:dual lagrangian}\n\\end{align*}\n\\end{definition}\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, the Lagrangian multipliers for Bayesian IC and IR constraints.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')\\geq 0$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, guaranteeing that the Lagrangian multipliers are non-negative.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\min_{\\lambda}\\max_{\\sigma\\in {P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)$.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{Partial Lagrangian of the Revenue Maximization LP.}\n\\label{fig:Lagrangian}\n\\end{figure}\n\n\n\\notshow{ \\begin{definition}[Useful Dual Variables~\\cite{CaiDW16}]\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$.\n\\end{definition}}\n\n\n\\begin{lemma}[Useful Dual Variables~\\cite{CaiDW16}]\\label{lem:useful dual}\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$. $\\lambda$ is useful iff for each bidder $i$, $\\lambda_{i}$ forms a valid flow, i.e., iff the following satisfies flow conservation at all nodes except the source and the sink:\n\n \\noindent\\textbf{\\emph{1.}} Nodes: A super source $s$ and a super sink $\\varnothing$, along with a node $t_{i}$ for every type $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{2.}} An edge from $s$ to $t_{i}$ with flow $f_i(t_{i})$, for all $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{3.}} An edge from $t_i$ to $t_i'$ with flow $\\lambda_i(t_i,t_i')$ for all $t_i\\in T_i$, and $t_i'\\in T_{i}^{+}$ (including the sink).\n\n\\end{lemma}\n\n\\begin{definition}[Virtual Value Function]\\label{def:virtual value}\nFor each flow $\\lambda$, we define a corresponding virtual value function $\\Phi(\\cdot)$, such that for every bidder $i$, every type $t_{i}\\in T_{i}$ and every set $S\\subseteq[m]$,\n$$\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right)$$.\n\\notshow{\n\\vspace{.1in}\\noindent$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-$\n\n$\\hspace{1.3cm}{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right).$\t\n}\n\\end{definition}\nThe proof of Theorem~\\ref{thm:revenue less than virtual welfare} is essentially the same as in~\\cite{CaiDW16}. We include it in Appendix~\\ref{sec:proof_duality} for completeness.\n\\begin{theorem}[Virtual Welfare $\\geq$ Revenue~\\cite{CaiDW16}]\\label{thm:revenue less than virtual welfare}\nFor any flow $\\lambda$ and any BIC mechanism $M=(\\sigma,p)$, the revenue of $M$ is $\\leq$  the virtual welfare of {$\\sigma$} w.r.t. the virtual valuation $\\Phi(\\cdot)$ corresponding to $\\lambda$.\n$$\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq \\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$$\n\n\\notshow{\n\\vspace{.05in} $\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq $\n\n$\\hspace{1cm}\\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$\n}\nLet $\\lambda^{*}$ be the optimal dual variables and $M^{*}=(\\sigma^{*},p^{*})$ be the revenue optimal BIC mechanism, then the expected virtual welfare with respect to $\\Phi^{*}$ (induced by $\\lambda^{*}$) under  $\\sigma^{*}$ equals to the expected revenue of $M^{*}$.\n\\end{theorem}\n\n\\section{Canonical Flow and Properties of the Virtual Valuations}\\label{sec:flow}\n\n\nIn this section, we present a canonical way of setting the dual variables\/flow that induces our benchmarks. A recap of the flow for additive valuations and the appealing properties of the corresponding virtual valuation functions can be found in Appendix~\\ref{sec:flow_additive}. We refer readers to that Section for more intuition about the flow.\n\nAlthough any flow can provide a finite upper bound of the optimal revenue, we focus on a particular class of flows, in which every flow $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n],j\\in[m]}\\in\\R^{nm}_{\\geq 0}$. Based on $\\beta$, we partition the type set $T_i$ of each buyer $i$ into $m+1$ regions: \\textbf{(i)} $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})<\\beta_{ij}$ for all $j\\in[m]$. \\textbf{(ii)} $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{V_i(t_{ik})-\\beta_{ik}\\}$. Intuitively, if we view $\\beta_{ij}$ as the price of item $j$ for bidder $i$, then $R^{(\\beta_i)}_0$ contains all types in $T_i$ that cannot afford any item, and any $R^{(\\beta_i)}_j$ with $j>0$ contains all types in $T_i$ whose ``favorite'' item is $j$. We first provide a {\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\n\\noindent\\textbf{1.} For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$,  the flow goes directly to $\\varnothing$ (the super sink).\n\n\\noindent \\textbf{2.}  For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from  $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n\n\\noindent \\textbf{3.} For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\n\\notshow{\n\\begin{figure}\n  \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n  \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n  \\label{fig:multiflow}\n\\end{figure}\n}\n\nFor additive valuations and any type $t_i \\in R_j^{(\\beta_i)}$ , the contribution to the virtual value function $\\Phi(t_i,S)$  from any type $t_i'\\in R_j^{(\\beta_i)}$ is either $0$ if $j\\notin S$, or {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))=\\lambda_i^{(\\beta)}(t_i', t_i)(t_{ij}'-t_{ij})$} if $t_i$, $t_i'$ only differs on the $j$-th coordinate and $j\\in S$. In either case, the contribution does not depend on $t_{ik}$ for any $k\\neq j$. This is the key property that allows~\\cite{CaiDW16} to choose a flow such that the value of the favorite item is replaced by the corresponding Myerson's ironed virtual value in the virtual value function $\\Phi_i(t_i,\\cdot)$.\nUnfortunately, this property no longer holds for subadditive valuations. When $j\\in S$ and $\\lambda_i^{(\\beta)}(t_i',t_i)>0$, the contribution {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))$} heavily depends on $t_{ik}$ of all the other item $k\\in S$. All we can conclude is that the contribution lies in the range {$[-\\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}), \\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}')]$}\\footnote{$v_i(t,\\cdot)$ is subadditive and monotone for every type $t\\in T_i$, therefore $v_i(t_i,S)\\in[v_i(t_i, S\\backslash\\{j\\}),v_i(t_i, S\\backslash\\{j\\})+V_{i}(t_{ij})]$ and $v_i(t'_i,S)\\in[v_i(t'_i, S\\backslash\\{j\\}),v_i(t'_i, S\\backslash\\{j\\})+V_{i}(t'_{ij})]$.}, but this is not sufficient for us to convert the value of item $j$ into the corresponding Myerson's ironed virtual value.\n\n\\subsection{Valuation Relaxation}\\label{sec:valuation relaxation}\nThis is the first major barrier for extending the duality framework to accommodate subadditive valuations. We overcome it by considering a relaxation of the valuation functions. More specifically, for any $\\beta$, we construct another function $v_i^{(\\beta_i)}(\\cdot,\\cdot): T_i\\times 2^{[m]}\\mapsto {\\mathbb{R}_{\\geq 0}}$ for every buyer $i$ such that: (i) for any $t_i$,  $v_i^{(\\beta_i)}(t_i,\\cdot)$ is subadditive and monotone, and for every bundle $S$ the new value $v_i^{(\\beta_i)}(t_i,S)$ is no smaller than the original value $v_i(t_i,S)$; (ii) for any BIC mechanism $M$ with respect to the original valuations, there exists another mechanism $M^{(\\beta)}$ that is BIC with respect to the new valuations and its revenue is comparable to the revenue of $M$; (iii) for the new valuations $v^{(\\beta)}$, there exists a flow whose induced virtual value functions have properties similar to those in the additive case.\nProperty (ii) implies that the optimal revenue with respect to $v^{(\\beta)}$ can serve as a proxy for the original optimal revenue. Moreover, due to Theorem~\\ref{thm:revenue less than virtual welfare}, the optimal revenue for $v^{(\\beta)}$ is upper bounded by the partial Lagrangian dual with respect to $v^{(\\beta)}$, which has an appealing format similar to the additive case by property (iii). Thus, we obtain a benchmark for subadditive bidders that resembles the benchmark for additive bidders in~\\cite{CaiDW16}\n\n\\begin{definition}[Relaxed Valuation]\\label{def:relaxed valuation}\n\tGiven $\\beta$, for any buyer $i$, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})$, if the ``favorite'' item is in $S$, i.e., $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$. Otherwise, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S)$.\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\nv_i^{(\\beta_i)}(t_i,S)=\n\\begin{cases}\nv_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})  &\\text{if the ``favorite'' item is in $S$, i.e., }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\\end{definition}\n\nIn the next Lemma, we show that for any BIC mechanism $M$ for $v$, there exists a BIC mechanism $M^{(\\beta)}$ for $v^{(\\beta)}$ such that its revenue is comparable to the revenue of $M$ (property (ii)). Moreover, the ex-ante probability for any buyer $i$ to receive any item $j$ in $M^{(\\beta)}$ is no greater than in $M$ (property (i)). We will see later that this is an important property for our analysis. The proof of Lemma~\\ref{lem:relaxed valuation} is similar to the $\\epsilon$-BIC to BIC reduction in~\\cite{HartlineKM11, BeiH11,DaskalakisW12} and can be found in Appendix~\\ref{sec:proof_relaxed_valuation}.\n\n\n\\begin{lemma}\\label{lem:relaxed valuation}\n\n\tFor any $\\beta$ and any BIC mechanism $M$ for subadditive valuation  $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\n \\vspace{.1in}\n  \\noindent \\emph{\\textbf{(i)}} $\\displaystyle\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\n  \\vspace{.1in}\n \\noindent \\emph{\\textbf{(ii)}} $\\displaystyle\\textsc{Rev}(M, v, D)\\leq2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\displaystyle+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).$\n\n\\vspace{0.05in}\n\\noindent$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$)\n\\end{lemma}\n\\notshow{\nFrom now on, we fix $M^{(\\beta)}$ to be the mechanism that is constructed by setting $\\eta$ to be $1\/2$ and $\\epsilon$ be a extremely tiny positive constant $\\epsilon_o$ in Lemma~\\ref{lem:relaxed valuation}.\n\\begin{corollary}\n\tFor any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that\n\t$$\\textsc{Rev}(M, v, D)\\leq 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\epsilon_0.$$\n\\end{corollary}}\n\\subsection{Virtual Valuation for the Relaxed Valuation}\\label{sec:virtual for relaxed}\nFor any $\\beta$, based on the same partition of the type sets as in the beginning of Section~\\ref{sec:flow}, we construct a flow $\\lambda^{(\\beta)}$ that respects the partial specification, such that the corresponding virtual valuation function for $v^{(\\beta)}$ has the same appealing properties as in the additive case.\nFor the relaxed valuation, as {$\\lambda_i^{(\\beta)}(t_i, t_i')$} is only positive for types $t_i$, $t_i'\\in R_j^{(\\beta_i)}$ that only differ in the $j$-th coordinate, the contribution from item $j$ to the virtual valuation solely depends on $t_{ij}$ and $t'_{ij}$ but not $t_{ik}$ for any other item $k\\in S$\n. Notice that this property does not hold for the original valuation, and it is the main reason why we choose the relaxed valuation as in Definition~\\ref{def:relaxed valuation}. Moreover, we can choose $\\lambda_i^{(\\beta)}$ carefully so that the virtual valuation of $v^{(\\beta)}$ has the following format:\n\n\n\n\n\\begin{lemma}\\label{lem:subadditive flow properties}\n\tLet $F_{ij}$ be the distribution of $V_i(t_{ij})$ when $t_{ij}$ is drawn from $D_{ij}$. For any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_i$ such that the corresponding virtual value function $\\Phi^{(\\beta_i)}_{i}(t_{i}, \\cdot)$ of valuation $v_i^{(\\beta_i)}(t_i,\\cdot)$ satisfies the following properties:\n\n\\vspace{.05in}\t\n\\noindent 1. For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi^{(\\beta_i)}_{i}(t_{i},S) = v_i(t_i, S)$.\n\n\\vspace{.05in}\n\\noindent 2. For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$,  $\\Phi_{i}^{(\\beta_i)}(t_{i},S)\\leq  v_i (t_{i}, S)\\cdot\\mathds{1}[j\\notin S]+\\left(v_i (t_{i}, S\\backslash\\{j\\})+{\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))\\right)\\cdot\\mathds{1}[j\\in S],$ where ${\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$ is the Myerson's ironed virtual value for  $V_i(t_{ij})$ with respect to $F_{ij}$.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem:subadditive flow properties} is postponed to Appendix~\\ref{sec:proof_virtual_relaxation}.\nNext, we use the virtual welfare of the allocation $\\sigma^{(\\beta)}$ to bound the revenue of $M^{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:upper bound the revenue of the relaxed mechanism}\n\tFor any $\\beta$, \\begin{align*} &\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot\\Phi^{(\\beta_i)}_i(t_i,S)\\\\\n \\leq &\t\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\backslash\\{j\\})+\\sum_{S:j\\notin S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\right)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\pi^{(\\beta)}_{ij}(t_i)\\cdot {\\tilde{\\varphi}}_{ij}(t_{ij}),\\end{align*}\n where $ \\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$. {\\bf \\textsc{Non-Favorite}$(M, \\beta)$} denotes the sum of the first two terms. {\\bf \\textsc{Single}$(M, \\beta)$} denotes the last term.  \\end{lemma}\n\n\\begin{proof}\nThe Lemma follows easily from the properties in Lemma~\\ref{lem:subadditive flow properties} and Theorem~\\ref{thm:revenue less than virtual welfare}.\n\\end{proof}\n\nWe obtain Theorem~\\ref{thm:revenue upperbound for subadditive} by combining Lemma~\\ref{lem:relaxed valuation} and~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n \\begin{theorem}\\label{thm:revenue upperbound for subadditive}\nFor any mechanism $M$ and any $\\beta$,\n$$\\textsc{Rev}{(M,v,D)}\\leq 4\\cdot\\textsc{Non-Favorite}(M, \\beta)+2\\cdot\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\n\\begin{prevproof}{Theorem}{thm:revenue upperbound for subadditive}\nFirst, let's look at the value of $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)$. When $t_i\\in R_j^{(\\beta_i)}$ for some $j>0$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)= v_i(t_i, S\\backslash\\{j\\})+V_i(t_{ij})-v_i(t_i,S)\\leq v_i(t_i, S\\backslash\\{j\\}),$ because $V_i(t_{ij})\\leq v_i(t_i,S)$. For the other cases, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)=0$. Therefore,\n\\begin{align*}\\label{eq:bounding delta}\n\t&\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\nonumber\\\\\n\\leq &\\sum_i\\sum_{t_i} f_i(t_i)\\sum_{j} \\mathds{1}[t_i\\in R_j^{(\\beta_i)}]\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\cdot v_i(t_i, S\\backslash\\{j\\})\\nonumber\\\\\n\\leq &\\textsc{Non-Favorite}(M,\\beta)~~~~~~~~~\\text{(Definition of $\\textsc{Non-Favorite}(M,\\beta)$)}\n\\end{align*}\n\nOur statement follows from combining Lemma~\\ref{lem:relaxed valuation}, Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism} with the inequality above.\n\\end{prevproof}\n\\begin{comment}\nNow we bound $\\textsc{Rev}(M, v, D)$. By Lemma~\\ref{lem:relaxed valuation},\n\t\\begin{align*}&\\textsc{Rev}(M, v, D)\\\\\n\t\\leq& 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\\\\n+&2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\\\\t\n\\leq &2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\textsc{Non-Favorite}(M,\\beta)~~~~\\text{(Equation~(\\ref{eq:bounding delta}))}\\\\\n\\leq & 4\\cdot\\textsc{Non-Favorite}(M,\\beta)+2\\cdot\\textsc{Single}(M,\\beta)~~~~\\text{(Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism})}.\n\t\\end{align*}\n\\end{comment}\n\n\n\n\\subsection{Upper Bound for the Revenue of Subadditive Buyers}~\\label{sec:choice of beta}\n In Section~\\ref{sec:valuation relaxation}, we have argued that for any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that its revenue with respect to the relaxed valuation $v^{(\\beta)}$ is comparable to the revenue of $M$ with respect to the original valuation. In Section~\\ref{sec:virtual for relaxed}, we have shown for any $\\beta$ how to choose a flow to obtain an upper bound for $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and also an upper bound for $\\textsc{Rev}(M,v,D)$. Now we specify our choice of $\\beta$.\n\nIn~\\cite{CaiDW16}, the authors fixed a particular $\\beta$, and shown that under any allocation rule, the corresponding benchmark can be bounded by the sum of the revenue of a few simple mechanisms. However, for valuations beyond additive and unit-demand, the benchmark becomes much more challenging to analyze\\footnote{Indeed, the difficulties already arise for valuations as simple as $k$-demand. A bidder's valuation is $k$-demand if her valuation is additive subject to a uniform matroid with rank $k$.}. We adopt an alternative and more flexible approach to obtain a new upper bound. Instead of fixing a single $\\beta$ for all mechanisms, we customize a different $\\beta$ for every different mechanism $M$. Next, we relax the valuation and design the flow based on the chosen $\\beta$ as specified in Section~\\ref{sec:valuation relaxation} and \\ref{sec:virtual for relaxed}.\n Then we upper bound the revenue of $M$ with the benchmark in Theorem~\\ref{thm:revenue upperbound for subadditive} and argue that for any mechanism $M$, the corresponding benchmark can be upper bounded by the sum of the revenue of a few simple mechanisms.  As we allow $\\beta$, in other words the flow $\\lambda^{(\\beta)}$, to depend on the mechanism, our new approach may provide a better upper bound. As it turns out, our new upper bound is indeed easier to analyze.\n\n Lemma~\\ref{lem:requirement for beta} specifies the two properties of our $\\beta$ that play the most crucial roles in our analysis. We construct such a $\\beta$ in the proof of Lemma~\\ref{lem:requirement for beta}, however the construction is not necessarily unique and any $\\beta$ satisfying these two properties suffices. Note that our construction heavily relies on property \\textbf{(i)} of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:requirement for beta}\n\tFor any constant $b\\in (0,1)$ and any mechanism $M$, there exists a $\\beta$ such that: for the mechanism $M^{(\\beta)} $ constructed in Lemma~\\ref{lem:relaxed valuation} according to $\\beta$, any $i\\in[n]$ and $j\\in[m]$,\n\n\\noindent\\emph{\\textbf{(i)}} $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]\\leq b$;\n\n\\noindent\\emph{\\textbf{(ii)}} $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\/ b$, where $\\pi_{ij}^{(\\beta)}(t_i) = \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)$.\n\\end{lemma}\n\nBefore proving Lemma~\\ref{lem:requirement for beta}, we provide some intuition behind the two required properties.\nProperty \\textbf{(i)} is used to guarantee that if item $j$'s price for bidder $i$ is higher than $\\beta_{ij}$ for all $i$ and $j$ in an RSPM, for any item $j'$ and any bidder $i'$, $j'$ is still available with probability at least $(1-b)$ when $i'$ is visited. As for any bidder $k\\neq i'$ to purchase item $j'$,  $V_k(t_{kj'})$ must be greater than her price for item $j'$. By the union bound, the probability that there exists such a bidder is upper bounded by the LHS of property (i), and therefore is at most $b$. With this guarantee, we can easily show that the RSPM achieves good revenue (Lemma~\\ref{lem:neprev}). Property \\textbf{(ii)} states that the ex-ante probability for bidder $i$ to receive an item $j$ in $M^{(\\beta)}$ is not much bigger than the probability that bidder $i$'s value is larger than item $j$. This is crucial for proving our key Lemma~\\ref{lem:hat Q}, in which we argue that two different valuations provide comparable welfare under the same allocation rule $\\sigma^{(\\beta)}$. With Lemma~\\ref{lem:hat Q}, we can show that the ASPE obtains good revenue.\n\n\\begin{prevproof}{Lemma}{lem:requirement for beta}\n\tWhen there is only one buyer, we can simply set every $\\beta_j$ to be $0$ and both conditions are satisfied.\n\tWhen there are multiple players, we let $$\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x\\right] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\},$$ where $ \\pi_{ij}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}(t_i)$. Clearly, when the distribution of $V_i(t_{ij})$ is continuous, then\n\\begin{equation}\\label{equ:beta_second_condition}\n\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i),\n\\end{equation}\nand therefore for any $j$, $$\\sum_i\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\leq b.$$\n\nSo the first condition is satisfied. The second condition holds because by the first property in Lemma~\\ref{lem:relaxed valuation}, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$.\n\nWhen the distribution for $V_i(t_{ij})$ is discrete, it is possible that Equation~\\ref{equ:beta_second_condition} does not hold, but this is essentially a tie breaking issue and not hard to fix. Let $\\epsilon>0$ be an extremely small constant that is smaller than $\\left|V_i(t_{ij})-V_i(t'_{ij})\\right|$ for any $t_{ij},  t'_{ij}\\in T_{ij}$, any $i$ and any $j$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[0,\\epsilon]$, and think of it as a random rebate that the seller gives to bidder $i$ when she purchases item $j$. Now we modify the definition of $\\beta_{ij}$ as $\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij},\\zeta_{ij}}[V_i(t_{ij})+\\zeta_{ij}\\geq x] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\}.$\n\n\\notshow{\\begin{equation}\n\\epsilon_1=\\epsilon\\cdot \\frac{\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)}{\\Pr_{t_{ij}}\\left[V_i(t_{ij})= \\beta_{ij}\\right]}\n\\end{equation}\n\nBy the definition of $\\beta_{ij}$, $\\epsilon_1\\in [0,\\epsilon)$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[\\epsilon_1-\\epsilon,\\epsilon_1]$.  It is not hard to verify that $\\Pr_{t_{ij},\\zeta_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+\\zeta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$. For those probabilities related to $\\beta_{ij}$ shown in the proofs below, which is written for simplification, we refer to this definition. With regard to this definition, we make essentially small changes for the mechanism described below. Instead of using fixed price, add a small disturbance $\\zeta_{ij}$ on item $j$'s price for bidder $i$. Since $\\epsilon$ can be chosen as small as possible, the revenue will only be affected by a small constant. All the argument maintain to be true.}\n\nBoth of the two properties in Lemma~\\ref{lem:requirement for beta} hold if we replace $V_i(t_{ij})$ with $V_i(t_{ij})+\\zeta_{ij}$. The only change we need to make in the mechanism is to actually give the bidders $\\zeta_{ij}$ as the corresponding rebate. Since we can choose $\\epsilon$ to be arbitrarily small, the sum of the rebate is also arbitrarily small.  For the simplicity of the presentation, we will omit $\\zeta_{ij}$ and $\\epsilon$ in the rest of the paper.\nThe random rebate indeed makes our mechanism randomized(according to the random variable $\\zeta_{ij}\\sim [0,\\epsilon]$). However, the randomized mechanism is a uniform distribution of deterministic DSIC mechanisms (after determining all $\\zeta_{ij}$), and the expected revenue of the randomized mechanism is simply the average revenue of all these deterministic mechanisms. Therefore, there must be one realization of the rebates such that the corresponding deterministic mechanism has revenue above the expectation, i.e., the expected revenue of the randomized one. Thus if the randomized mechanism is proved to achieve some approximation ratio, there must exist a deterministic one that achieves the same ratio. The deterministic mechanism will use a fixed value $z_{ij}\\in [0,\\epsilon]$ as the rebate.\n\nSimilarly, the same issue about discrete distributions arises when we define some other crucial parameters later, e.g., in the Definition of $c$, $c_i$ and $\\tau_i$. We can resolve all of them together using the trick (adding a random rebate) described above, and we will not include a detailed proof for those cases.\n\\end{prevproof}\n\n\n\n\n\\subsubsection{Bad Example for Chawla and Miller's Approach}\\label{sec:cs_ocrs}\nLet bidders be constrained additive and $\\mathcal{F}_i$ be bidder $i$ feasibility constraint. We use $P_{\\mathcal{F}_i}=conv(\\{1^S | S\\in \\mathcal{F}_i\\})$ to denote the feasibility polytope of bidder $i$. Let $\\{q_{ij}\\}_{i\\in[n],j\\in[m]}$ be a collection of probabilities that satisfy $\\sum_i q_{ij}\\leq 1\/2$ for all item $j$ and $\\boldsymbol{q}_i = (q_{i1},\\ldots, q_{im})\\in b\\cdot P_{\\mathcal{F}_i}$. Let $\\beta_{ij}=F_{ij}^{-1}(q_{ij})$. The analysis by Chawla and Miller~\\cite{ChawlaM16} needs to upper bound $\\sum_{i, j}\\beta_{ij}\\cdot q_{ij}$ using the revenue of some BIC mechanism. When $\\mathcal{F}_i$ is a matroid for every bidder $i$, this expression can be upper bounded by the revenue of a sequential posted price mechanism constructed using OCRS from~\\cite{FeldmanSZ16}. Here we show that if the bidders have general downward closed feasibility constraints, this expression is gigantic. More specifically, we prove that even when there is only one bidder, the expression could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare.\n\nConsider the following example.\n\\begin{example}\\label{ex:counterexample ocrs}\n\tThe seller is selling $m=k^2$ items to a single bidder. The bidder's value for each item is drawn i.i.d. from distribution $F$, which is the equal revenue distribution truncated at $k$, i.e.,\n\t\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x<k\\\\\n1,&\\text{if}~~x=k\n\\end{cases}\n\\]\n Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The bidder is additive subject to feasibility constraint $\\mathcal{F}=\\left\\{S\\subseteq [m]|\\exists i\\in[k], S\\subseteq A_i\\right\\}$.\n\\end{example}\n\n\\begin{lemma}\\label{lem:counter example}\nLet $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope for the bidder in Example~\\ref{ex:counterexample ocrs}. Let $SW$ be the optimal social welfare. Then for any constant $b>0$, there exists $q\\in b\\cdot P_{\\mathcal{F}}$ such that for sufficiently large $k$, $$\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{\\log k })\\cdot SW$$.\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule: w.p. $(1-b)$, don't allocate anything, and w.p. $b$, give the buyer one of the sets $A_i$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in b\\cdot P_{\\mathcal{F}}$. Since $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have $\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2$. We use $V_i$ to denote the random variable of the bidder's value for set $A_i$. It is not hard to see that $SW={\\mathbb{E}}[\\max_{i\\in[k]} V_i]$. \n\n\\begin{lemma}\nFor any $i\\in [k]$,\n\\[\\Pr\\left[V_i>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=2k$, and $|X|\\leq k$.\nFor every $i$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_i-k\\log(k)>t\\right]\\leq \\exp\\left(-\\frac{\\frac{1}{2}t^2}{2k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_i>3k\\log(k)\\right]\\leq \\exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nBy the union bound, $\\Pr[\\max_{i\\in[k]}V_i>3\\cdot k\\log(k)]\\leq k^{-2}$. Therefore, ${\\mathbb{E}}[\\max_{i\\in[k]} V_i]\\leq 3 k\\log k +k^2\\cdot k^{-2}\\leq 4 k\\log k$.\n\\end{proof}\n\n\\notshow{In the analysis of the paper by Chawla and Miller~\\cite{ChawlaM16}, they rely on the following lemma in single buyer auction.\n\\begin{lemma}\\label{lem:shuchi}\n~\\cite{ChawlaM16}~\\cite{FeldmanSZ16}Suppose the buyer is additive within a matroid constraint $\\mathcal{F}$ and let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. For any constant $b\\in (0,1)$ and ex-ante vector $q\\in bP_{\\mathcal{F}}$, let $\\beta$ be the corresponding ex-ante prices. In other words, $\\beta_j$ is chosen such that the probability that the value for item $j$\nexceeds this price is precisely $q_j$. Then the value $\\beta\\cdot q$ can be bounded within some constant factor by the revenue of a posted price mechanism with a more strict constraint, which guarantees that the ex-ante probability of the buyer getting item $j$ is at most $q_j$.\n\\end{lemma}\n\nThe result can be generated for $\\mathcal{F}$ beyond a matroid~\\cite{FeldmanSZ16}. However, Lemma~\\ref{lem:shuchi} does not hold for general downward-close $\\mathcal{F}$. In this section we provide a counterexample with some general downward-close constraint $\\mathcal{F}$, such that the term $\\beta\\cdot q$ cannot be upper bounded by any single buyer mechanism, within in a constant factor.\n\n\\begin{lemma}\nConsider the following single buyer auction with $m=k^2$ i.i.d. items. Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The value distribution $F$ for a single item is defined as the equal-revenue distribution truncated at value $k$, i.e.,\n\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x<k\\\\\n1,&\\text{if}~~x=k\n\\end{cases}\n\\]\nDefine a downward-close feasibility constraint $\\mathcal{F}$ for the buyer as $\\mathcal{F}=\\{S\\subseteq [m]|\\exists p, S\\subseteq A_p\\}$. Let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. Let $SW$ be the optimal social welfare among all BIC mechanisms in this auction. Then for any $b>0$, there exists $q\\in bP_{\\mathcal{F}}$ such that for sufficiently large $k$,\n\\begin{equation}\\label{equ:q times beta}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{log(k)})\\cdot SW\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule. With probability $(1-b)$, don't allocate anything. With probability $b$, give the buyer one of the sets $A_p$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in bP_{\\mathcal{F}}$.\n\nSince $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have\n\\begin{equation}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2\n\\end{equation}\n\nConsider the optimal social welfare. For every bundle $p$, denote $V_p$ the random variable of the buyer's value for bundle $p$. Notice $A_p\\in \\mathcal{F}$, $V_p$ is the sum of $k$ independent random variables with cdf $F$. With Bernstein Inequality, $V_p=O(klog(k))$ with high probability.\n\\notshow{\\begin{lemma}\\label{lem:bernstein}\n(Bernstein Inequality)~\\cite{bernstein1924modification}: Suppose $X_1,...,X_n$ are independent random variables with zero mean, and $|X_i|\\leq M$ almost surely for all $i$. Then for any $t>0$,\n\\[\\Pr\\left[\\sum_{i=1}X_i>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{\\sum_{i=1}^nE[X_i^2]+\\frac{1}{3}Mt}\\right)\\]\n\\end{lemma}}\n\\begin{lemma}\nFor any $p\\in [k]$,\n\\[\\Pr\\left[V_p>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=k-1$, and $|X|\\leq k$.\nFor every $p$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_p-k\\log(k)>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_p>3k\\log(k)\\right]\\leq exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nWith union bound,\n\\[\\Pr[\\max_{p\\in [k]}V_p\\geq 3k\\log(k)]\\leq \\sum_{p\\in [k]}\\Pr\\left[V_p>3k\\log(k)\\right]\\leq k^{-2}\\]\n\nNotice the social welfare for the mechanism is at most \\\\\n\\noindent$\\max_{p\\in [k]}V_p$ due to the feasible constraint $\\mathcal{F}$. Also notice that $\\max_{p\\in [k]}V_p\\leq k^2$, we have\n\\begin{align*}\nSW&\\leq E[\\max_{p\\in [k]}V_p]\\leq \\bigg(3k\\log(k)\\cdot \\Pr[\\max_{p\\in [k]}V_p\\leq 3k\\log(k)]\\\\\n&+k^2\\cdot\\Pr[\\max_{p\\in [k]}V_p> 3k\\log(k)]\\bigg)\\\\\n&\\leq 3k\\log(k)+k^2\\cdot k^{-2}=O(k\\log(k))\n\\end{align*}\n\n\\noindent When $k$ is sufficiently large, Equation~\\ref{equ:q times beta} holds.\n\n\\begin{comment}\nNow consider the optimal revenue in this auction. Let $REV_k$ be the optimal revenue selling the items in set $A_k$ to a single buyer. \\ref{HartN12} has shown that $REV_k=\\Theta(n\\log(n))$ for the equal-revenue distribution $F(x)$. Since people are only interested one set of items,\n\\begin{equation}\nREV\\leq \\sum_{k\\in [n]}REV_k=\\Theta(n^2\\log(n))\n\\end{equation}\n\\end{comment}\n\n\\end{proof}}\n\n\n\n\\section{Introduction}\nIn Mechanism Design, we aim to design a mechanism\/system such that a group of strategic participants, who are only interested in optimizing their own utilities, are incentivized to choose actions that also help achieve the designer's objective. Clearly, the quality of the solution with respect to the designer's objective is crucial. However, perhaps one should also pay equal attention to another criterion of a mechanism, that is, its simplicity. When facing a complicated mechanism, participants may be confused by the rules and thus unable to optimize their actions and react in unpredictable ways instead. This may lead to undesirable outcomes and poor performance of the mechanism. An ideal mechanism would be optimal and simple. However, such cases of simple mechanisms being optimal only exist in single-item auctions, with the seminal examples of auctions by Vickrey~\\cite{Vickrey61} and Myerson~\\cite{Myerson81}, while none has been discovered in broader settings. Indeed, we now know that even in fairly simple settings the optimal mechanisms suffer many undesirable properties including randomization, non-monotonicity, and others~\\cite{RochetC98, Tha04, Pavlov11a, HartN13, HartR12, BriestCKW10, DaskalakisDT13, DaskalakisDT14}.\nTo move forward, one has to compromise -- either settle with optimal but somewhat complex mechanisms or turn to simple but approximately optimal solutions.\n\nRecently, there has been extensive research effort focusing on the latter approach, that is, studying the performance of simple mechanisms through the lens of approximation. In particular, a central problem on this front is how to design simple and approximately revenue-optimal mechanisms in multi-item settings. For instance, when bidders have unit-demand valuations, we know sequential posted price mechanisms approximates the optimal revenue due to a line of work initiated by Chawla et al.~\\cite{ChawlaHK07, ChawlaHMS10, ChawlaMS15, CaiDW16}. When buyers have additive valuations, we know that either selling the items separately or running a VCG mechanism with per bidder entry fee approximates the optimal revenue due to a series of work initiated by Hart and Nisan~\\cite{HartN12, CaiH13, LiY13, BabaioffILW14, Yao15, CaiDW16}. Recently, Chawla and Miller~\\cite{ChawlaM16} generalized the two lines of work described above to matroid rank functions\\footnote{{Here is a hierarchy of the valuation functions. additive \\& unit-demand $\\subseteq$ matroid rank $\\subseteq$ constrained additive \\&  submodular\n $\\subseteq$ XOS $\\subseteq$ subadditive. A function is constrained additive if it is additive up to some downward closed feasibility constraints. The class of submodular functions is neither a superset nor a subset of the class of constrained additive functions.} See Definition~\\ref{def:valuation classes} for the formal definition. }. They show that a simple mechanism, the sequential two-part tariff mechanism, suffices to extract a constant fraction of the optimal revenue. For subadditive valuations beyond matroid rank functions, we only know how to handle a single buyer~\\cite{RubinsteinW15}\\footnote{All results mentioned above assume that the buyers' valuation distributions are over independent items. For additive and unit-demand valuations, this means a bidder's values for the items are independent. The definition is generalized to subadditive valuations by Rubinstein and Weinberg~\\cite{RubinsteinW15}. See Definition~\\ref{def:subadditive independent}.}. It is a major open problem to extend this result to multiple subadditive buyers.\n\nIn this paper, we unify and strengthen all the results mentioned above via an extension of the duality framework proposed by Cai et al.~\\cite{CaiDW16}. Moreover, we show that even when there are multiple buyers with XOS valuation functions, there exists a simple, deterministic and Dominant Strategy Incentive Compatible (DSIC) mechanism that achieves a constant fraction of the optimal Bayesian Incentive Compatible (BIC) revenue\\footnote{A mechanism is Bayesian Incentive Compatible (BIC) if it is in every bidder's interest to tell the truth, assuming that all other bidders' reported their values. A mechanism is Dominant Strategy Incentive Compatible (DSIC) if it is in every bidder's interest to tell the truth no matter what reports the other bidders make.}. For subadditive valuations, our approximation ratio degrades to $O(\\log m)$.\n\n\\begin{informaltheorem}\n\tThere exists a simple, deterministic and DSIC mechanism that achieves a constant fraction of the optimal BIC revenue in multi-item settings, when the buyers' valuation distributions are XOS over independent items. When the buyers' valuation distributions are subadditive over independent items, our mechanism achieves at least $\\Omega(\\frac{1}{\\log m})$ of the optimal BIC revenue, where $m$ is the number of items.\n\\end{informaltheorem}\n\nThe original paper by Cai et al.~\\cite{CaiDW16} provided a unified treatment  for additive and unit-demand valuations. However, it is inadequate to provide an analyzable benchmark for even a single subadditive bidder. In this paper, we show how to extend their duality framework to accommodate general subadditive valuations. Using this extended framework, we substantially improve the approximation ratios for many of the settings discussed above, and in the meantime generalize the results to broader cases. See Table~\\ref{table:comp} for the comparison between the best ratios reported in the literature and the new ratios obtained in this work.\n\n\\begin{table*}\n\n\\centering\n\\begin{tabular}{|c|l|p{2.1cm}|c|p{2cm}|c|c|}\n\\hline\n\t& &\\centering Additive or Unit-demand& \\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2.2cm}{\\centering Matroid-Rank}}}& \\centering Constrained Additive&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1cm}{\\centering XOS}}}&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2cm}{\\centering Subadditive}}} \\\\\n\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Single Buyer}}} & Previous&      6~\\cite{BabaioffILW14} or  4~\\cite{ChawlaMS15}& 31.1* &\\centering 31.1~\\cite{ChawlaM16} & 338* & 338~\\cite{RubinsteinW15} \\\\\\cline{2-7}\n\t\t&This Paper & \\centering - &{11*} &\\centering{11}&{40*}&  {40} \\\\\\cline{2-7}\n\n\\hline\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Multiple Buyer}}} & Previous&     8~\\cite{CaiDW16}  or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}&\\centering ? & ? &? \\\\ \\cline{2-7}\n\t\t& This Paper& \\centering - & 70*  &\\centering 70 & 268 &$O(\\log m)$  \\\\\n\\hline\n\\end{tabular}\\\\\n* The result is implied by another result for a more general setting.\n\\caption{Comparison of approximation ratios between previous and current work.}\n\t\t  \\label{table:comp}\n\\end{table*}\n\n\\notshow{\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\\hline\n\t\t                  & Additive or Unit-demand& Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer&      6~\\cite{BabaioffILW14} or  4~\\cite{ChawlaMS15}& 33.1~\\cite{ChawlaM16} &$\\rightarrow$ &$\\rightarrow$ & 338~\\cite{RubinsteinW15} \\\\\\hline\n\t\tMultiple Buyers&     8~\\cite{CaiDW16}  or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}& ? & ? &? \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t  \\caption{The best approximation ratios known prior to this work.}\n\t\t  \\label{table:old}\n\\end{table}\n\n\\begin{table}\\centering\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\\hline\n\t\t                   &Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer&  $\\rightarrow$ &14& $\\rightarrow$& 48 \\\\\\hline\n\t\tMultiple Buyers& $\\rightarrow$  & 70 & 268 &$O(\\log m)$ \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t  \\caption{New approximation ratios obtained in this work.}\n\t\t  \\label{table:new}\n\\end{table}\n\n}\nOur mechanism is either a \\emph{rationed sequential posted price mechanism} (\\textbf{RSPM}) or an \\emph{anonymous sequential posted price with entry fee mechanism} (\\textbf{ASPE}). In an RSPM, there is a price $p_{ij}$ for buyer $i$ if she wants to buy item $j$, and she is allowed to purchase at most one item. We visit the buyers in some arbitrary order and the buyer takes her favorite item among the available items given the item prices for her. Here we allow personalized prices, that is, $p_{ij}$ could be different from $p_{kj}$ if $i\\neq k$. In an ASPE, every buyer faces the same collection of item prices $\\{p_j\\}_{j\\in[m]}$. Again, we visit the buyers in some arbitrary order. For each buyer, we show her the available items and the associated price for each item. Then we ask her to pay the entry fee to enter the mechanism, which may depend on what items are still available and the identity of the buyer. If the buyer accepts the entry fee, she can proceed to purchase any item at the given prices; if she rejects the entry fee, then she will leave the mechanism without receiving anything. Given the entry fee and item prices, the decision making for the buyer is straightforward, as she only accepts the entry fee when the surplus for winning her favorite bundle is larger than the entry fee. Therefore, both RSPM and ASPE are DSIC and ex-post Individually Rational (ex-post IR).\n\n\n\\subsection{Our Contributions}\nTo obtain the new generalizations, we provide important extensions to the duality framework in~\\cite{CaiDW16}, as well as novel analytic techniques and new simple mechanisms.\n\n\\vspace{.05in}\n\\noindent \\textbf{1. Accommodating subadditive valuations:} the original duality framework in~\\cite{CaiDW16} already unified the additive case and unit-demand case by providing an approximately tight upper bound for the optimal revenue using a single dual solution. A trivial upper bound for the revenue is the social welfare, which may be arbitrarily bad in the worst case. The duality based upper bound in~\\cite{CaiDW16} improves this trivial upper bound, the social welfare, by substituting the value of each buyer's favorite item with the corresponding Myerson's virtual value. However, the substitution is viable only when the following condition holds -- the buyer's marginal gain for adding an item solely depends on her value for that item (assuming it's feasible to add that item\\footnote{WLOG, we can reduce any constrained additive valuation to an additive valuation with a  feasibility constraint (see Definition~\\ref{def:valuation classes})}), but not the set of items she has already received. This applies to valuations that are additive, unit-demand and more generally constrained additive, but breaks under more general valuation functions, e.g., submodular, XOS or subadditive valuations. As a consequence, the original dual solution from~\\cite{CaiDW16} fails to provide a nice upper bound for more general valuations. To overcome this difficulty, we take a different approach. Instead of directly studying the dual of the original problem, we first relax the valuations and argue that the optimal revenue of the relaxed valuation is comparable to the original one. Then, since we choose the relaxation in a particular way, by applying a dual solution similar to the one in~\\cite{CaiDW16} to the relaxed valuation, we recover an upper bound of the optimal revenue for the relaxed valuation resembling the appealing format of the one in~\\cite{CaiDW16}. Combining these two steps, we obtain an upper bound for subadditive valuations that is easy to analyze. Indeed, we use our new upper bound to improve the approximation ratio for a single subadditive buyer from $338$~\\cite{RubinsteinW15}  to $40$. See Section~\\ref{sec:valuation relaxation} for more details.\n\n\\vspace{.05in}\n\\noindent\\textbf{2. An adaptive dual:}  our second major change to the framework is that we choose the dual in an adaptive manner. In~\\cite{CaiDW16}, a dual solution $\\lambda$ is chosen up front inducing a virtual value function $\\Phi(\\cdot)$, then the corresponding optimal virtual welfare is used as a benchmark for the optimal revenue. Finally, it is shown that the revenue of some simple mechanism is within a constant factor of the optimal virtual welfare. Unfortunately, when the valuations are beyond additive and unit-demand, the optimal virtual welfare for this particular choice of virtual value function becomes extremely complex and hard to analyze. Indeed, it is already challenging to bound when the buyers' valuations are $k$-demand. In this paper, we take a more flexible approach. For any particular allocation rule $\\sigma$, we tailor a special dual $\\lambda^{(\\sigma)}$ based on $\\sigma$ in a fashion that is inspired by Chawla and Miller's ex-ante relaxation~\\cite{ChawlaM16}. Therefore, the induced virtual valuation $\\Phi^{(\\sigma)}$ also depends on $\\sigma$. By duality, we can show that the optimal revenue obtainable by $\\sigma$ is still upper bounded by the virtual welfare with respect to $\\Phi^{(\\sigma)}$ under allocation rule $\\sigma$. Since the virtual valuation is designed specifically for allocation $\\sigma$, the induced virtual welfare is much easier to analyze. Indeed, we manage to prove that for any allocation $\\sigma$ the induced virtual welfare is within a constant factor of the revenue of some simple mechanism, when bidders have XOS valuations. See Section~\\ref{sec:virtual for relaxed} and~\\ref{sec:choice of beta} for more details.\n\n\n\\vspace{.05in}\n\n\\noindent\\textbf{3. A novel analysis and new mechanism:} with the two contributions above, we manage to derive an upper bound of the optimal revenue similar to the one in \\cite{CaiDW16} but for subadditive bidders. The third major contribution of this paper is a novel approach to analyzing this upper bound. The analysis in~\\cite{CaiDW16} essentially breaks the upper bound into three different terms-- \\textsc{Single}, \\textsc{Tail}~ and \\textsc{Core}, and bound them separately. All three terms are relatively simple to bound for additive and unit-demand buyers, but for more general settings the $\\textsc{Core}$ becomes much more challenging to handle. Indeed, the analysis in~\\cite{CaiDW16} was  insufficient to tackle the $\\textsc{Core}$ even when the buyers have $k$-demand valuations\\footnote{The class of $k$-demand valuations is a generalization of unit-demand valuations, where the buyer's value is additive up to $k$ items.}-- a very special case of matroid rank valuations, which itself is a special case of XOS or subadditive valuations. Rubinstein and Weinberg~\\cite{RubinsteinW15} showed how to approximate the $\\textsc{Core}$ for a single subadditive bidder using grand bundling, but their approach does not apply to multiple bidders. Yao~\\cite{Yao15} showed how to approximate the $\\textsc{Core}$ for multiple additive bidders using a VCG with per bidder entry fee mechanism, but again it is unclear how his approach can be extended to multiple k-demand bidders. A recent paper by Chawla and Miller~\\cite{ChawlaM16} finally broke the barrier of analyzing the $\\textsc{Core}$ for multiple $k$-demand buyers. They showed how to bound the $\\textsc{Core}$ for matroid rank valuations  using a sequential posted price mechanism by applying the \\emph{online contention resolution scheme (OCRS)} developed by Feldman et al.~\\cite{FeldmanSZ16}. The connection with OCRS is an elegant observation, and one might hope the same technique applies to more general valuations. Unfortunately, OCRS is only known to exist for special cases of downward closed constraints, and as we show in Section~\\ref{sec:core comparison}, the approach by Chawla and Miller cannot yield any constant factor approximation for general constrained additive valuations.\n\nWe take an entirely different approach to bound the $\\textsc{Core}$. Here we provide some intuition behind our mechanism and analysis. The $\\textsc{Core}$ is essentially the optimal social welfare induced by some truncated valuation $v'$, and our goal is to design a mechanism that extracts a constant fraction of the welfare as revenue. Let $M$ be any sequential posted price mechanism. A key observation is that when bidder $i$'s valuation is subadditive over independent items, her utility in $M$, which is the largest surplus she can achieve from the unsold items, is also subadditive over independent items. If we can argue that her utility function is $a$-Lipschitz (Definition~\\ref{def:Lipschitz}) with some small $a$,  Talagrand's concentration inequality~\\cite{Talagrand1995concentration,Schechtman2003concentration} allows us to set an entry fee for the bidder so that we can extract a constant fraction of her utility just through the entry fee.\nIf we modify $M$ by introducing an entry fee for every bidder, according to Talagrand's concentration inequality, the new mechanism $M'$ should intuitively have revenue that is a constant fraction of the social welfare obtained by $M$~\\footnote{$M$'s welfare is simply its revenue plus the sum of utilities of the bidders, and $M'$ can extract some extra revenue from the entry fee, which is a constant fraction of the total utility from the bidders.}. Therefore, if there exists a sequential posted price mechanism $M$ that achieves a constant fraction of the optimal social welfare under the truncated valuation $v'$, the modified mechanism $M'$ can obtain a constant fraction of $\\textsc{Core}$ as revenue. Surprisingly, when the bidders have XOS valuations, Feldman et al.~\\cite{FeldmanGL15} showed that there exists an anonymous sequential posted price mechanism that always obtains at least half of the optimal social welfare. Hence, an anonymous sequential posted price with per bidder entry fee mechanism should approximate the $\\textsc{Core}$ well, and this is exactly the intuition behind our ASPE mechanism.\n\n To turn the intuition into a theorem, there are two technical difficulties that we need to address: (i) the Lipschitz constants of the bidders' utility functions turn out to be too large (ii) we deliberately neglected the difference in bidders' behavior under $M$ and $M'$ in hope to keep our discussion in the previous paragraph intuitive. However, due to the entry fee, bidders may end up purchasing completely different items under $M$ and $M'$, so it is not straightforward to see how one can relate the revenue of $M'$ to the welfare obtained by $M$.\n  See Section~\\ref{sec:core comparison} for a more detailed discussion on how we overcome these two difficulties.\n \n\n\n\n\n\n\n\n\n\n\\subsection{Related Work}\nIn recent years, we have witnessed several breakthroughs in designing (approximately) optimal mechanisms in multi-dimensional settings. The black-box reduction by Cai et al.~\\cite{CaiDW12a,CaiDW12b,CaiDW13a,CaiDW13b} shows that we can reduce any Bayesian mechanism design problem to a similar algorithm design problem via convex optimization. Through their reduction, it is proved that all optimal mechanisms can be characterized as a distribution of virtual welfare maximizers, where the virtual valuations are computed by an LP. Although this characterization provides important insights about the structure of the optimal mechanism, the optimal allocation rule is unavoidably randomized and might still be complex as the virtual valuations are only a solution of an LP.\n\nAnother line of work considers the ``Simple vs. Optimal'' auction design problem. For instance, a sequence of results~\\cite{ChawlaHK07,ChawlaHMS10,ChawlaMS10,ChawlaMS15} show that sequential posted price mechanism can achieve $\\frac{1}{33.75}$ of the optimal revenue, whenever the buyers have unit-demand valuations over independent items. Another series of results~\\cite{HartN12,CaiH13,LiY13,BabaioffILW14,Yao15} show that the better of selling the items separately and running the VCG mechanism with per bidder entry fee achieves $\\frac{1}{69}$ of the optimal revenue, whenever the buyers' valuations are additive over independent items. Cai et al.~\\cite{CaiDW16} unified the two lines of results and improved the approximation ratios to $\\frac{1}{8}$ for the additive case and $\\frac{1}{24}$ for the unit-demand case using their duality framework.\n\nSome recent works have shown that simple mechanisms can approximate the optimal revenue even when buyers have more sophisticated valuations. For instance, Chawla and Miller~\\cite{ChawlaM16} showed that the sequential two-part tariff mechanism can approximate the optimal revenue when buyers have matroid rank valuation functions over independent items. Their mechanism requires every buyer to pay an entry fee up front, and then run a sequential posted price mechanism on buyers who have accepted the entry fee. Our ASPE is similar to their mechanism, but with two major differences: (i) since buyers are asked to pay the entry fee before the seller visits them, the buyers have to make their decisions based on the expected utility (assuming every other buyer behaves truthfully) they can receive. Hence, the mechanism is only guaranteed to be BIC and interim IR. While in our mechanism, the buyers can see what items are still available before paying the entry fee, therefore the decision making is straightforward and the ASPE is DSIC and ex-post IR; (ii) the item prices in the ASPE are anonymous, while in the sequential two-part tariff mechanism, personalized prices are allowed. For valuations beyond matroid rank functions, Rubinstein and Weinberg~\\cite{RubinsteinW15} showed that for a single buyer whose valuation is subadditive over independent items, either grand bundling or selling the items separately achieves at least $\\frac{1}{338}$ of the optimal revenue.\n\nThe Cai-Devanur-Weinberg duality framework~\\cite{CaiDW16} has been applied to other intriguing Mechanism Design problems. For example, Eden et al. showed that the better of selling separately and bundling together gets an $O(d)$-approximation for a single bidder with ``complementarity-$d$ valuations over independent items''~\\cite{EdenFFTW16a}. The same authors also proved a Bulow-Klemperer result for regular i.i.d. and constrained additive bidders~\\cite{EdenFFTW16b}. Liu and Psomas provided a Bulow-Klemperer result for {dynamic auctions}~\\cite{LiuP16}. Finally, Brustle et al.~\\cite{BrustleCWZ17} extended the duality framework to two-sided markets and used it to design simple mechanisms for approximating the Gains from Trade.\n\nStrong duality frameworks have recently been developed for one additive buyer~\\cite{DaskalakisDT13,DaskalakisDT15,Giannakopoulos14a,GiannakopoulosK14,GiannakopoulosK15}. These frameworks show that the dual problem of revenue maximization can be viewed as an optimal transport\/bipartite matching problem. Hartline and Haghpanah provided an alternative duality framework in~\\cite{HartlineH15}. They showed that if certain paths exist, these paths provide a witness of the optimality of a certain Myerson-type mechanism, but these paths are not guaranteed to exist in general. Similar to the Cai-Devanur-Weinberg framework, Carroll~\\cite{Carroll15} independently made use of a partial Lagrangian over incentive constraints. These duality frameworks have been successfully provide conditions under which a certain type of mechanism is optimal when there is a single unit-demand or additive bidder. However, none of these frameworks succeeds in yielding any approximately optimal results in multi-buyer settings.\n\n\\input{prelim}\n\\input{roadmap}\n\\input{duality}\n\\input{single_subadditive}\n\\input{multi_XOS}\n\\input{example_ocrs}\n\\newpage\n\n\\section{Multiple Bidders}\\label{sec:multi}\n\nIn this section, we prove our main result -- simple mechanisms can approximate the optimal BIC revenue even when there are multiple XOS\/subadditive bidders.\nFirst, we need the definition of supporting prices.\n\\begin{definition}[Supporting Prices~\\cite{DobzinskiNS05}]\\label{def:supporting price}\nFor any $\\alpha\\geq 1$, a type $t$ and a subset $S\\subseteq[m]$, prices $\\{p_j\\}_{j\\in S}$\nare $\\alpha$-supporting prices for $v(t,S)$ if \\textbf{(i)}\t$v(t,S') \\geq \\sum_{j\\in S'} p_j$ for all $S'\\subseteq S$ and \\textbf{(ii)} $\\sum_{j\\in S}p_j\\geq \\frac{v(t,S)}{\\alpha}$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:multi}\nIf for any buyer $i$, any type $t_i\\in T_i$ and any bundle $S\\in [m]$, $v_i(t_i,S)$ has a set of $\\alpha$-supporting prices $\\{\\theta_j^{S}(t_i)\\}_{j\\in S}$, then for any BIC mechanism $M$ and any constant $b\\in (0, 1)$,\n\\begin{align*}\n\\textsc{Rev}(M,v,D)\\leq 32\\alpha \\cdot \\textsc{APostEnRev}\n+\\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev}\n\\end{align*}\n\n\\vspace{0.05in}\nIf $v_i(t_i,\\cdot)$ is an XOS valuation for all $i$ and $t_i\\in T_i$, then $\\alpha=1$. By setting $b$ to $\\frac{1}{4}$, we have $$\\textsc{Rev}(M,v,D)\\leq 236\\cdot\\textsc{PostRev}+32\\cdot\\textsc{APostEnRev}.$$ For general subadditive valuations, $\\alpha=O(\\log(m))$ by~\\cite{BhawalkarR11}, hence $$\\textsc{Rev}(M,v,D)\\leq O(\\log(m))\\cdot \\max\\{\\textsc{PostRev},\\textsc{APostEnRev}\\}.$$\n\\end{theorem}\nHere is a sketch of the proof for Theorem~\\ref{thm:multi}. We show how to upper bound $\\textsc{Single}(M,\\beta)$ in Lemma~\\ref{lem:multi_single}. Then, we decompose $\\textsc{Non-Favorite}(M,\\beta)$ into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ in Lemma~\\ref{lem:multi decomposition}. We show how to construct a simple mechanism to approximate $\\textsc{Tail}(M,\\beta)$ in Section~\\ref{subsection:tail} and how to approximate $\\textsc{Core}(M,\\beta)$ in Section~\\ref{subsection:core}.\n\n\\vspace{.1in}\n \\noindent\\textbf{Analysis of $\\textsc{Single}(M,\\beta)$:} \n\n\n\\begin{lemma}\\label{lem:multi_single}\nFor any mechanism $M$, $$\\textsc{Single}(M, \\beta)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one agent $(i,j)$ for each of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(M,\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M,\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$.\n\\end{proof}\n\n\n\\vspace{.05in}\n \\noindent\\textbf{Core-Tail Decomposition of $\\textsc{Non-Favorite}(M,\\beta)$:} we decompose $\\textsc{Non-Favorite}(M, \\beta)$ into two terms $\\textsc{Tail}(M, \\beta)$ and $\\textsc{Core}(M, \\beta)$\\footnote{In~\\cite{CaiDW16}, $\\textsc{Non-Favorite}$ is decomposed into four different terms $\\textsc{Under}$, $\\textsc{Over}$, $\\textsc{Core}$ and $\\textsc{Tail}$. We essentially merge the first three terms into $\\textsc{Core}(M, \\beta)$ in our decomposition.}. First, we need the following definition.\n\\begin{definition}\\label{def:c_i}\nFor every buyer $i$, let $c_i :=\\inf\\big\\{x\\geq 0:\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+x\\right]\\leq \\frac{1}{2}\\big\\}.$ For every $t_i \\in T_i$, let $\\mathcal{T}_i(t_i)=\\{j\\ |\\ V_i(t_{ij})\\geq \\beta_{ij}+c_i\\}$ and $\\mathcal{C}_i(t_i) = [m]\\backslash\\mathcal{T}_i(t_i)$.\n\\end{definition}\n Since $v_i(t_i,\\cdot)$ is subadditive for all $i$ and $t_i\\in T_i$, we have $v_i(t_i,S)\\leq v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. The term $\\textsc{Non-Favorite}(M,\\beta)$ can be decomposed into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ based on the inequality above. The complete proof of Lemma~\\ref{lem:multi decomposition} can be found in Appendix~\\ref{appx:multi}.\n\n \\begin{lemma}~\\label{lem:multi decomposition}\n\t\t\\begin{align*} &\\textsc{Non-Favorite}(M,\\beta)\\\\\n\t\t\t\\leq&  \\sum_i\\sum_{t_i}f_i(t_i) \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))~~~~~~~~~~(\\textsc{Core}(M,\\beta))\\\\\n\t\t\t+&\\sum_i\\sum_j  \\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot\\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]~~~~~~~(\\textsc{Tail}(M,\\beta))\n\t\t\\end{align*}\n\t\t\\end{lemma}\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\nIn this section we show how to bound $\\textsc{Tail}(M,\\beta)$ with the revenue of an RSPM.\n\\begin{lemma}\\label{lem:multi-tail}\n\tFor any BIC mechanism $M$, $\\textsc{Tail}(M, \\beta)\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{lemma}\n\nWe first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$\n\\begin{align*}\nr_{ij}&=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],\n\\end{align*}\n$r_i=\\sum_j r_{ij}$, and $r=\\sum_i r_i$. We show in the following Lemma that $r$ is an upper bound of $\\textsc{Tail}(M,\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M,\\beta)\\leq r.$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{proof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful for analyzing $\\textsc{Core}(M,\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be  a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by property (i) of Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$.\n\\end{proof}\n\n\\notshow{\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\n\\textsc{Tail}(M,\\beta)\\leq r\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n$$r= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{proof}\n}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n\\begin{equation}\\label{r and prev}\nr= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{prevproof}\n\n\n\nWe have done the analysis for $\\textsc{Tail}{(M,\\beta)}$. Before starting the analysis for $\\textsc{Core}{(M,\\beta)}$, we show that $r_i$ is within a constant factor of $c_i$. This Lemma is useful for bounding $\\textsc{Core}{(M,\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\n\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because when $c_i>0$,\n$\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]$ is at least $\\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Inequality~(\\ref{r and prev}), \n$\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\n\nIn this section we upper bound $\\textsc{Core}(M,\\beta)$. Recall that\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$$\nWe can view it as the welfare of another valuation function $v'$ under allocation $\\sigma^{(\\beta)}$ where $v'_i(t_i, S) = v_i(t_i,S\\cap\\mathcal{C}_i(t_i))$. In other words, we ``truncate'' the function at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold. The new function still satisfies monotonicity, subadditivity and no externalities.\n\nWe first compare existing methods for analyzing the $\\textsc{Core}$ with our approach before jumping into the proofs.\n\n\\subsubsection{Comparison between the Existing Methods and Our Approach}\\label{sec:core comparison}\nAs all results in the literature~\\cite{ChawlaHMS10, Yao15, CaiDW16,ChawlaM16} only study special cases of constrained additive valuations, we restrict our attention to constrained additive valuations in the comparison, but our approach also applies to XOS and subadditive valuations.\n\nWe compare our approach to the state of the art result by Chawla and Miller~\\cite{ChawlaM16}. They separate $\\textsc{Core}(M,\\beta) $ into two parts: (i) the welfare obtained from values below $\\beta$, and (ii) the welfare obtained from values between $\\beta$ and $\\beta+c$\\footnote{In particular, if bidder $i$ is awarded a bundle $S$ that is feasible for her, the contribution for the first part is $\\sum_{j\\in S} \\min\\left\\{\\beta_{ij},t_{ij}\\right\\}\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ and the contribution to the second part is $\\sum_{j\\in S} \\left(t_{ij}-\\beta_{ij}\\right)^+\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ }.\n It is not hard to show that the latter can be upper bounded by the revenue of a sequential posted price with per bidder entry fee mechanism.\n  Due to their choice of $\\beta$ (similar to the second property of Lemma~\\ref{lem:requirement for beta}), the former is upper bounded by $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$.\n   It turns out when every bidder's feasibility constraint is a matroid, one can use the OCRS from~\\cite{FeldmanSZ16} to design a sequential posted price mechanism to approximate this expression.\n    However, as we show in Example~\\ref{ex:counterexample ocrs}, $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$ could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare when the bidders have general downward closed feasibility constraints.\n     Hence, such approach cannot yield any constant factor approximation for general constrained additive valuations.\n\nAs explained in the intro, we take an entirely different approach. We first construct the posted prices $\\{Q_j\\}_{j\\in[m]}$ for our ASPE (Definition~\\ref{def:posted prices}), Feldman et al.~\\cite{FeldmanGL15} showed that the anonymous posted price mechanism with these prices achieves welfare $\\Omega\\left(\\textsc{Core}(M,\\beta)\\right)$. If all bidders have valuations that are subadditive over independent items, for any bidder $i$ and any set of available items $S$, $i$'s surplus for $S$ under valuation $v'_i(t_i, \\cdot)$ ($max_{S'\\subseteq S}~v'_i(t_i,S') -\\sum_{j\\in S'} Q_j$) is also subadditive over independent items. According to Talagrand's concentration inequality, the surplus concentrates and its expectation is upper bounded by its median and its Lipschitz constant $a$. One can extract at least half of the median by setting the median of the surplus as the entry fee. How about the Lipschitz constant $a$? Unfortunately, $a$ could be as large as $\\frac{1}{2}\\max_{j\\in[m]}\\{\\beta_{ij}+c_i\\}$, which is too large to be bounded.\n\nHere is how we overcome this difficulty. Instead of considering $v'$, we construct a new valuation $\\hat{v}$ that is always dominated by the true valuation $v$. We consider the social welfare induced by $\\sigma^{(\\beta)}$ under $\\hat{v}$ and define it as $\\widehat{\\textsc{Core}}(M,\\beta)$. In Section~\\ref{sec:proxy core}, we show that $\\widehat{\\textsc{Core}}(M,\\beta)$ is not too far away from $\\textsc{Core}(M,\\beta)$, so it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$ (Lemma~\\ref{lem:hat Q}). But why is $\\widehat{\\textsc{Core}}(M,\\beta)$ easier to approximate? The reason is two-fold. \\textbf{(i)} For any bidder $i$ and any set of available items $S$,  bidder $i$'s surplus for $S$ under $\\hat{v}_i(t_i,\\cdot)$ (defined as $\\mu_i(t_i,S)$ in Definition~\\ref{def:entry fee}, which is $max_{S'\\subseteq S}~ \\hat{v}_i(t_i,S') -\\sum_{j\\in S'} Q_j$), is not only subadditive over independent items, but also has a small Lipschitz constant $\\tau_i$ (Lemma~\\ref{lem:property of mu}). Indeed, these Lipschitz constants are so small that $\\sum_i \\tau_i$  and can be upper bounded by $\\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}). \\textbf{(ii)} If we set the entry fee of our ASPE to be the median of $\\mu_i(t_i,S)$ when $t_i$ is drawn from $D_i$, using a proof inspired by Feldman et al.~\\cite{FeldmanGL15}, we can show that our ASPE's revenue collected from the posted prices plus the expected surplus of the bidders (over the randomness of all bidders' types) approximates $\\widehat{\\textsc{Core}}(M,\\beta)$ (implied by Lemma~\\ref{lem:lower bounding mu}). Again by Talagrand's concentration inequality, we can bound bidder $i$'s expected surplus by our entry fee and $\\tau_i$ (Lemma~\\ref{lem:concentration entry fee}). As $\\hat{v}$ is always smaller than the true valuation $v$, thus for any type $t_i$ of bidder $i$ and any available items $S$, the surplus for $S$ under $v_i(t_i,\\cdot)$ must be larger than $\\mu_i(t_i,S)$, and the entry fee is accepted with probability at least $1\/2$. Putting everything together, we demonstrate that we can approximate $\\textsc{Core}(M,\\beta)$ with an ASPE or an RSPM (Lemma~\\ref{lem:upper bounding Q}).\n\\subsubsection{Construction of $\\widehat{\\textsc{Core}}(M,\\beta)$}\\label{sec:proxy core}\n\nWe first show that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{proof}\n\n\n\nNext, we define the prices of our ASPE.\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n{ $\\textsc{Core}(M,\\beta)$ can be upper bounded by $\\sum_{j\\in [m]}Q_j$. The proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).}\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta)$.\n\\end{lemma}\n\\begin{proof}\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\vspace{0.05in}\nIn the following definitions, we define $\\widehat{\\textsc{Core}}(M,\\beta)$ which is the welfare of another function $\\hat{v}$ under the same allocation $\\sigma^{(\\beta)}$.\n\n \n\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\}.$$\n\\end{definition}\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$. Moreover, let $$\\widehat{\\textsc{Core}}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{v}_i(t_i,S).$$\n\\end{definition}\n\nIn the next two Lemmas, we prove some useful properties of $\\tau_i$. In particular, we argue that $\\sum_{i\\in[n]} \\tau_i$ can be upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}).\n \\begin{lemma}\\label{lem:beta_ij}\n\\begin{align*}\n\\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\right]\n \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAccording to the definition of $\\tau_i$, for every buyer $i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$,\n and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. Our statement follows directly from Lemma~\\ref{lem:neprev}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n$$\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative,  \\begin{align*}\n  \\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\n  \\geq \\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right].\n  \\end{align*}\nAccording to the definition of $\\tau_i$, when $\\tau_i>0$, $$\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}.$\nTherefore, $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:beta_ij}.\n\\end{proof}\n\n\nIn the following two Lemmas, we compare $\\widehat{\\textsc{Core}}(M,\\beta)$ with $\\textsc{Core}(M,\\beta)$. The proof of Lemma~\\ref{lem:hat gamma} is postponed to Appendix~\\ref{appx:multi}.\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lem:hat gamma}, $\\sum_{j\\in[m]}\\hat{Q}_j\\leq \\widehat{\\textsc{Core}}(M,\\beta)\/2$. By Lemma~\\ref{lem:core and q_j}, $\\sum_{j\\in[m]}{Q}_j\\leq {\\textsc{Core}}(M,\\beta)\/2\\alpha$. Hence, Lemma~\\ref{lem:hat Q} shows that to approximate $\\textsc{Core}(M,\\beta)$, it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$. Indeed, we will use $\\sum_{j\\in[m]} \\hat{Q}_j$ as an proxy for $\\textsc{Core}(M,\\beta)$ in our analysis of the ASPE.\n\\subsubsection{Design and Analysis of Our ASPE}\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$ \\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\big(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\big).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n$\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\leq a]\\geq\\frac{1}{2}\\}$.\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nIn Lemma~\\ref{lem:property of mu}, we show that $\\tau_i$ is the Lipschitz constant for $\\mu_i(\\cdot,\\cdot)$ and the proof is postponed to Appendix~\\ref{appx:multi}. Moreover, $\\sum_i \\tau_i$ is upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:tau_i}.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\n\n\n\n\nThe following Lemma is crucial for our proof. We show that in expectation over all type profiles, we can lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i}))$ for all bidders. In particular, this lower bound plus our ASPE's revenue from the posted prices already approximates $\\widehat{\\textsc{Core}}(M,\\beta)$. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. For every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE. Then\n\n\\begin{align*}\n{\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]&\\geq \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot(2\\hat{Q}_j-Q_j)\\\\\n&\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\n\\end{align*}\n\n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq &\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\n\n\n\\vspace{0.05in}\nSince entry fee in the ASPE for every bidder as the median of her utility over the available items under $\\hat{v}$. Clearly, bidders accept the entry fee with probability at least $1\/2$, as their true utilities (under $v$) are always higher than their utilities under $\\hat{v}$. Combining the concentration property of the utility under $\\hat{v}$ and Lemma~\\ref{lem:lower bounding mu}, we can argue that the total revenue from our ASPE is comparable to $\\widehat{\\textsc{Core}}(M,\\beta)$, and therefore is comparable to $\\textsc{Core}(M,\\beta)$.\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j- \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $u_i(t_i,S) = \\max_{S'\\subseteq S} \\big( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\big).$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$\n\nFor any $i$ and $t_{<i}$, by Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}, we are able to derive a lower bound for $\\delta_i\\left(S_i(t_{<i})\\right)$, as shown in Lemma~\\ref{lem:concentration entry fee}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n\n\\begin{proof}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{proof}\n\n\\vspace{0.05in}\n\nBack to the proof of Lemma~\\ref{lem:entry fee acceptance probability}. According to Lemma~\\ref{lem:concentration entry fee}, the revenue from the entry fee is at least\n$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i$, which is equal to\n$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i$.\nCombining Lemma~\\ref{lem:tau_i} and Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $\\frac{1}{4}\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - (\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b(1-b)})\\textsc{PostRev}$. Since the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$, the total revenue of the ASPE is at least $\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$\n\\end{proof}\n\nCombining everything together, we have the main result of Section~\\ref{subsection:core}.\n\n\\begin{lemma}\\label{lem:upper bounding Q}\n\tFor any BIC mechanism $M$, {$\\textsc{Core}(M,\\beta)\\leq8\\alpha\\cdot\\textsc{APostEnRev}+4\\alpha\\left(\\frac{6}{1-b}+\\frac{1}{b(1-b)}\\right)\\textsc{PostRev}$.}\n\\end{lemma}\n\n\\begin{proof}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{proof}\n\nNow, we have upper bounded $\\textsc{Single}(M,\\beta)$, $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tThe proof is done by combining {Theorem~\\ref{thm:revenue upperbound for subadditive}}, Lemma~\\ref{lem:multi_single},~\\ref{lem:multi decomposition},~\\ref{lem:multi-tail} and~\\ref{lem:upper bounding Q}.\n\\end{prevproof}\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nWe focus on revenue maximization in the combinatorial auction with $n$ independent bidders and $m$ heterogenous items. Each bidder has a valuation that is \\textbf{subadditive over independent items} (see Definition~\\ref{def:subadditive independent}). We denote bidder $i$'s type $t_i$ as $\\langle t_{ij}\\rangle_{j=1}^m$, where $t_{ij}$ is bidder $i$'s private information about item $j$. For each $i$, $j$, we assume $t_{ij}$ is drawn independently from the distribution $D_{ij}$. Let $D_i=\\times_{j=1}^m D_{ij}$ be the distribution of bidder $i$'s type and $D=\\times_{i=1}^n D_i$ be the distribution of the type profile. We use $T_{ij}$ (or $T_i, T$) and $f_{ij}$ (or $f_i, f$) to denote the support and density function of $D_{ij}$ (or $D_i, D$). For notational convenience, we let  $t_{-i}$ to be the types of all bidders except $i$ and $t_{<i}$ (or $t_{\\leq i})$ to be the types of the first $i-1$ (or $i$) bidders. Similarly, we define $D_{-i}$, $T_{-i}$\n  and $f_{-i}$ for the corresponding  distributions, support sets and density functions. When bidder $i$'s type is $t_i$, her valuation for a set of items $S$ is denoted by $v_i(t_i,S)$.\n\n\\begin{definition}~\\cite{RubinsteinW15}\\label{def:subadditive independent}\nFor every bidder $i$, whose type is drawn from a product distribution $F_i=\\prod_j F_{ij}$, her distribution $\\mathcal{V}_i$ of valuation function $v_i(t_i,\\cdot)$ is \\textbf{subadditive over independent items} if:\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ has no externalities}, i.e., for each $t_i\\in T_i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ only depends on $\\langle t_{ij}\\rangle_{j\\in S}$, formally, for any $t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$, $v_i(t_i',S)=v_i(t_i,S)$.\n\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ is monotone}, i.e., for all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, $v_i(t_i,U)\\leq v_i(t_i,V)$.\n\n\\item \\textbf{- $v_i(\\cdot,\\cdot)$ is subadditive}, i.e., for all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, $v_i(t_i,U\\cup V)\\leq v_i(t_i,U)+ v_i(t_i,V)$.\n    \n\\end{itemize}\n\n We use $V_i(t_{ij})$ to denote $v_i(t_i,\\{j\\})$, as it only depends on $t_{ij}$. When $v_i(t_i,\\cdot)$ is XOS (or constrained additive) for all $i$ and $t_i\\in T_i$, we say $\\mathcal{V}_i$ is XOS (or constrained additive) over independent items.\n\\end{definition}\n\nWe first formally define various valuation classes.\n\\begin{definition}\\label{def:valuation classes}\nWe define several classes of valuations formally. Let $t$ be the type and $v(t,S)$ be the value for bundle $S\\in[m]$.\n\n\n\t\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item  \\textbf{Constrained Additive:}\n$v(t,S) =\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} v(t, \\{j\\})$, where $\\mathcal{I}\\subseteq 2^{[m]}$ is a downward closed set system over the items specifying the feasible bundles. In particular, when $\\mathcal{I}=2^{[m]}$, the valuation is an \\textbf{additive function}; when $\\mathcal{I}=\\{\\{j\\}\\ |\\ j\\in[m] \\}$, the valuation is a \\textbf{unit-demand function}; when $\\mathcal{I}$ is a matroid, the valuation is a \\textbf{matroid-rank function}. An equivalent way to represent any constrained additive valuations is to view the function as additive but the bidder is only allowed to receive bundles that are feasible, i.e., bundles in $\\mathcal{I}$. To ease notations, we interpret $t$ as an $m$-dimensional vector $(t_1, t_2,\\cdots, t_m)$ such that $t_j =  v(t, \\{j\\})$.\n\n\\item \\textbf{XOS\/Fractionally Subadditive:}\n $v(t,S) = \\max_{i\\in [{K}]} v^{(i)}(t, S)$, where ${K}$ is some finite number and $v^{(i)}(t,\\cdot)$ is an additive function for any $i\\in[{K}]$.\n\n\\item \\textbf{Subadditive:}\n$v(t,S_1\\cup S_2)\\leq v(t,S_1)+v(t,S_2)$ for any $S_1, S_2\\subseteq [m]$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{The following are a few examples of various valuation distributions which are over independent items (Definition~\\ref{def:subadditive independent}):\n\n\\begin{example}\\label{eg:valuation}\\cite{RubinsteinW15}\n$t=\\{t_j\\}_{j\\in[m]}$ where $t$ is drawn from $\\prod_j D_j$,\n\\begin{itemize}[leftmargin=0.7cm]\n\\item Additive: $t_j$ is the value of item $j$. $v(t,S)=\\sum_{j\\in S}t_j$.\n\\item Unit-demand: $t_j$ is the value of item $j$. $v(t,S)=\\max_{j\\in S}t_j$.\n\\item Constrained Additive: $t_j$ is the value of item $j$.\n $v(t,S)=$\\\\\n \\noindent$\\max_{R\\subseteq S, R\\in \\mathcal{I}}\\sum_{j\\in R} t_j$.\n\\item XOS\/Fractionally Subadditive: $t_j=\\{t_{j}^{(k)}\\}_{k\\in[K]}$ encodes all the possible values associated with item $j$, and $v(t,S)=$\\\\\n    \\noindent$\\max_{k\\in[K]}\\sum_{j\\in S}t_{j}^{(k)}$.\n\\end{itemize}\n\\end{example}\n\n\n}\n\n\nGiven  $D$ and $v=\\{v_i(\\cdot,\\cdot)\\}_{i\\in [n]}$, we use \\textbf{$\\textsc{Rev}(M,v,D)$} to denote the expected revenue of a BIC mechanism $M$. Throughout the paper, we use the following notations for the simple mechanisms we consider.\n\n\\vspace{.05in}\n\\noindent\\textbf{Single-Bidder Mechanisms:}\n\\vspace{.03in}\n\n\\noindent \\textbf{- $\\textsc{SRev}(v,D)$} denotes the optimal expected revenue achievable by any posted price mechanism that only allows the buyer to purchase at most one item, and we use $\\textsc{SRev}$ for short if there is no confusion\\footnote{The mechanism is slightly different from selling separately, as we only allow the buyer to purchase at most one item.}.\n\n\\noindent \\textbf{- $\\textsc{BRev}(v,D)$} denotes the optimal expected revenue achievable by selling a grand bundle and we use $\\textsc{BRev}$ for short if there is no confusion.\n\n\\vspace{.05in}\n\\noindent\\textbf{Multi-Bidder Mechanisms:}\n\\vspace{.03in}\n\n\\noindent\\textbf{- $\\textsc{PostRev}(v,D)$} denotes the optimal expected revenue achievable by selling the items via an RSPM to the bidders, and we use $\\textsc{PostRev}$ for short when there is no confusion.\n\n\\noindent\\textbf{- $\\textsc{APostEnRev}(v,D)$} denotes the optimal expected revenue achievable by selling the items via an ASPE to the bidders, and we use $\\textsc{APostEnRev}$ for short when there is no confusion.\n\n\\vspace{.05in}\n\\noindent\\textbf{Single-Dimensional Copies Setting:} In the analysis for unit-demand bidders in~\\cite{ChawlaHMS10, CaiDW16}, the optimal revenue is upper bounded by the optimal revenue in the single-dimensional copies setting defined in~\\cite{ChawlaHMS10}. We use the same technique. We construct $nm$ agents, where agent $(i,j)$ has value $V_i(t_{ij})$ of being served with $t_{ij}\\sim D_{ij}$, and we are only allow to use matchings, that is, for each $i$ at most one agent $(i,k)$ is served and for each $j$ at most one agent $(k,j)$ is served\\footnote{This is exactly the copies setting used in~\\cite{ChawlaHMS10}, if every bidder $i$ is unit-demand and has value $V_i(t_{ij})$ with type $t_i$. Notice that this unit-demand multi-dimensional setting is equivalent as adding an extra constraint, each buyer can purchase at most one item, to the original setting with subadditive bidders.}.\nNotice that this is a single-dimensional setting, as each agent's type is specified by a single number. Let $\\textsc{OPT}^{\\textsc{Copies-UD}}$ be the optimal BIC revenue in this copies setting.\n\n\\vspace{.05in}\n\\noindent\\textbf{Continuous vs. Discrete Distributions:} We explicitly assume that the input distributions are discrete. Nevertheless, it is known that every $D$ can be discretized into $D^{+}$ such that the optimal revenue for $D$ and $D^{+}$ are within $(1\\pm\\epsilon)$ of each other~\\cite{CaiDW16}. So our results also apply to  continuous distributions. \n\\subsection{Our Mechanisms}\nIn this section, we introduce a class of mechanisms called \\emph{Sequential Posted Price with Entry Fee}. For each bidder $i$, the mechanism first determines a posted price $\\xi_{ij}$ for each item $j$ and an entry fee function ${\\delta_i}(\\cdot):2^{[m]}\\to {\\mathbb{R}_{\\geq 0}}$ for each bidder $i$ that maps the set of available items to a real value entry fee. The seller visits the bidders sequentially in some arbitrary order. For simplicity, we assume the bidders are visited in the lexicographical order. When bidder $i$ is visited, let $S_i(t_{<i})$ be the set of items that are still available. Clearly, this set only depends on the types of bidders who are visited before $i$. The mechanism shows the set $S_i(t_{<i})$ to bidder $i$ and asks her for an entry fee ${\\delta_i}(S_i(t_{<i}))$. If she accepts the entry fee, she can enter the mechanism and take her favorite bundle $S_i^{*}$ by paying $\\sum_{j\\in S_i^{*}} \\xi_{ij}$.\n\n If there exist multiple bundles with the same maximum surplus, the bidder can break ties arbitrarily. Sometimes, there is a feasibility constraint $\\mathcal{F}$ on what items a buyer can purchase. In particular, if we say the mechanism is rationed, then $\\mathcal{F}=\\{\\emptyset\\} \\cup \\{\\{j\\}\\ |\\ j\\in[m] \\}$, i.e., a buyer can purchase at most one item. Formally, the favorite bundle $S_i^{*}$ is defined as follows:\n$S_i^{*}={\\argmax}_{S\\subseteq S_i(t_{<i})\\land S\\in \\mathcal{F}} v_i(t_i,S)-\\sum_{j\\in S}\\xi_{ij}$.\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n    \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n        \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n        \\STATE $S\\gets S\\backslash S_i^{*}$.\n    \\ELSE\n        \\STATE $i$ gets nothing and pays $0$.\n    \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n\nSee Algorithm \\ref{alg:seq-mech} for the formal specification of the above mechanism. Notice that before the bidder decides whether to pay the entry fee, she is aware of the set $S_i(t_{<i})$ which contains all available items. Thus, she can compute her favorite bundle $S_i^{*}$ and the corresponding utility if she chooses to enter the mechanism. She can then compare that utility with the entry fee and accept the entry fee if the former is greater than the latter. The mechanism described above is therefore deterministic and DSIC. Throughout this paper, we  focus on the following two special cases of this class of mechanisms:\n\n\n\\vspace{.05in}\n \\noindent\\textbf{-Rationed Sequential Posted Price Mechanism (RSPM):}  Every buyer can purchase at most one item and the mechanism always charges $0$ entry fee, i.e., $\\mathcal{F}=\\{\\emptyset\\} \\cup \\{\\{j\\}\\ |\\ j\\in[m] \\}$ and ${\\delta_i}(S)=0$ for all $i$ and $S$.\n\n \\vspace{.05in}\n\\noindent\\textbf{-Anonymous Sequential Posted Price with Entry Fee Mechanism (ASPE):}  The mechanism uses anonymous posted prices, i.e., ${\\xi}_{ij} = \\xi_{kj}$ for any item $j$ and bidders $i\\neq k$, but may charge positive and personalized entry fee. Also, any buyer can purchase any bundle available once she has paid the entry fee, i.e., $\\mathcal{F} = 2^{[m]}$.\n\n\n\n\\section{Proof of Corollary~\\ref{corollary:concentrate}}\\label{sec:concentration_proof}\n\n\n\\section{Proof of Theorem~\\ref{thm:revenue less than virtual welfare}}\\label{sec:proof_duality}\n\\begin{prevproof}{Theorem}{thm:revenue less than virtual welfare}\nWhen $\\lambda$ is useful, we can simplify function ${\\mathcal{L}}(\\lambda, \\sigma, p)$ by removing the term associated with $p$ and replacing $\\sum_{t_{i}'\\in T_{i}^{+}}\\lambda(t_{i},t_{i}')$ with $f_{i}(t_{i})+\\sum_{t_{i}'\\in T_{i}} \\lambda(t_{i}',t_{i})$. After the simplification, we have\n\\begin{align*}\n&{\\mathcal{L}}(\\lambda, \\sigma, p) = \\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i})\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot \\\\\n&\\left( v_i(t_{i},S)-{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right)\\right)\\\\\n&\\hspace{1.3cm}=\\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i})\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S),\t\n\\end{align*}\n which is exactly the virtual welfare of $\\sigma$ with respect to $\\lambda$. Now, we only need to prove that ${\\mathcal{L}}(\\lambda, \\sigma, p)$ is greater than the revenue of $M$. Let us think of ${\\mathcal{L}}(\\lambda, \\sigma, p)$ using Expression~(\\ref{eq:primal lagrangian}). Since $M$ is a BIC mechanism, $$\\sum_{S\\subseteq[m]}v_i(t_i,S)\\cdot\\left(\\sigma_{iS}(t_{i})-\\sigma_{iS}({t_{i}'})\\right)-\\left((p_{i}(t_{i})-p_{i}(t_{i}')\\right)\\geq 0$$ for any $i$, $t_{i}\\in T_{i}$ and $t_i'\\in T_i^+$. Also, all the dual variables $\\lambda$ are nonnegative. Therefore, it is clear that ${\\mathcal{L}}(\\lambda, \\sigma, p)$ is at least as large as the revenue of $M$.\n\nWhen $\\lambda^{*}$ is the optimal dual variable, by strong duality, we know $\\max_{\\sigma\\in{P(D)}, p}{\\mathcal{L}}(\\lambda^{*}, \\sigma, p)$ equals to the revenue of $M^{*}=(\\sigma^*,p^*)$. But we also know that ${\\mathcal{L}}(\\lambda^{*}, \\sigma^{*}, p^{*})$ is at least as large as the revenue of $M^{*}$, therefore $\\sigma^{*}$ maximizes the virtual welfare.\n\\end{prevproof}\n\n\n\\section{Analysis for the Multi-Bidder Case}\\label{sec:multi_proof}\n\n\\subsection{Analyzing $\\textsc{Single}(\\beta)$ in the Multi-Bidder Case}\n\n\\begin{prevproof}{Lemma}{lem:multi_single}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one copy of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$\n\\end{prevproof}\n\n\\subsection{\\textsc{Core}--\\textsc{Tail}~Decomposition}\\label{sec:multi_nf}\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\notshow{=}\\mingfeinote{\\leq}&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(\\beta))\n\\end{align*}\n\\mingfeinote{[Mingfei: $t_i$ in region k if $V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}$ and also other $j'\\not=k,j$ is no larger than k]}\n\\end{prevproof}\n\n\\subsection{Analyzing $\\textsc{Tail}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\nIn this section, we show how to upper bound $\\textsc{Tail}(\\beta)$ with the revenue of an RSPM.\n\tLet us first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$ $$r_{ij}=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$ $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. The next Lemma shows that $r$ is an upper bound of $\\textsc{Tail}(\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\n$$\\textsc{Tail}(\\beta)\\leq r.$$\t\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]=1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful when bounding $\\textsc{Core}(\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nIf $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ is  a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $$\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right].$$\n\\end{proof}\n\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\nr\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$,  it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$ (Definition~\\ref{def:c_i})). Therefore, $$r= \\sum_i\\sum_j(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\n\tIt follows from Lemma~\\ref{lem:tail and r} and Corollary~\\ref{cor:bound tail }.\n\\end{prevproof}\nFinally, we show that $r_i$ is within a constant factor of $c_i$. This Lemma will be useful for bounding $\\textsc{Core}{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\\\\\n&\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because if $c_i>0$, then $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]\\geq \\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Corollary~\\ref{cor:bound tail }, $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\subsection{Analyzing $\\textsc{Core}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$. Hence, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$.\\mingfeinote{Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$ for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then or any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nNext, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$  and any set $S$, then for any $t_i$ there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$ and any set $S$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. In particular, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\nAlso, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})>\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})\\leq\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)=\\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_j \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that   $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap X_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat},  $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\t\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n\\begin{equation*}\n\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\geq a]\\leq\\frac{1}{2}\\}.\n\\end{equation*}\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nWe separate the revenue of the ASPE into two different parts -- the entry fee part and the item pricing part. We first provide a lower bound of the entry fee part.\n We define another set of prices $\\{\\hat{Q}_j\\}_{j\\in[m]}$ similar to the way we define $\\{{Q}_j\\}_{j\\in[m]}$, and argue that $\\sum_j Q_j$ is not much larger than $\\sum_j \\hat{Q}_j$.\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:hat Q}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}). The second inequality is due to the choice of $\\beta_{ij}$ (Lemma~\\ref{lem:requirement for beta}). The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i})$ for all bidders. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}.\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. Then for every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE, then $${\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n=&\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\\notshow{\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i(t_i,t_{< i}) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\max_{S\\subseteq \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} \\hat{v}_i(t_i, S) - \\sum_{j\\in S} Q_j\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j}\\cdot \\mathds{1}[j\\in S_i(t_{<i})]\\cdot\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i, t'_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)~~~~~\\text{(Definition of $\\hat{Q}_j$ and $\\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq 1$ for all $j$)}\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{($\\forall j$, $Q_j-\\hat{Q}_j\\geq 0$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n{We apply the concentration bound in Corollary~\\ref{corollary:concentrate} to prove Lemma~\\ref{lem:concentration entry fee}. First, we argue that for any bidder $i$, $\\mu_i(\\cdot,\\cdot)$ is Lipschitz and for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{j\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{j\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all $U\\subseteq V\\subseteq S_i(t_{<i})$, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{prevproof}\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\n{\\mathbb{E}}_{t}\\left[\\mu_i(t_i,t_{< i})\\right]&= 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\int_{y=0}^{\\infty}\\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+y]dy\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot\\sum_{k=0}^m \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+k\\cdot \\tau_i] \\quad(\\mu_i(t_i,t_{< i})\\leq m\\cdot \\tau_i)\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot \\sum_{k=0}^2 \\Pr[\\mu_i(t_i,t_{< i})\\geq \\delta_i(t_{<i})]+ \\tau_i\\cdot\\sum_{k=3}^{m} \\frac{9}{4}\\cdot 2^{-k}\\\\\n&\\leq 3\\delta_i(t_{<i})+\\tau_i+\\frac{9}{16}\\tau_i\\\\\n&=3\\delta_i(t_{<i})+\\frac{25}{16}\\tau_i\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $$u_i(t_i,S) = \\max_{S'\\subseteq S} \\left( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\right).$$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $$\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$$ Due to Lemma~\\ref{lem:concentration entry fee}, this is no smaller than $$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i= \\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i.$$ Combining Lemma~\\ref{lem:tau_i} and by Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $$\\frac{1}{4}\\cdot\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\nOn the other hand, the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$. Therefore, the total revenue of the ASPE is at least $$\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\nFinally, we upper bound the $\\textsc{Core}(\\beta)$ by combining Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\n\n\\vspace{.1in}\n\\begin{prevproof}{Lemma}{lem:upper bounding Q}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{prevproof}\n\nNow, we have upper bounded $\\textsc{Single}(\\beta)$, $\\textsc{Tail}(\\beta)$ and $\\textsc{Core}(\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tCombining Lemma~\\ref{lem:multi_single}, Corollary~\\ref{cor:bound tail } and Lemma~\\ref{lem:upper bounding Q}, we have\n\t\\begin{equation*}\n\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b\\cdot(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev} + 32\\alpha \\cdot \\textsc{APostEnRev}.\n\\end{equation*}\n\\end{prevproof}\n\n\\notshow{\\yangnote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(\\beta)\\leq 120\\textsc{PostRev}+12\\textsc{APostEnRev}$ and $\\textsc{Tail}(\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 508\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf I optimize over $b$, I can get $\\textsc{Core}(\\beta)+\\textsc{Tail}(\\beta)\\leq 94\\textsc{PostRev}+12\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 388\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf we take $b=1\/4$, $\\textsc{Core}(\\beta)\\leq 18\\textsc{APostEnRev}+152\/3\\textsc{PostRev}$. $\\textsc{Tail}(\\beta)\\leq 8\/3\\textsc{PostRev}$. $\\textsc{Rev}\\leq (640\/3+12)\\textsc{PostRev}+72\\textsc{APostEnRev}\\leq 226\\textsc{PostRev}+72\\textsc{APostEnRev}.$\n}\\mingfeinote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(\\beta)\\leq 49\\textsc{PostRev}+18\\textsc{APostEnRev}$ and $\\textsc{Tail}(\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 224\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\nAfter optimizing over $b$($b=0.3$), I can get $\\textsc{Core}(\\beta)+\\textsc{Tail}(\\beta)\\leq 46\\textsc{PostRev}+18\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 196\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\n}}\n\n\n\n\\notshow{To bound $\\textsc{Core}(\\beta)$, notice the social welfare of new valuation$(\\textsc{Core})$ can be viewed as the sum of revenue and bidder's utility. With no entry fee, the revenue comes only from the posted price. We are going to show that with proper entry fee, the expected revenue from entry fee can approximately obtain the utility part.}\n\n\\notshow{For all $t_i$ and $S$, define $\\theta_j'(t_i)=\\gamma_j^{S}(t_i)(t_i)\\cdot \\mathds{1}\\left[V_i(t_{ij})\\leq Q_j+\\tau_i\\right]$. When each bidder is additive with item value $\\theta_j'(t_i)$, given t, the maximum utility can be written as\n\n\\begin{definition}\n\t$\\mu_i(t)=\\mathds{E}_{t_i\\sim T_i}\\left[\\max_{S\\subseteq S_i(t_{<i})}\\sum_{j\\in S}\\left(\\theta_j'(t_i)-Q_j\\right)\\right]$\n\\end{definition}\n\nDefine $\\textsc{Utility}=\\mathds{E}_{t\\in T}\\left[\\sum_i \\mu_i(t)\\right]$.\n\n\\begin{lemma}\n(citation)For every $i$ and every $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, for any $p\\in (0,1)$,\n\\begin{equation}\n\\Pr_{t_i}\\left[\\mu_i(t_i,t_{-i})\\leq (1-\\delta)\\cdot \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\right]\\leq e^{-\\frac{p^2}{2}}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIt comes directly from (citation) with the fact that $\\theta_j'(t_i)-Q_j\\leq \\tau_i$.\n\\end{proof}\n\n\nTo bound \\textsc{Utility}, for each $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, $\\mu_i(t)$ concentrates around its expectation $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})$. Define the entry fee as $\\delta_i(t_{\\geq i})=(1-p)\\cdot c\\cdot \\mathds{E}_{t_i}\\mu_i(t)$, where $p\\in (0,1)$ is a parameter to be decided. By the definition of supporting price, $v_i(t_i,S)\\geq c\\cdot \\sum_{j\\in S}\\theta_j'(t_i)$. Hence for any $t_{-i}$, if bidder $i$ with type $t_i$ accepts the entry fee, the maximum utility he can get is no less than $c\\cdot \\mu_i(t_i,t_{-i})$. Then with probability at least $(1-e^{-\\frac{p^2}{2}})$, bidder $i$ will accept the entry fee. When $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\leq 2\\tau_i$, we can directly use $2\\tau_i$ as the upper bound.\n\nGiven $t$, let $\\textsc{SOLD(t)}$ be the set of items sold during the auction. Then the expected revenue from posted price is $\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j$. Hence by analyzing the revenue of this sequential posted price mechanism, we have\n\\begin{equation}\n\\begin{aligned}\n\\textsc{APostEnRev}&\\geq \\mathds{E}_{t_{-i}}\\left[(1-e^{-\\frac{p^2}{2}})\\cdot \\sum_i\\delta_i(t_{\\geq i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&=\\mathds{E}_{t_{-i}}\\left[(1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot \\sum_i \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]\\quad(\\text{Definiton of $\\delta_i(t_{\\geq i})$})\\\\\n&+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&\\geq (1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot(\\textsc{Utility}-2\\sum_i\\tau_i)+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n\\end{aligned}\n\\end{equation}\n\t\nAfter picking a proper parameter $p=0.62$, by Lemma \\ref{lem:tau_i},\n\\begin{equation}\\label{equ:utility_upper_bound}\n\\textsc{Utility}+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq 2\\sum_i\\tau_i+8.25\\textsc{APostEnRev}\\leq (\\frac{8}{1-b}+8.25)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\n\nNotice that the term $\\textsc{Utility}$ is smaller than the real expect utility since item values $\\gamma_j^{S}(t_i)(t_i)$ are replaced by $\\theta_j'(t_i)$. We still need to make sure that these two parts are not far from each other to complete the proof. Lemma \\ref{lem:utility} describes this fact.\n\n\\begin{lemma}\\label{lem:utility}\n\\begin{equation}\n\\sum_{j\\in [m]}\\Pr_t\\left[j\\not\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq \\textsc{Utility}+\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nGiven a BIC mechanism $M$, let $M_i(t)$ be the set of items allocated to bidder $i$ when type profile is $t$. Notice that in the term $\\textsc{Utility}$, the favorite bundle just contains all items in $S_i(t_{<i})$ with positive surplus, we have\n\\begin{equation}\\label{equ:utility}\n\\begin{aligned}\n\\textsc{Utility}&=\\sum_i\\mathds{E}_{t\\sim T}\\left[\\sum_{j\\in S_i(t_{<i})}\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\right]\\\\\n&\\geq \\sum_i\\mathds{E}_{t_i,t_{-i},t_{-i}'}\\left[\\sum_{j\\in [m]}\\mathds{1}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t_{-i}'}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\mathds{E}_{t_i,t_{-i}}\\left[\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\\\\n\\end{aligned}\n\\end{equation}\n\nThe last inequality is because for each $t_i'$, the probability of $j\\in S_i(t_{<i}')$ can not exceed the probability of item $j$ is not sold after the sequential mechanism. Notice that for $j\\in A_i$, $\\theta_j'(t_i)$ and $\\gamma_j^{S}(t_i)(t_i)$ are different only if $V_i(t_{ij})\\geq Q_j+\\tau_i$. Using the fact that $\\gamma_j^{S}(t_i)(t_i)\\leq V_i(t_{ij})$, we have\n\n\\begin{equation}\n\\begin{aligned}\n&\\text{RHS of inequality \\ref{equ:utility}}\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\theta_j^{S\\cap\\mathcal{C}_i(t_i)}(t_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\left(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\gamma_j^{S}(t_i)(t_i)-Q_j\\right) \\quad(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)=1)\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot (\\beta_{ij}+c_i-Q_j) \\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\beta_{ij}+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\right)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\frac{1}{b}\\sum_i\\sum_{j\\not\\in A_i}\\beta_{ij}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-\\frac{1}{2}\\sum_i (c_i+\\tau_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev} \\quad (\\text{Lemma \\ref{lem:c_i},\\ref{lem:tau_i},\\ref{lem:beta_ij}})\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nThe inequality stated in Lemma \\ref{lem:core} comes directly from Equation \\ref{equ:core and q_j}, Inequality \\ref{equ:utility_upper_bound} and Lemma \\ref{lem:utility}.\n\nThe following lemma gives an upper bound for $\\textsc{Core}(\\beta)$.\n\\begin{lemma}\\label{lem:core}\nIf for all $i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ has a $c$-supporting price $\\{\\theta_j^{S}(t_i)\\}_{j=1}^m$, then\n\\[\\text{CORE }\\leq \\left(\\frac{24}{1-b}+\\frac{4}{b(1-b)}+16.5\\right)\\cdot c\\cdot \\textsc{APostEnRev}\\]\n\\end{lemma}\n}\n\n\\subsection{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nFor any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We describe the construction of the RSPM and the ASPE separately. Unfortunately, in order to approximate the optimal revenue, we need to know an (approximately) revenue-maximizing mechanism $M^*$. However, for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}, we show that we can directly construct the desired RSPM and ASPE that approximates the optimal revenue without knowing $M^*$.\n\n  Indeed, we can always directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue one can obtain from the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know the optimal revenue in the unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction does not depend on $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$'s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the computation requires sampling from the distribution, but it is not hard to argue that a polynomial number of samples suffices.\n\n  For symmetric bidders, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, select $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies the first property in Lemma~\\ref{lem:requirement for beta}. Notice that the optimal mechanism is symmetric, so for any item $j$ the ex-ante probability for any bidder to win it is the same, and therefore no more than $1\/n$. Hence, the second property is also satisfied. Given this $\\beta$, we can essentially follow the procedure mentioned above to construct the ASPE. The only difference is that when we construct the $Q_j$s, we no longer know the $\\sigma$. This can be resolved by using the welfare maximizing allocation with respect to $v'$.\n\n\n\n\n\\section{Analysis for the Multi-Bidder Case}\\label{sec:multi_proof}\n\n\\subsection{Analyzing $\\textsc{Single}(M,\\beta)$ in the Multi-Bidder Case}\n\n\n\\subsection{\\textsc{Core}--\\textsc{Tail}~Decomposition}\\label{sec:multi_nf}\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(M,\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(M,\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(M,\\beta))\n\\end{align*}\n\\end{prevproof}\n\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$. {Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$, for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then for any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nThen, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$  for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$  for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(M,\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that   $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$,  $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat},  $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{<i})) = \\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]\\footnote{Here $\\textsc{Median}_x[h(x)]$ denotes the median of a non-negative function $h(x)$ on random variable $x$, i.e.\n\\begin{equation*}\n\\textsc{Median}_x[h(x)]=\\inf\\{a\\geq 0:\\Pr_{x}[h(x)\\leq a]\\geq\\frac{1}{2}\\}.\n\\end{equation*}\n},$$ where $S_1(t_{<1})=[m]$ and $S_i(t_{<i})$ is the set of items that are not purchased by the first $i-1$ buyers in the ASPE. Notice that even though the seller does not know $t_{<i}$, she can compute the entry fee $\\delta_i(S_i(t_{<i}))$, as she observes $S_i(t_{<i})$ after visiting the first $i-1$ bidders.\n\\end{definition}\n\nWe separate the revenue of the ASPE into two different parts -- the entry fee part and the item pricing part. We first provide a lower bound of the entry fee part.\n We define another set of prices $\\{\\hat{Q}_j\\}_{j\\in[m]}$ similar to the way we define $\\{{Q}_j\\}_{j\\in[m]}$, and argue that $\\sum_j Q_j$ is not much larger than $\\sum_j \\hat{Q}_j$.\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:hat Q}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot  \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq  \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{<i}))$ for all bidders. The proof is inspired by Feldman et al.~\\cite{FeldmanGL15}.\n\\begin{lemma}\\label{lem:lower bounding mu}\n\tFor all $j$, let $Q_j$ (Definition~\\ref{def:posted prices}) be the price for item $j$ and every bidder's entry fee be described as in Definition~\\ref{def:entry fee}. For every type profile $t\\in T$, let $\\textsc{SOLD}(t)$ be the set of items sold in the corresponding ASPE. Then $${\\mathbb{E}}_{t}\\left[\\sum_{i\\in[n]} \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\geq \\sum_{j\\in[m]} \\Pr_t\\left[j\\notin \\textsc{SOLD}(t)\\right]\\cdot Q_j - \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\mathds{1}\\left[j\\in S_i(t_{<i})\\right]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+ \\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i,t'_{-i}}\\left[\\sum_{j\\in M^{(\\beta)}_i(t_i,t'_{-i})} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{S\\subseteq[m]} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\sum_{j\\in S} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i}\\left[\\sum_{j\\in [m]} \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i}\\left[\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}^{S}_j(t_i)-Q_j\\right)^+\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\n\\end{align*}\n\n$t_{-i}'$ are fresh samples drawn from $D_{-i}$. The first inequality is because the $\\mu_i(t_i,S)$ function is monotone in set $S$ for any $i$ and type $t_i\\in T_i$. We use $\\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)^+$ to denote $\\max\\left\\{\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j, 0\\right\\}$. If we let $S$ be the set of items that are in $S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})$ and satisfy that $\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\geq 0$, then $\\mu_i\\left(t_i,S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)\\geq \\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j\\geq \\sum_{j\\in S} \\left(\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j\\right)$ due to the definition of $\\mu_i(t_i,\\cdot)$ and Lemma~\\ref{lem:hat gamma}. This inequality is exactly the second inequality above. The next equality is because $S_i(t_{<i})$ only depends on the types of bidders other than $i$. The second last inequality is because $\\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\geq \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]$ for all $j$ and $i$, as the LHS is the probability that the item is not sold after the seller has visited the first $i-1$ bidders and the RHS is the probability that the item remains unsold till the end of the mechanism. Now, observe that $\\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\hat{\\gamma}_j^S(t_i)=2\\hat{Q}_j$ for any $j$ according to the definition in Lemma~\\ref{lem:hat Q}. Therefore,\n\n\\begin{align*}\n&  {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\cdot \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\cdot\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq &\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{(Due to Lemma~\\ref{lem:hat Q}, $Q_j-\\hat{Q}_j\\geq 0$ for all $j$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n\\end{proof}\n\n\\notshow{\\begin{align*}\n&\t{\\mathbb{E}}_{t}\\left[\\sum_i \\mu_i(t_i,t_{< i}) \\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\max_{S\\subseteq \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} \\hat{v}_i(t_i, S) - \\sum_{j\\in S} Q_j\\right]\\\\\n\\geq & \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j\\in \\left(S_i(t_{<i})\\cap M^{(\\beta)}_i(t_i,t'_{-i})\\right)} (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n=& \\sum_i {\\mathbb{E}}_{t_i, t_{-i}, t'_{-i}}\\left[\\sum_{j}\\cdot \\mathds{1}[j\\in S_i(t_{<i})]\\cdot\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n= & \\sum_i \\sum_j \\Pr_{t_{<i}}[j\\in S_i(t_{<i})]\\cdot {\\mathbb{E}}_{t_i, t'_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t'_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t'_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)^+\\right]\\\\\n\\geq & \\sum_i\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot {\\mathbb{E}}_{t_i, t_{-i}}\\left[\\mathds{1}[j\\in M^{(\\beta)}_i(t_i,t_{-i})] (\\hat{\\gamma}^{M^{(\\beta)}_i(t_i,t_{-i})}_j(t_i)-Q_j)\\right]\\\\\n\\geq & {\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)] \\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\left(\\hat{\\gamma}_j^S(t_i)-Q_j\\right)}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot (2\\hat{Q}_j-Q_j)~~~~~\\text{(Definition of $\\hat{Q}_j$ and $\\sum_i \\sum_{t_i} f_i(t_i)\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq 1$ for all $j$)}\\\\\n= &  \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j-  \\sum_j\\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot2(Q_j-\\hat{Q}_j)\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\sum_j 2(Q_j-\\hat{Q}_j )~~~~~{\\text{($\\forall j$, $Q_j-\\hat{Q}_j\\geq 0$)}}\\\\\n\\geq & \\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j- \\frac{(2b+2)}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}~~~~~\\text{(Lemma~\\ref{lem:hat Q})}\n\\end{align*}\n}\n\n\\begin{lemma}\\label{lem:concentration entry fee}\n\n\tFor all $i$ and $t_{<i}$,\n\t{$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$}\n\\end{lemma}\n{We apply the concentration bound in Corollary~\\ref{corollary:concentrate} to prove Lemma~\\ref{lem:concentration entry fee}. First, we argue that for any bidder $i$, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\\begin{lemma}\\label{lem:property of mu}\nFor any $i$, the function $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. Moreover, for any type $t_i\\in T_i$, $\\mu_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{k\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{k\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{<i}$, let $S_i(t_{<i})$ be the ground set $I$. Therefore, $\\mu_i(t_i,\\cdot)$  with $t_i\\sim D_i$ is a function drawn from a distribution that is subadditive over independent items. Since, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz and\n$\\delta_i(S_i(t_{<i}))=\\textsc{Median}_{t_i}\\left[\\mu_i\\left(t_i, S_i(t_{<i})\\right)\\right]$,\n$$2\\delta_i\\left(S_i(t_{<i})\\right)+\\frac{5\\tau_i}{2}\\geq {\\mathbb{E}}_{t_i}\\left[\\mu_i\\left(t_i,S_i(t_{< i})\\right) \\right].$$\\end{prevproof}\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\n{\\mathbb{E}}_{t}\\left[\\mu_i(t_i,t_{< i})\\right]&= 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\int_{y=0}^{\\infty}\\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+y]dy\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot\\sum_{k=0}^m \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})+k\\cdot \\tau_i] \\quad(\\mu_i(t_i,t_{< i})\\leq m\\cdot \\tau_i)\\\\\n&\\leq 3\\delta_i(t_{<i})\\cdot \\Pr[\\mu_i(t_i,t_{< i})\\geq 3\\delta_i(t_{<i})]+\\tau_i\\cdot \\sum_{k=0}^2 \\Pr[\\mu_i(t_i,t_{< i})\\geq \\delta_i(t_{<i})]+ \\tau_i\\cdot\\sum_{k=3}^{m} \\frac{9}{4}\\cdot 2^{-k}\\\\\n&\\leq 3\\delta_i(t_{<i})+\\tau_i+\\frac{9}{16}\\tau_i\\\\\n&=3\\delta_i(t_{<i})+\\frac{25}{16}\\tau_i\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem:entry fee acceptance probability}\n\tFor all $i$ and $t_{<i}$, bidder $i$ accepts $\\delta_i(t_{<i})$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. Moreover, $$\\textsc{APostEnRev}\\geq \\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nFor any bidder $i$, type $t_i\\in T_i$ and any set $S$, define bidder $i$'s utility as $$u_i(t_i,S) = \\max_{S'\\subseteq S} \\left( v_i(t_i, S') -\\sum_{j\\in S'} Q_j\\right).$$ Clearly, $u_i(t_i,S) \\geq \\mu_i(t_i,S)$ for any type $t_i$ and set $S$. For any $t_{<i}$, as long as  $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$, buyer $i$ accepts the entry fee. Since $\\delta_i(S_i(t_{<i}))$ is the median of $ \\mu_i(t_i,t_{< i})$, $u_i(t_i, S_i(t_{<i})) \\geq \\delta_i(S_i(t_{<i}))$ with probability at least $1\/2$ when $t_i$ is drawn from $D_i$. So the revenue from entry fee is at least $$\\frac{1}{2}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i}}\\left[\\delta_i\\left(S_i(t_{<i})\\right)\\right].$$ Due to Lemma~\\ref{lem:concentration entry fee}, this is no smaller than $$\\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t_{<i},t_i}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i= \\frac{1}{4}\\cdot\\sum_i {\\mathbb{E}}_{t}\\left[\\mu_i(t_i,S_i(t_{< i}))\\right]- \\frac{5}{8}\\cdot\\sum_i \\tau_i.$$ Combining Lemma~\\ref{lem:tau_i} and by Lemma~\\ref{lem:lower bounding mu}, we can further show that the revenue from the entry fee is at least $$\\frac{1}{4}\\cdot\\sum_j \\Pr_{t}[j\\notin \\textsc{SOLD}(t)]\\cdot Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}$$.\n\nOn the other hand, the revenue from the posted prices is exactly $\\sum_j \\Pr_t[j\\in \\textsc{SOLD}(t)]\\cdot Q_j$. Therefore, the total revenue of the ASPE is at least $$\\frac{1}{4}\\cdot\\sum_j Q_j - \\left(\\frac{5}{2(1-b)}+\\frac{(b+1)}{2b\\cdot(1-b)}\\right)\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\nFinally, we upper bound the $\\textsc{Core}(M,\\beta)$ by combining Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\n\n\\vspace{.1in}\n\\begin{prevproof}{Lemma}{lem:upper bounding Q}\nIt follows directly from Lemma~\\ref{lem:core and q_j} and~\\ref{lem:entry fee acceptance probability}.\n\\end{prevproof}\n\nNow, we have upper bounded $\\textsc{Single}(M,\\beta)$, $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ using the sum of the revenue of simple mechanisms (RSPM and ASPE). Combining these bounds, we complete the proof of Theorem~\\ref{thm:multi}.\n\n\\vspace{.1in}\n\\begin{prevproof}{Theorem}{thm:multi}\n\tCombining {Theorem~\\ref{thm:revenue upperbound for subadditive}}, Lemma~\\ref{lem:multi_single},~\\ref{lem:multi decomposition},~\\ref{lem:multi-tail} and~\\ref{lem:upper bounding Q}, we have\n\t\\begin{equation*}\n\\textsc{Rev}{(M,v,D)}\\leq \\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b\\cdot(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev} + 32\\alpha \\cdot \\textsc{APostEnRev}.\n\\end{equation*}\n\\end{prevproof}\n\n\\notshow{\\yangnote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(M,\\beta)\\leq 120\\textsc{PostRev}+12\\textsc{APostEnRev}$ and $\\textsc{Tail}(M,\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(M,\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 508\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf I optimize over $b$, I can get $\\textsc{Core}(M,\\beta)+\\textsc{Tail}(M,\\beta)\\leq 94\\textsc{PostRev}+12\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 388\\textsc{PostRev}+48\\textsc{APostEnRev}$.\n\nIf we take $b=1\/4$, $\\textsc{Core}(M,\\beta)\\leq 18\\textsc{APostEnRev}+152\/3\\textsc{PostRev}$. $\\textsc{Tail}(M,\\beta)\\leq 8\/3\\textsc{PostRev}$. $\\textsc{Rev}\\leq (640\/3+12)\\textsc{PostRev}+72\\textsc{APostEnRev}\\leq 226\\textsc{PostRev}+72\\textsc{APostEnRev}.$\n}\\mingfeinote{\nIf we choose $b=1\/2$, and when $v$ is XOS $\\textsc{Core}(M,\\beta)\\leq 49\\textsc{PostRev}+18\\textsc{APostEnRev}$ and $\\textsc{Tail}(M,\\beta)\\leq 4\\textsc{PostRev}$. $\\textsc{Single}(M,\\beta)\\leq 6\\textsc{PostRev}$. So totally, we have $\\textsc{Rev}\\leq 224\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\nAfter optimizing over $b$($b=0.3$), I can get $\\textsc{Core}(M,\\beta)+\\textsc{Tail}(M,\\beta)\\leq 46\\textsc{PostRev}+18\\textsc{APostEnRev}$. So totally we have $\\textsc{Rev}\\leq 196\\textsc{PostRev}+72\\textsc{APostEnRev}$.\n\n}}\n\n\n\n\\notshow{To bound $\\textsc{Core}(\\beta)$, notice the social welfare of new valuation$(\\textsc{Core})$ can be viewed as the sum of revenue and bidder's utility. With no entry fee, the revenue comes only from the posted price. We are going to show that with proper entry fee, the expected revenue from entry fee can approximately obtain the utility part.}\n\n\\notshow{For all $t_i$ and $S$, define $\\theta_j'(t_i)=\\gamma_j^{S}(t_i)(t_i)\\cdot \\mathds{1}\\left[V_i(t_{ij})\\leq Q_j+\\tau_i\\right]$. When each bidder is additive with item value $\\theta_j'(t_i)$, given t, the maximum utility can be written as\n\n\\begin{definition}\n\t$\\mu_i(t)=\\mathds{E}_{t_i\\sim T_i}\\left[\\max_{S\\subseteq S_i(t_{<i})}\\sum_{j\\in S}\\left(\\theta_j'(t_i)-Q_j\\right)\\right]$\n\\end{definition}\n\nDefine $\\textsc{Utility}=\\mathds{E}_{t\\in T}\\left[\\sum_i \\mu_i(t)\\right]$.\n\n\\begin{lemma}\n(citation)For every $i$ and every $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, for any $p\\in (0,1)$,\n\\begin{equation}\n\\Pr_{t_i}\\left[\\mu_i(t_i,t_{-i})\\leq (1-\\delta)\\cdot \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\right]\\leq e^{-\\frac{p^2}{2}}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIt comes directly from (citation) with the fact that $\\theta_j'(t_i)-Q_j\\leq \\tau_i$.\n\\end{proof}\n\n\nTo bound \\textsc{Utility}, for each $t_{-i}$, when $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i$, $\\mu_i(t)$ concentrates around its expectation $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})$. Define the entry fee as $\\delta_i(t_{\\geq i})=(1-p)\\cdot c\\cdot \\mathds{E}_{t_i}\\mu_i(t)$, where $p\\in (0,1)$ is a parameter to be decided. By the definition of supporting price, $v_i(t_i,S)\\geq c\\cdot \\sum_{j\\in S}\\theta_j'(t_i)$. Hence for any $t_{-i}$, if bidder $i$ with type $t_i$ accepts the entry fee, the maximum utility he can get is no less than $c\\cdot \\mu_i(t_i,t_{-i})$. Then with probability at least $(1-e^{-\\frac{p^2}{2}})$, bidder $i$ will accept the entry fee. When $\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\leq 2\\tau_i$, we can directly use $2\\tau_i$ as the upper bound.\n\nGiven $t$, let $\\textsc{SOLD(t)}$ be the set of items sold during the auction. Then the expected revenue from posted price is $\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j$. Hence by analyzing the revenue of this sequential posted price mechanism, we have\n\\begin{equation}\n\\begin{aligned}\n\\textsc{APostEnRev}&\\geq \\mathds{E}_{t_{-i}}\\left[(1-e^{-\\frac{p^2}{2}})\\cdot \\sum_i\\delta_i(t_{\\geq i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&=\\mathds{E}_{t_{-i}}\\left[(1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot \\sum_i \\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\cdot \\mathds{1}\\left[\\mathds{E}_{t_i}\\mu_i(t_i,t_{-i})\\geq2\\tau_i\\right]\\right]\\quad(\\text{Definiton of $\\delta_i(t_{\\geq i})$})\\\\\n&+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n&\\geq (1-p)(1-e^{-\\frac{p^2}{2}})\\cdot c\\cdot(\\textsc{Utility}-2\\sum_i\\tau_i)+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\\\\n\\end{aligned}\n\\end{equation}\n\t\nAfter picking a proper parameter $p=0.62$, by Lemma \\ref{lem:tau_i},\n\\begin{equation}\\label{equ:utility_upper_bound}\n\\textsc{Utility}+\\sum_j\\Pr_{t}\\left[j\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq 2\\sum_i\\tau_i+8.25\\textsc{APostEnRev}\\leq (\\frac{8}{1-b}+8.25)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\n\nNotice that the term $\\textsc{Utility}$ is smaller than the real expect utility since item values $\\gamma_j^{S}(t_i)(t_i)$ are replaced by $\\theta_j'(t_i)$. We still need to make sure that these two parts are not far from each other to complete the proof. Lemma \\ref{lem:utility} describes this fact.\n\n\\begin{lemma}\\label{lem:utility}\n\\begin{equation}\n\\sum_{j\\in [m]}\\Pr_t\\left[j\\not\\in \\textsc{SOLD}(t)\\right]\\cdot Q_j\\leq \\textsc{Utility}+\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nGiven a BIC mechanism $M$, let $M_i(t)$ be the set of items allocated to bidder $i$ when type profile is $t$. Notice that in the term $\\textsc{Utility}$, the favorite bundle just contains all items in $S_i(t_{<i})$ with positive surplus, we have\n\\begin{equation}\\label{equ:utility}\n\\begin{aligned}\n\\textsc{Utility}&=\\sum_i\\mathds{E}_{t\\sim T}\\left[\\sum_{j\\in S_i(t_{<i})}\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\right]\\\\\n&\\geq \\sum_i\\mathds{E}_{t_i,t_{-i},t_{-i}'}\\left[\\sum_{j\\in [m]}\\mathds{1}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t_{-i}'}\\left[j\\in S_i(t_{<i}')\\right]\\cdot \\mathds{E}_{t_i,t_{-i}}\\left[\\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[j\\in M_i(t_i,t_{-i})\\right]\\right]\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\theta_j'(t_i)-Q_j\\right)^{+}\\\\\n\\end{aligned}\n\\end{equation}\n\nThe last inequality is because for each $t_i'$, the probability of $j\\in S_i(t_{<i}')$ can not exceed the probability of item $j$ is not sold after the sequential mechanism. Notice that for $j\\in A_i$, $\\theta_j'(t_i)$ and $\\gamma_j^{S}(t_i)(t_i)$ are different only if $V_i(t_{ij})\\geq Q_j+\\tau_i$. Using the fact that $\\gamma_j^{S}(t_i)(t_i)\\leq V_i(t_{ij})$, we have\n\n\\begin{equation}\n\\begin{aligned}\n&\\text{RHS of inequality \\ref{equ:utility}}\\\\\n&\\geq \\sum_i\\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\gamma_j^{S}(t_i)(t_i)-Q_j\\right)^{+}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\theta_j^{S\\cap\\mathcal{C}_i(t_i)}(t_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\left(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\gamma_j^{S}(t_i)(t_i)-Q_j\\right) \\quad(\\sum_i\\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)=1)\\\\\n&-\\sum_i\\sum_{j\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot (\\beta_{ij}+c_i-Q_j) \\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\\\\n&-\\sum_i\\sum_{j\\not\\in A_i}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot \\sum_{t_i}f_i(t_i)\\pi_{ij}(t_i)\\cdot \\left(\\beta_{ij}+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]\\right)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\frac{1}{b}\\sum_i\\sum_{j\\not\\in A_i}\\beta_{ij}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-\\frac{1}{2}\\sum_i (c_i+\\tau_i)\\\\\n&\\geq \\sum_{j\\in [m]}\\Pr_{t'}\\left[j\\not\\in \\textsc{SOLD}(t')\\right]\\cdot Q_j-\\left(\\frac{2}{b(1-b)}+\\frac{4}{1-b}\\right)\\cdot \\textsc{APostEnRev} \\quad (\\text{Lemma \\ref{lem:c_i},\\ref{lem:tau_i},\\ref{lem:beta_ij}})\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nThe inequality stated in Lemma \\ref{lem:core} comes directly from Equation \\ref{equ:core and q_j}, Inequality \\ref{equ:utility_upper_bound} and Lemma \\ref{lem:utility}.\n\nThe following lemma gives an upper bound for $\\textsc{Core}(M,\\beta)$.\n\\begin{lemma}\\label{lem:core}\nIf for all $i$ and $S\\subseteq [m]$, $v_i(t_i,S)$ has a $c$-supporting price $\\{\\theta_j^{S}(t_i)\\}_{j=1}^m$, then\n\\[\\text{CORE }\\leq \\left(\\frac{24}{1-b}+\\frac{4}{b(1-b)}+16.5\\right)\\cdot c\\cdot \\textsc{APostEnRev}\\]\n\\end{lemma}\n}\n\n\\subsection{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n  Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n  When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\\section{Missing Proofs for the Multi-Bidder Case}\\label{appx:multi}\n\n\n\\begin{prevproof}{Lemma}{lem:multi decomposition}\nWe replace every $v_i(t_i,S)$ in $\\textsc{Non-Favorite}(M,\\beta)$ with $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+$\\\\\n\\noindent$\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. Let the contribution from $v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)$ be the first term and the contribution from  $\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$ be the second term.\n\\begin{align*}\n& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))+ \\\\\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\\\\n&~~~~~~~~~~~~~~~~~ \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)\\right)\\\\\n\\leq& \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))\\quad(\\textsc{Core}(M,\\beta))\n\\end{align*}\n\nThe inequality comes from the Monotonicity of $v_i(t_i,\\cdot)$ by replacing $v_i\\left(t_i,(S\\backslash\\{j\\})\\cap \\mathcal{C}_i(t_i)\\right)$ with $v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$.\n\nFor the second term, notice that when $t_i\\in R_0^{(\\beta_i)}$, $\\mathcal{T}_i(t_i)=\\emptyset$. It can be rewritten as:\n\\begin{align*}\n&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\left( \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})+\\sum_{S:j\\not\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ik})\\right)\\\\\n=&\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]\\cdot \\pi_{ij}^{(\\beta)}(t_i)~~~~~~~~\\text{(Recall $\\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$)}\\\\\n\\leq &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\mathds{1}\\left[t_i\\not\\in R_j^{(\\beta_i)}\\right]~~~~~~~~\\text{($\\pi_{ij}^{\\beta}(t_i)\\leq 1$)}\\\\\n= &\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in \\mathcal{T}_i(t_i)} V_i(t_{ij})\\cdot \\sum_{k\\neq j}\\mathds{1}\\left[t_i \\in R_k^{(\\beta_i)}\\right]~~~~~~~~\\text{($t_i\\notin R_0^{(\\beta_i)}$)}\\\\\n\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot \\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\quad(\\textsc{Tail}(M,\\beta))\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\\begin{prevproof}{Lemma}{lem:tail and r}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{Lemma}{lem:hat gamma}\n\tSince for every set $S\\subseteq[m]$ and every subset $S'$ of $S$,  $v_i(t_i, S')\\geq \\sum_{j\\in S'}\\theta_j^S(t_i)$. Therefore,\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\tfor all $j\\in S$, we have $\\min\\{t_{ij}^{(k)}, Q_j+\\tau_i\\}\\geq \\hat{\\gamma}^S_j(t_i)$. Therefore, $\\hat{v}_i(t_i,S)\\geq \\sum_{j\\in S}\\hat{\\gamma}^S_j(t_i)$.\n\t\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:property of mu}\nWe first prove that $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz. For any $t_i,t_i'\\in T_i$ and set $X,Y\\in[m]$, let $X^{*}\\in \\argmax_{S\\subseteq X} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S}Q_j \\right), Y^{*}\\in \\argmax_{S\\subseteq Y} \\left(\\hat{v}_i(t'_i,S)-\\sum_{j\\in S}Q_j \\right)$.   Recall that $\\hat{v}_i(t_i,X^{*}) =v_i\\left(t_i, \\left\\{j\\ |\\ (j\\in X^{*}) \\land (V_i(t_{ij})< Q_j+\\tau_i)\\right\\}\\right)$. This means that for every $k\\in X^{*}$, $V_i(t_{ik})$ must be less than $Q_k+\\tau_i$, because otherwise $\\mu_i(t_i,X^{*})<\\mu_i(t_i,X^{*}\\backslash \\{k\\})$. Therefore, $\\hat{v}_i(t_i,S)=v_i(t_i,S)$ for all $S\\subseteq X^{*}$. Since $v_i(t_i,\\cdot)$ is subadditive, $v_i(t_i,X^{*})\\leq v_i(t_i,X^{*}\\backslash \\{k\\})+V_i(t_{ik})$. So by the optimality of $X^*$, it must be that $V_i(t_{ik})\\geq Q_k$ for all $k\\in X^*$. Similarly, we can show that for every $k\\in Y^{*}$, $V_i(t_{ik}')\\in [Q_k,Q_k+\\tau_i]$.\n\nNow let set $H=\\{j\\ |\\ j\\in X\\cap Y \\land t_{ij}=t_{ij}'\\}$, if $\\mu_i(t_i,X)>\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\\end{prevproof}\n\n\\section{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n  Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n  Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n  When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\n\\section{Analysis for the Single-Bidder Case}\\label{sec:single_appx}\n\n\n\\begin{comment}\n\\begin{prevproof}[lemma]\\ref{lem:single-single}\nRecall that $\\textsc{Single}(M)=\\sum_{t\\in T}f(t)\\cdot$ \\\\\n\\noindent$\\sum_{j\\in[m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\pi^{(\\beta)}_{j}(t)\\cdot {\\tilde{\\varphi}}_{j}(V(t_{j}))$.\n\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to the buyer and $t\\in R_j^{(\\beta)}$, $M'$ serves the agent $j$. $M'$ is feasible in the copies setting as there is at most one agent being served, and $\\textsc{Single}(M)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M)$.\n\nAs shown in~\\cite{ChawlaHMS10}, when there is a single buyer, a simple posted-price mechanism with the constraint that the buyer can only purchase one item achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/2$ in the original setting. Therefore, by the definition of $\\textsc{SRev}$ we have $2\\textsc{SRev}\\geq\\textsc{OPT}^{\\textsc{Copies-UD}}$.\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{lemma}{lem:single decomposition}\n{Recall that for all $t\\in T$ and $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$.} We replace every $v(t,S)$ in $\\textsc{Non-Favorite}(M)$ with $v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. Also since $R^{\\beta}_0=\\emptyset$, the corresponding term is simply $0$. First, the contribution from $v\\left(t,S\\cap \\mathcal{C}(t)\\right)$ is upper bounded by the \\textsc{Core}(M).\n\n\\begin{align*}\n& \\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,(S\\backslash\\{j\\})\\cap \\mathcal{C}(t)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,S\\cap \\mathcal{C}(t)\\right)\\right)\\\\\n\\leq& \\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\\quad(\\textsc{Core}(M))\n\\end{align*}\n\nThe inequality comes from the monotonicity of $v(t,\\cdot)$.\n\nNext, we upper bound the contribution from $\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$ by the $\\textsc{Tail}(M)$.\n\\begin{align*}\n&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}(t)}V(t_k)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}(t)}V(t_k)\\right) \\\\\n=&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_j)\\cdot \\mathds{1}\\left[t\\not\\in R_j^{(\\beta)}\\right]\\cdot \\pi_j^{(\\beta)}(t)~~~~~~~\\text{{(Recall $\\pi_{j}^{(\\beta)}(t)=\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)$)}}\\\\\n\\leq &\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot\n{\\mathds{1}\\left[t \\not\\in R_j^{(\\beta)}\\right]}~~~\\text{($\\pi_j^{(\\beta)}(t)\\leq 1$)}\\\\\n\\leq &{\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot \\mathds{1}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~\\text{(Definition of $R_j^{(\\beta)}$)}}\\\\\n=&{\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]\\quad(\\textsc{Tail}(M))}\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsubsection{Analyzing $\\textsc{Tail}(M)$~in the Single-Bidder Case}\n\\begin{prevproof}[lemma]{lem:single-tail}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$, \\\\\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{prevproof}\n\\end{comment}\n\n\\begin{prevproof}{Lemma}{lem:single subadditive}\nWe argue the three properties one by one.\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\emph{Monotonicity:} For all $t\\in T$ and $U\\subseteq V\\subseteq [m]$, $U\\cap \\mathcal{C}(t)\\subseteq V\\cap \\mathcal{C}(t)$. Since $v(t,\\cdot)$ is monotone,\n$$v'(t,U)=v\\left(t,U\\cap \\mathcal{C}(t)\\right)\\leq v\\left(t,V\\cap \\mathcal{C}(t)\\right)=v'(t,V).$$ Thus, $v'(t,\\cdot)$ is monotone.\n\\item \\emph{Subadditivity:} For all $t\\in T$ and $U,V\\subseteq [m]$, notice $(U\\cup V)\\cap \\mathcal{C}(t)=\\left(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)$, we have\n$$v'(t,U\\cup V)=v\\left(\\left(t,(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)\\right)\\leq v\\left(t,U\\cap\\mathcal{C}(t)\\right)+v\\left(t,V\\cap\\mathcal{C}(t)\\right)=v'(t,U)+v'(t,V).$$\n\\item \\emph{No externalities:} For any $t\\in T$, $S\\subseteq [m]$, and any $t'\\in T$ such that $t_{j}=t_{j}'$ for all $j\\in S$, to prove $v'(t,S)=v'(t',S)$, it is enough to show $S\\cap \\mathcal{C}(t)=S\\cap \\mathcal{C}(t')$. Since $V(t_j)=V(t_j')$ for any $j\\in S$, $j\\in S\\cap \\mathcal{C}(t)$ if and only if $j\\in S\\cap \\mathcal{C}(t')$.\n\\end{itemize}\n\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:single Lipschitz}\nFor any $t,t'\\in T$, and set $X,Y\\subseteq [m]$, define set $H=\\left\\{j\\in X\\cap Y:t_j=t_j'\\right\\}$. Since $v'(\\cdot,\\cdot)$ has no externalities, $v'(t',H)=v'(t,H)$. Therefore,\n\\begin{align*}\n|v'(t,X)-v'(t',Y)|&=\\max\\left\\{v'(t,X)-v'(t',Y),v'(t',Y)-v'(t,X)\\right\\}\\\\\n&\\leq \\max\\left\\{v'(t,X)-v'(t',H),v'(t',Y)-v'(t,H)\\right\\}\\quad\\text{(Monotonicity)}\\\\\n&\\leq \\max\\left\\{v'(t,X\\backslash H),v'(t',Y\\backslash H)\\right\\}\\quad\\text{(Subadditivity)}\\\\\n& = \\max\\left\\{v\\left(t,(X\\backslash H)\\cap \\mathcal{C}(t)\\right),v\\left(t',(Y\\backslash H)\\cap\\mathcal{C}(t)\\right)\\right\\}\\quad\\text{(Definition of $v'(\\cdot,\\cdot)$)}\\\\\n&\\leq c\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\\\\n&\\leq c\\cdot (|X\\Delta Y|+|X\\cap Y|-|H|)\n\\end{align*}\nThe second last inequality is because both $v(t,\\cdot)$ and $v(t',\\cdot)$ are subadditive and for any item $j\\in \\mathcal{C}(t)$ ($\\mathcal{C}(t')$) the single-item valuation $V(t_j)$ ($V(t'_j)$) is less than $c$.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\\label{sec:proof_relaxed_valuation}\nWe first prove some properties of $v^{(\\beta)}$, which will be useful for proving Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed larger}\n\tFor any $\\beta_i$, $t_i\\in T_i$ and $S\\in[m]$, $v_i^{(\\beta_i)}(t_i,S)\\geq v_i(t_i,S)$.\n\\end{lemma}\n\\begin{proof}\n\tThis follows from the fact that $v_i(t_i,\\cdot)$ is a subadditive function over bundles of items for all $t_i$.\n\\end{proof}\n\n\\begin{lemma}\n\tFor any $\\beta_i$ and $t_i\\in T_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is a monotone, subadditive function over the items.\n\\end{lemma}\n\\begin{proof}\nMonotonicity follows directly from the monotonicity of $v_i(t_i,\\cdot)$. We only argue subadditivity here. If $t_i$ belongs to $R_0^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t_i,\\cdot)=v_i(t_i,\\cdot)$. So it is clearly a subadditive function. If $t_i$ belongs to $R_j^{(\\beta_i)}$ for some $j>0$ and $j$ is not in either $U$ or $V$, then clearly $v_i^{(\\beta_i)}(t_i,U\\cup V)\\leq v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)$. If $j$ is in one of the two sets, without loss of generality let's assume it is in $U$. Then $v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)=v_i(t_i,U\\backslash\\{j\\})+V_i(t_{ij})+v_i(t_i,V)\\geq v_i(t_i,V\\cup (U\\backslash\\{j\\}))+V_i(t_{ij})= v_i^{(\\beta_i)}(t_i,U\\cup V)$.\n\\end{proof}\n\n\nHere we prove a stronger version of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\nFor any $\\beta$, any absolute constant $\\eta\\in(0,1)$ and any BIC mechanism $M$ for subadditive valuations $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+\\frac{1}{\\eta}\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$).\n\\end{lemma}\n\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M^{(\\beta)}$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the buyer $i$ on the left and the surrogates on the right. The edge weight between a replica (or buyer $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i^{(\\beta_i)}(r_i,\\cdot)$ to receive buyer $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the interim payment of buyer $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(r_i,S) - (1-\\eta)p_i(s_i)$, where $p_i(s_i)$ is the interim payment for buyer $i$ if she reported $s_i$.\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M^{(\\beta)}$ truthfully, then the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type is sampled from the distribution, then we sampled $\\ell-1$ replicas and $\\ell$ surrogates i.i.d. from the same distribution. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M^{(\\beta)}$ is a BIC mechanism with respect to valuation $v^{(\\beta)}$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate.  If buyer $i$ has type $t_i$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is  $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(t_i,S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful. It is critical that the other buyers are reporting truthfully, otherwise we cannot invoke Lemma~\\ref{lem:same distribution} and buyer $i$'s value for winning a surrogate with type $s_i$ may be different from the weight on the corresponding edge.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M^{(\\beta)}$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, the chosen surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for  buyer $i$ to win the item in $M^{(\\beta)}$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}$ and $r_i^{k}$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. We will slightly abuse notation by using $s_i^k$ (or $r_i^j$) $\\in V(\\bold{s_i},\\bold{r_i})$ to denote that $s_i^k$  (or $r_i^j$) is matched in the VCG matching $V(\\bold{s_i},\\bold{r_i})$.\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M^{(\\beta)}$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M^{(\\beta)}$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, she pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M^{(\\beta)}$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider another matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from  $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles.  If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}[Adapted from~\\cite{DaskalakisW12}]\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is matched in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or\\\\ (b) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ in $V(\\bold{s_i},\\bold{r_i})$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P \\cap V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P\\cap X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\nFor any $\\beta$,\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$,  we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M^{(\\beta)}$ awards buyer $i$ (with type $t_i$)  bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq  &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\mathcal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon$ for any $\\epsilon$. Let $P^{(\\beta)}$ be the set of all BIC mechanisms that satisfy the first condition in Lemma~\\ref{lem:relaxed valuation stronger}. Clearly, $P^{(\\beta)}$ is a compact set and contains all $M^{(\\beta)}$ we constructed (by choosing different values for $\\ell$). Notice that both $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and $\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ are linear functions over the allocation\/price rules of mechanism $M^{(\\beta)}$. Therefore, \\begin{align*}\n \t&\\textsc{Rev}(M, v, D)\\\\\n \t\\leq &\\max_{M^{(\\beta)}\\in P^{(\\beta)}} \\left(\\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).\n \\end{align*}\nThis completes the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\n\tIn a $n$-player $m$-item combinatorial auction, for any absolute constant $\\eta\\in(0,1)$ and $\\epsilon>0$, any two type profile distributions $D, D'$ on type profile set $T$ and $T'$ accordingly($T$ and $T'$ might be different), any two valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$, $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$,  assume for every $i$ there exists a coupling $\\hat{D_i}$ for $D_i$ and $D_i'$ such that $\\forall t_i\\in T_i,t_i'\\in T_i', \\hat{D_i}(t_i,t_i')>0$, $v_i'(t_i',S)\\geq v_i(t_i,S)$ holds for subset $S$. Here $\\hat{D_i}(t_i,t_i')$ is the coupling probability. Then for any BIC mechanism $M$ for valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to $D$, there exists a BIC mechanism $M'$ for valuation functions $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to distribution $D'$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S}\\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M',v', D')}+\\frac{1}{\\eta}\\cdot \\sum_i\\sum_{t_i'\\in T_i'}\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]}\\hat{D_i}(t_i,t_i')\\cdot\\sigma'_{iS}(t_i')\\cdot \\left(v_i'(t_i', S)-v_i(t_i, S)\\right)+\\epsilon$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ is the revenue of the mechanism $M$ while the buyers' types are drawn from $D$ and their valuations are $v$ (similarly defined for $\\textsc{Rev}(M', v', D')$). $\\sigma'_{iS}(t_i')$ is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i'$ in mechanism $M'$ with respect to $D'$ and  $\\sigma_{iS}(t_i)$ is the probability for the same event in mechanism $M$ with respect to $D$\n\\end{lemma}\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M'$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} sampled i.i.d. from $D_i'$ and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i'$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the bidder $i$ on the left and the surrogates on the right. The edge weight between a replica (bidder $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i'(r_i,\\cdot)$ to receive bidder $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the expected payment of bidder $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(r_i,S) - (1-\\eta)p_i(s_i).$\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M'$ truthfully, then the distribution of types of the surrogate chosen for buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type and $\\ell-1$ replicas are sampled i.i.d. from the distribution $D_i'$, while $\\ell$ surrogates are sampled i.i.d. from the distribution $D_i$. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M'$ is a BIC mechanism with respect to valuation $v'$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate.  If buyer $i$ has type $t_i'$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is  $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(t_i',S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S} \\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M'$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, this surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for $i$ to win the item in $M'$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M',v', D')$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}\\in T_i$ and $r_i^{k}\\in T_i'$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. \t\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M'$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M'$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, $i$ pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M'$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider a different type of matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et. al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from  $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles.  If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or (b)$\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{r_i^{\\theta(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P$ and $V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P$ and $X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq  p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$,  we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i'(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means  whenever mechanism $M'$ awards buyer $i$ (with type $t_i$)  bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i'(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i E_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq  &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M',v',D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\cal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon\/(1-\\eta)$. Therefore, we finished the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\n\n\n\n\n\\section{Paper Organization}\\label{sec:roadmap}\nIn this section, we provide the roadmap to our paper. In Section~\\ref{sec:duality}, we review the Duality framework of~\\cite{CaiDW16}.\n\nIn Section~\\ref{sec:flow}, we derive an upper bound of the optimal revenue for subadditive bidders by combining the duality framework with our new techniques, i.e. valuation relaxation and adaptive dual variables. Our main result in this section,  Theorem~\\ref{thm:revenue upperbound for subadditive}, shows that the revenue can be upper bounded by two terms -- $\\textsc{Non-Favorite}$ and $\\textsc{Single}$ defined in Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n\nIn Section~\\ref{sec:single}, we use the single bidder case to familiarize the readers with some basic ideas and techniques used to bound $\\textsc{Single}$ and $\\textsc{Non-Favorite}$. The main result of this section, Theorem~\\ref{thm:single}, shows that the optimal revenue for a single subadditive bidder is upper bounded by $24\\textsc{SRev}$ and $16\\textsc{BRev}$.\n\nSection~\\ref{sec:multi} contains the main result of this paper. We show how to upper bound the optimal revenue for XOS (or subadditive) bidders with a constant number of (or $O(\\log m)$) $\\textsc{PostRev}$ (the optimal revenue obtainable by an RSPM) and $\\textsc{APostEnRev}$ ((the optimal revenue obtainable by an ASPE). In particular, $\\textsc{Single}$ can be upper bounded by the optimal revenue  $\\textsc{OPT}^{\\textsc{Copies-UD}}$ in the copies setting which is again upper bounded by $6\\textsc{PostRev}$. We further decompose $\\textsc{Non-Favorite}$ into two terms $\\textsc{Tail}$ and $\\textsc{Core}$, and show how to bound $\\textsc{Tail}$ in Section~\\ref{subsection:tail} and how to bound $\\textsc{Core}$ in Section~\\ref{subsection:core}.\n\n\n\n\\section{Warm Up: Single Bidder}\\label{sec:single}\nTo warm up, we first study the case where there is a single subadditive buyer and show how to improve the approximation ratio from $338$ to $40$. Since there is only one buyer, we will drop the subscript $i$ in the notations. As specified in Section~\\ref{sec:choice of beta}, we use a $\\beta$ that satisfies both properties in Lemma~\\ref{lem:requirement for beta}. For a single buyer, we can simply set $\\beta_{j}$ to be $0$ for all $j$. We use $\\textsc{Single}(M), \\textsc{Non-Favorite}(M)$ in the following proof to denote the corresponding terms in Theorem~\\ref{thm:revenue upperbound for subadditive} for $\\beta=\\textbf{0}$. Notice $R_0^{(\\textbf{0})}=\\emptyset$. Theorem~\\ref{thm:single} shows that the optimal revenue is within a constant factor of the better of selling separately and grand bundling.\n\n\\begin{theorem}\\label{thm:single}\nFor a single buyer whose valuation distribution is subadditive over independent items, \n\\[\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}\\]\nfor any BIC mechanism $M$.\n\\end{theorem}\n\nRecall that the revenue for mechanism $M$ is upper bounded by $4\\cdot \\textsc{Non-Favorite}(M)+2\\cdot\\textsc{Single}(M)$ (Theorem~\\ref{thm:revenue upperbound for subadditive}). We first upper bound $\\textsc{Single}(M)$ by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. Since $\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the original setting, $ \\mathds{1}[t\\in R_j^{(\\beta)}]\\cdot\\pi^{(\\beta)}_{j}(t)$ with $\\pi^{(\\beta)}_j(t)=\\sum_{S:j\\in S}\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the copies setting, and therefore $\\textsc{Single}(M)$ is the Myerson Virtual Welfare of a certain allocation in the copies setting, which is upper bounded by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. By~\\cite{ChawlaHMS10}, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is at most $2\\cdot\\textsc{SRev}$.\n\\begin{lemma}\\label{lem:single-single}\nFor any BIC mechanism $M$, $\\textsc{Single} (M)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2\\cdot\\textsc{SRev}.$\n\\end{lemma}\n\nWe decompose $\\textsc{Non-Favorite}(M)$ into two terms $\\textsc{Core}(M)$ and $\\textsc{Tail}(M)$, and bound them separately. For every $t\\in T$, define $\\mathcal{C}(t)=\\{j:V(t_j)< c\\}$, $\\mathcal{T}(t)=[m]\\backslash \\mathcal{C}(t)$. Here the threshold $c$ is chosen as\n\\begin{equation}\\label{equ:single-def of c}\nc:=\\inf\\left\\{x\\geq 0:\\  \\sum_j \\Pr_{t_j}\\left[V(t_j)\\geq x\\right]\\leq 2\\right\\}.\n\\end{equation}\nSince $v(t,\\cdot)$ is subadditive for all $t\\in T$ , we have for every $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. {We decompose $\\textsc{Non-Favorite}(M)$ based on the inequality above.} Proof of Lemma~\\ref{lem:single decomposition} can be found in Appendix~\\ref{sec:single_appx}.\n\n\n\n\n\\begin{lemma}\\label{lem:single decomposition}\n\\begin{align*}\t\\textsc{Non-Favorite}(M)\n\\leq &\t\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))~~~~~~~~~\\quad(\\textsc{Core}(M))\\\\\n+&\\sum_j\\sum_{t_{j}:V(t_{j})\\geq c}f_{j}(t_{j})\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~~\\quad(\\textsc{Tail}(M))\n\\end{align*}\n\\end{lemma}\n\n\nUsing the definition of $c$ and $\\textsc{SRev}$, we can upper bound $\\textsc{Tail}(M)$ with a similar argument as in~\\cite{CaiDW16}.  \n\\begin{lemma}\\label{lem:single-tail}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M)\\leq 2\\cdot\\textsc{SRev}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$,\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{proof}\n\n\nThe $\\textsc{Core}(M)$ is upper bounded by ${\\mathbb{E}}_{t}[v'(t,[m])]$ where $v'(t,S)$\n$= v(t,S\\cap \\mathcal{C}(t))$. We argue that $v'(t,\\cdot)$ is drawn from a distribution that is subadditive over independent items and $v'(\\cdot,\\cdot)$ is $c$-Lipschitz (see Definition~\\ref{def:Lipschitz}). Using a concentration bound by Schechtman~\\cite{Schechtman2003concentration}, we show ${\\mathbb{E}}_{t}[v'(t,[m])]$ is upper bounded by the median of random variable $v'(t,[m])$ and $c$, which are upper bounded by $\\textsc{BRev}$ and $\\textsc{SRev}$ respectively.\n\\begin{lemma}\\label{lem:single-core}\nFor any BIC mechanism $M$, $\\textsc{Core}(M) \\leq 3\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev}$.\n\\end{lemma}\n\nRecall that\n\\begin{equation}\n\\textsc{Core}(M)=\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq [m]}\\sigma_S^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\n\\end{equation}\n\nWe will bound $\\textsc{Core}(M)$ with a concentration inequality from~\\cite{Schechtman2003concentration}. It requires the following definition:\n\n\\begin{definition}\\label{def:Lipschitz}\nA function $v(\\cdot,\\cdot)$ is \\textbf{$a$-Lipschitz} if for any type $t,t'\\in T$, and set $X,Y\\subseteq [m]$,\n$$\\left|v(t,X)-v(t',Y)\\right|\\leq a\\cdot \\left(\\left|X\\Delta Y\\right|+\\left|\\{j\\in X\\cap Y:t_j\\not=t_j'\\}\\right|\\right),$$ where $X\\Delta Y=\\left(X\\backslash Y\\right)\\cup \\left(Y\\backslash X\\right)$ is the symmetric difference between $X$ and $Y$.\n\\end{definition}\n\nDefine a new valuation function for the bidder as $v'(t,S)=v(t,S\\cap \\mathcal{C}(t))$, for all $t\\in T$ and $S\\subseteq [m]$. Then $v'(\\cdot,\\cdot)$ is $c-$ Lipschitz, and when $t$ is drawn from the product distribution $D=\\prod_j D_j$, $v'(t,\\cdot)$ remains to be a valuation drawn from a distribution that is subadditive over independent items. See Appendix~\\ref{sec:single_appx} for the proof of Lemma~\\ref{lem:single subadditive} and Lemma~\\ref{lem:single Lipschitz}.\n\n\\begin{lemma}\\label{lem:single subadditive}\nFor all $t\\in T$, $v'(t,\\cdot)$ satisfies monotonicity, subadditivity and no externalities defined in Definition~\\ref{def:subadditive independent}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:single Lipschitz}\n$v'(\\cdot,\\cdot)$ is $c-$Lipschitz.\n\\end{lemma}\n\nNext, we apply the following concentration inequality to derive Corollary~\\ref{corollary:concentrate}, which is useful to analyze the $\\textsc{Core}(M)$. \n\n\\begin{lemma}~\\cite{Schechtman2003concentration}\\label{lem:schechtman}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is  subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then for all $a>0, k\\in \\{1,2,...,|I|\\}, q\\in \\mathbb{N}$,\n$$\\Pr_t[g(t,I)\\geq (q+1)a+k\\cdot c]\\leq \\Pr_t[g(t,I)\\leq a]^{-q}q^{-k}.$$\n\\end{lemma}\n\n\\begin{corollary}\\label{corollary:concentrate}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is  subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then if we let $a$ be the median of the value of the grand bundle $g(t,I)$, i.e. $a=\\inf\\left\\{x\\geq 0: \\Pr_t[g(t,I)\\leq x]\\geq \\frac{1}{2}\\right\\}$,\n$$\\mathds{E}_t[g(t,I)]\\leq 2a+\\frac{5c}{2}.$$\n\\end{corollary}\n\n\\begin{proof}\nLet $Y$ be $g(t,I)$. If we apply Lemma~\\ref{lem:schechtman} to the case where $a$ is the median and $q=2$, we have\n\n\\begin{align*}\n\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]&= 3a\\cdot \\Pr_t[Y\\geq 3a]+\\int_{y=0}^{\\infty}\\Pr_t[Y\\geq 3a+y]dy\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot\\sum_{k=0}^{|I|} \\Pr_t[Y\\geq 3a+k\\cdot c] \\quad(Y\\leq |I|\\cdot c)\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot \\sum_{k=0}^2 \\Pr_t[Y > a]+ c\\cdot\\sum_{k=3}^{|I|} 4\\cdot 2^{-k}\\quad(\\text{Lemma~\\ref{lem:schechtman}})\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n\\end{align*}\n\nWith the inequality above, we can upper bound the expected value of $Y$.\n\\begin{align*}\n{\\mathbb{E}}_{t}[Y]&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]\\\\\n&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n&= a+2a\\cdot \\Pr_{t}[Y>a]+\\frac{5}{2}c\\\\\n&\\leq 2a+\\frac{5}{2}c\n\\end{align*}\n\\end{proof}\n\n\nNow, we are ready to prove Lemma~\\ref{lem:single-core}.\n\n\\begin{prevproof}{Lemma}{lem:single-core}\nLet $\\delta$ be the median of $v'(t,[m])$ when $t$ is sampled from distribution $D$. Now consider the mechanism that sells the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $v(t,[m])\\geq v'(t,[m])$. Thus with probability at least $\\frac{1}{2}$,\n  the bidder purchases the bundle. Thus, $\\textsc{BRev}\\geq \\frac{1}{2}\\delta$.\n\nAccording to Corollary~\\ref{corollary:concentrate},\n\n\\begin{comment}\n\\begin{equation}\\label{equ:single-core}\n\\begin{aligned}\n\\textsc{Core}(M)&\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\\\\\n&< 4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev} \\text{(Lemma~\\ref{lem:single-bound for c}, Inequality~\\ref{equ:bound for delta})}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\\begin{equation}\\label{equ:single-core-prev}\n\\textsc{Core}(M)\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\n\\end{equation}\n\nIt remains to argue that the Lipchitz constant $c$ can be upper bounded using $\\textsc{SRev}$. Notice that by AM-GM Inequality,\n\\begin{align*}\n&\\Pr_t\\left[\\exists j\\in [m], V(t_j)\\geq c\\right]= 1-\\prod_{j}\\Pr_{t_j}[V(t_j)< c]\\\\\n\\geq& 1-(\\frac{\\sum_j \\Pr_{t_j}[V(t_j)< c]}{m})^m\n= 1-(1-\\frac{2}{m})^m\n\\geq 1-e^{-2}\n\\end{align*}\n\n\n\nConsider the mechanism that posts price $c$ for each item but only allow the buyer to purchase one item. Then with probability at least $1-e^{-2}$, the mechanism sells one item obtaining expected revenue $(1-e^{-2})\\cdot c$. Thus $c\\leq \\frac{1}{1-e^{-2}}\\cdot\\textsc{SRev}$. Inequality~\\eqref{equ:single-core-prev} becomes\n\n\\begin{equation}\\label{equ:single-core}\n\\textsc{Core}(M)\\leq 2\\delta+\\frac{5c}{2}<4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev}\n\\end{equation}\n\n\\end{prevproof}\n\n\\begin{prevproof}{Theorem}{thm:single}\nSince $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2 \\textsc{SRev}$ (Lemma~\\ref{lem:single-single}) and $\\textsc{Non-Favorite}(M)\\leq 5\\textsc{SRev}+4\\textsc{BRev}$ (Lemma~\\ref{lem:single-tail} and~\\ref{lem:single-core}), $\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}$ according to Theorem~\\ref{thm:revenue upperbound for subadditive}.\n\\end{prevproof}\n\n\n\n\n\n\n\n\n\n\\section{Sequentially Posted-Price Mechanism with Entry Fee}\\label{sec:spm}\nHere is the formal specification of the Sequential Posted Price with Entry Fee Mechanism.\\\\\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n    \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n        \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n        \\STATE $S\\gets S\\backslash S_i^{*}$.\n    \\ELSE\n        \\STATE $i$ gets nothing and pays $0$.\n    \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe development of sensor, peer-to-peer, and ad hoc wireless networks\nhas stimulated interest in distributed algorithms for data\naggregation, in which nodes in a network compute a function of local\nvalues at the individual nodes.  These networks typically do not have\ncentralized agents that organize the computation and communication\namong the nodes.  Furthermore, the nodes in such a network may not\nknow the complete topology of the network, and the topology may change\nover time as nodes are added and other nodes fail.  In light of the\npreceding considerations, distributed computation is of vital\nimportance in these modern networks.\n\nWe consider the problem of computing separable functions in a\ndistributed fashion in this paper.  A separable function can be\nexpressed as the sum of the values of individual functions.  Given a\nnetwork in which each node has a number, we seek a distributed\nprotocol for computing the value of a separable function of the\nnumbers at the nodes.  Each node has its own estimate of the value of\nthe function, which evolves as the protocol proceeds.  Our goal is to\nminimize the amount of time required for all of these estimates to be\nclose to the actual function value.\n\nIn this work, we are interested in {\\em totally distributed}\ncomputations, in which nodes have a local view of the state of the\nnetwork.  Specifically, an individual node does not have information\nabout nodes in the network other than its neighbors.  To accurately\nestimate the value of a separable function that depends on the numbers\nat all of the nodes, each node must obtain information about the other\nnodes in the network.  This is accomplished through communication\nbetween neighbors in the network.  Over the course of the protocol,\nthe global state of the network effectively diffuses to each\nindividual node via local communication among neighbors.\n\nMore concretely, we assume that each node in the network knows only\nits neighbors in the network topology, and can contact any neighbor to\ninitiate a communication.  On the other hand, we assume that the nodes\ndo not have unique identities (i.e., a node has no unique identifier\nthat can be attached to its messages to identify the source of the\nmessages).  This constraint is natural in ad-hoc and mobile networks,\nwhere there is a lack of infrastructure (such as IP addresses or\nstatic GPS locations), and it limits the ability of a distributed\nalgorithm to recreate the topology of the network at each node.  In\nthis sense, the constraint also provides a formal way to distinguish\ndistributed algorithms that are truly local from algorithms that\noperate by gathering enormous amounts of global information at all the\nnodes.\n\nThe absence of identifiers for nodes makes it difficult, without\nglobal coordination, to simply transmit every node's value throughout\nthe network so that each node can identify the values at all the\nnodes.  As such, we develop an algorithm for computing separable\nfunctions that relies on an {\\em order- and duplicate-insensitive}\nstatistic \\cite{ngsa} of a set of numbers, the minimum.  The algorithm\nis based on properties of exponential random variables, and reduces\nthe problem of computing the value of a separable function to the\nproblem of determining the minimum of a collection of numbers, one for\neach node.\n\nThis reduction leads us to study the problem of\n{\\em information spreading} or {\\em information dissemination} in a\nnetwork.  In this problem, each node starts with a message, and the\nnodes must spread the messages throughout the network using local\ncommunication so that every node eventually has every message.\nBecause the minimum of a collection of numbers is not affected by the\norder in which the numbers appear, nor by the presence of duplicates\nof an individual number, the minimum computation required by our\nalgorithm for computing separable functions can be performed by any\ninformation spreading algorithm.  Our analysis of the algorithm for\ncomputing separable functions establishes an upper bound on its\nrunning time in terms of the running time of the information spreading\nalgorithm it uses as a subroutine.\n\nIn view of our goal of distributed computation, we analyze a\n{\\em gossip} algorithm for information spreading.  Gossip algorithms\nare a useful tool for achieving fault-tolerant and scalable\ndistributed computations in large networks.  In a gossip algorithm,\neach node repeatedly iniatiates communication with a small number of\nneighbors in the network, and exchanges information with those\nneighbors.\n\nThe gossip algorithm for information spreading that we study is\nrandomized, with the communication partner of a node at any time\ndetermined by a simple probabilistic choice.  We provide an upper\nbound on the running time of the algorithm in terms of the\n{\\em conductance} of a stochastic matrix that governs how nodes choose\ncommunication partners.  By using the gossip algorithm to compute\nminima in the algorithm for computing separable functions, we obtain\nan algorithm for computing separable functions whose performance on\ncertain graphs compares favorably with that of known iterative\ndistributed algorithms \\cite{bgps} for computing averages in a\nnetwork.\n\n\\subsection{Related work}\n\\label{sec:related}\n\nIn this section, we present a brief summary of related work.\nAlgorithms for computing the number of distinct elements in a multiset\nor data stream \\cite{fm, streamdistinct} can be adapted to compute\nseparable functions using information spreading \\cite{clkb}.  We are\nnot aware, however, of a previous analysis of the amount of time\nrequired for these algorithms to achieve a certain accuracy in the\nestimates of the function value when the computation is totally\ndistributed (i.e., when nodes do not have unique identities).  These\nadapted algorithms require the nodes in the network to make use of a\ncommon hash function.  In addition, the discreteness of the counting\nproblem makes the resulting algorithms for computing separable\nfunctions suitable only for functions in which the terms in the sum\nare integers.  Our algorithm is simpler than these algorithms, and can\ncompute functions with non-integer terms.\n\n\n\nThere has been a lot of work on the distributed computation of\naverages, a special case of the problem of reaching agreement or\nconsensus among processors via a distributed computation.  Distributed\nalgorithms for reaching consensus under appropriate conditions have\nbeen known since the classical work of Tsitsiklis\n\\cite{tsitsiklis-thesis} and Tsitsiklis, Bertsekas, and Athans\n\\cite{tba} (see also the book by Bertsekas and Tsitsiklis\n\\cite{pardiscomp}).  Averaging algorithms compute the ratio of the sum\nof the input numbers to $n$, the number of nodes in the network, and\nnot the exact value of the sum.  Thus, such algorithms cannot be\nextended in general to compute arbitrary separable functions.  On the\nother hand, an algorithm for computing separable functions can be used\nto compute averages by separately computing the sum of the input\nnumbers, and the number of nodes in the graph (using one as the input\nat each node).\n\n\nRecently, Kempe, Dobra, and Gehrke showed the existence of a\nrandomized iterative gossip algorithm for averaging with the optimal\naveraging time \\cite{kempe}.  This result was restricted to complete\ngraphs.  The algorithm requires that the nodes begin the computation\nin an asymmetric initial state in order to compute separable\nfunctions, a requirement that may not be convenient for large networks\nthat do not have centralized agents for global coordination.\nFurthermore, the algorithm suffers from the possibility of oscillation\nthroughout its execution.\n\nIn a more recent paper, Boyd, Ghosh, Prabhakar, and Shah presented a\nsimpler iterative gossip algorithm for averaging that addresses some\nof the limitations of the Kempe et al. algorithm \\cite{bgps}.\nSpecifically, the algorithm and analysis are applicable to arbitrary\ngraph topologies.  Boyd et al. showed a connection between the\naveraging time of the algorithm and the mixing time (a property that\nis related to the conductance, but is not the same) of an appropriate\nrandom walk on the graph representing the network.  They also found an\noptimal averaging algorithm as a solution to a semi-definite program.\n\nFor completeness, we contrast our results for the problem of averaging\nwith known results.  As we shall see, iterative averaging, which has\nbeen a common approach in the previous work, is an order slower than\nour algorithm for many graphs, including ring and grid graphs.  In\nthis sense, our algorithm is quite different than (and has advantages\nin comparison with) the known averaging algorithms.\n\nOn the topic of information spreading, gossip algorithms for\ndisseminating a message to all nodes in a complete graph in which\ncommunication partners are chosen uniformly at random have been\nstudied for some time \\cite{frieze, rumor2, epidemic}.  Karp,\nSchindelhauer, Shenker, and V\\\"{o}cking presented a\n{\\em push and pull} gossip algorithm, in which communicating nodes\nboth send and receive messages, that disseminates a message to all $n$\nnodes in a graph in $O(\\log n)$ time with high probability\n\\cite{kssv}.  In this work, we have provided an analysis of the time\nrequired for a gossip algorithm to disseminate $n$ messages to $n$\nnodes for the more general setting of arbitrary graphs and non-uniform\nrandom choices of communication partners.  For other related results,\nwe refer the reader to \\cite{rumor3, gossip1, gossip2}.  We take note\nof the similar (independent) recent work of Ganesh, Massouli\\'{e}, and\nTowsley \\cite{gmt}, and Berger, Borgs, Chayes, and Saberi \\cite{bbcs},\non the spread of epidemics in a network.\n\n\n\\subsection{Organization}\n\nThe rest of the paper is organized as follows.  Section\n\\ref{sec:prelim} presents the distributed computation problems we\nstudy and an overview of our results.  In Section \\ref{sec:comp}, we\ndevelop and analyze an algorithm for computing separable functions in\na distributed manner.  Section \\ref{sec:infdis} contains an analysis\nof a simple randomized gossip algorithm for information spreading,\nwhich can be used as a subroutine in the algorithm for computing\nseparable functions.  In Section \\ref{sec:appl}, we discuss\napplications of our results to particular types of graphs, and compare\nour results to previous results for computing averages.  Finally, we\npresent conclusions and future directions in Section \\ref{sec:conc}.\n\n\\section{Preliminaries and Results}\n\\label{sec:prelim}\n\nWe consider an arbitrary connected network, represented by an\nundirected graph $G = (V, E)$, with $|V| = n$ nodes.  For notational\npurposes, we assume that the nodes in $V$ are numbered arbitrarily so\nthat $V = \\{1, \\dots, n\\}$.  A node, however, does not have a unique\nidentity that can be used in a computation.  Two nodes $i$ and $j$ can\ncommunicate with each other if (and only if) $(i, j) \\in E$.\n\nTo capture some of the resource constraints in the networks in which\nwe are interested, we impose a {\\em transmitter gossip} constraint on\nnode communication.  Each node is allowed to contact at most one other\nnode at a given time for communication.  However, a node can be\ncontacted by multiple nodes simultaneously.\n\nLet $2^{V}$ denote the power set of the vertex set $V$ (the set of all\nsubsets of $V$).  For an $n$-dimensional vector\n$\\vec{x} \\in \\mathbf{R}^{n}$, let $x_{1}, \\dots, x_{n}$ be the components\nof $\\vec{x}$.\n\\begin{definition}\nWe say that a function $f : \\mathbf{R}^{n} \\times 2^{V} \\to \\mathbf{R}$\nis {\\em separable} if there exist functions $f_{1}, \\dots, f_{n}$ such\nthat, for all $S \\subseteq V$,\n\\begin{equation}\nf(\\vec{x}, S) = \\sum_{i \\in S} f_{i}(x_{i}).\n\\label{sepsum}\n\\end{equation}\n\\label{sepfunc}\n\\end{definition}\n\n\\noindent\n{\\bf Goal.}  Let $\\cal{F}$ be the class of separable functions $f$ for\nwhich $f_{i}(x) \\geq 1$ for all $x \\in \\mathbf{R}$ and $i = 1, \\dots, n$.\nGiven a function $f \\in \\cal{F}$, and a vector $\\vec{x}$ containing\ninitial values $x_{i}$ for all the nodes, the nodes in the network are\nto compute the value $f(\\vec{x}, V)$ by a distributed computation,\nusing repeated communication between nodes.\n\n\\begin{note}\nConsider a function $g$ for which there exist functions\n$g_{1}, \\dots, g_{n}$ satisfying, for all $S \\subseteq V$, the\ncondition $g(\\vec{x}, S) = \\prod_{i \\in S} g_{i}(x_{i})$ in lieu of\n(\\ref{sepsum}).  Then, $g$ is {\\em logarithmic separable}, i.e.,\n$f = \\log_b g$ is separable.  Our algorithm for computing separable\nfunctions can be used to compute the function $f = \\log_{b} g$.  The\ncondition $f_{i}(x) \\geq 1$ corresponds to $g_{i}(x) \\geq b$ in this\ncase.  This lower bound of $1$ on $f_{i}(x)$ is arbitrary, although\nour algorithm does require the terms $f_{i}(x_{i})$ in the sum to be\npositive.\n\\end{note}\n\nBefore proceeding further, we list some practical situations where the\ndistributed computation of separable functions arises naturally.  By\ndefinition, the sum of a set of numbers is a separable function.\n\\renewcommand{\\labelenumi}{(\\arabic{enumi})}\n\\begin{enumerate}\n\\item\n{\\em Summation.} Let the value at each node be $x_{i} = 1$.  Then, the\nsum of the values is the number of nodes in the network.\n\n\\item\n{\\em Averaging.} According to Definition \\ref{sepfunc}, the average of\na set of numbers is not a separable function.  However, the nodes can\nestimate the separable function $\\sum_{i = 1}^{n} x_{i}$ and $n$\nseparately, and use the ratio between these two estimates as an\nestimate of the mean of the numbers.\n\nSuppose the values at the nodes are measurements of a quantity of\ninterest.  Then, the average provides an unbiased maximum likelihood\nestimate of the measured quantity.  For example, if the nodes are\ntemperature sensors, then the average of the sensed values at the\nnodes gives a good estimate of the ambient temperature.\n\\end{enumerate}\n\nFor more sophisticated applications of a distributed averaging\nalgorithm, we refer the reader to \\cite{distr_eigvec} and\n\\cite{MSZ}.\nAveraging is used for the distributed computation of the top $k$\neigenvectors of a graph in \\cite{distr_eigvec}, while in \\cite{MSZ}\naveraging is used in a throughput-optimal distributed scheduling\nalgorithm in a wireless network.\n\n\\noindent{\\bf Time model.} In a distributed computation, a time model\ndetermines when nodes communicate with each other.  We consider two\ntime models, one synchronous and the other asynchronous, in this\npaper.  The two models are described as follows.\n\\begin{enumerate}\n\\item\n{\\em Synchronous time model:} Time is slotted commonly across all\nnodes in the network.  In any time slot, each node may contact one of\nits neighbors according to a random choice that is independent of the\nchoices made by the other nodes.  The simultaneous communication\nbetween the nodes satisfies the transmitter gossip constraint.\n\n\\item\n{\\em Asynchronous time model:} Each node has a clock that ticks at the\ntimes of a rate $1$ Poisson process.  Equivalently, a common clock\nticks according to a rate $n$ Poisson process at times\n$C_{k}, k \\geq 1$, where $\\{C_{k + 1} - C_{k}\\}$ are i.i.d.\nexponential random variables of rate $n$.  On clock tick $k$, one of\nthe $n$ nodes, say $I_{k}$, is chosen uniformly at random.  We\nconsider this global clock tick to be a tick of the clock at node\n$I_{k}$.  When a node's clock ticks, it contacts one of its neighbors\nat random.  In this model, time is discretized according to clock\nticks.  On average, there are $n$ clock ticks per one unit of absolute\ntime.\n\\end{enumerate}\n\nIn this paper, we measure the running times of algorithms in absolute\ntime, which is the number of time slots in the synchronous model, and\nis (on average) the number of clock ticks divided by $n$ in the\nasynchronous model.  To obtain a precise relationship between clock\nticks and absolute time in the asynchronous model, we appeal to tail\nbounds on the probability that the sample mean of i.i.d. exponential\nrandom variables is far from its expected value.  In particular, we\nmake use of the following lemma, which also plays a role in the\nanalysis of the accuracy of our algorithm for computing separable\nfunctions.\n\\begin{lemma}\n\\label{discrete-to-cont}\nFor any $k \\geq 1$, let $Y_{1}, \\dots, Y_{k}$ be i.i.d. exponential\nrandom variables with rate $\\lambda$.  Let\n$R_{k} = \\frac{1}{k} \\sum_{i = 1}^{k} Y_{i}$.  Then, for any\n$\\epsilon \\in (0, 1\/2)$,\n\\begin{eqnarray}\n\\Pr \\left(\\left|R_k - \\frac{1}{\\lambda}\\right|\n\\geq \\frac{\\epsilon}{\\lambda} \\right)\n& \\leq & 2 \\exp\\left(-\\frac{\\epsilon^{2} k}{3}\\right).\n\\label{e:dtoc1}\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nBy definition,\n$E[R_{k}] = \\frac{1}{k}\\sum_{i = 1}^{k} \\lambda^{-1} = \\lambda^{-1}$.\nThe inequality in (\\ref{e:dtoc1}) follows directly from Cram\\'{e}r's\nTheorem (see \\cite{dembo}, pp. $30$, $35$) and properties of\nexponential random variables.\n\\end{proof}\n\nA direct implication of Lemma \\ref{discrete-to-cont} is the following\ncorollary, which bounds the probability that the absolute time $C_{k}$\nat which clock tick $k$ occurs is far from its expected value.\n\\begin{corollary}\\label{discrete-to-contc}\nFor $k \\geq 1$, $E[C_{k}] = k\/n$.  Further, for any\n$\\epsilon \\in (0, 1\/2)$,\n\\begin{eqnarray}\n\\Pr \\left( \\left|C_{k} - \\frac{k}{n}\\right|\n\\geq \\frac{\\epsilon k}{n} \\right)\n& \\leq & 2 \\exp\\left(-\\frac{\\epsilon^{2} k}{3}\\right).\n\\label{e:dtoc}\n\\end{eqnarray}\n\\end{corollary}\n\nOur algorithm for computing separable functions is randomized, and is\nnot guaranteed to compute the exact quantity\n$f(\\vec{x}, V) = \\sum_{i = 1}^{n} f_{i}(x_{i})$ at each node in the\nnetwork.  To study the accuracy of the algorithm's estimates, we\nanalyze the probability that the estimate of $f(\\vec{x}, V)$ at every\nnode is within a $(1 \\pm \\epsilon)$ multiplicative factor of the true\nvalue $f(\\vec{x}, V)$ after the algorithm has run for some period of\ntime.  In this sense, the error in the estimates of the algorithm is\nrelative to the magnitude of $f(\\vec{x}, V)$.\n\nTo measure the amount of time required for an algorithm's estimates to\nachieve a specified accuracy with a specified probability, we define\nthe following quantity.  For an algorithm ${\\cal C}$ that estimates\n$f(\\vec{x}, V)$, let $\\hat{y}_i(t)$ be the estimate of $f(\\vec{x}, V)$ at\nnode $i$ at time $t$.  Furthermore, for notational convenience, given\n$\\epsilon > 0$, let $A_{i}^{\\epsilon}(t)$ be the following event.\n\\[\nA_{i}^{\\epsilon}(t)\n= \\left\\{\\hat{y}_{i}(t) \\not\\in\n\\left[(1 - \\epsilon)f(\\vec{x}, V),\n(1 + \\epsilon)f(\\vec{x}, V) \\right] \\right\\}\n\\]\n\\begin{definition}\nFor any $\\epsilon > 0$ and $\\delta \\in (0, 1)$, the\n($\\epsilon$, $\\delta$)-computing time of $\\cal{C}$, denoted\n$T_{\\cal{C}}^{\\text{cmp}}(\\epsilon, \\delta)$, is\n\\[\nT_{\\cal{C}}^{\\text{cmp}}(\\epsilon, \\delta)\n= \\sup_{f \\in \\cal{F}} \\sup_{\\vec{x} \\in \\mathbf{R}^{n}}\n\\inf \\Big\\{\\tau : \\forall t \\geq \\tau,\n\\Pr \\big(\\cup_{i = 1}^{n} A_{i}^{\\epsilon}(t) \\big)\n\\leq \\delta \\Big\\}.\n\\]\n\\end{definition}\n\n\\noindent\nIntuitively, the significance of this definition of the\n$(\\epsilon, \\delta)$-computing time of an algorithm $\\cal{C}$ is that,\nif $\\cal{C}$ runs for an amount of time that is at least\n$T_{\\cal{C}}^{\\text{cmp}}(\\epsilon, \\delta)$, then the probability that the\nestimates of $f(\\vec{x}, V)$ at the nodes are all within a\n$(1 \\pm \\epsilon)$ factor of the actual value of the function is at\nleast $1 - \\delta$.\n\nAs noted before, our algorithm for computing separable functions is\nbased on a reduction to the problem of information spreading, which is\ndescribed as follows.  Suppose that, for $i = 1, \\dots, n$, node $i$\nhas the one message $m_{i}$.  The task of information spreading is to\ndisseminate all $n$ messages to all $n$ nodes via a sequence of local\ncommunications between neighbors in the graph.  In any single\ncommunication between two nodes, each node can transmit to its\ncommunication partner any of the messages that it currently holds.  We\nassume that the data transmitted in a communication must be a set of\nmessages, and therefore cannot be arbitrary information.\n\nConsider an information spreading algorithm $\\cal{D}$, which specifies\nhow nodes communicate.  For each node $i \\in V$, let $S_{i}(t)$ denote\nthe set of nodes that have the message $m_{i}$ at time $t$.  While\nnodes can gain messages during communication, we assume that they do\nnot lose messages, so that $S_{i}(t_{1}) \\subseteq S_{i}(t_{2})$ if\n$t_{1} \\leq t_{2}$.  Analogous to the $(\\epsilon, \\delta)$-computing\ntime, we define a quantity that measures the amount of time required\nfor an information spreading algorithm to disseminate all the messages\n$m_{i}$ to all the nodes in the network.\n\\begin{definition}\nFor $\\delta \\in (0, 1)$, the $\\delta$-information-spreading time\nof the algorithm $\\cal{D}$, denoted $T_{\\cal{D}}^{\\text{spr}}(\\delta)$, is\n\\[\nT_{\\cal{D}}^{\\text{spr}}(\\delta)\n= \\inf\n\\left\\{t : \\Pr \\left(\\cup_{i = 1}^{n} \\{S_{i}(t) \\neq V\\} \\right)\n\\leq \\delta \\right\\}.\n\\]\n\\end{definition}\n\nIn our analysis of the gossip algorithm for information spreading, we\nassume that when two nodes communicate, each node can send all of its\nmessages to the other in a single communication.  This rather\nunrealistic assumption of {\\em infinite} link capacity is merely for\nconvenience, as it provides a simpler analytical characterization of\n$T_{\\cal{C}}^{\\text{cmp}}(\\epsilon, \\delta)$ in terms of\n$T_{\\cal{D}}^{\\text{spr}}(\\delta)$.  Our algorithm for computing separable\nfunctions requires only links of unit capacity.\n\n\\subsection{Our contribution}\n\\label{ssec:contrib}\n\nThe main contribution of this paper is the design of a distributed\nalgorithm to compute separable functions of node values in an\narbitrary connected network.  Our algorithm is randomized, and in\nparticular uses exponential random variables.  This usage of\nexponential random variables is analogous to that in an \nalgorithm by Cohen\\footnote{We thank Dahlia Malkhi for pointing\nthis reference out to us.}\n for estimating the sizes of sets in a graph \\cite{cohen}.  The\nbasis for our algorithm is the following property of the exponential\ndistribution.\n\\begin{property}\n\\label{p1}\nLet $W_{1}, \\dots, W_{n}$ be $n$ independent random variables such\nthat, for $i = 1, \\dots, n$, the distribution of $W_{i}$ is\nexponential with rate $\\lambda_{i}$.  Let $\\bar{W}$ be the minimum of\n$W_{1}, \\dots, W_{n}$.  Then, $\\bar{W}$ is distributed as an exponential\nrandom variable of rate $\\lambda = \\sum_{i = 1}^{n} \\lambda_{i}$.\n\\end{property}\n\\begin{proof}\nFor an exponential random variable $W$ with rate $\\lambda$, for any\n$z \\in \\Reals_{+}$,\n\\[\n\\Pr(W > z) = \\exp(-\\lambda z).\n\\]\nUsing this fact and the independence of the random variables $W_{i}$,\nwe compute $\\Pr(\\bar{W} > z)$ for any $z \\in \\Reals_{+}$.\n\\begin{eqnarray*}\n\\Pr(\\bar{W} > z)\n& = & \\Pr \\left(\\cap_{i = 1}^{n} \\{W_{i} > z\\} \\right) \\\\\n& = & \\prod_{i = 1}^{n} \\Pr(W_{i} > z) \\\\\n& = & \\prod_{i = 1}^{n} \\exp(-\\lambda_{i} z) \\\\\n& = & \\exp\\left(-z \\sum_{i = 1}^{n} \\lambda_{i} \\right).\n\\end{eqnarray*}\nThis establishes the property stated above.\n\\end{proof}\n\nOur algorithm uses an information spreading algorithm as a subroutine,\nand as a result its running time is a function of the running time of\nthe information spreading algorithm it uses.  The faster the\ninformation spreading algorithm is, the better our algorithm performs.\nSpecifically, the following result provides an upper bound on the\n($\\epsilon$, $\\delta$)-computing time of the algorithm.\n\\begin{theorem}\n\\label{thm:main1}\nGiven an information spreading algorithm $\\cal{D}$ with\n$\\delta$-spreading time $T_{\\cal{D}}^{\\text{spr}}(\\delta)$ for\n$\\delta \\in (0, 1)$, there exists an algorithm ${\\cal{A}}$ for\ncomputing separable functions $f \\in \\cal{F}$ such that, for any\n$\\epsilon \\in (0, 1)$ and $\\delta \\in (0, 1)$,\n\\[\nT_{\\cal{A}}^{\\text{cmp}}(\\epsilon, \\delta)\n= O\\left( \\epsilon^{-2} (1 + \\log \\delta^{-1})\nT_{\\cal{D}}^{\\text{spr}}(\\delta\/2) \\right).\n\\]\n\\end{theorem}\n\nMotivated by our interest in decentralized algorithms, we analyze a\nsimple randomized gossip algorithm for information spreading.  When\nnode $i$ initiates a communication, it contacts each node $j \\neq i$\nwith probability $P_{ij}$.  With probability $P_{ii}$, it does not\ncontact another node.  The $n \\times n$ matrix $P = [P_{ij}]$\ncharacterizes the algorithm; each matrix $P$ gives rise to an\ninformation spreading algorithm $\\cal{P}$.  We assume that $P$ is\nstochastic, and that $P_{ij} = 0$ if $i \\neq j$ and $(i, j) \\notin E$,\nas nodes that are not neighbors in the graph cannot communicate with\neach other.  Section \\ref{sec:infdis} describes the data transmitted\nbetween two nodes when they communicate.\n\nWe obtain an upper bound on the $\\delta$-information-spreading time of\nthis gossip algorithm in terms of the {\\em conductance} of the matrix\n$P$, which is defined as follows.\n\\begin{definition}\nFor a stochastic matrix $P$, the conductance of $P$, denoted\n$\\Phi(P)$, is\n\\[\n\\Phi(P)\n= \\min_{S \\subset V, \\; 0 < |S| \\leq n\/2}\n\\frac{\\sum_{ i \\in S, j \\notin S} P_{ij}}{ |S|}.\n\\]\n\\end{definition}\n\n\\noindent\nIn general, the above definition of conductance is not the same as the\nclassical definition \\cite{sinclair}.  However, we restrict our\nattention in this paper to doubly stochastic matrices $P$.  When $P$\nis doubly stochastic, these two definitions are equivalent.  Note that\nthe definition of conductance implies that $\\Phi(P) \\leq 1$.\n\\begin{theorem}\n\\label{thm:main2}\nConsider any doubly stochastic matrix $P$ such that if $i \\neq j$ and\n$(i, j) \\notin E$, then $P_{ij} = 0$.  There exists an information\ndissemination algorithm $\\cal{P}$ such that, for any\n$\\delta \\in (0, 1)$,\n\\[\nT_{\\cal{P}}^{\\text{spr}}(\\delta)\n= O\\left(\\frac{\\log n + \\log \\delta^{-1}}{\\Phi(P)}\\right).\n\\]\n\\end{theorem}\n\\begin{note}\nThe results of Theorems \\ref{thm:main1} and \\ref{thm:main2} hold for\nboth the synchronous and asynchronous time models.  Recall that the\nquantities $T_{\\cal{C}}^{\\text{cmp}}(\\epsilon, \\delta)$ and\n$T_{\\cal{D}}^{\\text{spr}}(\\delta)$ are defined with respect to absolute time\nin both models.\n\\end{note}\n\n\n\\noindent\n{\\bf A comparison.} Theorems \\ref{thm:main1} and \\ref{thm:main2} imply\nthat, given a doubly stochastic matrix $P$, the time required for our\nalgorithm to obtain a $(1 \\pm \\epsilon)$ approximation with\nprobability at least $1 - \\delta$ is\n$O\\left(\\frac{\\epsilon^{-2} (1 + \\log \\delta^{-1})\n(\\log n + \\log \\delta^{-1})}{\\Phi(P)}\\right)$.\nWhen the network size $n$ and the accuracy parameters $\\epsilon$ and\n$\\delta$ are fixed, the running time scales in proportion to\n$1\/\\Phi(P)$, a factor that captures the dependence of the algorithm on\nthe matrix $P$.  Our algorithm can be used to compute the average of a\nset of numbers.  For iterative averaging algorithms such as the ones\nin \\cite{tsitsiklis-thesis} and \\cite{bgps}, the convergence time\nlargely depends on the mixing time of $P$, which is lower bounded by\n$\\Omega(1\/\\Phi(P))$ (see \\cite{sinclair}, for example).  Thus, our\nalgorithm is (up to a $\\log n$ factor) no slower than the fastest\niterative algorithm based on time-invariant linear dynamics.\n\n\n\n\\section{Function Computation}\n\\label{sec:comp}\n\nIn this section, we describe our algorithm for computing the value\n$y = f(\\vec{x}, V) = \\sum_{i = 1}^{n} f_{i}(x_{i})$ of the separable\nfunction $f$, where $f_{i}(x_{i}) \\geq 1$.  For simplicity of\nnotation, let $y_{i} = f_{i}(x_{i})$.  Given $x_{i}$, each node can\ncompute $y_{i}$ on its own.  Next, the nodes use the algorithm shown\nin Fig. \\ref{compalg}, which we refer to as COMP, to compute estimates\n$\\hat{y}_{i}$ of $y = \\sum_{i = 1}^{n} y_{i}$.  The quantity $r$ is a\nparameter to be chosen later.\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{\\textwidth}\n\\hrulefill\n\n\\noindent\n{\\bf Algorithm COMP}\n\n\\renewcommand{\\labelenumi}{{\\bf \\arabic{enumi}.}}\n\\renewcommand{\\labelenumii}{(\\alph{enumii})}\n\\begin{enumerate}\n\n\\item[{\\bf 0.}]\nInitially, for $i = 1, \\dots, n$, node $i$ has the value\n$y_{i} \\geq 1$.\n\n\\item\nEach node $i$ generates $r$ independent random numbers\n$W_{1}^{i}, \\dots, W_{r}^{i}$, where the distribution of each\n$W_{\\ell}^{i}$ is exponential with rate $y_{i}$ (i.e., with mean\n$1\/y_{i}$).\n\n\\item\n\\label{minstep}\nEach node $i$ computes, for $\\ell = 1, \\dots, r$, an estimate\n$\\hat{W}_{\\ell}^{i}$ of the minimum\n$\\bar{W}_{\\ell} = \\min_{i = 1}^{n} W_{\\ell}^{i}$.  This computation can be\ndone using an information spreading algorithm as described below.\n\n\\item\nEach node $i$ computes\n$\\hat{y}_{i} = \\frac{r}{\\sum_{\\ell = 1}^{r} \\hat{W}_{\\ell}^{i}}$ as its\nestimate of $\\sum_{i = 1}^{n} y_{i}$.\n\\end{enumerate}\n\\hrulefill\n\\end{minipage}\n\\caption{An algorithm for computing separable functions.}\n\\label{compalg}\n\\end{figure}\n\nWe describe how the minimum is computed as required by step\n{\\bf \\ref{minstep}} of the algorithm in Section \\ref{ssec:minima}.\nThe running time of the algorithm COMP depends on the running time of\nthe algorithm used to compute the minimum.\n\nNow, we show that COMP effectively estimates the function value $y$\nwhen the estimates $\\hat{W}_{\\ell}^{i}$ are all correct by providing a\nlower bound on the conditional probability that the estimates produced\nby COMP are all within a $(1 \\pm \\epsilon)$ factor of $y$.\n\\begin{lemma}\n\\label{lem:estaccuracy}\nLet $y_{1}, \\dots, y_{n}$ be real numbers (with $y_{i} \\geq 1$ for\n$i = 1, \\dots, n$), $y = \\sum_{i = 1}^{n} y_{i}$, and\n$\\bar{W} = (\\bar{W}_{1}, \\dots, \\bar{W}_{r})$, where the $\\bar{W}_{\\ell}$ are as\ndefined in the algorithm COMP.  For any node $i$, let\n$\\hat{W}^{i} = (\\hat{W}_{1}^{i}, \\dots, \\hat{W}_{r}^{i})$, and let $\\hat{y}_{i}$ be\nthe estimate of $y$ obtained by node $i$ in COMP.  For any\n$\\epsilon \\in (0, 1\/2)$,\n\\[\n\\begin{split}\n\\Pr & \\left( \\cup_{i = 1}^{n} \\left\\{\n\\left|\\hat{y}_{i} - y \\right| > 2 \\epsilon y \\right\\}\n\\mid \\forall i \\in V, \\: \\hat{W}^{i} = \\bar{W} \\right) \n \\leq 2\\exp\\left(-\\frac{\\epsilon^{2} r}{3} \\right).\n\\end{split}\n\\]\n\\end{lemma}\n\n\\begin{proof}\nObserve that the estimate $\\hat{y}_{i}$ of $y$ at node $i$ is a function\nof $r$ and $\\hat{W}^{i}$.  Under the hypothesis that $\\hat{W}^{i} = \\bar{W}$ for\nall nodes $i \\in V$, all nodes produce the same estimate\n$\\hat{y} = \\hat{y}_{i}$ of $y$.  This estimate is\n$\\hat{y} = r \\left(\\sum_{\\ell = 1}^{r} \\bar{W}_{\\ell} \\right)^{-1}$, and so\n$\\hat{y}^{-1} = \\left(\\sum_{\\ell = 1}^{r} \\bar{W}_{\\ell} \\right)r^{-1}$.\n\nProperty \\ref{p1} implies that each of the $n$ random variables\n$\\bar{W}_{1}, \\dots, \\bar{W}_{r}$ has an exponential distribution with rate\n$y$.  From Lemma \\ref{discrete-to-cont}, it follows that for any\n$\\epsilon \\in (0, 1\/2)$,\n\\begin{equation}\n\\begin{split}\n\\Pr & \\left( \\left|\\hat{y}^{-1} - \\frac{1}{y}\\right|\n> \\frac{\\epsilon}{y}\n\\;\\Big|\\; \\forall i \\in V, \\: \\hat{W}^{i} = \\bar{W} \\right) \n~ \\leq 2\\exp\\left(-\\frac{\\epsilon^2 r}{3}\\right).\n\\end{split}\n\\label{e1}\n\\end{equation}\nThis inequality bounds the conditional probability of the event\n$\\{\\hat{y}^{-1} \\not\\in\n[(1 - \\epsilon) y^{-1}, (1 + \\epsilon)y^{-1}]\\}$,\nwhich is equivalent to the event\n$\\{\\hat{y} \\not\\in [(1 + \\epsilon)^{-1}y, (1 - \\epsilon)^{-1}y]\\}$.\nNow, for $\\epsilon \\in (0, 1\/2)$,\n\\begin{equation}\n\\begin{split}\n(1 - \\epsilon)^{-1} & \\in\n\\left[ 1 + \\epsilon, 1 + 2\\epsilon \\right], \n~ (1 + \\epsilon)^{-1}\n~ \\in \\left[1 - \\epsilon, 1 - 2\\epsilon\/3\\right].\n\\end{split}\n\\label{e2}\n\\end{equation}\nApplying the inequalities in (\\ref{e1}) and (\\ref{e2}), we conclude\nthat for $\\epsilon \\in (0, 1\/2)$,\n\\[\n\\begin{split}\n\\Pr & \\left(\\left| \\hat{y} - y \\right| > 2 \\epsilon y\n\\mid \\forall i \\in V, \\: \\hat{W}^{i} = \\bar{W} \\right) ~\n \\leq 2 \\exp\\left(-\\frac{\\epsilon^2 r}{3}\\right).\n\\end{split}\n\\]\n\n\\noindent\nNoting that the event\n$\\cup_{i = 1}^{n} \\{|\\hat{y}_{i} - y| > 2 \\epsilon y\\}$ is equivalent to\nthe event $\\{|\\hat{y} - y| > 2 \\epsilon y\\}$ when $\\hat{W}^{i} = \\bar{W}$ for all\nnodes $i$ completes the proof of Lemma \\ref{lem:estaccuracy}.\n\\end{proof}\n\n\\subsection{Using information spreading to compute minima}\n\\label{ssec:minima}\n\nWe now elaborate on step {\\bf \\ref{minstep}} of the algorithm COMP.\nEach node $i$ in the graph starts this step with a vector\n$W^{i} = (W_{1}^{i}, \\dots, W_{r}^{i})$, and the nodes seek the vector\n$\\bar{W} = (\\bar{W}_{1}, \\dots, \\bar{W}_{r})$, where\n$\\bar{W}_{\\ell} = \\min_{i = 1}^{n} W_{\\ell}^{i}$.  In the information\nspreading problem, each node $i$ has a message $m_{i}$, and the nodes\nare to transmit messages across the links until every node has every\nmessage.\n\nIf all link capacities are infinite (i.e., in one time unit, a node\ncan send an arbitrary amount of information to another node), then an\ninformation spreading algorithm $\\cal{D}$ can be used directly to\ncompute the minimum vector $\\bar{W}$.  To see this, let the message\n$m_{i}$ at the node $i$ be the vector $W^{i}$, and then apply the\ninformation spreading algorithm to disseminate the vectors.  Once\nevery node has every message (vector), each node can compute $\\bar{W}$ as\nthe component-wise minimum of all the vectors.  This implies that the\nrunning time of the resulting algorithm for computing $\\bar{W}$ is the\nsame as that of the information spreading algorithm.\n\nThe assumption of infinite link capacities allows a node to transmit\nan arbitrary number of vectors $W^{i}$ to a neighbor in one time unit.\nA simple modification to the information spreading algorithm, however,\nyields an algorithm for computing the minimum vector $\\bar{W}$ using links\nof capacity $r$.  To this end, each node $i$ maintains a single\n$r$-dimensional vector $w^{i}(t)$ that evolves in time, starting with\n$w^{i}(0) = W^{i}$.\n\nSuppose that, in the information dissemination algorithm, node $j$\ntransmits the messages (vectors) $W^{i_{1}}, \\dots, W^{i_{c}}$ to node\n$i$ at time $t$.  Then, in the minimum computation algorithm, $j$\nsends to $i$ the $r$ quantities $w_{1}, \\dots, w_{r}$, where\n$w_{\\ell} = \\min_{u = 1}^{c} W_{\\ell}^{i_{u}}$.  The node $i$ sets\n$w_{\\ell}^{i}(t^{+}) = \\min(w_{\\ell}^{i}(t^{-}), w_{\\ell})$ for\n$\\ell = 1, \\dots, r$, where $t^{-}$ and $t^{+}$ denote the times\nimmediately before and after, respectively, the communication.  At any\ntime $t$, we will have $w^{i}(t) = \\bar{W}$ for all nodes $i \\in V$ if, in\nthe information spreading algorithm, every node $i$ has all the\nvectors $W^{1}, \\dots, W^{n}$ at the same time $t$.  In this way, we\nobtain an algorithm for computing the minimum vector $\\bar{W}$ that uses\nlinks of capacity $r$ and runs in the same amount of time as the\ninformation spreading algorithm.\n\nAn alternative to using links of capacity $r$ in the computation of\n$\\bar{W}$ is to make the time slot $r$ times larger, and impose a unit\ncapacity on all the links.  Now, a node transmits the numbers\n$w_{1}, \\dots, w_{r}$ to its communication partner over a period of\n$r$ time slots, and as a result the running time of the algorithm for\ncomputing $\\bar{W}$ becomes greater than the running time of the\ninformation spreading algorithm by a factor of $r$.  The preceding\ndiscussion, combined with the fact that nodes only gain messages as an\ninformation spreading algorithm executes, leads to the following\nlemma.\n\\begin{lemma}\n\\label{lem:mininfdis}\nSuppose that the COMP algorithm is implemented using an information\nspreading algorithm $\\cal{D}$ as described above.  Let $\\hat{W}^{i}(t)$\ndenote the estimate of $\\bar{W}$ at node $i$ at time $t$.  For any\n$\\delta \\in (0, 1)$, let $t_{m} = r T_{\\cal{D}}^{\\text{spr}}(\\delta)$.\nThen, for any time $t \\geq t_{m}$, with probability at least\n$1 - \\delta$, $\\hat{W}^{i}(t) = \\bar{W}$ for all nodes $i \\in V$.\n\\end{lemma}\n\nNote that the amount of data communicated between nodes during the\nalgorithm COMP depends on the values of the exponential random\nvariables generated by the nodes.  Since the nodes compute minima of\nthese variables, we are interested in a probabilistic lower bound on\nthe values of these variables (for example, suppose that the nodes\ntransmit the values $1\/W_{\\ell}^{i}$ when computing the minimum\n$\\bar{W}_{\\ell} = 1\/\\max_{i = 1}^{n} \\{1\/W_{\\ell}^{i}\\}$).\nTo this end, we use the fact that each $\\bar{W}_{\\ell}$ is an exponential\nrandom variable with rate $y$ to obtain that, for any constant\n$c > 1$, the probability that any of the minimum values $\\bar{W}_{\\ell}$\nis less than $1\/B$ (i.e., any of the inverse values $1\/W_{\\ell}^{i}$\nis greater than $B$) is at most $\\delta\/c$, where $B$ is proportional\nto $cry\/\\delta$.\n\n\\subsection{Proof of Theorem \\ref{thm:main1}}\n\nNow, we are ready to prove Theorem \\ref{thm:main1}.  In particular, we\nwill show that the COMP algorithm has the properties claimed in\nTheorem \\ref{thm:main1}. To this end, consider using an information spreading algorithm $\\cal{D}$ with\n$\\delta$-spreading time $T_{\\cal{D}}^{\\text{spr}}(\\delta)$ for\n$\\delta \\in (0, 1)$ as the subroutine in the COMP algorithm.  For any\n$\\delta \\in (0, 1)$, let $\\tau_{m} = rT_{\\cal{D}}^{\\text{spr}}(\\delta\/2)$.\nBy Lemma \\ref{lem:mininfdis}, for any time $t \\geq \\tau_{m}$, the\nprobability that $\\hat{W}^{i} \\neq \\bar{W}$ for any node $i$ at time $t$ is at\nmost $\\delta\/2$.\n\nOn the other hand, suppose that $\\hat{W}^{i} = \\bar{W}$ for all nodes $i$ at\ntime $t \\geq \\tau_{m}$.  For any $\\epsilon \\in (0, 1)$, by choosing\n$r \\geq 12 \\epsilon^{-2} \\log (4 \\delta^{-1})$ so that\n$r = \\Theta(\\epsilon^{-2}(1 + \\log \\delta^{-1}))$, we obtain from\nLemma \\ref{lem:estaccuracy} that\n\\begin{equation}\n\\begin{split}\n\\Pr & \\left(\\cup_{i = 1}^{n} \\left\\{\\hat{y}_{i}\n\\not\\in \\left[ (1 - \\epsilon) y, (1 + \\epsilon) y \\right] \\right\\}\n\\mid \\forall i \\in V, \\: \\hat{W}^{i} = \\bar{W} \\right) ~\n~ \\leq \\delta\/2.\n\\end{split}\n\\label{e3}\n\\end{equation}\nRecall that $T_{COMP}^{\\text{cmp}}(\\epsilon, \\delta)$ is the smallest time\n$\\tau$ such that, under the algorithm COMP, at any time $t \\geq \\tau$,\nall the nodes have an estimate of the function value $y$ within a\nmultiplicative factor of $(1 \\pm \\epsilon)$ with probability at least\n$1 - \\delta$.  By a straightforward union bound of events and\n(\\ref{e3}), we conclude that, for any time $t \\geq \\tau_{m}$,\n\\[\n\\Pr \\left(\\cup_{i = 1}^{n} \\left\\{\\hat{y}_{i} \\not\\in\n\\left[ (1 - \\epsilon) y, (1 + \\epsilon) y \\right] \\right\\} \\right)\n\\leq \\delta.\n\\]\nFor any $\\epsilon \\in (0, 1)$ and $\\delta \\in (0, 1)$, we now have, by\nthe definition of $(\\epsilon, \\delta)$-computing time,\n\\begin{eqnarray*}\nT_{COMP}^{\\text{cmp}}(\\epsilon, \\delta)\n& \\leq & \\tau_{m} \\\\\n& = & O \\left(\\epsilon^{-2} (1 + \\log \\delta^{-1})\nT_{\\cal{D}}^{\\text{spr}}(\\delta\/2) \\right).\n\\end{eqnarray*}\nThis completes the proof of Theorem \\ref{thm:main1}.\n\n\\section{Information spreading}\n\\label{sec:infdis}\n\nIn this section, we analyze a randomized gossip algorithm for\ninformation spreading.  The method by which nodes choose partners to\ncontact when initiating a communication and the data transmitted\nduring the communication are the same for both time models defined in\nSection \\ref{sec:prelim}.  These models differ in the times at which\nnodes contact each other: in the asynchronous model, only one node can\nstart a communication at any time, while in the synchronous model all\nthe nodes can communicate in each time slot.\n\n\nThe information spreading algorithm that we study is presented in\nFig. \\ref{spreadalg}, which makes use of the following notation.  Let\n$M_{i}(t)$ denote the set of messages node $i$ has at time $t$.\nInitially, $M_{i}(0) = \\{m_{i}\\}$ for all $i \\in V$.  For a\ncommunication that occurs at time $t$, let $t^{-}$ and $t^{+}$\ndenote the times immediately before and after, respectively, the\ncommunication occurs.\n\nAs mentioned in Section \\ref{ssec:contrib}, the nodes choose\ncommunication partners according to the probability distribution\ndefined by an $n \\times n$ matrix $P$.  The matrix $P$ is\nnon-negative and stochastic, and satisfies $P_{ij} = 0$ for any pair\nof nodes $i \\neq j$ such that $(i, j) \\not\\in E$.  For each such\nmatrix $P$, there is an instance of the information spreading\nalgorithm, which we refer to as SPREAD($P$).\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{\\textwidth}\n\\hrulefill\n\n\\noindent\n{\\bf Algorithm SPREAD($P$)}\n\n\\noindent\nWhen a node $i$ initiates a communication at time $t$:\n\n\\renewcommand{\\labelenumi}{{\\bf \\arabic{enumi}.}}\n\\begin{enumerate}\n\n\\item\nNode $i$ chooses a node $u$ at random, and contacts $u$.  The choice\nof the communication partner $u$ is made independently of all other\nrandom choices, and the probability that node $i$ chooses any node $j$\nis $P_{ij}$.\n\n\\item\nNodes $u$ and $i$ exchange all of their messages, so that\n\\[\nM_{i}(t^{+}) = M_u(t^+) =  M_{i}(t^{-}) \\cup M_{u}(t^{-}).\n\\]\n\\end{enumerate}\n\\hrulefill\n\\end{minipage}\n\\caption{A gossip algorithm for information spreading.}\n\\label{spreadalg}\n\\end{figure}\n\nWe note that the data transmitted between two communicating nodes in\nSPREAD conform to the {\\em push and pull mechanism}.  That is, when node $i$\ncontacts node $u$ at time $t$, both nodes $u$ and $i$ exchange all of\ntheir information with each other.  We also note that the\ndescription in the algorithm  assumes that the communication\nlinks in the network have infinite capacity.  As discussed in Section\n\\ref{ssec:minima}, however, an information spreading algorithm that\nuses links of infinite capacity can be used to compute minima using\nlinks of unit capacity.\n\nThis algorithm is simple, distributed, and satisfies the transmitter\ngossip constraint.  We now present analysis of the information\nspreading time of SPREAD($P$) for doubly stochastic matrices $P$ in\nthe two time models.  The goal of the analysis is to prove Theorem\n\\ref{thm:main2}.  To this end, for any $i \\in V$, let\n$S_{i}(t) \\subseteq V$ denote the set of nodes that have the message\n$m_{i}$ after any communication events that occur at absolute time $t$\n(communication events occur on a global clock tick in the asynchronous\ntime model, and in each time slot in the synchronous time model).  At\nthe start of the algorithm, $S_{i}(0) = \\{i\\}$.\n\n\\subsection{Asynchronous model}\n\nAs described in Section \\ref{sec:prelim}, in the asynchronous time\nmodel the global clock ticks according to a Poisson process of rate\n$n$, and on a tick one of the $n$ nodes is chosen uniformly at random.\nThis node initiates a communication, so the times at which the\ncommunication events occur correspond to the ticks of the clock.  On\nany clock tick, at most one pair of nodes can exchange messages by\ncommunicating with each other.\n\nLet $k \\geq 0$ denote the index of a clock tick.  Initially, $k = 0$,\nand the corresponding absolute time is $0$.  For simplicity of\nnotation, we identify the time at which a clock tick occurs with its\nindex, so that $S_{i}(k)$ denotes the set of nodes that have the\nmessage $m_{i}$ at the end of clock tick $k$.  The following lemma\nprovides a bound on the number of clock ticks required for every node\nto receive every message.\n\\begin{lemma}\n\\label{lem:asynchticks}\nFor any $\\delta \\in (0, 1)$, define \n\\[\nK(\\delta)\n= \\inf\\{k \\geq 0:\n\\Pr(\\cup_{i = 1}^{n} \\{S_{i}(k) \\neq V\\}) \\leq \\delta \\}.\n\\]\nThen,\n\\[\nK(\\delta)\n= O\\left( n \\frac{\\log n + \\log \\delta^{-1}}{\\Phi(P)}\\right).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nFix any node $v \\in V$.  We study the evolution of the size of the set\n$S_{v}(k)$.  For simplicity of notation, we drop the subscript $v$,\nand write $S(k)$ to denote $S_{v}(k)$.\n\nNote that $|S(k)|$ is monotonically non-decreasing over the course of\nthe algorithm, with the initial condition $|S(0)| = 1$.  For the\npurpose of analysis, we divide the execution of the algorithm into two\nphases based on the size of the set $S(k)$.  In the first phase,\n$|S(k)| \\leq n\/2$, and in the second phase $|S(k)| > n\/2$.\n\nUnder the gossip algorithm, after clock tick $k + 1$, we have either\n$|S(k + 1)| = |S(k)|$ or $|S(k + 1)| = |S(k)| + 1$.  Further, the size\nincreases if a node $i \\in S(k)$ contacts a node $j \\notin S(k)$, as\nin this case $i$ will push the message $m_{v}$ to $j$.  For each such\npair of nodes $i$, $j$, the probability that this occurs on clock tick\n$k + 1$ is $P_{ij}\/n$.  Since only one node is active on each clock\ntick,\n\\begin{equation}\nE[|S(k + 1)| - |S(k)| \\mid S(k)]\n\\geq \\sum_{i \\in S(k), j \\notin S(k)} \\frac{P_{ij}}{n}.\n\\label{expinc}\n\\end{equation}\n\n\\noindent\nWhen $|S(k)| \\leq n\/2$, it follows from (\\ref{expinc}) and the\ndefinition of the conductance $\\Phi(P)$ of $P$ that\n\\begin{eqnarray}\nE[|S(k + 1)| - |S(k)| \\mid S(k)]\n& \\geq & \\frac{|S(k)|}{n}\n\\frac{\\sum_{i \\in S(k), j \\notin S(k)} P_{ij} }{|S(k)|}\n\\notag \\\\\n& \\geq &\n\\frac{|S(k)|}{n} \\min_{S \\subset V, \\; 0 < |S| \\leq n\/2}\n\\frac{\\sum_{ i \\in S, j \\notin S} P_{ij}}{|S|}\n\\notag \\\\\n& = & \\frac{|S(k)|}{n} \\Phi(P)\n\\notag \\\\\n& = & |S(k)| \\hat{\\Phi},\n\\label{e4}\n\\end{eqnarray}\n\n\\noindent\nwhere $\\hat{\\Phi} = \\frac{\\Phi(P)}{n}$.\n\nWe seek an upper bound on the duration of the first phase.  To this\nend, let\n\\begin{equation*}\nZ(k) = \\frac{\\exp \\left(\\frac{\\hat{\\Phi}}{4} k \\right)}{|S(k)|}.\n\\end{equation*}\n\n\\noindent\nDefine the stopping time $L = \\inf \\{k : |S(k)| > n\/2\\}$, and\n$L \\land k = \\min(L, k)$.  If $|S(k)| > n\/2$, then\n$L \\land (k + 1) = L \\land k$, and thus\n$E[Z(L \\land (k + 1)) \\mid S(L \\land k)] = Z(L \\land k)$.\n\nNow, suppose that $|S(k)| \\leq n\/2$, in which case\n$L \\land (k + 1) = (L \\land k) + 1$.  The function $g(z) = 1\/z$ is\nconvex for $z > 0$, which implies that, for $z_{1}, z_{2} > 0$,\n\\begin{equation}\ng(z_{2}) \\geq g(z_{1}) + g'(z_{1})(z_{2} - z_{1}).\n\\label{convderunder}\n\\end{equation}\n\n\\noindent\nApplying \\eqref{convderunder} with $z_{1} = |S(k + 1)|$ and\n$z_{2} = |S(k)|$ yields\n\\begin{equation*}\n\\frac{1}{|S(k + 1)|}\n\\leq \\frac{1}{|S(k)|}\n- \\frac{1}{|S(k + 1)|^{2}} (|S(k + 1)| - |S(k)|).\n\\end{equation*}\n\n\\noindent\nSince $|S(k + 1)| \\leq |S(k)| + 1 \\leq 2|S(k)|$, it follows that\n\\begin{equation}\n\\frac{1}{|S(k + 1)|}\n\\leq \\frac{1}{|S(k)|}\n- \\frac{1}{4 |S(k)|^{2}} (|S(k + 1)| - |S(k)|).\n\\label{invsizeupper}\n\\end{equation}\n\nCombining \\eqref{e4} and \\eqref{invsizeupper}, we obtain that, if\n$|S(k)| \\leq n\/2$, then\n\\begin{equation*}\nE \\left[\\frac{1}{|S(k + 1)|} \\;\\Big|\\; S(k) \\right]\n\\leq \\frac{1}{|S(k)|} \\left(1 - \\frac{\\hat{\\Phi}}{4} \\right)\n\\leq \\frac{1}{|S(k)|} \\exp \\left(-\\frac{\\hat{\\Phi}}{4} \\right),\n\\end{equation*}\n\n\\noindent\nas $1 - z \\leq \\exp(-z)$ for $z \\geq 0$.  This implies that\n\\begin{eqnarray*}\nE[Z(L \\land (k + 1)) \\mid S(L \\land k)]\n& = & E \\left[\n\\frac{\\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land (k + 1)) \\right)}\n{|S(L \\land (k + 1))|} \\;\\bigg|\\; S(L \\land k) \\right]\n\\notag \\\\\n& = & \\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land k) \\right)\n\\exp \\left(\\frac{\\hat{\\Phi}}{4} \\right)\nE \\left[ \\frac{1}\n{|S((L \\land k) + 1)|} \\;\\Big|\\; S(L \\land k) \\right]\n\\notag \\\\\n& \\leq & \\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land k) \\right)\n\\exp \\left(\\frac{\\hat{\\Phi}}{4} \\right)\n\\exp \\left(-\\frac{\\hat{\\Phi}}{4} \\right)\n\\frac{1}{|S(L \\land k)|}\n\\notag \\\\\n& = & Z(L \\land k),\n\\end{eqnarray*}\n\n\\noindent\nand therefore $Z(L \\land k)$ is a supermartingale.\n\nSince $Z(L \\land k)$ is a supermartingale, we have the inequality\n$E[Z(L \\land k)] \\leq E[Z(L \\land 0)] = 1$ for any $k > 0$, as\n$Z(L \\land 0) = Z(0) = 1$.  The fact that the set $S(k)$ can contain\nat most the $n$ nodes in the graph implies that\n\\begin{equation*}\nZ(L \\land k)\n= \\frac{\\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land k) \\right)}\n{|S(L \\land k)|}\n\\geq \\frac{1}{n} \\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land k) \\right),\n\\end{equation*}\n\n\\noindent\nand so\n\\begin{equation*}\nE \\left[\\exp \\left(\\frac{\\hat{\\Phi}}{4} (L \\land k) \\right) \\right]\n\\leq n E[Z(L \\land k)] \\leq n.\n\\end{equation*}\n\n\\noindent\nBecause $\\exp(\\hat{\\Phi}(L \\land k)\/4) \\uparrow \\exp (\\hat{\\Phi}L\/4)$\nas $k \\to \\infty$, the monotone convergence theorem implies that\n\\begin{equation*}\nE \\left[\\exp \\left(\\frac{\\hat{\\Phi}L}{4} \\right) \\right] \\leq n.\n\\end{equation*}\n\n\\noindent\nApplying Markov's inequality, we obtain that, for\n$k_{1} = 4(\\ln 2 + 2 \\ln n + \\ln (1\/\\delta))\/\\hat{\\Phi}$,\n\\begin{eqnarray}\n\\Pr (L > k_{1})\n& = & \\Pr \\left(\\exp \\left(\\frac{\\hat{\\Phi} L}{4} \\right)\n> \\frac{2n^{2}}{\\delta} \\right)\n\\notag \\\\\n& < & \\frac{\\delta}{2n}.\n\\label{e6a}\n\\end{eqnarray}\n\nFor the second phase of the algorithm, when $|S(k)| > n\/2$, we study\nthe evolution of the size of the set of nodes that do not have the\nmessage, $|S(k)^{c}|$.  This quantity will decrease as the message\nspreads from nodes in $S(k)$ to nodes in $S(k)^{c}$.  For simplicity,\nlet us consider restarting the process from clock tick $0$ after $L$\n(i.e., when more than half the nodes in the graph have the message),\nso that we have $|S(0)^{c}| \\leq n\/2$.\n\nIn clock tick $k + 1$, a node $j \\in S(k)^{c}$ will receive the\nmessage if it contacts a node $i \\in S(k)$ and pulls the message from\n$i$.  As such,\n\\begin{equation*}\nE[|S(k)^{c}| - |S(k + 1)^{c}| \\mid S(k)^{c}]\n\\geq \\sum_{j \\in S(k)^{c}, i \\notin S(k)^{c}} \\frac{P_{ji}}{n},\n\\end{equation*}\n\n\\noindent\nand thus\n\\begin{eqnarray}\n\\notag\nE[|S(k + 1)^{c}| \\mid S(k)^{c}]\n& \\leq & |S(k)^c|\n- \\frac{\\sum_{j \\in S(k)^{c}, i \\notin S(k)^c} P_{ji}}{n}\n\\notag \\\\\n& = & |S(k)^c|\n\\left(1\n- \\frac{\\sum_{j \\in S(k)^c, i \\notin S(k)^c} P_{ji}}{n |S(k)^c|} \\right)\n\\notag \\\\\n& \\leq & |S(k)^c| \\left( 1 - \\hat{\\Phi} \\right).\n\\label{e:5}\n\\end{eqnarray}\n\nWe note that this inequality holds even when $|S(k)^{c}| = 0$, and as\na result it is valid for all clock ticks $k$ in the second phase.\nRepeated application of \\eqref{e:5} yields\n\\begin{eqnarray*}\nE[|S(k)^{c}|]\n& = & E[E[|S(k)^{c}| \\mid S(k - 1)^{c}]] \\\\\n& \\leq & \\left(1 - \\hat{\\Phi} \\right)E[|S(k - 1)^{c}|] \\\\\n& \\leq & \\left(1 - \\hat{\\Phi} \\right)^{k} E[|S(0)^{c}|] \\\\\n& \\leq & \\exp \\left(-\\hat{\\Phi} k \\right) \\left(\\frac{n}{2} \\right)\n\\end{eqnarray*}\n\nFor\n$k_{2} = \\ln (n^{2}\/\\delta)\/2\\hat{\\Phi} =\n(2\\ln n + \\ln (1\/\\delta))\/\\hat{\\Phi}$,\nwe have $E[|S(k_{2})^{c}|] \\leq \\delta\/(2n)$.  Markov's inequality now\nimplies the following upper bound on the probability that not all of\nthe nodes have the message at the end of clock tick $k_{2}$ in the\nsecond phase.\n\\begin{eqnarray}\n\\Pr(|S(k_{2})^{c}| > 0) & = & \\Pr(|S(k_{2})^{c}| \\geq 1)\n\\notag \\\\\n& \\leq & E[|S(k_{2})^{c}|]\n\\notag \\\\\n& \\leq & \\frac{\\delta}{2n}.\n\\label{e8}\n\\end{eqnarray}\n\nCombining the analysis of the two phases, we obtain that, for\n$k' = k_{1} + k_{2} = O((\\log n + \\log \\delta^{-1})\/\\hat{\\Phi})$,\n$\\Pr(S_{v}(k') \\neq V) \\leq \\delta\/n$.  Applying the union bound over\nall the nodes in the graph, and recalling that\n$\\hat{\\Phi} = \\Phi(P)\/n$, we conclude that\n\\begin{eqnarray*}\nK(\\delta)\n& \\leq & k' \n~ = ~ O\\left(n \\frac{\\log n + \\log \\delta^{-1}}{\\Phi(P)}\\right).\n\\end{eqnarray*}\nThis completes the proof of Lemma \\ref{lem:asynchticks}.\n\\end{proof}\n\nTo extend the bound in Lemma \\ref{lem:asynchticks} to absolute time,\nobserve that Corollary \\ref{discrete-to-contc} implies that the\nprobability that\n$\\kappa = K(\\delta\/3) + 27 \\ln (3\/\\delta) =\nO(n(\\log n + \\log \\delta^{-1})\/\\Phi(P))$\nclock ticks do not occur in absolute time\n$(4\/3) \\kappa\/n = O((\\log n + \\log \\delta^{-1})\/\\Phi(P))$ is at most\n$2 \\delta\/3$.  Applying the union bound now yields\n$T_{SPREAD(P)}^{\\text{spr}}(\\delta) =\nO((\\log n + \\log \\delta^{-1})\/\\Phi(P))$,\nthus establishing the  upper bound in Theorem \\ref{thm:main2} for the\nasynchronous time model.\n\n\\subsection{Synchronous model}\n\nIn the synchronous time model, in each time slot every node contacts a\nneighbor to exchange messages.  Thus, $n$ communication events may\noccur simultaneously.  Recall that absolute time is measured in rounds\nor time slots in the synchronous model.\n\nThe analysis of the randomized gossip algorithm for information\nspreading in the synchronous model is similar to the analysis for the\nasynchronous model.  However, we need additional analytical arguments\nto reach analogous conclusions due to the technical challenges\npresented by multiple simultaneous transmissions.\n\nIn this section, we sketch a proof of the time bound in Theorem\n\\ref{thm:main2},\n$T_{SPREAD(P)}^{\\text{spr}}(\\delta) =\nO((\\log n + \\log \\delta^{-1})\/\\Phi(P))$,\nfor the synchronous time model.  Since the proof follows a similar\nstructure as the proof of Lemma \\ref{lem:asynchticks}, we only point\nout the significant differences.\n\nAs before, we fix a node $v \\in V$, and study the evolution of the\nsize of the set $S(t) = S_{v}(t)$.  Again, we divide the execution of\nthe algorithm into two phases based on the evolution of $S(t)$: in the\nfirst phase $|S(t)| \\leq n\/2$, and in the second phase\n$|S(t)| > n\/2$. In the first phase, we analyze the increase in\n$|S(t)|$, while in the second we study the decrease in $|S(t)^{c}|$.\nFor the purpose of analysis, in the first phase we ignore the effect\nof the increase in $|S(t)|$ due to the {\\em pull} aspect of protocol:\nthat is, when node $i$ contacts node $j$, we assume (for the purpose\nof analysis) that $i$ sends the messages it has to $j$, but that $j$\ndoes not send any messages to $i$.  Clearly, an upper bound obtained\non the time required for every node to receive every message under\nthis restriction is also an upper bound for the actual algorithm.\n\nConsider a time slot $t + 1$ in the first phase.  For $j \\notin S(t)$,\nlet $X_{j}$ be an indicator random variable that is $1$ if node $j$\nreceives the message $m_{v}$ via a push from some node $i \\in S(t)$ in\ntime slot $t + 1$, and is $0$ otherwise.  The probability that $j$\ndoes not receive $m_{v}$ via a push is the probability that no node\n$i \\in S(t)$ contacts $j$, and so\n\\begin{eqnarray}\nE[X_{j} \\mid S(t)]\n& = & 1 - \\Pr(X_{j} = 0 \\mid S(t))\n\\notag \\\\\n& = & 1 - \\prod_{i \\in S(t)} (1 - P_{ij})\n\\notag \\\\\n& \\geq & 1 - \\prod_{i \\in S(t)} \\exp(-P_{ij})\n\\notag \\\\\n& = & 1 - \\exp \\left(-\\sum_{i \\in S(t)} P_{ij} \\right).\n\\label{pullproblower}\n\\end{eqnarray}\n\n\\noindent\nThe Taylor series expansion of $\\exp(-z)$ about $z = 0$ implies that,\nif $0 \\leq z \\leq 1$, then\n\\begin{equation}\n\\exp(-z) \\leq 1 - z + z^{2}\/2 \\leq 1 - z + z\/2 = 1 - z\/2.\n\\label{taylorexpsecondterm}\n\\end{equation}\n\n\\noindent\nFor a doubly stochastic matrix $P$, we have\n$0 \\leq \\sum_{i \\in S(t)} P_{ij} \\leq 1$, and so we can combine\n\\eqref{pullproblower} and \\eqref{taylorexpsecondterm} to obtain\n\\begin{equation*}\nE[X_{j} \\mid S(t)]\n\\geq \\frac{1}{2} \\sum_{i \\in S(t)} P_{ij}.\n\\end{equation*}\n\nBy linearity of expectation,\n\\begin{eqnarray*}\nE[|S(t + 1)| - |S(t)| \\mid S(t)]\n& = & \\sum_{j \\not\\in S(t)} E[X_{j} \\mid S(t)]\n\\notag \\\\\n& \\geq & \\frac{1}{2} \\sum_{i \\in S(t), j \\not\\in S(t)} P_{ij}\n\\notag \\\\\n& = & \\frac{|S(t)|}{2}\n\\frac{\\sum_{i \\in S(t), j \\not\\in S(t)} P_{ij}}{|S(t)|}.\n\\end{eqnarray*}\n\n\\noindent\nWhen $|S(t)| \\leq n\/2$, we have\n\\begin{equation}\nE[|S(t + 1)| - |S(t)| \\mid S(t)]\n\\geq |S(t)| \\frac{\\Phi(P)}{2}.\n\\label{synchexpinccond}\n\\end{equation}\n\nInequality \\eqref{synchexpinccond} is analogous to inequality\n\\eqref{e4} for the asynchronous time model, with $\\Phi(P)\/2$ in the\nplace of $\\hat{\\Phi}$.  We now proceed as in the proof of Lemma\n\\ref{lem:asynchticks} for the asynchronous model.  Note that\n$|S(t + 1)| \\leq 2 |S(t)|$ here in the synchronous model because of\nthe restriction in the analysis to only consider the push aspect of\nthe protocol in the first phase, as each node in $S(t)$ can push a\nmessage to at most one other node in a single time slot.  Repeating\nthe analysis from the asynchronous model leads to the conclusion that\nthe first phase of the algorithm ends in\n$O\\left(\\frac{\\log n + \\log \\delta^{-1}}{{\\Phi(P)}}\\right)$ time with\nprobability at least $1 - \\delta\/2n$.\n\nThe analysis of the second phase is the same as that presented for the\nasynchronous time model, with $\\hat{\\Phi}$ replaced by $\\Phi$.  As a\nsummary, we obtain that it takes at most\n$O\\left(\\frac{\\log n + \\log \\delta^{-1}}{{\\Phi(P)}}\\right)$ time for the\nalgorithm to spread all the messages to all the nodes with probability\nat least $1 - \\delta$.  This completes the proof of Theorem\n\\ref{thm:main2} for the synchronous time model.\n\n\n\\section{Applications}\n\\label{sec:appl}\n\nWe study here the application of our preceding results to several\ntypes of graphs.  In particular, we consider complete graphs,\nconstant-degree expander graphs, and grid graphs.  We use grid graphs\nas an example to compare the performance of our algorithm for\ncomputing separable functions with that of a known iterative averaging\nalgorithm.\n\nFor each of the three classes of graphs mentioned above, we are\ninterested in the $\\delta$-information-spreading time\n$T_{SPREAD(P)}^{\\text{spr}}(\\delta)$, where $P$ is a doubly stochastic\nmatrix that assigns equal probability to each of the neighbors of any\nnode.  Specifically, the probability $P_{ij}$ that a node $i$ contacts\na node $j \\neq i$ when $i$ becomes active is $1\/\\Delta$, where\n$\\Delta$ is the maximum degree of the graph, and\n$P_{ii} = 1 - d_{i}\/\\Delta$, where $d_{i}$ is the degree of $i$.\nRecall from Theorem \\ref{thm:main1} that the information dissemination\nalgorithm SPREAD($P$) can be used as a subroutine in an algorithm for\ncomputing separable functions, with the running time of the resulting\nalgorithm being a function of $T_{SPREAD(P)}^{\\text{spr}}(\\delta)$.\n\n\\subsection{Complete graph}\n\nOn a complete graph, the transition matrix $P$ has $P_{ii} = 0$ for\n$i = 1, \\dots, n$, and $P_{ij} = 1\/(n - 1)$ for $j \\neq i$.  This\nregular structure allows us to directly evaluate the conductance of\n$P$, which is $\\Phi(P) \\approx 1\/2$.  This implies that the\n($\\epsilon$, $\\delta$)-computing time of the algorithm for computing\nseparable functions based on SPREAD($P$) is\n$O(\\epsilon^{-2} (1 + \\log \\delta^{-1})(\\log n + \\log \\delta^{-1}))$.\nThus, for a constant $\\epsilon \\in (0, 1)$ and $\\delta = 1\/n$, the\ncomputation time scales as $O(\\log^{2} n)$.\n\n\\subsection{Expander graph}\n\nExpander graphs have been used for numerous applications, and explicit\nconstructions are known for constant-degree expanders \\cite{rvw}.  We\nconsider here an undirected graph in which the maximum degree of any\nvertex, $\\Delta$, is a constant.  Suppose that the edge expansion of\nthe graph is\n\\begin{equation*}\n\\min_{S \\subset V, \\; 0 < |S| \\leq n\/2}\n\\frac{|F(S, S^{c})|}{|S|} = \\alpha,\n\\end{equation*}\n\n\\noindent\nwhere $F(S, S^{c})$ is the set of edges in the cut $(S, S^{c})$, and\n$\\alpha > 0$ is a constant.  The transition matrix $P$ satisfies\n$P_{ij} = 1\/\\Delta$ for all $i \\neq j$ such that $(i, j) \\in E$, from\nwhich we obtain $\\Phi(P) \\geq \\alpha\/\\Delta$.  When $\\alpha$ and\n$\\Delta$ are constants, this leads to a similar conclusion as in the\ncase of the complete graph: for any constant $\\epsilon \\in (0, 1)$ and\n$\\delta = 1\/n$, the computation time is $O(\\log^{2} n)$.\n\n\\subsection{Grid}\n\\label{gridsec}\n\nWe now consider a $d$-dimensional grid graph on $n$ nodes, where\n$c = n^{1\/d}$ is an integer.  Each node in the grid can be represented\nas a $d$-dimensional vector $a = (a_{i})$, where\n$a_{i} \\in \\{1, \\dots, c\\}$ for $1 \\leq i \\leq d$.  There is one node\nfor each distinct vector of this type, and so the total number of\nnodes in the graph is $c^{d} = (n^{1\/d})^{d} = n$.  For any two nodes\n$a$ and $b$, there is an edge $(a, b)$ in the graph if and only if,\nfor some $i \\in \\{1, \\dots, d\\}$, $|a_{i} - b_{i}| = 1$, and\n$a_{j} = b_{j}$ for all $j \\neq i$.\n\nIn \\cite{isogrid}, it is shown that the isoperimetric number of this\ngrid graph is\n\\begin{equation*}\n\\min_{S \\subset V, \\; 0 < |S| \\leq n\/2}\n\\frac{|F(S, S^{c})|}{|S|}\n= \\Theta \\left(\\frac{1}{c} \\right)\n= \\Theta \\left(\\frac{1}{n^{1\/d}} \\right).\n\\end{equation*}\n\n\\noindent\nBy the definition of the edge set, the maximum degree of a node in the\ngraph is $2d$.  This means that $P_{ij} = 1\/(2d)$ for all $i \\neq j$\nsuch that $(i, j) \\in E$, and it follows that\n$\\Phi(P) = \\Omega \\left(\\frac{1}{dn^{1\/d}} \\right)$.  Hence, for any\n$\\epsilon \\in (0, 1)$ and $\\delta \\in (0, 1)$, the\n($\\epsilon$, $\\delta$)-computing time of the algorithm for computing\nseparable functions is\n$O(\\epsilon^{-2} (1 + \\log \\delta^{-1})(\\log n + \\log \\delta^{-1})\nd n^{1\/d})$.\n\n\\subsection{Comparison with Iterative Averaging}\n\nWe briefly contrast the performance of our algorithm for computing\nseparable functions with that of the iterative averaging algorithms in\n\\cite{tsitsiklis-thesis} \\cite{bgps}.  As noted earlier, the\ndependence of the performance of our algorithm is in proportion to\n$1\/\\Phi(P)$, which is a lower bound for the iterative algorithms based\non a stochastic matrix $P$.\n\nIn particular, when our algorithm is used to compute the average of a\nset of numbers (by estimating the sum of the numbers and the number of\nnodes in the graph) on a $d$-dimensional grid graph, it follows from\nthe analysis in Section \\ref{gridsec} that the amount of time required\nto ensure the estimate is within a $(1 \\pm \\epsilon)$ factor of the\naverage with probability at least $1 - \\delta$ is\n$O(\\epsilon^{-2} (1 + \\log \\delta^{-1})(\\log n + \\log \\delta^{-1})\ndn^{1\/d})$\nfor any $\\epsilon \\in (0, 1)$ and $\\delta \\in (0, 1)$.  So, for a\nconstant $\\epsilon \\in (0, 1)$ and $\\delta = 1\/n$, the computation\ntime scales as $O(dn^{1\/d} \\log^{2} n)$ with the size of the graph,\n$n$.  The algorithm in \\cite{bgps} requires $\\Omega(n^{2\/d} \\log n)$\ntime for this computation.  Hence, the running time of our algorithm\nis (for fixed $d$, and up to logarithmic factors) the {\\em square\nroot} of the runnning time of the iterative algorithm!  This\nrelationship holds on other graphs for which the spectral gap is\nproportional to the square of the conductance.\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:conc}\n\nIn this paper, we presented a novel algorithm for computing separable\nfunctions in a totally distributed manner.  The algorithm is based on\nproperties of exponential random variables, and the fact that the\nminimum of a collection of numbers is an order- and\nduplicate-insensitive statistic.\n\nOperationally, our algorithm makes use of an information spreading\nmechanism as a subroutine.  This led us to the analysis of a\nrandomized gossip mechanism for information spreading.  We obtained an\nupper bound on the information spreading time of this algorithm in\nterms of the conductance of a matrix that characterizes the algorithm.\n\nIn addition to computing separable functions, our algorithm improves\nthe computation time for the canonical task of averaging.  For\nexample, on graphs such as paths, rings, and grids, the performance of\nour algorithm is of a smaller order than that of a known iterative\nalgorithm.\n\nWe believe that our algorithm will lead to the following totally\ndistributed computations: (1) an approximation algorithm for convex\nminimization with linear constraints; and (2) a ``packet marking''\nmechanism in the Internet.  These areas, in which summation is a key\nsubroutine, will be topics of our future research.\n\n\\section{Acknowledgments}\n\nWe thank Ashish Goel for a useful discussion and providing\nsuggestions, based on previous work \\cite{Goel}, when we started this\nwork.\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nABCG~85{} is a very rich cluster located at a redshift z=0.0555. We\nperformed a detailed analysis of this cluster from the X-ray point of\nview, based on Einstein IPC data (Gerbal et al. 1992 and references\ntherein).  In the optical, no photometric data were available at that\ntime, except for an incomplete photometric catalogue by Murphy (1984),\nand about 150 redshifts were published in the literature only after we\ncompleted our first X-ray analysis (Beers et al. 1991, Malumuth et\nal. 1992). We therefore undertook a more complete analysis of this cluster,\nwith the aim of obtaining both photometric and redshift data at\noptical wavelengths and better X-ray data from the ROSAT data bank\n(Pislar et al. 1997, Lima--Neto et al. 1997). We present here our photometric\ndata. The redshift catalogue is published in a companion paper (Durret\net al. 1997a) and the analysis of all these combined optical data will be\npresented in Paper III (Durret et al. in preparation).\n\n\\section{The photographic plate data}\n\n\\subsection{Method for obtaining the catalogue}\n\nWe decided to obtain a photometric catalogue of the galaxies in the\ndirection of the Abell 85 cluster of galaxies by first processing the field\n681 in the SRC-J Schmidt atlas. This blue glass copy plate (IIIaJ$+$GG385)\nwas investigated with the MAMA (Machine \\`a Mesurer pour l'Astronomie)\nfacility located at the Centre d'Analyse des Images at the Observatoire de\nParis and operated by CNRS\/INSU (Institut National des Sciences de l'Univers).\nIn order to also get information on the neighbouring galaxy distribution, the\ncentral 5$^{\\circ}\\times$~5$^{\\circ}$ area has been searched for objects\nusing the on-line mode with the 10~$\\mu$m step size available at that time.\nThe involved algorithmic steps are well-known. They can be summarized as\nfollows~:\nfirst a local background estimate and its variance are computed from\npixel values inside a 256~$\\times$~256 window, then pixels with a\nnumber of counts higher than the background value plus three times the\nvariance are flagged, which leads to define an object as a set of\nconnected flagged pixels; an overlapping zone of 512 pixels is used in\nboth directions for each individual scan. Although this method may appear\nrather crude, its efficiency is nevertheless quite high for properly\ndetecting and measuring simple and isolated objects smaller than the\nbackground scale. The region where ABCG~85 is located is not crowded by\nstellar images ($b_{\\rm II}\\simeq-72^{\\circ}$), so that most of the objects\nlarger than a few pixels can indeed be detected this way. \nThe result was a list of more than 10$^5$ objects distributed over\nthe $\\sim$~25 square degrees of the field bounded by 0$^{\\rm\nh}$31$^{\\rm mn}$30.4$^{\\rm s}$ $<\\alpha<$ 0$^{\\rm h}$53$^{\\rm\nmn}$10.6$^{\\rm s}$ and $-$12$^{\\circ}$18'19.43\" $<\\delta<$\n$-$7$^{\\circ}$05'13.88\" (equinox 2000.0, as hereafter), with their\ncoordinates, their shape parameters (area, elliptical modelling) and\ntwo flux descriptors (peak density, sum of background-subtracted\npixel values).\n\nThe astrometric reduction of the whole catalogue was performed with\nrespect to 91 stars of the PPM star catalogue (Roeser \\& Bastian 1991)\nspread over the field, using a 3$^{\\rm rd}$-order polynomial fitting. The\nresiduals of the fit yielding the instrumental constants were smaller\nthan 0.25 arcsecond and the astrometry of our catalogue indeed appears \nto be very good, as confirmed by our multi-object fibre spectroscopy where\nthe galaxies were always found to be very close ($<$~2.0 arcseconds, i.e.\n3 pixels) to the expected positions.\n\nSince the required CCD observations were not available at that time, a\npreliminary photometric calibration of these photographic data has\nbeen done using galaxies with known total blue magnitude. The\nmagnitude of stars is certainly much easier to define, but such\nhigh-surface brightness objects suffer from severe saturation effects\non Schmidt plates when they are bright enough to be included in\navailable photometric catalogues. So, 83 galaxies were selected from\nthe Lyon Extragalactic Database (LEDA) in order to compare their\nmagnitude to their measured blue flux. A small region around each of\nthese objects was scanned and this image has been used: i) to identify\nthe object among its neighbours within the coordinate list and ii) to\nassess the quality of the flux value stored in the on-line catalogue\nwith respect to close, overlapping or merged objects. The 74 remaining\nundisturbed objects identified with no ambiguity came from eight\ndifferent catalogues in the literature. Whatever the intrinsic\nuncertainties about the integrated MAMA fluxes are, systematic\neffects were found with respect to the parent catalogue in a flux\nversus magnitude plot, as well as discrepancies for some objects\nbetween the LEDA and the Centre de Donn\\'ees Astronomiques de Strasbourg (CDS)\ndatabases.  Consequently, three catalogues including 12 objects were\nremoved and the LEDA magnitude of 5 objects was replaced by a CDS\nvalue which seems in better agreement with their aspect and with the overall\ntrend when compared to similar objects.  Later, 7 objects far from the overall\ntrend were discarded. These successive rejections resulted in a set of\n55 objects distributed over a six magnitude range. The magnitude\nzero-point for our photographic catalogue was obtained by plotting the\nflux of these objects against their expected magnitude. A rms scatter\nof 0.34 mag was computed around the linear fit.\n\n\\subsection{Classification of the objects}\n\nMost of the diffuse objects included in our main catalogue were\nautomatically selected according to their lower surface brightness\nwhen compared to stars.  As usual for glass copies of survey plates,\nthe discrimination power of this brightness criterion drops sharply\nfor objects fainter than approximately 19$^{\\rm th}$ magnitude, and so does\nthe completeness of the resulting catalogue if no\ncontamination is allowed for. The number of galaxy candidates brighter\nthan this limit within the investigated area appeared, however, to be\nalready large enough to get a much better view of the bright galaxy\ndistribution than using the deeper but very incomplete catalogue\npublished by Murphy (1984). Moreover, including faintest objects was\nnot necessary for the redshift survey of the Abell~85 cluster of\ngalaxies we were planning (see Durret et al. 1997). Hence, no attempt was\ndone to reach a fainter completeness limit. Nonetheless, in order to\nselect galaxies, the decision curve which has been computed in the Flux\nvs. Area parameter space was fitted to the data so that\nsome objects identified by Murphy from CCD frames as faint galaxies\nwere also classified as galaxies by us. Next, a further test based on\nthe elongation was performed in order to reject linear plate flaws or\nartefacts, as well as to pick bright elongated galaxies first\nclassified as stars due to strong saturation effects. Finally,\nspurious detections occuring around very bright stars (area greater\nthan 10$^3$ pixels) due to a wrong estimate of the local background were\ntentatively removed by checking their location with respect to these\nbright objects.\nIn this way, a list of more than 25,000 galaxy candidates over the 25\nsquare degrees of our SRC-J~681 blue field was obtained.\n\n\\begin{figure*}[ht!]\n\\centerline{\\psfig{figure=DS1413F1.ps,height=14cm}}\n\\caption[ ]{Spatial distribution of the 11,862 galaxies brighter than \nB$_{\\rm J}=$~19.75 in the SRC-J~681 field. The large overdensities are \nindicated by \nsuperimposed isopleths from a density map computed by the method introduced \nby Dressler (1980) with $N=$~50~; eleven isopleths are drawn from 850 to \n2,850 galaxies\/square degree.}\n\\protect\\label{allplate}\n\\end{figure*}\n\nThe distribution of these galaxies is displayed in Fig.~\\ref{allplate}\nfor objects brighter than B$_{\\rm J}=$~19.75. The Abell~85 cluster is\nclearly visible, as well as several other density enhancements which\nare mostly located along the direction defined by the cluster\nellipticity.\n\n\\subsection{Completeness and accuracy of the classification}\n\n\\begin{figure}[h!]\n\\centerline{\\psfig{figure=DS1413F2.ps,height=7cm,angle=-90}}\n\\caption[ ]{Differential magnitude distribution of the 25~10$^3$ galaxy \ncandidates in the SRC-J 681 field.}\n\\protect\\label{fdl}\n\\end{figure}\n\nThe differential luminosity distribution of the galaxy candidates\nindicates that the sample appears quite complete down to the\nB$_{\\rm J}=$~19.75 magnitude (see Fig.~\\ref{fdl}).  To go further, we first\ntested the completeness of this overall list by cross-identifying it\nwith three catalogues from the literature (Murphy 1984, Beers et\nal. 1991, Malumuth et al. 1992) with the help of images obtained from\nthe mapping mode of the MAMA machine. It appeared that: i) all but one\ngalaxy of the Malumuth et al. (1992) catalogue of 165 objects are actually\nclassified as galaxies, with a mean offset between individual\npositions equal to 1.10~$\\pm$~0.06 arcsecond ; ii) 94\\% of the 35\ngalaxies listed by Beers et al. (1991) inside the area are included in our\ncatalogue, only 2 bright objects which suffer from severe saturation\nbeing misclassified. Note that such an effect also caused 5 of the 83\ngalaxies chosen as photometric standards to be misclassified, which\ngives the same percentage as for the sample by Beers et al. The\ncomparison with the faint CCD catalogue built by Murphy (1984) in the so-called\n$r_{\\rm F}$ band (quite similar to that obtained using a photographic IIIaF\nemulsion with a R filter) was performed only for objects which were visible\non the photographic plate with secure identification (only uncertain X\nand Y coordinates are provided in the paper) and classified without any\ndoubt as galaxies from our visual examination. There remained 107\nobjects out of 170, among which 88 are brighter than B$_{\\rm J}\\sim$~19.75\n($r_{\\rm F}\\sim$~18.5).  Down to this flux limit, 82 objects ($\\sim$~93\\%)\nare in agreement, thereby validating the choice of our decision curve in the\nFlux vs. Area parameter space. These cross-identifications therefore indicate\nthat the completeness limit of our catalogue is about 95\\% for such objects,\nas expected from similar studies at high galactic latitude.\n\nIn order to confirm this statement and to study the homogeneity of our\ngalaxy catalogue, we then decided to verify carefully its reliability\ninside the region of the Abell~85 cluster of galaxies itself. The\ncentre of ABCG~85 was assumed to be located at the equatorial\ncoordinates given in the literature, $\\alpha=$~0$^{\\rm h}$41$^{\\rm\nmn}$49.8$^{\\rm s}$ and $\\delta=$~$-$9$^{\\circ}$17'33.\", and a square\nregion of $\\pm$1$^\\circ$ around this position was defined; such an\nangular distance corresponds to $\\sim$~2.7~Mpc~h$_{100}^{-1}$ at the\nredshift of the cluster ($z=$~0.0555). However, let us remark that the\nposition of the central D galaxy is slightly different,\n$\\alpha=$~0$^{\\rm h}$41$^{\\rm mn}$50.5$^{\\rm s}$ and\n$\\delta=$~$-$9$^{\\circ}$18'11.\", and so is the centre we found from\nour X-ray analysis of the diffuse component of this cluster, i.e.:\n$\\alpha=$~0$^{\\rm h}$41$^{\\rm mn}$51.9$^{\\rm s}$ and\n$\\delta=$~$-$9$^{\\circ}$18'17.\" (Pislar et al. 1997). For all our\nfuture studies, we then chose to define the cluster centre as that of\nthis X-ray component.\n\nThe distribution of the $\\sim$~4,100 candidates within the area has\nbeen first of all visually inspected to remove remaining conspicuous\nfalse detections around some stars as well as some defects mainly due to a\nsatellite track crossing the field. This cleaned catalogue contains a\nlittle more than 4,000 galaxy-like objects, half of which brighter\nthan B$_{\\rm J}=$~19.75.  The intrinsic quality of this list has then been\nchecked against a visual classification of all the recorded objects\nwithin a $\\pm$~11'25\" area covering the region already\nobserved by Murphy (1984) around the location $\\alpha=$~0$^{\\rm\nh}$41$^{\\rm mn}$57.0$^{\\rm s}$ and $\\delta=$~$-$9$^{\\circ}$23'05\". The\ninspection of the corresponding MAMA frame of 2048~$\\times$~2048\npixels enabled us to give a morphological code to each object, as well\nas to flag superimposed objects and to deblend manually 10 galaxies\n(new positions and flux estimates for each galaxy member). Of course,\nthe discrimination power of this visual examination decreases for\nstar-like objects fainter than B$_{\\rm J}=$~18.5 ($r_{\\rm F}\\sim$~17.3)\ndue to the sampling involved (pixel size of 0.67\"), and an exact\nclassification of such objects appeared to be hopeless above the {\\it\na priori} completeness limit of our automated galaxy list guessed to\nbe B$_{\\rm J}=$~19.75. Down to this limit, our results can be summarized as\nfollows~: i) $\\sim$94\\% of the selected galaxies are true galaxies\n(including 7 multiple galaxies and 2 mergers with stars), while 4\\%\nmay be galaxies~; ii) 7 genuine galaxies are missed (4\\%).  Since these\ncontamination and incompleteness levels of 5--6\\% were\nsatisfactory, we decided to set the completeness limit for our\nautomated galaxy catalogue at this magnitude B$_{\\rm J}=$~19.75.\n\n\\subsection{The photographic plate catalogue}\n\n\\begin{figure}[ht!]\n\\centerline{\\psfig{figure=DS1413F3.ps,height=8cm}}\n\\caption[ ]{Positions of the 4,232 galaxies detected on the photographic plate\nrelative to the centre of the cluster defined as the centre of the diffuse\nX-ray component. North is to the top and East to the left.}\n\\protect\\label{mamaxy}\n\\end{figure}\n\nFor objects fainter than our completeness limit, the visual check of\nthe inner ($\\pm$~11'25\") part of our object list has enabled us to\nconfirm the galaxy identification of 135 galaxy candidates as well as\nto select 214 misclassified faint galaxies.  The total number of\ngalaxies included in the visual sample down to the detection limit is\n541, whereas the initial list only contains 338 candidates within the\nsame area. Keeping in mind that both catalogues are almost identical\nfor objects brighter than B$_{\\rm J}=$~19.75, we decided to replace the\nautomated list by the visual one inside this $\\pm$~11'25\" central\narea. Note that about 150 objects remained unclassified, including 26\ngalaxies from the CCD list by Murphy. We added these 26 galaxies to\nthe final catalogue whose galaxies are plotted in Fig.~\\ref{mamaxy}.\n\nTable~1 lists the merged catalogue of 4,232 galaxies obtained from the \nSRC-J~681 plate in the $\\pm 1^{\\circ}$ field of ABCG~85, with V and R\nmagnitudes computed using the transformation laws obtained from our CCD\ndata (see \\S 3.3). This Table includes the following information~: running\nnumber~; equatorial coordinates (equinox 2000.0)~; ellipticity ; position\nangle of the major axis~; B$_{\\rm J}$, V, and R magnitudes~; X and Y\npositions in arcsecond relative to the centre defined as that of the diffuse\nX-ray emission of the cluster (see above)~; cross-identifications with the\nlists by Malumuth et al. (1992), Beers et al. (1991) and Murphy (1984).\n\n\\section{The CCD data}\n\n\\subsection{Description of the observations}\n\n\\begin{figure}[ht!]\n\\centerline{\\psfig{figure=DS1413F4.ps,height=5cm}}\n\\caption[ ]{Distribution of the fields observed with CCD imaging. The size\nof each field is 6.4$\\times$6.4~arcmin$^2$. Positions are drawn relatively to \nthe centre with equatorial coordinates\n$\\alpha=$~0$^{\\rm h}$41$^{\\rm mn}$46.0$^{\\rm s}$ and\n$\\delta=$~$-$9$^{\\circ}$20'10\".}\n\\protect\\label{champsccd}\n\\end{figure}\n\nThe observations were performed with the Danish 1.5m telescope at\nESO La Silla during 2 nights on November 2 and 3, 1994 (the third night was \ncloudy, and this accounts for the missing fields in Fig.~\\ref{champsccd}). \nA sketch of the observed fields is displayed in Fig.~\\ref{champsccd}. Field~1\nwas centered on the coordinates~: 00$^{\\rm h}$41$^{\\rm mn}$46.00$^{\\rm s}$, \n$-9^\\circ$20'10.0\" (2000.0). There was almost no overlap between the\nvarious fields (only a few arcseconds). The Johnson\nV and R filters were used. Exposure times were 10~mn for all fields; 1~mn \nexposures were also taken for a number of fields with bright objects in\norder to avoid saturation. The detector was CCD~\\#28 with 1024$^2$ pixels of \n24~$\\mu$m, giving a sampling on the sky of 0.377\"\/pixel, and a size of \n6.4$\\times$6.4~arcmin$^2$ for each field. \nThe seeing was poor the first night~: 1.5--2\" for fields 1 and 2, 2--3\" for \nfield 3 (in which consequently the number of galaxies detected is much \nsmaller), and good the second night~: 0.75--1.1\". On the \nother hand, the photometric quality of the first night was better than that\nof the second one. However, the observation of many standard stars per night\nmade a correct photometric calibration possible even for the second night as\nindicated by a comparison with an external magnitude list~: the photometric\ncatalogues from the six fields have the same behaviour for both nights\n(see e.g. Fig.~8).\n\n\n\\subsection{Data reduction}\n\nCorrections for bias and flat-field were performed in the usual way with\nthe IRAF software. Only flat fields obtained on the sky at twilight and\ndawn were used; dome flat fields were discarded because they showed too\nmuch structure.\n\nEach field was reduced separately.  The photometric calibration took\ninto account the exposure time, the time at which the exposure had\nbeen made, the color index (V-R), the airmass, and a second order term\nincluding both the color index and airmass. The photometric\ncharacteristics of both nights were estimated separately.\n\nObjects were automatically detected using the task\nDAOPHOT\/DAOFIND. This task performs a convolution with a gaussian\nhaving previously chosen characteristics, taking into account the\nseeing in each frame (FWHM of the star-like profiles in the image) as\nwell as the CCD readout noise and gain. Objects are identified as the\npeaks of the convolved image which are higher than a given threshold\nabove the local sky background (chosen as approximately equal to\n4~$\\sigma$ of the image mean sky level).  A list of detected objects\nis thus produced and interactively corrected on the displayed image so\nas to discard spurious objects, add undetected ones (usually close to\nthe CCD edges) and dispose of false detections caused by the events\nflagged in the previous section. Since exposure times were the same in\nV and R, the number of objects detected in the R band is of course much\nlarger.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=DS1413F5.ps,height=6cm}}\n\\caption[ ]{Positions of the galaxies detected in the R band relative to the\ncentre defined as the centre of the diffuse X-ray emission (see text).}\n\\protect\\label{ccdxy}\n\\end{figure}\n\nWe used the package developed by O.~Le F\\`evre (Le F\\`evre et al. \n1986) to obtain for each field a catalogue with the (x,y) galaxy positions, \nisophotal radii, ellipticities, major axis, position angles, and V and R \nmagnitudes within the 26.5 isophote.\nStar-galaxy separation was performed based on a compactness parameter q\ndetermined by Le F\\` evre et al. (1986, see also Slezak et al. 1988), as\ndescribed in detail e.g. by Lobo et al. (1997). We chose q=1.45 as the best\nseparation limit between galaxies and stars; very bright stars were \nclassified as galaxies with this criterion and had to be eliminated manually.\nAfter eliminating repeated detections of a few objects, we obtained a total \nnumber of 805 galaxies detected in R, among which 381 are detected in V. \nThe errors on these CCD magnitudes are in all cases smaller than 0.2 magnitude,\nand their rms accuracy is about 0.1 magnitude; these rather large values are\ndue to the bad seeing during the first night and to pretty poor photometric\nconditions during the second night.\n\nPositions of the galaxies detected in the R band relative to the\ncentre defined above are displayed in Fig.~\\ref{ccdxy}. Notice the smaller\nnumber of galaxies detected in field 3 due to a sudden worsening of the seeing \nduring the exposure on this field.\nThe astrometry of this CCD catalogue is accurate to 1.5--2.0 arcseconds as\nverified from the average mutual angular distance between CCD and MAMA\nequatorial coordinates for 174 galaxies included in both catalogues.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=DS1413F6.ps,height=7cm}}\n\\caption[ ]{Histogram of all the R magnitudes of the galaxies in the\nCCD catalogue.}\n\\protect\\label{ccdrmag}\n\\end{figure}\n\nThe histogram of the R magnitudes in the CCD catalogue is displayed in\nFig.~\\ref{ccdrmag}. It will be discussed in detail in Paper~III\n(Durret et al.  in preparation). The turnover value of this histogram\nis located between R=22 and R=23, suggesting that our catalogue is\nroughly complete up to R=22.\n\nThe (V-R) colours are plotted as a function of R for the 381 galaxies\ndetected in the V band in our CCD catalogue\n(Fig.~\\ref{coul}). Unfortunately, since the observed CCD field is\nsmall, there are only 50 of these galaxies with measured redshifts,\nand therefore it is not possible to derive a colour-magnitude relation\nfrom which to establish a membership criterion for the cluster.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=DS1413F7.ps,height=8cm}}\n\\caption[ ]{(V-R) colour as a function of R for the 381 galaxies detected\nin the V band in our CCD catalogue. The 50 galaxies indicated with a square are\nthose with redshifts in the interval 13,350 -- 20,000~km~s$^{-1}$  assumed to belong to\nABCG~85{}.}\n\\protect\\label{coul}\n\\end{figure}\n\n\\subsection{Transformation laws between the photometric systems}\n\n576 stars were also measured on the CCD images and used to calculate \ncalibration relations between our photographic plate B$_{\\rm J}$ \\ magnitudes and \nour V and R CCD magnitudes.\n\nFor stars:\\\\\n$$V_{CCD}=B_{\\rm J}- 40.8302+3.6656\\ B_{\\rm J} -0.082567\\ B_{\\rm J}^2 $$\n$$R_{CCD}=B_{\\rm J} -10.12663+0.430772\\ B_{\\rm J} $$\n\nFor galaxies where only R is detected:\\\\\n$$R_{CCD}=B_{\\rm J} -3.03532+0.121963\\ B_{\\rm J}$$\n\nFor galaxies where both V and R are detected:\\\\\n$$V_{CCD}=B_{\\rm J}-2.13942+0.108905\\ B_{\\rm J}$$\n$$R_{CCD}=B_{\\rm J}-0.566762(V-R)-2.29919+0.10482\\ B_{\\rm J}$$\n\nThe observed R band CCD magnitude $R_{CCD}$ as a function of the R\nmagnitude calculated from the photographic B$_{\\rm J}$  magnitude is plotted\nin Fig.~\\ref{rr5} for galaxies, showing the quality of the correlation\nfor the six different CCD fields, especially for objects brighter than R$=$19.\nAll the CCD fields appear to behave identically.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=DS1413F8.ps,height=8cm}}\n\\caption[ ]{Observed R band CCD magnitude $R_{CCD}$ as a function of the \nR magnitude calculated from the photographic B$_{\\rm J}$  magnitude. The six different \nsymbols correspond to the six CCD fields described above.}\n\\protect\\label{rr5}\n\\end{figure}\n\n\\subsection{The CCD catalogue}\n\nThe CCD photometric data for the galaxies in the field of ABCG~85{} are given \nin Table~2. This Table includes for each object the following information~:\nrunning number~; equatorial coordinates (equinox 2000.0)~; isophotal radius~;\nellipticity~; position angle of the major axis~; V and R magnitudes~; X and\nY positions in arcsecond relative to the centre assumed to have coordinates\n$\\alpha = 0^{\\rm h}41^{\\rm mn}51.90^{\\rm s}$ and \n$\\delta = -9^\\circ$18'17.0\" (equinox 2000.0) (this centre was chosen to\ncoincide with that of the diffuse X-ray gas component as defined by Pislar\net al. (1997) ).\\\\\n\n\\section{Conclusions} \n\nOur redshift catalogue is submitted jointly in a companion paper\n(Durret et al. 1997). Together with the catalogues presented here, it\nis used to give an interpretation of the optical properties of ABCG~85{}\n(Durret et al. in preparation, Paper~III), in relation with the X-ray\nproperties of this cluster (Pislar et al. 1997, Lima--Neto et al. 1997, \nPapers~I and II).\n\n\\acknowledgements {We are very grateful to the MAMA team at\nObservatoire de Paris for help when scanning the photographic plate,\nand to Cl\\'audia Mendes de Oliveira for her cheerful assistance at the\ntelescope.  CL is fully supported by the BD\/2772\/93RM grant attributed\nby JNICT, Portugal.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Overview}\n\\addcontentsline{toc}{section}{Overview}\n\nMany problems in high-dimensional statistics are believed to exhibit gaps between what can be achieved \\emph{information-theoretically} (or \\emph{statistically}, i.e., with unbounded computational power) and what is possible with bounded computational power (e.g., in polynomial time). Examples include finding planted cliques \\cite{J-clique,DM-clique,MPW-clique,pcal} or dense communities \\cite{block-model-1,block-model-2,HS-bayesian} in random graphs, extracting variously structured principal components of random matrices \\cite{BR-sparse,LKZ-mmse,LKZ-sparse} or tensors \\cite{HSS-tensor,sos-hidden}, and solving or refuting random constraint satisfaction problems \\cite{alg-barriers,refuting-any-csp}.\n\nAlthough current techniques cannot prove that such average-case problems require super-polynomial time (even assuming $P \\ne NP$), various forms of rigorous evidence for hardness have been proposed. These include:\n\\begin{itemize}\n    \\item failure of Markov chain Monte Carlo methods \\cite{J-clique,DFJ-mcmc};\n    \\item failure of local algorithms \\cite{GS-local,DM-hidden,tensor-local,subopt-local-maxcut};\n    \\item methods from statistical physics which suggest failure of belief propagation or approximate message passing algorithms \\cite{block-model-1,block-model-2,LKZ-mmse, LKZ-sparse} (see \\cite{stat-phys-survey} for a survey or \\cite{BPW-phys-notes} for expository notes);\n    \\item structural properties of the solution space \\cite{alg-barriers,KMRSZ,GS-local,GZ-reg,GZ-clique};\n    \\item geometric analysis of non-convex optimization landscapes \\cite{ABC,matrix-tensor};\n    \\item reductions from planted clique (which has become a ``canonical'' problem believed to be hard in the average case) \\cite{BR-sparse,HWX-reduction,WBS,hard-rip,bresler-sparse,bresler-pca};\n    \\item lower bounds in the statistical query model \\cite{sq-kearns,sq-half,sq-clique,sq-sat,sq-gaussian,sq-robust};\n    \\item lower bounds against the sum-of-squares hierarchy \\cite{grig-parity,sch-parity,DM-clique,MPW-clique,HSS-tensor,MW-sos,pcal,sos-hidden} (see \\cite{sos-survey} for a survey).\n\\end{itemize} \n\n\\noindent In these notes, we survey another emerging method, which we call the \\emph{low-degree method}, for understanding computational hardness in average-case problems. \nIn short, we explore a conjecture that the behavior of a certain quantity -- the second moment of the \\emph{low-degree likelihood ratio} -- reveals the computational complexity of a given statistical task. \nWe find the low-degree method particularly appealing because it is simple, widely applicable, and can be used to study a wide range of time complexities (e.g., polynomial, quasipolynomial, or nearly-exponential).\nFurthermore, rather than simply positing a certain ``optimal algorithm,'' the underlying conjecture captures an interpretable structural feature that seems to dictate whether a problem is easy or hard. \nFinally, and perhaps most importantly, predictions using the low-degree method have been carried out for a variety of average-case problems, and so far have always reproduced widely-believed results.\n\nHistorically, the low-degree method arose from the study of the sum-of-squares (SoS) semidefinite programming hierarchy. In particular, the method is implicit in the \\emph{pseudo-calibration} approach to proving SoS lower bounds \\cite{pcal}. Two concurrent papers \\cite{HS-bayesian,sos-hidden} later articulated the idea more explicitly. In particular, Hopkins and Steurer \\cite{HS-bayesian} were the first to demonstrate that the method can capture sharp thresholds of computational feasibility such as the Kesten--Stigum threshold for community detection in the stochastic block model. The low-degree method was developed further in the PhD thesis of Hopkins \\cite{sam-thesis}, which includes a precise conjecture about the complexity-theoretic implications of low-degree predictions. In comparison to sum-of-squares lower bounds, the low-degree method is much simpler to carry out and appears to always yield the same results for natural average-case problems.\n\nIn these notes, we aim to provide a self-contained introduction to the low-degree method; we largely avoid reference to SoS and instead motivate the method in other ways. We will briefly discuss the connection to SoS in Section~\\ref{sec:sos}, but we refer the reader to \\cite{sam-thesis} for an in-depth exposition of these connections.\n\nThese notes are organized as follows. In Section~\\ref{sec:decisiontheory}, we present the low-degree method and motivate it as a computationally-bounded analogue of classical statistical decision theory. In Section~\\ref{sec:agn}, we show how to carry out the low-degree method for a general class of additive Gaussian noise models. In Section~\\ref{sec:examples}, we specialize this analysis to two classical problems: the spiked Wigner matrix and spiked Gaussian tensor models. Finally, in Section~\\ref{sec:ldlr-conj-2}, we discuss various forms of heuristic and formal evidence for correctness of the low-degree method; in particular, we highlight a formal connection between low-degree lower bounds and the failure of spectral methods (Theorem~\\ref{thm:spectral-hard}).\n\n\n\\section{Towards a Computationally-Bounded Decision Theory}\\label{sec:decisiontheory}\n\n\\subsection{Statistical-to-Computational Gaps in Hypothesis Testing}\n\nThe field of \\emph{statistical decision theory} (see, e.g., \\cite{LR-sdt,LeCam-sdt} for general references) is concerned with the question of how to decide optimally (in some quantitative sense) between several statistical conclusions.\nThe simplest example, and the one we will mainly be concerned with here, is that of \\emph{simple hypothesis testing}: we observe a dataset that we believe was drawn from one of two probability distributions, and want to make an inference (by performing a statistical \\emph{test}) about which distribution we think the dataset was drawn from.\n\nHowever, one important practical aspect of statistical testing usually is not included in this framework, namely the \\emph{computational cost} of actually performing a statistical test.\nIn these notes, we will explore ideas from a line of recent research about how one mathematical method of classical decision theory might be adapted to predict the capabilities and limitations of \\emph{computationally bounded statistical tests}.\n\nThe basic problem that will motivate us is the following.\nSuppose $\\PPP = (\\PP_n)_{n \\in \\NN}$ and $\\QQQ = (\\QQ_n)_{n \\in \\NN}$ are two sequences of probability distributions over a common sequence of measurable spaces $\\sS = ((\\sS_n, \\sF_n))_{n \\in \\NN}$.\n(In statistical parlance, we will think throughout of $\\PPP$ as the model of the \\emph{alternative hypothesis} and $\\QQQ$ as the model of the \\emph{null hypothesis}. Later on, we will consider hypothesis testing problems where the distributions $\\PPP$ include a ``planted'' structure, making the notation a helpful mnemonic.)\nSuppose we observe $\\bY \\in \\sS_n$ which is drawn from one of $\\PP_n$ or $\\QQ_n$.\nWe hope to recover this choice of distribution in the following sense.\n\n\\begin{definition}\n    We say that a sequence of events $(A_n)_{n \\in \\NN}$ with $A_n \\in \\sF_n$ occurs with \\emph{high probability (in $n$)} if the probability of $A_n$ tends to 1 as $n \\to \\infty$.\n\\end{definition}\n\n\\begin{definition}\\label{def:stat-ind}\n    A sequence of (measurable) functions $f_n: \\sS_n \\to \\{ \\tp, \\tq \\}$ is said to \\emph{strongly distinguish}\\footnote{We will only consider this so-called \\emph{strong} version of distinguishability, where the probability of success must tend to 1 as $n \\to \\infty$, as opposed to the \\emph{weak} version where this probability need only be bounded above $\\frac{1}{2}$. For high-dimensional problems, the strong version typically coincides with important notions of estimating the planted signal (see Section~\\ref{sec:extensions}), whereas the weak version is often trivial.} $\\PPP$ and $\\QQQ$ if $f_n(\\bY) = \\tp$ with high probability when $\\bY \\sim \\PP_n$, and $f_n(\\bY) = \\tq$ with high probability when $\\bY \\sim \\QQ_n$. If such $f_n$ exist, we say that $\\PPP$ and $\\QQQ$ are \\emph{statistically distinguishable}.\n\\end{definition}\n\nIn our computationally bounded analogue of this definition, let us for now only consider polynomial time tests (we will later consider various other restrictions on the time complexity of $f_n$, such as subexponential time).\nThen, the analogue of Definition~\\ref{def:stat-ind} is the following.\n\n\\begin{definition}\n    $\\PPP$ and $\\QQQ$ are said to be \\emph{computationally distinguishable} if there exists a sequence of measurable \\textbf{and computable in time polynomial in $\\bm n$} functions $f_n: \\sS_n \\to \\{\\tp, \\tq\\}$ such that $f_n$ strongly distinguishes $\\PPP$ and $\\QQQ$.\n\\end{definition}\n\n\\noindent Clearly, computational distinguishability implies statistical distinguishability.\nOn the other hand, a multitude of theoretical evidence suggests that statistical distinguishability does not in general imply computational distinguishability.\nOccurrences of this phenomenon are called \\emph{statistical-to-computational (stat-comp) gaps}.\nTypically, such a gap arises in the following slightly more specific way. Suppose the sequence $\\PPP$ has a further dependence on a \\emph{signal-to-noise} parameter $\\lambda > 0$, so that $\\PPP_\\lambda = (\\PP_{\\lambda, n})_{n \\in \\NN}$.\nThis parameter should describe, in some sense, the strength of the structure present under $\\PPP$ (or, in some cases, the number of samples received). The following is one canonical example.\n\n\\begin{example}[Planted Clique Problem \\cite{J-clique,kucera-clique}]\nUnder the null model $\\QQ_n$, we observe an $n$-vertex Erd\\H{o}s-R\\'enyi graph $\\sG(n,1\/2)$, i.e., each pair $\\{i, j\\}$ of vertices is connected with an edge independently with probability $1\/2$. The signal-to-noise parameter $\\lambda$ is an integer $1 \\le \\lambda \\le n$. Under the planted model $\\PP_{\\lambda,n}$, we first choose a random subset of vertices $S \\subseteq [n]$ of size $|S| = \\lambda$ uniformly at random. We then observe a graph where each pair $\\{i, j\\}$ of vertices is connected with probability $1$ if $\\{i,j\\} \\subseteq S$ and with probability $1\/2$ otherwise. In other words, the planted model consists of the union of $\\sG(n,1\/2)$ with a planted \\emph{clique} (a fully-connected subgraph) on $\\lambda$ vertices.\n\\end{example}\n\n\\noindent As $\\lambda$ varies, the problem of testing between $\\PPP_\\lambda$ and $\\QQQ$ can change from statistically impossible, to statistically possible but computationally hard, to computationally easy.\nThat is, there exists a threshold $\\lambda_{\\mathsf{stat}}$ such that for any $\\lambda > \\lambda_{\\mathsf{stat}}$, $\\PPP_{\\lambda}$ and $\\QQQ$ are statistically distinguishable, but for $\\lambda < \\lambda_{\\mathsf{stat}}$ are not.\nThere also exists a threshold $\\lambda_{\\mathsf{comp}}$ such that for any $\\lambda > \\lambda_{\\mathsf{comp}}$, $\\PPP_\\lambda$ and $\\QQQ$ are computationally distinguishable, and (conjecturally) for $\\lambda < \\lambda_{\\mathsf{comp}}$ are not.\nClearly we must have $\\lambda_{\\mathsf{comp}} \\geq \\lambda_{\\mathsf{stat}}$, and a stat-comp gap corresponds to strict inequality $\\lambda_{\\mathsf{comp}} > \\lambda_{\\mathsf{stat}}$. For instance, the two models in the planted clique problem are statistically distinguishable when $\\lambda \\ge (2+\\varepsilon) \\log_2 n$ (since $2\\log_2 n$ is the typical size of the largest clique in $\\sG(n,1\/2)$), so $\\lambda_{\\mathsf{stat}} = 2 \\log_2 n$. However, the best known polynomial-time distinguishing algorithms only succeed when $\\lambda = \\Omega(\\sqrt{n})$ \\cite{kucera-clique,AKS-clique}, and so (conjecturally) $\\lambda_{\\mathsf{comp}} \\approx \\sqrt{n}$, a large stat-comp gap.\n\nThe remarkable method we discuss in these notes allows us, through a relatively straightforward calculation, to predict the threshold $\\lambda_{\\mathsf{comp}}$ for many of the known instances of stat-comp gaps.\nWe will present this method as a modification of a classical second moment method for studying $\\lambda_{\\mathsf{stat}}$.\n\n\n\\subsection{Classical Asymptotic Decision Theory}\n\nIn this section, we review some basic tools available from statistics for understanding statistical distinguishability.\nWe retain the same notations from the previous section in the later parts, but in the first part of the discussion will only be concerned with a single pair of distributions $\\PP$ and $\\QQ$ defined on a single measurable space $(\\sS, \\sF)$.\nFor the sake of simplicity, let us assume in either case that $\\PP_n$ (or $\\PP$) is absolutely continuous with respect to $\\QQ_n$ (or $\\QQ$, as appropriate).\\footnote{For instance, what will be relevant in the examples we consider later, any pair of non-degenerate multivariate Gaussian distributions satisfy this assumption.}\n\n\\subsubsection{Basic Notions}\n\nWe first define the basic objects used to make hypothesis testing decisions, and some ways of measuring their quality.\n\n\\begin{definition}\n    A \\emph{test} is a measurable function $f: \\sS \\to \\{\\tp, \\tq\\}$.\n\\end{definition}\n\n\\begin{definition}\n    The \\emph{type I error} of $f$ is the event of falsely rejecting the null hypothesis, i.e., of having $f(\\bY) = \\tp$ when $\\bY \\sim \\QQ$.\n    The \\emph{type II error} of $f$ is the event of falsely failing to reject the null hypothesis, i.e., of having $f(\\bY) = \\tq$ when $\\bY \\sim \\PP$.\n    The probabilities of these errors are denoted\n    \\begin{align*}\n        \\alpha(f) &\\colonequals \\QQ\\left(f(\\bY) = \\tp\\right), \\\\\n        \\beta(f) &\\colonequals \\PP\\left(f(\\bY) = \\tq\\right).\n    \\end{align*}\n    The probability $1 - \\beta(f)$ of correctly rejecting the null hypothesis is called the \\emph{power} of $f$.\n\\end{definition}\n\\noindent\nThere is a tradeoff between type I and type II errors.\nFor instance, the trivial test that always outputs $\\tp$ will have maximal power, but will also have maximal probability of type I error, and vice-versa for the trivial test that always outputs $\\tq$.\nThus, typically one fixes a tolerance for one type of error, and then attempts to design a test that minimizes the probability of the other type.\n\n\\subsubsection{Likelihood Ratio Testing}\n\nWe next present the classical result showing that it is in fact possible to identify the test that is optimal in the sense of the above tradeoff.\\footnote{It is important to note that, from the point of view of statistics, we are restricting our attention to the special case of deciding between two ``simple'' hypotheses, where each hypothesis consists of the dataset being drawn from a specific distribution. Optimal testing is more subtle for ``composite'' hypotheses in parametric families of probability distributions, a more typical setting in practice. The mathematical difficulties of this extended setting are discussed thoroughly in \\cite{LR-sdt}.}\n\n\\begin{definition}\n    Let $\\PP$ be absolutely continuous with respect to $\\QQ$. The \\emph{likelihood ratio}\\footnote{For readers not familiar with the Radon--Nikodym derivative: if $\\PP$, $\\QQ$ are discrete distributions then $L(\\bY) = \\PP(\\bY)\/\\QQ(\\bY)$; if $\\PP$, $\\QQ$ are continuous distributions with density functions $p$, $q$ (respectively) then $L(\\bY) = p(\\bY)\/q(\\bY)$.} of $\\PP$ and $\\QQ$ is\n    \\begin{equation*}\n        L(\\bY) \\colonequals \\frac{d\\PP}{d\\QQ}(\\bY).\n    \\end{equation*}\n    The \\emph{thresholded likelihood ratio test} with threshold $\\eta$ is the test\n    \\begin{equation*}\n        L_{\\eta}(\\bY) \\colonequals \\left\\{\\begin{array}{lcl} \\tp & : & L(\\bY) > \\eta \\\\ \\tq & : & L(\\bY) \\leq \\eta \\end{array}\\right\\}.\n    \\end{equation*}\n\\end{definition}\n\n\\noindent Let us first present a heuristic argument for why thresholding the likelihood ratio might be a good idea. Specifically, we will show that the likelihood ratio is optimal in a particular ``$L^2$ sense'' (which will be of central importance later), i.e., when its quality is measured in terms of first and second moments of a testing quantity.\n\n\\begin{definition}\\label{def:hilbert}\n    For (measurable) functions $f,g: \\sS \\to \\RR$, define the inner product and norm induced by $\\QQ$:\n    \\begin{align*}\n        \\la f, g \\ra &\\colonequals \\Ex_{\\bY \\sim \\QQ}\\left[ f(\\bY) g(\\bY) \\right], \\\\\n        \\|f\\| &\\colonequals \\sqrt{\\langle f,f \\rangle}.\n    \\end{align*}\n    Let $L^2(\\QQ)$ denote the Hilbert space consisting of functions $f$ for which $\\|f\\| < \\infty$, endowed with the above inner product and norm.\\footnote{For a more precise definition of $L^2(\\QQ_n)$ (in particular including issues around functions differing on sets of measure zero) see a standard reference on real analysis such as \\cite{SS-real}.}\n\\end{definition}\n\n\\begin{proposition}\n    \\label{prop:lr-optimal-l2}\n    If $\\PP$ is absolutely continuous with respect to $\\QQ$, then the unique solution $f^*$ of the optimization problem\n    \\begin{equation*}\n    \\begin{array}{ll}\n        \\text{maximize} & \\displaystyle \\Ex_{\\bY \\sim \\PP} [f(\\bY)] \\\\[10pt]\n        \\text{subject to} & \\displaystyle \\Ex_{\\bY \\sim \\QQ} [f(\\bY)^2] = 1\n    \\end{array}\n    \\end{equation*}\n    is the (normalized) likelihood ratio\n    \\[\n        f^\\star = L\/\\|L\\|,\n    \\]\n    and the value of the optimization problem is $\\|L\\|$.\n\\end{proposition}\n\\begin{proof}\n    We may rewrite the objective as\n    \\[\\Ex_{\\bY \\sim \\PP} f(\\bY) = \\Ex_{\\bY \\sim \\QQ} \\left[L(\\bY) f(\\bY) \\right] = \\langle L,f \\rangle,\\]\n    and rewrite the constraint as $\\|f\\| = 1$. The result now follows since $\\langle L, f \\rangle \\leq \\|L\\| \\cdot \\|f\\| = \\|L\\|$ by the Cauchy-Schwarz inequality, with equality if and only if $f$ is a scalar multiple of $L$.\n\\end{proof}\n\\noindent\nIn words, this means that if we want a function to be as large as possible in expectation under $\\PP$ while remaining bounded (in the $L^2$ sense) under $\\QQ$, we can do no better than the likelihood ratio.\nWe will soon return to this type of $L^2$ reasoning in order to devise computationally-bounded statistical tests.\n\nThe following classical result shows that the above heuristic is accurate, in that the thresholded likelihood ratio tests achieve the optimal tradeoff between type I and type II errors.\n\\begin{lemma}[Neyman--Pearson Lemma \\cite{N-P}]\n    \\label{lem:neyman-pearson}\n    Fix an arbitrary threshold $\\eta \\ge 0$. Among all tests $f$ with $\\alpha(f) \\leq \\alpha(L_\\eta) = \\QQ(L(\\bY) > \\eta)$, $L_{\\eta}$ is the test that maximizes the power $1 - \\beta(f)$.\n\\end{lemma}\n\\noindent \nWe provide the standard proof of this result in Appendix~\\ref{app:neyman-pearson} for completeness. (The proof is straightforward but not important for understanding the rest of these notes, and it can be skipped on a first reading.)\n\n\\subsubsection{Le Cam's Contiguity}\n\\label{sec:contig}\n\nSince the likelihood ratio is, in the sense of the Neyman--Pearson lemma, an optimal statistical test, it stands to reason that it should be possible to argue about statistical distinguishability solely by computing with the likelihood ratio.\nWe present one simple method by which such arguments may be made, based on a theory introduced by Le~Cam \\cite{lecam}.\n\nWe will work again with sequences of probability measures $\\PPP = (\\PP_n)_{n \\in \\NN}$ and $\\QQQ = (\\QQ_n)_{n \\in \\NN}$, and will denote by $L_n$ the likelihood ratio $d \\PP_n\/d \\QQ_n$. Norms and inner products of functions are those of $L^2(\\QQ_n)$.\nThe following is the crucial definition underlying the arguments to come.\n\n\\begin{definition}\n    A sequence $\\PPP$ of probability measures is \\emph{contiguous} to a sequence $\\QQQ$, written $\\PPP \\triangleleft \\QQQ$, if whenever $A_n \\in \\sF_n$ with $\\QQ_n(A_n) \\to 0$ (as $n \\to \\infty$), then $\\PP_n(A_n) \\to 0$ as well.\n\\end{definition}\n\n\\begin{proposition}\n    If $\\PPP \\triangleleft \\QQQ$ or $\\QQQ \\triangleleft \\PPP$, then $\\QQQ$ and $\\PPP$ are statistically indistinguishable (in the sense of Definition~\\ref{def:stat-ind}, i.e., no test can have both type I and type II error probabilities tending to 0).\n\\end{proposition}\n\\begin{proof}\n    We give the proof for the case $\\PPP \\triangleleft \\QQQ$, but the other case may be shown by a symmetric argument.\n    For the sake of contradiction, let $(f_n)_{n \\in \\NN}$ be a sequence of tests distinguishing $\\PPP$ and $\\QQQ$, and let $A_n = \\{\\bY: f_n(\\bY) = \\tp\\}$.\n    Then, $\\PP_n(A_n^c) \\to 0$ and $\\QQ_n(A_n) \\to 0$.\n    But, by contiguity, $\\QQ_n(A_n) \\to 0$ implies $\\PP_n(A_n) \\to 0$ as well, so $\\PP_n(A_n^c) \\to 1$, a contradiction.\n\\end{proof}\n\\noindent\nIt therefore suffices to establish contiguity in order to prove negative results about statistical distinguishability.\nThe following classical second moment method gives a means of establishing contiguity through a computation with the likelihood ratio.\n\\begin{lemma}[Second Moment Method for Contiguity]\n    \\label{lem:second-moment-contiguity}\n    If $\\|L_n\\|^2 \\colonequals \\EE_{\\bY \\sim \\QQ_n}[L_n(\\bY)^2]$ remains bounded as $n \\to \\infty$ (i.e., $\\limsup_{n \\to \\infty} \\|L_n\\|^2 < \\infty$), then $\\PPP \\triangleleft \\QQQ$.\n\\end{lemma}\n\\begin{proof}\n    Let $A_n \\in \\sF_n$.\n    Then, using the Cauchy--Schwarz inequality,\n    \\begin{equation*}\n        \\PP_n(A_n) = \\Ex_{\\bY \\sim \\PP_n} [\\One_{A_n}(\\bY)] = \\Ex_{\\bY \\sim \\QQ_n}\\left[ L_n(\\bY) \\One_{A_n}(\\bY)\\right] \\leq \\left(\\Ex_{\\bY \\sim \\QQ_n}[L_n(\\bY)^2]\\right)^{1\/2} \\left(\\QQ_n(A_n)\\right)^{1\/2},\n    \\end{equation*}\n    and so $\\QQ_n(A_n) \\to 0$ implies $\\PP_n(A_n) \\to 0$.\n\\end{proof}\n\n\\noindent This second moment method has been used to establish contiguity for various high-dimensional statistical problems (see e.g., \\cite{MRZ-spectral,BMVVX-pca,PWBM-pca,PWB-tensor}). Typically the null hypothesis $\\QQ_n$ is a ``simpler'' distribution than $\\PP_n$ and, as a result, $d\\PP_n\/d\\QQ_n$ is easier to compute than $d\\QQ_n\/d\\PP_n$. In general, and essentially for this reason, establishing $\\QQQ \\triangleleft \\PPP$ is often more difficult than $\\PPP \\triangleleft \\QQQ$, requiring tools such as the \\emph{small subgraph conditioning method} (introduced in \\cite{subgraph-1,subgraph-2} and used in, e.g., \\cite{MNS-rec,BMNN-community}). Fortunately, one-sided contiguity $\\PP_n \\triangleleft \\QQ_n$ is sufficient for our purposes.\n\nNote that $\\|L_n\\|$, the quantity that controls contiguity per the second moment method, is the same as the optimal value of the $L^2$ optimization problem in Proposition~\\ref{prop:lr-optimal-l2}:\n\\begin{equation*}\n    \\left\\{\\begin{array}{rl}\n        \\text{maximize} & \\EE_{\\bY \\sim \\PP_n} [f(\\bY)] \\\\[5pt]\n        \\text{subject to} & \\EE_{\\bY \\sim \\QQ_n} [f(\\bY)^2] = 1\n    \\end{array}\\right\\} = \\|L_n\\|.\n\\end{equation*}\nWe might then be tempted to conjecture that $\\PPP$ and $\\QQQ$ are statistically distinguishable \\emph{if and only if} $\\|L_n\\| \\to \\infty$ as $n \\to \\infty$. However, this is incorrect: there are cases when $\\PPP$ and $\\QQQ$ are not distinguishable, yet a rare ``bad'' event under $\\PP_n$ causes $\\|L_n\\|$ to diverge.\nTo overcome this failure of the ordinary second moment method, some previous works (e.g., \\cite{BMNN-community,BMVVX-pca,PWB-tensor,PWBM-pca}) have used \\emph{conditional} second moment methods to show indistinguishability, where the second moment method is applied to a modified $\\PPP$ that conditions on these bad events not occurring.\n\n\\subsection{Basics of the Low-Degree Method}\n\\label{sec:ldlr-conj}\n\nWe now describe the \\emph{low-degree} analogues of the notions described in the previous section, which together constitute a method for restricting the classical decision-theoretic second moment analysis to computationally-bounded tests.\nThe premise of this \\emph{low-degree method} is to take low-degree multivariate polynomials in the entries of the observation $\\bY$ as a proxy for efficiently-computable functions. The ideas in this section were first developed in a sequence of works in the sum-of-squares optimization literature \\cite{pcal,HS-bayesian,sos-hidden,sam-thesis}.\n\nIn the computationally-unbounded case, Proposition~\\ref{prop:lr-optimal-l2} showed that the likelihood ratio optimally distinguishes $\\PPP$ from $\\QQQ$ in the $L^2$ sense.\nFollowing the same heuristic, we will now find the low-degree polynomial that best distinguishes $\\PPP$ from $\\QQQ$ in the $L^2$ sense. In order for polynomials to be defined, we assume here that $\\mathcal{S}_n \\subseteq \\RR^N$ for some $N = N(n)$, i.e., our data (drawn from $\\PP_n$ or $\\QQ_n$) is a real-valued vector (which may be structured as a matrix, tensor, etc.).\n\n\\begin{definition}\\label{def:LDLR}\n    Let $\\sV^{\\leq D}_n \\subset L^2(\\QQ_n)$ denote the linear subspace of polynomials $\\sS_n \\to \\RR$ of degree at most $D$.\n    Let $\\sP^{\\leq D}: L^2(\\QQ_n) \\to \\sV^{\\leq D}_n$ denote the orthogonal projection\\footnote{To clarify, orthogonal projection is with respect to the inner product induced by $\\QQ_n$ (see Definition~\\ref{def:hilbert}).} operator to this subspace.\n    Finally, define the \\emph{$D$-low-degree likelihood ratio ($D$-LDLR)} as $L_n^{\\leq D} \\colonequals \\sP^{\\leq D} L_n$.\n\\end{definition}\n\\noindent We now have a low-degree analogue of Proposition~\\ref{prop:lr-optimal-l2}, which first appeared in \\cite{HS-bayesian,sos-hidden}.\n\\begin{proposition}\n    The unique solution $f^*$ of the optimization problem\n    \\begin{equation}\n    \\label{eq:l2-opt-low}\n    \\begin{array}{ll}\n        \\text{maximize} & \\displaystyle\\Ex_{\\bY \\sim \\PP_n} [f(\\bY)] \\\\[10pt]\n        \\text{subject to} & \\displaystyle\\Ex_{\\bY \\sim \\QQ_n} [f(\\bY)^2] = 1, \\\\[10pt] & f \\in \\sV_n^{\\leq D},\n    \\end{array}\n    \\end{equation}\n    is the (normalized) $D$-LDLR\n    \\[\n        f^\\star = L^{\\le D}_n\/\\|L^{\\le D}_n\\|,\n    \\]\n    and the value of the optimization problem is $\\|L_n^{\\leq D}\\|$.\n\\end{proposition}\n\\begin{proof}\nAs in the proof of Proposition~\\ref{prop:lr-optimal-l2}, we can restate the optimization problem as maximizing $\\langle L_n,f \\rangle$ subject to $\\|f\\| = 1$ and $f \\in \\sV_n^{\\le D}$. Since $\\sV_n^{\\leq D}$ is a linear subspace of $L^2(\\QQ_n)$, the result is then simply a restatement of the variational description and uniqueness of the orthogonal projection in $L^2(\\QQ_n)$ (i.e., the fact that $L_n^{\\le D}$ is the unique closest element of $\\sV_n^{\\le D}$ to $L_n$).\n\\end{proof}\n\nThe following informal conjecture is at the heart of the low-degree method. It states that a computational analogue of the second moment method for contiguity holds, with $L_n^{\\leq D}$ playing the role of the likelihood ratio. Furthermore, it postulates that polynomials of degree roughly $\\log(n)$ are a proxy for polynomial-time algorithms. This conjecture is based on \\cite{HS-bayesian,sos-hidden,sam-thesis}, particularly Conjecture~2.2.4 of \\cite{sam-thesis}.\n\n\\begin{conjecture}[Informal]\n    \\label{conj:low-deg-informal}\n    For ``sufficiently nice'' sequences of probability measures $\\PPP$ and $\\QQQ$, if there exists $\\varepsilon > 0$ and $D = D(n) \\ge (\\log n)^{1+\\varepsilon}$ for which $\\|L_n^{\\leq D}\\|$ remains bounded as $n \\to \\infty$, then there is no polynomial-time algorithm that strongly distinguishes (see Definition~\\ref{def:stat-ind}) $\\PPP$ and $\\QQQ$.\n\\end{conjecture}\n\\noindent We will discuss this conjecture in more detail later (see Section~\\ref{sec:ldlr-conj-2}), including the informal meaning of ``sufficiently nice'' and a variant of the LDLR based on \\emph{coordinate degree} considered by \\cite{sos-hidden,sam-thesis} (see Section~\\ref{sec:discuss-conj}). A more general form of the low-degree conjecture (Hypothesis~2.1.5 of \\cite{sam-thesis}) states that degree-$D$ polynomials are a proxy for time-$n^{\\tilde\\Theta(D)}$ algorithms, allowing one to probe a wide range of time complexities. We will see that the converse of these low-degree conjectures often holds in practice; i.e., if $\\|L_n^{\\le D}\\| \\to \\infty$, then there exists a distinguishing algorithm of runtime roughly $n^D$. As a result, the behavior of $\\|L_n^{\\le D}\\|$ precisely captures the (conjectured) power of computationally-bounded testing in many settings.\n\nThe remainder of these notes is organized as follows.\nIn Section~\\ref{sec:agn}, we work through the calculations of $L_n$, $L_n^{\\leq D}$, and their norms for a general family of additive Gaussian noise models. In Section~\\ref{sec:examples}, we apply this analysis to a few specific models of interest: the spiked Wigner matrix and spiked Gaussian tensor models.\nIn Section~\\ref{sec:ldlr-conj-2}, we give some further discussion of Conjecture~\\ref{conj:low-deg-informal}, including evidence (both heuristic and formal) in its favor.\n\n\n\n\\section{The Additive Gaussian Noise Model}\n\\label{sec:agn}\n\nWe will now describe a concrete class of hypothesis testing problems and analyze them using the machinery introduced in the previous section.\nThe examples we discuss later (spiked Wigner matrix and spiked tensor) will be specific instances of this general class.\n\n\\subsection{The Model}\n\n\\begin{definition}[Additive Gaussian Noise Model]\n\\label{def:agn}\nLet $N = N(n) \\in \\NN$ and let $\\bX$ (the ``signal'') be drawn from some distribution $\\sP_n$ (the ``prior'') over $\\RR^N$. Let $\\bZ \\in \\RR^N$ (the ``noise'') have i.i.d.\\ entries distributed as $\\sN(0,1)$. Then, we define $\\PPP$ and $\\QQQ$ as follows.\n\\begin{itemize}\n    \\item Under $\\PP_n$, observe $\\bY = \\bX + \\bZ$.\n    \\item Under $\\QQ_n$, observe $\\bY = \\bZ$.\n\\end{itemize}\n\\end{definition}\n\\noindent \nOne typical situation takes $\\bX$ to be a low-rank matrix or tensor. The following is a particularly important and well-studied special case, which we will return to in Section~\\ref{sec:spiked-matrix}.\n\n\\begin{example}[Wigner Spiked Matrix Model]\n\\label{ex:wig}\nConsider the additive Gaussian noise model with $N = n^2$, $\\RR^N$ identified with $n \\times n$ matrices with real entries, and $\\sP_n$ defined by $\\bX = \\lambda \\bx \\bx^{\\top} \\in \\RR^{n \\times n}$, where $\\lambda = \\lambda(n) > 0$ is a signal-to-noise parameter and $\\bx$ is drawn from some distribution $\\sX_n$ over $\\RR^n$. Then, the task of distinguishing $\\PP_{n}$ from $\\QQ_n$ amounts to distinguishing $\\lambda \\bx\\bx^\\top + \\bZ$ from $\\bZ$ where $\\bZ \\in \\RR^{n \\times n}$ has i.i.d.\\ entries distributed as $\\sN(0, 1)$. (This variant is equivalent to the more standard model in which the noise matrix is symmetric; see Appendix~\\ref{app:symm}.)\n\\end{example}\n\n\\noindent\nThis problem is believed to exhibit stat-comp gaps for some choices of $\\sX_n$ but not others; see, e.g., \\cite{LKZ-mmse,LKZ-sparse,mi-rank-one,BMVVX-pca,PWBM-pca}.\nAt a heuristic level, the typical \\emph{sparsity} of vectors under $\\sX_n$ seems to govern the appearance of a stat-comp gap.\n\n\\begin{remark}\nIn the spiked Wigner problem, as in many others, one natural statistical task besides distinguishing the null and planted models is to non-trivially estimate the vector $\\bx$ given $\\bY \\sim \\PP_n$, i.e., to compute an estimate $\\hat \\bx = \\hat \\bx (\\bY)$ such that $|\\langle \\hat \\bx, \\bx \\rangle|\/(\\|\\hat \\bx\\| \\cdot \\|\\bx\\|) \\ge \\varepsilon$ with high probability, for some constant $\\varepsilon > 0$. Typically, for natural high-dimensional problems, non-trivial estimation of $\\bx$ is statistically or computationally possible precisely when it is statistically or computationally possible (respectively) to strongly distinguish $\\PPP$ and $\\QQQ$; see Section~\\ref{sec:extensions} for further discussion.\n\\end{remark}\n\n\\subsection{Computing the Classical Quantities}\n\nWe now show how to compute the likelihood ratio and its $L^2$-norm under the additive Gaussian noise model. (This is a standard calculation; see, e.g., \\cite{MRZ-spectral,BMVVX-pca}.)\n\n\\begin{proposition}\n    \\label{prop:agn-L}\n    Suppose $\\PPP$ and $\\QQQ$ are as defined in Definition~\\ref{def:agn}, with a sequence of prior distributions $(\\sP_n)_{n \\in \\NN}$.\n    Then, the likelihood ratio of $\\PP_n$ and $\\QQ_n$ is\n    \\begin{equation*}\n        L_n(\\bY) = \\frac{d\\PP_n}{d\\QQ_n}(\\bY) = \\Ex_{\\bX \\sim \\sP_n}\\left[ \\exp\\left(-\\frac{1}{2}\\|\\bX\\|^2 + \\la \\bX, \\bY \\ra \\right)\\right].\n    \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n    Write $\\sL$ for the Lebesgue measure on $\\RR^N$.\n    Then, expanding the gaussian densities,\n    \\begin{align}\n        \\frac{d\\QQ_n}{d\\sL}(\\bY) \n        &= (2\\pi)^{-N \/ 2}\\cdot \\exp\\left(-\\frac{1}{2}\\|\\bY\\|^2\\right) \\label{eq:agn-ln-denom}\\\\\n        \\frac{d\\PP_n}{d\\sL}(\\bY) \n        &= (2\\pi)^{-N \/ 2}\\cdot \\Ex_{\\bX \\sim \\sP_n}\\left[\\exp\\left(-\\frac{1}{2}\\|\\bY - \\bX\\|^2\\right)\\right] \\nonumber \\\\\n        &= (2\\pi)^{-N \/ 2}\\cdot \\exp\\left(-\\frac{1}{2}\\|\\bY\\|^2\\right) \\cdot \\EE_{\\bX \\sim \\sP_n}\\left[\\exp\\left(-\\frac{1}{2}\\|\\bX\\|^2 + \\la \\bX, \\bY \\ra\\right)\\right]\\label{eq:agn-ln-num},\n    \\end{align}\n    and $L_n$ is given by the quotient of \\eqref{eq:agn-ln-num} and \\eqref{eq:agn-ln-denom}.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:agn-L-norm}\nSuppose $\\PPP$ and $\\QQQ$ are as defined in Definition~\\ref{def:agn}, with a sequence of prior distributions $(\\sP_n)_{n \\in \\NN}$.\nThen,\n\\begin{equation}\\label{eq:gaussian-2nd}\n\\|L_n\\|^2 = \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n} \\exp(\\langle \\bX^1, \\bX^2 \\rangle),\n\\end{equation}\nwhere $\\bX^1, \\bX^2$ are drawn independently from $\\sP_n$.\n\\end{proposition}\n\\begin{proof}\nWe apply the important trick of rewriting a squared expectation as an expectation over the two independent ``replicas'' $\\bX^1, \\bX^2$ appearing in the result:\n\\begin{align*}\n\\|L_n\\|^2 &= \\Ex_{\\bY \\sim \\QQ_n} \\left[\\left(\\Ex_{\\bX \\sim \\sP_n} \\exp\\left(\\langle \\bY,\\bX \\rangle - \\frac{1}{2} \\|\\bX\\|^2\\right)\\right)^2\\right] \\\\\n&= \\Ex_{\\bY \\sim \\QQ_n} \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n} \\exp\\left(\\langle \\bY,\\bX^1 + \\bX^2\\rangle - \\frac{1}{2} \\|\\bX^1\\|^2 - \\frac{1}{2} \\|\\bX^2\\|^2\\right),\n\\intertext{where $\\bX^1$ and $\\bX^2$ are drawn independently from $\\sP_n$. We now swap the order of the expectations,}\n&= \\Ex_{\\bX^1,\\bX^2 \\sim \\sP_n} \\left[\\exp\\left(-\\frac{1}{2} \\|\\bX^1\\|^2 - \\frac{1}{2} \\|\\bX^2\\|^2\\right)\\Ex_{\\bY \\sim \\QQ_n}\\exp\\left(\\langle \\bY,\\bX^1 + \\bX^2\\rangle \\right)\\right],\n\\intertext{and the inner expectation may be evaluated explicitly using the moment-generating function of a Gaussian distribution (if $y \\sim \\sN(0,1)$, then for any fixed $t \\in \\RR$, $\\EE[\\exp(ty)] = \\exp(t^2\/2)$),}\n&= \\Ex_{\\bX^1,\\bX^2} \\exp\\left(-\\frac{1}{2} \\|\\bX^1\\|^2 - \\frac{1}{2} \\|\\bX^2\\|^2 + \\frac{1}{2}\\|\\bX^1 + \\bX^2\\|^2\\right),\n\\end{align*}\nfrom which the result follows by expanding the term inside the exponential.\\footnote{Two techniques from this calculation are elements of the ``replica method'' from statistical physics: (1) writing a power of an expectation as an expectation over independent ``replicas'' and (2) changing the order of expectations and evaluating the moment-generating function. The interested reader may see \\cite{MPV-spin-glass} for an early reference, or \\cite{MM-IPC,BPW-phys-notes} for two recent presentations.}\n\\end{proof}\n\nTo apply the second moment method for contiguity, it remains to show that~\\eqref{eq:gaussian-2nd} is $O(1)$ using problem-specific information about the distribution $\\sP_n$. For spiked matrix and tensor models, various general-purpose techniques for doing this are given in \\cite{PWBM-pca,PWB-tensor}.\n\n\n\\subsection{Computing the Low-Degree Quantities}\n\\label{sec:agn-low-deg}\n\nIn this section, we will show that the norm of the LDLR (see Section~\\ref{sec:ldlr-conj}) takes the following remarkably simple form under the additive Gaussian noise model.\n\\begin{theorem}\n    \\label{thm:agn-ldlr-norm}\n    Suppose $\\PPP$ and $\\QQQ$ are as defined in Definition~\\ref{def:agn}, with a sequence of prior distributions $(\\sP_n)_{n \\in \\NN}$.\n    Let $L_n^{\\leq D}$ be as in Definition~\\ref{def:LDLR}.\n    Then,\n    \\begin{equation}\\label{eq:gaussian-2nd-low}\n        \\|L_n^{\\leq D}\\|^2 = \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n}\\left[ \\sum_{d = 0}^D \\frac{1}{d!} \\la \\bX^1, \\bX^2 \\ra^d \\right],\n    \\end{equation}\n    where $\\bX^1, \\bX^2$ are drawn independently from $\\sP_n$.\n\\end{theorem}\n\n\\begin{remark}\\label{rem:taylor}\nNote that~\\eqref{eq:gaussian-2nd-low} can be written as $\\EE_{\\bX^1,\\bX^2} [\\exp^{\\le D}(\\langle \\bX^1,\\bX^2 \\rangle)]$, where $\\exp^{\\le D}(t)$ denotes the degree-$D$ truncation of the Taylor series of $\\exp(t)$. This can be seen as a natural low-degree analogue of the full second moment~\\eqref{eq:gaussian-2nd}. However, the low-degree Taylor series truncation in $\\exp^{\\le D}$ is conceptually distinct from the low-degree projection in $L_n^{\\le D}$, because the latter corresponds to truncation in the Hermite orthogonal polynomial basis (see below), while the former corresponds to truncation in the monomial basis.\n\\end{remark}\n\n\\noindent Our proof of Theorem~\\ref{thm:agn-ldlr-norm} will follow the strategy of \\cite{HS-bayesian,sos-hidden,sam-thesis} of expanding $L_n$ in a basis of orthogonal polynomials with respect to $\\QQ_n$, which in this case are the \\emph{Hermite polynomials}.\n\nWe first give a brief and informal review of the multivariate Hermite polynomials (see Appendix~\\ref{app:hermite} or the reference text \\cite{Szego-OP} for further information).\nThe univariate Hermite polynomials\\footnote{We will not actually use the definition of the univariate Hermite polynomials (although we will use certain properties that they satisfy as needed), but the definition is included for completeness in Appendix~\\ref{app:hermite}.} are a sequence $h_k(x) \\in \\RR[x]$ for $k \\geq 0$, with $\\deg h_k = k$.\nThey may be normalized as $\\what{h}_k(x) = h_k(x) \/ \\sqrt{k!}$, and with this normalization satisfy the orthonormality conditions\n\\begin{equation}\n    \\label{eq:hermite-orth-1d}\n    \\Ex_{y \\sim \\sN(0, 1)}\\what{h}_k(y)\\what{h}_\\ell(y) = \\delta_{k\\ell}.\n\\end{equation}\nThe multivariate Hermite polynomials in $N$ variables are indexed by $\\bm\\alpha \\in \\NN^N$, and are merely products of the $h_k$: $H_{\\bm\\alpha}(\\bx) = \\prod_{i = 1}^N h_{\\alpha_i}(x_i)$.\nThey also admit a normalized variant $\\what{H}_{\\bm\\alpha}(\\bx) = \\prod_{i = 1}^N \\what{h}_{\\alpha_i}(x_i)$, and with this normalization satisfy the orthonormality conditions\n\\begin{equation*}\n    \\Ex_{\\bY \\sim \\sN(0, \\bm I_N)}\\what{H}_{\\bm\\alpha}(\\bY)\\what{H}_{\\bm\\beta}(\\bY) = \\delta_{\\bm\\alpha \\bm\\beta},\n\\end{equation*}\nwhich may be inferred directly from \\eqref{eq:hermite-orth-1d}.\n\nThe collection of those $\\what{H}_{\\bm\\alpha}$ for which $|\\bm\\alpha| \\colonequals \\sum_{i = 1}^N \\alpha_i \\leq D$ form an orthonormal basis for $\\sV_n^{\\leq D}$ (which, recall, is the subspace of polynomials of degree $\\le D$).\nThus we may expand\n\\begin{equation}\n    \\label{eq:agn-Ln-expansion}\n    L_n^{\\leq D}(\\bY) = \\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| \\leq D}} \\la L_n, \\what{H}_{\\bm\\alpha}\\ra \\what{H}_{\\bm\\alpha}(\\bY) = \\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| \\leq D}} \\frac{1}{\\prod_{i = 1}^N \\alpha_i!} \\la L_n, H_{\\bm\\alpha}\\ra H_{\\bm\\alpha}(\\bY),\n\\end{equation}\nand in particular we have\n\\begin{equation}\n    \\label{eq:agn-Ln-norm-expansion}\n    \\|L_n^{\\leq D}\\|^2 = \\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| \\leq D}} \\frac{1}{\\prod_{i = 1}^N \\alpha_i!} \\la L_n, H_{\\bm\\alpha}\\ra^2.\n\\end{equation}\nOur main task is then to compute quantities of the form $\\la L_n, H_{\\bm\\alpha} \\ra$. Note that these can be expressed either as $\\EE_{\\bY \\sim \\QQ_n}[L_n(\\by) H_{\\bm\\alpha}(\\bY)]$ or $\\EE_{\\bY \\sim \\PP_n}[H_{\\bm\\alpha}(\\bY)]$.\nWe will give three techniques for carrying out this calculation, each depending on a different identity satisfied by the Hermite polynomials. Each will give a proof of the following remarkable formula, which shows that the quantities $\\la L_n, H_{\\bm\\alpha} \\ra$ are simply the moments of $\\sP_n$.\n\n\\begin{proposition}\n    \\label{prop:agn-components}\n    For any $\\bm\\alpha \\in \\NN^N$,\n    \\begin{equation*}\n        \\langle L_n, H_{\\bm\\alpha} \\rangle = \\Ex_{\\bX \\sim \\sP_n}\\left[ \\prod_{i = 1}^N X_i^{\\alpha_i} \\right].\n    \\end{equation*}\n\\end{proposition}\n\n\\noindent Before continuing with the various proofs of Proposition~\\ref{prop:agn-components}, let us show how to use it to complete the proof of Theorem~\\ref{thm:agn-ldlr-norm}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:agn-ldlr-norm}]\n    By Proposition~\\ref{prop:agn-components} substituted into \\eqref{eq:agn-Ln-norm-expansion}, we have\n    \\begin{align*}\n        \\|L_n^{\\leq D}\\|^2\n        &= \\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| \\leq D}} \\frac{1}{\\prod_{i = 1}^N \\alpha_i!} \\left(\\Ex_{\\bX \\sim \\sP_n}\\left[\\prod_{i = 1}^N X_i^{\\alpha_i}\\right]\\right)^2, \\\\\n        \\intertext{and performing the ``replica'' manipulation (from the proof of Proposition~\\ref{prop:agn-L-norm}) again, this may be written}\n        &= \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n}\\left[\\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| \\leq D}} \\frac{1}{\\prod_{i = 1}^N \\alpha_i!} \\prod_{i = 1}^N (X^1_iX^2_i)^{\\alpha_i}\\right] \\\\\n        &= \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n}\\left[\\sum_{d = 0}^D \\frac{1}{d!}\\sum_{\\substack{\\bm\\alpha \\in \\NN^N \\\\ |\\bm\\alpha| = d}} \\binom{d}{\\alpha_1 \\cdots \\alpha_N} \\prod_{i = 1}^N (X^1_iX^2_i)^{\\alpha_i}\\right] \\\\\n        &= \\Ex_{\\bX^1, \\bX^2 \\sim \\sP_n}\\left[\\sum_{d = 0}^D \\frac{1}{d!}\\la \\bX^1, \\bX^2 \\ra^d \\right],\n    \\end{align*}\n    where the last step uses the multinomial theorem.\n\\end{proof}\n\nWe now proceed to the three proofs of Proposition~\\ref{prop:agn-components}. For the sake of brevity, we omit here the (standard) proofs of the three Hermite polynomial identities these proofs are based on, but the interested reader may review those proofs in Appendix~\\ref{app:hermite}.\n\n\\subsubsection{Proof 1: Hermite Translation Identity}\n\nThe first (and perhaps simplest) approach to proving Proposition~\\ref{prop:agn-components} uses the following formula for the expectation of a Hermite polynomial evaluated on a Gaussian random variable of non-zero mean.\n\n\\begin{proposition}\n    \\label{prop:hermite-translation}\n    For any $k \\geq 0$ and $\\mu \\in \\RR$,\n    \\[ \\Ex_{y \\sim \\sN(\\mu, 1)}\\left[h_k(y)\\right] = \\mu^k. \\]\n\\end{proposition}\n\n\\begin{proof}[Proof 1 of Proposition~\\ref{prop:agn-components}]\n    We rewrite $\\la L_n, H_{\\bm\\alpha} \\ra$ as an expectation with respect to $\\PP_n$:\n    \\begin{align*}\n    \\la L_n, H_{\\bm\\alpha} \\ra \n    &= \\Ex_{\\bY \\sim \\QQ_n}\\left[L_n(\\bY)H_{\\bm\\alpha}(\\bY)\\right] \\\\\n    &= \\Ex_{\\bY \\sim \\PP_n}\\left[H_{\\bm\\alpha}(\\bY)\\right] \\\\\n    &= \\Ex_{\\bY \\sim \\PP_n}\\left[\\prod_{i = 1}^N h_{\\alpha_i}(Y_i) \\right]\n    \\intertext{and recall $\\bY = \\bX + \\bZ$ for $\\bX \\sim \\sP_n$ and $\\bZ \\sim \\sN(\\bm 0, \\bm I_N)$ under $\\PP_n$,}\n    &= \\Ex_{\\bX \\sim \\sP_n}\\left[\\Ex_{\\bZ \\sim \\sN(\\bm 0, \\bm I_N)}\\prod_{i = 1}^N h_{\\alpha_i}(X_i + Z_i) \\right] \\\\\n    &= \\Ex_{\\bX \\sim \\sP_n}\\left[\\prod_{i = 1}^N\\Ex_{z \\sim \\sN(X_i, 1)}h_{\\alpha_i}(z)\\right] \\\\\n    &= \\Ex_{\\bX \\sim \\sP_n}\\left[\\prod_{i = 1}^N X_i^{\\alpha_i}\\right],\n    \\end{align*}\n    where we used Proposition~\\ref{prop:hermite-translation} in the last step.\n\\end{proof}\n\n\n\\subsubsection{Proof 2: Gaussian Integration by Parts}\n\nThe second approach to proving Proposition~\\ref{prop:agn-components} uses the following generalization of a well-known integration by parts formula for Gaussian random variables.\n\\begin{proposition}\n    \\label{prop:gaussian-ibp}\n    If $f: \\RR \\to \\RR$ is $k$ times continuously differentiable and $f(y)$ and its first $k$ derivatives are bounded by $O(\\exp(|y|^\\alpha))$ for some $\\alpha \\in (0, 2)$, then\n    \\begin{equation*}\n        \\Ex_{y \\sim \\sN(0, 1)}\\left[h_k(y)f(y)\\right] = \\Ex_{y \\sim \\sN(0, 1)}\\left[\\frac{d^k f}{dy^k}(y)\\right].\n    \\end{equation*}\n\\end{proposition}\n\\noindent\n(The better-known case is $k = 1$, where one may substitute $h_1(x) = x$.)\n\n\\begin{proof}[Proof 2 of Proposition~\\ref{prop:agn-components}]\nWe simplify using Proposition~\\ref{prop:gaussian-ibp}:\n\\begin{align*}\n    \\la L_n, H_{\\bm\\alpha} \\ra = \\Ex_{\\bY \\sim \\QQ_n}\\left[L_n(\\bY)\\prod_{i = 1}^N h_{\\alpha_i}(Y_i)\\right] = \\Ex_{\\bY \\sim \\QQ_n}\\left[ \\frac{\\partial^{|\\bm\\alpha|}L_n}{\\partial Y_1^{\\alpha_1} \\cdots \\partial Y_N^{\\alpha_N}}(\\bY)\\right].\n\\end{align*}\nDifferentiating $L_n$ under the expectation, we have\n\\[ \\frac{\\partial^{|\\bm\\alpha|}L}{\\partial Y_1^{\\alpha_1} \\cdots \\partial Y_N^{\\alpha_N}}(\\bY) = \\Ex_{\\bX \\sim \\sP_n}\\left[\\,\\prod_{i = 1}^N X_i^{\\alpha_i}\\, \\exp\\left(-\\frac{1}{2}\\|\\bX\\|^2 + \\la \\bX, \\bY \\ra\\right)\\right]. \\]\nTaking the expectation over $\\bY$, we have $\\EE_{\\bY \\sim \\QQ_n} \\exp(\\la \\bX, \\bY \\ra) = \\exp(\\frac{1}{2}\\|\\bX\\|^2)$, so the entire second term cancels and the result follows.\n\\end{proof}\n\n\n\\subsubsection{Proof 3: Hermite Generating Function}\n\nFinally, the third approach to proving Proposition~\\ref{prop:agn-components} uses the following generating function for the Hermite polynomials.\n\n\\begin{proposition}\n    \\label{prop:hermite-gf}\n    For any $x, y \\in \\RR$,\n    \\[ \\exp\\left(xy - \\frac{1}{2}x^2\\right) = \\sum_{k = 0}^\\infty \\frac{1}{k!}x^k h_k(y). \\]\n\\end{proposition}\n\n\\begin{proof}[Proof 3 of Proposition~\\ref{prop:agn-components}]\n    We may use Proposition~\\ref{prop:hermite-gf} to expand $L_n$ in the Hermite polynomials directly:\n    \\begin{align*}\n        L_n(\\bY)\n        &= \\Ex_{\\bX \\sim \\sP_n}\\left[\\exp\\left(\\la \\bX, \\bY \\ra - \\frac{1}{2}\\|\\bX\\|^2\\right)\\right] \\\\\n        &= \\Ex_{\\bX \\sim \\sP_n}\\left[\\,\\prod_{i = 1}^N\\left(\\sum_{k = 0}^\\infty\\frac{1}{k!}X_i^k h_k(Y_i)\\right)\\right] \\\\\n        &= \\sum_{\\bm\\alpha \\in \\NN^N} \\frac{1}{\\prod_{i = 1}^N \\alpha_i!}\\, \\EE_{\\bX \\sim \\sP_n}\\left[\\,\\prod_{i = 1}^N X_i^{\\alpha_i}\\right] H_{\\bm\\alpha}(\\bm Y).\n    \\end{align*}\n    Comparing with the expansion \\eqref{eq:agn-Ln-expansion} then gives the result.\n\\end{proof}\n\nNow that we have the simple form~\\eqref{eq:gaussian-2nd-low} for the norm of the LDLR, it remains to investigate its convergence or divergence (as $n \\to \\infty$) using problem-specific statistics of $\\bX$. In the next section we give some examples of how to carry out this analysis.\n\n\n\n\\section{Examples: Spiked Matrix and Tensor Models}\n\\label{sec:examples}\n\nIn this section, we perform the low-degree analysis for a particular important case of the additive Gaussian model: the \\emph{order-$p$ spiked Gaussian tensor model}, also referred to as the \\emph{tensor PCA (principal component analysis)} problem. This model was introduced by \\cite{RM-tensor} and has received much attention recently. The special case $p=2$ of the spiked tensor model is the so-called \\emph{spiked Wigner matrix model} which has been widely studied in random matrix theory, statistics, information theory, and statistical physics; see \\cite{leo-survey} for a survey.\n\nIn concordance with prior work, our low-degree analysis of these models illustrates two representative phenomena: the spiked Wigner matrix model exhibits a sharp computational phase transition, whereas the spiked tensor model (with $p \\ge 3$) has a ``soft'' tradeoff between statistical power and runtime which extends through the subexponential-time regime. A low-degree analysis of the spiked tensor model has been carried out previously in \\cite{sos-hidden,sam-thesis}; here we give a sharper analysis that more precisely captures the power of subexponential-time algorithms.\n\nIn Section~\\ref{sec:spiked-tensor}, we carry out our low-degree analysis of the spiked tensor model. In Section~\\ref{sec:spiked-matrix}, we devote additional attention to the special case of the spiked Wigner model, giving a refined analysis that captures its sharp phase transition and applies to a variety of distributions of ``spikes.''\n\n\n\\subsection{The Spiked Tensor Model}\n\\label{sec:spiked-tensor}\n\nWe begin by defining the model.\n\n\\begin{definition}\nAn $n$-dimensional order-$p$ \\emph{tensor} $\\bT \\in (\\RR^n)^{\\otimes p}$ is a multi-dimensional array with $p$ dimensions each of length $n$, with entries denoted $T_{i_1,\\ldots,i_p}$ where $i_j \\in [n]$. For a vector $\\bx \\in \\RR^n$, the \\emph{rank-one tensor} $\\bx^{\\otimes p} \\in (\\RR^n)^{\\otimes p}$ has entries $(\\bx^{\\otimes p})_{i_1,\\ldots,i_p} = x_{i_1} x_{i_2} \\cdots x_{i_p}$.\n\\end{definition}\n\n\\begin{definition}[Spiked Tensor Model]\n\\label{def:spiked-tensor}\nFix an integer $p \\ge 2$. The order-$p$ \\emph{spiked tensor model} is the additive Gaussian noise model (Definition~\\ref{def:agn}) with $\\bX = \\lambda \\bx^{\\otimes p}$, where $\\lambda = \\lambda(n) > 0$ is a signal-to-noise parameter and $\\bx \\in \\RR^n$ (the ``spike'') is drawn from some probability distribution $\\sX_n$ over $\\RR^n$ (the ``prior''), normalized so that $\\|\\bx\\|^2 \\to n$ in probability as $n \\to \\infty$. In other words:\n\\begin{itemize}\n    \\item Under $\\PP_n$, observe $\\bY = \\lambda \\bx^{\\otimes p} + \\bZ$.\n    \\item Under $\\QQ_n$, observe $\\bY = \\bZ$.\n\\end{itemize}\nHere, $\\bZ$ is a tensor with i.i.d.\\ entries distributed as $\\sN(0,1)$.\\footnote{This model is equivalent to the more standard model in which the noise is symmetric with respect to permutations of the indices; see Appendix~\\ref{app:symm}.}\n\\end{definition}\n\n\\noindent Throughout this section we will focus for the sake of simplicity on the Rademacher spike prior, where $\\bx$ has i.i.d.\\ entries $x_i \\sim \\Unif(\\{\\pm 1\\})$. We focus on the problem of strongly distinguishing $\\PP_n$ and $\\QQ_n$ (see Definition~\\ref{def:stat-ind}), but, as is typical for high-dimensional problems, the problem of estimating $\\bx$ seems to behave in essentially the same way (see Section~\\ref{sec:extensions}). \n\nWe first state our results on the behavior of the LDLR for this model.\n\n\\begin{theorem}\\label{thm:tensor-lowdeg}\nConsider the order-$p$ spiked tensor model with $\\bx$ drawn from the Rademacher prior, $x_i \\sim \\Unif(\\{\\pm 1\\})$ i.i.d.\\ for $i \\in [n]$. Fix sequences $D = D(n)$ and $\\lambda = \\lambda(n)$. For constants $0 < A_p < B_p$ depending only on $p$, we have the following.\\footnote{Concretely, one may take $A_p = \\frac{1}{\\sqrt{2}} p^{-p\/4-1\/2}$ and $B_p = \\sqrt{2} e^{p\/2} p^{-p\/4}$.}\n\\begin{enumerate}\n    \\item[(i)] If $\\lambda \\le A_p\\, n^{-p\/4} D^{(2-p)\/4}$ for all sufficiently large $n$, then $\\|L_n^{\\le D}\\| = O(1)$.\n    \\item[(ii)] If $\\lambda \\ge B_p\\, n^{-p\/4} D^{(2-p)\/4}$ and $D \\le \\frac{2}{p}n$ for all sufficiently large $n$, and $D = \\omega(1)$, then $\\|L_n^{\\le D}\\| = \\omega(1)$.\n\\end{enumerate}\n(Here we are considering the limit $n \\to \\infty$ with $p$ held fixed, so $O(1)$ and $\\omega(1)$ may hide constants depending on $p$.)\n\\end{theorem}\n\n\\noindent Before we prove this, let us interpret its meaning. If we take degree-$D$ polynomials as a proxy for $n^{\\tilde\\Theta(D)}$-time algorithms (as discussed in Section~\\ref{sec:ldlr-conj}), our calculations predict that an $n^{O(D)}$-time algorithm exists when $\\lambda \\gg n^{-p\/4} D^{(2-p)\/4}$ but not when $\\lambda \\ll n^{-p\/4} D^{(2-p)\/4}$. (Here we ignore log factors, so we use $A \\ll B$ to mean $A \\le B\/\\mathrm{polylog}(n)$.) These predictions agree precisely with the previously established statistical-versus-computational tradeoffs in the spiked tensor model! It is known that polynomial-time distinguishing algorithms exist when $\\lambda \\gg n^{-p\/4}$ \\cite{RM-tensor,HSS-tensor,tensor-hom,sos-fast}, and sum-of-squares lower bounds suggest that there is no polynomial-time distinguishing algorithm when $\\lambda \\ll n^{-p\/4}$ \\cite{HSS-tensor,sos-hidden}. \n\nFurthermore, one can study the power of subexponential-time algorithms, i.e., algorithms of runtime $n^{n^\\delta} = \\exp(\\tilde{O}(n^{\\delta}))$ for a constant $\\delta \\in (0,1)$. Such algorithms are known to exist when $\\lambda \\gg n^{-p\/4-\\delta(p-2)\/4}$ \\cite{strongly-refuting,BGG,BGL,kikuchi}, matching our prediction.\\footnote{Some of these results only apply to minor variants of the spiked tensor problem, but we do not expect this difference to be important.}\nThese algorithms interpolate smoothly between the polynomial-time algorithm which succeeds when $\\lambda \\gg n^{-p\/4}$, and the exponential-time exhaustive search algorithm which succeeds when $\\lambda \\gg n^{(1-p)\/2}$. (Distinguishing the null and planted distributions is information-theoretically impossible when $\\lambda \\ll n^{(1-p)\/2}$ \\cite{RM-tensor,PWB-tensor,tensor-phys,tensor-stat}, so this is indeed the correct terminal value of $\\lambda$ for computational questions.) The tradeoff between statistical power and runtime that these algorithms achieve is believed to be optimal, and our results corroborate this claim. Our results are sharper than the previous low-degree analysis for the spiked tensor model \\cite{sos-hidden,sam-thesis}, in that we pin down the precise constant $\\delta$ in the subexponential runtime. (Similarly precise analyses of the tradeoff between subexponential runtime and statistical power have been obtained for CSP refutation \\cite{strongly-refuting} and sparse PCA \\cite{subexp-sparse}.)\n\nWe now begin the proof of Theorem~\\ref{thm:tensor-lowdeg}. Since the spiked tensor model is an instance of the additive Gaussian model, we can apply the formula from Theorem~\\ref{thm:agn-ldlr-norm}: letting $\\bx^1,\\bx^2$ be independent draws from $\\sX_n$,\n\\begin{equation}\\label{eq:tensor-L}\n\\|L_n^{\\le D}\\|^2 = \\Ex_{\\bx^1,\\bx^2} \\exp^{\\le D}(\\lambda^2 \\langle \\bx^1,\\bx^2 \\rangle^p) = \\sum_{d=0}^{D} \\frac{\\lambda^{2d}}{d!} \\Ex_{\\bx^1,\\bx^2} [\\langle \\bx^1,\\bx^2 \\rangle^{pd}].\n\\end{equation}\nWe will give upper and lower bounds on this quantity in order to prove the two parts of Theorem~\\ref{thm:tensor-lowdeg}.\n\n\\subsubsection{Proof of Theorem~\\ref{thm:tensor-lowdeg}: Upper Bound}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:tensor-lowdeg}(i)]\n\nWe use the moment bound\n\\begin{equation}\\label{eq:subg-mom}\n\\Ex_{\\bx^1,\\bx^2}[|\\langle \\bx^1,\\bx^2 \\rangle|^k] \\le (2n)^{k\/2} k \\Gamma(k\/2)\n\\end{equation}\nfor any integer $k \\ge 1$. This follows from  $\\langle \\bx^1,\\bx^2 \\rangle$ being a \\emph{subgaussian} random variable with variance proxy $n$ (see Appendix~\\ref{app:subg} for details on this notion, and see Proposition~\\ref{prop:subg-mom} for the bound~\\eqref{eq:subg-mom}). Plugging this into~\\eqref{eq:tensor-L},\n\\[ \\|L_n^{\\le D}\\|^2 \\le 1 + \\sum_{d=1}^D \\frac{\\lambda^{2d}}{d!} (2n)^{pd\/2}pd\\,\\Gamma(pd\/2) =: 1 + \\sum_{d=1}^D T_d. \\]\nNote that $T_1 = O(1)$ provided $\\lambda = O(n^{-p\/4})$ (which will be implied by~\\eqref{eq:tensor-lam} below). Consider the ratio between successive terms:\n\\[ r_d := \\frac{T_{d+1}}{T_d} = \\frac{\\lambda^2}{d+1} (2n)^{p\/2} p \\,\\frac{\\Gamma(p(d+1)\/2)}{\\Gamma(pd\/2)}. \\]\nUsing the bound $\\Gamma(x+a)\/\\Gamma(x) \\le (x+a)^a$ for all $a, x > 0$ (see Proposition~\\ref{prop:gamma}), we find\n\\[ r_d \\le \\frac{\\lambda^2}{d+1} (2n)^{p\/2} p [p(d+1)\/2]^{p\/2} \\le \\lambda^2 p^{p\/2+1} n^{p\/2} (d+1)^{p\/2-1}. \\]\nThus if $\\lambda$ is small enough, namely if\n\\begin{equation}\\label{eq:tensor-lam}\n\\lambda \\le \\frac{1}{\\sqrt{2}}\\, p^{-p\/4-1\/2} n^{-p\/4} D^{(2-p)\/4},\n\\end{equation}\nthen $r_d \\le 1\/2$ for all $1 \\le d < D$.\nIn this case, by comparing with a geometric sum we may bound $\\|L_n^{\\le D}\\|^2 \\le 1 + 2 T_1 = O(1)$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Proof of Theorem~\\ref{thm:tensor-lowdeg}: Lower Bound}\n\\label{sec:tensor-lower}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:tensor-lowdeg}(ii)]\nNote that $\\langle \\bx^1,\\bx^2 \\rangle = \\sum_{i=1}^n s_i$ where $s_1, \\dots, s_n$ are i.i.d.\\ Rademacher random variables, so $\\EE_{\\bx^1,\\bx^2}[\\langle \\bx^1,\\bx^2 \\rangle^{2k + 1}] = 0$, and\n$$\\Ex_{\\bx^1,\\bx^2}[\\langle \\bx^1,\\bx^2 \\rangle^{2k}] = \\EE\\left[\\left(\\sum_{i=1}^n s_i\\right)^{2k}\\right] = \\sum_{i_1,i_2,\\ldots,i_{2k} \\in [n]} \\EE[s_{i_1} s_{i_2} \\cdots s_{i_{2k}}].$$\nBy counting only the terms $\\EE[s_{i_1} s_{i_2} \\cdots s_{i_{2k}}]$ in which each $s_i$ appears either 0 or 2 times, we have\n\\begin{equation}\\label{eq:mom-bound}\n\\Ex_{\\bx^1,\\bx^2} [\\langle \\bx^1,\\bx^2 \\rangle^{2k}] \\ge \\binom{n}{k}\\frac{(2k)!}{2^k}.\n\\end{equation}\nLet $d$ be the largest integer such that $d \\le D$ and $pd$ is even. By our assumption $D \\le \\frac{2}{p}n $, we then have $pd\/2 \\le n$.\nWe now bound $\\|L_n^{\\leq D}\\|^2$ by only the degree-$pd$ term of~\\eqref{eq:tensor-L}, and using the bounds $\\binom{n}{k} \\ge (n\/k)^k$ (for $1 \\le k \\le n$) and $(n\/e)^n \\le n! \\le n^n$, we can lower bound that term as follows:\n\\begin{align*}\n\\|L_n^{\\le D}\\|^2 &\\ge \\frac{\\lambda^{2d}}{d!} \\Ex_{\\bx^1,\\bx^2}[\\langle \\bx^1,\\bx^2 \\rangle^{pd}] \\\\\n&\\ge \\frac{\\lambda^{2d}}{d!} \\binom{n}{pd\/2} \\frac{(pd)!}{2^{pd\/2}} \\\\\n&\\ge \\frac{\\lambda^{2d}}{d^d} \\left(\\frac{2n}{pd}\\right)^{pd\/2} \\frac{(pd\/e)^{pd}}{2^{pd\/2}} \\\\\n&= \\left(\\lambda^2 e^{-p} p^{p\/2} n^{p\/2} d^{p\/2-1} \\right)^d.\n\\end{align*}\nNow, if $\\lambda$ is large enough, namely if\n$$\\lambda \\ge \\sqrt{2} e^{p\/2} p^{-p\/4} n^{-p\/4} D^{(2-p)\/4}$$\nand $D = \\omega(1)$, then $\\|L_n^{\\le D}\\|^2 \\ge (2-o(1))^d = \\omega(1)$.\n\\end{proof}\n\n\n\\subsection{The Spiked Wigner Matrix Model: Sharp Thresholds}\n\\label{sec:spiked-matrix}\n\nWe now turn our attention to a more precise understanding of the case $p = 2$ of the spiked tensor model, which is more commonly known as the \\emph{spiked Wigner matrix model}.\nOur results from the previous section (specialized to $p=2$) suggest that if $\\lambda \\gg n^{-1\/2}$ then there should be a polynomial-time distinguishing algorithm, whereas if $\\lambda \\ll n^{-1\/2}$ then there should not even be a subexponential-time distinguishing algorithm (that is, no algorithm of runtime $\\exp(n^{1-\\varepsilon})$ for any $\\varepsilon > 0$). In this section, we will give a more detailed low-degree analysis that identifies the precise value of $\\lambda \\sqrt{n}$ at which this change occurs. This type of sharp threshold has been observed in various high-dimensional inference problems; another notable example is the Kesten-Stigum transition for community detection in the stochastic block model \\cite{block-model-1,block-model-2,MNS-rec,massoulie,MNS-proof}. It was first demonstrated by \\cite{HS-bayesian} that the low-degree method can capture such sharp thresholds.\n\nTo begin, we recall the problem setup. Since the interesting regime is $\\lambda = \\Theta(n^{-1\/2})$, we define $\\hat\\lambda = \\lambda \\sqrt{2n}$ and take $\\hat\\lambda$ to be constant (not depending on $n$). With this notation, the spiked Wigner model is as follows:\n\\begin{itemize}\n    \\item Under $\\PP_n$, observe $\\bY = \\frac{\\hat\\lambda}{\\sqrt{2n}} \\bx\\bx^\\top + \\bZ$ where $\\bx \\in \\RR^n$ is drawn from $\\sX_n$.\n    \\item Under $\\QQ_n$, observe $\\bY = \\bZ$.\n\\end{itemize}\nHere $\\bZ$ is an $n \\times n$ random matrix with i.i.d.\\ entries distributed as $\\sN(0,1)$. (This asymmetric noise model is equivalent to the more standard symmetric one; see Appendix~\\ref{app:symm}.) We will consider various spike priors $\\sX_n$, but require the following normalization.\n\n\\begin{assumption}\\label{asm:spike-family}\nThe spike prior $(\\sX_n)_{n \\in \\NN}$ is normalized so that $\\bx \\sim \\sX_n$ satisfies $\\|\\bx\\|^2 \\to n$ in probability as $n \\to \\infty$.\n\\end{assumption}\n\n\n\\subsubsection{The Canonical Distinguishing Algorithm: PCA}\n\\label{sec:pca}\n\nThere is a simple reference algorithm for testing in the spiked Wigner model, namely \\emph{PCA (principal component analysis)}, by which we simply mean thresholding the largest eigenvalue of the (symmetrized) observation matrix.\n\n\\begin{definition}\n    The \\emph{PCA test} for distinguishing $\\PPP$ and $\\QQQ$ is the following statistical test, computable in polynomial time in $n$.\n    Let $\\overline{\\bY} \\colonequals (\\bY + \\bY^\\top) \/ \\sqrt{2n} = \\frac{\\hat\\lambda}{n} \\bx\\bx^\\top + \\bW$, where $\\bW = (\\bZ + \\bZ^\\top) \/ \\sqrt{2n}$ is a random matrix with the GOE distribution.\\footnote{Gaussian Orthogonal Ensemble (GOE): $\\bW$ is a symmetric $n \\times n$ matrix with entries $W_{ii} \\sim \\sN(0,2\/n)$ and $W_{ij} = W_{ji} \\sim \\sN(0,1\/n)$, independently.}\n    Then, let\n    \\begin{equation*}\n        f_{\\hat\\lambda}^{\\mathsf{PCA}}(\\bY) \\colonequals \\left\\{\\begin{array}{lcl} \\tp & : & \\lambda_{\\max}(\\overline{\\bY}) > t(\\hat\\lambda)  \\\\ \\tq & : & \\lambda_{\\max}(\\overline{\\bY}) \\leq t(\\hat\\lambda) \\end{array}\\right\\}\n    \\end{equation*}\n    where the threshold is set to  $t(\\hat\\lambda) \\colonequals 2 + (\\hat\\lambda + \\hat\\lambda^{-1} - 2) \/ 2$.\n\\end{definition}\n\\noindent\nThe theoretical underpinning of this test is the following seminal result from random matrix theory, the analogue for Wigner matrices of the celebrated ``BBP transition'' \\cite{BBP}.\n\\begin{theorem}[\\cite{FP-bbp,BGN}]\n    \\label{thm:wig-bbp}\n    Let $\\hat\\lambda$ be constant (not depending on $n$). Let $\\overline{\\bY} = \\frac{\\hat\\lambda}{n} \\bx\\bx^\\top + \\bW$ with $\\bW \\sim \\GOE(n)$ and arbitrary $\\bx \\in \\RR^n$ with $\\|\\bx\\|^2 = n$.\n    \\begin{itemize}\n        \\item If $\\hat\\lambda \\leq 1$, then $\\lambda_{\\max}(\\overline{\\bY}) \\to 2$ as $n \\to \\infty$ almost surely, and $\\la \\bv_{\\max}(\\overline{\\bY}), \\bx\/\\sqrt{n} \\ra^2 \\to 0$ almost surely (where $\\lambda_{\\max}$ denotes the largest eigenvalue and $\\bv_{\\max}$ denotes the corresponding unit-norm eigenvector).\n        \\item If $\\hat\\lambda > 1$, then $\\lambda_{\\max}(\\overline{\\bY}) \\to \\hat\\lambda + \\hat\\lambda^{-1} > 2$ as $n \\to \\infty$ almost surely, and $\\la \\bv_{\\max}(\\overline{\\bY}), \\bx\/\\sqrt{n} \\ra^2 \\to 1 - \\hat\\lambda^{-2}$ almost surely.\n    \\end{itemize}\n\\end{theorem}\n\n\\noindent Thus, the PCA test exhibits a sharp threshold: it succeeds when $\\hat\\lambda > 1$, and fails when $\\hat\\lambda \\le 1$. (Furthermore, the leading eigenvector achieves non-trivial estimation of the spike $\\bx$ when $\\hat\\lambda > 1$ and fails to do so when $\\hat\\lambda \\le 1$.)\n\n\\begin{corollary}\n    For any $\\hat\\lambda > 1$ and any spike prior family $(\\sX_n)_{n \\in \\NN}$ valid per Assumption~\\ref{asm:spike-family}, $f_{\\hat\\lambda}^{\\mathsf{PCA}}$ is a polynomial-time statistical test strongly distinguishing $\\PPP_{\\lambda}$ and $\\QQQ$.\n\\end{corollary}\n\n\\noindent For some spike priors $(\\sX_n)$, it is known that PCA is statistically optimal, in the sense that distinguishing (or estimating the spike) is information-theoretically impossible when $\\hat\\lambda < 1$.\nThese priors include the prior with $\\bx$ drawn uniformly from the sphere of radius $\\sqrt{n}$ and the priors with $\\bx$ having i.i.d.\\ $\\sN(0, 1)$ or Rademacher (uniformly $\\pm 1$) entries \\cite{MRZ-spectral,DAM,BMVVX-pca,PWBM-pca}. For these priors, we thus have $\\hat\\lambda_{\\mathsf{stat}} = \\hat\\lambda_{\\mathsf{comp}} = 1$, and there is no stat-comp gap.\n\nA different picture emerges for other spike priors, such as the \\emph{sparse Rademacher prior}\\footnote{In the sparse Rademacher prior, each entry of $\\bx$ is nonzero with probability $\\rho$ (independently), and the nonzero entries are drawn uniformly from $\\{\\pm 1\/\\sqrt{\\rho}\\}$.} with constant density $\\rho = \\Theta(1)$. If $\\rho$ is smaller than a particular small constant (roughly $0.09$ \\cite{mi-rank-one}), it is known that $\\hat\\lambda_{\\mathsf{stat}} < 1$. More precisely, an exponential-time exhaustive search algorithm succeeds in part of the regime where PCA fails \\cite{BMVVX-pca}. For any given $\\rho$, the precise threshold $\\hat\\lambda_{\\mathsf{stat}}$ can be computed using the \\emph{replica-symmetric formula} from statistical physics (see, e.g., \\cite{LKZ-mmse,LKZ-sparse,mi-rank-one,BDMKLZ-spiked,LM-symm,finite-size,replica-short,detection-wig,tensor-stat}, or \\cite{leo-survey} for a survey). However, for any constant $\\rho$, it is believed that $\\hat\\lambda_{\\mathsf{comp}} = 1$, i.e., that no \\emph{polynomial-time} algorithm can ``beat PCA.'' This has been conjectured in the same statistical physics literature based on failure of the \\emph{approximate message passing (AMP)} algorithm \\cite{LKZ-mmse,LKZ-sparse,mi-rank-one}.\n\nWe will next give a low-degree analysis corroborating that conjecture: for a large class of spike priors (including the sparse Rademacher prior with constant $\\rho$), our predictions suggest that any distinguishing algorithm requires nearly-exponential time whenever $\\hat\\lambda < 1$.\nOne interesting case we will \\emph{not} cover is that of the sparse Rademacher prior with $\\rho = o(1)$. In such models, there actually \\emph{are} subexponential-time algorithms that succeed for some $\\hat\\lambda < 1$; these are described in \\cite{subexp-sparse} along with matching lower bounds using the low-degree method. Furthermore, there are polynomial-time algorithms that can beat the PCA threshold once $\\rho \\lesssim 1\/\\sqrt{n}$; this is the much-studied ``sparse PCA'' regime (see, e.g., \\cite{JL04,JL09,AW-sparse,BR-sparse,KNV,DM-sparse,subexp-sparse}).\n\n\n\n\\subsubsection{Low-Degree Analysis: Informally, with the ``Gaussian Heuristic''}\n\\label{sec:spiked-wigner-gaussian-heuristic}\n\nWe first give a heuristic low-degree analysis of the spiked Wigner model which suggests $\\hat\\lambda_{\\mathsf{comp}} = 1$ (matching PCA) for sufficiently ``reasonable'' spike priors $(\\sX_n)$. In the next section, we will state and prove a rigorous statement to this effect.\n\nRecall from~\\eqref{eq:tensor-L} the expression for the norm of the LDLR:\n\\begin{equation*}\n\\|L_n^{\\le D}\\|^2 = \\sum_{d=0}^{D} \\frac{\\lambda^{2d}}{d!} \\Ex_{\\bx^1,\\bx^2} [\\langle \\bx^1,\\bx^2 \\rangle^{2d}] = \\sum_{d=0}^{D} \\frac{1}{d!}\\left(\\frac{\\hat\\lambda^2}{2n}\\right)^d \\Ex_{\\bx^1,\\bx^2} [\\langle \\bx^1,\\bx^2 \\rangle^{2d}].\n\\end{equation*}\n\n\\noindent To predict $\\hat\\lambda_{\\mathsf{comp}}$, it remains to determine whether $\\|L_n^{\\le D}\\|$ converges or diverges as $n \\to \\infty$, as a function of $\\hat\\lambda$.\nRecall that, per Assumption~\\ref{asm:spike-family}, $\\sX_n$ is normalized so that $\\|\\bx\\|^2 \\approx n$. Thus when, e.g., $\\bx$ has i.i.d.\\ entries, we may expect the following informal central limit theorem:\n\\begin{quote}\n    ``When $\\bx^1, \\bx^2 \\sim \\sX_n$ independently, $\\la \\bx^1, \\bx^2 \\ra$ is distributed approximately as $\\sN(0, n)$.''\n\\end{quote}\nAssuming this heuristic applies to the first $2D$ moments of $\\la \\bx^1, \\bx^2 \\ra$, and recalling that the Gaussian moments are $\\EE_{g \\sim \\sN(0,1)} [g^{2k}] = (2k-1)!! = \\prod_{i = 1}^k (2i - 1)$, we may estimate\n\\begin{equation*}\n    \\|L_n^{\\leq D}\\|^2 \\approx \\sum_{d=0}^{D} \\frac{1}{d!}\\left(\\frac{\\hat\\lambda^2}{2n}\\right)^d \\Ex_{g \\sim \\sN(0, n)} [g^{2d}] = \\sum_{d=0}^{D} \\frac{1}{d!}\\left(\\frac{\\hat\\lambda^2}{2n}\\right)^d n^d(2d - 1)!! =: \\sum_{d=0}^D T_d.\n\\end{equation*}\nImagine $D$ grows slowly with $n$ (e.g., $D \\approx \\log n$) in order to predict the power of polynomial-time algorithms. The ratio of consecutive terms above is $$\\frac{T_{d+1}}{T_d} = \\hat\\lambda^2\\cdot \\frac{2d+1}{2(d+1)} \\approx \\hat\\lambda^2,$$\nsuggesting that $\\|L_n^{\\le D}\\|$ should diverge if $\\hat\\lambda > 1$ and converge if $\\hat\\lambda < 1$.\n\nWhile this style of heuristic analysis is often helpful for guessing the correct threshold, this type of reasoning can break down if $D$ is too large or if $\\bx$ is too sparse. In the next section, we therefore give a rigorous analysis of $\\|L_n^{\\le D}\\|$.\n\n\n\n\\subsubsection{Low-Degree Analysis: Formally, with Concentration Inequalities}\n\\label{sec:wig-bound-L}\n\nWe now give a rigorous proof that $\\|L_n^{\\leq D}\\| = O(1)$ when $\\hat\\lambda < 1$ (and $\\|L_n^{\\leq D}\\| = \\omega(1)$ when $\\hat\\lambda > 1$), provided the spike prior is ``nice enough.'' Specifically, we require the following condition on the prior.\n\n\\begin{definition}\\label{def:local-chernoff}\nA spike prior $(\\sX_n)_{n \\in \\NN}$ admits a \\emph{local Chernoff bound} if for any $\\eta > 0$ there exist $\\delta > 0$ and $C > 0$ such that for all $n$,\n\\begin{equation*}\n\\Pr\\left\\{|\\la \\bx^1,\\bx^2 \\ra| \\ge t\\right\\} \\le C \\exp\\left(-\\frac{1}{2n}(1-\\eta)t^2\\right) \\quad \\text{for all } t \\in [0,\\delta n]\n\\end{equation*}\nwhere $\\bx^1,\\bx^2$ are drawn independently from $\\sX_n$.\n\\end{definition}\n\\noindent For instance, any prior with i.i.d.\\ \\emph{subgaussian} entries admits a local Chernoff bound; see Proposition~\\ref{prop:local-chernoff} in Appendix~\\ref{app:subg}. This includes, for instance, the sparse Rademacher prior with any constant density $\\rho$. The following is the main result of this section, which predicts that for this class of spike priors, any algorithm that beats the PCA threshold requires nearly-exponential time.\n\n\\begin{theorem}\\label{thm:wig-bound-L}\nSuppose $(\\sX_n)_{n \\in \\NN}$ is a spike prior that (i) admits a local Chernoff bound, and (ii) has then $\\|\\bx\\|^2 \\le \\sqrt{2}n$ almost surely if $\\bx \\sim \\sX_n$. Then, for the spiked Wigner model with $\\hat\\lambda < 1$ and any $D = D(n) = o(n\/\\log n)$, we have $\\|L_n^{\\leq D}\\| = O(1)$ as $n \\to \\infty$.\n\\end{theorem}\n\n\\begin{remark}\nThe upper bound $\\|\\bx\\|^2 \\le \\sqrt{2}n$ is without loss of generality (provided $\\|\\bx\\|^2 \\to n$ in probability). This is because we can define a modified prior $\\tilde \\sX_n$ that draws $\\bx \\sim \\sX_n$ and outputs $\\bx$ if $\\|\\bx\\|^2 \\le \\sqrt{2}n$ and $\\bm 0$ otherwise. If $(\\sX_n)_{n \\in \\NN}$ admits a local Chernoff bound then so does $(\\tilde \\sX_n)_{n \\in \\NN}$. And, if the spiked Wigner model is computationally hard with the prior $(\\tilde \\sX_n)_{n \\in \\NN}$, it is also hard with the prior $(\\sX_n)_{n \\in \\NN}$, since the two differ with probability $o(1)$.\n\\end{remark}\n\nThough we already know that a polynomial-time algorithm (namely PCA) exists when $\\hat\\lambda > 1$, we can check that indeed $\\|L_n^{\\le D}\\| = \\omega(1)$ in this regime. For the sake of simplicity, we restrict this result to the Rademacher prior.\n\\begin{theorem}\\label{thm:wig-above}\nConsider the spiked Wigner model with the Rademacher prior: $\\bx$ has i.i.d.\\ entries $x_i \\sim \\Unif(\\{\\pm 1\\})$. If $\\hat\\lambda > 1$, then for any $D = \\omega(1)$ we have $\\|L_n^{\\le D}\\| = \\omega(1)$.\n\\end{theorem}\n\\noindent The proof is a simple modification of the proof of Theorem~\\ref{thm:tensor-lowdeg}(ii) in Section~\\ref{sec:tensor-lower}; we defer it to Appendix~\\ref{app:wig-above}. The remainder of this section is devoted to proving Theorem~\\ref{thm:wig-bound-L}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:wig-bound-L}]\nStarting from the expression for $\\|L_n^{\\le D}\\|^2$ (see Theorem~\\ref{thm:agn-ldlr-norm} and Remark~\\ref{rem:taylor}), we split $\\|L_n^{\\leq D}\\|^2$ into two terms, as follows:\n$$\\|L_n^{\\leq D}\\|^2 = \\Ex_{\\bx^1, \\bx^2}\\left[\\exp^{\\leq D}\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right)\\right]\n=: R_1 + R_2,$$\nwhere\n\\begin{align*}\nR_1 &\\colonequals \\Ex_{\\bx^1, \\bx^2}\\left[\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\exp^{\\leq D}\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right)\\right], \\\\\nR_2 &\\colonequals \\Ex_{\\bx^1, \\bx^2}\\left[\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| > \\varepsilon n}\\,\\exp^{\\leq D}\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right)\\right].\n\\end{align*}\nHere $\\varepsilon > 0$ is a small constant to be chosen later. We call $R_1$ the \\emph{small deviations} and $R_2$ the \\emph{large deviations}, and we will bound these two terms separately.\n\\paragraph{Bounding the large deviations.}\nUsing that $\\|\\bx\\|^2 \\le \\sqrt{2}n$, that $\\exp^{\\le D}(t)$ is increasing for $t \\ge 0$, and the local Chernoff bound (taking $\\varepsilon$ to be a sufficiently small constant),\n\\begin{align*}\nR_2 &\\le \\Pr\\left\\{|\\langle \\bx^1,\\bx^2 \\rangle| > \\varepsilon n\\right\\} \\exp^{\\leq D}\\left(2 \\lambda^2 n^2\\right)\\\\\n&\\le C \\exp\\left(-\\frac{1}{3}\\varepsilon^2 n\\right) \\sum_{d=0}^D \\frac{(\\hat\\lambda^2 n)^d}{d!}\n\\intertext{and noting that the last term of the sum is the largest since $\\hat\\lambda^2 n > D$,}\n&\\le C \\exp\\left(-\\frac{1}{3}\\varepsilon^2 n\\right) (D+1) \\frac{(\\hat\\lambda^2 n)^D}{D!}\\\\\n&= \\exp\\left[\\log C - \\frac{1}{3} \\varepsilon^2 n + \\log(D+1) + 2D \\log \\hat\\lambda + D \\log n - \\log(D!)\\right]\\\\\n&= o(1)\n\\end{align*}\nprovided $D = o(n\/\\log n)$.\n\n\\paragraph{Bounding the small deviations.}\n\nWe adapt an argument from \\cite{PWB-tensor}. Here we do not need to make use of the truncation to degree $D$ at all, and instead simply use the bound $\\exp^{\\le D}(t) \\le \\exp(t)$ for $t \\ge 0$.\nWith this, we bound\n\\begin{align*}\nR_1 &= \\Ex_{\\bx^1, \\bx^2}\\left[\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\exp^{\\leq D}\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right)\\right]\\\\\n&\\le \\Ex_{\\bx^1, \\bx^2}\\left[\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\exp\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right)\\right]\\\\\n&= \\int_0^\\infty \\Pr\\left\\{\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\exp\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right) \\ge u\\right\\} du\\\\\n&= 1 + \\int_1^\\infty \\Pr\\left\\{\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\exp\\left(\\lambda^2\\la \\bx^1, \\bx^2 \\ra^2\\right) \\ge u\\right\\} du\\\\\n&= 1 + \\int_0^\\infty \\Pr\\left\\{\\One_{|\\langle \\bx^1,\\bx^2 \\rangle| \\le \\varepsilon n}\\,\\langle \\bx^1,\\bx^2 \\rangle^2 \\ge t\\right\\} \\lambda^2\\exp\\left(\\lambda^2 t\\right) dt \\tag{where $\\exp\\left(\\lambda^2 t\\right) = u$}\\\\\n&\\le 1 + \\int_0^\\infty C \\exp\\left(-\\frac{1}{2n}(1-\\eta)t\\right)  \\lambda^2\\exp\\left(\\lambda^2 t\\right) dt  \\tag{using the local Chernoff bound}\\\\\n&\\le 1 + \\frac{C \\hat\\lambda^2}{2n} \\int_0^\\infty \\exp\\left(-\\frac{1}{2n}(1 - \\eta - \\hat\\lambda^2)t\\right) dt\\\\\n&= 1 + C \\hat\\lambda^2 (1 - \\eta - \\hat\\lambda^2)^{-1} \\tag{provided $\\hat\\lambda^2 < 1-\\eta$}\\\\\n&= O(1).\n\\end{align*}\nSince $\\hat\\lambda < 1$, we can choose $\\eta > 0$ small enough so that $\\hat\\lambda^2 < 1 - \\eta$, and then choose $\\varepsilon$ small enough so that the local Chernoff bound holds. (Here, $\\eta$ and $\\varepsilon$ depend on $\\hat\\lambda$ and the spike prior, but not on $n$.)\n\\end{proof}\n\n\n\n\\section{More on the Low-Degree Method}\n\\label{sec:ldlr-conj-2}\n\nIn this section, we return to the general considerations introduced in Section~\\ref{sec:ldlr-conj} and describe some of the nuances in and evidence for the main conjecture underlying the low-degree method (Conjecture~\\ref{conj:low-deg-informal}). Specifically, we investigate the question of what can be concluded (both rigorously and conjecturally) from the behavior of the low-degree likelihood ratio (LDLR) defined in Definition~\\ref{def:LDLR}. We present conjectures and formal evidence connecting the LDLR to computational complexity, discussing various caveats and counterexamples along the way. \n\nIn Section~\\ref{sec:LDLR-poly}, we explore to what extent the $D$-LDLR controls whether or not degree-$D$ polynomials can distinguish $\\PPP$ from $\\QQQ$. Then, in Section~\\ref{sec:LDLR-alg}, we explore to what extent the LDLR controls whether or not \\emph{any} efficient algorithm can distinguish $\\PPP$ and $\\QQQ$.\n\n\n\\subsection{The LDLR and Thresholding Polynomials}\n\\label{sec:LDLR-poly}\n\nHeuristically, since $\\|L_n^{\\le D}\\|$ is the value of the $L^2$ optimization problem \\eqref{eq:l2-opt-low}, we might expect the behavior of $\\|L_n^{\\le D}\\|$ as $n \\to \\infty$ to dictate whether or not degree-$D$ polynomials can distinguish $\\PPP$ from $\\QQQ$: it should be possible to strongly distinguish (in the sense of Definition~\\ref{def:stat-ind}, i.e., with error probabilities tending to $0$) $\\PPP$ from $\\QQQ$ by \\emph{thresholding} a degree-$D$ polynomial (namely $L_n^{\\le D}$) if and only if $\\|L_n^{\\le D}\\| = \\omega(1)$. We now discuss to what extent this heuristic is correct.\n\n\\begin{question}\\label{q:thresh-poly-yes}\nIf $\\|L_n^{\\le D}\\| = \\omega(1)$, does this imply that it is possible to strongly distinguish $\\PPP$ and $\\QQQ$ by thresholding a degree-$D$ polynomial?\n\\end{question}\n\n\\noindent We have already mentioned (see Section~\\ref{sec:contig}) a counterexample when $D = \\infty$: there are cases where $\\PPP$ and $\\QQQ$ are not statistically distinguishable, yet $\\|L_n\\| \\to \\infty$ due to a rare ``bad'' event under $\\PP_n$. Examples of this phenomenon are fairly common (e.g., \\cite{BMNN-community,BMVVX-pca,PWBM-pca,PWB-tensor}). However, after truncation to only low-degree components, this issue seems to disappear. For instance, in sparse PCA, $\\|L_n^{\\le D}\\| \\to \\infty$ only occurs when either (i) there actually is an $n^{\\tilde{O}(D)}$-time distinguishing algorithm, or (ii) $D$ is ``unreasonably large,'' in the sense that there is a trivial $n^{t(n)}$-time exhaustive search algorithm and $D \\gg t(n)$ \\cite{subexp-sparse}. Indeed, we do not know any example of a natural problem where $\\|L_n^{\\le D}\\|$ diverges spuriously for a ``reasonable'' value of $D$ (in the above sense), although one can construct unnatural examples by introducing a rare ``bad'' event in $\\PP_n$. Thus, it seems that for natural problems and reasonable growth of $D$, the smoothness of low-degree polynomials regularizes $L_n^{\\le D}$ in such a way that the answer to Question~\\ref{q:thresh-poly-yes} is typically ``yes.'' This convenient feature is perhaps related to the probabilistic phenomenon of \\emph{hypercontractivity}; see Appendix~\\ref{app:hyp} and especially Remark~\\ref{rem:low-dom}.\n\nAnother counterexample to Question~\\ref{q:thresh-poly-yes} is the following. Take $\\PPP$ and $\\QQQ$ that are ``easy'' for degree-$D$ polynomials to distinguish, i.e., $\\|L_n^{\\le D}\\| \\to \\infty$ and $\\PPP, \\QQQ$ can be strongly distinguished by thresholding a degree-$D$ polynomial. Define a new sequence of ``diluted'' planted measures $\\PPP^\\prime$ where $\\PP^\\prime_n$ samples from $\\PP_n$ with probability $1\/2$, and otherwise samples from $\\QQ_n$. Letting $L'_n = d\\PP'_n\/d\\QQ_n$, we have $\\|(L'_n)^{\\le D}\\| \\to \\infty$, yet $\\PPP'$ and $\\QQQ$ cannot be strongly distinguished (even statistically). While this example is perhaps somewhat unnatural, it illustrates that a rigorous positive answer to Question~\\ref{q:thresh-poly-yes} would need to restrict to $\\PPP$ that are ``homogeneous'' in some sense.\n\nThus, while we have seen some artificial counterexamples, the answer to Question~\\ref{q:thresh-poly-yes} seems to typically be ``yes'' for natural high-dimensional problems, so long as $D$ is not unreasonably large. We now turn to the converse question.\n\n\\begin{question}\\label{q:thresh-poly-no}\nIf $\\|L_n^{\\le D}\\| = O(1)$, does this imply that it is impossible to strongly distinguish $\\PPP$ and $\\QQQ$ by thresholding a degree-$D$ polynomial?\n\\end{question}\n\n\\noindent Here, we are able to give a positive answer in a particular formal sense.\nThe following result addresses the contrapositive of Question~\\ref{q:thresh-poly-no}: it shows that distinguishability by thresholding low-degree polynomials implies exponential growth of the norm of the LDLR.\n\n\\begin{theorem}\\label{thm:thresh-poly-hard}\nSuppose $\\QQ$ draws $\\bY \\in \\RR^N$ with entries either i.i.d.\\ $\\sN(0,1)$ or i.i.d.\\ $\\Unif(\\{\\pm 1\\})$. Let $\\PP$ be any measure on $\\RR^N$ that is absolutely continuous with respect to $\\QQ$. Let $f: \\RR^N \\to \\RR$ be a polynomial of degree $\\le d$ satisfying\n\\begin{equation}\\label{eq:strong-thresh}\n\\Ex_{\\bY \\sim \\PP}[f(\\bY)] \\ge A \\quad \\text{and} \\quad \\QQ(|f(\\bY)| \\ge B) \\le \\delta\n\\end{equation}\nfor some $A > B > 0$ and some $\\delta \\le \\frac{1}{2} \\cdot 3^{-4kd}$. Then for any $k \\in \\NN$,\n$$\\|L^{\\le 2kd}\\| \\ge \\frac{1}{2}\\left(\\frac{A}{B}\\right)^{2k}.$$\n\\end{theorem}\n\n\\noindent We defer the proof to Appendix~\\ref{app:thresh-poly-hard}. To understand what the result shows, imagine, for example, that $A > B$ are both constants, and $k$ grows slowly with $n$ (e.g., $k \\approx \\log n$). Then, if a degree-$d$ polynomial (where $d$ may depend on $n$) can distinguish $\\PPP$ from $\\QQQ$ in the sense of \\eqref{eq:strong-thresh} with $\\delta = \\frac{1}{2} \\cdot 3^{-4kd}$, then $\\|L_n^{\\le 2kd}\\| \\to \\infty$ as $n \\to \\infty$. Note though, that one weakness of this result is that we require the explicit quantitative bound $\\delta \\leq \\frac{1}{2} \\cdot 3^{-4kd}$, rather than merely $\\delta = o(1)$.\n\nThe proof (see Appendix~\\ref{app:thresh-poly-hard}) is a straightforward application of \\emph{hypercontractivity} (see e.g., \\cite{boolean-book}), a type of result stating that random variables obtained by evaluating low-degree polynomials on weakly-dependent distributions (such as i.i.d.\\ ones) are well-concentrated and otherwise ``reasonable.''\nWe have restricted to the case where $\\QQ$ is i.i.d.\\ Gaussian or Rademacher because hypercontractivity results are most readily available in these cases, but we expect similar results to hold more generally.\n\n\n\\subsection{Algorithmic Implications of the LDLR}\n\\label{sec:LDLR-alg}\n\nHaving discussed the relationship between the LDLR and low-degree polynomials, we now discuss the relationship between low-degree polynomials and the power of \\emph{any} computationally-bounded algorithm.\n\nAny degree-$D$ polynomial has at most $n^D$ monomial terms and so can be evaluated in time $n^{O(D)}$ (assuming that the individual coefficients are easy to compute). However, certain degree-$D$ polynomials can of course be computed faster, e.g., if the polynomial has few nonzero monomials or has special structure allowing it to be computed via a spectral method (as in the \\emph{color coding} trick \\cite{color-coding} used by~\\cite{HS-bayesian}).\nDespite such special cases, it appears that for average-case high-dimensional hypothesis testing problems, degree-$D$ polynomials are typically as powerful as general $n^{\\tilde{\\Theta}(D)}$-time algorithms; this informal conjecture appears as Hypothesis~2.1.5 in \\cite{sam-thesis}, building on the work of \\cite{pcal,HS-bayesian,sos-hidden} (see also our previous discussion in Section~\\ref{sec:ldlr-conj}). We will now explain the nuances and caveats of this conjecture, and give evidence (both formal and heuristic) in its favor.\n\n\\subsubsection{Robustness}\n\\label{sec:robust}\n\nAn important counterexample that we must be careful about is XOR-SAT. In the random 3-XOR-SAT problem, there are $n$ $\\{\\pm 1 \\}$-valued variables $x_1,\\ldots,x_n$ and we are given a formula consisting of $m$ random constraints of the form $x_{i_\\ell} x_{j_\\ell} x_{k_\\ell} = b_\\ell$ for $\\ell \\in [m]$, with $b_\\ell \\in \\{\\pm 1\\}$. The goal is to determine whether there is an assignment $x \\in \\{\\pm 1\\}^n$ that satisfies all the constraints. Regardless of $m$, this problem can be solved in polynomial time using Gaussian elimination over the finite field $\\mathbb{F}_2$. However, when $n \\ll m \\ll n^{3\/2}$, the low-degree method nevertheless predicts that the problem should be computationally hard, i.e., it is hard to distinguish between a random formula (which is unsatisfiable with high probability) and a formula with a planted assignment. This pitfall is not specific to the low-degree method: sum-of-squares lower bounds, statistical query lower bounds, and the cavity method from statistical physics also incorrectly suggest the same (this is discussed in \\cite{sq-parity,stat-phys-survey,sos-notes}).\n\nThe above discrepancy can be addressed (see, e.g., Lecture~3.2 of \\cite{sos-notes}) by noting that Gaussian elimination is very brittle, in the sense that it no longer works to search for an assignment satisfying only a $1-\\delta$ fraction of the constraints (as in this case it does not seem possible to leverage the problem's algebraic structure over $\\mathbb{F}_2$). Another example of a brittle algorithm is the algorithm of \\cite{reg-LLL} for linear regression, which uses Lenstra-Lenstra-Lov\\'asz lattice basis reduction \\cite{LLL} and only tolerates an exponentially-small level of noise. Thus, while there sometimes exist efficient algorithms that are ``high-degree'', these tend not to be robust to even a tiny amount of noise. As with SoS lower bounds, we expect that the low-degree method correctly captures the limits of \\emph{robust} hypothesis testing \\cite{sos-hidden} for high-dimensional problems. (Here, ``robust'' refers to the ability to handle a small amount of noise, and should not be confused with the specific notion of \\emph{robust inference} \\cite{sos-hidden} or with other notions of robustness that allow adversarial corruptions \\cite{FK-semirandom,robust-est}.)\n\n\\subsubsection{Connection to Sum-of-Squares}\n\\label{sec:sos}\n\nThe sum-of-squares (SoS) hierarchy \\cite{parrilo-thesis,lasserre} is a hierarchy of increasingly powerful semidefinite programming (SDP) relaxations for general polynomial optimization problems. Higher levels of the hierarchy produce larger SDPs and thus require more time to solve: level $d$ typically requires time $n^{O(d)}$. SoS lower bounds show that certain levels of the hierarchy fail to solve a given problem. As SoS seems to be at least as powerful as all known algorithms for many problems, SoS lower bounds are often thought of as the ``gold standard'' of formal evidence for computational hardness of average-case problems. For instance, if any constant level $d$ of SoS fails to solve a problem, this is strong evidence that no polynomial-time algorithm exists to solve the same problem (modulo the robustness issue discussed above). \n\nIn order to prove SoS lower bounds, one needs to construct a valid primal certificate (also called a \\emph{pseudo-expectation}) for the SoS SDP. The \\emph{pseudo-calibration} approach \\cite{pcal} provides a strategy for systematically constructing a pseudo-expectation; however, showing that the resulting object is valid (in particular, showing that a certain associated matrix is positive semidefinite) often requires substantial work. As a result, proving lower bounds against constant-level SoS programs is often very technically challenging (as in~\\cite{pcal,sos-hidden}). We refer the reader to~\\cite{sos-survey} for a survey of SoS and pseudo-calibration in the context of high-dimensional inference.\n\nOn the other hand, it was observed by the authors of \\cite{pcal,sos-hidden} that the bottleneck for the success of the pseudo-calibration approach seems to typically be a simple condition, none other than the boundedness of the norm of the LDLR (see Conjecture~3.5 of \\cite{sos-survey} or Section~4.3 of \\cite{sam-thesis}).\\footnote{More specifically, $(\\|L_n^{\\le D}\\|^2 - 1)$ is the variance of a certain pseudo-expectation value generated by pseudo-calibration, whose actual value in a valid pseudo-expectation must be exactly 1. It appears to be impossible to ``correct'' this part of the pseudo-expectation if the variance is diverging with $n$.} Through a series of works \\cite{pcal,HS-bayesian,sos-hidden,sam-thesis}, the low-degree method emerged from investigating this simpler condition in its own right. It was shown in \\cite{HS-bayesian} that SoS can be used to achieve sharp computational thresholds (such as the Kesten--Stigum threshold for community detection), and that the success of the associated method also hinges on the boundedness of the norm of the LDLR. So, historically speaking, the low-degree method can be thought of as a ``lite'' version of SoS lower bounds that is believed to capture the essence of what makes SoS succeed or fail (see~\\cite{HS-bayesian,sos-hidden,sos-survey,sam-thesis}).\n\nA key advantage of the low-degree method over traditional SoS lower bounds is that it greatly simplifies the technical work required, allowing sharper results to be proved with greater ease. Moreover, the low-degree method is arguably more natural in the sense that it is not specific to any particular SDP formulation and instead seems to capture the essence of what makes problems computationally easy or hard. On the other hand, some would perhaps argue that SoS lower bounds constitute stronger evidence for hardness than low-degree lower bounds (although we do not know any average-case problems for which they give different predictions).\n\nWe refer the reader to~\\cite{sam-thesis} for more on the relation between SoS and the low-degree method, including evidence for why the two methods are believed to predict the same results.\n\n\n\n\\subsubsection{Connection to Spectral Methods}\n\\label{sec:spectral}\n\nFor high-dimensional hypothesis testing problems, a popular class of algorithms are the \\emph{spectral methods}, algorithms that build a matrix $\\bM$ using the data and then threshold its largest eigenvalue. (There are also spectral methods for estimation problems, usually extracting an estimate of the signal from the leading eigenvector of $\\bM$.) Often, spectral methods match the best\\footnote{Here, ``best'' is in the sense of strongly distinguishing $\\PP_n$ and $\\QQ_n$ throughout the largest possible regime of model parameters.} performance among all known polynomial-time algorithms. Some examples include the non-backtracking and Bethe Hessian spectral methods for the stochastic block model \\cite{spectral-redemption,massoulie,nb-spectrum,bethe-hessian}, the covariance thresholding method for sparse PCA \\cite{DM-sparse}, and the tensor unfolding method for tensor PCA \\cite{RM-tensor,HSS-tensor}. As demonstrated in \\cite{HSS-tensor,sos-fast}, it is often possible to design spectral methods that achieve the same performance as SoS; in fact, some formal evidence indicates that low-degree spectral methods (where each matrix entry is a constant-degree polynomial of the data) are as powerful as any constant-degree SoS relaxation \\cite{sos-hidden}\\footnote{In \\cite{sos-hidden}, it is shown that for a fairly general class of average-case hypothesis testing problems, if SoS succeeds in some range of parameters then there is a low-degree spectral method whose maximum \\emph{positive} eigenvalue succeeds (in a somewhat weaker range of parameters). However, the resulting matrix could \\emph{a priori} have an arbitrarily large (in magnitude) negative eigenvalue, which would prevent the spectral method from running in polynomial time. For this same reason, it seems difficult to establish a formal connection between SoS and the LDLR via spectral methods.}.\n\nAs a result, it is interesting to try to prove lower bounds against the class of spectral methods. Roughly speaking, the largest eigenvalue in absolute value of a polynomial-size matrix $\\bM$ can be computed using $O(\\log n)$ rounds of power iteration, and thus can be thought of as an $O(\\log n)$-degree polynomial; more specifically, the associated polynomial is $\\Tr(\\bM^{2k})$ where $k \\sim \\log(n)$. The following result makes this precise, giving a formal connection between the low-degree method and the power of spectral methods. The proof is given in Appendix~\\ref{app:spectral-hard} (and is similar to that of Theorem~\\ref{thm:thresh-poly-hard}).\n\n\\begin{theorem}\\label{thm:spectral-hard}\nSuppose $\\QQ$ draws $\\bY \\in \\RR^N$ with entries either i.i.d.\\ $\\sN(0,1)$ or i.i.d.\\ $\\Unif(\\{\\pm 1\\})$. Let $\\PP$ be any measure on $\\RR^N$ that is absolutely continuous with respect to $\\QQ$. Let $\\bM = \\bM(\\bY)$ be a real symmetric $L \\times L$ matrix, each of whose entries is a polynomial in $\\bY$ of degree $\\le d$. Suppose\n\\begin{equation}\\label{eq:strong-thresh-spec}\n\\Ex_{\\bY \\sim \\PP}\\|\\bM\\| \\ge A \\quad \\text{and} \\quad \\QQ(\\|\\bM\\| \\ge B) \\le \\delta\n\\end{equation}\n(where $\\|\\cdot\\|$ denotes matrix operator norm) for some $A > B > 0$ and some $\\delta \\le \\frac{1}{2} \\cdot 3^{-4kd}$. Then, for any $k \\in \\NN$,\n$$\\|L^{\\le 2kd}\\| \\ge \\frac{1}{2L}\\left(\\frac{A}{B}\\right)^{2k}.$$\n\\end{theorem}\n\\noindent For example, suppose we are interested in polynomial-time spectral methods, in which case we should consider $L = \\mathrm{poly}(n)$ and $d = O(1)$. If there exists a spectral method with these parameters that distinguishes $\\PPP$ from $\\QQQ$ in the sense of \\eqref{eq:strong-thresh-spec} for some constants $A > B$, and with $\\delta \\to 0$ faster than any inverse polynomial (in $n$), then there exists a choice of $k = O(\\log n)$ such that $\\|L^{\\le O(\\log n)}\\| = \\omega(1)$. And, by contrapositive, if we could show that $\\|L^{\\le D}\\| = O(1)$ for some $D = \\omega(\\log n)$, that would imply that there is no spectral method with the above properties. This justifies the choice of logarithmic degree in Conjecture~\\ref{conj:low-deg-informal}. Similarly to Theorem~\\ref{thm:thresh-poly-hard}, one weakness of Theorem~\\ref{thm:spectral-hard} is that we can only rule out spectral methods whose failure probability is smaller than any inverse polynomial, instead of merely $o(1)$.\n\n\\begin{remark}\nAbove, we have argued that polynomial-time spectral methods correspond to polynomials of degree roughly $\\log(n)$. What if we are instead interested in subexponential runtime $\\exp(n^\\delta)$ for some constant $\\delta \\in (0,1)$? One class of spectral method computable with this runtime is that where the dimension is $L \\approx \\exp(n^\\delta)$ and the degree of each entry is $d \\approx n^\\delta$ (such spectral methods often arise based on SoS \\cite{strongly-refuting,BGG,BGL}). To rule out such a spectral method using Theorem~\\ref{thm:spectral-hard}, we would need to take $k \\approx \\log(L) \\approx n^\\delta$ and would need to show $\\|L^{\\le D}\\| = O(1)$ for $D \\approx n^{2\\delta}$. However, Conjecture~\\ref{conj:low-deg-informal} postulates that time-$\\exp(n^\\delta)$ algorithms should instead correspond to degree-$n^\\delta$ polynomials, and this correspondence indeed appears to be the correct one based on the examples of tensor PCA (see Section~\\ref{sec:spiked-tensor}) and sparse PCA (see \\cite{subexp-sparse}).\n\nAlthough this seems at first to be a substantial discrepancy, there is evidence that there are actually spectral methods of dimension $L \\approx \\exp(n^\\delta)$ and \\emph{constant} degree $d = O(1)$ that achieve optimal performance among $\\exp(n^\\delta)$-time algorithms. Such a spectral method corresponds to a degree-$n^\\delta$ polynomial, as expected. These types of spectral methods have been shown to exist for tensor PCA \\cite{kikuchi}.\n\\end{remark}\n\n\n\n\\subsubsection{Formal Conjecture}\n\\label{sec:discuss-conj}\n\nWe next discuss the precise conjecture that Hopkins \\cite{sam-thesis} offers on the algorithmic implications of the low-degree method. Informally, the conjecture is that for ``sufficiently nice'' $\\PPP$ and $\\QQQ$, if $\\|L_n^{\\le D}\\| = O(1)$ for some $D \\ge (\\log n)^{1+\\varepsilon}$, then there is no polynomial-time algorithm that strongly distinguishes $\\PPP$ and $\\QQQ$. We will not state the full conjecture here (see Conjecture~2.2.4 in \\cite{sam-thesis}) but we will briefly discuss some of the details that we have not mentioned yet.\n\nLet us first comment on the meaning of ``sufficiently nice'' distributions. Roughly speaking, this means that: \n\\begin{enumerate}\n    \\item $\\QQ_n$ is a product distribution,\n    \\item $\\PP_n$ is sufficiently symmetric with respect to permutations of its coordinates, and\n    \\item $\\PP_n$ is then perturbed by a small amount of additional noise.\n\\end{enumerate}\nConditions (1) and (2) or minor variants thereof are fairly standard in high-dimensional inference problems. The reason for including (3) is to rule out non-robust algorithms such as Gaussian elimination (see Section~\\ref{sec:robust}).\n\nOne difference between the conjecture of \\cite{sam-thesis} and the conjecture discussed in these notes is that \\cite{sam-thesis} considers the notion of \\emph{coordinate degree} rather than \\emph{polynomial degree}. A polynomial has coordinate degree $\\le D$ if no monomial involves more than $D$ variables; however, each individual variable can appear with arbitrarily-high degree in a monomial.\\footnote{Indeed, coordinate degree need not be phrased in terms of polynomials, and one may equivalently consider the linear subspace of $L^2(\\QQ_n)$ of functions that is spanned by functions of at most $D$ variables at a time.} In \\cite{sam-thesis}, the low-degree likelihood ratio is defined as the projection of $L_n$ onto the space of polynomials of coordinate degree~$\\le D$. The reason for this is to capture, e.g., algorithms that preprocess the data by applying a complicated high-degree function entrywise. However, we are not aware of any natural problem in which it is important to work with coordinate degree instead of polynomial degree. While working with coordinate degree gives lower bounds that are formally stronger, we work with polynomial degree throughout these notes because it simplifies many of the computations.\n\n\n\\subsubsection{Empirical Evidence and Refined Conjecture}\n\nPerhaps the strongest form of evidence that we have in favor of the low-degree method is simply that it has been carried out on many high-dimensional inference problems and seems to always give the correct predictions, coinciding with widely-believed conjectures. These problems include planted clique \\cite{sam-thesis} (implicit in \\cite{pcal}), community detection in the stochastic block model \\cite{HS-bayesian,sam-thesis}, the spiked tensor model \\cite{sos-hidden,sam-thesis}, the spiked Wishart model \\cite{BKW-sk}, and sparse PCA \\cite{subexp-sparse}. In these notes we have also carried out low-degree calculations for the spiked Wigner model and spiked tensor model (see Section~\\ref{sec:examples}). Some of the early results \\cite{HS-bayesian,sos-hidden} showed only $\\|L_n^{\\le D}\\| = n^{o(1)}$ as evidence for hardness, which was later improved to $O(1)$ \\cite{sam-thesis}. Some of the above results \\cite{sos-hidden,sam-thesis} use coordinate degree instead of degree (as we discussed in Section~\\ref{sec:discuss-conj}). Throughout the above examples, the low-degree method has proven to be versatile in that it can predict both sharp threshold behavior as well as precise smooth tradeoffs between subexponential runtime and statistical power (as illustrated in the two parts of Section~\\ref{sec:examples}).\n\nAs discussed earlier, there are various reasons to believe that if $\\|L_n^{\\le D}\\| = O(1)$ for some $D = \\omega(\\log n)$ then there is no polynomial-time distinguishing algorithm; for instance, this allows us to rule out a general class of spectral methods (see Theorem~\\ref{thm:spectral-hard}). However, we have observed that in numerous examples, the LDLR actually has the following more precise behavior that does not involve the extra factor of $\\log(n)$.\n\\begin{conjecture}[Informal]\nLet $\\PPP$ and $\\QQQ$ be ``sufficiently nice.'' If there exists a polynomial-time algorithm to strongly distinguish $\\PPP$ and $\\QQQ$ then $\\|L_n^{\\le D}\\| = \\omega(1)$ for any $D = \\omega(1)$.\n\\end{conjecture}\n\n\\noindent In other words, if $\\|L_n^{\\le D}\\| = O(1)$ for some $D = \\omega(1)$, this already constitutes evidence that there is no polynomial-time algorithm.\nThe above seems to be a cleaner version of the main low-degree conjecture that remains correct for many problems of practical interest.\n\n\\subsubsection{Extensions}\n\\label{sec:extensions}\n\nWhile we have focused on the setting of hypothesis testing throughout these notes, we remark that low-degree arguments have also shed light on other types of problems such as \\emph{estimation} (or \\emph{recovery}) and \\emph{certification}.\n\nFirst, as we have mentioned before, non-trivial estimation\\footnote{Non-trivial estimation of a signal $\\bx \\in \\RR^n$ means having an estimator $\\hat{\\bm x}$ achieving $|\\langle \\hat \\bx, \\bx \\rangle|\/(\\|\\hat \\bx\\| \\cdot \\|\\bx\\|) \\ge \\varepsilon$ with high probability, for some constant $\\varepsilon > 0$.} typically seems to be precisely as hard as strong distinguishing (see Definition~\\ref{def:stat-ind}), in the sense that the two problems share the same $\\lambda_{\\mathsf{stat}}$ and $\\lambda_{\\mathsf{comp}}$. For example, the statistical thresholds for testing and recovery are known to coincide for problems such as the two-groups stochastic block model \\cite{MNS-rec,massoulie,MNS-proof} and the spiked Wigner matrix model (for a large class of spike priors) \\cite{finite-size,detection-wig}. Also, for any additive Gaussian noise model, any lower bound against hypothesis testing using the second moment method (Lemma~\\ref{lem:second-moment-contiguity}) or a conditional second moment method also implies a lower bound against recovery \\cite{BMVVX-pca}. More broadly, we have discussed (see Section~\\ref{sec:spectral}) how suitable spectral methods typically give optimal algorithms for high-dimensional problems; such methods typically succeed at testing and recovery in the same regime of parameters, because whenever the leading eigenvalue undergoes a phase transition, the leading eigenvector will usually do so as well (see Theorem~\\ref{thm:wig-bbp} for a simple example). Thus, low-degree evidence that hypothesis testing is hard also constitutes evidence that non-trivial recovery is hard, at least heuristically. Note, however, that there is no formal connection (in either direction) between testing and recovery (see \\cite{BMVVX-pca}), and there are some situations in which the testing and recovery thresholds differ (e.g., \\cite{planting-trees}).\n\nIn a different approach, Hopkins and Steurer \\cite{HS-bayesian} use a low-degree argument to study the recovery problem more directly. In the setting of community detection in the stochastic block model, they examine whether there is a low-degree polynomial that can non-trivially estimate whether two given network nodes are in the same community. They show that such a polynomial exists only when the parameters of the model lie above the problem's widely-conjectured computational threshold, the Kesten--Stigum threshold. This constitutes direct low-degree evidence that recovery is computationally hard below the Kesten--Stigum threshold.\n\nA related (and more refined) question is that of determining the optimal estimation error (i.e., the best possible correlation between the estimator and the truth) for any given signal-to-noise parameter $\\lambda$. Methods such as \\emph{approximate message passing} can often answer this question very precisely, both statistically and computationally (see, e.g., \\cite{AMP,LKZ-mmse,DAM,BDMKLZ-spiked}, or \\cite{stat-phys-survey,BPW-phys-notes,leo-survey} for a survey). One interesting question is whether one can recover these results using a variant of the low-degree method.\n\nAnother type of statistical task is \\emph{certification}. Suppose that $\\bY \\sim \\QQ_n$ has some property $\\sP$ with high probability. We say an algorithm \\emph{certifies} the property $\\sP$ if (i) the algorithm outputs ``yes'' with high probability on $\\bY \\sim \\QQ_n$, and (ii) if $\\bY$ does not have property $\\sP$ then the algorithm \\emph{always} outputs ``no.'' In other words, when the algorithm outputs ``yes'' (which is usually the case), this constitutes a proof that $\\bY$ indeed has property $\\sP$. Convex relaxations (including SoS) are a common technique for certification. In \\cite{BKW-sk}, the low-degree method is used to argue that certification is computationally hard for certain structured PCA problems. The idea is to construct a \\emph{quiet planting} $\\PP_n$, which is a distribution for which (i) $\\bY \\sim \\PP_n$ never has property $\\sP$, and (ii) the low-degree method indicates that it is computationally hard to strongly distinguish $\\PP_n$ and $\\QQ_n$. In other words, this gives a reduction from a hypothesis testing problem to a certification problem, since any certification algorithm can be used to distinguish $\\PP_n$ and $\\QQ_n$. (Another example of this type of reduction, albeit relying on a different heuristic for computational hardness, is \\cite{hard-rip}.)\n\n\n\\section*{Acknowledgments}\n\\addcontentsline{toc}{section}{Acknowledgments}\nWe thank the participants of a working group on the subject of these notes, organized by the authors at the Courant Institute of Mathematical Sciences during the spring of 2019.\nWe also thank Samuel B.\\ Hopkins, Philippe Rigollet, and David Steurer for helpful discussions. \n\n\n\\addcontentsline{toc}{section}{References}\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and motivation}\n\nThe work of this current paper is motivated by a question posed in a seminal paper by Mahler \\cite{Mahler_1984}; namely, how well can we approximate points in the middle-third Cantor set by:\n\\begin{enumerate}[(i)]\n\\itemsep-2pt\n\\item{rational numbers contained in the Cantor set, or} \\label{Mahler part 1}\n\\item{rational numbers not in the Cantor set?}\n\\end{enumerate}\n\nThe first contribution to this question was arguably made by Weiss \\cite{Weiss_2001}, who showed that almost no point in the middle-third Cantor set is \\emph{very well approximable} with respect to the natural probability measure on the middle-third Cantor set. Since this initial contribution, numerous authors have contributed to answering these questions, approaching them from many different perspectives. For example, Levesley, Salp, and Velani \\cite{LSV_2007} considered triadic approximation in the middle-third Cantor set, different subsets of the first named author, Baker, Chow, and Yu \\cite{ABCY, ACY, Baker4} studied dyadic approximation in the middle-third Cantor set, Kristensen \\cite{Kristensen_2006} considered approximation of points in the middle-third Cantor set by algebraic numbers, and Tan, Wang and Wu \\cite{TWW_2021} have recently studied part (\\ref{Mahler part 1}) by introducing a new notion of the ``height'' of a rational number. There has also been considerable effort invested in trying to generalise some of the above results to more general self-similar sets in $\\R$ and also to various fractal sets in higher dimensions. See, for example, \\cite{Allen-Barany, Baker1, Baker2, Baker3, Baker-Troscheit, BFR_2011, FS_2014, FS_2015, Khalil-Luethi, KLW_2005, PV_2005, WWX_Cantor, Yu_self-similar_2021} and references therein. The results in this paper can be thought of as a contribution to answering a natural $d$-dimensional weighted variation of part (\\ref{Mahler part 1}) of Mahler's question. In particular, we will be interested in weighted approximation in $d$-dimensional ``missing digit'' sets. \n\n Before we introduce the general framework we will consider here, we provide a very brief overview of some of the classical results on weighted Diophantine approximation in the ``usual'' Euclidean setting which provide further motivation for the current work. Fix $d \\in \\N$ and let $\\Psi=(\\psi_{1}, \\dots , \\psi_{d})$ be a $d$-tuple of approximating functions $\\psi_{i}:\\N \\to [0, \\infty)$ with $\\psi_{i}(r) \\to 0$ as $r \\to \\infty$ for each $1 \\leq i \\leq d$. The set of weighted simultaneously $\\Psi$-well-approximable points in $\\R^d$ is defined as\n\\begin{equation*}\nW_{d}(\\Psi):= \\left\\{ \\mathbf{x} = (x_{1}, \\dots , x_{d}) \\in [0,1]^{d}: \\left|x_{i}-\\frac{p_{i}}{q}\\right| < \\psi_{i}(q)\\, ,\\; 1 \\leq i \\leq d, \\text{ for i.m.} \\; (p_1, \\dots p_d, q) \\in \\Z^{d} \\times \\N \\right\\},\n\\end{equation*}\nwhere i.m. denotes infinitely many. Note that the special case  where each approximating function is the same, that is $\\Psi=(\\psi, \\dots , \\psi)$, is generally the more intensively studied set. The case where each approximating function is potentially different, usually referred to as \\emph{weighted simultaneous approximation}, is a natural generalisation of this. Simultaneous approximation (i.e. when the approximating function is the same in each coordinate axis) can generally be seen as a metric generalisation of Dirichlet's Theorem, whereas weighted simultaneous approximation is a metric generalisation of Minkowski's Theorem. Weighted simultaneous approximation has earned interest in the past few decades due to Schmidt  and natural connections to Littlewood's Conjecture, see for example \\cite{BRV_2016, BPV_2011, An_2013, An_2016, Schmidt_1983}. \n \nMotivated by classical works due to the likes of Khintchine \\cite{Khintchine_24, Khintchine_25} and Jarn\\'{i}k \\cite{Jarnik_31} which tell us, respectively, about the Lebesgue measure and Hausdorff measures of the sets of classical simultaneously $\\Psi$-well-approximable points (i.e. when $\\Psi=(\\psi, \\dots, \\psi)$), one may naturally also wonder about the ``size'' of sets of weighted simultaneously $\\Psi$-well-approximable points in terms of Lebesgue measure, Hausdorff dimension, and Hausdorff measures. Khintchine \\cite{Weighted_Khintchine} showed that if $\\psi: \\N \\to [0,\\infty)$ and $\\Psi(q)=(\\psi(q)^{\\tau_1}, \\dots, \\psi(q)^{\\tau_d})$ for some $\\boldsymbol{\\tau} = (\\tau_1,\\dots,\\tau_d) \\in (0,1)^d$ with $\\tau_1+\\tau_2+\\dots+\\tau_d=1$, then\n\\begin{equation*}\n\\lambda_d(W_d(\\Psi))=\\begin{cases}\n\\quad 0  \\quad  &\\text{ if } \\quad \\sum_{q=1}^{\\infty} q^d \\psi(q) < \\infty, \\\\[2ex]\n\\quad 1 \\quad &\\text{ if } \\quad \\sum_{q=1}^{\\infty} q^d \\psi(q) = \\infty, \\text{ and $q^d\\psi(q)$ is monotonic}.\n\\end{cases}\n\\end{equation*}\nThroughout we use $\\lambda_d(X)$ to denote the $d$-dimensional Lebesgue measure of a set \\mbox{$X \\subset \\R^d$}. For more general approximating functions $\\Psi(q)=(\\psi_1(q), \\dots, \\psi_d(q))$, with $\\prod_{i=1}^{d}\\psi_i(q)$ monotonically decreasing and $\\psi_{i}(q)<q^{-1}$ for each $1\\leq i \\leq d$, it has been proved, see \\cite{Cassels_1950, Weighted_Khintchine, Schmidt_1960, Gallagher_1962}, that\n\\begin{equation*}\n\\lambda_d(W_d(\\Psi))=\\begin{cases}\n\\quad 0  \\quad  &\\text{ if } \\quad \\sum_{q=1}^{\\infty} q^d \\psi_1(q)\\dots \\psi_d(q) < \\infty, \\\\[2ex]\n\\quad 1 \\quad &\\text{ if } \\quad \\sum_{q=1}^{\\infty} q^d \\psi_1(q)\\dots \\psi_d(q) = \\infty.\n\\end{cases}\n\\end{equation*}\nFor approximating functions of the form $\\Psi(q)=(\\psi_1(q),\\dots,\\psi_d(q))$ where\n\\[\\psi_i(q)=q^{-t_{i}-1}, \\quad \\text{for some vector } \\mathbf{t}=(t_{1}, \\dots , t_{d}) \\in \\R^{d}_{>0},\\] \nRynne \\cite{Rynne_1998} proved that if $\\sum_{i=1}^{d}t_{i} \\geq 1$, then \n\\begin{equation*}\n\\dimh W_{d}(\\Psi)=\\min_{1 \\leq k \\leq d} \\left\\{ \\frac{1}{t_{k}+1} \\left( d+1+\\sum_{i:t_{k} \\geq t_{i}}(t_{k}-t_{i})\\right) \\right\\}.\n\\end{equation*}\n Throughout, we write $\\dimh{X}$ to denote the \\emph{Hausdorff dimension} of a set $X \\subset \\R^d$, we refer the reader to \\cite{Falconer} for definitions and properties of Hausdorff dimension and Hausdorff measures. Rynne's result has recently been extended to a more general class of approximating functions by Wang and Wu \\cite[Theorem 10.2]{WW2019}.\n\nIn recent years, there has been rapidly growing interest in whether similar statements can be proved when we intersect $W_d(\\Psi)$ with natural subsets of $[0,1]^{d}$, such as submanifolds or fractals. The study of such questions has been further incentivised by many remarkable works of the recent decades, such as \\cite{KLW_2005, KM_1998, VV_2006}, and applications to other areas, such as wireless communication theory \\cite{ABLVZ_2016}.\n\n\\section{$d$-dimensional missing digit sets and main results}\n\nIn this paper we study weighted approximation in $d$-dimensional missing digit sets, which are natural extensions of classical missing digit sets (i.e. generalised Cantor sets) in $\\R$ to higher dimensions. A very natural class of higher dimensional missing digit sets included within our framework are the \\emph{four corner Cantor sets} (or \\emph{Cantor dust}) in $\\R^2$ with contraction ratio $\\frac{1}{n}$ for $n \\in \\N$.\n\nThroughout we consider $\\R^d$ equipped with the supremum norm, which we denote by $\\supn{\\cdot}$. For subsets $X,Y \\subset \\R^{d}$ we define $\\mathrm{diam}(X) = \\sup\\{\\|u-v\\|:u,v \\in X\\}$ and $\\operatorname{dist}(X,Y)= \\inf\\{\\|x-y\\|: x \\in X, y \\in Y\\}$. We define \\emph{higher-dimensional missing digit sets} via iterated function systems as follows. Let $b \\in \\N$ be such that $b \\geq 3$ and let $J_1,\\dots,J_d$ be proper subsets of $\\set{0,1,\\dots,b-1}$ such that for each $1 \\leq i \\leq d$, we have\n\\[N_i:=\\#J_i\\geq 2.\\]\nSuppose $J_i=\\set{a^{(i)}_1,\\dots,a^{(i)}_{N_i}}$. For each $1 \\leq i \\leq d$, we define the iterated function system \n\\[\\Phi^i=\\left\\{f_j:[0,1]\\to[0,1]\\right\\}_{j=1}^{N_i} \\quad \\text{where} \\quad f_j(x)=\\frac{x+a^{(i)}_j}{b}.\\]\nLet $K_i$ be the attractor of $\\Phi^i$; that is, $K_i \\subset \\R$ is the unique non-empty compact set which satisfies\n\\[K_i=\\bigcup_{j=1}^{N_i}{f_j(K_i)}.\\]\n We know that such a set exists due to work of Hutchinson \\cite{Hutchinson}. Equivalently $K_i$ is the set of $x \\in [0,1]$ for which there exists a base $b$ expansion of $x$ consisting only of digits from $J_i$. In view of this, we will also use the notation $K_{b}(J_i)$ to denote this set. For example, in this notation, the classical middle-third Cantor set is precisely the set $K_3(\\{0,2\\})$. We call the sets $K_b(J_i)$ \\emph{missing digit sets} since they consist of numbers with base-$b$ expansions missing specified digits. Note that, for each $1 \\leq i \\leq d$, the Hausdorff dimension of $K_i$, which we will denote by $\\gamma_i$, is given by \n\\[\\gamma_i = \\dimh{K_i} = \\frac{\\log{N_i}}{\\log{b}}.\\]\n\nWe will be interested in the \\emph{higher-dimensional missing digit set}\n\\[K:=\\prod_{i=1}^{d}{K_i}\\]\nformed by taking the Cartesian product of the sets $K_i$, $1 \\leq i \\leq d$. As a natural concrete example, we note that the \\emph{four corner Cantor set} in $\\R^2$ with contraction ratio $\\frac{1}{b}$ (with $b \\geq 3$ an integer) can be written in our notation as $K_{b}(\\{0,b-1\\}) \\times K_{b}(\\{0,b-1\\})$. \n\nWe note that $K$ is the attractor of the iterated function system \n\\[\\Phi=\\left\\{f_{(j_1,\\dots,j_d)}:[0,1]^d \\to [0,1]^d, (j_1,\\dots,j_d) \\in \\prod_{i=1}^{d}{J_i}\\right\\}\\]\nwhere\n\\[f_{(j_1,\\dots,j_d)}\\begin{pmatrix} x_1 \\\\ \\vdots \\\\ x_d\\end{pmatrix} = \\begin{pmatrix}\\frac{x_1+j_1}{b} \\\\ \\vdots \\\\ \\frac{x_d+j_d}{b}\\end{pmatrix}.\\]\nNotice that $\\Phi$ consists of \n\\[N:=\\prod_{i=1}^{d}{N_i}\\]\nmaps and so, for convenience, we will write \n\\[\\Phi = \\left\\{g_j:[0,1]^d\\to[0,1]^d\\right\\}_{j=1}^{N}\\]\nwhere the $g_j$'s are just the maps $f_{(j_1,\\dots,j_d)}$ from above written in some order. The Hausdorff dimension of $K$, which we denote by $\\gamma$, is\n\\[\\gamma = \\dimh{K} = \\frac{\\log{N}}{\\log{b}}.\\]\n\nWe will write\n\\[\\Lambda = \\set{1,2,\\dots,N} \\qquad \\text{and} \\qquad \\Lambda^*=\\bigcup_{n=0}^{\\infty}{\\Lambda^n}.\\]\nWe write $\\mathbf{i}$ to denote a word in $\\Lambda$ or $\\Lambda^*$ and we write $|\\mathbf{i}|$ to denote the length of $\\mathbf{i}$. For $\\mathbf{i} \\in \\Lambda^*$ we will also use the shorthand notation\n\\[g_{\\mathbf{i}} = g_{i_1} \\circ g_{i_2} \\circ \\dots \\circ g_{i_{|\\mathbf{i}|}}.\\]\nWe adopt the convention that $g_{\\emptyset}(x)=x$.\n\nLet $\\Psi: \\Lambda^* \\to [0,\\infty)$ be an approximating function. For each $x \\in K$, we define the set \n\\begin{align*} \nW(x,\\Psi)=\\left\\{y \\in K: \\supn{y-g_{\\mathbf{i}}(x)}<\\Psi(\\mathbf{i}) \\text{ for infinitely many } \\mathbf{i} \\in \\Lambda^*\\right\\}.\n\\end{align*}\nThe following theorem is a special case of \\cite[Theorem 1.1]{Allen-Barany}. \n\n\\begin{theorem} \\label{self-similar approx thm}\nLet $\\Phi$ and $K$ be as defined above. Let $x \\in K$ and let $\\varphi: \\N \\to [0,\\infty)$ be a monotonically decreasing function. Let $\\Psi(\\mathbf{i}) = \\mathrm{diam}(g_{\\mathbf{i}}(K))\\varphi(|\\mathbf{i}|)$. Then, for $s>0$, \n\\[\n{\\cal H}^{s}(W(x,\\Psi))=\n\\begin{cases}\n0 & \\text{if} \\quad \\sum_{\\mathbf{i} \\in \\Lambda^*}{\\Psi(\\mathbf{i})^s}<\\infty, \\\\[2ex]\n{\\cal H}^s(K) & \\text{if} \\quad \\sum_{\\mathbf{i} \\in \\Lambda^*}{\\Psi(\\mathbf{i})^s}=\\infty.\n\\end{cases}\n\\]\n\\end{theorem}\nOf particular interest to us here is the following easy corollary.\n\\begin{corollary} \\label{simultaneous corollary}\nLet $\\Phi$ and $K$ be as above and suppose that $\\mathrm{diam}(K)=1$. Let $\\psi: \\N \\to [0,\\infty)$  be such that $b^n\\psi(b^n)$ is monotonically decreasing and define $\\varphi: \\N \\to [0,\\infty)$ by $\\varphi(n)=b^n\\psi(b^n)$. Let $\\Psi(\\mathbf{i}) = \\mathrm{diam}(g_{\\mathbf{i}}(K))\\varphi(|\\mathbf{i}|)$. Recall that $\\gamma = \\dimh{K}$. Then, for $x \\in K$, we have\n\\[\n{\\cal H}^{\\gamma}(W(x,\\Psi))=\n\\begin{cases}\n0 & \\text{if} \\quad \\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}}<\\infty, \\\\[2ex]\n{\\cal H}^{\\gamma}(K) & \\text{if} \\quad \\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}}=\\infty.\n\\end{cases}\n\\]\n\\end{corollary}\n\n\\begin{proof}\nIt follows from Theorem \\ref{self-similar approx thm} that\n\\[\n{\\cal H}^{\\gamma}(W(x,\\Psi))=\n\\begin{cases}\n0 & \\text{if} \\quad \\sum_{\\mathbf{i} \\in \\Lambda^*}{\\Psi(\\mathbf{i})^{\\gamma}}<\\infty, \\\\[2ex]\n{\\cal H}^{\\gamma}(K) & \\text{if} \\quad \\sum_{\\mathbf{i} \\in \\Lambda^*}{\\Psi(\\mathbf{i})^{\\gamma}}=\\infty.\n\\end{cases}\n\\]\nHowever, in this case, by the definition of $\\varphi$ and our assumption that $\\mathrm{diam}(K)=1$, we have\n\\[\\sum_{\\mathbf{i} \\in \\Lambda^*}{\\Psi(\\mathbf{i})^{\\gamma}} = \\sum_{n=1}^{\\infty}{\\sum_{\\substack{\\mathbf{i} \\in \\Lambda^* \\\\ |\\mathbf{i}| = n}}{(\\mathrm{diam}(g_{\\mathbf{i}}(K))\\varphi(|\\mathbf{i}|))^{\\gamma}}} = \\sum_{n=1}^{\\infty}{\\sum_{\\substack{\\mathbf{i} \\in \\Lambda^* \\\\ |\\mathbf{i}| = n}}{\\psi(b^n)^{\\gamma}}} = \\sum_{n=1}^{\\infty}{N^n \\psi(b^n)^{\\gamma}} = \\sum_{n=1}^{\\infty}{(b^n \\psi(b^n))^{\\gamma}}. \\qedhere \\]\n\\end{proof}\n\nFor an approximating function $\\psi: \\N \\to [0,\\infty)$, define\n\\begin{align} \\label{W x psi}\nW(x,\\psi)=\\left\\{y \\in K: \\supn{y-g_{\\mathbf{i}}(x)} < \\psi(b^{|\\mathbf{i}|}) \\text{ for infinitely many } \\mathbf{i} \\in \\Lambda^*\\right\\}.\n\\end{align}\nIn essence, $W(x,\\psi)$ is a set of ``simultaneously $\\psi$-well-approximable'' points in $K$. The following statement regarding these sets can be deduced immediately from Corollary \\ref{simultaneous corollary}.\n\n\\begin{corollary} \\label{W x psi corollary}\nLet $\\Phi$ and $K$ be defined as above and let $\\psi: \\N \\to [0,\\infty)$ be such that $b^n\\psi(b^n)$ is monotonically decreasing. Suppose further that $\\mathrm{diam}(K)=1$. Then, \n\\[\n{\\cal H}^{\\gamma}(W(x,\\psi))=\n\\begin{cases}\n0 & \\text{if} \\quad \\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}}<\\infty, \\\\[2ex]\n{\\cal H}^{\\gamma}(K) & \\text{if} \\quad \\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}}=\\infty.\n\\end{cases}\n\\]\n\\end{corollary}\n\nHere we will be interested in weighted versions of the sets $W(x,\\psi)$. More specifically, for $\\mathbf{t}=(t_1,\\dots,t_d) \\in \\R^d_{\\geq 0}$ and for $x \\in K$, we define the \\emph{weighted approximation set}\n\\[W(x,\\psi,\\mathbf{t}) = \\left\\{\\mathbf{y}=(y_1,\\dots,y_d) \\in K: |y_j-g_{\\mathbf{i}}(x)_j|<\\psi(b^{|\\mathbf{i}|})^{1+t_i}, 1 \\leq j \\leq d, \\text{ for i.m. } \\mathbf{i} \\in \\Lambda^*\\right\\}.\\]\nHere we are using the notation $g_{\\mathbf{i}}(x)=(g_{\\mathbf{i}}(x)_1,\\dots,g_{\\mathbf{i}}(x)_d)$. Our main results relating to the Hausdorff dimension of sets of the form $W(x,\\psi,\\mathbf{t})$ are as follows.\n\n\\begin{theorem} \\label{lower bound theorem}\nLet $\\Phi$ and $K$ be defined as above. Recall that $\\gamma = \\dimh{K}$ and $\\gamma_i=\\dimh{K_i}$ for each $1 \\leq i \\leq d$. Let $\\psi: \\N \\to [0, \\infty)$ be such that $b^n\\psi(b^n)$ is monotonically decreasing. Further suppose that $\\mathrm{diam}(K)=1$ and\n\\[\\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}} = \\infty.\\]\nThen, for $\\mathbf{t} = (t_1,\\dots,t_d) \\in \\R^d_{\\geq 0}$, we have\n\\[\\dimh{W(x,\\psi,\\mathbf{t})} \\geq \\min_{1 \\leq k \\leq d}\\left\\{\\frac{1}{1+t_k}\\left(\\gamma + \\sum_{j:t_j \\leq t_k}{(t_k-t_j)\\gamma_{j}}\\right)\\right\\}.\\]\n\\end{theorem}\n\nIf $\\psi$ satisfies more stringent divergence conditions, then we an show that the lower bound given in Theorem \\ref{lower bound theorem} in fact gives an exact formula for the Hausdorff dimension of $W(x,\\psi,\\mathbf{t})$. More precisely, we are able to show the following.\n\n\\begin{theorem} \\label{upper bound theorem}\nLet $\\Phi$ and $K$ be as defined above. Let $x \\in K$ and let $\\psi:\\N \\to [0,\\infty)$ be such that:\n\\begin{enumerate}[(i)]\n    \\item{$b^n\\psi(b^n)$ is monotonically decreasing,}\n    \\item{$\\displaystyle{\\sum_{n=1}^{\\infty}{(b^n \\psi(b^n))^{\\gamma}}=\\infty}, \\quad$ and}\n    \\item{$\\displaystyle{\\sum_{n=1}^{\\infty}{(b^n \\psi(b^n)^{1+\\varepsilon})^{\\gamma}}<\\infty} \\quad$ for every $\\varepsilon>0$.}\n\\end{enumerate}\nThen, for $\\mathbf{t} = (t_1, \\dots, t_d) \\in \\R^d_{\\geq 0}$, we have \n\\[\\dimh{W(x,\\psi,\\mathbf{t})} = \\min_{1 \\leq k \\leq d}\\left\\{\\frac{1}{1+t_k}\\left(\\gamma + \\sum_{j:t_j \\leq t_k}{(t_k-t_j)\\gamma_{j}}\\right)\\right\\}.\\]\n\\end{theorem}\nAs an example of an approximating function which satisifies conditions $(i)-(iii)$, one can think of $\\psi(q)=\\left(q(\\log_{b}q)^{1\/\\gamma}\\right)^{-1}$. This function naturally appears when one considers analogues of Dirichlet's theorem in missing digit sets (see \\cite{BFR_2011, FS_2014}).\n As a corollary to Theorem \\ref{upper bound theorem} we deduce the following statement which can be interpreted as a higher-dimensional weighted generalisation of \\cite[Theorem 4]{LSV_2007}. In \\cite[Theorem 4]{LSV_2007}, Levesley, Salp, and Velani establish the Hausdorff measure of the set of points in a one-dimensional base-$b$ missing digit set (i.e. of the form $K_b(J)$ in our present notation) which can be well-approximated by rationals with denominators which are powers of $b$. Before we state our corollary, we fix one more piece of notation. Given an approximating function $\\psi: \\N \\to [0,\\infty)$, an infinite subset ${\\cal B} \\subset \\N$, and $\\mathbf{t} = (t_1,\\dots,t_d) \\in \\R_{\\geq 0}^d$, we define\n\\[W_{{\\cal B}}(\\psi,\\mathbf{t})=\\left\\{x \\in K: \\left|x_{i}-\\frac{p_i}{q}\\right|<\\psi(q)^{1+t_i}, 1 \\leq i \\leq d, \\text{ for i.m. } (p_1,\\dots,p_d,q) \\in \\Z^d \\times {\\cal B} \\right\\}.\\]\n\n\\begin{corollary} \\label{LSV_equivalent}\nFix $b \\in \\N$ with $b \\geq 3$ and let ${\\cal B}=\\{b^n: n=0,1,2,\\dots\\}$. Let $K$ be a  higher dimensional missing digit set as defined above (with base $b$) and write $\\gamma=\\dimh{K}$. Furthermore, suppose that $\\set{0,b-1} \\subset J_i$ for every $1 \\leq i \\leq d$. In particular, this also means that $\\mathrm{diam}{K}=1$. Let $\\psi: \\N \\to [0,\\infty)$ be an approximating function such that \n\\begin{enumerate}[(i)]\n\\item{$b^{n}\\psi(b^{n})$ is monotonically decreasing with $b^{n}\\psi(b^{n}) \\to 0$ as $n \\to \\infty$},\n\\item{$\\displaystyle{\\sum_{n=1}^{\\infty}{(b^n \\psi(b^n))^{\\gamma}}=\\infty}, \\quad$ and}\n\\item{$\\displaystyle{\\sum_{n=1}^{\\infty}{(b^n \\psi(b^n)^{1+\\varepsilon})^{\\gamma}}<\\infty} \\quad$ for every $\\varepsilon>0$.}\n\\end{enumerate}\nThen\n\\begin{equation*}\n\\dimh W_{{\\cal B}}(\\psi, \\mathbf{t}) = \\min_{1 \\leq k \\leq d}\\left\\{\\frac{1}{1+t_k}\\left(\\gamma + \\sum_{j:t_j \\leq t_k}{(t_k-t_j)\\, \\gamma_j}\\right)\\right\\}.\n\\end{equation*}\n\\end{corollary}\n\n\\begin{proof} \n\nObserve that the conditions imposed in the statement of Corollary \\ref{LSV_equivalent} guarantee that Theorem \\ref{upper bound theorem} is applicable. Furthermore, by our assumption that $b^{n}\\psi(b^{n}) \\to 0$ as $n \\to \\infty$, we may assume without loss of generality that $\\psi(b^n) < b^{-n}$ for all $n \\in \\N$.\n\nNext, we note that if $\\mathbf{p}=(p_1,\\dots,p_d) \\in \\Z^d$ and $\\frac{\\mathbf{p}}{b^n} = \\left(\\frac{p_1}{b^n},\\dots,\\frac{p_d}{b^n}\\right) \\notin K$, then we must have \n\\[\\operatorname{dist}\\left(\\frac{\\mathbf{p}}{b^n},K\\right) \\geq b^{-n}, \\quad \\text{where} \\quad \\operatorname{dist}(x,K)=\\inf\\set{\\|x-y\\|: y \\in K}.\\]  \n(Recall that we use $\\|\\cdot\\|$ to denote the supremum norm in $\\R^d$.)\nThus we need only concern ourselves with pairs $(\\mathbf{p},q) \\in \\Z^{d} \\times {\\cal B}$ for which $\\frac{\\mathbf{p}}{q} \\in K$.\n\nLet $G=\\left\\{x=(x_1,\\dots,x_d) \\in \\{0,1\\}^{d}\\right\\}$ and note that $G \\subset K$ by the assumption that $\\{0,b-1\\} \\subset J_{i}$ for each $1\\leq i \\leq d$. For any $x \\in G$ and any $\\mathbf{j} \\in \\Lambda^{n}$ it is possible to write $g_{\\mathbf{j}}(x)=\\frac{\\mathbf{p}}{b^{n}}$ for some $\\mathbf{p} \\in (\\N\\cup \\set{0})^{d}$. Hence\n\\[W(x, \\psi, \\mathbf{t}) \\subset W_{{\\cal B}}(\\psi, \\mathbf{t}).\\]\nFurthermore, the set of all rational points of the form $\\frac{\\mathbf{p}}{b^{n}}$ contained in $K$ is\n\\begin{equation*}\n\\bigcup_{x \\in G} \\bigcup_{\\mathbf{j} \\in \\Lambda^{n}} g_{\\mathbf{j}}(x).\n\\end{equation*}\nHence\n\\begin{equation*} \nW_{{\\cal B}}(\\psi, \\mathbf{t}) \\subset \\bigcup_{x \\in G}  W(x, \\psi, \\mathbf{t}).\n \\end{equation*}\nBy the finite stability of Hausdorff dimension (see \\cite{Falconer}), Corollary \\ref{LSV_equivalent} now follows from Theorem~\\ref{upper bound theorem}.\n\\end{proof}\n\n\nNotice that in Theorem \\ref{lower bound theorem}, Theorem \\ref{upper bound theorem}, and Corollary \\ref{LSV_equivalent}, we insist on the same underlying base $b$ in each coordinate direction. This is somewhat unsatisfactory and one might hope to be able to obtain results where we can have different bases $b_i$ in each coordinate direction. The first steps towards proving results relating to weighted approximation in this setting can be seen in \\cite[Section 12]{WW2019}. Proving more general results with different bases in different coordinate directions is likely to be a very challenging problem since such sets are \\emph{self-affine} and, generally speaking, self-affine sets are more difficult to deal with than self-similar or self-conformal sets. Indeed, very little is currently known even regarding non-weighted approximation in self-affine sets.\n\n\n\\noindent{\\bf Structure of the paper:} The remainder of the paper will be arranged as follows. In Section~\\ref{measure theory section} we will present some measure theoretic preliminaries which will be required for the proofs of our main results. The key tool required for proving Theorem \\ref{lower bound theorem} is a mass transference principle for rectangles proved recently by Wang and Wu \\cite{WW2019}. We introduce this in Section \\ref{mtp section}. In Section \\ref{lower bound section} we present our proof of Theorem \\ref{lower bound theorem} and we conclude in Section \\ref{upper bound section} with the proof of Theorem \\ref{upper bound theorem}.\n\n\\section{Some Measure Theoretic Preliminaries} \\label{measure theory section}\n\nRecall that $\\gamma = \\dimh{K}$ and that $\\gamma_i=\\dimh{K_i}$ for $1 \\leq i \\leq d$, where $K$ and $K_i$ are as defined above. Furthermore, note that $0 < {\\cal H}^{\\gamma}(K) < \\infty$ and $0< {\\cal H}^{\\gamma_{i}}(K_{i})<\\infty$ for each $1 \\leq i \\leq d$, see for example \\cite[Theorem 9.3]{Falconer}. Let us define the measures\n\\[\\mu:=\\frac{{\\cal H}^{\\gamma}|_{K}}{{\\cal H}^{\\gamma}(K)} \\qquad \\text{and} \\qquad \\mu_i:=\\frac{{\\cal H}^{\\gamma_i}|_{K_i}}{{\\cal H}^{\\gamma_i}(K_i)} \\quad \\text{for each } 1 \\leq i \\leq d.\\]\nSo, for $X \\subset \\R^d$, we have \n\\[\\mu(X) = \\frac{{\\cal H}^{\\gamma}(X \\cap K)}{{\\cal H}^{\\gamma}(K)}.\\]\nSimilarly, for $X \\subset \\R$, for each $1 \\leq i \\leq d$ we have \n\\[\\mu_i(X) = \\frac{{\\cal H}^{\\gamma_i}(X \\cap K_i)}{{\\cal H}^{\\gamma_i}(K_i)}.\\]\nNote that $\\mu$ defines a probability measure supported on $K$ and, for each $1 \\leq i \\leq d$, $\\mu_i$ defines a probability measure supported on $K_i$. \nNote also that the measure $\\mu$ is $\\delta$-Ahlfors regular with $\\delta = \\gamma$ and, for each $1 \\leq i \\leq d$, the measure $\\mu_i$ is $\\delta$-Ahlfors regular with $\\delta=\\gamma_i$ (see, for example, \\cite[Theorem 4.14]{Mattila_1999}).\n\nWe will also be interested in the product measure\n\\[\\Mu:=\\prod_{i=1}^{d}{\\mu_i}.\\]\nWe note that $\\Mu$ is $\\delta$-Ahlfors regular with $\\delta = \\gamma$. This fact follows straightforwardly from the Ahlfors regularity of each of the $\\mu_i$'s.\n\n\\begin{lemma} \\label{M regularity lemma}\nThe product measure $\\Mu = \\prod_{i=1}^{d}{\\mu_i}$ on $\\R^d$ is $\\delta$-Ahlfors regular with $\\delta=\\gamma$.\n\\end{lemma}\n\n\\begin{proof}\nLet $B=\\prod_{i=1}^{d}{B(x_i,r)}$, $r>0$, be an arbitrary ball in $\\R^d$. The aim is to show that \\mbox{$\\Mu(B) \\asymp r^{\\gamma}$}. Recall that for each $1 \\leq i \\leq d$, the measure $\\mu_i$ is $\\delta$-Ahlfors regular with \\mbox{$\\delta = \\gamma_i = \\dimh{K_i} = \\frac{\\log{N_i}}{\\log{b}}$}. Also recall that $N=\\prod_{i=1}^{d}{N_i}$ and $\\gamma = \\dimh{K} = \\frac{\\log{N}}{\\log{b}}$. Thus, we have\n\\[\\Mu(B) = \\prod_{i=1}^{d}{\\mu_i(B(x_i,r))} \\asymp \\prod_{i=1}^{d}{r^{\\gamma_i}} = r^{\\sum_{i=1}^{d}{\\gamma_i}}.\\]\nNote that\n\\[\\sum_{i=1}^{d}{\\gamma_i} = \\sum_{i=1}^{d}{\\frac{\\log{N_i}}{\\log{b}}} = \\frac{\\log(\\prod_{i=1}^{d}{N_i})}{\\log{b}} = \\frac{\\log{N}}{\\log{b}} = \\gamma.\\]\nHence, $\\Mu(B) \\asymp r^{\\gamma}$ as claimed.\n\\end{proof}\n\nWe also note that, up to a constant factor, the product measure $\\Mu$ is equivalent to the measure $\\mu = \\frac{{\\cal H}^{\\gamma}|_K}{{\\cal H}^{\\gamma}(K)}$.\n\n\\begin{lemma}\\label{measure equivalence lemma}\nLet $\\Mu=\\prod_{i=1}^{d}{\\mu_i}$. Then, up to a constant factor, $\\Mu$ is equivalent to $\\mu$; i.e. for any Borel set $F \\subset \\R^d$, we have $\\Mu(F) \\asymp \\mu(F)$.\n\\end{lemma}\n\nLemma \\ref{measure equivalence lemma} follows immediately upon combining Lemma \\ref{M regularity lemma} with \\cite[Proposition 2.2 (a) + (b)]{Falconer_techniques}.\n\nIn our present setting, where $K$ is a self-similar set with well-separated components, we can actually show the stronger statement that $\\mu=\\Mu$.\n\n\\begin{proposition} \\label{measures_are_equal}\nThe measures $\\mu$ and $\\Mu$ are equal, i.e. for every Borel set $F \\subset \\R^d$, we have $\\mu(F) = \\Mu(F)$.\n\\end{proposition}\n\n\\begin{proof}\nFor each $1 \\leq i \\leq d$, there exists a unique Borel probability measure (see, for example, \\cite[Theorem 2.8]{Falconer_techniques}) $m_i$ satisfying\n\\begin{align} \\label{measure uniqueness 1}\nm_i &= \\sum_{j=1}^{N_i}{\\frac{1}{N_i}m_i \\circ f_j^{-1}}.\n\\end{align}\nLikewise, there exists a unique Borel probability measure $m$ satisfying\n\\begin{align} \\label{measure uniqueness 2}\n    m &= \\sum_{j=1}^{N}{\\frac{1}{N}m \\circ g_j^{-1}}.\n\\end{align}\n\nWe begin by showing that $\\mu_i$ satisfies \\eqref{measure uniqueness 1} for each $1 \\leq i \\leq d$. Note that ${\\cal H}^{\\gamma_i}(f_{j_1}(K_i) \\cap f_{j_2}(K_i)) = 0$ for any $1\\leq j_1,j_2 \\leq N_i$ with $j_1 \\neq j_2$. Thus, for any Borel set $X \\subset \\R^d$, , we have\n\\begin{align*}\n    \\mu_i(X) &= \\frac{1}{{\\cal H}^{\\gamma_i}(K_i)}{\\cal H}^{\\gamma_i}(X \\cap K_{i}) \\\\\n             &= \\frac{1}{{\\cal H}^{\\gamma_i}(K_i)}\\sum_{j=1}^{N_i}{{\\cal H}^{\\gamma_i}(X \\cap f_j(K_i))} \\\\\n             &= \\frac{1}{{\\cal H}^{\\gamma_i}(K_i)}\\sum_{j=1}^{N_i}{{\\cal H}^{\\gamma_i}(f_j(f_j^{-1}(X) \\cap K_i))} \\\\\n             &= \\frac{1}{{\\cal H}^{\\gamma_i}(K_i)}\\sum_{j=1}^{N_i}{\\left(\\frac{1}{b}\\right)^{\\gamma_i}{\\cal H}^{\\gamma_i}(f_j^{-1}(X) \\cap K_i)} \\\\\n             &= \\frac{1}{{\\cal H}^{\\gamma_i}(K_i)}\\sum_{j=1}^{N_i}{\\frac{1}{N_i}{\\cal H}^{\\gamma_i}(f_j^{-1}(X) \\cap K_i)} \\\\\n             &= \\sum_{j=1}^{N_i}{\\frac{1}{N_i}\\mu_i \\circ f_j^{-1}(X)}.\n\\end{align*}\n\nBy an almost identical argument, it can be shown that $\\mu$ satisfies $\\eqref{measure uniqueness 2}$.\n\nFinally, we show that $\\Mu$ also satisfies \\eqref{measure uniqueness 2} and, hence, by the uniqueness of solutions to~\\eqref{measure uniqueness 2}, we conclude that $\\Mu$ must be equal to $\\mu$. Since $\\mu_i$ satisfies \\eqref{measure uniqueness 1} for each $1 \\leq i \\leq d$, we have \n\\begin{align*}\n\\Mu &= \\prod_{i=1}^{d}{\\mu_i} \\\\\n    &= \\prod_{i=1}^{d}{\\left(\\sum_{j=1}^{N_i}{\\frac{1}{N_i}\\mu_i \\circ f_j^{-1}}\\right)} \\\\\n    &= \\sum_{\\mathbf{j} = (j_1, \\dots, j_d) \\in \\prod_{i=1}^{d}{\\{1,\\dots,N_i\\}}}{\\frac{1}{N}\\prod_{i=1}^{d}{\\mu_i \\circ f_{j_i}^{-1}}} \\\\\n    &= \\sum_{j=1}^{N}{\\frac{1}{N}M\\circ g_j^{-1}}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\section{Mass transference principle for rectangles} \\label{mtp section}\n\nTo prove Theorem \\ref{lower bound theorem}, we will use the mass transference principle for rectangles established recently by Wang and Wu in \\cite{WW2019}. The work of Wang and Wu generalises the famous Mass Transference Principle originally proved by Beresnevich and Velani \\cite{BV_MTP}. Since its initial discovery in \\cite{BV_MTP}, the Mass Transference Principle has found many applications, especially in Diophantine Approximation, and has by now been extended in numerous directions. See \\cite{Allen-Beresnevich, Allen-Baker, BV_MTP, BV_Slicing, Koivusalo-Rams, WWX2015, WW2019, Zhong2021} and references therein for further information. Here we shall state the general ``full measure'' mass transference principle from rectangles to rectangles established by Wang and Wu in \\cite[Theorem~3.4]{WW2019}.\n\nFix an integer $d \\geq 1$. For each $1 \\leq i \\leq d$, let $(X,|\\cdot|_i,m_i)$ be a bounded locally compact metric space equipped with a $\\delta_i$-Ahlfors regular probability measure $m_i$. We consider the product space $(X, |\\cdot|, m)$ where\n\\[X = \\prod_{i=1}^{d}{X_i}, \\qquad |\\cdot|=\\max_{1 \\leq i \\leq d}{|\\cdot|_i}, \\quad \\text{and} \\quad m=\\prod_{i=1}^{d}{m_i}.\\]\nNote that a ball $B(x,r)$ in $X$ is the product of balls in $\\{X_i\\}_{1 \\leq i \\leq d}$;\n\\[B(x,r) = \\prod_{i=1}^{d}{B(x_i,r)} \\quad \\text{for} \\quad x=(x_1,\\dots,x_d).\\]\n\nLet $J$ be an infinite countable index set and let $\\beta: J \\to \\R_{\\geq 0}: \\alpha \\mapsto \\beta_{\\alpha}$ be a positive function such that for any $M > 1$, the set\n\\[\\{\\alpha \\in J: \\beta_{\\alpha} < M\\}\\]\nis finite. Let $\\rho: \\R_{\\geq 0} \\to \\R_{\\geq 0}$ be a non-increasing function such that $\\rho(u) \\to 0$ as $u \\to \\infty$.\n\nFor each $1 \\leq i \\leq d$, let $\\{R_{\\alpha,i}: \\alpha \\in J\\}$ be a sequence of subsets of $X_i$. Then, the \\emph{resonant sets} in $X$ that we will be concerned with are\n\\[\\left\\{R_{\\alpha}=\\prod_{i=1}^{d}{R_{\\alpha,i}:\\alpha \\in J}\\right\\}.\\]\nFor a vector $\\mathbf{a} = (a_1,\\dots,a_d) \\in \\R_{>0}^d$, write\n\\[\\Delta(R_{\\alpha}, \\rho(\\beta_{\\alpha})^{\\mathbf{a}}) = \\prod_{i=1}^{d}{\\Delta(R_{\\alpha,i},\\rho(\\beta_{\\alpha})^{a_i})},\\]\nwhere $\\Delta(R_{\\alpha,i},\\rho(\\beta_\\alpha)^{a_i})$ appearing on the right-hand side denotes the $\\rho(\\beta_\\alpha)^{a_i}$-neighbourhood of $R_{\\alpha,i}$ in $X_i$. We call $\\Delta(R_{\\alpha,i},\\rho(\\beta_{\\alpha})^{a_i})$ the \\emph{part of $\\Delta(R_{\\alpha},\\rho(\\beta_{\\alpha})^{\\mathbf{a}})$ in the $i$th direction.}\n\nFix $\\mathbf{a} = (a_1, \\dots, a_d) \\in \\R_{>0}^d$ and suppose $\\mathbf{t} = (t_1,\\dots,t_d) \\in \\R_{\\geq 0}^d$. We are interested in the set\n\\[W_{\\mathbf{a}}(\\mathbf{t}) = \\left\\{x \\in X: x \\in \\Delta(R_{\\alpha},\\rho(\\beta_{\\alpha})^{\\mathbf{a}+\\mathbf{t}}) \\quad \\text{for i.m. } \\alpha \\in J \\right\\}.\\]\nWe can think of $\\Delta(R_{\\alpha},\\rho(\\beta_{\\alpha})^{\\mathbf{a}+\\mathbf{t}})$ as a smaller ``rectangle'' obtained by shrinking the ``rectangle'' $\\Delta(R_{\\alpha},\\rho(\\beta_{\\alpha})^{\\mathbf{a}})$.\n\nFinally, we require that the resonant sets satisfy a certain \\emph{$\\kappa$-scaling} property, which in essence ensures that locally our sets behave like affine subspaces.\n\n\\begin{definition} \\label{kappa scaling}\nLet $0 \\leq \\kappa < 1$. For each $1 \\leq i \\leq d$, we say that $\\{R_{\\alpha,i}\\}_{\\alpha \\in J}$ has the \\emph{$\\kappa$-scaling property} if for any $\\alpha \\in J$ and any ball $B(x,r)$ in $X_i$ with centre $x_i \\in R_{\\alpha,i}$ and radius $r>0$, for any $ 0 < \\varepsilon < r$, we have\n\\[c_1 r^{\\delta_i \\kappa}\\varepsilon^{\\delta_i(1-\\kappa\n)} \\leq m_i(B(x_i,r) \\cap \\Delta(R_{\\alpha,i},\\varepsilon)) \\leq c_2 r^{\\delta_i \\kappa} \\varepsilon^{\\delta_i(1-\\kappa)}\\]\nfor some absolute constants $c_1, c_2 > 0$.\n\\end{definition}\nIn our case $\\kappa=0$ since our resonant sets are points. For justification of this, and calculations of $\\kappa$ for other resonant sets, see \\cite{Allen-Baker}. Wang and Wu established the following mass transference principle for rectangles in \\cite{WW2019}.\n\n\\begin{theorem}[Wang -- Wu, \\cite{WW2019}] \\label{Theorem WW}\nAssume that for each $1 \\leq i \\leq d$, the measure $m_i$ is $\\delta_i$-Ahlfors regular and that the resonant set $R_{\\alpha,i}$ has the $\\kappa$-scaling property for $\\alpha \\in J$. Suppose\n\\[m\\left(\\limsup_{\\substack{\\alpha \\in J \\\\ \\beta_{\\alpha} \\to \\infty}}{\\Delta(R_{\\alpha},\\rho(\\beta_{\\alpha})^{\\mathbf{a}}})\\right) = m(X).\\]\nThen we have\n\\[\\dimh{W_{\\mathbf{a}}(\\mathbf{t})} \\geq s(\\mathbf{t}):= \\min_{A \\in {\\cal A}}\\left\\{\\sum_{k \\in {\\cal K}_1}{\\delta_k} + \\sum_{k \\in {\\cal K}_2}{\\delta_k} + \\kappa \\sum_{k \\in {\\cal K}_3}{\\delta_k}+(1-\\kappa)\\frac{\\sum_{k \\in {\\cal K}_3}{a_k \\delta_k} - \\sum_{k \\in {\\cal K}_2}{t_k \\delta_k}}{A}\\right\\},\\]\nwhere\n\\[{\\cal A} = \\{a_i, a_i+t_i: 1 \\leq i \\leq d\\}\\]\nand for each $A \\in {\\cal A}$, the sets ${\\cal K}_1, {\\cal K}_2, {\\cal K}_3$ are defined as\n\\[{\\cal K}_1 = \\{k: a_k \\geq A\\}, \\quad {\\cal K}_2=\\{k: a_k+t_k \\leq A\\} \\setminus {\\cal K}_1, \\quad {\\cal K}_3=\\{1,\\dots,d\\}\\setminus ({\\cal K}_1 \\cup {\\cal K}_2)\\]\nand thus give a partition of $\\{1,\\dots,d\\}$.\n\\end{theorem}\n\n\\section{Proof of Theorem \\ref{lower bound theorem}} \\label{lower bound section}\n\nTo prove Theorem \\ref{lower bound theorem}, we will apply Theorem \\ref{Theorem WW} with $X_i = K_i$, $m_i=\\mu_i$ and $\\abs{\\cdot}_i = \\abs{\\cdot}$ (absolute value in $\\R$) for each $1 \\leq i \\leq d$. Then, in our setting, we will be interested in the product space $(X, \\supn{\\cdot}, \\Mu)$ where\n\\[X = \\prod_{i=1}^{d}{K_i} \\; = K, \\qquad \\Mu = \\prod_{i=1}^{d}{\\mu_i},\\]\nand $\\supn{\\cdot}$ denotes the supremum norm in $\\R^d$. Recall that for each $1 \\leq i \\leq d$, the measure $\\mu_i$ is $\\delta_i$-Ahlfors regular with \n\\[\\delta_i = \\gamma_i = \\dimh{K_i}\\]\nand the measure $\\Mu$ is $\\delta$-Ahlfors regular with\n\\[\\delta = \\gamma = \\dimh{K}.\\]\n\nFor us, the appropriate indexing set is \n\\[{\\cal J} = \\{\\mathbf{i} \\in \\Lambda^*\\}.\\]\nWe define our \\emph{weight function} $\\beta: \\Lambda^* \\to \\R_{\\geq 0}$ by\n\\[\\beta_{|\\mathbf{i}|} = \\beta(\\mathbf{i}) = |\\mathbf{i}|.\\]\nNote that $\\beta$ satisfies the requirement that for any real number $M > 1$ the set $\\set{\\mathbf{i} \\in \\Lambda^*: \\beta_{\\mathbf{i}} < M}$ is finite. Next we define $\\rho: \\R_{\\geq 0} \\to \\R_{\\geq 0}$ by\n\\[\\rho(u) = \\psi(b^u).\\]\nSince $b^n\\psi(b^n)$ is monotonically decreasing by assumption, it follows that $\\psi(b^n)$ is monotonically decreasing and $\\psi(b^n) \\to 0$ as $n \\to \\infty$.\n\nFor a fixed $x = (x_1,\\dots,x_d) \\in K$, we define the resonant sets of interest as follows. For each $\\mathbf{i} \\in \\Lambda^*$, take\n\\[R_{\\mathbf{i}}^x=g_{\\mathbf{i}}(x).\\]\nCorrespondingly, for each $1 \\leq j \\leq d$, \n\\[R_{\\mathbf{i},j}^x = g_{\\mathbf{i}}(x)_j,\\]\nwhere $g_{\\mathbf{i}}(x) = (g_{\\mathbf{i}}(x)_1, \\dots, g_{\\mathbf{i}}(x)_d)$. So, $R_{\\mathbf{i},j}^x$ is the coordinate of $g_{\\mathbf{i}}(x)$ in the $j$th direction. In each coordinate direction, the $\\kappa$-scaling property is satisfied with $\\kappa$=0, since our resonant sets are points.\n\nLet us fix $\\mathbf{a} = (1,1,\\dots,1) \\in \\R_{>0}^d$. Then, in this case, we note that \n\\[\\limsup_{\\substack{\\alpha \\in {\\cal J} \\\\ \\beta_\\alpha \\to \\infty}}{\\Delta(R_{\\alpha}^{x}, \\rho(\\beta_\\alpha)^{\\mathbf{a}})} = \\limsup_{\\substack{\\mathbf{i} \\in \\Lambda^* \\\\ |\\mathbf{i}| \\to \\infty}}{\\Delta(g_{\\mathbf{i}}(x), \\psi(b^{|\\mathbf{i}|})^{\\mathbf{a}})} = W(x,\\psi),\\]\nwhere $W(x,\\psi)$ is as defined in \\eqref{W x psi}. Moreover, it follows from Corollary \\ref{W x psi corollary} and Proposition \\ref{measures_are_equal} that $\\Mu(W(x,\\psi)) = \\Mu(K)$, since we assumed that $\\sum_{n=1}^{\\infty}{(b^n\\psi(b^n))^{\\gamma}} = \\infty$.\n\nNow suppose that $\\mathbf{t} = (t_1,\\dots,t_d) \\in \\R_{\\geq 0}^d$. Then, in our case, \n\\[W_{\\mathbf{a}}(\\mathbf{t}) = W(x,\\psi,\\mathbf{t}),\\]\nwhich is the set we are interested in. So, recalling that $\\kappa = 0$ in our setting, we may now apply Theorem \\ref{Theorem WW} directly to conclude that\n\\[\\dimh{W(x,\\psi,\\mathbf{t})} \\geq \\min_{A \\in {\\cal A}}\\left\\{\\sum_{k \\in {\\cal K}_1}{\\delta_k} + \\sum_{k \\in {\\cal K}_2}{\\delta_k}+\\frac{\\sum_{k \\in {\\cal K}_3}{\\delta_k}-\\sum_{k \\in {\\cal K}_2}{t_k \\delta_k}}{A}\\right\\} =: s(\\mathbf{t}),\\]\nwhere\n\\[{\\cal A} = \\set{1} \\cup \\set{1+t_i: 1 \\leq i \\leq d}\\]\nand for each $A \\in {\\cal A}$ the sets ${\\cal K}_1, {\\cal K}_2, {\\cal K}_3$ are defined as follows:\n\\[{\\cal K}_1 = \\set{k: 1 \\geq A}, \\quad {\\cal K}_2 = \\{k: 1+t_k \\leq A\\} \\setminus {\\cal K}_1, \\quad \\text{and} \\quad {\\cal K}_3 = \\set{1,\\dots,d} \\setminus ({\\cal K}_1 \\cup {\\cal K}_2). \\]\nNote that ${\\cal K}_1, {\\cal K}_2, {\\cal K}_3$ give a partition of $\\set{1,\\dots,d}$.\n\nTo obtain a neater expression for $s(\\mathbf{t})$, as given in the statement of Theorem \\ref{lower bound theorem}, we consider the possible cases which may arise. To this end, let us suppose, without loss of generality, that \n\\[0 < t_{i_1} \\leq t_{i_2} \\leq \\dots \\leq t_{i_d}.\\]\n\n\\smallskip\n\\underline{{\\bf Case 1: $A=1$}}\n\nIf $A = 1$, then ${\\cal K}_1 = \\set{1,\\dots,d}$, ${\\cal K}_2 = \\emptyset$, and ${\\cal K}_3 = \\emptyset$. In this case, the ``dimension number'' simplifies to\n\\[\\sum_{j=1}^{d}{\\delta_{j}} = \\sum_{j=1}^{d}{\\dimh{K_j}} = \\sum_{j=1}^{d}{\\frac{\\log{N_j}}{\\log{b}}} = \\frac{\\log{\\left(\\prod_{j=1}^{d}N_{j}\\right)}}{\\log{b}} = \\frac{\\log{N}}{\\log{b}} = \\dimh{K}.\\]\n\n\n\\smallskip\n\\underline{{\\bf Case 2:} $A=1+t_{i_k}$ with $t_{i_k}>0$}\n\\nopagebreak\n\nSuppose $A= 1+t_{i_k}$ for some $1 \\leq k \\leq d$ and that $t_{i_k}>0$ (otherwise we are in Case 1). Suppose $k \\leq k' \\leq d$ is the maximal index such that $t_{i_k} = t_{i_{k'}}$. In this case, \n\\[{\\cal K}_1 = \\emptyset, \\qquad {\\cal K}_2 = \\set{i_1, \\dots, i_{k'}}, \\quad \\text{and} \\quad {\\cal K}_3 = \\set{i_{k'+1},\\dots,i_d}\\]\nand the ``dimension number'' is\n\\begin{align*}\n    \\sum_{j=1}^{k'}{\\delta_{i_j}} + \\frac{\\sum_{j=k'+1}^{d}{\\delta_{i_j}}-\\sum_{j=1}^{k'}{t_{i_j}\\delta_{i_j}}}{1+t_{i_k}} &= \\frac{1}{1+t_{i_k}}\\left((1+t_{i_k})\\sum_{j=1}^{k'}{\\delta_{i_j}} + \\sum_{j=k'+1}^{d}{\\delta_{i_j}}-\\sum_{j=1}^{k'}{t_{i_j}\\delta_{i_j}}\\right) \\\\\n          &= \\frac{1}{1+t_{i_k}}\\left(\\sum_{j=1}^{d}{\\delta_{i_j}}+\\sum_{j=1}^{k'}{\\delta_{i_j}(t_{i_k}-t_{i_j})}\\right) \\\\\n          &= \\frac{1}{1+t_{i_k}}\\left(\\dimh{K} + \\sum_{j=1}^{k'}{(t_{i_k}-t_{i_j})\\dimh{K_j}}\\right).\n\\end{align*}\n\nPutting the two cases together, we conclude that \n\\[\\dimh{W(x,\\psi,\\mathbf{t})} \\geq \\min_{1 \\leq k \\leq d}\\left\\{\\frac{1}{1+t_k}\\left(\\gamma + \\sum_{j:t_j \\leq t_k}{(t_k-t_j)\\gamma_{j}}\\right)\\right\\},\\]\nas claimed. This completes the proof of Theorem \\ref{lower bound theorem}.\n\n\\section{Proof of Theorem \\ref{upper bound theorem}} \\label{upper bound section}\nLet \n\\begin{equation*}\n{\\cal A}_{n}(x, \\psi, \\mathbf{t}):= \\bigcup_{\\mathbf{i} \\in \\Lambda^{n}} \\Delta\\left(R^{x}_{\\mathbf{i}}, \\psi(b^{n})^{1+\\mathbf{t}}\\right)=\\bigcup_{\\mathbf{i} \\in \\Lambda^{n}}\\prod_{j=1}^{d}B\\left( R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{j}} \\right).\n\\end{equation*}\nThen\n\\begin{equation*}\nW(x, \\psi, \\mathbf{t}) = \\limsup_{n \\to \\infty} {\\cal A}_{n}(x, \\psi, \\mathbf{t}) \\, .\n\\end{equation*}\nFor any $m \\in \\N$ we have that \n\\begin{equation} \\label{cover_1}\nW(x, \\psi, \\mathbf{t}) \\subset \\bigcup_{n \\geq m} {\\cal A}_{n}(x, \\psi, \\mathbf{t}) \\, .\n\\end{equation}\nObserve that ${\\cal A}_{n}(x, \\psi, \\mathbf{t})$ is a collection of $N^{n}=(b^{n})^{\\gamma}$ rectangles with sidelengths $2\\psi(b^{n})^{1+t_{j}}$ in each $j$th coordinate axis.\n\nFix some $1 \\leq k \\leq d$. Throughout suppose that $n$ is sufficiently large such that $\\psi(b^{n})<1$. Condition $(i)$ of Theorem \\ref{upper bound theorem} implies that $\\psi(b^{n})^{1+t_{k}} \\leq \\psi(b^{n}) \\to 0$ as $n \\to \\infty$, and so for any $\\rho>0$ there exists a sufficiently large positive integer $n_{0}(\\rho)$ such that \n\\begin{equation*}\n\\psi(b^{n})^{1+t_{k}} \\leq \\rho \\quad \\text{ for all } n \\geq n_{0}(\\rho).\n\\end{equation*}\n Suppose $n \\geq n_0(\\rho)$ and that for each $1 \\leq j \\leq d$ we can construct an efficient finite $\\psi(b^{n})^{1+t_{k}}$-cover ${\\cal B}_{j}(\\mathbf{i},k,\\rho)$ for $B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{j}}\\right)$ with cardinality $\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho)$ for each $\\mathbf{i} \\in \\Lambda^{n}$. Then we can construct a $\\psi(b^{n})^{1+t_{k}}$-cover of\n$\\Delta\\left(R^{x}_{\\mathbf{i}}, \\psi(b^{n})^{1+\\mathbf{t}}\\right)$ for each $\\mathbf{i} \\in \\Lambda^{n}$ with cardinality $\\prod_{j=1}^{d}\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho)$ by considering the Cartesian product of the individual covers ${\\cal B}_j(\\mathbf{i}, k, \\rho)$ for each $1 \\leq j \\leq d$. By \\eqref{cover_1}\n\\begin{equation} \\label{cover_2}\n \\bigcup_{n \\geq n_{0}(\\rho)} {\\cal A}_{n}(x, \\psi, \\mathbf{t})\n\\end{equation}\nis a cover of $W(x, \\psi, \\mathbf{t})$. So, supposing that we can find such covers ${\\cal B}_{j}(\\mathbf{i},k,\\rho)$, we have that\n\\begin{equation*}\n\\bigcup_{n \\geq n_{0}(\\rho)}\\bigcup_{\\mathbf{i} \\in \\Lambda^{n}}\\prod_{j=1}^{d}{\\cal B}_{j}(\\mathbf{i},k,\\rho)\n\\end{equation*}\nis a $\\psi(b^{n})^{1+t_{k}}$-cover of $W(x,\\psi,\\mathbf{t})$.\n \nTo calculate the values $\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho)$ we consider two possible cases depending on the fixed $1\\leq k\\leq d$. Without loss of generality suppose that $0<t_{1}\\leq t_{2} \\leq \\dots \\leq t_{d}$. Then, since we are assuming that $\\psi(b^{n})<1$, we have that $\\psi(b^{n})^{1+t_{1}}\\geq \\dots \\geq \\psi(b^{n})^{1+t_{d}}$. \n\n\\smallskip\n\\underline{{\\bf Case 1: $t_{j} \\geq t_{k}$}}\n \nIn this case, $\\psi(b^{n})^{1+t_{k}} \\geq \\psi(b^{n})^{1+t_{j}}$ and so, for any $\\mathbf{i} \\in \\Lambda^{n}$, we have\n\\begin{equation*}\nB\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{k}}\\right) \\supset B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{|\\mathbf{i}|})^{1+t_{j}}\\right).\n\\end{equation*}\nHence, we may take our covers to be ${\\cal B}(\\mathbf{i},k, \\rho)=B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{k}}\\right)$, and so $\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho)=1$.\n\n\\smallskip\n\\underline{{\\bf Case 2: $t_{j} < t_{k}$}}\n\nIn this case, $\\psi(b^{n})^{1+t_{k}} < 2\\psi(b^{n})^{1+t_{j}}$. Let $u \\in \\N$ be the unique integer such that\n\\begin{equation} \\label{u_bound}\nb^{-u} \\leq 2\\psi(b^{n})^{1+t_{j}} < b^{-u+1},\n\\end{equation}\nand observe that, for any $\\mathbf{i} \\in \\Lambda^{n}$, we have \n\\begin{equation*}\nB\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{|\\mathbf{i}|})^{1+t_{j}}\\right) \\subset \\bigcup_{\\substack{\\mathbf{a} = (a_{1},\\dots,a_{u-1}) \\in \\Lambda_{j}^{u-1} \\\\ f_{a_i} \\in \\Phi^{j},\\, 1 \\leq i \\leq u-1}} f_{\\mathbf{a}}([0,1]),\n\\end{equation*}\nwhere $\\Lambda_j = \\{1,\\dots,N_j\\}$. Let $A$ denote the set of $\\mathbf{a} \\in \\Lambda_{j}^{u-1}$ such that\n\\begin{equation*}\n f_{\\mathbf{a}}([0,1]) \\cap B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{j}}\\right)\\neq \\emptyset.\n \\end{equation*}\nNote by the definition of $u$, and the fact that the mappings $f_{\\mathbf{a}}$ of the same length are pairwise disjoint up to possibly a single point of intersection, that $\\#A \\leq 2$ since \n\\begin{equation*}\n\\mathrm{diam} \\left(f_{\\mathbf{a}}([0,1])\\right)=b^{-(u-1)} > \\mathrm{diam} \\left( B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{n})^{1+t_{j}}\\right) \\right).\n\\end{equation*}\n Observe that $f_{\\mathbf{b}}([0,1]) \\subset f_{\\mathbf{a}}([0,1])$ if and only if $\\mathbf{b}=\\mathbf{a}\\mathbf{c}$ for $\\mathbf{c} \\in \\Lambda^{*}_{j}:=\\bigcup_{n=0}^{\\infty}{\\Lambda_j^n}$, where we write $\\mathbf{a}\\mathbf{c}$ to denote the concatenation of the two words $\\mathbf{a}$ and $\\mathbf{c}$. Let $v \\geq 0$ be the unique integer such that \n\\begin{equation} \\label{v_bound}\n b^{-u-v} \\leq \\psi(b^{n})^{1+t_{k}} < b^{-u-v+1}.\n \\end{equation}\nNote that $v$ is well defined since $\\psi(b^{n})^{1+t_{k}} < 2\\psi(b^{n})^{1+t_{j}} < b^{-u+1}$, and so $v \\geq 0$. Then \n\\begin{equation*}\n\\underset{\\mathbf{c} \\in \\Lambda_{j}^{v}}{\\bigcup_{\\mathbf{a} \\in A, } }f_{\\mathbf{a} \\mathbf{c}}([0,1]) \\supset B\\left(R^{x}_{\\mathbf{i},j}, \\psi(b^{|\\mathbf{i}|})^{1+t_{j}}\\right).\n\\end{equation*}\nNotice that the left-hand side above gives rise to a $\\psi(b^n)^{1+t_k}$-cover for the right-hand side and let us denote this cover by ${\\cal B}_{j}(\\mathbf{i},k,\\rho)$. By the above arguments an easy upper bound on $\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho)$ is seen to be $2N_{j}^{v}$. Furthermore, by \\eqref{u_bound} and \\eqref{v_bound} we have that\n\\begin{equation*}\n\\#{\\cal B}_{j}(\\mathbf{i},k,\\rho) \\leq 2N_{j}^{v} = 2 (b^{v})^{\\gamma_{j}} \\overset{\\eqref{v_bound}}{\\leq} 2\\left(b^{1-u}\\psi(b^{n})^{-1-t_{k}}\\right)^{\\gamma_{j}} \\overset{\\eqref{u_bound}}{\\leq}2^{1+\\gamma_j}b^{\\gamma_{j}}\\psi(b^{n})^{(t_{j}-t_{k})\\gamma_{j}}.\n\\end{equation*}\n\nSumming over $1 \\leq j \\leq d$ and $\\mathbf{i} \\in \\Lambda^n$ for each $n \\geq n_0(\\rho)$ we see that\n \\begin{align}\n{\\cal H}^{s}_{\\rho}(W(x, \\psi, \\mathbf{t})) \\, & \\ll \\sum_{n \\geq n_0(\\rho)}{\\left(\\left(\\psi(b^n)^{1+t_k}\\right)^s \\times \\sum_{\\mathbf{i} \\in \\Lambda^n}{\\prod_{j=1}^{d}{\\#{\\cal B}_j(\\mathbf{i},k,\\rho)}}\\right)} \\nonumber \\\\ \n&\\ll \\sum_{n\\geq n_{0}(\\rho)} \\left(\\psi(b^{n})^{1+t_{k}}\\right)^{s}N^{n} \\prod_{j:t_{j}<t_{k}} b^{\\gamma_{j}}\\psi(b^{n})^{(t_{j}-t_{k})\\gamma_{j}} \\nonumber \\\\\n& \\ll \\sum_{n\\geq n_{0}(\\rho)} \\psi(b^{n})^{s(1+t_{k})+\\sum_{j:t_{j}<t_{k}}(t_{j}-t_{k})\\gamma_{j} - \\gamma} \\left( \\psi(b^{n})b^{n} \\right)^{\\gamma}.\n\\end{align}\nThus, it follows from condition $(iii)$ in Theorem \\ref{upper bound theorem} that for any\n\\[s \\geq s_{0}=\\frac{\\gamma+\\sum_{j:t_j<t_k}(t_{k}-t_{j})\\gamma_{j}+\\delta\\gamma}{1+t_{k}} \\quad \\text{with } \\delta>0,\\] \nwe have\n\\[{\\cal H}^{s}_{\\rho}(W(x,\\psi,\\mathbf{t})) \\to 0 \\quad \\text{as } \\rho \\to 0.\\]\n This implies that $\\dimh W(x, \\psi, \\mathbf{t}) \\leq s_{0}$. The above argument holds for any initial choice of~$k$, and so we conclude that\n\\begin{equation*}\n\\dimh W(x, \\psi, \\mathbf{t}) \\leq \\min_{1 \\leq k \\leq d} \\left\\{ \\frac{1}{1+t_{k}}\\left(\\gamma+ \\sum_{j:t_j<t_k}(t_{k}-t_{j})\\gamma_{j}\\right) \\right\\}.\n\\end{equation*}\nCombining this upper bound with the lower bound result from Theorem \\ref{lower bound theorem} completes the proof of Theorem \\ref{upper bound theorem}.\n\n\\smallskip\n\\noindent{\\bf Acknowledgements.} The authors are grateful to Bal\\'{a}zs B\\'{a}r\\'{a}ny, Victor Beresnevich, Jason Levesley, Baowei Wang, and Wenmin Zhong for useful discussions.\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur main purpose here is to obtain generic properties, in the sense\nof Ma\\~n\\'e ~(see~\\cite{GoReMin},~\\cite{ManeGenProp}), for a convex\nand superlinear Lagrangian, in a fixed smooth and compact surface\n$M$ without boundary.\n\nOur main result is the  Kupka-Smale Theorem ,\nclaiming that, for each, $k\\in \\mathbb{R}$, \ngenerically, this level is regular, all periodic orbits in this level are\nnondegenerate at all orders and all heteroclinic intersections\nin this level are transversal. Where, Nondegeneracy  of order $m$, \nmeans that,  the linearized Poincar\\'e map, restricted to this energy level, \ndoes not have $m$-roots of the unity as eigenvalues.\n\nIn the proof  of the  Kupka-Smale Theorem , we will use the\nNondegeneracy Lemma (Lemma~\\ref{NondegeneracyLemma}) and a  Perturbation Lemma (Lemma~\\ref{PertTransvVariedEstInst}) for \nLagrangian submanifolds in order to\nget the transversality of heteroclinic \nintersections.\n\nThe  Kupka-Smale Theorem , in this formulation, resembles the Bumpy\nMetrics Theorem, for geodesic flows, formulated by R. Abraham in\n1968, and proved by D. V. Anosov in  1983\n(Anosov,~\\cite{Anos}).\n\nThe work of W. Klingenberg and F. Takens~\\cite{KgTk} in the Bumpy\nMetrics Theorem proof was corrected by Anosov~\\cite{Anos} using an\ninduction method similar to the one used by M. Peixoto~\\cite{Peix} in\nthe proof of the classical Kupka-Smale Theorem.\n\nIn this work, we will employ the same techniques used by  Anosov~\\cite{Anos},\nadapted to perturbations by potentials. In order to apply to the case of perturbation by\npotentials, it is necessary to introduce a \nmodification of the standard argument in the Control Theory for\ndifferential equations, initially used by Klingenberg~\\cite{Kg}, for\ngeodesic flow perturbation setting,  by J. A.\nMiranda~\\cite{Jan} for magnetic flows on surfaces and by\nContreras~\\cite{GonDomtd}  for the proof of Franks' Lemma for\ngeodesic flows.  In this case we do not  have especial coordinates, like Fermi\ncoordinates,  as in ~\\cite{Kg},~\\cite{Jan} and ~\\cite{GonDomtd}, thus\nwe introduce a new method without the use of tubular neighborhoods,\nthat solves this trouble. In the beginning of the proof\nwe use an argument similar to the one used by Robinson~\\cite{RobI}, Lemma\n19, Pg. 592, but the proof is quite different.\n\nObserve that the generic properties in Ma\\~n\\'e's sense cannot be\nobtained  from the pioneering work  of Robinson in the general Hamiltonian setting\n(see~\\cite{RobI} and~\\cite{RobII}) because the set of all\nHamiltonians is bigger than the set of all potentials in $M$.\n\nThe transversality is the easy part. Here we follow the approach of\nContreras \\& Paternain~\\cite{GonzaloPaternGenGeodFlowPositEntropy},\nLemma 2.6 or J.A. Miranda~\\cite{Jan}, Lemma 3.9. The problem in this\ncase is to construct an explicit potential that represents the\nperturbation.\n\nThe main obstacle in the proof of the Kupka-Smale Theorem  in dimension $n >2$ is\nthe nature of the perturbation constructed. As in\nContreras~\\cite{GonDomtd}, Lemma 7.3 and 7.4,  we need to solve a\nmatrix equation in the Lie algebra of the symplectic group $S_p(n)=\\{$Symplectic\nmatrices $2n \\times 2n\\}$. The solubility of this equation is\nstrongly related with the existence of repeated eigenvalues \nof the matrix $H_{pp}$ in local coordinates. The problem\nis that we can not change this characteristic  by adding a\npotential. Moreover, the equation involved is very complex too. \n\nHowever, we point out that the Kupka-Smale Theorem  in dimension 2 is a strong result in the study of generic Lagrangians because it works below the critical level. More than that, it can be combined with other results on the structure of Aubry-Mather sets in surfaces, like Haeflinger theorem for Mather measures with rational homology in an orientable surface, claiming that such measures are supported in periodic orbits, and  results on the nonexistence of conjugated points in the Aubry-Mather sets from Contreras and Iturriaga~\\cite{GoReConvex}, in order to guarantee  the hyperbolicity of the periodic orbits in this set.\n\n\\section{The  Kupka-Smale Theorem }\n\\vspace{0.3cm}\n\nWe consider $(M;g)$ a, $n$- dimensional, smooth and compact, Riemannian\nmanifold without boundary, $L: TM \\rightarrow\n\\mathbb{R} $, a Lagrangian in $M$, convex and fiberwise\nsuperlinear (see~\\cite{GoReMin} to definitions) and $H: T^{*} M\n\\rightarrow \\mathbb{R}$ the associated Hamiltonian obtained by\nLegendre transform.\n\n\nIn the study of generic properties of Lagrangians we use the\nconcept of genericity due to Ma\\~n\\'e. The idea is that, \nthe properties studied in the Aubry-Mather theory become much more\nstrongest in this generic setting. For more details and applications\nsee \\cite{GoReConvex}, \\cite{GoReMin}, \\cite{Jan} and\n\\cite{ManeGenProp}.\n\nWe will say that a property $\\mathcal{P}$ is generic, in Ma\\~n\\'e's\nsense,  for $L$,  if there exists a generic set\n$\\mathcal{O} \\subset C^{\\infty}(M;\\mathbb{R})$, in $C^{\\infty}$ topology,\nsuch that, for all $f \\in \\mathcal{O}$, $L+f$ has the property $\\mathcal{P}$.\n\nConsider $E_{L}(x,v)=\\frac{\\partial L}{\\partial v} (x,v)\\cdot v -\nL(x,v)$ the energy function associated to $L$ and $\\varepsilon\n^{k}_{L}=\\{(x,v) \\in TM\n\\mid E_{L}(x,v)=k\\}$ the set of all points in the energy level $k$.\n\nLet $\\theta \\in TM$ be a periodic\npoint of positive period, $T_{min}$ of the Euler-Lagrange \nflow $ \\phi_ {t} ^ {L}: TM \\to  TM$. \nFixed a local section transversal to this flow, $\\Sigma$\ncontained in the energy level of $\\theta$, there\nexists a smooth function $\\tau : U \\subset \\Sigma \\to\n\\mathbb{R}$, such that, $\\tau(\\theta)=T_{min} $ which is the\ntime of first return to $\\Sigma$,\nsuch that the map $P(\\Sigma, \\theta): U \\to\n\\Sigma$ given by\n$$P(\\Sigma, \\theta)(\\theta)=\\phi_{\\tau(\\theta)}^{L}(\\theta)$$\nis a local diffeomorphism and $\\theta$ is a fix point of $P(\\Sigma, \\theta)$.\nThis map is called Poincar\\'e  first return map.\nWe will say that $ \\theta $\n(or the orbit of $ \\theta $) is a\nnondegenerate orbit of order $m \\geq 1$ for $L$ if\n$$Ker((d_{\\theta} P(\\Sigma, \\theta))^{m}-Id)=0.$$\nThe property of Nondegeneracy of order $m$ means that $d_{\\theta} P(\\Sigma, \\theta)$\ndoes not have $m$-roots of the unity as eigenvalues.\n\nIf we are interested in the Hamiltonian viewpoint of the described\nLagrangian dynamics, then we consider the Hamiltonian $H$ associated to $L$  by the\nLegendre transform in the speed, that is, \n$$\\displaystyle H(x, p) = \\max_ {v \\in T_ {x} M} \\{pv - L (x, v) \\}.$$\nLet $X^ {H} $ be the Hamiltonian field, which is the unique field\n$X^ {H} $ in $T^{*} M$ such that $ \\omega_ {\\vartheta} (X^{H}\n(\\vartheta), \\xi) = d_ {\\vartheta} H \\xi$ for all $ \\xi \\in T_\n{\\vartheta} T^{*} M$. In the local coordinates $ (x, p) $, \n$X^ {H} = H_{p} \\, \\frac{\\partial \\,}{\\partial x} -\nH_{x} \\, \\frac{\\partial \\,}{\\partial p} $.  We denote by\n$ \\psi_ {t} ^ {H}: T^ {*} M \\to   T^{*} M$ the\nflow in $T^{*}M$ associated with the Hamiltonian field $X^ {H}: T^\n{*} M \\rightarrow TT^ {*} M$. This flow preserves the canonical symplectic\nform $\\omega$.  Since $L$ is a convex and superlinear Lagrangian  we have that $H$ is a\nconvex and superlinear Hamiltonian. Using the Legendre transform\n\\begin{center} $p=L_v (x, v) $ and $v=H_p (x, p) $, \\end{center}\nwe have that, $H_ {pp} (x, p) $ is positive defined in $T^ {*}_{x} M$,\nuniformly in $x \\in M$. Observe that the Legendre transform associates the energy\nlevel $\\varepsilon_{k}^{L} $ with the level set $H^{- 1}(k)$ of $H$.\nFrom the conjugation property between Lagrangian and Hamiltonian viewpoint, the nondegeneracy of an orbit is equivalent in both senses.\n\nOne can prove that the restriction of the symplectic form $\\omega$\nto $T_ {\\vartheta} \\Sigma$ is nondegenerate and closed form,\ntherefore the Poincar\\'e map is symplectic.Moreover, \n$$d_ {\\vartheta} \\psi_ {T_ {min}} ^ {H} (\\xi) = - d_\n{\\vartheta} \\tau (\\xi) X^ {H} + d_ {\\vartheta} P (\\Sigma, \\vartheta)\n(\\xi), \\; \\forall \\xi \\in T_ {\\vartheta} \\Sigma.$$ Therefore we have\nthat \\begin{center} \\( d_ {\\vartheta} \\psi_ {T_ {min}} ^ {H} \\mid_\n{T_ {\\vartheta} H^ {- 1} (k)} = \\left [ \\begin{matrix} 1 & d_\n{\\vartheta} \\tau   \\\\ 0   & d_ {\\vartheta} P (\\Sigma, \\vartheta)\n\\end{matrix} \\right], \\) \\end{center} in general for $T=m T_\n{min} $ \\\\ $d_ {\\vartheta} \\psi_ {T}^{H} (\\xi) = -d_ {\\vartheta}\n\\tau (\\sum_ {i=0} ^ {M-1} d_ {\\vartheta} P (\\Sigma, \\vartheta)^ {i})\n(\\xi) X^{H} + d_ {\\vartheta} P (\\Sigma, \\vartheta)^{m} (\\xi), \\;\n\\forall \\xi \\in T_ {\\vartheta} \\Sigma$.\n\nSo, the condition of that $ \\vartheta $ is nondegenerate of\norder $m \\geq 1$ is equivalent to say that the algebraic\nmultiplicity of $ \\lambda=1$ as eigenvalue of $d_ {\\vartheta} \\psi_\n{T}^{H} \\mid_{T_ {\\vartheta} H^ {- 1} (k)} $ is equal to 1, because\nthe characteristic polynomials are related by\n$\\mathfrak {p}_{d_{\\vartheta} \\psi_{T}^{H}} (\\lambda) = (1- \\lambda)\n\\cdot \\mathfrak{p}_{d_{\\vartheta} P(\\Sigma, \\vartheta)^{m}}\n(\\lambda) $.\n\nOur main result is the Kupka-Smale Theorem  that relies the \nBumpy Metrics Theorem proved by Anosov~\\cite{Anos}, but here\nfor the Lagrangian setting.\n\n\nWe state our main result just when $dim(M)=2$  because we do not know\nthe proof for Theorem~\\ref{PerturbacaoLocaldaOrbita} when $dim (M)=n\n\\geq 3$. This problem remains an open question.\n\n\\begin{theorem}\\label{KS}\\textnormal{(Kupka-Smale Theorem)}\n Suppose $dim (M)=2$. Let $L: TM \\to\n\\mathbb{R} $, be a Lagrangian in $M$, convex and fiberwise\nsuperlinear. Then, for each  $k \\in \\mathbb{R}$, the property\\\\\n i) $\\varepsilon ^{k}_{L}$ is regular;\\\\\n ii) Any periodic orbit in the level $\\varepsilon ^{k}_{L}$ is\n nondegenerate for all orders;\\\\\n iii) All heteroclinic intersections, in this level, are transversal.\\\\\n is generic for $L$.\n\\end{theorem}\n\\section{Proofs of the main results}\n\nGiven $k \\in \\mathbb{R}$, we define the set of the regular potentials for $k$, as being\n$$ \\mathcal{R}(k)=\\{ \\;f \\in C^{\\infty}(M;\\mathbb{R}) \\mid\n\\varepsilon ^{k}_{f}:=(H+f)^{-1}(k) \\; is \\; regular \\; \\},$$\n where $H$ is the associated Hamiltonian.\n\n\\begin{lemma}\\label{compacidade}\nConsider  $k \\in \\mathbb{R}$ and $f_{0} \\in C^{\\infty}(M;\\mathbb{R})$.\nFor each sequence\n$f_{n} \\to f_{0}$ in $C^{\\infty}(M;\\mathbb{R})$ topology and points\n$\\vartheta_{n}=(x_{n},p_{n}) \\in \\varepsilon ^{k}_{f_{n}}$\nthere exists a subsequence $\\vartheta_{n_{i}} \\to \\vartheta_{0} \\in \\varepsilon ^{k}_{f_{0}}$.\n\\end{lemma}\n\nIn fact, this lemma is an easy consequence of the compactness of the energy level.\n\n\\begin{theorem}\\label{PotRegAbertoDenso}\\textnormal{(Regularity of the energy level)}\n Given $k \\in \\mathbb{R}$, the subset\n $\\mathcal{R}(k)$ is open and dense in\n $C^{\\infty}(M;\\mathbb{R})$.\n\\end{theorem}\n\\begin{proof}\nThe openness of the set $\\mathcal{R}(k)$ follows directly of the\nLemma~\\ref{compacidade}. In order to obtain the density of $\\mathcal{R}(k)$ in\n$C^{\\infty}(M;\\mathbb{R})$, consider $f_{0} \\in C^{\\infty}(M;\\mathbb{R})$\nand $\\mathcal{U}$, a neighborhood  that contains a ball of radius $\\varepsilon > 0$, \nand center, $f_{0}$.  We claim that $\\mathcal{U} \\cap \\mathcal{R}(k) \\neq \\varnothing$. In fact, if it is not the case, we can achieve a contradiction by considering the Hamiltonian $H_{\\delta}:=H + (f_{0}+ \\delta)$, with $\\delta \\in (0,\\varepsilon)$.\n\\end{proof}\n\n{\\large \\textbf{The  Nondegeneracy Lemma }}\\\\\n\n\nGiven $k \\in \\mathbb{R}$ and $0< a \\leq b \\in \\mathbb{R}$,\nwe define the set $\\mathcal{G}_{k}^{a,b} \\subseteq \\mathcal{R}(k)$\nas   $\\mathcal{G}_{k}^{a,b}= \\{f \\in \\mathcal{R}(k) \\mid $\nall periodic points  $\\vartheta \\in (H+f)^{-1}(k)$, \nwith $T_{min}(\\vartheta) \\leq a$ are nondegenerate of order $m$ \nfor $H+f$, \\, $\\forall \\, m \\leq \\frac{b}{T_{min}} \\}$.  Observe that, \n   if we have $a,a',b,b' \\in \\mathbb{R}$, such that,\n   $0 < a, a'  < \\infty$, $a \\leq a'$ and $ b \\leq b'$, then\n   $\\mathcal{G}_{k}^{a',b'} \\subseteq \\mathcal{G}_{k}^{a,b}$.\nWe define $\\displaystyle\n\\mathcal{G}(k)=\\bigcap_{n=1}^{+ \\infty} \\mathcal{G}_{k}^{n,n}$. Then\n$\\mathcal{G}(k)$ is the set of all potentials $f \\in \\mathcal{R}(k)$\nsuch that, all periodic orbits with positive period in the\nenergy level $(H+f)^{-1}(k)$ are nondegenerate of all  orders\nfor $H+f$.\n\\begin{lemma}\\label{NondegeneracyLemma}\\textnormal{(Nondegeneracy Lemma)}\nGiven $k \\in \\mathbb{R}$ and $0< c \\in \\mathbb{R}$, the\nset $\\mathcal{G}_{k}^{c,c}$ is open and dense in\n$C^{\\infty}(M;\\mathbb{R})$.\n\\end{lemma}\nIf $\\mathcal{G}_{k}$ is generic, then  generically in $L$, the energy\nlevel $k$ is regular and all periodic orbits in this \nlevel  are nondegenerate of all  orders for $H+f$. \nThus, we must to prove that $\\mathcal{G}_{k}^{c,c}$ is open in\n $C^{\\infty}(M;\\mathbb{R})$ , $\\forall c \\in \\mathbb{R}_{+}$ and\ndense in $\\mathcal{R}(k)$, since Theorem~\\ref{PotRegAbertoDenso},\nimplies that $\\mathcal{R}(k)$ is dense in  $C^{\\infty}(M;\\mathbb{R})$. \nThe proof of this lemma requires a sequence of technical constructions.\n\\begin{lemma}\\label{MinimoPerioLema}\nGiven $k \\in \\mathbb{R}$ and $f_{0} \\in \\mathcal{R}(k)$ there exists a neighborhood, $\\mathcal{U}$,\nof $f_{0}$ in $C^{\\infty}(M;\\mathbb{R})$ and $0< \\alpha:=\\alpha(\\mathcal{U},f_{0})$\nsuch that, for all $f \\in \\mathcal{U}$,  the period of all periodic orbits of $H+f$, in the level $(H+f)^{-1}(k)$ , is bounded below by $\\alpha$.\n\\end{lemma}\n\\begin{proof}\nIf we suppose that our claiming  is false, we get the existence of sequences,\n$\\mathcal{U} \\ni f_{n} \\to f_{0}$,  $ T_{n} >\n0$ with $ T_{n} \\to  0$ and $\\vartheta_{n} \\in (H+f_{n})^{-1}(k)$\nsuch that $\\psi_{T_{n}}^{H+f_{n}}(\\vartheta_{n})=\\vartheta_{n}$.\n\nFrom Lemma~\\ref{compacidade}, we can choose a subsequence such that\n$$d_{T^{*}M}(\\psi_{t}^{H+f_{n}}(\\vartheta_{0}),\n\\vartheta_{0})=0, \\, \\forall t>0$$, that is, $\\vartheta_{0} \\in\n(H+f_{0})^{-1}(k)$ which is a fix point, contradicting the fact of $f_{0} \\in \\mathcal{R}(k)$. \n\\end{proof}\n\\begin{lemma}\\label{openlema}\nGiven $k \\in \\mathbb{R}$, $a,b \\in \\mathbb{R}$ with $0 < a \\leq b < \\infty$, the set\n$\\mathcal{G}_{k}^{a,b}$ is open in  $C^{\\infty}(M;\\mathbb{R})$.\n\\end{lemma}\n\\begin{proof}\nIf, $\\mathcal{G}_{k}^{a,b} \\neq\n\\varnothing$, take $f_{0} \\in\n\\mathcal{G}_{k}^{a,b}$. If $f_{0}$ is not an interior point we get the existence of a\nsequence  $f_{n} \\to f_{0}$ where\n$f_{n} \\not\\in \\mathcal{G}_{k}^{a,b}$. Therefore, there exists $\\vartheta_{n} \\in\n(H+f_{n})^{-1}(k)$, $ T_{n}=T_{min}(\\vartheta_{n}) \\in\n(0;a]$ and natural numbers $\\ell_{n} \\geq 1$ such that,\n$\\ell_{n}T_{n} \\leq b$, $\\psi_{\\ell_{n}T_{n}}^{H+f_{n}}(\\vartheta_{n})=\n\\vartheta_{n}$  and  $d_{\\vartheta_{n}}\n\\psi_{\\ell_{n}T_{n}}^{H+f_{n}}$ do not have 1 as eigenvalue\nwith algebraic multiplicity bigger than 1.\nConsider $\\mathcal{U}_{0}$ and $0 <\n\\alpha:=\\alpha(\\mathcal{U}_{0},f_{0}) <a$ as in\nLemma~\\ref{MinimoPerioLema}. Choosing a subsequence we can assume that\n$f_{n} \\in \\mathcal{U}_{0}$ and therefore $ T_{n} \\in [\\alpha;a]$,\nwith $\\vartheta_{n} \\to  \\vartheta_{0} \\in \\varepsilon^{k}_{f_{0}}$,\\, $T_{n} \\to\nT_{0}$,\\, $\\ell_{n} =  \\ell_{0}$,\\, $0< \\alpha \\leq T_{0} \\leq a$\\,\nand \\, $\\ell_{0}T_{0} \\leq b$.   Then $\\psi_{\\ell_{0}T_{0}}^{L+f_{0}}(\\vartheta_{0})=\\vartheta_{0}$, and\n$d_{\\vartheta_{0}}\\psi_{\\ell_{0}T_{0}}^{H+f_{0}}$ has 1 as eigenvalue\nwith algebraic multiplicity bigger than 1, that is, $\\vartheta_{0}$\nis a periodic orbit with minimal period $\\leq$ $a$, degenerate of order\n$\\ell_{0} \\leq \\frac{b}{T_{0}}$,  contradicting the fact of\n$f_{0} \\in  \\mathcal{G}_{k}^{a,b}$.\n\\end{proof}\n\n\nIn order to  prove that $\\mathcal{G}_{k}^{c,c}$ is dense in\n$C^{\\infty}(M;\\mathbb{R})$ , $\\forall c \\in \\mathbb{R}_{+}$, we observe that is enough  show that, $\\mathcal{G}_{k}^{c,c}$ is dense in\n$\\mathcal{R}(k)$. So, we can reduce this proof to a local approach.\nMore precisely, the claim is a direct consequence of the following\nReduction Lemma, whose proof we will  present in the Section~\\ref{ReducLema}.\n\\begin{lemma}\\label{ReduLocalDaPrimParte} \\textnormal{(Reduction Lemma)} \nFor each  $c \\in \\mathbb{R}_{+}$, and any $f_{0} \\in \\mathcal{R}(k)$,\nthere exists an open neighborhood $\\mathcal{U}_{f_{0}}$ of $f_0$, such that,\n$\\mathcal{G}_{k}^{c,c} \\cap \\mathcal{U}_{f_{0}}$ is dense in $\\mathcal{U}_{f_{0}}$.\n\\end{lemma}\nThus the Nondegeneracy Lemma is proven.\\\\\n\n{\\large \\textbf{Proof of  the Kupka-Smale Theorem }}\\\\\n\n   Consider a periodic orbit $\\gamma=\n\\{\\phi_{t}^{L}(\\theta_{0}), \\, 0 \\leq t \\leq T \\} \\subseteq\nH^{-1}(k)$, in $T^{*}M$, where $H$ is the Hamiltonian \nassociated to $L$ by the Legendre transform.\nWe will say that this orbit is hyperbolic if the Poincar\\'e map\nassociated does not have eigenvalue of norm 1. It is clear that the\nhyperbolicity implies in the nondegeneracy property. The\nconverse is not true. There exists  nondegenerate\norbits such that all eigenvalues has norm 1. Such orbits will be\ncalled elliptic orbits.\nWe define the  strong stable and strong unstable manifolds, of\n$\\gamma$ in $\\theta_{0}=\\gamma(0)$, as\n$$\\displaystyle W^{ss}(\\theta_{0})=\\{ \\theta \\in H^{-1}(k) \\mid\n\\lim_{t \\to +\\infty} d(\\phi_{t}^{L}(\\theta_{0}),\\phi_{t}^{L}(\\theta))=0 \\}$$\nand\n$$\\displaystyle W^{us}(\\theta_{0})=\\{ \\theta \\in H^{-1}(k) \\mid\n\\lim_{t \\to -\\infty} d(\\phi_{t}^{L}(\\theta_{0}),\\phi_{t}^{L}(\\theta))=0 \\}.$$\n\nRespectively we define the stable and unstable manifolds (weak) of\n$\\gamma$ as $$\\displaystyle W^{s}(\\gamma)=\\bigcup_{t \\in\n\\mathbb{R}}\\phi_{t}^{L}(W^{ss}(\\theta_{0}))  \\text{ and }  W^{u}(\\gamma)=\\bigcup_{t \\in\n\\mathbb{R}}\\phi_{t}^{L}(W^{us}(\\theta_{0})).$$\nFrom the general theory of the Lagrangians systems we know that,\n$W^{s}(\\gamma),$ $ W^{u}(\\gamma) \\subset H^{-1}(k)$ are Lagrangians submanifolds\nof $TM$, with the symplectic twist form, given by\n$\\omega(\\xi, \\zeta)= \\langle (\\xi_{h}, \\xi_{v})^{*}, J (\\zeta_{h},\n\\zeta_{v}) \\rangle $\nin local coordinates.\n\nA point $\\theta \\in  H^{-1}(k)$,  is heteroclinic if  $\\theta\n\\in W^{s}(\\gamma_{1}) \\cap W^{u}(\\gamma_{2})$. where $\\gamma_{1}, \\gamma_{2} \\subset\nH^{-1}(k)$ are hyperbolic periodic orbits. Additionally,\nif $T_{\\theta}W^{s}(\\gamma_{1})$ $ + T_{\\theta}W^{s}(\\gamma_{2}) =\nT_{\\theta}H^{-1}(k)$, that is, if  $W^{s}(\\gamma_{1}) \\pitchfork_{\\theta}\nW^{u}(\\gamma_{2})$  then $\\theta$  will be called a transversal heteroclinic point, same\nthing for homoclinics.\n\nA fundamental domain for $W^{s}(\\gamma)$ (or $W^{u}(\\gamma)$)\nis a compact subset $\\mathcal{D} \\subset W^{s}(\\gamma)$,\nsuch that, all orbits in $W^{s}(\\gamma)$  intercepts\n$\\mathcal{D}$ in one point, at least. One can show that there exists fundamental\ndomains arbitrarily small and arbitrarily close to $\\gamma$.\nFixed $a >0$ we define the local stable and local unstable\nsubmanifolds of $\\gamma$ as being\n$$W^{s}_{a}(\\gamma)=\\{ \\theta \\in W^{ss}(\\gamma) \\mid\nd_{W^{ss}(\\gamma)}(\\theta, \\gamma) < a \\}$$\nand\n$$W^{u}_{a}(\\gamma)=\\{ \\theta \\in W^{us}(\\gamma) \\mid\nd_{W^{us}(\\gamma)}(\\theta, \\gamma) < a \\}.$$\nThey are Lagrangians submanifolds of $TM$.\n\nIn order to prove the  Kupka-Smale Theorem , we define\n$\\mathcal{K}^{a}_{k}=\\{ f \\in \\mathcal{G}^{a,a}_{k} \\mid\n\\forall \\gamma_{1}, \\gamma_{2} \\subset  (H+f)^{-1}(k),$ hyperbolic periodic orbits\\\\\nfor $L+f$, with period $\\leq a$ we have $W^{s}_{a}(\\gamma_{1}) \\pitchfork\nW^{u}_{a}(\\gamma_{2}) \\}$   and\n$\\displaystyle \\mathcal{K}(k)=\\bigcap_{n \\in \\mathbb{N}} \\mathcal{K}^{n}_{k}$.\nIt is clear that the properties (i), (ii) and (iii) of the Kupka-Smale Theorem   \nare valid for all $f \\in \\mathcal{K}(k)$.\nThus, in order to prove the  Kupka-Smale Theorem , we must to show that $\\mathcal{K}(k)$\nis generic, or equivalent, that each, $\\mathcal{K}^{n}_{k}$\nis an open and dense set (in $C^{\\infty}$ topology).\nSince, the local stable and unstable  manifolds depends $C^{1}$ continuously \non compact parts, of the Lagrangian field,\nwe get the openness of $\\mathcal{K}^{n}_{k} $,\nbecause the transversality is an open property.\n\nThe next lemma can be found  in Paternain~\\cite{PaternainGeodFlows}, Proposition\n2.11, Pg.34, for the geodesic case, but here we present a Lagrangian\nversion.\n\n\\begin{lemma} \\label{PropTwistDoFibradoVertical} \\textnormal{(Twist Property of the vertical\nbundle)} Let $L$  be a smooth, convex and superlinear, Lagrangian\nin $M$, $\\theta \\in TM$ and $F \\subset T_{\\theta} TM$ an Lagrangian\nsubspace for the twist form in $T^{*}M$. Then, the set,\n$$\\mathcal{Z}_{F}=\\{ t \\in \\mathbb{R} \\mid d_{\\theta}\\phi_{t}^{L}(E)\n\\cap V(\\phi_{t}^{L}(\\theta))\n\\neq \\varnothing \\}$$ is discrete, where $V$ is the vertical bundle in $M$.\n\\end{lemma}\n\nThe next lemma allow us to make a local perturbation of a potential\n$f$ in such way that the correspondent stable and unstable manifolds\nbecome transversal in a certain heteroclinic point $\\theta$. The\ndensity of $\\mathcal{K}^{a}_{k}$ follows from\nLemma~\\ref{DensidadeTransvVariedEstInst}.\n\\begin{lemma}\\label{PertTransvVariedEstInst}\nLet $L$ be a Lagrangian, and $f \\in C^{\\infty}(M,\\mathbb{R})$. \nGiven $\\gamma_{1}, \\gamma_{2} \\subset\n(H-f)^{-1}(k)$ hyperbolic periodic orbits with period $\\leq a$ and\n$\\theta \\in W_{a}^{u}(\\gamma_{2})$, such that, the canonic\nprojection $\\pi \\mid_{W_{a}^{u}(\\gamma_{2})}$ is a local\ndiffeomorphism in $\\theta$ and $U, V$, are neighborhoods of \n$\\theta$ in $TM$ such that $\\theta \\in V \\subset\n\\bar{V} \\subset U$.\nThen, there exists $\\bar{f} \\in C^{\\infty}(M,\\mathbb{R})$, such that,\\\\\ni) $\\bar{f} $ is $ C^{\\infty}$ close to $f$;\\\\\nii) $\\mathop{\\rm supp}\\nolimits(  f - \\bar{f}) \\subset \\pi (U)$; \\\\\niii) $\\gamma_{1}, \\gamma_{2} \\subset  (H-\\bar{f})^{-1}(k)$ are\nhyperbolic periodic orbits to $\\bar{f}$, with the same period as to $f$; \\\\\niv) The connected component of $W_{a}^{u}(\\gamma_{2}) \\cap V$ that\ncontains $\\theta$ is transversal to $W^{s}(\\gamma_{1})$.\n\\end{lemma}\n\\begin{proof}\nInitially we consider the Hamiltonian $H-f$ associated to the Lagrangian\n$L+f$ by the Legendre transform $\\mathcal{L}$:\n$$ H-f(x,p)=\\sup_{v \\in T_{x}M} \\{ p(v)- (L+f)(x,v)\\} $$\nwith the canonic symplectic form of $T^{*}M$, $\\omega = \\sum dx_{i} \\wedge dp_{i}.$\n\nWe know, from the general theory of the Hamiltonian systems, that\n$\\gamma_{1}, \\gamma_{2}$ are in correspondence, by Legendre\ntransform with hyperbolic periodic orbits of same period,\n$\\tilde{\\gamma_{1}}, \\tilde{\\gamma_{2}} \\subset (H-f)^{-1}(k)$ for\nthe Hamiltonian flow $\\psi_{t}^{H-f}$. Consider\n$\\tilde{W}^{u}(\\tilde{\\gamma_{2}})$ and\n$\\tilde{W}^{s}(\\tilde{\\gamma_{1}})$, respectively, the invariant\nsubmanifolds, they will be Lagrangian submanifolds of $T^{*}M$, and\n$\\vartheta= \\mathcal{L}(\\theta) \\in\n\\tilde{W}^{u}(\\tilde{\\gamma_{2}})$.\nIf $\\pi:TM \\to M$ and $\\pi^{*}:T^{*}M \\to M$ are the\ncanonic projections, then\n$d_{\\vartheta}\\pi^{*}= d_{\\theta}\\pi  \\circ (d_{\\theta}\\mathcal{L})^{-1}$.\nTherefore the canonic projection $\\pi^{*}\n\\mid_{\\tilde{W}^{u}(\\tilde{\\gamma_{2}})}$ is a local diffeomorphism\nin $\\vartheta$. Moreover, $X^{H-f}(\\vartheta)=d_{\\theta}\\mathcal{L}\n\\circ X^{L+f}(\\theta) \\neq 0$. Thus we can prove the lemma in the\nHamiltonian setting.\nBy \\cite{GonzaloPaternGenGeodFlowPositEntropy}, Lemma A3,  we can\nfind a neighborhood $ U$ of $\\vartheta$, and $V \\subset U$, such\nthat, $ V \\subset \\bar{V} \\subset U$, and a Lagrangian submanifold,\n$\\mathcal{N}$, $C^{\\infty}$ close to\n$\\tilde{W}^{u}(\\tilde{\\gamma_{2}})$, satisfying the following conditions\\\\\n1) $\\vartheta \\in \\{ U \\backslash \\bar{V} \\}$\\\\\n2) $\\mathcal{N} \\cap \\{ U \\backslash \\bar{V} \\} =\n\\tilde{W}^{u}(\\tilde{\\gamma_{2}}) \\cap \\{ U \\backslash \\bar{V} \\}\n\\subset (H - f)^{-1}(k)$;\\\\\n3) $\\mathcal{N} \\cap \\bar{V}  \\pitchfork\n\\tilde{W}^{s}(\\tilde{\\gamma_{1}})  \\cap \\bar{V}$.\n\nAs, $\\mathcal{N}$ is  $C^{\\infty}$ close to\n$\\tilde{W}^{u}(\\tilde{\\gamma_{2}})$, we have that the canonic\nprojection $\\pi^{*} \\mid_{\\mathcal{N}}$ is a local\ndiffeomorphism in $\\vartheta$. If $U$ is\nsmall enough, then\n$\\mathcal{N} \\cap U =\\{ (x, p(x)) \\mid x \\in \\pi^{*}(u)\\}$ \nthat is, $\\mathcal{N}\\mid_{U}$ is a $C^{\\infty}(M,\\mathbb{R})$\ngraph.  We define the following potential, $\\bar{f} \\in\nC^{\\infty}(M,\\mathbb{R})$,\n$$\n\\bar{f}(x)= \\left \\{\n\\begin{array}{clcr}\nf(x) \\quad \\, \\quad \\quad \\quad \\quad if \\; x \\in \\pi^{*}(U)^{C}\\\\\nH(x,p(x))-k \\quad \\, if \\; x \\in \\pi^{*}(U) \\quad _{}\n\\end{array}\n\\right.\n$$\n\nObserve that, $\\mathop{\\rm supp}\\nolimits(  f - \\bar{f}) \\subset \\pi^{*}(U)$ and\n$\\vartheta \\not\\in \\mathop{\\rm supp}\\nolimits( f - \\bar{f})$, moreover, choosing $U$ small\nenough, we will have that $\\pi^{*}(U) \\cap \\{ \\tilde{\\gamma_{1}},\n\\tilde{\\gamma_{2}} \\} =\\varnothing$ and therefore\n$\\tilde{\\gamma_{1}}, \\tilde{\\gamma_{2}}$ still, hyperbolic periodic\norbits of same period for the Hamiltonian flow\n$\\psi_{t}^{H-\\bar{f}}$, contained in $(H-\\bar{f})^{-1}(k)$.\nWe denote $\\bar{W}^{u}(\\tilde{\\gamma_{2}})$  and\n$\\bar{W}^{s}(\\tilde{\\gamma_{1}})$, the invariant  manifolds for the\nnew flow $\\psi_{t}^{H-\\bar{f}}$.\nClearly $(H-\\bar{f})(\\mathcal{N})=k$. By\n\\cite{GonzaloPaternGenGeodFlowPositEntropy}, Lemma A1, we have that\n$\\mathcal{N}$ is $\\psi_{t}^{H-\\bar{f}}$ invariant. Since,\n$\\bar{W}^{u}(\\tilde{\\gamma_{2}})$ depends only of the negative times\nand, the connected component of $\\bar{W}^{u}(\\tilde{\\gamma_{2}})\n\\cap U$ that contains $\\vartheta$ and $\\mathcal{N}$ are coincident\nin a neighborhood of $ \\tilde{\\gamma_{2}}$ disjoint of  $\\mathop{\\rm supp}\\nolimits( f -\n\\bar{f})$, we have \n$\\mathcal{N}=\\bar{W}^{u}(\\tilde{\\gamma_{2}})$.\nOn the other hand, as $\\bar{W}^{s}(\\tilde{\\gamma_{1}})$ depends only\nof the positive times and $f=\\bar{f}$ in $\\{ U \\backslash \\bar{V}\n\\}$, we have $\\bar{W}^{s}(\\tilde{\\gamma_{1}}) =\n\\tilde{W}^{s}(\\tilde{\\gamma_{1}})$. Since $\\mathcal{N} \\cap \\bar{V}\n\\pitchfork \\tilde{W}^{s}(\\tilde{\\gamma_{1}})  \\cap \\bar{V}$,  we have\n$\\bar{W}^{u}(\\tilde{\\gamma_{2}}) \\cap \\bar{V} \\pitchfork\n\\tilde{W}^{s}(\\tilde{\\gamma_{1}})  \\cap \\bar{V}.$ From the\ninitial considerations we choose $L+ \\bar{f} $. The lemma is proven.\n\\end{proof}\n\n\\begin{lemma}\\label{DensidadeTransvVariedEstInst}\nThe set $\\mathcal{K}^{n}_{k}$, is dense in\n$C^{\\infty}(M,\\mathbb{R})$, for all $n \\in \\mathbb{N}$.\n\\end{lemma}\n\\begin{proof}\nTake $f_{0} \\in C^{\\infty}(M,\\mathbb{R})$, by the Nondegeneracy Lemma\nwe can  find $f_{0}$ arbitrarily  close to $f' \\in\n\\mathcal{G}_{k}^{n,n}$, which  is open and dense. Thus, is enough \nto find $f$ arbitrarily close to $f'$, such that, for any\n$\\gamma_{1}, \\gamma_{2} \\subset (H-f)^{-1}(k)$, hyperbolic periodic\norbits of period $\\leq n$, is valid  $W_{n}^{s}( \\gamma_{1})\n\\pitchfork W_{n}^{u}( \\gamma_{2})$. Then, $f \\in\n\\mathcal{K}^{n}_{k}$ and $f$ is arbitrarily close to $f_{0}$.\nGiven $\\gamma_{1}, \\gamma_{2} \\subset (H+f')^{-1}(k)$\nhyperbolic periodic orbits of period $\\leq n$, in order to conclude\nthat $W_{n}^{s}( \\gamma_{1}) \\pitchfork W_{n}^{u}( \\gamma_{2})$ we \nshould to prove that $W_{n}^{s}( \\gamma_{1})\n\\pitchfork_{\\mathcal{D}} W_{n}^{u}( \\gamma_{2})$ where $\\mathcal{D}$\nis a fundamental domain of $W^{u}( \\gamma_{2}) $ , because if\n$W^{s}( \\gamma_{1}) \\pitchfork_{\\theta} W^{u}( \\gamma_{2})$ then\n$W^{s}( \\gamma_{1}) \\pitchfork_{\\phi_{t}^{L+f'}(\\theta)} W^{u}(\n\\gamma_{2}), \\, \\forall t$.\nTake $\\mathcal{D}$ a fundamental domain of $W^{u}( \\gamma_{2}) $ and\n$\\theta \\in \\mathcal{D}$. By the inverse function theorem we know\nthat $\\pi|_{W^{u}( \\gamma_{2})}$ is a local diffeomorphism in\n$\\theta$ if, and only if, $T_{\\theta}W^{u}( \\gamma_{2}) \\cap Ker\nd_{\\theta} \\pi ={0}$. As $W^{u}( \\gamma_{2})$ is a Lagrangian\nsubmanifold we have, from Lemma~\\ref{PropTwistDoFibradoVertical},\nthat\n$\\{ t \\in \\mathbb{R} \\mid d_{\\theta}\\phi_{t}^{L}(T_{\\theta}W^{u}( \\gamma_{2}))\n\\cap Ker \\, d_{\\phi_{t}^{L+f'}(\\theta)} \\pi \\neq \\varnothing \\},$ is\ndiscrete. Then there exists $t(\\theta)$ arbitrarily close to $0$, such\nthat, $\\pi|_{W^{u}( \\gamma_{2})}$ is a local diffeomorphism in\n$\\tilde{\\theta}=\\phi_{t(\\theta)}^{L+f'}(\\theta)$.\nAs, $f' \\in \\mathcal{G}_{k}^{n,n}$, we can choose, $t(\\theta)$, such\nthat, $\\pi(\\tilde{\\theta})$ does not intercept any periodic orbit of\nperiod  $\\leq n$. Fix a  neighborhood $U$, of $\\tilde{\\theta}$,  arbitrarily small,\nsuch that, $\\pi(U)$ does not intercept any\nperiodic orbit of period  $\\leq n$. Taking $V$, a neighborhood of\n$\\tilde{\\theta} $, such that,  $V \\subset \\bar{V} \\subset U$, from\nLemma~\\ref{PertTransvVariedEstInst}, we can find $f_{1}=f'$ in\n$\\pi(U)^{C}$, such that, the connected component of\n$W_{n}^{u}( \\gamma_{2}) \\cap \\bar{V}$ (to the new flow)\ncontain  $\\tilde{\\theta}$, and is transversal to $W^{s}( \\gamma_{1})$ \n(to the new flow). Taking $\\bar{V}_{1} =\n\\phi_{t'}^{L+f_{1}}(\\bar{V})$ we will have that $W_{n}^{u}(\n\\gamma_{2}) \\pitchfork W^{s}( \\gamma_{1})$.\nWe can cover the fundamental domain $\\mathcal{D}$ with a finite number\nof neighborhoods like $\\bar{V}_{1}$, that is, $W_{1},..., W_{s}$.\nSince the transversality is an open condition and the local\nstable (unstable) manifold depends continuously on compact parts, we\ncan choose successively $W_{i+1}$ such that the transversality in\n$W_{j}, \\; j \\leq i$, is preserved. Thus,  \n$W_{n}^{u}( \\gamma_{2}) \\pitchfork W_{n}^{s}( \\gamma_{1})$.\n\\end{proof}\n\n\\section{Proof of the Reduction Lemma} \\label{ReducLema} \n\nFor the proof of the Reduction Lemma\n(Lemma~\\ref{ReduLocalDaPrimParte}) we will use an induction\nmethod similar to the one used by Anosov~\\cite{Anos}, using transversality arguments\nas described in Abraham \\cite{Ab} and \\cite{AbRb}. In this way, we remember a usefull theorem, the Parametric Transversality Theorem of Abraham.\n\nRemember that, if $\\mathcal{X}$ is a topological space. A subset $\\mathcal{R} \\subseteq\n\\mathcal{X}$ is said generic if $\\mathcal{R}$ is a countable\nintersection of open and dense sets. The space\n$\\mathcal{X}$ will be a Baire Space if all generic subsets are dense.  For additional results and definitions of Differential Topology, see~\\cite{AbRb}, \\cite{AbMardRat} or \\cite{KbMg}.\n\\begin{theorem}\\label{abraham}(\\cite{AbRb}, pg. 48,  Abraham's Parametric Transversality Theorem)\nConsider $\\mathbb{X}$ a submanifold finite dimensional (with\nboun\\-da\\-ry or boundaryless), $\\mathbb{Y}$ a boun\\-da\\-ry\\-less manifold and $S\n\\subseteq \\mathbb{Y}$ a submanifold with finite codimension.\nConsider $\\mathcal{B}$ boundaryless manifold, $\\rho : \\mathcal{B}\n\\rightarrow C^{\\infty}(\\mathbb{X};\\mathbb{Y})$ a smooth\nrepresentation and your evaluation $ev_{\\rho} : \\mathcal{B} \\times\n\\mathbb{X} \\rightarrow \\mathbb{Y}$. If $\\mathbb{X}$ and\n$\\mathcal{B}$ are Baire spaces and $ev_{\\rho} \\pitchfork S $ then\nthe set $\\mathcal{R}=\\{ \\varphi \\in \\mathcal{B} \\mid \\rho_{\\varphi}\n\\pitchfork S \\}$ is a generic subset (and obviously dense) of\n$\\mathcal{B}$.\n\\end{theorem}\n\n\n\nGiven a Hamiltonian $H$ we define the normal field associated,\nas being the gradient field, $Y^{H}=\\nabla H= H_{x} \\, \\frac{\\partial \\,}{\\partial x} + H_{p} \\, \\frac{\\partial \\,}{\\partial p}$ in $T^{*}M$. Observe that $JY^{H}=X^{H}$, where  $J$ is the canonic sympletic matrix. We denote $\\psi_{s}^{H^{\\perp}}: T^{*}M \\times (-\\varepsilon,\\varepsilon) \\rightarrow T^{*}M$ the flow in $T^{*}M$ generated by the normal field.\n  Let us briefly describe  the properties of the normal field. Initially\n  observe that $\\omega_{\\vartheta} (Y^{H}, X^{H})= H_x^{2} +\n  H_p^{2}, \\; \\forall \\vartheta \\in T^{*}M$. If $X^{H}(\\vartheta) \\neq\n  0$ then $0 \\neq Y^{H}(\\vartheta) \\not\\in T_{\\vartheta}H^{-1}(k)$ where\n  $k=H(\\vartheta)$, that is $Y^{H}$  points to the outside of the energy level.\n  From the compactness of the energy level $H^{-1}(k)$\n  we have that the flow of the normal field, restricted\n   to $H^{-1}(k)$ is defined in  $H^{-1}(k) \\times\n  (-\\varepsilon(H),\\varepsilon(H))$ where  $\\varepsilon(H) >0$ is uniformly defined in\n  $H^{-1}(k)$. Then the flow of the normal\n  field is defined in a neighborhood of the energy level $H^{-1}(k)$.\n  The action of the differential of the normal flow through an orbit is given by\n$$\n\\left \\{\n\\begin{array}{clcr}\n\\dot{Z}^{H}(s) = \\mathcal{H}(\\gamma(s)) Z^{H}(s) \\\\\nZ^{H}(0)=Y^{H}(\\vartheta) \\quad  \\quad \\, \\, \\; \\; \\; ^{}\n\\end{array}\n\\right .\n$$\nwhere $\\mathcal{H}$ is the hessian matrix of $H$.  The main property  of $Y^{H}$  is to establish a sympletic decomposition of $T_{\\vartheta}T^{*}M$ given by the next proposition.\n\n\\begin{proposition}\\label{BaseCampoNormal} Consider the normal field\n$Y^{H}$ associated to $H$ in the regular energy level, $H^{-1}(k)$.\nThen, for each $\\vartheta \\in H^{-1}(k) $, periodic of period $T\n>0$, there exists a symplectic base $\\{\nu_{1},...,u_{n},u_{1}^{*},...,u_{n}^{*} \\}$ of\n$T_{\\vartheta}T^{*}M $ verifying\\\\\n(i) $u_{1}= X^{H} $ and  $u_{1}^{*}=-\\frac{1}{H_x^{2} +\n  H_p^{2}} Y^{H} $;\\\\\n(ii) $\\mathcal{W}_{1}= \\langle u_{1}, u_{1}^{*} \\rangle ^{\\perp}\n\\subset T_{\\vartheta} H^{-1}(k) $ in particular, $T_{\\vartheta}\nH^{-1}(k) = \\langle u_1 \\rangle \\oplus \\mathcal{W}_{1}$;\\\\\n(iii) If $\\Sigma \\subset H^{-1}(k)$ is a section transversal to the\nflow, such that, $T_{\\vartheta} \\Sigma= \\mathcal{W}_{1}$, \nwe have that  $d_{\\vartheta} P(\\Sigma,\\vartheta)\n\\mathcal{W}_{1} \\subseteq \\mathcal{W}_{1} $;\\\\\n(iv) If $T=m T_{min}(\\vartheta)$, then\n$d_{\\vartheta} \\psi_{T}^{H} u_{1}=u_{1}$ and $d_{\\vartheta}\n\\psi_{T}^{H} u_{1}^{*}=c u_{1} + u_{1}^{*} +\\xi$, $\\xi \\in\n\\mathcal{W}_{1}$.\nIn particular, $(d_{\\vartheta} \\psi_{T}^{H} -Id)\n(T_{\\vartheta}T^{*}M) \\subseteq \\langle u_{1} \\rangle \\oplus\n\\mathcal{W}_{1}=T_{\\vartheta} H^{-1}(k) $.\\\\\n(v) There exists, $\\varepsilon >0$ uniform in $\\vartheta \\in H^{-1}(k)$, such\nthat, the map $e_{\\vartheta} : (-\\varepsilon, \\varepsilon) \\to~\\mathbb{R}$ given by\n$e_{\\vartheta}(s)=H \\circ \\psi_{s}^{H^{\\perp}}(\\vartheta)$ is\ninjective with $e_{\\vartheta}(0)=k$.\n\\end{proposition}\n\n\nUsing the normal field we are able to construct an representation, in order  to apply the Parametric Transversality Theorem.\n\n\\begin{proposition}\\label{repres}\n Given $k \\in \\mathbb{R}$, $0<a<b< + \\infty$, and $f_{0} \\in \\mathcal{R}(k)$,\n consider the normal field $Y^{H+f_0}$ as described before,\n $\\varepsilon=\\varepsilon(H+f_0) >0$ as in the Proposition~\\ref{BaseCampoNormal},~(v),\n and the sets, $\\mathcal{U}_{f_0} \\subset \\mathcal{R}(k)$ a $C^{\\infty}$ neighborhood\n  of $f_0$, $\\alpha=\\alpha(\\mathcal{U}_{f_0}) >0$\n as in Lemma~\\ref{MinimoPerioLema}, $\\mathbb{X}= T^{*}M \\times\n (a,b) \\times (-\\varepsilon, \\varepsilon)$ and $\\mathbb{Y}= T^{*}M \\times T^{*}M \\times\n \\mathbb{R}$. \n Then the map  $\\rho : \\mathcal{U}_{f_0} \\rightarrow\n C^{\\infty}(\\mathbb{X};\\mathbb{Y})$ given by\n $\\rho(f):=\\rho_{f}$, where\n \\begin{center}$\\rho_{f}(\\vartheta, t, s)=(\\psi_{s}^{H+f^{\\perp}}\n (\\vartheta), \\psi_{t}^{H+f}(\\vartheta),\n (H+f)(\\vartheta)-k)$ \\end{center}\n is an injective representation (see~\\cite{Ab}~or~\\cite{AbRb}).\n\\end{proposition}\n\n\\begin{proof}\nInitially  we point out that $\\rho$\nis well defined, therefore $\\mathbb{Y}$  has the structure of a product manifold. Writing\n$\\rho_{f}=(\\rho_{f}^{1},\\rho_{f}^{2},\\rho_{f}^{3})$, where\n $\\rho_{f}^{1}(\\vartheta, t,s)=\\psi_{s}^{H+f^{\\perp}}(\\vartheta) $, \n$\\rho_{f}^{2}(\\vartheta, t,s) = \\psi_{t}^{L+f}(\\vartheta) $,\n$\\rho_{f}^{3}(\\vartheta, t,s) = (H+f)(\\vartheta)-k$,\nwe can see that each coordinate is a smooth  function. Thus\n$\\rho_{f}\\in C^{\\infty}(\\mathbb{X};\\mathbb{Y})$.\nObserve that $\\rho$ is injective. Indeed, if $\\rho_{f_{1}}=\\rho_{f_{2}}$\nthen  $(H+f_{1})(\\vartheta)-k =\n(H+f_{2})(\\vartheta)-k$, for all $\\vartheta \\in T^{*}M$, so $f_{1}(x) = f_{2}(x)\n$, for all $x \\in M$.  Thus $f_{1} = f_{2}$.\nWe must to verify that $ev_{\\rho} : \\mathcal{U}_{f_0} \\times \\mathbb{X}\n\\to \\mathbb{Y}$  is smooth. Since $\\mathcal{U}_{f_0} \\times\n\\mathbb{X}$ have the  structure  of product\nmanifold , we can write\n$$d_{(f,x)} ev_{\\rho}:= \\frac{\\partial ev_{\\rho}}{\\partial f} (f,x)\n+ \\frac{\\partial ev_{\\rho}}{\\partial x} (f,x), \\; \\forall\n(f,x)\\in \\mathcal{U}_{f_0} \\times \\mathbb{X}$$\nIt is clear that $\\frac{\\partial ev_{\\rho}}{\\partial x}\n(f,x)$ is always defined as$ \\frac{\\partial ev_{\\rho}}{\\partial x} (f,x)=d_{x}\\rho_{f}$.\nMore precisely, given $(\\xi,\\dot{t},\\dot{s}) \\in T_{x=(\\vartheta,T,S)}\\mathbb{X}$\nwe have\n$$\\frac{\\partial ev_{\\rho}}{\\partial x}\n(f,x)(\\xi,\\dot{t},\\dot{s})=d_{x}\\rho_{f}(\\xi,\\dot{t},\\dot{s}) =\\frac{d \\,}{dr}\\rho_{f}(\\vartheta(r), t(r), s(r)) \\mid_{r=0}=$$\n$$=\\frac{d \\,}{dr} (\\psi_{s(r)}^{H+f^{\\perp}}(\\vartheta(r)), \\psi_{t(r)}^{H+f}(\\vartheta(r)),\n(H+f)(\\vartheta(r))-k)\\mid_{r=0}=$$\n$$( d_{\\vartheta} \\psi_{S}^{H+f^{\\perp}}(\\xi) +\n\\dot{s} Y^{H+f}(\\psi_{S}^{H+f^{\\perp}}(\\vartheta)), d_{\\vartheta}\\psi_{T}^{H+f}(\\xi) +\n\\dot{t} X^{H+f}(\\psi_{S}^{H+f}(\\vartheta)),$$ $$ d_{\\vartheta}(H+f)(\\xi) ).$$\nObserve that, if $S=0$ and $\\psi_{S}^{H+f}(\\vartheta)=\\vartheta$,\nthen  $$\\frac{\\partial ev_{\\rho}}{\\partial x}\n(f,x)(\\xi,\\dot{t},\\dot{s})=( \\xi + \\dot{s} Y^{H+f}(\\vartheta), d_{\\vartheta}\\psi_{T}^{H+f}(\\xi) +\n\\dot{t} X^{H+f}(\\vartheta),$$ $$ d_{\\vartheta}(H+f)(\\xi) ).$$ \nHowever we must to show that $\\frac{\\partial ev_{\\rho}}{\\partial f}\n(f,x)$ is always defined. By the structure of  $C^{\\infty}(M;\\mathbb{R})$\nwe know that this fact  is equivalent to show that there exists\n\\begin{center} $\\frac{d \\,}{dr} \\psi_{S}^{H + f + r h ^{\\perp}}(\\vartheta) \\mid_{r=0}$ ,\n$ \\frac{d \\,}{dr} \\psi_{T}^{H + f + r h }(\\vartheta)\n\\mid_{r=0}$ and $\\frac{d \\,}{dr} (H + f + r h ) (\\vartheta) \\mid_{r=0}$\n\\end{center}\nfor any $h \\in C^{\\infty}(M;\\mathbb{R})$  and\n$x=(\\vartheta,T, S) \\in \\mathbb{X}$.  From some straightforward  \ncalculations (see~\\cite{PhilHart}, pg.46), we get\\\\\n$\\frac{d \\,}{dr} (H + f + r h ) (\\vartheta) \\mid_{r=0}=h\\circ\\pi(\\vartheta)$,\\\\\n$ \\frac{d \\,}{dr} \\psi_{T}^{H + f + r h }(\\vartheta)\n\\mid_{r=0}=Z_{h}(T)= d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt$ and\\\\\n$ \\frac{d \\,}{dr} \\psi_{S}^{H + f + r h ^{\\perp} }(\\vartheta)\n\\mid_{r=0}=Z^{h}(S)= d_{\\vartheta}\\psi_{S}^{H + f^{\\perp} }\n\\int_{0}^{S} (d_{\\vartheta}\\psi_{s}^{H + f^{\\perp}  })^{-1}\nb^{h}(s)ds$.\\\\\nThus  $\\displaystyle  \\frac{\\partial ev_{\\rho}}{\\partial f} (f,x)\n(h)=$\n $$ ( d_{\\vartheta}\\psi_{S}^{H + f^{\\perp} }\n\\int_{0}^{S} (d_{\\vartheta}\\psi_{s}^{H + f^{\\perp}  })^{-1}\nb^{h}(s)ds \\; ,   \\; d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt  \\; ,$$ $$  \\; h\\circ\\pi(\\vartheta)).$$\n   If $S=0$ and $\\psi_{T}^{H+f}(\\vartheta)=\\vartheta$,\n   then $$\\displaystyle  \\frac{\\partial ev_{\\rho}}{\\partial f} (f,x)\n(h)=( 0  ,  \\; d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt  \\; ,  \\; h \\circ \\pi(\\vartheta)).$$\nThus $ev_{\\rho}$ is smooth\nand therefore $\\rho$ is a representation.\n\\end{proof}\n\nDefine the null diagonal $\\Delta_{0} \\subseteq\n\\mathbb{Y}$ given by $\\Delta_{0}=\\{(\\vartheta,\\vartheta,0) \\mid \\vartheta \\in T^{*}M\\}$.\nCombining, Propositions~\\ref{BaseCampoNormal} and \\ref{repres} we get\n\n\\begin{lemma}\\label{PeriodPontoSaoNaoDegenTrans}\nWith the same notations of the Proposition~\\ref{repres}, we have that,\n$\\forall f \\in \\mathcal{U}_{f_0}$, with $T \\in (a,b)$ and $S \\in (-\\varepsilon, \\varepsilon)$,\n\\begin{itemize}\n\\item[i)] If $\\vartheta$ is a periodic orbit of positive period $T$ for $H+f$\nin the level $(H+f)^{-1}(k)$ then,\n$\\rho_{f}(\\vartheta, T, 0) \\in \\Delta_{0}$.\nReciprocally, if $\\rho_{f}(\\vartheta, T, S) \\in \\Delta_{0}$\nthen, $S=0$ and $\\vartheta$ is a periodic orbit of positive period $T$\nfor $H+f$ in the level $(H+f)^{-1}(k)$.\n\n\\item[ii)] If $\\vartheta$ is a periodic orbit of positive period, for $H+f$\nin the level $(H+f)^{-1}(k)$. Then, $\\vartheta$ is\nnondegenerate of order $ m = \\frac{T}{T_{min}(\\vartheta)}$ if, and only if,\n$\\rho_{f} \\pitchfork _{(\\vartheta, T, 0)} \\Delta_{0}$.\n\\end{itemize}\n\\end{lemma}\n\nThe next corollary it is an easy consequence of the Lemma~\\ref{PeriodPontoSaoNaoDegenTrans}.\n\n\\begin{corollary}\\label{PeriodNivelSaoNaoDegenTrans}\nWith the same notations of the Lemma~\\ref{PeriodPontoSaoNaoDegenTrans} we have that,\ngiven $ f \\in \\mathcal{U}_{f_0}$, all periodic orbits $\\vartheta $, with positive period,\n$T_{min}(\\vartheta) \\in (a,b)$, in $(H+f)^{-1}(k)$,\nare nondegenerate for $H+f$ of order $m$, $\\forall m \\leq \\frac{b}{T_{min}}$\nif, and only if, $\\rho_{f} \\pitchfork \\Delta_{0}$.\n\\end{corollary}\n\nThe previous corollary shows that, the nondegeneracy of the periodic orbits\nof positive period in an interval $(a,b)$, for a given energy level $(H+f)^{-1}(k)$,\nis equivalent to the transversality of the map $\\rho_{f}$ in relation to the diagonal $\\Delta_{0}$. \nThe key element for the proof of the Lemma~\\ref{ReduLocalDaPrimParte}  is\nthe nest lemma.\n\\begin{lemma}\\label{evaluationtransverlema}\nConsider  the representation $\\rho$ as in the Proposition~\\ref{repres} and its evaluation\nin $\\mathcal{U}_{f_0}$, that is, $ev : \\mathcal{U}_{f_0} \\times \\mathbb{X} \\rightarrow\n\\mathbb{Y}$, given by $ev(f, \\vartheta, t , s)= \\rho_{f}(\\vartheta, t ,s)$.\nSuppose that $ev(f , \\vartheta, T , S) \\in \\Delta_{0}$ then,\n\\begin{itemize}\n\\item[i)] If $\\vartheta$ is  nondegenerate of order $m=\\frac{T}{T_{min}}$ for $H+f$\nthen $ev \\pitchfork _{(f, \\vartheta, T ,S)}\n\\Delta_{0}$;\n\\item[ii)] If $T = T_{min}(\\vartheta)$ then, $ev \\pitchfork _{(f, \\vartheta, T, S )}\n\\Delta_{0}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\n\ni) We know that $ev(f, \\vartheta, T ,S)=\\rho_{f}(\\vartheta, T ,S)$\ntherefore $\\rho_{f}(\\vartheta, T, S )\\in  \\Delta_{0}$, and  $S=0$.\nIf $\\vartheta$ is nondegenerate of order $m=\\frac{T}{T_{min}}$\nfor $H+f$, then, from the Lemma~\\ref{PeriodPontoSaoNaoDegenTrans}~(ii),\n$\\rho_{f} \\pitchfork _{(\\vartheta, T, 0 )} \\Delta_{0}$,\nin particular  $ev \\pitchfork _{(f, \\vartheta, T, 0)}\n\\Delta_{0}$.\\\\\n\nii) As $ev(f, \\vartheta, T , S) \\in \\Delta_{0}$ we must to show that\n\\begin{center}\n$d_{(f, \\vartheta, T , 0)} ev T_{(f, \\vartheta, T, 0 )} (\\mathcal{U}_{f_0} \\times\n\\mathbb{X})\n+ T_{(\\vartheta , \\vartheta, 0)} \\Delta_{0}= T_{(\\vartheta , \\vartheta, 0)}\\mathbb{Y} $.\n\\end{center}\nTake any $(u,v,w ) \\in T_{(\\vartheta,\\vartheta\n,0)}\\mathbb{Y}$, $(\\zeta, \\zeta, 0) \\in T_{(\\vartheta,\\vartheta\n,0)} \\Delta_{0} $ and \\\\ $(h, \\xi, \\dot{t}, \\dot{s}) \\in T_{(f, \\vartheta, T, 0 )}\n(\\mathcal{U}_{f_0} \\times \\mathbb{X})$. From Proposition~\\ref{repres} we have that\\\\\n$d_{(f,\\vartheta, T , 0)} ev_{\\rho}(h, \\xi, \\dot{t},\n\\dot{s})=$\\\\\n$=( 0  ,  \\; d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt  \\; ,  \\; h\\circ\\pi(\\vartheta))\n+ \\\\\n( \\xi + \\dot{s} Y^{H+f}(\\vartheta), d_{\\vartheta}\\psi_{T}^{H+f}(\\xi) +\n\\dot{t} X^{H+f}(\\vartheta), d_{\\vartheta}(H+f)(\\xi) )$\\\\\n$=( \\xi + \\dot{s} Y^{H+f}(\\vartheta) , \\;  d_{\\vartheta}\\psi_{T}^{H+f}(\\xi) +\n\\dot{t} X^{H+f}(\\vartheta)+ d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt  \\; , \\;\\\\ h\\circ\\pi(\\vartheta) +  d_{\\vartheta}(H+f)(\\xi) )$\\\\\nTherefore $ev_{\\rho} \\pitchfork _{(f, \\vartheta, T, 0 )}\n\\Delta_{0}$, if and only if, the system\n\\begin{center}\n\\(\n\\left \\{\n\\begin{array}{clcr}\nu=\\xi + \\dot{s} Y^{H+f}(\\vartheta) +\\zeta  \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad\n\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad  \\quad (1)\\\\\nv=d_{\\vartheta}\\psi_{T}^{H+f}(\\xi) +\n\\dot{t} X^{H+f}(\\vartheta)+ d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt + \\zeta \\; \\; \\, (2)\\\\\nw=h\\circ\\pi(\\vartheta) +d_{\\vartheta}(H+f)(\\xi)  \\quad \\quad \\quad \\quad\n\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad   \\quad(3)\n\\end{array}\n\\right.\n\\)\n\\end{center}\nhas a solution.Using the coordinates of the Proposition~\\ref{BaseCampoNormal}\nand  taking $\\zeta=u -\\xi - \\dot{s} Y^{H+f}(\\vartheta) $ we have that the equation (2)\nrestricted to the set of the solutions of (3),\n$$\\mathcal{V}_{w}=\\left\\{h \\in C^{\\infty}(M,\\mathbb{R}), \\, \\xi = a X^{H+f}(\\vartheta)\n + b_{0} Y^{H+f}(\\vartheta) + U \\; \\right\\} $$\nwhere\n$b_{0}=\\frac{w-h\\circ\\pi(\\vartheta)}{d_{\\vartheta}(H+f)(Y^{H+f}(\\vartheta))}$,\nwill have the expression\\\\\n$(\\dot{t}+b_{0}c+\\tau_{0}) X^{H+f}(\\vartheta)+ (c^{*}-  \\dot{s}) Y^{H+f}(\\vartheta) +\n(d_{\\vartheta} P(\\Sigma, \\vartheta)-Id)(U) +  b_{0}U_{0} + \\\\\nd_{\\vartheta}\\psi_{T}^{H + f}\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt=  \\tilde{a} X^{L+f}(\\vartheta) + \\tilde{b}  Y^{H+f}(\\vartheta) +\n\\bar{U}$,  where\n$$v-u= \\tilde{a} X^{L+f}(\\vartheta) + \\tilde{b}  Y^{H+f}(\\vartheta) + \\bar{U},$$\\\n$$(d_{\\vartheta}\\psi_{T}^{H+f} -id)(Y^{H+f}(\\vartheta))=c\nX^{H+f}(\\vartheta) + c^{*} Y^{H+f}(\\vartheta) + U_{0}$$ and \n$$(d_{\\vartheta}\\psi_{T}^{H+f} -id)(U)=\\tau_{0}\nX^{H+f}(\\vartheta) + (d_{\\vartheta} P(\\Sigma,\n\\vartheta)-Id)(U).$$\nThat is, the system  always has a solution, if the expression,\n\\begin{center}\n$(\\dot{t}+b_{0}c+\\tau_{0}) X^{H+f}(\\vartheta) + (c^{*}-  \\dot{s}) Y^{H+f}(\\vartheta) +\n(d_{\\vartheta} P(\\Sigma, \\vartheta)-Id)(U) +  b_{0}U_{0} +\nd_{\\vartheta}\\psi_{T}^{H + f}\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt$\n\\end{center}\nis surjective in $T_{\\vartheta}T^{*}M$.\nSo, we must to show that $$ d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt$$ generates  a $2n-2$ dimensional space complementary\nto  the space generated by $ X^{H+f}(\\vartheta)$ and $ Y^{H+f}(\\vartheta) $,  in\n$T_{\\vartheta}T^{*}M$, which is the claim of the next lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{sobrejetividade}\nWith the same notations as in Lemma~\\ref{evaluationtransverlema}, the map\n$ \\mathcal{B}: C^{\\infty}(M ; \\mathbb{R})$\n$\\to~T_{\\vartheta}T^{*}M$,\n\\begin{center}\n$\\displaystyle \\mathcal{B}(h)=- d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\nb_{h}(t)dt$\n\\end{center}\ngenerates a space complementary to\n$\\langle X^{H+f}(\\vartheta), Y^{H+f}(\\vartheta) \\rangle$.\n\\end{lemma}\n\\begin{proof}\nIn order to prove this claim is enough to restrict the map $\\displaystyle\n\\mathcal{B}$ to a subspace chosen in $C^{\\infty}(M ; \\mathbb{R})$.\nConsider $t_{0} \\in (0,T)$,  $\\varepsilon >0$, and\ndenote $\\mathcal{A}_{t_{0}}$, the  subspace of the smooth functions\n$$\\mathcal{A}_{t_{0}}=\\{\\alpha: \\mathbb{R} \\to \\mathbb{R}^{n-1} \\mid\n\\alpha(t)=(a_{1}(t),...,a_{n-1}(t)) \\neq 0, \\,\n\\forall t \\in (t_{0}-\\varepsilon, t_{0}+\\varepsilon) \\}. $$\nWe assume that, $x(t) = \\pi(\\gamma(t))$,\nwhere $\\gamma(t)=\\psi^{H+f}_{t}(\\vartheta)$, does not contain autointersections\nfor $t \\in (t_{0} -\\varepsilon, t_{0} + \\varepsilon)$, that is, that\n$H_{p}(\\gamma(t))=d\\pi X^{H+f}(\\gamma(t)) \\neq 0$. Then there exists a\nsystem of tubular coordinates $\\mathcal{V}$,  in a neighborhood of\n$\\pi(\\gamma(t_{0}))$, $F: \\mathcal{V} \\to \\mathbb{R}^{n}$, such that,\\\\\ni) $F(x)=(t,z_{1},...,z_{n-1})$;\\\\\nii) $F(x(t))=(t,0,...,0)$.\\\\\nObserve that, by construction, $d_{x(t)}F H_{p}(\\gamma(t))=(1,0,...0)$.\nConsider a bump function $ \\sigma :M \\to \\mathbb{R}$, such that, $\\mathop{\\rm supp}\\nolimits(\\sigma) \\subset \\mathcal{V}$,  $\\sigma \\mid_{\\mathcal{V}_{0}} \\equiv 1$,  with $x(t_{0}) \\in \\mathcal{V}_{0} \\subset \\mathcal{V}$.\nDefine the perturbation space\n$\\mathcal{F}_{t_{0}} \\subset C^{\\infty}(M ; \\mathbb{R})$ as being\n$$\\mathcal{F}_{t_{0}}=\\{h_{\\alpha,\\beta}(x)=\\tilde{h}_{\\alpha,\\beta}(x) \\cdot \\sigma(x)\n \\, \\mid \\,\\alpha, \\beta \\in \\mathcal{A}_{t_{0}}\\} $$\nwhere, $\\tilde{h}_{\\alpha,\\beta}(x)=\\langle \\alpha(t) \\delta_{t_{0}}(t) + \\beta(t)\n\\dot{\\delta}_{t_{0}}(t) \\, , \\,  z \\rangle$, $F(x)=(t,z)$ and\n$\\delta_{t_{0}}$ is a smooth approximation of the delta of Dirac in the point $t=t_{0}$.\nGiven $h_{\\alpha,\\beta} \\in \\mathcal{F}_{t_{0}}$ we get\n$d_{x}h_{\\alpha,\\beta}=d_{x} \\tilde{h}_{\\alpha,\\beta} \\cdot \\sigma(x) +\n\\tilde{h}_{\\alpha,\\beta} \\cdot d_{x} \\sigma(x)$. \nOn the other hand\n$$d_{x} \\tilde{h}_{\\alpha,\\beta}=(\\langle \\frac{d \\,}{dt}(\\alpha(t)\n\\delta_{t_{0}}(t) + \\beta(t)\n\\dot{\\delta}_{t_{0}}(t)),z \\rangle, \\alpha(t) \\delta_{t_{0}}(t) + \\beta(t)\n\\dot{\\delta}_{t_{0}}(t)) d_{x}F $$\nEvaluating $x(t) $ and using that\n$h_{\\alpha,\\beta}(x(t))=0$  and $\\sigma(x(t))=1$, we get\n$$d_{x(t)}h_{\\alpha,\\beta}=(0, \\alpha(t) \\delta_{t_{0}}(t) + \\beta(t)\n\\dot{\\delta}_{t_{0}}(t)) d_{x(t)}F.$$\nIn particular,\n$$d_{x(t)}h_{\\alpha,\\beta} H_{p}(\\gamma(t))=(0,\\langle \\alpha(t) \\delta_{t_{0}}(t) + \\beta(t)\n\\dot{\\delta}_{t_{0}}(t)) d_{x(t)}F H_{p}(\\gamma(t))=0$$\nfor any, $h_{\\alpha,\\beta} \\in \\mathcal{F}_{t_{0}}$.\n\nWe claim that,\\\\\n1) $\\mathcal{B}(\\mathcal{F}_{t_{0}}) \\subset T(H+f)^{-1}(k)$;\\\\\n2) $X^{H+f}(\\vartheta) \\not\\in \\mathcal{B}(\\mathcal{F}_{t_{0}})$;\\\\\n3) $\\dim(\\mathcal{B}(\\mathcal{F}_{t_{0}}))=2n-2$;\\\\\n4) In particular, $\\mathcal{B}(\\mathcal{F}_{t_{0}})$\ngenerates a space complementary to $\\langle X^{H+f}(\\vartheta), $ $Y^{H+f}(\\vartheta)\n\\rangle $.\\\\\nIn order to get (1) consider,\n $\\alpha_{0} = d_{x(t)}h_{\\alpha,0}=(0, \\alpha(t) \\delta_{t_{0}}(t))\nd_{x(t)}F = \\alpha_{1}  \\delta_{t_{0}}(t) $ and \n $\\beta_{0} = d_{x(t)}h_{0,\\beta}=(0, \\beta(t) \\dot{\\delta}_{t_{0}}(t))\nd_{x(t)}F = \\beta_{1} \\dot{\\delta}_{t_{0}}(t) $,\nthen,\n$$   \\mathcal{B}(h_{\\alpha})= d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\n\\left [\n\\begin{matrix}\n0 \\\\\n\\alpha_{0}\n\\end{matrix}\n\\right ]\ndt$$\n\\text{ and }   $$\\mathcal{B}(h_{\\beta})= d_{\\vartheta}\\psi_{T}^{H + f}\n\\int_{0}^{T} (d_{\\vartheta}\\psi_{t}^{H + f })^{-1}\n\\left [\n\\begin{matrix}\n0 \\\\\n\\beta_{0}\n\\end{matrix}\n\\right ]\ndt.$$\nObserve that,\n\\( \\omega( \\left [  \\begin{matrix}   0 \\\\   \\alpha_{0}  \\end{matrix}  \\right\n], X^{H+f}(\\gamma(t)))=  \\alpha_{0} H_{p}(\\gamma(t))=0, \\)\n\\text{ and }\\\\\n\\(\\omega( \\left [\\begin{matrix}   0 \\\\   \\beta_{0}  \\end{matrix}  \\right\n], X^{H+f}(\\gamma(t)))=  \\beta_{0} H_{p}(\\gamma(t))=0,\\)\ntherefore\n$\\left [  \\begin{matrix} 0 \\\\   \\alpha_{0}  \\end{matrix}  \\right\n]$ and $\\left [  \\begin{matrix}   0 \\\\   \\beta_{0}  \\end{matrix}  \\right\n]$ are in $T(H+f)^{-1}(k)$.\nThus, $\\mathcal{B}(\\mathcal{F}_{t_{0}})\\subset\nT(H+f)^{-1}(k)$.\n\nIn order to get (2),   we will make $\\delta_{t_{0}} \\to \\delta_{Dirac}$\nand will write $\\mathcal{B}(h_{\\alpha})$ and $\\mathcal{B}(h_{\\beta})$ as \n$$ \\displaystyle \\mathcal{B}(h_{\\alpha})= d_{\\vartheta}\\psi_{T}^{H + f}(d_{\\vartheta}\\psi_{t_{0}}^{H + f\n})^{-1} \\left [ \\begin{matrix} 0 \\\\ \\alpha_{1}(t_{0}) \\end{matrix} \\right].$$\nAnalogously,  \n$$ \\displaystyle \\mathcal{B}(h_{\\beta})= d_{\\vartheta}\\psi_{T}^{H + f}(d_{\\vartheta}\\psi_{t_{0}}^{H + f\n})^{-1}\n\\left\\{ J \\mathcal{H}^{H+f}(t_{0}) \\left[ \\begin{matrix} 0 \\\\ \\beta_{1}(t_{0})\n\\end{matrix} \\right] - \\left[ \\begin{matrix} 0 \\\\ \\dot{\\beta}_{1}(t_{0})\n\\end{matrix} \\right]\n\\right\\}. $$\n\nIf we assume, by contradiction, that $X^{H+f}(\\gamma(t))=\\mathcal{B}(h_{\\alpha})\n+\\mathcal{B}(h_{\\beta})$ then\n$$X^{H+f}(\\gamma(t_{0}))= \\left\\{ \\left [ \\begin{matrix}\n0 \\\\ \\alpha_{1}(t_{0}) \\end{matrix} \\right] +\nJ \\mathcal{H}^{H+f}(t_{0}) \\left [ \\begin{matrix} 0 \\\\ \\beta_{1}(t_{0})\n\\end{matrix} \\right ] - \\left [ \\begin{matrix} 0 \\\\ \\dot{\\beta}_{1}(t_{0})\n\\end{matrix} \\right ] \\right\\}.$$\nFrom this equality we have\n$H_{p}(\\gamma(t_{0}))= H_{pp}(\\gamma(t_{0})) \\beta_{1}(t_{0})$.\nSince $H_{p}(\\gamma(t_{0})) \\neq 0$   we have $n-1$ choices, linearly independent, for $\\beta_{1}(t_{0})$.  Indeed, $dF$ is an isomorphism and for all $\\beta (t_{0}) \\in  \\mathbb{R}^{n-1}$ we have $ \\beta_{1}(t_{0})H_{p}(\\gamma(t_{0}))=(0,\\beta (t_{0}))dFH_{p}(\\gamma(t_{0}))=0$.  Thus,\n $0=\\beta_{1}(t_{0})H_{p}(\\gamma(t_{0}))=\\beta_{1}(t_{0})\nH_{pp}(\\gamma(t_{0})) \\beta_{1}(t_{0})$,  contradicting the superlinearity of $H$.\nFor (3) observe that, in (2) we got the limit representation\n$$\\mathcal{B}(h_{\\alpha})\n+\\mathcal{B}(h_{\\beta})=  d_{\\vartheta}\\psi_{T}^{H + f}\n(d_{\\vartheta}\\psi_{t_{0}}^{H + f })^{-1} $$ $$ \\left\\{ \\left [ \\begin{matrix} 0 &\n\\quad H_{pp}(\\gamma(t_{0})) \\\\\nI_{n} & -H_{xp}(\\gamma(t_{0})) \\end{matrix} \\right]\n\\left [ \\begin{matrix} \\alpha_{1}(t_{0}) \\\\ \\beta_{1}(t_{0})  \\end{matrix} \\right ]-\n\\left [ \\begin{matrix} 0 \\\\ \\dot{\\beta}_{1}(t_{0})  \\end{matrix} \\right ] \\right\\}.$$\nFrom this equation we get  $\\dim( \\{ \\mathcal{B}(h_{\\alpha})\n+\\mathcal{B}(h_{\\beta}) \\})= \\dim(\\{\\alpha_{1}(t_{0}),$  $ \\beta_{1}\n(t_{0})\\}=2n-2$, since $\\left [ \\begin{matrix} 0 & \\quad H_{pp}(\\gamma(t_{0})) \\\\\nI_{n} & -H_{xp}(\\gamma(t_{0})) \\end{matrix} \\right]$ is an isomorphism.\n\nFinally, we observe that, the claim (1) is true independently of the\napproximation $\\delta_{t_{0}}$ of the delta of Dirac in the point  $t=t_{0}$.\nMoreover the claims (2) and (3) still true for $\\delta_{t_{0}}$, close enough to the delta of Dirac.\n\\end{proof}\n\n\nThe next theorem allow  us to  make a local perturbation  of a\nperiodic orbit nondegenerate of order $\\leq m$ in such way that it becomes\nnondegenerate of order $\\leq 2m$. The proof is just for dimension 2\nand the $n$-dimensional case is still open. Almost all th parts of the argument\nare true in the $n$-dimensional case, but we do not know how to show\nthe surjectivity of the representation in this case.\n\n\n\\begin{theorem}\\label{PerturbacaoLocaldaOrbita}\\textnormal{(Local perturbation of periodic orbits)}\nLet  $dim(M)=2$, $\\mathbb{H}:~T^{*}M \\to \\mathbb{R}$ be a smooth, convex\nand superlinear Hamiltonian and $\\gamma= \\{\n\\psi_{t}^{\\mathbb{H}}(\\vartheta_{0}) \\mid 0 \\leq t \\leq T \\}\n\\subseteq \\mathbb{H}^{-1}(k)$, where $\\mathbb{H}^{-1}(k)$ is a\nregular energy level, $T$ is the minimal period of $\\gamma$, and\n$\\gamma$ is isolated in this energy level, nondegenerate of order $\n\\leq m \\in \\mathbb{N}$. Then there exists a potential  $f_{0} \\in\nC^{\\infty}(M,\\mathbb{R})$ arbitrarily close to zero, with\n$\\mathop{\\rm supp}\\nolimits(f_{0}) \\subset \\mathcal{U} \\subset M$ such that, \n$\\gamma$ is nondegenerate of order $\\leq 2m$ to\n$\\mathbb{H}+f_{0}$. More over, $\\mathcal{U}$ can be chosen arbitrarily small.\n\\end{theorem}\n\n\\begin{proof}\nChoose $t_{0} \\in (0,T)$ and $E(0)=\\{e_{1}(0), e_{2}(0),\ne_{1}^{*}(0), e_{2}^{*}(0) \\}$ a symplectic frame in $\\gamma(t_{0})$\nwith $e_{1}(0)=X^{\\mathbb{H}}(\\gamma(t_{0}))$. Consider\n$$E(t)=\\{e_{1}(t), e_{2}(t), e_{1}^{*}(t), e_{2}^{*}(t) \\}$$, where\n$\\xi(t)= d_{\\gamma(t_{0})} \\psi_{-t}^{\\mathbb{H}} \\xi, \\quad \\forall\n\\xi \\in E(0)$, for $t \\in (0,r)$ with $r >0$ arbitrarily small.\n\nThen we can decompose the matrix of the differential of the flow, in\nthe base $E(0)$, $[d_{\\gamma(t_{0})}\n\\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} \\in Sp(2)$, as\n$ [d_{\\gamma(t_{0})} \\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} =\n[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}]_{E(0)}^{E(r)} \\cdot\n[d_{\\gamma(t_{0})} \\psi_{T-r}^{\\mathbb{H}}]_{E(r)}^{E(0)}.$ \nBy construction we have that $[d_{\\gamma(t_{0}-r)}\n\\psi_{r}^{\\mathbb{H}}]_{E(0)}^{E(r)}=I_{4}$, therefore \n$$  \\quad \\quad  \\quad   \\quad  \\quad \\quad  \\quad \\quad [d_{\\gamma(t_{0})} \\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} =\n[d_{\\gamma(t_{0})} \\psi_{T-r}^{\\mathbb{H}}]_{E(r)}^{E(0)}.  \\quad\\quad \\quad   \\quad \\quad   \\quad \\quad   \\quad \n\\quad (1)$$\nConsider $U$ an arbitrarily small neighborhood of\n$\\gamma(t_{0})$ in $T^{*}M$ and $r$ small enough, in  such way that,\n$\\hat{\\gamma}= \\{ \\psi_{t}^{\\mathbb{H}}(\\vartheta_{0}) \\mid t \\in\n(t_{0}-r,t_{0}) \\} \\subseteq \\mathbb{H}^{-1}(k) \\cap U$. Fix\n$t_{1} \\in (t_{0}-r,t_{0})$ and $V$ a neighborhood of\n$\\gamma(t_{1})$ in $T^{*}M$, small enough, in  such way that, $V \\subset U$\nand that $\\gamma(t_{0}),\\gamma(t_{0}-r) \\not\\in \\overline{V}$.\nSuppose  that we have $\\tilde{\\mathbb{H}}: T^{*}M \\to \\mathbb{R}$ a\nsmooth Hamiltonian re\\-pre\\-sen\\-ting a smooth perturbation of\n$\\mathbb{H}$, such that, $\\mathop{\\rm supp}\\nolimits(\\tilde{\\mathbb{H}}-\\mathbb{H})\n\\subset V$ and that $jet_{1}(\\tilde{\\mathbb{H}})\\mid_{\\gamma(t)}=\njet_{1}(\\mathbb{H})\\mid_{\\gamma(t)}$, and\n$ [d_{\\gamma(t_{0})} \\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)} =\n[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(r)}\n\\cdot [d_{\\gamma(t_{0})}\n\\psi_{T-r}^{\\tilde{\\mathbb{H}}}]_{E(r)}^{E(0)}$.\nSince $\\mathop{\\rm supp}\\nolimits(\\tilde{\\mathbb{H}}-\\mathbb{H}) \\subset V$, we have \n$[d_{\\gamma(t_{0})} \\psi_{T-r}^{\\tilde{\\mathbb{H}}}]_{E(r)}^{E(0)}=\n[d_{\\gamma(t_{0})} \\psi_{T-r}^{\\mathbb{H}}]_{E(r)}^{E(0)}$. By (1),\n$ [d_{\\gamma(t_{0})}\n\\psi_{T-r}^{\\mathbb{H}}]_{E(r)}^{E(0)}= [d_{\\gamma(t_{0})}\n\\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)}$, so\n$$\\quad \\quad  \\quad   \\quad \\quad \\quad[d_{\\gamma(t_{0})} \\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)} =\n[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(r)}\n\\cdot [d_{\\gamma(t_{0})} \\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} \\quad  \\quad \\quad\\quad\n\\quad (2)$$\nFrom the construction of the perturbation described above we have\nthat $[d_{\\gamma(t_{0})}\n\\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)}$ has the expression\n$$\n[d_{\\gamma(t_{0})}\n\\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)} =\n\\left [\n\\begin{matrix}\n1   & \\alpha & \\sigma & \\beta  \\\\\n0   & A & \\hat{\\alpha} & B  \\\\\n0   & 0 & 1 & 0  \\\\\n0   & C & \\hat{\\beta} & D  \\\\\n\\end{matrix}\n\\right ]\n\\, \\in Sp(2),\n$$\nbecause, the energy level in $\\gamma(t_{0})$  and\n$\\gamma(t_{0}-r)$ is the same for $\\tilde{\\mathbb{H}}$ and\n$\\mathbb{H}$. Thus it is invariant by the action of the flow of\nboth Hamiltonians. Let $\\hat{Sp(2)}$ be the following subgroup of $Sp(2)$,\n$$\n\\hat{Sp(2)} =\n\\left \\{\n\\left [\n\\begin{matrix}\n1   & \\alpha & \\sigma & \\beta  \\\\\n0   & A & \\hat{\\alpha} & B  \\\\\n0   & 0 & 1 & 0  \\\\\n0   & C & \\hat{\\beta} & D\n\\end{matrix}\n\\right ]\n\\in SL(4)\n\\,\n\\left |\n\\,\n\\left [\n\\begin{matrix}\n\\hat{\\alpha} \\\\\n\\hat{\\beta}\n\\end{matrix}\n\\right ]\n\\right.\n=\n\\left [\n\\begin{matrix}\nA &  B  \\\\\nC &  D\n\\end{matrix}\n\\right ]\nJ\n\\left [\n\\begin{matrix}\n\\alpha & \\beta\n\\end{matrix}\n\\right ]^{*}\n\\right \\}\n$$\nand consider the projection $\\pi : \\hat{Sp(2)} \\to Sp(1)$ given by\n$$\n\\pi \\left(\\left [\n\\begin{matrix}\n1   & \\alpha & \\sigma & \\beta  \\\\\n0   & A & \\hat{\\alpha} & B  \\\\\n0   & 0 & 1 & 0  \\\\\n0   & C & \\hat{\\beta} & D\n\\end{matrix}\n\\right ]\n\\right )=\n\\left [\n\\begin{matrix}\nA &  B  \\\\\nC &  D\n\\end{matrix}\n\\right ],\n$$\nwhich  is a homomorphism of Lie groups.\nObserve that $$[d_{\\gamma(t_{0}-r)}\n\\psi_{r}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(r)}, [d_{\\gamma(t_{0})}\n\\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} \\in \\hat{Sp(2)}$$ and\n$\\det([d_{\\gamma(t_{0})} \\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)} -\n\\lambda I_{4})=(\\lambda -1)^{2} \\det(\\pi([d_{\\gamma(t_{0})}\n\\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)}) - \\lambda I_{2}).\n$ Thus $\\gamma$ will be a nondegenerate\norbit of order $\\leq 2m$, to the perturbed Hamiltonian, if\n$\\pi([d_{\\gamma(t_{0})}\n\\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)})$ does not have roots\nof the unity of order~$\\leq~2m$ as eigenvalues. Since, the\nsymplectic matrices that are $2m$-elementary~\\footnote{A symplectic\nmatrix is N-elementary if its principal eigenvalues (the\neigenvalues $\\lambda$ such that $\\| \\lambda \\| < 1$ or $Re(\\lambda)\n\\geq 0$) are multiplicatively independent over the integer, that is,\nif $\\Pi \\lambda_{i}^{p_{i}}=1$, where $\\sum p_{i} =N$ then $p_{i}=0,\n\\, \\forall i $.} (in particular, does not have roots of the unity of\norder~$\\leq~2m$ as eigenvalues), forms an open and dense subset of\n$Sp(1)$,  we must to show that, for a choice of the perturbation\nspace , the correspondence\n$\\tilde{\\mathbb{H}} \\to \\pi([d_{\\gamma(t_{0})}\n\\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)})$ applied to a\nneighborhood of $\\mathbb{H}$, generate  an  open neighborhood of\n$\\pi([d_{\\gamma(t_{0})} \\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)})$ in $Sp(1)$.\nUsing the homomorphism property \n$$ \\pi( [d_{\\gamma(t_{0})} \\psi_{T}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(0)}) =\n\\pi( [d_{\\gamma(t_{0}-r)}\n\\psi_{r}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(r)}) \\cdot \\pi(\n[d_{\\gamma(t_{0})} \\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} ). $$\nWe define $\\mathbb{X}_{0}=\\pi( [d_{\\gamma(t_{0})}\n\\psi_{T}^{\\mathbb{H}}]_{E(0)}^{E(0)} )$ and\n$\\hat{S}(\\tilde{\\mathbb{H}})=\\pi( [d_{\\gamma(t_{0}-r)}\n\\psi_{r}^{\\tilde{\\mathbb{H}}}]_{E(0)}^{E(r)})$.\nSince the translation $\\mathbb{X} \\to \\mathbb{X} \\cdot\n\\mathbb{X}_{0}$ is an isomorphism of the of the Lie group $Sp(1)$,\nwe need to show that the map $\\tilde{\\mathbb{H}} \\to\n\\hat{S}(\\tilde{\\mathbb{H}})$ applied to a neighborhood of\n$\\mathbb{H}$ generates an open neighborhood of $I_{2}$ in $Sp(1)$.\nInorder to construct the perturbation space we will consider\n$\\mathcal{N} \\subset \\mathbb{H}^{-1}(k)$ a local Lagrangian\nsubmanifold in $\\gamma(t_{0})$. We can reduce, if necessary, the\nsize of the neighborhood $U$ of $\\gamma(t_{0})$ chosen previously\nin such way that $U$ admits the parameterization\n$(x=(x_{1},x_{2}),p=(p_{1},p_{2})):U \\to \\mathbb{R}^{2+2}$ as in\n\\cite{GonzaloPaternGenGeodFlowPositEntropy}, Lemma A3, that is,\\\\\na) $\\mathcal{N} \\cap U = \\{(x,0) \\}$; \\\\\nb) $\\omega=dx \\wedge dp$;\\\\\nc) $X^{\\mathbb{H}}|_{\\mathcal{N} \\cap U}= 1 \\frac{\\partial}{\\partial\nx_{1}}$.\\\\\nIn these coordinates we can see that $\\hat{\\gamma}=\\{ (t,0,0,0) \\mid\nt \\in (t_{0}-r,t_{0}) \\}$. Consider, the perturbation space,\n$$\\hat{\\mathcal{F}}=\\{ f:T^{*}M \\to \\mathbb{R} \\mid \\mathop{\\rm supp}\\nolimits(f) \\subset\n\\hat{W} \\subset W \\}$$ where $W = \\mathcal{N} \\cap V $ and $\\hat{W}$\nis a compact set contained in $W$ that contains $\\gamma(t_{1})$ in its\ninterior.  Observe that, $\\hat{\\mathcal{F}}$ can be identified with\n$C^{\\infty}(\\hat{W}, \\mathbb{R})$, therefore we can think\n$\\hat{\\mathcal{F}}$ as a vectorial space. Consider the\nfollowing finite dimensional subspace $\\mathcal{F} \\subset\n\\hat{\\mathcal{F}}$\n$$\\mathcal{F}=\\{ f \\mid f(x,p)=\\eta(x)(a \\delta_{t_{1}}(x_{1}) + b\n\\delta_{t_{1}}'(x_{1}) + c\n\\delta_{t_{1}}''(x_{1}))\\frac{1}{2}x_{2}^{2}, \\, a,b,c \\in\n\\mathbb{R} \\}$$ where $\\eta$ is a fix function with\n$\\mathop{\\rm supp}\\nolimits(\\eta) \\subset \\hat{W}$ and $\\eta \\equiv 1$ in some\nneighborhood of $\\gamma(t_{1})$, in $\\mathcal{N}$. Moreover,\n$\\delta_{t_{1}}$ is a smooth approximation of the delta of Dirac in\nthe point $t_{1}$.  Now we are able to define the differentiable map\n$S: \\mathcal{F} \\to Sp(1)$ given by\n$S(f)=\\hat{S}(\\mathbb{H}+f)=\\pi( [d_{\\gamma(t_{0}-r)}\n\\psi_{r}^{\\mathbb{H} +f }]_{E(0)}^{E(r)}) $. Observe that\n$dim(\\mathcal{F})=3=dim(Sp(1))$ and $S(0)=\\hat{S}(\\mathbb{H}+0)=\n\\pi( [d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H} +0 }]_{E(0)}^{E(r)})=\nI_{2}$. Thus we must to show that,\n$$d_{0}\\mathcal{F} :T_{0} \\mathcal{F} \\cong \\mathcal{F} \\to\nT_{Id_{2 \\times 2}}Sp(1) \\cong sp(1)$$ is surjective.\nGiven $h \\in \\mathcal{F}$ we have\n$\nd_{0}\\mathcal{F}(h)=\n\\pi( \\frac{d}{dl}  [d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H} +lh\n}]_{E(0)}^{E(r)} |_{l=0} ).\n$ \nConsider $\\xi \\in T_{\\gamma(t_{0}-r)}T^{*}M$, where $t \\in (0, r)$\n and define $$\\xi(t,l)=d_{\\gamma(t_{0}-r)}\n\\psi_{t}^{\\mathbb{H} +lh } \\xi.$$ For a fix $l$ we define a field\nthrough $\\gamma$ that verifies the equation\n$$\n\\left \\{\n\\begin{array}{clcr}\n\\dot{\\xi}(t,l) = J Hess(\\mathbb{H}+lh)(\\gamma(t)) \\xi(t,l) \\\\\n\\xi(0,l)=\\xi. \\quad \\quad \\quad \\quad \\; \\quad \\quad \\quad \\quad \\quad \\quad \\; ^{ }\n\\end{array}\n\\right.\n$$\nTaking the derivative of the equation above with respect to $l$ and\nusing the commutativity of the derivatives we get\n$$\\frac{d}{dt}( \\frac{d}{dl} \\xi(t,l)|_{l=0}) = J Hess(h)\n\\xi(t,l)|_{l=0} + J Hess(\\mathbb{H}) \\frac{d}{dl} \\xi(t,l)|_{l=0}$$\nDenote  $\\mathcal{H}= Hess(\\mathbb{H})$, $\\xi(t)=\\xi(t,l)|_{l=0}$ e\n$\\mathbb{Y}(t)= \\frac{d}{dl} \\xi(t,l)|_{l=0}$, then \n$$\n\\left \\{\n\\begin{array}{clcr}\n\\dot{\\mathbb{Y}}(t) =J \\mathcal{H} \\mathbb{Y}(t)+ J Hess(h) \\xi(t)\\\\\n\\mathbb{Y}(0)=0. \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\; ^{ }\n\\end{array}\n\\right.\n$$\nApplying the method of variation of constants and using\n$$\n\\left \\{\n\\begin{array}{clcr}\n\\dot{\\xi}(t) = J \\mathcal{H}(\\gamma(t)) \\xi(t) \\\\\n\\xi(0)=\\xi, \\quad \\quad \\quad \\quad \\quad \\; ^{ }\n\\end{array}\n\\right.\n$$\nwe get\n$$\\mathbb{Y}(t)=d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}\n\\int_{0}^{r} d_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}}\nJ Hess(h) d_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} \\xi dt.$$\nRemember that $\\mathbb{Y}(r)= \\frac{d}{dl} \\xi(r,l)|_{l=0}=\n\\frac{d}{dl} d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H} +lh }(\\xi)\n|_{l=0}$, so\n$$ \\frac{d}{dl} d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H} +lh }\n|_{l=0} =d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}\n\\int_{0}^{r} d_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}}\nJ Hess(h) d_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} dt.$$\nFrom this calculation\n$$\\;  d_{0}\\mathcal{F}(h)=\\pi \\left( [d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}\n\\int_{0}^{r} d_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}} J Hess(h)\nd_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} dt]_{E(0)}^{E(r)}  \\right ). \\; \n  (3)$$\nIn order to obtain the expression (3) we need to calculate $J Hess(h)$.\nAll the integrals will be calculated  with the delta of\nDirac and not with the approximations, however the same conclusions\nare true for an approximation, good enough.\nConsider $\\tilde{h}(x)=(a \\delta_{t_{1}}(x_{1}) + b\n\\delta_{t_{1}}'(x_{1}) + c\n\\delta_{t_{1}}''(x_{1}))\\frac{1}{2}x_{2}^{2}$ and\n$h(x)=\\eta(x) \\tilde{h}(x)$ then\n$d h= \\eta d \\tilde{h} + \\tilde{h} d \\eta$ and\n$d^{2}h= \\eta d^{2}\\tilde{h} +d \\tilde{h}^{*}  d \\eta + d \\eta^{*} d \\tilde{h} + \\tilde{h} d^{2}\n\\eta$. As, $Hess(h)(\\gamma)= d_{\\gamma}^{2}h$ and\n$jet_{1}(h)|_{\\gamma}=0$ we have that $Hess(h)(\\gamma)=\\eta(\\gamma)\nd_{\\gamma}^{2}\\tilde{h}$. On the other hand,\n$(d_{\\gamma}^{2}\\tilde{h})_{ij}= a \\delta_{t_{1}}(t_{0}-r +t) + b \\delta_{t_{1}}'(t_{0}-r +t)\n+ c \\delta_{t_{1}}''(t_{0}-r +t)$ if $ij=22$ and, and equal to 0 otherwise.\nTaking the $x_{1}$-support of $\\delta_{t_{1}}$ small enough, we can\nassume that\n$\nJ Hess(h)(\\gamma)=\n\\hat{A} \\delta_{t_{1}}(t_{0}-r +t) +\n\\hat{B} \\delta_{t_{1}}'(t_{0}-r +t) +\n\\hat{C} \\delta_{t_{1}}''(t_{0}-r +t)\n$\nwhere \n$(\\hat{A})_{ij}= -a $ if $ij=42$, and equal to 0 otherwise,\n$(\\hat{B})_{ij}= -b $ if $ij=42$, and equal to 0 otherwise and\n$(\\hat{C})_{ij}= -c $ if $ij=42$, and equal to 0 otherwise.\\\\\nDenote,\n\\begin{align*}\n\\hat{I}_{1}= d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\int_{0}^{r}\nd_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}} \\; \\hat{A} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} \\; \\delta_{t_{1}}(t_{0}-r +t) dt, \\\\\n\\hat{I}_{2}= d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\int_{0}^{r}\nd_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}} \\; \\hat{B} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} \\; \\delta_{t_{1}}'(t_{0}-r +t) dt, \\\\\n\\hat{I}_{3}= d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\int_{0}^{r}\nd_{\\gamma(t_{0}-r)}\\psi_{-t}^{\\mathbb{H}} \\; \\hat{C} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{t}^{\\mathbb{H}} \\; \\delta_{t_{1}}''(t_{0}-r +t) dt. \\\\\n\\end{align*}\nThus\\\\\n$\\hat{I}_{1} = d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}} \\; \\hat{A}\n\\; d_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}, $\\\\\n$\\hat{I}_{2} = - d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}}\n \\; [\\hat{B}, J \\mathcal{H}] \\;\nd_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}},\n$\\\\\nand\\\\\n$\\hat{I}_{3}=d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}} \\;(\n[[\\hat{C}, J \\mathcal{H}], J \\mathcal{H}] + [\\hat{C}, J\n\\dot{\\mathcal{H}}] ) \\; \\\\\nd_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}$.\\\\\nDefine $\\mathcal{Z}= \\hat{A} - [\\hat{B}, J \\mathcal{H}] +\n[\\hat{C}, J \\dot{\\mathcal{H}}] + [[\\hat{C}, J \\mathcal{H}], J\n\\mathcal{H}]$. Then,\n\n$$d_{0}\\mathcal{F}(h)=\\pi \\left( [d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}} \\;\n\\mathcal{Z} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(0)}^{E(r)}\n\\right ).$$\nWriting this matrix in the bases $E(0)$ and $E(r)$,  \nin each point of the curve, we get,\\\\\n$$[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}} \\;\n\\mathcal{Z} \\;\nd_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(0)}^{E(r)}=$$\n$$\n=[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}]_{E(0)}^{E(r)} \\;\n[d_{\\gamma(t_{0}-r)}\\psi_{-(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(r)}^{E(t_{0}-t_{1})}\n\\; [\\mathcal{Z}]_{E(t_{0}-t_{1})}^{E(t_{0}-t_{1})}$$ \\\\ $$\\;\n[d_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(t_{0}-t_{1})}^{E(r)}.$$\nMoreover $[d_{\\gamma(t_{0}-r)} \\psi_{r}^{\\mathbb{H}}]_{E(0)}^{E(r)}=\nI_{4}$ and there exists a symplectic conjugation $\\mathbb{G} \\in $ \n$Sp(2)$ between the base $E(t_{0}-t_{1})$ and the canonic symplectic\nbase,\n$\\left\\lbrace  \\frac{\\partial}{\\partial x_{1}}(\\gamma(t_{1})),\n\\frac{\\partial}{\\partial x_{2}}(\\gamma(t_{1})), \\right. $ \n$\\left. \\frac{\\partial}{\\partial p_{1}}(\\gamma(t_{1})),\n\\frac{\\partial}{\\partial p_{2}}(\\gamma(t_{1})) \\right\\rbrace $ such\nthat, \\\\ $[\\mathcal{Z}]_{E(t_{0}-t_{1})}^{E(t_{0}-t_{1})} = \n\\mathbb{G}^{-1} \\mathcal{Z} \\mathbb{G}$. Thus\n $d_{0}\\mathcal{F}(h)=$\n$$\\pi \\left( \\mathbb{G} [d_{\\gamma(t_{0}-r)}\n\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(r)}^{E(t_{0}-t_{1})} \\right)\n^{-1} \\pi \\left( \\mathcal{Z}\\right) \\, \\pi \\left( \\mathbb{G}\n[d_{\\gamma(t_{0}-r)}\\psi_{(t_{1}-t_{0}+r)}^{\\mathbb{H}}]_{E(r)}^{E(t_{0}-t_{1})}\n\\right)$$ that is, we need to show that $\\pi(\\mathcal{Z})$ is\nsurjective in $sp(1)$. A simple calculation shows that,\n$\n\\pi(\\mathcal{Z})=\n\\left [\n\\begin{matrix}\nz_{11}   & z_{12} \\\\\nz_{21}   & z_{22} \\\\\n\\end{matrix}\n\\right ]\n$\nwhere,\\\\\n$z_{11}=-b \\mathbb{H}_{p_{2}p_{2}} +2 c \\mathbb{H}_{p_{2}p_{2}}\n\\mathbb{H}_{x_{2}p_{2}} + \\dot{\\mathbb{H}}_{p_{2}p_{2}}$\\\\\n$z_{12}= 2c (\\mathbb{H}_{p_{2}p_{2}})^{2}$\\\\\n$z_{21}=-a + 2 b \\mathbb{H}_{x_{2}p_{2}}+ 2 c\n\\mathbb{H}_{p_{1}p_{2}} \\mathbb{H}_{x_{1}x_{2}} + 2 c\n\\mathbb{H}_{p_{2}p_{2}}\\mathbb{H}_{x_{2}x_{2}} -2 c\n\\mathbb{H}_{x_{2}p_{1}} \\mathbb{H}_{x_{1}p_{2}}\\\\ -4 c\n(\\mathbb{H}_{x_{2}p_{2}})^{2} -2 c \\dot{\\mathbb{H}}_{x_{2}p_{2}}$\\\\\n$z_{22}=b \\mathbb{H}_{p_{2}p_{2}} -2 c \\mathbb{H}_{p_{2}p_{2}}\n\\mathbb{H}_{x_{2}p_{2}} - \\dot{\\mathbb{H}}_{p_{2}p_{2}}$\\\\\n\nRemember that,\n$\nsp(1)=\n\\left \\{\n\\left [\n\\begin{matrix}\nB   & C \\\\\nA   & -B \\\\\n\\end{matrix}\n\\right ] \\; | A,B,C,D \\in\n\\mathbb{R}, \\;\n\\right \\}\n$\nand $\\mathbb{H}_{p_{2}p_{2}} \\neq 0$, thus we have the\nsurjectivity. In order to conclude the proof we must to find a potential in $M$ adapted to this perturbation. Consider $f \\in \\mathcal{F}$ arbitrarily close\nto zero such that $\\pi\n([d_{\\gamma(t_{0})}\\psi_{T}^{\\mathbb{H}+f}]_{E(0)}^{E(0)})$ is\nnondegenerate of order $\\leq 2m$. Let us remember that the\n$x$-support of $f$ is contained in $W$ which is an arbitrarily small\nneighborhood of $\\gamma(t_{1})$ in $\\mathcal{N}$.\nConsider $(\\hat{x},\\hat{p})$ the canonic symplectic coordinates in\n$\\gamma(t_{1})$, and $\\hat{\\pi} : T^{*}M \\to \\mathbb{R}$ given by\n$\\hat{\\pi}(\\hat{x},\\hat{p})=\\hat{x}$. As we are free  to\ndislocate the point $t_{1}$ by a $\\varepsilon$ arbitrarily small, we\ncan use the twist property of the vertical fiber bundle as in\nLemma~\\ref{PropTwistDoFibradoVertical} to conclude that\n$\\hat{\\pi}|_{\\mathcal{N}}$ is a local diffeomorphism in\n$\\gamma(t_{1})$. Take a diffeomorphism $q:W~\\subset~\\mathcal{N}~\\to~M$ given by\n$q(x)=\\hat{x}$, where $(x,0)\\equiv (\\hat{x},\\hat{p})$ in\n$\\mathcal{N}$. Choose the potential\n$$f_{0}(\\hat{x})=\n \\left \\{\n \\begin{matrix}\n f(q^{-1}(\\hat{x})) \\quad \\quad  x \\in \\hat{\\pi}(W)\\\\\n 0  \\quad \\quad  \\quad \\quad \\quad \\quad  x \\not\\in \\hat{\\pi}(W),\n \\end{matrix}\n \\right.\n$$\nby construction, $\\mathbb{H}(\\hat{x},\\hat{p}) + f_{0}(\\hat{x})$ has\nthe desired property. The lemma is proven.\n\\end{proof}\n\n\\textbf{Conjecture:}\\\\\n\\textit{If $\\dim(M)=n$ then $\\pi(\\mathcal{Z})$ is  surjective in $sp(n-1)$.}\\\\\n\nThe main obstruction to prove this conjecture is that if $\\mathcal{Z}= \\hat{A} - [\\hat{B}, J \\mathcal{H}] +\n[\\hat{C}, J \\dot{\\mathcal{H}}] + [[\\hat{C}, J \\mathcal{H}], J\n\\mathcal{H}]$ then we need to solve equations like $UX+XU=D$, in the space of simetric $n-1 \\times n-1$. But it is well known (see \\cite{GonDomtd}) that, in our case, the solving of this type of equations requires additional hypothesis on the eigenvalues of $\\mathbb{H}_{p p }$, which are not generic in Ma\\~n\\'e's sense. On the other hand, our approach it is essentially the only way to construct perturbations by potentials, thus we hope that in the future, we will be capable to solve this equation in higer dimensions.\n\n\n\\begin{lemma}\\label{a2a E denso Em aa}\nGiven $k \\in \\mathbb{R}$, $f_{0} \\in \\mathcal{R}(k)$,\n$\\mathcal{U}_{f_{0}} \\subseteq \\mathcal{R}$ a $C^{\\infty}$ neighborhood  of\n$f_{0}$, $\\alpha=\\alpha(\\mathcal{U}_{f_{0}})>0$, as in the\nLemma~\\ref{MinimoPerioLema}, $a \\in \\mathcal{R}$ such that $0 < a <\n\\infty$ and $\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}} \\neq\n\\varnothing$, we have that $\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$.\n\\end{lemma}\n\\begin{proof}\nTake $f \\in \\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$ and  $\\mathcal{U}$ an arbitrary  \nneighborhood of $f$. We must show that $\\mathcal{U} \\cap\n(\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}) \\neq  \\varnothing$.\nFrom the definition of $\\mathcal{G}_{k}^{a,a}$ we have that all periodic orbits\nof  $H+f$ in the level $k$ with  minimal period $\\leq\na$ are nondegenerate of order $m \\leq \\frac{a}{T_{min}}$.\nTake $\\rho_{f}: T^{*}M \\times (0,a) \\times (-\\varepsilon,\\varepsilon)\n\\to T^{*}M \\times T^{*}M \\times \\mathbb{R}$. From \nCorollary~\\ref{PeriodNivelSaoNaoDegenTrans} we have that\n$\\rho_{f} \\pitchfork \\Delta_{0}$.\nMoreover, by Lemma~\\ref{PeriodPontoSaoNaoDegenTrans},~(i),  \n$$\\rho_{f}^{-1}(\\Delta_{0})=\\{(\\vartheta, T, 0) \\mid \\vartheta\n\\in (H+f)^{-1}(k), \\; T \\in (0,a) ,\n\\psi_{T}^{H+f}(\\vartheta)=\\vartheta \\}$$\nObserve that $\\rho_{f}^{-1}(\\Delta_{0}) \\subset (H+f)^{-1}(k) \\times\n[0,a] \\times \\{0\\}$ which is a compact set. As $\\Delta_{0}$ is closed we have that\n$\\rho_{f}^{-1}(\\Delta_{0})$ is a submanifold of dimension 1, with a finite\nnumber of connected components. Since each periodic orbit,\n$\\{\\psi_{t}^{H+f}(\\vartheta) \\mid\nt \\in [0,T], \\; (\\vartheta, T, 0) \\in \\rho_{f}^{-1}(\\Delta_{0})\\}\n\\subset \\rho_{f}^{-1}(\\Delta_{0})$, is a connected component of\ndimension 1,the number of periodic orbits for\n$H+f$ in the level $k$ with minimal period $\\leq a$, distinct, is finite.\nDenote, $\\{\\psi_{t}^{H+f}(\\vartheta_{i}) \\mid\nt \\in [0, T_{i} = T_{min}(\\vartheta_{i})], \\; (\\vartheta_{i}, T_{i}, 0) \\in\n\\rho_{f}^{-1}(\\Delta_{0}) \\}$, for $i=1,..N$, the $N$ periodic orbits for\n$H+f$ in the level $k$, with its respective minimal periods.\nFrom Theorem~\\ref{PerturbacaoLocaldaOrbita} we can find a sum\nof $N$ potentials $f_{0}=f_{1}+...+f_{N}$\narbitrarily close to 0, such that, all orbits are nondegenerate of order $ \\leq 2m$\nfor $(H+f)+f_{0}$. The claim is proven because $f + f_{0} \\in \\mathcal{U} \\cap\n(\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}})$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{transa2a}\nWith the same notation of  the Lemma~\\ref{a2a E denso Em aa}, if\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}} \\neq \\varnothing$,\nthen $ev_{\\rho} : \\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}\n\\times T^{*}M \\times (0,2a) \\times (-\\varepsilon,\\varepsilon)\n\\to T^{*}M \\times T^{*}M \\times \\mathbb{R}$ is transversal to $\\Delta_{0}$.\n\\end{lemma}\n\\begin{proof}\nIndeed, given $(f, \\vartheta, T, S) \\in\n\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}\n\\times T^{*}M \\times (0,2a) \\times (-\\varepsilon,\\varepsilon)$,\nif $ev(f, \\vartheta , T, S) $ $\\not\\in \\Delta_{0}$,\nis done. So we can assume that $ev(f, \\vartheta, T, S)\n\\in \\Delta_{0}$, that is, $\\vartheta$ is a periodic orbit of\n $H+f$ in the level $k$ with minimal period,\n$T_{min}=T_{min}(\\vartheta)$ and $S=0$.\n\nIf $T=T_{min}$ then $ev \\pitchfork _{(f, \\vartheta, T, 0)}\n\\Delta_{0}$ by Lemma~\\ref{evaluationtransverlema}, (ii). On the\nother hand, if $T =m T_{min},\\; m \\geq 2$ we have that $  m \\leq  2a\/T_{min}$, that is,\n$ T_{min} \\leq  2a \/m \\leq a$ so $\\vartheta$ is nondegenerate of order $m$,\nbecause$f \\in \\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$.\nThus $ev \\pitchfork _{(f, \\vartheta, T, S)} \\Delta_{0}$ by\nLemma~\\ref{evaluationtransverlema}, (i).\n\\end{proof}\n\n\n\\begin{lemma}\\label{3meiosinter2a}\nWith the same notation of the Lemma~\\ref{a2a E denso Em aa}, if\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}} \\neq \\varnothing$,\nthen $(\\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap\n\\mathcal{U}_{f_{0}}) \\cap (\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}})$\nis dense in $\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$.\n\\end{lemma}\n\n\\begin{proof}\nConsider $\\mathcal{B}= \\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$,\nwhich is a submanifold of $C^{\\infty}(M;\\mathbb{R})$ because it is open.\nFrom the Lemma~\\ref{transa2a} we have that  $ev_{\\rho} : \\mathcal{G}_{k}^{a,2a}\n\\cap \\mathcal{U}_{f_{0}} \\times T^{*}M \\times (0,2a) \\times (-\\varepsilon,\\varepsilon)\n\\to T^{*}M \\times T^{*}M \\times \\mathbb{R}$ is transversal to $\\Delta_{0}$.\nThen,  Theorem~\\ref{abraham} implies that $\\mathfrak{R}=\\{ f \\in\n\\mathcal{G}_{k}^{a,2a} \\mid \\rho_{f} \\pitchfork \\Delta_{0} \\}$ is a generic\nsubset of $\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$. In\nparticular $\\mathfrak{R}$ is dense in\n$\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$.\nWe claim that $\\mathfrak{R} \\subset (\\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap\n\\mathcal{U}_{f_{0}}) \\cap (\\mathcal{G}_{k}^{a,2a} \\cap\n\\mathcal{U}_{f_{0}})$.\nIndeed, take $f \\in \\mathfrak{R}$, from Corollary~\\ref{PeriodNivelSaoNaoDegenTrans},\nall periodic orbits of the flow defined by $H+f$ in the energy level $k$,\nwith minimal period $T_{min}$ are nondegenerate of order\n$m \\leq \\frac{2a}{T_{min}}$ because $\\rho_{f} \\pitchfork \\Delta_{0}$.\nIf we have a periodic orbit for $H+f$ in the level $k$,\nwith minimal period $T_{min} \\leq 3a\/2$ take\n$m' \\leq \\frac{3a\/2}{T_{min}} = \\frac{2a}{4\/3 T_{min}} \\leq\n\\frac{2a}{T_{min}}$ then this orbit is nondegenerate of order $m'$\nin particular $f \\in \\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap \\mathcal{U}_{f_{0}}$.\nTherefore $(\\mathcal{G}_{k}^{3a\/2,3a\/2}\\cap \\mathcal{U}_{f_{0}}) \\cap\n(\\mathcal{G}_{k}^{a,2a}\\cap \\mathcal{U}_{f_{0}})$ is dense in\n$\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{3a23a2 denso Em aa}\nWith the same notation of the Lemma~\\ref{a2a E denso Em aa}, if\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}} \\neq \\varnothing$,\nwe have that $(\\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap\n\\mathcal{U}_{f_{0}})$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$.\n\\end{lemma}\n\\begin{proof}\nFrom Lemma~\\ref{3meiosinter2a} we have that\n$(\\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap \\mathcal{U}_{f_{0}}) \\cap\n(\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}})$ is dense in\n$\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$ and from\nLemma~\\ref{a2a E denso Em aa},  $\\mathcal{G}_{k}^{a,2a} \\cap \\mathcal{U}_{f_{0}}$\nis dense in $\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$ therefore\n$\\mathcal{G}_{k}^{3a\/2,3a\/2} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap\n\\mathcal{U}_{f_{0}}$.\n\\end{proof}\n\n\n{\\large \\textbf{  Proof of the Lemma~\\ref{ReduLocalDaPrimParte}:}} \\label{ProvaLemaReducao}\n\n\n\\begin{proof}\nGiven $k \\in \\mathbb{R}$, $f_{0} \\in \\mathcal{R}(k)$,\n$\\mathcal{U}_{f_{0}} \\subseteq \\mathcal{R}$ a $C^{\\infty}$ neighborhood \nof $f_{0}$, $\\alpha=\\alpha(\\mathcal{U}_{f_{0}})>0$ as in\nLemma~\\ref{MinimoPerioLema}. Take $c \\in \\mathbb{R}_{+}$, if  $c <\n\\alpha$ then $\\mathcal{G}_{k}^{c,c} \\cap\n\\mathcal{U}_{f_{0}}=\\mathcal{U}_{f_{0}}$ by Lemma~\\ref{MinimoPerioLema}.\n So we can assume that $c \\in \\mathbb{R}_{+}$, with  $c \\geq \\alpha > a >0$.\n\nWe claim that, $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell}a,\n(\\frac{3}{2})^{\\ell}a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$, $\\forall \\ell\n\\in \\mathbb{N}$. The proof is by induction in $\\ell$.\n\nFor $\\ell =1$ observe that, $\\mathcal{G}_{k}^{a,a} \\cap\n\\mathcal{U}_{f_{0}}=\\mathcal{U}_{f_{0}} \\neq \\varnothing $, because\n$\\alpha > a >0$. Therefore $\\mathcal{G}_{k}^{\\frac{3}{2} a,\n\\frac{3}{2} a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$ by\nLemma~\\ref{3a23a2 denso Em aa}. \n\nSuppose that, $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell}a,\n(\\frac{3}{2})^{\\ell}a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$, with $\\ell\n\\geq 1$. Then $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell}a,\n(\\frac{3}{2})^{\\ell}a} \\cap \\mathcal{U}_{f_{0}} \\neq \\varnothing\n$, from the density, and taking $a'=(\\frac{3}{2})^{\\ell}a$, we will have\nthat $\\mathcal{G}_{k}^{ \\frac{3}{2} a' , \\frac{3}{2} a'} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a',a'} \\cap \\mathcal{U}_{f_{0}}$ by Lemma~\\ref{3a23a2 denso Em aa}.\nSo $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell+1}a,\n(\\frac{3}{2})^{\\ell+1}a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$  concluding\nthe proof of the claim.\n\nConsider $\\ell_{0}$, such that, $(\\frac{3}{2})^{\\ell_{0}}a >\nc$. Then, $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell_{0}}a,\n(\\frac{3}{2})^{\\ell_{0}}a} \\cap \\mathcal{U}_{f_{0}} \\subset\n\\mathcal{G}_{k}^{c,c} \\cap \\mathcal{U}_{f_{0}} \\subset\n\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}=\\mathcal{U}_{f_{0}}$.\nSince $\\mathcal{G}_{k}^{(\\frac{3}{2})^{\\ell_{0}}a,\n(\\frac{3}{2})^{\\ell_{0}}a} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{G}_{k}^{a,a} \\cap \\mathcal{U}_{f_{0}}$, we conclude that\n$\\mathcal{G}_{k}^{c,c} \\cap \\mathcal{U}_{f_{0}}$ is dense in\n$\\mathcal{U}_{f_{0}}$. The lemma is proven.\n\\end{proof}\n\\vskip 10mm\n\n\n\n\\vspace{0.3cm} \\textbf{{\\Large Acknowledgements }} \\vspace{0.3cm}\n\nI wish to thank Dr. Gonzalo Contreras by the invitation to work in CIMAT, where part of this work was done. I thank to my thesis advisor, Dr. Artur O. Lopes for several important remarks and corrections, and  Patrick Z. A. by your help in translating this manuscript. This work is part\nof my PhD thesis in Programa de  P\\'os-Gradua\\c{c}\\~ao em Matem\\'atica - UFRGS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{sec:level1}First-level heading}\n\nSilicon Carbide has become the wide band gap (WBG) semiconductor with the most mature technology \\cite{Kimoto2015MaterialAnnealing} and it is finally ready to penetrate the power devices market after more than two decades elapsed as faint promise of next generation power electronics \\cite{Chelnokov1997OverviewElectronics,Willander2006SiliconApplications}. Indeed, it is expected that SiC will reach about 10\\% of the Si market by 2025 with a compound annual growth rate (CAGR) of about 40\\% from 2020 to 2022 \\cite{Bhalla2018StatusTechnology}. But, to continue the development of SiC technology and sustain the improvements in efficiency and performance of WBG based devices, research efforts need to be continued even at the level of material physical understanding. In fact, the material maturity process has been quite slow. An evident reason is the intrinsic complexity of this semiconductor compound: it occurs only rarely in nature and it has more than 200 polytypes \\cite{N.W.JeppsandT.F.Page1983NoTitle}. If properly understood and controlled, polytypism actually provides an added value. In fact, the most common SiC polytypes (3C-, 4H- and 6H-SiC) cover a range of band-gap from about 2.3 to 3.2 eV and thus they are suitable both for low and high-power devices.\\\\\nBesides the scientific interest, investigating SiC polytypism and understanding its driving force is crucial to correctly predict the energetics of extended defects in SiC, particularly stacking faults (SFs), which are a main concern of this WBG semiconductor since they cause deterioration and eventually failure of the devices after relative long operational time  \\cite{Ishida2002InvestigationEpilayers,Nagasawa2008FabricationDefects,Eriksson2011Electrical3C-SiC}.\\\\\nSiC polytypes consist of identical double layers with different stacking sequences, thus generating orders of SiC tetrahedrons with different orientation, as highlighted in blue and red color in Fig.\\ref{sym}. Truly, SFs are wrong sequences of the double layers or in other words, they can be seen as inclusions of few layers of a SiC polytype in the perfect layer stacking of another polytype (see inset in Fig.\\ref{sym}).  Due to the small-scale energy difference between the stacking sequences of double layers and hence between the different SiC polytypes, as discussed below, perturbations of the ideal stacking sequence during SiC crystal growth are very likely. This is another reason why SFs are so critical in this material. \\\\\n\\begin{figure}[b]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{SiCsymmDrawing4.png}\n\\caption{\\label{sym} Tetrahedral stacking sequences of 3C-, 2H-, 4H- and 6H-SiC. The red and blue triangles highlight the twinned or normal tetrahedra and correspond to down or up spin configurations of the SiC layers, according to the axial next-nearest neighbor Ising (ANNNI) model \\cite{Fisher1980InfinitelyModei}\nThe inset shows the stacking sequence of 3C-SiC polytype including an intrinsic, extrinsic and double extrinsic stacking fault (labeled ISF, ESF and ESFd respectively). \n}\n\\end{figure}\nThe literature on the thermodynamic stability and polytypism of SiC is abundant, encouraged by the physical and technological interests evidenced above. Still, the predictions on the free energy of SiC polytypes are misleading and the energetic hierarchy reported by different theoretical methods is often inconsistent with some experimental observations. In fact, a paradox concerning SiC polytypes \\cite{Heine1991NoTitle,Cheng1990AtomicPolytypes, Fan2014SiliconNanostructures,Zywietz1996InfluenceCarbide} has been often discussed, dealing with the theoretical predictions of hexagonal (6H) SiC as the most stable polytype and of the cubic (3C) one as not stable at any temperature \\cite{Cheng1990AtomicPolytypes,Bechstedt1997PolytypismCarbide,Feng2004SiCApplications}. Contrary, experiments have shown that the cubic (3C) structure does grow in preference to all others and only at very high temperatures hexagonal phases, i.e. 4H and 6H polytypes, have been observed to prevail \\cite{Boulle2010QuantitativeCrystals,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique,Heine1991NoTitle,Bechstedt1997PolytypismCarbide}. Thereby,  SiC polytype stability at different temperatures remains unclear. Different arguments has been proposed over the years to explain polytypism and polytypic transformation in SiC \\cite{Bechstedt1997PolytypismCarbide,Cheng1988Inter-layerPolytypes,Heine1991NoTitle,Ching2006TheSiC,Zywietz1996InfluenceCarbide,Bernstein2005Tight-bindingCarbide,Lindefelt2003StackingPo}, including  the motion of partial dislocations \\cite{Pirouz1993PolytypicTEM,Pirouz1997PolytypicSiC,Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations} and impurity effects \\cite{Heine1991NoTitle} on crystal growth. Nonetheless, inconsistencies between theory and experiments still remain. \n\nRecently it was shown that density functional theory (DFT) calculations \\cite{SM} including the van der Waals (vdW) correction do predict the 3C phase to have the lowest free energy at T=0K \\cite{Kawanishi2016EffectPolytypes}. This intriguing result points out the importance of considering long-range interactions when comparing different SiC polytypes.\n\nIn this Letter we show that, by consistently adding the entropic contribution to the free energy, within a DFT approach that includes vdW corrections, the full T-dependent hierarchy of polytypes is correctly predicted, showing a cross-over between 3C and 6H (or 4H) phases at typical experimental temperatures. As detailed below, the present results allow for better understanding of the physics behind SiC polytypism by simple thermodynamics considerations, highlighting the correlation between hexagonality, vibrational properties and cohesive energy.\nMoreover, calculations of the stability of SiC SFs reveal a T-dependent behavior, intimately correlated to the polytypic stability, and they provide insight into optimal growth temperatures for lowering the density of such defects.      \n\n\n\n\n\nIn Table \\ref{tab:table1} the lattice constants of 3C-, 6H-, 4H- and 2H-SiC are reported, both calculated at the generalized gradient approximation (GGA) level \\cite{Perdew1996GeneralizedSimple} and with the semiempirical Grimme's method, which employs GGA-type density functional constructed with a long-range dispersion correction \\cite{Barone2009RoleCases,Grimme2006SemiempiricalCorrection} and accounting for vdW interactions.\n\\begin{table}[!t\n\\caption{\\label{tab:table1}%\nLattice constants (\\text{\\AA}), total energy $\\Delta E_T$ (relative to 3C-SiC), hexagonality and heat of formation $\\Delta H_f$ (meV\/SiC) of 3C-, 6H-, 4H- and 2H-SiC.}\n\\begin{ruledtabular}\n\\begin{tabular}{ l  c c c}\n\\textrm{Polytype   \\hspace{15pt} hex} & \n\\textrm{a,c} &\n\\textrm{$\\Delta E_T$}&\n\\textrm{$\\Delta H_f$} \\\\\n\n\\colrule\n\\textbf{3C-SiC} \\hspace{18pt} 0\\%\\\\\nGGA & 4.377 & 0 & -402\\\\\n GGA(vdW) & 4.352 & 0 & -785\\\\\n exp\\cite{o1960silicon, Kleykamp1998GibbsModifications,Greenberg1970TheCalorimetry,2014CRCData}  & 4.3596  &  & -650, -758, -771\\\\\n\\colrule\n\n\\textbf{6H-SiC} \\hspace{18pt} 33\\%\\\\ \nGGA & 3.093,15.178 & -1.7 & -404 \\\\\n GGA(vdW) & 3.075,15.105 & 1.4 & -784\\\\\n exp\\cite{o1960silicon, Kleykamp1998GibbsModifications,Greenberg1970TheCalorimetry,Tairov1983ProgressCrystals} & 3.080,15.117 & & -676, -747, -771 \\\\\n\\colrule\n\n\\textbf{4H-SiC} \\hspace{18pt} 50\\%\n\\\\GGA & 3.092,10.123 & -1.8 & -404\\\\\n  GGA(vdW) & 3.074,10.079 & 2.9  & -783\\\\\n exp\\cite{Zemann1965iCrystalWyckoff,Tairov1983ProgressCrystals,2014CRCData} & 3.073,10.053 & & -650, -689\\\\\n\\colrule\n\n\\textbf{2H-SiC} \\hspace{18pt} 100\\%\\\\\n GGA & 3.090,5.072 & 5.8 & -396\\\\\n  GGA(vdW) & 3.072,5.056 & 15.1 & -770\\\\\n exp\\cite{Schulz1979STRUCTUREZnO} & 3.079,5.053 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nIt is evident that the agreement between theoretical and experimental lattice parameters improves considerably for vdW-corrected DFT simulations. These improvements are even more appreciable by looking at the heat of formation ($\\Delta H_f$) in Table \\ref{tab:table1}, which is severely underestimated by GGA but in very good agreement with experiments in the case of vdW-corrected simulations.\nNote that, not only the magnitude of the heat of formation but also the order of the values calculated for the different polytypes changes whether or not the simulations include the vdW correction. This is very clear looking at the total energy of the different polytypes ($\\Delta E_T$) in Table \\ref{tab:table1}, calculated as a relative value with respect to the total energy of 3C-SiC. While $\\Delta E_T$ values calculated by GGA are all negative except for 2H-SiC, thus predicting 3C-SiC as the least stable polytype after the 2H-SiC, the vdW-corrected simulations give all positive $\\Delta E_T$. Hence, 3C-SiC turns out to be the most stable polytype. This is well in agreement both with experimental evidences inferring 3C-SiC as the most stable SiC structure in the nuclear stage \\cite{Tairov1983ProgressCrystals} and with a recent theoretical work \\cite{Kawanishi2016EffectPolytypes}.\nNevertheless, the predicted energetic hierarchy at T=0K is still not sufficient to understand the competition in stability of SiC polytypes at higher temperature, as probed experimentally \\cite{Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique}. Thus, the variation of the entropic contributions with temperature for the different polytypes becomes crucial. This has been included in Fig.\\ref{HelmohotzF}, where the Helmholtz free energy for the SiC polytypes is plotted as a difference between the values of the 2H, 4H and 6H polytypes and that of 3C-SiC. \\\\\nThe typical expression of the Helmholtz free energy $F(T)=U-TS$, with $U$ the internal energy and $TS$ the product of temperature and entropy, can be also reformulated as $F(T)=U_0+U_{vib}-TS$, where the internal energy $U$ is split into the vibrational internal energy ($U_{vib}$) and static internal energy ($U_0$), with the latter corresponding to the total DFT energy of the SiC polytypes at their GGA-vdW equilibrium geometry. Also the other terms of the Helmholtz free energy can be conveniently calculated by DFT \\cite{Baroni2009ThermalPhonons}. In fact, one can additionally formulate $F(T)$ as a sum of the electronic and vibrational contributions, thus $F(T)=F_{el}(T)+F_{vib}(T)$. The former term $F_{el}(T)$ can be reasonably approximated by its zero-temperature limit $(U_0)$ \\cite{Bechstedt1997PolytypismCarbide}, by neglecting the electronic entropy; the vibrational contribution, which then correspond to $F_{vib}(T)=U_{vib}-TS$, can be calculated by the quasi-harmonic approximation as \\cite{Born1955DynamicalLattices,Feng2004SiCApplications}:\n\\begin{equation}\n\\nonumber \nF_{vib}(T)=\\int _{0}^{\\infty}g(\\omega)  \\left [  \\frac{\\hbar\\omega}{2 K_B T} + \\ln \\left ( 1-e^{  \\frac{-\\hbar\\omega}{K_B T } } \\right ) \\right ] d\\omega \\thinspace;\n\\end{equation}\nwhere $\\omega$ is the phonon frequency and $g(\\omega)$ is the phonon density of states. In our simulations, $\\omega$ and $g(\\omega)$ are calculated in the framework of the density functional perturbation theory (DFPT) \\cite{Baroni1987Greens-functionSolids,Baroni2009ThermalPhonons}.\\\\\n\\begin{figure}[!t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{HelmotzFinale.png}\n\\caption{\\label{HelmohotzF} Difference between the Helmholtz free energy of 3C-SiC with respect to the value of 6H, 4H and 2H polytypes.\n\\end{figure}\nThe values at T=0K of the three curves plotted in Fig.\\ref{HelmohotzF} reveal that the zero-point internal energy (ZPE), which is the main contribution to $F_{vib}$ in the low temperature range, only slightly affects the static internal energy (cf. $\\Delta E_T$ in Table \\ref{tab:table1}) and does not change the energetic hierarchy of the SiC polytypes. But for temperatures above 500K the vibrational contribution ($F_{vib}$) becomes considerable and the Helmohltz free energies of the hexagonal polytypes get closer to the cubic one. Particularly, at temperature of about 1750K the difference between the free energies $(F_{3C}-F_{6H})$ crosses the 0 energy line, meaning that 6H polytype becomes thermodynamically more stable than 3C. The energy crossing between 4H- and 3C-SiC is predicted a bit higher in temperature, at about 2400K. Contrary, 2H- never becomes more stable than 3C-SiC in the temperature range considered, albeit their free energies get closer at higher temperatures. The comparison between hexagonal polytypes reveals that 2H-SiC is the least thermodynamically stable structure: the difference of its free energy and that of 4H or 6H polytype marginally decreases with the temperature. Contrary, the free energies of 6H and 4H polytypes get very close at temperatures around 2500K. Our T-dependent diagram of the polytypic stability plotted in Fig.\\ref{HelmohotzF} is in excellent agreement with several experimental evidences such as the preferential growth at temperatures below $\\sim$1850K of the 3C polytype over all others \\cite{Heine1991NoTitle,Bechstedt1997PolytypismCarbide,Cheng1988Inter-layerPolytypes}, the higher stability of 4H and 6H polytypes at higher temperatures \\cite{Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique}, and the rare appearance of 2H-SiC \\cite{Pirouz1997PolytypicSiC,Imade2009Liquid-phaseMelt}.  \n\nThe correlation between hexagonality and the observed trends in the Helmholtz free energy is elucidated by the calculated entropy, $S=-(\\delta F_{vib}\/\\delta T)$, reported in Fig.\\ref{FigEntropy}.\nAn opposite hierarchy of the entropy with respect to the cohesive energy at all temperatures is found. This is further supported by the general decreasing trend of phonon frequencies with hexagonality, which is evident in the phonon density of states (PDOS) plotted for the different polytypes in the region of the longitudinal optical (LO) branch \\cite{SM} in the inset of Fig.\\ref{FigEntropy}. The shift of the phonon frequencies is associated to a different strength of the interactions between hexagonally and cubically stacked layers, thus a correspondence between the lower hexagonality and the higher cohesion of the structure is evident.\nThe difference in the free energy at T=0 between 3C- and 2H-SiC is so large compared to their entropy difference that 2H remains less stable even at high temperature. On the contrary the much smaller difference between the static energy  of 3C and 6H (or 4H) is overcompensated by the larger entropy contribution of the latter.\nThis provides an intuitive picture to understand SiC polytypism: cubic SiC polytype have higher cohesive energy, higher stiffness and lower entropy; contrary, hexagonal (2H) polytype has the lowest cohesive energy, lower stiffness but a higher entropy; in between, the trends of the other two hexagonal polytypes investigated follow their percentage of hexagonality, with their higher entropy, as compared to 3C-SiC, leading to changes of the energetic hierarchy with the temperature.\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{Entropy.png}\n\\caption{\\label{FigEntropy} Difference between the entropy of \\emph{x}H and 3C politypes. The inset shows the PDOS in the region near the LO band.}\n\\end{figure}\n\nThe correct prediction of the free energy of polytypes is a compelling need for the investigation of other essential aspects of SiC and it will be exploited below for studying the SFs stability. \nThe energetic cost of any error in the stacking sequence can be estimated by two different approaches: modeling a perturbed layer stacking by supercell structures, such as those illustrated in the inset of Fig.\\ref{sym}, and then calculating the total energy of the faulted supercell; alternatively one can calculate the SF energy according to the axial next-nearest neighbor Ising (ANNNI) model \\cite{Fisher1980InfinitelyModei}. In fact, the energy of the different polytypes, both pristine or faulted, can be expressed in terms of the interactions between SiC double layers. Accordingly, for the \\emph{x}H- (or \\emph{x}C-) SiC polytypes the total free energy is:\n\\begin{equation}\nE = E_0 - \\frac{1}{x}\\displaystyle\\sum_{i=1}^n \\displaystyle\\sum_{j=1}^{\\infty}J_j \\sigma_i \\sigma_{i+j} \\thinspace;\n\\label{eqn:Name}\n\\end{equation}\nwhere $E_0$ is a common reference energy and $J_j$ are the interaction energies between \\emph{i}th-neighbour double layers. The double layers are represented by a pseudospin $\\sigma_i$, which can be spin-up or spin-down (with value +1 or -1, respectively) according to the tetrahedron orientation that the layers form: $\\sigma_i=+1$ corresponds to a normal orientation (blue color in fig.\\ref{sym}) while $\\sigma_i=-1$ to a twinned one (red in fig.\\ref{sym}) \\cite{Cheng1988Inter-layerPolytypes, Pirouz1993PolytypicTEM}. For instance, 4H-SiC is represented by two spin-up and two spin-down, 6H- by three up and three down. In Eq.\\ref{eqn:Name}, spin coupling higher than third-order are usually neglected. The static free energies of Table \\ref{tab:table1} can be then used to obtain the $J_j$ values from Eq.\\ref{eqn:Name} and they allows one to calculate the free energies of faulted structures. Finally, stacking fault energies are estimated as the energy difference between the faulted and the pristine structure \\cite{Hong2000StackingCrystals,Boulle2013PolytypicSimulations,Lindefelt2003StackingPo} and they are listed in Table \\ref{tab:table2}. \n\\begin{table}[!t\n\\caption{\\label{tab:table2}%\nFormation energy (mJ m$^{-2}$) of SFs in 3C, 6H, and 4H-SiC. Values calculated by the ANNNI model (and GGA calculations with\/out vdW correction) or obtained by the supercell approach (with vdW) are reported. Experimental values from \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} and other theoretical values from \\cite{Umeno2012Ab3C-SiC, Kackell1998StackingStudy,Lindefelt2003StackingPo} are also listed.}\n\\begin{ruledtabular}\n\\begin{tabular}{l|ccc|c|c}\n        & \\multicolumn{3}{c|}{3C-SiC} & 6H-SiC & 4H-SiC \\\\\n        & ISF     & ESF    & ESFd    & ISF    & ISF    \\\\ \\hline\nGGA     & 4.35    & -16.7 & -19.5    & 2.84    & 18.23        \\\\\nGGA vdW & 40.70  & 19.6  & 16.85   &    2.77    &   18.35    \\\\\nSuperc.& 40.21 & 19.62 & 17.04 & & \\\\\nExp.    & 34 &  &  &2.9$\\pm$0.6&14.7$\\pm$2.5 \\\\\nCalc.\\cite{Lindefelt2003StackingPo}  & -6.27 &   &   & 3.14 & 18.3\\\\\nCalc.\\cite{Umeno2012Ab3C-SiC}  & 10.3 & -7.83 & -11.6 &  &  \\\\\nCalc.\\cite{Kackell1998StackingStudy}  & -3.4 & -28 &  &  &  \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nWe also checked the reliability of the SF energies calculated by the ANNNI model comparing them with the corresponding value obtained by simulating the defected supercells of 3C-SiC illustrated in the inset of Fig.\\ref{sym}. Interestingly, the calculated SF energy values are all positive and in excellent  agreement with experimental estimations \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} if the vdW correction is included in the DFT simulations. Contrary, SF energies obtained by bare GGA-DFT are very different, particularly for 3C-SiC. In fact, the formation energy of intrinsic stacking faults (ISFs) in 3C-SiC calculated by GGA is much lower than the corresponding value obtained including the vdW correction. For extrinsic stacking faults, both single (ESF) and double (ESFd), the SF energies turn even into negative values. This is not surprising if one goes through the literature of SFs in 3C-SiC, in which very small or even negative theoretical values of the ESF energy in 3C-SiC are well-accepted (see Table \\ref{tab:table2}). Instead, these SF energies are doubtful if compared with experiments \\cite{Ning1990ExperimentalCrystals}. Typically, SF energy is experimentally estimated by comparing the measured width of the stacking fault between the two terminating partial dislocations \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} and its expectation by means of the dislocation theory for anisotropic elastic media \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals,P.Hirth1982TheoryDislocations}. Accordingly, SFs with negative formation energies should not have finite width, thus in evident contradiction with experiments. \\\\%Note also that the discrepancy between bare GGA and vdW-corrected calculations is appreciable for the SF energies of 3C-SiC but not much for the hexagonal structure, confirming the higher effect of the long-range interactions on the energetics of the SiC structures with lower hexagonality \\cite{Kawanishi2016EffectPolytypes}.\nFinally, by exploiting the ANNNI model and the T-dependent free energies presented above, we plot in Fig.\\ref{defF} the SF formation energies for the 3C-, 6H-, and 4H-SiC as a function of the temperature. Different trends in temperature between hexagonal and cubic polytypes are found: while the formation energies of SF in 6H- and 4H-SiC slightly increase with the temperature, for 3C-SiC the SF energies decrease substantially with the temperature and particularly for the ESFs, they become negative at temperature above 1750K. \\\\\n\\begin{figure}[!t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{SFenergy2.png}\n\\caption{\\label{defF} Formation energy of ISF in 6H- and 4H-SiC, and of ISF, ESF and ESFd in 3C-SiC.}\n\\end{figure}\nIn conclusion, we have shown that DFT calculations including the vdW correction predict a T-dependent hierarchy of SiC \npolytypes in perfect agreement with the experimental results. 3C-SiC is predicted to have the highest cohesive energy but the lowest entropy. At high temperature, the higher entropic contribution  to the free energy of the hexagonal polytypes stabilizes their structures, with the 6H and 4H-SiC becoming thermodynamically more stable than 3C-SiC. These results demonstrate the key role of the thermodynamics in determining SiC polytypism and contribute to finally reconcile theory and experiments. They are also essential for understanding SF stability in SiC, yielding positive formation-energy values for both ESF and ISF in 3C-SiC that are at variance with previous theoretical results, but in accord with experimental evidences. Moreover, the  formation energy of 3C-SiC SFs is predicted to decrease substantially with temperature, becoming lower than that predicted for 6H-SiC and eventually negative. Importantly, this indicates that too high deposition temperatures should be avoided in order to decrease SF density in 3C-SiC.\n\n\\clearpage\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nAuthors acknowledge EU for founding the CHALLENGE project (3C-SiC Hetero-epitaxiALLy grown on silicon compliancE substrates and 3C-SiC substrates for sustaiNable wide-band-Gap powEr devices) within the EU's H2020 framework programme for research and innovation under grant agreement n. 720827.\n\\end{acknowledgements}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\t\n\t\\subsection{Dilating sets in diverse geometric contexts}\n\t\\label{sec:intro}\n\tIt is an intriguing geometric problem to understand the long-term distribution properties of a diversified range of progressively dilating shapes, when the ambient space in which they live is folded according to a prescribed rule. Formally, the framework underlying such a question can be laid down as follows. Consider a compact connected Riemannian manifold $S$ and a Riemannian covering space $N$ with covering projection $\\pi\\colon N\\to S$. The manifold $N$ plays the role of the ambient space, while $S$ is to be interpreted as the manifold resulting after a certain folding procedure operated on $N$. A homothety on $N$ is a diffeomorphism $h\\colon N\\to N$ transforming the Riemannian metric on $N$ into a scalar multiple thereof; if the rescaling ratio is equal to $1$, it simply reduces to a Riemannian isometry. Let now $(h_t)_{t\\in \\R_{>0}}$ be a collection of homotheties on $N$ whose scaling factor tends to infinity as $t$ does, and fix a Borel probability measure $\\mu$ on $N$. The latter should be thought of as describing quantitatively the original shape, whose dilations we wish to examine. For instance, $\\mu$ might be the renormalized volume measure on a finite-volume Riemannian submanifold of $N$, e.g.~a rectifiable curve. We then let $\\mu_t$ be the push-forward of $\\mu$  under the homothety $h_t$ and denote by $m_t$ the projection of $\\mu_t$ onto $S$, for any $t>0$.\n\t\n\tA mathematical formulation of our problem consists in asking for the limit points, as $t$ goes to infinity, of the family of measures $(m_t)_{t>0}$ in the topology of weak$^*$ convergence of probability measures on $S$. Somewhat less pretentiously, it is already interesting to determine sufficient geometric conditions on the initial measure $\\mu$ ensuring that the $m_t$ equidistribute as $t$ grows larger, that is, that they converge in the aforementioned topology to the renormalized volume measure $\\vol_S$ on $S$; this circumstance amounts pictorially to the fact that the measures $m_t$ (and hence, in an intuitive sense, their supports) fill up the folded space $S$ in the most uniform way.\n\t\n\t\\subsubsection*{The Euclidean case}\n\t\n\tTo the best of the authors' knowledge, the question formulated in the previous paragraph was first asked by Dennis Sullivan (cf.~\\cite{Randol}) in the zero-curvature setting of Euclidean spaces and tori, that is, when $S=\\T^{d}=\\R^{d}\/\\Z^d$ and $N=\\R^{d}$ for some integer $d\\geq 1$, and for $\\mu$ being the volume measure on a compact lower-dimensional submanifold of $\\R^{n}$. In this case, the transformations $h_t$ are the standard linear homotheties $x\\mapsto tx,\\;x\\in \\R^{d}$. The problem was originally addressed by Randol in~\\cite{Randol}, both in an Euclidean and in a hyperbolic setup. In the former case, the following equidistribution result is established.\n\t\n\t\\begin{thm}[{cf.~\\cite[Thm.~1]{Randol}}]\n\t\t\\label{thm:Randol}\n\t\tSuppose $C$ is the smooth boundary of a compact convex subset of $\\R^{d}$ with non-empty interior, and assume its Gaussian curvature is everywhere positive. Let $\\mu$ be the volume measure on $C$, renormalized to be a probability measure. Then the probability measures $m_t$ on $\\T^{d}$ defined by\n\t\t\\begin{equation*}\n\t\t\tm_t(A)=\\mu(\\{x\\in \\R^{n}:tx+\\Z^{d}\\in A \\})\\;, \\quad A\\subset \\T^{d} \\emph{ Borel }\n\t\t\\end{equation*} equidistribute towards the Haar measure on $\\T^{d}$ as $t\\to\\infty$.\n\t\\end{thm} \n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:quantitative}\n\t\t\\begin{itemize}\n\t\t\t\\item [(a)] The result in Theorem~\\ref{thm:Randol} is actually quantitative: for every sufficiently regular function $f$ on $\\T^{d}$, it holds that\n\t\t\t\\begin{equation*}\n\t\t\t\t\\biggl|\\int_{\\T^{d}}f\\;\\text{d}m_t-\\int_{\\T^{d}}f\\;\\text{d}m_{\\T^{d}}\\biggr|\\ll_{f,d}t^{-(d-1)\/2}\\;,\n\t\t\t\\end{equation*}\n\t\t\twhere $m_{\\T^d}$ denotes the Haar probability measure on $\\T^{d}$.\n\t\t\t\\item [(b)] As a special case of Theorem~\\ref{thm:Randol}, expanding spheres in $\\R^{d}$ equidistribute on the standard torus with a polynomial rate depending on the dimension $d$. The study of expanding spheres in other geometric settings shall be a driving theme of this manuscript. \n\t\t\t\\item [(c)] In~\\cite[Thm.~2]{Randol}, equidistribution is generalized to uniform measures supported on lower-dimensional rectilinear simplices in $\\R^{d}$.\n\t\t\\end{itemize}\n\t\\end{rmk}\n\t\n\tThe proof of Theorem~\\ref{thm:Randol} (more generally, of its quantitative version stated in Remark~\\ref{rmk:quantitative}(a)) relies on classical Fourier analysis on the $d$-dimensional torus; it is remarkably straightforward, once the decay properties at infinity of the Fourier transform of the measure $\\mu$\n\tare known. As such, it has been elaborated upon by Strichartz in~\\cite{Strichartz} to prove the following generalization of Theorem~\\ref{thm:Randol}. Let us say that the Fourier transform $\\hat{\\mu}\\colon \\R^{d}\\to \\C$ of $\\mu$ decays on rays if \n\t\\begin{equation}\n\t\t\\label{eq:raydecay}\n\t\t\\lim\\limits_{t\\to\\infty}\\hat{\\mu}(tx)=0\n\t\\end{equation}\n\tfor every nonzero vector $x\\in \\R^{d}$.\n\t\\begin{thm}[{cf.~\\cite[Lem.~1]{Strichartz}}]\n\t\t\\label{thm:Strichartz}\n\t\tLet $\\mu$ be a Borel probability measure on $\\R^{d}$ whose Fourier transform decays on rays. Then the conclusion of Theorem~\\ref{thm:Randol} holds true. \n\t\\end{thm}\n\t\n\tIt actually suffices that $\\hat{\\mu}$ decays on integral rays, that is, that~\\eqref{eq:raydecay} is verified, less restrictively, for any non-zero $x\\in \\Z^{d}$. On the other hand, it turns out (cf.~\\cite[Lem.~1]{Strichartz}) that decay on arbitrary rays is equivalent to equidistribution of the projections of the $\\mu_t$ onto any torus $\\R^{d}\/\\La$, where $\\La$ is a lattice in $\\R^{d}$.\n\t\n\t\n\t\n\tTo conclude this brief account of the state of the art on the problem in flat geometry,  we mention that the question of the limiting distribution of expanding circles has been recently examined in the setting of translation surfaces by Colognese and Pollicott~\\cite{Colognese-Pollicott}, who prove (non-effective) equidistribution towards a probability measure which is equivalent, though in general not proportional, to the area measure on the given surface.\n\t\n\t\n\t\\subsubsection*{The hyperbolic case}\n\tAs already mentioned, Randol's investigations in~\\cite{Randol} were not confined to the zero-curvature case. In the hyperbolic framework, namely when the sectional curvature is constantly equal to $-1$, $S$ is a compact connected hyperbolic $d$-manifold ($d\\geq 2$) and $N$ can be taken as its Riemannian universal covering manifold, that is, the $d$-dimensional hyperbolic space $\\Hyp^{d}$. A choice of the homotheties $(h_t)_{t>0}$ is determined by fixing a base point $x_0\\in \\Hyp^{d}$ and letting $h_t$ be the map which transforms\\footnote{This produces a well-defined assignment, as $\\Hyp^{d}$ is a uniquely geodesic metric space (cf.~\\cite[Part I, Chap.~1]{Bridson-Haefliger}) for the distance induced by the hyperbolic Riemannian metric.} each Riemannian geodesic $\\gamma(s)$ ($s\\in \\R$) passing through $x_0$ at time $0$ into the geodesic $\\gamma(ts)$. In this context, the result that can be elicited from the discussion in~\\cite{Randol} reads as follows.\n\t\\begin{thm}[{cf.~\\cite{Randol}}]\n\t\t\\label{thm:Randolhyp}\n\t\tLet $S$ be a compact connected hyperbolic $d$-manifold, $\\pi\\colon \\Hyp^{d}\\to S$ the universal covering map, $C$ a $(d-1)$-dimensional hyperbolic  sphere of unit radius centered at a point $x_0\\in \\Hyp^{d}$, $\\mu$ the unique isometrically-invariant\\footnote{Here we obviously intend invariance under isometries of $C$ equipped with the induced hyperbolic metric.} Borel probability measure on $C$, $(h_t)_{t>0}$ the family of homotheties $\\Hyp^{d}\\to \\Hyp^{d}$ with center $x_0$ defined as above. Then the probability measures $m_t$ on $S$ defined by\n\t\t\\begin{equation}\n\t\t\t\\label{eq:measureshyperbolic}\n\t\t\tm_t(A)=\\mu(\\{x\\in \\Hyp^{d}:\\pi\\circ h_t(x)\\in A \\})\\;, \\quad A\\subset S \\emph{ Borel}\n\t\t\\end{equation}\n\t\tequidistribute towards the renormalized volume measure on $S$ as $t\\to\\infty$.\n\t\\end{thm}\n\t\n\t\\begin{rmk}\n\t\tHere again the result takes on a quantitative form: the rate of equidistribution of the measures $m_t$ defined in~\\eqref{eq:measureshyperbolic} is exponential, as opposed to the Euclidean case, with the exponent depending on the spectral gap of the hyperbolic manifold $S$ (cf.~\\cite{Randol}).\n\t\\end{rmk}\n\t\n\tAkin in spirit to the proof of Theorem~\\ref{thm:Randol}, the argument leading to Theorem~\\ref{thm:Randolhyp} is based upon the harmonic analysis of locally symmetric spaces via techniques related to the Selberg trace formula; for those, the reader is referred to Selberg's original article~\\cite{Selberg-trace}.\n\t\n\t\n\t\n\t\\subsubsection*{Lifting the question to unit tangent bundles} \n\t\n\tLet us now consider the following upgraded version of the problem explored in the foregoing paragraphs. Suppose that $S$, $N$ and $(h_t)_{t>0}$ are as in the beginning of the present section, with $y_0\\in N$ being the common center of the homotheties $h_t$, and let $C$ be a compact Riemannian hypersurface in $N$.  Assume further that the Riemannian distance function on $N$ turns it into a uniquely geodesic metric space\\footnote{This is certainly the case for $N=\\R^d$ and $N=\\Hyp^{d}$, equipped respectively with the standard Euclidean metric and with the hyperbolic metric.}. If $T^{1}N$ denotes the unit tangent bundle of $N$, then $C$ identifies uniquely the subset $\\tilde{C}$ of $T^{1}N$ consisting of all pairs $(x,v)$, where $x$ is a point in $C$ and $v$ is the unit-length tangent vector to the unique geodesic connecting $y_0$ to $x$. Similarly, if $C_t$ indicates the image of $C$ under the homothety $h_t$ for any $t>0$, we denote by $\\tilde{C_t}$ its lift to $T^{1}N$ obtained in the previous fashion. A natural question thus arises as to the asymptotic distribution of the $\\tilde{C_t}$ when projected to the unit tangent bundle of $S$; in this respect, the natural choice of a measure on $\\tilde{C_t}$ is given by the push-forward of the renormalized volume measure on $C_t$ under the canonical identification of the latter submanifold with its lift $\\tilde{C_t}$. If the projections to $S$ of the hypersurfaces $C_t$ equidistribute towards the normalized volume measure on $S$, it may be expected that the projections of the lifts $\\tilde{C_t}$ equidistribute towards the corresponding Liouville measure on the unit tangent bundle $T^{1}S$ (cf.~\\cite[Part 1, Sec.~5.4]{Hasselblatt-Katok}). \n\t\n\tThe question lends itself to a description in the language of smooth dynamical systems. If $(g_t^{(N)})_{t\\in \\R}$ is the geodesic flow on $T^{1}N$ (cf.~\\cite[Part 1, Chap.~5]{Hasselblatt-Katok}), it takes a few moments to realize that, for any $t>1$, the set $\\tilde{C_t}$ is the image of the original lift $\\tilde{C}$ under the transformation $g_{t-1}^{(N)}$; the same relation holds for the natural measures carried by the latter sets, and carries over to their projections to $T^1S$, using the geodesic flow $(g_t^{(S)})_{t\\in \\R}$ defined on it in place of $(g_t^{(N)})_{t\\in \\R}$. \n\t\n\tThe equidistribution problem in this formulation is treated in Margulis' thesis~\\cite{Margulis-thesis}, which contains several striking developments and applications of the theory of Anosov systems to the large-scale geometry of negatively curved manifolds; among those, a proof is provided of equidistribution, towards the Liouville measure, of lifts of expanding circles on finite-volume hyperbolic surfaces. \n\n\tFor further comments thereupon, as well as for the connection to the hyperbolic circle problem, the reader is referred to Section~\\ref{sec:circleproblemhyperbolic}.\n\t\n\tIt is the chief purpose of the present work to provide a precise asymptotic expansion for the equidistribution rate of lifts of dilating hyperbolic circles, as well as of arbitrary sub-arcs thereof, on unit tangent bundles of compact hyperbolic surfaces; in the vein of the works of Randol~\\cite{Randol} and Strichartz~\\cite{Strichartz}, we resort to a spectral approach originating in the work of Ratner~\\cite{Ratner} on quantitative mixing of geodesic and horocycle flows on Riemann surfaces of finite volume. Section~\\ref{sec:quantitativeequidistribution} describes such results and expands on their connection to previous developments, whereas Sections~\\ref{sec:introductionCLT} and~\\ref{sec:circleproblemhyperbolic} discuss a number of applications to statistical limit theorems and the hyperbolic lattice point counting problem. \n\t\n\tWe conclude this historical overview by pointing out that    \n\tMargulis' groundbreaking contributions in~\\cite{Margulis-thesis}, together with the gradual emergence of conspicuous applications to counting and Diophantine problems, spawned intensive research aimed at understanding the asymptotic distribution properties of translates of finite-volume subgroup orbits, as well as of more general subsets, on homogeneous spaces\\footnote{It is worth noticing at this point that lifts of expanding hyperbolic circles represent a particular instance, as they are geodesic translates of orbits of the maximal compact subgroup $\\SO_2(\\R)$ on quotients of $\\SL_2(\\R)$: see Section~\\ref{sec:quantitativeequidistribution}.}; without purporting to provide an exhaustive list, we mention in this direction the works~\\cite{Benoist-Oh,Einsiedler-Margulis-Venkatesh,Eskin-McMullen,Eskin-Mozes-Shah,Kra-Shah-Sun,Shah,Shah-second,Shah-third,Shah-fourth,PYang}. \n\t\n\t\\subsection{The setup: circles in hyperbolic surfaces and in their unit tangent bundles}\n\t\n\tWe now set the stage for the main questions we address in the present manuscript, referring to Section~\\ref{sec:preliminaries} for the required background.\n\tLet $\\Gamma<\\SL_2(\\R)$ be a cocompact lattice, that is, a discrete subgroup of $\\SL_2(\\R)$ such that the quotient space $\\Gamma\\bsl \\SL_2(\\R)$ is compact; we indicate the latter homogeneous space with $M$. The group $\\Gamma$ acts properly discontinuously and isometrically on the Poincar\\'{e} upper half-plane $\\Hyp=\\{z=x+iy \\in \\C:y>0 \\}$, endowed with the standard hyperbolic Riemannian metric, by M\\\"{o}bius transformations; when the projection of $\\Gamma$ to $\\PSL_2(\\R)=\\SL_2(\\R)\/\\{\\pm I_2\\}$ is torsion-free, the quotient $S=\\Gamma\\bsl \\Hyp$ is a compact connected orientable smooth surface, inheriting a hyperbolic metric from $\\Hyp$. With respect to such a metric, there is a canonical identification of $M$ with (possibly, a double cover of) the unit tangent bundle\\footnote{More precisely, if $\\Gamma$ is the preimage under the canonical projection map $\\SL_2(\\R)\\to \\PSL_2(\\R)$ of a cocompact lattice in $\\PSL_2(\\R)$, then $M$ identifies with $T^{1}S$; else, it is a double cover thereof. \n\t\t\n\t\tIn the case where the image of $\\Gamma$ in $\\PSL_2(\\R)$ has non-trivial torsion elements, then $S$ has the structure of an orbifold (cf.~\\cite[Chap.~13]{Ratcliffe}). For the purposes of the paper, we shall never be concerned with this distinction.} $T^{1}S$.\n\t\n\tLet $(r_{s})_{s\\in \\R}$ be the one-parameter flow on $M$ defined by \n\t\\begin{equation}\n\t\t\\label{eq:rotationflow}\n\t\tr_s(\\Gamma g)=\\Gamma g \\begin{pmatrix}\n\t\t\t\\cos{s\/2}&\\sin{s\/2}\\\\\n\t\t\t-\\sin{s\/2}& \\cos{s\/2}\n\t\t\\end{pmatrix}\n\t\t=\\Gamma g \\exp{s\\Theta}, \\quad \\Theta=\\begin{pmatrix}\n\t\t\t0&1\/2\\\\\n\t\t\t-1\/2&0\n\t\t\\end{pmatrix}\n\t\t\\in \\sl_2(\\R)=\\operatorname{Lie}(\\SL_2(\\R)),\n\t\\end{equation}\n\tand denote by $(\\phi_t^{X})_{t\\in \\R}$ the geodesic flow on $M$, which is given algebraically by \n\t\\begin{equation}\n\t\t\\label{eq:geodesicflow}\n\t\t\\phi^{X}_{s}(\\Gamma g)=\\Gamma g \\begin{pmatrix} e^{t\/2}&0\\\\ 0&e^{-t\/2} \\end{pmatrix}=\\Gamma g \\exp{tX},\\quad X=\\begin{pmatrix}\n\t\t\t1\/2 &0\\\\\n\t\t\t0&-1\/2 \\end{pmatrix}\n\t\t\\in \\sl_2(\\R).\n\t\\end{equation}\n\t\n\tFor any point $p=\\Gamma g\\in M$, the orbit of $p$ under the flow $(r_s)_{s\\in \\R}$ is the preimage of $z=\\pi(p)$ under the fibration $\\pi\\colon M\\to S$. Therefore, if $M$ identifies with $T^{1}S$, then this set consists of all unit tangent vectors to $z\\in S$. For any real number  $t>0$, the time-$t$ geodesic evolution  $\\phi_t^{X}(\\{r_s(p):s\\in \\R \\})$ of the previous set coincides with the projection to $M$ of the subset of $T^{1}\\Hyp$ given by all outward-pointing normal vectors to the hyperbolic circle in $\\Hyp$ of radius $t$ centered at (a fixed representative in $\\Hyp$ of) $z$.\n\t\n\tWe indicate with $\\vol$ the Haar probability measure on $M$, that is, the unique $\\SL_2(\\R)$-invariant Borel probability measure on $M$; under the identification of $M$ with $T^1S$, it coincides with the Liouville measure projecting to the normalized hyperbolic area measure on $S$. For any $r\\in \\N\\cup \\{\\infty \\}$, we denote by $\\mathscr{C}^{r}(M)$ the set of complex-valued functions of class $\\cC^{r}$ defined on the smooth manifold $M$. The supremum norm of a continuous function $f\\colon M\\to \\C$ is denoted by $\\norm{f}_{\\infty}$. For any $m\\in \\N_{\\geq 1}$, $f\\in \\cC^{m}(M)$ and $j\\in \\{0,\\dots,m\\}$ let $\\nabla^{j}f$ be the $j^{\\text{th}}$ covariant derivative of $f$ and $|(\\nabla^{j}f)(p)|$ its norm at a point $p \\in M$. Define then the $\\cC^{m}$-norm of $f$ (cf.~\\cite[Chap.~1]{Aubin}) as\n\t\\begin{equation}\n\t\t\\label{eq:Cknorm}\n\t\t\\norm{f}_{\\cC^{m}}=\\sum_{j=0}^{m}\\sup\\limits_{p \\in M}|(\\nabla^{j}f)(p)|\\;.\n\t\\end{equation}\n\t\n\tLet $L^{2}(M)$ be the Hilbert space of complex-valued functions on $M$ whose modulus is square-integrable with respect to the measure $\\vol$, and denote by\n\t\\begin{equation*}\n\t\t\\langle \\phi,\\psi\\rangle=\\int_M \\phi\\;\\bar{\\psi}\\;\\text{d}\\vol\n\t\\end{equation*}\n\tthe inner product of two elements $\\phi,\\psi\\in L^{2}(M)$.\n\t\n\tDefine two additional elements\n\t\\begin{equation*}\n\t\tU=\n\t\t\\begin{pmatrix}\n\t\t\t0&1\\\\\n\t\t\t0&0\n\t\t\\end{pmatrix}\n\t\t\\;,\\quad \n\t\tV=\n\t\t\\begin{pmatrix}\n\t\t\t0&0\\\\\n\t\t\t1&0\n\t\t\\end{pmatrix}\n\t\\end{equation*}\n\tin the Lie algebra $\\sl_2(\\R)$.\n\tIdentifying elements of the universal enveloping algebra of $\\sl_2(\\R)$ with left-invariant differential operators on the space $\\cC^{\\infty}(M)$, we define the Casimir operator as the second-order differential operator $\\square=-X^{2}+X-UV\\colon \\cC^{2}(M)\\to \\cC^0(M)$. It admits a unique maximal extension to an unbounded self-adjoint operator on $L^{2}(M)$; in particular, its spectrum $\\text{Spec}(\\square)$ consists of real numbers. As $M$ is compact, it is well-known that $\\text{Spec}(\\square)$ is pure point, and is a discrete subset of $\\R$. The elements of $\\Spec(\\square)$ classify the irreducible representations strongly contained in the Koopman representation arising from the measure-preserving action of $\\SL_2(\\R)$ on the measure space $(M,\\vol)$, as belonging to the principal, complementary or discrete series representations if the corresponding eigenvalue $\\mu$ satisfies, respectively, $\\mu\\geq 1\/4$, $0<\\mu<1\/4$, $\\mu\\leq 0$. With the normalization we have chosen\\footnote{Since $\\sl_2(\\R)$ is a simple Lie algebra, Casimir elements in its universal enveloping algebra are uniquely determined up to real scalars.}, the action of $\\square$ on $\\cC^{2}$-functions defined on the surface $S$ is given by the Laplace-Beltrami operator $\\Delta_S$ associated to the hyperbolic structure on $S$.\n\t\n\tWe are interested in quantitative equidistribution properties of the uniform probability measures supported on the circle arcs\n\t\\begin{equation*}\n\t \\phi_t^{X}(\\{r_s(p):0\\leq s\\leq \\theta \\})\n\t \\end{equation*}\n \tas $t$ goes to infinity, for every fixed $p \\in M$ and $\\theta\\in (0,4\\pi]$ (cf.~our normalization of $\\Theta$ in~\\eqref{eq:rotationflow}, a full circle corresponds to $\\theta=4\\pi$). For any parameter $\\theta\\in (0,4\\pi]$ and any continuous function $f\\colon M\\to \\C$, we thus define the function $k_{f,\\theta}\\colon M\\times \\R\\to \\C$ as\n\t\\begin{equation}\n\t\t\\label{eq:ktheta}\n\t\tk_{f,\\theta}(p,t)\\coloneqq \\frac{1}{\\theta} \\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\; ,\\quad p \\in M,\\;t\\in \\R.\n\t\\end{equation}\n\tIn the forthcoming subsection, we provide a precise asymptotic expansion of $k_{f,\\theta}(p,t)$ as $t$ tends to infinity, first for joint eigenfunctions\\footnote{As we shall explain in Section~\\ref{sec:unitaryrepresentations}, there exists an orthonormal basis of $L^{2}(M)$ consisting of such joint eigenfunctions; Theorem~\\ref{thm:case_Theta_eigenfn} is thus to be regarded as a building block for the more general Theorem~\\ref{thm:mainexpandingtranslates}.} of the operators $\\square$ and $\\Theta$ (Theorem~\\ref{thm:case_Theta_eigenfn}), and then for arbitrary functions fulfilling a suitable regularity condition (Theorem~\\ref{thm:mainexpandingtranslates}).\n\t\n\t\n\t\n\t\\subsection{Quantitative equidistribution of expanding translates of circle arcs}\n\t\\label{sec:quantitativeequidistribution}\n\t\n\tWe begin with the case of joint eigenfunctions of the operators $\\square$ and $\\Theta$.\n\tObserve that the left-invariant vector field $\\Theta\\in \\sl_2(\\R)$ acts as an unbounded skew-symmetric operator on $L^{2}(M)$; if $\\Theta f=\\lambda f$ for some $f\\in \\mathscr{C}^{1}(M)$ and $\\lambda \\in \\C$, then $\\lambda=\\frac{i}{2}n$ for some $n\\in \\Z$. \n\t\n\tIn the following statement and throughout, we associate to each $\\mu\\in \\text{Spec}(\\square)$\n\t the unique complex number $\\nu\\in \\R_{\\geq 0}\\cup i\\R_{>0}$ satisfying $1-\\nu^2=4\\mu$.\n\t\n\t\n\t\\begin{thm}\\label{thm:case_Theta_eigenfn}\n\t\tThere exist real constants $\\kappa_0$ and $\\kappa(\\mu)$ for any positive Casimir eigenvalue $\\mu$ such that the following assertions hold. Let $\\theta\\in (0,4\\pi]$, $\\mu\\in \\emph{Spec}(\\square)$ and $n\\in \\Z$, and suppose $f\\in \\mathscr{C}^{2}(M)$ satisfies $\\square f=\\mu f$, $\\Theta f=\\frac{i}{2}nf$. Define $k_{f,\\theta}(p,t)$ as in~\\eqref{eq:ktheta}.  \n\t\t\\begin{enumerate}\n\t\t\t\\item If $\\mu>1\/4$, there exist H\\\"{o}lder-continuous functions $D^{+}_{\\theta,\\mu,n}f,D^{-}_{\\theta,\\mu,n}f\\colon M\\to \\C$ with H\\\"{o}lder exponent $1\/2$ and\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^2+1)\\norm{f}_{\\mathscr{C}^{1}} \n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$, \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estabovequarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}D^{+}_{\\theta,\\mu,n}f(p)+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}D^{-}_{\\theta,\\mu,n}f(p)+\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estremainderabovequarter}\n\t\t\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq \\frac{8\\kappa_0(n^{2}+1)}{\\theta\\; \\Im{\\nu}}\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu=1\/4$, there exist H\\\"{o}lder-continuous functions $D^{+}_{\\theta,1\/4,n}f\\colon M\\to \\C$, with H\\\"{o}lder exponent $1\/2-\\varepsilon$ for every $\\varepsilon>0$, and $D^{-}_{\\theta,1\/4,n}f\\colon M\\to \\C$, with H\\\"{o}lder exponent $1\/2$, and satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,1\/4,n}f}_{\\infty}\\leq \\frac{\\kappa(1\/4)}{\\theta}(n^2+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estquarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}D^{+}_{\\theta,1\/4,n}f(p)+te^{-\\frac{t}{2}}D^{-}_{\\theta,1\/4,n}f(p)+\\mathcal{R}_{\\theta,1\/4,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation*}\n\t\t\t\t|\\mathcal{R}_{\\theta,1\/4,n}f(p,t)|\\leq \\frac{4\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}(t+1)e^{-t}\\;.\n\t\t\t\\end{equation*}\n\t\t\t\\item If $0<\\mu<1\/4$, there exist functions $D^{+}_{\\theta,\\mu,n}f, D^{-}_{\\theta,\\mu,n}f \\colon M\\to \\C$, respectively H\\\"{older}-continuous with H\\\"{o}lder exponent $\\frac{1-\\nu}{2}$ and  of class $\\cC^{1}$, and satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estpositivebelowquarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{1+\\nu}{2}t}D^{+}_{\\theta,\\mu,n}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{\\theta,\\mu,n}f(p)+\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estremainderbelowquarter}\n\t\t\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq \\frac{4\\kappa_0}{\\theta\\nu(1-\\nu)(1+\\nu)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu=0$, there exists a function $G_{\\theta,n}f\\colon M\\times \\R_{>0}\\to \\C$, with $G_{\\theta,n}f(\\cdot,t)$ of class $\\cC^{1}$ for any $t>0$, $G_{\\theta,n}f(p,\\cdot)$ continuous for every $p \\in M$ and\n\t\t\t\\begin{equation*}\n\t\t\t\t\\sup\\limits_{p\\in M,\\;t>0}|G_{\\theta,n}f(p,t)|\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\cC^{1}}\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,  \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estzero}\n\t\t\t\tk_{f,\\theta}(p,t)=\\int_{M}f\\;\\emph{d}\\vol+e^{-t}\\int_1^{t}-G_{\\theta,n}f(p,\\xi)\\;\\emph{d}\\xi+\\mathcal{R}_{\\theta,0,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:remainderzero}\n\t\t\t\t|\\mathcal{R}_{\\theta,0,n}f(p,t)|\\leq\\frac{8e\\pi+\\kappa_0}{\\theta} (n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu<0$ then, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estimatediscreteseries}\n\t\t\t\t|k_{f,\\theta}(p,t)|\\leq \\frac{1}{\\theta}\\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2e\\pi(3+\\nu)}{\\nu}\\biggr)(n^{2}+1) \\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\t\n\t\n\t\\begin{rmk}\n\t\t\tThe asymptotic expansions for $k_{f,\\theta}(p,t)$ are easily transferred to the case of large negative values of the time parameter $t$, by noticing that\n\t\t\t\\begin{equation*}\n\t\t\t\tk_{f,\\theta}(p,-t)=k_{f \\circ r_{\\pi},\\theta}(r_{\\pi}(p),t) \n\t\t\t\\end{equation*}\n\t\tfor any $p \\in M$ and $t\\geq 1$.\n\t\\end{rmk}\n\t\n\tTaking advantage of Theorem~\\ref{thm:case_Theta_eigenfn} and of standard harmonic analysis on the group $\\SL_2(\\R)$, we establish an asymptotic expansion of $k_{f,\\theta}(p,t)$ for all sufficiently regular, but otherwise arbitrary test functions $f$.\n\tDefine the Laplacian on $M$ to be the second-order linear differential operator $\\Delta=\\square-2\\Theta^{2}$. \n\tFor any $s\\in \\R_{>0}$, let $W^{s}(M)$ be the Sobolev space of order $s$ on the manifold $M$, that is, the Hilbert-space completion of the complex vector space $\\cC^{\\infty}(M)$ of smooth functions on $M$ endowed with the inner product \n\t\\begin{equation*}\n\t\t\\langle \\phi,\\psi \\rangle_{W^s}=\\langle (1+\\Delta)^{s}\\phi,\\psi\\rangle\\;,\\quad \\phi,\\psi\\in \\cC^{\\infty}(M).\n\t\\end{equation*}\n\t\n\tAs $M$ is compact, the well-known Sobolev Embedding Theorem (which we recall in Theorem~\\ref{thm:Sobolevembedding}) ensures the existence of a continuous embedding $W^{s}(M)\\hookrightarrow \\cC^{r}(M)$ whenever $s\\in \\R_{>0}$ and $r\\in \\N$ are such that $s>r+3\/2$; explicitly, there is a constant $C_{r,s}\\in \\R_{>0}$ (which for definiteness we take equal to the operator norm of the corresponding embedding) such that $\\norm{f}_{\\cC^{r}}\\leq C_{r,s}\\norm{f}_{W^{s}}$ for any $f\\in W^{s}(M)$. Hereinafter, an element $f\\in W^{s}(M)$ for $s>3\/2$ is always identified with its unique continuous representative.\n\t\n\tSet\n\t\\begin{equation}\n\t\t\\label{eq:epszero}\n\t\t\\varepsilon_{0}=\n\t\t\\begin{cases}\n\t\t\t1 &\\text{ if }1\/4\\in \\text{Spec}(\\square)\\;,\\\\\n\t\t\t0 &\\text{ if }1\/4\\notin \\text{Spec}(\\square)\\;.\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\begin{thm}\n\t\t\\label{thm:mainexpandingtranslates}\n\t\tThere exist real constants $C_{\\emph{Spec}}$ and $\\;C_{\\emph{Spec}}'$, depending only on the spectrum of the Casimir operator on $L^{2}(M)$, such that the following holds. Let $\\theta\\in (0,4\\pi]$ and $s>11\/2$ be real numbers, and suppose $f\\in W^{s}(M)$.\n\t\tThen there exist continuous functions $D^{+}_{\\theta,\\mu}f,\\;D^{-}_{\\theta,\\mu}f\\colon M\\to \\C$ for any positive Casimir eigenvalue $\\mu$, with\n\t\t\\begin{equation}\n\t\t\t\\label{eq:boundDthetamu}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\norm{D^{+}_{\\theta,\\mu}f}_{\\infty}+\\norm{D^{-}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}},\n\t\t\\end{equation}\n\t\tsuch that, for every $p\\in M$ and $t\\geq 1$,\n\t\t\\begin{equation}\n\t\t\t\\label{eq:asymptoticgeneral}\n\t\t\t\\begin{split}\n\t\t\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_{t}\\circ r_s(p)\\;\\emph{d}s=&\\int_{M}f\\;\\emph{d}\\vol\\\\\n\t\t\t\t&+e^{-\\frac{t}{2}}\\biggl( \\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{+}_{\\theta,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{-}_{\\theta,\\mu}f(p)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t}D^{+}_{\\theta,\\mu}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{\\theta,\\mu}f(p)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t}{2}}D^{+}_{\\theta,1\/4}f(p)+te^{-\\frac{t}{2}}D^{-}_{\\theta,1\/4}f(p)\\bigr)+\\mathcal{R}_{\\theta}f(p,t)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere\n\t\t\\begin{equation}\n\t\t\t\\label{eq:globalremainderestimate}\n\t\t\t|\\mathcal{R}_{\\theta}f(p,t)|\\leq \\frac{C_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t} \\;.\n\t\t\\end{equation}\n\t\\end{thm}   \n\t\n\t\n\tWe record here below the ensuing effective equidistribution statement, in which we single out the two main terms of the asymptotic expansion, thereby highlighting the dependence of the latter on the spectral gap of the underlying hyperbolic surface $S=\\Gamma\\bsl \\Hyp$, defined as \n\t\\begin{equation*}\n\t\t\\mu_*=\\inf(\\text{Spec}(\\square)\\cap \\R_{>0})=\\inf(\\text{Spec}(\\Delta_S)\\setminus\\{0\\})\\;;\n\t\\end{equation*}\n\tits corresponding parameter is denoted by $\\nu_*$. \n\t\n\t\\begin{term}\n\t\tWe adopt the classical Landau notation $o(\\eta(t))$ for $\\eta\\colon \\R_{> 0}\\to \\R_{>0}$ tending to zero at infinity, to indicate a function $\\lambda\\colon \\R_{>0}\\to \\C$ such that $|\\lambda(t)|\/\\eta(t)\\to 0$ as $t\\to\\infty$. \n\t\\end{term}\n\t\n\t\n\t\\begin{cor}\n\t\t\\label{cor:effective}\n\t\tLet $\\theta,s,C_{1,s-3},C'_{\\emph{Spec}}$ and $f$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. Then there exists a function $D^{\\emph{main}}_{\\theta}f\\colon M\\to \\C$ with\n\t\t\\begin{equation}\n\t\t\t\\label{eq:boundDthetamain}\n\t\t\t\\norm{D^{\\emph{main}}_{\\theta}f}_{\\infty}\\leq \\frac{C'_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}\n\t\t\\end{equation}\n\t\tsuch that, for every $p \\in M$ and $t\\geq 1$, \n\t\t\\begin{equation}\n\t\t\t\\label{eq:effective}\n\t\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\emph{d}s=\\int_{M}f\\;\\emph{d}\\vol+\\;D^{\\emph{main}}_{\\theta}f(p)\\;t^{\\varepsilon_0}e^{-\\frac{1-\\Re{\\nu_*}}{2}t}+o(e^{-\\frac{1-\\Re{\\nu_*}}{2}t})\\;.\n\t\t\\end{equation}\n\t\\end{cor}\n\t\n\t\\begin{rmk}\n\t\tUsing the result of Dyatlov, Faure and Guillarmou in~\\cite{Dyatlov-Faure-Guillarmou} we can relate the coefficients $D^{-}_{\\theta,\\mu}f$, for $0< \\mu <1\/4$, to the resonant and co-resonant states in the first band for the geodesic flow (in other words, with the invariant distributions for the unstable and stable horocycle flow, respectively). \n\t\tLet $u_\\mu$ be a resonant state for the geodesic flow associated to the Pollicott-Ruelle resonance $-\\frac{1-\\nu}{2} \\in (-1\/2,0)$, and let $u^\\ast_\\mu$ be its dual (we refer to \\cite[Sec.~$1$ and $2$]{Dyatlov-Faure-Guillarmou} for the relevant definitions). \n\t\tIn the language of Flaminio-Forni's work~\\cite{Flaminio-Forni}, $u_\\mu$ is an invariant distribution for the unstable horocycle flow, and $u^\\ast_\\mu$ is an invariant distribution for the stable horocycle flow.\n\t\tBy virtue of \\cite[Thm.~4]{Dyatlov-Faure-Guillarmou}, we have, for any sufficiently small $\\varepsilon>0$ and for arbitrary $f,g \\in \\mathscr{C}^\\infty(M)$,\n\t\t\\begin{equation*}\n\t\t\t\\langle f \\circ \\phi^X_t, g\\rangle = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\langle f, u_\\mu \\rangle \\, \\langle u_\\mu^\\ast , g \\rangle \\, e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\\end{equation*}\n\t\tThus,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t&\\langle k_{f, \\theta}(\\cdot, t), g\\rangle = \\frac{1}{\\theta} \\int_0^\\theta \\langle f \\circ \\phi^X_t, g \\circ r_{-s}\\rangle \\;\\text{d}s \\\\\n\t\t\t\t& \\quad = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\left(\\frac{1}{\\theta} \\int_0^\\theta \\langle f, u_\\mu \\rangle \\, \\langle u_\\mu^\\ast , g \\circ r_{-s}\\rangle \\;\\text{d}s \\right) \\, e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tOn the other hand, by Theorem \\ref{thm:mainexpandingtranslates},\n\t\t\\begin{equation*}\n\t\t\t\\langle k_{f, \\theta}(\\cdot, t), g\\rangle = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\langle D^{-}_{\\theta,\\mu}f, g \\rangle \\,  e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\\end{equation*}\n\t\tEquating coefficients, we conclude that the functions $D^{-}_{\\theta,\\mu}f,\\; 0< \\mu <1\/4$, coincide as distributions with the corresponding averaged resonant states, namely\n\t\t\\begin{equation}\n\t\t\t\\label{eq:D-_resonant_states}\n\t\t\tD^{-}_{\\theta,\\mu}f = \\langle f, u_\\mu \\rangle \\cdot \\frac{1}{\\theta} \\int_0^\\theta u_\\mu^\\ast \\circ r_{s} \\;\\text{d}s\\;.\n\t\t\\end{equation}\n\t\tOf particular interest is the full average $\\theta = 4 \\pi$: in this case, $D^{-}_{4\\pi,\\mu}f$ is a multiple of $\\frac{1}{4\\pi} \\int_0^{4\\pi} u_\\mu^\\ast \\circ r_{s} \\;\\text{d}s$, which is an eigenfunction of the Laplacian of eigenvalue $\\mu$, see \\cite[p.~931]{Dyatlov-Faure-Guillarmou}.\n\t\t\n\t\tFrom \\eqref{eq:D-_resonant_states}, we also deduce that $D^{-}_{\\theta,\\mu}f $ is identically zero if and only if $\\langle f, u_\\mu \\rangle =0$, or, in other words, if and only if $f$ lies in the kernel of the invariant distribution for the unstable horocycle flow with Casimir parameter $\\mu$ (see \\cite[Sec.~3]{Flaminio-Forni}).\n\t\t\n\t\tIn order to prove an analogous relation between the other coefficients $D^{\\pm}_{\\theta,\\mu}f $ and the horocycle invariant distributions, we would need an asymptotic expansion of the correlations for the geodesic flow as in \\cite[Thm.~4.3]{Forni}, but with an explicit dependence of the coefficients in terms of the stable and unstable horocycle-invariant distributions.   \n\t\\end{rmk}\n\t\n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:equivalentmetrics}\n\t\tWe collect here below further comments about Theorem~\\ref{thm:case_Theta_eigenfn}, Theorem~\\ref{thm:mainexpandingtranslates} and Corollary~\\ref{cor:effective}.\n\t\t\\begin{enumerate}\n\t\t\\item The H\\\"{o}lder-continuity claims concerning the coefficients $D^{\\pm}_{\\theta,\\mu,n}$ appearing in Theorem~\\ref{thm:case_Theta_eigenfn} tacitly involve the choice of a distance function $d$ on $M$. It is intended that $d$ is the Riemannian distance function induced on the connected manifold $M$ by a Riemannian metric $g$. The H\\\"{o}lder-continuity property, as well as the H\\\"{o}lder exponent, of $D^{\\pm}_{\\theta,\\mu,n}$ is independent of the choice of such a $g$, as any two Riemannian metrics on a compact manifold induce Lipschitz-equivalent metrics (see \\cite[Lem.~13.28]{Lee}).\n\t\t\\item We point out the analogy of Theorem~\\ref{thm:case_Theta_eigenfn} with the asymptotics of horocycle ergodic integrals for Casimir eigenfunctions established in~\\cite[Thm.~1]{Rav} (in the same way, Theorem~\\ref{thm:mainexpandingtranslates} mirrors~\\cite[Thm.~2]{Rav}). This similarity stems computationally from the application of the exact same spectral method in both circumstances; as a matter of fact, it comes as no surprise from a geometric standpoint, for large hyperbolic circles approximate orbits of the unstable horocycle flow. \n\t\t\\item For $\\theta=4\\pi$, we recover the qualitative equidistribution of expanding circles towards the uniform measure $\\vol$ obtained by Margulis in~\\cite{Margulis-thesis}. \n\t\t\\item The equidistribution rate in Corollary~\\ref{cor:effective} matches exactly the mixing rate of the geodesic flow on $M$ obtained by Ratner in~\\cite[Thm.~2]{Ratner}.\n\t\t\\item As suggested by the underlying geometric picture, the various upper bounds in the statements of Theorem~\\ref{thm:case_Theta_eigenfn} and~\\ref{thm:mainexpandingtranslates} indicate that the speed of equidistribution improves as $\\theta$ increases to $4\\pi$, that is, as the length of the initial circle arc gets larger. \n\t\t\\end{enumerate}\n\t\\end{rmk}\n\t\n\tAs a special case of Theorem \\ref{thm:mainexpandingtranslates}, we obtain an asymptotic expansion for the equidistribution rate of $\\Theta$-invariant functions; in other words, we provide the precise asymptotic behaviour of large circles on the compact hyperbolic surface $S$, identified, here and afterwards whenever convenient, with the double coset space $\\Gamma\\bsl \\SL_2(\\R)\/\\SO_2(\\R)$.\n\t\n\tLet $d_{\\Hyp}$ denote the distance function on $\\Hyp$ arising from the hyperbolic Riemannian metric. For any $z\\in \\Hyp$ and $t>0$, let $C_{\\Hyp}(z,t)=\\{z'\\in \\Hyp:d_{\\Hyp}(z,z')=t \\}$ be the hyperbolic circle of radius $t$ centered at $z$, and denote by $C_{S}(z,t)$ its projection to $S$. With $m_{C_S(z,t)}$ we indicate the projection to $S$ of the unique isometrically-invariant Borel probability measure supported on $C_{\\Hyp}(z,t)$. Finally, let $m_{S}$ be the hyperbolic area measure on $S$, normalized to be a probability measure.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:expandingonsurface}\n\t\tLet $\\theta\\in (0,4\\pi],\\;s>9\/2$, $\\tilde{f}\\colon S \\to \\C$ a function such that the $\\SO_2(\\R)$-invariant function $f\\colon M\\to \\C$ defined by $f(\\Gamma g)=\\tilde{f}(\\Gamma g \\SO_2(\\R))$ for any $g\\in \\SL_2(\\R)$ is in $W^{s}(M)$. Then, for  every $z=\\Gamma g\\SO_2(\\R)\\in S$,  $p=\\Gamma g \\in M$ and $t\\geq 1$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t\\int_{S}\\tilde{f}\\;\\emph{d}m_{C_S(z,t)}&=\\int_{S}\\tilde{f}\\;\\emph{d}m_S\n\t\t\t\t+e^{-\\frac{t}{2}}\\biggl( \\sum_{\\mu\\in \\emph{Spec}(\\Delta_S),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{+}_{4\\pi,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{-}_{4\\pi,\\mu}f(p)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\Delta_S),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t}D^{+}_{4\\pi,\\mu}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{4\\pi,\\mu}f(p)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t}{2}}D^{+}_{4\\pi,1\/4}f(p)+te^{-\\frac{t}{2}}D^{-}_{4\\pi,1\/4}f(p)\\bigr)+\\mathcal{R}_{4\\pi}f(p,t)\\;.\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tIn particular, the functions $\\mathcal{R}_{4\\pi}f$ and $D^{\\pm}_{4\\pi,\\mu}f,\\;\\mu\\in \\emph{Spec}(\\Delta_S)\\cap \\R_{>0}$, defined as in Theorem~\\ref{thm:mainexpandingtranslates}, are $\\SO_2(\\R)$-invariant. \n\t\\end{thm}  \n\t\n\tClearly, an analogous statement holds for arbitrary sub-arcs of the circles $C_{S}(z,t)$.\n\t\n\t\\begin{rmk}\n\t\tIt is worth highlighting that, in the case of an $\\SO_2(\\R)$-invariant observable $f$, we require mildly less restrictive assumptions on its Sobolev regularity; this will become relevant in Section~\\ref{sec:latticepoint} when we deal with the error rate for the hyperbolic lattice point counting problem. \n\t\t\n\t\tFurthermore, we emphasize that Theorem~\\ref{thm:expandingonsurface} improves upon the equidistribution results in~\\cite{Randol} (which, on the other hand, apply to any dimension) in a twofold way: it demands less restrictive conditions on the regularity of test functions and, more importantly, it refines Randol's upper bound on the equidistribution rate by giving a precise asymptotic expansion.\n\t\\end{rmk}\n\t\n\tLeveraging the explicit dependence of the asymptotics in Theorem~\\ref{thm:mainexpandingtranslates} on the upper bound $\\theta$ of the domain of parametrization of the circle arc under consideration, we deduce a sufficient quantitative condition for the equidistribution of shrinking pieces of expanding circles; this is in the vein of Str\\\"{o}mbergsson's results in~\\cite{Strombergsson-closedhorocycles}, where the analogous question is investigated for shrinking portions of closed horocycles on non-closed hyperbolic surfaces of finite volume.\n\t\n\t\\begin{cor}\n\t\t\\label{cor:shrinkingarcs}\n\t\tLet $p \\in M$, $\\theta_1,\\theta_2\\colon \\R_{>0} \\to (0,4\\pi)$ two functions with $\\theta_1(t)\\leq \\theta_2(t)$ for any $t>0$, and consider the circle sub-arcs $\\gamma_{t}=\\{\\phi^{X}_t\\circ r_{s}(p):\\theta_1(t)\\leq s \\leq \\theta_2(t)\\}$. For any $t>0$, let $\\mu_t$ be the normalized restriction to $\\gamma_t$ of the unique isometrically-invariant measure on the corresponding full circle. Assume that there exist $t_0>0$ and a function $\\eta\\colon \\R_{>0}\\to \\R_{>0}$ with $\\eta(t)\\to\\infty$ as $t\\to\\infty$ such that $\\theta_2(t)-\\theta_1(t)\\geq\\eta(t) e^{-\\frac{1-\\Re{\\nu_*}}{2}t}$ for any $t\\geq t_0$. Then the circle arcs $\\gamma_t$ equidistribute as $t\\to\\infty$: more precisely, the measures $\\mu_t$ converge in the weak$^*$ topology, as $t\\to\\infty$, towards the uniform measure $\\vol$ on $M$. \n\t\\end{cor} \n\t\n\t\\subsection{Statistical limit theorems for deviations from the average}\n\t\\label{sec:introductionCLT}\n\t\n\tThe asymptotics in Theorem~\\ref{thm:mainexpandingtranslates} affords the means to examine the long-term statistical behaviour of the averages of a given observable along expanding circle arcs. Historically, a momentous discovery in the twentieth century was the realization that the long-term evolution of deterministic systems  frequently obeys the same statistical laws governing the asymptotic behaviour of random processes. Specifically, this feature is a typical characteristic of dynamical systems with hyperbolic behaviour, among which geodesic flows on negatively curved compact manifolds feature prominently; we refer the reader to the survey in the introduction to~\\cite{Dolgopyat-Sarig}, as well as to the references therein, for an extensive discussion of the topic.  \n\t\n\tBecause of the exponential mixing properties of the geodesic flow $(\\phi^{X}_t)_{t\\in \\R}$ on $M$ (cf.~\\cite[Thm.~2]{Ratner}),  at first it stands to reason to expect, for a real-valued function $f$ on $M$ with finite first moment with respect to the volume measure $\\vol$, the distribution of the deviations from the average\n\t\\begin{equation*}\n\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\;,\n\t\\end{equation*}\n\twhen the base point $p$ is randomly chosen according to the law $\\vol$, (something which is henceforth indicated with $p\\sim \\vol$), to mimic for large values of $T$ the law of the empirical mean\n\t\\begin{equation*} \\frac{1}{N}\\sum_{n=1}^{N}X_n\n\t\\end{equation*}\n\tof an increasing number $N$ of independent real-valued random variables $X_n$. More precisely, since the hyperbolic length of a circle of radius $T$ is proportional to $e^{T}$ (see the explanation below~\\eqref{eq:integralbounds}), a full analogoue of the classical Central Limit Theorem in this case would affirm that the random variables \n\t\\begin{equation*}\n\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol \\biggr)\\;,\\quad p\\sim \\vol\n\t\\end{equation*} \n\tconverge in law, as $T$ tends to infinity, to a normally distributed random variable. However, the geometric resemblance of large hyperbolic circles to orbits of the unstable horocycle flow, for which similar phenomena occur (cf.~\\cite[Thm.~1.4,~1.5]{BuFo} and~\\cite[Thm.~4]{Rav}) accounts both for the emergence of other types of limiting distributions, and for possibly different renormalization factors, depending on the spectral properties of the observable under consideration.\n\t\n\tIn order to minimize the difference with the classical probabilistic setup of sums of independent random variables, we shall state all the results in this subsection for real-valued observables, though the extension to complex-valued ones is immediate.   \n\t\\begin{thm}\n\t\t\\label{thm:CLT}\n\t\tLet $\\theta\\in (0,4\\pi]$, $s>11\/2$, $f\\in W^{s}(M)$ a real-valued function. Assume that \n\t\t\\begin{equation*}\n\t\t\t\\mu_f=\\inf\\{\\mu\\in \\emph{Spec}(\\square)\\cap \\R_{>0}:D^{-}_{\\theta,\\mu}f \\emph{ does not vanish identically on }M \\}\n\t\t\\end{equation*}\n\t\tis finite, and let $\\nu_f$ be the unique complex number in $\\R_{\\geq 0}\\cup i\\R_{>0}$ satisfying $1-\\nu_f^{2}=4\\mu_f$.\n\t\t\\begin{enumerate}\n\t\t\t\\item If $0<\\mu_f<1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\te^{\\frac{1-\\nu_f}{2}T}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution to $D^{-}_{\\theta,\\mu_f}f$ as $T\\to\\infty$.\n\t\t\t\\item If $\\mu_f=1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\tT^{-1}e^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution to $D^{-}_{\\theta,1\/4}f$ as $T\\to\\infty$.\n\t\t\t\\item If $\\mu_f>1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution, as $T\\to\\infty$,  to the quasi-periodic motion\n\t\t\t\\begin{equation*} \\varepsilon_0D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f+\\sum_{\\mu \\in \\emph{Spec}(\\square),\\;\\mu\\geq \\mu_f}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f\n\t\t\t\\end{equation*}\n\t\t\ton the set of real-valued random variables defined on the probability space $(M,\\vol)$.\n\t\t\\end{enumerate}\n\t\\end{thm}\n\t\n\tObserve the remarkable fact that the limiting distributions appearing in the statement of Theorem~\\ref{thm:CLT} are compactly supported on the real line, owing to the fact that the coefficients $D^{\\pm}_{\\theta,\\mu}f$ are bounded. This stands in stark contrast with the classical versions of the Central Limit Theorem in probability theory, where in non-trivial situations the distribution of errors is governed by the fully supported Gaussian distribution.\n\tFurthermore, it is straightforward to check, at least when the Casimir components of $f$ are eigenfunctions of $\\Theta$ and using the explicit expressions of the coefficients $D^{\\pm}_{\\theta,\\mu}f$ in~\\eqref{eq:Dplusabovequarter},~\\eqref{eq:Dminusabovequarter},~\\eqref{eq:Dquarter} and~\\eqref{eq:Dbelowquarter}, that the limit law is non-trivial\\footnote{As soon as $f$ is not almost-surely constant, obviously.}, that is, not a Dirac mass. In the general case, the coefficients are given by infinite superpositions of the previous ones; though we shall refrain from a detailed verification, there is no reason to expect cancellation phenomena to come about and produce limiting random variables which are constant almost-surely.\n\t\n\tTheorem~\\ref{thm:CLT} is a consequence of its quantitative version which we presently discuss. For $f$ fulfilling the conditions in Theorem~\\ref{thm:CLT}, let $\\mathbf{P}_{\\theta,f}$ denote the law of the random variable $D^{-}_{\\theta,\\mu_f}f$ when $\\mu_f\\leq 1\/4$, and $\\mathbf{P}_{\\theta,f}(T)$ the law of the random variable\n\t\\begin{equation*}\n\t\t\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu\\geq \\mu_f}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f\\;,\\quad T>0,\n\t\\end{equation*}\n\twhen $\\mu_f>1\/4$. Furthermore, for any $T\\geq 1$ we let $\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T)$ be:\n\t\\begin{enumerate} \n\t\t\\item the law of\n\t\t\\begin{equation*}\n\t\t\te^{\\frac{1-\\nu_f}{2}T}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr) \\quad \\text{if }0<\\mu_f<1\/4,\n\t\t\\end{equation*}\n\t\t\\item the law of\n\t\t\\begin{equation*}\n\t\t\tT^{-1}e^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr) \\quad \\text{if }\\mu_f=1\/4,\n\t\t\\end{equation*}\n\t\t\\item and the law of\n\t\t\\begin{equation*}\n\t\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr)\\quad \\text{if }\\mu_f>1\/4.\n\t\t\\end{equation*}\n\t\\end{enumerate}\n\t\n\tDenote by $d_{LP}$ the L\\'{e}vi-Prokhorov distance on the set of Borel probability measures on $\\R$ (cf.~Section~\\ref{sec:spatialDLT}), which induces on the latter the topology of weak convergence. Recall also that $\\mu_*$ denotes the spectral gap of $S=\\Gamma\\bsl \\Hyp$, with associated parameter $\\nu_*$.\n\t\n\n\t\\begin{prop}\n\t\t\\label{prop:CLT}\n\t\tLet the assumptions be as in Theorem~\\ref{thm:CLT}, and the constants $C_{\\emph{Spec}}$ and $C_{\\emph{Spec}}'$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. \n\t\t\\begin{enumerate}\n\t\t\t\\item If $0<\\mu_f<1\/4$, then there is an explicit constant $\\eta_f>0$, depending only on $\\mu_f$ and on $\\emph{Spec}(\\square)$, such that \n\t\t\t\\begin{equation*}\n\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C'_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}Te^{-\\eta_f T}\n\t\t\t\\end{equation*}\n\t\t\tfor every $T\\geq 1$.\n\t\t\t\\item If $\\mu_f=1\/4$, then there exists a constant $C_{\\emph{pos}}$, depending only on $\\emph{Spec}(\\square)\\cap \\R_{>0}$, such that \n\t\t\t\\begin{equation*}\n\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{\\emph{pos}}C_{1,s-3} }{\\theta}\\norm{f}_{W^{s}}T^{-1} \n\t\t\t\\end{equation*}\n\t\t\tfor every $T\\geq 1$.\n\t\t\t\\item If $\\mu_f>1\/4$, then:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item when $\\mu_*<1\/4$,\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor every $T\\geq 1$;\n\t\t\t\t\\item when $\\mu_*\\geq 1\/4$,\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{ C_{\\emph{Spec}}C_{1,s-3} }{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor every $T\\geq 1$.\n\t\t\t\\end{enumerate}\n\t\t\\end{enumerate}\t\n\t\\end{prop}\n\t\n\tEntirely analogous deductions, of which we omit the details (cf.~Section~\\ref{sec:spatialDLT}), can be made in the case where the $D^{-}_{\\theta,\\mu}f$ vanish everywhere for any Casimir eigenvalue $\\mu>0$ but $D^{+}_{\\theta,\\mu}f$ is not identically zero for at least one such $\\mu$.\n\t\n\t\\begin{rmk}\n\t\tAs the proof of Proposition~\\ref{prop:CLT} shall clearly illustrate (see, in particular, Lemma~\\ref{lem:LPnearbyvariables}), the assumption that the base point $p$ is sampled according to the uniform measure $\\vol$ is immaterial, as far as the validity of Proposition~\\ref{prop:CLT} and Theorem~\\ref{thm:CLT} is concerned. It is possible to replace the measure $\\vol$ with any other Borel probability measure $\\mu$ on $M$, without affecting the quantitative rate of convergence, provided that the laws of the limiting random variables are modified accordingly. \n\t\\end{rmk}\n\t\n\tIn the remaining case when the $D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any positive Casimir eigenvalue $\\mu$, the explicit expressions of the coefficients appearing in Theorem~\\ref{thm:case_Theta_eigenfn} and~\\ref{thm:mainexpandingtranslates} enable us to rule out the existence of a non-trivial distributional limit in the case of full circles, that is, when $\\theta=4\\pi$. More precisely, we establish the following result.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:noCLT}\n\t\tLet $s>11\/2$, $f\\in W^{s}(M)$ a real-valued function. Assume that, for any positive Casimir eigenvalue $\\mu$,  the functions $D^{\\pm}_{4\\pi,\\mu}f$ defined as in Theorem~\\ref{thm:mainexpandingtranslates} vanish identically on $M$. Then, for any collection $(B_T)_{T>0}$ of positive real numbers such that $B_T\\to\\infty$ as $T\\to\\infty$, the distributional limit of the random variables \n\t\t\\begin{equation*}\n\t\t\t\\frac{e^{T}\\bigl( \\frac{1}{4\\pi}\\int_0^{4\\pi}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\bigr)}{B_T}\\;,\\quad p\\sim \\vol\n\t\t\\end{equation*} \n\t\tas $T\\to\\infty$ exists and is almost surely equal to zero.\n\t\\end{thm}\n\t\n\tTheorem~\\ref{thm:noCLT} results from approximating averages of $f$ along expanding circle arcs with the difference of the ergodic integrals of $f$ along two geodesic orbits, which in the case of complete circles happen to coincide; details are carried out in Section~\\ref{sec:noCLT}.\n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:geodesiccoboundary}\n\t\tWe hasten to add that the argument we conduct enables to show that Theorem~\\ref{thm:noCLT} holds true for arbitrary circle arcs (that is, it is possible to replace $4\\pi$ with an arbitrary $\\theta\\in (0,4\\pi]$) provided that, in addition to the vanishing hypothesis on the $D^{\\pm}_{4\\pi,\\mu}$ for $\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}$, the derivative $Uf_0$ along the stable horocycle flow of the projection $f_0$ of $f$ onto the Casimir eigenspace of eigenvalue $0$ is assumed to be a coboundary for the geodesic flow (cf.~Section~\\ref{sec:noCLT}). \n\t\\end{rmk}\n\t\n\t\n\t\n\t\\subsection{Asymptotics for arbitrary translates of compact orbits and the circle problem in the hyperbolic plane}\n\t\\label{sec:circleproblemhyperbolic}\n\t\n\tThe celebrated Gauss circle problem asks for the precise asymptotic behaviour of the discrepancy between the number of integer points in a disk of radius $R$ in the Euclidean plane and the area of the disk, as $R$ tends to infinity. More precisely, define the integer-point counting function \n\t\\begin{equation*}\n\t\t\\cN(R)=|\\{(m,n)\\in \\Z^{2}:m^{2}+n^2\\leq R^{2}  \\}|,\\quad R\\in \\R_{>0},\n\t\\end{equation*}\n\twhere $|A|$ denotes, here and henceforth, the cardinality of a finite set $A$.\n\tTessellating the Euclidean plane with $\\Z^{2}$-translates of $[0,1)^{2}$, which is a fundamental domain for the $\\Z^{2}$-action by translations on $\\R^{2}$, leads to the main term $\\pi R^{2}$, equal to the area of the disk of radius $R$, for the asymptotics of $\\cN(R)$, as well as to the upper bound (due to Gauss himself)\n\t\\begin{equation*}\n\t\t |\\cN(R)-\\pi R^{2}|\\leq 2\\pi(\\sqrt{2}R+1)\n\t\\end{equation*} \n\t\t for the discrepancy. Despite considerable successive improvements on Gauss' original bound, for the history of which we refer to the comprehensive survey~\\cite{Ivic}, it is a notoriously unsolved problem to attain the conjectural sharpest upper bound, deemed to be of the order of $R^{1\/2+\\varepsilon}$ for any $\\varepsilon>0$.\n\t\n\tWe consider the analogous question in the hyperbolic plane. For $\\Gamma<\\SL_2(\\R)$ a cocompact lattice, we examine the asymptotics of the function\n\t\\begin{equation}\n\t\t\\label{eq:defcountingfunction}\n\t\tN(R)=|\\{z\\in \\Gamma\\cdot i:d_{\\Hyp}(z,i)\\leq R\\}|\n\t\\end{equation}\n\tas $R$ tends to infinity, where $\\Gamma\\cdot i$ denotes the (discrete) orbit\\footnote{We thus count the number of actual lattice-orbit elements; we remark that, in the literature, the count of lattice elements $\\gamma\\in \\Gamma$ such that $d_{\\Hyp}(\\gamma\\cdot i,i)\\leq R$ often appears instead; the two quantities are proportional by a factor $|\\text{Stab}_{\\Gamma}(i)|$.} of $i$ under the $\\Gamma$-action on the hyperbolic plane $\\Hyp$ and we recall that $d_{\\Hyp}$ is the hyperbolic distance function on $\\Hyp$.\n\t\n\t\\begin{rmk}\n\t\tUpon replacing $\\Gamma$ by a conjugate, there is no loss of generality in choosing $i\\in \\Hyp$ as the base point: an elementary algebraic computation, together with the fact that $\\SL_2(\\R)$ acts by $d_{\\Hyp}$-isometries, leads to the equality\n\t\t\\begin{equation*}\n\t\t\t|\\{z\\in \\Gamma\\cdot w:d_{\\Hyp}(z,w)\\leq R  \\}|=|\\{z\\in g^{-1}\\Gamma g\\cdot i:d_{\\Hyp}(z,i)\\leq R\\}|\n\t\t\\end{equation*}\n\t\tfor any $w=g\\cdot i \\in \\Hyp$ and  $g\\in \\SL_2(\\R)$.\n\t\\end{rmk}\t\n\tThere is a compact fundamental domain for the action of $\\Gamma$ on $\\Hyp$, and a transposition of Gauss' tesselation argument to this setup provides a rationale for the heuristics concerning the main term of the asymptotics, which once again should be proportional to the hyperbolic area measure of the ball $B_R=\\{z\\in \\Hyp:d_{\\Hyp}(z,i)\\leq R \\}$, which we denote by $m_{\\Hyp}(B_R)$. However, a consequence of the peculiar features of hyperbolic geometry is that boundary effects become relevant, as opposed to the Euclidean setting: more precisely, the growth rate of the length of the boundary $\\partial{B_R}$ turns out to be equal to the growth rate of $m_{\\Hyp}(B_R)$. The error rate resulting from the tesselation approach is consequently of the same order of the main term, and as such meaningless.\n\t\n\tAs for its Euclidean counterpart, the circle problem in the hyperbolic plane has been the subject of intensive research over the course of the twentieth century, with fundamental contributions due to Delsarte (\\cite{Delsarte}), Selberg (\\cite{Selberg}), Margulis (\\cite{Margulis}), Lax and Phillips (\\cite{Lax-Phillips}) and Phillips and Rudnick (\\cite{Phillips-Rudnick}, see its introduction for a detailed history of the problem). To a large extent, the state of the art concerning the best estimate on the error term $|N(R)-c_{\\Gamma}m_{\\Hyp}(B_R)|$,\n\twhere $c_{\\Gamma}$ is an explicit constant which we identify in Theorem~\\ref{thm:countingproblem}, \n\tis represented by Selberg's upper bound\n\t\\begin{equation}\n\t\t\\label{eq:Selbergbound} |N(R)-c_{\\Gamma}m_{\\Hyp}(B_R)|\\leq e^{(\\sup\\{2\/3,(1+\\Re{\\nu_*})\/2\\})R}\\;,\n\t\\end{equation}\n\twhere recall that $(1-\\nu_*^{2})\/4$ is the spectral gap of $S=\\Gamma\\bsl \\Hyp$. The estimate\\footnote{As a matter of fact, Selberg's asymptotics is more accurate than the one recorded here, featuring a main term which involves, beside the hyperbolic area of the balls, additional terms depending on Laplace eigenfunctions for small eigenvalues; see~\\cite[Eq.~1.13]{Phillips-Rudnick}.} in~\\eqref{eq:Selbergbound} (which is equally valid for non-uniform lattices $\\Gamma<\\SL_2(\\R)$) is obtained by means of a deep analysis of a  class of integral operators commuting with hyperbolic isometries (cf.~\\cite{Selberg}).\n\t\n\tIt was Margulis' realization (see~\\cite{Margulis-thesis}) that lattice point counting problems of the type we are examining are intimately interwoven with questions of equidistribution of translates of subgroup orbits on homogeneous spaces. Subsequently, this novel perspective was profitably pursued and vastly generalized in the works of Duke, Rudnick and Sarnak~\\cite{Duke-Rudnick-Sarnak} and Eskin and McMullen~\\cite{Eskin-McMullen}. Specializing to our current setup, it turns out that the distribution properties of translates of $\\SO_2(\\R)$-orbits\\footnote{We consider here the canonical left action $g\\cdot \\Gamma g'=\\Gamma g'g^{-1}$ of $\\SL_2(\\R)$ on $M$.} on $M= \\Gamma\\bsl \\SL_2(\\R)$ lead to meaningful information on the growth rate of $N(R)$, in a way that is amenable to quantitative refinements (see Section~\\ref{sec:latticepoint} for an extensive treatment of the connection in its quantitative form).    \n\t\n\tWe are thus lead to study effective equidistribution properties of $\\SO_2(\\R)$-orbits on $M$, which can be readily derived from Theorem~\\ref{thm:mainexpandingtranslates} via the standard Cartan decomposition for $\\SL_2(\\R)$.  For any $g_0\\in \\SL_2(\\R)$, define the right-translation map $R_{g_0}\\colon M\\to M$ by $R_{g_0}(\\Gamma g)=\\Gamma g g_0$ for any $g\\in \\SL_2(\\R)$. For every $p \\in M$, we indicate with $m_{\\SO_2(\\R)\\cdot p}$ the unique $\\SO_2(\\R)$-invariant Borel probability measure on $M$ which is fully supported on the (compact) $\\SO_2(\\R)$-orbit of the point $p$; furthermore, for any $g\\in \\SL_2(\\R)$, the notation $g_*m_{\\SO_2(\\R)\\cdot p}$ stands for the push-forward of $m_{\\SO_2(\\R)\\cdot p}$ under the action of $g$, which clearly depends only on the left coset of $g$ modulo $\\SO_2(\\R)$. Making use of the mixing properties of the global $\\SL_2(\\R)$-action on $M$ via a clever thickening argument, Margulis proved  (see~\\cite{Margulis,Margulis-thesis}) that arbitrary translates of $m_{\\SO_2(\\R)\\cdot p}$ equidistribute towards the $\\SL_2(\\R)$-invariant measure $\\vol$; this amounts to the fact that, for any continuous function $f\\colon M\\to \\C$, \n\t\\begin{equation*}\n\t\t\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}\\longrightarrow\\int_{M}\\;f\\;\\text{d}\\vol\n\t\\end{equation*} \n\tas $g\\SO_2(\\R)$ tends to infinity in the quotient $\\SL_2(\\R)\/\\SO_2(\\R)$.\n\t\n\tIn order to phrase a quantitative version of the previous statement conveniently, we introduce the notation $\\norm{g}_{\\text{op}}$ to indicate operator norm of an element $g\\in \\SL_2(\\R)$ with respect to the standard Euclidean norm on $\\R^{2}$; such a specific choice, while obviously immaterial, arises naturally over the course of the proof. \n\t\n\n\t\n\t\\begin{thm}\n\t\t\\label{thm:mainarbitrarytranslates}\n\t\tLet $C_{\\emph{Spec}},C'_{\\emph{Spec}}$ be as in Theorem~\\ref{thm:mainexpandingtranslates}, $s>11\/2$ a real number. There exists a real constant $C_{s,\\emph{rot}}$, depending only on $s$ and on $M$, such that the following holds:\n\t\tif $f\\in W^{s}(M)$, then there exist, for any positive Casimir eigenvalue $\\mu$, functions $D^{+}_{\\mu}f,\\;D^{-}_{\\mu}f\\colon M\\times \\SL_2(\\R)\\to \\C$ with\n\t\t\\begin{equation*}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\sup\\limits_{p \\in M,\\;g\\in \\SL_2(\\R)}|D^{\\pm}_{\\mu}f(p,g)|\\leq C_{s,\\emph{rot}}C_{\\emph{Spec}}' C_{1,s-3}\\norm{f}_{W^{s}}\\;,\n\t\t\\end{equation*}\n\t\tsuch that, for every $p \\in M$ and every $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\emph{op}}\\geq \\sqrt{e}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t\\int_{M}f\\;\\emph{d}&g_{*}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\;\\emph{d}\\vol\\\\\n\t\t\t\t&+\\norm{g}_{\\emph{op}}^{-1}\\biggl(\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\emph{op}}})}D^{+}_{\\mu}f(p,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\emph{op}}})}D^{-}_{\\mu}f(p,g)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\emph{op}}^{-(1+\\nu)}D^{+}_{\\mu}f(p,g)+\\norm{g}_{\\emph{op}}^{-(1-\\nu)}D^{-}_{\\mu}f(p,g)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\emph{op}}^{-1}D^{+}_{1\/4}f(p,g)+2\\norm{g}_{\\emph{op}}^{-1}\\log{\\norm{g}_{\\emph{op}}}D^{-}_{1\/4}f(p,g)\\bigr)+\\mathcal{R}f(p,g)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twhere \n\t\t\\begin{equation*}\n\t\t\t|\\mathcal{R}f(p,g)|\\leq C_{\\emph{Spec}}C_{1,s-3}C_{s,\\emph{rot}} \\norm{f}_{W^{s}}(2\\log{\\norm{g}_{\\emph{op}}}+1)\\norm{g}_{\\emph{op}}^{-2}\\;.\n\t\t\\end{equation*}\n\t\\end{thm} \n\t\n\t\\begin{rmk}\n\t\t\\begin{enumerate}\n\t\t\t\\item The problem of effective equidistribution of translates of finite-volume orbits, in the vastly more general context of affine symmetric spaces, was thoroughly explored by Benoist and Oh in~\\cite{Benoist-Oh}. Their approach relies crucially on effective bounds for the mixing rates\\footnote{Specializing to our setup, effective mixing for the $\\SL_2(\\R)$-action on finite-volume quotients $\\Gamma\\bsl \\SL_2(\\R)$ was first worked out in detail, to the best of our knowledge, by Kleinbock and Margulis in~\\cite[Sec.~2.4]{Kleinbock-Margulis} (see also \\cite{Ratner}).} of the relevant global action, and systematically developes quantitative versions of the geometric properties which play a major role in the original work~\\cite{Eskin-McMullen} of Eskin and McMullen.  In the specific instance of the affine symmetric space being the hyperbolic plane $\\Hyp$,  Theorem~\\ref{thm:mainarbitrarytranslates} improves upon~\\cite[Thm.~1.10]{Benoist-Oh} in that it quantifies the exponent governing the equidistribution rate and spells out additional terms in the asymptotic expansion.\n\t\t\t\\item Just as in the case of Theorem~\\ref{thm:mainexpandingtranslates} and Corollary~\\ref{cor:effective}, the asymptotic expansion in Theorem~\\ref{thm:mainarbitrarytranslates} delivers at once the (optimal) effective equidistribution bound\n\t\t\t\\begin{equation*}\n\t\t\t\t\\biggl|\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}-\\int_{M}f\\;\\text{d}\\vol\\biggr|\\leq D^{\\text{main}}f(p,g)\\;(\\log{\\norm{g}_{\\text{op}}})^{\\varepsilon_0}\\norm{g}_{\\text{op}}^{-(1-\\Re{\\nu_*})}\n\t\t\t\\end{equation*}\n\t\t\tfor every $p \\in M$ and $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$, \n\t\t\twhere the function $D^{\\text{main}}\\colon M\\times \\SL_2(\\R)\\to \\C$ is uniformly bounded in terms of an appropriate Sobolev norm of $f$ and of spectral data depending only on $M$.  Following the thread of the observations expressed in Remark~\\ref{rmk:equivalentmetrics}, it is instructive to  compare it with the decay rates for matrix coefficients of unitary representations of $\\SL_2(\\R)$ computed by Venkatesh in~\\cite[Sec.~9.1.2]{Venkatesh}.\n\t\t\\end{enumerate}\n\t\\end{rmk}\n\t\n\tTheorem~\\ref{thm:mainarbitrarytranslates} affords a precise asymptotic formula for the averaged counting of points on translates of $\\Gamma$-orbits inside balls of increasing radius, that is, for quantities of the form\\footnote{As it is customary in the literature addressing such themes, we shall choose to work with spaces of left cosets whenever dealing with the lattice point counting problem. The map $g\\Gamma\\mapsto \\Gamma g^{-1}$ establishes an $\\SL_2(\\R)$-equivariant diffeomorphism, between $\\SL_2(\\R)\/\\Gamma$ and $\\Gamma\\bsl \\SL_2(\\R)$; every object we have defined on $\\Gamma\\bsl \\SL_2(\\R)$ shall thus be identified (without altering notation) with the corresponding object in $\\SL_2(\\R)\/\\Gamma$ without further comment.}\n\t\\begin{equation}\n\t\t\\label{eq:averagecounting}\n\t\t\\int_{\\SL_2(\\R)\/\\Gamma}\\frac{|g\\Gamma\\cdot i\\cap B_R|}{m_{\\Hyp}(B_R)}\\psi(g\\Gamma )\\;\\text{d}\\vol(g\\Gamma)\n\t\\end{equation}\n\tas $\\norm{g}_{\\text{op}}$ tends to infinity, where $\\psi$ is a sufficiently regular function on $\\SL_2(\\R)\/\\Gamma$. Prior to the statement of the result, we shall fix advantageous normalizations for the various invariant measures involved (see Section~\\ref{sec:averagecounting} for the details).\n\t\n\tLet $m_{\\SL_2(\\R)}$ be the unique choice of Haar measure on $\\SL_2(\\R)$ such that, if $m_{\\SO_2(\\R)}$ is the probability Haar measure on the compact subgroup $\\SO_2(\\R)$, then $m_{\\SL_2(\\R)}$ is the (formal) product (cf.~Proposition~\\ref{prop:foldingunfolding}) of $m_{\\SO_2(\\R)}$ and of the  $\\SL_2(\\R)$-invariant measure on the homogeneous space $\\SL_2(\\R)\/\\SO_2(\\R)$ which corresponds, under the canonical identification of the latter space with $\\Hyp$, to the hyperbolic area measure $m_{\\Hyp}$. We then indicate with $m_{\\SL_2(\\R)\/\\Gamma}$ the unique $\\SL_2(\\R)$-invariant measure on $\\SL_2(\\R)\/\\Gamma$ such that $m_{\\SL_2(\\R)}$ is the product of $m_{\\SL_2(\\R)\/\\Gamma}$ and the counting measure on the discrete group $\\Gamma$. Observe that $m_{\\SL_2(\\R)\/\\Gamma}$ is a scalar multiple of the probability measure $\\vol$, the multiplying factor being equal to the covolume\n\t\\begin{equation*} \\text{covol}_{\\SL_2(\\R)}(\\Gamma)=m_{\\SL_2(\\R)\/\\Gamma}(\\SL_2(\\R)\/\\Gamma)\n\t\\end{equation*}\n\tof the lattice $\\Gamma$ inside $\\SL_2(\\R)$. We denote similarly by $\\text{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))$ the volume\\footnote{A straightfoward application of the formula in~\\eqref{eq:foldingunfolding} shows that this equals the reciprocal of the cardinality of the finite group $\\Gamma\\cap \\SO_2(\\R)$.} of the compact quotient $\\SO_2(\\R)\/(\\Gamma\\cap \\SO_2(\\R))$ with respect to the $\\SO_2(\\R)$-invariant measure induced by $m_{\\SO_2(\\R)}$ and the counting measure on $\\Gamma\\cap \\SO_2(\\R)$.\n\t\n\tLastly, recall that $B_R$ is the closed hyperbolic ball in $\\Hyp$ of radius $R>0$ centered at $i$.\n\t\n\t\\begin{prop}\n\t\t\\label{prop:averagedcounting}\n\t\tLet $C_{\\emph{Spec}},C_{\\emph{Spec}}'$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. Suppose given a real number $s>11\/2$ and a function $\\psi\\in W^{s}(\\SL_2(\\R)\/\\Gamma)$. There exist, for any positive Casimir eigenvalue $\\mu$, functions $\\beta_{\\psi,\\mu}^{+},\\beta_{\\psi,\\mu}^{-}\\colon \\R_{>0}\\to \\C$  with \n\t\t\\begin{equation}\n\t\t\t\\label{eq:betabound}\n\t\t\t\\sum_{\\mu\\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\sup_{R>0}|\\beta_{\\psi,\\mu}^{\\pm}(R)|\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3} }{2\\pi}\\norm{\\psi}_{W^{s}}\n\t\t\\end{equation}\n\t\tsuch that, for every $R\\geq 1$, \n\t\t\\begin{equation}\n\t\t\t\\label{eq:averagedcountingfunction}\n\t\t\t\\begin{split}\n\t\t\t\t\\frac{1}{\\emph{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))}&\\int_{\\SL_2(\\R)\/\\Gamma} \\frac{|g\\Gamma\\cdot i \\cap B_R |}{m_{\\Hyp}(B_R)}\\;\\psi(g\\Gamma )\\;\\emph{d}m_{\\SL_2(\\R)\/\\Gamma}(g\\Gamma)=\\\\\n\t\t\t\t&\\frac{1}{\\emph{covol}_{\\SL_2(\\R)}(\\Gamma)}\\int_{\\SL_2(\\R)\/\\Gamma}\\psi\\;\\emph{d}m_{\\SL_2(\\R)\/\\Gamma}\\\\\n\t\t\t\t&\n\t\t\t\t+ e^{-\\frac{R}{2}}\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\beta_{\\psi,\\mu}^{+}(R)+\\beta_{\\psi,\\mu}^{-}(R)\\\\\n\t\t\t\t&\n\t\t\t\t+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi,\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi,1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi,1\/4}(R)\\biggr)+\\gamma_{\\psi}(R)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere \n\t\t\\begin{equation}\n\t\t\t\\label{eq:gammabound}\n\t\t\t|\\gamma_{\\psi}(R)|\\leq \\frac{5C_{\\emph{Spec}}C_{1,s-3}}{4\\pi}\\norm{\\psi}_{W^{s}} (R+1)e^{-R}\\quad .\n\t\t\\end{equation}\n\t\\end{prop}\n\t\n\tRecall now our definition of the counting function $N(R)$ in~\\eqref{eq:defcountingfunction}. An optimization argument involving approximate identities on the homogeneous space $\\SL_2(\\R)\/\\Gamma$ enables us to derive information on the asymptotic behaviour of the error term in the pointwise counting problem discussed at the beginning of this subsection. We remind the reader that $\\nu_*$ is the unique complex number in $\\R_{\\geq 0}\\cup i\\R_{>0}$ such that $\\frac{1-\\nu_*^{2}}{4}$ equals the spectral gap of $S=\\Gamma\\bsl \\Hyp$.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:countingproblem}\n\t\tLet $\\Sigma\\colon \\R_{>0}\\to \\R_{>0}$ be the function defined by\n\t\t\\begin{equation*}\n\t\t\t\\Sigma(R)=\\frac{\\emph{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))}{\\emph{covol}_{\\SL_2(\\R)}(\\Gamma)}m_{\\mathbb{H}}(B_R)\\;, \\quad R>0.\n\t\t\\end{equation*}\n\t\tSet $\\eta_*=\\frac{1}{13}(1-\\Re{\\nu_*})$. Then,\n\t\tfor every $\\varepsilon>0$, \n\t\t\\begin{equation*}\n\t\t\tN(R)=\\Sigma(R)+o(e^{(1-\\eta_*+\\varepsilon)R})\n\t\t\\end{equation*}\n\t\tas $R$ tends to infinity.\n\t\\end{thm}\n\t\n\t\\begin{rmk}\n\t\tAs it will emerge in the proof of Theorem~\\ref{thm:countingproblem}, which is detailed in Section~\\ref{sec:countingproblem}, the appearance of the quantity $1\/13$ in the exponent is ultimately an outgrowth of the minimal Sobolev regularity of the test function $f$ we need to impose in Theorem~\\ref{thm:expandingonsurface}, which in turn is needed because of the upper bounds in~\\eqref{eq:boundDthetamu} and~\\eqref{eq:globalremainderestimate} depending on the Sobolev norm $\\norm{f}_{W^s}$ for some $s>9\/2$. In this regard, we observe the following: suppose that, in the latter two bounds, the norm $\\norm{f}_{W^{s}}$ can be replaced by $\\norm{f}_{W^1}$, as it is the case for joint eigenfunctions of $\\square$ and $\\Theta$ (cf.~Theorem~\\ref{thm:case_Theta_eigenfn}); then Theorem~\\ref{thm:countingproblem} would hold with $1\/6$ in place of $1\/13$. \n\t\\end{rmk}\n\t\n\t\\subsection{Outline of the proofs and layout of the article}\n\t\n\tThe method we employ to prove Theorem \\ref{thm:case_Theta_eigenfn} was originally devised by Ratner in~\\cite{Ratner}, who realized that the problem of finding mixing rates for geodesic and horocycle flows can be reduced to solving a family of linear second-order ordinary differential equations\\footnote{After the completion of a first draft of the present article, the authors were made aware of the unpublished manuscript~\\cite{Edwards-unpublished} by S.~Edwards, in which a weaker formulation of the quantitative equidistribution result in Theorem~\\ref{thm:mainexpandingtranslates} is provided. The strategy of proof is entirely analogous to the one pursued here, and is there applied, more generally, to the quantitative investigation of the equidistribution properties of translated orbits of symmetric subgroups on homogeneous spaces of semisimple Lie groups.}. \n\tThis ingenuous and yet fairly elementary approach has been further developed by Burger in~\\cite{Bur} to prove polynomial bounds for the equidistribution of horocycle orbits in compact quotients of $\\SL_2(\\R)$. Later, Str\\\"{o}mbergsson \\cite{Str} and Edwards \\cite{Edw} exploited the same idea to study effective equidistribution properties of unipotent orbits in more general settings.\n\tIn the same spirit, the second author recently provided (see~\\cite{Rav}), using Ratner's strategy, an alternative proof of Flaminio-Forni's asymptotic expansion for horocycle ergodic integrals \\cite{Flaminio-Forni}. \n\t\n\tWe begin in Section~\\ref{sec:preliminaries} with an overview of the required notions concerning hyperbolic surfaces, Sobolev spaces and harmonic analysis on the Lie group $\\SL_2(\\R)$, nailing down notation to be employed throughout the manuscript.\n\tAssuming that a $\\cC^{2}$-observable $f$ is a joint eigenfunction of the Casimir operator and of the vector field $\\Theta$, we then show in Section~\\ref{sec:reductiontoODE} that the behaviour of the circle-arc averages $k_{f, \\theta}(p, \\cdot)$ (cf.~\\eqref{eq:ktheta}), viewed as functions of the time $t$ for a fixed base point $p \\in \\Gamma\\bsl \\SL_2(\\R)$, fulfill a second order linear ODE, solving which explicitly leads to the proof of Theorem~\\ref{thm:case_Theta_eigenfn} presented in Section~\\ref{sec:jointeigenfn}; incidentally, we may arrange computations so that the latter takes on the same form of the ODE satisfied by time rescalings of horocycle averages in~\\cite{Rav} (see, in particular,~\\cite[Prop.~8]{Rav}), which accounts for the similarity between Theorem~\\ref{thm:case_Theta_eigenfn} with~\\cite[Thm.~1]{Rav}.\n\tIn Section~\\ref{sec:arbitraryfn}, the asymptotic expansion of Theorem \\ref{thm:mainexpandingtranslates} is deduced from Theorem \\ref{thm:case_Theta_eigenfn} taking advantage of a few elementary facts from the classical harmonic analysis of $\\SL_2(\\R)$. Additional regularity on $f$ is required in order to ensure the absolute convergence of the expansion in \\eqref{eq:asymptoticgeneral}; see, in particular, Section~\\ref{sec:sumestimates}. The asymptotics for arbitrary translates in Theorem~\\ref{thm:mainarbitrarytranslates} is derived from Theorem~\\ref{thm:mainexpandingtranslates} in Section~\\ref{sec:arbitrarytranslates}.\n\tBuilding on Theorem~\\ref{thm:mainexpandingtranslates} once more, we establish in Sections~\\ref{sec:spatialDLT} and~\\ref{sec:noCLT} the distributional limit Theorems~\\ref{thm:CLT} and~\\ref{thm:noCLT} for the random variable $k_{f,\\theta}(p,t)$, appropriately centered and normalized, when the initial point $p$ is taken randomly with respect to the uniform measure.\n\tThese limit theorems mirror those for horocycle ergodic integrals, for which we refer the reader to~\\cite{BuFo, Rav}. Section~\\ref{sec:temporalDLT} hosts a few considerations concerning the point of view of temporal distributional limit theorems (see~\\cite{Dolgopyat-Sarig}) on the problem of analyzing the statistical behaviour of circle averages. \n\tFinally, in Section~\\ref{sec:latticepoint}, we provide a quantitative treatment of the approach of Duke-Rudnick-Sarnak~\\cite{Duke-Rudnick-Sarnak} and Eskin-McMullen~\\cite{Eskin-McMullen}, which allows to prove both Proposition~\\ref{prop:averagedcounting} and Theorem~\\ref{thm:countingproblem} on the hyperbolic lattice point counting problem.\n\t\n\t\n\t\n\t\n\t\\section{Preliminaries on harmonic analysis on $\\SL_2(\\R)$}\n\t\\label{sec:preliminaries}\n\tIt is the aim of this section to carefully describe the setting of our main results as well as to review the required notions on the representation theory of the group $\\SL_2(\\R)$ which will play a central role throughout the article. \n\t\n\t\n\t\\subsection{Hyperbolic surfaces and their unit tangent bundles}\n\t\\label{sec:hyperbolic}\n\tThe subject of this subsection is classical: detailed treatments can be found, for instance, in~\\cite{Bekka-Mayer,Bergeron,Borel,Buser,Einsiedler-Ward,Iwaniec,Katok}.\n\t\n\tThe special linear group $\\SL_2(\\R)$ is the group of $2\\times 2$ real matrices with determinant $1$. It is a three-dimensional Lie group, whose Lie algebra we denote by $\\sl_2(\\R)$ and identify canonically with the Lie algebra of traceless $2\\times 2$  matrices with real entries. The identity matrix in $\\SL_2(\\R)$ is denoted by $I_2$. A basis of $\\sl_2(\\R)$ as a real vector space is given by the elements\n\t\\begin{equation*}\n\t\tX=\n\t\t\\begin{pmatrix}\n\t\t\t1\/2&0\\\\0&-1\/2\n\t\t\\end{pmatrix}\n\t\t, \\quad U=\n\t\t\\begin{pmatrix}\n\t\t\t0&1\\\\\n\t\t\t0&0\n\t\t\\end{pmatrix}\n\t\t, \\quad V=\n\t\t\\begin{pmatrix}\n\t\t\t0&0\\\\\n\t\t\t1&0\n\t\t\\end{pmatrix}\n\t\t.\n\t\\end{equation*}\n\tWith $\\exp\\colon \\sl_2(\\R)\\to \\SL_2(\\R)$ we indicate the exponential map, and with $\\Ad\\colon \\SL_2(\\R)\\to \\GL(\\sl_2(\\R))$, $g\\mapsto \\Ad_g$ the adjoint representation of $\\SL_2(\\R)$ onto $\\sl_2(\\R)$.\n\t\n\t\n\tLet $\\Hyp=\\{z=x+iy\\in \\C:y>0 \\}$ be the Poincar\\'{e} upper-half plane, endowed with the Riemannian metric\n\t\\begin{equation*}\n\t\tg_{(x,y)}=\\frac{(\\text{d}x)^{2}+(\\text{d}y)^{2}}{y^2}\\;, \\quad (x,y)\\in \\Hyp.\n\t\\end{equation*} \n\tThe Riemannian manifold $(\\Hyp,g)$ is a model of the hyperbolic plane, that is, of the unique complete simply connected two-dimensional Riemannian manifold of constant sectional curvature equal to $-1$ (cf.~\\cite[Part 1 Chap.~6]{Bridson-Haefliger}). \n\tThe Lie group $\\SL_2(\\R)$ acts smoothly by orientation-preserving\\footnote{Actually, the action is by analytic transformations of the Riemann surface $\\Hyp$.} isometries of the hyperbolic plane. The action is given by the M\\\"{o}bius transformations\n\t\\begin{equation*}\n\t\t\\begin{pmatrix}\n\t\t\ta&b\\\\\n\t\t\tc&d\n\t\t\\end{pmatrix}\n\t\t\\cdot z=\\frac{az+b}{cz+d}\\;,\\quad a,b,c,d\\in \\R, \\;ad-bc=1,\\;z\\in \\Hyp;\n\t\\end{equation*}\n\tit is transitive, and thus gives rise to an $\\SL_2(\\R)$-equivariant diffeomorphism between $\\Hyp$ and the quotient manifold $\\SL_2(\\R)\/\\SO_2(\\R)$, where the special orthogonal group $\\SO_2(\\R)$ is the stabilizer of the point $i\\in \\Hyp$.\n\t\n\tLet $\\Gamma<\\SL_2(\\R)$ be a cocompact lattice\\footnote{We recall that a lattice in a locally compact Hausdorff topological group $G$ is a discrete subgroup $\\La<G$ such that the quotient space $\\La\\bsl G$ admits a non-zero $G$-invariant Radon probability measure. The lattice $\\La$ is cocompact if $\\La\\bsl G$ is a compact topological space.}. If the projection of $\\Gamma$ to the projective special linear group $\\PSL_2(\\R)=\\SL_2(\\R)\/\\{\\pm I_2 \\}$ has no non-trivial torsion elements, then the quotient space $S=\\Gamma\\bsl \\Hyp$, homeomorphic to the double coset space $ \\Gamma\\bsl \\SL_2(\\R)\/\\SO_2(\\R)$, is a compact, connected, orientable smooth surface inheriting from $\\Hyp$ a Riemannian metric $g_{S}$ of constant sectional curvature equal to $-1$, that is, a hyperbolic Riemannian metric.  Conversely, it is a well-known consequence of Poincar\\'{e}-Koebe's Uniformization Theorem (cf.~\\cite[Chap.~IV]{Farkas-Kra}) that any compact, connected, orientable hyperbolic surface is isomorphic, as a Riemannian manifold, to a quotient $\\Gamma\\bsl\\Hyp$, with $\\Gamma<\\SL_2(\\R)$ a cocompact lattice with torsion-free projection on $\\PSL_2(\\R)$. In case the projection of $\\Gamma$ to $\\PSL_2(\\R)$ has non-trivial torsion elements, the quotient $S=\\Gamma\\bsl \\Hyp$ is, more generally, a hyperbolic \\emph{orbifold} (see~\\cite[Chap.~13]{Ratcliffe} and~\\cite[Chap.~13]{Thurston}). Unifying, though mildly abusing terminology, we shall throughout refer to $S$ as a surface in both the previous cases. \n\t\n\tThe simply transitive action of $\\PSL_2(\\R)$ on the unit tangent bundle of $\\Hyp$ allows to identify, as smooth manifolds, the unit tangent bundle $T^{1}S=\\{(p,v)\\in TS:\\norm{v}_{g_S}=1 \\}$ of $S$ with the homogeneous space $M=\\Gamma\\bsl \\SL_2(\\R)$. Throughout this manuscript, we shall solely appeal to the algebraic structure of $M$ as a homogeneous space of $\\SL_2(\\R)$, and not to its geometric nature of principal circle bundle over the surface $S$.\n\t\n\tFor any $r\\in \\N\\cup\\{\\infty\\}$, we denote by $\\mathscr{C}^r(M)$ the vector space of complex-valued functions of class $\\mathscr{C}^{r}$ defined on $M$. It is endowed with the norm $\\norm{\\cdot}_{\\cC^{r}}$ defined in~\\eqref{eq:Cknorm}. The $\\mathscr{C}^{0}$-norm is abbreviated with $\\norm{\\cdot}_{\\infty}$.\n\t\n\tA vector $W\\in \\sl_2(\\R)$ gives rise to the one-parameter subgroup $(\\exp(tW))_{t\\in \\R}$ of $\\SL_2(\\R)$, which in turn acts as a smooth flow on $M$ by right translations. Whenever convenient, we identify $W$ with the infinitesimal generator of such a flow, and denote its action as a derivation on the function space $\\mathscr{C}^{1}(M)$ by $f\\mapsto Wf$. This extends to an isomorphism of associative $\\R$-algebras between the universal enveloping algebra $U(\\sl_2(\\R))$ of $\\sl_2(\\R)$ and the algebra of $\\SL_2(\\R)$-invariant differential operators on $\\mathscr{C}^{\\infty}(M)$. More generally, any element $Y\\in U(\\sl_2(\\R))$ of order $k\\in \\N$ acts naturally on $\\mathscr{C}^{k}(M)$, the action being also denoted by $f\\mapsto Yf$. Observe that, for any $k\\in \\N$, the definition of the $\\mathscr{C}^k$-norm on $\\mathscr{C}^k(M)$ implies that $\\norm{Yf}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{k}}$ for any $Y\\in U(\\sl_2(\\R))$ of order at most $k$ and any $f\\in \\mathscr{C}^{k}(M)$.\n\t\n\t\\subsection{Unitary representations, the Casimir and Laplace operators, Weyl's law}\n\t\\label{sec:unitaryrepresentations}\n\t\n\tConvenient sources for the material presented in this subsection, in addition to those cited in Section~\\ref{sec:hyperbolic}, are~\\cite{Lang, Mackey,Sarnak, Terras}.\n\t\n\tLet us denote by $H$ be the complex Hilbert space $L^{2}(M)$ of functions which are square-integrable with respect to the $\\SL_2(\\R)$-invariant measure $\\vol$ on $M$; the inner product defining the Hilbert space structure on $H$ shall be denoted by $\\langle \\cdot, \\cdot \\rangle$. With $\\mathscr{U}(H)$ we indicate the group of unitary operators $T\\colon H\\to H$.\n\tThe standard smooth (left) action of $\\SL_2(\\R)$ on the homogeneous space $M$, given by $g_0\\cdot \\Gamma g=\\Gamma gg_0^{-1}$ for every $g_0,g\\in \\SL_2(\\R)$, preserves the measure $\\vol$, and hence gives rise to a unitary representation $\\rho\\colon \\SL_2(\\R)\\to \\mathscr{U}(H),\\;g\\mapsto \\rho_g$, called the quasi-regular representation of $\\SL_2(\\R)$ on $H$. \n\t\n\tFor any $r\\in \\N\\cup\\{\\infty\\}$, denote by $\\mathscr{C}^{r}(H)$ the linear subspace of $\\mathscr{C}^{r}$-vectors on $H$, that is,\n\t\\begin{equation*}\n\t\t\\mathscr{C}^{r}(H)=\\{v\\in H: \\text{ the map }\\SL_2(\\R)\\ni g\\mapsto \\rho_g(v)\\in H \\text{ is of class }\\mathscr{C}^{r}  \\}.\n\t\\end{equation*}\n\t\n\tAn element $W\\in \\sl_2(\\R)$ induces a linear operator $L_{W}\\colon \\mathscr{C}^{1}(H)\\to H$, called the Lie derivative with respect to $W$, defined by\n\t\\begin{equation}\n\t\t\\label{eq:Liederivative}\n\t\tL_W(v)=\\lim_{t\\to 0}\\frac{\\rho_{\\exp(tW)}(v)-v}{t}\\;, \\quad v\\in \\mathscr{C}^{1}(H).\n\t\\end{equation}\n\tObserve that, for a $\\mathscr{C}^{1}$-function $f$ on $M$, we have $L_W(f)=Wf$, regarding $\\mathscr{C}^{1}(M)$ as canonically embedded in $H$.\n\t\n\tThe center of the universal enveloping algebra $U(\\sl_2(\\R))$ is  generated, as an algebra, by any of its non-zero elements, called Casimir elements. For reasons to be clarified shortly, we choose the normalization of the Casimir element to be $\\square=-X^{2}+X+UV$, where $\\{X,U,V\\}$ is identified with a set of generators of the $\\R$-algebra $U(\\sl_2(\\R))$. By the universal property of the universal enveloping algebra, the Casimir element $\\square$ acts as a second-order linear differential operator $\\mathscr{C}^{2}(M)\\to \\mathscr{C}(M)$, and (in a compatible manner) as an unbounded operator on $H$, densely defined on the subspace $\\mathscr{C}^{2}(H)$. Slightly abusing notation, we shall again indicate with $\\square$ the Casimir operator arising in this fashion. It is a consequence of the Casimir element lying in the center of $U(\\sl_2(\\R))$ that the operator $\\square$ commutes with all unitary operators $\\rho_g,\\;g\\in \\SL_2(\\R)$ and all Lie derivatives $L_{W},\\;W\\in \\sl_2(\\R)$, wherever they are simultaneously defined.\n\t\n\tThe Casimir operator $\\square$ is an essentially self-adjoint operator on $H$; as $M$ is compact, its spectrum $\\text{Spec}(\\square)$ consists solely of eigenvalues and is a discrete subset of $\\R$. Our choice of the normalization of the Casimir element implies that, on the subspace $\\mathscr{C}^{2}(S)$ of continuously twice-differentiable functions defined on the surface $S=\\Gamma\\bsl \\Hyp$, the Casimir operator acts as the Laplace-Beltrami operator $\\Delta_{S}$ with respect to the hyperbolic Riemannian metric on $S$. As a consequence, the spectrum of $\\Delta_{S}$ is contained in the spectrum of $\\square$. As a matter of fact, it holds that $\\text{Spec}(\\Delta_S)=\\text{Spec}(\\square)\\cap \\R_{\\geq 0}$. \n\t\n\tA great deal of research has revolved around the properties of the spectrum of $\\Delta_S$; for the purposes of this article, we shall only need a weaker version (cf.~the proof of Lemma~\\ref{lem:finitespectralconstant}) of the celebrated Weyl's law concerning the asymptotics of the eigenvalues, which we shall now recall for completeness. \n\t\n\t\\begin{thm}[Weyl's law]\n\t\t\\label{thm:Weyllaw}\n\t\tLet $\\Gamma<\\SL_2(\\R)$ be as above,  $S=\\Gamma\\bsl \\Hyp$, and \n\t\t\\begin{equation*}\n\t\t\t\\lambda_0<\\lambda_1\\leq \\cdots \\leq \\lambda_n\\leq \\cdots \\to\\infty\n\t\t\\end{equation*} \n\t\tbe the eigenvalues, counted with multiplicity, of the Laplace-Beltrami operator $\\Delta_S$ associated to the hyperbolic Riemannian metric on $S$. Then\n\t\t\\begin{equation*}\n\t\t\t\\lim\\limits_{R\\to\\infty}\\frac{|\\{j\\in \\N:\\lambda_j\\leq R\\}|}{ R}=\\frac{\\emph{Area}(S)}{4\\pi}\\;,\n\t\t\\end{equation*}\n\t\twhere $\\emph{Area}(S)$ is the total mass of $S$ with respect to the hyperbolic area measure.\n\t\\end{thm}\n\tA proof of Theorem~\\ref{thm:Weyllaw} relying on Selberg's trace formula is given in~\\cite[Prop.~10]{Marklof}.\n\t\n\t\\medskip\n\tOn the other hand, the negative eigenvalues of $\\square$ are completely understood, being of the form $-m(m+2)\/4$, with $m$ ranging across the set $\\N^{\\ast}$ of strictly positive integers. \n\t\n\tIn general, any unitary representation of $\\SL_2(\\R)$ is unitarily equivalent to a direct integral of irreducible representations. Owing to compactness of $M$, the representation space $H$ of $\\rho$ actually decomposes as a Hilbert direct sum of (non-zero) irreducible $\\rho$-invariant subspaces:\n\t\\begin{equation*}\n\t\tH=\\bigoplus_{i \\in I}H_i\\;,\\quad \\rho(H_i)\\subset H_i,\\; \\rho|_{H_i} \\text{ irreducible},\\;I \\text{ countable index set}.\n\t\\end{equation*}\n\tAs the Casimir operator commutes with all the $\\rho_g$ for $g\\in \\SL_2(\\R)$, it restricts to an operator $\\mathscr{C}^2(H_i)\\to H_i$ for each $i \\in I$. By virtue of Schur's lemma for unbounded linear operators, there exists $\\mu \\in \\text{Spec}(\\square)$, depending on $i$, such that $\\square v=\\mu v$ for any $v\\in\\mathscr{C}^{2}(H_i)$. Regrouping all irreducible subspaces according to the corresponding Casimir eigenvalue, we obtain an orthogonal splitting into $\\rho$-invariant subspaces\n\t\\begin{equation}\n\t\t\\label{eq:deccasimireigenvalues}\n\t\tH=\\bigoplus\\limits_{\\mu \\in \\text{Spec}(\\square)}H_{\\mu}\\;,\\quad H_{\\mu}=\\{v\\in \\mathscr{C}^{2}(H):\\square v =\\mu v \\}.\n\t\\end{equation}\n\t\n\tLet $\\Theta\\in \\sl_2(\\R)$ be the element defined in~\\eqref{eq:rotationflow}, inducing the unbounded linear operator $L_{\\Theta}$ on $H$. As it is the case for any representation space of a unitary representation of $\\SL_2(\\R)$, each subspace $H_{\\mu}$ is the internal Hilbert direct sum \n\t\\begin{equation}\n\t\t\\label{eq:decthetaeigenvalues}\n\t\tH_{\\mu}=\\bigoplus_{n\\in \\Z}H_{\\mu,n}\\;,\\quad H_{\\mu,n}=\\biggl\\{v\\in \\mathscr{C}^{1}(H_{\\mu}):L_\\Theta v=\\frac{i}{2}nv \\biggr\\}.\n\t\\end{equation}  \n\t\n\t\\begin{rmk}\n\t\tObserve that, in the previous decomposition, it might occur (as it manifestly emerges in the equation~\\eqref{eq:differentSobolev} below) that $H_{\\mu,n}$ is the trivial subspace for some $\\mu\\in \\text{Spec}(\\square)$ and $n\\in \\Z$. For this reason, in the sequel, we let the index set of the direct sum in~\\eqref{eq:decthetaeigenvalues} be rather $I(\\mu)=\\{n\\in \\Z:H_{\\mu,n}\\neq \\{0\\} \\}\\subset \\Z$, bearing in mind this caveat.\n\t\\end{rmk}\n\t\n\t\\subsection{Sobolev spaces and the Sobolev Embedding Theorem}\n\t\\label{sec:Sobolev}\n\t\n\t\n\tThe classical theory of Sobolev spaces is thoroughly discussed in~\\cite{Adams}. In the context of arbitrary Riemannian manifolds, it is presented, for instance, in~\\cite{Aubin,Hebey}; the extension to fractional Sobolev orders is treated e.g.~in~\\cite{Strichartz-Sobolev,Triebel}. Here we shall confine ourselves to homogeneous spaces of $\\SL_2(\\R)$, placing emphasis on how this theory is brought to bear on the study of unitary representations of the special linear group. \n\t\n\tDefine the element\n\t\\begin{equation*}\n\t\tY=\n\t\t\\begin{pmatrix}\n\t\t\t0&-1\/2\\\\\n\t\t\t-1\/2&0\n\t\t\\end{pmatrix}\n\t\t\\in \\sl_2(\\R),\n\t\\end{equation*}\n\tso that $\\{X,\\Theta,Y \\}$ forms a basis of the real vector space $\\sl_2(\\R)$.\n\tDefine the second-order linear operator $\\Delta= -(X^{2}+Y^{2}+\\Theta^{2}) =\\square -2\\Theta^{2}$ on the subspace $\\mathscr{C}^{2}(H)$. As each Lie derivative $L_W$ ($W\\in \\sl_2(\\R)$) satisfies\\footnote{This follows at once from the definition of $L_W$ in~\\eqref{eq:Liederivative} and the fact that $\\rho_g$ is a unitary operator for any $g\\in \\SL_2(\\R)$.} $\\langle L_W u,v \\rangle =-\\langle u,L_Wv\\rangle$ for any $u,v \\in \\mathscr{C}^{1}(H)$, it is clear that $\\langle \\Delta u,v\\rangle =\\langle u, \\Delta v\\rangle$ for any $u,v\\in \\mathscr{C}^{2}(H)$, that is, $\\Delta$ is self-adjoint on its domain of definition. We shall refer to it as the Laplace operator on $H$. Akin to the Casimir operator, $\\Delta$ acts as the Laplace-Beltrami operator $\\Delta_S$ on the subspace $\\mathscr{C}^{2}(S)$. Indeed, any vector $u \\in \\mathscr{C}^{2}(S)$ is invariant under the action of $\\SO_2(\\R)$ by right translations, and hence lies in the kernel of $L_{\\Theta}$; as a consequence, \n\t\\begin{equation*}\n\t\t\\Delta u=(\\square -2\\Theta^{2})u=\\square u-2L_{\\Theta}^{2}u=\\square u=\\Delta_S u\\;. \n\t\\end{equation*}\n\t\n\tFor any $s\\in \\R_{>0}$, we define the Sobolev space of order $s$ on $H$, denoted by $W^{s}(H)$, as the maximal linear subspace of $H$ on which the unbounded linear operator $\\Delta^{s\/2}$ can be defined, and endow it with the inner product given by\n\t\\begin{equation}\n\t\t\\label{eq:Sobolevinnerproduct}\n\t\t\\langle u,v\\rangle_{W^{s}}=\\langle (I+\\Delta)^{s}u,v\\rangle\\;, \\quad u,v \\in W^{s}(H),\n\t\\end{equation}\n\twhere $I$ denotes the identity operator on $H$.\n\tThis assignment turns $W^{s}(H)$ into a Hilbert space, whose associated norm is denoted by $\\norm{\\cdot }_{W^{s}}$. Similarly, we define the Sobolev spaces $W^{s}(H_{\\mu})$ and $W^{s}(H_{\\mu,n})$ for any $\\mu \\in \\text{Spec}(\\Z)$ and $n\\in I(\\mu)$. It is a fact that the decompositions in~\\eqref{eq:deccasimireigenvalues} and~\\eqref{eq:decthetaeigenvalues} induce analogous decompositions on the level of Sobolev spaces, namely there are orthogonal\\footnote{Clearly, we intend that the closed subspaces $W^{s}(H_{\\mu,n})$ are orthogonal with respect to the $W^{s}$-inner product defined in~\\eqref{eq:Sobolevinnerproduct}. } splittings\n\t\\begin{equation}\n\t\t\\label{eq:Sobolevdecomposition}\n\t\tW^{s}(H)=\\bigoplus_{\\mu \\in \\text{Spec}(\\square)}W^{s}(H_{\\mu})=\\bigoplus_{\\mu \\in \\text{Spec}(\\square)}\\bigoplus_{n\\in I(\\mu)}\\;W^{s}(H_{\\mu,n})\\;.\n\t\\end{equation}\n\t\n\tThe argument in Section~\\ref{sec:arbitraryfn} crucially hinges upon the following elementary relationship between Sobolev norms of different order:\n\t\n\t\\begin{lem}\n\t\t\\label{lem:diffSobolevnorms}\n\t\tLet $s\\in \\R_{>0},\\;k\\in \\N$, and assume $u\\in W^{s+k}(H_{\\mu,n})$ for some $\\mu \\in \\emph{Spec}(\\square)$ and $n \\in I(\\mu)$. Then\n\t\t\\begin{equation}\n\t\t\t\\label{eq:differentSobolev}\n\t\t\t\\norm{u}_{W^{s+k}}^2=\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{k}\\norm{u}_{W^s}^2\\;.\n\t\t\\end{equation}\n\t\\end{lem} \n\t\\begin{proof}\n\t\tSuppose $k=1$. We may write, using self-adjointness of $\\Delta$ with respect to the $L^{2}$-inner product,\n\t\t\\begin{equation*}\n\t\t\t\\norm{u}_{W^{s+1}}^{2}=\\langle u,u\\rangle_{W^{s+1}}=\\langle (I+\\Delta)^{s+1}u,u\\rangle=\\langle (I+\\Delta)^{s}u,(I+\\Delta)u\\rangle=\\langle(1+\\Delta)^{s}u,u\\rangle +\\langle (1+\\Delta)^{s}u,\\Delta u \\rangle\\;.\n\t\t\\end{equation*}\n\t\tBy the assumption on $u$, it holds that $\\Delta u=(\\square-2\\Theta^{2})u=\\mu-2(\\frac{i}{2}n)^{2}u=(\\mu+\\frac{n^2}{2})u$. Therefore, we infer\n\t\t\\begin{equation*}\n\t\t\t\\norm{u}_{W^{s+1}}^{2}=\\norm{u}_{W^{s}}^2+\\biggl(\\mu+\\frac{n^{2}}{2}\\biggr)\\langle (1+\\Delta)^{s}u,u\\rangle = \\biggl(1+\\mu+\\frac{n^{2}}{2}\\biggr)\\norm{u}_{W^{s}}^{2}\\;,\n\t\t\\end{equation*}\n\t\tas desired.\n\t\t\n\t\tThe statement for an arbitrary $k\\in \\N$ is immediately achieved arguing by induction.\n\t\\end{proof}\n\t\n\tObserve that~\\eqref{eq:differentSobolev} readily implies the following: if $u \\in W^{s+k}(H)$ for some $s\\in \\R_{>0}$ and $k\\in \\N$,   and \n\t\\begin{equation*}\n\t\tu=\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}u_{\\mu,n}\n\t\\end{equation*}\n\tis its decomposition into joint eigenvectors for $\\square$ and $\\Theta$ provided by~\\eqref{eq:Sobolevdecomposition}, then the larger the integer $k$ is, the faster the decay of the Sobolev norms $\\norm{u_{\\mu,n}}_{W^{s}}$ as $|n|$ and $|\\mu|$ tend to infinity. This phenomenon\\footnote{The counterpart of this relationship in classical Fourier analysis is well-known; the regularity of a function is closely interwoven with the decay rate at infinity of its Fourier coefficients.} is going to be essential in our estimates over the course of the proof of Theorem~\\ref{thm:mainexpandingtranslates}.\n\t\n\t\\smallskip\n\tWe conclude this section recalling a version for compact three-manifolds of the celebrated Sobolev Embedding Theorem, which will be sufficient for our purposes.\n\t\n\t\n\t\n\t\\begin{thm}[Sobolev Embedding Theorem]\n\t\t\\label{thm:Sobolevembedding}\n\t\tFor any $r\\in \\N$ and $s\\in \\R_{>0}$ fulfilling the inequality $s-r>3\/2$, there is a continuous embedding of $W^{s}(M)$ into the Banach space $\\mathscr{C}^{r}(M)$: in particular, there exists a constant $C_{r,s}>0$ such that \n\t\t\\begin{equation*}\n\t\t\t\\norm{f}_{\\mathscr{C}^{r}}\\leq C_{r,s}\\norm{f}_{W^{s}}\n\t\t\\end{equation*}\n\t\tfor every $f\\in W^{s}(M)$.\n\t\\end{thm}\n\t\n\t\\section{Reduction to an ordinary differential equation}\n\t\\label{sec:reductiontoODE}\n\tThis section presents the gist of the approach we pursue in order to prove Theorem~\\ref{thm:case_Theta_eigenfn}, which concerns the asymptotic behaviour of circle-arc averages of joint eigenfunctions of the operators $\\square$ and $\\Theta$ (cf.~Section~\\ref{sec:unitaryrepresentations}); the partial differential equations (classically known in the literature as eigenvalue equations) expressing the eigenfunction condition are here shown to give rise to ordinary differential equations for the corresponding circle averages, when the latter are seen as functions of the time parameter.\n\t\n\tWe fix a function $f\\colon M\\to \\C$ of class $\\cC^{2}$ and a parameter $\\theta\\in (0,4\\pi]$. Recall from~\\eqref{eq:ktheta} the definition of the averages $k_{f,\\theta}(p,t)$, for $p \\in M$ and $t\\in \\R$. As we shall work with a fixed, arbitrary base point $p \\in M$, we shall abbreviate, for notational convenience, $k_{f,\\theta}(p,t)$ with $k_{\\theta}(t)$ in the computations that follow.\n\t\n\tOur goal is to show that the function $k_{\\theta}(t)$ satisfies a second-order linear ODE. In the upcoming computations, the following lemma will be of use. For any left-invariant vector field $W\\in \\sl_2(\\R)$, we indicate with $(\\phi_t^{W})_{t\\in \\R}$ be the one-parameter flow on $M$ defined by $\\phi^{W}_t(\\Gamma g)=\\Gamma g \\exp{tW}$ for any $t\\in \\R$ and $g\\in \\SL_2(\\R)$. For any pair $Y,W\\in \\sl_2(\\R)$ and any point $q\\in M$, the derivative of the smooth curve $s\\mapsto \\phi_{t}^{Y}\\circ \\phi^{W}_s(q)$ (seen as a function from $\\R$ to the tangent bundle of $M$), where $t\\in \\R$ is fixed, is denoted by $\\frac{\\text{d}}{\\text{d}s}\\;\\phi_t^{Y}\\circ \\phi_s^{W}(q)$.\n\tLastly, if $W\\in \\sl_2(\\R)$ and $q\\in M$, we denote by $W_q$ the value at $q$ of the infinitesimal generator of the smooth flow $(\\phi_t^{W})_{t\\in \\R}$ on $M$.\n\t\\begin{lem}\n\t\t\\label{lem:shiftedderivatives}\n\t\tFor every $Y,W\\in \\sl_2(\\R)\\setminus \\{ 0\\}$ and every $q\\in M$, it holds\n\t\t\\begin{equation*}\n\t\t\t\\frac{\\emph{d}}{\\emph{d}s}\\;\\phi_t^{Y}\\circ \\phi^{W}_s(q)=\\Ad_{\\exp{(-tY)}}(W)_{\\phi^{Y}_t\\circ \\phi_s^{W}(q)}\\;.\n\t\t\\end{equation*} \n\t\\end{lem}\n\t\\begin{proof}\n\t\tIt follows from elementary algebraic manipulations, see~\\cite[Lem.~4]{Rav-arcs}.\n\t\\end{proof}\n\t\n\tWe may now state the main result of this section.\n\t\n\t\\begin{prop}\n\t\t\\label{prop:ODE}\n\t\tLet $\\mu \\in \\emph{Spec}(\\square)$, $n\\in \\Z$ and $f\\in \\cC^{2}(M)$ be a function satisfying $\\square f=\\mu f$, $\\Theta f=\\frac{i}{2}n f$. For every $p \\in M$ and $\\theta\\in (0,4\\pi]$, there is a bounded continuous function $G_{\\theta,n}f(p,\\cdot)\\colon \\R_{>0}\\to \\C$ such that the function $k_{\\theta}(t)=\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\emph{d}s$ satisfies the linear ordinary differential equation\n\t\t\\begin{equation}\n\t\t\t\\label{eq:ode}\n\t\t\tk''_{\\theta}(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)=e^{-t}G_{\\theta,n}(p,t)\n\t\t\\end{equation}\n\t\tfor any $t>0$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tFix $f$, $p$ and $\\theta$ as in the assumptions. Since $f$ is of class $\\cC^{2}$ on $M$, differentiation under the integral sign gives\n\t\t\\begin{equation*}\n\t\t\tk'_{\\theta}(t)=\\frac{1}{\\theta} \\int_0^{\\theta}Xf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s,\\quad k''_{\\theta}(t)=\\frac{1}{\\theta} \\int_0^{\\theta}X^{2}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\n\t\t\\end{equation*}\n\t\tfor any $t\\in \\R$, as the geodesic flow $(\\phi_t^{X})_{t\\in \\R}$ on $M$ is generated by the vector field $X$. Therefore, the assumption $\\square f=\\mu f$, i.e.~$-X^{2}f+Xf-UVf=\\mu f$, translates readily into\n\t\t\\begin{equation*}\n\t\t\tk''_{\\theta}(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=-\\frac{1}{\\theta}\\int_0^{\\theta}UVf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\;.\n\t\t\\end{equation*}\n\t\tAs $V=U-2\\Theta$, we have \n\t\t\\begin{equation}\n\t\t\t\\label{ode}\n\t\t\tk''_{\\theta}(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=\t-\\frac{1}{\\theta}\\int_0^{\\theta}U^{2}f\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s+\\frac{in }{\\theta}\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\n\t\t\\end{equation}\n\t\tby the assumption $\\Theta f=\\frac{i}{2}nf$.\n\t\n\t\n\t\n\t\n\tNow, by Stokes' theorem\\footnote{Here, it really boils down to the fundamental theorem of calculus.}, we get that\n\t\\begin{equation*}\n\t\tf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)=\\int_{0}^{\\theta}\\frac{\\text{d}}{\\text{d}s}(f\\circ \\phi_t^{X}\\circ r_{s}(p))\\;\\text{d}s=\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\biggl(\\frac{\\text{d}}{\\text{d}s}(\\phi_t^{X}\\circ r_{s}(p))\\biggr)\\;\\text{d}s\\;,\n\t\\end{equation*}\n\tthe latter equality following from the chain rule for differentials. Recalling that the flow $(r_s)_{s\\in \\R}$ is generated by the vector field $\\Theta$, Lemma~\\ref{lem:shiftedderivatives} delivers\n\t\\begin{equation*}\n\t\t\\frac{\\text{d}}{\\text{d}s}\\phi_t^{X}\\circ r_s(q)=\\Ad_{\\exp(-tX)}(\\Theta)_{\\phi_{t}^{X}\\circ r_s(p)}\\;,\n\t\\end{equation*}\n\tso that\n\t\\begin{equation*} \n\t\t\\begin{split}\n\t\t\tf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)&=\n\t\t\t\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\bigl((\\Ad_{\\exp{(-tX)}}(\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\bigl(((-\\sinh{t})U+e^{t}\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=-\\sinh{t}\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s+\\frac{i}{2}n\\theta e^{t}k_{\\theta}(t)\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\tthe second equality being obtained by straightforward matrix multiplications.\n\t\n\tFrom now we choose $t$ strictly positive. We may thus write\n\t\\begin{equation}\n\t\t\\label{u}\n\t\t\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s=\\frac{1}{\\sinh{t}}\\biggl(\\frac{i}{2}n\\theta e^{t}k_{\\theta}(t)-A_{\\theta}(t)\\biggr)\n\t\\end{equation}\n\twhere\n\t\\begin{equation*} A_{\\theta}(t)=f\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)\\;,\\quad t>0.\n\t\\end{equation*}\n\tArguing as before, we also deduce\n\t\\begin{equation}\n\t\t\\label{dedusquare}\n\t\t\\begin{split}\n\t\t\tUf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi_t^{X}(p)&=\\int_{0}^{\\theta}\\frac{\\text{d}}{\\text{d}s}(Uf\\circ \\phi_t^{X}\\circ r_{s}(p))\\;\\text{d}s\\\\%=\\int_{0}^{\\theta}\\text{d}Uf_{\\phi_t^{X}\\circ r_s(p)}\\bigl((\\Ad_{\\exp{(-tX)}}(\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=\\frac{1}{2}e^{-t}\\int_0^{\\theta}U^{2}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s-\\frac{1}{2}e^{t}\\int_0^{\\theta}VUf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\;.\n\t\t\\end{split}\n\t\\end{equation}\n\tFrom $UV-VU=2X$ we get $VU=UV-2X=U(U-2\\Theta)-2X=U^{2}-2U\\Theta-2X$, so that \n\t\\begin{equation}\n\t\t\\label{vu}\n\t\t\\int_0^{\\theta} VUf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\int_0^{\\theta} U^2f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s-in\\int_0^{\\theta}Uf\\circ \\phi_t^{X}\\circ r_s(p)\\text{d}s-2\\theta k'_{\\theta}(t)\\;.\n\t\\end{equation}\n\tCombining~\\eqref{u},~\\eqref{dedusquare} and~\\eqref{vu} yields\n\t\\begin{equation}\n\t\t\\label{usquare}\n\t\t\\begin{split}\n\t\t\t\\int_0^{\\theta}U^{2}f\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s&=\\frac{1}{\\sinh{t}}\\biggl(\\frac{i}{2}ne^{t}\\int_0^{\\theta}Uf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s+\\theta e^{t}k'_{\\theta}(t) -B_{\\theta}(t)\\biggr)\\\\\n\t\t\t&=\\frac{1}{\\sinh{t}}\\biggl(\\frac{ine^{t}}{4\\sinh{t}}(in\\theta e^{t} k_{\\theta}(t)-A_{\\theta}(t)) +\\theta e^{t}k'_{\\theta}(t) -B_{\\theta}(t)\\biggr)\\;,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere\n\t\\begin{equation*} B_{\\theta}(t)=Uf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi_t^{X}(p)\\;,\\quad t>0.\n\t\\end{equation*}\n\tFrom~\\eqref{ode},~\\eqref{u} and~\\eqref{usquare} we infer that \n\t\\begin{equation*}\n\t\tk_{\\theta}''(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=\\frac{n^2 e^{2t} k_{\\theta}(t)}{4\\sinh^2{t}}+\\frac{ine^{t}A_{\\theta}(t)}{4\\theta\\sinh^2{t}} - \\frac{e^{t}k'_{\\theta}(t)}{\\sinh{t}} -\\frac{n^2 e^{t}k_{\\theta}(t)}{2\\sinh{t}}+\\frac{-inA_{\\theta}(t)+B_{\\theta}(t)}{\\theta \\sinh{t}}\n\t\\end{equation*}\n\tFinally, adding $2k_{\\theta}'(t)$ on both sides gives\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\tk_{\\theta}''(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)&=\\biggl(\\frac{n^{2}}{(1-e^{-2t})^{2}}-\\frac{n^{2}}{1-e^{-2t}}\\biggr)k_{\\theta}(t)+\\biggl(2-\\frac{2}{1-e^{-2t}}\\biggr)k'_{\\theta}(t)\\\\\n\t\t\t&+\\biggl(\\frac{in}{2\\theta \\sinh{t}(1-e^{-2t})}-\\frac{in}{\\theta\\sinh{t}}\\biggr)A_{\\theta}(t)+\\frac{B_{\\theta}(t)}{\\theta \\sinh{t}}\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\tso that $k_{\\theta}''(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)=e^{-t}G_{\\theta,n}f(p,t)$ for \n\t\\begin{equation}\n\t\t\\label{eq:constantterm}\n\t\tG_{\\theta,n}f(p,t)=\\frac{n^{2}e^{-t}}{(1-e^{-2t})^{2}}k_{\\theta}(t)-\\frac{2e^{-t}}{1-e^{-2t}}k'_{\\theta}(t)+\\frac{2ine^{-2t}}{\\theta(1-e^{-2t})^{2}}A_{\\theta}(t)+\\frac{2}{\\theta (1-e^{-2t})}B_{\\theta}(t)\\;.\n\t\\end{equation}\n\tThe function $f$ being of class $\\cC^{2}$, it is clear that the functions $k_{\\theta},k'_{\\theta},A_{\\theta}$ and $B_{\\theta}$ are continuous, hence so is the function $t\\mapsto G_{\\theta,n}f(p,t)$.  Furthermore, the trivial upper bounds\n\t\\begin{equation*}\n\t |k_{\\theta}(t)|\\leq \\norm{f}_{\\infty},\\;|k'_{\\theta}(t)|\\leq \\norm{Xf}_{\\infty},\\;|A_{\\theta}(t)|\\leq 2\\norm{f}_{\\infty},\\; |B_{\\theta}(t)|\\leq 2\\norm{Uf}_{\\infty}\n\t \\end{equation*}\n\t  imply that it is uniformly bounded on $\\R_{>0}$.\n\\end{proof}\nFor later purposes, we estimate explicitly the uniform norm of $G_{\\theta,n}f$. Using the bounds $e^{-t}\\leq 1$ and $1-e^{-2t}\\geq 1-e^{-1}$, valid for all $t\\geq 1\/2$, together with the fact that the three quantities $\\norm{f}_{\\infty},\\norm{Xf}_{\\infty},\\norm{Uf}_{\\infty}$ are bounded from above by $\\norm{f}_{\\mathscr{C}^{1}}$ (cf.~Section~\\ref{sec:hyperbolic}), we obtain that \n\\begin{equation*}\n\t\\sup_{t\\geq 1\/2}|G_{\\theta,n}f(p,t)|\\leq C_{\\theta,n}\\norm{f}_{\\mathscr{C}^{1}}\n\\end{equation*}\nwith\n\\begin{equation*}\n\tC_{\\theta,n}=\\biggl(\\frac{e}{e-1}\\biggr)^{2}\\frac{n(\\theta n+2)}{\\theta}+\\frac{e}{e-1}\\frac{2\\theta+2}{\\theta}\\;.\n\\end{equation*}\nSetting\n\\begin{equation*} \\kappa_0=\\frac{2e^{2}(1+4\\pi)}{(e-1)^{2}}\\;,\n\\end{equation*}\nwe may estimate\n\\begin{equation*}\n\tC_{\\theta,n}\\leq \\biggl(\\frac{e}{e-1}\\biggr)^{2}\\;\\frac{2(\\theta+1)}{\\theta}(n^{2}+1)\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\;,\n\\end{equation*}\nand hence conclude that \n\\begin{equation}\n\t\\label{eq:Gtheta}\n\t\\sup_{t\\geq 1\/2}|G_{\\theta,n}f(p,t)|\\leq\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\cC^{1}}\n\\end{equation}\nfor every choice of $p \\in M$, $\\theta\\in (0,4\\pi]$, $f\\in \\cC^{2}(M)$ and $n\\in \\Z$.\n\n\n\\section{Asymptotics for joint eigenfunctions}\n\\label{sec:jointeigenfn}\n\nThe purpose of this section is to prove Theorem~\\ref{thm:case_Theta_eigenfn} by explictly solving the ODE established in Proposition~\\ref{prop:ODE}. For definiteness, whe choose to impose initial conditions at time $t=1$ for the ensuing Cauchy problem. \n\nWe thus start with the following:\n\n\\begin{lem}\n\t\\label{lem:solution}\n\tLet $\\mu$ be an eigenvalue of the Casimir operator. If $G\\colon \\R_{>0}\\to \\C$ is a continuous function, then for any complex numbers $y_1$ and $y_1'$ the solution to the Cauchy problem\n\t\\begin{equation}\n\t\t\\label{eq:Cauchy}\n\t\t\\begin{cases}\n\t\t\ty''(t)+y'(t)+\\mu y(t)=e^{-t}G(t) & \\\\\n\t\t\ty(1)=y_1& \\\\\n\t\t\ty'(1)=y'_1\n\t\t\\end{cases}\n\t\\end{equation}\n\tis given by\n\t\\begin{equation}\n\t\t\\label{eq:solutionnotquarter}\n\t\t\\begin{split}\n\t\t\ty(t)=&e^{-\\frac{1-\\nu}{2}t}\\biggl(\\frac{(1+\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1-\\nu}{2}}} +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\emph{d}\\xi \\biggr)\\\\\n\t\t\t&-e^{-\\frac{1+\\nu}{2}t}\\biggl(\\frac{(1-\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1+\\nu}{2}}} +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\emph{d}\\xi \\biggr)\n\t\t\\end{split}\n\t\\end{equation}\n\tif $\\mu\\neq 1\/4$, and by\n\t\\begin{equation}\n\t\t\\label{eq:solutionquarter}\n\t\t\\begin{split}\n\t\t\ty(t)=&e^{-\\frac{t}{2}}\\biggl(\\frac{\\sqrt{e}(y_1-2y'_1)}{2}-\\int_1^{t}\\xi e^{-\\frac{\\xi}{2}}G(\\xi)\\;\\emph{d}\\xi\\biggr)\\\\\n\t\t\t& +te^{-\\frac{t}{2}}\\biggl(\\frac{\\sqrt{e}(y_1+2y'_1)}{2}+\\int_1^{t} e^{-\\frac{\\xi}{2}}G(\\xi)\\;\\emph{d}\\xi\\biggr)\n\t\t\\end{split}\n\t\\end{equation}\n\tif $\\mu=1\/4$.\n\\end{lem}\n\\begin{proof}\n\tLet $\\nu$ be the unique complex number in $\\R_{\\geq 0}\\cup i \\R_{>0}$ such that $1-\\nu^{2}=4\\mu$.\n\tThe characteristic polynomial of the homogeneous equation $y''(t)+y'(t)+\\mu y(t)=0$ is $P(Z)=Z^{2}+Z+\\mu$, having two distinct roots $-\\frac{1-\\nu}{2},-\\frac{1+\\nu}{2}$ if $\\mu\\neq 1\/4$, and a double root $-\\frac{1}{2}$ if $\\mu=1\/4$. We examine the case $\\mu\\neq 1\/4$; the case $\\mu=1\/4$ requires only minor modifications. A particular solution of the inhomogeneous equation is given by\n\t\\begin{equation*}\n\t\te^{-\\frac{1-\\nu}{2}t} +\\int_1^{t}\\frac{1}{\\nu}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi\n\t\t-e^{-\\frac{1+\\nu}{2}t}\\int_1^{t}\\frac{1}{\\nu}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi\n\t\\end{equation*}\n\tas direct computations allow to verify. Hence, the general solution of\n\t\\begin{equation*} y''(t)+y'(t)+\\mu y(t)=e^{-t}G(t)\n\t\\end{equation*}\n\tis given by\n\t\\begin{equation*}\n\t\ty(t)=e^{-\\frac{1-\\nu}{2}t}\\biggl(c_1 +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi \\biggr)\n\t\t+e^{-\\frac{1+\\nu}{2}t}\\biggl(c_2 -\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi \\biggr)\\;, \\quad c_1,c_2\\in \\C.\n\t\\end{equation*}\n\tImposing the conditions $y(1)=y_1,y'(1)=y'_1$ enables to determine the coefficients\n\t\\begin{equation*}\n\t\tc_1=\\frac{(1+\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1-\\nu}{2}}}\\;,\\quad c_2=-\\frac{(1-\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1+\\nu}{2}}}\\;.\n\t\\end{equation*} \n\\end{proof}\n\nWe may now apply Proposition~\\ref{prop:ODE} in conjunction with Lemma~\\ref{lem:solution} to determine the explicit analytic expression of the function $k_{f,\\theta}(p,t)$ (cf.~\\eqref{eq:ktheta}) in terms of the coefficient $G_{\\theta,n}f(p,t)$ (cf.~\\eqref{eq:constantterm}).\nBefore proceeding with this,  it will be convenient to set some useful notation first.\n\nLet $\\mu\\in \\text{Spec}(\\square)$, $f\\in \\cC^{2}(M)$ and $\\theta\\in (0,4\\pi]$. Define the functions $a_{\\theta,\\mu}^{+},a_{\\theta,\\mu}^{-}f\\colon M\\to \\C$ by\n\\begin{equation}\n\t\\label{eq:anquarter}\n\ta^{\\pm}_{\\theta,\\mu}f(p)= \\mp\\frac{(1\\mp\\nu)\\theta^{-1}\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s+2\\theta^{-1}\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s}{2\\nu e^{-\\frac{1\\pm\\nu}{2}}}\n\\end{equation}\nif $\\mu\\neq 1\/4$ and\n\\begin{equation}\n\t\\label{eq:aquarter}\n\ta^{\\pm}_{\\theta,1\/4}f(p)= \\frac{\\sqrt{e}\\bigl(\\theta^{-1}\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s\\mp 2\\theta^{-1}\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s\\bigr)}{2}\n\\end{equation}\nif $\\mu=1\/4$. When $\\mu\\neq 1\/4$, it holds \n\\begin{equation}\n\t\\label{eq:aestnquarter}\n\t\\norm{a^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{(1+|\\nu|)\\theta^{-1}\\int_0^{\\theta}\\norm{f}_{\\infty}\\text{d}s+2\\theta^{-1}\\int_0^{\\theta}\\norm{Xf}_{\\infty}\\text{d}s}{2e^{-1}|\\nu|}\\leq  \\frac{e(3+|\\nu|)}{2|\\nu|}\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\\end{equation}\nsince $\\norm{f}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{1}}$ and $\\norm{Xf}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{1}}$.\nIf $\\mu=1\/4$, similar estimates lead readily to\n\\begin{equation}\n\t\\label{eq:aestquarter}\n\t\\norm{a^{\\pm}_{\\theta,1\/4}f}_{\\infty}\\leq \\frac{3\\sqrt{e}}{2}\\norm{f}_{\\mathscr{C}^{1}}\\;.\n\\end{equation}\n\nWe are now in a position to start the proof of Theorem~\\ref{thm:case_Theta_eigenfn}, which will occupy the remainder of this section. We fix $\\theta\\in (0,4\\pi]$ and a function $f\\in \\cC^{2}(M)$ satisfying $\\Theta f=\\mu f$ and $\\Theta f=\\frac{i}{2}nf$ for some $\\mu \\in \\text{Spec}(\\square)$ and $n\\in \\Z$. For any $p \\in M$, the function $k_{f,\\theta}(p,\\cdot)\\colon \\R_{>0}\\to \\C$ we are interested in satisfies~\\eqref{eq:Cauchy} with initial conditions\n\\begin{equation*}\n\ty_1=\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s\\;,\\quad y'_1=\\frac{1}{\\theta}\\int_0^{\\theta}Xf\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s\\;.\n\\end{equation*} \n\nWe distinguish five cases as in the statement of Theorem~\\ref{thm:case_Theta_eigenfn}, that is, according to the value of the Casimir eigenvalue $\\mu$. Recall that $\\nu$ is the unique complex number in $\\R_{\\geq 0}\\cup i \\R_{>0}$ verifying $1-\\nu^{2}=4\\mu$.\n\n\\subsection{The case $\\mu>1\/4$}\n\\label{sec:caseabovequarter}\nSuppose $\\mu>1\/4$, so that $\\nu=i\\Im{\\nu}\\in i\\R_{>0}$. As follows from~\\eqref{eq:solutionnotquarter}, the solution to~\\eqref{eq:ode} with the prescribed initial conditions is given by\n\\begin{equation*}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)&=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(a^{+}_{\\theta,\\mu}f(p)+a^{-}_{\\theta,\\mu}f(p)-\\frac{2}{\\Im{\\nu}}\\int_1^{t}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(a^{-}_{\\theta,\\mu}f(p)-a^{+}_{\\theta,\\mu}f(p)-\\frac{2i}{\\Im{\\nu}}\\int_1^{t}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation*}\n   The functions\n   \\begin{equation*}\n    e^{-\\xi\/2}\\cos{(\\frac{\\Im{\\nu}}{2}\\xi)}G_{\\theta,n}f(p,\\xi),\\quad e^{-\\xi\/2}\\sin{(\\frac{\\Im{\\nu}}{2}\\xi)}G_{\\theta,n}f(p,\\xi) \n   \\end{equation*} \n    are integrable over the closed half-line $[1,+\\infty)$, as $G_{\\theta,n}f(p,\\cdot)$ is bounded thereon; we may therefore define functions $D^{+}_{\\theta,\\mu,n}f,D^{-}_{\\theta,\\mu,n}f\\colon M\\to \\C$ by setting\n\\begin{equation}\n\t\\label{eq:Dplusabovequarter}\n\tD^{+}_{\\theta,\\mu,n}f(p)=a^{+}_{\\theta,\\mu}f(p)+a^{-}_{\\theta,\\mu}f(p)-\\frac{2}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nand \n\\begin{equation}\n\t\\label{eq:Dminusabovequarter}\n\tD^{-}_{\\theta,\\mu,n}f(p)=a^{-}_{\\theta,\\mu}f(p)-a^{+}_{\\theta,\\mu}f(p)-\\frac{2i}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nfor every $p \\in M$. Then~\\eqref{eq:estabovequarter} is valid with \n\\begin{equation}\n\t\\label{eq:remainderabovequarter}\n\t\\begin{split}\n\t\t\\mathcal{R}_{\\theta,\\mu,n}f(p,t)&=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\int_t^{\\infty}\\frac{2}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\\\\n\t\t&+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\int_t^{\\infty}\\frac{2i}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;,\\quad t\\geq 1, \\;p \\in M.\n\t\\end{split}\n\\end{equation}\nLet us now estimate the uniform norms of $D^{\\pm}_{\\theta,\\mu,n}f$ and of $\\mathcal{R}_{\\theta,\\mu,n}f(\\cdot,t)$ for any $t\\geq 1$. From the explicit expressions in~\\eqref{eq:Dplusabovequarter} and~\\eqref{eq:Dminusabovequarter}, it follows at once that\n\\begin{equation}\n\t\\label{eq:kappamu}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}&\\leq \\norm{a^{+}_{\\theta,\\mu}f}_{\\infty}+\\norm{a^{-}_{\\theta,\\mu}}_{\\infty}+\\frac{2}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\sup_{p\\in M,\\;\\xi\\geq 1}|G_{\\theta,n}f(p,\\xi)|\\;\\text{d}\\xi\\\\\n\t\t&\\leq \\frac{1}{\\Im{\\nu}}\\biggl(e(3+\\Im{\\nu})+\\frac{4\\kappa_0(n^{2}+1)}{\\theta\\sqrt{e}}\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\\\\n\t\t&\\leq\\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\n\t\\end{split}\n\\end{equation}\napplying~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter} in the second inequality, with $\\kappa(\\mu)=\\frac{1}{\\Im{\\nu}}\\bigl(4e\\pi(3+\\Im{\\nu})+\\frac{4\\kappa_0}{\\sqrt{e}}\\bigr)$.\n\nThe remainder term defined in~\\eqref{eq:remainderabovequarter} is bounded from above by\n\\begin{equation*}\n\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq 2e^{-\\frac{t}{2}}\\sup_{p \\in M,\\;\\xi\\geq 1}|G_{\\theta,n}f(p,\\xi)|\\int_t^{\\infty}\\frac{2}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\leq\\frac{8\\kappa_0 (n^{2}+1)}{\\theta\\;\\Im{\\nu}}\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\n\\end{equation*}\nfor every $p\\in M$ and $t\\geq 1$, again relying on the upper bound in~\\eqref{eq:Gtheta}. \n\n\\smallskip\nUp to the regularity claims on the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$, which are the subject of Section~\\ref{sec:regularity}, the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(1) is complete.  \n\n\\subsection{The case $\\mu=1\/4$}\n\nSuppose $\\mu=1\/4$, whence $\\nu=0$. This time the solution to~\\eqref{eq:ode} with the given initial conditions has the expression (cf.~\\eqref{eq:solutionquarter}) \n\\begin{equation*}\n\n\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}\\biggl(a^{+}_{\\theta,1\/4}f(p)-\\int_1^{t}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\n\t+te^{-\\frac{t}{2}}\\biggl(a^{-}_{\\theta,1\/4}f(p)+\\int_1^{t} e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\\end{equation*}\nFollowing the steps carried out in Section~\\ref{sec:caseabovequarter} almost verbatim, define functions $D^{+}_{\\theta,1\/4,n}, D^{-}_{\\theta,1\/4,n}\\colon M\\to \\C$ via\n\\begin{equation}\n\t\\label{eq:Dquarter}\n\tD^{+}_{\\theta,1\/4,n}f(p)=a^{+}_{\\theta,1\/4}f(p)-\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi,\\quad D^{-}_{\\theta,1\/4,n}f(p)=a^{-}_{\\theta.1\/4}f(p)+\\int_1^{\\infty} e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nfor every $p \\in M$. Then~\\eqref{eq:estquarter} holds with \n\\begin{equation}\n\t\\label{eq:remainderquarter}\n\t\\mathcal{R}_{\\theta.1\/4,n}f(p,t)=e^{-\\frac{t}{2}}\\int_t^{\\infty}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi -te^{-\\frac{t}{2}}\\int_t^{\\infty}e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;, \\quad t\\geq 1,\\;p\\in M.\n\\end{equation}\nFrom~\\eqref{eq:Dquarter} we estimate, by virtue of~\\eqref{eq:Gtheta} and~\\eqref{eq:aestquarter}, \n\\begin{equation*}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,1\/4,n}f}_{\\infty}&\\leq \\frac{3\\sqrt{e}}{2}\\norm{f}_{\\mathscr{C}^{1}}+\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\\\\n\t\t&=\\biggl(\\frac{3\\sqrt{e}}{2}+\\frac{6\\kappa_0}{\\theta\\sqrt{e}}(n^{2}+1)\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\leq \\frac{\\kappa(1\/4)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we may choose $\\kappa(1\/4)=36\\pi \\sqrt{e}\\kappa_0$. Moreover, we deduce from~\\eqref{eq:remainderquarter} that, for every $p \\in M$ and $t\\geq 1$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta,1\/4,n}f(p,t)|&\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\biggl(e^{-\\frac{t}{2}}\\int_t^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi+te^{-\\frac{t}{2}}\\int_t^{\\infty}e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\biggr) \\\\ \n\t\t&=\\frac{4\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}(t+1)e^{-t}\\;.\n\t\\end{split}\n\\end{equation*}\nIn conjunction with Section~\\ref{sec:regularity}, this concludes the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(2).\n\n\\subsection{The case $0<\\mu<1\/4$} When $0<\\mu<1\/4$, we have $\\nu\\in (0,1)$. The solution to~\\eqref{eq:ode} is given, as in~\\eqref{eq:solutionnotquarter}, by\n\\begin{equation*}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)=&e^{-\\frac{1+\\nu}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&+e^{-\\frac{1-\\nu}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation*}\nSetting\n\\begin{equation}\n\t\\label{eq:Dbelowquarter}\n\t\\begin{split}\n\t\t&D^{+}_{\\theta,\\mu,n}f(p)=a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;, \\\\ &D^{-}_{\\theta,\\mu,n}f(p)=a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\t\\end{split}\n\\end{equation}\nand \n\\begin{equation*}\n\t\\mathcal{R}_{\\theta,\\mu,n}f(p,t)=\\frac{1}{\\nu}\\biggl(e^{-\\frac{1+\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi-e^{-\\frac{1-\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1+\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\n\\end{equation*}\nfor every $p \\in M$ and $t\\geq 1$, it is clear that~\\eqref{eq:estpositivebelowquarter} holds. As far as estimates on the supremum norm are concerned, we have\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}&\\leq \\frac{e(3+\\nu)}{2\\nu}\\norm{f}_{\\mathscr{C}^{1}}+\\frac{\\kappa_0}{\\theta\\nu}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_1^{\\infty}e^{-\\frac{1\\mp \\nu}{2}\\xi}\\;\\text{d}\\xi\\\\\n\t\t&=\\frac{1}{\\nu}\\biggl(\\frac{e(3+\\nu)}{2}+\\frac{\\kappa_0}{\\theta}\\frac{2}{1\\mp \\nu}e^{-\\frac{1\\mp \\nu}{2}}(n^{2}+1)\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\\\\n\t\t&\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\\end{split}\n\\end{equation*}\nfor $\\kappa(\\mu)=\\frac{2\\kappa_0e^{-\\frac{1-\\nu}{2}}}{\\nu(1-\\nu)}+\\frac{2e\\pi(3+\\nu)}{\\nu}$, and\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|&\\leq \\frac{\\kappa_0}{\\theta\\nu}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\biggl(e^{-\\frac{1+\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}\\;\\text{d}\\xi+e^{-\\frac{1-\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1+\\nu}{2}\\xi}\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&= \\frac{4\\kappa_0}{\\theta \\nu(1-\\nu)(1+\\nu)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;,\n\t\\end{split}\n\\end{equation*}\nwhich achieves the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(3) except for the regularity of the coefficients which is addressed separately in Section~\\ref{sec:regularity}.\n\n\\subsection{The case $\\mu= 0$} As $\\nu=1$ when $\\mu=0$,  equation~\\eqref{eq:solutionnotquarter} delivers the following expression for the solution to~\\eqref{eq:ode}:\n\\begin{equation}\n\t\\label{eq:kzero}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)&=a^{-}_{\\theta,0}f(p)+\\int_1^{t}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+e^{-t}\\biggl(a^{+}_{\\theta,0}f(p)-\\int_1^{t}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&= a^{-}_{\\theta,0}f(p)+\\int_1^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+\\biggl(-\\int_t^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\\\\n\t\t& +e^{-t}a^{+}_{\\theta,0}f(p)-e^{-t}\\int_1^{t}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nObserve that the term between parentheses in the last expression is infinitesimal as $t$ tends to infinity, so that $k_{f,\\theta}(p,t)$ has a well-defined limit, as $t$ tends to infinity, for every $p \\in M$.\nWe claim that\\footnote{The claim amounts to the qualitative equidistribution statement that circle-arc averages of $f$ converge to its spatial average with respect to the uniform measure. This has been shown by Margulis (for complete circles) in~\\cite{Margulis} via a thickening argument resting on the mixing properties of the geodesic flow. We prefer not to invoke Margulis' result here, and instead prove directly this special case of equidistribution using spectral considerations coupled with mixing.  }    \n\\begin{equation}\n\t\\label{eq:constanttermzero}\n\t\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)=\\int_{M}f\\;\\text{d}\\vol\n\\end{equation}\nfor any $p \\in M$. For a start, we show the equality in~\\eqref{eq:constanttermzero} holds on average with respect to the measure $\\vol$. Indeed, Fubini's theorem gives, for any $t>0$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)&=\\frac{1}{\\theta}\\int_{M}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\;\\text{d}\\vol(p)=\\frac{1}{\\theta}\\int_{0}^{\\theta}\\int_{M}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}\\vol(p)\\;\\text{d}s\\\\\n\t\t&=\\frac{1}{\\theta}\\int_0^{\\theta}\\int_{M}f\\;\\text{d}\\vol\\;\\text{d}s=\\int_{M}f\\;\\text{d}\\vol\\;,\n\t\\end{split}\n\\end{equation*}\nthe second-to-last equality following from the fact that the transformation $\\phi_t^{X}\\circ r_s\\colon M\\to M$ preserves the measure $\\vol$ for any $s,t\\in \\R$.  By dominated convergence,\n\\begin{equation}\n\t\\label{eq:sameaverage}\n\t\\int_{M}\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)=\\lim_{t\\to\\infty}\\int_{M}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)=\\lim_{t\\to\\infty}\\int_{M}f\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol\\;.\n\\end{equation}\n\nIn order to finish the proof of the claim, it remains to show that the limit $\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)$ does not depend on $p$. Choose a countable orthonormal basis $(u_{k})_{k\\in I}$ of $L^{2}(M)$ consisting of smooth eigenfunctions of the operator $\\Theta$ and containing a constant function $u_{k_0}$. If $\\Theta u_k=\\frac{i}{2}n_ku_k$ for $n_k\\in \\Z$, then $u_k\\circ r_{s}=e^{2\\pi i n_k s}u_k$ for every $s\\in \\R$ (cf.~Section~\\ref{sec:unitaryrepresentations}). Therefore, we can compute for every $k \\in I$ the $L^{2}$-inner product\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)\\;\\overline{u_k(p)}\\;\\text{d}\\vol(p)&=\\lim_{t\\to\\infty}\\int_{M}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\biggr) \\overline{u_k(p)}\\;\\text{d}p\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_{0}^{\\theta}\\int_{M}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\overline{u_k(p)}\\;\\text{d}p\\;\\text{d}s\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_0^{\\theta}\\int_{M}f\\circ\\phi_t^{X}(p)\\;\\overline{u_{k}}\\circ r_{-s}(p)\\;\\text{d}p \\;\\text{d}s\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_0^{\\theta}e^{-2\\pi i n_k s}\\int_{M}f\\circ\\phi_t^{X}(p)\\;\\overline{u_{k}(p)}\\;\\text{d}p \\;\\text{d}s\\\\\n\t\t&=\\frac{1}{\\theta}\\int_0^{\\theta}e^{-2\\pi i n_ks}\\text{d}s\\cdot \\lim_{t\\to\\infty}\\langle f\\circ \\phi_t^{X},u_k\\rangle\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we used, in successive order, the dominated convergence theorem, Fubini's theorem, invariance of the measure $\\vol$ under the transformation $r_{-s}$ and the definining property of $u_k$. Mixing of the geodesic flow $(\\phi_t^{X})_{t\\in \\R}$ on $M$ (cf.~\\cite[Cor.~2.3]{Bekka-Mayer}) delivers\n\\begin{equation*}\n\t\\lim_{t\\to\\infty}\\langle f\\circ \\phi_t^{X},u_k\\rangle =\\int_{M}f\\;\\text{d}\\vol\\int_{M}u_k\\;\\text{d}\\vol\n\t\\;.\n\\end{equation*}\n\nAs $u_k$ is orthogonal in $L^{2}(M)$ to the constant $u_{k_0}$ for any $k\\neq k_0$, the last expression vanishes for any such $k$. Therefore, we have just shown that the function $p\\mapsto \\lim_{t\\to\\infty}k_{f,\\theta}(p,t)$ is orthogonal to $u_k$ for every $k\\neq k_0$; necessarily, it must be constant. \n\n\\smallskip \n\nDefine now\n\\begin{equation*}\n\t\\mathcal{R}_{\\theta,0,n}f(p,t)=a_{\\theta,0}^{+}f(p)e^{-t}-\\int_t^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;;\n\\end{equation*}\nthen~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter} give\n\\begin{equation*}\n\t|R_{\\theta,0,n}f(p,t)|\\leq 2e \\norm{f}_{\\mathscr{C}^{1}}e^{-t}+\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_t^{\\infty}e^{-\\xi}\\;\\text{d}\\xi=\\frac{1}{\\theta}\\bigl(8e\\pi+\\kappa_0(n^{2}+1)\\bigr)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\n\\end{equation*}\nand, combining~\\eqref{eq:kzero} with~\\eqref{eq:constanttermzero}, we obtain\n\\begin{equation*}\n\tk_{f,\\theta}(p,t)=\\int_{M}f\\;\\text{d}\\vol+e^{-t}\\int_1^{t}-G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+\\mathcal{R}_{\\theta,0,n}f(p,t)\\;,\n\\end{equation*} \nwhich establishes Theorem~\\ref{thm:case_Theta_eigenfn}(4).\n\n\\subsection{The case $\\mu<0$}\nWe now turn to the case $\\mu<0$ or, equivalently, $\\nu>1$. The solution to~\\eqref{eq:ode} given in~\\eqref{eq:solutionnotquarter} becomes\n\\begin{equation}\n\t\\label{eq:kbelowzero}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)=& e^{\\frac{\\nu-1}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr) \\\\\n\t\t&+e^{-\\frac{\\nu+1}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\n\nIt follows that the quantity\n\\begin{equation*}\n\t\\begin{split}\n\t\te^{\\frac{\\nu-1}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)=&-e^{-\\frac{\\nu+1}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&-k_{f,\\theta}(p,t)\n\t\t+\\frac{e^{\\frac{\\nu-1}{2}t}}{\\nu}\\int_{t}^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\t\\end{split}\n\\end{equation*}\nis uniformly bounded in $t$, which forces \n\\begin{equation*}\n\ta^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi=0 \n\\end{equation*}\nfor every $p \\in M$. Therefore~\\eqref{eq:kbelowzero} results in\n\\begin{equation*}\n\tk_{f,\\theta}(p,t)=e^{-t}\\biggl(-\\frac{e^{\\frac{\\nu+1}{2}t}}{\\nu}\\int_t^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi +e^{-\\frac{\\nu-1}{2}t}a^+_{\\theta,\\mu}f(p)-\\frac{e^{-\\frac{\\nu-1}{2}t}}{\\nu}\\int_1^{t}e^{\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi \\biggr)\\;.\n\\end{equation*}\n\nWith the help of~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter}, we may estimate the three summands inside the parentheses. For the first, we have\n\\begin{equation*}\n\t\\biggl|\\frac{e^{\\frac{\\nu+1}{2}t}}{\\nu}\\int_t^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr|\\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu+1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\\end{equation*}\nwhile the second can be bounded as\n\\begin{equation*}\n\t\\bigl|e^{-\\frac{\\nu-1}{2}t}a^{+}_{\\theta,\\mu}f(p)\\bigr|\\leq \\frac{e(3+\\nu)}{2\\nu}\\norm{f}_{\\mathscr{C}^{1}}\\;;\n\\end{equation*}\nlastly,\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\biggl|\\frac{e^{-\\frac{\\nu-1}{2}t}}{\\nu}\\int_1^{t}e^{\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr| &\\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu-1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-\\frac{\\nu-1}{2}t}(e^{\\frac{\\nu-1}{2}t}-e^{\\frac{\\nu-1}{2}}) \\\\\n\t\t& \\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu-1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;.\n\t\\end{split}\n\\end{equation*}\n\nWe conclude that \n\\begin{equation*}\n\t|k_{f,\\theta}(p,t)|\\leq\\frac{1}{\\theta} \\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2 e\\pi(3+\\nu)}{\\nu}\\biggr)(n^{2}+1) \\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;,\n\\end{equation*}\nas desired. \n\n\\smallskip\nThis settles Theorem~\\ref{thm:case_Theta_eigenfn}(5).\n\n\n\n\n\n\n\n\n\n\n\\subsection{Regularity of the coefficients in the asymptotic expansion} \n\\label{sec:regularity}\nWe now turn to the examination of the regularity properties of the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$ appearing in the asymptotic expansion of $k_{f,\\theta}(p,t)$, as in Theorem~\\ref{thm:case_Theta_eigenfn}, for $f\\in \\mathscr{C}^{2}(M)$ satisfying $\\square f=\\mu f $ and $\\Theta f=\\frac{i}{2}nf$ for some $\\mu\\in \\text{Spec}(\\square)\\cap\\R_{>0}$ and $n\\in \\Z$. In so doing, we shall complete the proof of Theorem~\\ref{thm:case_Theta_eigenfn}.\n\nLet us fix $\\theta\\in (0,4\\pi]$ throughout this subsection. As follows readily from the definitions of the coefficients \\eqref{eq:anquarter}, \\eqref{eq:aquarter}, \\eqref{eq:Dplusabovequarter}, \\eqref{eq:Dminusabovequarter}, \\eqref{eq:Dquarter} and \\eqref{eq:Dbelowquarter}, it suffices to analyze the regularity of \n\\begin{equation*}\n\tk_{f,\\theta}(p,1)=\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s, \\; k'_{f,\\theta}(p,1)=\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s, \\;\\int_1^{\\infty}g(\\xi)G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation*}\nas functions of $p\\in M$, with $g(\\xi)$ being a function of the following forms:\n\\begin{equation}\n\t\\label{eq:g}\n\te^{-\\xi\/2},\\;\\xi e^{-\\xi\/2}, \\;e^{-\\xi\/2}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)},\\;e^{-\\xi\/2}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)},\\; e^{-\\frac{1\\pm\\nu}{2}\\xi}\\;(0<\\nu<1).\n\\end{equation}\n\nWe start with the following elementary lemma:\n\n\\begin{lem}\n\t\\label{lem:diffunderint}\n\tLet $\\theta$ be a positive real number. If $h\\colon M\\to \\C$ is of class $\\mathscr{C}^{1}$, then the function\n\t\\begin{equation*}\n\t\tp\\mapsto \\int_0^{\\theta}h\\circ r_s(p)\\;\\emph{d}s\\;, \\quad p\\in M\n\t\\end{equation*}\n\tis of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tFix a point $p_0\\in M$, and let $(\\partial_{x_i})_{i=1,2,3}$ be a local frame of the tangent bundle $TM$ around $p_0$. It suffices to prove that, for each $i=1,2,3$, the partial derivative \n\t\\begin{equation}\n\t\t\\label{eq:partialderivative}\n\t\tp\\mapsto\\partial_{x_i}|_p\\biggl(\\int_0^{\\theta}h\\circ r_s\\;\\text{d}s\\biggr)\n\t\\end{equation}\n\texists and is continuous in an open neighborhood of $p$. Upon passing to local smooth charts for $M$, the classical theorem of differentiation under the integral sign ensures the validity of the formal passage\n\t\\begin{equation*}\n\t\tp\\mapsto\\partial_{x_i}|_p\\biggl(\\int_0^{\\theta}h\\circ r_s\\;\\text{d}s\\biggr)=\\int_0^{\\theta}\\partial_{x_i}|_p(h\\circ r_s)\\;\\text{d}s\n\t\\end{equation*}\n\tprovided that there exists a positive real-valued function $\\varphi$ on $[0,\\theta]$, integrable with respect to the Lebesgue measure, such that $|\\partial_{x_i}|_q(h\\circ r_s)|\\leq \\varphi(s)$ for any $q$ in an open neighborhood of $p$. Notice that, by the dominated convergence theorem. this would yield continuity of the partial derivative in~\\eqref{eq:partialderivative} at the same time. The chain rule for the differential gives $\\mathrm{d}(h\\circ r_s)_q=(\\mathrm{d}h)_{r_s(q)}\\circ (\\mathrm{d}r_s)_q$ for any $q \\in M$ and $s\\in [0,\\theta]$. As follows readily from the explicit expression for $r_s$ in~\\eqref{eq:rotationflow} and direct computations, the operator norm\\footnote{Formally, we would need to specify a Riemannian metric on the compact manifold $M$. For the purposes of the proof however, only boundedness of the relevant quantities matters, so that any such metric would serve our goal (cf.~Remark~\\ref{rmk:equivalentmetrics}(1)).} of the linear operator $(\\mathrm{d}r_s)_q$ is uniformly bounded in $q$ and $s$, as the entries of any Jacobian matrix associated to it only involve finite linear combinations of the sine and cosine functions. Therefore, there exists a constant $C>0$ such that $|\\partial_{x_i}|_q(h\\circ r_s)|\\leq C\\norm{h}_{\\mathscr{C}^{1}}$ for any $s\\in [0,\\theta]$ and any $q$ in the domain of definition of the local frame $(\\partial_{x_i})_{i=1,2,3}$. The conclusion is thus achieved by setting $\\varphi$ to be constantly equal to $C\\norm{h}_{\\mathscr{C}^{1}}$. \n\\end{proof}\n\nAs $(\\phi_t^{X})_{t\\in \\R}$ is a smooth flow on $M$ and $f$ is of class $\\mathscr{C}^{2}$, the functions $f\\circ \\phi_1^{X},\\;Xf\\circ \\phi_1^{X}$ are of class $\\cC^{1}$. By virtue of Lemma~\\ref{lem:diffunderint}, the functions $p\\mapsto \\int_0^{\\theta}f\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s,\\;p\\mapsto \\int_0^{\\theta}Xf\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s$ are of class $\\cC^{1}$ on $M$. \n\nTherefore, it remains to deal with $\\int_1^{\\infty}g(\\xi)G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi$ as a function of $p \\in M$, $g$ being as in~\\eqref{eq:g}. Expanding out the expression from~\\eqref{eq:constantterm}, we obtain that it equals\n\\begin{equation*}\n\t\\begin{split} \\int_1^{\\infty}&g(\\xi)\\biggl(\\frac{n^{2}e^{-\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\int_0^{\\theta}f\\circ \\phi^{X}_\\xi\\circ r_s(p)\\text{d}s -\\frac{2e^{-\\xi}}{\\theta(1-e^{-2\\xi})}\\int_0^{\\theta}Xf\\circ \\phi^{X}_{\\xi}\\circ r_s(p)\\text{d}s\\\\\n\t\t&+\\frac{2ine^{-2\\xi}}{\\theta(1-e^{-2\\xi})^{2}}(f\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-f\\circ \\phi^{X}_{\\xi}(p))+\\frac{2}{\\theta (1-e^{-2\\xi})}(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p))\\biggr)\\;\\text{d}\\xi\\;.\n\t\\end{split}\n\\end{equation*}\n\nWe shall know treat the four summands separately. \n\n\\begin{lem}\n\t\\label{lem:regularterms}\n\tIf $g$ takes one of the forms in~\\eqref{eq:g}, the functions \n\t\\begin{align}\n\t\t\\label{eq:firstreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(\\int_0^{\\theta}f\\circ \\phi_{\\xi}^{X}\\circ r_s(p)\\;\\emph{d}s\\biggr)\\emph{d}\\xi\\;,\\\\\n\t\t\\label{eq:secondreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{1-e^{-2\\xi}}\\biggl(\\int_0^{\\theta}Xf\\circ \\phi_{\\xi}^{X}\\circ r_s(p)\\;\\emph{d}s\\biggr)\\emph{d}\\xi\\;, \\\\\n\t\t\\label{eq:thirdreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(f\\circ \\phi_{\\xi}^{X}\\circ r_\\theta(p)-f\\circ \\phi_{\\xi}^{X}(p)\\biggr)\\emph{d}\\xi\n\t\\end{align}\n\tare of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tThe proof proceeds along the same lines as the proof of Lemma~\\ref{lem:diffunderint}. The crucial point is that, for any point $q\\in M$ and any $\\xi\\geq 1$, the operator norm of the differential $(\\mathrm{d}\\phi_\\xi^{X})_{q}$ doesn't exceed (up to a constant factor depending only on the choice of a Riemannian metric on the tangent bundle $TM$) the quantity $e^{\\xi\/2}$, as direct computations allow to verify starting from the explicit expression of $\\phi_{\\xi}^{X}$ in~\\eqref{eq:geodesicflow}. As a consequence, there exists a constant $C>0$ such that\n\t\\begin{equation*}\n\t\t\\biggl|\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(\\int_0^{\\theta}\\partial_{x_i}|_q(f\\circ \\phi^{X}_{\\xi}\\circ r_s)\\;\\text{d}s\\biggr)\\biggr|\\leq C\\frac{|g(\\xi)|e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\int_0^{\\theta}\\norm{f}_{\\mathscr{C}^{1}}e^{\\frac{\\xi}{2}}\\;\\text{d}s=C\\norm{f}_{\\mathscr{C}^{1}}\\frac{|g(\\xi)|e^{-\\frac{\\xi}{2}}}{(1-e^{-2\\xi})^{2}}\n\t\\end{equation*}\n\tfor every $\\xi\\geq 1$ and $i=1,2,3$, where $(\\partial_{x_i})_{i=1,2,3}$ is a local frame of $TM$ around a given fixed point $p_0 \\in M$. Since the function $|g(\\xi)|e^{-\\xi\/2}\/(1-e^{-2\\xi})^{2}$ is integrable on the half-line $[1,\\infty)$, we deduce as in the proof of Lemma~\\ref{lem:diffunderint} that the function in~\\eqref{eq:firstreg} is of class $\\cC^{1}$ on $M$.\n\t\n\tThe same assertion for the remaining two functions in~\\eqref{eq:secondreg} and~\\eqref{eq:thirdreg} follows by a similar argument.  \n\\end{proof}\n\nWhat is left to investigate, up to multiplicative constants, is thus  the regularity of the function\n\\begin{equation}\n\t\\label{eq:holdercontinuousterm}\n\tp\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p)\\bigr) \\text{d}\\xi\n\\end{equation}\non the manifold $M$. As we shall presently see, the latter depends on the function $g$.\n\n\\begin{lem}\n\t\\label{lem:moreregular}\n\tIf $g(\\xi)=e^{-\\frac{1+\\nu}{2}\\xi}$, then the function in~\\eqref{eq:holdercontinuousterm} is of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tIt suffices to argue as in the proof of Lemma~\\ref{lem:regularterms} observing that, when $0<\\nu<1$, the function\n\t\\begin{equation*}\n\t\t\\frac{e^{-\\frac{1+\\nu}{2}\\xi}}{1-e^{-2\\xi}}\\;e^{\\frac{\\xi}{2}}=\\frac{e^{-\\frac{\\nu}{2}\\xi}}{1-e^{-2\\xi}}\n\t\\end{equation*}\n\tis integrable on the half-line $[1,\\infty)$.\n\\end{proof}\n\nIt is straightforward to realize that the same argument does not carry over to the other possible forms of $g(\\xi)$ listed in~\\eqref{eq:g}. For those remaining cases, we instead establish H\\\"{o}lder-continuity of the function in~\\eqref{eq:holdercontinuousterm} by a different argument.\n\nFix a Riemannian metric $g$ on the connected manifold $M$, inducing a Riemannian distance function $d$. The choice is immaterial for our purposes, as pointed out in Remark~\\ref{rmk:equivalentmetrics}. We start with the following well-known properties of the flows $(\\phi_t^{X})_{t\\in \\R},\\;(r_{s})_{s\\in \\R}$.\n\n\\begin{lem}\n\t\\label{lem:distanceflows}\n\tThere exist real constants $C_{X,d},C_{\\Theta,d}$, depending only on $d$, such that, for any pair of points $p,q\\in M$, it holds \n\t\\begin{equation}\n\t\t\\label{eq:geodesicdistance}\n\t\td(\\phi_t^{X}(p),\\phi^{X}_t(q))\\leq C_{X,d}\\;e^{|t|}d(p,q)\n\t\\end{equation}\n\tfor every $t\\in \\R$ and\n\t\\begin{equation}\n\t\t\\label{eq:rotationdistance}\n\t\td(r_{s}(p),r_s(q))\\leq C_{\\Theta,d}\\;d(p,q) \n\t\\end{equation}\n\tfor every $s\\in \\R$.\n\\end{lem}\n\\begin{proof}\n\tBy compactness of $M$, we have the freedom to prove the lemma for a judicious choice of $d$. To profit most from the algebraic description of the flows $(\\phi_t)_{t\\in \\R}$ and $(r_{s})_{s\\in \\R}$, we fix a left-invariant Riemannian metric $g_{\\SL_2(\\R)}$ on the Lie group $\\SL_2(\\R)$ and let $d$ be the Riemannian distance determined by the unique Riemannian metric $g$ on $M$ for which the projection map $(\\SL_2(\\R),g_{\\SL_2(\\R)})\\to (M,g)$ is a Riemannian submersion (cf.~\\cite[Cor.~2.29]{Lee-Riemannian}) or, equivalently for a covering map, a local isometry. As $M$ is compact, we can choose a finite open cover $(\\tilde{U_{i}})_{i\\in I}$ of $M$ and a collection $\\mathcal{U}=(U_i)_{i\\in I}$ of open subsets of $\\SL_2(\\R)$ such that, for any $i \\in I$, the restriction of the projection to $U_i$ is an isometry from $U_i$ onto $\\tilde{U_i}$. \n\t\n\tThe distance $d_{\\SL_2(\\R)}$ induced by $g_{\\SL_2(\\R)}$ is locally equivalent to the distance induced by the operator norm $\\norm{\\cdot}_{\\text{op}}$ on the vector space of $2\\times 2$ real matrices corresponding to the Euclidean norm on $\\R^{2}$ (cf.~\\cite[Lem.~9.12]{Einsiedler-Ward}): for every $g\\in \\SL_2(\\R)$, there exists an open neighborhood  $W_g$ of $g$ and a constant $C_{d,g}$ such that \n\t\\begin{equation*}\n\t\tC_{d,g}^{-1}\\norm{x-y}_{\\text{op}}\\leq d_{\\SL_2(\\R)}(x,y)\\leq C_{d,g}\\norm{x-y}_{\\text{op}}\n\t\\end{equation*}\n\tfor any $x,y\\in W_g$. Upon restricting the $U_i$'s (and the $\\tilde{U_i}$'s) if necessary, we may assume that each $\\tilde{U_i}$ is contained in $W_{g_i}$ for some $g_i \\in \\SL_2(\\R)$; define $C_{d}$ to be the supremum of the $C_{d,g_i},\\;i \\in I$. We also select a second finite open cover $\\mathcal{V}=(V_j)_{j\\in J}$ in such a way that the closure of each $V_j$ is compact and contained in some $U_{i(j)}$. Observe that, for every $j \\in J$, the function $\\tau_j\\colon \\overline{V_j}\\to (0,\\infty]$ defined as the first exit time\n\t\\begin{equation*}\n\t\t\\tau_j(p)=\\inf\\{t>0:\\phi_t^{X}(p)\\notin U_{i(j)} \\}\\;, \\quad  p\\in V_{j}\n\t\\end{equation*}\n\tis continuous, and as such attains a strictly positive minimal value $t_{j}$. Set $t_0=\\inf_{j\\in J}t_j$ and let $\\delta_{\\mathcal{V}}$ be a Lebesgue number for the covering $\\mathcal{V}$ (cf.~\\cite[Lem.~27.5]{Munkres}).\n\t\n\tConsider now two points $p,q\\in M$, and suppose first that $d(p,q)\\geq\\delta_{\\mathcal{V}}$; then \n\t\\begin{equation}\n\t\t\\label{eq:largedistance}\n\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\text{diam}_d(M)\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)d(p,q)\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)e^{|t|}d(p,q)\n\t\\end{equation}\n\tfor every $t\\in \\R$, where $\\text{diam}_{d}(M)=\\sup_{p',q'\\in M}d(p,q)$ is the diameter of $M$ with respect to the distance $d$.\n\t\n\tNow assume that $d(p,q)<\\delta_{\\mathcal{V}}$ so that $p$ and $q$ both lie in some $V_j$. Choose representatives $x,y$ of $p,q$, respectively, inside $\\tilde{U}_{i(j)}$; then, for every $0\\leq t<t_0$, we have\n\t\\begin{equation}\n\t\t\\label{eq:smalldistance}\n\t\t\\begin{split}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))&\\leq d_{\\SL_2(\\R)}(x\\exp{tX},y\\exp{tX})\\leq C_{d}\\norm{x\\exp{tX}-y\\exp{tX}}_{\\text{op}}\\\\\n\t\t\t&\\leq C_{d}\\norm{x-y}_{\\text{op}}\\norm{\\exp{tX}}_{\\text{op}}\\leq C_{d}^{2}\\;d_{\\SL_2(\\R)}(x,y) e^{|t|\/2}\\\\\n\t\t\t&\\leq C_d^{2}e^{t}d_{\\SL_2(\\R)}(x,y)=C_d^{2}e^{t}d(p,q)\\;,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere we used the fact that $p,q,\\phi_t^{X}(p),\\phi_t^{X}(q)$ all belong to $U_{i}^{(j)}$. If now $t_0\\leq t<2 t_0$, then we distinguishes two cases.\n\t\\begin{itemize}\n\t\t\\item If $d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))\\geq \\delta_{\\mathcal{V}}$, then~\\eqref{eq:largedistance} applies giving\n\t\t\\begin{equation*}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)e^{t-t_0}d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))\\leq C_{d}^{2}\\delta_{\\mathcal{V}}^{-1}\\text{diam}_d(M)e^{t}d(p,q)\\;.\n\t\t\\end{equation*}\n\t\tThis estimate is actually valid for any $t\\geq t_0$.\n\t\t\\item If $d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))< \\delta_{\\mathcal{V}}$, then the computations in~\\eqref{eq:smalldistance} are valid for the given $t$ and yield\n\t\t\\begin{equation*}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq C_d^{2}e^{t}d(p,q)\\;.\n\t\t\\end{equation*} \n\t\\end{itemize}\n\tSubdividing the half-line $\\R_{\\geq 0}$ into the intervals $[kt_0,(k+1)t_0),\\;k\\in \\N$, and arguing as above on each of them, we conclude that\n\t\\begin{equation*}\n\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\sup\\{ C_d^{2}, C_d^{2}\\delta_{\\mathcal{V}}^{-1}\\text{diam}_d(M)\\}e^{t}d(p,q)\n\t\\end{equation*}\n\tfor any $t>0$. \n\t\n\tThe same analysis can be performed, with the appropriate modifications, for times $t<0$. This shows~\\eqref{eq:geodesicdistance}.\n\t\n\tThe inequality in~\\eqref{eq:rotationdistance} is taken care of in an entirely analogous fashion, observing that $\\norm{\\exp(s\\Theta)}_{\\text{op}}=1$ for every $s\\in \\R$.\n\\end{proof}\n\n\nLet us now fix two points $p,q \\in M$. As in the previous proof, we denote by $\\text{diam}_d(M)$ the diameter of $M$; with\n\\begin{equation*} \\text{Lip}_d(Uf)=\\sup\\limits_{p',q'\\in M,\\; p'\\neq q'}\\frac{|Uf(p')-Uf(q')|}{d(p,q)}\n\\end{equation*}\nwe indicate the Lipschitz constant of the function $Uf$ with respect to the distance $d$.\n\nWe may then estimate\n\\begin{equation}\n\t\\label{eq:Holdercontinuity}\n\t\\begin{split}\n\t\t&\\biggl| \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p)\\bigr) \\text{d}\\xi - \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(q)-Uf\\circ \\phi^{X}_{\\xi}(q)\\bigr) \\text{d}\\xi \\biggr|\\\\\n\t\t&\\leq \\int_1^{\\infty}\\frac{|g(\\xi)|}{1-e^{-2\\xi}}\\bigl(|Uf\\circ \\phi_{\\xi}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(q)|+|Uf\\circ \\phi^{X}_{\\xi}(p)-Uf\\circ \\phi^X_{\\xi}(q)|\\bigr)\\text{d}\\xi\\\\\n\t\t&\\leq (1-e^{-2})^{-1}\\text{Lip}(Uf) \\int_1^{\\infty}|g(\\xi)|\\bigl(\\min\\{\\text{diam}_d(M),C_{X,d}\\;e^{\\xi}d(r_{\\theta}(p),r_{\\theta}(q) \\}+\\\\ &\\quad +\\inf\\{\\text{diam}_d(M),C_{X,d}\\;e^{\\xi}d(p,q) \\}\\bigr)\\text{d}\\xi\\\\\n\t\t&\\leq 2(1-e^{-2})^{-1}\\text{Lip}(Uf)\\int_1^{\\infty}|g(\\xi)|\\inf\\{\\text{diam}_d(M),Cd(p,q)e^{\\xi} \\}\\text{d}\\xi\\;,\n\t\\end{split}\n\\end{equation}\nwhere $C=C_{X,d}\\sup\\{1,C_{\\Theta,d} \\}$.\n\n\\smallskip\nThe following elementary estimates allow to finalize the argument.\n\\begin{lem}\n\t\\label{lem:integralbound}\n\tLet $r,K\\in \\R_{>0},\\;a\\in (0,1)$. Then \n\t\\begin{equation}\n\t\t\\label{eq:firstboundintegral}\n\t\t\\int_1^{\\infty}e^{-a\\xi}\\inf\\{K,re^{\\xi} \\} \\emph{d}\\xi\\leq \\frac{1}{a(1-a)K^{a-1}}\\;r^{a}\\;.\n\t\\end{equation}\n\tFurthermore,\n\t\\begin{equation}\n\t\t\\label{eq:secondboundintegral}\n\t\t\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\inf\\{K,re^{\\xi} \\}\\emph{d}\\xi= 4\\sqrt{K}r^{\\frac{1}{2}}(\\log{k}+\\log{r}^{-1})\\;.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tIt suffices to split the integral as\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\int_1^{\\infty}e^{-a\\xi}\\inf\\{K,re^{\\xi} \\} \\text{d}\\xi&=\\int_1^{\\log{\\frac{K}{r}}}e^{-a\\xi}\\cdot re^{\\xi} \\;\\text{d}\\xi+\\int_{\\log{\\frac{K}{r}}}^{\\infty}e^{-a\\xi}\\cdot K  \\;\\text{d}\\xi\\\\\n\t\t\t&=\\frac{r}{1-a}\\biggl(\\biggr(\\frac{r}{K}\\biggr)^{a-1}-e^{1-a}\\biggr)+\\frac{K}{a}\\biggl(\\frac{r}{K}\\biggr)^{a}\\\\\n\t\t\t&\\leq \\frac{r^{a}}{(1-a)K^{a-1}}+\\frac{r^{a}}{aK^{a-1}}= \\frac{1}{a(1-a)K^{a-1}}\\;r^{a} \\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\twhich delivers the inequality in~\\eqref{eq:firstboundintegral}. Analogous computations allow to establish~\\eqref{eq:secondboundintegral}.\n\\end{proof}\n\nCombining Lemmata~\\ref{lem:regularterms},~\\ref{lem:moreregular},~\\ref{lem:integralbound} together with the estimate in~\\eqref{eq:Holdercontinuity} and the explicit expressions for the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$ in~\\eqref{eq:Dplusabovequarter},~\\eqref{eq:Dminusabovequarter},~\\eqref{eq:Dquarter},~\\eqref{eq:Dbelowquarter} we deduce that:\n\\begin{itemize}\n\t\\item when $\\mu>1\/4$, $D^{\\pm}_{\\theta.\\mu,n}f$ are H\\\"{o}lder continuous with H\\\"{o}lder exponent $1\/2$;\n\t\\item when $\\mu=1\/4$, $D^{+}_{\\theta,1\/4,n}f$ and $D^{-}_{\\theta,1\/4,n}$ are H\\\"{o}lder continuous, the latter with H\\\"{o}lder exponent $1\/2$, while the former with H\\\"{o}lder exponent $1\/2-\\varepsilon$ for every $\\varepsilon>0$;\n\t\\item when $0<\\mu<1\/4$, $D^{+}_{\\theta,\\mu,n}f$ is H\\\"{o}lder continuous with H\\\"{o}lder exponent $\\frac{1-\\nu}{2}$, while $D^{-}_{\\theta,\\mu,n}f$ is of class $\\cC^{1}$ on $M$.\n\\end{itemize}\n\nThe proof of Theorem~\\ref{thm:case_Theta_eigenfn} is achieved.\n\n\\section{Asymptotics for arbitrary functions}\n\\label{sec:arbitraryfn}\n\nThe bulk of this section is devoted to the deduction of Theorem~\\ref{thm:mainexpandingtranslates}, which addresses the asymptotic equidistribution rate of sufficiently regular observables on $M$ not subject to any eigenvalue condition, from the special case of joint eigenfunctions of $\\square$ and $\\Theta$ phrased in Theorem~\\ref{thm:case_Theta_eigenfn}. The argument is crucially based upon the orthogonal decompositions of Sobolev spaces into joint eigenspaces of $\\square$ and $\\Theta$, which is recalled in detail in Section~\\ref{sec:unitaryrepresentations}. We then proceed by proving Theorem~\\ref{thm:mainarbitrarytranslates}, concerning the asymptotic behaviour of arbitrary translates of compact orbits inside $M$; in light of the classical Cartan decomposition of the Lie group $\\SL_2(\\R)$, the result follows from Theorem~\\ref{thm:mainexpandingtranslates} in a fairly straightforward manner. Along the way, we shall also clarify the steps needed to derive, from those two main results, Corollaries~\\ref{cor:effective} and~\\ref{cor:shrinkingarcs}.\n\n\\subsection{Sum estimates on Sobolev norms of eigenfunctions}\n\\label{sec:sumestimates}\n\nBefore proceeding with the proof of Theorem~\\ref{thm:mainexpandingtranslates}, we collect in this subsection a few  estimates relating sums of norms of Sobolev eigenfunctions with a higher-order Sobolev norm of the sum of such functions, which will prove to be instrumental in the sequel. The rationale for those is the fact that the Hilbert-sum decompositions in Section~\\ref{sec:unitaryrepresentations} only provide, by Bessel's inequality (cf.~\\cite[2,~XXIII,~6;~14]{Schwartz}), estimates on the sum of squares of the components' norms, while our approach necessitates bounds on the $\\ell^{1}$-norm (see Section~\\ref{sec:proofexpandingtranslates}).   \n\nNotation is as in Section~\\ref{sec:unitaryrepresentations}. \n\n\\begin{lem}\n\t\\label{lem:finitespectralconstant}\n\tLet $k$ be a natural number. \n\t\\begin{enumerate}\n\t\t\\item The infinite series \n\t\t\\begin{equation}\n\t\t\t\\label{eq:serieseigenvalues}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\n\t\t\\end{equation}\n\t\tis summable if and only if $k\\geq 2$.\n\t\t\\item The infinite series \n\t\t\\begin{equation}\n\t\t\t\\label{eq:serieseigenvaluessecond}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)} \\frac{(n^{2}+1)^{2}}{(1+\\mu+\\frac{n^2}{2})^{k}}\n\t\t\\end{equation}\n\t\tis summable if and only if $k\\geq 3$.\n\t\\end{enumerate}\n\\end{lem}\n\nObserve that the series in~\\eqref{eq:serieseigenvalues} and~\\eqref{eq:serieseigenvaluessecond} consist of nonnegative real numbers: see Lemma~\\ref{lem:diffSobolevnorms}.\n\\begin{proof}\n\tIt is convenient to examine separately the convergence properties of \n\t\\begin{equation}\n\t\t\\label{eq:negativeandpositive}\n\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{\\geq 0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\\;\\text{  and  } \\;\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{<0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\\;.\n\t\\end{equation}\n\tWe know (cf.~Section~\\ref{sec:unitaryrepresentations}) that negative eigenvalues of the Casimir operator are of the form $\\mu_m=-m(m+2)\/4$ for $m$ ranging over the set of positive natural numbers; therefore\n\t\\begin{equation}\n\t\t\\label{eq:negativecasimir}\n\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{<0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}=\\sum_{m \\in \\N^{\\ast}}\\sum_{n\\in I(\\mu_m)} \\frac{1}{(1-\\frac{m(m+2)}{4}+\\frac{n^2}{2})^{k}}\\;,\n\t\\end{equation}\n\twhich has the same convergence properties of the series\n\t\\begin{equation*}\n\t\t\\sum_{(m,n)\\in \\Z^{2}}\\frac{1}{(1+m^2+n^{2})^{k}}\\;.\n\t\\end{equation*}\n\tBy comparison with the integral\n\t\\begin{equation*}\n\t\t\\int_{\\R^{2}}\\frac{1}{(1+x^2+y^2)^{k}}\\;\\text{d}x\\text{d}y\\;,\n\t\\end{equation*}\n\twhich is convergent if and only if $k\\geq 2$, as it is well-known, we infer that:\n\t\\begin{enumerate}\n\t\t\\item the series in~\\eqref{eq:serieseigenvalues} cannot converge if $k=1$;\n\t\t\\item the series in~\\eqref{eq:negativecasimir} converges for any $k\\geq 2$.\n\t\\end{enumerate}\n\tAs to the first summation in~\\eqref{eq:negativeandpositive}, we may now suppose $k\\geq 2$ and appeal to the Weyl law  for the positive eigenvalues of the Casimir operator (see Theorem~\\ref{thm:Weyllaw}), which we list in increasing order as $\\mu^{(p)}_0=0<\\mu_1^{(p)}<\\cdots<\\mu^{(p)}_{m}<\\cdots$, without multiplicity. Recall that $\\text{area}(S)$ is the volume of the surface $S=\\Gamma\\bsl \\Hyp$ with respect to the hyperbolic area measure. Choose a real number $c>\\text{area}(S)\/4\\pi$; then, there exists $R_0\\in \\R_{>0}$ such that $\\mu^{(p)}_{m}>m\/c$ for any integer $m>cR_0$. On the one hand, the quantity\n\t\\begin{equation*}\n\t\t\\sum_{0\\leq m\\leq cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\mu_m^{(p)}+\\frac{n^2}{2})^{k}}\n\t\\end{equation*}\n\tis a finite sum of converging series. On the other hand, \n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\mu_m^{(p)}+\\frac{n^2}{2})^{k}}&<\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\frac{m}{c}+\\frac{n^2}{2})^{k}}\\\\\n\t\t\t&\\leq \\sum_{m>cR_0}\\frac{1}{(1+\\frac{m}{c})^{k}}+\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})\\setminus\\{0\\}}\\frac{1}{(1+\\frac{m}{c}+\\frac{n^2}{2})^{k}}\\;.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThe first summand in the last expression converges obviously whenever $k\\geq 2$, and so does the second by comparison with the integral\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\int_{cR_0}^{\\infty}\\int_0^{\\infty}\\frac{1}{\\bigl(1+\\frac{x}{c}+\\frac{y^2}{2}\\bigr)^{k}}\\;\\text{d}y\\;\\text{d}x&=\\int_{cR_0}^{\\infty}\\frac{1}{\\bigl(1+\\frac{x}{c}\\bigr)^{k}}\\int_0^{\\infty}\\frac{\\sqrt{2}\\bigl(1+\\frac{x}{c}\\bigr)^{1\/2}}{(1+u^{2})^{k}}\\;\\text{d}u\\;\\text{d}x\\\\\n\t\t\t&=\\biggl(\\int_{cR_0}^{\\infty}\\frac{\\sqrt{2}}{\\bigl(1+\\frac{x}{c}\\bigr)^{k-1\/2}}\\;\\text{d}x\\biggr)\\biggl(\\int_0^{\\infty}\\frac{1}{(1+u^{2})^{k}}\\;\\text{d}u\\biggr)<\\infty\\;.\n\t\t\\end{split}\n\t\\end{equation*} \n\tAs to the assertion for the series in~\\eqref{eq:serieseigenvaluessecond}, it follows readily by running the argument above with the appropriate modifications.\n\\end{proof}\n\nLeveraging the estimates in Lemma~\\ref{lem:finitespectralconstant}, we are now in a position to show:\n\n\\begin{prop}\n\t\\label{prop:L1bound}\n\tLet $k\\geq 2$ be an integer. There exists a constant $C_{\\emph{Spec},k}>0$ depending only on $k$ and on the spectrum of the Laplace-Beltrami operator on the hyperbolic surface $S$, such that the following holds. Let $s$ be a positive real number, $f$ a function in $W^{s+k}(H)$; for any $\\mu \\in \\emph{Spec}(\\square)$ and $n\\in I(\\mu)$, let $f_{\\mu,n}$ be the orthogonal projection of $f$, with respect to the inner product in $W^{s+k}(H)$, onto the closed subspace $W^{s+k}(H_{\\mu,n})$. Then\n\t\\begin{equation}\n\t\t\\label{eq:L1bound}\n\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}\\leq C_{\\emph{Spec},k}\\norm{f}_{W^{s+k}}\\;.\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tRecall from Lemma~\\ref{lem:diffSobolevnorms} that, for any $\\mu \\in \\text{Spec}(\\square)$ and $n\\in I(\\mu)$,\n\t\\begin{equation*}\n\t\t\\norm{f_{\\mu,n}}_{W^{s}}^2=\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{-k}\\norm{f_{\\mu,n}}_{W^{s+k}}^2\\;.\n\t\\end{equation*}\n\tUsing the Cauchy-Schwartz inequality, we get\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}&=\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{-k\/2}\\norm{f_{\\mu,n}}_{W^{s+k}}\\\\\n\t\t\t&\\leq \\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{k}}\\biggr)^{1\/2}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s+k}}^{2}\\biggr)^{1\/2}\\;.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThe inequality in~\\eqref{eq:L1bound} is thus a consequence of Parseval's identity (cf.~\\cite[2,~XXIII,~6;~17]{Schwartz}) \n\t\\begin{equation*}\n\t\t\\norm{f}_{W^{s+k}}^{2}=\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s+k}}^{2}\\;,\n\t\\end{equation*}\n\twhere we define the constant $C_{\\text{Spec},k}$ as \n\t\\begin{equation*}\n\t\tC_{\\text{Spec},k}=\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{k}}\\biggr)^{1\/2}\\;,\n\t\\end{equation*} \n\twhich is finite by Lemma~\\ref{lem:finitespectralconstant} and satisfies the dependence properties claimed in the statement (cf.~Sections~\\ref{sec:unitaryrepresentations},~\\ref{sec:Sobolev}).\n\\end{proof}\n\n\\subsection{Equidistribution of expanding translates}\n\\label{sec:proofexpandingtranslates}\n\nWe are now ready to prove Theorem~\\ref{thm:mainexpandingtranslates}.\nLet $\\theta$ be a real parameter in the interval $(0,4\\pi]$, $s$ a real number satisfying $s>11\/2$, $f$ a function in $W^{s}(M)$. Keeping with the notation introduced in the foregoing subsection, we denote by $f_{\\mu,n}\\in W^{s}(H_{\\mu,n})$ the orthogonal projection of $f$ onto $W^{s}(H_{\\mu,n})$, for any Casimir eigenvalue $\\mu$ and any $n\\in I(\\mu)$. In what follows, the equivalence classes $f$ and $f_{\\mu,n}$ are identified with their unique\\footnote{Uniqueness is a result of the fact that the uniform measure $\\vol$ on $M$ is fully supported.} continuous representatives. The asymptotic expansion in~\\eqref{eq:asymptoticgeneral} will result from the sum of the contributions of each component $f_{\\mu,n}$, which are provided by Theorem~\\ref{thm:case_Theta_eigenfn}; we now expose the details.\n\nChoose a real parameter $s'$ satisfying $3\/2<s'\\leq s-2$;\nthe Sobolev Embedding Theorem (Theorem~\\ref{thm:Sobolevembedding}) gives the bound $\\norm{f_{\\mu,n}}_{\\infty}\\leq C_{0,s'}\\norm{f_{\\mu,n}}_{W^{s'}}$ for any $\\mu$ and $n$ as before, for some constant $C_{0,s'}>0$ depending only on $s'$ and on the manifold $M$.\nFor any point $q \\in M$, we estimate\n\\begin{equation}\n\t\\label{eq:uniformbound}\n\t\\begin{split}\n\t\t\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}|f_{\\mu,n}(q)|&\\leq \\sum_{n\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{\\infty}\\leq C_{0,s'}\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s'}}\\\\\n\t\t\\leq & C_{0,s'}C_{\\text{Spec},2}\\norm{f}_{W^{s'+2}}\\;,\n\t\\end{split}\n\\end{equation}\nthe last inequality being given by Proposition~\\ref{prop:L1bound}. Select now a base point $p \\in M$, which will remain fixed until the end of this subsection. By virtue of~\\eqref{eq:uniformbound}, the dominated convegence theorem for infinite series yields\n\\begin{equation}\n\t\\label{eq:sumintegral}\n\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\int_0^{\\theta}f_{\\mu,n}\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\;.\n\\end{equation}\nObserve that, since $s>11\/2$, the components $f_{\\mu,n}$ are of class $\\cC^{2}$ by the Sobolev Embedding Theorem (Theorem~\\ref{thm:Sobolevembedding}); to each summand on the right-hand side of~\\eqref{eq:sumintegral}, we may thus apply Theorem~\\ref{thm:case_Theta_eigenfn}, which delivers on a formal level the equality\\footnote{Notice that $\\int_{M}f_{\\mu,n}\\;\\text{d}\\vol=0$ for every Casimir eigenvalue $\\mu\\neq 0$ and every $n\\in I(\\mu)$, as $f_{\\mu,n}$ is orthogonal to the joint eigenspace $H_{0,0}$ which contains the constant functions. For the same reason, $\\int_{M}f_{0,n}\\;\\text{d}\\vol=0$ for every $n\\in I(0)\\setminus\\{0\\}$. Therefore, dominated convergence gives $\\int_{M}f_{0.0}\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol$.}\n\\begin{equation}\n\t\\label{eq:fullexpansion}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta}&\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\int_{M}f\\;\\text{d}\\vol\\\\ \n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu\\geq 1\/4}e^{-\\frac{t}{2}}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(\\sum_{n\\in I(\\mu)}D^{+}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(\\sum_{n\\in I(\\mu)}D^{-}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu< 1\/4}e^{-\\frac{1+\\nu}{2}t}\\biggl(\\sum_{n\\in I(\\mu)}D^{+}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)+e^{-\\frac{1-\\nu}{2}t}\\biggl(\\sum_{n\\in I(\\mu)}D^{-}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr) \\\\\n\t\t&+ \\varepsilon_0\\biggl(e^{-\\frac{t}{2}}\\biggl(\\sum_{n\\in I(1\/4)}D^{+}_{\\theta,1\/4,n}f_{1\/4,n}(p)\\biggr)+te^{-\\frac{t}{2}}\\biggl(\\sum_{n\\in I(1\/4)}D^{-}_{\\theta,1\/4,n}f_{1\/4,n}(p)\\biggr)\\biggr)\\\\\n\t\t&+ \\mathcal{R}_{\\theta,\\text{pos}}f(p,t)+ e^{-t}\\mathcal{M}_{\\theta,0}f(p,t)+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t) +\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\n\t\\end{split}\n\\end{equation}\nfor every $t\\geq 1$,\nwhere $\\varepsilon_0$ is defined in~\\eqref{eq:epszero}, $G_{\\theta,n}f_{0,n}$ is as in~\\eqref{eq:constantterm} and the quantities\n\\begin{equation*}\n \\mathcal{R}_{\\theta,\\text{pos}}f(p,t),\\quad\\mathcal{M}_{\\theta,0}f(p,t),\\quad\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\n\\end{equation*} \n are defined as follows:\n\\begin{align}\n\t\\label{eq:remainderpos}\n\t&\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)=\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>0}\\;\\sum_{n\\in I(\\mu)}\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\\\\\n\t\\label{eq:remaindernull}\n\t&\\mathcal{M}_{\\theta,0}f(p,t)=\\sum_{n\\in I(0)}\\int_1^{t}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi \\;,\\\\\n\t\\label{eq:remainderneg}\n\t&\\mathcal{R}_{\\theta,\\text{d}}f(p,t)=\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu<0}\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\int_0^{\\theta}f_{\\mu,n}\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\;.\n\\end{align}\nThe equality in~\\eqref{eq:asymptoticgeneral} would follow directly from~\\eqref{eq:fullexpansion} by defining\n\\begin{equation}\n\t\\label{eq:defDthetamu}\n\tD^{\\pm}_{\\theta,\\mu}f(p)=\\sum_{n\\in I(\\mu)}D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\;, \\quad p\\in M,\\; \\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}\n\\end{equation} \nand\n\\begin{equation}\n\t\\label{eq:remainder}\n\t\\mathcal{R}_{\\theta}f(p,t)=\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)+e^{-t}\\mathcal{M}_{\\theta,0}f(p,t)+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t)+\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\\;,\\;p\\in M,\\;t\\geq 1.\n\\end{equation}\nIt is left to show that all the infinite sums we are considering with a formal meaning are actually convergent.\n\nLet us begin by examining the sums in~\\eqref{eq:defDthetamu}. Fix $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$; for any $p \\in M$ and $n\\in I(\\mu)$, we have from Theorem~\\ref{thm:case_Theta_eigenfn} that\n\\begin{equation*}\n\t|D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}(p)|\\leq \\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f_{\\mu,n}}_{\\mathscr{C}^{1}}\\;.\n\\end{equation*}\nChoose now a real parameter $s''$ so that $5\/2<s''\\leq s-3$; then Theorem~\\ref{thm:Sobolevembedding} allows to deduce\n\\begin{equation}\n\t\\label{eq:uniformboundD}\n\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq C_{1,s''}\\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s''}}\\;.\n\\end{equation}\nNow, in view of Lemma~\\ref{lem:diffSobolevnorms} and applying Cauchy-Schwartz's inequality and Parseval's identity (cf.~\\cite[2,~XXIII,~6;~17]{Schwartz}), we get\n\\begin{equation}\n\t\\label{eq:sameCasimir}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s''}}&=\\sum_{n\\in I(\\mu)}\\frac{n^{2}+1}{\\bigl(1+\\mu+\\frac{n^2}{2})^{3\/2}}\\norm{f_{\\mu,n}}_{W^{s''+3}}\\\\\n\t\t&\\leq \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^2+1)^{2}}{\\bigl(1+\\mu+\\frac{n^2}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}^{2}_{W^{s''+3}}\\biggr)^{1\/2}\\\\\n\t\t&= \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2} \\norm{f_{\\mu}}_{W^{s''+3}}\n\t\\end{split}\n\\end{equation}\nwhere $f_{\\mu}$ is the orthogonal projection of $f$ onto the closed subspace $W^{s}(H_{\\mu})$, and the infinite sum in the last expression converges (see Lemma~\\ref{lem:finitespectralconstant}).\n\nDefining\n\\begin{equation}\n\t\\label{eq:spectralconstant}\n\tC_{\\mu}= \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\;,\n\\end{equation}\nour argument thus leads, combining~\\eqref{eq:uniformboundD} and~\\eqref{eq:sameCasimir}, to the estimate\n\\begin{equation}\n\t\\label{eq:sumcoefficients}\n\t\\sum_{n\\in I(\\mu)}\\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq C_{1,s''}C_{\\mu} \\frac{\\kappa(\\mu)}{\\theta}\\norm{f_{\\mu}}_{W^{s''+3}}\\;,\n\\end{equation}\nwhich implies that the sums $\\sum_{n\\in I(\\mu)}D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}$ converge normally in the Banach space $\\mathscr{C}(M)$, hence absolutely and uniformly for all $p \\in M$. In particular, the functions $D^{\\pm}_{\\theta,\\mu}f$ are well-defined, continuous on $M$ and fulfill the upper bound\n\\begin{equation}\n\t\\label{eq:boundDmu}\n\t\\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq C_{1,s-3}C_{\\mu} \\frac{\\kappa(\\mu)}{\\theta}\\norm{f_{\\mu}}_{W^{s}},\n\\end{equation}\nobtained by picking $s''=s-3$ in~\\eqref{eq:sumcoefficients}. \n\nWe now consider sums over all positive Casimir eigenvalues. We have\n\\begin{equation*}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} |D^{\\pm}_{\\theta,\\mu}f(p)|\\leq \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} \\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{C_{1,s-3}}{\\theta}\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}C_{\\mu}\\kappa(\\mu)\\norm{f_{\\mu}}_{W^{s}}\\;.\n\\end{equation*}\nObserve that $\\kappa(\\mu)$ is uniformly bounded by a constant $C_{\\text{Spec},\\text{pos}}$ depending only on the infimum of the set $ \\text{Spec}(\\square)\\cap (1\/4,\\infty)$ (see~\\eqref{eq:kappamu}); recalling the definition of $C_{\\mu}$ in~\\eqref{eq:spectralconstant}, we apply once again Cauchy-Schwartz's inequality and Parseval's identity to infer\n\\begin{equation}\n\t\\label{eq:plustwo-three}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} \\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq C_{\\text{Spec},\\text{pos}}C_{1,s-3}\\norm{f}_{W^{s}}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\;\\sum_{n\\in I(\\mu)}\\frac{(n^2+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\;,\n\\end{equation}\nwhere the term between parentheses on the right-hand side is finite because of Lemma~\\ref{lem:finitespectralconstant}. \n\nSince the spectrum of the Casimir operator is discrete, there are only finitely many distinct eigenvalues in the interval $(0,1\/4)$, so that the series \n\\begin{equation*}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\n\\end{equation*}\ninvolves only finitely many additional terms with respect to~\\eqref{eq:plustwo-three}; each of those terms can be bounded with the help of~\\eqref{eq:boundDmu}. The claim in~\\eqref{eq:boundDthetamu} follows, by defining the constant $C'_{\\text{Spec}}$ appropriately in terms of $C_{\\text{Spec},\\text{pos}}$ and of the $C_{\\mu}$ for $0<\\mu<1\/4$.\n\n\\medskip\nIn order to finalize the proof of Theorem~\\ref{thm:mainexpandingtranslates}, we address now the remainder terms defined in~\\eqref{eq:remainderpos},\\linebreak\n\\eqref{eq:remaindernull}, ~\\eqref{eq:remainderneg} and~\\eqref{eq:remainder}.\n\nWe start with the term in~\\eqref{eq:remainderpos} stemming from positive Casimir eigenvalues. Define $\\mu_{\\text{princ}}$ to be the infimum of $\\text{Spec}(\\square)\\cap (1\/4,\\infty)$, and let  $\\nu_{\\text{princ}}$ be the corresponding parameter fulfilling $1-\\nu_{\\text{princ}}^{2}=4\\mu_{\\text{princ}}$. Using the bounds for the remainder terms $\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}$ corresponding to the single components $f_{\\mu,n}$, provided by Theorem~\\ref{thm:case_Theta_eigenfn}, we estimate\n\n\n\\begin{equation}\n\t\\label{eq:estimateremainder}\n\t\\begin{split}\n\t\t&\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\;\\sum_{n\\in I(\\mu)}|\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}(p,t)|\\\\\n\t\t&\\leq \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\frac{8\\kappa_0}{\\theta\\Im{\\nu}}e^{-t}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{\\cC^{1}}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta}e^{-t} \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\frac{1}{\\Im{\\nu}}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s-3}}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta\\; \\Im{\\nu_{\\text{princ}}}}e^{-t} \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}^{2}\\biggr)^{1\/2}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta\\; \\Im{\\nu_{\\text{princ}}}}e^{-t} \\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}^{2}\\biggr)^{1\/2}\\\\\n\t\t&\\leq \\frac{8\\kappa_0C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\frac{1}{\\Im{\\nu_{\\text{princ}}}}\\norm{f}_{W^{s}}e^{-t}\n\t\\end{split}\n\\end{equation}\nfor any $t\\geq 1$, applying the bound in~\\eqref{eq:estremainderabovequarter}, Theorem~\\ref{thm:Sobolevembedding}, the Cauchy-Schwartz's inequality (twice) and Bessel's inequality (cf.~\\cite[2,~XXIII,~6;~14]{Schwartz}) to $W^{s}(M)$. \nSimilarly, the bounds in~\\eqref{eq:estquarter} and~\\eqref{eq:estremainderbelowquarter} yield, respectively,\n\\begin{equation*}\n\t\\sum_{n\\in I(1\/4)}|\\mathcal{R}_{\\theta,1\/4,n}f_{1\/4,n}(p,t)|\\leq \\frac{4\\kappa_0 C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t}\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\;\\sum_{n\\in I(\\mu)}|\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}(p,t)|\\leq \\frac{4\\kappa_0 C_{1,s-3}C_{\\text{Spec},3}C_{\\text{comp}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\\end{equation*}\nfor any $t\\geq 1$, where we set \n\\begin{equation*}\n\tC_{\\text{comp}}=\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\frac{1}{\\nu(1-\\nu)(1+\\nu)}\\;.\\end{equation*}\nDefining thus\n\\begin{equation}\n\t\\label{eq:Cpos}\n\tC_{\\text{pos}}=\\frac{2}{\\Im{\\nu_{\\text{princ}}}}+C_{\\text{comp}}+1\n\\end{equation}\nand applying the triangular inequality for infinite sums, we get from~\\eqref{eq:remainderpos} that\n\\begin{equation}\n\t\\label{eq:estimateposremainder}\n\t|\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)|\\leq \\frac{4\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{pos}}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t}\\;, \\quad t\\geq 1.\n\\end{equation}\nAn entirely analogous argument, using the bound in~\\eqref{eq:remainderzero}, shows that \n\\begin{equation}\n\t\\label{eq:estimatezeroremainder}\n\t\\biggl|\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t)\\biggr|\\leq\\frac{ (8e\\pi+\\kappa_0)C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\\;,\\quad t\\geq 1.\n\\end{equation}\nDefine now \n\\begin{equation*}\n\tC_{\\text{disc}}=\\biggl(\\inf\\limits_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}|\\mu|\\biggr)^{-1}+\\frac{2e\\pi}{\\kappa_0}\\sup_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}\\frac{3+\\nu}{\\nu}\\;.\n\\end{equation*}\nThen, the bound in~\\eqref{eq:estimatediscreteseries} leads to\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu<0}\\;\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\biggl|\\int_0^{\\theta}f_{\\mu,n}\\circ \\phi^{X}_{t}\\circ r_{s}(p)\\;\\text{d}s\\biggr| \\\\\n\t\t&\\leq \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}\\frac{C_{1,s-3}}{\\theta}\\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2e\\pi(3+\\nu)}{\\nu}\\biggr)e^{-t}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s-3}}\\\\\n\t\t&\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{disc}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\t\\end{split}\n\\end{equation*}\nfor any $t\\geq 1$, arguing as in~\\eqref{eq:estimateremainder}. It follows at once from~\\eqref{eq:remainderneg} that\n\\begin{equation}\n\t\\label{eq:estimatenegremainder}\n\t|\\mathcal{R}_{\\theta,\\text{d}}f(p,t)|\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{disc}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\\end{equation}\nfor any $t\\geq 1$.\nWe finally come to the estimate of the term $\\cM_{\\theta,0}f(p,t)$, defined in~\\eqref{eq:remaindernull}.\nThe inequality in~\\eqref{eq:Gtheta} gives\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(0)}\\biggl|\\int_1^{t}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi\\biggr|&\\leq \\frac{\\kappa_0}{\\theta}\\sum_{n\\in I(0)}(n^{2}+1)\\int_1^{t}\\norm{f_{0,n}}_{\\mathscr{C}^{1}}\\;\\text{d}\\xi=\\frac{\\kappa_0}{\\theta}(t-1)\\biggl(\\sum_{n\\in I(0)}(n^{2}+1)\\norm{f_{0,n}}_{\\mathscr{C}^{1}}\\bigg)\\\\\n\t\t&\\leq \\frac{\\kappa_0C_{1,s-3}}{\\theta}(t-1)\\biggl(\\sum_{n\\in I(0)}(n^{2}+1)\\norm{f_{0,n}}_{W^{s}}\\biggr)\\;,\n\t\\end{split}\n\\end{equation*}\nso that \n\\begin{equation}\n\t\\label{eq:estimatezeromain}\n\t|\\mathcal{M}_{\\theta,0}f(p,t)|\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}(t-1)\n\\end{equation}\nfor any $t\\geq 1$.\nRecalling~\\eqref{eq:remainder} and combining the estimates in~\\eqref{eq:estimateposremainder},~\\eqref{eq:estimatezeroremainder},~\\eqref{eq:estimatenegremainder} and~\\eqref{eq:estimatezeromain} we conclude that, for any $t\\geq 1$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta}f(p,t)|&\\leq |\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)|+e^{-t}|\\mathcal{M}_{\\theta,0}f(p,t)|+\\sum_{n\\in I(0)}|\\mathcal{R}_{0,\\theta,n}f_{0,n}(p,t)|+|\\mathcal{R}_{\\theta,\\text{d}}f(p,t)|\\\\\n\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^s}(t+1)e^{-t}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we set\n\\begin{equation*}\n\tC_{\\text{Spec}}=C_{\\text{Spec},3}\\bigl(8e\\pi+\\kappa_0(2+C_{\\text{pos}}+C_{\\text{disc}})\\bigr)\\;,\n\\end{equation*}\nwhich ostensibly depends only on the spectrum of the Casimir operator.\n\nThe proof of Theorem~\\ref{thm:mainexpandingtranslates} is complete.\n\n\\subsubsection*{Effective equidistribution and shrinking circle arcs}\nIn this paragraph we briefly comment on the proof of Corollaries~\\ref{cor:effective} and~\\ref{cor:shrinkingarcs}.\n\nAs to Corollary~\\ref{cor:effective}, it suffices to define the function $D^{\\text{main}}_{\\theta}f\\colon M\\times \\R_{\\geq 0}\\to \\C$ as follows:\n\\begin{itemize}\n\t\\item $ D^{\\text{main}}_{\\theta}f=D^{-}_{\\theta,\\mu_*}f$\n\tif the spectral gap $\\mu_{*}\\leq 1\/4$;\n\t\\item \n\t$D^{\\text{main}}_{\\theta}f=D^{+}_{\\theta,\\mu_*}f+D^{-}_{\\theta,\\mu_*}f$ if $\\mu_*>1\/4$.\n\t\n\\end{itemize}\n\nThe effective equidistribution statement in~\\eqref{eq:effective} then follows directly from the asymptotics in~\\eqref{eq:asymptoticgeneral}.\n\nNow suppose that we let the boundaries of the parametrization depend on the time $t$, so as to deal with a collection of time-varying subarcs\n\\begin{equation*}\n \\gamma_t=\\{\\phi^{X}_{t}\\circ r_s(p):\\theta_1(t)\\leq s\\leq  \\theta_2(t) \\} \n\\end{equation*} \n as in the statement of Corollary~\\ref{cor:shrinkingarcs}. If, as in the assumptions to the latter, we suppose that $\\theta_2(t)-\\theta_1(t)\\geq \\eta(t)e^{-\\frac{1-\\Re{\\nu_*}}{2}t}$ for every sufficiently large $t$, where $\\nu_*$ corresponds to the spectral gap $\\mu_*$ and $\\eta\\colon \\R_{>0}\\to \\R_{>0}$ satisfies $\\eta(t)\\to\\infty$ as $t\\to\\infty$, then we obtain\n\\begin{equation}\n\t\\label{eq:varyingarc}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta_2(t)-\\theta_1(t)}&\\int_{\\theta_1(t)}^{\\theta_2(t)}f\\circ \\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s = \\frac{1}{\\theta_2(t)-\\theta_1(t)}\\int_{0}^{\\theta_2(t)-\\theta_1(t)}f\\circ \\phi^{X}_t\\circ r_{s+\\theta_1(t)}(p)\\;\\text{d}s\\\\\n\t\t&=\\int_{M}f\\;\\text{d}\\vol+D^{\\text{main}}_{\\theta_2(t) - \\theta_1(t)}f\\left( r_{\\theta_1(t)}(p)\\right)t^{\\varepsilon_0}e^{-\\frac{1-\\Re{\\nu_*}}{2}t}+o(e^{-\\frac{1-\\Re{\\nu_*}}{2}t})\\;\\;.\n\t\\end{split}\n\\end{equation}\nThe bound \\eqref{eq:boundDthetamain} results into \n\\begin{equation*}\n\t|D^{\\text{main}}_{\\theta_2(t) - \\theta_1(t)}f\\left( r_{\\theta_1(t)}(p)\\right)|\\leq C_{1,s-3}C_{\\text{Spec}}'\\norm{f}_{W^{s}}\\biggl(\\frac{1}{\\theta_2(t)-\\theta_1(t)}\\biggr)\\leq  C_{1,s-3}C'_{\\text{Spec}}\\norm{f}_{W^s}  \\eta(t)^{-1}e^{\\frac{1-\\Re{\\nu_*}}{2}t}\n\\end{equation*}\nfor any $t\\geq t_0$.  We deduce that the right-hand side of~\\eqref{eq:varyingarc} is equal to $\\int_{M}f\\;\\text{d}\\vol+o(t)$ as $t$ tends to infinity. An elementary application of the Stone-Weierstrass' theorem (cf.~\\cite[Thm.~4.51]{Folland}) gives that smooth functions are dense in the space of continuous functions on the compact manifold $M$; it follows that the convergence\n\\begin{equation*}\n\t\\frac{1}{\\theta_2(t)-\\theta_1(t)}\\int_{\\theta_1(t)}^{\\theta_2(t)}f\\circ \\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\overset{t\\to\\infty}{\\longrightarrow}\\int_{M}f\\;\\text{d}\\vol\n\\end{equation*}\ncan be upgraded to hold for every $f\\in \\mathscr{C}(M)$, whereby the desired equidistribution is shown.\n\n\n\\subsubsection*{Equidistribution of circle arcs on the surface}\nWe conclude this subsection with a few comments concerning the statement of Theorem~\\ref{thm:expandingonsurface}, which is nothing but a specialization of Theorem~\\ref{thm:mainexpandingtranslates} to the case of observables defined on the underlying surface $S=\\Gamma\\bsl \\Hyp$, except for the lower regularity assumed on the test function $f$. First, we remark that $\\SO_2(\\R)$-invariance of the functions $D^{\\pm}_{4\\pi,\\mu}f$ and $\\mathcal{R}_{4\\pi}f$ follows at once from their definition (see~\\eqref{eq:defDthetamu} and~\\eqref{eq:remainder}) and the fact that $f$ is assumed to be $\\SO_2(\\R)$-invariant. We are only left to show that we might take $s>9\/2$, less restrictively in comparison to an arbitrary $f$\ndefined on $M$. The relevant observation here is that, for any $\\SO_2(\\R)$-invariant function $f\\in L^{2}(M)$, the components $f_{\\mu}$ appearing in the decomposition\\footnote{Recall from Section~\\ref{sec:unitaryrepresentations} that the Casimir operator $\\square$ acts as the Laplace-Beltrami operator $\\Delta_S$ on $\\SO_2(\\R)$-invariant functions.}\n\\begin{equation*}\n\tf=\\sum_{\\mu \\in \\text{Spec}(\\Delta_S)}f_{\\mu}\\;,\\quad f_{\\mu}\\in H_{\\mu}\n\\end{equation*} \nare invariant under $\\SO_2(\\R)$, that is, they satisfy $\\Theta f_{\\mu}=0$. The estimate in~\\eqref{eq:plustwo-three} thus only requires $s>9\/2=11\/2-1$, as the sum\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\Delta_S)}\\frac{1}{(1+\\mu)^{k}}\n\\end{equation*}\nconverges already for $k=2$, and not only for $k=3$ as it is the case in~\\eqref{eq:plustwo-three}.\n\n\\subsection{Equidistribution of arbitrary translates}\n\\label{sec:arbitrarytranslates}\n\nIn light of Theorem~\\ref{thm:mainexpandingtranslates}, Theorem~\\ref{thm:mainarbitrarytranslates} is a rather straightforward consequence of the classical Cartan decomposition for the semisimple Lie group $\\SL_2(\\R)$, for which the reader is referred to~\\cite[Chap.~VI]{Knapp}. We present the details of the argument in this subsection. \n\nLet $A=\\{\\exp{tX}:t\\in \\R\\}$ be the subgroup of $\\SL_2(\\R)$ consisting of diagonal matrices with positive entries (recall that $X$ is defined as in~\\eqref{eq:geodesicflow}). The product map\n\\begin{equation*}\n\t\\SO_2(\\R)\\times A\\times \\SO_2(\\R)\\to \\SL_2(\\R),\\;(k_1,a,k_2)\\mapsto k_1ak_2\n\\end{equation*} \nis surjective. For any $g\\in \\SL_2(\\R)$, choose a decomposition $g=k_1(g)a(g)k_2(g)$, where $a(g)$ is the diagonal matrix having as entries the singular values of the matrix $g$, in decreasing order. In particular, if $t(g)\\in \\R_{\\geq 0}$ is (uniquely) determined by the condition\n\\begin{equation}\n\t\\label{eq:Cartanprojection}\n\ta(g)=\n\t\\begin{pmatrix}\n\t\te^{t(g)\/2}&0\\\\\n\t\t0&e^{-t(g)\/2}\n\t\\end{pmatrix}\n\t,\n\\end{equation}\nthen it clearly holds that\n\\begin{equation}\n\t\\label{eq:singvaluenorm} \\norm{g}_{\\text{op}}=e^{t(g)\/2}\\;, \\quad \\text{ or equivalently} \\quad  t(g)=2\\log{\\norm{g}_{\\text{op}}}\\;.\n\\end{equation}\n\nFix now  a real number $s>11\/2$ and a function $f$ in the Sobolev space $W^{s}(M)$. Recall that, for any $p \\in M$, we indicate with $m_{\\SO_2(\\R)\\cdot p}$ the unique $\\SO_2(\\R)$-invariant measure supported on the compact orbit $\\SO_2(\\R)\\cdot p$; furthermore, $g_{*}\\SO_2(\\R)$ denotes the push-forward of the latter measure under the right translation map $R_{g}(\\Gamma g')=\\Gamma g'g$ on $M$. For any $p \\in M$ and $g\\in \\SL_2(\\R)$, we resort to the Cartan decomposition of $g$ and write\n\\begin{equation}\n\t\\label{eq:rotationinvariance}\n\t\\begin{split}\n\t\t\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}&=\\int_{M}f\\circ R_{g}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\circ R_{k_2(g)}\\circ R_{a(g)}\\circ R_{k_1(g)}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}\\\\\n\t\t&=\\int_{M}f\\circ R_{k_2(g)}\\circ R_{a(g)}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}=\\frac{1}{4\\pi}\\int_0^{4\\pi}\\bigl(f\\circ R_{k_2(g)}\\bigr)\\circ \\phi^{X}_{t(g)}\\circ r_s(p)\\;\\text{d}s\\;,\n\t\\end{split}\n\\end{equation}\nusing the $R_{k_1(g)}$-invariance of $m_{\\SO_2(\\R)\\cdot p}$ and the fact that $R_{a(g)}=\\phi_{t(g)}^{X}$ in view of~\\eqref{eq:Cartanprojection}. \n\nWe may now make use of the asymptotic expansion provided by Theorem~\\ref{thm:mainexpandingtranslates} for the function $f\\circ R_{k_2(g)}$, which lies in the same Sobolev space $W^{s}(M)$ of $f$ since $R_{k_2(g)}$ is a smooth diffeomorphism of $M$. We thereby obtain, for a fixed base point $p \\in M$,\n\\begin{equation}\n\t\\label{eq:variabletranslates}\n\t\\begin{split}\n\t\t&\\frac{1}{4\\pi}\\int_0^{4\\pi}\\bigl(f\\circ R_{k_2(g)}\\bigr)\\circ \\phi^{X}_{t(g)}\\circ r_s(p)\\;\\text{d}s=\\int_{M}f\\circ R_{k_2(g)}\\;\\text{d}\\vol\\\\\n\t\t&+e^{-\\frac{t(g)}{2}}\\biggl( \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t(g)\\biggr)} D^{+}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t(g)\\biggr)} D^{-}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t(g)}D^{+}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)+e^{-\\frac{1-\\nu}{2}t(g)}D^{-}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t(g)}{2}}D^{+}_{4\\pi,1\/4}(f\\circ R_{k_2(g)})(p)+t(g)e^{-\\frac{t(g)}{2}}D^{-}_{4\\pi.1\/4}(f\\circ R_{k_2}(g))(p)\\bigr)+\\mathcal{R}_{4\\pi}(f\\circ R_{k_2(g)})(p,t(g))\n\t\\end{split}\n\\end{equation}\nfor any $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$.\nDefine now, for any Casimir eigenvalue $\\mu \\in \\R_{>0}$, the functions $D^{\\pm}_{\\mu}\\colon M\\times \\SL_2(\\R)\\to \\C$ by \n\\begin{equation}\n\t\\label{eq:differentcoefficients}\n\tD^{\\pm}_{\\mu}f(p,g)=D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\;,\\quad p \\in M,\\;g\\in \\SL_2(\\R),\n\\end{equation}\nand set also\n\\begin{equation*}\n\t\\mathcal{R}f(p,g)=\\mathcal{R}_{4\\pi}(f\\circ R_{k_2}(g))(p,t(g))\\;,\\quad p \\in M,\\;g\\in \\SL_2(\\R).\n\\end{equation*}\n\n\\smallskip\nThen, combining~\\eqref{eq:rotationinvariance} and~\\eqref{eq:variabletranslates} and recalling~\\eqref{eq:singvaluenorm} together with the fact that\\\\ $\\int_{M}f\\circ R_{k_2}\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol$, we deduce\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}f\\;\\text{d}&g_{*}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\;\\text{d}\\vol\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{g}^{-1}_{\\text{op}}\\bigl(\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{+}_{\\mu}f(p,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{-}_{\\mu}f(p,g)\\bigr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\text{op}}^{-(1+\\nu)}D^{+}_{\\mu}f(p,g)+\\norm{g}_{\\text{op}}^{-(1-\\nu)}D^{-}_{\\mu}f(p,g)\\\\\n\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\text{op}}^{-1}D^{+}_{1\/4}f(p,g)+2\\norm{g}_{\\text{op}}^{-1}\\log{\\norm{g}_{\\text{op}}}D^{-}_{1\/4}f(p,g)\\bigr)+\\mathcal{R}f(p,g)\n\t\\end{split}\n\\end{equation*}\nfor any $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$,\nwhich is precisely the asymptotic expansion appearing in the statement of Theorem~\\ref{thm:mainarbitrarytranslates}.\n\nAs stated in Theorem~\\ref{thm:mainexpandingtranslates}, the functions $D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})$ are continuous on $M$ for any fixed $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$ and $g\\in \\SL_2(\\R)$; equivalently, by~\\eqref{eq:differentcoefficients}, $D^{\\pm}_{\\mu}f(\\cdot,g)$ is continuous on $M$ for any fixed $g\\in \\SL_2(\\R)$.\n\nAlso, for any $p \\in M, \\;g\\in \\SL_2(\\R)$ and $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$, we have \n\\begin{equation*}\n\t|D^{\\pm}_{\\mu}f(p,g)|=|D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)|\\leq \\norm{D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})}_{\\infty}\\leq \\frac{C_{1,s-3}C_{\\mu}\\kappa(\\mu)}{4\\pi}\\norm{f\\circ R_{k_2(g)}}_{W^{s}}\\;,\n\\end{equation*} \nwhere the last inequality is given by~\\eqref{eq:boundDthetamu}. It remains to observe that compactness of $\\SO_2(\\R)$ implies that there exists a constant $C_{s,\\text{rot}}>0$ such that $\\norm{f\\circ R_{k}}_{W^{s}}\\leq C_{s,\\text{rot}}\\norm{f}_{W^s}$ for any $k\\in \\SO_2(\\R)$. The proof of this assertion runs along the same lines of the proof of Lemma~\\ref{lem:diffunderint}, with the appropriate modifications. Therefore, we get\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}}\\sup_{p \\in M, \\;g\\in \\SL_2(\\R)}|D^{\\pm}_{\\mu}f(p,g)|\\leq \\frac{C_{1,s-3}C_{s,\\text{rot}}C'_{\\text{Spec}}}{4\\pi}\\norm{f}_{W^{s}}\\;,\n\\end{equation*} \nwhere $C'_{\\text{Spec}}$ is as in Theorem~\\ref{thm:mainexpandingtranslates}.\n\nTo conclude the proof of Theorem~\\ref{thm:mainarbitrarytranslates}, it is left to take care of the remainder term $\\mathcal{R}f$. We easily estimate, from~\\eqref{eq:globalremainderestimate},\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}f(p,g)|&=|\\mathcal{R}_{4\\pi}(f\\circ R_{k_2}(g))(p,t(g))|\\leq \\frac{C_{\\text{Spec}}C_{1,s-3}}{4\\pi}\\norm{f\\circ R_{k_2(g)}}_{W^{s}}(t(g)+1)e^{-t(g)}\\\\\n\t\t&\\leq \\frac{C_{\\text{Spec}}C_{1,s-3}C_{s,\\text{rot}}}{4\\pi}\\norm{f}_{W^{s}}(2\\log{\\norm{g}_{\\text{op}}}+1)\\norm{g}_{\\text{op}}^{-2}\n\t\\end{split}\n\\end{equation*}\nfor any $p\\in M$ and $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$.\n\nThis achieves the proof of Theorem~\\ref{thm:mainarbitrarytranslates}.\n\n\\section{Distributional limit theorems for deviations from the average}\n\\label{sec:CLT}\n\nThe purpose of this section is threefold, articulated in three subsections. First, we establish the quantitative distributional convergence claimed in Proposition~\\ref{prop:CLT}, from which the qualitative statements in Theorem~\\ref{thm:CLT} follow directly; secondly, we prove absence of a central limit theorem as phrased in Theorem~\\ref{thm:noCLT}, and finally we explore further ways of examining the statistical behaviour of averages along circle arcs. \n\n\\subsection{Quantitative distributional convergence}\n\\label{sec:spatialDLT}\nLet us fix the length parameter $\\theta\\in (0,4\\pi]$, and consider a real-valued function $f$ lying in the Sobolev space $W^{s}(M)$ for some real $s>11\/2$. We are interested in the statistical behaviour of the deviations from the mean \n\\begin{equation*}\n\td_f(T,p)=\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_T\\circ r_{s}(p)\\;\\text{d}s -\\int_{M}f\\;\\text{d}\\vol\n\\end{equation*}\nappropriately renormalized, as the time parameter $T$ tends to infinity and when the base point $p$ is sampled according to the uniform probability measure $\\vol$ on $M$.\nDefine\n\\begin{equation*}\n\t\\mu_f=\\inf\\{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}:D^{-}_{\\theta,\\mu}f\\text { does not vanish identically on }M  \\}\\;.\n\\end{equation*}\nAs in the hypotheses of Proposition~\\ref{prop:CLT}, we assume that $\\mu_f$ is finite, that is, the set of Casimir eigenvalues over which the previous infimum is taken is non-empty. Let $\\nu_f$ be the corresponding parameter, namely $\\nu_f\\in \\R_{\\geq 0}\\cup i \\R_{>0}$ satisfies $1-\\nu_f^{2}=4\\mu_f$.\n\nIn order to quantify the rate of distributional convergence of the random variables under consideration, we make use of the L\\'{e}vy-Prokhorov metric $d_{LP}$ on the set $\\mathscr{P}(\\R)$ of Borel probability measures on $\\R$. We recall that this is defined as \n\\begin{equation*}\n\td_{LP}(\\lambda,\\rho)=\\inf\\{\\varepsilon>0:\\lambda(Y)\\leq \\rho(Y_{\\varepsilon})+\\varepsilon \\text{ and }\\rho(Y)\\leq \\lambda(Y_{\\varepsilon})+\\varepsilon \\text{ for every Borel set } Y\\subset \\R \\}\n\\end{equation*}\nfor any $\\lambda,\\rho \\in \\mathscr{P}(\\R)$, where $Y_{\\varepsilon}$ denotes the open $\\varepsilon$-neighborhood of $Y$ with respect to the Euclidean metric on $\\R$. The distance $d_{LP}$ induces the topology of weak convergence of probability measures on $\\mathscr{P}(\\R)$, namely the coarsest topology for which the maps \n\\begin{equation*}\n\t\\mathscr{P}(\\R)\\ni \\lambda \\mapsto \\int_{\\R}\\varphi\\;\\text{d}\\lambda \\in \\R\\;, \\quad \\varphi\\colon \\R \\to \\R \\text{ continuous and bounded}\n\\end{equation*}\nare continuous.\n\nIn the forthcoming estimates we shall make use of the following trivial upper bound for the L\\'{e}vy-Prokhorov distance between the laws of two random variables defined on the same probability space and taking on nearby values almost surely.\n\n\n\\begin{lem}\n\t\\label{lem:LPnearbyvariables}\n\tLet $(\\Omega,\\cF,\\mathbf{P})$ be a probability space, $\\varepsilon>0$. Suppose $X,X'\\colon \\Omega\\to \\R$ are random variables satisfying\n\t$|X(\\omega)-X'(\\omega)|< \\varepsilon$ for $\\mathbf{P}$-almost every $\\omega\\in \\Omega$. If $\\lambda_{X}$ and $\\lambda_{Y}$ denote the laws of $X$ and $X'$, respectively, then $d_{LP}(\\lambda_{X},\\lambda_{X'})\\leq \\varepsilon$.\n\\end{lem}\n\\begin{proof}\n\tLet $A\\subset \\R$ be a Borel subset. The event $\\{X\\in A \\}$ is contained in the event $\\{X'\\in A_{\\varepsilon} \\}$, up to a $\\mathbf{P}$-negligible subset, by the assumption on the distance between $X$ and $X'$. Therefore, \n\t\\begin{equation*}\n\t\t\\lambda_{X}(A)=\\mathbf{P}(X\\in A)\\leq \\mathbf{P}(X'\\in A_{\\varepsilon})=\\lambda_{X'}(A_{\\varepsilon})<\\lambda_{X'}(A_{\\varepsilon})+\\varepsilon\\;;\n\t\\end{equation*}\n\ta similar inequality holds reversing the role of $X$ and $X'$, whence $d_{LP}(\\lambda_X,\\lambda_{X'})\\leq \\varepsilon$.\n\\end{proof}\n\nWe now proceed with the proof of Proposition~\\ref{prop:CLT} by distinguishing the three different cases $0<\\mu_f<1\/4\\;,\\mu_f=1\/4$ and $\\mu_f>1\/4$.\n\n\\medskip\nSuppose first $0<\\mu_f<1\/4$. We would then like to show that the random variables\n\\begin{equation*}\n\te^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)\\;,\\quad p\\sim \\vol\n\\end{equation*}\nconverge in distribution, as $T$ tends to infinity, to the random variable $D^{-}_{\\theta,\\mu_f}f(p)$, $p\\sim \\vol$. Observe that, by virtue the asymptotic expansion in~\\eqref{eq:asymptoticgeneral} and the assumption on $\\mu_f$, we have\n\\begin{equation*}\n\t\\begin{split}\n\t\te^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)-D^{-}_{\\theta,\\mu_f}f(p)&=\\sum\\limits_{\\mu\\in \\text{Spec}(\\square),\\;\\mu_f<\\mu<1\/4}e^{-\\frac{\\nu_f-\\nu}{2}T}D^{-}_{\\theta,\\mu}f(p)\\\\\n\t\t&+e^{-\\frac{\\nu_f}{2}T}\\biggl( \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f(p)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu_f+\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{\\nu_f}{2}T}D^{+}_{\\theta,1\/4}f(p)+Te^{-\\frac{\\nu_f}{2}T}D^{-}_{\\theta.1\/4}f(p)\\bigr)+e^{\\frac{1-\\nu_f}{2}T}\\mathcal{R}_{\\theta}f(p,T)\\;,\n\t\\end{split}\n\\end{equation*}\nso that, because of the uniform bound in~\\eqref{eq:boundDthetamu}, we may estimate\n\\begin{equation*}\n\t\\biggl|e^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)-D^{-}_{\\theta,\\mu_f}f(p)\\biggr|\\leq \\frac{C_{1,s-3}C'_{\\text{Spec}}}{\\theta} \\norm{f}_{W^s}Te^{-\\frac{\\nu_f-\\Re{\\nu_f^{\\text{next}}}}{2}T}\n\\end{equation*}\nfor any $p \\in M$ and $T\\geq 1$, where $\\nu_f^{\\text{next}}$ is the parameter corresponding to the smallest eigenvalue $\\mu_f^{\\text{next}}$ of the Casimir operator exceeding\\footnote{Observe that we may dispense with the additional factor $T$ in the upper bound whenever $\\nu_f^{\\text{next}}\\in \\R$.} $\\mu_f$. \n\nBy Lemma~\\ref{lem:LPnearbyvariables}, and recalling the definitions of $\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T)$ and $\\mathbf{P}_{\\theta,f}$ introduced in Section~\\ref{sec:introductionCLT}, we get\n\\begin{equation*}\n\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta} \\norm{f}_{W^{s}}Te^{-\\eta_f T}\n\\end{equation*}\nfor $\\eta_f=\\frac{\\nu_f-\\Re{\\nu_f^{\\text{next}}}}{2}$.\n\n\\medskip\nSimilarly, if $\\mu_f=1\/4$, we readily obtain from~\\eqref{eq:asymptoticgeneral} that\n\\begin{equation*}\n\t\\begin{split}\n\t\tT^{-1}e^{\\frac{T}{2}}&\\;d_f(T,p)-D^{-}_{\\theta,1\/4}f(p)=T^{-1}\\biggl(D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\\\\\n\t\t&+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)+e^{\\frac{T}{2}}\\mathcal{R}_{\\theta}f(p,t)\\biggr)\n\t\\end{split}\n\\end{equation*}\nfor any $p \\in M$ and $T\\geq 1$; recalling the definition of the constant $C_{\\text{pos}}$ in~\\eqref{eq:Cpos}, we deduce the bound\n\\begin{equation*}\n\t\\bigl|T^{-1}e^{\\frac{T}{2}}\\;d_f(T,p)-D^{-}_{\\theta,1\/4}f(p)\\bigr|\\leq \\frac{C_{1,s-3}C_{\\text{pos}}}{\\theta} \\norm{f}_{W^{s}}T^{-1}\\;,\n\\end{equation*}\nso that, again by Lemma~\\ref{lem:LPnearbyvariables},\n\\begin{equation*}\n\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C_{1,s-3}C_{\\text{pos}}}{\\theta}\\norm{f}_{W^{s}}T^{-1}\n\\end{equation*} \nfor any $T\\geq 1$, as desired.\n\n\\medskip\nFinally, for $\\mu_f>1\/4$, we have from~\\eqref{eq:asymptoticgeneral} that\n\\begin{equation}\n\t\\label{eq:DLTabovequarter}\n\t\\begin{split}\n\t\t&e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\\\\\\n\t\t&= \\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)+e^{\\frac{T}{2}}\\mathcal{R}_{\\theta}f(p,T)\n\t\\end{split}\n\\end{equation} \nfor any $p \\in M$ and $T\\geq 1$, from which we deduce what follows. Let $\\mu_*=\\inf \\bigl(\\text{Spec}(\\square)\\cap \\R_{>0}\\bigr)$ be the spectral gap of $S=\\Gamma\\bsl \\Hyp$ and $\\nu_*$  the corresponding parameter:\n\\begin{itemize}\n\t\\item if $\\mu_{*}< 1\/4$, then~\\eqref{eq:DLTabovequarter} and~\\eqref{eq:boundDthetamu} give\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t&\\biggl|e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr|\\\\\n\t\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\twhence\n\t\\begin{equation*}\n\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\n\t\\end{equation*}\n\tfor any $T\\geq 1$.\n\t\\item if $\\mu_*\\geq 1\/4$, then~\\eqref{eq:DLTabovequarter} and~\\eqref{eq:globalremainderestimate} give\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t&\\biggl|e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr|\\\\\n\t\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\\;;\n\t\t\\end{split}\n\t\\end{equation*}\n\twe deduce that\n\t\\begin{equation*}\n\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq\\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\n\t\\end{equation*} \n\tfor any $T\\geq 1$.\n\\end{itemize}\n\n\nThis completes the proof of Proposition~\\ref{prop:CLT}.\n\n\\subsection{Failure of a distributional limit theorem}\n\\label{sec:noCLT}\nWe now turn to the proof of Theorem~\\ref{thm:noCLT}. Once again, we consider a fixed length parameter $\\theta\\in (0,4\\pi]$ and a function $f\\in W^{s}(M)$ for some real $s>11\/2$. This time, we suppose that the coefficients \n$D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any Casimir eigenvalue $\\mu>0$. As a result, the asymptotic expansion provided in~\\eqref{eq:fullexpansion} reduces to\n\\begin{equation}\n\t\\label{eq:expansionCLT}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\text{d}s=&\\int_{M}f\\;\\text{d}\\vol +e^{-T}\\int_1^{T}\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi\n\t\t\\\\\n\t\t&+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,T)+\\mathcal{R}_{\\theta,\\text{d}}f(p,T)\\;,\n\t\\end{split}\n\\end{equation}\nfor any $p \\in M$ and $T\\geq 1$, where $\\mathcal{R}_{\\theta,\\text{d}}$ is defined in~\\eqref{eq:remainderneg}. \n\nThe estimates carried out in Section~\\ref{sec:proofexpandingtranslates} lead to the bound\n\\begin{equation}\n\t\\label{eq:remainderCLT}\n\t\\biggl|\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,T)+\\mathcal{R}_{\\theta,\\text{d}}f(p,T)\\biggr|\\leq \\frac{(8e\\pi+\\kappa_0)C_{1,s-3}C_{\\text{Spec},3}\\sup\\{1,C_{\\text{disc}}\\}}{\\theta}\\norm{f}_{W^{s}}e^{-T}\\;.\n\\end{equation}\nOn the other hand, by means of~\\eqref{eq:constantterm} we expand \n\\begin{equation*}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)=&\\frac{2}{\\theta(1-e^{-2\\xi})}\\sum_{n\\in I(0)}(Uf_{0,n}\\circ \\phi_{\\xi}^{X}(p)-Uf_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\\\\n\t\t&+\\frac{e^{-2\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\sum_{n\\in I(0)}2in(f_{0,n}\\circ \\phi_{\\xi}^{X}(p)-f_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\\\\n\t\t&-\\frac{e^{-\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\sum_{n\\in I(0)}\\int_0^{\\theta}f_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{s}(p)\\;\\text{d}s\\\\\n\t\t&+\\frac{2e^{-\\xi}}{\\theta(1-e^{-2\\xi})}\\sum_{n\\in I(0)}\\int_0^{\\theta}Xf_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_s(p)\\;\\text{d}s\\;,\n\t\\end{split}\n\\end{equation*}\nfrom which\n\\begin{equation*}\n\t\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)=\\frac{2}{\\theta(1-e^{-2\\xi})}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))+\\mathcal{R}_{G}f(p,\\xi)\\;,\n\\end{equation*}\nwhere \n\\begin{equation}\n\t\\label{eq:remaindersecondCLT}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{G}f(p,\\xi)|&\\leq \\frac{1}{(1-e^{-2})^{2}}\\biggl(\\norm{f}_{W^{s}}e^{-2\\xi}+\\norm{f}_{\\infty}e^{-\\xi}+2\\norm{f}_{\\cC^{1}}e^{-\\xi}(1-e^{-2\\xi})\\biggr)\\\\\n\t\t&\\leq \\frac{1+C_{0,s}+2C_{1,s}}{(1-e^{-2})^{2}}\\norm{f}_{W^{s}}e^{-\\xi}\\;,\n\t\\end{split}\n\\end{equation}\nusing the bound $1-e^{-2\\xi}\\geq 1-e^{-2}$ valid for any $\\xi\\geq 1$.\n\nLet now $(B_T)_{T>0}$ be a collection of positive real numbers such that $B_T\\to\\infty$ as $T\\to\\infty$. In light of~\\eqref{eq:expansionCLT},~\\eqref{eq:remainderCLT} and~\\eqref{eq:remaindersecondCLT}, and because of the assumption on $(B_T)_{T>0}$, the distributional limits of the random variables\n\\begin{equation*}\n\t\\frac{e^{T}\\bigl( \\frac{1}{\\theta}\\int_0^{\\theta}f\\circ\\phi^{X}_{T}\\circ r_{s}(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\bigr)}{B_T}\\;,\\quad p\\sim \\vol,\n\\end{equation*}\nas $T$ tends to infinity coincide with the distributional limits of the random variables\n\\begin{equation}\n\t\\label{eq:sameCLT} \\frac{\\frac{2}{\\theta}\\int_1^{T}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi}{B_T}\\;,\\quad p\\sim \\vol\\;.\n\\end{equation}\nWhen $\\theta=4\\pi$, we have $r_{4\\pi}(p)=p$, so that the integrand in the numerator of the above expression vanishes. Therefore, the distributional limit we are seeking after equals to zero almost surely, which proves Theorem~\\ref{thm:noCLT}.\n\n\\medskip\nAs to Remark~\\ref{rmk:geodesiccoboundary}, suppose $\\theta\\in (0,4\\pi]$ is arbitrary, and that $Uf_0$ is a coboundary for $(\\phi^{X}_t)_{t\\in \\R}$, namely there exists a measurable function $g\\colon M\\to \\C$ with\\footnote{More accurately, this is the notion of a measurable coboundary; by the celebrated work of Livsic on the cohomological equation for Anosov flows (cf.~\\cite{Livsic}), the condition is actually equivalent to the seemingly more restrictive one of $Uf_0$ being a continuous coboundary, namely of requiring the transfer function $g$ to be continuous.} finite norm\n\\begin{equation*}\n\t\\norm{g}_{L^{\\infty}(M,\\vol)}=\\inf\\{\\lambda \\in \\R_{>0}:|g(p)|\\leq \\lambda \\text{ for $\\vol$-almost every }p \\in M  \\}\n\\end{equation*}\n\nsuch that, for all $T>0$, \n\\begin{equation*}\n\t\\int_0^{T}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi=g\\circ \\phi^{X}_T(p)-g(p) \\quad \\text{for $\\vol$-almost every }p \\in M\\;.\n\\end{equation*}\n\nIt follows trivially that, for every $T>0$,\n\\begin{equation*}\n\t\\biggl|\\int_1^{T}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi\\biggr|\\leq \\frac{4}{1-e^{-2}}\\norm{g}_{L^{\\infty}(M,\\vol)} \n\\end{equation*}\nfor $\\vol$-almost every $p \\in M$.\nAs a result, the distributional limit as $T\\to\\infty$ of the random variables in~\\eqref{eq:sameCLT} vanishes almost surely, since $B_T\\to\\infty$.\n\n\\begin{rmk}\n\tSlightly more generally, when $Uf_0$ is cohomologous to a constant function, namely it differs from a constant function by a coboundary, any distributional limit of the random variables in~\\eqref{eq:sameCLT} is almost surely constant.\n\t\n\tAssume now $Uf_0$ is not cohomologous to a constant function (and $\\theta\\neq 4\\pi$). The classical central limit theorem for geodesic ergodic integrals (see~\\cite{Sinai} for the constant curvature case, and~\\cite{Ratner} for variable negative curvature) gives that both\n\t\\begin{equation*}\n\t\t\\frac{\\int_1^{T}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi-\\int_{M}Uf_0\\;\\text{d}\\vol}{\\sqrt{T}}\\;,\\quad p\\sim \\vol\n\t\\end{equation*}\n\tand \n\t\\begin{equation*}\n\t\t\\frac{\\int_1^{T}Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi-\\int_{M}Uf_0\\;\\text{d}\\vol}{\\sqrt{T}}\\;,\\quad p\\sim \\vol\n\t\\end{equation*}\n\tconverge in distribution to a non-trivial centered Gaussian random variable as $T$ tends to infinity. \\textit{A priori}, the combination of these two distributional convergences doesn't provide any information on the distributional limits of the difference, which is what appears in~\\eqref{eq:sameCLT} up to the constant factor $2\/\\theta$; it would be desirable to reach a full understanding of this limiting distributional behaviour by carefully inspecting the dependence properties of the random variables $\\int_1^{T}Uf_0\\circ \\phi^{X}_\\xi(p)\\;\\text{d}\\xi$ and $\\int_1^{T}Uf_0\\circ \\phi^{X}_\\xi\\circ r_{\\theta}(p)\\;\\text{d}\\xi$ as $p$ is sampled according to the volume measure on $M$. \n\\end{rmk}\n\n\\subsection{Some reflections on temporal distributional limit theorems}\n\\label{sec:temporalDLT}\nAn upshot of the two foregoing subsections is the following consideration: examining the statistical behaviour, for large times $T$, of the (appropriately renormalized) averages \n\\begin{equation*}\n\t\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_{T}\\circ r_s(p)\\;\\text{d}s\n\\end{equation*}\nby randomly sampling the base point $p$ according to the uniform measure on $M$ leads to meaningful asymptotic results if and only if\\footnote{Possibly with the exception of the case examined at the end of Section~\\ref{sec:noCLT}.} at least one of the coefficients $D^{\\pm}_{\\theta,\\mu}f$ does not vanish identically on $M$. Irrespective of whether this is the case or not, it is natural to look for different sources of randomness, which might capture oscillatory behaviours more accurately. In accordance with the perspective of temporal distributional limit theorems, pioneered by Dolgopyat and Sarig~\\cite{Dolgopyat-Sarig} in the context of ergodic sums and integrals, we enquire about the existence of non-trivial distributional limits for the random variables\n\\begin{equation*}\n\t\\frac{e^{t}\\bigl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\bigr)-A_T}{B_T}\\;,\n\\end{equation*}\nwhere $p$ is a fixed base point in $M$, $(A_T)_{T>0}$ and $(B_T)_{T>0}$ are collections of real numbers, possibly depending on $p$, with $B_T>0$ and $B_T\\to\\infty$ as $T\\to\\infty$, and the time $t$ is chosen uniformly at random in the interval $[0,T]$. \n\n\\begin{rmk}\n\tIt is informative to compare this to the quest for temporal limit theorems for ergodic integrals along the orbits of a flow: see, in particular,~\\cite[Def.~1.3]{Dolgopyat-Sarig}. Observe notably that the rescaling of the circle-arc average $\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s$ by a factor of $e^{t}$ (the latter being asymptotically of the same order of the length of the expanding circle arc along which the average is taken) parallels the renormalization of ergodic averages by the linear factor $t$.\n\\end{rmk}\n\nLet us denote by $\\mathcal{U}_{[0,T]}$ the uniform probability measure on the compact interval $[0,T]$, for any $T>0$. If there is a non-identically vanishing coefficient $D^{\\pm}_{\\theta,\\mu}f$ for some Casimir eigenvalue $\\mu>0$, then a rather straightforward adaptation of the proof of~\\cite[Cor.~5.7]{Dolgopyat-Sarig} shows that, for $\\vol$-almost every $p \\in M$, any limiting distribution of \n\\begin{equation}\n\t\\label{eq:temporalDLT}\n\t\\frac{e^{t}\\bigl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\bigr)-A_T}{B_T}\\;,\\quad t\\sim \\mathcal{U}_{[0,T]}\n\\end{equation}\nis necessarily constant almost surely, no matter the choice of the constants $A_T$ and $B_T$.\n\nSuppose now that the coefficients $D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any  positive Casimir eigenvalue $\\mu$. The deduction in Section~\\ref{sec:noCLT} applies almost verbatim, showing  that the distributional limits of the random variables in~\\eqref{eq:temporalDLT} are the same as the limits of \n\\begin{equation}\n\t\\label{eq:sametemporal}\n\t\\frac{\\frac{2}{\\theta}\\int_1^{t}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim \\mathcal{U}_{[0,T]}\n\\end{equation}\nas $T$ tends to infinity. In the first place, this allows tu rule out the existence of any non-trivial (namely not almost surely constant) distributional limit whenever one of the following conditions is met:\n\\begin{itemize}\n\t\\item[(a)] $\\theta=4\\pi$;\n\t\\item[(b)] $Uf_0$ is cohomologous to a constant function for the geodesic flow.\n\\end{itemize} \nOn the other hand, when $Uf_0$ is not cohomologous to a constant function, then the geodesic ergodic integrals $\\int_1^{t}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi$ are well-approximated by Brownian trajectories. More precisely, the Almost Sure Invariance Principle (see~\\cite{Strassen,Strassen-two},~\\cite[Chap.~1]{Philipp-Stout} and~\\cite{Denker-Phillip}) for geodesic ergodic integrals asserts that there exist an auxiliary probability space $(\\Omega,\\cF,\\mathbf{P})$ and two continuous-time stochastic processes $(X_t)_{t\\geq 0}$ and $(B_t)_{t\\geq 0}$ defined on $(\\Omega,\\cF,\\mathbf{P})$ such that the following hold:\n\\begin{itemize}\n\t\\item the law of the process $(X_{t})_{t\\geq 0}$ under the probability measure $\\mathbf{P}$ coincides with the law of the process $\\bigl(\\int_0^{t}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi\\bigr)_{t\\geq 0}$ when $p$ is sampled according to the probability measure $\\vol$;\n\t\\item the process $(B_{t})_{t\\geq 0}$ is a standard one-dimensional Brownian motion (cf.~\\cite[Chap.~2]{LeGall});\n\t\\item there exists $\\sigma \\in \\R^{\\times}$ such that, for $\\mathbf{P}$-almost every $\\omega \\in \\Omega$,\n\t\\begin{equation}\n\t\t\\label{eq:Brownian}\n\t\t|X_t(\\omega)-B_{\\sigma^{2}t}(\\omega)|=o(\\sqrt{t}) \\quad \\text{as $t\\to\\infty$}\\;.\n\t\\end{equation}\n\\end{itemize}\nAs typical Brownian trajectories are of size $\\sim\\sqrt{t}$ at time $t$, the approximation in~\\eqref{eq:Brownian} enables to transfer classical results about the statistical behaviour of Brownian paths to analogous properties for geodesic ergodic integrals. In particular, there is no distributional limit\\footnote{Actually, when $Uf_0$ has zero average over $M$, $A_T=0$ and $B_T=\\sqrt{T}$,  any random variable may appear as distributional limit along an appropriate subsequence $(T_n)_{n\\in \\N}$ of times: see~\\cite[Thm.~3.2]{Dolgopyat-Sarig}.} for \n\\begin{equation}\n\t\\label{eq:temporalfirst}\n\t\\frac{\\int_0^{t}Uf_0\\circ\\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim\\mathcal{U}_{[0,T]}\n\\end{equation} \nas $T$ tends to infinity (cf.~\\cite[Sec.~3.1]{Dolgopyat-Sarig}). \nSince the process $\\bigl(\\int_0^{t}Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi\\bigr)_{t\\geq 0}$ has the same law, for $p\\sim \\vol$, as $(X_t(\\omega))_{t\\geq 0}$ for $\\omega\\sim \\mathbf{P}$, the same applies to the random variables \n\\begin{equation}\n\t\\label{eq:temporalsecond}\n\t\\frac{\\int_0^{t}Uf_0\\circ\\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim\\mathcal{U}_{[0,T]}\\;.\n\\end{equation}\nAs already argued in Section~\\ref{sec:noCLT} in the situation where the point $p$ is selected randomly and the time $T$ is fixed, here again the absence of distributional limits for each of the summands does not rule out, in principle, the possibility of non-trivial limits for the difference, hence for~\\eqref{eq:sametemporal}. Once more, a painstaking analysis of the dependence features of the two processes in~\\eqref{eq:temporalfirst} and~\\eqref{eq:temporalsecond} might clarify the seemingly elusive pathwise behaviour of their difference.\n\n\\section{The hyperbolic lattice point counting problem}\n\\label{sec:latticepoint}\n\nThis final section is consecrated to the applications of our equidistribution results to lattice-point counting problems in the hyperbolic plane; specifically, we shall first prove the precise asymptotics for the averaged counting function stated in Proposition~\\ref{prop:averagedcounting} and subsequently deduce Theorem~\\ref{thm:countingproblem} on the error estimate for the pointwise counting.  \n\nLet $\\Gamma$ be a cocompact lattice in $\\SL_2(\\R)$, and denote by $d_{\\Hyp}$ the hyperbolic distance function on the hyperbolic upper-half plane $\\Hyp$ (cf.~Section~\\ref{sec:hyperbolic}). For each real number $R>0$, let $B_R$ be the closed $d_{\\Hyp}$-ball of radius $R$ centered at the point $i\\in \\Hyp$, and define $N(R)=|\\Gamma\\cdot i \\;\\cap B_R|$, the cardinality of intersection of the $\\Gamma$-orbit of $i$ with $B_R$.\n\nRecall also from Section~\\ref{sec:hyperbolic} that $\\SL_2(\\R)$ acts on $\\Hyp$ by M\\\"{o}bius transformations.\nIn what follows, we identity the quotient manifold $\\SL_2(\\R)\/\\SO_2(\\R)$ with $\\Hyp$ whenever convenient, by means of the diffeomorphism $g\\SO_2(\\R)\\mapsto g\\cdot i,\\;g\\in \\SL_2(\\R)$. The hyperbolic area measure $m_{\\Hyp}$ (namely the volume measure arising from the hyperbolic structure on $\\Hyp$) is the Radon measure on $\\Hyp$ with density $\\text{d}m_{\\Hyp}(x,y)=y^{-2}\\text{d}x\\text{d}y$ with respect to the induced Lebesgue measure on $\\Hyp\\subset \\C$. \n\n\\begin{term}\n\tIn order not to overburden notation in the sequel, we shall denote $\\SL_2(\\R)$ by $G$ and $\\SO_2(\\R)$ by $K$.\n\\end{term}\n\n\\subsection{Asymptotics for the averaged counting function}\n\\label{sec:averagecounting}\n\nFor any subset $A\\subset G\/K$, we denote by $\\mathds{1}_{A}$ the indicator function of the set $A$. Define a function $F_R\\colon G\/\\Gamma\\to \\R_{\\geq 0}$\n\\begin{equation}\n\t\\label{eq:defaveragedcounting}\n\tF_R(g\\Gamma )=\\frac{|g\\Gamma \\cdot i \\cap B_R|}{m_{\\Hyp}(B_R)}=\\frac{1}{m_{\\Hyp}(B_R)}\\sum_{\\gamma \\Gamma\\cap K\\in \\Gamma\/\\Gamma\\cap K}\\mathds{1}_{B_R}(g\\gamma K)\\;, \\quad g\\in G;\n\\end{equation}\nobserve that the function $F_R$ is the subject of the averaged counting result in Proposition~\\ref{prop:averagedcounting}, which we now set out to prove. \n\n\\begin{rmk}\n\t\\label{rmk:leftrightcosets}\n\tWe choose to deal with spaces of left cosets in the sequel; in particular, we replace the homogeneous spaces $M=\\Gamma\\bsl G$ we have been considering so far with $G\/\\Gamma$, identifying them via the diffeomorphism $\\Gamma g\\mapsto g^{-1}\\Gamma$.\n\\end{rmk}\n\nWe follow the classical argument of Eskin and McMullen \\cite{Eskin-McMullen}, which relies on the well-known folding-unfolding formula for invariant measures on homogeneous spaces. For the sake of completeness, we recall it in the setting of the group $G=\\SL_2(\\R)$, referring the reader to~\\cite[Sec.~2.6]{Folland} or to~\\cite[Chap.~1]{Raghunathan} for the general statements and their proofs. \n\n\\begin{prop}\n\t\\label{prop:foldingunfolding}\n\t\\begin{enumerate}\n\t\t\\item Let $H<G$ be a unimodular closed subgroup. Then there exists a non-zero $G$-invariant positive Radon measure on the quotient space $G\/H $, which is uniquely determined up to positive real scalars. Moreover, if $\\mu_{G}$ and $\\mu_H$ are Haar measures\\footnote{The group $G$ is perfect, hence unimodular; thus $\\mu_{G}$ is also a right Haar measure.} on $G$ and $H$, respectively, there exists a unique normalization $\\mu_{G\/H}$ of the $G$-invariant measure on $G(H)$ such that, for any continuous compactly supported function $\\varphi\\colon G\\to \\C$, the following folding-unfolding formula holds:\n\t\t\\begin{equation}\n\t\t\t\\label{eq:foldingunfolding}\n\t\t\t\\int_{G}\\varphi\\;\\emph{d}\\mu_{G}=\\int_{G\/H}\\int_{H}\\varphi(gh)\\;\\emph{d}\\mu_{H}(h)\\;\\emph{d}\\mu_{G\/H}(gH)\\;.\n\t\t\\end{equation}\n\t\t\\item Let $H<L$ be closed subgroups of $G$, and suppose $G\/L$ admits a non-zero finite $G$-invariant measure $\\mu_{G\/L}$ and $L\/H$ admits a non-zero finite $L$-invariant measure $\\mu_{L\/H}$. Then $G\/H$ admits a non-zero finite $G$-invariant measure $\\mu_{G\/H}$. Moreover, if $\\mu_{G\/L},\\;\\mu_{L\/H}$ and $\\mu_{G\/H}$ are compatibly normalized, then for any continuous compactly supported function $\\varphi\\colon G\/H\\to \\C$, it holds\n\t\t\\begin{equation}\n\t\t\t\\label{eq:foldingunfoldingchain}\n\t\t\t\\int_{G\/H}\\varphi\\;\\emph{d}\\mu_{G\/H}=\\int_{G\/L}\\int_{L\/H}\\varphi(glH)\\;\\emph{d}\\mu_{L\/H}(lH)\\;\\emph{d}\\mu_{G\/L}(gL)\\;.\n\t\t\\end{equation} \n\t\\end{enumerate}\n\\end{prop}\nBy means of standard approximation arguments in measure theory, formula~\\eqref{eq:foldingunfolding} (resp.~ formula~\\eqref{eq:foldingunfoldingchain}) holds for any Borel-measurable function $\\varphi\\colon G\\to \\C$ (resp.~$\\varphi\\colon G\/H\\to \\C$) which either takes positive real values or is integrable with respect to $\\mu_{G}$ (resp.~$\\mu_{G\/H}$). \n\n\\smallskip\nLet now $m_{K}$ be the unique probability Haar measure on the compact group $K$, and normalize the Haar measure $m_{G}$ on $G$ so that, under the identification of $G\/K$ with $\\Hyp$, the resulting $G$-invariant measure on $G\/K$ (cf.~Proposition~\\ref{prop:foldingunfolding}) corresponds to the hyperbolic area measure $m_{\\Hyp}$. Let $m_{G\/\\Gamma}$ be the unique $G$-invariant finite Borel measure on $G$ corresponding, according to Proposition~\\ref{prop:foldingunfolding}, to the given choice of Haar measure on $G$ and to the counting measure on the discrete group $\\Gamma$. Similarly, endowing the finite discrete group $\\Gamma\\cap K$ with the counting measure, we indicate with $m_{G\/\\Gamma\\cap K}$, $m_{K\/\\Gamma\\cap K}$ and $m_{\\Gamma\/\\Gamma\\cap K}$ the induced measures on the respective homogeneous spaces. Recall also that with $m_{K\\cdot \\Gamma}$ we indicate the unique $K$-invariant probability measure supported on the compact $K$-orbit of the identity coset $\\Gamma$ inside $G\/\\Gamma$ (cf.~Theorem~\\ref{thm:mainarbitrarytranslates}).\n\nThe volumes of the homogeneous spaces $G\/\\Gamma$ and $K\/\\Gamma\\cap K$ with respect to the measures $m_{G\/\\Gamma}$ and $m_{K\/\\Gamma\\cap K}$ are indicated with $\\text{covol}_{K}(\\Gamma\\cap K)$ and $\\text{covol}_{G}(\\Gamma)$, respectively.\n\n\\medskip\nFix now a real parameter $s>11\/2$ and a test function $\\psi\\in W^{s}(G\/\\Gamma)$. We expand, for any $R>0$,\n\n\\begin{equation}\n\t\\label{eq:foldunfold}\n\t\\begin{split}\n\t\t&\\int_{M}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=\\int_{M}\\psi(g\\Gamma )\\biggl(\\frac{1}{m_{\\Hyp}(B_R)}\\sum_{\\gamma \\Gamma\\cap K\\in \\Gamma\/\\Gamma\\cap K}\\mathds{1}_{B_R}(g\\gamma K)\\biggr) \\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{M}\\int_{\\Gamma\/\\Gamma\\cap K}\\psi(g\\Gamma )\\mathds{1}_{B_R}(g\\gamma K)\\;\\text{d}m_{\\Gamma\/\\Gamma\\cap K}(\\gamma\\Gamma\\cap K) \\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{G\/\\Gamma\\cap K} \\psi(g\\Gamma)\\mathds{1}_{B_R}(gK) \\text{d}m_{G\/\\Gamma\\cap K}(g\\Gamma\\cap K)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{G\/K}\\int_{K\/\\Gamma\\cap K}\\psi(gk\\Gamma)\\mathds{1}_{B_R}(gK)\\;\\text{d}m_{K\/\\Gamma\\cap K}(k\\Gamma\\cap K)\\;\\text{d}m_{G\/K}(gK)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{B_R}\\int_{K\/\\Gamma\\cap K}\\psi\\;\\text{d}g_{*}m_{K\/\\Gamma\\cap K}\\;\\text{d}m_{\\Hyp}(gK)\\\\\n\t\t&=\\frac{\\text{covol}_{ K}(\\Gamma\\cap K )}{m_{\\Hyp}(B_R)}\\int_{B_R}\\int_{G\/\\Gamma}\\psi\\;\\text{d}g_{*}m_{K\\cdot \\Gamma}\\;\\text{d}m_{\\Hyp}(gK)\\;.\n\t\\end{split}\n\\end{equation}\nIn the previous chain of equalities, we applied in successive order:\n\\begin{itemize}\n\t\\item[(1)] the definition~\\eqref{eq:defaveragedcounting} of the function $F_R$;\n\t\\item[(2)] the fact that the invariant measure on the discrete space $\\Gamma\/\\Gamma\\cap K$ given by Proposition~\\ref{prop:foldingunfolding} is the counting measure;\n\t\\item[(3)] formula~\\eqref{eq:foldingunfoldingchain} to the tower of subgroups $\\Gamma\\cap K<\\Gamma<G$;\n\t\\item[(4)] formula~\\eqref{eq:foldingunfoldingchain} to the tower of subgroups $\\Gamma\\cap K<K<G$;\n\t\\item[(5)] the identification of $\\Hyp$ with $G\/K$ and of $m_{\\Hyp}$ with $m_{G\/K}$;\n\t\\item[(6)] the relationship $m_{K\/\\Gamma\\cap K}=\\text{covol}_{K}(\\Gamma\\cap K)m_{K\\cdot \\Gamma}$, derived from the fact that $m_{K\\cdot \\Gamma}$ is a probability measure, the definition of $\\text{covol}_{K}(\\Gamma\\cap K)$ and uniqueness up to scalars of the $K$-invariant measure on $K\/\\Gamma \\cap K\\simeq K\\cdot \\Gamma$.  \n\\end{itemize}\n\nWe may now replace the inner integral in the last expression of~\\eqref{eq:foldunfold} with the asymptotic expansion provided by Theorem~\\ref{thm:mainarbitrarytranslates} (with the caveat of Remark~\\ref{rmk:leftrightcosets}), thereby obtaining\n\\begin{equation}\n\t\\label{eq:asymptoticaverage}\n\t\\begin{split}\n\t\t\\int_{G\/\\Gamma}\\psi F_R&\\;\\text{d}m_{G\/\\Gamma}=\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma} +\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{m_{\\Hyp}(B_R)}\\\\\n\t\t&\\int_{B_R}\\biggl(\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{g}^{-1}_{\\text{op}}\\bigl(\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{+}_{\\mu}\\psi(\\Gamma,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{-}_{\\mu}\\psi(\\Gamma,g)\\bigr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\text{op}}^{-(1+\\nu)}D^{+}_{\\mu}\\psi(\\Gamma,g)+\\norm{g}_{\\text{op}}^{-(1-\\nu)}D^{-}_{\\mu}\\psi(\\Gamma,g)\\\\\n\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\text{op}}^{-1}D^{+}_{1\/4}\\psi(\\Gamma,g)+2\\norm{g}_{\\text{op}}^{-1}\\log{\\norm{g}_{\\text{op}}}D^{-}_{1\/4}\\psi(\\Gamma,g)\\bigr)+\\mathcal{R}\\psi(\\Gamma,g)\\biggr) \\text{d}m_{\\Hyp}(gK)\\;.\n\t\\end{split}\n\\end{equation}\n\nWe shall need the following analogue of the classical integration formula on spheres in Euclidean spaces: for any $r>0$, let $S_r=\\partial{B_r}=\\{z\\in \\Hyp:d_{\\Hyp}(z,i)=r \\}$ and $\\sigma_r$ the induced hyperbolic length measure on the circle $S_r$.\n\n\\begin{prop}\n\t\\label{prop:sphereintegration}\n\tLet $f\\colon \\Hyp\\to \\C$ be integrable with respect to $m_{\\Hyp}$. Then\n\t\\begin{equation*}\n\t\t\\int_{\\Hyp}f\\;\\emph{d}m_{\\Hyp}=\\int_{0}^{\\infty}\\int_{S_r}f(z)\\;\\emph{d}\\sigma_{r}(z)\\;\\emph{d}r\n\t\\end{equation*}\n\\end{prop} \n\nThe proof does not differ from the Euclidean case, for which we refer to~\\cite[Thm.~2.49]{Folland}.\n\n\\medskip\nDefine now, for any $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$ and $\\psi$ as above,\n\\begin{equation*}\n\t\\alpha^{\\pm}_{\\psi,\\mu}(r)=\\int_{S_r}D^{\\pm}_{\\mu}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;,\\quad r>0.\n\\end{equation*}\nFrom~\\eqref{eq:asymptoticaverage} we get, thanks to Proposition~\\ref{prop:sphereintegration}, \n\\begin{equation}\n\t\\label{eq:applyingsphereintegration}\n\t\\begin{split}\n\t\t&\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma}+\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{m_{\\Hyp}(B_R)}\\\\\n\t\t&\\biggl(\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\int_0^{R}e^{-\\frac{r}{2}}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha_{\\psi,\\mu}^{+}(r)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha^{-}_{\\psi,\\mu}(r)\\biggr)\\text{d}r\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\int_0^{R}e^{-\\frac{1+\\nu}{2}r}\\alpha^{+}_{\\psi,\\mu}(r)+e^{-\\frac{1-\\nu}{2}r}\\alpha^{-}_{\\psi,\\mu}(r)\\;\\text{d}r\\\\\n\t\t&+\\varepsilon_0\\biggl(\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{+}_{\\psi,1\/4}(r)+re^{-\\frac{r}{2}}\\alpha^{-}_{\\psi,1\/4}(r)\\;\\text{d}r\\biggr)+\\int_0^{R}\\int_{S_r}\\mathcal{R}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;\\text{d}r\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nLet $\\psi_{\\mu}$ be the orthogonal projection of $\\psi$ onto the closed subspace $W^{s}(H_{\\mu})$. By means of~\\eqref{eq:boundDthetamu}, we estimate \n\\begin{equation}\n\t\\label{eq:integralbounds}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}}|\\alpha_{\\psi,\\mu}^{\\pm}(r)|\\leq \\sum_{\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}}\\norm{D^{\\pm}_{\\mu}\\psi(\\Gamma,\\cdot )}_{\\infty}\\int_{S_r}\\text{d}\\sigma_{r}(z)\\leq 2\\pi \\frac{C_{1,s-3}C_{\\text{Spec}}'}{4\\pi}\\norm{\\psi}_{W^{s}}\\sinh{r}\n\\end{equation}\nfor any $r>0$,\nas the hyperbolic length  of $S_r$ equals\\footnote{This is an elementary verification in hyperbolic geometry, for instance approximating circles with regular $n$-gons; their hyperbolic perimeter can be easily computed by means of explicit formulas for the hyperbolic distance (cf.~\\cite[Thm.~1.2.6]{Katok}) and of the hyperbolic cosine law (cf.~\\cite[Thm.~1.5.2]{Katok}). \n\t\n\tSimilarly, the hyperbolic area of a ball is easily computed by approximation via the Gauss-Bonnet formula for the area of hyperbolic triangles (cf.~\\cite[Thm.~1.4.2]{Katok}).} $2\\pi\\sinh{r}$. Recalling that $m_{\\Hyp}(B_R)=2\\pi(\\cosh{R}-1)$ for any $R>0$, we deduce from~\\eqref{eq:applyingsphereintegration} that\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma}\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^{+}_{\\psi,\\mu}(R)+\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t&\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi,\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi,1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi,1\/4}(R)\\bigr)+\\gamma_{\\psi}(R)\\biggr)\n\t\\end{split}\n\\end{equation*}\nfor any $R\\geq 1$, where we have set\n\\begin{align*}\n\t&\\beta^+_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha_{\\psi,\\mu}^{+}(r)\\;\\text{d}r\\;,\\quad \\mu>1\/4\\;,\\\\\n\t&\\beta^-_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha^{-}_{\\psi,\\mu}(r)\\text{d}r\\;,\\quad\\mu>1\/4\\;,\\\\\n\t&\\beta^{\\pm}_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{1\\mp\\nu}{2}R}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{1\\pm\\nu}{2}r}\\alpha^{\\pm}_{\\psi,\\mu}(r)\\;\\text{d}r\\;,\\quad 0<\\mu<1\/4\\;,\\\\\n\t&\\beta^{+}_{\\psi,1\/4}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{+}_{\\psi,1\/4}(r)\\;\\text{d}r\\;,\\\\\n\t&\\beta^{-}_{\\psi,1\/4}(R)=\\frac{R^{-1}e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{-}_{\\psi,1\/4}(r)\\;\\text{d}r\\;,\\\\\n\t&\\gamma_{\\psi}(R)=\\frac{e^{-R}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}\\int_{S_r}\\mathcal{R}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;\\text{d}r\n\\end{align*}\n. Because of~\\eqref{eq:integralbounds}, we have the following estimates on the previous coefficients: for any $R\\geq 1$,\n\\begin{align*}\n\t&\\sum_{\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}}|\\beta^{\\pm}_{\\psi,\\mu}(R)|\\leq \\frac{5C_{1,s-3}C'_{\\text{Spec}}}{2\\pi}\\norm{\\psi}_{W^{s}}\\;,\\\\\n\t& |\\gamma_{\\psi}(R)|\\leq \\frac{5C_{1,s-3} C_{\\text{Spec}}}{4\\pi} \\norm{\\psi}_{W^{s}}(R+1)e^{-R}\\;,\n\\end{align*}\nusing  the (crude) bound $(1-2e^{-R}+e^{-2R})^{-1}\\leq 5$ in each of the previous inequalities.\n\nThis establishes Proposition~\\eqref{prop:averagedcounting} in its entirety.\n\n\\subsection{Error estimate for the pointwise counting problem}\n\\label{sec:countingproblem}\n\nThis subsection is devoted to the deduction of the estimate on the error for the counting problem stated in Theorem~\\ref{thm:countingproblem}, starting from the asymptotic expansion in~\\eqref{eq:averagedcountingfunction} for the averaged counting function. \n\nRecall from~\\eqref{eq:defaveragedcounting} that, for any real number $R>0$, the ratio $N(R)\/m_{\\mathbb{H}}(B_R)$ equals the value of the function $F_R$ at the identity coset $\\Gamma\\in G\/\\Gamma$. In order to find a convenient approximation for the latter, we shall compare it with the averages \n\\begin{equation*}\n\t\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}\n\\end{equation*}  \nwhere the function $\\psi$ ranges over a suitably defined approximate identity\\footnote{The terminology is common in the context of locally compact groups; see, for instance, \\cite[Sec.~2.5]{Folland}.} in $G\/\\Gamma$. \n\nWe now expose the details. Let us fix a parameter $\\delta\\in \\R_{>0}$, on which we shall subsequently impose conditions according to the needs of the argument; choose\n\\begin{itemize}\n\t\\item[(a)] an open symmetric\\footnote{Namely, $U_{\\delta}$ coincides with the set of inverses of its elements.} neighborhood $U_{\\delta}$ of the identity in $G$ such that, for any $R>0$, \n\t\\begin{equation}\n\t\t\\label{eq:neighborhoodcontainment}\n\t\tB_{R-\\delta}\\subset\\bigcap_{g\\in U_{\\delta}}g\\cdot B_R\\subset \\bigcup_{g\\in U_{\\delta}}g\\cdot B_R \\subset B_{R+\\delta}\n\t\\end{equation}\n\t\\item[(b)] and a smooth function $\\psi_{\\delta}\\colon G\/\\Gamma\\to \\R_{\\geq 0}$ with compact support contained in the open set  $U_{\\delta}\\Gamma=\\{g\\Gamma:g\\in U_{\\delta} \\}$ and satisfying\n\t\\begin{equation}\n\t\t\\label{eq:integralone}\n\t\t\\int_{G\/\\Gamma}\\psi_{\\delta}\\;\\text{d}m_{G\/\\Gamma}=1\\;.\n\t\\end{equation}\n\\end{itemize}\n\n\\begin{rmk}\n\tThe existence, for any $\\delta>0$, of a neighborhood $U_{\\delta}$ with the properties claimed above is routinely referred to in the literature (see, for instance,~\\cite{Eskin-McMullen}) as the \\emph{well-roundedness} property of the collection of balls $(B_R)_{R>0}$. A geometric condition of this sort affords to leverage equidistribution results to study lattice point counting problems.\n\\end{rmk}\n\nObserve that we may harmlessly replace $U_{\\delta}$ with\n\\begin{equation*} KU_{\\delta}=\\bigcup_{k\\in K}kU_{\\delta}\\;,\n\\end{equation*}\nand thus assume that $U_{\\delta}$ is saturated with respect to left translations by elements of $K$. Property~\\eqref{eq:neighborhoodcontainment} is unaffected: for any $k\\in K$ and $z\\in \\Hyp$, we have \n\\begin{equation*}\n\td_{\\Hyp}(k\\cdot z,i)=d_{\\Hyp}(k\\cdot z,k\\cdot i)=d_{\\Hyp}(z,i)\\;,\n\\end{equation*}\nas the subgroup $K$ fixes $i$ and acts by hyperbolic isometries; therefore  $k\\cdot B_{r}=B_{r}$ for any $k\\in K$ and any $r>0$. As a consequence of this, we might and shall assume that $\\psi_{\\delta}$ is $K$-invariant.\n\nWe now express, for any $R>0$, the ratio $N(R)\/m_{\\Hyp}(B_R)$ as\n\\begin{equation*}\n\t\\begin{split}\n\t\tF_R(\\Gamma)&=F_R(\\Gamma)-\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}+\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}\\\\\n\t\t&=\\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)(F_R(\\Gamma)-F_R(g\\Gamma))\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)+\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere the second inequality follows from the property in~\\eqref{eq:integralone}. Let us call $\\mathscr{E}_{\\delta}(R)$, for notational simplicity, the quantity\n\\begin{equation*}\n\t\\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)(F_R(\\Gamma)-F_R(g\\Gamma))\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\;;\n\\end{equation*}\nin view of~\\eqref{eq:averagedcountingfunction} applied to $\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}$, we may write \n\\begin{equation}\n\t\\label{eq:pointwiseaverage}\n\t\\begin{split}\n\t\tF_R(\\Gamma)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}+\\mathscr{E}_{\\delta}(R)\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl( e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{\\delta},\\mu}(R)+\\beta^-_{\\psi_{\\delta},\\mu}(R)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi_{\\delta},\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi_{\\delta},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi_{\\delta},1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi_{\\delta},1\/4}(R)\\biggr)+\\gamma_{\\psi_{\\delta}}(R)\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nWe estimate, for any $R>0$,\n\\begin{equation}\n\t\\label{eq:Edelta}\n\t|\\mathscr{E}_{\\delta}(R)|\\leq \\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)|F_R(\\Gamma)-F_R(g\\Gamma)|\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\n\t\\leq \\sup_{g\\in U_{\\delta}}|F_R(\\Gamma)-F_R(g\\Gamma)|\\;,\n\\end{equation}\nthe last inequality being a consequence of~\\eqref{eq:integralone} and the fact that $\\text{supp}\\;\\psi_{\\delta}\\subset U_{\\delta}\\Gamma$.\n\nNow, for any $g\\in U_{\\delta}$, we have\n\\begin{equation}\n\t\\label{eq:FR}\n\t\\begin{split}\n\t\t|F_R(\\Gamma)-F_R(g\\Gamma)|&=\\frac{\\bigl||\\Gamma\\cdot H\\cap B_R|-|g\\Gamma\\cdot H\\cap B_R|\\bigr|}{m_{\\mathbb{H}}(B_R)}=\\frac{\\bigl||\\Gamma\\cdot H\\cap B_R|-|\\Gamma\\cdot H\\cap g^{-1}\\cdot B_R|\\bigr|}{m_{\\mathbb{H}}(B_R)}\\\\\n\t\t&\\leq \\frac{\\bigl|\\Gamma\\cdot H\\cap \\bigl(\\bigl(\\bigcup_{g\\in U_{\\delta}}g\\cdot B_R\\bigr)\\setminus \\bigl(\\bigcap_{g\\in U_{\\delta}}g\\cdot B_{R}\\bigr)\\bigr)\\bigr|}{m_{\\mathbb{H}}(B_R)}\\leq\\frac{ N(R+\\delta)-N(R-\\delta)}{m_{\\mathbb{H}}(B_R)}\\\\\n\t\t&=F_{R+\\delta}(\\Gamma)\\frac{m_{\\mathbb{H}}(B_{R+\\delta})}{m_{\\mathbb{H}}(B_R)}-F_{R-\\delta}(\\Gamma)\\frac{m_{\\mathbb{H}}(B_{R-\\delta})}{m_{\\mathbb{H}}(B_R)}\\;,\n\t\\end{split}\n\\end{equation}\nwhere the second-to-last inequality follows from~\\eqref{eq:neighborhoodcontainment}. Choose $R_0=R_0(\\Gamma)>0$ such that the quantities\n\\begin{equation*}\n\tM=\\sup_{r\\geq R_0}F_{r}(\\Gamma) \\quad \\text{ and }\\quad m=\\inf_{r\\geq R_0}F_r(\\Gamma)\n\\end{equation*}\nare non-zero and finite\\footnote{A straightforward modification of the effective argument we are running leads to the well-known non-effective convergence \n\t\\begin{equation*}\n\t\tF_{R}(\\Gamma)\\overset{R\\to\\infty}{\\longrightarrow}\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\in \\R_{>0}\\;.\n\t\\end{equation*}\n}. Plugging~\\eqref{eq:FR} into~\\eqref{eq:Edelta}, we get that, for any $R\\geq 2R_0$ and $\\delta<R_0$, \n\\begin{equation}\n\t\\label{eq:neighboringvalues}\n\t\\begin{split}\n\t\t&|\\mathscr{E}_{\\delta}(R)|\\leq (M-m)\\biggl(\\frac{m_{\\mathbb{H}}(B_{R+\\delta})}{m_{\\mathbb{H}}(B_R)}-\\frac{m_{\\mathbb{H}}(B_{R-\\delta})}{m_{\\mathbb{H}}(B_R)}\\biggr)\\\\\n\t\t&=\\frac{(M-m)}{1-2e^{-R}+e^{-2R}}\\biggl(e^{\\delta}(1-2e^{-(R+\\delta)}+e^{-2(R+\\delta)})-e^{-\\delta}(1-2e^{-(R-\\delta)}+e^{-2(R-\\delta)})\\biggr)\\\\\n\t\t&\\leq(M-m)(ce^{\\delta}-e^{-\\delta})\\;,\n\t\\end{split}\n\\end{equation}\nwhere we might for example take $c=c_{\\Gamma}=\\frac{1+e^{-2R_0}}{1-2e^{-R_0}}$.\n\nWe now let the parameter $\\delta$ be a function of the radius $R$; for reasons which we will shortly elucidate (see Remark~\\ref{rmk:exponentialdecay}), we let $\\delta=\\delta(R)=e^{-\\eta R}$ in~\\eqref{eq:pointwiseaverage}, for a certain $\\eta>0$ to be determined later on. In this way, we obtain an expression of the form\n\\begin{equation}\n\t\\label{eq:Rfunctiondelta}\n\t\\begin{split}\n\t\tF_R(\\Gamma)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}+\\mathscr{E}_{e^{-\\eta R}}(R)\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+\\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+\\gamma_{\\psi_{e^{-\\eta R}}}(R)\\biggr)\\;,\n\t\\end{split}\n\\end{equation}\nwhich does not depend on the parameter $\\delta$ any longer.\n\nMultiplying by $m_{\\mathbb{H}}(B_R)$ on both sides of~\\eqref{eq:Rfunctiondelta} yields\n\\begin{equation}\n\t\\label{eq:pointwisecounting}\n\t\\begin{split}\n\t\tN(R)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}m_{\\mathbb{H}}(B_R)+\\pi(1-2e^{-R}+e^{-2R})e^{R}\\mathscr{E}_{e^{-\\eta R}}(R)\\\\\n\t\t&+\\pi(1-2e^{-R}+e^{-2R})\\;\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+ \\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t&\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{\\frac{1-\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)+e^{\\frac{1+\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+e^{R}\\gamma_{\\psi_{e^{-\\eta R}}}(R)\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\n\nIn order to reach an accurate upper bound for the error\n\\begin{equation*}\n\tE(R)=\\biggl|N(R)-\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}m_{\\mathbb{H}}(B_R)\\biggr|\\;,\n\\end{equation*}\nin our counting problem, it remains to determine which are the highest-order terms in the expansion~\\eqref{eq:pointwisecounting}. To this end, it is relevant to estimate the Sobolev norms of the functions $\\psi_{e^{-\\eta R}}$ for $R>0$, because of the bounds in~\\eqref{eq:betabound} and~\\eqref{eq:gammabound}.\n\n\n\\begin{lem}\n\t\\label{lem:growthSobolev}\n\tFor any $0<\\delta<1$, the function $\\psi_{\\delta}$ can be chosen to satisfy \n\t\\begin{equation}\n\t\t\\label{eq:decaySobolev}\n\t\t\\norm{\\psi_{\\delta}}_{W^{s}}\\leq \\delta^{-(1+s)}\\norm{\\psi_1}_{W^{s}}\n\t\\end{equation}\n\tfor any $s>0$.\n\\end{lem}\n\\begin{proof}\n\tRecall that $\\psi_\\delta$ is assumed to be $K$-invariant or, in other words, a smooth compactly supported function on the two-dimensional manifold $K\\bsl G\/\\Gamma$. Since any Riemannian metric on $K\\bsl G\/\\Gamma$ is equivalent, on a fixed compact coordinate ball containing the identity coset $Ke\\Gamma$, to the Euclidean metric on a compact neighborhood of the origin in $\\R^{2}$ (cf.~\\cite[Lem.~13.28]{Lee}), the problem of constructing $\\psi_{\\delta}$ so to meet our requirement can be transferred to the Euclidean plane. Specifically, we would like to construct a collection $(\\psi_{\\delta})_{0<\\delta\\leq1}$ of mollifiers (cf.~\\cite[Sec.~4.4]{Brezis}) so that~\\eqref{eq:decaySobolev} is satisfied, where $\\norm{\\cdot}_{W^{s}}$ are now the standard fractional Sobolev norms on $\\R^{2}$. A straightforward computation allows to ascertain that the customary choice \n\t\\begin{equation*}\n\t\t\\psi_{\\delta}(x)=\\frac{1}{\\delta^2}\\psi_1\\biggl(\\frac{x}{\\delta}\\biggr)\\;,\\quad x\\in \\R^{2},\n\t\\end{equation*}\n\twhere $\\psi_1$ is a fixed compactly supported smooth nonnegative function with unit average over $\\R^{2}$, fulfills~\\eqref{eq:decaySobolev}. \n\\end{proof}\n\nHenceforth, we assume that the collection $(\\psi_{\\delta})_{0<\\delta<1}$ satisfies the condition in Lemma~\\ref{lem:growthSobolev}.\n\nWe remind the reader that we indicate with $\\mu_*$ the spectral gap of the hyperbolic surface $S=\\Gamma\\bsl \\Hyp$, that is, the infimum of the set $\\text{Spec}(\\square)\\cap \\R_{>0}$. Also, we denote by $\\nu_*$ the complex number defined by the properties $\\nu_*\\in \\R_{\\geq 0}\\cup i\\R_{>0}$ and $1-\\nu_*^2=4\\mu_*$.\n\nSince $e^{\\delta}-e^{-\\delta}\\sim 2\\delta$ for $\\delta\\sim 0$, we deduce from~\\eqref{eq:neighboringvalues} that the term $e^{R}\\mathscr{E}_{e^{-\\eta R}}(R)$ is at most of order $e^{(1-\\eta)R}$. On account of Lemma~\\ref{lem:growthSobolev}, the highest-order term in the expression \n\\begin{equation*}\n\t\\begin{split}\n\t\t&e^{\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+ \\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{\\frac{1-\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +e^{\\frac{1+\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)+\\varepsilon_0\\biggl(e^{\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+e^{R}\\gamma_{\\psi_{e^{-\\eta R}}}(R)\n\t\\end{split}\n\\end{equation*}\nis $e^{\\frac{1+\\Re{\\nu_*}}{2}R}\\beta_{\\psi_{e^{-\\eta R}},\\mu_*}(R)$; because of Lemma~\\ref{lem:growthSobolev} and~\\eqref{eq:betabound}, the latter is at most of order\n\\begin{equation*}\n\te^{\\frac{1+\\Re{\\nu_*}}{2}R}e^{(1+s)\\eta R}=e^{\\frac{1+\\Re{\\nu_*}+2(1+s)\\eta}{2}R}\\;.\n\\end{equation*}\n\n\\begin{rmk}\n\t\\label{rmk:exponentialdecay}\n\tThe reason for choosing $\\delta$ to decay exponentially fast with $R$ becomes now apparent: it is the only way to get a sensible comparison between the orders of the two terms considered above. \t\n\\end{rmk}\n\nOstensibly the optimal choice of the parameter $\\eta$ for our purposes is\n\\begin{equation*} \\eta=\\frac{1-\\Re{\\nu_*}}{2(2+s)}\\;,\n\\end{equation*}\nwhich realizes the equality of exponents\n\\begin{equation*}\n\t1-\\eta=\\frac{1+\\Re{\\nu_*}+2(1+s)\\eta}{2}\\;.\n\\end{equation*}\nBearing in mind that the $K$-invariance of $\\psi_{e^{-\\eta R}}$ allows to choose $s$ can arbitrarily close to $9\/2$ (cf.~Theorem~\\ref{thm:expandingonsurface}), it is straightforward to deduce that, setting $\\eta_*=\\frac{1}{13}(1-\\Re{\\nu_{*}})$,  we have\n\\begin{equation*}\n\t\\lim\\limits_{R\\to\\infty}\\frac{E(R)}{e^{(1-\\eta_{*}+\\varepsilon)R}}=0\n\\end{equation*} \nfor any $\\varepsilon>0$, which establishes Theorem~\\ref{thm:countingproblem}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\footnotesize\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\sectlabel{intro}\n\nAmong the many exciting new applications of quantum physics\nin the realm of computation and information theory,\nI~am particularly fond of quantum cryptography,\nquantum computing and quantum teleportation~\\cite{LNCS}.\nQuantum cryptography allows for the confidential transmission\nof classical information under the nose of an eavesdropper,\nregardless of her computing power or technological\nsophistication~\\cite{BB84,BBBSS,sciam}.   Quantum computing allows for an\nexponential amount of computation to take place simultaneously in a single\npiece of hardware~\\cite{feynman,deutsch}; \nof~particular interest is the ability of quantum computers to factorize\nnumbers very efficiently~\\cite{shor}, with dramatic implications for\nclassical cryptography~\\cite{RSA}.  Quantum teleportation allows for the\ntransmission of quantum information to a distant location despite the\nimpossibility of measuring or broadcasting the information to be\ntransmitted~\\cite{BBCJPW}. Each of these concepts had a strong overtone of\nscience fiction when they were first introduced.\n\nIf~asked to rank these ideas on a scale of technological\ndifficulty, it is tempting to think that quantum cryptography\nis easiest while quantum teleportation is the most outrageous---especially\nwhen it comes to teleporting goulash~\\cite{IBM}!\nThis ranking is correct with respect to quantum cryptography,\nwhose feasibility has been demonstrated by several experimental\nprototypes capable of reliably transmitting confidential information\nover distances of tens of kilometres~\\cite{townsend,gisin,hughes}.\nThe situation is less clear when it comes to comparing the technological\nfeasibility of quantum computing with that of quantum teleportation.\n\nOn~the one hand, quantum teleportation can be implemented with a quantum\ncircuit that is much simpler than that required by any\nnontrivial quantum computational task: the~state of an\narbitrary qubit (quantum bit) can be teleported with as few as two quantum\nexclusive-or (controlled-not) gates.  Thus, quantum teleportation\nis significantly easier to implement than quantum computing if we\nare concerned only with the complexity of the required circuitry.\n\nOn~the other hand, quantum computing is meaningful\neven if it takes place very quickly---indeed its primary purpose\nis increased computational speed---and within a small region of space.\nQuite the opposite,\nthe interest of quantum teleportation would be greatly reduced if the actual\nteleportation had to take place immediately after the required preparation.\nThus, a working demonstration of quantum teleportation is likely to be seen\nbefore the quantum factorization of even a very small integer is achieved, but\nquantum teleportation across significant time and space will have to await a\ntechnology that allows for the efficient long-term storage of quantum\ninformation.\nNevertheless, it may be that short-distance quantum teleportation\nwill play a role in transporting quantum information inside quantum\ncomputers.  Thus we see that the fates of quantum computing and\nquantum teleportation are entangled!\n\n\\section{Quantum teleportation}\\sectlabel{teleport}\n\nRecall that any attempt at measuring quantum information\ndisturbs it irreversibly and yields incomplete information.\nThis makes it impossible to transmit quantum information\nthrough a classical channel.  Recall also that the purpose\nof quantum teleportation~\\cite{BBCJPW} is to circumvent this\nimpossibility so as to allow the faithful transmission of\nquantum information between two parties, conventionally\nreferred to as Alice and Bob.\n\nIn~order to achieve teleportation, Alice and Bob must\nshare prior quantum entanglement.  This is usually explained in terms\nof Einstein--Podolsky--Rosen nonlocal quantum states~\\cite{EPR}\nand Bell measurements, which makes the process seem very mysterious.\nThe~purpose of this note is to show how to achieve quantum teleportation\nvery simply in terms of quantum computation.  As~interesting side\nproduct, we obtain a quantum circuit with the unusual feature that\nthere are points in the circuit at which the quantum information can\nbe completely disrupted by a measurement---or~some types of interaction\nwith the environment---without ill effects:\nthe same final result is obtained whether or not measurement takes place.\nThis is true despite that fact that the qubits affected by these\nmeasurements are entangled with the other qubits carried by the\ncircuit, which should make these measurements even more damaging.\n\nOf~course, the uncanny power of quantum computation draws in parts\non nonlocal effects inherent to quantum mechanics.\nThe~quantum teleportation circuit described in~\\sect{circuit} is not really\ndifferent in principle from the original idea~\\cite{BBCJPW} since\nit uses quantum computation to create and measure nonlocal states.\nNevertheless it sheds new light on teleportation, at least from a\npedagogical point of view, since it makes the process completely\nstraightforward to anyone who believes that quantum computation is\na reasonable proposition.  Moreover, this circuit could genuinely be\nused for teleportation purposes inside a quantum computer.\nFinally, the surprising resilience of this circuit to measurements\nperformed while it is processing information\nmay turn out to have relevance to quantum error correction. \n\n\\section{The basic ingredients}\\sectlabel{ingredients}\n\nAs is often the case with quantum computation, we shall need two basic\ningredients: the exclusive-or gate (also known as controlled-not), which\nacts on two qubits at once, and arbitrary unitary operations on single qubits.\nLet~\\ket{0} and~\\ket{1} denote basis states for single qubits and recall\nthat pure states are given by linear combination of basis states\nsuch as \\mbox{$\\ket{\\psi}=\\alpha\\ket{0}+\\beta\\ket{1}$} where $\\alpha$ and\n$\\beta$ are complex numbers such that \\mbox{$\\norm{\\alpha}+\\norm{\\beta}=1$}.\n\nThe quantum exclusive-or (XOR), denoted as follows,\n\\begin{center}\n\\begin{picture}(180,60)\n\\thicklines\n\\put(0,40){\\makebox(20,20){\\sf a}}\n\\put(25,50){\\line(1,0){130}}\n\\put(90,50){\\circle*{10}}\n\\put(160,40){\\makebox(20,20){\\sf x}}\n\\put(0,0){\\makebox(20,20){\\sf b}}\n\\put(25,10){\\line(1,0){130}}\n\\put(90,10){\\circle{20}}\n\\put(160,0){\\makebox(20,20){\\sf y}}\n\\put(90,0){\\line(0,1){50}}\n\\end{picture}\n\\end{center}\nsends \\ket{00} to \\ket{00}, \\ket{01} to \\ket{01}, \\ket{10} to \\ket{11}\nand \\ket{11} to~\\ket{10}.  In~other words,\n{\\em provided the input states at {\\sf a} and {\\sf b} are in basis states},\nthe output state at {\\sf x} is the same as the input state at {\\sf a},\nand the output state at {\\sf y} is the exclusive-or of the two input states\nat {\\sf a} and~{\\sf b}.  This is also known as the controlled-not gate because\nthe state carried by the {\\em control} wire ``{\\sf ax}'' is not disturbed\nwhereas the state carried by the {\\em controlled} wire ``{\\sf by}'' is\nflipped if and only if the state on the control wire was~\\ket{1}.\nNote that the classical interpretation given above no longer holds \nif the input qubits are not in basis states: it is possible for\nthe output state on the control wire (at~{\\sf x}) to be different from its\ninput state (at~{\\sf a}).  Moreover, the joint state of the output\nqubits can be entangled even if the input qubits were not, and\nvice versa.\n\nIn addition to the quantum exclusive-or, we shall need two single-qubit\nrotations {\\sf L} and {\\sf R}, and two single-qubit conditional phase-shifts\n{\\sf S} and~{\\sf T}\\@.  Rotation {\\sf L} sends \\ket{0} to\n\\mbox{$(\\ket{0}+\\ket{1})\/\\sqrt{2}$} and \\ket{1}\nto \\mbox{$(-\\ket{0}+\\ket{1})\/\\sqrt{2}$},\nwhereas {\\sf R}~sends \\ket{0} to\n\\mbox{$(\\ket{0}-\\ket{1})\/\\sqrt{2}$} and \\ket{1}\nto \\mbox{$(\\ket{0}+\\ket{1})\/\\sqrt{2}$}.\nNote that \\mbox{$\\mbox{\\sf LR}\\ket{\\psi}=\\mbox{\\sf RL}\\ket{\\psi}=\\ket{\\psi}$}\nfor any qubit~\\ket{\\psi}.\nConditional phase-shift {\\sf S} sends \\ket{0} to $i\\ket{0}$\nand leaves \\ket{1} undisturbed, whereas {\\sf T} sends\n\\ket{0} to $-\\ket{0}$ and \\ket{1} to $-i\\ket{1}$.\nIn~terms of unitary matrices, the operations are\n\\[\n\\begin{array}{lll}\\multicolumn{2}{l}{\n\\mbox{\\sf L} = {\\displaystyle \\frac{1}{\\sqrt{2}}}\n\\left( \\begin{array}{rr}1&-1\\\\1&1\\end{array} \\right)} &\n\\mbox{\\sf R} = {\\displaystyle \\frac{1}{\\sqrt{2}}}\n\\left( \\begin{array}{rr}1&1\\\\-1&1\\end{array} \\right) \\\\[7mm]\n\\mbox{\\sf S} = \\left( \\begin{array}{rr}i&0\\\\0&1\\end{array} \\right)\n& \\mbox{and} &\n\\mbox{\\sf T} = \\left( \\begin{array}{rr}-1&0\\\\0&-i\\end{array} \\right)\n\\end{array}\n\\]\nif $\\alpha\\ket{0}+\\beta\\ket{1}$ is represented by vector \\state{\\alpha}{\\beta}.\nSimilarly the quantum exclusive-or operation is given by matrix\n\\[\\mbox{\\sf XOR} = \\left(\n\\begin{array}{llll}1&0&0&0\\\\0&1&0&0\\\\0&0&0&1\\\\0&0&1&0\\end{array}\\right)\\]\nif \\mbox{$\\alpha\\ket{00}+\\beta\\ket{01}+\\gamma\\ket{10}+\\delta\\ket{11}$} is\nrepresented by the transpose of vector \\mbox{$(\\alpha,\\beta,\\gamma,\\delta)$}.\n\n\\section{The teleportation circuit}\\sectlabel{circuit}\n\nConsider the following quantum circuit.  Please disregard the dashed line\nfor the moment.\n\\begin{center}\n\\begin{picture}(495,210)(0,-40)\n\\thicklines\n\\put(0,135){\\makebox(20,20){\\sf a}}\n\\put(25,145){\\line(1,0){155}}\n\\put(150,145){\\circle*{10}}\n\\put(180,130){\\framebox(30,30){\\sf R}}\n\\put(210,145){\\line(1,0){60}}\n\\put(270,130){\\framebox(30,30){\\sf S}}\n\\put(300,145){\\line(1,0){70}}\n\\put(335,145){\\circle{20}}\n\\put(370,130){\\framebox(30,30){\\sf S}}\n\\put(400,145){\\line(1,0){70}}\n\\put(435,145){\\circle{20}}\n\\put(475,135){\\makebox(20,20){\\sf x}}\n\\put(0,70){\\makebox(20,20){\\sf b}}\n\\put(25,80){\\line(1,0){25}}\n\\put(50,65){\\framebox(30,30){\\sf L}}\n\\put(80,80){\\line(1,0){390}}\n\\put(110,80){\\circle*{10}}\n\\put(150,80){\\circle{20}}\n\\put(285,80){\\circle*{10}}\n\\put(475,70){\\makebox(20,20){\\sf y}}\n\\put(0,5){\\makebox(20,20){\\sf c}}\n\\put(25,15){\\line(1,0){345}}\n\\put(110,15){\\circle{20}}\n\\put(285,15){\\circle{20}}\n\\put(335,15){\\circle*{10}}\n\\put(370,0){\\framebox(30,30){\\sf T}}\n\\put(400,15){\\line(1,0){70}}\n\\put(435,15){\\circle*{10}}\n\\put(110,5){\\line(0,1){75}}\n\\put(150,70){\\line(0,1){75}}\n\\put(285,5){\\line(0,1){75}}\n\\put(335,15){\\line(0,1){140}}\n\\put(435,15){\\line(0,1){140}}\n\\put(475,5){\\makebox(20,20){\\sf z}}\n\\thinlines\n\\put(240,-40){\\dashbox{7.5}(0,210){}}\n\\put(25,-40){\\makebox(215,30){\\sl Alice}}\n\\put(240,-40){\\makebox(230,30){\\sl Bob}}\n\\end{picture}\n\\end{center}\nLet \\ket{\\psi} be an arbitrary one-qubit state.  Consider\nwhat happens if you feed \\ket{\\psi00} in this circuit,\nthat is if you set upper input {\\sf a} to \\ket{\\psi} and both other\ninputs {\\sf b} and {\\sf c} to~\\ket{0}.  It~is a straightforward\nexercise to verify that state \\ket{\\psi} will be transferred\nto the lower output~{\\sf z}, whereas both other outputs {\\sf x} and {\\sf y}\nwill come out in state \\mbox{$\\ket{\\phi}=(\\ket{0}+\\ket{1})\/\\sqrt{2}$}.\nIn~other words the output will be~\\ket{\\phi\\phi\\psi}.\nIf~the two upper outputs are measured in the standard basis\n\\mbox{(\\ket{0} versus \\ket{1})}, two random classical bits will be obtained\nin addition to quantum state \\ket{\\psi} on the lower \\mbox{output}.\n\nNow, let us consider the state of the system at the dashed line.\nA~simple calculation shows that all three qubits are entangled.\nWe~should therefore be especially careful not to\ndisturb the system at that point.  Never\\-the\\-less, let us \nmeasure the two upper qubits, leaving the lower qubit undisturbed.\nThis~measurement results in two purely random classical bits $u$ and~$v$,\nbearing no correlation whatsoever with the original state~\\ket{\\psi}.\nLet~us now turn $u$ and $v$ back into quantum bits and reinject\n\\ket{u} and \\ket{v} in the circuit immediately after the dashed line.\n\nNeedless to say that the quantum state carried at the dashed line\nhas been completely disrupted by this measurement-and-resend process.\nWe~would therefore expect this disturbance to play havoc with the\nfinal output of the circuit.  Not~at all!  In~the end, the state\ncarried at {\\sf xyz} is~\\ket{uv\\psi}.  In~other words, \\ket{\\psi}\nis still obtained at~{\\sf z} and the other two qubits, if measured,\nare purely random provided we forget the measurement outcomes at the\ndashed line.  Another way of seeing this phenomenon is that the\noutcome of the circuit will not be altered if the state of the upper\ntwo qubits leaks to the environment (in the standard basis)\nat the dashed line.\n\nTo~turn this circuit into a quantum teleportation \\mbox{device}, we need\nthe ability to store qubits.  Assume Alice prepares two qubits in\nstate \\ket{0} and pushes them through the first two gates of the\ncircuit.\n\\begin{center}\n\\begin{picture}(325,80)(0,5)\n\\thicklines\n\\put(0,60){\\makebox(20,20){\\ket{0}}}\n\\put(25,70){\\line(1,0){25}}\n\\put(50,55){\\framebox(30,30){\\sf L}}\n\\put(80,70){\\line(1,0){75}}\n\\put(160,60){\\makebox(20,20){$\\sigma$}}\n\\put(110,70){\\circle*{10}}\n\\put(0,5){\\makebox(20,20){\\ket{0}}}\n\\put(25,15){\\line(1,0){300}}\n\\put(110,15){\\circle{20}}\n\\put(330,5){\\makebox(20,20){$\\rho$}}\n\\put(110,5){\\line(0,1){65}}\n\\end{picture}\n\\end{center}\nShe keeps the upper qubit $\\sigma$ in quantum memory and gives the other,\n$\\rho$, to Bob.  [We~do not denote these qubits by kets because\nthey are not individual pure states: \\mbox{together} they are in state\n\\mbox{$\\Phi^{+}=(\\ket{00}+\\ket{11})\/\\sqrt{2}$}.]\nAt~some later time, Alice receives a mystery qubit in \\mbox{unknown}\nstate~\\ket{\\psi}. In~order to teleport this qubit to Bob, she releases\n$\\sigma$ from her quantum memory and pushes it together with the mystery\nqubit through the next two gates of the circuit.  She measures both\noutput wires to turn them into classical bits $u$ and~$v$.\n\\begin{center}\n\\begin{picture}(200,80)(75,70)\n\\thicklines\n\\put(75,125){\\makebox(20,20){\\ket{\\psi}}}\n\\put(100,135){\\line(1,0){80}}\n\\put(150,135){\\circle*{10}}\n\\put(180,120){\\framebox(30,30){\\sf R}}\n\\put(210,135){\\line(1,0){40}}\n\\put(255,125){\\makebox(20,20){$u$}}\n\\put(75,70){\\makebox(20,20){$\\sigma$}}\n\\put(100,80){\\line(1,0){150}}\n\\put(150,80){\\circle{20}}\n\\put(255,70){\\makebox(20,20){$v$}}\n\\put(150,70){\\line(0,1){65}}\n\\end{picture}\n\\end{center}\nTo complete teleportation, Alice has to communicate $u$ and $v$ to Bob\nby way of a classical communication channel.  Upon reception of the\nsignal, Bob creates quantum states \\ket{u} and \\ket{v} from the\nclassical information received from Alice, he releases the qubit\n$\\rho$ he had kept in quantum memory, and he pushes all three\nqubits into his part of the circuit (on~the right of the dashed line).\nFinally Bob may wish to measure the two upper qubit at {\\sf x} and~{\\sf y}\nto make sure that he gets $u$ and~$v$; otherwise something went wrong\nin the teleportation apparatus.  At~this point, teleportation is\ncomplete as Bob's output {\\sf z} is in state~\\ket{\\psi}.\nNote that this process works equally well if Alice's mystery qubit\nis not in a pure state.  In~particular, Alice can teleport to Bob\nentanglement with an arbitrary auxiliary system, possibly outside both\nAlice's and Bob's labo\\-ra\\-tories.\n\nIn~practice, Bob need not use the quantum circuit shown right of\nthe dashed line at~all.  Instead, he may choose classically one of 4\npossible rotations to apply to the qubit he had kept in quantum memory,\ndepending on the 2 classical bits he receives from Alice.\n(This would be more in tune with the original teleportation\nproposal~\\cite{BBCJPW}.)  This explains the earlier claim\nthat quantum teleportation can be achieved at the cost of\nonly two quantum exclusive-ors: those of Alice.\nNever\\-theless, the unitary version of Bob's process\ngiven here may be more appealing than choosing classically\namong 4 courses of action if teleportation is used\ninside a quantum computer.\n\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nComposite supernova remnants (SNRs) are a subclass of core collapse SNRs where non-thermal radio emission is observed from both the expanding shell of the SNR, and from the PWN located inside it. The PWNe are powered by the relativistic outflows from the young neutron star and can be observed at all wavelengths, but mostly in the radio and X-rays.\nThe PWN morphology can provide crucial information on the properties of the outflow, the interacting ambient medium and the geometry of the pulsar powering it \\citep[see][and references therein for detailed understanding of the structure and evolution of PWNe]{gaensler2006}. The {\\it Chandra~\\\/} observatory with its excellent spatial resolution and high sensitivity has provided significant breakthroughs in the study of PWNe. Apart from steadily increasing the census on the number of PWNe discovered, it has provided a unique opportunity to study in detail their spatial and spectral structures and signatures of interaction with the surrounding medium \\citep{pwnchandra}. Most of these are however Galactic sources and there are no confirmed PWNe in our satellite galaxy the SMC, yet. To this day a total of 24 classified SNRs are known in the SMC \\citep[][and references therein]{haberl2012}. Out of them very few are PWN candidates. One such source, HFPK 334 has been recently dismissed as a background source \\citep{crawford2014}. Discovery of a PWN in the SMC will open a new window in the study of rotation powered pulsars in the satellite galaxy which has a rich history of active star formation and is therefore expected to host young energetic pulsars.\n\nIKT 16 is an X-ray and radio-faint SNR in the SMC first studied with {\\it XMM-Newton~\\\/} in a survey of known SNRs in the SMC \\citep{van2004}. In this study a region of harder \nX-ray emission was found at the centre of the remnant although it could not be probed further  due to poor statistics with a single exposure. \\cite{owen2011} further used eight more archival {\\it XMM-Newton~\\\/} {observations~\\\/} \n(total useful exposure 125 ks and off-axis angle 8-12$\\arcmin$) taken subsequently to carry out a multiwavelength study of the spatial and spectral properties of this SNR and the associated central source. The authors found substantial evidence that the unresolved source detected at the centre of the SNR by XMM-Newton, is a PWN associated with it.\nRadio images from the ATCA and MOST surveys displayed faint radio structures correlated with the X-ray throughout the remnant. The brightest feature was an extended radio emission \ncorresponding to the X-ray source and extending a distance of 40\\hbox{$^{\\prime\\prime}$}~ towards the centre of the SNR \\citep{owen2011}. This picture is consistent with a moving pulsar in the SNR.\n Further, the large size of the remnant suggested that the SNR is in the adiabatic Sedov phase of evolution. In the Sedov model the age of the SNR was $\\sim$ 14.7 kyr\nand implied the PWN may have interacted with and been compressed by the reverse shock. The bright central source was located 30\\hbox{$^{\\prime\\prime}$}~ from the SNR centre implying a transverse kick \nvelocity of $\\sim$ 580 km s$^{-1}$. \n\nTo further resolve the nature of the central source and establish the presence of the first PWN in the SMC, a 40 ks on-axis {\\it Chandra~\\\/} observation was solicited. In this paper we describe the results of the {\\it Chandra~\\\/} {observation~\\\/} of IKT 16. We have performed detailed spatial and spectral analysis of the hard X-ray emission near the centre of the SNR, previously unresolved. With the unprecedented spatial resolution of {\\it Chandra~\\\/} we have resolved the source into a central point source (a putative pulsar) and an extended emission surrounding it, indicating a PWN nature for the source. The {observations~\\\/} and analysis are described in section 2. Section 3 presents the spatial analysis including the imaging and morphological fitting of the source. Section  4 presents the detailed spectral analysis of the central source, the surrounding nebula and its decomposed regions. Section 5 presents the discussion  and section 6 the conclusions.\n\n\\begin{figure*}\n\\hspace*{-0.52cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.5]{src-col-new.ps}}\n\\hspace*{-1.5cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.45]{model-new-contour.ps}}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.52]{src_region-col-new.ps}}\n\\vspace*{2.0cm}\n\\hspace*{0.18 cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.42]{new_extended_smoothed_image.ps}}\n\\caption{a({\\it top left}): {\\it Chandra~\\\/} ACIS-S (0.5--8 keV) full resolution image of the PWN in IKT 16. The image size is 20\\hbox{$^{\\prime\\prime}$} x 25\\hbox{$^{\\prime\\prime}$}. The scale is in square root, the X and Y axis in degrees, and units are counts for all the images. The brightest pixel in the centre corresponds to the putative pulsar. b({\\it top right}): Best-fit model of the PWN (0.5--8 keV). Overlayed are contours from the data which have been smoothed with a Gaussian of $\\sigma = 3$\\hbox{$^{\\prime\\prime}$}. The contour levels are plotted at values of 0.7, 1.6 and 30 counts arcsecs$^{-2}$. \nc({\\it bottom left}): Same as the {\\it top left} figure showing the regions used for spectral extraction. The central circle corresponds to the pulsar, and the rectangular box the entire nebula with the central point source removed. The dashed circle corresponds to the outer boundary of the inner, and the inner boundary of the outer nebular extraction regions respectively. d({\\it bottom right}): Larger (0.5--2 keV) image centred on the PWN of IKT 16 (ACIS-S3), with the bottom right corner of the image corresponding to ACIS-S2. The box region used for nebular extraction is shown in black solid lines and the background annular region used for spectral extraction (inner radius 34\\hbox{$^{\\prime\\prime}$}) is shown in white solid lines. The white dashed circle indicates the position and extent of the SNR in \\cite{owen2011}. The point source contribution at the centre has been removed and replaced with values obtained by interpolation from the nebular region. The image has been smoothed using a Gaussian kernel of width 5\\hbox{$^{\\prime\\prime}$}.}\n\\label{images}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.62,angle=-90]{ikt16-new.eps}\n\\caption{ATCA 2.1\\,GHz image of PWN IKT\\,16 overlaid with Chandra contours (\\textbf{1.25, 5, 25 and 100 c\/arcsec$^{2}$}). The cross marked in green denotes the centre of the SNR in IKT 16.\nThe synthesis beam size of $3.38\\arcsec \\times 3.16\\arcsec$ at PA of $-17.8^{\\circ}$ is shown on the bottom left corner. The 2.1\\,GHz image grayscale bar is shown in units of mJy\/beam.}\n\\label{radio}\n\\end{figure*}\n\n\n\\section{Observations \\& Analysis}\n\\label{sec:obs}\n\n\\subsection{{\\it Chandra~\\\/} ACIS {observation~\\\/} and data reduction}\n\\label{sec:xobs}\n\nThe {\\it Chandra~\\\/} {observation~\\\/} (ObsID 13773) was carried out with the ACIS-S as the primary instrument in the timed exposure mode. The {observation~\\\/} was performed on 09-02-2013 with the central source of  IKT 16 positioned at the aimpoint of the S3 CCD with an exposure of 38.5 ks. \nIt is worthwhile to mention here that there was a previous {\\it Chandra~\\\/} {observation~\\\/} (obsID 2948, 9 ks) where the source was detected \\citep{evans2010}. But the short duration of this {observation~\\\/} and its large off axis\nangle prevented further investigation of the source from this data. Data reduction was performed using CIAO 4.6 using CALDB 4.6.3 and the standard analysis procedure prescribed \\footnote{http:\/\/cxc.harvard.edu\/ciao\/}.\nThe level 1 event file was reprocessed with \\textit{chandra$\\_$repro}, which incorporates the subpixel repositioning algorithm EDSER as a default for attaining better angular resolution for sources near the centre of the FOV. The effective exposure of the observation after filtering was 38.5 ks. We also checked for possible presence of pileup in the data. Using the tool {\\it pileup$\\_$map}, the estimated fraction of pileup in the centremost pixel is < $5\\%$. Hence the effect is not important for our {observation~\\\/}.\n\nImages were created using the task \\textit{dmcopy}. Spatial analysis was performed using the \\textit{Sherpa} analysis package 4.4 \\footnote{http:\/\/cxc.harvard.edu\/sherpa4.4\/}. The task {\\it specextract} was used for extracting source and background spectra and response files from regions of interest.\nThe selection of regions used for spectral extraction is described in the spectral analysis section. Spectra were fitted using \\textit{XSPEC v12.8.1} \\footnote{http:\/\/heasarc.gsfc.nasa.gov\/xanadu\/xspec\/}.\n \n\\subsection{Imaging}\n\\label{sec-im-an}\nFigure \\ref{images}a shows a full resolution {\\it Chandra~\\\/} image (0.5--8 keV) centred on the source near the centre of IKT 16, and zoomed into a region 20\\hbox{$^{\\prime\\prime}$} x 25\\hbox{$^{\\prime\\prime}$}~ wide. The source appears to be symmetrically elongated in the east-west direction with a bright point source located at its centre. The elongation measures about 5\\hbox{$^{\\prime\\prime}$}.\nTo accurately determine the position of the central source, we created a subpixel image of the same (at $\\frac{1}{5}$ of the ACIS pixel resolution) and applied the source detection algorithm {\\it celldetect}. The coordinates of the point source are RA(J2000)=$00^h58^m16.85^s$ Dec=$-72\\deg18\\arcm05.60\\hbox{$^{\\prime\\prime}$}$ considering an error of 0.6\\hbox{$^{\\prime\\prime}$}~ at 90 $\\%$ confidence level in absolute {\\it Chandra~\\\/} astrometry. The net count rate from the point source after subtracting the nebular component is 0.011 c\/s, and from the entire nebula after subtracting the point source contribution is 0.003 c\/s. The details of background subtraction are discussed in the spectral analysis section \\ref{sec-spec}. The point source is about three times\nbrighter than the extended emission. \nAfter removing the contribution from the bright point source at the centre, and smoothing the image with a Gaussian kernel of width 5\\arcsec, there is evidence of a diffuse emission extending further out up to $\\sim$ 30$\\hbox{$^{\\prime\\prime}$}$ (see Fig.~\\ref{images}d). This diffuse component is discussed in Sect.\\ref{diff-neb}. The same image shows hints of the presence of the SNR, especially the excess in the north coincident with that in the ${\\it XMM-Newton~\\\/}$ image of IKT 16 \\citep{owen2011}.\n\n\n\\subsection{Radio observations}\n\\label{sec:radioobs}\n\nOur new ATCA observations, project C2521 (CI: J. van Loon), used the Compact Array Broadband Backend (CABB) with the 6A array configuration at 2.1\\,GHz providing improved flux density estimates and resolution. These new images were acquired on 02-01-2012 with $\\sim10.48$~hours integration over the 12~hour observing session. The radio galaxy PKS 1934-638 was used as a primary flux calibration source for all observations, with the radio sources PKS 0230-790 and PKS 2353-686 used for phase calibration. A standard calibration process was carried out using the \\textsc{miriad} data reduction software package (Sault et al. 2011). In order to improve the fidelity and sensitivity of the final image, a single iteration of self-calibration was performed on the strongest sources in the field. A uniform weighting scheme was subsequently used throughout the imaging process, as it provided a balance between improving the theoretical rms noise while maintaining an adequate beam shape. Given the 2\\,GHz of bandwidth provided by CABB, images were formed using \\textsc{miriad} multi-frequency synthesis (\\textsc{mfclean}; Sault \\& Wieringa 1994). The same procedure was used for both {\\it U} and {\\it Q} Stokes parameter maps. However, there was no reliable detection in the {\\it U} or {\\it Q} intensity parameters associated with this object, implying a lack of polarisation at lower radio frequencies (2.1\\,GHz). The final image produced (after primary beam corrected), which possess a FWHM of $3.38\\arcsec \\times 3.16\\arcsec$ and a PA of $-17.8^{\\circ}$, has a 1$\\sigma$ rms noise level of 16\\,$\\mu$Jy. In Fig.~\\ref{radio} we show the ATCA 2.1\\,GHz surface brightness image of PWN IKT\\,16.\n   \n\\section{X-ray Spatial analysis}\n\\label{sec:xspec}\nTaking advantage of the excellent spatial resolution of {\\it Chandra}, we performed detailed spatial analysis of IKT 16 for the first time. To provide further evidence of the extended emission, and to probe the source extent we created a radial profile centred on the point source from the data.\n We compared this with the two dimensional model of the point spread function (PSF) to look for an excess indicating the extended emission. Further, we tested for signatures of asymmetry along the two halves of the nebular emission. Finally we performed a morphological fitting of the elongated structure with a simple model of the nebula centred on the point source, and determined its geometrical parameters like the size, ellipticity and the rotation angle.\n\\subsection{Radial profile}\nRadial profiles were created up to 1$\\hbox{$^\\prime$~\\\/}$ in two energy bands (0.5--2 keV, and 2--7 keV) by extracting net counts in circular annuli centred on the point source using the tool { \\it dmextract} and then rebinning to reach a reasonable statistical precision. The background was extracted from a circular region of the same area, away from the source region but inside the SNR.\nThe PSF of the {observation~\\\/} was simulated\nusing the {\\it Chandra~\\\/} ray tracer chaRT \\footnote{http:\/\/cxc.harvard.edu\/chart\/} which simulates the High Resolution Mirror Assembly based on the energy spectrum of the source and the {observation~\\\/} exposure. The output of chaRT was modelled with the software MARX \\footnote{http:\/\/cxc.harvard.edu\/chart\/threads\/marx\/} taking into account the instrumental effects and the EDSER subpixel algorithm to be consistent with the observational data. The best-fit spectrum of the point source (see section \\ref{spec-pnt}) was used. Fig.~\\ref{radial} shows the radial profile of the {observation~\\\/} along with the simulated PSF. The data at both the energy bands are consistent with a point source up to a radius < 1\\hbox{$^{\\prime\\prime}$}~ beyond which it clearly has higher net counts than expected from a point source simulated at the same position and with the same spectral parameters. This corresponds\nto the nebular component. \nAlthough the geometrical model (see section \\ref{sec:spa-mod}) indicates that the source FWHM is about 5\\hbox{$^{\\prime\\prime}$}, this exercise indicates that the source extends further beyond. This excess is best seen in the energy range of 0.5--2 keV. Its brightness decreases outwards like $r^{-1.5}$. Figure~\\ref{images}d indicates that this further extension is probably elongated in the east-west direction like the main X-ray nebula and the radio nebula.\n \n\\subsection{Investigating signatures of asymmetry in the nebula morphology}\n\\label{east-west-counts}\nVisual inspection of the extended source near the centre of IKT 16 from Fig.~\\ref{images}a, gives an impression of an elongated symmetric structure. However, we looked for the possibility of an east-west asymmetry\nby dividing the nebular structure into two halves along the axis going through the point source, and compared the total counts from the two regions. The number of counts in the east and west nebula are $50 \\pm 9$ and $67 \\pm 10$ respectively indicating that the west nebula may be slightly brighter than its other half. It cannot be determined conclusively as the values are consistent within their errors. We investigated this issue further through spectral analysis of the same regions in section \\ref{spec-east-west}. \n\n\n\n\n\\begin{table}\n\\centering\n\\caption{Parameters of the best-fit model to {pulsar wind nebula~\\\/} in IKT 16.\nModel used is a 2-D Gaussian function for the nebula and a constant background.\nTheta is the angle between the major axis and the north direction.}\n\\begin{tabular}{c c c}\n\\hline\n\nParameter  & Value & Units\\\\\n \\hline         \nFWHM   & 5.2$^{-0.9}_{+1.0}$ & arcsec \\\\\nEllipticity & 0.6 $\\pm 0.1$ & \\\\\nTheta & 82 $\\pm 7$ & degrees \\\\\nAmplitude & 2.9 $\\pm 1 $ & counts\/pixel \\\\\nBackground & 0.12 $\\pm 0.01$ & counts\/pixel \\\\\n\\hline    \n\\end{tabular}\n\\\\\n\\label{tabsherpa}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{IKT16BkgSubtractedProfile.eps}\n\\caption{Radial counts profile of the source (blue star: 0.5--2 keV; red triangle: 2--7 keV) plotted against the 2-D PSF (0.5--8 keV, in black crosses), clearly showing the extended nature of the source.}\n\\label{radial}\n\\end{figure}\n\n\\subsection{Spatial modelling}\n\\label{sec:spa-mod}\nSpatial fitting was performed on a larger region of size 35\\hbox{$^{\\prime\\prime}$} $\\times$ 45\\hbox{$^{\\prime\\prime}$} centring the source to detect the existence of a possible diffuse\n emission underlying the nebular one, and also constrain the background better. The PSF described in the \nprevious section was loaded as a table model in {\\it Sherpa} to model the point source emission. The remaining excess consisted of the background and the emission from the nebula. This was modelled with a constant background and a 2-D unnormalized Gaussian function (gauss2d model in {\\it Sherpa}), convolved with the PSF. In the initial iterations it was found that the position of the 2-D Gaussian used to model the nebula was consistent with the point source. In order to constrain the geometrical parameters of the nebula like its FWHM along its major axis, ellipticity and rotation angle better, its position was fixed to the point source henceforth.\nThe best-fit parameters were determined by the C-statistic \\citep{cash1979} and errors were estimated at 90$\\%$ confidence level. The residuals do not show any systematic pattern indicating that this analysis is not sensitive to any substructures apart from the elongated structure of the nebula. Fig.~\\ref{images}b shows the model with the contours from the data smoothed with a Gaussian of $\\sigma$ = 3\\hbox{$^{\\prime\\prime}$} overlayed on it. This highlights the fact that the model is a good description of the data. It is worth mentioning that we also tried to fit a 2-D Lorentz model with a varying power-law (beta2d model in {\\it Sherpa}) instead of the 2-D Gaussian used to model the nebula. We noticed that by freezing the rotation angle and ellipticity, the power-law index $\\alpha$ tends towards the maximum limit of 10 which closely approximates the 2-D Gaussian model. The difference in C-statistic between beta2d and gauss2d is 15 in the Lorentzian limit ($\\alpha$=1 in beta2d), and 2 in the Gaussian limit ($\\alpha$=10 in beta2d). We therefore concluded that the data does not favor broad wings and a 2-D Gaussian is preferred for the nebular emission.\n\nThe best fit parameters are listed in Table \\ref{tabsherpa}. The estimated FWHM of 5.2\\hbox{$^{\\prime\\prime}$}~is in agreement with that measured from the image of the source. It is inclined at an angle of 82\\hbox{$^\\circ$}~ between the long axis and the north direction and is aligned with the radio nebula (see Fig.~\\ref{radio}). \nAn ellipticity of 0.6 indicates a major to minor axis ratio of 0.8. Another important result is that there is no signature of displacement between the point source and the nebula.\nThe best-fit position of the centre of the nebula obtained corresponds to R.A.(J2000)=00:58:16.824 and DEC.(J2000)=-72.18:05.32 with an error of 0.1\\hbox{$^{\\prime\\prime}$} and 0.2\\hbox{$^{\\prime\\prime}$} on the x and y positions respectively. This is consistent with the position of the point source obtained with {\\it celldetect }.\n\n\n\\section{X-ray Spectral analysis}\n\\label{sec-spec}\nWe have performed detailed spectral analysis of the point source and the nebular component of IKT 16. In addition we have also investigated the diffuse component extending to larger scales.\nThe main deciding factors for the spectral analysis are the regions used for the spectral extraction and the background modelling, taking into account the contribution\nof the point source in the nebular spectrum and vice versa. This is described in the subsequent subsections. The analysis was performed in the energy range of 0.5--7 keV. C-statistic was used for spectral fitting and errors were estimated at 90$\\%$ confidence interval. To account for the photoelectric absorption by the interstellar gas, two absorption components were used as in \\cite{owen2011}. The first, {\\it phabs} component was fixed at the\nGalactic value of $6\\times10^{20}{\\rm ~erg~s^{-1}}$ (\\citealt{dickey90}), and a free absorption ({\\it vphabs}) component was used to account for absorption inside the SMC. This second component has metal abundances fixed at 0.2 solar, as is typical in the SMC (\\citealt{russell92}). \n\\subsection{Point source spectrum} \n\\label{spec-pnt}\nFor the point source, a circular region of radius 1.5 pixels was extracted centred on the best-fit coordinates of the source. The extraction region is shown in Fig.\n\\ref{images}c. A circular annulus outside the nebular extraction region was used for the background spectrum. Apart from this the\nastrophysical background due to the nebular contribution was also considered. For this we calculated the fraction of nebular emission contributing in the point source extraction region (from the best-fit morphological model of the source), and accounted for it as an additional model component in the spectral fitting, with parameters fixed to the best fit values obtained in section \\ref{spec-pwn}.\n\nWe tried to fit the spectrum with several models including an absorbed power-law, an absorbed blackbody and a combination of both models. The absorbed blackbody model leads to an unacceptable spectral fit with a difference of 54 in C-statistic value between the two models with respect to the power-law model. The absorbed power-law model provides a good fit as illustrated on Fig.~\\ref{spec-1}a. The best fit parameters along with the flux are tabulated in Table \\ref{table-specfit}. The addition of a blackbody component was not required and did not improve the fit significantly. Also, the additional absorption column density inside the SMC could not be constrained well and was prone to large error bars as can be seen from Table \\ref{table-specfit}. Its value was however\nconsistent with that obtained from the previous analysis of the {\\it XMM-Newton~\\\/} {observations~\\\/} by \\cite{owen2011} which had smaller errors associated with it probably due to the larger number of counts. \n\\begin{figure*}\n\\centering\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{pt_src_without_resi.ps}}\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{total_neb_new_resi.ps}}\n \\caption{The figures show the ACIS-S spectra for the putative pulsar (a,{\\it left}) and the entire PWN (b,{\\it right}).\nThe solid lines correspond to the respective best-fit spectral models. The plots have been rebinned for visual clarity.}\n\\label{spec-1}\n\\end{figure*}\n\\subsection{Nebula spectrum}\n\\label{spec-pwn}\nFor the spectral extraction of the extended source, a box region was used with the region corresponding to the point source excised (see Fig.~\\ref{images}c).\nIts size was optimized from the morphological modelling of the nebula, and its centre was made coincident with the point source. The background was taken from the same annular region that was used as a background for the point source spectral extraction. For the additional astrophysical background contributed by the point source leakage in the nebular region,\nwe extracted a spectrum from the events from the PSF simulated by chaRT with spectral parameters similar to those of the point source, in the same region that was used for the nebular spectral\nextraction, and added it to the internal background spectrum as an additional background component. \n\nWe used an absorbed power-law as the spectral model for all the nebular fits described henceforth. In contrast to the point source spectrum, the spectrum of the nebula is very soft. Keeping the absorption inside the SMC free in these fits leads to strong correlation between the absorption column density and the power-law index $\\Gamma$. To avoid this it is better to constrain this parameter from the point source spectrum, and freeze it to this value for the nebular spectral fits. However, the better capability of the {\\it XMM~\\\/} {observations~\\\/} of IKT 16 to constrain the local absorption density, as discussed earlier, led us to fix this value to that obtained from the previous analysis of IKT 16 with {\\it XMM-Newton~\\\/}\\citep{owen2011} in all the nebular fits. The spectrum along with its best-fit model and residuals is shown in Fig.~\\ref{spec-1}b, and the best fit parameters are tabulated in Table \\ref{table-specfit}.\n\\subsection{Outer and Inner nebula spectra}\n\\label{spec-pwn-inner}\nIn order to look for changes in the spectral parameters, particularly a spectral steepening with radius in the nebula, we divided the total nebular region into two parts, by choosing an inner annular region centred on the point source, and an outside region excluding it. The extraction regions are shown in Fig.~\\ref{images}c. The Fig.~\\ref{spec-2}a and \\ref{spec-2}b show the inner and outer spectrum respectively with their best-fit models and residuals, and Table \\ref{table-specfit} their best-fit parameters. Although the outer nebula is hinted to be softer then its inner counterpart by the absence of counts above 5 keV, we detect no steepening of the spectral index $\\Gamma$ which would be indicative of synchrotron cooling. The $\\Gamma$ values are consistent within errors at 90 $\\%$ confidence. \n\\subsection{East-west nebula spectra}\n\\label{spec-east-west}\nIn Sect.~\\ref{east-west-counts} where we report the extracted total number of counts in the east and west nebular region, we find evidence that the west of the nebula may be slightly brighter although the values are consistent within errors. In continuation, we also extract spectra from the same regions following the same procedure as in Secs.   \\ref{spec-pwn} and \\ref{spec-pwn-inner}. Figures~\\ref{spec-2}c and Fig.~\\ref{spec-2}d show the east and west spectrum respectively with their best-fit models and residuals, and Table \\ref{table-specfit} their best-fit parameters. As indicated by the earlier exercise in Sect. \\ref{east-west-counts}, the spectrum from the west of the nebula shows slightly higher value of flux but it is comparable to that measured from the east nebula within errors at 90 $\\%$ confidence level. We do not detect any significant change in the spectral index $\\Gamma$. \n\\subsection{Diffuse emission}\n\\label{diff-neb}\nFrom Figs.~\\ref{images}d and \\ref{radial}, we see evidence that the source extends further beyond the region adopted to study the nebula spectrum. To investigate further, we extracted the spectrum of the diffuse component by choosing an annular region excluding the nebular extraction region (box region in Fig.~\\ref{images}c) and extending up to the size of the {\\it XMM-Newton~\\\/} point source extraction region of 20\\hbox{$^{\\prime\\prime}$} \\citep{owen2011}.\nUsing the same spectral model as that for the nebular spectrum, we find that the diffuse component has a flux and spectral index (tabulated in Table \\ref{table-specfit}) comparable to that of the main nebular component. We also tried to fit the spectrum with a Sedov model, since that emission might be associated with the SNR. We fixed the parameters to those obtained by \\cite{owen2011} (the data quality does not allow fitting anything else than the normalisation). The resulting C-statistic is higher by 9 than that of the power-law model. This does not favor the thermal model, but does not allow ruling it out either. However, no other part of the SNR is as bright, and this diffuse emission is centred on the main X-ray nebula. So we favor the interpretation as a further PWN component.\n\nIn addition, we also investigated whether this emission could be a halo due to the scattering from the foreground dust in the interstellar medium. \\cite{predel1995} using \\emph{ROSAT} observations, studied X-ray scattering halos around 25 point sources including the three bright sources in the LMC, LMC X-1, LMC X-2 and LMC X-3. The local absorbing density around IKT 16 is lower that in LMC X-1, but higher than in LMC X-2 and LMC X-3, and thus lies in the range covered in this study. The authors found that the relative intensity of the scattering halos of the LMC sources w.r.t the point source are $\\sim$ 1\\%. This is much weaker than what we find for the diffuse component which is $\\sim$ 25\\% (< 2 keV) of the putative pulsar emission. Moreover, \\emph{ROSAT} observed a flat profile for the halos extending to > 100\\hbox{$^{\\prime\\prime}$}. Within our extraction region of the diffuse emission (20\\hbox{$^{\\prime\\prime}$}), the fraction is expected to be even lower than 1\\%. Hence the dust scattering origin for this diffuse emission is very unlikely.\n\nFinally, we compared our results with the spectral model of the point source seen with {\\it XMM-Newton~\\\/} \\citep[Table 3 in][counting the spillover of the point source into the SNR extraction region, and the SNR emission in the point source region]{owen2011}. The total flux measured in the energy range of 0.5--8 keV from the {\\it Chandra~\\\/} observation is $21.3 \\times 10^{-14}$\n${\\rm ~erg~cm^{-2}~s^{-1}}$ \n(putative pulsar + nebula + diffuse emission; see Table \\ref{table-specfit} for the obtained values) comparable to $24.9 \\times 10^{-14}$ ${\\rm ~erg~cm^{-2}~s^{-1}}$ measured from the {\\it XMM-Newton~\\\/} spectral model. The unabsorbed luminosity in the same energy range (0.5--8 keV) measured  from {\\it Chandra~\\\/}~ is \\textbf{1.0} $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$. In comparison, the {\\it XMM~\\\/} spectral model corresponds to an unabsorbed luminosity of 1.1 and 1.3 $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$ in the 0.5--8 keV and 0.5--10 keV energy ranges and is in agreement with the {\\it Chandra~\\\/} results. It should be noted that, although the total unabsorbed luminosity (0.5--10 keV, SNR + point source) in \\cite{owen2011} is consistent with the spectral model, there seems to be an error in quoting the unabsorbed luminosity of the point source as 1.6 $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$ in that paper.\n\n\n\\begin{figure*}[htp]\n\\centering\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{inner_new_resi.ps}}\\quad\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{outer_new_resi.ps}} \\\\\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{nebula_east_spec_new_resi.ps}} \\quad\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{nebula_west_spec_new_resi.ps}} \\\\\n\\caption{Same as in Fig.~\\ref{spec-1} but for the inner nebula {\\it a}, outer nebula {\\it b}, east nebula {\\it c} and west nebula {\\it d}. Please note that in the outer nebula, i.e. Fig.~{\\it b}, there are no events above 5 keV.}\n\\label{spec-2}\n\\end{figure*}\n\\begin{table*}\n\\caption{Parameters of the best-fit spectral model to putative pulsar and the nebula. Errors are quoted at 90 $\\%$ confidence.\nThe {\\it Chandra~\\\/} energy range is 0.5--8 keV.}\n\\centering\n\\begin{tabular}{lcccccc}\n\\hline\n\nRegion  & SMC absorption &  Power-law photon Index  & F$_{\\rm X}$$^{a}$ &  F$_{\\rm X}$$^{a}$ &  L$_{\\rm X}$$^{b}$\\\\\n   & $10^{21}$cm$^{-2}$  &  $\\Gamma$  & 0.5--8 keV &  2--10 keV &  0.5--8 keV \\\\\n\\hline\nPutative pulsar  & 5.3$^{+3.9}_{-3.5}$ & 1.11$\\pm0.23$ & 16$\\pm 1$  & 17$\\pm2$ & 7.20\\\\\nNebula & 3.4 (f)$^{c}$  & 2.21$^{+0.40}_{-0.37}$ & 2.2$\\pm0.2$  & 1.3$^{+0.4}_{-0.5}$ & 1.19 \\\\\nInner nebula & 3.4 (f)$^{c}$ & 2.22$^{+0.48}_{-0.45}$ & 1.6$\\pm0.3$  & 1.0$^{+0.5}_{-0.7}$ & 0.78\\\\\nOuter nebula & 3.4 (f)$^{c}$ & 2.18$^{+0.60}_{-0.54}$ & 0.75$^{+0.38}_{-0.41}$ & 0.47$\\pm0.63$ & 0.41\\\\\nWest nebula & 3.4 (f)$^{c}$  & 1.97$^{+0.52}_{-0.40}$  & 1.4$\\pm 0.3$ & 0.9$^{+0.5}_{-0.6}$ & 0.67 \\\\\nEast nebula & 3.4 (f)$^{c}$  & 2.23$^{+0.67}_{-0.47}$ & 0.93$^{+0.4}_{-0.36} $  & 0.55$\\pm0.67$ & 0.45 \\\\\nDiffuse emission & 3.4 (f)$^{c}$ & 2.20$^{+0.38}_{-0.30}$ & 3.1$\\pm0.2$ & 1.9$^{+0.3}_{-0.2}$ & \\textbf{1.70} \\\\\n\\hline \n\\label{tabspec}   \n\\end{tabular}\n\\\\\n$^{a}$ - Observed flux in units of $10^{-14}$ ${\\rm ~erg~cm^{-2}~s^{-1}}$ \\\\\n$^{b}$ - absorption corrected luminosity in units of $10^{34}$ ${\\rm ~erg~s^{-1}}$ assuming a distance of \\textbf{61} kpc. \\\\\n(f) - Parameter fixed for consistency between fit regions. \\\\ \n$^{c}$ - SMC absorption for all nebular spectra was frozen to the column density obtained from the {\\it XMM-Newton~\\\/} {observation~\\\/}. \\\\\n\\label{table-specfit}\n\\end{table*}\n\\section{Radio}\nThe PWN in the radio-continuum regime extends far beyond X-ray detection as can be seen in Fig.~\\ref{radio}. The radio extent of the source as measured visually, is $70\\pm10$\\hbox{$^{\\prime\\prime}$}.\nWe note that the X-ray point source correlates to one of the peaks in our RC image. However, no corresponding point source can be drawn anywhere near the X-ray point source.\n\nWe point to the new high resolution data (Fig.~\\ref{radio}) where two lobe like features, somewhat symmetric, each $20\\pm5$\\hbox{$^{\\prime\\prime}$}~in extent, outside of the X-ray nebula are seen. These radio-continuum features oppose our initial scenario that the radio nebula has been pushed aside by the reverse shock \\citep{owen2011}. It seems it is more likely that we are still seeing the nebular expansion in the cold ejecta. \n\nAcross the 2.1\\,GHz band, the radio spectral index is flat as reported in \\cite{owen2011}. Somewhat surprising, we did not detect any polarisation in either {\\it Q}, {\\it U} or {\\it V} stokes parameters even that our detection level is better than 1\\%. The small scale structure may play an important role in smoothing out any weak polarisation. At the same time, depolarisation may reveal regions of underlying turbulence and\/or compression and heating of thermal material at various shocks within the remnant system \\citep{anderson1995}. This depolarisation may also indicate somewhat older age of the remnant. The noticeably younger PWNe in the LMC such as N157B \\citep{lazendic}, SNR J0453-6829 \\citep{hab2012}, and 0540-69.3 \\citep{bran2014} all have strong polarisation. \n\\section{Discussion}\n\\label{sec:disc}\nConfirmation of the PWN nature of the source makes IKT 16 the first composite SNR to be discovered in the SMC.\nThe nebular structure is consistent with the presence of an elongated PWN around a putative pulsar, which extends further out in the form of a fainter diffuse emission.\nHowever, we cannot rule out the presence of an inner compact X-ray nebula of the order of 0.1 pc which cannot be resolved at the distance of the SMC. \nTherefore, the overall impression is that of the bright central point source corresponding to a putative pulsar or a putative pulsar + unresolved inner PWN component, and an  extended emission component representing the PWN.\n\n Although the present {\\it Chandra~\\\/} {observation~\\\/} in full-frame mode is not capable of detecting pulsations from the central putative pulsar, we can nevertheless infer its properties to a certain extent by assuming some basic characteristics of the neutron star as is described below. In the later subsections, we also discuss the evolutionary stage of the {pulsar wind nebula~\\\/} as expected from its morphology.\n\\subsection{Derived properties of the pulsar}\n\\label{sec:discsnr}\n\nThe nebular emission in X-rays reflects the youngest generation of the emitting particles. Therefore, a strong correlation is expected \nbetween the non-thermal X-ray luminosity of the PWN and its pulsar, with the spin down properties of the\npulsar itself. The properties of the putative pulsar in IKT 16 were estimated adopting the work of \\cite{pwnchandra}. In this paper the correlation between the luminosity of the PWN, the non-thermal luminosity of the pulsar, and its various properties are derived from a large sample of {\\it Chandra~\\\/} {observations~\\\/} of PWNe. The unabsorbed luminosities of the PWN and the pulsar in IKT 16 are reported in Table \\ref{tabspec}. Using these values and the tabulated properties of the PWNe from \\cite{pwnchandra}, we estimated the spin-down power of the putative pulsar by comparing the total non-thermal luminosity of the sources (PWN+pulsar component) with their spin-down power. This ensured our estimate was free from the assumption that the central source is exclusively the pulsar and does not contain a compact nebular component. It is noteworthy to mention that in order to perform this estimation, we have not accounted for the flux from the diffuse emission (described in section \\ref{diff-neb}) which is not as well measured and whose origin is less unambiguous. At the same time addition of this component would not alter our results in the limits of the correlation measured in \\cite{pwnchandra}. Fig.~\\ref{corr} plots the spin-down power against the total luminosity for all the objects from \\cite{pwnchandra}, with a line drawn to indicate the total luminosity of the central source in IKT 16. It corresponds to an expected spin-down power $\\dot{E} \\sim 10^{37} {\\rm ~erg~s^{-1}}$.  Further, assuming its age to be the same as that measured from the SNR \\citep{owen2011}, ${\\it i.e.~\\\/}$ 14.7 kyr, we overlayed the central source on the $P - \\dot{P}$ diagram from the ATNF catalogue \\citep{atnf}. The expected spin period is < 100 ms with $\\dot{P} \\sim 10^{-13}$ s s$^{-1}$. This points towards a young and energetic pulsar powering the PWN in IKT 16, in fact the youngest in the SMC. Future on-axis {\\it XMM~\\\/} {observations~\\\/} or {\\it Chandra~\\\/} {observations~\\\/} using operating modes with better timing resolution can be used to search for pulsations from this source.\n\\begin{figure}\n\\centering \n\\includegraphics[height=0.45\\textwidth,angle=-90]{correlation-psr-edot-new.ps} \n\\caption{ Dependence of the total non-thermal luminosity (nebula+pulsar) on the pulsar spin-down power for the PWNe observed with {\\it Chandra~\\\/}. Adapted from \\cite{pwnchandra}. The red dashed line indicates the luminosity of the putative pulsar + PWN in IKT 16.}\n\\label{corr}\n\\end{figure}\n\\subsection{Energy spectrum}\n\\label{discspec}\nThe spectra of both the central and the extended source can be satisfactorily fitted with a power-law model, expected from a synchrotron emission dominated spectrum. The hard power-law spectral index 1.1 for the point source resembles that of a typical young rotation powered pulsar where the non-thermal radiation is generated by the particles accelerated in the pulsar magnetosphere \\citep[][and references therein]{michel1992}.\nNo additional thermal component (which would denote the emission from the surface or the polar caps of the neutron star) was required to fit the pulsar spectrum. The point source is  also three times brighter than the surrounding nebula. If this comprises exclusively the pulsar emission, it is rather unusual since the average value of the nebular to pulsar emission is $\\langle\\frac{\\eta_{pwn}}{\\eta_{psr}}\\rangle \\sim 4 $ for a large sample of PWNe observed with {\\it Chandra~\\\/} \\citep{pwnchandra}. The $\\Gamma$ of the nebular emission is compatible with that usually observed for PWNe \\citep[1.5 to 2.1:][]{pwnchandra,li2008}, and is also consistent with particle wind models of Fermi shock acceleration \\citep{2001MNRAS.328..393A}.\n\\subsection{Morphology}\n\\label{sec:discsnrmorp}\nThe morphology of the PWN is elongated along the east-west direction as reported in section \\ref{spec-east-west}. \nThe obtained size of  5.2\\hbox{$^{\\prime\\prime}$}~of the X-ray nebula corresponds to a source extent of 1.5 pc at the distance of the SMC. The radio extent of the source is much larger corresponding to 21 pc with the brightest emission from the radio lobes, each $\\sim$ 6 pc in extent assuming the same distance.\nThis scenario is consistent with a longer cooling time for radio emission compared to that of X-ray emission.\nIn addition, the radio nebula extends even further than the estimated SNR centre, consistent with the fact that the earliest electrons were produced when the pulsar was much closer to the SNR centre. Further, the long axis of the radio and X-ray nebula are aligned with each other. This axis of alignment points to the centre of the SNR, so is presumably aligned with the pulsar's direction of motion.\n\\subsection{Interaction of the SNR with the PWN  and its stage of evolution}\n\\label{pwn-interact}\nThe SNR in IKT 16 is in the adiabatic Sedov phase of evolution with an age of $\\sim$ 14.7 kyr \\citep{owen2011}. The shell of the SNR has important implications on the PWN, and the physics of these systems is extremely complicated due to the rapid evolution of both the SNR and the central pulsar \\citep{gaensler2006,gelfand2007,gelfand2009}. The evolution of a PWN inside an SNR follows three important evolutionary phases: an initial free-expansion in the supernova ejecta, the collision between the PWN and SNR reverse shock, which crushes the PWN subjecting it to various instabilities, and eventually subsonic re-expansion of the PWN in the shock heated ejecta \\citep[][and references therein]{2001ApJ...563..806B,2003A&A...397..913V,2003A&A...404..939V,2004A&A...420..937V,gelfand2007}. After the reverse shock interaction has died off, ${\\it i.e.~\\\/}$ in the PWN re-expansion stage, the neutron star is often displaced from its PWN, leaving behind the \"relic\" PWN usually observed in radio and forms a \"new\" PWN comprising of freshly injected particles, and therefore observable in the X-rays. A later additional phase of \"bow-shock\" nebula can sometimes be identified, as the local sound velocity decreases with the pulsar's progression through the SNR. In this phase the pulsar's motion may become supersonic when it approaches the shell of the SNR, and it may acquire a bow-shock morphology \\citep{2003A&A...397..913V,2003A&A...404..939V,2004A&A...420..937V}.\n\nAdopting the total mechanical energy released in the explosion ($E_{51}$ in units of $10^{51}$ erg) and the ambient medium density ($n_{0}$) from the adiabatic Sedov modeling in \\cite{owen2011}, and assuming the mass of the ejecta, $M_{ej} = 10 M_{\\odot}$, we can calculate the reverse shock trajectory \\citep{mckee1995} at the present age of the SNR (14.7 kyr). At the distance of the SMC, we obtain that the reverse shock radius $R_{s}$ is 11 pc away from the centre of the SNR. This would imply that although it might not have encountered the smaller X-ray nebula yet \\citep[measured to be at 8 pc from the centre of the SNR][]{owen2011}, it would have crossed the larger radio extent of the nebula and interacted with the lobes. This picture is however not supported by the new high resolution radio data. The presence of two bright radio lobes more or less symmetric about the X-ray counterpart is consistent with an expansion of the PWN into cold ejecta in homologous expansion. \nWe however point out that there may be uncertainties in measurement of our Sedov model parameters for the SNR IKT 16 \\citep{owen2011} which may limit the estimation of $R_{s}$.\n\\section{Conclusions}\n\\label{sec:conc}\nThanks to the high resolution {\\it Chandra~\\\/} ACIS {observation~\\\/} we have been able to resolve the hard X-ray emission near the centre of the SNR IKT 16 into a PWN centred on a\nputative pulsar. We have imaged the PWN, constrained its geometrical parameters, and have measured all the  spectral components separately. With new high resolution radio data we have determined the radio morphology and its extent precisely. The main results can be summarized as follows:\n\\begin{itemize}\n\\item The putative PWN is elongated but centred on the point source. Comparison with the radio counterpart indicates that the reverse shock may not have yet met with the PWN surface.\n\\item The point source at the centre is about three times brighter than the elongated feature.\n\\item Morphological modelling of the X-ray nebula with a PSF and a 2D Gaussian resulted in an FWHM of 5.2\\hbox{$^{\\prime\\prime}$}~for the PWN with its axis aligned with the larger radio nebula.\n\\item The point source at the centre has a much harder spectrum than the extended emission from the nebula. This points to the presence of a \npulsar dominated by non-thermal emission. The expected $\\dot{E}$ is $\\sim 10^{37} {\\rm ~erg~s^{-1}}$ and spin period < 100 ms, with  \n$\\dot{P} \\sim 10^{-13}$ s s$^{-1}$. This points towards a young and energetic pulsar powering the PWN in IKT 16.\nHowever, the presence of a compact X-ray PWN unresolved by Chandra at the distance of the SMC cannot be ruled out.\n\\end{itemize} \n\n\n\n\n\n\n\\begin{acknowledgements}\nThis research has made use of data obtained from the {\\it Chandra~\\\/} X-ray observatory. We used the KARMA and MIRIAD software packages developed by the\nATNF. The Australia Telescope Compact Array is part of the Australia\nTelescope which is funded by the Commonwealth of Australia\nfor operation as a National Facility managed by CSIRO. CM acknowledges Fabio Acero for very useful discussions and comments on the paper. The authors would like to acknowledge the referee Satoru Katsuda for very useful comments which significantly improved the paper.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip\n\n\\subsection{The problem. Results and conclusions}\n\nThe clue issue in checking quantum electrodynamics (QED) is the measurement\nof the magnetic moment of electron with the subsequent comparison of its\nmeasured value with the anomalous electron magnetic moment calculated via\nthe Standard Model that is mostly QED in this case. Up to now, within every\nexperimental and theoretical accuracy achieved, these two values do\ncoincide. The allowed, within the errors, discrepancy between the\nexperimental and theoretical values of the electron magnetic moment is\nexpected to be diminishing on and on, as the precision grows, and hopefully\nthe coincidence between them will be maintained with better and better\naccuracy. On the other hand, as far as one is seeking for possible\ntheoretical amendments to the Standard Model, admissible within the above\nsituation, one should confine their impact on the electron magnetic moment\nto lie within the present experimental and theoretical indeterminacy. A\ncertain candidate for going beyond the standard QED is proposed by the\nnoncommutative (NC) electrodynamics. It was found recently \\cite{AGSV} that\nin the framework of that theory a static classical charge at rest \\ carries\na magnetic moment, called NC magnetic moment, whose smallness is determined\nby a noncommutativity parameter $\\theta,$ supplying the theory with the\nfundamental length \\footnote\nThe noncommutativity by no means is the only method of introducing the\nfundamental, or elementary, length into a theory. Throughout this paper,\nhowever, we shall mean namely NC fundamental length when using this notion.\nOn the other hand, we do not know whether the fundamental length as it is\nproposed by the noncommutativity mechanism is universal for all particles\nand fields. For this reason we shall discuss its values independently when\nwe deal with different particles.} $l=\\sqrt{\\theta}$. By demanding that, for\nthe electron, the NC magnetic moment be less than the existing error in\nmeasuring the electron magnetic moment we get an estimate from above on the\nparameter $\\theta$ and the associated fundamental length $l$ . Certain\nrestrictions on the fundamental length inherent in the NC theory also follow\nfrom the existence of the NC magnetic moment of heavier charged particles.\nHowever, the consideration of the noncommutative magnetic moment of the\nproton and of its contribution to the hyperfine splitting of the energy\nlevel $1S_{1\/2}$ in a hydrogen atom did not lead \\cite{AGSV} to any\nessentially new estimate for the maximum fundamental length. On the\ncontrary, consideration of leptons did.\n\nOnce the NC magnetic moment is found to be inversely proportional to the\nsize of the electric charge, an important role in getting this estimate is\nplayed by the size attributed to an electron, the smaller the size, the\nlarger the NC magnetic moment, the smaller the upper estimate on the NC\nparameter and the fundamental length. We probe different assumptions\nconcerning the \\textquotedblleft electron size\\textquotedblright , the\nultimate one being that it is restricted from below only by the fundamental\nlength $l$ itself, since neither object is supposed to be smaller than it.\nIn this way a hitherto lowest upper bound on the fundamental length, as it\nappears in the noncommutative field theory, was achieved in \\cite{AGSV}. On\nthe other hand, after we update the famous electron size estimate\\ due to\nBrodsky-Drell-Dehlmet \\cite{BrodDrell}, \\cite{Dehmelt} (not based on any\nnoncommutative mechanism, but only on a consideration of a possible\ncompositeness, or divisibility of the electron) by taking into account the\nmost recent measurements of the electron magnetic moment, we find the\nelectron size results to be two orders of magnitude smaller than the boldest\nestimate of the fundamental length obtained from speculations on\nnoncommutative magnetic moment. As far as in an NC field theory no size of\nany physical object is admitted to be smaller than the fundamental length,\nthis means that no more than 1\/100 part of the existing indeterminacy in the\nknowledge of the electron magnetic moment may be at the best ascribed to the\ncontribution of the noncomutative magnetic moment. Then, two options remain.\nEither there should be an extra extension beyond the standard QED, other\nthan NC electrodynamics, that may take responsibility for the main part of\nthe admitted, if any, deviation of the magnetic moment from the QED result,\nor, what is more probable, this admitted deviation will be essentially\nreduced by further more precise measurements.\n\nThe same analysis is repeated in the paper as applied to the $\\mu $-meson.\nThe crucial difference with the electron case is that the difference between\nthe theoretical and experimental values of the muon magnetic moment exceeds\nthe limits admitted by the errors. So, no further technical advancement is\nexpected to be able to remove this contradiction, and our results make us\nmore definite in claming that the noncommutativity cannot provide for the\nmissing part of the muon magnetic moment, a different way for extending the\nStandard Model remaining to be needed.\n\n\\bigskip\n\n\\subsection{Noncommutative magnetic moment}\n\nIn \\cite{AGSV}, classical field equations in $U(1)_{\\star}$-theory\n(noncommutative electrodynamics) were formulated that -- at least within the\nfirst order in the noncommutativity parameter $\\theta$ -- restrain the gauge\ninvariance in spite of the presence of external current, known to violate it\n(at least off-shell). By solving these equations electromagnetic field\nproduced by a finite-size static electric charge was found, and the fact\nthat this charge possesses a magnetic moment depending on its size was\nestablished. Let the external current in the field equations of NC\nelectrodynamics be just a static electric charge distributed inside a sphere\nof a finite radius $a$ with the uniform charge density \n\\begin{equation*}\n\\rho\\left( \\mathbf{r}\\right) =\\frac{3}{4\\pi}\\frac{Ze}{a^{3}}\\,,\\ \\ r<a,\\quad\nr=|\\mathbf{r}|\\,.\n\\end{equation*}\nOutside the sphere there is no charge: $\\rho\\left( \\mathbf{r}\\right) =0,$ if \n$r>a$. The above finite-size static total charge $Ze,$ where $e$ is the\nfundamental charge, produces not only the electrostatic field, but also\nbehaves itself as a magnetic dipole with the magnetic field given in the\nremote region $r\\gg a$ by the following vector-potentia\n\\begin{equation}\n\\hspace{0cm}\\mathbf{A}=\\frac{\\left[ \\mathbf{M}\\times\\mathbf{r}\\right] }{r^{3\n},\\ \\ \\mathbf{M}=\\boldsymbol{\\theta}(Ze)^{2}\\frac{2e}{5a}\\,,\n\\label{magnmoment}\n\\end{equation}\nwhere $\\mathbf{M}$ was called NC magnetic moment of the charged particle.\nHere the three spacial components of the vector $\\boldsymbol{\\theta}$ are\ndefined as $\\theta^{i}\\equiv(1\/2)\\varepsilon^{ijk}\\theta^{jk},$ $i,j,k=1,2,3$\nin terms of the antisymmetric noncommutativity tensor $\\theta^{\\mu\\nu}$ that\nfixes the commutation relations between the operator-valued coordinate\ncomponents $[X^{\\mu},X^{\\nu}]=i\\theta^{\\mu\\nu},$ and only the space-space\nnoncommutativity, i.e. the special case of $\\theta^{0\\nu}=0$ in a certain\nLorentz frame, was considered.\n\nThe extension (size) $a$ of the charge in (\\ref{magnmoment}) should be kept\nnonzero in the spirit of NC theory that does not admit objects with their\nsize smaller than the fundamental length $l=\\sqrt{\\theta}$, where $\\theta =\n\\boldsymbol{\\theta}|$. For a point charge a magnetic solution also exists \n\\cite{Stern}, although in this case it is not a magnetic dipole one. What is\nmore important is that that solution is too singular in the point $r=0$,\nwhere the charge is located, and hence it cannot be given a mathematical\nsense in terms of the distribution theory in a conventional way.\n\nIf we understand the radius $a$ in (\\ref{magnmoment}) as the size of an\nelectrically charged fundamental particle ($Z=1$), we can speculate on what\nthe contribution of the noncommutativity into its magnetic moment $\\mathbf{M}\n$ may be. Certainly, this is expected to be very small, because of the\nextreme smallness of the noncommutativity parameter $\\theta $. It is\nprimarily supposed \\cite{DFR} that the corresponding length $l=\\sqrt{\\theta }\n$ should be of the Plank scale of $l\\sim 10^{-33}$ cm (or $\\Lambda _{\\mathrm\nPl}}\\sim 4\\cdot 10^{19}$ Gev in energy units). The reason is that at so\nsmall distances unification of gravity with quantum mechanics requires\nquantization of space-time. Although the Plank scale is far beyond any\nexperimental reach, the everlasting problem is to estimate the upper limits\non $\\theta $ basing on the existing and advancing experimental preciseness.\nIn \\cite{AGSV} it was discussed what new restrictions on the extent of\nnoncommutativity may follow from the newly established fact that a charged\nfundamental particle is a carrier of the magnetic moment (\\ref{magnmoment})\nin an NC theory, irrespective of its orbital momentum or spin. In the\npresent article we shall further elaborate this matter addressing the\ncharged leptons $e$ and $\\mu $ as the \\textquotedblleft\nsmallest\\textquotedblright\\ -- and hence providing the maximum contribution\nof (\\ref{magnmoment}) -- particles, to leave alone quarks -- also small, but\nwhose magnetic moment is beyond measurements.\n\n\\section{Upper bounds for fundamental length from noncommutative magnetic\nmoment}\n\n\\subsection{Limitations based on high-energy scattering estimates of lepton\nsizes}\n\nIn high-energy electron-positron collisions leptons manifest themselves as\nstructureless particles (see e.g. \\cite{BrodDrell} for an early discussion\nof this point), described by a fundamental (local), not composite field. No\ndeviation from this rule has been up to now reported. Taking the LEP scale\nof 200 Gev as an upper limit, to which this statement may be thought of as\nchecked, we must accept that a possible structureness of these leptons is\nbelow the length (call it divisibility length) $r_{0}=10^{-3}Fm$. In our\nfurther consideration we identify the charge extension $a$ with the\ndivisibility length, because it is hard to imagine a region occupied by a\ncharge that extends above this length, but cannot be divided into parts. (If\nit could, either the resulting charge would acquire a continuous value,\nsmaller than $e$, which contradicts basic assumptions, or the resulting\ncharge would occupy a smaller space and we would be left again with smaller \na$, down to the divisibility length.)\n\n\\subsubsection{Electron}\n\nBearing in mind that, for electron, the existing local theory perfectly\nexplains the value of its magnetic moment $M_{\\mathrm{e}}$, we expect that\nthe noncommutativity might only contribute into the experimental and\ntheoretical uncertainty $\\delta M_{\\mathrm{e}}$ existing in measuring and\ncalculating this quantity. A recent direct measurement of the anomalous\nmagnetic moment of electron, using the magnetic resonance spectroscopy of an\nindividual electron in the Penning trap \\cite{Dehmelt}, gives the result \n\\cite{Hanneke}, \\cite{nakamura\n\\begin{equation}\n\\left. \\left( \\frac{M_{\\mathrm{e}}}{\\mu }-1\\right) \\right\\vert _{\\mathrm{MRS\n}=0.00115965218073\\pm 28\\cdot 10^{-14},  \\label{hanneke}\n\\end{equation\nwhere $\\mu =e\/2m$ is the Bohr magneton. On the other hand, a new report \\cit\n{Bouchendira} appeared on an \\textit{independent} experimental determination\nof the same magnetic moment with a matching accuracy, obtained with the use\nof a measurement of the ratio $h\/m_{\\mathrm{Rb}}$ between the Plank constant\nand the mass of the $^{87}$Rb atom. The result is \n\\begin{equation}\n\\left( \\left. \\frac{M_{\\mathrm{e}}}{\\mu }-1\\right) \\right\\vert _{\\mathrm{Rb\n}=0.00115965218113\\pm 84\\cdot 10^{-14}.  \\label{theor}\n\\end{equation\nAuthors of \\cite{Bouchendira} fit the value of the fine structure constant \n\\alpha $ in such a way as to make (\\ref{theor}) coincide with the\ntheoretical prediction for the electron anomalous magnetic moment,\ncalculated (see \\cite{Mohr} for a review) with the accuracy, including QED\ncalculations up to $(\\alpha \/\\pi )^{4}$, also electroweak and hadronic\ncontributions (this fit leads to the so far most precise value $\\alpha\n^{-1}=137.035999037(91)$). For this reason the value (\\ref{theor}) is\nreferred to as \\textquotedblleft theoretical\\textquotedblright . (Certainly,\nthe roles of (\\ref{theor}) and (\\ref{hanneke}) might be reversed.) The\ntheoretical, (\\ref{theor}), and experimental, (\\ref{hanneke}), values of the\nelectron magnetic moment do not contradict each other, demonstrating the\nhitherto best confirmation of QED. The discrepancy between the\n\\begin{equation}\n\\frac{\\delta M_{\\mathrm{e}}}{\\mu }\\sim 10^{-12}  \\label{delta}\n\\end{equation\nlies within the accuracy of measurements and calculations. We demand that a\npossible contribution of the noncommutative magnetic moment in (\\re\n{magnmoment}) should not exceed it\n\\begin{equation}\n\\frac{\\delta M_{\\mathrm{e}}}{\\mu }>\\alpha \\theta \\frac{4m}{5a}\\,,\\ \\alpha\n=e^{2}\\,.  \\label{delM}\n\\end{equation\nWith the high-energy restriction on the size $a<r_{0}$ accepted above, Eq.\n\\ref{delM}) implies $\\theta <\\frac{\\delta M_{\\mathrm{e}}}{\\mu }(5r_{\\mathrm{\n}}\/4m\\alpha )$. As $r_{0}\\sim 10^{-3}Fm,$ we get from here and from\\ (\\re\n{delta})\\ the restriction on the fundamental length $l=\\sqrt{\\theta }<7\\cdot\n10^{-6}Fm=(28~Tev)^{-1}.$\n\n\\subsubsection{Muon}\n\nThe matters stand differently with the $\\mu $-meson. In the literature, its\nanomalous magnetic moment is calculated via the Standard Model with the\ninclusion of the QED lowest-order $\\mu $-$\\gamma $ vertex, $Z$-boson,\nneutrino and hadron lines. The deviation of the measured magnetic moment $\\\nM_{\\mu \\text{ }}$from the result of calculations makes the value (see A.\nHocker's and W.J. Marciano's 2009 update in \\cite{nakamura}, also \\cite{Mohr}\nfor a later detailed account) \n\\begin{equation}\n\\frac{\\delta M_{\\mu }}{\\mu }\\simeq 25\\cdot 10^{-10}.  \\label{deltamu}\n\\end{equation\nThis exceeds about 3.2 times the estimated 1$\\sigma $ error \\cite{nakamura}.\nIt is believed that this discrepancy may be overcome by including\nsupersymmetry for amending the theoretical result. If, on the contrary, we\ntry to explain this discrepancy by the effect of NC magnetic moment of the\nmuon, we get in the way similar to the one described above in this\nSubsection, using (\\ref{deltamu}) and the same indevisability length \nr_{0}\\sim 10^{-3}Fm$ that $l$ is smaller than $2.8\\cdot 10^{-5}Fm=$ (7 $Tev$\n$^{-1}$ as the high-energy based estimate.\n\n\\subsection{Ultimate estimates}\n\nOnce there is no evidence for any electron extension, it is worth admitting\nthat it may be only restricted by the fundamental length. Then, using $a=l\n\\sqrt{\\theta}$ in (\\ref{delM}) and the indeterminacy (\\ref{delta}), we\nobtain the ultimate bound of $l<6.6\\cdot10^{-8}Fm=(3\\cdot10^{3}$ $Tev)^{-1}\n. Dealing with the muon in the same way, but referring to (\\ref{deltamu})\ninstead of (\\ref{delta}), we obtain the ultimate estimate of $8\\cdot\n10^{-7}Fm=(240$ $Tev)^{-1}$.\n\n\\section{Upper bounds on fundamental length versus compositeness sizes of\nleptons}\n\nThere are \\cite{BrodDrell} much stronger restrictions on the lepton sizes\nthan those following from the high-energy collision experiments. These\nextend to the energy scale far exceeding the accelerator means. The point is\nthat if one imagines a lepton as a bound state of much heavier particles so\nthat the binding energy compensates the most part of their masses to make\nthe resulting state light, the Bohr radius $R$ of the composite state -- to\nbe treated as its size -- would be much smaller than the Compton length of\nthe lepton $\\lambda_{\\mathrm{C}}$. According to the Drell-Hearn-Gerasimov\nsum rule (see \\cite{BrodDrell} for references) the deviation of the\nanomalous magnetic moment $(M\/\\mu-1)$ from its QED value is proportional to\nthe ratio $R\/\\lambda_{\\mathrm{C}},$which is the measure\\ of compositeness.\nBased on the experimental data on magnetic moments of the known composite\nparticles -- proton and triton -- plotted against their measured sizes, a\nconjecture was formulated by Dehmelt \\cite{Dehmelt} that the proportionality\ncoefficient should be of the order of unity. Then, $R=\\lambda_{\\mathrm{C\n}\\delta M\/\\mu$.\n\n\\subsection{Electron}\n\nReferring to Eqs. (\\ref{hanneke}, \\ref{theor}) and using (\\ref{delta}) we\nmay update Dehmelt's 1988 result for the electron of $R<4\\cdot10^{-8}Fm$ to \nR<4\\cdot10^{-10}Fm$. This is two orders of magnitude smaller than our\nultimate estimate of $6.6\\times10^{-8}Fm$ for the fundamental length\nobtained in Subsection \\textit{2.2}. (The use of the assertion $R=\\lambda _\n\\mathrm{C}}\\delta M\/\\mu$ together with (\\ref{delM}) would result in the\ncondition $l<\\sqrt{5\/8\\alpha}(\\delta M\/\\mu)\\lambda_{\\mathrm{C}}=9.25R$,\nweaker than the already accepted condition that the fundamental length \\\nshould be smaller than any size, including the composite electron radius $R\n, that is to $l<R$ $.$) So, Dehmelt's conjecture provides a stronger bound\non the fundamental length, than the noncommutative magnetic moment. This\nmeans that no more than $10^{-2}$ part of the measured difference (\\ref{delM\n) may be at the most attributed to noncommutative contribution.\n\n\\subsection{Muon}\n\nThe muon radius estimated analogously, basing on the compositeness arguments\nand on the theory-experiment discrepancy (\\ref{deltamu}), gives the result \nR_{\\mu }\\simeq 0.5\\cdot 10^{-8}Fm.$ This is smaller than the ultimate\nestimate of Subsection 2.2 based on muon data. Again, once the muon size\ncannot be smaller than the fundamental length, this result indicates that\nthe NC magnetic moment alone definitely cannot take on the responsibility\nfor the discrepancy (\\ref{deltamu}) between the theory and experiment, and\nhence deviations from the Standard Model other than NC electrodynamics are\nneeded. Unlike the electron case above, one cannot set one's hopes upon the\nfuture growth of precession of measurements to abandon this conclusion.\n\n\\section*{Acknowledgements}\n\nT. Adorno thanks FAPESP, D. Gitman thanks CNPq and FAPESP for permanent\nsupport, A. Shabad thanks FAPESP for support and USP for the kind\nhospitality extended to him during the fulfillment of this work. A. Shabad\nalso acknowledges the support by RFBR under the Project 11-02-00685-a.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intr}\n\nIn our previous paper \\cite{Inan_Kisselev:2021} we probed the\nanomalous quartic gauge couplings (QGCs) in the $\\gamma\\gamma\n\\rightarrow \\gamma\\gamma$ process at the Compact Linear Collider\n(CLIC) \\cite{Braun:2008,Boland:2016}. Both the unpolarized and\npolarized light-by-light scatterings were considered, and the bounds\non QGCs were obtained. The neutral anomalous quartic couplings are\nof particular interest. The anomaly interactions $\\gamma ZZZ$,\n$\\gamma\\gamma ZZ$, and $\\gamma\\gamma\\gamma Z$ at the LHC were\nanalyzed in \\cite{Chapon:2009}-\\cite{Eboli:2004}. The LHC\nexperimental bounds on QGCs were presented by the CMS\n\\cite{CMS:QGCs} and ATLAS \\cite{ATLAS:QGCs} Collaborations (see also\n\\cite{Schoeffel:2021}). The bounds on the anomalous\n$\\gamma\\gamma\\gamma Z$ vertex can be also derived from the\nconstraints on the $\\mathcal{B}(Z\\rightarrow\\gamma\\gamma\\gamma)$\nbranching ratio obtained at the LEP \\cite{L3:Z_decay} and LHC\n\\cite{ATLAS:Z_decay}. As for $e^+e^-$ colliders, they may operate in\n$e\\gamma$ and $\\gamma\\gamma$ modes~\\cite{Ginzburg:1981}. The bounds\non QGCs in $e^+e^-$, $e\\gamma$ and $\\gamma\\gamma$ collisions were\ngiven in \\cite{Eboli:1994}-\\cite{Koksal:2014}. In particular, the\nlimits on the quartic couplings for the vertex $\\gamma\\gamma\\gamma\nZ$ were derived in \\cite{Gutierrez:2014} using LEP~2 data for the\nreactions $e^+e^- \\rightarrow \\gamma\\gamma\\gamma, \\gamma\\gamma Z$. A\nsimilar analysis for the exclusive $\\gamma Z$ production with intact\nprotons at the LHC was done in \\cite{Baldenegro:2017}. The search\nfor virtual SUSY effects in the process $\\gamma\\gamma \\rightarrow\n\\gamma Z$ at high energies was presented in \\cite{Gounaris:1999_1}.\n\nAs one can see, the anomalous $\\gamma\\gamma\\gamma Z$ vertex urgently\nneeds to be examined in high energy $e^+e^-$ collisions. That is\nwhy, in the present paper we study the process (see\nFig.~\\ref{fig:3gammasZ})\n\\begin{equation}\\label{process}\n\\gamma(p_1, \\mu)+ \\gamma(p_2, \\nu) \\rightarrow \\gamma(p_3, \\rho) +\nZ(p_4, \\alpha) \\;,\n\\end{equation}\nwhere $p_1, p_2, p_3, p_4$ are boson momenta, $\\mu, \\nu, \\rho,\n\\alpha$ are boson Lorentz indices, and ingoing particles are real\npolarized photons generated at the CLIC by the laser Compton\nbackscattering~\\cite{Kramer:1994}-\\cite{Telnov:1998}.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{2_gammas_gamma_Z.eps}\n\\caption{The process $\\gamma + \\gamma \\rightarrow \\gamma + Z$.}\n\\label{fig:3gammasZ}\n\\end{center}\n\\end{figure}\nOur main goal is to derive bounds on anomaly couplings for the\nvertex $\\gamma\\gamma\\gamma Z$ which can be reached at the CLIC using\nboth polarized and unpolarized photon beams. The great potential of\nthe CLIC in probing new physics is well-known\n\\cite{Dannheim:2012}-\\cite{Franceschini:2020}. Let us underline that\na physical potential of a linear high energy $e^+e^-$ collider may\nbe significantly enhanced, provided the polarized beams are\nused~\\cite{polarized_beams,CLIC_lum}.\n\nLet $\\lambda_e$ be the helicity of the initial electron beam, while\n$\\lambda_0$ be the helicity of the ingoing laser photon beam. In our\ncalculations, we will consider two sets of these helicities, with\nopposite sign of $\\lambda_e$,\n\\begin{align}\\label{helicities}\n(\\lambda_e^{(1)}, \\lambda_0^{(1)}; \\lambda_e^{(2)}, \\lambda_0^{(2)})\n&= (0.8, 1; 0.8, 1) \\;, \\nonumber \\\\\n(\\lambda_e^{(1)}, \\lambda_0^{(1)}; \\lambda_e^{(2)}, \\lambda_0^{(2)})\n&= (-0.8, 1; -0.8, 1) \\;,\n\\end{align}\nwhere the superscripts 1 and 2 enumerate the beams. We will work in\nthe effective field theory framework. Previously effective\nLagrangians were used in \\cite{Stohr:1994}-\\cite{Novotny:1995} for\nexamining the $\\gamma\\gamma\\gamma Z$ interaction in the $Z\n\\rightarrow \\gamma\\gamma\\gamma$ decay, as well as in\n\\cite{Baillargeon:1996}, \\cite{Gutierrez:2014}, and\n\\cite{Baldenegro:2017}. Anomalous quartic gauge couplings (QGCs) are\ninduced at the dimension-six level already. However, they are not\nindependent of anomalous trilinear gauge couplings. That is why, in\nour paper, we study anomalous QGCs which enter the effective\nLagrangian at dimension-eight without contributing to anomalous\ntrilinear gauge interactions.\n\nThe paper is organized as follows. In the next section, the effective\nLagrangian is described, and Feynman rules for the anomalous\n$\\gamma\\gamma\\gamma Z$ vertex are presented. The helicity amplitudes\nare studied in Sec.~\\ref{sec:amplitudes}. In\nSec.~\\ref{sec:numerical_results}, both differential and total cross\nsections for the process \\eqref{process} are calculated, and bounds\non the QGCs are given. In Sec.~\\ref{sec:unit_const}, unitarity\nconstraints on anomalous quartic couplings are obtained. In\nAppendix~A, polarization tensors for the vertex $\\gamma\\gamma\\gamma\nZ$ are listed. The explicit expressions for the anomalous\ncontributions to the helicity amplitudes are given in Appendix~B.\nSome formulas for Wigner's $d$-function are collected in Appendix~C.\nFinally, in Sec.~\\ref{sec:concl}, we summarize our results and give\nconclusions.\n\n\n\\section{Effective Lagrangian}\n\\label{sec:Lagrangian}\n\nIt is appropriate to describe the anomalous $\\gamma\\gamma\\gamma Z$\ninteraction by means of an effective Lagrangian. Given parity is\nconserved and gauge invariance is valid, there are only two\nindependent operators with dimension 8. Following\n\\cite{Stohr:1994,Horejsi:1994}, we take the Lagrangian\n\\begin{equation}\\label{Lagrangian}\n\\mathcal{L}_{\\gamma\\gamma\\gamma Z} = g_1 O_1 + g_2 O_2 \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_G-R}\nO_1 = F^{\\rho\\mu} F^{\\alpha\\nu} \\partial_\\rho F_{\\mu\\nu} Z_\\alpha\n\\;, \\quad O_2 = F^{\\rho\\mu} F^\\nu_\\mu \\partial_\\rho F_{\\alpha\\nu}\nZ^\\alpha \\;,\n\\end{equation}\nwhere  $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$. The\noperators $O_{1,2}$ arise from a $SU(2)\\times U(1)_Y$ effective\nLagrangian with two operators like\n$B_{\\mu\\nu}B^{\\nu\\nu}B_{\\rho\\sigma}B^{\\rho\\varrho}$, four operators\nlike $W_{\\mu\\nu}W^{\\nu\\nu}W_{\\rho\\sigma}W^{\\rho\\varrho}$, and four\noperators like $B_{\\mu\\nu}B^{\\nu\\nu}W_{\\rho\\sigma}W^{\\rho\\varrho}$,\nwhere $W_\\mu$ and $B_\\mu$ are the $SU(2)$ and hypercharge gauge\nfields, respectively \\cite{Fichet:2014}. We consider only CP\nconserving operators hence the dual field strength tensors\n$\\tilde{W}_{\\mu\\nu}$ and $\\tilde{B}_{\\mu\\nu}$ are not used. The\ncoupling $g_{1,2}$ are linear combinations of ten coefficients of\ndimension-eight operators mentioned above. Note that certain\ncombinations of these coefficients must obey so-called positivity\nconstraints \\cite{Remmen:2019}-\\cite{Bi:2019}.\n\nAs one can see, this Lagrangian contains no derivatives of the $Z$\nboson field (correspondingly, no $p_4$ in the momentum space), that\nsimplifies a derivation of Feynman rules for the $\\gamma\\gamma\\gamma\nZ$ vertex.\n\nSome authors use the Lagrangian \\cite{Novotny:1995}\n\\begin{equation}\\label{Lagrangian_N}\n\\mathcal{L}^{(\\mathrm{N})}_{\\gamma\\gamma\\gamma Z} = G_1 \\bar{O}_1 +\nG_2 \\bar{O}_2 \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_B}\n\\bar{O}_1 =  F^{\\mu\\nu}F_{\\mu\\nu} F^{\\rho\\sigma}Z_{\\rho\\sigma} \\;,\n\\quad \\bar{O}_2 = F^{\\mu\\nu} F_{\\nu\\rho}F^{\\rho\\sigma}Z_{\\sigma\\mu}\n\\;,\n\\end{equation}\nwhere $Z_{\\mu\\nu} = \\partial_\\mu Z_\\nu - \\partial_\\nu Z_\\mu$, or the\nLagrangian \\cite{Baldenegro:2017}\n\\begin{equation}\\label{Lagrangian_B}\n\\mathcal{L}^{(\\mathrm{B})}_{\\gamma\\gamma\\gamma Z} = \\zeta O +\n\\tilde{\\zeta} \\tilde{O} \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_B}\nO = F^{\\mu\\nu}F_{\\mu\\nu} F^{\\rho\\sigma}Z_{\\rho\\sigma} \\;, \\quad\n\\tilde{O} = F^{\\mu\\nu}\n\\tilde{F}_{\\mu\\nu}F^{\\rho\\sigma}\\tilde{Z}_{\\rho\\sigma} \\;,\n\\end{equation}\nwhere $\\tilde{F}_{\\mu\\nu} =\n(1\/2)\\,\\varepsilon^{\\mu\\nu\\rho\\sigma}F_{\\varrho\\sigma}$, and\n$\\tilde{Z}_{\\mu\\nu} =\n(1\/2)\\,\\varepsilon^{\\mu\\nu\\rho\\sigma}Z_{\\varrho\\sigma}$.\n\nUsing integration by parts and equations of motion, one can easily\nobtain the following relations between two bases for the effective\nLagrangian \\cite{Novotny:1995},\n\\begin{equation}\\label{operators_N_vs_G-R}\n\\bar{O}_1 = -8O_1 \\;, \\quad \\bar{O}_2 = 2(O_2 - O_1) \\;.\n\\end{equation}\nWe also have the relations \\cite{Baldenegro:2017}\n\\begin{equation}\\label{operators_G-R_vs_B}\nO = \\bar{O}_1 = -8 O_1 \\;, \\quad \\tilde{O} = 4\\bar{O}_2 - 2\\bar{O}_1\n= 8(O_2 + O_1) \\;.\n\\end{equation}\nThe above listed equations enable us to relate anomalous coupling in\neqs.~\\eqref{Lagrangian}, \\eqref{Lagrangian_N}, and\n\\eqref{Lagrangian_B}. In particular, we find\n\\begin{equation}\\label{couplings_G-R_B}\ng_1 = 8(\\tilde{\\zeta} - \\zeta) \\;, \\quad g_2 = 8 \\tilde{\\zeta} \\;.\n\\end{equation}\n\nThe Feynman rules for the effective anomalous vertex, resulting from\nthe Lagrangian \\eqref{Lagrangian}, are given by \\cite{Horejsi:1994}\n\\begin{align}\\label{Feynman_rules}\nP^{\\mu\\nu\\rho\\alpha} = \\mathcal{P}&\\{ g_1[(p_1\\cdot p_2)(p_2\\cdot\np_3) g^{\\mu\\nu}g^{\\rho\\alpha} - (p_1\\cdot p_3) p_2^\\mu p_1^\\nu\ng^{\\rho\\alpha}\n\\nonumber \\\\\n&- (p_1\\cdot p_3)p_1^\\nu p_2^\\alpha g^{\\mu\\rho} + p_2^\\mu p_1^\\nu\np_1^ \\rho p_3^\\alpha ]\n\\nonumber \\\\\n&+ g_2[ -(p_1\\cdot p_2)(p_1\\cdot p_3) g^{\\mu\\alpha}g^{\\nu\\rho} +\n(p_2\\cdot p_3) p_1^\\nu p_1^\\alpha g^{\\mu\\rho}\n\\nonumber \\\\\n&- (p_2\\cdot p_3) p_1^\\nu p_1^\\rho g^{\\mu\\alpha} + (p_2\\cdot p_3)\np_1^\\nu p_2^\\alpha g^{\\mu\\rho} + 2(p_2\\cdot p_3) p_2^\\mu p_1^\\rho\ng^{\\nu\\alpha}\n\\nonumber \\\\\n&- (p_1\\cdot p_3) p_2^\\rho p_1^\\alpha g^{\\mu\\nu} + p_3^\\mu p_1^\\nu\np_2^\\rho p_1^\\alpha ] \\} \\;,\n\\end{align}\nwhere $\\mathcal{P}$ denotes possible permutations $(p_1,\\mu)\n\\leftrightarrow (p_2,\\nu) \\leftrightarrow (p_3, \\rho)$, and all\nmomenta in the $\\gamma\\gamma\\gamma Z$ vertex are assumed to be\nincoming ones. Correspondingly, the polarization tensor is equal to\n\\begin{equation}\\label{polarization_tensor}\nP_{\\mu\\nu\\rho\\alpha}(p_1, p_2, p_3) = g_1 \\sum_{i=1}^4\nP_{\\mu\\nu\\rho\\alpha}^{(1,i)}(p_1, p_2, p_3) + g_2 \\sum_{i=1}^7\nP_{\\mu\\nu\\rho\\alpha}^{(2,i)}(p_1, p_2, p_3) \\;.\n\\end{equation}\nElectromagnetic gauge invariance results in equations $p_1^\\mu\nP_{\\mu\\nu\\rho\\alpha} = p_2^\\nu P_{\\mu\\nu\\rho\\alpha} = p_3^\\rho\nP_{\\mu\\nu\\rho\\alpha} = 0$. Note that terms proportional to $p_1^\\mu,\np_2^\\nu, p_3^\\rho$ are omitted in \\eqref{Feynman_rules}, since they\ndo not contribute to the matrix element, see\neq.~\\eqref{matrix_element} below. Explicit expressions for the\ntensors $P_{\\mu\\nu\\rho\\alpha}^{(1,i)}$ and\n$P_{\\mu\\nu\\rho\\alpha}^{(2,i)}$ are presented in Appendix~A. To\ncalculate helicity amplitudes for the process \\eqref{process}, one\nhas to make the replacement $p_3 \\rightarrow -p_3$ in the Feynman\nrules for the $\\gamma\\gamma\\gamma Z$ vertex given by\neqs.~\\eqref{Feynman_rules}, \\eqref{polarization_tensor}, and\n\\eqref{P1.1}-\\eqref{P2.7}.\n\n\n\\section{Helicity amplitudes}\n\\label{sec:amplitudes}\n\nWe work in the c.m.s. of the colliding real photons, $\\vec{p}_1 +\n\\vec{p}_2 = 0$, where the momenta are given by\n\\begin{align}\\label{momenta}\np_1^\\mu &=(p,0,0,p) \\;,\n\\nonumber \\\\\np_2^\\mu &=(p,0,0,-p) \\;,\n\\nonumber \\\\\np_3^\\mu &=(k, 0, k\\sin\\theta, k\\cos\\theta) \\;,\n\\nonumber \\\\\np_4^\\mu &=(E, 0, -k\\sin\\theta, -k\\cos\\theta) \\;.\n\\end{align}\nHere $E = \\sqrt{k^2 + m_Z^2}$, with $m_Z$ being the mass of the $Z$\nboson. The Mandelstam variables of the process \\eqref{process} are\n\\begin{align}\\label{Mandelstam_var}\ns &= (p_1 + p_2)^2 = 4p^2 \\;,\n\\nonumber  \\\\\nt &= (p_1 - p_3)^2 = -2pk (1 - \\cos\\theta) \\;,\n\\nonumber  \\\\\nu &= (p_2 - p_3)^2 = -2pk (1 + \\cos\\theta) \\;,\n\\end{align}\nwhere $\\theta$ is a scattering angle in the c.m.s. Note that $s + t\n+ u = m_Z^2$.\n\nIn the chosen system the polarization vectors are equal to\n\\begin{align}\\label{pol_vectors_initial}\n\\varepsilon_\\mu^+(p_1) &= \\varepsilon_\\mu^-(p_2) =\n\\frac{1}{\\sqrt{2}}(0, 1, i, 0) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^-(p_1) &= \\varepsilon_\\mu^+(p_2) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i, 0) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^+(p_3) &= \\varepsilon_\\mu^-(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, i\\cos \\theta, -i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^-(p_3) &= \\varepsilon_\\mu^+(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i\\cos\\theta,\ni\\sin\\theta) \\;, \\nonumber \\\\\n\\varepsilon_\\mu^0(p_4) &= \\frac{1}{m_Z}(k, 0, -E\\sin\\theta, -\nE\\cos\\theta) \\;.\n\\end{align}\nThey obey the orthogonality condition $\\varepsilon_\\mu^\\lambda (k)\nk^\\mu = 0$. Correspondingly, we get the helicity vectors of the\nfinal photon and $Z$ boson,\n\\begin{align}\\label{pol_vectors_final}\n\\varepsilon_\\mu^{*+}(p_3) &= \\varepsilon_\\mu^{*-}(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i\\cos \\theta, i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^{*-}(p_3) &= \\varepsilon_\\mu^{*+}(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, i\\cos\\theta, -i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^{*0}(p_4) &= \\varepsilon_\\mu^0(p_4) =\n\\frac{1}{m_Z}(k, 0, -E\\sin\\theta, - E\\cos\\theta)  \\;.\n\\end{align}\n\nThe matrix element of the process \\eqref{process} with the definite\nhelicities of the incoming and outgoing bosons can be written as\n\\begin{equation}\\label{matrix_element}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(p_1, p_2, p_3) =\nP_{\\mu\\nu\\rho\\alpha}(p_1, p_2, p_3)\n\\,\\varepsilon_\\mu^{\\lambda_1}(p_1)\\varepsilon_\\nu^{\\lambda_2}(p_2)\n\\varepsilon_\\rho^{*\\lambda_3}(p_3)\\varepsilon_\\alpha^{*\\lambda_4}(p_4)\n\\;,\n\\end{equation}\nwhere the polarization tensor $P_{\\mu\\nu\\rho\\alpha}$ is given by\neq.~\\eqref{polarization_tensor}. We have calculated the anomalous\nhelicity amplitudes, and present their explicit expressions in\nAppendix~B. Using these expressions, we obtain the unpolarized\namplitude squared\n\\begin{equation}\\label{M2}\n\\sum_{\\lambda_1\\ldots\\lambda_4}|M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}|^2\n= \\frac{1}{4} [g_1^2(3A + 2B) - 4g_1g_2(A + B) + 4g_2^2(A + B)] \\;,\n\\end{equation}\nwhere\n\\begin{equation}\\label{A_B}\nA = s^2t^2 + t^2u^2 + u^2s^2 \\;, \\quad B = stu\\,m_Z^2 \\;.\n\\end{equation}\nWith a help of relations \\eqref{couplings_G-R_B}, we get from\n\\eqref{M2} the differential cross section\n\\begin{align}\\label{M2_B}\n\\frac{d\\sigma_{\\gamma\\gamma\\rightarrow\\gamma Z}}{d\\Omega} &=\n\\frac{\\beta}{64\\pi^2 s} \\frac{1}{4}\n\\sum_{\\lambda_1\\ldots\\lambda_4}|M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}|^2\n\\nonumber \\\\\n&= \\frac{\\beta}{16\\pi^2 s}\\,[(3\\zeta^2 + 3\\tilde{\\zeta}^2 - 2\\zeta\n\\tilde{\\zeta})A + 2(\\zeta^2 + \\tilde{\\zeta}^2)B] \\;,\n\\end{align}\nwhere $\\beta = 1 - m_Z^2\/s$, in a full agreement with eq.~(2.3) in\n\\cite{Baldenegro:2017}.\n\nTo estimate a SM background, we take analytical expressions for the\nSM helicity amplitudes from Appendix~A in \\cite{Gounaris:1999_1}.\nBoth $W$ boson loops \\cite{Bailargeon:1991,Jikia:1994} and charged\nfermion loops \\cite{Bailargeon:1991,Bij:1989} contribute to these\namplitudes. As shown in \\cite{Gounaris:1999_1}, for $s > (250 \\\n\\mathrm{GeV})^2$ the dominant SM amplitudes\n$A_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ are the $W$-loop non-flip\namplitudes $A^W_{++++}(s,t,u)$ and $A^W_{+-+-}(s,t,u) =\nA^W_{+--+}(s,u,t)$. Almost negligible are $A^W_{+++0}(s,t,u)$ and\n$A^W_{+-+0}(s,t,u) = A^W_{+--0}(s,u,t)$. The rest are even smaller.\nThe fermion-loop amplitudes are comparable only to very small\n$W$-loop amplitudes \\cite{Gounaris:1999_1}. Similar properties of\nthe SM helicity amplitudes are also valid for the process\n$\\gamma\\gamma\\rightarrow\\gamma\\gamma$ \\cite{Gounaris:1999_2}.\n\nAnother possible background comes from the SM process $\\gamma\\gamma\n\\rightarrow \\gamma l^+l^-$ where the invariant mass of the lepton\npair, $m_{l^+l^-}$, is close to the Z boson mass $m_Z$. We have\nobtained the cross section of the process to be of order 10$^{-3}$\nfb for $|m_{l^+l^-} - m_Z| < 10$ GeV. So, this background can be\nsafely ignored.\n\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\nThe differential cross section of the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ depends on spectra of the Compton\nbackscattered (CB) photons $f_{\\gamma\/e}(x_i)$, their helicities\n$\\xi(E_\\gamma^{(i)}, \\lambda_0)$ ($i = 1,2$), and helicity\namplitudes \\cite{Inan_Kisselev:2021,Sahin:2009},\n\\begin{align}\\label{diff_cs}\n\\frac{d\\sigma}{d\\cos \\theta} &= \\frac{\\beta}{128\\pi s}\n\\int\\limits_{x_{1 \\min}}^{x_{\\max}} \\!\\!\\frac{dx_1}{x_1}\n\\,f_{\\gamma\/e}(x_1) \\int\\limits_{x_{2 \\min}}^{x_{\\max}}\n\\!\\!\\frac{dx_2}{x_2} \\,f_{\\gamma\/e}(x_2)\n\\nonumber \\\\\n&\\times \\bigg\\{ \\left[ 1 + \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 + \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4}|M_{++\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 + \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 - \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4} |M_{+-\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 - \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 + \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4}|M_{-+\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 - \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 - \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4} |M_{--\\lambda_3\n\\lambda_4}|^2 \\bigg\\} ,\n\\end{align}\nwhere $\\lambda_3 = +, -$, $\\lambda_4 = +, -, 0$, $x_1 =\nE_{\\gamma}^{(1)}\/E_e$ and $x_2 = E_{\\gamma}^{(2)}\/E_e$ are the\nenergy fractions of the CB photon beams, $x_{1 \\min} =\np_\\bot^2\/E_e^2$, $x_{2 \\min} = p_\\bot^2\/(x_{1} E_e^2)$, and\n$p_{\\bot}$ is the transverse momentum of the outgoing particles.\nNote that $\\sqrt{s x_1 x_2}$ is the invariant energy of the\nbackscattered photons. The explicit expressions for\n$f_{\\gamma\/e}(x_i)$ and $\\xi(E_\\gamma^{(i)}, \\lambda_0)$ can be\nfound in \\cite{Inan_Kisselev:2021}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\hspace*{-0.4cm}\n\\includegraphics[scale=0.6]{WTDE750.eps}\n\\caption{The differential cross sections for the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ as functions of the invariant mass\nof the outgoing bosons for the CLIC energy $\\sqrt{s} = 1500$ GeV.\nThe left, middle and right panels correspond to the electron beam\nhelicities $\\lambda_e = 0.8, -0.8$, and 0, respectively. On each\nplot the curves denote (from the top downwards) the differential\ncross sections for the couplings $g_1 = 10^{-12} \\mathrm{\\\nGeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 = 10^{-12} \\mathrm{\\\nGeV}^{-4}$, the anomalous contributions for the same values of\ncouplings, the SM cross section.}\n\\label{fig:WTDE750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\hspace*{-0.4cm}\n\\includegraphics[scale=0.6]{WTDE1500F.eps}\n\\caption{The same as in Fig.~\\ref{fig:WTDE750}, but for the $e^+e^-$\ncollider energy $\\sqrt{s} = 3000$ GeV, and coupling sets $g_1 =\n10^{-13} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 =\n10^{-13} \\mathrm{\\ GeV}^{-4}$.} \\label{fig:WTDE1500F}\n\\end{center}\n\\end{figure}\n\nThe differential cross sections are shown in\nFigs.~\\ref{fig:WTDE750}, \\ref{fig:WTDE1500F} as functions of the\ninvariant mass of the $\\gamma Z$ system. We have imposed the cut on\nthe rapidity of the final bosons, $|\\eta| < 2.5$, and considered the\nregion $m_{\\gamma Z} > 250$ GeV. As one can see, the anomalous cross\nsections dominate the SM one for $m_{\\gamma Z} > 600$ GeV. The\neffect is more pronounced for the collision energy $\\sqrt{s} = 3000$\nGeV, especially as $m_{\\gamma Z}$ grows. Note that for $\\sqrt{s} =\n3000$ GeV the differential cross sections depend weakly on electron\nbeam helicity $\\lambda_e$. In Figs.~\\ref{fig:WCUTE750},\n\\ref{fig:WCUTE1500F} the total cross sections are presented\ndepending on $m_{\\gamma Z, \\mathrm{min}}$, minimal invariant mass of\ntwo outgoing bosons. The anomalous contribution dominates both the\ninterference one and SM cross section. The ratio of the total cross\nsection to the SM one grows with an increase of $m_{\\gamma Z}$,\nbeing more than one order of magnitude at large $m_{\\gamma Z}$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{WCUTE750.eps}\n\\caption{The total cross sections for the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ as functions of the minimal\ninvariant mass of the outgoing bosons for the $e^+e^-$ collider\nenergy $\\sqrt{s} = 1500$ GeV. The left, middle and right panels\ncorrespond to the electron beam helicities $\\lambda_e = 0.8, -0.8$,\nand 0, respectively. On each plot the curves denote (from the top\ndownwards) the total cross sections for the couplings $g_1 =\n10^{-12} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 =\n10^{-12} \\mathrm{\\ GeV}^{-4}$, the anomalous contributions for the\nsame values of couplings, the SM cross section.}\n\\label{fig:WCUTE750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{WCUTE1500F.eps}\n\\caption{The same as in Fig.~\\ref{fig:WCUTE750}, but for the\n$e^+e^-$ collider energy $\\sqrt{s} = 3000$ GeV, and coupling sets\n$g_1 = 10^{-13} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2\n= 10^{-13} \\mathrm{\\ GeV}^{-4}$.}\n\\label{fig:WCUTE1500F}\n\\end{center}\n\\end{figure}\n\nThe knowledge of the total cross sections and planned CLIC\nintegrated luminosities \\cite{CLIC_lum} enables us to calculate the\nexclusion regions for the QGCs. In our study we consider leptonic\n(electrons and muons) decays of the $Z$ boson. Let $s(b)$ be the\ntotal number of signal (background) events, and $\\delta$ the\npercentage systematic error. The number of events is defined as\n$\\sigma \\times L \\times \\mathcal{B}(Z \\rightarrow e, \\mu)$. The\nexclusion significance is given by \\cite{Zhang:2020}\n\\begin{equation}\\label{S_excl}\nS_{\\mathrm{excl}} = \\sqrt{ 2\\left[ s - b \\ln \\left( \\frac{b + s +\nx}{2b} \\right) - \\frac{1}{\\delta^2}\\ln \\left( \\frac{b - s + x}{2b}\n\\right) - (b + s -x) \\left( 1 + \\frac{1}{\\delta^2 b} \\right) \\right]\n} \\;,\n\\end{equation}\nwhere\n\\begin{equation}\\label{x}\nx = \\sqrt{(s+b)^2 - 4\\delta^2 s b^2\/(1 + \\delta^2 b)} \\;.\n\\end{equation}\nWe define the regions $S_{\\mathrm{excl}} \\leqslant 1.645$ as a\nregions that can be excluded at the 95\\% C.L. in the process\n$\\gamma\\gamma \\rightarrow \\gamma Z$ at the CLIC. To reduce the SM\nbackground, we impose the cut $m_{\\gamma Z} > 1000$ GeV, in addition\nto the bound $|\\eta| < 2.5$. The expected integrated luminosity at\nthe CLIC can be found, for instance, in \\cite{CLIC_lum}.\n\nIt is worth considering the unpolarized case first. One can obtain\nfrom eq.~\\eqref{M2} that the anomalous contribution to the\nunpolarized total cross section is proportional to the coupling\ncombination $3g_1^2 - 4g_1 g_2 + 4g_2^2$, provided terms\nproportional to $m_Z^2\/s \\ll 1$ are neglected in it. In such a case,\nthe exclusion regions are ellipses in the plane $(g_1 - g_2)$\nrotated clockwise through the angle $0.5 \\arctan 8 \\simeq\n41.4^\\circ$ around the origin. It is clear that our process is\nslightly more sensitive to the coupling $g_2$ rather than to $g_1$.\nOur 95\\% C.L. exclusion regions for anomalous QGCs for the\nunpolarized process $\\gamma\\gamma\\rightarrow\\gamma Z$ at the CLIC\nare shown in Figs.~\\ref{fig:excl_750}, \\ref{fig:excl_1500}. The\nresults are presented for $\\delta = 0$, $\\delta = 5\\%$, and $\\delta\n= 10\\%$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{exclusion_E750.eps}\n\\caption{The 95\\% C.L. exclusion regions for the couplings $g_1,\ng_2$ in the unpolarized reaction $\\gamma\\gamma \\rightarrow \\gamma Z$\nat the CLIC with the systematic errors $\\delta = 0\\%$ (black\nellipse), $\\delta = 5\\%$ (blue ellipse), and $\\delta = 10\\%$ (red\nellipse). The inner regions of the ellipses are inaccessible. The\ncollision energy is $\\sqrt{s} = 1500$ GeV, the integrated luminosity\nis $L = 2500$ fb$^{-1}$. The cut on the outgoing photon invariant\nmass $m_{\\gamma\\gamma} > 1000$ GeV was imposed.}\n\\label{fig:excl_750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{exclusion_E1500.eps}\n\\caption{The same as in Fig.~\\ref{fig:excl_750}, but for $\\sqrt{s} =\n3000$ GeV and $L=5000$ fb$^{-1}$.}\n\\label{fig:excl_1500}\n\\end{center}\n\\end{figure}\n\nIn Tabs.~\\ref{tab:excl_750}, \\ref{tab:excl_1500} we show the\nexclusion bounds on the couplings $g_1$ and $g_2$ for three values\nof the electron beam helicity $\\lambda_e$ and corresponding\nintegrated luminosity $L$. Let us underline that this time we did\nnot neglect the terms proportional to $m_Z^2$, both for unpolarized\nand for polarized reactions. As one can see, the best bound on the\ncouplings $g_{1,2}$ is approximately $5 \\times 10^{-15}$ GeV$^{-4}$\nfor the $e^+e^-$ energy $\\sqrt{s} = 3000$ GeV and electron beam\nhelicity $\\lambda_e = 0.8$.\n\n\\begin{table}[h]\n    \\centering \\caption{The 95\\% C.L. exclusion limits on the anomalous\n        quartic couplings $g_1$ and $g_2$  for the collision energy\n        $\\sqrt{s} = 1500$ GeV, and the cut $m_{\\gamma Z} > 1000$ GeV.\n        \\bigskip} \\label{tab:excl_750}\n    \\begin{tabular}{||c||c|c|c|c||}\n        \\hline\n        $\\lambda_e$ & & 0 & $-0.8$ & 0.8 \\\\\n        \\hline\n        \\quad $L$, fb$^{-1}$ & & 2500 & 2000 & 500 \\\\\n        \\hline\n        \\makecell{$|g_1|$, GeV$^{-4}$ \\\\ $(g_2=0)$} & \\makecell{$\\delta=0\\%$ \\\\  $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 4.19\\times10^{-14} \\\\ 5.32\\times10^{-14} \\\\ 6.81\\times10^{-14}}$ &\n        $\\makecell{ 6.25\\times10^{-14} \\\\ 7.91\\times10^{-14} \\\\ 1.02\\times10^{-13}}$ &\n        $\\makecell{ 4.42\\times10^{-14} \\\\ 5.38\\times10^{-14} \\\\ 6.78\\times10^{-14}}$ \\\\\n        \\hline\n        \\makecell{$|g_2|$, GeV$^{-4}$ \\\\  $(g_1=0)$} & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 3.61\\times10^{-14} \\\\ 4.63\\times10^{-14} \\\\ 5.91\\times10^{-14}}$ &\n        $\\makecell{ 5.47\\times10^{-14} \\\\ 6.94\\times10^{-14} \\\\ 8.87\\times10^{-14}}$ &\n        $\\makecell{ 4.53\\times10^{-14} \\\\ 5.51\\times10^{-14} \\\\ 6.94\\times10^{-14}}$ \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\\begin{table}[h]\n    \\centering \\caption{The same as in Tab.~\\ref{tab:excl_750}, but for\n        the energy $\\sqrt{s} = 3000$ GeV and different values of the\n        integrated luminosities. \\bigskip} \\label{tab:excl_1500}\n    \\begin{tabular}{||c||c|c|c|c||}\n        \\hline\n        $\\lambda_e$ & & 0 & $-0.8$ & 0.8 \\\\\n        \\hline\n        \\quad $L$, fb$^{-1}$ & & 5000 & 4000 & 1000 \\\\\n        \\hline\n        \\makecell{$|g_1|$, GeV$^{-4}$ \\\\ $(g_2=0)$ } & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 5.98\\times10^{-15} \\\\ 1.33\\times10^{-14} \\\\ 1.85\\times10^{-14}}$ &\n        $\\makecell{ 7.14\\times10^{-15} \\\\ 1.73\\times10^{-14} \\\\ 2.39\\times10^{-14}}$ &\n        $\\makecell{ 5.13\\times10^{-15} \\\\ 7.79\\times10^{-15} \\\\ 1.04\\times10^{-14}}$ \\\\\n        \\hline\n        \\makecell{$|g_2|$, GeV$^{-4}$ \\\\ $(g_1=0)$} & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 5.18\\times10^{-15} \\\\ 1.16\\times10^{-14} \\\\ 1.62\\times10^{-14}}$ &\n        $\\makecell{ 6.62\\times10^{-15} \\\\ 1.60\\times10^{-14} \\\\ 2.21\\times10^{-14}}$ &\n        $\\makecell{ 5.19\\times10^{-15} \\\\ 7.87\\times10^{-15} \\\\ 1.05\\times10^{-14}}$ \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\nRecently, the bounds on the anomalous quartic couplings for the\nvertex $\\gamma\\gamma\\gamma Z$ were obtained via $\\gamma Z$\nproduction with intact protons in the forward region at the LHC\n\\cite{Baldenegro:2017}. To examine this process, the effective\nLagrangian \\eqref{Lagrangian_B} was used with the anomalous\ncouplings $\\zeta, \\tilde{\\zeta}$. Both for integrated luminosity 300\nfb$^{-1}$ and high luminosity 3000 fb$^{-1}$ sensitivities were\nfound to be similar, $\\zeta, \\tilde{\\zeta} \\sim 1 \\times 10^{-13}$\nat the 95\\% C.L. Taking into account the relations between couplings\n$\\zeta, \\tilde{\\zeta}$ and our couplings $g_1, g_2$,\n\\eqref{couplings_G-R_B}, we expect that the sensitivities of $g_1,\ng_2 \\sim 8\\times 10^{-13}$ can be reached at the LHC (HL-LHC). These\nvalues should be compared with the CLIC bounds in\nTabs.~\\ref{tab:excl_750} and \\ref{tab:excl_1500}. Note that the\nexpected sensitivity from the $Z\\rightarrow\\gamma\\gamma\\gamma$ decay\nsearch at the LHC \\cite{ATLAS:Z_decay} is approximately three orders\nof magnitude smaller than that obtained in \\cite{Baldenegro:2017}.\n\n\n\\section{Unitarity constraints on anomalous quartic couplings}\n\\label{sec:unit_const}\n\nThe anomalous contribution to the total cross section rises as\n$s^3$. Thus, the contribution of the effective operators in\n\\eqref{Lagrangian} may lead to unitarity violation at high energies.\nThat is why we need to study bounds imposed by partial-wave\nunitarity. The partial-wave expansion of the helicity amplitude in\nthe center-of-mass system was derived in \\cite{Jacob:2000} and used\nin a number of papers \\cite{Gounaris:1994}. It looks like\n\\begin{align}\\label{helicity_ampl_expansion}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s, \\theta, \\varphi) &=\n16\\pi \\sum_J (2J + 1) \\sqrt{(1 + \\delta_{\\lambda_1\\lambda_2})(1 +\n\\delta_{\\lambda_3\\lambda_4})}\n\\nonumber \\\\\n&\\times \\,e^{i(\\lambda - \\mu)\\phi} \\,d^J_{\\lambda\\mu}(\\theta)\n\\,T^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) \\;,\n\\end{align}\nwhere $\\lambda = \\lambda_1 - \\lambda_2$, $\\mu = \\lambda_3 -\n\\lambda_4$, $\\theta(\\phi)$ is the polar (azimuth) scattering angle,\nand $d^J_{\\lambda\\mu}(\\theta)$ is the Wigner (small) $d$-function\n\\cite{Wigner}. Relevant formulas for the $d$-functions are given in\nAppendix~C. In our case $\\lambda, \\mu$ are even numbers, $\\lambda,\n\\,\\mu = 0, \\pm 2$ (see below). If we choose the plane $(x - z)$ as a\nscattering plane, then $\\phi = 0$ in\n\\eqref{helicity_ampl_expansion}. Parity conservation means that\n\\begin{equation}\\label{parity_conservation}\nT^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) = (-1)^{\\lambda_1 -\n\\lambda_2 - \\lambda_3 + \\lambda_4}\n\\,T^J_{-\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4}(s) \\;.\n\\end{equation}\nPartial-wave unitarity in the limit $s \\gg (m_1 + m_2)^2$ requires\nthat\n\\begin{equation}\\label{parity_wave_unitarity}\n\\left| T^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) \\right| \\leq 1\n\\;.\n\\end{equation}\nUsing orthogonality of the $d$-functions\n\\eqref{Wigner_func_orthogonality}, we find the partial-wave\namplitude\n\\begin{align}\\label{parity wave_func}\nT^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) &= \\frac{1}{32\\pi}\n\\frac{1}{\\sqrt{(1 + \\delta_{\\lambda_1\\lambda_2})(1 +\n\\delta_{\\lambda_3\\lambda_4})}} \\int\\limits_{-1}^1 \\!\\!\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s, z)\n\\,d^{J}_{\\lambda\\mu}(z) \\,dz \\;.\n\\end{align}\nHere and in what follows, $z = \\cos\\theta$. Note that\n$M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} = g_1\nM^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} + g_2\nM^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$, and the helicity\namplitudes $M^{(1,2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ are\ngiven in Appendix~B.\n\n\n\\subsection{Unitarity bounds on coupling $\\bs{g_1}$ ($\\bs{g_2 = 0}$)}\n\nTo obtain a unitarity bound on the coupling $g_1$, we put $g_2 = 0$.\nLet us note that due to eq.~\\eqref{parity_conservation}, it is\nsufficient to examine the helicity amplitudes with $\\lambda_1 = +1$\nonly. Moreover, it is enough to consider four amplitudes,\n$M^{(1)}_{++++}$, $M^{(1)}_{+-+-}$, $M^{(1)}_{+--+}$, and\n$M^{(1)}_{++--}$, since the rest are suppressed by small factor\n$m_Z\/\\sqrt{s}$ or zero. \\\\\n\n\\textbf{1.} $\\lambda_1 = \\lambda_2 = \\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$. The helicity amplitude is given by the\nfirst of equations \\eqref{independent_ampl_M1},\n\\begin{equation}\\label{M1_++++}\nM_{++++}(s,z) = g_1 M^{(1)}_{++++}(s) = -\\frac{g_1 }{4} \\,s(s -\nm_Z^2) \\;.\n\\end{equation}\nUsing eqs.~\\eqref{d-function_00}-\\eqref{integral_Legendre}, we find\nthat the partial-wave amplitude with $J=0$ is the only non-zero\namplitude,\n\\begin{equation}\\label{T1_++++}\nT^0_{++++}(s) = - \\frac{g_1}{128\\pi} \\,s(s - m_Z^2)\n\\int\\limits_{-1}^1 d^{\\,0}_{00}(z) \\,dz = - \\frac{g_1}{64\\pi} \\,s(s\n- m_Z^2) \\;.\n\\end{equation}\nCorrespondingly, we obtain from \\eqref{parity_wave_unitarity},\n\\eqref{T1_++++}\n\\begin{equation}\\label{T1++++_unitarity}\n|g_1| \\leq 64\\pi [s(s - m_Z^2)]^{-1} \\;.\n\\end{equation}\n\n\\textbf{2.} $\\lambda_1 = -\\lambda_2 = \\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 2$. According to \\eqref{independent_ampl_M1},\n\\begin{equation}\\label{M1_+-+-}\nM_{+-+-}(s,z) = g_1 M^{(1)}_{+-+-}(s,z) = -\\frac{g_1}{4}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 + z}{2}\\right)^{\\!2} .\n\\end{equation}\nIt follows from \\eqref{parity wave_func}, \\eqref{d-function_22} that\n\\begin{align}\\label{T1+-+-_J}\nT^J_{+-+-}(s) &= -\\frac{g_1}{128\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\int\\limits_{-1}^1 \\left( \\frac{1 + z}{2}\\right)^{\\!2}\n\\,d^{J}_{22}(z) \\,dz = -\\frac{g_1}{128\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\nonumber \\\\\n&\\times \\int\\limits_{-1}^1 \\left( \\frac{1 + z}{2}\\right)^{\\!4}\n{}_2F_1 \\left( 2-J, J+3; 1; \\frac{1-z}{2} \\right) dz \\;.\n\\end{align}\nLet $(1 - z)\/2 = x$, then $(1 + z)\/2 = 1 - x$, and we find\n\\begin{align}\\label{}\nT^J_{+-+-}(s) &= -\\frac{1}{64\\pi} \\,\\frac{g_1 s^3}{s - m_Z^2}\n\\int\\limits_0^1 (1 - x)^4 \\,{}_2F_1( 2-J, J+3; 1; x) \\,dx\n\\nonumber \\\\\n&= -\\frac{3g_1}{8\\pi\\Gamma(4+J)\\Gamma(3-J)} \\frac{s^3}{s - m_Z^2}\n\\;,\n\\end{align}\nwhere we used formulas 2.21.1.5 and 2.21.1.6 in \\cite{Prudnikov_v3}.\nThus, only three partial-waves amplitudes, $T^0_{+-+-}(s)$,\n$T^1_{+-+-}(s)$, and $T^2_{+-+-}(s)$, are non-zero. The most\nimportant for us is $T^0_{+-+-}(s)$, since it results in the\nstrongest constraint on the coupling $g_1$,\n\\begin{equation}\\label{T1+-+-_unitarity}\n|g_1| \\leq 32\\,\\pi (s - m_Z^2) s^{-3}\\;.\n\\end{equation}\n\n\\textbf{3.} $\\lambda_1 = -\\lambda_2 = -\\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = 2, \\,\\mu = -2$, and we have\n\\begin{equation}\\label{M1_+--+}\nM_{+--+}(s,z) = g_1 M^{(1)}_{+--+}(s,z) = -\\frac{g_1}{4}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 - z}{2}\\right)^{\\!2} .\n\\end{equation}\nUsing eq.~\\eqref{d-function_2-2} and first relation in\n\\eqref{d-function_symmetry}, after substitutions $(1 + z)\/2 = x$,\n$(1 - z)\/2 = 1 - x$, we reduce this case to the previous one. As a\nresult, we come again to the upper bound \\eqref{T1+-+-_unitarity}.\n\n\\textbf{4.} $\\lambda_1 = \\lambda_2 = -\\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$, and\n\\begin{equation}\\label{M1_++--}\nM_{++--}(s,z) = g_1 M^{(1)}_{++--}(s,z) = -\\frac{g_1}{8}\n\\,\\frac{s^3}{s - m_Z^2} (3 + z^2) \\;.\n\\end{equation}\nOnly two partial-waves amplitudes, $T^0_{++--}(s)$ and\n$T^2_{++--}(s)$, are non-zero,\n\\begin{equation}\\label{T1_++--}\nT^0_{++--}(s) =  -\\frac{5\\,g_1}{192\\pi} \\,\\frac{s^3}{s - m_Z^2} \\;,\n\\quad T^2_{++--}(s) =  -\\frac{g_1}{960\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\;.\n\\end{equation}\nThe strongest bound on $g_1$ comes from unitarity constraint on\n$T^0_{++--}(s)$,\n\\begin{equation}\\label{T1++--_unitarity}\n|g_1| \\leq \\frac{192\\,\\pi}{5} (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\n\n\n\\subsection{Unitarity bounds on coupling $\\bs{g_2}$ ($\\bs{g_1 = 0}$)}\n\nTo derive a unitarity bound on the coupling $g_2$, we take $g_1 =\n0$. It is sufficient to consider three amplitudes, $M^{(2)}_{++++}$,\n$M^{(2)}_{+-+-}$, and $M^{(2)}_{+--+}$. The rest are suppressed by\nsmall factor $m_Z\/\\sqrt{s}$ or zero. \\\\\n\n\\textbf{1.} $\\lambda_1 = \\lambda_2 = \\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$. The helicity amplitude is given by the\nfirst of equations \\eqref{independent_ampl_M2}\n\\begin{equation}\\label{M2_++++}\nM_{++++}(s,z) = g_2 M^{(2)}_{++++}(s) = \\frac{g_2}{2} \\,s(s - m_Z^2)\n\\;.\n\\end{equation}\nAs a result, we get\n\\begin{equation}\\label{T2++++_unitarity}\n|g_2| \\leq 32\\pi [s(s - m_Z^2)]^{-1} \\;.\n\\end{equation}\n\n\\textbf{2.} $\\lambda_1 = -\\lambda_2 = \\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 2$, and we find from\n\\eqref{independent_ampl_M2}\n\\begin{equation}\\label{M2_+-+-}\nM_{+-+-}(s,z) = g_2 M^{(2)}_{+-+-}(s,z) = \\frac{g_2}{2}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 + z}{2}\\right)^{\\!2} .\n\\end{equation}\nWe follow the derivation of eq.~\\eqref{T1+-+-_unitarity} and come to\nthe inequality\n\\begin{equation}\\label{T2+-+-_unitarity}\n|g_2|  \\leq 16\\pi (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\n\n\\textbf{3.} $\\lambda_1 = -\\lambda_2 = -\\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = 2, \\,\\mu = -2$, and we obtain\n\\begin{equation}\\label{M2_+--+}\nM_{+--+}(s,z) = g_2 M^{(2)}_{+--+}(s,z) = \\frac{g_2}{2}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 - z}{2}\\right)^{\\!2} .\n\\end{equation}\nUsing the first relation in \\eqref{d-function_symmetry} and\neq.~\\eqref{d-function_2-2}, we can reduce this case  to the previous\none to get eq.~\\eqref{T2+-+-_unitarity}.\n\n\\textbf{4.} $\\lambda_1 = \\lambda_2 = -\\lambda_3 = -\\lambda_4 = 1$.\nThe helicity amplitude $M^{(2)}_{++--}(s,z) = 0$.\n\n\n\\subsection{Unitarity bounds on couplings $\\bs{g_1}$ and $\\bs{g_2}$}\n\nNow we consider a general case with $g_1, g_2 \\neq 0$. Note that\n$M^{(2)}_{++++} = - 2M^{(1)}_{++++}$, $M^{(2)}_{+-+-} =\n-2M^{(1)}_{+-+-}$, and $M^{(2)}_{+--+} = - 2M^{(1)}_{+--+}$.\nCorrespondingly, $M_{++++} = (g_1 - 2g_2)M^{(1)}_{++++}$, etc. From\nformulas derived in two previous subsections we immediately get the\nfollowing bound on a linear combination of $g_1$ and $g_2$,\n\\begin{equation}\\label{g1_g2_unitarity}\n|g_1 - 2g_2| \\leq 32\\,\\pi (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\nLet us underline that $M_{++--}(s,z) = g_1 M^{(1)}_{++--}(s,z)$. It\nmeans that inequality \\eqref{T1++--_unitarity} holds for a general\ncase ($g_1, g_2 \\neq 0$). It enables us to obtain constraints\nseparately on each coupling. If the couplings $g_1$, $g_2$ have the\nsame sign, then\n\\begin{equation}\\label{g1_g2_same_signs}\n|g_1| \\leq  \\frac{192\\,\\pi}{5} (s - m_Z^2) s^{-3}  \\;, \\quad |g_2|\n\\leq \\frac{176\\,\\pi}{5}(s - m_Z^2) s^{-3}  \\;.\n\\end{equation}\nIf the signs of the couplings $g_1$, $g_2$ are opposite, we obtain\n\\begin{equation}\\label{g1_g2_opposite_signs}\n|g_1| \\leq  32\\,\\pi \\,(s - m_Z^2) s^{-3} \\;, \\quad |g_2| \\leq\n16\\,\\pi \\,(s - m_Z^2) s^{-3}  \\;.\n\\end{equation}\nThe bounds on the couplings $g_1, g_2$ along with their numerical\nvalues are collected in Tab.~\\ref{tab:unit_lim}. We have taken into\naccount that $m_Z^2\/s \\ll 1$ for the CLIC energies.\n\n{\\setlength{\\extrarowheight}{4pt}\n\\begin{table}[h]\n\\centering \\caption{Unitarity constraints on the anomalous couplings\nwhen just one coupling is non-zero (second and third columns), and\nwhen both couplings are non-vanishing (fourth and fifth columns for\nthe couplings of the same sign, sixth and seventh columns for the\ncouplings of opposite signs). The numerical values of the bounds are\ngiven for the collision energy $\\sqrt{s} = 1500(3000)$ GeV.\n\\bigskip} \\label{tab:unit_lim}\n\\begin{tabular}{||c||c|c||c|c||c|c||}\n  \\hline\n\\multicolumn{1}{||c||} {} & \\multicolumn{2}{c||} {1 operator ($g_2 =\n0$ or $g_1 = 0$)} & \\multicolumn{2}{c||} {2 operators ($g_1 g_2 >\n0$)} & \\multicolumn{2}{c||} {2 operators ($g_1 g_2 < 0$)}  \\\\\n  \\hline\n  $g_1$ & $32\\pi s^{-2}$ & 20(1.2) TeV$^{-4}$ & $\\frac{192}{5}s^{-2}$\n  & 24(1.5) TeV$^{-4}$ & $32\\pi s^{-2}$ & 20(1.2) TeV$^{-4}$ \\\\ [2pt]\n  \\hline\n  $g_2$  & $16\\pi s^{-2}$ & 10(0.6) TeV$^{-4}$ & $\\frac{176}{5}s^{-2}$\n  & 22(1.4) TeV$^{-4}$ & $16\\pi s^{-2}$ & 10(0.6) TeV$^{-4}$ \\\\ [2pt]\n  \\hline\n\\end{tabular}\n\\end{table}\n}\n\nTo summarize, in spite of the fact that the anomalous contribution\nto the total cross section is proportional to $s^3$, the unitarity\nis not violated in the region of the anomalous QGCs presented in\nTabs.~\\ref{tab:excl_750}, \\ref{tab:excl_1500}.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn the present paper, the CLIC discovery potential for exclusive\n$\\gamma Z$ production in the scattering of the Compton backscattered\nphotons at the $e^+e^-$ collision energies 1500 GeV and 3000 GeV is\nstudied. We have shown that such a process provides an opportunity\nof searching for the anomalous quartic neutral gauge couplings for\nthe $\\gamma\\gamma\\gamma Z$ vertex at the CLIC. Both unpolarized and\npolarized initial electron beams are examined. To describe the\nanomalous quartic gauge couplings we used the effective Lagrangian\nwhich conserves gauge invariance. Although quartic gauge couplings\nare already induced at the dimension-six level, we considered the\neffective Lagrangian with CP conserving dimension-eight operators\nwithout contributing to anomalous trilinear gauge interactions.\n\nWe have derived the explicit expressions for the anomalous\ncontributions to the helicity amplitudes of the process\n$\\gamma\\gamma \\rightarrow \\gamma Z$. After that the differential and\ntotal cross sections are calculated depending on $m_{Z\\gamma}$, the\ninvariant mass of the $\\gamma Z$ system. It is shown that the\nanomalous contribution dominates both the interference and SM cross\nsections. Moreover, the ratio of the total cross section to the SM\none grows with the increase of $m_{Z\\gamma}$, being more\napproximately one order of magnitude at large $m_{\\gamma Z}$.\n\nIt enabled us to obtain the exclusion regions for the anomalous\ncouplings with the systematic errors of 0\\%, 5\\%, and 10\\%. We have\nconsidered the $Z$ boson decay into leptons (electron and muons).\nFor both couplings, $g_{1,2}$, the best bounds are equal to\napproximately $4.4 \\times 10^{-14}$ GeV$^{-4}$ and $5.1 \\times\n10^{-15}$ GeV$^{-4}$, for the $e^+e^-$ energies 1500 GeV and 3000\nGeV, respectively. They are achieved when electron beam helicity is\nequal to 0.8. We have checked that the unitarity is not violated in\nthe region of the couplings considered in the paper. Our best bound\non the anomalous couplings for the collision energy 3000 GeV is\nroughly two orders of magnitude stronger than the limits which can\nbe reached at the LHC and HL-LHC. This points to a great potential\nof the CLIC and other future leptonic colliders to probe the\nanomalous $\\gamma\\gamma\\gamma Z$ couplings.\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\n\\section*{Appendix A}\n\\label{app:A}\n\nHere we present explicit expressions for components of the\npolarization tensor \\eqref{polarization_tensor}. They are the\nfollowing\n\\begin{align}\\label{P1.1}\nP_{\\mu\\nu\\rho\\alpha}^{(1.1)} &= (p_1\\cdot p_2)[ (p_1\\cdot p_3) +\n(p_2\\cdot p_3)] g_{\\mu\\nu}g_{\\rho\\alpha} + (p_1\\cdot p_3)[ (p_1\\cdot\np_2) + (p_2\\cdot p_3) ] g_{\\mu\\rho}g_{\\nu\\alpha}\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[ (p_1\\cdot p_2) + (p_1\\cdot p_3) ]\ng_{\\nu\\rho}g_{\\mu\\alpha} \\;,\n\\end{align}\n\\begin{align}\\label{P1.2}\nP_{\\mu\\nu\\rho\\alpha}^{(1.2)} &= - \\{ [ (p_1\\cdot p_2) + (p_1\\cdot\np_3)] p_{3\\nu} p_{2\\rho}g_{\\mu\\alpha} + [ (p_1\\cdot p_2) + (p_2\\cdot\np_3) ] p_{3\\mu}p_{1\\rho} g_{\\nu\\alpha}\n\\nonumber \\\\\n&\\quad + [ (p_1\\cdot p_3) + (p_2\\cdot p_3) ]\np_{2\\mu}p_{1\\nu}g_{\\rho\\alpha} \\} \\;,\n\\end{align}\n\\begin{align}\\label{P1.3}\nP_{\\mu\\nu\\rho\\alpha}^{(1.3)} &= -[ (p_1\\cdot p_2)( p_{1\\rho} +\np_{2\\rho} )p_{3\\alpha}g_{\\mu\\nu} + (p_1\\cdot p_3)( p_{1\\nu} +\np_{3\\nu} )p_{2\\alpha} g_{\\mu\\rho}\n\\nonumber \\\\\n&\\quad + (p_2\\cdot p_3)( p_{2\\mu} + p_{3\\mu} )p_{1\\alpha}g_{\\nu\\rho}\n] \\;,\n\\end{align}\n\\begin{align}\\label{P1.4}\nP_{\\mu\\nu\\rho\\alpha}^{(1.4)} &= p_{2\\mu}p_{1\\nu} ( p_{1\\rho} +\np_{2\\rho} ) p_{3\\alpha} + p_{3\\mu} ( p_{1\\nu} + p_{3\\nu} ) p_{1\\rho}\np_{2\\alpha}\n\\nonumber \\\\\n&+ p_{3\\nu} p_{2\\rho}( p_{2\\mu} + p_{3\\mu} ) p_{1\\alpha} \\;,\n\\end{align}\nand\n\\begin{align}\\label{P2.1}\nP_{\\mu\\nu\\rho\\alpha}^{(2.1)} = &- 2[(p_1\\cdot p_3)(p_2\\cdot p_3)\ng_{\\mu\\nu}g_{\\rho\\alpha} + (p_1\\cdot p_2)(p_2\\cdot p_3)\ng_{\\mu\\rho}g_{\\nu\\alpha}\n\\nonumber \\\\\n&+ (p_1\\cdot p_2)(p_1\\cdot p_3) g_{\\nu\\rho} g_{\\mu\\alpha}] \\;,\n\\end{align}\n\\begin{align}\\label{P2.2}\nP_{\\mu\\nu\\rho\\alpha}^{(2.2)} &= (p_1\\cdot\np_2)[p_{3\\mu}p_{3\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{3\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{2\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{2\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{1\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{1\\alpha}g_{\\mu\\nu}] \\;,\n\\end{align}\n\\begin{equation}\\label{P2.3}\nP_{\\mu\\nu\\rho\\alpha}^{(2.3)} = - 2[(p_1\\cdot p_2)p_{3\\mu}p_{3\\nu}\ng_{\\rho\\alpha} + (p_1\\cdot p_3)p_{2\\mu}p_{2\\rho} g_{\\nu\\alpha} +\n(p_2\\cdot p_3)p_{1\\nu}p_{1\\rho} g_{\\mu\\alpha}] \\;,\n\\end{equation}\n\\begin{align}\\label{P2.4}\nP_{\\mu\\nu\\rho\\alpha}^{(2.4)} &= (p_1\\cdot\np_2)[p_{3\\mu}p_{1\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{2\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{1\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{3\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{2\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{3\\alpha}g_{\\mu\\nu}] \\;,\n\\end{align}\n\\begin{align}\\label{P2.5}\nP_{\\mu\\nu\\rho\\alpha}^{(2.5)} &= 2\\{ (p_1\\cdot\np_2)[p_{3\\mu}p_{2\\rho}g_{\\nu\\alpha} +\np_{3\\nu}p_{1\\rho}g_{\\mu\\alpha}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{3\\nu}g_{\\rho\\alpha} +\np_{2\\rho}p_{1\\nu}g_{\\mu\\alpha}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{3\\mu}p_{1\\nu}g_{\\rho\\alpha} +\np_{2\\mu}p_{1\\rho}g_{\\nu\\alpha}] \\} \\;,\n\\end{align}\n\\begin{align}\\label{P2.6}\nP_{\\mu\\nu\\rho\\alpha}^{(2.6)} = &-\\{ (p_1\\cdot\np_2)[p_{3\\mu}p_{2\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{1\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{3\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{1\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{3\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{2\\alpha}g_{\\mu\\nu}] \\} \\;,\n\\end{align}\n\\begin{align}\\label{P2.7}\nP_{\\mu\\nu\\rho\\alpha}^{(2.7)} = - ( p_{2\\mu}p_{3\\nu}p_{1\\rho} +\np_{3\\mu}p_{1\\nu}p_{2\\rho} ) (p_{1\\alpha} + p_{2\\alpha} +\np_{3\\alpha}) \\;.\n\\end{align}\nNote that the last tensor does not contribute to the matrix element\n\\eqref{matrix_element}, since it is proportional to $p_{4\\alpha}$.\nOne can directly check that\n\\begin{align}\\label{polarization_tensor_gauge_inv}\np_1^\\mu \\sum_{i=1}^4 P_{\\mu\\nu\\rho\\alpha}^{(1.i)} &= p_2^\\nu\n\\sum_{i=1}^4 P_{\\mu\\nu\\rho\\alpha}^{(1.i)} = p_3^\\rho \\sum_{i=1}^4\nP_{\\mu\\nu\\rho\\alpha}^{(1.i)} = 0 \\;,\n\\nonumber \\\\\np_1^\\mu \\sum_{i=1}^7 P_{\\mu\\nu\\rho\\alpha}^{(2.i)} &= p_2^\\nu\n\\sum_{i=1}^7 P_{\\mu\\nu\\rho\\alpha}^{(2.i)} = p_3^\\rho \\sum_{i=1}^7\nP_{\\mu\\nu\\rho\\alpha}^{(2.i)} = 0 \\;.\n\\end{align}\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{B.\\arabic{equation}}\n\n\\section*{Appendix B}\n\\label{app:B}\n\nIn accordance with eq.~\\eqref{polarization_tensor}, any anomalous\nhelicity amplitude is the sum of two terms,\n\\begin{equation}\\label{helicity_ampl_sum}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} = g_1\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}^{(1)} + g_2\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}^{(2)} \\;.\n\\end{equation}\nThere are $2^3\\times3 = 24$ helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ and,\ncorrespondingly, 24 amplitudes\n$M^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ for the process\n\\eqref{process}. Bose-Einstein statistics and parity invariance\ndemand that there exist nine independent helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$, six for transverse $Z$ and three for longitudinal $Z$. Our\ncalculations resulted in the following helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$\n\\begin{align}\\label{independent_ampl_M1}\nM^{(1)}_{++++}(s,t,u) &= \\frac{1}{4} s(t + u) \\;,\n\\nonumber \\\\\nM^{(1)}_{+++-}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(1)}_{++-+}(s,t,u) &= \\frac{1}{2} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{++--}(s,t,u) &= \\frac{1}{2}\\frac{s(t^2 + tu + u^2)}{t + u}\n\\;,\n\\nonumber \\\\\nM^{(1)}_{+-++}(s,t,u) &= \\frac{1}{4} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{+-+-}(s,t,u) &= \\frac{1}{4}\\frac{su^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(1)}_{+++0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(1)}_{++-0}(s,t,u) &= \\frac{i}{2\\sqrt{2}}\\sqrt{stu} \\,\\frac{t -\nu}{t + u} \\,m_Z \\;,\n\\nonumber \\\\\nM^{(1)}_{+-+0}(s,t,u) &=  -\\frac{i}{2\\sqrt{2}}\\frac{u\\sqrt{stu}}{t +\nu} \\,m_Z \\;.\n\\end{align}\nThree more amplitudes $M^{(1)}_{+\\lambda_2\\lambda_3\\lambda_4}$ can\nbe obtained by exchanging Mandelstam variables $t$ and\n$u$~\\cite{Gounaris:1999_1,Glover:1993},\n\\begin{align}\\label{dependent_ampl_M1}\nM^{(1)}_{+--+}(s,t,u) &= M^{(1)}_{+-+-}(s,u,t) =\n\\frac{1}{4}\\frac{st^2}{t + u}\\;,\n\\nonumber \\\\\nM^{(1)}_{+---}(s,t,u) &= M^{(1)}_{+-++}(s,u,t) = \\frac{1}{4}\n\\frac{tu}{(t + u)} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{+--0}(s,t,u) &= M^{(1)}_{+-+0}(s,u,t) =\n-\\frac{i}{2\\sqrt{2}}\\frac{t\\sqrt{stu}}{t + u}\\,m_Z \\;.\n\\end{align}\n\nNine independent helicity amplitudes\n$M^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$ are\n\\begin{align}\\label{independent_ampl_M2}\nM^{(2)}_{++++}(s,t,u) &= -\\frac{1}{2} s(t + u) \\;,\n\\nonumber \\\\\nM^{(2)}_{+++-}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++-+}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++--}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-++}(s,t,u) &= - \\frac{1}{2} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-+-}(s,t,u) &= -\\frac{1}{2}\\frac{su^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(2)}_{+++0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++-0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-+0}(s,t,u) &= \\frac{i}{\\sqrt{2}}\\frac{u\\sqrt{stu}}{t +\nu}m_Z \\;.\n\\end{align}\nThe other three  helicity amplitudes\n$M^{(2)}_{+\\lambda_2\\lambda_3\\lambda_4}$ are given by\n\\begin{align}\\label{dependent_ampl_M2}\nM^{(2)}_{+--+}(s,t,u) &= M^{(2)}_{+-+-}(s,u,t) =\n-\\frac{1}{4}\\frac{st^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(2)}_{+---}(s,t,u) &= M^{(2)}_{+-++}(s,u,t) = - \\frac{1}{2}\n\\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(2)}_{+--0}(s,t,u) &= M^{(2)}_{+-+0}(s,u,t) =\n\\frac{i}{\\sqrt{2}}\\frac{t\\sqrt{stu}}{t + u}m_Z \\;.\n\\end{align}\nNote that all amplitudes $M^{(1,2)}_{\\lambda_1\\lambda_2\\lambda_30}$\nare equal to zero in the limit $m_Z = 0$.\n\nThe amplitudes with $\\lambda_1 = -1$ can be obtained from\nconstraints imposed by parity\ninvariance~\\cite{Gounaris:1999_1,Glover:1993},\n\\begin{equation}\\label{parity_relations}\nM^{(1,2)}_{-\\lambda_2\\lambda_3\\lambda_4}(s,t,u) = (-1)^{1 -\n\\lambda_4} M^{(1,2)}_{+-\\lambda_2-\\lambda_3-\\lambda_4}(s,t,u) \\;.\n\\end{equation}\nNote that we have directly calculated all 48 helicity amplitudes\nusing eq.~\\eqref{matrix_element}. Our calculations show that\nrelations \\eqref{dependent_ampl_M1}, \\eqref{dependent_ampl_M2}, and\n\\eqref{parity_relations} really hold.\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{C.\\arabic{equation}}\n\n\\section*{Appendix C}\n\\label{app:C}\n\nWigner's $d$-functions \\cite{Wigner} are related to the Jacobi\npolynomials $P^{(\\alpha, \\,\\beta)}_n(z)$ with nonnegative $\\alpha,\n\\beta$ \\cite{Varshalovich},\n\\begin{align}\\label{d-function}\nd^J_{\\lambda\\mu}(z) &= \\left[ \\frac{(J + \\lambda)!(J - \\lambda)!}{(J\n+ \\mu)!(J - \\mu)!}\\right]^{1\/2} \\left( \\frac{1 -\nz}{2}\\right)^{(\\lambda - \\mu)\/2} \\left( \\frac{1 +\nz}{2}\\right)^{(\\lambda + \\mu)\/2}\n\\nonumber \\\\\n&\\times P^{(\\lambda-\\mu, \\,\\lambda+\\mu)}_{J - \\lambda}(z) \\;,\n\\end{align}\nwhere $z = \\cos\\theta$. The $d$-functions obey the orthogonality\ncondition \\cite{Varshalovich}\n\\begin{equation}\\label{Wigner_func_orthogonality}\n\\int\\limits_{-1}^1 d^{J}_{\\lambda\\lambda'}(z)\n\\,d^{J'}_{\\lambda\\lambda'}(z) \\,dz = \\frac{2}{2J + 1} \\,\\delta_{JJ'}\n\\;.\n\\end{equation}\nIn its turn, the Jacobi polynomial is related to the hypergeometric\nfunction \\cite{Varshalovich},\n\\begin{equation}\\label{Jacobi_polynomial}\nP^{(\\rho, \\,\\sigma)}_n(z) = \\frac{\\Gamma(\\rho + 1 + n)}{\\Gamma(\\rho\n+ 1) \\,n!} \\,{}_2F_1 \\!\\left( -n, \\rho+\\sigma + n + 1; \\rho +1;\n\\frac{1-z}{2} \\right) .\n\\end{equation}\nNote that $P^{(\\alpha, \\,\\beta)}_n(-z) = (-1)^n P^{(\\beta,\n\\,\\alpha)}_n(z)$, and, correspondingly,\n\\begin{equation}\\label{d-function_symmetry}\nd^J_{\\lambda\\mu}(-z) = (-1)^{J - \\lambda}d^J_{\\mu -\\lambda}(z) \\;,\n\\quad d^J_{\\lambda\\mu}(z) = (-1)^{\\lambda - \\mu}d^J_{-\\lambda\n-\\mu}(z) \\;.\n\\end{equation}\nIn particular, we get\n\\begin{align}\nd^J_{22}(z) &= \\left( \\frac{1 + z}{2}\\right)^{\\!2} {}_2F_1 \\!\\left(\n2-J, J+3; 1; \\frac{1-z}{2} \\right) , \\label{d-function_22} \\\\\nd^J_{2-2}(z) &= (-1)^J \\left( \\frac{1 - z}{2}\\right)^{\\!2} {}_2F_1\n\\!\\left( 2-J, J+3; 1; \\frac{1+z}{2} \\right) . \\label{d-function_2-2}\n\\end{align}\nIn the simplest case, $\\lambda = \\mu = 0$, we find\n\\begin{align}\\label{d-function_00}\nd^J_{00}(z) &= P_J(z) \\;,\n\\end{align}\nwhere $P_J(z)$ being the Legendre polynomial. Using table integral\n7.231.1 in \\cite{Gradshteyn}, we derive the following formula\n\\begin{equation}\\label{integral_Legendre}\n\\int\\limits_{-1}^1 z^m P_J(z) \\,dz = \\frac{1}{2} \\left[ 1 + (-1)^J\n\\right] (-1)^{J\/2} \\frac{\\Gamma \\!\\left( \\frac{J-m}{2} \\right)\n\\Gamma \\!\\left( \\frac{1+m}{2}\\right)}{{\\Gamma \\!\\left(\n-\\frac{m}{2}\\right) \\Gamma \\!\\left( \\frac{J+m+3}{2}\\right)}} \\;,\n\\end{equation}\nwith integer $J$ and even number $m \\geq 0$. To obtain unitarity\nconstraints on the anomalous couplings, we need integral\n\\eqref{integral_Legendre} with $m=0, \\,2$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Selected topics on $A$-Connections}\nLet $(A,\\sharp_A,[\\,,\\,]_A)$ be a Lie algebroid and $V$ a vector\nbundle with the same base manifold $M^m$ ($m=dim\\,M$). By an\n$A$-connection we shall understand an $A$-covariant derivative\n$\\nabla:\\Gamma A\\times\\Gamma V\\rightarrow\\Gamma V$ ($\\Gamma$\ndenotes the space of cross sections of a vector bundle), written\nas $(a,v)\\mapsto\\nabla_av$, which is $\\mathds{R}$-bilinear and has\nthe properties\n\\begin{equation}\\label{propconex}\n\\nabla_{fa}v=f\\nabla_av,\\;\\nabla_a(fv)=f\\nabla_av+\\sharp_Aa(f)v\n\\hspace{2mm}(f\\in C^\\infty(M)).\\end{equation}\n\nAccordingly, the value $\\nabla_av(x)$ depends only on $a(x)$ and\non $v|_{U_x}$ where $U_x$ is a neighborhood of $x\\in M$. In order\nto write down the local expression of $\\nabla$, we take a local\nbasis $(b_i)_{i=1}^s$ $(s=rank\\,A)$ of $\\Gamma A$, with the dual\nbasis $(b^{*i})$ of $\\Gamma A^*$ and a local basis $(w_u)_{u=1}^r$\n$(r=rank\\,V)$ of $\\Gamma V$. Then, with the notation\n$\\Omega^k(A)=\\Gamma\\wedge^kA^*$ for the space of $A$-{\\it forms}\nof degree $k$ and using the Einstein summation convention, we get\n\\begin{equation}\\label{conexloc} \\nabla_{b_i}w_u=\\omega_u^t(b_i)w_t,\\;\n\\omega_u^t=\\gamma_{iu}^tb^{*i}\\in\\Omega^1(A),\\end{equation} and we\nsay that $(\\omega_u^t)$ is the {\\it local connection matrix}.\nCorrespondingly, the curvature\n$$R_\\nabla(a_1,a_2)w=\\nabla_{a_1}\\nabla_{a_2}w-\\nabla_{a_2}\\nabla_{a_1}w-\n\\nabla_{[a_1,a_2]_A}w$$\ngets the local expression\n\\begin{equation}\\label{curbloc}\nR_\\nabla(b_i,b_j)w_u=\\Omega_u^t(b_i,b_j)w_t,\\;\n\\Omega_u^t=d_A\\omega_u^t-\\omega_u^s\\wedge\\omega_s^t\\in\\Omega^2(A),\\end{equation}\nwhere $d_A$ denotes the $A$-exterior differential \\cite{Mk}. We\nwill say that $(\\Omega_u^t)$ is the {\\it local curvature matrix}\nand a change of the basis $(w_u)$ implies an\n$ad(Gl(r,\\mathds{R}))$-transformation of $(\\Omega_u^t)$. Like in\nclassical differential geometry, one has the covariant derivative\nmachinery of $V$-tensors and tensor valued $A$-forms and the\ncomputation of the $d_A\\Omega_u^t$ produces the Bianchi identity\nthat may be written under the form $\\nabla\\Omega_u^t=0$.\n\nIn the study of characteristic classes we shall need the direct\nproduct of two Lie algebroids $p_c:A_c\\rightarrow M_c$ $(c=1,2)$\nand we recall its definition given in \\cite{Mk}. Consider the\npullback bundles $\\pi_c^{-1}A_c$, where $\\pi_c$ is the projection\nof $M_1\\times M_2$ on $M_c$. Identify\n\\begin{equation}\\label{pullsections}\n\\Gamma(\\pi_c^{-1}A_c)\\equiv\\{\\sigma:M_1\\times M_2\\rightarrow\nA_c\\,\/\\,p_c\\circ\\sigma=\\pi_c\\}\\end{equation} and notice that local\nbases $(b_i^{(c)})$ of $\\Gamma A_c$ have natural lifts to local\nbases of $\\Gamma(\\pi_c^{-1}A_c)$, which will also be denoted by\n$(b_i^{(c)})$. Take local cross sections\n$$\\sigma_{(c)}=\\sigma_{(c)}^ib^{(c)}_i,\\,\n\\kappa_{(c)}=\\kappa_{(c)}^ib^{(c)}_i\\hspace{2mm}(\\sigma_{(c)}^i,\n\\kappa_{(c)}^i\\in C^\\infty(M_1\\times M_2))$$ (there is no summation\non indices in parentheses) and define the following anchors and\nbrackets\n$$\\sharp_{(c)}\\,\\sigma_{(c)}=\\sigma_{(c)}^i\\sharp_{A_c}\nb^{(c)}_i,[\\sigma_{(c)},\\kappa_{(c)}]_{(c)}=\n\\sigma_{(c)}^i\\kappa_{(c)}^j[b^{(c)}_i,b^{(c)}_j]_{A_c}$$\n$$+\\{\\sigma_{(c)}^j[(\\iota_{*c}\\sharp_{A_c}b^{(c)}_j)\\kappa_{(c)}^i]\n-\\kappa_{(c)}^j[(\\iota_{*c}\\sharp_{A_c}b^{(c)}_j)\\sigma_{(c)}^i]\\}b^{(c)}_i,$$\nwhere $\\iota_{*c}$ is the natural injection of $TM_c$ in\n$T(M_1\\times M_2)$. In these operations, the $M_{(c-1\\,{\\rm\nmod.}\\,2)}$-variable is just a passive parameter and, since the\nanchor and bracket of each $A_c$ are invariant, the results are\nindependent of the choice of the bases. Thus, the vector bundles\n$\\pi_c^{-1}A_c$ are Lie algebroids over $M_1\\times M_2$ and the\ndirect product of the Lie algebroids $A_c$ is the Whitney sum\n$\\mathcal{A}=\\pi_1^{-1}A_1\\oplus\n\\pi_2^{-1}A_2$ endowed with the direct sum of the anchors and brackets of\nthe two pullbacks (in particular, $[b_i^{(1)},b_j^{(2)}]=0$).\n\\begin{prop}\\label{conectlinks} Let $q:V\\rightarrow M_1$ be a vector\nbundle on $M_1$. Then, any $\\mathcal{A}$-connection\n$\\tilde{\\nabla}$ on the pullback $\\pi_1^{-1}(V)$ defines a\ndifferentiable family $\\nabla^{(x_2)}$ $(x_2\\in M_2)$ of\n$A_1$-connections on $V$. Conversely, any $x_2$-parameterized,\ndifferentiable family of $A_1$-connections on $V$ is induced by an\n$\\mathcal{A}$-connection on $\\pi_1^{-1}(V)$.\\end{prop}\n\\begin{proof} Assume that we have the\ncovariant derivatives $\\tilde{\\nabla}_{\\sigma^i_{(1)}b_i^{(1)}+\n\\sigma^j_{(2)}b_j^{(2)}}(\\nu^uw_u)$, where $\\sigma^i_{(c)},\\nu^u$\nare local, differentiable functions on $M_1\\times M_2$. Then, the\nrequired family of connections on $V$ is given by the covariant\nderivatives\n$$(\\nabla^{(x_2)}_{\\xi^ib_i^{(1)}}(\\eta^uw_u))(x_1)=\n(\\tilde{\\nabla}_{\\xi^ib_i^{(1)}}(\\eta^uw_u))(x_1,x_2),$$ where\n$\\xi^i,\\eta^u$ are local, differentiable functions on $M_1$,\n$x_1\\in M_1,x_2\\in M_2$ and we use an identification like\n(\\ref{pullsections}) for $V$. Notice that, if the local connection\nmatrices of $\\tilde{\\nabla}$ are\n\\begin{equation}\\label{eqfamilie}\n\\tilde\\omega_u^v=\\gamma_{(1)iu}^v(x_1,x_2)b_{(1)}^{*i}\n+\\gamma_{(2)ju}^v(x_1,x_2)b_{(2)}^{*j},\\end{equation} the\nconnection $\\nabla^{(x_2)}$ has the matrices\n$\\gamma_{(1)iu}^v(x_1,x_2)b_{(1)}^{*i}$ with the fixed value of\n$x_2$. Conversely, if the family $\\nabla^{(x_2)}$ is given, we get\nan $\\mathcal{A}$-connection $\\tilde{\\nabla}$ by adding the local\nequations $\\tilde{\\nabla}_{b_j^{(2)}}w_u=0$. The local matrices of\nthis connection $\\tilde{\\nabla}$ are the same as the matrices of\n$\\nabla^{(x_2)}$ where $x_2$ is allowed to vary in $M_2$.\n\\end{proof}\n\nIn particular, we may apply Proposition \\ref{conectlinks} for\n$M_1=M,M_2=I=\\{0\\leq \\tau\\leq1\\},A_1=A,A_2=TI$. Then, an $\n\\mathcal{A}$-connection $\\tilde{\\nabla}$ on $\\pi_1^{-1}(V)$ is called a\n{\\it link} between the $A$-connections $\\nabla^0,\\nabla^1$ on $V$.\nFormula (\\ref{eqfamilie}) shows that the local connection forms of\n$\\tilde{\\nabla}$ are given by\n\\begin{equation}\\label{eqlink}\n\\tilde{\\omega}_u^v=\\omega_{(\\tau)u}^v+\\lambda_u^v(x,\\tau)d\\tau,\\end{equation}\nwhere $\\omega_{(\\tau)}$ is the local connection matrix at the\nfixed value $\\tau$ and $\\lambda_u^v\\in C^\\infty(M\\times I)$. A\nsimple calculation gives the corresponding local curvature forms\n\\begin{equation}\\label{curblink}\n\\tilde{\\Omega}_u^v=\\Omega_{(\\tau)u}^v+\\Lambda_u^v\\wedge d\\tau,\n\\end{equation} where $$\n\\Lambda_u^v=d_A\\lambda_u^v+\n\\lambda_u^w\\omega_{(\\tau)w}^v-\\lambda_w^v\\omega_{(\\tau)u}^w+\n\\frac{\\partial\\omega_{(\\tau)u}^v}{\\partial \\tau}$$\n(the partial derivative with respect to $\\tau$ is applied to the\ncoefficients of the form).\n\nNow, we present another ``selected topic\". Let $V\\rightarrow M$ be\na vector bundle of rank $r$ endowed with either a positive,\nsymmetric tensor $g_+\\in\\Gamma\\odot^2V^*$ or a $2$-form\n$g_-\\in\\Gamma\\wedge^2V^*$. We shall say that $(V,g_{\\pm})$ is a\n{\\it quasi-(skew)-metric vector bundle}. Notice that we do not ask\n$rank\\,g_\\pm$ to be constant on $M$. An $A$-connection $\\nabla$ on\n$V$ such that $\\nabla g_\\pm=0$ will be called a {\\it\nquasi-(skew)-metric connection}. If $g_{\\pm}$ is non degenerate,\nthe particle ``quasi\" will not be used and the connection is\ncalled {\\it orthogonal} for $g_+$ and {\\it symplectic} for $g_-$.\nFor a Lie algebroid $A$ over $M$ we shall denote by $L$ a generic,\nintegral leaf of the distribution $im\\,\\sharp_A$ and by $L_x$ the\nleaf through the point $x\\in M$. In what follows we establish\nproperties of a quasi-(skew)-metric connection that are relevant\nto the construction of characteristic classes.\n\\begin{prop}\\label{theorem1} Assume that there exists a\nquasi-(skew)-metric connection $\\nabla$ on $(V,g_\\pm)$. Then, the\nfollowing properties hold. 1. If $x\\in M$ and $k\\in\\Gamma V$ is\nsuch that $k|_{L(x)}\\in K|_{L(x)}$ $(K=ann\\,g_\\pm)$, then\n$\\nabla_ak(x)\\in K_x$, $\\forall a\\in\\Gamma A$. 2.\n$q=rank\\,g_\\pm|_L$ is constant along each leaf $L$ and $\\forall\nx\\in M$ there exists an open neighborhood $U_x$ where $V$ has a\nlocal basis of cross sections of the form $(s_h,t_l)$\n$(h=1,...,q,\\,l=1,...,r-q)$ such that $t_l|_{U_x\\cap L_x}\\in\nK|_{U_x\\cap L_x}$ and the projections $[s_h]=s_h\\,({\\rm mod.}\\,K)$\ndefine a canonical basis of the (skew)-metric vector bundle\n$((V\/K)|_{U_x\\cap L_x},g'_\\pm)$, where $g'_\\pm$ is non-degenerate\nand induced by $g_\\pm$. 3. With respect to this basis, the\n$A$-connection $\\nabla$ has local equations\n\\begin{equation}\\label{eclocaleconex} \\nabla s_h=\\varpi_{(1)h}^ks_k\n+\\varpi_{(2)h}^pt_p,\\; \\nabla\nt_l=\\varpi_{(3)l}^ks_k+\\varpi_{(4)l}^pt_p,\\end{equation} where the\ncoefficients are local $1$-$A$-forms, $\\varpi_{(3)l}^k(x)=0$ and\n$(\\varpi_{(1)h}^k(x))\\in o(q)$, the orthogonal Lie algebra, in the\n$g_+$-case, $(\\varpi_{(1)h}^k(x))\\in sp(q,\\mathds{R})$, the\nsymplectic Lie algebra, in the $g_-$-case. 4. The curvature of\n$\\nabla$ has the local expression\n\\begin{equation}\\label{eclocalecurb} R_\\nabla s_h=\\Phi_{(1)h}^ks_k\n+\\Phi_{(2)h}^pt_p,\\; R_\\nabla\nt_l=\\Phi_{(3)l}^ks_k+\\Phi_{(4)l}^pt_p,\\end{equation} where the\ncoefficients are local $2$-$A$-forms and $\\Phi_{(3)l}^k(x)=0$,\n$(\\Phi_{(1)h}^k(x))\\in o(q)$ in the $g_+$-case,\n$(\\Phi_{(1)h}^k(x))\\in sp(q,\\mathds{R})$ in the\n$g_-$-case.\\end{prop}\n\\begin{proof} 1. For any $a\\in\\Gamma A,v\\in\\Gamma V$ one has\n\\begin{equation}\\label{conexsympl}\n(\\nabla_ag_\\pm)_x(v(x),k(x))=(\\sharp_Aa)_x(g_\\pm(v,k))-\ng_{\\pm,x}(\\nabla_av(x),k(x))\\end{equation}\n$$-g_{\\pm,x}(v(x),\\nabla_ak(x))=0.$$ Since\n$(\\sharp_Aa)_x(g_\\pm(v,k))$ depends only on $k|_{L(x)}\\in\nann\\,g_\\pm$, it vanishes, and we get the required result.\n\n2. $\\nabla g_\\pm=0$ is equivalent with the fact that $g_\\pm$ is\npreserved by parallel translations along paths in a leaf $L$\n\\cite{F}, therefore, $g_{\\pm}$ has a constant rank $q$ along\n$L_x$. This implies the existence of bases with the required\nproperties on a neighborhood $U_x\\cap L$ of $x$. (In the metric\ncase canonical means orthonormal and in the skew-metric case\ncanonical means symplectic.) Then, take any extension of such a\nbasis to $U_x$ and shrink the neighborhood $U_x$ as needed to\nensure the linear independence of the extended cross sections.\n\n3. The equality $\\varpi_{(3)l}^k(x)=0$ is an immediate consequence\nof part 1. Then, in (\\ref{conexsympl}), replace $v,k$ by\n$s_h,s_k$. Since the canonical character of the basis $(s_{h}|_L)$\nimplies $g_{\\pm}(s_{h}|_L,s_{k}|_L)= const.$, we get\n$$g_{\\pm,x}(\\nabla_as_h(x),s_k(x))\n+g_{\\pm,x}(s_h(x),\\nabla_as_k(x))=0,$$ whence,\n$(\\varpi_{(1)i}^j(x))\\in o(q),\\, sp(q,\\mathds{R})$, respectively.\n\n4. The (skew)-metric condition (\\ref{conexsympl}) also implies\n$$\\sharp_A\n[a_1,a_2]_A(g_\\pm(v_1,v_2))=g_\\pm(\\nabla_{[a_1,a_2]_A}v_1,v_2)+\ng_\\pm(v_1,\\nabla_{[a_1,a_2]_A}v_2),$$ where $a_1,a_2\\in\\Gamma A,\nv_1,v_2\\in\\Gamma V$, whence, after some obvious cancellations we\nget\n\\begin{equation}\\label{auxcurv}\\sharp_A[a_1,a_2]_A(\\omega(v_1,v_2)) =\n-\\omega(R_\\nabla(a_1,a_2)v_1,v_2)-\\omega(v_1,R_\\nabla(a_1,a_2)v_2).\\end{equation}\nLike in the proof of 3, (\\ref{auxcurv}) for $s_h,s_k$ implies\n$(\\Phi_{(1)i}^j(x))\\in o(q),\\, sp(q,\\mathds{R})$, respectively.\nThen, (\\ref{auxcurv}) for $s_h,t_l$ together with part 1 of the\nproposition implies $\\Phi_{(3)l}^k(x)=0$.\n\\end{proof}\n\nIn the theory of characteristic classes we need the Weil algebra\n$I(Gl(r,\\mathds{R}))$ $=\\oplus_{k\\geq0}I^k(Gl(r,\\mathds{R}))$,\nwhere $I^k(Gl(r,\\mathds{R}))$ is the space of real, ad-invariant,\nsymmetric, $k$-multilinear functions (equivalently, invariant,\nhomogeneous polynomials of degree $k$) on the Lie algebra of the\ngeneral, linear group $(r=rank\\,V)$. Using the exterior product\n$\\wedge$, such functions may be evaluated on arguments that are\nlocal matrices of $\\wedge$-commuting $A$-forms on $M$ with\ntransition functions of the adjoint type and the result is a\nglobal $A$-form on $M$ (e.g.,\n\\cite{V}). Secondary characteristic classes appear as a\nconsequence of vanishing phenomena encountered in the evaluation\nprocess described above. We shall need the following vanishing\nphenomenon (see \\cite{F}):\n\\begin{prop}\\label{theorem2} If the bundle $V$ endowed with the\nform $g_\\pm$ has a connection $\\nabla$ such that $\\nabla g_\\pm=0$\nand if $R_\\nabla(a_1,a_2)k_x=0$ for $x\\in M$, $a_1,a_2\\in\\Gamma\nA$, $k\\in ker\\,g_{\\pm,x}$, then, $\\forall\\phi\\in\nI^{2k-1}(Gl(r,\\mathds{R}))$, one has $\\phi(\\Phi)=0$, where $\\Phi$\nis the local curvature matrix of the connection $\\nabla$.\n\\end{prop}\n\\begin{proof} By $\\phi(\\Phi)$ we understand the\nevaluation of $\\phi$ where all the arguments are equal to $\\Phi$.\nIt is known that (with a harmless abuse of terminology and\nnotation) the required functions $\\phi$ are spanned by the Chern\npolynomials\n\\begin{equation}\\label{Chern}\nc_h(F)=\\frac{1}{h!}\\delta^{v_1...v_h}_{u_1...u_h}f_{v_1}^{u_1}\n...f_{v_h}^{u_h}\\end{equation} ($\\delta^{...}_{...}$ is the\nmulti-Kronecker index), which are the sums of the principal minors\nof order $h$ in $det(F-\\lambda Id)$ $(F\\in gl(r,\\mathds{R}))$.\nWith the notation of Proposition \\ref{theorem1} and since\n$R_\\nabla(a_1,a_2)k_x=0$, we have to take\n$$F=\\left(\\begin{array}{cc}\n\\Phi_{(1)}&0\\vspace{2mm}\\\\\\Phi_{(2)}&0\\end{array}\\right).$$ Therefore,\n$\\forall x\\in M$, we have $c_h(F)=c_h(\\Phi_{(1)x})$. It is known\nthat the polynomials $c_{2l-1}$ vanish on $o(q)$ and on\n$sp(q,\\mathds{R})$ (in the first case $\\Phi_{(1)}$ is\nskew-symmetric; for the second case see Remark 2.1.10 in \\cite{V},\nfor instance).\\end{proof}\n\\section{Secondary characteristic classes}\nA brief exposition of the classical theory of real characteristic\nclasses may be found in \\cite{V}. In this section, we present a\nLie algebroid version of the basic facts of the theory.\n\nConsider the direct product Lie algebroid $\\mathcal{A}=A\\times\nT\\Delta^k\\rightarrow M\\times\\Delta^k$, where\n$$\\Delta^k=\\{(t_0,t_1,...,t_k)\\in\\mathds{R}^{k+1}\\,\/\\,t_h\\geq0,\\,\n\\sum_{h=0}^k t_h=1\\}$$ is the standard $k$-simplex, $A$ is a Lie\nalgebroid over $M$ and $T\\Delta^k$ is the tangent bundle of\n$\\Delta^k$ endowed with the standard orientation\n$\\kappa=dt^1\\wedge...\\wedge dt^k$. Then,\n$\\forall\\Phi\\in\\Omega^*(\\mathcal{A})$, the fiber-integral\n$\\int_{\\Delta^k}\\Phi$ is defined as zero except for the case\n$$\\Phi=\\alpha\\wedge\\kappa,\\hspace{3mm}\\alpha=\n\\frac{1}{p!}\\alpha_{i_1...i_p}(x,t)b^{*i_1}\\wedge...\\wedge b^{*i_p}\n\\;\\;(x\\in M,t\\in\\Delta^k)$$ when\n$$\\int_{\\Delta^k}\\Phi=\\frac{1}{p!}\n\\left(\\int_{\\Delta^k}\\alpha_{i_1...i_p}(x,t)\\kappa\\right)\nb^{*i_1}\\wedge...\\wedge b^{*i_p}\\in\\Omega^p(A)$$ ($b_i$ is a local\nbasis of cross sections of $A$). The same proof as in the\nclassical case (e.g., \\cite{V}, Theorem 4.1.6) yields the Stokes\nformula:\n\\begin{equation}\\label{Stokes}\\int_{\\Delta^k}d_\\mathcal{A}\n\\Phi-d_{A}\\int_{\\Delta^k}\\Phi=\n(-1)^{deg\\,\\Phi-k}\\int_{\\partial{\\Delta^k}}\\iota^*\\Phi,\\hspace{5mm}\\iota:\n\\partial{\\Delta^k}\\subseteq{\\Delta^k}.\\end{equation}\n\nAssume that we have $k+1$ $A$-connections $\\nabla^{(s)}$ on the\nvector bundle $V\\rightarrow M$ that have the local connection\nmatrices $\\omega_{(\\alpha)}$ $(\\alpha=0,...,k)$ with respect to\nthe local basis $(w_u)$ of $V$. Then, the convex combination\n\\begin{equation}\\label{averagecon}\n\\nabla^{(t)}=\\sum_{\\alpha=0}^kt^\\alpha\\nabla^{\\alpha},\\hspace{3mm}\nt=(t^0,...,t^k)\\in\\Delta^k,\\end{equation} defines a family of\n$A$-connections parameterized by $\\Delta^k$ with the corresponding\n$\\mathcal{A}$-connection $\\tilde{\\nabla}$ on\n$\\pi_1^{-1}(V)\\rightarrow M\\times\\Delta^k$\n$(\\pi_1:M\\times\\Delta^k\\rightarrow M)$. The connection and\ncurvature matrices of $\\tilde{\\nabla}$ will be denoted by\n$\\tilde{\\omega},\\tilde{\\Omega}$; generally, the curvature matrix\nof a connection will be denoted by the upper case of the letter\nthat denotes the connection matrix. There exists a homomorphism\n$$\\Delta(\\nabla^0,...,\\nabla^k):I^h(Gl(r,\\mathds{R}))\n\\rightarrow\\Omega^{2h-k}(A),$$ defined by R. Bott in the classical\ncase, given by\n\\begin{equation}\\label{Bottmare}\n\\Delta(\\nabla^0,...,\\nabla^k)\\phi=(-1)^{\\left[\\frac{k+1}{2}\\right]}\n\\int_{\\Delta^k}\\phi(\\tilde{\\Omega}),\\hspace{2mm}\\phi\\in\nI^h(Gl(r,\\mathds{R})).\\end{equation} Moreover, Bott's proof in the\nclassical case (\\cite{V}, Proposition 4.2.3) also holds in the Lie\nalgebroid version and yields the following formula\n\\begin{equation}\\label{diffBott}\nd_A(\\Delta(\\nabla^0,...,\\nabla^k)\\phi) =\n\\sum_{\\alpha=0}^k(-1)^\\alpha\\Delta(\\nabla^0,...,\\nabla^{\\alpha-1},\n\\nabla^{\\alpha+1},...,\\nabla^k)\\phi.\\end{equation}\n\nLet $\\nabla$ be an $A$-connection on the  vector bundle\n$V\\rightarrow M$. As a consequence of the Bianchi identity,\n$\\forall\\phi\\in I^h(Gl(r,\\mathds{R}))$,\n$\\Delta(\\nabla)\\phi\\in\\Omega^{2h}(A)$ is a $d_A$-closed $A$-form\nand the $A$-cohomology classes defined by the $A$-forms\n$\\Delta(\\nabla)\\phi$ are called the $A$-{\\it principal\ncharacteristic classes of $V$}\n\\cite{F}. If $\\nabla^0,\\nabla^1$ are two $A$-connections, formula\n(\\ref{diffBott}) yields\n\\begin{equation}\\label{Bott2conex}\\Delta(\\nabla^1)\\phi-\\Delta(\\nabla^0)\\phi=\nd_A\\Delta(\\nabla^0,\\nabla^1)\\phi.\\end{equation} Therefore, the\nprincipal characteristic classes do not depend on the choice of\nthe connection.\n\nThe $\\mathcal{A}$-connection $\\tilde{\\nabla}$ to be used in\ndefinition (\\ref{Bottmare}) of $\\Delta(\\nabla^0,\\nabla^1)\\phi$ is\nthe link between $\\nabla^0,\\nabla^1$ given by the family of\n$A$-connections\n$$\\nabla^{(\\tau)}=(1-\\tau)\\nabla^0+\\tau\\nabla^1=\\nabla^0+\\tau D,\\hspace{3mm}\nD=\\nabla^1-\\nabla^0,\\;\\;(\\tau\\in I).$$ For this link, we have\n(\\ref{eqlink}) and (\\ref{curblink}) where\n$\\lambda_u^v=0,\\partial\\omega_{(\\tau)u}^v\/\\partial\\tau=\\alpha$,\nthe local matrix of the connection difference $D$, and formula\n(\\ref{Bottmare}) yields\n\\begin{equation}\\label{defDelta} \\Delta(\\nabla^0,\\nabla^1)\\phi\n=h\\int_0^1\\phi(\\alpha,\\underbrace{\\Omega_{(\\tau)},...,\\Omega_{(\\tau)}}_{(h-1)-{\\rm\ntimes}})d\\tau,\\end{equation} where\n\\begin{equation}\\label{Omegat}\n\\Omega_{(\\tau)}=(1-\\tau)\\Omega_{(0)}+\\tau\\Omega_{(1)}\n+\\tau(1-\\tau)\\alpha\\wedge\\alpha\\end{equation} is the local\ncurvature matrix of the connection $\\nabla^{(\\tau)}$.\n\nWe shall use the Lehmann version of the theory of secondary\ncharacteristic classes \\cite{{L},{V}}. Let $(J_0,J_1)$ be two\n(proper) homogeneous ideals of $I=I(Gl(r,\\mathds{R}))$. Define the\nalgebra\n\\begin{equation}\\label{Weil}W(J_0,J_1)=(I\/J_0)\\otimes(I\/J_1)\\otimes(\\wedge(I^+))\n\\hspace{3mm}(I^+=\\oplus_{k>0}I^k),\\end{equation} with the graduation\n$$deg\\,[\\phi]_{J_0}\n=deg\\,[\\phi]_{J_1}=2h,\\,deg\\,\\hat\\phi=2h-1,$$ and the differential\n$$d[\\phi]_{J_0}=d[\\phi]_{J_1}=0,\\,d\\hat\\phi=[\\phi]_{J_1}-[\\phi]_{J_0},$$\nwhere we refer to the three elements defined by $\\phi\\in I^h$ in\nthe factors of $W$.\n\nNow, take a vector bundle $V\\rightarrow M$ and two $A$-connections\n$\\nabla^0,\\nabla^1$ on $V$ such that $J_c\\subseteq\nker\\,\\Delta(\\nabla^c)$, $c=0,1$. By putting\n\\begin{equation}\\label{defrho}\\rho[\\phi]_{J_0}=\\Delta(\\nabla^0)\\phi,\\,\n\\rho[\\phi]_{J_1}=\\Delta(\\nabla^1)\\phi,\\,\n\\rho\\hat\\phi=\\Delta(\\nabla^0,\\nabla^1)\\phi,\\end{equation} we get a\nhomomorphism of differential graded algebras\n$$\\rho(\\nabla_0,\\nabla_1):W(J_0,J_1)\\rightarrow\\Omega(A)$$\nwith an induced cohomology homomorphism\n$$\\rho^{*}(\\nabla^0,\\nabla^1):H^*(W(J_0,J_1))\\rightarrow\nH^*(A).$$ The cohomology classes in $im\\,\\rho^{*}$ that are not\nprincipal characteristic classes are called $A$-{\\it secondary\ncharacteristic classes}.\n\nIf $J$ is a homogeneous ideal of $I(Gl(r,\\mathds{R}))$, two\n$A$-connections $\\nabla,\\nabla'$ on $V$ are called $J$-{\\it\nhomotopic connections} if there exists a finite chain of links\n$\\tilde{\\nabla}^0,...,\\tilde{\\nabla}^n$ that starts with $\\nabla$,\nends with $\\nabla'$ and is such that $J\\subseteq\\cap_{l=0}^n\nker\\,\\Delta(\\tilde{\\nabla}^l)$. By replacing the usual Stokes'\nformula by formula (\\ref{Stokes}) in the proof of Theorem 4.2.28\nof \\cite{V}, one gets\n\\begin{prop}\\label{invhomot} {\\rm \\cite{L}} The cohomology homomorphism\n$\\rho^{*}(\\nabla^0,\\nabla^1)$ remains unchanged if\n$\\nabla^0,\\nabla^1$ are replaced by $J_0,J_1$-homotopic\nconnections $\\nabla^{'0},\\nabla^{'1}$, respectively ($J_c\\subseteq\nker\\,\\Delta(\\nabla^c)$, $J_c\\subseteq ker\\,\\Delta(\\nabla^{'c})$,\n$c=0,1$).\\end{prop}\n\\begin{corol}\\label{corolhomotopie} The secondary characteristic\nclasses are invariant by any $J_0J_1$-homotopy of the\nconnections.\\end{corol}\n\nDenote by $J_{\\rm odd}\\subseteq I(Gl(r,\\mathds{R}))$ the ideal\nspanned by $\\{\\phi\\in I^{2h-1}(Gl(r,\\mathds{R})),\\\\h=1,2,...\\}$.\nAs explained in Proposition \\ref{theorem2}, if $\\nabla$ is an\northogonal connection for some metric $g$ on the vector bundle\n$V$, then $J_{\\rm odd}\\subseteq ker\\,\\Delta(\\nabla)$. Notice that\nthere always exist positive definite metrics $g$ on $V$ and\ncorresponding metric $A$-connections $\\nabla$, $\\nabla g=0$ (e.g.,\ntake $\\nabla_a=\\nabla'_{\\sharp_Aa}$, where $\\nabla'$ is a usual\northogonal connection on $(V,g)$). Furthermore, any two orthogonal\n$A$-connections on $V$ are $J_{\\rm odd}$-homotopic. Indeed, if\n$\\nabla,\\nabla'$ are orthogonal for the same metric $g$, then\n$(1-\\tau)\\nabla+\\tau\\nabla'$ $(0\\leq\\tau\\leq1)$ defines an\northogonal link. If orthogonality is with respect to different\nmetrics $g,g'$, then $(1-\\tau)g+\\tau g'$ is a metric on the\npullback of $V$ to $M\\times[0,1]$ and a corresponding metric\nconnection provides an orthogonal link between two orthogonal\nconnections $\\bar{\\nabla},\\bar{\\nabla}'$ with the metrics $g,g'$,\nrespectively. Thus, there exists a chain of three orthogonal links\nleading from $\\nabla$ to $\\bar{\\nabla}$, from $\\bar{\\nabla}$ to\n$\\bar{\\nabla}'$ and from $\\bar{\\nabla}'$ to $\\nabla'$, which\nproves the $J_{\\rm odd}$-homotopy of $\\nabla,\\nabla'$.\n\nNow, let $(V,g_\\pm)$ be a quasi-(skew)-metric vector bundle that\nhas a $K$-flat quasi-(skew)-metric connection $\\nabla^1$\n$(K=ann\\,g_\\pm)$. Then, Proposition \\ref{theorem2} tells us that\n$J_{\\rm odd}\\subseteq ker\\,\\Delta(\\nabla^1)$. Accordingly (like in\nthe case of the Maslov classes \\cite{V}), if we also take an\northogonal $A$-connection $\\nabla^0$ on $V$, we shall obtain\nsecondary characteristic classes corresponding to the ideals\n$J_0=J_1=J_{\\rm odd}$.\n\nFollowing \\cite{V}, Theorem 4.2.26, we may replace the algebra\n$W(J_0,J_1)$ by the algebra\n\\begin{equation}\\label{Maslov} \\tilde{W}=\n\\mathds{R}[c_2,c_4,...]\\otimes\\mathds{R}[c'_2,c'_4,...]\n\\otimes\\wedge(\\hat{c}_1,\\hat{c}_3,..),\\end{equation} where\n$c_\\centerdot$ are the Chern polynomials and the accent and hat\nindicate the place in the three factors of (\\ref{Maslov}); the\nhomomorphism $\\rho(\\nabla^0,\\nabla^1)$ is defined like on\n$W(J_0,J_1)$, while using orthogonal and quasi-(skew)-metric\n$A$-connections, respectively, and we get the same set of\ncharacteristic classes. Then, by the same argument like for\n\\cite{V}, Theorem 4.4.37  we get\n\\begin{prop}\\label{clsecFergen} The $A$-secondary characteristic\nclasses of $(V,g_\\pm)$ are the real linear combinations of\ncup-products of $A$-Pontrjagin classes of $V$ {\\rm\n\\cite{F}} and classes of the form\n\\begin{equation}\\label{clMV}\n\\mu_{2h-1}=[\\Delta(\\nabla^0,\\nabla^1)c_{2h-1}]\\in\nH^{4h-3}(A).\\end{equation}\\end{prop}\n\nThe classes $\\mu_{2h-1}$ will be called {\\it simple $A$-secondary\ncharacteristic classes}.\n\\begin{rem}\\label{obsclFern} {\\rm\nIf we start with an arbitrary vector bundle $(V,g_\\pm)$, a\n$K$-flat, quasi-(skew)-metric $A$-connection $\\nabla^1$ may not\nexist. Furthermore, if $\\nabla^1$ exists, it may happen that all\nthe secondary characteristic classes vanish. For instance, if we\nhave a non-degenerate form $g_-$, a usual connection on the bundle\nof $g_-$-canonical frames produces an $A$-connection $\\nabla^1$\nsuch that $\\nabla^1 g_-=0$ and, since $K=0$, we get $A$-secondary\ncharacteristic classes. Because of the $J_{\\rm odd}$-homotopy of\northogonal connections, these classes do not depend on the choice\nof the orthogonal connection $\\nabla^0$. Moreover, these classes\nare independent of the skew-metric connection $\\nabla^1$ because\nof the existence of the link $(1-\\tau)\\nabla^1+\\tau\\nabla^{'1}$\nbetween two such connections. But, the structure group of $V$ may\nbe reduced from the symplectic to the unitary group\n\\cite{V} and a unitary connection $\\bar\\nabla$ on $V$ will be skew-metric\nand orthogonal simultaneously. From (\\ref{defDelta}), and taking\n$\\nabla^0=\\nabla^1=\\bar\\nabla$, we see that the secondary\ncharacteristic classes above vanish.}\\end{rem}\n\\section{Characteristic classes of morphisms}\nLet $A$ be an arbitrary Lie algebroid on $M$, $V,W$ vector bundles\nwith the same basis $M$ and $\\varphi:V\\rightarrow W$ a morphism\nover the identity on $M$. The $A$-connections $\\nabla^V,\\nabla^W$\non $V,W$, respectively, will be called $\\varphi$-compatible if\n$\\nabla^W\\circ\\varphi=\\varphi\\circ\\nabla^V$. An equivalent way to\ncharacterize compatibility is obtained by considering the vector\nbundle $S=V\\oplus W^*$ , which is endowed with the $2$-forms\n\\begin{equation}\\label{omegainS} g_\\pm((v_1,\\nu_1),(v_2,\\nu_2))=\n<\\nu_2,\\varphi(v_1)>\\pm<\\nu_1,\\varphi(v_2)>,\\end{equation}\n$v_1,v_2\\in V,\\, \\nu_1,\\nu_2\\in W^*$. It suffices to work with one\nof these forms, but it is nice to mention that both may be used with\nthe same effect. The pair of $A$-connections $\\nabla^V,\\nabla^W$\nproduces an $A$-connection $\\nabla^S=\\nabla^V\\oplus\\nabla^{W^*}$ on\n$S$, where $\\nabla^{W^*}$ is defined by\n$$<\\nabla^{W^*}_a\\nu,v>=(\\sharp_Aa)<\\nu,v>-<\\nu,\\nabla^W_av>,\\hspace{3mm}\\nu\\in\nW^*,v\\in V.$$ A straightforward calculation shows that\n$\\nabla^V,\\nabla^W$ are $\\varphi$-compatible iff either\n$\\nabla^Sg_+=0$ or $\\nabla^Sg_-=0$. We also notice that the forms\n$g_\\pm$ have the same annihilator\n\\begin{equation}\\label{KinS} K=ker\\,\\varphi\\times\nker\\,^t\\varphi\\end{equation} where the index $t$ denotes\ntransposition.\n\\begin{prop}\\label{FerinS}\nIf $V=A$, if $W=A'$ is a second Lie algebroid and if $\\varphi$ is\na base-preserving Lie algebroid morphism, then there exist\n$K$-flat, $\\varphi$-compatible $A$-connections\n$(\\nabla,\\nabla')$.\\end{prop}\n\\begin{proof} We may proceed like in \\cite{F}. Take a neighborhood\nof $M$ where $\\Gamma A,\\Gamma A'$ have the fixed local bases\n$(b_i),(b'_u)$. Define local $A$-connections\n$\\nabla^U,\\nabla^{'U}$ by asking that\n\\begin{equation}\\label{conexpeU} \\nabla^U_{b_i}b_j=[b_i,b_j]_A,\\;\n\\nabla^{'U}_{b_i}b'_u=[\\varphi b_i,b'_u]_{A'},\\end{equation}\nthen, extending the operators to arbitrary local cross sections in\naccordance with the properties of a connection. Using the local\nexpression $\\varphi b_i=\\varphi_i^ub'_u$, it is easy to check that\n$\\varphi\\circ\\nabla^U=\\nabla^{'U}\\circ\\varphi$. If we consider a\nlocally finite covering $\\{U_\\sigma\\}$ of $M$ by such\nneighborhoods $U$ and glue up the local connections by a\nsubordinated partition of unity $\\{\\theta_\\sigma\\in\nC^\\infty(M)\\}$, we get $\\varphi$-compatible, global\n$A$-connections $\\nabla,\\nabla'$ defined by\n\\begin{equation}\\label{lipireconex}\\nabla_va(x)=\\sum_{x\\in\nU_\\sigma}\\theta_\\sigma(x)\\nabla^{U_\\sigma}_va(x),\\;\n\\nabla'_va'(x)=\\sum_{x\\in\nU_\\sigma}\\theta_\\sigma(x)\\nabla^{'U_\\sigma}_va'(x),\\end{equation}\nwhere $x\\in M,v\\in A_x,a\\in\\Gamma A,a'\\in\\Gamma A'$.\n\nNow, we notice that the local connections (\\ref{conexpeU}) satisfy\nthe following properties\n\\begin{equation}\\label{propcloc} \\nabla^U_{b_i}a=[b_i,a]_A,\\;\n\\nabla^{'U}_{b_i}a'=[\\varphi b_i,a']_{A'}.\\end{equation} Indeed, if\nwe put $a=f^jb_j,a'=h^ub'_u$, (\\ref{conexpeU}) and the properties of\nthe Lie algebroid bracket imply (\\ref{propcloc}). Furthermore, using\n(\\ref{propcloc}), it is easy to check the following properties of\nthe global compatible connections (\\ref{lipireconex})\n\\begin{equation}\\label{propluinabla} \\nabla_vk(x)=[\\tilde{v},k]_A(x),\n\\end{equation}\n\\begin{equation}\\label{propluinabla'} <\\nabla^{'*}_v\\alpha'(x),a'(x)>\n=(\\sharp_Av)<\\alpha',a'>-<\\alpha'(x),[\\varphi\\tilde{v},a']_{A'}(x)>,\\end{equation}\n$\\forall x\\in M,k\\in\\Gamma(ker\\,\\varphi),a'\\in\\Gamma\nA',\\alpha'\\in\\Gamma(ker\\,^t\\varphi)$ and $\\tilde{v}=\\nu^ib_i$ is a\ncross section of $\\Gamma A$ that extends $v\\in A_x$. The\nrestrictions put on $k,\\alpha'$ ensure the correctness of the\npassage from the covariant derivative to the Lie algebroid bracket\nand the independence of the result on the choice of $\\tilde{v}$.\nFormulas (\\ref{propluinabla}), (\\ref{propluinabla'}) imply\n$\\varphi(\\nabla_vk)=0$, $\\nabla^{'*}_v\\alpha'\\circ\\varphi=0$, which\nmeans that $ker\\,\\varphi$ and $ker\\,^t\\varphi$ are preserved by the\nconnections $\\nabla,\\nabla'$, respectively.\n\nFinally, if we denote $S=A\\oplus A^{'*}$ and\n$\\nabla^S=\\nabla\\oplus\\nabla^{'*}$, we can compute the curvature\n$[R_{\\nabla^S}(a_1,a_2)(k,\\alpha')](x)$, which has components on $A$\nand $A^{'*}$. The component on $A$ is\n$$(\\nabla_{a_1}\\nabla_{a_2}-\\nabla_{a_2}\\nabla_{a_1}\n-\\nabla_{[a_1,a_2]_A})\\tilde{k}(x)\\stackrel{(\\ref{propluinabla})}{=}\n([\\tilde{a}_1,[\\tilde{a}_2,\\tilde{k}]_A]_A$$ $$-\n[\\tilde{a}_2,[\\tilde{a}_1,\\tilde{k}]_A]_A-\n[[\\tilde{a}_1,\\tilde{a}_2]_A,\\tilde{k}]_A)(x)=0,$$ where tilde\ndenotes extensions to cross sections and the final result holds\nbecause of the Jacobi identity. For the component on $A^{'*}$ we\nget the following evaluation on any $a'\\in\\Gamma A'$:\n$$<(\\nabla^{'*}_{a_1}\\nabla^{'*}_{a_2}-\\nabla^{'*}_{a_2}\\nabla^{'*}_{a_1}\n-\\nabla^{'*}_{[a_1,a_2]_A})\\tilde{\\alpha}',a'>(x)\n\\stackrel{(\\ref{propluinabla'})}{=}<\\tilde{\\alpha}',\n[\\varphi\\tilde{a}_2,[\\varphi\\tilde{a}_1,\\tilde{k}]_{A'}]_{A'}$$ $$-\n[\\varphi\\tilde{a}_1,[\\varphi\\tilde{a}_2,\\tilde{k}]_{A'}]_{A'}-\n[[\\varphi\\tilde{a}_1,\\varphi\\tilde{a}_2]_{A'},\\tilde{k}]_{A'}>(x)=0,$$\nwhere the annulation is justified by the Jacobi identity again.\nTherefore, $$[R_{\\nabla^S}(a_1,a_2)(k,\\alpha')](x)=0,$$ which is the\nmeaning of $K$-flatness.\n\\end{proof}\n\\begin{rem}\\label{obsrankconst} {\\rm During the proof of Proposition \\ref{FerinS} we\nsaw that $ker\\,\\varphi$ is preserved by $\\nabla$, hence, it is\npreserved by the parallel translation along the paths in the leaves\n$L$ of $A$. This shows that $rank\\,\\varphi$ is constant along the\nleaves $L$.}\\end{rem}\n\\begin{rem}\\label{nouindistins} {\\rm If we use the\ndefinition of $\\nabla^{'*}$ in the left hand side of\n(\\ref{propluinabla'}) and take into account the relation\n$ann\\,ker\\,^t\\varphi=im\\,\\varphi$ we obtain the following equivalent\nform of (\\ref{propluinabla'}):\n\\begin{equation}\\label{eqpropluinabla'}\n\\nabla'_va'(x)=[\\varphi\\tilde v,a']_{A'}(x)\\;({\\rm\nmod.}\\,im\\,\\varphi_x)\\hspace{2mm}\\forall x\\in M,v\\in A_x.\n\\end{equation}}\\end{rem}\n\\begin{defin}\\label{defdistins} {\\rm\nA pair of $\\varphi$-compatible $A$-connections that satisfy the\nproperties (\\ref{propluinabla}), (\\ref{eqpropluinabla'}) will be\ncalled a {\\it distinguished pair} (in \\cite{F} one uses the term\nbasic connections).}\\end{defin}\n\nNow, we see that we may use Proposition \\ref{clsecFergen} in order\nto get secondary characteristic classes for the bundle $S=A\\oplus\nA^{'*}$ endowed with the quasi-(skew)-metrics (\\ref{omegainS}), with\na connection $\\nabla^1=\\nabla\\oplus\\nabla^{'*}$, where\n$(\\nabla,\\nabla')$ is a distinguished pair of $A$-connections, and\nwith an orthogonal connection\n$\\nabla^0=\\nabla^{g_A}\\oplus\\nabla^{g_{A'}*}$, where $g_A,g_{A'}$\nare metrics on the bundles $A,A'$ and $\\nabla^{g_A},\\nabla^{g_{A'}}$\nare corresponding orthogonal connections on $A,A'$.\n\\begin{defin}\\label{defclmor} {\\rm The above constructed\nsecondary characteristic classes of $A\\oplus A^{'*}$ will be called\nthe {\\it characteristic classes} of the base-preserving morphism\n$\\varphi$. In particular, one has the {\\it simple characteristic\nclasses} $\\mu_{2h-1}(\\varphi)\\in H^{4h-3}(A)$.}\\end{defin}\n\nThe secondary characteristic classes of the Lie algebroid $A$\ndefined in \\cite{F} are the simple characteristic classes of the\nmorphism $\\varphi=\\sharp_A:A\\rightarrow TM$.\n\\begin{prop}\\label{isomorfism} All the characteristic classes\nof a base-preserving isomorphism $\\varphi:A\\rightarrow A'$ are\nzero.\\end{prop} \\begin{proof} If $\\varphi$ is an isomorphism, then\n$g_-$ is non degenerate and we are in the situation discussed in\nRemark \\ref{obsclFern}.\\end{proof}\n\nThus, the characteristic classes of a morphism may be seen as a\nmeasure of its non-isomorphic character.\n\\begin{prop}\\label{invarlaconex} The characteristic\nclasses of a base preserving morphism $\\varphi:A\\rightarrow A'$ of\nLie algebroids do not depend on the choice of the orthogonal\nconnection and of the distinguished pair of compatible connections\nrequired by their definition.\\end{prop}\n\\begin{proof} The proposition is a consequence of Corollary\n\\ref{corolhomotopie}. In the previous section we have seen that two\northogonal $A$-connections are $J_{\\rm odd}$-homotopic. On the other\nhand, take two $\\varphi$-distinguished pairs of $A$-connections\n$\\nabla,\\nabla';\\tilde{\\nabla},\\tilde{\\nabla}'$. Then, it is easy to\ncheck that, $\\forall t\\in[0,1]$,\n$(1-t)\\nabla+t\\tilde{\\nabla},(1-t)\\nabla'+t\\tilde{\\nabla}'$ is a\n$\\varphi$-distinguished pair again. Therefore, $J_{\\rm\nodd}$-homotopy also holds for the corresponding quasi-(skew)-metric\nconnections on $S$ and we are done.\\end{proof}\n\nWe also have another consequence of Corollary \\ref{corolhomotopie}:\n\\begin{prop}\\label{morfismehomotope} Two homotopic, base-preserving\nmorphisms $\\varphi_0,\\varphi_1:A\\rightarrow A'$ of Lie algebroids\nhave the same secondary characteristic classes.\\end{prop}\n\\begin{proof} By homotopic morphisms we understand  morphisms\n$\\varphi_0,\\varphi_1$ that are linked by a differentiable family\nof morphisms $\\varphi_\\tau:A\\rightarrow A'$ $(0\\leq\\tau\\leq1)$.\nThe corresponding forms $g_{+,\\tau}$ on $S=A\\oplus A^{'*}$ are\ndifferent, but, still, all the connections $\\nabla^{1,\\tau}$\nrequired in the construction of the secondary classes have\nskew-symmetric local connection and curvature matrices. Therefore,\nthe $J_{\\rm odd}$-homotopy holds and we are done.\\end{proof}\n\\begin{rem}\\label{clasebicaract} {\\rm\nIn the case of an arbitrary pair of morphisms\n$\\varphi_0,\\varphi_1:A\\rightarrow A'$ we can measure the\ndifference between the secondary characteristic classes as\nfollows. Notice the existence of the {\\it bi-characteristic\nclasses}\n$\\bar{\\mu}_{2h-1}(\\varphi_1,\\varphi_2)=[\\Delta(\\nabla^1,\\nabla^2)c_{2h-1}]\\in\nH^{4h-3}(A)$ where $\\nabla^1,\\nabla^2$ are $A$-connections defined\non $S=A\\oplus A^{'*}$ by distinguished, $\\varphi_{1,2}$-compatible\nconnections respectively. Then, formula (\\ref{diffBott}) yields\n$$d_A\\Delta(\\nabla^0,\\nabla^1,\\nabla^2)c_{2h-1}=\\Delta(\\nabla^0,\\nabla^1)c_{2h-1}\n+\\Delta(\\nabla^1,\\nabla^2)c_{2h-1}+\\Delta(\\nabla^2,\\nabla^0)c_{2h-1},$$\nwhere $\\nabla^0$ is an orthogonal connection on $S$. Accordingly,\nwe get\n\\begin{equation}\\label{claseperechi} {\\mu}_{2h-1}(\\varphi_1)-\n{\\mu}_{2h-1}(\\varphi_2)=\n\\bar{\\mu}_{2h-1}(\\varphi_1,\\varphi_2).\\end{equation}}\\end{rem}\n\nIn what follows we give explicit local expressions of $A$-forms\nthat represent the characteristic classes $\\mu_{2h-1}(\\varphi)$.\nTake a point $x\\in M$ and an open neighborhood $U$ of $x$\ndiffeomorphic to a ball. Assume that $(\\nabla^U,\\nabla^{'U})$ and\n$(\\nabla,\\nabla')$ are pairs of local, respectively global,\ndistinguished, $\\varphi$-compatible $A$-connections on $A,A'$.\nThen, if $0\\leq\\chi\\in C^\\infty(M)$ is equal to $1$ on the compact\nclosure $\\bar V$ of the open neighborhood $V\\subseteq U$ of $x$\nand equal to $0$ on $M\\backslash U$, then the convex combinations\n$$\\bar{\\nabla}=\\chi\\nabla^U+(1-\\chi)\\nabla,\\,\n\\bar{\\nabla}'=\\chi\\nabla^{'U}+(1-\\chi)\\nabla'$$\ndefine a global pair of distinguished $A$-connections that\ncoincides with $(\\nabla^U,\\nabla^{'U})$ on $V$.\n\nAccordingly, in formula (\\ref{clMV}) for $S=A\\oplus A^{'*}$ we may\nalways use a connection $\\nabla^1$ such that the expressions\n(\\ref{conexpeU}) hold on the neighborhood $V$. Then, if we denote\n\\begin{equation}\\label{expresiiloc1} \\begin{array}{c}\n[b_i,b_j]_A=\\gamma_{ij}^kb_k,\\,\n[b'_u,b'_v]_{A'}=\\gamma_{uv}^{'w}b'_w,\\vspace{2mm}\\\\\n\\sharp_Ab_i=\\rho_i^j\\frac{\\partial}{\\partial x^j},\\,\n\\sharp_{A'}b'_u=\\rho_u^{'j}\\frac{\\partial}{\\partial x^j}\n\\,\\varphi(b_i)=\\varphi_i^sb'_s\\end{array}\\end{equation}\n(remember that we use the Einstein summation convention), where\n$x^i$ are local coordinates on $M$ and $(b_i)(,b'_u)$ are the\nbases used in (\\ref{conexpeU}), we get the following connection\nmatrix of $\\nabla^1$ on the neighborhood $V$\n\\begin{equation}\\label{expresiiloc2} \\left( \\begin{array}{cc}\n\\gamma_{ij}^kb^{*i}&0\\vspace{2mm}\\\\\n0&(-\\varphi_i^t\\gamma_{tu}^{'s}+\\rho^{'j}_u\n\\frac{\\partial\\varphi_i^s}{\\partial x^j})b^{*i}\\end{array}\\right)\\end{equation}\n(in (\\ref{expresiiloc2}), $b^{*i}$ is the dual basis of $b_i$).\n\nFurthermore, let $g^U,g^{'U}$ be local metrics on $A,A'$ such that\n$(b_i),(b'_u)$ are orthonormal bases and $g,g'$ arbitrary, global\nmetrics on $A,A'$. Then, define the metrics\n$$\\chi g^U+(1-\\chi)g,\\,\\chi g^{'U}+(1-\\chi)g'$$ and take\nan orthogonal connection $\\nabla^0$ whose components are\ncorresponding orthogonal connections. The connection matrix of\n$\\nabla^0$ on the neighborhood $V$, with respect to the same local\nbases like in (\\ref{expresiiloc2}), will be of the form\n\\begin{equation}\\label{expresiiloc3} \\left(\n\\begin{array}{cc}\\varpi_i^j&0\\vspace{2mm}\\\\ 0&-\\varpi_s^{'t}\n\\end{array}\\right),\\end{equation} where\n$(\\varpi_i^j),(\\varpi_s^{'t})$ are skew-symmetric matrices of\nlocal $1$-$A$-forms.\n\nIf these connections $\\nabla^0,\\nabla^1$ are used, then, along\n$V$, the difference matrix $\\alpha$ of formula (\\ref{defDelta}) is\nthe difference between the matrices (\\ref{expresiiloc2}) and\n(\\ref{expresiiloc3}). Furthermore, we can compute the matrix\n$\\Omega_{(\\tau)}$ by using formula (\\ref{Omegat}), where\n$\\Omega_{(0)}$ is a skew-symmetric matrix. The final result may be\nformulated as follows\n\\begin{prop}\\label{propexpresmu} If a point $x\\in M$ is fixed,\nthere exist global representative $A$-forms $\\Xi_{2h-1}\\in\n\\Omega^{4h-3}(A)$ of the characteristic classes $\\mu_{2h-1}$ such\nthat \\begin{equation}\\label{localmu} \\Xi_{2h-1}|_V=\n\\frac{1}{(2h-2)!}\\int_0^1\\left(\n\\delta^{\\sigma_1...\\sigma_{2h-1}}_{\\kappa_1...\\kappa_{2h-1}}\n\\alpha_{\\sigma_1}^{\\kappa_1}\\wedge\\Omega_{(\\tau),\\sigma_2}^{\\kappa_2}\n\\wedge...\\wedge\\Omega_{(\\tau),\\sigma_{2h-1}}^{\\kappa_{2h-1}}\\right)d\\tau,\n\\end{equation} for some neighborhood $V$ of $x$. In\n(\\ref{localmu}), the factors are the entries of the matrices\n$\\alpha,\\Omega_{(\\tau)}$ given by formulas (\\ref{expresiiloc2}),\n(\\ref{expresiiloc3}) and Greek indices run from $1$ to\n$dim\\,A+dim\\,A'$.\\end{prop}\n\\begin{proof} Use the expression (\\ref{Chern}) of the Chern\npolynomials and the connections $\\nabla^0,\\nabla^1$ constructed\nabove. \\end{proof}\n\nThe difficulty in using Proposition \\ref{propexpresmu}, besides\nits complexity in the case $h>1$, consists in the fact that\nformula (\\ref{localmu}) does not define global $A$-forms; for\nneighborhoods of different points $x_1\\neq x_2$ we have different\npairs of distinguished connections $\\bar{\\nabla},\\bar{\\nabla}'$.\nHowever, we can use Proposition \\ref{propexpresmu} in order to\nextend a result proven for a Lie algebroid $A$\n($\\varphi=\\sharp_A$) in\n\\cite{F}:\n\\begin{prop}\\label{comparmodul}\nThe secondary class $\\mu_1(\\varphi)$ is equal to the modular class\nof the morphism $\\varphi$.\\end{prop}\n\\begin{proof} Recall that the modular\nclass of a morphism is defined by\n$\\mu(\\varphi)=\\mu(A)-\\varphi^*\\mu(A')\\in H^1(A)$, where\n$\\mu(A),\\mu(A')$ are the modular classes of the Lie algebroids\n$A,A'$, respectively, \\cite{{GMM},{KW},{KGW}}. Furthermore, the\nmodular class $\\mu(A)$ is defined as follows\n\\cite{{ELW},{F},{H},{KGW}}. The line bundle\n$\\wedge^sA\\otimes\\wedge^mT^*M$ $(s=rank\\,A)$ has a flat\n$A$-connection defined, by means of local bases, as follows\n\\begin{equation}\\label{flattop}\n\\nabla_{b_i}((\\wedge_{j=1}^sb_j)\\otimes(\\wedge_{h=1}^mdx^h))\n=\\sum_{j=1}^sb_1\\wedge...\\wedge [b_i,b_j]_A\\wedge...\\wedge\nb_s\\otimes(\\wedge_{h=1}^mdx^h)\\end{equation}\n$$+(\\wedge_{j=1}^sb_j)\\otimes\nL_{\\sharp_Ab_i}(\\wedge_{h=1}^mdx^h),$$ where $L$ is the Lie\nderivative. Then, for\n$\\sigma\\in\\Gamma(\\wedge^sA\\otimes\\wedge^mT^*M)$ (which exists if\nthe line bundle is trivial; otherwise we go to its double\ncovering), one has $\\nabla_a\\sigma=\\lambda(a)\\sigma$ where\n$\\lambda$ is a $d_A$-closed $1$-$A$-form and defines the\ncohomology class $\\mu(A)$, which is independent on the choice of\n$\\sigma$.\n\nFrom (\\ref{flattop}) it follows easily that $\\mu(A),\\mu(A')$ are\nrepresented by the $A$-forms\n\\begin{equation}\\label{formemodulare}\n\\lambda=\\sum_{i,k,j}(\\gamma_{ik}^k+\\frac{\\partial\\rho_i^j}{\\partial\nx^j})b^{*i},\\,\n\\lambda'=\\sum_{s,t,h}(\\gamma_{st}^t+\\frac{\\partial\\rho_s^{'j}}{\\partial\nx^j})b^{'*s}\\end{equation} where the notation is that of\n(\\ref{expresiiloc1}). Notice that, even though the expressions\n(\\ref{formemodulare}) are local, the forms $\\lambda,\\lambda'$ are\nglobal $A$-forms because the connection that was used in their\ndefinition is global.\n\nOn the other hand, using formulas (\\ref{expresiiloc2}),\n(\\ref{expresiiloc3}) and since the trace of a skew-symmetric\nmatrix is zero, we may see that the $A$-form $\\Xi_1$ defined in\nProposition \\ref{propexpresmu} is such that\n$\\Xi|_{1V}=(\\lambda-\\varphi^*\\lambda')|_{V}$, where $V$ is a\nneighborhood of a fixed point $x\\in M$. Accordingly, there exists\na locally finite, open covering $\\{V_\\alpha\\}$ of $M$ and there\nexists a family of pairs of $A$-connections\n$(\\nabla^{0\\alpha},\\nabla^{1\\alpha})$ that provide representative\n$1$-$A$-forms $\\Xi_{1\\alpha}$ of the characteristic class\n$\\mu_1(\\varphi)$ such that\n\\begin{equation}\\label{auxmodular}\n\\Xi_{1\\alpha}|_{V_\\alpha}=(\\lambda-\\varphi^*\\lambda')|_{V_\\alpha}.\\end{equation}\nThen, if we take a partition of unity $\\{\\theta_\\alpha\\in\nC^\\infty(M)\\}$ subordinated to $\\{V_\\alpha\\}$ and glue up the\nfamilies $\\nabla^{0\\alpha},\\nabla^{1\\alpha}$, like in\n(\\ref{lipireconex}), we get connections $\\nabla^0,\\nabla^1$ that\ndefine the representative $A$-form\n$$\\Xi_1(x)=\\sum_{x\\in\nV_\\alpha}\\theta_\\alpha(x)\\Xi_{1\\alpha}(x)=(\\lambda-\\varphi^*\\lambda')(x),\n\\hspace{2mm}x\\in M$$ of $\\mu_1(\\varphi)$.\nThis justifies the required conclusion.\\end{proof}\n\\begin{example}\\label{Ps-Nij} {\\rm  An\ninteresting example appears on a Poisson-Nijenhuis manifold\n$(M,P,N)$, where $P$ is a Poisson bivector field and $N$ is a\nNijenhuis tensor. Then $^tN:(T^*M,N\\circ\\sharp_P)\n\\rightarrow (T^*M,\\sharp_P)$ is a morphism of cotangent Lie\nalgebroids. The modular class of the morphism $^tN$ was studied in\n\\cite{DF} and it would be interesting to get information about\nother characteristic classes of this morphism.}\\end{example}\n\nThe calculation of the classes $\\mu_{2h-1}$ for $h>1$ is much more\ncomplicated. One of the difficulties is the absence of a global\nconstruction of a distinguished pair of connections.\n\\begin{example}\\label{exdistins} {\\rm Let $\\varphi:A\\rightarrow A$ be\nan endomorphism of the Lie algebroid $A$ and assume that there\nexists an $A$-connection $\\nabla$ on $A$ that satisfies condition\n(\\ref{propluinabla}) and whose torsion\n$$T_\\nabla(a_1,a_2)=\\nabla_{a_1}a_2-\\nabla_{a_2}a_1-[a_1,a_2]_A,\\hspace{2mm}\na_1,a_2\\in\\Gamma A,$$ takes values in $K=ker\\,\\varphi$. Then, it\nis easy to check that the formula\n$$\\nabla'_{a_1}a_2=[\\varphi a_1,a_2]_A + \\varphi\\nabla_{a_2}a_1$$\ndefines a second $A$-connection that is $\\varphi$-compatible with\n$\\nabla$ and satisfies condition (\\ref{eqpropluinabla'}).\nTherefore, $(\\nabla,\\nabla')$ is a distinguished\npair.}\\end{example}\n\nAnother difficulty is produced by the complicated character of the\nexpression (\\ref{localmu}). A simple example follows.\n\\begin{example}\\label{exhmare} {\\rm If the Lie algebroids\n$A,A'$ have anchors zero, the $A$-connections are tensors and\nformula (\\ref{expresiiloc2}) gives the local connection matrices\nof a global, flat $A$-connection $\\nabla^1$ as required in the\ndefinition of the characteristic classes (flatness is just Jacobi\nidentity). In the simplest case\n$A=M\\times\\mathcal{G},A'=M\\times\\mathcal{G}'$ where\n$\\mathcal{G},\\mathcal{G}'$ are Lie algebras, we may take\n$\\varpi_i^j=0$ in (\\ref{expresiiloc3}), which gives a flat metric\nconnection $\\nabla^0$. Then, formula (\\ref{Omegat}) reduces to\n$$\\Omega_{(\\tau)}=\\tau(1-\\tau)\\alpha\\wedge\\alpha$$ where $\\alpha$\nis the matrix (\\ref{expresiiloc2}). Accordingly, like in \\cite{V},\nTheorem 4.5.11, we get the representative $A$-forms\n$$\\Xi_{2h-1}=\\frac{1}{(2h-2)!}\\nu_h\n\\delta^{\\sigma_1...\\sigma_{2h-1}}_{\\kappa_1...\\kappa_{2h-1}}\n\\alpha_{\\sigma_1}^{\\kappa_1}\\wedge\\alpha_{\\sigma_2}^{\\lambda_2}\n\\wedge\\alpha_{\\lambda_2}^{\\kappa_2}\\wedge...\\wedge\n\\alpha_{\\sigma_2h-1}^{\\lambda_{2h-1}}\n\\wedge\\alpha_{\\lambda_{2h-1}}^{\\kappa_{2h-1}}$$ of the classes\n$\\mu_{2h-1}$, where $\\alpha^{\\cdot}_{.}$ are the entries of the\nmatrix (\\ref{expresiiloc2}) and\n$$\\nu_h=\\int_0^1\\tau(1-\\tau)d\\tau=\\sum_{i=1}^{2h-2}(-1)^{h+i+1}\\frac{2^i}{4h-i-3}\n\\left(\\begin{array}{c}2h-2\\vspace{2mm}\\\\ i\\end{array}\\right).$$\n}\\end{example}\n\\begin{rem}\\label{bazediferite} {\\rm So far, we do not have\na good definition of characteristic classes of a morphism between\nLie algebroids over different bases. Using the terminology and\nnotation of\n\\cite{KGW}, let us\nconsider a morphism \\begin{equation}\\label{genmorf}\n\\begin{array}{ccc} A&\n\\stackrel{\\varphi}{\\rightarrow}&B\\vspace{2mm}\\\\ \\downarrow&\n&\\downarrow\\vspace{2mm}\\\\\nM&\\stackrel{f}{\\rightarrow}&N\\end{array}\\end{equation} between the\nLie algebroids $A,B$ and assume that the mapping $f$ is\ntransversal to the Lie algebroid $B$. Then, Proposition 3.11 of\n\\cite{KGW} tells us that $\\varphi=f_B^{!!}\\circ\\varphi'$, where\n$f_B^{!!}:f^{!!}B\\rightarrow B$, $\\varphi':A\\rightarrow f^{!!}B$\nare the natural projections of the pullback Lie algebroid\n$f^{!!}B$. Furthermore, Proposition 3.12 of \\cite{KGW}  tells that\nthe modular class of the non base preserving morphism $\\varphi$ is\nequal to the modular class of the base preserving morphism\n$\\varphi'$. This equality may be extended by definition to all the\ncharacteristic classes of $\\varphi$, but it is not clear whether\nthis definition is good (it does not loose information about\n$\\varphi$) even in the indicated particular case.}\\end{rem}\n\\section{Relative characteristic classes}\nFrom Proposition \\ref{comparmodul} and a known result on modular\nclasses (\\cite{KGW}, formula (2.5)) we see that the first class\n$\\mu_1(\\varphi)$ has a nice behavior with respect to the\ncomposition of morphisms namely, for the morphisms\n$\\varphi:A\\rightarrow A',\\psi:A'\\rightarrow A''$ one has\n\\begin{equation}\\label{modcompus} \\mu_1(\\psi\\circ\\varphi)=\n\\mu_1(\\varphi)+\\varphi^*(\\mu_1(\\psi)).\\end{equation} In this section\nwe give a proof of (\\ref{modcompus}) by means of the definition of\nthe characteristic classes of a morphism and we shall see why the\nresult does not extend to the higher classes $\\mu_{2h-1}$, $h>1$.\nThe proof will use a kind of relative characteristic classes that\nare interesting in their own right; in particular, we will show\nthat the relative classes defined by the jet Lie algebroid $J^1A$\n\\cite{CF} are cohomological images of the absolute characteristic\nclasses of a morphism $\\varphi:A\\rightarrow A'$.\n\nLike in the definition of the characteristic classes of $\\varphi$\nwe can produce characteristic classes of $\\psi:A'\\rightarrow A''$\nmodulo $\\varphi:A\\rightarrow A'$ as follows. Take the Lehmann\nmorphism $\\rho^*(D^0,D^1)$ for an orthogonal $A$-connection $D^0$\non the vector bundle $A'\\oplus A^{''*}$ associated with a sum of\nEuclidean metrics $g_{A'},g_{A''}$ and an $A$-connection $D^1$ on\n$A'\\oplus A^{''*}$, which is the sum of {\\it distinguished\n$A$-connections} $\\nabla',\\nabla''$ on $A',A''$, respectively.\nHere by a distinguished pair we mean a pair of $A$-connections\n$(\\nabla',\\nabla'')$ that satisfies the following properties\n\\begin{equation}\\label{distins2} \\begin{array}{l}\n\\psi\\nabla'_aa'=\\nabla''_a(\\psi a'),\\;\\;a\\in A_x\\,(x\\in M),\\,\na'\\in\\Gamma A',\\vspace{2mm}\\\\\n\\nabla'_ak(x)=[\\varphi\\tilde{a},k]_{A'}(x),\\;\\;k\\in\\Gamma ker\\,\\psi,\n\\vspace{2mm}\\\\\n\\nabla''_aa''(x)=[\\psi\\varphi\\tilde{a},a'']_{A''}(x)\\;\\;({\\rm\nmod.\\,}im\\,\\psi),\\end{array}\\end{equation} where the sign tilde\ndenotes the extension to a cross section. One can construct a\n$\\psi$-distinguished pair of $A$-connections $\\nabla',\\nabla''$ by\nreplacing the local formulas (\\ref{conexpeU}) by\n\\begin{equation}\\label{conexpeU1} \\nabla^{'U}_{b_i}b'_{j'}\n=[\\varphi b_i,b'_{j'}]_{A'},\\;\n\\nabla^{''U}_{b_i}b''_{j''}=[\\psi\\varphi b_i,b''_{j''}]_{A''},\n\\end{equation}\nthen gluing the local connections via a partition of unity. (In\n(\\ref{conexpeU1}) $(b_i),(b'_{i'}),(b''_{i''})$ are local bases of\n$\\Gamma A,\\Gamma A',\\Gamma A''$, respectively.)\n\\begin{defin}\\label{defclrel} {\\rm The\ncharacteristic $A$-cohomology classes in $im\\,\\rho^*(D^0,D^1)$\nwill be called {\\it relative characteristic classes} of $\\psi$\nmodulo $\\varphi$. In particular, $$\\mu_{2h-1}(\\psi\\,{\\rm\nmod.}\\,\\varphi) =[\\Delta(D^0,D^1)]\\in H^{4h-3}(A)$$ are the {\\it\nsimple relative characteristic classes}.}\\end{defin}\n\\begin{prop}\\label{1relativ} For $h=1$, the relative and absolute\ncharacteristic class $\\mu_1$ of the morphism $\\psi$ are related by\nthe equality \\begin{equation}\\label{pullbackmu1}\n\\mu_{1}(\\psi\\,{\\rm mod.}\\,\\varphi)\n=\\varphi^*\\mu_{1}(\\psi).\\end{equation}\n\\end{prop}\n\\begin{proof} By absolute classes we understand characteristic\nclasses $\\mu_{2h-1}(\\psi) \\in H^{4h-3}(A')$. The partition of\nunity argument given for (\\ref{expresiiloc2}) shows that we may\nassume the following local expressions of distinguished\n$A'$-connections on $A',A''$\n\\begin{equation}\\label{conexpeU2} \\bar{\\nabla}^{'U}_{b'_{i'}}b'_{j'}\n=[b'_{i'},b'_{j'}]_{A'},\\;\n\\bar{\\nabla}^{''U}_{b'_{i'}}b''_{j''}=[\\psi b'_{i'},b''_{j''}]_{A''}.\n\\end{equation}\nConnections (\\ref{conexpeU2}) induce $A$-connections\n$\\tilde{\\nabla}',\\tilde{\\nabla}''$ and we shall compute the local\nmatrices of the induced connections. By definition, we have\n$$\\tilde{\\nabla}^{'U}_{b_i}b'_{j'}=\n\\bar{\\nabla}^{'U}_{\\varphi b_i}b'_{j'},\\;\n\\tilde{\\nabla}^{''U}_{b_i}b''_{j''}=\n\\bar{\\nabla}^{''U}_{\\varphi b_i}b''_{j''}$$ and it is easy to\ncheck that the $A$-connections\n$\\tilde{\\nabla}^{'U},\\tilde{\\nabla}^{''U}$ satisfy conditions\n(\\ref{distins2}). Therefore,\n$\\tilde{\\nabla}^{'U},\\tilde{\\nabla}^{''U}$ may be used in the\ncalculation of the relative characteristic classes of $\\psi$ mod.\n$\\varphi$. If we denote $\\varphi b_i=\\varphi_i^{j'}b'_{j'}$ and\nuse expressions (\\ref{conexpeU2}) and the properties of the Lie\nalgebroid brackets we obtain the local connection matrices\n\\begin{equation}\\label{conexinduse} \\begin{array}{l}\n\\tilde{\\omega}^{'k'}_{j'}=\n\\varphi^*\\bar{\\omega}^{'k'}_{j'}-<d_{A'}\\varphi_i^{k'},b_{j'}>b^{*i},\\vspace{2mm}\\\\\n\\tilde{\\omega}^{''k''}_{j''}=\n\\varphi^*\\bar{\\omega}^{''k''}_{j''}-<d_{A''}\\varphi_i^{k'},\\psi b_{j'}>b^{*i}.\n\\end{array}\\end{equation}\nFormula (\\ref{conexinduse}) allows us to write down the local\nconnection matrix of the connection\n$D^1=\\tilde{\\nabla}'+\\tilde{\\nabla}^{''*}$ required by the\ndefinition of the relative classes. Furthermore, we may assume\nthat the local matrix of the orthogonal connection $D^0$ that we\nuse is skew-symmetric. Accordingly, and since $\\psi$ is a Lie\nalgebroid morphism, (\\ref{conexinduse}) yields\n$$\n\\Delta(D^0,D^1)c_1=tr\\,\\left(\\begin{array}{cc}\n\\tilde{\\omega}^{'k'}_{j'}&0\\vspace{2mm}\\\\0&\n-\\tilde{\\omega}^{''j''}_{k''}\\end{array}\\right) =\\varphi^*\ntr\\,\\left(\\begin{array}{cc}\n\\bar{\\omega}^{'k'}_{j'}&0\\vspace{2mm}\\\\0&\n-\\bar{\\omega}^{''j''}_{k''}\\end{array}\\right)=\n\\varphi^*\\Delta(\\bar{\\nabla}^0,\\bar{\\nabla}^1)c_1,$$\nwhere $\\bar{\\nabla}^1=\\bar{\\nabla}'+\\bar{\\nabla}^{''*}$ and\n$\\bar{\\nabla}^0$ is an orthogonal $A'$-connection on $A'\\oplus\nA^{''*}$. This result justifies (\\ref{pullbackmu1}).\\end{proof}\n\\begin{prop}\\label{2relativ}\nFor $h=1$, the relative and absolute characteristic class $\\mu_1$\nof the morphisms $\\varphi,\\psi$ are related by the equality\n\\begin{equation}\\label{eqKGW} \\mu_1(\\psi\\circ\\varphi)=\n\\mu_1(\\varphi)+\\mu_1(\\psi\\,{\\rm mod.}\\,\\varphi).\\end{equation}\\end{prop}\n\\begin{proof} In\nthe computation of $\\mu_1(\\psi\\circ\\varphi)$ we may use an\n$A$-connection $\\nabla+\\nabla^{''*}$ on $A\\oplus A^{''*}$ where,\non the specified neighborhood $U$, $\\nabla$ is given by\n(\\ref{conexpeU}) and $\\nabla''$ is given by (\\ref{conexpeU1}),\nwhile in the computation of $\\mu_{1}(\\psi\\,{\\rm mod.}\\,\\varphi)$\nwe shall use the connections $\\nabla',\\nabla''$ of\n(\\ref{conexpeU1}). Thus, the non-zero blocks of the local\ndifference matrix $\\alpha$ that enters into the expression of the\nrepresentative $1-A$-form of $\\mu_1(\\psi\\circ\\varphi)$ are given\nby the local matrix of\n\\begin{equation}\\label{blocuri}\n\\nabla''-\\nabla=\\nabla''-\\nabla'+\\nabla'-\\nabla\\end{equation} and\nthe opposite of its transposed matrix (in spite of the notation,\ncalculation (\\ref{blocuri}) is for the connection matrices not for\nthe connections). Then, if we use orthogonal connections of\nmetrics where the bases used in (\\ref{conexpeU1}) are orthonormal\nbases (therefore, with trace zero), formula (\\ref{blocuri})\njustifies (\\ref{eqKGW}).\\end{proof}\n\\begin{corol}\\label{corolarKGW} The characteristic class $\\mu_1$\nof a composed morphism $\\psi\\circ\\varphi,\\psi$ is given by formula\n(\\ref{modcompus}).\\end{corol} \\begin{proof} The result is an\nobvious consequence of formulas (\\ref{pullbackmu1}) and\n(\\ref{eqKGW}).\\end{proof}\n\\begin{rem}\\label{obsfinala} {\\rm Formulas (\\ref{modcompus}),\n(\\ref{pullbackmu1}), do not hold for $h>1$ because of the more\ncomplicated expression of the polynomials $c_{2h-1}$ (there is no\nnice formula for the determinant of a sum of matrices).}\\end{rem}\n\nWe finish by showing the relation between the characteristic\nclasses of the base-preserving Lie algebroid morphism\n$\\varphi:A\\rightarrow A'$ and the relative classes defined by the\nfirst jet Lie algebroid $J^1A$; for $\\varphi=\\sharp_A:A\\rightarrow\nTM$ this relation was established in \\cite{CF}.\n\nThe first jet bundle $J^1A$ may be defined as follows. Let $D$ be\na $TM$-connection on $A$ and let $Da$ denote the covariant\ndifferential of a cross section $a\\in\\Gamma A$ (i.e.,\n$Da(X)=D_Xa$, $X\\in\\Gamma TM$). The properties of a connection\ntell us that $Da\\in Hom(TM,A)$ and, if $a(x_0)=0$ for some point\n$x_0\\in M$, then $Da(x_0):T_{x_0}M\\rightarrow A_{x_0}$ is a linear\nmapping that is independent of the choice of the connection $D$.\n(This is not true if $a(x_0)\\neq0$.) If $(x^h)$ are local\ncoordinates of $M$ around $x_0$ and $(b_i)$ is a local basis of\n$\\Gamma A$, and if $a=\\xi^i(x^h)b_i$, the local matrix of\n$Da(x_0)$ is $(D_h\\xi^i(x_0))$ (the covariant derivative tensor),\nwhich reduces to $(\\partial\\xi^i\/\\partial x^h(x_0))$ if\n$\\xi^i(x_0)=0$.\n\nNow, for any point $x_0\\in M$, the space of $1$-jets of cross\nsections of $A$ at $x_0$ is\n\\begin{equation}\\label{spjetx}\nJ^1_{x_0}A=\\Gamma A\/\\{a\\in\\Gamma\nA\\,\/\\,a(x_0)=0,Da(x_0)=0\\}\\end{equation} and each $a\\in\\Gamma A$\ndefines an element $j_x^1a\\in J^1_{x_0}A$ called the $1$-jet of\n$a$ at $x_0$. With the local coordinates and basis considered\nabove, we may write\n$$a=\\xi^i(x^h)b_i=(\\xi^i(x_0)+\\frac{\\partial\\xi^i}{\\partial\nx^h}(x_0)(x^h-x^h(x_0))+o((x^h-x^h(x_0))^2))b_i.$$ Hence,\n$$j_{x_0}^1a=\\xi^i(x_0)j_{x_0}^1b_i+\n\\frac{\\partial\\xi^i}{\\partial x^h}(x_0)j_{x_0}^1((x^h-x^h(x_0))b_i)$$ and\n\\begin{equation}\\label{bazejet}\nj_{x_0}^1b_i,\\,j_{x_0}^1((x^h-x^h(x_0))b_i)=j_{x_0}^1\n(x^hb_i)-x^h(x_0)j_{x_0}^1b_i\\end{equation} is a basis of the\nvector space $J^1_{x_0}A$ such that\n$(\\xi^i(x_0),\\partial\\xi^i\/\\partial x^h(x)(x_0))$ are coordinates\nwith respect to this basis.\n\nA change of the local coordinates and basis of $A$ gives the\ntransition formulas\n\\begin{equation}\\label{schimbcoordjet}\n\\tilde{x}^h=\\tilde{x}^h(x^k),\\,\\tilde{\\xi}^i=\\lambda^i_j(x^k)\\xi^j,\\,\n\\frac{\\partial\\tilde{\\xi}^i}{\\partial\\tilde{x}^h}=\n\\frac{\\partial{x}^k}{\\partial\\tilde{x}^h}\\left(\n\\frac{\\partial\\lambda_j^i}{\\partial{x}^k}\\xi^j+\\lambda_j^i\n\\frac{\\partial{\\xi}^j}{\\partial{x}^k}\\right)\\end{equation}\nand may be seen as the composition of the change of the\ncoordinates $(x^h)$ with the change of the basis $(b_i)$, while\nthe order of the two changes is irrelevant. This remark allows for\nan easy verification of the fact that the change of the\ncoordinates discovered above in $J^1_{x_0}A$ has the cocycle\nproperty. Accordingly, (\\ref{schimbcoordjet}) shows that\n$J^1A=\\cup_{x\\in M}J^1_xA$ has a natural structure of a\ndifferentiable manifold and vector bundle $\\pi:J^1A\\rightarrow M$\nover $M$ called the first jet bundle of $A$.\n\nFrom (\\ref{bazejet}), we see that $(j^1b_i,j^1(x^hb_i))$ is a\nlocal basis of cross sections of $J^1A$ at each point of the\ncoordinate neighborhood where $x^h$ are defined. This basis\nconsists of $1$-jets of local cross sections of $A$, therefore,\nthe cross sections of $J^1A$ are locally spanned by $1$-jets of\ncross sections of $A$ over $C^\\infty(M)$. In the case of a Lie\nalgebroid $A$, the previous remark allows us to define a Lie\nalgebroid structure on $J^1A$ by putting\n\\begin{equation}\\label{strluiJ1A} \\sharp_{J^1A}(j^1a)=\\sharp_Aa,\\;\n[j^1a_1,j^1a_2]_{J^1A}=j^1[a_1,a_2]_A\\end{equation} and by\nextending the bracket to general cross sections via the axioms of\na Lie algebroid. We refer the reader to Crainic and Fernandes\n\\cite{CF} for details. A general, global expression of the Lie\nalgebroid bracket of $J^1A$ was given by Blaom \\cite{B}.\n\nMoreover, (\\ref{strluiJ1A}) shows that the natural projection\n$\\pi^1:J^1A\\rightarrow A$, $\\pi^1(j^1a)=a$ is a base-preserving\nmorphism of Lie algebroids and, if $\\varphi:A\\rightarrow A'$ is a\nmorphism of Lie algebroids, we may define relative characteristic\nclasses of $\\varphi$ modulo $\\pi^1$. Following \\cite{CF}, there\nexist flat $J^1A$-connections $\\nabla^{j^1},\\nabla^{'j^1}$ on\n$A,A'$, respectively, given by\n\\begin{equation}\\label{Jconex}\n\\nabla^{j^1}_{fj^1a_1}a_2=f[a_1,a_2]_A,\\, \\nabla^{'j^1}_{fj^1a}a'=f[\\varphi\na,a']_{A'},\\end{equation} where $a,a_1,a_2\\in\\Gamma A, a'\\in\\Gamma\nA',f\\in C^\\infty(M)$. These connections obviously satisfy\nconditions (\\ref{distins2}), hence,\n$D^1=\\nabla^{j^1}+\\nabla^{'*j^1}$ is a $J^1A$-connection on\n$A\\oplus A^{'*}$ that may be used in Definition \\ref{defclrel} for\nthe present case. We shall prove the following result\n\\begin{prop}\\label{propCF} The relative characteristic classes of\n$\\varphi$ modulo $\\pi^1$ are the images of the corresponding\nabsolute characteristic classes of $\\varphi$ by the homomorphism\n$\\pi^{1*}:H^*(A)\\rightarrow H^{*}(J^1A)$.\\end{prop}\n\\begin{proof} Here, we have the particular case of the situation\nthat existed in Proposition \\ref{1relativ} where $\\pi^1$ comes\ninstead of $\\varphi$ and $\\varphi$ comes instead of $\\psi$.\nTherefore, we may construct connections that correspond to\n(\\ref{conexpeU2}) and the induced $J^1A$-connections and get the\ncorresponding formulas (\\ref{conexinduse}). If we use the local\nbases (\\ref{bazejet}) of $\\Gamma J^1A$, the components\n$\\varphi_i^{k'}$ that appear in (\\ref{conexinduse}) are constant\nand (\\ref{conexinduse}) simply tell us that the local connection\nforms of the induced connections are the pullback of the\nconnection forms of the connections (\\ref{conexpeU2}) by $\\pi^1$.\nOf course, the same holds for the curvature forms, and, if we also\nuse a $J^1A$-orthogonal connection of $A\\oplus A^{'*}$ that is\ninduced by an $A$-orthogonal connection, we see that $\\pi^{1*}$\ncommutes with the Lehmann morphism, which implies the required\nresult.\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConsider a sample ${\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n$, where ${\\mathbf x}_i\\in\\RR^d$. We have a\nfeedforward neural network with a given architecture (but the weights\nare unknown). Each sample point ${\\mathbf x}_i$ has binary labels, either +1 or\n-1. Sauer's lemma provides an upper bound on the number of possible\nlabelings that could be generated by a hypothesis class (the\n\\emph{growth function}) in terms of the VC dimension of the hypothesis\nclass.\n\nWe are interested in hypothesis classes corresponding to neural\nnetworks with a fixed architecture but unspecified weights. While it\nis hard to exactly specify the VC dimension of this class, upper\nbounds on the VC dimension and the growth function are easily derived,\nsee for example~\\cite[Section 6.2]{anthony2009neural}. The growth\nfunction for a feedforward, linear threshold network is upper bounded\nby $(enk\/W)^W$, where $k$ is the number of neurons in the network, and\n$W$, the number of weights.\n\nOur goal in this paper is to generate labels of the sample uniformly\nat random from the set of all possible labelings that a given\nfeedforward architecture can provide. We obtain a polynomial time (in\nboth the number of samples and the size of the network), near uniform\nsampling from arbitrary feedforward networks.  In the special case of\na single neuron, we also provide a random walk based algorithm for\nperfectly uniform sampling, and with polynomial mixing time for the\nrandom walk.\n\nAside from the theoretical interest in generating labelings, we are\nalso motivated by questions in property testing. Namely, we want to\nestimate the statistics of all labelings generated by a given\narchitecture.  As an example, we may want to find out the the\nprobability that a subset of samples are all labeled the same if all\nlabels were generated at random from the given architecture.  In\nfuture work, we intend to leverage these insights into better\ninitializations of neural networks while training.\n\nWe obtain these results by developing insights on random walks between\nchambers of intersecting hyperplanes in high dimensions.  This is a\nwell studied area, see for example~\\cite{stanley2004introduction}.\nGeneral arrangements of these hyperplanes intersect in complicated\nways, as in our problem, and random walks between these chambers is\nnontrivial. It is common to visualize the geometry of these\narrangments by means of a \\emph{chamber graph}, see Chapter 7 of\n\\cite{ovchinnikov2011graphs} for a synopsis of such chamber graphs.\nRandom walks over hyperplane arrangements appears in contexts quite\ndifferent from ours. For example, Bidigere, Hanlon and Rockmore\nmodeled card shuffling in \\cite{bidigare1999combinatorial}, with such\nrandom walks. Some other applications are\nin~\\textit{e.g.,}\\xspace~\\cite{brown1998random,athanasiadis2010functions,pike2013eigenfunctions,bjorner2008random}.\n\nThe statistics of the random walks considered in the references\nabove is different from ours. Typically, these authors provide an\nexplicit expression to estimate the eigenvalues of the random walk to\nbound the mixing time. In our paper, we use conductance to\nunderstand the mixing properties of our random walk as\nin~\\cite{levin2017markov} and~\\cite{berestycki2016mixing}.\n\n\\ignore{The more general problem of uniformly sampling geometric objects is\nextensively studied in Markov Chain Monte Carlo (MCMC) literature,\ne.g. Dyer, Frieze and Kannan's work~\\cite{dyer1991random} on\nestimating the volume of high dimensional convex bodies.  }\n\n\\section{Setup and Notations}\n\\newcommand{\\W}{{\\textbf W}} We consider a feed-forward linear\nthreshold neural network with $L$ layers.  The input to the network is\n$d-$dimensional and there is a single binary output label. Namely,\ni.e. any neuron with parameters $\\textbf{w},b$, (${\\textbf w}\\in \\RR^d$,\n$b\\in\\RR$) outputs $\\sigma(\\textbf{x}^T\\textbf{w}+b)$ on an input\n${\\mathbf x}\\in\\RR^d$, where $\\sigma(u)=1$ if $u\\ge 0$ and $\\sigma(u)=0$\notherwise. In subsequent work, we extend our results to more general\nactivation functions.\n\nLet $N$ be the graph of the feedforward neural network with a fixed\narchitecture and $W$ different parameters (the weights and thresholds\nput together). Let $\\W\\in \\RR^W$, and let $N_{\\W}$ be the neural\nnetwork which assigns the parameters of $N$ to be $\\textbf{W}$. For\nany given architecture $N$, let\n$f_{\\W}:\\textbf{x}\\in\\mathbb{R}^d\\to\\{0,1\\}$ be the function expressed\nby $N_{\\W}$.\n\nThe vectors $\\textbf{x}\\in\\mathbb{R}^d$ are the input and\n$f_{\\W}(\\textbf{x})$ are the labels assigned to $\\textbf{x}$.\nFor a length $n$ sample\n$X=\\{\\textbf{x}_1,\\cdots,\\textbf{x}_n\\in \\mathbb{R}^d\\}$, let\n\\[\n  S_X=\\{(f_{\\W}(\\textbf{x}_1),\\cdots,f_{\\W}(\\textbf{x}_n))\\mid\n  \\textbf{W}\\in \\mathbb{R}^W\\}\n\\]\nbe the set of all labelings that can be generated on $X$ by the\narchitecture $N$. Note that the set $S_X\\subset \\sets{0,1}^n$ and for\n$W<n$,~\\cite[Section 6.2]{anthony2009neural}\n(or~\\cite{shalev2014understanding})\n\\[\n|S_X|\\le \\Paren{\\frac{enk}W}^W.\n\\]\nWhen $W\\ge n$, $|S_X|\\le 2^n$, or $X$ is potentially shattered.\n\n\\noindent\n\\textbf{Problem} For a given architecture $N$ and data $X$, how\ncan we randomly sample from $S_X$, in time polynomial in both $n$ and\n$W$, such that any labeling $v\\in S_X$ appears with\nprobability at least $\\Omega(1\/|S_X|)$?\n\n\\paragraph{Background}\nA hyperplane in $\\RR^d$ (or a hyperplane\nin $d$ dimensions) is the set of all points ${\\textbf w}\\in\\RR^d$ \nsatisfying ${\\mathbf x}^T {\\textbf w}=0$ for some fixed vector ${\\mathbf x}\\in\\RR^d$. \nLet $N$ be a single neuron with input dimension $d$. As before,\n$X=\\Sets{{\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n\\in \\RR^d}$ is a length $n$ sample.\n\nLet ${\\textbf w}\\in \\RR^d$ and $b\\in\\RR$.  Physically, the vector in\n$d+1$ dimensions, $({\\textbf w}, b)\\in\\RR^{d+1}$ defines the parameters of\nthe single neuron $N$.  For each sample point\n$\\textbf{x}_i\\in \\mathbb{R}^d$, define $P_i$ to be the hyperplane in\nthe parameter space $\\RR^{d+1}$:\n\\[\n\\textbf{x}_i^T\\textbf{w}+b=0.\n\\] \nWe start with a visualization from~\\cite{anthony2009neural}.\n\\begin{theorem} All parameter vectors that belong to the same connected\n  component of $\\mathbb{R}^{d+1}\\backslash \\bigcup_i P_i$ label $X$ in\n  the same way. Conversely, different components have different\n  labelings on $X$.\n\\end{theorem}\n\nWe recall a few standard terms regarding hyperplane arrangements\nformed by $P_1{,}\\ldots{,} P_n$.\n\\begin{list}{$\\bullet$}{\\setlength{\\listparindent}{0pt}\\setlength{\\leftmargin}{10pt}}\n\n\\item The connected components in\n  $\\mathbb{R}^{d}\\backslash \\bigcup_{i=1}^nP_i$ are called\n  \\emph{chambers} (or regions).\n\n\\item The \\emph{chamber graph} is constructed as follows: assign a\nvertex to every chamber. Two vertices are connected if their associated \nchambers share a common face. \n\n\\item Any hyperplane arrangement is \\emph{centered} if the\n  intersection of the component hyperplanes contains the origin. \n  In our case, $\\bigcap_{i=1}^nP_i$ always contains the origin,\n  \\ie the samples generate a \\emph{centered} arrangement\n  in the parameter space.\n\n\\item A collection of $n$ centered hyperplanes in $\\RR^{d+1}$ is\n  in \\emph{general} position, if for all $k\\le d+1$, every\n  intersection of $k$ distinct hyperplanes forms a $d+1-k$-dimension\n  linear space, and any intersection of more than $d+1$ hyperplanes is\n  contains only the origin. Randomly chosen\n  planes are in general position almost surely.\n\\end{list}\n\n\\paragraph{Psuedo polynomial optimal training algorithm}\nA theoretically useful framework was introduced in Theorem 4.1\nof~\\cite{arora2016understanding} for ReLU networks, where the network\nsize $W$ is treated as a constant, and we look at the dependency\npurely on the sample size $n$ (thereby treating $n^W$ as a\npolynomial).\n\nWe note that our near-uniform polynomial time sampling procedure\nimplies a probabilistic, psuedo-polynomial training algorithm that\nattains the global minimum for any feedforward linear threshold neural\nnetwork. This implication is immediate from the coupon collector\nproblem---since given any confidence, generating at most\n$O(n^W\\log n)$ samples guarantees that we have seen every possible\nlabeling that can be produced.\n\n\\section{Properties of hyperplane arrangements}\nWe summarize a few useful properties of hyperplane arrangments that\nwe will use in our arguments in the paper.\n\\begin{proposition}[{\\cite[Theorem 3.1]{anthony2009neural}}]\n  The number of chambers in a centered hyperplane arrangement formed\n  by $n$ hyperplanes in $d$ dimensions in the general position is\n$$2\\sum_{i=0}^{d-1}\\binom{n-1}{i}.$$\n\\end{proposition}\nIn fact, even sampling all labels of a sample of size $n$, even when\nthe network consists of a single neuron, in time polynomial in both\n$n$ and dimension $d$ of the data points, is non-trivial. The number\nof chambers by Theorem 1 is the number of labels on a size-$n$ sample,\nwhich from the above Proposition is roughly $O(n^d)$.  Clearly,\ntrivial enumeration of labels is out of question. As we will see later\nin Section~\\ref{s:rcr}, this is not the only difficulty even for a\nsingle neuron.\n\n\\begin{proposition}\n  Let $Q_1{,}\\ldots{,} Q_n$ form a centered hyperplane arrangement in $d$\n  dimensions. Let $\\v_i\\in\\RR^d$ be any vector normal to the \n  hyperplane $Q_i$. If $\\v_1{,}\\ldots{,} \\v_n$ have rank $r$,\n  then any chamber in the hyperplane arrangment has at least $r$\n  faces.\n\\end{proposition}\n\\begin{proof}\n  Let $V=\\sets{\\v_1{,}\\ldots{,} \\v_n, -\\v_1{,}\\ldots{,} -\\v_n}$.  Suppose the\n  proposition is false. Then there exists a chamber with exactly $b<r$ faces.\n  Without loss of generality, let $\\textbf{u}_1\\in V,\\cdots,\\textbf{u}_b\\in V$\n  be the normal vectors of the $b$ different faces of this chamber respectively,\n  such that for any point $\\textbf{x}$ within the chamber\n  \\[\n    \\u_i^T{\\mathbf x}>0.\n  \\] \n  Since the rank of $\\v_1{,}\\ldots{,} \\v_n$ is $r>b$, we can choose a vector\n  $\\u_{b+1}\\in V$ such that $\\u_{b+1}$ is linearly independent from\n  $\\u_1,\\cdots,\\u_b$. \n\n  We now show that the hyperplane that determined by $\\u_{b+1}$ is\n  also a face of the chamber by proving that there is a point\n  $\\textbf{x}'$ in the chamber satisfying\n  \\begin{equation}\n    \\label{eq:const}\n    \\u_i^T\\textbf{x}'>0\n    \\;\\;\\;\\; 1\\le i\\le b\n    \\quad\n    \\text{ and }\n    \\quad\n    \\u_{b+1}^T\\textbf{x}'=0. \n  \\end{equation}\n  Since $\\u_{b+1}$ is linearly independent of $\\u_1{,}\\ldots{,} \\u_b$, we\n  can choose a vector $\\textbf{y}$ such that $\\textbf{y}^T\\u_i=0$ for\n  $1\\le i\\le b$ but $\\textbf{y}^T\\u_{b+1}\\not=0$.\n  Now let $\\textbf{x}$ be any point in the chamber and set\n  $\\textbf{x}'=\\textbf{x}+t\\textbf{y}$ where\n   $t=-\\u_{b+1}^T{\\mathbf x}\/\\u_{b+1}^T\\textbf{y}$.\n   It is easy to verify now that ${\\mathbf x}'$ satisfies~\\eqref{eq:const}.\n   This contradicts the assumption that the chamber contained\n   $b$ faces, where $b<r$.\n\\end{proof}\n\\begin{proposition}\nThe chamber graph of any hyperplane arrangement $Q_1{,}\\ldots{,} Q_n$ in $\\RR^d$ in general position satisfies (i) the degree of any vertex is at least $d$ and at most $n$, and (ii) any pair of vertices has graph distance at most $n$.\n\\end{proposition}\n\\begin{proof}\n(i) from Proposition $2$, (ii) from \\cite[Lemma 7.15]{ovchinnikov2011graphs}.\n\\end{proof}\n\n\\section{Sampling labelings }\nFor the sample $X=\\sets{{\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n}$, where ${\\mathbf x}_i\\in\\RR^d$, \n$S_X$ is the set of all possible labels generated on $X$\nby the network $N$. We would like to sample from $S_X$ uniformly.\n\nIn Section~\\ref{s:rcr}, we let $N$ be a single neuron and even this\nturns out to be non-trivial. Inspired by the inductive approach for\ncomputing hyperplane partition number \\cite[Chapter\n2]{stanley2004introduction}, we derive Algorithm RS (for Recursive\nSampling) in Section~\\ref{s:rcr} that generates a label from $S_X$\nalmost uniformly.\n\nIn Sections~\\ref{s:rw} and~\\ref{s:an} we expand in two directions. In\nSection~\\ref{s:rw}, we provide means to perfectly sample from all\nlabelings of a single neuron using a random walk on $S_X$ with a perfectly\nuniform stationary distribution. This allows us to sample from $S_X$\nperfectly uniformly.  The mixing time of this random walk is as yet\nunproven, but we provide partial evidence (empirical as well as proofs\nfor small dimensions) that this random walk is fast mixing, with\nmixing time at most linear in the number of dimensions and at most\nquadratic in the number of samples.\n\nIn Section~\\ref{s:an}, we build on our RS approach to sample from\narbitrary feedforward networks in time (true) polynomial in the sample\nsize $n$, network size $W$ and input dimension ($d$), showing that\neven for arbitrary networks, we get near-uniform sampling of the\npossible labels that could be produced by the network.\n\n\\subsection{The recursive approach}\n\\label{s:rcr}\nFrom Theorem 1, to sample uniformly from $S_X$ we only need to sample the\nweights uniformly from the connected components of $\\mathbb{R}^{d+1}\\backslash \\bigcup_iP_i$.\n\n\nHowever, even for a single neuron, this is not trivial. As already\nnoted from Proposition 2, the basic combinatorial difficulty comes\nfrom the fact that there are roughly $n^d$ labelings for almost all\nsamples $X$---therefore the number of chambers is exponential in the\ndimension $d$. Clearly one can not simply enumerate all the possible\ncomponents.\n\nBut a bigger difficulty comes from the fact that the arrangement of\nthe hyperplanes can be very heterogeneous. The volume of some of the\nchambers can be arbitrarily small and therefore such chambers may be\ndifficult to find. We settle this problem by using a recursive\nsampling approach that is inspired by the inductive approach for\ncomputing hyperplane partition number.\n\nOur recursive algorithm RS($\\v_1{,}\\ldots{,} \\v_k$) (see Algorithm $1$) takes\nas its inputs $k$ unit vectors $\\v_1{,}\\ldots{,} \\v_k$, all from, say\n$\\RR^m$. The vectors $\\v_1{,}\\ldots{,} \\v_k$ are interpreted as normal\nvectors of $k$ distinct centered hyperplanes in $\\RR^m$. For\nsimplicity, the reader can assume that these hyperplanes are in the\ngeneral position, but they do not have to be. To sample from $S_X$, we\nwould therefore simply call RS($\\tilde{{\\mathbf x}}_1{,}\\ldots{,} \\tilde{{\\mathbf x}}_n$),\nwhere $\\tilde{{\\mathbf x}}_i= \\frac{({\\mathbf x}_i,-1)}{||({\\mathbf x}_i,-1)||_2}\\in\\RR^{d+1}$\nand ${\\mathbf x}_i\\in X$.\n\nThe call RS($\\v_1{,}\\ldots{,} \\v_k$) works recursively on the\ndimension $m$ of the vectors $\\v_i$ and $k$, by calling RS with a new set of\nvectors $\\u_1{,}\\ldots{,} \\u_{k'}$ in $\\RR^{m-1}$ with $k'\\le k-1$.\nThe base case is when RS is called with vectors in 1 dimension or when\n$k=1$. When RS is called with vectors in 1 dimension, the problem is\ntrivial since there is only one centered hyperplane arrangement in\n$1-$dimension, the origin. When RS is called with $k=1$ (no matter the\ndimension of the single input vector), the problem is also trivial\nsince there are only two chambers for one hyperplane.\n\nTo generate the vectors $\\u_i$ in $\\RR^{m-1}$, we choose a\nhyperplane at random from $\\v_1{,}\\ldots{,} \\v_k$, say $\\v_i$, and compute\nthe intersection of $\\v_i$ with all the remaining hyperplanes. These\nintersections are at most $k-1$  hyperplanes in $\\mathbb{R}^{m-1}$ \nand let $\\u_1{,}\\ldots{,} \\u_{k'}$ be the unit normal vectors of\nthese hyperplanes (written in the specific orthonormal basis indicated).\n\n\n\\begin{algorithm}[t]\n\\caption{RS($\\v_1{,}\\ldots{,} \\v_k$)}\\label{alg:1}\n\\textbf{Input:} $\\textbf{v}_1,\\cdots,\\textbf{v}_k\\in \\mathbb{R}^m$, interpreted\nas unit normal vectors of $k$ (distinct) centered hyperplanes in an $m-$dimensional space. \\\\\n\\textbf{Output:} point $\\textbf{y}\\in \\mathbb{R}^m$ representing a chamber\nin the hyperplane arrangement formed by $\\v_1{,}\\ldots{,} \\v_k$, \\\\\n\nLet $P_i$ be the hyperplane in $\\RR^m$ orthogonal to $\\v_i$.\n\\begin{itemize}\n\\item[1.]If $m=1$ output -1 or 1 with equal probability. If $m>1$ but $k=1$, output $\\v_1$ or $-\\v_1$ with equal probability.\n\\item[2.] Uniformly choose an index $I$ from $\\sets{1{,}\\ldots{,} k}$. \n\\item[3.] For hyperplane $P_I$, choose an arbitrary orthonormal basis\n  $B\\in\\RR^{(m-1)\\times m}$. Note that $P_I$ is a $(m-1)$-dimensional\n  linear space in $\\RR^m$, and the $m-1$ rows of $B$ contain the \n  orthonormal basis vectors, each being a vector in $\\RR^m$.\n\\item[4.] Compute the intersection of $P_I$ with $P_j$,\n  $j\\in \\sets{1{,}\\ldots{,} k}\\backslash \\sets{I}$. \n\\item[5.] Set $\\v_j'$ to be the unit vector in $P_I$ normal to $P_I\\cap P_j$\n  (written using the basis $B$),\n  $j\\in \\sets{1{,}\\ldots{,} k}\\backslash \\sets{I}$. Note $\\v_j'\\in\\RR^{m-1}$.\n\\item[6.]$\\textbf{x}=RS(\\u_1,\\cdots,\\u_{k'})$, where $\\u_1,\\cdots,\\u_{k'}$ are the distinct vectors among $\\{\\v_j'\\mid j\\not=I\\}$. Note $k'\\le k-1$.\n\\item[7.] Compute the smallest distance $\\delta$ of $\\textbf{x}^TB$ to\n  the planes $P_j$ with $j\\not=I$.\n\\item[8.] Let $t$ be -1 or 1 with equal probability, output\n  $\\textbf{y}=\\textbf{x}^TB+t\\delta\\textbf{v}_I$.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{theorem}\n  Let $V=\\sets{\\v_1{,}\\ldots{,} \\v_k}$ where $\\v_i\\in\\RR^m$ and rank of\n  $V$ is $m$. Let $C_V$ be the set of non-empty chambers induced by the $k$\n  centered hyperplanes orthogonal to the vectors in $V$. Algorithm\n  RS($\\v_1{,}\\ldots{,} \\v_k$) runs in $O(km^3)$ time and any chamber in the\n  hyperplane arrangment induced by $V$ is sampled with probability at\n  least\n  $$\n  \\frac{1}{2^m\\binom{k}{m}}\\ge \\left(\\frac{m}{2ek}\\right)^m,\\text{ where }e\\text{ is the base of nature logorithm}.\n  $$\n\\end{theorem}\n\\begin{proof} (Outline only)\n  The algorithm will run at most $m$ recursive iterations. For each\n  iteration, we need $O(m^2)$ to compute the base of the null space\n  (Step 3) and $O(km^2)$ time in Step 7 to compute the projection of\n  each input vectors to the plane chosen in Step 2. This yields the\n  total complexity to be $O(km^3)$.\n\n  To see the probability lower bound, define\n  $$p(m, k)=\\min_{V,c\\in C_V}\\mathrm{Pr}[RS(V)=c],\\text{ with }\\text{rank}(V)=m\\text{ and }|V|\\le k.$$\n  We now claim that\n  $$p(m, k)\\ge \\frac{m}{2k}p(m-1, k-1).$$\n  This is because any chamber $c\\in C_V$ has at least $m$ faces by\n  Proposition 2. For any chamber $c\\in C_V$, we therefore have\n  probability at least $\\frac{m}{2k}$ of choosing both a hyperplane\n  that forms the face of $c$ and the direction of the hyperplane that\n  faces the chamber $c$. Conditioned on this choice of hyperplane and\n  direction, we need to obtain the probability that the recursive call\n  in step in Step 6 returns a point in the face of $c$.\n\n  Observe that the face of $c$ is a $m-1$-dimensional linear space. In\n  Step 6, note that the rank of $\\sets{ \\u_1{,}\\ldots{,} \\u_{k'}}$ is exactly\n  $m-1$, but $k'$ can be less than $k-1$. The theorem follows by solving the recursive\n  inequality, standard approximations on binomial coefficient and by\n  noting that when $m=1$, there are two chambers, thus yielding\n  $p(1,k)=1\/2$ for all $k$.\n\\end{proof}\nNote that when $\\text{rank}(X)=d$ the above probability is\n${\\cal O}\\Paren{\\frac{d!}{2^dn^d}}$, a factor $\\frac{1}{2^d}$ off the\nhyperplane slicing bound $2\\sum_{i=0}^{d-1}\\binom{n-1}{i}$ in\nProposition 1. Note also that if the input vectors in $\\RR^d$ have\nrank $m<d$, the above approach still works. We can effectively project\ndown the inputs into $\\RR^m$ by choosing a basis for $\\RR^d$\nthat contains $d-m$ vectors that are orthogonal to the span of the\ninput vectors.\n\n\\subsection{A random walk approach}\n\\label{s:rw}\nTo mitigate the fact that the recursive approach above only yields\napproximately uniform sampling, We introduce a random walk based\nalgorithm that samples arbitrarily close to uniform. Specifically, we\nrun Algorithm NRW on a \\emph{lazy chamber graph}, both outlined\nbelow. One component of Algorithm NRW is Algorithm Chamber, that\ndetermines which chamber an input point belongs to.\n\\begin{algorithm}[t]\n\\caption{NRW}\\label{alg:2}\n\\textbf{Input:} walk length $T$ and hyperplanes $P_1,\\cdots,P_n$ in $\\mathbb{R}^m$\\\\\n\\textbf{Output:} point $\\textbf{w}\\in \\mathbb{R}^m$ and chamber $c$\n\\begin{itemize}\n\\item[1.] Initialize $\\textbf{w}_0=RS(\\textbf{v}_1,\\cdots,\\textbf{v}_n)$, where $\\textbf{v}_i$ is a normal vector of $P_i$\n\\item[2.] Set $c_0= $ Chamber$({\\textbf w}_0)$. $c_0$ will be the chamber\n  in the arrangement $\\sets{P_i}$ that contains\n  $\\textbf{w}_0$.\n\\item[3.] For $t=1$ through $T$, do\n  \\begin{itemize}\n  \\item[a.] Uniformly choose a face of $c_{t-1}$\n  \\item[b.] Set $c_t$ to be the chamber adjacent to $c_{t-1}$ and across the\n    face chosen in step (a.)\n  \\item[c.] Set $\\textbf{w}_t$ to any point in the chamber $c_t$\n    \\end{itemize}\n \\item[4.] Output ${\\textbf w}_T$ and $c_T$\n \\end{itemize}\n\\end{algorithm} \n\n\\begin{algorithm}[b]\n\\caption{Chamber}\\label{alg:3}\n\\textbf{Input:} point $\\textbf{w}\\in \\mathbb{R}^m$ and hyperplanes $P_1,\\cdots, P_n$\\\\\n\\textbf{Output:} The faces $P_{i_1},\\cdots,P_{i_k}$ \nof the chamber containing $\\textbf{w}$.\n\\begin{itemize}\n\\item[1.]Compute $\\sigma_i=\\text{sign}(\\textbf{w}^T\\textbf{v}_i)$.\n\\item[2.]For $1\\le i\\le n$ do:\n\\begin{itemize}\n\\item[3.]Define a linear program with $\\textbf{w}^T\\textbf{v}_i=0$ and $\\sigma_j(\\textbf{w}^T\\textbf{v}_j)>0$ for $j\\not =i$.\n\\item[4.]If the linear programming in step $3$ has a solution, add $P_i$ to the collection.\n\\end{itemize}\n\\end{itemize}\n\\end{algorithm} \n\n\\begin{theorem}\nAlgorithm Chamber runs in polynomial time both on $d$ and $n$.\n\\end{theorem}\n\\begin{proof}\nThe theorem follows since linear programming can be solved in polynomial time \\cite{megiddo1986complexity}.\n\\end{proof}\n\n\\paragraph{Analysis} We first analyze random walk defined by\nAlgorithm NRW over the simple chamber graph, assuming the hyperplanes are in general position.\nWith this assumption any vertex in the \\emph{chamber graph} has degree\nat least $d$ and at most $n$ from Proposition 3. Furthermore, from\nProposition 3 the graph is connected and the distance between any two\nvertexes is at most $n$.\n\nSince the random walk is a reversible Markov chain, the stationary\ndistribution $\\pi$ of the random walk will be proportional to the\ndegree of the vertices~\\cite[Chapter 1.6]{levin2017markov}. From our\nobservation on the bounds of degrees in Proposition 3,\nwe will therefore have for any two vertices $u$ and $v$\n\\[\n  \\frac{d}{n}\\le\\frac{\\pi(u)}{\\pi(v)}\\le \\frac{n}{d}.\n\\]\nThe more fundamental question is the mixing time of the\nrandom walk, or how quickly the walk generates stationary\nsamples. While there are several approaches to analyze the mixing time, we\nfocus on Cheeger's inequality \\cite[Theorem 13.14]{levin2017markov}\nthat bounds the spectral gap of the random walk's transition matrix \nusing the \\emph{conductance} of the graph. Recall that the\nconductance of a graph is\n$$\\min_{A\\subset V, \\text{vol}|A|\\le \\frac{1}{2}\\text{vol}|V|}\\frac{|\\partial A|}{\\text{vol}|A|},$$\nwhere $V$ is the vertex set, $\\partial A$ is size of the cut between\n$A$ and $V\\backslash A$, $\\text{vol}|A|$ is the sum of degrees of\nvertexes in $A$. The following theorem gives a lower bound on the\nconductance of chamber graph when dimension $d=2$.\n\\begin{theorem}\nThe chamber graph of $2$-dimensional hyperplane arrangement with size $n$ that is in the general position has conductance lower bounded by $\\frac{1}{2n}$.\n\\end{theorem}\n\\newcommand{\\text{vol}}{\\text{vol}}\n\\begin{proof}\n  For any set $A$ of vertices in the chamber graph with size no\n  greater than $\\frac{1}{2}|V|$, we will show that the conductance of $A$, \n  $\\frac{|\\partial A|}{\\text{vol}|A|}$, is lower bounded as follows\n  \\[\n    \\frac{|\\partial A|}{\\text{vol}|A|}\\ge \\frac{1}{2n}.\n  \\] \n  Let $X$ be the set with smallest volume satisfying\n  \\[ \n    X = \\arg \\min_{\\substack{A\\\\ \\text{vol}(A)\\le {\\frac12} \\text{vol}(V)}} \\frac{|\\partial A|}{\\text{vol}|A|}.\n  \\]\n  We first claim that $X$ must be connected. If not, we can write $X$\n  as the union of (maximally) connected components, \\ie\n  $X=\\bigcup_{i=1}^r X_i$, where $X_i$ are the maximally connected\n  components within $X$ (in particular, note that there are no edges\n  between distinct $X_i$). \n  Then, if $a_i=\\partial X_i$ and $b_i=\\text{vol}(X_i)$, then\n  \\[\n    \\frac{|\\partial X|}{\\text{vol}|X|} = \n    \\frac{a_1+a_2+\\cdots+a_r}{b_1+b_2+\\cdots+b_r}\\ge\n    \\min_{i\\in[r]}\\frac{a_i}{b_i},\n  \\]\n  implying that $X_i$ has lower conductance than $X$ and is smaller in\n  size than $X$, a contradiction.\n\n  Let $S$ be the boundary surface of the chambers corresponding to\n  vertexes in $X$. Since $X$ is connected, we must have $S$ to be\n  piece-wise line segments. \n\n  We now claim that $S$ will partition the chamber graph into two\n  connected components. Since $X$ is connected, we just have to show\n  that $V\\backslash X$ is also connected.\n\n  Suppose not, and let $V\\backslash X= \\bigcup_{i=1}^{m} Y_i$, where\n  $Y_i$ are maximally connected, and $V\\backslash X$ is the union of\n  $m$ different connected components.\n  Let $c_i=\\partial Y_i$ and $d_i=\\text{vol}(Y_i)$. Then we have\n  \\[\n    \\sum_{i=1}^m c_i=|\\partial X|,\n  \\]\n  and since $\\text{vol}(V\\backslash X)= \\text{vol}(V)-\\text{vol}(X)$ and\n  $\\text{vol}(X)\\le {\\frac12} \\text{vol}(V)$, we have\n  \\[\n    \\sum_{i=1}^m d_i\\ge {\\frac12} \\text{vol}(V)\\ge  \\text{vol}(X).\n  \\]\n  Therefore, there must be some component $i$ such that\n  \\[\n    \\frac{c_i}{d_i}\\le \\frac{\\sum c_i}{\\sum d_i} \\le \\frac{|\\partial\n      X|}{\\text{vol}|X|}.\n  \\] \n  If $Y_i$ satisfies $\\text{vol}(Y_i)\\le \\frac{1}{2}\\text{vol}|V|$, \n  then again we have a contradiction because of the following. If\n  $c_i\/d_i <  \\frac{|\\partial X|}{\\text{vol}|X|}$, we are done. \n  If   $c_i\/d_i = \\frac{|\\partial X|}{\\text{vol}|X|}$, it means\n  that every component in $V\\backslash X$ has conductance\n  $\\frac{|\\partial X|}{\\text{vol}|X|}$. But if there are more than\n  two components in $V\\backslash X$, then $X$ has a larger \n  cut $\\partial X$ than each of the components, and therefore\n  must have a larger volume as well, contradicting the assumption\n  on $X$.\n\n  If $\\text{vol}(Y_i)\\ge \\frac{1}{2}\\text{vol}|V|$, then consider the \n  set $Z= V \\backslash Y_i$. Note that\n  \\[\n    Z = X \\bigcup \\bigl( \\cup_{j\\ne i} Y_j \\bigr).\n  \\]\n  Now $|\\partial Z|=|\\partial Y_i| \\le |\\partial X|$. This follows\n  since there is no boundary between $Y_i$ and any of the other $Y_j$,\n  and the only boundary $Y_i$ has is with $X$. Furthermore,\n  $\\text{vol}(Z) > \\text{vol}(X)$, implying that $Z$ has lower conductance \n  than $X$, again a contradiction.\n\n\n  Now, we know that the boundary $S$ between the chambers in $X$ and\n  the rest of the hyperplane arrangement is exactly a piece-wise line\n  segment that separates $\\mathbb{R}^2$ into two connected\n  components. There are only 3 possibilities, as shown in Figure 1. We\n  now observe $\\text{vol}|X|$ is exactly the sum of the\n  $1$-dimensional faces in the arrangement that intersect with\n  $X$. Since there are at most $n$ lines in the arrangement, there\n  exist a line $P$ that intersect with $X$ (or $V\\backslash X$) by at\n  least $\\frac{\\text{vol}|X|}{n}$ many faces, see figure 1.  The\n  number of faces in $S$ is no less than the number of faces in $P$,\n  because any line that intersects with $P$ in $X$ must also intersect\n  with $S$, and at most two lines can intersect at the same point on\n  $S$ by our general position assumption. The theorem now follows.\n\\begin{figure}\n  \\centering \\vspace{-.5in}\n\\includegraphics[width=0.6\\textwidth]{part1.pdf}\n\\vspace{-1.5in}\n\\caption{Possibility of piece-wise linear partition}\n\\label{fig:slowmixing}\n\\end{figure}\\end{proof}\n\nFor the general dimension case, we have the following conjecture.\nSee Appendix for justification and partial proofs.\n\\begin{conj}\n  The conductance of any $d$-dimensional general position hyperplane\n  arrangement of size $n$ is lower bounded by\n$\\frac{1}{\\text{poly}(n,d)}$.\n\\end{conj}\n\n\\begin{remark}\n  Note that the requirement for general position of the hyperplanes is\n  necessary for fast mixing given by the Conjecture above. Else it is\n  easy to construct a hyperplane arrangement with mixing time lower\n  bounded by $O(\\frac{n^d}{2^d})$. As shown in Figure 2, the cut made\n  by the gray shaded top plane has only $4$ boundary chamber but the\n  total number of chambers below the plane is roughly $n^2$ (in two\n  dimensions, while in $d+1$ dimensions, we will have the cut and\n  volumne to be $2^{d+1}$ and $O(n^d)$ respectively).\n\\begin{figure}[!b]\n\\includegraphics[width=0.5\\textwidth]{hyperplane.pdf}\n\\vspace{-.7in}\n\\caption{Hyperplane arrangement with small conductance}\n\\label{fig:slowmixing}\n\\end{figure}\n\\end{remark}\n\n\\paragraph{Lazy Chamber graph}\nAlgorithm NRW on the regular chamber graph will not give an\n\\emph{exact} uniform sampling, but is off by a factor of $d\/n$ as\nmentioned above. This is easily fixed by adding dummy vertices and\ndummy edges to each vertex in the chamber graph raising the degree of\nevery vertex in the original chamber graph to $4n$. Call such a\ngraph to be \\emph{lazy chamber graph}.\n\nWe will call the vertex in the original chamber graph to be\n\\emph{chamber vertex} and the dummy vertices added to be\n\\emph{augmentation} vertices. The stationary probability of the new\nrandom walk, restricted on the chamber vertices, is exactly uniform.\nIf the Algorithm NRW on the chamber graph is fast mixing, we can\nshow that Algorithm NRW on the lazy chamber graph is also fast mixing:\n\n\\begin{lemma}\n  If the conductance of the chamber graph is $g$, the lazy chamber\n  graph has conductance $\\ge \\frac{g}{8n^2}$.\n\\end{lemma}\n\\begin{proof}\nWe only need to show that any subset $A$ of vertex in the lazy chamber\ngraph we have\n$\\frac{|\\partial A|}{|A|}\\ge \\frac{g}{8n}$.\nWe observe that if an augmentation vertex is in $A$, then the chamber\nvertex attached to it must also be included in $A$.  We denote\n$A'\\subset A$ to be the set of all chamber vertexes in $A$. \n\nThe vertexes in $A'$ can be partitioned into two classes,\n$A'=B'\\cup C'$ where $B'$ is the set of all chamber vertices that have\nall their attached augmentation vertices in $A$ and $C'$ is the\ncomplement of $B'$ in $A'$.\nSimilarly, $B$, $C$ to be the sets that contains\nalso the attached new vertex of $B'$ and $C'$ in $A$). We have\n$$\\frac{|\\partial{A'}|}{|A'|}=\\frac{|\\partial B'|+|\\partial C'|}{|B'|+|C'|}\\ge \\frac{|\\partial B'|}{|B'|},$$\nsince all vertexes in $C'$ are boundary vertexes. Note that\n$3n*|B'|\\le |A| \\le \\frac{4n*|V|}{2}$ since any vertex in $B'$ will\nattach at least $3n$ new vertex in order to make degree $4n$, we have\n$|B'|\\le \\frac{2}{3}|V|$. Now, by the definition of conductance we\nhave $\\frac{|\\partial B'|}{|B'|}\\ge g\/2$. This is because, if\n$|B'|\\le \\frac{|V|}{2}$ then $\\frac{|\\partial B'|}{|B'|}\\ge\ng$. Otherwise, we have\n$\\frac{1}{3}|V|\\le A'\\backslash B'|\\le \\frac{1}{2}|V|$, thus\n$|\\partial B'|\\ge g|A'\\backslash B'|\\ge \\frac{1}{3}g|V|$ and\n$|B'|\\le \\frac{2}{3}|V|$, we have\n$\\frac{|\\partial B'|}{|B'|}\\ge \\frac{g}{2}$. Therefore, we have\n$$\\frac{|\\partial A|}{|A|}\\ge \\frac{|\\partial B'|+|\\partial C|}{|B|+|C|}\\ge \\frac{|\\partial B'|}{|B|}\\ge \\frac{|\\partial B'|}{4n*|B'|}\\ge \\frac{g}{8n}.$$\nNow since $\\text{vol}|A|\\le n|A|$, the theorem follows.\n\\end{proof}\n\nCombining all the results, we have\n\\begin{theorem}\n  Assuming conjecture 1. For an given parameter $\\epsilon>0$ and $X$ in the general position,\n  Algorithm NRW run on the lazy chamber graph generated by $S_X$ can\n  generate labels from $S_X$ with distribution $\\epsilon$ close (in\n  variational distance) to uniform, and runs in time\n  $\\text{poly}(d,n,\\log(1\/\\epsilon))$.\n\\end{theorem}\n\\begin{proof}\nBy the relationship between mixing time and spectral gap \\cite[Theorem 2.2]{berestycki2016mixing}, we have\n$$t_{\\text{mix}}(\\epsilon)\\le \\frac{1}{g}\\log\\left(\\frac{1}{2\\epsilon n^d}\\right).$$\nThe theorem follows since the spectral gap is lower bounded by square\nof conductance by Cheeger's inequality \\cite[Theorem 13.14]{levin2017markov}.\n\\end{proof}\n\n\\subsection{Sampling for arbitrary neural networks}\n\\label{s:an}\nWe now consider the sampling for arbitrary neural networks. Let\n$X=\\{\\textbf{x}_1,\\cdots,\\textbf{x}_n\\}$ be the samples, we choose the\nweights of the network layer by layer. At layer $\\ell$ we use the\nprevious sampled weights in layers $1,\\cdots,\\ell-1$ to generate\noutputs $\\textbf{x}_1^{\\ell},\\cdots,\\textbf{x}_n^{\\ell}$, where\n$\\textbf{x}_i^{\\ell}$ is output of layer $\\ell-1$ with input\n$\\textbf{x}_i$, a binary vector. For each neuron in layer $\\ell$ we\n\\emph{independently} sample weights using Algorithm RS with input\n$\\{(1,\\textbf{x}_{1}^{\\ell}),\\cdots,(1,\\textbf{x}_{n}^{\\ell})\\}$.\n\nTo illustrate the idea more concretely, consider neural networks with\none hidden layer. Let $X$ to be the input samples of dimension $d$,\nfor each neuron in the hidden layer, we use Algorithm RS to generate\nthe weights \\emph{independently}. We now fix the weights we sampled\nfor the neuron in the hidden layer and view the function that\nexpressed by the hidden layer to be some function\n$h:=\\mathbb{R}^d\\rightarrow \\{0,1\\}^{u_2}$, where $u_2$ is the number\nof neurons in the hidden layer. We now define\n$\\textbf{x}_i'=h(\\textbf{x}_i)$ to be the new input sample for the\noutput layer, and again use Algorithm RS to sample the weights for the\noutput neuron with input $X'$.\n\\begin{theorem}\n  For a neural network with fixed architecture, $k$ neurons and $W$\n  parameters, the above sampling procedure runs in $O(nW^3)$\n  time. Given a sample $X$, each labeling in $S_X$ produced by this\n  architecture appears with probability at least\n$\\left(\\frac{W}{2enk}\\right)^W.$\n\\end{theorem}\n\\begin{proof}\nWe use induction on the layers. For any given labeling produced by\nweights $\\textbf{w}$, let $p(\\ell)$ to be the probability that the\noutput of layer $\\ell$ is consistent with the output on weight\n$\\textbf{w}$. We have\n$$p(\\ell)\\ge p(\\ell-1)\\prod_{i=1}^{u_{\\ell}}\\left(\\frac{d_i}{2e n}\\right)^{d_i},$$\nwhere $d_i$ is the input dimension of the $i$th neuron in layer\n$\\ell$, and the product term comes from Theorem 2 and\nindependence. Note that the rank of the outputs $X^{\\ell}$ may reduced\nafter passing the previous layers, however, this will only make the\nprobability larger than $\\left(\\frac{d_i}{2e n}\\right)^{d_i}$ by\nTheorem 2.  Now, the theorem follows with the same argument as\nin~\\cite[Theorem 6.1]{anthony2009neural} for bounding VC dimension of\nlinear threshold neural networks.\n\\end{proof}\n\n\\ignore{\\section{Simulations}\nWe run our recursive algorithm for randomly choosing samples with different dimension $d\\in \\{3,4,5,6,7,8,9,10\\}$ and size $n\\in\\{10,11,12,13,14,15\\}$, for each pair of $d,n$ we run the sampling procedure $30000$ times to count the empirical distribution on the different labeling. We then compute the ratio of the maximum and minimal probability that appears in the balling that we sampled, and rounding the ratio to be integers. One can see from figure 2 that, for each  sample size $n$ there is a peak for the probability ratio when the dimension of sample increases. For given dimension $d$, one can see that for $d$ is small the ratio will increase according the increasing of $n$, for $d$ is large the ratio will decrease when the $n$ is increasing. We also runs our sampling procedure on MNIST data set with $1000$ sample, the run time is around $5$ mins.\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=.6\\textwidth]{ratio.pdf}\n\\caption{Ratio of maximum and minimal empirical probability}\n\\label{fig:slowmixing}\n\\end{figure}}\n\n\n\n\n\\medskip\n\n\\small\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Great White Hope}\n\\label{sec:feynman}\n\n\nThe origins of John Wheeler's long and winding path to general relativity are hard to trace; all we have to go by are Wheeler's later recollections of the pipe dreams of a physicist in his mid-20s, or, as Wheeler liked to put it, his ``great white hopes.'' Of these there were two, which Wheeler recounted in two interviews, conducted by Charles Weiner and Gloria Lubkin (WL) in 1967 and by Finn Aaserud (FA) in 1988 (Session I),\\footnote{\\url{https:\/\/www.aip.org\/history-programs\/niels-bohr-library\/oral-histories\/5063-1}} respectively.\n\nThe first great white hope was electrons. After the discovery of the positron in 1932, the theory of electrons and positrons based on the Dirac equation, ``pair theory'', had been worked out by Dirac, Heisenberg, Oppenheimer and Furry. Wheeler felt that pair theory offered ``mechanisms for binding electrons in very small regions of space that never got a thorough discussion'' (WL), and that electrons might well be present in the nucleus after all (an assumption that had been dropped after the discovery of the neutron); in fact that electrons and positrons may form its fundamental constituents.\n\nThe second great white hope was scattering, which was to be viewed as the fundamental process from which all other characteristics of (primarily nuclear) interactions were to be derived. Both great white hopes were (at least in hindsight) also imbued with snappy and parallel slogans \\citep{wheeler_1989_the-young}, ``Everything as Electrons'' and ``Everything as Scattering,'' and even if these precise titles are not actually contemporary, they do show an essential characteristic of Wheeler's thinking: An extreme reductionism, a reduction to simple, catchy thoughts and a very small number of fundamental building blocks, a radical Ockhamism if you will. In particular, we see here what Wheeler would later call ``daring conservatism'': Taking (an element of) a well-established theory, and trying to use it beyond its usual domain of applicability, i.e., electrons in the nucleus or scattering theory to describe stationary states. But this unique approach did not show at the time. Feynman would later remark (perhaps apocryphally) that: \n\n\\begin{quote}\nSome people think Wheeler's gotten crazy in his later years, but he's always been crazy.\\footnote{Cited in \\citep{overbye_2002_peering} and in \\citep{wong_2008_remembering}. In the former, Kip Thorne is given as the reference for Feynman's supposed remark.}\n\\end{quote}\n\nWhile the later Wheeler would happily have publicized catchy slogans for crazy ideas, such as ``everything as electrons,'' there are no outward indications of his grand vision at the time. As he would remark in the interview with Weiner and Lubkin:\n\n\\begin{quote}\nNobody was as crazy as I was, to think that you could explain everything in terms of electrons. And this I think illustrates a weakness of my approach at that time, to have this secret hope nursed internally and talk about it occasionally with close friends but not feeling particularly at ease about bringing it out on a public platform...\n\\end{quote}\n\nSimilarly, a talk \\citep{wheeler_1934_interaction} he gave on (alpha particle) scattering at the APS meeting in Washington, DC, gave no indication (at least from the extant abstract) of the central role he was envisioning for scattering in fundamental physics, nor did his central paper on the subject, in which he famously introduced the S-Matrix \\citep{wheeler_1937_on-the-mathematical}.\n\nIt would take the meeting with a fellow eccentric to tickle Wheeler into presenting hints of his crazy ideas on ``everything as electrons.''\\footnote{We shall have no more to say about ``Everything as Scattering,'' which was here mentioned only to illustrate the early traces of daring conservatism in Wheeler's thinking. This notion was very influential in Feynman's later diagrammatic formulation of renormalized QED, which did effectively become a pure scattering theory, see \\citep{blum_2017_the-state}.} We refer, of course, to his PhD student, Richard Feynman, with whom he worked out what would later be known as Wheeler-Feynman electrodynamics \\citep{wheeler_1945_interaction}. In this theory, the field-mediated electromagnetic interaction of Maxwell's theory is replaced by a direct interaction at a (spatial) distance between charged point-like particles. This interaction is not instantaneous (in time), but is rather the sum of a retarded and an advanced component, corresponding to the two possible solutions of Maxwell's field equations and ensuring compliance with the special theory of relativity. When a given charged point particle (electron) exerts an advanced force on the other electrons in the universe, it will experience a retarded back-reaction, which will in fact be instantaneous. Imposing the ``absorber boundary condition'' (which in the corresponding field theory would imply that there are enough electrons in the universe to absorb all outgoing radiation, so that there is no radiation ``escaping to infinity'') then ensures that this instantaneous back-reaction is equal to the radiation reaction of the usual field theory. This then not only implies empirical equivalence with field theory, but also eliminates the possible difficulties with causality an advanced interaction might otherwise suggest.\n\nWheeler and Feynman give conflicting stories concerning the origins of their joint work on action-at-a-distance electrodynamics. Feynman in his 1965 Nobel lecture relates how, as an undergraduate, he had hit upon the idea of replacing the electromagnetic field with action-at-a-distance in order to eliminate the divergences of quantum electrodynamics (QED), how he had then learned as a graduate student that one cannot explain radiation reaction in this way, how he had tried to get radiation reaction into his action-at-a-distance framework as the backreaction of electrons, and how he had then presented this idea to Wheeler. Upon which Wheeler ``then went on to give a lecture as though he had worked this all out before and was completely prepared, but he had not, he worked it out as he went along,'' a lecture that ended in the conclusion that one would have to take into account advanced solutions of Maxwell's equations, in order to get an immediate radiation reaction. From this conversation then grew, as Feynman recalls it, their joint work on action-at-a-distance electrodynamics.\n\nWheeler's version of events is considerably younger, first published only a year after Feynman's death \\citep{wheeler_1989_the-young}. He recalls how Feynman had expressed some interest in Wheeler's idea of everything as electrons, how Wheeler had ``animated by the concept of everything as electrons' worked out one Sunday (up to a factor of 2) how one might get radiation reaction as a backreaction from the absorber even in a theory without fields, and how he had then presented his caculation to Feynman, who was able to sort out the missing factor of 2. But priority issues are not our concern here. In fact, in this case not only is the issue undecidable, the two stories are not even entirely incompatible: If we assume that both Feynman and Wheeler omitted substantial parts of the story, the two accounts might actually be merged together to form a coherent narrative.\n\nFor our story, another aspect evidenced by the two different accounts is much more important: It clearly shows that Wheeler and Feynman came to action-at-a-distance electrodynamics with very different motivations. For Feynman it was the divergence difficulties of QED, an issue which had somewhat dropped out of fashion in the late 1930s due to the great interest in meson physics, but an issue that would soon resurface in the late 1940s and ultimately earn Feynman the Nobel Prize. For Wheeler, it was his great white hope, his ``everything as electrons.''\n\nAccording to Feynman's recollections, it was he who got to write the first draft of their joint paper:\n\n\\begin{quote}\n[Wheeler] asked me to write the paper - I wrote this thing up in 27 pages, which we could have sent to a journal, but he began to think, ``No, it's too great a business, we'll write it good.'' And that of course made delays, and got interrupted with the war, and he got it so big that it was five parts - the whole reorientation of physics from a different point of view. I never went along with him on that. I mean, you know, with the idea that it's so marvelous, it's a reorientation of physics, you have to write five papers, and all of physics is turned upside down. But I felt that 27 pages were what it deserved. This was written mostly by him. See, it was a rewrite of the 27 pages, so to speak. I wouldn't say a rewrite because he didn't use the 27 pages as a basis, but the same ideas are developed, which I tried to write much more briefly, and which he tried to write in an historical context, about the arguments of Tetrode and Einstein - you see, it's a relatively long thing, and I didn't really write it, you understand.\n\\end{quote}\n\nThe manuscript that Feynman is referring to is probably identical with an untitled manuscript from 1941 in the Feynman Papers at Caltech (Box 6, Folder 1).\\footnote{The number of pages that this manuscript has depends on how exactly one counts, e.g., if one includes handwritten inserts, typed pages containing just one extraneous paragraph, and figure captions. A case can certainly be made for 27.} To assess the difference in tone, compare the first sentence of Feynman's draft:\n\n\\begin{quote}\nThe attempts to develop a satisfactory scheme of quantum electrodynamics have met with several difficulties, some of which are found not to be a result of the process of quantization, but to be contained in the classical electron theory itself.\n\\end{quote}\n\nwith the opening of the final, printed version:\n\n\\begin{quote}\nIt was the 19th of March in 1845 when Gauss described the conception of an action at a distance propagated with a finite velocity, the natural generalization to electrodynamics of the view of force so fruitfully applied by Newton and his followers. In the century between then and now what obstacle has discouraged the general use of this conception in the study of nature?\n\\end{quote}\n\nWheeler's grand vision clearly came out in the historical pathos. But as far as the motivation provided for reformulating classical electron theory in terms of action-at-a-distance, Wheeler fully adopted Feynman's motivation of solving the difficulties of QED. There is no hint of Wheeler's great white hope and so we need to ask the question, what did Wheeler mean when decades later he referred to his work as being ``animated by the concept of `everything as electrons'? Now, one might simply take this to mean that he was looking for a purely particulate description of nature, only electrons, no fields, and that action-at-a-distance was the implementation of this program. There is certainly something to this, and in his later recollections Wheeler certainly emphasized the ``everything as particles'' aspect of his early period, contrasting it with his later pure field approach. \n\nBut there is another aspect that appears to have been equally important, if not more so. Wheeler's program was specifically focused on electrons, and especially on their continuing role in the nucleus. It was thus not simply concerned with the electrons as generic particles, but with electrons as a very specific kind of particles, distinct from nucleons or mesons. Now, he was well aware that there were significant indications that electromagnetically interacting electrons were not the primary constituents of the nucleus, or even present there at all: The short range of the nuclear interactions and the huge kinetic energies obtained by electrons thus confined were the indications most typically cited. It seems, now, that Wheeler hoped that the action-at-a-distance formulation might help to put aside these objections to ``everything as electrons,'' as he also later stated in his autobiography \\citep[p. 164-165]{wheeler_1998_geons}:\n\n\\begin{quote}\nI had another motivation as well for pursuing action at a distance, for I clung to my hope that all of the matter in the world could be reduced to electrons and positrons. Yet I knew that if an electron and a positron were to be crowded together in subnuclear dimensions, some way would have to be found to get around the prediction of conventional theory that they would quickly radiate away their energy in the form of electromagnetic fields. Perhaps, I thought, an action-at-a-distance version of electromagnetic theory - one without fields - might explain the suppression of such radiation and permit the particles to live happily in such a confined space.\n\\end{quote}\n\nThere are only vague hints at this in the published paper with Feynman, which contains a section on ``Advanced effects associated with incomplete absorption,'' which discusses the physics of electrons in an incompletely absorbing cavity. This may be interpreted as Wheeler thinking about electrons in the nucleus, but this possibility is not mentioned explicitly. Stronger hints can be found in a lecture he gave, several months after the publication of the first Wheeler-Feynman paper, at a symposium of the American Philosophical Society on ``Atomic Energy and Its Implications.`` This symposium, conducted in November 1945, only three months after the use of atomic bombs on Hiroshima and Nagasaki, was a rather serious affair ``devoted to the atomic bomb'' (as per the opening words of Henry DeWolf Smyth), featuring talks by J. Robert Oppenheimer on ``Atomic Weapons'', Joseph H. Willits on ``Social Adjustments to Atomic Energy'' and Irving Langmuir on ``World Control of Atomic Energy'' (Proceedings of the American Philosophical Society, Vol. 90, No. 1, January 1946). Wheeler recalled that he ``didn't really want to talk about atomic energy'' but rather ``about what lay beyond it.'' \\citep[p. 168]{wheeler_1998_geons} So, inspired by his collaboration with Feynman and eager to do fundamental physics after the long war work, he chose this unusual venue to provide first hints on his grander schemes in a talk entitled ``Problems and Prospects in Elementary Particle Research''.\n\nThe hints towards his hope that action-at-a-distance electrodynamics might revolutionize nuclear physics are still very vague, merely by association \\citep[p. 45-46]{wheeler_1946_problems}:\n\n\\begin{quote}\n[T]he theory of [nuclear] mesons, may be said to be at present in a state of free experimentation with ideas and great uncertainty as to principle, both because of the incompleteness of our present experimental picture and because of difficulty in tying the proposed hypotheses to already existing theories. [...] The difficulties of inventing a new theory on the basis of  incomplete experimental evidence suggest that one possibility acceptable at this time is the conservative one of extending the range of applicability of already existing and well-established theories.\\\\\nThe second theory whose problems we consider is therefore the formalism of electron-positron pairs.\n\\end{quote}\n\nWe see here the notion of daring conservatism (without that name yet) appear in print for the first time, but as to how it is to be applied, there is nothing to go by but a lone and easily overlooked ``therefore.'' Wheeler would focus on cosmic ray physics in the following years, setting aside once again his great white hope (with the exception of some work with his student Gilbert Plass on counting degrees of freedom and thus doing thermodynamics - Planck's law - in an action-at-a-distance theory). But in June 1946, he submitted an application to the Guggenheim Foundation to pursue a project on an ``Analysis of the Problem of Measurement in Electron Theory.'' Through this scholarship, he hoped to obtain a leave of absence from Princeton to work on his foundational ideas. And he could think of no one better suited to help him in this endeavor than his old mentor Niels Bohr. As he wrote in his application:\n\n\\begin{quote}\nOnly a start could be made in the [...] program by the applicant himself and in collaboration with his student, R.P. Feynman; the war stopped further work. [...]\nThe primary reason for this proposal is the feeling that the stage has now come in the theoretical work where new concepts and points of view are essential. To develop and test such points of view it appears that by far the most effective course is to take up a close association with Niels Bohr. The writer's association with this scientist has convinced him that if there is hope of making advancement in the fundamental problems outlined above, this hope is best justified by Bohr's ability to see into the future, his courage and judgment in considering and testing new concepts. [...] [T]he Applicant therefore is proposing to go to work with Professor Bohr in Denmark.\n\\end{quote}\n\nEven in this proposal, Wheeler was still rather guarded about what he wanted to do exactly. The proposed title recalled Bohr and Rosenfeld's paper on the Problem of Measurement in QED \\citep{bohr_1933_zur-frage}, which to Wheeler was the model for an incisive theoretical analysis of current theory \\citep{hartz_2015_uses}, and which he referred to as ``a classic paper'' in his application. And when listing the tasks for his project, Wheeler remained excessively general:\n\n\\begin{quote}\n(1) The consistent formulation of the mathematical formalism of the theory of electrons and positrons.\\\\\n(2) A definition in terms of idealized experiments of the possibilities of measurement in the theory of the electrons and positrons.\\\\\n(3) The interpretation of these idealized experiments in terms of the formalism - and of the formalism in terms of the idealized experiments.\n\\end{quote}\n\nConsequently, Eugene Wigner, who acted as one of the references for Wheeler's application, was also rather non-committal about his views on Wheeler's project (received 24 June 1946):\n\n\\begin{quote}\nAs to his proposed study, I must admit that I find it quite impossible to make any predictions. [...] I know from personal contacts that Professor Wheeler is most deeply interested in the project which he outlined.\n\\end{quote}\n\nIn any case, while Wheeler's application was granted only five days after it was received (and five days before even receiving Wigner's letter of reference), Wheeler could not use the grant as soon as he had planned. As he wrote to Henry Allan Moe of the Guggenheim Foundation on 2 July 1946:\n\n\\begin{quote}\nI have now learned on my return to Princeton that there is a distinct possibility that the six-month leave of absence granted by the University to Professor Wigner may have to be extended through the second term of 1946-47 to allow him to accomplish most effectively his task to give a new direction to the work of the Oak Ridge Laboratory. I am afraid after examination of the situation with the Department here that it would create embarrassing difficulties for the work now underway to have me gone for the whole of the Spring term. However, it appears that I may be able to count on a period May 15 - October 1, 1947. Even this range of time can, however, not now be made entirely definite. [...] I am sorry to have to report to you that the situation is in this uncertain state. I should like to have your advice as to how I can best take into account these difficulties in a manner acceptable to the Foundation.\n\\end{quote}\n\nThere was no problem. Moe replied on 5 July 1946:\n\n\\begin{quote}\nIt is a pleasure to welcome you to the company of Guggenheim Fellows -- the distinguished company as I think.\\\\\nAs to the date of starting your Fellowship, there's no need to settle that now. [...] You may count on us to ``play ball.''\n\\end{quote}\n\nAfter some back and forth, Wheeler finally received his scholarship for the period from 1 July 1949 to 30 June 1950 (i.e., also for an extended time period). For family reasons \\citep[p. 183]{wheeler_1998_geons}, he also opted to make Paris his home during his stay in Europe, rather than Copenhagen. And even though he did visit Copenhagen on several occasions during his time in Europe, much of his work with Bohr was focused not on electron-positron theory, but rather on completing a paper with David Hill on the collective model of the nucleus, which Bohr ended up not co-signing (see Acknowledgments in \\citep{hill_1953_nuclear}). \n\nOne reason for this appears to have been a major advance that occurred in the years from 1946 to 1949: The problem of the electromagnetic self-energy of the electron had been solved, among others by Feynman, without the need to eliminate the self-interaction of the electron. Wheeler had always presented the elimination of the self-interaction as the prime motivation for action-at-a-distance. Consequently, when writing an updated version of his 1945 talk in 1949, he duly acknowledged the recent advances in quantum electrodynamics and used a much more cautious language when talking about Wheeler-Feynman electrodynamics. In 1945, he had concluded his brief elaborations on Wheeler-Feynman electrodynamics with the words:\n\n\\begin{quote}\nIt is too soon to say whether the translation of the revised classical theory into quantum mechanics will remove the outstanding divergences. To test this point is an important problem for the future.\n\\end{quote}\n\nIn 1949 he wrote:\n\n\\begin{quote}\nNaturally the ultimate complete equivalence of this approach to the usual field theoretical treatment makes it clear that nothing new can result so long as this equivalence is strictly maintained. What may come out by changes and reinterpretations of the existing theory of action at a distance is uncertain \\citep{wheeler_1949_elementary}.\n\\end{quote}\n\nThat he did not yet give up action at a distance entirely at this time (as Feynman did, see \\citep{blum_2017_the-state}) is clearly due to his great white hope of constructing the atomic nucleus from electrons, a conjecture that was largely unaffected by the advances in QED. Indeed, we have one bit of evidence that Wheeler did in fact discuss this idea with Bohr during his stay in Europe, a letter sent to Bohr from Paris on 21 January 1950,\\footnote{Archives for the History of Quantum Physics, Bohr Scientific Correspondence, Microfilm 34.} announcing his arrival in Copenhagen for the 27th. This letter is primarily concerned with their joint work on the collective nucleus, but then, on page two, Wheeler brings in ``everything as electrons'':\n\n\\begin{quote}\nThe other problem, about which I am very anxious to get your opinion, is the question: is it possible to exclude a picture of elementary particle constitution based entirely on positive and negative electrons?\n\\end{quote}\n\nWheeler discussed his hope how half-retarded, half-advanced action-at-a-distance might solve the problems usually brought forth:\n\n\\begin{quote}\n(1) \\emph{Localizability}. To localize an electron in a distance of $\\approx e^2\/mc^2$ it is sufficient to have a potential well of radius $(e^2\/mc^2)$ and depth $(137)^2 mc^2$ [...]\\\\\n(4) \\emph{Source of potential}. Acceleration and velocity of an electron bound in such a potential are so great that electrostatic forces are negligible in comparison with radiative forces. Radiational transfer of energy to the outer world is itself negligible owing to the symmetry of the charge-current distribution. The radiational forces within the system can therefore be considered in a good approximation as half-advanced, half-retarded [...]. [I]t does not seem impossible to suppose that the electronic system generates a self-consistent potential, in which however the correlations between movements of interacting electrons must have an altogether dominating importance, in contrast to the atomic case.\n\\end{quote}\n\nAll this was very qualitative. And even if Wheeler's hopes concerning the nuclear binding potential could be realized, he was aware of the fact that there was another difficulty, which he had only very vague ideas on how to address: That of spin and statistics. If a neutron, e.g., was really to be considered as consisting of electrons and positrons only, it would have to contain an equal number of electrons and positrons to ensure its neutrality, and thus would have to be an integer-spin boson. Wheeler was clearly interested in Bohr's judgment as to whether the project was worth pursuing despite this apparently insurmountable difficulty:\n\n\\begin{quote}\nThough I joke with you about my heresies, I am trying to be just as honest and as open as I can about the elementary particle problem. I know that there is no one who has your insight. So it will be a great privilege to talk with you over these and other problems...\n\\end{quote}\n\nAs this letter was discussed during Wheeler's visit to Copenhagen, we have no evidence as to Bohr's take on the matter. Soon after his visit to Copenhagen, Wheeler cut his sabbatical short to work on the hydrogen bomb. When Wheeler returned to foundational research after the hydrogen bomb interlude, his focus was now on a different, but related problem: action-at-a-distance gravity.\n\n\\section{Action-at-a-distance Gravity}\n\\label{sec:AAD}\n\nThe idea of action-at-a-distance gravity can be traced back to 1941, when Wheeler and Feynman had worked out the basics of their absorber theory and Feynman first presented their joint work at a seminar in Princeton. Feynman later recalled that Einstein had remarked \\citep[p. 80]{feynman_1985_surely}:\n\n\\begin{quote}\nI find [...] that it would be very difficult to make a corresponding theory for gravitational interaction...\n\\end{quote}\n\nWheeler appears to have been immediately intrigued by this challenge. Although soon mainly occupied by war work, he wrote a letter to Einstein on 3 November 1943 (Einstein Papers), requesting a meeting with Einstein to discuss ``where the force of gravitation fits into the point of view'' of action-at-a-distance theory. The meeting took place on 14 November in Wilmington, Delaware  (Letter from Einstein to Wheeler, 6 November 1943, Einstein Papers) and we can surmise some of its content from a letter which Wheeler wrote to Einstein after the meeting, on 2 December. Wheeler had developed a general framework for action-at-a-distance theories, which he referred to as ``the theory of world lines.'' In this framework not only was there no more talk of fields, there was even ``no reference to the concept of a space-time continuum.'' All that was left were the world lines of individual particles $a$, $b$, etc., the points on which were identified by  some parameter, $\\alpha$, $\\beta$, etc. The exact parameterization of the world lines was to be considered arbitrary, and all statements on physics were supposed to be independent of it.\n\nThe dynamics were now determined by functions that connect points on two worldlines. In his letter to Einstein, Wheeler referred to these functions as light cones, which most clearly reveals their physical interpretation. In his later notes, he would mainly use the term liaison. And when he finally wrote a paper on AAD gravity some 50 years later, together with Daniel Wesley, he called them associators \\citep{wesley_2003_towards-an-action-at-a-distance}.\\footnote{One reason for Wheeler's return to action-at-a-distance gravity after so many years appears to have been that it had by then become abundantly clear that quantum gravity would suffer from similar divergence difficulties as QED, difficulties that in electrodynamics the Wheeler-Feynman absorber theory had aimed to solve. Since these difficulties could not be solved by renormalization in the case of gravity, it appeared attractive to revisit the old AAD formulation. This was pointed out to us Daniel Wesley, who co-authored the AAD gravity paper with Wheeler as an undergraduate student, in an email to one of the authors (AB), 11 April 2019.} We will be referring to them as liaisons throughout this paper, as this is the term that Wheeler used for most of the period under study.\\footnote{We will also be using a notation that Wheeler introduced only after his letter to Einstein. This is additionally motivated by the fact that reproducing Wheeler's notation in the Einstein letter presents somewhat of a typesetting challenge.}\n\nThe liaison function $\\alpha_+ (\\beta) $ then returns the point $\\alpha$ that is on the forward lightcone of $\\beta$.\\footnote{\\label{fn:conjugate} In general, the forward lightcone might intersect the other world line more than once. The liaison was supposed to be a single-valued function that singles out one of those points and thus constituted an object somewhat more restricted than a lightcone.} Similarly, $\\beta_+ (\\alpha) $ returns the point $\\beta$ that is on the forward light cone of $\\alpha$. Wheeler also introduced liaisons $\\alpha_-$, $\\beta_-$ for the backward lightcone, which are the inverses of $\\beta_+$ and $\\alpha_+$ respectively. Wheeler now proposed to Einstein that one could construct physics from these functions by setting up non-trivial, parameterization-independent relations. He pitched the expression\n \n \\begin{equation}\n \\alpha' = \\alpha_- (\\gamma_+ (\\beta_+ (\\alpha))),\n \\end{equation}\n \nwhere $\\alpha'$ is the  point on world line $a$ that one reaches by moving along the forward light cone from a point $\\alpha$ to world line $b$, then further on the forward light cone to world line $c$ and then on the backward light cone back to world line $a$. As can easily be seen by imagining all three particles as being at rest in the (not yet constructed) three-dimensional background space, if $\\alpha'=\\alpha$ the three particles are on one line in three-space. If $\\alpha' \\neq \\alpha$ they are not. One could thus, merely from three one-dimensional world lines and the liaisons between them, distinguish between a line and a triangle through the (indeed parameterization-independent) statement that $\\alpha$ and $\\alpha'$ are equal or unequal, respectively. Wheeler further conjectured:\n \n \\begin{quote}\n With a number of particles greater than three, one can build up more complex geometrical concepts.\n \\end{quote}\n \nApparently, Wheeler had hoped that in this manner one would be able to construct a theory that would be fully equivalent to general relativity, just as Wheeler-Feynman absorber theory was equivalent to Maxwell. In their discussion, Wheeler and Einstein appear to have focused on a surprisingly specific difficulty with such an equivalence: A universe with two particles. In the world line theory, Wheeler asserted, ``no physics at all is possible'' in such a setup, because the only expression one can study, namely $\\alpha_- (\\beta_+(\\alpha))$, is trivially the identity, since $\\alpha_-$ is \\emph{defined} to be the inverse of $\\beta_+$.\\footnote{In general, it is of course possible to first move along the forward lightcone from worldline $a$ to worldline $b$ and then back, along the backward lightcone, to worldline $a$ and arrive at a point different from the point one started out from, when conjugate points are involved. In the liaison framework this issue is avoided: As mentioned in footnote \\ref{fn:conjugate}, the liaison has to be a single-valued function and thus it was natural on many levels for Wheeler to simply define the backward liaisons as the inverses of the forward ones.} In general relativity, on the other hand, there were, as Einstein pointed out, two-body solutions with a rich, non-trivial four-dimensional geometry.\\footnote{The two-body solutions by Weyl and Levi-Civit\\`a that Einstein was referring to are discussed in detail in an editorial footnote of the Collected Papers of Albert Einstein \\citep[p. 437]{buchwald_2018_the-collected}.} There was thus, as Wheeler remarked, ``an apparent discrepancy between the general theory of relativity and the general theory of action at a distance.''\n\nIn his letter to Einstein, Wheeler began to develop an understanding, which he would further develop over the course of the following years, that his world line theory was in fact right in implying that there should be ``no physics'' in a two-body universe. Wheeler would present first hints of his search for an AAD formulation of gravity in several talks given in the first postwar years, such as in his programmatic 1945 talk at the American Philosophical Society, already mentioned earlier, where he remarked:\n\n\\begin{quote}\nJust as the proper recognition of [...] atomicity requires in the electromagnetic theory a modification in the use of the field concept equivalent to the introduction of the concept of action at a distance, so it would appear that in the gravitational theory we should be able in principle to dispense with the concepts of space and time and take as the basis of our description of nature the elementary concepts of world line and light cone.\n\\end{quote}\n\nBut his stance on the two-body problem was only spelled out in the revised 1949 version of the talk. In defense of world line theory, Wheeler had adopted a relationalist view of space and time, in the tradition of Ernst Mach who had formulated such ideas in the late 19th century in opposition to the Newtonian notions of absolute space and time. Mach's ideas, in the shape of the more or less formalized ``Mach's Principle'' of the relativity of inertia, had played an important role in Einstein's application of general relativity to cosmology \\citep{smeenk_2014_einsteins}. But the existence of two-body solutions, Wheeler now argued, called into question the validity of Mach's principle in general relativity. In a strictly relationalist theory, an idealized two-body problem would be non-dynamical: There was only one length scale to relate things to, the distance between the two point masses, and hence that distance should not be representable as changing, since there was nothing that its change could be measured against. Any statement that general relativity claimed to be able to make about the time evolution of the distance between the two bodies was thus empty.\n\nWhile the centrality of the two-body problem was not to persist (already in the 1949 lecture, Wheeler voiced his doubts whether the paradox would necessarily be resolvable by action-at-a-distance gravity), the years 1943-49 see gravitation moving to the center of Wheeler's attention, with new interesting paths of inquiry popping up, such as the unresolved problem of gravitational radiation reaction. While his talks of the period, as mentioned earlier, contained somewhat defeatist language concerning the prospect of Wheeler-Feynman electrodynamics, this is not the case for action-at-a-distance gravity. As his hopes for explaining the nuclear forces through electrodynamics were waning, gravity was increasingly presenting itself as a worthwhile field of study. Wheeler was beginning to realize the untapped potential of general relativity, though it must be admitted that he does not yet appear to have had a definite program for using this potential. The most specific part of his new interest in gravitational theory was the focus on the role of Mach's principle (beyond the particular case of the two-body universe) in establishing the relation between the world line theory and general relativity. This became, as we shall see, a central theme in Wheeler's thinking about gravitation.\n\nWheeler appears to have worked on AAD gravity quite a bit during his 1949\/50 stay in Europe. In a letter to Gregory Breit of 28 December 1949 (Gregory Breit Papers, Yale University Archives), he wrote:\n\n\\begin{quote}\nI am working quietly, sometimes on the reconciliation of the individual particle model of the nucleus and the liquid drop model [i.e., collective models of the nucleus], sometimes constructing a description of nature which makes no use of the concepts of space and time (analogue in gravitation theory of electromagnetic action-at-a-distance).\n\\end{quote}\n\nSimilarly in a letter to Feynman of 10 November 1949 (Feynman Papers, Caltech). And also in the letter to Bohr of 21 January 1950, already cited earlier, Wheeler talked about action-at-a-distance gravity, explicitly connecting it with Machian ideas of making ``force dependent upon the number of particles in the universe'' and mentioning another letter (not extant) to Wilhelm Magnus, mathematician at the Courant Institute, who had provided Wheeler ``with some information about one of the group theoretical aspects of the problem.'' In later recollections, Wheeler even misremembered that he had proposed ``doing similar ideas [to Wheeler-Feynman] for gravitation theory'' (FA) already in his application to the Guggenheim Foundation in 1946. But at the time, his main focus was still on electromagnetism. With the success of renormalized QED, electrodynamics was on its way out for Wheeler, and Bohr appears to have disabused him of the last vestiges of ``everything as electrons.'' But gravitation, in Wheeler's mind, was up and coming! In April 1951, he returned to Princeton from Los Alamos, and while still chiefly concerned with the work on the hydrogen bomb \\citep[p. 218]{wheeler_1998_geons}, he did find some time to ponder these foundational questions. In a notebook entitled ``Action at a distance I'', we find an entry dated 10 November 1951, in which Wheeler considers two different pathways to AAD gravity:\n\n\\begin{quote}\nCan thus work towards desired [action] principle from either one of two directions---(1) math. convenience + naturalness; (2) correspondence.\n\\end{quote}\n\nAnd at this point the second pathway, establishing the theory through correspondence with the field theory of general relativity, clearly seemed the less favored, especially since the correspondence could not be exact for small numbers of particles, as he had established in his discussions with Einstein several years earlier. A few lines above the remark just quoted, Wheeler had written\n\n\\begin{quote}\nBut should satisfy the principle of correspondence to ordinary general relativity in the limit of infinitely many infinitely small masses (continuous mass distribution).\n\\end{quote}\n\nonly to then qualify this remark by a ``probably'' inserted after ``should.'' So how did Wheeler's pursuit of AAD gravity along the lines of mathematical convenience and naturalness look? Still using the liaisons as his central dynamical variables, Wheeler's idea was now to set up an action principle (similar to the Fokker action in Wheeler-Feynman electrodynamics), written as an integral over world line parameters, the integrand being some function of the liaisons. When varied, this action would return differential equations for determining the liaison functions. As to how this action should look, Wheeler thought that it might involve counting closed cycles of liaisons, as this could provide a notion of local world line density without having to invoke an underlying space-time. \n\nWhile these general ideas appear to have been present already in late 1951, Wheeler did not elaborate on it any further in his AAD notebook for quite a while. The next entry dealing with liaison theory dates from 17 March 1953. It would appear that Wheeler's purely mathematical approach turned out to be inadequate. For in 1952, he switched gears and started to engage general relativity head-on.\n\n\\section{Teaching Relativity}\n\\label{sec:course}\n\nEmbarking on his study of general relativity and the corresponding action-at-a-distance formulation, Wheeler asked to teach a course on relativity at Princeton in the academic year 1952\/53. His request was granted on 6 May 1952, the day on which Wheeler began his first in a long series of notebooks on Relativity.\\footnote{We will be citing frequently from Wheeler's first two relativity notebooks, which we will be abbreviating as WR1 and WR2, respectively. These notebooks are to be found in the John Wheeler Papers, held at the American Philosophical Society in Philadelphia, Section V, Volumes 39 and 40. } It is somewhat surprising that Wheeler was the first to teach a dedicated relativity course at the Princeton physics department. After all, Wheeler himself had clearly profited from the fact that Princeton was  \\emph{the} center for relativity in the US at the time. We have already mentioned his personal discussions with Einstein. And in his first writing on relativity, the 1945 talk at the American Philosophical Society, every author cited in the section on gravitation, aside from Ernst Mach, worked in Princeton. However, they all worked at the Institute of Advanced Study, resulting in a great divide between the accumulated expertise on relativity at Princeton and the lack of relativity teaching. There was expertise at the university and Wheeler tapped into that as well, as recalled by Churchill Eisenhart, son of Princeton professor Luther Eisenhart:\n\n\\begin{quote}\nAs I understand it, after [Luther Eisenhart] retired he and John Wheeler were working together at writing a book called Mathematics Essential for the Theory of Relativity. [...] Dad and Wheeler, as I understand it, were bringing together in their book the mathematics, from here and there in the various branches of mathematics, you need for the general field theory.\\footnote{Interview on 10 July 1984 with Churchill Eisenhart conducted by William Aspray, available at \\url{https:\/\/www.princeton.edu\/mudd\/finding_aids\/mathoral\/pmc09.htm}. In this interview, Churchill Eisenhart also recalls that the manuscript for Wheeler and Eisenhart's book disappeared under mysterious circumstances after Luther Eisenhart's death.}\n\\end{quote}\n\nEisenhart retired in 1945, around the time that Wheeler began thinking about AAD gravitation.\\footnote{For biographical information on Eisenhart, see \\citep{lefschetz_1969_luther}.} But Eisenhart (like Valentine Bargmann, another Princeton University expert on differential geomertry) was a mathematician, and indeed up until Wheeler's initiative general relativity was only taught in the mathematics department at Princeton \\citep{kaiser_1998_a-psi}.\\footnote{Kaiser erroneously gives the year of Wheeler's first course as 1954\/55. The 52\/53 course, which is well documented by Wheeler's notebook, indeed did not yet show up in the Princeton course catalogue. A course on relativity by Wheeler is listed for 1953\/54. This course catalogue had not been available to Kaiser at the time he wrote his paper.} So, despite the immense tradition and expertise that Wheeler could draw on in his exploration of general relativity, he was indeed the first one to teach it to Princeton physics graduate students.\n\nWheeler's notebook opens with his thoughts on his upcoming course:\n\n\\begin{quote}\n5:55 pm. Learned from [Allen] Shenstone [then head of the Princeton Physics Department] 1\/2 hour ago the great news that I can teach relativity next year. I wish to give the best possible course. To make the most of the opportunity, would be good to plan for a book on the subject. Points to be considered:\\\\\n(1) a short introductory outline of the whole\\\\\n(2) Emphasis on the Mach point of view \\\\\n(3) Many tie-ups with other fields of physics. Mention these in class; in the book put them in the ends of chapters as examples\n\\end{quote}\n\nThe last two remarks are especially noteworthy. In remark (3), we can already see a recurring theme in Wheeler's later work on and in relativity, both intellectually and institutionally,\\footnote{On Wheeler's role in ensuring that the institutionalization of research in general relativity would take place in the disciplinary context of physics, see \\citep{lalli_2017_building}.} namely to establish general relativity as a physical theory, rather than a mathematical or philosophico-cosmological one. Here we see the decidedly pedagogical aspect of this theme, as only in this manner could it legitimately be taught to and applied by physics students. This emphasis should of course also be viewed in light of the predominantly mathematical tradition in relativity at Princeton University.\n\nWheeler's personal intellectual perspective on general relativity shows in remark (2), which hints at how strongly Wheeler's interest in General Relativity was at this point tied up with the prospect of an action-at-a-distance formulation, in which space-time disappears as an independent entity. Indeed, action-at-a-distance was a defining element in Wheeler's subsequent course as documented by his notes. The first term, which dealt primarily with special relativity, saw frequent references to the Wheeler-Feynman papers, including a long discussion of Wheeler-Feynman electrodynamics itself, stretching from December 1952 to January 1953.\n\nWheeler's general relativity class began in February 1953 with a first class meeting on 5 February in which topics for seminar reports were discussed. Here Wheeler had already honed in on some key physical problems, problems that would be defining elements of the upcoming renaissance of relativity: Gravitationally collapsing stars, gravitational radiation, and empirical cosmology. These three problems were joined, in a list of topics for seminar reports discussed in the first class meeting (WR1, p. 47), by an `Assessment of Unified Theories.'' Here, Wheeler was clearly attempting to make contact with the general relativity scene as it presented itself to him at Princeton, as witnessed by the list of references for this report, which included not only the obvious Einstein (specifically his latest paper, which had just appeared in the January issue of the physical review \\citep{einstein_1953_a-comment}) but also work by Eisenhart, who had now, in retirement, turned to the study of non-symmetric metrics as they appeared in Einstein's Unified Field Theory \\citep{eisenhart_1951_generalized}. The suggested topics for seminar reports are followed by an unsorted list of further topics Wheeler wanted to cover, which included both the ``Mach point of view'' and, immediately afterward, ``our particulate point of view.'' The list also contains the entry ``Variational principle and connection with quantum theory,'' a clear reminiscence to the least-action (in modern parlance: path integral) formulation of quantum mechanics that Feynman had developed precisely in the attempt to quantize Wheeler-Feynman electrodynamics. So, also in his list of topics for the second half of the course, we see the two central foci of physical problems (where Wheeler's identification of the central ones was clearly very influential) and of Wheeler-Feynman-Mach action-at-a-distance gravity, now joined by a rising interest in the idea of a unified field theory stimulated by the Princeton milieu.\n\nOf course these questions were interrelated. The question of gravitational radiation, for example, was connected with the construction of an action-at-a-distance theory. The empirical adequacy of Wheeler-Feynman electrodynamics required imposing the so-called absorber condition that any electromagnetic radiation ultimately be absorbed, with nothing ever escaping to infinity. Is the gravitational world of GR ``non-absorptive'', Wheeler asks on the following page (WR1, p. 49), labeling it a ``\\emph{very} vital question to look at.'' Wheeler was thus following an intellectual trajectory typical for the renaissance of GR: In pursuing a speculative extension of GR (action-at-a-distance in this case), he was forced to reflect on fundamental questions of GR proper (gravitational radiation). The question of absorber boundary conditions was really more of a side issue in this study of GR for ulterior purposes, however. As we saw in the last section, the central challenge for Wheeler was to construct a least action formalism for AAD gravity using liaisons. Wheeler pursued this program in parallel to teaching the course, and his lecture notes are consequently interspersed with research notes, initially focussing on the construction of liaison theory. In the following, we will focus almost exclusively on the research notes, leaving the exact reconstruction of the curriculum of Wheeler's course aside.\n\nAfter his purely mathematical approach to this problem appears to have led nowhere, Wheeler's aim was now to construct liaison theory by studying its correspondence to regular GR. The first challenge here was to establish the locus of the correspondence, i.e., to identify the correct field quantity in GR that was to be reconstructed from the liaison formulation, the actual  ``gravitational field.'' Some two weeks into the course, Wheeler began to focus his attention on the Riemann tensor (WR1, p. 57). In an AAD theory, this would ultimately (via the liaisons) have to be reconstructed solely from the matter content (possibly merely in the form of singular worldlines), along with some sort of boundary conditions. Wheeler was thus led back to Einstein's original question concerning the realization of Mach's principle in GR: Is this sufficient to uniquely determine the Riemann tensor (up to coordinate transformations)? Wheeler ultimately reached a conclusion similar to that of \\citet{einstein_1917_kosmologische}, namely ``that there is a one to one correspondence between mass distribution and metric only when space closes up on itself.'' (WR1, p. 103)\n\nIt should be noted that Wheeler was aware that this statement was merely a plausible conjecture: ``Any proof of uniqueness of case where metric is made to close up on itself? Very important question of principle.'' (WR1, p. 105) His main source for this conjecture was  \\emph{The Meaning of Relativity} \\citep{einstein_1953_the-meaning}. He continued to discuss this matter at Princeton with Weyl (WR1, p. 111), Wigner and von Neumann (WR1, p. 120), all of whom disagreed with Wheeler's assessment. Wheeler took this aversion to Mach's principle to also be a result of unfortunate formulation of the principle and gave his class the task of coming up with a better ``presentation of Mach's principle in 2 pages for an elementary physics student.'' (WR1, p. 135). Despite these difficulties, Mach's principle remained central to Wheeler's research program as it provided an analog of the Wheeler-Feynman absorber boundary conditions in general relativity. For some time, closure of the universe became an unquestionable fact to Wheeler, as he explained to his students:\n\n\\begin{quote}\nQuestion raised in class whether mass density enough to permit open or closed universe, in view of expansion rate. Answer: [...] closure comes first, density knowledge too poor to permit proof of contradiction; closure so fundamental to whole Mach idea that in present state of knowledge think of density value having to yield precedence to Mach principle. (WR1, p. 104)\n\\end{quote}  \n\nWith the Riemann tensor identified as ``the field'' (WR1, p. 96), the possibility was now established to construct the action for liaison theory through correspondence with the usual field (Hilbert) action:\n \n \\begin{quote}\n Set up an experimental procedure to get $R_{ijkl}$ locally by liaisons between a number of particles. In this way tie up $R_{ijkl}$ with liaison picture. Hence express $R$ in terms of local liaisons. Hence get variation principle in terms of local liaisons. (WR1, p. 89)\n \\end{quote}\n \nThe ``experimental procedure'' was supposed to involve some sort of ``batting back and forth'' of light signals (WR1, p. 90) which would be a physical realization of the connection between two points established by a liaison. But Wheeler's attempts to tie up the Riemann tensor with the liaison picture ended inconclusively: He attempted to find the liaison function between two world lines from general relativity, where the light signals $\\kappa$ travel from one particle world line to another on light-like geodesics, soon focusing on the limiting flat space case, where the two world lines $x$ and $\\overline{x}$ are straight (WR1, p. 97), i.e. the system of equations:\n\n\\begin{eqnarray*}\nx^i (s) & = & x^i (0) + s \\left(\\frac{dx^i}{ds} \\right)_{s=0} \\\\\n\\overline{x}^i (\\overline{s}) & = & \\overline{x}^i (0) + \\overline{s} \\left(\\frac{d\\overline{x}^i}{d\\overline{s}} \\right)_{\\overline{s}=0} \\\\\n\\kappa^i &=& \\overline{x}^i - x^i \\\\\n\\kappa^{\\alpha} \\kappa_{\\alpha} &=& 0\n\\end{eqnarray*}\n\nwhich was supposed to give a relation between the parameters $s$ and $\\overline{s}$, i.e., the liaison function giving for any point on one world line the point on the other one that lies on the first point's light cone. But even this simplified, non-gravitational trial calculation (27 March 1953; WR1, p. 99) ended inconclusively. His simple idea of obtaining the liaison action merely by translating the Hilbert action into liaison language faltered. Although now fully immersed in general relativity and tensor calculus, he returned to his original mathematical approach and began to pursue (8 April) a new approach to the liaison action, no longer based on counting cycles, but rather on counting the number of (forward) liaisons entering and exiting a given volume element, a setup inspired (as he himself remarked) by the neutron balance in a nuclear chain reaction (WR1, p. 114).\n\nIn all this searching, Wheeler was well aware that he was pursuing an entirely new style of doing physics. On 18 March 1953, in the margins of notes on liaison theory (AAD notebook), he remarked:\n\n\\begin{quote}\nThis mushy thinking may in end be much better, if less attractive, to present than the usual 1,2, 3 type of argument with which one at the end so often presents his special conclusions.\n\\end{quote}\n\nWhat was driving him down this road of ``mushy thinking'' appears to have been the feeling of pursuing something grand, the ``great white hope'' feeling for which we here have the first contemporary archival evidence. Framing to himself his attempt to eliminate space and time, he wrote \n\n\\begin{quote} \nUndoing work of early man, that theoretical physicist who left no records. (26 March 1953; WR1, p. 97)\n\\end{quote}\n\nand also, for the first time in extant writing, coined one of his snappy slogans to describe his project, a ``universe of particles'' (WR1, p. 108). Indeed, though still bogged down in the attempt at formulating a liaison theory of gravitation alone, Wheeler always had in the back of his mind the further goal of combining this with electromagnetism and thereby achieving Einstein's goal of a unified theory (though without fields), and ultimately push on to include also the intrinsic properties of particles, such as spin:\n\n\\begin{quote}\nDon't feel discouraged about how much will still remain to do after expressing mere gravitation theory in liaison form. Should serve as guide in trying to put combined gravitation-electromagnetic theory in liaison form, and in later trying to put everything in neutrino language... (WR1, p. 113; 8 April 1953)\n \\end{quote}\n \nThe term ``neutrino'' appears here for the first time prominently in Wheeler's relativity notebook. Its significance for Wheeler is somewhat hard to grasp, as it can imply two distinct things: It appears as the barest possible point particle, carrying no charge or mass (only spin, possibly), or it can appear as a spinor field, the elementary carrier of spin and associated with the weak nuclear interaction, a reading that goes back to Wheeler's 1945 American Philosophical Society talk, where he referred to the neutrino as a ``field of interaction.'' This should be kept in mind in the following. What both notions have in common is that the neutrino is associated with the introduction of spin into the theory, which also appears to be the role in which it is invoked here. As the hope of recasting general relativity in liaison form faded, the fleshing out of the world line picture, i.e., the construction of a more sophisticated model of matter that would also include intrinsic properties such as spin, moved to the center of Wheeler's thinking.\n \n \\section{Particles as Singularities in the Field}\n \\label{sec:singularities}\n \n At some time in the spring of 1953, a shift began to occur in Wheeler's research agenda. Despite the day-to-day evidence we have from his notebooks, it is hard to date it exactly. It was rather a gradual shift, even though Wheeler's later use of religious metaphors to describe this tradition might rather imply an instantaneous conversion:\n \n\\begin{quote}\nThe idea of action at a distance I gave up, not because the action and the distance was complicated, but because the particle was complicated. It was just the wrong basic starting point for the description of physics, to think of a particle. Pair theory made clear, and renormalization theory, that what one thought was an electron was really an infinite number of pairs of positive and negative electrons indeterminate in number and that the whole of space is filled with pairs. [...] \\emph{And of course nobody gets religion like a reformed drunkard.} As I've often said about this subject, the fanaticism, if you would like to call it that, with which I pursued the opposite approach---that it's a pure field theory explanation of nature that one ought to work at---comes from having worked so hard at a pure particle explanation of what one sees. (LW, emphasis by us)\n\\end{quote}\n\nInterestingly the reasons that Wheeler gives for abandoning the particle approach (in particular the rise of renormalization theory) may well have been essential for his abandoning of the ``everything as electrons'' program, but played no role for his assessment of action-at-a-distance gravity, which, as we have seen, he was pursuing well into the 1950s. And his shift to field theory did not initially involve thinking of the particle as something ``complicated.'' Rather, he merely shifted from thinking of the particle world lines as the primary elements of the theory to thinking of them as secondary, derived objects, as singular lines in the field, whose equations of motion could be derived from the (vacuum) field equations simply by requiring consistent boundary conditions. This program goes back to the 1920s \\citep{einstein_1927_allgemeine}\\footnote{See \\citep{havas_1989_the-early} and \\citep{lehmkuhl_2017_general}.}. Wheeler focused primarily on the approach by Leopold Infeld, which was first formulated in a paper by Einstein, Infeld (then Einstein's assistant) and Banesh Hoffmann \\citep{einstein_1938_the-gravitational} and consequently goes by the name of EIH. It remained a major focus of Infeld's research all through the 1940s.\n\nWheeler had been interested in EIH early on and, in \\citep{infeld_1949_on-the-motion}, he is in fact credited with pointing out the fact that the EIH program has only a trivial (Minkowski) zero-mass limit, and that consequently a separate proof is needed in order to show that test particles follow geodesics in a non-trivial background field. The first reference to a paper by Infeld in Wheeler's notebook, however, appears only on 14 April 1953 (WR1, p. 125), several days after his last attempt to construct a liaison action (using the divergence of liaison lines in a small volume element) had ended inconclusively. Already in that attempt he had had to assume a pre-exisiting (though not necessarily metric) space in which to place the volume element. Wheeler was thus setting aside his ambitious goal of reconstructing space and time entirely from the world lines and liaisons, hoping that ``that deduction will come later'' (WR1, p. 113). Turning to the Einstein-Infeld-Hoffmann approach was a further step in this direction. After an intense study of Lichnerowicz's formulation of general relativity as initial value problem,\\footnote{He had been pointed to these mathematical works by Arthur Wightman; WR1, p.121.} which he hoped to combine with the EIH approach (the notes carry the header ``Geodesics from Field Eqns \\emph{or} Initial Conditions on Field Eqns''), he formulated, on the last pages of his first relativity notebook, a new research program on 1 May (WR1, p. 150).\n\nBefore we turn to this research program, we should briefly discuss the attraction of the EIH approach. For it is quite striking that only a few years earlier the EIH approach had been adopted as the basis for another attempt at a theory of everything, Peter Bergmann's construction of a theory of quantum gravity.\\footnote{For more details, see \\citep{blum_2016_quantum}.} Bergmann's hope had been that by transferring the EIH approach to quantum theory, the equations of the quantum mechanics for point particles might follow from the quantum field theory of general relativity in a similar manner as the classical equations of motion for point particles could be derived from the classical field theory. Even though Bergmann and Wheeler were pursuing quite different approaches, their common interest in EIH can be explained rather simply: EIH held the promise that general relativity might have something to contribute to the microphysics of particles. And for Wheeler, who had now been trying unsuccessfully to reconstruct general relativity from microscopic particle trajectories for quite some time, this prospect, which at the same time let him keep the central notion of the world line, was naturally very interesting.\n\nFor Einstein, the representation of matter particles as singular world lines in EIH had not been intended as final. It was a place holder for an ultimate (field theoretic) description of matter, no better (but also no worse) than the energy-momentum tensor on the right-hand side of the Einstein equations.\\footnote{This assessment is based on \\citep{lehmkuhl_2017_general}.} For Wheeler, on the other hand, coming from the pure world line approach, singular world lines appeared as a perfectly adequate description of material particles. The different status accorded to the world lines determined their assumed properties beyond mere approval or disapproval:  For Einstein the properties of the singularities could only be determined by the field equations. These did not determine the mass or the charge of the singularities, which were consequently free parameters, independently choosable for each individual singularity; much to Einstein's dismay, it should be added, as he hoped that the final theory would be able to explain why only two different masses (electron and proton) occur for the elementary particles \\citep{einstein_1935_the-particle}. For Wheeler, in contrast, the world lines were still entities in and of themselves, and the default assumption (at least in Wheeler's `everything as electrons' tradition) was that they would be identical:\n\n\\begin{quote}\n[T]here is no place for the $e\/m$ of a particle to enter, and all particles should have the same $e\/m$. (WR1, p. 150)\n\\end{quote}\n\nThis presented challenges of its own, since there was of course more than one type of particle in the world. Wheeler reported that his physicist colleague Hartland Snyder ``was inclined to pooh-pooh it all [on] acc't of existence of mesons, etc., in the world.'' (WR1, p. 150). Still, Wheeler was optimistic and had some ideas on how to produce a larger variety of particles with just one type of world line: Anti-particles were to be explained as world lines with the opposite orientation in time (an idea he had proposed to Feynman already a decade earlier); and he hoped to include spin in the picture by somehow taking into account the duality introduced by the two-sheeted Einstein-Rosen metric:\n\n\\begin{quote}\nTheir [Einstein and Rosen's] bridge idea is most intriguing -- two sheets of $g$ meeting at each singularity, get neutrino? (WR1, p. 151)\n\\end{quote}\n\nOn 13 May 1953, Wheeler then took his new idea of combining a (ideally unified, i.e., gravitational and electromagnetic) field theory with particles explicitly described as singular world lines to Einstein himself, when he visited him in his house on Mercer Street together with his entire relativity class. Ten years after his first discussion on AAD gravity with Wheeler, Einstein's reaction appears to have been mixed. As opposed to most of the others that Wheeler had spoken to, ``Einstein agreed [the] universe had to be closed to make [Mach's] principle valid'' (p. 11 of Wheeler's Notebook Relativity 2, henceforth referred to as WR2), but believed this to be merely a necessary but not a sufficient condition.\\footnote{According to the recollections of Wheeler's student Marcel Wellner, Einstein had apparently not thought about Mach's principle in a long time \\citep{wheeler_1979_mercer} when it came up during the visit of Wheeler's class. But less than a year after that visit, Einstein was asked about the matter again, by Felix Pirani. Einstein expressed his surprise at the renewed interest, opening his letter of 2 February 1954 (Einstein Papers, Jerusalem) with the words: ``There is a lot of talk about Mach's principle.\" By that time, apparently having rethought the matter following the meeting with Wheeler and his students, Einstein had convinced himself that the principle was obsolete, telling Pirani: ``In my opinion, one should not speak of Mach's principle at all any more.''}\n\nEinstein's reactions to the specifics of Wheeler's research plan were even more lukewarm: He declared that he ``was not interested in singularities'' (WR2, p. 11) and that the idea expressed of ``connecting [an Einstein-Rosen bridge] up with spin of electron, neutrino is no good.''\\footnote{Arthur Komar offered a more specific account of Einstein's dismissal of Einstein-Rosen bridges, recalling: ``John Wheeler asked him about the Einstein-Rosen bridge. Why had he first introduced it and then dropped it again? Einstein answered that he had initially believed that the bridge connects two almost plane surfaces in a unique manner. When he, however, discovered that they did not have a unique structure, the bridge seemed to him to be too cumbersome, unattractive, and ambiguous.'' (\\emph{John Wheeler fragte ihn \\\"uber die Einstein-Rosen-Br\\\"ucke. Warum habe er sie zun\\\"achst eingef\\\"uhrt und dann wieder fallengelassen? Einstein antwortete, dass er zun\\\"achst glaubte, die Br\\\"ucke verbinde zwei fast ebene Fl\\\"achen in eindeutiger Weise. Als er jedoch entdeckte, dass sie keine eindeutige Struktur war, schien ihm die Br\\\"ucke zu schwerf\\\"allig, unattraktiv und vieldeutig.}) These still rather vague recollections might be of Einstein referring to the fact that he had hoped that multi-bridge solutions to the Einstein equations might be so constrained as to enforce equal masses for the individual bridges, thereby addressing the problem discussed earlier of explaining why only a few different mass values for elementary particles were observed. He ultimately appears to have concluded that no such constraints would arise, as stated in a letter to Richard Tolman of 23 May 1935 (Einstein Papers): ``One does not see why the ponderable and electric masses cannot be arbitrarily large or different, when several are present.'' Many thanks to Dennis Lehmkuhl for discussions on the Einstein-Rosen paper and for making this letter available to us.} (WR2, p. 11) Wheeler's general relativity class ended two weeks later with a final exam on 28 May. His interest in general relativity was unbroken and his notebook contains notes on cosmology, gravitational radiation, and long passages in French copied from Lichnerowicz's 1948 lecture notes ``G\\'eom\\`etrie diff\\'erentielle et topologie''  before having to return them to the library (WR2, pp. 29-34). But for two months after the visit to Einstein, the notebook contains nothing new on Wheeler's foundational ideas and the question of how to turn singular world lines in general relativity into full-fledged particles. Wheeler did take Einstein's negative remarks with a grain of salt, in particular attributing Einstein's negative attitude toward singularities to the fact that recent work by Infeld had shown that applying the EIH method to Einstein's unified field theory did not return the correct equations of motion, i.e., the Lorentz force law in curved space-time \\citep{infeld_1950_the-new}. But it was his preparations to give a talk at the International Conference of Theoretical Physics in Japan, to be held in September 1953, that gave Wheeler a new impulse.\n\n\n\\section{Daring Conservatism and the Field Program}\n\\label{sec:fields}\n\nWheeler ended up giving three talks in Japan: two rather technical ones on the origin of cosmic rays \\citep{wheeler_1954_the-origin} and on collective models for nuclei \\citep{wheeler_1954_collective}, published in the conference proceedings; and one more programmatic talk, which he held on 10 September 1953, before the conference, at the Physical Society of Japan and which was only published in Japanese translation\\footnote{According to the notebook (Wheeler Papers) that Wheeler kept during his stay in Japan, the translation was done by Takahiko Yamanouchi; Japan Notebook p. 51.}  in the Proceedings of the Society. We provide a retranslation into English of this talk (the original manuscript and recording are lost) in the appendix. It is this talk which is of central importance to our story, and it is this talk that one finds Wheeler preparing in his notebook on 18 July 1953 under the heading: ``Philosophy of approach to elementary particle problem''. From the start, Wheeler was very eager to establish a clear connection to Japan in his talk, noting in the margins: ``Each one of us finds himself reflected in the countries he visits.'' But he also took the opportunity to reflect on his overall methodology. We have seen several times Wheeler's predilection for taking existing theories and using and extrapolating them outside their established domain of applicability, the paradigmatic example being his attempts to explain the nuclear forces electromagnetically. In the notes for the Tokyo talk, this methodology, which he would later characterize as ``daring conservatism'' is now, for the first time, made explicit as the ``Tokyo Program'':\n\n\\begin{quote}\nProposed Tokyo program: Be as conservative as possible about introducing new elements into description. Make basics as clear \\& simple as possible. Is only the consequences that are complicated: ice; elem. particles; meteorology; geology. [...] Strengths of this approach. Its weaknesses. Einstein's May '53 remark to JAW: `The Lord may have made the universe with five fields. I don't think so. But if he did, I am not interested in the universe.' Quote as a Princeton physicist, nameless. An extreme attitude, not fully open minded. Surely much good.\n\\end{quote}\n\nAs he outlined his guiding methodology explicitly for the first time, Wheeler was clearly becoming excited, referring to himself in the margins as ``Tokyo Wheeler'', drawing an admittedly somewhat bizarre analogy between his new ideas on elementary particle physics and the demoralizing propaganda spread to the American troops by ``Tokyo Rose'' (Iva Toguri), host of the WWII Japanese English-language radio show ``The Zero Hour''.\n\nAt this point Wheeler's notes shift away from a lecture sketch to an inner monologue about the foundations of his research program:\n\n\\begin{quote}\nEvidently have in mind something more fundamental. Out with it! Desert island philosophy: imagine selves cast up on Wake Island with library of all theory \\& exp[erimen]t up to now, to solve elem. particle problem -- What to use as starting points? -- Others not ambitious enough? Go whole hog now!\n\\end{quote}\n\nWhat follows is a long list of elements (of existing theory) that might be of importance in his attempts at crafting a theory of elementary particles. We explicitly see Wheeler assessing the potential of existing theory, in particular general relativity. The list contains familiar tropes (action at a distance -- point 2; Mach's principle -- point 7), but also some novel elements, indicating how Wheeler was reordering his vision of how to think of elementary particles. The central new element is an emphasis on fundamental masslessness, the vision of a theory without intrinsic mass parameters that would ideally include``\\emph{no natural constants}. Nothing but $e$, $\\pi$, etc.'' (point 1).\\footnote{In a manuscript entitled ``The Zero Rest Mass Fundamental Field Hypothesis'' (WR, p. 101), which we shall discuss later in more detail, Wheeler ascribes this vision of a theory with no free parameters to Einstein. We have not been able to find relevant statements in Einstein's work.} Where then was mass to come from? In point 8 of his list, Wheeler remarked on the ``Electromagnetic origin of mass and the self energy story'', jotting down the first order radiative corrections to the electron mass, as first derived by Weisskopf in 1939. These terms were, in modern theory, simply absorbed in a renormalization of the electron mass, ultimately implying total agnosticism about the origin of mass. But in view of the proposed masslessness of the fundamental pointlike particles, Wheeler was highlighting the electromagnetic origin of mass, advocating (point 11) that one ``should apply electrodynamics to very small distances''.\\footnote{Wheeler here also invoked, for the first time, Bohr as the godfather of daring conservatism, because Bohr had applied ``electrostatics to very small distances'' in his atomic model.} With mass externalized from the point-like singular particles to the surrounding field, Wheeler could consider all particles as composite (point 13), as ``structures held together by radiative, electrodynamic and gravitational forces.'' (point 14) Wheeler's new vision thus really amalgamated all existing theory by proposing particles with a singular point-like core and field-generated structure. \n\nThis new focus on masslessness temporarily moved the neutrino to the center of Wheeler's theorizing as he emphasized the ``importance of the \\emph{neutrino} in the scheme of things'' (point 3). We again encounter the ambiguity in the conceptualization of the neutrino: At one point it appears as the fundamental point-like entity, with the electron to be thought of as a ``neutrino with a charge loaded on its back''. At other times, it clearly appears as a field-like entity, possibly arising through ``spinorization'' of the metric of general relativity, that is taking the ``square root'' of the (vacuum) Einstein equations in a manner analogous to that which generates the Dirac from the Klein-Gordon equation. While Wheeler saw this as a major challenge, he was rather optimistic, remarking: ``Spinorize, fit all together, and listen for the harmony.''\n\nWheeler's novel emphasis on neutrinos was apparently also fueled by first results of the efforts by Frederick Reines and Clyde Cowan to directly detect these elusive particles. At this time, in the summer of 1953, Reines and Cowan were performing first background checks with their liquid scintillator detector at the nuclear reactor in Hanford, WA. They had found a source-independent background, which they thought might be due to ``natural neutrinos'' (what one might call cosmic relic neutrinos in big bang cosmology). Wheeler was aware of these results, referring in his notebook to ``Reines-Cowan radiation'' when emphasizing the importance of the neutrino. Wheeler was briefly envisioning the neutrino not only as the fundamental constituent of all particles, but also as the prime component of the energy density of the universe, and his notes of 6 August 1953 show him studying the Friedmann equations in a neutrino-dominated universe.\\footnote{No correspondence between Wheeler and Reines or Cowan from 1953 is extant, but Reines in turn was clearly aware of Wheeler's contemporaneous elevation of the neutrino to central stage. In his Nobel lecture, Reines makes an inside joke, remarking without mentioning Wheeler: ``While we were engaged in this background test, some theorists were rumored to be constructing a world made predominantly of neutrinos!''} These calculations were interrupted by a phone call from Reines, informing Wheeler that they had identified their source-independent background as due to nuclear capture of cosmic ray muons.\n\nStill, the neutrino kept an important role, also in the talk that Wheeler eventually held in September 1953 in Tokyo. The talk is set up as a dialogue between Wheeler and two figures from Japanese history, Saigo Takamori and Sugawara no Michizane.  Saigo, an important 19th Century Samurai, is given the role of the daring modernizer and presents the current state of the art in particle physics, the discovery of new particles at accelerators, and the meson theory of nuclear interaction. Sugawara no Michizane, a Ninth Century scholar and poet, is given the role of reflective traditionalist, who presents Wheelers Tokyo program, though the program is not actually named in the talk. It is merely characterized as ``the principle, which is the basis of the scientific method, of not introducing a new hypothesis until it is clearly and undoubtedly necessary.''\n\nSugawara begins by lauding general relativity as a model field theory: On the one hand, there\nis the point we have already discussed extensively, that it allows for the integration and the\nderivation of the equations of motion of point particles. But more importantly, general relativity, viewed as Einstein's formalization of Mach's principle, was supposed to provide an\naccount how a field theory (or more generally an interaction, which could also be a theory\nof action at a distance) could generate mass in a massless theory, or rather inertial mass in a\ntheory without inertial mass. The argument as presented in the talk (or at least as presented\nin the Japanese translation) is somewhat elliptic. It is formulated not in terms of general relativity, but in terms of an AAD theory. Clearly, such an AAD theory could not be equivalent\nto GR; we know that Wheeler had been searching for such an AAD formulation of GR for several years, but had not been able to construct one. Instead, the AAD theory he used in the\nTokyo talk was a slight modification of Newtonian theory, where the usual Coulomb field is supplemented by a second field that falls off only as $1\/r$ and thus dominates at long\ndistances. Wheeler gives this field explicitly as $G m_g a\/c^2 r$, where $m_g$ is the particle's \\emph{gravitational} mass and $a$ is its acceleration. This expression is analogous to the long-distance Li\\'enard-Wiechert field of an accelerating charge in electrodynamics, and since it was not to be expected that the analogy between electrodynamics and gravity would be that perfect, Wheeler\/Sugawara put the expression in scare quotes. With the long-distance interaction established, Wheeler then introduced what he called the \"whole idea of gravity theory\", namely that the total gravitational force on a particle is zero: a particle subject only to gravity is not moved by forces, but by the curvature of spacetime. In the modified Newtonian AAD theory the same idea is appropriate to express the expectation that inertia is provided by interaction and an \"intrinsic inertia\" term ($m_{\\mathrm{inert}}a$) is absent from the equation of motion:\n\n\\begin{equation}\n\\frac{G m_1 m_2}{r^2} - \\sum_k \\frac{G m_1 a m_k}{c^2 r_k} = 0\n\\end{equation}\n\nThe equation's second term can instead be understood as the reaction on mass $m_1$ to the force that $m_1$'s acceleration exerts on the masses $m_k$ through the new, long-distance, Li\\'enard-Wiechert-type interaction. One gets the usual (unmodified) Newtonian equation of motion for $m_1$ in gravitational interaction with $m_2$ (with $m_{\\mathrm{inert}} = m_g$), under the condition that\n\n\\begin{equation}\n\\label{eq:mach}\n\\frac{G}{c^2} \\sum_k \\frac{m_k}{r_k} = 1\n\\end{equation}\n\nwhere the sum extends over all of the distant masses $m_k$ which are at distances $r_k$ from the mass $m_1$.\n\nIt is appropriate at this point to point out the intimate relation between Wheeler's argument and a sketch of the origin of inertial mass published by Dennis \\citet{sciama_1953_on-the-origin} just a few months before Wheeler's talk. Sciama's argument was field-theoretical, but also built on the electromagnetic analogy, explicitly employed vector fields obeying the Maxwell equations as gravitational fields and obtaining long-distance Li\\'enard-Wiechert potentials that correspond to Wheeler's long-distance force. Sciama also introduced an analogous principle to Wheelers ``whole point'', which in his field-theory language reads that ``the total gravitational field at the body arising from all other matter in the universe is zero'', but Sciama explicitly labels this as a postulate and specifies that it holds in that body's rest frame. In this rest frame, the whole exterior universe is moving with acceleration $-a$, and the total field from the distant matter (the $1\/r$ term) should exactly cancel the short-distance gravitational field (the $1\/r^2$ term) of the particle with mass $m_2$. Rewriting this equation of cancellation, Sciama gets the usual Newtonian force law for the gravitational interaction between the masses $m_1$ and $m_2$  under the same condition as Wheeler (Equation \\ref{eq:mach}) obtained in field-theoretical terms (Equation 6\/7 of Sciama). It is unclear whether Wheeler knew of Sciama's argument and merely rephrased it\nin AAD terms, or whether he had found it independently in his attempts at constructing an AAD version of gravity, building on an AAD formulation of electrodynamics. Both stories seem plausible, and if Wheeler really did not mention Sciama in his talk (and this is not just an omission of the transcription that was then translated into Japanese) the second one\nseems the more likely. Wheeler did eventually learn of Sciama's paper, as he jotted the reference down on the last page of his second relativity notebook (which covers the period up to April\n1954), but since this last page appears to have served as a general place to note miscellaneous references, it is impossible to date. In any case, the Machian argument in the Japan talk was merely to serve as a proof of principle how mass might arise in a theory in which it is not a primary attribute of matter.\n\nA similar proof of principle was given for the electrodynamic generation of mass through the radiative corrections calculated by Weisskopf, which we have already mentioned above. Wheeler's treatment in the Tokyo talk is somewhat problematic. Following Weisskopf, he (or rather Sugawara) gave the radiative correction $\\delta_m$ to the electron mass as\n\n\\begin{equation}\n\\label{eq:weisskopf}\n\\frac{\\delta m}{m} = \\frac{3}{2 \\pi} \\frac{e^2}{\\hbar c} \\ln{\\frac{\\lambda_{max}}{\\lambda_{min}}}\n\\end{equation}\n\nLeaving the question of the infrared and ultraviolet cutoffs in the logarithm aside for the moment, the parameter $m$ is here the electron's bare mass, which should be zero according to Wheeler's assumptions. Wheeler, however, takes it to be the electron's physical mass, assumes this to arise entirely from radiative corrections (i.e., from the field), and thus sets the lefthand side of the equation to 1. Today it is well established that perturbatively a massless fermion cannot gain mass from its electromagnetic interaction, precisely because the radiative corrections are always proportional to the bare mass (due to chiral symmetry). However, chiral symmetry may well be broken through non-perturbative effects, so that the general idea of a purely electromagnetic mass is not implausible. And again, Wheeler appears to merely have been floating some rough ideas for how mass might arise in a fundamentally massless theory and how one might obtain a unique value for the fine structure constant.\\footnote{Here Wheeler was following in the footsteps of a number of famous physicists who had attempted to derive the value of the fine structure constant (which for a long time looked like it might be precisely 1\/137) in the preceding decades. See \\citep{kragh_2003_magic}.}\n\nAll of this was thus an elaboration of the program he had outlined in his preparatory notes. The conservative Tokyo Program was now personified by the measured statesman and poet who was filled with a ``love of Japanese beauty and harmony'', who wished to work only with well-established entities and theories and to introduce no free parameters, such as masses, into his considerations; though the end of the talk saw Sugawara reconciled with the audacious Samurai Saigo, already heralding the reformulation of Wheeler's program as not merely conservatism but ``daring conservatism'' several months later. The part of Wheeler's program that was most in flux, however, as witnessed not only by the Tokyo talk but also by the notebook entries of the time, was the exact role of the neutrino. While the talk clearly focussed on the field-theoretical aspect of the neutrino, it explicitly raised the question whether it was to be thought of as a massless field that joined the electromagnetic and gravitational fields in giving structure to the elementary particles, or whether it was only a derivative of the gravitational field, arising upon spinorization. \n\nThrough his study of the literature on spin in general relativity (specifically \\citep{pauli_1933_uber-die-formulierung}), and through discussions with the Princeton mathematician Oswald Veblen (30 October 1953), Wheeler reached the conclusion that the last point was true, but that this spinorization could only occur upon quantization:\n\n\\begin{quote}\nMy conclusion? I know that the neutrino obeys Pauli statistics, therefore cannot come into a classical theory, therefore ought to show up only after quantization, therefore I should look for the classical theory \\& then quantize it a la Feynman, but with a square root, antisym, spinor character all put in at that time.\n\\end{quote}\n\nThe neutrino and the issue of spin, which had temporarily been at the focus of Wheeler's interest and of the Japan talk, was thus temporarily set aside and relegated to the quantum realm. This further strengthened the focus on the classical fields of electrodynamics and gravitation, which, despite the persistence of singular point particles, were doing the work. It was the fields that had the potential to clarify the question of elementary particles, that would generate masses and define equations of motion, classically and in quantum theory. Wheeler's main focus was thus now on Einstein-Maxwell theory, a classical field theory that would, at least after quantization, give a full account of the physics of elementary particles:\n\n\\begin{quote}\nIf $\\nu$ is somehow contained in em+grav., and if we are right saying that only fields of zero mass count (no meson fields, etc.), and if we have the \\emph{right} theory of em+grav., and if Feynman procedure [path integral quantization] is legitimate for such fields, then \\emph{here's where we start}.\n\\end{quote}\n\nIn Einstein-Maxwell theory, the electromagnetic and gravitational fields appear as separate entities and are simply minimally coupled. Wheeler referred to it as the ``un-unified field theory''. The contrast with the unified field theory program of Einstein and others was clear and indeed these were to be viewed at the time as legitimate competitors of Einstein-Maxwell theory as classical descriptions of electrodynamics and gravitation. Wheeler thus felt the need to consider their merits, before further pursuing his program. \n\nHow now to judge these merits? Einstein's unified field theory \\citep{einstein_1950_the-meaning} was out, because, as we have already mentioned, one could not EIH-derive the Lorentz force from it. But Wheeler's student Arthur Komar (23 October) had pointed him to an alternative unified theory that gave, through the EIH method, the correct equations of motion, i.e., including the Lorentz force. This was the unified field theory of Behram \\citet{kursunoglu_1952_einsteins}. There was, however, a different problem with Kursunoglu's approach for Wheeler: It relied on the introduction of a fundamental length, i.e., a dimensionful parameter into the theory, which was of course in strict opposition to Wheeler's program of having no natural constants. Wheeler asserted that ``conservative me'' (30 October) had to try out what would happen in Kursunoglu's theory when one let the fundamental length go to zero: Would one still have a unified field theory or would one merely obtain general relativity without electrodynamics? The above quote thus continues:\n\n\\begin{quote}\nOnly one question \\emph{before we start} --- what about so-called unified field theory? Einstein's variety no good. Therefore try Kursunoglu's variety --- in case where his $p$ [inverse of fundamental length] is set equal to $\\infty$ --- just to test whether we have any \\emph{conservative} alternative to what we are doing.\n\\end{quote}\n\nWheeler was thus now explicitly using conservatism (in the sense of no natural constants) as a criterion for theory selection. On 1 November, he came to the conclusion that Kursunoglu's theory, in the limit where the fundamental length goes to zero, merely reproduced Einstein-Maxwell theory. His assessment of unified field theory thus ended with a ``bronze plaque'' in his notebook, reading: ``Unified Field Theories died here'' and a letter to Kursunoglu, on 3 November, in which Wheeler wrote:\n\n\\begin{quote}\nI am writing to ask if a conservative physicist who wants to deal with gravitation and electromagnetism within the framework of general relativity has nowadays any acceptable choice but to use as action the expression [action of Einstein-Maxwell theory]. By ``conservative'' I mean unwilling to  introduce new ideas, new concepts, and particularly unwilling to introduce any quantity with the character of a fundamental length except as called for by inescapable evidence.\\\\\nWill not one who adopts the conservative point of view, as just defined, have to abandon unified field theory as it stands at present?\n\\end{quote}\n\nWheeler had thus firmly convinced himself that the theory he needed to quantize was the conservative, minimal Einstein-Maxwell theory; he had found the new focus of his research in an attempt to quantize gravity, minimally coupled to electrodynamics. While quantum gravity nowadays, with all of the technical and conceptual difficulties it entails all too clear, hardly seems a conservative endeavor, to Wheeler it certainly seemed as such; it was based merely on a combination of the well-established principles of general relativity, Maxwell electrodynamics, and quantum theory. After 20 years of private speculations, he felt he was now ready to publicly elaborate on his vision for the foundation of physics, a vision that was built on general relativity, a theory that was not only coherent and well-established, but also, through its unique features, such as Mach's principle and the EIH determination of equations of motion, had the potential to resolve the great open questions of microscopic physics. On 4 November 1953, we thus find in Wheeler's notebook ``Points for proposed article `Elementary particles from Massless Fields --- An Assessment.''\n\nAround this time, Wheeler suddenly appears to have remembered a central point, which indeed was absent at least from his notebook entries for a while: the point particles. For November 8, we find the following short entry:\n\n\\begin{quote}\nThe big question\\\\\nLet's forget about electromagnetism for present. In quantum transcription of the pure gravitation theory with the variation principle based on $\\psi = \\sum e^{\\frac{i c^3}{16 \\pi G \\hbar} \\int \\int \\int \\int R \\sqrt{-g} dx^1 dx^2 dx^3 dx^4}$ [i.e., the path integral for the Hilbert action] how do we take into account the existence of singularities?\n\\end{quote}\n\nThe singular world lines had, over the course of the year 1953, been transformed from the central element of the theory into a problematic embarrassment in the promising program of quantizing general relativity. Like Bergmann several years earlier, Wheeler realized that quantization and point singularities in the field did not really mesh. Bergmann had resigned himself to studying pure general relativity, but this was hardly an option for Wheeler who was after all trying to solve the problem of elementary particles. And indeed, the fields in Wheeler's approach were still mainly meant to provide services to the point particles: give them mass, define their equations of motion. When he met with Einstein once more, in the morning of 13 November 1953, Einstein asked (WR2, p. 83): ``What about matter term in Lagrangian''  to which Wheeler replied that ``matter was to originate from singularities.'' However, when Wheeler then went on to explain Feynman quantization to Einstein, he remarked that in this setup ``the singularities in field get eliminated, never have to be talked about.'' This seems to be in reference to the assumption that singular field configurations would have measure zero in the path integral, which seems like a problem for describing matter by singularities, but is of course a good thing when talking about pathological singularities.\\footnote{Indeed, Einstein appears to have been impressed. While first remarking that he ``abhorred'' the idea of first constructing the classical field theory and then quantizing it, he then conceded (according to Wheeler's notes) that ``it was the first time he had ever heard describe a way that [quantum theory] might get through, found it very attractive.''}\n\nBut an even more severe difficulty with the singular point particle notion lay in its relation to the field concept. EIH determination of the equations of motion, of course, offered the prospect of reconciling  the notions of field and point particle; this fact had originally led Wheeler to reintroduce fields into his worldview and endorse a dualistic ontology. As soon, however, as mass generation through the field entered into the picture, the fundamental incompatibility of point particles and local fields, which had haunted fundamental physics ever since Hendrik Lorentz had first tried to think the two together in his electron theory, again became visible. Indeed, already in his Tokyo lecture, Wheeler had been forced to introduce an ultraviolet cutoff ($\\lambda_{\\mathrm{min}}$ of Equation \\ref{eq:weisskopf}) to make the field-generated mass finite. This essentially meant abandoning the idea of a point particle and introducing a finite size for the electron. It is important here that Wheeler (or Sugawara) had hypothesized that this finite size would be given by the gravitational (Schwarzschild) radius of the electron, and not the Planck length. So the necessary mass scale that one needed to make a length using the gravitational constant $G$ and the speed of light $c$ was provided by the mass $m$ of the electron and not by Planck's constant $h$. This clearly indicated that the cutoff was to arise not as a quantum effect, but due to the presence of the particle. By introducing the notion of field-generated masses, Wheeler had thus effectively abandoned the notion of a point particle that had been a mainstay of his research program for a long time. This was not a problem for the EIH determination of the equations of motion, as the use of point particles in that derivation could well be viewed as a mere approximation.\\footnote{While \\citet{wheeler_1961_geometrodynamics} would later conclude that point singularities were not a valid approximation for any reasonable model of matter (which by that time for him meant geons and wormholes), there is no indication that he (or anybody else) harbored such doubts in 1953\/54, given that the concepts and in particular the conception of matter that these conclusions were based on had not been developed yet.} But it ultimately undermined Wheeler's briefly-kept hopes for a dual theory of point particles and fields and forced him to consider novel conceptions of matter.\n\nWhile he spent the next weeks thinking about how to spinorize Einstein-Maxwell theory by taking the square root of the Lagrangian in the action (WR2, p. 88), the pressing question of the constitution of matter moved to the center in a working paper entitled ``The Zero Rest Mass Fundamental Field Hypothesis'' and dated 19 January 1954.\\footnote{The paper is included in WR2, p. 101, as an insert. This copy is noteworthy also for some remarks in the margins in which Wheeler explicitly connects his conservative heuristic in physics with conservatism in politics, noting: ``To defend well established physical ideas as unpopular as defending well established political parties. People like to criticize. Religion the great defender.'' In this connection it appears pertinent to mention that Wheeler's conservative stance (in physics), as outlined in the Tokyo talk, was explicitly criticized by the Japanese physicist Shoichi Sakata, an outspoken Marxist \\citep{staley_2004_lost}. In discussions on September 18 at the conference in Kyoto, a week after Wheeler's lecture, Sakata remarked: ``I am convinced the future theory should not be the progressive improvement of the present theory. At the Tokyo meeting Professor Wheeler pointed out that there are two methods of approaching the truth; that is Saigo Takamori's method and Sugawara Michizane's method. But in Japan Professor Tomonaga had pointed out that there are two ways, namely a non-reactionary conservative way and also a revolutionary way. This is our common sense.'' \\citep[p. 34-35]{japanproceedings}} Here, Wheeler addressed the central question that any theory of extended (i.e., not pointlike) particles would have to answer. While the spatial extension of the particles avoided the issue of divergent field strengths, it brought with it a different issue, which had a long tradition going back to first attempts at a solution by Poincar\\'e: the issue of stability. Given that there would be no more singular point-like cores, all that was left for constructing a particle were the electromagnetic, gravitational, and possibly neutrino fields (the ``zero rest mass fundamental fields'' of the manuscript's title), and ``an elementary particle is held together by the balance of gravitational, neutrino, and electromagnetic forces'' (p. 7 of the manuscript). But how to envision such an object? In the manuscript, Wheeler explored the possibility of comparing elementary particles with a (collapsing) star---the analogy being based on both objects (star and particle) being held together by gravitational forces.\n\nBut the big breakthrough for how to model elementary particles only occurred about a week later, when Wheeler attended the Fourth Rochester Conference on High Energy Nuclear Physics from 25-27 January 1954 \\citep{rochester_1954}. It is the last one of Wheeler's breakthroughs that we shall discuss in this paper, as it finally brings us to  Wheeler's geon paper \\citep{wheeler_1955_geons} and his embrace of a pure field theory, from which also the singularities representing matter had been removed. Up until now, Wheeler had mainly attempted to use the untapped potential of general relativity as it related to mass points: The ability to derive their equations of motion from the field equation, the possibility of generating mass for them from fields or interactions. In late January 1954, Wheeler seized upon a feature of general relativity, which he had hardly engaged with so far: the non-linearity of the field equations, which in principle allowed for solutions describing a localized and (meta)stable concentration of energy, an idea which had been in the back of Einstein's head for a long while. \n\nOn his manuscript of 19 January (which was never published), Wheeler had noted that he was distributing it to a small number of physicists, including Einstein, Bohr, and Wightman. Wightman was also attending the Rochester conference, and Wheeler appears to have discussed his ideas with him there, for on 25 January 1954, we find the notebook entry (WR2, p. 96):\n\n\\begin{quote}\nBall of light held together by gravitational forces as classical model for an elementary particle = fireball = (Wightman name) Kugelblitz\n\\end{quote}\n\nimmediately followed by calculations for a spherically symmetric graviational potential fulfilling the vacuum Einstein-Maxwell equations (i.e., the Einstein equations with only an electromagnetic energy-momentum tensor as a source), with all of the electromagnetic energy constrained to a sphere of finite radius. Such a field configuration, which could only exist in a non-linear field theory such as Einstein-Maxwell theory and which Wheeler would soon label a ``geon'' (first found in WR2, p. 104, in an entry dated 19 February 1954), was thus the new model for elementary particles that Wheeler would pursue for the next few years. Everything point-like had been expelled from the model, in favor of a spatially extended pure zero-mass-field configuration.\n\nThere were of course many open questions to tackle, some of which Wheeler listed in the entries of the next two days (WR2, p. 100ff), such as whether such entities really existed, how to incorporate charge,\\footnote{Here Wheeler already pondered the possibility of having ``outgoing lines of force [...] understood in terms of lines coming in from an `internal universe''', an idea that would later mature into his notion of a wormhole.} the still unsettled role of the neutrino and the square root of the Einstein-Maxwell Lagrangian, and the role of quantum theory and self energies,\\footnote{Here Wheeler encountered some conservative resistance from Wightman, who objected to Wheeler's predilection for path integrals, arguing instead that one should ``improve \\& understand present formalism'', i.e., pursue axiomatic quantum field theory. Even Feynman appears to have been doubtful about the ``general utility'' of the path integral, as he had not yet been able to properly accommodate fermions.} in particular concerning the quantization of general relativity, in which context Wheeler noted (WR2, p. 103):\n\n\\begin{quote}\nTry to understand whether Gupta or anyone else really know what he's talking about on the quantization of gravitation theory, esp. the comm'n. rel'ns at small distances.\n\\end{quote}\n\nBut while the new geon model of elementary particles brought with it a host of unanswered questions, an entire research program as it were, just days after the Rochester conference (where he had talked on charged meson decay) Wheeler certainly felt confident enough to publicly present his new idea in New York City, where he held the annual Richtmyer Memorial Lecture of the American Association of Physics Teachers (AAPT).\\footnote{\\label{fn:long} The AAPT was conducting its winter meeting in parallel with the American Physical Society, which conducted its annual meeting at Columbia University from 28-30 January 1954 (Physical Review, Volume 94, pp. 742ff), so that there were also many research physicists in the audience.} This Lecture, entitled ``Fields and Particles\", is the last text we shall be discussing and is, as we shall see, in many ways the sum of the development in Wheeler's thinking that we have reconstructed in this paper.\\footnote{The lecture was never published, but there is an extant transcript in the Wheeler Papers, in a folder entitled ``Fields and Particles.'' The Richtmyer Lecture Memorial Award had been established in 1941 to honor Floyd Richtmyer, one of the founders of the AAPT (\\url{https:\/\/www.aapt.org\/Programs\/awards\/richtmyer.cfm}). Many of the previous lectures had ben published in the AAPT's journal, the American Journal of Physics (e.g., \\citep{slater_1951_the-electron, vleck_1950_landmarks, dubridge_1949_the-effects}). Wheeler had plans to publish his lecture there as well, and the folder contains two revised versions of the original lecture transcripts, which were clearly supposed to lead up to a publication. The folder also contains some correspondence between Wheeler and Thomas Osgood, editor of the American Journal of Physics, such as a letter from Osgood of 28 January 1957, which begins: ``Here is my annual letter of inquiry about the manuscript of the paper ``Fields and Particles'' that you gave as Richtmyer Memorial Lecture during the meeting of the American Association of Physics Teachers in New York, January 28-30, 1954 It ought to be published without delay.'' Wheeler in fact cited the paper in the first footnote of the Geon paper as ``to be published''. That long footnote (a specialty of Wheeler, to which this footnote here is a sort of tribute) also contained a reference to Wheeler's Tokyo talk and ``the point of view ascribed by the author to Sugawara-no-Michizane,'' making the entire footnote rather enigmatic for the average American reader of the Physical Review.} \n\nThe Richtmyer Lecture began with Wheeler's most explicit elaboration of his conservative methodology, which he now labelled ``daring conservatism'' and couched in religious terms, citing the apostle Paul:\n\n\\begin{quote}\n``Whatsoever things are true, whatsoever things are honored, whatsoever things are judged, whatsoever things are pure, whatsoever things are lovely, whatsoever things are of good repute. If there be any virtue and if there be any praise, think on these things.''\\footnote{This passage is from Philippians 4:8, where it reads ``honest'' instead of ``honored'', ``just'' instead of ``judged'', and ``report'' instead of ``repute.'' We have given the quote as it appears in the lecture transcript, and it is to be assumed that the transcriber simply misheard these three words. Wheeler corrected all three in the later manuscripts of the Richtmyer Lecture mentioned in Footnote \\ref{fn:long}.} Following these words of Paul, I would like to dedicate this occasion [...] to an appreciation of the great truth of physics in the saying that from them we will receive guidance in this elementary particle problem beyond anything that we now imagine.\n\\end{quote}\n\n Wheeler then went on to highlight the role of general relativity among the ``already well established ideas'' of physics on which the conservative physicist should build by daringly ``following out [its] consequences'' to the ``utter most extreme.'' He then went on to outline the great potential (``exciting new possibilities'') of general relativity both ``in the realm of what might be called astrophysics'' and for the ``elementary particle problem'', introducing his geon\\footnote{Then still referred to as a ``Kugelblitz'' or, in the words of the person who transcribed the lecture, ``cugoflix''.} idea to the world and presenting it as a new research program:\n \n \\begin{quote}\n In my view following out the philosophy of the conservative daring [sic], it's an inescapable obligation of our present-day physics to continue the investigation of these objects and to see what boundary line if any separates them from the elementary particle problem. The full investigation of both electromagnetism and gravitation of course has to take place within the frame work of quantum theory.\n \\end{quote}\n \nWheeler had thus publicly outlined his new research program in general relativity, which consisted of studying stable, localized solutions of the Einstein-Maxwell equations, their modification through quantum theory and their relation to elementary particles, as well as the inclusion of further elements into this picture, such as charge and the neutrino\/spin. Wheeler's transition to a full-blown ``relativist'' was completed, and the research program outlined in the Richtmyer lecture would occupy him and his graduate students for years to come. So fruitful was this approach that Princeton and the Wheeler School, despite being the youngest of the relativity centers soon to be connected in the Renaissance, became one of the central hubs of that process.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper we have reconstructed John Wheeler's turn to general relativity in the years ca. 1941-1954 and how it was driven by what we have called the untapped potential of general relativity, thereby corroborating and filling with meaning the claim of \\citep{blum_2015_the-reinvention} that this untapped potential was one of the motors of the renaissance of general relativity. Our reconstruction has shown that Wheeler's general methodology, ultimately branded ``daring conservatism'', precisely consisted in seeking out the potential of existing theories, rather than constructing new ones. It should, however, be added that the general notion of daring conservatism can be read in two ways, both of which Wheeler endorsed. One is to extrapolate existing theory in order to make predictions for new, unexpected phenomena and then trust those predictions, even though they are made outside the domain for which the theory has been experimentally corroborated. This view of daring conservatism is to be found in an example that Wheeler gave in the Richtmyer lecture, where he claimed that he could have predicted nuclear fission two years before its experimental discovery, had he only trusted the extreme predictions of 1930s nuclear modelling. This view also applies to the use of general relativity in making novel predictions for astrophysics.\n\nBut as we have seen, it was another reading of daring conservatism that was initially more central to Wheeler's thinking: Using existing theory not to predict novel phenomena, but to solve existing (theoretical) problems and paradoxes that one might otherwise have been tempted to solve by introducing new theories. The central issue that Wheeler came to believe general relativity had the potential to solve was what he called the ``elementary particle problem''. A precise definition of this ``problem'' is hard to come by, but it meant something along the lines of obtaining a consistent description of the internal structure of elementary particles (which originally of course implied finding a consistent theory of point-like particles without structure). The solutions that Wheeler considered to this problem were shaped by several convictions, in particular that (i) the general idea of the solution should be expressible in classical language, (ii) the solution should be monistic, or at least not gratuitously introduce various types of particles, and (iii) the solution should ideally not involve any free parameters. All of these three conditions favored Wheeler's turn to GR, which was (i) a classical theory, (ii) dealing in universal substance (space-time), (iii) involving no free parameters beside the gravitational constant (which could be set to 1 in what Wheeler would later call Planck units).\n\nWe thus see that also the further development of Wheeler's career in relativity closely paralleled the overall development, as questions relativistic astrophysics (and thus the first reading of daring conservatism) gradually supplanted (or merged with) his original foundationalist aspirations, in what Roberto Lalli, J\\\"{u}rgen Renn and one of the authors (AB) have called the astrophysical turn of the late renaissance \\citep[p. 540f]{blum_2018_gravitational}. It turns out then that an important factor in assessing the relevance of the epistemic potential of GR in the renaissance is the question of ``potential for what?''. This is true not only with regards to what problems to solve, but also to what kind of work to generate. For we have clearly seen the strong pedagogical bent in the way in which Wheeler tackled general relativity, and the focus on problems to be solved; the general relativity that Wheeler was exploring was swarming with future PhD theses, theses in physics, that is, connecting the heretofore isolated field of general relativity to particle physics, quantum theory, and astrophysics. It was this aspect which turned Princeton from a research center among several to the home of the ``Wheeler School'' \\citep{christensen_2009_john, misner_2010_john}\n\nThis brings us to a final paradox: How to explain the great impact of Wheeler's approach to general relativity, given that the various solutions to the elementary particle problem that we have discussed in this paper were all eventually viewed as misguided. Neither worldlines and liaisons nor geons are nowadays regarded as fruitful ways for thinking about the structure of particles, and also the quantization of gravity did not yield to Wheeler's simple path integral vision. Our study at least suggests an answer to this paradox: The important thing was not so much the specific manner(s) in which Wheeler tried to resolve the elementary particle problem, but rather his keen sense for which elements of general relativity would turn out to be the most fruitful. \n\nLooking at Wheeler's trajectory thus also provides insight into where exactly the epistemic potential of general relativity lay, namely in its unique features as a theory: the determination of the equations of motion through the field equations, the non-linearity of the field equations, and that its quantization will lead to non-trivial new physics. Conceptual studies on the role of point particles in GR could thus segue into studies on the so-called problem of motion, studies on geons into studies of exact solutions of the full Einstein equations, studies on path integral quantization would come to be regarded as important puzzle pieces in the ongoing search for a quantum theory of gravity. Here too, we observe Wheeler's trajectory closely mirroring general trends, where isolated research centers originally focusing on GR-based speculative theorizing move, in the course of the Renaissance, to the study of important conceptual questions within general relativity, relevant to the emerging community at large. The question remains to what extent Wheeler's original interests actually shaped the problems considered important in the GR community of the renaissance and beyond. But this question is beyond the scope of our study, which focused on an individual intellectual trajectory and on a conversion from particle to field theory that turned out to be far more gradual than expected. If the reader thus takes home just one fact from our story, it might be this: For a few months there, in late 1953, John Wheeler believed in both particles and fields.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIt is generally believed that QCD undergoes chiral restoration\nat sufficiently high temperatures. This is supported by lattice\nsimulations \\cite{K95}, as well as by a variety of model \ncalculations. As the temperature grows, the value\nof the quark condensate increases from its negative $T=0$ value \nand approaches zero. As shown in Ref. \\cite{GL}, in the exact chiral \nlimit (zero current quark masses) chiral symmetry dictates the form of the\nfirst two terms ($\\sim T^2$ and $T^4$) in the low-temperature expansion \nof the quark condensate. At higher temperatures we do not have\nfundamental knowledge of the behavior of $\\langle \\overline{q}q \\rangle$, \nhowever most of model calculations show a phase transition at\ntemperatures $T \\sim {\\rm 150 - 200}$~MeV. Lattice calculations\nalso show a dramatic change of $\\langle \\overline{q}q \\rangle$ \nat similar temperatures. \n\nIn this letter we study the temperature dependence of the quark condensate \nin the two-flavor Nambu--Jona-Lasinio (NJL) model \\cite{NJL}. There have \nalready been several studies \\cite{HK85,BMZ87,Lutz92} of chiral restoration \nin this model. Our investigation brings a new important element: it includes \n{\\em meson loops} in a self-consistent way. \nPrevious studies have been performed at the quark-loop level only. \nAttempts have been made to include \nmeson loops in the NJL model, but self-consistency was not \ncompletely \nfulfilled \\cite{Heid}. \nOur approximation is symmetry-conserving \\cite{DSTL95,NBCRG96}, \nhence it is consistent \nwith all requirements of chiral symmetry.\nThe key ingredient is the self-consistency \nin solving the equation for the scalar density with meson loops present.\nThis makes the approach consistent with the requirements of \nchiral symmetry, such as the Goldstone theorem, Gell-Mann--Oaks--Renner\nand Goldberger-Treiman relations, or one-loop chiral expansions.\n\nWe find important qualitative and quantitative differences in\nthe temperature dependence of the quark condensate \nin our calculation with meson loops \ncompared to the case with quark loops only.\nWith quark loops only, at low temperatures the condensate\nremains flat, whereas in our case it changes considerably. We show that  \nin the exact chiral limit the change agrees with the prediction\nof the chiral perturbation theory \\cite{GL}. We also find that \nmeson loops decrease the temperature of chiral restoration by about 10\\%. \n\n\n\n\\section{Definition of the model}\n\\label{sec:model}\n\nThe Lagrangian of the two-flavor NJL model with scalar-isoscalar and\npseu\\-do\\-sca\\-lar-isovector interactions is\n\\begin{equation}\n{\\cal L}=\\bar q({\\rm i}\\partial^\\mu \\gamma_\\mu - m)q + \n {\\frac{1}{2 a^2}}\\left( (\\bar q q)^2 + \n (\\bar q{\\rm i}\\gamma_5 \\mbox{\\boldmath $\\tau$} q)^2\\right) \\; ,  \n \\label{eq:lagr}\n\\end{equation}\nwhere $q$ is the quark field, $m$ is the current quark mass, and \n$1\/a^2$ is the coupling constant. It is convenient to apply the \nformalism of effective action \\cite{ItzZub} to Lagrangian (\\ref{eq:lagr}). \nDetails of this procedure are given in Ref. \\cite{NBCRG96}. Meson fields \nare introduced in the usual way (partial bosonization), with $\\Phi = \n(\\Phi_0, \\mbox{\\boldmath $\\Phi$})$ related to the sigma and \nthe pion mean field. At the quark-loop level the \neffective action is\n\\begin{equation}\n{I}(\\Phi) = \\int d^4x \\left ( \\half{a^2} \\Phi^2 \n - a^2 m \\Phi_0 + \\half{a^2} m^2 \\right )\n - \\half{\\rm Tr}\\,\\ln (D^{\\dagger }D)  \\;,  \n\\label{eq:Seffq}\n\\end{equation}\nwhere  $D$ is the Dirac operator , \n $D = \\partial_\\tau - {\\rm i}{\\mbox{\\boldmath $\\alpha$}\n  \\cdot \\mbox{\\boldmath $\\nabla$} } \n    + \\beta \\Phi_0 + {\\rm i} \\beta \\gamma_5 \n   {\\mbox{\\boldmath $\\tau$}} \\cdot {\\mbox{\\boldmath $\\Phi$}}$.\nWe work in Euclidean space-time ($\\tau$, ${\\mbox{\\boldmath $x$}}$). \nIn Eq.~(\\ref{eq:Seffq}) we have replaced the usual ${\\rm Tr}\\,\\ln D$ term \nwith $\\half{\\rm Tr}\\,\\ln (D^{\\dagger }D)$, which is allowed in the \nabsence of anomalies. In fact, this replacement is necessary for the\nintroduction of the proper-time regulator \\cite{pt} used in many NJL \ncalculations, and also in this paper.\n\nMeson loops bring an additional term to the effective action \n\\cite{NBCRG96,ItzZub}\n\\begin{equation}\n{\\Gamma}(\\Phi) =  {I}(\\Phi) + \\half {\\rm Tr}\\,\\ln ({K}^{-1})  \\; . \n \\label{eq:Seffm}\n\\end{equation}\nThe inverse {\\em meson propagator} matrix $K$ is defined as \n $K^{-1}_{ab}(x,y) = \\frac{\\delta^2 I \\left( \\Phi \\right)}\n  {\\delta \\Phi_a(x) \\delta \\Phi_b(y)}$.\nIn Eqs.~(\\ref{eq:Seffq},\\ref{eq:Seffm}) \n${\\rm Tr}$ denotes the full trace, including functional space, \nisospin, and in addition color and spinor trace for quarks.\nIn the $N_c$-counting scheme, the quark loop term ${I}(\\Phi)$\nis the leading contribution of order ${\\cal O}(N_c)$, and the \nmeson loop term $\\frac{1}{2}{\\rm Tr}\\,\\ln {K}$ is of order ${\\cal O}(1)$.\nThus the one-meson-loop contributions give the first correction to\nthe leading-$N_c$ results. \n\n\nUsing standard methods, Green's functions can be obtained from \nEq.~(\\ref{eq:Seffm}) via differentiation with respect to mean \nmeson fields.\nOf particular importance is the one-point function, which gives the\nexpectation value of the sigma field.  The condition\n\\begin{eqnarray}\n\\label{GAP} \n\\frac{\\delta \\Gamma(\\Phi)} {\\delta \\Phi_0(x)}_{\\mid \\Phi_0(x)=S} \n= && a^2 (S-m) - \\half {\\rm Tr}\n\\left( (D^{\\dagger} D)^{-1} \\frac{\\delta (D^{\\dagger }D)}{\\delta \\Phi_0(x)}\n\\right )_{\\Phi_0(x)=S}  \\nonumber \\\\ \n&& + \\half {\\rm Tr} \\left ( K \\frac{\\delta K^{-1}}\n{\\delta \\Phi_0(x)} \\right)_{\\Phi_0(x)=S} = 0\n\\end{eqnarray}\nyields the equation for the vacuum expectation value of $\\Phi_0$, \nwhich we denote by $S$. \nIntroducing \n\\begin{eqnarray}\n\\label{prop}\nK_\\sigma(S,Q^2) & = & \\left ( 4 N_c f(S,Q^2)(Q^2+4S^2) + \na^2 m\/S \\right )^{-1} \\; , \\nonumber \\\\\nK_\\pi(S,Q^2) & = & \\left ( 4 N_c f(S,Q^2) Q^2 + a^2 m\/S  \\right )^{-1} \\;,\n\\end{eqnarray}\nand retaining terms up to order ${\\cal O}(N_c^0)$, \nEq.~(\\ref{GAP}) can be written in the form \\cite{NBCRG96}\n\\begin{eqnarray}\n\\label{gap0}\n& & a^2 \\left(S - m \\right) - 8 N_c \\, S g(S) \\nonumber \\\\\n& & + S \\frac{N_c}{4 \\pi^4} \\int d^4 Q \n   \\left\\{ \\left [2 f(S,0) + \\frac{d}{dS^2} \n \\left (f(S,Q^2)(Q^2 + 4 S^2) \\right ) \\right]\n   K_\\sigma(S,Q^2) \\right. \\nonumber \\\\\n& & + \\left. 3 \\left [2 f(S,0) + \\frac{d}{dS^2} f(S,Q^2) Q^2 \\right]  \n K_\\pi(S,Q^2) \\right\\} = 0.\n\\end{eqnarray}\nFunctions $g$ and $f$ in the above expressions are the {\\em quark \nbubble functions}. Their form is very simple if no cut-offs were present.\nIn this case we would have \n\\mbox{$g(S) = \\int {d^4k \\over (2\\pi)^4} {1 \\over k^2 + S^2}$} and\n\\mbox{$f(S,Q^2) = \\int {d^4k \\over (2\\pi)^4} {1 \\over k^2 + S^2}\n{1 \\over (k+Q)^2 + S^2 }$}, and Eq.~(\\ref{gap0}) could be interpreted\nvia standard Feynman diagrams (see Fig.~\\ref{fig:0}).\nIn the presence of a cut-off these functions are complicated. \nIn the case of the proper-time cut-off \\cite{pt} used here\nwe have \\cite{NBCRG96}\n\\begin{equation}\n\\label{g0r}\ng(S) = \\int {d^4k \\over (2\\pi)^4} \\int\\limits_{\\Lambda_f^{-2}}^{\\infty}\nds \\, \\exp\\left\\{-s [k^2+S^2]\\right\\} = {\\Lambda_f^2 \\over 16 \\pi^2} \n\\, E_2\\left[{S^2 \\over \\Lambda_f^2} \\right]\n\\end{equation}\nand\n\\begin{eqnarray}\n\\label{f0r}\nf(S,Q^2) &=& \\int {d^4k \\over (2\\pi)^4} \\int\\limits_{\\Lambda_f^{-2}}^{\\infty}\nds \\, s \\int\\limits_0^1 du \\exp\\left\\{-s [k^2 + S^2 + u(1-u) Q^2] \\right\\} \n \\nonumber \\\\\n&=& {1 \\over 16 \\pi^2} \\int\\limits_0^1 du \\, E_1\\left[{S^2 \\over \\Lambda_f^2}\n+ u (1-u) {Q^2 \\over \\Lambda_f^2} \\right],\n\\end{eqnarray}\nwhere $\\Lambda_f$ is the quark cut-off, and \nthe exponential integral is defined as \n\\mbox{$E_n(x) \\equiv \\int\\limits_1^{\\infty} dt \\, {e^{-xt} \\over t^n}$}.\n\nThe one meson-loop gap equation (\\ref{gap0}) requires also the\nintroduction of a regulator for meson momenta. In other words,  \nwe have to regularize the divergent integral over $d^4Q$.\nIn Ref. \\cite{NBCRG96} this was achieved by the substitution\n\\mbox{$\\int d^4Q  \\longrightarrow \\pi^2 \\int\\limits_0^{\\Lambda_b^2} \n dQ^2 \\, Q^2$}, where $\\Lambda_b$ was the four-dimensional Euclidean meson\nmomentum cutoff. \nIn the present study at finite temperatures, \nwe employ the\nthree-dimensional cutoff procedure, i.e., we make the replacement\n\\begin{equation}\n\\label{3dc}\n\\int d^4Q  \\longrightarrow 4 \\pi \\int d\\omega \n\\int\\limits_0^{\\Lambda_b} dq \\, q^2 \\,\\,\\,\\,\\,\\, ,\n\\end{equation}\nwhere $Q=(\\omega,{\\bf q})$ and $q = |{\\bf q}|$. The form\n(\\ref{3dc}) is convenient for the implementation of the boundary\nconditions satisfied by temperature Green's functions.\n\n\\vfill\n\n\\begin{figure}[b]\n\\xslide{fig0.ps}{3cm}{30}{370}{560}{490}\n\\caption{Diagramatic representation of Eq.~(\\ref{gap0}). The cross\nrepresents the first term, the one-quark-loop contribution \ncorresponds to the second term, and meson-loop terms \nrepresent subsequent terms. The solid lines represent the \nquark propagator $1\/(D^{\\dagger} D)$, the dased lines correspond to \nthe meson propagators $K$ of Eqs.~(\\ref{prop}), the external \ndased line represents scalar-isoscalar coupling, and the vertices follow\nfrom the form of $(D^{\\dagger} D)$.} \n\\label{fig:0}\n\\end{figure}\n\n\n\\section{Finite temperature}\n\\label{sec:gap}\n\nFor calculations at finite temperature $T$ we shall adopt the imaginary time\nformalism \\cite{Kapusta}. \nThis can be done by making the following replacement in the\nquark momentum integrals\n\\begin{equation}\n\\label{itf}\n\\int {d^4k \\over (2\\pi)^4} F(k) = \\int {dE \\over 2\\pi} \\int\n{d^3k \\over (2\\pi)^3} F(E,{\\bf k}) \\rightarrow T \\sum_{j=-\\infty}^{\\infty}\n\\int {d^3k \\over (2\\pi)^3} F(E_j,{\\bf k}).\n\\end{equation}\nHere $F(k)=F(E,{\\bf k})$ is an arbitrary integrand, \nand the sum runs over the fermionic \nMatsubara frequencies $E_j = (2j+1)\\pi T$. The integral over the meson\nfour-momenta should be also replaced by the sum of the form (\\ref{itf}).\nIn this case, however, the sum runs over the bosonic Matsubara\nfrequencies $\\omega_n = 2\\pi n T$.\nWith this prescription we can turn to\nthe calculation of the functions which are the finite temperature\nanalogs of $g(S)$ and  $f(S,Q^2)$.\nWe find\n\\begin{eqnarray}\ng(S,T) &=& T \\sum_j \\int {d^3k \\over (2\\pi)^3} \n\\int\\limits_{\\Lambda_f^{-2}}^{\\infty} ds \\, \n\\exp\\left\\{ -s \\left[ E_j^2\n+ {\\bf k}^2 + S^2 \\right] \\right\\} \\nonumber \\\\\n&=& {T \\Lambda_f \\over 8 \\pi^{{3 \\over 2}} } \\sum_j\nE_{3 \\over 2} \\left[{{S^2+E_j^2} \\over \\Lambda_f^2} \\right]\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& &f(S,n,q,T) = \nT \\sum_j \\int {d^3k \\over (2\\pi)^3} \n\\int_{\\Lambda_f^{-2}}^{\\infty} ds \\, s \\int\\limits_0^1 du \\times \n\\nonumber \\\\ \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& & \\exp\\left\\{ -s \\left[ \nS^2 + u(1-u)(\\omega_n^2 + {\\bf q}^2) + \\left[ {\\bf k}\n- {\\bf q} (1-u) \\right]^2 + \\left[E_j - \\omega_n (1-u) \\right]^2\n\\right] \\right\\}  \\nonumber \\\\ \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& &=  {T \\over 8 \\pi^{{3 \\over 2}} \\Lambda_f} \\sum_j\n\\int\\limits_0^1 du \\, E_{1\\over 2}\\left[{S^2 \\over \\Lambda_f^2} +\nu(1-u) {\\omega_n^2 + {\\bf q}^2 \\over \\Lambda_f^2} \\right. \n  +  \\left. {[E_j-\\omega_n(1-u)]^2 \\over \\Lambda_f^2} \\right] \\;.\n\\end{eqnarray}\nAnalogously, the inverse meson propagators become\n\\begin{eqnarray}\n\\label{propT}\n K_\\sigma(S,n,q,T) & = & \\left ( \n4 N_c f(S,n,q,T)(\\omega_n^2 + {\\bf q}^2+4S^2) + \n  a^2 m\/S \\right )^{-1} \\; , \\nonumber \\\\ \n K_\\pi(S,n,q,T) & = & \\left \n ( 4 N_c f(S,n,q,T)(\\omega_n^2 + {\\bf q}^2) + a^2 m\/S  \\right )^{-1} \\;.\n\\end{eqnarray}\nFinally, we can write the finite-temperature analog\nof Eq.~(\\ref{gap0}):\n\\begin{eqnarray}\n\\label{gapT}\n& & a^2 \\left(S - {m} \\right) - 8 N_c S \\, g(S,T) + \n {2 S N_c T \\over \\pi^2} \\sum_n \\int\\limits_0^{\\Lambda_b} dq \\, q^2 \\times \n\\nonumber \\\\\n& & \\left\\{ \\left [2 f(S,0,0,T) + \\frac{d}{dS^2} \\left (f(S,n,q,T)\n(\\omega_n^2 + {\\bf q}^2 + 4S^2) \\right ) \\right]\n K_\\sigma(S,n,q,T) \\right. \\nonumber \\\\\n& & + \\left. 3 \\left [2 f(S,0,0,T) + \\frac{d}{dS^2} f(S,n,q,T) \n(\\omega_n^2 + {\\bf q}^2) \\right]  \n K_\\pi(S,n,q,T) \\right\\} = 0 \\;.\\nonumber \\\\\n\\end{eqnarray}\nIf chiral symmetry is broken, then the above equation has a \nnontrivial solution for $S$. \nThe quark condensate and $S$ are related by the formula\n\\begin{equation}\n\\label{qq}\n\\langle \\overline{q}q \\rangle = - a^2 (S - m) \\;,\n\\end{equation}\nwhich follows immediately from the fact that \n $\\langle \\overline{q}q \\rangle = \\delta \\Gamma(\\Phi)\/\\delta m$\nand Eq.~(\\ref{eq:Seffm}).\n\n\n\\section{Low-temperature expansion in the chiral limit}\n\\label{sec:lowT}\n\nBefore presenting our numerical results for $\\langle {\\overline q} q\n\\rangle_T$\nlet us consider the low-temperature expansion. As shown by\nGasser and Leutwyler \\cite{GL}, {\\em in the chiral limit}\nthe low-temperature expansion of the quark condensate has the form\n\\begin{equation}\n\\label{eq:gl}\n\\langle \\overline{q} q \\rangle_T = \\langle \\overline{q} q \\rangle_0 \n\\left ( 1 - \\frac{T^2}{8 F_\\pi^2} - \\frac{T^4}{384 F_\\pi^4} + ... \\right ) .\n\\end{equation}\nFirst, let us do the $N_c$ counting in this formula. Since\n $F_\\pi \\sim {\\cal O}(\\sqrt{N_c})$, subsequent terms in the expansion are \nsuppressed by $1\/N_c$. Since our one-meson-loop \ncalculation accounts for first\nsubleading effects in the  $1\/N_c$ expansion, we can hope for reproducing \nonly the $T^2$ term in Eq.~(\\ref{eq:gl}). Further terms would require\nmore loops.\n\nUsing standard techniques \\cite{Kapusta}, \nthe sum over the bosonic Matsubara\nfrequencies in Eq.~(\\ref{gapT}) can be converted to a contour integral\nin the complex energy plane. By deforming this contour\nwe collect all contributions from the singularities of the\nintegrand, weighted with the thermal Bose distribution. At low\ntemperatures, the dominant contribution comes from the lowest\nlying pion pole, and other singularities are negligible. \nThus, the third term in\n(\\ref{gapT}) becomes\n\\begin{equation}\n\\label{ae1}\n{3T \\over \\pi^2} \\sum_n \\int\\limits_0^{\\Lambda_b} dq \\, q^2\n{1 \\over \\omega_n^2 + q^2} = {3 \\over 2\\pi^2 }\n\\int\\limits_0^{\\Lambda_b} dq \\, q \\left[ 1 + {2 \\over e^{q\/T} - 1} \\right].\n\\end{equation}\nWriting Eq. (\\ref{ae1}) we have approximated the function \n$f(M,n,q,T)$, appearing in the pion propagator, by its value at $n=q=0$. \nFor sufficiently large cutoff $\\Lambda_b$, the integral over the thermal \ndistribution function in (\\ref{ae1}) can be expressed by the Riemann\nzeta function $\\zeta(2) = \\pi^2\/6$. Thus, the final result for\n(\\ref{ae1}) is $ 3\\Lambda^2_b\/4 \\pi^2 + T^2\/2$.\nInserting the above result into the gap equation (\\ref{gapT}) we find,\nwith $m=0$, the following equality:\n\\begin{equation}\n\\label{ae2}\nh(S,T) \\equiv a^2 - 8N_c g(S,T) + {3\\Lambda_b^2 \\over\n4 \\pi^2} + \\half T^2 = 0.\n\\end{equation}\nEq. (\\ref{ae2}) defines implicitly the function $S(T)$,\nwhich satisfies the equation\n\\begin{equation}\n\\label{ae3}\n{dS \\over dT^2} = - \n{ \\partial h(S,T) \/ \\partial T^2 \\over \\partial h(S,T)   \/  \\partial S}\n= \\left[ 16 N_c\n{\\partial g(S,T) \\over \\partial S} \\right]^{-1}.\n\\end{equation}\nHere we have neglected the term $\\partial g(S,T) \/ \n\\partial T^2$, since it is exponentially suppressed by the factor\n $\\exp(-S\/T)$. Furthermore, the leading-$N_c$ term on the \nright hand side of (\\ref{ae3}) can be rewritten using the \nrelations \\cite{NBCRG96}\n $\\partial g(S,T)\/ \\partial S = -2S f(S,0)$ and \n \\mbox{$4 N_c f(S,0) = \\overline{F}_{\\pi}^2\/S^2$}, \n where $\\overline{F}_{\\pi}$ is the leading-$N_c$ \npiece of the pion decay constant. \nCollecting these equalities we \narrive at $dS\/dT^2 = -S\/(8\\overline{F}_{\\pi}^2)$, which finally gives\n\\begin{equation}\n\\label{ae4}\nS(T) = S(0) \\left[ 1 - {T^2 \\over 8\\overline{F}_{\\pi}^2} \\right].\n\\end{equation}\nProportionality (\\ref{qq}) implies that the above expression\ncoincides (in the large $N_c$ limit) with Eq.~(\\ref{eq:gl}).\nHence our method is consistent with a basic requirement of chiral\nsymmetry at the one-meson-loop level.\n\n\\section{Results}\n\\label{sec:res}\n\nIn the exact chiral limit the \nmodel has 3 parameters: $a$, $\\Lambda_f$, and $\\Lambda_b$. \nIn this paper we fix arbitrarily $\\Lambda_b\/\\Lambda_f = \\half$.\nThe remaining\n2 parameters are fixed by reproducing the physical value of\n $F_\\pi=93{\\rm MeV}$ and \na chosen value for $\\langle \\overline{q} q \\rangle_0$. \nFor the case of $m \\neq 0$ we have an extra parameter, $m$, which is fitted\nby requiring that the pion has its physical mass.\nWe compare results with meson loops to results with the quark loop\nonly ($\\Lambda_b = 0$).  Parameters for the two calculations\nare adjusted in such a way, that the values of $F_\\pi$, \n $\\langle \\overline{q} q \\rangle_0$, and $m_\\pi$ are the same.\n\nThe calculation of $F_\\pi$ with meson loops, although \nstraightforward, is rather tedious, so we \ndo not present it here. The method has been presented in detail in \nRef.~\\cite{NBCRG96,thesis}. The only difference in our calculation \nis that the three-momentum cut-off (\\ref{3dc}) rather than\nthe four-momentum cut-off of Ref.~\\cite{NBCRG96} is used.\n\n\\begin{figure}[b]\n\\xslide{fig1.ps}{13.5cm}{45}{160}{550}{680}\n\\caption{Dependence of of the quark condensate on $T^2$ \nin the chiral limit $m_\\pi=0$. The curves correspond to \nthe calculation\nwith meson loops (solid line), with quark loops only (dashed line),\nand the lowest-order chiral expansion (dotted line). The parameters\nfor the solid line and dashed line are adjusted in such a way that\n$F_\\pi = 93{\\rm MeV}$ and $\\langle \\overline{q} q \\rangle_0 \n =-(184{\\rm MeV})^3$. For the solid line $a=175{\\rm MeV}$, \n $\\Lambda_f = 723{\\rm MeV}$, and $\\Lambda_b = \\half \\Lambda_f$, \nwhereas for the\ndashed line $a=201{\\rm MeV}$, $\\Lambda_f = 682{\\rm MeV}$, and\n $\\Lambda_b = 0$. }\n\\label{fig:1}\n\\end{figure}\nFigure \\ref{fig:1} shows the dependence of $\\langle \\overline{q} q \\rangle$\non $T^2$. The solid line represents the case with meson loops. We note that\nat low temperatures the curve has a finite slope, as requested by\nEq.~(\\ref{ae4}). The slope is close to the leading-order \nGasser-Leutwyler result (dotted curve). As explained earlier, \nthe slopes would overlap in the large-$N_c$ limit. \nThis behavior is radically different from the case with quark loops \nonly (dashed curve). In this case at low temperatures \n \\mbox{$\\langle \\overline{q} q \\rangle_T - \n  \\langle \\overline{q} q \\rangle_0 \\sim e^{-M\/T}$}, \nwhere $M$ is the \nmass of the constituent quark. All derivatives of this function vanish\nat $T=0$, and $\\langle \\overline{q} q \\rangle$ is flat at the origin.\nWe can also see from the figure that the fall-off of the condensate\nis faster when the meson loops are included. \nIn fact, for the parameters of Fig.~\\ref{fig:1} we have an interesting\nphenomenon. At $T = 162{\\rm MeV}$ the condensate abruptly jumps to 0. There is \na first-order phase transition, with a latent heat necessary to melt\nthe quark condensate. Such a behavior is not present \nin the case of calculations without meson loops \\cite{HK85,BMZ87,Lutz92}.\nWe note that with meson loops present the chiral restoration \ntemperature is $162{\\rm MeV}$, {\\em i.e.} about 10\\% \nless than $176{\\rm MeV}$ of the quark-loop-only case.\n\\begin{figure}[b]\n\\xslide{fig2.ps}{13.5cm}{45}{160}{550}{680}\n\\caption{Same as Fig.~\\ref{fig:1} for $m_\\pi=139~{\\rm MeV}$, \n $F_\\pi = 93{\\rm MeV}$, and $\\langle \\overline{q} q \\rangle_0 \n =-(174{\\rm MeV})^3$.\nFor the solid line $a=164{\\rm MeV}$, \n $\\Lambda_f = 678{\\rm MeV}$, $\\Lambda_b = \\half \\Lambda_f$, and \n $m = 15{\\rm MeV}$, whereas for the\ndashed line $a=175{\\rm MeV}$, \n $\\Lambda_f = 645{\\rm MeV}$, $\\Lambda_b = 0$, and $m = 15{\\rm MeV}$. }\n\\label{fig:2}\n\\end{figure}\n\nFigure \\ref{fig:2} shows the same study, but for the physical\nvalue of $m_\\pi$. We note that now $\\langle \\overline{q} q \\rangle$ \n(solid line) is also flat at the origin, since the pion is\nno more massless, and at low $T$ we have \n \\mbox{$\\langle \\overline{q} q \\rangle_T - \\langle \\overline{q} q \\rangle_0\n \\sim e^{-m_\\pi\/T}$}.\nNevertheless, the region of this flatness is small, and at \nintermediate temperatures the curve remains close to the\nGasser-Leutwyler expansion. We note again that meson loops \nconsiderably speed up\nthe melting of the condensate compared to the case of quark loops only.\nHowever, there is no first-order phase transition such as in \nFig.~\\ref{fig:1}. Instead, we observe a smooth cross-over typical for the case\nof $m \\neq 0$. \n\nThe faster change of the quark condensate in our study\nis not surprising.\nIt is caused by the presence of light pions which \nare known to play a dominant role\nat low-temperatures \\cite{Heid}. \nThe behavior of  $\\langle \\overline{q} q \\rangle$ \nreflects this general feature.\nConcluding, we stress that the inclusion of meson loops \nin the NJL model \nqualitatively and quantitatively changes the results \nin comparison to the \ncalculations at the quark-loop level. In particular, we find finite slope of\n $\\langle \\overline{q} q \\rangle$ vs. $T^2$ at the origin in the chiral\nlimit, faster melting of the condensate, and lower chiral restoration \ntemperature. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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products and the algebraic structure of diffeomorphism groups}{{Free products and the algebraic structure of diffeomorphism groups}}\n\\def{Sang-hyun Kim and Thomas Koberda}{{Sang-hyun Kim and Thomas Koberda}}\n\\usepackage{hyperref}\n\\hypersetup{\n  \n   colorlinks=false,\n   plainpages,\n   urlcolor=black,\n   linkcolor=black\n   pdftitle=   {Free products and the algebraic structure of diffeomorphism groups},\n   pdfauthor=  {{Sang-hyun Kim and Thomas Koberda}}\n}\n\n\n\n\\theoremstyle{theorem}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{con}[thm]{Conjecture}\n\\newtheorem{que}[thm]{Question}\n\\newtheorem*{claim*}{Claim}\n\\newtheorem*{fact}{Fact}\n\\newtheorem{claim}{Claim}\n\\newtheorem{thmA}{Theorem}\n\\renewcommand*{\\thethmA}{\\Alph{thmA}}\n\\newtheorem{corA}[thmA]{Corollary}\n\\renewcommand*{\\thecorA}{\\Alph{corA}}\n\n\n\\theoremstyle{remark}\n\\newtheorem{exmp}[thm]{Example}\n\\newtheorem{rem}[thm]{Remark}\n\n\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\newtheorem{prob}{Problem}[section]\n\\newtheorem{exc}{Exercise}[section]\n\n\\begin{document}\n\\title{Free products and the algebraic structure of diffeomorphism groups}\n\\date{\\today}\n\\keywords{free product; metabelian group; Thompson's group; right-angled Artin group; smoothing; co-graph}\n\\subjclass[2010]{Primary: 57M60; Secondary: 20F36, 37C05, 37C85, 57S05}\n\n\\author[S. Kim]{Sang-hyun Kim}\n\\address{Department of Mathematical Sciences, Seoul National University, Seoul, Korea}\n\\email{s.kim@snu.ac.kr}\n\\urladdr{http:\/\/cayley.kr}\n\n\\author[T. Koberda]{Thomas Koberda}\n\\address{Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA}\n\\email{thomas.koberda@gmail.com}\n\\urladdr{http:\/\/faculty.virginia.edu\/Koberda}\n\n\n\n\n\n\\begin{abstract}\nLet $M$ be a compact one--manifold,\nand let $\\Diffb(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation.\nWe prove that if $G$ is a group which is not virtually metabelian, then $(G\\times\\Z)*\\Z$ is not realized as a subgroup of $\\Diffb(M)$.\nThis  gives the first examples of finitely generated groups $G,H\\le \\Diff_+^\\infty(M)$ such that $G\\ast H$ does not embed into $\\Diffb(M)$.\nBy contrast, \nfor all countable groups $G,H\\le\\Homeo^+(M)$ there exists an embedding $G\\ast H\\to \\Homeo^+(M)$.\nWe deduce that many common groups of homeomorphisms do not embed into $\\Diffb(M)$, for example the free product of $\\bZ$ with Thompson's group $F$.\nWe also complete the classification of right-angled Artin groups which can act smoothly on $M$\nand in particular, recover the main result of a joint work of the authors with Baik~\\cite{BKK2014}. \nNamely, a right-angled Artin group $A(\\gam)$ either admits a faithful $C^{\\infty}$ action on $M$, or $A(\\gam)$ admits no faithful $\\Cb$ action on $M$.  In the former case, $A(\\gam)\\cong\\prod_i G_i$\nwhere $G_i$ is a free product of free abelian groups.\nFinally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.\n\\end{abstract}\n\n\\maketitle\n\n\n\n\\section{Introduction}\n\nLet $M$ be a compact one--manifold. In this article, we study the algebraic structure of the group $\\Diffb(M)$, where here $\\Diffb(M)$ denotes the group of $C^1$ diffeomorphisms of $M$ whose derivatives have bounded variation.\nSpecifically, we consider the restrictions placed on subgroups of $\\Diffb(M)$ by the $\\Cb$ regularity assumption. Our main result implies that there is a large class $\\AA_0 $ of finitely generated subgroups of $\\Diff_+^\\infty(M)$ such that for all $G,H\\in\\AA_0$, the free product $G\\ast H$ can never be realized as a subgroup of $\\Diffb(M)$; see Corollary~\\ref{cor:classification}.\n\nAs a corollary, we complete a program initiated by Baik and the authors in~\\cite{BKK2014,BKK2016} to decide which right-angled Artin groups admit faithful $C^{\\infty}$ actions on a compact one--manifold, and exhibit many classes of finitely generated subgroups of $\\Homeo^+(M)$ which cannot be realized as subgroups of $\\Diffb(M)$.\n\n\\subsection{Statement of results}\n\nUnless otherwise noted, $M$ will denote a compact one--manifold.\n That is to say, $M$ is a finite union of disjoint closed intervals $I=[0,1]$ and circles $S^1=\\bR\/\\bZ$.\nAn \\emph{action} on a one--manifold is always assumed to mean \nan orientation preserving action. \nFor each $0\\le r\\le\\infty$ or $r=\\omega$, we let $\\Diff^r_+(M)$ denote the group of orientation preserving $C^r$ (analytic if $r=\\omega$) diffeomorphisms.\nWe write $\\Homeo^+(M)=\\Diff^0_+(M)$.\nOur main result is the following:\n\n\\begin{thm}\\label{thm:main}\nIf $G$ is a group which is not virtually metabelian, then the group $(G\\times\\Z)*\\Z$ admits no faithful $C^{1+\\mathrm{bv}}$ action on $M$.\n\\end{thm}\n\n\nTheorem~\\ref{thm:main} clearly implies that $(G\\times\\Z)*\\Z$ does not embed into $\\Diff^r_+(M)$ for every level of regularity $r\\geq 2$.\n A key step in our proof of Theorem~\\ref{thm:main} is the following result on $C^1$--smoothability:\n\\begin{thm}[Theorem~\\ref{t:tech-main}]\\label{thm:tech-main-intro}\nLet $X\\in\\{I,S^1\\}$,\nand let\n $a,b,t\\in\\Diff^1_+(X)$.\n If\n\\[\\supp a\\cap\\supp b=\\varnothing,\\] \nthen the group $\\form{a,b,t}$  is not isomorphic to $\\bZ^2\\ast\\bZ$.\n\\end{thm}\n\n\nWe have stated Theorem~\\ref{thm:main} as above for clarity and concision, though several stronger statements can be deduced from the proof we give. In the particular case of finitely generated groups, we note the following, which should be compared to Corollary~\\ref{cor:cnt}:\n\n\n\\begin{cor}\\label{cor:fg1}\nIf $G$ is a finitely generated group which is not virtually abelian, then $(G\\times\\Z)*\\Z$ admits no faithful $\\Cb$ action on $M$. \n\\end{cor}\nThe motivation for proving Theorem~\\ref{thm:main} came from investigating right-angled Artin subgroups of $\\Diffb(M)$ (cf. Corollary~\\ref{cor:classification} below). The simplest right-angled Artin group to which Theorem~\\ref{thm:main} applies is $(F_2\\times\\Z)*\\Z$:\n\n\n\\begin{cor}\\label{cor:FtimesZ}\nThe group $(F_2\\times\\Z)*\\Z$ is not a subgroup of $\\Diffb(M)$.\n\\end{cor}\n\nSince the hypotheses on Theorem~\\ref{thm:main} are relatively weak, there are many finitely generated subgroups of $\\Homeo^+(M)$ which can be thus shown to admit no faithful $\\Cb$ actions on a compact manifold. Recall that \\emph{Thompson's group $F$} is a group of piecewise linear homeomorphisms of the interval, with dyadic break points and with all slopes being powers of two. \\emph{Thompson's group $T$} is the analogous group defined for the circle. A famous result of Ghys--Sergiescu~\\cite{GS1987} says that the standard actions of $T$ and $F$ are topologically conjugate into a group of $C^{\\infty}$ diffeomorphisms of the circle and of the interval, respectively. However, we have the following:\n\n\\begin{cor}\\label{cor:Thompson}\nThe groups $F*\\Z$ and $T*\\Z$ are not subgroups of $\\Diffb(M)$.\n\\end{cor}\n\nIn the vein of Corollary~\\ref{cor:Thompson}, we do not know the answer to the following:\n\n\\begin{que}\\label{que:thompson-critical}\nAre the groups $F*\\Z$ and $T*\\Z$ subgroups of $\\Diff_+^1(I)$ and $\\Diff_+^1(S^1)$ respectively? If so, what is their optimal regularity, i.e. the supremum of the H\\\"older continuity exponent $\\tau\\in [0,1)$ such that these groups embed in $\\Diff_+^{1+\\tau}$ for the relevant manifolds (cf.~\\cite{JNR2018,KK2017crit})?\n\\end{que}\n\n\n\n\nReturning to the original motivation, Theorem \\ref{thm:main} combined with results of Farb--Franks and Jorquera completes the classification of right-angled Artin groups admitting actions of various regularities on compact one--manifolds. Before stating the result, we define some terminology. We write $\\gam$ for a finite simplicial graph with vertex set $V(\\gam)$ and edge set $E(\\gam)$. The \\emph{right-angled Artin group} (or \\emph{RAAG}, for short) on $\\gam$ is defined as \\[A(\\gam)=\\langle V(\\gam)\\mid [v,w]=1 \\textrm{ if and only if }\\{v,w\\}\\in E(\\gam)\\rangle.\\]\n\nA subgraph $\\Lambda$ of a graph $\\gam$ is called a \\emph{full} subgraph if $\\Lambda$ is spanned by the vertices of $\\Lambda$. That is, two vertices in $\\Lambda$ are adjacent if and only if they are adjacent in $\\gam$.\nA simplicial graph $\\gam$ is called \\emph{$P_4$--free} if no full subgraph of $\\gam$ is isomorphic to a path $P_4$ on four vertices. Such graphs are often called \\emph{cographs}~\\cite{CLB1981,KK2013}. We write $\\mathcal{K}$ for the class of cographs. It is well--known that cographs can be fit into a hierarchy which is defined as follows:\n\\begin{enumerate}\n\\item\nThe class $\\mathcal{K}_0$ consists of a single vertex;\n\\item\nIf $n\\geq 1$ is odd then $\\mathcal{K}_n$ is obtained by taking $\\mathcal{K}_{n-1}$ together with finite joins of elements in $\\mathcal{K}_{n-1}$;\n\\item\nIf $n\\geq 2$ is even then $\\mathcal{K}_n$ is obtained by taking $\\mathcal{K}_{n-1}$ together with finite disjoint unions of elements in $\\mathcal{K}_{n-1}$.\n\\end{enumerate}\nHere, a \\emph{join} of two simplicial graphs $X$ and $Y$ is a simplicial graph consisting of the disjoint union of $X$ and $Y$, together with an edge of the form $\\{x,y\\}$ for every vertex $x$ of $X$ and every vertex $y$ of $Y$.\n\n\nWe have that $\\mathcal{K}_0\\subset\\mathcal{K}_1\\subset\\KK_2\\subset\\cdots$ and \\[\\mathcal{K}=\\bigcup_i\\mathcal{K}_i.\\] \nNote that join and disjoint union correspond to direct product and free product respectively, so that right-angled Artin groups on cographs are exactly the smallest class of groups containing $\\Z$, which is closed under finite direct products, and which is closed under finite free products. The reader will observe that if $\\gam\\in\\mathcal{K}_0$ then $A(\\gam)\\cong\\Z$. Similarly, if $\\gam\\in\\mathcal{K}_1$ then $A(\\gam)$ is free abelian, and if $\\gam\\in\\mathcal{K}_2$ then $A(\\gam)$ is a free product of free abelian groups.\nIf $\\gam\\in\\mathcal{K}_3$ then $A(\\Gamma)$ can be written as \\[A(\\Gamma)=\\prod_{i=1}^m G_i,\\]\n  where each $G_i$ is a free product of free abelian groups.\nIn~\\cite{BKK2016}, Baik and the authors proved that if $A(\\gam)$ admits an injective homomorphism into $\\Diffb(M)$, then $\\gam\\in\\KK$. We will deduce a strengthening of this result, using Theorem~\\ref{thm:main}.\n\n\n\n\n\n\\begin{cor}[cf. \\cite{BKK2016}]\\label{cor:classification}\nLet $A(\\gam)$ be a right-angled Artin group.\n\\begin{enumerate}\n\\item (see \\cite{FF2003,Jorquera})\nThere exists an injective homomorphism $A(\\gam)\\to\\Diff_+^1(M)$; thus, $A(\\gam)$ admits a faithful $C^1$ action of $M$.\n\\item\nIf there exists an injective homomorphism $A(\\gam)\\to\\Diffb(M)$ then $\\gam\\in\\mathcal{K}_3$; conversely, if $\\gam\\in\\mathcal{K}_3$ then $A(\\gam)\\le\\Diff_+^{\\infty}(M)$.\n\\end{enumerate}\n\\end{cor}\n\nIn particular, if we define\n\\[\n\\AA_0=\\{A(\\Gamma)\\mid \\Gamma\\in\\KK_3\\setminus\\KK_2\\}\\]\nthen each $G\\in\\AA_0$ embeds into $\\Diff_+^\\infty(M)$,\nbut for all $G,H\\in\\AA_0$ the group $G\\ast H$ never embeds into $\\Diff_+^{1+\\mathrm{bv}}(M)$.\nBy contrast, we have the following, which is well--known from several contexts.\n\n\\begin{prop}[cf. \\cite{KM1996,Rivas2012JAlg,BS2015}]\\label{prop:free prod}\nThe class of countable subgroups of $\\Homeo^+(M)$ is closed under finite free products.\n\\end{prop}\n\nIn the same spirit of Question~\\ref{que:thompson-critical} and the authors' paper~\\cite{KK2017crit}, we have the following:\n\n\\begin{que}\nLet $\\gam\\notin\\mathcal{K}_3$. What is the supremum of $\\tau\\in [0,1)$ for which $A(\\gam)$ embeds in $\\Diff_+^{1+\\tau}(M)$? Does $\\tau$ depend on $\\gam$?\n\\end{que}\n\nLet $S$ be an orientable surface of genus $g$ and with $n$ punctures or boundary components. We say that $S$ is \\emph{sporadic} if \\[3g-3+n\\leq 1.\\] We write $\\Mod(S)$ for the mapping class group of $S$, i.e. $\\Mod(S)=\\pi_0(\\Homeo^+(S))$. Using the main result of~\\cite{Koberda2012}, we immediately recover the following result as a corollary of Theorem~\\ref{thm:main}:\n\n\\begin{cor}[cf.~\\cite{BKK2016}]\nLet $M$ be a compact one--manifold, and let $S$ be an orientable finite-type surface. Then there exists a finite index subgroup $G\\le\\Mod(S)$ such that $G\\le\\Diffb(M)$ if and only if $S$ is sporadic.\n\\end{cor}\n\n\n\nTheorem \\ref{thm:main} allows us to build a hierarchy on right-angled Artin groups, whose levels correspond to right-angled Artin groups with more or fewer ``dynamically different\" actions on the circle. Roughly speaking, two group actions \\[\\rho_1,\\rho_2\\colon G\\to\\Homeo^+(S^1)\\] are \\emph{semi--conjugate} (or, \\emph{monotone-equivalent}) if there exists \nanother action \\[\\rho\\co G\\to\\Homeo^+(S^1)\\]\nand monotone degree one maps $h_i\\colon S^1\\to S^1$ such that\n\\[h_i\\circ \\rho=\\rho_i\\circ h_i\\] for each $i=1,2$. See~\\cite{Ghys1987,CD2003IM,Ghys2001,Mann-hb,BFH2014,KKMj2016} for instance, and the many references therein.\nA \\emph{projective action} of a group $G$ is a representation\n\\[\n\\rho\\co G\\to \\PSL_2(\\bR),\\]\nwhere $\\PSL_2(\\R)$ sits inside of $\\Homeo^+(S^1)$ as the group of projective analytic diffeomorphisms of $S^1$. \n\n\n\\begin{cor}\\label{cor:conj}\nLet $A(\\gam)$ be a right-angled Artin group.\n\\begin{enumerate}\n\\item\nIf $\\gam\\in\\mathcal{K}_2$, then $A(\\gam)$ admits uncountably many distinct semi--conjugacy classes of faithful orientation preserving projective actions on $S^1$;\n\\item\nIf $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then any faithful orientation preserving $\\Cb$ action of $A(\\gam)$ on $S^1$ has a periodic point and no dense orbits and hence admits at most countably many distinct semi--conjugacy classes of $\\Cb$ actions on $S^1$;\n\\item\nIf $\\gam\\notin\\mathcal{K}_3$ then $A(\\gam)$ admits no faithful $\\Cb$ action on $S^1$.\n\\end{enumerate}\n\\end{cor}\n\nIn the case of analytic actions on a compact connected one--manifold $M$, one has the following result of Akhmedov and Cohen:\n\n\\begin{thm}[See~\\cite{AC2015TA}]\nThe right-angled Artin group $A(\\gam)$ embeds into $\\Diff^\\omega(M)$ if and only if $\\Gamma\\in\\KK_2$, i.e. $A(\\gam)$ decomposes as a free product of free abelian groups.\n\\end{thm}\n\n\\subsection{Notes and references}\n\nThis paper\nreveals some of the subtlety of the interplay between algebra and regularity in diffeomorphism groups of one--manifolds. \nOur paper arose during the effort to complete the classification of right-angled Artin subgroups of $\\Diff_+^{\\infty}(S^1)$\nin the spirit of~\\cite{BKK2016}.\nThe essential content of this paper is Lemma~\\ref{l:main}, which exhibits an explicit element of the kernel of any given $\\Cb$ action of $(G\\times\\Z)*\\Z$ action on a compact one--manifold.\nSince a group of the form $(G\\times\\Z)*\\Z$ is generally simpler than the right-angled Artin groups considered in~\\cite{BKK2016}, it is more difficult to find elements in the kernel of a given action, and therefore we develop more sophisticated tools here. We note that our main result does subsume the main result of~\\cite{BKK2016}. Indeed, the main result of~\\cite{BKK2016} is that there is no injective homomorphism $A(P_4)\\to\\Diffb(M)$, where here \\[A(P_4)=\\langle a,b,c,d\\mid [a,b]=[b,c]=[c,d]=1\\rangle.\\] The group $A(P_4)$ contains a copy of $(F_2\\times\\Z)*\\Z$, which cannot embed in $\\Diffb(M)$ by Corollary~\\ref{cor:FtimesZ}. An explicit embedding of $(F_2\\times\\Z)*\\Z$ into $A(P_4)$ is given by $\\langle a,b,c,dad^{-1}\\rangle\\le A(P_4)$ (see~\\cite{KK2013} for a discussion of this fact).\n\nThe program completed by Corollary~\\ref{cor:classification} fully answers a question raised in a paper of M. Kapovich (attributed to Kharlamov) as to which right-angled Artin groups admit faithful $C^{\\infty}$ actions on the circle~\\cite{Kapovich2012}.\n\nRight-angled Artin subgroups of diffeomorphism groups of one--manifolds find an analogue in right-angled Artin subgroups of linear groups. It is well--known that right-angled Artin groups are always linear over $\\Z$ and hence admit injective homomorphisms into $\\SL_n(\\Z)$~\\cite{Humphries1994,HW1999,DJ2000}. For $\\SL_3(\\Z)$, it is still unclear which right-angled Artin groups appear as subgroups. Long--Reid~\\cite{LR2011} showed that $F_2\\times\\Z$ is not a subgroup of $\\SL_3(\\Z)$, which implies that any right-angled Artin subgroup of $\\SL_3(\\Z)$ is a free product of free abelian groups of rank at most two. It is currently unknown whether or not $\\Z^2*\\Z$ is a subgroup of $\\SL_3(\\Z)$.\n\nA consequence of the technical work behind Theorem~\\ref{thm:main} is a certain criterion to prove that a group contains a lamplighter subgroup. See Lemma~\\ref{l:recursive} and Proposition~\\ref{t:tech-main2} for precise statements.\n\nA crucial step in our proof of the main theorem is a $C^1$--rigidity result, which is Theorem~\\ref{t:tech-main}. \nWe note that Thurston~\\cite{Thurston1974Top}, Calegari~\\cite{Calegari2008AGT}, Navas~\\cite{Navas2008GAFA} and Bonatti, Monteverde, Navas and Rivas~\\cite{BMNR2017MZ} explored various remarkable $C^1$--rigidity results. \nIn particular, a $C^1$--rigidity result on Baumslag--Solitar groups in~\\cite{BMNR2017MZ} was employed in a very recent paper by Bonatti, Lodha and Triestino~\\cite{BLT2017},\nto produce certain piecewise affine homeomorphism groups of $\\bR$ which do not embed into $\\Diff^1_+(I)$.\n\nAs for other classes of groups of homeomorphisms which cannot be realized as groups of $\\Cb$ diffeomorphisms, Corollary~\\ref{cor:Thompson} is a complement to a result of the second author with Lodha~\\cite{KL2017}, in which they show that certain ``square roots\" of Thompson's group $F$ may fail to act faithfully by $\\Cb$ diffeomorphisms on a compact one--manifold, even though they are manifestly groups of homeomorphisms of these manifolds. Thus, the Ghys--Sergiescu Theorem appears to place $F$ and $T$ at the cusp of smoothability in the sense that even relatively minor algebraic variations on $F$ and on $T$ fail to be smoothable.\n\nFinally, we remark on the optimality of the differentiability hypothesis. \nCorollary~\\ref{cor:classification} shows that every right-angled Artin group can act faithfully by $C^1$ diffeomorphisms, but nearly none of them can act by $\\Cb$ diffeomorphisms. \nAs for Corollary~\\ref{cor:not closed} and Proposition~\\ref{prop:free prod}, we have the following result (based on~\\cite{BMNR2017MZ} and on a suggestion by Navas), which is proved in the authors' recent manuscript~\\cite{KK2017crit}:\n\n\\begin{prop}[\\cite{KK2017crit}]\\label{prop:c1freeproduct}\nLet $M\\in\\{I,S^1\\}$, and let $BS(1,2)$ be the Baumslag--Solitar group with the presentation $\\langle s,t\\mid sts^{-1}=t^2\\rangle$. Then the group $(BS(1,2)\\times\\Z)*\\Z$ is not a subgroup of $\\Diff_+^1(M)$. In particular, the class of finitely generated subgroups of $\\Diff_+^1(M)$ is not closed under taking finite free products.\n\\end{prop}\n\n\\section{Background on one--dimensional smooth dynamics}\nWe very briefly summarize the necessary background from one--dimensional dynamics. The reader may also consult~\\cite{BKK2016}, parts of which we repeat here, and where we direct the reader for proofs.\n\n\\subsection{Poincar\\'e's theory of rotation numbers}\n\nLet $f\\in\\Homeo^+(S^1)$, and let $\\tilde f\\co \\bR\\to\\bR$ be  an arbitrary lift of $f$.\nThen the \\emph{rotation number} of $f$ is defined as\n\\[\n\\rot f = \\lim_{n\\to\\infty}\\frac{\\tilde f^n(x)}{n}\\in \\bR\/\\bZ=S^1\\]\nwhere $x\\in \\bR$. Then $\\rot f$ is well-defined, and independent of the choice of a lift $\\tilde f$ and a base point $x\\in \\bR$; see~\\cite{Navas2011} for instance. \nThe set of periodic points of $f$ is denoted as $\\Per f$.\nLet us record some elementary facts.\n\\begin{lem}\\label{l:inv}\nFor $f\\in\\Homeo^+(S^1)$, the following hold.\n\\be\n\\item\n$\\rot(f) =0$ if and only if $\\Fix f\\ne\\varnothing$.\n\\item\n$\\rot(f)\\in\\bQ$ if and only if $\\Per f\\ne\\varnothing$.\n\\item\\label{p:inv}\nIf $x\\in S^1$ and $g\\in\\Homeo^+(S^1)$ satisfy\n\\[\nf^n(x)=g^n(x)\\]\nfor all $n\\in\\bZ$, then $\\rot(f)=\\rot(g)$.\n\\ee\n\\end{lem}\n\nThe rotation number is a continuous class function (that is, constant on each conjugacy class)\n\\[\n\\rot\\co \\Homeo^+(S^1)\\to S^1.\\]\nMoreover, the rotation number restricts to a group homomorphism on each amenable subgroup of $\\Homeo^+(S^1)$; see~\\cite{Ghys2001}.\n\nLet us recall the following classical result.\n\n\n\\begin{thm}[H\\\"older's Theorem~\\cite{Holder1901}; see~\\cite{Navas2011}]\\label{t:hoelder}\nA group acting freely on $\\bR$ or on $S^1$ by orientation preserving homeomorphisms is abelian.\n\\end{thm}\n\nWe will use the following variation, which is similar to \\cite[Theorem 2.2]{FF2001}.\n\\begin{cor}[cf.~\\cite{FF2001}]\\label{c:hoelder-s1}\nLet  $X$ be a nonempty closed subset of $S^1$,\nand let $G$ be a group acting freely on $X$  by orientation preserving homeomorphisms.\nThen  the action of $G$ extends to a free action $\\rho\\co G\\to\\Homeo^+(S^1)$ such that \n\\[\\rot\\circ\\rho\\co G\\to S^1\\] is an injective group homomorphism.\n\\end{cor}\n\\bp\nIf $(a,b)$ is a component of $S^1\\setminus X$ and $g \\in  G$, then $(ga,gb)$ is also a component of $S^1\\setminus X$. So, $G$ extends to some action $\\rho$  on $S^1$ in an affine manner.  \nPut \\[S^1\\setminus X = \\coprod_{i\\ge1} I_i.\\] \nIf $\\rho (g)y = y$ for some $g \\in  G$ and for some $y \\in  S^1$, then $y \\in  I_i$ for some $i$. We have that $ \\rho (g)$ restricts to the identity on $ I_i$ by definition. This would imply $\\rho (g)\\partial I_i = g \\partial I_i = \\partial I_i$, and so, $g = 1$. That is, the action $\\rho$  of $G$ on $S^1$ is free.\n\nBy H\\\"older's theorem, we see $\\rho(G)\\cong G$ is abelian. Since abelian groups are amenable, \nwe have that $\\rot \\circ \\rho$ is a group homomorphism. The freeness of the action $\\rho$ implies that $\\rot\\circ\\rho(g)\\ne0$ for all nontrivial $g$.\\ep\n\n\n\n\\subsection{Kopell--Denjoy theory}\nLet $M\\in\\{I,S^1\\}$.\nWe denote by $\\var(g;M)$ the total variation of a map $g\\co M\\to\\bR$:\n\\[\\var(g;M) = \\sup\\left\\{\\sum_{i=0}^{n-1}\\left| g(a_{i+1})- g(a_i)\\right|\n\\co (a_i\\co 0\\le i\\le n)\\text{ is a partition of }M\\right\\}.\\]\nIn the case $M=S^1$, we require $a_n=a_0$ in the above definition.\nFollowing \\cite{Navas2011},\nwe say a $C^1$ diffeomorphism $f$ on $M$ is $\\Cbv$ if $\\var(f';M)<\\infty$.\nWe let $\\Diffb(M)$ denote\nthe group of orientation preserving $\\Cbv$ diffeomorphisms of $M$.\nThe following two results play a fundamental role on the study of $\\Cbv$ diffeomorphisms.\n\n\\begin{thm}[Denjoy's Theorem {\\cite{Denjoy1958}, \\cite{Navas2011}}]\\label{t:denjoy}\nIf $a\\in\\Diffb(S^1)$ and $\\Per a=\\emptyset$, then $a$ is topologically conjugate to an irrational rotation.\n\\end{thm}\n\n\\begin{thm}[Kopell's Lemma {\\cite{Kopell1970}, \\cite{Navas2011}}]\\label{t:kopell}\nSuppose $a\\in\\Diffb[0,1)$, $b\\in\\Diff_+^1[0,1)$, and $[a,b]=1$.\nIf $\\Fix a\\cap (0,1)=\\varnothing$ and $b\\ne 1$,\nthen $\\Fix b\\cap (0,1)=\\varnothing$.\n\\end{thm}\nWe remark that the original statement by Kopell was for $C^2$--regularity. Navas extended her result to $C^{1+\\mathrm{bv}}$ case~\\cite[Theorem 4.1.1]{Navas2011}.\n\nLet $X$ be a topological space. Then we define the \\emph{support} of $h\\in \\Homeo(X)$ as\n\\[\\supp h = X\\setminus\\Fix h.\\] \nIt is convenient for us to consider the non--standard \\emph{open support} of a homeomorphism as defined here, whereas many other authors use the closure of the open support. We will consistently mean the  open support unless otherwise noted.\n\nFor a subgroup $G\\le\\Homeo(X)$,  we put\n\\[\\supp G = \\bigcup_{g\\in G}\\supp g.\\]\nWe say that $f\\in\\Homeo(X)$ is \\emph{grounded} if $\\Fix f\\neq\\varnothing$. We note that every $f\\in\\Homeo^+(I)$ is grounded by definition.\n\nThe following important observations on commuting $\\Cbv$ diffeomorphisms essentially builds on Kopell's Lemma and H\\\"older's Theorem. \n\n\n\\begin{lem}\\label{l:disj-abel}\nThe following hold:\n\\be\n\\item (Disjointness Condition,~\\cite{BKK2016})\nLet $M\\in\\{I,S^1\\}$,\nand let $a,b\\in\\Diffb(M)$ be commuting grounded diffeomorphisms.\nIf $A$ and $B$ are components of $\\supp a$ and $\\supp b$ respectively,\nthen either $A=B$ or $A\\cap B=\\varnothing$.\n\\item (Abelian Criterion, cf.~\\cite{FF2001})\nIf $a,b,c\\in\\Diffb(I)$ satisfy $\\Fix a =\\partial I$ and that\n$[a,b]=1=[a,c]$, then $[b,c]=1$.\n\\ee\n\\end{lem}\n\n\\begin{rem}\nThe abelian criterion as given by Farb and Franks is a straightforward consequence of H\\\"older's Theorem and Kopell's Lemma. We thank one of the referees for pointing out  another simple proof using Szekeres' Theorem~\\cite{Navas2011}.\n\\end{rem}\n\nThe notation $f\\restriction_A$ for a function $f$ (or set of functions) and a set $A$ means the restriction to $A$.\nWe will need the following properties of centralizer groups.\n\n\n\\begin{lem}\\label{l:circle}\nLet $a\\in \\Diffb(S^1)$ be an infinite order element,\nand let $Z(a)$ be the centralizer of $a$ in $\\Diffb(S^1)$.\nThen the following hold.\n\\be\n\\item\\label{p:zz}\nIf $a$ is grounded and if a group $H\\le Z(a)$ is generated by grounded elements, then every element in $H$ is grounded and moreover,\n\\[\\supp a\\cap \\supp [H,H]=\\varnothing.\\]\n\\item\\label{p:rot-irr}\nIf $\\rot a\\not\\in\\bQ$, then $Z(a)$ is topologically conjugate to a subgroup of $\\operatorname{SO}(2,\\bR)$.\n\\item\\label{p:rot-q}\nIf $\\rot a\\in\\bQ$, then $\\rot Z(a)\\sse\\bQ$.\n\\item\\label{p:hom} \n({cf. \\cite[Lemma 3.4]{FF2001}})\nThe rotation number restricts to a homomorphism on $Z(a)$;\nin particular, every element of $[Z(a),Z(a)]$ is grounded.\n\\ee\n\\end{lem}\n\\bp\n(\\ref{p:zz})\nLet $J$ be a component of $\\supp a$.\nBy the Disjointness Condition, the group $H$ acts on the open interval $J$. Since $H$ fixes $\\partial J$, every element of $H$ is grounded.\nThe Abelian Criterion implies that\n\\[[H,H]\\restriction_J=1.\\]\nSo, we have that $J\\cap \\supp[H,H]=\\varnothing$.\n\n(\\ref{p:rot-irr}) By Denjoy's Theorem, the map $a$ is topologically conjugate to an irrational rotation. The centralizer of an irrational rotation in $\\Homeo^+(S^1)$ is $\\operatorname{SO}(2,\\bR)$;\nsee~\\cite[Proposition 2.10]{FF2001} or~\\cite[Exercise 2.2.12]{Navas2011}. \n\n\n\n(\\ref{p:rot-q})\nSuppose some element $b$ in $Z(a)$ has an irrational rotation number. \nSince $a\\in Z(b)$, part (\\ref{p:rot-irr}) implies that $a$ is conjugate to a rotation. This is a contradiction, for a rotation with a rational rotation number must have a finite order.\n\n(\\ref{p:hom})\nIf $\\rot a$ is irrational, then the conclusion follows from part (\\ref{p:rot-irr}). So we may assume $\\rot a\\in\\bQ$. Then $a^p$ is grounded for some $p\\ne0$.\nSince $Z(a)\\le Z(a^p)$, it suffices to prove the lemma for $a^p$. In other words, we may further suppose that $a$ is grounded.\n\nLet us put \n\\[G = Z(a),\\quad G_0 = \\rot^{-1}(0)\\cap G.\\]\nPart (\\ref{p:zz}) implies that \n\\[\nG_0=\\bigcup_{x\\in S^1}\\stab_G(x)=\\form{G_0}\\]\nis a group.\nSince $\\rot$ is a class function on $\\Homeo^+(S^1)$,\nwe see that $G_0\\unlhd G$.\n\nFor each $x\\in \\Fix a$ and $g\\in G$, we note\n\\[\nag(x) = ga(x) = g(x).\\]\nSo, $\\Fix a$ is $G$--invariant. \nWe have a nonempty proper closed $G$--invariant set\n\\[\nX=\\partial \\Fix a.\\]\n\\begin{claim}\\label{claim:HG}\nFor all $x\\in X$, the group $G_0$ fixes $x$.\n\\end{claim}\nFor each $J\\in\\pi_0\\supp a$ and $g\\in G_0$, \nwe have seen in part (\\ref{p:zz}) that $\\partial J\\sse \\Fix g$.\nSince we can write\n\\[\nX = \\overline{\\bigcup\\{\\partial J\\mid J\\in\\pi_0\\supp a\\}}\\]\nwe see that $X\\sse \\Fix g$. This proves the claim.\n\nLet $p\\co G\\to G\/G_0$ denote the quotient map.\nBy Claim~\\ref{claim:HG}, \n the natural action\n\\[G\/G_0\\to \\Homeo^+(X),\\quad p(g).x=g(x)\\] \nis well-defined and free. \nBy Corollary~\\ref{c:hoelder-s1}, this free action extends to a free action \\[\\rho\\co G\/G_0\\to \\Homeo^+(S^1)\\] such that $\\rot\\circ\\rho$ is an injective homomorphism.\n\nFor each $g\\in G$, $x\\in X$ and $n\\in\\bZ$, we have\n\\[\\rho\\circ p(g^n)(x)=g^n(x).\\]\nBy Lemma~\\ref{l:inv} (\\ref{p:inv}) we see that\n\\[\\rot\\circ\\rho\\circ p = \\rot\\restriction_G,\\]\nand hence, that $\\rot\\restriction_G$ is a group homomorphism.\nAs $S^1$ is abelian, we also obtain \n\\[\n[G,G]\\le \\ker(\\rot\\restriction_G)=G_0.\\qedhere\\]\n\\ep\n\n\n\nNote that a finitely generated subgroup of $\\operatorname{SO}(2,\\bR)$ consisting of elements with rational rotation numbers is necessarily finite. So in Lemma~\\ref{l:circle} (\\ref{p:rot-q}), if $G$ is a finitely generated subgroup of $Z(a)$ then $\\rot(G)$ is a finite subgroup of $S^1\\cong\\operatorname{SO}(2,\\bR)$.\n\n\n\n\n\\begin{rem}\nPart (\\ref{p:hom}) of Lemma~\\ref{l:circle} appears in the unpublished work of Farb and Franks \\cite{FF2001}, on which our argument is based. We included here a detailed, self-contained proof for readers' convenience. We also remark the necessity of the infinite--order hypothesis, which was omitted in~\\cite{FF2001}. For example, let us consider $a,b,c\\in\\Diff^\\infty_+(S^1)$ such that \nfor each $x\\in S^1=\\bR\/\\bZ$ \nwe have\n\\[\na(x)=x+1\/2,\\quad\nb(x)=x+1\/4,\\quad\nc(x+1\/2)=c(x)+1\/2\\]\nand such that\n\\[\nc(0) = 0,\\quad\nc(1\/8) = 1\/4,\\quad\nc(1\/4) = 3\/8.\\]\nThen $b,c\\in Z(a)$ and $(bc)^3(0)=0$. Hence we have\n\\[\\rot (b)+\\rot(c) = 1\/4 + 0 \\ne 1\/3 =\\rot(bc).\\]\n\\end{rem}\n\n\n\\begin{lem}\\label{l:interval}\nLet $a\\in \\Diffb(I)$,\nand let $Z(a)$\nbe the centralizer of $a$ in $\\Diffb(I)$.\nThen we have\n\\[\\supp a\\cap \\supp [Z(a),Z(a)]=\\varnothing.\\]\n\\end{lem}\nThe proof is almost identical to that of Lemma~\\ref{l:circle} (\\ref{p:zz}).\n\n\n\n\n\\subsection{The Two--jumps Lemma}\nThe Two--jumps Lemma was developed by Baik and the authors in~\\cite{BKK2016} and is the second essential analytic result needed to establish Theorem~\\ref{thm:main}.\n\n\n\\begin{lem}[Two--jumps Lemma,~\\cite{BKK2016}]\\label{l:fg}\nLet $M\\in\\{I,S^1\\}$ and let $f,g\\co M\\to M$ be continuous maps.\nSuppose  $(s_i), (t_i)$ and $(y_i)$ are infinite sequences of points in $M$\nsuch that for each $i\\ge1$, one of the following two conditions hold:\n\\be[(i)]\n\\item\n$f(y_i)\\le s_i = g(s_i) < y_i < t_i = f(t_i) \\le g(y_i)$;\n\\item\n$g(y_i)\\le t_i = f(t_i) < y_i < s_i = g(s_i) \\le f(y_i)$.\n\\ee\nIf $|g(y_i)-f(y_i)|$ converges to $0$ as $i$ goes to infinity,\nthen $f$ or $g$ fails to be $C^1$.\n\\end{lem}\n\nFigure~\\ref{f:fg} illustrates the case (i) of Lemma~\\ref{l:fg}.\nThe reader may note that the homeomorphisms $f$ and $g$ above are \\emph{crossed elements}~\\cite[Definition 2.2.43]{Navas2011}.\nIndeed,  the Two--jumps Lemma generalizes a unpublished lemma of Bonatti--Crovisier--Wilkinson regarding crossed $C^1$--diffeomorphisms,\nwhich can be found in \\cite[Proposition 4.2.25]{Navas2011}.\n\n\\begin{figure}[h!]\n  \\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]\n\\begin{tikzpicture}[>=stealth',auto,node distance=3cm, thick]\n\\draw  (-4,0) node (1) [bv] {} node [below]  {\\small $f(y_i)$} \n-- (-2,0)  node  (2) [bv] {} node [below] {\\small $s_i$}\n-- (0,0) node (3) [bv] {} node [below] {\\small $y_i$} \n-- (2,0)  node (4) [bv] {} node [below] {\\small $t_i$}\n-- (4,0) node (5) [bv] {} node [below]  {\\small $g(y_i)$};\n\\path (3) edge [->,bend right,red] node  {} (1);\n\\path (3) edge [->>,bend right,blue] node  {} (5);\n\\draw [->>, blue] (2)  edge [out = 200,in=-20,looseness=50] (2);\n\\draw [->, red] (4)  edge [out = 20,in=160,looseness=50] (4);\n\\end{tikzpicture}%\n\\caption{Two--jumps Lemma.}\n\\label{f:fg}\n\\end{figure}\n\n\n\n\n\\section{The $C^1$--Smoothability of $\\bZ^2\\ast\\bZ$}\nThis section establishes the main technical result of the paper as below.\n\n\n\\begin{thm}\\label{t:tech-main}\nLet $M\\in\\{I,S^1\\}$,\nand let\n $a,b,t\\in\\Diff^1_+(M)$. If \n\\[\\supp a\\cap\\supp b=\\varnothing,\\] \nthen the group $\\form{a,b,t}$  is not isomorphic to $\\bZ^2\\ast\\bZ$.\n\\end{thm}\n\n\n\\begin{rem}\n\\be\n\\item\nWe emphasize that this theorem is about $C^1$, rather than $\\Cbv$, diffeomorphisms. \n\\item The regularity hypothesis of $C^1$ cannot be replaced by $C^0$; see Proposition~\\ref{prop:ab-homeo}.\n\\item The proof of Theorem~\\ref{t:tech-main} is relatively easy and standard if  $\\supp a$ or $\\supp b$ is assumed to have finitely many components.\n\\ee\n\\end{rem}\n\nLet us prove Theorem~\\ref{t:tech-main} through a sequence of lemmas in this section.\n\n\\subsection{Finding a lamplighter group from compact support}\nA crucial step in the proof of Theorem~\\ref{t:tech-main} is the following construction,\nwhich generalizes a result of Brin and Squier in the PL setting~\\cite{BS1985}.\nThe same idea to find vanishing words from successive commutators goes back even to the Zassenhaus Lemma~\\cite{Raghunathan1972};\nthe authors thank an anonymous referee for suggesting us to further prove the existence of a lamplighter subgroup.\n\n\\begin{lem}\\label{l:recursive}\nLet $1\\ne g\\in H\\le\\Homeo^+(I)$. If the closure of $\\supp g$ is contained in $\\supp H$, then $H$ contains the lamplighter group $\\bZ\\wr\\bZ$.\n\\end{lem}\n\nMore precisely, we will show that for $g_1=g$, there exists a positive integer $m$ and elements $u_1,\\ldots,u_m\\in H$ such that the recursively defined sequence \\[ g_{i+1}=[g_i,u_i g_i u_i^{-1}],\\quad i=1,2,\\ldots,m,\\] satisfies that\n$\\form{g_m,u_m}\\cong\\bZ\\wr\\bZ$\nand that $g_{m+1}=1$.\nHere and throughout this paper, when $1$ refers to a homeomorphism or a group element then it means the identity, and otherwise it refers to the real number $1$.\n\n\n\n\\bp[Proof of Lemma~\\ref{l:recursive}]\nSince $\\overline{\\supp g_1}$ is a compact subset of the open set $\\supp H$, we can enumerate\n\n\\[ I_1,I_2,\\ldots,I_N\\in\\pi_0(\\supp H)\\]\nsuch that $\\supp g_1\\cap I_i\\ne\\varnothing$ for each $i$ and such that\n\n \\[\\overline{\\supp g_1}\\sse \\bigcup_{i=1}^N I_i.\\]\nNote that each $I_i$ is an open, $H$--invariant interval contained in $(0,1)$.\n\nLet us inductively construct the elements $u_1,\\ldots,u_{k-1}$ satisfying the required properties.\nAs a base case, we put $r(1)=1\\in\\bN$ and \n\\[ K_1 :=\\overline{\\supp g_1}\\cap I_1.\\]\nSince $K_1$ is a nonempty compact subset of $I_1\\sse\\supp H$, we have that\n\\[\n\\sup\\{ u(\\inf K_1)\\mid u\\in H\\}=\\sup I_1>\\sup K_1.\\]\nSo, there exists $u_1\\in H$ such that $K_1\\cap u_1^j K_1=\\varnothing$ for all $j\\in\\bN$. Note that \n\\[\\form{g_1,u_1}\\restriction_{I_1}\\cong\\form{s,t\\mid \\left[s,t^j st^{-j}\\right]=1\\text{ for all }j\\in\\bN}=\\bZ\\wr\\bZ.\\]\nLet us set \\[ r(2) = 1+\\sup\\{ s\\in [1,N]\\mid \\supp [g_1, u_1^j g_1 u_1^{-j}]\\restriction_{I_s}=1\\text{ for all }j\\in\\bN\\}\\ge 1+r(1)=2.\\]\nIf $r(2)>N$, then we have a sequence of surjections\n\\[\n\\bZ\\wr\\bZ\\twoheadrightarrow\n\\form{g_1,u_1}\\twoheadrightarrow\n\\form{g_1,u_1}\\restriction_{I_1}\\twoheadrightarrow\\bZ\\wr\\bZ,\n\\]\nwhich composes to the identity. In particular, $\\form{g_1,u_1}\\cong\\bZ\\wr\\bZ$.\nIn the case where $r(2)\\le N$, we pick $j\\in\\bN$ such that the element\n\\[g_2:= \\left[ g_1, u_1^j g_1 u^{-j}\\right]\\]\nsatisfies $\\supp g_2\\cap I_{r(2)}\\ne\\varnothing$,\nand apply the same argument to $g_2$.\n\nBy a straightforward induction, we eventually find $1\\le m\\le r(m)\\le N$\nand $g_m, u_m\\in G$ such that the following hold:\n\\begin{align*}\n&\\overline{\\supp g_m} \\sse I_{r(m)}\\cup\\cdots\\cup I_N,\\\\\n&\\supp g_m\\cap I_{r(m)}\\ne\\varnothing,\\\\\n&\\supp g_m\\cap u_m^j \\supp g_m\\cap I_{r(m)}=\\varnothing,\\quad\\text{ for all }j\\in\\bN,\\\\\n&\\left[g_m, u_m^j g_m u_m^{-j}\\right] =1\\quad\\text{ for all }j\\in\\bN.\n\\end{align*}\nIt follows that $\\form{g_m,u_m}\\cong \\bZ\\wr\\bZ$.\n\\ep\n\n\n\nLemma~\\ref{l:recursive} implies the following for circle homeomorphisms.\n\n\\begin{lem}\\label{l:global}\nLet $a,b,c,d\\in\\Homeo^+(S^1)$ be nontrivial elements such that\n\\[\n\\supp a \\cap \\supp b = \\varnothing\\quad\\text{ and }\\quad\n\\supp c \\cap \\supp d = \\varnothing.\\]\nIf $\\supp G= S^1$, then $G$ contains $\\bZ\\wr\\bZ$.\n\\end{lem}\n\n\\bp\nFor simplicity, let us abbreviate\n$\\AA  = \\pi_0\\supp a$, and similarly define \n$\\BB$,  $\\CC$ and $\\DD$.\nSince $S^1$ is compact, there exists a finite open covering $\\mathcal{V}$ of $S^1$ such that \n\\[\n\\mathcal{V}\\sse\\AA\\cup \\BB\\cup\\CC\\cup\\DD.\\]\n\nBy minimizing the cardinality,  we can require that $\\mathcal{V}$ forms a \\emph{chain of intervals}.\nMore precisely, this means that\n$\\mathcal{V}=\\{V_1,\\ldots,V_k\\}$ for some $k\\ge1$\nand that \\[\\inf V_i<\\sup V_{i-1}\\le\\inf V_{i+1}\\]\nfor $i=1,2,\\ldots,k$, where here the indices are taken cyclically.\n\nWithout loss of generality, let us assume $V_1\\in\\AA $.\nThen we have $V_{2i-1}\\in\\AA \\cup \\BB$\nand $V_{2i}\\in\\CC\\cup\\DD$ for each $i$. Note that $k$ is an even number\nand that $x=\\inf V_2$ is a global fixed point of $H=\\form{b,c,d}$.\nIn particular, we can regard $H$ as acting on $I$, which is a two-point compactification of $S^1\\setminus \\{x\\}$.\nNote that\n\\[\n\\varnothing\\ne \\overline{\\supp b}\\sse S^1\n\\setminus \\bigcup(\\mathcal{V}\\cap\\AA ) \\sse \\bigcup(\\mathcal{V}\\cap(\\BB\\cup\\CC\\cup\\DD))\n\\sse\\supp H.\\]\nThe desired conclusion follows from Lemma~\\ref{l:recursive}.\n\\ep\n\n\n\\subsection{Supports of commutators}\nWe will need rather technical estimates of supports as given in this subsection.\nIn order to prevent obfuscation of the ideas, we have included some intuition behind the proofs when appropriate. \n\\begin{lem}\\label{l:comm-supp}\nIf $f$ and $g$ are homeomorphisms of a topological space $X$, then\n\\[\\overline{\\supp[f,g]}\\sse \\supp f \\cup \\supp g \\cup \\overline{\\supp f\\cap\\supp g}.\\]\n\\end{lem}\n\\bp\nSuppose \n\\[x\\not\\in \\supp f\\cup \\supp g\\cup \\overline{\\supp f\\cap \\supp g}.\\]\nThen $f(x)=x=g(x)$. Moreover, for some open neighborhood $U$ of $x$ we have \n\\[U\\cap \\supp f\\cap \\supp g =\\varnothing.\\]\nWe can find an open neighborhood $V\\sse U$ of $x$ such that\n\\[f^{\\pm1}(V)\\cup g^{\\pm1}(V)\\sse U.\\]\nLet $y\\in V$. We see $[f,g](y)=y$, by considering the following three cases separately:\n\\[\ny\\in V\\cap \\supp f,\\quad y\\in V\\cap \\supp g,\\quad y\\in V\\cap\\Fix f\\cap \\Fix g.\\]\nSo we obtain that\n\\[[f,g]\\restriction_V=1.\\]\nThis implies  \\[x\\not\\in\\overline{\\supp [f,g]}.\\qedhere\\]\n\\ep\n\n\\begin{lem}\\label{lem:c0} \nLet $X$ be a topological space.\nIf $b,c,d\\in\\Homeo(X)$ satisfy \\[\\supp c\\cap\\supp d= \\varnothing,\\]\nthen for $\\phi=[c,bdb^{-1}]$ we have that\n\\[\n\\supp\\phi \\sse \\supp b \\cup cb(\\supp b \\cap\\supp d)\\cup db^{-1}(\\supp b \\cap \\supp c).\\]\n\\end{lem}\n\nLet us briefly explain the key idea behind the statement of this lemma. The support of the homeomorphism $bdb^{-1}$ is exactly $b(\\supp d)$. The homeomorphism $\\phi$ may be viewed as a composition of $bd^{-1}b^{-1}$ and the conjugate of $bdb^{-1}$ by $c$, and the latter of these has support $cb(\\supp d)$. \nBy exhaustively checking the possible images of points under $bdb^{-1}$ and $cbdb^{-1}c^{-1}$, we see that every $x\\in\\supp \\phi$ belongs to one of the three sets as stated in the lemma.\n\n\\bp[Proof of Lemma~\\ref{lem:c0}]\nFor brevity, let us write \n\\[\n\\tilde b = \\supp b,\\quad\n\\tilde c = \\supp c,\\quad\n\\tilde d = \\supp d.\\]\nLet us consider three equivalent expressions for $\\phi$:\n\\[\n [c,bdb^{-1}]=cb d (cb)^{-1}\\cdot bd^{-1}b^{-1}= c\\cdot b\\cdot db^{-1}c^{-1}(db^{-1})^{-1}\\cdot b^{-1}.\\]\nAfter some set theoretic computation, one sees the following.\n\\begin{align*}\\supp  \\phi&\\sse\\left(\\tilde c\\cup b\\tilde d\\right)\\cap \\left(cb \\tilde d\\cup b\\tilde d\\right)\\cap\\left(\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c\\right)\\\\&\\sse\\left((\\tilde c\\cap cb \\tilde d)\\cup b\\tilde d\\right)\\cap\\left(\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c\\right)\\\\&\\sse(\\tilde c\\cap cb \\tilde d) \\cup \\left( b\\tilde d\\cap (\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c)\\right)\\\\&\\sse\\left(\\tilde c\\cap cb \\tilde d\\right) \\cup \\left( (\\tilde b\\cup \\tilde d)\\cap (\\tilde b\\cup\\tilde c\\cup db^{-1}\\tilde c) \\right)\\\\ &\\sse\\left(\\tilde c\\cap cb \\tilde d\\right)\\cup\\tilde b\\cup \\left(\\tilde d\\cap (\\tilde c\\cup db^{-1}\\tilde c)\\right)\\\\ &\\sse\\left(\\tilde c\\cap cb \\tilde d\\right)\\cup\\tilde b \\cup\\left(\\tilde d\\cap db^{-1}\\tilde c\\right). \\end{align*}\nNote that we used $b\\tilde d\\sse \\tilde b\\cup \\tilde d$, and also $\\tilde c\\cap\\tilde d=\\varnothing$.\nIt now suffices for us to prove the following claim:\n\\begin{claim*}\\label{claim:cbbd}\nWe have the following:\n\\begin{align*}\n\\tilde c\\cap cb\\tilde d&\\sse cb\\left(\\tilde b\\cap\\tilde d\\right),\\\\\n\\tilde d\\cap db^{-1}\\tilde c&\\sse db^{-1}\\left(\\tilde b\\cap\\tilde c\\right).\n\\end{align*}\n\\end{claim*}\nTo see the first part of the claim, let us consider \n$x\\in X$ satisfying\n\\[cb(x)\\in \\tilde c\\cap cb \\tilde d.\\]\nThen we have $x\\in \\tilde d$ and $cb(x)\\in \\tilde c$.\nSince\n $\\tilde c\\cap \\tilde d=\\varnothing$\nand $b(x)\\in c^{-1}\\tilde c=\\tilde c$, we see $x\\ne b(x)$.\nIn particular, we have $x\\in\\tilde b$\n and $cb(x)\\in cb(\\tilde b\\cap \\tilde d)$. This proves the first part of the claim. The second part follows by symmetry.\\ep\n \n\\begin{lem}\\label{lem:c1} \nIf $b,c,d\\in\\Diff^1_+(I)$ are given such that \\[\\supp c\\cap\\supp d= \\varnothing,\\]\nthen for $\\phi=[c,bdb^{-1}]$ we have that\n\\[\\overline{\\supp\\phi\\setminus \\supp b}\\sse\\supp c\\cup\\supp d.\\]\n\\end{lem}\n\\bp\nAs in the proof of Lemma~\\ref{l:global}, we let $\\BB=\\pi_0\\supp b$ and $\\CC=\\pi_0\\supp c$.\nLet\n\\[J_B=B\\cup cb(B\\cap\\supp d)\\cup db^{-1}(B\\cap \\supp c)\\]\nfor each $B\\in\\BB$.\nBy Lemma~\\ref{lem:c0}, we have that\n\\[\\supp\\phi\\sse \\bigcup\\{J_B\\mid B\\in\\BB\\}\n=  \\bigcup\\{J_B\\setminus B\\mid B\\in \\BB\\}\\cup \\supp b.\\]\nMoreover, for each $B\\in\\BB$ we note that\n\\[\n\\overline{\nJ_B\\setminus B}\n\\sse \\overline{c(B)\\setminus B}\\cup \\overline{d(B)\\setminus B}\n\\sse\\supp c \\cup\\supp d.\\]\n\n\\begin{claim*}\\label{claim:jbb}\nThe following set is a finite collection of intervals:\n\\[ \\BB_0=\\{B\\in\\BB\\mid J_B\\ne B\\}.\\]\n\\end{claim*}\nWe will employ the $C^1$--hypothesis for this claim. \nLet us write\n\\[\\BB_1=\\{B\\in\\BB\\mid cb(B\\cap \\supp d)\\setminus B\\ne\\varnothing\\},\\ \n\\BB_2=\\{B\\in\\BB\\mid db^{-1}(B\\cap \\supp c)\\setminus B\\ne\\varnothing\\}.\\]\nAssume for a contradiction that $\\BB_0=\\BB_1\\cup\\BB_2$ is infinite. \nWe may suppose $\\BB_1$ is infinite, as the proof is similar when $\\BB_2$ is infinite.\nThere are infinitely many distinct $B_1,B_2,\\ldots\\in \\BB_1$ \nand $x_i\\in B_i\\cap\\supp d$ such that $cb (x_i)\\not\\in B_i$. \nThen we have $C_1,C_2,\\ldots\\in \\CC$ such that\n$b (x_i), cb(x_i)\\in C_i$.\nSince $x_i\\in\\supp d$, we have $x_i\\not\\in C_i$; see Figure~\\ref{f:JB}.\n\nLet us consider the interval $J_i = [x_i,cb (x_i)]$ which contains $b(x_i)$ in the interior, up to switching the endpoints of this interval.\nThen we have\n\\[(b(x_i),cb (x_i)]\\cap \\partial B_i\\ne\\varnothing,\\quad [x_i,b (x_i))\\cap\\partial C_i\\ne\\varnothing.\\]\nWe now apply the Two--jumps Lemma (Lemma~\\ref{l:fg}) to the following parameters\n\\[f=b^{-1},\\ g=c,\\ s_i=\\partial C_i\\cap B_i,\\  t_i=\\partial B_i\\cap C_i,\\ y_i=b(x_i).\\]\nWe deduce that $b$ or $c$ is not $C^1$.\nThis is a contradiction and the claim is proved.\n\nFrom the claim above, we deduce the conclusion as follows.\n\\[\n\\overline{\\supp\\phi\\setminus\\supp b}\n\\sse\\overline{\\bigcup\\{J_B\\setminus B\\mid B\\in\\BB_0\\}}=\\bigcup\\{\\overline{J_B\\setminus B}\\mid B\\in\\BB_0\\}\n\\sse\\supp c\\cup\\supp d.\\qedhere\\]\n\\ep\n\n\\begin{figure}[h!]\n  \\tikzstyle {a}=[black,postaction=decorate,decoration={%\n    markings,%\n    mark=at position 1 with {\\arrow[black]{stealth};}    }]\n  \\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1.5pt]\n{\n\\begin{tikzpicture}[thick,scale=.7]\n\\draw [red,ultra thick] (-4,0.5) -- (2,0.5);\n\\draw [blue,ultra thick] (0,0) -- (6,0);\n\\draw [teal,ultra thick] (4,.5) -- (10,.5);\n\\draw (-3,0) node [] {\\small $\\supp d$}; \n\\draw (9,0) node [] {\\small $C_i$}; \n\\draw (3,-.5) node [] {\\small $B_i$}; \n\\draw [dashed] (1,1.4) -- (1,-.3) node [below]  {\\small $x_i$};\n\\draw [dashed] (5,1.4) -- (5,-.3) node [below]  {\\small $b(x_i)$};\n\\draw [dashed] (7.5,1.4) -- (7.5,-.3) node [below]  {\\small $cb(x_i)$};\n\\draw (1,1) -- (7.5,1);\n\\draw (3,1.4) node [] {\\small $J_i$}; \n\\end{tikzpicture}%\n}\n\\caption{Lemma~\\ref{lem:c1}.}\n\\label{f:JB}\n\\end{figure}\n\n\\subsection{Finding compact supports}\nWe will deduce Theorem~\\ref{t:tech-main} from the following, seemingly weaker result.\n\\begin{lem}\\label{l:main}\nLet $M\\in\\{I,S^1\\}$ and let\n$a,b,c,d\\in\\Diff^1_+(M)$.\nIf \n\\[\n\\supp a\\cap\\supp b=\\varnothing,\\quad\n\\supp c\\cap\\supp d=\\varnothing,\\]\nthen the group $\\form{a,b,c,d}$ is not isomorphic to $\\bZ^2\\ast\\bZ^2$.\n\\end{lem}\n\nLet us note two properties of RAAGs. First, a RAAG does not contain a subgroup isomorphic to $\\bZ\\wr\\bZ$. The reason is that, every two--generator subgroup of a RAAG is either free or free abelian~\\cite{Baudisch1981}; see also~\\cite[Corollary 1.3]{KK2015GT}.\nSecond, a RAAG is \\emph{Hopfian}; that is, every endomorphsm of a RAAG is an isomorphism. This follows from a general fact that every finitely generated residually finite group is Hopfian~\\cite{LS2001}.\n\n\n\\bp[Proof of Theorem~\\ref{t:tech-main} from Lemma~\\ref{l:main}]\nAssume $\\form{a,b,t}\\cong\\bZ^2\\ast\\bZ$. \nSince the RAAG $\\bZ^2\\ast\\bZ$ is Hopfian, the natural surjection between groups\n\\[ \\form{A,B,T\\mid [A,B]=1}\\to \\form{a,b,t}\\]\nis actually an isomorphism. It follows that\n\\[\\form{a,b,tat^{-1},tbt^{-1}}\\cong\\form{A,B,TAT^{-1},TBT^{-1}}\\cong\\bZ^2\\ast\\bZ^2.\\]\nThis contradicts Lemma~\\ref{l:main}, since the four diffeomorphism $a,b,tat^{-1},tbt^{-1}$ satisfy the conditions of the lemma.\n\\ep\n\n\\bp[Proof of Lemma~\\ref{l:main}]\nWe put $G=\\form{a,b,c,d}$ and  consider an abstract group\n\\[ G_0 = \\form{a_0,b_0,c_0,d_0\\mid [a_0,b_0]=1=[c_0,d_0]}\\cong \\bZ^2\\ast\\bZ^2.\\]\nThere is a natural surjection $p\\co G_0\\to G$ defined by \\[(a_0,b_0,c_0,d_0)\\mapsto(a,b,c,d).\\] \n\nAssume for a contradiction that $G\\cong G_0$. By the Hopficity of $\\bZ^2\\ast\\bZ^2$,  we see that $p$ is an isomorphism. Since $G$ does not contain $\\bZ\\wr\\bZ$,  Lemma~\\ref{l:global} implies that $G$ has a global fixed point.\nIn other words, we may assume $M=I$.\n\nLet us define \n$\\phi=[c,bdb^{-1}]$ and $\\psi=[\\phi,a]$.\nLemma~\\ref{l:comm-supp} implies that\n\\[\\overline{\\supp\\psi}\\sse\\supp\\phi\\cup\\supp a\\cup\\overline{\\supp\\phi\\cap\\supp a}.\\]\nWe see from Lemma~\\ref{lem:c1} that\n\\[\\overline{\\supp\\phi\\cap\\supp a}\n\\sse\\overline{\\supp\\phi\\setminus\\supp b}\n\\sse\\supp c\\cup \\supp d.\\]\nSo, it follows that\n\\[\n\\overline{\\supp \\psi}\\sse\\supp G.\\]\n\nAs we are assuming $p$ is injective, we have \n\\[\\psi = [\\phi,a]=\\left[ [c, bdb^{-1}],a\\right]\\ne1.\\]\nLemma~\\ref{l:recursive} implies that $G$ contains $\\bZ\\wr\\bZ$, which is a contradiction. This completes the proof.\\ep\n\nLet us conclude this section by describing one generalization of Theorem~\\ref{t:tech-main}.\n\n\\begin{prop}\\label{t:tech-main2}\nLet $M\\in\\{I,S^1\\}$.\nIf $a,b,c,d\\in\\Diff^1_+(M)$ satisfy \n\\[\\supp a\\cap\\supp b=\\varnothing,\\quad\n\\supp c\\cap\\supp d=\\varnothing.\\]\nand \n\\[\n\\left[[c,bdb^{-1}],a\\right]\\ne1,\n\\]\nthen $\\form{a,b,c,d}$ contains the lamplighter group $\\bZ\\wr\\bZ$.\n\\end{prop}\n\nIn particular, the group $\\form{a,b,c,d}$ does not embed into a RAAG.\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:main}}\\label{s:main-thm}\n\nIn this section, we apply the facts we have gathered to complete the proof of the main result.\n\n\\subsection{Reducing to the connected case}\nWe will reduce the proof of Theorem~\\ref{thm:main} to the case $M\\in\\{I,S^1\\}$, using the following group theoretic observations.\n\n\\begin{lem}\\label{lem:L1}\nSuppose $A,B,C,D$ are groups, and suppose that $A\\times B$ is a normal subgroup of $C*D$. Then at least one of these four groups is trivial.\n\\end{lem}\n\\begin{proof}\nIf $A\\times B$ is a subgroup of $C*D$ then the Kurosh Subgroup Theorem implies that there is a free product decomposition \\[A\\times B\\cong F\\ast\\bigast_{i} H_i,\\] where $F$ is a free group (possibly of infinite rank) and where each $H_i$ is conjugate into $C$ or into $D$. By analyzing centralizers of elements, it is easy to show that a nontrivial free product is never isomorphic to a nontrivial direct product (cf.~\\cite[p.177]{LS2001}). It follows that $A\\times B$ is conjugate into $C$ or $D$, which contradicts the normality of $A\\times B$.\n\\end{proof}\n\nAn alternative proof of Lemma~\\ref{lem:L1} can be given using Bass--Serre theory (see~\\cite{Serre1977}).\n\n\\begin{lem}\\label{lem:L2}\nSuppose $A,B,C,D$ are nontrivial groups, and that $A*B\\le C\\times D$. Then there is an injective homomorphism from $A*B$ into either $C$ or $D$.\n\\end{lem}\n\\begin{proof}\nSuppose the contrary, so that $K_C$ and $K_D$ are the (nontrivial) kernels of the inclusion of $A*B$ into $C\\times D$ composed with the projections onto $C$ and $D$. Then $K_C\\cap K_D=1$ and $K_C$ and $K_D$ normalize each other, so that $K_CK_D\\cong K_C\\times K_D\\le A*B$. This contradicts Lemma~\\ref{lem:L1}.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{lem}\\label{lem:connected}\nSuppose \\[M=\\coprod_{i=1}^n M_i\\] is a compact one--manifold, and suppose that $A\\ast B$ embeds into $\\Diffb(M)$. Then \n for some finite index subgroups $A_0\\le A$ and $B_0\\le B$,\n  and for some $i$, we have an embedding of $A_0\\ast B_0$ into $\\Diffb(M_i)$.\n\\end{lem}\n\\begin{proof}\nThis follows immediately from Lemma~\\ref{lem:L2}, using the fact that $\\Diffb(M)$ is commensurable with \\[\\prod_{i=1}^n\\Diffb(M_i).\\]\n\nNote that passage to  finite index subgroups is necessary, since $M$ may consist of a union of diffeomorphic manifolds which are permuted by the action of $A\\ast B$.\n\\end{proof}\n\n\\subsection{Taming supports}\n\nLet us denote the center of a group $G$ by $Z_G$.\nIf $M$ is a one--manifold and if $s\\in\\Diffb(M)$, then\nwe denote by $Z(s)$ the centralizer of $s$ in $\\Diffb(M)$.\nThe following lemma is crucial for applying Lemma~\\ref{l:main}:\n\n\n\\begin{lem}\\label{lem:Z2}\nAssume one of the following:\n\\be[(i)]\n\\item\n$M=I$ and $G$ is a nonabelian group such that $Z_G\\ne1$.\n\\item\n$M=S^1$ and $G$ is a non-metabelian group such that $\\bZ\\le Z_G$.\n\\item\n$M=S^1$ and $G$ is a finitely generated group such that $\\bZ\\le Z_G$\nand such that $G$ is not abelian-by-finite cyclic.\n\\ee\nIn each of the cases, if $G\\le\\Diffb(M)$, then there is a subgroup $\\bZ^2\\le G$ generated by diffeomorphisms $a$ and $b$ such that $\\supp a\\cap\\supp b=\\varnothing$.\n\\end{lem}\n\n\n\\bp\n\n\\emph{Case (i).} Let us pick $s\\in Z_G\\setminus1$ and $b\\in[G,G]\\setminus1$. \nSince $G\\le Z(s)$,  Lemma~\\ref{l:interval} implies that \n\\[\\supp s\\cap \\supp b=\\varnothing.\\]\nSince $\\Homeo^+(I)$ is torsion-free, we have $\\form{b,s}\\cong\\bZ^2$ as desired.\n\n\\emph{Case (ii).}\nWe are given with some $s\\in Z_G$ such that $\\form{s}\\cong\\bZ$. As $G$ is nonabelian, Lemma~\\ref{l:circle} (\\ref{p:rot-irr}) implies that $\\rot(s)\\in\\bQ$; in particular, $s^n$ is grounded for some $n\\ge1$.\nFrom part (\\ref{p:hom}) of the same lemma\nand from that $G\\le Z(s)$, we see \nevery element of $[G,G]$ is grounded.\nSince $[G,G]\\le Z(s^n)$, we note from Lemma~\\ref{l:circle} (\\ref{p:zz}) that \n\\[\\supp(s^n)\\cap \\supp G''=\\varnothing.\\]\nFrom the metabelian hypothesis, we can find $b\\in G''\\setminus1$.\nAs $s^n$ and $b$ are grounded, they have infinite orders. It follows that $\\form{s^n,b}\\cong\\bZ^2$.\n\n\n\\emph{Case (iii).}\nLet us proceed similarly to the case (ii). Namely, pick $s\\in Z_G$ such that $\\form{s}\\cong\\bZ$. \nBy Lemma~\\ref{l:circle}, we have a homomorphism\n\\[\n\\rot\\restriction_G\\co G\\to \\bQ.\\]\nWe fix $n\\ge1$ such that $s^n$ is grounded.\nAs $G$ is finitely generated, we see $\\rot(G)$ is finite cyclic.\nThe hypothesis implies that $G_0=\\ker(\\rot\\restriction_G)$ is not abelian.\nSince every element of $G_0$ is grounded, we can apply\n Lemma~\\ref{l:circle} (\\ref{p:zz}) and deduce \n\\[\\supp s^n\\cap \\supp [G_0,G_0]=\\varnothing.\\]\nEach $b\\in [G_0,G_0]\\setminus1$ then yields the desired subgroup\n$\\form{b,s^n}\\cong\\bZ^2$.\n\\ep\n\n\\subsection{The main result}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nSuppose $(G\\times\\Z)*\\Z\\le \\Diffb(M)$ for some compact one--manifold $M$. Replacing $G$ by a finite index subgroup if necessary, we may assume that $M$ is connected, by Lemma~\\ref{lem:connected}. \nBy applying the cases (i) and (ii) of Lemma~\\ref{lem:Z2} to the group $G\\times \\bZ$,\nwe can find a subgroup $\\langle a,b\\rangle\\cong\\bZ^2\\le G\\times\\Z$ such that $\\supp a\\cap\\supp b=\\varnothing$.\nIf we write the $\\bZ$--free factor of $(G\\times \\bZ)\\ast\\bZ$ as $\\form{t}$, then\n\\[\\form{a,b,t}\\cong\\form{a,b}\\ast\\form{t}\\cong\\bZ^2\\ast\\bZ.\\]\nThis contradicts Theorem~\\ref{t:tech-main}.\n\\end{proof}\n\nOne can now deduce Corollary~\\ref{cor:fg1} as well as Corollary~\\ref{cor:cnt} below from Lemma~\\ref{lem:Z2}, in the exact same fashion as Theorem~\\ref{thm:main}. \n\\begin{cor}\\label{cor:cnt}\nLet $G$ be a group.\n\\be\n\\item \nIf $G$ is nonabelian and if the center of $G$ is nontrivial,\nthen\n$G*\\Z$ admits no faithful $\\Cb$ action on $I$. \n\\item Suppose $G$ is finitely generated.\nIf $G$ is not abelian-by-finite cyclic\nand if the center of $G$ contains a copy of $\\bZ$,\nthen \n$G*\\Z$ admits no faithful $\\Cb$ action on $S^1$. \n\\end{enumerate}\n\\end{cor}\n\n\n\nHere, a group $G$ is $\\mathcal{X}$--by--$\\mathcal{Y}$ for group theoretic properties $\\mathcal{X}$ and $\\mathcal{Y}$ if there is an exact sequence \\[1\\to K\\to G\\to Q\\to 1\\] such that $K$ has property $\\mathcal{X}$ and $Q$ has property $\\mathcal{Y}$. We allow both $K$ and $Q$ to be trivial.\n\nNote that $G\\times \\bZ$ often occurs as a subgroup of $\\Diffb(M)$, where $G$ is not virtually metabelian. We have the following immediate consequence:\n\n\\begin{cor}\\label{cor:not closed}\nLet $\\mathcal{G}$ denote the class of finitely generated subgroups of $\\Diffb(M)$. The class $\\mathcal{G}$ is not closed under taking finite free products.\n\\end{cor}\n\n\n\n\n\\section{Smooth right-angled Artin group actions on compact one--manifolds}\n\nIn this and the remaining sections, we deduce several corollaries from Theorem \\ref{thm:main}. We first complete the classification of right-angled Artin groups which admit faithful $C^{\\infty}$ actions on a compact one--manifold (Corollary \\ref{cor:classification}).\n\n\\begin{lem}\\label{lem:dichotomy}\nLet $A(\\gam)$ be a right-angled Artin group. Then one of the following mutually exclusive conclusions holds:\n\\begin{enumerate}\n\\item\nWe have $(F_2\\times\\Z)*\\Z\\le A(\\gam)$;\n\\item\nThe graph $\\gam$ lies in $\\mathcal{K}_3$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nLet us consider a stratification of graph classes:\n\\[ \\KK_2\\sse\\KK_3\\sse\\KK.\\]\n\nSuppose $\\gam\\in\\KK_2$. Then $A(\\gam)$ is the free product of free abelian groups, and hence contains no copy of $(F_2\\times\\Z)*\\Z$.\n\nLet $\\gam\\in\\KK_3\\setminus\\KK_2$. Then $\\gam$ is the join of at least two graphs $\\gam_1,\\gam_2$ in $\\KK_2$. \nWe write\n\\[\nA(\\gam)=A(\\gam_1)\\times A(\\gam_2).\\]\nIf $A(\\gam)$ contains a copy of $(F_2\\times\\bZ)\\ast\\bZ$,\nthen so does $A(\\gam_1)$ or $A(\\gam_2)$\nby Lemma~\\ref{lem:L2}; this would contradict the previous paragraph.\n\nAssume $\\gam\\in\\KK\\setminus\\KK_3$.\nFirst consider the case that $\\gam\\in\\KK_{2i}\\setminus\\KK_{2i-1}$ for some $i\\ge2$. \nWe can write\n\\[\n\\gam=\\coprod_{j=1}^k \\gam_j\\]\nfor some $k\\ge2$ and for some nonempty connected graphs $\\gam_j\\in\\KK_{2i-1}$. \nThese graphs $\\Gamma_j$ cannot all be complete graphs, for otherwise $\\gam\\in\\KK_2$. \nSo at least one graph $\\Gamma_j$ contains $P_3$, the path on three vertices, as a full subgraph. \nThis implies that $A(\\gam)$ contains a copy of $(F_2\\times\\bZ)\\ast\\bZ$.\n\nWe then consider the case that $\\gam\\in\\KK_{2i+1}\\setminus\\KK_{2i}$ for some $i\\ge2$. \nNote $\\gam$ is the join of some graphs $\\gam_1,\\ldots,\\gam_k$ in $\\KK_{2i}$. \nBy the previous graph, each $A(\\gam_j)$ contains $(F_2\\times\\bZ)\\ast\\bZ$.\n\nFinally assume $\\gam\\notin\\KK$, so that $\\gam$ is not a cograph. \nThen we have that $P_4$ is a full subgraph of $\\gam$, so that $A(P_4)\\le A(\\gam)$. The group $A(P_4)$ contains every right-angled Artin group $A(F)$, where $F$ is a finite forest (see~\\cite{KK2013}). Since the defining graph of $(F_2\\times\\Z)*\\Z$ is a copy of a path $P_3$ on three vertices together with an isolated vertex, its defining graph is a finite forest. We see that $(F_2\\times\\Z)*\\Z\\le A(\\gam)$.\n\\end{proof}\n\n\nWe complete the proof of Corollary \\ref{cor:classification} with the following proposition:\n\n\\begin{prop}\\label{prop:cinfty}\nLet $\\gam\\in\\mathcal{K}_3$ and let $M$ be a compact one--manifold. Then there is an embedding of $A(\\gam)$ into $\\Diff_+^{\\infty}(M)$.\n\\end{prop}\n\n\nIn the case when $M=S^1$, we will prove more precise facts in Section \\ref{sec:semiconj}.\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:cinfty}]\nLet $\\Diff^{\\infty}_0(I)$ denote the group of $C^{\\infty}$ diffeomorphisms of the interval which are infinitely tangent to the identity at $\\{0,1\\}$. It suffices to find a copy $A(\\gam)\\le\\Diff^{\\infty}_0(I)$, since $I$ is a submanifold of every compact one--manifold $M$ and every element of $\\Diff^{\\infty}_0(I)$ can, by definition, be extended to all of $M$ by the identity map. \n\nIn the case when $\\gam\\in\\mathcal{K}_2$, we can write \n$A(\\gam)=\\Z^{n_1}\\ast\\cdots\\ast\\Z^{n_k}$.\nAs we have an embedding $A(\\gam)\\hookrightarrow\\Z*\\Z^N$ for $N=\\max_i n_i$, it suffices to prove the proposition for $\\Z*\\Z^N$ in this case. \nTo do this, we first find a copy of $\\Z^N\\le \\Diff^{\\infty}_0(I)$ such that the support of each nontrivial element of $\\Z^N$ is all of $(0,1)$. The existence of such a copy of $\\Z^N$ follows from choosing a $C^{\\infty}$ vector field on $I$ which vanishes only at $\\partial I$ and integrating it to get a flow, which gives an $\\R$--worth of commuting elements of $\\Diff^{\\infty}_0(I)$.\nNow, choosing a generic (in the sense of Baire) element $\\psi$ of $\\Diff^{\\infty}_0(I)$, we have that $\\psi$ and this copy of $\\Z^N$ generate a copy of $\\Z*\\Z^N\\le\\Diff^{\\infty}_0(I)$ (cf.~\\cite{KKMj2016}).\n\n\n\nFinally, if $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then $A(\\gam)$ is a finite direct product of $k$ right-angled Artin groups with defining graphs in $\\mathcal{K}_2$. Write again $N$ for the maximal rank of an abelian subgroup of $A(\\gam)$. We choose a finite collection of disjoint intervals $\\{J_1,\\ldots,J_k\\}$ with nonempty interior inside of $I$, and realize a copy of $\\Z*\\Z^N$ on each $J_i$, extending by the identity outside of $J_i$. It is clear that $A(\\gam)$ is thus realized as a subgroup of $\\Diff^{\\infty}_0(I)$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Lower regularity}\n\nIn this section, we prove Proposition \\ref{prop:free prod}. \nRecall a \\emph{left order} on a group $G$ is a total order $\\prec$ on $G$ such that for all triples $a,b,g\\in G$ we have $a\\prec b$ if and only if $ga\\prec gb$. \nA group is \\emph{left orderable} if it admits a left order.\n\n\nEvery subgroup of $\\Homeo^+(\\bR)$ is left orderable. Conversely, if $G$ is countable and left orderable,  then there is a faithful action $G\\to\\Homeo^+(\\bR)$;\nit can be further required from the action that for some fixed point $x_0\\in\\bR$,\nwhenever $g\\prec h$ we have $g(x_0)<h(x_0)$. There exists a standard example of such an action, called a \\emph{dynamical realization} of the given left order~\\cite{Navas2011}.\n\n\nNote that if $\\prec$ is a left order on a group $G$, then $G$ also admits the opposite order $\\prec^{opp}$, where $g\\prec h$ if and only if $h\\prec^{opp} g$.  It follows that if $g$ is a nontrivial element of a countable left orderable group $G$, then there exists a faithful action of $G$ on $\\R$ such that $x_0<g(x_0)$ for some $x_0\\in \\R$. We will call an action coming from the opposite order as an \\emph{opposite action}.\n\nLet us now establish Proposition \\ref{prop:free prod} for the case $M=I$:\n\n\\begin{prop}\\label{prop:homeo closed}\nLet $\\mathcal{G}$ denote the class of countable subgroups of $\\Homeo^+(I)$. Then $\\mathcal{G}$ is closed under countable free products.\n\\end{prop}\n\nProposition \\ref{prop:homeo closed} also follows from the general fact that if $G$ and $H$ are countable left orderable groups then so is $G\\ast H$, as was shown in~\\cite{KM1996,Rivas2012JAlg}. \nWe are including a proof as the idea will be needed for Proposition~\\ref{prop:ab-homeo}.\nUnlike a dynamical realization of $G\\ast H$ coming from its natural left order~\\cite{DNR2014}, the construction below may not have a point $x_0$ with trivial stabilizer.\n\n\\bp\nThe proof is based on the idea of~\\cite{BKK2014}. \nBy induction, it suffices to show that if $G_1,G_2,\\ldots\\in\\mathcal{G}$ then $\\bigast_{i\\ge1}G_i\\in\\mathcal{G}$. Since $K=\\langle \\{G_i\\mid i\\ge1\\}\\rangle\\in\\mathcal{G}$ and since $\\bigast_{i\\ge1}K\\le \\Z*K$ \nby the normal form theorem for free products,\nit suffices to show that if $G\\in\\mathcal{G}$ then $\\Z*G\\in\\mathcal{G}$.\nWe let $H=G\\ast \\bZ$, and choose countably many disjoint subintervals $\\{I_h\\}_{1\\neq h\\in H}$ of $I$, each with nonempty interior. \n\nLet $h\\in H\\setminus1$ be arbitrary.\nWe claim that there exists an action $\\rho_h\\co H\\to\\Homeo^+(I_h)$ such that $h\\not\\in\\ker\\rho_h$.\nFor some $g_i\\in G$ and $r_i\\in\\bZ$, we can write\n\\[ h =g_\\ell t^{r_\\ell} \\cdots g_1t^{r_1} .\\]\nIf $h\\in\\form{t}$ or $h\\in G$, then the claim is obvious.\nSo, possibly after conjugation we may assume that $r_i\\ne0$ and $g_i\\ne1$ for each $i$. \n\nWe choose disjoint closed intervals \\[\\{J_1,\\ldots,J_\\ell\\}\\subset I_h\\] with nonempty interior, $J_i=[p_i,q_i]$ and $q_i<p_j$ for $1\\leq i<j\\leq \\ell$.\nFor each $1\\leq i\\leq \\ell$, we choose an action of $G$ on $J_i$ such that\n $x_i<g_i(x_i)$ for some $x_i\\in J_i$.\n\nWe now define an action of $t$ on $I_h$. We choose $\\ell$ disjoint closed intervals \\[\\{L_1,\\ldots,L_\\ell\\}\\subset I_h\\] of the form $L_i=[c_i,d_i]$. We choose $c_1<x_0<p_1$ and $x_1<d_1<g_1(x_1)$. For $i>1$, we choose points $d_{i-1}<c_i<g_{i-1}(x_{i-1})$ and $x_i<d_i<g_i(x_i)$. We now define $t$ on \\[\\bigcup_{i=1}^\\ell L_i\\] so that $t^{r_1}(x_0)=x_1$ and so that $t^{r_i}(g_{i-1}(x_{i-1}))=x_i$. Since the $\\{L_1,\\ldots,L_\\ell\\}$ are disjoint, such a choice for the definition of $t$ is possible. It is routine to check that $x_0<h(x_0)=g_\\ell(x_\\ell)$. \nSince $h$ is not in the kernel of this action, the claim is proved.\n\nBy taking disjoint subintervals $\\{I_h\\}_{h\\in H\\setminus 1}$ of $\\R$ with each $I_h$ equipped with an action of $H$ by $\\rho_h$, we obtain the desired embedding\n\\[\\rho = \\prod_{h\\in H\\setminus1}\\rho_h \\co H\\to\\prod_{h\\in H\\setminus1}\\Homeo^+(I_h)\\le  \\Homeo^+(\\bR)\\cong\\Homeo^+(I).\\qedhere\\]\n\\ep\n\nWe note from the above proof that the restriction of $G$ on each $J_i=[p_i,q_i]$ corresponds to the original action of $G$, or the opposite action of $G$. \nBy taking $G=\\bZ^2$, we obtain the following.\n\n\\begin{prop}\\label{prop:ab-homeo}\nThere exists an embedding\n\\[ \\rho\\co \\form{a,b,t\\mid [a,b]=1}\\cong\\bZ^2\\ast\\bZ\\to\\Homeo^+(I)\\]\nsuch that $\\supp\\rho(a)$ and $\\supp\\rho(b)$ are disjoint.\n\\end{prop}\nIn particular, the $C^1$ hypothesis in Theorem~\\ref{t:tech-main} cannot be lowered to $C^0$.\n\n\n\n\\section{Complexity of right-angled Artin groups versus diversity of circle actions}\\label{sec:semiconj}\n\nIn this section, we prove Corollary~\\ref{cor:conj}. The proof follows easily from the results in~\\cite{KKMj2016}, together with Corollary~\\ref{cor:classification}.\n\n\nIn a joint work with Mj~\\cite{KKMj2016}, the authors defined a class of finitely generated groups $\\mathcal{F}$ and called each group in the class as \\emph{liftable--flexible}.\nLet us extract the necessary facts which are demonstrated therein.\nRecall from the introduction that a \\emph{projective} action of a group is a representation into $\\PSL(2,\\bR)$.\n\n\\begin{thm}[cf. Theorem 1.1 of~\\cite{KKMj2016}]\\label{thm:KKMj}\nThere exists a class of finitely generated groups $\\FF$ satisfying the following.\n\\be\n\\item For each $G\\in\\FF$, there exist uncountably many distinct semiconjugacy classes of faithful projective actions on $S^1$.\n\\item\nEvery limit group is in $\\FF$; in particular, every free group and every free abelian group is in $\\FF$.\n\\item\nIf $G,H\\in\\FF$, then $G*H\\in\\FF$.\n\\end{enumerate}\n\\end{thm}\n\n\n\n\\begin{lem}\\label{l:ab}\nLet $A$ and $B$ be nontrivial torsion-free finitely generated subgroups such that $B$ is nonabelian. \nWe assume that $A\\times B\\le \\Diffb(S^1)$.\nThen there exist finite index normal subgroups $A_1\\unlhd A$ and $B_1\\unlhd B$ such that \n $A_1\\times B_1$ has a global fixed point on $S^1$.\nFurthermore, the closure of $\\supp A_1$ is a proper subset of $S^1$.\n\\end{lem}\n\\bp\nChoose an arbitrary element $a\\in A\\setminus 1$.\nSince the centralizer $Z(a)$ is nonabelian and $A$ is torsion-free, \nLemma~\\ref{l:circle} implies that $\\rot Z(a)\\sse\\bQ$.\nSince $B$ and $a\\times B$ are subsets of $Z(a)$, we see that $\\rot(B), \\rot(a\\times B)\\sse\\bQ$.\nThis shows that $\\rot(A\\times B)\\sse\\bQ$.\n \nFor each $a\\in A$ and $b\\in B$, note that \n \\[a,b,ab\\in Z(a).\\]\nBy Lemma~\\ref{l:circle} (\\ref{p:hom}), it follows that $\\rot(a)+\\rot(b)=\\rot(ab)$.\nIn particular, we have a homomorphism \n\\[\\rot\\co A\\times B\\to \\bQ.\\]\nSince $A\\times B$ is finitely generated, the image of the above map is finite.\nSo we can find finite index normal subgroups $A_1\\unlhd A$ and $B_1\\unlhd B$ such that every element of $A_1\\times B_1$ is grounded. \n\nLet $a\\in A_1\\setminus1$.\nSince $B_1$ centralizes $a$, it acts on each component of $\\supp a$ as an abelian group; see Lemma~\\ref{l:disj-abel}. It follows that $B_1$ preserves the set $\\supp A_1$ and\n\\[\\supp A_1\\cap\\supp [B_1,B_1]=\\varnothing.\\]\nSince $B_1$ is nonabelian, we see $\\overline{\\supp A_1}\\ne S^1$. \nIt follows that $\\partial\\supp A_1$ is a global fixed point of $A_1\\times B_1$.\n\\ep\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:conj}]\n\n(1) \nIf $\\gam\\in\\mathcal{K}_2$, then $A(\\gam)$ is a finite free product of free abelian groups. Combining parts (2) and (3) of Theorem~\\ref{thm:KKMj}, we see $A(\\gam)$ is in the class $\\FF$. The  first part of the same theorem implies the desired conclusion.\n\n(2) Suppose that $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$, so that $A(\\gam)$ decomposes as a nontrivial direct product $A(\\gam_1)\\times A(\\gam_2)$, where at least one of $A(\\gam_1)$ and $A(\\gam_2)$ is nonabelian. Say $A(\\gam_2)$ is nonabelian. \nBy Lemma~\\ref{l:ab}, there exist finite index normal subgroups $G\\unlhd A(\\gam_1)$\nand $H\\unlhd A(\\gam_2)$ \nsuch that $G\\times H$ has a global fixed point; in particular, $A(\\gam)$ has a finite orbit. \nWe put\n\\[X = \\overline{\\supp G}, \\quad Y = S^1\\setminus X.\\]\nSince $G$ is normal in $A(\\gam)$ we see that \n$X$ and $Y$ are $A(\\gam)$--invariant sets of $S^1$, which are proper by the same lemma.\nThis implies that $X$ and $Y$ have nonempty interiors, and hence $A(\\gam)$ does not have a dense orbit. \n\n\n\n(3) This follows immediately from Corollary~\\ref{cor:classification}.\n\\end{proof}\n\nWe briefly remark that if a group acts on $S^1$ with a global fixed point then it is semi--conjugate to a trivial action, and if it acts with a periodic point then the action is semi--conjugate to a rational rotation group. Corollary~\\ref{cor:conj} implies that if $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then $A(\\gam)$ admits only countably many semi--conjugacy classes of faithful actions, and it is not difficult to realize one semi--conjugacy class for each rational rotation.\n\nObserve that since every right-angled Artin group surjects to $\\Z$, every right-angled Artin group admits uncountably many distinct semi--conjugacy classes of non--faithful actions on $S^1$, so only faithful actions are interesting for our purposes.\n\n\n\n\n\n\n\n\n\n\n\\section{Thompson's groups}\nLet $F$ and $T$ be the Thompson's groups acting on the closed interval and on the circle, respectively~\\cite{CFP1996}. Recall that these are the groups of piecewise linear homeomorphisms of the interval and circle respectively, with dyadic breakpoints and all slopes given by powers of two. It is known that the standard action of $T$ is conjugate to a $C^\\infty$ action~\\cite{GS1987}.\nThe restriction of such a smooth action yields a smooth action of $F$ on a closed interval.\nOn the other hand, the groups $F*\\Z$ and $T\\ast\\bZ$ do not admit any smooth actions on the circle, or indeed on any compact one--manifold (cf. Corollary~\\ref{cor:Thompson}):\n\n\\begin{cor}\nIf $M$ is a compact one-manifold,\nthen $F\\ast\\bZ$ and $T\\ast\\bZ$ are not subgroups of $\\Diff^{1+\\mathrm{bv}}(M)$.\n\\end{cor}\n\n\\bp\nSince $F\\le T$, it suffices to prove the corollary for $F*\\Z$ only. In order to apply Theorem~\\ref{thm:main}, it suffices to show that $F\\times\\Z\\le F$ and that $F$ is not virtually metabelian, whence the conclusion will be immediate. These claims follow immediately from the well--known facts that $F$ is not virtually solvable and that $F\\times F\\le F$, and we make these details explicit below for the reader's convenience.\n\nSince $F$ is a group of homeomorphisms of the interval, it is immediate that it is torsion--free. Moreover, conjugating $F$ by the homeomorphism of $\\R$ given by $x\\mapsto x\/2$ scales $F$ to be the group of piecewise linear homeomorphisms with dyadic breakpoints and slopes given by powers of two, only scaled to act on the interval $[0,1\/2]$. It follows that we may realize $F\\times F\\le F$, since we can realize one copy of $F$ on $[0,1\/2]$ and a second one on the interval $[1\/2,1]$, with the points $\\{0,1\/2,1\\}$ globally invariant. It follows that $\\Z\\times F\\le F$, since $F$ is torsion--free.\n\nTo see that $F$ is not virtually metabelian, we use the standard fact that $[F,F]$ is an infinite simple group and that $Z(F)=\\{1\\}$ (cf.~\\cite{CFP1996}). It follows that if $H\\le F$ is a finite index subgroup then $[F,F]\\le H$. Since $H$ contains an infinite simple group, it cannot be solvable, much less metabelian.\n\\ep\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThe authors thank the anonymous referees for many careful comments which improved the exposition of the paper. The first author is supported by Samsung Science and Technology Foundation (SSTF-BA1301-06).\nThe second author is partially supported by Simons Foundation Collaboration Grant number 429836.\n\n\n\\def\\cprime{$'$} \\def\\soft#1{\\leavevmode\\setbox0=\\hbox{h}\\dimen7=\\ht0\\advance\n  \\dimen7 by-1ex\\relax\\if t#1\\relax\\rlap{\\raise.6\\dimen7\n  \\hbox{\\kern.3ex\\char'47}}#1\\relax\\else\\if T#1\\relax\n  \\rlap{\\raise.5\\dimen7\\hbox{\\kern1.3ex\\char'47}}#1\\relax \\else\\if\n  d#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.9ex \\char'47}}#1\\relax\\else\\if\n  D#1\\relax\\rlap{\\raise.5\\dimen7 \\hbox{\\kern1.4ex\\char'47}}#1\\relax\\else\\if\n  l#1\\relax \\rlap{\\raise.5\\dimen7\\hbox{\\kern.4ex\\char'47}}#1\\relax \\else\\if\n  L#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.7ex\n  \\char'47}}#1\\relax\\else\\message{accent \\string\\soft \\space #1 not\n  defined!}#1\\relax\\fi\\fi\\fi\\fi\\fi\\fi}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nBerkovich analytic spaces and rigid analytic spaces \\cite{Ber1990}, \\cite{Ber2009} have the advantage that both schemes of finite type over a field and formal completions of such schemes along their closed subschemes can be thought of as living in the same category of analytic spaces. Punctured tubular neighborhoods of algebraic subvarieties inside ambient varieties over any field can be defined \\cite{BeTe}, \\cite{Th} as Berkovich analytic spaces.  In this article, we consider non-Archimedean analytic spaces from the perspective of algebraic geometry relative to the closed symmetric monoidal categories of Banach spaces. This language is very universal and provides a place to compare different geometries (rigid analytic spaces, Berkovich spaces and others). To\\\"{e}n and Vaqui\\'{e} introduced in \\cite{TV} algebraic geometry relative to any closed symmetric monoidal category. This idea had also been pursued by Hakim \\cite{Ha} and Deligne \\cite{Del}. We examine one of To\\\"{e}n and Vezzosi's \\cite{TVe5} topologies (which we call the homotopy Zariski topology) restricted to the opposite category of affinoid algebras over a non-Archimedean field and show that Berkovich and rigid analytic geometry embed fully and faithfully into the resulting category of schemes. \nWe use the framework of To\\\"{e}n, Vaqui\\'{e} and Vezzosi to suggest a new approach to analytic geometry which will extend in a natural way to the setting of higher and derived analytic stacks over Banach rings. This of course produces the difficult task of comparing the abstract definitions to the existing ones.  In order to make this article accessible to a broad range of mathematicians, we have included as many details as was possible. In forthcoming work \\cite{BaBe}, \\cite{BaBeKr}, we incorporate complex analytic geometry and give a more general approach that covers dagger analytic geometry and Stein geometry over a general valuation field (Archimedean or non-Archimedean). Some of the properties of the localization maps which we look at were proven from a complex analytic or differential geometric point of view in \\cite{Block, Pir, Tay}. Some similar work to ours was done \\cite{Mac} by A. Macpherson who uses an abstract, categorical notion of localizations instead of the condition on derived categories which we use. See also \\cite{Yu} and the references therein for the relationship with mirror symmetry. \n\nUsing the homological algebra from the work of Schneiders \\cite{SchneidersQA}, we give a new interpretation of the homotopy Zariski topology from \\cite{TVe5} which is suited to deal with quasi-abelian categories. Two of the theorems which we prove (Theorems \\ref{thm:TensLocs} and \\ref{thm:localForm}) show that that the finitely presented morphisms of affinoid algebras $\\mathcal{A}\\to \\mathcal{B}$ which are \\emph{homotopy epimorphisms} (meaning that $\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\cong \\mathcal{B}$) correspond geometrically to the affinoid subdomains (affinoids which are the unions of rational domains). In terms of modules, this says that a morphism of affinoids $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ is a domain embedding if and only if the corresponding morphism $D^{-}(\\mathcal{B})\\to D^{-}(\\mathcal{A})$ is fully faithful. Our categorical characterization of affinoid domains answers an open question poised by Soibelman in section 1.4 of \\cite{So}. He identifies this question as being a key step in the development of non-commutative analytic geometry. The other main feature of the G-topology is that the covers in this topology have finite subcovers $\\pi:\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ of an affinoid by affinoid subdomains which are surjective. In order to understand this surjectivity via modules, we introduce the notion of a RR-quasicoherent module for a commutative monoid $A$ relative to a symmetric monoidal quasi-abelian category in Definition \\ref{defn:RRqc}. We call this category of modules ${\\text{\\bfseries\\sf{Mod}}}^{RR}(A)$.   We prove in Lemmas \\ref{lem:ConsImpliesSur} and \\ref{lem:CoversRconservative} that $\\pi:\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ is surjective if and only if a morphism $f:M \\to N$ of RR-quasicoherent modules is an isomorphism if and only $\\pi^{*}f: \\pi^{*}M \\to \\pi^{*}N$ is an isomorphism. These modules satisfy a version of Tate acyclicity (see Remark \\ref{rem:TateAcylicity}) for the version we use of the Tate complex which uses only completed tensor products. More work on the category of quasi-coherent sheaves in analytic geometry will appear in \\cite{BaBeKr, BeKr2}.\n\nThe condition we use on morphisms of algebras: homotopy epimorphism, is the key idea behind the holomorphic functional calculus studied by \\cite{Tay} by Taylor in complex analytic geometry, who called these morphisms absolute localizations. Pirkovskii in \\cite{Pir} shows that open embeddings of complex Stein manifolds satisfy the same abstract condition. \n\n\n In \\cite{BeKr2} we construct monoidal model structures on categories of simplicial objects and on categories of complexes (negatively graded or unbounded) in a quasi-abelian category ${\\tC}$ satisfying certain extra conditions. We prove in \\cite{BeKr2} that these structures form HAG contexts in the sense of \\cite{TVe3}.  A HAG context is essentially a monoidal model category $M$ satisfying a long list of axioms that make it easy to work with. The model structure is induced from the model structure on simplicial sets. For instance, a morphism $X_{\\bullet} \\to Y_{\\bullet}$ in the category $M={\\text{\\bfseries\\sf{sC}}}$ of simplicial objects in ${\\tC}$  is a fibration (resp. weak equivalence) if for all projectives $P$ in ${\\tC}$ (defined in \\ref{defn:Projective}) that $\\Hom(P, X_{\\bullet}) \\to \\Hom(P, Y_{\\bullet})$ is a fibration (resp. weak equivalence) of simplicial sets. In particular, our main application is when the quasi-abelian category ${\\tC}={\\text{\\bfseries\\sf{Ind}}}({\\text{\\bfseries\\sf{Ban}}}_{R})$ is the category of Ind-Banach spaces over a Banach ring. In \\cite{TVe3} derived and homotopy algebraic geometry is developed relative to a HAG context.  Derived algebraic geometry (the case where ${\\tC}$ is the abelian category of abelian groups) is only one special case of their work. Our goal is to show that derived analytic geometry is another. A key feature in their work is definitions of (Grothendieck) topologies on the categories of affine schemes relative to $M$. Affine schemes are defined to be opposite category to ${\\text{\\bfseries\\sf{Comm}}}(M)$, the category of commutative monoids relative to $M$. The main way to describe these topologies is to explain which morphisms should appear in the cover, and describe when such a collection of morphisms is a cover. Both of these can be explained in terms of geometrically induced functors on the homotopy categories of modules. One topology that can be defined from their point of view says \n\\begin{defn}\\label{defn:TVtopologyIntro}\n A cover of $\\spec(A) \\in {\\text{\\bfseries\\sf{Comm}}}(M)^{op}$ is a collection of morphisms $\\spec(B_i) \\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Comm}}}(M)^{op}$ for $i \\in I$ such that for some finite set $J \\subset I$ we have \n\\begin{enumerate}\n\\item the push forward functor from the derived category of modules on $B_{i}$ to that on $A$ is fully faithful for all $i \\in J$\n\\item a morphism in the derived category of modules on $A$ is an isomorphism if and only if it becomes an isomorphism when pulled back in the derived sense to a morphism in the derived category of modules on $B_i$ for all $i \\in J$.\n\\end{enumerate} \nFor more details see Definition 1.2.6.1 (3) of \\cite{TVe3} for the first item and Definition 1.2.5.1 of \\cite{TVe3} for the second.\n\\end{defn}\nIn our type of derived analytic geometry, the category of affine schemes over a valuation field is opposite to the category of commutative monoids relative to $M={\\text{\\bfseries\\sf{sInd}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Theorem \\ref{thm:BigRestriction} shows that on the subcategory opposite to affinoid algebras, the topology in Definition \\ref{defn:TVtopologyIntro} restricts to the weak G-topology on affinoids. This theorem relies on \\cite{BeKr2}.\n\nWhile condition (1) of Definition \\ref{defn:TVtopologyIntro} restricts by our results to something completely classical and well understood, Condition (2) of Definition \\ref{defn:TVtopologyIntro} seems difficult to check on modules. Therefore, one might try to use a similar condition on the (underived) quasi-abelian category of modules using the underived pullback. However, this does not correspond to the surjectivity condition on covers, forcing us to reconsider which modules we care about and introduce the RR-quasicoherent modules.  For this reason, we defined the formal homotopy Zariski topology on a subcategory ${\\text{\\bfseries\\sf{A}}}$ of the category of affine schemes (${\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$) of a closed symmetric monoidal quasi-abelian category ${\\tC}$ where ${\\text{\\bfseries\\sf{A}}}$ and ${\\tC}$ satisfy a few simple conditions. The covers are a collection of morphisms $A\\to B_{i}$ of in ${\\text{\\bfseries\\sf{A}}}$ for $i \\in I$ such that for some finite set $J \\subset I$\n\\begin{enumerate}\n\\item $A \\to B_{i}$  are homotopy epimorphisms in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ for all $i \\in J$\n\\item a morphism $f:M \\to N$ in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(A)$ is an isomorphism if and only if the induced morphisms $M\\overline{\\otimes}_{A}B_i \\to N\\overline{\\otimes}_{A}B_i$ are for all $i \\in J$.\n\\end{enumerate}\nIn the special case that ${\\tC}={\\text{\\bfseries\\sf{Ban}}}_{k}$ and ${\\text{\\bfseries\\sf{A}}}$ is the category of affinoids our Theorem \\ref{thm:covers} says that this topology restricts to the weak G-topology on affinoids. Our work, put together with the work of To\\\"{e}n and Vezzosi establishes foundations for derived (and homotopy) analytic geometry and the study of higher and derived stacks over general Banach rings. There is another approach due to F. Paugam. In the complex analytic case, J. Lurie and M. Porta have a different approach to derived analytic geometry which should be compared to ours. \n\n\n\\section{Algebras and modules in closed symmetric monoidal categories} We assume that the reader is comfortable with the idea of a symmetric monoidal category. In particular the bifunctor $\\overline{\\otimes}$ admits an adjoint  $\\underline{\\Hom} :{\\tC}^{op} \\times {\\tC} \\to {\\tC}$ in the sense that there are natural isomorphisms\n\\[\\Hom(U \\overline{\\otimes} V,W) \\cong \\Hom(U, \\underline{\\Hom}(V,W))\\] for any $U,V,W \\in {\\tC}.$\nConsider a category ${\\tC}$ which has all finite limits and colimits equipped with extra data $(\\underline{\\Hom}, \\overline{\\otimes}, \\text{id})$ making it into a closed, symmetric monodial category. \nIn such a category one always has the following natural isomorphisms for any $U,V,W \\in {\\tC}$\n\n\\begin{enumerate}\n\\item \\[\\Hom(U,V) \\cong \\Hom(\\text{id}, \\underline{\\Hom}(U,V))\\]\n\\item \\[\\underline{\\Hom}(U \\overline{\\otimes} V,W) \\cong \\underline{\\Hom}(U, \\underline{\\Hom}(V,W))\\]\nwhich lifts by taking the $\\Hom$ from $\\text{id}$ the corresponding isomorphisms for $\\Hom$ instead of $\\underline{\\Hom}$\n\\item \\[ U \\cong \\underline{\\Hom}(\\text{id},U).\\]\n\\end{enumerate}\n\n\nAlso, there are natural morphisms for any $T,U,V,W \\in {\\tC}$\n\n\\begin{enumerate}\n\\item \\[\\underline{\\Hom}(U,U) \\overline{\\otimes} U \\to U\\]\n\\item \\[\\underline{\\Hom}(V,W) \\overline{\\otimes} \\underline{\\Hom}(U,V) \\to \\underline{\\Hom}(U,W) \\] satisfying associativity \n\\item \\[\\underline{\\Hom}(T,U) \\overline{\\otimes} \\underline{\\Hom}(V,W) \\to \\underline{\\Hom}(T \\overline{\\otimes} V, U \\overline{\\otimes} W)  \\]\n\\item \\[\\text{id} \\to \\underline{\\Hom}(U,U)\\]\n\\end{enumerate}\n\nwhich are compatible with the corresponding morphisms for $\\Hom$ instead of $\\underline{\\Hom}$. \n\\begin{rem}\nThe morphisms in the second item are adjoint to morphisms \n\\[\\underline{\\Hom}(V,W) \\to \\underline{\\Hom}( \\underline{\\Hom}(U,V) ,\\underline{\\Hom}(U,W))\n\\]\nand\n\\[\\underline{\\Hom}(U,V) \\to \\underline{\\Hom}( \\underline{\\Hom}(V,W) ,\\underline{\\Hom}(U,W))\n\\]\nand hence maps\n\\[\\Hom(V,W) \\to \\Hom( \\underline{\\Hom}(U,V) ,\\underline{\\Hom}(U,W))\n\\]\nand\n\\[\\Hom(U,V) \\to \\Hom( \\underline{\\Hom}(V,W) ,\\underline{\\Hom}(U,W)).\n\\]\n\\end{rem}\nSuch categories admit clear definitions of categories of commutative unital algebras objects in them. We denote these categories by ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. The opposite category to ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ we will be denoted ${\\text{\\bfseries\\sf{Aff}}}({\\tC})= {\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$ and if $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ we use $\\spec(A)$ to indicate this object in the opposite category.\n\n\nFor any fixed commutative unital algebra object $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ there is a category of unital left modules over such an algebra which we denote ${\\text{\\bfseries\\sf{Mod}}}(A)$. See Chapter 1.3 of \\cite{M} for details.  \nBecause we will be considering the unital notions only, we drop the word unital from our descriptions. This category is also a  closed symmetric monoidal category and has all finite limits and colimits. We write $\\overline{\\otimes}_{A}$ for the monoidal structure and $\\Hom_{A}$ for the morphisms in this category.  Given a morphism $p:\\spec(B) \\to \\spec(A)$, we sometimes write $p^{*}$ for the functor \n\\[{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)\n\\]\ngiven by\n\\[\nM \\mapsto B \\overline{\\otimes}_{A} M. \n\\]\n\\begin{defn}\\label{defn:TensA}Let us consider $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ and $E,F \\in {\\text{\\bfseries\\sf{Mod}}}(A)$. Let $a_{E}:A \\overline{\\otimes} E \\to E$ and $a_{F}:A \\overline{\\otimes} F \\to F$ denote the action morphisms. \nThe set of morphisms $\\Hom_{A}(E,F)$ is defined as the limit of the diagram\n\n\\[\n\\xymatrix{\\Hom(E,F) \\ar@\/^2pc\/[rr]^{f \\mapsto f \\circ a_{E}} \\ar[dr]_{h \\mapsto \\text{id}_{A} \\otimes h} &   & \\Hom(A \\overline{\\otimes} E,  F) \\\\\n& \\Hom(A \\overline{\\otimes} E, A \\overline{\\otimes} F)\\ar[ur]_{g \\mapsto a_{F} \\circ g} & \n}.\n\\] \nIn order to describe the tensor product, let us use $\\sigma:A \\overline{\\otimes} F \\to F \\overline{\\otimes} A$ to denote the symmetric structure. Let $E \\overline{\\otimes}_{A} F\\in {\\text{\\bfseries\\sf{Mod}}}(A)$ be the element of ${\\tC}$ given as the colimit of the diagram \n\\begin{equation}\\label{equation:TensDef}\\xymatrix{E \\overline{\\otimes} A \\overline{\\otimes} F \\ar@\/^\/[rr]^{\\text{id}_{E} \\otimes a_{F}} \\ar@\/_\/[rr]_{(a_{E} \\otimes \\text{id}_{F})\\circ (\\sigma \\otimes \\text{id}_{F})} & & E \\overline{\\otimes} F}\n\\end{equation}\nendowed with the obvious action of $A$.\n\n\\end{defn}\nNotice that for any $E\\in {\\text{\\bfseries\\sf{Mod}}}(A)$ and $F \\in {\\tC}$ can define \n\\[L_{E,F}:A\\overline{\\otimes} \\underline{\\Hom}(E,F) \\to  \\underline{\\Hom}(E,F)\n\\]\nas the composition \n\\[A\\overline{\\otimes} \\underline{\\Hom}(E,F) \\to  \\underline{\\Hom}(E,E) \\overline{\\otimes} \\underline{\\Hom}(E,F)\\to  \\underline{\\Hom}(E,F)\n\\]\nwhere the morphism $A \\to \\underline{\\Hom}(E,E)$ is adjoint to the action morphisms $A \\overline{\\otimes} E \\to E.$ Similarly for any $E\\in {\\tC}$ and $F \\in {\\text{\\bfseries\\sf{Mod}}}(A)$, one can define a morphism \n\\[R_{E,F}:A \\overline{\\otimes} \\underline{\\Hom}(E,F)  \\to  \\underline{\\Hom}(E,F)\n\\]\nas the composition \n\\[ A \\overline{\\otimes} \\underline{\\Hom}(E,F)  \\to  \\underline{\\Hom}(F,F) \\overline{\\otimes} \\underline{\\Hom}(E,F) \\to   \\underline{\\Hom}(E,F) \\overline{\\otimes} \\underline{\\Hom}(F,F)  \\to \\underline{\\Hom}(E,F).\n\\]\nBoth $L_{E,F}$ and $R_{E,F}$ endow  $\\underline{\\Hom}(E,F)$ with the structure of an element of ${\\text{\\bfseries\\sf{Mod}}}(A)$ satisfying various natural properties and we use these structures without further comment.\n\\begin{lem}\\label{lem:ALinHom}Suppose now that ${\\tC}$ is a closed, symmetric monoidal additive category with all finite limits and colimits and $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. Then ${\\text{\\bfseries\\sf{Mod}}}(A)$ is a closed symmetric monoidal category with all finite limits and colimits as well. These limits and colimits can be computed in ${\\tC}$.\n\\end{lem}\n{\\bf Proof.}\nBy tensoring the morphisms $\\text{id} \\to \\underline{\\Hom}(A,A)$ with the identity of $\\underline{\\Hom}(E,F)$ one can form \n\\[\\underline{\\Hom}(E,F) \\to \\underline{\\Hom}(A,A) \\overline{\\otimes} \\underline{\\Hom}(E,F) \\to \\underline{\\Hom}(A\\overline{\\otimes} E,A\\overline{\\otimes} F). \n\\] Now by using the functor $\\underline{\\Hom}(A\\overline{\\otimes} E, \\;)$ and the contravariant functor $\\underline{\\Hom}(\\;,F)$ applied to $A\\overline{\\otimes} F \\to F$ and $A\\overline{\\otimes} E \\to E$ respectively one can define $\\underline{\\Hom}_{A}(E,F)$ with the same type of limit as in \\ref{defn:TensA} replacing $\\Hom$ with $\\underline{\\Hom}$. Both $L_{E,F}$ and $R_{E,F}$ induce well defined (and equal) morphisms on  $\\underline{\\Hom}_{A}(E,F)$ and give  $\\underline{\\Hom}_{A}(E,F)$ the structure of an element of ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\ \\hfill $\\Box$\n\nWe have natural isomorphisms \n\\begin{equation}\\label{equation:ClosedPlus}\\underline{\\Hom}(M \\overline{\\otimes}_{A} N, L) \\cong \\underline{\\Hom}(M, \\underline{\\Hom}_{A}(N,L))\n\\end{equation}\nwhich satisfy the relevant axioms of a closed, symmetric monoidal category.\nIt is easy to see that (as in) \\cite{MeyerDerived} for any $E\\in {\\text{\\bfseries\\sf{Mod}}}(A)$\n\\begin{lem}\\label{lem:MeyerProperties}\\[\\underline{\\Hom}_{A}(A,E) \\to E \\]\n\\[f \\mapsto f(1)\n\\]\nis an isomorphism and so for any modules $M,N \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ and any $V \\in {\\tC}$ we have \n\\begin{enumerate}\n\\item $A \\overline{\\otimes}_{A} M \\cong M$\n\\item $\\underline{\\Hom}_{A}(A \\overline{\\otimes} V,E) \\cong \\underline{\\Hom}(V,E)$\n\\item $\\underline{\\Hom}_{A}(E, \\underline{\\Hom}(A,V)) \\cong \\underline{\\Hom}(E,V)$\n\\item $\\Hom_{A}(A, \\underline{\\Hom}_{A}(M,N)) \\cong \\Hom_{A}(M,N)$\n\\end{enumerate} \n\\end{lem}\n\nWe will discuss in this article various closed symmetric monoidal categories whose objects are vector spaces over Archimedean or non-Archimedean fields equipped with extra structures. The unit object in these categories is the field itself and it is easy to check the associativity, commutativity, and unit constraints. It is also clear that all of our closed symmetric monoidal categories are $k$-linear additive categories, as are the categories ${\\text{\\bfseries\\sf{Mod}}}(A)$ for $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. Because ${\\tC}$ has all finite limits and colimits, ${\\text{\\bfseries\\sf{Mod}}}(A)$ does as well. \n\n\\begin{defn}A functor $\\straightF:{\\tT}\\to {\\tS}$ is \nsaid to be conservative under that \nthe condition that a morphism $f$ in \n${\\tT}$ is an isomorphism if and only \nif $\\straightF(f)$ is an isomorphism.\n\\end{defn}\nNotice that for every $p:\\spec (B) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ the functor $p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ has a conservative right adjoint given by considering a $B$ module as an $A$ module \n\\[p_{*}:{\\text{\\bfseries\\sf{Mod}}}(B) \\to {\\text{\\bfseries\\sf{Mod}}}(A).\\]\n\n\\begin{defn}\\label{Symmetric}\n\nIf ${\\tC}$ has countable coproducts, the forgetful functor $\\straightF:{\\text{\\bfseries\\sf{Comm}}}({\\tC})\\to {\\tC}$ has a left adjoint $\\straightS:{\\tC}\\to {\\text{\\bfseries\\sf{Comm}}}({\\tC})$, given on objects \nby \n\\begin{equation*}\n\\straightS(V)=\\coprod_{n\\geq 0} \\straightS^n(V)\n\\end{equation*}\n\nwhere $\\straightS^n(V)$ are the co-invariants for the symmetric group action on $V^{\\overline{\\otimes} n}$. If $A$ is a commutative algebra in ${\\tC}$ we get in a similar way a functor \n$\\straightS_A:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Mod}}}(A))$ which is left adjoint to the forgetful functor. \n\\end{defn}\n\n\\begin{defn}\\label{defn:morphismSymAlg}\nLet ${\\tC}$ be a closed symmetric monoidal category with countable coproducts. Let $E$ and $E'$ be two objects, and assume given a map of algebras $\\alpha:\\straightS(E')\\to \\straightS(E)$. Let us denote by $\\straightS(E,E',\\alpha)$ the quotient of $\\straightS(E)$ by the ideal generated by the image of $\\alpha$ restricted to the augmentation ideal. In the case that $A\\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ and $E$ and $E'$ are in ${\\text{\\bfseries\\sf{Mod}}}(A)$ we use the notation $\\straightS_{A}(E,E',\\alpha)$ to denote the same construction taken in the closed symmetric monoidal category ${\\text{\\bfseries\\sf{Mod}}}(A)$ in place of ${\\tC}$.\nAn equivalent way of describing it is as the quotient of $\\straightS(E)$ by the ideal generated by the image of the object $E'$ under the map $\\alpha$.\n\\end{defn}\n \n \\hfill $\\Box$\n\\section{Algebraic geometry relative to closed symmetric monoidal categories}\\label{TV}\n\nIn this section, we review several notions from the article \\cite{TV} by To\\\"{e}n and Vaqui\\'{e}. We review their definitions of a flat morphism, a Zariski open immersion, and the Zariski and fpqc topologies on the opposite of the category of algebra objects in a closed, symmetric monoidal category admitting all finite limits and colimits. We also include definitions of a formal Zariski open immersion and the formal Zariski topology. We also review the definitions of schemes and (higher) stacks in this setting.\n\\begin{defn}\nThe category of presheaves of sets on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the category whose objects are contravariant functors ${\\text{\\bfseries\\sf{Aff}}}({\\tC}) \\to {\\text{\\bfseries\\sf{Set}}}$ and whose morphisms are natural transformations. It will be denoted ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC}))$. Given a Grothendieck topology $T$ on the category ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ we will often consider the full subcategories of sheaves of sets ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{T}) \\subset {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC}))$, where ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{T}$ is the site consisting of the underlying category ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ along with the Grothendieck topology $T$.\n\\end{defn}\n\\begin{defn}\nA family of functors $\\{\\straightF_{i}:{\\tC} \\to \\text{\\bfseries\\sf{D}}_{i}\\}_{i \\in I}$ \nis said to be conservative if a morphism $f$ in ${\\tC}$ \nis an isomorphism if and only if the \nmorphisms $\\straightF_{i}(f)$ in $\\text{\\bfseries\\sf{D}}_i$ are \nisomorphisms for all $i \\in I$.\n\\end{defn}\nSince we do not assume the existence of arbitrary limits and colimits, we need to define finite presentation in a non-standard way.\n\\begin{defn}\\label{defn:AbstractFinitePres}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. An object $W\\in {\\tC}$ is called of finite presentation if for every filtered diagram of objects $V_{i} \\in {\\tC}$ such that $\\colim_{i}V_{i}$ exists, the natural morphism\n\\begin{equation}\\label{eqn:AbstractFinitePres}\\colim_{i}\\Hom_{{\\tC}}(W,V_{i}) \\to \\Hom_{{\\tC}}(W,\\colim_{i}V_{i})\n\\end{equation}\nis an isomorphism.\n\\end{defn}\n\\begin{rem}Notice that any finite colimit of finitely \npresented objects is finitely presented.\n\\end{rem}\n\\begin{defn}\\label{defn:FinitePres}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. A morphism $A\\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is of finite presentation if for every filtered diagram of objects $A'_{i}\\in A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$ such that $\\colim_{i}A_{i}'$ exists in $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$, the natural morphism\n\\begin{equation}\\label{eqn:FinitePres}\\colim_{i}\\Hom_{A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,A_{i}') \\to \\Hom_{A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,\\colim_{i}A_{i}')\n\\end{equation}\nis an isomorphism.\n\\end{defn}\n\\begin{rem}Notice that a morphism $A\\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is of finite presentation if and only if $B$ is of finite presentation with respect to the category $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$.\n\\end{rem}\n\\begin{rem}\\label{rem:epi}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. We will need to pay special attention to the {\\it epimorphisms} in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$: those morphisms $p:A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ such that for all $C \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$, the induced map $\\Hom_{{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,C) \\to \\Hom_{{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(A,C)$ is injective. These correspond to monomorphisms in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. Notice that $p:A \\to B$ is an epimorphism if and only if the multiplication $B\\overline{\\otimes}_{A}B \\to B$ is an isomorphism.\n\\end{rem}\n\\begin{defn}\\label{defn:AbstractFlat}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. An object $V\\in {\\tC}$ is called flat when the functor ${\\tC}\\to{\\tC}$ given by $W \\mapsto V\\overline{\\otimes} W$ is exact (commutes with finite limits).\n\\end{defn}\n\\begin{rem}\\label{rem:SummandOfFlat} Suppose that ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits and that finite products and coproducts agree in ${\\tC}$. Then a finite coproduct of elements of ${\\tC}$ is flat if and only if each of them is individually flat.\n\\end{rem}\n\\begin{defn}\\label{defn:TVf}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. A morphism $p:A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is flat when the morphism \\[p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)\\] is exact (commutes with finite limits). This precisely says that $B$ is flat in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Mod}}}(A))$.\n\\end{defn}\nSay that $q:\\spec(C) \\to \\spec(A)$ is arbitrary. Consider the Cartesian diagram \n\\begin{equation}\\label{basechange}\\xymatrix{  \\spec(C\\otimes_{A}B) \\ar[r]^{q'} \\ar[d]_{p'} & \\spec(B) \\ar[d]^{p} \\\\\n\\spec(C) \\ar[r]_{q}& \\spec(A). \n}\n\\end{equation}\n\n\n\nUsing the notation of diagram (\\ref{basechange}) There is a natural equivalence \\cite{TV}\n\\begin{equation}\\label{eqn:basechangeequation}p^{*}q_{*} \\Longrightarrow q'_{*}p'^{*}\n\\end{equation}\ncalled base change. \n\n \\begin{defn}A base change of a morphism $p:\\spec(B) \\to \\spec(A)$ is the morphism $p'$ appearing in diagram (\\ref{basechange}) for some $q$.\n\\end{defn}\n\n\\begin{lem}\\label{InvBaseChange}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Suppose that $p:A \\to B$ is a flat morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ then any base change $p'$ of $p$ is also a flat morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. \n\\end{lem}\n\n{\\bf Proof.}\nBy assumption, $p^{*}: {\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ is exact. Let ${\\tL}$ be a finite category and consider the categories ${\\text{\\bfseries\\sf{Mod}}}(D)^{{\\tL}},$ the category of functors from ${\\tL}$ to ${\\text{\\bfseries\\sf{Mod}}}(D)$. Consider the functor \n\\[\\lim_{{\\tL},D}: {\\text{\\bfseries\\sf{Mod}}}(D)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(D)\n\\]\nwhich takes a diagram in ${\\text{\\bfseries\\sf{Mod}}}(D)$ to its limit.\nConsider the following commutative diagram of natural transformations of functors ${\\text{\\bfseries\\sf{Mod}}}(C)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(B)$: \n\\[\n\\xymatrix{q'_{*}\\lim_{{\\tL},C\\otimes_{A}B} p'^{*} \n\\ar@{=>}@\/_2pc\/[rrrr] \n& \\ar@{=>}[l] \\lim_{{\\tL},B} q'_{*} p'^{*} & \\ar@{=>}[l] \\lim_{{\\tL},B} p^{*}q_{*} \\ar@{=>}[r] & p^{*}q_{*} \\lim_{{\\tL},C} \\ar@{=>}[r] & q'_{*} p'^{*}\\lim_{{\\tL},C}.}\n\\]\n\\\\\nAll the natural transformations except the curved one on the bottom are invertible. This follows from the base change natural equivalences and the fact that the pushforward functors are right adjoints and so commute with limits. Therefore, the natural transformation on the bottom is also invertible.\nSince $q'_{*}$ is conservative, the natural transformation $\\lim_{{\\tL},C\\otimes_{A}B} p'^{*} \\Longrightarrow p'^{*}\\lim_{{\\tL},C}$ of functors ${\\text{\\bfseries\\sf{Mod}}}(C)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(C \\otimes_{A} B)$ is also invertible and so $p'^{*}$ commutes with finite limits. Therefore, $p'$ is flat.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:left}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Let $p$ be a morphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and let $p'$ be a base change of $p$.  If $p$ is a monomorphism then so is $p'$.  If $p$ is of finite presentation, so is $p'$.\n\\end{lem}\n{\\bf Proof.} Left to the reader.\n\\ \\hfill $\\Box$\n\nConsider the following very slight modification of Proposition 2.4 and its proof from \\cite{TV} (we only require finite limits and colimits and consider only a special case of their proposition).\n\\begin{lem}\\label{lem:BaseChangeConserv}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}(X) \\to {\\text{\\bfseries\\sf{Mod}}}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ has the same property.\n\\end{lem}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$. Choose a finite set $J \\subset I$ such that $\\prod_{i \\in J} p^{*}_{i}$ is conservative. Consider the functor $\\prod_{i \\in J} p^{'*}_{i}$ where $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). In order to show it is conservative, its enough to show that $\\prod_{i \\in J} q'_{i*}p^{'*}_{i}$ is conservative but using equation (\\ref{eqn:basechangeequation}) this is isomorphic to $(\\prod_{i \\in J} p^{*}_{i}) q_{*}$ which is conservative since $q_{*}$ is conservative.\n\\ \\hfill $\\Box$\n\\begin{prop}\\label{TV1}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Consider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}(X) \\to {\\text{\\bfseries\\sf{Mod}}}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}^{*}$ is left exact. These families define a pretopology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p_{i}^{*}$ are exact follows from Lemma \\ref{InvBaseChange}.\n\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:fpqc}\nFor any closed symmetric monoidal category ${\\tC}$ which has all finite limits and colimits, the topology coming from Proposition \\ref{TV1} is called the fpqc topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})= {\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}).$\n\\end{defn}\n\\begin{defn}\\label{defn:fTVZ}\n The morphism $\\spec(B) \\to \\spec(A)$ is called a formal Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a flat epimorphism (defined in Remark \\ref{rem:epi} and Definition \\ref{defn:TVf}).\n\\end{defn}\n\\begin{defn}\\label{defn:TVZ}\n The morphism $\\spec(B) \\to \\spec(A)$ is called a Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a flat epimorphism of finite presentation (defined in Remark \\ref{rem:epi} and Definitions \\ref{defn:FinitePres} and \\ref{defn:TVf}).\n\\end{defn}\n\\begin{lem}\\label{lem:ConsAgain}If a family $\\{A\\to B_{i}\\}_{i\\in I}$ is conservative and $A'$ is any $A$-algebra then the family $\\{A'\\to B_{i}\\otimes_{A} A'\\}_{i\\in I}$ is conservative.\n\\end{lem}\n{\\bf Proof.}\nThis has already been shown in Proposition \\ref{TV1}.\n\\ \\hfill $\\Box$\n\n\\begin{prop}\\label{prop:formalZarPretop}\nThere is a pretopology whose covering families $\\{A\\to B_{i}\\}_{i\\in I}$ are those families where each $A\\to B_{i}$ is a formal Zariski open immersion and the family $\\{A\\to B_{i}\\}_{i\\in I}$ has a finite conservative subfamily.\n\\end{prop}\n{\\bf Proof.}\nThis follows immediately from Lemmas \\ref{InvBaseChange}, \\ref{lem:ConsAgain} and \\ref{lem:left}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:fTV2} The formal Zariski topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the topology associated to the pretopology from Proposition \\ref{prop:formalZarPretop}. When equipped with this topology, we denote the category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fZar}$. The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fZar}).$\n\\end{defn}\n\\begin{prop}\\label{prop:ZarPretop}\nThere is a pretopology whose covering families $\\{A\\to B_{i}\\}_{i\\in I}$ are those families where each $A\\to B_{i}$ is a Zariski open immersion and the family $\\{A\\to B_{i}\\}_{i\\in I}$ has a finite conservative subfamily.\n\\end{prop}\n{\\bf Proof.}\nThis follows immediately from Lemmas \\ref{InvBaseChange}, \\ref{lem:ConsAgain} and \\ref{lem:left}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:TV2} The Zariski topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the topology associated to the pretopology from Proposition \\ref{prop:ZarPretop}. When equipped with this topology, we denote the category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{Zar}$. The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{Zar}).$\n\\end{defn}\n\\begin{defn}\\label{defn:hRep} For any affine object $\\spec(A),$ $A \\in {\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}}),$ the presheaf of sets $\\straighth_{A}$ is given by\n\\[{\\text{\\bfseries\\sf{Aff}}}({\\tC}) \\to {\\text{\\bfseries\\sf{Set}}}\n\\]\n\\[\\spec(B) \\mapsto \\Hom_{{\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}})}(A,B).\n\\]\n\\end{defn}\nCor. 2.11 of \\cite{TV} implies that \n\\begin{prop}\\label{TV3}\nFor any $A \\in {\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}})$, the preseheaf $\\straighth_{A}$ is a sheaf for fpqc, the formal Zariski and the Zariski topologies. \n\\end{prop}\n\n\\begin{defn}\\label{defn: ZarOpenSch}\\cite{TV} Let $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and $F\\in {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ be a subsheaf of $X$. Then $F$ is a called a Zariski open of $X$ if there is a family of Zariski opens $\\{X_i \\to X\\}_{i \\in I}$ such that $F$ is the image of the morphism of sheaves $\\coprod_{i \\in I} X_i \\to X$. A morphism $F \\to G$ in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ is called is a Zariski open immersion if for every $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and every $X \\to G$ the induced morphism $F \\times_{G} X \\to X$ is a monomorphism whose image is a Zariski open in $X$.\nThe category ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ of schemes is defined to be the full subcategory of ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ of sheaves $F$ such that there exists a family of $X_{i} \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ for $i \\in I$ and a morphism $p: \\coprod_{i \\in I} X_i \\to F$ such that $p$ is an epimorphism of sheaves and for each $i$ the morphism $X_i \\to F$ is a Zariski open.\n\\end{defn}\n\n\n\\begin{defn}\\label{defn:GeneralSch} Suppose that ${\\text{\\bfseries\\sf{A}}}$ is a full subcategory of ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and suppose that $\\tau$ is a subcategory of ${\\text{\\bfseries\\sf{A}}}$ with all objects and such that all morphisms in $\\tau$ are monomorphisms and so that the base change of a morphism in $\\tau$ by an arbitrary morphism of ${\\text{\\bfseries\\sf{A}}}$ is in $\\tau$. Say that $T$ is a pre-topology whose covers consist of families of morphisms where each morphism in the cover belongs to $\\tau$.  Let $X \\in {\\text{\\bfseries\\sf{A}}}$ and $F\\in {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ be a subsheaf of $X$. Then $F$ is a called a $\\tau$-open if there is a family of morphisms in $\\tau$ written  $\\{X_i \\to X\\}_{i \\in I}$ such that $F$ is the image of the morphism of sheaves $\\coprod_{i \\in I} X_i \\to X$. A morphism $F \\to G$ in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ is called is a $\\tau$-open immersion if for every $X \\in {\\text{\\bfseries\\sf{A}}}$ and every $X \\to G$ the induced morphism $F \\times_{G} X \\to X$ is a monomorphism whose image is a $\\tau$-open in $X$.\nThe category of schemes ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}, T, \\tau)$ is defined to be the full subcategory of ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ of sheaves $F$ such that there exists a family of $X_{i} \\in {\\text{\\bfseries\\sf{A}}}$ for $i \\in I$ and a morphism $p: \\coprod_{i \\in I} X_i \\to F$ such that $p$ is an epimorphism of sheaves and for each $i$ the morphism $X_i \\to F$ is a $\\tau$-open immersion.\n\\end{defn}\n\\begin{example}Two interesting general categories of schemes that we have in mind are ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}, T,\\tau)$ where ${\\text{\\bfseries\\sf{A}}}={\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$. First, the case where $\\tau$ of Zariski open immersions and $T$ is the Zariski pre-topology. Second, the category $\\tau$ of formal Zariski open immersions and $T$ is the formal Zariski pre-topology. \n\\end{example}\n\n\n\\begin{example}\nLet $k$ be any field and consider the category ${\\tC}= {\\text{\\bfseries\\sf{Vect}}}_{k}$. Then ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is the category of $k$-algebras. Recall that a morphism $A \\to B$ of $k$-algebras is of finite presentation in the usual sense when $B$ is finitely generated as an $A$ algebra and the ideal of relations is also finitely generated. Let us temporarily call a morphism TVfp if it satisfies the condition from Definition \\ref{defn:FinitePres}. Consider the functor\n\\[\\straightF : {\\text{\\bfseries\\sf{Set}}} \\to A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC}) \n\\]\nwhich sends each set to the $A$-algebra freely generated by it. Then Lawvere's work on finitary algebraic theories and Corollary 3.13 and the remark following it in \\cite{AR} show that $A \\to B$ is TVfp if and only if there exists finite sets $S_g$ and $S_r$ and an isomorphism in $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$ of the form\n\\[\\colim[\\straightF S_r \\rightrightarrows \\straightF S_g ] \\to B.\n\\]\nSo a morphism $f:A \\to B$ of $k$-algebras is of finite presentation in the categorical sense if and only it is of finite presentation in terms of generators and relations. This fact also appears in the algebraic geometry literature. The implication that finite presentation in terms of generators and relations implies finite presentation in the categorical sense was shown in this case in Lemma III.8.8.2.3 of \\cite{EGA4}. For the opposite implication see \\cite{stacks-project}. \nNow \\cite{EGA4} IV.17.9.1 tells us that a morphism of schemes is a flat monomorphism, locally of finite presentation if and only if it is an open immersion. Since a morphism of affine schemes $\\spec(B) \\to \\spec(A)$ is locally of finite presentation if and only if the corresponding morphism $A \\to B$ realizes $B$ as an $A$-algebra of finite presentation we can conclude that the Zariski open immersions are precisely the standard (Zariski) open immersions in algebraic geometry. The Zariski topology in the sense of relative algebraic geometry agrees with the Zariski topology in the standard sense in the case ${\\tC}= {\\text{\\bfseries\\sf{Vect}}}_{k}$. We should remark that this is the only example in this article for which we can use Lawvere's theory and \\cite{AR} in a straightforward way. Another way to characterize the Zariski open immersion is by replacing the flat epimorphism condition with a homotopy epimorphism condition, i.e. that the natural morphism in the derived category $B\\otimes^{\\mathbb{L}}_{A}B \\to B$ is an isomorphism. It is this condition that we examine this article for the category of Banach spaces, and not the flatness condition which would give a different answer. We comment on the case of vector spaces again in Remark \\ref{rem:recover}.\n\\end{example}\n\\begin{defn}\\label{defn:PresheafZarOpIm} A morphism $f:F \\to G$ in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is a Zariski open immersion if for every affine scheme $X$ and every morphism $X \\to G$, the induced morphism $F \\times_{G}X \\to X$ is a monomorphism of presheaves and its image agrees with the image of the map of sheaves $\\coprod_{i \\in I} X_{i } \\to X$ corresponding to a family of Zariski opens $X_{i} \\to X$.  \n\\end{defn}\n\\begin{defn}\\label{defn:AffToPreFlat}\nSuppose  $ F \\in {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ and $X \\in {\\text{\\bfseries\\sf{Aff}}}({\\tC})$. A morphism $X \\to F$ is flat if for every $Y \\in {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and every morphism $Y \\to F$ there is a Zariski open cover $\\coprod Z_{i} \\to X \\times_{F} Y$ such that the combined morphisms $Z_{i} \\to  X \\times_{F} Y \\to Y$ are flat. \n\\end{defn}\n\\begin{defn}\\label{defn:PresheafFlatMor} A morphism $f:F \\to G$ in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is flat if for every affine scheme $X$ and every morphism $X \\to G$ and every flat morphism $W \\to X \\times_{G} F$ the composition $W \\to X \\times_{G} F \\to X$ is flat.\n\\end{defn}\n\nConsider the Grothendieck site ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc}$. The category of simplicial objects in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is denoted ${\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})).$ This category comes with a (local) model structure as explained in \\cite{T}.\n\\begin{defn}\\cite{T} An object $F \\in {\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc})$ is called a pre-Stack. An object $F \\in {\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc})$ is called an fpqc stack if for any $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and any hypercovering $H_{\\bullet} \\to X,$ the natural morphism\n\\[F(X) \\to \\text{holim}_{[n] \\in \\Delta} F(H_n)\n\\]\nis an equivalence of simplicial sets. The category ${\\text{\\bfseries\\sf{St}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}) =Ho({\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc}))$ is the category of stacks.\n\\end{defn}\n\nAn equivalent definition to the above is to define pre-stacks as functors ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{op}\\to \\infty-{\\text{\\bfseries\\sf{Gpd}}}$, where $\\infty-{\\text{\\bfseries\\sf{Gpd}}}$ is the category of infinity groupoids. This is a full subcategory of the category of $(\\infty,1)$-categories for which one could use quasi-categories. $\\infty$-groupoids in this model are Kan simplicial sets. Stacks would be pre-stacks which satisfy descent with respect to hypercovers \\cite{DHI, L1,L2, TVe2,TVe3}. Similarly we could define (pre-)stacks valued in other categories, for instance a pre-stack in categories would be a functor ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{op} \\to (\\infty,1)-{\\text{\\bfseries\\sf{Cat}}}$. Note that the category of $(1,1)$-categories embeds into the category of $(\\infty,1)$-categories. Using quasi-categories to model $(\\infty,1)$-categories, the nerve of a $1$-category is a quasi-category. We can also view dg-categories as stable quasi-categories tensored over complexes \\cite{Coh}. We will use this later to view categories of quasi-coherent $\\mathcal{O}$-modules, and $\\mathcal{D}$-modules (derived or underived) as pre-stacks valued in categories. There is an inductive definition of an $n$-algebraic stack for $n=0,1,2,\\dots$. An algebraic fpqc stack on ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ is an fpqc stack on ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ which is $n$-algebraic for some $n$.\n\n\nOne can study schemes and stacks using with the fpqc or Zariski topology we have defined above. These could be useful in the analytic context as well when ${\\tC}$ is the category of Banach spaces and one can study faithfully flat descent in this context. However, the usual G-topology that is usually studied in non-Archimedean geometry as well as the classical metric topology of complex analytic geometry are finer topologies and have smaller ``open sets\". The localizations from Definition \\ref{defn:AffLocalization} are not flat with respect to the monoidal structure on ${\\text{\\bfseries\\sf{Ban}}}_{k}$ and for this reason we need to introduce new abstractly defined topologies which will fit in well with these facts. To do so, we need to use quasi-abelian categories.\n\n\n\n\\section{Quasi-abelian categories}We review some of Schneiders' theory of quasi-abelian categories. These are special cases of Palamodov's semi-abelian categories and of pseudo-abelian categories. They also have the  structure of a (Quillen) exact category in one natural way.\nThe main reference for this section is \\cite{SchneidersQA}.\n\\begin{defn}\nLet $\\mathcal{E}$ be an additive category with kernels and cokernels. A morphism $f:E\\to F$ is $\\mathcal{E}$ is called strict if the induced morphism \n\\[\\coim(f)\\to \\im(f)\n\\] is an isomorphism. \n\\end{defn}\nHere the image of $f$ is the kernel of the canonical map\n$F \\to \\coker(f)$, and the coimage of $f$ is the cokernel of the canonical map $\\ker (f)\\to E$. \n\n\\begin{defn}\\label{defn:CartCoCart}\nLet $\\mathcal{E}$ be an additive category with kernels and cokernels. We say that $\\mathcal{E}$ is quasi-abelian if it satisfies the following two conditions:\n\\begin{itemize}\n\\item In a cartesian square \n\\begin{equation*}\n\\xymatrix{  E' \\ar[r]^{f'} \\ar[d] & F' \\ar[d] \\\\\nE \\ar[r]_{f}& F}\n\\end{equation*}\n\nIf $f$ is a strict epimorphism then $f'$ is a strict epimorphism.\n\n\\item In a co-cartesian square\n\\begin{equation*}\n\\xymatrix{  E \\ar[r]^{f} \\ar[d] & F \\ar[d] \\\\\nE' \\ar[r]_{f'}& F'}\n\\end{equation*}\nIf $f$ is a strict monomorphism then $f'$ is a strict monomorphism.\n\n\\end{itemize}\n\\end{defn}\n\\begin{rem}\\label{rem:LeftRightInverse} Any morphism in a quasi-abelian category with a right inverse is a strict epimorphism.  Any morphism in a quasi-abelian category with a left inverse is a strict monomorphism. \\end{rem}\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. \nLet $\\xymatrix{ E'\\ar[r]^{e'} & E\\ar[r]^{e''} & E''}$ be a sequence of maps such that $e''\\circ e'=0$. We call such a sequence strictly exact (resp. strictly coexact) if $e'$ (resp. $e''$) is strict and the canonical map \n$\\im(e')\\to \\ker(e'')$ is an isomorphism. A complex \n$E_1\\to \\cdots \\to E_n$ is strictly exact (resp. strictly coexact) if each subsequence $E_{i-1}\\to E_i\\to E_{i+1}$ is strictly exact (resp. strictly coexact).\n\\end{defn}\n\n\\begin{rem}\\label{rem:threetermstrict}\nNote that the sequence \\begin{equation}\\label{equation:threetermstrict}\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]^{v}& G\\ar[r]& 0}\\end{equation} is strictly exact if and only if $u$ is the kernel of $v$ and $v$ is the cokernel of $u$. Any strict monomorphism or strict epimorphism can be completed to a strictly exact sequence in the form of Equation \\ref{equation:threetermstrict}. \nThis implies that such a sequence is strictly exact if and only if it is strictly coexact.\n\\end{rem}Let ${\\text{\\bfseries\\sf{E}}}$ be a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Then an object is flat if and only if tensoring with it preserves strict short exact sequences.\n\n\\begin{defn} Call a sequence  $\\xymatrix{ E'\\ar[r]^{e'} & E\\ar[r]^{e''} & E''}$  exact (resp. coexact) if the canonical map \n$\\im(e')\\to \\ker(e'')$ is an isomorphism. A sequence \n$E_1\\to \\cdots \\to E_n$ is exact (resp. coexact) if each subsequence $E_{i-1}\\to E_i\\to E_{i+1}$ is exact (resp. coexact).\n\\end{defn}\n\\begin{rem}\\label{rem:threeterm}\nNote that the sequence \\begin{equation}\\label{equation:threeterm}\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]^{v}& G\\ar[r]& 0}\\end{equation} is  exact if and only if $\\ker(u)=0$, $\\im(v)=G$ and $\\im(u) \\to \\ker(v)$ is an isomorphism.  Any  monomorphism or epimorphism can be completed to a exact sequence in the form of Equation \\ref{equation:threeterm}. \n\\end{rem}\n\nThe following is remark $1.1.11$ in \\cite{SchneidersQA}:\n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. The class of strictly exact short exact sequences endows ${\\text{\\bfseries\\sf{E}}}$ with the structure of an exact category.\n\\end{thm}\n\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. Let $\\straightK({\\text{\\bfseries\\sf{E}}})$ be its homotopy category. The derived category of ${\\text{\\bfseries\\sf{E}}}$ is $\\straightD({\\text{\\bfseries\\sf{E}}})=\\straightK({\\text{\\bfseries\\sf{E}}})\/\\straightN({\\text{\\bfseries\\sf{E}}})$ where $\\straightN({\\text{\\bfseries\\sf{E}}})$ is the full subcategory of strictly exact sequences.\n\\end{defn}\n\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. Let $\\straightK({\\text{\\bfseries\\sf{E}}})$ be its homotopy category. A morphism in \n$\\straightK({\\text{\\bfseries\\sf{E}}})$ is called a strict quasi-isomorphism if its mapping cone is strictly exact. \n\\end{defn}\n\nThe following is 1.2.17, 1.2.19, 1.2.20, 1.2.27 and 1.2.31 in \\cite{SchneidersQA}: \n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category.\n\\begin{enumerate}\n\\item $\\straightD({\\text{\\bfseries\\sf{E}}})$ has a canonical t-structure (the left t-structure). A complex $E$ belongs to $D^{\\leq 0}$ if and only if it is strictly exact in strictly positive degrees. $E$ belongs to $D^{\\geq 0}$ if and only if it is strictly exact in strictly negative degrees.\n\\item The heart of this t-structure $\\LH({\\text{\\bfseries\\sf{E}}})$, is equivalent to the localization of the full subcategory of $\\straightK({\\text{\\bfseries\\sf{E}}})$ consisting of complexes E of the form \n\\begin{equation}\n\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]& 0}\n\\end{equation}\nwhere $u$ is a monomorphism and $F$ is in degree $0$, by the multiplicative system formed by morphisms which are both cartesian and cocartesian. \n\\item There is a canonical fully faithful functor $\\sI:{\\text{\\bfseries\\sf{E}}}\\to LE({\\text{\\bfseries\\sf{E}}})$. A sequence $E'\\to E\\to E''$ is strictly exact in ${\\text{\\bfseries\\sf{E}}}$ if and only if $\\sI(E')\\to \\sI(E)\\to \\sI(E'')$ is exact in $\\LH({\\text{\\bfseries\\sf{E}}})$.\n\\item The functor $I$ induces an equivalence between $\\straightD({\\text{\\bfseries\\sf{E}}})$ and $\\straightD(\\LH({\\text{\\bfseries\\sf{E}}}))$. This equivalence sends the (left) t-structure on $\\straightD({\\text{\\bfseries\\sf{E}}})$ to the standard t-structure on $\\straightD(\\LH({\\text{\\bfseries\\sf{E}}}))$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\nThe embedding $\\sI:{\\text{\\bfseries\\sf{E}}}\\to \\LH({\\text{\\bfseries\\sf{E}}})$ is universal \nin the sense that induces an equivalence for any abelian category $\\mathcal{F}$ between left strictly exact functors from ${\\text{\\bfseries\\sf{E}}}$ to $\\mathcal{F}$ and left exact functors \nfrom $\\LH({\\text{\\bfseries\\sf{E}}})$ to $\\mathcal{F}$. In this sense $\\LH({\\text{\\bfseries\\sf{E}}})$ is the (left) abelian envelope of ${\\text{\\bfseries\\sf{E}}}$. See 1.2.33 in \\cite{SchneidersQA}.\n\\end{rem}\n\n\\begin{defn}\\label{defn:Projective}\nLet ${\\text{\\bfseries\\sf{E}}}$ be an additive category with kernels and cokernels. An object $I$ is called injective (resp. strongly injective) if the functor $E\\mapsto \\Hom(E,I)$ is exact (resp. strongly exact), i.e. for any strict (resp. arbitrary) monomorphism $u:E\\to F$, the induced map $\\Hom(F,I)\\to \\Hom(E,I)$ is surjective. Dually, $P$ is called projective (resp. strongly projective) if the functor $E\\mapsto \\Hom(P,E)$ is exact (resp. strongly exact), i.e. for any strict (resp. arbitrary) epimorphism $u:E\\to F$, the associated map $\\Hom(P,E)\\to \\Hom(P,F)$ is surjective.     \n\\end{defn}\n\n\\begin{defn}\\label{defn:enough}\nA quasi-abelian category ${\\text{\\bfseries\\sf{E}}}$ has enough projectives if for any object $E$ there is a strict epimorphism $P\\to E$ where $P$ is projective. A quasi-abelian category ${\\text{\\bfseries\\sf{E}}}$ has enough injectives if for any object $E$ there is a strict monomorphism $E \\to I$ where $I$ is injective.\n\\end{defn}\n\nThe following is 1.3.24 in \\cite{SchneidersQA}:\n\\begin{lem}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. \n\\begin{enumerate}\n\\item An object $P$ of ${\\text{\\bfseries\\sf{E}}}$ is projective if and only if $\\sI(P)$ is projective in $\\LH({\\text{\\bfseries\\sf{E}}})$.\n\\item ${\\text{\\bfseries\\sf{E}}}$ has enough projectives if and only if $\\LH({\\text{\\bfseries\\sf{E}}})$ has enough projectives. In this case an object of $\\LH({\\text{\\bfseries\\sf{E}}})$ is projective if it is isomorphic  to $\\sI(P)$ where $P$ is projective in ${\\text{\\bfseries\\sf{E}}}$.\n\\end{enumerate}\n\\end{lem}\n\nThe following is 1.3.22 in \\cite{SchneidersQA}:\n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category with enough projectives (resp. injectives). Let $\\text{\\bfseries\\sf{P}}$ be the full additive subcategory of projective objects (resp. $\\text{\\bfseries\\sf{I}}$ the category of injective objects). The canonical functor $\\sK^-(\\text{\\bfseries\\sf{P}})\\to \\sD^-({\\text{\\bfseries\\sf{E}}})$ (resp. $\\sK^+(\\text{\\bfseries\\sf{I}})\\to \\sD^+({\\text{\\bfseries\\sf{E}}})$) is an equivalence.\n\\end{thm}\n\n\n\n\n\\subsection{Closed symmetric monoidal quasi-abelian categories.}\n\nThe following is 1.5.1 in \\cite{SchneidersQA}:\n\\begin{prop}\\label{prop:StrictIFF}\nSuppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Suppose that $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$.  The category ${\\text{\\bfseries\\sf{Mod}}}(A)$ \nis quasi-abelian and the forgetful functor ${\\text{\\bfseries\\sf{Mod}}}(A)\\to {\\tC}$ preserves limits and colimits. A morphism in ${\\text{\\bfseries\\sf{Mod}}}(A)$ is strict if and only if it is strict in ${\\tC}$. \n\\end{prop}\n\n\n\\begin{lem}Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Using Remark \\ref{rem:LeftRightInverse} we see that for any $V  \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ the canonical morphism\n\\[V \\overline{\\otimes} A \\to V\n\\]\nis a strict epimorphism and the canonical morphism\n\\[V \\to \\underline{\\Hom}(A,V)\n\\]\nis a strict monomorphism.\n\\end{lem}\n\n\n\\begin{defn}\\label{defn:finiteAmodAbstract}\nSuppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. An object $V$ is called finite if there is a strict epimorphism $\\coprod_{i=1}^{n} \\text{id}_{{\\tC}} \\to V$ in ${\\tC}$ for some finite non-negative integer $n$. In the case that ${\\tC}={\\text{\\bfseries\\sf{Mod}}}(A)$ for $A$ a commutative monoid in a closed symmetric monoidal quasi-abelian category, we denote the full subcategory of finite objects by ${\\text{\\bfseries\\sf{Mod}}}^{f}(A)$.\n\\end{defn}\n\n\\begin{lem}\\label{lem:FreeProjCofreeInj} Suppose that $0 \\to L \\to M \\to N \\to 0$ is a strictly exact sequence in ${\\text{\\bfseries\\sf{Mod}}}(A)$ and $F =A\\overline{\\otimes} P \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ is free and $C=\\underline{\\Hom}(A,I) \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ is cofree. Then the sequences \n\\[0 \\to \\Hom_{A}(F,L) \\to  \\Hom_{A}(F,M)\\to  \\Hom_{A}(F,N) \\to 0\n\\]\nand\n\\[0 \\to \\Hom_{A}(N,C) \\to  \\Hom_{A}(M,C)\\to  \\Hom_{A}(L,C) \\to 0\n\\]\nare exact. If the sequence $0 \\to L \\to M \\to N \\to 0$ is only exact and $F$ is strictly free and $C$ is strictly cofree we can make the same conclusion.\n\\end{lem}\n{\\bf Proof.}\nUsing Lemma \\ref{lem:MeyerProperties} these sequences are isomorphic to the sequences\n\\[0 \\to \\Hom(P,L) \\to  \\Hom(P,M)\\to  \\Hom(P,N) \\to 0\n\\]\nand \n\\[0 \\to \\Hom(N,I) \\to  \\Hom(M,I)\\to  \\Hom(L,I) \\to 0\n\\]\nwhich are exact by definition of projectivity and injectivity (or the strict versions).\n\\hfill $\\Box$\n\\begin{lem}\\label{lem:MaintainProjInj}If $P$ is projective in ${\\tC}$ then $P\\overline{\\otimes} A$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A)$. Similarly, if $I$ is injective in ${\\tC}$ then $\\underline{\\Hom}(A,I)$ is injective in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\end{lem}\n{\\bf Proof.}\nBoth of these facts are immediately implied by Lemma \\ref{lem:FreeProjCofreeInj} together with Remark \\ref{rem:threetermstrict}.\n\\hfill $\\Box$\n\\begin{lem} \\label{lem:MonoEpiPreserved} The functor $E \\mapsto E\\overline{\\otimes} A$ takes epimorphisms in ${\\tC}$ to epimorphisms in ${\\text{\\bfseries\\sf{Mod}}}(A)$. The functor $E \\mapsto \\underline{\\Hom}(A,E)$ takes monomorphisms in ${\\tC}$ to monomorphisms in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\end{lem}\n{\\bf Proof.}\nThis follows directly from the definitions and Lemma \\ref{lem:MeyerProperties}.\n\\hfill $\\Box$\n\n\\begin{defn}\\label{defn:KerFlat} For $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ a module $M$ in ${\\text{\\bfseries\\sf{Mod}}}(A)$ is called kernel flat if for any morphism $f:E \\to F$ in ${\\text{\\bfseries\\sf{Mod}}}(A)$  the natural morphism \n\\begin{equation}\\label{equation:want4flatAbstr}B\\overline{\\otimes} \\ker(f) \\to \\ker(f_{M})\n\\end{equation}\nis an isomorphism where $f_{M}$ is defined as $\\text{id}_{M} \\overline{\\otimes}_{A}f:M \\overline{\\otimes}_{A}E \\to M \\overline{\\otimes}_{A}F.$ A morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is called kernel flat if it makes $B$ kernel flat over $A$.\n\\end{defn}\n\n\\begin{lem}\\label{lem:FlatkerFlat}Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category.  An object $V\\in {\\tC}$ is kernel flat if and only it is flat (see Definition \\ref{defn:AbstractFlat}). Therefore, a morphism of algebras $p: A \\to B$ is kernel flat (see Definition \\ref{defn:KerFlat}) if and only if it is flat (Definition \\ref{defn:AbstractFlat}) in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. \n\n\\end{lem}\n\n{\\bf Proof.}\nFirst of all if $p$ is flat it is clearly kernel flat since a kernel is a type of limit. In the other direction suppose that $p$ is kernel flat. It means that tensoring with $B$ commutes with kernels. Note that every limit over a finite diagram can be written as a combination of finite products and kernels. Finite products are isomorphic to finite coproducts and the functor given by tensoring with $B$ commutes with coproducts and hence it commutes with finite products. Therefore, tensoring with $B$ commutes with finite limits and hence $p$ is flat. \n\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:HelpWithEnough} Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Suppose that $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. If the category ${\\tC}$ has enough projectives \nthen the category ${\\text{\\bfseries\\sf{Mod}}} (A)$ has enough projectives. If the category ${\\tC}$ has enough injectives \nthen the category ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough injectives. \n\\end{lem}\n{\\bf Proof.} Suppose that ${\\tC}$ has enough projectives. Suppose that $V \\in {\\text{\\bfseries\\sf{Mod}}}(A)$. Choose a strict epimorphism in ${\\tC}$ of the form $P \\to V$ where $P$ is projective in ${\\tC}$.  Lemma \\ref{lem:MaintainProjInj} implies that $P\\overline{\\otimes} A$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A).$ Consider the morphism $P\\overline{\\otimes} A \\to V.$ We need to show it is a strict epimorphism in ${\\text{\\bfseries\\sf{Mod}}}(A).$ It factorizes as \n\\begin{equation}\nP \\overline{\\otimes} A \\to V \\overline{\\otimes} A  \\to V.\n\\end{equation}\nThe second morphism is a strict epimorphism because it admits a right inverse. The arrow $P \\overline{\\otimes} A \\to V \\overline{\\otimes} A$  is an epimorphism by Lemma \\ref{lem:MonoEpiPreserved} and in fact a strict epimorphism because the monoidal product with $A$ is a left adjoint functor and preserves cokernels. Therefore ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough projectives.\nSuppose that ${\\tC}$ has enough injectives. Choose a strict monomorphism in ${\\tC}$ of the form $V \\to I$ where $I$ is injective in ${\\tC}$. Lemma \\ref{lem:MaintainProjInj} implies that $\\underline{\\Hom}(A,I)$ is injective in ${\\text{\\bfseries\\sf{Mod}}}(A).$ Consider the morphism $V \\to \\underline{\\Hom}(A,I).$ We need to show that it is a strict monomorphism in ${\\text{\\bfseries\\sf{Mod}}}(A).$ It factorizes as \\begin{equation}\nV \\to \\underline{\\Hom}(A,V) \\to \\underline{\\Hom}(A,I).\n\\end{equation}\nNotice that here, we are considering $\\underline{\\Hom}(A,V)$ and $\\underline{\\Hom}(A,I)$ as elements of \n${\\text{\\bfseries\\sf{Mod}}}(A)$ using the action of $A$ on itself. The first arrow is a strict monomorphism because it admits a left inverse. Using Lemma \\ref{lem:MonoEpiPreserved}, $\\underline{\\Hom}(A,V) \\to \\underline{\\Hom}(A,I)$ is a monomorphism in ${\\text{\\bfseries\\sf{Mod}}}(A)$ and in fact a strict monomorphism because the internal Hom from $A$ is a right adjoint functor and preserves kernels. Therefore ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough injectives.\n\\hfill $\\Box$\n\\begin{lem}\\label{lem:BaseChangeInj}Suppose that $\\sF: {\\tC} \\to \\text{\\bfseries\\sf{D}}$ is a functor which has a right adjoint $\\sR$ and which \npreserves strict monomorphisms (preserves monomorphisms). Then an injective (strongly injective) in $\\text{\\bfseries\\sf{D}}$ is an injective (strongly injective) when considered in ${\\tC}$ via $\\sR$ .\n\\end{lem}\n{\\bf Proof.}\n\nSuppose that $I \\in \\text{\\bfseries\\sf{D}}$ is injective (strongly injective). Then consider a strict monomorphism (monomorphism) $E\\to F$ in ${\\tC}.$ Then $\\sF(E) \\to \\sF(F)$ is a strict monomorphism (monomorphism) in ${\\tC}.$ We have a commutative diagram\n\\[\\xymatrix{{\\tC}(F,\\sR (I))  \\ar[r]& {\\tC} (E,\\sR (I)) \\\\\n\\text{\\bfseries\\sf{D}}(\\sF (F),I) \\ar[r] \\ar[u]& \\text{\\bfseries\\sf{D}} (\\sF (E),I). \\ar[u]\n}\n\\]\nBecause the upwards arrows are isomorphisms and the lower horizontal arrow is surjective, the upper horizontal arrow is surjective as well. Therefore, $I$ is  injective (strongly injective) when considered as an object of ${\\tC}$. \n\\hfill $\\Box$\n\n\\begin{lem}\\label{lem:BaseChangeProj}Suppose that $\\sG: {\\tC} \\to \\text{\\bfseries\\sf{D}}$ is a functor which has a left adjoint $\\sL$ and which preserves strict epimorphisms (preserves epimorphisms). Then a projective (strongly projective) in $\\text{\\bfseries\\sf{D}}$  is projective (strongly projective) when considered in ${\\tC}$.\n\\end{lem}\n{\\bf Proof.}\nSuppose that $P \\in \\text{\\bfseries\\sf{D}}$ is projective (strongly projective). Consider a strict epimorphism (epimorphism) $E\\to F$ in $\\text{\\bfseries\\sf{D}}(A).$ Then by $\\sG(E) \\to \\sG(F)$ is a strict epimorphism (epimorphism) in $\\text{\\bfseries\\sf{D}}.$ We have a commutative diagram\n\\[\\xymatrix{{\\tC}(\\sL (P),E)  \\ar[r]& {\\tC} (\\sL (P),F)\\\\\n\\text{\\bfseries\\sf{D}}(P,\\sG (E))\\ar[u] \\ar[r]& \\text{\\bfseries\\sf{D}}(P,\\sG (F)) \\ar[u].\n}\n\\]\nBecause the upwards arrows are isomorphisms and the lower horizontal arrow is surjective, the upper horizontal arrow is surjective as well. Therefore, $P$ is projective (strongly projective) when considered in ${\\tC}$. \n\\ \\hfill $\\Box$\n\nLet ${\\text{\\bfseries\\sf{E}}}$ be a closed symmetric monoidal quasi-abelian category and let $A\\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{E}}})$. \nThe following is contained in 2.1.18 in \\cite{SchneidersQA}: If $P$ is projective in ${\\text{\\bfseries\\sf{E}}}$ then $A\\overline{\\otimes} P$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\n\n\n \n\\subsection{Derived Functors}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor betwen quasi-abelian categories ${\\tC}$ and ${\\tD}$. Schneiders gave the following definitions in 1.3.2 of \\cite{SchneidersQA}\n\\begin{defn} A full additive subcategory ${\\tP}$ of ${\\tC}$ is called $\\sF$-projective if:\n\\begin{enumerate}\n\\item for any object $V$ of ${\\tC}$ there is an object $P$ of ${\\tP}$ and a strict epimorphism $P\\to V$\n\\item in any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V$ and $V''$ are objects of ${\\tP}$, $V'$ is as well\n\\item for any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V, V'$ and $V''$ are objects of ${\\tP}$, the sequence \n\\[0 \\to \\sF(V') \\to \\sF(V) \\to \\sF(V'') \\to 0\n\\]\nis strictly exact in ${\\tD}.$\n\\end{enumerate}\n A full additive subcategory ${\\tI}$ of ${\\tC}$ is called $\\sF$-injective if:\n\\begin{enumerate}\n\\item for any object $V$ of ${\\tC}$ there is an object $I$ of ${\\tI}$ and a strict monomorphism $V\\to I$\n\\item in any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V$ and $V''$ are objects of ${\\tI}$, $V'$ is as well\n\\item for any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V, V'$ and $V''$ are objects of ${\\tI}$, the sequence \n\\[0 \\to \\sF(V') \\to \\sF(V) \\to \\sF(V'') \\to 0\n\\]\nis strictly exact in ${\\tD}.$\n\\end{enumerate}\n\\end{defn}\nSchneiders also includes the following (Lemma 1.3.3 \\cite{SchneidersQA})\n\\begin{lem}\nLet ${\\tC}$ be a quasi-abelian category and let ${\\tP}$ be a subset of the objects of ${\\tC}$. Assume that for any object $V$ of ${\\tC}$ there is a strict epimorphism $P \\to V$ with $P \\in {\\tP}$. Then for each object $V$ of $C^{-}({\\tC})$ there is a quasi-isomorphism $u:P \\to V$ with $P$ in $C^{-}({\\tP})$ and such that each $u^{k}:P^{k} \\to V^{k}$ is a strict epimorphism.\n\\end{lem}\n\nFrom this we get (proposition 1.3.5 \\cite{SchneidersQA}):\n\\begin{prop}\\label{lem:derivable}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$.\n\\begin{enumerate}\n\\item Assume that ${\\tC}$ has an $\\sF$-projective subcategory. Then $\\sF$ has a left derived functor $L\\sF:D^-({\\tC})\\to D^-({\\tD})$.\n\\item Assume that ${\\tC}$ has an $\\sF$-injective subcategory. Then $\\sF$ has a right derived functor $R\\sF:D^+({\\tC})\\to D^+({\\tD})$.\n\\end{enumerate}\n\\end{prop}\n\\begin{defn}In the situations of Lemma \\ref{lem:derivable}, $\\sF$ is called explicitly left derivable or explicitly right derivable.\n\\end{defn}\nHere, derived functors are defined as usual by their universal property.\n\n\\begin{rem}\\label{rem:sse2sse}Note that if $\\sF$ is exact (sends strict short exact sequences to strict short exact sequences) then the full subcategory ${\\tC}$ itself is an $\\sF$-projective (and injective) subcategory. Hence exact functors are always derivable.   \n\\end{rem}\n\n\n\nAs in the abelian case, projective and injectives form $\\sF$-projective and $\\sF$-injective subcategories (remark 1.3.21 \\cite{SchneidersQA}):\n\\begin{prop}\\label{prop:ProjFProj}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$. \n\\begin{enumerate}\n\\item Assume that ${\\tC}$ has enough projectives. Then the full subcategory of projective objects is a $\\sF$-projective subcategory and therefore can be used to explicitly left derive the functor $\\sF.$\n\\item Assume that ${\\tC}$ has enough injectives. Then the full subcategory of injective objects is a $\\sF$-injective subcategory  and therefore can be used to explicitly right derive the functor $\\sF.$\n\\end{enumerate}\n\\end{prop}\nWe also have the following (remark 1.3.7 \\cite{SchneidersQA}):\n\\begin{lem}\\label{lem:acyclic}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$. Assume that $\\sF$ has a right derived functor $R\\sF:D^+({\\tC})\\to D^+({\\tD})$. Call an object $I$ $\\sF$-acyclic if $R\\sF(I)\\cong \\sF(I)$. Assume that for any object \n$A$, there is an $\\sF$-acyclic object $I$ and a monomorphism \n$A\\to I$. Then the $\\sF$-acyclic objects form a $\\sF$-injective subcategory.  Assume that $\\sF$ has a left derived functor $L\\sF:D^-({\\tC})\\to D^-({\\tD})$. Call an object $P$ $\\sF$-acyclic if $L\\sF(P)\\cong \\sF(P)$. Assume that for any object \n$A$, there is an $\\sF$-acyclic object $P$ and a epimorphism \n$P \\to A$. Then the $\\sF$-acyclic objects form a $\\sF$-projective subcategory. \n\\end{lem}\n\\begin{defn}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelian category with monoidal structure $\\overline{\\otimes}$. An object $V$ of ${\\tC}$ is called $\\overline{\\otimes}$-acyclic if $V$ is $\\sF$-acyclic for all of the functors $\\sF:{\\tC} \\to {\\tC}$ given by $U \\mapsto U \\overline{\\otimes} W$ for any object $W$ in ${\\tC}$.\n\\end{defn}\n\\subsection{Topologies based on homological algebra}\nUsing the homological algebra in this section, we now introduce some more classes of morphisms and Grothendieck topologies on a closed symmetric monoidal quasi-abelian category $\\text{\\bfseries\\sf{C}}$ with all finite limits and colimits.\n\\begin{lem}For any morphism $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC}),$ the induced morphism $p_{*}:{\\text{\\bfseries\\sf{Mod}}}(B)\\to {\\text{\\bfseries\\sf{Mod}}}(A)$ is derivable to a functor $\\sD^{-}(B) \\to \\sD^{-}(A).$ \n\\end{lem}\n{\\bf Proof.} This functor sends strict exact sequences to strict exact sequences so this follows from Remark \\ref{rem:sse2sse}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:homotopyEpi}A morphism $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy monomorphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ if the induced morphism $p_{*}:\\sD^{-}(B) \\to \\sD^{-}(A)$ is fully faithful.\n\\end{defn}\nNotice that by considering $i=0$ in Definition \\ref{defn:homotopyEpi} we see that a homotopy epimorphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is in particular an epimorphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC}).$\n\n\\begin{lem}\\label{lem:ComposHom}The composition of homotopy monomorphisms in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is a homotopy monomorphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$.\n\\end{lem}\n{\\bf Proof.}\nThis follows from the fact that the composition of fully faithful functors is fully faithful.\n\\ \\hfill $\\Box$\n\\begin{lem}\n\nAssume that $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and that the functor $p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ given by tensoring with $B$ over $A$ is explicitly left derivable to a functor $\\mathbb{L}p^{*}:\\sD^{-}(A)\\to \\sD^{-}(B)$. Then $p$ is homotopy monomorphism if and only if the natural morphism of functors $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is an isomorphism.\n\n\\end{lem}\n{\\bf Proof.}\nWe have natural isomorphisms for any objects $M,N \\in \\sD^{-}(B)$ \n\\[\\Hom_{\\sD^{-}(B)}(\\mathbb{L}p^{*}p_{*}M,N)   \\cong \\Hom_{\\sD^{-}(A)}(p_{*}M,p_{*}N) .\n\\]\nTherefore, if $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is a isomorphism then $p$ is a homotopy epimorphism. The converse follows from a simple application of the Yoneda lemma.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:HomotopyMon}\n Assume that $p:\\spec(B)\\to \\spec(A)$ is a morphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and that the functor ${\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ given by tensoring with $B$ over $A$ is explicilty left derivable to a functor $\\sD^{-}(A)\\to \\sD^{-}(B)$. Then $p$ is homotopy monomorphism if and only if $B\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong B$.\n\\end{lem}\n{\\bf Proof.} \nFor any object $M$ of $\\sD^{-}(B)$ we have \n\\begin{equation*}\nM\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong M\\overline{\\otimes}^{\\mathbb{L}}_{B}(B\\overline{\\otimes}^{\\mathbb{L}}_{A}B).\n\\end{equation*}\n\nHence $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is an isomorphism if and only if  we have natural isomorphisms $M\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong M$ for any $M \\in \\sD^{-}(B)$ which happens if and only if $B\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong B$.\n\\ \\hfill $\\Box$\n\n\n\\begin{defn}\\label{defn:fhTVZ} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.\n The morphism $\\spec(B) \\to \\spec(A)$ of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy formal Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a homotopy epimorphism.\n\\end{defn}\n\\begin{defn}\\label{defn:hTVZ} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.\n The morphism $\\spec(B) \\to \\spec(A)$ of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a homotopy epimorphism of finite presentation.\n\\end{defn}\n\n\\begin{defn}\\label{defn:Amitsur}The Amitsur complex of a morphism $f:A \\to B$ in $ {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is the complex $\\mathscr{A}(f)$ given by \n\\[0 \\to A \\to B \\to B\\overline{\\otimes}_{A}B \\to  B\\overline{\\otimes}_{A}B\\overline{\\otimes}_{A}B \\to \\cdots\n\\]\nwhere the morphism $B^{\\overline{\\otimes}_{A}^{m}}\\to B^{\\overline{\\otimes}_{A}^{m+1}}$ is defined by \n\\[d(b_{1} \\overline{\\otimes} b_{2} \\overline{\\otimes} \\cdots \\overline{\\otimes} b_{m})= \\sum_{i=1}^{m+1}(-1)^{i}b_{1} \\overline{\\otimes} \\cdots  \\overline{\\otimes} b_{i-1}\\otimes 1 \\overline{\\otimes} b_{i} \\overline{\\otimes} \\cdots \\overline{\\otimes} b_{m}.\n\\]\nIn the case that we can chose a decomposition $f= \\prod_{i=1}^{n} f_{i}:A \\to \\prod_{i=1}^{n} B_{i}=B$ there is a strictly included subcomplex $\\mathscr{A}^{a}(f)\\to \\mathscr{A}(f)$ where $\\mathscr{A}^{a}(f)$ is given by the finite complex \n\\[0 \\to A \\to \\prod_{1 \\leq i_1 \\leq n} B_{i}  \\to \\prod_{1\\leq i_1<i_2\\leq n} B_{i_1}\\overline{\\otimes}_{A} B_{i_2} \n\\to \\cdots \\to B_{1} \\overline{\\otimes}_{A} \\cdots \\overline{\\otimes}_{A} B_{n} \\to 0\\]\nwith the induced differentials. \n\\end{defn}\n\\begin{defn}\\label{defn:hZt} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. We call a full subcategory ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ \\emph{homotopy Zariski transversal} if it is closed under fiber products and for any homotopy monomorphism $\\spec (B) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{A}}}$ and for any morphism $\\spec (C) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{A}}}$ \n\\[\\text{the natural morphism} \\ \\ B \\overline{\\otimes}_{A}^{\\mathbb{L}}C \\to  B \\overline{\\otimes}_{A}C \\ \\ \\text{is an isomorphism.}\n\\]\n\\end{defn}\n\\begin{lem}\\label{lem:IfThen} If ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory then the base change of a homotopy monomorphism is a homotopy monomorphism.\n\\end{lem}\n{\\bf Proof.} If $A \\to B$ is a homotopy epimorphism in ${\\text{\\bfseries\\sf{A}}}$ and $A \\to C$ is any morphism in ${\\text{\\bfseries\\sf{A}}}$ then \n\\[(B \\overline{\\otimes}_{A} C) \\overline{\\otimes}^{\\mathbb{L}}_{C} (B \\overline{\\otimes}_{A} C) \\cong (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C) \\overline{\\otimes}^{\\mathbb{L}}_{C} (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C) \\cong B \\overline{\\otimes}^{\\mathbb{L}}_{A} (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C)\\cong (B \\overline{\\otimes}^{\\mathbb{L}}_{A} B) \\overline{\\otimes}^{\\mathbb{L}}_{A} C \\cong B \\overline{\\otimes}^{\\mathbb{L}}_{A} C\\cong B \\overline{\\otimes}_{A} C.\n\\]\n\\ \\hfill $\\Box$\n\nThe following definition comes from work \\cite{RR} of Ramis-Ruget on quasi-coherent sheaves in complex analytic geometry. Based on their work, quasi-coherent modules in the complex analytic context were discussed in the book \\cite{EP}. Our definition is inspired by that one. \n\\begin{defn}\\label{defn:RRqc}Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. For $\\spec(A) \\in {\\text{\\bfseries\\sf{A}}}$ define ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(A)$ to be the full subcategory of ${\\text{\\bfseries\\sf{Mod}}}(A)$ consisting of modules $M$ such that $M$ is transversal to all homotopy epimorphisms in ${\\text{\\bfseries\\sf{A}}}$. That is, we consider modules $M$ such that the natural morphism \\[M \\overline{\\otimes}_{A}^{\\mathbb{L}}B \\to  M \\overline{\\otimes}_{A}B \\] is an isomorphism for all homotopy epimorphisms $A \\to B$. We call these RR-quasi-coherent modules.\n\\end{defn}\n\\begin{lem}Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. If $\\spec(C) \\to \\spec(A)$ is any morphism in ${\\text{\\bfseries\\sf{A}}}$ then the morphism $A \\to C$ gives $C$ the structure of an object of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(A)$.\n\\end{lem}\n{\\bf Proof.} This is an obvious consequence of Definition \\ref{defn:hZt}.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:RRpreserve} If ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory then pushforwards by morphisms in ${\\text{\\bfseries\\sf{A}}}$ preserve the category of RR-quasi-coherent modules. If a morphism is transverse to a RR-quasi-coherent module, then the pull-back of this module by that morphism is also RR-quasi-coherent. Therefore, pullbacks by homotopy monomorphisms or flat morphisms in ${\\text{\\bfseries\\sf{A}}}$ also preserve this category. \n\\end{lem}\n{\\bf Proof.} Say we are given a morphism $ \\spec(C) \\to \\spec(A)$ in  ${\\text{\\bfseries\\sf{A}}}$. Given any object $M$ of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(C)$ and a homotopy monomorphism $\\spec(B) \\to \\spec(A)$, we have \n\\[M \\overline{\\otimes}^{\\mathbb{L}}_{A} B \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} (C \\overline{\\otimes}_{A}^{\\mathbb{L}}B) \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} (C \\overline{\\otimes}_{A}B) \\cong M \\overline{\\otimes}_{C} (C \\overline{\\otimes}_{A}B) \\cong M \\overline{\\otimes}_{A} B.\n\\]\nThis proves the statement about push-forwards. For the statements about pullbacks, fix a morphism $\\spec(D) \\to \\spec(C)$ in ${\\text{\\bfseries\\sf{A}}}$ (which we will pullback with) which is transverse to $M$ and a homotopy monomorphism $\\spec(E) \\to \\spec(D)$ in ${\\text{\\bfseries\\sf{A}}}$. Then\n\\[(M \\overline{\\otimes}_{C} D)\\overline{\\otimes}^{\\mathbb{L}}_{D}E \\cong (M \\overline{\\otimes}^{\\mathbb{L}}_{C} D)\\overline{\\otimes}^{\\mathbb{L}}_{D}E \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} E \\cong M \\overline{\\otimes}_{C} E \\cong (M \\overline{\\otimes}_{C} D)\\overline{\\otimes}_{D}E.\n\\]\n\\ \\hfill $\\Box$\n\nAs a corollary, we can get a helpful analogue of Lemma \\ref{lem:BaseChangeConserv}: \n\\begin{cor}\\label{cor:BaseChangeConservHzAR}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelian category. Suppose that ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ in ${\\text{\\bfseries\\sf{A}}}$ of homotopy monomorphisms is such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ has the same property.\n\\end{cor}\n{\\bf Proof.} The proof of Lemma \\ref{lem:BaseChangeConserv} uses only the functors $q_{*}$, $q'_{*}$, $p_{i}^{*}$ and ${p'}_{i}^{*}$. The first three preserve the categories of RR-quasi-coherent modules by Lemma \\ref{lem:RRpreserve} and the last one  does by Lemmas \\ref{lem:IfThen} and \\ref{lem:RRpreserve}. Therefore, that same proof works here.\n\\ \\hfill $\\Box$\n\n\\begin{prop}\\label{hfTV1}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}}$ be a homotopy transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$.  Consider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{A}}}$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}$ is a homotopy monomorphism. These families define a pretopology on ${\\text{\\bfseries\\sf{A}}}$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{A}}}$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p'_{i}$ are homotopy monomorphisms follows from the assumption.\n\n\\ \\hfill $\\Box$\n\nLet ${\\tC}$ be a closed symmetric monoidal quasi-abelain category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. Say that $q:\\spec(C) \\to \\spec(A)$ is arbitrary. Say we are given $A, B, C \\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{sC}}})$.  Consider a Cartesian diagram \n\\begin{equation}\\label{Derivedbasechange}\\xymatrix{  \\spec(C\\otimes^{\\mathbb{L}}_{A}B) \\ar[r]^{q'} \\ar[d]_{p'} & \\spec(B) \\ar[d]^{p} \\\\\n\\spec(C) \\ar[r]_{q}& \\spec(A). \n}\n\\end{equation}\n\n\n\nUsing the notation of diagram (\\ref{Derivedbasechange}) there is a natural equivalence \n\\begin{equation}\\label{eqn:basechangeequation}(\\mathbb{L}p^{*})q_{*} \\Longrightarrow q'_{*}(\\mathbb{L}p'^{*})\n\\end{equation}\ncalled base change. \n\n\\begin{lem}\\label{lem:DerivedBaseChangeConserv}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelain category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ of homotopy monomorphisms in ${\\text{\\bfseries\\sf{A}}}$ is such that the family $\\{\\mathbb{L}p^{*}_{i}:D^{-}(X) \\to D^{-}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ in ${\\text{\\bfseries\\sf{A}}}$ has the same property.\n\\end{lem}\n{\\bf Proof.}  In order to show the base change property, consider $q:Y \\to X$. Choose a finite set $J \\subset I$ such that $\\prod_{i \\in J} \\mathbb{L}p^{*}_{i}$ is conservative. Consider the functor $\\prod_{i \\in J} \\mathbb{L}{p'}^{*}_{i}$ where $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). In order to show it is conservative, its enough to show that $\\prod_{i \\in J} q'_{i*}\\mathbb{L}{p'}^{*}_{i}$ is conservative. By the definition of homotopy Zariski transversal, the natural morphism $X_{i}\\times_{X} Y \\to  X_{i}\\times^{h}_{X} Y$ is an isomorphism. Therefore, we can use  (\\ref{eqn:basechangeequation}) to conclude that $\\prod_{i \\in J} q'_{i*}\\mathbb{L}{p'}^{*}_{i}$ is natrually equivalent to $(\\prod_{i \\in J} \\mathbb{L}p^{*}_{i}) q_{*}$ which is conservative since $q_{*}$ is conservative.\n\\ \\hfill $\\Box$\n\\begin{rem}Notice that the {\\it derived} base change of a homotopy monomorphism is always a homotopy monomorphism but there is no reason that such a thing would work for ordinary base changes. In order to construct a Gronthendieck topology, we need the admissible opens to be preserved by base change. Because the current article is not about derived geometry which will be discussed in \\cite{BeKr2}, we defined our Gronthendieck topology on a homotopy transversal subcategory ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$. The main application in this article is the case where ${\\tC}$ is the category ${\\text{\\bfseries\\sf{Ban}}}_{k}$ for a non-Archimedean valued field $k$ and ${\\text{\\bfseries\\sf{A}}}$ is the opposite to the category of affinoid algebras. In the preprints \\cite{BaBe} and \\cite{BaBeKr} we take ${\\tC}$ to be the category of Ind-Banach spaces and ${\\text{\\bfseries\\sf{A}}}$ to be categories of dagger affinoid and Stein algebras which we define. In these upcoming articles we do not assume the field is non-Archimedean so they apply to complex analytic geometry as well. In the article on derived analytic geometry \\cite{BeKr2}, we take ${\\tC}$ to be the monoidal model category of simplicial Ind-Ban spaces.\n\\end{rem}\n\n\\begin{defn}\\label{defn:fhZar}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.  Let ${\\text{\\bfseries\\sf{A}}}$ be homotopy Zariski transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ . The topology coming from Proposition \\ref{hfTV1} is called the formal homotopy Zariski topology on ${\\text{\\bfseries\\sf{A}}}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{A}}}^{fhZar}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{fhZar}).$ The category of schemes is denoted by ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}^{fhZar}).$\n\\end{defn}\n\n\\begin{prop}\\label{hTV1}\nConsider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{A}}}$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}$ is a homotopy monomorphism of finite presentation. These families define a pretopology on ${\\text{\\bfseries\\sf{A}}}$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{A}}}$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p'_{i}$ are homotopy monomorphisms follows from the assumption.\n\n\\ \\hfill $\\Box$\n\n\\begin{defn}\\label{defn:hZar}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}}$ be homotopy Zariski transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. The topology coming from Proposition \\ref{hTV1} is called the homotopy Zariski topology on ${\\text{\\bfseries\\sf{A}}}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{A}}}^{hZar}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{hZar}).$ The category of schemes is denoted by ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}^{hZar}).$\n\\end{defn}\n\n\n\n\n\n\\section{Main Theorems}\n\\subsection{From Berkovich geometry to Banach algebraic geometry}\n Let $k$ be a non-Archimedean valuation field. \nWe now introduce the full subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k} \\subset {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. The objects in this category are the $k$-affinoid algebras from the literature on Berkovich analytic spaces.\n\\begin{defn}\\label{defn:NiceAlg} For any finite ordered set of positive real numbers $r=(r_1, \\dots, r_n)$, let \\[k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} \\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})\\] be the completion of $k[x_{1}, \\dots,  x_{n}]$ with respect to the norm \n\\[\\|\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} a_{I} x^{I}\\|_{r}=\\text{max}_{I}\\{|a_{I}|r^{I}\\}\n\\] \nwhere for each $I= (i_1, \\dots, i_n) \\in \\mathbb{Z}^{n}_{\\geq 0}$,  $a_{I}= a_{i_1, \\dots, i_n}$ and $x^{I}=x^{i_1}_{1} \\cdots x^{i_n}_{n}$. \n\\end{defn}\n\\begin{lem} More concretely, we have\n\\begin{equation}\\label{equation:concrete}\nk \\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} = \\{ \\sum_{} a_{I}x^{I} \\in k[[x_1, \\dots, x_n]] \\ \\ | \\ \\ \\lim_{|I| \\to \\infty} |a_{I}| r^{I} = 0 \\}\n\\end{equation}\nwhere $|I|= i_{1} + \\cdots +i_{n}$ and we equip the right hand side with the norm \n\\[\\|\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} a_{I} x^{I}\\|_{r}=\\sup_{I}\\{|a_{I}|r^{I}\\}\n\\]\n\\end{lem}\n{\\bf Proof.} First, notice that the right hand side is complete: given a Cauchy sequence $\\{f^{(j)}\\}$ in the right hand side of (\\ref{equation:concrete}), note that for each $I$, the sequence of coefficients $f^{(j)}_{I}$ of $x^I$ must be Cauchy, and hence converge in $k$, to some $f_{I}.$ Define $f=\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} f_{I} x^{I}.$ It is easy to check that $f$ lies in the right hand side using the inequality\n\\[|f_{I}|r^{I} \\leq \\max \\{|f_{I} - f^{(j)}_{I}|r^{I},|f^{(j)}_{I}|r^{I} \\}.\n\\]\n \nIf $f$ is an element of the right hand side, let $f^{(j)} = \\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}, |I| \\leq j} f_{I} x^{I}.$ These polynomials $f^{(j)}$ converge to $f$ so the polynomials are dense in the right hand side.\n\\ \\hfill $\\Box$\n\\begin{rem}Notice by Lemma \\ref{lem:sumsConverge} that the right hand side consists precisely of the formal power series which can be evaluated on elements $(x_1, \\dots, x_n) \\in k^{n}$ such that $|x_{i}| \\leq r_i$ for $i=1, \\dots, n$.\n\\end{rem}\n\n\\begin{defn} A $k$-affinoid algebra $\\mathcal{A}$ is an object in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ which admits an admissible surjection \n\\[k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} \\to \\mathcal{A}.\n\\]\nwhose multiplication is contracting (see Definition \\ref{defn:nonexpanding}). The full category of ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ consisting of such objects is denoted ${\\text{\\bfseries\\sf{Afnd}}}_{k}$.\n\\end{defn}\n Notice that the completed symmetric algebra in ${\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}$ on $k_{r_1} \\oplus k_{r_2} \\oplus \\cdots \\oplus k_{r_n}$ is just $k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$ from Definition \\ref{defn:NiceAlg}. So these algebras are precisely the free commutative monoids in ${\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}$. In fact, we have\n\\begin{lem}Affinoid algebras are precisely the finitely presented objects in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$. This means that they are exactly those objects which are the cokernel of a morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ of free commutative algebra objects: \n\\[k\\{s_{1}^{-1}y_{1}, \\dots, s_{m}^{-1} y_{m}\\} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\n\\end{lem}\n{\\bf Proof.} Let $\\mathcal{A} = k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\/I$ be an affinoid algebra. It is known \\cite{Te} that $k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$ is Noetherian. Therefore, $I$ must be finitely generated by some finite set of non-zero elements $f_1, \\dots, f_m \\in k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$. Let $s_{j} = \\|f_{j}\\|$ for $j= 1, \\dots, m$. Consider the morphism of algebras\n\\[k\\{s_{1}^{-1}y_{1}, \\dots, s_{m}^{-1} y_{m}\\} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\ndetermined by the (contracting) morphism of Banach spaces \n\\[k_{s_1} \\oplus \\cdots \\oplus k_{s_m} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\ndetermined by the maps sending $1 \\in k_{s_j}$ to $f_j$. Its image is the ideal generated by $f_1, \\dots, f_m$.\n\n\\ \\hfill $\\Box$\n\n\n\\begin{defn}\\label{defn:AffLocalization} A $k$-affinoid localization is a morphism $\\mathcal{A} \\to \\mathcal{D}$ of $k$-affinoid algebras such that the morphism $|\\mathcal{M}(\\mathcal{D})|\\to |\\mathcal{M}(\\mathcal{A})|$ is injective, the image of $|\\mathcal{M}(\\mathcal{D})|$ is closed in $|\\mathcal{M}(\\mathcal{A})|$ and any morphism of $k$-affinoid algebras $\\mathcal{A} \\to \\mathcal{B}$ such that $|\\mathcal{M}(\\mathcal{B})|$ lands in the image of  $|\\mathcal{M}(\\mathcal{D})|$ factors as $\\mathcal{A} \\to \\mathcal{D} \\to \\mathcal{B}$. \n\\end{defn}\nNote that $k$-affinoid localizations correspond to the subspaces known as affinoid domain embeddings. They are usually written as $\\mathcal{A} \\to \\mathcal{A}_{V}$ where $V$ is the closed image and they satisfy the property (proven in 2.2.2 (iv) of \\cite{Ber1990}) that \n\\[\\mathcal{A}_{V_1 \\cap V_2} \\cong \\mathcal{A}_{V_1} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_2} \n\\]\nfor affinoid domains $V_1$ and $V_2$.\n\nThree important examples are the localizations corresponding to the rational, Weirstrass and Laurent domains. Any affinoid domain is a union of a finite collection of rational domains.\n\n\\begin{defn}\\label{defn:RatDom}A rational localization is a morphism of affinoid algebras of the form \n\\[\\mathcal{A} \\to \\mathcal{A} \\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\nfor some $g, f_j \\in \\mathcal{A}$ such that $(f_1,\\dots, f_m,g)=1$.\n\\end{defn}\nNotice that the norm of this morphism is one. \n\n\\begin{rem}\n\nProposition 2.2.4 (ii) of \\cite{Ber1990} tells us that $\\mathcal{A}_{V}$ is a flat $\\mathcal{A}$-algebra with respect to the algebraic (non-completed) tensor product, however it is not flat with respect to $\\widehat{\\otimes}_{\\mathcal{A}}$. Note that since completions commute with colimits, a rational localization $\\mathcal{A}_{V}$ is the completion of \n$\\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_n - f_n)$ with respect to the residue semi-norm coming from the semi-norm in Definition \\ref{def:NDiskrelA}. We have an \nisomorphism of $\\mathcal{A}$ modules \n\\[\\mathcal{A}_g= \\mathcal{A}[S]\/(g S-1) \\to \\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\ngiven by $S = b+\\sum_{i=1}^{n}a_i T_i$ where $bg + \\sum_{i=1}^{n}a_if_i = 1$. The inverse is given by $T_i=f_iS$. Therefore $\\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_n - f_n)$ is flat over $\\mathcal{A}$ in the algebraic sense. Given any $\\mathcal{M} \\in {\\text{\\bfseries\\sf{Mod}}}(\\mathcal{A})$ we have a morphism $\\mathcal{M}_{g} \\to \\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_V$ and $\\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_V$ is the completion of $\\mathcal{M}_{g}=\\mathcal{M}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_m - f_m)$ with the residue semi-norm of the algebraic tensor product $\\mathcal{M}[T_1, \\dots, T_n]$.\n\\end{rem}\n\\begin{defn}\\label{defn:WDom}A Weirstrass localization is a rational localization for which $g=1$.\n\\end{defn}\n\\begin{defn}\\label{defn:LDom}A Laurent loaclization is a localization of the form \n\\[\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T_1}{p_1}, \\dots, \\frac{T_n}{p_n}, q_1S_1, \\dots, q_mS_m\\}(T_1 - f_1, \\dots, T_n-f_n, g_1S_1-1, \\dots, g_mS_m-1)\n\\]\nwhere the $p_i$ and $q_j$ are positive reals and $f_i,g_j \\in \\mathcal{A}$.\n\\end{defn}\n\\begin{defn}\\label{defn:CAcomplex}\nFix a system $\\mathcal{A} \\to \\mathcal{A}_{V_i}$ which give a cover of $\\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i$. The \\v{C}ech-Amitsur complex is the complex \n\\begin{equation}\\label{equation:CA} 0 \\to \\mathcal{M} \\to \\prod_{i} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i} \\to \\prod_{i,j}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_j} \\to \\cdots.\n\\end{equation}\nTo any morphism $\\mathcal{M} \\to \\mathcal{N}$ we have the obvious morphism of \\v{C}ech-Amitsur complexes. \nThe complex written here differs from the standard long exact sequence defined in section 8.2 of \\cite{BGR} in that we consider the completed tensor products everywhere whereas they only complete the tensor products between the various $\\mathcal{A}_{V_i}$ terms. The standard complex mentioned is exact as proven in the Acyclicity Theorem (Proposition 2.2.5 of \\cite{Ber1990}). The above complex is not always exact, however.\n\\end{defn}\n\\begin{lem}\\label{lem:CAcplxISexact} The complex in definition \\ref{defn:CAcomplex} is strictly exact for modules $\\mathcal{M}$ such that the natural morphism \n\\[\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\prod_{i}\\mathcal{A}_{V_i}\\to \\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}}\\prod_{i}\\mathcal{A}_{V_i}\n\\] is an isomorphism.\n\\end{lem}\n\n{\\bf Proof.}\n\nThis follows from an easy double complex argument using free resolutions.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:easyLW} Let $\\mathcal{A}$ be an affinoid algebra over a non-Archimedean  valuation field $k$. Let $\\mathcal{A}_{V}$ be the localization $\\mathcal{A}_{V}= \\mathcal{A}\\{\\frac{T}{s}\\}\/(T-f)$ for some $f\\in \\mathcal{A}$ or $\\mathcal{A}_{V}= \\mathcal{A}\\{\\frac{T}{s}\\}\/(gT-1)$ for some $g\\in \\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $\\mathcal{A}$-algebra. Then the natural morphism\n\\[\\mathcal{B} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V} \\to   \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V}\\] is an isomorphism in $\\sD^{-}(\\mathcal{A})$. In particular,  by taking $\\mathcal{B}=\\mathcal{A}_{V}$ we see that the morphisms $\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\/(gT-1)$ and  the morphism $\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\/(T-f)$ homotopy epimorphisms.\n\\end{lem}\n{\\bf Proof.} \nNotice that it enough to show that for any affinoid algebra $\\mathcal{C}$ and any element $f\\in \\mathcal{C}$ the morphism \n\\begin{equation}\\label{mono1}\\mathcal{C}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{C}\\{\\frac{T}{s}\\}\n\\end{equation}\nis a strict monomorphism and for any $g \\in \\mathcal{C}$ the morphism\n\\begin{equation}\\label{mono2}\\mathcal{C}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{C}\\{\\frac{T}{s}\\}\n\\end{equation}\nis a strict monomorphism. Indeed, we can use in the case $\\mathcal{C}=\\mathcal{A}$\n\\[\\mathcal{A}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\n\\]\nor\n\\[\\mathcal{A}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\n\\]\nas resolutions of $\\mathcal{A}_{V}$ whose terms are projective and $\\widehat{\\otimes}_{\\mathcal{A}}$-acyclic. Then by taking the completed tensor product over $\\mathcal{A}$ with $\\mathcal{B}$ we find a representative for $\\mathcal{B} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V}$ which looks like \n\\[\\mathcal{B}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{B}\\{\\frac{T}{s}\\}\n\\]\nor\n\\[\\mathcal{B}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{B}\\{\\frac{T}{s}\\}\n\\]\nwhere the $f$ and $g$ here are the images of the original ones in $\\mathcal{B}$. These complexes are immediately recognized as also being special cases of (\\ref{mono1}) and (\\ref{mono2}) in the case $\\mathcal{C}=\\mathcal{B}.$ This shows that the operation of taking the completed tensor product over $\\mathcal{A}$ with $\\mathcal{B}$ is (strictly) exact.\n\nWe start by showing that (\\ref{mono1}) and (\\ref{mono2}) are monomorphisms which using Lemma \\ref{lem:BanProps} simply means to show that they are injective. Multiplication by $gT-1$ is clearly injective as can be seen by looking at the lowest order term in $T$. Consider the ascending sequence of ideals of $\\mathcal{C}$ given by the kernel of the morphisms of $\\mathcal{C}$ to itself defined by multiplication by $f^{i}$: \\[\\ker(f) \\subset \\ker(f^{2}) \\subset \\ker(f^{3}) \\subset \\cdots. \\] Since $\\mathcal{C}$ is Noetherian by Proposition 2.1.3 of \\cite{Ber1990} this sequence must terminate at some $\\ker(f^{N}).$ Suppose that \\[(T-f)\\left(\\sum_{j=0}^{\\infty} a_{j} T^{j} \\right)= 0\\] with $\\sum_{j=0}^{\\infty} a_{j} T^{j} \\neq 0$ and let $a_{i}$ be the first non-zero coeficient. Then $f^{N}a_{i+N}= a_{i}$ and so $f^{N+1}a_{i+N}= fa_{i}=0$ and so $a_{i+N} \\in \\ker(f^{N+1})= \\ker(f^{N}).$ This shows that $a_{i}=0,$ a contradiction. Therefore, multiplication by $T-f$ is injective.\nLet us assume for a moment that $k$ is non-trivially valued. To show that these morphisms are strict one must simply show that the set-theoretic image is closed. However, this set-theoretic image in both cases is an ideal and all ideals in affinoid algebras are closed by Proposition 2.1.3 of \\cite{Ber1990}. Therefore, they are strict monomorphisms in this case. Now if $k$ is arbitrary we find that these two morphisms become strict after tensoring with the non-trivially valued field introduced in Proposition 2.1.2 of \\cite{Ber1990}. Therefore, by this same proposition, they were strict monomorphisms all along.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:LCSrHoEpis}Let $\\mathcal{A}_{V}$ be a rational localization of an affinoid $k$-algebra $\\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Then the natural morphism\n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\] \nis an isomorphism in $\\sD^{-}(\\mathcal{A})$. In particular taking $\\mathcal{B}=\\mathcal{A}_{V}$, any rational, Weirstrass, or Laurent localization $\\mathcal{A}\\to \\mathcal{A}_{V}$ is a homotopy epimorphism.\n\\end{lem}\n\n{\\bf Proof.}\n\nAssume now that $k$ is non-trivially valued and consider the rational localization \n\\[\\mathcal{A}_{V} = \\mathcal{A} \\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\nwhere $r_{i}>0$ and $(f_1, \\dots, f_m,g)=1.$\nNotice that following Proposition 1 of 7.2.4 of \\cite{BGR} $|g|$ cannot be arbitrarily small for $|\\;| \\in \\mathcal{M}(\\mathcal{A}_{V})$ and in fact we can realize the rational localization as an Laurent localization of a Weierstrass localization. That is, there is an $\\epsilon>0$ such that\n\\[\\mathcal{A}_{V}\n\\cong \\left(\\mathcal{A}\\{ \\frac{S}{\\epsilon^{-1}}\\}\/(gS-1)\\right)\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/( T_1 - \\frac{f'_1}{g'}, \\dots, T_m -\\frac{f'_m}{g'})\n\\]\nwhere the $f'_i$ and $g'$ are the image of $f_i$ and $g$ in $\\mathcal{A}\\{\\epsilon S\\}\/(gS-1).$ This corresponds to the fact that we can choose $\\epsilon >0$ such that the conditions $|f_{i}|\\leq r_{i}|g|$ and $(f_1, \\dots, f_m,g)=1$ are equivalent to $|g|>\\epsilon$ and $|\\frac{f_{i}}{g}|\\leq r_{i}$. Therefore Lemma \\ref{lem:easyLW} implies that the natural morphism\n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\] \nis an isomorphism in $\\sD^{-}(\\mathcal{A})$. \nNow if $k$ is trivially valued, consider, using Proposition 2.1.2 of \\cite{Ber1990}, a field $K_r$  containing $k$ which is flat over $k$ with respect to the completed tensor product and non-trivially valued. When we take the tensor product of the canonical morphism \\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B}\\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\\] with $K_{r}$ over $k$ we get, using the flatness of $K_r$, that\n\\[(\\mathcal{A}_{V} \\widehat{\\otimes}_{k} K_r)\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}  \\widehat{\\otimes}_{k} K_r}(\\mathcal{B} \\widehat{\\otimes}_{k} K_r) \\to \\mathcal{A}_{V} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\widehat{\\otimes}_{k} K_r\n\\] is an isomorphism in $\\sD^{-}(\\mathcal{A}\\widehat{\\otimes}_{k}K_r)$ and hence the original morphism $\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V}\\to \\mathcal{A}_{V} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}$ is an isomorphism in $\\sD^{-}(\\mathcal{A}).$\n\n\\ \\hfill $\\Box$\n\\begin{lem}Let $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ be affinoid localizations of an affinoid $k$-algebra $\\mathcal{A}$ corresponding to subdomains $W_1$ and $W_2$. Assume also that $W_1 \\cup W_2$ is an affinoid subdomain. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Assume that the morphisms \n\\[\\mathcal{A}_{W_i}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_i}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nare isomorphisms for $i=1,2$. Assume also that the morphism\n\\[\\mathcal{A}_{W_1 \\cap W_2}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_1 \\cap W_2}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nis an isomorphism. Then the morphism \n\\[\\mathcal{A}_{W_1 \\cup W_2}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_1 \\cup W_2}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nis an isomorphism.\n\\end{lem}\n{\\bf Proof.}\nThis follows immediately from considering the strict short exact sequence \n\\[0 \\to \\mathcal{A}_{W_1 \\cup W_2} \\to \\mathcal{A}_{W_1} \\times \\mathcal{A}_{W_2} \\to \\mathcal{A}_{W_1 \\cap W_2} \\to 0\n.\\]\n\n\\ \\hfill $\\Box$\nFinally, we are able to show in the following theorem that affinoid subdomains of affinoids give examples homotopy monomorphisms of affine schemes in the abstract sense.\n\\begin{thm} \\label{thm:TensLocs}Let $\\mathcal{A}_{V}$ be an affinoid localization of an affinoid $k$-algebra $\\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Then the natural morphism  \n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\]\nis an isomorphism in $\\sD^{-}(\\mathcal{A})$.\nIn particular let $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ be affinoid localizations of an affinoid algebra $\\mathcal{A}$. Then $\\mathcal{A}_{W_1} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{W_2} \\cong \\mathcal{A}_{W_1} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{W_2}.$ Therefore taking $W_1=W_2$, any affinoid localization is a homotopy epimorphism.\n\\end{thm}\n\n{\\bf Proof.}\nFor Weirstrass or Laurent localizations this follows immediately from using induction on Lemma \\ref{lem:easyLW} or the fact that it holds for the more general set of rational localizations which we now consider. When $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ are rational localizations this has been shown already in Lemma \\ref{lem:LCSrHoEpis}. Suppose now that $V=W_1$ is an rational domain and $W=W_2=V_1 \\cup \\cdots \\cup V_{N}$ is an affinoid domain written as a union of rational domains. For $N=1$ the claim is true. The induction step follows from considering the derived tensor product of $\\mathcal{A}_{V}$ over $\\mathcal{A}$ with the short exact sequence \n\\begin{equation}\\label{equation:ind}0\\to \\mathcal{A}_{W} \\to \\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\times \\mathcal{A}_{V_N} \\to \\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_N} \\to 0.\n\\end{equation}\nAssume by induction that the for some $N>2$ the derived tensor product of the rational localization $\\mathcal{A}_V$ and the affinoid algebra corresponding to the union of $N-1$ or fewer rational domains is equivalent to the ordinary tensor product. This implies that the derived tensor product of $\\mathcal{A}_{V}$ with $\\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\times \\mathcal{A}_{V_N}$ and with $\\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_N} =\\mathcal{A}_{(V_1 \\cap V_{N})\\cup \\dots \\cup (V_{N-1} \\cap V_{N})}$ are equivalent to the ordinary tensor products. Hence the same holds with $\\mathcal{A}_{W}$. Finally, one takes $W$ as $W_1$ in (\\ref{equation:ind}) and considers the derived tensor product with $\\mathcal{A}_{W_2}$ and uses a similar induction to do the general case.\n\n\n\\ \\hfill $\\Box$\n\n\n\n\\begin{rem}\\label{rem:recover}This is analogous to Lemma 2.1.4 (1) of \\cite{TVe5} where it is shown that homotopy Zariski open immersions in the category of affine schemes relative to the closed symmetric monoidal category of abelian groups give precisely the ordinary notion of a Zariski open imemrsions.  In that reference first a special case (inverting a single element of the ring) was shown and the general case follows by their descent formalism.\n\\end{rem}\n\\begin{defn}\nFor any topological space $X$, a {\\it quasi-net} is a set $T$ of subsets of $X$ such that any point $x \\in T$ has a neighborhood of the form $\\cup_{i=1}^{n}V_{i}$ with $x \\in V_{i} \\in T$ for $1 \\leq i \\leq n$. A {\\it net} is a quasi-net $T$ such that such that for every $U,V \\in T$ the set $\\{W \\in T | W \\subset U \\cap V\\}$ is a quasi-net of subsets of $U \\cap V$.  A $k$-analytic space is a locally Hausdorff topological space $X$, a net $\\tau_{0}$ on $X$, a functor $\\phi: \\tau_{0} \\to {\\text{\\bfseries\\sf{Afnd}}}_{k},$ and an invertible natural transformation ${\\text{\\bfseries\\sf{Top}}} \\Longrightarrow {\\text{\\bfseries\\sf{Top}}} \\circ \\phi$. \n\\end{defn}\n\\begin{defn} \nFor any $k$-affinoid algebra $\\mathcal{A}$, the topological space $|\\mathcal{M}(\\mathcal{A})|$ is defined to be the set of non-archimedean bounded semivaluations $| \\ |$ on $\\mathcal{A}$ equipped with the weakest topology such that for each $f \\in \\mathcal{A}$, the maps $|\\mathcal{M}(\\mathcal{A})| \\to \\mathbb{R}_{+}$ defined by sending $| \\ |$ to $|f|$ is continuous.\n\\end{defn}\n\n\\begin{defn}A {\\it k-affinoid space} is a locally ringed space of the form $\\mathcal{M}(\\mathcal{A})=(|\\mathcal{M}(\\mathcal{A})|,\\mathcal{O}_{\\mathcal{M}(\\mathcal{A})})$ where $\\mathcal{A}$ is a $k$-affinoid algebra and $\\mathcal{O}_{\\mathcal{M}(\\mathcal{A})}(U)$ is the limit over $\\mathcal{A}_{V}$, where $V \\subset U$ is a finite union of affinoid domains. The category of $k$-affinoid spaces defined to be a full subcategory of the category of locally ringed spaces of the given form. \n\\end{defn}\nThe category of $k$-affinoid spaces is equivalent to the category  ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$. So we treat $\\mathcal{M}$ as a functor giving this equivalence from ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ to the category of  $k$-affinoid spaces.\n\\begin{defn}\nA {\\it k-analytic space} consists of a triple $(X,\\tau,\\mathcal{A})$ where $X$ is a locally Hausdorff topological space, $\\tau$ is a net on $X$, and for each $V \\in \\tau$, $\\mathcal{A}(V)$ is a k-affinoid algebra along with a homeomorphism $|\\mathcal{M}(\\mathcal{A}(V))| \\cong V$ (functorially assigned to the elements of $\\tau$) such that if $V, V' \\in \\tau$ and $V' \\subset V$ then $V'$ is an affinoid subdomain of $\\mathcal{M}(\\mathcal{A}(V))$ with coordinate ring $\\mathcal{A}(V')= \\mathcal{A}(V)_{V'}$.  In the event that for every $U \\in \\tau_{2},$ $\\tau_{1}$ restricted to $g^{-1}(U)$ is an atlas of $g^{-1}(U),$ a morphism  $(X_{1},\\tau_{1},\\mathcal{A}_{1}) \\to (X_{2},\\tau_{2},\\mathcal{A}_{2})$ consists of a continuous and $G$-continuous map $g:X_{1} \\to X_{2}$ along with bounded homomorphisms $g^{\\#}_{U,V}:\\mathcal{A}_{2}(U) \\to \\mathcal{A}_{1}(V)$ for every $U \\in \\tau_{2}, V  \\in \\tau_{1}$ with $g(V) \\subset U$ such that for every $V, V' \\in \\tau_{1}$ with $V'\\subset V$ and $U, U' \\in \\tau_{2}$ with $U' \\subset U$ such that $g(V) \\subset U$ and $g(V') \\subset U'$ the diagram \n\\begin{equation}\n\\xymatrix{ \\mathcal{A}_{2}(U) \\ar[d] \\ar[r] & \\mathcal{A}_{1}(V) \\ar[d] \\\\  \\mathcal{A}_{2}(U') \\ar[r] &  \\mathcal{A}_{1}(V') }\n\\end{equation}\ncommutes. We use the terms $k$-analytic space and Berkovich analytic spaces interchangeably. Let ${\\text{\\bfseries\\sf{An}}}_{k}$ denote the category of $k$-analytic spaces.\n\\end{defn}\n\n\n\\begin{defn}\\label{quasinet}A quasi-net on a topological space $X$ is a collection $T$ of subsets of $X$ such that for every $x \\in X$ there is a subset $T_x \\subset T$ with $|T_x| < \\infty$ such that $x \\in \\cap_{V \\in T_x}V$ and there is an open set $U \\subset X$ such that $x \\in U \\subset \\cup_{V \\in T_x} V.$\n\\end{defn}\n\\begin{lem}\\label{easyquasinet}Any finite set $T=\\{V_1,V_2, \\dots V_m \\}$ of closed subsets of a topological space $X$ which cover $X$ is a quasi-net.\n\\end{lem}\n{\\bf Proof.} Given $x \\in X,$ consider the subset $T_x \\subset T$ defined by those subsets in $T$ which contain $x$.  Then $x \\in \\cap_{V \\in T_x}V$. Let $U=X-\\cup_{V \\in T-T_x}V,$ this satisfies the required property.\n\\ \\hfill $\\Box$\n\nConsider ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X$, the category of affinoid $k$-analytic spaces over $X$. This means that the objects are pairs $(\\mathcal{M}(\\mathcal{A}),f)$ where $\\mathcal{M}(\\mathcal{A})$ is an affinoid $k$-analytic space and $f:\\mathcal{M}(\\mathcal{A}) \\to X$ is a morphism of $k$-analytic spaces.  The morphisms from $(\\mathcal{M}(\\mathcal{A}_1),f_1)$ to $(\\mathcal{M}(\\mathcal{A}_2),f_2)$ are morphisms to $X$ commuting with the $f_i$.    \n\\begin{lem}\\label{EverythingColimit} Any $k$-analytic space $X$ is a colimit of the category ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X$ when considered as a subcategory of ${\\text{\\bfseries\\sf{An}}}_{k}$.\n\\end{lem}\n{\\bf Proof.}\nConsider the family $\\hat{\\tau}$ of all affinoid domains in $|X|$. It is a net and $|X|$ has a maximal $k$-affinoid atlas $\\hat{\\mathcal{A}}$. For each $V \\in \\hat{\\tau}$, $\\hat{\\mathcal{A}}$ assigns $\\mathcal{M}(\\mathcal{A}_{V})\\to X.$  In particular, $\\hat{\\mathcal{A}}$ assigns homeomorphisms $|\\mathcal{M}(\\mathcal{A}_{V})| \\cong V\\subset|X|$ and such that these homeomorphisms satisfy obvious compatibilities. For any other $k$-analytic space $X'$ we have by Exercise 3.2.2 of \\cite{Ber2009} an isomorphism\n\\begin{equation}\n\\Hom(X,X') \\rightarrow \\eq[\\prod_{V \\in  \\hat{\\tau}} \\Hom(\\mathcal{M}(\\mathcal{A}_{V}), X')\\rightrightarrows \\prod_{(V,W) \\in \\hat{\\tau}^{2}}  \\Hom(\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes_{k}}\\mathcal{A}_{W}), X')\n].\\end{equation}\nTogether with the factorization of this isomorphism as\n\\footnotesize\n\\[\\Hom(X,X') \\to \\lim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X}\\Hom(\\mathcal{M},X') \\hookrightarrow  \\eq[\\prod_{V \\in  \\hat{\\tau}} \\Hom(\\mathcal{M}(\\mathcal{A}_{V}), X')\\rightrightarrows \\prod_{(V,W) \\in \\hat{\\tau}^{2}}  \\Hom(\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes_{k}}\\mathcal{A}_{W}),X')]\n\\]\n\\normalsize\nthis implies that the natural morphism \n\\[\\Hom(X,X') \\to \\lim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X}\\Hom(\\mathcal{M},X')\n\\]\nis an isomorphism. Therefore, $X=\\colim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X} \\mathcal{M}.$\n\\ \\hfill $\\Box$\n\\subsection{From Banach algebraic geometry to Berkovich geometry}\\label{BSSOCBA}\nConsider the category $\\text{\\bfseries\\sf{C}}={\\text{\\bfseries\\sf{Ban}}}_{k}$ for some valuation field $k$. We have shown in Section \\ref{BanachSpaces} that it is a closed symmetric monoidal quasi-abelian categories with $\\underline{\\Hom}=\\underline{\\Hom}_{k}$ and $\\overline{\\otimes} =\\widehat{\\otimes}_{k}$ with all finite limits and colimits and enough projectives so that we can do algebraic geometry relative to ${\\text{\\bfseries\\sf{Ban}}}_{k}$. In particular the categories of affine schemes over it has certain distinguished morphisms and topologies and we have notions of (Archimedean\/non-Archimedean) Banach schemes Banach (infinity) stacks and $n$-algebraic Banach stacks over an (Archimedean\/non-Archimedean) valuation field $k$.  \nIn the Archimedean case we could compare this geometry to the geometry of complex varieties covered by Stein compact subsets. However, we focus here on the non-Archimedean case. In this section $k$ will be a non-Archimedean valuation field.\n\n\\begin{defn}\\label{def:NDiskrelA}\nLet $\\mathcal{A}$ be a $k$-affinoid algebra. Given $r_{1}, \\cdots, r_{n} \\in \\mathbb{R}$, we can define an $\\mathcal{A}$ algebra \n\\begin{equation}\\mathcal{A}\\{r_1^{-1}T_1, \\dots, r_n^{-1}T_n\\}\n\\end{equation}\nas the completion of $\\mathcal{A}[T_1, \\dots, T_n]$ with respect to the norm \n\\[\\|\\sum a_{I} T^{I}\\|_{r}=\\text{max}_{I}\\{\\|a_{I}\\|_{\\mathcal{A}}r^{I}\\}.\n\\] \n\\end{defn}\nLet $r$ be a real number greater than zero, denote by $\\mathcal{A}_{r}$ the $\\mathcal{A}$ module with norm $\\|a\\|= r\\|a\\|_{\\mathcal{A}}$.\n\\begin{lem}\\label{lem:DiskAlg}$\\mathcal{A}\\{r_1^{-1}T_1, \\dots, r_n^{-1}T_n\\}$ is the symmetric algebra (see subsection \\ref{Symmetric}) on $\\mathcal{V}=\\mathcal{A}_{r_1} \\oplus \\cdots \\oplus \\mathcal{A}_{r_n}$ in ${\\text{\\bfseries\\sf{Mod}}}^{\\leq 1}(\\mathcal{A})$. It can also be seen as the filtered colimit in ${\\text{\\bfseries\\sf{Mod}}}^{\\leq 1}(\\mathcal{A})$ of \n\\[\\sS^{0}(\\mathcal{V}) \\hookrightarrow \\sS^{0}(\\mathcal{V}) \\oplus \\sS^{1}(\\mathcal{V})  \\hookrightarrow \\sS^{0}(\\mathcal{V}) \\oplus \\sS^{1}(\\mathcal{V}) \\oplus \\sS^{2 }(\\mathcal{V})\\hookrightarrow \\cdots \n\\]\nwhere \n\\[\\sS^{m}(\\mathcal{V}) = \\{ \\sum_{|I|=m} a_{I} T^{I} | a_{I} \\in \\mathcal{A}\\}\\]\nequipped with the norm \n\\[\\|\\sum_{|I|=m} a_{I} T^{I}  \\|= \\max_{|I|=m} \\|a_I\\| r^{I}.\n\\]\n\\end{lem}\n{\\bf Proof.} Left to the reader.\n\\ \\hfill $\\Box$\n\n\n\n\n\\begin{lem}\\label{lem:FinTypePres} \n\nA morphism $p: \\mathcal{A} \\to \\mathcal{B}$ in  ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ induces a presentation \n\\begin{equation}\\label{eqn:NiceForm} \\mathcal{B} \\cong \\mathcal{A}\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_g}{r_g}\\}\/(P_1, \\dots, P_r) \n\\end{equation}\nwhere $P_{i} \\in \\mathcal{A}\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_g}{r_g}\\}$\nif and only if $p$ is of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ as defined in Definition \\ref{defn:FinitePres}.\n\\end{lem}\n\n\\ \\hfill $\\Box$\n\nThe following two lemmas are technical results that will be used only in the proof of Theorem \\ref{thm:localForm}.\n\\begin{lem}\\label{SplitProdStr2}Let $\\mathcal{A}, \\mathcal{C}$ be $k$-affinoid algebras considered as objects in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ and let $f:\\mathcal{A} \\to \\mathcal{C}$ be morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ which is a strict epimorphism when considered in ${\\text{\\bfseries\\sf{Ban}}}_{k}$ such that there is a $k$-affinoid algebra $\\mathcal{B}$ and the post-composition of $f$ with some homotopy epimorphism $g:\\mathcal{C} \\to \\mathcal{B}$ in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ is a homotopy epimorphism $h:\\mathcal{A}\\to \\mathcal{B}$ in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Then there is a $k$-affinoid algebra $\\mathcal{A}'$ and a isomorphism $\\mathcal{A} \\cong \\mathcal{C} \\times \\mathcal{A}'$ such that the projection to $\\mathcal{C}$ corresponds to $f$ under this isomorphism.\\end{lem}\n{\\bf Proof.} \nThe morphisms \n\\[\\mathcal{A}\\to \\mathcal{C} \\to \\mathcal{B}\n\\]\ninduce morphisms on the derived categories \n\\[\\sD^{-}(\\mathcal{B}) \\to \\sD^{-}(\\mathcal{C}) \\to \\sD^{-}(\\mathcal{A}).\n\\]\nSince the composition is fully faithful and the first morphism is as well, the second morphism must be fully faithful and so by Lemma \\ref{lem:HomotopyMon} we have $\\mathcal{C} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C} \\cong \\mathcal{C}$. Let $I=\\ker (f)$. There is a strict, short exact sequence \n\\begin{equation}\\label{eqn:idealAC} 0 \\to I \\to \\mathcal{A} \\to \\mathcal{C} \\to 0.\n\\end{equation}\nIf we consider the derived completed tensor product of (\\ref{eqn:idealAC}) over $\\mathcal{A}$ with $\\mathcal{C}$ we find an exact triangle \n\\[I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C} \\to \\mathcal{C} \\to \\mathcal{C} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}\n\\]\nand because the second morphism is an isomorphism, we see that \n$I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}$ is isomorphic to $0$. If we now consider the derived completed tensor product over $\\mathcal{A}$ of (\\ref{eqn:idealAC}) with $I$ we an exact triangle \n\\[I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} I \\to I \\to I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}\n\\]\nand so we get an isomorphism $I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}I \\to I$.\nSo we have $I=\\text{image}[I \\widehat{\\otimes}_{\\mathcal{A}}I \\to I]$ and in fact this implies that $I= I^2:= \\text{image}[I \\otimes_{\\mathcal{A}}I \\to I]$. Therefore, there exists an element $e \\in \\mathcal{A}$ such \nthat $e^{2}=e$ and $e\\mathcal{A} =I$. This gives the structure of a $k$-affinoid algebra \nto $I$, which we denote \nby $\\mathcal{A}'=\\mathcal{A}\/(1-e)\\mathcal{A}$. Now because $f$ is \na strict epimorphism, there is a strict short exact sequence \n\\[0 \\to I \\to \\mathcal{A} \\stackrel{f}\\to \\mathcal{C} \\to 0\n\\]\nwhich in fact is split by the morphism of algebras $e:\\mathcal{A} \\to \\mathcal{A}'.$ Therefore, \\[(e,f):\\mathcal{A} \\to \\mathcal{A}' \\times \\mathcal{C}\\] is \nan isomorphism.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{CoverOfAff}Let $\\mathcal{A}, \\mathcal{B}$ be $k$-affinoid algebras and let $f:\\mathcal{A} \\to \\mathcal{B}$ be a morphism in the category of $k$-affinoid algebras with the property that $|\\mathcal{M}(\\mathcal{A})|$ has a finite covering by affinoid domains $V_{j}$ corresponding to affinoid domain embeddings $\\mathcal{M}(\\mathcal{A}_{V_j})\\to \\mathcal{M}(\\mathcal{A}).$ Suppose also that morphisms $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A}_{V_j})$ are affinoid domain embeddings. Then the morphism $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ is an affinoid domain embedding.\n\\end{lem}\n{\\bf Proof.}\nLet us denote by $U$ the image of $|\\mathcal{M}(\\mathcal{B})|$ inside $|\\mathcal{M}(\\mathcal{A})|.$ We have $\\overline{U} \\cap V_{j} = \\overline{U \\cap V_j} $ for all $j$. Since $U \\cap V_j$ is closed in $V_j$ and $V_j$ is closed in $|\\mathcal{M}(\\mathcal{A})|$ we see that  $U \\cap V_j$ is closed in $|\\mathcal{M}(\\mathcal{A})|$.  Therefore $\\overline{U} \\cap V_{j} = U \\cap V_{j} $ for all $j$. Hence $\\overline{U}=U$ and so $U$ is closed inside $|\\mathcal{M}(\\mathcal{A})|.$\nLet $\\mathcal{A} \\to \\mathcal{C}$ be a bounded homomorphism of affinoid $k$-algebras such that the image of $|\\mathcal{M}(\\mathcal{C})|$ lies in $U$.  We wish to show that the morphism $\\mathcal{M}(\\mathcal{C}) \\to \\mathcal{M}(\\mathcal{A})$ factors through a morphism $\\mathcal{M}(\\mathcal{C}) \\to \\mathcal{M}(\\mathcal{B})$. Notice that $\\mathcal{A}_{V_j} \\to \\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} $ is a bounded homomorphism of affinoid $k$-algebras such that the image of $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )$ lies in $U\\cap V_j$. Therefore, the morphisms $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ) \\to \\mathcal{M}(\\mathcal{A}_{V_j} )$ factor in a unique way through morphisms  $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}).$ When thought of as morphisms   $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{B})$ they agree when pulled back to $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j \\cap V_k} ). $ The preimages of $V_j$ in $|\\mathcal{M}(\\mathcal{C})|$ are analytic domains by \\cite{Te} Exercise 3.2.2 (v). These preimages are the pullback of a quasi-net and therefore form a quasi-net by Lemma \\ref{easyquasinet} and therefore by Exercise 3.2.2 (v) of \\cite{Ber2009} we have a unique morphism $\\mathcal{M}(\\mathcal{C})  \\to \\mathcal{M}(\\mathcal{B})$ which restricts to the morphisms $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{B}).$ \nIndeed, this follows from the commutative diagram of exact sequences \n\\footnotesize\n\\begin{equation}\n\\xymatrix{\n0 \\ar[r] & \\Hom(\\mathcal{M}(\\mathcal{C}),\\mathcal{M}(\\mathcal{B})) \\ar[d] \\ar[r]  & \\prod_{j}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ),\\mathcal{M}(\\mathcal{B})) \\ar[r] \\ar[d] & \n\\prod_{j,k}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j\\cap V_k} ),\\mathcal{M}(\\mathcal{B})) \\ar[d] \\\\ \n0 \\ar[r] &\\Hom(\\mathcal{M}(\\mathcal{C}),\\mathcal{M}(\\mathcal{A})) \\ar[r] & \\prod_{j}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ),\\mathcal{M}(\\mathcal{A})) \\ar[r] &  \\prod_{j,k}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j\\cap V_k} ),\\mathcal{M}(\\mathcal{A})). }\n\\end{equation}\n\\normalsize\nThis clearly provides the required factorisation. \n\\ \\hfill $\\Box$\n\nIf also, the $\\mathcal{M}(\\mathcal{A}_{V_j})$ are rational in $\\mathcal{M}(\\mathcal{A})$ and  $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B})$ is rational in $\\mathcal{M}(\\mathcal{A}_{V_j})$ notice that $\\mathcal{M}(\\mathcal{B})$ is a union of the rational domains $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B})$ in $\\mathcal{M}(\\mathcal{A}).$\n \n\\begin{lem}\\label{lem:NonExpFP} Let $\\mathcal{A}$ be a $k$-affinoid algebra and suppose there is a morphism $f: \\mathcal{A} \\to \\mathcal{B}$ of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}).$ Then $\\mathcal{B}$ is a $k$-affinoid algebra.\n\\end{lem}\n{\\bf Proof.}\nBy combining a presentation for $\\mathcal{B}$ over $\\mathcal{A}$ and a presentation for $\\mathcal{A}$ over $k$ one can write $\\mathcal{B}$ as a finite colimit of objects of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$. Therefore, $\\mathcal{B}$ has finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$.\n\\ \\hfill $\\Box$\n\\begin{thm}\\label{thm:localForm}Let $\\mathcal{A}, \\mathcal{B}$ be $k$-affinoid algebras and let $f:\\mathcal{A} \\to \\mathcal{B}$ be a morphism in the category of $k$-affinoid algebras. Assume that $f$ is a homotopy epimorphism (see Definition \\ref{defn:homotopyEpi}) when considered in the category ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Then the morphism $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ corresponding to $f$ is an affinoid domain embedding.\\end{thm}\n{\\bf Proof.} \nWe refer here to Temkin's proof \\cite{Te2} of the Gerritzen-Grauert Theorem for morphisms of affinoid algebras.  This theorem, assuming only the epimorphism condition on $f$, produces a finite collection of morphisms of $k$-affinoid algebas $\\mathcal{A} \\to \\mathcal{A}_{V_i}$ corresponding to rational domain embeddings $\\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ covering $|\\mathcal{M}(A)|$ with the images $V_i.$  The theorem further ensures that the morphisms $\\mathcal{A}_{V_i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ induced from $f$ admit factorzations \\[\\mathcal{A}_{V_i} \\twoheadrightarrow \\mathcal{C}_{i} \\hookrightarrow \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}.\\] These factorizations correspond to the composition of the morphism of $k$-affinoid algebras $\\mathcal{C}_{i} \\to (\\mathcal{C}_{i})_{W_i}=\\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ corresponding to Weierstrass domain embeddings  with the surjective morphisms of affinoid $k$-algebras $\\mathcal{A}_{V_i} \\to \\mathcal{C}_{i}$ corresponding to closed immersions. \nTherefore, by Lemma \\ref{lem:LCSrHoEpis} the morphism  $\\mathcal{C}_{i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ is a homotopy epimorphism in the category ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Notice that \n\\[(\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i})\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}_{V_i}}(\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}) \\cong \\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nbecause homotopy epimorphisms are closed under derived base change. However, applying Lemma \\ref{lem:LCSrHoEpis} we have \n\\[\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i} \\cong \\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nand so we see that \n\\[(\\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i})\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}_{V_i}}(\\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i}) \\cong \\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nand so by Lemma \\ref{lem:HomotopyMon} the morphisms $\\mathcal{A}_{V_i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ are homotopy epimorphisms. Also, the morphisms of affinoid algebras corresponding to Weierstrass domain embeddings are injective. Therefore, Lemma \\ref{SplitProdStr2} can applied by choosing the $f$ from that lemma to be the morphism $\\mathcal{A}_{V_i} \\to \\mathcal{C}_{i}$ and $g$ from that lemma to be the morphism $\\mathcal{C}_{i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}.$ The lemma then tells us that the morphism $\\mathcal{M}(\\mathcal{C}_i) \\to \\mathcal{M}(\\mathcal{A}_{V_i})$ is simply the inclusion of a connected component in a disjoint union of affinoids. Therefore,  $\\mathcal{M}(\\mathcal{C}_i) \\to \\mathcal{M}(\\mathcal{A}_{V_i})$ is an affinoid domain embedding. Because the composition of affinoid domain embeddings is an affinoid domain embedding, we conclude that the morphisms $\\mathcal{M}(\\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i})\\to \\mathcal{M}(\\mathcal{A}_{V_i})$ are affinoid domain embeddings as well. By Lemma \\ref{CoverOfAff} we conclude that the original morphism gives an affinoid domain embedding $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$.\n\n\\ \\hfill $\\Box$\n\n\n\nFrom now on we write ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ in place of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}}^{RR}(\\mathcal{A})$.\n\\begin{lem}\\label{lem:ConsImpliesSur}\nLet $\\mathcal{A}$ be a $k$-affinoid algebra. Let $\\{f_{i}:\\mathcal{A} \\to \\mathcal{A}_{V_i}\\}_{i \\in I}$ be a family of affinoid localizations such that for some finite set $J \\subset I$ the corresponding family of functors \n\\[{\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})\n\\]\nfor $i \\in J$ is conservative. Then the morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ is surjective.\n\\end{lem}\n{\\bf Proof.}\nWe argue by contradiction.  First assume that $k$ is non-trivially valued and $\\mathcal{A}$ is strictly affinoid. Suppose that the family of functors is conservative and some point $x\\in \\mathcal{M}(\\mathcal{A})$ is not in the image. By Proposition 2.1.15 of \\cite{Ber1990} the subset of points of $y\\in \\mathcal{M}(\\mathcal{A})$ such that $\\ker(|\\;|_{y})$ is a maximal ideal is a dense subset of $\\mathcal{M}(\\mathcal{A})$. Therefore, since the image is the closed set $\\cup_{i\\in J}V_{i}$ we may assume (by changing the point $x$) that $x\\in \\mathcal{M}(\\mathcal{A})$ is not in the image and $\\ker(|\\;|_{x})$ is a maximal ideal. Chose using Proposition 2.2.3 (iii) of \\cite{Ber1990} an affinoid subdomain $W$ of $\\mathcal{M}(\\mathcal{A})$ such that $x \\in W$ and $W \\cap V_{i}$ is empty for all $i \\in J$.  Consider the \n morphism $0 \\to \\mathcal{A}_W$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$. It is not an isomorphism but for each $i \\in J$, the pullback to each $\\spec(\\mathcal{A}_{V_i})$ is the isomorphism $0 \\to 0=\\mathcal{A}_{W} \\widehat{\\otimes}_{A} \\mathcal{A}_{V_i}$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})$. This gives a contradiction. For the general case, choose using Proposition 2.1.2 of \\cite{Ber1990}, a valuation field extension $k \\to K$ such that the valuation on $K$ is non-trivial and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is a strictly $K$-affinoid algebra. Notice that the conservativity assumption on the original family implies by Lemma \\ref{cor:BaseChangeConservHzAR} applied to the base change $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ that the family of functors $\\{{\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K)\\}_{i \\in J}$ is also conservative. The morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k} K) \\to \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ cannot be surjective because in the commutative diagram, \n\\[\n\\xymatrix{ \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K) \\ar[d] \\ar[r] & \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\ar[d] \\\\  \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\ar[r] &  \\mathcal{M}(\\mathcal{A}) }\\]\nthe horizonal arrows are surjective.  Therefore, we have reduced to the previous case and so the proof is complete.\n\\ \\hfill $\\Box$\n\n\\begin{lem}\\label{lem:AlternatingTate}Consider a (surjective) cover of $X= \\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i= \\mathcal{M}(A_{V_{i}})$. Then the complex \n\\[\n0 \\to \\mathcal{A} \\to \\prod_{i_1} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to  \\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0. \n\\]\nis strictly exact.\n\\end{lem}\n{\\bf Proof.} By Proposition 1, section 8.1 of \\cite{BGR}, the inclusion (which is strict) of alternating cochains inside all cochains is a quasi-isomorphism. On the other hand, the complex of all cochains is strictly exact by Proposition 2.2.5 of \\cite{Ber1990}. The conclusion follows.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:CoversRconservative}Consider a (surjective) cover of $X= \\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i= \\mathcal{M}(A_{V_{i}})$. Then the corresponding family of functors ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})$ is conservative.\n\\end{lem}\n{\\bf Proof.}\nLet $f: \\mathcal{M} \\to \\mathcal{N}$ in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ be any morphism such that $f_{i}: \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i} \\to \\mathcal{N} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i} $ are isomorphisms for all $i$. \nThe alternating version of the \\v{C}ech-Amitsur complex (see Definition \\ref{defn:Amitsur}) corresponding to the morphism $\\mathcal{A} \\to \\prod_{i=1}^{n} \\mathcal{A}_{V_i}$ is a strictly exact bounded above complex by \\ref{lem:AlternatingTate} and so defines an element of $D^{-}(\\mathcal{A})$ (in fact the $0$ element!).\n\\[0 \\to \\mathcal{A} \\to \\prod_{i_1} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to  \\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0. \n\\]\nEach object in this complex is acyclic for the functor $\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}(-)$ since $\\mathcal{M}$ (being RR-quasi-coherent) is transversal to localizations of $\\mathcal{A}$. Therefore by Proposition \\ref{prop:ProjFProj}, if we apply the derived functor $\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}^{\\mathbb{L}}(-)$ we are left with a strictly exact complex \n\\[0 \\to \\mathcal{M} \\to \\prod_{i_1} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1<i_2}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0.\n\\]\nWe can do the same thing for $\\mathcal{N}$. The $f_{i}$ extend uniquely to morphisms of the complexes resolving $\\mathcal{M}$ and $\\mathcal{N}$. Therefore $f$ is an isomorphism. Conversely, if $f$ is an isomorphism, the $f_i$ obviously are as well.\n\\ \\hfill $\\Box$\n\\begin{rem}\\label{rem:TateAcylicity} In the proof of Lemma \\ref{lem:CoversRconservative} we showed the interesting fact that for any $\\mathcal{M} \\in {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$, and any finite cover $\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ the complex  \n\\[0 \\to \\mathcal{M} \\to \\prod_{i_1} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1<i_2}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0.\n\\]\nis strictly exact (a version of Tate acyclicity). \n\\end{rem}\n\\begin{example}This example was suggested via a correspondence with V. Berkovich. The analogue of Corollary 4.48 is false for finite modules since they are not preserved by push-forward. On the other hand,  RR-quasi-coherent modules are preserved by push-forward (see Lemma \\ref{lem:RRpreserve}) and do detect surjectivity. If $k$ is a field equipped with the trivial valuation and $\\mathcal{A}= k\\{\\frac{x}{r}\\}$ for $0<r<1$ and $\\mathcal{A}_{V}= k\\{\\frac{x}{r'}\\}$ for $0<r'<r$ then $\\mathcal{M}(\\mathcal{A}_{V})\\to \\mathcal{M}(\\mathcal{A})$ is not surjective while the natural morphism $\\mathcal{A} \\to \\mathcal{A}_{V}$ induces an equivalence ${\\text{\\bfseries\\sf{Mod}}}^{f}(\\mathcal{A})\\to {\\text{\\bfseries\\sf{Mod}}}^{f}(\\mathcal{A}_{V})$, as both categories are equivalent to the category of finite modules in the algebraic sense over $k[[x]]$.  On the other hand, it is easy to find objects of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ which go to zero in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V})$. For instance, choose a valued extension $K$ of $k$ such that $K$ is non-trivialy valued and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is strictly affinoid. Chose a point $x \\in \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) - \\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)$ such that $\\ker(|\\ |_{x})$ is a closed maximal ideal of $\\mathcal{A}\\widehat{\\otimes}_{k}K$. Choose using Proposition 2.2.3 (iii) of \\cite{Ber1990} an affinoid subdomain $W$ of $\\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ such that $W$ does not intersect $\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)$ and $x \\in W$. Then we have the non-zero object $(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W}$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$. It is the push-forward of $(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W} \\in {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ along $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ and therefore RR-quasicoherent by Lemma \\ref{lem:RRpreserve}.  It goes to zero after applying the functor $(-)\\widehat{\\otimes}_{\\mathcal{A}}A_{V}$ because using (\\ref{eqn:basechangeequation}) we have \\[(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W}\\widehat{\\otimes}_{\\mathcal{A}}A_{V} \\cong (\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W} \\widehat{\\otimes}_{\\mathcal{A}\\widehat{\\otimes}_{k}K}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)\\cong 0.\n\\]\n\\end{example}\n\\subsection{Topologies in the Banach algebraic geometry setting}\n\n\n\n\\begin{thm}\\label{thm:covers}Consider the Berkovich space $\\mathcal{M}(\\mathcal{A})$ for $\\mathcal{A}\\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$. The covers in the weak $G$-topology on $\\mathcal{M}(\\mathcal{A})$, as defined on page 30 of \\cite{Ber1990}, coincide precisely with the covers of $\\spec(\\mathcal{A})$ by affinoids in the formal homotopy Zariski topology on the homotopy transversal subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op} \\subset{\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ (using the terminology of Definition \\ref{defn:fhTVZ} and Proposition  \\ref{hfTV1}).\n\\end{thm}\n{\\bf Proof.}\nSay that we are given a cover of $\\mathcal{M}(\\mathcal{A})$ by rational domains which has a finite subcover.  Lemma \\ref{lem:LCSrHoEpis} tells us that the affinoid domains correspond to homotopy epimorphisms $\\mathcal{A} \\to \\mathcal{B}_{i}$ for some $i \\in I$ in ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}$. Lemma \\ref{lem:CoversRconservative} tells us that the family of functors indexed by the finite subset $J \\subset I$ corresponding to the finite subcover is conservative. In the other direction suppose we are given a cover of $X$ in the formal homotopy Zariski topology. It has a finite conservative sub-cover which must be surjective by Lemma \\ref{lem:ConsImpliesSur}. Every element of it must be a subset of $\\mathcal{M}(\\mathcal{A})$  because every morphism in the cover is a monomorphism. In fact, Theorem \\ref{thm:localForm} implies that every morphism in the cover is an affinoid domain embedding. This concludes the proof.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:IsSheaf}\nLet $\\mathcal{A}, \\mathcal{B}_{i}$ be $k$-affinoid algebras and let $\\{f_{i}:\\mathcal{A} \\to \\mathcal{B}_{i}\\}_{i\\in I}$ be a family of morphisms of $k$-affinoid algebras which is a formal homotopy Zariski open cover in the category ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$. Chose a finite subset $J \\subset I$ with the conservative property. Then for any $k$-analytic space $X$ we have \n\\begin{equation}\\label{equation:gluingdiagram}\\Hom(\\mathcal{M}(\\mathcal{A}),X)=\\eq [\\prod_{i \\in J}\\Hom(\\mathcal{M}(\\mathcal{B}_{i}),X) \\rightrightarrows \\prod_{i \\in J,j \\in J}\\Hom(\\mathcal{M}(\\mathcal{B}_{i} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{B}_{j}), X)].\n\\end{equation}\nTherefore, the presheaf $\\straighth_{X}$ is a sheaf in the \nformal homotopy Zariski topology on the homotopy transversal subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op} \\subset{\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$.\n\\end{lem}\n{\\bf Proof.}\nLemma \\ref{lem:ConsImpliesSur} and Theorem \\ref{thm:localForm} imply that the family of morphisms indexed by $J$ corresponds to a finite cover by affinoid domains. Lemma \\ref{easyquasinet} implies that it is a quasinet. Exercise 3.2.2 (v) of \\cite{Ber2009} now tells us that Equation \\ref{equation:gluingdiagram} is valid.\n\n\\ \\hfill $\\Box$\n\\noindent\n\nUnlike the rest of this article, the following theorem is only significant given the results in \\cite{BeKr2}.\n\\begin{thm}\\label{thm:BigRestriction} The topology of Definition \\ref{defn:TVtopologyIntro} restricted to the category ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ agrees with the weak G-topology. \n\\end{thm}\n\n{\\bf Proof.}\n\nIn light of Theorems \\ref{thm:TensLocs} and \\ref{thm:localForm} we only need to understand why condition (2) of Definition \\ref{defn:TVtopologyIntro} is equivalent to a finite collection of affinoid subdomains $V_i$ covering every point of an affinoid $\\mathcal{M}(\\mathcal{A})$. Suppose first that the union of the $V_i$ is all of $\\mathcal{M}(\\mathcal{A})$. Say we have a morphism $\\mathcal{M} \\to \\mathcal{N}$ in $D^{-}(\\mathcal{A})$ and we know that the induced morphism $\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\\to \\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}$ is an isomorphism in  $D^{-}(\\mathcal{A})$ for all $i$. For any $i_1 < i_2 < \\cdots < i_p$ we get an isomorphism $\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_{i_1, \\dots, i_p}}\\to \\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_{i_1, \\dots, i_p}}$ where \\[\\mathcal{A}_{V_{i_1, \\dots, i_p}}= \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\cdots  \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}}= \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\cdots  \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}}.\\] Therefore, we get an isomorphism from \n\\begin{equation}\\label{eqn:Mres}\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}( \\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0) \n\\end{equation}\nto\n\\begin{equation}\\label{eqn:Nres}\\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}( \\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0) \n\\end{equation}\nin $D^{-}(\\mathcal{A})$. Since by Lemma \\ref{lem:AlternatingTate} the natural morphism from $\\mathcal{A}$ \nto \n\\[\n\\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0\n\\]\nis an isomorphism in $D^{-}(\\mathcal{A})$,  (\\ref{eqn:Mres}) and (\\ref{eqn:Nres}) are equivalent to $\\mathcal{M}$ and $\\mathcal{N}$ respectively. Therefore, we can conclude that the original morphism $\\mathcal{M} \\to \\mathcal{N}$ is an isomorphism. Conversely, suppose that there is a point $x\\in \\mathcal{M}(\\mathcal{A})$ which is not in any of the $V_{i}$. Suppose first that $k$ is non-trivially valued and that $\\mathcal{A}$ is strictly $k$-affinoid. Chose as in Lemma \\ref{lem:ConsImpliesSur} a subdomain $W$ of $\\mathcal{M}(\\mathcal{A})$ such that $x \\in W$ and $W$ does not intersect any of the $V_i$. Then $\\mathcal{A}_{W}\\to 0$ is a morphism in $D^{-}(\\mathcal{A})$ but for each $V_{i}$ the derived pullback gives an isomorphism $\\mathcal{A}_{W} \\widehat{\\otimes}_{A}^{\\mathbb{L}}\\mathcal{A}_{V_{i}} \\cong  \\mathcal{A}_{W} \\widehat{\\otimes}_{A}\\mathcal{A}_{V_{i}} \\cong 0 \\to 0$. For the general case, choose using Proposition 2.1.2 of \\cite{Ber1990}, a valuation field extension $k \\to K$ such that the valuation on $K$ is non-trivial and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is a strictly $K$-affinoid algebra. Notice that the conservativity assumption on the original family implies by Lemma \\ref{lem:DerivedBaseChangeConserv} applied to the base change $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ that the family of functors $\\{D^{-}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to D^{-}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K)\\}_{i \\in J}$ is also conservative. The morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k} K) \\to \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ cannot be surjective because in the commutative diagram, \n\\[\n\\xymatrix{ \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K) \\ar[d] \\ar[r] & \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\ar[d] \\\\  \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\ar[r] &  \\mathcal{M}(\\mathcal{A}) }\\]\nthe horizonal arrows are surjective.  Therefore, we have reduced to the previous case and so the proof is complete. \n\\ \\hfill $\\Box$\n\nFor the next lemma, we will need the notion of continuous and cocontinuous functors. These are defined in the appendix in Definition \\ref{defn:ContCocont}. \n\\begin{lem}\\label{lem:inclprops}The inclusion functor \n\n\\[{\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k}) \n\\]\nis fully faithful and continuous with respect to the Zariski, formal Zariski, homotopy Zariski, formal homotopy Zariski and fpqc topologies.\n\\end{lem}\n{\\bf Proof.}\nMost of these properties are more or less obvious at this point. The fiber products correspond to completed tensor products of affinoid algebras or commutative algebra objects in ${\\text{\\bfseries\\sf{Ban}}}_{k}$. The fact that these functors take covers to covers in the various topologies follows from the definition of covers on the domain of the functors. The fully faithful property is discussed in \\ref{lem:extension} of the appendix.\n  \n\\hfill $\\Box$\n\nRecall that ${\\text{\\bfseries\\sf{An}}}_{k}$ is the category of Berkovich spaces over a non-Archimedean valuation field $k$.\n\\begin{thm}\\label{thm:embedding} There is a fully faithful embedding from ${\\text{\\bfseries\\sf{An}}}_{k}$ to the categories of presheaves of sets on ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}$ and ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})^{op}.$\nThis embedding induces a fully faithful embedding \n\\[{\\text{\\bfseries\\sf{An}}}_{k} \\hookrightarrow  {\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}) \n\\].\nFor any affinoid algebras $\\mathcal{A}$ and $\\mathcal{B}$, the scheme assigned to\n$\\mathcal{M}(\\mathcal{A})$ evaluates on $\\spec(\\mathcal{B})$ to $\\Hom_{{\\text{\\bfseries\\sf{Afnd}}}_{k}}(\\mathcal{A},\\mathcal{B})$.\n\\end{thm}\n{\\bf Proof.}\nLet $X$ be a $k$-analytic space. Consider the contravariant functors \n\\[\\straighth_{X}:{\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Set}}}\n\\]\ndefined by \n\\[\\straighth_{X}(\\spec(\\mathcal{A}))=\\Hom_{{\\text{\\bfseries\\sf{An}}}_{k}}(\\mathcal{M}(\\mathcal{A}),X)\n\\]\nfor $\\mathcal{A} \\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$.\nThey form a functor\n\\begin{equation}\\label{equation:AnToPre}{\\text{\\bfseries\\sf{An}}}_{k} \\stackrel{\\straighth}\\to {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})\n\\end{equation}\nwhich can be seen the Yoneda embedding for ${\\text{\\bfseries\\sf{An}}}_{k}$ followed by the restriction functor ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{An}}}_{k}) \\to {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})$. \nWe would like to know that for every $X_{1},X_{2} \\in {\\text{\\bfseries\\sf{An}}}_{k}$ that \n\\[\\Hom_{{\\text{\\bfseries\\sf{An}}}_{k}}(X_{1},X_{2}) \\to \\Hom_{{\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})}(\\straighth_{X_1},\\straighth_{X_2})\n\\]\nis an isomorphism. The fact that the functor $\\straighth$ is fully faithfull follows immediately from Lemma \\ref{EverythingColimit} which says that $X$ is the final object in the category of morphisms $\\spec(\\mathcal{A}) \\to X$ where $\\mathcal{A} \\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$ and the fact that for any full subcategory ${\\tC} \\subset {\\tD}$ such that every object $d$ in ${\\tD}$ is a colimit of objects in ${\\tC}$ mapping to $d$, the functor $\\straighth$ embeds ${\\tC}$ fully and faithfully into the category of presheaves of sets on ${\\tD}$.  This fact can be found in \\cite{SGA4}, Exp I, Prop 7.2. \nConsider the fully faithful, continuous and cocontinuous functor from Lemma \\ref{lem:inclprops}\n\\[\\su: {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k}). \n\\]\n\nwhich is left adjoint to the restriction functor. \n\nIt is clear by the definition of Berkovich analytic spaces that we actually get a fully faithful embedding \n\\[{\\text{\\bfseries\\sf{An}}}_{k} \\hookrightarrow  {\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}) \n\\]\nand when we restrict the scheme to the subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ this assignment agrees with the standard functor of points in the category ${\\text{\\bfseries\\sf{An}}}_{k}$.\n\\ \\hfill $\\Box$\n\n\\begin{rem}\nNote that exactly the same argument shows that the category of rigid analytic spaces (which contains $k$-analytic spaces as a full subcategory) embeds fully faithfully into ${\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}).$ \n\\end{rem}\n\n\\begin{rem}\nIn order to embed $k$-analytic spaces and rigid analytic spaces into a catgeory of Banach schemes, ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$, we need to work in the derived setting as the homotopy Zariski topology is only well defined on simplicial Banach algebras. As ${\\text{\\bfseries\\sf{Afnd}}}_{k}$ is homotopy Zariski transversal this topology restricts to the non-derived site of affinoids. The derived approach is pursued in \\cite{BeKr2}.  \n\\end{rem}\n\n\\subsection{Huber Points}\n\nLet $X$ be an object of ${\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}_{k})^{op})^{hZar})$. Define a category ${\\text{\\bfseries\\sf{hZar}}}(X)$ as the full sub-category of ${\\text{\\bfseries\\sf{Sh}}}(({\\text{\\bfseries\\sf{Afnd}}}_{k})^{op})^{hZar})\/X$ whose objects are $u:Y\\to X$ with $u$ a homotopy Zariski open immersion. This category \nis a locale and inherits a topology: a family $\\{Y_i\\to Y\\}$ is a covering if and only if $\\coprod Y_i\\to Y$ is an epimorphism of sheaves. We have the full subcategory ${\\text{\\bfseries\\sf{hZarAff}}}(X)$ whose objects are $u:Y\\to X$ where $Y$ is affine. \nThe continuous inclusion ${\\text{\\bfseries\\sf{hZarAff}}}(X)\\to {\\text{\\bfseries\\sf{hZar}}}(X)$ induces an equivalence of categories of sheaves:\n${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZar}}}(X))\\cong {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZarAff}}}(X))$. \n\nThe fact that ${\\text{\\bfseries\\sf{hZar}}}(X)$ is a locale and that the topology on it \nis generated by a quasi-compact  pre-topology (as covering families in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZarAff}}}(X))$ are finite), implies that \n${\\text{\\bfseries\\sf{hZar}}}(X)$ is equivalent as a locale to the locale of open subsets of a topological space $|X|$. Using this we can view $X$ a ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$-valued ringed space:\n$(|X|,\\mathcal{O}_X)$, where $\\mathcal{O}_X$ is a sheaf valued in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k}).$\n\n\n\\begin{lem}\nLet $X$ be a Berkovich space. ${\\text{\\bfseries\\sf{hZarAff}}}(X)$ is the rigid analytic site of $X$.\n\\end{lem}\n\n{\\bf Proof.} This is a rephrasing of Theorem \\ref{thm:covers}.\n\\hfill $\\Box$\n\n\\begin{thm}\nLet $X$ be a Berkovich space. The space $|X|$ is the space of Huber points.\n\\end{thm}\n{\\bf Proof.} From the previous lemma we know that the locale\ncorresponding to $X$ is the rigid analytic one. From Huber \\cite{Hu}, we see that the topological space corresponding to this locale is the space of Huber points of $X$.\n\\hfill $\\Box$\n\n  \n\n\\section{Work in progress}\nIn future work, we intend to embed rigid analytic spaces and Huber spaces into the categories of schemes over the opposite category to Banach algebras. We also intend develop a variant which will work with complete, convex bornological algebras, instead of Banach algebras or Fr\\'echet algebras or other types of topological vector spaces. Similarly to Paugam \\cite{Pa} we would like to handle the Archimedean and non-Archimedean cases with a single language. Also with the same goal, Bambozzi has defined and studied affinoid bornological algebras and dagger algebras in his thesis \\cite{Bam}. The categories based on convex bornological spaces are nicer than those based on ${\\text{\\bfseries\\sf{Ban}}}_{k}$ in that they have arbitrary limits and colimits. The bornological categories contain the categories based on Banach spaces or Fr\\'echet spaces via fully faithful embeddings. A lot of work in this direction of bornological geometry and representation theory has been done by Meyer \\cite{M} in the Archimedean case over $\\mathbb{C}$ and by Houzel and others \\cite{H,H2} in the case of a valuation field where the valuation is not the trivial valuation. The extension to the case of a trivially valued field is more difficult, especially in the definition of completeness. Therefore, we could have done our constructions for convex bornological vector spaces over an arbitrary field or for complete convex bornological vector spaces over a non-discretely valued field.  In another direction we think that categories of topological spaces, topological manifolds, differentiable manifolds and real analytic manifolds could also be described in terms of this style of algebraic geometry. This could fit in well with the algebraic analysis of Kashiwara, Schapira, Saito and others. Non-commutative versions may be possible as well along the lines of \\cite{So,KR}. It would also be nice to be able to work over rings instead of fields, for instance the rings of $p$-adic intgers $\\mathbb{Z}_{p}$ and the integers equipped with either the discrete or the standard norm and do types of analytic geometry over these rings. We are developing a formalism of derived analytic stacks along the lines of following To\\\"{e}n and Vezzosi's approach \\cite{TVe} or Lurie's approach \\cite{L1,L2}. It will work simultaneously in the Archimedean and non-Archimedean setting. In particular we need to define a cotangent complex and the notion of smooth and \\'{e}tale morphisms and compare these notions with the notions defined by Berkovich.\nWe also believe that a non-Archimedean versions of both Lurie's Tanakian duality theorem \\cite{L3} and (after developing differential graded categories of modules) the algebrization theorem \\cite{TV2} of To\\\"{e}n and Vaqui\\'{e} should be provable in our framework. We plan to prove a version of the Beilinson-Bernstein localization theorem over certain non-Archimedean fields in our context which would be a generalisation of the work of \\cite{AW} over $\\mathbb{Q}_{p}$. Finally, we believe that when working over a finite field with trivial valuation, our formalism will be useful for studying $G$-bundles over an algebraic surface or Kac-Moody bundles over an algebraic curve with an eye towards representation theory as in \\cite{BrKa}, \\cite{K}, \\cite{GP}. This is based on the relationship between $G$-bundles on a punctured formal neighborhood of a curve in surface and twisted $G((t))$-bundles on the curve. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe classical Monge's transport problem refers to the problem of moving one distribution of mass onto another as efficiently as possible, where the efficiency criterion is expressed in terms of the average distance transported. It originates from a paper by G. Monge, \\textit{M\\'emoire sur la th\\'eorie des d\\'eblais et des remblais}, in 1781. Rephrased and generalized in modern mathematical terms, we are given two Borel probability measures $\\mu$ and $\\nu$ on a metric space $(X,d)$ and we want to minimize\n\\begin{equation*}\n T \\mapsto \\int_X d(x,T(x))\\,d\\mu(x)\n\\end{equation*}\namong all transport maps $T$ from $\\mu$ to $\\nu$, i.e., all $\\mu$-measurable maps $T:X\\to X$ such that $T_\\#\\mu=\\nu$, meaning that $\\nu(B)=\\mu(T^{-1}(B))$ for all Borel set $B$. \n\nIn this paper, we are interested in Monge's transport problem in the Heisenberg group $(\\mathbb{H}^n,d)$ equipped with its Carnot-Carath\\'eodory distance. We prove the existence of an optimal transport map between two compactly supported Borel probability measures $\\mu$ and $\\nu$ on $\\mathbb{H}^n$ assuming that the first measure $\\mu$ is absolutely continuous with respect to the Haar measure $\\mathcal{L}^{2n+1}$ of $\\mathbb{H}^n$. \n\n\\begin{thm} \\label{mainthm}\nLet $\\mu$ and $\\nu$ be two compactly supported Borel probability measures on $\\mathbb{H}^n$. Assume that $\\mu \\ll \\mathcal{L}^{2n+1}$. Then there exists an optimal transport map solution to Monge's transport problem between $\\mu$ and $\\nu$, i.e., a $\\mu$-measurable map $T:\\mathbb{H}^n\\rightarrow\\mathbb{H}^n$ such that $T_\\#\\mu=\\nu$ and\n\\begin{equation*}\n\\int_{\\mathbb{H}^n} d(x,T(x))\\,d\\mu(x) = \\inf_{S_\\#\\mu=\\nu} \\int_{\\mathbb{H}^n} d(x,S(x))\\,d\\mu(x).\n\\end{equation*}\n\\end{thm}\n\nMonge's transport problem in $\\mathbb{R}^n$ equipped with a distance induced by a norm has already been widely investigated. A first attempt to solve this problem goes back to the work of Sudakov \\cite{sudakov}. It was however discovered some years later that the proof in \\cite{sudakov} was not completely correct. In \\cite{evans-gangbo} a PDE-based alternative to Sudakov's approach has been developed. The authors prove the existence of an optimal transport map in $\\mathbb{R}^n$ equipped with the Euclidean norm under the assumptions that $\\operatorname{spt} \\mu \\cap \\operatorname{spt} \\nu = \\emptyset$, $\\mu$, $\\nu \\ll \\mathcal{L}^{n}$ with Lipschitz densities with compact support. Existence results for general absolutely continuous measures $\\mu$, $\\nu$ with compact support have been obtained independently in \\cite{cfm} and \\cite{TrudWang} and have been extended to a Riemannian setting in \\cite{FeldMcC}. The existence of a solution to Monge's transport problem assuming only that the initial measure $\\mu$ is absolutely continuous has been proved in \\cite{ambrosio}, see also \\cite{ap}, \\cite{akp}, \\cite{caravenna1}. All these later proofs roughly involve a Sudakov-type dimension reduction argument, via different technical implementations though, and require some regularity assumptions about the norm $\\mathbb{R}^n$ is endowed with. For some time, it seemed that there were indeed some borderline cases about the norms that could not be attacked through these techniques. \n\nRecently another approach that does not go through Sudakov-type arguments and in particular does not require disintegration of measures has been developed in \\cite{champion-dePascale}, see also \\cite{champion-dePascale-first}, to solve Monge's transport problem for general norms in $\\mathbb{R}^n$. This approach relies on rather simple but powerful density results. In the present paper we follow closely this approach. We basically show that a very similar strategy can be implemented in the context of the Heisenberg group equipped with its Carnot-Carath\\'eodory distance. The main features that play a role in this approach are that $(\\mathbb{H}^n, d, \\mathcal{L}^{2n+1})$ is a doubling polish metric measure space, a non-branching geodesic space and satisfies a so-called Measure Contraction Property. It is very likely that this approach can be extended to more general metric measure spaces, see Section~\\ref{extensions}. We have chosen however to present the particular case of the Heisenberg group for simplicity, this space being moreover an instructive explicit example of non Riemannian space.~\\footnote{The preprint~\\cite{bianc-cav} which appeared during the completion of the present paper also addresses Monge's transport problem in metric spaces with a geodesic distance using a Sudakov-type dimension reduction argument and disintegration of measures.} \n\nThe strategy starts by considering the nowadays classical relaxation of Monge's transport problem proposed by Kantorovich. In Kantorovich' formulation one considers transport plans, i.e., Borel probability measures on $X \\times X$ with first and second marginals $\\mu$ and $\\nu$ respectively. Denoting by $\\Pi(\\mu,\\nu)$ the class of all transport plans, one wants to minimize\n\\begin{equation*}\n \\gamma \\mapsto \\int_{X\\times X} d(x,y)\\,d\\gamma(x,y)\n\\end{equation*}\namong all $\\gamma \\in \\Pi(\\mu,\\nu)$. Due to the linearity of the constraint $\\gamma \\in \\Pi(\\mu,\\nu)$, weak topologies provide existence of optimal transport plans. As a classical fact it turns out that whenever an optimal transport plan is induced by a $\\mu$-measurable map $T$, i.e., can be written in the form $(I \\otimes T)_\\sharp\\mu$ where $(I \\otimes T)(x) := (x,T(x))$, then $T$ is an optimal transport map between $\\mu$ and $\\nu$ solution to Monge's transport problem. We follow here this scheme seeking after optimal transport plans that will be shown to be eventually induced by $\\mu$-measurable maps.\n\nIn our present context we first prove that any optimal transport plan is concentrated on a set of pairs of points that are connected by a unique minimal curve and that these transport rays cannot bifurcate (see Section~\\ref{sect:optplanning}). Next, following ideas already introduced in the literature and more specifically here inspired by~\\cite{sant}, one introduces variational approximations (see Section~\\ref{varapprox}). This procedure allows to select optimal transport plans with specific properties. These transport plans will eventually be proved to be induced by $\\mu$-measurable maps. This procedure is here essentially twofold. On one hand it allows to select optimal transport plans that are solution to a secondary variational problem. This secondary variational problem prescribes the geometry of transport rays. The selected transport plans are indeed shown to be monotonic along transport rays (see Lemma~\\ref{pi2}). On the other hand, given a transport plan, one can in our context interpolate between its first and second marginal in a natural way (see Subsection~\\ref{sect-interpolation}). Absolute continuity and more importantly $L^\\infty$-estimates on the density of the interpolations will play an important role and one can indeed prove $L^\\infty$-estimates on the interpolations in the approximating variational problems (see Proposition~\\ref{e:Linftydensityestimates}). These estimates rely on the so-called Measure Contraction  Property of $\\mathbb{H}^n$. In the limit one will  eventually get suitable $L^\\infty$-estimates on the interpolations constructed from optimal transport plans selected through the variational approximation procedure. Next we note that some properties of plans with absolutely continuous first marginal proved in~\\cite{champion-dePascale} can be easily generalized to our setting (see Section~\\ref{lebpoints}). These properties are independent of the transport problem. They rely on the notion of Lebesgue points of functions and Lebesgue points of sets, notions which make sense for instance in any doubling metric measure space. Together with the above mentioned $L^\\infty$-estimates on the interpolations, one can in particular prove density estimate on the transport set of selected optimal transport plans in the same way as in~\\cite{champion-dePascale} (see Section~\\ref{main}). All together, namely combining this later density estimate on the transport set (Lemma~\\ref{mainlemma}) with Lemma~\\ref{dens2} and remembering the monotonicity along transport rays (Lemma~\\ref{pi2}) it turns out that the selected transport plans are necessarily induced by a transport map as eventually proved in Theorem~\\ref{mainthmbis}.\n\nThe paper is organized as follows. In Section~\\ref{opttrans} we recall classical facts about optimal transportation for later use. In Section~\\ref{prelimhn} we describe the Heisenberg group focusing on the features that will be needed in this paper. In Section~\\ref{sect:optplanning} we prove geometric properties of optimal transport plans and the monotonicity along transport rays of solutions to  the secondary variational problem. The variational approximations are introduced and studied in Section~\\ref{varapprox}. In Section~\\ref{lebpoints} we state in our framework properties of plans with absolutely continuous first marginal proved in~\\cite{champion-dePascale} in $\\mathbb{R}^n$. This section, independent of the transport problem, contains density results that play an essential role in the strategy followed here. In Section~\\ref{main} we prove lower bounds on the density, in some suitable sense, of the transport set of optimal transport plans selected through the variational approximations. We conclude in Section~\\ref{conclusion} proving that the selected transport plans are induced by a transport map. We discuss in the final Section~\\ref{extensions} some possible extensions of this approach to other spaces.\n\n\n\\section{Preliminaries on optimal transportation} \\label{opttrans}\n\nWe recall some well-known facts about optimal transportation confining ourselves to statements that will fit our needs in the rest of the paper. More general versions of these results hold in more general contexts. We refer to e.g. \\cite{villani} and the references therein.\n\nLet $(X,d)$ be a Polish space, i.e., a complete and separable metric space. We denote by $\\mathcal P(X)$ the set of all Borel probability measures on $X$ and by $\\mathcal P_c(X)$ the set of Borel probability measures on $X$ with compact support. The weak topology we consider on $\\mathcal P(X)$ is the topology induced by convergence against bounded and continuous test functions (or narrow topology).\n\n\\subsection{Kantorovich transport problem} \nLet $\\mu$, $\\nu\\in\\mathcal P(X)$. We denote by \n\\begin{equation*}\n \\Pi(\\mu,\\nu) := \\{ \\gamma \\in \\mathcal P(X \\times X);\\, (\\pi_1)_\\sharp\\gamma = \\mu,\\, (\\pi_2)_\\sharp\\gamma = \\nu \\}\n\\end{equation*}\nthe set of all transport plans between $\\mu$ and $\\nu$. Here $\\pi_1$, $\\pi_2: X\\times X \\rightarrow X$ denote the canonical projections on the first and second factor respectively.\n\nGiven $c:X\\times X \\rightarrow [0,+\\infty]$ a lower semicontinuous cost function, we look at Kantorovich transport problem between $\\mu$ and $\\nu$ with cost $c$:\n\\begin{equation} \\label{e:MK}\n \\min_{\\gamma \\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} c(x,y)\\,d\\gamma(x,y).\n\\end{equation}\nAs a classical fact, existence of solutions to \\eqref{e:MK} follows from the weak compactness of $\\Pi(\\mu,\\nu)$ together with the lower semicontinuity of the functional to be minimized. We call them optimal transport plans.\n\n\\subsection*{Cyclical monotonicity} We say that a set $\\Gamma\\subset X\\times X$ is $c$-cyclically monotone if \n\\begin{equation*}\n\\sum_{i=1}^N c(x_i,y_i) \\leq \\sum_{i=1}^N c(x_{i+1},y_i)\n\\end{equation*}\nwhenever $N\\geq 2$ and $(x_1,y_1), \\dots, (x_N,y_N)\\in\\Gamma$.\n\n\\begin{thm} \\label{ccycl}\nLet $\\gamma \\in \\Pi(\\mu,\\nu) $ be an optimal transport plan and assume that $\\int_{X\\times Y} c(x,y)\\,d\\gamma <+\\infty$. Then $\\gamma$ is concentrated on a $c$-cyclically monotone Borel set.\n\\end{thm}\n\n\\subsection*{Dual formulation - Kantorovich potentials}\nLet $\\psi:X\\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$. We say that $\\psi$ is $c$-concave if $\\psi\\not\\equiv -\\infty$ and if there exists $\\varphi:X \\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$, $\\varphi\\not\\equiv -\\infty$, such that \n\\begin{equation*}\n\\psi(x) = \\inf_{y\\in X} c(x,y) - \\varphi(y).\n\\end{equation*}\n\n\\begin{thm}  \\label{duality}\nIn addition to the previous assumptions, assume that $c$ is real-valued and that \n\\begin{equation*}\n \\forall\\, (x,y)\\in X\\times X, \\qquad c(x,y) \\leq a(x) + b(y)\n\\end{equation*}\nfor some $a\\in L^1(\\mu)$ and $b\\in L^1(\\nu)$. Then one has\n\\begin{equation} \\label{e:duality}\n \\min_{\\gamma\\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} c(x,y)\\,d\\gamma(x,y) \n= \\max \\int_{X} \\psi(x)\\,d\\mu(x) + \\int_{X} \\psi^c(y)\\,d\\nu(y)\n\\end{equation}\nwhere the above maximum is taken among all $c$-concave functions $\\psi$ and $ \\psi^c(y) := \\inf_{x\\in X} c(x,y) - \\psi(x)$. \n\\end{thm}\n\n\\begin{defi} [Kantorovich potentials]\n We say that $\\psi:X\\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$ is a Kantorovich potential if $\\psi$ is a $c$-concave maximizer for the right-hand side of \\eqref{e:duality}.\n\\end{defi}\n\n\\begin{thm} \\label{caracterisation_opt_planning}\nWith the same assumptions as in Theorem~\\ref{duality}, let $\\psi$ be a Kantorovich potential. Then $\\gamma\\in \\Pi(\\mu,\\nu)$ is an optimal transport plan if and only if \n\\begin{equation*}\n c(x,y) = \\psi(x) + \\psi^c(y) \\qquad \\gamma-\\text{a.e. in } X\\times X.\n\\end{equation*}\n\\end{thm}\n\nWe will use these results for various cost functions. In the particular case which is the core of this paper and where $c(x,y) = d(x,y)$ and $\\mu$, $\\nu \\in \\mathcal P_c(X)$, one can rephrase these results in terms of 1-Lipschitz Kantorovich potentials. More precisely, set \n\\begin{equation*}\n \\text{Lip}_1(d) := \\{u:X\\rightarrow \\mathbb{R};\\quad |u(x) - u(y)| \\leq d(x,y) \\quad \\forall\\, x, y \\in X\\}.\n\\end{equation*}\n\n\\begin{thm} \\label{1lip_potential}\nLet $\\mu$, $\\nu \\in \\mathcal P_c(X)$. Then one can find a Kantorovich potential $u\\in \\text{Lip}_1(d) $ so that\n\\begin{equation*}\n \\min_{\\gamma\\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} d(x,y)\\,d\\gamma(x,y) \n= \\int_{X} u(x)\\,d\\mu(x) - \\int_{X} u(y)\\,d\\nu(y)\n\\end{equation*}\nand \n$\\gamma\\in \\Pi(\\mu,\\nu)$ is an optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ if and only if \n\\begin{equation*}\nu(x) - u(y) = d(x,y)  \\qquad \\gamma-\\text{a.e. in } X\\times X.\n\\end{equation*}\n\\end{thm}\n\n\\subsection{Transport problem} Let $\\mu$, $\\nu\\in\\mathcal P(X)$. We say that a $\\mu$-measurable map $T:X\\rightarrow X$ is a transport map between $\\mu$ and $\\nu$ if $T_\\sharp\\mu=\\nu$, i.e., $\\nu(B)=\\mu(T^{-1}(B))$ for\nall Borel set $B$. \n\nGiven $c:X\\times X \\rightarrow [0,+\\infty[$ a continuous cost function, we look at the transport problem between $\\mu$ and $\\nu$ with cost $c$:\n\\begin{equation}  \\label{e:M}\n \\min_{T_\\#\\mu=\\nu} \\int_{X} c(x,T(x))\\,d\\mu(x).\n\\end{equation}\n\nWe say that a transport plan $\\gamma\\in \\Pi(\\mu,\\nu)$ is induced by a transport if there exists a $\\mu$-measurable map $T:X\\rightarrow X$ such that $(I \\otimes T)_\\sharp\\mu = \\gamma$ where $(I \\otimes T)(x) := (x,T(x))$. Such a map is automatically a transport map between $\\mu$ and $\\nu$. We also recall that if a transport plan $\\gamma$ is concentrated on a $\\gamma$-measurable graph then $\\gamma$ is induced by a transport.\n\n\\begin{thm} [Optimal transport plans versus optimal transport maps] \\hfill \\label{plan-transport}\n\n(i) Assume that $\\gamma$ is an optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} and that $\\gamma$ is induced by transport $T$. Then $T$ is an optimal transport map solution to the transport problem~\\eqref{e:M}.\n\n(ii) Assume that any optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} is induced by transport. Then there exists a unique optimal transport map solution to the transport problem~\\eqref{e:M}.\n \\end{thm}\n\n\n\\section{Preliminaries on $\\mathbb{H}^n$} \\label{prelimhn}\n\nWe consider the Heisenberg group $\\mathbb{H}^n$ equipped with its Carnot-Carath\\'eodory distance. Endowed with this distance $\\mathbb{H}^n$ is a polish geodesic and non-branching metric space and a doubling metric measure space when equipped with its Haar measure.\n\n\\subsection{The Heisenberg group}\nThe Heisenberg group $\\mathbb{H}^n$ is a connected, simply connected Lie group with stratified Lie algebra. We identify it with $\\mathbb{C}^n \\times \\mathbb{R}$ equipped with the group law \n\\begin{equation*}\n[\\zeta,t] \\cdot [\\zeta',t'] := [\\zeta + \\zeta', t+t'+ 2 \\sum_{j=1}^n \\operatorname{Im} \\zeta_j \\overline\\zeta'_j]\n\\end{equation*}\nwhere $\\zeta = (\\zeta_1,\\dots,\\zeta_n)$, $\\zeta' = (\\zeta'_1,\\dots,\\zeta'_n)\\in\\mathbb{C}^n$ and $t$, $t'\\in \\mathbb{R}$. The unit element is 0 and the center of the group is \n\\begin{equation*}\nL := \\{[0,t] \\in \\mathbb{H}^n ;\\; t \\in \\mathbb{R}\\}.\n\\end{equation*}\nThere is a natural family of dilations $\\delta_r$ on $\\mathbb{H}^n$ defined by $\\delta_r([\\zeta,t]):= [r\\zeta,r^2 t]$. These dilations are group homomorphisms. \n\nWe may also identify $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$ via the correspondence $[\\zeta,t]= (\\xi,\\eta,t)$ where $\\xi=(\\xi_1,\\dots,\\xi_n)$, \n$\\eta=(\\eta_1,\\dots,\\eta_n) \\in \\mathbb{R}^n$, $t\\in \\mathbb{R}$ and $\\zeta = (\\zeta_1,\\dots,\\zeta_n)\\in\\mathbb{C}^n$ with $\\zeta_j = \\xi_j + i \\eta_j$. The horizontal subbundle of the tangent bundle is defined by\n\\begin{equation*}\n\\mathcal{H} := \\operatorname{span} \\left\\{X_j; \\, j=1,\\dots,n\\right\\} \\oplus \\operatorname{span} \\left\\{Y_j; \\, j=1,\\dots,n\\right\\}\n\\end{equation*}\nwhere the left invariant vector fields $X_j$ and $Y_j$ are given by\n\\begin{equation*}\nX_j := \\partial_{\\xi_j} + 2 \\eta_j \\partial_t \\,, \\quad Y_j := \\partial_{\\eta_j} - 2 \\xi_j \\partial_t.\n\\end{equation*}\nVector fields in $\\mathcal{H}$ will be called horizontal vector fields. Setting $T:=\\partial_t$, the only non trivial bracket relations are $[X_j,Y_j] = -4 T$ hence $\\operatorname{span} \\{T\\} = [ \\mathcal{H},\\mathcal{H}]$ and the Lie algebra $\\mathcal{H}^n$ of $\\mathbb{H}^n$ admits the stratification $\\mathcal{H}^n = \\mathcal{H} \\oplus \\operatorname{span} \\{T\\}$.\n\nThe Lebesgue measure $\\mathcal{L}^{2n+1}$ on $\\mathbb{H}^n \\thickapprox \\mathbb{R}^{2n+1}$ is a Haar measure of the group. It is $(2n+2)$-homogeneous with respect to the dilations, \n\\begin{equation*}\n \\mathcal{L}^{2n+1}(\\delta_r(A)) = r^{2n+2} \\mathcal{L}^{2n+1}(A)\n\\end{equation*}\nfor all Borel set $A$ and all $r>0$.\n\n\\subsection{Carnot-Carath\\'eodory distance} The Carnot-Carath\\'eodory distance on $\\mathbb{H}^n$ is defined by \n\\begin{equation} \\label{e:cc-dist}\n d(x,y) = \\inf \\{ length_{g_0} (\\gamma); \\; \\gamma \\text{ horizontal } C^1 \\text{-smooth curve joining } x \\text{ to } y\\},\n\\end{equation}\nwhere a $C^1$-smooth curve is said to be horizontal if, at every point, its tangent vector belongs to the horizontal subbundle of the tangent bundle and $g_0$ is the left invariant Riemannian metric which makes $(X_1,\\dots,X_n,Y_1,\\dots,Y_n,T)$ an orthonormal basis. For a general presentation of Carnot-Carath\\'eodory spaces, see e.g. \\cite{bel}, \\cite{mont}. \n\nThe topology induced by this distance is the original (Euclidean) topology on $\\mathbb{H}^n \\thickapprox (\\mathbb{R}^{2n+1},g_0)$ and $(\\mathbb{H}^n,d)$ is a  complete metric space. The distance is left invariant and 1-homogeneous with respect to the dilations,\n\\begin{equation*}\nd(x \\cdot y, x\\cdot z) = d(y,z) \\quad \\text{and} \\quad d(\\delta_r(y), \\delta_r(z)) = r\\,d(y,z)\n\\end{equation*}\nfor all $x$, $y$, $z\\in\\mathbb{H}^n$ and all $r>0$. It follows in particular that $B(x,r) = x\\cdot \\delta_r (B(0,1))$ and hence \n\\begin{equation} \\label{e:measball}\n \\mathcal{L}^{2n+1}(B(x,r)) = c_n \\, r^{2n+2}\n\\end{equation}\nfor all $x\\in \\mathbb{H}^n$, all $r>0$ and where $c_n := \\mathcal{L}^{2n+1}(B(0,1))>0$. The measure $\\mathcal{L}^{2n+1}$ is in particular a doubling measure on $(\\mathbb{H}^n,d)$. For more details about doubling metric measure spaces, see e.g. \\cite{heinonen}.\n\nEndowed with its Carnot-Carath\\'eodory distance $\\mathbb{H}^n$ is a geodesic space, i.e., for all $x$, $y\\in\\mathbb{H}^n$, there exists a curve $\\sigma \\in C([a,b],\\mathbb{H}^n)$ such that $\\sigma(a)=x$, $\\sigma(b)=y$ and $d(x,y) = l(\\sigma)$ where \n\\begin{equation*}\n l(\\sigma) = \\sup_{N\\in  \\mathbb{N}^*} \\sup_{a=t_0\\leq \\dots \\leq t_N = b} \\sum_{i=0}^{N-1} d(\\sigma(t_i),\\sigma(t_{i+1})).\n\\end{equation*}\nUp to a reparameterization one can always assume that length minimizing curves $ \\sigma$ are parameterized proportionally to arc-length, i.e., \n\\begin{equation*}\nd(\\sigma(s),\\sigma(s')) = v \\,(s'-s)\n\\end{equation*}\nfor all $s<s' \\in [a,b]$, where $v:=d(\\sigma(a),\\sigma(b))\/(b-a)$ is the (constant) speed of the curve. As a convention we will use throughout this paper the terminology \\textit{minimal curves} to denote length minimizing curves parameterized proportionally to arc-length.\n\n\\begin{defi}[Minimal curves] \nWe say that a continuous curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ is a minimal curve if $l(\\sigma) = d(\\sigma(a),\\sigma(b))$ and $\\sigma$ is parameterized proportionally to arc-length.\n \\end{defi}\n\nIn general Carnot-Carath\\'eodory spaces, issues about uniqueness and regularity of minimal curves between any two points as well as issues about the regularity of the distance function to a given point could be delicate. In the specific case of the Heisenberg group, equations of all minimal curves can be explicitly computed and exploited to overcome these difficulties. We recall below the description of minimal curves in $\\mathbb{H}^n$, see e.g. \\cite{gaveau}, \\cite{ar}. We set\n\\begin{equation} \\label{e:omega}\n \\Omega := \\{(x,y)\\in \\mathbb{H}^n\\times \\mathbb{H}^n;\\,\\, x^{-1}\\cdot y \\not \\in L\\}.\n\\end{equation}\n\n\\begin{thm} [Minimal curves in $\\mathbb{H}^n$] \\label{geod} \nMinimal curves in $\\mathbb{H}^n$ are horizontal $C^1$-smooth curves such that the infimum in \\eqref{e:cc-dist} is achieved. One has the more precise description:\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\n\\item Non trivial minimal curves starting from 0 and parameterized on $[0,1]$ are all curves $\\sigma_{\\chi,\\varphi}$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and $\\varphi\\in [-2\\pi,2\\pi]$ where \n\\begin{equation*}\n\\sigma_{\\chi,\\varphi}(s) = [i \\dfrac{(e^{-i\\varphi s} -1) \\chi}{\\varphi}, 2 |\\chi|^2 \\,\\dfrac{\\varphi s - \\sin (\\varphi s)}{\\varphi^2}]\n \\end{equation*}\nif $\\varphi\\in [-2\\pi,2\\pi]\\backslash \\{0\\}$ and \n\\begin{equation*}\n \\sigma_{\\chi,\\varphi}(s) = [\\chi s, 0]\n\\end{equation*}\nif $\\varphi =0$. Moreover one has $|\\chi| = d(0,\\sigma_{\\chi,\\varphi}(1))$.\n\n\\item For all $(x,y)\\in \\Omega$, there is a unique minimal curve $x\\cdot \\sigma_{\\chi,\\varphi}$ between $x$ and $y$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and some $\\varphi\\in\\, (-2\\pi,2\\pi)$ and one has $|\\chi| = d(x,y)$.\n\n\\item \nIf $(x,y)\\not\\in \\Omega$, $x^{-1}\\cdot y = [0,t]$ for some $t\\in \\mathbb{R}^*$, there are infinitely many minimal curves between $x$ and $y$. These curves are all curves of the form $x \\cdot \\sigma_{\\chi,2\\pi}$ if $t>0$, $x \\cdot \\sigma_{\\chi,-2\\pi}$ if $t<0$, for all $\\chi \\in \\mathbb{C}^n$ such that $|\\chi| = \\sqrt{\\pi |t|}$.\n\\end{enumerate}\n\\end{thm}\n\nHere and in the following, $|\\chi| = (\\sum_{j=1}^n |\\chi_j|^2)^{1\/2}$ for $\\chi = (\\chi_1,\\dots,\\chi_n)\\in\\mathbb{C}^n$. In particular it follows from this description that $(\\mathbb{H}^n,d)$ is non-branching.\n\n\\begin{prop} [Non-branching property of $\\mathbb{H}^n$] \\label{nonbranching}\n The space $(\\mathbb{H}^n,d)$ is non-branching, i.e., any two minimal curves which coincide on a non trivial interval coincide on the whole intersection of their intervals of definition. \n\\end{prop}\n\nEquivalently for any quadruple of points $z$, $x$, $y$, $y'\\in \\mathbb{H}^n$, if $z$ is a midpoint of $x$ and $y$ as well as a midpoint of $x$ and $y'$, then $y=y'$. \n\nThe next lemma collects some differentiability properties of the distance function to a given point to be used later. For $y\\in \\mathbb{H}^n$, we set $L_y:= y\\cdot L$.\n\n\\begin{lem} \\label{prop-distcc} Let $y\\in \\mathbb{H}^n$ and set $d_y(x) := d(x,y)$. Then the function $d_y$ is of class $C^{\\infty}$ on $\\mathbb{H}^n \\backslash L_y$ (equipped with the usual differential structure when identifying $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$). Moreover one has \n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\n\\item $|\\nabla_H d_y(x)| = 1$ for all $x \\in \\mathbb{H}^n \\backslash L_y$ where $$\\nabla_H d_y(x) := (X_1 d_y(x)+i Y_1 d_y(x), \\dots, X_n d_y(x)+i Y_n d_y(x)).$$ \n\n\\item If $\\nabla d_y(x) =  \\nabla d_{y'}(x)$ and $d(x,y) = d(x,y')$ for some $x \\in \\mathbb{H}^n \\backslash ( L_y\\cup L_{y'})$, then $y=y'$. Here $\\nabla = (\\partial_{\\xi_1},\\cdots,\\partial_{\\xi_n},\\partial_{\\eta_1},\\cdots,\\partial_{\\eta_n}, \\partial_t)$ denotes the classical gradient when identifying $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$. \n \\end{enumerate}\n\\end{lem}\n\n\\begin{proof} Set $\\Phi(\\chi,\\varphi):=\\sigma_{\\chi,\\varphi}(1)$ where $\\sigma_{\\chi,\\varphi}$ is given in Theorem~\\ref{geod}. This map is a $C^\\infty$-diffeomorphism from $\\mathbb{C}^n\\setminus\\{0\\} \\times (-2\\pi,2\\pi)$ onto $\\mathbb{H}^n\\setminus L$ (see e.g. \\cite{monti}, \\cite{ar}, \\cite{juillet}). If $x = \\Phi(\\chi,\\varphi) \\in \\mathbb{H}^n\\setminus L$ with $(\\chi,\\varphi) \\in \\mathbb{C}^n\\setminus\\{0\\} \\times (-2\\pi,2\\pi)$, one has $d_0(x) = |\\chi|$ and \n\\begin{equation*}\n\\nabla_H d_0(x) = \\dfrac{\\chi}{|\\chi|} e^{-i\\varphi} \\quad \\text{and} \\quad \\partial_t d_0(x) = \\dfrac{\\varphi}{4|\\chi|},\n\\end{equation*}\nsee \\cite[Lemma 3.11]{ar}. Next, by left invariance, we have $d_y(x) = d_0(y^{-1}\\cdot x)$, $\\nabla_H d_y(x) = \\nabla_H d_0(y^{-1}\\cdot x)$ and $\\partial_t d_y(x) = \\partial_t d_0(y^{-1}\\cdot x)$ if $x\\in \\mathbb{H}^n \\backslash L_y$ and the lemma follows easily.\n\\end{proof}\n\n\\subsection{Interpolation between measures} \\label{sect-interpolation} The notion of interpolation constructed from a transport plan between any two measures will be one of the key notion to be used later. To define it in our geometrical context, we first fix a measurable selection of minimal curves, i.e., a Borel map $S:\\mathbb{H}^n\\times\\mathbb{H}^n \\rightarrow C([0,1], \\mathbb{H}^n)$ such that for all $x$, $y\\in\\mathbb{H}^n$, $S(x,y)$ is a minimal curve joining $x$ and $y$. The existence of such a measurable recipe to join any two points in $\\mathbb{H}^n$ by a minimal curve follows from general theorems about measurable selections, see e.g. \\cite[Chapter 7]{villani}. Next we set $e_t(\\sigma) := \\sigma(t)$ for all $\\sigma \\in C([0,1],\\mathbb{H}^n)$ and $t\\in [0,1]$. In particular $e_t(S(x,y))$ denotes the point lying at distance $t \\, d(x,y)$ from $x$ on the selected minimal curve $S(x,y)$ between $x$ and $y$. \n\n\\begin{defi} \\label{interpolation} \n Let $\\mu$, $\\nu\\in \\mathcal P(\\mathbb{H}^n)$ and let $\\gamma \\in \\Pi(\\mu,\\nu)$. The interpolations between $\\mu$ and $\\nu$ constructed from $\\gamma$ are defined as the family $((e_t \\circ S)_\\sharp \\gamma))_{t\\in [0,1]}$ of Borel probability measures on $\\mathbb{H}^n$. \n\\end{defi}\n\nNote that these interpolations depend a priori on the measurable selection $S$ of minimal curves. This is actually not a serious issue for our purposes. We will moreover always consider interpolations constructed from transport plans that are concentrated on the set $\\Omega$ on which $S(x,y)$ is nothing but the unique minimal curve between $x$ and $y$. Note also for further reference that $S\\lfloor_\\Omega$ is continuous.\n\n\\subsection{Intrinsic differentiability} Intrinsic differentiability properties of real-valued Lipschitz functions on $\\mathbb{H}^n$, namely a Rademacher's type theorem, will be useful when considering 1-Lipschitz Kantorovich potentials. This theorem is a particular case of a more general result due to P. Pansu. We say that a group homomorphism $g :\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is homogeneous if $g(\\delta_r(x)) = r\\,g(x)$ for \nall $x\\in\\mathbb{H}^n$ and all $r >0$. \n\n\\begin{defi}\n We say that a map $f:\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is Pansu-differentiable at $x \\in \\mathbb{H}^n$ if there exists \nan homogeneous group homomorphism $g :\\mathbb{H}^n \\rightarrow \\mathbb{R}$ such that\n\\begin{equation*}\n\\lim_{y \\rightarrow x} \\frac{f(y) - f(x) - g(x^{-1} \\cdot y)}{d(y,x)} = 0.\n\\end{equation*}\nThe map $g$ is then unique and will be denoted by $D_H f(x)$.\n\\end{defi}\n\nIf $f:\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is Pansu-differentiable at $x \\in \\mathbb{H}^n$ then the maps $s \\mapsto f(x \\cdot \\delta_s[e_j, 0])$, resp. $s \\mapsto f(x\\cdot \\delta_s[e_{n+j}, 0])$, are differentiable at $s=0$ and if we denote the corresponding derivatives by $X_jf(x)$, resp. $Y_jf(x)$, then \n\\begin{equation*}\nD_H f(x)(\\xi,\\eta,t) = \\sum_{j=1}^n \\xi_j X_jf(x) + \\eta_j Y_jf(x).\n\\end{equation*}\nHere $e_j = (\\delta_{1}^{j},\\dots,\\delta_{n}^{j})\\in \\mathbb{C}^n$ and $e_{n+j} = (i\\delta_{1}^{j},\\dots,i\\delta_{n}^{j})\\in \\mathbb{C}^n$. Using similar notations as in the classical smooth case, we then set $\\nabla_H f(x) := (X_1 f(x)+i Y_1 f(x), \\dots, X_n f(x)+i Y_n f(x))$.\n\n\\begin{thm}[Pansu-differentiability theorem] \\label{PansuRademacher} \\cite{pansu}\nLet $f:(\\mathbb{H}^n,d) \\rightarrow \\mathbb{R}$ be a $C$-Lipschitz function. Then, for $\\mathcal{L}^{2n+1}$-a.e. $x\\in\\mathbb{H}^n$, the function $f$ is Pansu-differentiable at $x$ and $|\\nabla_H f(x)|\\leq C$.\n\\end{thm}\n\nThe next lemma will be used to prove that any optimal transport plan is concentrated on the set $\\Omega$.\n\n\\begin{lem} \\label{uniquegeod}\n Let $u\\in \\text{Lip}_1(d)$, $x\\in\\mathbb{H}^n$ be such that $u$ is Pansu-differentiable at $x$ with $|\\nabla_H u(x)| \\leq 1$ and let $y\\in\\mathbb{H}^n$ be such that $u(x) - u(y) = d(x,y)$. Then there exists a unique minimal curve between $x$ and $y$.\n\\end{lem}\n\n\\begin{proof}\n Let $\\sigma:[0,1]\\rightarrow\\mathbb{H}^n$ be a minimal curve between $x$ and $y$. Then $\\sigma$ is a horizontal $C^1$-smooth curve and if $\\sigma(t) = (\\sigma_1(t),\\dots,\\sigma_{2n+1}(t)) \\in \\mathbb{H}^n \\thickapprox \\mathbb{R}^{2n+1}$, one has for all $t\\in [0,1]$,\n\\begin{equation*}\n \\dot\\sigma(t) = \\sum_{j=1}^n \\dot\\sigma_j(t)\\, X_j(\\sigma(t)) + \\dot\\sigma_{n+j}(t)\\, Y_j(\\sigma(t))\n\\end{equation*}\nand $|\\dot\\sigma_H(t)| = d(x,y)$ where $\\dot\\sigma_H(t) := (\\dot\\sigma_1(t) +i\\,\\dot\\sigma_{n+1}(t),\\dots,\\dot\\sigma_n(t) +i\\,\\dot\\sigma_{2n}(t))\\in\\mathbb{C}^n$. On the other hand, one has \n\\begin{equation*}\n u(x) - u(\\sigma(t)) = d(x,\\sigma(t)) = t\\, d(x,y)\n\\end{equation*}\nfor all $t\\in [0,1]$. Differentiating this equality with respect to $t$, we get\n\\begin{equation*}\n \\sum _{j=1}^n \\dot\\sigma_j(0)\\, X_ju(x) + \\dot\\sigma_{n+j}(0) \\, Y_ju(x)= \\dfrac{d}{dt}\\,u(\\sigma(t))_{|_{t=0}} = - d(x,y).\n\\end{equation*}\nAll together, it follows that \n\\begin{equation*}\nd(x,y) = |\\sum _{j=1}^n \\dot\\sigma_j(0)\\, X_ju(x) + \\dot\\sigma_{n+j}(0) \\, Y_ju(x)| \\leq  |\\nabla_H u(x)|\\, |\\dot\\sigma_H(0)| \\leq d(x,y).\n\\end{equation*}\nIn particular, there is equality in all the previous inequalities which implies in turn that $\\dot\\sigma_H(0) = - \\,d(x,y) \\,\\nabla_H u (x)$. On the other hand one knows from Theorem~\\ref{geod} that $\\sigma = x\\cdot \\sigma_{\\chi,\\varphi}$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and $\\varphi\\in\\, [-2\\pi,2\\pi]$. In particular one has $\\dot\\sigma_H(0) = \\chi$. It follows that $\\chi = - \\,d(x,y) \\,\\nabla_H u (x)$ is uniquely determined hence there is a unique minimal curve joining $x$ and $y$ according once again to the description given in Theorem~\\ref{geod}.\n\\end{proof}\n\n\n\\section{Properties of $\\Pi_1(\\mu,\\nu)$ and $\\Pi_2(\\mu,\\nu)$} \\label{sect:optplanning}\n\nLet $\\mu$, $\\nu \\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. We denote by $\\Pi_1(\\mu,\\nu)$ the set of optimal transport plans solution to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$. \n\nWe first prove some geometric properties of optimal transport plans. These properties follow from the behavior of minimal curves in $(\\mathbb{H}^n,d)$. In the next lemma, we prove that any optimal transport plan is concentrated on the set $\\Omega$ (see \\eqref{e:omega}) of pair of points that are connected by a unique minimal curve.\n\n\\begin{lem} \\label{pi1.1}\n Let $\\gamma\\in \\Pi_1(\\mu,\\nu)$ and assume that $\\mu\\ll\\mathcal{L}^{2n+1}$. Then for $\\gamma$-a.e. $(x,y)$, there exists a unique minimal curve between $x$ and $y$.\n\\end{lem}\n\n\\begin{proof}\nLet $u\\in \\text{Lip}_1(d)$ be a Kantorovich potential associated to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ (see Section~\\ref{opttrans} and Theorem~\\ref{1lip_potential} there). Since $u\\in \\text{Lip}_1(d)$, we know from Theorem \\ref{PansuRademacher} that for $\\mathcal{L}^{2n+1}$-a.e., and  hence $\\mu$-a.e., $x\\in\\mathbb{H}^n$, $u$ is Pansu-differentiable at $x$ with $|\\nabla_H u(x)|\\leq 1$. Then the conclusion follows from Lemma~\\ref{uniquegeod} since $u(x) - u(y) = d(x,y)$ for $\\gamma$-a.e. $(x,y)$ (see Theorem~\\ref{1lip_potential}). \n\\end{proof}\n\nThe next lemma says that minimal curves used by an optimal transport plan cannot bifurcate. It follows essentially from the non-branching property of $(\\mathbb{H}^n,d)$.\n\n\\begin{lem} \\label{pi1.2}\n Let $\\gamma\\in \\Pi_1(\\mu,\\nu)$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that the following holds. For all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ such that $x\\not=y$ and $x\\not=x'$, if $x'$ lies on a minimal curve between $x$ and $y$ then all points $x$, $x'$, $y$ and $y'$ lie on the same minimal curve. More precisely, there exists a minimal curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ such that $x=\\sigma(a)$, $y=\\sigma(t)$ for some $t\\in \\,(a,b]$, $x'=\\sigma(s)$ for some $s\\in\\, (a,t]$ and $y'=\\sigma(t')$ for some $t'\\in [s,b]$.\n\\end{lem}\n\n\\begin{proof} Let $(x,y)\\in \\mathbb{H}^n \\times \\mathbb{H}^n$ and $(x',y')\\in \\mathbb{H}^n \\times \\mathbb{H}^n$ such that $x\\not=y$ and $x'\\not=x$. Assume that $x'\\in \\sigma((0,d(x,y)])$ where $\\sigma:[0,d(x,y)] \\rightarrow \\mathbb{H}^n$ is a unit-speed minimal curve between $x$ and $y$. Let $\\sigma'$ be a unit-speed minimal curve between $x'$ and $y'$ parameterized on $[d(x,x'),d(x,x')+d(x',y')]$. Assume moreover that \n\\begin{equation*} \n d(x,y) + d(x',y') \\leq d(x,y')+d(x',y).\n\\end{equation*}\nRecall that this holds true for $\\gamma$-a.e. $(x,y)$ and $(x',y')$ by Theorem \\ref{ccycl}. Then the curve $\\tilde \\sigma :[0, d(x,x')+d(x',y')]\\rightarrow\\mathbb{H}^n$ which coincides with $\\sigma$ on $[0,d(x,x')]$ and $\\sigma'$ on $[d(x,x'),d(x,x')+d(x',y')]$ is a length minimizing curve between $x$ and $y'$. Indeed, otherwise we would have \n\\begin{equation*}\n d(x,y') < l(\\tilde\\sigma) = l(\\sigma_{|[0,d(x,x')]}) + l(\\sigma'_{|[d(x,x'),d(x,x')+d(x',y')]}) = d(x,x') + d(x',y').\n\\end{equation*}\nSince $x'$ lies on a minimal curve between $x$ and $y$, we have $d(x,x') + d(x',y) = d(x,y)$ and we get \n\\begin{equation*}\n  d(x,y') + d(x',y) < d(x,y) + d(x',y')\n\\end{equation*}\nwhich gives a contradiction. It follows that $\\sigma$ and $\\tilde\\sigma$ are unit-speed minimal curves that coincide on the non trivial interval $[0,d(x,x')]$. Since $\\mathbb{H}^n$ is non-branching (see Proposition~\\ref{nonbranching}), this implies that $\\sigma$ and $\\tilde \\sigma$ are sub-arcs of the same minimal curve, namely $\\sigma$ if $d(x,y') \\leq d(x,y)$ and $\\tilde\\sigma$ otherwise, on which all points $x$, $x'$, $y$ and $y'$ lie. And the conclusion follows.\n\\end{proof}\n\nWe denote by $\\Pi_2(\\mu,\\nu)$ the set of transport plans solution to the secondary variational problem:\n\\begin{equation*}\n\\min_{\\gamma \\in \\Pi_1(\\mu,\\nu)} \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y).\n\\end{equation*}\n\nOptimal transport plans selected through the variational approximations to be introduced in Section~\\ref{varapprox} will be solution to this secondary variational problem. The next lemma gives a one-dimensional monotonicity condition along minimal curves used by optimal transport plans in $\\Pi_2(\\mu,\\nu)$. This follows essentially from a constrained version of $d^2$-cyclical monotonicity.\n\n\\begin{lem} \\label{pi2}\nLet $\\gamma\\in \\Pi_2(\\mu,\\nu)$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that the following holds. For all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ such that $x\\not=y$ and $x\\not=x'$, if $x'$ lies on a minimal curve between $x$ and $y$ then all points $x$, $x'$, $y$ and $y'$ lie on the same minimal curve ordered in that way.\n\\end{lem}\n\nIn other words, there exists a minimal curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ such that $\\sigma(a) = x$, $\\sigma(t)=y$ for some $t\\in \\,(a,b]$, $\\sigma(s) = x'$ for some $s\\in\\, (a,t]$ and $\\sigma(t') = y'$ for some $t'\\in [t,b]$.\n\n\\begin{proof} First, as a classical fact, one can rephrase the secondary variational problem as a classical Kantorovich transport problem \\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = \\beta(x,y)$ with \n\\begin{equation*}\n\\beta(x,y) = \\begin{cases}\n              d(x,y)^2 \\quad \\text{if } u(x)-u(y) = d(x,y),\\\\\n              +\\infty \\qquad \\phantom{\\text{if}} \\text{otherwise},\n             \\end{cases}\n\\end{equation*}\nwhere $u\\in \\text{Lip}_1(d)$ is a Kantorovich potential associated to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ (see Section~\\ref{opttrans} and Theorem~\\ref{1lip_potential} there). Since $\\beta$ is lower semicontinuous and $\\int_{\\mathbb{H}^n\\times \\mathbb{H}^n} \\beta(x,y) \\,d\\gamma(x,y)<+\\infty$ for all $\\gamma \\in \\Pi_2(\\mu,\\nu)$, it follows from Theorem \\ref{ccycl} that any $\\gamma \\in \\Pi_2(\\mu,\\nu)$ is concentrated on a $\\beta$-cyclically monotone set. So, taking into account the fact that $\\Pi_2(\\mu,\\nu)\\subset\\Pi_1(\\mu,\\nu)$, we know that $\\gamma \\in \\Pi_2(\\mu,\\nu)$ is concentrated on a set $\\Gamma$ such that \n\\begin{equation*}\n u(x) - u(y) = d(x,y)\n\\end{equation*}\nfor all $(x,y)\\in \\Gamma$,\n\\begin{equation*}\n \\beta(x,y) +\\beta(x',y') \\leq \\beta(x,y')+\\beta(x',y)\n\\end{equation*}\nfor all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ and the conclusion of Lemma \\ref{pi1.2} holds.\n\nThen let $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ be as in the statement. By Lemma \\ref{pi1.2}, the conclusion will follow if we show that $d(x',y) \\leq d(x',y')$. First we check that $\\beta(x',y) = d(x',y)^2$ and $\\beta(x,y') = d(x,y')^2$. We have \n\\begin{equation*}\nu(x) \\leq u(x') + d(x,x') \\leq u(y) + d(x',y) + d(x,x') = d(x,y) + u(y) = u(x)\n\\end{equation*}\nhence all these inequalities are equalities. In particular, we get that $u(x') = u(y) + d(x',y)$ hence $\\beta(x',y) = d(x',y)^2$. We also get that  \n\\begin{equation*}\n u(x) = d(x,x') + u(x') = d(x,x') + u(y') + d(x',y') = u(y') + d(x,y')\n\\end{equation*}\nhence $\\beta(x,y') = d(x,y')^2$. If $d(x',y')< d(x',y)$, we  get\n\\begin{equation*}\n\\begin{split}\n \\beta(x,y') + &\\beta(x',y) - \\beta(x',y') - \\beta(x,y)\\\\\n&= d(x,y')^2  + d(x',y)^2 - d(x',y')^2 - d(x,y)^2 \\\\\n&= (d(x,x')+d(x',y'))^2 + d(x',y)^2 - d(x',y')^2 - (d(x,x')+d(x',y))^2\\\\\n&= 2 \\, d(x,x') (d(x',y') - d(x',y))<0\n\\end{split}\n\\end{equation*}\nwhich gives a contradiction.\n\\end{proof}\n\n\n\\section{Variational approximations} \\label{varapprox}\n\nWe introduce variational approximations in the spirit of \\cite{ap} (see also \\cite{cfm}, \\cite{akp}) by rephrasing in our geometrical context the variational approximations considered recently in \\cite{sant}. This approximation procedure will be used to select optimal transport plans that will be eventually proved to be induced by transport maps.\n\nLet $\\mu$, $\\nu \\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Let $K$ be a compact subset of $\\mathbb{H}^n$ such that $\\operatorname{spt}{\\mu}\\cup\\operatorname{spt}{\\nu}\\subset K$ and set\n\\begin{equation*}\n \\Pi := \\{\\gamma \\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n);\\, (\\pi_1)_\\sharp\\gamma=\\mu,\\, \\operatorname{spt}{(\\pi_2)_\\sharp\\gamma} \\subset K\\}.\n\\end{equation*}\nFor $\\varepsilon>0$ fixed and $\\gamma \\in \\Pi$, we set\n\\begin{multline*}\n C_\\varepsilon(\\gamma) := \n\\frac{1}{\\varepsilon}\\, W_1((\\pi_2)_\\sharp\\gamma,\\nu) + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y) \\\\ +  \\varepsilon \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y) + \\varepsilon^{6n+8} \\operatorname{card}{(\\operatorname{spt}{(\\pi_2)_\\sharp\\gamma})}\n\\end{multline*}\nand consider the family of minimization problems:\n\\begin{equation} \\label{e:varapprox}\n \\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}. \\tag{$P_\\varepsilon$}\n\\end{equation}\nHere $W_1$ denotes the 1-Wasserstein distance defined for any two probability measures $\\mu_1$, $\\mu_2 \\in \\mathcal P(\\mathbb{H}^n)$ by \n\\begin{equation*}\n W_1(\\mu_1,\\mu_2) := \\min_{\\gamma \\in \\Pi(\\mu_1,\\mu_2)} \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y).\n\\end{equation*}\nFirst we note that \\eqref{e:varapprox} always admits solutions.\n\n\\begin{thm}\n For any $\\varepsilon>0$, the problem \\eqref{e:varapprox} admits at least one solution and $\\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}<+\\infty$.\n\\end{thm}\n\n\\begin{proof}\nFirst note that since $K$ is compact, $C_\\varepsilon(\\gamma)<+\\infty$ for any $\\gamma\\in \\Pi$ such that $(\\pi_2)_\\sharp\\gamma$ is finitely atomic. Next the existence of solutions to \\eqref{e:varapprox} follows from the weak compactness of $\\Pi$, the lower semicontinuity of the three first terms to be minimized and the Kuratowski convergence of the supports of weakly converging probability measures (see \\cite[Chapter~5]{ags}).\n\\end{proof}\n\nNext, weak limits of solutions to \\eqref{e:varapprox} are optimal transport plans that are solutions to the secondary variational problem introduced in Section~\\ref{sect:optplanning} to which we refer for the definition of $\\Pi_2(\\mu,\\nu)$. Modulo minor modifications due to our geometrical context, this can be proved with the same arguments as those given in \\cite{sant}. \n\n\\begin{lem} \\label{optpi2}\n Let $\\varepsilon_k$ be a sequence converging to 0 and $\\gamma_{\\varepsilon_k}$ a sequence of solutions to $(P_{\\varepsilon_k})$ which is weakly converging to some $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$. Then $\\gamma \\in \\Pi_2(\\mu,\\nu)$.\n\\end{lem}\n\n\\begin{proof} First we note that for any $m\\geq 1$, one can find a finite set $F_m\\subset K$ such that $\\operatorname{card} F_m \\leq C\\, m^{2n+2}$ for some constant $C>0$ which depends only on $n$ and $\\operatorname{diam} K$ and a Borel map $p_m:K \\rightarrow F_m$ such that\n\\begin{equation*}\n d(p_m(x),x) < 1\/m  \n\\end{equation*}\nfor all $x\\in K$. Indeed choose $x_1 \\in K$. For $i\\geq 2$, choose by induction $x_i \\in K\\setminus \\cup_{j<i} B(x_j,1\/m)$ as long as $K\\setminus \\cup_{j<i} B(x_j,1\/m) \\not=\\emptyset$. Let $F_m$ denote the set of all these points. The balls $B(x_i,1\/(2m))$ are mutually disjoint and $\\cup_{i} B(x_i,(1\/(2m)) \\subset B(x_1,\\operatorname{diam} K + 1)$. Remembering \\eqref{e:measball}, it follows that\n\\begin{equation*}\n\\begin{split}\n c_n (2m)^{-2n-2} \\operatorname{card} F&\\leq \\mathcal{L}^{2n+1}(\\cup_i B(x_i,1\/(2m)))\\\\\n&\\leq \\mathcal{L}^{2n+1}(B(x_1,\\operatorname{diam} K +1)) = c_n (\\operatorname{diam} K + 1)^{2n+2}\n\\end{split}\n\\end{equation*}\nfor any finite subset $F\\subset F_m$, hence $F_m$ is a finite set with $\\operatorname{card} F_m \\leq C\\, m^{2n+2}$ where $C$ depends only on $n$ and $\\operatorname{diam} K$. Next, by construction, for any $x \\in K$, there exists a unique $x_i\\in F_m$ such that $x \\in B(x_i,1\/m)\\setminus \\cup_{j<i} B(x_j,1\/m)$ and we then set $p_m(x) := x_i$. \n\nThe proof of the lemma can now be completed following the same arguments as those in \\cite{sant}. For sake of completeness, we sketch these arguments below. Let $\\gamma_{\\varepsilon_k}$ and $\\gamma$ be as in the statement. For $m\\geq 1$, one sets $\\nu_m:= (p_m)_\\sharp\\nu$. Note that by construction of $F_m$ and $p_m$, one has $\\operatorname{card}{(\\operatorname{spt}{\\nu_m})} \\leq  C\\, m^{2n+2}$ and $W_1(\\nu_m,\\nu) \\leq 1\/m$. To check that $\\gamma \\in \\Pi(\\mu,\\nu)$, one takes some $\\gamma_m \\in \\Pi(\\mu,\\nu_m)\\subset \\Pi$ and uses the optimality of $\\gamma_{\\varepsilon_k}$ which implies \n\\begin{equation*}\n\\begin{split}\n W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu) &\\leq \\varepsilon_k \\, C_{\\varepsilon_k}(\\gamma_m) \\\\\n&\\leq \\dfrac{1}{m} + \\varepsilon_k \\operatorname{diam}(K) + \\varepsilon_k^2 \\operatorname{diam}(K)^2 + C \\varepsilon_k^{6n+9} m^{2n+2}.\n\\end{split}\n\\end{equation*}\nThen one lets $\\varepsilon_k \\rightarrow 0$ with $m\\geq 1$ fixed and then $m \\rightarrow +\\infty$ to get that $(\\pi_2)_\\sharp\\gamma_{\\varepsilon_k}$ converges weakly to $\\nu$. Since it also converges weakly to $(\\pi_2)_\\sharp\\gamma$, it follows that $\\gamma \\in \\Pi(\\mu,\\nu)$. \n\nTo check that $\\gamma \\in \\Pi_1(\\mu,\\nu)$, one takes $\\overline\\gamma \\in \\Pi_1(\\mu,\\nu)$ and sets $\\overline\\gamma_m := (Id,p_m)_\\sharp\\overline\\gamma \\in \\Pi(\\mu,\\nu_m)$ where $(Id,p_m)(x,y) = (x,p_m(y))$. By optimality of $\\gamma_{\\varepsilon_k}$, one has \n\\begin{equation*}\n\\begin{split}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)&\\,d\\gamma_{\\varepsilon_k}(x,y) \\leq  C_{\\varepsilon_k}(\\overline\\gamma_m)\\\\\n&\\leq \\dfrac{1}{m \\,\\varepsilon_k} + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma_m(x,y) + \\varepsilon_k \\operatorname{diam}(K)^2 + C \\varepsilon_k^{6n+8} m^{2n+2}.\n\\end{split}\n\\end{equation*}\nChoosing $m$ of the order of $\\varepsilon_k^{-2}$ and letting $\\varepsilon_k \\rightarrow 0$, one gets \n\\begin{equation*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y) \\leq  \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma(x,y)\n\\end{equation*}\nhence $\\gamma \\in \\Pi_1(\\mu,\\nu)$. \n\nFinally, to check that $\\gamma \\in \\Pi_2(\\mu,\\nu)$, one uses once again the optimality of $\\gamma_{\\varepsilon_k}$ in the following way,\n\\begin{multline*}\n  W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu) + \\varepsilon_k \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma_{\\varepsilon_k}(x,y) + \\varepsilon_k^2 \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma_{\\varepsilon_k}(x,y) \\\\\\leq \\varepsilon_k \\, C_{\\varepsilon_k}(\\overline\\gamma_m) \n\\end{multline*}\nOn the other hand, one has \n\\begin{gather*}\n\\begin{split}\n W_1(\\mu,\\nu) &\\leq W_1(\\mu,(\\pi_2)_\\sharp\\gamma_{\\varepsilon_k}) + W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu)\\\\\n& \\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma_{\\varepsilon_k}(x,y)  + W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu)\n\\end{split}\\\\\n\\intertext{and}\n\\begin{split}\n\\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma_m(x,y) &= \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,p_m(y))\\,d\\overline\\gamma(x,y)\\\\\n&\\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma(x,y) + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(y,p_m(y))\\,d\\overline\\gamma(x,y) \\\\\n&\\leq  W_1(\\mu,\\nu) + \\frac{1}{m}.\n\\end{split}\n\\end{gather*}\nIt follows that \n\\begin{multline*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma_{\\varepsilon_k}(x,y) \\\\ \\leq \\frac{1}{m\\, \\varepsilon_k^2} + \\frac{1}{m\\, \\varepsilon_k} + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\overline\\gamma_m(x,y) + C \\varepsilon_k^{6n+7} m^{2n+2}\n\\end{multline*}\nprovided $\\varepsilon_k\\leq 1$. Choosing $m$ of the order of $\\varepsilon_k^{-3}$ and letting $\\varepsilon_k \\rightarrow 0$, one gets\n\\begin{equation*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y) \\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\overline\\gamma(x,y).\n\\end{equation*}\nSince $\\overline\\gamma \\in \\Pi_1(\\mu,\\nu)$ was arbitrary, it follows that $\\gamma \\in \\Pi_2(\\mu,\\nu)$.\n\\end{proof}\n\\medskip\n\nThe rest of this section is devoted to the study of the solutions to ($P_\\varepsilon$). We fix $\\varepsilon>0$ and set \n\\begin{equation*}\n c_\\varepsilon (x,y) := d(x,y) + \\varepsilon\\, d(x,y)^2.\n\\end{equation*}\nWe first recall the following classical fact.\n\n\\begin{lem} \\label{restrictions}\nLet $\\gamma_\\varepsilon$ be a solution to ($P_\\varepsilon$). Then for any Borel set $U\\subset\\mathbb{H}^n\\times\\mathbb{H}^n$, $(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U)$ is finitely atomic and  $\\gamma_\\varepsilon\\lfloor U$ is a solution to Kantorovich transport problem \\eqref{e:MK} between $(\\pi_1)_\\sharp (\\gamma_\\varepsilon\\lfloor U)$ and $(\\pi_2)_\\sharp (\\gamma_\\varepsilon\\lfloor U)$ with  cost $c_\\varepsilon$.\n\\end{lem}\n\n\\begin{proof}\n The fact that $(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U)$ is finitely atomic obviously follows from the fact that $C_\\varepsilon(\\gamma_\\varepsilon) = \\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}<+\\infty$. Next it is also immediate that $\\gamma_\\varepsilon$ is a solution to Kantorovich transport problem \\eqref{e:MK} between $\\mu$ and $(\\pi_2)_\\sharp (\\gamma_\\varepsilon)$ with cost $c_\\varepsilon$. Then as a classical fact, the claim follows from the linearity of the functional to be minimized with respect to the transport plan. If $\\gamma \\in \\Pi((\\pi_1)_\\sharp (\\gamma_\\varepsilon\\lfloor U),(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U))$, one indeed simply compare $C_\\varepsilon(\\gamma_\\varepsilon)$ with $C_\\varepsilon(\\hat\\gamma)$ where $ \\hat\\gamma = \\gamma_\\varepsilon\\lfloor (\\mathbb{H}^n\\times\\mathbb{H}^n)\\setminus U + \\gamma \\in \\Pi(\\mu,(\\pi_2)_\\sharp(\\gamma_\\varepsilon))$ to get the conclusion.\n\\end{proof}\n\nNext in this section we consider interpolations between two measures $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ that are constructed from a transport plan solution to Kantorovich transport problem \\eqref{e:MK} between these two measures with cost $c_\\varepsilon$. We prove absolute continuity and, more importantly, $L^\\infty$-estimates on the density with respect to $\\mathcal{L}^{2n+1}$ of these interpolations whenever $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic, see Proposition~\\ref{e:Linftydensityestimates}. We divide the arguments into several steps. First we prove that any solution to this Kantorovich transport problem is induced by a transport.\n\n\\begin{thm}  \\label{existenceceps}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Assume that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and that $\\overline \\nu$ is finitely atomic. Then any solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ is induced by a transport. In particular there exists a unique optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$.\n\\end{thm}\n\n\\begin{proof} Let $\\psi$ be a Kantorovich potential for Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ given by Theorem~\\ref{duality}. Let $\\{y_i\\}_{i=1}^k$ denote the atoms of $\\overline \\nu$. We prove that for $\\mathcal{L}^{2n+1}$-a.e. $x\\in\\mathbb{H}^n$, there is at most one point $y_i$ for some $i\\in\\{1,\\dots,k\\}$ such that\n\\begin{equation*} \n \\psi(x) + \\psi^c(y_i) = c_\\varepsilon (x,y_i).\n\\end{equation*}\nSince $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$, it will follow that any transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ is concentrated on a $\\overline\\mu$-measurable graph and hence induced by a transport. This implies in turn existence and uniqueness of the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$ (see Theorem~\\ref{plan-transport}). \n\nFor $i\\not= j$, set $h_{ij}(x) := c_\\varepsilon (x,y_i) - c_\\varepsilon (x,y_j) + \\psi^c(y_j) - \\psi^c(y_i)$. It follows from Lemma~\\ref{prop-distcc} that $h_{ij}$ is of class $C^\\infty$ on the open set $\\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j})$ with $\\nabla h_{ij} \\not= 0$. Indeed assume on the contrary that $\\nabla h_{ij}(x) =0$ for some $x\\in \\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j})$. Then, differentiating along the horizontal vector fields $X_j$ and $Y_j$, we would have\n\\begin{equation*} \n \\nabla_H d_{y_i} (x) \\,(1+2\\varepsilon\\, d_{y_i}(x)) =  \\nabla_H d_{y_j} (x) \\,(1+2\\varepsilon \\,d_{y_j}(x)).\n\\end{equation*}\nSince $|\\nabla_H d_{y_i} (x) | = |\\nabla_H d_{y_j} (x) |$ (see Lemma~\\ref{prop-distcc}(i)), this would imply that $d_{y_i}(x) = d_{y_j}(x)$ and in turn that $\\nabla_H d_{y_i} (x) = \\nabla_H d_{y_j} (x)$. Since we also have by assumption $\\partial_t d_{y_i} (x) = \\partial_t d_{y_j} (x)$, Lemma~\\ref{prop-distcc}(ii) would give $y_i=y_j$. It follows that the set $\\{x\\in\\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j});\\, h_{ij}(x) =0\\}$ is a $C^\\infty$-smooth submanifold of dimension $2n$ in $\\mathbb{R}^{2n+1}$ and hence has Lebesgue measure 0. Since $\\mathcal{L}^{2n+1}(L_{y_i}) =0$, it follows that \n\\begin{equation*} \n \\mathcal{L}^{2n+1} (\\bigcup_{i\\not=j} \\{x\\in\\mathbb{H}^n;\\,h_{ij}(x) =0\\})  = 0\n\\end{equation*}\nand the claim follows.\n\\end{proof}\n\nIf $T:\\mathbb{H}^n \\rightarrow \\mathbb{H}^n$, we set $T_t = e_t \\circ S \\circ (I \\otimes T)$, i.e., $T_t(x)$ is the point lying at distance $t \\, d(x,T(x))$ from $x$ on the selected minimal curve $S(x,T(x))$ between $x$ and $T(x)$ (see Subsection~\\ref{sect-interpolation} for the definition of $S$ and $e_t$).\n\n\\begin{prop} \\label{injectivity} \\cite[Chapter 7]{villani}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed such that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic. Let $T^\\varepsilon$ be the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$. Then there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) = 1$ and such that for each $t\\in\\, [0,1)$, $T^\\varepsilon_t\\lfloor_A$ is injective.\n\\end{prop}\n\nThe cost $c_\\varepsilon$ can be recovered as coming from a so-called coercive Lagrangian action. Since $(\\mathbb{H}^n,d)$ is non-branching, the proposition essentially follows from~\\cite[Chapter 7, Theorem 7.30]{villani}. However one does not need the full strength of the theory developed in~\\cite[Chapter 7]{villani} to get the conclusion of Proposition~\\eqref{injectivity} and we sketch below the arguments for the reader's convenience. \n\n\\begin{proof} Let $0\\leq s<t \\leq 1$ and $x$, $y\\in \\mathbb{H}^n$. Set \n\\begin{equation*} \nc_\\varepsilon^{s,t} (x,y) = d(x,y) + \\varepsilon \\, \\dfrac{d(x,y)^2}{t-s}.\n\\end{equation*}\nThe space $(\\mathbb{H}^n,d)$ being a geodesic space and $u \\mapsto u + \\varepsilon\\,u^2$ being strictly increasing and strictly convex on $[0,+\\infty )$, one has \n\\begin{equation*} \n c_\\varepsilon (x,y) \\leq c_\\varepsilon^{0,t} (x,z) + c_\\varepsilon^{t,1} (z,y)\n\\end{equation*}\nfor all $x$, $y$, $z\\in \\mathbb{H}^n$ and $t\\in (0,1)$, with equality if and only if any curve in $C([0,1],\\mathbb{H}^n)$ obtained by concatenation of a minimal curve $\\sigma_{x,z}:[0,t]\\rightarrow\\mathbb{H}^n$ between $x$ and $z$ and a minimal curve $\\sigma_{z,y}:[t,1]\\rightarrow\\mathbb{H}^n$ between $z$ and $y$ is a minimal curve between $x$ and $y$.\n\nOn the other hand, by $c_\\varepsilon$-cyclical monotonicity, one knows that there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) =1$ and \n\\begin{equation*} \n c_\\varepsilon(x,T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x)) \\leq c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(x))\n\\end{equation*}\nfor all $x$, $\\tilde x \\in A$ (see Theorem~\\ref{ccycl}).\n\nNow let $t\\in (0,1)$ be fixed and let $x$, $\\tilde x \\in A$. Assume that $T^\\varepsilon_t(x) = T^\\varepsilon_t(\\tilde x)$. Then \n\\begin{equation} \n c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) \\leq c_\\varepsilon^{0,t} (x,z) + c_\\varepsilon^{t,1} (z,T^\\varepsilon(\\tilde x)),\\label{e:injectivity}\n\\end{equation}\nand similarly,\n\\begin{equation*} \nc_\\varepsilon(\\tilde x,T^\\varepsilon(x)) \\leq c_\\varepsilon^{0,t} (\\tilde x,z) + c_\\varepsilon^{t,1} (z,T^\\varepsilon(x))\n\\end{equation*}\nwhere $z = T^\\varepsilon_t(x) = T^\\varepsilon_t(\\tilde x)$. It follows that \n\\begin{equation*} \n\\begin{split}\nc_\\varepsilon(x,& T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x))\\\\ &\\leq c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(x))\\\\\n&\\leq c_\\varepsilon^{0,t} (x,T^\\varepsilon_t(x)) + c_\\varepsilon^{t,1} (T^\\varepsilon_t(x),T^\\varepsilon(x)) + c_\\varepsilon^{0,t} (\\tilde x,T^\\varepsilon_t(\\tilde x)) + c_\\varepsilon^{t,1} (T^\\varepsilon_t(\\tilde x),T^\\varepsilon(\\tilde x))\\\\\n&= c_\\varepsilon(x,T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x)).\n\\end{split}\n\\end{equation*}\nHence equality has to hold in all these inequalities. In particular equality holds in \\eqref{e:injectivity}. It follows that the curve obtained by concatenation of the minimal curve $s\\in [0,t] \\mapsto e_s(S(x,T^\\varepsilon(x)))$ between $x$ and $z$ with the minimal curve $s \\in [t,1] \\mapsto e_s(S(\\tilde x,T^\\varepsilon(\\tilde x)))$ between $z$ and $T^\\varepsilon(\\tilde x)$ is a minimal curve. Since this curve coincide with the minimal curve $\\sigma: s\\in [0,1] \\mapsto e_s(S(x,T^\\varepsilon(x)))$ on the non trivial interval $[0,t]$ and $\\mathbb{H}^n$ is non-branching (see Proposition~\\ref{nonbranching}), we get that it coincides with $\\sigma$ on the whole interval $[0,1]$. Similarly, it coincides with the minimal curve $\\tilde \\sigma: s\\in [0,1] \\mapsto e_s(S(\\tilde x,T^\\varepsilon(\\tilde x)))$ on the whole interval $[0,1]$. Hence $\\sigma = \\tilde \\sigma$ and in particular $x=\\sigma(0) = \\tilde \\sigma (0) = \\tilde x$.\n\\end{proof}\n\nWe turn now to the main estimate that will lead to Proposition~\\ref{e:Linftydensityestimates}.\n\n\\begin{prop} \\label{estimsup}\n Let $\\overline\\mu \\in \\mathcal P(\\mathbb{H}^n)$, $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$, $\\overline\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$, and $T:\\mathbb{H}^n\\rightarrow\\mathbb{H}^n$ be a $\\overline\\mu$ - measurable map such that $T_\\sharp\\overline\\mu$ is finitely atomic. Let $t\\in\\,(0,1)$ and set $\\overline\\mu_t := T_{t\\,\\sharp}\\overline\\mu$. Assume that there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) = 1$ and $T_t\\lfloor_A$ is injective. Then $\\overline\\mu_t \\ll \\mathcal{L}^{2n+1}$, $\\overline\\mu_t = \\rho_t \\,d\\mathcal{L}^{2n+1}$ with \n\\begin{equation*}\n \\rho_t \\leq \\frac{{\\rm 1\\mskip-4mu l}_{T_t(A)}}{(1-t)^{2n+3}} \\,\\rho \\circ T_t^{-1}\\lfloor_{T_t(A)} \\qquad \\mathcal{L}^{2n+1}-a.e.\n\\end{equation*}\n\\end{prop}\n\nArguments for the proof of this proposition can be found in \\cite[Section~3]{figjuil} even-though not explicitly stated in the same way in that paper. They rely on the following estimate:\n\\begin{equation*}\n \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}} \\,\\mathcal{L}^{2n+1}((e_t \\circ S)(E,y))\n\\end{equation*}\nfor any $y\\in\\mathbb{H}^n$ and $E\\subset \\mathbb{H}^n$ which is proved in \\cite[Section~2]{juillet} and which roughly means that $(\\mathbb{H}^n,d,\\mathcal{L}^{2n+1})$ satisfies a  so-called Measure Contraction Property. We detail the proof below for the reader's convenience.\n\n\\begin{proof} \n Let $\\{y_i\\}_{i=1}^k$ denote the atoms of $T_\\sharp\\overline\\mu$. Set $A_i = T^{-1}(\\{y_i\\}) \\cap A$ and $\\hat A = \\cup_i A_i$. The sets $A_i$ are mutually disjointed and $\\overline\\mu(\\hat A)=1$ by hypothesis. For any $x\\in A_i$, $T_t(x) = (e_t \\circ S) (x,y_i)$, hence\n\\begin{equation*}\n \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}} \\,\\mathcal{L}^{2n+1}(T_t(E))\n\\end{equation*}\nfor any $E\\subset A_i$. Next if $E\\subset \\hat A$, writing $E = \\cup_i (E\\cap A_i)$ where the sets $A_i$ are mutually disjointed and remembering that $T_t$ is injective on $\\hat A\\subset A$ by hypothesis, one gets\n\\begin{equation*}\n  \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}}\\, \\mathcal{L}^{2n+1}(T_t(E)).\n\\end{equation*}\nIt follows that for any $F\\subset \\mathbb{H}^n$,\n\\begin{equation*} \\label{e:1}\n \\mathcal{L}^{2n+1}(T_t^{-1}(F)\\cap \\hat A) \\leq \\frac{1}{(1-t)^{2n+3}} \\mathcal{L}^{2n+1}(F\\cap T_t(\\hat A)).\n\\end{equation*}\n\nAssume that $F\\subset \\mathbb{H}^n$ is such that $\\mathcal{L}^{2n+1}(F)=0$. We get $\\mathcal{L}^{2n+1}(T_t^{-1}(F)\\cap \\hat A) = 0$ from the previous inequality. On the other hand $\\overline\\mu_t(F) = \\overline \\mu (T_t^{-1}(F)) = \\overline \\mu (T_t^{-1}(F)\\cap \\hat A)$ hence $\\overline\\mu_t(F)=0$ since $\\overline\\mu\\ll\\mathcal{L}^{2n+1}$ and it follows that $\\overline\\mu_t \\ll \\mathcal{L}^{2n+1}$. \n\nNext, to prove the estimate on the density of $\\overline\\mu_t$ with respect to $\\mathcal{L}^{2n+1}$, we note that the inequality above implies that \n\\begin{equation*}\n \\int_{\\hat A} g(T_t) \\, d\\mathcal{L}^{2n+1} \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} g \\, d\\mathcal{L}^{2n+1}\n\\end{equation*}\nfor any non negative measurable map $g:\\mathbb{H}^n\\rightarrow [0,+\\infty]$. Let $h:\\mathbb{H}^n\\rightarrow [0,+\\infty]$ be a non negative measurable map and set \n\\begin{equation*}\n g(x) = {\\rm 1\\mskip-4mu l}_{T_t(\\hat A)}(x) \\, h(x) \\, \\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)).\n\\end{equation*}\nThen \n\\begin{equation*}\n \\int_{\\hat A} g(T_t(x)) \\, d\\mathcal{L}^{2n+1}(x) \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} h(x) \\,\\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)) \\, d\\mathcal{L}^{2n+1}(x)\n\\end{equation*}\nOn the other hand\n\\begin{equation*}\n \\int_{\\hat A}  g(T_t) \\, d\\mathcal{L}^{2n+1} = \\int_{\\hat A} h(T_t)\\, \\rho \\, d\\mathcal{L}^{2n+1} = \\int_{\\mathbb{H}^n} h(T_t)\\, d\\overline\\mu = \\int_{\\mathbb{H}^n} h\\, d\\overline\\mu_t,\n\\end{equation*}\nhence\n\\begin{equation*}\n \\int_{\\mathbb{H}^n} h(x)\\, d\\overline\\mu_t(x) \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} h(x) \\, \\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)) \\, d\\mathcal{L}^{2n+1}(x).\n\\end{equation*}\nRemembering that $\\hat A \\subset A$, this concludes the proof.\n\\end{proof}\n\nFinally, combining Theorem~\\ref{existenceceps}, Propositions~\\ref{injectivity} and \\ref{estimsup}, we get the following proposition which gives the absolute continuity of the interpolations together with an $L^\\infty$-estimate on their density. Note that if $\\gamma_\\varepsilon$ is the transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$ and $T^\\varepsilon$ the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$, which hence induces $\\gamma_\\varepsilon$, then $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon=T_{t\\,\\sharp}^\\varepsilon\\overline\\mu$.\n\n\\begin{prop} \\label{e:Linftydensityestimates}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Assume that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ with $\\overline\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic. Let $\\gamma_\\varepsilon$ be the transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$. Then for any $t\\in [0,1)$, the interpolation $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon$ is absolutely continuous with respect to $\\mathcal{L}^{2n+1}$, $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon = \\rho_t^\\varepsilon \\,d\\mathcal{L}^{2n+1}$, and one has \n\\begin{equation*}\n \\|\\rho_t^\\varepsilon\\|_{L^\\infty} \\leq \\frac{1}{(1-t)^{2n+3}}\\, \\|\\rho\\|_{L^\\infty}.\n\\end{equation*}\n\\end{prop}\n\n\n\n\n\\section{Properties of measures $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ with $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$} \\label{lebpoints}\n\nThis section is independent of the transport problem. We state some properties of measures in $\\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ with first marginal absolutely continuous with respect to $\\mathcal{L}^{2n+1}$. These properties are essential steps in the strategy adopted here to solve Monge's transport problem. They are the exact counterpart in the framework of $(\\mathbb{H}^n, d, \\mathcal{L}^{2n+1})$ of similar properties proved in~\\cite{champion-dePascale} in $\\mathbb{R}^n$. These properties hold actually true in more general settings, for instance in any separable doubling metric measure space.\n\nWe first recall some facts about Lebesgue points of Borel functions and density of absolutely continuous measures. Since the measure $\\mathcal{L}^{2n+1}$ is a doubling measure on $(\\mathbb{H}^n,d)$, see \\eqref{e:measball}, if $\\rho:\\mathbb{H}^n \\rightarrow [0,+\\infty]$ is a $\\mathcal{L}^{2n+1}$-locally summable Borel function then for $\\mathcal{L}^{2n+1}$-a.e. $x\\in \\mathbb{H}^n$, one has \n\\begin{equation} \\label{e:lebpoint}\n \\lim_{r\\rightarrow 0} \\dfrac{1}{\\mathcal{L}^{2n+1}(B(x,r))} \\, \\int_{B(x,r)} |\\rho(y) - \\rho(x)|\\,d\\mathcal{L}^{2n+1}(y) = 0,\n\\end{equation}\nsee e.g. \\cite{heinonen}. A point $x\\in \\mathbb{H}^n$ where \\eqref{e:lebpoint} holds is called a Lebesgue point of $\\rho$ and we denote by $\\operatorname{Leb} \\rho$ the set of all Lebesgue points of $\\rho$. \n\nIn rest of this paper, and especially in the lemma to follow, it will be technically convenient to consider the density $\\rho$ of an absolutely continuous measure $\\mu \\ll \\mathcal{L}^{2n+1}$ as a $\\mathcal{L}^{2n+1}$-summable Borel function, i.e., a function well-defined everywhere, so that one can speak about its Lebesgue points and its value at any arbitrary point without any ambiguity. If $\\mu\\in \\mathcal P(\\mathbb{H}^n)$, we set \n\\begin{equation} \\label{e:density}\n \\rho(x) := \\limsup_{r\\rightarrow 0} \\dfrac{\\mu(B(x,r))}{\\mathcal{L}^{2n+1}(B(x,r))}\\,.\n\\end{equation}\nThis map $\\rho:\\mathbb{H}^n \\rightarrow [0,+\\infty]$ is a Borel map and if $\\mu\\ll \\mathcal{L}^{2n+1}$ then $\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$. By a slight abuse of terminology, when speaking about the density of an absolutely continuous measure $\\mu\\in \\mathcal P(\\mathbb{H}^n)$ with respect to $\\mathcal{L}^{2n+1}$, we will thus always refer in the following to the Borel function $\\rho$ defined above.\n\nThe next lemma will be an essential ingredient in the proof of Lemma~\\ref{mainlemma}. \n\n\\begin{lem} \\label{dens1}\n Let $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ be such that $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that, for all $(x,y)\\in  \\Gamma$ and all $r>0$, there exist $y'\\in\\mathbb{H}^n$ and $r'>0$ such that\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n \\item $y \\in B(y',r') \\subset\\subset B(y,r)$,\n\n \\item $x\\in \\operatorname{Leb}{\\rho}$ and $\\rho(x) <+\\infty$,\n\n \\item $x\\in \\operatorname{Leb}{\\rho'}$ and $\\rho'(x)>0$,\n\\end{enumerate}\nwhere $\\rho$ denotes the density of $(\\pi_1)_\\sharp\\gamma$ and $\\rho'$ the density of $(\\pi_1)_\\sharp\\gamma\\lfloor(\\mathbb{H}^n\\times B(y',r'))$ with respect to $\\mathcal{L}^{2n+1}$.\n\\end{lem}\n\n\\begin{proof} Let $(y_m)_{m\\geq 1}$ be a dense sequence in $\\mathbb{H}^n$.\nFor each $m,k \\in \\mathbb{N}^*$, set $\\gamma_{m,k}:=\\gamma \\lfloor (\\mathbb{H}^n \\times B(y_m, r_k))$ where $r_k:=1\/k$. Let $\\rho_{m,k}$ denote the density of $(\\pi_1)_\\sharp\\gamma_{m,k}$ with respect to $\\mathcal{L}^{2n+1}$. Set $A_{m,k} := \\mathbb{H}^n \\setminus (\\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho_{m,k} \\cap \\{\\rho < +\\infty\\})$. We have $\\mathcal{L}^{2n+1}(A_{m,k})=0$. Since $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$, it follows that $\\gamma(A_{m,k}\\times B(y_m, r_k)) \\leq (\\pi_1)_\\sharp\\gamma (A_{m,k}) =0$. Next\n \\begin{equation*}\n  \\gamma(\\{\\rho_{m,k}=0\\} \\times B(y_m,r_k)) = (\\pi_1)_\\sharp\\gamma_{m,k}(\\{\\rho_{m,k}=0\\}) = 0.\n \\end{equation*}\nIt follows that $\\gamma(D_{m,k})=0$ for all $m,k \\in \\mathbb{N}^*$ where\n\\begin{equation*}\n D_{m,k}:=\\left[\\mathbb{H}^n \\setminus (\\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho_{m,k} \\cap \\{\\rho < +\\infty\\} \\cap \\{\\rho_{m,k}>0\\}) \\right] \\times B(y_m,r_k)\n\\end{equation*}\nhence $\\gamma(\\cup_{m,k} D_{m,k})=0$ and $\\gamma$ is concentrated on $\\mathbb{H}^n \\setminus \\cup_{m,k} D_{m,k}$. Then the conclusion follows noting that for each $(x,y) \\in \\mathbb{H}^n\\times\\mathbb{H}^n$ and $r>0$, one can find $m,k\\in \\mathbb{N}^*$ such that $y \\in B(y_m,r_k)\\subset \\subset B(y,r)$.\n\\end{proof}\n\nWe say that $x\\in E$ is a Lebesgue point of a Borel set $E$ if $x\\in \\operatorname{Leb} {\\rm 1\\mskip-4mu l}_E$, i.e., if $x\\in E$ and \n\\begin{equation*}\n \\lim_{r\\rightarrow 0} \\dfrac{\\mathcal{L}^{2n+1}(E\\cap B(x,r))}{\\mathcal{L}^{2n+1}(B(x,r))} = 1,\n\\end{equation*}\nand we denote by $\\operatorname{Leb} E:= \\operatorname{Leb} {\\rm 1\\mskip-4mu l}_E$ the set of all Lebesgue points of $E$. Note that $\\mathcal{L}^{2n+1}(E\\setminus \\operatorname{Leb} E) = 0$.\n\nThe next lemma together with Lemma~\\ref{mainlemma} and Lemma~\\ref{pi2} is one of the key ingredients of the proof of Theorem~\\ref{mainthmbis} and eventually of the existence of a solution to Monge's transport problem. It can be recovered as a consequence of Lemma~\\ref{dens1}. However, for sake of clarity, we state and prove it independently.\n\n\\begin{lem}  \\label{dens2}\n Let $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ be such that $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$. Assume that $\\gamma$ is concentrated on a $\\sigma$-compact set $\\Gamma$. For $y\\in\\mathbb{H}^n$ and $r>0$, set \n\\begin{equation*}\n \\Gamma^{-1} (B(y,r)) = \\pi_1(\\Gamma \\cap (\\mathbb{H}^n \\times B(y,r))).\n\\end{equation*}\nThen $\\Gamma^{-1} (B(y,r))$ is a Borel set and $\\gamma$ is concentrated on a set $\\Gamma'\\subset\\Gamma$ such that for all $(x,y)\\in\\Gamma'$ and all $r>0$, $x\\in \\operatorname{Leb}{\\Gamma^{-1} (B(y,r)})$.\n\\end{lem}\n\n\\begin{proof}\n Since $\\Gamma$ is  $\\sigma$-compact, $\\Gamma^{-1} (B(y,r))$ is also $\\sigma$-compact hence a Borel set. Set $A := \\{(x,y) \\in \\Gamma;\\,x\\notin \\operatorname{Leb}\\Gamma^{-1}(B(y,r)) \\text{ for some } r>0 \\}$\nand let us show that $\\gamma(A)=0$. For each $k\\in\\mathbb{N}^*$, consider a countable covering of \n$\\mathbb{H}^n$ by balls $(B(y_i^k,r_k))_{\\,i \\geq 1}$ of radius $r_k:=1\/(2k)$. If $(x,y) \\in \\Gamma$ and $x \\notin \\operatorname{Leb}\\Gamma^{-1}(B(y,r))$ then for any $k \\geq 1\/r$ and $y_i^k$ such that\n$d(y_i^k,y)<r_k$, one has $x\\in \\Gamma^{-1}(B(y_i^k,r_k)) \\setminus \\operatorname{Leb}\\Gamma^{-1}(B(y_i^k,r_k))$. It follows that \n\\begin{equation*}\n\\pi_1(A) \\,\\, \\subset \\,\\, \\bigcup_{k\\geq 1} \\, \\bigcup_{\\,i \\geq 1} \n\\Gamma^{-1}(B(y_i^k,r_k)) \\setminus \\operatorname{Leb}\\Gamma^{-1}(B(y_i^k,r_k)) .\n\\end{equation*}\nThe set on the right-hand side has $\\mathcal{L}^{2n+1}$-measure 0. Since $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$, it follows that $\\gamma(A) \\leq  (\\pi_1)_\\sharp\\gamma(\\pi_1(A))=0$.\n\\end{proof}\n\n\n\\section{Lower density of the transport set} \\label{main}\n\nWe consider optimal transport plans in $\\Pi_1(\\mu,\\nu)$ that are obtained as weak limit of solutions to the variational approximations introduced in Section~\\ref{varapprox}. We prove that if $\\gamma$ is such a transport plan then it is concentrated on a set $\\Gamma$ whose related transport set has positive lower density at each point $x\\in \\pi_1(\\Gamma)$ for some suitable notion of lower density. As already mentioned, this is one of the main ingredient in the proof of Theorem~\\ref{mainthmbis}. Following the notion of transport set introduced in e.g.~\\cite{ap}, we define in our geometrical context the transport set related to a set $\\Gamma\\subset \\mathbb{H}^n\\times\\mathbb{H}^n$ as \n\\begin{equation*}\n T(\\Gamma) := \\{ (e_t \\circ S)(x,y);\\, (x,y)\\in \\Gamma, \\, t\\in (0,1) \\}.\n\\end{equation*}\nRecall that $S$ is a measurable selection of minimal curves and that $(e_t \\circ S)(x,y)$ denotes the point at distance $t \\,d(x,y)$ from $x$ on the selected minimal curve $S(x,y)$ between $x$ and $y$, see Subsection~\\ref{sect-interpolation}.\n\n\n\\begin{lem}  \\label{mainlemma}\n Let $\\gamma\\in\\Pi(\\mu,\\nu)$ obtained as a weak limit of solutions to $(P_{\\varepsilon_k})$ for some sequence $\\varepsilon_k$ converging to 0. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that for all $(x,y)\\in\\Gamma$ such that $x\\not=y$ and all $r>0$, we have\n\\begin{equation*}\n \\liminf_{\\delta\\downarrow 0} \\, \\dfrac{\\mathcal{L}^{2n+1}(T(\\Gamma \\cap [B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))}{\\mathcal{L}^{2n+1}(B(x,\\delta))}\\, > 0.\n\\end{equation*}\n\\end{lem}\n\nThe proof below follows the line of the proof of the similar property in~\\cite{champion-dePascale}. In our context it requires however some technical refinement. \n\n \\begin{proof}  We consider the set $\\Gamma$ obtained by Lemma~\\ref{dens1}, $(x,y) \\in \\Gamma$ with $x \\neq y$ and $r>0$. Then let $y'\\in\\mathbb{H}^n$ and $r'>0$ be given by Lemma~\\ref{dens1} so that Lemma~\\ref{dens1}(i), (ii) and (iii) hold. Using the same notations as in this lemma, we set \\begin{equation*}\n G:=\\{z \\in \\mathbb{H}^n;\\, \\ \\frac{1}{2}\\rho'(x) \\leq \\rho' (z) \\,\\, \\mbox{and} \\,\\,\n\\rho(z) \\leq  2\\rho(x) \\}.\n\\end{equation*}\nThen $G$ is a Borel set. We have $0< \\rho'(x) \\leq \\rho(x)$ (remember the convention about densities of absolutely continuous measure, see~\\eqref{e:density}). Since $x\\in \\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho'$, see Lemma~\\ref{dens1}(ii) and (iii), it follows that $x\\in\\operatorname{Leb} G$.\n\nFix $\\delta>0$ such that $\\delta < d(x,y)+r$ and \n\\begin{equation} \\label{e:reg-delta}\n \\frac{1}{2}\\mathcal{L}^{2n+1}(B(x,s)) \\leq \\mathcal{L}^{2n+1}(G \\cap B(x,s))\n\\end{equation}\nfor all $s \\in (0,\\delta)$ and fix $t>0$ such that $4t(d(x,y)+r)< \\delta$.\n\nWe set $G_\\delta:=G\\cap B(x,\\frac{\\delta}{2})$, $A_\\delta:=G_\\delta \\times B(y',r')$ and \n$\\gamma_{\\delta}:= \\gamma \\lfloor A_\\delta$.\nWe shall prove that\n\\begin{equation} \\label{e:main1}\n \\frac{\\rho'(x)}{4} \\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) \n\\end{equation} \nand\n\\begin{equation} \\label{e:main2}\n (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) \\leq 2^{2n+4} \\rho(x) \\mathcal{L}^{2n+1} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta)).\n\\end{equation}\nThen \\eqref{e:main1} and \\eqref{e:main2} will yield \n\\begin{equation*}\n2^{-(2n+6)} \\frac{\\rho' (x)}{\\rho(x)}\\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq \\mathcal{L}^{2n+1}(T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))\n\\end{equation*}\nfor any $\\delta>0$ small enough which completes the proof.\n\nTo prove \\eqref{e:main1}, we note that $(\\pi_1)_\\sharp \\gamma_{\\delta} \\ll \\mathcal{L}^{2n+1}$ with density bounded below by $\\frac{1}{2} \\rho'(x) $ $\\mathcal{L}^{2n+1}$-a.e. on $G_\\delta$. Together with \\eqref{e:reg-delta}, it follows that\n\\begin{equation*}\n \\frac{\\rho'(x)}{4} \\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq (\\pi_1)_\\sharp \\gamma_{\\delta} (B(x,\\frac{\\delta}{2})).\n\\end{equation*}\nNext, by choice of $\\delta$ and $t$, we have $(e_t \\circ S)(z,w) \\in B(x,\\delta)$ for all $z \\in B(x,\\frac{\\delta}{2})$ and $w \\in B(y,r)$, hence\n\\begin{equation*}\n B(x,\\frac{\\delta}{2}) \\times B(y',r') \\subset B(x,\\frac{\\delta}{2}) \\times B(y,r) \\subset (e_t \\circ S)^{-1} (B(x,\\delta))\n\\end{equation*}\nand it follows that \n\\begin{equation*}\n (\\pi_1)_\\sharp \\gamma_{\\delta} (B(x,\\frac{\\delta}{2})) = \\gamma_{\\delta} (B(x,\\frac{\\delta}{2}) \\times B(y',r')) \\leq (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta))\n\\end{equation*}\nand this completes the proof of \\eqref{e:main1}.\n\nWe prove now \\eqref{e:main2}. By hypothesis, $\\gamma$ is a weak limit of solutions $\\gamma_k$ to $(P_{\\varepsilon_k})$ for some sequence $\\varepsilon_k$ converging to 0. For each fixed $k\\in\\mathbb{N}$, we apply Lemma~\\ref{restrictions} with $U = G_\\delta \\times \\mathbb{H}^n$ and Proposition~\\ref{e:Linftydensityestimates} with $\\overline \\mu = (\\pi_1)_\\sharp (\\gamma_k \\lfloor U)$ and $\\overline \\nu = (\\pi_2)_\\sharp (\\gamma_k \\lfloor U)$. Taking into account the fact that $(\\pi_1)_\\sharp (\\gamma_k \\lfloor U) = \\mu \\lfloor G_\\delta$, we get that $(e_t \\circ S)_\\sharp (\\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n) \\ll \\mathcal{L}^{2n+1}$ with density in $L^\\infty$ and whose $L^\\infty$-norm is bounded by \n\\begin{equation} \\label{e:rhok}\n \\frac{1}{(1-t)^{2n+3}} \\, \\|\\rho \\lfloor_{G_\\delta} \\|_{L^\\infty} \\leq 2^{2n+4} \\rho(x).\n\\end{equation}\nNext we check that $(e_t \\circ S)_\\sharp (\\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n)$ converges weakly to $(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$. First it follows from Lemma \\ref{weakres} (to be proved below) that $\\gamma_k\\lfloor G_\\delta \\times \\mathbb{H}^n$ converges weakly to $\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n$. Then, noting that $\\gamma$ and each $\\gamma_k$ are concentrated on $\\Omega$ and that $e_t \\circ S$ is continuous on $\\Omega$, the claim follows from Lemma \\ref{conv} (to be proved below) applied with $\\overline\\gamma = \\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n$, $\\overline\\gamma_k = \\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n$, $B=\\Omega$ and $f = \\varphi \\circ e_t \\circ S$ where $\\varphi \\in C_b(\\mathbb{H}^n)$. The fact that $\\gamma$ is concentrated on $\\Omega$ follows from Lemma \\ref{pi1.1}. To check that $\\gamma_k$ is concentrated on $\\Omega$, denote by $\\{y_i^k\\}_i$ the finite set of the atoms of $(\\pi_2)_\\sharp \\gamma_k$. We have that $\\gamma_k$ is concentrated on $\\mathbb{H}^n\\times \\{y_i^k\\}_i$. On the other hand  $\\gamma_k(L_{ y_i^k} \\times \\{y_i^k\\}) \\leq \\gamma_k(L_{ y_i^k} \\times \\mathbb{H}^n) = \\mu(L_{ y_i^k}) = 0$ since $\\mu \\ll \\mathcal{L}^{2n+1}$. It follows that $\\gamma_k$ is concentrated on $\\cup_i [(\\mathbb{H}^n \\setminus L_{ y_i^k}) \\times \\{y_i^k\\}] \\subset \\Omega$. Then, taking into account \\eqref{e:rhok}, we get\n\\begin{equation*}\n |\\int_{\\mathbb{H}^n} \\varphi \\,\\, d(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)| \\leq 2^{2n+4} \\rho(x) \\, \\|\\varphi\\|_{L^1}\n\\end{equation*}\nfor every $\\varphi \\in C_b(\\mathbb{H}^n)$. It follows that $(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$ is in $(L^1)'$ with density in $L^\\infty$ and whose $L^\\infty$-norm is bounded by $2^{2n+4} \\rho(x)$. Since $(e_t \\circ S)_\\sharp \\gamma_\\delta \\leq (e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$, the same holds true for $(e_t \\circ S)_\\sharp \\gamma_\\delta$. Finally we note that $\\gamma_\\delta$ being concentrated on $\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y',r')] \\subset \\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]$, the measure $(e_t \\circ S)_\\sharp \\gamma_\\delta$ is concentrated on $T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y',r')]) \\subset T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)])$. All together we get\n\\begin{equation*}\n\\begin{split}\n (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) &= (e_t \\circ S)_\\sharp \\gamma_{\\delta} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2})\n\\times B(y,r)]) \\cap B(x,\\delta) )\\\\\n&\\leq  2^{2n+4} \\rho(x) \\mathcal{L}^{2n+1} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))\n\\end{split}\n\\end{equation*}\nwhich proves \\eqref{e:main2}.\n\\end{proof}\n\n\\begin{lem} \\label{weakres} \nLet $X$ be a separable and locally compact Hausdorff metric space in which every open set is $\\sigma$-compact. Let $(\\gamma_k)_k$ be a sequence in $\\mathcal P (X\\times X)$\nwhich converges weakly to some $\\gamma \\in \\mathcal P (X\\times X)$ and\nsuch that $(\\pi_1)_\\sharp \\gamma_k = (\\pi_1)_\\sharp \\gamma$ for every $k\\in \\mathbb{N}$. Then for any Borel set $G \\subset X$, the sequence \n$(\\gamma_k \\lfloor  G\\times X)_k$ converges weakly to $\\gamma \\lfloor G\\times X$.\n\\end{lem}\n\n\\begin{proof} We have to prove that for any $\\varphi \\in C_b(X)$,\n\\begin{equation*}\n\\lim_{k\\rightarrow +\\infty} \\int_{X\\times X} {\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y) \\,d \\gamma_k(x,y) = \\int_{X \\times X} {\\rm 1\\mskip-4mu l}_G(x) \\varphi(x,y) \\,d \\gamma(x,y).\n\\end{equation*}\n\nIt follows from Lusin's Theorem that for any \n$\\varepsilon>0$ there exists a closed set $F_\\varepsilon$ such that ${{\\rm 1\\mskip-4mu l}_G} \\lfloor_{F_\\varepsilon}$ is continuous and $(\\pi_1)_\\sharp \\gamma(X \\setminus\nF_\\varepsilon) < \\varepsilon$. As a consequence, for every $\\varepsilon >0$, the restriction of $(x,y) \\mapsto {\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y)$ to $F_\\varepsilon \\times X$ is continuous and \n$$\\limsup_{k\\rightarrow +\\infty} \\gamma_k ((X\\setminus F_\\varepsilon) \\times X)\n= (\\pi_1)_\\sharp \\gamma (X \\setminus F_\\varepsilon) < \\varepsilon.$$\nThen since $(x,y)\\mapsto |{\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y)|$ is\nbounded and hence uniformly integrable with respect to\n$(\\gamma_k)_k$, the claim follows from \\cite[Proposition 5.1.10]{ags}.\n\\end{proof}\n\n\\begin{lem} \\label{conv}\nLet $X$ be a separable metric space and $(\\overline\\gamma_k)_k$ be a sequence in $\\mathcal P(X)$ which converges weakly \nto some $\\overline\\gamma \\in\\mathcal P(X)$. Let $f:X\\to \\mathbb{R}$ be a measurable and bounded function which is continuous \nin $B$ for some Borel set $B\\subset X$ such that $\\overline\\gamma_k(X\\setminus B)=0$ for every $k\\in \\mathbb{N}$ and $\\overline\\gamma(X\\setminus B)=0$, then\n\\begin{equation*}\n \\lim_{k \\to \\infty} \\int_X f d\\gamma_k= \\int_X f d\\gamma.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof} Let $\\overline{f}$ and $\\tilde{f}$ be respectively the lower and\n upper semicontinuous envelope of $f$. We have $\\overline{f}=f=\\tilde{f}$ on $B$ and hence $\\gamma$-a.e. and $\\gamma_k$-a.e. for every $k\\in\\mathbb{N}$. It follows that\n\\begin{multline*}\n \\int_X f \\, d\\gamma = \\int_X \\overline f \\, d\\gamma \\leq \\liminf_{k \\to \\infty} \\int_X \\overline{f} \\,d\\gamma_k \n= \\liminf_{k \\to \\infty} \\int_X f  \\,d\\gamma_k \\\\\n\\leq \\limsup_{k \\to \\infty} \\int_X f d\\gamma_k \n= \\limsup_{k \\to \\infty} \\int_X \\tilde f d\\gamma_k \\leq \\int_X \\tilde f d\\gamma = \\int_X f \\, d\\gamma\n\\end{multline*}\nwhich proves the claim.\n\\end{proof}\n\n\\section{Solution to Monge's problem} \\label{conclusion}\n\nWe prove that optimal transport plans in $\\Pi_1(\\mu,\\nu)$ that are obtained as weak limit of solutions of the variational approximations introduced in Section~\\ref{varapprox} are induced by a transport, hence giving a solution to Monge's transport problem as stated in Theorem~\\ref{mainthm}. Note that due to the fact that $\\Pi$ is relatively compact in $\\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$, such optimal transport plans do exist.\n\n\\begin{thm} \\label{mainthmbis}\n Let $\\varepsilon_k$ be a sequence converging to 0 and $\\gamma_{\\varepsilon_k}$ a sequence of solutions to $(P_{\\varepsilon_k})$ which is weakly converging to some $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$. Then $\\gamma$ is concentrated on a $\\mu$-measurable graph and hence induced by a transport.\n\\end{thm}\n\n\\begin{proof}\nFirst we know from Lemma~\\ref{optpi2} that $\\gamma\\in \\Pi_2(\\mu,\\nu)$. From the previous sections and using inner regularity of Borel probability measures, one can then find $\\sigma$-compact sets $\\Gamma$ and $\\Gamma'$ such that $\\Gamma'\\subset \\Gamma \\subset \\Omega$ and the conclusions of Lemma~\\ref{pi2}, Lemma~\\ref{dens2} and Lemma~\\ref{mainlemma} hold. We prove here that for any $x\\in \\pi_1(\\Gamma')$ there is a unique $y\\in \\mathbb{H}^n$ such that $(x,y)\\in \\Gamma'$.\n\nBy contradiction, assume that one can find $x_0 \\in \\pi_1(\\Gamma')$ and $(x_0,y_0)\\in \\Gamma$, $(x_0,y_1)\\in \\Gamma$ with $y_0\\not=y_1$. Without loss of generality one can assume that $d(x_0,y_0) \\leq d(x_0,y_1)$ and $x_0\\not=y_1$. Then, by Lemma \\ref{dens2} and Lemma \\ref{mainlemma}, for all $r>0$ and for all $\\delta>0$ small enough, one can find $x'\\in B(x_0,\\delta) \\cap \\Gamma^{-1} (B(y_0,r)) \\cap T(\\Gamma \\cap [B(x_0,\\frac{\\delta}{2}) \\times B(y_1,r)])$. It follows that one can find $y' \\in B(y_0,r)$ such that $(x',y') \\in \\Gamma$ and $(x,y) \\in \\Gamma \\cap (B(x_0,\\frac{\\delta}{2})\\times B(y_1,r))$ such that $x\\not=y$, $x'\\not=x$ and $x'$ lie on the minimal curve between $x$ and $y$. Then it follows from Lemma~\\ref{pi2} that $x$, $x'$, $y$ and $y'$ lie on the same minimal curve ordered in that way.\n\nAssume first that $ d(x_0,y_0) < d(x_0,y_1)$. We know from Lemma \\ref{pi2} that $d(x,y)\\leq d(x,y')$. On the other hand, we have\n\\begin{equation*}\n\\begin{split}\n d(x,y')&\\leq d(x,x_0) + d(x_0,y_0) + d(y_0,y')\\\\\n& \\leq d(x_0,y_0) + \\dfrac{\\delta}{2} + r \\\\\n&= d(x_0,y_1) + d(x_0,y_0) - d(x_0,y_1) + \\dfrac{\\delta}{2} + r \\\\\n&\\leq  d(x_0,x) + d(x,y) + d(y,y_1) + d(x_0,y_0) - d(x_0,y_1) + \\dfrac{\\delta}{2} + r\\\\\n&\\leq d(x,y) + d(x_0,y_0) - d(x_0,y_1) + \\delta + 2r.\n\\end{split}\n\\end{equation*}\nIt follows that $d(x,y')< d(x,y)$ provided we take $r>0$ and $\\delta>0$ small enough which gives a contradiction. If $ d(x_0,y_0) = d(x_0,y_1)$, we have\n\\begin{equation*}\n\\begin{split}\nd(x_0,y_1) &\\leq d(x_0,x) + d(x,y) + d(y,y_1) \\\\\n& = d(x_0,x) + d(x,y') - d(y',y) +  d(y,y_1)\\\\\n& \\leq d(x,y') - d(y',y) + \\dfrac{\\delta}{2} + r ,\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n d(x,y') \\leq d(x,x_0) + d(x_0,y_0) + d(y_0,y') \\leq d(x_0,y_0) + \\dfrac{\\delta}{2} + r,\n\\end{equation*}\n\\begin{equation*}\n d(y',y) \\geq d(y_0,y_1) - d(y_0,y') - d(y_1,y)\\geq d(y_0,y_1) -2r,\n\\end{equation*}\nhence,\n\\begin{equation*}\n d(x_0,y_1) \\leq d(x_0,y_0) - d(y_0,y_1) + 4r +\\delta. \n\\end{equation*}\nIt follows that $d(x_0,y_1) < d(x_0,y_0)$ provided we take $r>0$ and $\\delta>0$ small enough which gives also a contradiction.\n\\end{proof}\n\n\n\\section{Extension to more general metric measure spaces} \\label{extensions}\n\nFirst we note that a major part of intermediate steps in the strategy adopted in the present paper can be naturally extended to Polish and non-branching geodesic spaces equipped with a reference measure for which the Lebesgue's differentiation theorem holds. \n\nNext our choice of approximating costs $c_\\varepsilon$ in the approximation procedure is not the only possible one. This choice could in particular be adapted to fit other contexts (for instance concerning the relevant properties of solutions to the transport problem associated to the approximating cost).\n\nFinally the Measure Contraction Property is here technically very convenient. We note however that this property is unnecessarily too strong for what is actually needed in the proof about the lower density of the transport set. Much local and weaker versions about the behavior of the measure of sets transported along minimal curves are indeed sufficient as clearly shows up from the proof.\n\nThis approach can in particular be adapted to give an alternative proof of the existence of solutions to Monge's transport problem in the Riemannian setting without using Sudakov's type arguments.\n\nFor the reasons listed above it is furthermore very likely that the present strategy could be adapted and extended to other geodesic metric spaces.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\currenttime}\n\n\\title{Minimum degree conditions for containing an $r$-regular $r$-connected subgraph}\n\n\\author[M.~Hahn-Klimroth]{Max Hahn-Klimroth} \n\\address{hahnklim@math.uni-frankfurt.de, Goethe University Frankfurt, Robert-Mayer-Str. 10, 60235 Frankfurt, Germany }\n\\thanks{MHK is supported by DFG grant CO 646\/5.}\n\n\\author[O.~Parczyk]{Olaf Parczyk}\n\\address{parczyk@mi.fu-berlin.de, FU Berlin, Arnimallee 3, 14195 Berlin, Germany}\n\\thanks{OP is supported by DFG grant PA 3513\/1-1.}\n\n\n\\author[Y.~Person]{Yury Person}\n\\address{yury.person@tu-ilmenau.de, TU Ilmenau, Weimarer Str. 25, 98684 Ilmenau, Germany} \n\\thanks{YP is supported by the Carl Zeiss Foundation and by DFG grant PE 2299\/3-1.}\n\n\\begin{abstract}\nWe study  optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. \nWe prove for $r$ fixed and $n$ large the condition to be $\\delta(G) \\ge \\frac{n+r-2}{2}$ when $nr \\equiv 0 \\pmod 2$. This answers a question of M.~Kriesell.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\n\\@ifstar{\\origsection*}{\\mysection}{Introduction}\n\nA typical question in extremal graph theory is to determine (asymptotically) optimal minimum degree conditions for a graph $G$ on $n$ vertices to contain a given copy of some spanning graph. \nThe classical theorem of Dirac~\\cite{dirac} asserts the optimal minimum degree condition to contain a Hamilton cycle to be $\\tfrac n2$. There are numerous generalisations of this result to higher connected cycles (powers of Hamilton cycles)~\\cite{KSS_Seynmour}, which in turn generalise  the theorems of Corradi and Hajnal~\\cite{CH63} and  Hajnal and Szemer\\'edi~\\cite{HS_erdos} about clique factors in graphs. The most comprehensive result which asymptotically subsumes all of the mentioned results is the bandwidth theorem of B\\\"ottcher, Schacht and Taraz~\\cite{BST09}. This theorem provides a sufficient condition, which asymptotically depends only on the chromatic number of a bounded degree graph with sublinear bandwidth to be contained in a given dense graph. We also refer to the excellent survey~\\cite{KO09} by K\\\"uhn and Osthus for more results.\n\nThe present work is motivated by a question of Matthias Kriesell~\\cite{MKcomm} about optimal minimum degree condition sufficient to assert the existence of a $4$-regular $4$-connected spanning subgraph. This question in turn was motivated by the work of Bang-Jensen and Kriesell on good acyclic orientations of $4$-regular $4$-connected graphs~\\cite{BJK19}.\n\nWe answer Kriesell's question by proving the following general result about $r$-connected $r$-regular subgraphs of $G$.\n\\begin{theorem}\n     \\label{thm:main}\n    For any $r \\ge 2$ there exists an $n_0$ such that any $n$-vertex graph $G$ with minimum degree $\\delta(G) \\ge \\frac{n+r-2}{2}$, $n \\ge n_0$, and $nr \\equiv 0 \\pmod 2$ contains a spanning $r$-regular $r$-connected subgraph.   \n\\end{theorem}\nNote that for $r \\ge 2$ an $n$-vertex graph $G$ with minimum degree $\\delta(G) \\ge \\frac{n+r-2}{2}$ always is $r$-connected, whereas one can easily come up with examples certifying the optimality of this result (e.g.\\ two $K_{(n+r)\/2}$'s sharing $r$ vertices).\nThe theorem above asserts that  there are minimal $r$-connected subgraphs of $G$ which are in fact $r$-regular.\nObserve that for $r=2$ this follows immediately from Dirac's theorem~\\cite{dirac} with $n_0=3$, as  a Hamilton cycle  is $2$-regular and $2$-connected.\nOwing to the use of the regularity lemma the $n_0$ given by Theorem~\\ref{thm:main} will be very large.\n\nIn the following we briefly introduce some notation and discuss possible candidates for $r$-regular $r$-connected subgraphs that will be found  in $G$ by Theorem~\\ref{thm:main}.  \nThe \\emph{$t$-blow-up of a graph $F$} is obtained by replacing every vertex by $t$ vertices and every edge by a complete bipartite graph $K_{t,t}$.\nLet $C_n$ be the cycle on $n$ vertices and $P_n$ the $n$-vertex path.\nWe denote by $C_n(t)$ and $P_n(t)$ the $t$-blow-up of $C_n$ and $P_n$, respectively.\nWe use a similar definition for odd values of $t$.\nWe denote by $C_n(t-\\tfrac12)$ the $t$-blow-up of $C_n$ for $n$ even, where every other edge only gets a $K_{t,t}$ minus a perfect matching.\nSimilarly, $P_n(t-\\tfrac12)$ is the $t$-blow-up of the $n$-vertex path, where every other edge (starting with the first) only gets a $K_{t,t}$ minus a perfect matching.\nWe also call these the \\emph{$(t-\\tfrac 12)$-blow-ups}.\nNote that $C_n(t)$ is $2t$-regular and $C_n(t-\\tfrac 12)$ is $(2t-1)$-regular.\n\nIn most cases in our proof of Theorem~\\ref{thm:main} we will be able to find a spanning copy of an $\\tfrac r2$-blow-up of a cycle, while allowing other structures with all but a small fraction of vertices in $\\tfrac r2$-blow-ups of paths (see Section~\\ref{sec:o_constructions} for more details).\nHowever, when $n$ is even and not divisible by $4$, the graph $G$ obtained by taking the disjoint union of two cliques $K_{n\/2-2}$ and adding four additional vertices that are connected to all previous $n-4$ vertices cannot contain a copy of $C_n(4)$. \nFinally, observe, that the bandwidth theorem~\\cite{BST09} guarantees the asymptotically best minimum degree condition $\\tfrac n2 +o(n)$. Thus, Theorem~\\ref{thm:main} improves this asymptotic bound to the exact one.\n\nBeyond these blow-ups it would be interesting to study the minimum degree threshold for other spanning structures that can be obtained by identifying vertices or edges of copies of a small graph on a cycle.\nIn particular, when the small graph is not bipartite, this threshold can depend on its chromatic number or critical chromatic number similarly as when taking disjoint copies (see~\\cite{KO09}).\n\n\n\\subsection{Organisation of the paper}\nThe paper is structured as follows. In Section~\\ref{sec:tools} we collect the essential tools (regularity and blow-up lemmas), while Section~\\ref{sec:overview} provides a proof overview, which consists of three cases (extremal case I, extremal case II and non-extremal case). These cases are dealt with in the subsequent Sections~\\ref{sec:extremal1},~\\ref{sec:extremal2} and~\\ref{sec:non-extremal}.\n\n\\@ifstar{\\origsection*}{\\mysection}{Tools and Notation}\\label{sec:tools}\nFor standard graph theoretic definitions we refer to Bollob\\'as~\\cite{Bolbook98}.  \nThe main tools are Szemer\u00e9di's regularity lemma~\\cite{Sze_regularity} and the blow-up lemma by Koml\u00f3s, S\u00e1rk\u00f6zy, and Szemer\u00e9di~\\cite{KSS_Blowup}.\nFor this let $G=(V,E)$ be a graph.\nFor any two sets $A,B \\subseteq V$ we denote by $e_G(A,B)$ the number of edges of $G$ with one endpoint in $A$ and one in $B$.\nThen the \\emph{density} $d(A,B)$ between these sets is $\\frac{e(A,B)}{|A||B|}$.\n\n\\begin{definition}\n    The pair $(A,B)$ is \\emph{$\\varepsilon$-regular} if for all $X \\subseteq A$, $Y \\subseteq B$ with $|X| \\ge \\varepsilon |A|$, $|Y| \\ge \\varepsilon |B|$ we have $|d(X,Y)-d(A,B)| \\le \\varepsilon$.    \n\\end{definition}\n\nThe following lemma guarantees that (not too small) induced subgraphs of  $\\varepsilon$-regular pairs are still regular (although with a slightly worse parameter).\n\\begin{lemma}[Slicing lemma]\n\\label{lem:slicing}\nLet $(A,B)$ be an $\\eps$-regular pair with $d(A,B)=d$, let $\\tfrac 12 \\ge \\gamma > \\eps$, and $A' \\subseteq A$ and $B' \\subseteq B$ be of size $|A'| \\ge \\gamma |A|$ and $|B'|\\ge \\gamma |B|$.\nThen $(A',B')$ is $2\\eps$-regular pair with $d(A,B) \\ge d'$, where $|d-d'|\\le \\eps$.\\qed\n\\end{lemma}\n\nWhen working with the regular pairs, one often needs a somewhat stronger concept of super-regularity.\n\\begin{definition}\n    The pair $(A,B)$ is an \\emph{$(\\varepsilon,\\delta)$-super-regular} pair if it is $\\varepsilon$-regular and $\\deg(a,B) \\ge \\delta |B|$, $\\deg(b,A) \\ge \\delta |A|$ for all $a \\in A$, $b \\in B$.\n\\end{definition}\n\nThe next lemma asserts that there every $\\varepsilon$-regular pair contains an almost spanning super-regular pair.\n\\begin{lemma}\n\\label{lem:superreg}\nLet $(A,B)$ be an $\\eps$-regular pair with $d(A,B)=d$.\nThen there exists $A'\\subseteq A$ and $B' \\subseteq B$ with $|A'|\\ge (1-\\eps) |A|$ and $|B'| \\ge (1-\\eps) |B|$ such that $(A',B')$ is a $(2\\eps,d-3\\eps)$-super-regular pair. \\qed\n\\end{lemma}\n\nWe will use the following degree form of the regularity lemma by Koml\u00f3s and Simonovits~\\cite{koml_simon}.\n\n\\begin{lemma}[Regularity lemma, degree version]\n    \\label{lem:regularity}\n    For every $\\varepsilon>0$ there exists an integer $M$ such that for any graph $G$ and $d \\in [0,1]$ there is a partition of $V(G)$ into $\\ell+1 \\le M$ clusters $V_0,\\dots,V_{\\ell}$ and a subgraph $G'$ of $G$ such that\n    \\begin{enumerate}[label=\\upshape(P\\arabic*)]\n        \\item \\label{reg:size} $|V_0| \\le \\varepsilon |V(G)|$ and  $|V_i| = L \\le \\varepsilon |V(G)|$ for all $1 \\le i \\le \\ell$.\n        \\item \\label{reg:deg} $\\deg_{G'}(v) \\ge \\deg_G(v) - (d+\\varepsilon) |V|$ for all $v \\in V$.\n        \\item \\label{reg:ind} For $1 \\le i \\le \\ell$ the set $V_i$ is independent in $G'$.\n        \\item \\label{reg:reg} For $1 \\le i < j \\le \\ell$ the pair $(V_i,V_j)$ is $\\varepsilon$-regular in $G'$ and has density $0$ or $d$.\n    \\end{enumerate}\n\\end{lemma}\n\nThe blow-up lemma allows us to embed spanning subgraphs with bounded degree. We will use the following special case deduced from~\\cite[Remark~13]{KSS_Blowup}.\n\n\\begin{lemma}[Bipartite blow-up lemma]\n    \\label{lem:blowup}\n    For each $d,c>0$ and integer $\\Delta$ there exist $\\varepsilon>0$, $\\alpha>0$ and integer $n_0$ such that the following holds for any $n \\ge n_0$.\n    Let $H$ be a bipartite graphs on classes $A$ and $B$ with $|A|=|B|=n$ such that $(A,B)$ is a $(\\varepsilon,d)$-super-regular pair and let $G$ be a bipartite graph on classes $X$ and $Y$ with $|X|=|Y|=n$ that has maximum degree bounded by $\\Delta$.\n    Moreover, for any $X' \\subseteq X$ and $Y' \\subseteq Y$ with $|X'|,|Y'| \\le \\eps n$ let $A_x \\subseteq A$ and $B_y \\subseteq B$ for each $x \\in X'$ and $y \\in Y'$ with $|A_x|,|B_x| \\ge cn$.\n    Then there exists an embedding of $G$ into $H$ such that all $x \\in X'$ and $y \\in Y'$ are embedded into $A_x$ and $B_y$, respectively.\n\\end{lemma}\n\nWe remark that in our application $X'$ and $Y'$ will be of constant size and all $A_x$ and all $B_y$ will be the same.\n\n\\@ifstar{\\origsection*}{\\mysection}{Proof overview}\\label{sec:overview}\n\nThe proof of Theorem~\\ref{thm:main} will be split into three cases.\nWe now explain this case distinction and then give an overview of the proof for each of these cases.\nLet $G$ be a graph with minimum degree $\\tfrac{n+r-2}{2}$.\nFor $\\alpha>0$ we call $G$ \\emph{$\\alpha$-extremal} if there are two sets $A,B \\subseteq V(G)$ of size $(\\frac{1}{2}-\\alpha)n \\le |A|,|B| \\le \\frac{n}{2}$ such that $d(A,B) < \\alpha$.\nWith the help of the regularity lemma we will cover the case that $G$ is not $\\alpha$-extremal for any $\\tfrac{1}{32}>\\alpha>0$ in Section~\\ref{sec:non-extremal}.\n\nSo we can assume that $G$ is $\\alpha$-extremal for some $\\alpha>0$.\nUsing the minimum degree condition in $G$ it is easy to see that the sets $A$ and $B$ have to be almost disjoint or almost the same.\nThis implies that $G$ contains a large set that is 'almost' independent or it is 'close' to the disjoint union of two cliques $K_{n\/2}$.\nMore precisely, there exists $\\alpha'>0$ such that one of the following holds:\nEither, there are two disjoint sets $A,B \\subseteq V(G)$ with $(\\tfrac 12 -\\alpha')n \\le |A|,|B| \\le (\\tfrac 12 +\\alpha')n$ such that $G[A]$ and $G[B]$ have minimum degree $(\\tfrac 12 -3\\alpha')n$ and every vertex outside of $A \\cup B$ has degree at least $\\alpha'n$ into $A$ and $B$ -- this will be the first extremal case treated in Section~\\ref{sec:extremal1}.\nOr, there is one set $A \\subseteq V(G)$ with $(\\tfrac12 -\\alpha')n \\le |A| \\le (\\tfrac12 +\\alpha')n$ is such that any vertex in $A$ has degree at least $(\\tfrac12 -3\\alpha')n$ into $V(G) \\setminus A$ and every vertex outside of $A$ has degree at least $3 \\alpha' n$ into $A$ -- this is the second extremal case treated in Section~\\ref{sec:extremal2}.\n\nTherefore, when choosing $0<\\alpha< \\tfrac{1}{32}$ sufficiently small for both extremal cases and the remaining cases will be `non-extremal'.\nThis implies Theorem~\\ref{thm:main}.\nIn the remainder of this section we sketch the argument for each of the three cases and afterwards explain why our constructions are indeed $r$-connected.\n\n\\subsection{Non-Extremal Case}\n\\label{sec:o_non-extremal}\n\nWe would like to find a spanning copy of $C_k(\\tfrac r2)$ in $G$, but an obvious necessary condition for this is that $v(G) \\equiv 0 \\pmod{2 \\lceil \\tfrac r2 \\rceil}$.\nIf this condition is satisfied, we will succeed, and, otherwise, find a slightly locally modified version.\nFor the proof we will have constants\n\\begin{align*}\n    \\varepsilon \\ll \\nu \\ll d \\ll \\beta \\ll \\alpha < \\frac{1}{32}\n\\end{align*}\nand $s = \\lceil \\tfrac r2 \\rceil$.\nWe follow similar arguments as in~\\cite{KSS_square}, which can be summarised by the following procedure:\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n    \\item Apply regularity lemma (Lemma~\\ref{lem:regularity}) with $\\varepsilon$ and $d$ to obtain a regular partition of $G$.\\label{step:regularise}\n    \\item Find $\\ell$ $\\varepsilon$-regular pairs $(X_i,Y_i)$ covering all but a small set $V_0$ with $|V_0| \\le 20 dn$.\\label{step:matching}\n    \\item For $i=1,\\dots,\\ell$ connect $Y_i$ to $X_{i+1}$ with the $\\tfrac r2$-blow-up of a path that we denote by $P_i$.\\label{step:connect}\n    \\item For $i=1,\\dots,\\ell$ turn $(X_i,Y_i)$ into an $(\\varepsilon,d-\\varepsilon)$-super-regular pair with $|X_i|=|Y_i|$, slightly increasing $V_0$ to $|V_0| \\le 23 dn$.\\label{step:superreg}\n    \\item Repeatedly take $\\nu n$ vertices from $V_0$ and append them to the paths $P_i$.\\label{step:absorb}\n    \\item For $i=1,\\dots,\\ell$ use blow-up lemma (Lemma~\\ref{lem:blowup}) to find a spanning copy of an $\\tfrac r2$-blow-up of a path in $(X_i,Y_i)$ connecting $P_{i-1}$ with $P_{i}$.\\label{step:spanning}\n\\end{enumerate}\n\nThe index $\\ell+1$ corresponds to $1$.\n\\ref{step:regularise} is natural and for~\\ref{step:matching} it is enough to find a large matching in a graph with minimum degree close to $\\frac{n}{2}$.\nDuring the performance of~\\ref{step:absorb} the degree of some vertices might get too small.\nIn this case we add them to a set $Q$ that we take care of before the next round.\nThis terminates as in every execution there are at most $3 \\varepsilon n \\ll \\nu n$ vertices added to $Q$.\nApart from this~\\ref{step:absorb} is very similar to~\\ref{step:connect}, which we now sketch with more details.\n\nLet $X$, $Y$ be the clusters that we want to connect with the $\\tfrac r2$-blow-up of a path $P$.\nIf there is a cluster $Z$ such that $(X,Z)$ and $(Z,Y)$ are $\\varepsilon$-regular pairs with density at least $d$ then we can easily find this path.\nOtherwise, let $A$ be the union of all clusters $Z$ such that $(X,Z)$ is an $\\varepsilon$-regular pair with density at least $d$ and $B$ the union of all clusters $Z$ for $(Y,Z)$ analogously.\nBy the minimum degree property in the cluster graph we get $|A|,|B| \\ge (\\frac{1}{2}-\\alpha)n$.\nAs $G$ is not $\\alpha$-extremal we have $d(A,B) > \\alpha$.\nTherefore, there exist two clusters $Z_1 \\in A$ and $Z_2 \\in B$ with $d(Z_1,Z_2) \\ge \\alpha$ and then $(X,Z_1)$, $(Z_1,Z_2)$, and $(Z_2,Y)$ are $\\varepsilon$-regular pairs with density at least $d$.\nThen it is again easy to find the path that we are interested in by following these three regular pairs.\n\nWe have to ensure that the end vertices of the paths always have high degree into the other cluster of the respective super-regular pair, because we want to connect them later and keep them through~\\ref{step:superreg}.\nFurthermore, we have to ensure that in~\\ref{step:absorb} the sizes of the $(\\varepsilon,d-\\varepsilon)$-super-regular pairs remain balanced.\nWe will give the details in Section~\\ref{sec:non-extremal}.\n\n\\subsection{Extremal Case I}\n\\label{sec:o_extremal1}\n\nIn this extremal case we will not use the regularity lemma, but the blow-up lemma will be helpful.\nRecall that in this case $G$ is `close' to the union of two disjoint cliques of size roughly $\\tfrac n2$ on vertex sets $A$ and $B$.\nThe main challenge is to find a bridge that connects both these cliques.\nIt is then easy to find the desired structure using the high degrees.\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n\\item \\label{stepd:bridge} In the case when $r$ is even the bridge will be a matching of size $r$ between $A$ and $B$ such that the end-vertices are well connected on their side.\nThe odd case is a little more delicate and we will find a matching of size $r+1$ or $r$ depending on the size of $V(G) \\setminus \\bc{A \\cup B}$ and the parity of $A$ and $B$.\n\\item \\label{stepd:absorb} Absorb all vertices that do not not belong to $A$ or $B$ by extending both ends of the path.\nWe can ensure that the left-over on each side has size divisible by $2r$.\n\\item \\label{stepd:cover} It is easy to see that the left-over on both sides can be split into a super-regular pair and that we can cover both with the $\\tfrac r2$-blow-up of a path using Lemma~\\ref{lem:blowup}.\n\\end{enumerate}\nIf we take care of the end-tuples between each of the steps this gives an $r$-regular $r$-connected path-structure covering $G$.\nIn Section~\\ref{sec:extremal1} we will give the details of the even and odd case separately.\n\n\\subsection{Extremal Case II}\n\\label{sec:o_extremal2}\n\nAgain, we will not use the regularity lemma in this part, but the blow-up lemma will still be helpful.\nWe can assume that we have a partition of $V(G)$ into $A$ and $B$ of size $(\\tfrac 12 \\pm \\alpha)n$ such that between these sets we have minimum degree $\\alpha n$ and all but at most $\\alpha n$ vertices from $A$ (or $B$) have degree $|B|-\\alpha n$ (or $|A|-\\alpha n$) into $B$ (or $A$).\nW.l.o.g.~assume that $|A|+m=\\tfrac 12 n=|B|-m$, where $0 \\le m \\le \\alpha n$.\nNote that in $G[B]$ we have minimum degree at least $m+\\tfrac{r-2}{2}$.\nLet $s=\\lceil \\tfrac r2 \\rceil$.\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n    \\item \\label{stepe:find} If $\\Delta(G[B]) \\le 2 r \\alpha n$ find $m$ copies of $K_{1,s}$, such that all vertices are well connected to the other side. Otherwise, separate the vertices with higher degrees, then find copies of $K_{1,s}$, and afterwards find additional copies of $K_{1,r}$, such that the leaves are well connected.\n    \\item \\label{stepe:absorb} Absorb these copies of $K_{1,s}$ and $K_{1,r}$ into an $r$-regular path-structure and then connect these together into one longer path-structure.\n    After removing the path that we constructed we are left with sets $A_1\\subseteq A$ and $B_1 \\subseteq B$ with $|A_1|=|B_1|$.\n    \\item \\label{stepe:absorb2} Absorb all vertices that do not have large degree to the other side into the path by alternating between both sides.\n    After removing these vertices we are left with sets $A_2\\subseteq A_1$ and $B_2 \\subseteq B_1$ with $|A_2|=|B_2|$ and the property that all vertices have large degree to the other side.\n    \\item \\label{stepe:cover} It is easy to see that $(A_2,B_2)$ is a super-regular pair and that we can cover it with the $\\tfrac r2$-blow-up of a path using Lemma~\\ref{lem:blowup}.\n\\end{enumerate}\nIf we take care of the end-tuples between each of the steps this gives an $r$-regular $r$-connected path-structure covering $G$.\nFor the first step we use the following.\n\\begin{lemma}\n\\label{lem:stars}\n    For any integer $s$ there exists $\\alpha>0$ such that the following holds.\n    Let $G$ be an $n$ vertex graph with maximum degree $\\Delta(G) \\le 4 s \\alpha n$ and minimum degree $\\delta(G) \\ge m + s - 1$, where $1 \\le m \\le \\alpha n$.\n    Then there are $2m$ pairwise disjoint copies of $K_{1,s}$ in $G$.\n\\end{lemma}\n\nThe proof of this lemma and the second extremal case will be given in Section~\\ref{sec:extremal2}.\n\n\\subsection{Constructions}\n\\label{sec:o_constructions}\nFirst recall that the $\\tfrac r2$-blow-up of a cycle is $r$-regular and also $r$-connected.\nIt will not always be possible to construct this, but it will be the basic building block.\nWe might need to absorb some exceptional vertices, for example, when $n$ is not divisible by $r$.\nIn the case when $r$ is even we then remove a perfect matching from one $K_{s,s}$ and add one vertex that is connected to all $2s=r$ vertices that just lost one neighbour (c.f.~Figures~\\ref{fig:absorber_even},~\\ref{fig:absorbK1s}, and~\\ref{fig:Step54}).\nThe resulting graph is still $r$-connected, because we can not disconnect this part of the cycle by removing less than $\\tfrac r2$ vertices.\nA similar construction will be used in the case when $r$ is odd (c.f.~Figures~\\ref{fig:absorber_odd},~\\ref{fig:absorbK1s}~and~\\ref{fig:Step55}) that also preserves $r$-connectivity.\nApart from this, we also have to connect to $\\tfrac r2$-blow-ups of cycles by using at most $r$ edges between them (c.f.~Step~\\ref{stepd:bridge} of Section~\\ref{sec:o_extremal1}).\nWe will only need to take care of a small linear fraction of the vertices from $G$ and, therefore, almost all vertices are in the $\\tfrac r2$-blow-up of a path.\n\n\\@ifstar{\\origsection*}{\\mysection}{Extremal Case I}\n\\label{sec:extremal1}\nIn this section we deal with the first extremal case.\nWe will not use the regularity lemma in this part, but the blow-up lemma will still be helpful.\n\n\\begin{proof}[Proof of Extremal Case I]\nLet $r \\geq 3$ be an integer, let $\\eps>0$ be given by Lemma~\\ref{lem:blowup} on input $\\tfrac 12$, $\\tfrac 12$, and $r$ and let $0<\\alpha \\le \\eps (1000 r^2)^{-1}$.\nLet $G$ be an $n$-vertex graph with $\\delta(G) \\geq \\tfrac {n+r-2}{2}$ and let $A,B \\subseteq V(G)$ with $(\\tfrac 12-\\alpha)n \\le |A|,|B| \\le (\\tfrac 12+\\alpha)n$ such that $G[A]$ and $G[B]$ have minimum degree $(\\tfrac 12-3\\alpha)n$ and every vertex in $C=V(G) \\setminus (A \\cup B)$ has degree at least $\\alpha n$ into $A$ and $B$.\nOur goal is to find an $r$-regular, $r$-connected spanning subgraph in $G$ provided that $n$ is large enough.  \n\n\\subsection{The even case}\n\nAssume that $r$ is even.\nWe begin by constructing $\\tfrac r2$ bridges of size 2 between $A$ and $B$ (\\ref{stepd:bridge} of Section \\ref{sec:o_extremal1}). A visualisation can be found in Figure~\\ref{fig:bridge_graph_h}.\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 3) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 1.5) {$a_{12}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 3) {$a_{21}$};\n        \\node[circle, draw=green] (xa1) at (2, 1.5) {$x_{a_1}$};\n        \\node[circle, draw=green] (xb1) at (2, -1.5) {$x_{b_1}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 3) {$a_{31}$};\n        \\node[circle, draw=green] (xa2) at (4, 1.5) {$x_{a_2}$};\n        \\node[circle, draw=green] (xb2) at (4, -1.5) {$x_{b_2}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 3) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 1.5) {$a_{42}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \n   \n        \\path[-] (a_11) edge[draw=black] (a_21);\n        \\path[-] (a_12) edge[draw=black] (xa1);\n        \\path[-] (a_11) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (a_21);\n        \n        \\path[-] (xa1) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (xa2);\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \n        \\path[-] (a_31) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \n \n        \\path[-] (b_11) edge[draw=black] (xb1);\n        \\path[-] (b_12) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \n        \\path[-] (b_22) edge[draw=black] (xb2);\n        \\path[-] (xb1) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \n        \\path[-] (xb2) edge[draw=black] (b_41);\n        \\path[-] (b_32) edge[draw=black] (b_42);\n        \\path[-] (xb2) edge[draw=black] (b_42);\n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \n        \\path[-] (xb2) edge[draw=green] (xa2);\n        \\path[-] (xb1) edge[draw=green] (xa1);\n\n    \\end{tikzpicture}\n    \\caption{Bridge between the sets $A$ and $B$ in the special case $r = 4$.}\n    \\label{fig:bridge_graph_h}\n\\end{figure}\n\n\n\\begin{claim}\\label{lemma_matching_edges}\nSuppose $\\delta(G) \\geq \\tfrac {n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_r}x_{b_r})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r$.\n\\end{claim}\n\\begin{claimproof}\n In order to construct the matching, it suffices to find $r$ edges from $A$ to $V \\setminus A$. Indeed, suppose we find the $r$ edges $a_1c_1, \\ldots, a_rc_r$. If $c_i \\in B$, we take this edge. If $c_i \\in V \\setminus (A \\cup B)$, $c_i$ has either $\\tfrac n5$ edges into $B$ (in this case, take edge $a_ic_i$), or it has $\\tfrac n5$ edges into $A$. Let $i_1, \\ldots, i_l$ be the indices such that $c_{i_j}$ does not have $\\tfrac n5$ edges into $B$. By definition of $\\alpha$-extremity, each $c_{i_j}$ has $\\alpha n$ neighbors in $B$. Select $b_{i_1}, \\ldots, b_{i_l}$ s.t. $b_{i_j} \\in N(c_{i_j}) \\cap B$ and $b_{i_j} \\neq b_{i_k}$ for all $j \\neq k$ (and being disjoint from those $c_i \\in B$ (this is clearly possible). We add edges $b_{i_j}c_{i_j}$ to the matching. \n \nIt remains to show that these edges exist. \nFirst, suppose that $n$ is even. If $\\abs{A} \\leq \\tfrac{n-r}{2}$, the minimum degree of $ \\tfrac{n+r-2}{2}$ guarantees that each vertex of $A$ needs to find at least $\\tfrac{n+r-2}{2} - (\\tfrac{n-r}{2} - 1) = r$ neighbors outside of $A$, hence the assertion follows.\nSuppose $\\abs{A} = \\tfrac{n-r}{2} + i$ with $i = 1, \\ldots, \\tfrac r2$. In this case, $\\abs{V \\setminus A} = \\tfrac{n+r}{2} - i$. Each vertex of $A$ finds at least $r-i$ neighbors outside of $A$. Suppose that $N(A) \\setminus A$ has size at most $r-1$ (thus, all edges from $A$ into the rest of the graph belong to $r-1$ vertices). Now pick a different vertex in the complement (which exists, as $\\tfrac{n+r}{2} - i \\gg r$). This vertex requires $\\tfrac{n+r-2}{2} - (\\tfrac{n+r}{2} - i - 1) = i$ neighbors in $A$, which is a contradiction.\n\nNow, if $n$ is odd, because the minimum degree needs to be an integer, it is at least $\\tfrac{n+1+r-2}{2}$, hence upon removal of one vertex, we are left with a graph on $n' = n-1$ vertices and minimum degree at least $\\tfrac{n+1+r-2}{2}-1 = \\tfrac{n'+r-2}{2}$ (and $n'$ being even). Hence the assertion follows from the previous discussion.\n\\end{claimproof}\n\nTherefore, Claim~\\ref{lemma_matching_edges} gives us the green sub-structure of Figure \\ref{fig:bridge_graph_h}. \nNow, we take two of those matching edges (think of them as being $\\tfrac r2$ pairs of $2$ edges). Denote the vertices that are connected to at least $\\tfrac n5$ vertices in $A$ as $x_{a_1}, x_{a_2}$. We next prove that the black structure around $x_{a_1}, x_{a_2}$ shown Figure \\ref{fig:bridge_graph_h} exists.\n\n\\begin{claim}\\label{bridge_graph_existence}\nThere are distinct vertices $a_{i,1}, \\ldots, a_{i, r\/2} \\in A$ for $i=1,4$ and $a_{i,1}, \\ldots, a_{i, r\/2-1} \\in A$ for $i=2,3$ with the following properties.\n\\begin{enumerate}\n    \\item The edges $a_{i, j}a_{i+1, k}$ for $i=1, 2 ,3$ and $j = 1 , \\ldots, \\tfrac r2$ (or $\\tfrac r2 - 1$, respectively) exist,\n    \\item the edges $x_{a_1}a_{1, j}$ and $x_{a_2}a_{4, j}$ exist for $j = 1 , \\ldots,\\tfrac r2$,\n    \\item the edges $x_{a_1}a_{3, j}$ and $x_{a_2}a_{1, j}$ exist for $j = 1 , \\ldots,\\tfrac r2 -1$.\n\\end{enumerate}\n\\end{claim}\n\n\\begin{claimproof}\nWe select $r - 1$ vertices $a_{1,i}$, $a_{3, j} \\in N(x_{a_1}) \\cap A$ arbitrarily (but disjoint from $x_{a_2}$). Those exist as $x_{a_1}$ has at least $\\tfrac n5$ neighbors in $A$. Each of those vertices is connected to at least $(\\tfrac 12 - 3 \\alpha)n$ vertices in $A$, hence each vertex has at least $(\\tfrac 12 - 3 \\alpha)n - r - 1$ neighbors in $A$ that do not belong to $a_{1,i}$, $a_{3, j}$ or $x_{a_1}, x_{a_2}$.\nTherefore, the joint neighborhood \n\\[N := \\bc{A \\cap \\bigcap_{i = 1}^{r\/2} N(a_{1,i}) \\bigcap_{j=1}^{r\/2 - 1} N(a_{1,j})} \\setminus \\bc{\\bigcup_{i=1}^{r\/2} \\cbc{ a_{4, i} } \\bigcup_{j=1}^{r\/2-1} a_{3, j} \\cup \\cbc{ x_{a_1}, x_{a_2} }}\\]\nhas size at least $(\\tfrac 12 - 4 r \\alpha)n$. Therefore, we find \n\\[\\abs{ N(x_{a_2}) \\cap N } \\geq \\frac{n}{100} \\, , \\]\nthus the claim follows as the same token holds in $B$ as well.\n\\end{claimproof}\nWe denote the resulting collection of vertices in $A$ by $X_{A, 1}, \\ldots, X_{A, r\/2}$.\nClearly, each vertex in $A$ stays connected to at least $(\\tfrac 12 - 4 \\alpha)n$ vertices in\n\\[A'_{0} =  A \\setminus \\bc{ V(X_{A, 1}) \\cup \\ldots \\cup V(X_{A, r\/2}) } \\, .\\]\n\nNext, we introduce a \\textit{gluing operation} GE.\n\\begin{claim}[Gluing operation GE]\nGiven two disjoint sets $D_1, D_2 \\subset A'_0$ of size exactly $\\tfrac r2$, we find two disjoint sets $D, D' \\subset A'_0 \\setminus (D_1 \\cup D_2)$ of size $\\tfrac r2$ such that\n\\[G[D_1, D] \\equiv K_{r\/2, r\/2}, \\quad G[D, D'] \\equiv K_{r\/2, r\/2}\\quad \\text{and} \\quad G[D', D_2] \\equiv K_{r\/2, r\/2} \\, .\\]\n\\end{claim}\n\\begin{claimproof}\nAs the joint $A'_0$ - neighborhood of $D_1$ and $D_2$ has size at least $(\\tfrac 12 - 10r \\alpha)n$, the assertion follows.\n\\end{claimproof}\n\nUsing GE, we glue the $\\tfrac r2$ bridges in $A$ and $B$ respectively together using mutually disjoint vertex sets $D^A_1, \\ldots,D^A_{r\/2 - 1}$ and $D^B_1, \\ldots,D^B_{r\/2 - 1}$ and are left with path-like structures $P_A$ and $P_B$. After gluing the bridges together, we let $A' = A'_0 \\setminus V(P_A), B' = B'_0 \\setminus V(P_B)$. \n\nIn a next step we need to absorb left-over vertices (\\ref{stepd:absorb} of Section \\ref{sec:o_extremal1}). To this end define two absorber-graphs for a vertex $u$: $\\xi_r(u)$ and $\\xi'_r(u)$ (see Figure \\ref{fig:absorber_even}).\n\\begin{definition}\n    Let $D \\in \\cbc{A', B'}$ and $u$ a vertex such that $\\abs{ N(u) \\cap D } \\geq \\tfrac n6$. Define $\\xi_r(u)$ as follows.\n    \\begin{itemize}\n        \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{r\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{r\/2}} \\subset N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n        \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{r\/2 - 1}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n    \\end{itemize}\n    Define $\\xi_r(u)$ as the graph containing $D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$ and from $D_2$ to $D' \\cup \\cbc{u}$.\n    Furthermore, define $\\xi'_r(u)$ via\n    \\begin{itemize}\n        \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{r\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{r\/2}} \\subset N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n        \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{r\/2 - 1}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n        \\item Select $\\tfrac r2$ vertices $E_0 = \\cbc{e_0, \\ldots, e_{r\/2-1}}$ and an additional disjoint vertex $e_{r\/2}$ from $D \\setminus \\bc{ D_1 \\cup D_2 \\cup D' }$ such that $E_0 \\cup \\cbc{e_2} \\subset N(d_2) \\cap \\ldots \\cap N(d_{r\/2}) \\cap D$ and $G[E_0, \\cbc{e_{r\/2}}] \\equiv K_{r\/2, 1}$. \n    \\end{itemize}\n    Define $\\xi'_r(u)$ as the graph containing $D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$ and from $D_2$ to $D' \\cup \\cbc{u}$.\n\\end{definition}\nClearly, by the sizes of $A', B'$ and the minimum-degree condition, given at most $100 \\alpha n$ pairwise different vertices, there is an absorber for each vertex which is disjoint from all other absorbers. Furthermore, by the minimum degree condition inside of $A'$ and $B'$, this family of absorbers exists.\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}\n    \n\n     \\node[circle, draw=green!40] (d1) at (2, 1) {$d_1$};\n     \\node[circle, draw=green!40] (d2) at (2, -1) {$d_2$};\n\n     \\node[circle, draw=green!40] (dp1) at (6, 1) {$d'_{1}$};\n     \\node[circle, draw=green!40] (dp2) at (6, -1) {$d'_{2}$};\n     \n     \\node[circle, draw=blue!40] (up1) at (4, 1) {$u'_{1}$};\n     \\node[circle, draw=green] (u) at (4, -1) {$u$};\n\n     \n     \\path[-] (u) edge[draw=green] (d1);\n     \\path[-] (u) edge[draw=green] (d2);\n     \\path[-] (u) edge[draw=green] (dp1);\n     \\path[-] (u) edge[draw=green] (dp2);\n     \n     \\path[-] (d1) edge[draw=blue] (up1);\n     \\path[-] (d2) edge[draw=blue] (up1);\n     \\path[-] (dp1) edge[draw=blue] (up1);\n     \\path[-] (dp2) edge[draw=blue] (up1);\n     \n    \n     \n     \\node[circle, draw=blue!40] (e0) at (10, 1) {$e_0$};\n     \\node[circle, draw=blue!40] (e1) at (10, -1) {$e_1$};\n     \\node[circle, draw=blue!40] (e2) at (11, 0) {$e_2$};\n     \n     \\node[circle, draw=green!40] (ld1) at (12, 1) {$d_1$};\n     \\node[circle, draw=green!40] (ld2) at (12, -1) {$d_2$};\n\n     \\node[circle, draw=green!40] (ldp1) at (16, 1) {$d'_{1}$};\n     \\node[circle, draw=green!40] (ldp2) at (16, -1) {$d'_{2}$};\n     \n     \\node[circle, draw=blue!40] (lup1) at (14, 1) {$u'_{1}$};\n     \\node[circle, draw=green] (lu) at (14, -1) {$u$};\n\n     \n     \\path[-] (lu) edge[draw=green] (ld1);\n     \\path[-] (lu) edge[draw=green] (ld2);\n     \\path[-] (lu) edge[draw=green] (ldp1);\n     \\path[-] (lu) edge[draw=green] (ldp2);\n     \n     \\path[-] (ld1) edge[draw=blue] (lup1);\n     \\path[-] (ld2) edge[draw=blue] (lup1);\n     \\path[-] (ldp1) edge[draw=blue] (lup1);\n     \\path[-] (ldp2) edge[draw=blue] (lup1);\n     \n     \\path[-] (ld1) edge[draw=blue] (e2);\n     \\path[-] (ld2) edge[draw=blue] (e2);\n     \\path[-] (e1) edge[draw=blue] (e2);\n     \\path[-] (e0) edge[draw=blue] (e2);\n     \n     \\path[-] (e0) edge[draw=blue, bend left=30] (ld2);\n     \\path[-] (e1) edge[draw=blue, bend left=30] (ld1);\n    \n    \\end{tikzpicture}\n    \n    \\caption{Absorbers $\\xi_4(u)$ (left) and $\\xi'_4(u)$ (right) with $r=4$, where $u$ is the green vertex, the green vertices are inside the $A'$-neighborhood of $u$ and the blue vertices are vertices chosen from $A'$.}\n    \\label{fig:absorber_even}\n\\end{figure}\n\nWe are now in position to absorb the exceptional set $C$ (of course, without the bridging vertices on $P_A$ and $P_B$. For all up to $6 \\alpha n$ vertices in $C$ create a disjoint absorber $\\xi_r(u)$ as above. Chose as $D$ either $A'$ (if a vertex has $\\tfrac n5$ neighbors in $A$), or $B'$ otherwise. Next, using the gluing operation GE up to $6 \\alpha n$ times, glue the absorbers inside of $A'$ and $B'$ together, always using only vertices that did not get used in a previous gluing step or are part of the absorbers. As one only requires at most $24 r \\alpha n$ vertices during this procedure, it is clearly possible by above discussion.\n\nFinally, use GE again to glue the series of absorbers to $P_A$ and $P_B$ respectively (which is clearly possible, as this is only one operation on each set).\n\nAfter the repetitive gluing, we are left with sets $A''$ and $B''$ (hence, $A'$ without the glued structures $P'_A$) and the path-like subgraph $P'_A$ of size at most $25r \\alpha n$, hence $\\abs{A''} \\geq (\\tfrac 12 - 30 r \\alpha)n$. Furthermore, each vertex in $A''$ is connected to at least $(\\tfrac 12 - 40 r \\alpha)n$ vertices in $A''$. Clearly, the same holds for $B''$. \n\nBefore closing the path in both sets (hence, creating a cycle which contains all vertices that are not part of $P'_A$ or $P'_B$), which is a standard application of the blow-up lemma, we need to make sure that certain divisibility conditions hold. \nAs we wish to close the cycle by appending blocks of two layers of size $r$ (thus, $K_{{r\/2}, {r\/2}}$), we require that $\\abs{A''} \\equiv \\abs{B''} \\equiv 0 \\pmod r$. If this is the case, set $A''' = A''$ and proceed. Otherwise, if there is $0 < i < r$ such that $\\abs{A''} \\equiv i \\pmod r$, select $i$ vertices $a_1, \\ldots,a_{i} \\in A''$ and absorb them using disjoint instances $\\xi'_r(a_1), \\ldots, \\xi'_r(a_{i})$ with $D = A''$. Clearly, as this requires only finitely many vertices, such a disjoint family exists. Further, because $a_j \\in A''$, each absorber consumes $2r + 1$ vertices of $A''$, hence afterwards, the divisibility condition holds. Now, glue the absorbers sequentially to $P'_A$ using GE and sets $G_1, \\ldots,G_{i-1}$. As each gluing operation consumes $r$ vertices, the divisibility does not change hence we are left with a set $A''' = A'' \\setminus \\bc{ V(\\xi'_r(a_1)) \\cup \\ldots \\cup V(\\xi'_r(a_{i})) \\cup G_1 \\cup \\ldots \\cup G_{i-1} }$.\n\nNow it is easy to check that $(A''',B''')$ is $(\\eps,\\tfrac 12)$-super-regular and by Lemma~\\ref{lem:blowup} we find an $\\tfrac r2-$blowup of the path on all remaining vertices of $A'''$ and $B'''$ (\\ref{stepd:cover} of Section \\ref{sec:o_extremal1}). \nMoreover, the end-tuples of the path-like structure constructed before have at least $\\tfrac 12 |A'''|$ and $\\tfrac 12 |B'''|$ common neighbours in $A'''$ and $B'''$ respectively.\nTherefore, we may choose the start- and end-tuples of this path-blow-up to connect to these end-tuples.\n\nWe are left to argue that the constructed subgraph is $r$-connected and $r$-regular.\n\\begin{claim}\nThe constructed subgraph is $r$-connected and $r$-regular.\n\\end{claim}\n\\begin{claimproof}\nWhile $r$-regularity follows obviously, the $r$-connected part needs a short argument. Upon removal of up to $r-1$ bridge-vertices, the parts do not fall apart. Furthermore, removing up to $r-1$ vertices in the $\\tfrac r2$-blow-up of the path part of the subgraph does not disconnect the structure. Finally, the absorbing structure $\\xi_r$ itself is isomorphic to an $\\tfrac r2$-blowup of the path on three vertices. Moreover, disconnecting the graph by removing up to $r-1$ vertices in $\\xi_r'$ is not possible.\n\\end{claimproof}\n\n\n\n\\subsection{The odd case}\n\n\nAssume that $r$ is odd.\nThe argument in the odd case is a bit more delicate as in the even case. Indeed, while in the process above all divisibility conditions could be easily established, in the odd case, we might end with two almost cliques of odd size. If there is a set $C$, we can easily absorb those vertices in a way that after absorbing both parts of the graph contain an even number of vertices - which we require to embed a regular graph. If on the other hand there is no such set $C$, we need to be much more careful. We will tackle this problem by having two different types of bridges between $A$ and $B$, one consuming an even number of vertices of each set, one consuming an odd number - thus, depending on the size of $C$ and the parity of $A$ and $B$, we need to use two different constructions. The two types of bridges are visualised in Figure~\\ref{fig:bridge_graph_h_odd} for the special case $ r = 5$.\n\n\n\\begin{figure}\n    \\centering\n    \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 4.5) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 3) {$a_{12}$};\n        \\node[circle, draw=black] (a_13) at (0, 1.5) {$a_{13}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \\node[circle, draw=black] (b_13) at (0, -4.5) {$b_{13}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 4.5) {$a_{21}$};\n        \\node[circle, draw=black] (a_22) at (2, 3) {$a_{22}$};\n        \\node[circle, draw=green] (xa1) at (2, 1.5) {$x_{a_1}$};\n        \\node[circle, draw=green] (xb1) at (2, -1.5) {$x_{b_1}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \\node[circle, draw=black] (b_23) at (2, -4.5) {$b_{23}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 4.5) {$a_{31}$};\n        \\node[circle, draw=black] (a_32) at (4, 3) {$a_{32}$};\n        \\node[circle, draw=green] (xa2) at (4, 1.5) {$x_{a_2}$};\n        \\node[circle, draw=green] (xb2) at (4, -1.5) {$x_{b_2}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \\node[circle, draw=black] (b_33) at (4, -4.5) {$b_{33}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 4.5) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 3) {$a_{42}$};\n        \\node[circle, draw=black] (a_43) at (6, 1.5) {$a_{43}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \\node[circle, draw=black] (b_43) at (6, -4.5) {$b_{43}$};\n        \n   \n       \n        \\path[-] (a_11) edge[draw=black] (a_22);\n        \\path[-] (a_11) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (a_21);\n       \n        \\path[-] (a_13) edge[draw=black] (a_21);\n        \\path[-] (a_13) edge[draw=black] (a_22);\n\n        \\path[-] (xa1) edge[draw=black] (a_31);\n        \\path[-] (xa1) edge[draw=black] (a_32);\n        \\path[-] (a_21) edge[draw=black] (xa2);\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (a_32);\n        \\path[-] (a_22) edge[draw=black] (xa2);\n        \\path[-] (a_22) edge[draw=black] (a_31);\n        \\path[-] (a_22) edge[draw=black] (a_32);\n        \n       \n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_43);\n        \\path[-] (a_32) edge[draw=black] (a_41);\n       \n        \\path[-] (a_32) edge[draw=black] (a_43);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n       \n        \n \n       \n       \n        \\path[-] (b_12) edge[draw=black] (b_23);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_23);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \\path[-] (b_13) edge[draw=black] (b_22);\n       \n        \\path[-] (b_13) edge[draw=black] (xb1);\n        \n        \n        \\path[-] (b_22) edge[draw=black] (xb2);\n        \\path[-] (xb1) edge[draw=black] (b_32);\n        \\path[-] (xb1) edge[draw=black] (b_33);\n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_33);\n        \\path[-] (b_23) edge[draw=black] (b_32);\n        \\path[-] (b_23) edge[draw=black] (xb2);\n        \\path[-] (b_23) edge[draw=black] (b_33);\n        \n       \n        \\path[-] (xb2) edge[draw=black] (b_43);\n       \n        \\path[-] (b_32) edge[draw=black] (b_43);\n        \\path[-] (xb2) edge[draw=black] (b_42);\n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_42);\n       \n        \n        \\path[-] (xb2) edge[draw=green] (xa2);\n        \\path[-] (xb1) edge[draw=green] (xa1);\n\n    \\end{tikzpicture}\n    \\end{minipage} \\hfill\\vline\\hfill\n    \\begin{minipage}{0.45\\textwidth}\n        \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 4.5) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 3) {$a_{12}$};\n        \\node[circle, draw=black] (a_13) at (0, 1.5) {$a_{13}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \\node[circle, draw=black] (b_13) at (0, -4.5) {$b_{13}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 4.5) {$a_{21}$};\n        \\node[circle, draw=black] (a_22) at (2, 3) {$a_{22}$};\n        \\node[circle, draw=black] (a_23) at (2, 1.5) {$a_{23}$};\n        \\node[circle, draw=black] (b_21) at (2, -1.5) {$b_{21}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \\node[circle, draw=black] (b_23) at (2, -4.5) {$b_{23}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 4.5) {$a_{31}$};\n        \\node[circle, draw=black] (a_32) at (4, 3) {$a_{32}$};\n        \\node[circle, draw=black] (a_33) at (4, 1.5) {$a_{33}$};\n        \\node[circle, draw=black] (b_31) at (4, -1.5) {$b_{31}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \\node[circle, draw=black] (b_33) at (4, -4.5) {$b_{33}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 4.5) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 3) {$a_{42}$};\n        \\node[circle, draw=black] (a_43) at (6, 1.5) {$a_{43}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \\node[circle, draw=black] (b_43) at (6, -4.5) {$b_{43}$};\n        \n        \\node[circle, draw=green] (xa) at (3, 2.25) {$x_{a}$};\n        \\node[circle, draw=green] (xb) at (3, -2.25) {$x_{b}$};\n        \n        \n       \n        \\path[-] (a_12) edge[draw=black] (a_23);\n        \\path[-] (a_11) edge[draw=black] (a_23);\n       \n        \\path[-] (a_11) edge[draw=black] (a_22);\n       \n       \n        \\path[-] (a_12) edge[draw=black] (a_21);\n       \n        \\path[-] (a_13) edge[draw=black] (a_21);\n        \\path[-] (a_13) edge[draw=black] (a_22);\n\n\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (a_32);\n        \\path[-] (a_22) edge[draw=black] (a_31);\n        \\path[-] (a_22) edge[draw=black] (a_32);\n        \n       \n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_43);\n        \\path[-] (a_32) edge[draw=black] (a_41);\n       \n        \\path[-] (a_32) edge[draw=black] (a_43);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n       \n        \n \n       \n       \n        \\path[-] (b_12) edge[draw=black] (b_23);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_23);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \\path[-] (b_13) edge[draw=black] (b_22);\n       \n        \\path[-] (b_13) edge[draw=black] (xb1);\n        \n        \n       \n      \n       \n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_33);\n        \\path[-] (b_23) edge[draw=black] (b_32);\n     \n        \\path[-] (b_23) edge[draw=black] (b_33);\n        \n       \n      \n       \n        \\path[-] (b_32) edge[draw=black] (b_43);\n      \n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_42);\n       \n        \n        \\path[-] (b_31) edge[draw=black] (b_42);\n        \\path[-] (b_31) edge[draw=black] (b_43);\n        \n        \\path[-] (b_21) edge[draw=black, bend right=10] (b_33);\n        \\path[-] (b_31) edge[draw=black, bend left=10] (b_23);\n        \\path[-] (b_31) edge[draw=black] (b_21);\n        \n        \\path[-] (xb) edge[draw=black] (b_21);\n        \\path[-] (xb) edge[draw=black] (b_22);\n        \\path[-] (xb) edge[draw=black] (b_31);\n        \\path[-] (xb) edge[draw=black] (b_32);\n        \n        \\path[-] (a_23) edge[draw=black, bend left=10] (a_31);\n        \\path[-] (a_33) edge[draw=black, bend right=10] (a_21);\n        \\path[-] (a_23) edge[draw=black] (a_33);\n        \n        \\path[-] (xa) edge[draw=black] (a_22);\n        \\path[-] (xa) edge[draw=black] (a_32);\n        \\path[-] (xa) edge[draw=black] (a_23);\n        \\path[-] (xa) edge[draw=black] (a_33);\n        \n        \n        \\path[-] (xa) edge[draw=green] (xb);\n     \n\n    \\end{tikzpicture}\n    \\end{minipage}\n    \\caption{The two types of connections between the sets $A$ and $B$ in the special case $r = 5$.}\n    \\label{fig:bridge_graph_h_odd}\n\\end{figure}\n\nWe begin by showing that we find three pairs of bridges of the first type, using an even number of vertices of both classes (\\ref{stepd:bridge} of Section \\ref{sec:o_extremal1}).\n\n\\begin{claim}\\label{lemma_matching_edges_odd}\nSuppose $\\delta(G) \\geq \\tfrac{n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. Furthermore, let $n$ be large enough. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_{r+1}}x_{b_{r+1}})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r+1$.\n\\end{claim}\n\\begin{claimproof}\nAs in the proof of Claim~\\ref{lemma_matching_edges}, it suffices to find $r+1$ edges from $A$ to $V \\setminus A$. \n\nFirst, suppose that $n$ is odd. If $\\abs{A} \\leq \\tfrac{n-r-2}{2}$, the minimum degree of $ \\tfrac{n+r-2}{2}$ guarantees that each vertex of $A$ needs to find at least $\\tfrac{n+r-2}{2} - (\\tfrac{n-r-2}{2} - 1) = r+1$ neighbors outside of $A$, hence the assertion follows.\nSuppose $\\abs{A} = \\tfrac{n-r}{2} + i$ with $i = 1, \\ldots,\\floor{\\tfrac r2}$. In this case, $\\abs{V \\setminus A} = \\tfrac{n+r}{2} - i$. Each vertex of $A$ finds at least $r-i$ neighbors outside of $A$. Suppose that $N(A) \\setminus A$ has size at most $r$ (thus, all edges from $A$ into the rest of the graph belong to at most $r$ vertices). Now pick a different vertex in the complement (which exists, as $\\tfrac{n+r}{2} - i \\gg r$). This vertex requires $\\tfrac{n+r-2}{2} - (\\tfrac{n+r}{2} - i - 1) = i$ neighbors in $A$, which is a contradiction.\nIf finally $\\abs{A} = \\tfrac{n-r}{2}$, each vertex of $A$ has at least $r$ neighbors in the complement of $A$. If all vertices of $A$ share those $r$ vertices (hence, we only find $r$ matching edges), those vertices are connected to all vertices in $A$, hence can be moved to $A$ by only increasing $\\alpha$-extremity, thus the assertion follows from the previous case.\n\nNow, if $n$ is even, because the minimum degree needs to be an integer, it is at least $\\tfrac{n+1+r-2}{2}$, hence upon removal of one vertex, we are left with a graph on $n' = n-1$ vertices and minimum degree at least $\\tfrac{n+1+r-2}{2} -1= \\tfrac{n'+r-2}{2}$ (and $n'$ being odd). Hence the assertion follows from the previous discussion.\n\\end{claimproof}\n\nSimilarly as in the even case, Claim~\\ref{lemma_matching_edges_odd} gives us $\\tfrac{r+1}{2}$ pairs of bridge-edges as in Figure \\ref{fig:bridge_graph_h_odd} (the green part). Clearly, the rest of the bridge graph can be created completely analogously to Claim~\\ref{bridge_graph_existence}.\n\nBy the same token we get the following claim immediately.\n\\begin{claim}\\label{cor_matching_edges_odd}\nSuppose $\\delta(G) \\geq \\tfrac{n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. Furthermore, let $n$ be large enough. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_{r-1}}x_{b_{r-1}})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq  \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r-1$. Furthermore, there are different vertices $x_a \\in A, x_b \\in B$ such that $x_ax_b \\in E(G)$, $\\abs{N(x_a) \\cap A} \\geq r-1$ and $\\abs{ N(x_b) \\cap B} \\geq r-1$.\n\\end{claim}\n\\begin{claimproof}\nThis follows directly from the proof of Claim~\\ref{lemma_matching_edges_odd}.\n\\end{claimproof}\n\nNext, we re-define the gluing operation GE to GO as follows.\n\\begin{claim}[Gluing operation GO]\nGiven two disjoint sets $D_1, D_2 \\subset A'_0$ of sizes exactly $\\tfrac{r+1}{2}$, we find two disjoint sets $D'_1, D'_2 \\subset A'_0 \\setminus (D_1 \\cup D_2)$ of size $\\tfrac{r+1}{2}$ such that\n\\[G[D_1, D'_1] \\equiv G[D'_1, D'_2] \\equiv G[D'_2, D_2] \\equiv K_{(r+1)\/2, (r+1)\/2} \\, .\\]\n\\end{claim}\n\\begin{claimproof}\nThis follows directly from the fact that each vertex in $A'_0$ is connected to at least $(\\tfrac 12 - 4 \\alpha)n$ vertices in $A'_0$ and GO uses only finitely many vertices of the neighborhoods.\n\\end{claimproof}\nWe stress at this point that GO can be applied to blocks whose end-vertices $D_1$ have currently degree $\\tfrac{r-1}{2}$ (then we chose all edges from the connecting graphs $K_{(r+1)\/2, (r+1)\/2}$) or degree $\\tfrac{r+1}{2}$ (then we chose the complete bipartite graph between $D'_1$ and $D'_2$ and at the other connections, we remove one matching of size $\\tfrac{r+1}{2}$.\nFurthermore observe, that gluing consumes $r+1$ vertices from the underlying set.\n\nWe proceed as follows. If $\\abs{C} > 0$ or $\\abs{C} = 0$ and $\\abs{A'_0}, \\abs{B'_0} \\equiv 0 \\pmod 2$, we create $\\tfrac{r+1}{2}$ pairs of bridge vertices by Claim~\\ref{lemma_matching_edges_odd}. Otherwise, we create $\\tfrac{r-1}{2}$ pairs of bridge vertices and one additional bridge by Claim~\\ref{cor_matching_edges_odd}. In both cases, we glue the bridges in $A$ and $B$ respectively together using mutually disjoint vertex sets $D^A_1, \\ldots,D^A_{(r-1)\/2}$ and $D^B_1, \\ldots,D^B_{(r-1)\/2}$ constructing $P_A, P_B$ and, similarly as in the even case, we set $A' = A'_0 \\setminus V(P_A), B' = B'_0 \\setminus V(P_B)$. \nClearly, the parity of $A'$ and $B'$ are both even.\n\nNext, we define absorbing structures for the left-over vertices and for absorbing vertices in order to guarantee divisibility (\\ref{stepd:absorb} of Section \\ref{sec:o_extremal1}). They need to be defined slightly differently as in the even case (Figure \\ref{fig:absorber_odd}).\n\n\\begin{definition}\n    Let $D \\in \\cbc{A', B'}$ and $u$ a vertex such that $\\abs{ N(u) \\cap D } \\geq \\tfrac n6$. Define $\\xi_r(u)$ as follows.\n    \\begin{itemize}\n         \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{(r+1)\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{(r+1)\/2}} \\subset  N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n         \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{(r-1)\/2}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n         \\item Select $E_0 = \\cbc{ e_1, \\ldots, e_{(r+1)\/2} }$ in the joint $D$-neighborhood of $D_1 \\setminus (D_2 \\cup D' \\cup \\cbc{u})$.  \n    \\end{itemize}\n    Define $\\xi_r(u)$ as the graph containing $E_0, D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$. Furthermore, take all edges from $D_2$ to $D' \\cup \\cbc{u}$ removing one matching of size $\\tfrac{r+1}{2}$ and the edges from $E_0$ to $D_1$ removing a matching as well.\n    Furthermore,  for two adjacent vertices $u_1, u_2 \\in D$ define $\\xi'_r(u_1, u_2)$ via\n    \\begin{itemize}\n         \\item Select $F_1 = \\cbc{f_1, f_2, \\ldots, f_{(r+1)\/2}}, F' = \\cbc{f'_1, \\ldots, f'_{(r+1)\/2}} \\subset N(u_1) \\cap N(u_2) \\cap D$, hence $r+1$ pairwise disjoint vertices in the joint neighborhood of $u_1$ and $u_2$.\n         \\item Select the following edges\n         \\begin{itemize}\n             \\item $f_1u_1, \\ldots, f_{(r-1)\/2}u_1$,\n             \\item $f'_1u_1, \\ldots, f'_{(r-1)\/2}u_1$,\n             \\item $f_2u_2, \\ldots, f_{(r+1)\/2}u_2$,\n             \\item $f'_2u_2, \\ldots, f'_{(r+1)\/2}u_2$ and\n             \\item $f_1f'_1$ as well as $f_{(r+1)\/2}f'_{(r+1)\/2}$.\n         \\end{itemize}\n         \\item Finally, draw $(r-2)$ half-edges at each $f_i, f'_i$ and match them such that a simple graph is induced by the matching.\n     \\end{itemize}\n\\end{definition}\nAs $u_1, u_2 \\in D$, hence the neighborhood of $u_1$, $u_2$ contains only vertices that, themselves, have $(\\tfrac 12 - 30 r \\alpha)n$ vertices from $D$, the joint neighborhood of those two vertices has size at least $(\\tfrac 12 - 60 r \\alpha )n$, $\\xi'(u_1, u_2)$ is well defined. Observe that absorbing a vertex $u \\not \\in D$ consumes $2r + 1$ vertices from $D$ while absorbing $u_1, u_2 \\in D$ consumes $r+3$ vertices (including $u_1, u_2$) in $D$.\n\nAs in the even case, we find a family of disjoint structures to absorb at least $100 \\alpha n$ different vertices.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n    \n    \\node[circle, draw=blue] (e1) at (0, 3) {$e_1$};\n     \\node[circle, draw=blue] (e2) at (0, 1) {$e_2$};\n     \\node[circle, draw=blue] (e3) at (0, -1) {$e_3$};\n\n     \\node[circle, draw=green!40] (d1) at (2, 3) {$d_1$};\n     \\node[circle, draw=green!40] (d2) at (2, 1) {$d_2$};\n     \\node[circle, draw=green!40] (d3) at (2, -1) {$d_3$};\n\n     \\node[circle, draw=green!40] (dp1) at (6, 3) {$d'_{1}$};\n     \\node[circle, draw=green!40] (dp2) at (6, 1) {$d'_{2}$};\n     \\node[circle, draw=green!40] (dp3) at (6, -1) {$d'_{3}$};\n     \n     \\node[circle, draw=blue!40] (up1) at (4, 3) {$u'_{1}$};\n     \\node[circle, draw=blue!40] (up2) at (4, 1) {$u'_{2}$};\n     \\node[circle, draw=green] (u) at (4, -1) {$u$};\n\n     \n     \\path[-] (u) edge[draw=green] (d1);\n     \\path[-] (u) edge[draw=green] (d2);\n     \\path[-] (u) edge[draw=green] (d3);\n     \\path[-] (u) edge[draw=green] (dp1);\n     \\path[-] (u) edge[draw=green] (dp2);\n    \n     \n     \\path[-] (d1) edge[draw=blue] (up1);\n     \\path[-] (d2) edge[draw=blue] (up1);\n     \\path[-] (d1) edge[draw=blue] (up2);\n     \\path[-] (d2) edge[draw=blue] (up2);\n     \\path[-] (d3) edge[draw=blue] (up1);\n     \\path[-] (d3) edge[draw=blue] (up2);\n     \n    \n     \\path[-] (d2) edge[draw=blue] (e1);\n     \\path[-] (d3) edge[draw=blue] (e1);\n     \\path[-] (d1) edge[draw=blue] (e2);\n    \n     \\path[-] (d3) edge[draw=blue] (e2);\n     \\path[-] (d1) edge[draw=blue] (e3);\n     \\path[-] (d2) edge[draw=blue] (e3);\n    \n     \n     \n    \n     \\path[-] (dp2) edge[draw=blue] (up1);\n     \\path[-] (dp3) edge[draw=blue] (up1);\n     \\path[-] (dp1) edge[draw=blue] (up2);\n    \n     \\path[-] (dp3) edge[draw=blue] (up2);\n     \n    \n     \n     \\node (h1) at (7, 4) {};\n     \\node (h2) at (7, -2) {};\n     \n  \n     \\node[circle, draw=green!40] (f1) at (8, 3) {$f_1$};\n     \\node[circle, draw=green!40] (f2) at (8, 1) {$f_2$};\n     \\node[circle, draw=green!40] (f3) at (8, -1) {$f_3$};\n     \n     \\node[circle, draw=green!40] (u1) at (10, 2) {$u_1$};\n     \\node[circle, draw=green!40] (u2) at (10, 0) {$u_2$};\n     \n     \\node[circle, draw=green!40] (fp1) at (12, 3) {$f'_1$};\n     \\node[circle, draw=green!40] (fp2) at (12, 1) {$f'_2$};\n     \\node[circle, draw=green!40] (fp3) at (12, -1) {$f'_3$};\n\n     \\path[-] (h1) edge[draw=black] (h2);\n     \n     \\path[-] (fp1) edge[draw=green] (u1);\n     \\path[-] (fp2) edge[draw=green] (u2);\n     \\path[-] (u2) edge[draw=green] (u1);\n     \n     \\path[-] (fp2) edge[draw=green] (u1);\n     \\path[-] (fp3) edge[draw=green] (u2);\n     \\path[-] (f1) edge[draw=green] (u1);\n     \\path[-] (f2) edge[draw=green] (u2);\n     \n     \\path[-] (f2) edge[draw=green] (u1);\n     \\path[-] (f3) edge[draw=green] (u2);\n     \n     \\path[-] (fp1) edge[draw=green] (f1);\n    \n    \n    \n    \n    \n    \n    \n     \\path[-] (fp3) edge[draw=green] (f3);\n     \n    \n    \\end{tikzpicture}\n    \n    \\caption{Absorbers $\\xi_5(u)$ (left) and $\\xi'_5(u_1, u_2)$ (right) with $r=5$, where $u, u_1, u_2$ are the green vertices, the green vertices are inside the $A'$-neighborhood of $u$ (or $u_1, u_2$ respectively) and the blue vertices are vertices chosen from $A'$.}\n    \\label{fig:absorber_odd}\n\\end{figure}\n\n\nSubsequently, we absorb $C$ using independent copies of $\\xi_r(\\cdot)$ such that the parity of the remaining vertices in the almost cliques is even. Then, as above, we glue the absorbers together by GO and are left with sets $A''$ and $B''$ (hence, $A'$ without the glued structures) and the path-like subgraph $P'_A$ of size at most $25r \\alpha n$, hence $\\abs{A''} \\geq (\\tfrac 12 - 30 r \\alpha n)$. As each vertex of $C$ has degree at least $\\tfrac{\\alpha}{100} n$ into $A''$ and $B''$, we can absorb one vertex such that in the end $A''$ and $B''$ contain an even number of vertices. Furthermore, each vertex in $A''$ is connected to at least $(\\tfrac 12 - 40 r \\alpha)n$ vertices in $A''$ and the same applies to $B''$. \n\nAgain, as in the even case, we need to make sure that $\\abs{A''} \\equiv 0 \\pmod{2r}$, as we want to close the cycle by blocks of subsequently fully connected layers of sizes $\\tfrac{r+1}{2}$ where the second block misses one matching of size $r+1$. If the divisibility condition holds, set $A''' = A''$ and proceed. Otherwise, if there is $0 < i < 2r$ such that $\\abs{A''} \\equiv i \\pmod{2r}$, $i$ has to be even as the parity of $A''$ guarantees. Select $\\tfrac i2$ pairs vertices $(a_{11}, a_{12}), \\ldots,(a_{i\/2, 1} a_{i\/2, 2}) \\in A'' \\times A''$ and absorb them using disjoint instances $\\xi'_r(a_{11}, a_{12}), \\ldots, \\xi'_r(a_{i\/2, 1} a_{i\/2, 2})$ with $D = A''$. As each absorber consumes $r + 3$ vertices, the divisibility condition now holds. Finally, as in the even case, glue the absorbed parts together with GO which does not change the divisibility by $r+1$. Thus, we are left with a set $A'''$ which consists of the vertices of $A''$ without the absorbed vertices and the gluing structures. Analogously, the same applies for $B'''$. Now, as in the even case, the result directly follows from Lemma~\\ref{lem:blowup} and the following claim (\\ref{stepd:cover} of Section \\ref{sec:o_extremal1}).\n\n\\begin{claim}\nThe constructed subgraph is $r$-connected and $r$-regular.\n\\end{claim}\n\\begin{claimproof}\nAs in the even case, $r$-regularity as well as $r$-connectivity on the $\\tfrac{r+1}{2}$-blow-up of the path part is obvious.\nThe first type of bridge (build with Claim~\\ref{lemma_matching_edges_odd}, see Figure~\\ref{fig:bridge_graph_h_odd} on the left) does not harm connectivity as before. In the second type of bridge (build with Claim~\\ref{cor_matching_edges_odd}, see Figure~\\ref{fig:bridge_graph_h_odd} on the right) only the special vertices $x_a$ and $x_b$ need our attention. But as they are of degree $r$ and connected to $(\\tfrac{r-1}{2}$ vertices on both sides of the $K_{(r+1)\/2), (r+1)\/2}$, isolating a part of the graph is not possible either. The absorbing structures clearly sustain the connectivity property.\n\\end{claimproof}\nThis finishes the proof of the first extremal case.\n\\end{proof}\n\n\\@ifstar{\\origsection*}{\\mysection}{Extremal Case II}\n\\label{sec:extremal2}\nIn this section we deal with the second extremal case and follow~\\ref{stepe:find}--\\ref{stepe:cover} as outlined in Section~\\ref{sec:o_extremal2}.\nWe start by proving the auxiliary lemma for finding stars.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:stars}]\nLet $\\alpha = \\tfrac{1}{32s^2(s+1)}$.\nAssume we have already found $0 \\le t < 2m$ copies of $K_{1,s}$ and let $V'$ be the remaining vertices.\nThen, by the maximum degree condition in $G$,\n\\[e(G[V']) \\ge \\tfrac 12 n (m+s-1) - t (s+1) 4 s \\alpha n \\,.\\]\nIf $m \\ge s+1$ this is at least $\\tfrac 14 n (m+s-1) \\ge \\tfrac 12 ns$ and gives a vertex of degree at least $s$ in $G[V']$.\nOn the other hand, if $m \\le s$ the above is at least $(\\tfrac 12 s - \\tfrac 14) n > \\tfrac{s-1}{2}n$ and again this gives a vertex of degree at least $s$ in $G[V']$.\n\\end{proof}\n\n\\begin{proof}[Proof of Extremal Case II]\nLet $r \\ge 2$ and $s = \\lceil \\tfrac{r}{2} \\rceil \\ge 1$.\nLet $\\eps>0$ be given by Lemma~\\ref{lem:blowup} on input $\\tfrac 12$, $\\tfrac 12$, and $r$.\nWe obtain $\\alpha>0$ from Lemma~\\ref{lem:stars} and additionally assume that $40 s^2 \\alpha < \\eps$.\nThen let $G$ be an $n$-vertex graph with minimum degree $\\delta(G) \\ge \\tfrac{n+r-2}{2}$ and $nr \\equiv 0 \\pmod 2$.\nFurther assume, that there is a partition of $V(G)$ into $A$ and $B$ of size $|A|+m=\\tfrac 12 n=|B|-m$, where $0 \\le m \\le \\alpha n$ such that between these sets we have minimum degree $\\alpha n$ and all but at most $\\alpha n$ vertices from $A$ (or $B$) have degree $|B|-\\alpha n$ (or $|A|-\\alpha n$) into $B$ (or $A$).\n\n\\noindent \\textbf{\\ref{stepe:find}.}\nNote $\\delta(G[B]) \\ge m+s-1$.\nLet $B' \\subseteq B$ be the vertices of degree at most $2 s \\alpha n$ in $G[B]$ and let $m'=|B \\setminus B'|$.\nIf $m'<m$, then $\\delta(G[B']) \\ge (m-m')+s-1$ and we apply Lemma~\\ref{lem:stars} to find $2(m-m')$ copies of $K_{1,s}$.\nBy choice of $B'$ each vertex from these copies of $K_{1,s}$ has degree at least $|A|-2s\\alpha n$ into $A$.\nLet $W$ be the union of the vertices from these copies of $K_{1,s}$\n\nAfterwards, for $i=1,\\dots,\\min\\{m', m\\}$ we can pick a vertex $x_i \\in B \\setminus (B' \\cup \\{x_1,\\dots,x_{i-1} \\})$ and neighbours $y_{i,1},\\dots,y_{i,r}$ of $x_i$ from $A \\setminus (W \\cup \\bigcup_{j=1}^{i-1} \\{y_{j,1},\\dots,y_{j,r} \\}$ such that $y_{i,1},\\dots,y_{i,r}$ have degree at least $|A|-\\alpha n$  into $A$.\nFor some $0 \\le m' \\le m$ we have obtained $m'$ copies of  $K_{1,r}$ and $2(m-m')$ copies of $K_{1,s}$ such that all vertices in the copies of $K_{1,s}$ and the leaves in the copies of $K_{1,r}$ have degree at least $|A|-2s\\alpha n$ into $A$.\n\n\\noindent \\textbf{\\ref{stepe:absorb}.}\nWe will now iteratively absorb these copies of $K_{1,s}$ and $K_{1,r}$ into an $r$-regular path-structure.\nLet $W$ be the set of vertices in the union of these copies.\nWe start with $r$ vertices $\\mathbf{v}=(v_1,\\dots,v_s)$ from $A \\setminus W$ that have degree at least $|A|-\\alpha n$ into $A$ and $s$ vertices $\\mathbf{u}_0=(u_{0,1},\\dots,u_{0,s})$ from $B \\setminus W$ that have degree at least $|B|-\\alpha n$ into $B$ such that $v_ju_{0,j'}$ is an edge for $1 \\le j, j' \\le s$.\nWe add the vertices in $\\mathbf{v}$ and $\\mathbf{u}_0$ to $W$.\nNow for some $i=0,\\dots,m$ let $W$ be the vertices used for the structure and in the remaining copies of $K_{1,s}$ and $K_{1,s}$ with $|W| \\le 6 s i + 2s$ and let $\\mathbf{u}_i=(u_{i,1},\\dots,u_{i,r})$ be one end of the structure in $B$ such that $u_{i,j}$ has at least $|A|-2s\\alpha n$ neighbours in $A$ for $j=1,\\dots,s$.\n\nIf there are two copies of $K_{1,s}$ left then by alternating between $A$ and $B$ we find the structure shown in Figure~\\ref{fig:absorbK1s} where the vertices on the left are $\\mathbf{u}_i$ and the vertices on the right we denote by $\\mathbf{u}_{i+1}=(u_{i+1,1},\\dots,u_{i+1,s})$.\nNote that $u_{i+1,1},\\dots,u_{i+1,s}$ are exactly the leaves of one of the $K_{1,s}$ and, therefore, have at least degree $|A|-2s\\alpha n$ into $A$.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1.5) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b1) at (-3, -0.75) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a21) at (-0, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1.5) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b2) at (1, -0.75) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1.5) {};\n\n\n     \n     \\path[-] (b11) edge[draw=black] (a11);\n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b12) edge[draw=black] (a11);\n     \\path[-] (b12) edge[draw=black] (a12);\n    \n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b1);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b1);\n     \n     \\path[-] (b1) edge[draw=green!60] (b21);\n     \\path[-] (b1) edge[draw=green!60] (b22);\n     \n     \\path[-] (b21) edge[draw=black] (a21);\n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b22) edge[draw=black] (a21);\n     \\path[-] (b22) edge[draw=black] (a22);\n    \n     \\path[-] (a21) edge[draw=black] (b31);\n     \\path[-] (a21) edge[draw=black] (b2);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \\path[-] (a22) edge[draw=black] (b2);\n     \n     \\path[-] (b2) edge[draw=green!60] (b31);\n     \\path[-] (b2) edge[draw=green!60] (b32);\n\n   \n   \n    \n   \n    \\end{tikzpicture}\n    \\qquad \n    \\begin{tikzpicture}[scale=0.8]\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b13) at (-6, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a13) at (-4, -3) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b1) at (-3, -1) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b23) at (-2, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a21) at (-0, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a23) at (-0, -3) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b2) at (1, -1) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b33) at (2, -3) {};\n\n\n    \n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b11) edge[draw=black] (a13);\n     \\path[-] (b12) edge[draw=black] (a11);\n    \n     \\path[-] (b12) edge[draw=black] (a13);\n     \\path[-] (b13) edge[draw=black] (a11);\n     \\path[-] (b13) edge[draw=black] (a12);\n    \n    \n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a11) edge[draw=black] (b1);\n     \\path[-] (a11) edge[draw=black, bend left = 15] (b23);\n     \n     \\path[-] (a12) edge[draw=black] (b21);\n     \\path[-] (a12) edge[draw=black] (b1);\n     \\path[-] (a12) edge[draw=black] (b23);\n     \n     \\path[-] (a13) edge[draw=black, bend left = 15] (b21);\n     \\path[-] (a13) edge[draw=black] (b1);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \n     \\path[-] (b1) edge[draw=green!60] (b21);\n     \\path[-] (b1) edge[draw=green!60] (b22);\n     \n    \n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b21) edge[draw=black] (a23);\n     \\path[-] (b22) edge[draw=black] (a21);\n    \n     \\path[-] (b22) edge[draw=black] (a23);\n     \\path[-] (b23) edge[draw=black] (a21);\n     \\path[-] (b23) edge[draw=black] (a22);\n     \\path[-] (b23) edge[draw=black] (a23);\n    \n     \\path[-] (a21) edge[draw=black] (b32);\n     \\path[-] (a21) edge[draw=black] (b2);\n     \\path[-] (a21) edge[draw=black, bend left = 15] (b33);\n    \n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (a22) edge[draw=black] (b2);\n     \\path[-] (a22) edge[draw=black] (b33);\n     \n     \\path[-] (a23) edge[draw=black, bend left = 15] (b31);\n     \\path[-] (a23) edge[draw=black] (b32);\n    \n     \\path[-] (b2) edge[draw=green!60] (b31);\n     \\path[-] (b2) edge[draw=green!60] (b32);\n     \\path[-] (b2) edge[draw=green!60] (b33);\n     \\end{tikzpicture}\n    \n    \\caption{Including two copies of $K_{1,s}$ into an $r$-regular path structure in the even ($r=4$) and odd case ($r=5$) with blue vertices in $A$ and green vertices in $B$.}\n    \\label{fig:absorbK1s}\n\\end{figure}\n\nOtherwise, we pick a copy of $K_{1,r}$ and find the structure shown in Figure~\\ref{fig:absorbK1r} (this is an $\\tfrac r2$-blow-up of a path), where the vertices on the left are $\\mathbf{u}_i$ and the vertices on the right we denote by $\\mathbf{u}_{i+1}=(u_{i+1,1},\\dots,u_{i+1,s})$.\nWe do not need any assumption on the degree of the centre vertex of the $K_{1,r}$ as we do not need any additional edges containing it.\nAgain $u_{i+1,1},\\dots,u_{i+1,r}$ have at least degree $|A|- 2s\\alpha n$ into $A$.\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1.5) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1.5) {};\n\n     \\node[circle, draw=green!60, fill=green!60] (b1) at (-0, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1.5) {};\n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1.5) {};\n\n\n     \n     \\path[-] (b11) edge[draw=black] (a11);\n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b12) edge[draw=black] (a11);\n     \\path[-] (b12) edge[draw=black] (a12);\n    \n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b21);\n\n     \\path[-] (b21) edge[draw=green!60] (b1);\n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b22) edge[draw=green!60] (b1);\n     \\path[-] (b22) edge[draw=black] (a22);\n    \n     \\path[-] (b1) edge[draw=green!60] (b31);\n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (b1) edge[draw=green!60] (b32);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \n    \\end{tikzpicture}\n    \\qquad \n    \\begin{tikzpicture}[scale=0.8]\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b13) at (-6, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a13) at (-4, -3) {};\n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b23) at (-2, -3) {};\n\n     \\node[circle, draw=green!60, fill=green!60] (b1) at (-0, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a23) at (-0, -3) {};\n     \n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b33) at (2, -3) {};\n\n\n    \n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b11) edge[draw=black] (a13);\n     \\path[-] (b12) edge[draw=black] (a11);\n    \n     \\path[-] (b12) edge[draw=black] (a13);\n     \\path[-] (b13) edge[draw=black] (a11);\n     \\path[-] (b13) edge[draw=black] (a12);\n    \n    \n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b23);\n     \n     \\path[-] (a12) edge[draw=black] (b21);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b23);\n     \n     \\path[-] (a13) edge[draw=black] (b21);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \n    \n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b21) edge[draw=black] (a23);\n     \\path[-] (b22) edge[draw=green!60] (b1);\n    \n     \\path[-] (b22) edge[draw=black] (a23);\n     \\path[-] (b23) edge[draw=green!60] (b1);\n     \\path[-] (b23) edge[draw=black] (a22);\n    \n    \n     \\path[-] (b1) edge[draw=green!60] (b32);\n     \\path[-] (b1) edge[draw=green!60] (b31);\n     \\path[-] (b1) edge[draw=green!60] (b33);\n    \n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \\path[-] (a22) edge[draw=black] (b33);\n     \n     \\path[-] (a23) edge[draw=black] (b31);\n     \\path[-] (a23) edge[draw=black] (b32);\n     \\path[-] (a23) edge[draw=black] (b33);\n\n     \\end{tikzpicture}\n    \n    \\caption{Including one copy of $K_{1,r}$ into an $r$-regular path structure in the even ($r=4$) and odd case ($r=5$) with blue vertices in $A$ and green vertices in $B$.}\n    \\label{fig:absorbK1r}\n\\end{figure}\n\nWe add all new vertices in the structure to $W$. We can repeat this until all copies of $K_{1,s}$ and $K_{1,r}$ are covered, because in each step $W$ will increase by at most $6r$ vertices and all vertices have large enough neighbourhoods.\nWe let $\\mathbf{u}'=(u_{m,1},\\dots,u_{m,s})$ be the final end of this construction.\nNow let $A_1=A \\setminus W$ and $B_2=B \\setminus W$ and note that $|A_1|=|B_1|$ because our construction uses $2m$ vertices more from $B$ than from $A$.\n\n\\noindent \\textbf{\\ref{stepe:absorb2}.}\nFor the next step let $A' \\subseteq A_1$ be the vertices of degree at most $|A_1|-5 s \\alpha n$ into $B_1$ and  $B' \\subseteq B_1$ be the vertices of degree at most $|A_1|-5 s \\alpha n$ into $A_1$.\nNote that $|A'|,|B'| \\le \\alpha n$, because we removed at most $4 s \\alpha n$ vertices from each of $A$ and $B$ to get $A_1$ and $B_1$.\nBy using an $\\tfrac r2$-blow-up of a path (similar as in Figure~\\ref{fig:absorbK1r}) we cover the vertices in $A'$ and $B'$ with our $r$-regular path structure covering in total at most $12 s \\alpha n$ additional vertices.\nFor this we iteratively extend from $\\mathbf{u}'$.\nLet $W$ be the vertices used for covering $A'$ and $B'$ and let $\\mathbf{u}=(u_1,\\dots,u_s)$ be the last vertices of this construction in $B$ and note that we can assume that they have degree at least $|A|-20 s \\alpha n$ into $A$.\nWe let $A_2=(A_1 \\setminus W)$ and $B_2=(B_1 \\setminus W)$ and note that $|A_2|=|B_2|$.\n\n\\noindent \\textbf{\\ref{stepe:cover}.}\nWe have that every vertex from $A_2$ (or $B_2$) has degree at least $|A_2|-20 s \\alpha n$ into $B_2$ (or into $A_2$).\nThen it is easy to see that $(A_2,B_2)$ is $(\\eps,\\tfrac 12)$-super-regular as $40 s \\alpha < \\eps$.\nMoreover, the vertices in $\\mathbf{v}$ have $|B_2| - 20s^2\\alpha n \\ge \\tfrac 12 |B_2|$ common neighbours in $B_2$ and similarly for $\\mathbf{u}$ with $A_2$.\nWe apply Lemma~\\ref{lem:blowup} to cover the remaining vertices of $A_2$ and $B_2$ with the $\\tfrac r2$-blow-up of a path such that the ends connect to $\\mathbf{v}$ and $\\mathbf{u}$.\nThis completes the construction of a spanning $r$-regular structure in $G$.\nTo see that it is also $r$-connected it suffices to note that we have the $\\tfrac r2$-blow-up of a path except for some parts replaced by the graphs from Figure~\\ref{fig:absorbK1s}, which do not harm this property.\n\\end{proof}\n\n\\@ifstar{\\origsection*}{\\mysection}{Non-Extremal Case}\n\\label{sec:non-extremal}\n\nIn this section we deal with the case that $G$ is not $\\alpha$-extremal.\nRecall, that the assumption implies that for any two sets $A,B \\subseteq V(G)$ of size $(\\tfrac12 - \\alpha)n \\le |A|,|B| \\le \\tfrac n2$ we have $d(A,B) \\ge \\alpha$.\nWe will follow~\\ref{step:regularise}--\\ref{step:spanning} as outlined in Section~\\ref{sec:o_non-extremal}.\n\n\\begin{proof}[Proof of Non-Extremal Case]\nGiven $r \\ge 3$ and $0<\\alpha<\\tfrac{1}{32}$ we choose constants such that\n\\begin{align*}\n    \\varepsilon \\ll \\nu \\ll d \\ll \\beta \\ll \\alpha\\,,\n\\end{align*}\nwhere, in particular,\n\\begin{align*}\n    \\eps \\le \\nu, \\qquad \\nu \\le d^{s+1}, \\qquad 10 s d \\le \\beta, \\qquad 500 s \\beta \\le \\alpha \\,\n\\end{align*}\nand $2 \\eps$ is small enough for Lemma~\\ref{lem:blowup} with input $\\tfrac d2$, $\\tfrac 14 d^s$, and $r$.\nLet $M$ be given by Lemma~\\ref{lem:regularity} on input $\\eps$ and let $s=\\lceil \\tfrac r2 \\rceil$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:regularise}.}\nLet $G$ be an $n$-vertex graph with minimum degree $\\delta(G) \\ge (\\tfrac 12-\\beta)n$.\nFrom Lemma~\\ref{lem:regularity} we get a partition of the vertex set $V(G)$ into $\\ell+1 \\le M$ clusters $V_0,\\dots,V_\\ell$ of size $L$ and a subgraph $G' \\subseteq G$ such that~\\ref{reg:size}--\\ref{reg:reg} hold.\nWe denote by $R$ the graph on vertex set $[\\ell]$ with edges $ij$ if and only if the pair $(V_i,V_j)$ is $(\\eps,d)$-regular.\nIn $R$ we have minimum degree $\\delta(R) \\ge (\\tfrac 12-\\beta-2d)\\ell$, because otherwise with~\\ref{reg:ind} there would be a vertex with degree in $G'$ at most $(\\tfrac 12-\\beta-2d) \\ell \\cdot \\tfrac{n}{\\ell} + \\eps n < (\\tfrac 12-\\beta) n - (d+\\eps)n$ contradicting~\\ref{reg:deg}.\nSimilarly, we can deduce that $R$ is not $\\tfrac \\alpha 2$-extremal.\nOtherwise, there would be two sets of vertices $\\mathcal{A}$, $\\mathcal{B}$ in $R$ such that $(\\tfrac 12-\\tfrac \\alpha2 )\\ell \\le |\\mathcal{A}|,|\\mathcal{B}| \\le \\tfrac 12 \\ell$ and $d(\\mathcal{A},\\mathcal{B})< \\tfrac \\alpha2$.\nThen $A=\\bigcup_{i \\in \\mathcal{A}} V_i$ and $B=\\bigcup_{i \\in \\mathcal{B}} V_i$ both have size at most $\\tfrac \\ell 2 \\cdot \\tfrac{n}{\\ell} = \\tfrac n2$ and at least $(\\tfrac 12- \\tfrac \\alpha 2) \\ell \\cdot (1-\\eps) \\tfrac{n}{\\ell} \\ge (\\tfrac 12-\\alpha)n$ and we have\n\\[ d(A,B) = \\frac{e(A,B)}{|A||B|} \\le \\frac{\\tfrac \\alpha 2 \\cdot (\\tfrac n \\ell)^2 \\cdot |\\mathcal{A}||\\mathcal{B}| + |A| (d+\\eps) n}{|A| |B|} \\le \\frac{\\tfrac \\alpha 2}{(1-\\eps)^2} + \\frac{(d+\\eps)}{\\tfrac 12-\\alpha} \\le \\alpha\\,,\\]\nwhich contradicts our assumption that $G$ is not $\\alpha$-extremal.\nWe will repeatedly use the following fact that holds as $R$ is not $\\tfrac \\alpha2$-extremal.\n\\begin{fact}\n\\label{fact:non-extremal}\n    For any two sets $\\mathcal{A}, \\mathcal{B} \\subseteq V(R)$ of size at least $(\\tfrac 12 - \\tfrac \\alpha 2)\\ell$ there is an edge $AB \\in E(R)$ with $A \\in \\mathcal{A}$ and $B \\in \\mathcal{B}$.\n\\end{fact}\nIn the following we will abuse notation and also treat the clusters as vertices of $R$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:matching}.}\nNext let $M$ be a largest matching in $R$ and $M_0 \\subseteq [\\ell]$ be the clusters not covered by $M$.\nNaturally, $M_0$ is an independent set in $R$ and if there are at least two vertices $u$ and $v$ in $M_0$ then no neighbour of $u$ is connected to a neighbour of $v$ by an edge of $M$.\nTherefore, $2|M| \\ge \\deg_R(u)+\\deg_R(v)$ and $|M | \\ge (\\tfrac 12-\\beta-2d) \\ell$.\nWe let $\\ell'=|M|$ and denote the regular pairs corresponding to edges of $M$ by $(X_i,Y_i)$ for $i=1,\\dots,\\ell'$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:connect}.}\nFor any clusters $Z$ and $W$ we call an $s$-tuple $\\mathbf{z}=(z_1,\\dots, z_s)$ from $Z$ \\emph{well-connected into $W$} if the vertices $z_1,\\dots,z_s$ have at least $\\tfrac 12 d^s L$ common neighbours in $W$.\nNow fix any $i \\in [\\ell']$.\nWe want to connect $Y_i$ and $X_{i+1}$ by the $\\tfrac r2$-blow-up of a path.\nFor this we consider the neighbours $\\mathcal{W}$ and $\\mathcal{Z}$ in $R$ of $X_{i+1}$ and $Y_i$, respectively.\nIt follows from the minimum degree in $R$ that $\\mathcal{W}$ and $\\mathcal{Z}$ have size at least $(\\tfrac 12 - \\tfrac \\alpha 2)\\ell$.\nBy Fact~\\ref{fact:non-extremal} there is $W \\in \\mathcal{W}$ and $Z \\in \\mathcal{Z}$ such that $WZ$ is an edge in $R$.\n\nAll but at most $2 s \\eps L^s$ $s$-tuples $\\mathbf{x}=(x_1,\\dots, x_s)$ from $X_{i+1}$ are well connected into $W$ and $Y_{i+1}$.\nThe same holds for tuples from $Y_i$, $W$, and $Z$ with respect to the neighbouring clusters.\nWe fix tuples $\\mathbf{x}=(x_1,\\dots, x_s)$, $\\mathbf{w}=(w_1,\\dots, w_s)$, $\\mathbf{z}=(z_1,\\dots, z_s)$, and $\\mathbf{y}=(y_1,\\dots, y_s)$ from $X$, $W$, $Z$, and $Y$, respectively, such that $\\mathbf{x}\\mathbf{w}\\mathbf{z}\\mathbf{y}$ gives the $\\tfrac r2$-blow up of a path on $4$ vertices.\nWe denote this path by $P_i$ and remove any internal vertices (those in $\\mathbf{w}$ and $\\mathbf{z}$) from the clusters.\nWe can repeat this for all $i$, because we need only $4 s \\ell'$ vertices in total.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:superreg}.}\nTo make the matching edges super-regular, we let $i \\in [\\ell']$ and apply Lemma~\\ref{lem:superreg} to the pair $(Y_i,X_i)$.\nAfter removing a few additional vertices we arrive at sets $Y_i$ and $X_i$ such that $|Y_i|=|X_i| = L' \\ge (1-2\\eps)L$, where $L' \\equiv 0 \\pmod{s}$, and the pair $(Y_i,X_i)$ is $(2 \\eps, d-2\\eps)$-super-regular.\nNote that we can do this such that the end-tuple $\\mathbf{x}$ of $P_{i-1}$ is contained in $X_i$ and the end-tuple $\\mathbf{y}$ of $P_{i}$ is contained in $Y_i$.\nWe add the vertices removed during this procedure and also the vertices that belong to clusters of $M_0$ to $V_0$ and note that $|V_0| \\le \\eps n + \\ell' 4 \\eps L + \\beta n + 2dn \\le 2 \\beta n$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:absorb}. Setup.}\nWe want to absorb $V_0$ by extending the paths $P_i$.\nAfter each extension we need to maintain the location of the end-tuples and also ensure that they are well-connected.\nDuring the procedure we will have to deal with sets of already covered vertices.\nFor this let $W_0$ be a set of size at most $18 s^2 \\nu n$ and let $W$ be a set of size at most $20 s \\beta n$.\nThese will be the sets of vertices that we already used.\nThere are at most $18 s^2 \\nu n \/ (\\tfrac 14 d^s \\tfrac n\\ell) = 72 s^2 \\nu d^{-s}  \\ell \\le 8 s \\beta \\ell$ clusters that intersect $W_0$ in at least $\\tfrac 14 d^s \\tfrac n\\ell$ vertices and at most $20 s \\beta n \/ (\\tfrac 12 \\cdot \\tfrac n\\ell) \\le 40 s \\beta \\ell$ clusters that intersect $W$ in at least $\\tfrac 12 \\cdot \\tfrac n\\ell$ vertices.\nWe denote by $H$ the set of all clusters that do not have this property.\n\nNow consider a vertex $v \\in V_0$.\nThere are at most $(\\tfrac 12 + 50 s \\beta) \\ell$ clusters that intersect $N_G(v) \\setminus (W \\cup W_0)$ in less than $d \\tfrac n \\ell$ vertices.\nTherefore, there are at least $(\\tfrac 12 - 100 s \\beta) \\ell$ clusters in $H$ that intersect $N_G(v) \\setminus (W \\cup W_0)$ in at least $d \\tfrac n \\ell$ vertices.\nWe denote this set of clusters by $H(v)$.\nSimilarly, let $H_M(v)$ be the clusters form $H$, which share an edge of $M$ with another cluster from $H(v)$ and note that we have the same lower bound as $M$ is a matching.\nSumming up we have $|H| \\ge (1-50 s \\beta)\\ell$ and $|H(v)|,|H_M(v)| \\ge (\\tfrac 12 - 100 s \\beta) \\ell$ for all $v \\in V_0$.\nNote that $H(v)$ and $H_M(v)$ are large enough for Fact~\\ref{fact:non-extremal}.\n\n\\smallskip\n\\textbf{Covering $2s$ vertices.}\nWe let $W=\\emptyset$ and $W_0$ be all internal vertices (not in end-tuples) of the paths $P_1,\\dots,P_\\ell$.\nWe pick any $2s$ vertices $v_1,\\dots,v_{2s}$ from $V_0 \\setminus (W \\cup W_0)$ and our goal is to embed them, such that in any pair $(X_i,Y_i)$ we still have $|X_i|=|Y_i| \\equiv 0 \\pmod s$.\nLet $i_1$ be such that $X_{i_1} \\in H(v_1)$ and $Y_{i_1} \\in H$.\nAs the end-tuple of $P_{i_1}$ in $X_{i_1}$ is well-connected into $Y_{i_1}$ and $(N_G(v)\\cap X_{i_1}) \\setminus (W \\cup W_0)$ is of size at least $d \\tfrac n \\ell$, we can greedily pick tuples $\\mathbf{x}_1$, $\\mathbf{x}_2$ in $X_{i_1}$ and $\\mathbf{y}_1$, $\\mathbf{y}_2$ in $Y_{i_1}$ with the exception that $\\mathbf{y}_1$ contains $v_1$ and such that $\\mathbf{x}_1\\mathbf{y}_1\\mathbf{x}_2\\mathbf{y}_2P_{i_1}$ gives the $\\tfrac r2$-blow-up of a path and $\\mathbf{x_1}$ is well-connected into $Y_{i_1}$.\nNow we remove the internal vertices (those in $\\mathbf{x_2}$, $\\mathbf{y_1}$, $\\mathbf{y_2}$ and the end-tuple $\\mathbf{x}$ of $P_{i_1}$) from the clusters and add them to $W_0$ (this adds $4s$ vertices; see Figure~\\ref{fig:Step51}).\nWe note that $|Y_{i_1}|-1=|X_{i_1}|\\equiv 0 \\pmod s$ and let $P_{i_1}$ be the longer path.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n\n    \\node[circle, draw=red, fill = red, minimum width = 6, font=\\tiny] (v1) at (0,0) {};\n    \\node[] at (0.5, 0) {$v_1$};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x11) at (-2.7,1) {};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x12) at (-2.7,1.5) {};\n    \\node[] at (-3.2, 1.25) {$\\mathbf{x_1}$};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x21) at (-2.25,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x22) at (-2.25,2.75) {};\n    \\node[] at (-2.7, 2.5) {$\\mathbf{x_2}$};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (y1) at (1,1) {};\n    \\node[] at (1.5, 1) {$y_1$};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y21) at (1,2.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y22) at (1,3) {};\n    \\node[] at (1.5, 2.75) {$\\mathbf{y_2}$};\n\n    \n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x1) at (-2.7,3.5) {};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x2) at (-2.7,4) {};\n    \\node[] at (-3.2, 3.75) {$\\mathbf{x}$};\n    \n    \\node[ellipse, draw = black, minimum width = 60, minimum height = 100, label = above:{$X_{i_1}$}] (Xi) at (-2.75,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 60, minimum height = 100, label = above:{$Y_{i_1}$}] (Yi) at (1.25,2.375) {};\n    \n    \\path[-] (Xi) edge[draw=blue!20, line width=20pt] (Yi);\n    \n    \\path[-] (v1) edge[draw=red] (x11);\n    \\path[-] (v1) edge[draw=red] (x21);\n    \\path[-] (v1) edge[draw=red] (x12);\n    \\path[-] (v1) edge[draw=red] (x22);\n    \n    \\path[-] (y1) edge[draw=black] (x11);\n    \\path[-] (y1) edge[draw=black] (x21);\n    \\path[-] (y1) edge[draw=black] (x12);\n    \\path[-] (y1) edge[draw=black] (x22);\n    \n    \\path[-] (y21) edge[draw=black] (x11);\n    \\path[-] (y22) edge[draw=black] (x21);\n    \\path[-] (y22) edge[draw=black] (x12);\n    \\path[-] (y21) edge[draw=black] (x22);\n    \n    \\path[-] (y21) edge[draw=black] (x1);\n    \\path[-] (y22) edge[draw=black] (x2);\n    \\path[-] (y21) edge[draw=black] (x2);\n    \\path[-] (y22) edge[draw=black] (x1);\n    \n\n\n    \n     \\end{tikzpicture}\n    \n    \\caption{Absorbing vertex $v_1$ in the case $r=2s=4$ if $\\mathbf{x}$ is the current end of path $P_i$ and $\\mathbf{x_1}$ is the new end.}\n    \\label{fig:Step51}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi) at (-11.2, 2.5) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj1) at (-7.9, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj2) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj3) at (-7.9, 2.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj4) at (-7.9, 3.25) {};\n    \n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj) at (-4.75,1.875) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj1) at (-5.2, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj2) at (-5.2, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj3) at (-5.2, 2.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj4) at (-5.2, 3.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj5) at (-5.2, 4) {};\n    \\node[] at (-4.7, 3.75) {$P_{i_j}$};\n    \n    \n    \n    \\node[circle, draw=red, fill = red, minimum width = 6, font=\\tiny] (v1) at (0,0) {};\n    \\node[] at (0.5, 0) {$v_t$};\n    \\node[circle, draw=black, fill=black, font=\\tiny, minimum width = 6] (x11) at (-1.7,1) {};\n    \\node[circle, draw=black, fill=black, font=\\tiny, minimum width = 6] (x12) at (-1.7,1.5) {};\n\n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x21) at (-1.7,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x22) at (-1.7,2.75) {};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x31) at (-1.7,3.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x32) at (-1.7,4) {};\n\n    \n    \\node[circle, draw=orange, fill=orange, minimum width = 6, font=\\tiny] (y11) at (1,1) {};\n    \\node[circle, draw=orange, fill=orange, minimum width = 6, font=\\tiny] (y12) at (1,1.5) {};\n\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y21) at (1,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y22) at (1,2.75) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (y31) at (1,3.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (y32) at (1,4) {};\n    \n    \\node[] at (1.5, 3.75) {$P_{i_t}$};\n\n\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_{t}-1}$}] (Yi) at (-11.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_j}$}] (Yj) at (-5,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{i_j}$}] (Xj) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{i_t}$}] (Xt) at (-1.7,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_t}$}] (Yt) at (1.2,2.375) {};\n    \n    \\path[-] (Xt) edge[draw=blue!20, line width=20pt] (Yt);\n    \\path[-] (Xt) edge[draw=black!20, line width=20pt] (Yj);\n    \\path[-] (Xj) edge[draw=blue!20, line width=20pt] (Yj);\n    \\path[-] (Xj) edge[draw=black!20, line width=20pt] (Yi);\n    \n    \\path[-] (v1) edge[draw=red] (x11);\n    \\path[-] (v1) edge[draw=red] (x21);\n    \\path[-] (v1) edge[draw=red] (x12);\n    \\path[-] (v1) edge[draw=red] (x22);\n    \n    \\path[-] (y11) edge[draw=black] (x11);\n    \\path[-] (y12) edge[draw=black] (x11);\n    \\path[-] (y11) edge[draw=black] (x12);\n    \\path[-] (y12) edge[draw=black] (x12);\n    \n    \\path[-] (y21) edge[draw=black] (x11);\n    \\path[-] (y21) edge[draw=black] (x12);\n    \\path[-] (y21) edge[draw=black] (x21);\n    \\path[-] (y21) edge[draw=black] (x22);\n    \n    \\path[-] (y22) edge[draw=black] (x31);\n    \\path[-] (y22) edge[draw=black] (x32);\n    \\path[-] (y22) edge[draw=black] (x21);\n    \\path[-] (y22) edge[draw=black] (x22);\n    \n    \\path[-] (yj) edge[draw=black] (x31);\n    \\path[-] (yj) edge[draw=black] (x32);\n    \\path[-] (yj) edge[draw=black] (x21);\n    \\path[-] (yj) edge[draw=black] (x22);\n    \n    \\path[-] (xj1) edge[draw=black] (yj1);\n    \\path[-] (xj1) edge[draw=black] (yj2);\n    \\path[-] (xj1) edge[draw=black] (yj3);\n    \n    \\path[-] (xj2) edge[draw=black] (yj1);\n    \\path[-] (xj2) edge[draw=black] (yj2);\n    \\path[-] (xj2) edge[draw=black] (yj3);\n    \n    \\path[-] (xj3) edge[draw=black] (yj4);\n    \\path[-] (xj3) edge[draw=black] (yj5);\n    \\path[-] (xj3) edge[draw=black] (yj3);\n    \n    \\path[-] (xj4) edge[draw=black] (yj4);\n    \\path[-] (xj4) edge[draw=black] (yj5);\n    \\path[-] (xj4) edge[draw=black] (yj3);\n    \n    \\path[-] (yi) edge[draw=black] (xj1);\n    \\path[-] (yi) edge[draw=black] (xj2);\n    \\path[-] (yi) edge[draw=black] (xj3);\n    \\path[-] (yi) edge[draw=black] (xj4);\n    \n\n\n    \n     \\end{tikzpicture}\n    \n    \\caption{Absorbing vertex $v_t$ in the case $r=2s=4$. The blue connection between classes indicates that those edges belong to the fixed matching in the cluster graph while gray connections indicate using additional edges.}\n    \\label{fig:Step52}\n\\end{figure}\n\nWe continue in a similar fashion to cover $v_2,\\dots,v_s$.\nFor this let $t=2,\\dots,s$ and assume that $i_{t-1}$ is such that $|Y_{i_{t-1}}|-t=|X_{i_{t-1}}| \\equiv 0 \\pmod s$.\nLet $\\mathcal{A}$ be the neighbours of $X_{i_{t-1}}$ in $R$ and let $\\mathcal{B}$ be those clusters which share an edge of $M$ with a cluster from $\\mathcal{A}$.\nBy Fact~\\ref{fact:non-extremal} applied to $\\mathcal{B}$ and $H(v_t)$ there are indices $i_t,j$ such that $X_{i_t} \\in H(v_t)$, $Y_{i_t},X_j,Y_j \\in H$ and $Y_{i_t}Y_j$ and $X_jX_{i_{t-1}}$ are edges of $R$.\nThis gives the path $Y_{i_{t-1}},X_J,Y_j,X_{i_t},Y_{i_t}$ in $R$.\nWe extend the paths $P_{i_t}$ and $P_j$, such that after removing the internal vertices we have $|Y_{i_{t-1}}|=|X_{i_{t-1}}| \\equiv 0 \\pmod s$, $|Y_j|=|X_j|\\equiv 0 \\pmod s$, and $|Y_{i_{t}}|-(t+1)=|X_{i_{t}}|\\equiv 0 \\pmod s$.\nThis can be done similarly as for $P_{i_1}$ above and we add the internal vertices to $W_0$ (this adds $6s$ vertices, see Figure~\\ref{fig:Step52}).\n\nWe repeat the same procedure to cover $v_{2s},\\dots,v_{s+1}$ and are left do deal with $|Y_{i_{s}}|-s=|X_{i_{s}}|\\equiv 0 \\pmod s$ and $|Y_{i_{s+1}}|-s=|X_{i_{s+1}}|\\equiv 0 \\pmod s$.\nFor this let $\\mathcal{A}$ be the clusters in $R$ that share an edge in $M$ with a neighbour of the cluster $Y_{i_j}$ and, similarly, let $\\mathcal{B}$ be the clusters in $R$ that share an edge in $M$ with a neighbour of the cluster $Y_{i_{j+1}}$.\nBy Fact~\\ref{fact:non-extremal} applied to $\\mathcal{A}$ and $\\mathcal{B}$ we find indices $j_1,j_2$ such that $X_{j_1},Y_{j_1},X_{j_2},Y_{j_2} \\in H$ and $Y_{i_{s}}X_{j_1}$, $Y_{j_1}Y_{j_2}$, and $X_{j_2}Y_{i_{s+1}}$ are edges of $R$.\nWe extend the paths $P_{j_1}$ and $P_{j_2}$ such that after removing the internal vertices we have $|X_{j_1}|=|Y_{j_1}|\\equiv 0 \\pmod s$, $|X_{j_2}|=|Y_{j_2}|\\equiv 0 \\pmod s$, $|Y_{i_{s}}|=|X_{i_{s}}|\\equiv 0 \\pmod s$, and $|Y_{i_{s+1}}|=|X_{i_{s+1}}|\\equiv 0 \\pmod s$.\nWe add the internal vertices to $W_0$ (this adds $10s$ vertices, see Figure~\\ref{fig:Step53}).\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi1) at (-11.2, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi2) at (-11.2, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj13) at (-7.9, 3) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj14) at (-7.9, 3.5) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-5.3, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-5.3, 2) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-5.3, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-5.3, 3.5) {};\n    \\node[] at (-4.7, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj15) at (-4.5, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj16) at (-4.5, 2) {};\n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (-1.4, 1.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (-1.4, 1.75) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj23) at (-2, 2.25) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj24) at (-2, 2.75) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj25) at (-1.4, 3.25) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj26) at (-1.4, 3.75) {};\n    \\node[] at (-1.9, 3.5) {$P_{i_{j_2}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (1.2, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (1.2, 2) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj23) at (1.2, 3) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj24) at (1.2, 3.5) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (ys1) at (4.7, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (ys2) at (4.7, 2.75) {};\n  \n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_{s}}$}] (Yi) at (-11.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (-1.7,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (1.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_s+1}$}] (Ys) at (4.7,2.375) {};\n    \n    \\path[-] (Yi) edge[draw=black!20, line width=20pt] (Xj1);\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \\path[-] (Yj1) edge[draw=black!20, line width=20pt] (Yj2);\n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Ys);\n    \n    \\path[-] (yi1) edge[draw=black] (xj11);\n    \\path[-] (yi1) edge[draw=black] (xj12);\n    \\path[-] (yi1) edge[draw=black] (xj13);\n    \\path[-] (yi1) edge[draw=black] (xj14);\n    \n    \\path[-] (yi2) edge[draw=black] (xj11);\n    \\path[-] (yi2) edge[draw=black] (xj12);\n    \\path[-] (yi2) edge[draw=black] (xj13);\n    \\path[-] (yi2) edge[draw=black] (xj14);\n    \n    \\path[-] (xj11) edge[draw=black] (yj11);\n    \\path[-] (xj12) edge[draw=black] (yj11);\n    \\path[-] (xj11) edge[draw=black] (yj12);\n    \\path[-] (xj12) edge[draw=black] (yj12);\n    \n    \\path[-] (xj13) edge[draw=black] (yj13);\n    \\path[-] (xj14) edge[draw=black] (yj13);\n    \\path[-] (xj13) edge[draw=black] (yj14);\n    \\path[-] (xj14) edge[draw=black] (yj14);\n    \n    \\path[-] (yj15) edge[draw=black] (yj21);\n    \\path[-] (yj15) edge[draw=black] (yj22);\n    \\path[-] (yj15) edge[draw=black] (yj23);\n    \\path[-] (yj15) edge[draw=black] (yj24);\n    \n    \\path[-] (yj16) edge[draw=black] (yj21);\n    \\path[-] (yj16) edge[draw=black] (yj22);\n    \\path[-] (yj16) edge[draw=black] (yj23);\n    \\path[-] (yj16) edge[draw=black] (yj24);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj21) edge[draw=black] (xj22);\n    \\path[-] (yj22) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj25) edge[draw=black] (xj23);\n    \\path[-] (yj25) edge[draw=black] (xj24);\n    \\path[-] (yj26) edge[draw=black] (xj23);\n    \\path[-] (yj26) edge[draw=black] (xj24);\n\n    \\path[-] (xj21) edge[draw=black] (ys1);\n    \\path[-] (xj21) edge[draw=black] (ys2);\n    \\path[-] (xj22) edge[draw=black] (ys1);\n    \\path[-] (xj22) edge[draw=black] (ys2);\n    \\path[-] (xj23) edge[draw=black] (ys1);\n    \\path[-] (xj23) edge[draw=black] (ys2);\n    \\path[-] (xj24) edge[draw=black] (ys1);\n    \\path[-] (xj24) edge[draw=black] (ys2);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Balancing $Y_{i_{s}}$ and $Y_{i_{s+1}}$ in the case $r=2s=4$. As before, we indicate the fixed matching in the cluster graph by blue connections and additional edges by gray connections.}\n    \\label{fig:Step53}\n\\end{figure}\n\nIt is easy to ensure that in each of these steps the new end-tuples are always well-connected.\nDuring this procedure when absorbing $2s$ vertices into the paths we added at most $18 s^2$ vertices to $W_0$.\n\n\\smallskip\n\\textbf{Reset after $\\nu n$ iterations.}\nWe can repeat this for $\\nu n$ steps as this still guarantees that $|W_0| \\le 18 s^2 \\nu n$ as required.\nIt follows from the bound on $\\nu$ that no degree in the regular pairs in $M$ dropping below $\\tfrac d2 \\cdot \\tfrac{n}{\\ell}$.\nMoreover, for $i \\in [\\ell']$ the end-tuples of the path $P_i$ that were well-connected at some point still have at least $\\tfrac 14 d^s L$ common neighbours into the respective sets.\nAfter $\\nu n$ steps we want to get close enough to the original situation such that we can continue for another $\\nu n$ steps.\nBy Lemma~\\ref{lem:superreg} for any $i \\in [\\ell']$ we need to remove at most $2 \\eps \\tfrac n\\ell$ vertices from the $X_i,Y_i$ to get that  $(X_i,Y_i)$ is $(2\\eps,\\tfrac{3d}{4})$-super-regular.\nWe will greedily absorb these vertices into the path $P_i$ by alternating between $X_i$ and $Y_i$ without any degree dropping below $\\tfrac{3d}{4} \\cdot \\tfrac n\\ell$.\nHere we use that the original pair was $(\\eps,d)$-regular and that it intersects $W$ in at most $\\tfrac 12 \\tfrac n\\ell$ vertices.\nWhile doing this, we can ensure that, for each $i \\in [\\ell']$, both ends of the path $P_i$ are extended at least two steps and the ends are well-connected again.\nWe add the vertices used in these paths to $W$, also move the vertices from $W_0$ to $W$, and set $W_0=\\emptyset$.\nWe continue by covering $2s$ vertices from $V_0$ as explained above.\n\n\\smallskip\n\\textbf{Covering the last vertices.}\nWe can repeat this until $|V_0|<2s$ and if $|V_0|=0$ we are done with this step.\nOtherwise, we have $|V_0|=t\\not=0$, $n \\not\\equiv 0 \\pmod{2s}$, and we can not find the $\\tfrac r2$-blow-up of a cycle.\nIf $r$ is odd $nr \\equiv 0 \\pmod 2$ implies that $n$ and also $t$ are even.\nWe need to absorb the last $t$ vertices in a different way.\nIf $r$ is even, let $v \\in V_0$ and with Fact~\\ref{fact:non-extremal} pick $j_1,j_4$ such that $X_{j_1},X_{j_4} \\in H(v)$, $Y_{j_1},Y_{j_4} \\in H$, and $X_{j_1}X_{j_4}$ is an edge of $R$.\nThen we consider those clusters that share an edge of $M$ with a neighbour of $Y_{j_1}$ and $Y_{j_4}$ respectively and with Fact~\\ref{fact:non-extremal} pick $j_2,j_3$ such that $X_{j_2},Y_{j_2},X_{j_3},Y_{j_3} \\in H$, and $Y_{j_1}Y_{j_2}$, $Y_{j_4}Y_{j_3}$, and $X_{j_2}X_{j_3}$ are edges of $R$.\nWe extend the path $P_{j_1}$ by following $Y_{j_1},X_{j_1},X_{j_4},Y_{j_4},Y_{j_3},X_{j_3},X_{j_2},Y_{j_2},Y_{j_1}$ such that the vertex $v$ is in the neighbourhood of the new vertices from $X_{j_1}$ and $X_{j_4}$.\nThis allows us to include $v$ into the path.\nNote that this is no longer the $\\tfrac r2$-blow-up of a path (see Figure~\\ref{fig:Step54}).\nIf $r$ is odd let $u,v \\in V_0$ and we proceed similarly to include both vertices (see Figure~\\ref{fig:Step55}).\nHere we find $X_{j_1},X_{j_4} \\in H(v)$ and $X_{i_2},X_{j_3} \\in H(u)$ such that $X_{j_1}X_{j_4}$ and $X_{i_2}X_{j_3}$ are edges of $R$ and then connect $Y_{j_1}$ to $Y_{j_2}$ and $Y_{j_4}$ to $Y_{j_3}$ as in the even case by using four additional clusters for each connection.\nNote that here the path structure on the lower half also is not an $\\tfrac r2$-blow-up (alternating $K_{s,s}$ and $K_{s,s}-K_{1,1}^{(s)}$), but has one edge shifted (alternating $K_{s,s}-K_{1,1}$ and $K_{s,s}-K_{1,1}^{(s-1)}$).\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n\n    \n \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-5.25, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-5.25, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-4.75, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-4.75, 3.5) {};\n    \\node[] at (-5.25, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (-1.7, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (-1.7, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (1.2, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (1.2, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj31) at (1.2, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj32) at (1.2, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj31) at (-1.7, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj32) at (-1.7, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj41) at (-5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj42) at (-5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj41) at (-7.9, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj42) at (-7.9, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2.75) {};\n\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (-1.7,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (1.2,2.375) {};\n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_4}$}] (Xj4) at (-7.9,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_4}$}] (Yj4) at (-5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_3}$}] (Yj3) at (-1.7,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_3}$}] (Xj3) at (1.2,-4) {};\n    \n    \n    \n\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \\path[-] (Yj1) edge[draw=black!20, line width=20pt] (Yj2);\n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \n    \\path[-] (Xj4) edge[draw=blue!20, line width=20pt] (Yj4);\n    \\path[-] (Yj4) edge[draw=black!20, line width=20pt] (Yj3);\n    \\path[-] (Yj3) edge[draw=blue!20, line width=20pt] (Xj3);\n    \n    \\path[-] (Xj1) edge[draw=black!20, line width=20pt] (Xj4);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Xj3);\n    \n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (v) at (-10, -0.875) {};\n    \\node[] at (-10.5, -0.875) {$v$};\n    \n    \\path[-] (v) edge[draw=red] (xj11);\n    \\path[-] (v) edge[draw=red] (xj12);\n    \\path[-] (v) edge[draw=red] (xj41);\n    \\path[-] (v) edge[draw=red] (xj42);\n    \n    \\path[-] (xj11) edge[draw=black] (xj41);\n    \\path[-] (xj12) edge[draw=black, bend left = 25] (xj42);\n    \n    \\path[-] (xj21) edge[draw=black, bend right = 20] (xj31);\n    \\path[-] (xj21) edge[draw=black, bend right = 25] (xj32);\n    \\path[-] (xj22) edge[draw=black, bend left = 20] (xj31);\n    \\path[-] (xj22) edge[draw=black, bend left = 25] (xj32);\n    \n    \\path[-] (xj11) edge[draw=black] (yj11);\n    \\path[-] (xj12) edge[draw=black] (yj11);\n    \\path[-] (xj11) edge[draw=black] (yj12);\n    \\path[-] (xj12) edge[draw=black] (yj12);\n    \n    \\path[-] (xj41) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj41);\n    \\path[-] (xj41) edge[draw=black] (yj42);\n    \\path[-] (xj42) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (yj41);\n    \\path[-] (yj32) edge[draw=black] (yj41);\n    \\path[-] (yj31) edge[draw=black] (yj42);\n    \\path[-] (yj32) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj31);\n    \\path[-] (yj31) edge[draw=black] (xj32);\n    \\path[-] (yj32) edge[draw=black] (xj32);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj21);\n    \\path[-] (yj21) edge[draw=black] (xj22);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj21) edge[draw=black] (yj13);\n    \\path[-] (yj21) edge[draw=black] (yj14);\n    \\path[-] (yj22) edge[draw=black] (yj13);\n    \\path[-] (yj22) edge[draw=black] (yj14);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Absorbing $v$ in the even case, where $r=2s=4$. The blue and gray connections represent the matching edges and non-matching edges in the cluster graph again.}\n    \\label{fig:Step54}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n\n    \n \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-4.75, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-4.75, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-5.25, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-5.25, 3.5) {};\n    \\node[] at (-4.75, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (1.5, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (1.5, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (4, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (4, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj31) at (4, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj32) at (4, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj31) at (1.5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj32) at (1.5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj41) at (-5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj42) at (-5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj41) at (-7.9, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj42) at (-7.9, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2.75) {};\n\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (1.5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (4,2.375) {};\n    \n    \n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_4}$}] (Xj4) at (-7.9,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_4}$}] (Yj4) at (-5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S1) at (-3.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S2) at (-2.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S3) at (-1.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S4) at (-0.25,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T1) at (-3.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T2) at (-2.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T3) at (-1.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T4) at (-0.25,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_3}$}] (Yj3) at (1.5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_3}$}] (Xj3) at (4,-4) {};\n    \n    \n    \n\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \n    \n    \\path[-] (Yj1) edge[draw=black!20, line width=10pt] (T1);\n    \\path[-] (T1) edge[draw=blue!20, line width=10pt] (T2);\n    \\path[-] (T2) edge[draw=black!20, line width=10pt] (T3);\n    \\path[-] (T3) edge[draw=blue!20, line width=10pt] (T4);\n    \\path[-] (Yj2) edge[draw=black!20, line width=10pt] (T4);\n    \n    \n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \n    \\path[-] (Xj4) edge[draw=blue!20, line width=20pt] (Yj4);\n    \n    \\path[-] (Yj4) edge[draw=black!20, line width=10pt] (S1);\n    \\path[-] (S1) edge[draw=blue!20, line width=10pt] (S2);\n    \\path[-] (S2) edge[draw=black!20, line width=10pt] (S3);\n    \\path[-] (S3) edge[draw=blue!20, line width=10pt] (S4);\n    \\path[-] (Yj3) edge[draw=black!20, line width=10pt] (S4);\n    \n    \n    \\path[-] (Yj3) edge[draw=blue!20, line width=20pt] (Xj3);\n    \n    \\path[-] (Xj1) edge[draw=black!20, line width=20pt] (Xj4);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Xj3);\n    \n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (v) at (-9.5, -0.875) {};\n    \\node[] at (-10, -0.875) {$v$};\n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (u) at (5.8, -0.875) {};\n    \\node[] at (6.3, -0.875) {$u$};\n    \n    \\path[-] (v) edge[draw=red] (xj11);\n    \\path[-] (v) edge[draw=red] (xj12);\n    \\path[-] (v) edge[draw=red] (xj41);\n\n   \n    \\path[-] (u) edge[draw=red] (xj32);\n    \\path[-] (u) edge[draw=red] (xj21);\n    \\path[-] (u) edge[draw=red] (xj22);\n    \n    \\path[-] (xj11) edge[draw=black] (xj41);\n    \\path[-] (xj12) edge[draw=black, bend left = 25] (xj42);\n    \n    \\path[-] (xj21) edge[draw=black] (xj31);\n    \\path[-] (xj22) edge[draw=black, bend right = 25] (xj32);\n    \n    \\path[-] (xj12) edge[draw=black] (yj14);\n    \\path[-] (xj11) edge[draw=black] (yj13);\n\n    \\path[-] (xj41) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black, dotted] (yj41);\n    \\path[-] (yj31) edge[draw=black, dotted] (yj42);\n    \\path[-] (yj32) edge[draw=black, dotted] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj32);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj21) edge[draw=black, dotted] (yj11);\n    \\path[-] (yj21) edge[draw=black, dotted] (yj12);\n    \\path[-] (yj22) edge[draw=black, dotted] (yj11);\n    \\path[-] (yj22) edge[draw=black, dotted] (yj12);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Absorbing $v$ and $u$ in the odd case, where $r=3, s=2$. The blue and gray connections represent the matching edges and non-matching edges in the cluster graph again. The dotted edges indicate an $r-$regular $r-$connected path.}\n    \\label{fig:Step55}\n\\end{figure}\n\n\\smallskip\n\\textbf{Summary.}\nWe need to estimate the number of vertices added to $W$ throughout the whole procedure of covering $V_0$, which are at most $2 \\beta n$ vertices.\nThere are at most $\\lceil \\tfrac{2\\beta n}{2s\\nu n} \\rceil \\le \\tfrac{\\beta}{s \\nu} + 1$ iterations of the argument for covering $2s\\nu n$ vertices and for covering $2s$ vertices of $V_0$ we need at most $18 s^2$ vertices.\nDuring these iterations we will always have \n\\[|W| \\le \\frac{18s^2}{2s} 2 \\beta n + \\left(\\frac{\\beta}{s\\nu}+1\\right) \\cdot \\ell' 2 \\eps \\frac n\\ell \\cdot 2s \\le 20 s \\beta n\\,,\\]\nwhere the second term comes from the vertices we need to absorb after each iteration.\nCovering the last $t<2s$ vertices does not change anything and, therefore, we can indeed repeat this until all vertices from $V_0$ are covered by the paths.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:spanning}.}\nWe fully absorbed $V_0$ into the connecting paths such that the end-tuples are well-connected to the other side of the matching edge.\nLet $i \\in [\\ell']$ and denote by $\\mathbf{x}=(x_1,\\dots, x_s)$ and $\\mathbf{y}=(y_1,\\dots, y_s)$ the end-tuples of the paths $P_{i-1}$ and $P_{i}$, respectively.\nRemove $\\mathbf{x}$ from $X_i$ and $\\mathbf{y}$ from $Y_i$, note that $|X_i|=|Y_i| \\equiv 0 \\pmod s$ and that $(X_i,Y_i)$ is $(2\\eps,\\tfrac d2)$-super-regular.\nDenote the common neighbours of $\\mathbf{x}$ in $Y_i$ by $Y'$, the common neighbours of $\\mathbf{y}$ in $X_i$ by $X'$, and note that $|X'|,|Y'| \\ge \\tfrac 14 d^s |X|$.\nTherefore, we can apply Lemma~\\ref{lem:blowup} to cover $X_i$ and $Y_i$ with the $\\tfrac r2$-blow-up of a path and end-tuples within $X'$ and $Y'$, which then connects $P_{i-1}$ to $P_{i}$.\n\nTogether this gives an $r$-regular subgraph in $G$.\nIn the case when $n \\equiv 0 \\pmod{2s}$ we have constructed the $\\tfrac r2$-blow-up of a path, which is $r$-connected.\nTo see that it is also $r$-connected in the other cases it suffices to observe that in the case when $r$ is even removing a perfect matching from a $K_{s,s}$ and adding a vertex $v$ to all these $r$ vertices (see Figure~\\ref{fig:Step54}) preserves this property.\nSimilarly, in the case when $r$ is odd, removing a perfect matching from two copies of $K_{s,s}$, connecting vertices $u,v$ two $r$ of these vertices, and shifting the path in between as described (see Figure~\\ref{fig:Step55}) also preserves this property.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nAttack trees allow for an effective security analysis by systematically organizing the different \nways in which a system can be attacked into a tree. \nThe root node of an attack tree \nrepresents the {\\em attacker's goal} and the children of a given node represent \nits refinement into {\\em sub-goals}. A refinement is typically either disjunctive \n(denoted by $\\OR$) or conjunctive (denoted by $\\AND$). \nThe \nleaves of an attack tree represent the attacker's actions and are called {\\em basic \nactions}. \n\nSince their inception by Schneier~\\cite{Schn}, attack trees\nhave quickly become a popular modeling tool for security \nanalysts.\nHowever, the limitations of this formalism,\nin particular with respect to expressing the order in which the\nvarious attack steps are executed, have been recognized by\nmany authors (see \\eg,~\\cite{KoPiSc_CSR14}).\nIn practice, modeling of security scenarios often requires constructs\nwhere conditions on the execution order of the attack components can\nbe clearly specified. This is for instance the case when the time or\n(conditional) probability \nof an attack is considered, as\nin~\\cite{Arnold-POST14,LL2011}. Consequently, several studies have\nextended attack trees informally with sequential conjunctive\nrefinements. Such extensions have resulted in improved modeling and\nanalyses (e.g., \\cite{LL2011,PB2010,JW2009}) and\nsoftware tools, e.g., ATSyRA~\\cite{pinchinat:hal-01064645}.\n\nEven though the sequential conjunctive refinement, \nthat we denote by \\SAND, \nis well understood at a conceptual level \nand even applied to real world scenarios~\\cite{pinchinat:hal-01064645}, \nnone of the existing solutions have provided \na rigorous mathematical formalization of attack trees with \\SAND. \nIndeed, the extensions found in the literature \nare rather diverse in terms of application domain, interpretation, and \nformality. Thereby, it is infeasible\n to answer fundamental questions such as: \nWhat is the precise expressibility of \\SAND~attack trees?\nWhen do two such trees \nrepresent the same security \nscenario? Or what type of attributes can be synthesized \non \\SAND~attack trees in the standard bottom-up way?\nThese questions can only be precisely answered if \\SAND~attack trees are \nprovided with a formal, general, and explicit interpretation, that is to say, \nif \\SAND~attack trees are given a formal foundation. \n\n\\noindent{\\em Contributions:} \nIn this article we formalize \nthe meaning of a $\\SAND$ attack tree by defining its semantics. Our \nsemantics \nis based on series-parallel (SP) graphs, which is a well-studied branch of \ngraph theory. We provide a complete \naxiomatization \nfor the SP semantics and show that the \nSP semantics for $\\SAND$ attack trees are a conservative extension of the \nmultiset semantics for standard attack trees~\\cite{MaOo} (i.e., our \nextension \ndoes not \nintroduce unexpected equivalences w.r.t.\\ the multiset semantics). \nTo do so, \nwe define a term \nrewriting system that is terminating and confluent\nand obtain normal forms for $\\SAND$ attack trees.\nAs a consequence, we achieve the rather surprising result that the\ndomains of $\\SAND$ attack trees\nand sets of SP graphs are isomorphic. \nWe also extend the notion of attributes \nfor $\\SAND$ attack trees which enable \nthe quantitative analysis of attack scenarios using the standard bottom-up \nevaluation algorithm. \n\n{\nOne of the goals of our work is to provide a level of abstraction that\nencompasses most of the existing approaches from literature. For example,\noperators, \nsuch as the priority-based and the\ntime-based connectors~\\cite{WaWhPhPa}, are indeed captured by the\n$\\SAND$ operator defined in this article. Moreover, other published\nsemantics, such as those based on cumulative distribution\nfunctions~\\cite{Arnold-POST14}, conditional\nprobabilities~\\cite{WaWhPhPa}, or boolean algebra~\\cite{Khan}, can be\nexpressed as an attribute in our formalism. Last but not least, even\nthough we make the distinction between $\\AND$ and $\\SAND$ refinements\nexplicit, our semantics satisfies backward compatibility with the\nwell-known multiset semantics of attack trees~\\cite{MaOo}.\nThis stresses the, much needed, unifying character of our approach.\n}\n\n\\noindent{\\em Organization:} Section~\\ref{sec:related-work} summarizes the \nrelated work and puts our work in context. \nSection~\\ref{sec:sand-atree} provides a formal definition of \\SAND~attack trees\nand its semantics using series-parallel graphs. Section~\\ref{sec:axioms}\ndefines a complete set of axioms for \\SAND~attack trees and presents \na term rewriting system which allows identification of\nsemantically equivalent \\SAND~attack trees. Section~\\ref{sec:attributes}\noutlines an approach to quantitatively analyze \\SAND~attack trees using \nattributes. Finally, Section~\\ref{sec:conclusions} concludes with an outlook \non future work.\n\n\n\n\n\\section{Related Work and Motivation}\n\\label{sec:related-work}\n\n\n\n{}\nSeveral extensions of attack trees with temporal or \ncausal dependencies between \nattack steps have been proposed.\nWe observe that there are three different approaches to achieve this\ngoal.\nThe first approach is to use standard attack trees with the added\nassumption that the children of an \\AND\\ node are sequentially\nordered. This approach is mostly applied to the design\nof algorithms or tools for the analysis of attack trees under the\nassumption of ordered events. \n\nThe second approach is to introduce a mechanism for ordering\nevents in an attack tree, for instance by adding a\nnew type of edge to express causality or conditionality. In its most\ngeneral case, any partial order on the events in an attack tree can be\nspecified.\nThe third approach consists of the introduction of a new type of\nnode for sequencing. Most extensions fall in this category.\nThis approach is used by authors who require their formalism to be\nbackward compatible, or who need standard, as well as ordered\nconjunction. We discuss for each of these approaches \nthe most relevant papers with respect to the present article.\n{\nThat is, we only consider approaches that still have the main\ncharacteristics of attack trees, being the presence of \\AND\\ and \\OR\\\nnodes and the interpretation of the edges as a refinement relation.\nThus, we consider approaches such as attack\ngraphs~\\cite{SHJLW2002,OBM2006} and Bayesian\nnetworks~\\cite{QL2004,JW2007,LM2005} as out of scope\nfor this paper.\n}\n\n\n\\subsubsection{Approaches with a sequential interpretation of \\AND.}\nIn their work on \\emph{Bayesian networks for security}, Qin and\nLee define a transformation from attack trees to Bayesian\nnetworks~\\cite{QiLe}. They state that ``there always exists\nan implicit dependent and sequential relationship between \\AND\\ nodes\nin an attack tree.'' Most literature on attack trees seem to\ncontradict this statement, implying that there is a need to explicitly\nidentify such sequential relationships.\n\nJ\\\"urgenson and Willemson developed an algorithm to calculate the\n\\emph{expected outcome} of an attack tree~\\cite{JW2009}. The goal of\nthe algorithm is to determine a permutation of leaves for which the\noptimal expected outcome for an attacker can be achieved. In essence,\ntheir input is an attack tree where an \\AND\\ node represents all\npossible sequences of its children.\nA peculiarity of their interpretation is that multiple occurrences of\nthe same node are considered only once, implying that the execution of\ntwice the same action cannot be expressed. \n\n\\subsubsection{Approaches introducing a general order.}\n{\nPeine, Jawurek, and Mandel \nintroduce \\emph{security goal indicator trees}~\\cite{PJM2008}\nin which nodes can be related by a notion of \\emph{conditional\ndependency} and Boolean connectors. \nThe authors, \nhowever, do not formally specify the syntax and \nsemantics of the model. \n} \nA more general approach is proposed by Pi\\`etre-Cambac\\'ed\\`es and \nBouissou \n\\cite{PB2010}, who apply \n\\emph{Boolean logic driven Markov\nprocesses} to security modeling. Their formalism does not introduce new\ngates, but a (trigger-)relation on the nodes of the attack tree.\nAlthough triggers \ncan express a more general sequential relation than the $\\SAND$\noperator, they lack the readability of standard attack tree operators. \n\n\\emph{Vulnerability cause graphs}~\\cite{ArBySh,ByArShDu} combine properties of attack \ntrees ($\\AND$ and $\\OR$ nodes) and attack graphs (edges express order \nrather \nthan refinement). The interaction between the $\\AND$ nodes and the order \nrelation is defined through a graph transformation called \\emph{conversion of \nconjunctions}, which ignores the order between nodes. This \ndiscrepancy could be solved by considering distinct\nconjunctive and sequential conjunctive nodes, as we do in this paper.\n\n\\subsubsection{Approaches introducing sequential \\AND.}\nAs noted by Arnold et \nal.~\\cite{Arnold-POST14}, the analysis of time-dependent \nattacks requires attack trees to be extended with a sequential operator. This \nis\naccomplished by defining sequential nodes as conjunctive nodes with a\nnotion of progress of time.\nThe authors define a formal semantics for this extension based on cumulative\ndistribution functions (CDFs), where a CDF denotes the probability that a\nsuccessful attack occurs within time $t$. The main difference with our work is \nthat their approach is based on\nan explicit notion of time, while we have a more abstract approach\nbased on causality. In their semantics, the meaning of an extended\nattack tree is a CDF, in which the relation to the individual basic\nattacks is not explicit anymore. In contrast, in our semantics\nthe individual basic attacks and their causal ordering remain visible.\nAs such, our semantics can be considered more abstract,\nand indeed, we can formulate their semantics as an \\emph{attribute} in our\napproach.\n\n\n\\emph{Enhanced attack trees}~\\cite{AY2006} (EATs) distinguish between $\\OR$, $\\AND$ \nand $\\OAND$ (Ordered $\\AND$). Similarly to the approach of Arnold et\nal.~\\cite{Arnold-POST14}, ordered $\\AND$ nodes are used to express \ntemporal dependencies between attack components. The authors evaluate EATs by \ntransforming them into tree automata. Intermediate states in the automaton \nsupport the task of reporting partial attacks. However, because every \nintermediate node of the tree corresponds to a state in the tree automaton, \ntheir approach does not scale well. This problem can be addressed by \nconsidering \nthe normal form of attack trees, as proposed in this article. \n\nNot every extension of attack trees with $\\SAND$ refinements concerns \ntime-dependent attack scenarios; some aim at supporting risk \nanalyses with conditional\nprobabilities. For that purpose, Wen-Ping and Wei-Min \nintroduce \\emph{improved attack trees}~\\cite{LL2011}. The concepts, \nhowever, are described at an intuitive level only. \n\n\\emph{Unified parameterizable attack trees}~\\cite{WaWhPhPa} unify different extensions \nof attack trees (structural, computational, and hybrid). The authors \nconsider two types of ordered $\\AND$ connectors: priority-based connectors\nand time-based connectors. \nThe children of the former are ordered from\nhighest to lowest priority, whereas the children of the latter are\nordered temporally. \nOur formalism gives a single interpretation to the \n$\\SAND$ operator, yet it can capture both connectors. \n\n{\nDue to obvious similarities, we also review approaches that \nintroduce the $\\SAND$ operator in fault \ntrees. For example, Brooke and Paige include\nfive fault tree gates: $\\AND$, $\\OR$, priority $\\AND$, exclusive $\\OR$,\nand an inhibit gate~\\cite{BP2003}. The authors do not\ndiscuss the semantics of their model for security, though. Another fault tree \nbased approach is discussed by\nKhand~\\cite{Khan}, who proposes to extend attack trees with a set of gates\nfrom dynamical fault tree modeling that overlaps with the gates used\nby Brooke and Paige~\\cite{BP2003} and in particular contains the priority \n$\\AND$ gate. \nKhand assigns truth values to his \nattack trees by giving truth tables for all gates. Khand's\ntruth tables, when restricted to $\\AND$, $\\OR$, and priority $\\AND$, \nconstitute an attribute domain \nwhich is compatible (in the sense\nof~\\cite{KoMaRaSc_JLC}) with the SP semantics for \n$\\SAND$ attack trees as defined in this paper.\n}\n\nWe observe that the extensions of attack trees with sequential conjunction are \nrather diverse in terms of application domain, interpretation, and formality. \nIn order to give a clear and unambiguous interpretation of the $\\SAND$ operator \nand capture different application domains, it is necessary to give a formal \nsemantics as a translation to a well-understood domain. Note that, neither the \nmultiset~\\cite{MaOo} nor the propositional \nsemantics~\\cite{KoPoSc} can express ordering of attack components. \nTherefore, a richer semantical domain needs to be defined. \nThe purpose of this article is to address this problem. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Attack Trees with Sequential Conjunction}\n\\label{sec:sand-atree}\n\nWe extend the attack tree formalism so that a refinement of a\n(sub-)goal of an attacker can be a sequential conjunct (denoted by\n\\SAND) in addition to disjuncts and conjuncts. We first give a \ndefinition of attack trees with the new sequential operator and then \ndefine series-parallel graphs on which the semantics for the new attack\ntrees is based.\n\n\\subsection{\\SAND \\ Attack Trees}\\label{sec:atree-sand}\n\nLet $\\basicact$ denote the set of all possible basic actions of an attacker. \nWe formalize standard attack trees introduced by Schneier in~\\cite{Schn} \nand call them simply \\emph{attack trees} in the rest of this\npaper. Attack trees are closed terms \nover the signature $\\basicact\\cup \\{\\OR, \\AND\\}$, \ngenerated \nby the following grammar, where $b\\in\\basicact$ is a terminal symbol. \n\\begin{equation} \n\\label{gram:AT}\nt ::= b \\mid \\OR(t, \\dots, t) \\mid \\AND(t, \\dots, t).\n\\end{equation}\nThe universe of attack trees is denoted by $\\atree$. \n\\emph{$\\SAND$ attack trees} are closed terms over the signature\n$\\basicact\\cup \\{\\OR, \\AND, \\SAND\\}$, where \n$\\SAND$ is a non-commutative operator called \\emph{sequential conjunction},\nand are generated by the grammar\n\\begin{equation} \n\\label{gram:SAT}\nt ::= b \\mid \\OR(t, \\dots, t) \\mid \\AND(t, \\dots, t) \\mid \\SAND(t, \\dots, t\n). \n\\end{equation}\nThe universe of $\\SAND$ attack trees is denoted by $\\sandtree$. The \npurpose of \\OR\\ and \\AND\\ refinements in $\\SAND$ attack trees\nis the same as in \nattack trees. The sequential conjunctive refinement \\SAND\\ allows us to model \nthat a certain goal is reached if and only if all \nits subgoals are reached in a precise order. \n\nThe following attack scenario \nmotivates the need for extending attack trees with sequential\nconjunctive refinement. \n\\begin{example}\n\\label{eg:def-attack-tree}\nConsider \na file server $\\server$, offering\nftp, ssh, and rsh services. \nThe attack tree in Figure~\\ref{fig:attack-tree}\nshows how an attacker can gain root privileges on \n$\\server$ (\\becomeroot), in two ways: \neither without providing any user credentials \n(\\noauth)\nor by breaching the authentication mechanism\n(\\auth).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=4.5cm]{atree1}\n\\caption{An attack tree with sequential and parallel conjunctions}\n\\label{fig:attack-tree}\n\\end{figure}\n\nIn the first case,\nthe attacker must first gain user privileges on $\\server$ \n(\\textit{gain user pri\\-vi\\-leges}) \nand then perform a local buffer overflow attack (\\localbofx). \nSince the attack steps must be executed in this particular order, \nthe use of \\SAND~refinement is substantial.\nTo gain user privileges, the attacker \nmust exploit an ftp vulnerability \nto anonymously upload a list of trusted hosts to $\\server$ \n(\\ftprhostsx).\\footnote{For readability, attack actions are named after\nthe services that are exploited.} Finally, she can use the\nnew trust condition to remotely execute shell commands on $\\server$ (\\rshx). \n \nThe second way is to abuse a buffer overflow in both the \nssh daemon (\\sshbofx) and the RSAREF2 library (\\rsarefbofx) used for authentication.\nThese attacks can be executed in any order, which is modeled with \nthe standard \\AND\\ refinement. \n\nUsing the term notation introduced in this section, \nwe can represent the \\SAND\\ attack tree in Figure~\\ref{fig:attack-tree} \nas\n\\[\nt = \\OR\\Big(\\SAND\\big(\\SAND(\\ftprhostsx, \\rshx), \\localbofx\\big), \\\\\n \\AND(\\sshbofx, \\rsarefbofx)\\Big),\n\\]\nwhere $\\ftprhostsx, \\rshx, \\localbofx, \\sshbofx, \\rsarefbofx\\in \\basicact$\nare basic actions.\n\\end{example}\n\n\\subsection{Series-Parallel Graphs}\n\\label{sec:spgraphs}\n\nA \\emph{series-parallel graph} (SP graph) is an edge-labeled directed graph that has \ntwo unique, distinct vertices, called \\emph{source} and \\emph{sink}, \nand that can be constructed with\nthe two operators for sequential and parallel composition of graphs\nthat we formally define below. \nA source is a vertex which has no incoming edges and a \nsink is a vertex without outgoing edges. \n\nOur formal definition of SP graphs is based on \\emph{multisets},\ni.e., sets in which members are allowed to occur more than once. \nWe use $\\msl \\cdot \\msr$ to denote multisets and $\\powerset(\\cdot)$\nto denote powersets. \nThe \\emph{support} $M^{\\star}$ of a multiset $M$ is the set of \ndistinct elements in $M$. \nFor instance, the support of the multiset $M = \\msl b_1, b_2, b_2 \\msr$\nis $M^{\\star} = \\{b_1, b_2\\}$. \n\nIn order to define SP graphs, we first introduce the notion of \nsource-sink graphs labeled by the elements of $\\basicact$. \n\n\\begin{definition}\nA \\emph{source-sink graph} over $\\basicact$ is a tuple\n$G=(V,E,s,z)$, where $V$ is the set of vertices,\n$E$ is a multiset of labeled edges with support $E^{\\star} \\subseteq V \\times \n\\basicact \\times V$, \n$s\\in V$ is the unique source, $z\\in V$ is the unique sink, \nand $s\\not =z$. \n\\end{definition}\nThe sequential composition of a source-sink graph $G=(V,E,s,z)$ with a \nsource-sink graph \n$G'=(V',E',s',z')$, denoted by $G\\seqop G'$, is the\ngraph resulting from taking the \ndisjoint union of $G$ and $G'$ and identifying the\nsink of $G$ with the source of $G'$. \nMore precisely,\nlet $\\dot\\cup$ denote the disjoint union operator\nand $E^{[s\/z]}$ denote the multiset of edges in $E$, where all occurrences of \nvertex $z$ are replaced by vertex $s$. \nThen we define\n\\[G\\seqop G' = (V\\setminus\\{z\\}\\dot\\cup V', E^{[s'\/z]}\\dot\\cup E', s,\nz').\\]\nThe parallel composition, denoted by $G\\parop G'$, is defined\nsimilarly, except that the two sources are identified \nand the two sinks are identified. \nFormally, we have\n\\[G\\parop G' = (V\\setminus\\{s,z\\}\\dot\\cup V', E^{[s'\/s,z'\/z]}\\dot\\cup\nE', s', z').\\] \nIt follows directly from the definitions that \nthe sequential composition is associative and that the \nparallel composition is associative and\ncommutative.\n\nWe write $\\xrightarrow{b}$ for the graph with a single edge labeled with \n$b$ and define SP graphs as follows. \n\\begin{definition} \\label{def:sp-graph}\nThe set $\\spset$ of \\emph{series-parallel graphs} (SP graphs) over \n$\\basicact$ is defined inductively by the following two rules\n\\begin{itemize}\n\\item For $b\\in \\basicact$, $\\xrightarrow{b}$ is an SP graph.\n\\item If $G$ and $G'$ are SP graphs, then so are \n$G\\seqop G'$ and $G\\parop G'$.\n\\end{itemize}\n\\end{definition}\nIt follows directly from Definition~\\ref{def:sp-graph} \nthat SP graphs are connected and acyclic. Moreover, \nevery vertex of an SP graph lies on a path from the source to\nthe sink. \nWe consider two SP graphs to be \\emph{equal} \nif there is a bijection between their\nsets of vertices that preserves the edges and edge labels.\n{\n}\n\\begin{example}\nFigure~\\ref{fig:example-spgraph} shows an example of an SP graph with\nthe source $s$ and the sink $z$.\nThis graph corresponds to the construction\n\\[\\big(\\xrightarrow{a}\\parop \\xrightarrow{b} \\parop \\xrightarrow{b}\\big) \\seqop\n\\xrightarrow{c} \\seqop \\Big(\\big(\\xrightarrow{d} \\seqop (\\xrightarrow{e}\\parop\n\\xrightarrow{f})\\big) \\parop \\xrightarrow{g}\\Big).\\]\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=2.2cm]{sp-graph}\n\\caption{A series-parallel graph}\n\\label{fig:example-spgraph}\n\\end{figure}\n\n\\end{example}\n\n\\subsection{SP Semantics for $\\SAND$ Attack Trees}\n\\label{sec:atree-sand-semantics}\n\nNumerous semantics have been proposed to \ninterpret attack trees, \nincluding \npropositional logic~\\cite{KoPoSc_iFM'14}, multisets~\\cite{MaOo}, \n De Morgan lattices~\\cite{KoPoSc}, tree automata~\\cite{AY2006},\nand \nMarkov processes~\\cite{PB2010,Arnold-POST14}. \nThe choice of a semantics allows us to accurately represent the\nassumptions made in a security scenario, e.g., \nwhether actions can be repeated or resources reused, and \nto decide which trees represent the same security scenario. \nThe advantages of formalizing attack trees and the need for various\nsemantics have been discussed in~\\cite{KoMaRaSc_JLC}.\nSince attack trees are \\AND\/\\OR~trees, the most natural interpretation \nis based on propositional logic. However, because the logical operators \nare idempotent, this interpretation assumes \nthat the multiplicity of an action is irrelevant. \nAs a consequence, \nthe propositional semantics is not well suited to reason about \nscenarios with multiple occurrences of the same action. \nDue to this lack of expressivity a semantics was\nproposed~\\cite{MaOo} in which \nthe multiplicity of actions is taken into account. This was achieved\nby interpreting an attack tree as a set of multisets that represent\ndifferent ways of reaching the root goal. \nThis multiset semantics is compatible\nwith computations \nthat depend on the number of occurrences of an action in the tree, such as \nthe minimal time to carry out the attack represented by the root goal. \n \nWe now extend the multiset semantics to $\\SAND$ attack trees. \nSince SP graphs naturally extend multisets with \na partial order, they supply a formalism in which we can interpret \ntrees using both --- commutative and sequential --- conjunctive refinements. \nSP graphs therefore provide a canonical semantics for \\SAND\\ trees in which \nmultiplicity and ordering of goals and actions are significant.\nThe idea is to interpret an attack tree $t$ as a set of SP graphs. \nThe semantics $\\sem{t}=\\{G_1,\\dots,G_k\\}$ \nof a tree $t$ corresponds to the set of possible \nattacks $G_i$, where each attack is\ndescribed by an SP graph labeled by the basic actions of $t$. \n{\n}\n\n\\begin{definition} \\label{def:sp-sem}\nThe \\emph{SP semantics} for $\\SAND$ attack trees is given by the function \n$\\sem{\\cdot}: \\sandtree \\to \\power{\\spset}$, \nwhich is defined recursively as follows:\nfor $b\\in\\basicact$, $t_i \\in \\sandtree$, $1\\leq i \\leq k$, \n\\[\n\\def1.3{1.3}\n\\begin{array}{l}\n\\sem{b} = \\{\\xrightarrow{b}\\}\\\\\n\\sem{\\OR(t_1, \\dots, t_k)} = \\bigcup_{i=1}^{k}{\\sem{t_i}}\\\\\n\\sem{\\AND(t_1, \\dots, t_k)} =\n \\{ G_1 \\parop \\dots \\parop G_k \\ \\mid \\ \n (G_1, ..., G_k) \\in \\sem{t_1} \\times ... \\times \\sem{t_k}\n \\}\\\\\n\\sem{\\SAND(t_1, \\dots, t_k)} =\n \\{ G_1 \\seqop \\dots \\seqop G_k\\ \\mid \\ \n (G_1, ..., G_k) \\in \\sem{t_1} \\times ... \\times \\sem{t_k}\n \\}.\\\\\n\\end{array}\n\\]\n\\end{definition} \n{\n The SP semantics maps $\\SAND$ attack trees to sets of SP graphs as follows. \n A leaf corresponding to a basic action $b$\n is translated into a singleton set containing the SP graph \n which consists of a single edge labeled with $b$.\n The semantics of a disjunctive node is the set \n of all the alternative attacks described by the node's children.\n The semantics of a conjunctive node is \n the parallel composition of every attack\n alternative from each of its children. \n Finally, the semantics of a sequential conjunctive node is \n a sequential composition of \nattack alternatives for the children. \n}\n\n\\begin{example}\n\\label{ex:sem(t)}\nThe SP semantics of the attack tree $t$ \ndepicted in Figure~\\ref{fig:attack-tree} is\n\\begin{equation*}\n\\sem{t}=\\{\\xrightarrow{\\ftprhostsx}\\xrightarrow{\\rshx}\n\\xrightarrow{\\localbofx}\\ , \\ \n\\xrightarrow{\\sshbofx} \\parop \\xrightarrow{\\rsarefbofx}\\}.\n\\end{equation*}\n\\end{example}\nAs shown in Example~\\ref{ex:sem(t)}, the SP semantics provides an alternative \ngraph representation for attack trees \nand therefore contributes\na different perspective on an attack scenario. \nThe \\SAND~attack tree emphasizes the refinement of goals, whereas \nSP graphs highlight the sequential aspect of attacks.\n{\n}\n\nThe SP semantics provides a natural partition of\n$\\sandtree$ into equivalence classes. \n\\begin{definition}\\label{def:equivalent}\nTwo \\SAND\\ attack trees $t_1$ and $t_2$ are \\emph{equivalent with respect \nto the SP semantics}\nif and only if they are interpreted by the same set of SP graphs, i.e., \n$\\sem{t_1}=\\sem{t_2}$.\n\\end{definition}\nBy Definition~\\ref{def:equivalent}, if the SP semantics provides \naccurate assumptions for an attack scenario, then \ntwo $\\SAND$ attack trees \nrepresent the same attack scenario \nif and only if they are equivalent \nwith respect to the SP semantics. \n\n{We finish this section by noticing that in the case of \\SAND\\ attack \ntrees without any \n$\\SAND$ refinement, the SP semantics coincides with the multiset semantics \nintroduced in~\\cite{MaOo}. Indeed, it suffices to identify the multiset \n$\\msl b_1, \\dots, b_k \\msr$ with the SP \ngraph $\\xrightarrow{b_1}\\parop \\dots \\parop \\xrightarrow{b_k}$. \nWe discuss this issue more in details in Section~\\ref{sec:SP_vs_M}.\n\n}\n\n\n\n\n\n\\section{Axiomatization of the SP Semantics} \n\\label{sec:axioms}\n{\nIn order to provide efficient analysis methods \nfor attack tree-like models, we need to be able to decide whether two \ntrees are equivalent with respect to a given semantics. \nIdeally, we would like to find the most efficient (e.g., the smallest)\nrepresentation of a given security scenario. However, \nin the case of the \\SAND\\ semantics, \nthere exists an infinite number of trees $t'$ equivalent to a given tree $t$.\n\nIn this section we study the mathematical implications of \nusing sets of SP graphs as an interpretation domain for \\SAND\\ attack trees. \nWe introduce an axiomatization of $\\SAND$ attack\ntrees which is complete with respect to the SP semantics.\nThis allows us to reason directly on $\\SAND$ attack trees, without having to\nmove to the semantical domain. Further, we derive a term\nrewriting system from the axiomatization as a means to effectively\ndecide whether two $\\SAND$ attack trees are equivalent with respect to the \nSP semantics. As a\nconsequence, we obtain a canonical representation of $\\SAND$\nattack trees which we prove to be isomorphic to sets of SP graphs.\n}\n\n\n\n\\subsection{A complete set of axioms for the SP semantics}\n\nLet $\\var$ be a set of variables denoted \nby capital letters. \nFollowing the approach developed in~\\cite{KoMaRaSc_JLC}, we \naxiomatize\n$\\SAND$ attack trees \nwith equations \n$l=r$, where $l$ and $r$ are terms over \nvariables in $\\var$, \nconstants in $\\basicact$, and the operators \\AND, \\OR, and \\SAND. \nThe equations formalize the intended properties of refinements\nand provide semantics-preserving transformations of \n\\SAND\\ attack trees. \n\\begin{example}\n\\label{ex:axiom}\nLet $\\mathrm{Sym}_{\\ell}$ denote the set of all bijections\nfrom $\\{1,\\dots,{\\ell}\\}$ to itself. \nThe axiom \n\\[\\AND(Y_1,\\dots,Y_{\\ell})=\\AND(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})})\\] \nexpresses that the order between children refining a parallel conjunctive node \nis not relevant. \nIn other words, the operator \\AND\\ is commutative. This \nimplies that any two trees of the form \n$\\AND(t_1,\\dots,t_l)$ and $\\AND(t_{\\sigma(1)},\\dots,t_{\\sigma(l)})$\nrepresent the same scenario.\n\\end{example}\n\nOur goal is to define a complete set of axioms, denoted by $\\ESP$, for\nthe SP semantics for \\SAND\\ attack trees. Intuitively, $\\ESP$\nis a set of equations\nthat can be applied to transform \na \\SAND\\ attack tree \ninto any equivalent \\SAND~attack tree \nwith respect to the SP semantics. \nBefore defining the set $\\ESP$, \nwe formalize the notion of a complete set of axioms for a given \nsemantics for (\\SAND) attack trees, following~\\cite{KoMaRaSc_JLC}. \n\nLet $T(\\var,\\Sigma)$ be the free term algebra\nover the set of variables\n$\\var$ and a signature $\\Sigma$, \nand let $E$ be a set of equations over $T(\\var,\\Sigma)$. \nThe equation $t=t'$, where $t,t'\\in T(\\var,\\Sigma)$,\nis a \\emph{syntactic consequence} of $E$ \n(denoted by $E\\vdash t=t'$) \nif it can be derived from $E$ by application \nof the following rules. \nFor all $t,t',t''\\in T(\\var,\\Sigma)$, $\\rho\\colon \\var\\to T(\\var,\\Sigma)$,\n and $X\\in \\var$: \n\\begin{itemize}\n\\item $E\\vdash t=t$, \n\\item if $t= t' \\in E$, then $E\\vdash t=t'$,\n\\item if $E\\vdash t=t'$, then $E\\vdash t'=t$, \n\\item if $E\\vdash t=t'$ and $E\\vdash t'=t''$, then $E\\vdash t=t''$. \n\\item if \n$E\\vdash t=t'$, then\n$E\\vdash \\rho(t)=\\rho(t')$,\n\\item if $E\\vdash t=t'$, then \n$E \\vdash t'' [t\/X] = t'' [t'\/X]$, \nwhere $t'' [t\/X]$ is the term obtained from $t''$\nby replacing all occurrences of variable $X$ with $t$. \n\\end{itemize}\n\n Let $\\sandtreevar$ denote the set of terms constructed from the set of \nvariables $\\var$, the set of basic actions $\\basicact$ (treated as\nconstants), and operators $\\OR$, $\\AND$ and $\\SAND$.\nLet $\\atreevar$ be the set of terms constructed \nfrom the same parts, except for the operator \\SAND. \nUsing the notion of syntactic consequence, \nwe define a complete set of axioms for a semantics for attack trees. \n\\begin{definition}\n\\label{def\/compl_set_axioms}\nLet $\\semantic{\\cdot}$ be a semantics for attack trees \n(resp. \\SAND\\ attack trees)\nand let $E$ be a set of equations over $\\atree^{\\var}$ \n(resp. $\\sandtreevar$). \nThe set $E$ is a \\emph{complete set of axioms} for $\\semantic{\\cdot}$\nif and only if, for all $t,t'\\in \\atree$ (resp. $\\sandtree$)\n\\[\\semantic{t}=\\semantic{t'} \\iff E\\vdash t=t'. \\]\n\\end{definition}\n\nWe are now ready to give a complete set of axioms for the SP semantics\nfor \\SAND\\ attack trees. These axioms \nallow us to determine whether two visually\ndistinct trees represent the same security \nscenario according to the SP semantics. \n\n\\begin{theorem}\n\\label{th:axioms}\nGiven $k,m\\geq 0$, and $\\ell\\geq 1$, let\n$\\Xvec = X_1, \\ldots, X_k$,\n$\\Yvec = Y_1, \\ldots, Y_{\\ell}$, and\n$\\Zvec = Z_1, \\ldots, Z_m$\nbe sequences of variables.\nLet $\\mathrm{Sym}_{\\ell}$ be the set of all bijections\nfrom $\\{1,\\dots,{\\ell}\\}$ to itself. The following \nset of equations over $\\sandtreevar$, denoted by $\\ESP$, is a complete \nset of axioms\\footnote{Note that the axioms are in fact \\emph{axiom schemes}. \nThe operators $\\OR$, $\\AND$ and $\\SAND$ are \\emph{unranked},\nrepresenting infinitely many $k$-ary function symbols ($k\\geq1$).} for \nthe SP semantics for $\\SAND$ attack trees.\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n& \\OR(Y_1,\\dots,Y_{\\ell})=\\OR(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})}),\n\\quad \\forall \\sigma\\in \\mathrm{Sym}_{\\ell} \\tag{$E_{1}$} \\label{E1}\\\\\n& \\AND(Y_1,\\dots,Y_{\\ell})=\\AND(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})}),\n\\quad \\forall \\sigma\\in \\mathrm{Sym}_{\\ell} \n\\tag{$E_{2}$}\\label{E2}\\\\ \n& \\OR\\big(\\Xvec,\\OR(\\Yvec)\\big)=\n\\OR(\\Xvec,\\Yvec) \\tag{$E_{3}$} \\label{E3}\\\\\n& \\AND\\big(\\Xvec,\\AND(\\Yvec)\\big)=\n\\AND(\\Xvec,\\Yvec) \\tag{$E_{4}$} \\label{E4}\\\\\n& \\SAND\\big(\\Xvec,\\SAND(\\Yvec),\\Zvec\\big)=\n\\SAND(\\Xvec,\\Yvec,\\Zvec) \\tag{$E_{4'}$} \\label{E4'}\\\\\n& \\OR(A)= A \\tag{$E_{5}$} \\label{E5}\\\\\n& \\AND(A)= A \\tag{$E_{6}$} \\label{E6}\\\\\n& \\SAND(A)= A \\tag{$E_{6'}$} \\label{E6'}\\\\\n& \\AND\\big(\\Xvec,\\OR(\\Yvec)\\big)= \n\\OR\\big(\\AND(\\Xvec,Y_1),\\dots,\\AND(\\Xvec,Y_{\\ell})\\big) \\tag{$E_{10}$} \\label{E10}\\\\\n& \\SAND\\big(\\Xvec,\\OR(\\Yvec),\\Zvec\\big)= \n\\OR\\big(\\SAND(\\Xvec,Y_1,\\Zvec),\\dots,\\SAND(\\Xvec,Y_{\\ell},\\Zvec)\\big) \\tag{$E_{10'}$} \n\\label{E10'}\\\\\n& \\OR(A,A,\\Xvec)= \\OR(A,\\Xvec). \\tag{$E_{11}$} \n\\label{E11}\n\\end{align}\n\\endgroup\n\\end{theorem}\nThe numbering of the axioms in $\\ESP$ corresponds to the numbering \nof the axioms for the multiset semantics for standard attack trees, \nas presented in~\\cite{KoMaRaSc_JLC}, while new axioms (involving \\SAND) are \nmarked with primes. \n\\begin{proof}\nThe proof of this theorem follows the same\nline of reasoning as the proofs of Theorems~4.2 and~4.3 of\nGischer~\\cite{Gischer1988199}, where series--parallel pomsets are axiomatized.\n{\n To prove the theorem, we remark that \n SP graphs form a visual representation of \n series-parallel \n partially ordered multisets (SP pomsets). \n A complete, finite axiomatization of pomsets under \n concatenation, parallel composition and union\n has been provided in~\\cite{Gischer1988199}, where \n sets of series-parallel pomsets have been used to represent processes. \n In our case, sets of series-parallel pomsets (\\ie, sets of SP graphs)\n represent attack trees constructed using \n \\AND\\ (having the same properties as the parallel composition of processes), \n \\SAND\\ (having the same properties as concatenation),\n and \\OR\\ (having the same properties as choice). \n The set $\\ESP$ corresponds to the axioms from~\\cite{Gischer1988199}. \n The axioms involving the identity elements (\\ie, $1$ -- the empty pomset, and \n$0$ --\n the empty process) have been omitted because they can only be used \n for transforming processes involving $0$ or $1$ and \n such identity elements do not exist in the case of attack trees. \n Furthermore, our axioms are written using unranked operators \n contrary to the binary operators of concatenation, parallel composition, and \n choice. \n}\n\n\\qed\n\\end{proof}\n\n\n\\subsection{\\SAND~Attack Trees in Canonical Form}\n\\label{sec:NF}\nLet $\\semantic{\\cdot}$ be a semantics for (\\SAND) attack trees.\nA complete axiomatization of $\\semantic{\\cdot}$\ncan be used to derive a canonical form of trees \ninterpreted with $\\semantic{\\cdot}$. \nSuch canonical forms provide the most concise \nrepresentation for equivalent trees\nand are the natural representatives \nof equivalence classes defined by \n$\\semantic{\\cdot}$. \n\nWhen \\SAND~attack trees are interpreted using the SP semantics, \ntheir canonical forms \nconsist of either a single basic action, \nor of a root node labeled with \\OR\\ and subtrees \nwith nested, alternating \noccurrences of \\AND\\ and \\SAND\\ nodes. \nCanonical forms correspond exactly to \nthe sets of SP graphs labeled by $\\basicact$\nand they depict all attack alternatives in a straightforward \nway.\n\n{\nCanonical representations of \\SAND~attack trees\nunder the SP semantics\ncan be defined using the complete set of axioms \n$\\ESP$. \nBy orienting the equations \n\\eqref{E3}, \\eqref{E4}, \\eqref{E4'}, \\eqref{E5}, \\eqref{E6},\n\\eqref{E6'}, \\eqref{E10}, \\eqref{E10'}, and \\eqref{E11}\nfrom left to right, we obtain a term rewriting system, \ndenoted by $\\thetrs$. \nThe canonical representations of \\SAND~attack trees \ncorrespond to normal forms with respect to $\\thetrs$. \nIn the rest of this section we show that \nthe normal forms with respect to $\\thetrs$\nare exactly the terms generated by the following grammar, \nwhere $k\\geq 2$ and $b\\in\\basicact$\n\\begin{align} \n\\label{grammar:NF}\n\\nfterm ::= \\quad& \\asterm \\mid \\OR(\\asterm_1, \\dots, \\asterm_k)\n \\quad\\mbox{for } \\asterm_i \\neq \\asterm_j\n \\mbox{ if } i\\neq j\\\\\\notag\n\\asterm ::= \\quad & \\andterm \\mid \\sandterm\\\\\\notag\n\\andterm ::= \\quad & b \\mid \\AND(\\sandterm_1, \\ldots, \\sandterm_k)\\\\\\notag\n\\sandterm ::= \\quad& b \\mid \\SAND(\\andterm_1, \\ldots, \\andterm_k).\\\\\\notag\n\\end{align}\nThe non-terminal $\\andterm$ produces all trees that consist of\na single basic action or being \na nested alternation of \n$\\AND$ and $\\SAND$ operators,\nwhere the outer operator is $\\AND$.\nSimilarly, $\\sandterm$ produces all such trees \nwhere the outer operator is $\\SAND$.\nThe non-terminal $\\asterm$ generates the two \npreviously described types of trees. \nFinally, $\\nfterm$ \ncombines the trees generated by $\\asterm$ using \nthe \\OR\\ refinement. \nWe denote the sets of terms \ngenerated by $\\nfterm$, $\\asterm$, $\\andterm$, and $\\sandterm$, by $\\nfterms$, \n$\\asterms$, $\\andterms$, and $\\sandterms$, respectively. \n\nWe first observe that the terms generated by the non-terminal $\\nfterm$\ncorrespond exactly to all sets of SP graphs labeled by the elements of \n$\\basicact$. \n\\begin{lemma}\\label{lem:bijection}\nThe restriction of function $\\sem{.}$ to $\\nfterms$\nis a bijection from $\\nfterms$ to $\\power{\\spset}$.\n\\end{lemma}\n\\begin{proof}\nThe proof consists of two steps. First we prove that the terms from\n$\\asterms$ exactly correspond to SP graphs, after which we extend this\nresult to the correspondence between $\\nfterms$ and the sets of SP graphs.\n\\begin{enumerate}\n\\item $\\sem{.}$ is a bijection from $\\asterms$ to $\\spset$.\\\\\n For injectivity we prove by induction  that $\\sem{\\asterm_1} = \\sem{\\asterm_2}$\n implies $\\asterm_1 = \\asterm_2$. Given that for every $t \\in \\asterms\n $ the set $\\sem{t}$ contains a single SP graph, we abuse notation and \n refer to \n $\\sem{t}$ as the SP graph contained in $\\sem{t}$. Let us assume that \n $\\asterm_1$ is a basic action $b$, then $\\sem{\\asterm_1}$ is a single \n edge graph $\\xrightarrow{b}$, it follows that $\\asterm_2 = \n b$ considering the edge labels preservation property of two isomorphic SP \n graphs. Let us consider now that $\\asterm_1 = \\AND(\\sandterm_1^1, \\cdots, \\sandterm_k^1)$ for \n some $k \\geq 2$, which means that $\\sem{\\asterm_1} = \n \\sem{\\sandterm_1^1} \\parop \\cdots \\parop\n \\sem{\\sandterm_k^1}$. Given that $\\sem{\\asterm_1} = \\sem{\\asterm_2}$,\n for every $i \\in \\{1, \\cdots, k\\}$, $\\sem{\\asterm_2}$ contains a subgraph \n $G_i$ which is isomorphic to $\\sem{\\sandterm_i^1}$. Thus, considering that \n $\\sem{\\sandterm_i^1}$ is either a basic action or a sequential composition, \n we have that \n $\\sem{\\asterm_2} = G_1 \\parop \\cdots \\parop G_k$ and, according to grammar\n \\eqref{grammar:NF}, \n $\\asterm_2 = \\AND(\\sandterm_1^2, \\cdots, \\sandterm_k^2)$, for some terms \n $\\sandterm_1^2, \\cdots, \\sandterm_k^2 \\in \\sandterms$. By the \n induction hypothesis, it \n follows that $\\sem{\\sandterm_i^1} = G_i = \n \\sem{\\sandterm_i^2}$ implies $\\sandterm_i^1 = \\sandterm_i^2$, which leads to \n $\\asterm_1 = \\asterm_2$. A similar \n proof can be obtained for the case where $\\asterm_1 = \\SAND(\\andterm_1^1, \n \\cdots, \\andterm_k^1)$. \n \n\n Surjectivity follows from the fact that every SP graph has a unique\n (modulo associativity) decomposition in terms of \n the operators for \n sequential and parallel composition.\n Such a decomposition naturally corresponds to terms from $\\asterms$.\n\\item $\\sem{.}$ is a bijection from $\\nfterms$ to $\\powerset(\\spset)$.\\\\\n For injectivity, we assume that $\\sem{\\nfterm_1} = \\sem{\\nfterm_2}$.\n This implies that the sets $\\sem{\\nfterm_1}$ and $\\sem{\\nfterm_2}$\n have the same size and the same elements. If this size is $1$, then\n they both contain the same element, which uniquely corresponds to\n a term $\\asterm$, so $\\nfterm_1 = \\asterm = \\nfterm_2$.\n If the size of the sets is larger than $1$, then $\\nfterm_i$ \n are of the form $\\OR(\\asterm_1^i,\\dots, \\asterm_k^i)$, for \n $i\\in\\{1,2\\}$. Since $\\sem{\\nfterm_1} = \\sem{\\nfterm_2}$, \n the elements of these two\n sets are pairwise identical.\n Moreover, by definition, \n all arguments in $\\nfterm_i$ are different, which implies \n $k_1=k_2$.\n From the previous item, it follows\n that the elements of $\\sem{\\nfterm_i}$ correspond uniquely \n to $\\asterm_1^i,\\dots, \\asterm_k^i$. \n From the pairwise equality between the\n arguments of the two terms, it follows that $\\nfterm_1$ and\n $\\nfterm_2$ are identical.\n \n For surjectivity, \n let $\\{G_1, \\dots, G_k\\}$ \n be a set of SP graphs. \n It follows from the previous item\n that there exist trees $\\asterm_1, \\dots, \\asterm_k$, \n such that $\\sem{\\asterm_i}=G_i$, for $i\\in\\{1,\\dots, k\\}$. \n This implies that $\\{G_1, \\dots, G_k\\}= \\sem{\\OR(\\asterm_1, \\dots, \\asterm_k)}$, \n which finishes the proof of surjectivity. \n\\end{enumerate}\n\\qed\n\\end{proof}\nLemma~\\ref{lem:bijection} shows that \nthe grammar \\eqref{grammar:NF} \ngenerates all \\SAND~attack trees in canonical form. \nIt remains to be proven that\nthe set of trees generated by the grammar \\eqref{grammar:NF}\nis equal to the set of normal forms of the term rewriting system\n$\\thetrs$ and that these normal forms are unique. \n\n\\begin{theorem}\n\\label{th:trs}\nThe term rewriting system $\\thetrs$ is strongly terminating and confluent.\n\\end{theorem}\n\\begin{proof}\nWe show that the term rewriting system $\\thetrs$ is terminating and confluent\nwith help of the grammar~\\eqref{grammar:NF}, \nin four steps. \n\\begin{enumerate}\n\\item \nFirst, we show with\nstandard methodology that the\nterm rewriting system is terminating.\nWe define the following norm which assigns natural numbers to terms:\n\\[\n\\begin{array}{lll}\n\\norm{b} & = & 1\\\\\n\\norm{\\OR(X_1, \\ldots, X_k)} & = &\n \\norm{X_1} + \\ldots + \\norm{X_k} + 2\\\\\n\\norm{\\AND(X_1, \\ldots, X_k)} & = &\n 2\\cdot \\norm{X_1} \\cdot \\ldots \\cdot \\norm{X_k}\\\\\n\\norm{\\SAND(X_1, \\ldots, X_k)} & = &\n 2\\cdot \\norm{X_1} \\cdot \\ldots \\cdot \\norm{X_k}\\\\\n\\end{array}\n\\]\n\nIt can be easily verified that for every rewrite rule $l\\rightarrow r\\in \\thetrs$,\nwe have $\\norm{l} > \\norm{r}$. Consequently, there are no\ninfinite reduction sequences, or, in other words, the term rewriting\nsystem is strongly terminating.\nNotice that because we consider term rewriting modulo commutativity of\n$\\OR$ and $\\AND$, we have to verify that the left-hand side and the right-hand\nside of equations \\eqref{E1} and \\eqref{E2}\nhave equal norms~\\cite{KBV2001}. This is clearly the\ncase.\n\\item \nNow we prove that the terms \nproduced by the grammar~\\eqref{grammar:NF} \nare exactly the normal forms with respect to $\\thetrs$.\nFor the terms in $\\asterms$, none of the rewrite rules can be applied,\nbecause these terms do not contain $\\OR$, have no occurrences of\n$\\AND$ containing an argument of type $\\AND$, have no occurrences of\n$\\SAND$ containing an argument of type $\\SAND$, and do not contain\noperators with a single argument.\nWe extend this to terms $\\nfterms$ by observing that all $\\OR$\noperators occurring in such terms have at least two arguments and that\nall these arguments are different.\n\nConversely, consider a term $t$ in normal form that \ncontains an $\\OR$ operator. Then $t=\\OR(t_1,\\ldots,t_n)$, where the $t_i$\ndo not contain an $\\OR$ operator, else \\eqref{E3}, \\eqref{E10}, or\n\\eqref{E10'} can be applied. \nIt remains to show that normal form terms without occurrence of an\n$\\OR$ operator are in $\\asterms$. Such terms are basic terms or have\n$\\SAND$ or $\\AND$ as their top-level operator. The last two cases are\nsymmetric and we therefore only consider the case\n$\\AND(t_1,\\ldots,t_n)$. We must show that each $t_i$ is a basic term\nor in the form $t_i=\\SAND(t_1',\\ldots,t_m')$. Suppose not, then there\nexists a $t_i$ that has $\\AND$ as its top-level operator. It follows\nthat the term is not in normal form because \\eqref{E4} can be applied.\n\n\\item The normal forms are unique. \nTo show that the normal forms are unique, assume that \n$\\nfterm_1$ and $\\nfterm_2$ are both normal forms for a\n \\SAND~attack tree $t$. \nSince the rewrite system $\\thetrs$ \nwas constructed by orienting the axioms from $\\ESP$, \nwe have that $\\ESP\\vdash \\nfterm_1=\\nfterm_2$. \nThis means that $\\sem{\\nfterm_1}=\\sem{\\nfterm_2}$.\nFrom bijectivity proven in Lemma~\\ref{lem:bijection}, we obtain \n$\\nfterm_1 = \\nfterm_2$.\n\n\\item Now that we have proven termination and uniqueness of normal forms, it\nimmediately follows that the term rewriting system is\nconfluent~\\cite{D2005}.\n\\end{enumerate} \n\\qed\n\\end{proof}\n \n\n\n\n\nExample~\\ref{ex:nf} illustrates the notion of \ncanonical form for \\SAND~attack trees.\n}\n\n\\begin{example}\n\\label{ex:nf}\nThe canonical form of the \\SAND~attack tree $t$ \nin Figure~\\ref{fig:attack-tree} is the tree \n\\[\nt' = \\OR\\Big(\\SAND\\big(\\ftprhostsx, \\rshx, \\localbofx\\big), \\\\\n \\AND\\big(\\sshbofx, \\rsarefbofx\\big)\\Big)\n\\]\nshown in Figure~\\ref{fig:attack-tree2}. \nIt \nis easily seen to be in normal form with respect to $\\thetrs$.\n\\end{example}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=3.4cm]{atree3}\n\\caption{\\SAND\\ attack tree $t'$ equivalent to \\SAND\\ attack tree $t$ from \nFigure~\\ref{fig:attack-tree}}\n\\label{fig:attack-tree2}\n\\end{figure}\n\n\\subsection{SP Semantics as a Generalization of the Multiset Semantics}\n\\label{sec:SP_vs_M}\n\nHaving a complete set of axioms for the SP semantics\nallows us to formalize the relation between \n\\SAND\\ attack trees under the SP semantics and \nattack trees under the multiset semantics, denoted by $\\msem{\\cdot}$. \nThis is achieved by extracting \na complete set of axioms for \nthe multiset semantics for \nattack trees from \nthe set $\\ESP$. \nLet $\\ESM$ be the subset of axioms from $\\ESP$ that\ndo not contain the $\\SAND$ operator, \\ie, \n$\\ESM=\\{$\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref{E4}, \\eqref{E5},\n\\eqref{E6}, \\eqref{E10}, \\eqref{E11}$\\}$.\n\\begin{theorem}\n\\label{th:multiset-axioms}\nThe axiom system $\\ESM$\nis a complete set of axioms for the multiset\nsemantics for attack trees.\n\\end{theorem}\n{\n\\begin{proof}\nIn \\cite[Theorem 4.9]{KoMaRaSc_JLC}, a complete axiomatization of\nthe multiset semantics for an extention of attack trees \ncalled attack--defense trees (ADTrees) is given. In the following, we\ncall that axiomatization $\\EADT$.\nADTrees are a superset of attack trees.\nThey may contain defender's nodes \nmodeled by the so called\nopponent's functions and countermeasures. \nWe claim that $\\ESM$\nis a complete axiomatization of the multiset semantics for attack trees.\nObviously if two attack trees\nare equal with respect to $\\ESM$, then they\nare also equal with respect to $\\EADT$.\nThis is clear, because $\\ESM\\subset \\EADT$.\n\nConversely, we prove that if two attack trees are equal with respect to \n$\\EADT$, they are equal with respect to $\\ESM$. This\nfollows from the following syntactical reasoning. \n$\\EADT$ contains function symbols which we call\n\\emph{countermeasures}. \nObserve by inspecting the axioms of $\\EADT$ that if a\ncountermeasure occurs at the left-hand side of an equation, then it also\noccurs at the right-hand side, and vice versa. Therefore, axioms \n$(E_{13}),\n(E_{16}), (E_{17}), (E_{18}), (E_{19}), (E_{20})$ from $\\EADT$\ncan never be used in a derivation of equality of\ntwo standard attack trees. Further, observe that the remaining \naxioms $(E_9)$ and $(E_{12})$ from $\\EADT$ make use of \n\\emph{opponent's functions}.\nIn these axioms, \nan opponent function occurs on the left-hand side if and only\nif it occurs on the right hand side. Thus \nthese axioms are never used to equate\ntwo attack trees which do not contain opponent's nodes. \nThe remaining axioms are precisely \n\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref{E4}, \\eqref{E5},\n\\eqref{E6}, \\eqref{E10}, \\eqref{E11}.\nSo, we can only use these axioms \nto derive equalities\nof attack trees with respect to $\\EADT$, which implies that such a\nderivation is also possible using axioms from $\\ESM$.\n\\qed\n\\end{proof}\n}\n\n\nBy comparing the complete sets of axioms \n$\\ESP$ and $\\ESM$ we obtain that two attack trees are equivalent under\nthe multiset semantics if and only if they are equivalent under the SP \nsemantics. This is formalized in the following theorem. \n\\begin{theorem}\n\\label{th:SP_vs_M}\n$\\SAND$ attack trees under the SP semantics are a \\emph{conservative\nextension} of attack trees under the multiset semantics.\n\\end{theorem}\n\\begin{proof}\n{\nLet $t$ and $t'$ be standard attack trees.\nLet $\\msem{t}$ and $\\msem{t'}$ be their interpretation in the \nmultiset semantics and $\\sem{t}$ and\n$\\sem{t'}$ be their interpretation in the SP semantics.\nWe prove that $\\msem{t} = \\msem{t'}$ if and only if $\\sem{t} =\n\\sem{t'}$.\n\nBy Theorem~\\ref{th:multiset-axioms}, a complete axiomatization of \nthe multiset semantics for attack \ntrees \nconsists of axioms~\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref\n{E4}, \\eqref{E5}, \\eqref{E6}, \\eqref{E10}, \\eqref{E11}.\nThe complete axiomatization of the SP semantic for \n$\\SAND$ attack trees additionally \ncontains axioms \\eqref{E4'}, \\eqref{E6'},\nand \\eqref{E10'}. Thus, every equivalence of attack\ntrees under the multiset semantics \nis clearly an equivalence of $\\SAND$ attack trees\nunder the SP semantics.\n\nTo see the converse, \nwe show that the additional axioms do not introduce new\nequalities on standard attack trees. \nFirst inspect the three additional axioms and note that all of them\ncontain the $\\SAND$ operator.\n\nNext, observe that for all axioms, the set of variables occurring on the\nleft-hand side is equal to the set of variables occurring on the\nright-hand side. Thus, there is no axiom eliminating all\noccurrences of a variable. \nIn particular, we claim that \nall axioms transform \nterms containing a $p$-ary $\\SAND$ expression, where $p\\geq 2$, into\nterms containing a $q$-ary $\\SAND$ expression, for some $q\\geq 2$. \nThis is evident for equations without the $\\SAND$ operator (since no\nvariables are eliminated) and remains\nto be shown for equations \\eqref{E4'}, \\eqref{E6'},\nand \\eqref{E10'}. \nAxiom~\\eqref{E6'} introduces and removes unary $\\SAND$, but does not\nmodify the single variable $A$ and therefore satisfies the claim. \nThe arities of the two left-hand side $\\SAND$ operators \nin equation~\\eqref{E4'} are $l$ and $k+l+m$ and the arity of the \nright-hand side operator is $k+l+m$, where $k,m\\geq 0$ and $l\\geq 1$.\nSince $1\\leq l\\leq k+l+m$ and both sides contain a $\\SAND$ operator of\narity $k+l+m$, if either of the two sides\ncontains a $\\SAND$ operator with two or more arguments, then so does\nthe other side.\nFinally, since $l\\geq 1$, \nthe arity of the $\\SAND$ operator on the left-hand side of\nequation~\\eqref{E10'} is equal to the arities of the $\\SAND$ operators\non its right-hand side and at least one $\\SAND$ operator\noccurs on the right-hand side.\n\nWe can now show that none of the three axioms~\\eqref{E4'}, \\eqref{E6'}, \\eqref\n{E10'} introduces new equalities on\nstandard attack trees. \nIn particular, axiom~\\eqref{E6'} introduces and removes unary\n$\\SAND$, but this does not introduce new equalities on standard attack\ntrees. Equations~\\eqref{E4'} and~\\eqref{E10'} match unary $\\SAND$, but\nrequire a further $\\SAND$ with $2$ or more arguments to add a new equality.\nSince, by the above claim, no $p$-ary $\\SAND$ for $p\\geq 2$ \ncan be introduced with any of\nthe equations, the additional equations do not introduce new\nequalities on standard attack trees.\n\\qed\n}\n\n\\end{proof}\n\n{\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Attributes}\n\\label{sec:attributes}\n\n\nAttack trees do not only serve to represent security scenarios \nin a graphical way. They can also be used to \nquantify such scenarios with respect to \na given parameter, called an \\emph{attribute}. \nTypical examples of attributes include\nthe likelihood that the attacker's goal is satisfied and \nthe minimal time or cost of an attack. \nSchneier described~\\cite{Schn} \nan intuitive bottom-up algorithm for calculating attribute values\non attack trees:\nattribute values are assigned to the leaf nodes \nand two functions\\footnote{These are actually \nfamilies of functions representing infinitely many $k$-ary function symbols,\nfor all $k \\geq 2$.} (one for the \\OR\\ and one for the \\AND\\ refinement)\nare used to propagate the attribute value up to the root node. \nMauw and Oostdijk showed~\\cite{MaOo} that \nif the binary operations induced by the two functions \ndefine a semiring, \nthen the \nevaluation of the attribute on two attack trees \nequivalent with respect to the multiset semantics \nyields the same value. This result has \nbeen generalized to any semantics and attribute that satisfy a notion of \n\\emph{compatibility}~\\cite{KoMaRaSc_JLC}. \nWe briefly discuss it for \\SAND~attack trees \nat the end of this section. \nWe start with a demonstration on how \nthe bottom-up evaluation \nalgorithm \ncan naturally be extended to \\SAND\\ attack trees.\n \n \nAn {\\em attribute domain for an attribute $\\fullattr$ on \\SAND\\ attack trees} \nis a tuple $\\attrdomain_\\attr = (\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr)$\nwhere $\\attrval_\\attr$ is a set of values and \n $\\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr$ are families of \n$k$-ary functions of the\n form $\\attrval_\\attr\\times\\dots\\times\\attrval_\\attr\\to\\attrval_\\attr$, \nassociated to \\OR, \\AND, and \\SAND\\ refinements, respectively. \nAn {\\em attribute for \\SAND~attack trees} is a pair \n$\\fullattr = (\\attrdomain_\\attr, \\basicassign_\\attr)$ \nformed by an attribute domain $\\attrdomain_\\attr$ and a function \n$\\basicassign_\\attr:\\basicact\\to\\attrval_\\attr$,\ncalled {\\em basic assignment} for $\\fullattr$, which associates a value from \n$\\attrval_\\attr$ with each basic action $b \\in \\basicact$. \n\\begin{definition}\n\\label{def:attr}\nLet $\\fullattr = \\big((\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\n\\attr), \\basicassign_\\attr\\big)$\nbe an attribute. \nThe attribute evaluation function \n$\\attr: \\sandtree \\to \\attrval_\\attr$ which calculates the value of \nattribute \n$\\fullattr$ \nfor every \\SAND \\ attack tree $t \\in \\sandtree$\nis defined recursively as follows\n\\[ \\attr(t) = \\left\\{ \n \\begin{array}{l l}\n \\basicassign_\\attr(t) & \\quad \\text{if $t=b,\\ b\\in\\basicact$}\\\\\n \\attror_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\OR(t_1, \\dots, t_k)$}\\\\\n \\attrand_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\AND(t_1, \\dots, t_k)$}\\\\\n \\attrsand_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\SAND(t_1, \\dots, t_k)$}\n \\end{array} \\right.\\]\n\\end{definition}\n\n\nThe following example illustrates the bottom-up evaluation\nof the attribute \\emph{minimal attack time} on the \\SAND~attack trees \ngiven in Example~\\ref{eg:def-attack-tree}.\n\\begin{example}\n\\label{ex:min_cost}\nLet $\\attr$ denote the minimal time \nthat the attacker needs to achieve her goal.\nWe make the following assignments to the basic actions: \n$\\ftprhostsx\\mapsto 3$, $\\rshx\\mapsto 5$, $\\localbofx\\mapsto 7$, $\\sshbofx\\mapsto \n8$, $\\rsarefbofx\\mapsto 9$. \nSince we are interested in the minimal attack time, \nthe function for an \\OR\\ node \nis defined by \n$\\attror_\\attr(x_1,\\dots,x_k)=\\min\\{x_1,\\dots, x_k\\}$. \nThe function for an \\AND\\ node \nis $\\attrand_\\attr(x_1,\\dots,x_k)=\\max\\{x_1,\\dots, x_k\\}$, \nwhich models that the children \nof a conjunctively refined node are executed in parallel.\nFinally, in order to model that the children \nof a \\SAND\\ node need to be executed \nsequentially, \nwe let \n$\\attrsand_\\attr(x_1,\\dots,x_k)=\\sum_{i=1}^{k} x_i$. \nAccording to Definition~\\ref{def:attr}, \nthe minimal attack time for our running scenario $t$ is\n\\[\\attror_\\attr\\Big(\\attrsand_\\attr\\big(\\attrsand_\\attr(3,5),7\\big),\n\\attrand_\\attr(8, 9)\\Big)=\n\\min\\Big({\\mathrm\\Sigma}\\big({\\mathrm\\Sigma}(3,5),7\\big), \\max(8, 9)\\Big)=9.\\]\n\\end{example}\n\nIn the case of standard attack trees, \nthe bottom-up procedure uses only two functions to propagate \nthe attribute values to the root -- one for conjunctive and one for \ndisjunctive nodes. This means that the same function is employed \nto calculate the value of every conjunctively refined node, \nindependently of whether its children need to be executed sequentially or can \nbe executed simultaneously. \nEvidently, with \\SAND~attack trees, we can apply different propagation \nfunctions for \\AND\\ and \\SAND\\ nodes, as in \nExample~\\ref{ex:min_cost}. \nTherefore, \\SAND~attack trees can be evaluated over a larger set of \nattributes, and hence may provide more accurate evaluations of \nattack \nscenarios than \nstandard attack trees.\n\n\n\nTo guarantee that the evaluation of an attribute on equivalent \nattack trees yields\nthe same value, \nthe attribute domain must be \n\\emph{compatible} with a considered semantics~\\cite{KoMaRaSc_JLC}. \nOur complete set of axioms is a useful tool \nto check for compatibility. \nConsider an attribute domain\n$\\attrdomain_\\attr = (\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr)$, \nand let $\\sigma$ be a mapping\n$\\sigma=\\{\\OR\\mapsto \\attror_\\attr, \\AND \\mapsto \\attrand_\\attr, \n\\SAND \\mapsto \\attrsand_\\attr\\}$. \nGuaranteeing that $\\attrdomain_\\attr$ is compatible with \na semantics axiomatized by $E$ \namounts to verifying that the equality \n$\\sigma(l)=\\sigma(r)$ holds in $\\attrval_\\attr$, for every axiom\n$l=r\\in E$.\nIt is an easy exercise to show that the attribute domain \nfor minimal attack time, considered in Example~\\ref{ex:min_cost},\nis compatible with the SP semantics for \\SAND\\ attack trees.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have formalized the extension of attack trees \nwith sequential conjunctive refinement, \ncalled \\SAND, \nand given a semantics to \\SAND~attack trees in terms of\nsets of series-parallel graphs.\nThis SP semantics \nnaturally extends the multiset semantics \nfor attack trees from~\\cite{MaOo}.\nWe have shown that the notion of \na complete set of axioms for a semantics and \nthe bottom-up evaluation procedure can be generalized \nfrom attack trees to \\SAND~attack trees, \nand have proposed a complete axiomatization of the SP semantics. \n \n\nA number of recently proposed solutions\nfocus on extending attack trees with defensive\nmeasures~\\cite{RoKiTr2,KoMaRaSc_JLC}.\nThese extensions support reasoning about security scenarios\ninvolving two players -- an attacker and a defender --\nand the interaction between them.\nIn future work, we intend to add\nthe $\\SAND$ refinement to such trees.\nAfterwards, we plan to investigate\nsequential disjunctive refinement, as used\nfor instance in~\\cite{Arnold-POST14}. Our goal is to propose\na complete formalization\nof trees with attack and defense nodes,\nthat have parallel and sequential, conjunctive and disjunctive\nrefinements. \nThe findings will be implemented in the software \napplication ADTool~\\cite{adtool}. \n\n\n\n\\paragraph{\\textbf{Acknowledgments}} The research leading\nto these results has received funding from the European Union Seventh Framework\nProgramme under grant agreement number 318003 (TREsPASS) and from\nthe Fonds National de la Recherche Luxembourg under grant \nC13\/IS\/5809105.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}  Let $\\S$ be a co-dimension two  spacelike submanifold \nof a spacetime $M$.  Under suitable orientation assumptions, there exists two families\nof future directed null geodesics issuing orthogonally from $\\S$.  If one of the families has vanishing expansion along $\\S$ then\n$\\S$ is called a marginally outer trapped surface (or an apparent horizon).  The notion of a\nmarginally outer trapped surface  (MOTS) was introduced early on in the development of the\ntheory of black holes, and plays a fundamental role in quasi-local descriptions of \nblack holes; see e.g.,  \\cite{AK}.  MOTSs arose in a more purely mathematical context \nin the work of Schoen and Yau \\cite{SY2} concerning the existence\nof solutions to the Jang equation, in connection with their proof\nof positivity of mass.   \n\nMathematically, MOTSs may be viewed as spacetime\nanalogues of minimal surfaces in Riemannian manifolds.  Despite the\nabsence of a variational characterization for MOTSs\nlike that for minimal surfaces,  MOTSs have recently\nbeen shown to satisfy a number of analogous properties; see for example,\n\\cite{AMS0,AMS,AM1,AM2, AG, E, GS}.   Of importance to many of these developments\nis the fact, first discussed in \\cite{AMS0}, that MOTSs admit a notion of stability analogous, in the analytic sense, to that of minimal \nsurfaces (cf., Section 2).  \n\nIn this paper we consider  applications of stable MOTSs to two problems\nin general relativity.   In Section 3 we address the issue of how the size of a\nmaterial body tends to be restricted by the amount of matter contained within it.  \nMore specifically, we consider an extension of a result of Schoen and Yau \\cite{SY}\nconcerning the size of material bodies to nonmaximal initial data sets. \nIn Section 4 we discuss a higher dimensional version of  the lower area (entropy) bounds\nobtained by Gibbons \\cite{Gi} and Woolgar \\cite{Wo}  for  ``topological black holes\" which\ncan arise in spacetimes with negative cosmological constant.   This extends a\nresult in \\cite{CG} to the general nontime-symmetric setting.  We defer \nfurther discussion of these problems until Sections 3 and 4.   In the next section we \npresent some basic background material on MOTSs relevant to our needs.\n\n\\section{Marginally outer trapped surfaces}\n\nWe recall here some  basic definitions and facts about marginally outer\ntrapped surfaces.   We refer the reader to \\cite{AMS, AM1, GS,G} for further\ndetails.  \n\nLet $V$ be a spacelike hypersurface in an $n+1$ dimensional, $n \\ge 3$, spacetime $(M,g_M)$. \nLet $g = \\<\\,,\\,\\>$ and $K$ denote the induced metric and second fundamental form \nof $V$, respectively.  To set sign conventions, for vectors $X,Y \\in T_pV$, $K$ is defined\nas, $K(X,Y) = \\<\\D_X u,Y\\>$, where $\\D$ is the Levi-Civita connection of $M$ and $u$ is the future directed timelike unit vector field to $V$.  Note that we are using the `Wald',\nrather than the `ADM\/MTW', convention for the extrinsic curvature, i.e., positive ${\\rm tr}\\,K$ implies expansion.\n\nLet $\\S$ be a smooth compact hypersurface in $V$, perhaps with boundary $\\delta\\S$, \nand assume $\\S$ is two-sided  in $V$.   Then $\\S$ admits a smooth unit normal field\n$\\nu$ in $V$, unique up to sign.  By convention, refer to such a choice as outward pointing. \nThen $l = u+\\nu$ is a future directed outward pointing null normal vector field along $\\S$, unique\nup to positive scaling.   \n\nThe null second fundamental form of $\\S$ with respect to $l$ is, for each $p \\in \\S$,\nthe bilinear form defined by,\n\\beq\n\\chi : T_p\\S \\times T_p\\S \\to \\Bbb R , \\qquad \\chi(X,Y) = g_M(\\D_Xl, Y) \\,.\n\\eeq\nThe null expansion  $\\th$ of $\\S$ with respect to $l$  is obtained by tracing the\nnull second fundamental form, \n$\\theta = {\\rm tr}_h \\chi = h^{AB}\\chi_{AB} = {\\rm div}\\,_{\\S} l$, where\n$h$ is the induced metric on $\\S$.  In terms of the initial data $(V,g,K)$,\n$\\th = {\\rm tr}_h K + H$, where $H$ is the mean curvature of $\\S$ within\n$V$.  It is well known that the sign of $\\th$ is invariant under positive scaling \nof the null vector field $l$.\n\nIf $\\th$ vanishes then $\\S$ is called a marginally outer trapped surface (MOTS).   As mentioned in the introduction, MOTSs may be viewed as  spacetime analogues of minimal\nsurfaces in Riemannian geometry.  In fact in the time-symmetric case ($K=0$)\na MOTS $\\S$ is simply a minimal surface in $V$.  Of particular relevance\nfor us is the fact that MOTSs admit a notion of stability analogous to that of minimal \nsurfaces, as we now discuss.\n\nLet $\\S$ be a MOTS in $V$ with outward unit normal $\\nu$.   We consider\nvariations $t \\to \\S_t$   of  $\\S = \\S_0$, \n$- \\epsilon < t  <  \\epsilon,$ with variation vector field \n$\\calV = \\left . \\frac{\\delta}{\\delta t}\\right |_{t=0} = \\phi \\nu$,  $\\phi \\in C_0^{\\infty}(\\S)$, where\n$C_0^{\\infty}(\\S)$ denotes the space of  smooth functions on $\\S$ that vanish on the boundary of $\\S$, if there is one.  \n Let $\\th(t)$ denote\nthe null expansion of $\\S_t$ with respect to $l_t = u + \\nu_t$, where $u$ is the future directed timelike unit normal to $V$ and $\\nu_t$ is the\nouter unit normal  to $\\S_t$ in $V$.   A computation shows,\n\\beq\\label{op}\n\\left . \\frac{\\delta\\th}{\\delta t} \\right |_{t=0}   =   L(\\f) \\nonumber \n\\eeq\nwhere $L : C_0^{\\infty}(\\S) \\to C_0^{\\infty}(\\S)$ is the operator,\n\\beq\nL(\\phi)  = -\\triangle \\phi + \\<X,\\D\\phi\\>  + \n\\left( \\frac12 S - (\\mu + \\<J,\\nu\\>) - \\frac12 |\\chi|^2+{\\rm div}\\, X - |X|^2 \\right)\\phi \\,.\n\\eeq\nIn the above, $S$ is the scalar curvature of $\\S$, $\\mu = G(u,u)$, where $G = \\ric_M -\\frac12R_Mg_M$ is  the Einstein tensor of spacetime, $J$ is the vector field on $V$ dual to the one form $G(u,\\cdot)$, and $X$ is the vector field  on $\\S$  defined by taking the tangential part of $\\D_{\\nu}u$ along $\\S$.\n In terms of initial data, the Gauss-Codazzi equations\nimply, $\\mu = \\frac12\\left(S_V + ({\\rm tr}\\,K)^2 - |K|^2\\right)$  and \n$J = (\\div K)^{\\sharp} - \\D({\\rm tr}\\, K)$.   \n\nIn the time-symmetric case,  $\\th$ becomes the mean curvature $H$, the vector field\n$X$ vanishes and $L$ reduces to the classical stability operator of minimal surface theory.\nIn analogy with the minimal surface case, we refer to $L$ in \\eqref{op} as the stability\noperator associated with variations in the null expansion $\\th$.  Although in general $L$ is not self-adjoint,  its principal eigenvalue\\footnote{If $\\S$ has nonempty boundary, \nwe mean the principal Dirichlet eigenvalue.} (eigenvalue with smallest real part) \n$\\l_1(L)$ is real.   Moreover there exists\nan associated eigenfunction $\\phi$ which is positive on $\\S \\setminus \\delta\\S$. \nContinuing the analogy with the minimal surface case,\nwe say that a MOTS is stable provided $\\l_1(L) \\ge 0$.  (In the minimal surface case\nthis is equivalent to the second variation of area being nonnegative.) Note that if $\\phi$ is positive, we are moving `outwards' from the MOTS $\\S$, and if  there are no outer trapped surfaces outside of $\\S$,  then there shall exist no positive $\\phi$ for which $L(\\phi) < 0$.  It follows in this case that $\\S$ is stable \\cite{AMS,AM1,G}.\n\nAs it turns out, stable MOTSs share a number of properties in common with  minimal surfaces.  This sometimes  depends on the following fact.   Consider the \n``symmetrized\" operator\n$L_0: C_0^{\\infty}(\\S) \\to C_0^{\\infty}(\\S)$,\n\\beq\\label{symop}\nL_0(\\phi)  = -\\triangle \\phi  + \\left( \\frac12 S - (\\mu + \\<J,\\nu\\>) - \\frac12 |\\chi|^2\\right)\\phi \\,.\n\\eeq\nformally obtained  by  setting $X= 0$ in \\eqref{op}.   Then arguments in \\cite{GS}\nshow the following (see also \\cite{AMS}, \\cite{G}).\n\n\\begin{prop}\\label{eigen}  \n$\\l_1(L_0) \\ge \\l_1(L)$.\n\\end{prop}\n\nWe will say that a  MOTS is symmetric-stable if  $\\l_1(L_0) \\ge 0$; hence ``stable\"\nimplies ``symmetric-stable\". \n\n\\section{On the size of material bodies}\n\nIn this section we restrict attention to four dimensional spacetimes $M$,\nand hence three dimensional initial date sets $(V,g,K)$, $\\dim V = 3$. \n\nIt is a long held view in general relativity that  the size of a material body\nis limited by the amount of matter contained within it.  There are several\nprecise results in the literature supporting this point of view.   In \\cite{FG},\nit was shown, roughly, that the size of a stationary fluid body is bound by the reciprocal\nof the difference of the density and rotation of the fluid.  In this case ``size\" refers to the\nradius of the largest distance ball contained in the body.  \n\nMore closely related to the considerations of the present paper is the result of Schoen and Yau \\cite{SY} which asserts that\nfor a maximal (${\\rm tr}\\, K = 0$) initial data set $(V,g,K)$, the size of a body \n$\\Omega \\subset V$ is bound  by the reciprocal of the square root of the\nminimum  of the energy density $\\mu$ on $\\Om$.\nIn this case ``size\" refers to the radius of the largest tubular neighborhood in $\\Omega$\nof a loop contractible in $\\Om$ but not contractible in the tubular neighborhood.\nAs was discussed in \\cite{OM1},\nthis notion of size can be replaced by a notion based on the size of the largest\nstable minimal surface contained in $\\Omega$.\\footnote{This is formulated most simply\nwhen $\\Omega$ is bounded and {\\it mean convex}, meaning that the boundary\nof $\\Omega$ has mean curvature $H > 0$.    Then geometric measure theory\nguarantees the existence of many smooth least area surfaces contained in $\\Omega$.} As argued there, this in general\ngives a larger measure of the size of a body, but must still satisfy the same \nSchoen-Yau bound.   The aim of this section is to observe that a similar result holds\nwithout the maximality assumption if one replaces minimal surfaces with MOTS.\n\nLet $V$ be a $3$-dimensional spacelike hypersurface, which gives rise to the\ninitial data set $(V,g,K)$, as in Section 2.  Consider a {\\it body} in $V$ by which we mean a\nconnected open set $\\Om \\subset V$ with smooth boundary $\\delta\\Om$. \n We  describe a precise\nmeasure of the size of $\\Omega$ in terms of MOTSs contained within $\\Omega$.  \nLet  $\\S$ be a compact connected surface with boundary $\\delta\\S$ contained in $\\Omega$. Let $x$ be a\npoint in $\\S$ furthest from $\\delta\\S$ in $\\Omega$, i.e., $x$ satisfies,  $d_{\\Omega}(x, \\delta\\S) \n= sup_{y \\in \\S}\\, d_{\\Omega}(y, \\delta\\S)$, where $d_{\\Omega}$ is distance measured\nwithin $\\Omega$.  Then the (ambient) radius of $\\S$, $R(\\S)$, is defined as $R(\\S) =  d_{\\Omega}(x, \\delta\\S)$.  \n\nWe then define the radius of $\\Om$, $R(\\Om)$ as follows,\n\\beq\nR(\\Om) = \\sup_{\\S}  R(\\S)  \\,,\n\\eeq\nwhere the sup is taken over all compact connected symmetric-stable MOTSs with boundary contained\nin $\\Om$.    Now this can only be a reasonable measure of the size of\n$\\Om$ if there are a plentiful number of large symmetric-stable MOTSs contained\nin $\\Omega$.   But in fact a recent result of Eichmair \\cite{E} guarantees the existence of such MOTS, subject to  a natural convexity condition on the body $\\Om$.  \nWe say that $\\Om$\nis a {\\it null mean convex body} provided its boundary $\\delta\\Om$ has positive \noutward null expansion, $\\th_+ > 0$, and negative inward null expansion, $\\th_-< 0$.\nThe following is an immediate consequence of Theorem 5.1 in \\cite{E}.\n\n\\begin{thm} Let $\\Omega$ be a relatively compact null mean convex body, with connected boundary, in the\n$3$-dimensional initial data set $(V,g,K)$.   Let $\\s$ be a closed curve on $\\delta\\Om$\nthat separates $\\delta\\Om$ into two connected components.   Then there exists a smooth \nsymmetric-stable MOTS $\\S$ contained in $\\Om$ with boundary~$\\s$.\n\\end{thm}\n\nThe fact that $\\S$ is symmetric-stable follows from a straight forward modification of arguments in \\cite[p. 254]{SY2};  see also the discussion   at the end of Section 4 in \\cite{E}. \nIn fact, a variation of the arguments in \\cite[Section 4]{AM2},  may well imply that the MOTS \n$\\S$ constructed in Eichmair's theorem  is actually stable.  If that were the case, then\n$R(\\Omega)$ could be defined in terms of stable, rather than symmetric-stable, MOTS, which we believe would be conceptually preferable.  \n\nWe now state our basic result about the size of bodies.\n\n\\begin{thm}\\label{bound}  Let $\\Om$ be a body in the initial data set $(V,g,K)$, and suppose \nthere exists $c> 0$ such that $\\mu - |J| \\ge c$ on $\\Om$.  Then, \n\\beq\nR(\\Om) \\le  \\frac{2\\pi}{\\sqrt {3}}\\cdot \\frac1{\\sqrt{c}} \\,.\n\\eeq\n\\end{thm}  \n\n\\noindent \\emph{Proof:\\ }  The proof is similar to the proof  of Theorem 1 in \\cite{SY}.   The latter\nfollows essentially as a special case of the more general Proposition 1 in \\cite{SY}.\nFor the convenience of the reader we present here a simple direct proof of Theorem\n\\ref{bound}, which involves  a variation of the arguments in \\cite{SY}.\n\nLet $\\S$ be a symmetric-stable MOTS with boundary $\\delta\\S$ in $\\Om$; hence\n$\\l_1 = \\l_1(L_0) \\ge~0$.    Choose an associated eigenfunction $\\psi$ such that\n$\\psi > 0$ on $\\S\\setminus \\delta\\S$.  In fact, by perturbing the boundary $\\delta\\S$ ever so slightly\ninto $\\S$, we may assume without loss of generality that $\\psi > 0$ on $\\S$.   Substituting\n$\\phi = \\psi$ into Equation \\eqref{symop}, we obtain,\n\\beq\\label{laplace}\n\\triangle \\psi = - (\\mu + \\<J,\\nu\\>  +  \\frac12 |\\chi|^2 + \\l_1 - \\kappa) \\psi\n\\eeq\nwhere $\\kappa= \\frac12 S$ is the Gaussian curvature of $\\S$ in the induced metric $h$.\n\nNow consider $\\S$ in the conformally related metric $\\tilde h = \\psi h$.   The Gaussian\ncurvature of $(\\S, \\tilde h)$ is related to the Gaussian curvature of $(\\S, h)$ by,\n\\beq\\label{relate}\n\\tilde \\kappa = \\psi^{-2} \\kappa - \\psi^{-3} \\triangle \\psi + \\psi^{-4} |\\psi|^2 \\,.\n\\eeq\nCombining \\eqref{laplace} and \\eqref{relate} we obtain,\n\\beq\\label{gauss}\n\\tilde \\kappa = \\psi^{-2}(Q + \\psi^{-2} |\\D\\psi|^2)   \\,,\n\\eeq\nwhere, \n\\beq\\label{Q}\nQ = \\mu + \\<J,\\nu\\> + \\frac12 |\\chi|^2 + \\l_1  \\,\n\\eeq \n\nNow let $x$ be a point  in $\\S$ furthest from $\\delta\\S$ in $\\Om$, as in the \ndefinition of $R(\\S)$.  Let $\\g$ be a shortest curve in $(\\S, \\tilde h)$ from $x$\nto $\\delta\\S$.   Then $\\g$ is a geodesic in $(\\S, \\tilde h)$, and by Synge's formula \\cite{ON}\nfor the second variation of arc length, we have along $\\g$,\n\\beq\\label{ineq}\n\\int_0^{\\tilde \\ell} \\left(\\frac{df}{d\\tilde s}\\right)^2 - \\tilde \\kappa f^2\\, d \\tilde s \\ge 0 \\,,\n\\eeq\nfor all smooth functions $f$ defined on $[0,\\tilde \\ell]$ that vanish at the end points, where\n$\\tilde \\ell$ is the $\\tilde h$-length of $\\g$ and $\\tilde s$ is $\\tilde h$-arc length along $\\g$.\nBy making the change of variable $s  = s(\\tilde s)$, where $s$ is $h$-arc length along $\\g$,\nand using Equation \\eqref{gauss}, we arrive at,\n\\beq\\label{ineq2}\n\\int_0^{\\ell} \\psi^{-1}(f')^2 - (Q + \\psi^{-2} |\\D\\psi|^2)\\psi^{-1}  f^2  \\, d s \\ge 0 \\,,\n\\eeq\nfor all smooth functions $f$ defined on $[0,\\ell]$ that vanish at the endpoints, where\n$\\ell$ is the $h$-length of $\\g$, and $' = \\frac{d}{ds}$.\n\nSetting $k= \\psi^{-1\/2}f$ in \\eqref{ineq2},  we obtain after a small\nmanipulation,\n\\beq\\label{ineq3}\n\\int_0^{\\ell}  (k')^2  -  Q\\,k^2+ \\psi^{-1}\\psi'kk' -\\frac34\\psi^{-2}(\\psi')^2k^2 \\, ds \\ge 0  \\,,\n\\eeq\nwhere $\\psi'$ is shorthand for $(\\psi \\circ \\g)'$, etc.\nCompleting the square on the last two terms of the integrand,\n\\beq\n\\frac34\\psi^{-2}(\\psi')^2k^2 - \\psi^{-1}\\psi'kk' = \\left(\\frac{\\sqrt{3}}2 \\psi^{-1}\\psi'k -\\frac1{\\sqrt{3}} k'\\right)^2 - \\frac13(k')^2 ,  \\nonumber\n\\eeq\nwe see that \\eqref{ineq3} implies,\n\\beq\\label{ineq4}\n\\int_0^{\\ell}  \\frac43(k')^2  -  Q\\,k^2 \\, ds \\ge 0 \\,.\n\\eeq\nSince,  from \\eqref{Q}, we have that  $Q \\ge \\mu - |J| \\ge c$,\n\\eqref{ineq4}  implies,\n\\beq\\label{ineq5} \n\\frac43\\int_0^{\\ell}  (k')^2 \\,ds \\ge  c  \\int_0^{\\ell} k^2 \\, ds \\,.\n\\eeq\nSetting $k = \\sin \\frac{\\pi s}{\\ell}$  in \\eqref{ineq5} then gives,\n\\beq\n\\ell \\le \\frac{2\\pi}{\\sqrt {3}}\\cdot \\frac1{\\sqrt{c}} \\,.\n\\eeq \nSince $R(\\S) \\le \\ell$, the result follows. \\hfill $\\Box$ \\medskip\n\n\\section{On the area of black holes in asymptotically anti-de Sitter spacetimes}\n\nA basic step in the classical black hole uniqueness theorems is Hawking's theorem\non the topology of black holes \\cite{HE} \nwhich asserts that cross sections of the event horizon\nin $3+1$-dimensional asymptotically flat stationary black hole spacetimes obeying the\ndominant energy condition are topologically 2-spheres.  As shown by Hawking \\cite{Hawking},\nthis conclusion also holds for outermost MOTSs in spacetimes that\nare not necessarily stationary.    In \\cite{GS, G}  a natural\ngeneralization of these results to higher dimensional spacetimes  was \nobtained by showing that cross sections of the event horizon (in the\nstationary case) and outermost MOTSs (in the general case)\nare of positive Yamabe type, i.e., admit metrics of positive\nscalar curvature.  This implies many well-known restrictions on\nthe topology, and is consistent with recent examples of five\ndimensional stationary black hole spacetimes with horizon topology\n$S^2 \\times S^1$ \\cite{Emp}.  \n\nThese results on black hole topology depend crucially on the dominant\nenergy condition.  Indeed, there is a well-known class of  $3+1$-dimensional static locally anti-de Sitter black hole spacetimes which are solutions to the vacuum Einstein equations \nwith negative cosmological constant $\\Lambda$\nhaving horizon topology of arbitrary genus $g$ \\cite{Brill, Mann}.  Higher dimensional versions\nof these topological black holes have been considered in \\cite{Bir, Mann}.    However,\nas Gibbons  pointed out in \\cite{Gi}, although Hawking's theorem does not hold in the asymptotically locally anti-de Sitter setting,  his basic argument still\nleads to an interesting conclusion.  Gibbons showed that for $3$-dimensional time-symmetric initial data sets that\ngive rise to spacetimes satisfying the Einstein equations with $\\Lambda <0$,  \noutermost MOTSs  $\\S$ (which are  stable minimal surfaces in this case)  must satisfy the area bound,\n\\beq\\label{areabound}\n{\\rm Area}(\\S)\\ge \\frac{4\\pi(g-1)}{|\\Lambda|} \\, ,\n\\eeq\nwhere $g$ is the genus of $\\S$.   Woolgar \\cite{Wo}  obtained a similar bound in  the general, nontime-symmetric, case.\nHence, at least for stationary black holes, black hole entropy has a lower bound depending on a global topological invariant.  \n\n\nIn \\cite{CG} Cai and Galloway considered  an extension of Gibbon's result to higher dimensional spacetimes.  There  it was shown, for time-symmetric initial data, that a bound similar to that obtained by Gibbons still holds, but where the genus is replaced by the\nso-called $\\s$-constant (or Yamabe invariant).  The $\\s$-constant is a diffeomorphism invariant of smooth compact manifolds that in dimension two reduces to a multiple of the \nEuler characteristic; see \\cite{CG} and references therein for further details.  The\naim of this section is to observe that this result extends to the general, nontime-symmetric case.\n\nWe begin by recalling the definition of the $\\s$-constant.  \nLet $\\S^{n-1}$, $n\\ge 3$, be a smooth compact (without boundary) $(n-1)$-dimensional\nmanifold.  If $g$ is a Riemannian metric on $\\S^{n-1}$, let $[g]$ denote the conformal\nclass of $g$.  The Yamabe constant with respect to $[g]$, which we denote by $\\calY[g]$, is the number, \n\\beq\\label{yam}\n\\calY[g]  = \\inf_{\\tilde g\\in [g]} \n\\frac{\\int_{\\S}S_{\\tilde g}d\\mu_{\\tilde g}}\n{(\\int_{\\S}d\\mu_{\\tilde g})^{\\frac{n-3}{n-1}}}\\, ,\n\\eeq  \nwhere $S_{\\tilde g}$ and $d\\mu_{\\tilde g}$ are respectively the scalar curvature and volume measure of $\\S^{n-1}$ \nin the metric $\\tilde g$.  The  expression involving integrals is just the volume-normalized total\nscalar curvature of $(\\S,\\tilde g)$.\nThe solution to the Yamabe problem, due to Yamabe, Trudinger, Aubin and Schoen, \nguarantees that the infimum \nin (\\ref{yam}) is  achieved by a metric of constant scalar curvature.  \n\nThe $\\s$-constant  of $\\S$ is \n defined by taking the supremum of the Yamabe constants over all conformal\nclasses,\n\\beq\n\\s(\\S) = \\sup_{[g]} \\calY[g] \\, .\n\\eeq    \nAs observed by Aubin, the supremum is finite, and in fact bounded above in terms of the volume\nof the standard unit $(n-1)$-sphere $S^{n-1} \\subset \\Bbb R^n$.  The $\\s$-constant divides\nthe family of compact manifolds into three classes according to: (1) $\\s(\\S) > 0$, (2) $\\s(\\S) = 0$,\nand (3) $\\s(\\S) < 0$.  \n\nIn the case $\\dim \\S =2$, the Gauss-Bonnet theorem implies $\\s(\\S) = 4\\pi\\chi(\\S)=8\\pi(1-g)$.  \nNote that the inequality \\eqref{areabound} only gives information when $\\chi(\\S) < 0$.\nCorrespondingly, in higher dimensions, we shall only be interested in the case \nwhen $\\s(\\S) < 0$.  It follows from the resolution of the Yamabe problem that\n$\\s(\\S) \\le 0$ if and only if $\\S$ does not carry a metric of positive scalar curvature.\nIn this case, and with $\\dim \\S = 3$, Anderson \\cite{An} has shown, as an application of Perlman's work on\nthe geometrization conjecture, that $\\s(\\S)$  is \ndetermined by the volume of the ``hyperbolic part'' of $\\S$, which when present\nimplies $\\s(\\S) < 0$.  In particular, all closed\nhyperbolic $3$-manifolds have negative $\\s$-constant.\n\nWe now turn to the spacetime setting.  In what follows, all MOTSs are compact\nwithout boundary.  The following theorem extends Theorem 5 in \\cite{CG}\nto the nontime-symmetric case.  \n\n\\begin{thm}\\label{volbound}  \nLet $\\S^{n-1}$ be a stable MOTS in the initial data set $(V^n,g,K)$, $n \\ge 4$,\nsuch that $\\s(\\S) < 0$.  Suppose there exists $c > 0$, such that  $\\mu +\\<J,\\nu\\> \\ge -c$.\nThen the $(n-1)$-volume of $\\S$ satisfies,\n\\beq\n{\\rm vol}(\\S^{n-1}) \\ge \\left(\\frac{|\\s(\\S)|}{2c}\\right)^{\\frac{n-1}2} \\, .\n\\eeq   \n\\end{thm}\n\nWe make some comments about the assumptions.  Suppose $V$ is a spacelike hypersurface in a spacetime $(M,g_M)$, satisfying the Einstein equation with cosmological\nterm,\n\\beq\nG + \\Lambda g_M = \\calT\n\\eeq\nwhere, as in Section 2, $G = \\ric_M -\\frac12R_Mg_M$ is  the Einstein tensor, and $\\calT$\nis the energy-momentum tensor. \nThus,\nsetting $\\ell = u+\\nu$, we have along $\\S$ in $V$,\n\\begin{align}\n\\mu +\\<J,\\nu\\> &= G(u,\\ell) = \\calT(u,\\ell) + \\Lambda \\nonumber \\\\\n& \\ge - |\\Lambda| \\,,\n\\end{align}\nprovided $\\Lambda < 0$ and $\\calT(u,\\ell) \\ge 0$.  Hence, when $\\Lambda < 0$ and the fields giving rise to $\\calT$ obey the dominant energy condition, the energy condition in Theorem \\ref{volbound} is satisfied with $c = |\\Lambda|$.   \n\nWe briefly comment on the stability assumption. As defined in \\cite{G}, a MOTS $\\S$ is {\\it weakly outermost} in $V$ provided there are no strictly outer trapped surfaces  outside of, and  homologous to $\\S$ in $V$.  Weakly outermost MOTSs are necessarily stable, as noted in Section 2, \nand arise naturally in a number of physical\nsituations.  For example,  smooth compact cross sections of the\nevent horizon in \nstationary black hole spacetimes obeying the null energy condition, are \nnecessarily weakly outermost MOTSs.\nMoreover, results of Andersson and Metzger \\cite{AM2} provide natural criteria for the existence of weakly outermost MOTSs in general black hole spacetimes containing\ntrapped regions. \n\n\\noindent \\emph{Proof:\\ }[Proof of Theorem \\ref{volbound}] The proof is a simple  modification of the\nproof of Theorem 5 in \\cite{CG}.\nBy the stability assumption and Proposition\n\\ref{eigen}, we have $\\l_1(L_0) \\ge 0$, where $L_0$ is the operator  given in  \\eqref{symop}. \nThe Rayleigh formula,\n$$\n\\l_1(L_0) = \\inf_{\\phi \\ne 0} \\frac{ \\int_{\\S} \\phi L_0(\\phi)d\\mu}{\\int_{\\S} \\phi^2 d\\mu}\n$$ \ntogether with an integration by parts yields the {\\it stability inequality},\n\\beq\\label{stab}\n\\int_{\\S} (|\\D \\f|^2 + \\left( \\frac12 S - (\\mu + \\<J,\\nu\\>) - \\frac12 |\\chi|^2\\right)\\phi^2 \\, d\\mu \\ge 0 \\, ,\n\\eeq\nfor all $\\phi \\in C^{\\infty}(\\S)$.\n\nThe Yamabe constant $\\calY[h]$, where $h$ is the induced metric on $\\S$, can\nbe expressed as \\cite{Be},\n\\beq\\label{yam2}\n\\calY[h] = \\inf_{\\f\\in C^{\\8}(\\S), \\f>0} \\frac{\\int_{\\S} (\\frac{4(n-2)}{n-3}|\\D\\f|^2 +S\\f^2)\\, d\\mu}\n{(\\int_{\\S}\\f^{\\frac{2(n-1)}{n-3}}\\, d\\mu)^{\\frac{n-3}{n-1}}} \\, .\n\\eeq\n\nNoting that $\\frac{4(n-2)}{n-3}> 2$, the stability inequality implies,\n\\begin{align}\\label{ineqc}\n\\int_{\\S} \\frac{4(n-2)}{n-3}|\\D \\f|^2 + S\\f^2) \\, d\\mu & \\ge  \\int_{\\S}\n2( \\mu + \\<J,\\nu\\> )\\phi^2\\,d\\mu\n\\nonumber\\\\\n& \\ge  - 2c \\int_{\\S} \\f^2\\,d\\mu \\, .\n\\end{align}\n\nBy H\\\"older's inequality we have,\n\\beq\n\\int_{\\S} \\f^2\\,d\\mu \\le \\left(\\int_{\\S} \\f^{\\frac{2(n-1)}{n-3}}\\, d\\mu\\right)^{\\frac{n-3}{n-1}}\n\\left(\\int_{\\S} 1 \\, d\\mu\\right)^{\\frac2{n-1}}  \\, ,\n\\eeq\nwhich, when combined with  (\\ref{ineqc}), gives,\n\\beq\n\\frac{\\int_{\\S} (\\frac{4(n-2)}{n-3}|\\D\\f|^2 + \\hat S\\f^2)\\, d\\mu}\n{(\\int_{\\S}\\f^{\\frac{2(n-1)}{n-3}}\\, d\\mu)^{\\frac{n-3}{n-1}}} \\ge - 2c\\,(\\mbox{vol($\\S$)})^{\\frac2{n-1}} \\, .\n\\eeq\nMaking use of this  inequality in (\\ref{yam2}) gives,\n$\\calY[h] \\ge - 2c\\, (\\mbox{vol($\\S$)})^{\\frac2{n-1}}$, or, equivalently,\n\\beq\n{\\rm vol}(\\S^{n-1}) \\ge \\left(\\frac{|\\calY[h]|}{2c}\\right)^{\\frac{n-1}2} \\, .\n\\eeq  \nSince $|\\s(\\S)| \\le |\\calY[h]|$, the result follows.\\hfill $\\Box$ \\medskip\n\n\\section*{Acknowledgements}\n\n\\vspace{-.1in}\nThis work was supported in part by NSF grant\nDMS-0708048 (GJG) and SFI grant 07\/RFP\/PHYF148 (NOM). \n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\n\n\n\n \n\nStock Movement Prediction (SMP) \nis a hot topic in \\textit{Fintech} area since investors continuously\nattempt to predict the stock future trend of listed companies for seeking maximized profit in the volatile financial market \\cite{Li2017Web,Hu2018Listening,Cheng2021Modeling,Wang2021Coupling}. \nThe task has spurred the interest of researchers over the years to develop better predictive models \\cite{Wang2021Coupling}. \nIn particular, the application of machine learning approaches yields a promising performance for SMP task \\cite{Feng2019Temporal,Ye2020Multi-Graph}. \nPrevious studies in both finance and AI research fields predicting a stock movement\nrely on time-series analysis techniques \nusing its own historical prices \n(e.g. \\textit{opening price, closing price, volume}, etc)\n\\cite{Lin2017Hybrid,Feng2019Enhancing}. \nAccording to the Efficient Market Hypothesis (ETH) that implies financial market is informationally efficient \\cite{Malkiel1970Efficient}, therefore, besides these stock trading factors, other researchers \nmine more indicative features from its outside-market data such as web media \\cite{Li2017Web}, including news information \\cite{Ming2014Stock,Liu2018Hierarchical,Li2020A} and social media \\cite{Bollen2011Twitter,Si2013Exploiting,Nguyen2015Sentiment}, while ignoring the stock fluctuation diffusion influence from its related companies, which is also known as momentum spillover effect \\cite{Ali2020Shared} in finance. \n\n\nRecent studies attempt to model stock momentum spillover via Graph Neural Networks (GNN) \\cite{Velickovic2018Graph}. However, most of them only consider the simple explicit relations among related companies \\cite{Feng2019Temporal,Ye2020Multi-Graph,Sawhney2020Spatiotemporal,Li2020Modeling}, which inevitably fail to model the complex connections of listed companies in real financial market, such as the implicit relation, and the associated executives-based meta-relations \\cite{cai2016price,jing2021online}.\n\n \n\n\n\n\n\nTo address this issue, we construct a more comprehensive {M}arket {K}nowledge {G}raph (MKG), which consists of a considerable amount of triples in the form of ({\\emph{head entity}}, {\\emph{relation}}, {\\emph{tail entity}}),\nindicating that there exists a relation between the two entities. \nDifferent from previous graphs in other SOTA works \\cite{Chen2018Incorporating,Feng2019Temporal,Ye2020Multi-Graph,Li2020Modeling,Sawhney2021Stock,Cheng2021Modeling}, the newly constructed MKG develops two essential characteristics: \\textbf{(1)} \\textit{\\textbf{Bi-typed}}, i.e. containing the significant associated executive entities aside from the ordinary company entities; \\textbf{(2)} \\textit{\\textbf{Hybrid-relational}}, i.e. providing an additional implicit relation among listed companies aside from their typical explicit relations. \nFigure $\\ref{figure-instance}$ shows a toy example of MKG (See Section \\ref{section-mkgc} for more details).\n\n\n\n\n\n\nAfterward, to learn the stock\\footnote{The term ``listed company\" and ``stock\" are used interchangeably.} momentum spillover signals on such bi-typed hybrid-relational MKG for stock movement prediction, we pertinently propose a novel \\textbf{D}ual \\textbf{A}ttention \\textbf{N}etworks (\\textsc{DanSmp}), as shown in Figure \\ref{figure-DanSmp-Model}-II.\nSpecifically, the proposed model \\textsc{DanSmp} is equipped with dual attention modules that are able to learn the inter-class interaction among listed companies and associated executives, and their own complex intra-class interaction alternately. \nDifferent from previous methods that can only model homogeneous stock graph \\cite{Chen2018Incorporating,Cheng2021Modeling} or heterogeneity of stock explicit relations \\cite{Nelson2017Stock,Chen2019Investment,Sawhney2021Stock}, our method is able to learn bi-typed heterogeneous entities and hybrid-relations in newly constructed market graph of stock for its spillover effects. \nThe comprehensive comparison between the existing state-of-the-art (SOTA) methods with our newly proposed \\textsc{DanSmp} model in terms of used market signals and main ideas is shown in Table \\ref{table-model-comparison-characteristics}, demonstrating the distinguished advantage of our work.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{Figure1.pdf}\n    \\caption{Example of a bi-typed (i.e. \\textit{listed companies}, and \\textit{executives}) hybrid-relational (i.e. \\textit{explicit relations}, and \\textit{implicit relation}) market knowledge graph (MKG).\n   \n    The relational information in MKG is essential for stock prediction but has not been well utilized in previous works.\n    }\n    \\label{figure-instance}\n\\end{figure}\n\n\n\nWe collect public data and construct two new SMP datasets (named \\textbf{CSI100E}\\footnote{\"E\" denotes an extension version.} and \\textbf{CSI300E}) based on Chinese Stock Index to evaluate the proposed method, since no existing benchmark datasets can satisfy our need. \nAside from the typical stock historical prices and media news, our newly published benchmark datasets CSI100E and CSI300E also provide rich market knowledge graph as mentioned above.\nThe empirical experimental results on CSI100E and CSI300E against nine SOTA methods demonstrate the better performance of our model \\textsc{DanSmp} with MKG. \nThe ablation studies reaffirm that the performance gain mainly comes from the use of the associated executives, and additional implicit relation among companies in MKG via the proposed \\textsc{DanSmp}.\n\nThe {contributions} of this paper are threefold:\n\n\\begin{itemize}\n    \\item To model stock momentum spillover via the complex relations among companies in real market, we first construct a novel market knowledge graph. To the best of our knowledge, this study is the first attempt to explore such bi-typed hybrid-relational knowledge graph of stock via heterogeneous GNNs for its spillover effects.\n   \n    \n    \\item We then propose \\textsc{DanSmp}, a novel Dual Attention Networks to learn the stock momentum spillover features based on the newly constructed bi-typed hybrid-relational MKG for stock prediction, which is also a non-trivial and challenging task.\n   \n   \n   \n    \n    \\item We propose two new benchmark datasets (CSI100E and CSI300E) to evaluate our method, which are also expected to promote Fintech research field further. The empirical experiments on our constructed datasets demonstrate our method can successfully improve stock prediction with bi-typed hybrid-relational MKG via the proposed \\textsc{DanSmp}\\footnote{The source code and our newly constructed benchmark datasets (CSI100E and CSI300E) will be released on \\url{Github}: {https:\/\/github.com\/trytodoit227\/\\textsc{DANSMP}}}.\n       \n\\end{itemize}\n\n\n\n\nThe rest of the paper is organized as follows. In Section \\ref{section-related-work}, we summarize and compare the related work. In Section \\ref{section-market-signal}, we introduce the market signals for stocks prediction. Section \\ref{section-method} introduces the details of the proposed methodology. \nExtensive experiments are conducted to evaluate the effectiveness of the proposed model in Section \\ref{section-experiments}. Finally, we conclude the paper in Section \\ref{section-conclusion}.\n\n\n\n\n\n        \n\n            \n            \n            \n\n\n\n\n\n\\section{Related Work}\n\\label{section-related-work}\n\nIn this section, we evaluate the existing relevant research on stock prediction. \nStock movement prediction (SMP) has received a great deal of attention from both investors and researchers since it helps investors to make good investment decisions \\cite{Rather2015Recurrent,Ding2016Knowledge-driven,Li2016A,Deng2019Knowledge-Driven}. In general, traditional SMP methods mainly can be categorized into two classes: {technical analysis} and {fundamental analysis}, according to the different types of the available stock own information they mainly used. Another major aspect for yielding better stock prediction is to utilize the stock connection information \\cite{Chen2018Incorporating,Ye2020Multi-Graph,Sawhney2020Spatiotemporal,Cheng2021Modeling}. We review them in the following.\n\n\n\\subsection{Technical Analysis}\nTechnical analysis takes time-series historical market data of a stock, such as trading price and volume, as features to make prediction \\cite{Edwards2018Technical,Chen2019Investment}. The basic idea behind this type of approach is to discover the hidden trading patterns that can be used for SMP. Most recent methods of this type predict stock movement trend using deep learning models \\cite{Nelson2017Stock,Bao2017A,Lin2017Hybrid}. \nTo further capture the long-term dependency in time series, the Recurrent Neural Networks (RNN) especially Long Short-Term Memory networks (LSTM) have been usually leveraged for prediction \\cite{Gao2016Stock}. \n\\citet{Bao2017A} presented a deep learning framework for stock forecasting using stacked auto-encoders and LSTM. \n\\citet{Nelson2017Stock} studied the usage of LSTM networks to predict future trends of stock prices based on the price history, alongside with technical analysis indicators. \n\\citet{Lin2017Hybrid} proposed an end-to-end hybrid neural networks that leverage convolutional neural networks (CNNs) and LSTM to learn local and global contextual features respectively for predicting the trend of time series. \n\\citet{Zhang2017Stock} proposed a state frequency memory recurrent network to capture the multi-frequency trading patterns for stock price prediction. \n\\citet{Feng2019Enhancing} proposed to employ adversarial training and add perturbations to simulate the stochasticity of price variable, and train the model to work well under\nsmall yet intentional perturbations. \nDespite their achieved progress, however, technical analysis faces an issue that it is incapable of unveiling the rules that govern the fluctuation of the market beyond stock price data.\n\n\\subsection{Fundamental Analysis}\nOn the contrary, fundamental analysis takes advantage of information from outside market price data, such as economic and financial environment, and other qualitative and quantitative factors \\cite{Hu2018Listening,Zhang2018Improving,Xu2018Stock}. Many methods are proposed to explore the relation between stock market and web media, e.g., news information, and social media opinion \\cite{Li2017Web,Akita2016Deep,Zhang2018Improving,Wu2018Hybrid}. For instance, \n\\citet{Ming2014Stock} mined text information from Wall Street Journal for SMP. \n\\citet{Akita2016Deep} presented a deep learning method for stock prediction using numerical and textual information. \n\\citet{Vargas2017Deep} proposed a deep learning method for stock market prediction from financial news articles. \n\\citet{Xu2018Stock} put forward a novel deep generative model jointly exploiting text and price signals for this task. \n\\citet{Liu2018Hierarchical} presented a hierarchical complementary attention network for predicting stock price movements with news. \n\\citet{Li2020A} proposed a multimodal event-driven LSTM model for stock prediction using online news. \nSome researchers mined market media news via analyzing its sentiment, and used it for SMP \\cite{Sedinkina2019Automatic,Qin2019What}. For instance, \\citet{Bollen2011Twitter} analyzed twitter mood to predict the stock market. \n\\citet{Si2013Exploiting} exploited topics based on twitter sentiment for stock prediction. \n\\citet{Nguyen2015Sentiment} incorporated the sentiments of the specific topics of the company into the stock prediction model using social media. \n\\citet{Rekabsaz2017Volatility} investigated the sentiment of annual disclosures of companies in stock markets to forecast volatility. \n\\citet{Qin2019What} proposed a multimodal method that takes CEO's vocal features, such as emotions and voice tones, into consideration. \nIn this paper, we extract the signals from stock historical prices and media news sentiments as the sequential embeddings of stocks.\n\n\n\n\n\n\n\n\n\\subsection{Stock Relations Modeling}\n\\label{Section-srm}\nRecent SMP studies take stock relations into consideration \\cite{Chen2018Incorporating,Sawhney2020Spatiotemporal,Li2020Modeling,Cheng2021Modeling}.\nFor instance, \n\\citet{Chen2018Incorporating} proposed to incorporate corporation relationship via graph convolutional neural networks for stock price prediction. \n\\citet{Feng2019Temporal} captured the stock relations in a time-sensitive manner for stock\nprediction. \n\\citet{Kim2019HATS} proposed a hierarchical attention network for stock prediction using relational data. \n\\citet{Li2019Multi-task} presented a multi-task recurrent neural network (RNN) with high-order Markov random fields (MRFs) to predict stock price movement direction using stock's historical records together with its correlated stocks. \n\\citet{Li2020Modeling} proposed a LSTM Relational Graph Convolutional Network (LSTM-RGCN) model, which models the connection among stocks with their correlation matrix. \n\\citet{Ye2020Multi-Graph} encoded multiple relationships among stocks into graphs based on financial domain knowledge and utilized GCN to extract the cross effect based on these pre-defined graphs for stock prediction. \n\\citet{Sawhney2020Spatiotemporal} proposed a spatio-temporal hypergraph convolution network for stock movement forecasting. \n\\citet{Cheng2021Modeling} proposed to model the momentum spillover effect for stock prediction via attribute-driven graph attention networks. \nDespite the substantial efforts of these SOTA methods, surprisingly, most of them only focus on modeling the momentum spillover via the explicit relations among stocks, while ignoring their complex relations in real market.\n\n\nTable \\ref{table-model-comparison-characteristics} summarizes the key advantages of our model, comparing with a variety of previous state-of-the-art (SOTA) studies in terms of the used market signals, their methods and GNN types. (1) Different from previous studies, our method takes advantage of all three types of stock market signals, including stock historical data, media news, and market knowledge graph. In particular, we construct a more comprehensive heterogeneous market graph that contains explicit relations, implicit relations and executive relations. (2) Different from most existing models that can only {model homogeneous stock graph} \\cite{Chen2018Incorporating,Cheng2021Modeling}, or heterogeneity of stock explicit relations \\cite{Feng2019Temporal,Sawhney2020Deep,Ye2020Multi-Graph,Sawhney2021Stock}, which fall down in modeling heterogeneity of entities in real market, we propose a novel dual attention networks that is able to model bi-typed heterogeneous entities and hybrid-relations in newly constructed market graph of stock for its spillover effects. (3) To the best of our knowledge, this work is the first attempt to study stock movement prediction via heterogeneous GNNs. \n\n\n\\begin{sidewaystable}[thp]\n~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\ ~\\\\~\\\\ \n    \\begin{center}\n         \\caption{Comparison between several SOTA methods and the proposed model in terms of used market signals and main ideas.}\n         \\label{table-model-comparison-characteristics}\n         \\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n        \\resizebox{\\textwidth}{!}{\n          \\begin{tabular}{l|l|c|c|c|c|c|c|c|c|c}\n            \\toprule\n            \\multirow{3}*{\\bf{Literature} } & \\multirow{3}*{\\bf{Main ideas} }& \\multirow{3}*{\\bf{Market} } & \\multirow{3}*{\\bf{Metrics} } & \\multirow{3}*{\\bf{Method} }&\n            \\multirow{3}*{\\bf{GNN Types}}&\n            \\multicolumn{5}{c}{\\textbf{Stock Market Signals}}    \\\\\n            \\cline{7-11}\n             &  & & & & &\\multirow{2}*{\\tabincell{c}{\\textbf{Historical}\\\\ \\textbf{Data}} }&\\multirow{2}*{\\tabincell{c}{\\textbf{Media}\\\\ \\textbf{News}} } & \\multicolumn{3}{c}{\\textbf{Market Knowledge Graph}} \\\\\n             \\cline{9-11}\n             &  & & & & & & &\\tabincell{c}{\\textbf{Explicit}\\\\ \\textbf{Relation}} & \\tabincell{c}{\\textbf{Implicit}\\\\ \\textbf{Relation}} &\\tabincell{c}{\\textbf{Executives}\\\\ \\textbf{Relation}}\\\\\n            \\midrule\n            \\midrule\n            \\tabincell{l}{\\textbf{EB-CNN}\\\\ \\cite{ding2015deep}}&\\tabincell{l}{$\\bullet$ Neural tensor network for \\\\ \\quad learning event embedding \\\\ $\\bullet$ Deep CNN to model the \\\\ \\quad combined influence} &\\textbf{S\\&P500} & \\tabincell{c}{\\textbf{DA},\\\\ \\textbf{MCC},\\\\ \\textbf{Profit}}&\\tabincell{c}{\\textbf{Open IE},\\\\ \\textbf{CNN}} & \\textbf{-}&\\XSolidBrush & \\Checkmark & \\XSolidBrush & \\XSolidBrush &\\XSolidBrush \\\\     \n            \\midrule\n       \n           \n            \\tabincell{l}{\\textbf{SFM}\\\\ \\cite{Zhang2017Stock}} & \\tabincell{l}{$\\bullet$ Extending LSTM by decomposing \\\\ \\quad the hidden memory states \\\\ $\\bullet$ Modeling the latent trading \\\\ \\quad patterns with multiple frequencies } &\\textbf{NASDAQ} & \\textbf{Average square error}& \\tabincell{c}{\\textbf{LSTM},\\\\ \\textbf{DFT}} &\\textbf{-}&\\Checkmark & \\XSolidBrush & \\XSolidBrush & \\XSolidBrush &\\XSolidBrush \\\\\n            \\midrule\n            \\tabincell{l}{\\textbf{Chen's}\\\\ \\cite{Chen2018Incorporating}} & \\tabincell{l} {$\\bullet$ Single investment relation} &\\textbf{CSI300} & \\textbf{DA}&\\tabincell{c}{\\textbf{LSTM},\\\\ \\textbf{GCN}} &\\textbf{Homogeneous GNNs}& \\Checkmark & \\XSolidBrush & \\Checkmark & \\XSolidBrush &\\XSolidBrush \\\\\n            \\midrule\n            \\tabincell{l}{\\textbf{TGC}\\\\ \\cite{Feng2019Temporal}} &\\tabincell{l}{$\\bullet$ Single historical data \\\\ $\\bullet$ Dynamically adjust predefined\\\\ \\quad firm relations } &\\tabincell{c}{\\textbf{NASDAQ},\\\\ \\textbf{NYSE}} & \\tabincell{c}{\\textbf{MSE},\\\\ \\textbf{MRR},\\\\ \\textbf{IRR}}& \\tabincell{c}{\\textbf{LSTM},\\\\ \\textbf{Temporal Graph}\\\\ \\textbf{Convolution}}&\\textbf{Homogeneous GNNs}& \\Checkmark & \\XSolidBrush & \\Checkmark &\\XSolidBrush & \\XSolidBrush \\\\\n            \\midrule\n            \\tabincell{l}{\\textbf{HATS}\\\\ \\cite{Kim2019HATS}}& \\tabincell{l}{$\\bullet$ Hierarchical aggregate different \\\\ \\quad types of firm relational data } &\\textbf{S\\&P500} & \\tabincell{c}{\\textbf{SR},\\\\ \\textbf{F1},\\\\ \\textbf{DA},\\\\ \\textbf{Return}}& \\textbf{GAT}&\\textbf{Homogeneous GNNs}& \\Checkmark & \\XSolidBrush & \\Checkmark & \\XSolidBrush &\\XSolidBrush\\\\\n           \n           \n           \n           \n            \\midrule\n\n             \\tabincell{l}{\\textbf{ALBERT+eventHAN}\\\\ \\cite{wu-2020-event}}&\\tabincell{l}{$\\bullet$ ALBERT enhanced event\\\\ \\quad  representations \\\\ $\\bullet$ Event-enhanced hierarchical\\\\ \\quad  attention network }&\\tabincell{c}{\\textbf{S\\&P500},\\\\ \\textbf{DOW},\\\\ \\textbf{NASDAQ}} & \\tabincell{c}{\\textbf{DA},\\\\ \\textbf{Annualized return}}&\\tabincell{c}{\\textbf{Open IE},\\\\ \\textbf{ALBERT},\\\\ \\textbf{HAN}} & \\textbf{\n            -}& \\XSolidBrush & \\Checkmark & \\XSolidBrush & \\XSolidBrush &\\XSolidBrush \\\\     \n            \\midrule\n           \n           \n            \\tabincell{l}{\\textbf{MAN-SF}\\\\ \\cite{Sawhney2020Deep}}  & \\tabincell{l}{$\\bullet$ Multi-modal market information \\\\ $\\bullet$ Hierarchical graph attention method } & \\textbf{S\\&P500} & \\tabincell{c}{\\textbf{F1},\\\\ \\textbf{MCC}}& \\tabincell{c}{\\textbf{GAT},\\\\ \\textbf{GRU}}& \\textbf{Homogeneous GNNs}& \\Checkmark & \\Checkmark & \\Checkmark & \\XSolidBrush &\\XSolidBrush\\\\\n            \\midrule\n            \\tabincell{l}{\\textbf{STHAN-SR}\\\\ \\cite{Sawhney2021Stock}} & \\tabincell{l}{$\\bullet$ A neural hypergraph architecture \\\\ \\quad for stock selection \\\\ $\\bullet$ Temporal hawkes attention \\\\ \\quad mechanism } & \\tabincell{c}{\\textbf{NASDAQ},\\\\ \\textbf{NYSE},\\\\ \\textbf{TSE}} & \\tabincell{c}{\\textbf{SR},\\\\ \\textbf{IRR},\\\\ \\textbf{NDCG\\@5}} & \\tabincell{c}{\\textbf{Hypergraph} \\\\ \\textbf{Convolution}} & \\textbf{Homogeneous GNNs}& \\Checkmark & \\Checkmark & \\Checkmark & \\XSolidBrush &\\XSolidBrush\\\\\n            \n            \\midrule\n            \\tabincell{l}{\\textbf{AD-GAT}\\\\ \\cite{Cheng2021Modeling}} & \\tabincell{l}{$\\bullet$ Multi-modal market information \\\\ $\\bullet$ Attribute-driven graph \\\\ \\quad attention network }&\\textbf{S\\&P500} & \\tabincell{c}{\\textbf{DA},\\\\ \\textbf{AUC}}&  \\textbf{GAT}&\\textbf{Homogeneous GNNs}&  \\Checkmark & \\Checkmark & \\XSolidBrush & \\Checkmark &\\XSolidBrush\\\\\n            \\midrule\n            \n            \\textbf{\\textsc{DanSmp} (ours)} &\\tabincell{l}{$\\bullet$ Multi-modal market information \\\\ $\\bullet$ Bi-typed hybrid-relation data \\\\ $\\bullet$ Dual attention network} &\\tabincell{c}{\\textbf{CSI100E},\\\\ \\textbf{CSI300E}} & \\tabincell{c}{\\textbf{DA},\\\\ \\textbf{AUC},\\\\ \\textbf{SR},\\\\ \\textbf{IRR}}& \\tabincell{c}{\\textbf{GRU},\\\\ \\textbf{Dual Attention Network}} &\\textbf{Heterogeneous GNNs}& \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark  &\\Checkmark \\\\\n            \n            \\bottomrule\n          \\end{tabular}\n         }\n    \\end{center}\n\\end{sidewaystable}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n   \n   \n   \n   \n    \n    \n   \n   \n    \n   \n    \n   \n   \n   \n    \n   \n   \n   \n   \n    \n   \n   \n   \n   \n   \n    \n    \n   \n   \n   \n   \n   \n   \n   \n    \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \n\n   \n   \n   \n   \n   \n    \n   \n   \n\n\n\n\n\n\n\n\n\\section{Market Signals}\n\\label{section-market-signal}\nIn this section, we introduce the significant signals of stocks in real financial market. We first give the details of the newly proposed bi-typed hybrid-relational market knowledge graph. Afterward, we introduce the  historical price data and media news of stocks. Most of previous works focus on partial financial market information, which makes their modeling insufficient. In this work, we take advantage of all three types of market data, fusing numerical, textual, and relational data together, for stock prediction. The features of each data used in this study are summarized in Table \\ref{Features-statistics}.\n\n\\begin{table*}[htb]\n    \\begin{center}\n         \\caption{Features}\n         \\label{Features-statistics}\n         \\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n        \\resizebox{0.95\\textwidth}{!}{\n           \\begin{tabular}{l|l}\n            \\toprule\n            \\multicolumn{1}{c|}{\\textbf{Information data}} &\n            \\multicolumn{1}{c}{\\textbf{Features}}    \\\\\n            \\midrule\n            \\textbf{Entities}& companies, executives.\\\\\n            \\midrule\n            \\textbf{Relations}& explicit relations (industry category, supply chain, business partnership, investment), implicit relation.\\\\\n            \\midrule\n            \\textbf{Historical price}& opening price (op), closing price (cp), highest price (hp), lowest price (lp), trade volume (tv).\\\\\n            \\midrule\n            \\textbf{Media news}& positive media sentiment of a stock $Q(i)^{+}$, negative media sentiment of a stock  $Q(i)^{-}$, media sentiment divergence of a stock $D(i)$. \\\\\n            \\bottomrule\n           \\end{tabular}\n         }\n    \\end{center}\n\\end{table*}\n\n\n\\subsection{Bi-typed Hybrid-relational Market Knowledge Graph (MKG)}\n\\label{section-mkgc}\n\\subsubsection{Bi-typed Entities}\nMost existing SMP methods solely learn from company relationships in market \\cite{Kim2019HATS,Ye2020Multi-Graph,Li2020Modeling}. \n{In fact, in most stock markets there are also significant associated executives for listed companies, with rich associative information about these companies \\cite{cai2016price,jing2021online}. Hence, our constructed MKG contains not only company entities but also executive entities.}\nThe executive entities can act as the intermediary among companies to build the meta-relations involved in company entities (e.g. \\textit{Company-Executive-Company (CEC)}, \\textit{Company-Executive-Executive-Company (CEEC)}). \nFor example, in Fig. \\ref{figure-instance} we show the associated executives of the companies sampled from Chinese Stock Index. \nThe stock spillover signals can pass from neighboring companies to a target company through the meta-relations established by their connected executives, such as Company 4-Executive C-Company 1; \nCompany 2-Executive A$\\stackrel{classmate}{\\longleftrightarrow}$Executive B-Company 1.\nIn sum, the newly constructed MKG contains bi-typed entities, i.e. \\textbf{\\textit{listed companies}} and their \\textbf{\\textit{associated executives}}.\n\n\n\n\n\\subsubsection{Hybrid-relations}\nMost existing methods only take the explicit relations among companies, such as \\textit{industry category, supply chain, business partnership} and \\textit{investment}, into consideration \\cite{Chen2018Incorporating}. However, the limited explicit company relationships are always insufficient for market knowledge graph construction due to the complex nature of financial market. To solve the MKG incompleteness issue, here we propose an attribute-driven method to conduct MKG completion by inferring missing implicit correlative relation among stocks, which employs stocks attributions. \n\n\nSpecifically, the attribute-driven implicit unobserved relation is calculated based on the features from both its historical prices and news information filtered by a normalized threshold. \nA single-layer feed-forward neural network is adopted to calculate the attention value $\\alpha_{ij}^{t}$ between company $i$ and $j$ for inferring their implicit relation.\n\n\\begin{equation}\n\\alpha_{ij}^{t}=\\mathop{\\text{LeakyRelu}} \\Big ({\\bf u} ^{\\top} [ {\\bf s}_i ^t \\parallel {\\bf s}_j^t  ]\\Big)  \\ ,\n\\end{equation}\nwhere \n${\\bf s}_{i}^{t}$ and ${\\bf s}_{j}^{t}$ are fused market signals of $i$ and $j$, which are calculated by the Equation \\ref{equaion-seq-learn} (Section \\ref{section-se}). $\\|$ denotes the concatenate operation. ${\\bf u}$ denotes the learnable matrix and LeakyReLU is a nonlinearity activation function. Borrowing gate mechanism in \\cite{Cheng2021Modeling}, we set an implicit relation between $i$ and $j$ if $\\alpha_{ij}^{t} > \\eta$. $\\eta$ denotes a pre-defined threshold. \nIn short, the constructed MKG contains hybrid-relations, i.e. \\textbf{\\textit{explicit relations}} and \\textbf{\\textit{implicit relation}}.\n \n\n\n\\subsection{Historical Price and Media News}\n\\label{section-msr}\n\\subsubsection{Technical Indicators}\nTransactional data is the main manifestation of firms' intrinsic value and investors' expectations. We collect the daily stock price and volume data, including \\textit{opening price (op), closing price (cp), highest price (hp), lowest price (lp), and trade volume (tv)}. \nIn order to better compare and observe the fluctuation of stock price, the stock price is transferred to the return ratio, and the trade volume is transferred to the turnover ratio before being fed into our model. The return ratio is an index reflecting the level of stock return, the higher the return ration is; the better the profitability of the stock is. The turnover is the total value of stocks traded over a period of time; the higher the share turnover could indicate that the share haves good liquidity. ${\\bf p}_i  \\in {\\mathbb R}^5$ indicates the {technical indicators} of company $i$, as follows:\n\\begin{equation}\n\\label{equation-technical-indicator}\n    {\\bf p}_i = [op(i),\\ cp(i),\\  hp(i),\\  lp(i),\\  tv(i)]^{\\top} \\ .\n\\end{equation} \n\n\n\n\\subsubsection{Sentiment Signals}\nModern behavioral finance theory \\cite{Li2017Web} believes that investors are irrational, tending to be influenced by the opinions expressed in the media. Media sentiment reflects investors' expectations concerning the future of a company or the whole stock market, resulting in the fluctuations of stock price. To capture media sentiment signals, we extract the following characteristics: \\textit{positive media sentiment, negative media sentiment} and \\textit{media sentiment divergence} \\cite{Li2020A}. They are denoted respectively as follows: \n   \n   \n\n\\begin{equation}\n\\begin{aligned}\n    Q(i)^{+} &=\\frac{N(i)^{+}}{N(i)^{+}+N(i)^{-}} \\ , \\\\\n    Q(i)^{-} &=\\frac{N(i)^{-}}{N(i)^{+}+N(i)^{-}}\\ , \\\\\n    D(i)& =\\frac{N(i)^{+}-N(i)^{-}}{N(i)^{+}+N(i)^{-}} \\ ,\n\\end{aligned}\n\\end{equation}\nwhere $N(i)^{+}$ and $N(i)^{-}$ are the sum of the frequency of each positive and negative sentiment word found in the financial news articles of company $i$, respectively. $D(i)$ denotes the sentiment divergence. Since many negative sentiment words in the general sentiment dictionary no longer express negative emotional meanings\nin the financial field, we resort to a finance-oriented sentiment dictionary created in previous study \\cite{Li2016A}. ${\\bf q}_i  \\in {\\mathbb R}^3 $ indicates the news sentiment signals of company $i$.\n\\begin{equation}\n\\label{equation-sentiment-feature}\n    {\\bf q}_i =[Q(i)^{+},\\ Q(i)^{-},\\ D(i)]^{\\top}\\ .\n\\end{equation}\n\nNote that we do not have everyday news for all companies since the randomness of the occurrence of media news. In order to make the technical indicators aligned with the media sentiment signals and keep pace with the real situation, the sentiment feature ${\\bf q}_i$ of the firm $i$ is assigned to zero on the day when there are no any media news about it.\n\n\n\n\n\\section{Methodology}\n\\label{section-method}\nIn this section, we introduce the details of our proposed method. \nFigure \\ref{figure-DanSmp-Model} gives an overview of the proposed framework. \n(I) First, the stock sequential embeddings are learned with historical price and media news via multi-modal feature fusion and sequential learning. \n(II) Second, a Dual Attention Networks is proposed to learn the stock relational embeddings based upon the constructed MKG. \n(III) Last, the combinations of sequential embeddings and relational embeddings are utilized to make stock prediction. \n\n   \n\n\n\n\n\\subsection{Learning Stock Sequential Embeddings}\n\\label{section-se}\nThe stocks are influenced by multi-modal time-series market signals. \nConsidering the strong temporal dynamics of stock markets, the historical state of the stock is useful for predicting its future trend. \nDue to the fact that the influence of market signals on the stock price would last for some time, we should consider the market signals in the past couple of days when predicting stock trend $\\hat{y}_{i}^{t}$ of company $i$ at date $t$. \nWe first capture the multimodal interactions of technical indicators and sentiment signals. \nWe then feed the fused features into a one-layer GRU and take the last hidden state as the sequential embedding of stock $i$ which preserves the time dependency, as shown in Figure \\ref{figure-DanSmp-Model}-I.\n\n\\subsubsection{Multimodal Features Fusion} \n\nTo learn the fusion of the technical indicators vector ${\\bf p}_i$ and media news sentiment features ${\\bf q}_i$, we adopt a Neural Tensor Network (NTN) which replaces a standard linear neural network layer with a $M$-dimensional bilinear tensor layer that can directly relate the two features across multiple dimensions. \nThe fused daily market signals\\footnote{Here, the superscript $t$ is omitted for simplicity.} of stock $i$, ${\\bf x}_{i} \\in {\\mathbb R}^M$, are calculated by the tensor-based formulation as follows:\n\\begin{equation}\n{\\bf x}_{i}=\\sigma \\left ( {\\bf p}_{i}W_{{\\mathcal{T}}}^{\\left [ 1: M\\right ]}{\\bf q}_{i}+{\\mathcal{V}}\\begin{bmatrix}\n{\\bf p}_{i}\\\\ {\\bf q}_{i}\n\\end{bmatrix}+{\\bf b}\\right ) \\ ,\n\\end{equation}\nwhere $\\sigma$ is an activation function, $W_{{\\mathcal{T}}}^{\\left [ 1: M \\right ]}\\in {\\mathbb R}^{5\\times 3 \\times M}$ is a trainable tensor, ${\\mathcal{V}} \\in {\\mathbb R} ^{ 8\\times M}$ is the learned parameters matrix and ${\\bf b}\\in {\\mathbb R}^M$ is the bias vector. Three parameters are shared by all stocks. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{Figure2_model_framework.pdf}\n    \\caption{The overall framework of the proposed method. \\textbf{(I) Learning Stock Sequential Embeddings} based on Tensor Fusion and GRU. Tensor Fusion is the Neural Tensor Network (NTN) to learn the fusion of the technical indicators vector ${\\bf p}_i$ and media news sentiment features ${\\bf q}_i$. The GRU is designed to learn the sequential embedding ${\\bf s}_{i}^{t}$. \\textbf{(II) Learning Stock Relational Embeddings} by a dual mechanism to model the mutual affects and inner interaction among the bi-typed entities (i.e. companies and executives) alternately, including: \\textbf{(a)} inter-class attention, and \\textbf{(b)} intra-class attention. The former aims to deal with the interaction between listed companies and their associated executives and the latter aims to learn the interaction among the same type of entities. \\textbf{(III) Stock Movement Prediction} via Feed-forward Neural Network (FNN) with the learned firm embeddings.\n   \n    }\n    \\label{figure-DanSmp-Model}\n\\end{figure} \n\n\\subsubsection{Sequential Learning} \nWe feed the fused daily market signals in the past $T$ days into the GRU to learn {its} sequential embedding ${\\bf s}_{i}^{t}$, as follows:\n\\begin{equation}\n\\label{equaion-seq-learn}\n{\\bf s}_{i}^{t}=\\textbf{GRU}\\left ({\\bf x}_{i}^{t-T}, {\\bf x}_{i}^{t-T+1}, \\dots ,{\\bf x}_{i}^{t-1} \\right ) \\ ,\n\\end{equation}\nwhere ${\\bf s}_{i}^{t}\\in {\\mathbb R}^{F}$ denotes the last hidden state of GRU. ${F}$ is the hidden size of GRU.\n\n\\subsection{Learning Stock Relational Embeddings via Dual Attention Networks}\nIn real market, the stock fluctuation is partially affected by its related stocks which is known as momentum spillover effect in finance \\cite{Ali2020Shared}. \nIn this section, based upon our newly constructed bi-typed hybrid-relational MKG, we propose a Dual Attention Networks to learn the relational embeddings of stocks that represent their received spillover signals. Specifically, we employ a \\textit{\\textbf{dual mechanism}} to model the mutual affection and inner influence among the \\textit{bi-typed} entities (i.e. companies and executives) alternately, including {inter-class interaction} and {intra-class interaction}, as shown in Figure \\ref{figure-DanSmp-Model}-II.\n\n\n\\subsubsection{Inter-class Attention Networks}\nThe inter-class attention aims to deal with the interaction between listed companies and their associated executives, as shown in Figure \\ref{figure-DanSmp-Model} (II-a). Since they are different types of entities, their features usually lie in different space. Hence, we first project their embeddings into a common spaces. Specifically, for a company entity $u \\in {\\mathcal{E}}_1$ with type $\\tau(u)$ and an executive entity $v \\in {\\mathcal{E}}_2$ with type $\\tau(v)$, we design two type-specific matrices\n${\\bf W}^{\\tau(\\cdot)}$ \nto map their features ${\\bf h}_{u},{\\bf h}_v$ into a common space.\n\\begin{equation}\n\\begin{array}{l}\n\\begin{aligned}\n    \\label{equation-att-node}\n     {\\bf h}_u'&={\\bf W}^{\\tau(u)}{\\bf h}_u \\ , \\quad \\\\\n     {\\bf h}_v'&={\\bf W}^{\\tau(v)}{\\bf h}_v \\ ,\n\\end{aligned}\n\\end{array}\n\\end{equation}\nwhere ${\\bf h}_u'\\in {\\mathbb R}^{F'}$ and ${\\bf h}_u\\in {\\mathbb R}^{F}$ denote the original and transformed features of the entity $u$, respectively. \n${\\mathcal{E}}_1$ and ${\\mathcal{E}}_2$ are the sets of listed companies and the executives, respectively.\nHere, the original vectors of company entities (${\\bf h}_u,{\\bf h}_v$) are initialized by learned sequential embeddings (${\\bf s}_u,{\\bf s}_v$) learned in Section \\ref{section-se}, which can bring rich semantic information in downstream learning. The initial features of executives are then simply an average of the features of the companies which they work for.\n\nWe assume that the target company entity $u$ connects with other executives via a relation ${\\theta_i \\in \\Theta_{\\text{inter} }}$ which denotes the set of inter-class relations, so the neighboring executives of a company $u$ with relation ${\\theta_i}$ can be defined as ${\\mathcal{N}}_\\text{inter}^{\\theta_i}(u)$. For entity $u$, different types of inter-class relations contribute different semantics to its embeddings, and so do different entities with the same relation. Hence, we employ attention mechanism here in entity-level and relation-level to hierarchically aggregate signals from other types of neighbors to target entity $u$.\n\nWe first design an entity-level attention to learn the importance of entities within a same relation. \nThen, to learn the importance $e_{u\\upsilon}^{\\theta_i}$ which means how important an executive $v$ for a company $u$ under a specific relation $\\theta_i$, we perform {self-attention} \\cite{Vaswani2017Attention} on the entities as follows:\n\\begin{equation}\n\\begin{array}{l}\n\\begin{aligned}\n    \\label{equation-att-node}\n     e_{uv}^{\\theta_i}  &=  att_{node}({\\bf h}_u',{\\bf h}_v';\\theta_i) \\\\\n     &= \\text{LeakyRelu}({\\bf a}_{\\theta_i}^\\top \\cdot [ {\\bf h}_u' \\|  {\\bf h}_v']) \\ ,\n\\end{aligned}\n\\end{array}\n\\end{equation}\nwhere ${\\bf h}_u'$ and ${\\bf h}_\\upsilon'$ are the transformed representations of the node $u$ and $\\upsilon$. ${\\bf a}_{\\theta_i}\\in {\\mathbb R}^{2F'}$ is a trainable weight vector. $\\|$ denotes the concatenate operation. LeakyReLU is a nonlinearity activation function. \nTo make $e_{u\\upsilon}^{\\theta_i}$ comparable over different entities, we normalize it using the softmax function.\n\\begin{equation}\n    \\gamma_{uv}^{\\theta_i} =\\text{softmax}_\\upsilon (e_{uv}^{\\theta_i})= \\frac{\\exp{(e_{uv}^{\\theta_i}})}{\\sum\\limits_{\\bar{v} \\in {\\mathcal{N}}_\\text{inter}^{\\theta_i}(u) }\\exp{(e_{u\\bar{v}}^{\\theta_i})}} \\ ,\n\\end{equation}\nwhere $\\gamma_{uv}^{\\theta_i}$ denotes the attention value of entity $v$ with relation $\\theta_i$ to entity $u$. ${\\mathcal{N}}_\\text{inter}^{\\theta_i}(u)$ denotes the specific relation-based neighbors with the different type.\nWe apply entity-level attention to fuse inter-class neighbors with a specific relation $\\theta_i$:\n\\begin{equation}\n    \\label{equation-node-level-aggregation}\n    {\\bf h}_u^{\\theta_i} = \\sigma \\Big(\\sum_{v \\in {\\mathcal{N}}_\\text{inter}^{\\theta_i}(u)}\\gamma_{uv}^{\\theta_i} \\cdot {\\bf h}_v' \\Big ) \\ ,\n\\end{equation}\nwhere $\\sigma$ is a nonlinear activation, and ${\\bf h}_v'$ is the projected feature of entity $v$.  \n\n\n\n\n\n\n\nOnce we learned all relation embeddings $\\{{\\bf h}_u^{\\theta_i}\\}$, we utilize relation-level attention to fuse them together to obtain the inter-class relational embedding $z_u$ for entity $u$.\nWe first calculate the importance of each relation $w^{\\theta_i}$ as follows:\n\\begin{equation}\n   w^{\\theta_i}=\\frac{1}{|{\\mathcal{E}}_1|}\\sum\\limits_{u\\in {\\mathcal{E}}_1} {\\bf q}^{\\tau(u)} \\cdot {\\bf h}_u^{\\theta_i} + \\frac{1}{|{\\mathcal{E}}_2|}\\sum\\limits_{v\\in {\\mathcal{E}}_2} {\\bf q}^{\\tau(v)} \\cdot {\\bf h}_v^{\\theta_i} \\ ,\n\\end{equation}\n\n\\begin{equation}\n    \\epsilon^{\\theta_i} = \\frac{\\exp{(w^{\\theta_i})}}{\\sum\\limits_{\\theta_j \\in \\Theta_{\\text{inter} }}\\exp{(w^{\\theta_j})}} \\ ,\n\\end{equation}\nwhere ${\\bf q}^{\\tau(\\cdot)}\\in {\\mathbb R}^{F'\\times 1}$ is learnable parameter. We fuse all relation embeddings to obtain the inter-class relational embedding ${\\bf z}_u\\in {\\mathbb R}^{F'}$ of entity u. \n\\begin{equation}\n    \\label{equation-node-level-aggregation}\n    {\\bf z}_u  = \\sum_{\\theta_i \\in \\Theta_{\\text{intra} }} \\epsilon^{\\theta_i }\\cdot {\\bf h}_u^{\\theta_i} \\ .\n\\end{equation}\nIn inter-class attention, the aggregation of different entities' embedding are seamlessly integrated, and they are mingled and interactively affected each other in nature, as shown in Figure \\ref{figure-DanSmp-Model} (II-a).\n\n\\subsubsection{Intra-class Attention Networks}\nThe intra-class attention aims to learn the interaction among the same type of entities, as shown in Figure \\ref{figure-DanSmp-Model} (II-b).\nSpecifically, given a relation $\\phi_k \\in \\Phi_\\text{intra}^{\\tau(u)}$ that starts from entity $u$, we can get the intra-class relation based neighbors ${\\mathcal{N}}_\\text{intra}^{\\phi_k}(u)$.  $\\Phi_\\text{intra}^{\\tau(u)}$ indicates the set of all intra-class relations of $u$. For instance, as shown in Figure \\ref{figure-instance}, Company 5 is a neighbor of Company 3 based on an implicit relation, and Company 4 is a neighbor of Company 1 based on meta-relation \\textit{CEC}. Each intra-class relation represents one semantic interaction, and we apply relation-specific attention to encode this characteristic. We first calculate the attention value of entity $\\tilde{u}$ with relation $\\phi_k$ to entity $u$ as follows:\n\n\n\n\n\n\\begin{equation}\n    \\alpha_{u\\tilde{u}}^{\\phi_k} =\n   \n    \\frac{\\exp{(\\text{LeakyRelu}({\\bf a}_{\\phi_k}^\\top \\cdot [{\\bf W} {\\bf z}_u \\| {\\bf W} {\\bf z}_{\\tilde{u}}]))}}{\\sum\\limits_{u' \\in {\\mathcal{N}}_\\text{intra}^{\\phi_k}(u) }\\exp{(\\text{LeakyRelu}({\\bf a}_{\\phi_k}^\\top \\cdot [{\\bf W} {\\bf z}_u \\| {\\bf W} {\\bf z}_{u'}]))}} \\ ,\n\\end{equation}\nwhere ${\\bf z}_u$ and ${\\bf z}_{\\tilde{u}}$ are output representations of the inter-class attention, respectively. ${\\bf W}\\in {\\mathbb R}^{F'\\times F'}$ is a trainable weight matrix which is shared to every node of the same type.\n${\\bf a}_{\\phi_k}\\in {\\mathbb R}^{2F'}$ is the node-level attention weight vector for relation $\\phi_k$.\n${\\mathcal{N}}_\\text{intra}^{\\phi_k}(u)$ denotes the intra-class neighbors of $u$ under relation $\\phi_k$. \nThe embedding ${\\bf h}_u^{\\phi_k}$ of entity $u$ for the given relation $\\phi_k$ is calculated as follows.\n\\begin{equation}\n    \\label{equation-node-level-aggregation}\n    {\\bf h}_u^{\\phi_k}  = \\sigma \\Big(\\sum_{\\tilde{u} \\in {\\mathcal{N}}_\\text{intra}^{\\phi_k}(u)}\\alpha_{u\\tilde{u}}^{\\phi_k} \\cdot {\\bf W}{\\bf z}_{\\tilde{u}} \\Big ) \\ ,\n\\end{equation}\nwhere $\\sigma$ is a non-linear activation. In total, we can get $|\\Phi_\\text{intra}^{\\tau(u)}|$ embeddings for entity $u$. Then, we conduct relation-level attentions to fuse them into the relational embedding ${\\bf h}_u\\in {\\mathbb R}^{F'}$:\n\n\\begin{equation}\n    \\label{equation-node-level-aggregation}\n    {\\bf h}_u  = \\sum_{\\phi_k \\in \\Phi_{\\text{intra}}^{\\tau(u)}} \\beta^{\\phi_k} \\cdot {\\bf h}_u^{\\phi_k} \\ ,\n\\end{equation}\nwhere $\\Phi_\\text{intra}^{\\tau(u)}$ denotes the set of all intra-class relationships of entity $u$. $\\beta^{\\phi_k}$ denotes the importance of intra-class relation $\\phi_k$, which is calculated as follows:\n\\begin{equation}\n\\begin{array}{l}\n\\begin{aligned}\n   {\\bf g}^{\\phi_k}&=\\frac{1}{|{\\mathcal{E}}_1|}\\sum\\limits_{u\\in {\\mathcal{E}}_1} {\\bf q}^{\\tau(u)} \\cdot {\\bf h}_u^{\\phi_k} \\ ,\\\\\n   \\beta^{\\phi_k} &= \\frac{\\exp{({\\bf g}^{\\phi_k})}}{{\\sum\\limits_{\\phi_l \\in \\Phi_\\text{intra}^{\\tau(u)} }}\n    \\exp{({\\bf g}^{\\phi_l})}} \\ .\n\\end{aligned}\n\\end{array}\n\\end{equation}\nwhere ${\\bf q}^{\\tau(u)}\\in {\\mathbb R}^{F'}$ is a learnable parameter.\n\n\n\n\n\n\n    \n\n\n\n\n\n\n\n\\subsection{SMP with Stock Final Embeddings}\nFinally, the stock final embeddings by combining learned sequential embeddings and relational embeddings are utilized to make stock prediction by a dense layer feed-forward neural network (FNN) and a softmax function, as shown in Figure \\ref{figure-DanSmp-Model}-III.\n\\begin{equation}\n\\begin{aligned}\n    \\hat{y}_{i}^{t}&=\\textbf{SMP} \\Big({\\bf s}_{i}^{t} \\parallel {\\bf h}_{i}^{t} \\Big) \\\\ \n    &= \\text{Softmax} \\Big ({\\bf W}_{smp} [ {\\bf s}_{i}^{t} \\parallel {\\bf h}_{i}^{t} ]+ b_{smp} \\Big )  \\ ,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf W}_{smp}$ is a trainable weight matrix, and $b_{smp}$ is the bias vector. \nWe leverage the Adam algorithm \\cite{Kingma2014Adam} for optimization by minimizing the cross entropy loss function ${\\mathcal{L}}$.\n\n\\begin{equation}\n{\\mathcal{L}}=-\\sum_{i=1}^{\\left | N\\right |}\\sum_t {y}_{i}^t ln\\left ( \\hat{y}_{i}^t\\right ) \\ ,\n\\end{equation}\nwhere ${y}_{i}^t$ and $\\hat{y}_{i}^t$ represent the ground truth and predict stock trend of stock $i$ at $t$ day, respectively. $\\left | N\\right |$ is the total number of stocks.\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{section-experiments}\n\nIn this section, we present our experiments, mainly focusing on the following research questions:\n\n$\\bullet$ RQ1: Can our model achieve better performance than the state-of-the-art stock prediction methods?\n\n$\\bullet$ RQ2: Can our model achieve a higher investment return and lower risk in the investment simulation on real-world datasets?\n\n$\\bullet$ RQ3: How is the effectiveness of different components in our model?\n\n$\\bullet$ RQ4: Are all firm relations equally important for SMP? How do different parameters influence our model's performance?\n\nIn the following, we first present the experimental settings and then answer these research questions by analyzing the experimental results.\n\n\\subsection{Experimental Settings} \n\\subsubsection{Data Collection}\nSince no existing stock prediction benchmark datasets can satisfy our need to evaluate the effectiveness of our method, we collect public available data about the stocks from the famous China Securities Index (CSI) and construct two new datasets. \nWe name them \\textbf{CSI100E} and \\textbf{CSI300E}\nwith different number of listed companies, respectively. \n185 stocks in CSI300E index without missing transaction data and having at least 60 related news articles during the selected period are kept. Similarly, 73 stocks in CSI100E index are kept.\nFirst, we get historical price of stocks\\footnote{We collect daily stock price and volume data from \\url{https:\/\/www.wind.com.cn\/}} from November 21, 2017 to December 31, 2019 which include 516 transaction days.  \nSecond, we collect web news published in the same period from four financial mainstream sites, including \\textit{Sina\\footnote{\\url{http:\/\/www.sina.com}}, Hexun\\footnote{\\url{http:\/\/www.hexun.com}}, Sohu\\footnote{\\url{http:\/\/www.sohu.com}}} and \\textit{Eastmoney}\\footnote{\\url{http:\/\/www.eastmoney.com}}.\nLast, we collect four types of company relations\\footnote{We collect four types of company relations by a publicly available API tushare: \\url{https:\/\/tushare.pro\/}.} and the connections of executives\\footnote{We collect executives relationships from : \\url{http:\/\/www.51ifind.com\/}.} for CSI300E and CSI100E.\nThe basic statistics of the datasets are summarized in Table \\ref{data-stcs}.\nThe usage details of the multimodal market signals are described in Section \\ref{section-msr}. \n\n\n\\begin{table}[t]\n\\caption{Statistics of datasets.}\n\\label{data-stcs}\n\\centering\n\\begin{tabular}[t]{l||c|c}\n\\toprule\n{ }  &\\textbf{CSI100E} & \\textbf{CSI300E}\\\\\n\\midrule\n\\textbf{\\textit{\\#Companies(Nodes)}}  & 73 & 185  \\\\\n\\textbf{\\textit{\\#Executives(Nodes)}}  & 163& 275  \\\\\n\\midrule\n\\textbf{\\textit{\\#Investment(Edges)}}  & 7& 44  \\\\\n\\textbf{\\textit{\\#Industry category(Edges)}}  & 272& 1043  \\\\\n\\textbf{\\textit{\\#Supply chain(Edges)}}  & 27& 37  \\\\\n\\textbf{\\textit{\\#Business partnership(Edges)}}  & 98& 328  \\\\\n\\textbf{\\textit{\\#Implicit relation(Edges)}}  & \\textit{dynamic}& \\textit{dynamic}  \\\\\n\\textbf{\\textit{\\#meta-relation CEC}}  &  18&  42 \\\\\n\\textbf{\\textit{\\#meta-relation CEEC}}  &  134&  252 \\\\\n\\midrule\n\\textbf{\\textit{\\#Classmate(Edges) }}  & 338& 592  \\\\\n\\textbf{\\textit{\\#Colleague(Edges)}}  & 953& 2224  \\\\\n\\midrule\n\\textbf{\\textit{\\#Management(Edges)}}  & 166& 275  \\\\\n\\textbf{\\textit{\\#Investment(Edges)}}  & 1& 8  \\\\\n\\midrule\n\\textbf{\\textit{\\bf\\#Train Period}}  & 21\/11\/2017-05\/08\/2019 & 21\/11\/2017-05\/08\/2019  \\\\\n\\textbf{\\textit{\\bf\\#Valid Period}}  & 06\/08\/2019-22\/10\/2019 & 06\/08\/2019-22\/10\/2019  \\\\\n\\textbf{\\textit{\\bf\\#Test Period}}  & 23\/10\/2019-31\/12\/2019 & 23\/10\/2019-31\/12\/2019  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n  \n\\subsubsection{Evaluation Protocols}\nSMP is usually treated as a binary classification problem. \nIf the closing price of a stock $i$ is higher than its opening price at day $t$, the stock movement trend is defined as \"upward\" $\\left ( y_{i}^{t}=1\\right )$, otherwise as \"downward\" $\\left ( y_{i}^{t}=0\\right )$. According to statistics, there are 46.7$\\%$ \"upward\" {stocks} and 53.3$\\%$ \"downward\" ones in CSI100E, and 47.8$\\%$ \"upward\" and 52.2$\\%$ \"downward\" {stocks} in CSI300E. Hence, the datasets are roughly balanced. \n\nSome indicators \\cite{sousa2019bert,Ye2020Multi-Graph} are selected to demonstrate the effectiveness of the proposed method, i.e. Directional Accuracy (DA), Precision,  (AUC), Recall, F1-score. \nWe use the \\textbf{Directional Accuracy (DA)} and \\textbf{AUC} (the area under the precision-recall curve) \nto evaluate classification performance in our experiments, \nwhich are widely adopted in previous works \\cite{Li2020A,Cheng2021Modeling}. \nSimilar to \\cite{Sawhney2020Spatiotemporal,Sawhney2021Stock}, \nto evaluate \\textsc{DanSmp}'s applicability to real-world trading, we assess its profitability on CSI100E and CSI300E using metrics: cumulative investment return rate (\\textbf{IRR}) and \\textbf{Sharpe Ratio} \\cite{sharpe1994sharpe}.\nSimilar to previous method \\cite{Li2020A,Cheng2021Modeling}, we use the market signals of the past $T$ trading days (also called lookback window size) to predict stock movement on $t^{th}$ day. The DA, IRR and SR are defined as follows:\n\\begin{equation}\n\\begin{aligned}\n    DA &=\\frac{n}{N} \\ , \\\\\n    IRR^{t} &=\\sum_{i\\in S^{t-1}}\\frac{p_{i}^{t}-p_{i}^{t-1}}{p_{i}^{t-1}} \\ ,  \\\\\n    SR_{a}& =\\frac{E\\left [ R_{a}-R_{f}\\right ]}{std\\left [ R_{a}-R_{f}\\right ]} \\ ,\n\\end{aligned}\n\\end{equation}\nwhere $n$ is the number of predictions, which witness the same direction of stock movements for the predicted trend and the actual stock trend and $N$ is the total number of predictions. $S^{t-1}$ denotes the set of stocks on day $t-1$, and  $p_{i}^{t}$ is the price of stock i at day $t$. $R_{a}$ denotes an asset return and $R_{f}$ is the risk-free rate. In this study, the risk-free rate is set as the one-year deposit interest rate of the People's Bank of China in 2019, i.e. $R_{f}=1.5\\%$.\n\nNote that, to ensure the robustness of the evaluation, we repeat the testing procedure 10 times with different initialization for all the experimental results and the average performance is reported as the final model result.\n\n\n\n\\subsubsection{Parameter Settings}\nAll trainable parameters vectors and matrices are initialized using the Glorot initialization \\cite{glorot2010understanding}.\nIn our \\textsc{DanSmp}, we set the lookback window size $T$ among [10, 15, 20, ... , 40]. We search the learning rate from\n[0.00005, 0.0001, 0.00015, ... , 0.002]. The slice size of NTN $M$ and attention layer hidden size ${F}'$ are determined in [5,10,15, ... ,50] and [10, 11, 12, ... , 50], respectively. The GRU hidden size $F$ is set from [20, 22, 24, ... , 100]. In our model, all hyperparameters were optimized with the validation set, and Table \\ref{table-hyperparameters} shows the hyper-parameter settings of our method. The proposed\n\\textsc{DanSmp} is implemented with PyTorch\\footnote{\\url{https:\/\/pytorch.org\/}.} \nand PyTorch Geometric\\footnote{\\url{https:\/\/pytorch-geometric.readthedocs.io\/en\/latest\/}.}, \nand each training process costs 1.5hrs averagely using a GTX 1080 GPU. To prevent overfitting, we use early stopping based on AUC (the area under the precision-recall curve) over the validation set.\n\n\\begin{table}[htb]\n\\caption{The hyper-parameter settings on two datasets.}\n\\label{table-hyperparameters}\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n\\centering\n\\begin{tabular}[t]{l||c c}\n\\toprule\n\\bf Parameter & \\bf CSI100E & \\bf CSI300E\\\\\n\\midrule\nLookback window size $T$& 20 & 20 \\\\ \nThe slice size of NTN $M$ & 10 & 10\\\\\nAttention layer hidden size ${F}'$ & 39 & 22\\\\\nGRU hidden size $F$& 78 & 44 \\\\\nLearing rate & 0.0008 & 0.00085\\\\\nImplicit relation threshold $\\eta$ & 0.0054 & 0.0052 \\\\\nMaximum number of epochs  & 400 & 400 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsubsection{Baselines}\n\nTo demonstrate the effectiveness of our proposed model \\textsc{DanSmp}, we compare the results with the following baselines. \n\n\n\n\n$\\bullet$ LSTM \\cite{Hochreiter1997Long}: a typical RNN model that has promising performance on time-series data. In the evaluation, two-layer LSTM networks are implemented. \\textcolor{blue}{}\n\n$\\bullet$ GRU \\cite{Cho2014Learning}: a simpler RNN that achieves similar performance with LSTM. In the comparison, two-layer GRU networks are implemented.\n\n\n$\\bullet$ GCN \\cite{Kipf2017Semi-supervised}: It performs graph convolutions to linearly aggregate the attributes of the neighbor nodes. In this study, two-layer GCN network was implemented.  \\textcolor{blue}{}\n\n$\\bullet$ GAT \\cite{Velickovic2018Graph}: It introduces attention mechanism which assigns different importance to the neighbors adaptively.  Two-layer GAT networks are implemented.\n\n$\\bullet$ RGCN \\cite{Schlichtkrull2018Modeling}: It designs specialized mapping matrices for each relations.  Two-layer RGCN network was implemented.\n\n$\\bullet$ HGT \\cite{Hu2020Heterogeneous}: It uses transformer architecture to capture features of different nodes based on type-specific transformation matrices.\n\n\n$\\bullet$ MAN-SF \\cite{Sawhney2020Deep}: It fuses chaotic temporal signals from financial data, social media and stock relations in a hierarchical fashion to predict future stock movement.\n\n$\\bullet$ STHAN-SR \\cite{Sawhney2021Stock}: It uses hypergraph and temporal Hawkes attention mechanism to rank stocks with only historical price data and explicit firm relations. We only need to slightly modify the objective function of MAN-SF to predict future stock movement.\n\n$\\bullet$ AD-GAT \\cite{Cheng2021Modeling}: a SOTA method to use an attribute-driven graph attention network\nto capture attribute-sensitive momentum spillover of stocks, which can modeing market information space with feature interaction to further improve stock movement prediction\n\n\nThese baselines cover different model characters. \nSpecifically, the sequential-based LSTM \\cite{Hochreiter1997Long} and GRU \\cite{Cho2014Learning} can capture the time dependency of stock data, and the fused market signals were used as the input to the LSTM and GRU model. \nThe homogeneous GNNS-based GCN \\cite{Kipf2017Semi-supervised}, GAT \\cite{Velickovic2018Graph}, RGCN \\cite{Schlichtkrull2018Modeling}, HGT \\cite{Hu2020Heterogeneous}, MAN-SF \\cite{Sawhney2020Deep}, STHAN-SR \\cite{Sawhney2021Stock} and AD-GAT \\cite{Cheng2021Modeling} can capture the influence of related stocks based on the fused market signals and simple firm relations. \nNote that, for fair comparison, we do not select the methods that are incapable of dealing with all fused multi-modal market signals (i.e. historical price, media news and stock relations) as baselines. \n\n\n\n\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Stock prediction results of different models.}\n   \n    \\label{table-model-comparison}\n    \\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n    \\centering\n   \n    \\begin{tabular}{l||cc|cc}\n    \\toprule    \n    \\multirow{2}*{\\bf{Methods} } &\\multicolumn{2}{c|}{\\bf CSI100E} &\\multicolumn{2}{|c}{\\bf CSI300E}\\\\\n   \n       &\\textbf{Accuracy}&\\textbf{AUC}&\\textbf{Accuracy} &\\textbf{AUC} \\\\\n    \\midrule\n     \\midrule\n        LSTM \\cite{Hochreiter1997Long} & 51.14 & 51.33&51.78  &52.24  \\\\\n   \n        GRU \\cite{Cho2014Learning} & 51.66 & 51.46&51.11  &52.30  \\\\\n    \\midrule\n    GCN \\cite{Kipf2017Semi-supervised} & 51.58 &52.18&51.68 & 51.81   \\\\\n   \n    GAT \\cite{Velickovic2018Graph} & 52.17 &  52.78&51.40 & 52.24 \\\\\n   \n    RGCN \\cite{Schlichtkrull2018Modeling} & 52.33 &  52.69&51.79 & 52.59 \\\\\n   \n    HGT \\cite{Hu2020Heterogeneous} & 53.01 &52.51&51.70 & 52.19  \\\\\n   \n   \n   \n   \n    MAN-SF \\cite{Sawhney2020Deep}& 52.86 & 52.23 &51.91 & 52.48\\\\\n   \n    STHAN-SR \\cite{Sawhney2021Stock}& 52.78 & 53.05 & \\underline{52.89}& 53.48\\\\\n   \n    AD-GAT \\cite{Cheng2021Modeling} &\\underline{54.56} & \\underline{55.46}&52.63 &\\underline{54.29}\\\\\n    \\midrule\n    \\textbf{\\textsc{DanSmp} (ours)} & \\textbf{57.75}  & \\textbf{60.78} & \\textbf{55.79}& \\textbf{59.36} \\\\\n    \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\\subsection{Experimental Results and Analysis (RQ1)}\nTable \\ref{table-model-comparison} shows the evaluation results of the two datasets against nine state-of-the-art (SOTA) baselines, from which we observe that our proposed method outperforms all baselines for stock movement prediction in terms of all metrics on CSI100E and CSI300E. \nIt confirms the capability of our method in modeling the comprehensive market signal representations via dual attention networks.\n\n\\paragraph{\\textbf{Analysis.}}\n\n(1) The LSTM and GRU, which only consider historical prices and media news, perform largely worse than our method. The results indicate that the relation datas contribute to stock movement prediction and the proposed method can take full advantage of the relational information in MKG to improve performance.  \n(2) The graph-based methods, such as GCN and GAT, are homogeneous GNNs which are incapable of modeling heterogeneous market graph. Although being able to model multi-relational graph, RGCN can not sufficiently encode bi-typed heterogeneous graph become of the fact that it ignores the heterogeneity of node attributes and calculates the importance of neighbors within the same relation based on predefined constants.\nHGT focuses on handling web-scale heterogeneous graphs via graph sampling strategy, which thus is prone to overfitting when dealing with relative sparse MKG. HGT can not learns multi-level representation by sufficiently utilize interactions between two types of nodes.\nWe believe that is the reason they perform worse than our model \\textsc{DanSmp} which is pertinently designed to model bi-typed hybrid-relational MKG.\n(3) The proposed \\textsc{DanSmp} consistently outperforms three other SMP competitors, including AD-GAT, STHAN-SR and MAN-SF.\nSpecifically, it exceeds the second place by approximately 3.19$\\%$ and 5.32$\\%$ in terms of Accuracy and AUC in CSI100E, and 3.16$\\%$ and 5.07$\\%$ in CSI300E. The results clearly demonstrate the effectiveness of \\textsc{DanSmp} and the explicit relation and executives relation are meaningful for stock movement prediction. \n\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Profitability of all methods in back-testing.}\n   \n    \\label{table-as-four}\n    \\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n    \\centering\n   \n    \\begin{tabular}{l||rr|rr}\n    \\toprule    \n    \\multirow{2}*{\\bf{Methods} } &\\multicolumn{2}{c|}{\\bf CSI100E} &\\multicolumn{2}{|c}{\\bf CSI300E}\\\\\n   \n       &\\textbf{IRR}&\\textbf{SR}&\\textbf{IRR} &\\textbf{SR} \\\\\n    \\midrule\n     \\midrule\n        LSTM \\cite{Hochreiter1997Long} &-4.57\\% & -2.1713&-0.38\\%  &-0.326  \\\\\n   \n        GRU\\cite{Cho2014Learning} & -2.55\\% & -1.053&-3.73\\%  &-1.197  \\\\\n    \\midrule\n    GCN \\cite{Kipf2017Semi-supervised} & 1.59\\% &0.719&3.55\\% & 1.873   \\\\\n   \n    GAT \\cite{Velickovic2018Graph} & 0.3\\% &  0.050&-1.82\\% & -1.121 \\\\\n   \n    RGCN \\cite{Schlichtkrull2018Modeling}& 6.41\\% &  3.789&-3.64\\% & -1.905 \\\\\n   \n    HGT\\cite{Hu2020Heterogeneous} & 2.54\\% &1.716&0.36\\% & 0.076  \\\\\n   \n   \n   \n   \n    MAN-SF\\cite{Sawhney2020Deep}& -2.91\\% & -1.590 &1.38\\% & 0.604\\\\\n   \n    STHAN-SR\\cite{Sawhney2021Stock}& -0.12\\% & -0.092 & 5.41\\%& 1.565\\\\\n   \n    AD-GAT \\cite{Cheng2021Modeling} &2.34\\% & 1.190&15.12\\% &4.081\\\\\n    \\midrule\n    \\textbf{\\textsc{DanSmp} (ours)} & \\textbf{10.18\\%}  & \\textbf{4.112} & \\textbf{16.97\\%}& \\textbf{4.628} \\\\\n    \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{simulation.png}\\\\\n    \\caption{Profitability analysis on CSI100E and CSI300E.}\n   \n   \n    \\label{figure-investment-simulation}\n\\end{figure}\n\n\n\\subsection{Investing simulation (RQ2)}\nTo test whether \na model can make a profit, we set up a back-testing via simulating the stock investment in CSI100E and CSI300E over the test period, during which the CSI100 and CSI300 index increased by 4.20\\% and 5.14\\% (from 4144.05 to 4317.93 and 3896.31 to 4096.58), respectively. Specifically, the top-15 stocks with the highest predicted ranking score in each model are bought and held for one day. We choose RMB 10,000 as the investment budget, and take into account a transaction cost of 0.03\\% when calculating the investment return rate, which is in accordance with the stock market practice in China. The cumulative profit will be invested into the next trading day.\nFrom Table \\ref{table-as-four} and Figure \\ref{figure-investment-simulation}, we can find that \\textsc{DanSmp} achieves approximately stable and continuous positive returns throughout the back-testing. Particularly, the advantage of \\textsc{DanSmp} over all baselines mainly lies in its superior performance when the stock market is in a bear stage. The proposed \\textsc{DanSmp} achieves significantly higher returns than all baselines with the cumulative rate of 10.18\\% and 16.97\\% in CSI100E and CSI300E. In addition, \\textsc{DanSmp} results in a more desirable risk-adjusted return with Sharpe Ratio of 4.113 and 4.628 in CSI100E and CSI300E, respectively. These results further demonstrate the superiority of the proposed method in terms of the trade-off between the return and the risk.\n\n\n\n\n\n\n\n\n\n\\subsection{Ablation Study (RQ3)}\nTo examine the usefulness of each component in \\textsc{DanSmp}, we conduct ablation studies on CSI100E and CSI300E. We design four variants: \n(1) \\textbf{\\textsc{DanSmp} w\/o executives}, which deletes the executive entities. MKG is degraded into a simple uni-type knowledge graph.\n(2) \\textbf{\\textsc{DanSmp} w\/o implicit relation}, which removes the implicit relation when we model the stock momentum spillover effect.\n(3) \\textbf{\\textsc{DanSmp} w\/o explicit relation}, which deletes the explicit relations and only use the implicit relation to predict stock movement.\n(4) \\textbf{\\textsc{DanSmp} w\/o dual}, which replaces the dual attention module by conventional attention mechanism and does not distinguish the node intra-class and inter-class relation. \n\nFrom Table \\ref{table-as-three}, we observe that \nremoving any component of \\textsc{DanSmp} would lead to worse results. \nThe effects of the four components vary in different datasets, but all of them contribute to improving the prediction performance.  \nSpecifically, removing executives relations and implicit relations leads to the most performance drop, compared to the other two, which means a company can influence the share price of other companies through interactions between executives. In contrast, using the conventional attention mechanism produces the least performance drop. Compared with conventional attention mechanism, the dual attention module enables \\textsc{DanSmp} to adaptively select more important nodes and relations.\nThis finding further proves that the proposed \\textsc{DanSmp} fully leverages bi-typed hybrid-relational information in MKG via dual mechanism for better stock prediction.\n\n\\begin{table}[t]\n    \\centering\n    \\caption{The ablation study over \\textsc{DanSmp}.}\n    \\label{table-as-three}\n    \\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n    \\centering\n   \n    \\begin{tabular}{l||cc|cc}\n    \\toprule    \n    \\multirow{2}*{\\bf{Variants} } &\\multicolumn{2}{c|}{\\bf CSI100E} &\\multicolumn{2}{|c}{\\bf CSI300E}\\\\\n   \n       &\\textbf{Accuracy}&\\textbf{AUC}&\\textbf{Accuracy} &\\textbf{AUC} \\\\\n   \\midrule\n   \n\n    \\textsc{DanSmp} & \\textbf{57.75} &\\textbf{60.78}& \\textbf{55.79}& \\textbf{59.36} \\\\\n    \\midrule\n    \\  w\/o executives &52.71 & 52.62&53.80&55.88 \\\\\n    \\  w\/o implicit rel. &53.52 & 54.38&52.13&53.37 \\\\\n    \\  w\/o explicit rel. &55.12 & 57.05&54.10&55.49 \\\\\n    \\midrule\n    \\  w\/o dual &56.12 & 58.60&55.43&57.85 \\\\\n\n     \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{2.png}\\\\\n    \\caption{The presentation of the learned attention scores of \\textsc{DanSmp} on CSI100E and CSI300E. Here, IC denotes the industry category; BP stands for the business partnership; IV denotes the investment; SC is the supply chain; IR denotes the implicit relation.}\n   \n   \n    \\label{figure-attention-scores}\n\\end{figure}\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{Figure4_paremeters.pdf}\n    \\caption{Sensitivity to parameters $T$ and $\\eta$.}\n   \n   \n    \\label{figure-parameter-analysis-1}\n\\end{figure}\n\n\\subsection{Analysis of Firm Relation (RQ4)}\nTo investigate the impact of using different types of relations for stock prediction, we show the learned attention scores of our model in Figure \\ref{figure-attention-scores}. The attention score is learned parameter for different firm relations. Some main findings are as follows: \n(1) We can observe Figure \\ref{figure-attention-scores} that the learned attention score of implicit relation gains more weight than other relations, and the implicit relation which contains a lot of valuable information proved to be helpful for stock movement prediction. \n(2) The industry category, supply chain and investment get almost the same attention scores, and those can improve the\nperformance of model. \n(3) Compared with other relations, the business partnership has the lowest score. Although the number of business partnership relation is greater than that of investment and supply chain, the relatively dense business partnership relation may carry some noise, which adds irrelevant information to the representations of target nodes. The results further demonstrate the necessity of considering the implicit relation in modeling the momentum spillover effect. In addition, our model can adaptively weight important company relations to obtain better representations of target nodes, which can improve the performance of the model for stock movement prediction.\n\n\\subsection{Parameter Sensitivity Analysis (RQ4)}\nWe also investigate on the sensitivity analysis of two parameters in \\textsc{DanSmp}. We report the results of \\textsc{DanSmp}  under different parameter settings on CSI100E and CSI300E and experimental results are shown in Figure \\ref{figure-parameter-analysis-1}.\n\n\\noindent \\textbf{Lookback window size $T$.} \nWe analyze the performance variation with different lookback window size $T$ in Figure \\ref{figure-parameter-analysis-1} (a). Our model performs best when $T$ is set to about 20 in both datasets.\n\n\\noindent \\textbf{Implicit relation threshold $\\eta$.} \nThe results of our model with different implicit relation thresholds are reported in Figure \\ref{figure-parameter-analysis-1} (b). \nThe performance of our proposed model grows with the increment of $\\eta$ and achieves the best performance when $\\eta$ is set to 0.0054 in CSI100E. With the increment of $\\eta$, the performance raises at first and then drops gradually in the dataset CSI300E. When the $\\eta$ becomes bigger, the performance decreases  possibly because some meaningful implicit edges are neglected.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\n\\label{section-conclusion}\nIn this paper, we focus on stock movement prediction task. To model stock momentum spillover in real financial market, we first construct a novel bi-typed hybrid market knowledge graph. Then, we propose a novel Dual Attention Networks, which are equipped with both inter-class attention module and intra-class attention module, to learn the stock momentum spillover features on the newly constructed MKG. To evaluate our method, we construct two new datasets CSI100E and CSI300E. The empirical experiments on the constructed datasets demonstrate our method can successfully improve stock prediction with bi-typed hybrid-relational MKG via the proposed \\textsc{DanSmp}.\nThe ablation studies reaffirm that the performance gain mainly comes from the use of the associated executives, and additional implicit relation between companies in MKG.\n\nAn interesting future work direction is to explore web media about the executives including: (i) the negative facts from news, such as accusation of crime, health issue, etc; (ii) the improper speech on social media, such as Twitter and Weibo. We believe these factual event information of executives can be detected and utilized to feed into graph-based methods for better SMP performance.\n\n\\begin{acks}\nThe authors would like to thank all anonymous reviewers in advance.\nThis research has been partially supported by grants from the National Natural Science Foundation of China under Grant No. 71725001, 71910107002, 61906159, 62176014, U1836206, 71671141, 71873108, 62072379, the State key R \\& D Program of China under Grant No. 2020YFC0832702, the major project of the National Social Science Foundation of China under Grant No. 19ZDA092, and the Financial Intelligence and Financial Engineering Key Laboratory of Sichuan Province.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPhysiological information about neural structure and activity \nwas employed from the very beginning to construct effective mathematical models of \nbrain functions. Typically, \nneural networks were introduced as assemblies of elementary dynamical units, that interact with \neach other through a graph of connections \\cite{boccaletti}. Under the stimulus of experimental investigations,  \nthese models have been including finer and finer details.\nFor instance, the combination of complex single--neuron  dynamics, delay and plasticity in\nsynaptic evolution, endogenous noise and specific  network topologies revealed quite crucial\nfor reproducing experimental observations, like the spontaneous emergence of synchronized\nneural activity,  both {\\sl in vitro} (see, e.g.,  \\cite{volman}) and  {\\sl in vivo},  and the appearance \nof peculiar fluctuations, the so--called ``up--down\" states, in\ncortical sensory areas \\cite{miguel,torres}. \n\n\nSince the brain activity is a dynamical process, its statistical\ndescription needs to take into account time as an intrinsic variable. \nAccordingly, non--equilibrium statistical mechanics should be the proper \nconceptual frame, where effective models of collective brain activity should be casted in. \nMoreover, the large number of units and the redundancy of connections suggest that\na mean--field approach can be the right mathematical tool for\nunderstanding the large--scale dynamics of neural network\nmodels. Several analytical and numerical investigations have been devoted to mean field approaches to \nneural dynamics. In particular, stability analysis of asynchronous states in globally coupled networks and collective observables in highly connected sparse network can be deduced in relatively simple neural network models through mean field techniques  \\cite{mfbrunel,mfcessac,mfbress,polmf,millman}.\n\nIn this paper we provide a detailed account of a mean--field approach, that\nhas been inspired by the    \n``heterogeneous   mean--field\" (HMF) formulation,\nrecently introduced for general interacting networks \\cite{vespignani,mendes}.\nThe overall method is applied here to the simple case \nof random networks of leaky integrate--and--fire (LIF) excitatory neurons in the\npresence of synaptic plasticity.  On the other hand,  it can be applied\nto a much wider class of neural network models, based on a similar mathematical\nstructure.\n\nThe main advantages of the HMF method are the following:  ({\\sl i}) it can identify  the \nrelation between the dynamical properties of the global ({\\sl synaptic}) activity \nfield and the network topology, ({\\sl ii})  it allows one to establish under which conditions  partially\nsynchronized or irregular firing events may appear , ({\\sl iii}) it provides a solution to the inverse \nproblem of recovering the network structure from the features of the global activity\nfield.\n\nIn Section \\ref{sec2}, we describe the network model of excitatory LIF neurons with\nshort--term plasticity. The  dynamical properties of the model are discussed at\nthe beginning of  Section \\ref{sec3}. In particular, we recall that the random structure of the\nnetwork is responsible for  the spontaneous organization of  neurons in two\nfamilies of {\\sl locked} and {\\sl unlocked} ones \\cite{DLLPT}. In the rest of this Section  \nwe summarize how to define a {\\sl heterogeneous thermodynamic limit}, that\npreserves the effects of the network randomness and allows one to transform\nthe original dynamical model into its HMF representation\n\\cite{BCDLV}). The HMF equations provide a relevant computational advantage with respect\nto the original system. Actually, they describe the dynamics\nof classes of equal--in--degree neurons, rather than that of individual neurons.\nIn practice, one can take advantage of a suitable sampling, according to its probability\ndistribution, of the continuous\nin--degree parameter present in the HMF formulation.\nFor instance, by properly \"sampling\" the HMF model into  300 equations one can \nobtain an effective description of the dynamics engendered by a random\nErd\\\"os--Renyi network made of ${\\mathcal O} (10^4)$ neurons.\n\nIn Section  \\ref{sec4} we show that the HMF  formulation\nallows also for a clear interpretation of the presence of classes of {\\sl locked} and {\\sl unlocked}\nneurons in QSE: they correspond to the presence of a {\\sl fixed point} or of an {\\sl intermittent-like} map of the \nreturn time of firing events, respectively. Moreover, we analyze in details the stability properties of the model and we find that any finite sampling of the\nHMF dynamics is chaotic, i.e. it is characterized by a positive maximum Lyapunov exponent,\n$\\lambda_{\\mathrm max}$. Its value depends indeed on the finite sampling\nof the in--degree parameter. On the other hand,  chaos is found to be relatively weak and, when the number \nof samples, $M$,  is increased,\n$\\lambda_{\\mathrm max}$ vanishes with a power--law decay, $M^{-\\gamma}$, with $\\gamma \\sim 1\/2$.\nThis is  consistent with the mean--field like nature of the HMF equations: in fact, it\ncan be argued that, in the thermodynamic limit, any chaotic component of the dynamics\nshould eventually disappear, as it happens for the original LIF model, when a naive\nthermodynamic limit is performed \\cite{DLLPT}.\n\nIn Section \\ref{sec5} we analyze the HMF dynamics for networks with different topologies (e.g., Erd\\\"os--Renyi\nand in particular scale free). We find that the dynamical phase characterized by QSE is robust with\nrespect to the network topology and it can be observed only if the variance of the considered\nin--degree distributions is sufficiently small. In fact, quasi-synchronous events are\nsuppressed for too broad in--degree distributions, thus yielding a transition between \na fully asynchronous dynamical phase  and a quasi-synchronous one, controlled by the\nvariance of the in--degree distribution. In all the cases analyzed in this Section, we find\nthat the global synaptic--activity field characterizes completely\nthe dynamics in any network topology. \n\nAccordingly,  the HMF formulation appears as an\neffective algorithmic tool \nfor solving the following {\\sl inverse problem}: given a global\nsynaptic--activity field, which kind of network topology \nhas generated it? In Section \\ref{sec6}, after a summary of the\nnumerical procedure used to solve such an inverse problem, we analyze \nthe  robustness of the method in two circumstances: $a)$ when a noise is \nadded to the average synaptic--activity field, and $b)$ when there are\nnoise and disorder in the external currents.\n\nSuch robustness studies are particularly relevant in view of applying this strategy to\nreal data  obtained from experiments.  \nFinally, in Section \\ref{sec7} we show that a HMF formulation can be \nstraightforwardly extended to non--massive networks, i.e. random networks, where the in--degree does not\nincrease proportionally to the number of neurons. In this case the relevant quantity\nin the HMF-like formulation is the average value of the in--degree\ndistribution, and  the HMF equations are expected to reproduce confidently the dynamics of\nnon--massive networks, provided this average is sufficiently large.\nConclusions and perspectives are contained in Section \\ref{sec8}.\n\n\n\n\\section{The model}\\label{sec2}\nWe consider a network of $N$ excitatory LIF neurons\ninteracting via a synaptic current and regulated by short--term plasticity, \naccording to a model introduced in \\cite{tsodyksnet}. \nThe membrane potential $V_j$ of each neuron evolves in time following the\ndifferential equation\n\\begin{equation}\n\\label{eq1}\n\\tau_\\mathrm{m} \\dot V_j= E_{\\mathrm{c}} -V_j + R_\\mathrm{in}I_{\\mathrm{syn}}(j)\\, ,\n\\end{equation}\nwhere $\\tau_\\mathrm{m}$ is the membrane time constant, \n$R_{\\mathrm{in}}$ is the membrane resistance,\n$I_{\\mathrm{syn}}(j)$ is the synaptic current received by neuron $j$ from \nall its presynaptic neurons (see below\nfor its mathematical definition) and \n$E_{\\mathrm{c}}$ is the contribution of an  external current \n(properly multiplied by a unit resistance). \n \nWhenever the potential $V_j(t)$ reaches the threshold value $V_{\\mathrm{th}}$, it is \nreset to $V_{\\mathrm{r}} $, and a spike is sent towards the postsynaptic neurons. \nFor the sake of simplicity the spike is assumed to be a $\\delta$--like function of time. \nAccordingly, the spike--train $S_j(t)$ produced by neuron $j$, is defined as,\n\\begin{equation}\n\\label{eq2}\nS_j(t)=\\sum_m \\delta(t-t_{j}(m)),\n\\end{equation}\nwhere $t_{j}(m)$ is the time when neuron $j$ fires its $m$-th spike.\n\nThe transmission of the  spike--train $S_j(t)$ is mediated by the\nsynaptic dynamics.\nWe assume that all efferent synapses of a given neuron follow \nthe same evolution (this is justified in so far as no inhibitory \ncoupling is supposed to be present).  The state of the $i$-th synapse is\ncharacterized by three variables, $x_i$, $y_i$, and $z_i$, which represent the\nfractions of synaptic transmitters in the recovered, active, and inactive state, \nrespectively ($x_i+y_i+z_i=1$) \\cite{plast1,plast2,tsodyksnet}. \nThe evolution equations are\n\\begin{align}\n\\label{dynsyn}\n& \\dot y_{i} = -\\frac{y_{i}}{\\tau_{\\mathrm{in}}} +ux_{i}S_i\\\\\n\\label{contz}\n& \\dot z_{i} = \\frac{y_{i}}{\\tau_{\\mathrm{in}}}  -   \\frac{z_{i}}{\\tau_{\\mathrm{r}}} \\ .\n\\end{align} \nOnly the active transmitters react to the incoming spikes: the parameter $u$ \ntunes their effectiveness. Moreover, $\\tau_{\\mathrm{in}}$ is the characteristic decay time of the\npostsynaptic current, while $\\tau_{\\mathrm{r}}$ is the recovery time from synaptic depression. \nFor the sake of simplicity, we assume also that all parameters appearing in the above \nequations are independent of the neuron indices. \nThe model equations are finally closed, by representing the synaptic current \nas the sum of all the active transmitters delivered to neuron $j$ \n\\begin{equation}\nI_{\\mathrm{syn}}(j) = \\frac{ G}{N}\\sum_{i\\ne j}  \\epsilon_{ij}y_i,\n\\label{input}\n\\end{equation}\nwhere $G$ is the strength of the synaptic coupling (that we assume \nindependent of both $i$ and $j$), while $\\epsilon_{ij}$ is the directed connectivity\nmatrix whose entries are set equal to 1 or 0 if the presynaptic neuron\n$i$ is connected or disconnected with the postsynaptic neuron $j$, respectively. \nSince we suppose  the input resistance\n$R_{\\mathrm{in}}$  independent of $j$, it can be included into $G$.\nIn this paper we study the case of excitatory coupling between neurons,\ni.e. $G > 0$.  We assume that each neuron\nis connected to a macroscopic number, \n${\\mathcal O}(N)$, of pre-synaptic neurons: this is the reason why the sum is divided by the \nfactor $N$. \nTypical values of the parameters contained in the model have phenomenological\norigin \\cite{volman,tsodyksnet}. Unless otherwise stated,  we adopt the following set of values: \n$\\tau_\\mathrm{in} = 6$ ms, \n$\\tau_\\mathrm{m} = 30$ ms,  $\\tau_\\mathrm{r} = 798$ ms, \n$V_{\\mathrm{r}} = 13.5$ mV, $V_{\\mathrm{th}} = 15$ mV, \n$E_{\\mathrm{c}} =15.45$ mV,  ${ G} = 45$ mV and $u = 0.5$.\nNumerical simulations can be performed much more effectively by introducing\ndimensionless quantities,\n\\begin{align}\n& a = \\frac{E_c-V_{\\mathrm{r}}}{V_{\\mathrm{th}}-V_{\\mathrm{r}}}\\\\\n& g = \\frac{G}{V_{\\mathrm{th}}-V_{\\mathrm{r}}}\\\\\n&  v=\\frac{V-V_\\mathrm{r}}{V_{\\mathrm{th}}-V_\\mathrm{r}},\n\\end{align}  \nand by rescaling time, together with all the other temporal parameters, in units of the membrane time \nconstant $\\tau_\\mathrm{m}$ \n(for simplicity, we leave the notation unchanged after rescaling). The values of the\nrescaled parameters are:\n$\\tau_\\mathrm{in} = 0.2$,  $\\tau_{\\mathrm{r}} = 133\\tau_{\\mathrm{in}}$, $v_{\\mathrm{r}} = 0$, \n$v_{\\mathrm{th}} = 1$, $a= 1.3$,  $g = 30$ and $u = 0.5$. \nAs to the normalized external current $a$, its value for the first part of our analysis corresponds to the firing regime for neurons.\nWhile the rescaled Eqs. (\\ref{dynsyn}) and (\\ref{contz}) keep the same form, Eq.~(\\ref{eq1}) \nchanges to,\n\\begin{equation}\n\\label{eq1n}\n\\dot v_j= a -v_j + \\frac{g}{N} \\sum_{i \\ne j} \\epsilon_{ij} y_i \\, .\n\\end{equation}\nA major advantage for numerical simulations comes from the possibility of transforming \nthe set of differential equations (\\ref{dynsyn})--(\\ref{input}) and (\\ref{eq1n})\ninto an event--driven map (for details see \\cite{DLLPT} and also  \\cite{brette,zill}). \n\n\n\\section{Dynamics and heterogeneous mean field limit}\\label{sec3}\n \n\nThe dynamics of the fully coupled neural network (i.e., $\\epsilon_{ij}=1, \\, \\forall i,j$),\ndescribed by Eq.s (\\ref{eq1n}) and  (\\ref{eq2})--(\\ref{input}), converges to a periodic synchronous state, \nwhere all neurons fire simultaneously and the period depends on the model parameters \\cite{DLLPT}.   \nA more interesting dynamical regime appears when some disorder is introduced in the network structure.\nFor instance, this can be obtained by maintaining each link between neurons with probability\n$p$, so that the in-degree of a neuron (i.e.  the number of  presynaptic connections acting on it)\ntakes the average value $\\langle k_i \\rangle = p N$, and the\nstandard deviation of the corresponding in-degree distribution is given by the relation $\\sigma_{k} = \\sqrt{Np(1-p)}$.\nIn such an Erd\\\"os-Renyi random network one typically  \nobserves quasi--synchronous events (QSE), where a large fraction of neurons  fire in a short\ntime interval of a few milliseconds, separated by an irregular firing activity lasting over some tens of ms\n(e.g., see \\cite{DLLPT}). \nThis dynamical regime emerges as a collective phenomenon, where neurons separate spontaneously into\ntwo different families: the {\\it locked}  and the {\\sl unlocked} ones. \nLocked neurons determine the QSE and exhibit a periodic behavior, with a common period but different phases.\nTheir in--degree $k_i$  ranges over a finite interval below the  average value $\\langle k_i \\rangle$.\nThe unlocked ones participate to the irregular firing activity and exhibit a sort of intermittent evolution \\cite{DLLPT}.\nTheir in-degree is either very small or higher than  $\\langle k_i \\rangle$.\n\n\nAs the dynamics is very sensitive to the different values of of $k_i $, in a recent publication \\cite{BCDLV} we have  shown that one can design a\n{\\it heterogeneous mean-field} (HMF) approach by  a suitable\nthermodynamic limit preserving, for increasing values of $N$,  the main features associated with topological disorder.  \nThe basic step of this approach is the\nintroduction of a  probability distribution,  $P(\\tilde k)$, for the\nnormalized in-degree variable ${\\tilde k} = k\/N$, where the average $\\langle \\tilde k \\rangle$\nand the variance $\\sigma_{\\tilde k}^2 = \\langle {\\tilde k}^2 \\rangle - \\langle \\tilde k \\rangle^2$ \nare fixed independently of $N$.  \nA realization of the random network containing $N$ nodes (neurons) \nis obtained by extracting for each neuron $i$ ($i =1, \\cdots , N$) a value $\\tilde k_i$ from $P(\\tilde k)$, and \nby connecting the neuron $i$ with $\\tilde k_i  N$ randomly chosen neurons (i.e., $\\epsilon_{i,j} = 1$,  $j(i)= 1, \\cdots , \\tilde k_i N $). \nFor instance, one can consider a suitably normalized Gaussian--like  distribution \ndefined on the compact support, $\\tilde k \\in (0,1]$,\ncentered around  $\\langle \\tilde k \\rangle$ with a sufficiently small value of the standard deviation  \n$\\sigma_{\\tilde k}$, so that the tails of the distribution vanish at the boundaries of the support.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{newrp.eps}\n\\caption{\nRaster plot of a randomly diluted network containing 500 \nneurons, ordered along the\nvertical axis according to their in--degree. The distribution $P(\\tilde k)$ is a Gaussian with  $\\langle \\tilde k \\rangle =0.7$, standard deviation $\\sigma_{\\tilde k} =0.077$. A black dot in the raster plot indicates that\nneuron $s$ has fired at time $t$. The red line is the global field $Y(t)$ and the green curve is its analytic fit by the function $Y_f(t)=\nAe^{-\\frac{t}{\\tau_1}}+B(e^{\\frac{t}{\\tau_2}}-1)$, that repeats over each \nperiod of $Y(t)$; the parameter values are $A=2 \\cdot 10^{-2}$, $B=3.56\\cdot 10^{-6}$, $\\tau_1=0.268$ and $\\tau_2=0.141$. Notice that the amplitude of both $Y(t)$ and $Y_f(t)$ has been suitably rescaled to be appreciated on the same scale of the Raster plot.}\n\\label{rp1}\n\\end{figure}\n\nIn  Fig.\\ref{rp1} we show the raster plot for a network of $N=500$ neurons and a Gaussian distribution $P(\\tilde k)$ \nwith  $\\langle \\tilde k \\rangle=0.7$ and  $\\sigma_{\\tilde k}=0.077$. One can observe\na quasi-synchronous dynamics characterized by the presence of locked\nand unlocked neurons, and such a distinctive dynamical feature is preserved\nin the thermodynamic limit \\cite{BCDLV}. For example the time average of the inter--spike time interval between firing events of each neuron, (in formulae $ISI_m=t_m-t_{m-1}$, where the integer $m$ labels the $m$-th firing event)  as a function of the connectivity $\\tilde k$ is, apart from fluctuations, the same for each network size $N$. This confirms that the main features of the dynamics are maintained for increasing values of $N$. \n\n\n\n\n\n\n The main advantage of this approach is that one can \nexplicitly perform the limit $N\\to \\infty$ on the set of equations\n(\\ref{eq1n}) and (\\ref{eq2})--(\\ref{input}), thus\nobtaining the corresponding HMF equations:\n\n\\begin{align}\n\\label{vk}\n&\\dot v_{\\tilde k}(t)= a -v_{\\tilde k}(t) + g\\tilde kY(t)\\\\\n\\label{sk}\n&S_{\\tilde k}(t) = \\sum_m \\delta(t-t_{\\tilde k}(m)) \\\\\n\\label{yk}\n& \\dot y_{\\tilde k}(t) = -\\frac{y_{\\tilde k}(t)}{\\tau_{\\mathrm{in}}} +u(1-y_{\\tilde k}(t)-z_{\\tilde k}(t))S_{\\tilde k}(t)\\\\\n\\label{zk}\n& \\dot z_{\\tilde k}(t) = \\frac{y_{\\tilde k}(t)}{\\tau_{\\mathrm{in}}}  -   \\frac{z_{\\tilde k}(t)}{\\tau_{\\mathrm{r}}}\\\\\n\\label{meanfield}\n&Y(t)=\\int_{0}^{1}P(\\tilde k) y_{\\tilde k}(t)d\\tilde k .\n\\end{align}\nThe dynamical variables depend now on the continuous  in--degree index $\\tilde k$, and\nthis set of equations represents the dynamics of equivalence classes of neurons. In fact, in this HMF formulation,\nneurons with the same $\\tilde k$  follow the same evolution\n\\cite{vespignani, mendes}. In practice,  Eq.s   (\\ref{vk})--(\\ref{meanfield}) can be integrated numerically by  sampling  the \nprobability distribution $P(\\tilde k)$: one can subdivide  the support $(0,1]$ of $\\tilde k$ \nby $M$ values  $\\tilde k_i \\,\\,\\, (i=1,\\cdots , M)$, in such a way that $\\int_{\\tilde k_i}^{\\tilde k_{i+1}}P(\\tilde k)d\\tilde k$ is \nconstant (importance sampling).  Notice that the integration of the discretized  HMF equations is much less\ntime consuming than the simulations performed on a random network.\nFor instance, numerical tests indicate that \nthe dynamics of a network with $N=10^4$ neurons can be confidently reproduced by an importance sampling with $M= 300$.\n\nThe effect of the discretization of ${\\tilde k}$ on the HMF dynamics can be analyzed\nby  considering the distance $d(Y_{M_1}(t),Y_{M_2}(t))$ between the global\nactivity fields $Y_{M_1}(t)$ and $Y_{M_2}(t)$ (see Eq.(\\ref{meanfield})) obtained for two different values $M_1$ and $M_2$ of the sampling,\n i.e.:\n\\begin{equation}\nd(Y_{M_1}(t),Y_{M_2}(t))=\\Bigg(\\frac{1}{T}\\sum_{i =1}^T\\frac{(Y_{M_1}(t_i)-Y_{M_2}(t_i))^2}{Y_{M_1}(t_i)^2}\\Bigg)^{\\frac{1}{2}}.\n\\end{equation}\nIn general $Y(t)$ exhibits a quasi periodic behavior and  $d(Y_{M_1}(t),Y_{M_2}(t))$ is evaluated over a  time interval equal to its period $T$.\nIn order to avoid an overestimation of  $d(Y_{M_1}(t),Y_{M_2}(t))$\ndue to different initial conditions, the field  $Y_2(t)$ is suitably translated in time in order to make its\nfirst maximum coincide with the first maximum of  $Y_1(t)$ in the time interval $[1,T]$. \nIn Fig. \\ref{scarto}  we plot $d_M=d(Y_M,Y_{M\/2})$ as a function of $M$. We  find that $d_M\\sim 1\/\\sqrt M$,\nthus confirming that the finite size simulation of the HMF dynamics is consistent with the HMF model ($M\\to \\infty$).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{scarto.eps}\n\\caption{(Color online) The effect of sampling the probability distribution\n$P(\\tilde k)$ with $M$ classes of neurons in the HMF dynamics. Finite size\neffects are controlled by plotting the distance\nbetween the activity fields obtained for two sampling values $M$ and $M\/2$, $d_M=d(Y_M(t),Y_{M\/2}(t))$ (defined in the text), vs. $M$. The red dashed line is \nthe power law $1\/\\sqrt M$. Data is obtained for a Gaussian distribution $P(\\tilde k)$, with  $\\langle \\tilde k \\rangle=0.7$ and  $\\sigma_{\\tilde k}=0.077$.}\n\\label{scarto}\n\\end{figure} \n \nAs a final remark, notice that the presence of  short--term synaptic plasticity \nplays a fundamental role in determining the partially synchronized regime.\nIn fact, numerical simulations show that the discretized  HMF dynamics without plasticity, \ni.e. $Y(t) = \\int_{0}^{1}P(\\tilde k) S_{\\tilde k}(t)d\\tilde k$, \nconverges to a synchronous periodic dynamics for any value of $M$  \\cite{DL} .\n\n\\section{Stability analysis of the HMF dynamics}\n\\label{sec4}\n\nIn the HMF equations (\\ref{vk})--(\\ref{meanfield}) the dynamics of each neuron is\ndetermined by its in--degree $\\tilde k$ and by the global synaptic activity field $Y(t)$.\nFor the stability analysis of these equations, we follow  a procedure \nintroduced in \\cite{tso_locked} and employed also in \\cite{BCDLV}.\nFor  sufficiently large $M$ the  discretized  HMF dynamics allows one to \nobtain a precise fit of the periodic function $Y(t)$ and to estimate its period $T$. \nAs an instance of its  periodic behavior,  \nin  Fig.\\ref{rp1} we report also $Y(t)$ (red line)  and its fit (green line and the\nformula in the caption). The fitted field is exactly periodic and is a good approximation of the global field that\n one expects to observe in the mean field model corresponding to an infinite discretization $M$. As a result, the analysis performed using this periodic field are relative to the dynamics of the HMF model, i.e. in the limit  $M\\to \\infty$.\nUsing this fit, one can represent the dynamics of  each class $\\tilde k$ of neurons \nby the discrete--time map \n\\begin{equation}\n\\label{mappa}\n\\tau_{\\tilde k}(n+1)=R_{\\tilde k}[ \\tau_{\\tilde k}(n)],\n\\end{equation}\nwhere $\\tau_{\\tilde k}(n) = | t_{\\tilde k}(n) - nT |$ is  the modulus of the time difference \nbetween the  $n$-th  spike  of neuron $\\tilde k$ and $nT$, i.e. the $n$-th QSE, that \nis conventionally identified by  the corresponding maximum of  $Y(t)$ (see Fig. \\ref{rp1}). \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{newmap1.eps}\n\\caption{The return map $R_{\\tilde k}$ of the rescaled variable \n$\\tau_{\\tilde k}\/T$ (see Eq.(\\ref{mappa})) for different values of \n$\\tilde k$, corresponding \nto lines of different colors (see the legend in the inset: the black line \nis the bisector of the square).\n}\n\\label{maps}\n\\end{figure}\n\nIn Fig. \\ref{maps} we show $R_{\\tilde k} $ for different values of $\\tilde k$. The map\nof each class of {\\sl locked} neurons has a stable fixed point, whose value decreases with \n$\\tilde k$.  As a consequence, different classes of {\\sl locked} neurons share the\nsame periodic behavior, but  exhibit different phase shifts\nwith respect to the maximum of $Y(t)$. This analysis describes in a clear \nmathematical language what is observed in simulations (see Fig \\ref{rp1}):\nequally periodic classes of locked neurons determine\nthe QSE by firing sequentially, over a very short time interval, that depends on\ntheir relative phase shift.\nIn general, the values of $\\tilde k$ identifying the family of locked neurons\nbelong to a subinterval $(\\tilde k_1, \\tilde k_2)$ of $(0,1]$: the values of \n$\\tilde k_1$  and $\\tilde k_2$ mainly depend on $P(\\tilde k)$ and on its \nstandard deviation $\\sigma_{\\tilde k}$ (more details are reported in \\cite{BCDLV}).\nFor what concerns {\\sl unlocked} neurons,  $R_{\\tilde k} $ exhibits the features of an intermittent-like dynamics. In fact, unlocked neurons with $\\tilde k$\nclose to  $\\tilde k_1$  and $\\tilde k_2$ spend a long time in an almost periodic\nfiring activity, contributing to a QSE, then they depart from it,  firing  irregularly\nbefore possibly coming back again close to a QSE. The duration of the irregular\nfiring activity of unlocked neurons typically increases for values of $\\tilde k$ far from\nthe interval $ (\\tilde k_1, \\tilde k_2 )$.\n\nUsing the deterministic map (\\ref{mappa}), one can tackle in full\nrigor the stability problem of the HMF model. The existence of stable\nfixed points for the locked neurons implies that they yield a negative\nLyapunov exponent associated with their periodic evolution.\n\nAs for the unlocked neurons,  their  Lyapunov\nexponent, $\\lambda_{\\tilde k}$,  can be calculated numerically by \n the time-averaged  expansion rate \nof nearby orbits  of  map (\\ref{mappa}):\n\\begin{equation}\n\\label{lyn}\n\\lambda_{\\tilde k}(n)=  \\frac{1}{n} \\sum_{j=1}^n \\mathrm{log}\\Bigg[\\frac{|\\delta(j)|}{|\\delta(0)|}\\Bigg],\n\\end{equation}\nwhere $\\delta(0)$ is the initial distance between nearby orbits and\n$\\delta(j)$ is their distance at the $j$--th iterate,  so that\n\\begin{equation}\n\\label{lyap}\n\\lambda_{\\tilde k}= \\lim_{n \\to \\infty} \\lambda_{\\tilde k}(n)\n\\end{equation}\nif this limit exists. The Lyapunov exponents for the unlocked component vanish as\n$\\lambda_{\\tilde k} (n) \\sim 1\/n$.\n According to these results, one expects that the maximum Lyapunov exponent\n $\\lambda_{\\mathrm{max}}(M)$  goes to zero in the limit $M \\to \\infty$. \nIn fact, at each finite $M$,  $\\lambda_{\\mathrm{max}}$ can be evaluated by using the standard algorithm by Benettin et al.\n\\cite{BGGS}. \nIn Fig.\\ref{lyup_max} we plot $\\lambda_{\\mathrm{max}}$  as a function of the discretization parameter $M$. \nThus, $\\lambda_{\\mathrm{max}}(M)$ is positive,\nbehaving approximately  as  $M^{-\\gamma}$, with $\\gamma \\sim 1\/2$\n(actually,  we find $\\gamma = 0.55$).\n\nThe  scenario in any discretized version of the HMF dynamics is the following: \n{\\sl (i)}  all {\\sl unlocked neurons} exhibit positive Lyapunov exponents, i.e. they represent\nthe chaotic component of the dynamics; {\\sl (ii)}  $\\lambda_{\\mathrm{max}}$ is typically \nquite small, and its value depends on the discretization parameter\n$M$ and on $P(\\tilde k)$; {\\sl (iii)} in the limit $M\\to \\infty$  $ \\lambda_{\\mathrm{max}}$ and\nall $\\lambda_{\\tilde k}$'s of unlocked neurons vanish, thus converging to a quasi periodic\ndynamics, while the {\\sl locked neurons} persist in their periodic behavior.\n\nThe same scenario is observed in the dynamics of random networks built with the\nHMF strategy, where the variance of the distribution $P(\\tilde k)$ is kept independent of the \nsystem size $N$, so that  the fraction of locked neurons is constant.\n\nFor the LIF dynamics in an E\\\"ordos--Renyi random network with $N$ neurons, it was found that \n$ \\lambda_{\\mathrm{max}}(N) \\approx N^{- 0.27}$  in the limit $N\\to\\infty$ \\cite{DLLPT}. According to the argument proposed in \\cite{DLLPT},  the value of the power-law exponent is associated to the\n scaling of the  number of unlocked neurons, $N_u$  with the system size $N$, namely $N_u \\sim N^{0.9}$.\nThe same argument applied to HMF dynamics indicates that the exponent \n$\\gamma \\sim 1\/2$, ruling the vanishing of $\\lambda_{\\mathrm{max}} (M)$ in the limit $M\\to\\infty$,\nstems from the fact that the HMF dynamics keeps the fraction of unlocked neurons constant.\n\n\\vskip 30pt\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{lmnew.eps}\n\\caption{(Color online) The maximum Lyapunov exponent $\\lambda_{\\mathrm{max}}$ \nas a function of the sampling parameter $M$: $\\lambda_{\\mathrm{max}}$ has\nbeen averaged also over ten different realizations of the network (the error bars refer to the maximum deviation from the average). The dashed red line is the powerlaw  $M^{-\\gamma}$, with $\\gamma=0.55$. }\n\\label{lyup_max}\n\\end{figure}\n\n\nWhen the distribution $P(\\tilde k)$ is sufficiently broad, the system becomes asynchronous and locked neurons disappear. The global field  $Y(t)$ exhibits  fluctuations due to finite size effects and in the thermodynamic limit \nit tends to a constant value  $Y^*$. From Eq.s (\\ref{vk})--(\\ref{zk}), one obtains that in this regime each neuron\nwith in--degree $\\tilde k$ fires  periodically with a period \n $$\n T_{\\tilde k}=\\mathrm{ln}\\Bigg[\\frac{b+g\\tilde k Y^*}{b+g\\tilde k Y^* -1}      \\Bigg]~,\n $$\nwhile its  phase depends  on the initial conditions. In this case all the Lyapunov exponents \n$\\lambda_{\\tilde k}$ are negative.\n\n\n\\section{Topology and collective behavior}\n\\label{sec5}\nFor a  given in--degree probability distribution\n$P(\\tilde k)$, the fraction of locked neurons (i.e.,  $f_{\\mathrm{l}}=\\int_{\\tilde k_1 }^{\\tilde k_2} P(\\tilde k)d\\tilde k$)  decreases \nby increasing $\\sigma_{\\tilde k}$ \\cite{BCDLV}. In particular, there is a critical value  $\\sigma^*$ at which $f_{\\mathrm{l}}$ vanishes.\nThis signals a very interesting dynamical transition between the quasi-synchronous phase ($\\sigma_{\\tilde k} < \\sigma^*$)\nto a multi-periodic phase ($\\sigma_{\\tilde k} > \\sigma^*$), where all neurons are periodic with different periods. Here we focus on the different  collective dynamics that may emerge for choices of  $P(\\tilde k)$\nother than the Gaussian case, discussed in the previous section.\n \nFirst, we consider a power--law distribution\n\\begin{equation}\nP(\\tilde k)=A\\tilde k^{-\\alpha},\n\\label{eqplaw}\n\\end{equation}\nwhere  the constant $A$ is given  by the normalization condition $\\int_{\\tilde k_m}^{1}P(\\tilde k)d\\tilde k=1$. \nThe lower bound $\\tilde k_m$ is introduced  in order to maintain $A$ finite. \nFor simplicity, we fix  the parameter $\\tilde k_m$ and analyze the dynamics by varying $\\alpha$. \nNotice that the standard deviation  $\\sigma_{\\tilde k}$ of  distribution (\\ref{eqplaw})  decreases  for increasing\nvalues of  $\\alpha$.  The dynamics for relatively high $\\alpha$  is very similar to the quasi--synchronous regime \nobserved for $\\sigma_{\\tilde k} < \\sigma^*$  in the Gaussian case (see Fig. \\ref{rp1}). \nBy decreasing $\\alpha$ one can observe again a transition to  the  asynchronous phase observed \nfor  $\\sigma_{\\tilde k} > \\sigma^*$ in the Gaussian case.  Accordingly, also for the power--law distribution (\\ref{eqplaw})\na phase with locked neurons may set in only when there is  a sufficiently large group of neurons sharing  close values of $\\tilde k$.\nIn fact, the group of locked neurons is concentrated at values of $\\tilde k$ quite close to the lower bound  \n $\\tilde k_m$,  while in the Gaussian case they concentrate at values smaller than  $\\langle \\tilde k\\rangle $.\n \nAnother distribution, generating an  interesting dynamical phase, is\n\\begin{equation}\nP(\\tilde k)=B \\mathrm{exp}\\Bigg(-\\frac{(\\tilde k-p_1)^2}{2\\sigma_s^2}\\Bigg) +    B\\mathrm{exp}\\Bigg(-\\frac{(\\tilde k-p_2)^2}{2\\sigma_s^2}\\Bigg)   ,\n\\label{dgauss}\n\\end{equation}\ni.e. the sum of two Gaussians  peaked around different values, $p_1$ and $p_2$, of $\\tilde k$, \nwith the same variance  $\\sigma_s^2$.  $B$ is the  normalization constant such that $\\int_{0}^{1}P(\\tilde k)=1$.\nWe fix $p_1=0.5$ and vary both the variance,\n$\\sigma_s$, and the distance between the peaks, $\\Delta= |p_2-p_1|$.\n \nIf  $\\sigma_s$ is very large ($\\sigma \\gtrsim 0.1$), the situation is the same observed for a single Gaussian with large variance, yielding  \na multi--periodic asynchronous dynamical phase.\n\nFor intermediate values of  $\\sigma_s$ i.e. $0.05\\lesssim\\sigma\n\\lesssim 0.1$, the dynamics of the network can exhibit \na quasi--synchronous phase or  a multi--periodic asynchronous phase, depending on the value of $\\Delta$.\nIn fact, one can easily realize that this parameter tunes the standard deviation of the overall distribution: small separations\namount to broad distributions. \n \nFinally, when $\\sigma_s\\lesssim 0.05$, a new dynamical phase appears.  \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{isikerp.eps}\n\\caption{ The time average of the inter--spike interval \n$\\overline{ ISI_{\\tilde k}}$ vs. $\\tilde k$ for the probability distribution\n$P(\\tilde k)$ defined in Eq.(\\ref{dgauss}), with  $\\Delta= |p_2- p_1|= 0.4$, and $\\sigma_{\\mathrm{s}}=0.03$. We have obtained the global field $Y(t)$ simulating the HMF dynamics with a discretization with $M=300$ classes of neurons. We have then used $Y(t)$ to calculate the $ISI$ of neurons evolving Eq. (\\ref{vk}). In the inset we show the raster plot of the dynamics: as\nin Fig.1, neurons are ordered along the vertical axis according to their \nin--degree.\n}\n\\label{rpdue}\n\\end{figure} \nFor small values of $\\Delta$ (e.g. $\\Delta \\approx 0.1$) , we observe the usual QSE scenario with one family of locked neurons\n(data not shown). However, when $\\Delta$ is sufficiently large\n(e.g. $\\Delta \\approx  0.4$), each peak of the distribution generates its own group of locked neurons.  More precisely, neurons separate into three different sets: \ntwo locked groups, that evolve with different periods, $T_1$ and  $T_2$, and the unlocked group. In Fig.\\ref{rpdue} we show the dependence of $\\overline{ ISI_{\\tilde k} }$ on $\\tilde k$  and the raster plot of the dynamics (see the inset)\nfor $\\sigma_s=0.03$ . Notice that  the plateaus of locked neurons extend over values\nof $\\tilde k$ on the left of  $p_1$ and $p_2$.   \nIn the inset of Fig. \\ref{fourier} we plot  the global activity field $Y(t)$:  the peaks signal the quasi-synchronous firing events  of the two groups of locked neurons. One can\nalso observe that very long oscillations are present over  a time scale much larger than $T_1$ and $T_2$. They are the effect of the {\\sl firing synchrony} of the of two locked families.  In fact, the two frequencies $\\omega_1=2\\pi\/T_1$ and $\\omega_2=2\\pi\/T_2$ are in general not commensurate, and the resulting global field is a quasi--periodic function. \nThis can be better appreciated by looking at Fig.\\ref{fourier},  where we report the frequency spectrum of the signal $Y(t)$ (red curve). We observe peaks at frequencies $\\omega=n\\omega_1+m\\omega_2$, for integer values of  $n$ and $m$. For comparison, we report also the spectrum of a periodic $Y(t)$, generated by the HMF with power law probability distribution \n(\\ref{eqplaw}), with  $\\alpha = 4.9$ (black curve):  in this case the peaks are  located at frequencies multiples of the frequency of the locked group of neurons.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{fourierecampo.eps}\n\\caption{The frequency spectra of the global activity field $Y(t)$ for\ndifferent in--degree probability distributions. The \nblack spectrum has been obtained for the HMF dynamics with $M=350$, generated \nby the power law probability distribution $P(\\tilde k)\\sim\\tilde k^{-4.9}$\n(see Eq.(\\ref{eqplaw})),\nwith $\\tilde k_m=0.1$: in this case there is a unique family of locked neurons\ngenerating a periodic global activity field $Y(t)$.\nThe red spectrum has been obtained for a random network of $N = 300$ neurons\ngenerated by the double Gaussian distribution\n(see Eq.(\\ref{dgauss})) described in Fig.s 6 and 7: in this case two families\n of locked neurons are present while, as reported in the inset, $Y(t)$ exhibits a quasi--periodic \nevolution.}\n\\label{fourier}\n\\end{figure} \nOn the basis of this analysis, we can conclude that slow oscillations of the global activity field $Y(t)$ may signal the presence \nof more than one group of topologically homogeneous (i.e. locked) neurons. Moreover, we have also learnt\n that one can generate a large variety of global synaptic activity fields by selecting suitable\nin-degree distributions $P(\\tilde k)$, thus unveiling unexpected perspectives for exploiting a sort of  {\\sl  topological engineering} of the \nneural signals. For instance, one could investigate which kind of $P(\\tilde k)$ could give rise to an almost resonant dynamics,\nwhere  $\\omega_2$ is close to a multiple of  $\\omega_1$. \n\n\\section{HMF and the Inverse problem in presence of noise}\n\\label{sec6}\nThe HMF  formulation allows  one to define and solve the following\nglobal inverse problem: how to recover  the \nin--degree distribution $P(\\tilde k)$ from the knowledge of  the global synaptic activity field  $Y(t)$ \\cite{BCDLV}.  \n\nHere we just sketch the basic steps of the procedure.\nGiven $Y(t)$, each class of neurons of in-degree $\\tilde k$ evolves according to the HMF equations:\n\\begin{align}\n\\label{vktil}\n&\\dot {\\mathcal{ V}}_{ \\tilde k}(t)= a -\\mathcal{ V}_{ \\tilde k}(t) + g \\tilde kY(t)\\\\\n\\label{yktil}\n& \\dot {\\mathcal{ Y}}_{ \\tilde k}(t) = -\\frac{\\mathcal{Y}_{\\tilde k}(t)}{\\tau_{\\mathsf{in}}} +u(1-\\mathcal{ Y}_{ \\tilde{k}}(t)-\\mathcal{ Z}_{ \\tilde{k}}(t))\\tilde S_{ \\tilde{k}}(t)\\\\\n\\label{zktil}\n& \\dot {\\mathcal{ Z}}_{ \\tilde k}(t) = \\frac{\\mathcal{ Y}_{ \\tilde k}(t)}{\\tau_{\\mathsf{in}}}  -   \\frac{\\mathcal{ Z}_{ \\tilde{k}}(t)}{\\tau_{\\mathsf{r}}} \\,\\,\\, .\n\\end{align}\nThe different fonts used here, with respect to \nEq.s   (\\ref{vk})--(\\ref{meanfield}), point out that\nin this framework the choice of the initial conditions is arbitrary and the dynamical variables $\\mathcal{V}(t)$, $\\mathcal{Y}(t)$, $\\mathcal{Z}(t)$ \nin general may take different values  from those assumed by  $v(t)$,\n$y(t)$, $z(t)$, i.e. the variables   generating $Y(t)$ in (\\ref{vk})--(\\ref{meanfield}).  \nHowever,  one can exploit the self consistent relation for the global field $Y(t)$:\n\\begin{equation}\\label{global}\nY(t)=\\int_0^1 P({\\tilde k}) \\mathcal{Y}_{\\tilde k}(t)d{\\tilde k} \\,\\,\\, .\n\\end{equation}\nIf $Y(t)$ and $\\mathcal{Y}_{\\tilde k}(t)$ are known, this is a Fredholm\nequation of the first kind for the unknown $P(\\tilde k)$ \\cite{kress}.\nIf $Y(t)$ is a periodic signal,  Eq. (\\ref{global}) can be easily solved by a functional Montecarlo minimization\nprocedure, yielding a faithful reconstruction of  $P(\\tilde k)$\n\\cite{BCDLV}.\nThis method applies successfully also  when $Y(t)$ is a quasi-periodic signal,  like the\none generated by in--degree distribution (\\ref{dgauss}).\n\n \nIn this section we want to study the robustness of the HMF equations and \nof the corresponding inverse problem procedure in the presence of noise. \nThis is quite an important test for the reliability of the overall HMF approach. \nIn fact, a real neural structure is always affected by some level of noise, that,\nfor instance, may emerge in the form of fluctuations of ionic or synaptic currents.  \nMoreover, it has been observed that noise is crucial for reproducing  dynamical phases,\nthat exhibit  some peculiar synchronization patterns observed in  {\\it in vitro}  experiments \\cite{volman,DL}.\n\nFor the sake of simplicity, here we introduce noise by turning the\nexternal current  $a$, in Eq. (\\ref{vk}), from a  constant to a time  and neuron dependent stochastic processes\n$a_{\\tilde k}(t)$. Precisely, the $a_{\\tilde k}(t)$ are assumed to be \ni.i.d.  stochastic variables, that evolve in time as a random walk with boundaries, \n$a_{\\mathrm{min}}$ and $a_{\\mathrm{max}}$ (the same rule adopted in \\cite{DL}).\nAccordingly, the average value, $\\bar a$ of  $a_{\\tilde k}(t)$ is given by the expression $\\bar\na=(a_{\\mathrm{min}}+a_{\\mathrm{max}})\/2$, while the amplitude of fluctuations is\n$\\delta = a_{\\mathrm{max}}-a_{\\mathrm{min}}$.\nAt each step of the walk, the values of $a_{\\tilde k}(t)$ are independently updated by adding or subtracting, with equal\nprobability, a fixed increment $\\Delta a$. Whenever the value of $a_{\\tilde k}(t)$ crosses one of the boundaries, it is reset to the boundary value.\n\nSince the dynamics has lost its deterministic character,  its numerical integration cannot exploit an event driven algorithm, and \none has to integrate Eq.s (\\ref{vk}) --(\\ref{zk}) by a scheme based on\nexplicit time discretization. The results reported hereafter \nrefer to an integration time step  $\\Delta t=9\\cdot 10^{-4}$, that guarantees an effective sampling of the dynamics over the whole\nrange of parameter values that we have explored. We have assumed that $\\Delta t$  is also the time step of the stochastic evolution of $a_{\\tilde k}(t)$.\n\nHere we consider the case of uncorrelated noise, that can be obtained by a suitable choice of $\\Delta a$\n\\cite{DL}. In our simulations $\\Delta a = 10^{-2}$, that yields \na value $\\mathcal{O}(10^{-2})$ of the  correlation time of the random walk with boundaries. This value, \nmuch smaller than the value  $\\mathcal{O}(1)$ typical of the  ISI of neurons, makes the stochastic evolution\nof the external currents, $a_{\\tilde k}(t)$, an effectively  uncorrelated process with respect to the typical time scales of the neural\ndynamics.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{campi_rum.eps}\n\\caption{ The global activity field $Y(t)$ of the HMF dynamics, sampled by\n$M=4525$ classes of neurons, for a gaussian\nprobability distribution $P(\\tilde k)$, with $\\langle \\tilde k\\rangle=0.7$ \nand $\\sigma_{\\tilde k}=0.0455$. Lines of different colors correspond to\ndifferent values of the noise amplitude, $\\delta$, added to the external \ncurrents $a_{\\tilde k}(t)$: $\\delta = 0$ (black line), $\\delta = 0.1$ \n(red line), $\\delta = 0.15$ (green line), $\\delta = 0.2$ (blue line) and\n$\\delta = 0.3$ (orange line).}\n\\label{camponoise}\n\\end{figure} \nIn Fig. \\ref{camponoise} we show  $Y(t)$, produced by the discretized HMF dynamics with \n$M=4525$ and for a Gaussian distribution $P(\\tilde k)$, with  $\\langle \\tilde k \\rangle=0.7$ and $\\sigma_{\\tilde k}=0.0455$. \nCurves of different colors correspond to different values of $\\delta$. \n We have found that up to  $\\delta\\simeq 0.1$, i.e. also for non negligible noise\namplitudes ($\\bar a =1$),  the HMF dynamics is practically unaffected by noise.\nBy further increasing $\\delta$, the amplitude of $Y(t)$   decreases, as a result\nof the desynchronization of the network induced by large amplitude noise.\n\n\nAlso the inversion procedure exhibits the same robustness\nwith respect to noise. As a crucial test, we have solved the inverse problem \nto recover $P(\\tilde k)$ by injecting the noisy signal $Y(t)$  in the noiseless equations \n(\\ref{vktil})--(\\ref{zktil}), where $a = \\bar a$  (see  Fig.\\ref{camponoise}). \nThe reconstructed distributions $P(\\tilde k)$, for different $\\delta$, are shown in  Fig. \\ref{noise_invert_1}.\nFor relatively small noise amplitudes ($\\delta< 0.1$) the recovered form of $P(\\tilde k)$ is quite\nclose to the original one, as expected because the noisy $Y(t)$ does\nnot differ significantly from the noiseless one.\n On the contrary, for relatively large noise amplitudes\n ($\\delta>0.1$),  the recovered  distribution\n $P(\\tilde k)$  is broader than the  original one and centered around\n a  shifted average value $\\langle \\tilde k \\rangle$.\nThe dynamics exhibits much weaker synchrony effects, the same \nindeed one could observe for the noiseless dynamics on the\nlattice built up  with this broader  $P(\\tilde k)$ given by the inversion method.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{inv_rum1.eps}\n\\caption{\nSolution of the inverse problem by the HMF equations in the presence\nof noise added to the external currents. We consider the same setup of Fig. 9 \nand we compare, for different\nvalues of the noise amplitude $\\delta$, the \nreconstructed probability distribution $P(\\tilde k)$ (red circles) with the\noriginal gaussian distribution (black line): the upper--left panel \ncorresponds to the\nnoiseless case ($\\delta =0$), while the upper--right, the lower--left and \nand the lower--right correspond to $\\delta = 0.1, 0.2, 0.3$, respectively.\n}\n\\label{noise_invert_1}\n\\end{figure}\n\nAs a matter of fact, the global neural activity fields obtained by  experimental\nmeasurements are unavoidably affected by some level of noise. \nAccordingly, it is  worth investigating the robustness of the inversion\nmethod also in the case of noise acting directly on $Y(t)$. \nIn order to tackle this problem, we have considered a simple\nnoisy version of the global synaptic activity field,  defined as\n$Y_{\\delta} (t) =(1+\\eta(t))Y(t) $, where the random number $\\eta(t)$  is\nuniformly extracted, at each integration time step,  in the interval $[-\\frac{\\delta}{2},\\frac{\\delta}{2}]$.\nIn Fig. \\ref{noise_invert_2} we show \nthe distributions $P(\\tilde k)$ obtained for different values of $\\delta$. \nWe can conclude  that the inversion method is quite stable with respect to this  \nadditive noise. In fact, even for very large signal--to--noise ratio (e.g. low--right panel of Fig. \\ref{noise_invert_2},\nwhere $\\delta = 0.8$) the main features of the original distribution are still recovered,\nwithin a reasonable approximation.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.35]{rumsuy.eps}\n\\caption{\nSolution of the inverse problem by the HMF equations in the presence\nof noise added to the activity field. We consider the same setup of \nFig. 9, where now $a = 1$ and $Y_{\\delta}(t) = (1 - \\eta(t))Y(t)$ (the random\nvariable $\\eta(t)$ is extracted from a uniform probability distribution \nin the interval $[-\\delta\/2, \\delta\/2]$).\nWe compare, for different\nvalues of the noise amplitude $\\delta$, the \nreconstructed probability distribution $P(\\tilde k)$ (red circles) with the\noriginal gaussian distribution (black line): the upper--left, \nthe upper--right, the lower--left and \nand the lower--right panels correspond to $\\delta = 0.1, 0.4, 0.8, 1.2$, \nrespectively.\n}\n\\label{noise_invert_2}\n\\end{figure} \n\n\n\n\\section{HMF in sparse networks}\n\\label{sec7}\n\nIn this section we analyze the effectiveness of the HMF\napproach for sparse networks, i.e. networks where the neurons\ndegree does not scale linearly with $N$ and, in particular, the average degree\n$\\langle k \\rangle$ is independent of the system size. \nIn this context, the coupling term describing\nthe membrane potential of a generic neuron $i$, in a network of $N$ neurons, \nevolves according to the following equation:\n\\begin{equation}\n\\label{vsparsa}\n\\dot v_j= a -v_j + \\frac{g}{\\langle k\\rangle} \\sum_{i \\ne j} \\epsilon_{ij} y_i, \\, \n\\end{equation} \nwhile the dynamics of $y_i$ is the same of Eq.s\n(\\ref{dynsyn})--(\\ref{contz}). The coupling therm is now independent of\n$N$, and the normalization factor, \n$\\langle k\\rangle$, has been introduced in order to compare models with different average connectivity. \nThe  structure of the adjacency matrix $\\epsilon_{ij}$ is determined \nby choosing for each neuron $i$ its  in-degree $k_i$  from a probability distribution $P(k_i)$\n(with support over positive integers)      independent of the system size.\n\n\n\\begin{figure}[htbp]\n\\centering\\\n\\includegraphics[scale=0.35]{sparsa_mf.eps}\n\\vskip 30pt\n\\includegraphics[scale=0.35]{scale_confr.eps}\n\\caption{\nComparison of the global synaptic activity field $Y(t)$ from \nsparse random networks with the same quantity generated by the corresponding\nHMF dynamics. We have considered sparse random networks with \n$N= 10^4$ neurons. In the upper panel  we consider  a\nGaussian probability distributions $P(k)$ with different averages\n$\\langle k \\rangle$ and \nvariances $\\sigma_k$, such that $ \\sigma_k\/\\langle k \\rangle = 0.06$:\n$\\langle k \\rangle = 10, 20, 60, 100$ correspond to the violet, orange, red and blue\nlines, respectively. The black line represents $Y(t)$ from the HMF dynamics\n($M=10^3$), where $\\hat P(\\hat k)$ is a Gaussian probability distribution\nwith $\\langle \\hat k \\rangle = 1$ and $\\sigma_{\\hat k} = \\sigma_k\/\\langle k \n\\rangle = 0.06$.  In the lower panel  we consider  \nthe scale free case with fixed power exponent $\\alpha$ and different $k_m$:\n$k_m = 10, 30, 70$ correspond to the orange, red and blue\nlines, respectively. The black line represents $Y(t)$ from the HMF dynamics\n($M=10^3$), where $\\hat P(\\hat k)=(\\alpha-1)\\hat{k}^{-\\alpha} $\nwith cutoff $\\hat k_m = 1$.}\n\\label{campi_sparsa}\n\\end{figure}\n\n\nOn sparse networks the HMF model is not recovered in the thermodynamic limit, \nas the fluctuations of the field received by each neuron of in--degree $k_i$ \ndo not  vanish for  $N\\to \\infty$.\nNevertheless, for large enough values of $k_i$, one can expect that the\nfluctuations become negligible in such a limit,\ni.e. the synaptic activity field received by different neurons with the same \nin-degree is approximately the same. \nEq. (\\ref{vsparsa})  can be turned into a mean--field like\nform as follows\n\\begin{equation}\n\\label{vsparsa_mean}\n\\dot v_j= a -v_j + \\frac{g}{\\langle k\\rangle} k_jY ~,  \n\\end{equation} \nwhere $Y(t)$ represents the global field, averaged over all neurons in the\nnetwork. This implies that the equation  is the same for all neurons with \nin--degree $k_j$, depending only on the ratio $\\hat{k}_j=k_j\/\\langle\nk\\rangle$.  \nConsequently, also in this case one can read Eq. (\\ref{vsparsa_mean}) as a HMF formulation of Eq. (\\ref{vsparsa}), \nwhere each class of  neurons $\\hat{k}$ evolves according to\nto Eq.s (\\ref{vk})--(\\ref{zk}), with $\\hat{k}$ replacing $\\tilde k$, while the global activity field is given by\nthe relation \n$Y(t)=\\int_0^{\\infty}\\hat{P}(\\hat{k})y_{\\hat{k}}(t)d\\hat{k}$.  \n\nIn order to analyze the validity of the HMF as an approximation of models defined \non sparse networks, we consider two main cases: ({\\sl i})   $\\hat{P}(\\hat{k})$ is a truncated Gaussian\nwith average $ \\langle \\hat{k}\\rangle=1$ \nand standard deviation $\\sigma_{\\hat{k}}$; ({\\sl ii}) $\\hat{P}(\\hat{k})=(\\alpha-1)\\hat{k}^{-\\alpha}$ is a \npower--law (i.e., scale free) distribution with a lower cutoff $\\hat{k}_m=1$.\nThe Gaussian case ({\\sl i}) is  an approximation of\nany sparse model, where $P(k_j)$ is a discretized Gaussian distribution \nwith parameters  $\\langle k\\rangle$ and $\\sigma_k$, chosen in such a way that\n$\\sigma_{\\hat{k}} = \\sigma_k\/\\langle k\\rangle$. The scale free case ({\\sl ii}) approximates\nany sparse model, where $P(k_j)$ is a power law with exponent $\\alpha$ and a generic cutoff.  Such an\napproximation is expected to provide better results the larger is $\\langle k \\rangle$, i.e.  \nthe larger is the cutoff $k_m$ of the scale free distribution.\nIn Fig. \\ref{campi_sparsa} we plot the global field emerging from \nthe HMF model, superposing those coming from a large finite size realization \nof the sparse network, with different values of $\\langle k\\rangle$ for the Gaussian case (upper panel) and of $k_m$ for the scale free case (lower panel).\nThe HMF equations exhibit a remarkable agreement with models on sparse \nnetwork,  even for relatively small values of $ \\langle k\\rangle$ and $k_m$. \nThis analysis indicates  that the HMF approach works also for\nnon--massive topologies, provided the typical connectivities\nin the network are large enough, e.g. $\\langle k\\rangle \\sim {\\mathcal O} (10^2)$\nin a Gaussian random network with $N=10^4$ neurons (see Fig. (\\ref{campi_sparsa})).\n \n \n \n\\section{Conclusions}\n\\label{sec8}\n\n\nFor systems with a very large number of components, the effectiveness\nof a statistical approach, paying the price of some necessary approximation, has been\nextensively proven, and  mean--field methods are typical in this sense. \nIn this paper we discuss how such a method, in the specific form of\nHeterogeneous Mean--Field, can be defined in order to fit an \neffective description of neural dynamics on random networks. \n\nThe relative simplicity of the model studied here, \nexcitatory  leaky--integrate--and fire neurons with short term\nsynaptic plasticity, is also a way of providing a pedagogical\ndescription of the HMF  and of its potential interest in similar contexts \\cite{BCDLV}.  \n\nWe have reported a detailed study of the HMF approach including\ninvestigations on \n{\\sl (i)} its stability properties,  \n{\\sl (ii)} its effectiveness in describing the dynamics and in solving\nthe associated inverse problem for different network topologies,\n {\\sl (iii)} its robustness with respect to noise, \nand {\\sl (iv)} its adaptability to different formulations of the model\nat hand. In the light of {\\sl (ii)}  and  {\\sl (iii)}, the HMF approach\nappears quite a promising tool to match\nexperimental situations, such as the identification of topological  features\nof real neural structures, through the inverse analysis of signals\nextracted as  time series  from small, but not microscopic, domains.\nOn a mathematical  ground, the HMF approach is a simple and effective\nmean--field formulation, that can be extended to other neural network models\nand also to a wider class of dynamical models on random graphs.\nThe first step in this direction could be the extension of  the HMF method\nto the more interesting case, where the random network contains \nexcitatory and inhibitory neurons, according to distributions of interest\nfor neurophysiology \\cite{abeles, bonif} . This will be the subject of our future work.\n\n\n\n\\begin{acknowledgments}\nR.L. acknowledges useful discussions\nwith A. Pikovsky and L. Bunimovich.\n\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn $(n,k,\\lambda)-$\\textit{design} (or $(n,k,\\lambda)$-BIBD) is a pair $(P, \\mathcal{B})$ where $P$ is a  finite set of $n$ \\textit{points} and $\\mathcal{B}$ is a collection of $k-$subsets of $P$, called  \\textit {blocks},  such that every two distinct points in $P$  is contained in exactly $\\lambda$ blocks. \nIn case $|P|={|\\cal B|}$, it is called a \\textit{symmetric design}. For positive integer $ q $,  \na $(q^2+q+1, q+1, 1)$-BIBD and a $(q^2,q,1)$-BIBD are called a \\textit{projective plane} and an \\textit{affine plane} of order $q$, respectively.\nA design is called \\textit{resolvable}, if there exists a  partition  of the set of blocks\n$\\mathcal{B}$ into \\textit{parallel classes}, each of which is a partition of $P$.\n\nA \\textit{pairwise balanced design} (PBD) is a pair $(P,\\mathcal{B})$, where $P$ is a finite set of $n$ points and $\\mathcal{B}$ is  a family of subsets of $P$, called \\textit {blocks}, such that every two distinct points in $P$, appear in exactly one block. A \\textit{nontrivial} PBD is a PBD where $P\\not \\in \\mathcal{B}$. A PBD $(P,\\mathcal{B})$ on $n$ points with one block of size $n-1$ and the others of size two is called \\textit{near-pencil}.\n\nThe problem of determining the minimum number of blocks in a pairwise balanced design when the size of its largest block is specified or the size of a particular block is specified, has been the subject of many researches in recent decades.\nThe most important and well-known result about this problem is due to de Bruijn and Erd\\H{o}s~\\cite{deBruijn48} which states that every nontrivial PBD on $n$ points has at least $n$ blocks and the only nontrivial PBDs on $n$ points with exactly $n$ blocks  are near-pencil and projective plane.\nFor every positive integers $n,m$, where $m\\leq n$, let $\\mathscr{G}(n,m)$ be the minimum number of blocks in a PBD on $n$ points whose largest block has size $m$. Also, let  $\\mathscr{G}'(n,m)$ be the minimum number of blocks in a PBD on $n$ points which has a block of size $m$.\nA classical result known as Stanton-Kalbfleisch Bound \\cite{stanton70} states that $\\mathscr{G}'(n,m)\\geq 1+(m^2(n-m))\/(n-1)$ and equality holds if and only if there exists a resolvable $(n-m,(n-1)\/m,1)$- BIBD. Also, a corollary of Stanton-Kalbfleisch is that $\\mathscr{G}(n,m)\\geq \\max\\{n(n-1)\/m(m-1), 1+(m^2(n-m))\/(n-1)\\}$. For a survey on these and more bounds, see \\cite{Rees90,stanton97}.\n\n\nIn this paper, we are interested in minimizing the  sum of block sizes in a PBD, where there are some constraints on the size of one block or the size of the largest block. For every positive integers $n,m$, where $m\\leq n$, let $\\mathscr{S}(n,m)$ be the smallest integer $s$ for which there exists a PBD on $n$ points whose largest block has size $m$ and  the sum of its block sizes is equal to $s$. Also, let  $\\mathscr{S}'(n,m)$ be the smallest integer $s$ for which there exists a PBD on $n$ points which has a block of size $m$ and the  sum of it block sizes is equal to $s$. \n In Section~\\ref{sec:pbd}, we prove some lower bounds for $\\mathscr{S}(n,m)$ and $\\mathscr{S}'(n,m)$. In particular, we show that $\\mathscr{S}(n,m)\\geq 3n-3$, for every $m$, $2\\leq m\\leq n-1$. Also, we prove that, for every $2\\leq m\\leq n $,\n \\[\\mathscr{S}'(n,m)\\geq \\max\\left\\{(n+1)m-\\frac{m^2(m-1)}{n-1}, m+\\frac{(n-m)(n-5m-1)}{2}\\right\\},\\]\n where equality holds for $ m\\geq n\/2 $. Furthermore, we prove that if $ n\\geq 10 $ and $2\\leq m\\leq n-\\frac{1}{2} ( \\sqrt{n} +1)$, then\n $\\mathscr{S}(n,m)\\geq n(\\lfloor\\sqrt{n}\\rfloor+1)-1$.\n\nThe connection of pairwise balanced designs and clique partition of graphs is already known in the literature. Given a simple graph $ G $, by a \\textit{clique} in $G$ we mean a subset of mutually adjacent vertices. A \\textit{clique partition } $\\mathcal{C}$ of $G$ is a family of cliques in $G$ such that the endpoints of every edge of $G$ lie in exactly one member  of $\\mathcal{C}$. The minimum size of a clique partition of $G$ is called the  \\textit{clique partition number} of $G$ and is denoted by $\\cp(G)$.  \n\nFor every graph $G$ with $ n $ vertices, the union of a clique partition of $G$ and a clique partition of its complement, $\\overline{G}$, form a PBD on $n$ points. This connection has been deployed to estimate $\\cp(G)$, when $G$ is some special graph such as $K_n-K_m$~\\cite{WallisAsymp,erdos88,pullman82,Rees}, Cocktail party graphs and complement of paths and cycles \\cite{Wallis82,Wallis84,Wallis87}. \n\nOur motivation for study of the above mentioned problem is a weighted version of clique partition number. The \\textit{sigma clique partition number} of a graph $G$, denoted by $ \\scp(G) $, is defined as the smallest integer $s$ for which there exists a clique partition of $G$ where  the sum of  the sizes of its cliques is equal to $s$. It is shown that for every graph $G$ on $n$ vertices, $\\scp(G)\\leq \\lfloor n^2\/2\\rfloor$, in which equality holds if and only if $ G $ is the complete bipartite graph $K_{\\lfloor n\/2\\rfloor, \\lceil n\/2\\rceil } $  \\cite{chung81,kahn81,gyori80}. \n\n\nGiven a clique partition $\\mathcal{C}$  of a graph $G$, for every vertex $x\\in V(G)$, the \\textit{valency} of $x$ (with respect to $\\mathcal{C}$), denoted by $v_\\mathcal{C}(x)$, is defined to be the number of cliques in $\\mathcal{C}$ containing $x$. In fact,\n\\[\\scp(G)=\\min_{\\mathcal{C}} \\sum_{C\\in \\mathcal{C}}|C|= \\min_{\\mathcal{C}} \\sum_{x\\in V(G)} v_{\\mathcal{C}}(x),\\]\nwhere the minimum is taken over all possible clique partitions of $G$.\n\n\n In Section~\\ref{sec:Kn-Km}, we apply the results of Section~\\ref{sec:pbd} to determine the asymptotic behaviour of the sigma clique partition number of the graph $K_n-K_m$. In fact, we prove that if $m\\leq {\\sqrt{n}}\/{2}$, then $\\scp(K_n-K_m)\\sim (2m-1)n$, if ${\\sqrt{n}}\/{2}\\leq m\\leq \\sqrt{n}$, then $\\scp(K_n-K_m)\\sim n\\sqrt{n}$ and if $m\\geq \\sqrt{n}$ and $m=o(n)$, then $\\scp(K_n-K_m)\\sim mn$. Also, if $G$ is Cocktail party graph, complement of path or cycle on $n$ vertices, then we prove that $\\scp(G)\\sim n\\sqrt{n}$.\n\n\n\n\\section{Pairwise balanced designs}\\label{sec:pbd}\n\nA celebrated result of de Bruijn and Erd\\H{o}s states that for every nontrivial PBD  $(P,\\mathcal{B})$, we have $|\\mathcal{B}|\\geq |P|$ and equality holds if and only if  $(P,\\mathcal{B})$ is near-pencil or projective plane~\\cite{deBruijn48}. In this section, we are going to answer the question that what is the minimum sum of block sizes in a PBD.\n\n The following theorem can be viewed as a de Bruijn-Erd\\H{o}s-type bound, which shows that $\\mathscr{S}(n,m)\\geq 3n-3$, for every $m$, $2\\leq m\\leq n-1$.\n\n\n\\begin{theorem}\\label{thm:PBDsigma}\nLet $(P,\\mathcal{B})$ be a nontrivial PBD with $n$ points, then we have\n\\begin{equation}\\label{eq:pbd}\n\\sum_{B\\in\\mathcal{B}} |B| \\geq 3n-3,\n\\end{equation}\nand equality holds if and only if $(P,\\mathcal{B})$ is near-pencil.\n\\end{theorem}\n\\begin{proof}\nWe use induction on the number of points. Let $(P,\\mathcal{B})$ be a nontrivial PBD with $n$ points. Inequality~\\eqref{eq:pbd} clearly holds when $n=3$. So assume that $n\\geq 4$ and for every $x\\in P$, let $r_x$ be the number of blocks containing $x$. First note that for every block $B\\in\\mathcal{B}$ and every $x\\in P\\setminus B$, we have $r_x\\geq |B|$.  \n\nIf there is a block $B_0\\in \\mathcal{B}$ of size $n-1$ and $x_0$ is the unique point in \n$P\\setminus B_0$, then for every $x\\in B_0$, $x$ and $x_0$ appear within a block of size two. Therefore,  $(P,\\mathcal{B})$ is near-pencil and $\\sum_{B\\in\\mathcal{B}}|B|=(n-1)+2(n-1)=3n-3$. \n\nOtherwise,  all blocks are of size at most $n-2$. First we prove that there exists some point $x\\in P$ with $r_x\\geq 3$. Since there is no block of size $n$, $r_x\\geq 2$ for all $x\\in P$. Now for some $y\\in P$, assume that $B_1, B_2$ are the only two blocks containing $y$. Since $n\\geq 4$, the size of at least one of these blocks, say $B_1$, is greater than two. Let $x\\neq y$ be an element of $B_2$. Then,  $r_x\\geq |B_1|\\geq 3$. \nHence, there exists some point $x\\in P$ which appears in at least three blocks.\n\nNow, remove $x$ from all blocks to obtain the nontrivial PBD $(P',\\mathcal{B'})$, where $P'=P\\setminus \\{x\\}$ and $\\mathcal{B'}=\\{B\\setminus\\{x\\}\\ :\\  B\\in\\mathcal{B}\\}$. Therefore, \n\\begin{equation}\\label{eq:case2}\n\\sum_{B\\in \\mathcal{B}}|B| = r_x+\\sum_{B'\\in\\mathcal{B'}}|B'|\\geq 3+3(n-2),\n\\end{equation}\nwhere the last inequality follows from the induction hypothesis.\n\nNow, assume that for a  PBD $(P,\\mathcal{B})$ equality holds in \\eqref{eq:pbd}. If $(P,\\mathcal{B})$ is not a near-pencil, then equality holds in \\eqref{eq:case2} as well and thus  we have $2\\leq r_x\\leq 3$, for every $x\\in P$. On the other hand, $\\sum_{B\\in\\mathcal{B}}|B|=\\sum_{x\\in P} r_x=3n-3$. Therefore, there are exactly $3$ points, say $x,y,z$, each of which appears in exactly two blocks and each of the other points appears in exactly three blocks. Also, let $B_1, B_2$ be the only two blocks containing $y$ and assume that $x\\in B_1$. Therefore, $2=r_x\\geq |B_2|$ and then $|B_1|=n-1$, which is a contradiction.\n\\end{proof}\nSince  the union of every clique partition of $G$ and $\\overline{G}$ forms a clique partition for $K_n$ which is equivalent to a PBD on $n$ points, the following corollaries are straightforward.\n\\begin{cor}\nLet $\\mathcal{C}$ be a clique partition of $K_n$ whose cliques are of size at most $n-1$. Then,  $\\sum_{C\\in \\mathcal{C}} |C|\\geq 3n-3$.\n\\end{cor}\n\n\\begin{cor}\nFor every graph $G$ on $n$ vertices except the empty and  complete graph, we have \n\\[\\scp(G)+\\scp(\\overline{G})\\geq 3n-3,\\]\nand equality holds if and only if $G$ or $\\overline{G}$ contains a clique of size $n-1$.\n\\end{cor}\n\nIn the same vein, one can prove the following theorem which states a lower bound on the maximum number of appearance of the points in a PBD.\n\n\\begin{theorem}\nLet $(P,\\mathcal{B})$ be a nontrivial PBD with $n$ points, and for every $x\\in P$, let $r_x$ be the number of blocks containing $x$. Then,  we have\n\\begin{equation}\\label{eq:pbd2}\n\\max_{x\\in P}{r_x}\\geq \\frac{1+\\sqrt{4n-3}}{2},\n\\end{equation}\nand equality holds if and only if $(P,\\mathcal{B})$ is a projective plane or near-pencil.\n\\end{theorem}\n\\begin{proof}\nLet $(P,\\mathcal{B})$ be a nontrivial PBD with $n$ points and define $r=\\max_{x\\in P} r_x$. Fix a point $x\\in P$ and let $\\mathcal{B}_x \\subset \\mathcal{B}$ be the set of blocks containing $x$. The family of sets $\\{B\\setminus \\{x\\} \\ : \\  B\\in \\mathcal{B}_x\\}$ is a partition of the set $P\\setminus \\{x\\}$. Thus, \n\\begin{equation}\\label{eq:PBDproof1}\nn-1= \\sum_{B\\in\\mathcal{B}_x}(|B|-1)\\leq r_x(\\max_{B\\in\\mathcal{B}_x} |B|-1).\n\\end{equation}\nTherefore,  there exists some block $B_0$ containing $x$, where $r_x (|B_0|-1)\\geq n-1$. Now, let $y$ be a point not in $B_0$. By a note within the proof of Theorem~\\ref{thm:PBDsigma}, we have $r_y\\geq |B_0|$ and then\n\\begin{equation}\\label{eq:PBDproof2}\nr(r-1)\\geq r_x(r_y-1)\\geq r_x(|B_0|-1)\\geq n-1.\n\\end{equation}\nThis yields the assertion.\n\n Now, assume that  equality holds in $\\eqref{eq:pbd2}$. Then,  we have  equalities in \\eqref{eq:PBDproof1} and \\eqref{eq:PBDproof2}. Thus, all valencies $r_x$ are equal and all blocks have the same size, say $k$, which shows that $(P,\\mathcal{B})$ is an $(n,k,1)-$design. Also by \\eqref{eq:PBDproof2}, we have $r=k$, i.e. $(P,\\mathcal{B})$ is a symmetric design.\n\\end{proof}\n\nAlthough the given bound in \\eqref{eq:pbd} is sharp, it can be improved if the PBD avoids blocks of large sizes. The following theorem, as an improvement of Theorem~\\ref{thm:PBDsigma}, provides some lower bounds on the sum of block sizes, when there are some constraints on the size of a block.\n\n\\begin{theorem} \\label{thm:PBD}\nIf  $(P,\\mathcal{B})$ is  a PBD with $n$ points where $\\tau$ is the maximum size of blocks in $\\mathcal{B}$, then\n\\begin{equation}\\label{eq:pbdA}\n\\sum_{B\\in \\mathcal{B}} |B| \\geq \\frac{n(n-1)}{\\tau-1}.\n\\end{equation}\nAlso if there is a block of size $\\kk$, then \n\\begin{equation}\\label{eq:pbdB}\n\\sum_{B\\in \\mathcal{B}} |B| \\geq (n+1) \\kk - \\frac{\\kk^2(\\kk-1)}{n-1},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:pbdC}\n\\sum_{B\\in \\mathcal{B}} |B| \\geq \\kk-\\frac{(n-\\kk)(n-5\\kk-1)}{2}.\n\\end{equation}\nMoreover, if $\\kk\\geq n\/2$, then there exists a PBD on $n$ points with a block of size $\\kk$, for which equality holds in {\\em (\\ref{eq:pbdC})}.\n\\end{theorem}\n\n\\begin{proof}\nFor every $x\\in P$, let $r_x$ be the number of blocks containing $x$. By Inequality~(\\ref{eq:PBDproof1}), we have\n\\[\\sum_{B\\in \\mathcal{B}} |B| =\\sum_{x\\in P} r_x\\geq \\sum_{x\\in P} \\frac{n-1}{\\tau-1}= \\frac{n(n-1)}{\\tau-1}.\\]\nIn order to prove \\eqref{eq:pbdB}, let $B_0\\in\\mathcal{B}$ and $|B_0|=\\kk$. Define,\n\\[\\tilde{\\mathcal{B}}= \\{B\\setminus B_0 \\ : \\  B\\in\\mathcal{B}, B\\cap B_0\\neq \\emptyset\\}.\\]\nWe have\n\\[ \\sum_{B\\in \\tilde{\\mathcal{B}} }  |B|=\\kk (n-\\kk).\\]\nNow, consider the following set\n\\[S=\\{(x,y) \\ : \\    x\\neq y, x,y \\in B, B\\in \\tilde{\\mathcal{B}}\\}.\\]\nWe have\n\\begin{equation}\\label{eq:S1}\n|S|=\\sum_{B\\in \\tilde{\\mathcal{B}} } |B| (|B|-1)\\geq \\frac{1}{|\\tilde{\\mathcal{B}}|} \\left(\\sum_{B\\in \\tilde{\\mathcal{B}} } |B|\\right)^2 - \\sum_{B\\in \\tilde{\\mathcal{B}} } |B|=\\frac{1}{|\\tilde{\\mathcal{B}}|} \\kk^2(n-\\kk)^2 - \\kk(n-\\kk).\n\\end{equation}\n\nOn the other hand,  $S\\subseteq \\{(x,y) \\ : \\  x,y\\in P\\setminus B_0\\}$. Thus, \n\\begin{equation}\\label{eq:S2}\n|S|\\leq (n-\\kk)(n-\\kk-1).\n\\end{equation}\nInequalities (\\ref{eq:S1}) and (\\ref{eq:S2}) yield\n\\[|\\tilde{\\mathcal{B}}|\\geq \\frac{\\kk^2(n-\\kk)}{n-1}. \\]\nFinally, \n\\[\\sum_{B\\in\\mathcal{B}} |B|\\geq |B_0|+\\sum_{B\\in \\tilde{\\mathcal{B}}} (|B|+1)\\geq \\kk+\\kk(n-\\kk)+\\frac{\\kk^2(n-\\kk)}{n-1}.\\]\nThus, we conclude \n\\[\\sum_{B\\in \\mathcal{B}} |B| \\geq (n+1) \\kk - \\frac{\\kk^2(\\kk-1)}{n-1}\\]\n\nTo prove Inequality \\eqref{eq:pbdC}, let $B_0\\in\\mathcal{B}$ and $|B_0|=\\kk$ and assume that $\\mathcal{B}$ has $u$ blocks of size 2 intersecting $B_0$. \nDefine, \n\\[\\hat{\\mathcal{B}}= \\{B\\setminus B_0 \\ :\\  B\\in\\mathcal{B}, \\ B\\cap B_0\\neq \\emptyset, \\ |B|\\geq 3\\}.\\]\nThus,\n\\begin{equation*}\n\\binom{n-\\kk}{2}\\geq \\sum_{B\\in\\hat{\\mathcal{B}}} {\\binom{|B|}{2}}\\geq\\sum_{B\\in\\hat{\\mathcal{B}}}(|B|-1).\n\\end{equation*}\nAlso,\n\\begin{equation*}\n\\kk(n-\\kk)= u+\\sum_{B\\in \\hat{\\mathcal{B}}} |B|.\n\\end{equation*}\nHence, \n\\begin{align*}\n\\sum_{B\\in \\mathcal{B}}|B| &\\geq  |B_0|+ 2u+ \\sum_{B\\in \\hat{\\mathcal{B}}} (|B|+1) = \\kk+ 2\\kk(n-\\kk)-  \\sum_{B\\in \\hat{\\mathcal{B}}} (|B|-1) \\\\\n& \\geq  \\kk+ 2\\kk(n-\\kk)- \\binom{n-\\kk}{2}.\n\\end{align*}\n\nNow, assume that $\\kk\\geq n\/2$ and $B_0=\\{x_1,\\ldots, x_k\\}$. We provide a PBD with a block $B_0$  for which\nequality holds in (\\ref{eq:pbdC}). Consider a proper edge coloring of $K_{n-\\kk}$ by $n-\\kk$ colors and let $C_1,\\ldots, C_{n-\\kk}$ be color classes. Each $C_i$ is a collection of subsets of size $2$. For every $i$,  $1\\leq i\\leq n-\\kk$, add $x_i$ to each member of $C_i$. Now, we have exactly $(n-\\kk)(n-\\kk-1)\/2$ blocks of size $3$. By adding missing pairs as blocks of size $2$, we get a PBD  $(P,\\mathcal{B})$ on $n$ points, with blocks of size $2$ and $3$ and a block of size $\\kk$. In fact, each block of size $3$ contains two pairs from the set $\\{(x,y) \\ :\\  x\\in B_0, y\\not\\in B_0\\}$. Hence,\n\n\\begin{align*}\n\\sum_{B\\in\\mathcal{B}}|B| &= \\kk+\\frac{3(n-\\kk)(n-\\kk-1)}{2}+ 2 (\\kk(n-\\kk)- (n-\\kk)(n-\\kk-1))\\\\\n& =\\kk-\\frac{(n-\\kk)(n-5\\kk-1)}{2}.\n\\end{align*}\n\\end{proof}\n\\begin{remark}\nLet $(P,\\mathcal{B})$ be a PBD with $n$ points where $\\tau$ is the maximum size of blocks in $\\mathcal{B}$. It is easy to check that among the lower bounds (\\ref{eq:pbdA}), (\\ref{eq:pbdB}) and (\\ref{eq:pbdC}), if  $1\\leq \\tau\\leq  (\\sqrt{4n-3}+1)\/2$, then (\\ref{eq:pbdA}) is the best one, if $ (\\sqrt{4n-3}+1)\/2\\leq \\tau\\leq (n-1)\/2$, then (\\ref{eq:pbdB}) is the best one and if $(n-1)\/2\\leq \\tau \\leq n-1$, then (\\ref{eq:pbdC}) is the best one. The diagram of the lower bounds in terms of $\\tau$ are depicted in Figure~\\ref{fig:PBD} for $n=21$. \n\n\\begin{center}\n\\begin{tikzpicture}\n\\pgfdeclareimage[width=12cm]{main}{PBD.pdf}\n\\pgftext{\\pgfuseimage{main}}\n\\node at (-6.1cm,.2cm)[rotate=90]{\\small ${\\mathscr{S}}(21,k)$ };\n\\node at (-.2cm,-3.7cm){\\small $k=\\tau$ };\n\\end{tikzpicture}\n\\captionof{figure}{Diagram of the lower bounds in (\\ref{eq:pbdA}), (\\ref{eq:pbdB}) and (\\ref{eq:pbdC}) for $n=21$.}\\label{fig:PBD}\n\\end{center}\n\n\\end{remark}\n\nNow, we apply Theorem \\ref{thm:PBD} to improve the bound in (\\ref{eq:pbd}), whenever the PBD does not contain large blocks.\n\\begin{theorem}\\label{thm:LargeBlock}\nLet $n\\geq 10$ and $(P,\\mathcal{B})$ be a PBD on $n$ points and assume that $\\mathcal{B}$ contains no block of size larger than $n-\\frac{1}{2} ( \\sqrt{n} +1)$. Then,  we have\n\\[\\sum_{B\\in \\mathcal{B}} |B|\\geq n(\\lfloor\\sqrt{n}\\rfloor+1)-1.\\]\nAlso, the bound is tight in the sense that equality occurs for infinitely many $n$.\n\\end{theorem}\n\\begin{proof}\nLet $\\tau$ be the maximum size of the blocks in $\\mathcal{B}$. If $\\tau\\leq \\sqrt{n}$, then by (\\ref{eq:pbdA}),\n\\[\\sum_{B\\in\\mathcal{B}}|B| \\geq \\frac{n(n-1)}{\\tau-1}\\geq \\frac{n(n-1)}{\\sqrt{n}-1}\\geq n(\\sqrt{n}+1).\\]\nNow, suppose that $\\tau\\geq \\lfloor\\sqrt{n}\\rfloor+1$. Then,  $\\mathcal{B}$ contains a block of size larger than or equal $\\lfloor\\sqrt{n}\\rfloor+1$. First assume that $\\mathcal{B}$ contains a block of size $\\kk$, where $\\lfloor\\sqrt{n}\\rfloor+1\\leq \\kk\\leq \\frac{n}{2}$. Then, by (\\ref{eq:pbdB}),\n\\[\\sum_{B\\in\\mathcal{B}}|B|\\geq (n+1) \\kk - \\frac{\\kk^2(\\kk-1)}{n-1}.\\]\nThe right hand side of the above inequality as a function of $\\kk$ takes its minimum on the interval $[\\lfloor\\sqrt{n}\\rfloor+1, \\frac{n}{2}]$ at $\\lfloor\\sqrt{n}\\rfloor+1$. Thus, \n\\begin{align*}\n\\sum_{B\\in\\mathcal{B}}|B|&\\geq (n+1) (\\lfloor\\sqrt{n}\\rfloor+1) - \\frac{(\\lfloor\\sqrt{n}\\rfloor+1)^2\\lfloor\\sqrt{n}\\rfloor}{n-1}\\\\\n&\\geq n (\\lfloor\\sqrt{n}\\rfloor+1) + (\\lfloor\\sqrt{n}\\rfloor+1)(1- \\frac{(\\sqrt{n}+1)\\sqrt{n}}{n-1}) \\\\\n& = n (\\lfloor\\sqrt{n}\\rfloor+1) - \\frac{\\lfloor\\sqrt{n}\\rfloor+1}{\\sqrt{n}-1} \\\\\n& > n (\\lfloor\\sqrt{n}\\rfloor+1) - 2.\n\\end{align*}\nThe last inequality is due to the fact that $n\\geq 10$.\nFinally, assume that $\\mathcal{B}$ contains a block of size $\\kk$, where $\\frac{n}{2}< \\kk\\leq n-\\frac{1}{2} ( \\sqrt{n} +1)$. Then, by (\\ref{eq:pbdC})\n\\[\\sum_{B\\in \\mathcal{B}} |B| \\geq \\kk-\\frac{(n-\\kk)(n-5\\kk-1)}{2}.\\]\n Again, the right hand side of the above inequality as a function of $\\kk$ takes its minimum on the interval $[\\frac{n}{2},n-\\frac{1}{2} ( \\sqrt{n} +1)]$ at $n-\\frac{1}{2} (\\sqrt{n}+1)$. Hence,\n \\begin{align*}\n\\sum_{B\\in \\mathcal{B}} |B| & \\geq  n-\\frac{1}{2}(\\sqrt{n} +1) -\\frac{(\\sqrt{n}+1)(-4n+\\frac{5}{2} (\\sqrt{n}+1)-1)}{4}  \\\\\n&=  n(\\sqrt{n}+1)+\\frac{3n-7}{8}-\\frac{3}{2}\\sqrt{n}\\\\\n&>  n(\\sqrt{n}+1)-2,\n \\end{align*}\n where the last inequality is because $ n\\geq 10 $. This completes the proof. \n\nFinally, in order to prove tightness of the bound, let $q$ be a prime power and $(P,\\mathcal{B})$ be an affine plane of order $q$. Suppose that $\\{B_1,\\ldots, B_{q}\\}$ is a parallel class. Add a single new point to all the blocks $B_1,\\ldots, B_q$. The new PBD has $n=q^2+1$ points, $q^2$ blocks of size $q$ and $q$ blocks of size $q+1$. Hence,  the sum of its block sizes is \n\\[q^3+q^2+q=(q^2+1)(q+1)-1= n(\\lfloor \\sqrt{n}\\rfloor+1)-1. \\]\n\\end{proof}\n\\section{Sigma clique partition of complement of  graphs }\\label{sec:Kn-Km}\nGiven a graph $G$ and its subgraph $H$, the complement of $H$ in $G$ denoted by $G-H$ is obtained from $G$ by removing all edges (but no vertices) of $H$. If $H$ is a graph on $n$ vertices, then $K_n-H$ is called the complement of $H$ and is denoted by $\\overline{H}$.\n\n In this section, applying  the results of Section~\\ref{sec:pbd}, we are going to determine the asymptotic  behaviour of the sigma clique partition number of the graph $K_n-K_m$, when $ m $ is a function of $ n $, as well as Cocktail party graph, the complement of path and cycle on $n$ vertices.\n\nThe clique partition number of the graph $K_n-K_m$, for $m\\leq n$, has been studied by several authors. \nIn order to notice the hardness of determining the exact value of $\\cp(K_n-K_m)$, note  that if we could show that $\\cp(K_{111}-K_{11})\\geq 111$, then we could determine whether there exists a projective plane of order $10$~\\cite{pullman82}.\nWallis in \\cite{WallisAsymp}, proved that $\\cp(K_n-G)\\sim n$, if $G$ has $o(\\sqrt{n})$ vertices. Also, Erd\\H{o}s et al. in \\cite{erdos88} showed that $\\cp(K_n-K_m)\\sim m^2$, if $\\sqrt{n}< m< n$ and $m=o(n)$. Moreover, if $m=cn$ and $1\/2\\leq c\\leq 1$, then Pullman et al. in \\cite{pullmanII} proved that $\\cp(K_n-K_m)=1\/2 (n-m) (3m-n-1)$.\n\n \nIn the following theorem, we present upper and lower bounds for $ \\scp(K_n-K_m) $ and then we improve these bounds in order to determine asymptotic behaviour of $ \\scp(K_n-K_m) $.  \n\\begin{theorem}\\label{thm:kn-km}\nFor every $m,n$, $1\\leq m\\leq n$, we have \n\\begin{equation}\\label{eq:kn-km}\nmn-\\dfrac{m^2(m-1)}{n-1}\\leq\\scp(K_n-K_m)\\leq(2m-1)(n-m)+1.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nAdding the clique $K_m$ to every clique partition of $K_n-K_m$ forms a PBD on $n$ points. Thus,  the lower bound is obtained from Inequality~(\\ref{eq:pbdB}).\n\nFor the upper bound, let $V(K_n)=\\{x_1,\\ldots, x_n\\}$ and $V(K_m)=\\{x_{n-m+1},\\ldots, x_n\\}$. Note that the clique $\\{x_1,\\ldots, x_{n-m+1}\\}$ along with $(m-1)(n-m)$ remaining edges form a clique partition of $K_n-K_m$. Hence,  $\\scp(K_n-K_m)\\leq (n-m+1)+2(m-1)(n-m)$.  \n\\end{proof}\n\nIn the following theorem, for $m\\leq\\frac{\\sqrt{n}}{2}$, we improve\nthe lower bound in (\\ref{eq:kn-km}).\n\\begin{theorem}\\label{thm:kn-km2}\nIf $m\\leq \\frac{\\sqrt{n}}{2}$, then \n\\[(2m-1) n - O(m^2) \\leq \\scp(K_n-K_m)\\leq  (2m-1) n - \\Omega(m^2).\\]\n\\end{theorem}\n\\begin{proof}\nThe upper bound holds by (\\ref{eq:kn-km}).\nFor the lower bound, consider an arbitrary clique partition of $K_n-K_m$, say $\\mathcal{C}$, and add the clique $K_m$ to obtain a PBD $(P,\\mathcal{B})$ with $n$ points. Let $\\tau$ be the size of maximum block in $\\mathcal{B}$. It is clear that $m\\leq \\tau\\leq n-m+1$. We give the lower bound in the following cases. First note that since $m\\leq \\sqrt{n}\/2$, we have $(2m-1)^2\\leq n-1$.\n\nIf $\\tau\\leq \\frac{n-1}{2m-1}$, then by (\\ref{eq:pbdA}), we have\n\\[\\sum_{C\\in\\mathcal{C}} |C|\\geq (2m-1)n-m.\\]\n\nIf  $\\frac{n-1}{2m-1}\\leq \\tau\\leq n\/2$, then  $2m-1\\leq \\tau\\leq n\/2$, and  by (\\ref{eq:pbdB}),\n\\[\\sum_{C\\in\\mathcal{C}} |C|\\geq (n+1) \\tau - \\frac{\\tau^2(\\tau-1)}{n-1}-m. \\]\nThe right hand side of this inequality is increasing as a function of $\\tau$ within the interval $[2m-1,n\/2]$. Hence, \n\\[\\sum_{C\\in\\mathcal{C}} |C|\\geq (n+1) (2m-1) - \\frac{(2m-1)^2(2m-2)}{n-1}-m\\geq  (2m-1) n-m. \\]\nFinally, if  $n\/2\\leq \\tau\\leq n-m+1$, then,  by (\\ref{eq:pbdC}),\n\\[\\sum_{C\\in\\mathcal{C}} |C|\\geq\\tau-\\frac{(n-\\tau)(n-5\\tau-1)}{2}-m.\\]\nConsider the right hand side of this inequality as a function of $\\tau$ within the interval $[n\/2,n-m+1]$. It attains its minimum at $\\tau=n-m+1$. Hence, \n\\[\\sum_{C\\in\\mathcal{C}} |C|\\geq n-2m+1-\\frac{(m-1)(5m-4n-6)}{2} = (2m-1) n - O(m^2).\\]\n\\end{proof}\n\nThe following lemma is a direct application of Theorem~\\ref{thm:LargeBlock} that gives a lower bound for $ \\scp(K_n-H) $ in terms of $ \\scp(H) $. Here, $\\omega(G)$ stands for the clique number of graph $G$. \n\n\\begin{lemma}\\label{lem:lower bound K_n-H}\nLet $ H $ be a graph on $ m $ vertices. If $  \\omega(H)\\leq n-\\frac{1}{2}(\\sqrt{n}+1) $ and $\\omega(\\overline{H})\\leq m-\\frac{1}{2}(\\sqrt{n}+1)$, then $$ \\scp(K_n-H)+\\scp(H)\\geq n(\\lfloor\\sqrt{n}\\rfloor+1)-1. $$\n\\end{lemma}\n\\begin{proof}\nAssume that $ \\cal C $ is an arbitrary clique partition for $ K_n-H $ and $ \\tau $ is the size of largest clique in $ \\cal C $. Then,  $ \\tau\\leq n-m+\\omega(\\overline{H})\\leq n-m+m-\\frac{1}{2}(\\sqrt{n}+1)=n-\\frac{1}{2}(\\sqrt{n}+1)$. Also, by assumption, $H$ has no clique of size larger than $n-\\frac{1}{2}(\\sqrt{n}+1)$. Moreover, every clique partition of $H$ along with every clique partition for $K_n-H$ form a PBD. Hence, by Theorem~\\ref{thm:LargeBlock}, $ \\scp(K_n-H)+\\scp(H)\\geq n(\\lfloor\\sqrt{n}\\rfloor+1)-1$.\n\\end{proof}\n\n\nWe need the following lemma in order to improve the upper bound in \\eqref{eq:kn-km} whenever $ \\sqrt{n}\\leq m\\leq n $. The idea is similar to \n\\cite{WallisAsymp} that uses a projective plane of appropriate size to give a clique partition for the graph $K_n-K_m$.\n\\begin{lemma}\\label{lem:design}\nLet $ H $ be a graph on $ m $ vertices. If there exists a $(v,k,1)-$design, such that $k\\geq m$ and $v-k\\geq n-m$, then $\\scp(K_n-H)\\leq n(v-1)\/(k-1)+\\scp(\\overline{H})-m$.\n\\end{lemma}\n\\begin{proof}\nLet $ (P,\\mathcal{B}) $ be a $ (v,k,1)-$design. \nSelect a block $ B_1\\in \\mathcal{B} $ and delete $ k-m $ points from it. Also, delete $ v-k-(n-m) $ points not in $ B_1 $. Now, consider the remaining points as vertices of $ K_n-H $ and each block except $ B_1 $ as a clique in $ K_n-H $. Thus, $ \\scp(K_n-H)\\leq r(n-m)+(r-1)m+\\scp(\\overline{H})=nr-m+\\scp(\\overline{H})$, where $ r=(v-1)\/(k-1) $ is the number of blocks containing a single point.\n\\end{proof}\nWe are going to apply Lemma~\\ref{lem:design} to projective planes and provide a clique covering for $ K_n-H $. Since the existence of projective planes of order $ q $ is only known for prime powers, we need the following well-known theorem to approximate an integer by a prime.  \n\\begin{thm}{\\em\\cite{baker}}\\label{lem:prime gap}\nThere exists a constant $ x_0 $ such that for every integer $ x > x_0 $, the interval \u2030$ [x , x+x^{.525}] $ contains  prime numbers.\n\\end{thm}\nThe following two theorems determine asymptotic behaviour of $ \\scp(K_n-K_m) $, when $\\sqrt{n}\/2\\leq m$ and $ m=o(n) $. \n\\begin{theorem}\\label{thm:kn--km}\nLet $ H $ be a graph on $ m $ vertices. If $\\frac{\\sqrt{n}}{2}\\leq m\\leq \\sqrt{n}$, then $\\scp(K_n-H)\\leq (1+o(1))\\, n\\sqrt{n}$. Moreover,  $ \\scp(K_n-K_m)= (1+o(1))\\, n\\sqrt{n}$.\n\\end{theorem}\n\\begin{proof}\nLet $q$ be the smallest prime power greater than or equal to $\\sqrt{n}$. By Theorem~\\ref{lem:prime gap}, we have $\\sqrt{n}\\leq q\\leq \\sqrt{n}+\\sqrt{n}^{.525}$. Thus, \n$ q\\geq\\sqrt{n}>m-1 $\nand\n$ q^2\\geq n\\geq n-m $. Since there exists a projective plane of order $q$, by Lemma~\\ref{lem:design}, we have\n\\[\\scp(K_n-H)\\leq n (q+1)-m+\\scp(\\overline{H}) \\leq n (q+1)-m+\\frac{m^2}{2},\\]\nwhere the last inequality is due to the fact that sigma clique partition number of every graph on $ n $ vertices is at most $ n^2\/2 $ \\cite{chung81,gyori80}. Hence,\n\\[\\scp(K_n-H)\\leq n^{1.5}+n^{1.2625}+1.5\\ n= (1+o(1))\\, n\\sqrt{n}.\\]\nAlso, by Lemma \\ref{lem:lower bound K_n-H}, $ \\scp(K_n-K_m)\\geq (1+o(1))\\, n\\sqrt{n} $.\n\\end{proof}\nIn the following theorem, for $\\sqrt{n}\\leq m\\leq n$, we improve\nthe upper bound in (\\ref{eq:kn-km}).\n\\begin{theorem}\\label{thm:kn-km1}\nIf $\\sqrt{n}\\leq m\\leq n$, then $\\scp(K_n-K_m)\\leq (1+o(1))\\, nm$ and if in addition $ m=o(n) $, then $\\scp(K_n-K_m)= (1+o(1))\\,nm$.\n\\end{theorem}\n\\begin{proof}\nLet $\\sqrt{n}\\leq m\\leq n$, and also let $q$ be the smallest prime power which is greater than or equal to $m$. By Lemma \\ref{lem:prime gap}, $m\\leq q\\leq m+m^{.525}$. Thus,  $q=(1+o(1))\\, m$. Since there exists a projective plane of order $q$, by Lemma~\\ref{lem:design}, we have\n\\[\\scp(K_n-K_m)\\leq n (q+1)-m = (1+o(1))\\, nm.\\]\nOn the other hand, when $ m=o(n) $, Inequality~(\\ref{eq:kn-km}) yields  $ \\scp(K_n-K_m)\\geq (1+o(1))\\, nm$, which completes the proof.\n\\end{proof}\nTheorems \\ref{thm:kn-km2}, \\ref{thm:kn--km} and \\ref{thm:kn-km1}  make clear asymptotic behaviour of $K_n-K_m$ in case $m=o(n)$.\n\\begin{cor}\\label{cor: K_n-K_m}\nLet $m$ be a function of $n$. Then\n\\begin{itemize}\n\\item[\\rm i)]  If $m\\leq \\frac{\\sqrt{n}}{2}$, then $\\scp(K_n-K_m)\\sim (2m-1)n$.\n\\item[\\rm ii)] If $\\frac{\\sqrt{n}}{2}\\leq m\\leq \\sqrt{n}$, then $\\scp(K_n-K_m)\\sim n\\sqrt{n}$.\n\\item[\\rm iii)] If $m\\geq \\sqrt{n}$ and $m=o(n)$, then $\\scp(K_n-K_m)\\sim mn$.\n\\end{itemize}\n\\end{cor}\nIn what follows, we consider the case $m=cn$, where $c$ is a constant.\n First note that if $1\/2\\leq c\\leq 1$, then by Theorem~\\ref{thm:PBD}, since $m\\geq n\/2$, there exists a PBD on $n$ points with a block of size $m$, for which equality holds in (\\ref{eq:pbdC}). Hence, we have $\\scp(K_n-K_m)= \\frac{(1-c)}{2}\\bigg((5c-1)n^2+n\\bigg)$.\n In order to deal with the case $c<1\/2$, we need the following well-known existence theorem of resolvable designs.\n \\begin{thm}{\\em \\cite{Lu}}\\label{thm:resol}\n Given any integer $k\\geq 2$, there exists an integer $v_0(k)$ such that for every $ v\\geq v_0(k) $, a $(v,k,1)-$resolvable design exists if and only if $v \\overset{k}{\\equiv} 0$ and $v-1\\overset{k-1}{\\equiv} 0$.\n \\end{thm}\n\\begin{theorem}\nLet  $0<c<1\/2$ be a constant and $m,n$ be some integers satisfying $m=cn$. Then\n\\begin{equation}\\label{eq:cn}\nc(1-c^2)n^2+ \\Omega(n)\\leq \\scp(K_n-K_m) \\leq  \\frac{(1-c)(\\lfloor 1\/c\\rfloor-c)}{\\lfloor 1\/c\\rfloor(\\lfloor 1\/c\\rfloor-1)} n^2+O(n).\n\\end{equation}\nIn particular, if $1\/c$ is integer, then $\\scp(K_n-K_m) \\sim c(1-c^2)n^2$.\n\\end{theorem}\n\\begin{proof}\nThe lower bound in (\\ref{eq:cn}) is obtained from the lower bound in (\\ref{eq:kn-km}). For the upper bound, let $k=\\lfloor 1\/c\\rfloor$ and define $v$ as the smallest number greater than or equal to $n-m$ which satisfies the conditions of Theorem~\\ref{thm:resol}. Without loss of generality we can assume that $n$ is sufficiently large, i.e. $ n\\geq v_0(k) $. Thus, we have $v\\leq n-m+k^2$ and by Theorem~\\ref{thm:resol}, there exists a $(v,k,1)-$resolvable design. Remove $v-n+m$ points from such a design to obtain a PBD $ (P,\\mathcal{B}) $ on $n-m$ points whose blocks are partitioned into  $t=(v-1)\/(k-1)$ parallel classes. First, we show that $m\\leq t$. Note that\n\\[m-t=cn-\\frac{v-1}{k-1}\\leq cn-\\frac{(1-c)n-1}{k-1}=\\frac{(ck-1)n+1}{k-1}.\\]\nIf $k=2$, then $ck<1$ and $m-t<1$. Also, if $k>2$, then $ck\\leq 1$ and thus $m-t\\leq 1\/(k-1)<1$. Therefore, $m\\leq t$.\n\n\nNow, let $v_1,\\ldots, v_m$ be $m$ new points and for every $i$, $1\\leq i\\leq m$, add point $v_i$ to all blocks of $i$-th parallel class. These blocks form a clique partition $\\mathcal{C}$ for $K_n-K_m$, where\n\\begin{equation*}\n\\sum_{C\\in\\mathcal{C}}|C|\\leq \\sum_{B\\in\\mathcal{B}}|B|+\\frac{v}{k} m= (n-m)\\frac{v-1}{k-1}+\\frac{mv}{k}.\n\\end{equation*}\nHence,\n\\begin{align*}\n\\sum_{C\\in\\mathcal{C}}|C|&\\leq \\left(\\frac{(1-c)^2}{k-1}+ \\frac{c(1-c)}{k}\\right)n^2+O(n)\\\\\n&= \\frac{(1-c)(k-c)}{k(k-1)} n^2+O(n).\n\\end{align*}\n\\end{proof}  \nWe close the paper by proving that if $ G $ is Cocktail party graph, complement of path or cycle on $ n $ vertices, then $ \\scp(G)\\sim n\\sqrt{n} $.  \nGiven an  even positive integer $n$, Cocktail party graph $T_n$ is obtained from the complete graph $K_{n}$ by removing a perfect matching. If $n$ is an odd positive integer, then $T_n$ is obtained from $T_{n+1}$ by removing a single vertex. \nIn \\cite{Wallis87,gregory86} it is proved that if $G$ is Cocktail party graph or complement of a path or a cycle on $n$ vertices, then $n\\leq \\cp(G)\\leq (1+o(1))\\, n \\log\\log n$ and it is conjectured that for such a graph, $\\cp(G)\\sim n$.\n\n\\begin{theorem}\\label{thm:complment of path}\nLet $ P_n $ be the path on $ n $ vertices. Then,  $ \\scp(\\overline{P_n})\\sim n^{3\/2} $.\n\\end{theorem}\n\\begin{proof}\nBy Lemma~\\ref{lem:lower bound K_n-H}, we have $ \\scp(\\overline{P_n})\\geq n^{3\/2}-2n-3$. Now, by induction on $n$, we prove that there exists a constant $c$, such that $\\scp(\\overline{P_n})\\leq n^{3\/2}+c\\, n^{13\/10}$.  The idea is similar to \\cite{Wallis87}.\\\\\nLet $ d=\\lfloor\\sqrt{n}\\rfloor $, $ e=\\lceil\\frac{n}{d}\\rceil $ and $ q $ be the smallest prime greater than $ \\sqrt{n} $. By Lemma~\\ref{lem:prime gap}, $q\\leq \\sqrt{n}+n^{3\/10}$. In an affine plane of order $ q $, choose a parallel class, say $ C_1 $, and delete $ q-d $ blocks in $C_1$. Then,  remove $ q-e $ blocks in a second parallel class, say $ C_2 $. The collection of remaining blocks is a PBD on $de$ points.\n\nAssume that $ a_{ij} $ is the intersection point of block $ i $ of $ C_1 $ and block $ j $ of $ C_2 $ in the remaining PBD. Thus, \n $C_1=\\{\\{a_{i1}, a_{i2},\\ldots , a_{ie}\\} \\ : \\  1\\leq i\\leq d\\}$ and $C_2=\\{\\{a_{1j}, a_{2j},\\ldots , a_{dj}\\} \\ : \\  1\\leq j\\leq e \\}$.\nNow, replace each block in $ C_2 $ by members of a clique partition of a copy of $\\overline{P_d }$ on the same vertices. Also, replace each of the blocks $ \\{a_{11}, a_{12},\\ldots , a_{1e}\\} $ and $ \\{a_{d1}, a_{d2},\\ldots , a_{de}\\} $ in $ C_1 $ by members of a clique partition of a copy of $\\overline{P_e}$ on the same vertices. In fact, we have replaced $ e+2 $ blocks by some clique partitions of complement of paths and $ q(q+1)-(e+2) $ blocks are left unchanged. It can be seen that the resulting collection, is a partition of all edges of $\\overline{P_{de}}$ except $(e-1)$ edges namely $a_{11}a_{12}, a_{d2}a_{d3}, a_{13}a_{14}, a_{d4}a_{d5},\\dots$ . Adding these $e-1 $ edges to this collection comprise a clique partition for $\\overline{P_{de}}$. Hence,\n\n\\begin{align*}\n\\scp(\\overline{P_n})\\leq\\scp(\\overline{P_{de}})&\\leq qde -2e+e\\scp(\\overline{P_d})+2\\scp(\\overline{P_e})+2(e-1)\n\\end{align*}\nSince $ e\\leq d+3 $, $ \\scp(\\overline{P_e})\\leq\\scp(\\overline{P_d})+6d $. Thus,\n\\begin{align*}\n\\scp(\\overline{P_n})&\\leq qd(d+3)+(d+5)\\scp(\\overline{P_d})+12d.\n\\end{align*}\nTherefore, by the induction hypothesis, we have\n\\begin{align*}\n\\scp(\\overline{P_n})&\\leq (\\sqrt{n}+n^{3\/10}) \\sqrt{n}(\\sqrt{n}+3)+(\\sqrt{n}+5)(n^{3\/4}+c\\, n^{13\/20})+12\\sqrt{n}\\\\\n&\\leq n^{3\/2}+ (1+o(1))\\, n^{13\/10}\\\\\n& \\leq n^{3\/2}+c\\, n^{13\/10}.\n\\end{align*}\n\\end{proof}\nAsymptotic behavior of $\\scp(T_n)$ and $\\scp(\\overline{C_n})$ can be easily determined using $\\scp(\\overline{P_n})$, as follows.\n\\begin{cor}\nLet $ T_n $ and $ C_n $ be Cocktail party graph and cycle on $ n $ vertices, respectively. Then,  $ \\scp(\\overline{C_n})\\sim n^{3\/2} $ and $ \\scp(T_n)\\sim n^{3\/2} $.\n\\end{cor}\n\\begin{proof}\nBy Lemma \\ref{lem:lower bound K_n-H}, $ \\scp(\\overline{C_n})\\geq n^{3\/2}-2n-1 $ and $ \\scp(T_n)\\geq n^{3\/2}-n-1 $.\\\\\nNote that $\\overline{P_n}$ is obtained from $\\overline{C_{n+1}}$ by removing an arbitrary vertex $v$. Adding $n-2$ edges incident with $v$ to any clique partition of $ \\overline{P_{n}} $ forms a clique partition for $ \\overline{C_{n+1}} $. Therefore, $ \\scp(\\overline{C_{n+1}})\\leq\\scp(\\overline{P_n})+2(n-1) $. Also, adding at most $n\/2$ edges to any clique partition for $\\overline{P_n}$ forms a clique partition for $T_n$. Thus, $ \\scp(T_n)\\leq\\scp(\\overline{P_n})+2\\frac{n}{2} $. Hence, by Theorem~\\ref{thm:complment of path}, $ \\scp(\\overline{C_n}), \\scp(T_n)\\leq (1+o(1))\\, n^{3\/2}$. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\footnotetext[1]{The authors contributed to this work when they were at NVIDIA.}\n\t\n\tMulti-view stereo algorithms \\cite{Bleyer11bmvc, Campbell08eccv, Furukawa07cvpr, Galliani15iccv, schoenberger2016mvs, Seitz2006mvs} have played an important role in 3D reconstruction and scene understanding for applications such as augmented reality, robotics, and autonomous driving. However, if a scene contains motion, such as non-stationary rigid objects or non-rigid surface deformations, the assumption of an epipolar constraint is violated~\\cite{Hartley04book}, causing algorithms to fail in most cases. \n\tScenes with rigidly-moving objects have been reconstructed by segmenting foreground objects from the background, and treating these regions independently \\cite{Zhang11-multibody, MulVSLAM_Abhijit_ICCV2011, Wang15}. However, the reconstruction of scenes with surfaces undergoing deformation is still a challenging task.\n\t\n\tTo solve this problem for sparse point pairs, various non-rigid structure from motion (NRSfM) methods have been introduced~\\cite{Jensen18}. \n\tThese methods often require either dense views (video frames)~\\cite{Ansari17} for the acquisition of dense correspondences with flow estimation, or prior information to constrain the problem~\\cite{Dai17}.\n\tNewcombe et al.~\\cite{Newcombe15cvpr} and Innmann et al.~\\cite{Innmann16eccv} recently demonstrated solutions for the 3D reconstruction of arbitrary, non-rigid, dynamic scenes using a dense stream of known metric depths captured from a commercial depth camera. \n\tHowever, there are many common scenarios one may encounter for which this method cannot be used, for example, when capturing scenes that contain non-rigid changes that are neither acquired as video nor captured by stereo or depth cameras, but rather come from independent single views.\n\t\n\tIn this paper, we are specifically interested in \\emph{dense 3D scene reconstruction with dynamic non-rigid deformations} acquired from images with wide spatial baselines, and sparse, unordered samples in time. This requires two solutions: (1) a method to compute the most plausible deformation from millions of potential deformations between given wide-baseline views, and (2) a dense surface reconstruction algorithm that satisfies a photometric consistency constraint between images of surfaces undergoing non-rigid changes.\n\t\n\tIn our solution, we first compute a \\emph{canonical surface} from views that have minimal non-rigid changes (Fig.~\\ref{fig:teaser}(a)), and then estimate the deformation between the canonical pose and other views by joint optimization of depth and photometric consistency. This allows the expansion of 3D points of canonical views (Fig.~\\ref{fig:teaser}(b)). \n\tThen, through the individual deformation fields estimated from each view to the canonical surface, we can reconstruct a dense 3D point cloud of each single view  (Fig.~\\ref{fig:teaser}(c)).\n\tA brief overview of our entire framework is described in Fig.~\\ref{fig:overview}.\\\\\n\t\n\t\\noindent Our contributions are as follows:\n\t\\begin{itemize}[topsep=0pt,noitemsep]\n\t\t\\item The first non-rigid MVS pipeline that densely reconstructs dynamic 3D scenes with non-rigid changes from wide-baseline and sparse RGB views.\n\t\t\\item A new formulation to model non-rigid motion using a deformation graph~\\cite{sumner2007embedded} and the approximation of the inverse-deformation used for the joint optimization with photometric consistency.\n\t\t\\item Patchmatch-based~\\cite{Bleyer11bmvc} dense sample propagation on top of an existing MVS pipeline~\\cite{schoenberger2016mvs}, which allows flexible implementation depending on different MVS architectures.\n\t\\end{itemize}\n\t\n\t\n\t\n\t\\section{Related Work}\n\t\\label{sec:related}\n\t\n\t\\noindent \\textbf{Dynamic RGB-D Scene Reconstruction.}\n\tA prior step to full dynamic scene reconstruction is dynamic template tracking of 3D surfaces.\n\tThe main idea is to track a shape template over time while non-rigidly deforming its surface \\cite{deAguiar2008,allain2015efficient,l0Norigid2015,hernandez2007non,li2008global,li2009robust,li2012temporally,gall2008driftfree,zollhoefer2014deformable}.\n\tJointly tracking and reconstructing a non-rigid surface is significantly more challenging.\n\tIn this context, researchers have developed an impressive line of works based on RGB-D or depth-only input~\\cite{zeng2013templateless,mitra2007dynamic,tevs2012animation,bojsen2012tracking,dou2013scanning,Dou_2015_CVPR,Malleson2014,wang2016capturing,Li13}.\n\tDynamicFusion~\\cite{Newcombe15cvpr} jointly optimizes a Deformation Graph \\cite{sumner2007embedded}, then fuses the deformed surface with the current depth map.\n\tInnmann et al.~\\cite{Innmann16eccv} follows up on this work by using an as-rigid-as-possible regularizer to represent deformations~\\cite{sorkine2007rigid}, and incorporate RGB features in addition to a dense depth tracking term.\n\tFusion4D~\\cite{dou2016fusion4d} brings these ideas a level further by incorporating a high-end RGB-D capture setup, which achieves very impressive results.\n\tMore recent RGB-D non-rigid fusion frameworks include KillingFusion~\\cite{slavcheva2017killingfusion} and \n\tSobolevFusion~\\cite{slavcheva2018sobolevfusion}, which allow for implicit topology changes using advanced regularization techniques.\n\tThis line of research has made tremendous progress in the recent years; but given the difficulty of the problem, all these methods either rely on depth data or calibrated multi-camera rigs.\n\t\n\t\\noindent \\textbf{Non-Rigid Structure from Motion.} \n\tSince the classic structure from motion solutions tend to work well for many real world applications~\\cite{Triggs:1996, Kanade1153, Tomasi92}, many recent efforts have been devoted to computing the 4D structure of sparse points in the spatio-temporal domain, which we call non-rigid structure from motion (NRSfM)~\\cite{Jensen18, kumar17, Rabaud08, garg2013dense}. \n\tHowever, most of the NRSfM methods consider the optimization of sparse points rather than a dense reconstruction, and often require video frames for dense correspondences~\\cite{Ansari17} or prior information~\\cite{Dai17}. \n\tScenes with rigidly moving objects (e.g., cars or chairs) have been reconstructed by segmenting rigidly moving regions~\\cite{Zhang11-multibody, MulVSLAM_Abhijit_ICCV2011, Ladick10, Wang15}.\n\tIn our work, we focus on a new scenario of reconstructing scenes with non-rigid changes from a few images, and estimate deformations that satisfy each view.\n\t\n\t\\noindent \\textbf{Multi-View Stereo and Dense Reconstruction.} \n\tVarious MVS approaches for dense 3D scene reconstruction have been introduced in the last few decades~\\cite{Furukawa07cvpr, Galliani15iccv, schoenberger2016mvs, Seitz2006mvs}. \n\tWhile many of these methods work well for static scenes, they often reject regions that are not consistent with the epipolar geometry~\\cite{Hartley04book}, e.g., if the scene contains changing regions. \n\tReconstruction failure can also occur if the ratio of static to non-rigid parts present in the scene is too low~\\cite{Lv18eccv}. \n\tA recent survey~\\cite{Schoeps2017CVPR} on MVS shows that COLMAP~\\cite{schoenberger2016mvs} performs the best among state-of-the-art methods. Therefore, we adopt COLMAP's Patchmatch framework for dense photometric consistency.\n\t\n\t\n\t\n\t\\section{Approach}\n\tThe input to our non-rigid MVS method is a set of images of a scene taken from unique (wide-baseline) locations at different times. An overview of our method is shown in Fig.~\\ref{fig:overview}.\n\tWe do not assume any knowledge of temporal order, i.e., the images are an unorganized collection. \n\tHowever, we assume there are at least two images with minimal deformation, and the scene contains sufficient background in order to measure the ratio of non-rigidity and to recover the camera poses. We discuss the details later in Sec~\\ref{sec:implementation}.\n\tThe output of our method is an estimate of the deformation within the scene from the canonical pose to every other view, as well as a depth map for each view.\n\tAfter the canonical view selection, we reconstruct an initial canonical surface that serves as a template for the optimization.\n\tGiven another arbitrary input image and its camera pose, we estimate the deformation between the canonical surface and the input.\n\tFurthermore, we compute a depth map for this processed frame using a non-rigid variant of PatchMatch.\n\tHaving estimated the motion and the geometry for every input image, we recompute the depth for the entire set of images to maximize the growth of the canonical surface.\n\t\n\t\\subsection{Modeling Deformation in Sparse Observations}\n\t\\label{sec:deformation_model}\n\t\n\tTo model the non-rigid motion in our scenario, we use the well known concept of deformation graphs \\cite{sumner2007embedded}.  Each graph node represents a rigid body transform, similar to the as-rigid-as-possible deformation model \\cite{sorkine2007rigid}. These transforms are locally blended to deform nearby space.\n\t\n\tGiven a point $\\mathbf{v} \\in \\mathbb{R}^3$, the deformed version $\\hat{\\mathbf{v}}$ of the point is computed as:\n\t\\[\n\t\\hat{\\mathbf{v}} = \\sum_{i=1}^k w_i(\\mathbf{v}) \\left[ \\mathbf{R}_i (\\mathbf{v} - \\mathbf{g}_i) + \\mathbf{g}_i + \\mathbf{t}_i \\right],\n\t\\]\n\twhere $\\mathbf{R}_i$ and $\\mathbf{t}_i$ represent the rotation and translation of a rigid body transform about position $\\mathbf{g}_i$ of the $i$-nearest deformation node, and $k$ is the user-specified number of nearest neighbor nodes (we set to $k = 4$ throughout our paper).\n\tThe weights $w_i$ are defined as:\n\t\\[\n\tw_i(\\mathbf{v}) = \\frac{1}{\\sum_{j=1}^k w_j (\\mathbf{v})} \\left( 1 - \\frac{\\|\\mathbf{v} - \\mathbf{g}_i \\|_2}{\\| \\mathbf{v} - \\mathbf{g}_{k+1} \\|_2}\\right)^2.\n\t\\]\n\tFor a complete description of deformation graphs, we refer to the original literature~\\cite{sumner2007embedded}. \n\t\n\tWhen projecting points between different images, we also need to invert the deformation. \n\tThe exact inverse deformation can be derived given known weights:\n\t\\[\n\t\\mathbf{v} = \\left(\\sum_{i=1}^k w_i(\\mathbf{v}) \\mathbf{R}_i \\right)^{-1} \\left[ \\hat{\\mathbf{v}} + \\sum_{i=1}^k w_i(\\mathbf{v}) \\left[ \\mathbf{R}_i \\mathbf{g}_i - \\mathbf{g}_i - \\mathbf{t}_i \\right] \\right]\n\t\\]\n\tHowever, because we do not know the weights a priori, which requires the nearest neighbor nodes and their distances, this becomes a non-linear problem. \n\tSince this computationally expensive step is necessary at many stages of our pipeline, we introduce an approximate solution:\n\t\\[\n\t\\mathbf{v} \\approx \\left(\\sum_{i=1}^k \\hat{w}_i(\\mathbf{\\hat v}) \\mathbf{R}_i \\right)^{-1} \\left[ \\hat{\\mathbf{v}} + \\sum_{i=1}^k \\hat{w}_i(\\mathbf{\\hat v}) \\left[ \\mathbf{R}_i \\mathbf{g}_i - \\mathbf{g}_i - \\mathbf{t}_i \\right] \\right],\n\t\\]\n\twhere the weights $\\hat{w}_i$ are given by\n\t\\[\n\t\\hat{w}_i(\\hat{\\mathbf{v}}) = \\frac{1}{\\sum_{j=1}^k \\hat{w}_i (\\hat{\\mathbf{v}})} \\left( 1 - \\frac{\\|\\mathbf{\\hat v} - (\\mathbf{g}_i + \\mathbf{t}_i) \\|_2}{\\|\\mathbf{\\hat v} - (\\mathbf{g}_{k+1} + \\mathbf{t}_{k+1}) \\|_2}\\right)^2.\n\t\\]\n\tNote that our approximation can be computed directly and efficiently, without leading to any error of observable influence in our synthetic experiments. \n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/node-visualization}\n\t\t\\caption{\\textbf{Deformation nodes and correspondences:} (Left) shows the deformation nodes at $t_0$ (orange), and another set of nodes at $t_1$ (red) overlaid in the canonical view. (Right) we show the relationship between deformation nodes from two views and sparse 3D matches (after lifting) in the context of a non-rigid change. Note that we only show the sparse matching for simpler visualization while there is also a dense term for photometric consistency that drives the displacement of deformation nodes together with the sparse matches.}\n\t\t\\label{fig:node-viz}\n\t\t\\vspace{-.2cm}\n\t\\end{figure}\n\t\n\t\\subsection{Non-rigid Photometric Consistency and Joint Optimization}\n\t\\label{sec:photometric-conssitency}\n\tWith the deformation model in hand, we next estimate the depth of the other views by estimating deformations that are photometrically consistent with the collection images  and subject to  constraints on the geometry. \n\tThis entire step can be interpreted as a non-rigid version of a multi-view stereo framework.\\\\\n\t\\noindent \\textbf{Canonical View Selection} \n\tFrom the set of input images, we select two views with a minimal amount of deformation.\n\tWe run COLMAP's implementation of PatchMatch \\cite{schoenberger2016mvs} \n\tto acquire an initial temple model of the canonical pose.\n\tBased on this template, we compute the deformation graph by distributing a user-specified number of nodes on the surface.\n\tTo this end, we start with all points of the point cloud as initial nodes.\n\tWe iterate over all nodes, and for each node remove all its neighbors within a given radius.\n\tThe process is repeated with a radius that is increased by 10\\%, until we have reached the desired number of nodes.\n\tIn our experiments, we found that 100 to 200 nodes are sufficient to faithfully reconstruct the motion. Fig.~\\ref{fig:node-viz}(left) shows an example of the node distribution.\\\\\n\t\\noindent \\textbf{Correspondence Association}\n\tFor sparse global correspondences, we detect SIFT keypoints \\cite{lowe1999object} in each image and match descriptors for every pair of images to compute a set of feature tracks $\\{ \\mathbf{u}_i \\}$. A \\textit{feature track} represents the same 3D point and is computed by connecting each keypoint with each of its matches. We reject inconsistent tracks, i.e.,~if there is a path from a keypoint $\\mathbf{u}^{(j)}_i$ in image $i$ to a different keypoint $\\mathbf{u}^{(k)}_i$ with $j \\neq k$ in the same image. \n\t\n\tWe lift keypoints $\\mathbf{u}_i$ to 3D points $\\mathbf{x}_i$, if there is a depth value in at least one processed view, compute its coordinates in the canonical pose $\\mathbf{D}_i^{-1}(\\mathbf{x}_i)$ and apply the current estimate of our deformation field $\\mathbf{D}_j$ for frame $j$ to these points. \n\tTo establish a sparse 3D-3D correspondence $(\\mathbf{D}_i^{-1}(\\mathbf{x}_i), \\mathbf{x}_j)$ between the canonical pose and the current frame $j$ for the correspondence set $S$, we project $\\mathbf{D}_j(\\mathbf{D}_i^{-1}(\\mathbf{x}_i))$ to the ray of the 2D keypoint $\\mathbf{u}_j$ (see Fig.~\\ref{fig:concept1}).\n\tTo mitigate ambiguities and to constrain the problem, we also aim for dense photometric consistency across views.\n\tThus, for each point of the template of the canonical pose, we also add a photometric consistency constraint with a mask $C_i \\in \\{ 0, 1 \\}$.\n\t\n\t\n\t\n\t\\noindent \\textbf{Deformation and Depth Estimation}\n\tIn our main iteration (see also Algorithm \\ref{alg:algorithm1}), we estimate the deformation $\\hat{\\mathbf{D}}$ between the canonical pose and the currently selected view by minimizing the joint optimization problem:\n\t\\begin{align}\n\t&E = w_\\text{sparse} E_\\text{sparse} + w_\\text{dense} E_\\text{dense} + w_\\text{reg} E_\\text{reg} \\\\\n\t&E_\\text{sparse} = \\sum_{(i, j) \\in S} \\| \\hat{\\mathbf{D}}(\\mathbf{x}_i) - \\mathbf{x}_j \\|_2^2 \\nonumber\\\\ \n\t&E_\\text{dense} = \\sum_r \\sum_s \\sum_i C_i \\cdot (1 - \\rho_{r, s}(\\hat{\\mathbf{D}}(\\mathbf{x}_i), \\hat{\\mathbf{D}}(\\mathbf{n}_i), \\mathbf{x}_i, \\mathbf{n}_i))^2  \\nonumber \\\\\n\t&E_\\text{reg} = \\sum_{j=1}^m \\sum_{k \\in N(j)} \\| \\mathbf{R}_j (\\mathbf{g}_k - \\mathbf{g}_j) + \\mathbf{g}_j + \\mathbf{t}_j - (\\mathbf{g}_k + \\mathbf{t}_k) \\|_2^2 \\nonumber\n\t\\end{align}\n\t\n\tTo measure photometric consistency $\\rho_{r,s}$ between a reference image $r$, i.e.~the canonical pose, and a source view $s$, we use the bilaterally weighted adaption of normalized cross-correlation (NCC) as defined by Schoenberger et al. \\cite{schoenberger2016mvs}. Throughout our pipeline, we employ COLMAP's default settings, i.e. a window of size $11 \\times 11$. \n\tThe regularizer $E_\\text{reg}$ as defined in \\cite{sumner2007embedded} ensures a smooth deformation result.\n\tTo ensure non-local convergence, we solve the problem in a coarse-to-fine manner using an image pyramid with 3 levels in total. \n\t\n\tBoth the sparse and dense matches are subject to outliers. \n\tIn the sparse case, these outliers manifest as incorrect keypoint matches across images.\n\tFor the dense part, outliers mainly occur due to occlusions, either because of the camera pose or because of the observed deformation.\n\t\n\tTo reject outliers in both cases, we reject correspondences with the highest residuals calculated from the result of the non-linear solution.\n\tWe re-run the optimization until a user-specified maximum error is satisfied.\n\tThis rejection is run in a 2-step process.\n\tFirst, we only solve for the deformation considering the sparse 3D-3D matches.\n\tSecond, we fix the retained 3D-3D matches and solve the joint optimization problem, discarding only dense correspondences, resulting in a consistency map $C_i \\in \\{ 0, 1 \\}$.\n\t\n\tWe iterate this process (starting with the correspondence association) until we reach convergence.\n\tIn our experiments, we found that 3 to 5 iterations suffice to ensure a converged state.\n\t\n\tTo estimate the depth for the currently processed view, we then run a modified, non-rigid variant of COLMAP's PatchMatch \\cite{schoenberger2016mvs}.\n\tInstead of simple homography warping, we apply the deformation to the point and its normal.\n\t\n\t\\begin{figure}[t]\n\t\t\\def2.1cm{0.42\\linewidth}\n\t\t\\centering\n\t\t\\subfloat[Iteration 1]{\\includegraphics[width=2.1cm]{concept12_newa}}\n\t\t\\hfill\n\t\t\\subfloat[Iteration 2]{\\includegraphics[width=2.1cm]{concept12_newb}}\n\t\t\\caption{\\textbf{Sparse correspondence association}: In iteration $i$, we transform the 3D point $\\mathbf{x}_0$ according to the previous estimate of the deformation $\\mathbf{D}_2^{(i-1)}$ and project $\\mathbf{D}_2^{(i - 1)}(\\mathbf{x}_0)$ onto the ray defined by $\\mathbf{u}_2$. The projection is used to define a force $F$ pulling the point towards the ray.}\n\t\t\\label{fig:concept1}\n\t\t\\vspace{-.3cm}\n\t\\end{figure}\n\t\n\t\n\t\\subsection{Implementation Details}\n\t\\label{sec:implementation}\n\tIn this section, we provide more details on our implementation of the NRMVS framework (Fig.~\\ref{fig:overview}).\n\tAlgorithm~\\ref{alg:algorithm1} shows the overall method, introduced in Sec.~\\ref{sec:deformation_model}, and Sec.~\\ref{sec:photometric-conssitency}.\n\t\n\tGiven input RGB images, we first pre-process the input.\n\tTo estimate the camera pose for the images, we use the SfM implementation of Agisoft Photoscan \\cite{photoscan}. Our tests showed accurate results for scenes containing at least 60\\% static background. A recent study~\\cite{Lv18eccv} shows that 60$\\sim$90\\% of static regions in a scene results in less than $0.02$ degree RPE~\\cite{Sturm12iros} error for standard pose estimation techniques (see more discussion in the appendix~\\ref{sec:pose}\n\tGiven the camera pose, we triangulate sparse SIFT matches \\cite{lowe1999object}, i.e.,~we compute the 3D position of the associated point by minimizing the reprojection error. \n\tWe consider matches with a reprojection error of less than 1 pixel to be successfully reconstructed (static inliers).\n\tThe ratio of static inliers to the number of total matches is a simple yet effective indication of the non-rigidity in the scene.\n\tWe pick the image pair with the highest ratio to indicate the minimum amount of deformation and use these as the canonical views to bootstrap our method.\n\t\n\tTwo important aspects of our main iteration are described in more detail:\n\tOur method to filter sparse correspondences (line 16 in Algorithm~\\ref{alg:algorithm1}) is given in Algorithm~\\ref{alg:filter}.\n\tThe hierarchical optimization algorithm (line 17 in Algorithm~\\ref{alg:algorithm1}) including filtering for dense correspondences is given in Algorithm~\\ref{alg:optimization}.\n\t\n\tThe joint optimization in our framework is a computationally expensive task. \n\tThe deformation estimation, which strongly dominates the overall run-time, is CPU intensive, while the depth computation runs on the GPU. \n\tSpecifically, for the face example shown in Fig.~\\ref{fig:teaser} (6 images with 100 deformation nodes) the computation time needed is approximately six hours (Intel i7-6700 3.4 GHz, NVIDIA GTX 980Ti).\n\tMore details about the computational expense will be discussed in the appendix~\\ref{sec:performance}.\n\t\n\t\\begin{algorithm}\n\t\t\\KwData{RGB input images $\\{ \\mathbf{I}_k \\}$}\n\t\t\\KwResult{Deformations $\\{ \\mathbf{D}_k \\}$, depth $\\{ d_k \\}$}\n\t\t$P := \\{ 1, \\ldots, k \\}, Q := \\emptyset$ \\;\n\t\t$\\{ \\mathbf{C}_k \\}$ = PhotoScanEstimateCameraPoses()\\;\n\t\t$(i, j)$ = selectCanonicalViews()\\;\n\t\t$(d_i^{(0)}, \\mathbf{n}_i^{(0)}, d_j^{(0)}, \\mathbf{n}_j^{(0)})$ = ColmapPatchMatch($\\mathbf{I}_i, \\mathbf{I}_j$)\\;\n\t\t$\\mathbf{D}_i^{(0)} = \\mathbf{D}_j^{(0)}$ = initDeformationGraph($d_i^{(0)}, d_j^{(0)}$)\\;\n\t\t$\\{ \\mathbf{u}_k \\}$ = computeFeatureTracks()\\;\n\t\t$Q := Q \\cup \\{ i, j \\}$\\;\n\t\t\\While{$Q \\neq P$}{\n\t\t\t$l$ = nextImage($P \\setminus Q$)\\;\n\t\t\t$\\{ \\mathbf{x}_k \\}$ = liftKeyPointsTo3D($\\{ \\mathbf{u}_k \\}_{k \\in Q}$) \\;\n\t\t\t$\\{ \\mathbf{x}_i \\} = \\mathbf{D}_k^{-1}(\\{ \\mathbf{x}_k \\})$ \\;\n\t\t\t$\\mathbf{D}_l^{(1)} = \\mathbf{Id}$ \\;\n\t\t\t\\For{$m = 1$ \\KwTo $N$}{\n\t\t\t\t$\\{ \\mathbf{\\hat{x}}_l^{(m)} \\} = \\mathbf{D}_l^{(m)}(\\{ \\mathbf{x}_i \\})$ \\;\n\t\t\t\t$\\{ \\mathbf{x}_l^{(m)} \\}$ = projToRays($\\{ \\mathbf{\\hat{x}}_l^{(m)} \\}, \\{ \\mathbf{u}_l \\})$ \\;\n\t\t\t\t$\\{ (\\mathbf{\\tilde{x}}_i, \\mathbf{\\tilde{x}}_l^{(m)}) \\}$ = filter($\\mathbf{D}_l^{(m)}, \\{ (\\mathbf{x}_i, \\mathbf{x}_l^{(m)}) \\}$)\\;\n\t\t\t\t$\\mathbf{D}_l^{(m + 1)}$ = solve($\\mathbf{D}_l^{(m)}, \\{ (\\mathbf{\\tilde{x}}_i, \\mathbf{\\tilde{x}}_l^{(m)}) \\}$, $\\mathbf{I}_i, d_i^{(0)}, \\mathbf{n}_i^{(0)}, \\mathbf{I}_j, d_j^{(0)}, \\mathbf{n}_j^{(0)}, \\mathbf{I}_l$)\n\t\t\t}\n\t\t\t$\\mathbf{D}_l = \\mathbf{D}_l^{(m + 1)}$ \\;\n\t\t\t$Q := Q \\cup \\{ l \\}$\\;\n\t\t\t$(d_l^{(0)}, \\mathbf{n}_l^{(0)})$ = NRPatchMatch($\\{ \\mathbf{I}_k, \\mathbf{D}_k \\}_{k \\in Q}$)\\;\n\t\t}\n\t\t$\\{ (d_k, \\mathbf{n}_k) \\}_{k \\in Q}$ = NRPatchMatch($\\{ \\mathbf{I}_k, \\mathbf{D}_k \\}_{k \\in Q}$)\\;\n\t\t\\caption{Non-rigid multi-view stereo}\n\t\t\\label{alg:algorithm1}\n\t\\end{algorithm}\n\t\\vspace{-0.2cm}\n\t\\begin{algorithm}\n\t\t\\SetKwFunction{FMain}{filter}\n\t\t\\SetKwProg{Fn}{Function}{:}{}\n\t\t\\KwData{Threshold $d_\\text{max}$, Ratio $\\tau \\in (0, 1)$}\n\t\t\\Fn{\\FMain{$\\mathbf{D}_l, \\{ (\\mathbf{x}_i, \\mathbf{x}_l) \\}$}}{\n\t\t\t\\While{true}{\n\t\t\t\t$\\mathbf{D}_l^\\ast$ = solve($\\mathbf{D}_l, \\{ (\\mathbf{x}_i, \\mathbf{x}_l) \\}$)\\;\n\t\t\t\t$\\{ r_k \\} = \\{ \\| \\mathbf{D}_l^\\ast(\\mathbf{x}_i) - \\mathbf{x}_l \\|_2 \\}$\\;\n\t\t\t\t$e_\\text{max} = \\max \\{ r_k \\} $\\;\n\t\t\t\t\\If{$e_\\text{max} < d_\\text{max}$}{\n\t\t\t\t\tbreak\\;\n\t\t\t\t}\n\t\t\t\t$d_\\text{cut} := \\max \\{ d_\\text{max}, \\tau \\cdot e_\\text{max} \\}$ \\;\n\t\t\t\t$\\{ (\\mathbf{x}_i, \\mathbf{x}_l) \\} := \\{ (\\mathbf{x}_i, \\mathbf{x}_l) : r_k < d_\\text{cut} \\} $\\;\n\t\t\t}\n\t\t\t\\Return $\\{ (\\mathbf{x}_i, \\mathbf{x}_l) \\}$\\;\n\t\t}\n\t\t\\caption{Filtering of sparse correspondences}\n\t\t\\label{alg:filter}\n\t\\end{algorithm}\n\t\\vspace{-0.2cm}\n\t\\begin{algorithm}\n\t\t\\SetKwFunction{FMain}{solve}\n\t\t\\SetKwProg{Fn}{Function}{:}{}\n\t\t\\KwData{Threshold $\\rho_\\text{max}$, Ratio $\\tau \\in (0, 1)$}\n\t\t\\Fn{\\FMain{$\\mathbf{D}_l, \\{ (\\mathbf{x}_i, \\mathbf{x}_l) \\}, \\mathbf{I}_i, d_i,     \\mathbf{n}_i, \\mathbf{I}_j, d_j, \\mathbf{n}_j, \\mathbf{I}_l$}}{\n\t\t\t$\\hat{\\mathbf{D}}_l = \\mathbf{D}_l$\\;\n\t\t\t\\For{$m = 1$ \\KwTo levels}{\n\t\t\t\t$\\rho_\\text{cut} := \\tau \\cdot (1 - \\text{NCC}_\\text{min}) = \\tau \\cdot 2$ \\;\n\t\t\t\t$C_p := 1 \\quad \\forall p$\\;\n\t\t\t\t\\While{true}{\n\t\t\t\t\t$\\mathbf{D}_l^\\ast$ = solveEq1($\\hat{\\mathbf{D}_l}$) \\;\n\t\t\t\t\t$\\{ r_p \\} = \\{ C_p \\cdot~(1~-~\\rho(\\mathbf{D}_l^\\ast(\\mathbf{x}_p), \\mathbf{D}_l^\\ast(\\mathbf{n}_p), \\mathbf{x}_p, \\mathbf{n}_p)) \\}$\\;\n\t\t\t\t\t$e_\\text{max} = \\max \\{ r_p \\} $\\;\n\t\t\t\t\t\\If{$e_\\text{max} < \\rho_\\text{max}$}{\n\t\t\t\t\t\t$\\hat{\\mathbf{D}}_l := \\mathbf{D}_l^\\ast$\\;\n\t\t\t\t\t\tbreak\\;\n\t\t\t\t\t}\n\t\t\t\t\t\\If{$m = levels$}{\n\t\t\t\t\t\t$\\hat{\\mathbf{D}}_l := \\mathbf{D}_l^\\ast$\\;\n\t\t\t\t\t}\n\t\t\t\t\t$C_p := 0 \\quad \\forall p : r_p > \\rho_\\text{cut}$\\;\n\t\t\t\t\t$\\rho_\\text{cut} := \\max \\{ \\rho_\\text{max}, \\tau \\cdot \\rho_\\text{cut} \\}$\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\Return $\\hat{\\mathbf{D}}_l$\n\t\t}\n\t\t\\caption{Solving the joint problem}\n\t\t\\label{alg:optimization}\n\t\\end{algorithm}\n\t\\vspace{-0.2cm}\n\t\n\t\n\t\\begin{table*}[t]\n\t\t\\caption{\\textbf{Evaluation for ground truth data:} (a) using COLMAP, i.e., assuming a static scene, (b) applying our dense photometric optimization on top of an implementation of non-rigid ICP (NRICP), and (c) using different variants of our algorithm. S denotes \\emph{sparse}, D denotes \\emph{dense}, photometric objective. $N$ equals the number of iterations for sparse correspondence association (see paper for more details). We compute the mean relative error (MRE) for all reconstructed values as well as the overall completeness. The last row (w\/o filter) shows the MRE, with disabled rejection of outlier depth values, i.e., a completeness of 100 \\%.\n\t\t}\n\t\t\\vspace{-.2cm}\n\t\t\\begin{tabularx}{\\textwidth}{@{}rccccccc@{}}\n\t\t\t\\toprule\n\t\t\t& & \\multicolumn{6}{c}{Ours} \\\\\n\t\t\t\\cmidrule{4-8}\n\t\t\t& COLMAP \\cite{schoenberger2016mvs} & NRICP~\\cite{Li09} & S ($N=1$) & S ($N=10$) & D & S ($N=1$) + D &  S ($N=10$) + D \\\\ \n\t\t\t\\midrule \n\t\t\tCompleteness & 68.74 \\% & 99.30 \\%  & 97.24 \\%  & 97.71 \\% & 96.41 \\% & 98.76 \\% &  \\textbf{98.99} \\% \\\\\n\t\t\t\\hline\n\t\t\tMRE & 2.11 \\% & 0.53 \\%  & 1.48 \\% & 1.50 \\% & 2.37 \\% & 1.12 \\% & \\textbf{1.11} \\%  \\\\ \n\t\t\t\\hline \n\t\t\t\\hline\n\t\t\tMRE w\/o filter & 6.78 \\% & 0.74 \\%  & 2.16 \\% & 2.05 \\% & 3.32 \\% & 1.63 \\% & \\textbf{1.34} \\%  \\\\ \n\t\t\t\\bottomrule \n\t\t\\end{tabularx} \n\t\t\\label{tab:eval}\n\t\t\\vspace{-0.2cm}\n\t\\end{table*}\n\t\n\t\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\newlength\\exlen\n\t\t\\setlength\\exlen{.15\\linewidth}\n\t\t\\def1.5cm{1.5cm}\n\t\t\\captionsetup[subfloat]{labelformat=empty}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\, \\, \\, \\,  Input\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=11cm 7cm 11cm 4cm, clip=true,width=\\exlen]{figures\/gt2\/Cam0_0}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm 6cm 10cm 5cm, clip=true,width=\\exlen]{figures\/gt2\/Cam0_1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=7.7cm 4.9cm 7.7cm 2.2cm, clip=true,width=\\exlen]{figures\/gt2\/Cam4_6}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12.5cm 5.5cm 9.5cm 5.5cm, clip=true,width=\\exlen]{figures\/gt2\/Cam7_1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=11cm 7cm 11cm 4cm, clip=true,width=\\exlen]{figures\/gt2\/Cam8_2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=11cm 7cm 11cm 4cm, clip=true,width=\\exlen]{figures\/gt2\/Cam13_2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[height=1.5cm]{gt_eval_colorbar_dummy}}  \\\\\n\t\t\\vspace{-0.3cm}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\quad \\, 3D points\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=9cm 7cm 8cm 5cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot0_0}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 6cm 7cm 6cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot0_1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=9cm 6cm 8cm 6cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot4_6}} %\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=9cm 6cm 8cm 6cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot7_1-2}} %\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=8.5cm 6cm 8.5cm 6cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot8_2}} %\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=7cm 6cm 10cm 6cm, clip=true,width=\\exlen]{figures\/gt2\/snapshot13_2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[height=1.5cm]{gt_eval_colorbar_dummy}} \\\\ %\n\t\t\\vspace{-0.3cm}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize Relative error \\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[0.51 \\% \/\/ 97.4 \\%]}]{\\includegraphics[trim=5.5cm 3.5cm 5.5cm 2cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_0_0}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[0.08 \\% \/\/ 100 \\%]}]{\\includegraphics[trim=6cm 3cm 5cm 2.5cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_0_1}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[1.23 \\% \/\/ 99.25 \\%]}]{\\includegraphics[trim=3.85cm 2.45cm 3.85cm 1.1cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_4_6}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[1.31 \\% \/\/ 99.44 \\%]}]{\\includegraphics[trim=6.25cm 2.75cm 4.75cm 2.75cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_7_1}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[0.97 \\% \/\/ 99.98 \\%]}]{\\includegraphics[trim=5.5cm 3.5cm 5.5cm 2cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_8_2}}\n\t\t\\hfill\n\t\t\\subfloat[\\scriptsize{[1.00 \\% \/\/ 99.14 \\%]}]{\\includegraphics[trim=5.5cm 3.5cm 5.5cm 2cm, clip=true,width=\\exlen]{figures\/gt2\/error_rel_depth_13_2}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[height=1.5cm]{gt_eval_colorbar}}\n\t\t\\caption{\\textbf{Quantitative evaluation with synthetic data:} We created images of a deforming surface with 10 different views. For the evaluation, we randomly chose six examples from the set and reconstructed the surface. The first row shows the input images. The first two columns show the chosen canonical views. The results of the reconstructed surface (with the point cloud propagated to each view) are shown in the second row. In the third row, we visualize the relative depth error compared to the ground truth. We also show the mean relative depth error value (\\%) and the completeness (\\%). The overall quantitative evaluation including a comparison to other baselines are shown in Table~\\ref{tab:eval}.}\n\t\t\\label{fig:results-synth}\n\t\t\\vspace{-.2cm}\n\t\\end{figure*}\n\t\n\t\\begin{figure*}\n\t\t\\def\\vspace{-1em}{\\vspace{-1em}}\n\t\t\\def\\vspace{-0.2cm}{\\vspace{-0.2cm}}\n\t\t\\centering\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\qquad Input images\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=3cm 0.6cm 1.8cm 0cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/1.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2cm 0.3cm 2.8cm 0.3cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/0.jpg}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.6cm 2.3cm 0cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/2.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.3cm 2.3cm 0.3cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/3.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.6cm 2.3cm 0cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/4.jpg}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.6cm 2.3cm 0cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/5.jpg}} \n\t\t\\\\\n\t\t\\vspace{-1em}\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\qquad \\, 3D point cloud\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/canonical\/canno-kihwan2.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/snapshot00_L00.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/snapshot00_L01.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/snapshot00_L02.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/snapshot00_L03.jpg}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=14cm 6cm 13cm 4cm, clip=true,width=.16\\linewidth]{figures\/res-kihwan1\/snapshot00_L04.jpg}} \n\t\t\\\\\n\t\t\\vspace{-0.2cm}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\, \\, Input images\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g2}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g3}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g4}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g5}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=1cm   0cm 1cm  0cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/g6}} \n\t\t\\\\\n\t\t\\vspace{-1em}\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\, 3D point cloud \\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-cano}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-3}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-4-2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=12cm   8cm 12cm  8cm, clip=true,width=.16\\linewidth]{figures\/globe-v3\/globe-5-2}}\n\t\t\\\\\n\t\t\\vspace{-0.2cm}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\quad\\ Input images\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.4cm 0.5cm 1.6cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.3cm 0.5cm 1.7cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s2}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.2cm 0.5cm 1.8cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s3}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2cm 0.5cm 2cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s5}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.5cm 1.5cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s6}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.4cm 1cm 1.6cm 0cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/s7}}\n\t\t\\\\\n\t\t\\vspace{-1em}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize 3D point cloud \\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=11cm 5cm 8cm 5cm, clip=true,width=.16\\linewidth]{figures\/canonical\/shirt-cano}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/snapshot00_L00}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/snapshot00_L01}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=11cm 5cm 8cm 5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/snapshot00_L03}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/snapshot00_L04}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 5cm, clip=true,width=.16\\linewidth]{figures\/shirt-v2\/snapshot00_L05}}\n\t\t\\\\\n\t\t\\vspace{-0.2cm}\n\t\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize \\quad Input images\\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.5cm 1.5cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/paper\/0_0}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.5cm 0.5cm 1.5cm 0.5cm, clip=true,width=.16\\linewidth]{figures\/paper\/0_1}} \n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2cm 0cm 2cm 1cm, clip=true,width=.16\\linewidth]{figures\/paper\/1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2.3cm 0.8cm 1.7cm 0.2cm, clip=true,width=.16\\linewidth]{figures\/paper\/2}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2cm 0cm 2cm 1cm, clip=true,width=.16\\linewidth]{figures\/paper\/3}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=2cm 0.8cm 2cm 0.2cm, clip=true,width=.16\\linewidth]{figures\/paper\/4}}\n\t\t\\\\\n\t\t\\vspace{-1em}\n\t\t\\subfloat{\\parbox[t]{.02\\linewidth}{\\begin{sideways}\\centering \\footnotesize 3D point cloud \\end{sideways}}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 9cm, clip=true,width=.16\\linewidth]{figures\/paper\/canno}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 5cm 9cm 9cm, clip=true,width=.16\\linewidth]{figures\/paper\/p1}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 8cm 9cm 6cm, clip=true,width=.16\\linewidth]{figures\/paper\/p3}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 9cm 9cm 5cm, clip=true,width=.16\\linewidth]{figures\/paper\/p4}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 6cm 9cm 8cm, clip=true,width=.16\\linewidth]{figures\/paper\/p5}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[trim=10cm 8.5cm 9cm 5.5cm, clip=true,width=.16\\linewidth]{figures\/paper\/p6}}\n\t\t\n\t\t\\caption{\\textbf{Qualitative evaluation with real data}: In each row, the first two columns show the views used to create each canonical surface. The first column of each result row (even row) shows the original canonical surface. The remaining views from the second column of each result row shows the propagated version of reconstructed surfaces for each view. }\n\t\t\\label{fig:results}\n\t\t\\vspace{-.2cm}\n\t\\end{figure*}\n\t\n\n\t\\begin{figure}\n\t\t\\centering\n\t\n\t\t\\includegraphics[clip=true,width=.9\\linewidth]{figures\/interpolation1\/eye-interp-final}\n\t\t\\includegraphics[clip=true,width=.9\\linewidth]{figures\/interpolation1\/globe-interp-final}\n\t\t\\parbox[h]{.29\\linewidth}{\\centering \\scriptsize Source}\n\t\t\\parbox[h]{.29\\linewidth}{\\centering \\scriptsize Intermediate scene}\n\t\t\\parbox[h]{.29\\linewidth}{\\centering \\scriptsize Target}\n\t\t\\caption{\\textbf{Dynamic 3D scene interpolation with in-between deformations:} We interpolate a point cloud between two reconstructed views from their depth and deformation. We show two key-frames, source and target, denoted as red and yellow frames respectively, and then demonstrate the interpolated \\emph{intermediate point cloud} in the middle column. For the top row, zoomed in-set images of the eye region show how the deformation is applied to the intermediate point cloud. More interpolated frames and 4D animations created from our deformation estimation are shown in the supplementary video with various views\\textsuperscript{\\ref{ftn:video}}.}\n\t\t\\label{fig:results-interpolation}\n\t\t\\vspace{-.2cm}\n\t\\end{figure}\n\t\n\t\\section{Evaluation}\n\tFor existing non-rigid structure from motion methods, different types of datasets are used to evaluate sparse points~\\cite{Jensen18, Dai17}, and dense video frames with small baseline (including actual camera view variation)~\\cite{Ansari17}. \n\tSince our problem formulation is intended for dense reconstruction of scenes with sufficient variation in both camera view and deformation, there are only few examples applicable to our scenario~\\cite{Li13, Wang15, Innmann16eccv}. Unfortunately, these datasets are either commercial and not available~\\cite{Li13}, or only exhibit rigid changes~\\cite{Wang15}. Few depth-based approaches share the input RGB as well~\\cite{Innmann16eccv}, but the quality of the images is not sufficient for our method (i.e., severe motion blur, low resolution (VGA) that does not provide sufficient detail for capturing non-rigid changes). \n\tThus, we created both synthetic data and captured real-world examples for the evaluation.\n\tTo quantitatively evaluate how our method can accurately capture a plausible deformation and reconstruct each scene undergoing non-rigid changes, we rendered several synthetic scenes with non-rigid deformations as shown in the first row of Fig.~\\ref{fig:results-synth}. \n\tWe also captured several real-world scenes containing deforming surfaces from different views at different times. Some examples (face, rubber globe, cloth and paper) appear in Fig.~\\ref{fig:results}, and several more viewpoints are contained in the supplementary video\\footnote{\\url{https:\/\/youtu.be\/et_DFEWeZ-4}\\label{ftn:video}}.\n\t\n\t\\subsection{Quantitative Evaluation with Synthetic Data}\n\tFirst, we evaluate the actual depth errors of the reconstructed depth of each time frame (i.e., propagated\/refined to a specific frame), and of the final refined depth of the \\emph{canonical view}. \n\tBecause we propose the challenging new problem of reconstructing non-rigid dynamic scenes from a small set of images, it is not easy to find other baseline methods. \n\tThus, we conduct the evaluation with an existing MVS method, COLMAP~\\cite{schoenberger2016mvs}, as a lower bound, and use as an upper bound a non-rigid ICP method similar to Li~et~al.~\\cite{Li09} based on the ground truth depth. \n\tThe non-rigid ICP using the point-to-plane error metric serves as a geometric initialization.\n\tWe refine the deformation using our dense photometric alignment (see Sec.~\\ref{sec:photometric-conssitency}).\n\t\n\tTo compare the influence of our proposed objectives for deformation estimation, i.e. sparse 3D-3D correspondences and dense non-rigid photometric consistency, we evaluate our algorithm with different settings. \n\tThe relative performance of these variants can be viewed as an ablation study. \n\tWe perform evaluation on the following variants: 1) considering only the sparse correspondence association using different numbers of iterations (see Sec.~\\ref{sec:photometric-conssitency}), 2) considering only the dense photometric alignment, and 3) the combination of sparse and dense.\n\tThe results of the quantitative evaluation can be found in Table~\\ref{tab:eval}.\n\tAs can be seen, all methods\/variants obtain a mean relative error $< 2.4 \\%$, overall resulting in faithfully reconstructed geometry.\n\tOur joint optimization algorithm considerably improves the reconstruction result both in terms of accuracy (by a factor of $1.9$) and completeness (by $30$ pp, a factor of $1.4$).\n\tAdditionally, we compute the mean relative depth error (MRE) without rejecting outliers; i.e., resulting in depth images with a completeness of 100\\%.\n\t\n\t\\subsection{Qualitative Evaluation with Real Data}\n\tFig.~\\ref{fig:results} shows results of our non-rigid 3D reconstruction. \n\tFor each pair of rows, we show six input images and the corresponding deformed 3D point clouds. \n\tNote that the \\emph{deformed} surfaces belong to the collection of 3D reconstructed points propagated by the computed deformations using the other views as described in Sec.~\\ref{sec:photometric-conssitency}. \n\tThe point cloud of each first column of Fig.~\\ref{fig:results} shows the first canonical surface (triangulated points from two views with minimal deformation).\n\tFor evaluation purposes, we visualize each reconstructed scene from a similar viewpoint as one of the canonical views. \n\tMore viewpoints of the reconstructed 3D results can be found in the supplementary video\\textsuperscript{\\ref{ftn:video}}.\n\t\n\t\\subsection{Dynamic Scene Interpolation} \n\tSince we estimate deformations between each view and the canonical surface, once all deformation pairs have been created, we can easily interpolate the non-rigid structure.\n\tTo blend between the deformations, we compute interpolated deformation graphs by blending the rigid body transform at each node using dual-quaternions \\cite{kavan2007skinning}.\n\t\n\tIn Fig.~\\ref{fig:results-interpolation}, we show interpolated results from reconstructed scenes of the face example and the globe example shown in Fig.~\\ref{fig:results}. \n\tThe scene deformations used for this interpolation (like key-frames) are framed in as red and yellow. \n\tNote that even though the estimated deformation is defined between each view and the canonical pose, any combination of deformation interpolation is possible. \n\tMore examples and interpolated structures from various viewpoints can be found in the supplementary video\\textsuperscript{\\ref{ftn:video}}.\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\def2.1cm{2.1cm}\n\t\t\\vspace{-0.3cm}\n\t\t\\subfloat[Bad canonical views selection]{\\includegraphics[height=2.1cm]{temp-fail}}\n\t\t\\hfill\n\t\t\\subfloat[Ambiguity along view direction]{\\includegraphics[height=2.1cm]{globe_broken}}\n\t\t\\caption{\\textbf{Failure cases:} (a) shows the result of canonical surface reconstruction from two views that are incorrectly selected (large deformation between two views: images in first and third column in top row of Fig.~\\ref{fig:results}). While the camera pose is successfully computed, since there are large portions of non-rigid changes happening in the upper part of face and near the mouth, there are many holes on the face, which is not the best case if we choose this pair.  (b) shows a failure case when deformation (red circles) occurs along the view direction, which causes the ambiguity.}\n\t\t\\label{fig:failure}\n\t\t\\vspace{-.2cm}\n\t\\end{figure}\n\t\n\t\n\t\n\t\\section{Conclusion and Discussion}\n\tWe propose a challenging new research problem for dense 3D reconstruction of scenes containing deforming surfaces from sparse, wide-baseline RGB images. \n\tAs a solution, we present a joint optimization technique that optimizes over depth, appearance, and the deformation field in order to model these non-rigid scene changes. \n\tWe show that an MVS solution for non-rigid change is possible, and that the estimated deformation field can be used to interpolate motion in-between views.\n\t\n\tIt is also important to point out the limitations of our approach (Fig.~\\ref{fig:failure}). We first assume that there is at least one pair of images that has minimal deformation for the initial canonical model. \n\tThis can be interpreted as the first step used by many SLAM or 3D reconstruction algorithms for the initial triangulation. \n\tFig.~\\ref{fig:failure}(a) shows an example of a canonical surface created from two views that contain too much deformation, only leading to a partial triangulation.  \n\tFig.~\\ref{fig:failure}(b) shows an example where the deformation occurs mostly along the view direction. \n\tWhile we successfully estimate the deformation and reconstruct a similar example shown in Fig.~\\ref{fig:results}, depending on the view this can cause an erroneous estimation of the deformation.\n\tOn the other hand, we believe that recent advances in deep learning-based approaches to estimate depth from single RGB input~\\cite{Fu2018DeepOR} or learning local rigidity~\\cite{Lv18eccv} for rigid\/non-rigid classification can play a key role for both the initialization and further mitigation of these ambiguities. \n\t\n\t{\\small\n\t\t\\bibliographystyle{ieee}\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Generating Experimental Configurations, Extended}\n\\label{apdx:data-generation}\n\nIn this section we detail how we use \\texttt{SpotCheck}{} to generate random experimental configurations.  \n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item In Section \\ref{apdx:semantic-features}, we define the different types of semantic features that can appear in each image. \n    \\item In Section \\ref{apdx:dataset}, we define a synthetic image dataset, how we generate random datasets, and how we sample images from a dataset.\n    \\item In Section \\ref{apdx:blindspots}, we define a blindspot for a synthetic image dataset, how we generate a random blindspot, and how we generate an unambiguous set of blindspots.  \n   \n\\end{itemize} \n\n\\newpage\n\\subsection{Semantic Features}\n\\label{apdx:semantic-features}\n\nTable \\ref{table:layers+attributes+values} defines all of the semantic features that \\texttt{SpotCheck}{} uses to generate synthetic images.\nWe call these semantic features \\emph{Attributes} and group them into \\emph{Layers} based on what part of an image they describe.\nEach Attribute has two possible \\emph{Values}: a Default and Alternative Value.\nEach synthetic image has an associated list of (Layer, Attribute, Value) triplets that describes the image.  \nFigure \\ref{fig:apdx-rollable} shows this triplet list for two synthetic images.\n\nWe sometimes refer to the Square\/Rectangle\/Circle\/Text Layers as \\emph{Object Layers} because they all describe a specific object that can be present in an image.  \nThe location of each object within an image is chosen randomly, subject to the constraint that each object doesn't overlap with any other object.  \n\n\\input{tables\/tab-attributes}\n\n\\newpage\n\\subsection{Defining a Dataset using these Semantic Features}\n\\label{apdx:dataset}\n\nAt a high level, \\texttt{SpotCheck}{} defines a \\emph{Dataset} by deciding whether or not each Attribute of each Layer is \\emph{Rollable} (\\textit{i.e.,}{} the Attribute can take either its Default or Alternative Value, uniformly at random) or not Rollable (\\textit{i.e.,}{} the Attribute only takes its Default Value).\nWe measure a Dataset's complexity using the number of Rollable Attributes it has.\nFigure \\ref{fig:apdx-rollable} describes the Rollable and Not Rollable Attributes for an example Dataset.\n\n\\textbf{Generating a Random Dataset.}\nWe start by picking which Layers will be part of the Dataset:  \n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item Images need a background, so all Datasets have the Background Layer.  \n    \\item The task is to predict whether there is a square in the image, so all Datasets have the Square Layer.  \n    \\item We add 1-3 (chosen uniformly at random) of the other Object Layers (chosen uniformly at random without replacement from the set \\{Rectangle, Circle, Text\\}) to the Dataset.  \n\\end{itemize}\n\nOnce the Layers are chosen, we make 6-8 (chosen uniformly at random) of the Attributes Rollable:\n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item Each Object Layer has its Presence Attribute made Rollable.\n    \\item Then, the remaining Rollable Attributes are chosen by iteratively:\n    \\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n        \\item Selecting a Layer uniformly at random from those that have at least one Not Rollable Attribute.\n        \\item Selecting an Attribute from that Layer uniformly at random from those that are Not Rollable.\n    \\end{itemize}\n\\end{itemize}\n\n\\textbf{Sampling an Image from a Dataset.}\nOnce a Dataset's Rollable Attributes have been defined, generating a random image is straightforward:\n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item For each Attribute from each Layer in the Dataset, we pick a random Value if the Attribute is Rollable.  \n    Attributes that are Not Rollable will take their Default Value.\n    \\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item If the Layer is an Object Layer:\n        \\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n            \\item If the Presence Attribute is True, the location of the object is chosen randomly (subject to the non-overlapping constraint).  \n            \\item If the Presence Attribute is False, the object will not be rendered (regardless of the Values chosen for the other Attributes of this Layer).  \n        \\end{itemize}\n    \\end{itemize}\n    \\item We then use the resulting (Layer, Attribute, Value) triplet list and the list of object locations to render a 224x224 RGB image. \n    \\item Finally, we calculate any MetaAttributes (explained next) and append these (Layer, MetaAttribute, Value) triplets to the image's definition list.  \n\\end{itemize}\n\n\\textbf{Calculating MetaAttributes.}  \nWhile each Attribute corresponds to a semantic feature, there are a potentially infinite number of MetaAttributes that one could calculate as semantically meaningful functions of an image.\nWe list the MetaAttributes that we calculate in our experiments in Table \\ref{table:metaattributes}.  \nBecause this space is infinitely large and grows with the number of Attributes, we exclude MetaAttributes from our measure of Dataset complexity.\n\n\\newpage\n\\input{figures\/fig-rollable}\n\\paragraph{ }\n\\paragraph{ }\n\\input{tables\/tab-metaattributes}\n\n\n\\newpage\n\\subsection{Defining the Blindspots for a Dataset}\n\\label{apdx:blindspots}\n\n\\texttt{SpotCheck}{} defines a \\emph{Blindspot} using a list of (Layer, (Meta)Attribute, Value) triplets.\nWe measure a Blindspot's specificity using the length of its definition list.  \nAn image belongs to a blindspot if and only if the Blindspot's definition list is a subset of the image's definition list.\nFigure \\ref{fig:apdx-blindspots} shows two example Blindspots.\n\n\\textbf{Generating a Random Blindspot.}  \n\\texttt{SpotCheck}{} generates a random Blindspot consisting of 5-7 (chosen uniformly at random) (Layer, (Meta)Attribute, Value) triplets for a Dataset by iteratively:\n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item Selecting a Layer (uniformly at random from those that have at least one Rollable Attribute\\footnote{%\n    All MetaAttributes are considered to be ``Rollable'' when generating a random Blindspot.}%\n    \\ that is not already in this Blindspot)\n    \\item Selecting a Rollable Attribute from that Layer:\n    \\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n        \\item Object Layers:  \n        If the Layer's Presence Attribute is not in this Blindspot, select its Presence Attribute.  \n        Otherwise, select an Attribute uniformly at random from those that are not already in this Blindspot and set the Layer's Presence Attribute Value to True for this Blindspot.\n        \\item Background Layers:  Select an Attribute uniformly at random from those that are not already in this Blindspot.\n    \\end{itemize}\n    \\item Selecting a Value for that Attribute (uniformly at random)\n\\end{itemize}\n\nNotice that, if an Object Layer is selected more than once, then we ensure that the Object's Presence Attribute has a Value of True in the Blindspot definition.  \nWe enforce this \\emph{Feasibility Constraint} to ensure that every triplet in the Blindspot's definition list correctly describes the images belonging to the Blindspot (\\textit{e.g.,}{} [(Circle, Presence, False), (Circle, Color, Blue)] is infeasible because an image with a blue circle must have a circle in it).  \n\n\\textbf{Generating an Unambiguous Set of Blindspots.}\nFor each Dataset, we generate 1-3 (chosen uniformly at random) Blindspots using the process described above.  \nHowever, when generating multiple blindspots, they can be \\emph{ambiguous} which causes problems when using them to evaluate BDM{}s.\n\n\\emph{\\textbf{Definition.}}\nA set of Blindspots, $S_1$, is ambiguous if there exists a different set of Blindspots, $S_2$, such that both:\n\\begin{enumerate}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item The union of images belonging to $S_1$ is equivalent to the union of images belonging to $S_2$.\n    As a result, $S_1$ and $S_2$ would both correctly describe the model's blindspots.\n    \\item An evaluation that uses Discovery Rate (Equation \\ref{eq:dr}) would penalize a BDM{} if it returns $S_2$ instead of $S_1$.  \n    More precisely, $\\text{DR}(S_2, S_1) < 1$ for $\\lambda_p = \\lambda_r = 1$.\n   \n   \n   \n   \n   \n\\end{enumerate}\n\n\\emph{\\textbf{Example.}}\nSuppose that we have a very simple Dataset with two Rollable Attributes, $X$ and $Y$ which are uniformly distributed and independent, and consider two different sets of Blindspots for this Dataset: \n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item $S_1 = \\{B_1, B_2\\}$ where $B_1 = [(X = 1)]$ and $B_2 = [(X = 0), (Y = 1)]$\n    \\item $S_2 = \\{B_1^{'}, B_2^{'}\\}$ where $B_1^{'} = [(X = 1), (Y = 0)]$ and $B_2^{'} = [(Y = 1)]$\n\\end{itemize}\n\nThen, $S_1$ is ambiguous because:\n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item $S_1$ and $S_2$ induce the same behavior in the model: they both mislabel an image if $X = 1 \\vee Y = 1$.\n    \\item A BDM{} would be penalized for returning $S_2$: \n    $$\\text{BP}(B_1^{'}, B_1) = 1.0 \\wedge \\text{BP}(B_1^{'}, B_2) = 0 \\wedge \\text{BP}(B_2^{'}, B_1) = \\text{BP}(B_2^{'}, B_2) = 0.5 \\implies $$\n    $$ \\text{BR}(S_2, B_1) = 0.5 \\wedge \\text{BR}(S_2, B_2) = 0 \\implies $$\n    $$\\text{DR}(S_2, S_1) = 0$$\n\\end{itemize}\n\nIn fact, for this example, there are only two sets of two unambiguous Blindspots, ($\\{[(X = 0), (Y = 0)], [(X = 1), (Y = 1)]\\}$ and $\\{[(X = 0), (Y = 1)], [(X = 1), (Y = 0)]\\}$), and there exists no unambiguous set of three Blindspots.\n\n\\emph{\\textbf{Preventing Ambiguity.}}\nIn general, ambiguity occurs whenever the union of two blindspots forms a contiguous region in the discrete space defined by the Rollable Attributes.  \nConsequently, we prevent ambiguity by ensuring that any pair of blindspots has at least two of the \\emph{same} Rollable Attributes with \\emph{different} Values in their definition lists.  \nWe call this the \\emph{Ambiguity Constraint}. \n\n\\emph{\\textbf{Implications of the Ambiguity and Feasibility Constraints.}}  \nIn our experiments, our goal is to generate  experimental configurations with a diverse set of Datasets and associated Blindspots.  \nHowever, the Ambiguity Constraint (AC) and Feasibility Constraint (FC) limit the number of valid Blindspots for any specific Dataset.  \n\nTo see this, notice that the AC places more constraints on each successive Blindspot added to an experimental configuration.\nThis has two implications.  \nFirst, that generating an experimental configuration with more Blindspots requires a Dataset with more Rollable Attributes (more complexity) and Blindspots with more triplets (more specificity).  \nFurther, because we cannot set the Attribute Values of a Blindspot's triplets independently of each other [FC], we need more complexity and specificty than a simple analysis based only on the AC suggests.\nSecond, that each successive Blindspot is more closely related to the previous ones which means that larger sets of Blindspots are ``less diverse'' or ``less random'' in some sense.  \n\nWith these trade-offs in mind, we generated experimental configurations with:\n\\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]\n    \\item Background, Square, and 1-3 other Object Layers\n    \\item A total of 6-8 Rollable Attributes\n    \\item 1-3 Blindspots\n    \\item 5-7 triplets per Blindspot\n\\end{itemize}\nbecause an experimental configuration with any combination of these values is able to satisfy the AC and the FC while still having a diverse set of Blindspots.  \n\n\\input{figures\/fig-example-blindspots}\n\n\n\\section{Introduction}\n\\input{sec-intro}\n\n\\section{Synthetic Evaluation for BDM{}s}\n\\input{sec-eval-framework}\n\n\\section{Evaluation Metrics}\n\\input{sec-eval-metrics}\n\n\\section{\\texttt{\\RealMethodName}}\n\\input{sec-method}\n\n\\section{Experiments}\n\\input{sec-experiments}\n\n\\section{Related Work} \n\\input{sec-related-work}\n\n\\section{Conclusion}\n\\input{sec-discussion}\n\n\\bibliographystyle{unsrtnat}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet \\(X\\) be an \\(n\\)-element set, and let \\(X^{(k)}\\) denote the collection of all \\(k\\)-element subsets of \\(X\\). We say a family \\(\\mathcal{A} \\subset X^{(k)}\\) is \\(t\\)-{\\em intersecting} if any two sets in $\\mathcal{A}$ share at least \\(t\\) elements, i.e. \\(|x\\cap y| \\geq t\\) for any \\(x,y \\in \\mathcal{A}\\). Erd\\H{o}s, Ko and Rado \\cite{tekr} proved in 1961 that if \\(n\\) is sufficiently large depending on \\(k\\) and \\(t\\), and \\(\\mathcal{A} \\subset X^{(k)}\\) is \\(t\\)-intersecting, then \\(|\\mathcal{A}| \\leq {n-t \\choose k-t}\\), with equality holding only if \\(\\mathcal{A}\\) is the family of all \\(k\\)-sets containing some fixed \\(t\\)-element subset of \\(X\\).\n\nIn \\cite{erdosconj}, Erd\\H{o}s asked what happens if we weaken the condition, and just forbid an intersection of size {\\em exactly} \\(t-1\\). Frankl and F\\\"uredi \\cite{franklfuredi} proved that for \\(k \\geq 2t\\) and for \\(n\\) sufficiently large depending on \\(k\\), if $\\mathcal{A} \\subset X^{(k)}\\) such that no two sets in $\\mathcal{A}$ have intersection of size exactly \\(t-1\\), then $|\\mathcal{A}| \\leq {n-t \\choose k-t}$, with equality holding only if \\(\\mathcal{A}\\) is the family of all \\(k\\)-sets containing some fixed \\(t\\)-element subset of \\(X\\).\n\nIn this paper, we consider analogues of these problems for the symmetric group \\(S_{n}\\), the group of all permutations of \\(\\{1,2,\\ldots,n\\}=:[n]\\). We say that a family of permutations \\(\\mathcal{A} \\subset S_{n}\\) is \\(t\\)-{\\em intersecting} if any two permutations in $\\mathcal{A}$ agree on at least $t$ points --- in other words, for all \\(\\sigma,\\tau \\in \\mathcal{A}\\), we have \\(\\#\\{i:\\ \\sigma(i)=\\tau(i)\\} \\geq t\\).\n\nDeza and Frankl \\cite{dezafrankl} proved in 1977 that if $\\mathcal{A} \\subset S_n$ is 1-intersecting, then $|\\mathcal{A}| \\leq (n-1)!$. The case of equality turned out to be somewhat harder than one might expect; this was resolved in 2003 by Cameron and Ku \\cite{cameron}, and independently by Larose and Malvenuto \\cite{larose}, who proved that if $\\mathcal{A} \\subset S_n$ is an intersecting family of size $(n-1)!$, then $\\mathcal{A}$ is a coset of the stabiliser of a point.\n\nDeza and Frankl conjectured in \\cite{dezafrankl} that for any $t \\in \\mathbb{N}$, if \\(n\\) is sufficiently large depending on \\(t\\), and \\(\\mathcal{A} \\subset S_{n}\\) is \\(t\\)-intersecting, then \\(|\\mathcal{A}| \\leq (n-t)!\\). This was proved in 2008, by the author and independently by Friedgut and Pilpel, using very similar techniques (specifically, eigenvalue methods, combined with the representation theory of $S_n$); we have written a joint paper, \\cite{jointpaper}. We also proved that equality holds only if $\\mathcal{A}$ is a {\\em $t$-coset} of $S_n$ (meaning a coset of the stabiliser of $t$ points), again provided $n$ is sufficiently large depending on $t$.\n\nCameron and Ku \\cite{cameron} conjectured that if \\(\\mathcal{A} \\subset S_{n}\\) is 1-intersecting, and $\\mathcal{A}$ is not contained in any 1-coset, then $\\mathcal{A}$ is no larger than the family\n\\[\\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\ \\sigma(j)=j \\textrm{ for some }j>2\\} \\cup \\{(1\\ 2)\\},\\] \nwhich has size \\((1-1\/e+o(1))(n-1)!\\). This was proved by the author in \\cite{cameronkuconj}, using the representation theory of \\(S_{n}\\) combined with some combinatorial arguments. It can be seen as an analogue of the Hilton-Milner Theorem \\cite{hiltonmilner} on 1-intersecting families of \\(r\\)-subsets of \\(\\{1,2,\\ldots,n\\}\\). In \\cite{dezafranklstability}, the author proved a generalization of the Cameron-Ku conjecture for \\(t\\)-intersecting families, namely that if \\(\\mathcal{A} \\subset S_{n}\\) is a \\(t\\)-intersecting family which is not contained within a coset of the stabilizer of \\(t\\) points, then \\(\\mathcal{A}\\) is no larger than the family\n\\[\\{\\sigma: \\sigma(i)=i\\ \\forall i \\leq t,\\ \\sigma(j)=j\\ \\textrm{for some}\\ j > t+1\\} \\cup \\{(1\\ t+1),\\ldots,(t \\ t+1)\\}\\]\nwhich has size \\((1-1\/e+o(1))(n-t)!\\). The proof uses similar ideas to in \\cite{cameronkuconj}, but both the representation theory and the combinatorial arguments are somewhat more involved.\n\nIn this paper, we consider problem of forbidding just one intersection-size, for families of permutations. We say that two permutations $\\sigma,\\pi \\in S_n$ {\\em agree on exactly $k$ points} if $\\#\\{i \\in [n]:\\ \\sigma(i)=\\pi(i)\\}=k$. We make the following conjecture.\n\\begin{conjecture}\n\\label{conj:all-t}\nFor any \\(t \\in \\mathbb{N}\\), and for \\(n\\) sufficiently large depending on \\(t\\), if $\\mathcal{A} \\subset S_n$ with no two permutations in $\\mathcal{A}$ agreeing on exactly \\(t-1\\) points, then\n\\[|\\mathcal{A}| \\leq (n-t)!,\\]\nand equality holds only if \\(\\mathcal{A}\\) is a \\(t\\)-coset of \\(S_{n}\\).\n\\end{conjecture}\nThis is a natural permutation analogue of the above-mentioned conjecture of Erd\\H{o}s on families of $k$-element sets with a forbidden intersection.\n\nOf course, a family of permutations in which no two permutations disagree everywhere is precisely a 1-intersecting family, so the $t=1$ case of Conjecture \\ref{conj:all-t} holds for all $n$, by the above-mentioned results of Deza and Frankl and of Cameron and Ku.\n\nIn this paper, we prove the $t=2$ case of Conjecture \\ref{conj:all-t}.\n\\begin{theorem}\n\\label{thm:main}\nIf $n$ is sufficiently large, and $\\mathcal{A} \\subset S_n$ is a family of permutations with no two permutations in $\\mathcal{A}$ agreeing at exactly one point, then\n$$|\\mathcal{A}| \\leq (n-2)!,$$\nand equality holds only if $\\mathcal{A}$ is a $2$-coset of $S_n$.\n\\end{theorem}\n\nWe also prove a corresponding stability result and a Hilton-Milner type result. We use similar techniques to in \\cite{cameronkuconj} and \\cite{jointpaper} --- namely, eigenvalue techniques, combined with the representation theory of $S_n$ --- but these techniques do not work quite so cleanly in the present case. Indeed, they are only capable of proving asymptotic results, and must be combined with stability arguments to prove the exact bound in Theorem \\ref{thm:main}.\n\nUnfortunately, for each $t \\geq 3$ in Conjecture \\ref{conj:all-t}, our techniques yield only a bound of $O((n-2)!)$. This remains true even if one uses `weighted' versions of Hoffman's bound, such as Lov\\'asz' theta-function bound (see for example \\cite{lovasz}), together with conjugation-invariant weightings. (While Lov\\'asz' theta-function concerns arbitrary weightings, not just conjugation-invariant ones, conjugation-invariance is necessary if one wishes to use representation theory to analyse eigenvalues and eigenspaces, via Theorem \\ref{thm:normalcayley} below.) It seems that new techniques will be required to solve the problem for \\(t \\geq 3\\).\n\n\\begin{comment}\nWe prove that for \\(n\\) sufficiently large, any family of permutations \\(\\mathcal{A} \\subset S_{n}\\) in which no two intersect at exactly one point has size at most \\((n-2)!\\), with equality only if \\(\\mathcal{A}\\) is a 2-coset of \\(S_{n}\\). (From now on, we abbreviate the condition `no two permutations in \\(\\mathcal{A}\\) agree at exactly one point' to `\\(\\mathcal{A}\\) has no singleton intersection'.)\n\nWe first use very similar techniques to in \\cite{jointpaper} to obtain an asymptotic result, namely that any family of permutations \\(\\mathcal{A} \\subset S_{n}\\) with no singleton intersection has size at most \\((1+O(1\/n))(n-2)!\\). (Interestingly, one cannot prove the exact result by applying the Delsarte-Hoffman bound using the conjugacy-class association scheme.) We then prove a rough stability result, namely that for any \\(c > 0\\), if \\(\\mathcal{A} \\subset S_{n}\\) is any family of permutations with no singleton intersection with size \\(|\\mathcal{A}| \\geq c(n-2)!\\), then there exists a \\(2\\)-coset of \\(S_{n}\\) containing all but \\(O((n-3)!)\\) of the permutations in \\(\\mathcal{A}\\). The proof of this makes use of a result proved by the author in \\cite{cameronkuconj}, namely that for any \\(c > 0\\), if \\(\\mathcal{B} \\subset S_{n}\\) is a 1-intersecting family of permutations with size \\(|\\mathcal{B}| \\geq c(n-2)!\\), then there exists a 1-coset of \\(S_{n}\\) containing all but \\(O((n-2)!)\\) of the permutations in \\(\\mathcal{B}\\).\n\nWe deduce from our rough stability result that for \\(n\\) sufficiently large, any family of permutations \\(\\mathcal{A} \\subset S_{n}\\) with no singleton intersection has size at most \\((n-2)!\\), with equality only if \\(\\mathcal{A}\\) is a 2-coset of \\(S_{n}\\).\n\nWe also obtain a Hilton-Milner type result for families of permutations with no singleton intersection, namely that if \\(\\mathcal{A} \\subset S_{n}\\) is such a family which is not contained within a coset of the stabilizer of two points, then it is no larger than the family\n\\begin{eqnarray*}\n\\mathcal{B} & = & \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\sigma(2)=2,\\ \\#\\{\\textrm{fixed points of }\\sigma \\geq 5\\} \\neq 1\\}\\\\\n&& \\cup \\{(1\\ 3)(2\\ 4),(1\\ 4)(2\\ 3),(1\\ 3\\ 2\\ 4),(1\\ 4\\ 2\\ 3)\\}.\n\\end{eqnarray*}\n\nWe conjecture that for any \\(t \\in \\mathbb{N}\\), for \\(n\\) sufficiently large depending on \\(t\\), any family of permutations \\(\\mathcal{A} \\subset S_{n}\\) in which no two permutations intersect at exactly \\(t-1\\) points has size at most \\((n-t)!\\). Further, we conjecture that equality holds only if \\(\\mathcal{A}\\) is a \\(t\\)-coset of \\(S_{n}\\). Interestingly, for general $t$, our techniques yield only a bound of $O((n-2)!). This remains true even if one uses `weighted' versions of Hoffman's bound, such as Lov\\'asz' theta-function bound, together with conjugation-invariant weightings. It seems that new techniques will be required to solve the problem for \\(t \\geq 3\\).\n\\end{comment}\n\n\\section{Notation, background and tools}\nIn this section, we outline our notation, recall the tools we will use to prove our main results, and give some background on the representation theory of $S_n$.\n\nFrom now on, we will often abbreviate the condition `no two permutations in \\(\\mathcal{A}\\) agree at exactly one point' to `\\(\\mathcal{A}\\) has no singleton intersection'.\n\nLet $\\Gamma_n$ be the graph on $S_n$ where two permutations \\(\\sigma\\) and \\(\\tau\\) are joined if and only if they agree at exactly one point. This is the Cayley graph on \\(S_{n}\\) generated by the set\n\\[\\mathcal{E}_{n}=\\{\\sigma \\in S_{n}:\\ \\sigma \\textrm{ has exactly one fixed point}\\}.\\]\nFor \\(n \\in \\mathbb{N}\\), let \\(d_{n}\\) denote the number of derangements of \\(n\\), i.e. permutations in \\(S_{n}\\) without fixed points; the inclusion-exclusion formula yields the familiar identity\n\\[d_{n}=\\sum_{i=0}^{n}(-1)^{i}\\frac{n!}{i!} = (1\/e+o(1))n!.\\]\nNote that\n$$|\\mathcal{E}_{n}|=nd_{n-1} = (1\/e+o(1))n!,$$\nso \\(\\Gamma_n\\) is \\(nd_{n-1}\\)-regular.\n\nObserve that a family \\(\\mathcal{A} \\subset S_{n}\\) in which no two permutations agree at exactly one point, is precisely an {\\em independent set} in \\(V(\\Gamma_n)\\), meaning a set of vertices of \\(\\Gamma_n\\) with no edges of \\(\\Gamma_n\\) between them. \n\nOur first step will be to apply Hoffman's eigenvalue bound to \\(\\Gamma_n\\).\n\\begin{theorem}[Hoffman, \\cite{hoffman}]\n\\label{thm:hoffman}\nLet \\(\\Gamma\\) be a \\(d\\)-regular graph on \\(N\\) vertices, whose adjacency matrix \\(A\\) has eigenvalues \\(d=\\lambda_{1} \\geq \\lambda_{2} \\geq \\ldots \\geq \\lambda_{N}\\). Then if \\(X \\subset V(\\Gamma)\\) is an independent set, we have\n\\[|X| \\leq \\frac{-\\lambda_{N}}{d-\\lambda_{N}}N.\\]\n\\end{theorem}\n\nThis will yield an approximate version of Theorem \\ref{thm:main}. To apply Theorem \\ref{thm:hoffman}, we will of course need to calculate the minimum eigenvalue of $\\Gamma_n$. Since \\(\\mathcal{E}_{n}\\) is a union of conjugacy-classes of \\(S_{n}\\), we may analyse the eigenvalues of $\\Gamma$ using representation-theoretic techniques. Before outlining these, we first give some background on the representation theory of finite groups.\n\n\\subsection*{Background on the representation theory of finite groups}\n\nIf $G$ is a finite group, let $\\mathbb{C}[G]$ denote the Euclidean space of all complex-valued functions on $G$, equipped with the inner product\n$$\\langle f,g \\rangle = \\frac{1}{|G|} \\sum_{\\sigma \\in G} f(\\sigma) \\overline{g(\\sigma)}\\quad (f,g \\in \\mathbb{C}[G]),$$\nand the corresponding Euclidean norm\n$$||f||_2 = \\sqrt{\\frac{1}{|G|} \\sum_{\\sigma \\in G} |f(\\sigma)|^2} \\quad (f \\in \\mathbb{C}[G]).$$\nRecall that a {\\em representation} of $G$ over $\\mathbb{C}$ is a pair $(\\rho,V)$, where $V$ is a finite-dimensional complex vector space, and $\\rho$ is a homomorphism from $G$ to $GL(V)$, the group of all invertible linear endomorphisms of $V$. The {\\em dimension} of the representation is the dimension of $V$. A representation $(\\rho,V)$ is said to be {\\em irreducible} if it has no proper subrepresentation, i.e. there is no proper subspace $V' \\leq V$ such that $V'$ is $\\rho(\\sigma)$-invariant for all $\\sigma \\in G$. We say that two representations \\((\\rho,V)\\) and \\((\\rho',V')\\) are {\\em isomorphic} if there is an invertible linear map \\(\\phi\\colon V \\to V'\\) such that such that \\(\\phi(\\rho(g)(v)) = \\rho'(g)(\\phi(v))\\) for all \\(g \\in G\\) and all \\(v \\in V\\). In this case, we write $(\\rho,V) \\cong (\\rho',V')$. \n\nRecall that, as a vector space, $\\mathbb{C}[G]$ may be equipped with the {\\em left-regular representation}, defined by\n$$(\\rho_{\\textrm{reg}}(\\pi) (f))(\\sigma) = f(\\pi^{-1} \\sigma)\\quad (f \\in \\mathbb{C}[G],\\ \\sigma,\\pi \\in G).$$\n\nFor any finite group $G$, we can choose a complete set \\(\\mathcal{R}\\) of non-isomorphic complex irreducible representations of \\(G\\) --- i.e., a set of complex irreducible representations of $G$ containing exactly one member of each isomorphism class of complex irreducible representations of $G$. For each $\\rho \\in \\mathcal{R}$, let $U_{\\rho}$ denote the subspace of $\\mathbb{C}[G]$ spanned by all isomorphic copies of $\\rho$ in the left-regular representation. Then we have an orthogonal direct-sum decomposition\n$$\\mathbb{C}[G] = \\bigoplus_{\\rho \\in \\mathcal{R}} U_{\\rho},$$\nand $\\dim(U_{\\rho}) = (\\dim(\\rho))^2$ for all $\\rho$.\n\nWe will use the following classical result to analyse the eigenvalues of $\\Gamma$.\n\n\\begin{theorem}[Frobenius \/ Schur \/ Diaconis-Shahshahani \\cite{diaconis}]\n\\label{thm:normalcayley}\nLet \\(G\\) be a finite group, let \\(X \\subset G\\) be an inverse-closed, conjugation-invariant subset of \\(G\\), let \\(\\Gamma = \\Cay(G,X)\\) be the Cayley graph on \\(G\\) with generating set \\(X\\), and let \\(A\\) be the adjacency matrix of \\(\\Gamma\\). Let \\(\\mathcal{R}\\) be a complete set of non-isomorphic complex irreducible representations of \\(G\\). Then each $U_{\\rho}$ is an eigenspace of $\\Gamma$, with corresponding eigenvalue \n\\begin{equation}\\label{eq:normalcayley}\\lambda_{\\rho} = \\frac{1}{\\dim(\\rho)} \\sum_{\\sigma \\in X} \\chi_{\\rho}(\\sigma).\\end{equation}\n\\end{theorem}\nHere, $\\chi_{\\rho}$ denotes the {\\em character} of $\\rho$, i.e.\n$$\\chi_{\\rho}(\\sigma) = \\textrm{Trace}(\\rho(\\sigma)).$$\n\n\\subsection*{Background and tools from the representation theory of $S_n$}\n\nWe will now give some brief background on the representation theory of $S_n$. For more detail, the reader may consult for example \\cite{sagan}, or the exposition in \\cite{jointpaper}.\n\nAs is well-known, there is an explicit one-to-one correspondence between irreducible representations of $S_n$ (up to isomorphism) and {\\em partitions} of $n$.\n\n\\begin{definition} A \\emph{partition} of \\(n\\) is a non-increasing sequence of positive integers summing to \\(n\\), i.e. a sequence $\\alpha = (\\alpha_1, \\ldots, \\alpha_l)$ with \\(\\alpha_{1} \\geq \\alpha_{2} \\geq \\ldots \\geq \\alpha_{l} \\geq 1\\) and \\(\\sum_{i=1}^{l} \\alpha_{i}=n\\).\n\\end{definition}\nIf $\\alpha$ is a partition of $n$, we write \\(\\alpha \\vdash n\\). For example, \\((3,2,2) \\vdash 7\\). We sometimes use the shorthand \\((3,2,2) = (3,2^{2})\\).\n\\begin{definition} The \\emph{Young diagram} of $\\alpha = (\\alpha_1,\\ldots,\\alpha_l)$ is an array of $n$ cells, having $l$ left-justified rows where row $i$ contains $\\alpha_i$ cells.\n\\end{definition}\nFor example, the Young diagram of the partition \\((3,2^{2})\\) is\\\\\n$$\\yng(3,2,2).$$\nFor each partition \\(\\alpha\\) of \\(n\\), we may define an irreducible representation \\(\\rho_{\\alpha}\\) of \\(S_{n}\\) called the {\\em Schur module} of \\(\\alpha\\); the Schur modules \\(\\{\\rho_{\\alpha}:\\ \\alpha \\vdash n\\}\\) form a complete set of non-isomorphic irreducible complex representations of \\(S_{n}\\). We write the corresponding character as \\(\\chi_{\\alpha}\\), and the corresponding dimension \\(\\dim(\\rho_{\\alpha}) = f^{\\alpha}\\).\n\nAn {\\em \\(\\alpha\\)-tableau} is a Young diagram of shape \\(\\alpha\\), each of whose cells contains a different number between 1 and \\(n\\). For example,\n$$\\young(713,52,46)$$\nis a $(3,2^2)$-tableau. We say that two \\(\\alpha\\)-tableaux are {\\em row-equivalent} if they contain the same numbers in each row. A row-equivalence-class of \\(\\alpha\\)-tableaux is called a {\\em \\(\\alpha\\)-tabloid}. Consider the natural action of \\(S_n\\) on the set of \\(\\alpha\\)-tabloids, and let \\(M^{\\alpha}\\) denote the induced permutation representation. We write \\(\\xi_{\\alpha}\\) for the character of \\(M_{\\alpha}\\); the \\(\\xi_{\\alpha}\\) are called the {\\em permutation characters} of \\(S_n\\).\n\nWe can express the irreducible characters in terms of the permutation characters using the {\\em determinantal formula}: for any partition \\(\\alpha\\) of \\(n\\),\n\\begin{equation}\\label{eq:determinantalformula} \\chi_{\\alpha} = \\sum_{\\pi \\in S_{n}} \\sgn(\\pi) \\xi_{\\alpha - \\textrm{id}+\\pi}.\\end{equation}\nHere, if \\(\\alpha = (\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})\\), \\(\\alpha - \\textrm{id}+\\pi\\) is defined to be the sequence\n\\[(\\alpha_{1}-1+\\pi(1),\\alpha_{2}-2+\\pi(2),\\ldots,\\alpha_{l}-l+\\pi(l)).\\]\nIf this sequence has all its entries non-negative, we let \\(\\overline{\\alpha-\\textrm{id}+\\pi}\\) be the partition of \\(n\\) obtained by reordering its entries, and we define \\(\\xi_{\\alpha - \\textrm{id}+\\pi} = \\xi_{\\overline{\\alpha-\\textrm{id}+\\pi}}\\). If the sequence has a negative entry, we define \\(\\xi_{\\alpha - \\textrm{id}+\\pi} = 0\\).\n\nFor any partition $\\alpha \\vdash n$, and any permutation $\\sigma \\in S_n$,  $\\xi_{\\alpha}(\\sigma)$ is the trace of the permutation representation $M^{\\alpha}$ at $\\sigma$, which is simply the number of $\\alpha$-tabloids fixed by $\\sigma$. For example, $\\xi_{(n-1,1)}(\\sigma)$ is the number of $(n-1,1)$-tabloids fixed by $\\sigma$, which is precisely the number of fixed points of $\\sigma$. It will be be convenient for us to express certain irreducible characters in terms of permutation characters, via the determinantal formula.\n\nWe need one final representation-theoretic tool.\n\n\\begin{definition}\nLet $\\alpha$ be a partition of $n$. The {\\em transpose} of $\\alpha$ is the partition $\\alpha^t$ whose Young diagram is obtained by transposing that of $\\alpha$. In other words, if the Young diagram of $\\alpha$ has $k$ columns of heights $c_1 \\geq c_2 \\geq \\ldots \\geq c_k$, then $\\alpha^{t} = (c_1,\\ldots,c_k)$.\n\\end{definition}\n\\begin{lemma}\n\\label{lemma:sign}\nFor any partition $\\alpha \\vdash n$, we have\n$$\\rho_{\\alpha^{t}} \\cong \\rho_{\\alpha} \\otimes \\sgn,$$\nwhere $\\sgn$ denotes the (1-dimensional) sign-representation of $S_n$, and $\\otimes$ denotes tensor product of representations. Hence,\n$$\\chi_{\\alpha^{t}}(\\sigma) = \\chi_{\\alpha}(\\sigma) \\sgn(\\sigma) \\quad \\forall \\sigma \\in S_n.$$\n\\end{lemma}\nThis will also be a useful tool for calculations.\n\n\\subsection*{Preliminary results}\nIn this section, we list some preliminary results which will be useful in our proofs. \n\nTheorem \\ref{thm:normalcayley} implies that if \\(\\Gamma = \\Cay(S_{n},X)\\) is a normal Cayley graph on $S_n$, then the eigenvalues of its adjacency matrix are given by\n\\begin{equation}\\label{eq:evals}\\lambda_{\\alpha} = \\frac{1}{f^{\\alpha}} \\sum_{\\sigma \\in X} \\chi_{\\alpha}(\\sigma) \\quad (\\alpha \\vdash n).\\end{equation}\nNote that an eigenvalue \\(\\lambda\\) has geometric multiplicity\n\\[\\sum_{\\alpha \\vdash n:\\ \\lambda_{\\alpha}=\\lambda}(f^{\\alpha})^{2}.\\]\n\nIn our case of $\\Gamma = \\Gamma_n$, we can bound all but 8 of these eigenvalues by $O((n-3)!)$, using only the fact that their geometric multiplicities are large. We will do this using the following simple identity.\n\\begin{lemma}\n\\label{lemma:trace}\nIf \\(G\\) is an \\(N\\)-vertex graph whose adjacency matrix \\(A\\) has eigenvalues \\(\\lambda_{1} \\geq \\ldots \\geq \\lambda_{N}\\) (repeated with their multiplicities), then\n\\[\\sum_{i=1}^{N} \\lambda_{i}^{2} = 2e(G).\\]\n\\end{lemma}\n\\begin{proof}\nWe have \n$$ \\sum_{i=1}^{N} \\lambda_{i}^{2} = \\textrm{Trace}(A^2) = \\textrm{Trace}(A^{\\top}A) = \\sum_{i,j}A_{i,j}^2 = \\sum_{i,j} A_{i,j} = 2e(G).$$\n\\end{proof}\nCombined with the fact that $\\lambda_{\\alpha}$ has multiplicity $\\geq (f^{\\alpha})^2$, this immediately implies the following.\n\\begin{lemma}\n\\label{lemma:cayley}\nIf $\\Gamma = \\Cay(S_n,X)$ is a normal Cayley graph on $S_n$, then its eigenvalues satisfy\n$$|\\lambda_{\\alpha}| \\leq \\frac{\\sqrt{|X|n!}}{f^{\\alpha}}\\quad (\\alpha \\vdash n).$$\n\\end{lemma}\nWe now recall a bound on the dimensions $f^{\\alpha}$ from \\cite{jointpaper}.\n\\begin{definition}\nWe say that a partition of \\(n\\) is \\(k\\)-{\\em fat} if its Young diagram has first row of length at least \\(n-k\\), \\(k\\)-{\\em tall} if its Young diagram has first column of height least \\(n-k\\), and \\(k\\)-{\\em medium} otherwise.\n\\end{definition}\n\\begin{lemma}(See \\cite{jointpaper}.)\nFor any \\(k \\in \\mathbb{N}\\), there exists a positive constant \\(c_{k}\\) such that for any \\(n \\in \\mathbb{N}\\), and any \\(k\\)-medium partition \\(\\alpha \\vdash n\\), we have\n\\[f^{\\alpha} \\geq c_{k} n^{k+1}.\\]\n\\end{lemma}\nIn particular, any 2-medium $\\alpha$ has $f^{\\alpha} \\geq c_2 n^3$. Combined with Lemma \\ref{lemma:cayley}, and the fact that $|\\mathcal{E}_n| = (1\/e+o(1))n!$, this yields a bound on all the eigenvalues corresponding to 2-medium partitions.\n\\begin{corollary}\n\\label{corr:2medium}\nIf $\\alpha$ is a 2-medium partition of $n$, then\n$$|\\lambda_{\\alpha}| \\leq \\frac{\\sqrt{1\/e+o(1)}n!}{c_2n^3} \\leq C (n-3)!,$$\nwhere $C$ is an absolute constant.\n\\end{corollary} \n\n\\section{An asymptotic result}\nWe are now ready to prove our asymptotic version of Theorem \\ref{thm:main}.\n\\begin{proposition}\n\\label{prop:asymptoticbound}\nLet \\(\\mathcal{A} \\subset S_{n}\\) be such that no two permutations in \\(\\mathcal{A}\\) agree on exactly one point. Then\n\\[|\\mathcal{A}| \\leq (1+O(1\/n))(n-2)!.\\]\n\\end{proposition}\n\\begin{proof}\nTo prove this, we will simply calculate the minimum eigenvalue of $\\Gamma_n$ and apply Theorem \\ref{thm:hoffman} (Hoffman's bound). By Corollary \\ref{corr:2medium}, all the 2-medium partitions $\\alpha$ of $n$ have\n\\begin{equation} \\label{eq:uniformbound} |\\lambda_{\\alpha}| \\leq C(n-3)!.\\end{equation}\n\nThere are 8 other partitions of $n$. Using equation (\\ref{eq:evals}), combined with the determinantal formula (\\ref{eq:determinantalformula}), one finds that these have eigenvalues as follows.\\\\\n\n\\begin{tabular}{l|l}\n\\(\\alpha\\) & \\(\\lambda_{\\alpha}\\)\\\\\n\\hline\n\\((n)\\) & \\(nd_{n-1}\\)\\\\\n\\((1^{n})\\) & \\((-1)^{n-2}n(n-2)\\)\\\\\n\\((n-1,1)\\) & \\(0\\)\\\\\n\\((2,1^{n-2})\\) & \\(0\\)\\\\\n\\((n-2,2)\\) & \\(-\\frac{nd_{n-1}}{(n-1)(n-2)-2} (1+(-1)^{n}(n-2)\/d_{n-1})\\)\\\\\n\\((2,2,1^{n-4})\\) & \\((-1)^{n-1}(n-2)^{2}\\)\\\\\n\\((n-2,1,1)\\) & \\(-\\frac{nd_{n-1}}{(n-1)(n-2)}(1-(-1)^{n}(n-2)\/d_{n-1})\\)\\\\\n\\((3,1^{n-3})\\) & \\((-1)^{n}n(n-4)\\)\\\\\n\\end{tabular}\\\\\n\\\\\n\nNote that, since $\\chi_{(n)} \\equiv 1$ is the trivial character, by (\\ref{eq:evals}) we have\n$$\\lambda_{(n)} = |\\mathcal{E}_n| = nd_{n-1}.$$\n\nIn the case $\\alpha = (n-2,2)$, the determinantal formula yields\n$$\\chi_{(n-2,2)} = \\xi_{(n-2,2)} - \\xi_{(n-1,1)}.$$\nNote that $\\xi_{(n-2,2)}(\\sigma) = \\#\\{x \\in [n]^{(2)}:\\ \\sigma(x)=x\\}$ is simply the number of pairs which are fixed by $\\sigma$, and $\\xi_{(n-1,1)}(\\sigma)$ is the number of fixed points of $\\sigma$. We have\n$$f^{(n-2,2)} = \\chi_{(n-2,2)}(\\textrm{Id}) = \\xi_{(n-2,2)}(\\textrm{Id}) - \\xi_{(n-1,1)}(\\textrm{Id}) = {n \\choose 2} - n.$$\nTherefore, by (\\ref{eq:evals}), we have\n\\begin{align*}\n\\lambda_{(n-2,2)} & = \\frac{1}{{n \\choose 2}-n} \\sum_{\\sigma \\in \\mathcal{E}_n} (\\xi_{(n-2,2)}(\\sigma)-\\xi_{(n-1,1)}(\\sigma))\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left(\\sum_{ij \\in [n]^{(2)}} \\sum_{\\sigma \\in \\mathcal{E}_n} 1\\{\\sigma\\{i,j\\} = \\{i,j\\}\\} - nd_{n-1}\\right)\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left({n \\choose 2} (n-2)d_{n-3} - nd_{n-1}\\right)\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left({n \\choose 2} (n-2) \\frac{d_{n-1}+ (-1)^{n-1}(n-2)}{(n-1)(n-2)} -nd_{n-1}\\right)\\\\\n& = -\\frac{n}{2\\left({n \\choose 2}-n\\right)} (d_{n-1}+(-1)^{n}(n-2))\\\\\n& = -\\frac{nd_{n-1}}{2\\left({n\\choose 2}-n\\right)}(1+(-1)^{n}(n-2)\/d_{n-1})\\\\\n& = -\\frac{nd_{n-1}}{(n-1)(n-2)-2} (1+(-1)^{n}(n-2)\/d_{n-1}).\n\\end{align*}\n\nThe calculations of $\\lambda_{(n-1,1)}$ and $\\lambda_{(n-2,1^2)}$ are very similar.\n\nFor $\\alpha = (1^n)$, we have $\\chi_{(1^n)} = \\sgn$, so by (\\ref{eq:evals}), we have\n$$\\lambda_{(1^n)} = \\sum_{\\sigma \\in \\mathcal{E}_n} \\sgn(\\sigma) = n(e_{n-1} - o_{n-1}),$$\nwhere $e_n$ and $o_n$ denote the number of respectively even and odd derangements in $S_n$. It is well-known (and can easily be proved by induction) that $e_n - o_n = (-1)^{n-1}(n-1)$ for all $n$. This yields\n\\[\\lambda_{(1^{n})} = n(e_{n-1}-o_{n-1}) = (-1)^{n-2}n(n-2).\\]\n\nFor the calculations of $\\lambda_{(2,1^{n-2})}$, $\\lambda_{(2,2,1^{n-4})}$ and $\\lambda_{(3,1^{n-3})}$, we use Lemma \\ref{lemma:sign}, combined with (\\ref{eq:evals}) and the determinantal formula. For example,\n\n \\begin{align*}\n\\lambda_{(2,2,1^{n-4})} & = \\frac{1}{{n \\choose 2}-n} \\sum_{\\sigma \\in \\mathcal{E}_n} \\sgn(\\sigma)(\\xi_{(n-2,2)}(\\sigma)-\\xi_{(n-1,1)}(\\sigma))\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left(\\sum_{ij \\in [n]^{(2)}} \\sum_{\\sigma \\in \\mathcal{E}_n} \\sgn(\\sigma)1\\{\\sigma\\{i,j\\} = \\{i,j\\}\\} - n(e_{n-1}-o_{n-1})\\right)\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left({n \\choose 2} (n-2)(o_{n-3}-e_{n-3}) - n(n-2)(-1)^{n-2}\\right)\\\\\n& = \\frac{1}{{n \\choose 2}-n} \\left({n \\choose 2} (n-2) (-1)^{n-3}(n-4)+ n(n-2)(-1)^{n-1}\\right)\\\\\n& = -\\frac{(-1)^{n-1}}{2\\left({n \\choose 2}-n\\right)} (n(n-1)(n-2)(n-4)+2n(n-2))\\\\\n& = -\\frac{(-1)^{n-1}n(n-2)}{(n-1)(n-2)-2} ((n-1)(n-4)+2)\\\\\n& = -\\frac{(-1)^{n-1}n(n-2)}{n(n-3)} (n-2)(n-3)\\\\\n& =  (-1)^{n-1}(n-2)^{2}.\n\\end{align*}\n\nWe see from equation (\\ref{eq:uniformbound}) and the table following it that $\\Gamma_n$ has\n\\[\\lambda_N = -(1+O(1\/n))\\frac{nd_{n-1}}{n^{2}}.\\]\nApplying Theorem \\ref{thm:hoffman} (Hoffman's bound) with \\(d = nd_{n-1}\\), \\(N=n!\\), and the above bound on $\\lambda_N$, we see that any independent set $\\mathcal{A} \\subset V(\\Gamma_n)$ has\n\\[|\\mathcal{A}| \\leq (1+O(1\/n))(n-2)!,\\]\nproving Proposition \\ref{prop:asymptoticbound}.\n\\end{proof}\n\nFrom the above proof, we can also read off a two-family version of Proposition \\ref{prop:asymptoticbound}. Recall the following `cross-independent' version of Hoffman's theorem.\n\\begin{theorem}\n\\label{thm:crosshoffman}\nLet \\(\\Gamma\\) be a \\(d\\)-regular graph on \\(N\\) vertices, whose adjacency matrix \\(A\\) has eigenvalues \\(\\lambda_{1}=d \\geq \\lambda_{2} \\geq \\ldots \\geq \\lambda_{N}\\). Let \\(\\nu = \\max(|\\lambda_{2}|,|\\lambda_{N}|)\\). Then if \\(X,Y \\subset V(\\Gamma)\\) with \\(xy \\notin E(\\Gamma)\\) for any $x \\in X$ and any $y \\in Y$, we have:\n\\[\\sqrt{|X||Y|} \\leq \\frac{\\nu}{d+\\nu}N.\\]\n\\end{theorem}\n(For a proof, see for example \\cite{jointpaper}.) By equation (\\ref{eq:uniformbound}) and the table following it, our graph $\\Gamma_n$ has $\\nu = |\\lambda_N|$ for $n$ sufficiently large. Hence, Theorem \\ref{thm:crosshoffman} immediately gives the following.\n\n\\begin{proposition}\n\\label{prop:cross}\nIf \\(\\mathcal{A},\\mathcal{B} \\subset S_n\\) are families of permutations such that no permutation in \\(\\mathcal{A}\\) agrees with any permutation in \\(\\mathcal{B}\\) at exactly one point, then\n\\[|\\mathcal{A}||\\mathcal{B}| \\leq (1+O(1\/n))((n-2)!)^{2}.\\]\n\\end{proposition}\n\nThis proposition will be used in the proof of our stability result in the next section.\n\n\\section{A stability result}\nOur main aim in this section is to prove the following stability result, which says that a `large' family of permutation with no singleton intersection, is `almost' contained within a 2-coset.\n\\begin{theorem}\n\\label{thm:roughstability}\nLet \\(c >0\\). Let \\(\\mathcal{A} \\subset S_{n}\\) with no singleton intersection, and with \\(|\\mathcal{A}| \\geq c(n-2)!\\). Then there exist \\(i,j,k,l \\in [n]\\) such that\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{i \\mapsto j,k \\mapsto l}| \\leq K_{c}(n-3)!,\\]\nwhere $K_c >0$ depends only upon $c$.\n\\end{theorem}\n\nIn our proof, we will use the following additional notation. If $f = f(n,c)$ and $g = g(n,c)$ are functions of $n$ and $c$, we will write $f = O_c (g)$ to mean that for each $c >0$, there exists $K_c >0$ depending upon $c$ alone, such that $f(n,c) \\leq K_c g(n,c)$ for all $n \\in \\mathbb{N}$. \n\nFor two permutations \\(\\sigma,\\tau \\in S_{n}\\), we write \\(\\sigma \\cap \\tau\\) for the set \\(\\{i:\\ \\sigma(i)=\\tau(i)\\}\\), i.e. for the set of points at which they agree. For \\(\\mathcal{A} \\subset S_{n}\\), and for distinct \\(i_{1},\\ldots,i_{l} \\in [n]\\) and distinct \\(j_{1},\\ldots,j_{l} \\in [n]\\), we write\n\\[\\mathcal{A}_{i_{1} \\mapsto j_{1}, i_{2} \\mapsto j_{2}, \\ldots, i_{l} \\mapsto j_{l}} := \\{\\sigma \\in \\mathcal{A}: \\sigma(i_{k})=j_{k}\\ \\forall k \\in [l]\\}.\\]\nSimilarly, for \\(\\mathcal{A} \\subset S_{n}\\), and for \\(S,T \\subset [n]\\), we write\n\\[\\mathcal{A}_{S \\to T} = \\{\\sigma \\in \\mathcal{A}:\\ \\sigma(S) \\subset T\\}\\]\nfor the subset of permutations in \\(\\mathcal{A}\\) which map \\(S\\) into \\(T\\).\n\nIf $H$ is an inner product space, and $W$ is a subspace of $H$, we write $P_W:H \\to H$ for orthogonal projection onto $W$.\n\nIf $V$ is a finite set, and $X \\subset V$, we write $\\chi_{X} \\in \\mathbb{R}^V$ for the {\\em characteristic vector} of $X$, meaning the vector whose $i$th coordinate is $1$ if $i \\in X$ and 0 if $i \\notin X$. We use $\\mathbf{f}$ to denote the all-1's vector in $\\mathbb{R}^V$. We regard $\\mathbb{R}^V$ as an inner product space, equipped with the inner product\n$$\\langle x,y \\rangle = \\frac{1}{|V|} \\sum_{i \\in V} x_i y_i;$$\nwe let\n$$||x||_2 = \\sqrt{\\frac{1}{|V|} \\sum_{i \\in V} x_i^2 }$$\ndenote the corresponding Euclidean norm.\n\nWe will need the following `stability version' of Hoffman's theorem.\n\n\\begin{lemma}\n\\label{lemma:stability-hoffman}\nLet \\(\\Gamma = (V,E)\\) be an $N$-vertex, \\(d\\)-regular graph whose adjacency matrix $A$ has eigenvalues \\(d=\\lambda_{1} \\geq \\lambda_{2} \\geq \\ldots \\geq \\lambda_{N}\\). Let $M \\in \\{1,2,\\ldots,N-1\\}$, and let\n\\[U = \\textrm{Span}\\{\\mathbf{f}\\} \\oplus \\bigoplus_{i > M}\\\\ker(A-\\lambda_i I).\\]\nLet $X \\subset V$ be an independent set, and let \\(\\alpha = |X|\/N\\). Define\n$$D = ||\\chi_X - P_{U} (\\chi_X)||_2,$$\ni.e. $D$ is the Euclidean distance from $\\chi_X$ to $U$. Then\n\\[D^2 \\leq \\frac{(1-\\alpha)|\\lambda_{N}| - d \\alpha}{|\\lambda_{N}|-|\\lambda_{M}|}\\alpha.\\]\n\\end{lemma}\n\nFor completeness, we include a proof.\n\\begin{proof}\nLet \\(u_{1} = \\mathbf{f}, u_{2},\\ldots,u_{N}\\) be an orthonormal basis of real eigenvectors of \\(A\\) corresponding to the eigenvalues \\(\\lambda_{1},\\lambda_2,\\ldots,\\lambda_{N}\\). Write\n\\[\\chi_{X}=\\sum_{i=1}^{N} \\xi_{i} u_{i}\\]\nas a linear combination of the eigenvectors of \\(A\\). Note that\n\\begin{equation}\\label{eq:measure} \\xi_{1}=\\langle \\chi_X, \\mathbf{f} \\rangle = |X|\/N = \\alpha\\end{equation}\nand\n\\begin{equation} \\label{eq:parseval} \\sum_{i=1}^{N} \\xi_{i}^{2} = \\langle \\chi_X, \\chi_X \\rangle = |X|\/N = \\alpha.\\end{equation}\nThen we have the crucial property:\n\\[0=\\frac{1}{N} \\sum_{x,y \\in X}A_{x,y}= \\langle \\chi_{X},A \\chi_{X} \\rangle = \\sum_{i=1}^{N} \\lambda_{i} \\xi_{i}^{2} \\geq \\lambda_{1}\\xi_{1}^{2} + \\lambda_{N} \\sum_{i=M+1}^{N} \\xi_{i}^{2} + \\lambda_{M} \\sum_{i=2}^{M}\\xi_{i}^{2}.\\]\nNote that\n\\[\\sum_{i=2}^{M} \\xi_{i}^{2} = D^{2}\\]\nand\n\\[\\sum_{i=M+1}^{N} \\xi_{i}^{2} = \\alpha - \\alpha^{2} - D^{2},\\]\nusing (\\ref{eq:measure}) and (\\ref{eq:parseval}). Hence,\n\\[0 \\geq \\lambda_{1} \\alpha^{2} + \\lambda_{N} (\\alpha -\\alpha^{2} -D^{2}) + \\lambda_{M} D^{2}.\\]\nRearranging, we obtain:\n\\[D^{2} \\leq  \\frac{(1-\\alpha)|\\lambda_{N}| - \\lambda_{1} \\alpha}{|\\lambda_{N}|-|\\lambda_{M}|}\\alpha,\\]\nas required.\n\\end{proof}\n\nWe will also need the following two technical lemmas, which we prove using Proposition \\ref{prop:cross}.\n\n\\begin{lemma}\n\\label{lemma:twoposvalues}\nLet \\(\\mathcal{A},\\mathcal{B} \\subset S_{n}\\) such that for any \\(\\sigma \\in \\mathcal{A}\\) and \\(\\tau \\in \\mathcal{B}\\),\n\\begin{itemize}\n\\item If \\(\\sigma(1)=\\tau(1)\\), then \\(|\\sigma \\cap \\tau| \\neq 2\\);\n\\item If \\(\\sigma(1)\\neq \\tau(1)\\), then \\(|\\sigma \\cap \\tau| \\neq 1\\).\n\\end{itemize}\nThen\n\\[|\\mathcal{A}||\\mathcal{B}| \\leq 4(1+O(1\/n))((n-2)!)^{2}.\\]\n\\end{lemma}\n\\begin{proof}\nLet\n\\[J = \\{j \\in [n]:\\ |\\mathcal{A}_{1 \\mapsto j}| \\geq \\frac{|\\mathcal{A}|}{2n}\\}.\\]\nThen \\(|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\to J}| < |\\mathcal{A}|\/2\\), so \\(|\\mathcal{A}_{1 \\to J}| > |\\mathcal{A}|\/2\\).\n\nFix \\(j \\in [n]\\), and consider the families\n\\[\\mathcal{A}_{1 \\mapsto j},\\ \\mathcal{B}_{1 \\mapsto j}.\\]\nNotice that\n\\[(1\\ j)\\mathcal{A}_{1 \\mapsto j},(1\\ j)\\mathcal{B}_{1 \\mapsto j}\\]\nare families of permutations fixing 1 and with \\(|\\sigma \\cap \\tau| \\neq 2\\) for any \\(\\sigma \\in (1\\ j)\\mathcal{A}_{1 \\mapsto j}\\) and any \\(\\tau \\in (1\\ j)\\mathcal{B}_{1 \\mapsto j}\\). Restricting them to \\(\\{2,\\ldots,n\\}\\) yields a pair of families \\(\\mathcal{E},\\mathcal{F} \\subset S_{\\{2,\\ldots,n\\}}\\) with \\(|\\sigma \\cap \\tau| \\neq 1\\ \\forall \\sigma \\in \\mathcal{E},\\ \\tau \\in \\mathcal{F}\\). Applying Proposition \\ref{prop:cross} to \\(\\mathcal{E}\\) and \\(\\mathcal{F}\\) gives\n\\[|\\mathcal{E}||\\mathcal{F}| \\leq (1+O(1\/n))((n-3)!)^{2},\\]\nso\n\\[|\\mathcal{A}_{1 \\mapsto j}||\\mathcal{B}_{1 \\mapsto j}| \\leq (1+O(1\/n))((n-3)!)^{2}.\\]\nHence,\n\\[\\frac{|\\mathcal{A}|}{2n}|\\mathcal{B}_{1 \\mapsto j}| < (1+O(1\/n))((n-3)!)^{2}\\ \\forall j \\in J,\\]\nand therefore\n\\[|\\mathcal{A}||\\mathcal{B}_{1 \\mapsto j}| < 2n(1+O(1\/n))((n-3)!)^{2} \\ \\forall j \\in J.\\]\nSumming over all \\(j \\in J\\) gives:\n\\begin{equation}\n\\label{eq:crossbound1}\n|\\mathcal{A}||\\mathcal{B}_{1 \\to J}| < 2(1+O(1\/n))((n-2)!)^{2}.\n\\end{equation}\nNotice that no permutation in \\(\\mathcal{A}_{1 \\to J}\\) can have singleton intersection with any permutation in \\(\\mathcal{B}_{1 \\nrightarrow J}\\), and therefore by Proposition \\ref{prop:cross} again, \n\\[|\\mathcal{A}_{1 \\to J}||\\mathcal{B}_{1 \\nrightarrow J}| \\leq (1+O(1\/n))((n-2)!)^{2}.\\]\nHence,\n\\begin{equation}\n\\label{eq:crossbound2}\n|\\mathcal{A}||\\mathcal{B}_{1 \\nrightarrow J}| < 2(1+O(1\/n))((n-2)!)^{2}.\n\\end{equation}\nCombining (\\ref{eq:crossbound1}) and (\\ref{eq:crossbound2}) gives:\n\\[|\\mathcal{A}||\\mathcal{B}| \\leq 4(1+O(1\/n))((n-2)!)^{2},\\]\nas required.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:3posvalues}\nLet \\(\\mathcal{A},\\mathcal{B} \\subset S_{n}\\) such that for any \\(\\sigma \\in \\mathcal{A}\\) and any \\(\\tau \\in \\mathcal{B}\\),\n\\begin{itemize}\n\\item If exactly one of \\(\\sigma(1)=\\tau(1),\\ \\sigma(2)=\\tau(2)\\) holds, then \\(|\\sigma \\cap \\tau| \\neq 2\\);\n\\item If both hold, then \\(|\\sigma \\cap \\tau| \\neq 3\\);\n\\item If neither holds, then \\(|\\sigma \\cap \\tau| \\neq 1\\).\n\\end{itemize}\nThen\n\\[|\\mathcal{A}||\\mathcal{B}| \\leq 16(1+O(1\/n))((n-2)!)^{2}.\\]\n\\end{lemma}\n\\begin{proof}\nLet\n\\[K = \\{k \\in [n]: |\\mathcal{A}_{1 \\mapsto k}| \\geq \\frac{|\\mathcal{A}|}{2n}\\}.\\]\nAs in the proof of Lemma \\ref{lemma:twoposvalues}, we have \\(|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\to K}| < |\\mathcal{A}|\/2\\), so \\(|\\mathcal{A}_{1 \\to K}| > |\\mathcal{A}|\/2\\).\n\nFix \\(k \\in [n]\\), and consider the pair of families\n\\[\\mathcal{A}_{1 \\mapsto k},\\ \\mathcal{B}_{1 \\mapsto k}.\\]\nNotice that\n\\[(1\\ k)\\mathcal{A}_{1 \\mapsto k},\\ (1\\ k)\\mathcal{B}_{1 \\mapsto k}\\]\nare families of permutations fixing 1, such that for any \\(\\sigma \\in (1\\ j)\\mathcal{A}_{1 \\mapsto j},\\) and any \\(\\tau \\in (1\\ j)\\mathcal{B}_{1 \\mapsto j}\\),\n\\begin{itemize}\n\\item if \\(\\sigma(2) = \\tau(2)\\) we have \\(|\\sigma \\cap \\tau|\\neq 3\\);\n\\item if \\(\\sigma(2)\\neq \\tau(2)\\) we have \\(|\\sigma \\cap \\tau| \\neq 2\\).\n\\end{itemize}\nRestricting this pair of families to \\(\\{2,\\ldots,n\\}\\) yields a pair of families \\(\\mathcal{E},\\mathcal{F} \\subset S_{\\{2,\\ldots,n\\}}\\) such that for any \\(\\sigma \\in \\mathcal{E}\\) and any \\(\\tau \\in \\mathcal{F}\\),\n\\begin{itemize}\n\\item if \\(\\sigma(2) = \\tau(2)\\) we have \\(|\\sigma \\cap \\tau|\\neq 2\\);\n\\item if \\(\\sigma(2)\\neq \\tau(2)\\) we have \\(|\\sigma \\cap \\tau| \\neq 1\\).\n\\end{itemize}\nApplying Lemma \\ref{lemma:twoposvalues} to \\(\\mathcal{E}\\) and \\(\\mathcal{F}\\) gives\n\\[|\\mathcal{E}||\\mathcal{F}| \\leq 4(1+O(1\/n))((n-3)!)^{2},\\]\nso\n\\[|\\mathcal{A}_{1 \\mapsto k}||\\mathcal{B}_{1 \\mapsto k}| \\leq 4(1+O(1\/n))((n-3)!)^{2}.\\]\nHence,\n\\[\\frac{|\\mathcal{A}|}{2n}|\\mathcal{B}_{1 \\mapsto k}| < 4(1+O(1\/n))((n-3)!)^{2}\\quad \\forall k \\in K,\\]\ni.e.\n\\[|\\mathcal{A}||\\mathcal{B}_{1 \\mapsto k}| < 8n(1+O(1\/n))((n-3)!)^{2}\\quad \\forall k \\in K.\\]\nSumming over all \\(k \\in K\\) gives\n\\begin{equation}\n\\label{eq:crossbound3}\n|\\mathcal{A}||\\mathcal{B}_{1 \\to K}| < 8(1+O(1\/n))((n-2)!)^{2}.\n\\end{equation}\nObserve that \\(\\mathcal{A}_{1 \\to K},\\ \\mathcal{B}_{1 \\nrightarrow K}\\) are families of permutations satisfying the hypotheses of Lemma \\ref{lemma:twoposvalues}, and therefore\n\\[|\\mathcal{A}_{1 \\to K}||\\mathcal{B}_{1 \\nrightarrow K}| \\leq 4(1+O(1\/n))^{2}((n-2)!)^{2}.\\]\nHence,\n\\begin{equation}\n\\label{eq:crossbound4}\n|\\mathcal{A}||\\mathcal{B}_{1 \\nrightarrow K}| \\leq 8(1+O(1\/n))^{2}((n-2)!)^{2}\n\\end{equation}\nCombining equations (\\ref{eq:crossbound3}) and (\\ref{eq:crossbound4}) gives\n\\[|\\mathcal{A}||\\mathcal{B}| \\leq 16(1+O(1\/n))((n-2)!)^{2}\\]\nas required.\n\\end{proof}\n\nWe can now prove Theorem \\ref{thm:roughstability}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:roughstability}.]\nLet \\(\\mathcal{A} \\subset S_{n}\\) with no singleton intersection, and with \\(|\\mathcal{A}| \\geq c(n-2)!\\).\n\nNote that by taking $K_c \\geq 1\/c^{26}$, we may assume that $c \\geq n^{-1\/13}$ (otherwise the conclusion of the theorem holds trivially). Note also that by taking $K_c \\geq n_0(c)^2$, we may assume throughout that $n \\geq n_0(c)$ for any integer $n_0(c)$ depending only on $c$.\n\nWe first apply Lemma \\ref{lemma:stability-hoffman} to our graph $\\Gamma=\\Gamma_n$, with $X = \\mathcal{A}$ and $M = n! - (f^{(n-2,2)})^2 - (f^{(n-2,1^2)})^2$, so that\n$$\\{\\lambda_{M+1},\\lambda_{M+2} \\ldots, \\lambda_N\\} = \\{\\lambda_{(n-2,2)}, \\lambda_{(n-2,1^2)}\\},$$\nand\n$$U = \\textrm{Span}\\{\\mathbf{f}\\} \\oplus U_{(n-2,2)} \\oplus U_{(n-2,1^2)}.$$\nBy (\\ref{eq:uniformbound}) and the table following it, we have\n$$|\\lambda_M| \\leq C(n-3)! \\leq \\frac{C'}{n}|\\lambda_N|,$$\nwhere $C'$ is an absolute constant. Hence, by Lemma \\ref{lemma:stability-hoffman}, we have\n\\[||\\chi_{\\mathcal{A}} - P_{U}(\\chi_{\\mathcal{A}})||_2^{2} \\leq (1-c)(1+O(1\/n))||\\chi_{\\mathcal{A}}||_2^{2}.\\]\n\nNow let\n\\[U_{t} = \\bigoplus_{\\alpha \\vdash n:\\ \\alpha \\textrm{ is }t\\textrm{-fat}} U_{\\alpha}.\\]\nSince the partitions $(n)$, $(n-2,2)$ and $(n-2,1^2)$ are all 2-fat, we have \\(U \\leq U_{2}\\), so\n\\[||\\chi_{\\mathcal{A}} - P_{U_{2}}(\\chi_{\\mathcal{A}})||_2^{2} \\leq ||\\chi_{\\mathcal{A}} - P_{U}(\\chi_{\\mathcal{A}})||_2^{2} \\leq (1-c)(1+O(1\/n))||\\chi_{\\mathcal{A}}||_2^{2}.\\]\n\nIt follows from the proof of Lemma 7 in \\cite{dezafranklstability} that for any fixed \\(c > 0\\) and \\(t \\in \\mathbb{N}\\), if\n\\[||\\chi_{\\mathcal{A}} - P_{U_{t}}(\\chi_{\\mathcal{A}})||_2^{2} \\leq (1-c)(1+O(1\/n))||\\chi_{\\mathcal{A}}||_2^{2},\\]\nthen there exist \\(i_{1},\\ldots,i_{t}\\) and \\(j_{1},\\ldots,j_{t}\\) such that\n\\[|A_{i_{1}\\mapsto j_{1},\\ldots,i_{t} \\mapsto j_{t}}| \\geq f(n) c(n-2t)!,\\]\nwhere \\(f(n) = \\Theta(\\sqrt{n\/\\log n})\\), provided $c = \\Omega(n^{-1\/12})$ (which holds by assumption).\n\nApplying this with \\(t=2\\) shows that there exist \\(i,j,k\\) and \\(l \\in [n]\\) such that\n\\[|\\mathcal{A}_{i \\mapsto j,k \\mapsto l}| \\geq f(n) c(n-4)!.\\]\n\nWithout loss of generality, we may assume \\(i=j=1,k=l=2\\). Hence, we have\n\\begin{equation}\n\\label{eq:lowerbound1122}\n|\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\geq f(n)c(n-4)!.\n\\end{equation}\n\nOur aim is now to show that all but at most \\(o((n-2)!)\\) of the permutations in \\(\\mathcal{A}\\) fix either \\(1\\) or \\(2\\). To show this, it will suffice to prove the following.\n\n\\begin{claim}\nIf \\(j \\neq 1\\) and \\(l\\neq 2\\), then \\(|\\mathcal{A}_{1 \\mapsto j,2 \\mapsto l}| \\leq o((n-4)!)\\).\n\\end{claim}\n\n\\begin{proof}[Proof of claim.]\nFirst, we deal with the case \\(j=2,\\ l=1\\). Clearly, restricting the families\n\\[\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 1},\\ \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\]\nto \\(\\{3,4,\\ldots,n\\}\\) (`deleting 1 and 2') produces two families \\(\\mathcal{C},\\mathcal{D} \\subset S_{\\{3,\\ldots,n\\}}\\) such that $|\\sigma \\cap \\tau| \\neq 1\\ \\forall \\sigma \\in \\mathcal{C},\\tau \\in \\mathcal{D}$. Applying Proposition \\ref{prop:cross} to $\\mathcal{C},\\mathcal{D}$ gives\n\\[|\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 1}||\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| = |\\mathcal{C}||\\mathcal{D}| \\leq (1+O(1\/n))((n-4)!)^{2}.\\]\nTherefore, by (\\ref{eq:lowerbound1122}),\n\\[|\\mathcal{A}_{1 \\mapsto 2,2\\mapsto 1}| \\leq o((n-4)!),\\]\nas required.\n\nNext, we deal with the case $j=2,\\ l \\notin \\{1,2\\}$. Without loss of generality, we may assume that \\(l=3\\); we need to show that \\(|\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3}| \\leq o((n-4)!)\\).\n\nConsider the pair of families\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2},\\ (1\\ 3\\ 2)\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3}.\\]\nLet \\(\\sigma \\in \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2},\\ \\tau' = (1 \\ 3\\ 2)\\tau\\) where \\(\\tau \\in \\mathcal{A}_{1 \\mapsto 2, 2 \\mapsto 3}\\). Let \\(a = \\sigma^{-1}(3)\\). Observe that\n\\begin{itemize}\n\\item If \\(\\tau(a)=1\\), i.e. \\(\\tau'^{-1}(3) = a\\), then \\(\\sigma(a) = \\tau'(a) = 3\\), so \\(\\sigma\\) and \\(\\tau'\\) agree wherever \\(\\sigma\\) and \\(\\tau\\) agree, and also at \\(1,2\\) and \\(a\\), but nowhere else. Hence, \\(\\sigma\\) and \\(\\tau'\\) cannot agree at exactly 4 points.\n\\item If \\(\\tau(a) \\neq 1\\), i.e. \\(\\tau'^{-1}(3) \\neq a\\), then \\(\\sigma(a) \\neq \\tau'(a)\\), and therefore \\(\\sigma\\) and \\(\\tau'\\) agree wherever \\(\\sigma\\) and \\(\\tau\\) agree, and also at \\(1\\) and \\(2\\), but nowhere else. Hence, \\(\\sigma\\) and \\(\\tau'\\) cannot agree at exactly 3 points.\n\\end{itemize}\nLet\n\\[\\mathcal{C} = (\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2})^{-1},\\quad \\mathcal{D} = ((1 \\ 3 \\ 2)\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3})^{-1}.\\]\nThen \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) are families of permutations fixing both \\(1\\) and \\(2\\). Moreover, for any \\(\\rho \\in \\mathcal{C}\\) and any \\(\\pi \\in \\mathcal{D}\\),\n\\begin{itemize}\n\\item if \\(\\rho(3)=\\pi(3)\\), then \\(|\\rho \\cap \\pi| \\neq 4\\);\n\\item if \\(\\rho(3) \\neq \\pi(3)\\), then \\(|\\rho \\cap \\pi| \\neq 3\\).\n\\end{itemize}\nRestrict \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) to \\(\\{3,4,\\ldots,n\\}\\) (`delete 1 and 2') to obtain the families \\(\\mathcal{C}',\\mathcal{D}' \\subset S_{\\{3,\\ldots,n\\}}\\). Observe that for any \\(\\rho' \\in \\mathcal{C}'\\) and any \\(\\pi' \\in \\mathcal{D}'\\),\n\\begin{itemize}\n\\item if \\(\\rho'(3)=\\pi'(3)\\), then \\(|\\rho' \\cap \\pi'| \\neq 2\\);\n\\item if \\(\\rho'(3) \\neq \\pi'(3)\\), then \\(|\\rho' \\cap \\pi'| \\neq 1\\).\n\\end{itemize} \nClearly, \\(|\\mathcal{C}'| = |\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}|\\) and \\(|\\mathcal{D}'| = |\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3}|\\). Applying Lemma \\ref{lemma:twoposvalues} to \\(\\mathcal{C}'\\) and \\(\\mathcal{D}'\\) (with ground set \\(\\{3,\\ldots,n\\}\\)) shows that\n\\[|\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}||\\mathcal{A}_{1 \\mapsto 2,2\\mapsto 3}| = |\\mathcal{C}'||\\mathcal{D}'| \\leq 4(1+O(1\/n))^{2} ((n-4)!)^{2}.\\]\nTherefore, by (\\ref{eq:lowerbound1122}),\n\\[|\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3}| \\leq o((n-4)!)\\]\nas required.\n\nThe case $l=1,\\ j \\notin \\{1,2\\}$ is the same as the previous case. It remains to deal with the case \\(\\{j,l\\} \\cap \\{1,2\\} = \\emptyset\\). Without loss of generality, we may assume that \\(j=3\\) and \\(l=4\\); we just need to show that \\(|\\mathcal{A}_{1 \\mapsto 3,2 \\mapsto 4}| \\leq o((n-4)!)\\).\n\nConsider the pair of families\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2},\\ (1\\ 3)(2 \\ 4)\\mathcal{A}_{1 \\mapsto 3,2 \\mapsto 4}.\\]\nLet \\(\\sigma \\in \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2},\\tau' = (1 \\ 3)(2 \\ 4)\\tau\\) where \\(\\tau \\in \\mathcal{A}_{1 \\mapsto 3, 2 \\mapsto 4}\\). Let $a = \\sigma^{-1}(3)$, and let $b=\\sigma^{-1}(4)$.\n\nIf \\(\\tau(a)=1\\), i.e. \\(\\tau'^{-1}(3) = a\\), then \\(\\sigma(a) = \\tau'(a) = 3\\). Similarly, if \\(\\tau(b) = 2\\), i.e. \\(\\tau'^{-1}(4)=b\\), then \\(\\sigma(b) = \\tau'(b)=4\\). Hence, \\(\\sigma\\) and \\(\\tau'\\) agree wherever \\(\\sigma\\) and \\(\\tau\\) agree, and also at \\(1\\) and \\(2\\), and possibly at \\(a\\) or \\(b\\), but nowhere else.\n\nLet\n\\[\\mathcal{C} = (\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2})^{-1},\\ \\mathcal{D} = ((1 \\ 3)(2 \\ 4)\\mathcal{A}_{1 \\mapsto 3,2 \\mapsto 4})^{-1}.\\]\nThen \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) are families of permutations fixing both \\(1\\) and \\(2\\). Moreover, for any \\(\\rho \\in \\mathcal{C}\\) and any \\(\\pi \\in \\mathcal{D}\\),\n\\begin{itemize}\n\\item if exactly one of \\(\\rho(3)=\\pi(3)\\) and \\(\\rho(4)=\\pi(4)\\) holds, then \\(|\\rho \\cap \\pi| \\neq 4\\);\n\\item if both hold, then \\(|\\rho \\cap \\pi| \\neq 5\\);\n\\item if neither hold, then \\(|\\rho \\cap \\pi| \\neq 3\\).\n\\end{itemize}\nRestrict \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) to \\(\\{3,4,\\ldots,n\\}\\) (`delete 1 and 2') to obtain the families \\(\\mathcal{C}',\\mathcal{D}' \\subset S_{\\{3,\\ldots,n\\}}\\); note that for any \\(\\rho' \\in \\mathcal{C}'\\) and any \\(\\pi' \\in \\mathcal{D}'\\),\n\\begin{itemize}\n\\item if exactly one of \\(\\rho'(3)=\\pi'(3)\\) and \\(\\rho'(4)=\\pi'(4)\\) holds, then \\(|\\rho' \\cap \\pi'| \\neq 2\\);\n\\item if both hold, then \\(|\\rho' \\cap \\pi'| \\neq 3\\);\n\\item if neither hold, then \\(|\\rho' \\cap \\pi'| \\neq 1\\).\n\\end{itemize}\nClearly, \\(|\\mathcal{C}'| = |\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}|\\) and \\(|\\mathcal{D}'| = |\\mathcal{A}_{1 \\mapsto 3,2 \\mapsto 4}|\\).\n\nApplying Lemma \\ref{lemma:3posvalues} to \\(\\mathcal{C}'\\) and \\(\\mathcal{D}'\\) (with ground set \\(\\{3,4,\\ldots,n\\}\\)) shows that\n\\[|\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}||\\mathcal{A}_{1 \\mapsto 3,2\\mapsto 4}| = |\\mathcal{C}'||\\mathcal{D}'| \\leq 4(1+O(1\/n))^{2} ((n-4)!)^{2}.\\]\nTherefore, by (\\ref{eq:lowerbound1122}),\n\\[|\\mathcal{A}_{1 \\mapsto 3,2 \\mapsto 4}| \\leq o((n-4)!)\\]\nas required.\n\\end{proof}\n\nSumming the inequality in the above claim over all \\((n-1)^{2}\\) possible pairs $(j,l)$ gives:\n\\[|\\mathcal{A}\\setminus (\\mathcal{A}_{1 \\mapsto 1} \\cup \\mathcal{A}_{2 \\mapsto 2})| = \\sum_{(j,l):\\ j \\neq 1, l \\neq 2} |\\mathcal{A}_{1 \\mapsto j,2 \\mapsto l}| \\leq o((n-2))!.\\]\n\nIt follows that \\(|\\mathcal{A}_{1 \\mapsto 1} \\cup \\mathcal{A}_{2 \\mapsto 2}| \\geq c(n-2)!-o((n-2)!)\\), and therefore\n\\[\\max(|\\mathcal{A}_{1 \\mapsto 1}|,|\\mathcal{A}_{2 \\mapsto 2}|) \\geq \\tfrac{1}{2}(1-o(1))c(n-2)! > c'(n-2)!,\\]\nwhere \\(c' := c\/4\\), provided $n$ is sufficiently large depending on $c$. Without loss of generality, we may assume that\n\\begin{equation}\n\\label{eq:lowerbound1}\n|\\mathcal{A}_{1 \\mapsto 1}| \\geq c'(n-2)!.\n\\end{equation}\n\nOur next aim is to show that all but at most \\(O((n-3)!)\\) of the permutations in \\(\\mathcal{A}\\) fix 1. To show this, it suffices to prove the following.\n\n\\begin{claim}\nFor each $j \\neq 1$, we have \\(|\\mathcal{A}_{1 \\mapsto j}| \\leq O(1\/c)(n-5)!\\).\n\\end{claim}\n\\begin{proof}[Proof of claim.]\nFix \\(j \\neq 1\\), and consider the pair of families\n\\[\\mathcal{A}_{1 \\mapsto 1},\\quad (1\\ j)\\mathcal{A}_{1 \\mapsto j}.\\]\nLet \\(\\sigma \\in \\mathcal{A}_{1 \\mapsto 1},\\tau' = (1 \\ j)\\tau\\) where \\(\\tau \\in \\mathcal{A}_{1 \\mapsto j}\\). Let \\(a = \\sigma^{-1}(j)\\). If \\(\\tau(a)=1\\), i.e. \\(\\tau'^{-1}(j) = a\\), then \\(\\sigma(a) = \\tau'(a) = j\\), so \\(\\sigma\\) and \\(\\tau'\\) agree wherever \\(\\sigma\\) and \\(\\tau\\) agree, and also at \\(1\\) and \\(a\\), but nowhere else. Hence, \\(\\sigma\\) and \\(\\tau'\\) cannot agree at exactly 3 points. If \\(\\tau(a) \\neq 1\\), i.e. \\(\\tau'^{-1}(j) \\neq a\\), then \\(\\sigma(a) \\neq \\tau'(a)\\), so \\(\\sigma\\) and \\(\\tau'\\) agree wherever \\(\\sigma\\) and \\(\\tau\\) agree, and also at \\(1\\), but nowhere else. Hence, \\(\\sigma\\) and \\(\\tau'\\) cannot agree at exactly 2 points.\n\nLet\n\\[\\mathcal{C} = (\\mathcal{A}_{1 \\mapsto 1})^{-1},\\ \\mathcal{D} = ((1 \\ j)\\mathcal{A}_{1 \\mapsto j})^{-1}.\\]\nThen \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) are families of permutations fixing \\(1\\). Moreover, for any \\(\\rho \\in \\mathcal{C}\\) and any \\(\\pi \\in \\mathcal{D}\\),\n\\begin{itemize}\n\\item if \\(\\rho(j)=\\pi(j)\\), then \\(|\\rho \\cap \\pi| \\neq 3\\);\n\\item if \\(\\rho(j) \\neq \\pi(j)\\), then \\(|\\rho \\cap \\pi| \\neq 2\\).\n\\end{itemize}\nRestrict \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) to \\(\\{2,3,\\ldots,n\\}\\) (`delete 1') to obtain the families \\(\\mathcal{C}',\\mathcal{D}' \\subset S_{\\{2,\\ldots,n\\}}\\). Observe that for any \\(\\rho' \\in \\mathcal{C}'\\) and any \\(\\pi' \\in \\mathcal{D}'\\),\n\\begin{itemize}\n\\item if \\(\\rho'(j)=\\pi'(j)\\), then \\(|\\rho' \\cap \\pi'| \\neq 2\\);\n\\item if \\(\\rho'(j) \\neq \\pi'(j)\\), then \\(|\\rho' \\cap \\pi'| \\neq 1\\).\n\\end{itemize}\n\nClearly, \\(|\\mathcal{C}'| = |\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}|\\) and \\(|\\mathcal{D}'| = |\\mathcal{A}_{1 \\mapsto 2,2 \\mapsto 3}|\\). Applying Lemma \\ref{lemma:twoposvalues} to \\(\\mathcal{C}',\\mathcal{D}'\\) yields\n\\[|\\mathcal{C}'||\\mathcal{D}'|\\leq 4(1+O(1\/n))((n-3)!)^{2},\\]\nand therefore\n\\[|\\mathcal{A}_{1 \\mapsto 1}||\\mathcal{A}_{1 \\mapsto j}| \\leq 4(1+O(1\/n))((n-3)!)^{2}.\\]\nHence, by (\\ref{eq:lowerbound1}),\n\\[|\\mathcal{A}_{1 \\mapsto j}| \\leq \\frac{4(1+O(1\/n))((n-3)!)^{2}}{c'(n-2)!} = O(1\/c)(n-5)!,\\]\nproving the claim.\n\\end{proof}\n\nSumming the inequality of the above claim over all \\(j \\neq 1\\) gives\n\\begin{equation}\\label{eq:almost} |\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1}| = \\sum_{j \\neq 1} |\\mathcal{A}_{1 \\mapsto j}| \\leq O(1\/c)(n-4)!.\\end{equation}\n\nObserve that \\(\\mathcal{A}_{1 \\mapsto 1} \\subset S_n\\) is a 2-intersecting family of permutations all fixing 1. Restricting $\\mathcal{A}_{1 \\mapsto 1}$ to \\(\\{2,\\ldots,n\\}\\) yields a 1-intersecting family of permutations \\(\\mathcal{A}_{1 \\mapsto 1}' \\subset S_{\\{2,\\ldots,n\\}}\\), with\n$$|\\mathcal{A}_{1 \\mapsto 1}'| \\geq c'(n-2)!.$$\n\nRecall the following stability theorem for 1-intersecting families of permutations, proved by the author in \\cite{cameronkuconj}.\n\n\\begin{theorem}\n\\label{thm:roughstabilityintersecting}\nLet \\(c > 0\\) be a positive constant. If \\(\\mathcal{A}\\subset S_{n}\\) is a 1-intersecting family of permutations with \\(|\\mathcal{A}|\\geq c(n-1)!\\), then there exist \\(i,j \\in [n]\\) such that all but at most \\(O_{c}((n-2)!)\\) permutations in \\(\\mathcal{A}\\) map \\(i\\) to \\(j\\).\n\\end{theorem}\n\nApplying this theorem to \\(\\mathcal{A}_{1\\mapsto 1}'\\) (with ground set \\(\\{2,\\ldots,n\\}\\)), we see that there exist \\(i,j \\geq 2\\) such that all but at most \\(O_{c}((n-3)!)\\) permutations in \\(\\mathcal{A}_{1 \\mapsto 1}\\) map \\(i\\) to \\(j\\). Without loss of generality, we may assume that \\(i=j=2\\), so\n$$|\\mathcal{A}_{1 \\mapsto 1} \\setminus \\mathcal{A}_{1 \\mapsto 1,2\\mapsto 2}| \\leq O_c((n-3)!).$$\nCombining this with (\\ref{eq:almost}) yields\n$$|\\mathcal{A} \\setminus \\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| \\leq O_c((n-3)!),$$\ncompleting the proof of Theorem \\ref{thm:roughstability}.\n\\end{proof}\n\nWe immediately obtain the following corollary.\n\\begin{corollary}\n\\label{corr:stab}\nFor any \\(c > 1-1\/e\\), and any \\(n\\) sufficiently large depending on \\(c\\), if \\(\\mathcal{A} \\subset S_{n}\\) is a family of permutations with no singleton intersection and with \\(|\\mathcal{A}| \\geq c(n-2)!\\), then there exist \\(i,j,k\\) and \\(l\\) such that every permutation in \\(\\mathcal{A}\\) maps \\(i\\) to \\(j\\) and \\(k\\) to \\(l\\), i.e. \\(\\mathcal{A}\\) is contained within a 2-coset of \\(S_{n}\\).\n\\end{corollary}\n\\begin{proof}\nBy Theorem \\ref{thm:roughstability}, there exist \\(i,j,k\\) and \\(l\\) such that\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{i \\mapsto j,k \\mapsto l}| \\leq O((n-3)!).\\]\nWithout loss of generality, \\(i=j=1\\) and \\(k=l=2\\), so\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\leq O((n-3)!).\\]\nSuppose for a contradiction that \\(\\mathcal{A}\\) contains a permutation \\(\\tau\\) not fixing both \\(1\\) and \\(2\\). It is easy to check that the number of permutations fixing both \\(1\\) and \\(2\\) and agreeing with \\(\\tau\\) at exactly one point is\n\\[(1\/e+o(1))(n-2)!.\\]\nNo such permutation can be in \\(\\mathcal{A}\\), so\n\\[|\\mathcal{A}| = |\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}|+|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\leq (1-1\/e+o(1))(n-2)!+O((n-3)!),\\]\na contradiction provided \\(n\\) is sufficiently large depending on \\(c\\).\n\\end{proof}\n\nCorollary \\ref{corr:stab} clearly implies our main theorem, Theorem \\ref{thm:main}.\n\n\\section{A Hilton-Milner type result}\nIn this section, we will use Theorem \\ref{thm:roughstability} to prove the following Hilton-Milner type result.\n\\begin{theorem}\n\\label{thm:hm}\nFor \\(n\\) sufficiently large, if \\(\\mathcal{A} \\subset S_{n}\\) is a family of permutations with no singleton intersection, and \\(\\mathcal{A}\\) is not contained in a 2-coset of \\(S_{n}\\), then \\(|\\mathcal{A}| \\leq |\\mathcal{B}|\\), where\n\\begin{align*}\n\\mathcal{B} = & \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\sigma(2)=2,\\ \\#\\{\\textrm{fixed points of }\\sigma \\geq 5\\} \\neq 1\\}\\\\\n& \\cup \\{(1\\ 3)(2\\ 4),(1\\ 4)(2\\ 3),(1\\ 3\\ 2\\ 4),(1\\ 4\\ 2\\ 3)\\}.\n\\end{align*}\nEquality holds only if \\(\\mathcal{A}\\) is a double translate of \\(\\mathcal{B}\\), meaning that there exist \\(\\pi,\\tau \\in S_{n}\\) such that \\(\\mathcal{A}=\\pi \\mathcal{B} \\tau\\).\n\\end{theorem}\n\n\\begin{proof}\nLet \\(\\mathcal{A}\\) be a family of maximum size satisfying the conditions of Theorem \\ref{thm:hm}. Observe that \\(\\mathcal{B}\\) satisfies them, and has size\n\\[(n-2)!-(n-4)(d_{n-3}+2d_{n-4}+d_{n-5})+4 = (1-1\/e+o(1))(n-2)!.\\]\n\nTo see this, note that the number of permutations fixing 1 and 2 and with exactly one fixed point \\(\\geq 5\\) is\n\\[(n-4)(d_{n-3}+2d_{n-4}+d_{n-5}).\\]\nIndeed, there are \\(n-4\\) choices of a number \\(p \\geq 5\\) to fix, and \\(d_{n-3}+2d_{n-4}+d_{n-5}\\) permutations whose fixed points are 1, 2, \\(p\\) and the numbers in some subset of \\(\\{3,4\\}\\).\n\nTherefore, by our assumption of the maximality of \\(|\\mathcal{A}|\\),\n\\begin{equation}\n\\label{eq:lowerboundA}\n|\\mathcal{A}| \\geq |\\mathcal{B}| = (1-1\/e+o(1))(n-2)!.\n\\end{equation}\nBy Theorem \\ref{thm:roughstability}, there exist \\(i,j,k\\) and \\(l\\) such that\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{i \\mapsto j,k \\mapsto l}| \\leq O((n-3)!).\\]\nBy double translation, we may assume that \\(i=j=1\\) and \\(k=l=2\\), so\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\leq O((n-3)!).\\]\nBy assumption, \\(\\mathcal{A}\\) must contain some permutation \\(\\tau\\) not fixing both 1 and 2. There are four possibilities:\n\\begin{enumerate}\n\\item \\(\\tau(1)=2,\\ \\tau(2)=1\\);\n\\item \\(\\tau\\) fixes 1 or 2;\n\\item \\(\\tau\\) does not fix 1 or 2, but \\(|\\tau\\{1,2\\} \\cap \\{1,2\\}|=1\\);\n\\item \\(\\tau\\{1,2\\} \\cap \\{1,2\\} = \\emptyset\\);\n\\end{enumerate}\n\nFirst, we obtain bounds on \\(|\\mathcal{A}_{1\\mapsto 1,2 \\mapsto 2}|\\) in each of the cases 1-3.\n\nIn case 1, by considering the translated family \\(\\mathcal{A}\\tau^{-1}(1\\ 2)\\), we may assume that \\(\\tau = (1\\ 2)\\). Hence, \\(\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}\\) cannot contain any permutation in the family\n\\[\\mathcal{F}_{1} = \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\ \\sigma(2)=2,\\ \\#\\{\\textrm{fixed points of }\\sigma \\geq 3\\} = 1\\},\\]\nsince every \\(\\sigma \\in \\mathcal{F}_{1}\\) has \\(|\\sigma \\cap (1\\ 2)|=1\\). Observe that\n\\[|\\mathcal{F}_{1}|=(n-2)d_{n-3},\\]\nthe number of permutations of \\(\\{3,4,\\ldots,n\\}\\) with exactly one fixed point. Let\n\\[\\mathcal{G}_{1} = \\{\\sigma \\in S_{n}: \\sigma(1)=1,\\ \\sigma(2)=2\\} \\setminus \\mathcal{F}_{1};\\]\nthen\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\mathcal{G}_{1},\\]\nand therefore\n\\[|\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| \\leq |\\mathcal{G}_1| = (n-2)!-(n-2)d_{n-3}.\\]\n\nIn case 2, we may assume that \\(\\tau(2)=2,\\ \\tau(1)=3\\). Then, by considering the translated family \\( \\mathcal{A}\\tau^{-1}(1\\ 3)\\), we may assume that \\(\\tau = (1\\ 3)\\). Hence, \\(\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}\\) cannot contain any permutation in the family\n\\[\\mathcal{F}_{2} = \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\ \\sigma(2)=2,\\ \\sigma\\textrm{ has no fixed points}\\geq 4\\},\\]\nsince every \\(\\sigma \\in \\mathcal{F}_{2}\\) has \\(|\\sigma \\cap (1\\ 3)|=1\\). Observe that\n\\[|\\mathcal{F}_{2}|=d_{n-2}+d_{n-3},\\]\nsince there are \\(d_{n-2}\\) permutations whose fixed points are 1 and 2, and \\(d_{n-3}\\) permutations whose fixed points are 1, 2 and 3. Let\n\\[\\mathcal{G}_{2} = \\{\\sigma \\in S_{n}: \\sigma(1)=1,\\ \\sigma(2)=2\\} \\setminus \\mathcal{F}_{2};\\]\nthen\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\mathcal{G}_{2},\\]\nand therefore\n\\[|\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| \\leq |\\mathcal{G}_{2}| = (n-2)!-d_{n-2}-d_{n-3}.\\]\n\nIn case 3, we may assume that \\(\\tau(1)=2\\) and \\(\\tau(2)=3\\). Then, by considering the translated family \\(\\mathcal{A}\\tau^{-1}(1\\ 2\\ 3)\\), we may assume that \\(\\tau = (1\\ 2\\ 3)\\). Hence, \\(\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}\\) cannot contain any permutation in the family\n\\[\\mathcal{F}_{3} = \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\ \\sigma(2)=2,\\ \\sigma\\textrm{ has exactly one fixed point}\\geq 4\\},\\]\nsince every \\(\\sigma \\in \\mathcal{F}_{3}\\) has \\(|\\sigma \\cap (1\\ 2\\ 3)|=1\\). Observe that\n\\[|\\mathcal{F}_{3}|=(n-3)(d_{n-3}+d_{n-4}),\\]\nsince there are \\(n-3\\) choices of a point \\(i \\geq 4\\) to fix, \\(d_{n-3}\\) permutations whose fixed points are 1, 2 and \\(i\\), and \\(d_{n-4}\\) permutations whose fixed points are 1, 2, 3 and \\(i\\). Let\n\\[\\mathcal{G}_{3} = \\{\\sigma \\in S_{n}: \\sigma(1)=1,\\ \\sigma(2)=2\\} \\setminus \\mathcal{F}_{3};\\]\nthen\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\mathcal{G}_{3},\\]\nand therefore\n\\[|\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| \\leq |\\mathcal{G}_{3}| = (n-2)!-(n-3)(d_{n-3}+d_{n-4}).\\]\n\nIt is easily checked that for each \\(j \\leq 3\\),\n\\begin{equation}\\label{eq:boundG} |\\mathcal{G}_{j}| \\leq |\\mathcal{B}|-4-d_{n-4} = (1-1\/e+o(1))(n-2)!.\\end{equation}\nSince in case $j$, we have shown that \\(\\mathcal{A} \\subset \\mathcal{G}_{j}\\) (for $j=1,2,3$), it follows from (\\ref{eq:lowerboundA}) that\n\\begin{equation}\n\\label{eq:toomanyothers}\n|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\geq d_{n-4}+4 = (1\/e+o(1))(n-4)!.\n\\end{equation}\nWe now show that this leads to a contradiction. The number of permutations in \\(S_{n}\\) with at least \\(\\lfloor n\/2 \\rfloor\\) fixed points is at most\n\\[{n \\choose \\lfloor n\/2 \\rfloor} (\\lceil n\/2 \\rceil)! = n!\/(\\lfloor n \/ 2 \\rfloor)! = o((n-4)!),\\]\nso the number of permutations in \\(\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\) with less than \\(\\lfloor n \/ 2 \\rfloor\\) fixed points is at least \\((1\/e-o(1))(n-4)!\\). Choose one such permutation \\(\\pi \\in \\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\). We now prove the following.\n\n\\begin{claim}\n\\label{claim:H}\nIn each of the cases 1-3, if there exists $\\pi \\in \\mathcal{A}\\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}$ with at most $\\lfloor n\/2 \\rfloor$ fixed points, then\n$$|\\mathcal{A}_{1\\mapsto 1,2 \\mapsto 2}| \\leq (1-1\/e - 1\/(8e)+o(1))(n-2)!.$$\n\\end{claim}\n\\begin{proof}[Proof of claim.]\nLet\n\\[N(\\pi) = \\{i \\geq 5:\\ \\pi(i) \\neq i\\};\\]\nthen \\(|N(\\pi)| \\geq n-4-\\lfloor n \/2 \\rfloor = \\lceil n\/2 \\rceil -4.\\)\n\nLet\n\\[\\mathcal{H} = \\{\\sigma \\in S_{n}:\\ \\sigma \\textrm{ fixes 1, 2, and at least two points in } N(\\pi),\\ \\textrm{and}\\ |\\sigma \\cap \\pi|=1\\}.\\]\nObserve that \\(\\mathcal{H} \\subset \\mathcal{G}_{j}\\), since each \\(\\sigma \\in \\mathcal{H}\\) has at least two fixed points \\(\\geq 4\\), and that \\(\\mathcal{A} \\cap \\mathcal{H} = \\emptyset\\). If \\(\\pi\\) fixes 1 or 2, then \\(\\mathcal{H}\\) is the set of permutations fixing 1,2, and at least two points in \\(N(\\pi)\\), and disagreeing with \\(\\pi\\) at every point \\(\\geq 3\\), so has size  \n\\[|\\mathcal{H}| \\geq {|N(\\pi)|\\choose 2} d_{n-4} \\geq \\tfrac{1}{8}(1\/e-o(1))(n-2)!.\\]\nIf \\(\\pi\\) does not fix 1 or 2, then \\(\\mathcal{H}\\) is the set of permutations fixing 1,2 and at least two points in \\(N(\\pi)\\), and agreeing with \\(\\pi\\) at exactly one point \\(\\geq 3\\), so we have\n\\[|\\mathcal{H}| \\geq {|N(\\pi)|\\choose 2} (n-6)d_{n-5} \\geq \\tfrac{1}{8}(1\/e-o(1))(n-2)!.\\]\nSince $\\mathcal{A}_{1 \\mapsto 1,2\\mapsto 2} \\subset \\mathcal{G}_j$, it follows that\n\\begin{align*} |\\mathcal{A}_{1 \\mapsto 1,2\\mapsto 2}| & \\leq |\\mathcal{G}_j| - |\\mathcal{H}|\\\\\n& \\leq (1-1\/e+o(1))(n-2)! - \\tfrac{1}{8}(1\/e - o(1))(n-2)!\\\\\n& = (1-1\/e-1\/(8e)+o(1))(n-2)!,\\end{align*}\nas required.\n\\end{proof}\n\nSince \\(|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\leq O((n-3)!)\\), it follows from the above claim that\n\\begin{align*} |\\mathcal{A}| & \\leq (1-1\/e-1\/(8e)+o(1))(n-2)!+O((n-3)!)\\\\\n& = (1-1\/e-1\/(8e)+o(1))(n-2)!,\\end{align*}\ncontradicting (\\ref{eq:lowerboundA}).\n\nHence, we may assume that none of the cases 1-3 occur, so case 4 must occur, i.e.\n\\begin{equation}\n\\label{eq:12condition}\n\\tau\\{1,2\\} \\cap \\{1,2\\} = \\emptyset\\quad \\forall \\tau \\in \\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}.\n\\end{equation}\nTake any \\(\\tau_0 \\in \\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\). Without loss of generality, we may assume that \\(\\tau_0(1)=3\\) and \\(\\tau_0(2)=4\\). Then, by considering the translated family \\( \\mathcal{A}\\tau_0^{-1}(1\\ 3)(2 \\ 4)\\), we may assume that \\(\\tau_0 = (1\\ 3)(2\\ 4)\\). Hence, \\(\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}\\) cannot contain any permutation in the family\n\\[\\mathcal{F}_{4} = \\{\\sigma \\in S_{n}:\\ \\sigma(1)=1,\\ \\sigma(2)=2,\\ \\#\\{\\textrm{fixed points of }\\sigma \\geq 5\\} = 1\\},\\]\nsince every \\(\\sigma \\in \\mathcal{F}_{4}\\) has \\(|\\sigma \\cap (1\\ 3)(2\\ 4)|=1\\). As observed above,\n\\[|\\mathcal{F}_{4}|=(n-4)(d_{n-3}+2d_{n-4}+d_{n-5}).\\]\nLet\n\\[\\mathcal{G}_{4} = \\{\\sigma \\in S_{n}: \\sigma(1)=1,\\ \\sigma(2)=2\\} \\setminus \\mathcal{F}_{4};\\]\nthen\n\\begin{equation}\n\\label{eq:containmentfor1122}\n\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\mathcal{G}_{4},\n\\end{equation}\nand therefore\n\\begin{align*} |\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| & \\leq |\\mathcal{G}_4|\\\\\n& = (n-2)!-(n-4)(d_{n-3}+2d_{n-4}+d_{n-5})\\\\\n& = (1-1\/e+o(1))(n-2)!.\\end{align*}\nWe now prove the following.\n\n\\begin{claim}\n\\label{claim:fix}\nEvery permutation in \\(\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\) must fix \\(\\{5,6,\\ldots,n\\}\\) pointwise.\n\\end{claim}\n\\begin{proof}[Proof of claim.]\nSuppose for a contradiction that some permutation \\(\\rho \\in \\mathcal{A}\\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\) does not fix \\(\\{5,6,\\ldots,n\\}\\) pointwise; without loss of generality, we may assume that \\(\\rho(5) \\neq 5\\). We assert that this forces \\(\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\) to contain a permutation \\(\\pi\\) with at most \\(\\lfloor n \/ 2\\rfloor\\) fixed points.\n\nIndeed, if \\(\\rho\\) itself has at most \\(\\lfloor n \/ 2\\rfloor\\) fixed points, we are done already, so we may assume that $\\rho$ has more than \\(\\lfloor n \/ 2\\rfloor\\) fixed points. Let\n\\[F(\\rho) = \\{i \\geq 5:\\ \\rho(i)=i\\}.\\]\nWe have \\(|F(\\rho)| > \\lfloor n\/2\\rfloor -4\\).\n\nConsider the family\n\\[\\mathcal{M} = \\{\\sigma \\in S_{n}:\\ \\sigma \\textrm{ fixes 1, 2, 5 and some } i \\in F(\\rho),\\ \\sigma(i) \\neq \\rho(i)\\ \\textrm{for all other }i \\geq 3\\}.\\]\nObserve that \\(\\mathcal{M} \\subset \\mathcal{G}_{4}\\), since each permutation in \\(\\mathcal{G}\\) has at least two fixed points \\(\\geq 5\\). However, each \\(\\sigma \\in \\mathcal{M}\\) has \\(|\\sigma \\cap \\rho|=1\\), since by (\\ref{eq:12condition}), \\(\\rho\\) cannot fix 1 or 2. Hence,\n\\[\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\mathcal{G}_{4} \\setminus \\mathcal{M}.\\]\nObserve that\n\\[|\\mathcal{M}| \\geq |F(\\rho)| d_{n-4} \\geq \\tfrac{1}{2}(1\/e-o(1))(n-3)!,\\]\nso by (\\ref{eq:lowerboundA}), we see that\n\\[|\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}| \\geq \\tfrac{1}{2}(1\/e-o(1))(n-3)!.\\]\nAs observed above, there are at most \\(n!\/(\\lfloor n \/ 2 \\rfloor)!\\) permutations with at least \\(\\lfloor n\/2 \\rfloor\\) fixed points, so there must be some permutation \\(\\pi \\in \\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}\\) with at most \\(\\lfloor n \/ 2\\rfloor\\) fixed points, as asserted.\n\nBy exactly the same argument as in the proof of Claim \\ref{claim:H}, this implies that\n\\begin{align*} |\\mathcal{A}_{1\\mapsto 1,2\\mapsto 2}| & \\leq |\\mathcal{G}_4| - \\tfrac{1}{8} (1\/e +o(1))(n-2)!\\\\\n& = (1-1\/e+o(1))(n-2)! - \\tfrac{1}{8}(1\/e+o(1))(n-2)!\\\\\n& = (1-1\/e-1\/(8e)+o(1))(n-2)!,\\end{align*}\nand therefore\n\\begin{align*} |\\mathcal{A}| & \\leq (1-1\/e-1\/(8e)+o(1))(n-2)!+|\\mathcal{A}\\setminus\\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2}|\\\\\n& \\leq (1-1\/e-1\/(8e)+o(1))(n-2)! + O((n-3)!)\\\\\n& \\leq (1-1\/e-1\/(8e)+o(1))(n-2)!,\\end{align*}\ncontradicting (\\ref{eq:lowerboundA}). The claim is proved.\n\\end{proof}\n\nClaim \\ref{claim:fix} says that\n\\[\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1, 2\\mapsto 2} \\subset S_{[4]}.\\]\nBy (\\ref{eq:12condition}), we have\n\\[\\mathcal{A} \\setminus \\mathcal{A}_{1 \\mapsto 1,2 \\mapsto 2} \\subset \\{(1 \\ 3)(2\\ 4),(1\\ 4)(2\\ 3),(1\\ 3\\ 2\\ 4),(1\\ 4\\ 3\\ 2)\\}.\\]\nCombining this with (\\ref{eq:containmentfor1122}), we see that\n\\[\\mathcal{A} \\subset \\mathcal{G}_{4} \\cup \\{(1 \\ 3)(2\\ 4),(1\\ 4)(2\\ 3),(1\\ 3\\ 2\\ 4),(1\\ 4\\ 3\\ 2)\\} = \\mathcal{B}.\\]\nHence, by the maximality of \\(\\mathcal{A}\\), \\(\\mathcal{A} = \\mathcal{B}\\), proving the theorem.\n\\end{proof}\n\n\\begin{comment}\n\\section{Conclusion and Open Problems}\nIt is natural to conjecture the following generalization of our result:\n\\begin{conjecture}\nFor any \\(t \\in \\mathbb{N}\\), if \\(n\\) is sufficiently large depending on \\(t\\), then any family of permutations \\(\\mathcal{A} \\subset S_{n}\\) in which no two intersect on exactly \\(t-1\\) points has size\n\\[|\\mathcal{A}| \\leq (n-t)!\\]\nwith equality only if \\(\\mathcal{A}\\) is a \\(t\\)-coset of \\(S_{n}\\).\n\\end{conjecture}\nAs mentioned in the introduction, applying the Delsarte-Hoffman bound using the conjugacy-class association scheme, as in \\cite{jointpaper}, one only obtains\n\\[|\\mathcal{A}| \\leq O((n-2)!)\\]\nfor general \\(t\\), so it seems that new techniques are required.\n\\end{comment}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nEfficient and accurate methods for uncertainty quantification (UQ) are topics of much interest in the field of computational mathematics and engineering.\nWhile there are many UQ methods that solve a variety of stochastic inverse problems, the most commonly used methods Bayesian inversion \\cite{CDS10, Stuart_Bayesian} and ensemble Kalman filtering \\cite{Kalman1960, Evensen03}) use Monte Carlo \\cite{Robert_Casella_book, Gentle_book} or Markov Chain Monte Carlo  \\cite{Gilks_book, Geyer92} sampling techniques to evaluate a Quantity of Interest (QoI) map, which introduces error due to finite sampling.\nThis error is exacerbated because numerical techniques are used to solve the model instead of solving it exactly.\n\nA common strategy for reducing the effect of finite sampling error is to construct a surrogate to the QoI response surface.\nEvaluating the surrogate is done at a greatly reduced computational cost.\nSurrogate modeling is a large topic, so a full review of such techniques is not possible.\nA widely used class of surrogate approaches involve global polynomial approximations based on stochastic spectral methods \\cite{XK_PC_2002, WK_PC_2006, LeMaitre_UQ_04a, LeMaitre_UQ_04b, MNR07, BCW12, BDW13, Prudhomme_Bryant_2015, BPW_2015}.\nAnother popular approach is using tensor grid and sparse grid stochastic collocation methods for building surrogates \\cite{Almeida_Oden_2010, NTW_SparseGrid_2008}, including adaptive methods \\cite{ma2009}.\nThere are also approaches using stochastic optimization to construct global polynomial \\cite{li2014} and local \\cite{conrad2016} approximations over sequences of distributions adaptively determined from the data.\nThe surrogate modeling approach considered in this work most closely resembles techniques that exploit derivative information for building piecewise low-order surrogate approximations to improve pointwise accuracy in propagations of uncertainties \\cite{MNR07, Estep_Neckels_2006}.\n\nThe surrogate response surface is polluted by two sources of error affecting local accuracy \\cite{BCW12, BDW13, Prudhomme_Bryant_2015, BPW_2015}. First, there is approximation error introduced by the type of surrogate. Second, there is numerical error in the evaluation of the numerical model used to construct the surrogate.\nBoth of these are types of discretization errors.\nThus, using a surrogate can represent a trade-off between the reduction in finite sampling error at the expense of an overall increase in the discretization error.\nThe end result is that our ability to accurately quantify uncertainties by solution of a stochastic inverse problem may be compromised by the use of surrogates unless additional steps are taken to reduce the discretization errors.\n\nAdjoint techniques for finding computable and accurate a posteriori estimates of discretization errors have existed  for decades \\cite{Babuska_Rheinboldt_1978, Estep95, Ainsworth_Oden_1997}.\nSuch techniques have served as the basis of the error estimates for polynomial chaos and pseudospectral based surrogates derived in \\cite{BCW12}.\nSubsequently, in \\cite{BDW13, Prudhomme_Bryant_2015}, such error estimates were used as part of a Bayesian inference to quantify uncertainties on parameters to evolutionary partial differential equations where QoI response surfaces were approximated with polynomial chaos techniques and enhanced by the error estimates.\n\n\nWe present an adaptive method for updating the surrogate to accurately make predictions under uncertainty in a stochastic inverse problem setting.\nThis adaptivity includes increasing the local polynomial order of the approximation, adding more sample points, and increasing the fidelity of the model for certain samples.\nThe refinement is guided by two levels of surrogate models, with one incorporating adjoint-based a posteriori error estimates to reduce the effect of numerical errors.\nThe method is designed to simultaneously decrease the effects of both types of discretization errors on the prediction at each iteration.\n\nThis paper is organized as follows.\nWe provide some general notation, terminology, and assumptions used in this work in Section~\\ref{sec:notation}, as well as a brief summary of the theory behind stochastic inversion and the various contributions of errors. \nIn Section 3, we describe the abstract process of constructing surrogate approximations, identify the various sources of error in the surrogate, and describe the implicit construction of a general piecewise low-order surrogate.\nWe subsequently provide the conditions relating the exact and surrogate response surfaces for which the approximation solution to the stochastic inverse problem is in fact exact.\nA brief review of adjoint based a posteriori error and derivative estimates along with a list of useful references are provided.\nIn Section 4, we also describe how we use such error estimates to enhance surrogates by correcting for persistent local biases due to discretization errors.\nSuch enhanced surrogates are used to derive local error indicators which can be used for local refinement in a variety of ways.\nThe enhanced surrogates, error indicators, and refinement strategies are combined in an adaptive strategy for surrogate construction.\nIn Section 5, the method is applied to a number of example problems of varying complexity, including realistic engineering problems.\nConclusions are discussed in Section 6.\n\n\n\n\n\n\\section{Notation, Terminology, and Assumptions}\n\\label{sec:notation}\n\n\nWe present some notation, terminology, and general assumptions for stochastic inversion of a physical system.\nSuppose there is a model $\\mathcal{M}(u;\\lambda)=0$ of the system, where $u$ denotes a vector of state variables determined by the solution of the model for a specified vector of parameters $\\lambda$.\nThese parameters may include coefficients, initial conditions, boundary conditions, source terms, etc.\nWe assume the space of possible parameters, denoted by $\\mathbf{\\Lambda}$, is known, and these parameters explicitly determine the solution to the model.\n\nA quantity of interest (QoI) map, $Q$, is defined as a vector of linear functions on the model solution, $Q(u(\\lambda))$.\nNote that $Q$ explicitly depends on $\\lambda$\nLet $\\mathbf{\\mathcal{D}} :=Q(\\mathbf{\\Lambda})$ denote the space of model QoI.\nIn general, $\\mathbf{\\Lambda}$ and $\\mathbf{\\mathcal{D}}$ should be Banach spaces.\nAssume that the QoI map defined by $Q$ is piecewise smooth.\n\n\nNormally, the model is solved using a numerical approximation, resulting in an approximate solution $u_h(\\lambda)$ to the model.\nUsing the approximate solution $u_h(\\lambda)$ introduces error into the computation of QoI.\nThere may be other numerical errors introduced in the QoI being calculated approximately (e.g. using quadrature to approximate an integral quantity or using an iterative solver).\nDefine $Q_h(\\lambda) := Q(u_h(\\lambda)))$ as the computed QoI map incorporating these numerical errors and \n$\\epsilon_{Q,h}(\\lambda) := Q_h(\\lambda)-Q(\\lambda)$ as the error.\nThe numerical solution to the stochastic inverse problem involves a number of (approximate) evaluations of the map $Q$ (i.e. $Q_h$) and gives an approximation to $P_{\\mathbf{\\Lambda}}$.\n\n\n\n\n\n\nThe goal of stochastic inversion could be to approximate the entire probability measure $P_{\\mathbf{\\Lambda}}$, but often the goal is to make a prediction under uncertainty.\nSuppose $f: \\mathbf{\\Lambda} \\rightarrow \\mathbb{R}$ is an integrable function.\nThe problem of prediction under uncertainty is to estimate \n\\begin{equation} \\label{eq:integral}\nI = \\int_A f(\\lambda) dP_{\\mathbf{\\Lambda}}\n\\end{equation}\nfor some $A \\subset \\mathbf{\\Lambda}$.\nThere are a variety of different functions $f$ which could be considered.\nFor instance, values of $f$ representing characteristic functions could be used to calculate probabilities of sets of interest of parameters, and\nvalues of $f$ representing unobserved quantities of interest could be useful for predictions and decision-making.\nIn many cases, $I$ can be approximated well without fully resolving $P_{\\mathbf{\\Lambda}}$.\n\n\nIn this paper, the stochastic inverse problem is posed as a Bayesian inverse problem; however, the method is generally applicable to other types of stochastic inversion.\nBayesian inversion is an increasingly popular approach.\nIt has the benefit of generally being well-posed, but is often expensive to implement due to the large number of model evaluations required, so informed adaptivity has the potential to greatly reduce computational costs.\nWe focus on the applicability of the methods to Bayesian inversion because of its importance in the field and the great potential benefits of such adaptive methods.\n\n\nA general formation of the Bayesian stochastic inverse problem is described by Stuart \\cite{Stuart_Bayesian}.\nSuppose that $\\mathbf{\\Lambda}$ and $\\mathbf{\\mathcal{D}}$ are Banach spaces, and that $Q: \\mathbf{\\Lambda} \\rightarrow \\mathbf{\\mathcal{D}}$ represents the QoI.\nIn Bayesian inversion, $Q$ is often called the observation operator.\nSuppose that $y \\in \\mathbf{\\mathcal{D}}$ is given data.\nThe classical inverse problem of finding $\\lambda \\in \\mathbf{\\Lambda}$ such that\n\\begin{equation}\ny = Q(\\lambda)\n\\end{equation}\nis typically ill-posed.\nHowever, suppose that the observations $y$ are subject to observational noise.\nA more appropriate model is\n\\begin{equation}\ny = Q(\\lambda) + \\eta,\n\\end{equation}\nwhere $\\eta$ is a mean zero random variable with known statistical properties.\n\nThe prior beliefs about $\\lambda$ are described in terms of a probability measure $P_0$ on $\\mathbf{\\Lambda}$.\nAssuming that $\\mathbf{\\Lambda} \\subset \\mathbb{R}^n$ and $\\mathbf{\\mathcal{D}} \\subset \\mathbb{R}^m$,\nthe probability of $y$ given $\\lambda$ has the density\n\\begin{equation}\n\\rho(y | \\lambda) := \\rho(y - Q(\\lambda)).\n\\end{equation}\nThis is called the data likelihood.\nWe are interested in the posterior measure $P_{\\mathbf{\\Lambda}}$ the probability measure of $\\lambda$ given $y$.\nSuppose that $\\pi_0$ and $\\pi_{\\mathbf{\\Lambda}}$ are the probability densities associated with $\\mu_0$ and $\\mu$ respectively.\nUsing Bayes' formula, \n\\begin{equation}\n\\pi_{\\mathbf{\\Lambda}}(\\lambda) \\propto \\rho(y - Q(\\lambda)) \\pi_0(\\lambda).\n\\end{equation}\nThere are many methods for sampling from the posterior measure $P_{\\mathbf{\\Lambda}}$ using Bayes' formula.\nOne of the most common is Markov Chain Monte Carlo (MCMC) \\cite{Stuart_Bayesian}.\nThe basic idea of MCMC is to design a Markov chain with the property that a single sequence of output of the chain $\\set{\\lambda_i}_{i=1}^{\\infty}$ is distributed according to $P_{\\mathbf{\\Lambda}}$.\n\n\n \n\\section{Surrogate Models}\n\\label{sec:Surrogate}\n\\subsection{Surrogate Models and Error}\nA well-known challenge of stochastic inversion is that it often is extremely computationally expensive.\nMethods for sampling from and\/or describing probability measures such as Markov Chain Monte Carlo (MCMC) and filtering require a \\textit{large} number of evaluations of the (approximate) QoI map.\nEach evaluation of $Q_h$ requires an evaluation of the model $\\mathcal{M}(u; \\lambda)$ which is often computationally expensive.\nA common approach for reducing the computational cost is by using a surrogate model.\nConstructing a surrogate map $Q_s(\\lambda)$ often requires using some particular set of samples of $Q_h(\\lambda)$ based on a specific type of sampling in $\\mathbf{\\Lambda}$, e.g., using a possibly different set of random samples or using deterministic sampling approaches such as sparse grids \\cite{Almeida_Oden_2010, NTW_SparseGrid_2008}.\nLet $Q_s(\\lambda)$ denote a computationally inexpensive surrogate approximation to $Q(\\lambda)$.\nThe map $Q_s$ is then used for sampling from and\/or describing the probability measures of interest.\nFor the (normal) case where $Q_s$ is constructed from approximate numerical evaluations of $Q$ ($Q_h$) denote the surrogate $Q_{s,h}(\\lambda)$.\nLet $\\epsilon_{s,h}(\\lambda)$ denote the error $Q(\\lambda) - Q_{s,h}(\\lambda)$.\nWe decompose the error as  $$\\epsilon_{s,h}(\\lambda) := \\epsilon_s(\\lambda) + \\epsilon_h(\\lambda),$$ \nwhere $\\epsilon_s(\\lambda)$ is the error in the choice of surrogate due to limited approximation properties of the surrogate, and $\\epsilon_h(\\lambda)$ is the error in the surrogate from numerical solution of the model.\nWe now describe a surrogate of piecewise polynomials on Voronoi tessellations.\n\n\\subsection{A Piecewise Polynomial Surrogate on an Implicit Voronoi Discretization}\nVoronoi tessellations are a convenient way to discretize domains with moderate dimensions, and can be used to define a class of piecewise-defined surrogate models \\cite{VPS_Rushdi}.\nThe space of uncertain parameters $\\mathbf{\\Lambda}$ can be discretized by an implicit Voronoi tessellation simply by sampling the space.\nSuppose that $\\{ \\lambda^{(i)}\\}_{i=1}^N$ is a finite set of $N$ distinct points in $\\mathbf{\\Lambda}$ that we will call ``samples.\"\nTake a metric $d(\\cdot, \\cdot)$ defined on $\\mathbf{\\Lambda}$.\nThere is a \\textbf{Voronoi tessellation} of $\\mathbf{\\Lambda}$ denoted by $\\{ \\mathcal{V}_{i,N} \\}_{i=1}^N \\subset \\mathbf{\\Lambda}$ defined by\n$$\\mathcal{V}_{i,N} :=  \\{ \\lambda \\in \\mathbf{\\Lambda} : d(\\lambda^{(i)}, \\lambda) \\leq  d(\\lambda^{(j)}, \\lambda),  \\forall j=1,2,..., N \\}.$$\nEach set $\\mathcal{V}_{i,N} $ is called \\textbf{Voronoi cell}.\nNote that two Voronoi cells $\\mathcal{V}_{i,N}$ and $\\mathcal{V}_{j,N}$ may intersect, but only on a set of measure zero.\nThis is an implicit tessellation, i.e. the Voronoi cells do not have to be explicitly constructed.\nIt is generally only necessary to identify which cell $\\mathcal{V}_{i,N}$ contains a point $\\lambda$ via a nearest neighbor search amongst $\\{ \\lambda^{(i)}\\}_{i=1}^N$.\n\nSuppose that $\\set{\\lambda_j}_{j=1}^{M}$ is a Markov chain distributed with respect to the probability measure $P_{\\mathbf{\\Lambda}}$ (e.g. the output of an MCMC algorithm).\nSuch a chain can be used to approximate the probability measure of Voronoi cells in $\\mathbf{\\Lambda}$.\nGiven a chain $\\set{\\lambda_j}_{j=1}^{M}$ and a Voronoi tessellation of $\\mathbf{\\Lambda}$, $\\set{\\mathcal{V}_{i,N}}_{i=1}^N$, the probability measures are given by\n\\begin{equation}\nP_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N}) \\approx \\frac{1}{M} \\sum_{j=1}^M \\chi_{\\mathcal{V}_{i,N}}(\\lambda_j),\n\\end{equation}\nwhere $\\chi_{\\mathcal{V}_{i,N}}$ is the characteristic function\n \\[\n    \\chi_{\\mathcal{V}_{i,N}}(\\lambda)=\\left\\{\n                \\begin{array}{lll}\n                  1 & \\lambda \\in \\mathcal{V}_{i,N} \\\\\n                  0 & \\lambda \\not\\in \\mathcal{V}_{i,N}.\n                \\end{array}\n              \\right.\n  \\]\nSuch local probability measure estimates can be used for error estimation and adaptivity.\n\n\nLocal polynomial approximations of $Q(\\lambda)$ on each Voronoi cell could be formed in a variety of ways including interpolating or fitting evaluations of $Q_h$ or using local Taylor expansions.\nInterpolants could be useful in relatively low dimensions; however, in higher dimensions a large number of evaluations of $Q_h$ could be required for an accurate approximation, and the polynomials are highly sensitive to the choice of sets of parameters for which the model is solved.\nApproximation has some of the same issues as interpolation as well as the additional problem that error is possibly added at points where the model is actually evaluated, i.e. $Q_{s,h}(\\lambda_i) \\neq Q_h(\\lambda_i)$, where $\\lambda_i$ are parameters for which the model has been evaluated.\nLocal Taylor expansions avoid these problems.\n\n\nTo calculate a local Taylor approximation of $Q$ on $\\mathcal{V}_{i,N}$, $Q(\\lambda)$ and partial derivatives of $Q$ with respect to $\\lambda$ must be approximated at some $\\lambda_i \\in \\mathcal{V}_{i,N}$.\nAn obvious choice of $\\lambda_i$ is the generating point of the Voronoi cell $\\mathcal{V}_{i,N}$, $\\lambda^{(i)}$.\nThe centroid of $\\mathcal{V}_{i,N}$ could be another choice of $\\lambda_i$; however, the centroid is not trivial (and possibly quite computationally expensive) to calculate in high dimensions.\nAlso, if the Voronoi tessellation is refined by adding more generating samples, the centroids would change, and the old centroids may no longer be contained in the same Voronoi cell.\nHowever, for a low dimensional, non-adaptive problem it could be an advantage to use the centroid because of the smaller effective radius.\nFor an adaptive scheme, approximating the model and derivatives at the generating points of the Voronoi cells is the natural choice.\nThe simplest local Taylor approximation on Voronoi cells is a piecewise constant approximation.\nThe piecewise constant surrogate $Q^{(N)}_{s,0}(\\lambda)$ is defined as a simple function on a set of Voronoi cells $\\set{\\mathcal{V}_{i,N}}_{1\\leq i\\leq N}$ defined by a set of generating samples $\\set{\\lambda^{(i)}}_{1\\leq i\\leq N}$, i.e., \n\\begin{equation}\\label{eq:0surrogate}\n\tQ^{(N)}_{s,0}(\\lambda) = \\sum_{1\\leq i\\leq N}Q_h(\\lambda^{(i)})\\chi_{\\mathcal{V}_{i,N}}(\\lambda).\n\\end{equation}\nConstructing $Q^{(N)}_{s,0}$ only requires $Q_h$ to be evaluated at the generating samples $\\set{\\lambda^{(i)}}_{1\\leq i\\leq N}$.\nThe piecewise linear surrogate is defined as\n\\begin{equation}\\label{eq:1surrogate}\n\tQ^{(N)}_{s,1}(\\lambda) = \\sum_{1\\leq i\\leq N} \\left[Q_h(\\lambda^{(i)}) + \t\\nabla_\\lambda Q_h(\\lambda^{(i)}) (\\lambda-\\lambda^{(i)})\\right] \\chi_{\\mathcal{V}_{i,N}}(\\lambda).\n\\end{equation}\nConstructing $Q^{(N)}_{s,1}$ requires $Q_h$ and $\\nabla_\\lambda Q_h$ to be evaluated at the points $\\set{\\lambda^{(i)}}_{1\\leq i\\leq N}$.\nCalculating the Jacobian $\\nabla_\\lambda Q_h$ is discussed in Section 3.4.\nHigher-order piecewise polynomial surrogates may be constructed similarly; however, it is generally computationally prohibitive to explicitly calculate higher-order partial derivatives of $Q_h$ with respect to $\\lambda$.\nThe action of higher order derivative tensors (notably Hessians) on parameters may be computationally viable, but such calculations would have to be performed each time the surrogate is evaluated, so they are not particularly viable for stochastic inversion.\n\n\\subsection{Enhancing Surrogates with Error Estimates}\nTraditionally, a posteriori error estimates of QoI from differential equation models derived by variational analysis and adjoints were used to guide local $h$- or $p$-adaptivity, i.e., mesh or order refinement, respectively, in the numerical solution to the model (e.g., see \\cite{Becker_Rannacher_2001} and the references therein).\nSuppose that for each model evaluation $Q_h(\\lambda^{(i)})$ used to generate the surrogate, there exists a reliable error estimate $e_{Q,h}(\\lambda^{(i)})$.\nThere is a set of error estimates$\\set{e_{Q,h}(\\lambda^{(i)})}_{1\\leq i\\leq N}$ corresponding with the set of samples $\\set{\\lambda^{(i)}}_{1\\leq i\\leq N}$.\n\n\nWe can correct for the persistent local bias due to the error $\\epsilon_{Q,h}(\\lambda^{(k)})$ polluting the evaluation of the surrogate model any $\\lambda\\in\\mathcal{V}_{i,N}$ by enhancing the surrogate model with the error estimates.\nWe define the enhanced surrogate by\n\\begin{equation}\n\\widehat{Q}^{(N)}_{s,h}(\\lambda) = Q^{(N)}_{s,h}(\\lambda) + \\sum_{1\\leq i\\leq N} e_{Q,h}(\\lambda^{(i)}) \\chi_{\\mathcal{V}_{i,N}}(\\lambda).\n\\label{eq:enhanced_surrogate}\n\\end{equation}\nThe error enhanced surrogate has a reduced amount of error due to the numerical solution of the model $\\epsilon_h(\\lambda)$.\nAdjoint problems may be useful for calculating reliable error estimates for QoI and also for calculating derivatives.\n\\subsection{Adjoint-based a posteriori Error Estimates and Derivatives}\n\\label{sec:adj}\n\nSuppose the solution to the model $\\mathcal{M}(u;\\lambda)=0$ is defined by the solution to the finite dimensional parameterized linear system\n\\begin{equation}\\label{eq:pls}\n\tA(\\lambda)\\mathbf{u}(\\lambda) = \\mathbf{b}(\\lambda),\n\\end{equation}\nwhere for each $\\lambda\\in\\mathbf{\\Lambda}\\subset\\mathbb{R}^m$, $\\mathbf{b}(\\lambda)\\in\\mathbb{R}^n$ and $A(\\lambda)\\in\\mathbb{R}^{n\\times n}$ is invertible.\nThen, for each $\\lambda\\in\\mathbf{\\Lambda}$, there exists a solution $\\mathbf{u}(\\lambda)\\in\\mathbb{R}^n$.\nSuppose that the QoI map is given by a scalar functional defined by $Q(\\lambda)=\\left\\langle \\mathbf{u}(\\lambda), \\pmb{\\psi} \\right\\rangle$ where $\\pmb{\\psi} \\in\\mathbb{R}^n$ and $\\left\\langle \\cdot, \\cdot \\right\\rangle$ denotes the inner product.\nThe adjoint problem to Eq.~\\eqref{eq:pls} is \n\\begin{equation}\\label{eq:pls_adj}\n\tA(\\lambda)^\\top\\pmb{\\phi}(\\lambda) = \\pmb{\\psi},\n\\end{equation} \nwhere $\\pmb{\\phi}(\\lambda)$ is the adjoint solution, and $\\pmb{\\psi}$ is determined by the QoI.\nSuppose for a fixed $\\lambda\\in\\mathbf{\\Lambda}$ we numerically solve Eq.~\\eqref{eq:pls} to obtain $\\mathbf{u}_h(\\lambda)\\approx \\mathbf{u}(\\lambda)$ and subsequently compute $Q_h(\\lambda)\\approx Q(\\lambda)$.\nThe exact error representation is given by\n\\begin{eqnarray}\n\t\\epsilon_{Q,h}(\\lambda)\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t &=& \\left\\langle A(\\lambda)\\mathbf{u}_h(\\lambda) - \\mathbf{b}(\\lambda), \\pmb{\\phi}(\\lambda) \\right\\rangle. \\label{eq:pls_QoI_error}\n\\end{eqnarray}\nGenerally, $\\pmb{\\phi}(\\lambda)$ is replaced by an approximation $\\pmb{\\phi}_h(\\lambda)$  in Eq.~\\eqref{eq:pls_QoI_error}, giving a computable a posteriori error estimate, which we denote by $e_{Q,h}(\\lambda)$.\nTypically, we compute $\\pmb{\\phi}_h(\\lambda)$ using a higher order method than used to compute $\\mathbf{u}_h(\\lambda)$.\n\nLet $\\lambda_i$ denote the $i$th component of the vector $\\lambda$ for $1\\leq i\\leq m$.\nThen, differentiating Eq.~\\eqref{eq:pls} with respect to $\\lambda_i$ and following a similar set of steps, we arrive at\n\\begin{equation}\\label{eq:pls_QoI_deriv}\n\t\\partial_{\\lambda_i} Q_h(\\lambda) = \\left\\langle \\partial_{\\lambda_i} \\mathbf{b}(\\lambda) - \\left[\\partial_{\\lambda_i}A(\\lambda)\\right] \\mathbf{u}(\\lambda), \\pmb{\\phi}(\\lambda) \\right\\rangle.\n\\end{equation}\nThe partial derivatives of $\\mathbf{b}(\\lambda)$ and $A(\\lambda)$ can often be determined by algorithmic\/automatic differentiation, e.g., see \\cite{Bartlett_AD_2006}.\nSubsequently, this implies that the gradient of the QoI with respect to the parameter $\\lambda$, denoted by $\\nabla_\\lambda Q(\\lambda)$, can be approximated by solving both the model and adjoint model exactly once and then computing a finite number of inner products given by Eq.~\\eqref{eq:pls_QoI_deriv}.\n\n\n\tThis  adjoint-based approach can be applied to most models defined by a linear operator where only a few specific details change.\n\tFor example, when the model is given by a partial differential equation, and a finite element method is used to compute $u_h(\\lambda)$, then we generally solve $\\phi_h(\\lambda)$ either on a refined mesh or using higher order elements to avoid negative effects of Galerkin orthogonality.\n\tTwo comprehensive references on this subject are \\cite{Bangerth_book} and \\cite{Becker_Rannacher_2001}.\n\n\n\nWhen the operator defining the model is nonlinear, one must linearize the model operator prior to defining the adjoint problem.\n\n\\subsection{General Surrogates}\nOur goal is to adaptively form a surrogate $Q^{(N)}_{s,h}$ for use in stochastic inversion.\nWe want to minimize the number of computationally expensive model and adjoint solves, while still providing an accurate solution to a stochastic inverse problem.\nSuppose there are $M$ possible levels to the numerical model.\nOrder the levels so that with increasing index $j$ the model fidelity increases.\nFor example, suppose the forward model $\\mathcal{M}(u;\\lambda)=0$ represents a 1-D steady-state partial differential equation that we solve numerically using the finite element method.\nSuppose we have three different levels of meshes with 10, 100, and 1000 degrees of freedom (DOF) respectively.\nThen, $j=1$ is the level associated with the 10 DOF mesh, $j=2$ is the level associated with the 100 DOF mesh, and $j=3$ is the mesh associated with the 1000 DOF mesh.\nThe error in the QoI due to the numerical solution of the model $\\epsilon_h(\\lambda)$, should generally decrease with increasing $j$.\nLet $Q_{h,j}(\\lambda)$ denote the QoI computed using the numerical solution to the model with level $j$ at $\\lambda$.\nWe can construct the piecewise polynomial surrogate using different levels of model evaluations on different Voronoi cells.\nLet $l=\\set{l_i}_{i=1}^N$ be a set of levels associated with samples $\\set{\\lambda^{(i)}}_{i=1}^N$, i.e. $1 \\leq l_j \\leq M$, for $j=1,2,..,N$.\nWe also can allow different orders of Taylor approximations on different Voronoi cells.\nLet $p=\\set{p_i}_{i=1}^N$ be a set of local polynomial orders associated with samples $\\set{\\lambda^{(i)}}_{i=1}^N$.\nIn practice, we will only allow $p_i=0$ or $p_i=1$ because of the computational cost of using higher local polynomial orders.\nWe can use these to define a general piecewise polynomial surrogate on Voronoi cells\n\\begin{equation}\\label{eq:general_surrogate}\n\tQ^{(N)}_{l,p}(\\lambda) = \\sum_{1\\leq i\\leq N} \\left[Q_{h,l_i}(\\lambda^{(i)}) + \tp_i \\nabla_\\lambda Q_{h,l_i}(\\lambda^{(i)}) (\\lambda-\\lambda^{(i)})\\right] \\chi_{\\mathcal{V}_{i,N}}(\\lambda).\n\\end{equation}\nCorrespondingly, if there are error estimates, one can have an enhanced general surrogate\n\\begin{equation}\n\\widehat{Q}^{(N)}_{l,p}(\\lambda) = Q^{(N)}_{l,p}(\\lambda) + \\sum_{1\\leq i\\leq N} e_{Q,h}(\\lambda^{(i)}) \\chi_{\\mathcal{V}_{i,N}}(\\lambda).\n\\label{eq:enhanced_general_surrogate}\n\\end{equation}\n\n\n\n\n\n\n\n\\section{Error Estimation and Adaptivity}\n\\label{sec:Error}\nComputed QoI error estimates and derivatives from adjoint methods can be combined  to estimate the error in the solution to the stochastic inverse problem both globally and locally.\nLocal error estimates can be used to guide a local adaptive scheme to locally improve the accuracy of the surrogate in its relation to the solution of the stochastic inverse problem.\nWe derive such error estimates, explain how to calculate them numerically, and develop an adaptive refinement scheme based on them.\n\n\\subsection{Derivation of Error Indicators}\n\\label{sec:error_id}\nWe can use the piecewise polynomial on Voronoi tessellation surrogates on the types of stochastic inverse problems above and in the numerical methods for solving them.\nIt may be computationally prohibitive to evaluate $Q_h$ enough times to get an accurate approximation of $P_{\\mathbf{\\Lambda}}$.\nThe surrogate $Q^{(N)}_{l,p}$ as presented above is extremely cheap to evaluate.\nEvaluating $Q^{(N)}_{l,p}(\\lambda)$ only requires performing a nearest neighbor search among $\\set{\\lambda^{(i)}}_{i=1}^N$ and performing some floating point operations.\nBy replacing the map $Q$ with $Q^{(N)}_{l,p}$ when numerically solving the stochastic inverse problem, it can be solved relatively cheaply.\n\n\nLet $P_{\\mathbf{\\Lambda}}(A)$ be the exact probability of an event $A$ based on the exact map $Q$, and let $P_{\\mathbf{\\Lambda}, N,h}(A)$ be the probability of $A$ using the numerical solution to a stochastic inverse problem using the surrogate $Q^{(N)}_{l,p}$ instead of $Q$.\nFor prediction under uncertainty, the goal is to compute the integral $\\int_{A} f d P_{\\mathbf{\\Lambda}}$, given a measurable function $f$ and $A$.\nExtend $f$ by zero outside of $A$\nso that $\\int_A f dP_{\\mathbf{\\Lambda}} = \\int_{\\mathbf{\\Lambda}} f dP_{\\mathbf{\\Lambda}}.$\n\n\nLet $\\mathcal{P}$ be the space of all probability measures on $\\mathbf{\\Lambda}$.\nWe want to find  $Z_\\mathbf{\\Lambda} \\in \\mathcal{P}$ that exactly calculates the integral and also approximates $P_\\mathbf{\\Lambda}$ well.\nDefine the absolute global error with reference to $P_\\mathbf{\\Lambda}$  of $Z_\\mathbf{\\Lambda}$ as\n\\begin{equation} \\label{eq:global_error}\nE_P(Z_\\mathbf{\\Lambda}) := \\left| \\sum_{i=1}^N   \\int_{\\mathcal{V}_{i,N}} f dP_{\\mathbf{\\Lambda}} - \\int_{\\mathcal{V}_{i,N}} f d Z_{\\mathbf{\\Lambda}} \\right| + \\gamma \\left|\\sum_{i=1}^N  P_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N}) - Z_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N} ) \\right|,\n\\end{equation}\nwhere $\\gamma$ is a Lagrange multiplier.\nAny $Z_\\mathbf{\\Lambda}$ such that $E_P(Z_\\mathbf{\\Lambda})=0$ is an adequate approximation of $P_\\mathbf{\\Lambda}$ for the goal of computing the integral.\nSuppose that $\\rho_{\\mathbf{\\Lambda}}$ is the probability density associated with $P_{\\mathbf{\\Lambda}}$ i.e. the Radon-Nikodym derivative of $P_{\\mathbf{\\Lambda}}$ with respect to the Lebesgue measure $\\mu_{\\mathbf{\\Lambda}}$.\nIf $z_\\mathbf{\\Lambda}$ is the Radon-Nikodym derivative of $Z_\\mathbf{\\Lambda}$ with respect to $\\mu_{\\mathbf{\\Lambda}}$, then\n\\begin{equation} \\label{eq:global_error2}\nE_P(Z_\\mathbf{\\Lambda})=E_{\\rho}(z_\\mathbf{\\Lambda}):= |E_1(z_\\mathbf{\\Lambda})| + \\gamma |E_2(z_\\mathbf{\\Lambda})|,\n\\end{equation}\nwhere \n\\begin{equation}\nE_1(z_\\mathbf{\\Lambda}) := \\sum_{i=1}^N   \\int_{\\mathcal{V}_{i,N}} \\left( f \\rho_{\\mathbf{\\Lambda}} -  f z_{\\mathbf{\\Lambda}} \\right) d \\mu_{\\mathbf{\\Lambda}}\n\\end{equation}\nand\n\\begin{equation}\nE_2(z_\\mathbf{\\Lambda}) := \\sum_{i=1}^N  \\int_{\\mathcal{V}_{i,N}} \\left(  \\rho_{\\mathbf{\\Lambda}} -   z_{\\mathbf{\\Lambda}} \\right) d \\mu_{\\mathbf{\\Lambda}}.\n\\end{equation}\nTo balance the error contributions of $E_1$ and $E_2$ in an error reduction algorithm, we want $\\left| \\frac{\\partial E_1}{\\partial z_\\mathbf{\\Lambda}} \\right| \\approx \\gamma \\left| \\frac{\\partial E_2}{\\partial z_\\mathbf{\\Lambda}} \\right|$.\nLet $\\mathbf{\\Lambda}^* = \\set{\\bigcup\\limits_{i=1}^N \\mathcal{V}_{i,N} | P_\\mathbf{\\Lambda}(\\mathcal{V}_{i,N}) \\neq  Z_\\mathbf{\\Lambda}(\\mathcal{V}_{i,N} )}$.\nIf $\\mathbf{\\Lambda}^*$ is bounded  then in $\\mathbf{\\Lambda}^*$\n\\begin{equation}\n\\left| \\frac{\\partial E_1}{\\partial z_\\mathbf{\\Lambda}} \\right| = \\left| \\int_{\\mathbf{\\Lambda}^*} f d \\mu_\\mathbf{\\Lambda} \\right| \\leq \\int_{\\mathbf{\\Lambda}^*} |f| d \\mu_\\mathbf{\\Lambda} \\text{, and } \\left| \\frac{\\partial E_2}{\\partial z_\\mathbf{\\Lambda}} \\right| = \\mu_\\mathbf{\\Lambda}(\\mathbf{\\Lambda}^*).\n\\end{equation}\nNo variation is done in $\\mathbf{\\Lambda} \\setminus \\mathbf{\\Lambda}^*$ since it already has no direct effect on the calculated integral.\nThus, to balance the error contributions\n\\begin{equation}\n\\gamma = \\frac{1}{\\mu_\\mathbf{\\Lambda}(\\mathbf{\\Lambda}^*)} \\int_{\\mathbf{\\Lambda}^*} |f| d \\mu_\\mathbf{\\Lambda}.\n\\end{equation}\nIf $(\\mathbf{\\Lambda}^* \\cap \\text{supp}_{\\mu_\\mathbf{\\Lambda}} (f))$ \nhas zero measure, then $\\gamma=0$.\n$\\gamma$ is theoretically undefined if $\\mathbf{\\Lambda}^*$ is unbounded, but for most reasonable cases a bounded $\\mathbf{\\Lambda}^*$ can be found be refining the Voronoi tessellation.\n\nThe global error $E_P$ can be bounded by a sum of local errors $E_i$ on each Voronoi cell using the triangle inequality:\n\\begin{equation}\\label{eq:global_error3}\nE_P(Z_\\mathbf{\\Lambda}) \\leq  \\sum_{i=1}^N E_i(P_{\\mathbf{\\Lambda}, N,h}) := E(Z_{\\mathbf{\\Lambda}}),\n\\end{equation}\nwhere\n\\begin{equation}\nE_i(P_{\\mathbf{\\Lambda}, N,h}) := \\left|   \\int_{\\mathcal{V}_{i,N}} f dP_{\\mathbf{\\Lambda}} - \\int_{\\mathcal{V}_{i,N}} f d Z_{\\mathbf{\\Lambda}} \\right| + \\gamma \\left| P_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N}) - Z_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N} ) \\right|.\n\\end{equation}\nSuppose the stochastic inverse problem is solved with the surrogate $Q^{(N)}_{l,p}$ and the corresponding enhanced surrogate $\\widehat{Q}^{(N)}_{l,p}$.\nDenote the probability measure calculated by solving the stochastic inverse problem with $Q_{l,p}$ as $P_{\\mathbf{\\Lambda}, N, h}$ and with $\\widehat{Q}^{(N)}_{l,p}$ as $\\widehat{P}_{\\mathbf{\\Lambda}, N, h}$\nThe effect of the deterministic error is smaller in $\\widehat{Q}^{(N)}_{l,p}$ which causes less error pollution in $\\widehat{P}_{\\mathbf{\\Lambda}, N, h}$.\nEstimates $\\widehat{E}_i$ of $E_i(P_{\\mathbf{\\Lambda}, N,h})$ can be computed by replacing the exact probability measure $P_{\\mathbf{\\Lambda}}$ with $\\widehat{P}_{\\mathbf{\\Lambda}, N, h}$ in Equation \\ref{eq:global_error3}.\nto define the local error indicators\n\\begin{equation}\\label{eq:local_error}\n\\widehat{E}_i :=\\widehat{E}_{int, i} + \\widehat{E}_{prob, i},\n\\end{equation}\nwhere\n\\begin{equation}\n\\widehat{E}_{int, i} := \\left| \\int_{\\mathcal{V}_{i,N}} f d\\widehat{P}_{\\mathbf{\\Lambda}, N, h} - \\int_{\\mathcal{V}_{i,N}} f d P_{\\mathbf{\\Lambda}, N, h} \\right|\n\\end{equation}\nand\n\\begin{equation}\n\\widehat{E}_{prob, i} = \\gamma \\left| \\widehat{P}_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N}) - P_{\\mathbf{\\Lambda}, N, h}(\\mathcal{V}_{i,N} ) \\right|.\n\\end{equation}\nIn general, these local integrals and the calculation of $\\gamma$ must be approximated, and the method of approximation may depend on the model.\n\n\n\n\n\n\n\n\n\n\\subsection{Approximation of Integrals} \n\nWe approximate integrals in several ways, depending on how computationally expensive $f$ is to evaluate and the measure that we are integrating with respect to.\nWe look at three cases.\n\\subsection*{Calculation of $\\gamma$ by Emulation}\nEmulation is the best method to estimate $\\gamma$, the weighting factor for $\\widehat{E}_{prob, i}$, which is integrated with respect to $\\mu_{\\mathbf{\\Lambda}}$, the volume measure on $\\mathbf{\\Lambda}$.\nLet $\\set{\\lambda_{em}^{(j)}}_{j=1}^{N_{em}}$ be $N_{em}$ uniform (with respect to the volume measure $\\mu_{\\mathbf{\\Lambda}}$) i.i.d. samples in $\\mathbf{\\Lambda}$, and let $\\set{\\lambda_{em,i}^{(j)}}_{j=1}^{N_{em,i}} = \\left( \\set{\\lambda_{em}^{(j)}}_{j=1}^{N_{em}} \\cap \\mathcal{V}_{i,N} \\right)$ be the $N_{em,i}$ points in $\\mathcal{V}_{i,N}$.\nWe call these ``emulation points\" and use them for Monte Carlo integration over $\\mathcal{V}_{i,N}$.\n\nIn the case where $f$ is very cheap to evaluate compared to solving the numerical model, e.g. a characteristic function, a polynomial, algebraic function, estimate $\\gamma$, by Monte Carlo integration:\n\\begin{equation} \\label{eq:gamma}\n\\gamma \\approx \\left( \\sum_{\\substack{i=1 \\\\ \\widehat{P}_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N}) \\not= P_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N})}}^N \\sum_{j=1}^{N_{em,i}} \\left| f(\\lambda^{(j)}_{em,i}) \\right| \\right)\/ \\left( \\sum_{\\substack{i=1 \\\\ \\widehat{P}_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N}) \\not= P_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N})}}^N N_{em,i} \\right).\n\\end{equation}\nIf $f$ is expensive to solve (e.g. requires solving an expensive model) then replace $f(\\lambda^{(j)}_{em,i})$ in  (\\ref{eq:gamma}) with $f(\\lambda_i)$.\nWe now can calculate $\\widehat{E}_{prob,i}$ for each Voronoi cell using its definition in (\\ref{eq:local_error}).\n\n\n\n\n\\subsection*{Monte Carlo Estimation of $\\widehat{E}_{int,i}$ for Cheap Models}\nSuppose that $f$ is cheap to evaluate and that the solution of the stochastic inverse problem is a set of points $\\set{\\lambda_c^{(j)}}_{j=1}^M$ that are distributed according to $P_{\\mathbf{\\Lambda}}$, e.g. a chain from an MCMC solution to a Bayesian inverse problem.\nLet $\\set{\\lambda_c^{(j)}}_{j=1}^M$ and $\\set{\\widehat{\\lambda}_c^{(j)}}_{j=1}^{\\widehat{M}}$ be the sets distributed according to $P_{\\mathbf{\\Lambda}, N,h }$ and $\\widehat{P}_{\\mathbf{\\Lambda}, N,h }$ respectively.\nWe approximate the integral \n\\begin{equation} \\label{eq:mc_int}\n\\int_{\\mathcal{V}_{i,N}} f dP_{\\mathbf{\\Lambda},N,h} \\approx \\frac{1}{M} \\sum_{j=1}^M f(\\lambda_c^{(j)})\\chi_{\\mathcal{V}_{i,N}}(\\lambda_c^{(j)}),\n\\end{equation}\nand likewise for $\\widehat{P}_{\\mathbf{\\Lambda}, N,h }$.\nSo\n\\begin{equation}\n\\widehat{E}_{int,i} = \\left|\\frac{1}{\\widehat{M}} \\sum_{j=1}^{\\widehat{M}} f(\\widehat{\\lambda}_c^{(j)})\\chi_{\\mathcal{V}_{i,N}}(\\widehat{\\lambda}_c^{(j)}) - \\frac{1}{M} \\sum_{j=1}^M f(\\lambda_c^{(j)})\\chi_{\\mathcal{V}_{i,N}}(\\lambda_c^{(j)}) \\right|.\n\\end{equation}\nThe estimate of the integral is\n\\begin{equation}\nI_N = \\sum_{j=1}^{M} f(\\lambda_c^{(j)}).\n\\end{equation}\n$\\widehat{I}_N$ can correspondingly be calculated using $\\set{\\widehat{\\lambda}_c^{(j)}}_{j=1}^{\\widehat{M}}$.\n\n\\subsection*{Estimation of $\\widehat{E}_{int,i}$ for Expensive Models}\nIt is possible that the function $f$ might be computationally expensive to evaluate.\nIt may involve solving another model or may depend on the solution to the same model that the stochastic inverse problem is based on.\nThis is common if $f$ represents some kind of model prediction.\nIn this case, we will approximate $f$ with a simple function approximation on the Voronoi tessellation, i.e.\n\\begin{equation}\nf_N(\\lambda) \\approx \\sum_{i=1}^N f(\\lambda^{(i)}) \\chi_{\\mathcal{V}_{i,N}}(\\lambda), \\text{   }\\forall \\lambda \\in A.\n\\end{equation}\nThe approximation to the integral over a Voronoi cell is\n\\begin{equation}\n\\int_{\\mathcal{V}_{i,N}} f d P_{\\mathbf{\\Lambda},N,h} = \\int_{\\mathcal{V}_{i,N} \\cap A} f d P_{\\mathbf{\\Lambda},N,h} \\approx \\int_{\\mathcal{V}_{i,N} \\cap A} f_N d P_{\\mathbf{\\Lambda},N,h} = f(\\lambda^{(i)}) P_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N} \\cap A)=I_{i,N}.\n\\end{equation}\nLikewise, the enhanced integral estimate $\\widehat{I}_{i,N}$ can be calculated with $\\widehat{P}_{\\mathbf{\\Lambda},N,h}$.\nUsing the triangle inequality, the local integration error indicator can be expanded:\n$$\\widehat{E}_{int,i} = \\left| \\int_{\\mathcal{V}_{i,N}} f d \\widehat{P}_{\\mathbf{\\Lambda},N,h} - I_{i,N} \\right| \\leq  \\underbrace{\\left| \\int_{\\mathcal{V}_{i,N}} f d \\widehat{P}_{\\mathbf{\\Lambda},N,h} - \\widehat{I}_{i,N}\\right|}_{E_{int,a,i}} + \\underbrace{\\left|\\widehat{I}_{i,N} -  I_{i,N}\\right|}_{E_{int,b,i}}$$\n$E_{int,b,i}$ is easily computable.\nBy Butler et al. \\cite{butler2017}, $E_{int,a,i}$ can be bounded\n\\begin{equation}\nE_{int,a,i} \\leq \\frac{\\widehat{P}_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N} \\cap A)}{\\mu_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N} \\cap A)} \\left| \\int_{\\mathcal{V}_{i,N} \\cap A} f(\\lambda) - f(\\lambda^{(i)}) d \\mu_{\\mathbf{\\Lambda}} \\right| \\leq C_{i,f} \\left[2 \\sup_{\\lambda \\in \\mathcal{V}_{i,N}} d(\\lambda, \\lambda^{(i)})\\right]\n\\end{equation}\nwhere  $C_{i,f}$ is a decreasing function of dimension $n$.\nIf $f$ is Lipschitz continuous with local Lipschitz constant $L_i$, then $C_{i,f}$ is given by\n\\begin{equation} \\label{eq:cif}\nC_{i,f} = \\frac{L_i \\pi^{n\/2}\\widehat{P}_{\\mathbf{\\Lambda},N,h}(\\mathcal{V}_{i,N} \\cap A)}{2^n \\Gamma(\\frac{n}{2}+1)\\mu_{\\mathbf{\\Lambda}}(\\mathcal{V}_{i,N} \\cap A)}.\n\\end{equation}\nIf the local Lipschitz constant is not known and for (possibly) discontinuous $f$, then $L_i$ in (\\ref{eq:cif}) can be estimated by\n\\begin{equation} \\label{eq:Li}\nL_i \\approx \\sup_{\\lambda, \\gamma \\in \\mathcal{V}_{i,N} \\cap A} \\left| \\frac{f(\\lambda) - f(\\gamma)}{\\lambda - \\gamma} \\right|,\n\\end{equation}\nwhich can approximately be solved by taking $\\lambda$ and $\\gamma$ from a set of proposal points in $\\mathcal{V}_{i,N} \\cap A$.\nMore details can be found in Butler et al. \\cite{butler2017}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Types of Refinement}\n\nThe calculated local error indicators can be used to refine the surrogate to increase its accuracy and the accuracy for solving the stochastic inverse problem.We consider three types of refinement.\n\nThe first type of refinement is \\emph{p-refinement}.\nIn this type of refinement, the local polynomial order $p_i$ of the surrogate on a Voronoi cell is increased.\nThis should decrease the local effect of $\\epsilon_s(\\lambda)$, the error in the approximate QoI map due to the increase in quality of the surrogate model.\nBecause of the computational cost of calculating higher-order derivatives,\nwe only consider $p_i=0$ or $p_i=1$, so this refinement can only happen if $p_i=0$.\np-refinement is outlined in Algorithm \\ref{alg:p_refine}.\n\n\n\n\n\n\n\nThe second type of refinement is \\emph{level-refinement}.\nIn this type of refinement, the model level $l_i$ of the surrogate on a Voronoi cell $\\mathcal{V}_{i,N_k}$ is increased.\nThis should decrease the local effect of $\\epsilon_h(\\lambda)$, the error in the approximate QoI map due to numerical error in solving the model.\nA higher-level solve of the model should generally decrease the error in the approximate computations of QoI.\nLevel refinement is outlined in Algorithm \\ref{alg:l_refine}.\n  \n\n\n\n\nThe third type of refinement is \\emph{h-refinement}.\nIn this type of refinement, new samples (generating points for the Voronoi tessellation) are added.\nAdding new samples should locally decrease both $\\epsilon_s(\\lambda)$ and $\\epsilon_h(\\lambda)$ on the approximate QoI map.\nThe local Taylor approximations become better because of the decreases radius and the numerical error is extrapolated less.\nh-refinement is outlined in Algorithm \\ref{alg:h_refine}.\n\n\n\n\n\n\n\n\\begin{algorithm}[h]\\caption{p-refinement}\n\\label{alg:p_refine}\nInput: Set of indices $\\mathcal{I}^p$ of cells to p-refine.\n\n\n\\For{$i \\in \\mathcal{I}_p$}{\nCalculate gradient $\\nabla_{\\lambda}Q_{h,l_i^{(k)}}(\\lambda^{(i)})$.\n\nUpdate $p^{(k+1)}_i = p^{(k)}_i + 1.$\n\n}\n\n\\end{algorithm}\n\n\\begin{algorithm}[h]\\caption{level-refinement}\n\\label{alg:l_refine}\nInput: Set of indices $\\mathcal{I}^l$ to level-refine.\n\n\n\n\\For{$i \\in \\mathcal{I}^l$}{\nUpdate $l^{(k+1)}_i = l^{(k)}_i + 1.$ \\\\\nSolve model $\\mathcal{M}(u; \\lambda)$ numerically at $\\lambda^{(i)}$ at level $l_i^{(k+1)}$, and calculate corresponding QoIs $Q_{h,l^{(k+1)}_i}(\\lambda^{(i)})$ and error estimates $e_{Q,h}(\\lambda^{(i)})$. If $p_i^{(k+1)} > 0$, calculate gradient, $\\nabla_{\\lambda}Q_{h,l_i^{(k+1)}}(\\lambda^{(i)})$.\n\n}\n\n\\end{algorithm}\n\n\\begin{algorithm}[h]\\caption{h-refinement}\n\\label{alg:h_refine}\nInput: New samples $\\mathbf{\\Lambda}^h$.\n\n\n\nSet $N_{k+1} = N_k +J$.\n\n\n\\For{$i=N_k+1,..., N_k + J$}{\nSet $\\lambda^{(i)} = \\lambda_{new}^{i-N_k}$.\n\nSet $l_i^{(k+1)} = l_I^{(k)}$ and $p_i^{(k+1)} = p_I^{(k)}$.\n\nIdentify $I$, such that $\\lambda^{(i)} \\in \\mathcal{V}_{I,N_k}$ (via nearest neighbor search).\n\nSolve model $\\mathcal{M}(u; \\lambda)$ numerically at $\\lambda^{(i)}$ at level $l_i^{(k+1)}$, and calculate corresponding QoIs $Q_{h,l^{(k+1)}_i}(\\lambda^{(i)})$ and error estimates $e_{Q,h}(\\lambda^{(i)})$. If $p_i^{(k+1)} > 0$, calculate gradient $\\nabla_{\\lambda}Q_{h,l_i^{(k_1)}}(\\lambda^{(i)})$.\n}\n\n\\end{algorithm}\n\n\n\n\\subsection{Goal-Oriented Adaptive Refinement}\n\\begin{algorithm}\\caption{Goal-Oriented Adaptive Surrogate Construction}\\label{alg:ad_samp}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\n\\Input{Tolerance $\\epsilon$ and maximum iterations $its_{max}$.}\n\\Output{Integral estimate $\\widehat{I}_N$.}\n\nChoose initial samples $\\set{\\lambda^{(i)}}_{i=1}^{N_0}$ defining $\\set{\\mathcal{V}_{i,N_0}}_{i=1}^{N_0}$, initial polynomial orders $p^{(0)}$, and initial levels $l^{(0)}.$\n\nSolve model $\\mathcal{M}(u; \\lambda)$ numerically at each sample $\\lambda^{(i)}$ at level $l_i^{(0)}$ and calculate corresponding QoIs, $Q_{h,l^{(0)}_i}(\\lambda^{(i)})$, and error estimates, $e_{Q,h}(\\lambda^{(i)})$. If $p_i^{(0)} > 0$, calculate gradient $\\nabla_{\\lambda}Q_{h,l_i^{(0)}}(\\lambda^{(i)})$.\n\n\nConstruct  surrogate $Q^{(N_0)}_{l^{(0)}, p^{(0)}}$ and enhanced surrogate $\\widehat{Q}^{(N_0)}_{l^{(0)}, p^{(0)}}$, solve the stochastic inverse problem with both. Calculate error estimate $I_E$,  local error indicators $\\widehat{E}_i$, and integral estimate $\\widehat{I}_N$.\n\n\n\n\n\\While{$\\left| I_E \\right| > \\epsilon$ \\text{ and } $k < its_{max}$}{\n\nAdd cells with $p_i = 0$ and nonzero probability to $\\mathcal{I}^p$ and do   p-refinement (Alg. \\ref{alg:p_refine}).\n\n\n\nIdentify the max. local error, $\\widehat{E}_{max}$, and cells to refine, $\\mathcal{I}^{ref}$\n\n\\For{$i \\in \\mathcal{I}^{ref}$}{\nCalculate $\\widehat{E}^l_{\\mathcal{J}_i}(i)$,  $\\widehat{E}^h_{\\mathcal{J}_i}(i)$, and $\\lambda^{opt}$.\n\n\\eIf{$\\widehat{E}^l_{\\mathcal{J}_i}(i) \\leq \\widehat{E}^h_{\\mathcal{J}_i}(i)$ and $l_i^{(k)} < l_{max} $}{Add index $i$ to $\\mathcal{I}^l$.}{\nAdd $\\lambda^{opt}$ to $\\mathbf{\\Lambda}^h$.}\n}\n\nLevel-refine (Alg. \\ref{alg:l_refine}) for $\\mathcal{I}^l$, h-refine (Alg. \\ref{alg:h_refine}) for $\\mathbf{\\Lambda}^h$, and update iteration number $k = k +1$\n\nConstruct  surrogate $Q^{(N_k)}_{l^{(k)}, p^{(k)}}$ and enhanced surrogate $\\widehat{Q}^{(N_k)}_{l^{(k)}, p^{(k)}}$ and solve the stochastic inverse problem with both. Calculate error estimate $I_E$,  local error indicators $\\widehat{E}_i$, and integral estimate $\\widehat{I}_N$.\n\n}\n\\end{algorithm}\n\nThe computed probabilities, enhanced probabilities, and local error indicators can be used to guide adaptive refinement for the goal of accurately calculating the integral.\nConsider an initial discretization via an implicit Voronoi tessellation $\\set{\\mathcal{V}_{i,N_0}}_{i=1}^{N_0}$ of $\\mathbf{\\Lambda}$ by $N_0$ points $\\set{\\lambda^{(i)}}_{i=1}^{N_0}$.\nSolve the model with these input parameters, at the lowest level and calculate error estimates (and possibly derivatives).\nThe initial levels are $l^{(0)} = \\set{1}_{i=1}^{N_0}$.\nChoose the polynomial order $p_0$ (0 or 1) with which to define the surrogate depending on whether or not derivatives are known.\nThe initial polynomial orders are $p^{(0)} = \\set{p_0}_{i=1}^{N_0}$.\nThus, we can construct the initial surrogate $Q^{(N_0)}_{l^{(0)}, p^{(0)}}(\\lambda)$ and enhanced surrogate $\\widehat{Q}^{(N_0)}_{l^{(0)}, p^{(0)}}(\\lambda)$ on $\\mathbf{\\Lambda}$.\nStarting with the initial surrogate and enhanced surrogate, an iterative procedure is performed for goal-oriented adaptive refinement.\nWe denote each iteration with an index $k$.\n\nIf an adjoint approach is being used to calculate error estimates and to enhance the surrogate, then derivative information is computationally cheap to obtain as discussed in Section \\ref{sec:adj}.\nA locally piecewise linear surrogate has much more accuracy compared to the piecewise constant surrogate, so if derivative information is known, p-refinement (from p=0 to p=1) should be performed for all cells with nonzero probability at each iteration.\n\n\nThe local error indicators $\\set{E_i}_{i=1}^{N_k}$ are used to guide h- and level-refinement.\nIn practice, the exact error indicators $\\set{E_i}_{i=1}^{N_k}$ are not known, so we use  the approximations $\\set{\\widehat{E}_i}_{i=1}^{N_k}$ as described in Section \\ref{sec:error_id}.\nWe want to perform h- or level-refinement on the Voronoi cells with the highest local error.\nThe maximum local error indicator is $\\widehat{E}_{max} = \\max\\limits_{i=1,..,N_k} \\widehat{E}_i$.\nGiven a parameter $\\alpha \\leq 1$, we want to refine all cells $i$ such that $\\widehat{E}_i > \\alpha \\widehat{E}_{max}$.\nDenote the set of these indices as $\\mathcal{I}^{ref}$.\nThere are alternative techniques for marking cells for refinement (such as the mean strategy or D\\\"{o}rfler strategy which are common in adaptive finite element methods \\cite{demkowicz2006}).\nThe type of refinement for each cell $i \\in \\mathcal{I}^{ref}$ must be determined.\n\n\nDetermining between h- and level-refinement for a sample should be done by determining which type (potentially) reduces the error in a neighborhood of the cell the most.\nLet $\\mathcal{J}_i \\subset \\set{1,2, .., N_k} $ be a set of indices of cells in a neighborhood of cell $i$.\nThese are cells whose local error indicators would likely be changed if refinement is done for cell $i$.\nObvious choices of $\\mathcal{J}_i$ are $i$ and its direct neighbors or cells within some distance of $\\lambda^{(i)}$.\nLet $\\widehat{E}_j^l(i)$ be defined as the local error indicator for cell $j$ after level-refinement is done for cell $i$.\nThe total local error sum of over $\\mathcal{J}_i$ under level-refinement is \n\\begin{equation}\n \\widehat{E}^l_{\\mathcal{J}_i}(i) =  \\sum_{j \\in \\mathcal{J}_i} \\left(\\widehat{E}_j^l(j)\\right).\n\\end{equation}\n\n\nLet $\\widehat{E}_j^h(\\lambda)$ be defined as the local error indicator for cell $j$ after a sample $\\lambda$ is added to the discretization via h-refinement.\nThe total local error sum over $\\mathcal{J}_i$ under h-refinement with $\\lambda$ is \n\\begin{equation}\n \\widehat{E}^h_{\\mathcal{J}_i}(\\lambda) =  \\sum\\limits_{j \\in \\mathcal{J}_i} \\left(\\widehat{E}_j^h(\\lambda)\\right) + \\widehat{E}_{N_k+1}^h(\\lambda),\n\\end{equation}\nand the optimal sample to add is\n\\begin{equation} \\label{eq:opt_h_point}\n\\lambda^{opt} = \\arg \\min_{\\lambda \\in \\mathbf{\\Lambda}} \\widehat{E}^h_{\\mathcal{J}_i}(\\lambda).\n\\end{equation}\nIf $\\widehat{E}^l_{\\mathcal{J}_i}(i) \\leq \\widehat{E}^h_{\\mathcal{J}_i}(\\lambda^{opt})$ level-refinement should be done for cell $i$, and otherwise h-refinement should be done by adding $\\lambda^{opt}$ to the surrogate.\n\n\nIn practice, these optimization problems are unfeasible to solve directly, but simple approximations can be used to estimate $\\widehat{E}^l_{\\mathcal{J}_i}(i)$, $\\lambda^{opt}$, and $\\widehat{E}^h_{\\mathcal{J}_i}(\\lambda^{opt})$.\nThe effect of level-refinement can be approximated by locally replacing $P_{\\mathbf{\\Lambda}, N,h}$ by $\\widehat{P}_{\\mathbf{\\Lambda}, N,h}$, rescaling (assuring it integrates to the original value) $P_{\\mathbf{\\Lambda}, N,h}$ on the rest of the neighborhood, and recalculating the local error indicators.\nThe effect of h-refinement can approximated by adding a reasonable number of proposal points to the Voronoi tessellation and recalculating the local error indicators.\nThe proposal with the smallest local error indicator is the best proposal point for h-refinement.\n\nAfter all of the cells that will undergo refinement and the type of refinements are identified, the model is solved correspondingly. For p-refinement, derivatives are calculated.\nFor level-refinement, the model and adjoints are solved with the higher level model and corresponding QoIs, error estimates, and derivatives are calculated for the parameters corresponding with the generating point of the Voronoi cell.\nFor h-refinement the model and adjoints are solved and corresponding QoIs, error estimates, and derivatives are calculated for the new parameter.\nThis new information is used to update the surrogate and enhanced surrogate to stage $k+1$.\nThe stochastic inverse problem is solved, local error indicators are calculated, and new iteration of refinements are performed.\nThe process terminates when a stopping criterion is met.\nA reasonable stopping criterion is when the approximation of the integral has not varied within some tolerance for several steps.\nIf computational resources are limited the process should also be stopped after some computational budget is met.\nThe adaptive method, including surrogate construction, was implemented using the open-source Python package BET \\cite{pyBET}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical Results}\n\\label{sec:Numerics}\nWe have applied our adaptive scheme for goal-oriented surrogate construction to a variety of example problems.\nThe first example is a simple 1D second order PDE system with two uncertain parameters.\nThis simple low-dimensional problem helps illustrate the algorithm and results can be displayed visually.\nThis problem is cheap to solve and has a analytical solution; however, the relationships between parameters and QoIs are highly nonlinear so it useful for illustrating the benefits of the method.\nThe second example is an 2-D elliptic boundary value problem with a complicated conductivity field parameterized by eight coefficients.\nThe third is a nonlinear system of ordinary differential equations where two initial conditions and four coefficients are uncertain.\nThe fourth and final example is a complicated engineering problem from vibroacoustics involving the deformation of a violin bridge.\n\\subsection{1D Elliptic PDE}\nThe first example is a one-dimensional elliptic boundary value problem with homogeneous Dirichlet boundary conditions\n\\begin{eqnarray}\n-\\lambda_1 v''(x) &=& \\exp(\\lambda_2 x), \\text{ for } x\\in (0,1) \\\\\nv(0)&=&v(1)=0, \\nonumber\n\\end{eqnarray}\nwhere $\\lambda =[\\lambda_1, \\lambda_2]$ are the uncertain parameters.\nLet $u(\\lambda, x) = v(x)$ be the solution to the problem with given parameters $\\lambda$.\nSuppose that that QoI map is $Q(\\lambda) = [Q_1(\\lambda), Q_2(\\lambda)]$ where $Q_1(\\lambda) = \\int_{0.1}^{0.4} u(\\lambda, x) dx$ and $Q_2(\\lambda) = \\int_{0.6}^{0.9} u(\\lambda, x) dx$.\nLet $\\psi_1$ and $\\psi_2$ be the Riesz representors of $Q_1$ and $Q_2$ in $L^2(0,1)$.\nFor reference, $u(\\lambda,x)$, $Q_1(\\lambda)$ and $Q_2(\\lambda)$ can be expressed exactly by \n\\begin{equation}\nu(\\lambda,x) = {\\frac {-{{\\rm e}^{\\lambda_2\\,x}}-x+x{{\\rm e}^{\\lambda_2}}+1}{\\lambda_1{\\lambda_2}^{2}},\n}\n\\end{equation}\n\\begin{equation}\nQ_1(\\lambda) = {\\frac { 0.075\\,{{\\rm e}^{\\lambda_2}}\\lambda_2-{{\\rm e}^{ 0.4\\,\\lambda_2}}+{\n{\\rm e}^{ 0.1\\,\\lambda_2}}+ 0.225\\,\\lambda_2}{\\lambda_1{\\lambda_2}^{3}}},\n\\end{equation}\nand\n\\begin{equation}\nQ_2(\\lambda) = {\\frac { 0.225\\,{{\\rm e}^{\\lambda_2}}\\lambda_2-{{\\rm e}^{ 0.9\\,\\lambda_2}}+{\n{\\rm e}^{ 0.6\\,\\lambda_2}}+ 0.075\\,\\lambda_2}{\\lambda_1{\\lambda_2}^{3}}}.\n\\end{equation}\nThese exact solutions will be used as a reference for comparison with results using a numerical method.\n\n\nThe problem can easily be discretized with a standard centered finite difference approximation with uniform spacing $h$ forming a linear system\n\\begin{equation}\nA_h(\\lambda) \\mathbf{u}_h(\\lambda) = \\mathbf{b}_h(\\lambda)\n\\end{equation}\nfor given parameters $\\lambda$.\nThis system can be efficiently solved directly using a sparse solver.\n$\\psi_{1}$ and $\\psi_{2}$ can be discretized as $\\psi_{h,1}$ and $\\psi_{h,2}$ and can be used to evaluate the approximate QoI map $Q_h(\\lambda) = [Q_{h,1}(\\lambda), Q_{h,2}(\\lambda)]$, where $Q_{h,1}(\\lambda) = \\left\\langle \\mathbf{u}_h(\\lambda), \\psi_{h,1} \\right\\rangle$ and $Q_{h,2}(\\lambda) = \\left\\langle \\mathbf{u}_h(\\lambda), \\psi_{h,2} \\right\\rangle$.\nWe solve the adjoint problems on a mesh that is twice as fine\n\\begin{equation}\nA_{h\/2}(\\lambda)^T \\boldsymbol{\\phi}_{h\/2,1}(\\lambda) = \\boldsymbol{\\psi}_{h\/2,1}\n\\end{equation}\n\\begin{equation}\nA_{h\/2}(\\lambda)^T \\boldsymbol{\\phi}_{h\/2,2}(\\lambda) = \\boldsymbol{\\psi}_{h\/2,2}.\n\\end{equation}\nThe adjoint solutions $\\boldsymbol{\\phi}_{h\/2,1}(\\lambda)$ and $\\boldsymbol{\\phi}_{h\/2,2}(\\lambda)$ are used to calculate $\\epsilon_{Q,h}(\\lambda)$ and $\\partial_{\\lambda} Q_h(\\lambda)$ as shown in (\\ref{eq:pls_QoI_error}) and (\\ref{eq:pls_QoI_error}) respectively.\nThere are five levels of model resolution corresponding with meshes with $h=0.2$, $h=0.1$, $h=0.05$, $h=0.025$, and $h=0.0125$ respectively.\nThis provides all of the ingredients necessary to form regular and enhanced piecewise constant, piecewise linear, and general surrogates $Q^{(N)}_{l,p}$.\n\n\nThe stochastic inverse problem is a Bayesian inverse problem with the forward map $Q: \\mathbf{\\Lambda} \\rightarrow \\mathbb{R}^2$, where $\\mathbf{\\Lambda} = [1,5]^2$ and $Q$ is defined above.\nThe data is $y = [0.22, 0.15]$.\nWe assume a uniform prior on $\\mathbf{\\Lambda}$ and mean-zero Gaussian noise $\\eta \\backsim \\mathcal{N}([0,0], [0.0025, 0.0025])$.\nThe function $f$ that we are interested in integrating with respect to the posterior is $f(\\lambda) = \\frac{dv}{dx}\\vert_{0.83}$.\nA standard Metropolis-Hastings MCMC method is used for solving the stochastic inverse problem.\nA reference calculation of the posterior $P_{\\mathbf{\\Lambda}}$ is calculated using the exact evaluation of $Q$ and an MCMC solution with $10^7$ samples and is shown in Figure \\ref{fig:true_post}.\nUsing the true posterior, the true value of the integral is $\\int_{\\mathbf{\\Lambda}} f P_{\\mathbf{\\Lambda}} = -0.60178$ calculated using Monte Carlo integration.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{hist_exact.png}\n\\caption{Reference posterior distribution.}\n\\label{fig:true_post}\n\\end{figure}\n\n\nFor comparison with the adaptive algorithm, the integral was calculated using the posterior calculated using piecewise constant and piecewise linear surrogates with uniformly distributed generating samples.\n10, 100, 1000, and 10000 uniform i.i.d. generating points were used and the model was solved with all five levels of the mesh.\nTables \\ref{tab:ex1_const} and \\ref{tab:ex1_lin} show the average absolute errors in the calculation of the integral using piecewise constants and piecewise linears respectively with the different numbers of generating points and model levels.\nWe see that as the mesh level is increased the calculations usually become more accurate because $\\epsilon_h$ is decreasing.\nHowever, when there are a small number of generating samples, the effect of the surrogate error $\\epsilon_s$ dominates and the convergence slows down or end completely.\nThere is also the tendency for the error to decrease as the number of generating samples increases, because $\\epsilon_s$ is decreasing.\nHowever, at the low levels there is more deterministic model error and the convergence slows or stops.\nThis is because $\\epsilon_h$ is polluting the calculation, and the effect cannot be removed without reducing the error in the solution of the QoIs.\nThe similarity in errors between the piecewise constant and piecewise linear surrogates is explained in an analogous way to why the trapezoidal rule and midpoint rules for quadrature in one-dimension are the same order.\nWe see this tendency go away in higher dimensional problems.\nThis illustrates that for the solution of the integral to be truly accurate, the surrogate needs to reduce the effects of both of these types of error.\nThe adaptive strategy is designed to reduce both of these effects efficiently.\n\n\nWe initialize the surrogate with $N_0 =50$ uniform i.i.d. samples in $\\mathbf{\\Lambda}$.\nThe model is initially solved at level 1 with piecewise constants everywhere.\nThe adaptive algorithm (Alg. \\ref{alg:ad_samp}) was used to adaptively update the surrogate and calculate the integral, with a relative tolerance of $\\epsilon = 0.01$.\nThe algorithm was run until the convergence criterion was met.\nThe Monte Carlo method for computing error indicators for cheap numerical models was used.\nTable \\ref{tab:ex1_adapt_levels} shows the cumulative number of model evaluations at each level that were performed through each iteration for the first 16 iterations as well as the absolute error in the computation of the integral.\nFigure \\ref{fig:ex1_int} shows the calculated value of the integral at each iteration  calculated with the normal and enhanced surrogate.\nFigure \\ref{fig:ex1_levels} shows the Voronoi tessellation for the initial discretization and iterations 3, 6, 9, 12, and 15.\n\n\nNotice that the first several iterations identify the cells of importance for the computation of the integral and mostly does level-refinement on these cells.\nThe calculated value of the integral changes a great deal at each iteration as the local deterministic error is removed from the surrogate.\nCancellations of error and errors not being removed at the same iteration at different locations of the domain cause the errors to oscillate for a few iterations.\nBy iteration 8, a small amount of h-refinement and much level-refinement has reduced the effect of much of the deterministic error by refining at the highest level in the regions that most influence the integral.\nAt this point, the integral is relatively well-approximated, and h-refinement in these important regions from iteration 8 onward causes the error to rapidly decrease.\nThe cells in the regions with little or no influence on the computation of the integral remain coarse and involve only solving the model at low levels, requiring little computational cost.\nNotice that the error at iteration 12 (with 313 model solves at the highest level) gives a more accurate computation of the integral than with uniform refinement with 10000 model solves at the highest level.\n\n\nThis illustrates the performance of the adaptive surrogate construction.\nRegions highly impacting the computation of the integral are slowly refined with a mixture of refinement methods.\nEventually, the effect of the deterministic error $\\epsilon_h$ is mostly removed  and h-refinement takes over and the error rapidly decreases as the surrogate error $\\epsilon_s$ has an increasingly smaller effect.\nOur method avoids computationally expensive high-level model solves in areas with little influence on the integral and strategically performs these high-level model solves in areas with much influence on the computation of the integral.\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r||r|r|r|r|r|}\n\\hline\n$N$ & Level 1 & Level 2 & Level 3 & Level 4 & Level 5 \\\\ \\hline \\hline\n 10 & 1.15e-01 & 1.22e-01 & 1.21e-01 & 1.03e-01 & 9.90e-02\\\\ \\hline \n 100 & 4.55e-02 & 1.80e-02 & 1.30e-02 & 1.20e-02 & 1.27e-02\\\\ \\hline  \n 1000 & 4.41e-02 & 9.59e-03 & 3.24e-03 & 2.32e-03 & 2.92e-03\\\\ \\hline \n 10000 & 4.45e-02 & 9.37e-03 & 3.51e-03 & 2.62e-03 & 2.76e-03\\\\  \n\\hline\n\\end{tabular}\n\\caption{Average errors (over 20 runs) in the computed integrals using piecewise constant surrogates created with $N$ uniform generating points in $\\mathbf{\\Lambda}$ using the five levels of the mesh.}\n\\label{tab:ex1_const}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r||r|r|r|r|r|}\n\\hline\n$N$ & Level 1 & Level 2 & Level 3 & Level 4 & Level 5 \\\\ \\hline \\hline\n10 &  1.56e-01 & 3.71e-01 & 2.27e-01 & 2.24e-01 & 9.26e-02 \\\\ \\hline \n100 &  5.87e-02 & 2.44e-02 & 3.80e-02 & 2.07e-02 & 1.62e-02 \\\\ \\hline \n1000 &  4.41e-02 & 8.48e-03 & 2.60e-03 & 2.50e-03 & 2.78e-03 \\\\ \\hline \n 10000 & 4.38e-02 & 9.98e-03 & 2.96e-03 & 2.43e-03 & 2.27e-03 \\\\  \n\\hline\n\\end{tabular}\n\\caption{Average errors (over 20 runs) in the computed integrals using piecewise linear surrogates created with $N$ uniform generating points in $\\mathbf{\\Lambda}$ using the five levels of the mesh.}\n\\label{tab:ex1_lin}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r||r|r|r|r|r|r|}\n\\hline\nIteration & Level 1 & Level 2 & Level 3 & Level 4 & Level 5 & Error \\\\\n\\hline \\hline\n0 & 50 &  0 &  0 &  0 &    0 & 4.30e-02 \\\\ \\hline\n1 & 54 & 11 &  0 &  0 &    0 &  2.32e-02\\\\ \\hline\n2 & 58 & 21 &  6 &  0 &    0 &  1.65e-02\\\\ \\hline\n3 & 58 & 32 & 12 &  4 &    0 &  3.68e-02 \\\\ \\hline\n4 & 58 & 32 & 23 &  9 &    2 & 8.22e-03 \\\\ \\hline\n5 & 59 & 32 & 25 & 20 &    6 & 7.25e-02\\\\ \\hline\n6 & 59 & 32 & 26 & 24 &   15 & 6.08e-02 \\\\ \\hline\n7 & 59 & 33 & 26 & 27 &   30 &  6.30e-02\\\\ \\hline\n8 & 60 & 35 & 31 & 34 &   55 & 9.88e-03\\\\ \\hline\n9 & 61 & 37 & 31 & 34 &   70 & 5.03e-03 \\\\ \\hline\n10 & 62 & 49 & 35 & 38 &  115 & 9.78e-03 \\\\ \\hline\n11 & 68 & 51 & 44 & 47 &  190 & 3.67e-03  \\\\ \\hline\n12 & 70 & 57 & 52 & 55 &  307 & 3.73e-03 \\\\ \\hline\n13 & 70 & 63 & 58 & 66 &  457 & 2.51e-03 \\\\ \\hline\n14 & 70 & 65 & 64 & 76 &  726 &  1.13e-03\\\\ \\hline\n15 & 70 & 65 & 67 & 79 &  854 & 4.90e-03 \\\\ \\hline\n16 & 70 & 73 & 80 & 95 & 1487 & 1.27e-03\\\\ \\hline\n\\hline\n\\end{tabular}\n\\caption{Number of cumulative model evaluations at each level and absolute errors in the computation of the integral for 16 iterations of the adaptive algorithm for the 1D problem.}\n\\label{tab:ex1_adapt_levels}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{error_plot.png}\n\\caption{Integral estimate at each iteration of the adaptive algorithm using the normal surrogate and the enhanced surrogate.}\n\\label{fig:ex1_int}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{levels_0.png}\n\\includegraphics[width=0.45\\textwidth]{levels_3.png}\n\\includegraphics[width=0.45\\textwidth]{levels_6.png}\n\\includegraphics[width=0.45\\textwidth]{levels_9.png}\n\n\\includegraphics[width=0.45\\textwidth]{levels_12.png}\n\\includegraphics[width=0.45\\textwidth]{levels_15.png}\n\\caption{Voronoi cells and model levels at iteration 0 (top left), 3 (top right), 6 (center left), 9 (center right), 12 (bottom left), and 15 (bottom right).}\n\\label{fig:ex1_levels}\n\\end{figure}\n\n\n\n\n\\subsection{Elliptic PDE with Uncertain Conductivity}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r||r|r|r|r|r|}\n\\hline\nIteration & Level 1 & Level 2 & Level 3 & Level 4 & Error \\\\ \\hline \\hline\n   0 &  539 &   8 &   2 &   2 &  5.44e-02 \\\\ \\hline\n 100 &  696 &  33 &  10 &  12 & 2.40e-02\\\\   \\hline\n 150 &  764 &  49 &  16 &  22 & 2.29e-02 \\\\ \\hline\n 200 &  823 &  67 &  21 &  40 & 2.44e-02 \\\\ \\hline\n 250 &  884 &  85 &  28 &  54 & 2.48e-02 \\\\ \\hline\n 300 &  942 & 100 &  43 &  66 & 2.33e-02 \\\\ \\hline\n 350 & 1003 & 108 &  60 &  80 & 2.36e-02 \\\\ \\hline\n 400 & 1067 & 117 &  74 &  94 & 2.24e-02\\\\ \\hline\n 450 & 1126 & 126 &  93 & 106 &  2.18e-02\\\\ \\hline\n 500 & 1194 & 132 & 100 & 125 & 8.78e-03\\\\ \\hline\n 550 & 1272 & 148 & 104 & 131 & 2.88e-03  \\\\ \\hline\n 600 & 1345 & 169 & 111 & 140 &  2.96e-03 \\\\\n\\hline\n\\end{tabular}\n\\caption{Cumulative number of model evaluations at each level and absolute errors in the calculations of integrals for iterations of the adaptive scheme applied to the 2D PDE.}\n\\label{tab:ex2}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{PDE_2D_conv-eps-converted-to.pdf}\n\\caption{Integral error in every 50th iteration for the 2D elliptic problem.}\n\\label{fig:ex2}\n\\end{figure}\n\n \n\nWe consider the elliptic boundary value problem on the unit square\n\\begin{eqnarray} \\label{eq:ex2}\n- \\nabla \\cdot \\left( K(x,y) \\nabla u(x,y) \\right) &=& 0, \\text{ for }  (x,y) \\in (0,1)^2  \\nonumber \\\\\nu(0,y)  &=0&, \\nonumber \\\\\n u(1,y) &=& 1, \\nonumber \\\\\n\\nabla u(x,0) \\cdot \\mathbf{n} =\\nabla u(x,1) \\cdot \\mathbf{n} &=& 0,\n\\end{eqnarray}\nwhere $K(x,y)$ is a conductivity field\n which we treat as a random function.\n$K$ belongs to an infinite-dimensional space, but truncating a Karhunen-Lo\\`eve (K-L) Expansion is a classical option for deriving finite-dimensional parameterizations for $\\log(K)$.\nWe  construct the K-L Expansion of $Y(x,y)$ where $Y(x,y) := \\log [K(x,y)]$.\nLet $\\bar{Y}(x,y)$ be the mean value of $Y(x)$, and that it has a exponential covariance $C$ with correlation lengths of 0.1 in both directions.\nHence, $Y(x,y)$ can be written as\n\\begin{equation} \\label{eq:KLE_infty}\nY(x,y) = \\bar{Y}(x,y) + \\sum_{n=0}^{\\infty} \\xi_n \\sqrt{\\lambda_n} f_n(x,y),\n\\end{equation}\nwhere $\\lambda_n$ and $f_n(x,y)$ are eigenpairs determined by $C$, and $\\xi_n$ are standard normal random variables.\nTruncating the series in Eq. \\eqref{eq:KLE_infty} at the $N${th} term gives the finite-dimensional approximation\n\\begin{equation}\\label{eq:KLE_truncate}\nY(x,y) \\approx \\bar{Y}(x,y) + \\sum_{n=0}^{N} \\xi_n \\sqrt{\\lambda_n} f_n(x,y).\n\\end{equation}\nWe use the first eight K-L terms (i.e. $N=8$) because the eigenvalues above this are observed to be negligible for this correlation length and take $\\bar{Y}(x,y) = 0.05$.\n\nGiven K-L coefficients $\\xi_i$, the system is discretized with the continuous Galerkin finite element method using linear elements on structured triangular grids using the open-source software FEniCS \\cite{FEniCS_book, ans20553}. There are four levels of refinement of the mesh, which correspond to the model levels for the adaptive scheme, with 15x15, 21x21, 30x30, and 42x42 elements respectively.\nThe assembled linear systems are solved with a direct solver.\nThe uncertain parameters are the K-L coefficients $\\lambda = [\\xi_1, \\xi_2,..., \\xi_8]$, and thus the parameter \nspace is $\\mathbf{\\Lambda} = \\mathbb{R}^8.$\nThe stochastic inverse problem is defined with the forward map $Q: \\mathbf{\\Lambda} \\rightarrow \\mathbb{R}^4$, where  $Q(\\lambda) = [Q_1(\\lambda), Q_2(\\lambda),Q_3(\\lambda),Q_4(\\lambda)]$, and $Q_1(\\lambda) = u(0.25, 0.25)$, $Q_2(\\lambda) = u(0.25 ,0.75)$, $Q_3(\\lambda) = u(0.75, 0.25)$, $Q_4(\\lambda) = u(0.75,0.75)$.\nThe corresponding adjoint problems are\n\\begin{eqnarray} \\label{eq:ex2_adj}\n- \\nabla \\cdot \\left( K(x,y) \\nabla \\phi(x,y) \\right) &=& \\psi_i(x,y), \\text{ for }  (x,y) \\in [0,1]^2  \\nonumber \\\\\n\\phi(0,y)  &=0&, \\nonumber \\\\\n \\phi(1,y) &=& 0, \\nonumber \\\\\n\\nabla \\phi(x,0) \\cdot \\mathbf{n} =\\nabla \\phi(x,1) \\cdot \\mathbf{n} &=& 0,\n\\end{eqnarray}\nwhere $\\psi_i(x,y)$ are steep Gaussians approximating a Dirac delta at the evaluation points of $Q_i$.\nThe adjoint problems are solved on the same finite element meshes, but with quadratic elements.\nUsing an enriched space for the adjoint problem is required with finite elements for performing error estimates.\nError estimates and gradients can be calculated using Equations \\ref{eq:pls_QoI_error} and \\ref{eq:pls_QoI_deriv}.\n\nWe pose the stochastic inverse problem as a Bayesian inverse problem.\nThe data is $y_{data} = [0.2803, 0.2693, 0.8114, 0.6506]$.\nWe assume a standard normal prior on $\\mathbf{\\Lambda}$ and mean-zero Gaussian noise $\\eta \\backsim \\mathcal{N}(0,0.0025)$ in each component.\nThe function $f$ that we want to integrate is \n$$f(\\lambda) = \\int_{0.4}^{0.6} \\int_{0.4}^{0.6} K(x,y, \\lambda) dx dy$$\n the upscaled (through volume averaging) conductivity over the block $[0.4, 0.6]^2$.\nThe method calculating error indicators that is used is the method for ``expensive\" models.\nThe problem is solved with the Metropolis-Hastings algorithm.\nA reference solution using $10^5$ model evaluations at the highest level is used for comparison with results from the adaptive scheme.\n\n\nThe adaptive scheme was initialized with 500 i.i.d. samples in $\\mathbf{\\Lambda}$ with respect to the standard normal distribution.\nThe model was solved at the lowest level for the initial discretization.\nThe adaptive scheme was run until it met the stopping criterion with $\\epsilon = 10^{-4}$.\nThe adaptive algorithm terminated after 600 iterations with an absolute error of $2.96 \\times 10^{-3}$.\nTable \\ref{tab:ex2} shows the number of average number of cumulative model evaluations at each level for every 50 iterations as well as the average absolute error in the calculation of the integral.\nWe see that in the early iterations, much h-refinement and a small amount of level-refinement occurs.\nIn the moderately high dimensional space $\\mathbf{\\Lambda}$, this is adding new generating points in the areas where the posterior has larger values.\nThe error quickly decreases as samples are added in these regions.\nIn the middle steps of the process, the error fluctuates as the surrogate is refined, sometimes with cancellations of error occurring.\nAfter around 450 iterations, h-refinement at the lowest levels continue and level-refinement increases in the areas where deterministic model error has a larger effect and a great reduction in error occurs.\nThe process terminates with 1765  model evaluations, with the grand majority at the lowest level.\nThis indicates that in this problem, error due to the surrogate was generally more important than the deterministic error in the calculation of the QoI.\nThis result is likely different from the previous example because of the dimension of $\\mathbf{\\Lambda}$.\nUsing a surrogate in higher dimensions can introduce more error than in lower ones.\nh-refinement at the lowest level was mostly able resolve the surrogate enough for an accurate solution.\nExpensive level 4 evaluations only had to be done 140 times, in the areas where the deterministic model error greatly affects the computation of the integral.\nThe adaptive scheme was able to provide a highly accurate computation of the integral with a relatively small number of model evaluations.\nMost of the model evaluations could be done with the cheap low-level model.\nThis illustrates the methods ability to make predictions under uncertainty accurately with much lower computational costs than a non-adaptive method.\n\n\n\n\n\\subsection{Predator-Prey Model}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{NLODE_conv-eps-converted-to.pdf}\n\\caption{Integral error in every 5th iteration for the predator-prey model.}\n\\label{fig:ex3}\n\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r||r|r|r|r|r|r|}\n\\hline\nIteration & Level 1 & Level 2 & Level 3 & Level 4 & Error & Run Time (s) \\\\ \\hline \\hline\n  0 & 100 &  0 &  0 &    0 &  3.63e-01 & 1.16e+00\\\\ \\hline\n  5 & 103 & 15 & 12 &    7 & 3.08e-01  & 1.82e+01 \\\\ \\hline\n 10 & 103 & 20 & 16 &   23 & 3.60e-01 &  5.13e+01 \\\\ \\hline\n 15 & 104 & 23 & 18 &   45 &  6.78e-02 & 9.60e+01 \\\\ \\hline\n 20 & 104 & 26 & 19 &   79 & 1.61e-01 & 1.65e+02 \\\\ \\hline\n 25 & 105 & 26 & 20 &  114 &  2.23e-01 &  2.35e+02\\\\ \\hline\n 30 & 105 & 27 & 20 &  159 & 2.57e-01  &  3.26e+02 \\\\ \\hline\n 35 & 105 & 27 & 20 &  245 & 1.13e-01 & 4.99e+02\\\\ \\hline\n 40 & 105 & 27 & 20 &  385 & 1.10e-01 &  7.80e+02 \\\\ \\hline\n 45 & 105 & 27 & 20 &  599 &  1.40e-01 & 1.21e+03  \\\\ \\hline\n 50 & 105 & 27 & 20 &  833 & 1.33e-01 & 1.68e+03 \\\\ \\hline\n 55 & 106 & 27 & 20 & 1098 & 1.10e-01  & 2.21e+03 \\\\ \\hline\n 60 & 106 & 27 & 20 & 1436 & 9.89e-02  & 2.89e+03  \\\\ \\hline\n 65 & 106 & 27 & 20 & 2128 & 6.77e-02 & 4.28e+03 \\\\ \\hline\n 70 & 106 & 27 & 20 & 3322 & 3.33e-02  & 6.68e+03  \\\\ \\hline\n 75 & 106 & 27 & 20 & 5472 & 1.31e-02  & 1.10e+04 \\\\ \\hline\n 80 & 106 & 27 & 20 & 6875 & 3.29e-03 & 1.38e+04  \\\\ \\hline\n\\end{tabular}\n\\caption{Cumulative number of model evaluations at each level and absolute errors in the calculations of integrals for iterations of the adaptive scheme applied to the predatory-prey model.}\n\\label{tab:ex3}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|r|r|}\n\\hline\nNum. of Evals. & Error & Run Time (s) \\\\ \\hline\n100 &  2.56e-01 & 2.01e+02 \\\\ \\hline\n1000 &  2.53e-01 & 2.01e+03 \\\\ \\hline\n10000 &  2.32e-01 & 2.01e+04 \\\\ \\hline\n\\end{tabular}\n\\caption{Absolute errors and run times for the predator-prey model with uniform refinement at the finest level.}\n\\label{tab:ex3:2}\n\\end{table}\n\n\nWe consider the classical predator-prey model, a nonlinear system of ordinary differential equations defined by the Lotka-Volterra Equations\n\\begin{equation} \\label{eq:predprey}\n{\\begin{aligned}{\\frac {dx}{dt}}&=\\alpha x-\\beta xy\\\\[6pt]{\\frac {dy}{dt}}&=\\delta xy-\\gamma y, \\end{aligned}}\n\\end{equation}\nfor $t \\in [0,T]$ and with initial conditions $x(0)=x_0$ and $y(0)=y_0$.\n$x$ and $y$ represent the population of species of prey and predators respectively at time $t$, and the other parameters describe the population dynamics.\nThe six parameters $\\alpha$, $\\beta$, $\\delta$, $\\gamma$, $x_0$, and $y_0$ are unknown and uncertain.\nFor solving the model, the backward Euler method is used with Newton's method being used to solve the nonlinear system at each step, ensuring stability even with large time steps.\nThe calculated solutions are $x_h$ and $y_h$.\nThe QoIs are the populations of both species at times $t=5$ and $t=10$.\nThe adjoint problems associated with these QoI involve solving the linearized system\n\\begin{equation}\n\\frac{d\\boldsymbol{\\phi}}{dt} = J(x_h(t),y_h(t))\\boldsymbol{\\phi}\n\\end{equation}\nbackward in time for $t \\in [T,0]$ where $J$ is the Jacobian of the RHS of Equation \\ref{eq:predprey}, with $T$ and initial conditions corresponding to the respective QoI.\nWe solve the adjoint system using the Crank-Nicolson method for time integration.\nNote that each time step of the adjoint problem only involves solving a 2x2 linear system.\nError estimates and gradients can be calculated using Equations \\ref{eq:pls_QoI_error} and \\ref{eq:pls_QoI_deriv}, where the inner product $\\langle \\cdot, \\cdot \\rangle$, is the space-time inner product $\\langle \\mathbf{z}_1, \\mathbf{z}_2 \\rangle = \\int_0^T \\mathbf{z}_2^T \\mathbf{z}_1 dt$.\nThe time integrals are approximated with the midpoint rule over each time step.\nWe consider four levels of the model with different time steps $\\Delta t = 0.25, 0.1, 0.01,$ and  $0.001$ respectively.\n\nThe parameter domain is $\\mathbf{\\Lambda} = [1,2]^6$ with $\\lambda \\in \\mathbf{\\Lambda}$ defined by $\\lambda = [\\alpha, \\beta, \\delta, \\gamma, x_0, y_0]$.\nThe stochastic inverse problem is a Bayesian inverse problem with the map $Q: \\mathbf{\\Lambda} \\rightarrow \\mathbb{R}^4$,\n where $Q(\\lambda) = [Q_1(\\lambda), Q_2(\\lambda),Q_3(\\lambda),Q_4(\\lambda)]$, and $Q_1(\\lambda) = x(5)$, $Q_2(\\lambda) = y(5)$, $Q_3(\\lambda) = x(10)$, $Q_4(\\lambda) = y(10)$.\nThe data is $y_{data} = [1, 1.8, 0.5, 1.4]$.\nWe assume a uniform prior on $\\mathbf{\\Lambda}$ and mean-zero Gaussian noise $\\eta \\backsim \\mathcal{N}(0,0.065)$ in each direction.\nThe function that we want to integrate with respect to the posterior is $f(\\lambda) = x_0\/y_0$, the ratio of the initial populations.\n\nThe adaptive scheme was initialized with 100 samples chosen uniformly in $\\mathbf{\\Lambda}$.\nThe model was solved at level 1 at each sample, and the iterative process was run until it converged with $\\epsilon = 10^{-3}$, which took 80 iterations.\nThe errors are calculated with a reference value coming from the solution to the Bayesian inverse problem with $10^5$ samples at the highest model level (with a corresponding run time of $2 \\times 10^6$ s).\nTable \\ref{tab:ex3} shows the number of cumulative model evaluations at each level for every 5th iteration as well as the absolute error in the calculation of the integral.\nFigure \\ref{fig:ex3} shows the error at each of these iterations.\nAs with the first example, notice that for the first 50 iterations the error oscillates without reducing substantially.\nDuring these iterations, a small amount of h-refinement and a large amount of level-refinement is being performed, slowly identifying the regions in $\\mathbf{\\Lambda}$ with significant impact on the computation of the integral.\nThese refinements are done with relatively few model evaluations at any level.\nAfter approximately 50 iterations, the effect of the model discretization error $\\epsilon_h$ on the computation of the integral has mostly been removed.\nThe dominating error contribution is from the local inaccuracy of the surrogate model, so the adaptive scheme continues with h-refinement at the highest model level in the important regions.\nThe error in the integral approximations steadily decreases as the surrogate error is reduced with the h-refinement.\nThe method terminates at the 80th iteration with approximately 7000 model evaluations.\nTable \\ref{tab:ex3:2} shows errors and run times for uniform refinement using the highest level model.\nUsing 10,000 fine model evaluations at a much higher computational cost than the adaptive method results in a much greater error (almost two orders of magnitude).\nThis illustrates the computational savings of the adaptive method.\n\n\\subsection{A Vibroacoustics Application}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{Geigensteg_04.jpg}\n\\caption{A standard violin bridge.}\n\\label{fig:e4}\n\\end{figure}\n\nVibroacoustic applications often involve complex domains such as bridges, mechanical devices, stereo equipment, and musical instruments.\nWe consider the bridge of a violin (shown in Figure \\ref{fig:e4}), which plays a critical role in transmitting the vibration of strings to the body of the instrument.\nViolin bridges have complex curved geometries that are difficult to mesh with classical methods.\nBecause of this, we utilize an isogemetric mortar discretization.\nAlso, for computational efficiency we use a reduced basis method with saddle point approach which allows efficient construction of the linear systems given material parameters.\nA full description and analysis of the reduced basis isogeometric mortar discretization is done by Horger et al. \\cite{horger2017}.\n\n\nOn the three-dimensional violin bridge we consider a problem of linear elasticity\n\\begin{equation}\n-div (\\sigma(u))= f,\n\\end{equation}\nwhere the strain $\\sigma(u)$ depends on the material laws of the bridge and f are body forces.\nThe stress-strain relationship by Hooke's law $\\sigma(u) = \\mathbb{C} \\epsilon(u)$, where $\\epsilon(u) = \\left( \\nabla u + \\nabla u^T \\right)\/2$.\nThe material is orthotropic, and the stiffness tensor is given by\n$$\\mathbb{C} = \\left[ \\begin {array}{cccccc} A_{11}&A_{12}&A_{13}&0&0&0\\\\ \\noalign{\\smallskip}A_{21}&A_{22}&A_{23}&0&0&0\\\\ \\noalign{\\smallskip}\nA_{31}&A_{32}&A_{33}&0&0&0\\\\ \\noalign{\\smallskip}\n0&0&0&G_{yz}&0&0\\\\ \\noalign{\\smallskip}\n0&0&0&0&G_{zx}&0\\\\ \\noalign{\\smallskip}\n0&0&0&0&0&G_{xy}\n\\end {array}\n \\right],$$\nwhere $G_{yz}$, $G_{zx}$, and $G_{xy}$ are the shear moduli, and the entries $A_{ij}$ depend on the elastic moduli $E_x$, $E_y$, $E_z$ and the Poisson's ratios $\\nu_{xy}$, $\\nu_{yz}$, $\\nu_{zx}$.\nThe exact formulation of $\\mathbb{C}$ is shown in \\cite{rand2007}.\nAssuming a known thickness and mass density of the bridge, there are thus nine material parameters for the structure:\n$\\lambda = [E_x, E_y, E_z, \\nu_{xy}, \\nu_{yz}, \\nu_{zx}, G_{yz}, G_{zx}, G_{xy}]$.\nWe assume that the bridge is made out of \\textit{Fagus sylvatica}, the common beech.\nThere are known ranges for the material parameters which are shown in Table \\ref{tab:violin}.\nLet the domain of possible parameters $\\mathbf{\\Lambda}$ be the tensor product of these intervals.\n\n\\begin{table}\n\\small\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|} \n\\hline\n & $E_x$ [MPa] & $E_y$ [MPa] & $E_z$ [MPa] & $\\nu_{xy}$& $\\nu_{yz}$& $\\nu_{zx}$& $G_{yz}$ [MPa] & $G_{zx}$ [MPa]  & $G_{xy}$ [MPa]  \\\\ \\hline \\hline\nmin & 13,000 & 1,500 & 750 & 0.3 & 0.03 & 0.4 & 100 & 500 & 1000 \\\\ \\hline \nmax & 15,000 & 3,000 & 1,500 & 0.4 & 0.06 & 0.5 & 1,000 & 1,500& 2,000 \\\\ \\hline\n\\end{tabular}\n\\caption{Ranges of material parameters for the violin bridge.}\n\\label{tab:violin}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|r|r|r|r|r|} \\hline\nIter. & Level 1 & Level 2 & Level 3 & Rel. Error & Run Time (s)\\\\ \\hline \\hline\n0 & 100 & 0 & 0 & 1.04e-02 & 6,544 \\\\ \\hline\n1 & 102 & 70 & 0 & 5.69e-03 & 20,572\\\\ \\hline\n2 & 103 & 73 & 49 & 4.70e-04 & 35,708 \\\\ \\hline\n\\end{tabular}\n\\caption{Results from two iterations of the adaptive scheme for the violin bridge problem.} \\label{tab:violin2}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|r|r|} \\hline\nNum. of Evals. & Rel. Error & Run Time (s)  \\\\ \\hline \\hline\n10 &  2.93e-02 &  601 \\\\ \\hline\n100 & 3.45e-02 & 6,010 \\\\ \\hline\n1000 &1.37e-02  &  60,100 \\\\ \\hline\n\\end{tabular}\n\\caption{Results with uniform refinement at level 1 for the violin bridge problem.} \\label{tab:violin3} \n\\end{table}\n\nWe assume a known homogeneous force on the top of the bridge from the strings, homogeneous Dirichlet conditions where the bridge attaches to the body, and Neumann conditions on the remaining boundaries.\nThe reduced basis isogeometric mortar discretization results in a large sparse linear saddle point system.\nWe consider three mesh levels and a maximally smooth $p=3$ discretization and a $p=4$ discretization containing the $p=3$ space on each of these meshes.\nThe adjoint problems, error representations, and derivative calculations are constructed by the methods described in Section \\ref{sec:adj}.\nThe forward problems are solved with the $p=3$ discretizations and the adjoint problems are usually solved with the $p=4$ discretizations.\nFor the coarsest mesh, the $p=3$ system has 9,132 degrees of freedom (DOF) and the $p=4$ system has 22,635 DOF.\nFor the middle mesh, the $p=3$ system has 15,468 DOF and the $p=4$ system has 47,985 DOF.\nFor the finest mesh, we only consider the $p=3$ system which has 47,985 DOFs.\nBecause the solution is already so accurate, we do not calculate error estimates and solve the adjoint problems with the same matrix to calculate derivatives.\n\n\nThe QoI map $Q$ has five components.\n$Q_1$ and $Q_2$ are the average x-displacements on the left and right boundaries of the bridge respectively.\n$Q_3$, $Q_4$ and $Q_5$ are the average displacements on the front face of the bridge in the x, y, and z directions respectively.\nThis QoI map is used to define a Bayesian inverse problem with artificial data generated by solving the fine model with reference parameters and adding noise, $y = [1.79 \\times 10^{-4}, -6.57\\times 10^{-4}, -4.17\\times 10^{-3}, -1.31\\times 10^{-3}, -6.61\\times 10^{-2}]$, where the displacement data is measured in cm.\nThe solution is a posterior measure $P_{\\mathbf{\\Lambda}}$ on $\\mathbf{\\Lambda}$.\nWe assume a uniform prior on $\\mathbf{\\Lambda}$ and mean-zero Gaussian noise at a level of 20\\%.\nThe goal is to predict the expected average vertical displacement on the top of the bridge, i.e. $f(x,y,z, \\lambda) = \\int_{\\Omega_{top}} u_z(x,y,z, \\lambda) ds$.\n\n\nOur adaptive algorithm was initialized with 100 uniform i.i.d. samples in $\\mathbf{\\Lambda}$ with a piecewise linear surrogate ($p_i=1$).\nThe model is computationally expensive, with one full evaluation including adjoint solves taking 67.7 s, for level 1, 198.5 s for level 2, and 295.4 s for level 3 approximately, so the error indicators $\\widehat{E}_{int,i}$ are calculated using the method for expensive models.\nA reference computation of the integral of -0.01412 cm was calculated using an error-corrected piecewise linear surrogate generated from 1000 uniform samples solved at level 2, which relates to a run time of 55.1 hours.\nThe adaptive scheme was run for two iterations before converging.\nTable \\ref{tab:violin2} shows results from each iteration including the cumulative number of model evaluations, the relative error compared to the reference, and the cumulative run time.\nWith the initial surrogate, the relative error in the calculation of the integral is already quite small.\nThis small error is due to the fact that for this problem, the QoI response with respect the the parameters are close to linear locally, which is not evident a priori.\nHence, the piecewise linear surrogate does not introduce much error into the prediction directly.\nMost of the error is because the error in the numerical computation of the QoI with the coarse discretization.\nBetween iterations 0 and 1, level-refinement is performed on 70 cells to reduce the effect of the discretization error, and a small amount of h-refinement refines the Voronoi tessellation.\nThese refinements cause a large reduction in the error in the predicted value.\nBetween iterations 1 and 2, more level-refinement is performed on 49 cells requiring the fine model to be solved, and a small amount of h-refinement occurs at levels 1 and 2.\nThese refinements decrease the effect of deterministic model errors even more, and the relative error decreases by an order of magnitude, and the algorithm terminates.\n\n\nThe piecewise polynomial surrogate, combined with our adaptive algorithm have proven to be highly successful for this problem.\nThe piecewise linear surrogate does a good job of approximating the QoI response surface with is locally close to linear.\nThe adaptive scheme identifies the areas where deterministic model error is polluting the prediction and refines accordingly.\nIn a relatively small computation time (approximately 10 hours of CPU time), a highly accurate calculation of the predicted value.\nTable \\ref{tab:violin3} shows corresponding errors and run times for uniform h-refinement of the surrogate using model level 1.\nThe error does slightly decrease as the the number of model evaluations increases, but at a slow rate.\nThe effect of deterministic model error is not being reduced as it is with the adaptive method.\nEven with 1000 model evaluations and almost double the CPU time as with the adaptive method, the error in the prediction is still two orders of magnitude higher than with the adaptive scheme.\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:Conclusions}\nWe have presented a method for goal-oriented adaptive surrogate construction for prediction under uncertainty.\nA general class of surrogate models for response surfaces based on piecewise polynomial approximations on Voronoi tessellations forms the basis of this adaptive strategy.\nThe solution of adjoint problems is used to enhance these surrogates via derivative information which is used to increase the local polynomial order of the approximation and via a posteriori error estimates for QoIs.\nThese enhancements are used to create two levels of surrogates from which local error indicators are derived.\nComputational algorithms for estimating these error indicators are also presented.\nThe error indicators are used to guide p-refinement, level-refinement, and h-refinement of the surrogates.\nSuch refinements improve both the regular and enhanced surrogates.\nThe surrogates and refinement strategies are combined in an iterative method for surrogate construction which reduces the effect of various types of discretization and surrogate errors on the computations of integrals corresponding to predictions under uncertainty.\n\nThe presented method is applied to four example problems of varying complexity; however, even for relatively simple forward models, the map between parameters and QoIs is often highly nonlinear and quite complex.\nThe results show that the method is successful in accurate computations of the integrals of interest with a relatively cheap computational cost.\nIt is important to note that the method was tested on problems of moderate parameter dimension.\nCertain attributes of the algorithm are not tenable for very high-dimensional problems (the ``curse of dimensionality\"), which is certainly a drawback.\nHowever, there has been much work recently in techniques for effectively reducing the dimension of stochastic inverse problems (e.g. active subspaces \\cite{CDW_ActiveSubspace_2014} and reduced basis methods \\cite{le2010}) which can potentially be applied to higher dimensional problems to reduce the effective dimension to a moderate values for which the presented method is feasible.\nThe combination of the method presented here with such dimension reduction techniques is left to future work.\n\n\n\n\\section*{Acknowledgments}\nFinancial support was provided by the the German Research Foundation (DFG, Project WO 671\/11-1).\nThe authors acknowledge Linus Wunderlich for his help in setting up the violin bridge problem and for the violin bridge photograph.\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe balls-and-bins paradigm (see, e.g., \\cite{Feller,JK}) describes\nthe process where $b$ balls are placed independently and uniformly at\nrandom in $n$ bins. Many variants of this classical occupancy problem\nwere intensively studied, having a wide range of applications in\ncomputer science.\n\nIt is well known that when $b = \\lambda n$ for $\\lambda$ fixed and\n$n\\to\\infty$, the load of each bin tends to Poisson with mean\n$\\lambda$ and the bins are asymptotically independent. In particular,\nfor $b=n$, the typical number of empty bins at the end of the process\nis $(1\/\\mathrm{e}+o(1))n$. The typical maximal load in that case is\n$(1+o(1))\\frac{\\log n}{\\log\\log n}$ (cf.~\\cite{Gonnet}). In what\nfollows, we say that an event holds with high probability (w.h.p.) if\nits probability tends to $1$ as $n\\to\\infty$.\n\nThe extensive study of this model in the context of load balancing was\npioneered by the celebrated paper of Azar et al. \\cite{ABKU} (see\nthe survey \\cite{MRS}) that analyzed the effect of a choice between\n$k$ independent uniform bins on the maximal load, in an online\nallocation of $n$ balls to $n$ bins. It was shown in \\cite{ABKU} that\nthe \\textsc{Greedy} algorithm (choose the least loaded bin of the\n$k$) is\noptimal and achieves a maximal-load of $\\log_k\\log n$ w.h.p., compared\nto a load of $\\frac{\\log n}{\\log\\log n}$ for the original case $k=1$.\nThus, $k=2$ random choices already significantly reduce the maximal\nload, and as $k$ further increases, the maximal load drops until it\nbecomes constant at \\mbox{$k = \\Omega(\\log n)$}.\n\nIn the context of online bipartite matchings, the process of\ndynamically matching each client in a group $A$ of size $n\/2$ with one\nof $k$ independent uniform resources in a group $B$ of size $n$\nprecisely corresponds to the above generalization of the balls-and-bins\nparadigm: Each ball has $k$ options for a bin, and is assigned to one\nof them by an online algorithm that should avoid collisions (no two\nballs can share a bin). It is well known that the threshold for\nachieving a perfect matching in this case is $k = \\log_2 n$: For $k\n\\geq(1+\\varepsilon)\\log_2 n$, w.h.p. every client can\nbe exclusively matched to a target resource, and if $k \\leq\n(1-\\varepsilon\n)\\log_2 n$ then $\\Omega(n)$ requests cannot be satisfied.\n\nIn this work, we study the above models in the presence of a constraint\non the memory that the online algorithm has at its disposal. We find\nthat a tradeoff between the choice and the memory governs the ability\nto achieve a perfect allocation as well as a constant maximal load.\nSurprisingly, the threshold separating the subcritical regime from the\nsupercritical regime takes a simple form, in terms of the product of\nthe number of choices $k$, and the size of the memory in bits $m$:\n\\begin{itemize}\n\\item If $km \\gg n$, then one can allocate $(1-\\varepsilon)n$ balls in\n$n$ bins without any collisions w.h.p., and consequently achieve\na load\nof $2$ for $n$ balls.\n\\item If $km \\ll n$, then \\textit{any} algorithm for allocating\n$\\varepsilon n$ balls w.h.p. creates $\\Omega(n)$ collisions\nand an\nunbounded maximal load.\n\\end{itemize}\nRoughly put, when $km \\gg n$ the amount of choice and memory at hand\nsuffices to guarantee an essentially best-possible performance. On the\nother hand, when $km \\ll n$, the memory is too limited to enable the\nalgorithm to make use\nof the extra choice it has, and no substantial improvement can be\ngained over the case $k=1$, where no choice is offered whatsoever.\n\nNote that rigorous lower bounds for space, and in particular tradeoffs\nbetween space and performance (time, communication, etc.), have been\nstudied intensively in the literature of Algorithm Analysis, and are\nusually highly nontrivial. See, for example, \\cite\n{Ajtai,Beame,BJSK,BFMUW,BSSV,BC,Fortnow,FLMV} for some notable examples.\n\nOur first main result establishes the exact threshold of the\nchoice-memory tradeoff for achieving a constant maximal-load. As\nmentioned above, one can verify that when there is unlimited memory,\nthe maximal load is w.h.p. uniformly bounded iff $k = \\Omega\n(\\log n)$.\nThus, assuming that $k=\\Omega(\\log n)$ is a prerequisite for\ndiscussing the effect of limited memory on this threshold.\n\n\\begin{maintheorem}\\label{thm-const-load}\nConsider $n$ balls and $n$ bins, where each ball has $k=\\Omega(\\log\nn)$ uniform choices for bins, and $m=\\Omega(\\log^2 n)$ bits of memory\nare available.\nIf $k m = \\Omega(n)$, one can achieve a maximal-load of $O(1)$ w.h.p.\nConversely, if $k m = o(n)$,\nany algorithm w.h.p. creates a load that exceeds any constant.\n\\end{maintheorem}\n\nConsider the case $k = \\Theta(\\log n)$. The na\\\"{\\i}ve algorithm\nfor achieving a constant maximal-load in this setting requires roughly\n$n$ bits of memory ($2n$ bits of memory always suffice; see\nSection~\\ref{subsec:m-size}). Surprisingly, the above theorem implies\nthat $O(n\/\\log n)$ bits of memory already suffice, and this is tight.\n\nAs we later show, one can extend the upper bound on the load, given in\nTheorem~\\ref{thm-const-load}, to $O(\\frac{n}{km})$\n(useful when $\\frac{n}{km} \\leq\\frac{\\log n}{\\log\\log n}$), whereas\nthe lower bound tends to~$\\infty$ with $\\frac{n}{km}$. This further\ndemonstrates how the quantity $\\frac{n}{km}$ governs the value of the\noptimal maximal load. Indeed, Theorem~\\ref{thm-const-load} will follow\nfrom Theorems \\ref{thm-matching-lower} and~\\ref{thm-matching-upper}\nbelow, which\ndetermine that the threshold for a perfect matching is $km = \\Theta(n)$.\n\nAgain consider the case of $k = \\Theta(\\log n)$, where\nan online algorithm with unlimited memory can achieve an $O(1)$ load\nw.h.p. While the above theorem settles the memory threshold for\nachieving a\nconstant load in this case, one can ask what the optimal maximal load\nwould be below the threshold. This is answered by the next theorem, which\nshows that in this case, for example, $m=n^{1-\\delta}$ bits of memory yield\nno significant improvement over an algorithm which makes random allocations.\n\n\\begin{maintheorem}\\label{thm-large-load}\nConsider $n\/k$ balls and $n$ bins, where each ball has $k$ uniform\nchoices for bins, and $m\\geq\\log n$ bits of memory are available.\nThen for any algorithm, the maximal load is at least\n$(1+o(1))\\frac{\\log(n\/m)}{\\log\\log(n\/m)+\\log k}$ w.h.p.\n\nIn particular, if $m = n^{1-\\delta}$ for some $\\delta> 0$ fixed\nand $2 \\leq k \\leq\\operatorname{polylog}(n)$, then the maximal load\nis $\\Theta\n(\\frac{\\log n}{\\log\\log n})$ w.h.p.\n\\end{maintheorem}\n\nRecall that a load of order $\\frac{\\log n}{\\log\\log n}$ is what one\nwould obtain using a random allocation of $n$ balls in $n$ bins. The\nabove theorem states that, when $m=n^{1-\\delta}$ and $k\\leq\n\\operatorname{polylog}\n(n)$, any algorithm would create such a load already after $n\/k$ rounds.\n\nBefore describing our other results, we note that the lower bounds in\nour theorems in fact apply to a more general setting. In the original\nmodel, in each round the online algorithm chooses one of $k$ uniformly\nchosen bins, thus inducing a distribution on the location of the next\nball. Clearly, this distribution has the property that no bin has a\nprobability larger than $k\/n$.\n\nOur theorems apply to a relaxation of the model, where the algorithm\nis allowed to dynamically choose a distribution $Q_t$ for each round\n$t$, which is required to satisfy the above property (i.e., $\\|Q_t\\|\n_\\infty\\leq k\/n$). We refer to these distributions as \\textit{strategies}.\n\nObserve that indeed this model gives more power to the online\nalgorithm. For instance, if $k=2$ (and the memory is unlimited), an\nalgorithm in the relaxed model can allocate $n\/2$ balls perfectly (by\nassigning $0$ probability to the occupied bins), whereas in the\noriginal model collisions occur already with $n^{2\/3}w(n)$ balls\nw.h.p.,\nfor any $w(n)$ tending to $\\infty$ with $n$.\n\nFurthermore, we also relax the memory constraint on the model. Instead\nof treating the algorithm as an automaton with $2^m$ states, we only impose\nthe restriction that there are at most $2^m$ different strategies to\nchoose from. In other words, at time~$t$, the algorithm knows the\nentire history (the exact location of each ball so far), and needs to\nchoose one of its $2^m$ strategies for the next round. In this sense,\nour lower bounds are for the case of limited communication complexity\nrather than limited space complexity.\n\nWe note that all our bounds remain valid when each\nround offers $k$ choices with repetitions.\n\n\\subsection{Tradeoff for perfect matching}\n\nThe next two theorems address the threshold for\nachieving a perfect matching when allocating $(1-\\delta) n$ balls in\n$n$ bins for some fixed $0<\\delta< 1$ [note that for $\\delta= 0$,\neven with unlimited memory, one needs $k = \\Omega(n)$ choices to avoid\ncollisions w.h.p.].\nThe upper and lower bounds obtained for this threshold are tight up to\na multiplicative constant, and again\npinpoint its location at $k m = \\Theta(n)$. The constants below were\nchosen to simplify the proofs and could be optimized.\n\n\\begin{maintheorem}\\label{thm-matching-lower}\nFor $\\delta>0$ fixed, consider $(1-\\delta) n $ balls and $n$ bins:\nEach ball has $k$ uniform choices for bins, and there are $m\\geq\\log\nn$ bits of memory.\nIf\n\\[\nk m \\leq\\varepsilon n\\qquad \\mbox{for some small constant $\\varepsilon>\n0$} ,\n\\]\nthen \\textit{any} algorithm has $\\Omega(n)$ collisions w.h.p.\n\nFurthermore, the maximal load is w.h.p. $\\Omega(\\log\\log\n(\\frac{n}{km}))$.\n\\end{maintheorem}\n\n\\begin{maintheorem}\\label{thm-matching-upper}\nFor $\\delta>0$ fixed, consider $(1-\\delta)n $ balls and $n$ bins,\nwhere each ball has $k$ uniform choices for bins, and $m$ bits of\nmemory are available. The following holds for any $k \\geq(3\/\\delta)\n\\log n$ and $m \\geq\\log n \\cdot\\log_2\\log n$. If\n\\[\nk m \\geq C n\\qquad \\mbox{for some $C = C(\\delta) > 0$},\n\\]\nthen a perfect allocation (no collisions) can be achieved w.h.p.\n\\end{maintheorem}\n\nIn light of the above, for any value of $k$, the online allocation\nalgorithm given by Theorem~\\ref{thm-matching-upper} is optimal with\nrespect to its memory\nrequirements.\n\n\\subsection{Nonadaptive algorithms}\n\nIn the nonadaptive case, the algorithm is again allowed to choose a fixed\n(possibly randomized) strategy for selecting the placement of\nball number $t$ in one of the $k$ possible randomly chosen bins\ngiven in step~$t$. Therefore, each such algorithm consists of\na sequence $Q_1, Q_2, \\ldots,Q_n$ of $n$ predetermined strategies,\nwhere $Q_t$\nis the strategy for selecting the bin in step number~$t$.\n\nHere, we show\nthat even if $k= n\\frac{\\log\\log n}{\\log n}$, the maximum load\nis w.h.p.  at least $(1-o(1)) \\frac{\\log n}{ \\log\n\\log n}$,\nthat is, it is essentially as large as in the case $k=1$.\nIt is also possible\\vspace*{1pt} to obtain tight bounds for larger values of $k$. We\nillustrate this by considering the case $k=\\Theta(n)$.\n\n\\begin{maintheorem}\n\\label{thm-nonadaptive}\nConsider the problem of allocating $n$ balls into $n$ bins, where each\nball has $k$ uniform choices for bins,\nusing a nonadaptive algorithm.\n\\begin{longlist}[(ii)]\n\\item[(i)]\\label{item-thm-nonadapt-1}\nThe maximum load in any nonadaptive algorithm with\n$k \\leq n\\frac{\\log\\log n}{\\log n}$ is w.h.p. at least $(1-o(1)) \\frac{\\log n}{ \\log\\log n}$.\n\n\\item[(ii)]\\label{item-thm-nonadapt-2}\nFix $0 < \\alpha< 1$. The maximum load in any nonadaptive algorithm with\n$k=\\alpha n$ is w.h.p.  $\\Omega( \\sqrt{\\log\nn})$. This is tight, that is, there exists a nonadaptive algorithm\nwith $k=\\alpha n$ so that the maximum load in it is $O( \\sqrt{\\log\nn})$ w.h.p.\n\\end{longlist}\n\\end{maintheorem}\n\n\\subsection{Range of parameters}\\label{subsec:m-size}\nIn the above theorems and throughout the paper, the parameter $k$ may\nassume values up to $n$. As for the memory, one may na\\\"{\\i}vely use\n$n\\log_2 L$ bits\nto store the status of $n$ bins, each containing at most $L$ balls.\nThe~next observation shows that the $\\log_2 L$ factor is redundant.\n\n\\begin{observation*}\\label{obs-memory-req}\nAt most $n+b-1$ bits of memory suffice to keep track of the number of\nballs in each bin when allocating $b$ balls in $n$ bins.\n\\end{observation*}\n\nIndeed, one can maintain the number of balls in each bin using a vector\nin $\\{0,1\\}^{n+b-1}$, where $1$-bits stand for separators between the bins.\nIn light of this, the original case of unlimited memory corresponds to\nthe case $m = 2n$.\n\n\\subsection{Main techniques}\nThe key argument in the lower bound on the performance of the algorithm\nwith limited memory is analyzing the expected number of new collisions\nthat a given step introduces. We wish to estimate this value with an\nerror probability smaller than $2^{-m}$, so it would hold w.h.p. for\nall of the $2^m$ possible strategies for this step.\n\nTo this end, we apply a large deviation inequality, which relates the\nsum of a sequence of dependent random variables $(X_i)$ with the sum of\ntheir ``predictions''~$(Y_i)$, where $Y_i$ is the expectation of $X_i$\ngiven the history up to time $i$. Proposition~\\ref{prop-predictions}\nessentially shows that if the sum of the predictions $Y_i$ is large\n(exceeds some $\\ell$), then so is the sum of the actual random\nvariables $X_i$, except with probability $\\exp(-c \\ell)$. In the\napplication, the variable $X_i$ measures the number of new collisions\nintroduced by the $i$th ball, and $Y_i$ is determined by the strategy\n$Q_i$ and the history so far.\n\nThe key ingredient in proving this proposition is a\nBernstein--Kolmogorov type inequality for martingales, which appears in\na paper of Freedman \\cite{Freedman} from 1975, and bounds the\nprobability of deviation of a martingale in terms of\nits cumulative variance. We reproduce its elegant proof for\ncompleteness. Crucially, that theorem does not require a uniform bound on\nindividual variances (such as the one that appears in standard versions\nof Azuma--Hoeffding), and rather treats them as random variables.\nConsequently, the quality of our estimate in Proposition~\\ref\n{prop-predictions} is unaffected by the number of random variables involved.\n\nFor the upper bounds, the algorithm essentially partitions the bins\ninto blocks, where for different blocks it maintains an accounting\nof the occupied bins with varying resolution. Once a block exceeds\na certain threshold of occupied bins, it is discarded and a new block\ntakes its place.\n\n\\subsection{Related work} The problem of balanced allocations with\nlimited memory is due to Itai Benjamini. In a recent independent work,\nBenjamini and Makarychev \\cite{BM} studied the special case of the\nproblem for $k=2$ (i.e., when there are two choices for bins at each\nround). While our focus was mainly the regime\n$k=\\Omega(\\log n)$ (where one can readily achieve a constant maximal\nload when there is unlimited memory), our results\nalso apply for smaller values of $k$. Namely, as a by-product, we\nimprove the lower bound of \\cite{BM} by a factor of $2$,\nas well as extend it from $k=2$ to any $k\\leq\\operatorname{polylog}(n)$.\n\nA different notion of memory was introduced to load balancing balls\ninto bins in \\cite{MPS}, where one has the option of placing the\ncurrent ball in the least loaded bin offered in the previous round.\nIn that setting, one could indeed improve the asymptotics (yet not the\norder) of the maximal load.\nNote that in our case we consider the original balls and bins model (as\nstudied in \\cite{ABKU}) and just impose restrictions on the space\ncomplexity of the algorithm.\n\nSee, for example, \\cite{MU}, Chapter 5, for more on the vast literature of\nload balancing balls into bins and its applications in computer science.\n\nA modern application for the classical online perfect matching problem\nhas advertisers (or \\textit{bidders}) play the role of the bins and\ninternet search queries (or \\textit{keywords}) play the role of the\nballs. Upon receiving a search query, the search engine generates the\nlist of related advertisements (revealing the choices for this ball)\nand must decide which of them to present in response to the query\n(where to allocate the ball). Note that in the classical papers that\nanalyze online perfect matching one assumes a worst-case graph rather\nthan a random bipartite graph, and the requests are randomly permuted;\nsee \\cite{KVV} for a fundamental paper in this area.\n\n\\subsection{Organization}\nThis paper is organized as follows. In\nSection~\\ref{sec:martingale}, we prove the large deviation inequality\n(Proposition~\\ref{prop-predictions}). Section~\\ref{sec:lower-bound}\ncontains the lower bounds on the collisions and load, thus proving\nTheorem~\\ref{thm-matching-lower}. Section~\\ref{sec:upper-bound}\nprovides algorithms for achieving a perfect-matching\nand for achieving a constant load, respectively proving Theorem~\\ref\n{thm-matching-upper} and completing the proof of Theorem~\\ref\n{thm-const-load}. In Section~\\ref{sec:large-q},\nwe extend the analysis of the lower bound to prove Theorem~\\ref\n{thm-large-load}. Section~\\ref{sec:nonadaptive}\ndiscusses nonadaptive allocations, and contains the proof of\nTheorem~\\ref{thm-nonadaptive}.\nFinally, Section~\\ref{sec:conclusion} is devoted to concluding remarks.\n\n\\section{A large deviation inequality}\\label{sec:martingale}\n\nThis section contains a large deviation result,\nwhich will later be one of the key ingredients in proving our lower\nbounds for the load.\nOur proof will rely on a Bernstein--Kolmogorov-type inequality of\nFreedman \\cite{Freedman}, which extends the standard Azuma--Hoeffding\nmartingale concentration inequality.\nGiven a sequence of bounded (possibly dependent) random variables\n$(X_i)$ adapted to some filter $(\\mathcal{F}_i)$,\none can consider the sequence $(Y_i)$ where $Y_i = \\mathbb{E}\n[X_i\\mid\n\\mathcal{F}_{i-1}]$, which can be viewed as\npredictions for the $(X_i)$'s. The following proposition essentially\nsays that, if the sum of the predictions $Y_i$ is large, so is the\nsum of the actual variables $X_i$.\n\n\\begin{proposition}\\label{prop-predictions}\nLet $(X_i)$ be a sequence of random variables adapted to the filter\n$(\\mathcal{F}_i)$ so that $0 \\leq X_i \\leq M$ for all $i$, and let\n$Y_i = \\mathbb{E}\n[X_i\\mid\\mathcal{F}_{i-1}]$. Then\n\\[\n\\mathbb{P}\\biggl(\\biggl\\{\\bigg|\\frac{\\sum_{i\\leq t} X_i}{\\sum\n_{i\\leq t}\nY_i}-1\\bigg| \\geq\\frac12 \\mbox{ and } \\sum_{i\\leq t} Y_i\n\\geq h\\biggr\\}\\mbox{ for some $t$}\\biggr)\\le\n\\exp\\biggl(-\\frac{h}{20M} + 2\\biggr).\n\\]\n\\end{proposition}\n\n\\begin{pf}\nAs mentioned above, the proof hinges on a tail-inequality for sums of\nrandom variables, which appears\nin the work of Freedman \\cite{Freedman} from 1975 (see also~\\cite\n{Steiger}), and extends such inequalities of Bernstein and Kolmogorov\nto the setting of martingales. See \\cite{Freedman} and the references\ntherein for more background on these inequalities, as well as \\cite{Burkholder}\nfor similar martingale estimates.\nWe include the short proof of Theorem~\\ref{thm-martingale} for completeness.\n\n\\begin{theorem}[(\\cite{Freedman}, {Theorem 1.6})]\\label{thm-martingale}\nLet $(S_0,S_1,\\ldots)$ be a martingale with respect to the filter\n$(\\mathcal{F}_i)$.\nSuppose that $S_{i+1}-S_i \\leq M$ for all $i$, and write $V_t = \\sum\n_{i=1}^t \\var(S_i\\mid\\mathcal{F}_{i-1})$.\nThen for any $s,v>0$ we have\n\\[\n\\mathbb{P}(S_n \\geq S_0 + s , V_n \\leq v\\mbox{ for some\n$n$})\n\\leq\\exp\\biggl[- \\frac{s^2}{2(v + M s )}\\biggr].\n\\]\n\\end{theorem}\n\n\\begin{pf\nWithout loss of generality, suppose $S_0=0$, and put $X_i \\stackrel\n{\\triangle}{=}S_i -\nS_{i-1}$. Re-scaling $S_n$ by $M$, it clearly suffices to treat the\ncase $X_i \\leq1$. Set\n\\[\nV_t \\stackrel{\\triangle}{=}\\sum_{i=1}^t \\var(S_i\\mid\\mathcal\n{F}_{i-1}) = \\sum_{i=1}^t \\mathbb{E}\n(X_i^2 \\mid\\mathcal{F}_{i-1}),\n\\]\nand for some $\\lambda> 0$ to be specified later, define\n\\[\nZ_t \\stackrel{\\triangle}{=}\\exp\\bigl(\\lambda S_t - (\\mathrm\n{e}^\\lambda- 1 - \\lambda\n)V_t\\bigr).\n\\]\nThe next calculation will show that $(Z_t)$ is a super-martingale with\nrespect to the filter $(\\mathcal{F}_t)$.\nFirst, notice that the function\n\\[\nf(z) \\stackrel{\\triangle}{=}\\frac{\\mathrm{e}^{z}-1-z}{z^2}\\qquad \\mbox{for $z \\neq\n0$}, f(0)\\stackrel{\\triangle}{=}\\frac12,\n\\]\nis monotone increasing [as $f'(z) > 0$ for all $z \\neq0$], and in\nparticular, $ f(\\lambda z) \\leq f(\\lambda)$\nfor all $ z \\leq1$. Rearranging,\n\\[\n\\exp(\\lambda z) \\leq1 + \\lambda z + (\\mathrm{e}^{\\lambda} - 1\n- \\lambda)z^2\n\\qquad \\mbox{ for all $z \\leq1$}.\n\\]\nNow, since $X_i\\leq1$ and $\\mathbb{E}[X_i\\mid\\mathcal{F}_{i-1}]=0$\nfor all $i$, it\nfollows that\n\\begin{eqnarray*}\n\\mathbb{E}[ \\exp(\\lambda X_i) \\mid\\mathcal{F}_{i-1}]\n&\\leq&\n1 + (\\mathrm{e}^{\\lambda} - 1 - \\lambda)\\mathbb\n{E}[X_i^2\n\\mid\\mathcal{F}_{i-1}] \\\\\n&\\leq&\\exp\\bigl((\\mathrm{e}^{\\lambda} - 1 - \\lambda\n)\\mathbb{E}[X_i^2 \\mid\\mathcal{F}_{i-1}]\\bigr).\n\\end{eqnarray*}\nBy definition, this precisely says that $\\mathbb{E}[Z_i \\mid\\mathcal\n{F}_{i-1}] \\leq\nZ_{i-1}$. That is, $(Z_t)$ is a super-martingale,\nand hence by the Optional Stopping Theorem so is $(Z_{\\tau\\wedge n})$,\nwhere $n$ is some integer and $\\tau= \\min\\{t \\dvtx  S_t \\geq s\\}$. In particular,\n\\[\n\\mathbb{E}Z_{\\tau\\wedge n} \\leq Z_0 = 1,\n\\]\nand (noticing that $V_{t+1} \\geq V_t$ for all $t$) Markov's inequality\nnext implies that\n\\begin{eqnarray*}\n\\mathbb{P}\\biggl(\\bigcup_{t\\leq n}(S_t \\geq s,\\ V_t \\leq v)\\biggr)\n\\leq\\exp\n[-\\lambda s + (\\mathrm{e}^{\\lambda}-1-\\lambda)v ].\n\\end{eqnarray*}\nA choice of $\\lambda= \\log(\\frac{s+v}v) \\geq\\frac\n{s}{s+v} + \\frac12 (\\frac{s}{s+v})^2$ therefore yields\n\\begin{eqnarray*}\n\\mathbb{P}\\biggl(\\bigcup_{t\\leq n}(S_t \\geq s ,\\ V_t \\leq v)\n\\biggr)&\\leq&\n\\exp\\biggl[s-(s+v)\\log\\biggl(\\frac{s+v}{v}\\biggr)\\biggr]\\\\\n& \\leq&\\exp\n\\biggl[-\\frac{s^2}{2(s+v)}\\biggr],\n\\end{eqnarray*}\nand taking a limit over $n$ concludes the proof.\n\\end{pf}\n\n\\begin{remark*}\nNote that Theorem~\\ref{thm-martingale} generalizes the\nwell-known version of the Azuma--Hoeffding inequality, where each of the\nterms $\\var(X_{i}\\mid\\mathcal{F}_{i-1})$ is bounded by some \\textit{constant}\n$\\sigma_i^2$ (cf., e.g., \\cite{McDiarmid}).\n\\end{remark*}\n\n\nWe now wish to infer Proposition~\\ref{prop-predictions} from\nTheorem~\\ref{thm-martingale}. To this end, define\n\\[\nZ_t \\stackrel{\\triangle}{=}\\sum_{i=1}^t Y_i - X_i ,\\qquad   V_t\n\\stackrel{\\triangle}{=}\\sum\n_{i=1}^t\\var( Z_i\\mid\\mathcal{F}_{i-1}),\n\\]\nand observe that $(Z_t)$ is a martingale by the definition $Y_i =\n\\mathbb{E}\n[X_i\\mid\\mathcal{F}_{i-1}]$. Moreover, as the $X_i$'s are uniformly bounded,\nso are the increments of $Z_t$:\n\\[\n| Z_i - Z_{i-1} | = | Y_i - X_i | \\leq M.\n\\]\nFurthermore, crucially, the variances of the increments are bounded as\nwell in terms\nof the conditional expectations:\n\\[\n\\var(Y_i - X_i \\mid\\mathcal{F}_{i-1}) = \\var(X_i\n\\mid\n\\mathcal{F}_{i-1}) \\leq M \\cdot\\mathbb{E}[\nX_i \\mid\\mathcal{F}_{i-1}] = M \\cdot Y_i,\n\\]\ngiving that $V_t \\leq M \\sum_{i=1}^t Y_i$.\n\nFinally, for any integer $j\\geq1$ let $A_j$ denote the event\n\\[\nA_j = \\biggl( \\biggl\\{\\sum_{i \\leq t} X_i \\leq\\frac12 \\sum\n_{i\\leq t}Y_i \\mbox{ and }jh \\leq\\sum_{i \\leq t} Y_i \\leq(j+1)h\n\\biggr\\}\\mbox{ for some $t$}\\biggr).\n\\]\nNote that the event $A_j$ implies that $Z_t \\geq jh\/2$. Hence, applying\nTheorem~\\ref{thm-martingale} to the martingale $(Z_t)$ along with its\ncumulative variances\n$(V_t)$ we now get\n\\begin{eqnarray*}\n\\mathbb{P}(A_j) &\\leq&\\mathbb{P}\\biggl( Z_t \\geq\\frac\n{j}2 h  ,  V_t\n\\leq(j+1)hM\\biggr)\\\\\n& \\leq&\\exp\\biggl[-\\frac{(({j}\/2) h)^2}{2\n((j+1)hM+M(({j}\/2)h))}\\biggr]\\\\\n&=& \\exp\\biggl[-\\frac{j^2}{4(3j+2)}(h\/M)\\biggr] \\leq\\exp\n\\biggl(-\\frac{h}{20M} j\\biggr).\n\\end{eqnarray*}\nSumming over the values of $j$, we obtain that if $h \\geq20M$ then\n\\[\n\\mathbb{P}\\biggl(\\bigcup_{j\\geq1} A_j\\biggr) \\leq\\frac{\\mathrm\n{e}}{\\mathrm\n{e}-1} \\exp\\biggl(-\\frac{h}{20M} \\biggr) \\leq\\exp\\biggl(-\\frac\n{h}{20M} j + 1\\biggr),\n\\]\nwhile for $h\\leq20M$ the above inequality holds trivially. Hence, for\nall $h>0$,\n\\begin{equation}\\label{eq-prob-x\/y-1}\n\\quad \\mathbb{P}\\biggl(\\exists t \\dvtx  \\biggl\\{\\sum_{i \\leq t} X_i \\leq\\frac\n12 \\sum_{i\\leq t}Y_i \\mbox{ and }\\sum_{i \\leq t} Y_i \\geq h \\biggr\\}\n\\biggr)\n\\leq\\exp\\biggl(-\\frac{h}{20M}+1\\biggr).\n\\end{equation}\nTo complete the proof of the proposition, we repeat the above analysis for\n\\[\nZ'_t \\stackrel{\\triangle}{=}-Z_t = \\sum_{i=1}^t X_i - Y_i\n,\\qquad  V'_t \\stackrel{\\triangle}{=}\\sum\n_{i=1}^t\\var( Z'_i\\mid\\mathcal{F}_{i-1}) = V_t.\n\\]\nClearly, we again have $|Z'_i - Z'_{i-1}| \\leq M$ and $V'_t \\leq M \\sum\n_{i=1}^t Y_i$. Defining\n\\[\nA'_j = \\biggl( \\biggl\\{\\sum_{i \\leq t} Y_i \\leq\\frac23 \\sum\n_{i\\leq t}X_i \\mbox{ and }jh \\leq\\sum_{i \\leq t} Y_i \\leq(j+1)h\n\\biggr\\}\\mbox{ for some $t$}\\biggr),\n\\]\nit follows that the event $A'_j$ implies that $Z'_t \\geq\\frac12 \\sum\n_{i}Y_i \\geq jh\/2$. Therefore, as before, we have that\n\\[\n\\mathbb{P}(A'_j) \\leq\\mathbb{P}\\biggl( Z'_t \\geq\\frac\n{j}2h  ,  V'_t\n\\leq(j+1)hM\\biggr) \\leq\\exp\\biggl(-\\frac{h}{20M} j\\biggr),\n\\]\nand thus for all $h>0$\n\\begin{equation}\\label{eq-prob-x\/y-2}\n\\mathbb{P}\\biggl(\\exists t\\dvtx  \\biggl\\{\\sum_{i \\leq t} Y_i \\leq\\frac\n23 \\sum_{i\\leq t}X_i \\mbox{ and }\\sum_{i \\leq t} Y_i \\geq h \\biggr\\}\n\\biggr)\n\\leq\\exp\\biggl(-\\frac{h}{20M}+1\\biggr).\n\\end{equation}\nSumming the probabilities in \\eqref{eq-prob-x\/y-1} and \\eqref\n{eq-prob-x\/y-2} yields the desired result.\n\\end{pf}\n\nWe note that essentially the same proof yields the following\ngeneralization of Proposition~\\ref{prop-predictions}.\nAs before, the constants can be optimized.\n\n\\begin{proposition}\\label{prop-predictions-2}\nLet $(X_i)$ and $(Y_i)$ be as given in Proposition~\\textup{\\ref\n{prop-predictions}}. Then for any $0 < \\varepsilon\\leq\\frac12$,\n\\begin{eqnarray*}\n&&\\mathbb{P}\\biggl(\\biggl\\{\\bigg|\\frac{\\sum_{i\\leq t} X_i}{\\sum\n_{i\\leq t}\nY_i} -1 \\bigg| \\geq\\varepsilon\\mbox{ and } \\sum_{i\\leq t} Y_i \\geq\nh\\biggr\\}\\mbox{ for some $t$}\\biggr)\\\\\n&&\\qquad  \\leq\\exp\\biggl(-\\frac\n{h\\varepsilon\n^2}{5M} + 2\\biggr).\n\\end{eqnarray*}\n\\end{proposition}\n\n\n\\begin{remark}\\label{rem-prop-stopping-time}\nThe statements of Propositions~\\ref{prop-predictions} and~\\ref{prop-predictions-2} hold also in conjunction with any\nstopping time $\\tau$ adapted to the filter $(\\mathcal{F}_i)$. That\nis, we get\nthe same bound on the probability of the mentioned event happening at\nany time $t < \\tau$. This follows easily, for instance, by altering\nthe sequence of increments to be identically $0$ after $\\tau$.\nSuch statements become useful when the uniform bound on the increments\nis only valid before $\\tau$.\n\\end{remark}\n\n\\section{Lower bounds on the collisions and load}\\label{sec:lower-bound}\n\nIn this section, we prove Theorem~\\ref{thm-matching-lower} as well as\nthe lower bound in Theorem~\\ref{thm-const-load}, by showing that if\nthe quantity $km\/n$\nis suitably small, then any allocation would necessarily produce nearly\nlinearly many bins with arbitrarily large load.\n\nThe main ingredient in the proof is a bound for the number of\ncollisions, that is, pairs of balls that share a bin, defined next.\nLet $N_t(i)$ denote the number of balls in bin $i$ after performing $t$\nrounds; the number of collisions at time $t$ is then\n\\[\n\\operatorname{Col}_2(t) \\stackrel{\\triangle}{=}\\sum_{i=1}^n \\pmatrix{ N_t(i)\\cr 2}.\n\\]\nThe following theorem provides a lower bound on $\\operatorname\n{Col}_2(t)$ for $t\n\\geq c\\cdot km$\nfor some absolute $c > 0$.\n\n\\begin{theorem}\\label{thm-collisions-lower-bound}\nConsider $n $ balls and $n$ bins,\nwhere each ball has $k$ uniform choices for bins, and $m\\geq\\log n$\nbits of memory are available.\n\\begin{longlist}[(ii)]\n\\item[(i)]\\label{item-thm-col-1}For all $t \\geq500\\cdot k m $, we have\n\\[\n\\mathbb{E}\\operatorname{Col}_2(t) \\geq t^2 \/ (9n).\n\\]\n\\item[(ii)]\\label{item-thm-col-2}Furthermore, with probability\n$1-O(n^{-4})$, for all $L=L(n)$ and any $t \\geq(  500 k m\n\\vee  30 \\sqrt{Ln\\log n} )$, either the maximal load is at\nleast $L$ or\n\\[\n\\operatorname{Col}_2(t) \\geq t^2 \/ (16n).\n\\]\n\\end{longlist}\n\\end{theorem}\n\nNote that the main statement of Theorem~\\ref{thm-matching-lower}\nimmediately follows from the above theorem,\nby choosing $t = (1-\\delta)n$ and $L = \\sqrt{n}$. Indeed, recalling\nthe assumption in Theorem~\\ref{thm-matching-lower} that $m \\geq\\log\nn$, we obtain\nthat, except with probability $O(n^{-4})$,\neither the algorithm creates a load of $\\sqrt{n}$, or\nit has $\\operatorname{Col}_2(n) \\geq\\frac{(1-\\delta)^2}{16} n$.\nObserving that a\nload of $L$ immediately induces ${L\\choose 2}$ collisions, we deduce\nthat either way there are at least $\\Omega(n)$ collisions w.h.p.\n\nWe next prove Theorem~\\ref{thm-collisions-lower-bound}; the\nstatement of Theorem~\\ref{thm-matching-lower} on unbounded maximal\nload will follow from an iterative\napplication of a more general form of this theorem (namely,\nTheorem~\\ref{thm-collisions-gen}), which appears\nin Section~\\ref{subsec:unbounded-maxload}.\n\n\\begin{pf*}{Proof of Theorem~\\ref{thm-collisions-lower-bound}}\nAs noted in the \\hyperref[sec1]{Introduction}, we relax the model by allowing the\nalgorithm to choose any distribution $\\mu= (\\mu(1),\\ldots,\\mu(n))$\nfor the location of the next ball, as long as it satisfies\n$\\|\\mu\\|_\\infty\\leq k\/n$.\n\nWe also relax the memory constraint as follows. The algorithm has a\npool of at most $2^m$ different strategies, and may choose any of them\nat a given step without any restriction (basing its dynamic decision on\nthe entire history).\n\nTo summarize, the algorithm has a pool of at most $2^m$ strategies, all\nof which have an $L^\\infty$-norm of at most $k\/n$. In each given\nround, it adaptively chooses a strategy $\\mu$ from this pool based on\nthe entire history, and a ball then falls to a bin distributed\naccording to $\\mu$.\n\nThe outline of the proof is as follows: consider the sequence\n$Q_1,\\ldots,Q_n$, chosen adaptively out of the pool of $2^m$ of strategies.\nThe large deviation inequality of Section~\\ref{sec:martingale}\n(Proposition~\\ref{prop-predictions})\nwill enable us to show the following. The expected number of collisions\nencountered in the above process\nis well approximated by the expected number of collisions between $n$\nindependent balls, placed according to $Q_1,\\ldots,Q_n$\n(i.e., equivalent to the result of the nonadaptive algorithm with\nstrategies $Q_1,\\ldots,Q_n$).\n\nHaving reduced the problem to the analysis of a nonadaptive algorithm,\nwe may then\nderive a lower bound on $\\mathbb{E}\\operatorname{Col}_2(t)$ by\nanalyzing the structure of\nthe above strategies.\nThis bound is then translated to a bound on $\\operatorname{Col}_2(t)$\nusing another\napplication\nof the large deviation inequality of Proposition~\\ref{prop-predictions}.\n\nLet $\\nu= (\\nu(1),\\ldots,\\nu(n))$ be an arbitrary probability\ndistribution on $[n]$ satisfying\n$\\|\\nu\\|_\\infty\\leq k\/n$, and denote by $Q_s = (Q_s(1),\\ldots\n,Q_s(n))$ the\nstrategy of the algorithm at time $s$. It will be convenient from time\nto time to treat these distributions as vectors in $\\mathbb{R}^n$.\n\nBy the above discussion, $Q_s$ is a random variable\nwhose values belong to some a priori set $\\{\\mu_1,\\ldots\n,\\mu_{2^m}\\}$.\nWe further let $J_s$ denote the actual position of the ball at time $s$\n(drawn according\nto the distribution $Q_s$).\n\nGiven the strategy at time $s$, let $x_s$ denote the probability of a\ncollision between\n$\\nu$ and $Q_s$ given $J_s$, that is, that the ball that is distributed\naccording to $\\nu$\nwill collide with the one that arrived in time $s$. We let $v_s$ be the\ninner product of $Q_s$ and~$\\nu$, which\nmeasures the expectation of these collisions:\n\\begin{eqnarray*}\nx_s^\\nu&\\stackrel{\\triangle}{=}&\\nu( J_s),\\\\\nv_s^{\\nu} &\\stackrel{\\triangle}{=}&\\langle Q_s,\\nu\\rangle  = \\sum\n_{i=1}^n Q_s(i) \\nu(i)\n= \\mathbb{E}[x_s^\\nu\\mid\\mathcal{F}_{s-1}].\n\\end{eqnarray*}\nFurther define the cumulative sums of $v_s^\\nu$ and $x_s^\\nu$ as follows:\n\\begin{eqnarray*}\nX_t^{\\nu} &\\stackrel{\\triangle}{=}&\\sum_{s=1}^t x_s^\\nu,\\\\\nV_t^{\\nu} &\\stackrel{\\triangle}{=}&\\sum_{s=1}^t v_s^\\nu.\n\\end{eqnarray*}\nTo motivate these definitions, notice that given the history up to time\n$s-1$ and any possible strategy for the next round, $\\nu$,\nwe have\n\\[\nX_{s-1}^{\\nu} = \\sum_{i=1}^{s-1} \\nu(J_i) = \\sum_{i=1}^n \\nu(i) |\\{\nr<s\\dvtx J_r = i\\}|\n= \\sum_{i=1}^n \\nu(i) N_{s-1}(i),\n\\]\nand so $X_{s-1}^{Q_s}$ is the expected number of collisions that will\nbe contributed by the ball $J_s \\sim Q_s$\ngiven the entire history $\\mathcal{F}_{s-1}$. Summing over $s$, we\nhave that\n\\[\n\\mathbb{E}\\operatorname{Col}_2(t) = \\mathbb{E}\\Biggl[\\sum_{s=1}^t\nX_{s-1}^{Q_s}\\Biggr],\n\\]\nthus estimating the quantities $X_{s-1}^{Q_s}$ will provide a bound on\nthe expected number of collisions.\nOur aim in the next lemma is to show that w.h.p., whenever\n$V_{s-1}^{Q_s}$ is large, so is~$X_{s-1}^{Q_s}$. This will reduce\nthe problem to the analysis of the quantities~$V_{s-1}^{Q_s}$, which\nare deterministic functions of~$Q_1,\\ldots,Q_n$.\nThis is the main conceptual ingredient in the lower bound, and its\nproof will follow directly\nfrom the large deviation estimate given in Proposition~\\ref{prop-predictions}.\n\n\\begin{lemma}\\label{lem-Xt-Wt-approx}\nLet $Q_1,\\ldots,Q_n$ be a sequence of strategies adapted to the filter~$(\\mathcal{F}_{i})$, and let $X^\\nu_s$ and $V^\\nu_s$ be defined as above.\nThen with probability at least $1-O(\\mathrm{e}^{-4m})$, for every $\\nu\n\\in\\{\\mu_1,\\ldots,\\mu_{2^m}\\}$ and every $s$\nwe have that $V^\\nu_s \\geq100\\|\\nu\\|_\\infty m$ implies $X^\\nu_s\n\\geq V^\\nu_s\/2$.\n\\end{lemma}\n\n\\begin{pf\nBefore describing the proof, we wish to emphasize a delicate point.\nThe lemma holds for any sequence of strategies $Q_1,Q_2,\\ldots,Q_n$\n(each $Q_i$ is an arbitrary function of $\\mathcal{F}_{i-1}$). No restrictions\nare made here on the way each\nsuch $Q_i$ is produced (e.g., it does not even need to belong to the\npool of $2^m$ strategies), as long as it satisfies $\\|Q_i\\|_\\infty\\leq k\/n$.\nThe reason that such a general statement is possible is the following:\nOnce we specify how each $Q_i$ is determined from $\\mathcal{F}_{i-1}$ (this\ncan involve extra random bits, in case the adaptive algorithm is\nrandomized), the process of exposing the positions\nof the balls, $J_i \\sim Q_i$, defines a martingale. Hence, for each\nfixed $\\nu$, we would be able to show that the desired event occurs except\nwith probability $O(\\mathrm{e}^{-5m})$. A union bound over the\nstrategies $\\nu$ (which, crucially, do belong to the pool of size $2^m$)\nwill then complete the proof.\n\nFix a strategy $\\nu$ out of the pool of $2^m$ possible strategies, and\nrecall the definitions of $x_s^\\nu$ and $v_s^\\nu$, according to which\n\\[\n0 \\leq x_s^\\nu\\leq\\|\\nu\\|_\\infty,\\qquad   v_s^{\\nu} = \\mathbb\n{E}[x_s^\\nu\n\\mid\\mathcal{F}_{s-1}].\n\\]\nBy applying Proposition~\\ref{prop-predictions} to the sequence\n$(x_s^\\nu)$ (with the cumulative sums $X_s^\\nu$ and cumulative\nconditional expectations $V_s^\\nu$), we obtain that for all $h$,\n\\[\n\\mathbb{P}( X_s^\\nu\\leq V_s^\\nu\/2, V_s^\\nu\\geq h\\mbox{ for some $s$})\n\\leq\\exp\\biggl(-\\frac{h}{20\\|\\nu\\|_\\infty} + 2\\biggr).\n\\]\nThus, taking $h = 100\\|\\nu\\|_\\infty m$ we obtain that\n\\[\n\\mathbb{P}( X_s^\\nu\\leq V_s^\\nu\/2, V_s^\\nu\\geq100\\|\\nu\\|_\\infty\nm\\mbox\n{ for some $s$}) \\leq\\exp(-5m+2).\n\\]\nSumming over the pool of at most $2^m$ predetermined strategies, $\\nu$\ncompletes the proof.\n\\end{pf}\n\nHaving shown that $X_t^\\nu$ is well approximated by $V_t^\\nu$, and\nrecalling that we are interested\nin estimating $X_{s-1}^{Q_s}$, we now turn our attention to the\npossible values of~$V_{s-1}^{Q_s}$.\n\n\\begin{claim}\\label{clm-large-Ws-sum}\nFor any sequence of strategies $Q_1,\\ldots,Q_t$, we have that\n\\[\n\\sum_{s=1}^{t} V_{s-1}^{Q_s} \\geq\\frac{t(t-k)}{2n}.\n\\]\n\\end{claim}\n\n\\begin{pf}\nBy our definitions, for the strategies $Q_1,\\ldots,Q_t$ we have\n\\begin{eqnarray}\\label{eq-Ws-sum-estimate}\n\\sum_{s=1}^t V_{s-1}^{Q_s} &=& \\sum_{s=1}^t \\sum_{r=1}^{s-1}\n\\langle Q_r,Q_s\\rangle  = \\sum_{i=1}^n \\sum_{r < s \\leq t} Q_r(i) Q_s(i)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&=& \\frac12 \\sum_{i=1}^n \\Biggl[\\Biggl(\\sum_{s=1}^t Q_s(i)\\Biggr)^2 -\n\\sum_{s=1}^t Q_s(i)^2\\Biggr].\\nonumber\n\\end{eqnarray}\nRecalling the definition of the strategies $Q_i$, we have that\n\\[\n\\cases{\n0 \\leq Q_s(i) \\leq k\/n, & \\quad for all $i$ and $s$,\\vspace*{2pt}\\cr\n\\displaystyle\\sum_{i=1}^n Q_s(i) = 1, &\\quad for all $s$.\n}\n\\]\nTherefore,\n\\[\n\\sum_{i=1}^n\\sum_{s=1}^t Q_s(i)^2 \\leq\\frac{k}n \\sum_{i=1}^n \\sum\n_{s=1}^t Q_s(i) = \\frac{k t}n.\n\\]\nOn the other hand, by Cauchy--Schwarz,\n\\[\n\\sum_{i=1}^n\\Biggl(\\sum_{s=1}^t Q_s(i)\\Biggr)^2 \\geq\\frac{1}n\n\\Biggl(\\sum_{i=1}^n \\sum_{s=1}^t Q_s(i)\\Biggr)^2 = \\frac{t^2}n.\n\\]\nPlugging these two estimates in \\eqref{eq-Ws-sum-estimate}, we deduce that\n\\[\n\\sum_{s=1}^t V_{s-1}^{Q_s} \\geq\\frac{t(t-k)}{2n},\n\\]\nas required.\n\\end{pf}\n\nWhile the above claim tells us that the average size of $V_{s-1}^{Q_s}$\nis fairly large [has order at least $(t-k)\/n$], we wish to obtain\nbounds corresponding to individual distributions $Q_s$.\nAs we next show, this sum indeed enjoys a significant contribution from\nindices $s$ where $V_{s-1}^{Q_s} = \\Omega(km\/n)$.\nMore precisely, setting $h = 100km\/n$, we claim that\nfor large enough $n$,\n\\begin{equation}\\label{eq-many-large-Ws}\n\\sum_{s=1}^t V_{s-1}^{Q_s} \\bolds{1}_{\\{V_{s-1}^{Q_s} > h\\}} \\geq\n\\frac\n{t^2}{4n} .\n\\end{equation}\nTo see this, observe that if\n\\[\nt \\geq t_0 \\stackrel{\\triangle}{=}5hn = 500 k m,\n\\]\nthen\n\\[\n\\sum_{s=1}^t V_{s-1}^{Q_s} \\bolds{1}_{\\{V_{s-1}^{Q_s} \\leq h\\}} \\leq\nt h\n\\leq\\frac{t^2}{5n}.\n\\]\nCombining this with Claim~\\ref{clm-large-Ws-sum} [while noting that\n$\\frac{t(t-k)}{2n} = (1-o(1))\\frac{t^2}{2n}$] yields \\eqref\n{eq-many-large-Ws} for any sufficiently large $n$.\n\nWe may now apply Lemma~\\ref{lem-Xt-Wt-approx}, and obtain that, except\nwith probability $O(\\mathrm{e}^{-4m})$, whenever\n$ V_{s-1}^{Q_s} > h$ we have $X_{s-1}^{Q_s} \\geq\\frac12\nV_{s-1}^{Q_s}$, and so\n\\begin{equation}\\label{eq-Xs-sum-t^2\/8n}\n\\sum_{s=1}^t X_{s-1}^{Q_s}\\geq\\frac12 \\sum_{s=1}^t V_{s-1}^{Q_s}\n\\bolds{1}_{\\{V_{s-1}^{Q_s} > h\\}} \\geq\\frac{t^2}{8n} \\qquad \\mbox{for\nall $t\n\\geq t_0$}.\n\\end{equation}\nAltogether, since $X_{s-1}^{Q_s} \\geq0$, we infer that\n\\begin{equation}\n\\label{eq-E-Col(t)}\n\\quad \\mathbb{E}\\operatorname{Col}_2(t) = \\mathbb{E}\\Biggl[\\sum_{s=1}^t\nX_{s-1}^{Q_s}\\Biggr] \\geq\\frac\n{t^2}{8n}\\bigl(1- O(n^{-4})\\bigr) \\geq\\frac{t^2}{9n}\\qquad \\mbox{for all\n$t \\geq t_0$},\n\\end{equation}\nwhere the last inequality holds for large enough $n$. This proves\npart~(i) of Theorem~\\ref{thm-collisions-lower-bound}.\n\n\n\nIt remains to establish concentration for $\\operatorname{Col}_2(t)$\nunder the\nadditional assumption that $t \\geq30 \\sqrt{L n\\log n}$\nfor some $L = L(n)$. First, set the following stopping-time for\nreaching a maximal-load of $L$:\n\\[\n\\tau_L \\stackrel{\\triangle}{=}\\min\\Bigl\\{t \\dvtx  \\max_j N_t(j) \\geq\nL\\Bigr\\}.\n\\]\nNext, recall that\n\\[\n\\operatorname{Col}_2(t) = \\sum_{s=1}^t N_{s-1}(J_{s}),\n\\]\nand notice that\n\\[\n \\mathbb{E}[ N_{s-1}(J_s) \\mid\\mathcal{F}_{s-1}] =\n\\sum_{i=1}^n\nQ_s(i)N_{s-1}(i) = X_{s-1}^{Q_s}.\n\\]\nTherefore, we may apply our large deviation estimate given in\nSection~\\ref{sec:martingale} (Proposition~\\ref{prop-predictions}),\ncombined with the stopping-time $\\tau_L$ (see Remark~\\ref\n{rem-prop-stopping-time}):\n\\begin{itemize}\n\\item The sequence of increments is $(N_{s-1}(J_s))$.\n\\item The sequence of conditional expectations is $(X_{s-1}^{Q_s})$.\n\\item The bound on the increments is $L$, as $N_{s-1}(J_s) \\leq\\max_i\nN_{s-1}(i) \\leq L$ for all $s < \\tau_L$.\n\\end{itemize}\nIt follows that\n\\begin{eqnarray*}\n&&\\mathbb{P}\\biggl(\\biggl\\{\\operatorname{Col}_2(t) \\leq\\frac\n12\\sum_{s\\leq t}\nX_{s-1}^{Q_s} \\mbox{ and } \\sum_{s\\leq t} X_{s-1}^{Q_s} \\geq\\frac\n{t^2}{8n}\\biggr\\}\\mbox{ for some $t < \\tau_L$}\\biggr) \\\\\n&&\\qquad \\leq\n\\exp\\biggl(-\\frac{t^2\/8n}{20L} + 2\\biggr) \\leq O(n^{-5}),\n\\end{eqnarray*}\nwhere the last inequality is by the assumption $t \\geq30\\sqrt{Ln\\log n}$.\nFinally, by~\\eqref{eq-Xs-sum-t^2\/8n}, we also have\nthat $\\sum_{s\\leq t} X_{s-1}^{Q_s} \\geq t^2\/(8n)$ for all $t \\geq\nt_0$, except with probability $O(n^{-4})$.\nCombining these two statements, we deduce that for any $t \\geq(t_0\n\\vee 30\\sqrt{L n\\log n})$,\n\\[\n\\mathbb{P}\\biggl(\\operatorname{Col}_2(t) < \\frac{t^2}{16n},~\\tau\n_L>t \\biggr) = O\n(n^{-4}),\n\\]\nconcluding the proof of Theorem~\\ref{thm-collisions-lower-bound}.\n\\end{pf*}\n\n\\subsection{Boosting the subcritical regime to unbounded maximal\nload}\\label{subsec:unbounded-maxload}\nWhile Theorem~\\ref{thm-collisions-lower-bound} given above provides a\ncareful analysis for the number of $2$-collisions, that is, pairs of balls\nsharing a bin, one can iteratively apply this theorem, with very few\nmodifications, in order to obtain that the number of $q$-collisions (a\nset of $q$ balls sharing a bin) has order $\\Omega(n^{1-o(1)})$ w.h.p.\nThe proof of this result hinges on Theorem~\\ref{thm-collisions-gen}\nbelow, which is a generalization of Theorem~\\ref{thm-collisions-lower-bound}.\n\nRecall that in the relaxed model studied so far, at any given time $t$\nthe algorithm adaptively selects a strategy $Q_t$ (based on the entire\nhistory $\\mathcal{F}_{t-1}$), after which a ball is positioned in a\nbin $J_t\n\\sim Q_t$. We now introduce an extra set of random variables,\nin the form of a sequence of increasing subsets, $A_1 \\subset\\cdots\n\\subset A_n \\subset[n]$. The set $A_t$ is determined\nby $\\mathcal{F}_{t-1}$, and has the following effect: If $J_t \\in\nA_t$, we add\na ball to this bin as usual, whereas if\n$J_t \\notin A_t$, we ignore this ball (all bins remain unchanged). That\nis, the number of balls in bin $i$ at time $t$ is now given by\n\\[\nN_t(i) \\stackrel{\\triangle}{=}\\sum_{s=1}^t \\bolds{1}_{\\{J_s = i\\}}\n\\bolds{1}_{A_s}(i),\n\\]\nand as before we are interested in a lower bound for the number of collisions:\n\\[\n\\operatorname{Col}_2(t) \\stackrel{\\triangle}{=}\\sum_{i=1}^n \\pmatrix{ N_t(i)\\cr 2}.\n\\]\nThe idea here is that, in the application, the set $A_t$ will consists of\nthe bins that already contain $\\ell$ balls at time $t$. As such, they indeed\nform an increasing sequence of subsets determined by $(\\mathcal\n{F}_i)$. In\nthis case, any collision\ncorresponds to $2$ balls placed in some bin which already has $\\ell$\nother balls, and thus immediately implies a load of $\\ell+2$.\n\n\\begin{theorem}\\label{thm-collisions-gen}\nConsider the following balls and bins setting:\n\\begin{enumerate}[(3)]\n\\item[(1)] The online adaptive algorithm has a pool of $2^m$ possible\nstrategies, where each strategy $\\mu$ satisfies $\\|\\mu\\|_\\infty\\leq k\/n$.\nThe algorithm selects a (random) sequence of strategies $Q_1,\\ldots\n,Q_n$ adapted to the filter $(\\mathcal{F}_i)$.\n\\item[(2)] Let $A_1\\subset\\cdots\\subset A_n \\subset[n] $ denote a random\nincreasing sequence of subsets adapted to the filter $(\\mathcal\n{F}_i)$, that is, $A_i$\nis determined by $\\mathcal{F}_{i-1}$.\n\\item[(3)] There are $n$ rounds, where in round $t$ a new potential location\nfor a ball is chosen according to $Q_t$. If this location\nbelongs to $A_t$, a ball is positioned there (otherwise, nothing happens).\n\\end{enumerate}\nDefine $T = \\sum_{s=1}^n Q_s(A_s)$. Then for any $L=L(n)$,\n\\[\n\\mathbb{P}\\biggl(T \\geq30\\bigl( \\sqrt{k m n}  \\vee\\sqrt{L n \\log\nn}\n\\bigr),  \\operatorname{Col}_2(n) < \\frac{T^2}{16n}, \\max_{j}N_n(j) \\leq\nL\\biggr) \\leq\nO(n^{-4}).\n\\]\n\\end{theorem}\n\n\\begin{pf}\nAs the proof follows the same arguments of Theorem~\\ref\n{thm-collisions-lower-bound}, we restrict our attention to describing\nthe modifications that are required for the new statement to hold.\n\nDefine the following subdistribution of $Q_s$ with respect to $A_s$:\n\\[\nQ'_s \\stackrel{\\triangle}{=}Q_s \\bolds{1}_{A_s}.\n\\]\nAs before, given $Q_s$, the strategy at time $s$, define the following\nparameters:\n\\[\nx_s^\\nu\\stackrel{\\triangle}{=}\\nu( J_s),\\qquad v_s^{\\nu} \\stackrel\n{\\triangle}{=}\\sum_{i=1}^n Q'_s(i) \\nu(i),\n\\]\nand let the cumulative sums of $v_s^\\nu$ and $x_s^\\nu$ be denoted by\n\\[\nX_t^{\\nu} \\stackrel{\\triangle}{=}\\sum_{s=1}^t x_s^\\nu,\\qquad V_t^{\\nu\n} \\stackrel{\\triangle}{=}\\sum_{s=1}^t\nv_s^\\nu.\n\\]\nWe claim that a statement analogous to that of Lemma~\\ref\n{lem-Xt-Wt-approx} holds as is with respect to the above definitions,\nfor any choice of increasing subsets $A_1\\subset\\cdots\\subset A_n$\n[adapted to the filter $(\\mathcal{F}_i)$]. As we soon argue, the martingale\nconcentration argument is valid without any changes, and the only\ndelicate point is the identity of the target strategy $\\nu$.\n\n\\begin{lemma}\\label{lem-Xt-Wt-approx-gen}\nLet $Q_1,\\ldots,Q_n$ and $A_1\\subset\\cdots\\subset A_n$ be strategies and\nsubsets respectively, adapted to the filter $(\\mathcal{F}_{i})$, and let\n$X^\\nu_s$\nand $V^\\nu_s$ be defined as above.\nThen with probability at least $1-O(\\mathrm{e}^{-4m})$, for every $\\nu\n\\in\\{\\mu_1,\\ldots,\\mu_{2^m}\\}$ and every~$s$\nwe have that $V^{\\nu'}_s \\geq100\\|\\nu\\|_\\infty m$ implies $X^{\\nu\n'}_s \\geq V^{\\nu'}_s\/2$, where $\\nu' = \\nu\\bolds{1}_{A_{s+1}}$.\n\\end{lemma}\n\n\\begin{pf}\nLet $\\nu$ be a strategy. Previously (in the proof of Lemma~\\ref\n{lem-Xt-Wt-approx}), we compared $X_s^\\nu$ to $V_s^\\nu$\nusing the large deviation inequality of Section~\\ref{sec:martingale}.\nNow, for each~$s$, our designated $\\nu'$ is a function of $\\nu$ and\n$A_{s+1}$, and hence depends on $\\mathcal{F}_s$.\nIn particular, there are potentially more than $2^m$ different\nstrategies to consider as $\\nu'$,\ndestroying our union bound.\nThe crucial observation that resolves this issue is the following.\n\n\\begin{observation}\nLet $r > s$ and let $\\nu$ be a strategy. Then $V_{s}^\\nu= V_{s}^{\\nu\n'}$ and $X_{s}^\\nu= X_{s}^{\\nu'}$\nfor any increasing sequence $A_1,\\ldots,A_r$, where $\\nu'=\\nu\\bolds\n{1}_{A_r}$.\n\\end{observation}\n\nTo see this, first consider $X_s^\\nu$ and $X_s^{\\nu'}$. If $x_i^\\nu$\nfor some $1 \\leq i \\leq s$ had a nonzero\ncontribution to $X_s^\\nu$, then by definition \\mbox{$J_i \\in A_i$}.\nSince~$A_i\\subset A_r$, we also have \\mbox{$J_i \\in A_r$}, and so $x_i^{\\nu'}\n= \\nu(J_i)\\bolds{1}_{A_r}(J_i) = x_i^{\\nu}$.\nThe statement $V_s^{\\nu'}=V_s^{\\nu}$ now follows from the fact that\n$V_s^\\nu$ is the sum of $v_i^\\nu= \\mathbb{E}[x_i^\\nu\\mid\\mathcal\n{F}_{i-1}]$.\n\nUsing the above observation, it now suffices to prove the statement of\nLemma~\\ref{lem-Xt-Wt-approx-gen} directly on the strategies $\\nu$\n(rather than on $\\nu'$). Hence, the only difference between this\nsetting and that of Lemma~\\ref{lem-Xt-Wt-approx} is that\nhere some of the rounds are forfeited (as reflected in the new\ndefinition of the $v_s^\\nu$'s). The proof of\nLemma~\\ref{lem-Xt-Wt-approx} therefore holds unchanged for this case.\n\\end{pf}\n\nSimilarly, the following claim is the analogue of Claim~\\ref\n{clm-large-Ws-sum}, with $t$ (the number of balls\nin the original version) replaced by $T=\\sum_s \\sum_i Q'_s(i)$ (the\nexpected number of balls actually positioned).\n\n\\begin{claim}\\label{clm-large-Ws-sum-gen}\nFor any $Q_1,\\ldots,Q_n$ and $A_1\\subset\\cdots\\subset A_n$, we have that\n\\[\n\\sum_{s=1}^{n} V_{s-1}^{Q'_s} \\geq\\frac{T(T-k)}{2n}.\n\\]\n\\end{claim}\n\nThe proof of the above claim follows from the exact same argument as in\nClaim~\\ref{clm-large-Ws-sum}.\nNotice that the bound there, given as a function of $t$, was\nactually a bound in terms of $\\sum_{s=1}^t \\sum_i Q_s(i)$, and so\nreplacing $Q_s$ by $Q'_s$ yields the desired bound as a function of $T$.\n\nWith this in mind, set $h = 100 km\/n$ and note that, clearly,\n\\[\n\\sum_{s=1}^n V_{s-1}^{Q'_s} \\bolds{1}_{\\{V_{s-1}^{Q'_s} \\leq h\\}}\n\\leq h n.\n\\]\nTherefore, if\n\\[\nt_0 \\stackrel{\\triangle}{=}n \\sqrt{5 h} \\leq25 \\sqrt{kmn},\n\\]\nthen\n\\[\nh n \\leq\\frac{T^2}{5n}\\qquad \\mbox{ for any $T \\geq t_0$},\n\\]\nand so for such $T$ and any large enough $n$\n\\[\n\\sum_{s=1}^n V_{s-1}^{Q'_s} \\bolds{1}_{\\{V_{s-1}^{Q'_s} > h\\}} \\geq\n\\frac\n{T^2}{4n} .\n\\]\nBy following the next arguments from the proof of Theorem~\\ref\n{thm-collisions-lower-bound},\nit now follows that, as long as $T \\geq t_0$,\n\\[\n\\mathbb{E}\\operatorname{Col}_2(n) = \\mathbb{E}\\Biggl[\\sum_{s=1}^t\nX_{s-1}^{Q'_s}\\Biggr] \\geq\\frac\n{T^2}{8n}\\bigl(1- O(n^{-4})\\bigr) \\geq\\frac{T^2}{9n}.\n\\]\nSimilarly, using the argument as in the proof of Theorem~\\ref\n{thm-collisions-lower-bound}, which defines the stopping-time\n$\\tau_L$ and applies Proposition~\\ref{prop-predictions} on the\nsequence of increments given by\n\\[\n\\operatorname{Col}_2(t)-\\operatorname{Col}_2(t-1) = N_{s-1}(J_s)\n\\bolds{1}_{A_s}(J_s),\n\\]\nwe deduce that, if $T \\geq(t_0  \\vee 30\\sqrt{L n\\log n})$ then\n\\[\n\\mathbb{P}\\biggl(\\operatorname{Col}_2(n) < \\frac{T^2}{16n},\\tau_L\n> n\\biggr) = O\n(n^{-4}),\n\\]\nas required.\n\\end{pf}\n\nWe next show how to infer the results regarding an unbounded maximal load\nfrom Theorem~\\ref{thm-collisions-gen}. For each integer $\\ell\n=0,1,2,\\ldots,$ we define\nthe increasing sequence $(A_t)$ by\n\\[\nA_t^{\\ell} \\stackrel{\\triangle}{=}\\{ i \\in[n] \\dvtx  N_t(i) \\geq\\ell\\}.\n\\]\nFurther define\n\\[\nT_\\ell\\stackrel{\\triangle}{=}\\sum_{s=1}^n Q_s (A_s^{\\ell}),\n\\]\nwhich is the expected number of balls that are placed in bins which\nalready hold at least $\\ell$ balls.\nThe proof will follow from an inductive argument,\nwhich bounds the value of $T_{\\ell+1}$ in terms of $T_\\ell$.\n\nFor some $L=L(n)$ to be specified later, our bounds will be meaningful\nas long as the maximal load is at most $L$, and\n\\begin{equation}\n\\label{eq-T-ell-cond}\nT_\\ell\\geq30\\bigl( \\sqrt{k m n}  \\vee  \\sqrt{L n\\log n} \\bigr).\n\\end{equation}\nUsing Theorem~\\ref{thm-collisions-gen}, we will show that, if \\eqref\n{eq-T-ell-cond} holds then\n\\begin{equation}\n\\label{eq-T-ell-induction}\nT_{\\ell+1} \\geq\\frac{T_{\\ell}^2}{20n L}.\n\\end{equation}\nTo this end, define\n\\[\nR_\\ell\\stackrel{\\triangle}{=}\\sum_{i=1}^n \\pmatrix{ N_n(i) - \\ell\\cr 2},\n\\]\nthat is, $R_\\ell$ denotes the number of collisions between all pairs\nof balls that were placed in a bin, that already held at least $\\ell$ balls.\n\nTo infer \\eqref{eq-T-ell-induction}, apply Theorem~\\ref\n{thm-collisions-gen} with respect to the subsets~$(A_s^\\ell)$.\nThe assumption \\eqref{eq-T-ell-cond} implies that, except with\nprobability $O(n^{-4})$, either the load is\nat least~$L$, or\n\\[\nR_\\ell\\geq\\frac{T_\\ell^2}{16n}.\n\\]\nNotice that any ball that is placed in a bin, which contains at most\n$L$ balls, can contribute at most $L$ collisions to the count of\n$R_\\ell$.\nTherefore, if the maximal load is less than $L$, the following holds:\nThe number of balls placed in bins that already\ncontain at least $\\ell$ balls, is at least\n\\begin{equation}\n\\label{eq-R-ell-bound}\nR_\\ell\/ L \\geq\\frac{T_\\ell^2}{16nL}\\qquad \\mbox{with probability\n$1-O(n^{-4})$}.\n\\end{equation}\nRecalling that $T_{\\ell+1}$ is the expected number of such balls, we\ninfer that\n\\[\nT_{\\ell+1} \\geq\\bigl(1-O(n^{-4})\\bigr) \\frac{R_\\ell}{L}\n\\geq\\frac{T_\\ell^2}{20nL},\n\\]\nwhere the last inequality holds for large enough $n$ (with room to spare).\n\n\nThis establishes that \\eqref{eq-T-ell-cond} implies \\eqref\n{eq-T-ell-induction}. Since by definition $T_0 = n$, we deduce\nthat the decreasing series $(T_0,T_1,\\ldots)$ satisfies\n\\[\nT_{\\ell+1} \\geq\\frac{n}{(20 L)^{2^{\\ell+1}-1}}\\qquad  \\mbox{if\n$T_{\\ell}$ satisfies \\eqref{eq-T-ell-cond}}.\n\\]\nRearranging, it follows that, in particular, \\eqref{eq-T-ell-cond} is\nsatisfied if\n\\begin{equation}\\label{eq-suff-T-ell-cond}\n30 \\cdot(20L)^{2^\\ell-1} \\leq\\sqrt{\\frac{n}{km}}  \\wedge  \\sqrt\n{\\frac{n}{L\\log n}}.\n\\end{equation}\nIt is now easy to verify that, for any fixed $\\varepsilon> 0$, choosing\n\\[\nL = \\ell= (1-\\varepsilon)\\log_2\\log\\biggl(\\frac{n}{km}\\biggr)\n\\]\nsatisfies \\eqref{eq-suff-T-ell-cond} for large enough $n$.\nBy \\eqref{eq-R-ell-bound}, we can then infer that $R_\\ell> 0$\nwith probability $1-O(n^{-4})$, hence the maximal load is at least\n$\\ell$.\nThis concludes the proof of Theorem~\\ref{thm-matching-lower}. \\qed\n\n\n\n\\section{Algorithms for perfect matching and constant load}\\label\n{sec:upper-bound}\n\nIn this section, we prove Theorem~\\ref{thm-matching-upper} by\nproviding an algorithm that avoids collisions w.h.p. using only\n$O(n\/k)$ bits of memory,\nwhich is the minimum possible by Theorem~\\ref{thm-matching-lower}.\nThe case $k m = \\Omega(n)$ of Theorem~\\ref{thm-const-load} will then\nfollow from repeated applications of this algorithm.\n\n\\begin{tttt*}\n\\begin{enumerate}[ 1.]\n\\item For $\\ell= \\lfloor\\frac{n}{\\lfloor m\/2\\rfloor} \\rfloor$,\npartition the bins into contiguous blocks $B_1,\\ldots,B_\\ell$ each\ncomprising $\\lfloor m\/2\\rfloor$ bins.\nIgnore any remaining unused bins.\n\\item Set $d = \\lceil\\log_2(\\frac{5}{C\\delta}\\log\nn)\\rceil$, and define the arrays $A_0,\\ldots,A_{d-1}$:\n\\begin{itemize}\n\\item$A_j$ comprises $2^j$ contiguous blocks (a total of $\\sim\n2^{j-1}m$ bins).\n\\item For each contiguous (nonoverlapping) $4^j$-tuple of bins in\n$A_j$, we keep a single bit that holds whether any of its bins is occupied.\n\\item All blocks currently or previously used are contiguous.\n\\end{itemize}\n\\item Repeat the following procedure until exhausting all rounds:\n\\begin{itemize}\n\\item Let $j$ be the minimal integer so that a bin of $A_j$, marked as\nempty, appears in the current selection of $k$ bins. If no such $j$\nexists, the algorithm announces failure.\n\\item Allocate the ball into this bin, and mark its $4^j$-tuple as occupied.\n\\item If the fraction of empty $4^j$-tuples remaining in $A_j$ just\ndropped below $\\delta\/2$, relocate the array $A_j$ to a fresh\ncontiguous set of empty $2^j$ blocks (immediately beyond the last\nallocated block).\nIf there are less than $2^j$ available new blocks, the algorithm fails.\n\\end{itemize}\n\\item Once $(1-\\delta)n$ rounds are performed, the algorithm stops.\n\\end{enumerate}\n\\end{tttt*}\n\nWe proceed to verify the validity of the algorithm\nin stages: First, we discuss a more basic version of the algorithm suited\nfor the case where $k m = \\Omega(n \\log n)$; then, we examine an\nintermediate version\nwhich extends the range of the parameters to $km\\log m = \\Omega(n\\log\nn)$; finally, we study the actual algorithm, which features the tight\nrequirement $km = \\Omega(n)$.\n\nThroughout the proof of the algorithm, assume that in each round\nwe are presented with $k$ independent uniform indices of bins, possibly\nwith repetitions.\nClearly, an upper bound for the maximal load in this relaxed model\ntranslates into one for\nthe original model ($k$ choices without repetitions).\n\n\\subsection{Basic version of the algorithm}\nWe begin with a description and a proof of a simpler version of the\nabove algorithm, suited for the case where\n\\begin{equation}\\label{eq-kn-nlogn}\nk m \\geq(3\/\\delta) n \\log n.\n\\end{equation}\nThis version will serve as the base for the analysis.\nFor simplicity, assume first that $m \\mid n$.\n\n\\begin{bbb*}\n\\begin{enumerate}\n\\item Let $B_1,\\ldots,B_\\ell$ be an arbitrary partition of the $n$\nbins into $\\ell\\stackrel{\\triangle}{=}n\/m $ blocks,\neach containing $m$ bins.\nPut $r \\stackrel{\\triangle}{=}\\lfloor(1-\\delta) m \\rfloor$.\n\\item Throughout stage $j\\in[\\ell]$, only the $m$ bins belonging to\n$B_j$ are tracked.\nAt the beginning of the stage, all bins in the block are marked empty.\n\\item Stage $j$ comprises $r$ rounds, in each of which:\n\\begin{itemize}\n\\item The algorithm attempts to place a ball in an arbitrary empty bin\nof $B_j$ if possible.\n\\item If no empty bin of $B_j$ is offered, the algorithm declares failure.\n\\end{itemize}\n\\item Once $(1-\\delta)n$ rounds are performed, the algorithm stops.\n\\end{enumerate}\n\\end{bbb*}\n\nTo verify that this algorithm indeed produces a perfect allocation\nw.h.p., examine a specific round of stage $j$, and condition on the event\nthat so far\nthe algorithm did not fail. In particular, its accounting of which bins\nare occupied in $B_j$ is accurate,\nand at least $m-r = (\\delta-o(1))m$ bins in $B_j$ are still empty\n[notice that by our\nassumption $m =\\Omega(\\log n)$, and so $m\\to\\infty$ with $n$].\n\nLet $\\textsf{Miss}_j$ denote the event that\nthe next ball precludes all of the empty bins of $B_j$ in its $k$\nchoices, we have\n\\begin{equation}\n\\label{eq-alg-upper-prob}\n\\mathbb{P}(\\textsf{Miss}_j) \\leq\\biggl(1-\\frac{m-r}n\\biggr)^k \\leq\n\\mathrm\n{e}^{-(\\delta-o(1))({km}\/n)} \\leq n^{-3+o(1)},\n\\end{equation}\nby assumption \\eqref{eq-kn-nlogn}. A union bound over the $n$ rounds\nnow yields (with room to spare)\nthat the algorithm succeeds w.h.p.\n\nThe case where $m$ does not divide $n$ is treated similarly: Set $\\ell\n= \\lfloor\\frac{n}{\\lfloor m\/2 \\rfloor} \\rfloor$,\nand\\vspace*{1pt} partition the bins into blocks that now hold $\\lfloor m\/2\\rfloor$\nbins each,\nexcept for the final block $B_\\ell$ which would have between $\\lfloor\nm\/2\\rfloor$ and $m-1$ bins.\nAs before, in stage $j$ we attempt to allocate $\\lfloor(1-\\delta\n)|B_j| \\rfloor$ balls into $B_j$, while relying on the property\nthat $B_j$ has at least $(\\delta-o(1))|B_j| \\geq(\\delta-o(1))m\/2$\nempty bins. This gives\n\\[\n\\mathbb{P}(\\textsf{Miss}_j) \\leq\\mathrm{e}^{-(\\delta-o(1))\n({(km\/2)}\/n)} \\leq\nn^{-3\/2+o(1)},\n\\]\nas required.\n\n\\subsection{Intermediate version of the algorithm}\nWe now wish to adapt the above algorithm to the following case:\n\\begin{equation}\\label{eq-km-n-m-poly}\nk m \\log_2 m \\geq(20\/\\delta)\\log(5\/\\delta) n \\log n,\\qquad  \\log\n^3 n \\leq m \\leq\\frac{n}{\\log n}.\n\\end{equation}\nNotice that if $m \\geq n^\\varepsilon$, the above requirement is essentially\nthat\n\\[\nk m = \\Omega(n\/\\varepsilon).\n\\]\nThe full version of the algorithm will eliminate\nthis dependency on $\\varepsilon$.\n\n\\begin{iii*}\\vspace*{-9pt}\n\\begin{enumerate}\n\\item For $\\ell= \\lfloor\\frac{n}{\\lfloor m\/2\\rfloor} \\rfloor$,\npartition the bins into contiguous blocks $B_1,\\ldots,B_\\ell$ each\\vspace*{1pt}\ncomprising $\\lfloor m\/2\\rfloor$ bins.\nIgnore any remaining unused bins.\n\\item Set $d = \\lfloor\\frac14 \\log_2 m \\rfloor$, and define the\narrays $A_0,\\ldots,A_{d-1}$:\n\\begin{itemize}\n\\item$A_j$ is one of the blocks $B_1,\\ldots,B_\\ell$.\n\\item For each contiguous (non-overlapping) $2^j$-tuple of bins in\n$A_j$, we keep a single bit that holds whether any of its bins is occupied.\n\\end{itemize}\n\\item Repeat the following procedure until exhausting all rounds:\n\\begin{itemize}\n\\item Let $j$ be the minimal integer so that a bin of $A_j$, marked as\nempty, appears in the current selection of $k$ bins. If no such $j$\nexists, the algorithm announces failure.\n\\item Allocate the ball into this bin, and mark its $2^j$-tuple as occupied.\n\\item If the fraction of empty $2^j$-tuples remaining in $A_j$ just\ndropped below $\\delta\/2$, relocate the array $A_j$ to a fresh block\n(immediately beyond the last allocated block).\nIf no such block is found, the algorithm fails.\n\\end{itemize}\n\\item Once $(1-\\delta)n$ rounds are performed, the algorithm stops.\n\\end{enumerate}\n\\end{iii*}\n\nSince the array $A_j$ contains $2^{-j}(m\/2)$ different $2^j$-tuples,\nthe amount of memory required to maintain the status of all tuples is\n\\[\n\\frac{m}2 \\sum_{j=0}^{d-1} 2^{-j} = (1- 2^{-d})m \\leq m - m^{3\/4}.\n\\]\nIn addition, we keep an index for each $A_j$, holding its position\namong the $\\ell$ blocks. By definition of $d$ and $\\ell$, this\namounts to at most\n\\[\nd \\log_2\\ell\\leq(\\log_2 n)^2 < m^{3\/4}\n\\]\nbits of memory, where the last inequality holds for any large $n$ by\n\\eqref{eq-km-n-m-poly}.\n\nWe first show that the algorithm does not fail to find a bin of $A_j$\nmarked as~empty. At any given point, each $A_j$ has a fraction of at\nleast $\\delta\/2$ bins marked~as empty. Hence, recalling \\eqref\n{eq-alg-upper-prob}, the probability of missing all the bins marked as~empty in $A_0,\\ldots,A_{d-1}$ is at most\n\\begin{eqnarray}\\label{eq-miss-Aj}\n&&\\exp\\biggl[-\\biggl(\\frac\\delta2-o(1)\\biggr)\\frac{km}{2n}\nd\\biggr]\\nonumber\\\\\n&&\\qquad \\leq\n\\exp\\biggl[-\\biggl(\\frac\\delta2-o(1)\\biggr)\\frac{10\\log n}{\\delta\\log\n_2 m} \\log\\biggl(\\frac{20}\\delta\\biggr) \\frac14\\log_2 m\n\\biggr]\\\\\n&&\\qquad \\leq n^{-\\log(5\/\\delta)5\/4-o(1)} < n^{-5\/4},\\nonumber\n\\end{eqnarray}\nwhere the last inequality holds for large $n$.\nTherefore, w.h.p. the algorithm never fails to find an array\n$A_j$ with\nan empty bin among the $k$ choices.\n\nIt remains to show that, whenever the algorithm relocates an array\n$A_j$, there is always a fresh block available.\n\nBy the above analysis, the probability that a ball is allocated in\n$A_j$ for $j\\geq1$ at a given round is at most\n\\begin{eqnarray*}\n\\exp\\biggl[-\\biggl(\\frac\\delta2-o(1)\\biggr)\\frac{km\/2}n j\\biggr] &\\leq&\n\\exp\\biggl[-\\biggl(\\frac\\delta2-o(1)\\biggr)\\frac{10\\log n}{\\delta\\log\n_2 m} \\log\\biggl(\\frac{20}\\delta\\biggr) j \\biggr]\\\\\n&\\leq&\\exp\\bigl(-3\\log(5\/\\delta) j\\bigr) \\stackrel{\\triangle}{=}p_j,\n\\end{eqnarray*}\nwhere the last inequality holds for any sufficiently large $n$.\n\nLet $N_j$ denote the number of balls that were allocated in blocks of\ntype $j$ throughout the run of the algorithm. Clearly, $N_j$ is\nstochastically dominated by a binomial random variable $\\operatorname\n{Bin}(n,p_j)$.\nHence, known estimates for the binomial distribution (see, e.g., \\cite\n{AS}) imply that for all $j$,\n\\[\n\\mathbb{P}\\bigl( N_j > n p_j + C \\sqrt{n}\\log n\\bigr) \\leq n^{-C}.\n\\]\nThe total number of blocks needed for $A_j$ is at most\n\\[\n\\bigg\\lceil\\frac{2^j N_j}{(1-\\delta\/2 )({m}\/2)}\\bigg\\rceil,\n\\]\nand hence the total number of blocks needed is w.h.p. at most\n\\begin{eqnarray*}\n&&\\Bigg\\lceil\\sum_{j=0}^{d-1}\\frac{2^j (1-\\delta)n p_j + C2^j\\sqrt\n{n}\\log n}{(1-\\delta\/2 )({m}\/2)} \\Bigg\\rceil\\\\\n&&\\qquad \\leq\n\\sum_{j=0}^{d-1}\\frac{2^j (1-\\delta)n p_j}{(1-\\delta\/2 )({m}\/2)} + O\\biggl(\\frac{n^{3\/4}\\log n}m\\biggr).\n\\end{eqnarray*}\nSince\n\\[\n\\sum_{j=1}^{d-1} 2^j p_j = \\sum_{j=1}^{d-1}\\exp\\bigl( j \\bigl(\\log2 -\n3\\log(5\/\\delta)\\bigr) \\bigr) < 2\\cdot2(\\delta\/5)^3 < \\delta\/5\n\\]\n(with room to spare), the total number of blocks needed is w.h.p. at most\n\\[\n\\frac{(1+\\delta\/5)(1-\\delta)n} {(1-\\delta\/2 )({m}\/2)} +\nO\\biggl(\\frac{n^{3\/4}\\log n}m\\biggr) < \\bigg\\lfloor\\frac{n}{\\lfloor\nm\/2\\rfloor} \\bigg\\rfloor= \\ell\n\\]\nfor any sufficiently large $n$.\n\n\\subsection{Final version of the algorithm}\n\nThe main disadvantage in the intermediate version of the algorithm is that\nthe size of each $A_j$ was fixed at $m\/2$ bins. Since the resolution\nof each $A_j$ is in $2^j$-tuples, we are limited to at most $\\log_2 m$\narrays. However, the probability of missing all the arrays $A_0,\\ldots\n,A_{d-1}$ has to compete with $n$, hence the requirement that $m$ would\nbe polynomial in $n$.\n\nTo remedy this, the algorithm uses arrays with increasing sizes, namely\n$2^j$ blocks for $A_j$.\nThe resolution of each array is now in $4^j$-tuples, that is, tracking the\nstatus of $A_j$ now requires at most $2^j \\lfloor m\/2\\rfloor\/ 4^j \\leq\nm\/2^{j+1}$ bits.\nRecalling that $d = \\lceil\\log_2(\\frac{5}{C\\delta}\\log\nn)\\rceil$,\nthe number of memory bits required for all arrays is at most\n\\begin{equation}\n\\label{eq-mem-req}\n\\frac{m}2 \\sum_{j=0}^{d-1} 2^{-j} = (1- 2^{-d})m \\leq m - O(m\/\\log n).\n\\end{equation}\nThe following calculation shows that indeed there are sufficiently\nmany blocks to initially accommodate all the arrays:\n\\[\n(2^d-1)\\lfloor m\/2\\rfloor\\leq\\frac{5}{2C\\delta}m \\log n \\leq\\frac\n{5km}{6C} = \\frac56 n,\n\\]\nwhere we used the assumptions $k \\geq(3\/\\delta)\\log n$ and $k m = C n$.\n\nEach of the arrays comes along with a pointer to its starting block,\nand the total number of memory bits required for this is at most\n\\[\nd \\log_2 (2n\/m) \\leq\\bigl(\\log_2 \\log n + O(1)\\bigr)\\log_2 n = \\bigl(1+o(1)\\bigr)\\log\n_2 n \\cdot\\log_2 \\log n.\n\\]\nWhen $m = \\Omega(\\log^3 n)$, the space for these pointers clearly\nfits among the $O(m\/\\log n)$ bits remaining according to \\eqref{eq-mem-req}.\nFor smaller values of $m$, as before we can apply the algorithm for,\nsay, $m'=m\/3$ (after tripling the constant $C_\\delta$ to reflect this change),\nthus earning $2m\/3$ bits for the pointers (recall the requirement that\n$m \\geq\\log_2 \\log n \\cdot\\log_2 n$).\n\nAs final evidence that the choice of parameters for the algorithm is\nvalid, note\\vspace*{1pt} that each $A_j$ indeed contains many $4^j$-tuples. It suffices\nto check $A_{d-1}$, which indeed comprises about\n\\[\n\\bigl(1+o(1)\\bigr)\\frac{m}2 \\frac{2^{d-1}}{4^{d-1}} = \\bigl(1+o(1)\\bigr)m\/2^d =\n\\biggl(\\frac{C\\delta}5+o(1)\\biggr)\\frac{m}{\\log n}=\\Omega(\\log\\log n)\n\\]\n$4^{d-1}$-tuples, where the last equality is by the assumption on the\norder of $m$.\n\nIt remains to verify that the algorithm succeeds w.h.p. This will\nfollow from\nthe same argument as in the intermediate version of the algorithm. In\nthat version, each $A_j$\ncontained at least a fraction of $(\\delta\/2)$ empty bins, and $|A_j|$\nwas about $m\/2$ for all $j$.\nIn the final version of the algorithm, each $A_j$ again contains at\nleast a fraction of $(\\delta\/2)$ empty bins,\nbut crucially, now $A_j$ contains $2^j$ bins. Thus, recalling \\eqref\n{eq-miss-Aj}, the probability to miss\n$A_0,\\ldots,A_{d-1}$ in a given round is now at most\n\\begin{eqnarray*}\n\\exp\\Biggl[-\\biggl(\\frac\\delta2-o(1)\\biggr)\\frac{km}{2n} \\sum\n_{j=0}^{d-1}2^j \\Biggr]\n&\\leq&\\exp\\biggl(-\\bigl(1-o(1)\\bigr)\\frac{C\\delta}4 (2^d-1) \\biggr)\\\\\n& =& n^{-5\/4-o(1)},\n\\end{eqnarray*}\nwhere the last inequality is by the definition of $d$.\nA union bound over the $n$ rounds gives that, w.h.p., an array $A_j$\nwith an empty bin is found for every ball.\n\nTo see that w.h.p. there are always sufficiently many available fresh\nblocks to relocate an array,\none essentially repeats the argument from the intermediate version of\nthe algorithm. That is,\nwe again examine the probability that a ball is allocated in $A_j$, to\nobtain that this time\n\\[\np_j = \\exp\\biggl(-\\bigl(1-o(1)\\bigr)\\frac{C\\delta}4 (2^j-1)\\biggr).\n\\]\nA choice of $C \\geq K (1\/\\delta)\\log(1\/\\delta)$ with\nsome suitably large $K > 0$ would give\n\\[\n\\sum_{j=1}^{d-1}4^j p_j < \\delta\/ 5,\n\\]\nand the rest of that argument unchanged now implies that the algorithm\nnever runs out of fresh blocks w.h.p.\n\nThis completes the proof of Theorem~\\ref{thm-matching-upper}.\n\n\\subsection[Proof of upper bound in Theorem 1]{Proof of upper bound in Theorem~\\protect\\ref{thm-const-load}}\nWe now wish to apply the algorithm from Theorem~\\ref\n{thm-matching-upper} in order to obtain a\nconstant load in the case where $k m \\geq c n$ for some $c> 0$. To\nachieve this, consider the perfect\nmatching algorithm for, say, $\\delta=\\frac12$, and let $C_\\delta$ be\nthe constant that appears in Theorem~\\ref{thm-matching-upper}.\nNext, join every consecutive $\\lceil C_\\delta\/c\\rceil$-tuple of bins\ntogether and write $n'$ for the number of such tuples.\nAs $km \\geq C n'$, we may apply the perfect-matching algorithm for\n$n'\/2$ balls with respect to the $n'$ tuples of bins, keeping\nin mind that the algorithm is valid also for the model of repetitions.\nThis gives a perfect matching w.h.p., and repeating this process\ngives a\ntotal load of at most $2C_\\delta\/c = O(1)$ for all $n$ balls. \\qed\n\n\\eject\n\\section{Improved lower bounds for poly-logarithmic choices}\\label\n{sec:large-q}\n\\subsection[Proof of Theorem 2]{Proof of Theorem~\\protect\\ref{thm-large-load}}\nOur proof of this case is an extension of the proof of Theorem~\\ref\n{thm-matching-lower}. We now wish to estimate the number of\n$q$-collisions for general $q$:\n\\[\n\\operatorname{Col}_q(t) \\stackrel{\\triangle}{=}\\sum_{i=1}^n \\pmatrix{ N_t(i)\\cr q}.\n\\]\nThe analysis hinges on a recursion on $q$, for which we need to achieve\nbounds on a generalized quantity, a linear function of the\n$q$-collisions vector:\n\\begin{eqnarray}\n\\qquad X_t^{f;q} &\\stackrel{\\triangle}{=}&\\sum_{s_1<\\cdots< s_{q} \\leq t}\n\\sum_i f(i) \\bolds{1}_{\\{\nJ_{s_1}=i\\}}\\cdots\\bolds{1}_{\\{J_{s_{q}}=i\\}} = \\sum_i f(i) \\pmatrix{ N_{t}(i)\\cr q},\\label{eq-Xt-def}\\\\\nV_t^{f;q} &\\stackrel{\\triangle}{=}&\\sum_{s_1<\\cdots< s_{q} \\leq t}\n\\sum_i f(i)\nQ_{s_1}(i)\\cdots Q_{s_{q}}(i) \\label{eq-Vt-def}.\n\\end{eqnarray}\nOur objective is to obtain lower bounds for $X_t^{f;q}$ with $f \\equiv\n1$, as clearly \\mbox{$\\operatorname{Col}_q(t) = X_t^{\\bolds{1};q}$}.\nNotice that the parameters $X_t^\\nu,V_t^\\nu$ from Section~\\ref\n{sec:lower-bound} are exactly\\vspace*{1pt} $X_t^{\\nu;1},V_t^{\\nu;1}$ defined above.\nThere, $\\nu$ was a strategy, whereas now\nour $f$ will be the product of different strategies. This fact will\nallow us to formulate\na recursion relation between the $V_t^{f;q}$'s and an approximate\nrecursion for the $X_t^{f;q}$.\nWe achieve this using the next lemma, where here in and throughout the\nproof we let\n\\begin{equation}\n\\label{eq-L-def}\nL \\stackrel{\\triangle}{=}\\log(n\/m)\n\\end{equation}\ndenote a maximal load we do not expect to reach (except if the\nalgorithm is far from optimal).\nWe further define\n\\[\n\\Gamma\\stackrel{\\triangle}{=}\\Biggl\\{ \\prod_{i=1}^L f_i\n \\dvtx   f_i\\in\\{\n\\bolds{1},\\mu_1,\\ldots,\\mu_{2^m}\\}\\mbox{ for all $i$}\\Biggr\\}\n\\]\nto be the set of all point-wise products of at most $L$ strategies from\nthe pool.\n\n\\begin{lemma}\\label{lem-q-collisions-induction}\nEither the maximal load exceeds $L$, or\nthe following holds for all $q < L$, every $t \\leq n\/k$ and every $f\n\\in\\Gamma$,\nexcept with probability $\\mathrm{e}^{-3mL}$:\n\\begin{equation}\n\\label{eq-Vt-large-Xt-large}\n\\mbox{If }\nV_t^{f;q} \\geq100 \\frac{(3L)^{q+1}}{q!} m \\|f\\|_\\infty\\qquad  \\mbox{then }\nX_t^{f;q} \\geq3^{-q} V_t^{f;q}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{pf}\nThe key property of the quantities $V_t^{f;q}$, which justified the\ninclusion of the inner products with $f$, is the following\nrecursion relation, whose validity readily follows from definition\n\\eqref{eq-Vt-def}:\n\\begin{equation}\n\\label{eq-Vt-recursion}V_t^{f;q+1} = \\sum_{s < t}\nV_{s}^{(Q_{s+1}\\cdot f) ; q}\\qquad  \\mbox{ for any $q\\geq1$ and any $t$}.\n\\end{equation}\nWe now wish to write a similar recursion for the variables $X_t^{f;q}$.\nAs opposed to the variables $V_t^{f;q}$,\nwhich satisfied the above recursion combinatorially, here the recursion\nwill only be stochastic. Notice that\n\\begin{eqnarray*}\nX_{t+1}^{f;q+1} - X_t^{f;q+1}\n&=& f(J_{t+1})\\biggl(\\pmatrix{ N_{t}(J_{t+1})+1\\cr q+1}-\\pmatrix{ N_{t}(J_{t+1})\\cr q+1}\\biggr)\\\\\n&= &f(J_{t+1})\\pmatrix{ N_{t}(J_{t+1})\\cr q},\n\\end{eqnarray*}\nand hence\n\\[\n\\mathbb{E}[X_{t+1}^{f;q+1}-X_t^{f;q+1} \\mid\\mathcal{F}_t\n] = \\sum_i\nQ_{t+1}(i)f(i)\\pmatrix{ N_{t}(i)\\cr q} = X_{t}^{(Q_{t+1}\\cdot f);q}.\n\\]\nWe may therefore apply Proposition~\\ref{prop-predictions} as follows:\n\\begin{itemize}\n\\item The sequence of increments we consider is\n$(X_{t+1}^{f;q+1}-X_t^{f;q+1})$ (that results in a telescopic sum).\n\\item The sequence of conditional expectations is $(X_t^{(Q_{t+1}\\cdot\nf); q})$.\n\\item The bound on the increment is $M = \\|f\\|_\\infty{L\\choose q}$,\nwhere $L$ is an upper bound for the maximal load (if we encounter\na load of $L$, we stop the process).\n\\end{itemize}\nThis implies that\n\\begin{eqnarray*}\n&&\\mathbb{P}\\biggl(\\exists t \\dvtx  \\biggl\\{X_t^{f;q+1} \\leq\\frac12 \\sum\n_{s<t}X_s^{(Q_{s+1}\\cdot f);q},\\sum_{s<t}X_s^{(Q_{s+1}\\cdot\nf);q} \\geq100 m L \\| f\\|_\\infty\\pmatrix{ L\\cr q}\\biggr\\}\\biggr) \\\\\n&&\\qquad \\leq\\exp\\biggl(-\\frac{100 m L \\| f\\|_\\infty{ L\\choose q}}{20\\|f\\|\n_\\infty{ L\\choose q}}+2\\biggr) = O(\\exp(-5mL)).\n\\end{eqnarray*}\nAs a result, the above event does not occur for any $f \\in\\Gamma$\n(since there are at most $2^{mL}$ such functions)\nexcept with probability $\\mathrm{e}^{-4mL}$. Therefore, setting\n\\[\nh_{f;q} \\stackrel{\\triangle}{=}100 \\frac{(3L)^{q+1}}{q!}m\\|f\\|\n_\\infty,\n\\]\nwe have that, except with probability $\\mathrm{e}^{-4mL}$,\n\\begin{equation}\n\\label{eq-Xt-large-sum-Xt-large}\n\\mbox{if }\n\\sum_{s<t}X_s^{(Q_{s+1}\\cdot f);q} > 3^{-q}h_{f;q}\\qquad   \\mbox{then\n}\nX_t^{f;q+1} \\geq\\frac12 \\sum_{s<t}X_s^{(Q_{s+1}\\cdot f);q}.\n\\end{equation}\n\nWe now proceed to prove \\eqref{eq-Vt-large-Xt-large} by induction on\n$q$. For $q=1$, notice that\n\\begin{eqnarray*}\nX_t^{f;1} &=& \\sum_{s \\leq t} \\sum_i f(i) \\bolds{1}_{\\{J_{s}=i\\}}=\n\\sum_i\nf(i) N_{t}(i),\\\\\nV_t^{f;1} &=& \\sum_{s \\leq t} \\sum_i f(i) Q_{s}(i).\n\\end{eqnarray*}\nFurthermore, as the definition of $X_t^{f;q}$ also applies to the case\n$q=0$, we obtain that\n\\begin{eqnarray*}\nX_t^{f;0} &=& \\sum_i f(i)\\qquad \\mbox{and so}\\\\\nV_t^{f;1} &=& \\sum_{s < t} \\sum_i Q_{s+1}(i)f(i) = \\sum\n_{s<t}X_s^{(Q_{s+1}\\cdot f);0}.\n\\end{eqnarray*}\nHence, combining the assumption $V_t^{f;1} \\geq100 (3L)^2 m \\|f\\|\n_\\infty= h_{f;1} $\nwith statement \\eqref{eq-Xt-large-sum-Xt-large} yields that $X_t^{f;1}\n\\geq\\frac12 V_t^{f;1} \\geq\\frac13 V_t^{f;1}$, except with\nprobability $\\mathrm{e}^{-4mL}$.\n\nIt remains to establish the induction step. The induction hypothesis\nfor $q$ states that whenever $V_t^{f;q} \\geq h_{f;q}$ we also have\n$X_t^{f;q}\\geq3^{-q} V_t^{f;q}$\nexcept with probability~$\\mathrm{e}^{-4mL}$. Therefore,\n\\begin{eqnarray}\\label{eq-sum-Xs-1}\n\\sum_{s<t}X_s^{(Q_{s+1}\\cdot f) ; q} &\\geq&\n3^{-q} \\sum_{s<t} V_s^{(Q_{s+1}\\cdot f) ; q} \\cdot\\bolds{1}_{\\{\nV_s^{(Q_{s+1}\\cdot f) ; q} > h_{(Q_{s+1}\\cdot f); q}\\}} \\nonumber\\\\\n&\\geq&3^{-q} \\biggl(\\sum_{s<t} V_s^{(Q_{s+1}\\cdot f) ; q} - t \\cdot\nh_{(Q_{s+1}\\cdot f); q}\\biggr)\\\\\n&\\geq&3^{-q} \\biggl(V_t^{f;q+1} - t \\cdot100 \\frac{(3L)^{q+1}}{q!}m \\|\nQ_{s+1}\\cdot f\\|_\\infty\\biggr),\\nonumber\n\\end{eqnarray}\nwhere in the last inequality we applied the recursion relation \\eqref\n{eq-Vt-recursion}.\nRecalling that $Q_{s+1}$ is a strategy, the following holds for all $t\n\\leq n\/k$:\n\\[\nt\\|Q_{s+1}\\cdot f\\|_\\infty\\leq t\\|Q_{s+1}\\|_\\infty\\|f\\|_\\infty\\leq\nt\\frac{k}n\\|f\\|_\\infty\\leq\\|f\\|_\\infty.\n\\]\nPlugging this into \\eqref{eq-sum-Xs-1}, we obtain that for all $t \\leq n\/k$,\n\\begin{eqnarray}\\label{eq-sum-Xs-2}\n\\sum_{s<t}X_s^{(Q_{s+1}\\cdot f) ; q} &\\geq&3^{-q} \\biggl(V_t^{f;q+1} -\n100 \\frac{(3L)^{q+1}}{q!}m \\|f\\|_\\infty\\biggr)\\nonumber\\\\[-8pt]\\\\[-8pt]\n&=& 3^{-q} (V_t^{f;q+1} - h_{f;q}).\\nonumber\n\\end{eqnarray}\nNow, if $V_t^{f;q+1} \\geq h_{f;q+1} = 100 \\frac{(3L)^{q+2}}{(q+1)!}m\\|\nf\\|_\\infty$, then in particular\n\\[\nV_t^{f;q+1} \\geq3\\cdot100 \\frac{(3L)^{q+1}}{q!}m\\|f\\|_\\infty=\n3h_{f;q} \\qquad  \\mbox{for all $q \\leq L-1$.}\n\\]\nThus, under this assumption, \\eqref{eq-sum-Xs-2} takes the following form:\n\\[\n\\sum_{s<t}X_s^{(Q_{s+1}\\cdot f) ; q} \\geq3^{-q} \\cdot\\frac23\nV_t^{f;q+1} \\geq3^{-q}\\cdot2 h_{f;q} .\n\\]\nSince this satisfies the condition of \\eqref{eq-Xt-large-sum-Xt-large}\n(where we actually only needed a lower\nbound of $3^{-q} h_{f;q}$), we obtain that except with probability\n$\\mathrm{e}^{-4mL}$,\n\\[\nX_t^{f;q+1} \\geq\\frac12 \\sum_{s<t}X_s^{(Q_{s+1}\\cdot f) ; q} \\geq\n3^{-(q+1)} V_t^{f;q+1},\n\\]\ncompleting the induction step.\n\nSumming the error probabilities over the induction steps for every $q <\nL$ concludes the proof of the lemma.\n\\end{pf}\n\nIt remains to apply the above lemma to deduce the maximal load of\n$\\Omega(\\frac{\\log n}{\\log\\log n})$ for $k=\\operatorname{polylog}(n)$.\nRecalling that $m \\leq n^{1-\\delta}$ for some fixed $\\delta> 0$, let\n$0<\\varepsilon< \\delta\/2$ and choose the following parameters:\n\\[\nq = (1-\\varepsilon)\\frac{\\log(n\/m)}{\\log k + \\log\\log(n\/m)},\\qquad f =\n\\bolds{1},t = n\/k.\n\\]\nLemma~\\ref{lem-q-collisions-induction} now gives that, either the\nmaximal load exceeds $L = \\log(n\/m)$, or w.h.p. the following statements holds:\n\\begin{equation}\n\\label{eq-Vn\/k-Xn\/k-implication}\n\\mbox{If }\nV_{n\/k}^{\\bolds{1};q} \\geq100 \\frac{(3L)^{q+1}}{q!} m \\qquad  \\mbox{then\n}\nX_{n\/k}^{\\bolds{1};q} \\geq3^{-q} V_{n\/k}^{\\bolds{1};q}.\n\\end{equation}\nNotice that for the above value of $q$, we have $3^{-q} =\n(n\/m)^{o(1)}$, and therefore, showing that the condition of \\eqref\n{eq-Vn\/k-Xn\/k-implication}\nis satisfied and that $V_{n\/k}^{\\bolds{1};q} \\geq\n(n\/m)^{\\varepsilon\/2}$ would immediately imply that the maximal load\nexceeds $q$ w.h.p.\n\nThe following lemma, which\nprovides a lower bound on $V_t^{\\bolds{1};q}$, is thus the final ingredient\nrequired for the proof of the theorem:\n\n\\begin{lemma}\\label{lem-Vt-packing}\nFor all $t$, $k$ and $q$, all $Q_1,\\ldots,Q_t$ and any fixed $\\alpha>\n0$ we have\n\\[\nV_{t}^{\\bolds{1};q} \\geq\\frac{(t-(1+\\alpha)kq\n)^q}{\\mathrm\n{e}^{q\/(2\\alpha)}n^{q-1}q!}.\n\\]\n\\end{lemma}\n\n\\begin{pf}\nRecall that\n\\begin{eqnarray*}\nV_t^{\\bolds{1};q} &=& \\sum_{s_1<\\cdots<s_q\\leq t} \\sum_{i=1}^n\n(Q_{s_1}\\cdots Q_{s_q})(i)\\nonumber\\\\\n&=&\\frac1{q!}\\sum_{i=1}^n \\sum_{s_1\\leq t} Q_{s_1}(i) \\mathop{\\sum\n_{s_2\\leq t}}_{s_2\\neq s_1} Q_{s_2}(i) \\cdots\\mathop{\\sum_{s_q \\leq\nt}}_{s_q\\notin\\{s_1,\\ldots,s_{q-1}\\}} Q_{s_q}(i).\n\\end{eqnarray*}\nDefining\n\\[\nr_i \\stackrel{\\triangle}{=}\\sum_{s \\leq t} Q_s(i),\n\\]\nand recalling that $\\|Q_{s}\\|_\\infty\\leq k\/n$ for all $s$, it follows\nthat for all $i$ and $j\\geq1$,\n\\[\n\\mathop{\\sum_{s_j\\leq t}}_{s_j\\neq\\{s_1,\\ldots,s_{j-1}\\}}\nQ_{s_j}(i) \\geq r_i - (j-1)k\/n.\n\\]\nConsequently,\n\\begin{eqnarray}\\label{eq-Vt1q-bound-1}\nV_t^{\\bolds{1};q} &\\geq&\\frac1{q!}\\sum_{i=1}^n \\prod\n_{j=1}^{q} \\bigl(r_i - (j-1) k\/n\\bigr)\\nonumber\\\\[-8pt]\\\\[-8pt]\n&\\geq&\\frac1{q!}\\sum_{i=1}^n \\prod_{j=1}^{q} \\bigl(r_i - (j-1)\nk\/n\\bigr) \\bolds{1}_{\\{r_i > (1+\\alpha){(k(q-1)}\/n)\\}}.\\nonumber\n\\end{eqnarray}\nNext, notice that for all $1 \\leq j\\leq q-1$,\n\\begin{eqnarray*}\n1-\\frac{j}{(1+\\alpha)(q-1)} &\\geq&\\exp\\biggl[ -\\frac{j}{(1+\\alpha\n)(q-1)} \\Big\/ \\biggl(1- \\frac{j}{(1+\\alpha)(q-1)}\\biggr) \\biggr]\\\\\n&\\geq&\\exp\\biggl[ -\\frac{j}{\\alpha(q-1)}\\biggr].\n\\end{eqnarray*}\nThus, in case $r_i > (1+\\alpha)\\frac{k(q-1)}n$ we have the following\nfor all $1 \\leq j \\leq q$:\n\\[\nr_i - \\frac{(j-1) k}n > r_i\\biggl( 1 - \\frac{j-1}{(1+\\alpha\n)(q-1)}\\biggr) \\geq r_i \\exp\\biggl[ -\\frac{j-1}{\\alpha(q-1)}\\biggr].\n\\]\nCombining this with \\eqref{eq-Vt1q-bound-1}, we deduce that\n\\begin{eqnarray*}\nV_t^{\\bolds{1};q} &\\geq&\\frac1{q!}\\sum_{i=1}^n \\prod\n_{j=1}^{q} r_i \\exp\\biggl[ -\\frac{j-1}{\\alpha(q-1)}\\biggr]\n\\bolds{1}_{\\{r_i > (1+\\alpha)(k(q-1)\/n)\\}}\\\\\n&=& \\frac1{q!}\\sum_{i=1}^n \\bigl( \\mathrm{e}^{-1\/(2\\alpha)} r_i\n\\bolds{1}_{\\{r_i > (1+\\alpha)({k(q-1)}\/n)\\}}\\bigr)^q.\n\\end{eqnarray*}\nApplying Cauchy--Schwarz, we infer that\n\\begin{eqnarray*}\nV_t^{\\bolds{1};q} &\\geq&\\frac{n}{q!} \\biggl(\\frac{\\mathrm\n{e}^{-1\/(2\\alpha)} \\sum_i r_i\n\\bolds{1}_{\\{r_i > (1+\\alpha){{k(q-1)}\/n}\\}}}n \\biggr)^q \\\\\n&=&\n\\frac{(\\sum_i r_i \\bolds{1}_{\\{r_i > (1+\\alpha)\n({k(q-1)}\/n)\\}}\n)^q}{\n\\mathrm{e}^{q\/(2\\alpha)}n^{q-1}q!}.\n\\end{eqnarray*}\nThe proof of the lemma now follows from noticing that\n\\[\n\\sum_i r_i \\bolds{1}_{\\{r_i \\leq(1+\\alpha)({k(q-1)}\/n)\\}} \\leq\n(1+\\alpha) k(q-1) < (1+\\alpha)k q,\n\\]\nwhereas $\\sum_i r_i = \\sum_{s \\leq t} \\sum_i Q_s(i) = t$.\n\\end{pf}\n\nTo complete the proof using Lemma~\\ref{lem-Vt-packing}, apply this\nlemma for $\\alpha=1$, $t = n\/k$, and $kq=n^{o(1)}$, giving that\n\\[\nV_{n\/k}^{1;q} \\geq\\frac{n}{((\\mathrm{e}^{1\/2}-o(1)\n)k)^{q} q!} \\geq(2k)^{-q} n \/ q!,\n\\]\nwhere the last inequality holds for any sufficiently large $n$. Consequently,\n\\[\n\\frac{100(3L)^{q+1}m\/q!}{V_{n\/k}^{\\bolds{1};q}} \\leq100(3L)^{q+1} (2k)^q\n\\frac{m}n \\leq\n100(6kL)^{q+1} \\frac{m}n .\n\\]\nSince our choice of $q$ is such that\n\\[\n(6kL)^{q+1} = \\mathrm{e}^{(1+o(1))q (\\log L + \\log k)} =\n(n\/m)^{1-\\varepsilon-o(1)},\n\\]\nwe have that\n\\[\n\\frac{100(3L)^{q+1}m\/q!}{V_{n\/k}^{\\bolds{1};q}} \\leq\n(n\/m)^{-\\varepsilon+o(1)}.\n\\]\nThis implies both that $V_{n\/k}^{\\bolds{1};q} \\geq(n\/m)^{\\varepsilon\n\/2}$ for\nany large $n$ (recall that $L > q$), and that the condition of\n\\eqref{eq-Vn\/k-Xn\/k-implication} is satisfied for any large $n$.\nAltogether, the\nmaximal load is w.h.p. at least $q$, concluding the proof of\nTheorem~\\ref{thm-large-load}. \\qed\n\n\\subsection{A corollary for nonadaptive algorithms}\nWe end this section with a corollary of Theorem~\\ref{thm-large-load}\nfor the case\nof non-adaptive algorithms, that is, the strategies $Q_1,\\ldots,Q_n$ are fixed\nahead of time. Namely, we show that for $k = O(n\\frac{\\log\\log\nn}{\\log n})$ the optimal maximal\nload is w.h.p. $\\Theta(\\frac{\\log n}{\\log\\log n})$, that is, of the\nsame order as the one for $k=1$.\nTheorem~\\ref{thm-nonadaptive}, whose proof appears in Section~\\ref\n{sec:nonadaptive}, includes\na different approach that proves this result more directly.\n\n\\begin{corollary}\nConsider the allocation problem of $n$ balls\ninto $n$ bins, where each ball has $k$ independent uniform choices.\nIf $k \\leq C n\\frac{\\log\\log n}{\\log n}$, then any nonadaptive algorithm\nw.h.p. has a maximal-load of at least $\\frac{1-o(1)}{C \\vee\n1}\\cdot\n\\frac{\\log n}{\\log\\log n}$.\nIn particular, if $k \\leq n\\frac{\\log\\log n}{\\log n}$ then the load is\nat least $(1-o(1))\\frac{\\log n}{\\log\\log n}$ w.h.p.\n\\end{corollary}\n\n\\begin{pf}\nLet $Q_1,\\ldots,Q_n$ be the optimal sequence of strategies for the\nproblem. Using definitions \\eqref{eq-Xt-def} and \\eqref{eq-Vt-def}\nwith $f \\equiv1$, we have the following for all $q$:\n\\begin{eqnarray*}\nX_t^{\\bolds{1};q} &=& \\sum_i \\pmatrix{ N_t(i)\\cr q} = \\operatorname\n{Col}_q(t)\\quad \\mbox{and\n}\\\\\nV_t^{\\bolds{1};q} &=& \\mathbb{E}X_t^{\\bolds{1};q}.\n\\end{eqnarray*}\nFix $0 < \\varepsilon< \\frac12$. Applying Lemma~\\ref{lem-Vt-packing}\nwith $t=n$ and $\\alpha= \\varepsilon\/[2(1-\\varepsilon)]$,\n\\begin{equation}\n\\label{eq-V1n-lower-bound-eps}\nV_n^{\\bolds{1};q} \\geq\\frac{(n-(1+\\alpha)kq)^q}{\\mathrm\n{e}^{q\/(2\\alpha\n)} n^{q-1}q!} =\n\\biggl(1-\\frac{(2-\\varepsilon)kq}{2(1-\\varepsilon)n}\\biggr)^q \\cdot\n\\frac\n{n}{\\mathrm{e}^{(1-\\varepsilon)q\/\\varepsilon}q!}.\n\\end{equation}\nRecalling that $k \\leq C n \\frac{\\log\\log n}{\\log n}$ for some fixed\n$C > 0$, set\n\\begin{equation}\n\\label{eq-q-value-non-adaptive}\nq = \\frac{1-\\varepsilon}{C \\vee  1}\\cdot\\frac{\\log n}{\\log\\log n}.\n\\end{equation}\nThis choice has $kq\/n \\leq1-\\varepsilon$ and $q! \\leq\nn^{1-\\varepsilon\n+o(1)}$. Combined with \\eqref{eq-V1n-lower-bound-eps},\n\\begin{eqnarray}\\label{eq-non-adapt-E-q-col}\n\\mathbb{E}\\operatorname{Col}_q(n) &=& V_n^{\\bolds{1};q}\\nonumber\\\\\n & \\geq&\\exp\n\\biggl(-\\frac{(2-\\varepsilon\n)kq^2}{2(1-\\varepsilon)n} \\Big \/  \\biggl(1-\\frac{(2-\\varepsilon\n)kq}{2(1-\\varepsilon)n}\\biggr)\\biggr)\n\\frac{n}{\\mathrm{e}^{(1-\\varepsilon)q\/\\varepsilon} q!} \\\\\n& \\geq&\\exp\\biggl(-\\frac{2-\\varepsilon}{\\varepsilon}q\\biggr) \\frac\n{n}{n^{1-\\varepsilon+ o(1)}} = n^{\\varepsilon- o(1)} .\\nonumber\n\\end{eqnarray}\nTo translate the number of $q$-collisions to the number of bins with\nload $q$, consider\nthe case where for some bin $j$ we have $\\sum_{s=1}^n Q_s(j) \\geq100\n\\log n$. Proposition~\\ref{prop-predictions} (applied\nto the Bernoulli variables $\\bolds{1}_{\\{J_s = j\\}}$) then implies\nthat $N_n(j) \\geq50\\log n$ except with probability $O(n^{-5})$, and in\nparticular the maximal load exceeds $q$ w.h.p.\nWe may therefore assume from this point on that $\\sum_{s=1}^n Q_s(j)\n\\leq100 \\log n$ for all~$j$.\n\nSet $L \\stackrel{\\triangle}{=}150 \\log n$. Clearly, upon increasing\n$Q_s(j)$ for some $1\n\\leq s\\leq n$, the load in bin $j$ will stochastically\ndominate the original one. Thus, for any integer $r \\geq1$ we may\nincrease $\\sum_{s=1}^n Q_s(j)$ to $\\frac23 r L$, and by\nProposition~\\ref{prop-predictions} obtain that\n\\mbox{$N_n(j) \\leq r L$} except with probability $O(\\exp(-rL\/30))$. Defining\n\\[\nA_r \\stackrel{\\triangle}{=}\\Bigl\\{ rL \\leq\\max_{1\\leq j \\leq n}\nN_n(j) < (r+1)L\\Bigr\\}\\qquad\n\\mbox{(for $r=0,1,\\ldots$)},\n\\]\nwe in particular get $\\mathbb{P}(A_r) = O(n\\exp(-rL\/30))$. However, clearly\non this event $\\operatorname{Col}_q(n) \\leq n {(r+1)L\\choose q}$,\nand since $n^2{(r+1)L\\choose q} \\leq O(\\exp(rL\/50))$, we have\n\\[\n\\mathbb{E}[\\operatorname{Col}_q(n) \\mid\\overline{A_0}\n]\\mathbb{P}(\\overline\n{A_0}) \\leq\\sum_{r\\geq1} O\\bigl(\\exp(-rL\/100)\\bigr) = O(n^{-3\/2}) = o(1).\n\\]\nThus, by \\eqref{eq-non-adapt-E-q-col}, we have $\\mathbb\n{E}[\\operatorname{Col}_q(n) \\mid\nA_0] \\geq n^{\\varepsilon-o(1)}$.\nFinally, since any given bin can contribute at most ${L\\choose q} =\nn^{o(1)}$ collisions to $\\operatorname{Col}_q(n)$ given $A_0$,\n\\[\n\\mathbb{E}\\Biggl[\\sum_{j=1}^n \\bolds{1}_{\\{N_n(j) \\geq q\\}}\\Biggr]\n\\geq\\mathbb{E}\n\\biggl[\\operatorname{Col}_q(n)\\Big\/\\pmatrix{ L\\cr q} \\Big| A_0\\biggr]\n\\mathbb{P}(A_0) =\nn^{\\varepsilon-o(1)}.\n\\]\n\nAs demonstrated in the next section (see Lemma~\\ref\n{lem-negative-correlation}), one can now use the fact that the events\n$\\{N_n(j) \\geq q\\}$ are negatively\ncorrelated to establish concentration for the variable $\\sum_{j=1}^n\n\\bolds{1}_{\\{N_n(j) \\geq q\\}}$.\nAltogether, we deduce that the maximal load w.h.p. exceeds $q$,\nas required.\n\\end{pf}\n\n\\section{Tight bounds for nonadaptive allocations}\\label{sec:nonadaptive}\n\nIn this section, we present the proof of Theorem~\\ref{thm-nonadaptive}.\nThroughout the proof we assume, whenever this is needed, that\n$n$ is sufficiently large. To simplify the presentation, we omit all\nfloor and ceiling signs whenever these are not crucial.\nWe need the following lemma.\n\n\\begin{lemma}\n\\label{l61}\nLet $p_1,p_2, \\ldots,p_n$ be reals satisfying $0 \\leq p_i \\leq\n\\frac{\\log\\log n}{\\log n}$ for all~$i$, such that $\\sum_{i=1}^n p_i\n\\geq\n1-\\varepsilon$, where $\\varepsilon=\\varepsilon(n) \\in[0,1]$. Let\n$X_1, X_2,\n\\ldots,X_n$\nbe independent indicator random variables, where\n$\\mathbb{P}(X_i=1)=p_i$ for all $i$, and put $X=\\sum_{i=1}^n X_i$. Then\n\\[\n\\mathbb{P}\\biggl(X \\geq(1-\\varepsilon) \\frac{\\log n}{\\log\\log n}\n\\biggr) \\geq\n\\frac{1}{n^{1-\\varepsilon}}.\n\\]\n\\end{lemma}\n\n\\begin{pf}\nWithout loss of generality, assume that $p_1 \\geq p_2 \\geq\\cdots\n\\geq p_n$. Define a family\nof $k$ pairwise disjoint blocks $B_1,B_2, \\ldots,B_k \\subset\\{1, 2,\n\\ldots,n\\}$, where\n$k \\geq(1-\\varepsilon) \\frac{\\log n}{\\log\\log n} $\nso that for each $i$, $1 \\leq i \\leq k$,\n\\[\n\\frac{2}{\\log n} \\leq\\sum_{j \\in B_i} p_j \\leq\\frac{\\log\\log\nn}{\\log n}.\n\\]\nThis can be easily done greedily; the first block consists of\nthe indices $1,2, \\ldots,r$ where $r$ is the smallest integer\nso that $\\sum_{j=1}^r p_j \\geq\\frac{2}{\\log n}$. Note that it is\npossible that $r=1$, and that since the sequence $p_j$ is monotone\ndecreasing, $\\sum_{j=1}^r p_j \\leq\\frac{\\log\\log n}{\\log n}$.\nAssuming we have already partitioned the indices $\\{1,\\ldots\n,r\\}$ into blocks, and assuming we still do not have $(1-\\varepsilon)\n\\frac{\\log n}{\\log\\log n}$ blocks, let the next block\nbe $\\{r+1, \\ldots,s\\}$ with $s$ being the smallest integer\nexceeding $r$ so that \\mbox{$\\sum_{j=r+1}^s p_j \\geq\n\\frac{2}{\\log n}$}. Note that if $p_{r+1} \\geq\\frac{2}{\\log n}$\nthen $s=r+1$, that is, the block consists of a single element,\nand otherwise $\\sum_{j=r+1}^s p_j < \\frac{4}{\\log n} <\n\\frac{\\log\\log n}{\\log n}$. Thus,\nin any case the sum above is at least $\\frac{2}{\\log n}$ and at\nmost $\\frac{\\log\\log n}{\\log n}$. Since the total sum of the reals\n$p_j$ is at least $1-\\varepsilon$ this process does not terminate\nbefore generating $k \\geq(1-\\varepsilon) \\frac{\\log n}{\\log\\log n}$\nblocks, as needed.\n\nFix a family of $k=(1-\\varepsilon) \\frac{\\log n}{\\log\\log n}$ blocks\nas above. Note that for each fixed block $B_i$ in the family, the\nprobability that $\\sum_{j \\in B_i} X_j \\geq1$ is at least\n\\[\n\\sum_{j \\in B_i} p_j -\\sum_{j,q \\in B_i, j<q} p_j p_q\n\\geq\\sum_{j \\in B_i}p_j -\\frac{1}{2} \\biggl(\\sum_{j \\in B_i} p_j\\biggr)^2\n\\geq\\frac{2}{\\log n} -\\frac{2}{\\log^2 n} > \\frac{1}{\\log n}.\n\\]\nIt thus follows that the probability that for each of the $k$\nblocks $B_i$ in the family $\\sum_{j \\in B_i} X_j \\geq1$ is\nat least $(\\frac{1}{\\log n})^k=\\frac{1}{n^{1-\\varepsilon}}$,\ncompleting the proof of the lemma.\n\\end{pf}\n\n\n\n\\begin{pf*}{Proof of Theorem~\\ref{thm-nonadaptive} [\\normalfont{Part~(i)}]\nAs before, our framework is the relaxed model where there are strategies\n$Q_1,Q_2,\\ldots,Q_n$, where $Q_t$ is the distribution of the bin\nto be selected for ball number $t$, satisfying $\\|Q_t\\|_\\infty\\leq k\/n$.\nHowever, since now we consider nonadaptive algorithms, the strategies\nare no longer random variables, but rather a predetermined sequence.\nWe therefore let $P = (p_{i t})$ denote the $n\\times n$ matrix of probabilities,\nwhere $p_{i t}$ is the probability that the ball at time $t$ would be\nplaced in\nbin $i$.\nClearly,\n\\[\n0 \\leq p_{it} \\leq k\/n =\\frac{\\log\\log n}{\\log n}\\qquad \\mbox{for all\n$i$ and $t$}\n\\]\nand\n\\[\n\\sum_{1 \\leq i \\leq n} p_{it}=1\\qquad \\mbox{for all $t$}.\n\\]\nThe sum of entries of each column\nof the $n$ by $n$ matrix\n$p_{it}$ is $1$, and hence the total sum of its entries is $n$.\nIf it contains a row $i$ so that the sum of entries in this row\nis at least, say, $\\log n$, then the expected number of balls in\nbin number $i$ by the end of the process is $\\sum_{t=1}^n p_{it}\n\\geq\\log n$. As the variance is\n\\[\n\\sum_{t+1}^n p_{it} (1-p_{it})\n\\leq\\sum_{t=1}^n p_{it},\n\\]\nit follows by Chebyshev's inequality (or by Hoeffding's inequality)\nthat with high probability the actual number of balls placed in bin\nnumber $i$ exceeds $\\frac{\\log n}{2} > \\frac{\\log n}{\\log\\log\nn}$, showing that in this case the desired result holds.\n\nWe thus assume that the sum of entries in each row is at most $\\log\nn$. As the average sum in a row is $1$, there is a row whose total\nsum is at least $1$. Omit this row, and note that since its total\nsum is at most $\\log n$, the sum of all remaining entries of the\nmatrix is still at least $n-\\log n$, and hence the average sum\nof a row in it is at least $\\frac{n-\\log n}{n-1} > 1-\\frac{\\log\nn}{n}$. Therefore, there is another row of total sum at least this\nquantity. Omitting this row and proceeding in this manner we can\ndefine a set of rows so that the sum in each of them is large. Note\nthat as long as we defined at most $\\frac{n}{\\log^2 n}$ rows, the\ntotal sum of the remaining elements of the matrix is still at least\n$n-\\frac{n}{\\log n}$, and hence there is another row of total sum\nat least $1-\\frac{1}{\\log n}$. We have thus shown that there is a\nset $I$ of $\\frac{n}{\\log^2 n}$ rows such that\n\\[\n\\sum_{t=1}^n p_{it} \\geq1-\\frac{1}{\\log n}\\qquad \\mbox{for each $i \\in I$}.\n\\]\nFor each $i\\in I$, let $A_i$ denote the event\n\\[\nA_i \\stackrel{\\triangle}{=}\\biggl( \\mbox{There are at most }\n\\biggl(\\frac{\\log\nn}{\\log\\log n}-4\\biggr)\\mbox{ balls in bin $i$ }\\biggr).\n\\]\nApplying Lemma \\ref{l61} with $\\varepsilon=\\frac{4 \\log\\log n}{\\log\nn}$, we get\n\\[\n\\mathbb{P}(A_i) \\leq1- \\frac{ \\log^4 n}{ n}\\qquad \\mbox{for each $i\\in I$}.\n\\]\nWe will next show that, as the events $A_i$ are negatively correlated,\nthe probability that all of\nthese events occurs is at most the product of these probabilities\n(which is negligible).\n\n\\begin{lemma}\\label{lem-negative-correlation}\nDefine the events $\\{A_i \\dvtx  i \\in I\\}$ as above. Then\n\\[\n\\mathbb{P}\\biggl(\\bigcap_{i\\in S} A_i \\biggr) \\leq\\prod_{i\\in S}\\mathbb\n{P}(A_i)\\qquad \\mbox{for any subset $S \\subset I$}.\n\\]\n\\end{lemma}\n\n\\begin{pf}\nThe proof proceeds by induction on $|S|$. For the empty set this is\ntrivial, and we will prove\nthat for every set $S$ and $j \\in I \\setminus S$\n\\[\n\\mathbb{P}\\biggl(\\bigcap_{i\\in S} A_i\\cap A_j \\biggr) \\leq\\mathbb{P}\n\\biggl(\\bigcap_{i\\in S}\nA_i \\biggr) \\mathbb{P}(A_j).\n\\]\nDefine the following independent random variables for every time $t$:\n\\begin{eqnarray*}\n\\mathbb{P}(B_t = 1) &=& p_{tj},\\qquad \\mathbb{P}(B_t = 0) = 1-p_{tj},\\\\\n\\mathbb{P}(H_t = i) &=& \\frac{p_{ti}}{1-p_{tj}}\\qquad \\mbox{for each $i\n\\neq j$}.\n\\end{eqnarray*}\nWe may now define $J_t$, the position of the ball at time $t$, as a\nfunction of $B_t$ and~$H_t$,\nsuch that indeed $\\mathbb{P}(J_t = i) = p_{ti}$ for all $i$:\n\\[\nJ_t = \\cases{\nj & \\quad $B_t = 1$,\\vspace*{2pt}\\cr\nH_t & \\quad $B_t = 0$.}\n\\]\nCrucially, the event $A_j$ depends only on the values of $\\{B_t\\}$, and\nis a monotone decreasing in them.\nFurther notice that the function\n\\[\nf(b_1,\\ldots,b_n) \\stackrel{\\triangle}{=}\\mathbb{P}\\biggl(\\bigcap\n_{i\\in S} A_i \\mid B_1=b_1,\\ldots\n,B_n=b_n \\biggr)\n\\]\nis monotone increasing in the $b_i$'s. Therefore, applying the\nFKG-inequality (see, e.g., \\cite{AS}, {Chapter 6}, and also \\cite\n{Grimmett}, {Chapter 2}) on $Y = f(B_1,\\ldots,B_n)$ and $\\bolds\n{1}_{A_j}$ gives\n\\[\n\\mathbb{P}\\biggl(\\bigcap_{i\\in S} A_i\\cap A_j \\biggr)= \\mathbb{E}[Y\n\\bolds{1}_{A_j}] \\leq\\mathbb{E}\n[Y] \\mathbb{P}(A_j) = \\mathbb{P}\\biggl(\\bigcap_{i\\in S} A_i \\biggr)\n\\mathbb{P}(A_j),\n\\]\nas required.\n\\end{pf}\nAltogether, we obtain that the probability that all of the bins with\nindices in $I$\nhave at most $(\\frac{\\log n}{\\log\\log n} - 4)$ balls is\n\\[\n\\mathbb{P}\\biggl(\\bigcap_{i\\in I} A_i\\biggr) \\leq\\biggl(1-\\frac{ \\log^4 n}{\nn}\\biggr)^{n\/\\log^2 n} \\leq\\mathrm{e}^{-\\log^2 n}.\n\\]\nThis completes the proof of part~(i).\n\\end{pf*}\n\n\\begin{pf*}{Proof of Theorem~\\ref{thm-nonadaptive} [\\normalfont{Part~(ii)}]}\nIt is convenient to describe the proof of this part\nfor a slightly different model instead of the one considered\nin the previous sections. Namely, in the variant model, in every round\neach bin among the $n$ bins is chosen randomly and independently\nas one of the\noptions with probability $\\alpha$. By Chernoff's bounds, our results\nin this model will carry into\nthe original one, since obtaining $\\alpha n$ uniform bins is dominated\nby getting each bin independently with probability $(1+\\varepsilon\n)\\alpha\n$, and dominates a probability of $(1-\\varepsilon)\\alpha$ for each bin.\n\nFor the simplicity of the notations, we will henceforth consider the\ncase \\mbox{$k=n\/2$}, noting that our proofs hold for $k=\\alpha n$ with any $0\n< \\alpha< 1$ fixed.\n\nAs noted in the \\hyperref[sec1]{Introduction}, the relaxed\nmodel of strategies $Q_t$ such that $\\|Q_t\\|_\\infty\\leq k\/n$\nis stronger than the model where there are $k$ uniform options for bins.\nIn fact, the results of this part (an optimal maximal load of order\n$\\sqrt{\\log n}$)\ndo not hold for the relaxed model. For instance, if $Q_t$ assigns probability\n$k\/n = \\frac12$ to $i=t$ and $i=(t+1)$ (with the indices reduced\nmodulo $n$),\nthe maximum load will be at most $2$.\n\n\nHowever, it is easy to see\nthat in fact each strategy $Q_t$ is more restricted.\nIndeed, the total probability that $Q_t$ can assign to any $r$ bins\ndoes not\nexceed $1-2^{-r}$, as for each fixed set $I$ of $r$ bins,\nthe probability that none of the members of $I$ is an optional\nchoice for ball number $t$ is $2^{-r}$.\n\nWe start with the simple\nproof of the upper bound, obtained by the natural algorithm which places\nthe ball in round $t$ in the first possible bin\n(among the $k$ given choices)\nthat follows bin number $t$ in the cyclic\norder of the bins.\n\n\\begin{lemma}\n\\label{l62}\nThere exists a nonadaptive strategy ensuring that, w.h.p., the maximum\nload in the above model is\nat most $O(\\sqrt{\\log n})$.\n\\end{lemma}\n\n\\begin{pf}\nOrder the bins cyclically $b_1,b_2, \\ldots,b_n,b_{n+1}=b_1$.\nFor each round~$t$, $1 \\leq t \\leq n$, place the ball number $t$\nin the first possible bin $b_i$ that follows $b_t$ in our cyclic order\nand is one of the given options for this round. Note, first, that\nthe probability that the ball in round $t$ is placed in a bin whose\ndistance from $b_t$ exceeds $2 \\log n$, is precisely the\nprobability that none of the $2 \\log n$ bins following $b_t$ is\nchosen in round $t$, which is\n\\[\n2^{-2 \\log n} < n^{-5\/4}.\n\\]\nTherefore, with high probability, this does not happen for any $t$.\nIn addition, the probability that a fixed bin $b_i$\ngets a load of $ \\sqrt{\\log n}$ from balls placed in the\n$2 \\log n- 2 \\sqrt{\\log n}$\nrounds $\\{i-2 \\log n +1, i-2 \\log n+2 , \\ldots,i-2 \\sqrt{\\log n}\\}$,\ndoes\nnot exceed\n\\[\n\\pmatrix{ 2 \\log n -2 \\sqrt{\\log n}\\vspace*{3pt}\\cr \\sqrt{\\log n}}\n\\biggl(\\frac{1}{2^{2 \\sqrt{\\log n}}}\\biggr)^{\\sqrt{\\log n}}\n\\leq\\frac{1}{n^{2-o(1)}}.\n\\]\nIndeed, for each fixed value of $t \\in[i-2 \\log n+1,\ni-2 \\sqrt{\\log n} ]$, if the ball placed in\nround number $t$ ends in bin number $i$, then none of the\n$2 \\sqrt{\\log n}$ bins preceding $b_i$ is chosen as an optional\nbin for ball number $t$, and the probability of this event is\n$2^{-2 \\sqrt{\\log n}}$. There are ${2 \\log n -2 \\sqrt{\\log n}\\choose \\sqrt{\\log n}}$ possibilities to select\n$ \\sqrt{\\log n}$ rounds in the set\n$\\{i-2 \\log n +1, i-2 \\log n+2 , \\ldots,i-2 \\sqrt{\\log n}\\}$,\nand as the choices of options for each round are independent, the\ndesired estimate follows.\n\nWe conclude that with high probability no bin $b_i$ gets any\nballs from round $t$ with $t \\leq i-2 \\log n$, and no bin $b_i$ gets\nmore than $\\sqrt{\\log n}$ balls from rounds $t$ with\n$i-2 \\log n < t \\leq i-2\\sqrt{\\log n}$. As $b_i$ can get at\nmost $2 \\sqrt{\\log n}$ balls from all other rounds $t$, (as there\nare only $2 \\sqrt{\\log n}$ such rounds), it follows\nthat with high probability the maximum load does not exceed $3\n\\sqrt{\\log n}$, completing the proof of the lemma. Note that it is\neasy to improve the constant factor $3$ in the estimate proved\nhere, but we make no attempt to optimize it.\n\\end{pf}\n\nWe proceed with the proof of the lower bound. As in the proof of\npart~(i) of the theorem,\nlet $P=(p_{it})$ be the $n\\times n$ matrix of probabilities\ncorresponding to our nonadaptive strategy, where\n$p_{it}$ is the probability that\nthe ball in round $t$ will be placed in bin number $i$. Recall that\nfor each\nfixed round $t$, the sum of the largest $r$ numbers $p_{it}$ cannot\nexceed $1-2^{-r}$.\nThis fact will be the only property of the distribution\n$p_{it}$ used in the proof.\n\nCall an entry\n$p_{it}$ of the matrix $P$ \\textit{large} if\n$p_{it} \\geq2^{-\\sqrt{\\log n}}$, otherwise, $p_{it}$ is \\textit{small}. Call a column $t$\nof $P$ \\textit{concentrated} if it has at least\n$\\frac{\\sqrt{\\log n}}{2}$ large elements. We consider two possible\ncases.\n\n\\begin{itemize}\n\\item \\textit{Case 1}:  There are at least $n\/2$ concentrated columns.\n\\end{itemize}\n\n\nIn this case, there are at least $\\frac{n \\sqrt{\\log n}}{4}$ large\nentries in $P$. If there is a row, say row number $i$\nof $P$, containing at least, say,\n$2^{2 \\sqrt{\\log n}}$ large entries, then the expected number of\nballs in the corresponding bin is $\\sum_{t=1}^n p_{it} > 2^{\\sqrt\n{\\log n}} $, and, as the variance of this quantity is\nsmaller than the expectation, it follows that in this case with high\nprobability this bin will have a load that exceeds\n$\\Omega( 2^{\\sqrt{\\log n}}) > \\sqrt{\\log n}$. We thus assume that no row\ncontains more than $2^{2 \\sqrt{\\log n}}$ large elements.\nTherefore, there are at least $\\frac{n}{2^{2 \\sqrt{\\log n}}}\n=n^{1-o(1)}$ rows,\neach containing at least, say, $\\frac{\\sqrt{\\log n}}{8}$ large\nelements. Indeed, we can select such rows one by one. As long as the\nnumber of selected rows does not exceed $\\frac{n}{2^{2 \\sqrt\n{\\log}}}$, the total number of large elements in them is\\vspace*{-2pt} at most\n$n$, and hence the remaining rows still contain at least\n$\\frac{n \\sqrt{\\log n}}{4}-n > \\frac{n \\sqrt{\\log n}}{8}$ large\nelements, implying that there is still another row containing at\nleast $\\frac{\\sqrt{\\log n}}{8}$ large elements.\n\nFix a bin corresponding to a\nrow with at least $\\frac{\\sqrt{\\log n}}{8}$ large\nelements, and fix $\\frac{\\sqrt{\\log n}}{8}$ of them.\nThe probability that all balls corresponding to these\nlarge elements\\vspace*{1pt} will be placed in this bin is at least\n\\[\n\\biggl(\\frac{1}{2^{\\sqrt{\\log n}}}\\biggr)^{{\\sqrt{\\log n}}\/{8}}\n=\\frac{1}{n^{1\/8}}.\n\\]\nAs there are at least $n^{1-o(1)}$ such bins, and the events of\nno large load in distinct bins are negatively correlated (see\nLemma~\\ref{lem-negative-correlation}), we\nconclude that the probability that none of these bins has a load\nat least $\\frac{\\sqrt{\\log n}}{8}$ is at most\n\\[\n\\biggl(1-\\frac{1}{n^{1\/8}}\\biggr)^{n^{1-o(1)}}=o(1),\n\\]\nshowing that in this case\nthe maximum load is indeed $\\Omega(\\sqrt{\\log n})$ with high\nprobability.\n\n\\begin{itemize}\n\\item \\textit{Case 2}:  There are less than $n\/2$ concentrated columns.\n\\end{itemize}\n\n\nIn this case, the sum of all small entries of the matrix $P$ is at\nleast\n\\[\n\\frac{n}{2 \\cdot2^{0.5 \\sqrt{\\log n}}},\n\\]\nsince each of the $n\/2$ nonconcentrated columns has less\nthan $0.5 \\sqrt{\\log n}$ large elements, and hence in each such\ncolumn the sum of all small elements is at least\n$2^{-0.5 \\sqrt{\\log n}}$.\n\nCall a small entry $p=p_{ij}$ of $P$ an entry of \\textit{type} $r$\n(where $\\sqrt{\\log n} \\leq r \\leq2 \\log n$), if\n$\\frac{1}{2^{r+1}} \\leq p < \\frac{1}{2^r}$. Since the sum of all\nentries of $P$ that are smaller than \\mbox{$2^{-2 \\log n} =1\/n^2$} is at\nmost $1$, there is a value of $r$ in the above range, so that the\nsum of all entries of $P$ of type $r$ is at least\n\\[\n\\frac{n}{4 \\log n \\cdot2^{0.5 \\sqrt{\\log n}}}\n> \\frac{n}{2^{0.75 \\sqrt{\\log n}}}.\n\\]\nPut\n\\[\nx\\stackrel{\\triangle}{=}2^{0.75 \\sqrt{\\log n}},\n\\]\nand note that there are at least\n$\\frac{n2^r}{x}$ entries of type $r$ in $P$ (since otherwise their\ntotal sum cannot be at least $n\/x$). We now restrict our attention\nto these entries.\n\nWe can assume that there is no row containing more than\n$2^{r+1} \\log n$ of these entries. Indeed, otherwise the expected\nnumber of balls in the corresponding bin is at least $\\log n$, the\nvariance is smaller, and hence by Chebyshev with high probability\nthe load in this bin will exceed\n$\\Omega(\\log n) >\\sqrt{\\log n}$. We can now apply\nagain\\vspace*{1pt} our greedy procedure and conclude that there are at least\n$\\frac{n}{4x \\log n} =n^{1-o(1)}$ rows, each containing at least\n$\\frac{2^r}{2x}$ entries of type $r$; indeed, a set of less\nthan $\\frac{n}{4x \\log n}$ rows contains a total of at most\n\\[\n\\frac{n}{4x \\log n} 2^{r+1} \\log n =\\frac{n 2^r}{2x}\n\\]\nelements\nof type $r$, leaving at least $\\frac{n 2^r}{2x}$ such elements in\nthe remaining rows, and hence ensuring the existence of an\nadditional row with at least $\\frac{2^r}{2x}$ such entries.\n\nFix a bin corresponding to a\nrow with at least $\\frac{2^r}{2x}$ entries of type $r$.\nThe probability that exactly $t$ balls corresponding to these entries\nwill be placed in this bin is at least\n\\[\n\\pmatrix{ \\dfrac{2^r}{2x}\\vspace*{3pt}\\cr t} \\biggl(\\frac{1}{2^{r+1}}\\biggr)^t\n\\biggl(1-\\frac{1}{2^r}\\biggr)^{{2^r}\/{(2x)}}\n\\geq\n\\biggl(\\frac{2^r}{2xt}\\biggr)^t \\biggl(\\frac{1}{2^{r+1}}\\biggr)^t\n\\biggl(1-\\frac{1}{2x}\\biggr)\n> \\frac{1}{2}(4xt)^{-t}.\n\\]\nfor $t=\\sqrt{\\log n}$ the last quantity is at least\n\\[\n\\tfrac{1}{2} \\bigl(4 \\cdot2^{0.75\n\\sqrt{\\log n}} \\sqrt{\\log n}\\bigr)^{-\\sqrt{\\log n}}\n=n^{-3\/4-o(1)}.\n\\]\nThis, the fact that there are $n^{1-o(1)}$ such rows, and the\nnegative correlation implies that in this case, too, with high\nprobability there is a bin with load at least\n$\\Omega(\\sqrt{\\log n})$. This completes the proof of part~(ii) of the theorem.\n\\end{pf*}\n\n\\section{Concluding remarks and open problems}\\label{sec:conclusion}\n\n\\begin{itemize}\n\\item We have established a sharp choice-memory tradeoff for achieving\na constant maximal load in the balls-and-bins experiment,\nwhere there are $n$ balls and $n$ bins, each ball has $k$ uniformly\nchosen options for bins, and there are $m$ bits of memory available. Namely:\n\\begin{enumerate}[1.]\n\\item[1.] If $k m = \\Omega(n)$ for $k = \\Omega(\\log n)$ and $m = \\Omega\n(\\log n \\log\\log n)$, then there exists an algorithm that achieves an\n$O(1)$ maximal load w.h.p.\n\\item[2.] If $k m = o(n)$ for $m = \\Omega(\\log n)$, then any algorithm\nw.h.p. creates an unbounded maximal load. For this case we\nprovide two lower bounds on the load: $\\Omega(\\log\\log\n(\\frac{n}{km}))$ and\n$(1+o(1))\\frac{\\log(n\/m)}{\\log\\log(n\/m)+\\log k}$.\n\\end{enumerate}\n\n\\item In particular, if $m = n^{1-\\delta}$ for some $\\delta> 0$ fixed\nand $2 \\leq k \\leq\\operatorname{polylog}(n)$, we obtain\na lower bound of $\\Theta(\\frac{\\log n}{\\log\\log n})$ on the maximal\nload. That is, the typical maximal load in \\textit{any} algorithm has\nthe same order as the typical maximal load in a \\textit{random}\nallocation of $n$ balls in $n$ bins.\n\n\\item Given our methods, it seems plausible and interesting to improve\nthe above lower bounds to $(1+o(1))\\frac{\\log({n}\/{(km)})}{\\log\n\\log({n}\/{(km)})}$, analogous to the load of $(1+o(1))\\frac{\\log\nn}{\\log\\log n}$\nin a completely random allocation.\n\n\\item Note that, when $k m = n^{1-\\delta}$ for some fixed $\\delta> 0$,\neven the above conjectured lower bound is still a factor of $\\delta$\naway from the upper bound given by a random allocation.\nIt would be interesting to close the gap between these two bounds.\nConcretely, suppose that $k m = \\sqrt{n}$; can one outperform\nthe typical maximal load in a random allocation?\n\n\\item To prove our main results, we study the problem of achieving a\nperfect allocation (one that avoids collisions, i.e., a matching) of\n$(1-\\delta)n$ balls into $n$ bins. We show that there exist constants\n$C > c > 0$ such that:\n\\begin{enumerate}[1.]\n\\item If $k m > C n$ for $k = \\Omega(\\log n)$ and $m = \\Omega(\\log n\n\\cdot\\log\\log n)$, then there exists an algorithm that achieves\na perfect allocation w.h.p.\n\\item If $k m < c n$ for $m = \\Omega(\\log n)$, then any algorithm\ncreates $\\Omega(n)$ collisions w.h.p.\n\\end{enumerate}\n\n\\item In light of the above, it would be interesting to show that there\nexists a critical $c > 0$ such that, say for $k,m \\geq\\log^2 n$, the\nfollowing holds: If $k m \\geq(c+o(1))n$ then there is an algorithm\nthat achieves a perfect allocation w.h.p., whereas if\n$k m \\leq(c-o(1))n$ then any algorithm has $\\Omega(n)$ collisions\nw.h.p.\n\n\\item The key to proving the above results is a combination of\nmartingale analysis and a Bernstein--Kolmogorov type large deviation\ninequality. The latter, Proposition~\\ref{prop-predictions}, relates a\nsum of a sequence of random variables to the sum of its conditional\nexpectations, and crucially does \\textit{not} involve the length of the\nsequence. We believe that this inequality may have other applications\nin combinatorics and the analysis of algorithms.\n\n\\item We also analyzed the case of nonadaptive algorithms, where we\nshowed that for every $k = O( n \\frac{\\log\\log n}{\\log n}\n)$, the best\npossible maximal load w.h.p. is $\\Theta(\\frac{\\log\nn}{\\log\\log\nn})$, that is, the same as in a random allocation. For $k = \\alpha n$\nwith $0 < \\alpha< 1$, we proved that the best possible maximal load is\n$\\Theta(\\sqrt{\\log n})$. Hence, one can ask what the minimal order of\n$k$ is, where an algorithm can outperform the order of the maximal load\nin the random allocation.\n\\end{itemize}\n\n\\section*{Acknowledgments}\n\nWe thank Yossi Azar and Allan Borodin for helpful discussions, as well\nas Yuval\nPeres for pointing us to the reference for Theorem~\\ref{thm-martingale}.\nWe also thank Itai Benjamini for proposing the problem of balanced\nallocations with limited memory.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\vspace{-3mm}\n\nThe task of semantic segmentation is formatted as predicting \nthe semantic category for each pixel in an image.\nBased on the pioneering fully convolutional network~\\cite{long2015fully},\nprevious studies have achieved great success as reflected by increasing the performance on various challenging semantic segmentation benchmarks~\\cite{caesar2018coco,cordts2016cityscapes,zhou2017scene}.\n\n\n{\n\\begin{figure}\n\\centering\n\\normalsize\n\\begin{minipage}[c]{0.6\\textwidth}\n{\n\t\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/1\/gt.png}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/1\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/1\/gscnn.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/1\/hrnet48.png}} \\\\\n\t{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/2\/gt.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/2\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/2\/gscnn.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/2\/hrnet48.png}} \\\\\n\t{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/3\/gt.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/3\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/3\/gscnn.png}}\n\t\\frame{\\includegraphics[width=.2\\textwidth]{images\/intro\/fig1\/3\/hrnet48.png}} \\\\\n}\n\\end{minipage}\n\\hspace{-7mm}\n\\begin{minipage}[c]{0.43\\textwidth}\n\\caption{\\small{\n{\\textbf{Qualitative analysis of the segmentation error maps.}}\nThe \\nth{1} column presents the ground-truth segmentation maps, \nand the \\nth{2}\/\\nth{3}\/\\nth{4} column presents the \nthe error maps of DeepLabv3\/HRNet\/Gated-SCNN separately.\nThese examples are cropped from Cityscapes \\texttt{val} set.\nWe can see that there exist many errors along the thin boundary for all three methods.}}\n\\label{fig:boundary_pixel_example}\n\\end{minipage}\n\\vspace{-6mm}\n\\end{figure}\n}\n\nMost of the existing works mainly addressed semantic segmentation\nthrough (i) increasing the resolution of feature maps~\\cite{chen2017rethinking,chen2018encoder,sun2019high}, (ii) constructing\nmore reliable context information~\\cite{zhao2017pyramid,yuan2018ocnet,fu2019dual,huang2019interlaced,zhang2019co}\nand (iii) exploiting boundary information~\\cite{bertasius2016semantic,chen2016semantic,liu2018devil,takikawa2019gated}.\nIn this work,\nwe follow the \\nth{3} line of work and focus on improving segmentation\nresult on the pixels located within the thinning boundary\\footnote{\nIn this paper,\nwe treat the pixels with neighboring pixels belonging to different \ncategories as the boundary pixels.\nWe use the distance transform to generate the ground-truth boundary map\nwith any given width in our implementation.\n}\nvia a simple but effective\nmodel-agnostic boundary refinement mechanism.\n\nOur work is mainly motivated by the observation that\n\\emph{most of the existing state-of-the-art segmentation models\nfail to deal well with the error predictions along the boundary.}\nWe illustrate some examples of the segmentation error maps with DeepLabv3~\\cite{chen2017rethinking},\nGated-SCNN~\\cite{takikawa2019gated} and HRNet~\\cite{sun2019high}\nin Figure~\\ref{fig:boundary_pixel_example}.\nMore specifically, we illustrate the statistics on the \nnumbers of the error pixels \\emph{vs.} the distances\nto the object boundaries in Figure~\\ref{fig:boundary_distance_map}.\nWe can observe that, for all three methods, the number of error pixels significantly decrease \nwith larger distances to the boundary.\nIn other words, \npredictions of the interior pixels are more reliable.\n\nWe propose a novel model-agnostic post-processing\nmechanism to reduce boundary errors by \nreplacing labels\nof boundary pixels with the labels of \ncorresponding interior pixels\nfor a segmentation result.\nWe estimate the pixel correspondences by processing the input image\n(without exploring the segmentation result)\nwith two steps.\nThe first step aims to \nlocalize the pixels along the object boundaries.\nWe follow the contour detection methods~\\cite{Bertasius_2015_ICCV,arbelaez2010contour,DollarARXIV14edges}\nand simply use a convolutional network\nto predict a binary mask indicating the boundary pixels.\nIn the second step,\nwe learn a direction away from the boundary pixel to an interior pixel\nand identify the corresponding interior pixel\nby moving from the boundary pixel\nalong the direction \nby a certain distance.\nEspecially,\nour SegFix can reach nearly real-time speed with high resolution inputs.\n\n\\begin{figure}\n\\centering\n\\hspace{-10mm}\n\\begin{subfigure}[b]{0.32\\textwidth}\n{\\includegraphics[width=\\textwidth]{images\/intro\/error_hist_deeplabv3.pdf}}\n\\caption{DeepLabv3~\\cite{chen2017rethinking}}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.32\\textwidth}\n{\\includegraphics[width=\\textwidth]{images\/intro\/error_hist_hrnet.pdf}}\n\\caption{\\small{HRNet~\\cite{sun2019high}}}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.32\\textwidth}\n{\\includegraphics[width=\\textwidth]{images\/intro\/error_hist_gscnn.pdf}}\n\\caption{Gated-SCNN~\\cite{takikawa2019gated}}\n\\end{subfigure}\n\\hspace{-6mm}\n\\caption{\n\\small{\n\\textbf{Histogram statistics of errors:}\nthe number of error pixels \\emph{vs.} their (Euclidean) distances to the boundaries on Cityscapes \\texttt{val} based on DeepLabv3\/HRNet\/Gated-SCNN.\nWe can see that pixels with larger distance tend to be well-classified with higher probability and \nthere exist many errors distributing within $\\sim5$ pixels width along the boundary.\n}\n}\n\\label{fig:boundary_distance_map}\n\\vspace{-5mm}\n\\end{figure}\n\nOur SegFix is a general scheme that consistently \nimproves the performance\nof various segmentation models\nacross multiple benchmarks without any prior information.\nWe evaluate the effectiveness of SegFix\non multiple semantic segmentation benchmarks\nincluding Cityscapes, ADE20K and GTA5.\nWe also extend SegFix to\ninstance segmentation task on Cityscapes.\nAccording to the Cityscapes leaderboard,\n``HRNet + OCR + SegFix\" and ``PolyTransform + SegFix\" achieve\n$84.5\\%$ and $41.2\\%$, which rank the \\nth{1} and \\nth{2} place\non the semantic and instance segmentation track separately\nby the ECCV 2020 submission deadline.\n\n\t\n\\vspace{-1mm}\n\\section{Related Work}\n\\vspace{-1mm}\n\n\\noindent \\textbf{Distance\/Direction Map for Segmentation:}\nSome recent work~\\cite{bai2017deep,hayder2017boundary,wang2019object}\nperformed distance transform to compute distance maps for instance segmentation task.\nFor example, \n\\cite{bai2017deep,hayder2017boundary} proposed to train the model to \npredict the truncated distance maps within each cropped instance.\nThe other work~\\cite{dangi2019distance,bischke2019multi,chen2018masklab,papandreou2018personlab}\nproposed to regularize the semantic or instance segmentation predictions\nwith distance map or direction map in a multi-task mechanism.\nCompared with the above work,\nthe key difference is that \nour approach does not perform any segmentation predictions\nand instead predicts the direction map from only the image, \nand then we refine the segmentation results of the existing approaches.\n\n\n\\noindent \\textbf{Level Set for Segmentation:} Many previous efforts ~\\cite{osher1988fronts,caselles1997geodesic,kass1988snakes} have used \nthe level set approach to\naddress the semantic segmentation problem\nbefore the era of deep learning.\nThe most popular formulation of level set is\nthe signed distance function,\nwith all the zero values corresponding to predicted boundary positions.\nRecent work~\\cite{acuna2019devil,chen2019learning,wang2019object,kim2019cnn} extended the conventional level-set scheme to deep network for regularizing the boundaries of predicted segmentation map.\nInstead of representing the boundary with a level set function directly,\nwe implicitly encode the relative \ndistance information of the boundary pixels\nwith a boundary map and a direction map.\n\n\\noindent \\textbf{DenseCRF for Segmentation:}\nPrevious work~\\cite{chen2017deeplab,yu2015multi,Zheng_2015_ICCV,Lin_2017_CVPR} improved their segmentation results\nwith the DenseCRF~\\cite{krahenbuhl2011efficient}.\nOur approach is also a kind of general post processing scheme\nwhile being simpler and more efficient for usage.\nWe empirically show that our approach\nnot only outperforms but also is complementary with \nthe DenseCRF.\n\n\\noindent \\textbf{Refinement for Segmentation:}\nExtensive studies~\\cite{fieraru2018learning,gidaris2017detect,islam2017label,kuo2019shapemask,li2016iterative} have proposed various mechanisms to\nrefine the segmentation maps from coarse to fine.\nDifferent from most of the existing refinement approaches\nthat depend on the segmentation models,\nto the best of our knowledge,\nour approach is the first model-agnostic segmentation refinement\nmechanism that can be applied to refine the segmentation results of \nany approach without any prior information.\n\n{\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{images\/approach\/segfix_pipeline_v3.pdf}\n\\vspace{-4mm}\n\\caption{\\small{\n\\textbf{\nIllustrating the SegFix framework:\n}\nIn the training stage,\nwe first send the input image into a backbone to predict a feature map.\nThen we apply a boundary branch to predict a binary boundary map \nand a direction branch to predict a direction map and\nmask it with the binary boundary map. \nWe apply boundary loss and direction loss \non the predicted boundary map and direction map separately.\nIn the testing stage,\nwe first convert the direction map to offset map\nand then refine the segmentation results of any existing methods \naccording to the offset map.\n}}\n\\vspace{-6mm}\n\\label{fig:pipeline}\n\\end{figure*}\n}\n\n\\noindent \\textbf{Boundary for Segmentation:}\nSome previous efforts~\\cite{acuna2019devil,yu2017casenet,liu2017richer,yu2018simultaneous} focused on localizing semantic boundaries.\nOther studies~\\cite{bertasius2016semantic,takikawa2019gated,ding2019boundary,liu2018devil,liu2017learning,ke2018adaptive,ding2019semantic,ding2018context} also exploited the boundary information to improve the segmentation. \nFor example,\nBNF~\\cite{bertasius2016semantic} introduced a global energy model to\nconsider the pairwise pixel affinities based on the boundary predictions. \nGated-SCNN~\\cite{takikawa2019gated} exploited the\nduality between the segmentation predictions and the\nboundary predictions with a two-branch mechanism and a regularizer.\n\nThese methods~\\cite{bertasius2016semantic,ding2019boundary,takikawa2019gated,ke2018adaptive} \nare highly dependent on the segmentation models\nand require careful re-training or fine-tuning.\nDifferent from them,\nSegFix does not perform either segmentation prediction or \nfeature propagation and we instead refine the \nsegmentation maps with an offset map\ndirectly.\nIn other words,\nwe only need to train a single unified SegFix model once w\/o any\nfurther fine-tuning the different segmentation models (across multiple different datasets).\nWe also empirically verify that our approach is complementary with the above methods,\ne.g., Gated-SCNN~\\cite{takikawa2019gated} and Boundary-Aware Feature Propagation~\\cite{ding2019boundary}.\n\n\\noindent \\textbf{Guided Up-sampling Network:}\nThe recent work~\\cite{mazzini2018guided,Mazzini_2019_CVPR_Workshops}\nperformed a segmentation guided offset scheme \nto address boundary errors caused by the bi-linear up-sampling.\nThe main difference is that they do not apply any explicit supervision on their offset maps and require re-training for different models, while \nwe apply explicit semantic-aware supervision on the offset maps and \nour offset maps can be applied to\nvarious approaches directly without any\nre-training.\nWe also empirically verify the advantages of our approach.\n\n\\noindent \\textbf{Semantically Thinned Edge Alignment Learning (STEAL):}\nThe previous study STEAL~\\cite{acuna2019devil} is the most similar work\nas it also predicts both boundary maps and direction maps (simultaneously) to refine the \nboundary segmentation results. \nTo justify the main differences between STEAL and our SegFix, \nwe summarize several key points as following: \n(i) STEAL predicts $K$ independent boundary maps (associated with $K$ categories) \nwhile SegFix only predicts a single boundary map w\/o differentiating the different categories.\n(ii) STEAL first predicts the boundary map and then applies a fixed convolution on the boundary map to estimate the direction map while SegFix uses two parallel branches to predict them independently.\n(iii) STEAL uses mean-squared-loss on the direction branch while SegFix uses cross-entropy loss (on the discrete directions).\nBesides,\nwe empirically compare STEAL and our SegFix in the ablation study.\n\n\n\n\\vspace{-3mm}\n\\section{Approach}\n\\label{methods}\n\\vspace{-2mm}\n\n\\subsection{Framework}\nThe overall pipeline of \nSegFix is illustrated in Figure~\\ref{fig:pipeline}.\nWe first train a model to pick out boundary pixels \n(with the boundary maps) and\nestimate their corresponding interior pixels\n(with offsets derived from the direction maps)\nfrom only the image.\nWe do not perform segmentation \ndirectly during training.\nWe apply this model to generate offset maps from the images\nand use the offsets to get the corresponding pixels which should\nmostly be the more confident interior pixels,\nand thereby \nrefine segmentation results \nfrom any segmentation model.\nWe mainly describe SegFix scheme for semantic segmentation\nand we illustrate the details for instance segmentation in the Appendix.\n\n\n\\noindent\\textbf{Training stage.}\nGiven an input image $\\mathbf{I}$ of shape $H\\times W\\times 3$, \nwe first use a backbone network to \nextract a feature map $\\mathbf{X}$,\nand then send $\\mathbf{X}$ in parallel to\n(1) the \\emph{boundary branch} to predict\na binary map $\\mathbf{B}$,\nwith $1$ for the boundary pixels\nand $0$ for the interior pixels,\nand \n(2) the \\emph{direction branch}\nto predict a direction map $\\mathbf{D}$\nwith each element storing the direction \npointing from the boundary pixel to the interior pixel.\nThe direction map $\\mathbf{D}$ is then\nmasked by the binary map $\\mathbf{B}$\nto yield the input for the offset branch.\n\nFor model training, we use a binary cross-entropy loss\nas the boundary loss\non $\\mathbf{B}$ and a categorical cross-entropy loss\nas the direction loss on\n$\\mathbf{D}$ separately.\n\n\\noindent\\textbf{Testing stage.}\nBased on the predicted boundary map $\\mathbf{B}$ and direction map $\\mathbf{D}$, \nwe apply the \\emph{offset branch} to generate a offset map $\\Delta{\\mathbf{Q}}$.\nA coarse label map ${\\mathbf{L}}$ output by any semantic segmentation model\nwill be refined as:\n\n\\begin{ceqn}\n\\begin{align}\n\\widetilde{\\mathbf{L}}_{\\mathbf{p}_i} &= {\\mathbf{L}}_{\\mathbf{p}_i+\\Delta{\\mathbf{q}_i}},\n\\end{align}\n\\end{ceqn}\n\n\\noindent where $\\widetilde{\\mathbf{L}}$ is refined label map,\n$\\mathbf{p}_i$ represents the coordinate of the boundary pixel $i$, $\\Delta{\\mathbf{q}_i}$ is the generated offset vector pointing to an interior pixel,\nwhich is indeed an element of $\\Delta{\\mathbf{Q}}$.\n$\\mathbf{p}_i+\\Delta{\\mathbf{q}_i}$ is the position of \nthe identified interior pixel.\n\nConsidering that there might be some ``fake\" interior pixels\n\\footnote{\nWe use ''fake\" interior pixels to represent\npixels (after offsets)\nthat still lie on the boundary\nwhen the boundary is thick.\nNotably, we identify an pixel as interior pixel\/boundary pixel if its value in the predicted boundary map $\\mathbf{B}$ is $0$\/$1$.\n}\nwhen the boundary is thick,\nwe propose two different schemes as following:\n(i) re-scaling all the offsets by a factor, e.g., 2.\n(ii) iteratively applying the offsets (of the ``fake\" interior pixels) until finding an interior pixel.\nWe choose (i) by default for simplicity as their performance is close. \n\n\nDuring testing stage,\nwe only need to generate the offset maps on test set\n\\emph{for once},\nand could \napply the same offset maps to refine the segmentation results from\nany existing segmentation model without requiring any prior information.\nIn general,\nour approach is agnostic to any existing segmentation models.\n\n\n{\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{images\/approach\/shift-improve-v3.pdf}\n\\caption{\\small{\n\\textbf{Illustrating the refinement mechanism of our approach:}\nwe refine the coarse label map based on the offset map by \nreplacing the labels of boundary pixels with the labels of \n(more) reliable interior pixels.\nWe represent different offset vectors with different arrows.\nWe mark the error positions in the coarse label map with $\\textcolor{red}{\\Box}$\nand the corresponding corrected positions in the refined label map with $\\textcolor{green}{\\Box}$.\nFor example, the top-most error pixel (class road) in the coarse label map is associated with a direction $\\rightarrow$.\nWe use the label (class car) of the updated position based on offset ($1,0$)\nas the refined label.\nOnly one-step shifting based on the offset\nmap already refines several boundary errors.\n}}\n\\label{fig:approach:shift-improve}\n\\vspace{-8mm}\n\\end{figure*}\n}\n\n\n\\vspace{-3mm}\n\\subsection{Network Architecture}\n\\noindent\\textbf{Backbone.}\nWe adopt the recently proposed\nhigh resolution network (HRNet)~\\cite{sun2019high} \nas backbone,\ndue to its strengths at maintaining high resolution feature maps\nand \nour need to apply full-resolution boundary maps and direction maps \nto refine full-resolution coarse label maps.\nTo further increase the resolution of the output feature map,\nwe modify the original HRNet through\nadding an extra branch that maintains higher output resolution, i.e., $2\\times$,\ncalled HRNet-${2\\times}$.\nWe directly perform the boundary branch and the direction branch\non the output feature map with the highest resolution.\nThe resolution is $\\frac{H}{s}\\times \\frac{W}{s}\\times D$,\nwhere $s=4$ for HRNet and $s=2$ for HRNet-${2\\times}$.\nWe empirically verify that our approach consistently \nimproves the coarse segmentation results\nfor all variations of our backbone choices in Section~\\ref{sec:ablation},\ne.g., HRNet-W$18$ and HRNet-W$32$.\n\n\n\\noindent\\textbf{Boundary branch\/loss.}\nWe implement the boundary branch as\n$1\\times 1~\\operatorname{Conv} \\rightarrow \\operatorname{BN} \\rightarrow \\operatorname{ReLU}$ with $256$ output channels.\nWe then apply a linear classifier \n($1\\times 1~\\operatorname{Conv}$) and up-sample \nthe prediction to generate the final\nboundary map $\\mathbf{B}$ of size $H\\times W\\times 1$.\nEach element of $\\mathbf{B}$ records\nthe probability of the pixel belonging to the boundary.\nWe use binary cross-entropy loss as the boundary loss.\n\n\n\\noindent\\textbf{Direction branch\/loss.}\nDifferent from the previous approaches~\\cite{acuna2019devil,bai2017deep}\nthat perform regression on continuous directions in $[0^{\\circ}, 360^{\\circ})$ as \nthe ground-truth,\nour approach directly predicts discrete directions\nby evenly dividing the entire direction range\nto $m$ partitions (or categories) as our ground-truth ($m=8$ by default).\nIn fact, we empirically find that our discrete categorization scheme\noutperforms the regression scheme, e.g., mean squared loss in the angular domain~\\cite{bai2017deep}, measured by the final segmentation performance improvements.\nWe illustrate more details for the discrete direction map in Section~\\ref{sec:gtdir}.\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{0.45\\textwidth}\n\\vspace{-2mm}\n{\\includegraphics[width=\\textwidth]{images\/approach\/angle2offset.pdf}}\n\\caption{Illustrating the directions $\\to$ offsets.}\n\\end{subfigure}\n\\hspace{3mm}\n\\begin{subfigure}[b]{0.47\\textwidth}\n{\\includegraphics[width=\\textwidth]{images\/approach\/approach_gt.pdf}}\n\\caption{Ground-truth generation procedure.}\n\\end{subfigure} \\\\\n\\vspace{-3mm}\n\\caption{\n\\small{\n(a) We divide the entire direction value range\n$[0^{\\circ}, 360^{\\circ})$\nto $m$ partitions or categories (marked with different colors), \nFor example, when $m=4$, \nwe have $[0^{\\circ}, 90^{\\circ})$, $[90^{\\circ}, 180^{\\circ})$, $[180^{\\circ}, 270^{\\circ})$ and $[270^{\\circ}, 360^{\\circ})$\ncorrespond to $4$ different categories separately. \nThe above $4$ direction categories\ncorrespond to offsets $(1,1)$, $(-1,1)$, $(-1,-1)$\nand $(1,-1)$ respectively. \nThe situation for $m=8$ is similar.\n(b) Binary maps $\\to$ Distance maps $\\to$ Direction maps.\nThe ground-truth binary maps are of category car, road and side-walk.\nWe first apply distance transform on each binary map to compute the\nground-truth distance maps.\nThen we use Sobel filter on the distance maps to compute the \nground-truth direction maps.\nWe choose different colors to represent different distance values \nor the direction values.\n}\n}\n\\label{fig:approach:angle2offset_gtdirection}\n\\vspace{-7mm}\n\\end{figure*}\n\nWe implement the direction branch as\n$1\\times 1~\\operatorname{Conv} \\rightarrow \\operatorname{BN} \\rightarrow \\operatorname{ReLU}$ with $256$ output channels. \nWe further apply a linear classifier ($1\\times 1~\\operatorname{Conv}$)\nand up-sample the classifier prediction to generate the final direction map $\\mathbf{D}$\nof size $H\\times W\\times m$.\nWe mask the direction map $\\mathbf{D}$ by multiplying\nby the (binarized) boundary map $\\mathbf{B}$ to ensure that we only apply direction \nloss on the pixels identified as boundary by the boundary branch.\nWe use the standard category-wise cross-entropy loss\nto supervise the discrete directions in this branch.\n\n\n\\noindent\\textbf{Offset branch.}\nThe offset branch is used to convert the predicted direction\nmap $\\mathbf{D}$\nto the offset map $\\Delta{\\mathbf{Q}}$ of size $H\\times W\\times 2$.\nWe illustrate the mapping mechanism in Figure~\\ref{fig:approach:angle2offset_gtdirection} (a).\n\\textcolor{black}\n{\nFor example, the ``upright'' direction category (corresponds to the\nvalue within range $[0^{\\circ}, 90^{\\circ})$)\nwill be mapped to offset $(1, 1)$ when $m=4$.\n}\nLast, we generate the refined label map\nthrough shifting the coarse label map \nwith the grid-sample scheme~\\cite{jaderberg2015spatial}.\nThe process is shown\nin Figure~\\ref{fig:approach:shift-improve}.\n\n\\vspace{-3mm}\n\\subsection{Ground-truth generation and analysis}\n\\label{sec:gtdir}\n\nThere may exist many different mechanisms to\ngenerate ground-truth for the boundary maps\nand the direction maps.\nIn this work, we mainly exploit the \nconventional distance transform~\\cite{kimmel1996sub}\nto generate ground-truth\nfor both semantic segmentation task and\nthe instance segmentation task.\n\nWe start from the ground-truth segmentation label\nto generate the ground-truth distance map, followed by\nboundary map and direction map.\nFigure~\\ref{fig:approach:angle2offset_gtdirection} (b) illustrates the overall procedure.\n\n\n\\noindent\\textbf{Distance map.}\nFor each pixel,\nour distance map records its \nminimum (Euclidean) distance to the pixels belonging to other object category.\nWe illustrate how to compute the distance map as below.\n\nFirst, we decompose the ground-truth label into \n$K$ binary maps associated with different semantic categories,\ne.g., car, road, sidewalk.\nThe $k^{th}$ binary map records the pixels belonging \nto the $k^{th}$ semantic category as $1$ and $0$ otherwise.\nSecond, we perform distance transform~\\cite{kimmel1996sub}\n\\footnote{\nWe use \\texttt{scipy.ndimage.morphology.distance$\\_$transform$\\_$edt}\nin implementation.\n} \non each binary map independently to compute the distance map.\nThe element of $k^{th}$ distance map encodes the\ndistance from a pixel belonging to $k^{th}$ category\nto the nearest pixel belonging to other categories.\nSuch distance can be treated as the distance to the object boundary. \nWe compute a fused distance map through \naggregating all the $K$ distance maps.\n\nNote that the values in our distance map are (always positive)\ndifferent from the conventional signed distances that represent \nthe interior\/exterior pixels\nwith positive\/negative distances separately.\n\n\\noindent\\textbf{Boundary map.}\nAs the fused distance map represents the distances to the object boundary,\nwe can construct the ground-truth boundary map\nthrough setting all the pixels with distance value smaller than a threshold\n$\\gamma$ as boundary\\footnote{\nWe define the boundary pixels and interior pixels\nbased on their distance values.\n}.\nWe empirically choose small $\\gamma$ value, e.g., $\\gamma=5$,\nas we are mainly focused on the thin boundary refinement.\n\n\\noindent\\textbf{Direction map.} \nWe perform the Sobel filter (with kernel size $9\\times 9$) \non the $K$ distance maps independently to compute the \ncorresponding $K$ direction maps respectively.\nThe Sobel filter based direction is in \nthe range $[0^{\\circ}, 360^{\\circ})$,\nand each direction points to the \ninterior pixel (within the neighborhood) \nthat is furthest away from the object boundary.\nWe divide the entire direction range to $m$ categories (or partitions) and \nthen assign the direction of each pixel to the corresponding category.\nWe illustrate two kinds of partitions \nin Figure~\\ref{fig:approach:angle2offset_gtdirection} (a)\nand we choose the $m=8$ partition by default.\nWe apply the evenly divided direction map as our ground-truth for training.\nBesides, we also visualize some examples of direction map in Figure~\\ref{fig:approach:angle2offset_gtdirection} (b).\n\n\\noindent\\textbf{Empirical Analysis.}\nWe apply the generated ground-truth\non the segmentation results of three state-of-the-art methods including DeepLabv3~\\cite{chen2017rethinking}, HRNet~\\cite{sun2019high} and Gated-SCNN~\\cite{takikawa2019gated} \nto investigate the potential of our approach.\nSpecifically, \nwe first project the ground-truth direction map to offset map\nand then refine the segmentation results on Cityscapes \\texttt{val} based on our generated ground-truth offset map.\nTable~\\ref{table:ideal_segfix} summarizes the related results. \nWe can see that our approach significantly improves both the overall\nmIoU and the boundary F-score.\nFor example, our approach ($m=8$) improves the mIoU of Gated-SCNN by $3.1\\%$.\nWe may achieve higher performance through re-scaling \nthe offsets for different pixels adaptively,\nwhich is not the focus of this work.\n\n\\noindent\\textbf{Discussion.}\nThe key condition for ensuring the effectiveness of our approach\nis that \\emph{segmentation predictions of the interior pixels are\nmore reliable empirically.}\nGiven accurate boundary maps and direction maps,\nwe could always improve the segmentation \nperformance in expectation.\nIn other words, the segmentation performance ceiling \nof our approach is also determined by the interior pixels' prediction accuracy.\n\n\n\\begin{table}[t]\\centering\n\\footnotesize\n\\resizebox{0.75\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|c|c|cc@{}}\n\\shline\n\\multirow{2}{*}{metric} & \\multirow{2}{*}{method} & \\multirow{2}{*}{w\/o SegFix} &\\multicolumn{2}{c}{w\/ SegFix} \\\\\n& & & $m=4$ & $m=8$   \\\\\n\\shline\n\\multirow{3}{*}{mIoU} \n& DeepLabv3 (Our impl.)   & $79.5$ & $82.6$ (+$3.1$) & $82.4$ (+$2.9$) \\\\\n& HRNet-W$48$ (Our impl.) & $81.1$ & $84.1$ (+$3.0$) & $84.1$ (+$3.0$)  \\\\\n& Gated-SCNN  (Our impl.) & $81.0$ & $84.2$ (+$3.2$) & $84.1$ (+$3.1$)  \\\\\n\\hline\n\\multirow{3}{*}{F-score} \n& DeepLabv3 (Our impl.)   & $56.6$ & $68.6$ (+$12.0$) & $68.4$ (+$11.8$) \\\\\n& HRNet-W$48$ (Our impl.) & $62.4$ & $73.8$ (+$11.4$) & $73.8$ (+$11.4$) \\\\\n& Gated-SCNN (Our impl.)  & $61.4$ & $72.3$ (+$10.9$) & $72.3$ (+$10.9$) \\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\textbf{\nImprovements with ground-truth boundary offset on Cityscapes} \\texttt{val}.\nWe report both the segmentation performance mIoU and the boundary performance F-score (1px width).\n}\n\\label{table:ideal_segfix}}\n\\vspace{-9mm}\n\\end{table}\n\n\\vspace{-2mm}\n\\section{Experiments: Semantic Segmentation}\n\\vspace{-2mm}\n\n\\subsection{Datasets \\& Implementation Details}\n\n\\noindent\\textbf{Cityscapes}~\\cite{cordts2016cityscapes}\nis a real-world dataset that consists of $2,975$\/$500$\/$1,525$ images with resolution $2048\\times 1024$ for training\/validation\/testing respectively. \nThe dataset contains $19$\/$8$ semantic categories for semantic\/instance segmentation task.\n\n\\noindent\\textbf{ADE20K}~\\cite{zhou2017scene}\nis a very challenging benchmark \nconsisting of around $20,000$\/$2,000$ images for training\/validation respectively.\nThe dataset contains $150$ fine-grained semantic categories.\n\n\\noindent\\textbf{GTA5}~\\cite{richter2016playing}\nis a synthetic \ndataset that consists of $12,402$\/$6,347$\/$6,155$ images with\nresolution $1914\\times 1052$ for training\/validation\/testing respectively.\nThe dataset contains $19$ semantic categories which are compatible with Cityscapes.\n\n\\noindent\\textbf{Implementation details}.\nWe perform the same training and testing settings on Cityscapes and GTA5 \nbenchmarks as follow.\nWe set the initial learning rate as $0.04$, weight decay as $0.0005$, crop size as $512\\times 512$ and batch size as $16$, and train for $80$K iterations.\nFor the ADE20K benchmark, we set the initial learning as $0.02$ and \nall the other settings are kept the same as on Cityscapes.\nWe use ``poly'' learning rate policy with power $=0.9$.\nFor data augmentation, we all apply random flipping horizontally, random cropping and random brightness jittering within the range of $[-10, 10]$.\nBesides, we all apply syncBN~\\cite{inplaceabn} across multiple GPUs to\nstabilize the training.\nWe simply set the loss weight as $1.0$ for both the boundary loss and direction loss\nwithout tuning.\n\n\n\nNotably, our approach does not require extra training or fine-tuning any semantic segmentation models.\nWe only need to predict the boundary mask and the direction map \nfor all the test images\n\\emph{in advance} and refine the segmentation results of any existing approaches accordingly.\n\n\\noindent\\textbf{Evaluation metrics}.\nWe use two different metrics including:\n\\emph{mask F-score} and top-$1$ \\emph{direction accuracy}\nto evaluate the performance of our approach during the training stage.\nMask F-score is performed on \nthe predicted binary boundary map and \ndirection accuracy is performed on\nthe predicted direction map.\nEspecially, we only measure the direction accuracy within the regions identified as boundary by the boundary branch.\n\nTo verify the effectiveness of our approach for semantic segmentation,\nwe follow the recent Gated-SCNN~\\cite{takikawa2019gated} and \nperform two quantitative measures including:\n\\emph{class-wise mIoU} to measure the overall segmentation performance on regions;\n\\emph{boundary F-score} to measure the boundary quality of predicted mask with a small slack in distance.\nIn our experiments,\nwe measure the boundary F-score using thresholds $0.0003$, $0.0006$ and $0.0009$ corresponding to $1$, $2$ and $3$ pixels respectively.\nWe mainly report the performance with threshold as $0.0003$ for most\nof our ablation experiments.\n\n\\subsection{Ablation Experiments}\n\\label{sec:ablation}\nWe conduct a group of ablations to analyze the influence of various factors within SegFix.\nWe report the improvements over the segmentation baseline DeepLabv3 (mIoU\/F-score\nis $79.5\\%$\/$56.6\\%$) if not specified.\n\n\n\\vspace{.1cm}\n\\noindent\\textbf{Backbone.}\nWe study the performance of our SegFix based on\nthree different backbones with increasing complexities, i.e.,\nHRNet-W$18$, HRNet-W$32$ and HRNet-${2\\times}$.\nWe apply the same training\/testing settings for all three backbones.\nAccording to the comparisons in Table~\\ref{table:segfix_backbone},\nour SegFix consistently improves both the segmentation performance \nand the boundary quality with different backbone choices.\nWe choose HRNet-${2\\times}$ in the following experiments \nif not specified as it performs best.\nBesides, we also report their running time in Table~\\ref{table:segfix_backbone}.\n\n\n\n\\begin{table}[bpt]\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{1\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|ccccccc@{}}\n\\shline\nbackbone         & $\\#$param (M) &  runtime (ms) & {mask F-score} & {direction accuracy}  & mIoU$\\triangle$ & F-score$\\triangle$ \\\\\\shline\nHRNet-W18              & 9.6    & 16 & 71.44 & 64.44 & +0.8 & +3.7 \\\\\nHRNet-W32              & 29.4   & 20 & 72.24 & 65.10 & +0.9 & +3.9 \\\\\nHRNet-${2\\times}$           & 47.3   & 69 & 73.67 & 66.87 & +1.0 & +4.4 \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Influence of backbones.}\nThe runtime is tested with an input image of resolution\n$2048\\times1024$ on a single V100 GPU (PyTorch1.4 + TensorRT).\nSegFix reaches real-time speed with light-weight backbone, e.g., HRNet-W18 or HRNet-W32.\n}\n\\label{table:segfix_backbone}}\n\\vspace{-2mm}\n\\end{minipage}\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{0.8\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|cccc|ccc@{}}\n\\shline\n& \\multicolumn{4}{c|}{boundary width} & \\multicolumn{3}{c}{\\# directions} \\\\\n& $\\gamma=3$ & $\\gamma=5$ & $\\gamma=10$ & $\\gamma=\\infty$ & $m=4$ & $m=8$ & $m=16$\\\\\\shline\nmIoU$\\triangle$           &  +0.94  & +0.96 & +0.95 & +0.84 &  +0.97  & +0.96 & +0.96  \\\\\nF-score$\\triangle$        &  +4.1  & +4.2  & +4.1 & +3.6 &  +4.1 &  +4.2 & +4.2  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Influence of the boundary width and direction number.}\nSegFix is robust to boundary width and direction number.\nWe choose $\\gamma=5$ and $m=8$ according to their F-scores.\n}\n\\label{table:segfix_boundary_width}}\n\\end{minipage}\n\\vspace{-5mm}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}[c]{0.5\\textwidth}\n\t\\includegraphics[width=0.85\\textwidth]{images\/exp\/boundary_visualize_v2.pdf}\n\t\\end{minipage}\n\t\\hspace{-7mm}\n\t\\begin{minipage}[c]{0.4\\textwidth}\n\t\\caption{\n\t\\small{\n\t\\textbf{\n\tQualitative results of our boundary branch prediction.\n\t}\n\tThe $2$ example images are selected from Cityscapes \\texttt{val}.\n\tWe can see that their predicted boundaries are of high quality.\n\t}\n\t} \\label{fig:boundary_map}\n\t\\end{minipage}\n\t\\vspace{-4mm}\n\\end{figure}\n\n\n{\n\\vspace{-3mm}\n\\begin{figure*}\n\\centering\n\\resizebox{\\linewidth}{!}\n{\n\t\\includegraphics[width=0.95\\textwidth]{images\/exp\/direction_example.pdf}\n}\n\\vspace{-6mm}\n\\caption{\n\\small{\n\\textbf{\nQualitative results of our direction branch predictions.\n}\nThe \\nth{1} and \\nth{3} columns represent the ground-truth segmentation map.\nThe \\nth{2} and \\nth{4} columns illustrate the predicted directions \nwith the segmentation map of HRNet as the background.\nWe mark the directions that fix errors with \\textcolor{blue}{blue arrow} and directions that lead to extra errors with \\textcolor{red}{red arrow}.\nOur predicted directions addresses boundary errors for various object categories such as bicycle, traffic light and traffic sign.\n(Better viewed zoom in)\n}}\n\\label{fig:direction_example}\n\\vspace{-2mm}\n\\end{figure*}\n}\n\n\n\\vspace{1mm}\n\\noindent\\textbf{Boundary branch.}\nWe verify that SegFix is robust to\nthe choice of hyper-parameter $\\gamma$ within the boundary branch \nand illustrate some qualitative results.\n\n\\noindent$\\Box$ \\textbf{boundary width}:\nTable~\\ref{table:segfix_boundary_width} shows the \nperformance improvements based on boundary with different widths. \nWe choose different $\\gamma$ values to control the boundary width, \nwhere smaller $\\gamma$ leads to thinner boundaries.\nWe also report the performance with $\\gamma=\\infty$,\nwhich means all pixels is identified as boundary.\nWe find their improvements are close and we choose $\\gamma=5$ by default.\n\n\\noindent$\\Box$ \\textbf{qualitative results}:\nFigure~\\ref{fig:boundary_map} shows \nthe qualitative results\nwith our boundary branch.\nWe find that the predicted boundaries are\nof high quality.\nBesides, we also compute the F-scores between \nthe boundary computed from the segmentation map of the\nexisting approaches, e.g., Gated-SCNN and HRNet, \nand the predicted boundary from \nour boundary branch.\nThe F-scores are\naround $70\\%$, which (in some degree) means\nthat their boundary maps are well aligned and \nensures that more accurate direction\npredictions bring larger performance gains.\n\n\n\\vspace{1mm}\n\\noindent\\textbf{Direction branch.}\nWe analyze the influence of the direction number $m$ and \nthen present some qualitative results of our predicted directions.\n\n\n\n\n\\vspace{1mm}\n\\noindent$\\Box$ \\textbf{direction number}:\nWe choose different direction numbers to perform different direction partitions\nand control the generated offset maps that are used to refine the coarse label map.\nWe conduct the experiments with $m=4$, $m=8$ and $m=16$.\nAccording to the reported results on the right $3$ columns in Table~\\ref{table:segfix_boundary_width},\nwe find different direction numbers all lead to significant improvements\nand we choose $m=8$ if not specified as our SegFix is \nless sensitive to the choice of $m$.\n\n\\vspace{1mm}\n\\noindent$\\Box$ \\textbf{qualitative results}:\nIn Figure~\\ref{fig:direction_example},\nwe show some examples to illustrate that our predicted boundary directions\nimprove the errors.\nOverall, the improved pixels (marked with \\textcolor{blue}{blue arrow}) \nare mainly distributed along  the very thin boundary.\n\n\\vspace{1mm}\n\\noindent\\textbf{Comparison with GUM.}\nWe compare SegFix with the previous model-dependent guided up-sampling mechanism~\\cite{mazzini2018guided,Mazzini_2019_CVPR_Workshops} based on DeepLabv3 \nas the baseline.\nWe report the related results in Table~\\ref{table:compare_to_guide_upsample}.\nIt can be seen that our approach significantly outperforms GUM\nmeasured by both mIoU and F-score.\nWe achieve higher performance through combining GUM with our approach,\nwhich achieves $5.0\\%$ improvements on F-score compared to the baseline.\n\n\n\\begin{table}[bpt]\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{0.8\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|C{2cm}C{3cm}C{2cm}c@{}}\n\\shline\n& baseline  &  GUM (Our impl.)  &  SegFix  &  GUM+SegFix   \\\\\\shline\nmIoU            & 79.5 & 79.8 (+0.3) & 80.5 (+1.0)  &  80.6 (+1.1) \\\\\nF-score         & 56.6 & 57.7 (+1.1) & 60.9 (+4.3)  &  61.6 (+5.0)  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Comparison with GUM}~\\cite{mazzini2018guided}.\nSegFix not only outperforms GUM but also is complementary with GUM.\n}\n\\label{table:compare_to_guide_upsample}}\n\\end{minipage}\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{0.8\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|C{2cm}C{2cm}C{2cm}c@{}}\n\\shline\n& baseline & DenseCRF &  SegFix  &  DenseCRF+SegFix   \\\\\\shline\nmIoU            & 79.5 & 79.7 (+0.2) & 80.5 (+1.0) & 80.5 (+1.0) \\\\\nF-score         & 56.6 & 60.9 (+4.3) & 61.0 (+4.4) & 64.1 (+7.5) \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Comparison with DenseCRF}~\\cite{krahenbuhl2011efficient}.\nSegFix achieves comparable F-score improvements and\nmuch larger mIoU gains.\n}}\n\\label{table:compare_to_crf}\n\\end{minipage}\n\n\\begin{minipage}[t]{0.46\\linewidth}\n\t\\centering\n\t\\scriptsize\n    \\begin{tabular}{@{}l|cc@{}}\n    \\shline\n     & Gated-SCNN &  Gated-SCNN+SegFix   \\\\\\shline\n    mIoU           & 81.0 & 81.5 (+0.5)  \\\\\n    F-score        & 61.4 & 63.1 (+1.7)  \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{\n    \\small{\n    \\bd{Comparison with Gated-SCNN}~\\cite{takikawa2019gated}\n    The result of Gated-SCNN is based on multi-scale testing.\n    }\n    \\setlength{\\tabcolsep}{5pt}\n    }\n    \\label{table:compare_to_Gated-SCNN}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{0.5\\linewidth}\n\t\\centering\n\t\\scriptsize\n    \\begin{tabular}{@{}l|cc|cc@{}}\n    \\shline\n    & \\multicolumn{2}{c|}{ADE20K} & \\multicolumn{2}{c}{GTA5} \\\\\n    & baseline & +SegFix & baseline & +SegFix \\\\\n    \\shline\n    mIoU    & $44.8$ & $45.4 (+0.6)$ & $77.8$ & $80.6 (+2.8)$ \\\\\n    F-score & $16.4$ & $19.3 (+2.9)$ & $50.2$ & $61.7 (+11.5)$  \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{\n    \\small{\n    \\bd{DeepLabv3 with SegFix on ADE20K and GTA5.}\n    We all choose DeepLabv3 as the baseline.\n    }\n    \\label{table:gta5_synthia}\n    }\n\\end{minipage}\n\\vspace{-5mm}\n\\end{table}\n\n\n\\vspace{1mm}\n\\noindent\\textbf{Comparison with DenseCRF.}\nWe compare our approach with the conventional well-verified\nDenseCRF~\\cite{krahenbuhl2011efficient} based on \nthe DeepLabv3 as our baseline.\nWe fine-tune the hyper-parameters of DenseCRF and set\nthem empirically following~\\cite{chen2017deeplab}.\nAccording to Table~\\ref{table:compare_to_crf},\nour approach not only outperforms DenseCRF\nbut also is complementary with DenseCRF.\nThe possible reasons for the limited mIoU improvements \nof DenseCRF might be that it brings more extra errors\non the interior pixels.\n\n\\vspace{1mm}\n\\noindent\\textbf{Application to Gated-SCNN.}\nConsidering that Gated-SCNN~\\cite{takikawa2019gated} \nintroduced multiple components to \nimprove the performance,\nit is hard to compare our approach with Gated-SCNN fairly to a large extent.\nTo verify the effectiveness of our approach to some extent,\nwe first take the open-sourced Gated-SCNN (multi-scale testing) segmentation results\non Cityscapes validation set as the coarse segmentation maps,\nthen we apply the SegFix offset maps to refine the results.\nWe report the results in Table~\\ref{table:compare_to_Gated-SCNN}\nand SegFix improves the boundary F-score by $1.7\\%$,\nsuggesting that SegFix is complementary with the strong\nbaseline that also focuses on improving the segmentation boundary quality.\nBesides, we also report the detailed category-wise improvements measured by both mIoU and boundary F-score in Table~\\ref{table:cityscapes_val}.\n\n\n\\vspace{1mm}\n\\noindent\\textbf{Comparison with STEAL.}\nDue to the training code of STEAL~\\cite{acuna2019devil} is not open-sourced, \nwe simply apply the released checkpoints\\footnote{\n\\rm{STEAL:\\space https:\/\/github.com\/nv-tlabs\/STEAL}\n} to predict $K$ semantic boundary maps and convert them to binary boundary map. We empirically find that the boundary quality of our SegFix ($35.54\\%$) is comparable with the carefully designed STEAL ($35.86\\%$) measured by F-score along the ground-truth boundary with $1$-px width, suggesting that our method achieves nearly the state-of-the-art boundary detection performance.\nTo verify whether SegFix can benefit from the more accurate boundary maps predicted by STEAL,\nwe also train a SegFix model to only predict the direction map while using the (fixed) pre-computed boundary maps with STEAL.\nWe find the result becomes slightly worse ($80.5\\%\\to 80.32\\%$) based on the coarse results with DeepLabv3.\n\n\n{\n\\begin{figure*}\n\\centering\n\\tiny{\n\\begin{tabularx}{\\textwidth}{YYYYYY}\nImage & Ground-Truth & DeepLabV3 & DeepLabV3+SegFix & HRNet & HRNet+SegFix \\\\\n\\end{tabularx}\n}\n\\normalsize\n{\n\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/rgb.png}\n\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/gt_label.png}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/deeplabv3.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/deeplabv3_segfix.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/hrnet48.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/1\/hrnet48_segfix.png}}\n\\\\\n\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/rgb.png}\n\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/gt_label.png}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/deeplabv3.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/deeplabv3_segfix.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/hrnet48.png}}\n\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/2\/hrnet48_segfix.png}}\n}\n\\vspace{-6mm}\n\\caption{\\small{\n\\textbf{\nQualitative comparison in terms of errors on Cityscapes} \\texttt{val}.\nOur approach well addresses the existing boundary errors of various categories, e.g., car, bicycle and traffic sign, for both DeepLabv3 and HRNet.\n}}\n\\label{fig:improve_hrnet}\n\\vspace{-3mm}\n\\end{figure*}\n}\n\n\\vspace{1mm}\n\\subsection{Application to State-of-the-art}\n\nWe generate the boundary maps and the direction maps in advance and apply\nthem to the segmentation results of various state-of-the-art approaches \nwithout extra training or fine-tuning.\n\n\\vspace{1mm}\n\\noindent\\textbf{Cityscapes val}:\nWe first apply our approach on various state-of-the-art approaches (on\nCityscapes \\texttt{val}) including DeepLabv3, Gated-SCNN and HRNet.\nWe report the category-wise performance improvements in Table~\\ref{table:cityscapes_val}.\nIt can be seen that our approach significantly improves the segmentation quality along the boundaries of all the evaluated approaches.\nFigure~\\ref{fig:improve_hrnet} provides some qualitative examples of\nthe improvements with our approach along the thin boundaries based on \nboth DeepLabv3 and HRNet.\n\n\\vspace{1mm}\n\\noindent\\textbf{Cityscapes test}:\nWe further apply our approach on several recent state-of-the-art methods on Cityscapes \\texttt{test} including PSANet~\\cite{zhao2018psanet}, DANet~\\cite{fu2019dual}, BFP~\\cite{ding2019boundary}, HRNet~\\cite{sun2019high}, Gated-SCNN~\\cite{takikawa2019gated}, VPLR~\\cite{zhu2019improving}\nand HRNet + OCR~\\cite{yuan2019object}.\nWe directly apply the same model that are trained with only the $2,975$ training images without any other tricks, e.g., training with validation set or Mapillary Vistas~\\cite{neuhold2017mapillary}, online hard example mining.\n\nNotably, the state-of-the-art methods have applied various advanced techniques, e.g., multi-scale testing, multi-grid, performing boundary supervision or utilizing extra training data such as Mapillary Vistas or Cityscapes video, to improve their results.\nIn Table~\\ref{table:segfix_sota},\nour model-agnostic boundary refinement scheme consistently \nimproves all the evaluated approaches. \nFor example, with our SegFix, ''HRNet + OCR\" achieves $84.5\\%$ on Cityscapes \\texttt{test}.\nThe improvements of our SegFix is in fact already significant\nconsidering the baseline is already very strong and the performance gap between top ranking methods is just around $0.1\\% \\sim 0.3\\%$.\nWe believe that lots of other advanced approaches might also benefit from our approach. \n\n\n\\begin{table}[bpt]\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{\\linewidth}{!}\n{\n\\begin{tabular}{@{}l|l|ccccccccccccccccccc|c@{}} \n\\shline\nwidth & method & \\rotatebox{90}{road} & \\rotatebox{90}{sidewalk} & \\rotatebox{90}{building} & \\rotatebox{90}{wall} & \\rotatebox{90}{fence} &\\rotatebox{90}{pole} & \\rotatebox{90}{traffic light}&\\rotatebox{90}{traffic sign}&\\rotatebox{90}{vegetation}&\\rotatebox{90}{terrian}&\\rotatebox{90}{sky}&\\rotatebox{90}{person}&\\rotatebox{90}{rider}&\\rotatebox{90}{car}&\\rotatebox{90}{truck}&\\rotatebox{90}{bus} & \\rotatebox{90}{train} & \\rotatebox{90}{motorcycle} & \\rotatebox{90}{bicycle} & mean\\\\\\shline\n\\multirow{6}{*}{1px} & DeepLabV3 & 70.7 & 44.4 & 50.0 & 45.9 & 42.3 & 48.3 & 45.8 & 46.5 & 49.5 & 45.4 & 60.5 & 43.0 & 55.9 & 56.9 & 76.6 & 84.5 & 92.3 & 70.5 & 45.9 & 56.6 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{73.9} & \\textbf{49.1} & \\textbf{55.5} & \\textbf{47.8} & \\textbf{43.7} & \\textbf{57.6} & \\textbf{52.7} & \\textbf{58.3} & \\textbf{54.7} & \\textbf{47.4} & \\textbf{64.7} & \\textbf{50.2} & \\textbf{59.7} & \\textbf{64.6} & \\textbf{77.4} & \\textbf{86.0} & \\textbf{92.6} & \\textbf{72.0} & \\textbf{51.5} & \\textbf{61.0} \\\\\n\\cline{2-22}\n& HRNet-W48 & 73.1 & 48.9 & 55.4 & 49.2 & 49.0 & 58.9 & 59.0 & 55.5 & 54.0 & 51.0 & 65.1 & 52.0 & 62.0 & 63.4 & 79.0 & 87.5 & 95.0 & 77.4 & 51.0 & 62.4 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{74.8} & \\textbf{51.9} & \\textbf{58.2} & \\textbf{50.9} & \\textbf{49.7} & \\textbf{63.6} & \\textbf{64.0} & \\textbf{61.6} & \\textbf{57.1} & \\textbf{52.5} & \\textbf{66.8} & \\textbf{56.8} & \\textbf{64.4} & \\textbf{67.5} & \\textbf{79.7} & \\textbf{88.7} & \\textbf{95.2} & \\textbf{77.7} & \\textbf{55.0} & \\textbf{65.1} \\\\\n\\cline{2-22}\n& Gated-SCNN & 73.5 & 49.8 & 55.5 & 46.7 & 43.0 & 59.9 & 61.8 & 57.4 & 54.4 & 45.7 & 65.9 & 51.4 & 61.9 & 64.0 & 72.5 & 84.8 & 92.4 & 71.9 & 53.6 & 61.4 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{74.2} & \\textbf{51.3} & \\textbf{57.7} & \\textbf{47.2} & \\textbf{45.3} & \\textbf{64.0} & \\textbf{63.8} & \\textbf{61.2} & \\textbf{56.7} & \\textbf{46.9} & \\textbf{66.6} & \\textbf{55.6} & \\textbf{64.0} & \\textbf{66.9} & 72.0 & \\textbf{85.0} & \\textbf{92.6} & 71.8 & \\textbf{55.9} & \\textbf{63.1} \\\\\n\\hline\n\\multirow{6}{*}{2px} & DeepLabV3 & 79.1 & 57.5 & 62.2 & 49.3 & 45.5 & 64.1 & 54.5 & 61.3 & 62.6 & 49.8 & 72.2 & 54.8 & 62.4 & 71.6 & 78.0 & 86.5 & 92.7 & 72.3 & 54.7 & 64.8 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{81.2} & \\textbf{60.9} & \\textbf{66.3} & \\textbf{51.1} & \\textbf{46.6} & \\textbf{69.6} & \\textbf{59.7} & \\textbf{69.3} & \\textbf{66.6} & \\textbf{51.6} & \\textbf{75.0} & \\textbf{60.4} & \\textbf{65.6} & \\textbf{76.6} & \\textbf{78.8} & \\textbf{87.7} & \\textbf{93.0} & \\textbf{73.5} & \\textbf{59.5} & \\textbf{68.1} \\\\\n\\cline{2-22}\n& HRNet-W48 & 81.1 & 61.7 & 67.4 & 52.5 & 52.5 & 73.2 & 67.7 & 69.4 & 66.9 & 55.4 & 76.3 & 63.7 & 68.2 & 77.3 & 80.4 & 89.6 & 95.5 & 79.1 & 60.3 & 70.4 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{82.1} & \\textbf{63.7} & \\textbf{69.1} & \\textbf{54.0} & \\textbf{52.8} & \\textbf{75.2} & \\textbf{71.1} & \\textbf{72.5} & \\textbf{69.1} & \\textbf{56.7} & \\textbf{77.2} & \\textbf{66.9} & \\textbf{70.4} & \\textbf{79.5} & \\textbf{80.9} & \\textbf{90.3} & \\textbf{95.6} & 79.1 & \\textbf{63.3} & \\textbf{72.1} \\\\\n\\cline{2-22}\n& Gated-SCNN & 80.9 & 61.9 & 67.1 & 50.0 & 46.4 & 73.9 & 70.3 & 70.1 & 67.1 & 50.0 & 76.7 & 62.8 & 68.5 & 77.3 & 74.0 & 86.8 & 92.9 & 73.8 & 62.5 & 69.1 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{81.5} & \\textbf{63.0} & \\textbf{68.6} & \\textbf{50.5} & \\textbf{48.5} & \\textbf{75.9} & \\textbf{71.1} & \\textbf{72.1} & \\textbf{68.6} & \\textbf{51.2} & \\textbf{77.1} & \\textbf{65.9} & \\textbf{70.3} & \\textbf{78.9} & 73.4 & 86.7 & \\textbf{93.0} & 73.6 & \\textbf{64.6} & \\textbf{70.2} \\\\\n\\hline\n\\multirow{6}{*}{3px} & DeepLabV3 &84.1 & 65.8 & 70.7 & 52.0 & 47.9 & 72.5 & 60.8 & 70.2 & 72.2 & 53.2 & 79.9 & 62.9 & 67.3 & 79.8 & 79.0 & 87.8 & 93.0 & 73.7 & 61.6 & 70.2 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{85.2} & \\textbf{67.8} & \\textbf{73.0} & \\textbf{53.3} & \\textbf{48.6} & \\textbf{74.8} & \\textbf{64.0} & \\textbf{74.5} & \\textbf{74.6} & \\textbf{54.5} & \\textbf{81.4} & \\textbf{66.1} & \\textbf{69.5} & \\textbf{82.2} & \\textbf{79.5} & \\textbf{88.6} & \\textbf{93.3} & \\textbf{74.6} & \\textbf{65.0} & \\textbf{72.1} \\\\\n\\cline{2-22}\n& HRNet-W48 & 85.5 & 69.1 & 74.7 & 54.9 & 54.9 & 79.0 & 72.9 & 75.6 & 75.5 & 58.6 & 83.0 & 70.4 & 72.6 & 84.3 & 81.3 & 90.8 & 95.7 & 80.3 & 66.9 & 75.1 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{86.0} & \\textbf{70.3} & \\textbf{75.4} & \\textbf{55.8} & 54.8 & \\textbf{79.5} & \\textbf{74.9} & \\textbf{77.0} & \\textbf{76.8} & \\textbf{59.5} & \\textbf{83.3} & \\textbf{72.0} & \\textbf{74.0} & \\textbf{84.9} & \\textbf{81.6} & \\textbf{91.2} & \\textbf{95.8} & 80.1 & \\textbf{68.6} & \\textbf{75.9} \\\\\n\\cline{2-22}\n& Gated-SCNN & 85.0 & 68.8 & 74.2 & 52.2 & 48.7 & 79.7 & 75.0 & 75.9 & 75.4 & 53.0 & 83.1 & 69.3 & 73.1 & 83.6 & 74.9 & 87.8 & 93.2 & 75.2 & 68.8 & 73.5 \\\\\n& \\textcolor{blue}{+ SegFix} & \\textbf{85.3} & \\textbf{69.6} & \\textbf{74.9} & \\textbf{52.5} & \\textbf{50.6} & \\textbf{80.3} & 75.0 & \\textbf{76.7} & \\textbf{76.3} & \\textbf{54.0} & \\textbf{83.3} & \\textbf{71.1} & \\textbf{74.2} & \\textbf{84.2} & 74.1 & 87.6 & 93.2 & 74.9 & \\textbf{70.0} & \\textbf{74.1} \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n{\n\\textbf{Boundary F-score with SegFix.}\nWe illustrate the category-wise comparison with various baselines in terms of boundary F-score on Cityscapes \\texttt{val}.\n}\n}\n}\n\\label{table:cityscapes_val}\n\\end{minipage}\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|ccccccccccccccccccc|c@{}} \n\\shline\nmethod & \\rotatebox{90}{road} & \\rotatebox{90}{sidewalk} & \\rotatebox{90}{building} & \\rotatebox{90}{wall} & \\rotatebox{90}{fence} &\\rotatebox{90}{pole} & \\rotatebox{90}{traffic light}&\\rotatebox{90}{traffic sign}&\\rotatebox{90}{vegetation}&\\rotatebox{90}{terrian}&\\rotatebox{90}{sky}&\\rotatebox{90}{person}&\\rotatebox{90}{rider}&\\rotatebox{90}{car}&\\rotatebox{90}{truck}&\\rotatebox{90}{bus} & \\rotatebox{90}{train} & \\rotatebox{90}{motorcycle} & \\rotatebox{90}{bicycle} & mean\\\\ \\shline\nPSANet & 98.7 & 87.0 & 93.5 & 58.9 & 62.5 & 67.8 & 76.0 & 80.0 & 93.7 & 72.6 & 95.4 & 86.9 & 73.0 & 96.2 & 79.3 & 91.2 & 84.9 & 71.1 & 77.9 & 81.4  \\\\\n\\textcolor{blue}{+ SegFix} & 98.7 & \\textbf{87.4} & \\textbf{93.7} & \\textbf{59.3} & \\textbf{62.8} & \\textbf{69.5} & \\textbf{77.6} & \\textbf{81.4} & \\textbf{93.9} & \\textbf{73.0} & \\textbf{95.6} & \\textbf{88.0} & \\textbf{73.9} & \\textbf{96.5} & \\textbf{79.6} & \\textbf{91.5} & \\textbf{85.1} & \\textbf{71.8} & \\textbf{78.6} & \\textbf{82.0} \\\\ \n\\hline\nDANet & 98.6 & 86.1 & 93.5 & 56.1 & 63.3 & 69.7 & 77.3 & 81.3 & 93.9 & 72.9 & 95.7 & 87.3 & 72.9 & 96.2 & 76.8 & 89.4 & 86.5 & 72.2 & 78.2 & 81.5 \\\\\n\\textcolor{blue}{+ SegFix} & \\textbf{98.7} & \\textbf{86.6} & \\textbf{93.7} & \\textbf{56.5} & \\textbf{63.5} & \\textbf{71.4} & \\textbf{78.7} & \\textbf{82.4} & \\textbf{94.1} & \\textbf{73.2} & \\textbf{95.9} & \\textbf{88.2} & \\textbf{73.7} & \\textbf{96.5} & \\textbf{77.0} & \\textbf{89.7} & \\textbf{86.8} & \\textbf{72.8} & \\textbf{78.8} & \\textbf{82.0} \\\\ \n\\hline\nBFP & 98.7 & 87.0 & 93.5 & 59.8 & 63.4 & 68.9 & 76.8 & 80.9 & 93.7 & 72.8 & 95.5 & 87.0 & 72.1 & 96.0 & 77.6 & 89.0 & 86.9 & 69.2 & 77.6 & 81.4 \\\\\n\\textcolor{blue}{+ SegFix} & 98.7 & \\textbf{87.5} & \\textbf{93.7} & \\textbf{60.2} & \\textbf{63.7} & \\textbf{71.1} & \\textbf{78.4} & \\textbf{82.4} & \\textbf{94.0} & \\textbf{73.2} & \\textbf{95.7} & \\textbf{88.1} & \\textbf{72.9} & \\textbf{96.3} & \\textbf{77.8} & \\textbf{89.3} & \\textbf{87.2} & \\textbf{69.9} & \\textbf{78.4} & \\textbf{82.0} \\\\\n\\hline\nHRNet & 98.8 & 87.5 & 93.7 & 55.6 & 62.3 & 71.8 & 79.3 & 81.8 & 94.0 & 73.1 & 95.8 & 88.5 & 76.1 & 96.5 & 72.2 & 86.5 & 84.7 & 73.8 & 79.4 & 81.8 \\\\\n\\textcolor{blue}{+ SegFix} & 98.8 & \\textbf{87.9} & \\textbf{93.9} & \\textbf{56.0} & \\textbf{62.5} & \\textbf{73.6} & \\textbf{80.7} & \\textbf{83.2} & \\textbf{94.1} & \\textbf{73.4} & \\textbf{95.9} & \\textbf{89.3} & \\textbf{76.7} & \\textbf{96.6} & \\textbf{72.4} & \\textbf{86.7} & \\textbf{85.0} & \\textbf{74.3} & \\textbf{80.2} & \\textbf{82.2} \\\\ \n\\hline\nVPLR & 98.8 & 87.8 & 94.2 & 64.1 & 65.0 & 72.4 & 79.0 & 82.8 & 94.2 & 74.0 & 96.1 & 88.2 & 75.4 & 96.5 & 78.8 & 94.0 & 91.6 & 73.8 & 79.0 & 83.5 \\\\\n\\textcolor{blue}{+ SegFix} & 98.8 & \\textbf{88.0} & \\textbf{94.3} & \\textbf{64.4} & \\textbf{65.3} & \\textbf{73.3} & \\textbf{80.0} & \\textbf{83.5} & \\textbf{94.3} & \\textbf{74.3} & \\textbf{96.2} & \\textbf{89.0} & \\textbf{76.2} & \\textbf{96.7} & \\textbf{79.0} & \\textbf{94.2} & \\textbf{92.0} & \\textbf{74.4} & \\textbf{79.7} & \\textbf{83.9} \\\\\n\\hline\nHRNet + OCR & 98.9 & 88.3 & 94.3 & 66.8 & 66.6 & 73.6 & 80.3 & 83.7 & 94.3 & 74.4 & 96.0 & 88.7 & 75.4 & 96.6 & 82.5 & 94.0 & 90.8 & 73.8 & 79.7 & 84.2 \\\\\n\\textcolor{blue}{+ SegFix} & 98.9 & 88.3 & \\textbf{94.4} & \\textbf{68.0} & \\textbf{67.8} & \\textbf{73.6} & \\textbf{80.6} & \\textbf{83.9} & \\textbf{94.4} & \\textbf{74.5} & \\textbf{96.1} & \\textbf{89.2} & \\textbf{75.9} & \\textbf{96.8} & \\textbf{83.6} & \\textbf{94.2} & \\textbf{91.3} & \\textbf{74.0} & \\textbf{80.1} & \\textbf{84.5} \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\textbf{Segmentation mIoU with SegFix:}\n{Category-wise improvements of SegFix based on various state-of-the-art methods on Cityscapes \\texttt{test}.\nNotably, ``HRNet + OCR + SegFix\" ranks the first place on the Cityscapes\nsemantic segmentation leaderboard by the ECCV 2020 submission deadline.\n}\n}\n\\label{table:segfix_sota}}\n\\end{minipage}\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\resizebox{0.7\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|cccccccc|l@{}} \n\\shline\nmethod & \\rotatebox{90}{person}&\\rotatebox{90}{rider}&\\rotatebox{90}{car}&\\rotatebox{90}{{truck}}&\\rotatebox{90}{bus} & \\rotatebox{90}{train} & \\rotatebox{90}{motorcycle} & \\rotatebox{90}{bicycle} & mean ($\\%$) \\\\\\shline\nMask-RCNN  & 36.0 & 28.8 & 51.6 & 30.0 & 38.7 & 27.3 & 23.9 & 19.4 & 32.0  \\\\\n\\textcolor{blue}{+ SegFix} & \\bf{37.9} & \\bf{30.3} & \\bf{54.1} & \\bf{31.0} & \\bf{40.0} & \\bf{27.9} & \\bf{25.1} & \\bf{20.5} & \\bf{33.3} (+1.3)  \\\\\n\\hline\nPointRend & 36.6 & 29.7 & 53.7 & 29.9 & 40.4 & 33.3 & 23.6 & 19.6 & 33.3 \\\\\n\\textcolor{blue}{+ SegFix} & \\bf{38.7} & \\bf{31.1} & \\bf{56.2} & \\bf{31.1} & \\bf{41.6} & \\bf{34.1} & \\bf{24.6} & \\bf{20.7} & \\bf{34.8} (+1.5)\\\\ \\hline\nPANet  & 41.5 & 33.6 & 58.2 & 31.8 & 45.3 & 28.7 & 28.2 & 24.1 & 36.4 \\\\\n\\textcolor{blue}{+ SegFix} & \\bf{43.3} & \\bf{34.9} & \\bf{60.4} & \\bf{32.9} & \\bf{47.0} & \\bf{30.1} & \\bf{29.1} & \\bf{24.7} & \\bf{37.8} (+1.4)\\\\ \\hline\nPolyTransform  & 42.4 & 34.8 & 58.5 & 39.8 & 50.0 & 41.3 & 30.9 & 23.4 & 40.1 \\\\\n\\textcolor{blue}{+ SegFix} & \\bf{44.3} & \\bf{35.9} & \\bf{60.5} & \\bf{40.5} & \\bf{51.2} & \\bf{41.6} & \\bf{31.7} & \\bf{24.1} & \\bf{41.2} (+1.1)\\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Results on Cityscapes Instance Segmentation task.}\nOur SegFix significantly improves the mask AP\nof Mask-RCNN~\\cite{he2017mask}, PointRend~\\cite{kirillov2019pointrend}, PANet~\\cite{liu2018path}\nand PolyTransform~\\cite{liang2019polytransform} on Cityscapes \\texttt{test} (w\/ COCO pre-training).\nNotably, ``PolyTransform + SegFix\" ranks the second place on the Cityscapes\ninstance segmentation leaderboard by the ECCV 2020 submission deadline..\n}\n\\label{table:segfix-instance-seg}}\n\\end{minipage}\n\\end{table}\n\n\n\\begin{table*}[htb]\n\\begin{minipage}[t]{0.9\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{1\\linewidth}{!}\n{\n\\tablestyle{5pt}{1.0}\n\\begin{tabular}{@{}l|C{2cm}C{2cm}C{3cm}c@{}}\n\\shline\n& DeepLabv3 & HRNet-W18 &  DeepLabv3+SegFix  &  DeepLabv3+HRNet-W18  \\\\\\shline\nmIoU            & 79.5 & 79.4  & 80.3 (+0.8) & 79.9 (+0.5) \\\\\nF-score         & 56.6 & 57.0  & 60.3 (+3.7) & 58.2 (+1.6) \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\n\\small{\n\\bd{Comparison with model ensemble.}\n``DeepLabv3+HRNet-W18\" reports the results based on model ensemble\nand ``DeepLabv3+SegFix\" reports the results based on our SegFix.\nOur SegFix outperforms the model ensemble on both mIoU and F-score metrics.\nWe report the improvements compared to the performance with DeepLabv3.\n}}\n\\label{table:compare_to_ensemble}\n\\end{minipage}\n\\vspace{-9mm}\n\\end{table*}\n\n\\vspace{-3mm}\n\\subsection{Experiments on ADE20K \\& GTA5}\n\\vspace{-1.5mm}\nWe evaluate our SegFix scheme on two other challenging semantic segmentation \nbenchmarks including ADE20K and GTA5.\nWe choose DeepLabv3\nas our baseline on both datasets.\nAs illustrated in Table~\\ref{table:gta5_synthia},\nour approach also achieves significant performance\nimprovements along the boundary on both benchmarks,\ne.g., the boundary F-score of DeepLabv3 gains $2.9\\%$\/$11.5\\%$\non ADE20K \\texttt{val}\/GTA5 \\texttt{test} separately.\n\n\\vspace{-3mm}\n\\subsection{Unified SegFix Model}\n\\vspace{-1.5mm}\n\\label{unify_segix}\nWe propose to train a single unified SegFix model on Cityscapes and ADE20K,\nand we  report the improvements over DeepLabv3 as below:\nwith a single unified SegFix model,\nthe performance gains are $0.9\\%$\/$3.8\\%$ on Cityscapes\nand $0.5\\%$\/$2.7\\%$ on ADE20K measured by mIoU\/F-score.\nWe can see these improvements are comparable with \nthe SegFix trained on each dataset independently.\nMore experimental details are illustrated in the Appendix.\n\nIn general, we only need to train a single unified SegFix model\nto improve the boundary quality of various segmentation \nmodels across different datasets,\nthus SegFix is much more training friendly (and saves a lot of energy consumption) \ncompared to the previous \nmethods~\\cite{bertasius2016semantic,takikawa2019gated,ding2019boundary,liu2018devil,liu2017learning,ke2018adaptive} that require re-training \nthe existing segmentation models on each dataset independently.\n\n\\subsection{Comparison with Model Ensemble}\n\\label{sec:vs_ensemble}\nTo investigate whether our SegFix mainly benefits from model ensemble,\nwe conduct a group of experiments to compare our method with the standard model ensemble\n(that ensembles two segmentation models with the same compacity)\nunder fair settings and report the results in Table~\\ref{table:compare_to_ensemble}.\nSpecifically speaking, when processing a single image with resolution $1024\\times2048$,\nthe overall computation cost of DeepLabv3+SegFix\/DeepLabv3+HRNet-W18 is $2054$\/$2060$ GFLOPs separately. \nWe can see that SegFix outperforms the model ensemble,\ne.g., DeepLabv3+SegFix gains $1.9\\%$ (on F-score) over model ensemble method DeepLabv3+HRNet-W18,\nsuggesting that our SegFix is capable to fix that boundary errors\nthat the model ensemble fails to address.\nBesides, another advantage of our method lies at that we can use a single unified SegFix model\nacross multiple datasets while the model ensemble requires training multiple different \nsegmentation models on different datasets independently.\n\n\n\n\n\n\n\\section{Experiments: Instance Segmentation}\n\nIn Table~\\ref{table:segfix-instance-seg},\nwe illustrate the results of SegFix on Cityscapes instance segmentation task.\nWe can find that the SegFix consistently improves the\nmean AP scores over Mask-RCNN~\\cite{he2017mask}, PANet~\\cite{liu2018path},\nPointRend~\\cite{kirillov2019pointrend} and PolyTransform~\\cite{liang2019polytransform}.\nFor example,\nwith SegFix scheme,\nPANet gains $1.4\\%$ points on the Cityscapes \\texttt{test} set.\nWe also apply our SegFix on the very recent PointRend and PolyTransform.\nOur SegFix consistently improves the performance of \nPointRend and PolyTransform \nby $1.5\\%$ and $1.1\\%$ separately,\nwhich further verifies the effectiveness of our method.\n\nWe use the public available checkpoints from Dectectron2\\footnote{\n\\rm{Detectron2: \\space https:\/\/github.com\/facebookresearch\/detectron2}\n}\nand PANet\\footnote{\n\\rm{PANet:\\space https:\/\/github.com\/ShuLiu1993\/PANet}\n} to generate the predictions of Mask-RCNN, PointRend and PANet.\nBesides,\nwe use the segmentation results of PolyTransform directly. \nMore training\/testing details of SegFix on Cityscapes instance segmentation \ntask are illustrated in the Appendix.\nWe believe that SegFix can be used to improve\nvarious other state-of-the-art instance segmentation methods directly w\/o any prior requirements.\n\nNotably, the improvements on the instance segmentation tasks ($+1.1\\%\\sim1.5\\%$) are more significant\nthan the improvements on semantic segmentation task ($+0.3\\%\\sim0.5\\%$).\nWe guess the main reason is that the instance segmentation evaluation (on Cityscapes) only considers $8$ object categories without including the stuff categories. \nThe performance of stuff categories is less sensitive to the boundary errors due to that their area is (typically) larger than the area of object categories. According to the category-wise results in Table 9, we can also find that the improvements on several object categories, e.g., person, rider, and truck, is more significant than the stuff categories, e.g., road, building.\n\n\\vspace{-2mm}\n\\section{Conclusion}\n\\vspace{-2mm}\nIn this paper, \nwe have proposed a novel model-agnostic approach\nto refine the segmentation maps\npredicted by an unknown segmentation model.\nThe insight is that \nthe predictions of \nthe interior pixels are more reliable.\nWe propose to replace the predictions of the boundary pixels\nusing the predictions of the corresponding interior pixels.\nThe correspondence is learnt only from the input image.\nThe main advantage of our method is that SegFix\ngeneralizes well on various strong segmentation models.\nEmpirical results show that the effectiveness of our approach\nfor both semantic segmentation and instance segmentation tasks.\nWe hope our SegFix scheme can become a strong baseline\nfor more accurate segmentation results along the boundary.\n\n\n\\noindent \\textbf{Acknowledgement:}\nThis work is partially supported by Natural Science Foundation of China under contract No. 61390511, and Frontier Science Key Research Project CAS No. QYZDJ-SSW-JSC009.\n\n\n\n\\section{Appendix}\n\nFirst of all,\nwe need to clarify that all of our semantic segmentation ablation experiments\nchoose the DeepLabv3 as baseline if not specified.\nIn Section~\\ref{sec:boundary_pixel},\nwe illustrate the statistics of the proportions of boundary pixels\nover different categories as their scales vary so much.\nIn Section~\\ref{sec:val_miou}, \nwe report the category-wise mIoU improvements of our approach\non Cityscapes \\texttt{val}.\nIn Section~\\ref{sec:unified_segfix_model}, we provide\nmore details of our experiments with unified SegFix scheme.\nIn Section~\\ref{sec:instance_details}, we present more details\nof our experiments on Cityscapes instance segmentation task.\nLast,\nin Section~\\ref{sec:vis_result}, we illustrate more qualitative \nresults of our approach.\n\n\\begin{table*}[htb]\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{\\linewidth}{!}\n{\n\\begin{tabular}{c|ccccccccccccccccccc|c} \n\\toprule[1.2pt]\n\\pbox{10cm}{boundary width}\n & \\rotatebox{90}{road} & \\rotatebox{90}{{sidewalk}} & \\rotatebox{90}{building} & \\rotatebox{90}{{wall}} & \\rotatebox{90}{{fence}} &\\rotatebox{90}{pole} & \\rotatebox{90}{traffic light}&\\rotatebox{90}{traffic sign}&\\rotatebox{90}{vegetation}&\\rotatebox{90}{{terrain}}&\\rotatebox{90}{sky}&\\rotatebox{90}{person}&\\rotatebox{90}{rider}&\\rotatebox{90}{car}&\\rotatebox{90}{{truck}}&\\rotatebox{90}{bus} & \\rotatebox{90}{train} & \\rotatebox{90}{motorcycle} & \\rotatebox{90}{bicycle} & mean\\\\\\hline\n1 & 1.0 & 4.2 & 2.7 & 4.4 & 4.3 & 20.1 & 12.2 & 9.0 & 2.8 & 5.2 & 3.5 & 8.1 & 10.9 & 2.5 & 2.2 & 2.4 & 3.2 & 8.4 & 8.3 & 2.6 \\\\\n2 & 1.9 & 8.2 & 5.2 & 8.5 & 8.3 & 38.1 & 23.4 & 17.5 & 5.6 & 10.1 & 6.8 & 15.8 & 21.0 & 4.9 & 4.4 & 4.7 & 6.2 & 16.3 & 16.0 & 5.2 \\\\\n3 & 2.7 & 11.4 & 7.2 & 12.0 & 11.7 & 51.6 & 32.3 & 24.2 & 7.8 & 14.0 & 9.5 & 21.6 & 28.2 & 6.8 & 6.2 & 6.7 & 8.6 & 21.9 & 21.4 & 7.2 \\\\\n4 & 3.5 & 14.5 & 9.2 & 15.3 & 14.9 & 61.8 & 40.3 & 30.5 & 9.9 & 17.7 & 12.0 & 26.9 & 34.8 & 8.7 & 7.9 & 8.5 & 10.9 & 27.1 & 26.5 & 9.1 \\\\\n5 & 4.4 & 18.0 & 11.4 & 18.8 & 18.2 & 69.8 & 48.5 & 37.1 & 12.4 & 21.7 & 15.0 & 33.4 & 42.6 & 11.0 & 9.8 & 10.7 & 13.6 & 33.2 & 32.4 & 11.2 \\\\\n\\bottomrule[0.8pt]\n\\end{tabular}\n}\n\\caption{\\small{The proportion of boundary pixels (with different widths) over different categories on Cityscapes \\texttt{val} (\\%).}}\n\\label{table:boundary_pixel_hist}\n\\end{minipage}\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\scriptsize\n\\resizebox{\\linewidth}{!}\n{\n\\begin{tabular}{l|ccccccccccccccccccc|c} \n\\toprule[1.2pt]\nmethod & \\rotatebox{90}{road} & \\rotatebox{90}{sidewalk} & \\rotatebox{90}{building} & \\rotatebox{90}{wall} & \\rotatebox{90}{fence} &\\rotatebox{90}{pole} & \\rotatebox{90}{traffic light}&\\rotatebox{90}{traffic sign}&\\rotatebox{90}{vegetation}&\\rotatebox{90}{terrian}&\\rotatebox{90}{sky}&\\rotatebox{90}{person}&\\rotatebox{90}{rider}&\\rotatebox{90}{car}&\\rotatebox{90}{truck}&\\rotatebox{90}{bus} & \\rotatebox{90}{train} & \\rotatebox{90}{motorcycle} & \\rotatebox{90}{bicycle} & mean\\\\\n\\toprule[1.2pt]\nDeepLabV3 & 98.4 & 86.5 & 93.1 & 63.9 & 62.6 & 66.1 & 72.2 & 80.0 & 92.8 & 66.3 & 95.0 & 83.3 & 65.5 & 95.3 & 74.5 & 89.0 & 80.0 & 67.4 & 78.4 & 79.5 \\\\\n\\textcolor{blue}{+ SegFix} & \\textbf{98.5} & \\textbf{87.1} & \\textbf{93.5} & \\textbf{64.6} & \\textbf{63.1} & \\textbf{69.0} & \\textbf{74.9} & \\textbf{82.4} & \\textbf{93.2} & \\textbf{66.7} & \\textbf{95.3} & \\textbf{84.9} & \\textbf{66.9} & \\textbf{95.8} & \\textbf{75.0} & \\textbf{89.6} & \\textbf{80.7} & \\textbf{68.4} & \\textbf{79.7} & \\textbf{80.5} \\\\ \n\\hline\nHRNet-W48 & 98.5 & 87.0 & 93.5 & 58.5 & 64.7 & 71.4 & 75.6 & 82.8 & 93.2 & 64.8 & 95.3 & 84.7 & 66.9 & 95.8 & 82.9 & 91.5 & 82.9 & 69.8 & 80.1 & 81.1 \\\\\n\\textcolor{blue}{+ SegFix} & 98.5 & \\textbf{87.4} & \\textbf{93.7} & \\textbf{59.0} & \\textbf{65.1} & \\textbf{72.5} & \\textbf{77.0} & \\textbf{84.0} & \\textbf{93.4} & \\textbf{65.1} & \\textbf{95.4} & \\textbf{85.7} & \\textbf{67.7} & \\textbf{96.1} & \\textbf{83.1} & \\textbf{91.9} & \\textbf{83.4} & \\textbf{70.8} & \\textbf{81.0} & \\textbf{81.6}\\\\ \n\\hline\nGated-SCNN & 98.3 & 86.4 & 93.3 & 56.5 & 64.2 & 70.8 & 75.8 & 83.1 & 93.0 & 65.4 & 95.3 & 85.3 & 67.8 & 96.0 & 81.3 & 91.4 & 84.6 & 69.9 & 80.5 & 81.0 \\\\\n\\textcolor{blue}{+ SegFix} & \\textbf{98.4} & \\textbf{86.7} & \\textbf{93.4} & \\textbf{56.8} & \\textbf{64.4} & \\textbf{72.0} & \\textbf{77.0} & \\textbf{84.1} & \\textbf{93.2} & \\textbf{65.7} & \\textbf{95.4} & \\textbf{86.0} & \\textbf{68.8} & \\textbf{96.2} & \\textbf{81.5} & \\textbf{91.5} & \\textbf{84.8} & \\textbf{70.6} & \\textbf{81.1} & \\textbf{81.5} \\\\ \n\\bottomrule[0.8pt]\n\\end{tabular}\n}\n\\caption{\\small{Category-wise mIoU improvements of SegFix based on various methods on Cityscapes \\texttt{val}}.}\n\\label{table:segfix_val}\n\\end{minipage}\n\\vspace{-8mm}\n\\end{table*}\n\n\\subsection{Statistics of Boundary Pixels}\n\\label{sec:boundary_pixel}\nWe collect some statistics of the proportion of the boundary pixels\nover different categories in Table~\\ref{table:boundary_pixel_hist}.\nWe can find that the boundary pixels occupy large\nproportions for three (small-scale) categories including\n\\emph{pole}, \\emph{traffic light} and \\emph{traffic sign}.\nIn fact, the performance improvements (measured by mIoU)\nalso mainly come from these three categories.\nFor example, in Table~\\ref{table:segfix_val},\nour SegFix improves the DeepLabv3's mIoUs of these three categories\nby $3.1\\%$, $2.7\\%$ and $2.4\\%$ separately.\n\n\\subsection{Category-wise mIoU Improvements}\n\\label{sec:val_miou}\nWe perform the SegFix on the \nCityscapes \\texttt{val} segmentation results\nbased on  DeepLabv3~\\cite{chen2017deeplab}, \nGated-SCNN~\\cite{takikawa2019gated}\nand HRNet~\\cite{sun2019high}.\nWe report the category-wise mIoU improvements\nin Table~\\ref{table:segfix_val}\nand we can see that our approach significantly\nimproves the performance on object categories including\n\\emph{pole}, \\emph{traffic light} \nand \\emph{traffic sign}.\nThe key reason might be that the objects \nbelonging to\nthese categories tend to be of small scale,\nwhich benefit more from the accurate boundary.\n\n\n\\subsection{Details of Unified SegFix Experiments}\n\\label{sec:unified_segfix_model}\nIn our implementation, we use the same backbone HRNet-${2\\times}$ for SegFix\nand we illustrate the training policy as below:\nwe set the batch size as $16$ and construct each mini-batch by sampling $8$ images from Cityscapes\nand $8$ images from ADE20K.\nWe choose the initial learning rate as $0.02$ and all the other training\nsettings are kept the same.\nthe same learning rate policy,\nthe crop size as $512$ (for images from both datasets) and the same augmentation policy.\nAs illustrated in the paper, the performance of unified SegFix\nis comparable with the performance of SegFix trained on each dataset separately.\nIn general, the proposed unified SegFix is a general scheme\nthat well addresses the boundary errors across multiple benchmarks.\n\n\n\\subsection{Details of Experiments on Instance Segmentation}\n\\label{sec:instance_details}\n\nWe generate the instance segmentation results of Mask-RCNN\/PointRend\nbased on the open-sourced Detectron2~\\cite{wu2019detectron2},\nand we get the results of PANet~\\cite{liu2018path} and PolyTransform~\\cite{liang2019polytransform} from the authors directly\nas our approach does not require training any segmentation models.\n\nTo predict suitable offset maps for instance segmentation,\nwe start from the instance masks and re-compute the ground-truth\ndistance maps, boundary maps and direction maps.\nSpecifically, for the instance pixels, we first estimate a distance map based on\neach instance map\nand then merge all the instance based distance maps as the final distance map.\nWe generate their direction maps and boundary maps following the same manner\nas the manner for semantic segmentation.\nWe apply the predicted offset map on each predicted instance map\nseparately during the testing stage.\nAccording to the experimental results on Cityscapes instance segmentation task,\nwe can see that SegFix consistently improves\nthe performance of various methods on Cityscapes \\texttt{test}.\nWe also believe the recent state-of-the-art methods\nmight benefit from our SegFix.\n\n\n\\subsection{More Qualitative Results}\n\\label{sec:vis_result}\nWe illustrate more qualitative examples of the improvements (on semantic segmentation task)\nwith our approach in Figure~\\ref{fig:improve_hrnet_supply}.\nWe can see that our approach well \naddresses the errors along thin boundary.\nThere still exist some errors located in the interior regions \nthat our approach fail to address as we are mainly focused on \nthe thin boundary refinement.\n\n\n{\n\\begin{figure*}\n\\centering\n\\tiny{\n\\begin{tabularx}{\\textwidth}{YYYYYY}\n  {Image} & {Ground-Truth} & {DeepLabV3} & {DeepLabV3+SegFix} & {HRNet} & {HRNet+SegFix}\n\\end{tabularx}\n}\n\\normalsize\n{\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/rgb.png}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/gt_label.png}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/deeplabv3_segfix.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/hrnet48.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/4\/hrnet48_segfix.png}}\n\t\\\\\n\t\\vspace{.05cm}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/rgb.png}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/gt_label.png}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/deeplabv3_segfix.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/hrnet48.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/6\/hrnet48_segfix.png}}\n\t\\\\\n\t\\vspace{.05cm}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/rgb.png}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/gt_label.png}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/deeplabv3_segfix.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/hrnet48.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/3\/hrnet48_segfix.png}}\n\t\\\\\n\t\\vspace{.05cm}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/rgb.png}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/gt_label.png}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/deeplabv3_segfix.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/hrnet48.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/5\/hrnet48_segfix.png}}\n\t\\\\\n\t\\vspace{.05cm}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/7\/rgb.png}\n\t\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/7\/gt_label.png}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/7\/deeplabv3.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/7\/deeplabv3_segfix.png}}\n\t\\frame{\\includegraphics[width=.158\\textwidth]{images\/exp\/segfix_example\/7\/hrnet48.png}}\n\t\\frame{\\includegraphics[width=.160\\textwidth]{images\/exp\/segfix_example\/7\/hrnet48_segfix.png}}\n}\n\\vspace{-0.5cm}\n\\caption{\\small{\nQualitative comparison in terms of errors on Cityscapes \\texttt{val}.\nOur approach well addresses the existing boundary errors of various categories,\ne.g., person, pole and traffic sign, for both DeepLabv3 and HRNet.\n}}\n\\label{fig:improve_hrnet_supply}\n\\end{figure*}\n}\n\n\n\\bibliographystyle{splncs04}\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nSetting goals for yourself in different aspects of your life and publicly announcing those goals is nothing new. All year long, our social media feeds are filled with various instances of month, or year-long fitness \\cite{site:fitness_challenge,ehrlen2020shared}, coding \\cite{site:coding_challenge}, or art \\cite{site:art_challenge} challenges, where users announce they are going to be doing a certain activity for a duration of time or share their progress. Sharing of new years resolutions is also a common occurrence these days \\cite{site:instagram_new_years}. Hashtags such as \\#paintingchallenge and \\#codingchallenge, each have tens of thousands of posts on Instagram, showing many partake in such public advertisements of their goals and progress. But does the setting and announcement of goals on public platforms help or hurt your actual progress?\\\\\nThe media and numerous individuals \\cite{site:should_you_talk_about_goals,site:dont_talk_about_goals,site:talk_about_your_goals,sivers2010keep} have weighed in with their thoughts on the effects of sharing your goals with others. With some supporting the act and some being strictly against it. There are also many scientific studies on the matter, which we will thoroughly review in Section \\ref{sec:related_work}. In this study, we will be focusing on a specific instance of this question: reading goals. And to this end, we will be analyzing Goodreads.\\\\\nGoodreads is a social media platform for readers. Users can form social connections and add their readings on the website. One feature of the website, which is of great interest to this study, is \"\\textit{Goodreads Reading Challenge}\". The challenges are yearly events where users can set a certain number of books they plan to read that year (known as \"\\textit{pledging}\" on the platform) and monitor their progress throughout the year. Goodreads makes it a point for users to see their challenges by showing your progress on your homepage, as well as adding it to your profile. Additionally, users can view the pledged counts and advances of other participants.\\\\\nIn this study, we aim to take a closer look at these challenges and answer the following questions:\n\n\\begin{itemize}\n    \\item How has challenge participation changed throughout the years? Have pledged counts and success rates changed since the feature was first introduced?\n    \\item How do demographic variables influence reading habits?\n    \\item Does challenge participation (and the public commitment to reading a certain number of books) increase the number of books read?\n\\end{itemize}\nWe further extend the study by looking at discussion of the topic on more mainstream social networks, such as Instagram and Twitter, to see how users feel about these challenges and what they share with people outside their reading community.\\\\\nThe rest of this paper is structured as follows: In Section \\ref{sec:related_work}, a brief overview of related studies is presented. Next, our data collection methods and dataset statistics are discussed in Section \\ref{sec:data}. Our results are presented in Section \\ref{sec:results}, and finally Section \\ref{sec:conclusion} concludes the paper. \n\\section{Related Work}\n\\label{sec:related_work} \nWe will begin this section by first providing a review of studies conducted on Goodreads. Then we will briefly review studies on goal setting and sharing and the effects they could have on performance.\\\\\nAs Goodreads is primarily a platform for users to add and talk about the books they have read, a large proportion of the body of work on Goodreads is on the analysis of book reviews. In \\cite{kousha2017goodreads}, the viability of using Goodreads as a means to assess book impact is put to the test. With a focus on academic books, the impact factor and Goodreads engagement of the books were compared. The authors report that the engagements on the website, while prone to manipulations, could be used to assess books' impact. Additionally, book reading behavior on Goodreads has been shown to predict how well the book will do regarding sales \\cite{maity2017book}. The content of reviews are also extensively studied \\cite{driscoll2019faraway,parksepp2019sentiment,reisler2019cognitive}. \\cite{hajibayova2019investigation} investigates the lingual features of reviews, demonstrating that user-generated reviews are unreliable as they are mostly positive and attempt to persuade others to read the book as well. \\cite{shahsavari2020automated} use aggregate reviews as a means for story-graph creation and find their method to be quite accurate compared to the book's true network.\\\\\nMoreover, some studies have looked at specific instances and events concerning Goodreads. For instance, its acquisition by Amazon \\cite{albrechtslund2017negotiating}, or some policy changes made by the platform \\cite{matthews2016professionals}. \\\\\nOther studies have also investigated the social aspect of Goodreads by examining how users behave on the platform and who they form friendships with \\cite{nakamura2013words,thelwall2017goodreads,thelwall2019reader}. \\cite{thelwall2017goodreads} finds that men and woman mostly have similar behaviors, with women usually adding more books and usually rating them less favorably. Another study reports significant gender differences in rating books of different genres, finding that users usually rate books by authors of their own gender more favorably \\cite{thelwall2019reader}. \\cite{sabri2020cross} is another study that takes advantage of the website's users' multinational nature to investigate cross-country reading preferences and the factors that influence these preferences. However, to the best of our knowledge, no studies have been conducted on Goodreads challenges.\\\\\nGoals are defined as \"\\textit{Desired states that people seek to obtain, maintain, or avoid}\" \\cite{emmons1989personal,klein2008goal}. Goal-setting theory and goal-choice have been studied extensively, but most prominently in the context of organizations and work. Research has found that more challenging and specific goals result in higher levels of performance \\cite{lunenburg2011goal,locke2006new}. Goal-choice is significantly affected by gender, and self-esteem \\cite{levy1991effects}. \\\\\nSharing your goals with others is often viewed through a couple of different lenses. The first is premature praise and the intention-behavior gap. Praise is defined as \"\\textit{positive evaluations made by a person of another's products, performances, or attributes, where the evaluator presumes the validity of the standards on which the evaluation is based}\" \\cite{kanouse1981semantics,delin1994praise}. \\cite{haimovitz2011effects} explores how person and process praise could affect performance and reports that more generally, process praise increases motivation while personal praise decreases it. \\cite{gollwitzer2009intentions} similarly finds that identity-related behavioral intentions that were noticed by others would result in less intense actions. However, since reading is not an identity-related behavior, the finding of \\cite{gollwitzer2009intentions} might not apply, and the sharing of progress could potentially result in process praise, which could help improve performance.\\\\ \nAnother view is that of accountability. Accountability is defined as \"\\textit{stewardship with responsibility for creation and use of resources and a public reckoning of how they are used}\" \\cite{hubbell2007quality}. A thorough review of the literature on accountability is available in \\cite{lerner1999accounting}. While some studies have found accountability to help performance, others show that it is not always the case. To better perform appears to be connected to who you decide to share your goals with \\cite{klein2020goals,site:share_but_be_careful_with_who}. Reporting that the group with whom you share your goal must be perceived by you to have a higher status, for the sharing to be effective \\cite{klein2020goals}. \\\\\nSocial attention is another matter to consider. Research has shown that people act differently when they know they could be observed compared to when they are alone \\cite{herman2003effects,kurzban2007audience}. In more detail, studies have demonstrated that performance of simple tasks is improved (in terms of speed and accuracy) in the presence of an audience \\cite{zajonc1965social}, while the performance of complex tasks are worsened \\cite{bond1983social}. \\cite{steinmetz2017beyond} provides an in-depth review of studies on the topic. So whether we consider reading to be a simple or a complex task, the effects based on this theory would be different. \n\\section{Data Collection and Preparation}\n\\label{sec:data}\nIn this section, the data collection process is first described, then the steps we took to clean and extract various features from the data to prepare it for further analysis are explained.\n\\subsection{Challenges}\nAnnual Goodreads challenges are one of the features that help the Goodreads community define a goal, specifying how many books they want to read in the following year. This number is known as the \"\\textit{pledged}\" number of books. These challenges begin every January and finish when the year comes to an end. Users can keep track of the number of books they read during this time, and every book they read will get them closer to their goal. The Goodreads reading challenge data and the associated books are the main datasets used in this research to help us understand how reading challenges affect users' reading habits. The exact features available in this dataset are depicted in Table \\ref{tab:challenge}. The data is collected through Goodreads' public API \\footnote{https:\/\/www.goodreads.com\/api}. This dataset includes 5,523,896 instances of challenge data for 4,363,093 unique users and 289,078 books associated with these challenges. We query the API with random challenge identifiers to retrieve this data. Since identifiers are assigned incrementally, with higher numbers corresponding to newer years, we make sure to request numbers distantly apart. With this method, at least 25,549 challenge entries were retrieved for each year from 2011 to 2020. At the time of writing this paper, the users' challenges in 2020 were still in progress, and we decided not to use their information, thus omitted them from the dataset. Moreover, we excluded the information of challenges with more than 500 pledged books or more than 200 read books, as they tend to be outliers. Also, as Goodreads users are members of a reading community, it is logical for them to read at least one book while participating in a challenge; therefore, we did not consider challenges with 0 read books. One possible explanation is that users with 0 read books did not know that they had to record the finish time of the corresponding books or update their read books as Goodreads is not as popular as other social platforms such as Instagram or Twitter. 2,233,517 out of 5,523,896 entries were deleted for this reason; therefore, 3,254,382 challenges for 2,251,574 users remained. The information regarding the number of read and pledged books during these challenges can be seen in Table \\ref{tab:challenge_pledged_read}. This information indicates that users tend to overestimate their abilities in reading books. \\\\\nIn addition to the data explained above, we collected the data from users' profiles indicating all the challenges they have participated in, including the number of pledged books and read books during each one. This data was then used in order to compare users' performance while participating in a yearly challenge versus the years they were not part of an annual challenge. This dataset has 10,649 entries belonging to 4,558 unique users; the users were selected randomly from the aforementioned pool of users. This data also contained 283 challenges with 0 read books, which were deleted due to the same reason mentioned above. \\\\\n\\begin{table*}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{Reading Challenge Information}\n  \\label{tab:challenge}\n  \\begin{tabular}{c c}\n    \\hline\n    Feature&Description\\\\\n    \\hline\n    Challenge ID&Unique identifier for a challenge \\\\\n    \\hline\n    User ID&Unique identifier for the user associated with this challenge \\\\\n    \\hline\n    Read Count&Number of books this user completely read in this challenge \\\\\n    \\hline\n    Pledged Count&Number of books this user planned to read in this challenge \\\\\n    \\hline\n    Year & The year this challenge took place \\\\\n    \\hline\n    Book IDs& A list of unique identifiers for the books read in this challenge\\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{Reading Challenge Pledged and Read Counts}\n  \\label{tab:challenge_pledged_read}\n  \\begin{tabular}{c c c}\n    \\hline\n    &Pledged&Read\\\\\n    \\hline\n    mean&36.59&23.30\\\\ \\hline\n    median&25.0&13.0\\\\ \\hline\n    standard deviation&34.13&29.25\\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Users and Books}\nTo analyze demographic features and how such features affect users' participation and reading habits, users' personal information is needed. For this purpose, Goodreads' public API was used again. In order to use the API, user IDs were extracted from corresponding challenges and used to retrieve the aforementioned data. Moreover, books' information, such as their format, is needed for further analysis. This information can also be retrieved similarly. User information and book information dataset columns are shown in Table \\ref{tab:user} and Table \\ref{tab:book}, respectively.\nIn total, 35,695 instances of user information were retrieved in Users dataset, among which, only 186 users' \\textit{gender}, 1,977 users' \\textit{age}, and 5,735 users' \\textit{location} were initially available. This should not be confused with the Challenges dataset. It was previously mentioned that 3,254,382 instances of challenge data (after deleting challenges with 0 read books) for 2,251,574 unique users were retrieved, but we do not have these user's personal information. We know that these challenges belonged to 2,251,574 unique users based on their \\textit{user ID}. As mentioned, only a small number of users' \\textit{gender}, \\textit{age}, and \\textit{location} were initially available; therefore, these features for the rest of these 35,695 users were extracted by analyzing their profile pictures and personal details. Particularly, for extracting \\textit{countries}, both \\textit{location} and \\textit{about} features were analyzed and the user's country, city, or state name were retrieved in case of availability using \\textit{geotext}\\footnote{https:\/\/pypi.org\/project\/geotext\/} and \\textit{pycountry}\\footnote{https:\/\/pypi.org\/project\/pycountry\/} libraries. This data was then checked manually, and the required corrections were made. Gender detection was done using each user's \\textit{name}, \\textit{about}, and \\textit{image} columns. Moreover, age was detected from the \\textit{about} column by finding age keywords such as \\textit{age}, \\textit{years}, \\textit{y\/o}, \\textit{year} and then categorized in ranges. After doing so, users with ages less than 9 and more than 100 were deleted from the dataset as they are unlikely to be real values. In total, 10\\% of users' \\textit{age} was detected. Others had not mentioned anything regarding their age. \\\\\nThere is also one more dataset that shows which users have read which books, regardless of their participation in reading challenges. This data is then used to compare users' reading habits while they are not participating in any challenges as opposed to the time they are. The columns for this dataset are depicted in Table \\ref{tab:userBook}. \n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{User Information}\n  \\label{tab:user}\n  \\begin{tabular}{c c}\n    \\hline\n    Column&Description\\\\\n    \\hline\n    User ID&Unique Identifier for a User \\\\ \\hline\n    Name&Name of this user \\\\ \\hline\n    Image&Profile picture of this user \\\\ \\hline\n    Location&User's location - can be country, city, or state \\\\ \\hline\n    Gender&Specified as male, female, or unknown \\\\ \\hline\n    Age&This is between 9 to 110 \\\\ \\hline\n    About&User's \"\\textit{about me}\" information \n    \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{Book Information}\n  \\label{tab:book}\n  \\begin{tabular}{c c}\n    \\hline\n    Column&Description\\\\\n    \\hline\n    Book ID & Unique identifier for a book\\\\ \\hline\n    Name & Name of this book\\\\ \\hline\n    Format & Format which can be Hardcover, Paperback, Audio, etc.\\\\ \\hline\n    Number of Pages & This book's page count\\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{User-Book Information}\n  \\label{tab:userBook}\n  \\begin{tabular}{c c}\n    \\hline\n    Column&Description\\\\\n    \\hline\n    User ID & Unique identifier for a user\\\\ \\hline\n    Book ID & Unique identifier for a book this user read \\\\ \\hline\n    Read At & Date the book was marked as read\\\\ \\hline\n    Read Count & Number of times the user read the book \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Twitter and Instagram}\nTo analyze how people talk about their reading challenges outside their reading community, we collected Instagram posts and tweets which contained \\#readingChallenge and \\#goodreadsChallenge hashtags. These hashtags were used in different years, and therefore, the posts belonged to various years. For Instagram, specifically, we also searched for hashtags such as \\#2015goodreadsChallenge and \\#2015readingChallenge, which included the years. This was not done for Twitter as we had enough data for each year. Furthermore, since our analysis was made on these posts' distributions through months, we did not necessarily need data for all years. In total, 418,913 Instagram posts and 48,320 tweets were collected.  By manually studying these posts and tweets, we realized that they contained book reviews and challenge updates. Using the timestamp of each entry, the month they were posted was extracted, and using the text or caption, the sentiment for each was retrieved. For extracting the sentiments, \\textit{TextBlob} library \\footnote{https:\/\/textblob.readthedocs.io\/en\/dev\/} was used. Texts with sentiment between -0.1 and 0.1 were considered neutral, whereas sentiments between -1 and -0.1 and between 0.1 and 1 were considered negative and positive, respectively. Several examples for the tweets and posts' captions are shown in Table \\ref{tab:sentiment_example}. This data was then used to analyze the posts' distribution through the months of the year. The result of which has been reported in the following sections.\n\n\\begin{table*}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{Posts and Tweets' Sentiment Examples}\n  \\label{tab:sentiment_example}\n  \\begin{tabular}{c c}\n    \\hline\n    Text&Sentiment\\\\\n    \\hline\n    I woke up with a cold!! Oh no!!! At least it got me out of work so now I can work on the& -0.65\\\\\n     \\#YASavesChallenge \\#readingchallenge \\\\ \\hline\n    October is the perfect month to read a book that takes place at night! Check out our recommendations & 0.75 \\\\\n    to fulfill this reading challenge prompt. Our picks are seasonally appropriate with witches \\\\and vampires! \\#booklist \\#readingchallenge \\#bookclubbelles\\\\ \\hline\n    I have read and reviewed 6 books from my TBR list. I am woefully behind schedule when it comes & -0.5\\\\\n     to meeting my Goodreads 2020 challenge, but I'm alright with it because these reads were fantast-\\\\\n    ic! \\#ReadingChallenge2020 \\#readersofinstagram \\#romancereadersofinstagram \\#wonderfulbook\\\\ \\#AmReadingRomance \\#slowreader \\#amreading \\#readingromance \\\\ \\hline\n    The Perfect Girlfriend\" is a pretty good read so far. I usually only say this if I'm 50 pages & 0.45 \\\\\n     in but at 31 pages it's got a pace set. So far, so good. \\#readingtime \\#readingchallenge2020 \\#bookworm \\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Results}\n\\label{sec:results}\n\\subsection{Throughout the years} \nOur goal is to find how reading habits have changed through the years and whether people read less or more compared to the past. For this matter, we tried to find the trend in the read and pledged counts during challenges. By studying the average count of pledged books and read books, we realized that the average pledged count in challenges is 36.59 per challenge, and since each of these challenges correspond to a year, it means finishing 36.59 books every year, or 3.04 books every month. Conversely, the average number of books read during challenges is 23.30, which means finishing 1.94 books per month. This difference shows that people tend to read less but aim higher. It is crucial to bear in mind that this number of read books is slightly higher in a reading community and cannot be generalized to the entire society. For instance, based on a research \\cite{site:average_number_of_read_books}, a person reads 12 books a year on average, or in other words, one book a month.\\\\ \nWe also wanted to see how this overestimation has changed throughout the years. In order to find out, we grouped the challenge data by their year and calculated the median and mean pledged and read count for each year separately. The difference between these two counts can be seen in Figure \\ref{fig:pledged_read_years}. \nBased on these figures, the difference between the pledged count and read count has mainly decreased, and people tend to better estimate their abilities compared to the past. The reason to plot both the mean and the median is that the median is less sensitive to outliers. It is also apparent that people are generally reading and pledging less than before.\n\n\\begin{figure}[ht]  \n  \\centering\n   \\includegraphics[scale=0.3]{Figures\/Fig1.png}\n  \\captionsetup{justification=centering}\n  \\caption{The left-side plots show the median (top) and mean (bottom) count of pledged and read books for each year from 2011 to 2019. The right plots show the difference between the two for each year.}\n  \\label{fig:pledged_read_years}\n\\end{figure}\n\n\\subsection{Do challenges make people read more?}\nIn this part, the data collected from users' profiles containing all the challenges they had participated in was used. Each book that these users had read during different years was also retrieved and collected in a separate dataset. This way, we could find out how many books these users had read through the years and also had the information about which year they were participating in an annual challenge. After calculating the average number of books each user reads in years that they are not participating in a yearly challenge and the years they are and comparing them, it was concluded that 81\\% of people read more on average while being part of a reading challenge. This information belongs to the users that we have information on their readings both in years they were participating in challenges and years they were not, namely 787 unique users. Users read 298\\% more books on average while participating in a challenge than when they are not. In order to find out whether this difference is significant, we performed a hypothesis test: $H_0: \\mu_1 - \\mu_2 = 0$ and $H_A: \\mu_1 - \\mu_2 \\ne 0$, where $\\mu_1$ and $\\mu_2$ are the average read books during challenges and outside them, respectively. \n\nThis test's p-value is almost 0, which shows that the observed difference is statistically significant, and people read more books while taking part in yearly challenges. The reason could be that users feel more motivated to reach their goal when publicly announcing it or are more likely to update their Goodreads profile and their reading progress while participating in a challenge.\n \\\\\n Another interesting result is the number of users who have read a specific number of books in a challenge. This data is depicted in Figure \\ref{fig:zipf_vs_powerlaw}. Before plotting this histogram, we assumed that it would have a normal distribution as it corresponds to human performance. However, the result showed a completely different distribution. We tried to fit power law and Zipf distributions, and they had the sum of squared errors 0.070 and 0.0027, respectively.\n \\\\\n \n \\begin{figure}[ht]  \n  \\centering\n   \\includegraphics[scale=0.25]{Figures\/Fig2.png}\n  \\captionsetup{justification=centering}\n  \\caption{Powerlaw vs Zipf Distributions for the Read Count Histogram}\n  \\label{fig:zipf_vs_powerlaw}\n\\end{figure}\n\n\n\\subsection{Are demographic variables influential in reading habits?}\nThis section investigates whether people's demographic identities, including gender and place of residence, significantly influence their reading habits and preferences. \\\\\nThe questions below were studied in this section:\n\\begin{itemize}\n    \\item What are the reading challenge success rates in different countries?\n    \\item Is gender statistically significant in whether people read audiobooks or not?\n    \\item Is gender statistically significant in succeeding at Goodreads reading challenges?\n\\end{itemize}\n\nAfter extracting the country names for users based on the location and personal details, each country's success rate was calculated. To avoid bias, the countries with less than 100 challenges were not considered in this calculation. After the filtering, 27 countries were left. The success rate for these countries is shown in Figure~\\ref{fig:success_countries}. The most successful countries were Poland, Portugal, and Italy, with success rates of 0.4656, 0.4552, and 0.4522. Furthermore, the number of challenges in each of these countries is shown in Figure \\ref{fig:count_countries}. This figure shows how much of our data belonged to each country. The top three countries are the United States, Canada, and the United Kingdom, with 3557, 888, and 790 challenges, respectively. \\\\\n\nAnother question that we will address here is whether gender has a significant role in users' tendency to read audiobooks during challenges. To find each user's gender belonging to the dataset containing each user and their books, \\textit{gender-guesser}\\footnote{https:\/\/pypi.org\/project\/gender-guesser\/} library was used; this dataset was created by performing inner join on the challenge dataset and the books they had read during these challenges. There were 124,263 male and 575,406 female users in this data, and 303,245 users' gender could not be detected. Proportion difference hypothesis test is used to investigate whether gender has a significant role in choosing audiobooks over other book formats during challenges. (See Table \\ref{tab:audiobook_gender}). \n\nBased on the results of a hypothesis test we performed where $H_0: p_{male} - p_{female} = 0$ and $H_1: p_{male} - p_{female} > 0$, the p-value for one-tailed hypothesis is $<$ .00001 and the result is significant at the significance level 0.05. As a result, We can conclude that male users are more likely to choose audiobooks over other forms of books during challenges.\\\\\nThe other question was whether gender is significant in reading challenge success rate (See Table \\ref{tab:success_gender}).\nBased on a hypothesis test we performed where $H_0: p_{female} - p_{male} = 0$ and $H_1: p_{female} - p_{male} > 0$, the p-value is $<$ .00001 and the result is significant at the significance level 0.05. To be more specific, considering the count of challenge participants of each gender and their success rate, it can be concluded that gender has a significant influence on Goodreads users' success in annual reading challenges, and women are more successful in these challenges.\\\\ \\\\\n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{People Reading Audiobooks}\n  \\label{tab:audiobook_gender}\n  \\begin{tabular}{c c c}\n    \\hline\n    Gender&People reading audiobooks count&All count\\\\\n    \\hline\n    Male & 1731 & 124263  \\\\ \\hline\n    Female & 6424 & 575406 \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{People Success Count}\n  \\label{tab:success_gender}\n  \\begin{tabular}{c c c}\n    \\hline\n    Gender&Successful count&All count\\\\\n    \\hline\n    Male & 154,010 & 533,153  \\\\ \\hline\n    Female & 424,566 & 1,394,773 \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[ht]  \\centering\n   \\includegraphics[scale=0.255]{Figures\/Fig3.png}\n  \\captionsetup{justification=centering}\n  \\caption{Success Rate in Countries}\n  \\label{fig:success_countries}\n\\end{figure}\n\n\\begin{figure}[ht]  \\centering\n   \\includegraphics[scale=0.255]{Figures\/Fig4.png}\n  \\captionsetup{justification=centering}\n  \\caption{Challenge Count in Countries}\n  \\label{fig:count_countries}\n\\end{figure}\n\n\\subsection{Other social networks}\nWe analyzed the number of reading challenge tweets and posts through different months of the year. Generally, people announce their goals at the beginning of the year as these reading challenges are annual events, and people usually set goals as their new year's resolution. For instance, 19\\% of Instagram posts and 22\\% of tweets regarding reading challenges were posted in January. They will keep posting less and less in the following months until December when the challenge is close to the end, and a sudden increase in the number of posts is observed then. The number of reading challenge Instagram posts and tweets through months are depicted in Figure \\ref{fig:insta_count_months} and Figure \\ref{fig:twitter_count_months}, respectively, which confirm the information given above.\n\nThe sentiment analysis results show that users' posts regarding reading challenges are rarely negative. Users might merely post a report on their progress, express their feeling about a specific book, or how successful they are in their challenge. They might also express their disappointment in falling behind during the challenge. Some users also post reviews about books and add their opinions about them, which can be positive or negative. The average sentiment through months in Instagram and Twitter are shown in Figure \\ref{fig:insta_sentiment_months} and Figure \\ref{fig:twitter_sentiment_months}, respectively. In Instagram, the general attitude towards reading challenges is similar through months, but Twitter data shows some fluctuations in the tweets' sentiments. However, the difference between the highest and lowest average sentiments in months is merely 0.123, and therefore, we can assume that the general attitude is similar.\n\n\nThe percentage of posts with negative and positive sentiments on both social platforms are almost identical. This is depicted in Table \\ref{tab:sentiment}. This shows that people on both platforms have similar attitudes toward their reading challenges.\n\n\\begin{figure}[ht]  \n  \\centering\n  \\captionsetup{justification=centering}\n   \\includegraphics[scale=0.25]{Figures\/Fig5.png}\n  \\caption{Instagram Posts Count through Months}\n  \\label{fig:insta_count_months}\n\\end{figure}\n\n\\begin{figure}[ht]  \n  \\centering\n  \\captionsetup{justification=centering}\n   \\includegraphics[scale=0.25]{Figures\/Fig6.png}\n  \\caption{Tweets Count through Months}\n  \\label{fig:twitter_count_months}\n\\end{figure}\n\n\\begin{figure}[ht]  \n  \\centering\n  \\captionsetup{justification=centering}\n   \\includegraphics[scale=0.25]{Figures\/Fig7.png}\n  \\caption{Average Sentiment of Instagram Posts through Months}\n  \\label{fig:insta_sentiment_months}\n\\end{figure}\n\n\\begin{figure}[ht]  \n  \\centering\n  \\captionsetup{justification=centering}\n   \\includegraphics[scale=0.25]{Figures\/Fig8.png}\n  \\caption{Average Sentiment of Tweets through Months}\n  \\label{fig:twitter_sentiment_months}\n\\end{figure}\n\n\\begin{table}[htbp]\n  \\centering\n  \\captionsetup{justification=centering}\n  \\caption{Sentiment Percentage}\n  \\label{tab:sentiment}\n  \\begin{tabular}{c c c c}\n    \\hline\n    Platform&Positive&Negative&Neutral\\\\\n    \\hline\n   \n   \n    Instagram &50.54\\% & 4.77\\%&44.67\\% \\\\ \\hline\n    Twitter & 47.74\\% & 4.64\\%&47.60\\% \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this study, we examined the effects of public reading challenges on how much people read. Specifically, we showed that people are significantly more likely to read more once they have taken part in a challenge in comparison to their normal performance. We further show how gender is a significant factor in how well people perform in their challenges. \\\\\nWe would like to discuss one limitation of our study now. This analysis was conducted on Goodreads, and while the selection process was conducted completely at random, the website itself tends to attract avid readers mostly. Hence, the sample is out of people with a keen interest in reading, introducing bias into our analysis. \n\n\\section{Declarations}\n\\subsection{Funding}\nNot applicable \/ No funding was received.\n\n\\subsection{Conflicts of interest}\nThe authors declare that they have no competing interests.\n\n\\subsection{Availability of data and material}\n Although all data was collected from public accounts, we have decided not to make this data publicly available due to the availability of user information in our data. \n \n\\subsection{Code availability}\nNot applicable\n\n\n\\bibliographystyle{spmpsci}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSpin polarization experimental measurements of $\\Lambda$ hyperons recently taken by the STAR Collaboration \\cite{STAR:2017ckg,Adam:2018ivw,Niida:2018hfw,Adam:2019srw} have created a huge interest in the spin polarization studies and in studies correlating between the vorticity and particle spin polarization in relativistic heavy-ion collisions\\cite{Shi:2020htn,Weickgenannt:2020aaf,Becattini:2009wh,Becattini:2013fla,Montenegro:2017rbu,Montenegro:2017lvf,Becattini:2018duy,Boldizsar:2018akg,Prokhorov:2018bql,Yang:2018lew,Florkowski:2019voj,Weickgenannt:2019dks,Hattori:2019lfp,Ambrus:2019ayb,Sheng:2019kmk,Prokhorov:2019cik,Ivanov:2019wzg,Hattori:2019ahi,Xie:2019jun,Liu:2019krs,Wu:2019eyi,Becattini:2019ntv,Zhang:2019xya,Li:2019qkf,Florkowski:2019gio,Prokhorov:2019yft,Fukushima:2020qta,Liu:2020ymh,Bhadury:2020puc,Tabatabaee:2020efb,Liu:2020flb,Yang:2020hri,Deng:2020ygd,Taya:2020sej,Gao:2020lxh,Bhadury:2020cop,Montenegro:2020paq}; for reviews see\\cite{Becattini:2020ngo,Tinti:2020gyh,Speranza:2020ilk,Gao:2020vbh,Becattini:2020sww}. Thermal-based models~\\cite{Becattini:2016gvu,Karpenko:2016jyx,Li:2017slc,Xie:2017upb} which precisely explain the global polarization of particles, does able to explain differential results correctly\\cite{Adam:2019srw}, these models assume the condition that particle spin polarization emitted at freeze-out hypersurface is defined by the thermodynamical quantity which is named as thermal vorticity~\\cite{Becattini:2007sr,Becattini:2013fla}, not considering the fact that it may evolve independently during the expansion of the fluid. In this article, we follow scheme proposed in Refs.~\\cite{Florkowski:2017ruc,Florkowski:2017dyn,Florkowski:2018myy,Florkowski:2018fap,Florkowski:2018ahw,Florkowski:2019qdp}, and analyze such possibility of spin polarization evolution using relativistic hydrodynamics framework with spin.\n\\section{Distribution functions in equilibrium}\nIf we know phase space distribution function for the system's equilibrium state, then it is possible to derive relativistic hydrodynamics from the underlying kinetic theory definitions\\cite{Florkowski:2017olj}. Following ideas developed by Becattini \\textit{et al}.\\cite{Becattini:2013fla}, we take into consideration the following distribution functions for the relativistic systems of spin $\\onehalf$ massive particles (and antiparticles) in the local equilibrium state.\n\\bel{fplusrsxp}\nf^+_{rs}(x,p) =  {\\bar u}_r(p) X^+ u_s(p), \\qquad\nf^-_{rs}(x,p) = - {\\bar v}_s(p) X^- v_r(p),\n\\end{eqnarray}\nwhere $x$ and $p$ is the space-time position coordinate and the four-vector momentum, respectively, with $u_r(p)$ and $v_r(p)$ being the Dirac bispinors ($r,s = 1,2$). Dirac bispinors follow the normalization conditions as ${\\bar u}_r(p) u_s(p)=\\,\\delta_{rs}$ and ${\\bar v}_r(p) v_s(p)=-\\,\\delta_{rs}$, here $\\delta_{rs}$ is the Dirac delta function and, $X^{\\pm}$ have the following form in terms of relativistic Boltzmann distributions\n\\bel{XpmM}\nX^{\\pm} =  \\exp\\left[\\pm \\xi(x) - \\beta_\\mu(x) p^\\mu  \\pm \\f{1}{2} \\omega_{\\mu\\nu}(x)  {\\Sigma}^{\\mu\\nu} \\right], \\nonumber\n\\end{eqnarray}\nwhere $\\beta^\\mu \\equiv U^\\mu\/T$ and $\\xi \\equiv \\mu\/T$, here $T$, $\\mu$ and $U^\\mu$ is the temperature, baryon chemical potential and four-vector velocity,  respectively, and $\\omega_{\\mu\\nu}$ is the second rank asymmetric tensor known as spin polarization tensor with ${\\Sigma}^{\\mu\\nu}  \\equiv  \\f{i}{4} [\\gamma^\\mu,\\gamma^\\nu]$ being the spin operator.  \n\nWith the help of the expressions from Ref.~\\cite{DeGroot:1980dk} and Eqs. \\rfn{fplusrsxp} we can obtain the Wigner functions in equilibrium as follows\n\\begin{eqnarray}\n{\\cal W}^\\pm_{\\rm eq}(x,k) &=& \\frac{e^{\\pm \\xi}}{4 m}  \\int dP\n\\,e^{-\\beta \\cdot p }\\,\\, \\delta^{(4)}(k \\mp p) \\label{eq:Weqpxk2} \\\\\n&& \\times  \\left[2m (m \\pm \\slashed{p}) \\cosh(\\zeta) \\pm \\f{\\sinh(\\zeta)}{2\\zeta}  \\, \\omega_{\\mu\\nu} \\,(\\slashed{p} \\pm m) {\\Sigma}^{\\mu\\nu} (\\slashed{p} \\pm m) \\right],\\nonumber\n\\end{eqnarray}\nwhere $k$ being the off mass-shell particle four-momentum, \n$dP = d^3p\/((2 \\pi )^3 E_p)$ is the invariant measure with $E_p = \\sqrt{m^2 + {\\boldsymbol p}^2}$ denoting the on-shell energy of the particle, and $\\zeta =  \\f{1}{2 \\sqrt{2}} \\sqrt{ \\omega_{\\mu\\nu} \\omega^{\\mu\\nu}}$ in terms of spin polarization tensor.\\\\\nWigner function can also be expanded using Clifford-algebra expansion \\rfn{eq:Weqpxk2}\n\\begin{eqnarray}\n{\\cal W}^{\\pm}_{\\rm eq}(x,k) &=& \\f{1}{4} \\left[ {\\cal F}^{\\pm}_{\\rm eq}(x,k) + i \\gamma_5 {\\cal P}^{\\pm}_{\\rm eq}(x,k) + \\gamma^\\mu {\\cal V}^\\pm_{{\\rm eq}, \\mu}(x,k) \\right. \\nonumber \\\\\n&& \\left.  \\hspace{1cm} + \\gamma_5 \\gamma^\\mu {\\cal A}^\\pm_{{\\rm eq}, \\mu}(x,k)\n+ {\\Sigma}^{\\mu\\nu} {\\cal S}^\\pm_{{\\rm eq}, \\mu \\nu}(x,k) \\right], \\nonumber\n\\label{eq:wig_expansion}\n\\end{eqnarray}\nwhere $\\mathcal{X} \\in\\left\\{\\mathcal{F}, \\mathcal{P}, \\mathcal{V}_{\\mu}, \\mathcal{A}_{\\mu}, \\mathcal{S}_{\\nu \\mu}\\right\\}$ are the coefficient functions of the Wigner function, which can obtained from the trace of ${\\cal W}^{\\pm}_{\\rm eq}(x,k)$ multiplying first by: $\\left\\{\\mathbf{1},-i \\gamma_{5}, \\gamma_{\\mu}, \\gamma_{\\mu} \\gamma_{5}, 2 \\Sigma_{\\mu \\nu}\\right\\}$.\n\\section{Kinetic and hydrodynamical equations}\nThe kinetic equation to be followed by Wigner function is\n\\bel{eq:eqforWC}\n\\left(\\gamma_\\mu K^\\mu - m \\right) {\\cal W}(x,k) = C[{\\cal W}(x,k)],\n\\end{eqnarray}\nwith $K^\\mu = k^\\mu + \\frac{i \\hbar}{2} \\,\\partial^\\mu$. \nFor global equilibrium state, the Wigner function follows exactly \\rf{eq:eqforWC} with the collision term $C[{\\cal W}(x,k)] = 0$. The widely used method of treating \\rf{eq:eqforWC} is the semi-classical expansion method of the coefficient functions of the Wigner function \n\\bel{eq:semi}\n\\mathcal{X}=\\mathcal{X}^{(0)}+\\hbar \\mathcal{X}^{(1)}+\\hbar^{2} \\mathcal{X}^{(2)}+\\cdots. \\nonumber\n\\end{eqnarray}\nUp to the first order (i.e. next-to-leading order) in $\\hbar$ the treatment of \\rf{eq:eqforWC} gives the following kinetic equations to be followed by the two independent coefficients which are: $\\mathcal{F}_{\\mathrm{eq}}$ and $\\mathcal{A}_{\\mathrm{eq}}^{\\nu}$, \n\\bel{eq:ke}\nk^{\\mu} \\partial_{\\mu} \\mathcal{F}_{\\mathrm{eq}}(x, k)=0, \\quad k^{\\mu} \\partial_{\\mu} \\mathcal{A}_{\\mathrm{eq}}^{\\nu}(x, k)=0, \\quad k_{\\nu} \\mathcal{A}_{\\mathrm{eq}}^{\\nu}(x, k)=0.\n\\end{eqnarray}\nIn the case of global equilibrium  \\rfm{eq:ke} are satisfied exactly which in-turn yields the conditions that $\\beta_{\\mu}$ is a Killing vector, whereas, $\\xi$ and $\\omega_{\\mu \\nu}$ are constant, but $\\omega_{\\mu \\nu}$ does not necessarily be equal to thermal vorticity $\\varpi_{\\mu \\nu} = -\\frac{1}{2} \\left(\\partial_\\mu \\beta_\\nu - \\partial_\\nu \\beta_\\mu \\right) = \\hbox{const}$.\nBut in the case of local equilibrium \\rfm{eq:ke} are not exactly followed, here we follow\\cite{Denicol:2012cn} and by permitting $\\beta$, $\\xi$ and $\\omega$ dependence on $x$, we need to have only certain moments in momentum space of the kinetic equations \\rfn{eq:ke} which are satisfied, which lead to conservation laws for charge, energy-linear momentum and spin\\cite{Florkowski:2018ahw}\n\\begin{eqnarray} \n\\quad\\partial_\\mu N^\\mu = 0, \\label{Ncons} \\\\\n\\partial_\\mu T^{\\mu\\nu}_{\\rm GLW} = 0, \\label{Tcons}\\\\\n\\partial_\\lambda  S_{\\rm GLW}^{\\lambda, \\alpha \\beta} =0,\n\\label{Scons}\n\\end{eqnarray}\nhere the baryon current, the energy-momentum and the spin tensors are based on the forms by the de Groot - van Leeuwen - van Weert (GLW)~\\cite{DeGroot:1980dk}\n\\begin{eqnarray}\nN^\\alpha &=& n U^\\alpha,\\\\\n T^{\\alpha\\beta}_{\\rm GLW} &=& (\\varepsilon + P ) U^\\alpha U^\\beta - P g^{\\alpha\\beta},\\\\\n S^{\\alpha , \\beta \\gamma }_{\\rm GLW}\n&=& \\cosh(\\xi) \\LS n_{(0)} U^\\alpha \\omega^{\\beta\\gamma}  +  {\\cal A}_{(0)} \\, U^\\alpha U^\\d U^{[\\beta} \\omega^{\\gamma]}_{\\HP\\d}  \\right. \\\\\n&& +\\left. \\, {\\cal B}_{(0)} \\, \\Big( \nU^{[\\beta} \\Delta^{\\alpha\\d} \\omega^{\\gamma]}_{\\HP\\d}\n+ U^\\alpha \\Delta^{\\d[\\beta} \\omega^{\\gamma]}_{\\HP\\d}\n+ U^\\d \\Delta^{\\alpha[\\beta} \\omega^{\\gamma]}_{\\HP\\d}\\Big) \\right],\n\\end{eqnarray}\nwhere $\\Delta^{\\alpha\\beta} =g^{\\alpha\\beta} - U^\\alpha U^\\beta$ is the spatial projection operator which is orthogonal to the hydrodynamic flow 4-vector $U$.\\\\\nFor the case of polarization tensor in the leading order,  the baryon number density, the energy density and, the pressure are expressed respectively as\n\\begin{eqnarray} \nn &=& \\sinh(\\xi)\\, n_{(0)}(T), \\label{n0small} \\\\\n\\varepsilon &=& \\cosh(\\xi) \\, \\varepsilon_{(0)}(T),\\label{e0small}\\\\ \nP &=& \\cosh(\\xi) \\, P_{(0)}(T), \\label{P0small} \n\\end{eqnarray}\nwhere for spin-less and neutral massive Boltzmann particles, thermodynamical properties are defined by\\cite{Florkowski:2010zz}\n\\begin{eqnarray}\nn_{(0)}(T) &=&  \\f{2~T^3}{\\pi^2}\\,  \\hat{m}^2 K_2\\left( \\hat{m}\\right), \\label{n0c}\\\\\n\\varepsilon_{(0)}(T) &=& \\f{2~T^4 }{\\pi^2}  \\, \\hat{m} ^2\n \\Big[ 3 K_{2}\\left( \\hat{m} \\right) + \\hat{m}  K_{1} \\left( \\hat{m}  \\right) \\Big],  \\label{e0c}\\\\\n %\nP_{(0)}(T) &=& T \\, n_{(0)}(T) . \\label{P0c}\n\\end{eqnarray}\n %\n Here, $K_{1} \\left( \\hat{m}  \\right)$ and $K_{2} \\left( \\hat{m}  \\right)$ are modified Bessel functions of 1st and 2nd kind respectively. The thermodynamical quantities ${\\cal B}_{(0)} $ and ${\\cal A}_{(0)}$ are expressed as\n %\n\\begin{eqnarray}\n{\\cal B}_{(0)} =-\\frac{2}{\\hat{m}^2} s_{(0)}(T) ,\\qquad\n{\\cal A}_{(0)}  = -3{\\cal B}_{(0)} +2 n_{(0)}(T) \n\\end{eqnarray}\nwith entropy density $s_{(0)} =   \\LR\\varepsilon_{(0)}+P_{(0)}\\right) \/ T$ and $\\hat{m}=m\/T$. \n\\section{Bjorken expansion set-up}\nSince the spin polarization tensor $\\omega_{\\mu\\nu}$ is a 2nd rank asymmetric tensor, so in analogy to the Faraday electromagnetic field strength tensor, it can be written into electric-like ($\\kappa$) and magnetic-like ($\\omega$) components\n\\begin{eqnarray}\n\\omega_{\\mu\\nu} &=& \\kappa_\\mu U_\\nu - \\kappa_\\nu U_\\mu + \\epsilon_{\\mu\\nu\\alpha\\beta} U^\\alpha \\omega^{\\beta},\\label{spinpol1}\n\\end{eqnarray}\nwhere $\\kappa$ and $\\omega$ are 4-vectors, orthogonal to fluid flow vector $U_\\mu$. \nFor longitudinal boost-invariant and transversely homogeneous systems\\cite{Bjorken:1982qr,Rezzolla:2013zz}, one can write the following basis vectors \n\\begin{eqnarray}\nU^\\alpha &=& \\frac{1}{\\tau}\\LR t,0,0,z \\right) = \\LR \\cosh(\\eta), 0,0, \\sinh(\\eta) \\right), \\nonumber \\\\\nX^\\alpha &=& \\LR 0, 1,0, 0 \\right),\\nonumber\\\\\nY^\\alpha &=& \\LR 0, 0,1, 0 \\right), \\nonumber\\\\\nZ^\\alpha &=& \\frac{1}{\\tau}\\LR z,0,0,t \\right) = \\LR \\sinh(\\eta), 0,0, \\cosh(\\eta) \\right), \n\\label{BIbasis}\n\\end{eqnarray}\nwhere longitudinal proper time is defined as $\\tau = \\sqrt{t^2-z^2}$ and, the space-time rapidity is defined as $\\eta = \\half \\ln((t+z)\/(t-z))$.\nThe normalization conditions satisfied by the basis vectors \\rfn{BIbasis} are\n\\begin{eqnarray}\n &&U \\cdot U = 1,\\nonumber\\\\\nX \\cdot X &=&   Y \\cdot Y \\,\\,=\\,\\, Z \\cdot Z \\,\\,=\\,\\, -1,\\nonumber \\\\ \\label{orthXYZ}\nX \\cdot U  \\,\\,&=& Y \\cdot U\\,\\, \\,\\,=\\,\\, Z \\cdot U \\,\\,=\\,\\, 0,   \\\\\nX \\cdot Y  &=&  Y \\cdot Z \\,\\,=\\,\\, Z \\cdot X \\,\\,=\\,\\, 0.  \\nonumber\n\\end{eqnarray}\nUsing the fact that  $\\kappa$ and $\\omega$ are orthogonal to $U_\\mu$ and \\rfm{orthXYZ}, $\\kappa^{\\mu}$ and $\\omega^{\\mu}$ can be written as\n\\begin{eqnarray}\n\\kappa^\\alpha &=&  C_{\\kappa X}(\\tau) X^\\alpha + C_{\\kappa Y}(\\tau) Y^\\alpha + C_{\\kappa Z}(\\tau) Z^\\alpha,  \\nonumber\\\\\n\\omega^\\alpha &=&  C_{\\omega X}(\\tau) X^\\alpha + C_{\\omega Y}(\\tau) Y^\\alpha + C_{\\omega Z}(\\tau) Z^\\alpha, \\label{decom}\n\\end{eqnarray}\nwhere one can notice that the scalar functions depend only on proper time $(\\tau)$.\\\\\nPutting \\rfm{decom} in \\rf{Scons} and then using the projection method, we project the resulting tensor on different combination of basis vectors $U_\\alpha X_\\beta$, $U_\\alpha Y_\\beta$, $U_\\alpha Z_\\beta$, $Y_\\alpha Z_\\beta$, $X_\\alpha Z_\\beta$ and $X_\\alpha Y_\\beta$, we get the six equations of motions as\n\\begin{equation}\n{\\rm diag}\\LR\n\\cal{L}, \\cal{L}, \\cal{L}, \\cal{P}, \\cal{P}, \\cal{P}\\right) \\,\\,\n\\Dot{{\\boldsymbol C}} ={\\rm diag}\\LR\n{\\cal{Q}}_1, {\\cal{Q}}_1, {\\cal{Q}}_2, {\\cal{R}}_1, {\\cal{R}}_1, {\\cal{R}}_2 \\right)\\,\\,\n{\\boldsymbol C}, \\label{cs}\n\\end{equation} \nwhere ${\\boldsymbol C} = \\LR C_{\\kappa X}, C_{\\kappa Y}, C_{\\kappa Z},  C_{\\omega X}, C_{\\omega Y}, C_{\\omega Z} \\right)$, $\\dot{(\\dots)} \\equiv U \\cdot \\partial = \\partial_\\tau$ and  \n\\begin{eqnarray}\n{\\cal L}(\\tau)&=&{\\cal A}_1-\\frac{1}{2}{\\cal A}_2-{\\cal A}_3,\\nonumber\\\\\n{\\cal P}(\\tau)&=&{\\cal A}_1,\\nonumber\\\\\n{\\cal{Q}}_1(\\tau)&=&-\\left[\\dot{{\\cal L}}+\\frac{1}{\\tau}\\left( {\\cal L}+ \\frac{1}{2}{\\cal A}_3\\right)\\right],\\nonumber\\\\\n {\\cal{Q}}_2(\\tau)&=&-\\left(\\dot{{\\cal L}}+\\frac{{\\cal L}}{\\tau}   \\right),\\nonumber\\\\\n  {\\cal{R}}_1(\\tau)&=&-\\left[\\Dot{\\cal P}+\\frac{1}{\\tau}\\left({\\cal P} -\\frac{1}{2} {\\cal A}_3 \\right) \\right],\\nonumber\\\\\n {\\cal{R}}_2(\\tau)&=&-\\left(\\Dot{{\\cal P}} +\\frac{{\\cal P}}{\\tau}\\right).\\nonumber\n \\label{LPQR}\n \\end{eqnarray}\n %\n with \n \\begin{eqnarray}\n{\\cal A}_1 &=& \\cosh(\\xi) \\LR n_{(0)} -  {\\cal B}_{(0)} \\right) \\label{A1} ,\\nonumber\\\\ \n{\\cal A}_2 &=& \\cosh(\\xi) \\LR {\\cal A}_{(0)} - 3{\\cal B}_{(0)} \\right) \\nonumber \\label{A2} , \\\\ \n{\\cal A}_3 &=& \\cosh(\\xi)\\, {\\cal B}_{(0)}\\label{A3},\\nonumber\n\\end{eqnarray}\n\\rfm{cs} implies that the ${C}$ functions evolve independently of each other for the case of Bjorken flow and, ${C}_{\\kappa X}$ and ${C}_{\\kappa Y}$ (similarly ${C}_{\\omega X}$ and ${C}_{\\omega Y}$) follows the same form of evolution equations due to the rotational invariance.\\\\\nCharge current conservation \\rfn{Ncons} for Bjorken type flow is expressed as\n\\begin{eqnarray}\n\\frac{dn}{d\\tau}+\\frac{n}{\\tau}=0\\label{ns}\n\\end{eqnarray}\nwhereas the energy and linear momentum conservation law \\rfn{Tcons} (after projecting on $U$) yields\n\\begin{eqnarray}\n\\frac{d\\varepsilon}{d\\tau}+\\frac{(\\varepsilon+P)}{\\tau}=0.\\label{ts}\n\\end{eqnarray} \n\\section{Particle spin polarization at freeze-out}\nTo calculate the mean spin polarization per particle, the following formula is used~\\cite{Florkowski:2018ahw}\n\\begin{eqnarray}\n\\langle\\pi_{\\mu}\\rangle=E_p\\frac{d\\Pi _{\\mu }(p)}{d^3 p}\/E_p\\frac{d{\\cal{N}}(p)}{d^3 p}, \\label{pi}\n\\end{eqnarray}\nwith $E_p\\frac{d\\Pi _{\\mu }(p)}{d^3 p}$ being the total value of the Pauli-Luba\\'nski vector (after integrating over the freeze-out hypersuface, $\\Delta \\Sigma_{\\lambda }$),\n\\begin{eqnarray}\nE_p\\frac{d\\Pi _{\\mu }(p)}{d^3 p} = -\\f{ \\cosh(\\xi)}{(2 \\pi )^3 m}\n\\int\n\\Delta \\Sigma _{\\lambda } p^{\\lambda } \\,\ne^{-\\beta \\cdot p} \\,\n\\tilde{\\omega }_{\\mu \\beta }p^{\\beta },  \\nonumber\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nE_p\\frac{d{\\cal{N}}(p)}{d^3 p}&=&\n\\f{4 \\cosh(\\xi)}{(2 \\pi )^3}\n\\int\n\\Delta \\Sigma _{\\lambda } p^{\\lambda } \n\\,\ne^{-\\beta \\cdot p}, \\nonumber\n\\end{eqnarray}\nis the total momentum density of both particles and antiparticles with four-momentum given as $p^\\lambda = \\left( m_T\\ch y_p,p_x,p_y,m_T\\sh y_p \\right)$.\\\\\nAfter performing the the canonical boost \\cite{Leader:2001gr} of \\rfn{pi}, we obtain the polarization vector $\\langle\\pi^{\\star}_{\\mu}\\rangle$ in the local rest frame of the particle as\n\\begin{eqnarray}\n\\langle\\pi^{\\star}_{\\mu}\\rangle=-\\frac{1}{8m }\\left[\\begin{array}{c}\n0 \\\\ \\\\\n\\left(\\frac{p_x\\sh y_p }{b}\\right) a_i +\n  \\LR\\frac{ \\chi \\,p_x \\ch y_p  }{b} \\right) a_j\\!+\\!2 C_{\\kappa Z} p_y  \\!-\\!\\chi C_{\\omega X}{m}_T \\\\ \\\\\n %\n\\left(\\frac{p_y\\sh y_p }{b}\\right)a_i+   \\LR\\frac{ \\chi \\,p_y \\ch y_p  }{b} \\right) a_j\\!-\\!2 C_{\\kappa Z} p_x \\!-\\!\\chi C_{\\omega Y}{m}_T \\\\ \\\\\n %\n -\\left(\\frac{m\\ch y_p+m_T}{b}\\right)a_i\n-\\LR\\frac{\\chi \\,m\\,\\sh y_p}{b}\\right) a_j \\nonumber\\\\\n\\end{array}\n\\right],\\\\\n\\label{PLVPPLPRF}  \n\\end{eqnarray} \nwith $a_i=\\chi\\left(C_{\\kappa X} p_y-C_{\\kappa Y} p_x\\right)+2 C_{\\omega Z} m_T$, $a_j=C_{\\omega X} p_x+C_{\\omega Y} p_y$, $b=m_T \\ch y_p+m$, and $\\chi=\\left( K_{0}\\left( \\hat{m}_T \\right)+K_{2}\\left( \\hat{m}_T \\right)\\right)\/K_{1}\\left( \\hat{m}_T \\right)$ and  $\\hat{m}_T=m_T\/T$.\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{Tmu.pdf} \n\\caption{Dependence of the temperature re-scaled by its initial value (solid black line) and the ratio of baryon chemical potential over temperature re-scaled by the initial ratio (dotted blue line) on the proper-time.}\n\\label{fig:Tmu}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{C.pdf} \n\\caption{Dependence of scalar functions $C_{\\kappa X}$ (solid black line), $C_{\\kappa Z}$ (dashed-dotted blue line), $C_{\\omega X}$ (dotted red line) and $C_{\\omega Z}$ (dashed green line) on the proper-time.}\n\\label{fig:C}\n\\end{figure}\n\\section{Results}\nHere we show the solutions of the differential equations \\rfn{cs}, \\rfn{ns}, and \\rfn{ts}. System is initialized at the initial proper time $\\tau_0 = 1$ fm with initial temperature and the initial baryon chemical potential as $T_0=T(\\tau_0)=150$ MeV and $\\mu_0=\\mu(\\tau_0)=800$ MeV, respectively. Here the system is assumed to be formed with $\\Lambda$ particles having mass $m=1116$ MeV. In Fig.~\\ref{fig:Tmu}, proper-time dependence of temperature and baryon chemical potential is depicted, where the temperature decreases with proper-time, whereas the ratio of baryon chemical potential and temperature increases with proper-time. From Fig.~\\ref{fig:C}, proper time dependence of the $C$ functions can be known describing the spin polarization evolution of the system.\\\\\nUsing the information of the thermodynamic parameters and $C$ coefficients evolution, we can calculate the different components of the mean polarization vector in the rest frame of the particles $\\langle\\pi^{\\star}_{\\mu}\\rangle$ at freeze-out, see Fig.~\\ref{fig:polarization1}.\nWe note that $\\langle\\pi^{\\star}_{y}\\rangle$ is negative reflecting the system's initial spin polarization. Because of the Bjorken symmetry which we have assumed in our calculations in this article, the longitudinal component ($\\langle\\pi^{\\star}_{z}\\rangle$) of the mean polarization vector is vanishing which is not in agreement with the quadrupole structure of the longitudinal component of the spin polarization seen in the experiment. But we note here that $\\langle\\pi^{\\star}_{x}\\rangle$ shows quadrupole structure. We however see and note that the Bjorken set-up is very simple to address the measurements done by the experiment.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.32\\textwidth]{px.pdf}\n\\includegraphics[width=0.32\\textwidth]{py.pdf}\n \\includegraphics[width=0.32\\textwidth]{pz.pdf}\n\\caption{Different components of the mean polarization of $\\Lambda$ particles in the rest frame of the particle obtained with the initial values $\\mu_0=800$~MeV,\n$T_0=155$~MeV, $C_{\\kappa, 0}=(0,0,0)$ and $ C_{\\omega, 0}=(0,0.1,0)$ for $y_p=0$.}\n\\label{fig:polarization1}\n\\end{figure} \n\\section{Summary}\nWe briefly presented the key ingredients of relativistic perfect-fluid hydrodynamics with spin framework initiated recently. From the definitions of kinetic theory for the equilibrium phase space distribution functions in the local equilibrium we obtained the equations of motions for the expansion of the the system. For the case of Bjorken type of flow we investigated the system's spin polarization dynamics, which in turn show that the scalar functions describing the dynamics of the spin polarization evolve independently of each other. These results are used to obtain the particle spin polarization at the freeze-out hypersurface. We however note that within the current simple set-up of Bjorken symmetry experimental measurements cannot be addressed properly.\n\\section*{Acknowledgments}\nI thank Wojciech Florkowski, Radoslaw Ryblewski and Avdhesh Kumar for inspiring discussions and clarifications. This research is supported in part by the Polish National Science Center Grants No.   2016\/23\/B\/ST2\/00717 and No. 2018\/30\/E\/ST2\/00432.  \n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMost SNe can be assigned to two main physical classes \\citep[e.g.,][]{Turatto03,Tur+07}: the gravitational\ncollapse of young massive stellar cores (Types Ib, Ic and II SNe) and the\nthermonuclear explosions of a white dwarf in close binary systems (Type Ia\nSNe). While SNe with no or weak hydrogen lines were classified\ninto type I, it is now understood that there are actually three\nspectroscopically and photometrically\ndistinct subclasses of SNe~I. Type Ia SNe are characterized by spectra with no\nhydrogen lines and strong \\ion{Si}{II} lines \\citep[e.g.,][]{Leibundgut2000,Liv01}.\nThese SNe appear in galaxies of all morphological types\n\\citep[e.g.,][]{BBCT99,vdBLF05}.\nType Ib SNe are characterized by spectra with no\nevident hydrogen lines, weak or absent \\ion{Si}{II} and strong \\ion{He}{I} lines. The third\nsubclass, type Ic SNe, discovered later, shows weak or absent hydrogen, helium\nlines, and \\ion{Si}{II}. Type II SNe are characterized by the obvious presence of\nhydrogen lines. This SN type displays a wide variety of properties:\ntype II Plateau SNe (SNe~IIP) with flat light curves in the first few months;\ntype II Linear SNe (SNe~IIL) with a rapid, steady decline in the same period;\nthe narrow-lined SNe (SNe~IIn), dominated by emission lines with narrow components,\na sign of energetic interaction between the SN ejecta and the circumstellar material;\nand type IIb SNe (SNe~IIb), a transitional type, which have early time spectra\nsimilar to SNe~II and late time spectra similar\nto SNe~Ib\/c \\citep[e.g.,][]{Filippenko+90,Filippenko+94}.\n\nWith the exception of SNe~Ia, all types of SNe\nare rare in early-type galaxies \\citep[e.g.,][]{vdBLF05}.\nSurprisingly, among morphologically classified hosts of SNe~Ib\/c and SNe~II \\citet{HPM+2008}\nfound 22 cases where the host has been classified as an Elliptical or S0 galaxy.\nHowever, all the early-type hosts of SNe~Ib\/c and SNe~II\ndisplay independent indicators of\nrecent star formation due to merging or gravitational interaction.\n\nAccording to theoretical models, the progenitors of SNe~Ib\/c are\nmassive O type stars that have lost most\nor all of their hydrogen (and perhaps their helium) envelopes, either by strong winds\nas in Wolf-Rayet stars \\citep[e.g.,][]{Eldridge+04}, or through the\ntransfer of material to a binary companion via Roche lobe overflow \\citep[e.g.,][]{Heger+03}.\nProgenitors of SNe~II are massive stars that retain their hydrogen\nenvelopes \\citep[e.g.,][]{WW86,Hamuy03rev}.\n\nMany observational studies confirm these theoretical\ninterpretations of\nthese classes of core-collapse SNe (CCSNe).\nThe rates of all CCSN types depend on the morphology of the\nhost galaxies. The rate of SNe~Ib\/c and II per unit stellar mass increases by\nfactors of 3 and 5 respectively from early- to late-type spirals host\ngalaxies \\citep[e.g.,][]{Mannucci+05}. Given the short lifetime of their\nmassive progenitors, the rate of CCSNe in host galaxies directly traces the\ncurrent star formation rate \\citep[e.g.,][]{Mal03}. Conversely, when the\nstar formation rate is known, it can be used to verify the consistency\nof the progenitor scenario \\citep[e.g.,][]{HB06}.\n\nThe observation of progenitor stars of CCSNe in archival pre-explosion images\nprovides a direct test of the theoretical predictions. Since\nmassive evolved stars are the most luminous objects in a galaxy, the\nprogenitors of CCSNe should be directly detectable on pre-explosion\nimages of nearby host galaxies. The advent of data archives of large telescopes with\nhigh image quality, most importantly that of the\nHubble Space Telescope, has allowed to extend the progenitor detection to\ndistances larger than 20 Mpc \\citep[e.g.,][]{vD+03,Mau+05}.\nThe observations have generally confirmed  the theoretical predictions mentioned above\n\\citep[e.g.,][]{Crockett+07,Pastorello+07,Smartt+08}.\n\nThe spatial distribution of SNe in host galaxies provides another strong constraint on the nature of SN progenitors.\nVarious studies show that CCSNe are tightly connected to the disk \\citep[e.g.,][]{JM63}\nand to the spiral arm structure (\\citeauthor{JM63}; \\citealp{MvdB76,BTF94,vDHF96}),\nwhich dramatically differs from the SN~Ia distribution \\citep[e.g.,][]{FoSch08}. In addition, CCSNe are well\nassociated with star-forming sites: OB-associations, and \\ion{H}{ii} regions\n\\citep{BTF94,vDHF96,TBP01}.\nSNe~Ib\/c show a higher degree of association with \\ion{H}{ii} regions\nthan those of type II SNe, suggesting that they arise from a higher mass\nprogenitor population than the SNe~II \\citep{AndeJa08}.\nMost recently, \\citet{AndeJa09} found that the radial positions of a sample of 177 CCSNe closely follow the\nradial distribution of H$\\alpha$ emission, implying that these SNe are excellent tracers of star formation\nwithin galaxies.\n\nOne can better constrain the nature of the progenitors of CCSNe by\nconsidering quantitative measures of\ntheir radial distribution within their host galaxies,\nafter correcting for the inclination of the disk.\nIn their pioneering study, \\citeauthor{JM63} found a rapidly decreasing\nsurface density distribution of SNe, except for an important lack of SNe in the\ncentral regions of spiral galaxies.\nThe surface density of SNe is known to be exponential \\citep{BCC75,VZ77},\nalthough two studies \\citep{IK75,GKK80} suggested a ring like distribution.\n\\citeauthor{BCC75} quantified the\nexponential to have a slope that amounts to a scale length of $0.46\\pm0.03$\ntimes the optical radius, while \\citeauthor{VZ77} found\na scale length of $0.12\\pm0.01$ times the optical radius.\nThe difference in scale length arises from the different definitions of\noptical radius.\nAll SN types, even the SNe Ia, were included in these early studies.\nUsing a large sample of CCSNe (of known types), i.e. 74 SNe~Ib\/c and SNe~II,\n\\citet{vdB97} found  a scale length of $0.22\\,R_{25}$\n(with no error bar given), where  $R_{25}$ is the isophotal radius for the\nblue-band surface brightness of $\\mu_B=25\\,\\rm mag\\,arcsec^{-2}$.\nAlso, \\citet{Bar+92} found that their 99 SNe~II have an exponential\ndistribution with scale length that amounts to $(0.27\\pm0.08)\\,R_{25}$,\nextending far beyond the optical radius of galaxies.\nMoreover, \\citeauthor{Bar+92} noticed a sharp decrease in slope of\nboth SNe~II and SNe~Ib at $1.4\\,R_{25}$.\n\nIt has rapidly become apparent that SNe~Ib\/c are more centrally concentrated\nwithin galaxy disks than are SNe~II.\n\\citeauthor{Bar+92} noticed that their 22 SNe~Ib show a non-exponential\ndistribution that increases slope with radius so that the scale\nlength varies from $(0.15\\pm0.02)\\,R_{25}$ inside\n$0.5\\,R_{25}$ to $(0.10\\pm0.02)\\,R_{25}$ between 0.5 and $1.0\\,R_{25}$.\nThe more centrally concentrated distribution of SNe~Ib\/c in comparison to that\nof SNe~II was also noticed by \\citet{vdB97} and \\citet{Wang+97},\nbut, in contrast to the study of \\citeauthor{Bar+92},\nthe difference with the distribution of SNe~II was not statistically\nsignificant.\n\nFinally, \\citet{TPB04} found that, while in the central regions,\nSNe~Ib\/c are more concentrated than are the\nSNe~II, in the outer regions the two distributions are similar.\nThis result is at odds with the results of \\citet{Bar+92}, who found\nincreasingly dissimilar slopes for the surface densities of SNe~II and of\nSNe~Ib\/c in the outer regions of spirals.\n\nAdditional insight is obtained by comparing the radial distribution of CCSNe\nin galaxies of different activity or environment.\n\\citet{PT90} found that their sample of 8 SNe~II and SNe~Ib within galaxies\nhosting AGN were significantly more radially concentrated in\ntheir galaxy hosts than analogous CCSNe in galaxies without active nuclei.\n\\citet{Petr+05} studying a sample of 12 SNe~II and SNe~Ib\/c\nin galaxies hosting AGN,  confirmed the above result and found that\nSNe~Ib\/c in active\/star-forming galaxies are more centrally concentrated\nthan are the SNe~II, but given the small sample, this difference was not\nstatistically significant. These results were confirmed with larger samples\nof CCSNe by \\cite{Hak08}, who used\nboth one-dimensional and multivariate statistics.\n\nThe locations of SN explosions in multiple systems have also been studied.\nIn interacting galaxies, CCSNe are not preferentially located towards the\ncompanion galaxy \\citep{Petr+95,Nav+01}. Similarly, the\nazimuthal distributions inside the host members of galaxy groups are consistent\nwith being isotropic (\\citealp{Nav+01}).\n\nIn this paper, we make use of a considerably larger sample of CCSNe to\nreanalyze the distribution of deprojected SN radii normalized both to $R_{25}$ and, for the first time,\nto indirect estimates of individual stellar disk scale lengths.\nWe make statistical comparisons between the radial distributions of CCSNe and\nthe distribution of blue light and \\ion{H}{ii} regions in the disks of spiral galaxies.\n\nThe plan of the paper is as follows. The samples of CCSNe and host galaxies\nare presented in Sect.~\\ref{sample}. We discuss the normalizations in Sect.~\\ref{norm}.\nThe results, given in Sect.~\\ref{results}, are discussed and summarized in\nSect.~\\ref{discus}. Throughout this article we have assumed a value of\n$H_0=75 \\,\\rm km \\,s^{-1} \\,Mpc^{-1}$ for the Hubble constant.\n\n\n\\section{The CCSN and host galaxy sample}\n\\label{sample}\n\nThe present investigation is based upon the Asiago Supernova Catalogue (ASC)\n\\citep{BBCT99}\\footnote{http:\/\/web.oapd.inaf.it\/supern\/cat}, updated on\n30~September 2008.\nThis version contains 4730 SNe (86 SNe~I,\n2196 SNe~Ia, 304 SNe~Ib\/c, 1141 SNe~II and 1003 unclassified SNe) and data for\ntheir host galaxies.\nNote that for SNe~II it includes also subtypes IIP, IIL, IIn, IIb.\nSNe classified as type~I or type~Ia, or for which the classification was uncertain\n(marked as : and ?) or SNe classified  from the light curve only\n(labelled as *) were excluded from the present study.\nThe last SN included in the present investigation is SN~2008fq,\ndiscovered on 15~September 2008.\n\nThe analysis of the radial distribution of SNe in\nhost galaxies requires a  well-defined sample.\nWe have selected all CCSNe and their hosts with the following criteria:\n\\begin{enumerate}\n\\item existing SN spectroscopic classification;\n\\item host galaxy with morphological type between Sa and Sd,\n      excluding peculiar morphologies;\n\\item host galaxy inclination $i \\leq 50^\\circ$ to minimize absorption and\n      projection effects ($0^\\circ$ means face-on);\n\\item known position angle of the major axis of the host galaxy;\n\\item known host galaxy heliocentric radial velocity;\n\\item known integrated $B$ magnitude of the host galaxy;\n\\item known position of the SN with respect to the galaxy center.\n\\end{enumerate}\n\nThese criteria led to the selection of 239 CCSNe within 216 host galaxies, of which 61 are of type Ib\/c\n(in 61 hosts) and 178 are of type II (in 162 hosts).\nOur sample includes galaxies with multiple SNe, in particular 19 galaxies with 2 SNe and 1 with 5 SNe.\nIn order to use homogeneous, updated  information\nfor all these galaxies, we re-extracted all the main galaxy data from\nHyperLeda\\footnote{http:\/\/leda.univ-lyon1.fr} \\citep{Paturel+03}\nsuch as absolute magnitudes and isophotal diameters,\nboth corrected for Galactic and host galaxy internal extinction\n(\\citealp{SFD98,BGPT95}, respectively)\nand the former corrected for Virgocentric infall \\citep{TPE02}.\n\nThe distribution of\nmorphological types of the 216 host galaxies is peaked around Sbc and Sc.\n130 host galaxies are not  barred with\n38 SNe of type Ib\/c and 105 SNe of type II. 86 hosts are barred and\nhave 23 SNe of type Ib\/c and 73 SNe of type II.\n\nBecause of projection effects, some of the SNe showing small\ndistances to the center may lie quite far away from the center.\nWith the assumption that the CCSNe have young progenitors\nthat are located in the disks of the galaxies, a more realistic separation\nmay be derived, using the inclinations of the host galaxies.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\hsize]{sngeometry1.eps}\n\\caption{Location of the SN within its host galaxy. The\ncenter of the galaxy is at the origin of coordinate systems\nand the \\emph{star symbol} is the projected location of the SN.\n$\\Delta \\alpha$ and $\\Delta \\delta$ are offsets of the SN in equatorial coordinate\nsystem, \\textsc{u} and \\textsc{v} are coordinates of the SN in host galaxy coordinate system\nalong the major (U) and the minor (V) axis, respectively.\nThe \\emph{inset in the upper-left corner} illustrates the inclination of\nthe polar axis of the galaxy with respect to the line of sight.\n\\label{snloc}}\n\\end{figure}\nFig.~\\ref{snloc} illustrates the\ngeometrical location of a SN in the plane of the host galaxy.\nThe coordinates of the SN in its host galaxy coordinate system are\n\\begin{equation}\n\\begin{split}\n \\textsc{u} = \\Delta \\alpha \\,\\sin {\\rm PA} + \\Delta \\delta \\,\\cos {\\rm PA} \\ ,\\\\\n \\textsc{v} = \\Delta \\alpha \\,\\cos {\\rm PA} - \\Delta \\delta \\,\\sin {\\rm PA} \\ , \\\\\n\\end{split}\n\\label{uvdefs}\n\\end{equation}\nand the true radial distance of the SN to the galactic center satisfies\n\\begin{equation}\nR_{\\rm SN}^2 = {\\textsc{u}}^2+\\left ({{\\textsc{v}}\\over \\cos i} \\right)^2 \\ .\\\\\n\\label{RSN}\n\\end{equation}\n\n\n\\section{Radius normalization}\n\\label{norm}\n\nFollowing \\citet{PT90}, \\citet{Bar+92} and \\citet{vdB97},\nwe first normalize the SN radius to the $25^{\\rm th}$ magnitude isophotal radius,\n$R_{25}$ (corrected for dust extinction).\n\nThe radial surface brightness distribution of spiral disks are well\ndescribed by an exponential law \\citep[e.g.,][]{Free70}\n$\\Sigma^d(R)=\\Sigma_0^d \\exp(-R\/h)$,\nwhere $R$ is radius measured along the disk from the center,\n$h$ is the scale length of the exponential disk,\nand $\\Sigma_0^d$ is the central surface brightness.\nIn terms of $r=R\/R_{25}$, the surface brightness distribution of disks is\n$\\Sigma^d(r)=\\Sigma_0^d \\exp(-r\/\\tilde{h})$, where the scale length follows\nthe relation\n\\begin{equation}\n\\tilde{h} = {h\\over R_{25}} = {2.5\/\\ln 10\\over 25-\\mu_0^{\\rm disk}} \\ , \\\\\n\\label{mju}\n\\end{equation}\nwhere $\\mu_0^{\\rm disk}$ is the \\emph{disk} central $B$-band surface brightness.\n\n\\citet{Free70} asserted that the extrapolated disk central surface blue brightness,\ncorrected for inclination and Galactic absorption,\nis almost constant and equal to $\\mu_0^{\\rm Freeman}=21.65\\pm0.30$ $\\rm mag \\, arcsec^{-2}$.\nInserting $\\mu_0^{\\rm Freeman}$ into equation~(\\ref{mju}),\nwe obtain the stellar disk scale length of $\\tilde{h}_{\\rm Freeman}=0.32\\pm0.03$.\nHereafter, we denote this stellar distribution as the \\emph{Freeman disk}.\n\nHowever, the Freeman disk is an oversimplified model:\nthe central disk surface\nbrightness is fainter for late-type spirals, and also decreases with\nincreasing  scale length $h$, as found by \\citet{GdB01} in the $B$ and $R$\nbands, as well as \\citet{Graham01} in the $K$ band.\n\nTherefore, rather than assuming that the surface density profile of SNe has a\ncharacteristic scale that is proportional to the isophotal radius of their\nhost galaxy, as above, we can alternatively assume  that this SN scale is proportional\nto the stellar disk scale length of their host galaxy, and thus normalize the\nSN galactocentric radii to the scale lengths of the disks of their host galaxies.\n\nIdeally, we could perform a bulge\/disk decomposition for each of our galaxies\nto obtain the individual disk scale length. However, this is beyond the scope of the\npresent paper, and we have chosen instead to adopt a statistical approach, by\nrelying on the fits of $\\log h$ as a function of disk\nabsolute magnitude $M_{\\rm disk}$ by \\citet{GW08}.\nNoting that the slope in the $\\log h$ vs. $M_{\\rm disk}$ relation of types Sc and earlier\ndepends little on the waveband (Table~9 of \\citeauthor{GW08}), we assume that\nthe same holds for later morphological types, and thus take the slopes in\nthe $B$ band from those established by \\citeauthor{GW08} (their Table~10)\nin the $K$ band for\neach morphological type. We also assume that the difference\nin $\\log h$ normalizations between the $B$ and $K$ bands is independent of\nmorphological type. This finally gives us\n\\begin{equation}\n\\log h = a - 0.4\\, b \\left (M_{\\rm disk}+20\\right) \\ ,\n\\label{heq}\n\\end{equation}\nwhere $h$ is in kpc and $M_{\\rm disk}$ is the disk absolute blue magnitude.\nThe values for the normalization $a$ and slope $b$ are given in\nTable~\\ref{hparstab}. Note that \\citeauthor{GW08} obtained their parameters\nafter correcting for extinction.\n\n\\begin{table}[ht]\n\\begin{center}\n\\caption{Parameters of the scale length vs. luminosity relation (eq.~[\\ref{heq}]).\n\\label{hparstab}}\n\\begin{tabular}{lcc}\n\\hline\n\\hline\nMorphological type & $a$ & $b$ \\\\\n\\hline\nSa & 0.35 & 0.70 \\\\\nSb & 0.39 & 0.58 \\\\\nSc & 0.42 & 0.39 \\\\\nSd & 0.46 & 0.23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe estimate the disk magnitudes by adopting the median\n$B$ band bulge-to-disk ratios $\\log({\\rm B\/D})$ from Fig.~7 of \\citet{GW08}, i.e.\n$M_{\\rm disk} = M +2.5\\,\\log \\left (1+ {\\rm B\/D} \\right )$.\n\nFig.~\\ref{RsnR25h} illustrates the comparison of the two normalized radius\nindicators. Also shown is the best fit line\n\\begin{equation}\n \\log \\left ({h \\over R_{\\rm 25}}\\right ) = (-0.30\\pm0.04) +\n (-0.27\\pm0.04) \\, \\log \\left ({R_{25}\\over {\\rm kpc}}\\right) \\ .\n\\label{RSNoverhfit}\n\\end{equation}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\hsize]{RsnR25_h.eps}\n\\caption{Comparison of $h \/ R_{25}$ and $R_{25}$, where the scale\nlength $h$ is computed according to equation~(\\ref{heq}) and\nTable~\\ref{hparstab}.\n\\emph{Red filled circles} and \\emph{blue crosses} respectively show the\nearly-type (Sa-Sbc) and late-type (Sc-Sd) galaxies.\nThe \\emph{solid line} is a best fit to the full sample of CCSNe.}\n\\label{RsnR25h}\n\\end{figure}\nThe galaxies with large isophotal radii have a smaller ratio of disk scale\nlength to isophotal radius, suggesting (eq.~[\\ref{mju}]) that large spiral\ngalaxies\nhave higher central disk surface brightness, as found by\n\\citet{Driver+06}. Moreover, since\nthe central disk surface brightness can be written\n\\begin{equation}\n\\mu_0^{\\rm disk} = M_{\\rm disk} + 5\\,\\log h + 38.74 \\ ,\n\\label{mu01}\n\\end{equation}\nwhere $h$ is in kpc, and combining equations~(\\ref{heq}) and (\\ref{mu01}),\none finds that\n$\\mu_0^{\\rm disk} = 2.5\\,b\/(2\\,b-1)\\,\\log h + constant$, i.e. higher central surface\nbrightness for larger scale length when $b<1\/2$, which, according to\n\\citeauthor{GW08}, occurs for late-type\nspirals (see Table~\\ref{hparstab} above).\n\nOn average, we find $\\langle h \/R_{25} \\rangle = 0.26\\pm0.02$ (where the errors here\nand below are on the mean).\nUsing equation~(\\ref{mju}), one deduces $\\mu_0^{\\rm disk} = 20.82\\pm0.30$,\ni.e. about 0.8 magnitude brighter than the Freeman disk.\n\nFrom a statistical point of view the central disk surface brightness\nfor the galaxies of our sample can be obtained in either two ways:\n\\begin{enumerate}\n\\item Given the galaxy isophotal radius $R_{25}$ and \\citeauthor{GW08}'s\nstatistical estimate of $h$, using equation~(\\ref{mju});\n\\item Given the galaxy absolute magnitude $M$ and\n\\citeauthor{GW08}'s statistical estimates of $M_{\\rm disk}-M$\n(given the galaxy type) and $h$\n(given the galaxy type and disk luminosity),\nusing equation~(\\ref{mu01}).\n\\end{enumerate}\nThe first method yields\n$\\mu_0^{\\rm disk}=20.91^{+1.36}_{-1.00}$, 3\/4\nof a magnitude  brighter than the Freeman disk,\nwith some scatter, while the second method yields\n$\\mu_0^{\\rm disk}=20.79^{+0.15}_{-0.26}$, which is even brighter, but with very little scatter.\nThe distribution of the central disk surface brightness is given in\nFig.~\\ref{mu0dist}.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\hsize]{mu0hist.eps}\n\\caption{Distribution of central disk surface brightness obtained from\nequations~(\\ref{mju}) (\\emph{open bars}) and (\\ref{mu01}) (\\emph{filled\n  bars}),\nusing our statistical measurement of disk scale length.\n\\label{mu0dist}}\n\\end{figure}\n\nWhich measure of $\\mu_0^{\\rm disk}$ is more accurate?\nThe uncertainty on the first measure of $\\mu_0^{\\rm disk}$ (eq.~[\\ref{mju}]) is\n\\[\n\\sigma_{\\mu_0^{\\rm disk}}^2 = \\left ({2.5\\over \\ln 10}\\,{R_{25}\\over h}\\right )^2\\,\n\\left [\\left ({\\sigma_h\\over h}\\right )^2 + \\left ({\\sigma_{R_{25}}\\over\n    R_{25}} \\right )^2 \\right ] \\ .\n\\]\nUsing equation~(\\ref{mu01}), the uncertainty on the second measure\nof $\\mu_0^{\\rm disk}$ is\n\\[\n\\sigma_{\\mu_0^{\\rm disk}}^2 = \\left ({5\\over \\ln 10}\\right )^2\\,\\left ({\\sigma_h\\over\n  h}\\right )^2 + \\sigma_{M_{\\rm disk}-M}^2 + \\sigma_M^2 \\ .\n\\]\nHence the difference in the square uncertainties\non the first (eq.~[\\ref{mju}]) and second (eq.~[\\ref{mu01}]) measures of\n$\\mu_0^{\\rm disk}$ is equal to\n\\[\n\\left ({2.5\\over \\ln 10}\\right)^2\n\\left[\\left ({R_{25}\\over h}\\right  )^2\\!-\\!4\\right]\n\\left ({\\sigma_h\\over  h}\\right )^2\n+ \\left ({2.5\\over \\ln 10}\\right)^2\\left ({\\sigma_{R_{25}}\\over R_{25}}\n\\right )^2  -  \\sigma_{M_{\\rm disk}-M}^2 - \\sigma_M^2\\,.\n\\]\nGiven that, on average, $R_{25}\/h \\simeq 3.6$, i.e. $(R_{25}\/h)^2 = 12.8$,\nand given typical uncertainties $\\sigma_h\/h \\approx 0.25$ (from Fig.~10a of\n\\citeauthor{GW08}), $\\sigma_{M_{\\rm disk}-M} = 2.5 ({\\rm B\/D})\/(1+{\\rm B\/D})\\,\n\\sigma \\log ({\\rm B\/D}) \\la\n0.25$ (from Fig.~6 of \\citeauthor{GW08}), then even with negligible errors on\n$R_{25}$ the uncertainty on the first measure of $\\mu_0^{\\rm disk}$ (eq.~[\\ref{mju}]) is\nconsiderably greater than that of the second measure of $\\mu_0^{\\rm disk}$\n(eq.~[\\ref{mu01}]).\nWe therefore adopt equation~(\\ref{mu01}) to estimate $\\mu_0^{\\rm disk}$.\n\nIn comparison, repeating the same analysis for the 1500 galaxies in a\nreference sample found in HyperLeda with\nthe same selection as in points 2 to 6 of Sect.~\\ref{sample} (plus a limit on\nredshift), we find (with eq.~[\\ref{mu01}]) a mean central disk\nsurface brightness of $\\left\\langle \\mu_0^{\\rm disk} \\right\\rangle = 20.78$,\nvery similar to the central surface brightness inferred from\nequation~(\\ref{mu01}),\nindicating that SN hosts do not have centrally brighter disks than\nother galaxies selected in the same way.\n\n\n\\section{Data analysis and results}\n\\label{results}\n\n\\subsection{Isophotal radius normalization}\n\\label{resisonorm}\n\nFig.~\\ref{his1} presents the histograms of the relative radial distributions\nof types Ib\/c and II SNe in the sample of spiral host galaxies.\n45 out of 61 SNe~Ib\/c (74\\%) are found to be\nlocated within relative distance $R_{\\rm SN}\/R_{25}=0.5$ from  the nuclei of\ntheir host galaxies, compared to 79 out of 178 (44\\%) of all SNe~II.\nThe average relative radial distances\nfor types SNe~Ib\/c, SNe~II, and all CCSNe are\n$0.45\\pm0.04$ ($N_{\\rm SN}=61$), $0.62\\pm0.03$ ($N_{\\rm SN}=178$) and\n$0.58\\pm0.03$ ($N_{\\rm SN}=239$), respectively.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\hsize]{hist1.eps}\n\\caption{Distribution of deprojected isophotal-normalized radial distances\n  for SNe of types Ib\/c (\\emph{open bars}) and II (\\emph{filled bars}).}\n\\label{his1}\n\\end{figure}\n\nThe observational deficit of SNe\nin the central regions of remote galaxies relative to nearer\ngalaxies, the \\cite{Shaw79} effect, is found to be\nmore important for deep photographic SN searches and negligible\nfor visual and CCD searches in nearby galaxies \\citep[e.g.,][]{CTTB+97}.\nThe relative lack of CCSNe at the largest relative galactocentric distances\nfor nearby\ngalaxies might be caused by the limited fields of view of\nvarious SN search programs.\n\nFig.~\\ref{RsnR25V}, similar to Fig.~1 of\n\\cite{Wang+97}, illustrates the relative\ndistance of CCSNe from the centers of their hosts\nvs. the radial velocity of the host galaxy.\nSpearman rank correlation tests indicate that SNe~II show no trend for\n$R_{\\rm SN}\/R_{25}$ vs. host galaxy radial velocity (rank correlation\ncoefficient $r_{\\rm S} = -0.002$). However, these tests indicate strong\npositive trends between galaxy radial velocity and $R_{\\rm SN}\/h$ ($r_{\\rm S}\n= 0.13$ with probability $P=0.04$ of a stronger trend occurring by chance) for SNe~II\non one hand, and both $R_{\\rm SN}\/R_{25}$ ($r_{\\rm S}=0.19, P=0.06$) and $R_{\\rm\n  SN}\/h$ ($r_{\\rm S}=0.20, P=0.05$) for SNe~Ib\/c\non the other hand.\nHowever, if one removes the nearest galaxies ($V_{\\rm r} < 1000 \\, \\rm km \\,\ns^{-1}$),\nthe trends become weaker and no longer statistically significant:\n$r_{\\rm S}=0.10, P=0.10$ for SNe~II with scale length\nnormalization, and $r_{\\rm S}=0.08, P=0.27$ for SNe~Ib\/c with either\nnormalization.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\hsize]{RR25Vr.eps}\n\\caption{Deprojected isophotal-normalized SN-to-host distances\nvs. radial velocity of host galaxy (as a proxy for distance).\nSNe~Ib\/c and SNe~II are shown as \\emph{green filled circles} and\n\\emph{red plus signs}, respectively.}\n\\label{RsnR25V}\n\\end{figure}\n\nWe therefore limit our CCSN sample to distances $V_{\\rm r} > 1000 \\, \\rm km \\,\ns^{-1}$, thus retaining 224 SNe of the original 239, living in 204 host galaxies.\nTable~\\ref{tabl1} displays the matrix of SNe type versus host galaxy\nmorphological type in this reduced sample.\nThe SNe~Ib\/c are equally split into 25 early-type spiral hosts (Sa to Sbc) and 30\nlate-type (Sc to Sd), while among the SNe~II, 102 are in early-type spirals and only\n67 in late-types. But this preference for late-type spirals of SNe~Ib\/c is\nnot statistically significant (from binomial statistics).\n\n\\begin{table}[ht]\n\\begin{center}\n  \\caption{Distribution of SN types and host galaxy morphological types.}\n  \\label{tabl1}\n  \\begin{tabular}{l r r r r r r r r} \\hline\n  \\hline\n  &\\multicolumn{1}{c}{  Sa} &\\multicolumn{1}{c}{  Sab} &\\multicolumn{1}{c}{  Sb} &\\multicolumn{1}{c}{  Sbc} &\\multicolumn{1}{c}{  Sc} &\\multicolumn{1}{c}{  Scd} &\\multicolumn{1}{c}{  Sd} &\\multicolumn{1}{c}{ Total} \\\\ \\hline\n  SNe~Ib\/c & 2 & 3 & 10 & 10 & 26 & 4 & 0 & 55 \\\\\n  SNe~II & 14 & 7 & 34 & 47 & 38 & 17 & 12 & 169 \\\\\n  $N_{\\rm gal}$ & 16 & 10 & 42 & 48 & 57 & 20 & 11 & 204 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\noindent\nNotes: The bottom row lists the number of unique host galaxies.\n\\end{table}\n\nThe distribution of radial distances can be better understood by considering the\nnormalized surface density profile of SNe within their host\ngalaxies. We have determined the\nsurface density of CCSNe, $\\Sigma_{j}^{\\rm SN} = n_{j} \/ \\pi (r_{j+1}^2 - r_{j}^2)$ where $r_j$\nis the relative inner radius of the circular annuli of width $r_{j+1} - r_{j}$ ($j$-th bin),\nand $n_{j}$ indicates the number of CCSNe in the $j$-th bin.\nThe top left panel of Fig.~\\ref{sdensall} shows that the stacked surface density\ndistribution of all 224 CCSNe appears consistent with an exponential law, except for a lack of\nSNe in the central regions ($R_{\\rm SN} < 0.2\\,R_{25}$).\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\hsize]{Figgrid.eps}\n\\caption{\\emph{Top}: surface density profiles (with arbitrary normalization)\nof all CCSNe, with host galaxy isophotal\nradius normalization (\\emph{left}), and with disk scale length normalization\n(\\emph{right}).\nThe \\emph{thick solid line} shows the maximum likelihood exponential surface\ndensity profile of CCSNe.\nThe \\emph{thin solid line} represents the stellar\nexponential disk of the host galaxies.\n\\emph{Bottom}: same as top panels, for SNe~Ib\/c\n(\\emph{green open circles}) and SNe~II (\\emph{red filled circles}).\nThe \\emph{thick lines} indicate the maximum likelihood\nexponential surface density profiles for the SNe~Ib\/c (\\emph{green dashed}) and SNe~II\n(\\emph{red solid}).\nThe error bars  assume a Poisson distribution\n(with $\\pm1$ object if none is found).}\n\\label{sdensall}\n\\end{figure*}\n\nIf the surface density of CCSNe is an exponential\nfunction of $\\tilde{r}=R_{\\rm SN}\/R_{25}$\n(i.e. $\\Sigma^{\\rm SN}(\\tilde{r})=\\Sigma_0^{\\rm SN} \\exp(-\\tilde{r}\/\\tilde{h}_{\\rm SN})$,\nwhere $\\tilde{h}_{\\rm SN}$ is the scale length\nfor the distribution on disk, and $\\Sigma_0^{\\rm SN}$ is the central surface\ndensity of CCSNe), then\nthe probability that a SN is observed at normalized radius $\\tilde{r}_i=R_{\\rm SN}^i\/R_{25}^i$ is\n\\begin{equation}\n\\begin{split}\n p(\\tilde{r}_i | \\tilde{h}_{\\rm SN}) =\n {2\\pi \\tilde{r}_i\\Sigma_0^{\\rm SN}\\exp\\left(-\\tilde{r}_i\/\\tilde{h}_{\\rm SN}\\right) \\over\n \\int_{0}^\\infty 2\\pi \\tilde{r} \\Sigma_0^{\\rm SN}\\exp\\left(-\\tilde{r} \/\\tilde{h}_{\\rm SN}\\right) d\\tilde{r}} \\\\\n =  {\\tilde{r}_i\\over \\tilde{h}_{\\rm SN}^2}\\, \\exp\\left(-\\tilde{r}_i\/\\tilde{h}_{\\rm SN}\\right) \\ . \\\\\n\\end{split}\n\\label{probr}\n\\end{equation}\n\nEquation~(\\ref{probr}) assumes no truncation in the normalized SN radii, and\nindeed we consider all SN, whatever their distance to the center of their\nhost galaxy. The likelihood of the set of $\\{\\tilde{r}_i\\}$ is\n\\begin{equation}\n{\\cal L}(\\tilde{h}_{\\rm SN})=\\prod_{i=1}^{N_{\\rm SN}} p(\\tilde{r}_i | \\tilde{h}_{\\rm SN}) \\ ,\n\\end{equation}\nand maximizing $\\ln {\\cal L}$ amounts to maximizing\n\\[\n{\\sum_{i=1}^{N_{\\rm SN}} \\ln \\tilde{r}_i} - {1\\over{\\tilde{h}_{\\rm SN}}}\n\\sum_{i=1}^{N_{\\rm SN}} \\tilde{r}_i - 2 \\,N_{\\rm SN} \\ln \\tilde{h}_{\\rm SN}\n\\]\nfor $\\tilde{h}_{\\rm SN}$ (where we made use of eq.~[\\ref{probr}]),\nwhich yields\n\\begin{equation}\n\\tilde{h}_{\\rm SN} = {{\\sum_{i=1}^{N_{\\rm SN}} \\tilde{r}_i}\\over {2 \\,N_{\\rm SN}}} =\n      {1\\over2}\\,\\left\\langle \\tilde{r}_i\\right \\rangle \\ ,\n\\label{htildeML}\n\\end{equation}\ni.e., the maximum likelihood exponential scale length is half the arithmetic\nmean of the radial distances.\nThe integral of the denominator of equation~(\\ref{probr})\nyields the normalization\n\\begin{equation}\n\\Sigma_0^{\\rm SN} = {{N_{\\rm SN}}\\over {2 \\pi \\tilde{h}_{\\rm SN}^2}} \\ .\n\\end{equation}\n\nWe check the goodness of fit using the Kolmogorov-Smirnov (KS) test on the\nnormalized cumulative distributions of all SN types, where the exponential model has a\ncumulative normalized distribution\n$E(\\tilde{r}) = 1 - (1 + {\\tilde{r}\/\\tilde{h}_{\\rm SN}}) \\exp (- {\\tilde{r}\/ \\tilde{h}_{\\rm SN}})$.\n\n\\begin{table}[t]\n\\begin{center}\n  \\caption{\nMaximum likelihood fits of an exponential surface density profile to the\nradial distribution of\nCCSNe.}\n\\label{hMLall}\n\\tabcolsep 3.3pt\n  \\begin{tabular}{lrcccc} \\hline\n    \\hline\n\\multicolumn{1}{c}{Sample}&$N_{\\rm SN}$&$\\tilde{h}_{\\rm SN}$&$\\hat{h}_{\\rm SN}$&$\\tilde{P}_{\\rm KS}$&$\\hat{P}_{\\rm KS}$\\\\\n\\multicolumn{1}{c}{(1)}&(2)&(3)&(4)&(5)&(6)\\\\\n\\hline\n    Full sample&224&$0.29\\pm0.01$&$1.25\\pm0.06$&0.194&0.646\\\\\n\\hline\n    Sa-Sbc&127&$0.31\\pm0.02$&$1.35\\pm0.09$&0.235&0.412\\\\\n    Sc-Sd&97&$0.28\\pm0.02$&$1.13\\pm0.07$&0.890&0.872\\\\\n    Sa-Sbc (SNe~Ib\/c)&25&$0.23\\pm0.04$&$0.93\\pm0.19$&0.098&0.213\\\\\n    Sc-Sd \\,\\ (SNe~Ib\/c)&30&$0.24\\pm0.03$&$1.03\\pm0.13$&1.000&0.986\\\\\n    Sa-Sbc (SNe~II)&102&$0.32\\pm0.02$&$1.46\\pm0.10$&0.441&0.965\\\\\n    Sc-Sd \\,\\ (SNe~II)&67&$0.29\\pm0.02$&$1.17\\pm0.09$&0.783&0.793\\\\\n    Sa-Sd \\,\\ (without bars)&133&$0.29\\pm0.02$&$1.18\\pm0.08$&0.273&0.707\\\\\n    Sa-Sd \\,\\ (with bars)&91&$0.30\\pm0.02$&$1.37\\pm0.10$&0.453&0.931\\\\\n    Sa-Sbc (without bars)&58&$0.32\\pm0.03$&$1.26\\pm0.14$&\\textbf{0.043}&0.523\\\\\n    Sa-Sbc (with bars)&69&$0.29\\pm0.02$&$1.43\\pm0.13$&0.899&0.973\\\\\n    Sc-Sd \\,\\ (without bars)&75&$0.26\\pm0.02$&$1.11\\pm0.09$&0.977&0.997\\\\\n    Sc-Sd \\,\\ (with bars)&22&$0.32\\pm0.04$&$1.19\\pm0.14$&0.884&0.829\\\\\n    $-23 < M_{\\rm disk} \\leq -20.5$&123&$0.27\\pm0.02$&$1.06\\pm0.07$&0.586&0.933\\\\\n    $-20.5 < M_{\\rm disk} < -18$&101&$0.32\\pm0.02$&$1.49\\pm0.10$&0.308&0.603\\\\\n    $19.65 < \\mu_0^{\\rm disk} \\leq 20.60$&102&$0.28\\pm0.02$&$1.35\\pm0.10$&0.895&0.840\\\\\n    $20.60 < \\mu_0^{\\rm disk} < 21.85$&122&$0.30\\pm0.02$&$1.18\\pm0.08$&0.203&0.803\\\\\n    $0^\\circ \\leq i \\leq 30^\\circ$&68&$0.26\\pm0.02$&$1.11\\pm0.10$&0.912&0.453\\\\\n    $30^\\circ < i \\leq 50^\\circ$&156&$0.31\\pm0.02$&$1.32\\pm0.08$&0.153&0.719\\\\\n\\hline\n    SNe~Ib&20&$0.23\\pm0.03$&$0.90\\pm0.11$&0.765&0.993\\\\\n    SNe~Ic&27&$0.22\\pm0.03$&$0.97\\pm0.18$&0.671&0.644\\\\\n    SNe~Ib\/c&55&$0.24\\pm0.02$&$0.99\\pm0.11$&0.476&0.707\\\\\n    SNe~II&169&$0.31\\pm0.02$&$1.34\\pm0.07$&0.328&0.746\\\\\n\\hline\n\\multicolumn{2}{l}{stars (in Freeman disk)}&$0.32\\pm0.03$&$1.23\\pm0.17$&&\\\\\n\\multicolumn{2}{l}{stars (in SN host disks)}&$0.26\\pm0.02$&\\multicolumn{1}{l}{1.00}&&\\\\\n\\multicolumn{2}{l}{\\ion{H}{ii} regions (Freeman disk)}&$0.26\\pm0.13$&$1.00\\pm0.50$&&\\\\\n\\multicolumn{2}{l}{\\ion{H}{ii} regions (SN host disk)}&$0.21\\pm0.11$&$0.80\\pm0.40$&&\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\\noindent\nNotes: Column~1 gives the CCSN sample, Col.~2 number of CCSNe in the sample,\nCol.~3 the maximum likelihood value of $\\tilde h_{\\rm SN} = h_{\\rm SN}\/R_{25}$,\nCol.~4 the maximum likelihood value of $\\hat{h}_{\\rm SN} = h_{\\rm SN}\/h$, and\nCols.~5 and 6 give the KS test probabilities that\nthe surface density distribution of CCSNe is consistent with an exponential\nlaw with the isophotal radius (Col.~5) and scale length (Col.~6)\nnormalizations.\nThe four last lines are not for the SNe, but for the stars (first two\nof these lines) and \\ion{H}{ii} regions (last two lines),\nwith the\ncentrally extrapolated surface brightness taken from the Freeman disk  (first and third of these lines)\nand the hosts (second and fourth of these lines).\nThe value of unity in Col.~4 in the second of these lines is a direct consequence of\n$h_{\\rm SN} = h$ and thus has no uncertainty associated with it.\nThe values of Col.~4 in the first of these lines is scaled relative to that\nof the the second of these lines, according to the mean\nvalue of $\\langle h \/R_{25} \\rangle$ (Sect.~\\ref{norm}).\nSimilarly the last two lines of Col.~4 are scaled to the corresponding values in Col.~3.\nThe statistically significant deviation from an exponential law is highlighted\nin bold.\n\\end{table}\n\nTable~\\ref{hMLall} presents the maximum likelihood fits of the exponential\nsurface density distributions. One sees that \\emph{all SN samples show stacked\nsurface number density distributions that are consistent with an exponential\ndistribution}, with the exception of non-barred early-type spiral hosts (among\nwhich are the three galaxies of our 224 SN host galaxy sample with CCSNe\nbeyond $2\\,R_{25}$).\n\nHowever, Table~\\ref{hMLall} indicates that \\emph{the distribution of SNe~Ib\/c is significantly\nmore centrally concentrated than the distribution of stars in a Freeman disk.}\nMore precisely, \\emph{SNe Ib\/c appear 30\\% more centrally\nconcentrated than SNe~II.}\nThe scale length of SNe~Ib\/c is consistent with the\nscale length of stars in the host galaxies, while the SNe~II appear 20\\% less\nconcentrated than the stars in the host galaxies, but as concentrated as the stars in\nFreeman disks.\nThe host galaxy morphological type appears to play no role in the\ndistribution of SNe. However, the SNe in nearly face-on host galaxies appear\nmore centrally concentrated than the SNe in more inclined host galaxies.\nThis indicates that observational selection effects (dust extinction and\nconfusion) prevent the observation of SNe near the centers of inclined disks.\n\nThe bottom left panel of Fig.~\\ref{sdensall} highlights the differences\nin the radial distributions of  SNe~Ib\/c and SNe~II.\nWhile SNe~II have a 30\\% larger scale length,\nboth classes of SN types show an important\n(statistically significant for the SNe~II) drop in\nthe center ($R_{\\rm SN}<0.2\\,R_{25}$).\n\nThe surface density distributions of \\ion{H}{ii} regions in spiral galaxy disks can\nalso be represented reasonably well by an exponential function\n\\citep[e.g.,][]{HK83,Ath+93,GGAB02}, the exceptions being mainly due to\nunsatisfactory fits of the maxima near the center. \\citeauthor{Ath+93} found\nthat the \\ion{H}{ii} regions are marginally more concentrated to the centers of spiral galaxies\nthan are the disk stars, with\na ratio of scale lengths of $h^{\\rm \\ion{H}{ii}} \/ h$ is $0.8\\pm0.4$.\nIn units of $R_{25}$, the ionized gas scale length is then\n$\\tilde{h}_{\\rm Freeman}^{\\rm \\ion{H}{ii}} = 0.26\\pm0.13$\nif we use the Freeman disk normalization, i.e.\nFreeman disk scale length of $\\tilde{h}_{\\rm Freeman}=0.32\\pm0.03$.\nIf we use the CCSN host disk normalization, i.e.\nCCSN host disk average scale length of\n$\\tilde{h}_{\\rm hosts} = \\left\\langle h\/R_{25}\\right\\rangle = 0.26\\pm0.02$,\nthen $\\tilde{h}_{\\rm hosts}^{\\rm \\ion{H}{ii}} = 0.21\\pm0.11$.\nGiven the large uncertainty on the ratio of gas to stellar disk scale\nlength, \\emph{the SN distribution is not inconsistent with that of the\nionized gas}.\n\nIn reality, the surface brightness distribution of massive young stars in galaxies\nis more complex than a simple exponential disk model: indeed, the surface\ndensity of the star\nformation rate (SFR) has a hole in the center of our galaxy \\citep{Rana+86},\nand in other galaxies the SFR follows the surface density of the total (atomic and\nmolecular) gas \\citep{Buat+89,Kennicutt98}, which is known to present a hole\nin the central regions \\citep[e.g.,][]{Kenn98}.\n\n\\begin{table}[h]\n\\begin{center}\n  \\caption{Kolmogorov-Smirnov tests of the consistency of different CCSN distributions\nwith two exponential disk models.\n\\label{SNvsdisks}}\n  \\begin{tabular}{lcccc} \\hline\n    \\hline\n\\multicolumn{1}{c}{Sample}&$\\tilde{P}_{\\rm KS}^{\\rm Freeman}$ &$\\hat{P}_{\\rm KS}^{\\rm Freeman}$& $\\tilde{P}_{\\rm KS}^{\\rm hosts}$&$\\hat{P}_{\\rm KS}^{\\rm hosts}$\\\\\n\\multicolumn{1}{c}{(1)}&(2)&(3)&(4)&(5)\\\\\n\\hline\nFull sample&\\textbf{0.002}&\\textbf{0.010}&0.895&\\textbf{0.010}\\\\\n\\hline\n    Sa-Sbc&\\textbf{0.040}&\\textbf{0.028}&0.749&\\textbf{0.012}\\\\\n    Sc-Sd&\\textbf{0.044}&0.602&0.328&0.166\\\\\n    Sa-Sbc (SNe~Ib\/c)&\\textbf{0.000}&\\textbf{0.016}&\\textbf{0.003}&0.091\\\\\n    Sc-Sd \\,\\ (SNe~Ib\/c)&0.111&0.944&0.335&0.973\\\\\n    Sa-Sbc (SNe~II)&0.434&\\textbf{0.021}&0.229&\\textbf{0.001}\\\\\n    Sc-Sd \\,\\ (SNe~II)&0.219&0.252&0.693&0.106\\\\\n    Sa-Sd \\,\\ (without bars)&\\textbf{0.006}&0.194&0.302&0.208\\\\\n    Sa-Sd \\,\\ (with bars)&0.157&\\textbf{0.036}&0.814&\\textbf{0.013}\\\\\n    Sa-Sbc (without bars)&\\textbf{0.037}&0.162&0.703&0.387\\\\\n    Sa-Sbc (with bars)&0.298&0.296&0.390&\\textbf{0.006}\\\\\n    Sc-Sd \\,\\ (without bars)&\\textbf{0.036}&0.965&0.428&0.566\\\\\n    Sc-Sd \\,\\ (with bars)&0.873&0.265&0.686&0.259\\\\\n    $-23 < M_{\\rm disk} \\leq -20.5$&\\textbf{0.003}&0.733&\\textbf{0.039}&0.947\\\\\n    $-20.5 < M_{\\rm disk} < -18$&0.359&\\textbf{0.007}&0.055&\\textbf{0.000}\\\\\n    $19.65 < \\mu_0^{\\rm disk} \\leq 20.60$&0.064&0.396&0.940&0.055\\\\\n    $20.60 < \\mu_0^{\\rm disk} < 21.85$&\\textbf{0.023}&\\textbf{0.049}&0.676&0.125\\\\\n    $0^\\circ \\leq i \\leq 30^\\circ$&\\textbf{0.032}&0.889&0.059&0.532\\\\\n    $30^\\circ < i \\leq 50^\\circ$&\\textbf{0.029}&\\textbf{0.005}&0.797&\\textbf{0.003}\\\\\n\\hline\n    SNe~Ib&\\textbf{0.019}&0.302&0.160&0.840\\\\\n    SNe~Ic&\\textbf{0.002}&0.109&\\textbf{0.042}&0.523\\\\\n    SNe~Ib\/c&\\textbf{0.000}&0.105&\\textbf{0.014}&0.621\\\\\n    SNe~II&0.102&\\textbf{0.006}&0.496&\\textbf{0.001}\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\\noindent Notes: Col.~1: sample;\nCol.~2: KS test probabilities that\nCCSN subsamples have  isophotal radius - normalized distributions consistent\nwith the Freeman disk;\nCol.~3: same as Col.~2, but with scale-length normalization;\nCol.~4: same as Col.~2, but for the host stellar disk;\nCol.~5: same as Col.~4, but with scale-length normalization.\nCCSN distributions\ninconsistent with these exponential models are highlighted in bold.\n\\end{table}\n\nTable~\\ref{SNvsdisks} presents, for the full 224 CCSN sample and\ndifferent subsamples, the  KS test probabilities that the\nisophotal-radius-normalized\ndistributions are consistent with those of the stars in a Freeman disk ($\\tilde{P}_{\\rm\n  KS}^{\\rm Freeman}$), and of the CCSN host disks ($\\tilde{P}_{\\rm KS}^{\\rm hosts}$).\nWhile there are several departures from the Freeman exponential disk,\nmost subsamples are consistent with the exponential disk that we\ninfer for the galaxy hosts. This is most significant for\nthe full CCSN sample compared to the Freeman disk,\nbecause of the central drop and of the few SNe lying very far out.\n\n\\begin{table}[ht]\n\\begin{center}\n  \\caption{Kolmogorov-Smirnov tests of the consistency of the\nradial distributions of CCSNe among different pairs of subsamples.}\n\\label{KSsubsamp}\n\\tabcolsep 5pt\n  \\begin{tabular}{llcc} \\hline\n    \\hline\n\\multicolumn{1}{c}{Subsample 1} &\\multicolumn{1}{c}{Subsample\n  2}&$\\tilde{P}_{\\rm KS}$&$\\hat{P}_{\\rm KS}$\\\\\n\\multicolumn{1}{c}{(1)} & \\multicolumn{1}{c}{(2)} & (3) & (4) \\\\\n    \\hline\n    Sc-Sd & Sa-Sbc&0.935&0.300\\\\\n    Sa-Sbc (SNe~Ib\/c)&Sc-Sd \\,\\ (SNe~Ib\/c)&0.197&0.110\\\\\n    Sc-Sd \\,\\ (SNe~II)&Sa-Sbc (SNe~II)&0.926&0.268\\\\\n    Sa-Sd \\,\\ (without bars)&Sa-Sd \\,\\ (with bars)&0.398&0.214\\\\\n    Sa-Sbc (with bars)&  Sa-Sbc (without bars)&0.711&\\,\\ 0.313*\\\\\n    Sc-Sd \\,\\ (without bars)&Sc-Sd \\,\\ (with bars)&0.294&0.282\\\\\n    $-23 < M_{\\rm disk} \\leq -20.5$&$-20.5 < M_{\\rm disk} < -18$&0.114&\\textbf{0.006}\\\\\n    $19.65 < \\mu_0^{\\rm disk} \\leq 20.60$&$20.60 < \\mu_0^{\\rm disk} < 21.85$&0.469&\\,\\ 0.723*\\\\\n    $0^\\circ \\leq i \\leq 30^\\circ$&$30^\\circ < i \\leq 50^\\circ$&0.273&0.050\\\\\n    $V_{\\rm r}\\leq5000$&$V_{\\rm r}>5000$&\\,\\ 0.992*&0.568\\\\\n    $V_{\\rm r}\\leq5000$ (SNe~Ib\/c)&$V_{\\rm r}>5000$ (SNe~Ib\/c)&\\,\\ 0.444*&\\,\\ 0.475*\\\\\n    $V_{\\rm r}\\leq5000$ (SNe~II)&$V_{\\rm r}>5000$ (SNe~II)&\\,\\ 0.924*&0.674\\\\\n    SNe~Ic&SNe~Ib&0.946&\\,\\ 0.751*\\\\\n    SNe~Ib&SNe~II&0.055&0.156\\\\\n    SNe~Ic&SNe~II&\\textbf{0.006}&\\textbf{0.035}\\\\\n    SNe~Ib\/c&SNe~II&\\textbf{0.002}&\\textbf{0.015}\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\\noindent Notes:\nCol.~1: more radially concentrated subsample;\nCol.~2: less radially concentrated subsample\n(the contrary cases are marked by an asterisks).\nCCSN distributions inconsistent within two subsamples are highlighted in bold.\n\\end{table}\n\nKS tests comparing different subsamples of CCSNe are given in Table~\\ref{KSsubsamp}.\nThese tests confirm our conclusions based upon the maximum likelihood scale\nlengths: 1) the surface density distribution of SNe~Ib\/c is highly inconsistent with that\nof SNe~II, because the former are more centrally concentrated (Table~\\ref{hMLall});\n2) the distribution of CCSNe is not significantly affected by the morphological\ntype of the host galaxy, the presence of bars, the luminosity of the host\ngalaxy or its disk inclination.\n\n\n\\subsection{Scale length normalization}\n\\label{resscalelennorm}\n\nWhen we normalize the deprojected radial distances of CCSNe to the indirectly\ndetermined scale lengths of the disks of their host galaxies, we find that\ntheir distribution is even better described by an exponential law.\nWith the exponential model of\n$\\Sigma^{\\rm SN}(\\hat{r})=\\Sigma_0^{\\rm SN} \\exp(-\\hat{r}\/\\hat{h}_{\\rm SN})$,\nwhere $\\hat{r}=R_{\\rm SN}\/h$, the maximum likelihood estimate of\n$\\hat{h}_{\\rm SN} = h_{\\rm SN}\/h$ will satisfy $\\hat{h}_{\\rm SN} = {1\\over 2}\\,\\left \\langle\n\\hat{r}_i \\right \\rangle$, as can be found in the same way as\nequation~(\\ref{htildeML}).\nThe top right panel of Fig.~\\ref{sdensall} shows that the stacked surface density\ndistribution of all 224 CCSNe appears consistent with an exponential law,\nwith only a very small drop in the central regions ($R_{\\rm SN} < 0.5\\,h$).\n\nTable~\\ref{hMLall} indicates that all subsamples have surface density profiles that\nare consistent with the exponential model. The Table also confirms that\n\\emph{the mean scale length of SNe~II is 35\\% greater than that of SNe~Ib\/c,\nwith the scale length of stars in between}.\n\nThe bottom right panel of Fig.~\\ref{sdensall} illustrates the different\nradial distributions of the SNe~Ib\/c and SNe~II.\nFor both SN types, the drop in the center is much less pronounced\nthan with the isophotal radius normalization, and not statistically significant.\n\nTable~\\ref{SNvsdisks} indicates that nearly all subsamples have\nsurface densities distributions that\nare consistent with the exponential disks of their host galaxies.\nThe exceptions are early-type, barred, and\nlow luminosity and highly inclined galaxies, as well as SNe~II\n(there are 1 SNe~Ic and 4 SNe~II lying beyond 7 scale lengths).\n\nTable~\\ref{KSsubsamp} shows that the distributions of subsamples are consistent when coupled in pairs,\nexcept that \\emph{also with scale-length normalization,\nSNe~Ib\/c are significantly more centrally concentrated than are SNe~II\nand that CCSNe are more concentrated in high luminosity host galaxies}.\n\nWe also find marginal trends for the CCSNe to be more centrally concentrated\nin face-on host galaxies, and no significant difference between\nthe radial distributions of SNe~Ib and SNe~Ic.\n\nIn addition, we performed KS tests for dividing our subsamples of both types of\nCCSNe into different distance bins, of the half most nearby and half most distant\nhost galaxies. These tests confirm that the distribution of CCSNe in our sample\n(both isophotal and scale-length normalizations) is not significantly affected\nby the distance of the host galaxy (see Table~\\ref{KSsubsamp}).\n\n\n\\section{Discussion and Conclusions}\n\\label{discus}\n\nOur results reported in Sect.~\\ref{results} (the top panels of Fig.~\\ref{sdensall},\nand Table~\\ref{hMLall}) indicate\nthat the global surface density of CCSNe within their host galaxies is exponential\nwith a scale length of $h_{\\rm SN} = 0.29\\pm0.01\\,R_{25}$, which is 30\\%\nlarger than the scale length found by \\cite{vdB97} with a 5 times smaller\nsample.\n\nThere are several reasons why the observed\nradial distribution of CCSNe might not be a perfect exponential:\n\\begin{enumerate}\n\\item the total radial distribution may be the combination of several components\n  with different distributions;\n\\item the inner disk may be destroyed or perturbed by an important bulge;\n\\item the inner CCSNe may be confused with the light of the bulge;\n\\item dust extinction may prevent the observation of CCSNe in the more opaque\n      inner regions of edge-on spiral galaxies;\n\\item the progenitors of CCSNe may follow a truncated exponential disk.\n\\end{enumerate}\n\nIndeed, the exponential surface density profile\nis the combination of a dominant exponential surface\ndensity profile of SNe~II and a secondary steeper exponential surface density\nprofile of SNe~Ib\/c\n(see bottom panels of Fig.~\\ref{sdensall}),\nso it is more prudent to consider the SN types separately.\nFor the SNe~II alone, we find a scale length of $0.31\\pm0.02\\,R_{25}$ which is\nconsistent with the result of $0.27\\pm0.08\\,R_{25}$ found by \\cite{Bar+92} with a\nsample half the size of ours.\nBut contrary to \\citeauthor{Bar+92}, we see no signs of the break in the\nexponential slope for SNe~Ib\/c at $0.5\\,R_{25}$ with\na sample nearly three times as large (bottom left panel of Fig.~\\ref{sdensall}).\nAn exponential model also fits well the surface density\nof both types of CCSNe with radii normalized to the\n(statistically-determined) disk scale lengths\n(bottom right panel of Fig.~\\ref{sdensall}).\n\nThere is a small loss of CCSNe in\nthe central regions of host galaxies\n(bottom panels of Fig.~\\ref{sdensall}).\nThis is not unexpected since the observed radial distribution of CCSNe can be\naffected by the internal dust extinction of their hosts as well as the\nconfusion with the high surface brightness of their bulges.\nIn our Galaxy, the star formation rate\nas a function of galactocentric radius does not follow a pure exponential\ndisk, but is vigorous\nnear the center and is strongly peaked around $R \\sim 5 \\,\\rm kpc$ \\citep[e.g.,][]{Kenn89}.\nSince the mean normalized\ndistances of CCSNe in inclined galaxies turn out to be\nlarger than those in face-on\nones (Tables~\\ref{hMLall} and \\ref{KSsubsamp}), dust extinction\nplays a quantitative role, and should also explain\nthe central dip in the CCSN surface density profile, mainly seen with the\nisophotal normalization\n(bottom left panel of Fig.~\\ref{sdensall}).\nThis dip cannot be explained by confusion with the light from the galaxy bulge, since\nCCSNe appear more centrally concentrated in the more luminous galaxies.\n\nThe dip in the inner distribution of SNe~II has also been recently reported by\n\\citet{AndeJa09}. These authors found that this central deficit of SNe~II is\noffset by a central excess of SNe~Ib\/c. Instead, our analysis\n(Fig.~\\ref{sdensall}) shows no central excess for SNe~Ib\/c.  In fact with\nthe isophotal normalization one sees fairly significant central dips in the\ndistributions of both SNe~II and SNe~Ib\/c, while with the scale length\nnormalization there is a small dip for the SNe~Ib\/c, and only a weak dip at\nbest for the SNe~II (neither being statistically significant).\n\nWhereas several authors have suggested that stellar disks are sharply\ntruncated at several scale lengths\n(\\citealp[e.g.,][]{Ruphy+96} found that the Milky Way is truncated at 6.5 scale lengths),\nthere are no signs of\nsuch a truncation in the distribution of CCSNe out to\n7 scale lengths (right panels of Fig.~\\ref{sdensall})\nand SNe are observed out to nearly 11 scale lengths.\nIn fact, with our fairly large sample, we find weak signs of\na shallower slope at large radii, which does not occur at\n$1.4\\,R_{25}$ as reported by \\citeauthor{Bar+92}, but at\n$R \\simeq 2.0\\,R_{25}$ for SNe~II, as we have two SNe~II near $2.4\\,R_{25}$\n(see bottom left panel of Fig.~\\ref{sdensall}).\n\nSince CCSNe are believed to originate from massive stars, one expects\nthat the radial distribution of CCSNe should resemble that of\ntracers of recent star formation\nsuch as ionized gas or perhaps even of\nfuture star formation such as molecular gas.\nThe values of scale length $\\tilde{h}_{\\rm SN}$ for CCSNe in spiral host galaxies\n(see Col.~3 of Table~\\ref{hMLall}) are in good agreement with the data reported by\n\\citet{Ath+93} on scale lengths of \\ion{H}{ii} regions on exponential disks in spiral galaxies of\ndifferent types. But the huge uncertainty on the scale length of the ionized gas prevents us\nfrom making a firm conclusion here.\n\nWe thus focused our analysis of the\nsurface density profile of CCSNe by comparing them to the surface density\nprofile of\nstars in spiral disks, which admittedly is a poorer proxy for CCSNe\nprogenitors than are the \\ion{H}{ii} regions.\nWith the isophotal radius normalization, the stacked surface density profile\nof SNe~II is similar to that of the Freeman stellar disk, while the surface\ndensity profile of the SNe~Ib\/c has a smaller scale length than that of the\nFreeman disk. Given the uncertainties on the relevance of the Freeman disk\nfor spiral galaxies, we also scaled the SN galactocentric distances to our\nstatistical estimate of the scale length of each galaxy. Then, we find that\nthe disk scale length is in between the small scale length of the\nSNe~Ib\/c and the larger scale length of the SNe~II.\n\nAs mentioned above, we also find that SNe~Ib\/c\nare significantly more concentrated towards the centers of\ntheir host galaxies than are the SNe~II, for the isophotal radius\nnormalization (bottom left panel of Fig.~\\ref{sdensall}, as well as\nTable~\\ref{KSsubsamp}).\nThis result has been known for some time\n\\citep[e.g.,][]{vdB97,Wang+97,Petr+05,Hak08} for the distribution of radii normalized to\n$R_{25}$, but we confirm the more centrally concentrated distribution of\nSNe~Ib\/c with our normalization to the disk scale lengths\n(Fig.~\\ref{sdensall} and Table~\\ref{KSsubsamp}). For both\nnormalizations, the scale length of the SN~Ib\/c distribution is roughly\n30\\% smaller than that of the SN~II distribution.\n\nWe find no statistically significant difference in the radial\ndistributions of SNe~Ib and SNe~Ic, regardless of the normalization used.\nThis result may appear in\nconflict with \\cite{AndeJa09}, who found that SNe~Ic are more centrally\nconcentrated than SNe~Ib, while our measures of the concentration are\nconsistent (see values of $\\tilde h_{\\rm SN}$ and $\\hat h_{\\rm SN}$ in\nTable~\\ref{hMLall}).\nHowever \\citeauthor{AndeJa09} admit that their result\nis not statistically significant. Moreover, the mean distance of SNe~Ic in our\nsample is pushed up by one SN~Ic that lies very far from its host galaxy\n($1.8\\,R_{25}$ and over 10 scale lengths). Still, even if this apparent\noutlier is omitted from our sample, the normalized radial distributions of\nSNe~Ib and SNe~Ic are similar enough that the KS test fails to distinguish\nthem with statistical significance. Admittedly, the analysis of\n\\citeauthor{AndeJa09} had the advantage of a much more precise measurement of the\nnormalization, as they actually went through the trouble of measuring the\nlight distribution around each host galaxy.\n\nIt is a well established fact that the metallicity\nin spiral disks decreases with increasing\ngalactocentric distance \\citep[e.g.,][]{hw99}.\nThe obvious physical explanation for the more peaked radial distribution of SNe~Ib\/c with respect to SNe~II\nis the effect of metallicity of SN progenitor environment.\nThe local metallicity of the SN progenitor environment \\citep{BP09} as well\nas the global\nmetallicity of the host galaxy \\citep[e.g.,][]{pb03,mm05,PSB08} are approximately correlated\nwith the SN progenitor metallicity: the ratio of the number of SNe~Ib\/c to SNe~II\nincreases with increasing local metallicity in CCSN hosts.\nIn this respect, \\citet{EMM02} found\nthat the number ratio of the blue to\nred supergiants and the local metallicity in the Milky Way decreases with increasing\ngalactocentric radius.\n\\cite{AndeJa09} also concluded that the more concentrated distribution\nof SNe~Ib\/c must be  related to a strong metallicity dependence on\nthe relative production of these CCSNe, with SNe~Ib\/c arising from higher\nmetallicity progenitors than SNe~II.\nNote that, since higher luminosity spirals have higher metallicity at\nthe effective radius \\citep{Garnett02}, the more concentrated distribution (with scale-length\nnormalization) of CCSNe in high luminosity galaxies may also be a sign of the\nhigher metallicity of these hosts.\n\nIn summary, our reanalysis of a considerably large sample of CCSNe has allowed us to\nderive precise surface density profiles for different subsamples of CCSNe.\nThese in turn led us to the following statistically based conclusions:\n\\begin{enumerate}\n\\item The surface density of CCSNe in all studied subsamples (different CCSN type or galaxy\n      host type or luminosity or inclination) falls exponentially with relative\n      radius $R_{\\rm SN} \/ R_{25}$ and  $R_{\\rm SN} \/h$, with no signs of\n      truncation out to 7 disk scale lengths;\n\\item The radial distribution of CCSNe is not significantly affected by the\n      host galaxy type, or by the presence of a bar in the host;\n\\item The radial distribution of CCSNe, measured with scale-length\n      normalization, is more concentrated in high luminosity host galaxies;\n\\item The radial distribution of SNe~Ib\/c is significantly more concentrated than the Freeman\n      disk distribution and consistent with the radial distribution of \\ion{H}{ii} regions;\n\\item The radial distribution of SNe~II is consistent with both the Freeman\n      disk and the \\ion{H}{ii} regions distributions, but significantly less\n      concentrated than the host disks;\n\\item There is a small lack of CCSNe\n      within one-fifth of the isophotal radius ($R_{\\rm SN} < 0.2\\,R_{25}$),\n      not well visible with scale-length normalization;\n\\item The radial distribution of type Ib\/c SNe in their host galaxies is more centrally\n      concentrated than that of type II SNe, the ratio of scale lengths is $0.77\\pm0.03$,\n      probably because of a metallicity effect.\n\\end{enumerate}\n\n\nIt would be worthwhile to extend these analyses, by comparing the radial\ndistribution of CCSNe with recent accurate measures of the distribution of\nmolecular gas (i.e. CO), as well as ionized gas (see compilation by\n\\citealp{BPBG03}) and even neutral gas, and by extending the analysis to two\ndimensions (extending previous analyses such as those by \\citealp{JM63},\n\\citealp{BTF94}, \\citealp{vDHF96}, and \\citealp{TBP01}, making use of the\nmuch larger CCSN sample used here).\n\\begin{acknowledgements}\n\nWe thank F. Matteucci, N. Prantzos, and J. Silk for useful discussions.\nThis research was supported in part by a grant\nfrom the PICS France-Arm\\'enie and a scholarship to A.A.H. from the French Government.\nA.R.P. wishes to thank the Institut d'Astrophysique de Paris\n(France) for the hospitality and support (via the EARA network) during the beginning\nstage of this work and the Osservatorio Astrofisico di Catania (Italy) for hospitality during the\nlast stage of this work.\nThis research has made use of the Asiago Supernova Catalogue, which is\navailable at http:\/\/web.oapd.inaf.it\/supern\/cat and the HyperLeda database,\navailable at http:\/\/leda.univ-lyon1.fr\/. Finally, we\nare especially grateful to our referee for his\/her constructive comments.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{\n\\section*{Acknowledgments}\n\\textcolor{black}{Many thanks to Annika Reinke (DKFZ, Germany) for designing Fig. 1}. We further thank all participants of the workshop, in particular those who filled out the questionnaire including Max Allan (Intuitive Surgical Inc., United States), Mark Asselin (Queen's University, Canada), Steven Bishop (CMR Surgical Ltd., United Kingdom), Sebastian Bodenstedt (National Center for Tumor Diseases (NCT), Germany), Harold Jay Bolingot (Kyushu Institute of Technology, Japan), Elvis Chen (Robarts Research Institute, Canada), Bijan Dastgheib (International Centre for Surgical Safety ICSS, Canada), Roger Daglius Dias (Brigham Health \/ Harvard Medical School, United States), Luc Duong (Ecole de technologie superieure, Canada), Ulrich Eck (Technical University of Munich, Germany), Isabel Funke (National Center for Tumor Diseases (NCT) Dresden, Germany), Cong Gao (Johns Hopkins University, United States), Pablo Garcia Kilroy (Verb Surgical Inc., United States), Matthias Grimm (Technische Universit\u00e4t M\u00fcnchen, Germany), Tamas Haidegger (Obuda University, Hungary), Georges Hattab (National Center for Tumor Diseases (NCT), Germany), Changyan He (Johns Hopkins University, United States), Enes Hosgor (surgical.ai, United States), Hassan Ismail Fawaz (Universit\u00e9 Haute-Alsace, France), Anthony Jarc (Intuitive Surgical, United States), Leo Joskowicz (The Hebrew University of Jerusalem, Israel), Ertugrul Karademir (German Aerospace Center (DLR), Germany), Tae Soo Kim (Johns Hopkins University, United States), Kirsten Klein (KARL STORZ SE \\& Co. KG, Germany), Michael Kranzfelder (Klinikum rechts der Isar, TU M\u00fcnchen, Germany), Shlomi Laufer (Technion, Israel), Greg Nelson (Aesculap AG, Germany), Chinedu Nwoye (University of Strasbourg, France), Molly O'Brien (Johns Hopkins University, United States), Daniel Ostler (Klinikum rechts der Isar of Technical University of Munich, Germany), Micha Pfeiffer (National Center for Tumor Diseases Dresden (NCT), Germany), Mohammad Rahbari (University Hospital Mannheim, Medical Faculty Mannheim of the University of Heidelberg, Germany), Wolfgang Reiter (Wintegral GmbH, Germany), Nicola Rieke (NVIDIA, Germany), Tobias Ro\u00df (German Cancer Research Center (DKFZ), Germany), Roozbeh Shams (\u00c9cole Polytechnique de Montreal, Canada), Amber Simpson (Memorial Sloan Kettering Cancer Center, United States), Vinkle Srivastav (University of Strasbourg, France), Sarina Thomas (German Cancer Research Center (DKFZ), Germany), Liset Vazquez Romaguera (Polytechnique Montreal, Canada), Tong Yu (University of Strasbourg, France). We further thank Tim R\u00e4dsch (German Cancer Research Center (DKFZ), Germany) for his contribution to the data annotation section \\textcolor{black}{and Alexander Jenke (NCT Dresden, Germany) for his contribution to the data repository table}.\n\nThis work was supported by the European Research Council (ERC) starting grant COMBIOSCOPY under the New Horizon Framework Programme grant agreement [ERC-2015-StG-637960]; the Helmholtz Imaging Platform (HIP); the NCT Heidelberg; BPI France (project CONDOR); the Johns Hopkins Science of Learning Institute Research Grant; the National Institutes of Health [NIDCR R01 DE025265, P41 EB015902, P41 EB015898, R01 CA235589]; the National Center For Tumor Diseases (NCT) Surgical Oncology Program; KARL STORZ SE \\& Co. KG; the Royal Society (UF140290) and NIHR Imperial BRC (Biomedical Research Centre); the ERC - H2020 Autonomous Robotic Surgery (ARS) grant agreement [ERC-2016-ADG-742671]; the Surgical Metrics Project - American College of Surgeons (National Society Contract); the Quantified Physician - 7-SIGMA Simulation Systems, Minnesota, MN (Industry Contract); the Tourniquet Master Training - DOD SBIR Phase IIb \u2013 Continuation Award [W81XWH-13-C-0021]; the Ontology for Human Motion and Psychomotor Performance - Stanford University Media-X; Motion Analysis for Microvascular Anastomosis - University of Wisconsin (Academic Contract); the Precision Learning Initiative - American Medical Association (National Society Grant); Quantifying the Metrics of Surgical Mastery: An Exploration in Data Science (NIH) [R01DK123445]; the Wellcome\/EPSRC Centre for Interventional and Surgical Sciences (WEISS) [203145Z\/16\/Z]; Engineering and Physical Sciences Research Council (EPSRC) [EP\/P027938\/1, EP\/R004080\/1, EP\/P012841\/1]; the Royal Academy of Engineering Chair in Emerging Technologies; the St. Michael's Hospital; the University of Toronto; the Grant-in-Aid for Scientific Research on Innovative Area from MEXT, Japan; the National Cancer Data Ecosystem, contract number 19X037Q under Task Order HHSN26100071 from NCI; the project ProteCT [BMBF 16SV8568]; the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) as part of Germany's Excellence Strategy - EXC 2050\/1 - Project ID 390696704 - Cluster of Excellence \\enquote{Centre for Tactile Internet with Human-in-the-Loop} (CeTI); the Canada Research Chair in Computer Integrated Surgery, Natural Sciences and Engineering Research Council of Canada; \\textcolor{black}{the ANR with grants ANR-16-CE33-0009 (project DeepSurg), ANR-10-IAHU-02 (IHU Strasbourg) and ANR-20-CHIA-0029-01 (Chair AI4ORSafety); and the Federal Ministry of Economics and Energy (grant number BMWI 01MT17001C) within the OP 4.1 project.}\n\n\\section*{Conflicts of interest}\nAnand Malpani is a future employee at Mimic Technologies Inc. (Seattle, WA, US). Johannes Fallert and Lars M\u00fcndermann are employed at KARL STORZ SE \\& Co. KG (Tuttlingen, Germany). Hirenkumar Nakawala is employed at CMR Surgical Ltd (Cambridge, UK). \\textcolor{black}{Nicolas Padoy is a scientific advisor of Caresyntax (Berlin, Germany). Daniel A. Hashimoto is a consultant for Johnson \\& Johnson (New Brunswick, NJ, USA), Verily Life Sciences (San Francisco, CA, USA), and Activ Surgical (Boston, MA, USA). He has received research support from Olympus Corporation and the Intuitive Foundation.} Carla Pugh is the founder of 10 Newtons Inc. (Madison, WI, US). Danail Stoyanov is employed at Digital Surgery Ltd (London, UK) and Odin Vision Ltd (London, UK). Teodor Grantcharov is the founder of Surgical Safety Technologies Inc. (Toronto, Ontario, Canada). All other authors do not declare any conflicts of interest.\n\n\n\\bibliographystyle{cas-model2-names.bst}\n\n\\section{Clinical translation}\n\\label{sec:clinicalTranslation}\n\nThe process of clinical translation from bench to bedside has been described as a valley of death, not only for surgical (software) products, but biomedical research in general \\citep{butler_translational_2008}. \nIn this section, we will begin by describing current practice and key initiatives in clinical translation of SDS. We elaborate on the concept of \\enquote{low-hanging fruit} that may be reached in a comparatively straightforward manner through collaboration of surgeon scientists, computer scientists and industry leaders. Finally, we will outline current challenges and next steps for those low-hanging fruit to cross the valley of death, rendering SDS applications from optional translational research projects to key elements of the product portfolio for modern operating room vendors, which in turn will increase engagement on the part of researchers, industry, funding agencies and regulatory bodies alike.\n\n\\subsection{Current practice}\n\nClinical translation of products developed through SDS is regulated under existing rules and guidelines. Ultimately, systems or products using SDS components must be able to provide value before, during or after surgery or interventions. Validating such capabilities requires prospective clinical trials in real treatment practices, which require ethics and safety approval by relevant bodies as well as adherence to software standards described in Sec.~\\ref{sec:dataAnalytics_currentChallengesAndNextSteps}. System documentation and reliability is critical to pass through such approval procedures, which can however also exceptionally be obtained for research purposes without proof of code stability.\n\nFrom a clinical research perspective, meta-analyses of RCTs are considered the gold standard. However, the field of surgery exhibits a notable lack of high-quality clinical studies as compared to other medical disciplines \\citep{mcculloch_randomised_2002}. While long-term clinical studies are a common prerequisite for clinical translation, despite intense research, the number of existing clinical studies in AI-based medicine is extremely low \\citep{nagendran_artificial_2020}. As a result, most current clinical studies in the field are based on selected data that are retrospectively analyzed, leading to a lack of high quality evidence that in turn hampers clinical progress. A recent scoping review on AI-based intraoperative decision support in particular named the small size, single-center provenance and questionable representability of the data sets, the lack of accounting for variability among human comparators, the lack of quantitative error analysis, and a failure to segregate training and test data sets as the prevalent methodological shortcomings \\citep{navarrete-welton_current_2020}. \n\nDespite these shortcomings, it should be noted that not all questions that arise in the process of clinical translation of an algorithm necessarily need to be addressed by RCTs. For example, a recent deep learning algorithm to diagnose diabetic retinopathy was approved by the FDA based on a pivotal cross-sectional study \\citep{abramoff_pivotal_2018}. Translational research on SDS products for prognosis also leverages existing methodology on prospective and retrospective cohort studies for the purposes of internal and external validation.\n\nGenerally speaking, the field of SDS still faces several domain-specific impediments. For instance, digitalization has not percolated the OR and the surgical community in the same way as other areas of medicine. A lack of standardization of surgical procedures hampers the creation of standardized annotation protocols, an important prerequisite for large-scale multi-center studies. Pioneering clinical success stories are important motivators to help set in motion a virtuous circle of advancement in the OR and beyond.\n\n\n\\subsection{Key initiatives and achievements}\n\nThe following section will provide an overview of existing SDS products and clinical studies in SDS.\n\n\\textbf{SDS products:} Over the past few years, modest success in clinical translation and approval of SDS products has been achieved, as summarized in Table~\\ref{tab:products}. This predominantly includes decision support in endoscopic imaging. Endoscopic AI (AI Medical Service, Tokyo, Japan) and GI Genius\\texttrademark{} (Medtronic, Dublin, Ireland) support gastroenterologists in the detection of cancerous lesions, the former albeit struggling with a low positive predictive value \\citep{hirasawa_application_2018}. Other successful applications include OR safety algorithms or computer vision-based data extraction.\n\n\\input{tables\/products}\n\n\\textbf{Translational progress in academia:}  While most of the work has focused on preoperative decision support, here, we place a particular focus on intraoperative assistance. Table~\\ref{tab:clinicalStudies} shows several exemplary studies in academia that illustrate how far SDS products have been translated to clinical practice in this regard. \n\n\\textit{Intraoperative assistance:} A recent review on AI for surgery mainly found studies that use ML to improve intraoperative imaging such as hyperspectral imaging or optical coherence tomography \\citep{navarrete-welton_current_2020}. Further notable intraoperative decision support efforts have focused on hypoxemia prevention \\citep{lundberg_explainable_2018}, sensor monitoring to support anesthesiologists with proper blood pressure management \\citep{wijnberge_effect_2020} and intelligent spinal cord monitoring during spinal surgery \\citep{fan_intelligent_2016}. A number of models have been developed to promote safety in laparoscopic cholecystectomy, a very common and standardized minimally invasive abdominal procedure. For instance, a model for bounding box detection of hepatocystic anatomy was recently tested in the operating room \\citep{Tokuyasu2021_LCanatomydet}. Another example of SDS for safe cholecystectomy is DeepCVS, a neural network trained to semantically segment hepatocystic anatomy and assess the criteria defining the CVS \\citep{MascagniVardazaryanAlapatt2021_DeepCVS}. A recent study based on 290 laparoscopic cholecystectomy videos from 37 countries showed that DL-based image analysis may be able to identify safe and dangerous zones of dissection~\\citep{madani_artificial_2021}. Finally, a cross-sectional study using deep learning algorithms developed on videos of the surgical field from more than 1000 cholecystectomy procedures from two institutions showed an association between disease severity and surgeons' ability to verify the CVS \\citep{korndorffer_situating_2020}.\nAnother example of intraoperative decision support is a study by \\citet{harangi_recognizing_2017}, who developed a neural network-based method to classify a structure specified by a surgeon (by drawing a line in the image) into the uterine artery or ureter. The authors reported a high accuracy, but the study was a cross-sectional design with a convenience sample. In fact, convenience samples are the norm in most existing studies in SDS addressing recognition of objects or anatomical structures in the surgical field. This sampling mechanism makes the findings susceptible to selection bias, which affects generalizability or external validation of the methods.\n\n\\textit{Perioperative decision support and prediction:} A selection of studies on perioperative assistance can be found in Appendix~\\ref{app:clinicalStudies-peri-op}. One important application of academic SDS is clinical decision support systems (CDSS) that integrate various information sources and compute a recommendation for surgeons about the optimal treatment option for a certain patient. Many of these CDSS are prediction systems that integrate into a mathematical model clinical, radiological and pathological attributes collected in a routine setting and weigh these parameters automatically to achieve a novel risk stratification \\citep{shur_clinical-radiomic_2020}. Trained with a specifically selected subpopulation of patients, these prediction systems may help improve current classification systems in guiding surgical decisions \\citep{tsilimigras_utilizing_2020}. Relevant information like overall- and recurrence-free survival \\citep{schoenberg_novel_2020} or the likelihood of intra- and postoperative adverse events to occur \\citep{bhandari_predicting_2020} can be assessed and obtained quickly via online applications such as the \\textit{pancreascalculator.com} \\citep{van_roessel_international_2020}. In contrast to these score-based prediction systems, ML-based systems are more flexible. The most prominent ML-based system, IBM's Watson for Oncology, is based on natural language processing and iterative features and demonstrated good accordance with treatments selected by a multidisciplinary tumor board in hospitals in India \\citep{somashekhar_watson_2018} and South Korea \\citep{lee_assessing_2018}. Weaknesses of this system include the necessity of skilled oncologists to operate the program, low generalizability to different regions, and the fact that not all subtypes of a specific cancer can be processed \\citep{yao_real_2020, strickland_ibm_2019}.\n\nAnother important application besides decision support is prediction of adverse events. A widely discussed work showed that DL may predict kidney failure up to 48 hours in advance \\citep{tomasev_clinically_2019}. In the intensive care unit (ICU), where surgeons face enormous quantities of clinical measurements from multiple sources, such as monitoring systems, laboratory values, diagnostic imaging and microbiology results, data-driven algorithms have demonstrated the ability to predict circulatory failure \\citep{hyland_early_2020}.\n\nTable~\\ref{tab:registeredClinicalStudies} provides an overview of currently registered SDS clinical studies. While most aim for evaluation of specific applications, a number of ongoing clinical trials focus on data collection for the original development of future CDSS or other SDS applications.\n\n\\input{tables\/clinicalStudies}\n\n\\subsection{Low-hanging fruit}\n\nIn light of the lack of a critical number of clinical success stories, a viable approach to clinical translation initially should focus on \\enquote{low-hanging fruit}. We believe the following criteria influence the likelihood of successful translation of an SDS application: high patient safety, technical feasibility - especially regarding data needs and performance requirements -  easy workflow integration, high clinical value and high business value to encourage industry adoption.\nLow-hanging fruit typically also avoid being classified as a high-risk medical product, thereby reducing regulatory demands and development barriers. However, it is difficult to satisfy all of these often conflicting criteria simultaneously. For example, applications of significant clinical value such as real-time decision support are highly technically challenging. By contrast, low-level video processing applications such as uninformative frame detection are technically simple but of limited clinical value. SDS applications that are low-hanging fruit are ones that offer a good balance between most or all of these criteria.\n\nAn example for a low-risk medical device in the broader scope of SDS is the aforementioned GI Genius that uses AI for real-time detection and localization of polyp detection during  colonoscopy, supporting the examination but not replacing the clinical decision making and diagnostics by clinicians. Considering the low risk to patients, GI Genius is classified as a Class II medical device (with special controls) by the FDA~\\citep{fda_news_release_2021}. \n\n\nIn surgery, a framework that may help determine the next steps for low-hanging fruit is the digital technology framework that categorizes data-centric product innovations in \\textit{descriptive}, \\textit{diagnostic}, \\textit{predictive} and \\textit{prescriptive}, as detailed in section~\\ref{sec:dataAnalytics}. Currently, the overwhelming focus for SDS researchers is in the prescriptive technology area \u2013 for example on tools that provide surgical decision support or predict adverse events. Changing the development lens from prescriptive to descriptive SDS applications, however, may open up entirely new avenues. For instance, a low-hanging fruit may lie in a descriptive decision support tool that informs surgeons on how many surgeons performed certain steps within an intervention and the consequences. Such a data-centric SDS product would not require embedded surgical expertise in order to provide value to the surgeon, but only a database of surgical videos and automated recognition of anatomical structures and surgical instruments, which is technically feasible. In essence, instead of the very difficult automation of surgical decisions, value can be found in providing surgeons and surgical teams with moment-to-moment risk stratification data to facilitate their decisions. An additional benefit of this approach is that it can be combined with real-time data acquisition regarding \\textit{how} surgeons interact with the risk stratification data, which would greatly facilitate the development of both predictive and prescriptive decision support tools.\n\nImportantly, presenting statistical data and evidence-based risk stratification information to the surgeon would also have a different regulatory path than a prescriptive SDS product that offers surgical decisions based on an AI database grounded in surgical decision making.  The data-focused product leaves the surgeon fully responsible, while the decision based product makes it questionable who is fully responsible if the surgeon followed an AI-based decision and there was a poor outcome. Another benefit of focusing on descriptive technologies is there is a much smaller technology adoption hurdle for the surgeon when faced with trusting descriptive statistics compared to an AI-based prescriptive decision support tool.\n\nAn ML-based descriptive low-hanging fruit could be data-driven surgical reporting and documentation. Surgical procedures are currently documented as one to two pages of text. While a six to eight hour video will not serve as a report in itself, SDS may help extract relevant information from this video by automatically documenting important steps in the procedure.\nHere, computer vision algorithms for recognition of surgical phases and instruments may be used to extract metainformation from videos~\\citep{MasacagniAlapatt2021_EndoDigest}. \n\nAn ML-based predictive low-hanging fruit could lie in the optimization of OR logistics. Prediction of procedure time either preoperatively or utilizing intraoperative sensor data may not improve patient outcome, but could provide value to hospital managers if it helps cut down costs in the OR by optimizing patient volume \\citep{aksamentov_deep_2017, bodenstedt_prediction_2019, twinanda_rsdnet_2019}.\nThis, too, harbors low risk for patients and has a low barrier for market entry. Furthermore, the reference information, i.e., time between incision and suture, is already documented in most hospitals and no laborious annotation by surgical experts is necessary to train the respective ML algorithms. Since OR management tools already exist, SDS applications could even yield success stories within existing tools without having to establish entirely new software tools.\n Improvements in patient safety may already result from a simple tool that combines SDS algorithms for object recognition in laparoscopic video (e.g. gauze, specimen bag or suture needle) with a warning for surgeons and scrub nurses if these objects are introduced into the patient's abdomen but not removed afterwards. Since such an SDS application warns clinical staff but does not perform an action on the patient itself, the risk for the patient is inherently low. Here, a combination of surgical knowledge (which objects are at what time introduced into the patient's body?) with SDS algorithms (which objects can robustly be detected?) and an unobtrusive user interface with a low false alarm rate may result in a low-hanging fruit. Along these lines, automation of the surgical checklist \\citep{conley_effective_2011} would be a technically feasible SDS application with high clinical value.\n\nThe impending success of next-generation surgical ro\\-bot\\-ics in the OR may bring further opportunities to the clinical translation of SDS. The da Vinci\\textsuperscript{\\textregistered} surgical system (Intuitive Surgical Inc., Sunnyvale, CA, USA)  and its upcoming competitors lay the foundation for systematic data capture as well as surgical guidance by information augmentation in the OR. A relatively low-hanging fruit with benefit to the surgeon in the domain of surgical robotics may be an automated camera guidance system, as suggested by Wagner et al.~\\citep{wagner_learning_2021}. On the one hand, the risk of poor camera positioning for the patient is low compared to that of invasive tasks such as suturing. On the other hand, correcting the camera position is currently a highly disruptive task to the surgeon. The first products for autonomous endoscopic camera control are now emerging in robotic surgery, such as the FDA-approved system from \\citeauthor{transenterix_transenterix_nodate} (Morrisville, NC, USA). %\n\n\n\\subsection{Current challenges and next steps}\n\nAs highlighted in several previous publications~\\citep{maier-hein_surgical_2017, maier-hein_surgical_2018, hager_chapter_2020}, clinical applications for SDS are manifold, ranging from pre- and intraoperative decision support to context-aware assistance and surgical skills training. The clinical translation-related goals generated by the consortium as part of the Delphi process\nare provided in Tab.~\\ref{tab:goalsMission4}. \n\\input{tables\/goalsMission4}\n The following aspects deserve particular attention:\\\\\n\n\\textbf{How to catalyze clinical translation of SDS?} (goals 4.1\/4.2)  Clinical data is recognized as \\enquote{the resource most central to healthcare progress} \\citep{institute_of_medicine_us_roundtable_on_value__science-driven_health_care_clinical_2010}. What is needed is thus a cultural shift towards data acquisition, annotation and analysis within a well-defined data governance framework as a primary clinical \ntask~\\citep{august_value_2021}. \nThe allocation of economic, infrastructural and personnel resources within hospitals for this appears as a non-negotiable requirement for the purpose.\nThe need for creating value from large amounts of representative data, both for de novo development\/validation and external validation studies, further necessitates multi-institutional collaborations. Researchers in other domains have achieved such collaborations, for example in genomics and bioinformatics; SDS would benefit from adopting relevant aspects of these domains' research cultures. In addition, enabling explicit academic recognition for developing rigorously annotated data sets can facilitate data resources for research in SDS, as discussed in Sec.~\\ref{sec:dataAnnotation}. Paving the way for short-term clinical success stories as well as long-term clinical translation further requires SDS applications to be integrated into clinical workflows. In fact, the sparsity of studies on SDS solutions for intraoperative care illustrate the challenge of conducting multi-disciplinary research while prioritizing the patient. Therefore, research on SDS products should consider the impact on workflow early in product development and closely engage relevant stakeholders (see Tab.~\\ref{tab:stakeholders}). Impactful success stories could then be generated by focusing on low-hanging fruit presented in the previous section. These, in turn, would contribute to building public trust in SDS and boost public enthusiasm to spark patient demand.\\\\\n\n\n\n\n\\textbf{How to improve knowledge transfer among different stakeholders?} (goal 4.3)  The creation of interdisciplinary networks involving the different stakeholders and the regular organization of SDS events in conjunction with both technical and medical conferences is key to improving knowledge transfer between the groups. Such events should, in part, be dedicated to specific questions, such as annotation guidelines, data structures or good practices with respect to external validation.  As a means for actively disseminating, discussing, and promoting new insights in the field of SDS, a well-curated community web platform should be established as the central information hub.\nOne could even go further and offer e.g. a prize for clinical trials demonstrating SDS success.\nA good means for public outreach could be the hosting of public days focused on a particular topic at major conferences in the field, as a way of creating awareness for that topic, or campaigns e.g. in the vein of \"Stop the Bleed\"~\\citep{acs_committee_on_trauma_stop_nodate}.\\\\\n\n\\textbf{How to train key SDS personnel?} (goal 4.4)\nIn order to facilitate clinical translation of SDS in the long term, it will further be crucial to promote the transdisciplinary training of future surgical data scientists and thereby establish SDS as a career path. Computer scientists will have to enter operating rooms on a regular basis to understand real clinical problems and to get an impression of the obstacles in clinical translation. Similarly, surgeons will have to  understand the basic principles, capabilities and limits of data science techniques to identify solvable clinical problems and proper applications for SDS. A viable path to improve knowledge transfer would be to establish SDS as a commonly respected career path in hospitals. In this context, both technical and clinical disciplines should be complemented by knowledge and expertise in clinical research methodology, i.e., epidemiology and biostatistics. Moreover, human factors engineering and human computer interaction researchers should be integrated into the community. Setting up such an SDS career path should also involve the definition of specifics and skills an 'AI-ready' clinician should meet. A curriculum should put a specific focus on medical statistics covering confounding variables, risk correction and data biases, as well as on regulatory issues (e.g. SaMD). On top of the research-oriented positions, we should further seek to establish SDS-related jobs for data acquisition, management and annotation, specifically in university hospitals. \\\\\n\n\n\\textbf{How to ensure high-quality external validation of SDS applications?} (goal 4.5-4.7) A critical pitfall with clinical prediction models, which include models for diagnosis and prognosis, is unbridled proliferation of de novo development and validation studies, but scant external validation studies \\citep{adibi_validation_2020}. Research to support regulatory approval of SDS products, i.e., in order to market these products, would typically address external validation. However, advances in clinical care are not restricted to marketed products. Therefore, it is necessary for the research community to not only conduct de novo development and validation studies but also well designed external validation studies. Past experience with clinical prediction models shows the need for creative solutions. While some solutions, such as \\enquote{living registries}, have been proposed \\citep{adibi_validation_2020}, proactive effort by the SDS community to develop effective solutions that allow for consistent and uniform external validation can be a transformative contribution. The status quo, summarized in a review of existing literature in AI-based intraoperative decision-making, shows that the SDS community has not addressed the pitfall of inadequate external validation studies \\citep{navarrete-welton_current_2020}. This challenge is systematically addressed when the end-goal for the translational research is regulatory approval to market a SDS product; the regulatory agency serves as a steward in this case. Similar stewardship may benefit translational research in SDS that is not intended to support regulatory approval. Finally, it is important to develop new performance metrics for AI algorithms that quantify clinically relevant parameters currently not accounted for in outcome validation studies. ML techniques offer the possibility to capture data patterns that may serve as potential surrogate measures of long-term outcomes. For example, many established metrics, such as 5-year-survival, are not immediately available after a surgical intervention for cancer. Surgical video or motion data localized to anatomy through imaging studies may be used to identify activities or events that increase the risk of cancer cell seeding (and subsequent metastasis).\\\\\n\n\n\n\\textbf{How to ensure ethical and legal guidance?} (goals 4.8\/ 4.9) With the face of data-driven clinical practice about to change in a vast manner, unprecedented ethical and legal questions pertaining to both the regulation of medical AI as well as its practical use will be raised. Moving forward, liability and medical negligence\/insurance regulations need to be adapted for data-driven clinical practice. A recent survey among Dutch surgeons revealed privacy and liability concerns as significant grounds for objection to video and audio recording of surgical procedures \\citep{van_de_graaf_current_2020}, reinforcing the importance of clear regulatory frameworks towards better clinical acceptance. New regulations will have to go much further than these current considerations, with a particular focus to be placed on cases of AI failure, human rejection of AI recommendations, or potentially the omission of AI \\citep{directorate-general_for_parliamentary_research_services_european_parliament_ethics_2020}. Notably, the FDA recently put forth an \\textit{Artificial Intelligence and Machine Learning (AI\/ML) Software as a Medical Device Action Plan} \\citep{health_artificial_2021}. These regulatory issues strongly interconnect with previously raised issues of trust in as well as transparency and explainability of AI models, which have also been raised in the very recent WHO report \\textit{Ethics \\& Governance of Artificial Intelligence for Health} \\citep{health_ethics__governance_ethics_2021}. An ethical and human rights-based framework intended to guide the development and use of AI was further proposed  by \\cite{fjeld2020principled}, taking eight key themes such as privacy, accountability, safety\/security, transparency\/explainability, fairness and non-discrimination, human control of technology, professional responsibility, and promotion of human values into account.\nMoreover, ethical and moral considerations regarding the democratization of data and\/or AI model access will be necessary. \nIn the specific context of surgery, first guidance on the ethical implications of integrating AI algorithms into surgical training workflows has recently become available \\citep{collins_ethical_2021}. Similarly, new concepts for obtaining patient consent to data sharing that take into account the dynamics and unforeseeability of data usage in future SDS applications need to be established. One way to go might be the introduction of a data donor card, analogously to organ donor cards, as suggested in Sec.~\\ref{sec:dataAnnotation_currentChallengesAndNextSteps}. Both patient- and healthcare professional-centric ethical and legal considerations are likely to have a large impact on the public perception of and trust in SDS, which needs to be boosted for higher patient demand. Above all, patient safety must be supported by the development of contemplative regulatory frameworks.\n\n\nIn summary, a multi-pronged approach to address challenges that can catalyze rapid advances in SDS and to develop solutions to problems considered low-hanging fruit will be crucial to the future of SDS as a scientific field. The introduction of initial features that provide clear benefits can facilitate advanced changes. To this end, a compositional approach may be pursued wherein complex SDS products reuse simpler AI models that have been previously approved and adopted in clinical care. Once a number of high value applications are established and there is hospital buy-in, a virtuous circle of SDS can be expected to begin, enabling more applications, higher volume data collection, stronger models, streamlined regulation, and better acceptance.\n\n\\section{Data analytics}\n\\label{sec:dataAnalytics}\n\nData analytics (addressing the \\textit{interpretation} task in Fig.~\\ref{fig:sdsComponents}) is often regarded as the core of any SDS system. The perioperative data is processed to derive information addressing  a specific clinical need, where applications may range from prevention and training to interventional diagnosis, treatment assistance and follow-up \\citep{maier-hein_surgical_2017}. \n\n\n\\subsection{Current practice}\n\nSurgical practice has traditionally been based on observational learning, and decision making before, during and after surgical procedures highly depends on the domain knowledge and past experiences of the surgical team \\citep{maier-hein_surgical_2017}. SDS has the potential to initiate a paradigm shift with a data-driven approach \\citep{hager_chapter_2020, vercauteren_cai4cai_2020}. Bishop and others classify data analytics tools as descriptive, diagnostic, predictive, and prescriptive  \\citep{bishop_pattern_2006, tukey_exploratory_1977}:\\\\\n\n\\textbf{Descriptive analytics tools - what happened?} Descriptive analytics primarily provide a global, comprehensive summary of data made available through data communication such as simple reporting features. \n\\citeauthor{syus_syus_nodate}' Periop Insight (Syus, Inc., Nashville, TN, USA) is an example of how descriptive analytics are used to access data, view key performance metrics, and support operational decisions through documentation and easy interpretation of historical data on supply costs, delays, idle time etc., relating overall operating room efficiency and utilization. Business Intelligence (BI) \\citep{chen_business_2012} tools are a typical form of descriptive analysis tools which comprise an integrated set of IT tools to transform data into information and then into knowledge, and have been used in healthcare settings \\citep{ward_applications_2014} (e.g. Sisense\\texttrademark{} (Sisense Ltd., New York City, NY, USA), Domo\\texttrademark{} (Domo, Inc., American Fork, UT, USA), MicroStrategy\\texttrademark{} (MicroStrategy Inc., Tysons Corner, VA, USA), Looker\\texttrademark{} (Looker Data Sciences Inc., Santa Cruz, CA, USA), Microsoft Power BI\\texttrademark{} (Microsoft Corporation, Redmond, WA, USA) and Tableau\\texttrademark{} (Tableau Software Inc., Seattle, WA, USA)). These tools often incorporate features such as interactive dashboards \\citep{upton_heart_2019} that provide customized graphical displays of key metrics, historical trends, and reference benchmarks and can be used to assist in tasks such as surgical planning, personalized treatment, and postoperative data analysis. \\\\\n\n\n\\textbf{Diagnostic analytics tools - why did it happen?} Diagnostic analytics tools, on the other hand, explore the data, address the correlations and dependencies between variables, and focus on interpreting the factors that contributed to a certain outcome through data discovery and data mining. These tools can facilitate the understanding of complex processes and reveal relationships between variables, or find root causes. For example, clinicians can use data on postoperative care to assess the effectiveness of a treatment \\citep{bowyer_importance_2016, kehlet_evidence-based_2008}. \\\\\n\n\n\\textbf{Predictive and prescriptive analytics tools - What will happen? How can we make it happen?} \nPredictive analytics uses historical data, performs an in-depth analysis of historical key trends underlying patterns and correlations, and uses the insights gained to make predictions about what will likely happen next (\\textit{What will happen?}).\nPrescriptive analytics complement predictive analytics by offering insights into what actions can be taken to achieve target outcomes (\\textit{How can we make it happen?}).\nML can meet these needs, but the challenges specific to surgery are manifold, as detailed in \\cite{maier-hein_surgical_2017}. Importantly, the preoperative, intraoperative and postoperative data processed are potentially highly heterogeneous, consisting of 2D\/3D\/4D imaging data (e.g. diagnostic imaging data), video data (e.g. from medical devices or room cameras), time series data (e.g. from medical devices or microphones), and more (e.g. laboratory results, patient history, genome information). Furthermore, while the diagnostic process follows a rather regular flow of data acquisition, the surgical process varies significantly and is highly specific to patient and procedure. Finally, team dynamics play a crucial role. In fact, several studies have demonstrated a correlation between nontechnical skills, such as team communication, and technical errors during surgery \\citep{hull_impact_2012}.\nWhile first steps have been taken to apply ML in open research problems with applications ranging from decision support (e.g. determining surgical resectability \\citep{marcus_improved_2020}) to data fusion for enhanced surgical vision (e.g. \\citet{akladios_augmented_2020}), and OR logistics (e.g. \\citet{twinanda_rsdnet_2019, bodenstedt_prediction_2019, hager_chapter_2020}), the vast majority of research has not yet made it to clinical trial stages. Sec.~\\ref{sec:dataAnalytics_currentChallengesAndNextSteps} highlights several challenges that need to be addressed in order to effectively adopt ML as an integral part of surgical routine.\n\n\\subsection{Key initiatives and achievements}\n\nThis section reviews some key initiatives and achievements from both an industrial and an academic perspective,\n\n\\textbf{Industry initiatives:} Commercial platforms and projects have conventionally focused on analysing multidimensional patient data for clinical decision-making - primarily outside the field of surgery. The most widely discussed initiative so far is probably IBM\\textsuperscript{\\textregistered} Watson\\texttrademark{} Health\\textsuperscript{\\textregistered} (International Business Machines Corporation (IBM), Armonk, NY, USA), which initiated several projects such as \\textit{Watson Medical Sieve}, \\textit{Watson For Oncology} or \\textit{Watson Clinical Matching} that apply the Watson cognitive computing technology to different challenges in healthcare \\citep{chen_ibm_2016}. The goal of \\textit{Watson Medical Sieve}, for example, is to filter relevant information from patient records consisting of multimodal data to assist clinical decision making in radiology and cardiology. \\textit{Watson Clinical Matching} finds clinical studies that match the conditions of individual patients. With its vast capability to reach patient records and medical literature, Watson was believed to be the future of medicine. However, after it was put to use in the real world, it quickly became clear that the powerful technology has its limitations, as reported by Strickland: It performed poorly in India for breast cancer, where only 73\\% of the treatment recommendations were in concordance with the experts. \nAnother critical example is the \\textit{Watson-powered Oncology Expert Advisor} which had only around 65\\% accuracy in extracting time-dependent information like therapy timelines from text documents in medical records \\citep{strickland_ibm_2019}. Despite its limitations, Watson Health has shown to be efficient in certain, narrow and controlled applications. For example, \\textit{Watson for Genomics} is used by genetics labs that generate reports for practicing oncologists. Given the information on a patient's genetic mutations, it can generate a report that describes all relevant drugs and clinical trials \\citep{strickland_ibm_2019}. \nOther companies, societies and initiatives, such as Google (Mountain View, CA, USA) DeepMind Health~\\citep{graves_hybrid_2016, tomasev_clinically_2019}, Intel (Santa Clara, CA, USA) \\citep{healthcare_it_news_leveraging_2012} and the American Society of Clinical Oncology (ASCO) CancerLinQ\\textsuperscript{\\textregistered} \\citep{sledge_cancerlinq_2013} have also been focusing on clinical data, and industrial success stories in surgery at scale are still lacking, as detailed in Sec.~\\ref{sec:clinicalTranslation}.\n\n\\textbf{Academic initiatives:} In academia, interdisciplinary collaborative large-scale research projects have developed data analytics tools to address different aspects of SDS. The Transregional Collaborative Research Center \"Cognition Guided Surgery\" focused on the development of a technical-cognitive assistance system for surgeons that explores new methods for knowledge-based decision support for surgery \\citep{marz_toward_2015} as well as intraoperative assistance \\citep{katic_bridging_2016}. First steps towards the operating room of the future have recently been taken, focusing on different aspects like advanced imaging and robotics, multidimensional data modelling, acquisition and interpretation, as well as novel human-machine interfaces for a wide range of surgical and interventional applications (e.g. \\citeauthor{advanced_multimodality_image-guided_operating_amigo_advanced_nodate}, \\citeauthor{computer-integrated_surgical_systems_and_technology_cisst_computer-integrated_nodate} Engineering Research Center, \\citeauthor{hamlyn_centre_hamlyn_nodate}, \\citeauthor{university_college_london_ucl_ucl_nodate}, \\citeauthor{innovation_center_computer_assisted_surgery_iccas_iccas_nodate}, \\citeauthor{ihu_strasbourg_ihu_nodate}, \\citeauthor{national_center_for_tumor_diseases_dresden_nctucc_national_nodate} and \\citeauthor{national_center_for_tumor_diseases_heidelberg_surgical_nodate}).\n\nBroadly speaking, much of the academic work in SDS is currently focusing on the application of ML methods in various contexts \\citep{navarrete-welton_current_2020, zhou_artificial_2019, alapatt_neural_2020}, but clinical impact remains to be demonstrated (see Sec.~\\ref{sec:clinicalTranslation}).\n\n\\subsection{Standards, platforms and tools}\n\\label{sec:dataAnalaytics_standardsAndTools}\n\n A broad range of software tools are used by the SDS community each day, reflecting the interdisciplinary nature of the field. Depending on the SDS application, tools may be required from the following technical disciplines that intersect with SDS: classical statistics, general ML, deep learning, data visualization, medical image processing, registration and visualization, computer vision, Natural Language Processing (NLP), signal processing, surgery simulation, surgery navigation and Augmented Reality (AR), robotics, BI and software engineering. Many established and emerging software tools exist within each discipline and a comprehensive list would be vast and continually growing. In Table~\\ref{tab:commonTools}, we have listed software tools that are commonly used by SDS practitioners today, organized by the technical disciplines mentioned above. In this section, we focus on ML frameworks and the regulatory aspects of software development for SDS.\n\n\\textbf{ML frameworks and model standards:} ML is today one of the central themes of SDS analytics, and many frameworks are used by the SDS community. The \\citeauthor{scikit-learn_scikit-learn_nodate} library in Python is the most widely used framework for ML-based classification, regression and clustering using non-DL models such as Support Vector Machines (SVMs), decision trees and multi-layer perceptron (MLPs). DL, the sub-field of ML that uses Artificial Neural Networks (ANNs) with many hidden layers, has exploded over the past 5 years,\nalso due to the mature DL frameworks. The dominating open-source frameworks today are \\citeauthor{tensorflow_tensorflow_nodate} by Google and \\citeauthor{pytorch_pytorch_nodate} by Facebook (Menlo Park, CA, USA). These provide mechanisms to construct, train and test ANNs with comprehensive and ever-growing APIs and they are backed up by large industrial investment and community involvement. Other important, but less widely used frameworks include \\citeauthor{caffe_caffe_nodate}, \\citeauthor{caffe2_caffe2_nodate} (now a part of PyTorch), \\citeauthor{apache_mxnet_apache_nodate}, \\citeauthor{flux_fluxmlfluxjl_2020}, \\citeauthor{chainer_chainer_nodate}, MATLAB's \\citeauthor{deep_learning_toolbox_deep_nodate} and Microsoft's \\citeauthor{microsoft_cognitive_toolkit_cntk_microsoft_nodate}. Wrapper libraries have been constructed on top of several frameworks with higher level APIs that simplify DL model design and promote reusable components. These include TensorFlow's \\citeauthor{keras_keras_nodate} (now native to TensorFlow), \\citeauthor{tensorlayer_tensorlayer_nodate}, \\citeauthor{tflearn_tflearn_nodate} and \\citeauthor{niftynet_niftynet_nodate} (specifically for medical image data), and PyTorch's TorchVision \\citep{nguyen_machine_2019}. Other useful tools include training progress visualization with Tensorboard, and AutoML systems for efficient automatic hyperparameter and model architecture search, such as \\citeauthor{h2o_automl_nodate}, \\citeauthor{auto-sklearn_auto-sklearn_nodate}, \\citeauthor{autokeras_autokeras_nodate} and Google \\citeauthor{cloud_automl_cloud_nodate}. \\citeauthor{nvidia_digits_nvidia_nodate} takes framework abstraction a step further with a web application to train DL models for image classification, segmentation and object detection, and a graphical user interface (GUI) suitable for non-programmers. \nSuch tools are relevant in SDS where clinical researchers can increasingly train standard DL models without any programming or ML experience \\citep{faes_automated_2019}. On the one hand this is beneficial for technology democratization, but on the other hand it elevates known risks of treating ML and DL systems as \\enquote{black boxes} \\citep{phg_foundation_black_2020}. \nRecently NVIDIA has released \\citeauthor{nvidia_clara_nvidia_nodate}, a software infrastructure to develop DL models specifically for healthcare applications with large-scale collaboration and federated learning.\n\nEach major framework has its own format for representing and storing ML models and associated computation graphs. There are now efforts to standardize formats to improve interoperability, model sharing, and to reduce framework lock-in. Examples include the Neural Network Exchange Format (NNEF), developed by the \\citeauthor{khronos_group_khronos_2020} with participation from over 30 industrial partners, \\citeauthor{onnx_onnx_nodate} and Apple's (Cupertino, CA, USA) \\citeauthor{core_ml_core_nodate} for sharing models, and for sharing source code to train and test these models. \\citeauthor{github_github_nodate} is undeniably the most important sharing platform, used extensively by SDS practitioners, which greatly helps to promote research code reusability and reproducibility. \\enquote{Model Zoos} (e.g. \\citeauthor{model_zoo_model_nodate}, \\citeauthor{onnxmodels_onnxmodels_2020}) are also essential online tools to allow easy discovery and curation of many of the landmark models from research literature.\n\n\\textbf{Regulatory software standards:} The usual research and development pipeline for an SDS software involves software developed at various stages including data collection and curation, model training, model testing, application deployment, distribution, monitoring, model improvement, and finally a medically approved product. For the classification as a medical product, the intended purpose by the manufacturer is more decisive than the functions of the software. Software is a \\enquote{medical device software} (or \\enquote{software as a medical device} (SaMD)) if \\enquote{intended to be used, alone or in combination, for a purpose as specified in the definition of a medical device in the medical devices regulation or in vitro diagnostic medical devices regulation} (\\citeauthor{mdcg_2019-11_guidance_2019}), i.e. if intended to diagnose, treat or monitor diseases and injuries. The manufacturer of an SDS software application as SaMD needs to ensure that the safety of the product is systematically guaranteed, prove that they have sufficient competencies to ensure the relevant safety and performance of the product according to the state of the art (and keep evidence for development, risk management, data management, verification and validation, post-market surveillance and vigilance, service, installation, decommissioning, customer communication, monitoring applicable new or revised regulatory requirements).\n\nYet, ML-based software requires particular considerations \\citep{gerke_need_2020}. For example, the fact that models can be improved over time with more training data (often called the \\enquote{virtuous cycle}) is not well handled by these established standards. In 2019, the FDA published a \\enquote{Proposed Regulatory Framework for Modifications to Artificial Intelligence\/Machine Learning (AI\/ML)-Based Software as a Medical Device (SaMD)}, specifically aimed to clarify this subject \\citep{fda_proposed_2019}. In contrast to the previously \\enquote{locked} algorithms and models, this framework formulates requirements on using Continuous Learning Systems (CLS) and defines a premarket submission to the FDA when the AI\/ML software modification significantly affects device performance, or safety and effectiveness; the modification is to the device's intended use; or the modification introduces a major change to the SaMD algorithm. The implementation of these requirements, especially with regard to the actual product development, is an unsolved problem.\n\n\n\n\n\n\\input{tables\/goalsMission3}\n\n\\subsection{Current challenges and next steps}\n\\label{sec:dataAnalytics_currentChallengesAndNextSteps}\n\n\nThe data analytics-related mission as well as corresponding goals generated by the consortium\nare provided in Tab.~\\ref{tab:goalsMission3}. This section elaborates on the most important research questions from a ML methodological perspective:\\\\\n\n\n\n\\textbf{How to ensure robustness and generalization?} (goal 3.1) Models trained on the data from one clinical site may not necessarily generalize well to others due to variability in devices, individual practices of the surgical team or the patient demographic.\nWhile data augmentation~ \\citep{itzkovich_using_2019} can address this issue to some extent, an alternative promising approach is to develop\narchitectures designed to generalize across domains. \nEarly approaches focused on \\textit{domain adaptation} \\citep{heimann_learning_2013, wirkert_physiological_2017} or more generically \\textit{transfer learning} \\citep{pan_survey_2010} to compensate for domain shifts in the data. Other attempts have focused on converting data into a domain-invariant representation and on decoupling generic task-relevant features from domain-specific ones \\citep{dai_deformable_2017, mitchell_artificial_2019, sabour_dynamic_2017, sarikaya_towards_2020}. Generally speaking, however, ML methods trained in a specific setting (e.g. hospital) still tend to fail to generalize to new settings.\\\\\n\n\\textbf{How to improve transparency and explainability?}\\\\ (goal 3.2) \nThe WHO document on \\textit{Ethics \\& Governance of Artificial Intelligence for Health} \\citep{health_ethics__governance_ethics_2021} (see Sec.~\\ref{sec:technicalInfrastructure}) states that ``AI technologies should be intelligible [...] to developers, medical professionals, patients, users and regulators'' and that ``two broad approaches to intelligibility are to improve the transparency of AI technology and to make AI technology explainable.'' In this context, transparency also relates to the requirement that ``sufficient information be published or documented before the design or deployment of an AI technology and that such information facilitate meaningful public consultation and debate on how the technology is designed and how it should or should not be used''. Explainability stems from the urge to understand why an algorithm produced a certain output.\n In fact, the complexity of neural network architectures with typically millions of parameters poses a difficulty for humans to understand how these models reach their conclusions \\citep{reyes_interpretability_2020}. As a result, the EU's GDPR, implemented in 2018, also discourages the use of black-box approaches, thus providing explicit motivation for the development of models that provide human-interpretable information on how conclusions were reached.\nInterpretable models are still in their infancies and are primarily studied by the ML community~\\citep{adebayo_sanity_2018, bach_pixel-wise_2015, koh_understanding_2017, shrikumar_learning_2017}.\nThese advances are being adopted within medical imaging communities in applications that are used to make a diagnosis (e.g. detecting\/segmenting cancerous tissue, lesions on MRI data) \\citep{gallego-ortiz_interpreting_2016}, and to generate reports that are on par with human radiologists \\citep{gale_producing_2018}, for example. Open research questions are related to how to validate the explanation of the models (lack of ground truth) and how to best communicate the results to non-experts. \nA concept related to explainability is causality.\nTo date, it is generally unknown how a given intervention or change is likely to affect outcome, which is influenced by many factors even beyond the surgeon and the patient. Furthermore, randomized controlled trials (RCTs) to evaluate surgical interventions are difficult to perform \\citep{mcculloch_randomised_2002}. Thus, it is hard to provide the same quality of evidence and understanding of surgery as, for example, for a drug treating a common non-life-threatening condition \\citep{hager_chapter_2020}. While large-scale data may help reveal relationships among many factors in surgery, correlation does not equal causation. Recent work on causal analysis \\citep{peters_elements_2017,scholkopf_causality_2019, castro_causality_2020}, however, may help in this regard.\\\\\n\n\n\\textbf{How to address data sparsity?} (goal 3.3) One of the most crucial problems in SDS is the data sparsity (see Sec.~\\ref{sec:lackOfSuccessStories}). Several complementary approaches have been proposed to address this bottleneck. \nThese include \\textit{crowdsourcing}~\\citep{maier-hein_can_2014, maier-hein_crowdtruth_2015, malpani_study_2015,heim_clickstream_2018,albarqouni_aggnet_2016, maier-hein_crowd-algorithm_2016} and \\textit{synthetic data generation} \\citep{pfeiffer_generating_2019, ravasio_learned_2020, wirkert_physiological_2017, rivoir2021long} briefly mentioned above. Unlabeled data can also be exploited by using \\textit{self-supervised} (see e.g. \\citep{ross_exploiting_2018}) and \\textit{semi-supervised} learning (see e.g. \\citep{yu_learning_2019, srivastav_self-supervision_2020}). Self-supervised methods solve an alternate, pretext or auxiliary task, the result of which is a model or representation that can be used in the solution of the original problem. Semi-supervised methods can exploit the unlabelled data in many different ways. In \\citep{yu_learning_2019, srivastav_self-supervision_2020}, for example, pseudo-annotations are generated on the unlabelled data using a teacher model, and the resulting pseudo-annotated dataset is then used to train another (student) model. \nRecent studies have further shown that exploiting the relationship across different tasks with the concept of  \\textit{multi-task learning} \\citep{twinanda_endonet_2017} may be used to address data sparsity as well. It has been demonstrated to be beneficial to jointly reason across multi-tasks \\citep{kokkinos_ubernet_2017, long_learning_2017, yao_describing_2012, sarikaya_joint_2018} and take advantage of a combination of shared and task-specific representations \\citep{misra_cross-stitch_2016}. However, the performance of some tasks may also worsen through such a paradigm \\citep{kokkinos_ubernet_2017}. A possible solution to this problem might lie in the approach of attentive single-tasking \\citep{maninis_attentive_2019}. Finally, \\textit{meta-learning} \\citep{vanschoren_meta-learning_2018, godau2021} and more generally \\textit{lifelong learning} \\citep{parisi_continual_2019} are further potential paradigms for addressing the problem of data sparsity in the future. Progress in this field will, at any rate, crucially depend on the availability of more public multi-task data sets, such as~\\cite{maier-hein_heidelberg_2021}\\\\\n\n\n\n\n\\textbf{How to detect, represent and compensate for uncertainties and biases?} (goal 3.4) A common criticism of ML-based solutions is the way that they handle \\enquote{anomalies}. If a measurement is out-of-distribution (ood; i.e. it does not resemble the training data), the algorithm cannot make a meaningful inference, and the probability of failure (error) is high. This type of \\textit{epistemic uncertainty} \\citep{kendall_what_2017} is particularly crucial in medicine as not all anomalies\/pathologies can be known beforehand. As a result, current work is dedicated to this challenge of anomaly\/novelty\/ood detection \\citep{adler_uncertainty-aware_2019}. Even if a sample is in the support of the training distribution, a problem may not be uniquely solvable \\citep{ardizzone_analyzing_2018} or the solution may be associated with high uncertainty. Further research has therefore been directed at estimating and representing the certainty of AI algorithms \\citep{adler_uncertainty-aware_2019, nolke_invertible_2021}. Future work should focus on making use of the uncertainty estimates in clinical applications and increasing the reliability of ood methods, as well as systematically understanding and addressing the issue of biases and confounders (see Sec.~\\ref{sec:dataAnnotation_currentChallengesAndNextSteps}). In this context the increased involvement of statisticians and experts from clinical epidemiology, such as in the biomedical image analysis initiative~\\citep{maier-hein_bias_2020,ros_how_2021}, would be desirable. Adopting the necessity of reporting data biases and confounders in publications should be a natural progression for the field of SDS.\n\\\\\n\n\\textbf{How to address data heterogeneity and complexity?} (goal 3.5) The surgeons and surgical team dynamics play a significant role in intraoperative care. While the main surgeon has the lead and makes decisions based on domain knowledge, experience and skills, anesthesiologists, assistant surgeons, nurses and further staff play crucial roles at different steps of the workflow. Their smooth, dynamic collaboration and coordination is a crucial factor for the success of the overall process. Data analytics can play a key role in quantifying these intangibles by modeling workflows and processes. Surgeon skill evaluation, personalized and timely feedback during surgical training, optimal surgeon and patient\/case or surgeon and surgical team matches are among the issues that can benefit from data analytics tools.\n Furthermore, data collected from multiple sources such as vital signs from live monitoring devices, electronic health records, patient demographics, or preoperative imaging modalities require analysis approaches that can accommodate their heterogeneity. Recent approaches in fusion of heterogeneous information include the use of specialized frameworks such as iFusion \\citep{guo_ifusion_2019}. Other work has specifically focused on handling incomplete heterogeneous data with Variational Autoencoders (VAEs) \\citep{nazabal_handling_2020}. Graph neural networks \\citep{zhou_graph_2019} appear to be another particularly promising research direction in this regard. Here as well, however, the lack of large amounts of annotated data is a limiting factor. \\citep{raghu_transfusion_2019}. Heterogeneity may also occur in labels~\\citep{joskowicz_inter-observer_2019}. This could potentially be addressed with fuzzy output\/references as well as with probabilistic methods capable of representing multiple plausible solutions in the output, as suggested by some early work on the topic~\\citep{kohl_probabilistic_2018, adler_uncertainty-aware_2019, trofimova_representing_2020}.\\\\\n\n\n\\textbf{How to enable real-time assistance?} (goal 3.6) Fast inference in an interventional setting relies on (1) an adequate hardware and communication infrastructure (covered in Sec.~\\ref{sec:technicalInfrastructure}) and on (2) fast algorithms. The trade-off between algorithm and software optimization should be finely balanced between the available edge compute power and the latency requirements of the specific application. Moving high resolution video between devices or displays inherently adds delays and should be minimized for dynamic assistance applications or whether data inference links to control systems. This means that edge compute solutions should carefully consider the input to the display pipeline and the size of the inference models that can be loaded into an edge processor. Where latency is less critical, cloud execution of AI models has already been shown to be viable in assistive systems (e.g. Cydar EV from Cydar Medical (Cambridge, UK) for endovascular navigation, or CADDIE \/ CADDU from Odin Vision Ltd (London, UK) for AI assisted endoscopy). Cloud computing for real-time assistance relies on good connectivity to move data but offers the possibility of running potentially large inference models and returning results for assistance to the OR. Recent advances in the emerging research field of Tactile Internet with Human-in-the-Loop (TaHiL) \\citep{fitzek2021tactile}, which involves intelligent telecommunication networks and secure computing infrastructure is an enabling technology for real-time remote SDS application. To trigger progress in the field, specific clinical applications requiring real-time support should be identified and focused on. Dedicated benchmarking competitions in the context of these applications could further guide methodological development.\\\\\n\n\n\n\\textbf{How to train and apply algorithms under regulatory constraints?} (goal 3.7) When an SDS data set contains personal medical data, an open challenge lies in how to perform data analytics and train ML models without sensitive information being exposed in the results or models. A general solution that is gaining increasing traction in ML is differential privacy \\citep{dwork_calibrating_2006}. This offers a strong protection mechanism against linkage, de-anonymization and data reconstruction attacks, with rigorous privacy guarantees from cryptography theory. \nA limitation of differential privacy can be seen in the resulting compromise in terms of model accuracy, which may conflict with accuracy targets. \nDifferential privacy may ultimately be mandatory for federated learning \\citep{li_privacy-preserving_2019} and publicly releasing SDS models built from personal medical data. Since patients have the right to delete their data, privacy questions also arise regarding models that were trained on their data. In addition, it might be an attractive business model for companies to sell their annotated data or make them publicly available for research purposes. This requires methods to detect whether specific data has been used to train models, e.g. using concepts of \\enquote{radioactive data} \\citep{sablayrolles_radioactive_2020}, or methods that detect whether a model has forgotten specific data \\citep{liu_have_2020}. A complementary approach to preserving privacy is to work with a different representation of the data. For example, \\citep{twinanda_data-driven_2015, SharghiHOM20} evaluate the use of depth images rather than RGB images to recognize human activity in the hospital, while \\cite{haque2018activity,srivastav_human_2019} performs the analysis on low-resolution images. \\\\\\\\\n\n\n\\textbf{How to ensure meaningful validation and evaluation?} (all goals) Validation - defined as the demonstration that a system does what it has been designed to do - as well as evaluation - defined as the demonstration of the short-, mid- and long-term added values of the system - are crucial for the development of SDS solutions. The problem with the assessment of ML methods today is that models trained on a particular data set are evaluated on new data \ntaken from the same distribution as the training data. Although recent efforts have been made in healthcare \\citep{mckinney_international_2020} to include test data from different clinical sites, these still remain limited. This situation poses a challenge particularly for healthcare applications, as real-world test data, after the model is deployed for clinical use, will typically  not have ground-truth annotation, making its assessment difficult \\citep{castro_causality_2020}. A recent example of this is Google Health's deep learning system that predicts whether a person might be at risk for diabetic retinopathy. In this case, after its deployment at clinics in rural Thailand, despite having high theoretical accuracy, the tool was reported to be impractical in real-world testing \\citep{techcrunch_google_2020}. In the future, evaluation of methods should be performed increasingly in multi-center settings and incorporate the important aspects of robustness to domain shifts, data imbalance and bias. Global initiatives such as \\citeauthor{mlcommons_2018} and its Medical Working Group will play a central role in designing benchmarks and propose best practices in this regard. Furthermore, matching performance metrics to the clinical goals should be more carefully considered, as illustrated in recent work~\\citep{reinke_common_2021}. Finally, specific technical aspects (e.g. explainability, generalization) should be comparatively benchmarked with international challenges and covered at dedicated workshops. In this context, acquiring dedicated sponsor money for annotations could help generate more high-quality public data sets.\n\n\\section{Data annotation and sharing}\n\\label{sec:dataAnnotation}\n\nThe access to annotated data is one of the most important prerequisites for SDS.\nThere are different requirements that impact the quality of the annotated data sets. Ideally, they should include multiple centers to capture possible variations using defined protocols regarding acquisition and annotation, preferably linked to patient outcome. In addition, the data set has to be representative for the task to be solved and combined with well-defined criteria for validation and replication of results. Broadly, the key considerations when generating an annotated data set include reliability, accuracy, efficiency, scalability, cost, representativeness and correct specification.\n\n\\subsection{Current practice}\nA comprehensive list of available curated data sets that are relevant to the field of SDS is provided in appendix \\ref{app:publicDataRepositories}. In general, they serve as a good starting point, but are still relatively small, often tied to a single institution, and extremely diverse in structure, nomenclature, and target procedure.\n\nSurgical data such as video involves diverse annotations with different granularity depending on the clinical use case to be solved. It can be distinguished between spatial, temporal or spatio-temporal annotations. Examples for spatial annotations include image-level classification (e.g. what tissue\/tools\/events are visible in an image), semantic segmentation (e.g. which pixels belong to which tissue\/tools\/events in an image) and numerical regression (e.g. what is the tissue oxygenation at a certain location). Temporal annotations involve the surgical workflow and can have different levels of granularity, e.g. surgical phases at the highest level, which consist of several steps, which are in turn composed of activities such as suturing or knot-tying \\citep{lalys_surgical_2014}. In addition, specific events such as complications, performance or quality assessment of specific tasks complement temporal annotations. Spatio-temporal annotations involve both spatial and temporal information. While simple annotation tasks such as labeling surgical instruments may be accomplished by non-experts~\\citep{maier-hein_can_2014}, more complex tasks such as tissue labeling or quality assessment of anastomoses most likely require domain experts.\n\n\nThe major bottleneck for data annotation in surgical applications is access to expert knowledge.\nReducing the annotation effort is therefore of utmost importance, and various methods have been proposed. Crowdsourcing \\citep{maier-hein_can_2014} has proven to be a successful method, but designing the task such that non-experts are able to provide meaningful annotations is still one of the biggest challenges. Recently, active learning approaches that determine which unlabeled data points would provide the most information and  thus reduce the annotation effort to these samples have been proposed \\citep{bodenstedt_active_2019}. Similarly, error detection methods reduce the annotation effort to  erroneous samples only \\citep{lecuyer_assisted_2020}. Data can also be annotated directly during acquisition\n\\citep{padoy_statistical_2012, sigma_surgical_corporation_sigma_nodate}.\n\n\\subsection{Key initiatives and achievements}\n\n\nOne of the most successful initiatives fostering access to open data sets is \\citeauthor{grand_challenge_grand_nodate} which provides infrastructure and tools for organizing challenges in the context of biomedical image analysis. The platform hosts several challenges including data sets and also serves as a framework for end-to-end development of ML solutions. Notably, the Endoscopic Vision Challenge \\citeauthor{endovis_endovis_2015}, an initiative that takes place at the international conference hosted by the Medical Image Computing and Computer Assisted Intervention (MICCAI) Society, is the largest source of SDS data collections~\\citep{bernal_comparative_2017, endovis15_instrument_subchallenge_dataset_endovis15_nodate, endovis-giana_gastrointestinal_nodate, allan_2017_2019, hattab2020kidney, ac97-8m18-21, allan_2018_2020, maier-hein_heidelberg_2021, allan_stereo_2021, endovis-workflowandskill_endovissub-workflowandskill_nodate, ros_comparative_2021, zia_surgical_2021, huaulme_micro-surgical_2021, EndoVis-HeiSurf_nodate, EndoVis-GIANA21_nodate, EndoVis-CholecTriplet2021_nodate, EndoVis-FetReg_nodate, EndoVis-PETRAW_nodate, EndoVis-SimSurgSkill_nodate}. It consists of several sub-challenges every year which support the availability of new public data sets for developing and benchmarking methods. Generally speaking, however, quality control in biomedical challenges and data sharing is still an issue \\citep{maier-hein_why_2018, maier-hein_bias_2020}. \n\n\nThe importance of public data sets in general is illustrated through new journals dedicated to only publishing high quality data sets, such as\n\\textit{Nature Scientific Data}.\nAn important contribution in this context are the FAIR data principles\\citep{wilkinson_fair_2016}, already introduced in the \\textit{context statement} above.  \nRecently, the Journal of the American Medical Association (JAMA) Surgery partnered with the \\citeauthor{surgical_outcomes_club_surgical_nodate} and launched a series consisting of statistical methodology articles and a checklist that aims to elevate the science of surgical database research \\citep{haider_checklist_2018}. It also includes an overview of the most prominent surgical registries and databases, e.g. the National Cancer Database \\citep{merkow_practical_2018}, the National Trauma Data Bank \\citep{hashmi_practical_2018} or the National Surgical Quality Improvement Program \\citep{raval_practical_2018}.\n\nAnnotation of data sets requires consistent ontologies for SDS. The OntoSPM project \\citep{gibaud_ontospm_2014} is the first initiative whose goal is to focus on the modeling of the entities of surgical process models, as well as the derivation LapOntoSPM \\citep{katic_erratum_2016} for laparoscopic surgery. OntoSPM is now organized as a collaborative action associating a dozen research institutions in Europe, with the primary goal of specifying a core ontology of surgical processes, thus gathering the basic vocabulary to describe surgical actions, instruments, actors, and their roles. An important endeavor that builds upon current initiatives was recently initiated by SAGES, which hosted an international consensus conference on video annotation for surgical AI. The goal was to define standards for surgical video annotation based on different working groups regarding temporal models, actions and tasks, tissue characteristics and general anatomy as well as software and data structure \\citep{meireles2021sages}.\n\n\\subsection{Standards, platforms and tools}\n\nIn SDS, images or video are typically the main data sources since they are ubiquitous and can be used to capture information at different granularities ranging from cameras observing the whole interventional room or suite to cameras inserted into the body endoscopically or observing specific sites through a microscope \\citep{chadebecq_computer_2020}. Different image\/video annotation tools regarding spatial, temporal and spatio-temporal annotations already exist (Table~\\ref{tab:annotationTools}), but to date no gold standard framework enabling different annotation types combined with AI-assisted annotation methods exists in the field of SDS.\n\nConsistent annotation requires well-defined standards and protocols taking different clinical applications into account. Current initiatives are working on the topic of standardized annotation, but no widely accepted standards have resulted from the efforts yet. Notable exceptions can be seen in the fields of skill assessment, where annotations have been required for a long time to rate students and can serve as an example for different kinds of SDS annotation protocols \\citep{vedula_objective_2017}, and in  cholecystectomy, where methods for consistent assessment of photos \\citep{Sanford2014_Doublet} and videos \\citep{mascagni_formalizing_2020} of the Critical View of Safety (CVS) were developed to favour documentation of this important safety step.\n\nData annotation also requires a consistent vocabulary, preferable modeled as ontology (Sec.~\\ref{sec:technicalInfrastructure}). Several relevant ontologies with potential use in surgery such as the Foundational Model of Anatomy (FMA), SNOMED CT or RadLex \\citep{langlotz_radlex_2006} are already available. Existing initiatives like the OBO Foundry project that focuses on biology and biomedicine provide further evidence that building and sharing interoperable ontologies stimulate data sharing within a domain. In biomedical imaging, ontologies have been successfully used to promote interoperability and sharing of heterogeneous data through consistent tagging \\citep{gibaud_neurolog_2011, smith_biomedical_2015}. \n\nThe challenges and needs for gathering large-scale, representative and high-quality annotated data sets are certainly not limited to SDS. In response, a new industry branch has emerged offering online data set annotation services through large organized human workforces. A listing of the major companies is provided in Table~\\ref{tab:annotationServices}. Interestingly, the market was estimated to grow to more than USD~1~billion by 2023 in 2019 \\citep{cognilytica_data_2019}, but the consecutive annual report this year estimates the market to grow to more than USD~4.1~billion by 2024 \\citep{cognilytica_data_2020}. Most companies recruit non-specialists who can perform conceptually simple tasks on image and video data, such as urban scene segmentation and pedestrian detection for autonomous driving. Recently, several companies such as Telus International (Vancouver, BC, CA) and Edgecase AI LLC (Hingham, MA, US) have started offering medical annotation services performed by networks of medical professionals. However, it is unclear to what extent medical image data annotation can be effectively outsourced to such companies, particularly in the case of surgical data, where important context information may be lost. Furthermore, the associated costs of medical professionals as annotators and annotation reviewers for quality assurance may render these services out of reach for many academic institutes and small companies.\n\n\\subsection{Current challenges and next steps}\n\\label{sec:dataAnnotation_currentChallengesAndNextSteps}\n\n\nThe data annotation-related mission as well as corresponding goals generated by the consortium  \nare provided in Tab.~\\ref{tab:goalsMission2}. This section elaborates on some of the most fundamental aspects:\n\n\\input{tables\/goalsMission2}\n\n\n\n\\textbf{How to develop standardized ontologies for surgical data science?} (goal 2.1)\nAs current  practices and standards differ greatly between different countries, clinical sites, and healthcare professionals, publicly available surgical data sets generally display vast variation in terms of their annotations. The field, however, is in need of standardized annotations based on a common vocabulary which can be achieved by shared ontologies.  For example, evaluating the efficacy of a particular procedure requires a standardized definition and nomenclature for the different hierarchy levels, e.g. the phases, steps\/tasks and activities\/actions. A standardized nomenclature along with specifics such as beginning and end of temporal events does not exist yet. Studies can help standardize these definitions and reach a consensus. This is for instance demonstrated by \\citet{kaijser_delphi_2018} who conducted a Delphi consensus study to standardize the definitions of crucial steps in the common procedures of gastric bypass and sleeve gastrectomy. Such processes could be adopted for other domains, with the Delphi methods being a particularly useful tool to agree on terminology. Once available and broadly adopted, a shared ontology would stimulate the community as well as boost data and knowledge exchange in the entire domain of SDS. Less formal options such as terminologies are also an alternative but may risk to reach some limits in the long term.\\\\\n\n\n\\textbf{How to account for biases?} (goal 2.2)\nVarious sources and types of bias with potential relevance to SDS have been identified in the past \\citep{ho_biases_2020}. Among the most critical are \\textit{selection bias} and \\textit{confounding bias}. Selection bias, also called \\textit{sample bias}, refers to a selection of contributing data in a way that does not allow for proper randomization or representativeness to be achieved. Crucially, in the context of SDS, representativeness refers to numerous factors including variances related to patients (e.g. age, gender, origin), the surgical procedure (e.g. adverse events), input data (e.g. device type, protocol; preprocessing methods), and surgeons (e.g. level of expertise). Creating a fully representative data set is thus highly challenging and only possible in a multi-center setting. Unrepresentative data, on the other hand, leads to biased algorithms. A recent study published in the context of radiological data science \\citep{larrazabal_gender_2020}, for example, showed that the performance of AI algorithms for a specific sex (e.g. female) crucially depends on the ratio of samples from the respective sex in the training data set. Another source of overestimation regarding algorithm performance is confounding bias. Confounding ``arises when variables that are not mediators of the effect under study, and that can explain part or all of the observed association between the study exposure and the outcome, are not measured and controlled for during study design or analysis''~\\citep{arah_bias_2017}. Recent work in biomedical image analysis \\citep{badgeley_deep_2019, roberts_common_2021, dietrich_machine_2021} showed that knowledge of confounding variables is crucial to the development of successful predictive models. Conversely, a striking recent example of a confounder rendering results meaningless can be seen in the many papers using a particular pneumonia dataset as a control group in the development of COVID-19 detection and prognostication models. Since this dataset solely consists of young paediatric patients, any model using adult COVID-19 patients and these patients as a control group would likely overperform merely by detecting children \\citep{roberts_common_2021}. Other examples of confounders (also called \\textit{hidden variables}) are chest drains and skin markings in the context of melanoma \\citep{Winkler2019_markedmelanoma} and pneumothorax diagnosis \\citep{Oakden-Rayner2020_hidden}. Recognizing and minimizing potential biases in SDS by enhancing data sets with, for example, relevant metadata is thus of eminent importance.  \\\\\n\n\\textbf{How to make data annotation more efficient?} (goal~2.3)\nOvercoming the lack of experienced observers might be possible through embedding clinical data annotation in the education and curricula of medical students. In fact, early evidence suggests that annotating surgical skills during video-based training improves the learning experience \\citep{delagarza2019_eduannot}. The annotation process could also involve several stages, starting with annotations by non-experts that are reviewed by experts. In a similar fashion, active learning methods reduce the annotation effort to the most uncertain samples \\citep{bodenstedt_active_2019, maier-hein_crowd-algorithm_2016}.\n An alternative approach to overcome the lack of annotated data sets is to generate realistic synthetic data based on simulations. A challenge in this context is to bridge the domain gap, so that models trained on synthetic data generalize well to real data. Promising approaches already studied in the context of SDS are for example generative adversarial networks (GANs) for image-to-image translation of laparoscopic images \\citep{pfeiffer_generating_2019, rivoir2021long} or transfer learning-based methods for physiological parameter estimation \\citep{wirkert_physiological_2017}. In the context of photoacoustic imaging, recent work has further explored the GAN-based generation of plausible tissue geometries from available imaging data~\\citep{schellenberg_data-driven_2021}.\\\\\n\n\n\\textbf{How to establish common standards, protocols and best practices for quality-assured data annotation?} (goals 2.3-2.6\/2.9)\nStandardized open-source protocols that include well-defined guidelines for data annotation are needed to provide accurate labels. Ideally, the annotations should be generated by multiple observers and the protocol should be defined to reduce inter-observer variability and bias. A recent study in the context of CT image annotation concluded that more than three annotators might be necessary to establish a reference standard \\citep{joskowicz_inter-observer_2019}. Comprehensive labeling guides and extensive training are necessary to ensure consistent annotation. \\citet{shankar_evaluating_2020}, for example, proposed a 400-page labeling guide in the context of ImageNet annotations to reduce common human failure modes such as fine-grained distinction of classes.  In SDS, a protocol with checklists and examples on how to consistently segment hepatocystic anatomy and assess the CVS in laparoscopic cholecystectomy was recently published to favour reproducibility and trust in the clinical relevance of annotations \\citep{MascagniAlapatt2021_AnnotationProtocol}. Such detailed annotation protocols and extensive user training supported by adequate training material are now required.\nHowever, establishing annotation guides for surgical video data is a particularly challenging task since it involves complex actions that require understanding of the surgical intent based on visual cues. In particular, temporal annotations such as phase transitions are often challenging as the start and end of a specific phase is hard to define.  \\cite{ward_2021} provide a comprehensive list regarding challenges associated with surgical video annotation. Taking into account the variety of surgical techniques this may lead to annotation inconsistencies even amongst experts, but these could also be used as a hint to estimate the difficulty associated with a surgical situation \\citep{ward_2021}. In this context, research on the needs with respect to data and annotation quality in the context of the clinical goals is also required. \nAs data sets and annotations evolve over time, another aspect to be taken into account involves versioning of data sets and annotations, similar to code, which is a non-trivial task \\citep{marzahl2021exact}. For all tasks related to data annotation, it will be prudent to establish and enforce best practices that can easily be integrated into the surgical workflow. This could be achieved by journal editors explicitly requesting annotation protocols to be submitted along with a paper that is based on annotated data. Journals could also allow for the explicit publication of annotation protocols in analogy to study protocols. \nFinally, platforms that enable spatial as well as temporal annotation in a collaborative manner and  share common annotation standards and protocols as well as ML based methods to facilitate automatic annotations are crucial.One means is to adapt already existing annotation platforms (see Table~\\ref{tab:annotationTools}) to fit the specific needs of SDS. Funding agencies should explicitly support efforts to make progress in this regard.\nOverall, a particularly promising approach to generating progress with respect to annotation standards is to start from the respective societies, such as SAGES. Alternatively or additionally, international working groups, similar to the one developing the DICOM standard, should be established. Such working groups should collaborate with existing initiatives, such as DICOM or HL7.\nIn the end, standards will only be successful if enough resources are invested into the actual data annotation. In this case various non-monetary incentives should be considered, including gamification and the issuing of certificates (e.g. for \\textit{Certified Professional for Medical Data Annotation} in analogy to \\textit{Certified Professional for Medical Software})\\\\\n\n\n\\textbf{How to incentivize and facilitate data sharing across institutions?} (goals 2.7-2.9)\nData anonymization is a key enabler for sharing medical data and advancing the SDS field. By definition, anonymized data cannot be traced back to the individual and in both the USA and EU, anonymized data are not considered personal data, rendering them out of the scope of privacy regulation such as the GDPR. However, achieving truly anonymized data is usually difficult, especially when multiple data sources from an individual are linked in one data set. Removing identifiable metadata such as sensitive DICOM fields linking the patient to the medical image is necessary but not always sufficient for anonymization. For example, removing DICOM fields in a Magnetic Resonance Imaging (MRI) scan of a patient's head is not sufficient because the individual may be identified from the image data through facial recognition \\citep{schwarz_identification_2019}. Pseudonymization is a weaker form of anonymization where data cannot be attributed to an individual unless it is linked with other data held separately \\citep[Article 4, Definitions]{general_data_protection_regulation_gdpr_regulation_2016}. This is often easier to achieve compared to true anonymization, however, pseudonymized data are still defined as personal data, and as such remain within the scope of the GDPR. The public data sets used in SDS research such as endoscopic videos recorded within the patient's body are generally assumed to be anonymized but clear definitions and regulatory guidance are needed.\nRecent advances in federated learning could reduce security and privacy concerns since they rely on sharing machine learning models rather than the data itself~\\citep{kaissis2020secure} (see Sec.~\\ref{sec:technicalInfrastructure}).\nA complementary strategy for bypassing current hurdles related to data sharing is \\textit{data donation}. \\textit{Medical Data Donors e.V.}, for example, is a registered German non-profit organization, designed to build a large annotated image database which will serve as a basis for medical research. It can be supported by the public via donation of medical imaging data or by shopping at Amazon Smile. In the broader context of data donation, the SDS initiative discussed the concept of a \\textit{data donation  card} in analogy to the existing \\textit{organ donation card}. With such a card, patients could explicitly state which kind of data they are willing to share with whom and under which circumstances. \n Overall, making progress on large public databases will require establishing an interlocking set of standards, technical methods, and data analysis tools tied to metrics to support reproducible SDS \\citep{nichols_best_2017} and provide value for the community. Clinical registries provide a good example of such a mechanism. In a registry, a specific area of practice agrees on data to be shared, outcome measures to be assessed, and standardized formats as well as quality measures for the data \\citep{arts_defining_2002}. Identifying areas of SDS where the value proposition exists to drive the use of registries would provide much-needed impetus to create data archives. So would creating more monetary and non-monetary incentives for institutions, clinical staff and patients to share and annotate data, although particularly the issue of incentivizing patients to share data presents an ethical gray area.\\\\\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n15 years have passed since the vision of the operating room of the future was sketched for the year 2020 \\citep{cleary_or2020_2004}. A central goal of the SDS 2019 workshop was to revisit the paper and report produced by \\citet{cleary_or_2005} and \\citet{mun_operating_2005} and investigate where we stand, what has hindered us to achieve some of the goals envisioned and what are new trends that had not been considered at the time.\n\nWhen asked: \\enquote{What has really changed when you are entering the OR of today as compared to the setting in 2004?}, participants came to the conclusion that they do not perceive any disruptive changes. Improvements were stated to be of rather incremental nature including advances in visualization (e.g. 3D visualization and 4K video imaging \\citep{ceccarelli_evolving_2018, dunkin_3d_2015, rigante_preliminary_2017}) and improvements in tissue dissection, which is now safer, easier and faster to perform due to ultrasound scissors and impedance controlled electrosurgery, for example. None of these innovations includes a relevant AI or ML aspect. And some developments did not even come with the envisioned benefits. For instance, staplers of today are by far more sophisticated than 10 years ago, but the problem of anastomotic leakage is still relevant \\citep{stamos_anastomotic_2018}. The following paragraphs put the main (six) topics of the 2004 workshop into today's perspective.\n\n\\textbf{Operational efficiency and workflow:} Core problems identified in 2004 were the \\enquote{absence of a standard, computerized medical record for patients that documents their histories and their needs} as well as \\enquote{multiple and disparate systems for tracking related work processes}. While these problems have remained until today (see Sec.~\\ref{sec:technicalInfrastructure}), the challenge of integrating the different information sources related to the entire patient pathway has meanwhile been widely acknowledged. Emerging standards like HL7 FHIR and the maturing efforts of IHE form a solid base for future developments. However, standards alone are not sufficient to solve the problem; hospitals need to make data acquisition, exchange and accessibility a requirement. HIT that enables fast deployment of tools for data acquisition, annotation and processing should be seen as a core service to enable cutting edge research. By centralizing such efforts, data pools can be maintained over the scope of many projects instead of creating isolated databases. This brings with it the need to standardize regulatory workflows. Getting access to data for research is often highly challenging. By outlining clear guidelines and codes of conduct, time spent on formalities can be cut while reducing uncertainties regarding what is the right or wrong way to handle sensitive data. Finally, the prevalence of unstructured data needs to be decreased in order to increase data accessibility. At this point, this also seems to be a matter of user interfaces - by providing clinicians with tools to rapidly create structured reports, reliance on free text can be reduced. This, however, requires training and acceptance by clinical personnel - which could be increased through education in data science topics.\n\n\\textbf{Systems integration and technical standards:} OR integration was the aim of multiple international initiatives, such as OR.NET, the Smart Cyber Operating Theater (SCOT) project \\citep{iseki_scot_2012} and the Medical Device \\enquote{Plug-and-Play} (MD PnP) Interoperability Program. Despite these ongoing efforts we are, however, still far from an OR in which \\enquote{all machines and imaging modalities can talk to each other}, as postulated in 2004. Again, interoperability with intraoperative devices should be viewed as a prerequisite by clinical management, and as an investment in future workflow and cost optimization. Emerging standards like SDC provide a means to enable data exchange; however, more work needs to be invested in the creation of platforms that enable dynamic reactions to events and complex interactions.\n\n\\textbf{Telecollaboration:} While the OR of the twenty-first century connects many different individuals from various disciplines, telecollaboration has only slightly evolved during the last one and a half decades, and a genuine breakthrough has not yet been achieved \\citep{choi_telesurgery_2018}. Many of the impediments can be seen in missing technical developments (e.g. regarding data compression and latency), coordination issues and knowledge gaps on the part of the prospective users as well as the aforementioned lack of data standardization \\citep{mun_operating_2005}. It is to be hoped that coming improvements in  intelligent telecommunication networks (e.g. 5G) might trigger future progress in telecollaboration.\n\n\\textbf{Robotics and surgical instrumentation:} In 2020, numerous surgical procedures, including major surgery on the esophagus, pancreas or rectum, are feasible to be performed using surgical robots. In striking contrast, the actual use of surgical robotics is still marginal. A number of high-quality controlled trials failed to prove superiority, making the use of surgical robotics in many cases difficult to justify \\citep{roh_robot-assisted_2018}. Another reason for the poor progress may lie in the lack of competition in hardware. Since the discontinuation of the development of the ZEUS device in 2003, the field has been clearly dominated by the da Vinci system. Only in recent times, truly competitive systems such as the Senhance\\texttrademark{} (TransEnterix) or the Versius\\textsuperscript{\\textregistered} (Cambridge Medical Robotics Ltd., Cambridge, UK) system have begun to emerge \\citep{peters_review_2018}. It will be exciting to see whether a broader range of technical solutions, along with, perhaps, a stronger interlocking with next-generation intraoperative imaging, will stimulate this particular aspect of the next OR.\n\n\\textbf{Intraoperative diagnosis and imaging:} While intraoperative imaging appeared very promising in 2004, the modest successes that have been made in that area are mostly related to mobile X-Ray based devices and drop-in devices in robotics \\citep{diana_prospective_2017, goyal_new_2018}. The pivotal problem of matching pre- and intraoperative images still remains, as does the unsolved issue of adaptive real-time visualization during intraoperative deformation of soft tissue. One emerging and very promising field is the field of biophotonics (see Sec.~\\ref{sec:technicalInfrastructure}). Benefiting from a lack of ionizing radiation, low hardware complexity and easy integrability into the surgical workflow, biophotonics has yielded an increasing number of success stories in intraoperative imaging \\citep{bruins_vascular_2020, neuschler_pivotal_2017}.\n\n\\textbf{Surgical informatics:} In 2004, the term SDS had not been invented. At that time, surgical informatics was defined as the collection, storage\/organization, retrieval, sharing, and rendering of biomedical information that is relevant to the care of the surgical patient, with an aim to provide comprehensive support to the entire healthcare team \\citep{mun_operating_2005}. Since the beginnings of the field of computer-aided surgery, however, artificial intelligence and in particular ML have arisen as new enabling techniques that were not in the focus 15 years ago. While these techniques have begun revolutionizing other areas of medicine, in particular radiology \\citep{kickingereder_automated_2019, shen_deep_2017}, SDS still suffers from a notable absence of success stories. This can be attributed to a number of various challenges, specifically related to high quality and high volume data annotation, as well as intraoperative data acquisition and analysis and surgical workflow integration, as detailed in Sec.~\\ref{sec:technicalInfrastructure}-~\\ref{sec:clinicalTranslation}.\\\\\n\nOverall, the comparison between the workshop topics discussed in 2004 and 2019 revealed that the most fundamental perceived difference is related to how the future of surgery is envisioned by experts in the field. While discussions in 2004 were mainly centered around devices, AI is now seen as a key enabling technique for the future OR. This article has therefore been centered around technical challenges related to applying AI\/ML techniques to surgery.\n\nA core challenge now is to put the vision of SDS into clinical practice, but the hurdles appear high. A recent query among surgeons clearly demonstrated that \\enquote{digitalization} in general and typical SDS topics in particular are still far removed from their focus of interest \\citep{wilhelm_digitalization_2020}. Accordingly, their readiness to get involved in scientific contributions to SDS is still low. As evidenced in a recent systematic review, even in the context of medical image diagnosis, only few prospective deep learning studies and randomized clinical trials exist, with the vast majority of the latter at a high risk of bias and suffering from reproducibility issues, small human comparator groups as well as deviation from existing reporting standards \\citep{nagendran_artificial_2020}. Striving for more and higher quality studies on the impact of AI thus becomes imperative for the advancement of the field towards clinical applicability as well as for precluding the rise of false expectations or unattainable promises for the future.\n\nLast but not least, in order to position SDS as a thriving research field in the 21st century, it is of the utmost importance to recognize and enhance the interdisciplinary nature of the field. The establishment of SDS as an independent transdisciplinary career path between the fields of surgery and informatics, which is currently being initiated, is one step on a potential roadmap towards restructuring SDS. Further steps will include enhancement of visibility and clearer communication of challenges to the computer vision and ML communities. Since many SDS applications could be beyond the capabilities of today's ML models (especially time-series data), it becomes crucial to enlist and bring together as many and highly qualified AI researchers of different backgrounds as possible. In a similar vein, clinical researchers who have worked independently on surgical process modeling and objective skill assessment need to be featured more prominently in the development of SDS applications. Clinicians need to become actively involved in driving applications that will be of value to them, and common goals between clinicians and researchers need to be exploited, for instance in the shape of shared task forces. A concrete example for this can be seen in the integration of the SDS community with the video-based assessment program SAGES, where clinicians and researchers alike benefit from shared goals regarding data collection and annotation for surgical education \\citep{feldman_sages_2020}.\n\nSDS is a multidisciplinary field that requires motivation, discipline and involvement from a number of different stakeholders and scientific or clinical backgrounds. Together, it becomes viable to strive for fast and efficient development of better models that accurately represent the complexities of SDS issues, thereby moving closer to clinical translation.\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nMore than 15 years ago, in 2004, leading researchers in the field of computer aided surgery (CAS) organized the workshop \\enquote{OR2020: Operating Room of the Future}. Around 100 invited experts including physicians, engineers, and operating room (OR) personnel attended the workshop \\citep{cleary_or2020_2004} to define the OR of the future, with 2020 serving as target time frame. Interestingly, many of the problems and challenges identified back in 2004 do not differ substantially from those we are facing today. Already then, researchers articulated the need for \\enquote{integration of technologies and a common set of standards}, \\enquote{improvements in electronic medical records and access to information in the operating room}, as well as \\enquote{interoperability of equipment}. In the context of data-driven approaches, they criticized the lack of an \\enquote{ontology or standard} for \\enquote{high-quality surgical informatics systems} and underlined the need for \\enquote{clear understanding of surgical workflow and modeling tools}. Broadly speaking, the field has not made progress as quickly as researchers had hoped for at the time.\n\nMore recently, the renaissance of data science techniques in general and deep learning (DL) in particular has given new momentum to the field of CAS. In response to the general artificial intelligence (AI) hype, a consortium of international experts joined forces to discuss the role of data-driven methods for the OR of the future. Based on a workshop held in 2016 in Heidelberg, Germany, the consortium defined Surgical Data Science (SDS) as a scientific discipline with the objective of improving \\enquote{the quality of interventional healthcare and its value through capture, organization, analysis, and modelling of data} \\citep{maier-hein_surgical_2017}. In this context, \\enquote{data may pertain to any part of the patient care process (from initial presentation to long-term outcomes), may concern the patient, caregivers, and\/or technology used to deliver care, and are analyzed in the context of generic domain-specific knowledge derived from existing evidence, clinical guidelines, current practice patterns, caregiver experience, and patient preferences}. Importantly, SDS involves the physical \\enquote{manipulation of a target anatomical structure to achieve a specified clinical objective during patient care} \\citep{maier-hein_surgical_2018}. In contrast to general biomedical data science, it also includes procedural data as depicted in Fig.~\\ref{fig:sdsComponents}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/sds_building_blocks.pdf}\n\\caption{Building blocks of a surgical data science (SDS) system. \\textit{Perception}: Relevant data is perceived by the system (Sec.~\\ref{sec:technicalInfrastructure}). In this context, \\textit{effectors} include humans and\/or devices that manipulate the patient including surgeons, operating room (OR) team, anesthesia team, nurses and robots. \\textit{Sensors} are devices for perceiving patient- and procedure-related data such as images, vital signals and motion data from effectors. Data about the \\textit{patient} includes preoperative images and laboratory data, for example. Domain \\textit{knowledge} serves as the basis for data interpretation (Sec.~\\ref{sec:dataAnnotation}). It comprises \\textit{factual knowledge}, such as previous findings from studies, clinical guidelines or hospital-specific standards related to the clinical workflow as well as \\textit{practical knowledge} from previous procedures. \\textit{Interpretation}: The perceived data is interpreted in a context-aware manner (Sec.~\\ref{sec:dataAnalytics}) to provide real-time \\textit{assistance} (Sec.~\\ref{sec:clinicalTranslation}). Applications of SDS are manifold, ranging from surgical education to various clinical tasks, such as early detection, diagnosis, and therapy assistance.}\n\\label{fig:sdsComponents}\n\\end{figure*}\n\nThree years later, in 2019, an international poll revealed that no commonly recognized surgical data science success stories exist to date, while success stories in other fields have been dominating media reports for years, as detailed in Sec.~\\ref{sec:lackOfSuccessStories}. The purpose of this paper was therefore to go beyond the broad discussion of the \\textit{potential} of SDS by providing an extensive review of the field and identifying concrete measures to pave the way for clinical success stories. The paper is based on an international workshop that took place in June 2019 in Rennes, France, and structured according to core topics discussed at the workshop. In Section~\\ref{sec:lackOfSuccessStories}, we will review the questionnaire that served as the basis for the workshop as well as the international 4-round Delphi process \\citep{hsu_delphi_2007} we conducted with clinical and technical stakeholders from more than 50 institutions to present concrete goals for the future. In the ensuing sections, we will present the current practice, key initiatives and achievements, standards, platforms and tools as well as current challenges and next steps for the main building blocks of SDS, namely technical infrastructure for data acquisition, storage and access (Sec.~\\ref{sec:technicalInfrastructure}), methods for data annotation and sharing (Sec.~\\ref{sec:dataAnnotation}) as well as data analytics (Sec.~\\ref{sec:dataAnalytics}). A section about achievements, pitfalls and current challenges related to clinical translation of SDS (Sec.~\\ref{sec:clinicalTranslation}) and a discussion of our findings (Sec.~\\ref{sec:discussion}) will close the manuscript. While, by definition, SDS encompasses multiple interventional disciplines, such as interventional radiology and gastroenterology, the present paper puts a strong focus on surgery.\n\n\\section{Lack of success stories in surgical data science}\n\\label{sec:lackOfSuccessStories}\n\nMachine learning (ML) has begun to revolutionize almost all areas of healthcare. Success stories cover a wide variety of application fields ranging from radiology and dermatology to gastroenterology and mental health applications \\citep{miotto_deep_2018, topol_high-performance_2019}. Strikingly, such success stories appear to be lacking in surgery.\n\nThe international \\citeauthor{surgical_data_science_initiative_surgical_2015} \\citep{maier-hein_surgical_2017} was founded in 2015 with the mission to pave the way for AI success stories in surgery. Key result of the first workshop, which was inspired by current open space and think tank formats, was a common definition of SDS~\\citep{maier-hein_surgical_2017} and a thorough description of the challenges in applying AI in interventional healthcare. The second edition of the workshop in 2019 focused on a comprehensive overview of the field including key research initiatives, industrial perspectives and first success stories. Prior to the workshop, the registered participants were asked to fill out a questionnaire, covering various aspects related to SDS. 43\\% of the 77~participants were professors\/academic group leaders (clinical or engineering), while the remaining were mostly either from industry (14\\%) or PhD students \/ Postdocs (36\\%). The majority of participants (61\\%) agreed that the most important developments since the last workshop in 2016 were related to advances in AI. Notably, however, when participants were asked about the most impressive SDS paper, only a single paper \\citep{maier-hein_surgical_2017} (the position paper from the first workshop) was mentioned more than twice (primarily by non-co-authors). The majority of participants agreed that the lack of representative annotated data is the main obstacle in the field and the main reason for the failure of previous SDS projects. Also, when referring to their personal experience, 33\\% associated the main reason of failure of an SDS project with lack of data, followed by underestimation of the problem complexity (29\\%). \\citeauthor{endovis_endovis_2015} (28\\%), Cholec80 \\citep{twinanda_endonet_2017} (21\\%) and JIGSAWS \\citep{gao_jhu-isi_2014} (17\\%) were mentioned as the most useful publicly available data sets but the small size\/limited representativeness of the data set was identified as a core issue~(45\\%).\n\n\n\nBased on the replies to the questionnaire and the subsequent workshop discussions, we identified four areas that are essential for moving the field forward: (1) Technical infrastructure for data acquisition, storage and access, (2) data annotation and sharing, (3) data analytics, and (4) aspects related to clinical translation. These are reflected in the four main sections of this paper. We then conducted a Delphi process involving a consortium of medical and technical experts from more than 50 institutions (see list of co-authors) to formulate a mission statement along with a set of goals that are necessary to accomplish the respective mission (see Tab.~\\ref{tab:goalsMission1}, ~\\ref{tab:goalsMission2}, ~\\ref{tab:goalsMission3} and ~\\ref{tab:goalsMission4}) for each of the four areas.\nMore specifically, the coordinating team of the Delphi process put forth an initial mission statement and an initial set of goals for each of the four missions based on the workshop discussions. In a 4-round Delphi process, the remaining consortium members then iteratively refined the phrasing of the missions statements and goals and added further proposals for goals. This process yielded a set of 6-9 goals per mission that received support by at least two thirds of the voting members. Finally, the consortium collaboratively compiled a list of relevant stakeholders (Tab.~\\ref{tab:stakeholders}) and then rated their importance for the four missions (Appendix~\\ref{app:stakeholderImportance}). To avoid redundancy, the consortium further agreed on the following:\n\n   Context statement: \\textit{Unless otherwise specified, in all of the following text, a) surgical data science (SDS) represents the general context of the suggested phrases and b) \"data\" may pertain to any part of the patient care process (from initial presentation to long-term outcomes), may concern the patient, caregivers and\/or technology used to deliver care and must be acquired, stored, and shared in accordance with both local and international regulatory constraints. In general, c) data handling should comply with the FAIR (\\textbf{F}indability, \\textbf{A}ccessibility, \\textbf{I}nteroperability, and \\textbf{R}euse) principles and d) user-friendliness should be a guiding principle in all processes related to data handling. Finally, e) the term SDS stakeholders refers to clinical, research, industrial, regulatory, public and private stakeholders.}\\\\\n\n\n\n\\input{tables\/stakeholders}\n\n\nBased on the international questionnaire, the on-site workshop and the subsequent Delphi process, the following sections present the perspective of the members of the international data science initiative on the identified key aspects for generating SDS success stories.\n\n\\section{Technical infrastructure for data acquisition, storage and access}\n\\label{sec:technicalInfrastructure}\n\nTo date, the application of data science in interventional medicine (e.g. surgery, interventional radiology, endoscopy, radiation therapy) has found comparatively limited attention in the literature. This can partly be attributed to the fact that only a fraction of patient-related data and information is being digitized and stored in a structured manner \\citep{hager_chapter_2020} and that doing so is often an infeasible challenge in modern ORs. This section focuses on current hurdles in creating an environment that can record and structure highly heterogeneous surgical data for long-term usage. \n\n\n\\subsection{Current practice}\n\nDifferent types of data pose different types of challenges:\\\\\n\n\\textbf{Not all data can currently be acquired:} The OR is a highly dynamic environment where a team of health workers with varying specializations (e.g. surgeons, anesthesia team) continuously makes decisions based on device data, observation of the patient, and the outcome of previous actions. However, a lot of information that the healthcare workers perceive by interacting with the patient and each other is currently not at all acquired although it crucially affects decision making. This information relates to different human senses including vision, touch (e.g. palpation and tactile feedback from tissue) and hearing (e.g. acoustic signals resulting from instrument-tissue interaction \\citep{ostler_acoustic_2020}, communication in the OR, etc.). First initiatives have begun addressing these issues (see Sec.~\\ref{sec:technicalInfrastructure_keyInitiatives}) but the infrastructure is not yet widely available.\\\\\n\n\\textbf{Not all data that can be acquired is recorded and permanently stored:} Surgical data in minimally invasive surgery (MIS) routinely involves live image data of high resolution and frame rate. \nModern stereoscopic endoscopes create two Full High Definition (HD) video streams at 60~Hz. If this data is to be stored uncompressed, it can quickly exceed 50~GB per video, with much larger file sizes possible depending on the situation and additional sensory input, and even larger again considering 4K resolutions.\nHealthcare information technology (HIT) is currently not designed to prospectively record and store such large data files. \\\\\n\n\n\\textbf{Not all acquired data is digitized and stored in a structured manner:} A large proportion of documentation in the hospital is still unstructured. Reports, doctors' letters, transcripts from examinations, treatment strategy plans and many more need to be documented in their original form for legal reasons \\citep{kilian_method_2015}. When creating such documents, it is not uncommon to use printouts or Portable Document Format (PDF) documents that then form the basis of discussions between healthcare personnel or with patients. Resulting decisions are subsequently entered into the most relevant information systems as scans, unstructured, or semi-structured documents. As a result, all processes are documented in a manner satisfactory for legal aspects, but largely inaccessible to computation. This is especially true for information related to the surgical procedure, where the decision process leading up to the final operation strategy may not be stored at all (in simple cases) or only in the form of handwritten plans (in complex cases). \nAdditionally, the exact parameters recorded for a specific intervention may differ between hospitals, leading to missing values if such data sets are merged. \nA host of information is potentially available from the actual surgery, including the exact steps taken, instruments used, information exchanged between personnel, haptic feedback, distractions, adaptations of the strategy plan, etc., many of which are not documented at all in OR reports, or documented incompletely. Evidence of this are e.g. similarly sized reports of the same procedures while the corresponding surgeries have radically different lengths. Additionally, problems during surgery may systematically be underreported \\citep{hamilton_are_2018}. \\\\\n\n\\textbf{Not all data that is stored can be exchanged between systems:} Perioperative data is distributed over varying information systems. For example, Picture Archiving and Communication Systems (PACS) contain image data and videos, Radiology Information Systems (RIS) contain reports, findings and radiotherapy plans, and Laboratory Information Systems (LIS) contain laboratory data. Information systems that focus on a single aspect, e.g. laboratory data, can implement efficient storage, manipulation and retrieval methods specific to the given data types. At the same time, user interaction can be kept as simple as possible, with a large degree of workflow optimization for relevant personnel interfacing with the information systems. Linking data from several systems effectively complicates these models. The more data types are incorporated in a model, the more special cases need to be considered, making the model less accessible and harder to query. However, a strict semantic annotation is a prerequisite for guaranteeing retrievability and interoperability \\citep{lehne_why_2019}. As a result, data exchange between information systems is rare. A positive example has been set in radiology, where the \\citeauthor{digital_imaging_and_communications_in_medicine_dicom_nema_nodate} standard has enabled the structured exchange of imaging data. OR data recording systems have also started to offer connection to other hospital infrastructure systems like electronic medical records (EMR), e.g. NUCLeUS\\texttrademark{} (Sony Corporation, Tokyo, Japan).\nAt present, however, this connectivity is typically not utilized widely or effectively. Also, stored OR data is generally not labelled and hence has limited utilization for SDS projects without significant efforts to restructure it.\\\\\n\\newpage\n\\textbf{Regulatory constraints make data acquisition, storage and access challenging:} SDS data collection, management and use must comply with standards in security and fidelity which typically vary depending on the data type and level of patient-specific information. Data governance in healthcare and specifically in surgery is still challenging and less mature compared to other domains \\citep{tse_challenges_2018}. In the European Union (EU), the \\textit{General Data Protection Regulation (GDPR)} covers issues pertaining to personal data both within the EU and its entry to or exit out of the EU since 2018 \\citep{general_data_protection_regulation_gdpr_regulation_2016}. Similarly, in the United States of America (USA) the healthcare-specific \\textit{Health Insurance Portability and Accountability Act of 1996 (HIPAA)} protects the confidentiality and integrity of patient data. In the United Kingdom (UK), the \\textit{Data Protection Act (2018)} was put in place for the National Health Service (NHS). In other countries, equivalents for data protection exist and are related to the legal frameworks of the respective healthcare system. \n\nFrom an ethico-legal perspective, it is worth noting that companies commonly obtain surgical data either through contracts with individual consulting surgeons, licensing agreements with hospitals or in exchange for discounted pricing of their products. This current practice raises important issues regarding power imbalances and the democratization of data access \\citep{august_value_2021}.\n\n\n\\subsection{Key initiatives and achievements}\n\\label{sec:technicalInfrastructure_keyInitiatives}\n\nThis section presents prominent SDS initiatives with a specific focus on data acquisition, access and exchange.\n\n\\textbf{Data acquisition:} Several industrial and academic initiatives have been proposed to overcome the bottleneck of prospective surgical data acquisition.\n\nThe \\textit{DataLogger} (KARL STORZ SE \\& Co. KG, Tuttlingen, Germany) is a technical platform for synchronously capturing endoscopic video and device data from surgical devices, such as the endoscopic camera, light source, and insufflator \\citep{wagner_big_2017}.\nThe DataLogger has served as a basis for the development of a Smart Data Platform as part of the \\textit{InnOPlan} project \\citep{roedder_digital_2016} and has been continuously expanded to support an increasing number of medical devices and clinical information systems. It has also been used to collect data for Endoscopic Vision challenges (e.g. \\citeauthor{endovis-_endovissub-workflow_nodate, endovis-workflowandskill_endovissub-workflowandskill_nodate, endovis-robust-mis_endovis_nodate}).\n\nThe OR Black Box\\textsuperscript{\\textregistered} \\citep{goldenberg_using_2017} is a platform that allows healthcare professionals to identify, understand, and mitigate risks that impact patient safety. It combines input from video cameras, microphones, and other sensors with human and automated processing to produce insights that lead to improved efficiency and reduced adverse events. \nThe OR Black Box has been in operation in Canada since 2014, in Europe since 2017 and in the USA since 2019. An early analysis of the OR Black Box use in laparoscopic procedures of over 100 patients has demonstrated that errors and distractions as annotated by experts viewing the procedures took place in every case, and often went unnoticed  or were at least not recalled by the surgeon at the time \\citep{jung_first-year_2020}. \n\nIn Strasbourg, France, the Nouvel H\u00f4pital Civil (NHC), the Institut de Recherche contre les Cancers de l'Appareil Dig\u00e9stif (IRCAD) and the Institut hospitalo-universitaire (IHU) record surgery videos for education purposes and research. These are curated and used mainly for IRCAD's WebSurg \\citep{mutter_websurg_2011}, a free online reference for video-based surgery training with over 370,000 members.\n\nThe \\textit{Surgical Metrics Project} began in October 2019 at the Annual Clinical Congress meeting of the American College of Surgeons (ACS). Over 200 board certified surgeons were equipped with wearable technology while they performed a simulated open bowel repair on porcine intestines. Multi-modal data, including electroencephalography (EEG) audio and video data were acquired to quantify efficient and successful operative approaches \\citep{pugh_how_2020}.\n\nThe CDEGenerator is an online platform that addresses the need to create and share definitions of joint Core Data Elements (CDE) \\citep{Varghese2018-gb}. These definitions combine a list of recorded parameters together with an exact semantic description. By agreeing on a common CDE, two hospitals can guarantee that the collected data is compatible to the degree of the described acquisition processes.\n\n\n\\textbf{Data access and exchange:} In the perioperative environment, the nonprofit organization Integrating the Healthcare Enterprise (IHE, Oak Brook, Illinois, USA) has been a driving force in forming a set of standards that facilitate data exchange \\citep{grimes_challenge_2005}. It identifies clinical use cases, their requirements and relevant standards, and publishes guidelines (called \\enquote{profiles}) on how to fulfill such use cases. IHE does not publish standards by itself, but rather identifies sets of standards (e.g. DICOM for image exchange and Logical Observation Identifiers Names and Codes (LOINC) \\citep{forrey_logical_1996} for nomenclature) that are best suited to solve specific aspects of healthcare interoperability. Additionally, IHE regularly hosts \\enquote{Connectathons}, where vendors present services with IHE profile implementations and test their systems against those of other vendors, verifying correct data exchange.\n\nInside the OR, efforts for transmitting and centralizing data have been explored for some time, particularly with integrated OR solutions provided by endoscopic device manufacturers and medical technology providers (KARL STORZ: OR1\\texttrademark; Olympus Medical Systems (Tokyo, Japan): ENDOALPHA; Stryker (Michigan, USA): iSuite; Getinge AB (Getinge, Sweden): Tegris\\textsuperscript{\\textregistered}; Richard Wolf GmbH (Knittlingen, Germany): core nova; STERIS plc (Derby, UK): Harmony iQ\\textsuperscript{\\textregistered}; Brainlab AG (Munich, Germany): Digital O.R.; caresyntax GmbH (Berlin, Germany): PRIME365; Medtronic plc (Dublin, Ireland): Touch Surgery\\texttrademark{} Enterprise; Sony: NUCLeUS\\texttrademark; General Electric Company (Boston, USA): Edison\\texttrademark; EIZO Corporation (Hakusan, Japan): \\newline CuratOR\\textsuperscript{\\textregistered}). The wide availability of such systems should be an enabling technology for SDS efforts, not only allowing capturing of data from the OR but also setting a precedent on data management, security, storage and transmission.\n\nFrequently, integrated ORs only provide technical interoperability for connecting image sources with displays (sinks) by using video and broadcasting standards such as Video Graphics Array (VGA), Digital Visual Interface (DVI), High-Definition Multimedia Interface (HDMI) or DisplayPort (DP). \nHigher levels of interoperability are easier to achieve with Internet Protocol (IP)-based data exchange standards (see Sec.~\\ref{sec:technicalInfrastructure_standardsAndTools}).\n\nAdditionally to video routing and capturing, the integration of data from further devices in the OR is relevant. The German Federal Ministry of Education and Research (BMBF) lighthouse project OR.NET \\citep{rockstroh_ornet_2017}, now continued as a nonprofit organization OR.NET e.V., worked on cross-manufacturer concepts and standards for the dynamic and secure networking of medical devices and information technology (IT) systems in the operating room and clinics \\citep{kricka_history_2019, miladinovic_nfv_2018}. Initial results laid important foundations in the shape of a service-oriented communication protocol for the dynamic cross-vendor networking of medical devices and resulted in the International Organization for Standardization (ISO)\/Institute of Electrical and Electronics Engineers (IEEE) 11073 Service-oriented Device Connectivity (SDC) series of standards (see Sec.~\\ref{sec:technicalInfrastructure_standardsAndTools}). The projects InnOPlan \\citep{roedder_digital_2016} (see paragraph \"Data acquisition\") and \\citeauthor{op_41_op_nodate} also used SDC as the basis for device communication. InnOPlan's Smart Data platform enables real-time provision and analysis of medical device data to enable data-driven services in the operating room. The project \\textit{OP 4.1} aimed at developing a platform for the OR - in analogy to an operating system for smartphones - that allows for integration of new technical solutions via apps.\n\nThe project \\citeauthor{connected_optimized_network__data_in_operating_rooms_condor_project_nodate} is another collaborative endeavor that aims to build a video-driven Surgical Control Tower \\citep{padoy_machine_2019, mascagni2021_ORblackTower} within the new surgical facilities of the IRCAD and IHU Strasbourg hospital by developing a novel video standard and new surgical data analytics tools. A similar initiative is The Operating Room of the Future (ORF) that researches device integration in the OR, workflow process improvement, as well as decision support by combining patient data and OR devices for MIS \\citep{stahl_introducing_2005}.\n\n\n\\subsection{Standards, platforms and tools}\n\\label{sec:technicalInfrastructure_standardsAndTools}\nStandards, platforms and tools have focused on the topics of interoperability as well as data storage and exchange.\n\\subsubsection{Interoperability}\n\nInteroperability is defined by IEEE as \\enquote{the ability of two or more systems or components to exchange information and to use the information that has been exchanged} \\citep{ieee_ieee_1991} or by the Association for the Advancement of Medical Instrumentation (AAMI) as \\enquote{the ability of medical devices, clinical systems, or their components to communicate in order to safely fulfill an intended purpose} \\citep{aami_medical_2020}.\n\nNumerous standards have been introduced to provide interoperability including Health Level 7 (HL7), IEEE 11073, American Society for Testing\nand Materials (ASTM) F2761 (Integrated Clinical Environment (ICE)), DICOM, ISO TC215, European Committee for Standardization (CEN) TC251 and International Electrotechnical Commission (IEC) 62A. Different levels of interoperability can be distinguished, for example through the 7 Level Conceptual Interoperability Model (LCIM) from \\citet{tolk_applying_2007}, which is defined as follows \\citep{wang_levels_2009}:\n\n\\begin{itemize}\n\\item Level 0 -- No interoperability: \\textit{Two systems cannot interoperate.}\n\\item Level 1 -- Technical interoperability: Two systems have the means to communicate, but neither has a shared understanding of the structure nor meaning of the data communicated. \\textit{The systems have common physical and transport layers.}\n\\item Level 2 -- Syntactic interoperability: Two systems communicate using an agreed-upon protocol with structure but without any meaning. \\textit{The systems exchange data using a common format.}\n\\item Level 3 -- Semantic interoperability: Two systems communicate with structure and have agreed on the meaning of the exchanged terms. \\textit{The meaning of only the exchanged data is understood.}\n\\item Level 4 -- Pragmatic interoperability: Two systems communicate with a shared understanding of data, the relationships between elements of the data, and the context of the data but these systems do not support changing relationships or context over time. \\textit{The meaning of the exchanged data and the relationships between pieces of information is understood.}\n\\item Level 5 -- Dynamic interoperability: Two systems are able to adapt their information models based on changing meaning and context of data over time. \\textit{Evolving semantics are understood.}\n\\item Level 6 -- Conceptual interoperability: Includes the understanding and exchange of complex concepts. \\textit{Systems are aware of each other's underlying assumptions, models and processes.}\n\\end{itemize}\n\nThe number of interoperability levels varies from model to model and depends on the goal of the intended classification. For example, \\citet{lehne_why_2019} use only four levels, the first two being identical to those listed above; the third, also called \\enquote{semantic interoperability} addresses the complexities mentioned in levels 3 to 5 here, and the fourth puts forth the concept of \\enquote{Organisational Interoperability}, which includes aspects of level 5 and 6.\nThe following paragraphs use the LCIM to classify the standards of interest to this paper.\n\n\\textbf{(1) Technical interoperability:} Modern hospitals typically have sophisticated networks, which makes technical interoperability the most achievable level \\citep{lehne_why_2019}. The main challenge inside the OR, where real-time capability is often critical, is the available bandwidth. An uncompressed Full HD video stream at 60~fps in a color depth of 24~bit requires a bandwidth of 2.98~Gigabit per second (Gbps, not to be confused with Gigabyte per second (GBps), which is eight times larger). Available Ethernet ports typically have a data transfer rate of 1~Gbps. While more modern installations may reach Ethernet data transfer rates of 10~Gbps, this technology is still expensive and typically reserved for networks in data centers. Wireless networks are even slower: Modern devices often support theoretical speeds between 0.45~Gbps and 1.3~Gbps, which results in an effective bandwidth of around 50\\% of the theoretical limit. The newest Wi-Fi-6 Standard, released late 2019, increases this theoretical limit to over 10~Gbps under laboratory conditions, but the effective speeds and adoption rate remain to be seen. In general, Wi-Fi (Wireless Fidelity) suffers from a higher rate of associated uncertainties as well as latency, depending on a number of environment factors. Critically, Wi-Fi packets may get lost if interference between networks is too high, causing latency spikes of potentially several hundreds of milliseconds, which may negatively affect real-time applications. The new 5G standard for wireless communication can potentially ease some of these problems by reaching theoretical speeds of 20~Gbps and avoiding conflicts with other networks since the relevant frequencies are licensed for specific areas. Additionally, 5G as a method of internet access could enable the transfer of large amounts of data to and from the hospital in relatively short time, something which previously required not readily available fast physical connections like glass fibre. While limitations of available bandwidth can be mitigated by using data compression, importantly, \\enquote{losses imperceptible to humans} can still impede algorithm performance. \n\nIt is worth noting that, especially inside the OR, devices still exist that are entirely unable to connect to networks (from basic technical infrastructure like doors or lights to routine medical equipment like certain anesthesia systems) or are not in the network due to missing capacities (e.g. Ethernet sockets) or software add-ons (e.g. a proprietary application programming interface (API)).\n\n\\textbf{(2) Syntactic interoperability:} At this level, the structure of exchanged data is defined with basic semantic information. This level is arguably where most of today's efforts in medical data interoperability take place, and where a number of standards compete. \nA major player in the standardization is HL7 \\citep{kalra_openehr_2005}, which has developed standards for the exchange of patient data since 1987. The eponymous HL7 standard has been continuously updated and most notably includes the Version 3 Messaging Standard, which specifies interoperability for health and medical transactions. HL7 has been criticized for the complexity of its implementation \\citep{goldenberg_using_2017}, resulting in the proposal of HL7 Fast Healthcare Interoperability Resources (FHIR). HL7 FHIR simplifies implementation through the use of widely applied web technologies. Another important standard is provided by the openEHR foundation. In contrast to HL7, openEHR is not only a standard for medical data exchange, but an architecture for a data platform that provides tools for data storage and exchange. With this, however, come added complexity and challenges.\n\nHL7 and openEHR provide the broadest scope of medical data exchange, but both build on standards that solve specific subtasks. While a complete listing is out of scope for this article, one notable exception is DICOM, which today is the undisputed standard for the management of medical imaging information. In 2019, DICOM was extended to include real-time video (DICOM Real-Time Video (DICOM-RTV)). This extension is an IP-based DICOM service for transmitting and broadcasting real-time video, with synchronized metadata, to subscribers (e.g. a monitor or SDS application server) with a quality comparable to standard OR video cables. \n\nThe previously mentioned standards focus on enabling the exchange of patient-individual data between Hospital Information Systems (HIS). Inside the OR, requirements differ, since a host of devices create a real-time data stream that focuses on sensoric input instead of direct patient information (diagnosis, habits, morbidity). Accordingly, data exchange standards inside the OR are geared towards these data types. OpenIGTLink \\citep{tokuda_openigtlink_2009}, for example, started as a communication protocol for Image Guided Therapy (IGT) applications. Today, OpenIGTLink has been expanded to exchange arbitrary types of data by providing a general framework for data communication. However, it does not define broad standards for the data format, instead relying on users to implement details according to their needs. Through this model, OpenIGTLink enabled data exchange inside the OR long before broad standards were feasible. Similarly, for the field of robotics, the Robot Operating System (ROS) \\citep{koubaa_robot_2016} has been proposed.\n\nMore recent efforts by the OR.NET initiative (see Sec.~\\ref{sec:technicalInfrastructure_keyInitiatives}) produced the IEEE 11073 SDC ISO standard which provides a means for general data and command exchange for devices and enables users to control devices in the OR. Standards less specific to the healthcare environment are also available. Similar to OpenIGTLink, The Internet of Things (IoT), for example, defines a standard for device communication without defining standards for the communicated data. While it has been used for data exchange between information systems \\citep{xie_open_2018}, and between devices in the OR \\citep{miladinovic_nfv_2018}, it has elicited mixed reactions.\n\n\\textbf{(3) Semantic interoperability:} This is the domain of clinical nomenclatures, terminologies and ontologies.\nWhile modern standards like HL7 FHIR and openEHR already define basic semantics in data exchange, extending these annotations to more powerful nomenclatures like SNOMED CT (Systematized Nomenclature of Medicine - Clinical Terms) \\citep{cornet_forty_2008} (see Sec.~\\ref{sec:dataAnnotation}) enables systems to not only share data, but also their exact meaning and scope (i.e. what kind of data exactly falls under the given definition). To illustrate the difference between this level and the previous: HL7 FHIR defines less than 200 healthcare concepts (i.e. terms with a well-defined meaning) \\citep{bender_hl7_2013}, while SNOMED CT defines more than 340,000 concepts \\citep{minarro-gimenez_quantitative_2019}. Today, semantic interoperability is largely defined by terminologies (systematic lists of vocabulary), ontologies (definitions of concepts and categories along with their relationships) and taxonomies (classifications of entities, especially organisms) - the borders between which are often fluid. \nStandard languages such as the Resource Description Framework (RDF), Resource Description Framework Schema (RDFS) and the Web Ontology Language (OWL) \\citep{bechhofer_owl_2009} have been defined by the World Wide Web Consortium (W3C), guaranteeing interoperability between ontology resources and data sets based on these ontologies. The aforementioned SNOMED CT is arguably the most complete terminology, spanning the whole field of clinical terms with a wide set of available translations. However, specialized alternatives may perform better on their respective field. Additionally, a host of medical ontologies are available. Most notable is the family of ontologies gathered under the OpenBiological and Biomedical Ontologies (OBO) Foundry \\citep{smith_obo_2007}, which cover a wide array of topics from the biomedical domain and share the Basic Formal Ontology (BFO) \\citep{grenon_snap_2004} as a common top-level ontology. Intraoperatively, the OntoSPM \\citep{gibaud_toward_2018} provides terminology for the annotation of intraoperative processes, and has spawned efforts for the annotation of binary data \\citep{katic_what_2017}. Common to all these efforts is that they serve best in combination with a standard addressing syntactic interoperability, where they can add semantic information to the data exchange. Semantic interoperability goes hand in hand with data annotation, and is expanded upon in Sec.~\\ref{sec:dataAnnotation}.\n\nIt is important to note that semantic interoperability does not guarantee the availability of data. If two hospitals have agreed on a detailed semantic model but record different parameters for a specific procedures, then the two resulting data sets will contain well-defined but empty fields. Two avoid this, it is necessary to agree on lists of recorded parameters, e.g. in the form of CDE.\n\n\\textbf{(4) Pragmatic interoperability:} In order to define context, additional modeling is required to capture data context and involved processes. This can in part be achieved by extending modeling efforts from the semantic interoperability level to include these concepts. Furthermore, efforts to formalize the exchange processes themselves are required. In IEEE 11073 descriptions for architecture and protocol (IEEE 11073-20701) and in HL7 the IHE Patient Care Device (PCD) implementation guide and the conformance model are provided. \n\nFor the remaining two levels, developments are more recent and less formalized. For Level \\textbf{(5) Dynamic interoperability}, it is required to model how the meaning of data changes over time. This can range from simple state changes (planned operations becoming realized, proposed changes becoming effective) to new data types being introduced and old data types changing meaning or being deprecated. In IEEE 11073 the participant key purposes and in HL7 the workflow descriptions are created for supporting these aspects. Finally, Level \\textbf{(6) Conceptual interoperability} allows for exchanging and understanding complex concepts. This requires a means to share the conceptual model of the system, its processes, state, architecture and use cases. This level can be achieved through defining use cases and profiles (e.g. IHE Services-oriented Device Point-of-care Interoperability (SDPi) Profiles) and\/or provisioning reference architecture and frameworks.\n\n\\subsubsection{Data storage and distribution}\n\nWhile current standards have focused on data exchange, they typically do not address data distribution and storage. Typically, data is exchanged between two defined endpoints (e.g. a tracking device and an IGT application, or a computed tomography (CT) scanner and a PACS system). To achieve a system that can be dynamically expanded with regard to its communication capabilities, it is necessary to implement messaging technology. Such tools allow arbitrary devices to take part in communication by registering via a message broker, where messages can typically be filtered by their origin, type, destination, for instance. Examples include Apache Kafka \\citep{kim_reliable_2017, spangenberg_implementation_2018} or RabbitMQ\\textsuperscript{\\textregistered} \\citep{ongenae_semantic_2016, trinkunas_research_2018}. Such systems enable developers to create flexible data exchange architectures using technologies that are mature and usually well documented thanks to their wide application outside the field of healthcare. However, they also create a level of indirection which introduces additional delay (which may be negligible with only a few milliseconds in local networks, or significant with several tens or even hundreds of milliseconds over the internet or wireless networks).\n\nFinally, recording of the exchanged data requires distinct solutions as well. High-performance, high-reliability databases form an essential requirement for many modern businesses. Thanks to this demand, a large body of established  techniques exists, from which users can select the right tool for their specific needs. Binary medical data (images, videos, etc.) can be stored on premise in modern PACS systems, which provide extensive support for data annotation, storage and exchange. For clinical metadata, the selection of technology typically depends on the level of standardization of the recorded data. Highly standardized data can possibly be stored directly through interfaces of e.g. the IHE family of standards. If the target data are not standardized, but homogeneous, then a database model for classical database languages (e.g. Structured Query Language (SQL)) may be suitable. Use cases where a wide array of highly heterogeneous data is recorded may choose modern NoSQL databases. These databases do not (or not exclusively) rely on classical tabular data models, but instead allow the storage and querying of tree-like structures. The JavaScript Object Notation (JSON) Format is a popular choice for NoSQL databases for its wide support in toolkits and the immediate applicability with regard to Representational State Transfer (REST)-APIs. While initially applications of these databases were geared towards data lakes because of the relative ease of application, NoSQL databases have recently seen widespread application in big data and ML \\citep{dasgupta_practical_2018}. A notable example is Elasticsearch (Elastic NV, Amsterdam, the Netherlands), which has achieved widespread distribution and is ranked among the most used search servers \\citep{db-engines_db-engines_2020}.\n\nThrough the rising relevance of web technology, storing data in the cloud is increasingly becoming a viable option. A vast array of services are available and have been applied in the medical domain (e.g. Amazon Web Services (AWS) \\citep{holmgren_health_2017}, Microsoft Azure \\citep{hussain_cloud-based_2013}, and others). Storing data in the cloud has the potential to save money on HIT by eliminating the need to reduce the locally required storage capacity and maintenance personnel, but brings with it privacy concerns and slower local access to data than from local networks, which may be noticeable especially for large binary data like medical images and video streams. While data privacy options are available for all major services, the implementing personnel have to understand these options and align with them  the privacy needs of the institution and the respective data. Since answering these questions is complex, the privacy requirements  strict, and the consequences for failing to comply with the law severe, the created solutions are often conservative in nature with regard to privacy. Additionally, downloading large data sets may be costly, as in general, cloud storage providers incentivize performing computations in the cloud.\n\nFinally, solutions to facilitate local storage have been proposed. Commercially available systems such as SCENARA\\textsuperscript{\\textregistered} .STORE (KARL STORZ) compress surgical images and video data over time to decrease storage needs.\nAlternatively, SDS tools can be used to selectively store critical video sequences instead of entire procedural videos, as recently proposed \\citep{MasacagniAlapatt2021_EndoDigest}\n\n\n\\input{tables\/goalsMission1}\n\n\n\\subsection{Current challenges and next steps}\n\nThe infrastructure-related mission as well as the corresponding goals generated by the consortium as part of the Delphi process \nare provided in Tab.~\\ref{tab:goalsMission1}. This section elaborates on some of the most fundamental aspects:\\\\\n\n\n\n\n\\textbf{How to enable prospective capturing and storing of relevant perioperative data?} (goals 1.1\/1.2): \nA major challenge we face is to capture all relevant perioperative data. While several initiatives and standards are already dedicated to this problem, a particular focus should be put on the recording and integration of patient outcome measures, including measures that need to be captured long after the patient has left the hospital (e.g. 5-year-survival).\nThe field of SDS stands in contrast to the field of radiology, where the DICOM standard now covers the exchange of medical images and related data. This standard can be seen as a direct result of market pressure: Early medical imaging devices did not prioritize communication standards, instead relying on manufacturer-supplied software specific to the hardware purchased. This behaviour did not change until PACS systems became widespread, providing specialized software that offered a benefit to clinical workflows, and the ability to transmit images to them became a driving requirement for the purchase of new imaging hardware. However, the previously mentioned domain complexity also affects standard development. For example, the DICOM specification document alone consists of 6,864 pages\\footnote{\\url{http:\/\/dicom.nema.org\/medical\/dicom\/current\/} (accessed 2020-07-30)}, indicating the effort to develop and maintain such a standard.\nEvolving standards for the exchange of medical data like IEEE 11073 SDC and HL7 FHIR are a step in the right direction, but in order to create a driving force, incentivizing  the industry to enable widespread interconnection appears useful.\n\nStoring acquired data is, in theory, largely possible with modern technologies. \n Missing, however, are standards for storage format, duration and data quality. These should be developed with the involvement of industrial stakeholders and the respective clinical\/technical societies and should specifically include recommendations with respect to minimum standards for storage and annotation. The international Society of American Gastrointestinal and Endoscopic Surgeons (SAGES), for example, created an AI task force with the mission to propose and establish best practices for structured video data acquisition and storage, including recommendations for resolution and compression. Generally speaking, a clear distribution of roles between different stakeholders, particularly regarding who takes the initiative, as well as a clear definition of the subject matter to be standardized are now needed.\\\\\n\n\\textbf{How to link data from different sources and sites?} (goal 1.3) The need for exchanging data between different sources and sites calls for semantic operability (Sec.~\\ref{sec:technicalInfrastructure_standardsAndTools}): Simply storing all data in a data lake without sufficient metadata management poses the risk of creating a data swamp that makes data extraction hard to impossible \\citep{hai_constance_2016}. \nData distribution among several systems is a healthy approach since it reduces load on a single system and enables engineers to choose the system best suited for the specific types of data stored within. As long as metadata models \\citep{gibaud_toward_2018, marz_toward_2015, soualmia_efficient_2016} exist that are able to sufficiently describe the data and where to find them, retrieval will be possible through querying the model. Accordingly, efforts should focus on enhancing current clinical information infrastructures from the level of syntactic operability to semantic interoperability. Metadata also becomes essential for data sharing. \nAn increasingly popular approach to data sharing is federated learning \\citep{konecny_federated_2016, rieke_future_2020}. Instead of sharing data between institutions, the training of algorithms is distributed among participants. While this presumably reduces the ethical and legal complications associated with large-scale data sharing, it is still necessary to achieve semantic interoperability, and the regulatory issues regarding the exchange of models that contain encoded patient data are not fully understood yet.\\\\\n\n\\textbf{How to perceive relevant tissue properties dynamically?} (goal 1.4) \nSurgical imaging modalities should provide discrimination of local tissue with a high contrast-to-noise-ratio, should be quantitative and digital, ideally be radiation- and contrast agent-free, enable fast image acquisition and be easy to integrate into the clinical workflow. The approach of registering 3D medical image data sets to the current patient anatomy for augmented reality visualization of subsurface anatomical details has proven ill-suited for handling tissue dynamics such as perfusion or oxygenation (e.g. for ischemia detection).\nThe emerging field of biophotonics refers to techniques that take advantage of the fact that different tissue components feature unique optical properties for each wavelength. \nSpecifically, spectral imaging uses multiple bands across the electromagnetic spectrum~\\citep{clancy_surgical_2020} to extract relevant information on tissue morphology, function and pathology (see e.g. \\citet{wirkert_robust_2016, moccia_uncertainty-aware_2018, ayala_video-rate_2021}). Benefiting from a lack of ionizing radiation, low hardware complexity and easy integrability into the surgical workflow, spectral imaging could be leveraged to inform surgical operators directly or be used for the generation of relevant input for SDS algorithms \\citep{Mascagni2018_ORimaging}. Open research questions are, among others, related to reproducibility of measurements, possible confounders in the data~\\citep{dietrich_machine_2021}, inter-patient variability and the robust quantification of tissue parameters in clinical settings.\\\\\n\n\\textbf{How to enable real-time inference in interventional settings?} (goal 1.5)\nWhile processing times of several seconds or even minutes may be acceptable in some scenarios, other SDS applications, such as autonomous robotics, require real-time inference. Real-time inference requires a number of complex prerequisites to be fulfilled. Relevant data needs to be streamed to a common endpoint where it can be processed; data streams need to be sufficiently formalized to enable fully automatic decoding; the hardware and networks receiving these streams must be sufficiently fast to decode the streams with minimal latency and high resilience, and the algorithms that provide inference need to be implemented efficiently and run on sufficiently fast hardware to enable real-time execution. If additional data (e.g. preoperative imaging, patient-specific data, etc.) is required, the algorithms need to be able to access this data, and inferred information needs to be relayed to the OR team in an adequate manner. These problems can potentially be addressed in a variety of ways, however, it seems prudent to integrate the necessary infrastructure (acquisition, computation, display) directly on site in or near the OR. In a first step, test environments such as experimental operating rooms can serve as platforms where technical concepts for real-time interference can be developed, validated and evaluated in a realistic setting.\\\\\n\n\n\\textbf{How to overcome regulatory and political hurdles?} (goal 1.6) Timelines and associated costs of data privacy management (discussed further in Sec.~\\ref{sec:dataAnnotation_currentChallengesAndNextSteps}) and regulatory processes need to be supported in both academic and commercial projects: Academic work requires funding and appropriate provision for delays in the project timeline. \nNotably, the COVID-19 pandemic may have stimulated rapid response from both academic and regulatory bodies in response to urgent needs, and perhaps some of this expedience will remain (examples in Continuous Positive Airway Pressure (CPAP) devices such as \\citeauthor{ucl-ventura_ucl-ventura_2020}). Industry also needs to allocate costs, adhere and maintain standards, cover liability and have clear expectations on the required resources. While these processes are well developed and supported in large organizations, smaller companies, in particular startups,  have less capacities for them at their disposal. A variety of additional standards would also need to be met since a prospective SDS system approaches a medical device as defined by The U.S. Food and Drug Administration (FDA) (USA) or the Medical Device Regulation (MDR) (EU). These may be ISO-certified or require audits and approval from regulatory agencies and notified bodies, compliance with data protection regulations (e.g. GDPR), more stringent (cyber-)security features and testing adherence.\nAs the field of AI and its regulation is increasingly discussed in public venues, political visibility is rising. By clearly identifying the limiting effects of insufficient infrastructure on the one hand, and potential benefits of improving it on the other, it should become possible to convince political and clinical stakeholders that an investment in HIT as well as dedicated data management and processing personnel is key to exploiting the potential of AI for interventional healthcare. Furthermore, industrial engagement in creating the necessary infrastructure needs to be fostered within the boundaries of global standardization while considering the specific market needs. Healthcare institutions thus need to engage globally with industry to put forth common standards and processes enabling SDS applications compatible with strategic business needs. Of note, existing infrastructures can be leveraged and enhanced in this process. The SDS community should be aware of the complexity of the topic and the messages that are publicized (i.e. premature success stories) and create constructive proposals with realistic outlooks on potential benefits, focusing on long-term investments with the potential to drive change. Specifically, market studies could identify for each individual stakeholder the benefits of SDS solutions compared to their expected costs. Consider for instance a \"number needed to treat\" type of example, where for every X number of patients for which data insights are applied, one complication costing USD Y may be avoided. By providing estimated returns on investment for improvements to clinical delivery based on reducing person-hours, complications, or duplicative work, such studies would in turn provide key arguments for future investments. \n\nOverall, local and international collaborations and partnerships involving clinical, patient, academic, industry and political stakeholders are needed (see Tab.~\\ref{tab:stakeholders}). Policies and procedures regarding data governance within an institution have to be defined that involve all stakeholders within  the SDS data lifecycle. Already existing multinational political entities or governing bodies, as exemplified by the EU, can be leveraged in a first step towards international collaboration and standardization. When implementing the goals put forth in Tab.~\\ref{tab:goalsMission1}, internationally agreed standards should be respected. These include, but are not limited to, ethical guidelines. In fact, the world health organization (WHO) recently put forth  a guidance document on \\textit{Ethics \\& Governance of Artificial Intelligence for Health} \\citep{health_ethics__governance_ethics_2021}, which was compiled by a multidisciplinary team of experts from the fields of ethics, digital technology, law and human rights, as well as experts from Ministries of Health. The report identifies the ethical challenges and risks associated with the use of AI in healthcare and puts forth several internationally agreed on best practices for both the public and the private sector.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nRecently, wireless architectures based on the notion of cell-free have gained much interest \\cite{Ngo2015,Ngo2017}. In a cell-free system set-up, the cell-boundaries can be relaxed, and thus, a vast number of access-points (APs) can be  spatially distributed to serve all users with a uniformly better  quality-of-service (QoS)  over a much larger geographical region \\cite{Ngo2015,Ngo2017}.  Moreover, cell-free set-ups may render spectral\/energy efficiency gains, mitigate  impediments caused by spatial-correlated fading in  compact\/co-located antenna arrays, and circumvent shadow fading impairments    \\cite{Ngo2015,Ngo2017}. Thus, cell-free architecture is a foundation for practically realizing extremely large antenna arrays for next-generation wireless standards. \n\nAn intelligence reflective surface (IRS) consists of a large number of  passive reflectors, whose  reflective coefficients can be  adjusted to attain desired propagation effects for the  \nimpinging electromagnetic (EM) waves \\cite{Renzo2019,Liaskos2018}. \nThe feature of intelligently adjustable phase-shifts at an IRS can be used to boost the signal-to-noise ratio (SNR)  and  to mitigate co-channel interference at an intended destination through  constructive and destructive  signal combining, respectively \\cite{Diluka2020}. This leads to the notion of  recycling of EM waves within a propagation medium, and thereby, spectral\/energy efficiency gains and implementation cost reduction  can be realized as IRSs are made out of low-cost meta-atoms without active  radio-frequency (RF) chains\/amplifiers \\cite{Liaskos2018}.\n\n\n\\subsection{Our motivation} \nIn this paper, we aim to investigate the feasibility of embedding an IRS within a cell-free set-up. Specifically,  our objective is to investigate the performance of an IRS-assisted cell-free set-up, and thereby, we explore the feasibility of jointly reaping the aforementioned   benefits of cell-free architectures and IRS-assisted wireless channels. Moreover, to the best of the authors knowledge, the fundamental performance metrics for an IRS-assisted cell-free set-up have not yet been reported in open literature. To this end, we aim to fill this important gap in IRS literature by presenting a performance analysis for an IRS-assisted cell-free set-up.\n\n\n\\subsection{A literature survey for cell-free architecture and performance analysis of IRS-assisted channels}\\label{sec:literature}\nIn \\cite{Ngo2015,Ngo2017}, the basic concept of cell-free architectures is investigated, and thereby, the performance metrics are compared against those of the  co-located antenna arrays. The analyses in \\cite{Ngo2015,Ngo2017,Ngo2018} reveal that the cell-free  set-ups can outperform the co-located counterparts by serving users with a uniformly better QoS,  minimizing the impediments of spatial-correlation, and shortening  the end-to-end transmission distances to boost the overall energy\/spectral efficiency  \\cite{Ngo2015,Ngo2017}. \nReference \\cite{Nayebi2017} proposes max-min power optimization algorithms for cell-free massive  multiple-input multiple-output (MIMO). In \\cite{Galappaththige2019},  the performance of cell-free massive MIMO with underlay spectrum sharing is investigated.\n\nReferences \\cite{Renzo2019,Liaskos2018} present core architectural design principles of IRSs for wireless communications.\nRay-tracing techniques are used in \\cite{Ozdogan2020} to generate a novel path-loss model for IRS-assisted wireless channels.\nIn \\cite{Wu2019}, joint optimization of precoder at the base-station (BS) and phase-shifts at the IRS is studied  through semi-definite relaxation and alternative optimization techniques.  \nReference \\cite{Diluka2020} studies the fundamental performance limits of distributed IRS-assisted end-to-end channels  with Nakagami-$m$ fading channels. In \\cite{Han2019}, by using the statistical channel state information (CSI), an optimal phase-shift design framework is developed to maximize the achievable rates of  IRS-assisted wireless channels. \nIn \\cite{Chen2019}, joint beamforming and reflecting coefficient designs are investigated for IRSs  to provision physical layer security. Reference \\cite{Abeywickrama2020} proposes a practical IRS phase-shift adjustment model, and thereby, the achievable rate is maximized through jointly optimizing the transmit power and the BS beamformer by using alternative optimization techniques. \n\n\n\\subsection{Our contribution}\\label{sec:Motivation}\nIn  above-referred prior research \\cite{Wu2019,Diluka2020,Han2019,Abeywickrama2020,Chen2019} for IRS-assisted communications, a BS with either a single-antenna or  a co-located antenna  array is used.\nHaving been inspired by this  gap in IRS\/cell-free literature, in this paper, we investigate an IRS-assisted wireless channel embedded within a cell-free set-up over Rayleigh fading,  and thereby, we present  fundamental performance metrics. \nTo this end, first, we invoke the central limit theorem (CLT)  to tightly approximate the end-to-end optimal SNR  to facilitate a mathematically tractable probabilistic characterization. Then, we derive  the probability density function (PDF) and the cumulative density function (CDF) of this approximated optimal SNR in closed-form. Thereby, we present a tight approximation to the outage probability. Moreover, we derive tight upper\/lower bounds for the achievable rate. In particular, we investigate the   impediments of discrete phase-shifts in the presence of phase-shift quantization errors. Finally, we present a set of rigorous  numerical results to explore the performance gains of  the proposed system, and we validate the  accuracy of our analysis through Monte-Carlo simulations. From our numerical results, we observe that  by using an  IRS with  controllable phase-shift adjustments, the performance of cell-free wireless set-ups  can be enhanced. \n\n\n\n \n\\noindent\n\\textbf{Notation:} The transpose of vector $\\mathbf y$ is denoted as $\\mathbf y^{T}$. The expectation and variance of a random variable  $Y$ are represented by \n$\\E[]{Y}$ and $\\Var{Y}$, respectively.\n$Y\\sim\\mathcal {CN}\\left(\\mu_Y,  \\sigma_Y^{2}  \\right) $ denotes that $Y$ is  complex-valued circularly symmetric Gaussian distributed with $\\mu_Y$ mean and $\\sigma_Y^{2}$ variance. Moreover, $C_n=\\{0,1,\\cdots,n\\}$ and $C_n'= C_n\/\\{0\\}$.\n \n \n\n\\vspace{0mm}\n\\begin{figure}[!t]\\centering \\vspace{0mm}\n\t\\def\\svgwidth{180pt} \n\t\t\\fontsize{8}{3}\\selectfont \n\t\\input{system_fig1.pdf_tex} \\vspace{-3mm}\n\t\\caption{System model - IRS-aided cell-free communication set-up }\\vspace{-6mm} \\label{fig:system_model}\n\\end{figure}\n\n\n\\section{System, Channel and Signal Models  }\\label{sec:system_model}\n\n\\subsection{System and channel model}\\label{sec:system_and_channel}\n\nWe consider a cell-free communication set-up consisting of $M$   single-antenna APs ($\\mathrm{AP}_m$ for $m = 1, \\cdots, M$)  and a single-antenna destination $(D)$. An IRS having $N$ passive reflective elements is embedded within this cell-free set-up as shown in  Fig. \\ref{fig:system_model}. \nFor the sake of exposition, we denote the set of APs as $\\mathcal M=\\{1,\\cdots, M\\}$ and the set of reflective elements at the IRS as $\\mathcal N=\\{1,\\cdots, N\\}$.\n\nThe direct link between the $m$th AP and $D$ is represented by $u_m$, while  $h_{mn}$ denotes the channel between the $m$th AP and the $n$th reflective element of the IRS. Moreover, $g_n$ is used to represent the channel between the $n$th reflective element of the IRS and $D$.  We model the envelops of  all aforementioned channels to be independent Rayleigh distributed \\cite{Ding2020}, and the corresponding polar-form of these channels is given by \n\\begin{eqnarray}\\label{eqn:channels}\n\tv = \\lambda_v \\exp{j\\theta_{v}},\n\\end{eqnarray}\nwhere $v\\in \\{u_m,h_{mn},g_{n}\\}$ for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$. In \\eqref{eqn:channels}, the envelop and the phase of $v$ are given by $\\lambda_v$ and $\\theta_{v}$, respectively. The  PDF of $\\lambda_v$ is given by\n\\cite{papoulis02}\n\\begin{eqnarray}\\label{eqn:chnl_pdf}\n\tf_{\\lambda_v}(x) = \\left({x}\/{\\xi_v}\\right) \\Exp{{- x^2}\/{\\left(2\\xi_v\\right)}},\n\\end{eqnarray} \nwhere $\\xi_v = \\zeta_v\/2$ is the Rayleigh parameter, and $\\zeta_v$ captures  the large-scale fading\/path-loss of the channel $v$. Since all reflective elements are co-located within the IRS, it is assumed that all large-scale fading parameters are the same.\n\n\n\\subsection{Signal model}\\label{sec:sgnl_modl}\nThe  signal transmitted by the $m$th AP reaches $D$ through the  direct and IRS-assisted reflected channels. Thus, we can write the signal received at $D$ as\n\\begin{eqnarray}\\label{eqn:rx_signl}\nr = \\sqrt{P} \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\mathbf{g}^T  \\mathbf{\\Theta} \\mathbf{h}_{m} \\right) x + w,\n\\end{eqnarray}\nwhere $x$ is the transmit signal from $S$ satisfying $\\E{|x|^2} =1$,  $P$ is the  transmit power at each AP, and $w$ is an additive white Gaussian noise (AWGN) at $D$ with zero mean and variance of  $\\sigma_{w}^2$ such that  $w\\sim \\mathcal{CN}(0,\\sigma_{w}^2)$. In \\eqref{eqn:rx_signl}, $\\mathbf{h}_m = [h_{m1},\\cdots, h_{mn}, \\cdots, h_{mN}]^T\\in \\mathbb C^{N\\times 1}$ is the channel vector between the $m$th AP and the IRS. Moreover,  $\\mathbf{g}^T = [g_{1},\\cdots, g_{n}, \\cdots, g_{N}]\\in \\mathbb C^{1\\times N}$ denotes the channel vector between the IRS and $D$. The diagonal matrix, $\\mathbf \\Theta = \\diag{\\beta_{1} \\exp{j\\theta_{1}}, \\cdots, \\beta_{n} \\exp{j\\theta_{n}}, \\cdots, \\beta_{N} \\exp{j\\theta_{N}}}\\in \\mathbb C^{N\\times N}$, captures the reflective properties of the IRS through complex-valued reflection coefficients  $\\beta_{n} \\exp{j\\theta_{n}}$ for $n\\in \\mathcal{N}$, where $\\beta_{n}$ and $\\theta_{n}$  are the magnitude of attenuation and phase-shift of the $n$th reflective element of the IRS, respectively. Thus, we can rewrite the received signal at $D$ in \\eqref{eqn:rx_signl} as \n\\begin{eqnarray}\\label{eqn:rx_signl_rearng}\nr = \\sqrt{P}  \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n  {h}_{mn} \\exp{j\\theta_{n}} \\right) x + w.\n\\end{eqnarray}\nThereby, we derive the SNR at $D$ from \\eqref{eqn:rx_signl_rearng} as\n\\begin{eqnarray}\\label{eqn:snr}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\gamma &=&  \\bar{\\gamma} \\left| \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n  {h}_{mn} \\exp{j\\theta_{n}} \\right) \\right|^2 \\nonumber \\\\\n&=& \\bar{\\gamma} \\left|  \\sum\\nolimits_{m \\in{\\mathcal{M}}} \\!u_m  \\!+\\! \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n \\!  \\left(\\sum\\nolimits_{m \\in{\\mathcal{M}}} {h}_{mn}\\right) \\exp{j\\theta_{n}} \\right|^2\\!\\!,\n\\end{eqnarray}\nwhere the average transmit SNR is denoted by $\\bar{\\gamma}=P\/\\sigma_{w}^2$. Then, we define $u= \\sum_{m \\in{\\mathcal{M}}} u_m$ and $h_n=\\sum_{m \\in{\\mathcal{M}}} h_{mn}$. Since $u_m$ and $h_{mn}$ are independent complex Gaussian distributed for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$, the polar-form of $u$ and $h_n$ can be also expressed similar to \\eqref{eqn:channels}, where $\\lambda_{u}$ and $\\lambda_{{h}_{n}}$ are the envelops of $u$ and $h_n$, respectively. Thus, $\\lambda_{u}$ and $\\lambda_{{h}_{n}}$ are  independent Rayleigh distributed with    parameters $\\xi_u=\\sum_{m \\in{\\mathcal{M}}} \\zeta_{u_m}\/2$ and $\\xi_{h_n}=\\sum_{m \\in{\\mathcal{M}}} \\zeta_{h_{mn}}\/2$, respectively. From \\eqref{eqn:channels}, we can rewrite the SNR in \\eqref{eqn:snr} in terms of the  channel phases as \n\\begin{eqnarray}\\label{eqn:snr_phase}\n\\gamma =  \\bar{\\gamma} \\left| \\lambda_{u} \\exp{j\\theta_{u}} + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\exp{j\\left(\\theta_{n} + \\theta_{{g}_n} + \\theta_{{h}_{n}}\\right)} \\right|^2.\n\\end{eqnarray}\nIt can be seen from  \\eqref{eqn:snr_phase} that the received SNR at $D$  can be maximized by smartly adjusting  the phase-shifts at each IRS reflecting elements $(\\theta_{n})$. Thus, it enables a constructive addition of the received signals through the direct channels and IRS-aided reflected channels  \\cite{Wu2019,Wu2020}. To this end, the optimal choice of $\\theta_{n}$ is given by $\\theta_{n}^* =\\underset{-\\pi\\leq \\theta_{n} \\leq \\pi}{ \\mathrm{argmax}} \\;{\\gamma} = \\theta_u - \\left(\\theta_{g_{n}} + \\theta_{h_{n}}\\right)$. Then,  we can derive the optimal SNR at $D$  as\n\\begin{eqnarray}\\label{eqn:snr_opt}\n\\gamma^* =  \\bar{\\gamma} \\left| \\lambda_{u} + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}}  \\right|^2.\n\\end{eqnarray}\n\n\n\n\n\\section{Preliminaries}\\label{sec:Preliminary_analysis}\nIn this section, we present a probabilistic  characterization of the optimal received SNR at $D$ in (\\ref{eqn:snr_opt}). First, we denote the weighted sum of the product of  random variables in (\\ref{eqn:snr_opt}) by  $Y=\\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} $. Then,  we use the fact that $\\lambda_{{g}_n}$ and  $\\lambda_{{h}_{n}}$ for $n\\in \\mathcal{N}$  are independently distributed Rayleigh random variables to tightly approximate $Y$ through an one-sided  Gaussian distributed random variable $(\\tilde{Y})$ by invoking the CLT \\cite{papoulis02} as \\cite{Diluka2020}\n\\begin{eqnarray}\\label{eqn:pdf_Y}\n\\!\\!\\!\\!\\! f_Y(y) \\approx \tf_{\\tilde Y}(y) =\n\\frac{\\psi}{\\sqrt{2 \\pi \\sigma_{Y}^2}} \\Exp{\\!\\frac{-(y-\\mu_Y)^2}{2 \\sigma_{Y}^2}\\!}, \\,  \\text{for}  \\,\\, y\\geq 0,\n\\end{eqnarray}\nwhere  $\\psi \\triangleq 1\/\\mathcal{Q}\\left(-\\mu_Y\/\\sigma_{Y}\\right)$ is a  normalization factor, which is used to ensure that  $\\int_{-\\infty}^{\\infty} f_{\\tilde Y}(x) dx=1$, and $\\mathcal{Q}(\\cdot)$ is the Gaussian-$\\mathcal{Q}$ function \\cite{papoulis02}. In (\\ref{eqn:pdf_Y}), $\\mu_Y$ and   $\\sigma_{Y}^2$  are given by  \n\\begin{subequations}\n\t\\begin{eqnarray} \\label{eqn:mean_&_var}\n\t\\mu_Y &=& \\sum\\nolimits_{n \\in{\\mathcal{N}}}  \\pi \\beta_{n} \\left(\\xi_{g_n} \\xi_{h_n}\\right)^{1\/2}\/2,\\label{eqn:mean}\\\\\n\t\\sigma_{Y}^2 &=& \\sum\\nolimits_{n \\in{\\mathcal{N}}}  \\beta_{n}^2 \\xi_{g_n} \\xi_{h_n} \\left(16-\\pi^2\\right)\/4. \\label{eqn:var}\n\t\\end{eqnarray}  \t\t \n\\end{subequations}\nNext, we derive a tight approximation for the PDF of $R=\\lambda_{u}+Y$ as (see Appendix \\ref{app:Appendix1})\n\\begin{eqnarray}\\label{eqn:pdf_R}\n\\!\\!\\!\\! f_R(x) \\!&\\approx&\\! f_{\\tilde R}(x)  \\!=\\! \\sqrt{\\pi} \\rho \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\right) \\Exp{-\\Delta \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\right)^2} \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\! \\!\\!\\!\\! \\times \\left(\\err{\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}}+1\\right) +\\rho \\Exp{- \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 }\\right)^2}\\!, \n\\end{eqnarray} \n where $\\err{x} = 2\/\\sqrt{\\pi}\\int_{0}^{x} \\exp{-t} dt$ is the error function \\cite[Eqn. 8.250.1]{Gradshteyn2007}. Here,  $a$, $\\rho$,  and $\\Delta$ are  given by\n\\begin{subequations}\n\\begin{eqnarray} \\label{eqn:def_1}\n\\!\\!\\!\\!\\!\\! a &=& {1}\/{2\\xi_u} + {1}\/{2\\sigma_{Y}^2},  \\label{eqn:def_a} \\qquad \n\\rho = { \\psi }\\Big\/\\left({ 2a \\xi_u  \\sqrt{2\\pi \\sigma_{Y}^2}}\\right), \\label{eqn:def_digamma}\\\\\n\\!\\!\\!\\!\\!\\! \\Delta &=& \\left(1-{1}\/{2\\sigma_{Y}^2} \\right) 2\\sigma_{Y}^2 a. \\label{eqn:def_delta}\t\n\\end{eqnarray}  \t\t \n\\end{subequations}\nIn particular, (\\ref{eqn:pdf_R}) serves as the exact PDF of $\\tilde{R}=\\lambda_{u}+\\tilde{Y}$, where $\\tilde Y$ is the one-sided Gaussian approximated random variable for $Y$ in (\\ref{eqn:snr_opt}).\nThen, we derive an approximated PDF for $\\gamma^* = \\bar \\gamma R^2 $ as\n\\begin{eqnarray}\\label{eqn:pdf_gamma}\nf_{\\gamma^*}(y) &\\approx& \nf_{\\tilde R}\\left(\\sqrt{{y}\/{\\bar{\\gamma}}}\\right) \\times {1}\\big\/{2 \\sqrt{\\bar{\\gamma} y}}.\n\\end{eqnarray} \nSpecifically, (\\ref{eqn:pdf_gamma}) serves as the the exact PDF of $ \\gamma^* \\approx  \\tilde{\\gamma}^* = \\bar \\gamma \\tilde {R}^2$.\nFrom \\eqref{eqn:pdf_R}, we derive the CDF of $\\tilde{R}$ as (see Appendix \\ref{app:Appendix2})\n\\begin{eqnarray}\\label{eqn:cdf_R}\nF_{\\tilde{R}}(x) &=&  1 - \\int_{x}^{\\infty} f_{\\tilde{R}}(u) du= 1 - \\left(I_a+I_b\\right) ,\n\\end{eqnarray} \n \n\n\\noindent where $I_a$ and $I_b$ are given by\n\\begin{subequations}\n\t\\begin{eqnarray} \\label{eqn:I_a_I_b}\n\t\\!\\!\\!\\!\\!\\!\\! I_a &=& \\frac{\\lambda \\exp{-\\Delta d} \\left(\\err{d+1}\\right)}{2\\Delta}   + \\frac{\\lambda \\left(1-\\err{d \\sqrt{\\Delta+1}}\\right)}{2\\Delta \\sqrt{\\Delta+1}}, \\label{eqn:I_a}\\\\\n\t\\!\\!\\!\\!\\!\\!\\!I_b &=&  \\sqrt{\\frac{\\pi \\sigma_{Y}^2}{2} }\\rho \\left(1-\\err{\\sqrt{2 \\sigma_{Y} a} d} \\right), \\label{eqn:I_b}\t\n\t\\end{eqnarray}  \t\t \n\\end{subequations}\nwhere $\\lambda = 2 \\sigma_{Y}^2 \\rho\\sqrt{\\pi a}$, \n$\\rho$ is given in \\eqref{eqn:def_digamma}, \nand $d=(x-\\mu_Y)\/(2\\sigma_{Y}^2\\sqrt{a})$. From \\eqref{eqn:cdf_R}, we approximate the CDF of $\\gamma^* = \\bar \\gamma R^2$  as\n\\begin{eqnarray}\\label{eqn:cdf_SNR}\nF_{\\gamma^*}(y) &=&  \\mathrm{Pr}\\left(\\gamma^*\\leq y\\right) \\approx F_{\\tilde R}\\left(\\sqrt{y\/\\bar{\\gamma}} \\right).\n\\end{eqnarray}\n\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{PDF_CDF_M_N}\\vspace{-3mm}\n\t\\caption{PDF and CDF of  SNR ($\\gamma^*$) for $\\bar{\\gamma}= -10$dB. The combinations of $M$ and $N$ for Case 1 to Case 4 are set to $\\{M=64, N=32\\}$, $\\{M=64, N=64\\}$, $\\{M=144, N=64\\}$, and $\\{M=64, N=128\\}$. }\n\t\\label{fig:PDF_CDF_M_N}\\vspace{-6mm}\n\\end{figure}\n\n\\noindent  \\textbf{\\textit{Remark 1:}}  \n\tWe plot the  approximated PDF and CDF of $\\gamma^*$ by using the analysis in \\eqref{eqn:pdf_gamma} and \\eqref{eqn:cdf_SNR}, respectively, in Fig. \\ref{fig:PDF_CDF_M_N}. Monte-Carlo simulations are also plotted in the same figure for various $M$ and $N$ to verify the accuracy of our approximations.\n\tFrom Fig. \\ref{fig:PDF_CDF_M_N}, we   observe that  our analytical approximations for the  PDF \\eqref{eqn:pdf_gamma} and CDF \\eqref{eqn:cdf_SNR} of $\\gamma^*$  are accurate even for moderately large values for  $M$ and $N$.\n \n\n\n\n\n\n \\setcounter{mycounter}{\\value{equation}}\n \\begin{figure*}[!t] \n \t\\addtocounter{equation}{1}\n \t\\vspace{-0mm}\n \t\\begin{small}\n \t\t\\begin{eqnarray} \\label{eqn:rate_lb_sub}\n \t\t\\mathcal{R}_{lb} = \\log[2]{1+ \\frac{\\bar{\\gamma} \\left(\\xi_u +\\sigma_{Y}^2 + 2\\mu_{u} \\mu_{Y} + \\mu_{u}^2 + \\mu_{Y}^2 \\right)^3} {\\sum\\nolimits_{n\\in C_4} \\binom{4}{n} \\left(2 \\xi_u\\right)^{n\/2} \\Gamma\\left(n\/2+1\\right) \\frac{\\psi}{2\\sqrt{\\pi}} \\sum\\nolimits_{i \\in C_n}  \\binom{n}{i} \\left( {2 \\sigma_{Y}^2}\\right)^{(n-i)\/2}  \\mu_{Y}^i I\\left(n-i, \\frac{-\\mu_{Y}}{2 \\sigma_{Y}^2}\\right) }} \n \t\t\\end{eqnarray}\n \t\\end{small} \n \t\\vspace{-7mm}\n \\end{figure*}\n \\setcounter{equation}{\\value{mycounter}}\n\n \n \n \\setcounter{mycounter}{\\value{equation}}\n \\begin{figure*}[!t] \n \t\\addtocounter{equation}{3}\n \t\\vspace{-0mm}\n \t\\begin{small}\n \t\t\\begin{eqnarray} \\label{eqn:rate_ub_sub_q}\n \t\t\\hat{\\mathcal{R}}_{ub} =   \\log[2]{1+ \\bar{\\gamma} \\left(\\xi_u + {(\\mu_{Y} \\sin(\\tau))}\/{\\tau} \\left[2\\mu_{u} + {(\\mu_{Y} \\sin(\\tau))}\/{\\tau}\\right] + {4 \\sigma_{Y}^2}\/({16-\\pi^2}) \\left[4- {\\pi^2 \\sin(\\tau)^2}\/{(4 \\tau^2)}\\right] \\right)}\n \t\t\\end{eqnarray}\n \t\\end{small} \n \t\\vspace{-7mm}\n \t\n \t\\hrulefill\n \t\n \t\\vspace{-7mm}\n \\end{figure*}\n \\setcounter{equation}{\\value{mycounter}}\n\n \n \n \n\\section{Performances Analysis}\\label{sec:Performance_analysis}\n\\subsection{Outage probability }\\label{sec:outage_prob}\nAn outage event occurs when the optimal received  SNR \\eqref{eqn:snr_opt} falls below a threshold SNR ($\\gamma_{th}$). To this end, we define the the outage probability of the proposed system model as \n$P_{out} = P_{r}\\left(\\gamma \\leq \\gamma_{th}\\right)$. From \\eqref{eqn:cdf_SNR}, we can compute the a tight approximation for the outage probability as $P_{out}   \\approx F_{\\gamma^*}(\\gamma_{th})$. \n\n \n\n\n\\subsection{Average achievable rate }\\label{sec:achvble_rate}\nThe average achievable rate of the proposed system can be defined as $\n\\mathcal{R} =   \\E{\\log[2]{1+\\gamma^*}}$. \nThe exact derivation of this expectation in $\\mathcal{R}$   appears mathematically intractable. Thus,  we   resort to  tight upper\/lower bounds for $\\mathcal R$ as  $\\mathcal{R}_{lb} \\lesssim \\mathcal{R} \\lesssim \\mathcal{R}_{ub}$ by invoking the Jensen's inequality \\cite{Zhang2014}. \nNext, we derive  $\\mathcal{R}_{ub}$  as (see Appendix \\ref{app:Appendix3})\n\\addtocounter{equation}{0}\n\\begin{eqnarray}\\label{eqn:rate_ub_sub}\n\\mathcal{R}_{ub} =   \\log[2]{1+ \\bar{\\gamma} \\left(\\xi_u +\\sigma_{Y}^2 + 2\\mu_{u} \\mu_{Y} + \\mu_{u}^2 + \\mu_{Y}^2 \\right)}.\n\\end{eqnarray}\nWe derive $\\mathcal{R}_{lb}$ as given in \\eqref{eqn:rate_lb_sub} at the top of the next page. \n\n\n\n\\section{Impact of discrete phase-shift adjustments}\\label{sec:Q_phase}\nDue to the hardware limitation, the adoption of continuous phase-shift adjustments for passive reflective elements at the IRS is practically challenging. Thus,  we investigate the feasibility of adopting discrete phase-shifts for the proposed set-up via phase-shift quantization. It is assumed that a limited number of discrete phase-shifts is available to select at the $n$th  reflector such that $\\hat{\\theta}_{n}^*= \\pi \\varsigma\/2^{B-1}$, where $B$ denotes the number of quantization bits,    $\\varsigma =\\underset{q\\in \\{0,\\pm 1, \\cdots, \\pm 2^{B-1} \\} }{ \\mathrm{argmin}} |{\\theta}_{n}^* - \\pi q\/2^{B-1}| $, and $\\theta_{n}^*$ is the optimal phase-shift in Section \\ref{sec:sgnl_modl}.\nThen, we can define the error of the continuous and quantized phase-shifts as $\\varepsilon_n = {\\theta}_{n}^*-\\hat{\\theta}_{n}^*$. For a large number of quantization levels, $\\varepsilon_n$ can shown to be uniformly distributed as $\\varepsilon_n \\sim \\mathcal{U} \\left[-\\tau,\\tau \\right) $ with $\\tau=\\pi\/2^B$ \\cite{Haykin2009}. The signal and   error $\\varepsilon_n$ becomes uncorrelated for a high number of quantization levels \\cite{Haykin2009}. Thus, the optimal SNR in \\eqref{eqn:snr_opt} can be rewritten with discrete phase-shift as  \n\\begin{eqnarray}\\label{eqn:snr_opt_q}\n \t\\addtocounter{equation}{1}\n\\!\\!\\!\\!\\!\\! \\hat{\\gamma}^* \\!=\\!  \\bar{\\gamma} \\left| \\lambda_{u} \\!+\\! \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\exp{j \\varepsilon_n} \\right|^2 \\!=\\!  \\bar{\\gamma} \\left( (\\lambda_{u} \\!+\\! Y_R)^2 \\!+\\!Y_I^2\\right)\\!,\n\\end{eqnarray}\nwhere $Y_R = \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\cos(\\varepsilon_n)$ and $Y_I = \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\sin(\\varepsilon_n)$. By following steps similar to those in Appendix \\ref{app:Appendix3},  an upper bound for the achievable rate with phase-shift quantization errors $(\\hat{\\mathcal{R}}_{ub})$ can be derived by using  \\eqref{eqn:snr_opt_q} as shown in \\eqref{eqn:rate_ub_sub_q}.\n\n\n\\section{Numerical Results}\\label{sec:Numerical}\nThe system parameters for  our simulations are given below: \n$\\zeta_{v} = \\left(d_0\/d_{v}\\right)^{\\kappa} \\times 10^{\\varphi_{v}\/10}$ is used to model large-scale fading, where  $v\\in \\{u_m,h_{mn},g_{n}\\}$ for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$. The transmission distance between nodes is denoted by $d_v$,  $d_0=1$\\,m is a  reference distance, the path-loss exponent is $\\kappa=2.8$, and log-normal shadow fading is captured by $10^{\\varphi_{v}\/10}$  with $\\varphi_{v} \\sim (0,8)$ \\cite{Marzetta2016_Book}. In our system topology, the IRS and $D$ are in positioned at fixed locations and  $250\\,$m apart, while the APs  are uniformly distributed over an area of $1000 \\times 1000$\\,$\\mathrm m^2$. The amplitudes of reflection coefficients are set to $\\beta_{n}=0.9$ for $n \\in \\mathcal{N}$, which is a typical assumption for IRSs \\cite{Wu2019,Wu2020}.\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{outage_M_N}\\vspace{-4mm}\n\t\\caption{The outage probability for different $M$ and $N$ and $\\gamma_{th}=0$ dB. The combinations of $M$ and $N$ for Case-1 to Case-6 are set to $\\{M=36,N=16\\}$, $\\{M=36,N=32\\}$, $\\{M=16,N=64\\}$, $\\{M=36,N=64\\}$, $\\{M=64,N=64\\}$, and $\\{M=36,N=128\\}$.}\n\t\\label{fig:outage_M_N}\\vspace{-5mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:outage_M_N}, we plot the outage probability as a function of the average transmit SNR ($\\bar{\\gamma}$) for different combinations of distributed APs $(M)$ and reflective elements  $(N)$ at the IRS. For comparison purposes, we also plot  the outage probability for the APs-to-$D$ direct transmission (without using an IRS) for $M=64$  in the same figure. \nWe use our closed-form derivation in \\eqref{eqn:cdf_SNR}  to plot the analytical outage probability approximations, and we plot the exact counterparts through Monte-Carlo simulation. The latter is used to verify the accuracy\/tightness of our outage probability approximations.\nAccording to Fig. \\ref{fig:outage_M_N}, the tightness of our outage analysis  improves with as  $M$ or\/and $N$ increase. \nThe reason for this is that large $M$ or\/and $N$ improves the accuracy of CLT.\nMoreover, the outage probability can be reduced by either increasing $M$ or\/and $N$. For example, at an average SNR of $-5\\,$dB, the outage probability can be reduced by $99.9$\\% by doubling $N$ from $16$ (Case-1) to $32$ (Case-2) while keeping $M=36$.\nMoreover, by increasing  $M,N$ from $\\{M=36,N=32\\}$ in Case-2 to $\\{M=64,N=64\\}$ in Case-4, the average SNR required to achieve an outage probability of $10^{-3}$ can be reduced by $155.6\\%$\\,dB. \nFrom Fig. \\ref{fig:outage_M_N}, we observe that the proposed IRS-aided cell-free  set-up  outperforms the APs-to-$D$ direct transmission. For instance,  the set-up without IRS needs an average transmit SNR of $18$\\,dB to reach an outage probability of $10^{-2}$, which is about $177.6\\%$  increase over the transmit SNR requirement for the Case-5 with the IRS-aided set-up for the same number of APs  $(M=64)$. Thus, the co-existence of IRSs within a cell-free set-up can be beneficial in reducing the system outage probability. \n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{rate_N_0_16_32_64_128_256}\\vspace{-3mm}\n\t\\caption{The average achievable   rate for $N \\in \\{16,32,64,128,256\\}$ and $M=64$.}\n\t\\label{fig:rate_N_0_16_32_64_128_256}\\vspace{-6mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:rate_N_0_16_32_64_128_256}, we study the   average achievable rate of the proposed system as a function of  the average transmit SNR ($\\bar{\\gamma}$)  for $N \\in \\{16,32,64,128,256\\}$.\nWe also compare the achievable   rates of APs-to-$D$ direct transmission and the IRS-aided transmission. \nThe upper and lower bounds for the achievable rates are plotted by using our analysis in \\eqref{eqn:rate_ub_sub} and \\eqref{eqn:rate_lb_sub}, respectively. We again validate the accuracy of our analysis through Monte-Carlo simulations of the exact achievable rate. The tightness of our upper\/lower rate bounds is clearly depicted in enlarged portion of Fig.  \\ref{fig:rate_N_0_16_32_64_128_256}. We observe that the rate gains  can be achieved by increasing the  number of reflective elements in the IRS.  Fig.  \\ref{fig:rate_N_0_16_32_64_128_256} also illustrates that an IRS can be embedded within a cell-free set-up to boost the achievable gains. For instance, an IRS with $N=16$  provides a rate gain of about $180$\\,\\% compared to the APs-to-$D$   transmission  without an IRS at an average transmit SNR of $0$\\,dB.\n\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{rate_gain_M_N}\\vspace{-3mm}\n\t\\caption{The impact of discrete phase-shifts with  phase-shift quantization on the average achievable rate for different $M$ and $N$. The combinations of $M$ and $N$ for Case-1 to Case-4 are set to $\\{M=36,N=32\\}$, $\\{M=64,N=32\\}$, $\\{M=36,N=64\\}$, and $\\{M=64,N=64\\}$.}\n\t\\label{fig:rate_gain_M_N}\\vspace{-6mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:rate_gain_M_N}, we investigate the impact of discrete phase-shifts and the  number of quantization bits ($B$) by plotting the  percentage rate ratio $(\\mathcal{R}_{ub}^{per})$ against the average transmit SNR for different combinations of $M$ and $N$. The phase-shift quantization errors are uniformly distributed: $\\mathcal{U} \\left[-\\pi\/2^B, \\pi\/2^B \\right)$.\nThe percentage rate ratio is defined as follows: $\\mathcal{R}_{ub}^{per} = \\hat{\\mathcal{R}}_{ub}\/\\mathcal{R}_{ub} \\times 100\\%$, where $\\hat{\\mathcal{R}}_{ub}$ and $\\mathcal{R}_{ub}$ are the upper bounds of the average achievable rate with and without phase-shift quantization errors given in \\eqref{eqn:rate_ub_sub_q} and \\eqref{eqn:rate_ub_sub}, respectively. Monte-Carlo simulation curves are also generated to validate our analysis. Fig. \\ref{fig:rate_gain_M_N} shows  that the impact of phase-shift quantization errors vanishes when a higher $B$ is used. For instance, we can recover more than $98\\%$ of the average rate when $4$ bit quantization is used at the IRS compared to the system with continuous phase-shift adjustments. As per Fig. \\ref{fig:rate_gain_M_N}, $\\mathcal{R}_{ub}^{per}$ improves in the high  SNR regime. For example, by varying $B$ as 1, 2, and 4 bits,  the average rate can be recovered more than $90\\%$, $98\\%$, and almost $100\\%$, respectively, at a transmit SNR of $20\\,$dB. Fig. \\ref{fig:rate_gain_M_N} shows that a higher number of $M,N$ is also beneficial for recovering the achievable rate in the moderate-to-large transmit SNR regime.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, the feasibility of adopting an IRS embedded within a   cell-free  set-up has been explored.\nThe optimal received SNR through multiple distributed APs with an IRS-aided channel has been  statistically characterized by deriving the tight PDF and CDF approximations. \nThis probabilistic SNR analysis has been  used to derive tight approximations\/bounds for  the  outage probability and the average achievable rate in closed-form. \nThe impairments of discrete phase-shifts with equalization errors \nhave been explored.\nThe accuracy  of our performance analysis of the proposed system set-up has been verified  by providing  Monte-Carlo simulations.\nWe observe from our numerical results that  IRS-aided cell-free system set-ups may be  used to reduce the outage probability and boost the achievable rates of next-generation wireless systems.\n\n\n\n\\appendices\n\n\\section{The derivation of PDF of $\\tilde{R}$ in \\eqref{eqn:pdf_R} }\\label{app:Appendix1}\nBy using the fact that $\\lambda_u$ and $\\tilde{Y}$ are independent random variables, we derive the PDF of $\\tilde{R}$   as\n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_1}\nf_{\\tilde R}(x) &=& \\int_{0}^{\\infty} f_u(u) f_{\\tilde Y}(x-u) du \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \n= 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\int_{0}^{\\infty} u \\exp{-au^2+bu} du \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n= 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\exp{\\frac{b^2}{4a}}  \\int_{0}^{\\infty} u \\exp{-a \\left(u-\\frac{b}{2a}\\right)^2} du \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\stackrel{(a)}{=} 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\exp{\\frac{b^2}{4a}} \\!\\! \\left(\\underbrace{\\int_{-b\/2a}^{\\infty} \\!\\!\\! t \\exp{-at^2}dt  }_{I_1} + \\frac{b}{2a} \\underbrace{\\int_{-b\/2a}^{\\infty}  \\!\\!\\!\\exp{-at^2}dt  }_{I_2}\\right), \\nonumber\n\\end{eqnarray} \nwhere $b = (x-\\mu_{Y})\/\\sigma_{Y}^2$. The step $(a)$ is obtained by letting $t=u-b\/2a$. Then, we can evaluate $I_1$ in \\eqref{eqn:Apx_1_eqn_1} as \n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_2}\n\\!\\!\\!\\!\\!\\!\\!\\!  I_1 &=& \\int_{-b\/2a}^{\\infty} \\!\\!\\! t \\exp{-at^2}dt \\stackrel{(b)}{=} \\left[-\\exp{-at^2}\/2a\\right]_{-b\/2a}^{\\infty} =\\exp{-b^2\/2a},\n\\end{eqnarray} \nwhere the step $(b)$ is computed by using \\cite[Eqn. 2.33.12]{Gradshteyn2007}.  Next, we evaluate $I_2$  as \n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_3}\nI_2 &=& \\int_{-b\/2a}^{\\infty}  \\exp{-at^2}dt \\stackrel{(c)}{=} \\left[\\frac{\\sqrt{\\pi}\\err{\\sqrt{a}t}}{2\\sqrt{a}}\\right]_{-b\/2a}^{\\infty} \\nonumber \\\\\n&=& \\frac{\\sqrt{\\pi}}{2\\sqrt{a}} \\left(1-\\err{\\frac{-b}{2\\sqrt{a}}}\\right),\n\\end{eqnarray}\nwhere the step $(c)$ is due to \\cite[Eqn. 2.33.16]{Gradshteyn2007}. We substitute \\eqref{eqn:Apx_1_eqn_2} and \\eqref{eqn:Apx_1_eqn_3} into \\eqref{eqn:Apx_1_eqn_1} to obtain the PDF of $\\tilde{R}$  in \\eqref{eqn:pdf_R}.\n\n\\section{The derivation of CDF of $\\tilde{R}$ in \\eqref{eqn:cdf_R} }\\label{app:Appendix2}\nWe substitute  \\eqref{eqn:pdf_R} into  \\eqref{eqn:cdf_R} to derive $I_a$  as \n\\begin{eqnarray}\\label{eqn:Apx_2_eqn_1}\nI_a \\!&=&\\! \\sqrt{\\pi} \\rho \\!\\!\\! \\int_{x}^{\\infty} \\!\\!\\! \\left( \\!\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}} \\!\\right) \\! \\exp{\\!-\\Delta \\left(\\!\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\!\\right)^{\\!2}} \\!\\! \\left( \\!\\err{\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}}\\!+ \\!1 \\!\\right) du \\nonumber \\\\\n&\\stackrel{(d)}{=}& \\lambda \\int_{d}^{\\infty} t \\Exp{-\\Delta t^2} \\left(\\err{t}+1\\right) dt \\nonumber \\\\\n&\\stackrel{(e)}{=}& \\lambda \\left[\\frac{-\\exp{-\\Delta t^2} (\\err{t}+1)}{2 \\Delta}\\right]_{d}^{\\infty} + \\lambda \\int_{d}^{\\infty} \\frac{\\exp{-t^2(\\Delta+1)}}{2 \\Delta} dt \\nonumber \\\\\n&\\stackrel{(f)}{=}& \\frac{\\lambda \\exp{-\\Delta d} \\left(\\err{d+1}\\right)}{2\\Delta}   + \\frac{\\lambda \\left(1-\\err{d \\sqrt{\\Delta+1}}\\right)}{2\\Delta \\sqrt{\\Delta+1}},\n\\end{eqnarray}\nwhere $\\lambda=2 \\sigma_{Y}^2 \\rho\\sqrt{\\pi a}$ and $d=(x-\\mu_Y)\/(2\\sigma_{Y}^2\\sqrt{a})$. The step $(d)$ is  obtained by through $t=(u-\\mu_{Y})\/2\\sigma_{Y}^2 \\sqrt{a}$. The step $(e)$ is written by invoking  part-by-part integration, while the step $(f)$ is due to \\cite[Eqn. 2.33.16]{Gradshteyn2007}. Next, we compute $I_b$ as    \n\\begin{eqnarray}\\label{eqn:Apx_2_eqn_2}\nI_b &=&  \\rho  \\int_{x}^{\\infty} \\exp{- \\left(\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 }\\right)^2} du \\stackrel{(g)}{=} \\sqrt{2 \\sigma_{Y}^2} \\rho \\int_{\\sqrt{2 \\sigma_{Y} a} d}^{\\infty} \\exp{-t^2} dt \\nonumber \\\\\n&\\stackrel{(h)}{=}&  \\sqrt{\\frac{\\pi \\sigma_{Y}^2}{2} }\\rho \\left(1-\\err{\\sqrt{2 \\sigma_{Y} a} d} \\right),\n\\end{eqnarray}\nwhere the step $(g)$  is due to a changing of dummy variable as $t=(u-\\mu_{Y})\/(2\\sigma_{Y}^2)$, and the step $(h)$ is resulted due to  \\cite[Eqn. 2.33.16]{Gradshteyn2007}.\n\n\\section{The derivation of $\\mathcal{R}_{lb}$ and $\\mathcal{R}_{ub}$ in \\eqref{eqn:rate_lb_sub} and \\eqref{eqn:rate_ub_sub}}\\label{app:Appendix3}\nFirst,  by invoking Jensen's inequality, $\\mathcal{R}_{lb}$ and $ \\mathcal{R}_{ub}$  can be defined  as \n\\begin{subequations}\n\t\\begin{eqnarray} \n\t\\mathcal{R}_{lb} &=& \\log[2]{1 + \\left(\\E{1\/\\tilde{\\gamma}^*}\\right)^{-1}}, \\label{eqn:rate_lb}\\\\\n\t\\mathcal{R}_{ub} &=& \\log[2]{1+ \\E{\\tilde {\\gamma}^*}}. \\label{eqn:rate_ub}\t\n\t\\end{eqnarray}  \n\\end{subequations}\nThen, we evaluate the expectation term in  \\eqref{eqn:rate_ub}  as\n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_1}\n\\E{\\tilde{\\gamma}^*} &=& \\E{\\bar{\\gamma} \\tilde R^2} = \\bar{\\gamma}\\E{(\\lambda_u + \\tilde Y)^2} \\nonumber\\\\\n&=& \\bar{\\gamma}  \\sum_{n \\in C_2} \\!\\! \\binom{2}{n} \\E{\\lambda_u^{(2-n)}} \\E{\\tilde Y^n} \\nonumber\\\\\n&=& \\bar{\\gamma} \\left(\\xi_u + \\mu_{u}^2 + \\sigma_{Y}^2 + \\mu_{Y}^2 + 2 \\mu_{u}\\mu_{Y}  \\right), \n\\end{eqnarray}\nwhere $\\mu_{u} = \\sqrt{\\pi \\xi_u\/2}$. Moreover, $\\mu_{Y}$ and $\\sigma_{Y}^2$ are given in \\eqref{eqn:mean} and \\eqref{eqn:var}, receptively. By substituting \\eqref{eqn:Apx_3_eqn_1} into \\eqref{eqn:rate_ub}, $\\mathcal{R}_{ub}$ can be computed as \\eqref{eqn:rate_ub_sub}.\nNext, we can write the expectation term in \\eqref{eqn:rate_lb} as  \n\\begin{eqnarray}\\label{eqn:E_gamma_lb}\n\\E{1\/\\tilde{\\gamma}^*} = {1}\/{\\E{\\tilde\\gamma^*}} + {\\Var{\\tilde\\gamma^*}}\/{\\left(\\E{\\tilde\\gamma^*}\\right)^3},\n\\end{eqnarray}\nwhere $\\E{\\tilde{\\gamma}^*}$ is defined in \\eqref{eqn:Apx_3_eqn_1} and $\\Var{\\tilde\\gamma^*}= \\bar{\\gamma}^2\\E{\\tilde R^4} - \\left(\\E{\\tilde\\gamma^*}\\right)^2$. Then, we can compute $\\E{\\tilde R^4}$ as follows: \n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_2}\n\\!\\!\\!\\!\\!\\!\\!\\! \\E{\\tilde R^4} &=& \\E{(\\lambda_u+ \\tilde Y)^4}\\! = \\sum_{n \\in C_4} \\! \\binom{4}{n} \\E{\\lambda_u^{(4-n)}} \\E{\\tilde Y^n},\n\\end{eqnarray}\nwhere the $n$th moment of $\\lambda_u^n$ is denoted by  $\\E{\\lambda_u^n}$. We compute $\\E{\\lambda_u^n}$ as\n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_3}\n\\E{\\lambda_u^n} &=& \\int_{0}^{\\infty} x^n f_{u}(x) dx   =\\int_{0}^{\\infty} \\frac{x^{n+1}}{\\xi_u} \\Exp{-\\frac{x^2}{2\\xi_u}} dx \\nonumber \\\\\n&\\stackrel{(i)}{=}& \\left(2\\xi_u\\right)^{n\/2} \\Gamma\\left(n\/2+1\\right),  \n\\end{eqnarray}\nwhere the step $(m)$ is evaluated from \\cite[Eqn. 2.33.10]{Gradshteyn2007} and $\\Gamma(t) =\\int_{0}^{\\infty} x^t \\exp{-x} dx$ is the Gamma function \\cite[Eqn. 8.310.1]{Gradshteyn2007}. Then, we  evaluate  $\\E{\\tilde Y^n}$  for $n \\in C_4'$ as \n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_4}\n\\E{\\tilde Y^n} &=& \\frac{\\psi}{\\sqrt{2\\pi \\sigma_{Y}^2}}  \\int_{0}^{\\infty} y^{n}  \\exp{-\\frac{(y-\\mu_{Y})^2}{2\\sigma_{Y}^2}} dy  \\\\\n&\\stackrel{(j)}{=}& \\frac{\\psi}{\\sqrt{\\pi }} \\int_{{-\\mu_{Y}}\/{\\sqrt{2\\sigma_{Y}^2}}}^{\\infty} \\left(\\sqrt{2\\sigma_{Y}^2}t +\\mu_{Y}\\right)^n \\exp{-t^2} dt \\nonumber \\\\\n&\\stackrel{(k)}{=}& \\!\\frac{\\psi}{2\\sqrt{\\pi}} \\sum\\limits_{i\\in C_n} \\!\\! \\binom{n}{i} \\left( {2 \\sigma_{Y}^2}\\right)^{\\frac{n-i}{2}} \\mu_{Y}^i I\\!\\left(n\\!-\\!i, \\frac{-\\mu_{Y}}{2 \\sigma_{Y}^2}\\right), \\nonumber\n\\end{eqnarray}\nwhere the step $(j) $ is due to a changing  of the dummy variable,  the step $(k)$ is obtained by expanding $\\left( \\!\\sqrt{2\\sigma_{Y}^2}t \\!+\\! \\mu_{Y} \\!\\right)^{\\!\\!n}\\!$ based on $n\\!$ value. Moreover, $\\!I(\\!\\cdot,\\cdot\\!)\\!$ is given as\n\\begin{eqnarray}\\label{eqn:I}\n\\!\\!\\!\\!\\!\\!\\!\\!\\! I\\!\\left(m,t\\right) \\!=\\! \\begin{cases}\n(-1)^m \\gamma\\left(\\frac{m+1}{2}, t^2\\right) + \\Gamma\\left(\\frac{m+1}{2}\\right),  &\\text{for} \\,\\, t \\leq 0,  \\\\\n\\Gamma\\left(\\frac{m+1}{2}, t^2\\right),& \\text{otherwise},\n\\end{cases}\n\\end{eqnarray} \nwhere $\\gamma(\\lambda,x) =  \\int_{0}^{x} \\exp{-t} t^{\\lambda-1}dt$ is the lower incomplete Gamma function \\cite[Eqn. 8.350.1]{Gradshteyn2007}. Finally, $\\mathcal{R}_{lb}$ is derived  as  \\eqref{eqn:rate_lb_sub}.\n\n\n\n\n  \\linespread{1.0}\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\\label{sec:introduction}\n\nRecently, wireless architectures based on the notion of cell-free have gained much interest \\cite{Ngo2015,Ngo2017}. In a cell-free system set-up, the cell-boundaries can be relaxed, and thus, a vast number of access-points (APs) can be  spatially distributed to serve all users with a uniformly better  quality-of-service (QoS)  over a much larger geographical region \\cite{Ngo2015,Ngo2017}.  Moreover, cell-free set-ups may render spectral\/energy efficiency gains, mitigate  impediments caused by spatial-correlated fading in  compact\/co-located antenna arrays, and circumvent shadow fading impairments    \\cite{Ngo2015,Ngo2017}. Thus, cell-free architecture is a foundation for practically realizing extremely large antenna arrays for next-generation wireless standards. \n\nAn intelligence reflective surface (IRS) consists of a large number of  passive reflectors, whose  reflective coefficients can be  adjusted to attain desired propagation effects for the  \nimpinging electromagnetic (EM) waves \\cite{Renzo2019,Liaskos2018}. \nThe feature of intelligently adjustable phase-shifts at an IRS can be used to boost the signal-to-noise ratio (SNR)  and  to mitigate co-channel interference at an intended destination through  constructive and destructive  signal combining, respectively \\cite{Diluka2020}. This leads to the notion of  recycling of EM waves within a propagation medium, and thereby, spectral\/energy efficiency gains and implementation cost reduction  can be realized as IRSs are made out of low-cost meta-atoms without active  radio-frequency (RF) chains\/amplifiers \\cite{Liaskos2018}.\n\n\n\\subsection{Our motivation} \nIn this paper, we aim to investigate the feasibility of embedding an IRS within a cell-free set-up. Specifically,  our objective is to investigate the performance of an IRS-assisted cell-free set-up, and thereby, we explore the feasibility of jointly reaping the aforementioned   benefits of cell-free architectures and IRS-assisted wireless channels. Moreover, to the best of the authors knowledge, the fundamental performance metrics for an IRS-assisted cell-free set-up have not yet been reported in open literature. To this end, we aim to fill this important gap in IRS literature by presenting a performance analysis for an IRS-assisted cell-free set-up.\n\n\n\\subsection{A literature survey for cell-free architecture and performance analysis of IRS-assisted channels}\\label{sec:literature}\nIn \\cite{Ngo2015,Ngo2017}, the basic concept of cell-free architectures is investigated, and thereby, the performance metrics are compared against those of the  co-located antenna arrays. The analyses in \\cite{Ngo2015,Ngo2017,Ngo2018} reveal that the cell-free  set-ups can outperform the co-located counterparts by serving users with a uniformly better QoS,  minimizing the impediments of spatial-correlation, and shortening  the end-to-end transmission distances to boost the overall energy\/spectral efficiency  \\cite{Ngo2015,Ngo2017}. \nReference \\cite{Nayebi2017} proposes max-min power optimization algorithms for cell-free massive  multiple-input multiple-output (MIMO). In \\cite{Galappaththige2019},  the performance of cell-free massive MIMO with underlay spectrum sharing is investigated.\n\nReferences \\cite{Renzo2019,Liaskos2018} present core architectural design principles of IRSs for wireless communications.\nRay-tracing techniques are used in \\cite{Ozdogan2020} to generate a novel path-loss model for IRS-assisted wireless channels.\nIn \\cite{Wu2019}, joint optimization of precoder at the base-station (BS) and phase-shifts at the IRS is studied  through semi-definite relaxation and alternative optimization techniques.  \nReference \\cite{Diluka2020} studies the fundamental performance limits of distributed IRS-assisted end-to-end channels  with Nakagami-$m$ fading channels. In \\cite{Han2019}, by using the statistical channel state information (CSI), an optimal phase-shift design framework is developed to maximize the achievable rates of  IRS-assisted wireless channels. \nIn \\cite{Chen2019}, joint beamforming and reflecting coefficient designs are investigated for IRSs  to provision physical layer security. Reference \\cite{Abeywickrama2020} proposes a practical IRS phase-shift adjustment model, and thereby, the achievable rate is maximized through jointly optimizing the transmit power and the BS beamformer by using alternative optimization techniques. \n\n\n\\subsection{Our contribution}\\label{sec:Motivation}\nIn  above-referred prior research \\cite{Wu2019,Diluka2020,Han2019,Abeywickrama2020,Chen2019} for IRS-assisted communications, a BS with either a single-antenna or  a co-located antenna  array is used.\nHaving been inspired by this  gap in IRS\/cell-free literature, in this paper, we investigate an IRS-assisted wireless channel embedded within a cell-free set-up over Rayleigh fading,  and thereby, we present  fundamental performance metrics. \nTo this end, first, we invoke the central limit theorem (CLT)  to tightly approximate the end-to-end optimal SNR  to facilitate a mathematically tractable probabilistic characterization. Then, we derive  the probability density function (PDF) and the cumulative density function (CDF) of this approximated optimal SNR in closed-form. Thereby, we present a tight approximation to the outage probability. Moreover, we derive tight upper\/lower bounds for the achievable rate. In particular, we investigate the   impediments of discrete phase-shifts in the presence of phase-shift quantization errors. Finally, we present a set of rigorous  numerical results to explore the performance gains of  the proposed system, and we validate the  accuracy of our analysis through Monte-Carlo simulations. From our numerical results, we observe that  by using an  IRS with  controllable phase-shift adjustments, the performance of cell-free wireless set-ups  can be enhanced. \n\n\n\n \n\\noindent\n\\textbf{Notation:} The transpose of vector $\\mathbf y$ is denoted as $\\mathbf y^{T}$. The expectation and variance of a random variable  $Y$ are represented by \n$\\E[]{Y}$ and $\\Var{Y}$, respectively.\n$Y\\sim\\mathcal {CN}\\left(\\mu_Y,  \\sigma_Y^{2}  \\right) $ denotes that $Y$ is  complex-valued circularly symmetric Gaussian distributed with $\\mu_Y$ mean and $\\sigma_Y^{2}$ variance. Moreover, $C_n=\\{0,1,\\cdots,n\\}$ and $C_n'= C_n\/\\{0\\}$.\n \n \n\n\\vspace{0mm}\n\\begin{figure}[!t]\\centering \\vspace{0mm}\n\t\\def\\svgwidth{180pt} \n\t\t\\fontsize{8}{3}\\selectfont \n\t\\input{system_fig1.pdf_tex} \\vspace{-3mm}\n\t\\caption{System model - IRS-aided cell-free communication set-up }\\vspace{-6mm} \\label{fig:system_model}\n\\end{figure}\n\n\n\\section{System, Channel and Signal Models  }\\label{sec:system_model}\n\n\\subsection{System and channel model}\\label{sec:system_and_channel}\n\nWe consider a cell-free communication set-up consisting of $M$   single-antenna APs ($\\mathrm{AP}_m$ for $m = 1, \\cdots, M$)  and a single-antenna destination $(D)$. An IRS having $N$ passive reflective elements is embedded within this cell-free set-up as shown in  Fig. \\ref{fig:system_model}. \nFor the sake of exposition, we denote the set of APs as $\\mathcal M=\\{1,\\cdots, M\\}$ and the set of reflective elements at the IRS as $\\mathcal N=\\{1,\\cdots, N\\}$.\n\nThe direct link between the $m$th AP and $D$ is represented by $u_m$, while  $h_{mn}$ denotes the channel between the $m$th AP and the $n$th reflective element of the IRS. Moreover, $g_n$ is used to represent the channel between the $n$th reflective element of the IRS and $D$.  We model the envelops of  all aforementioned channels to be independent Rayleigh distributed \\cite{Ding2020}, and the corresponding polar-form of these channels is given by \n\\begin{eqnarray}\\label{eqn:channels}\n\tv = \\lambda_v \\exp{j\\theta_{v}},\n\\end{eqnarray}\nwhere $v\\in \\{u_m,h_{mn},g_{n}\\}$ for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$. In \\eqref{eqn:channels}, the envelop and the phase of $v$ are given by $\\lambda_v$ and $\\theta_{v}$, respectively. The  PDF of $\\lambda_v$ is given by\n\\cite{papoulis02}\n\\begin{eqnarray}\\label{eqn:chnl_pdf}\n\tf_{\\lambda_v}(x) = \\left({x}\/{\\xi_v}\\right) \\Exp{{- x^2}\/{\\left(2\\xi_v\\right)}},\n\\end{eqnarray} \nwhere $\\xi_v = \\zeta_v\/2$ is the Rayleigh parameter, and $\\zeta_v$ captures  the large-scale fading\/path-loss of the channel $v$. Since all reflective elements are co-located within the IRS, it is assumed that all large-scale fading parameters are the same.\n\n\n\\subsection{Signal model}\\label{sec:sgnl_modl}\nThe  signal transmitted by the $m$th AP reaches $D$ through the  direct and IRS-assisted reflected channels. Thus, we can write the signal received at $D$ as\n\\begin{eqnarray}\\label{eqn:rx_signl}\nr = \\sqrt{P} \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\mathbf{g}^T  \\mathbf{\\Theta} \\mathbf{h}_{m} \\right) x + w,\n\\end{eqnarray}\nwhere $x$ is the transmit signal from $S$ satisfying $\\E{|x|^2} =1$,  $P$ is the  transmit power at each AP, and $w$ is an additive white Gaussian noise (AWGN) at $D$ with zero mean and variance of  $\\sigma_{w}^2$ such that  $w\\sim \\mathcal{CN}(0,\\sigma_{w}^2)$. In \\eqref{eqn:rx_signl}, $\\mathbf{h}_m = [h_{m1},\\cdots, h_{mn}, \\cdots, h_{mN}]^T\\in \\mathbb C^{N\\times 1}$ is the channel vector between the $m$th AP and the IRS. Moreover,  $\\mathbf{g}^T = [g_{1},\\cdots, g_{n}, \\cdots, g_{N}]\\in \\mathbb C^{1\\times N}$ denotes the channel vector between the IRS and $D$. The diagonal matrix, $\\mathbf \\Theta = \\diag{\\beta_{1} \\exp{j\\theta_{1}}, \\cdots, \\beta_{n} \\exp{j\\theta_{n}}, \\cdots, \\beta_{N} \\exp{j\\theta_{N}}}\\in \\mathbb C^{N\\times N}$, captures the reflective properties of the IRS through complex-valued reflection coefficients  $\\beta_{n} \\exp{j\\theta_{n}}$ for $n\\in \\mathcal{N}$, where $\\beta_{n}$ and $\\theta_{n}$  are the magnitude of attenuation and phase-shift of the $n$th reflective element of the IRS, respectively. Thus, we can rewrite the received signal at $D$ in \\eqref{eqn:rx_signl} as \n\\begin{eqnarray}\\label{eqn:rx_signl_rearng}\nr = \\sqrt{P}  \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n  {h}_{mn} \\exp{j\\theta_{n}} \\right) x + w.\n\\end{eqnarray}\nThereby, we derive the SNR at $D$ from \\eqref{eqn:rx_signl_rearng} as\n\\begin{eqnarray}\\label{eqn:snr}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\gamma &=&  \\bar{\\gamma} \\left| \\sum\\nolimits_{m \\in{\\mathcal{M}}}  \\left(u_m + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n  {h}_{mn} \\exp{j\\theta_{n}} \\right) \\right|^2 \\nonumber \\\\\n&=& \\bar{\\gamma} \\left|  \\sum\\nolimits_{m \\in{\\mathcal{M}}} \\!u_m  \\!+\\! \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} {g}_n \\!  \\left(\\sum\\nolimits_{m \\in{\\mathcal{M}}} {h}_{mn}\\right) \\exp{j\\theta_{n}} \\right|^2\\!\\!,\n\\end{eqnarray}\nwhere the average transmit SNR is denoted by $\\bar{\\gamma}=P\/\\sigma_{w}^2$. Then, we define $u= \\sum_{m \\in{\\mathcal{M}}} u_m$ and $h_n=\\sum_{m \\in{\\mathcal{M}}} h_{mn}$. Since $u_m$ and $h_{mn}$ are independent complex Gaussian distributed for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$, the polar-form of $u$ and $h_n$ can be also expressed similar to \\eqref{eqn:channels}, where $\\lambda_{u}$ and $\\lambda_{{h}_{n}}$ are the envelops of $u$ and $h_n$, respectively. Thus, $\\lambda_{u}$ and $\\lambda_{{h}_{n}}$ are  independent Rayleigh distributed with    parameters $\\xi_u=\\sum_{m \\in{\\mathcal{M}}} \\zeta_{u_m}\/2$ and $\\xi_{h_n}=\\sum_{m \\in{\\mathcal{M}}} \\zeta_{h_{mn}}\/2$, respectively. From \\eqref{eqn:channels}, we can rewrite the SNR in \\eqref{eqn:snr} in terms of the  channel phases as \n\\begin{eqnarray}\\label{eqn:snr_phase}\n\\gamma =  \\bar{\\gamma} \\left| \\lambda_{u} \\exp{j\\theta_{u}} + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\exp{j\\left(\\theta_{n} + \\theta_{{g}_n} + \\theta_{{h}_{n}}\\right)} \\right|^2.\n\\end{eqnarray}\nIt can be seen from  \\eqref{eqn:snr_phase} that the received SNR at $D$  can be maximized by smartly adjusting  the phase-shifts at each IRS reflecting elements $(\\theta_{n})$. Thus, it enables a constructive addition of the received signals through the direct channels and IRS-aided reflected channels  \\cite{Wu2019,Wu2020}. To this end, the optimal choice of $\\theta_{n}$ is given by $\\theta_{n}^* =\\underset{-\\pi\\leq \\theta_{n} \\leq \\pi}{ \\mathrm{argmax}} \\;{\\gamma} = \\theta_u - \\left(\\theta_{g_{n}} + \\theta_{h_{n}}\\right)$. Then,  we can derive the optimal SNR at $D$  as\n\\begin{eqnarray}\\label{eqn:snr_opt}\n\\gamma^* =  \\bar{\\gamma} \\left| \\lambda_{u} + \\sum\\nolimits_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}}  \\right|^2.\n\\end{eqnarray}\n\n\n\n\n\\section{Preliminaries}\\label{sec:Preliminary_analysis}\nIn this section, we present a probabilistic  characterization of the optimal received SNR at $D$ in (\\ref{eqn:snr_opt}). First, we denote the weighted sum of the product of  random variables in (\\ref{eqn:snr_opt}) by  $Y=\\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} $. Then,  we use the fact that $\\lambda_{{g}_n}$ and  $\\lambda_{{h}_{n}}$ for $n\\in \\mathcal{N}$  are independently distributed Rayleigh random variables to tightly approximate $Y$ through an one-sided  Gaussian distributed random variable $(\\tilde{Y})$ by invoking the CLT \\cite{papoulis02} as \\cite{Diluka2020}\n\\begin{eqnarray}\\label{eqn:pdf_Y}\n\\!\\!\\!\\!\\! f_Y(y) \\approx \tf_{\\tilde Y}(y) =\n\\frac{\\psi}{\\sqrt{2 \\pi \\sigma_{Y}^2}} \\Exp{\\!\\frac{-(y-\\mu_Y)^2}{2 \\sigma_{Y}^2}\\!}, \\,  \\text{for}  \\,\\, y\\geq 0,\n\\end{eqnarray}\nwhere  $\\psi \\triangleq 1\/\\mathcal{Q}\\left(-\\mu_Y\/\\sigma_{Y}\\right)$ is a  normalization factor, which is used to ensure that  $\\int_{-\\infty}^{\\infty} f_{\\tilde Y}(x) dx=1$, and $\\mathcal{Q}(\\cdot)$ is the Gaussian-$\\mathcal{Q}$ function \\cite{papoulis02}. In (\\ref{eqn:pdf_Y}), $\\mu_Y$ and   $\\sigma_{Y}^2$  are given by  \n\\begin{subequations}\n\t\\begin{eqnarray} \\label{eqn:mean_&_var}\n\t\\mu_Y &=& \\sum\\nolimits_{n \\in{\\mathcal{N}}}  \\pi \\beta_{n} \\left(\\xi_{g_n} \\xi_{h_n}\\right)^{1\/2}\/2,\\label{eqn:mean}\\\\\n\t\\sigma_{Y}^2 &=& \\sum\\nolimits_{n \\in{\\mathcal{N}}}  \\beta_{n}^2 \\xi_{g_n} \\xi_{h_n} \\left(16-\\pi^2\\right)\/4. \\label{eqn:var}\n\t\\end{eqnarray}  \t\t \n\\end{subequations}\nNext, we derive a tight approximation for the PDF of $R=\\lambda_{u}+Y$ as (see Appendix \\ref{app:Appendix1})\n\\begin{eqnarray}\\label{eqn:pdf_R}\n\\!\\!\\!\\! f_R(x) \\!&\\approx&\\! f_{\\tilde R}(x)  \\!=\\! \\sqrt{\\pi} \\rho \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\right) \\Exp{-\\Delta \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\right)^2} \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\! \\!\\!\\!\\! \\times \\left(\\err{\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}}+1\\right) +\\rho \\Exp{- \\left(\\frac{x-\\mu_Y}{2\\sigma_{Y}^2 }\\right)^2}\\!, \n\\end{eqnarray} \n where $\\err{x} = 2\/\\sqrt{\\pi}\\int_{0}^{x} \\exp{-t} dt$ is the error function \\cite[Eqn. 8.250.1]{Gradshteyn2007}. Here,  $a$, $\\rho$,  and $\\Delta$ are  given by\n\\begin{subequations}\n\\begin{eqnarray} \\label{eqn:def_1}\n\\!\\!\\!\\!\\!\\! a &=& {1}\/{2\\xi_u} + {1}\/{2\\sigma_{Y}^2},  \\label{eqn:def_a} \\qquad \n\\rho = { \\psi }\\Big\/\\left({ 2a \\xi_u  \\sqrt{2\\pi \\sigma_{Y}^2}}\\right), \\label{eqn:def_digamma}\\\\\n\\!\\!\\!\\!\\!\\! \\Delta &=& \\left(1-{1}\/{2\\sigma_{Y}^2} \\right) 2\\sigma_{Y}^2 a. \\label{eqn:def_delta}\t\n\\end{eqnarray}  \t\t \n\\end{subequations}\nIn particular, (\\ref{eqn:pdf_R}) serves as the exact PDF of $\\tilde{R}=\\lambda_{u}+\\tilde{Y}$, where $\\tilde Y$ is the one-sided Gaussian approximated random variable for $Y$ in (\\ref{eqn:snr_opt}).\nThen, we derive an approximated PDF for $\\gamma^* = \\bar \\gamma R^2 $ as\n\\begin{eqnarray}\\label{eqn:pdf_gamma}\nf_{\\gamma^*}(y) &\\approx& \nf_{\\tilde R}\\left(\\sqrt{{y}\/{\\bar{\\gamma}}}\\right) \\times {1}\\big\/{2 \\sqrt{\\bar{\\gamma} y}}.\n\\end{eqnarray} \nSpecifically, (\\ref{eqn:pdf_gamma}) serves as the the exact PDF of $ \\gamma^* \\approx  \\tilde{\\gamma}^* = \\bar \\gamma \\tilde {R}^2$.\nFrom \\eqref{eqn:pdf_R}, we derive the CDF of $\\tilde{R}$ as (see Appendix \\ref{app:Appendix2})\n\\begin{eqnarray}\\label{eqn:cdf_R}\nF_{\\tilde{R}}(x) &=&  1 - \\int_{x}^{\\infty} f_{\\tilde{R}}(u) du= 1 - \\left(I_a+I_b\\right) ,\n\\end{eqnarray} \n \n\n\\noindent where $I_a$ and $I_b$ are given by\n\\begin{subequations}\n\t\\begin{eqnarray} \\label{eqn:I_a_I_b}\n\t\\!\\!\\!\\!\\!\\!\\! I_a &=& \\frac{\\lambda \\exp{-\\Delta d} \\left(\\err{d+1}\\right)}{2\\Delta}   + \\frac{\\lambda \\left(1-\\err{d \\sqrt{\\Delta+1}}\\right)}{2\\Delta \\sqrt{\\Delta+1}}, \\label{eqn:I_a}\\\\\n\t\\!\\!\\!\\!\\!\\!\\!I_b &=&  \\sqrt{\\frac{\\pi \\sigma_{Y}^2}{2} }\\rho \\left(1-\\err{\\sqrt{2 \\sigma_{Y} a} d} \\right), \\label{eqn:I_b}\t\n\t\\end{eqnarray}  \t\t \n\\end{subequations}\nwhere $\\lambda = 2 \\sigma_{Y}^2 \\rho\\sqrt{\\pi a}$, \n$\\rho$ is given in \\eqref{eqn:def_digamma}, \nand $d=(x-\\mu_Y)\/(2\\sigma_{Y}^2\\sqrt{a})$. From \\eqref{eqn:cdf_R}, we approximate the CDF of $\\gamma^* = \\bar \\gamma R^2$  as\n\\begin{eqnarray}\\label{eqn:cdf_SNR}\nF_{\\gamma^*}(y) &=&  \\mathrm{Pr}\\left(\\gamma^*\\leq y\\right) \\approx F_{\\tilde R}\\left(\\sqrt{y\/\\bar{\\gamma}} \\right).\n\\end{eqnarray}\n\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{PDF_CDF_M_N}\\vspace{-3mm}\n\t\\caption{PDF and CDF of  SNR ($\\gamma^*$) for $\\bar{\\gamma}= -10$dB. The combinations of $M$ and $N$ for Case 1 to Case 4 are set to $\\{M=64, N=32\\}$, $\\{M=64, N=64\\}$, $\\{M=144, N=64\\}$, and $\\{M=64, N=128\\}$. }\n\t\\label{fig:PDF_CDF_M_N}\\vspace{-6mm}\n\\end{figure}\n\n\\noindent  \\textbf{\\textit{Remark 1:}}  \n\tWe plot the  approximated PDF and CDF of $\\gamma^*$ by using the analysis in \\eqref{eqn:pdf_gamma} and \\eqref{eqn:cdf_SNR}, respectively, in Fig. \\ref{fig:PDF_CDF_M_N}. Monte-Carlo simulations are also plotted in the same figure for various $M$ and $N$ to verify the accuracy of our approximations.\n\tFrom Fig. \\ref{fig:PDF_CDF_M_N}, we   observe that  our analytical approximations for the  PDF \\eqref{eqn:pdf_gamma} and CDF \\eqref{eqn:cdf_SNR} of $\\gamma^*$  are accurate even for moderately large values for  $M$ and $N$.\n \n\n\n\n\n\n \\setcounter{mycounter}{\\value{equation}}\n \\begin{figure*}[!t] \n \t\\addtocounter{equation}{1}\n \t\\vspace{-0mm}\n \t\\begin{small}\n \t\t\\begin{eqnarray} \\label{eqn:rate_lb_sub}\n \t\t\\mathcal{R}_{lb} = \\log[2]{1+ \\frac{\\bar{\\gamma} \\left(\\xi_u +\\sigma_{Y}^2 + 2\\mu_{u} \\mu_{Y} + \\mu_{u}^2 + \\mu_{Y}^2 \\right)^3} {\\sum\\nolimits_{n\\in C_4} \\binom{4}{n} \\left(2 \\xi_u\\right)^{n\/2} \\Gamma\\left(n\/2+1\\right) \\frac{\\psi}{2\\sqrt{\\pi}} \\sum\\nolimits_{i \\in C_n}  \\binom{n}{i} \\left( {2 \\sigma_{Y}^2}\\right)^{(n-i)\/2}  \\mu_{Y}^i I\\left(n-i, \\frac{-\\mu_{Y}}{2 \\sigma_{Y}^2}\\right) }} \n \t\t\\end{eqnarray}\n \t\\end{small} \n \t\\vspace{-7mm}\n \\end{figure*}\n \\setcounter{equation}{\\value{mycounter}}\n\n \n \n \\setcounter{mycounter}{\\value{equation}}\n \\begin{figure*}[!t] \n \t\\addtocounter{equation}{3}\n \t\\vspace{-0mm}\n \t\\begin{small}\n \t\t\\begin{eqnarray} \\label{eqn:rate_ub_sub_q}\n \t\t\\hat{\\mathcal{R}}_{ub} =   \\log[2]{1+ \\bar{\\gamma} \\left(\\xi_u + {(\\mu_{Y} \\sin(\\tau))}\/{\\tau} \\left[2\\mu_{u} + {(\\mu_{Y} \\sin(\\tau))}\/{\\tau}\\right] + {4 \\sigma_{Y}^2}\/({16-\\pi^2}) \\left[4- {\\pi^2 \\sin(\\tau)^2}\/{(4 \\tau^2)}\\right] \\right)}\n \t\t\\end{eqnarray}\n \t\\end{small} \n \t\\vspace{-7mm}\n \t\n \t\\hrulefill\n \t\n \t\\vspace{-7mm}\n \\end{figure*}\n \\setcounter{equation}{\\value{mycounter}}\n\n \n \n \n\\section{Performances Analysis}\\label{sec:Performance_analysis}\n\\subsection{Outage probability }\\label{sec:outage_prob}\nAn outage event occurs when the optimal received  SNR \\eqref{eqn:snr_opt} falls below a threshold SNR ($\\gamma_{th}$). To this end, we define the the outage probability of the proposed system model as \n$P_{out} = P_{r}\\left(\\gamma \\leq \\gamma_{th}\\right)$. From \\eqref{eqn:cdf_SNR}, we can compute the a tight approximation for the outage probability as $P_{out}   \\approx F_{\\gamma^*}(\\gamma_{th})$. \n\n \n\n\n\\subsection{Average achievable rate }\\label{sec:achvble_rate}\nThe average achievable rate of the proposed system can be defined as $\n\\mathcal{R} =   \\E{\\log[2]{1+\\gamma^*}}$. \nThe exact derivation of this expectation in $\\mathcal{R}$   appears mathematically intractable. Thus,  we   resort to  tight upper\/lower bounds for $\\mathcal R$ as  $\\mathcal{R}_{lb} \\lesssim \\mathcal{R} \\lesssim \\mathcal{R}_{ub}$ by invoking the Jensen's inequality \\cite{Zhang2014}. \nNext, we derive  $\\mathcal{R}_{ub}$  as (see Appendix \\ref{app:Appendix3})\n\\addtocounter{equation}{0}\n\\begin{eqnarray}\\label{eqn:rate_ub_sub}\n\\mathcal{R}_{ub} =   \\log[2]{1+ \\bar{\\gamma} \\left(\\xi_u +\\sigma_{Y}^2 + 2\\mu_{u} \\mu_{Y} + \\mu_{u}^2 + \\mu_{Y}^2 \\right)}.\n\\end{eqnarray}\nWe derive $\\mathcal{R}_{lb}$ as given in \\eqref{eqn:rate_lb_sub} at the top of the next page. \n\n\n\n\\section{Impact of discrete phase-shift adjustments}\\label{sec:Q_phase}\nDue to the hardware limitation, the adoption of continuous phase-shift adjustments for passive reflective elements at the IRS is practically challenging. Thus,  we investigate the feasibility of adopting discrete phase-shifts for the proposed set-up via phase-shift quantization. It is assumed that a limited number of discrete phase-shifts is available to select at the $n$th  reflector such that $\\hat{\\theta}_{n}^*= \\pi \\varsigma\/2^{B-1}$, where $B$ denotes the number of quantization bits,    $\\varsigma =\\underset{q\\in \\{0,\\pm 1, \\cdots, \\pm 2^{B-1} \\} }{ \\mathrm{argmin}} |{\\theta}_{n}^* - \\pi q\/2^{B-1}| $, and $\\theta_{n}^*$ is the optimal phase-shift in Section \\ref{sec:sgnl_modl}.\nThen, we can define the error of the continuous and quantized phase-shifts as $\\varepsilon_n = {\\theta}_{n}^*-\\hat{\\theta}_{n}^*$. For a large number of quantization levels, $\\varepsilon_n$ can shown to be uniformly distributed as $\\varepsilon_n \\sim \\mathcal{U} \\left[-\\tau,\\tau \\right) $ with $\\tau=\\pi\/2^B$ \\cite{Haykin2009}. The signal and   error $\\varepsilon_n$ becomes uncorrelated for a high number of quantization levels \\cite{Haykin2009}. Thus, the optimal SNR in \\eqref{eqn:snr_opt} can be rewritten with discrete phase-shift as  \n\\begin{eqnarray}\\label{eqn:snr_opt_q}\n \t\\addtocounter{equation}{1}\n\\!\\!\\!\\!\\!\\! \\hat{\\gamma}^* \\!=\\!  \\bar{\\gamma} \\left| \\lambda_{u} \\!+\\! \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\exp{j \\varepsilon_n} \\right|^2 \\!=\\!  \\bar{\\gamma} \\left( (\\lambda_{u} \\!+\\! Y_R)^2 \\!+\\!Y_I^2\\right)\\!,\n\\end{eqnarray}\nwhere $Y_R = \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\cos(\\varepsilon_n)$ and $Y_I = \\sum_{n \\in{\\mathcal{N}}} \\beta_{n} \\lambda_{{g}_n}  \\lambda_{{h}_{n}} \\sin(\\varepsilon_n)$. By following steps similar to those in Appendix \\ref{app:Appendix3},  an upper bound for the achievable rate with phase-shift quantization errors $(\\hat{\\mathcal{R}}_{ub})$ can be derived by using  \\eqref{eqn:snr_opt_q} as shown in \\eqref{eqn:rate_ub_sub_q}.\n\n\n\\section{Numerical Results}\\label{sec:Numerical}\nThe system parameters for  our simulations are given below: \n$\\zeta_{v} = \\left(d_0\/d_{v}\\right)^{\\kappa} \\times 10^{\\varphi_{v}\/10}$ is used to model large-scale fading, where  $v\\in \\{u_m,h_{mn},g_{n}\\}$ for $m\\in \\mathcal{M}$ and $n\\in \\mathcal{N}$. The transmission distance between nodes is denoted by $d_v$,  $d_0=1$\\,m is a  reference distance, the path-loss exponent is $\\kappa=2.8$, and log-normal shadow fading is captured by $10^{\\varphi_{v}\/10}$  with $\\varphi_{v} \\sim (0,8)$ \\cite{Marzetta2016_Book}. In our system topology, the IRS and $D$ are in positioned at fixed locations and  $250\\,$m apart, while the APs  are uniformly distributed over an area of $1000 \\times 1000$\\,$\\mathrm m^2$. The amplitudes of reflection coefficients are set to $\\beta_{n}=0.9$ for $n \\in \\mathcal{N}$, which is a typical assumption for IRSs \\cite{Wu2019,Wu2020}.\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{outage_M_N}\\vspace{-4mm}\n\t\\caption{The outage probability for different $M$ and $N$ and $\\gamma_{th}=0$ dB. The combinations of $M$ and $N$ for Case-1 to Case-6 are set to $\\{M=36,N=16\\}$, $\\{M=36,N=32\\}$, $\\{M=16,N=64\\}$, $\\{M=36,N=64\\}$, $\\{M=64,N=64\\}$, and $\\{M=36,N=128\\}$.}\n\t\\label{fig:outage_M_N}\\vspace{-5mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:outage_M_N}, we plot the outage probability as a function of the average transmit SNR ($\\bar{\\gamma}$) for different combinations of distributed APs $(M)$ and reflective elements  $(N)$ at the IRS. For comparison purposes, we also plot  the outage probability for the APs-to-$D$ direct transmission (without using an IRS) for $M=64$  in the same figure. \nWe use our closed-form derivation in \\eqref{eqn:cdf_SNR}  to plot the analytical outage probability approximations, and we plot the exact counterparts through Monte-Carlo simulation. The latter is used to verify the accuracy\/tightness of our outage probability approximations.\nAccording to Fig. \\ref{fig:outage_M_N}, the tightness of our outage analysis  improves with as  $M$ or\/and $N$ increase. \nThe reason for this is that large $M$ or\/and $N$ improves the accuracy of CLT.\nMoreover, the outage probability can be reduced by either increasing $M$ or\/and $N$. For example, at an average SNR of $-5\\,$dB, the outage probability can be reduced by $99.9$\\% by doubling $N$ from $16$ (Case-1) to $32$ (Case-2) while keeping $M=36$.\nMoreover, by increasing  $M,N$ from $\\{M=36,N=32\\}$ in Case-2 to $\\{M=64,N=64\\}$ in Case-4, the average SNR required to achieve an outage probability of $10^{-3}$ can be reduced by $155.6\\%$\\,dB. \nFrom Fig. \\ref{fig:outage_M_N}, we observe that the proposed IRS-aided cell-free  set-up  outperforms the APs-to-$D$ direct transmission. For instance,  the set-up without IRS needs an average transmit SNR of $18$\\,dB to reach an outage probability of $10^{-2}$, which is about $177.6\\%$  increase over the transmit SNR requirement for the Case-5 with the IRS-aided set-up for the same number of APs  $(M=64)$. Thus, the co-existence of IRSs within a cell-free set-up can be beneficial in reducing the system outage probability. \n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{rate_N_0_16_32_64_128_256}\\vspace{-3mm}\n\t\\caption{The average achievable   rate for $N \\in \\{16,32,64,128,256\\}$ and $M=64$.}\n\t\\label{fig:rate_N_0_16_32_64_128_256}\\vspace{-6mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:rate_N_0_16_32_64_128_256}, we study the   average achievable rate of the proposed system as a function of  the average transmit SNR ($\\bar{\\gamma}$)  for $N \\in \\{16,32,64,128,256\\}$.\nWe also compare the achievable   rates of APs-to-$D$ direct transmission and the IRS-aided transmission. \nThe upper and lower bounds for the achievable rates are plotted by using our analysis in \\eqref{eqn:rate_ub_sub} and \\eqref{eqn:rate_lb_sub}, respectively. We again validate the accuracy of our analysis through Monte-Carlo simulations of the exact achievable rate. The tightness of our upper\/lower rate bounds is clearly depicted in enlarged portion of Fig.  \\ref{fig:rate_N_0_16_32_64_128_256}. We observe that the rate gains  can be achieved by increasing the  number of reflective elements in the IRS.  Fig.  \\ref{fig:rate_N_0_16_32_64_128_256} also illustrates that an IRS can be embedded within a cell-free set-up to boost the achievable gains. For instance, an IRS with $N=16$  provides a rate gain of about $180$\\,\\% compared to the APs-to-$D$   transmission  without an IRS at an average transmit SNR of $0$\\,dB.\n\n\n\\begin{figure}[!t]\\centering\\vspace{-0mm}\n\t\\includegraphics[width=0.38\\textwidth]{rate_gain_M_N}\\vspace{-3mm}\n\t\\caption{The impact of discrete phase-shifts with  phase-shift quantization on the average achievable rate for different $M$ and $N$. The combinations of $M$ and $N$ for Case-1 to Case-4 are set to $\\{M=36,N=32\\}$, $\\{M=64,N=32\\}$, $\\{M=36,N=64\\}$, and $\\{M=64,N=64\\}$.}\n\t\\label{fig:rate_gain_M_N}\\vspace{-6mm}\n\\end{figure}\n\nIn Fig. \\ref{fig:rate_gain_M_N}, we investigate the impact of discrete phase-shifts and the  number of quantization bits ($B$) by plotting the  percentage rate ratio $(\\mathcal{R}_{ub}^{per})$ against the average transmit SNR for different combinations of $M$ and $N$. The phase-shift quantization errors are uniformly distributed: $\\mathcal{U} \\left[-\\pi\/2^B, \\pi\/2^B \\right)$.\nThe percentage rate ratio is defined as follows: $\\mathcal{R}_{ub}^{per} = \\hat{\\mathcal{R}}_{ub}\/\\mathcal{R}_{ub} \\times 100\\%$, where $\\hat{\\mathcal{R}}_{ub}$ and $\\mathcal{R}_{ub}$ are the upper bounds of the average achievable rate with and without phase-shift quantization errors given in \\eqref{eqn:rate_ub_sub_q} and \\eqref{eqn:rate_ub_sub}, respectively. Monte-Carlo simulation curves are also generated to validate our analysis. Fig. \\ref{fig:rate_gain_M_N} shows  that the impact of phase-shift quantization errors vanishes when a higher $B$ is used. For instance, we can recover more than $98\\%$ of the average rate when $4$ bit quantization is used at the IRS compared to the system with continuous phase-shift adjustments. As per Fig. \\ref{fig:rate_gain_M_N}, $\\mathcal{R}_{ub}^{per}$ improves in the high  SNR regime. For example, by varying $B$ as 1, 2, and 4 bits,  the average rate can be recovered more than $90\\%$, $98\\%$, and almost $100\\%$, respectively, at a transmit SNR of $20\\,$dB. Fig. \\ref{fig:rate_gain_M_N} shows that a higher number of $M,N$ is also beneficial for recovering the achievable rate in the moderate-to-large transmit SNR regime.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, the feasibility of adopting an IRS embedded within a   cell-free  set-up has been explored.\nThe optimal received SNR through multiple distributed APs with an IRS-aided channel has been  statistically characterized by deriving the tight PDF and CDF approximations. \nThis probabilistic SNR analysis has been  used to derive tight approximations\/bounds for  the  outage probability and the average achievable rate in closed-form. \nThe impairments of discrete phase-shifts with equalization errors \nhave been explored.\nThe accuracy  of our performance analysis of the proposed system set-up has been verified  by providing  Monte-Carlo simulations.\nWe observe from our numerical results that  IRS-aided cell-free system set-ups may be  used to reduce the outage probability and boost the achievable rates of next-generation wireless systems.\n\n\n\n\\appendices\n\n\\section{The derivation of PDF of $\\tilde{R}$ in \\eqref{eqn:pdf_R} }\\label{app:Appendix1}\nBy using the fact that $\\lambda_u$ and $\\tilde{Y}$ are independent random variables, we derive the PDF of $\\tilde{R}$   as\n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_1}\nf_{\\tilde R}(x) &=& \\int_{0}^{\\infty} f_u(u) f_{\\tilde Y}(x-u) du \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \n= 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\int_{0}^{\\infty} u \\exp{-au^2+bu} du \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n= 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\exp{\\frac{b^2}{4a}}  \\int_{0}^{\\infty} u \\exp{-a \\left(u-\\frac{b}{2a}\\right)^2} du \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\stackrel{(a)}{=} 2a \\rho \\exp{-\\frac{(x-\\mu_{Y})^2}{2\\sigma_{Y}^2}} \\exp{\\frac{b^2}{4a}} \\!\\! \\left(\\underbrace{\\int_{-b\/2a}^{\\infty} \\!\\!\\! t \\exp{-at^2}dt  }_{I_1} + \\frac{b}{2a} \\underbrace{\\int_{-b\/2a}^{\\infty}  \\!\\!\\!\\exp{-at^2}dt  }_{I_2}\\right), \\nonumber\n\\end{eqnarray} \nwhere $b = (x-\\mu_{Y})\/\\sigma_{Y}^2$. The step $(a)$ is obtained by letting $t=u-b\/2a$. Then, we can evaluate $I_1$ in \\eqref{eqn:Apx_1_eqn_1} as \n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_2}\n\\!\\!\\!\\!\\!\\!\\!\\!  I_1 &=& \\int_{-b\/2a}^{\\infty} \\!\\!\\! t \\exp{-at^2}dt \\stackrel{(b)}{=} \\left[-\\exp{-at^2}\/2a\\right]_{-b\/2a}^{\\infty} =\\exp{-b^2\/2a},\n\\end{eqnarray} \nwhere the step $(b)$ is computed by using \\cite[Eqn. 2.33.12]{Gradshteyn2007}.  Next, we evaluate $I_2$  as \n\\begin{eqnarray}\\label{eqn:Apx_1_eqn_3}\nI_2 &=& \\int_{-b\/2a}^{\\infty}  \\exp{-at^2}dt \\stackrel{(c)}{=} \\left[\\frac{\\sqrt{\\pi}\\err{\\sqrt{a}t}}{2\\sqrt{a}}\\right]_{-b\/2a}^{\\infty} \\nonumber \\\\\n&=& \\frac{\\sqrt{\\pi}}{2\\sqrt{a}} \\left(1-\\err{\\frac{-b}{2\\sqrt{a}}}\\right),\n\\end{eqnarray}\nwhere the step $(c)$ is due to \\cite[Eqn. 2.33.16]{Gradshteyn2007}. We substitute \\eqref{eqn:Apx_1_eqn_2} and \\eqref{eqn:Apx_1_eqn_3} into \\eqref{eqn:Apx_1_eqn_1} to obtain the PDF of $\\tilde{R}$  in \\eqref{eqn:pdf_R}.\n\n\\section{The derivation of CDF of $\\tilde{R}$ in \\eqref{eqn:cdf_R} }\\label{app:Appendix2}\nWe substitute  \\eqref{eqn:pdf_R} into  \\eqref{eqn:cdf_R} to derive $I_a$  as \n\\begin{eqnarray}\\label{eqn:Apx_2_eqn_1}\nI_a \\!&=&\\! \\sqrt{\\pi} \\rho \\!\\!\\! \\int_{x}^{\\infty} \\!\\!\\! \\left( \\!\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}} \\!\\right) \\! \\exp{\\!-\\Delta \\left(\\!\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}\\!\\right)^{\\!2}} \\!\\! \\left( \\!\\err{\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 \\sqrt{a}}}\\!+ \\!1 \\!\\right) du \\nonumber \\\\\n&\\stackrel{(d)}{=}& \\lambda \\int_{d}^{\\infty} t \\Exp{-\\Delta t^2} \\left(\\err{t}+1\\right) dt \\nonumber \\\\\n&\\stackrel{(e)}{=}& \\lambda \\left[\\frac{-\\exp{-\\Delta t^2} (\\err{t}+1)}{2 \\Delta}\\right]_{d}^{\\infty} + \\lambda \\int_{d}^{\\infty} \\frac{\\exp{-t^2(\\Delta+1)}}{2 \\Delta} dt \\nonumber \\\\\n&\\stackrel{(f)}{=}& \\frac{\\lambda \\exp{-\\Delta d} \\left(\\err{d+1}\\right)}{2\\Delta}   + \\frac{\\lambda \\left(1-\\err{d \\sqrt{\\Delta+1}}\\right)}{2\\Delta \\sqrt{\\Delta+1}},\n\\end{eqnarray}\nwhere $\\lambda=2 \\sigma_{Y}^2 \\rho\\sqrt{\\pi a}$ and $d=(x-\\mu_Y)\/(2\\sigma_{Y}^2\\sqrt{a})$. The step $(d)$ is  obtained by through $t=(u-\\mu_{Y})\/2\\sigma_{Y}^2 \\sqrt{a}$. The step $(e)$ is written by invoking  part-by-part integration, while the step $(f)$ is due to \\cite[Eqn. 2.33.16]{Gradshteyn2007}. Next, we compute $I_b$ as    \n\\begin{eqnarray}\\label{eqn:Apx_2_eqn_2}\nI_b &=&  \\rho  \\int_{x}^{\\infty} \\exp{- \\left(\\frac{u-\\mu_Y}{2\\sigma_{Y}^2 }\\right)^2} du \\stackrel{(g)}{=} \\sqrt{2 \\sigma_{Y}^2} \\rho \\int_{\\sqrt{2 \\sigma_{Y} a} d}^{\\infty} \\exp{-t^2} dt \\nonumber \\\\\n&\\stackrel{(h)}{=}&  \\sqrt{\\frac{\\pi \\sigma_{Y}^2}{2} }\\rho \\left(1-\\err{\\sqrt{2 \\sigma_{Y} a} d} \\right),\n\\end{eqnarray}\nwhere the step $(g)$  is due to a changing of dummy variable as $t=(u-\\mu_{Y})\/(2\\sigma_{Y}^2)$, and the step $(h)$ is resulted due to  \\cite[Eqn. 2.33.16]{Gradshteyn2007}.\n\n\\section{The derivation of $\\mathcal{R}_{lb}$ and $\\mathcal{R}_{ub}$ in \\eqref{eqn:rate_lb_sub} and \\eqref{eqn:rate_ub_sub}}\\label{app:Appendix3}\nFirst,  by invoking Jensen's inequality, $\\mathcal{R}_{lb}$ and $ \\mathcal{R}_{ub}$  can be defined  as \n\\begin{subequations}\n\t\\begin{eqnarray} \n\t\\mathcal{R}_{lb} &=& \\log[2]{1 + \\left(\\E{1\/\\tilde{\\gamma}^*}\\right)^{-1}}, \\label{eqn:rate_lb}\\\\\n\t\\mathcal{R}_{ub} &=& \\log[2]{1+ \\E{\\tilde {\\gamma}^*}}. \\label{eqn:rate_ub}\t\n\t\\end{eqnarray}  \n\\end{subequations}\nThen, we evaluate the expectation term in  \\eqref{eqn:rate_ub}  as\n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_1}\n\\E{\\tilde{\\gamma}^*} &=& \\E{\\bar{\\gamma} \\tilde R^2} = \\bar{\\gamma}\\E{(\\lambda_u + \\tilde Y)^2} \\nonumber\\\\\n&=& \\bar{\\gamma}  \\sum_{n \\in C_2} \\!\\! \\binom{2}{n} \\E{\\lambda_u^{(2-n)}} \\E{\\tilde Y^n} \\nonumber\\\\\n&=& \\bar{\\gamma} \\left(\\xi_u + \\mu_{u}^2 + \\sigma_{Y}^2 + \\mu_{Y}^2 + 2 \\mu_{u}\\mu_{Y}  \\right), \n\\end{eqnarray}\nwhere $\\mu_{u} = \\sqrt{\\pi \\xi_u\/2}$. Moreover, $\\mu_{Y}$ and $\\sigma_{Y}^2$ are given in \\eqref{eqn:mean} and \\eqref{eqn:var}, receptively. By substituting \\eqref{eqn:Apx_3_eqn_1} into \\eqref{eqn:rate_ub}, $\\mathcal{R}_{ub}$ can be computed as \\eqref{eqn:rate_ub_sub}.\nNext, we can write the expectation term in \\eqref{eqn:rate_lb} as  \n\\begin{eqnarray}\\label{eqn:E_gamma_lb}\n\\E{1\/\\tilde{\\gamma}^*} = {1}\/{\\E{\\tilde\\gamma^*}} + {\\Var{\\tilde\\gamma^*}}\/{\\left(\\E{\\tilde\\gamma^*}\\right)^3},\n\\end{eqnarray}\nwhere $\\E{\\tilde{\\gamma}^*}$ is defined in \\eqref{eqn:Apx_3_eqn_1} and $\\Var{\\tilde\\gamma^*}= \\bar{\\gamma}^2\\E{\\tilde R^4} - \\left(\\E{\\tilde\\gamma^*}\\right)^2$. Then, we can compute $\\E{\\tilde R^4}$ as follows: \n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_2}\n\\!\\!\\!\\!\\!\\!\\!\\! \\E{\\tilde R^4} &=& \\E{(\\lambda_u+ \\tilde Y)^4}\\! = \\sum_{n \\in C_4} \\! \\binom{4}{n} \\E{\\lambda_u^{(4-n)}} \\E{\\tilde Y^n},\n\\end{eqnarray}\nwhere the $n$th moment of $\\lambda_u^n$ is denoted by  $\\E{\\lambda_u^n}$. We compute $\\E{\\lambda_u^n}$ as\n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_3}\n\\E{\\lambda_u^n} &=& \\int_{0}^{\\infty} x^n f_{u}(x) dx   =\\int_{0}^{\\infty} \\frac{x^{n+1}}{\\xi_u} \\Exp{-\\frac{x^2}{2\\xi_u}} dx \\nonumber \\\\\n&\\stackrel{(i)}{=}& \\left(2\\xi_u\\right)^{n\/2} \\Gamma\\left(n\/2+1\\right),  \n\\end{eqnarray}\nwhere the step $(m)$ is evaluated from \\cite[Eqn. 2.33.10]{Gradshteyn2007} and $\\Gamma(t) =\\int_{0}^{\\infty} x^t \\exp{-x} dx$ is the Gamma function \\cite[Eqn. 8.310.1]{Gradshteyn2007}. Then, we  evaluate  $\\E{\\tilde Y^n}$  for $n \\in C_4'$ as \n\\begin{eqnarray}\\label{eqn:Apx_3_eqn_4}\n\\E{\\tilde Y^n} &=& \\frac{\\psi}{\\sqrt{2\\pi \\sigma_{Y}^2}}  \\int_{0}^{\\infty} y^{n}  \\exp{-\\frac{(y-\\mu_{Y})^2}{2\\sigma_{Y}^2}} dy  \\\\\n&\\stackrel{(j)}{=}& \\frac{\\psi}{\\sqrt{\\pi }} \\int_{{-\\mu_{Y}}\/{\\sqrt{2\\sigma_{Y}^2}}}^{\\infty} \\left(\\sqrt{2\\sigma_{Y}^2}t +\\mu_{Y}\\right)^n \\exp{-t^2} dt \\nonumber \\\\\n&\\stackrel{(k)}{=}& \\!\\frac{\\psi}{2\\sqrt{\\pi}} \\sum\\limits_{i\\in C_n} \\!\\! \\binom{n}{i} \\left( {2 \\sigma_{Y}^2}\\right)^{\\frac{n-i}{2}} \\mu_{Y}^i I\\!\\left(n\\!-\\!i, \\frac{-\\mu_{Y}}{2 \\sigma_{Y}^2}\\right), \\nonumber\n\\end{eqnarray}\nwhere the step $(j) $ is due to a changing  of the dummy variable,  the step $(k)$ is obtained by expanding $\\left( \\!\\sqrt{2\\sigma_{Y}^2}t \\!+\\! \\mu_{Y} \\!\\right)^{\\!\\!n}\\!$ based on $n\\!$ value. Moreover, $\\!I(\\!\\cdot,\\cdot\\!)\\!$ is given as\n\\begin{eqnarray}\\label{eqn:I}\n\\!\\!\\!\\!\\!\\!\\!\\!\\! I\\!\\left(m,t\\right) \\!=\\! \\begin{cases}\n(-1)^m \\gamma\\left(\\frac{m+1}{2}, t^2\\right) + \\Gamma\\left(\\frac{m+1}{2}\\right),  &\\text{for} \\,\\, t \\leq 0,  \\\\\n\\Gamma\\left(\\frac{m+1}{2}, t^2\\right),& \\text{otherwise},\n\\end{cases}\n\\end{eqnarray} \nwhere $\\gamma(\\lambda,x) =  \\int_{0}^{x} \\exp{-t} t^{\\lambda-1}dt$ is the lower incomplete Gamma function \\cite[Eqn. 8.350.1]{Gradshteyn2007}. Finally, $\\mathcal{R}_{lb}$ is derived  as  \\eqref{eqn:rate_lb_sub}.\n\n\n\n\n  \\linespread{1.0}\n\\bibliographystyle{IEEEtran}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe purpose of this paper will be to introduce a way to reconstruct objects from their grey-scale digital images. More specifically, we focus on objects that are small compared to the image resolution and satisfy a certain regularity constraint called \\emph{$r$-regularity}. The notion of $r$-regularity was developed independently by Serra \\cite{Serra} and Pavlidis \\cite{Pavlidis} to describe a class of objects for which reconstruction from digital images preserved certain topological features. They both consider \\emph{subset digitisation}, that is, digitisation formed by placing an image grid on top of an object and then colouring an image cell black if its midpoint is on top of the object, and white if the cell midpoint is on top of the complement of the object. This way a binary image is produced, and they consider the set of black cells as the reconstructed set. Serra showed that if the grid is hexagonal and the object satisfies certain constraints, the original and reconstructed sets have the same homotopy, and Pavlidis showed that for a square grid and for certain $r$-regular sets, the set and its reconstruction are homeomorphic. Later on, Stelldinger and K\u00f6the \\cite{StelldingerKothe},\\cite{SK} argued that the concepts of homotopies or homeomorphisms were not strong enough to fully capture human perception of shape similarity. Instead they proposed two new similarity criterions called \\emph{weak} and \\emph{strong $r$-similarity}, and showed that under certain conditions, an $r$-regular set and its reconstruction by a square grid are both weakly and strongly $r$-similar. {\\color{black}We, too, will consider the notion of weak $r$-similarity in this paper.}\n\nHowever, Serra, Pavlidis, Stelldinger and K\u00f6the were modelling images using subset digitisation, which outputs a binary image. In contrast to this approach, Latecki et al. \\cite{LateckiConradGross} modelled an image by requiring that the intensity in each pixel be a monotonic function of the fraction of that object covered by that pixel. This way they seek to model a pixel intensity as the light intensity measured by a sensor in the middle of the pixel, and the result is a grey-level image much like the ones obtained in real situations. They show that after applying any threshold to such an image of an $r$-regular object with certain constraints, the set of black pixels has the same homotopy type as the original object and, in the case where the original object is a manifold with boundary, the two are even homeomorphic. They also conjecture that all $r$-regular objects are manifolds with boundary. This was later proven by Duarte and Torres in \\cite{DTs}.\n\nWe will model our images in the same way as Latecki et al. did, namely by requiring each pixel intensity to be a monotonic function of the fraction of the pixel covered by the object. In contrast to the above reconstruction approaches, we do not wish to use a set of pixels as our reconstructed set, but rather to construct a new set with smooth boundary that we may then use as the reconstruction. Also in contrast to the above, we will not consider binary images, but keep the information stored in the grey values in our endeavour to make a more precise reconstruction.\n\nWhen reconstruction, one should decide which properties one wishes the reconstructed object to share with the original one. Should the reconstructed set have the same topological features as the original one? Should the reconstructed set be close to the original one? Should a digitisation of the reconstructed set yield the same image as the original set? Should the reconstructed set be $r'$-regular for some $r'$ close to $r$? Though all of these comparison criteria are interesting to work with and  an ideal reconstruction should satisfy them all, it is hard to construct such a set. In this paper, we will therefore focus on constructing a set that is close to the original one in Hausdorff distance (which will be introduced in the following), has a smooth boundary{\\color{black}, and is homeomorphic to the original set. This means that we show that our reconstructed set and the original are weakly $r$-similar in the sense of \\cite{SK}}.\n\n\\section{Basic definitions and theorems about \\texorpdfstring{$r$}{r}-regular sets}\\label{SecRBasics}\nLet us start by establishing some terminology.\nLet $X\\subset \\R^2$ be a set. We will denote the closure of $X$ by $\\overline{X}$, the interior of $X$ by $\\text{Int}(X)$ and the boundary of $X$ by $\\partial X$. The complement $\\R^2\\backslash X$ will be denoted by $X^C$. The set $X$ is compact if and only if $X$ is closed and bounded.\n\nThe Euclidean distance between two points $x$ and $y$ in $\\R^2$ will be denoted by $d(x,y)$ or, occasionally, by $\\norm{x-y}$.\n\nFor an $s>0$, we let $B_s(x)=\\sett{y\\in \\R^2\\mid d(x,y)<s}$ be the open ball with centre $x$ and radius $s$. For a line segment $L$ we will denote the length of $L$ by $\\abs{L}$.\n\n{\\color{black}A part of the \n goal will be to construct a set from a digital image whose boundary is close to the boundary of the original set. The intuitive concept of closeness between two sets is captured by the Hausdorff distance: For $X,Y\\subseteq \\R^2$, the Hausdorff distance $d_H$ between $X$ and $Y$ is given by\n\\begin{equation*}\nd_H(X,Y)=\\max\\sett{\\sup_{x\\in X} \\inf_{y\\in Y}d(x,y), \\sup_{y\\in Y} \\inf_{x\\in X}d(x,y)}.\n\\end{equation*}\nThe set of compact sets of $\\R^2$ equipped with the Hausdorff metric is a complete metric space.\n\nThe digital images that we will be working with in this paper are formed in the following way:\n\n\\begin{defn}\nLet $X\\subseteq \\R^2$ be a set and $d\\Z^2\\subseteq\\R^2$ a grid\nwith side length $d$. To each grid square $C$, we assign an \\emph{intensity} $\\lambda$ given by\n\\begin{equation*}\n  \\lambda=\\phi\\left(\\frac{\\text{area}(X\\cap C)}{d^2}\\right)\\in[0,1],\n\\end{equation*}\nwhere $\\phi:[0,1]\\to [0,1]$ is a monotonic function with $\\phi(0)=0$, $\\phi(1)=1$ and $\\phi((0,1))\\subseteq (0,1)$.\n\nThe \\emph{digitisation} of $X$ is the matrix of\nintensities. We will visualise it as the collection of pixels of side length $d$, each coloured a shade of grey corresponding to the value of $\\lambda$.\n\nLet $V(X)$ denote \\emph{the black pixels} of this\ndigitisation of $X$. We will sometimes refer to $V(X)$ as the black\ndigitisation pixels of $X$.\n\\end{defn}\n\nTo make sure that the objects in the images we are considering are not arbitrarily strange, we will follow in the footsteps of previous approaches and only consider $r$-regular sets:\n\n\\begin{defn}\nLet $r>0$. A closed set $X\\subseteq \\R^n$ is said to be \\emph{$r$-regular} if for each $x\\in\\partial X$ there exists two $r$-balls $B_r(x_b)\\subseteq X$ and $B_r(x_w)\\subseteq X^C$ such that $\\overline{B_r(x_b)}\\cap \\overline{B_r(x_w)}=\\sett{x}$, see \\cref{FigR-reg}.\n\\end{defn}\n\n\\begin{figure}\n\\includegraphics[scale=1]{R-regularCircles}\n\\caption{An $r$-regular set $X$ is a set where each boundary point belongs to both the boundary of an $r$-ball contained in $X$ and the boundary of an $r$-ball contained in $X^C$}\n\\label{FigR-reg}\n\\end{figure}\n\nIn general, we believe that a reconstruction can be made more accurately by taking the intensities of the grey pixels into account, and we are currently working on this idea. However, in this paper we restrict ourselves to looking at images where each pixel is considered to be either black, grey or white, without taking the exact intensities of the grey pixels into account:\n\n\\begin{defn}\nA \\emph{trinary} digital image is a digital image where the intensities of all grey pixels are set to 0.5.\n\\end{defn}\n\nThese trinary images will be our main interest in this paper. Note that the colour of a pixel (black, grey or white) does not depend on the monotonic function $\\phi$ used for calculating the pixel intensities - in fact, a pixel in a trinary image of an object $X$ is black if it is contained in $X$, white if it is contained in $X^C$ and grey if $\\partial X$ passes through it.\n\nWhen we make the digital image of an $r$-regular object by a lattice $d\\Z^2$, we can in general not be certain that there are any black or white pixels in the image - for instance, if $d$ is large compared to $r$, all pixels could contain an $r$-ball, which would mean that the image would be all grey. Since we cannot hope to make a very good reconstruction in this case, we will put a restriction on the relationship between the $r$ and $d$:\n\n\\begin{Convention}\nThroughout the following, we assume that $X$ is a bounded $r$-regular set and that $d\\sqrt{2}<r$. We also assume that $\\partial X$ does not pass through  a pixel corner.\n\\end{Convention}\n\nNote that the boundedness condition on $X$ implies that $X$ is compact.\n\nPavlidis \\cite{Pavlidis} defines a grid $d\\Z^2$ and a set $X$ to be \\emph{compatible} if $X$ is $r$-regular with $d\\sqrt{2}<r$. With this restriction, since $d\\sqrt{2}$ is the diameter of a pixel, each black $r$-ball contains the pixel that its centre belongs to, meaning that each black $r$-ball is centered in a black pixel. Similarly the centre of each white $r$-ball is contained in a white pixel. This means that for each component of $X$ yields at least one black pixel, and each component of $X^C$ yields at least one white pixel. Latecki et al. showed that for a compatible grid $d\\Z^2$ and set $X$, the set $V(X)$ of black pixels is homeomorphic to $X$. Hence $X$ and $V(X)$ have the same topological features. Furthermore, the above conditions ensure that we do not get too large grey areas, as will be clear in the following section. We will only concern ourselves with images that capture all of the objects photographed, and not just a part of them.\n\n{\\color{black}Let us introduce the notion of \\emph{weak $r$-similarity}, as introduced in \\cite{SK},\\cite{SLS}.\n\n\\begin{defn}\nLet $A, B\\subseteq \\R^2$ be bounded sets and $r>0$. We call $A$ and $B$ weakly $r$-similar if there exists a homeomorphism $f:\\R^2\\to\\R^2$ such that $x\\in A\\iff f(x)\\in B$ and the Hausdorff distance between the set boundaries satisfies $d_H(\\partial A, \\partial B)<r$\n\\end{defn}\n\nThe overall purpose of this paper will be to show the following:\n\\begin{thm}\\label{ThmMainResult}\nLet $I$ be a digital image of an $r$-regular set $X$ by a lattice $d\\Z^2$ with $d\\sqrt{2}<r$. We may construct an object $\\Gamma$ from $I$ such that $\\Gamma$ and $X$ are weakly $d$-similar, where $d$ is the pixel side length.\n\\end{thm}\n\nWe believe that the above result may be strengthened to prove strong $d+\\varepsilon$-similarity between the two for a suitable $\\varepsilon$, but such a result is beyond the scope of this paper. \n\nA large part of the proof of Theorem \\ref{ThmMainResult} will be to prove the following:\n}\n\n\\begin{thm}\nLet $I$ be a digital image of an $r$-regular set by a lattice $d\\Z^2$ with $d\\sqrt{2}<r$. We may construct an object $\\Gamma$ from $I$ such that $d_H(\\partial \\Gamma,\\partial X)<d$, where $d_H$ is the Hausdorff distance.\n\\end{thm}\n\nTo start working with $r$-regular sets, we first sum up some basic statements about them:\n\n\\begin{prop}[Tang Christensen and du Plessis, \\cite{TC}, Proposition A.1]\\label{PropEquivalentRregDefinitions}\nLet $A\\subseteq \\R^n$ be a closed set and $r>0$. Then the following are equivalent:\n\\begin{enumerate}\n\\item At any point $x\\in\\partial X$ there exist two closed $r$-balls $B_r\\subseteq A$ and $B_r'\\subseteq\\overline{A^C}$ such that $B_r\\cap B_r'=\\sett{x}$.\n\\item The sets $A$ and $\\overline{A^C}$ are equal to unions of closed $r$-balls.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{defn}\n  For $\\delta>0$, we denote the $\\delta$-tubular neighbourhood of\n  $\\partial X$ in $\\R^2$ by\n  $N_\\delta=\\sett{x\\in\\R^2\\mid d(x,\\partial X)<\\delta}$.\n\\end{defn}\n\n\\begin{lem}[Duarte \\& Torres, \\cite{DTs}, Lemma 5]\\label{LemProjection}\n  Let $X$ be an $r$-regular set. For each $x\\in N_{r}$ there is a\n  unique point $\\pi(x)\\in\\partial X$ such that $d(x,\\partial\n  X)=d(\\pi(x),x)$. Hence there is a well-defined projection\n  $\\pi:N_{r}\\to\\partial X$.\n\\end{lem}\n\n\\begin{thm}[Duarte and Torres, \\cite{DTr}]\nThe projection map $\\pi: N_r\\to\\partial X$ is continuous.\n\\end{thm}\n\n\nAnother important fact that we will be using heavily is the following:\n\nThere is a retraction $\\rho_{X^C}:N_r\\to X^C\\cup\\partial X$ (that we will\nsometimes just denote by $\\rho$) defined by\n\\begin{equation*}\n  \\rho_{X^C}(x)=\\begin{cases}\n    x & \\text{if $x\\in X^C\\cup\\partial X$},\n    \\\\\n    \\pi(x) & \\text{otherwise},\n  \\end{cases}\n\\end{equation*}\nand likewise a retraction $\\rho_{X}:N_r\\to X$ defined by\n\\begin{equation*}\n  \\rho_{X}(x)=\\begin{cases}\n    x & \\text{if $x\\in X$},\n    \\\\\n    \\pi(x) & \\text{otherwise}.\n  \\end{cases}\n\\end{equation*}\nThese retractions will prove to be crucial in later arguments, since\nthey have some nice properties.\n\nWe now state some results about $\\rho=\\rho_{X^C}$. However, the\nsimilar results for $\\rho_X$ also hold.\n\n\\begin{prop}[Stelldinger et al., \\cite{SLS}] Let $x, y\\in X^C$ with\n  $d(x,y)<2r$ and let $L\\subseteq\\R^n$ be the line segment between\n  them. Then\n  \\begin{enumerate}[(i)]\n  \\item The line segment $L$ is a subset of $ X^C\\cup N_r$, and\n    $\\rho\\vert_L$ is injective,\n  \\item For $s<r$ and $B_s$ any $s$-ball containing $x$ and $y$,\n    $\\rho(L)$ is a subset of $B_s$.\n  \\end{enumerate}\n\\end{prop}\n\n\\begin{defn}\n  Let $L\\subseteq\\R^n$ be a closed line segment of length\n  $\\vert L\\vert<2r$. Then the $r$-spindle $S(L,r)$ around $L$ is the intersection\n  of all closed balls of radius $r$ whose boundaries contain both endpoints of $L$. If $x$ and $y$ are the endpoints of $L$, we will sometimes write $S(x,y,r)$ in stead of $S(L,r)$.\n\\end{defn}\n\n\\begin{lem}[du Plessis, A.20 \\cite{TC}]\\label{LemSpindleWidth}\nLet $L$ be a closed line segment in $\\R^2$ of length $\\abs{L}<2r$. Then the maximal distance from a point in the $r$-spindle $S(L,r)$ to $L$ is $r-\\sqrt{r^2-\\frac{L^2}{4}}$.\n\\end{lem}\n\n\\begin{lem}[du Plessis, A.13 \\cite{TC}]\\label{LemSpindleIsIntersection}\n  Let $L\\subseteq\\R^n$ be a closed line segment of length\n  $\\vert L\\vert<2r$. Then the $r$-spindle $S(L,r)$ is the intersection\n  of all balls of radius at most $r$ that contain $L$.\n\\end{lem}\n\n\\begin{cor}[du Plessis, A.16 \\cite{TC}]\\label{CorPathInSpindle}\n  Let $x, y\\in X^C$ with $d(x,y)<2r$ and let $L\\subseteq\\R^n$ be the\n  line segment between them. Then $\\rho(L)$ is a subset of the\n  $r$-spindle $S(L,r)$.\n\\end{cor}\n\n\\begin{rem}\nSince $\\pi=\\rho_X\\circ\\rho_{X^C}=\\rho_{X^C}\\circ\\rho_X$, the above\ncorollary is also true for $\\pi$.\n\\end{rem}\n\n\n\\section{Impossible configurations at a resolution satisfying \\texorpdfstring{$d\\sqrt{2}<r$}{dsqrt(2)<r}}\\label{SecConfigurations}\nBefore we start reconstructing the original $r$-regular object, we need to discuss which configurations of $3\\times 3$ pixels of grey, black and white pixels can occur in the digital image of an $r$-regular object by a lattice $d\\Z^2$ where $d\\sqrt{2}<r$. We can make a computer put together all possible configurations of $3\\times 3$ pixels by telling it that the only possible configurations of $2\\times 2$ pixels are the ones in Figure \\ref{Fig2x2Configurations}, up to rotation and interchanging of black and white. We can then make a MatLab programme that combines these configurations in all possible ways.\nIf we do this, we get (up to rotation, mirroring and switching of black and white pixels) the configurations in Figure \\ref{FigMany3x3Configurations}.\n\n\\begin{figure}\n\\includegraphics[scale=0.3]{InputConfigurations2}\n\\caption{The only possible configurations of $2\\times2$ pixels, up to rotation and switching of black and white. Note that we have used Lemma \\ref{LemSkalfarvessorte1}, part ii), which is stated below.}\n\\label{Fig2x2Configurations}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[scale=0.7]{Many3x3Configurations.png}\n\\caption{All possible combinations of the allowed $2\\times2$ pixel configurations, up to rotation, mirroring and interchanging of black and white pixels.}\n\\label{FigMany3x3Configurations}\n\\end{figure}\n\nNote that not all these configurations can occur in the image of some $r$-regular object by a lattice $(d\\Z)^2$ with $d\\sqrt{2}<r$. We would like to remove configurations that do not occur from the list in Figure \\ref{FigMany3x3Configurations}. To do so, we need to prove a series of lemmas. Their proofs are mainly geometric and rather technical, so we will put them in the appendix instead of presenting them here.\n\nFirst of all, let us start with a definition, borrowed from Pavlidis' book \\cite{Pavlidis}.\n\n\\begin{defn}\nTwo pixels are \\emph{direct neighbours} (abbreviated \\emph{d-neighbours}) if the respective cells share a side. Two pixels are \\emph{indirect neighbours} (abbreviated \\emph{i-neighbours}) if those cells touch only at a corner. The term \\emph{neighbour} denotes either type.\n\\end{defn}\n\nIn the following lemmas, we will only be considering pixel configurations in images of $r$-regular objects by lattices $d\\Z^2$ with $d\\sqrt{2}<r$ according to our convention, but for brevity we will omit this requirement from the lemma statements.\n\n\\begin{lem}\\label{LemTwoIntersections}\n  Consider four pixels as in Figure \\ref{FigToSkaeringer}. Suppose\n  $\\partial X$ intersects the edge between the two pixels $B$ and $C$ more\n  than once. Then one of the pixels $A$ and $D$ is black, and the\n  other one is white.\n  The same result is true if $L$ is tangent to $\\partial X$ in a point.\n\\end{lem}\n\n\\begin{figure}\n  \\includegraphics[height=5cm]{ToSkaeringer}\n  \\caption{Consider four pixels as in the figure, where the boundary\n    $\\partial X$ intersects the edge between the pixels $B$ and $C$\n    twice. The proof consists of showing that there must be two $\\sqrt{2}d$-balls with\n    centres in $A$ and $D$, respectively, and that one of the balls is\n    black and the other one white.}\n  \\label{FigToSkaeringer}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{YieldsTwoIntersections}\n\\caption{If a configuration as the left one occurs in a digital image of an $r$-regular object with $d\\sqrt{2}<r$, then pixel $A$ must be black, and pixel $D$ must be white.}\n\\label{FigYieldsTwoIntersections}\n\\end{figure}\n\n\\begin{lem}\\label{LemYieldsTwoIntersections}\nIn a configuration as the one in Figure \\ref{FigYieldsTwoIntersections} left, the pixel named $A$ in Figure \\ref{FigYieldsTwoIntersections} right must be black, and the the pixel named $D$ must be white.\n\\end{lem}\n\n\\begin{lem}\\label{LemNo9greys}\n  Consider a configuration of $3\\times3$ pixels with the middle one grey. Then one of its 8 neighbour pixels is not grey.\n\\end{lem}\n\n\\begin{figure}\n  \\includegraphics[scale=0.5]{5greysAdd}\n  \\caption{In Theorem \\ref{Thm5greys}, we consider 9 pixels, of which\n    the middle one is grey, as in the figure.}\n  \\label{Fig5greysAdd}\n\\end{figure}\n\n\\begin{lem}\\label{Thm5greys}\n  Consider a configuration of $3\\times3$ pixels as in Figure\n  \\ref{Fig5greysAdd}, where the middle one is grey and has centre\n  $c$. Let $p=\\pi(c)$ be the point of $\\partial X$ that is nearest to $c$, and\n  suppose the centre of the black ball $B_r(x_b)$ tangent to $\\partial X$\n  at $p$ is closer than the centre of the white ball $B_r(x_w)$\n  tangent to $\\partial X$ at $c$, and that $x_b$ belongs to the lower left\n  pixel (which is hence black).\n\n  Then the upper right pixel is white.\n\\end{lem}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{Remark}\n\\caption{If a grey pixel has four grey d-neighbours as in the left figure, it must have a black and a white i-neighbour sitting diagonally across from each other. Equivalently, if a grey pixel does not have a black and a white i-neighbour sitting across from each other as in the right figure, it cannot have four grey d-neighbours.}\n\\label{FigRemark}\n\\end{figure}\n\n\\begin{rem}\\label{Rem3x3Pix}\n  Theorem \\ref{Thm5greys} and Lemma \\ref{LemNo9greys} combined tell us\n  that a grey pixel $C$ with four grey d-neighbours must always have a black and a white i-neighbour whose common vertices with $C$ sit diagonally across from each other, see Figure \\ref{FigRemark}, left.\n  Equivalently, if a grey pixel $C$ does not have a black and a white neighbour sitting opposite of each other, then at least one of its d-neighbours must not be grey.\n\\end{rem}\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{0.3\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.4]{Skalfarvessorte1}\n    \\subcaption{}\n    \\label{FigSkalfarvessorte1}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.3\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.4]{Skalfarvessorte3a}\n    \\subcaption{}\n    \\label{FigSkalfarvessorte3a}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.3\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.4]{Skalfarvessorte2}\n    \\subcaption{}\n    \\label{FigSkalfarvessorte2}\n  \\end{minipage}\n  \\caption{Consider configurations of two black and a grey pixel (we\n    do not assume anything about the colour of the dotted pixels). If\n    a grey and two black pixels sit in a configuration\n    as in Figure \\ref{FigSkalfarvessorte1} or\n    \\ref{FigSkalfarvessorte3a}, then the pixels $A$ and $B$ must also\n    be black. If a grey and two black (or two\n    white) pixels sit in a configuration as in Figure\n    \\ref{FigSkalfarvessorte2}, then either the pixels $A_1$, $A_2$,\n    $A_3$ are all black, or the pixels $B_1$,\n    $B_2$, $B_3$ are all black.}\n\\end{figure}\n\n\\begin{lem}\\label{LemSkalfarvessorte1}\\label{LemSkalfarvessorte2}\nThe following holds:\n  \\begin{enumerate}[(i)]\n  \\item Consider $2\\times3$ pixels as in the lower part of Figure\n    \\ref{FigSkalfarvessorte1} with the grey and black pixels placed\n    relatively to each other as in the figure. Then pixels $A$ and $B$\n    must necessarily be black.\n\n  \\item Consider $2\\times 2$ pixels as in Figure\n    \\ref{FigSkalfarvessorte3a}, with the grey and black pixels placed\n    relatively to each other as in the figure. Then $A$ must necessarily\n    be black.\n\n  \\item Consider $3\\times 3$ pixels as in Figure\n    \\ref{FigSkalfarvessorte2}, with the grey and black pixels placed\n    relatively to each other as in the figure. Then either the pixels\n    $A_1$, $A_2$, $A_3$ are all black, or the pixels $B_1$, $B_2$,\n    $B_3$ are all black.\n  \\end{enumerate}\n  \\medskip\n  \\noindent\n  The similar result is also true if we replace the black pixels with\n  white ones in the figures.\n\\end{lem}\n\n\\begin{figure}\n \\includegraphics[height=5cm]{6greys2}\n \\caption{We are considering 6 grey pixels in a $2\\times3$\n    combination, and we have shown that $\\partial X\\cap(G_1\\cup G_2)$\n    belongs to the red set in the figure, and that one of the pixels\n    $K_1$, $K_2$ must be black, and the other one white.}\n \\label{Fig6greys}\n\\end{figure}\n\n\\begin{lem}\\label{Lem2x3grey}\n  Suppose we have a configuration of 6 grey pixels as in Figure\n  \\ref{Fig6greys}, with pixels $G_1$, $G_2$, $K_1$ and $K_2$ as in the\n  figure. Then the following holds:\n  \\begin{enumerate}[(i)]\n  \\item One of the pixels $K_1$, $K_2$ must be black, and the other\n    one white,\n  \\item The set $\\partial X\\cap(G_1\\cup G_2)$ belongs to the set of points\n    in $G_1\\cup G_2$ that are no further than $(\\sqrt{2}-1)d$ from the\n    common edge of $G_1\\cup G_2$ (i.e. the red set in the figure).\n  \\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{LemImpossible2x3}\nA configuration as the one in Figure \\ref{Fig2x3Impossible} left cannot occur.\n\\end{lem}\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{Impossible}\n\\caption{The configuration to the left cannot occur in the digital image of an $r$-regular object with $d\\sqrt{2}<r$, for if it did, one of the pixels that are coloured red to the right would have a colour that would not be compatible with any legal configuration.}\n\\label{Fig2x3Impossible}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{Impossible3x3}\n\\caption{The configuration in this figure cannot occur in the image of an $r$-regular object with $d\\sqrt{2}<r$.}\n\\label{FigImpossible3x3A}\n\\end{figure}\n\n\\begin{lem}\\label{LemTrickyConfiguration}\nA configuration as the one in Figure \\ref{FigImpossible3x3A}, left cannot occur.\n\\end{lem}\n\n\\begin{thm}\\label{ThmConfigurationsUsualResolution}\nUp to rotation, mirroring and interchanging of black and white, any $3\\times 3$ configuraton of pixels is one of those shown in Figure \\ref{FigConfigurationsUsualResolution}.\n\\end{thm}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{3x3-3}\n\\caption{Up to rotation, mirroring and switching of black and white colours, these are the only $3\\times 3$-configurations that can occur in the digital image of an $r$-regular object by a lattice $(d\\Z)^2$ with $d\\sqrt{2}<r$.}\n\\label{FigConfigurationsUsualResolution}\n\\end{figure}\n\nIn the following, it will also be useful to know which $4\\times4$-configurations with a centre of $2\\times 2$ grey pixels that may occur in a digital image of an $r$-regular object by a lattice $d\\Z^2$. We may have a computer find these by combining all possible $3\\times 3$-configurations from Figure \\ref{FigConfigurationsUsualResolution}, together with the rotations, mirror images and inverses of these configurations.  After removing configurations that violate Lemma \\ref{Lem2x3grey}, this yields the configurations in Figure \\ref{FigKonfigurationer} (up to rotations, mirror images and interchanging of black and white). We aim to remove configurations from this list if they cannot occur in a digital image like the ones we are considering.\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{Konfigurationer}\n\\caption{These are all possible combinations of the configurations in Figure \\ref{FigConfigurationsUsualResolution}, their inverses, rotations and mirror images.}\n\\label{FigKonfigurationer}\n\\end{figure}\n\n\\begin{lem}\\label{LemLeftoverconfiguration}\n  The configuration in Figure \\ref{FigLeftoverconfiguration} cannot\n  occur.\n \n\\end{lem}\n\n\\begin{figure}\n  \\includegraphics[height=4cm]{Leftoverconfiguration}\n  \\caption{We show that this configuration cannot\n    occur in the digitisation of an $r$-regular object.}\n  \\label{FigLeftoverconfiguration}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n  \\includegraphics[scale=0.3]{2x4Configuration}\n  \\hspace{1cm}\n  \\includegraphics[scale=0.3]{2x4ConfigurationHelp}\n  \\caption{The right configuration does not occur in the\n    digitisation. If it did, then one of the red pixels (right part of\n    the figure) would have to be non-grey by Theorem\n    \\ref{Lem2x3grey}.}\n  \\label{Fig2x4Configuration}\n\\end{figure}\n\n\\begin{lem}\\label{Lem2x4Configuration}\n  The left configuration in Figure \\ref{Fig2x4Configuration}\n  cannot occur.\n\\end{lem}\n\n\\begin{lem}\\label{Lem4Intersections2x2}\nThe boundary $\\partial X$ cannot intersect all four boundary edges of a configuration of $2\\times 2$ grey pixels.\n\\end{lem}\n\n\\begin{thm}\\label{Thm2x2}\nThe only possible configurations of $4\\times 4$-configurations with $2\\times 2$ grey pixels in the middle are the ones shown in Figure \\ref{FigKonfigurationer13}.\n\\end{thm}\n\n  \\begin{figure}\n    \\begin{minipage}[b]{\\linewidth}\n      \\centering\n      \\includegraphics[width=0.9\\linewidth]{Konfigurationer13-3}\n      \\caption{The 33 possible configurations of $4\\times4$ pixels\n        with the middle ones grey, up to rotation, reflection and\n        switching of black and white pixels.}\n      \\label{FigKonfigurationer13}\n    \\end{minipage}\n  \\end{figure}\n  \n  Note that the converse is not true: There are configurations in Figure \\ref{FigKonfigurationer13} that does not occur in any image of an $r$-regular object by a lattice $d\\Z^2$ with $d\\sqrt{2}<r$. But since the proofs of this is rather technical and the result is not relevant to our further progress, we will not discuss them here.\n\n\n\\section{Reconstruction of the boundary of the set}\nAll the work done in the previous section was leading up to the development of a reconstruction algorithm, which we will introduce in this section. The idea will be to use circle arcs to approximate the boundary of the edge. The reconstructed set will not in general be $r$-regular.\n\nBefore we start, we will introduce some points, called auxiliary points, that our reconstructed boundary must pass through. These are defined differently for different grey pixels. Hence we define\n\n\\begin{defn}\nA grey pixel sitting in a $2\\times 2$ configuration of grey pixels is called complex. A grey pixel that is not sitting in a $2\\times 2$ configuration of grey pixels is called simple. \n\\end{defn}\n\n\n\n\nWe now introduce the auxiliary points needed for the reconstruction. \n\n\nConsider a pixel edge shared by two grey pixels $A$ and $B$. If the two grey pixels sit in the same $2\\times 2$ configuration of grey pixels, we introduce an auxiliary point at the midpoint of this configuration. If they do not, we introduce a point on the midpoint of their common edge, see Figure \\ref{FigAuxiliaryPoints}. (Note that the two grey pixels may be part of two different $2\\times 2$ configurations of grey pixels. In that case, we introduce two auxiliary points, on at the centre of each of the two $2\\times2$ configurations).\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{AuxiliaryPoints}\n\\caption{If two grey pixels share an edge and are not a part of the same $2\\times 2$ configuration of grey pixels as in the left figure, we introduce an auxiliary point (red) at the midpoint of their common edge. If on the other hand the two pixels are a part of the same $2\\times2$ configuration of grey pixels, we introduce an auxiliary point at the centre of this $2\\times2$ configuration.}\n\\label{FigAuxiliaryPoints}\n\\end{figure}\n\n\\begin{lem}\nAll simple grey pixels have between one and three auxiliary points on their boundary. All complex grey pixels have either one or two grey auxiliary points on their boundary.\n\\end{lem}\t\t\n\n\n\\begin{proof}\nFor the simple pixels, look at all possible configurations in Figure \\ref{FigConfigurationsUsualResolution}. For the complex pixels, look at all possible configurations in Figure \\ref{FigKonfigurationer13}.\n\\end{proof}\n\n\\begin{lem}\\label{LemRemovedPoints}\nA simple pixel with just one auxiliary point on its boundary must share this point with a simple pixel with three auxiliary points on its boundary. On the other hand, a simple pixel with three auxiliary points on its boundary must share exactly one of these points with a simple pixel with just one auxiliary point on its boundary.\n\\end{lem}\n\n\n\\begin{proof}\nCheck all cases as presented in Figure \\ref{FigConfigurationsUsualResolution}.\n\\end{proof}\n\nWe will now remove the auxiliary point of all simple pixels that has only one auxiliary point. By the above lemmas, this now means that all simple pixels have zero or two auxiliary points on their boundary, and all complex pixels have one or two auxiliary points on their boundary.\n\n\\begin{lem}\\label{LemAuxiliaryPoints2x2}\nIn each $2\\times 2$ configuration of grey pixels there are exactly 3 auxiliary points - one at the centre and two on the configuration boundary\n\\end{lem}\n\n\\begin{proof}\nEach $2\\times 2$-configuration of grey pixels sits in one of the configurations in Figure \\ref{FigKonfigurationer13} (up to rotation, mirroring and switching of colours). Hence we get the above theorem by checking all possible cases.\n\\end{proof}\n\n\\begin{thm}\nFor each auxiliary point $p$, there are exactly two auxiliary points with the property that there is a pixel having both $p$ and that auxiliary point on its boundary\n\\end{thm}\n\n\\begin{proof}\nConsider an auxiliary point $p$, sitting on the boundary of pixel $C$. If $p$ is the centre of some $2\\times 2$ configuration of black pixels, then by Lemma \\ref{LemAuxiliaryPoints2x2} there are only two auxiliary points on the boundary of this configuration as claimed.\n\nIf $p$ is the midpoint of some pixel edge, consider a pixel $C$ having this edge. It is either simple (and hence has two auxiliary points on its boundary), or it is complex and hence has an auxiliary point in a corner of $C$. In both cases, there is exactly one other auxiliary point on the boundary of each of the pixels having $p$ on their boundaries.\n\\end{proof}\n\nThe above theorem means that there is a natural way of defining 'neighbouring auxiliary points': Two auxiliary points are neighbours if they sit on the boundary of the same pixel.\n\nThe next step is to approximate the boundary of $X$ with curve segments: Consider an auxiliary point and its two neighbour auxiliary points. We approximate $\\partial X$ by circle arc segments through these three points (or, if the points are collinear, by line segments). Hence there are two curve segments through each two neighbouring auxiliary points sitting on the boundary of the same pixel $C$, one starting in one of the points, the other ending in the other point.\nEach of these curve segments are graphs over the straight line $L$ through the points, so we may write them as $\\gamma_1:[0,\\abs{L}]\\to \\R$ and $\\gamma_2:[0,\\abs{L}]\\to \\R$. Then choosing a bump function $\\phi:[0,\\abs{L}]\\to [0,1]$ with $\\phi(0)=1$ and $\\phi(\\abs{L})=0$, we may patch a connected curve $\\gamma$ together by putting $\\gamma_C(t)=(1-\\phi(t))\\gamma_1(t)+\\phi(t)\\gamma_2(t)$ in each pixel $C$, see Figure \\ref{FigGamma}. The resulting curve $\\gamma_C$ is then also a graph over $L$, and the curve $\\gamma$ is a smooth embedded submanifold of $\\R^2$.\n\n\\begin{figure}\n\\includegraphics[scale=0.7]{Gamma}\n\\caption{The curve $\\gamma_1$ (blue) and the curve $\\gamma_2$ (green) are patched together inside $C$ using a bump function. This produces the curve $\\gamma_C$ (dashed red curve).}\n\\label{FigGamma}\n\\end{figure}\n\n\\begin{lem}\\label{LemGammaInConvexHull}\nThe path $\\gamma_C$ is contained in the area bounded by $\\gamma_1$ and $\\gamma_2$.\n\\end{lem}\n\n\\begin{proof}\nSince $\\gamma_C$ is a graph over a line $L$ and it is a convex combination of a point on $\\gamma_1$ and a point on $\\gamma_2$, the curve $\\gamma_C$ must lie between these two points, and hence also between the curves $\\gamma_1$ and $\\gamma_2$. \n\\end{proof}\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{Exceptions}\n\\caption{Auxiliary points sitting in one of these configurations around the red pixel are exemptions to Lemma \\ref{LemGammaInC}.}\n\\label{FigExceptions}\n\\end{figure}\n\n\\begin{lem}\\label{LemGammaInC}\nConsider two neighbour auxiliary points on the boundary of some pixel $C$ not sitting in a configuration like the ones in \\ref{FigExceptions}. \n\\begin{itemize}\n\\item If the two neighbour auxiliary points do not sit at the endpoints of some edge of $C$, then the curve $\\gamma_C$ is contained in $C$.\n\\item If the two neighbour auxiliary points do sit at the endpoints of the edge shared by $C$ and some other pixel $C'$, then the curve $\\gamma_C$ is contained in $C\\cup C'$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{CircleArcs2}\n\\caption{The figure shows all possible positions of two auxiliary points on the boundary of some pixel $C$ (the red ones in the figure), and all circle arcs through these two and a third auxiliary point. The radii of the circle arcs are also calculated.}\n\\label{FigCircleArcs}\n\\end{figure}\n\n\\begin{proof}\nWe start by proving the lemma for the arc segments $\\gamma_1$ and $\\gamma_2$: We can consider all possible positions of three neighbour auxiliary points, see Figure \\ref{FigCircleArcs}. Look at auxiliary points $p_1$, $p_2$ sitting in configurations around a red pixel like any of the five figures to the left. A calculation (or a look at the figures!) then show that with the exception of auxiliary points in configurations as the ones shown in Figure \\ref{FigExceptions}, all possible circle arcs through two auxiliary points sitting on a red pixel are contained in that red pixel.\n\nNow, lets argue that $\\gamma_C$ is contained in $C$: By Lemma \\ref{LemGammaInConvexHull}, $\\gamma_C$ is contained in the area $A_C$ bounded by $\\gamma_1$ and $\\gamma_2$. Since $\\gamma_1$ and $\\gamma_2$ are both contained in $C$ which is convex, $A_C$ is also contained in $C$. Hence $\\gamma_C$ must also be contained in $C$. This proves $i)$. Part $ii)$ is proved in the same way, but now looking at the right figure in Figure \\ref{FigCircleArcs}. \n\\end{proof}\n\n\\begin{prop}\nThe curves $\\gamma_C$ have no self-intersections, and do not intersect each other. Hence each component of $\\gamma$ is a simple closed curve.\n\\end{prop}\n\n\\begin{proof}\nSince each segment $\\gamma_C$ is a graph over some straight line $L$, we only need to show that two segments $\\gamma_C$ and $\\gamma_{C'}$ do not intersect. By Lemma \\ref{LemGammaInConvexHull} it suffices to show that the area $A_C$ bounded by two arcs $\\gamma_1$ and $\\gamma_2$ in a pixel $C$ does not intersect the area $A_{C'}$ bounded by two arcs $\\gamma'_1$ and $\\gamma'_2$ in another pixel $C'$. \n\nConsider the possible circle arcs shown in Figure \\ref{FigCircleArcs}. With the exception auxiliary points sitting in configurations like the ones in Figure \\ref{FigExceptions}, all arc segments stays inside the pixel(s) containing both of the auxiliary points they join. Since no pixel can have more than two auxiliary points on their boundary, the only possible way that two curve segments can intersect is if one of them, say $\\gamma_C$, is made using a curve $\\gamma_1$ connecting points in a configuration like the one in Figure \\ref{FigExceptions}. But going through all possible configurations where such a $\\gamma_1$ could occur one can conclude that $\\gamma$ cannot intersect itself in this case either.\n\nThat each segment of $\\gamma$ is a simple closed curve follows from the fact that the segments $\\gamma_C$ always connect two neighbour auxiliary points, and all auxiliary points have two neighbours. If a component of $\\gamma$ were not a closed curve, it would have an endpoint (since all components of $\\gamma$ are bounded) - but $\\gamma$ is the join of curve segments between neighbour auxiliary points, so such an endpoint can only occur in one of the auxiliary points. But since all auxiliary points have two neighbour auxiliary points, this is impossible.\n\n\\end{proof}\n\n\\begin{thm}\\label{ThmGammaProperties}\nFor each component of $\\partial X$, there is exactly one component of $\\gamma$. Each component of $\\gamma$ separates the boundary components of a connected component $A$ of the set of grey pixels.\n\\end{thm}\n\n\\begin{proof}\nLet $\\partial X'$ be a component of $\\partial X$, and let $A$ be the set of grey pixels containing points of $\\partial X'$. Note that $A$ cannot have any grey neighbour pixel $B$, because this would imply that $B$ contained a point from another component of $\\partial X$, which would mean that there were two points on different components of $\\partial X$ closer than $2d\\sqrt{2}$ - a contradiction by Corollary \\ref{CorPathInSpindle} applied to $\\pi$.\nHence $A$ is a connected component of the set of grey pixels.\n\nConsider any chain of grey pixels in $A$, where each pixel in the chain is a neighbour of the previous and the next pixel in the chain, and each pixel appear in the chain no more than once. Assume that the start pixel and end pixel of the chain has at least two grey d-neighbours. We aim to show that the first and last pixel in such a chain is connected by a segment of $\\gamma$. \n\nBy construction, each pixel in such a chain has at least two grey d-neighbours, hence has at least one auxiliary point on its boundary. If a pixel $C$ in the chain has only one auxiliary point on its boundary, its two grey d-neighbours must sit in a $2\\times2$ configuration with $C$, and hence one of its grey d-neighbours must have two auxiliary points on its boundary. Hence if we replace $C$ by its d-neighbours with two auxiliary points on their boundary, we still get a chain of pixels in $A$. Repeating this, we end up with a chain of pixels where all pixels in the chain have two auxiliary points on their boundary. The construction of $\\gamma$ then yields a segment of $\\gamma$ through this pixel chain.\n\nNow by $r$-regularity of $X$, $A$ must be larger than 2 pixels and hence have at least one pixel with two grey d-neighbours. Hence $A$ contains at least one component of $\\gamma$. If $A$ contained two components $\\gamma'$, $\\gamma''$ of $\\gamma$, we could pick a chain of grey pixels connecting a pixel containing a point of $\\gamma'$ with a pixel containing a point of $\\gamma''$. Then by the above, the auxiliary points on the first pixel would be connected to the auxiliary points on the last pixel by a segment of $\\gamma$. But then $\\gamma'$ and $\\gamma''$ would be connected - a contradiction. Thus for any component of $\\partial X$, there is exactly one component of $\\gamma$.\n\nFor the second part of the statement, consider a chain of pixels following a boundary component $\\partial A'$ of $A$. By the above, this chain yields a segment of $\\gamma'$ which is a closed curve containing $\\partial A'$, but not containing any other components of the boundary of $A$. Hence $\\gamma'$ separates any component of $\\partial A$ from the others - in particular, there can be at most two boundary components of $A$, and $\\gamma$ separates them. In fact, there are always two components of $\\partial A$: Any point $x\\in\\partial X'$ has a black and a white $\\sqrt{2}d$-ball osculating at $x$, and these balls contain the pixels in which they are centred. Since $\\partial X'$ separates the two balls and hence the two pixels where they are centred, so does $A$. But then $A$ must have two different boundary components, and both $\\partial X'$ and $\\gamma'$ separates these two components.\n\\end{proof}\n\nSince the set of grey pixels separates the white pixels from the black, the above theorem actually implies that $\\gamma$ also separates the white pixels from the black (in the sense that any curve from a black to a white pixel must intersect $\\gamma$).\nWe may conclude (via the Jordan Curve Theorem) that each component of $\\gamma$ separates $\\R^2$ into two sets, a bounded and an unbounded. From now on, $\\gamma$ is the boundary of the reconstructed set, which we define as follows:\n\n\\begin{defn}\nWe define \\emph{the reconstructed set} $\\Gamma$ to be the bounded set having $\\gamma$ as boundary.\n\\end{defn\n\n\\section{Hausdorff distance between the boundaries of the original set and the reconstruction}\n\nWe are now ready to look at the Hausdorff distance between our reconstruction and the original object. Let us start by proving a lemma:\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{CornerPixel}\n\\caption{A point $q\\in\\partial X$ that belongs to a pixel with two black neighbour pixels sharing an edge must belong to the ball $B_d(p)$.}\n\\label{FigCornerPixel}\n\\end{figure}\n\n\\begin{lem}\\label{LemCornerPixel}\nConsider a grey pixel $C$ as the one in Figure \\ref{FigCornerPixel}, with two black (or two white) neighbour pixels sharing a vertex. Let $p$ be the vertex of $C$ that does not belong to the two black (or white) neighbour pixels. If $q\\in\\partial X\\cap C$, then $q\\in B_d(p)$.\n\\end{lem}\n\n\\begin{proof}\nLet $q$ be as above. Then $q$ belongs to a component of $\\partial X$ that must enter and leave $C$ in two places, say in points $x_1$ and $x_2$. These points belong to one of the edges of $C$ having $p$ as a corner. Let $L$ be the line segment between $x_1$ and $x_2$. \n\nNow $q$ must belong to $\\pi(L)$, which in turn must belong to $S(L,r)$ by Corollary \\ref{CorPathInSpindle}, which again is contained in any ball of radius less than $r$ containing $L$ by Lemma \\ref{LemSpindleIsIntersection}. The ball $B_d(p)$ contains both of the edges of $C$ that have $p$ as an endpoint, hence it also contains $x_1$ and $x_2$ and consequently $L$, $S(L,r)$ and $q$.\n\\end{proof}\n\nNow that we have a suggestion for the boundary of the original set, we aim to show how good this approximation is. The first step will be to prove the following:\n\n\\begin{thm}\\label{ThmHausdorffA}\nAny point of $\\partial X$ has distance at most $d$ to the curve $\\gamma$ consisting of curve segments $\\gamma_C$. Hence $\\sup _{y\\in \\partial X}\\inf _{x\\in \\gamma} d(x,y)\\leq d$.\n\\end{thm}\n\nThis theorem, however, requires some additional lemmas:\n\n\n\n\\begin{lem}\\label{LemGammaPassing2x3Grey}\nIf two neighbour auxiliary points sit on the common boundary edge of two grey pixels $C_1$ and $C_2$, then the curve $\\gamma_{C_1}=\\gamma_{C_2}$ is contained in the set $C'$ of points in $C_1\\cup C_2$ that are at a distance $0.133d$ from the common edge of $C$.\n\\end{lem}\n\n\\begin{proof}\nIf two auxiliary points sit at the common boundary edge $e$ of $C_1$ and $C_2$, they must sit on the ends of $e$, i.e. they are the two common vertices of $C_1$ and $C_2$.\n\n\nBy Lemma \\ref{LemGammaInC}, part ii), $\\gamma_{C_1}=\\gamma_{C_2}$ belongs to $C_1\\cup C_2$. Let $\\gamma_1$ and $\\gamma_2$ be the two arc segments whose merge is $\\gamma_{C_1}$. Then they are both circle arcs of radius no smaller than $s=\\frac{\\sqrt{65}}{8}$ (see Figure \\ref{FigCircleArcs}), hence they are contained in the spindle $S(e,s)$ whose height is $(s-\\sqrt{s^2-\\frac{1}{4}})d\\approx0.133d$. Thus no point of $\\gamma_1$ or $\\gamma_2$ is further away than $0.133d$ from $e$. Since the curve $\\gamma_{C_1}$ belongs to he area bounded by $\\gamma_1$ and $\\gamma_2$ by Lemma \\ref{LemGammaInConvexHull}, we must also have that $\\gamma_{C_1}$ belongs to $C'$.\n\\end{proof}\n\n\\begin{lem}\\label{LemGammaPassingThroughBall}\nIf two auxiliary points sit on two edges of a pixel $C$ sharing a corner $p$, then $\\gamma_C$ is contained in $B_d(p)\\cap C$.\n\\end{lem}\n\n\\begin{proof}\nWe already know from Lemma \\ref{LemGammaInC} that $\\gamma_C$ belongs to $C$. Hence we only need to show that $\\gamma_C$ belongs to $B_d(p)$. By Lemma \\ref{LemGammaInConvexHull} it suffices to show that the area $A$ between the two curves $\\gamma_1$ and $\\gamma_2$ belongs to $B_d(p)$ and, since $B_d(p)$ is convex, it is even enough to show that $\\gamma_1$ and $\\gamma_2$ both belong to $B_d(p)$. This can be done by a calculation for all possible cases, or by looking at Figure \\ref{FigGammaPassingThroughC}.\n\\end{proof}\n\n\\begin{lem}\\label{LemGammaPassingCWithPointsOnOppositeEdges}\nIf two neighbour auxiliary points on the boundary of some pixel $C$ do not sit on the same edge of $C$ and neither in a configuration as the ones in Figure \\ref{FigExceptions}, then the distance between any point of $\\partial X$ in $C$ and the curve $\\gamma_C$ is less than ${d}$.\n\\end{lem}\n\n\\begin{proof}\nConsider two auxiliary points on the boundary of pixel $C$. Suppose they sit on two opposite edges $e_1$, $e_2$ of $C$. If furthermore the two points do not sit in one of the configurations of Figure \\ref{FigExceptions}, then by Lemma \\ref{LemGammaInC}, the curve $\\gamma_C$ belongs to $C$. Hence the curve $\\gamma_C$ must run from one side of $C$ to the other without leaving $C$, see Figure \\ref{FigGammaPassingThroughC} left. Thus given any point in $C$, projecting it to $\\gamma_C$ along a line parallel to $e_1$ moves it no further than a distance $d$. Hence all points of $C$ is closer than $d$ to $\\gamma_C$.\n\n\\begin{figure}\n\\includegraphics[scale=0.7]{GammaPassingThroughC}\n\\caption{Two different ways that a curve $\\gamma$ can pass through a pixel $C$ with two auxiliary points on its boundary.}\n\\label{FigGammaPassingThroughC}\n\\end{figure}\n\nOn the other hand, suppose the two auxiliary points on the boundary of $C$ sit on the midpoint of two edges $e_1$, $e_2$ sharing a vertex $p$, see Figure \\ref{FigGammaPassingThroughC} right. Then $C$ is a simple pixel, and since its auxiliary points do not sit on opposite edges, it cannot be one of the simple pixels that we removed auxiliary points from (by the proof of Lemma \\ref{LemRemovedPoints}). So $C$ must have two grey d-neighbour pixels sharing the vertex $p$, and two non-grey d-neighbour pixels sharing the vertex opposite of $p$, as in Figure \\ref{FigGammaPassingThroughC} right. Let us assume these two non-grey pixels to be black.\n\nConsider at point $q\\in C\\cap\\partial X$. By Lemma \\ref{LemCornerPixel}, $q$ must belong to $B_d(p)$.\nThen since the path $\\gamma_C$ is also contained in this ball by Lemma \\ref{LemGammaPassingThroughBall} and runs from $e_1$ to $e_2$, we hit $\\gamma_C$ somewhere if we move a point in $B_d(p)\\cap C$ along a radius of $B_d(p)$. Such a movement displaces the point a distance of at most $d-\\frac{1}{2\\sqrt{2}}d$, since this is the maximal distance between a point on $\\gamma_C$ and a point on $\\partial B_d(p)$ on the same radius of $B_d(p)$. Hence a point of $\\partial X\\cap C$ is at most a distance $d$ from $\\gamma_C$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{ThmHausdorffA}]\nBy Lemma \\ref{LemGammaPassingCWithPointsOnOppositeEdges} the theorem holds for any point of $\\partial X$ contained in a pixel with two auxiliary points not sitting on the same edge, and not sitting in one of the configurations of Figure \\ref{FigExceptions}. Hence we need to show the result for points on $\\partial X$ contained in i) grey pixels with two auxiliary points sitting on the same edge, ii) the special cases in Figure \\ref{FigExceptions}, iii) grey pixels with one auxiliary point on their boundary and iv) grey pixels with zero auxiliary points on their boundaries.\n\n\\emph{Ad i)}:\nBy Lemma \\ref{LemGammaPassing2x3Grey}, $\\gamma_{C_1}$ must belong to the set $C'$ of points in $C_1\\cup C_2$ closer than $0.133d$ to $e$, and by Lemma \\ref{Lem2x3grey}, all points of $\\partial X$ in $C_1\\cup C_2$ must be closer than $(\\sqrt{2}-1)d$ to $e$. Since the curve $\\gamma_{C_1}$  runs from one side of $C'$ to the other, then pushing a point $p\\in\\partial X\\cap C'$ orthogonally to $e$ inside $C_1\\cup C_2$, we must hit $\\gamma_{C_1}$ at some point. The displacement made in this manner can be no larger than $(\\sqrt{2}-1+0.133)d\\approx 0.55d$, hence any point of $(C_1\\cup C_2)\\cap\\partial X$ is closer than $0.55d$ to $\\gamma_{C_1}=\\gamma_{C_2}$.\n\n\\begin{figure}\n\\includegraphics[scale=0.7]{Exception1}\n\\caption{Both of the configurations excepted from Lemma \\ref{LemGammaInC} must sit in a configuration like the one shown above. We aim to show that the red rectangle cannot contain any points of $\\partial X$.}\n\\label{FigExceptedConfigurationsTriangle}\n\\end{figure}\n\n\\emph{Ad ii): } Now consider instead either of the cases from Figure \\ref{FigExceptions}. Such a configuration must necessarily sit in a configuration like in Figure \\ref{FigExceptedConfigurationsTriangle}, by looking at the possible configurations involving $2\\times 2$ grey pixels in Figure \\ref{FigKonfigurationer13}. We will aim to show that the rectangle $T$ in the figure, which shares two vertices with pixel $C$ and has the other two vertices at the midpoints of the vertical pixel edges of $C$, does not contain any points of $\\partial X$.\n\nLook at the blue line $L$ separating the two upper grey pixels from the lower. Since there are grey pixels on both sides of $L$, $\\partial X$ must pass it somewhere, and since both endpoints of $L$ are black, $\\partial X$ must pass $L$ at least twice. Then there must be some point $p$ in one of the two upper pixels where $\\partial X$ has horisontal tangent, and hence the centres of the black and white $\\sqrt{2}d$-balls meeting at this point sit on the vertical line through $p$. Since the pixels above the $2\\times 2$ grey are black in the figure, the upper ball osculating $\\partial X$ at $p$ must be black, and the lower must be white.\n\nBy Corollary \\ref{CorPathInSpindle} applied to $\\pi$ the part of $\\partial X$ between two points in $\\partial X\\cap L$ must be contained in the spindle $S(L,r)$ (shown in the figure), which contains points no further from $L$ than $(\\sqrt{2}-1)d$. So $p$ cannot be further above $L$ than $(\\sqrt{2}-1)d$.\n\nNow if $p$ belonged to the right upper grey pixel, the centre of the white ball osculating $\\partial X$ at $p$ would belong to a grey pixel and hence colour that pixel white. So this is not possible. Therefore $p$ must belong to the upper left pixel, and be no further from $L$ then $(\\sqrt{2}-1)d$. The centre of the white $\\sqrt{2}d$-ball osculating $\\partial X$ at $p$ must therefore lie in the white pixel, no further than $\\sqrt{2}d$ from $p$ and hence no further from the common edge between the white pixel and $C$ than $(\\sqrt{2}-1)d$ (so somewhere in the light blue part of this white pixel). But any $\\sqrt{2}d$-ball centred centred in the top half of the white pixel must contain $T$, the bottom half of pixel $C$, since $T$ and the top half of $C$ form a square with side length $d\n Hence the white ball osculating $\\partial X$ at $p$ must contain all of $T$, so $T$ cannot contain any points of $\\partial X$.\n\nA calculation shows that no point of $\\gamma_C$ lies further above $L$ than $0.041d$.\nHence if we take any point $q$ in $C\\backslash T$ and push it along a vertical line to $\\gamma_C$, we can do this without moving $q$ more than a distance $0.541d$ away. So any point of $\\partial X\\cap C$ is closer than $d$ to $\\gamma_C$.\n\n\\begin{figure}\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.7]{OneAuxiliary}\n\\caption{A pixel $C$ with just one auxiliary point on its boundary must sit in a configuration as the above.}\n\\label{FigOneAuxiliary}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.7]{ZeroAuxiliaries}\n\\caption{A grey pixel with zero auxiliary points on its boundary must sit in this configuration of pixels. Then any circle arc through the auxiliary points of $D$ and one of its neighbours must look like one of the above}\n\\label{FigZeroAuxiliaries}\n\\end{minipage}\n\\end{figure}\n\n\\emph{Ad iii):} Consider a grey pixel $C$ with only one auxiliary point $p$ on its boundary. By construction $C$ must be complex and have two grey neighbour pixels, and two non-grey neighbour pixels, see Figure \\ref{FigOneAuxiliary}. By Lemma \\ref{LemCornerPixel} all points of $\\partial X\\cap C$ must belong to the ball $B_d(p)$. Hence the distance from a point in $\\partial X\\cap C$ to $p$ is less than $d$. Since $p$ belongs to $\\gamma$, this shows the claim in this case.\n\n\n\\emph{Ad iv):} Consider a grey pixel $C$ without any auxiliary points on its boundary. By construction, it means that $C$ is a simple grey pixel with one grey d-neighbour pixel $D$, and three non-grey d-neighbour pixels, see Figure \\ref{FigZeroAuxiliaries}.\n\nNow, the boundary $\\partial X$ must pass the common edge $e$ of $C$ or $D$ in order to get into and out of $C$. Hence the part of $\\partial X$ that is in $C$ must be contained in $S(e,r)$\nBut $S(e,r)$ contains no points in $C$ that are further from $e$ than $\\sqrt{2}d-\\sqrt{2d^2-\\frac{d^2}{4}}<0.1d$.\n\nFurthermore, $D$ must have one auxiliary point on the midpoint of each vertical edge - let us call these $p_1$ and $p_2$. Then an arc segment $\\gamma_1$ through $p_1$ and $p_2$ and a third auxiliary point of one of the grey pixels neighbouring $C$ must lie above the straight line connecting $p_1$ and $p_2$ (just look at all possible cases, as is done in Figure \\ref{FigZeroAuxiliaries}).\n\nHence any point $p$ in $C\\cap\\partial X$ is closer than $0.1d$ to $e$, and any point in $e$ is closer than $\\frac{d}{2}$ to $\\gamma_D$. So the distance from $p$ to $\\gamma$ is less than $0.6d<d$. This finishes the proof that any point of $\\partial X$ is closer than $d$ to $\\gamma$.\n\\vspace{0.5cm}\n\nFor any point $x$ in $\\partial X$, there is a point $y'$ in $\\gamma$ that is no further than a distance $d$ from $x$, meaning that\n\\begin{equation*}\n\\inf_{y\\in \\gamma} d(x,y)\\leq d(x,y')\\leq d.\n\\end{equation*}\nThus we get\n\\begin{equation*}\n\\sup_{y\\in\\partial X}\\inf_{x\\in \\gamma} d(x,y)\\leq d.\n\\end{equation*}\n\\end{proof}\n\nThis proof is the first step on our way to show that $\\partial X$ and $\\gamma$ are close to each other in Hausdorff distance. The second step is taken when we prove the following\n\n\\begin{thm}\\label{ThmHausdorffB}\nAny point of $\\gamma$ has distance at most $d$ to the boundary $\\partial X$ of the original set $X$. Hence $\\sup _{y\\in \\gamma}\\inf _{x\\in \\partial X} d(x,y)\\leq d$.\n\\end{thm}\n\nThe proof of this theorem is very similar to the proof of Theorem \\ref{ThmHausdorffA}. Again we split the proof in a couple of lemmas.\n\n\\begin{lem}\\label{LemSimplePixelOppositeEdges}\nConsider a simple pixel $C$ with two auxiliary points on two opposite edges. Then any point of $\\gamma_C$ is closer than $d$ to some point of $\\partial X$.\n\\end{lem}\n\n\\begin{figure}\n\\begin{minipage}{0.55\\linewidth}\n\\includegraphics[scale=0.4]{SimplePixelOppositeEdges}\n\\caption{A simple pixel $C$ with auxiliary points on opposite edges must sit in one of the configurations above}\n\\label{FigSimplePixelOppositeEdges}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.40\\linewidth}\n\\begin{center}\n\\includegraphics[scale=0.4]{SimplePixelVertexAdjacent}\n\\caption{A simple pixel $C$ with auxiliary points on vertex-adjacent edges must sit in the configuration above}\n\\label{FigSimplePixelVertexAdjacent}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\\begin{proof}\nNotice that $C$ must sit in one of the two configurations of Figure \\ref{FigSimplePixelOppositeEdges}. In the first case, pick a point $p\\in\\gamma_C\\subseteq C$. The horisontal line in $C$ through $p$ has a black and a white endpoint, hence it must contain a point of $\\partial X$. Since $p$ is no further than $0.62d$ from the endpoints of this line,\nthere must be a point in $\\partial X$ that is closer than $0.62d<d$ to $p$.\n\nIn the second case, pick again a point $p$ in $\\gamma_C\\subseteq C$. Let $D$ be the pixel above $C$. Notice that $\\partial X$ must enter and leave $D$ by crossing $e$ in order for $D$ to be grey and both endpoints of $e$ to be black. Then by Corollary \\ref{CorPathInSpindle} with $\\rho$ replaced by $\\pi$, any point of $\\partial X$ in $D$ must belong to the spindle $S(e,r)$ which contains points no further from $e$ than $\\sqrt{2}d-\\sqrt{2d^2-\\frac{d^2}{4}}<0.1d$ by Lemma \\ref{LemSpindleWidth}. So any point of $\\partial X\\cap(C\\cup D)$ must either belong to $C$ or be no further from $e$ than $0.1d$. On the other hand, a calculation\nshows that the path $\\gamma_C$ is closer than $0.5d$ to $e$.\n\nLook at a vertical line in $C\\cup D$ through $p$. There must be a point on this line belonging to $\\partial X$, since its endpoints have different colours. Either this point belongs to $C$ (in which case they can be no further apart than $0.62d$ by the first part of the proof), or it belongs to $D$. If it belongs to $D$, it is no further from $e$ than $0.1d$, and since $p$ is no further than $0.5d$ from $e$, this point of $\\partial X$ must be closer than $d$ to $p$.\n\\end{proof}\n\n\\begin{lem}\\label{LemSimplePixelVertexAdjacent}\nConsider a simple pixel $C$ with two auxiliary points, located at the midpoint of two vertex-adjacent edges of $C$. Then no point of $\\gamma_C$ is further away from $\\partial X$ than $d$.\n\\end{lem}\n\n\\begin{proof}\nA pixel $C$ as the described must sit in a configuration as the one in Figure \\ref{FigSimplePixelVertexAdjacent}. Let $q$ denote the vertex of $C$ where the two edges containing auxiliary points meet. Let $p\\in \\gamma_C\\subseteq C$. \n\nConsider then line in $C$ through $p$ and $q$. Since it has endpoints of different colours, it must contain a point $s$ in $\\partial X$. By Lemma \\ref{LemCornerPixel}, any point of $\\partial X\\cap C$ must belong to the ball $B_d(q)$, and by Lemma \\ref{LemGammaPassingThroughBall}, so must $p$. Hence $p$ and $s$ both sit on a radius of the ball $B_d(q)$. By looking at the possible curves $\\gamma_C$, such two points cannot be further apart than a distance $d-\\frac{1}{2\\sqrt{2}}d$. So any point $p\\in \\gamma_C$ is closer than $d$ to $\\partial X$.\n\\end{proof}\n\n\\begin{lem}\\label{LemComplexPixelSameEdge}\nConsider a complex pixel $C$ with two auxiliary points located at the endpoints of some edge $e$ of $C$. Then no point of $\\gamma_C$ is further away from $\\partial X$ than $d$.\n\\end{lem}\n\n\\begin{figure}\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.4]{ComplexPixelSameEdge}\n\\caption{A complex pixel $C$ with auxiliary points at the endpoints of one of its edges $e$ must sit in a configuration as the one above.}\n\\label{FigComplexPixelSameEdge}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.4]{ComplexPixelOppositeVertices}\n\\caption{A complex pixel $C$ with auxiliary points at two vertices of $C$ located opposite of one another must sit in a configuration as the one above.}\n\\label{FigComplexPixelOppositeVertices}\n\\end{minipage}\n\\end{figure}\n\n\\begin{proof}\nThe pixel $C$ must sit in a configuration as the one in Figure \\ref{FigComplexPixelSameEdge}, by means of Lemma \\ref{Lem2x3grey}. By Lemma \\ref{LemGammaInC}, part ii) $\\gamma_C$ must belong to $C\\cup D$.\n\nNow pick a point $p\\in\\gamma_C$, and look at the vertical line in $C\\cup D$ through $p$. Since this line has endpoints of different colours, it must contain a point $q\\in\\partial X$. By Lemma \\ref{Lem2x3grey} again, $q$ must belong to the set of points in $C\\cup D$ that are no further than $\\frac{d}{2}$ from the common edge $e$ of $C$ and $D$, and by Lemma \\ref{LemGammaPassing2x3Grey} $p$ is no further than $0.133d$ from $e$. Hence\n$p$ and $q$ cannot be further than $0.55d<d$ from each other.\n\\end{proof}\n\n\\begin{lem}\\label{LemComplexPixelOppositeVertices}\nConsider a complex pixel $C$ with two auxiliary points located at vertices of $C$ diagonally opposite each other. Then no point of $\\gamma_C$ is further away from $\\partial X$ than $d$.\n\\end{lem}\n\n\\begin{proof}\nA pixel $C$ as in this lemma must sit in a configuration as the one in Figure \\ref{FigComplexPixelOppositeVertices}, by means of Lemma \\ref{Thm5greys}. Let $p_1$ and $p_2$ denote the two auxiliary points on the boundary of $C$.\n\nA calculation shows that any circle arc through $p_1, p_2$ and an auxiliary point neighbouring $p_2$ has radius greater than $d$. Hence any such circle arc is contained in the spindle $S(L,d)$ where $L$ is the line segment between $p_1$ and $p_2$, by Lemma \\ref{LemSpindleIsIntersection}. By the same lemma, this means that any such circle arc $\\gamma_1$ is contained in any ball of radius $d$ containing $L$. The same holds for the area bounded by the two circle arcs $\\gamma_1$ and $\\gamma_2$ (since $S(L,d)$ is also convex), and hence also for $\\gamma_C$. Thus, if we can find some point $q$ in $\\partial X$ such that the $d$-ball around $q$ contains $L$ and hence $S(L,d)$ and $\\gamma_C$, then any point of $\\gamma_C$ must be closer than $d$ to $\\partial X$.\n\nConsider the line $M$ connecting the black and white vertex of $C$. Since its endpoints have different colours, it must contain some point $q\\in \\partial X$. Since the distance between points of $M$ and $p_1$, $p_2$ is less than $d$ everywhere, the ball $B_d(q)$ contains $p_1$ and $p_2$ and hence the spindle $S(L,d)$ between them, and we are done.\n\\end{proof}\n\n\\begin{lem}\\label{LemComplexPixelRemaining}\nSuppose $C$ is a complex pixel with an auxiliary point on one of its edge midpoints and another auxiliary point at one of the vertices of $C$. Then any point of $\\gamma_C$ is closer than $d$ to $\\partial X$.\n\\end{lem}\n\n\\begin{figure}\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.4]{ComplexPixelRemaining}\n\\caption{A complex pixel as the one in Lemma \\ref{LemComplexPixelRemaining} must sit in a configuration as the one above. Then $\\partial X$ must either intersect the upper edge of $C$ (the upper blue dashed line), or it must intersect the right vertical edge of $C$ (the lower blue line).}\n\\label{FigComplexPixelRemaining}\n\\end{minipage}\n\\begin{minipage}{0.45\\linewidth}\n\\includegraphics[scale=0.4]{ComplexPixelRemainingGammaOutside}\n\\caption{The only case where $\\gamma_C$ may not stay inside $C$ is when $C$ sit in a configuration as the one above. In this case, the part of $\\gamma_C$ inside $D$ must belong to the red set.}\n\\label{FigComplexPixelRemainingGammaOutside}\n\\end{minipage}\n\\end{figure}\n\n\\begin{proof}\nA pixel as the one in this lemma must sit in a configuration as the one shown in Figure \\ref{FigComplexPixelRemaining}.\n\nThere are two cases: Either $\\gamma_C$ is contained in $C$ or it is not.\n\nSuppose first that $\\gamma_C\\subseteq C$. Note that $\\partial X$ must intersect the left vertical edge of $C$ once, since the endpoints of this edge has different colours. It cannot intersect the edge multiple times by Lemma \\ref{LemTwoIntersections}. Let $q_1$ be the intersection between the left vertical edge of $C$ and $\\partial X$. \n\nNow, $\\partial X$ must intersect the boundary of $C$ in at least two points, one of which is $q_1$. Suppose that $\\partial X$ intersects $C$ somewhere on the upper edge of $C$, say in a point $q_2$. Let $L$ be the line between $q_1$ and $q_2$. Then by Corollary \\ref{CorPathInSpindle} there is a path $\\pi(L)$ in $\\partial X$ from $q_1$ to $q_2$ contained in $S(L,r)$, and by changing $q_2$ if necessary, we may assume that $\\pi(L)$ does not intersect the upper edge of $C$ except at $q_2$, hence it stays inside $C$. Let $s$ be the upper left vertex of $C$.\n\nSince $B_d(s)$ contains the left and upper edge of $C$, it contains both $q_1$, $q_2$ and hence $L$. Since $d<r$, this also means that it contains $S(L,r)$ by Lemma \\ref{LemSpindleIsIntersection}, hence it contains the path $\\pi(L)$. It also contains $\\gamma_C$, which can be seen by considering the possible cases in Figure \\ref{FigCircleArcs}.\n\nNow, take any point $p$ on $\\gamma_C$. Then it belongs to a radius of $B_d(s)$. Since $\\pi(L)$ runs from one side of $C$ to another inside $C\\cap B_d(s)$, there must also be a point $q$ on $\\pi(L)$ lying on the same radius of $B_d(s)$ as $p$. But then $q$ and $p$ can be no further than $d$ apart, so the lemma is true in this case.\n\nIf on the other hand $\\gamma_C$ belongs to $C$, but there are no points of $\\partial X$ on the upper edge of $C$, then there must be a point $q_2\\in \\partial X$ on the right edge of $C$. Let $L$ be the line between $q_1$ and $q_2$. By Corollary \\ref{CorPathInSpindle}, there must be a path $\\pi(L)$ in $\\partial X$ connecting $q_1$ and $q_2$, and this path can nowhere intersect other edges of $C$.\n\nAgain pick a point $p$ in $\\gamma_C$, and look at the vertical line in $C$ containing $p$. This line must be intersected by $\\pi(L)$ in some point $q$, since $\\pi(L)$ connects the two sides of $C$. But then $p\\in \\gamma_C$ and $q\\in\\partial X$ both lie on the same line of length $d$, hence they can be no further than $d$ apart, as claimed. This concludes the proof in the case where $\\gamma_C$ is contained in $C$.\n\nFinally, assume $\\gamma_C$ is not contained in $C$. Then one of the curve segments $\\gamma_1$ and $\\gamma_2$ are not contained in $C$ - let us say it is $\\gamma_1$. Copying the results from before, we see that all points of $\\gamma_C$ inside $C$ are closer than $d$ to some point of $\\partial X$, so it remains to show this for points of $\\gamma_C$ outside $C$. Such points must lie in the set $A$ bounded by $\\gamma_1$ and one of the edges of $C$ (the red set in Figure \\ref{FigComplexPixelRemainingGammaOutside}).\n\nNotice that the only case where the curve segment $\\gamma_1$ is not contained in $C$ is when $C$ sit in a configuration as the one in Figure \\ref{FigComplexPixelRemainingGammaOutside}. Let $D$ be the pixel above $C$ in this configuration.\n\nThe boundary $\\partial X$ must intersect the boundary of $D$ at least twice in order for $D$ to be grey. By Lemma \\ref{LemTwoIntersections}, $\\partial X$ cannot intersect the right boundary of $D$ twice. Hence it must intersect the common edge $e$ of $C$ and $D$ at least once, say in a point $q$.\n\nNow, a calculation shows for any point $q'\\in e$, the ball $B_d(q')$ contains all of $A$. In particular, the ball $B_d(q)$ contains all of $A$ and hence any point of $\\gamma_C$ in $D$, so any such point can be no further away from $\\partial X$ than $d$. This concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{ThmHausdorffB}]\nThe curve $\\gamma$ consists of a curve segments $\\gamma_C$ for each pixel $C$ with two auxiliary points on its boundary. Hence the theorem follows from the Lemmas \\ref{LemSimplePixelOppositeEdges}, \\ref{LemSimplePixelVertexAdjacent}, \\ref{LemComplexPixelSameEdge}, \\ref{LemComplexPixelOppositeVertices} and \\ref{LemComplexPixelRemaining}.\n\nFurthermore, for any point $x$ in $\\gamma$, there is a point $y'$ in $\\partial X$ that is no further than a distance $d$ from $x$, meaning that\n\\begin{equation*}\n\\inf_{y\\in \\partial X} d(x,y)\\leq d(x,y')\\leq d.\n\\end{equation*}\nThus we get\n\\begin{equation*}\n\\sup_{y\\in\\gamma}\\inf_{x\\in \\partial X} d(x,y)\\leq d.\n\\end{equation*}\n\\end{proof}\n\n\\begin{cor}\\label{CorHausdorff}\nThe reconstructed boundary $\\gamma$ is closer than $d$ to the boundary of $X$.\n\\end{cor}\n\n\\begin{proof}\nCombining Theorems \\ref{ThmHausdorffA} and \\ref{ThmHausdorffB}, we get that\n\\begin{equation*}\n\\sup_{y\\in\\gamma}\\inf_{x\\in \\partial X}d(x,y)\\leq d\\text{ and }\\sup_{x\\in \\partial X}\\inf_{y\\in\\gamma} d(x,y)\\leq d,\n\\end{equation*}\nhence\n\\begin{equation*}\nd_H(\\gamma,\\partial X)=\\text{max}\\left(\\sup_{y\\in\\gamma}\\inf_{x\\in \\partial X} d(x,y),\\sup_{x\\in \\partial X}\\inf_{y\\in\\gamma} d(x,y))\\right)\\leq d.\n\\end{equation*}\n\\end{proof}\n\n{\\color{black}\\section{Homeomorphism between Object and Reconstruction}\nLet us now finish the proof of Theorem \\ref{ThmMainResult}. We need to show that there is homeomorphism taking the reconstructed set $\\Gamma$ to the original set $X$. To do so, let us start with a lemma:\n\n\\begin{lem}\\label{LemHomeomorphism}\nLet $M\\subseteq \\R^2$ be a set homeomorphic to $S^1\\times [-1,1]$, and let $m\\subseteq M$ be the subset homeomorphic to $S^1\\times\\sett{0}$. Let $\\gamma: S^1\\to M$ be a closed curve. Then there is a homeomorphism $f:\\R^2\\to\\R^2$ taking $\\gamma$ to $m$ fixing points in the unbounded component of $\\R^2\\backslash M$.\n\\end{lem}\n\n\\begin{proof}\nSince there exists a homeomorphism of $\\R^2$ taking the outer boundary component $M_o$ of $M$ to the unit circle by Schoenflies' Theorem, we may assume that $M_o$ is the unit circle. By the Annulus Theorem, the set $A$ between $M_o$ and $\\gamma$ is homeomorphic to the annulus $S^1\\times[\\frac{1}{2},1]$ - let $g$ denote this homeomorphism. We may assume that $g$ is the identity on $M_o$ - if this is not the case, then after reversing the orientation of the map $g\\vert_{M_o}$ if necessary there is an isotopy from $g(M_o)$ to $M_o$ which we may extend to an ambient isotopy of $A$ in a small tubular neighbourhood of $g(M_o)$ in $M$, and composing the result of this isotopy with $g$ we get a homeomorphism that is the identity on $M_o$.\n\nWe may continuously extend $g$ to a map $g_1$ of all of $\\R^2$ by extending it by the identity on the unbounded component of $M_o^C$ (since the map $g\\vert_\\gamma\\to g(\\gamma)$ may be extended to a map of the disc bounded by $\\gamma$) . Thus we get a map $g_1:\\R^2\\to \\R^2$ taking $\\gamma$ to $\\frac{1}{2}S^1$ and fixing points in the unbounded component of $\\R^2\\backslash M_o$.\n\nRepeating the above with $\\gamma$ replaced by $m$, we also get a map $g_2:\\R^2\\to\\R^2$ taking $S^1\\times\\sett{0}$ to $\\frac{1}{2}S^1$. Hence the composition $g_1^{-1}\\circ g_2:\\R^2\\to\\R^2$ takes $S^1\\times\\sett{0}$ to $\\gamma$ and fixes points in the unbounded component of $M_o$.\n\\end{proof}\n\n\\begin{thm}\nThere is a homeomorphism $H:\\R^2\\to\\R^2$ taking $X$ to $\\Gamma$. Hence $X$ and $\\Gamma$ are weakly $d$-similar.\n\\end{thm}\n\n\\begin{proof}\nNote that by Theorem \\ref{ThmGammaProperties} $\\gamma$ separates black pixels from white ones, and there is a 1-1 correspondence between components of $\\partial X$ and components of $\\gamma$.\n\nConsider an outermost component $\\partial X'$ of $\\partial X$. Since $\\partial X'$ is a manifold of dimension $1$, it is homeomorphic to $S^1$. Thus its tubular neighbourhood $N_{d\\sqrt{2}}(\\partial X')$ is homeomorphic to $S^1\\times [-1,1]$ (see \\cite{TC}, Proposition A.10) via a map $h$ that takes the points of each normal line of length $2\\sqrt{2}$ to a fiber $\\sett{x}\\times[-1,1]$ in $S^1\\times[-1,1]$, and takes $\\partial X'$ to $S^1\\times\\sett{0}$. Moreover, $h^{-1}(S^1\\times\\sett{-1})$ is a subset of the set of white pixels, and $h^{-1}(S^1\\times\\sett{-1})$ is a subset of the black pixels. Since the boundary $\\gamma$ of the reconstructed set $\\Gamma$ separates black and white pixels, this means that there is a component $\\gamma'$ of $\\gamma$ in $N_{d\\sqrt{2}}(\\partial X')$.\n\nThen by Lemma \\ref{LemHomeomorphism} there is a homeomorphism $f_1:\\R^2\\to R^2$ taking $\\gamma'$ to $\\partial X'$ and fixing points in the unbounded component of $h^{-1}(S^1\\times\\sett{1})$. Since $\\partial X'$ and $\\gamma'$ both separates black pixels from white, any component of $\\partial X$ inside $\\partial X'$ also lies inside $\\gamma'$. Hence $f_1$ also takes any component of $\\partial X$ inside $\\partial X'$ to the inside of $\\gamma'$.\n\nApplying the above technique to the other components of $\\partial X$, we thus get a series of homeomorphisms $f_1,f_2,\\dots, f_n$ that each takes one component $\\partial X_i$ of $\\partial X$ to a component $\\gamma_i$ of $\\gamma$. Since each homeomorphism fixes the points of the unbounded component of $\\partial X_i^C$, the composition $f_n\\circ\\dots\\circ f_1:\\R^2\\to \\R^2$ that starts by mapping the outer component(s) of $\\partial X$ to $\\gamma$ and then works its way in, sends $\\partial X$ to $\\gamma$. Since it also sends bounded sets to bounded sets and $\\Gamma$ was the bounded set bounded by $\\gamma$ and $X$ was compact, this means that $H:=f_n\\circ\\dots\\circ f_1$ takes $X$ to $\\Gamma$.\n\nSince there is a map of $\\R^2$ taking $X$ to $\\Gamma$, and since $d_H(\\partial X,\\gamma)\\leq d$ by Corollary \\ref{CorHausdorff}, they are weakly $d$-similar.\n\\end{proof}\n}\n\n\\section{Example of the reconstruction algorithm}\n\\begin{example}\nAn example of this algorithm is shown in Figure \\ref{FigExampleAlgorithm}. In this figure, we used the bump function\n\\begin{equation*}\n\\phi(t)=\\begin{cases}\n1 & \\text{if $x=0$}\\\\\n0 & \\text{if $x=1$}\\\\\n1-\\frac{1}{1+exp(\\frac{6}{7x}-\\frac{6}{7-7x})} & \\text{otherwise}\n\\end{cases}.\n\\end{equation*}\nIt seems from our example that the curves $\\partial X$ and $\\gamma$ may be even closer than $d$.\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{ExampleV2}\n\\caption{Example of the algorithm: The thin black line is the outline of the original $r$-regular set, from which the image came. The yellow line is the reconstructed boundary of the original set.}\n\\label{FigExampleAlgorithm}\n\\end{figure}\n\\end{example}\n\n\\section{Conclusion}\nIn this paper we have presented restrictions on pixel configurations in digital images of $r$-regular objects at a reasonable resolution. We have used these restrictions to reconstruct the original object by constructing an object with smooth boundary that is {\\color{black}weakly $d$-similar to \nthe original object (where $d$ is the side length of each pixel). This tells us that our reconstruction is not far from the original object {\\color{black}and has the right topology, and though that ensures that the sets are not fundamentally different}, sadly {\\color{black}it is not quite enough to cover all aspects of human perception of similarity, as discussed in \\cite{SK}, \\cite{SLS}.\nOngoing work is aiming at showing that the reconstructed object is in fact strongly $s$-similar to the original one for a suitable $s$.\n\n\nWe do by no means believe that a Hausdorff distance of $d$ between the original object and our reconstructed object is the optimal  - in fact, we are working on obtaining even stronger bounds on their Hausdorff difference. We also believe that taking the actual intensities of each pixel into account can result in even more precise reconstruction, though there is still a lot of work to be done before we are ready to prove this.\n\nThe object that our reconstruction method outputs will in general not be $r$-regular. We may, since the boundary of the reconstructed set consists of the join of finitely many curve segments, calculate the maximal curvature of a reconstructed set - but note that a maximal curvature or $\\frac{1}{s}$ is not enough to ensure that our reconstructed set is $s$-regular. Thus we have left the question of regularity of the reconstruction out of this paper, though it is also be an interesting aspect of the reconstruction process.\n\\section{Proofs of the lemmas from Section \\ref{SecConfigurations}}\nWe here include the proofs that were omitted in Section \\ref{SecConfigurations}. To lighten the notation, we will measure distances in units of $d$, so that each grid square has side length 1, and the assumption $r>d\\sqrt{2}$ becomes $r>\\sqrt{2}$.\n\n\n\\begin{proof}[Proof of Lemma \\ref{LemTwoIntersections}]\n  Let $x$ and $y$ be two points on the common edge of $B$ and $C$ that belongs to $\\partial X$, and let $L$ be the line segment\n  between them. Note that since $X$ is $r$-regular and $r>\\sqrt{2}$,\n  $X$ is in particular $\\sqrt{2}$-regular (cf. \\cite{TC}, Proposition\n  A.2).\n\n  Since the distance from $x$ to $y$ is less that $\\sqrt{2}$, there\n  must be a path $\\pi(L)$ in $\\partial X$ between them, where $\\pi$ is the\n  projection onto $\\partial X$, see Section \\ref{SecRBasics}. Since the\n  projection is continuous and fixed at the endpoints, there must be\n  a point $p$ on $\\pi(L)$ such that the tangent to $\\partial X$ at $p$ is\n  horisontal. Let $p=(p_1,p_2)$. Since $p\\in\\partial X$ and $X$ is an\n  $\\sqrt{2}$-regular set, there are balls\n  $B_{\\sqrt{2}}(x_b)\\subseteq X$ and $B_{\\sqrt{2}}(x_w)\\subseteq X^c$\n  such that\n  $\\overline{B_{\\sqrt{2}}(x_b)}\\cap\\overline{B_{\\sqrt{2}}(x_w)}=\\sett{p}$,\n  and since the tangent to $\\partial X$ at $p$ is horisontal, the centres\n  $x_b$ and $x_w$ must lie on the vertical line through $p$.\n\n  Note that $p\\in\\pi(L)\\subseteq S(L,\\sqrt{2})$. By Lemma\n  \\ref{LemSpindleWidth}, the thickness of $S(L,\\sqrt{2})$ is\n  $\\sqrt{2}-\\sqrt{2-\\frac{L^2}{4}}\\leq\n  \\sqrt{2}-\\sqrt{2-\\frac{1}{4}}$. So $d(p, L)\\leq \\sqrt{2}-\\sqrt{2-\\frac{1}{4}}$. Then \n  \\begin{equation*}\nd(x_b,L)\\leq d(x_b,p)+d(p,L)\\leq 2\\sqrt{2}-\\sqrt{2-\\frac{1}{4}}<1.51\n  \\end{equation*}\n  and\n  \\begin{equation*}\nd(x_b,L)> d(p,L)-d(p,x_b)>\\sqrt{2}-\\sqrt{2-\\frac{1}{4}}-\\sqrt{2}=\\sqrt{2-\\frac{1}{4}}>1\n  \\end{equation*}\n  So $x_b$ belongs to either $A$ or $D$ - let us say $D$. Then $D$ must be black. In fact, since the first\n  and last inequality are sharp, the common edge of $B$ and $C$ must\n  be interior points of $X$, and hence it contains no intersection\n  points. A symmetric argument for $x_w$ shows that $x_w$ must belong to $A$, hence\n  $A$ must be white, and that the common edge of $A$ and $B$ cannot\n  contain any points of $\\partial X$.\n  \n  If $\\partial X$ is tangent to $L$ at a point $p'$, replacing $p$ with $p'$ in the above argument shows the result.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{LemYieldsTwoIntersections}]\nLet us name the two grey pixels in the configuration $B$ and $C$, as in the right part of Figure \\ref{FigYieldsTwoIntersections}. Choose boundary points $x_C\\in C$ and $x_D\\in D$. Both of these points are contained in a ball $B_{\\sqrt{5}\/2}(p)$, where $p$ is the midpoint of the common edge $e$ of pixel $B$ and $C$. By Corollary \\ref{CorPathInSpindle} and Lemma \\ref{LemSpindleIsIntersection} applied to the projection $\\pi$ in stead of $\\rho$, there is a path $\\gamma$ in $\\partial X$ from $x_C$ to $x_D$ contained in $B_{\\sqrt{5}\/2}(p)$. This path must pass the line containing $e$, and since it cannot do so if passing this line means entering a black pixel, it must in fact pass the edge $e$. The endpoints of $e$ are both black, hence if $\\partial X$ passes $e$ once, it must also pass $e$ a second time, as $\\partial X$ separates black points from white ones.\n\nBut then by Lemma \\ref{LemTwoIntersections} $A$ must be black and $D$ must be white (it cannot be the other way around, because then a black and a white pixel would share a corner, meaning that $\\partial X$ passes through that corner - which is against our assumptions).\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{LemNo9greys}]\nLet $c$ be the centre point of the grey pixel $C$. Assume $c\\in X$ (the other case is similar). Then $c$ belongs to a black ball of radius $\\sqrt{2}$ by Proposition \\ref{PropEquivalentRregDefinitions}, hence the centre of this black ball belongs to $B_{\\sqrt{2}}(c)$ and thus to either $C$ or one of its neighbours. Since the pixel containing the centre of the black ball must be entirely contained in the black ball, said pixel must be black. Hence one of the neighbours of $C$ must be black.\n\\end{proof}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{5greys}\n\\caption{We aim to show that the centre of the white ball tangent to $\\partial X$ at $p$ belongs to the red set $Y$.}\n\\label{Fig5greys}\n\\end{figure}\n\n\\begin{proof}[Proof of Lemma \\ref{Thm5greys}]\nPlace the configuration in a coordinate system as in the figure,\n  such that the pixels has side length 1. The aim will be to show that\n  $x_w$ lies so close to the upper right pixel that the white ball\n  $B_r(x_w)$ contains the pixel.\n\n  Note first that the line $l$ through $x_b$ and $c$ passes an edge of\n  the black pixel, say the right edge. Let $a$ be this intersection\n  point, see Figure~\\ref{Fig5greys}. To study the limit case, we\n  will first assume that $x_b=a$.\n\n  Since $d(a,c)<r$, then rotating the line segment from $a$ to $c$\n  about $c$ with an angle $\\pi$ one sees that the line segment of\n  length $2r$ from $a$ through $c$ has an endpoint $x_w=(x_1,x_2)$\n  with $x_1\\geq 0$.\n\n  Let $b$ be a point directly above $a$ such that\n  $\\angle abc=\\frac{\\pi}{2}$, and let $d$ be a point directly above\n  $a$ such that $\\angle adx_w=\\frac{\\pi}{2}$, see\n  Figure~\\ref{Fig5greysAdd}. Finally, let\n  $t=d(a,b)-\\frac{1}{2}\\geq 0$.\n\n  Now,\n  $\\tan(\\angle bac)=\\frac{d(b,c)}{d(a,b)}=\\frac{1\/2}{1\/2+t}\\leq 1$, so\n  $\\angle bac\\leq\\frac{\\pi}{4}$. Thus we have that\n  $\\frac{d(d,x_w)}{2\\sqrt{2}}=\\sin(\\angle bac)\\leq\n  sin\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\sqrt{2}}$, so\n  $x_1=d(d,x_w)-1\\leq 1$. So $0\\leq x_1\\leq1$. Also, by rotational\n  symmetry and since $d(a,c)\\leq r$, we must have $x_2\\geq -1+t$.\n\n  Let $Y$ be the intersection of the four $r$-balls with centres\n  $(0,0)$, $(1,0)$, $(0,-1)$ and $(1,-1)$ (the corners of the upper\n  right pixel) - this is the red set in Figure~ \\ref{FigCloseup5greys}. Then\n  $Y$ is a convex set containing $(0,0)$, $(1,0)$, $(0,-1)$ and\n  $(1,-1)$, and hence the entire upper right pixel. Any\n  point in $Y$ is closer than $r$ to all the points $(0,0)$, $(1,0)$,\n  $(0,-1)$ and $(1,-1)$, so an $r$-ball with centre in $Y$ contains\n  all of the upper right pixel. Hence we aim to show that $x_w\\in Y$.\n\n  Notice that the two triangles $abc$ and $adx_w$ are\n  equiangular. Hence\n  \\begin{equation*}\n    \\frac{1+x_1}{2\\sqrt{2}}=\\frac{\\frac{1}{2}}{d(a,c)}=\\frac{\\frac{1}{2}}{\\sqrt{(t+\\frac{1}{2})^2+\\frac{1}{4}}}\\Rightarrow t=\\sqrt{\\frac{2}{(x_1+1)^2}-\\frac{1}{4}}-\\frac{1}{2}.\n  \\end{equation*}\n  Furthermore, we have that\n  \\begin{equation*}\n    8=(2+t+x_2)^2+(1+x_1)^2,\n  \\end{equation*}\n  so\n  \\begin{equation*}\n    x_2=\\sqrt{8-(1+x_1)^2}-2-t=\\sqrt{8-(1+x_1)^2}-\\frac{3}{2}-\\sqrt{\\frac{2}{(x_1+1)^2}-\\frac{1}{4}}\n  \\end{equation*}\n  So we may express $x_2$ as a function of $x_1$. Since\n  $0\\leq x_1\\leq 1$ and $x_2\\geq -1$, we need only check that $x_w$\n  lies under the upper border of $Y$ on the interval $[0,1]$, i.e. we\n  must check that $x_2(x_1)$ lies under the function\n  \\begin{equation*}\n    \\tilde{f}(x_1)=\\begin{cases}\n      \\sqrt{2-(x_1-1)^2}-1 & x_1\\leq\\frac{1}{2},\n      \\\\\n      \\sqrt{2-x_1^2)}-1 & x_1\\geq\\frac{1}{2},\n    \\end{cases}\n  \\end{equation*}\n  on $[0,1]$. However, it turns out to be easier to check that\n  $x_2(x_1)$ lies under the graph of the function\n  \\begin{equation*}\n    f(x_1)=\\begin{cases}\n      (\\sqrt{7}-\\frac{1}{2})x_1 & 0\\leq x_1\\leq\\frac{1}{2},\n      \\\\\n      -(\\sqrt{7}-\\frac{1}{2})x_1+(\\sqrt{7}-\\frac{1}{2}) & \\frac{1}{2}\\leq x_1\\leq1.\n    \\end{cases}\n  \\end{equation*}\n  Note that $f([0,1])\\subseteq Y$, since the image of $f$ is just the\n  union of two line segments, both of which have endpoints in the\n  convex set $Y$.\n\n  To show that $x_2$ lies somewhere below $f$, note first that\n  \\begin{align*}\n    \\frac{d^2}{dx_1^2}x_2={} &-\\sqrt{8-(x_1+1)^2}-\\frac{(x_1+1)^2}{\\sqrt{8-(x_1+1)^2}^3}\\\\\n    &+\\frac{4}{\\sqrt{2(x_1+1)^2-\\frac{1}{4}(x_1+1)^4}^3}-\\frac{6}{\\sqrt{2(x_1+1)^6-\\frac{1}{4}(x_1+1)^8}}\\\\\n    \\leq {} & \\frac{4}{\\sqrt{2(x_1+1)^2-\\frac{1}{4}(x_1+1)^4}^3}-\\frac{6}{\\sqrt{2(x_1+1)^6-\\frac{1}{4}(x_1+1)^8}}\\\\\n    = {} &\\frac{1}{2(x_1+1)^3}\\left( \\frac{3x_1^2+6x_1-13}{\\sqrt{2(x_1+1)^2-\\frac{1}{4}(x_1+1)^4}^3}\\right)\\\\\n    \\leq {}  & 0,\n  \\end{align*}\n  where the first inequality comes from the fact that the first two\n  terms of the derivative is negative, and the last inequality comes\n  from observing that $3x_1^2+6x_1-13\\leq 0$ on $[0,1]$.\n\n  Now, we want to show that $f-x_2\\geq 0$. Note that since $f$ is\n  (piecewise) linear, we get that\n  \\begin{equation*}\n    \\frac{d^2}{dx_1^2}(f-x_2)=\\frac{d^2}{dx_1^2}(-x_2)\\geq 0\n  \\end{equation*}\n  on $[0,1\/2]$ and $[1\/2,1]$, so $\\frac{d}{dx_1}(f-x_2)$ is\n  increasing. Now,\n  \\begin{equation*}\n    \\frac{d}{dx_1}(x_2)=-\\frac{1+x_1}{\\sqrt{8-(1+x_1)^2}}+\\frac{2}{\\sqrt{\\frac{2}{(1+x_1)^2}-\\frac{1}{4}}}\\frac{1}{(1+x_1)^3},\n  \\end{equation*}\n  so on $[0,1\/2]$\n  \\begin{align*}\n    \\frac{d}{dx_1}(f-x_2)&=\\sqrt{7}-\\frac{1}{2}+\\frac{1+x_1}{\\sqrt{8-(1+x_1)^2}}-\\frac{2}{\\sqrt{\\frac{2}{(1+x_1)^2}-\\frac{1}{4}}}\\frac{1}{(1+x_1)^3}\n    \\\\\n    &\\geq \\frac{d}{dx_1}(f-x_2)\\vert_{x_1=0}\n    \\\\\n    &=\\sqrt{7}-\\frac{1}{2}+\\frac{1}{\\sqrt{7}}-\\frac{4}{\\sqrt{7}}\n    \\\\\n    &>0,\n  \\end{align*}\n  and on $[1\/2,1]$,\n  \\begin{align*}\n    \\frac{d}{dx_1}(f-x_2)&=-\\sqrt{7}+\\frac{1}{2}+\\frac{1+x_1}{\\sqrt{8-(1+x_1)^2}}-\\frac{2}{\\sqrt{\\frac{2}{(1+x_1)^2}-\\frac{1}{4}}}\\frac{1}{(1+x_1)^3}\\\\\n    &\\leq \\frac{d}{dx_1}(f-x_2)\\vert_{x_1=1}\\\\\n    &=-\\sqrt{7}+\\frac{1}{2}+{1}-\\frac{1}{2}\\\\\n    &<0.\n  \\end{align*}\n  So $f-x_2$ is increasing on $[0,1\/2]$ and decreasing on\n  $[1\/2,1]$. Hence, on $[0,1\/2]$\n  \\begin{equation*}\n    (f-x_2)(x_1)\\geq (f-x_2)(0)=-\\frac{\\sqrt{7}}{2}+\\frac{3}{2}>0\n  \\end{equation*}\n  and similarly, on $[1\/2,1]$\n  \\begin{equation*}\n    (f-x_2)(x_1)\\geq (f-x_2)(1)=0.\n  \\end{equation*}\n  Putting the last two equations together we see that\n  $f(x_1)-x_2(x_1)\\geq 0$ everywhere on $[0,1]$, so $x_2\\leq f$ as\n  claimed. So if $x_b=a$, then $x_w$ belongs to the set $Y$.\n\n  \\begin{figure}\n    \\includegraphics[scale=0.5]{Closeup5greys}\n    \\caption{Close-up on the upper right pixel. The red graph is the\n      graph of $x_2(x_1)$, and the blue graph is the graph of\n      $f(x_1)$. The point $(x_1,x_2)$ on $l$ is chosen such that\n      $d((x_1,x_2),a)=2r$, and hence $x_b$ lies closer to $c$ than\n      $(x_1,x_2)$ does.}\n    \\label{FigCloseup5greys}\n  \\end{figure}\n\n  Suppose now that $x_b$ is just any point in the lower left pixel\n  that is closer than $\\sqrt{2}$ to $c$, and suppose that the line $l$\n  through $x_b$ and $c$ leaves the lower left pixel in a point $a$ on\n  the right pixel edge.  Then $(0,1-t)$ lies on $l$ and inside $Y$. By\n  what we just showed, the point $(x_1,x_2)$, $x_1\\geq0$, on $l$ that\n  is at a distance $2r$ from $a$ is also in $Y$, hence the entire line\n  segment from $(0,1-t)$ to $(x_1,x_2)$ is in $Y$, since $Y$ was\n  convex, see Figure~\\ref{FigCloseup5greys}.  But noticing that\n  $r\\geq d(a,c)=d(c,(0,1-t))$, we get that\n  \\begin{equation*}\n    d(x_b,(0,1-t))=d(x_b,c)+d(a,c)\\leq 2r\n  \\end{equation*}\n  and\n  \\begin{equation*}\n    d(x_b,(x_1,x_2))=d(x_b,a)+d(a,x_w)\\geq d(a,x_w)=2r.\n  \\end{equation*}\n  Combining these equations, we see that\n  $d(x_b,(0,1-t))\\leq d(x_b,x_w)\\leq d(x_b,(x_1,x_2))$. Hence $x_w$\n  belongs to the line segment between $(0,1-t)$ and $(x_1,x_2)$ which\n  was contained in $Y$, so $x_w\\in Y$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{LemSkalfarvessorte1}]\nLet us show $(i)$. Let $x$ and $y$ be\n  corner points of the two pixels as in Figure\n  \\ref{FigSkalfarvessorte1}, and let $L$ denote the line between them.\n\n  If there are points of $ X^C$ in $L$, then $\\partial X$ must either be tangent to $X$ or intersect $L$ in several points (since the endpoints of $L$ clearly all\n  belong to $X$). By Lemma \\ref{LemTwoIntersections} this means that\n either the pixel above $C$ or the pixel below pixel $B$ is white. Both of these pixels share a corner with a black pixel. But by the proof of Lemma \\ref{LemTwoIntersections}, the black corner point must be an interior point of $X^C$ and hence white - a contradiction.\nSo $L\\subseteq\\text{Int} (X)$\n\n  If $B$ is not black, pick white points $b\\in \\text{Int} (B)$ and\n  $c\\in\\text{Int} (C)$. Let $L_{bc}$ denote the line between them. Then\n  there is a path $\\rho_{X^C}(L_{bc})$ in $X^C\\cup\\partial X$ connecting\n  $b$ and $c$, and this path belongs to all balls of radius less than\n  $r$ that contains both $b$ and $c$, (cf. Section\n  \\ref{SecRBasics}). In particular,\n  $\\rho_{X^C}(L_{bc})\\subseteq {B_{\\sqrt{2}}(x)}$, since this ball\n  contains all of $B$ and all of $C$.\n\n  Let $\\gamma$ be the piecewise linear path from $z$ though $x$ and\n  $y$ to $w$. Then $\\gamma$ is contained in $\\text{Int}(X)$ and separates\n  $B$ from $C$ inside $B_{\\sqrt{2}}(x)$. Hence $\\rho_{X^C}(L_{bc})$\n  must intersect $\\gamma$ somewhere, but this is impossible, since\n  $\\gamma\\subseteq\\text{Int}(X)$ and\n  $\\rho_{X^C}(L_{bc})\\subseteq X^C\\cup\\partial X$. So $B$ cannot contain\n  any white points, and hence it must be black.\n\n  A similar reasoning can be applied to $A$: If $A$ is not black, pick\n  a white point $a\\in A$, and let $L_{ac}$ denote the line between $a$\n  and $c$ (where $c\\in C$ is the point we chose earlier). Then there\n  is a path $\\rho_{X^C}(L_{ac})$ in $X^C\\cup\\partial X$ connecting $a$ and\n  $c$, and this path must belong to the ball $B_{\\sqrt{2}}(x)$. But\n  since $\\gamma$ also separates $A$ from $C$ inside\n  $B_{\\sqrt{2}}(x)$, $\\rho_{X^C}(L_{ac})$ must intersect $\\gamma$\n  somewhere. But this is impossible, since $\\gamma\\subseteq\\text{Int}(X)$\n  and $\\rho_{X^C}(L_{ac})\\subseteq X^C\\cup\\partial X$ as before. So $A$\n  cannot contain any white points, and hence it must also be black.\n  \n  The second part of this proof also proves $(ii)$. To prove $(iii)$, we apply Lemma \\ref{Thm5greys} to argue that one of the pixels $A_1$, $A_2$, $A_3$, $B_1$, $B_2$, $B_3$ is black.\n  \nIndeed, suppose none of the pixels $A_1$, $A_2$, $A_3$, $B_1$, $B_2$, $B_3$ were black. Then $A_1$, $A_3$, $B_1$ and $B_3$ would have to be grey, since black and white pixels cannot be neighbours by assumption. But then by Remark \\ref{Rem3x3Pix}, either $A_2$ or $B_2$ would have to be black - a contradiction. So one of the pixels has to be black.\n   If $A_1$, $A_3$, $B_1$ or $B_3$ is black, we are in situation $(i)$ and may use this proof to complete the proof of $(iii)$. If $A_2$ is black, and neither $A_1$ or $A_3$ is black Lemma \\ref{LemYieldsTwoIntersections} shows that both $B_1$ and $B_3$ is black, and we are again in the situation of case $(i)$. If $A_2$ and either $A_1$ or $A_3$ is black, we are in the situation of case $(i)$. The proof works equivalently if $B_2$ is black. So $(iii)$ is true when one of the pixels $A_1$, $A_2$, $A_3$, $B_1$, $B_2$, $B_3$ are black.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{subfigure}{0.3\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.4]{6greysallnamed2}\n    \\caption{We are considering 6 grey pixels in a $2\\times3$\n      combination, and we wish to show that $\\partial X\\cap(G_1\\cup G_2)$\n      belongs to the red set in the figure, and that one of the pixels\n      $K_1$, $K_2$ must be black, and the other one white.}\n    \\label{Fig6greysAdd}\n  \\end{subfigure}\\hspace{0.5cm}\n  \\begin{subfigure}{0.3\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.4]{6greys2spindles2}\n    \\caption{If both $K_1$ and $K_2$ were white, then points $x_a$\n      and $x_c$ would be joined by path in $X^C\\cup \\partial X$, and so\n      would points $x_b$, $x_d$ (these are the white paths in the\n      figure).}\n    \\label{Fig6greys2spindlesAdd}\n  \\end{subfigure}\n  \\hspace{0.5cm}\n  \\begin{subfigure}{0.3\\linewidth}\n    \\includegraphics[scale=0.4]{6greysWithSpindles2}\n    \\caption{The projection $\\pi$ yields a path $\\gamma$ in\n      $\\partial X$ from $p_{34}$ through $p_{12}$ to $p_{56}$, and this\n      path lives inside the spindles between points $p_{34}$ and\n      $p_{12}$, and points $p_{12}$ and $p_{56}$.}\n    \\label{Fig6greysWithSpindlesAdd}\n  \\end{subfigure}\n\\end{figure}\n\n\\begin{proof}[Proof of Lemma \\ref{Lem2x3grey}]\n  Let us start by discussing (i).\n\n  Consider $G_1$ as in Figure \\ref{Fig6greys2spindlesAdd}, and look at the configuration of $3\\times3$ pixel with $G_1$ as the centre pixel. Then all but the three upper neighbours of $G_1$ are grey. By Remark \\ref{Rem3x3Pix}, the upper d-neighbour $K_1$ of $G_1$ cannot be grey, hence it must be either black or white. By the same reasoning, $K_2$ must be either black or white\n\n\n  It remains to prove that $K_1$ and $K_2$ cannot have the same\n  colour, so suppose that $K_1$, $K_2$ are both white. Let $x_a$,\n  $x_b$ be the lower corners of $K_1$ and $x_c$, $x_d$ the upper\n  corners of $K_2$, as in Figure~\\ref{Fig6greys2spindlesAdd}. Note\n  that these points are all elements of $X^C\\cup\\partial X$.\n\n  Let $L_{ac}$ be the line segment between $x_a$ and $x_c$, and let\n  $L_{bd}$ be the line segment between $x_b$ and $x_d$. Since\n  $d(x_a,x_c)=d(x_b,x_d)=2<2\\sqrt{2}$, the map $\\rho$ maps these line\n  segments to continuous paths in $X^C\\cup\\partial X$ by projecting points\n  of $\\text{Int}(X)$ to $\\partial X$ and fixing all other points.\n\n  Now, $L_{ac}$ and $L_{bd}$ cannot both be contained entirely in\n  $\\overline{X^C}$, since this would imply that $\\rho_{X^C}$ kept them\n  fixed. But since $G_1$, $G_2$ were grey, they must contain a point\n  of $\\text{Int}(X)$ which in turn would belong to some black $r$-ball\n  $B_r(x)\\subseteq X$. However, an interior point of such an $r$-ball would\n  have to intersect the boundary of $G_1\\cup G_2$, which hence cannot\n  be a subset of $X^C\\cup\\partial X$.\n\n  So assume that $\\rho_{X^C}$ does not fix $L_{ac}$. Then there is a\n  point $q$ on $\\rho_{X^C}(L_{ac})$ that is furthest from $L_{ac}$ and hence is a point on $\\partial X$ with a vertical tangent. This point must\n  belong to $S(L_{ac},r)$ since all of $\\rho_{X^C}(L_{ac})$ does, and there\n  must be a black and a white ball that are tangent to $\\partial X$ at\n  $q$. Since the thickness of $S(L_{ac},r)\\leq\\sqrt{2}-1$, we must have $d(q,L_{ac})\\leq\\frac{1}{2}$. But then the centre $x$ of the left $\\sqrt{2}$-ball tangent to $\\partial X$ at $q$ must satisfy \n  \\begin{equation*}\n  1<d(x,q)-d(q,L_{ac})\\leq d(x,L_{ac})\\leq d(x,q)+d(q,L_{ac})<2\\sqrt{2}-1\n  \\end{equation*}  \n   So the centre $x$ of a white or black $\\sqrt{2}$-ball belongs to one of the grey pixels $G_5$, $G_6$ left of $L_{ac}$, which is\n  impossible, since that pixel would thus be entirely contained in the ball, and hence not grey. So we cannot have that both $K_1$ and $K_2$ have the\n  same colour, completing the proof of (i).\n\n  For (ii), let $N$ be the line separating the upper three grey pixels\n  from the lower three grey pixels.  We wish to prove that $\\partial X$\n  intersects $N$ at least two times inside the four leftmost grey pixels,\n  and also at least two times in the four rightmost grey pixels.\n\n  Let $L_{ac}$ be the vertical line separating $G_3$ and $G_4$ from the\n  other grey pixels, and similarly, let $L_{bd}$ be the vertical line\n  separating $G_5$ and $G_6$ from the others. Pick a boundary point\n  $x_i$ in each of the grey pixels $G_i$, $i=1,\\dots, 6$, and let\n  $L_{ij}$ be the line segment joining $x_i$ and $x_j$,\n  $i,j=1,\\dots,6$.\n\n  Using the projection $\\pi:N_r\\to\\partial X$, we know that there is a\n  path $\\pi(L_{12})$ in $\\partial X$ from $x_1$ to $x_2$, and this path\n  must necessarily cross $N$ somewhere. If it passes $L_{ac}$ on the way,\n  it must do so at least twice. By an argument similar to the one we\n  used to prove part (i), there must then be a boundary point $q$ with\n  vertical tangent line. Since $q$ belongs to $S(L_{12},r)$, the right\n  osculating ball at $q$ must belong to either $G_5$ or $G_6$, which\n  yields a contradiction. Hence $\\pi(L_{12})$ does not intersect\n  $G_L$, and by a symmetric argument, it does not intersect $L_{bd}$\n  either. So it must intersect $N$ at a point $p_{12}$ on the common\n  edge of $G_1$ and $G_2$.\n\n  A similar argument shows that the line segments $\\pi(L_{34})$ must intersect $N$ in a point $p_{34}$ the common edge\n  of $G_3$ and $G_4$, and hat $\\pi(L_{56})$ intersects $N$ in a point $p_{56}$ the common edge of $G_5$\n  and $G_6$, respectively.\n\n  So we have three points of $\\partial X$ on $N$. Using the projection on the\n  line segments between them, we get a path $\\gamma$ in $\\partial X$ from\n  $p_{34}$ through $p_{12}$ to $p_{56}$, and this path must live\n  inside the spindles $S(\\sqrt{2},p_{12},p_{34})$, and\n  $S(\\sqrt{2},p_{12},p_{56})$, see\n  Figure~\\ref{Fig6greysWithSpindlesAdd}.\n\n  Since the maximum height of such a spindle is $\\sqrt{2}-1$,\n  then $\\gamma$ must belong to the red part of the pixels\n  $G_1\\cup G_2$. Note that there cannot be any other elements of\n  $\\partial X$ in $G_1\\cup G_2$ than those of $\\gamma$, for suppose $y_1$\n  was a point in $G_1\\cap \\partial X$ that did not belong to $\\gamma$, and\n  let $y_2$ be a point on $\\gamma$ on the same vertical line as\n  $y_1$. Let $L_y$ be the line segment between $y_1$ and $y_2$. Since\n  $y_1$ and $y_2$ would be closer than $2r$ to each other, they would\n  be connected through the path $\\pi(L_y)$, and by an argument similar\n  to the former ones, there would be a point $y$ on $\\pi(L_y)$ where\n  $X$ has horisontal normal vector. Suppose $y$ is located to the left\n  of the vertical line through the centre of $G_1$. There is a black\n  and a white ball that are tangent to $\\partial X$ at $y$, and one of\n  them would have a centre $x$ at a distance $\\sqrt{2}$ to the right\n  of $y$. But since $y$ belonged to the left half of the\n  configuration, this means that the centre $x$ would belong to one of\n  the grey pixels $G_1$, $G_2$, $G_5$ or $G_6$ - which would give a\n  contradiction.\n\n  So since the path $\\gamma$ belongs to the red part of $G_1\\cup G_2$,\n  the proof is complete.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{LemImpossible2x3}]\nSuppose this configuration did occur, and look at the $3\\times3$-configuration that also includes the three pixels to the right of it (see Figure \\ref{Fig2x3Impossible}, right). Not all of the red pixels can be grey, since this would violate Lemma \\ref{Thm5greys} and Lemma \\ref{LemNo9greys}. Hence one of them must be another colour, say black.\n\nIf the upper red pixel were black, Lemma \\ref{LemYieldsTwoIntersections} would require the bottom grey pixel to be white - a contradiction.\n\nIf the middle red pixel were black, then by the first part of Lemma \\ref{LemSkalfarvessorte1} one of the grey ones in the middle column of the configuration would be so, too - a contradiction again.\n\nIf the bottom red pixel were black, then by the third part of Lemma \\ref{LemSkalfarvessorte1} one of the grey ones in the middle column of the configuration would be so, too - yet another contradiction.\n\nHence there can be no legal way to colour the red pixels, so this configuration cannot occur.\n\\end{proof}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{Impossible3x3}\\hfill\n\\includegraphics[scale=0.4]{Impossible3x3A}\\hfill\n\\includegraphics[scale=0.4]{Impossible3x3B}\n\\caption{The configuration to the left in this figure cannot occur in the image of an $r$-regular object with $d\\sqrt{2}<r$. As a first part of the proof, we aim to show that the configuration is part of a larger configuration looking like the one on the right.}\n\\label{FigImpossible3x3}\n\\end{figure}\n\n\\begin{proof}[Proof of Lemma \\ref{LemTrickyConfiguration}]\nBefore we go on to the proof, we will state and prove the following lemma for later use:\n\\begin{lem}\\label{LemCircleIntersections}\nLet $A\\subseteq\\R^2$ be a convex polygon, $r>0$. The intersection of all $r$-balls centred in $A$ is equal to the intersection of all $r$-balls centred at the vertices of $A$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to show the theorem for lines, since if it holds for lines, it holds for any edge of $A$, and hence for all of $A$ by convexity.\n\nSo let $A$ be a line with endpoints $(x_1,0)$, $(x_2,0)$. Let $(p_1,p_2)\\in B_r((x_1,0))\\cap B_r((x_2,0))$ and $(x,0)\\in A$. Assume $p_1\\leq x$ - the other case is symmetrical. Then $\\norm{(x,0)-(p_1,p_2)}^2=(x-p_1)^2+p_2^2<(x_2-p_1)^2+p_2^2=\\norm{(x_2,0)-(p_1,p_2)}^2<r^2$, because $x_2$ must be further from $p_1$ than $x$ when $p_1<x<x_2$. But this means that $B_r((x_1,0))\\cap B_r((x_2,0))\\subseteq B_r((x,0))$, so $B_r((x_1,0))\\cap B_r((x_2,0))\\cap B_r((x,0))=B_r((x_1,0))\\cap B_r((x_2,0))$.\n\\end{proof}\n\nNow we turn to the proof of Lemma \\ref{LemTrickyConfiguration}.\n\nSuppose the configuration did occur. Let us first argue that then it must be a part of a larger configuration looking like the one in Figure \\ref{FigImpossible3x3}, right.\n\nSince the top 6 pixels of the configuration to the left in the figure are grey and the lower middle one is white, the middle pixel just above the configuration must be black by Lemma \\ref{Lem2x3grey}, as indicated in Figure \\ref{FigImpossible3x3} middle.\n\nSimilarly, look at the left neighbour configuration of the one in the figure (this is the one with the blue frame in Figure \\ref{FigImpossible3x3}, middle). This configuration has a grey middle and a white corner. Combining Remark \\ref{Rem3x3Pix} and Lemma \\ref{LemSkalfarvessorte1}, this means that either the top left or the middle left pixel (the two darker red pixels in the figure) must also be black. The same is true for the right neighbour configuration. By Lemma \\ref{LemSkalfarvessorte1} $(i)$ and $(iii)$, if one of the two dark red pixels to the left is black, then the upper dark red pixels must be black, and so must the 4 red pixels in the top row in Figure \\ref{FigImpossible3x3}, too. So if the configuration did occur in a digital image, it would have to sit in a configuration like the one in Figure \\ref{FigImpossible3x3}, right.\n\nNow, consider the two red points $p_1$ and $p_2$ at the top corners of the centre pixel in Figure \\ref{FigImpossible3x3} right. These cannot be black: If they were, then $\\partial X$ must intersect one of the edges of the top left and right grey pixel at least twice, which would violate Lemma \\ref{LemTwoIntersections}. So they must both be white.\n\nNow, since all corners of the centre pixel $C$ are white and the pixel itself is grey, the boundary $\\partial X$ must intersect at least one edge of $C$ at least twice. It cannot be the bottom edge, and it cannot be either of the two vertical edges either by Lemma \\ref{LemTwoIntersections}, so $\\partial X$ must intersect the line between $p_1$ and $p_2$ at least twice. We now aim to show that this line is in fact contained in the set of white points, so that it cannot contain any points of $\\partial X$, giving us a contradiction.\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{Impossible3x3C}\n\\hfill\n\\includegraphics[scale=0.4]{Impossible3x3D}\n\\caption{Left: Since $p_1$ is white, it must lie in a white $\\sqrt{2}$-ball $B$ centred somewhere inside $B_{\\sqrt{2}}(p_1)$ (the blue circle. Since $B$ must contain neither $p_2$ nor $p_3$, the centre of $B$ must lie between the two dashed lines. Right: A point in the left part of the blue set $T'$ must belong to the triangle with vertices $p$, $v$ and $w$.}\n\\label{FigImpossible3x3B}\n\\end{figure}\n\nSince $p_1$ is white, it is contained in some white $\\sqrt{2}$-ball $B$. The centre of $B$ lies somewhere inside $B_{\\sqrt{2}}(p_1)$. Since $B$ cannot contain the black corner $p_3$ (see Figure \\ref{FigImpossible3x3B} left), its center must lie closer to $p_1$ than to $p_2$, hence it must lie to the right of the vertical line midway between the two points. Likewise, $B$ cannot contain both $p_1$ and $p_2$ without also containing the entire line between them, so the centre of $B$ must also lie closer to $p_1$ than to $p_2$, that is, to the left of the vertical line midway between the two points.\n\nFinally, the centre of $B$ can only belong to a white pixel. Hence it must belong to the bright blue part of the white pixel in Figure \\ref{FigImpossible3x3B}, left. Let us call this set $T$.\n\nNext, consider the bottom left grey pixel $C$, and let $p_4$ be its lower left corner. Since $T$ is a part of the upper left quarter of the white pixel next to it, any point in $T$ is further from $p_4$ than from any of the other three corners of $C$. Hence if $B$ contained $p_4$, it would also contain all the corners of $C$ that are closer to its centre, and hence it would contain all of $C$ which would then not be grey. So the centre of $B$ must lie further away from $p_4$ than $\\sqrt{2}$, hence outside the ball $B_{\\sqrt{2}}(p_4)$. Let $T'=T\\backslash B_{\\sqrt{2}}(p_4)$, the blue set in Figure \\ref{FigImpossible3x3B}, right.\n\nBy calculating the intersection $p$ between the boundaries of $B_{\\sqrt{2}}(p_1)$ and $B_{\\sqrt{2}}(p_4)$ inside the white pixel, we find that they intersect in a point that is a distance $\\frac{1}{10}(2\\sqrt{15}-5)\\approx 0.27$ from the left edge of the white pixel and a distance $\\sqrt{3\/20}\\approx 0.39$ from the top edge of the white pixel.\n\nLet $m$ be the midpoint of the line between $p_1$ and $p_2$. A calculation shows that $d(p,m)<\\sqrt{2}$, so $m\\in B_{\\sqrt{2}}(p)$. Any point of $T'$ to the right of $p$ is closer to $m$ than $p$ is, so any ball centred here must also contain $m$.\n\nConsider a point in $T'$ left of $p$. Such a point must be contained in the triangle with corners $p$, $v$ and $w$ as seen in Figure \\ref{FigImpossible3x3B}, right. Here $v$ is the upper left vertex of the white pixel, and $w$ is the point on the edge of the white pixel that is directly above $p$. A calculation shows that $m\\in B_{\\sqrt{2}}(p)\\cap B_{\\sqrt{2}}(v)\\cap B_{\\sqrt{2}}(w)$, which by Lemma \\ref{LemCircleIntersections} means that $m$ belongs to $B_{\\sqrt{2}}(x)$ for any $x$ in the left part of $T'$. Since the same was true for any point right of $p$, $m$ belongs to $B_{\\sqrt{2}}(x)$ for any $x\\in T'$. But since $p_1$ also belongs to $B_{\\sqrt{2}}(x)$ for any $x$ in $T'$, any white ball containing $p_1$ also contains $m$, and hence the line segment from $p_1$ to $m$.\n\nRepeating this argument for $p_2$, any white ball containing $p_2$ also contains $m$, and hence it contains the entire line segment from $m$ to $p_2$.\n\nBut then each point on the line segment from $p_1$ to $p_2$ is contained in a white ball - a contradiction.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{ThmConfigurationsUsualResolution}]\nCombining Lemmas \\ref{LemSkalfarvessorte1}, \\ref{LemTwoIntersections}, \\ref{LemYieldsTwoIntersections},  \\ref{LemNo9greys}, \\ref{Thm5greys}, \\ref{LemImpossible2x3} and \\ref{LemTrickyConfiguration}, we get the result with the exception of the configuration located at $(23,17)$ in Figure \\ref{FigMany3x3Configurations}. But this configuration is also impossible: Let $C$ be the grey centre pixel. $C$ must contain some boundary point $p\\in \\partial X$, so there would have to be a white ball of radius $r$ tangent to $\\partial X$ at $p$, and in particular, there would have to be a white ball of radius $1$ with $p$ on its boundary. But any point that is closer than $1$ to a point $p$ in $C$ would either have to belong to either $C$ or to one of the neighbouring black pixels, so the same must be true for the centre of the white $1$-ball tangent to $p$. If the centre belonged to a black pixel it would not be black, and if it belonged to $C$, the white ball of radius $d$ would contain some set of interior points of a pixel adjacent to $C$ - but these are all black. Hence we reach a contradiction.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{LemLeftoverconfiguration}]\n  Look at the centre point $c$ of the $4\\times4$ pixels. Suppose\n  it is white (the case where it is black is symmetric). Then one of the grey\n  pixels having $c$ as a vertex has only white vertices (in the\n  figure, it would be the lower left pixel). Call this pixel $A$.\n\n  Since $A$ s grey, the boundary $\\partial X$ must intersect one of its\n  edges, and since all of its corners are white, an edge intersected\n  by $\\partial X$ must be intersected at least twice. Note that only the\n  edges of $A$ that are shared with another grey pixel can be\n  intersected by $\\partial X$. But then by Lemma\n  \\ref{LemTwoIntersections}, one of the grey pixels in the figure\n  would have to be non-grey - a contradiction. So this configuration\n  cannot occur.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{Lem2x4Configuration}]\n  The proof follows from Lemma \\ref{Lem2x3grey}: Look at the two\n  pixels in the column to the right of the configuration (the red ones\n  in Figure \\ref{Fig2x4Configuration}, right). These cannot both be\n  grey, since that would violate Theorem \\ref{Lem2x3grey}, so at least\n  one of them must have another colour. Say that one of them is black\n  (the other case is symmetric). Depending on which one of the red\n  pixels is black, some part of Lemma \\ref{LemSkalfarvessorte2} tells\n  us that the $2\\times4$ configuration in Figure\n  \\ref{Fig2x4Configuration} must have more black pixels than what is\n  the case - a contradiction. So the configuration cannot occur.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{Lem4Intersections2x2}]\n  Let $C$ denote the $2\\times 2$ configuration of grey pixels. By a proof copying the proof of Lemma \\ref{LemTwoIntersections}, if any edge of $C$ is intersected\n  by $\\partial X$ multiple times, then one of the pixels in $C$ would not be grey - a contradiction. So if $\\partial X$ intersects all edges of\n  $C$, it only intersects each edge once. Hence $C$ has two black\n  vertices on one diagonal and two white vertices on the other. Let\n  $L$ be the line connecting the black vertices and $M$ the line\n  connecting the white vertices.\n\n  There is a black path $\\rho_X(L)$ in $X$ connecting the two\n  black corners of the pixel. By Corollary \\ref{CorPathInSpindle} and Lemma \\ref{LemSpindleIsIntersection}, this path must belong to $B_{\\sqrt{2}}(p)$, where $p$ is the centre of $C$.\n\n  Similarly, there is a path $\\rho_{X^C}(M)$ in $X^C\\cup\\partial X$\n  connecting the two white vertices and contained in $B_{\\sqrt{2}}(p)$. If\n  $\\rho_{X^C}(M)$ contains points of $\\partial X$, we may push these\n  points a little along the normal vector field of $\\partial X$ to get a\n  path $\\tilde{\\rho}$ in $X^C$ connecting the white vertices of\n  $C$. This alteration can be made in the interior of $B_{\\sqrt{2}}(p)$ since the endpoints of $M$ are not boundary points, hence $\\tilde{\\rho}$ is also contained in $C$.\n\n  But then we have a black path in $B_{\\sqrt{2}}(p)$ separating the white vertices\n  of $C$, and a white path in $B_{\\sqrt{2}}(p)$ connecting them. This means that the\n  two paths must necessarily intersect each other in a point that must\n  be both black and white -- a contradiction. So $\\partial X$ cannot\n  intersect all four edges of $C$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Thm2x2}]\nCombining Lemmas \\ref{LemLeftoverconfiguration}, \\ref{Lem2x4Configuration} and \\ref{Lem4Intersections2x2} yields most of the result. The only configuration remaining is the one centred at $(45,22)$ in Figure \\ref{FigKonfigurationer}. But this is also not possible: If it where, the middle grey pixels would contain some boundary point $p\\in\\partial X$. Then there would be a white $\\sqrt{2}$-ball with $p$ in its boundary, and such a ball would either be centred inside the $4\\times4$ pixels of the configuration, or in one of the pixels neighbouring the configuration. Since the pixel containing the centre of the white ball is white itself, the white ball cannot be centred inside the configuration. But it also cannot be centred in one of the pixels neighbouring the configuration, since this would mean that a white pixel and a black one were sharing boundary points, which is against our assumption.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Cascade Size Prediction}\n\\label{sec: applications}\nTo demonstrate the effectiveness of our $M^4C$ model in quantitatively characterizing cascades, we use it to investigate an unexplored but fundamental problem in online social networks - \\emph{cascade size prediction}:\n\\emph{given the first $\\tau_1$ edges in a cascade, we want to predict whether the cascade will have a total of at least $\\tau_2$ ($\\tau_2>\\tau_1$) edges over its lifetime.}\nBesides serving the purpose of validating the relevance of our $M^4C$ model, this prediction has many real-world applications.\nFor instance, it is useful for media organizations to forecast popular news stories \\cite{gruhl05predictiveonlinechatter}.\nLikewise, popular videos on social media -- if predicted early -- can be cached by content distribution networks at their servers to achieve better performance \\cite{rodrigues11wordofmouth}.\nFurthermore, solving this problem enables the early detection of epidemic outbreaks and political crisis.\n\nTo the best of our knowledge, this problem has not been investigated in prior literature.\nThe closest effort is that Galuba \\etal analyzed the cascades of URLs on Twitter to predict URLs that users will tweet \\cite{galuba10twitters}.\nTheir proposed approach achieved about $50\\%$ true positive rate with about $15\\%$ false positive rate.\nUnfortunately, this accuracy is not much useful in practice.\n\nWe compare the prediction performance of $M^4C$ based scheme with a baseline scheme that uses the following $8$ cascade graph features with Na\\\"ive Bayes classifier: (1) edge growth rate, (2) number of nodes, (3) degree of the root node, (4) average shortest path length, (5) diameter, (6) number of spanning trees, (7) clustering coefficient, and (8) clique number.\nWe evaluate the effectiveness of these schemes in terms of the following decision sets.\n\\begin{enumerate}\n  \\item \\emph{True Positives (TPs)}: The set of cascades that are correctly predicted to have a total of at least $\\tau_2$ edges over their lifetime.\n  \\item \\emph{False Positives (FPs)}: The set of cascades that are incorrectly predicted to have a total of at least $\\tau_2$ edges over their lifetime.\n  \\item \\emph{True Negatives (TNs)}: The set of cascades that are correctly predicted to have a total of less than $\\tau_2$ edges over their lifetime.\n  \\item \\emph{False Negatives (FNs)}: The set of cascades that are incorrectly predicted to have a total of less than $\\tau_2$ edges over their lifetime.\n\\end{enumerate}\nWe further quantify the effectiveness of both cascade size prediction schemes in terms of the following three Receiver Operating Characteristic (ROC) metrics \\cite{fawcett04ROC}.\n\\begin{equation}\n  \\emph{\\text{Detection Rate}} = \\frac{|TPs|}{|TPs| + |FNs|}\n\\end{equation}\n\\begin{equation}\n  \\emph{\\text{False Positive Rate}} = \\frac{|FPs|}{|FPs| + |TNs|}\n\\end{equation}\n\\begin{equation}\n  \\emph{\\text{Precision}} = \\frac{|TPs| + |TNs|}{|TPs| + |TNs| + |FPs| + |FNs|}\n\\end{equation}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=.9\\columnwidth]{number_edges_distribution_tau2_10}\n\\precaption\n\\caption{Evaluation setup for varying $\\tau_1$.} \\label{fig: variation tau1}\n\\postcaption\n\\end{figure}\n\n\nTo ensure that the classification results are generalizable, we divide the data set into $k$ folds and use $k-1$ of them for training and the left over for testing.\nWe repeat these experiments $k$ times and report the average results in the following text.\nThis setup is called stratified $k$-fold cross-validation procedure \\cite{dataminingbook}.\nFor all experimental results reported in this paper, we use the value of $k=10$.\n\n\nIn this paper, we treat the cascade size prediction problem to an equivalent cascade classification problem: given a cascade with $\\tau_1$ edges, classify it into two classes: the class of cascades that will have less than $\\tau_2$ edges over their lifetime and the class of cascades that will have greater than or equal to $\\tau_2$ edges over their lifetime.\nWe use the initial $\\tau_1$ edges to train both the cascade size prediction scheme based on our $M^4C$ model and the baseline scheme that is based on the known cascade graph features.\nFor thorough evaluation, we vary the values of $\\tau_1$ and $\\tau_2$.\nBecause the distribution of the number of edges in our data set is skewed, that is, most cascades having only a few edges over their lifetime, the larger the values of $\\tau_1$ and $\\tau_2 - \\tau_1$ are, the more imbalanced the two classes are.\nTo mitigate the potential adverse effect of class imbalance \\cite{japkowicz02classimbalance}, we employ instance re-sampling to ensure that both classes have equal number of instances before the cross-validation evaluations.\nBelow we discuss the classification accuracies of both schemes as we vary the values of $\\tau_1$ and $\\tau_2$.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=.9\\columnwidth]{roc_tau1}\n\\precaption\n\\caption{ROC plot of $M^4C$ based scheme for varying $\\tau_1$.} \\label{fig: roc tau1}\n\\postcaption\n\\end{figure}\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{number_edges_distribution_tau1_10}\n\\precaption\n\\caption{Evaluation setup for varying $\\tau_2 - \\tau_1$.} \\label{fig: variation tau2}\n\\postcaption\n\\end{figure}\n\n\n\n\n\\begin{figure*}[!t]\n\\centering\n     \\subfigure[Detection Rate]{\n          \\includegraphics[width=.66\\columnwidth]{tp_rate_tau2}\n          }\n     \\subfigure[False Positive Rate]{\n          \\includegraphics[width=.66\\columnwidth]{fp_rate_tau2}\n          }\n     \\subfigure[Precision]{\n          \\includegraphics[width=.66\\columnwidth]{precision_tau2}\n          }\n\\precaption\n\\caption{Classification results of $M^4C$ and baseline schemes for varying values of $\\tau_2 - \\tau_1$, at $\\tau_1 = 10$.} \\label{fig: accuracy tau2}\n\\postcaption\n\\end{figure*}\n\n\n\\subsection{Impact of Varying $\\tau_1$}\nFigure \\ref{fig: variation tau1} shows the evaluation setup as we vary the values of $\\tau_1 \\in \\{10, 50, 100\\}$, while keeping $\\tau_2 - \\tau_1$ fixed at $10$.\nThe solid, dashed, and dotted vertical black lines corresponds to $\\tau_1 = 10, 50,$ and $100$.\nThe solid, dashed, and dotted vertical grey lines all correspond to $\\tau_2 - \\tau_1 = 100$.\nThe value of $\\tau_1$ impacts the classification results because it determines the number of edges in each cascade that are available for training.\nTherefore, larger values of $\\tau_1$ generally improve training quality of both cascade size prediction schemes and lead to better prediction accuracy.\n\n\nFigure \\ref{fig: accuracy tau1} plots the detection rate, false positive rate, and precision of $M^4C$ and baseline schemes for varying $\\tau_1 \\in \\{10, 50, 100\\}$, while keeping $\\tau_2 - \\tau_1$ fixed at $10$.\nOverall, we observe that $M^4C$ consistently outperforms the baseline scheme with peak precision of $96\\%$ at $\\tau_1 = 100, \\tau_2 - \\tau_1 = 10s$.\nWith some exceptions, we generally observe that the effectiveness of both schemes decreases as the value of $\\tau_1$ is increased.\nThe standard ROC threshold plots of $M^4C$ shown in Figure \\ref{fig: roc tau1} also confirm this observation.\n\n\n\n\n\n\n\\subsection{Impact of Varying $\\tau_2 - \\tau_1$}\n\\vspace{0.2in}\nFigure \\ref{fig: variation tau2} shows the evaluation setup as we vary the values of $\\tau_2 - \\tau_1 \\in \\{10, 50, 100\\}$, while keeping $\\tau_1$ fixed at $10$.\nThe solid vertical black line corresponds to $\\tau_1 = 10$.\nThe solid, dashed, and dotted vertical grey lines correspond to $\\tau_2-\\tau_1 = 10, 50,$ and $100$, respectively.\nThe value of $\\tau_2 - \\tau_1$ also impacts the classification results because it determines the separation or distance between the two classes.\nTherefore, larger values of $\\tau_2 - \\tau_1$ generally lead to better prediction accuracy.\n\n\nFigure \\ref{fig: accuracy tau2} plots the detection rate, false positive rate, and precision of $M^4C$ and baseline schemes for varying values of $\\tau_2 - \\tau_1$.\nOnce again, we observe that $M^4C$ consistently outperforms the baseline scheme with peak precision of $99\\%$ at $\\tau_2 - \\tau_1 = 100, \\tau_1 = 10$.\nWe also observe that the classification performance of both methods improves as the value of $\\tau_2 - \\tau_1$ is increased.\nThe standard ROC threshold plots of $M^4C$ shown in Figure \\ref{fig: roc tau2} also confirm this observation.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions and Future Work}\n\\label{sec: conclusions}\n\\postsub\nIn this paper, we first propose $M^4C$, a multi-order Markov chain based model to represent and quantitatively characterize the morphology of cascades with arbitrary structures, shapes, and sizes.\nWe then demonstrate the relevance of our $M^4C$ model in solving the cascade size prediction problem.\nThe experimental results using a real-world Twitter data set showed that $M^4C$ significantly outperforms the baseline scheme in terms of prediction accuracy.\nIn summary, our $M^4C$ model allows us to formally and rigorously study cascade morphology, which is otherwise difficult.\n\n\n\n\n\n\nIn this paper, we applied our $M^4C$ model in the context of online social networks; however, our model is generally applicable to cascades in other contexts as well such as sociology, economy, psychology, political science, marketing, and epidemiology.\nApplications of our model in these contexts are interesting future work to pursue.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{roc_tau2}\n\\precaption\n\\caption{ROC plot of $M^4C$ based scheme for varying $\\tau_2 - \\tau_1$.} \\label{fig: roc tau2}\n\\postcaption\n\\end{figure}\n\n\n\\Comment\nWe have planned future work along the following two directions.\nFirst, we plan to explore randomized graph encoding methods such as those based on random walks on graphs \\cite{rosvall08random,figueiredo12randomwalks}.\nSecond, we plan to apply $M^4C$ to solve other important cascade classification problems that can use morphological features.\nFor example, $M^4C$ can be used to differentiate spam and normal activity cascades in online social networks.\n}\n\n\n\n\n\n\n\\section{Data set}\n\\label{sec: data set}\n\\postsec\n\n\\presub\n\\subsection{Data Collection}\n\\postsub\nAmong the popular online social networks, Twitter is one of the social networks that allows systematic collection of public data from its site.\nTherefore, we chose to study the morphology of cascades appearing on Twitter.\nTo collect data from Twitter, we focused on tweets related to the Arab Spring event, which represents an ideal case study because it spans several months.\nFor countries involved in the Arab Spring event, we collected data from Twitter during one complete week in March $2011$.\nWe provide more details of the data collection process in the following text.\n\n\nFor our study, we separately collected two data sets from Twitter.\nThe first data set was collected using Twitter's \\textit{streaming API}, which allows the realtime collection of public tweets matching one or more filter predicates \\cite{twitterapi}.\nTo collect tweet data pertaining to a given country, we provided relevant keywords as filter predicates.\nFor example, we used the keywords `Libya' and `Tripoli' to collect tweets related to Libya.\nIn total, we collected tweets for $8$ countries over a period of a week in March $2011$.\nUsing Twitter's streaming API, we collected more than $8$ million tweets involving more than $200$ thousand unique users.\n\n\n\nAs mentioned in Section \\ref{subsec: cascade construction}, we cannot accurately construct cascade graphs without information about whom the users are following.\nThe one-way following policy of Twitter results in three types of relationships between two given users: (1) both follow each other, (2) only one of them follows the other, and (3) they do not follow each other.\nTwitter provides follower information for a given user via a separate interface called REST API \\cite{twitterapi}.\nREST API employs aggressive rate limiting by allowing clients to make only a limited number of API calls in an hour.\nTwitter applies this limit based on the public IP address or authentication token from the client who issues the request.\nCurrently, rate limiting for REST API permits only $150$ requests per hour for unauthenticated users and $350$ requests per hour for authenticated users.\nIn our tweet data set, we encountered more than $200,000$ unique users and we were required to make at least one request per user to get the follower list.\nFor each user who follows more than $5000$ users, we had to make a separate call to get each subset of $5000$ users.\nHere it is noteworthy that some users were following or were being followed by millions of users, requiring thousands of separate calls for each user.\nIt would take us several months to collect this data if we were to use a single authentication token or a single external IP address.\nTo overcome this limitation, we utilized dozens of public proxy servers to parallelize calls to Twitter's REST API \\cite{wang04codeen}.\nUsing this methodology, we collected follower lists of all users in less than a month.\n\n\nTwitter provides a ``re-tweet'' functionality which allows users to re-post the tweet of other users to their profiles.\nThe reference to the user with original tweet is maintained in all subsequent re-tweets.\nThere is no information on intermediate users in re-tweets.\nUsing the follower graph, we constructed cascade graphs for all sets of re-tweets which are essentially cascades.\nTherefore, the overall graph is a union of all cascades in our data.\nIn Figure \\ref{fig: tweet visualizations}, we visualize two cascades in our data set using radial and circular layout methods in Graphviz \\cite{graphviz}.\nIn a radial layout, we choose the user with original tweet as a center vertex (or root vertex in general) and the remaining vertices are put in concentric circles based on their proximity to the center vertex.\nIn a circular layout, all components are plotted separately with their respective vertices in a circular format.\nVisualization of two example cascades provides us interesting insights about their morphology.\nFrom the first example, we observe that the degree of vertices typically decreases as their distance from the root vertex increases.\nHowever, for the second example, we observe that subsequent vertices have degrees comparable to the root vertex.\nIn this paper, our aim is to capture such differences in an automated fashion using our proposed model.\n\n\n\\begin{figure*}[!t]\n\\precaption\n\\centering\n         \\subfigure[Edge and node counts]{\n          \\label{fig: url degree}\n          \\includegraphics[width=1\\columnwidth]{scatter_fig1}\n          }\n       \\subfigure[Root node degree and average path length]{\n          \\label{fig: url cc}\n          \\includegraphics[width=1\\columnwidth]{scatter_fig2}\n          }\n         \\subfigure[Diameter]{\n          \\label{fig: url degree}\n          \\includegraphics[width=1\\columnwidth]{cdf_fig4}\n          }\n         \\subfigure[Number of spanning trees]{\n          \\label{fig: url spanning}\n          \\includegraphics[width=1\\columnwidth]{cdf_fig6}\n          }\n         \\subfigure[Clustering coefficient]{\n          \\label{fig: url spanning}\n          \\includegraphics[width=1\\columnwidth]{cdf_fig3}\n          }\n       \\subfigure[Clique number]{\n          \\label{fig: url cc}\n          \\includegraphics[width=1\\columnwidth]{cdf_fig5}\n          }\n\\postcaption\n\\caption{Distributions of various cascade graph attributes in the Twitter data set.}\n\\label{fig: distribution properties}\n\\postcaption\n\\postcaption\n\\end{figure*}\n\n\\subsection{Data Analysis}\n\\label{subsec: data analysis}\nWe now analyze the structural features of the cascades in our collected data set in terms of degree, path, and connectivity.\nLater in Section \\ref{sec: applications}, we will use these features for baseline comparison with our proposed model in terms of classification accuracy.\nFor structural features that can only be computed from undirected graphs, such as clustering coefficient and diameter, we compute them on the undirected versions of cascade graphs.\n\n\n\\subsubsection{Degree Properties}\nWe first jointly study the number of edges and the number of nodes for all cascades in our data set.\nThe cascade graphs in our data set are connected and each user in the cascade graph has at least one inward or outward edge.\nTherefore, the number of edges in a cascade graph $|E|$ has the lower bound: $|E| \\ge |V| - 1$, where $|V|$ is the number of users participating in the cascade.\nFigure \\ref{fig: distribution properties}(a) shows the scatter plot between edge and node counts for all cascades in our data set.\nNote that we use the logarithmic scale for both axes.\nFrom this figure, we observe that the scatter plot takes the form of a strip whose thickness represents the average number of additional edges for each node.\nThe average thickness of this strip approximately corresponds to having twice the number of edges compared to the number of nodes.\n\n\\vfill\\eject\n\\subsubsection{Path Properties}\nAnother important characteristic of a cascade is the degree of the root node (user who initiated the cascade), which typically has the highest degree compared to all other nodes in a cascade graph.\nIn our data set, the root node has the highest degree compared to all other nodes in cascade graphs for more than $92\\%$ of the cascades.\nThe degree of the root node essentially represents the number of different routes through which cascade propagates in an online social network.\nNote that these paths may merge together after the first hop; however, we expect some correlation between the degree of root node and the number of unique routes through which a cascade propagates.\nOne relevant characteristic of a graph is average (shortest) path length ($APL$), which denotes the average of all-pair shortest paths \\cite{bondaygraphtheory}.\n\\[\nAPL = \\sum_{\\forall i,j \\in V, i\\neq j} \\frac{d(i,j)}{|V|(|V|-1)},\n\\]\nwhere $d(i,j)$ is the shortest path length between users $i$ and $j$.\nWe expect the average path length of a cascade to be proportional to the degree of the root node.\nFigure \\ref{fig: distribution properties}(b) shows the scatter plot of the root node degree and the average path length.\nAs expected, we observe that cascades with higher root node degrees tend to have larger average path lengths.\nWe have changed the x-axis to logarithm scale to emphasize this relationship.\n\n\nAnother fundamental characteristic of a graph is called diameter, which denotes the largest value of all-pair shortest paths \\cite{bondaygraphtheory}.\nFigure \\ref{fig: distribution properties}(c) shows the distribution of diameter of cascades in our data set.\nThe bars represent the probability mass function and the line represents the cumulative density function (CDF).\nThe minimum diameter is $1$ because the minimum number of nodes in a cascade is $2$.\nCascades with more than $2$ nodes can have a diameter of $1$ only if they are cliques.\nIn our data set, approximately $40\\%$ cascades have a diameter of $1$.\nThe largest cascades in our data set have a diameter of $9$.\n\n\nFinally, we can characterize the number of unique paths that connect nodes in a graph by using the notion of spanning trees.\nFor a given graph, the number of unique paths between nodes is proportional to the number of spanning trees.\nThe number of spanning trees of a graph $G$, denoted by $t(G)$, is given by the product of non-zero eigenvalues of the Laplacian matrix and the reciprocal of the number of nodes \\cite{bondaygraphtheory}.\n\\[\nt(G) = \\frac{1}{n}\\lambda_1\\lambda_2...\\lambda_{n-1},\n\\]\nwhere $n$ is the number of nodes of the graph and $\\lambda_i$ is the $i$-th eigenvalue of the Laplacian matrix of the graph and $\\lambda_i \\neq 0, \\forall i$.\nFigure \\ref{fig: distribution properties}(d) shows the CDF of the number of spanning trees for cascades in our data set.\nNote that the x-axis is converted to logarithm scale.\nWe observe that only a small fraction $(< 15\\%)$ of cascades have more than one spanning tree in our data set, which highlights their sparsity.\n\n\n\\subsubsection{Connectivity Properties}\nThe clustering coefficient of a vertex $v_i$ is denoted by $c_i$ and is defined as the ratio of the number of existing edges among $v_i$ and $v_i$'s neighbors and the number of all possible edges among them \\cite{bondaygraphtheory}.\nUsing $\\Delta_i$ to denote the number of triangles containing vertex $v_i$ and $d_i$ to denote the degree of vertex $v_i$, the clustering coefficient of vertex $v_i$ is defined as:\n\\[\nc_i = \\frac{\\Delta_i}{{d_i\\choose 2}} = \\frac{2\\Delta_i}{d_i(d_i - 1)}\n\\]\nThe average clustering coefficient of a graph $G$ with $n$ nodes is simply the mean of clustering coefficients of individual nodes.\n\\[\nC_{avg} = \\frac{1}{n}\\sum_{\\forall i}c_i\n\\]\nFigure \\ref{fig: distribution properties}(e) shows the CDF of the average clustering coefficient for all cascades in our data set.\nWe note that approximately $86\\%$ of all cascades in our data set have average clustering coefficient value equal to $0$, \\ie, they do not have a single triangle.\nOnly a small fraction (less than $2\\%$) of cascades in our data set have clustering coefficient values greater than $0.5$, which again highlights their sparsity.\n\nWe are also interested in investigating the sizes of cliques in cascades that have one or more triangles.\nTowards this end, we study the clique numbers of all cascade graphs in our data set.\nThe clique number of a graph is the number of vertices in its largest clique \\cite{bondaygraphtheory}.\nFigure \\ref{fig: distribution properties}(f) shows the distribution of clique number for all cascades in our data set.\nSimilar to our observation in Figure \\ref{fig: distribution properties}(e), we observe that approximately $86\\%$ of cascades have a clique number of $2$, which means that they do not have a triangle.\nA little more than $10\\%$ of cascades have at least one triangle.\nThe largest clique number observed in our data set is $6$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{sec: introduction}\n\\postsec\n\n\\subsection{Background and Motivation}\n\\postsub\nThe term \\emph{cascade} describes the phenomenon of something propagating along the links in a social network.\nThat something can be information such as a URL, action such as a monetary donation, influence such as buying a product, discussion such as commenting on a blog article, and a resource such as a torrent file.\nBased on what is being propagated, we can categorize cascades into various classes such as information cascades \\cite{cha09flickr}, action cascades \\cite{dave11icwsm}, influence cascades \\cite{kempe03kddinfluence}, discussion cascades \\cite{gomez11ht}, and resource cascades \\cite{starr92resourcecascade}.\nConsider a toy example where user $A$, connected to users $B$ and $C$ in a social network, broadcasts a piece of information (\\eg a picture or a news article) to his neighbors.\nUsers $B$ and $C$, after receiving it from user $A$, may further rebroadcast it to their neighbors resulting in the formation of a cascade.\n\n\nCascade phenomenon has been a fundamental topic in many disciplines such as sociology, economy, psychology, political science, marketing, and epidemiology with research literature tracing back to the 1950s \\cite{Rogers03Diffusion}.\nA key challenge in these studies is the lack of large scale cascade data.\nAs online social networks have recently become a primary way for people to share and disseminate information, the massive amount of data available on these networks provides unprecedent opportunities to study cascades at a large scale.\nRecent events, such as the Iran election protests, Arab Spring, Japanese earthquake, and London riots, have been significantly impacted by campaigns via cascades in online social networks \\cite{zhou10iran,ray11arab,londonsocial}.\nStudying cascades in online social networks will benefit a variety of domains such as social campaigns \\cite{zhou10iran}, product marketing and adoption \\cite{li04adoption}, online discussions \\cite{gomez11ht}, sentiment flow \\cite{miller11sentiment}, URL recommendation \\cite{rodrigues11wordofmouth}, and meme tracking \\cite{rod10meme}.\n\n\\subsection{Problem Statement}\n\\label{subsec: problem statement}\nThe goal of this paper is to study the morphology of cascades in online social networks.\nCascade morphology encompasses many aspects of cascades such as their structures, shapes, and sizes.\nSpecifically, we aim to develop a model that allows us to \\emph{represent} and \\emph{quantitatively characterize} cascade morphology; which are extremely difficult without a model.\nThere are two important requirements on the desired model of cascade morphology.\nFirst, this model should have enough expressivity and scalability to allow us to represent and describe cascades with arbitrary structures, shapes, and sizes.\nReal-world cascades sometimes have large sizes, containing thousands of nodes and edges \\cite{kwak10twitter}.\nSecond, this model should allow us to quantitatively characterize and rigorously analyze cascades based on the features extracted from this model.\n\n\n\\presub\n\\subsection{Limitations of Prior Art}\n\\postsub\nDespite the numerous publications regarding different aspects of online social networks, little work has been done on the morphology of cascades.\nRecently some researchers have studied the structure of cascades \\cite{leskovec07cascadingbehavior, kwak10twitter, zhou10iran, gomez11ht}; however, their analysis of cascade structures is limited to basic structural properties such as degree distribution, size, and depth.\nThese structural properties of cascades are important; however, they are far from being sufficient to precisely describe and represent cascade morphology.\n\n\n\\presub\n\\subsection{Proposed Model}\n\\postsub\nIn this paper, we propose a Multi-order Markov Model for the Morphology of Cascades ($M^4C$) that can represent and quantitatively characterize the morphology of cascades with arbitrary structures, shapes, and sizes.\n$M^4C$ has two key components: a cascade encoding algorithm and a cascade modeling method.\nThe cascade encoding algorithm uniquely encodes the morphology of a cascade for quantitative representation.\nIt encodes a cascade by first performing a depth-first traversal on the cascade graph and then compressing the traversal results using run-length encoding.\nThe cascade modeling method models the run-length encoded sequence of a cascade as a discrete random process.\nThis random process is further modeled as a Markov chain, which is then generalized into a multi-order Markov chain model.\n$M^4C$ satisfies the aforementioned two requirements.\nFirst, this model can precisely represent cascades with arbitrary structures, shapes, and sizes.\nSecond, this model allows us to quantitatively characterize cascades with different attributes using the state information from the underlying multi-order Markov chain model.\n\n\\presub\n\\subsection{Experimental Evaluation}\n\\postsub\nTo demonstrate the effectiveness of our $M^4C$ model in quantitatively characterizing cascades, we use it to investigate an unexplored but important problem in online social networks -- \\emph{cascade size prediction}:\n\\emph{given the first $\\tau_1$ edges in a cascade, we want to predict whether the cascade will have a total of at least $\\tau_2$ ($\\tau_2>\\tau_1$) edges over its lifetime.}\nThis prediction has many real-world applications.\nFor example, media companies can use it to predict social media stories that can potentially go viral \\cite{gruhl05predictiveonlinechatter,rodrigues11wordofmouth}.\nFurthermore, solving this problem enables early detection of epidemic outbreaks and political crisis.\nDespite its importance, this problem has not been addressed in prior literature.\n\nWe validate the effectiveness of $M^4C$ based cascade size prediction scheme on a real-world data set collected from Twitter containing more than $8$ million tweets, involving more than $200$ thousand unique users.\nThe results show that our $M^4C$ based cascade size prediction scheme consistently achieves more than $90\\%$ classification accuracy under different experimental scenarios.\nWe also compare our $M^4C$ based cascade size prediction scheme with a baseline prediction scheme based on cascade graph features such as edge growth rate, degree distribution, clustering, and diameter.\nThe results show that $M^4C$ allows us to achieve significantly better classification accuracy than the baseline method.\n\n\\vfill\\eject\n\\presub\n\\subsection{Key Contributions}\n\\postsub\nIn this paper, we not only propose the first cascade morphology model, but also propose the first cascade size prediction scheme based on our model. In summary, we make the following key contributions in this paper.\n\\begin{enumerate}\n\\item We propose $M^4C$ for representing and quantitatively characterizing the morphology of cascades with arbitrary structures, shapes, and sizes.\n\n\\item To demonstrate the effectiveness of our $M^4C$ model in quantitatively characterizing cascades, we develop a cascade size prediction scheme based on $M^4C$ features and compare its performance with that based on non-$M^4C$ features.\n\\end{enumerate}\n\n\nThe rest of this paper proceeds as follows.\nWe first review related work in Section \\ref{sec: related work}.\nWe then introduce our proposed model in Section \\ref{sec: model}.\nWe describe the details of our Twitter data set in Section \\ref{sec: data set}.\nWe present the experimental results of the aforementioned application in Section \\ref{sec: applications}.\nFinally, we conclude in Section \\ref{sec: conclusions} with an outlook to our future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proposed Model}\n\\label{sec: model}\n\\postsec\nIn this section, we present $M^4C$ for quantitatively representing the morphology of cascades in online social networks.\nIt consists of two major components.\nThe first component encodes a given cascade graph for quantitative representation such that its morphological information is retained.\nThe second component models the encoded sequence using a multi-order Markov chain.\nBefore we describe these two components, we first present the details of the cascade graph construction process.\n\n\n\\presub\n\\subsection{Cascade Graph Construction}\n\\label{subsec: cascade construction}\n\\postsub\nA social network can be represented using two graphs, a relationship graph and a cascade graph.\nBoth graphs share the same set of nodes (or vertices) $V$, which represents the set of all users in a social network.\nA \\emph{relationship graph} represents the relationships among users in a social network.\nIn this graph, nodes represent users and edges represent the relationship among users.\nIf the edges are directed, where a directed edge from user $u$ to user $v$ denotes that $v$ is a follower of $u$, then this graph is called a \\emph{follower graph}, denoted as $(V, \\overrightarrow{E_f})$, where $V$ is the set of users and $\\overrightarrow{E_f}$ is the set of directed edges.\nIf the edges are undirected, where an undirected edge between user $u$ and user $v$ denotes that $u$ and $v$ are friends, then this graph is called a \\emph{friendship graph}, denoted as $(V, E_f)$, where $V$ is the set of users and $E_f$ is the set of undirected edges.\nBy the nature of our study, we focus on the follower graph denoted as $G_f = (V, \\overrightarrow{E_f})$.\nThe \\emph{cascade graph} represents the dynamic activities that are taking place in a social network (such as users sharing a URL or joining a group).\nA cascade graph is an acyclic directed graph denoted as $G_c = (V, \\overrightarrow{E_c}, T)$ where $V$ is the set of users, $\\overrightarrow{E_c}$ is a set of directed edges where a directed edge $e = (u,v)$ from user $u$ to user $v$ represents the propagation of something from $u$ to $v$, and $T$ is a function whose input is an edge $e \\in \\overrightarrow{E_c}$ and output is the time when the propagation along edge $e$ happens.\n\n\nWhile the static relationship graph is easy to construct from a social network, the dynamic cascade graph is non-trivial to construct because there maybe multiple propagation paths from the cascade source to a node.\nSo far there is no consensus on cascade graph construction in prior literature.\nIn this paper, we use a construction method that is similar to the method described in \\cite{sadikov11missing}.\nWe next explain our construction method through a Twitter example.\nConsider the follower graph in Figure \\ref{fig: cascade construction encoding}(a).\nLet $(u, t)$ denote a user $u$ performing an action, such as posting a URL on $u$'s Twitter profile, at time $t$.\nSuppose the following actions happen in the increasing time order: $(A,t_1)$, $(B,t_2)$, $(D,t_3)$, $(C,t_4)$, $(E,t_5)$, where $t_1 < t_2 < t_3 < t_4 < t_5$.\nSuppose $(A,t_1)$ denotes that $A$ posts a URL on his Twitter profile, and all other actions (namely $(B,t_2)$, $(D,t_3)$, $(C,t_4)$, and $(E,t_5)$) are reposting the same URL from $A$.\n\n\nThe cascade graph regarding the propagation of this URL is constructed as follows.\nFirst, $A$ is the root of the cascade graph because it is the origin of this cascade.\nSecond, $B$ reposting $A$'s tweet (which is a URL in this example) at time $t_2$ must be under $A$'s influence because there is only one path from $A$ to $B$ in the follower graph in Figure \\ref{fig: cascade construction encoding}(a).\nTherefore, in the cascade graph in Figure \\ref{fig: cascade construction encoding}(b), there is an edge from $A$ to $B$ with time stamp $t_2$.\nNote that each repost (or retweet in Twitter's terminology) contains the origin of the tweet ($A$ in this example).\nThird, however, $D$ reposting $A$'s tweet at time $t_3$ could be under either $A$'s influence (because there is a path from $A$ to $D$ in the follower graph in Figure \\ref{fig: cascade construction encoding}(a) and $t_1 < t_3$) or $B$'s influence (because there is a path from $B$ to $D$ in the follower graph as well and $t_2 < t_3$).\nNote that even if $D$ sees $A$'s tweet through $B$'s retweet, the repost of $A$'s tweet on $D$'s profile does not contain any information about $B$ and only shows that the origin of the tweet is $A$.\nIn this scenario, we assume that $D$ is partially influenced by both $A$ and $B$, instead of assuming that $D$ is influenced by either user $B$ or $A$, because this way we can retain more information with respect to the corresponding follower graph.\nTherefore, there is an edge from $A$ to $D$ and another edge from $B$ to $D$ in the cascade graph shown in Figure \\ref{fig: cascade construction encoding}(b), where the time stamps of both edges are $t_3$.\nSimilarly, we add the edge from $B$ to $C$ with a time stamp $t_4$ and the edge from $D$ to $E$ with a time stamp $t_5$ in the cascade graph.\n\n\n\\presub\n\\subsection{Cascade Encoding}\n\\label{subsection: cascade encoding}\nThe first step in cascade encoding is to encode the constructed cascade graph as a binary sequence that uniquely represents the structure of the cascade graph.\nGraph encoding has been studied for a wide range of problems across several domains such as image compression, text and speech recognition, and DNA profiling \\cite{reid97imagecoding, biem06handwriting, hsieha08dnagraph}.\nThe typical goal of graph encoding is to transform large geometric data into a succinct representation for efficient storage and processing.\nHowever, our goal here is to encode a given cascade graph in a way that its morphological information is captured.\nTowards this end, we use the following graph encoding algorithm.\n\nWe first conduct a depth-first traversal of the constructed cascade graph starting from the root node, which results in a spanning tree.\nTo result in a unique spanning tree, at each node in the cascade graph, we sort the outgoing edges in the increasing order of their time stamps, \\ie, sort the outgoing edges $e_1, e_2, \\cdots, e_k$ of a node so that $T(e_1) < T(e_2) < \\cdots < T(e_k)$; and then traverse them in this order.\nFor each edge, we use 1 to encode its downward traversal and 0 to encode its upward traversal.\nFigure \\ref{fig: cascade construction encoding}(c) shows the traversal of the cascade graph in Figure \\ref{fig: cascade construction encoding}(b) and the encoding of each downward or upward traversal.\nThe binary encoding results from this traversal process is \\textcolor{red}{11}0\\textcolor{red}{11}000.\nLet $C$ represent the binary code of a cascade graph $G = (V,\\overrightarrow{E})$.\nThen the length of the binary code $|C|$ is always twice the size of the edge set $|\\overrightarrow{E}|$, \\ie, $|C|=2|\\overrightarrow{E}|$.\nFurthermore, let $C[i]$ be the $i$-th element of the binary code and $I(C[i])$ be an indicator function so that $I(C[i]) = 1$ if $C[i] = 1$, and $I(C[i]) = -1$ if $C[i] = 0$.\nBecause each edge is exactly traversed twice, one downward and one upward, we have:\n\\[\n\\sum_{i=1}^{|C|}I(C[i])  = 0.\n\\]\n\n\nThe second step in cascade encoding is to convert the binary sequence, which is obtained from the depth-first traversal of the cascade graph, into the corresponding run-length encoding.\nA \\emph{run} in a binary sequence is a subsequence where all bits in this subsequence are 0s (or 1s) but the bits before and after the subsequence are 1s (or 0s), if they exist.\nBy replacing each run in a binary sequence with the length of the run, we obtain the run-length encoding of the binary sequence \\cite{jaynat84coding}.\nFor example, for the binary sequence \\textcolor{red}{11}0\\textcolor{red}{11}000, the corresponding run-length encoding is \\textcolor{red}{2}1\\textcolor{red}{2}3.\nSince the binary sequence obtained from our depth-first traversal of a cascade graph always starts with 1, the run-length encoding uniquely and compactly represents the binary sequence.\n\n\n\\presub\n\\subsection{Markov Chain Model of Cascades}\n\\label{subsec: markov model}\nWe want to model cascade encoding to capture characteristics of cascades so that they can be used to identify the similarities and differences among cascades.\nThis model should allow us to extract morphological features for different classes of cascades and then use these features to classify them.\nWe first present our model, and then demonstrate its usefulness in classifying cascades.\n\n\nConsider the run-length encoded sequence $\\hat{C}$ of a cascade graph $G$.\nWe can model this sequence using a discrete random process $\\{\\hat{C}_k\\}$, $k = 1,2,...,|\\hat{C}|$.\nBasic analysis of this process reveals that there is some level of dependencies among the consecutive symbols emitted by the random process.\nIn other words, it would be unreasonable to assume that the process is independent or memoryless.\nMeanwhile, to balance between capturing some of the dependencies within the process and to simplify the mathematical treatment of this encoded sequence, we resort to invoking the Markovian assumption \\cite{pierre08markovchains}.\nAs we show later, this assumption can be reasonably justified (to some extent) by analyzing the autocorrelation function of the underlying process $\\{\\hat{C}_k\\}$.\nFor a first order Markov process, this implies the following assumption:\n$Pr[\\hat{C}_n = c_n | \\hat{C}_1 = c_1, \\hat{C}_2 = c_2, ..., \\hat{C}_{n-1} = c_{n-1}] = Pr[\\hat{C}_n = c_n | \\hat{C}_{n-1} = c_{n-1}]$.\nEquivalently:\n\\begin{equation}\nPr[c_1, c_2, ..., c_n] = Pr[c_1]Pr[c_2|c_1]...Pr[c_n|c_{n-1}].\n\\label{equation: markov}\n\\end{equation}\nIn other words, we invoke the Markovian assumption about the underlying cascade process and its morphology, which is represented by the encoded sequence $\\hat{C}$.\n\nGiven the Markovian assumption with homogeneous time-invariant transition probabilities, $\\hat{C}$ can be represented using a traditional Markov chain.\nFigure \\ref{fig: markov chain} shows the Markov chain corresponding to the toy example in Figure \\ref{fig: cascade construction encoding}, where each unique symbol in $\\hat{C}$ is represented as a state.\nThe Markov chain in Figure \\ref{fig: markov chain} has $3$ states because there are $3$ unique symbols in its run-length encoding.\n\n\\begin{figure}[htbp]\n\\precaption\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{figure_encoding_markovchain1}\n\\caption{Markov chain model for the toy example.}\n\\label{fig: markov chain}\n\\end{figure}\n\nA Markov chain can also be specified in terms of its state transition probabilities, denoted as $T$.\nHence, for the toy example of Figure 2, we have:\n\\[ T = \\left( \\begin{array}{ccc}\nP_{1|1} & P_{1|2} & P_{1|3} \\\\\nP_{2|1} & P_{2|2} & P_{2|3} \\\\\nP_{3|1} & P_{3|2} & P_{3|3} \\end{array} \\right),\\]\nwhere $P_{i|j}$ represents the conditional probabilities $Pr[\\hat{C}_n = i | \\hat{C}_{n-1} = j]$.\nThe Markov chain framework allows us to quantify the probability of an arbitrary sequence of states by using Equation \\ref{equation: markov}.\nThis will help us to identify sequences that are more (or less) probable in one class of cascades.\nWe next further generalize the above basic Markov chain model.\n\n\n\\subsection{Multi-order Generalization}\nEach element of the state transition matrix of a Markov chain is equivalent to a sub-sequence of $\\hat{C}$, which in turn is equivalent to a subgraph of the corresponding cascade.\nWe can generalize a Markov chain model by incorporating multiple consecutive transitions as a single state in the state transition matrix, which will allow us to specify arbitrary sized subgraphs of cascades.\nSuch generalized Markov chains are called multi-order Markov chains and are sometimes referred to as full-state Markov chains \\cite{khayam03markov}.\nThe order of a Markov chain represents the extent to which past states determine the present state.\nThe basic Markov chain model introduced earlier is of order $1$.\n\n\nAutocorrelation is an important statistic for selecting appropriate order for a Markov chain model \\cite{pierre08markovchains}.\nFor a given lag $t$, the autocorrelation function of a stochastic process, $X_m$ (where $m$ is the time or space index), is defined as:\n\\begin{equation}\n\\rho[t] = \\frac{E\\{ X_0 X_t\\} - E\\{ X_0\\} E\\{X_t\\}}{\\sigma_{X_0}\\sigma_{X_t}},\n\\end{equation}\nwhere $E(\\cdot)$ represents the expectation operation and $\\sigma_{X_i}$ is the standard deviation of the random variable at time or space lag $i$.\nThe value of the autocorrelation function lies in the range $[-1,1]$, where $|\\rho[t]|=1$ indicates perfect correlation at lag $t$ and $\\rho[t]=0$ means no correlation at lag $t$.\nFigure \\ref{fig: acf} plots the sample autocorrelation function of the run-length encoding of an example cascade.\nThe dashed horizontal lines represent the $95\\%$ confidence envelope.\nFor this particular example, we observe that sample autocorrelation values jump outside the confidence envelope at lag $= 3$.\nThis indicates that the underlying random process has the third order dependency.\nThus, we select the third order for Markov chain model for this particular cascade.\nThe autocorrelation-based analysis of more complex cascades can lead to even higher order Markov chains.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1\\columnwidth]{untitled}\n\\vspace{0.1in}\n\\caption{Sample autocorrelation function for the toy example.}\n\\label{fig: acf}\n\\vspace{0.1in}\n\\end{figure}\n\n\nThe number of possible states of a Markov chain increase exponentially with an increase in the order of the Markov chain model.\nFor the $n$-th order extension of a Markov chain with $k$ states, the total number of states is $k^n$.\nFigure \\ref{fig: multiorder} shows the plot of the second order extension of the $3$-state, $1$-st order Markov chain model shown in Figure \\ref{fig: markov chain}.\nThis second order Markov chain contains a total of $3^2 = 9$ states, $4$ of which are shown in the figure due to space limitations.\nIn this second order Markov chain model, the conditional probabilities are in the form $P_{i,j|k,l}$ and the state transition matrix is now defined as follows.\n\\[\n\\vspace{0.2in}\nT_2 = \\left( \\begin{array}{ccccc}\nP_{1,1|1,1} & P_{1,1|1,2} & P_{1,1|1,3} & ... & P_{1,1|3,3}\\\\\nP_{1,2|1,1} & P_{1,2|1,2} & P_{1,2|1,3} & ... & P_{1,2|3,3}\\\\\nP_{1,3|1,1} & P_{1,3|1,2} & P_{1,3|1,3} & ... & P_{1,3|3,3}\\\\\n. & . & . & \\ddots & .\\\\\n. & . & . & \\ddots & .\\\\\nP_{3,2|1,1} & P_{3,2|1,2} & P_{3,2|1,3} & ... & P_{3,2|3,3}\\\\\nP_{3,3|1,1} & P_{3,3|1,2} & P_{3,3|1,3} & ... & P_{3,3|3,3}\\\\\n\\end{array} \\right)\n\\vspace{0.1in}\n\\]\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1\\columnwidth]{figure_encoding_markovchain2}\n\\caption{Multi-order generalization of the Markov chain model for the toy example.}\n\\label{fig: multiorder}\n\\end{figure}\n\nFor a set of cascade encoding sequences, let $\\mathbb{T}$ denote the set of selected orders as per the aforementioned criterion.\nWe select the maximum value in $\\mathbb{T}$, denoted by $T_{max}$, as the order of a single Markov chain model that we want to employ.\n\n\n\\subsection{Cascade Classification}\n\\label{subsec: cascade classification}\nAs mentioned in Section \\ref{subsec: problem statement}, an important desirable property for our proposed model is to identify differentiating features of cascade morphology that can be potentially leveraged for automated classification of cascades.\nWe now show how to use the aforementioned Markov chain model to classify cascades.\n\n\n\\subsubsection{Feature Selection}\nThe essence of our modeling approach is to capture the morphology of a cascade through the states of the multi-order Markov model.\nEach state in the Markov chain represents a likely sub-structure of cascades' morphology.\nThus, we can use these states to serve as underlying features that can be used to characterize a given cascade and to determine the class that it might belong to.\nHowever, as mentioned earlier, the number of states in a Markov chain increase exponentially for higher orders and so does the complexity of the underlying model.\nFurthermore, higher order Markov chains require a large amount of training data to identify a subset of states that actually appear in the training data.\nIn other words, a Markov chain model trained with limited data is typically sparse.\nTherefore, we use the following two approaches to systematically reduce the number of states in the Markov chain of order $T_{max}$.\n\n\nFirst, we can combine multiple states in the Markov chain to reduce its number of states.\nBy combining states in a multi-order Markov chain, we are essentially using states from lower order Markov chains.\nWe need to establish a criterion to combine states in the Markov chain.\nTowards this end, we use the concept of \\emph{typicality} of Markov chain states.\nTypicality allows us to identify a typical subset of Markov chain states by generating its realizations \\cite{pierre08markovchains}.\nBefore delving into further details, we first state the well-known typicality theorem below:\nFor any stationary and irreducible Markov process $X$ and a constant $c$, the sequence $x_1, x_2, ..., x_m$ is almost surely $(n, \\epsilon)$-typical for every $n \\le c \\log m$ as $m \\rightarrow \\infty$.\nA sequence $x_1, x_2, ..., x_m$ is called $(n, \\epsilon)$-typical for a Markov process $X$ if $\\hat{P}(x_1,x_2,..., x_n) = 0$, whenever $P(x_1,x_2,..., x_n) = 0$, and\n\\[\n\\bigg|\\frac{\\hat{P}(x_1,x_2,..., x_n)}{P(x_1,x_2,..., x_n)} - 1\\bigg| < \\epsilon \\mbox{, when } P(x_1,x_2,..., x_n)>0.\n\\]\nHere $\\hat{P}(x_1,x_2,..., x_n)$ and $P(x_1,x_2,..., x_n)$ are the empirical relative frequency and the actual probability of the sequence $x_1,x_2,..., x_n$, respectively.\nIn other words,\n\\[\n\\hat{P}(x_1,x_2,..., x_n) \\approx {P(x_1,x_2,..., x_n)}.\n\\]\nThis theorem shows us a way of empirically identifying typical sample paths of arbitrary length for a given Markov process.\nBased on this theorem, we generate realizations (or sample paths) of arbitrary lengths from the transition matrix of the Markov process.\nBy generating a sufficiently large number of sample paths of a given length, we can identify a relatively small subset of sample paths that are typical.\nUsing this criterion, we select a subset of up to top-$100,000$ typical states as potential features, whose lengths vary in the range $[0,\\mathbb{T}_{max}]$.\nIn what follows, we further short-list the Markov states from the top-$100,000$ typical subset and use them as features to classify cascades.\n\n\nSecond, to further reduce the number of features to be employed in a classifier, we need to prioritize the aforementioned typical Markov states.\nThe prioritization of features can be based on their differentiation power.\nAn information theoretic measure that can be used to quantify the differentiation power of features (Markov states in our case) is information gain \\cite{cover91infotheory}.\nIn this context, information gain is the mutual information between a given feature $X_i$ and the class variable $Y$.\nFor a given feature $X_i$ and the class variable $Y$, the information gain of $X_i$ with respect to $Y$ is defined as:\n\\[\nIG(X_i;Y) = H(Y) - H(Y|X_i),\n\\]\nwhere $H(Y)$ denotes the marginal entropy of the class variable $Y$ and $H(Y|X_i)$ represents the conditional entropy of $Y$ given feature $X_i$.\nIn other words, information gain quantifies the reduction in the uncertainty of the class variable $Y$ given that we have complete knowledge of the feature $X_i$.\nNote that, in this paper, the class variable $Y$ is $\\{0,1\\}$ because we apply our morphology modeling framework to problems that require differentiating between two classes of cascades (as described later).\nIn this study, we eventually only select the top-$100$ features with highest information gain.\n\n\n\\subsubsection{Classification}\n\\vspace{0.1in}\nLet us assume that the presence of a state $i$ is represented by a binary random variable $X_i, i = 1,2, ..., 100$.\nHence, $P(X_i = 1)$ represents the probability for the presence of state $X_i$.\nWe can think of the $X_i$s as the variables representing potential features.\nThus, our training process proceeds as follows.\nFor a given class $Y$ of cascades, we evaluate the presence of a given feature (state) $X_i$ in $Y$ by analyzing a sufficiently large number of sample cascades that belong to the class $Y$.\nSubsequently, we are able to evaluate the a-priori conditional probability $P(X_i|Y)$ for each class $Y \\in \\{1,2,..., k\\}$, where the number of classes $k$ is usually very small.\nIn our case, we are interested in the traditional binary classifier with $k = 2$.\nHowever, note that this classification methodology can be extended to the cases with $k > 2$ using the well-known one-against-one (pairwise) or multiple one-against-all formulations \\cite{hsu02multiclass}.\n\n\nWe can jointly use multiple features to differentiate between two sets of cascades belonging to different classes.\nIn particular, given the top-$100$ features with respect to information gain, we can classify cascades by deploying a machine learning classifier.\nIn this study, we use a Bayesian classifier to jointly utilize the selected features to classify cascades.\nNa\\\"ive Bayes is a popular probabilistic classifier that has been widely used in the text mining and bio-informatics literature, and is known to outperform more complex techniques in terms of classification accuracy \\cite{dataminingbook}.\nIt trains using two sets of probabilities: the prior, which represents the marginal probability $P(Y)$ of the class variable $Y$; and the a-priori conditional probabilities $P(X_i|Y)$ of the features $X_i$ given the class variable $Y$.\nAs previously explained, these probabilities can be computed from the training set.\n\n\nNow, for a given test instance of a cascade with observed features $X_i$, $i = 1,2,..., n$, the \\textit{a-posteriori} probability $P(Y|X^{(n)})$ can be computed for both classes $Y \\in \\{0,1\\}$, where $X^{(n)} = (X_1, X_2, ..., X_n)$ is the vector of observed features in the test cascade under consideration:\\\\\\\\\n\\begin{equation}\n\\vspace{0.1in}\nP(Y|X^{(n)}) = \\frac{P(X^{(n)},Y)}{P(X^{(n)})} = \\frac{P(X^{(n)}|Y)P(Y)}{P(X^{(n)})}\n\\vspace{0.1in}\n\\end{equation}\nThe na\\\"ive Bayes classifier then combines the a-posteriori probabilities by assuming conditional independence (hence the ``na\\\"ive'' term) among the features.\\\\\n\\begin{equation}\n\\vspace{0.1in}\nP({X^{(n)}}|Y) = \\prod_{i=1}^{n} P(X_i|Y).\n\\vspace{0.1in}\n\\end{equation}\nAlthough the independence assumption among features makes it feasible to evaluate the a-posteriori probabilities with much lower complexity, it is unlikely that this assumption truly holds all the time.\nFor our study, we mitigate the effect of the independence assumption by pre-processing the features using the well-known Karhunen-Loeve Transform (KLT) to uncorrelate them \\cite{dony01klt}.\n\n\nIn the following section, we provide details of the data set that we have collected to demonstrate the usefulness of our $M^4C$ model.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\label{sec: related work}\n\\postsec\nCascades in online social networks have attracted much attention and investigation; however, little work has been done on cascade morphology.\nBelow we summarize the prior work related to cascade morphology.\n\n\n\\subsection{Shape} Zhou \\etal studied Twitter posts (\\ie, tweets) about the Iranian election \\cite{zhou10iran}.\nIn particular, they studied the frequency of pre-defined shapes in cascades.\nTheir experimental results showed that cascades tend to have more width than depth.\nThe largest cascade observed in their data has a depth of seven hops.\nLeskovec \\etal studied patterns in the shapes and sizes of cascades in blog and recommendation networks \\cite{leskovec06pakdd,leskovec07cascadingbehavior}.\nTheir work is also limited to studying the frequency of fixed shapes in cascades.\n\n\\subsection{Structure} Kwak \\etal investigated the audience size, tree height, and temporal characteristics of the cascades in a Twitter data set \\cite{kwak10twitter}.\nTheir experimental results showed that the audience size of a cascade is independent of the number of neighbors of the source of that cascade.\nThey found that about $96\\%$ of the cascades in their data set have a height of $1$ hop and the height of the biggest cascade is $11$ hops.\nThey also found that about $10\\%$ of cascades continue to expand even after one month since their start.\nRomero \\etal specifically studied Twitter cascades with respect to hashtags in terms of degree distribution, clustering, and tie strengths \\cite{romero11crosstopics}.\nThe results of their experiments showed that cascades from diverse topics (identified using hashtags), such as sports, music, technology, and politics, have different characteristics.\nSimilarly, Rodrigues \\etal studied structure-related properties of Twitter cascades containing URLs \\cite{rodrigues11wordofmouth}.\nThey studied cascade properties like height, width, and the number of users for cascades containing URLs from different web domains.\nSadikov \\etal investigated the estimation of the sizes and depths of information cascades with missing data \\cite{sadikov11missing}.\nTheir estimation method uses multiple features including the number of nodes, the number of edges, the number of isolated nodes, the number of weakly connected components, node degree, and non-leaf node out-degree.\nTheir empirical evaluation using a Twitter data set showed that their method accurately estimates cascade properties for varying fractions of missing data.\n\n\n\\subsection{Simulation}\nGomez \\etal studied the structure of discussion cascades in Wikipedia, Slashdot, Barrapunto, and Meneame using features solely based on the depth and degree distribution of cascades \\cite{gomez11ht}.\nThey also developed a generative model based on the maximum likelihood estimation of preferential attachment process to simulate synthetic discussion cascades.\nHowever, their model does not capture morphological properties of cascades and is limited to generation of synthetic discussion cascades.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuite often in theoretical work, approximation schemes for some\nquantities converge rather slowly.   Thus, there is a need for means to\naccelerate convergence or, equivalently, to extrapolate from few\nmembers of a sequence to its limit. Fortunately, the development of\nsuch methods has become a rather active field at the borderline between\nmathematics and the sciences in recent years. Brezinski and Redivo\nZaglia \\cite{BrezinskiRedivoZaglia91} have given an excellent\nmathematical introduction to such methods. There are many methods that\ncan be used to accelerate slowly convergent (or to sum divergent) power\nseries in terms of rational approximations, e.g., Pad\\'e approximants\n\\cite{Baker65,BakerGravesMorris81a,BakerGravesMorris81b} that are\nrelated to the famous epsilon algorithm \\cite{Wynn56a}, Levin-type\nmethods \\cite{Levin73,Weniger89,Homeier93,HomeierWeniger95}, and\niterative methods \\cite{Weniger91} like the recently developed\n$\\mathcal{J}$ transformation\n\\cite{Homeier94a,Homeier94d,Homeier94e}. There are also methods\nthat can be used to accelerate the convergence of Fourier\n\\cite{KieferWeiss81,Homeier92,Homeier93} and other orthogonal\nseries \\cite{Longman87,Homeier94b}. Onedimensional iteration\nsequences can be accelerated very effectively as is demonstrated\nin \\cite{Homeier95} for the case of the inverse Dyson equation.\nThere is also a growing literature on extrapolation of matrix\nand vector sequences (see \\cite{BrezinskiRedivoZaglia91} for an\nintroduction) that have found applications to the computation of\nmatrix functions \\cite{Homeier94c} and the iterative\nsolution of fixed-point equations \\cite{HomeierRastKrienke95}.\nThe full potential for application of these methods in the\nsciences has still to be explored.\n\nOne of the fields where these methods may be applied is\nMany-Body Perturbation Theory (MBPT), that is  one of the standard methods to\nobtain the correlation energy in molecular {\\em ab initio}\ncalculations. The convergence acceleration of many-body perturbation\nseries has recently become a topic of increasing interest\n\\cite{SchmidtWarkenHandy93,DietzSchmidtWarkenHess92,%\nDietzSchmidtWarkenHess93a,DietzSchmidtWarkenHess93b,%\nDietzSchmidtWarkenHess93c,DietzSchmidtWarkenHess94a,%\nDietzSchmidtWarkenHess94b}, also in the context of time-dependent\nphenomena \\cite{DietzSchmidtWarken95}. Here, we restrict attention to\napproaches to correlation energy estimation that are based on the\nM{\\o}ller-Plesset (MP) series since the latter is commonly and\nroutinely used in quantum chemistry for closed-shell systems. For\nopen-shell systems, the restricted MP (RMP) method has been developed\n\\cite{KnowlesAndrewsAmosHandyPople91} that is based on an restricted\nopen-shell Hartree-Fock (ROHF) determination of the MP unperturbed\nHamiltonian. In this way, the RMP approach largely avoids spin\ncontaminations that are characteristic for unrestricted MP (UMP) based\non an unrestricted HF (UHF) zero-order calculation. For smaller\nmolecules, calculations up to fourth or even fifth order do not pose\nlarge problems, and MPn (n=2,4) calculations are a popular approach to\nthe correlation problem. However, the computational effort increases\nsteeply with the order of the perturbation series, and with the size of\nthe molecular system. Therefore, there is a need to make the best use\nof the lower-order terms since higher terms are difficult to obtain.\nOrder-by-order summation of the perturbation expansion as given by\n\\begin{equation}\\label{eq0}\nE = E_{0} + E_{1} + E_{2} + E_{3} +  E_{4}\n+  E_{5} + \\dots\\>,\n\\end{equation}\ni.e., using the $n$th order estimate\n\\begin{equation}\\label{eq0a}\nE^{(n)} = \\sum_{j=0}^{n} E_j\\>,\n\\end{equation}\nis not the best way to exploit the information content\nof its terms. It has been shown by Schmidt, Warken and Handy\n\\cite{SchmidtWarkenHandy93} that a specific variant of a method\noriginally proposed by Goldhammer and Feenberg\n\\cite{GoldhammerFeenberg56,Feenberg56} for the Brillouin-Wigner\nperturbation expansion allows to obtain better estimates for the\ncorrelation energy than order-by-order summation of the usual MP series.\nThis variant was called the {\\em Feenberg series} in\n\\cite{SchmidtWarkenHandy93}. It is also a special case of the so-called\n{\\em Geometric Approximation}\n\\cite{Amos70,Bhattacharyya81,SchulmanMusher68,WilsonSilverFarrell77}.\nSimilar to the original approach of Goldhammer and Feenberg\n\\cite{GoldhammerFeenberg56,Feenberg56}, the computation of the Feenberg\nseries requires only the terms $E_{j}$ of the perturbation series.\n\n\n\nAlternatively, one may use Pad\\'e approximants\nthat provide rational approximations $[p,q]$ to power series,\nwhere $p$ denotes the order of the numerator polynomial, and $q$\nthat of the denominator polynomial.  Pad\\'e approximants may be\ncalculated for the original perturbation series, and also for\nthe renormalized perturbation series. As shown by Wilson,\nSilver, and Farrell \\cite{WilsonSilverFarrell77}, the special\nPad\\'e approximants $[n+1,n]$ have the property that they are\ninvariant under the scaling of the unperturbed Hamilton operator\nand, thus, are identical for the original and the renormalized\ncase. This invariance is an important property of correlation\nenergy estimators since the true correlation energy is\nindependent of our choice of the unperturbed Hamiltonian.\n\nRecently, a method based on effective characteristic polynomials\nhas been applied to correlation energy computations of some model\nsystems\n\\cite{Bracken94,BrackenCizek94,TakahashiBrackenCizekPaldus95,%\nBrackenCizek95a,BrackenCizek95b,CizekBracken95,%\nDowningMichlCizekPaldus79}\nand for the summation of perturbation expansions of anharmonic\noscillators \\cite{CizekWenigerBrackenSpirko96}. We will see that\nresults based on low-order effective characteristic polynomials\nalso have the desirable invariance property under rescaling of\nthe unperturbed Hamiltonian.\n\nAll these methods require only the terms $E_{i}$ of the\nM{\\o}ller-Plesset perturbation series. The additional effort to\ncalculate them besides the usual perturbation series is very\nlow. As will be shown, these methods allow to obtain much better\nestimates of the correlation energy in many cases, and allow the\nidentification of cases where standard perturbation theory\nfails. In these cases, computationally more demanding\ncorrelation energy estimators have to be used \\cite{%\nSchmidtWarkenHandy93,DietzSchmidtWarkenHess92,%\nDietzSchmidtWarkenHess93a,DietzSchmidtWarkenHess93b,%\nDietzSchmidtWarkenHess93c,DietzSchmidtWarkenHess94a,%\nDietzSchmidtWarkenHess94b,%\nCizek69,Paldus74,%\nPaldus76,%\nKutzelnigg77,%\nShavitt77,%\nHoseKaldor79,%\nBartlett81,%\nJeziorskiMonkhorst81,%\nHoseKaldor82,%\nBartlettDykstraPaldus84,%\nWilson84,%\nHoffmanSchaefer86,%\nHose89,%\nPaldus88,%\nMukherjeePal89,%\nSzaboOstlund89,%\nFulde91,%\nKarwowski92,%\nMcWeeny92,%\nPaldus92,%\nRoos92,Roos94,%\nHandy94,%\nRychlewski94,%\nMeissner95,%\nSteinbornSilver96}.\n\n                       \\section{Methods}\nThe Goldhammer-Feenberg approach \\cite{GoldhammerFeenberg56,Feenberg56}\nrenormalizes the unperturbed Hamiltonian $H_0$ by a constant factor\naccording to\n\\begin{equation}\\label{eq1}\nH_0(\\alpha) = (1-\\alpha) H_0\\>.\n\\end{equation}\nThis leads to a repartitioning of the total Hamiltonian\n$H=H_0+H_1$ as\n\\begin{equation}\\label{eq0alpha}\nH = H_0(\\alpha) + H_1(\\alpha), \\quad H_1(\\alpha)= H_1 + \\alpha H_0\\>.\n\\end{equation}\nIt also leads to a renormalized perturbation series\n\\begin{equation}\\label{eq3}\nE(\\alpha) = E_{0}(\\alpha) + E_{1}(\\alpha) + E_{2}(\\alpha) +\nE_{3}(\\alpha) +  E_{4}(\\alpha) +  E_{5}(\\alpha) +\n\\dots\\>\n\\end{equation}\nwith partial sums --- i.e., renormalized $n$th order energy\nestimates --- given by\n\\begin{equation}\\label{eq3a}\nE^{(n)}(\\alpha) = \\sum_{j=0}^{n} E_j(\\alpha)\\>\n\\end{equation}\ndepending on renormalized $j$th order contributions \\cite[Eq.\n(12)]{Feenberg56}\n\\begin{equation}\\label{eq5}\n\\begin{array}{rl}\nE_{0}(\\alpha) =&\\displaystyle (1-\\alpha) E_{0}\\>, \\qquad\nE_{1}(\\alpha) =\\displaystyle E_{1} + \\alpha E_{0}\\>,\\\\\nE_{n}(\\alpha) =&\\displaystyle\n\\frac{1}{(1-\\alpha)^{n-1}} \\,\\sum_{j=2}^{n}\n{{n-2}\\choose{j-2}}(-\\alpha)^{n-j} \\, E_{j}\\>,\\qquad (n\\ge 2)\\>.\n\\end{array}\n\\end{equation}\nFor the Feenberg series, the factor $\\alpha$ is determined by\nrequiring that the third order energy $E^{(3)}(\\alpha)$\nof the renormalized perturbation expansion is stationary with respect\nto variations of the factor $\\alpha$. This leads to an optimized\nvalue based on the third order result given by\n$\\alpha^{(3)}=E_{3}\/E_{2}$.\nIn this way, the partitioning of the Hamiltonian is fixed, and the\nFeenberg series is obtained as the usual Rayleigh-Schr\\\"odinger series\nfor the unperturbed Hamilton operator $H_0(\\alpha^{(3)})$.\nThe total energies are\n\\begin{equation}\\label{eqFn}\nF_n = E^{(n)}(\\alpha^{(3)}) = E^{(n)}(E_{3}\/E_{2})\\>.\n\\end{equation}\n\nThe stationarity of the eigenvalue is based on the observation\nthat the exact value of the energy, i.e., the infinite order\nresult should be independent of the value of $\\alpha $ that is\nused. When applying this to an approximation obtained in some\nfinite order, that value of $\\alpha$ is best where the\nderivative of the approximation is as small as possible in\nabsolute value, preferably zero. We remark that this is related\nto the concept of order-dependent mappings as discussed in\n\\cite[Sec. 18]{ArtecaFernandezCastro90}. Since order-by-order\nsummation of the $\\alpha$ dependent Rayleigh-Schr\\\"odinger\nexpansion leads to the $n$th order estimate $E^{(n)}(\\alpha)$\ndefined in Eq.~(\\ref{eq3a}), the optimal value $\\alpha^{(n)}$ of\n$\\alpha$ in $n$th order is determined from the equation $(n>\n1)$\n\\begin{equation}\\label{eq8}\n0=\\frac{d E^{(n)}} {d \\alpha} (\\alpha^{(n)})\\>, \\qquad \\frac{d\nE^{(n)}} {d \\alpha} (\\alpha)\n = (n-1)\\, E_n(\\alpha) \/ (1-\\alpha) \\>.\n\\end{equation}\nThe second equality here follows from an explicit calculation.\nA solution of this equation leads to an approximation\n\\begin{equation}\\label{eqGFn}\nGF_n=E^{(n)}(\\alpha^{(n)})\n\\end{equation}\nfor the total energy. Thus, in each order of the\nrenormalized perturbation series, different values of $\\alpha$\nare\nchosen. This approach has been proposed already by Feenberg. We will\ncall its results the total {\\em Goldhammer-Feenberg\nenergies}\nin order to distinguish it from the Feenberg total energies. Obviously,\nthere can be several solutions of Eq. (\\ref{eq8}), and the\nGoldhammer-Feenberg energies are not guaranteed to be real.\n\nIn the case\nof fifth order, the condition (\\ref{eq8}) reduces in combination\nwith Eq.~(\\ref{eq5}) to requiring that\n$\\alpha^{(5)}$ is a root of the third order polynomial\n$(1-\\alpha)^4\\,E_5(\\alpha)$. The latter has\nreal coefficients and, thus, is guaranteed to have a real\nsolution $\\alpha^{(5)}_r$. The corresponding value $E^{(5)}(\\alpha^{(5)}_r)$ will be called\nGF5 later. Alternatively, one can use the average of the two (in\nthe present case always) complex energies obtained from the\nother roots of the third order polynomial. This average will be\ncalled GF5b later.\n\nAs is well-known (see for instance \\cite{Kutzelnigg77,Hose89}),\nRayleigh-Schr\\\"odinger MBPT is size-extensive order by order,\ni.e., for a super-molecule build up from $N$ non-interacting\nidentical systems, the perturbation energies are linear in $N$\nin each order. Thus, if $E_j$ is the $j$th term of the\nperturbation series of one of the $N$ subsystems, the $j$th\norder term of the perturbation series for the super-molecule is\n$N\\,E_j$.\n\nIn the case of the Feenberg scaling, we note that Eq.\\\n(\\ref{eq5}) implies that for $E_j\\to N\\,E_j$, we also have\n$E_j(\\alpha)\\to N\\,E_j(\\alpha)$. Thus, for any $\\alpha$ that\nis independent of $N$, also\nthe renormalized perturbation series is size-extensive in each\norder. Since\n$\\alpha^{(3)}=E_3\/E_2$ is invariant under $E_j\\to N\\,E_j$, all Feenberg\nenergies $F_n$ are size-extensive as a consequence of Eq.\\\n(\\ref{eqFn}).\n\nThe Goldhammer-Feenberg energies\n$GF_n$ for $n>1$ are also size-extensive. To prove this, we note that\nunder $E_n\\to N\\,E_n$, we have $d E_n\/d \\alpha\\to N\\,d\nE_n\/d\\alpha$. This follows from the last equality in Eq.\\ (\\ref{eq8}), since\n$E_n(\\alpha)\\to N\\,E_n(\\alpha)$ under $E_n\\to N\\,E_n$.\nThis implies that the positions of the zeros of $d E_n\/d \\alpha$,\nand hence the positions $\\alpha^{(n)}$ of the extrema of\n$E_n(\\alpha)$ are invariant under $E_n\\to N\\,E_n$. Since the\n$\\alpha^{(n)}$ are used to define the Goldhammer-Feenberg energies, the\nlatter  are size-extensive. In particular, this applies to GF5\nand GF5b.\n\n\n  Now, we sketch the method of the effective characteristic polynomial\n  that has recently been applied to the summation of divergent\n  perturbation series \\cite{CizekWenigerBrackenSpirko96}. In the linear\n  variation method with $n$ orthonormal basis functions\n  $\\{\\phi_j\\}_{j=1}^{n}$ applied to a Hamiltonian $H$, the\n  characteristic polynomial $P_n(E)$ of degree $n$ in the unknown energy\n  $E$  has the form\n\\begin{equation}\\label{eqC1}\nP_n(E)={\\rm det}\\,\\left\\vert \\left\\langle \\phi_j \\vert H \\vert\n\\phi_k \\right\\rangle -E\\,\\delta_{j,k}\\right\\vert \\>.\n\\end{equation}\n  If $H=H_0+\\beta V$, the polynomial has the form\n  (\\cite{CizekWenigerBrackenSpirko96}, Eq. (3.2))\n\\begin{equation}\\label{eqC2}\nP_n(E) = \\sum_{j=0}^{n} E^j \\,\\sum_{k=0}^{n-j} f_{n,j,k} \\beta^k\n\\end{equation}\n  with $f_{n,n,0}=1$. Thus, $N=n(n+3)\/2$ coefficients $f_{n,j,k}$ have\n  to be determined. They could be obtained\n  from the matrix elements of $H_0$ and $V$. In the method of the\n  characteristic polynomial, they are obtained from the coefficients of\n  the perturbation series for $E$\n\\begin{equation}\\label{eqC3}\nE = \\sum_{j=0}^{\\infty} E_j \\, \\beta^j\\>.\n\\end{equation}\n  For this end, one uses (\\ref{eqC3}) in (\\ref{eqC2}) and does a Taylor\n  expansion in $\\beta$ with the result\n\\begin{equation}\\label{eqC4}\nP_n\\left(\\sum_{j=0}^{\\infty} E_j \\, \\beta^j\\right) = \\sum_{k=0}^{N-1}\nA_k \\beta^k + O\\left(\\beta^{N}\\right)\\>.\n\\end{equation}\nThe $A_k$ depend on the $f_{n,j,k}$. Since $P_n(E)=0$ for an eigenvalue\n$E$, one demands\n\\begin{equation}\\label{eqC5}\nA_k = 0\\>, \\qquad  0\\le k \\le N-1\\>.\n\\end{equation}\nThis yields a linear equation system for the unknown\n$f_{n,j,k}$, and thus, these coefficients can be determined.\nAfter the determination, the effective characteristic equation\n$P_n(E)=0$ is\nsolved for $E$. If only perturbation coefficients $E_j$ up to $j=5$ are\navailable, only a second degree effective characteristic\npolynomial can be used. In our case, one finally puts $\\beta=1$. In this way,\none obtains an explicit solution of $P_2(E)=0$ as\n\\begin{equation}\\label{eqP2}\n \\Pi_2 = E_0 + E_1  +\n {\\displaystyle \\frac {E_2^2}{2}}\n \\,\\frac{\\displaystyle\n    E_2 -E_3 +\n   \\sqrt {(E_2- E_3)^2 - 4 \\,(E_2\\,E_4- E_3^2)}\n   }\n   {\\displaystyle  E_2\\,E_4 - E_3^2\n   }\n\\end{equation}\nA further solution (with a minus sign of the square root) only\nyields the correct result for small $\\beta$ if $E_2>0$\nholds which does not occur in perturbation theory calculations of\nground states.\n\nDirect calculation shows that the estimate $\\Pi_2$ is\nindependent under a scaling of $H_0$, i.e., we have\n\\begin{equation}\\label{eqP2a}\n\\Pi_2(E_0,\\dots,E_4) = \\Pi_2(E_0(\\alpha),\\dots,E_4(\\alpha))\\>.\n\\end{equation}\nSince the true characteristic polynomials --- depending only on\nthe total Hamiltonian --- are invariant under Feenberg scaling,\nit may be conjectured that this invariance also holds for\nestimates obtained as roots of effective characteristic\npolynomials of higher degree. A proof of this conjecture is\nunder investigation.\n\nWe denote ${\\Pi}_{2}$  also as estimate $\\Pi$2 for the total\nenergy in the following.\n\nIt is easy to see from Eq.\\ (\\ref{eqP2}) that $\\Pi_2\\to\nN\\,\\Pi_2$ if $E_j\\to N\\,E_j$ for all $j$ with $0\\le j\\le 4$. Thus, the\n$\\Pi_2$ estimator is size-extensive.\n\n\n\n\nPad\\'e approximants\n\\cite{Baker65,BakerGravesMorris81a,BakerGravesMorris81b} are defined\nwith respect to a given power series as ratios of two polynomials.\nGiven numerator and denominator polynomial degrees $p$ and $q$, the\ncoefficients of these polynomials in the Pad\\'e approximant $[p,q]$ are\ndetermined by requiring that up to the order $p+q$, the coefficients in\nthe Taylor expansion of the ratio of polynomials are equal to the\ncoefficients of the given power series. In the present contribution, we\ntake as this power series the perturbation expansion (\\ref{eqC3}) in\nthe parameter $\\beta$ that is put equal to one in the final formulas.\nWe note that a different power series that is not explicitly defined,\nseems to have been used for the Pad\\'e approximants in\n\\cite{KucharskiNogaBartlett89}. For the application of rational\napproximants to the M{\\o}ller-Plesset series  see also Ref.\\\n\\cite{HandyKnowlesSomasundram85}.\n\n\n                     \\section{Numerical Results}\nFortunately, excellent data for the test of the methods\ndescribed in the previous section are available in\n\\cite{SchmidtWarkenHandy93}. This paper also includes results given in\n\\cite{KucharskiNogaBartlett89}. In these references, a large\nnumber of M{\\o}ller-Plesset results up to fifth order, and FCI\n(Full Configuration Interaction) or CCSDT (Coupled Cluster\nSingles Doubles Triples) results are given for the ground states\nof benchmark molecules (BH, HF, CH${}_2$, H${}_2$O, NH${}_2$,\nNH${}_3$, CO, C${}_2$H${}_2$, O${}_3$, CN). The results of the\nreanalysis of these data is presented in Table \\ref{tab1}. For\ncompleteness, the MP data are also plotted. If not stated\notherwise, MPn means RMPn in open shell cases. Apart from case\nn (NH${}_3$), the left half of the data in Table \\ref{tab1} is obtained\nfrom the data up to fourth order, while the right half also\ndepends on the fifth order.\n\nIt is seen that in many cases, the correlation energy estimators\nprovide excellent results. Problematic cases are s, t, and u. In case\ns corresponding to CN, the perturbation series is divergent, being based on doubly occupied\nROHF orbitals where for alpha and beta spins the same orbitals are\nused, unlike the RMP orbitals where occupied alpha and beta set both\nare rotated. \\cite{SchmidtWarkenHandy93,Handy94} In cases t and\nu corresponding to H${}_2$O at stretched geometries, the\napproach is based on an UMP series that is monotonously and very slowly\nconvergent \\cite{SchmidtWarkenHandy93,HandyKnowlesSomasundram85}.\n\n\n\\setlongtables\n\\setlength{\\LTleft}{0pt}\n\\setlength{\\LTright}{0pt}\n\\begin{longtable}{l@{\\extracolsep{\\fill}}rr|lrr}\n\\caption{Comparison of Correlation Energy Estimators}\\label{tab1}    \\\\\n Method & \\multicolumn{1}{c}{Energy} & \\%Corr\n& Method & \\multicolumn{1}{c}{Energy} &\n\\%Corr \\\\\n\\hline\n\\endfirsthead\n\\multicolumn{6}{l}{(Table \\ref{tab1} -- continued)}\n\\endhead\n\\multicolumn{6}{c}{Case a: BH (${}^1\\Sigma$, $r=2.329\\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,HarrisonHandy83,%\nBartlettSekinoPurvis83})} \\\\\n SCF    &  -25.125260 &   0.00   & MP5    &  -25.225101 &  97.53 \\\\\n MP2    &  -25.198988 &  72.02   & F5     &  -25.226881 &  99.27 \\\\\n MP3    &  -25.216566 &  89.19   & GF5    &  -25.226971 &  99.36 \\\\\n MP4    &  -25.222567 &  95.06   & GF5b   &  -25.227088 &  99.47 \\\\\n F4     &  -25.226167 &  98.57   & [3,2]  &  -25.227299 &  99.68 \\\\\n $[2,2]$&  -25.225294 &  97.72   & [2,3]  &  -25.227478 &  99.85 \\\\\n $\\Pi$2 &  -25.226555 &  98.95   & FCI    &  -25.227627 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case b: BH (${}^1\\Sigma$, $r=1.5 \\times 2.329\\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,NogaBartlett87})} \\\\\n SCF    &  -25.062213 &   0.00   & MP5    &  -25.172372 &  96.83 \\\\\n MP2    &  -25.139869 &  68.26   & F5     &  -25.174484 &  98.69 \\\\\n MP3    &  -25.160249 &  86.18   & GF5    &  -25.174544 &  98.74 \\\\\n MP4    &  -25.168745 &  93.64   & GF5b   &  -25.177010 & 100.91 \\\\\n F4     &  -25.175345 &  99.45   & [3,2]  &  -25.175078 &  99.21 \\\\\n $[2,2]$&  -25.173623 &  97.93   & [2,3]  &  -25.175106 &  99.24 \\\\\n $\\Pi$2 &  -25.176791 & 100.72   & FCI    &  -25.175976 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case c: BH (${}^1\\Sigma$, $r=2 \\times 2.329\\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,NogaBartlett87})} \\\\\n SCF    &  -24.988201 &   0.00   & MP5    &  -25.121278 &  95.65 \\\\\n MP2    &  -25.074503 &  62.03   & F5     &  -25.126844 &  99.65 \\\\\n MP3    &  -25.100221 &  80.51   & GF5    &  -25.126983 &  99.75 \\\\\n MP4    &  -25.114005 &  90.42   & GF5b   &  -25.130104 & 101.99 \\\\\n F4     &  -25.128829 & 101.08   & [3,2]  &  -25.129407 & 101.49 \\\\\n $[2,2]$&  -25.124953 &  98.29   & [2,3]  &  -25.129475 & 101.54 \\\\\n $\\Pi$2 &  -25.137084 & 107.01   & FCI    &  -25.127333 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case d: HF ($r=1.733 \\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherLanghoffTaylorHandyKnowles86})} \\\\\n SCF    & -100.047087 &   0.00   & MP5    & -100.250158 &  99.60 \\\\\n MP2    & -100.243165 &  96.17   & F5     & -100.250099 &  99.57 \\\\\n MP3    & -100.245531 &  97.33   & GF5    & -100.250276 &  99.66 \\\\\n MP4    & -100.251232 & 100.13   & GF5b   & -100.251988 & 100.50 \\\\\n F4     & -100.251443 & 100.23   & [3,2]  & -100.250468 &  99.75 \\\\\n $[2,2]$& -100.251547 & 100.28   & [2,3]  & -100.250481 &  99.76 \\\\\n $\\Pi$2 & -100.251820 & 100.42   & FCI    & -100.250969 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case e: HF ($r=1.5 \\times 1.733 \\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherLanghoffTaylorHandyKnowles86})} \\\\\n SCF    &  -99.933230 &   0.00   & MP5    & -100.158121 &  99.00 \\\\\n MP2    & -100.149756 &  95.32   & F5     & -100.158152 &  99.01 \\\\\n MP3    & -100.148543 &  94.78   & GF5    & -100.158247 &  99.05 \\\\\n MP4    & -100.159627 &  99.66   & GF5b   & -100.161609 & 100.53 \\\\\n F4     & -100.159443 &  99.58   & [3,2]  & -100.158750 &  99.28 \\\\\n $[2,2]$& -100.160091 &  99.87   & [2,3]  & -100.158757 &  99.28 \\\\\n $\\Pi$2 & -100.160708 & 100.14   & FCI    & -100.160395 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case f: HF ($r=2 \\times 1.733 \\,a_0$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherLanghoffTaylorHandyKnowles86})} \\\\\n SCF    &  -99.817571 &   0.00   & MP5    & -100.073004 &  96.93 \\\\\n MP2    & -100.057062 &  90.88   & F5     & -100.073139 &  96.98 \\\\\n MP3    & -100.054148 &  89.77   & GF5    & -100.073301 &  97.04 \\\\\n MP4    & -100.076267 &  98.16   & GF5b   & -100.079678 &  99.46 \\\\\n F4     & -100.075480 &  97.86   & [3,2]  & -100.075064 &  97.71 \\\\\n $[2,2]$& -100.077899 &  98.78   & [2,3]  & -100.075072 &  97.71 \\\\\n $\\Pi$2 & -100.080476 &  99.76   & FCI    & -100.081107 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case g: CH${}_2$ (${}^1A_1$, $r=2.11 \\,a_0$,\n$\\theta=102.4\\,{}^{\\circ}$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherTaylor86b})} \\\\\n SCF    &  -38.886297 &   0.00   & MP5    &  -39.024234 &  97.91 \\\\\n MP2    &  -38.996127 &  77.96   & F5     &  -39.025336 &  98.69 \\\\\n MP3    &  -39.016593 &  92.48   & GF5    &  -39.025450 &  98.77 \\\\\n MP4    &  -39.022203 &  96.47   & GF5b   &  -39.025413 &  98.74 \\\\\n F4     &  -39.024615 &  98.18   & [3,2]  &  -39.025674 &  98.93 \\\\\n $[2,2]$&  -39.024049 &  97.78   & [2,3]  &  -39.025895 &  99.09 \\\\\n $\\Pi$2 &  -39.024791 &  98.30   & FCI    &  -39.027183 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case h: H${}_2$O (${}^1A_1$, $r=1.88973 \\,a_0$,\n$\\theta=104.5\\,{}^{\\circ}$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherTaylor86a})} \\\\\n SCF    &  -76.040542 &   0.00   & MP5    &  -76.255924 &  99.68 \\\\\n MP2    &  -76.243660 &  94.00   & F5     &  -76.255918 &  99.67 \\\\\n MP3    &  -76.249403 &  96.66   & GF5    &  -76.255929 &  99.68 \\\\\n MP4    &  -76.255706 &  99.58   & GF5b   &  -76.257338 & 100.33 \\\\\n F4     &  -76.256262 &  99.83   & [3,2]  &  -76.256134 &  99.77 \\\\\n $[2,2]$&  -76.256282 &  99.84   & [2,3]  &  -76.256135 &  99.77 \\\\\n $\\Pi$2 &  -76.256729 & 100.05   & FCI    &  -76.256624 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case i: H${}_2$O (${}^1A_1$, $r=1.5\\times 1.88973 \\,a_0$,\n$\\theta=104.5\\,{}^{\\circ}$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherTaylor86a})} \\\\\n SCF    &  -75.800494 &   0.00   & MP5    &  -76.066422 &  98.16 \\\\\n MP2    &  -76.048095 &  91.40   & F5     &  -76.066368 &  98.14 \\\\\n MP3    &  -76.045081 &  90.28   & GF5    &  -76.066442 &  98.17 \\\\\n MP4    &  -76.065641 &  97.87   & GF5b   &  -76.068395 &  98.89 \\\\\n F4     &  -76.064909 &  97.60   & [3,2]  &  -76.068528 &  98.94 \\\\\n $[2,2]$&  -76.066937 &  98.35   & [2,3]  &  -76.068533 &  98.94 \\\\\n $\\Pi$2 &  -76.068954 &  99.10   & FCI    &  -76.071405 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case j: H${}_2$O (${}^1A_1$, $r=2\\times 1.88973 \\,a_0$,\n$\\theta=104.5\\,{}^{\\circ}$, DZP,\n\\cite{KucharskiNogaBartlett89,BauschlicherTaylor86a})} \\\\\n SCF    &  -75.582286 &   0.00   & MP5    &  -75.935304 &  95.41 \\\\\n MP2    &  -75.898603 &  85.50   & F5     &  -75.934525 &  95.20 \\\\\n MP3    &  -75.877664 &  79.84   & GF5    &  -75.935353 &  95.43 \\\\\n MP4    &  -75.937410 &  95.98   & GF5b   &  -75.923566 &  92.24 \\\\\n F4     &  -75.927115 &  93.20   & [3,2]  &  -75.949379 &  99.22 \\\\\n $[2,2]$&  -75.941045 &  96.97   & [2,3]  &  -75.949401 &  99.22 \\\\\n $\\Pi$2 &  -75.954930 & 100.72   & FCI    &  -75.952269 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case k: NH${}_2$ (${}^2B_1$, $r=1.013 \\,\\mbox{\\AA}$,\n$\\theta=103.2\\,{}^{\\circ}$, 6-31G,\n\\cite{HandyKnowlesSomasundram85,KnowlesAndrewsAmosHandyPople91})} \\\\\n SCF    &  -55.530177 &   0.00   & MP5    &  -55.632426 &  99.18 \\\\\n MP2    &  -55.617272 &  84.48   & F5     &  -55.632818 &  99.56 \\\\\n MP3    &  -55.627501 &  94.40   & GF5    &  -55.632834 &  99.57 \\\\\n MP4    &  -55.631220 &  98.01   & GF5b   &  -55.633280 & 100.00 \\\\\n F4     &  -55.632525 &  99.27   & [3,2]  &  -55.633011 &  99.74 \\\\\n $[2,2]$&  -55.632204 &  98.96   & [2,3]  &  -55.633022 &  99.75 \\\\\n $\\Pi$2 &  -55.632825 &  99.56   & FCI    &  -55.633276 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case l: NH${}_2$ (${}^2B_1$, $r=1.5\\times 1.013 \\,\\mbox{\\AA}$,\n$\\theta=103.2\\,{}^{\\circ}$, 6-31G,\n\\cite{HandyKnowlesSomasundram85,KnowlesAndrewsAmosHandyPople91})} \\\\\n SCF    &  -55.367729 &   0.00   & MP5    &  -55.520522 &  96.14 \\\\\n MP2    &  -55.489967 &  76.91   & F5     &  -55.521721 &  96.89 \\\\\n MP3    &  -55.504270 &  85.91   & GF5    &  -55.521724 &  96.90 \\\\\n MP4    &  -55.516470 &  93.59   & GF5b   &  -55.523319 &  97.90 \\\\\n F4     &  -55.521456 &  96.73   & [3,2]  &  -55.523696 &  98.14 \\\\\n $[2,2]$&  -55.521125 &  96.52   & [2,3]  &  -55.523706 &  98.14 \\\\\n $\\Pi$2 &  -55.526202 &  99.71   & FCI    &  -55.526658 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case m: NH${}_2$ (${}^2B_1$, $r=2\\times 1.013 \\,\\mbox{\\AA}$,\n$\\theta=103.2\\,{}^{\\circ}$, 6-31G,\n\\cite{HandyKnowlesSomasundram85,KnowlesAndrewsAmosHandyPople91})} \\\\\n SCF    &  -55.181593 &   0.00   & MP5    &  -55.418215 &  91.36 \\\\\n MP2    &  -55.357617 &  67.96   & F5     &  -55.420149 &  92.11 \\\\\n MP3    &  -55.375463 &  74.85   & GF5    &  -55.420173 &  92.12 \\\\\n MP4    &  -55.409165 &  87.87   & GF5b   &  -55.412429 &  89.13 \\\\\n F4     &  -55.421427 &  92.60   & [3,2]  &  -55.432093 &  96.72 \\\\\n $[2,2]$&  -55.426946 &  94.73   & [2,3]  &  -55.432101 &  96.72 \\\\\n $\\Pi$2 &  -55.478348 & 114.58   & FCI    &  -55.440593 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case n: NH${}_3$ ($r=1.91165 \\,a_0$,\n$\\theta=106.7\\,{}^{\\circ}$, DZ,\n\\cite{HarrisonHandy83,BartlettSekinoPurvis83})} \\\\\n SCF    &  -56.165931 &   0.00   & F4     &  -56.291937 &  99.47 \\\\\n MP2    &  -56.277352 &  87.95   & $[2,2]$&  -56.291782 &  99.35 \\\\\n MP3    &  -56.285281 &  94.21   & $\\Pi$2 &  -56.292636 & 100.02 \\\\\n MP4    &  -56.290692 &  98.48   & FCI    &  -56.292612 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case o: CO (${}^1\\Sigma$,\nDZ,\n\\cite{KucharskiNogaBartlett89})} \\\\\n SCF    & -112.760093 &   0.00   & MP5    & -113.059117 &  98.36 \\\\\n MP2    & -113.045824 &  93.99   & F5     & -113.059254 &  98.41 \\\\\n MP3    & -113.044659 &  93.61   & GF5    & -113.060859 &  98.93 \\\\\n MP4    & -113.067749 & 101.20   & GF5b   & -113.073579 & 103.12 \\\\\n F4     & -113.067469 & 101.11   & [3,2]  & -113.062479 &  99.47 \\\\\n $[2,2]$& -113.069566 & 101.80   & [2,3]  & -113.062539 &  99.49 \\\\\n $\\Pi$2 & -113.072074 & 102.62   & CCSDT  & -113.064100 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case p: C${}_2$H${}_2$ (${}^1\\Sigma_g$,\nDZP,\n\\cite{KucharskiNogaBartlett89})} \\\\\n SCF    &  -76.831819 &   0.00   & MP5    &  -77.118892 & 102.18 \\\\\n MP2    &  -77.085307 &  90.23   & F5     &  -77.120192 & 102.65 \\\\\n MP3    &  -77.097232 &  94.47   & GF5    &  -77.122141 & 103.34 \\\\\n MP4    &  -77.111732 &  99.63   & GF5b   &  -77.117205 & 101.58 \\\\\n F4     &  -77.113928 & 100.42   & [3,2]  &  -77.127079 & 105.10 \\\\\n $[2,2]$&  -77.114110 & 100.48   & [2,3]  &  -77.127731 & 105.33 \\\\\n $\\Pi$2 &  -77.116235 & 101.24   & CCSDT  &  -77.112760 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case q: O${}_3$ (${}^1A_1$,\nDZP,\n\\cite{KucharskiNogaBartlett89})} \\\\\n SCF    & -224.295920 &   0.00   & MP5    & -224.929902 &  97.54 \\\\\n MP2    & -224.931924 &  97.86   & F5     & -224.933812 &  98.15 \\\\\n MP3    & -224.888104 &  91.11   & GF5    & -224.934513 &  98.25 \\\\\n MP4    & -224.952784 & 101.07   & GF5b   & -224.952167 & 100.97 \\\\\n F4     & -224.941418 &  99.32   & [3,2]  & -224.938301 &  98.84 \\\\\n $[2,2]$& -224.950280 & 100.68   & [2,3]  & -224.938367 &  98.85 \\\\\n $\\Pi$2 & -224.952387 & 101.00   & CCSDT  & -224.945859 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case r: CN (${}^2\\Sigma$, $r=1.1619\\, \\,\\mbox{\\AA}$,\nSTO-3G, RMP\n\\cite{KnowlesAndrewsAmosHandyPople91})} \\\\\n SCF    &  -90.99752 &   0.00    & MP5    &  -91.16157 &  95.07 \\\\\n MP2    &  -91.15437 &  90.90    & F5     &  -91.16165 &  95.12 \\\\\n MP3    &  -91.14799 &  87.20    & GF5    &  -91.16166 &  95.12 \\\\\n MP4    &  -91.16300 &  95.90    & GF5b   &  -91.16360 &  96.24 \\\\\n F4     &  -91.16133 &  94.93    & [3,2]  &  -91.16297 &  95.88 \\\\\n $[2,2]$&  -91.16321 &  96.02    & [2,3]  &  -91.16297 &  95.88 \\\\\n $\\Pi$2 &  -91.16426 &  96.63    & FCI    &  -91.17008 & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case s: CN (${}^2\\Sigma$, $r=1.1619\\, \\,\\mbox{\\AA}$,\nSTO-3G, Hubac-Carsky,\n\\cite{Handy94,HubacCarsky80})} \\\\\n SCF    &  -90.99752  &   0.00   & MP5    &  -91.12039  &  71.20 \\\\\n MP2    &  -91.17762  & 104.37   & F5     &  -91.15212  &  89.59 \\\\\n MP3    &  -91.14160  &  83.50   & GF5    &  -91.15998  &  94.15 \\\\\n MP4    &  -91.19422  & 113.99   & GF5b   &  -91.18190  & 106.85 \\\\\n F4     &  -91.17389  & 102.21   & [3,2]  &  -91.16350  &  96.19 \\\\\n $[2,2]$&  -91.18753  & 110.11   & [2,3]  &  -91.16359  &  96.24 \\\\\n $\\Pi$2 &  -91.19152  & 112.42   & FCI    &  -91.17008  & 100.00 \\\\\n\\hline\n\\multicolumn{6}{c}{Case t: H${}_2$O  ($r=1.5 \\times 0.967\\, \\,\\mbox{\\AA}$,\n$\\theta=107.6\\,{}^{\\circ}$,\n6-21G,\\cite{HandyKnowlesSomasundram85})} \\\\\n RHF    &  -75.707206 &   0.00        & UMP5    &  -75.853895 & 76.41 \\\\\n UHF    &  -75.735012 &  14.48        & F5      &  -75.855560 & 77.28 \\\\\n UMP2   &  -75.829388 &  63.65        & GF5     &  -75.856608 & 77.82 \\\\\n UMP3   &  -75.836823 &  67.52        & GF5b    &  -75.850870 & 74.84 \\\\\n UMP4   &  -75.848211 &  73.45        & [3,2]   &  -75.862349 & 80.81 \\\\\n F4     &  -75.851276 &  75.05        & [2,3]   &  -75.862421 & 80.85 \\\\\n $[2,2]$&  -75.851994 &  75.42        & FCI     &  -75.899180 &100.00 \\\\\n $\\Pi$2 &  -75.857074 &  78.07        &         &             & \\\\\n\\hline\n\\multicolumn{6}{c}{Case u: H${}_2$O  ($r=2 \\times 0.967\\, \\,\\mbox{\\AA}$,\n$\\theta=107.6\\,{}^{\\circ}$,\n6-21G,\\cite{HandyKnowlesSomasundram85})} \\\\\n RHF    &  -75.491406 &  0.00         & UMP5    &  -75.763370 & 90.72 \\\\\n UHF    &  -75.699298 & 69.35         & F5      &  -75.763704 & 90.83 \\\\\n UMP2   &  -75.754669 & 87.82         & GF5     &  -75.763826 & 90.88 \\\\\n UMP3   &  -75.760219 & 89.67         & GF5b    &  -75.763657 & 90.82 \\\\\n UMP4   &  -75.762422 & 90.41         & [3,2]   &  -75.764089 & 90.96 \\\\\n F4     &  -75.763098 & 90.63         & [2,3]   &  -75.764104 & 90.97 \\\\\n $[2,2]$&  -75.762941 & 90.58         & FCI     &  -75.791180 &100.00 \\\\\n $\\Pi$2 &  -75.763281 & 90.69         &         &             & \\\\\n\\hline\n\\hline\n\\end{longtable}\n\n\nApart\nfrom these problematic cases, it is seen that in case\nm corresponding to NH${}_2$ at twice the equilibrium distances, the errors are\nrather high. Excluding this case also, one may study the performance of\nthe correlation energy estimators statistically as shown in Table\n\\ref{tab1a}. Plotted are the maximal error, the mean absolute error,\nthe root mean square (rms) absolute error, and the mean percentage of\nthe correlation energy as obtained with the various methods. In\ncases o, p, and q corresponding to the molecules CO,\nC${}_2$H${}_2$, O${}_3$, respectively, no FCI result is\navailable. The statistical comparison is done\nonce excluding these cases, and once including these cases where as\nreference for the error calculation the CCSDT result is taken.\nFor these cases, the given correlation energies\nshould thus be taken with care. Carefully designed fourth order methods like\n$\\Pi$2 yield correlation energy estimates that can compete with fifth\norder results. As regards the fifth order methods, it seems that the\nGoldhammer-Feenberg estimator GF5 is slightly superior to the Feenberg\nenergy F5, and the somewhat {\\em ad hoc} estimator GF5b performs\nsurprisingly well. Among the Pad\\'e approximants, the $[3,2]$ approximant (that\nis invariant under the Feenberg scaling) is a rather successful\ncorrelation estimator while the $[2,3]$ approximant performs very\nsimilarly. Other Pad\\'e approximants (not displayed in Table\n\\ref{tab1}) do not perform as well as the ones given in this table when\napplied to the same data.\n\n\\begin{table}\n\\caption{Statistical comparison of various correlation energy\nestimators}\\label{tab1a}\n\\begin{tabular*}{\\linewidth}{@{}l@{\\extracolsep{\\fill}}rrrr@{}}\n\\hline\nMethod   & max $\\vert error\\vert$   & mean $\\vert error\\vert$   & rms\n$\\vert error\\vert$   & mean $\\%$Corr\\\\\n\\hline\n\\multicolumn{5}{@{}c@{}}{Sampling 14 cases (a-l,n,r)}\\\\\nF4      & 0.02515   & 0.00433 & 0.00767 &  98.3\\\\\n$[2,2]$ & 0.01122   & 0.00319 & 0.00433 &  98.3\\\\\n$\\Pi$2  & 0.00975   & 0.00199 & 0.00329 & 100.1\\\\\n\\hline\n\\multicolumn{5}{@{}c@{}}{Sampling 17 cases (a-l,n-r)}\\\\\nF4      & 0.02515   & 0.00409 & 0.00710 &  98.6\\\\\n$[2,2]$ & 0.01122   & 0.00329 & 0.00430 &  98.8\\\\\n$\\Pi$2  & 0.00975   & 0.00269 & 0.00398 & 100.3\\\\\n\\hline\n\\multicolumn{5}{@{}c@{}}{Sampling 13 cases (a-l,r)}\\\\\nF5      & 0.01774   & 0.00407 & 0.00628 &  98.2\\\\\nGF5     & 0.01692   & 0.00394 & 0.00607 &  98.2\\\\\nGF5b    & 0.02870   & 0.00400 & 0.00834 &  99.0\\\\\n$[3,2]$ & 0.00711   & 0.00228 & 0.00308 &  99.1\\\\\n$[2,3]$ & 0.00711   & 0.00224 & 0.00307 &  99.1\\\\\n\\hline\n\\multicolumn{5}{@{}c@{}}{Sampling 16 cases (a-l,o-r)}\\\\\n F5      & 0.01774 & 0.00483 & 0.00678 & 98.5\\\\\n GF5     & 0.01692 & 0.00470 & 0.00664 & 98.6\\\\\n GF5b    & 0.02870 & 0.00452 & 0.00811 & 99.6\\\\\n $[3,2]$ & 0.01432 & 0.00332 & 0.00492 & 99.4\\\\\n $[2,3]$ & 0.01497 & 0.00332 & 0.00503 & 99.5\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\nA careful analysis of the data in Table \\ref{tab1} reveals that the\ncorrelation energy estimation based on MP perturbation theory is the\nbetter the closer one is to the optimal geometries of the molecule\nunder consideration. This is not very much surprising since it is\nwell-known that the quality of the MP series deteriorates with\nincreasing separations from the equilibrium geometries. Compare for\ninstance the triples of cases (a,b,c) for BH, (d,e,f) for HF,\n(h,i,j) for H${}_2$O, and (k,l,m) for NH${}_2$,\nwith ratios 1:1.5:2 of the relevant distances. The values away from the\nequilibrium geometries may or may not be reliable. The data, however,\nsuggest that then the correlation energy estimates are reliable if ---\nas in cases f for HF at $2\\times r_e$ and i for H${}_2$O at\n$1.5\\times r_e$--- the values of $\\Pi$2, F4 and $[2,2]$ do not\ndiffer too much from each other. In this situation, the $\\Pi$2\nestimator seems to provide the best results. On the other hand, large\ndifferences between the estimates $\\Pi$2, F4 and $[2,2]$ --- as in the\ncases j for H${}_2$O at $2\\times r_e$ and m for NH${}_2$ at\n$2\\times r_e$ --- clearly indicate that in these cases more\nsophisticated methods (for instance the $\\Lambda$ transformation\n\\cite{SchmidtWarkenHandy93,DietzSchmidtWarkenHess92,%\nDietzSchmidtWarkenHess93a,DietzSchmidtWarkenHess93b,%\nDietzSchmidtWarkenHess93c,DietzSchmidtWarkenHess94a,%\nDietzSchmidtWarkenHess94b} or multi-reference methods\n\\cite{JeziorskiMonkhorst81,Hose89,SteinbornSilver96,%\nLindgren74,WahlDas77,%\nWerner87,MurphyMessmer93}) are needed to\ncalculate the correlation energies reliably. As regards the fifth order\nestimates, it is similarly seen that a large spread of the values of\nthe various estimates reveals that the MP based methods do not provide\nsufficiently accurate results. Reversely, a small spread of the various\nestimates indicates that with a high probability, the (R)MP based\ncorrelation energy estimates are reliable.\n\nComparing fourth and fifth order based estimators, it is seen that the\nlatter do not always provide better estimates of the correlation\nenergy. In many cases, the $\\Pi$2 estimate that is based on fourth\norder, provides results of comparable quality.\n\n\n\\begin{table}\n\\caption{Dissociation barrier (kJ\/mol) of H${}_2$CO$\\longrightarrow$H${}_2+{}$CO\nusing a TZ2P basis at MP2 geometries ${}^{a}$}\\label{tab2}\n\\begin{tabular*}{\\linewidth}{@{}l@{\\extracolsep{\\fill}}...l@{}}\n\\hline\n Method &  \\multicolumn{1}{c}{Minimum}     &\n \\multicolumn{1}{c}{Transition state} & \\multicolumn{1}{c@{}}{Barrier\n} & Ref. \\\\\n\\hline\n SCF     &  -113.912879 & -113.748693 &  431.1 & \\cite{SchmidtWarkenHandy93}\\\\\n MP2     &  -114.329202 & -114.182435 &  385.3 & \\cite{SchmidtWarkenHandy93}\\\\\n MP3     &  -114.334186 & -114.185375 &  390.7 & \\cite{SchmidtWarkenHandy93}\\\\\n MP4     &  -114.359894 & -114.219892 &  367.6 & \\cite{SchmidtWarkenHandy93}\\\\\n F4      &  -114.360838 & -114.220603 &  368.2 & \\cite{SchmidtWarkenHandy93}\\\\\n $[2,2]$ &  -114.362267 & -114.223409 &  364.6 & This work\\\\\n $\\Pi$2  &  -114.364840 & -114.227767 &  359.9 & This work\\\\\n BE${}^{b}$\n         &              &             &  360   & \\cite{DupuisLesterLengsfieldLiu83}\\\\\n\\hline\n\\end{tabular*}\n\\\\\n${}^{a}$ \\cite{SchmidtWarkenHandy93} \\\\\n${}^{b}$ Best estimate \\cite{DupuisLesterLengsfieldLiu83}\n\\end{table}\n\n\\begin{table}%\n\\caption{Barrier height and heat of reaction (kJ\/mol) for\nCH${}_3+{}$C${}_2$H${}_4\\longrightarrow{}$C${}_3$H${}_7$ with a\n6-31G${}^{*}$\nbasis${}^{a}$}\\label{tab3}%\n\\begin{tabular*}{\\linewidth}{@{}l@{\\extracolsep{\\fill}}rrrrl@{}}\n\\hline\nMethod    &  \\multicolumn{1}{l}{Reactants}      &\n\\multicolumn{1}{l}{TS${}^{b}$} & \\multicolumn{1}{l}{Product}         &\n\\multicolumn{1}{c}{Barrier} & \\multicolumn{1}{c@{}}{HR${}^{c}$}\\\\\n\\hline\n RHF      &  $-$117.585674  &  $-$117.553736  &  $-$117.626572  & 83.8 & $-$107.4 \\\\\n RMP2     &  $-$117.967150  &  $-$117.952092  &  $-$118.014126  & 39.5 & $-$123.3 \\\\\n RMP3     &  $-$118.004259  &  $-$117.986543  &  $-$118.049999  & 46.5 & $-$120.1 \\\\\n RMP4     &  $-$118.022888  &  $-$118.008072  &  $-$118.066816  & 38.9 & $-$115.3 \\\\\n F4       &  $-$118.028674  &  $-$118.014137  &  $-$118.071720  & 38.2 & $-$113.0 \\\\\n $[2,2]$  &  $-$118.027529  &  $-$118.013226  &  $-$118.070703  & 37.6 & $-$113.3 \\\\\n $\\Pi$2   &  $-$118.030923  &  $-$118.017302  &  $-$118.073432  & 35.8 & $-$111.6 \\\\\n exp.${}^{d}$     &                 &                 &                 & 33.1 & $-$107   \\\\\n\\hline\n\\end{tabular*}\n\\\\\n${}^{a}$ \\cite{SchmidtWarkenHandy93} \\\\\n${}^{b}$ Transition state\\\\\n${}^{c}$ Heat of reaction\\\\\n${}^{d}$ \\cite{Kerr72,XXX85,CastelhanoGriller82,SchmidtWarkenHandy93}\n\\end{table}\n\n\n\n\n\n\n\n\n\nIn Tables \\ref{tab2} and \\ref{tab3} the correlation energy\nestimators are used to calculate the dissociation barrier for\nH${}_2$CO$\\longrightarrow$H${}_2+{}$CO, and the barrier height\nand the heat of reaction for\nCH${}_3+{}$C${}_2$H${}_4\\longrightarrow{}$C${}_3$H${}_7$.\n\n\nIn both examples, the calculation is based on known M{\\o}ller-Plesset\nenergies up to fourth order \\cite[Tab. 2-4]{SchmidtWarkenHandy93}. The\nresults show that reliable correlation energy estimates as provided by\nthe Feenberg energy F4 \\cite{SchmidtWarkenHandy93}, the Pad\\'e\napproximant $[2,2]$, and the effective characteristic polynomial\nestimate $\\Pi$2 lead to good agreement with experimental data. The\n$\\Pi$2 estimator yields in both cases the best results.\n\nIn summary, it has been shown that the availability of various\nestimators based on (R)MP results allows in many cases the\naccurate calculation of the correlation energy at negligible\nadditional computational costs. Also, larger deviations between\nthe values indicate clearly cases where further work is\nnecessary.\n\nFinally, we note that the above estimators are expected to be\nuseful to improve convergence of perturbation series for the\nenergies also for the multi-reference case. This conjecture is\na promising topic for further investigations.\n\n\\begin{ack}\nI thank Dr. E. J. Weniger and Prof. Dr. J. {{\\v C}{\\' \\i}{\\v\nz}ek} for discussions regarding the effective characteristic\npolynomial approach. I am grateful to Prof.\\ Dr.\\\nE.\\ O.\\ Steinborn for his support and the excellent working\nconditions at Regensburg. Help of the staff of the computing\ncentre of the University  of Regensburg is thankfully\nacknowledged.\n\\end{ack}\n\n\n\n\n\\input{varpt.bbl}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum general relativity is known to experience problems with renormalizability. At the one-loop level pure general relativity is renormalizable only on-shell \\cite{tHooft:1974toh}. When matter is added to the model, then even on-shell renormalization does not take place \\cite{tHooft:1974toh}. At the two-loop level even pure general relativity is completely non-renormalizable \\cite{Goroff:1985th}. Because of this it can hardly be considered as a fundamental theory.\n\nEffective field theory technique provides a way to account for quantum corrections to general relativity consistently \\cite{Burgess:2003jk,Donoghue:2012zc,Donoghue:1994dn} and avoid problems associated with non-renor\\-malizability. The technique is based on a premise that general relativity describes gravitational interaction at some energy scale $\\mu$ below the Planck scale. From the normalization scale $\\mu$ the theory is extended to the low energy regime via standard loop corrections. Applicability of the theory is limited only to energies below the normalization scale and it is assumed that all divergences in loop corrections can be renormalized within the complete theory.\n\nThe effective theory for general relativity allows one to obtain some verifiable predictions, for instance to recover the low energy effective action \\cite{Donoghue:1994dn,Burgess:2003jk}, to obtain corrections to the Newton law \\cite{BjerrumBohr:2002kt,Bjerrum-Bohr:2016hpa}, etc \\cite{Calmet:2018qwg,Calmet:2018uub}.\n\nIn this paper we highlight a phenomenon that we call ``graviton mixing''. It is similar to the fermion mixing in the standard model. In the electroweak sector of the standard model neutrino free states which are eigenstates of the mass operator $(\\nu_1,\\nu_2,\\nu_3)$ are given as a superposition of eigenstates of an interaction operator $(\\nu_e,\\nu_\\mu,\\nu_\\tau)$. Their relation is described by the PMNS matrix \\cite{Pontecorvo:1957qd,Maki:1962mu,Tanabashi:2018oca}. To put it otherwise, fermion mixing takes place, when a superposition of fermion state with well-defined mass is coupled to gauge bosons \\cite{Tanabashi:2018oca}.\n\nA similar phenomenon takes place in gravity. General relativity contains massless spin-2 degree of freedom on shell, while off-shell it receives an additional massless scalar degree of freedom \\cite{Fierz:1939ix,Hinterbichler:2011tt,RN7}. In the classical theory these degrees of freedom are coupled to an energy-momentum source $T^{\\mu\\nu}$ in the same way:\n\\begin{align}\n  T^{\\mu\\nu} h_{\\mu\\nu} = T^{\\mu\\nu} h_{\\mu\\nu}^{(s=2)} + T^{\\mu\\nu} h_{\\mu\\nu}^{(s=0)}.\n\\end{align}\nWithin an effective theory this is not so, as the correspondent couplings are modified due to quantum effects. In full analogy with the case of the standard model one should introduce a mixing matrix $\\mathcal{M}_{\\mu\\nu\\alpha\\beta}$ that couples a linear combination of spin-2 and spin-0 states to an energy-momentum source:\n\\begin{align}\n  &T^{\\mu\\nu} h_{\\mu\\nu} \\to T^{\\mu\\nu} \\mathcal{M}_{\\mu\\nu\\alpha\\beta} h^{\\alpha\\beta}= \\nonumber \\\\\n  &=T^{\\mu\\nu} ~\\left[ M_{\\mu\\nu\\alpha\\beta} h^{(s=2)}{}^{\\alpha\\beta} + M_{\\mu\\nu\\alpha\\beta} h^{(s=0)}{}^{\\alpha\\beta}\\right].\n\\end{align}\n\nThe main focus of the paper is to draw an attention to the graviton mixing and to highlight its role within effective gravity. In the next section we discuss a motivation behind the graviton mixing in more details and provide a way to construct a graviton mixing matrix. Then we show that the mixing has a non-trivial influence on processes involving matter. Namely, the structure of virtual graviton exchange processes is strongly affected by the mixing. We conclude with a discussion of the role of graviton mixing.\n\n\\section{Graviton Mixing}\n\nWithin general relativity an interaction between a week gravitational field $h_{\\mu\\nu}$ and an energy-momentum source is given by the following Lagrangian:\n\\begin{align}\n  \\mathcal{L}_\\text{int} = T^{\\mu\\nu} ~I_{\\mu\\nu\\alpha\\beta} h^{\\alpha\\beta}.\n\\end{align}\nHere $I_{\\mu\\nu\\alpha\\beta}$ is the gauge-invariant generalization of a rank-$2$ unit tensor\n\\begin{align}\n  I_{\\mu\\nu\\alpha\\beta} \\overset{\\text{def}}{=} \\cfrac12 \\left( \\theta_{\\mu\\alpha}\\theta_{\\nu\\beta}+\\theta_{\\mu\\beta} \\theta_{\\nu\\alpha} \\right) .\n\\end{align}\nAnd the standard projectors $\\theta_{\\mu\\nu} = \\eta_{\\mu\\nu} - k_\\mu k_\\nu\/k^2$ are used. The interaction contains spin-$2$ and spin-$0$ parts:\n\\begin{align}\\label{interaction_gauge-invariant_form}\n  L_\\text{int}&= T^{\\mu\\nu} P^2_{\\mu\\nu\\alpha\\beta} h^{\\alpha\\beta} + T^{\\mu\\nu} P^0_{\\mu\\nu\\alpha\\beta} h^{\\alpha\\beta} \\nonumber \\\\\n  &=T^{\\mu\\nu} h^{(s=2)}_{\\mu\\nu} + T^{\\mu\\nu} h^{(s=0)}_{\\mu\\nu}.\n\\end{align}\nWhere $P^2$ and $P^0$ are Nieuwenhuizen operators \\cite{VanNieuwenhuizen:1973fi,Accioly:2000nm}:\n\\begin{align}\n  \\begin{cases}\n    P^2_{\\mu\\nu\\alpha\\beta} &\\overset{\\text{def}}{=} \\cfrac12 \\left( \\theta_{\\mu\\alpha} \\theta_{\\nu\\beta} + \\theta_{\\mu\\beta} \\theta_{\\nu\\alpha}\\right) - \\cfrac13 ~ \\theta_{\\mu\\nu} \\theta_{\\alpha\\beta}, \\\\\n    P^0_{\\mu\\nu\\alpha\\beta} &\\overset{\\text{def}}{=} \\cfrac13 ~ \\theta_{\\mu\\nu} \\theta_{\\alpha\\beta}.\n  \\end{cases}\n\\end{align}\nThey are orthogonal projectors on spin-2 and spin-0 states:\n\\begin{align}\n  \\begin{cases}\n    P^2_{\\mu\\nu\\rho\\sigma} P^2{}^{\\rho\\sigma}{}_{\\alpha\\beta} &= P^2_{\\mu\\nu\\alpha\\beta}, \\\\\n    P^0_{\\mu\\nu\\rho\\sigma} P^0{}^{\\rho\\sigma}{}_{\\alpha\\beta} &= P^0_{\\mu\\nu\\alpha\\beta}, \\\\\n    P^2_{\\mu\\nu\\rho\\sigma} P^0{}^{\\rho\\sigma}{}_{\\alpha\\beta} &= 0 .\n  \\end{cases}\n\\end{align}\nThus, in general relativity spin-$2$ and spin-$0$ graviton states share the matter coupling constant.\n\nWithin a more general setup of effective theory this feature can hardly holds. Even if general relativity does describe gravity at some energy scale $\\mu$ below the Planck scale, then loop corrections can still induce a non-minimal coupling between graviton spin states and matter. In the most general case one can account for such effects by introducing the graviton mixing matrix:\n\\begin{align}\n  \\mathcal{M}_{\\mu\\nu\\alpha\\beta} = I_{\\mu\\nu\\alpha\\beta} + A P^2_{\\mu\\nu\\alpha\\beta} + B P^0_{\\mu\\nu\\alpha\\beta}\n\\end{align}\nwith $A$ and $B$ being mixing parameters. The matrix redefines the form of the interaction between gravity and matter:\n\\begin{align}\n  \\mathcal{L}_\\text{int}\\to T^{\\mu\\nu} \\mathcal{M}_{\\mu\\nu\\alpha\\beta} h^{\\alpha\\beta}.\n\\end{align}\nA few comment on the mixing are due.\n\nFirstly, the mixing parameters $A$ and $B$ should be given in terms of momentum expansions. Loop corrections can be consistently taken into account within effective field theory \\cite{Donoghue:1994dn,Burgess:2003jk}. The form of the mixing parameters can be recovered via dimension reasoning:\n\\begin{align}\\label{mixing_parameters_expansion}\n  \\begin{cases}\n    A (\\kappa^2\\square) = a_1 \\kappa^2 \\square + a_2 (\\kappa^2 \\square)^2 + \\cdots \\\\\n    B (\\kappa^2\\square)= b_1 \\kappa^2 \\square + b_2 (\\kappa^2 \\square)^2 + \\cdots\n  \\end{cases}.\n\\end{align}\nThese expansions are due, as the $n$-loop graviton correction is suppressed by the factor $\\kappa^{2n}$. Thus, constants $a_1$, $b_1$ contain data on one-loop corrections, $a_2$ and $b_2$ describe two-loop corrections, etc. The explicit values of these constants cannot be evaluated within the effective theory, but can be calculated within quantum gravity models.\n\nSecondly, mixing parameters are defined by the particle content of the effective theory. Alongside graviton loop corrections the energy-momentum tensor is also affected by matter loop corrections. It is safe to consider only renormalizable interactions with dimensionless couplings. Corrections from such matter loops do not influence the structure of the momentum expansions \\eqref{mixing_parameters_expansion} and only contribute to coefficients $a_i$, $b_i$. Cases of non-renormalizable interactions and interactions with dimensionful couplings require a special treatment and lie beyond the scope of this paper.\n\nSummarizing all of the above, within an effective theory for general relativity the graviton mixing appears naturally due to loop corrections. The mixing parameters are defined by the structure of the underlying fundamental theory, but their form can be recovered via dimension reasoning. But most importantly, the mixing must be taken into account for processes involving matter states.\n\nThe importance of the graviton mixing can be easily illustrated via a simple example of a virtual graviton exchange. To proceed with the goal one should use the gauge-invariant part of the graviton propagator \\cite{Accioly:2000nm}:\n\\begin{align}\n  G_{\\mu\\nu\\alpha\\beta} = \\cfrac{i}{k^2} \\left( P^2_{\\mu\\nu\\alpha\\beta} -\\cfrac12 ~ P^0_{\\mu\\nu\\alpha\\beta} \\right).\n\\end{align}\nTo account for the graviton self-energy the following graviton polarization operator should be used:\n\\begin{align}\n  \\Pi_{\\mu\\nu\\alpha\\beta} = i \\mathcal{N} \\kappa^2 k^4 \\left[ P^2_{\\mu\\nu\\alpha\\beta} + \\zeta P^0_{\\mu\\nu\\alpha\\beta} \\right].\n\\end{align}\nIn this expression $\\mathcal{N}$ and $\\zeta$ are numerical coefficients defined by the structure of the complete quantum model. We do not specify their values, as they are irrelevant for the reasoning to be presented. Moreover, this approach goes in line with classical results \\cite{tHooft:1974toh,Goroff:1985th}.\n\nThis allows one to recover the resummed graviton propagator:\n\\begin{align}\\label{resummed_propagator}\n  \\mathcal{G}_{\\mu\\nu\\alpha\\beta}=&G_{\\mu\\nu\\alpha\\beta} + (G \\Pi G)_{\\mu\\nu\\alpha\\beta}+\\cdots\\nonumber\\\\\n  =&\\cfrac{i}{k^2} \\left(P^2_{\\mu\\nu\\alpha\\beta}-\\cfrac12~P^0_{\\mu\\nu\\alpha\\beta}\\right) -\\cfrac{i~P^2_{\\mu\\nu\\alpha\\beta}}{k^2-\\cfrac{-1}{\\mathcal{N}\\kappa^2}} \\\\\n  & +\\cfrac{i~\\frac12\\,P^0_{\\mu\\nu\\alpha\\beta}}{k^2-\\cfrac{1}{\\frac12 ~\\zeta\\mathcal{N}\\kappa^2}} \\nonumber ~.\n\\end{align}\nIn full agreement with classical results \\cite{tHooft:1974toh,Stelle:1976gc} the propagator contains additional poles corresponding to spin-$0$ massive states and spin-$2$ massive ghosts. These poles mark the applicability limits of the effective theory.\n\nIn such a setup an exchange of a virtual graviton between two energy-momentum sources is described by the following expression:\n\\begin{align}\\label{effective_propagator_definition}\n  T^{\\mu\\nu}\\mathcal{M}_{\\mu\\nu\\alpha\\beta} \\left( G^{\\alpha\\beta\\rho\\sigma} + \\left( G\\,  \\mathcal{M} \\, \\Pi \\,  \\mathcal{M} \\,  G \\right)^{\\alpha\\beta\\rho\\sigma + \\cdots}\\right) \\mathcal{M}_{\\rho\\sigma\\lambda\\tau} T^{\\lambda\\tau}=T^{\\mu\\nu} \\overline{\\mathcal{G}}_{\\mu\\nu\\alpha\\beta}T^{\\mu\\nu},\n\\end{align}\n\\begin{align}\\label{effective_propagator}\n  \\overline{\\mathcal{G}}_{\\mu\\nu\\alpha\\beta} =\\cfrac{i}{k^2} \\left( (1+A)^2~ P^2_{\\mu\\nu\\alpha\\beta}-\\cfrac12 ~(1+B)^2 ~P^0_{\\mu\\nu\\alpha\\beta} \\right) \\\\\n  -\\cfrac{i~(1+A)^2~P^2_{\\mu\\nu\\alpha\\beta}}{k^2 -\\cfrac{-1}{\\mathcal{N}\\kappa^2\\,(1+A)^2} } +\\cfrac12~\\cfrac{i~(1+B)^2~P^0_{\\mu\\nu\\alpha\\beta}}{k^2 -\\cfrac{1}{\\frac12 \\zeta\\mathcal{N} \\kappa^2\\,(1+B)^2}}\\nonumber ~.\n\\end{align}\nIn this expression $\\overline{\\mathcal{G}}$ is not a resummed graviton propagator, but a quantity that accounts both for graviton propagation and for the mixing. To be short we will call $\\overline{\\mathcal{G}}$ the effective propagator.\n\nIt may seem that effective and resummed propagators have a similar structure, but this is not so. The mixing coefficients $A$ and $B$ are functions of the transferred momentum $k^2$, so they alter the pole structure. For instance, if $A^2$ is proportional to $k^2+1\/(\\mathcal{N} \\kappa^2)$, then the corresponding matrix element is free from the ghost pole. At the same time, if either $A^2$ or $B^2$ admit additional poles, then these poles are to appear in an expression for the matrix element.\n\nThe difference between the pole structure of \\eqref{resummed_propagator} and \\eqref{effective_propagator} shows the following feature of the mixing. The graviton mixing alters the pole structure of the amplitudes involving matter states. This feature has two immediate corollaries.\n\nFirstly, as the graviton mixing influence the pole structure of amplitudes ivolving matter, it also influence the applicability of the effective model. Poles of the resummed graviton propagator \\eqref{resummed_propagator} indicate the energy scale at which the effective theory applicability should be put under question, as the ghost instability can be triggered \\cite{Solomon:2017nlh}. Poles of the effective propagator \\eqref{effective_propagator} play the same role and indicate the limits of the effective theory applicability. Due to the mixing the position of the pole corresponding to ghost states can be altered, which changes the area of applicability for the effective theory.\n\nSecondly, the graviton mixing can be probed empirically. The matrix element of a virtual graviton exchange can be studied empirically, as it defines the form of the Newton law. Namely, it can be studied in the terrestrial environment via experiments of E\\\"ot-Wash type \\cite{Hoyle:2004cw}. The structure of the corresponding matrix element is defined via the effective propagator \\eqref{effective_propagator} which contains data on the graviton mixing. Therefore the graviton mixing can be put to a direct empirical verification.\n\nThis conclusion is also independently supported with well-known results considering corrections to the Newton potential \\cite{BjerrumBohr:2002kt,Bjerrum-Bohr:2016hpa}. In these papers the graviton mixing is not separated explicitly, but it is taken into account. Without the mixing corrections to the Newton potential would have a Yukawa-like form due to the new poles associated with massive states. As the mixing is accounted for, the one-loop effective non-relativistic potential has corrections of a different form.\n\nFinally, one can argue that the graviton mixing is the reason behind the fact the it is impossible to introduce a universal definition of a running gravitational coupling \\cite{Anber:2011ut}. As it was highlighted before, within general relativity graviton spin components share the matter coupling. Because of this one can introduce a single coupling constant, the Newton constant, to describe these interactions. When loop corrections are taken into account so the spin components are mixed, the corresponding coupling receive different correction and can no longer be described by a single coupling.\n\n\\section{Conclusion}\n\nIn this paper we discussed a mechanism of graviton mixing and its possible application within effective field theory for gravity.\n\nWe define the graviton mixing in the following way. Within general relativity interaction between matter and spin-2 gravitons (that exist on- and off-shell) alongside with spin-0 gravitons (that exist only off-shell) is defined uniquely. Within effective theory this feature does not hold, as loop corrections can change the corresponding couplings of graviton spin states. We introduce the graviton mixing matrix that accounts for possible mixing of spin-2 and spin-0 graviton states in interaction with an energy-momentum source.\n\nWe have shown that graviton mixing strongly influence amplitudes containing matter states. A virtual graviton exchange process was used as an illustrative example. Formulae \\eqref{effective_propagator_definition} and \\eqref{effective_propagator} show that the poles structure of such a process is defined by the graviton mixing coefficients.\n\nThree conclusions can be made about the graviton mixing. First of all, the mixing can change the area of applicability of the effective theory due to the pole structure of the mixing parameters. Secondly, the mixing can be directly probed empirically. Namely, experiments of E\\\"ot-Wash type are sensitive to the graviton mixing. Thirdly, the graviton mixing is the reason behind an inability to define a universal running gravitational coupling. The mixing is due loop corrections that has different influence on spin-$2$ and spin-$0$ graviton states coupling to matter. Consequently the corresponding couplings can no longer be described by a single Newton constant.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Young Dense Clusters in the Galaxy and Beyond}\n\nAmong the many massive young clusters now known throughout the local\nuniverse, perhaps the most interesting to dynamicists are those in\nwhich stellar dynamical time scales are short enough that the cluster\ncan undergo significant structural change during the lifetimes of the\nmost massive stars.  In such clusters, dynamical evolution opens up\nnovel avenues for stellar and binary evolution, making possible the\ncreation of entirely new stellar species.  One obvious modification to\nstandard stellar evolutionary tracks arises from collisions and\nmergers between stars, and we focus on that here.  From this\nperspective, the clusters listed by Portegies Zwart et al.~(2004b;\nTable 1) represent an ideal combination of properties, having ages of\nless than a few million years and relaxation times of less than a few\ntens of millions of years.  In these clusters, dynamical evolution,\ntraditionally regarded as a ``slow'' process, actually occurs much\nmore rapidly than the stellar evolution of even the most massive\nstars.  In fact, cluster dynamics controls the early phases of these\nstars' lives.\n\nPortegies Zwart et al.~(2004b) are primarily concerned with the\nlifetimes and global structural evolution of young dense clusters in\nthe vicinity of the Galactic center.  In this paper we consider mainly\nthe stellar evolutionary aspects of life in such an extreme\nenvironment.  We start by investigating the circumstances under which\ncollisions are likely to occur, and how a cluster might find its way\ninto such a state.  We then present a scenario which may plausibly\nlead to the formation of very massive stars and (perhaps)\nintermediate-mass black holes (IMBHs) in sufficiently young, dense\nsystems.  Our results are based in large part on detailed $N$-body\nsimulations of model clusters.  Finally, we apply this scenario to\nrecent observations of the starburst galaxy M82.  In a companion\ncontribution, Baumgardt et al.~(2004) extend these ideas to the\nsubsequent evolution of a cluster containing a massive, compact\nobject.\n\n\n\\section{Stellar Collisions and Cluster Structure}\n\nWe are interested here in the possibility of runaway collisions\nleading to ultramassive stars.  To appreciate the conditions under\nwhich such runaways can occur, consider a massive object moving\nthrough a field of background stars of total mass density $\\rho$ and\nvelocity dispersion $v$.  We assume that the mass $M$ and radius $R$\nof the object are large compared to the masses and radii of other\nstars, and that all velocities are small enough that gravitational\nfocusing dominates the total cross section.  In that case, the\nobject's collision cross section is\n\\begin{equation}\n \t\\sigma \\approx 2\\pi G M R \/ v^2\\,,\n\\end{equation}\nnearly independent of the properties of the other stars.  The rate of\nincrease of the object's mass due to collisions is therefore\n\\begin{eqnarray}\n\t\\frac{dM}{dt} &\\approx& \\rho\\sigma v \\nonumber\\\\\n\t\t      &\\approx& 2\\pi G M R \\rho \/ v \\nonumber\\\\\n\t\t      &=& 6\\times10^{-11}\n\t\t\t\\left(\\frac{M}{M_\\odot}\\right)\n\t\t\t\\left(\\frac{R}{R_\\odot}\\right)\\nonumber\\\\\n\t\t&~&~~~~~~~~~~~~~~~\\times\\,\n\t\t\t\\left(\\frac{\\rho}{10^6\\,M_\\odot\/{\\rm pc}^3}\\right)\n\t\t\t\\left(\\frac{v}{10\\, {\\rm km\/s}}\\right)^{-1}\n\t\t\tM_\\odot\/{\\rm yr}\\,.\\ \\ \\ \\ \\ \n\\end{eqnarray}\nThus, if the object initially has $M=100 M_\\odot$ and $R=30 R_\\odot$,\nwe fix $v$ at 10 km\/s, and adopt a mass-radius relation $R\\propto\nM^{1\/2}$, we find that in order for the object to accrete\n$10^3M_\\odot$ of material in 5 Myr (to form an IMBH within the typical\nlifetime of a massive star), the local density must satisfy\n\\begin{equation}\n \t\\rho \\ \\ga\\  5\\times10^8\\, M_\\odot\/{\\rm pc}^3 \\ \\ =\\ \\\n\t \t\\rho_{crit},\\ \\  {\\rm say}\\,.\n\\end{equation}\nSuch a density is much higher than the mean density of any known star\ncluster, young or old.  For comparison, the average density of the\nArches cluster is $\\sim$$6\\times10^5\\,M_\\odot\/{\\rm pc}^3$, that of a\nfairly compact globular cluster is $\\sim$$10^4\\,M_\\odot\/{\\rm pc}^3$,\nwhile even the most concentrated globular cluster cores have densities\n$\\la 10^{6-7}\\, M_\\odot\/{\\rm pc}^3$.\n\nMight we be able to generate conditions more conducive to mergers by\nassuming that a cluster is born very centrally concentrated\n(e.g.~Portegies Zwart et al.~2004, Merritt et al.~2004)?  As a simple\nlimiting model of a very condensed cluster, consider the nearly\nisothermal system of total mass $M_c$ and half-mass radius $r_h$,\ndescribed by the density profile\n\\begin{eqnarray}\n\t\\rho(r) &=& \\frac{M_c}{8\\pi r_h r^2}\\,,\\\\\n\tM(r)    &=& {\\textstyle\\frac12}M_c\\left(\\frac{r}{r_h}\\right)\\,,\n\\end{eqnarray}\nfor $0 \\le r \\le 2r_h$.  Densities exceeding $\\rho_{crit}$ are found\nfor $r<r_{crit}$, where\n\\begin{eqnarray}\n\tr_{crit} &=& \\sqrt{\\frac{M_c}{8\\pi r_h\\rho_{crit}}} \\nonumber\\\\\n\t\t &=& \n\t\t\t1.8\\times10^{-3}\n\t\t\t\\left(\\frac{v}{10\\,{\\rm km\/s}}\\right)\n\t\t\t{\\rm pc}\\,,\n\\end{eqnarray}\nwhere $v = \\sqrt{GM_c\/2r_h} \\sim 10$ km\/s for all the clusters of\ninterest here.  However, the total mass contained within this radius\nis just\n\\begin{equation}\n\tM_{crit} = \n\t\t\t40 \\left(\\frac{v}{10\\,{\\rm km\/s}}\\right)^3\n\t\t\tM_\\odot\\,.\n\\end{equation}\nGiven the highly optimistic assumptions needed to accrete even a small\nfraction of this mass onto the original object, it is clear that, for\nreasonable cluster parameters, there is far too little initial mass in\nthe high-density region to accomplish the task of forming a\n$1000\\,M_\\odot$ object in the time available.\n\n\n\\section{Cluster Dynamics}\n\nThus it seems that collisions in a static cluster core cannot lead to\nthe formation of an ultramassive object.  However, it is well known\nthat cluster dynamical evolution can result in conditions much more\nfavorable for a runaway merger to occur.  Here we briefly describe the\nrelevant processes and their consequences.  (We note in passing that\nessentially the same result could be achieved if there were\nsignificant initial mass segregation in the cluster, but there is\ncurrently no firm evidence to support this assumption.)\n\nThe evolution of a cluster is governed by its half-mass relaxation\ntime, the time scale on which two-body encounters transport energy\naround the system:\n\\begin{equation}\n\tt_{rh}\\ \\approx\\ \\frac{0.14\\, M_c^{1\/2} r_h^{3\/2}}\n\t\t\t\t{G^{1\/2} m \\ln\\Lambda}\n\t      \\ \\approx\\ 5\\, \\left(\\frac{v}{10\\,{\\rm km\/s}}\\right)^3\n\t\t\t  \\left(\\frac{\\bar{\\rho}}\n\t\t\t\t{10^5\\, M_\\odot\\,{\\rm\n\t\t\t\t\tpc}^3}\\right)^{-1}\\ {\\rm Myr}\n\\label{trelax}\n\\end{equation}\n(Heggie \\& Hut 2003).  Here, $N$ is the number of stars in the system,\n$m = M_c\/N$ is the mean stellar mass, $\\bar{\\rho} = 3M_c\/8\\pi r_h^3$\nis the mean cluster density, and $\\ln\\Lambda\\sim \\ln(0.1 N)\\sim10$.\nFor an equal-mass system, the time scale for dynamical evolution---the\ncore collapse time---is about $15 t_{rh}$, too long to cause\nsignificant structural change in a few million years as required here.\nHowever, the presence of even a modest range in masses greatly\naccelerates the process of core collapse (Spitzer 1987).  The time\nscale for a star of mass $M$ to sink to the cluster center as\nequipartition reduces its velocity is\n\\begin{equation}\n      t_s(M) \\ \\sim\\  \\frac{m}{M}\\, t_{rh}\\,,\\label{tseg}\n\\end{equation}\nwhere we note that typical mass spectra (e.g.~Scalo 1986 or Kroupa\n2001) have $m \\sim 0.4$--0.6 $M_\\odot$ ($0.5\\,M_\\odot$ is used in\nEq.~\\ref{trelax}) and maximum mass $\\sim$50--100 $M_\\odot$.\n\nFrom $N$-body simulations, Portegies Zwart \\& McMillan (2002) find\nthat the most massive ($\\ga 20 M_\\odot$) stars segregate rapidly to\nthe cluster center, forming a dense stellar subcore on a time scale\n$t_{cc}\\sim0.2 t_{rh}$.  A central density increase of 2--3 orders of\nmagnitude is typical, boosting even a relatively low-density core into\nthe range where collisions become common, and greatly increasing the\nreserve of raw material available to form a collision runaway.  The\ncollisions naturally involve the most massive stars in the cluster,\nand the low relative velocities typical of these systems ensures that\nthe colliding stars merge with minimal mass loss (J.~Lombardi 2004,\nprivate communication).\n\nIn systems having $t_{cc} \\la 5$ Myr, corresponding to $t_{rh}\\la 25$\nMyr, essentially all the massive stars in the cluster can reach the\ncenter before exploding as supernovae and hence participate in the\nrunaway process.  Using the parameters presented by Portegies Zwart et\nal.~(2004b; Table 1), we find that the Arches and Westerlund I fall\ninto this category, while the Quintuplet, NGC 3603, and R\\,136 all\ncome close.  In these cases, the maximum mass of the runaway is\nlimited primarily by the total number of massive stars in the\nsystem---a few tenths of 1 percent of the total cluster mass.  In less\ndense or more massive clusters, such as MGG-11, the relaxation time is\nlonger and only a fraction of the massive stars initially present in\nthe system can reach the center in the time available, but, as we will\nsee, their greater number may still ensure that a runaway can occur.\n\nIn small systems (containing less than a few tens of thousands of\nsolar masses), collision rates are significantly enhanced by the fact\nthat the massive object tends to form binaries, which are then\nperturbed into eccentric orbits by encounters with other stars\n(Portegies Zwart \\& McMillan 2002).  Binary-induced mergers increase\nthe collision cross section, but they still require high central\ndensities before the (three-body) binary formation rates become\nsignificant.  In larger systems, unbound collisions appear to be the\nnorm, and these have been observed in both direct N-body\n(NBODY4\/Starlab\/treecode) and Monte-Carlo (G\\\"urkan et al.~2003)\ncluster simulations.\n\n\n\\section{X-ray Sources and Dense Clusters in M82}\n\nRecently a bright X-ray point source (M82 X-1) has been observed some\n200 pc from the center of the starburst galaxy M82 (Matsumoto et\nal.~2001; Kaaret et al.~2001).  With a luminosity exceeding $10^{41}$\nerg\/s, it is too bright to be an ordinary X-ray binary, while its\noff-center location in the galaxy argues against its being a\nsupermassive black hole.  The luminosity is consistent with an\naccreting compact object of at least 350 solar masses, raising the\nintriguing possibility that it might be an IMBH.  The discovery of\n$54.4\\pm0.9$\\,mHz quasi-periodic oscillations (Strohmayer et al.~2003)\nsupports this assertion.  Follow-up observations indicate that M82 X-1\nis apparently located in the star cluster MGG-11, one of several\nmassive young clusters in the central region of M82.  This prompts the\nobvious question: Could the observed X-ray source be the result of the\nrunaway collision process just outlined?\n\n\\begin{figure}[!htp]\n\\psfig{figure=.\/mcmillan_fig1.ps,width=\\columnwidth} \n\\caption{Region of interest in M82.  The background X-ray image is\nfrom Matsumoto et al.~(2001); M82 X-1 is near the center.  Star\nclusters from Table 3 of McCrady et al.~(2003) are indicated by\ncircles and squares, the latter marking the positions of the two star\nclusters MGG-9 and MGG-11.  Magnified near-infrared images of these\nclusters from McCrady et al.~are presented at the upper right (MGG-11)\nand lower left (MGG-9).}\n\\end{figure}\n\nMcCrady et al.~(2003) have made accurate measurements of the bulk\nparameters of several clusters in the central regions of M82.  Figure\n1 shows a composite X-ray (Matsumoto et al.) and near-infrared\n(McCrady et al.) map of the few hundred parsecs around MGG-11.  The\nrelative positional accuracies of both the X-ray and the infrared\nobservations are better than 1 arcsecond.  However, the absolute\npointing accuracy is much poorer, for both telescopes.  Although\napparently off-center, the positions of the bright X-ray source and\nthe star cluster MGG-11 are in fact consistent with one another\n(D.~Pooley 2004, private communication).  Curiously, the brighter,\nmore massive, and apparently coeval cluster MGG-9, lying just a few\narc seconds from MGG-11 in projection, shows no X-ray emission.\n\nThe clusters MGG-9 and MGG-11 have quite similar ages, in the range\n7--12\\,Myr (McCrady et al.~2003).  The line-of-sight velocity\ndispersion of MGG-11 ($\\sigma_r = 11.4\\pm0.8$ km\/s) is somewhat\nsmaller than that of MGG-9 ($\\sigma_r = 15.9\\pm0.8$ km\/s).  Combining\nthese numbers with the projected half-light radii, 1.2 pc for MGG-11\nand 2.6 pc for MGG-9, McCrady et al.~estimate total cluster masses of\n$\\sim (3.5\\pm 0.7) \\times 10^5 M_\\odot$ for MGG-11, and about four\ntimes higher for MGG-9.  The mean density of MGG-9 is just under half\nthat of MGG-11, raising a second question: Are such seemingly small\ndifferences in cluster parameters sufficient to explain the presence\nof a $>350 M_\\odot$ IMBH in MGG-11 and the absence of a similar object\nin MGG-9?\n\nPortegies Zwart et al.~(2004a; PZBHMM) have addressed this issue using\ndetailed $N$-body simulations.  Starting with MGG-11, they first\ndemonstrate that IMBH formation is a natural outcome of that cluster's\ndynamical evolution, and then go on to show that the same processes\nwould have failed to create a runaway in MGG-9.  Their calculations\nwere carried out using two independently developed $N$-body codes,\n{\\tt Starlab} (see Portegies Zwart et al.~2001) and {\\tt NBODY4}\n(Aarseth 1999, Baumgardt 2003).  Initial conditions for the model\nclusters were chosen so that at the present time they have mass\nfunctions, luminosities, half-mass radii and velocity dispersions in\nagreement with the McCrady et al.~observations.\n\nSince the initial and the current central densities of both clusters\nare unknown, the concentration parameter $c$ (the logarithm of the\nratio of the tidal radius to the core radius) is treated as a free\nparameter controlling the initial central density of the models.\nPZBHMM find that, for $c > 2$ (which for ``King'' 1966 models is\nequivalent to a dimensionless central potential $W_0 \\ga 9$) the\nMGG-11 models show runaway growth via repeated collisions.  The\nmass-segregation time scale of a $50 M_\\odot$ star in MGG-11 is\n$t_s\\sim 4$\\,Myr (Eq.~\\ref{tseg}).  Thus, massive stars in MGG-11 can\neasily reach the center of the cluster before leaving the main\nsequence.  Given the high central density of MGG-11, once those stars\nhave accumulated in the center, a runaway collision is inevitable,\nleading to IMBHs with masses in the range 800--3000 $M_\\odot$.  No\nepisode of runaway growth occurs in the MGG-11 models with $c < 2$,\nnor in any of the MGG-9 simulations, regardless of initial\nconcentration.  In MGG-9, $t_s\\ga 15 $\\,Myr even for $100 M_\\odot$\nstars, so mass segregation cannot occur in the time available and no\nrunaway is seen.\n\n\\begin{figure}[!ht]\n\\psfig{figure=.\/mcmillan_fig2.ps,width=\\columnwidth,angle=-90} \n\\caption[]{Growth in mass $M_r(t)$ of the collision runaway for some\nof the simulations of PZBHMM, performed using {\\tt Starlab} and {\\tt\nNBODY4}.  The choice of initial concentration parameter $W_0$ is\nindicated.  The star symbols indicate the moment when the runaway\nexperiences a supernova, typically around 3\\,Myr. Open and filled\nstars indicate simulations performed with {\\tt NBODY4} and {\\tt\nStarlab}, respectively.  The observed age range of MGG-11 and MGG-9 is\nindicated by the horizontal bar near the bottom of the figure.  }\n\\label{fig:Mbh}\n\\end{figure}\n\nFigure 2 presents a representative sample of results from a number of\nsimulations performed for a broad range of cluster parameters.  It\nshows the growth in mass of the star that will ultimately become the\nmost massive object in the cluster.  Following detailed supernova\ncalculations by Heger et al.~(2003), stars having masses greater than\n$260\\,M_\\odot$ are assumed to collapse to black holes without\nsignificant mass loss.  The stellar evolution models for stars with\nmasses between 50 and $1000\\,M_\\odot$ are based on work by Stothers \\&\nChin (1997) and Ishii et al.~(1999). The quantitative differences\nbetween the simulations performed with {\\tt Starlab} and those using\n{\\tt NBODY4} are due mainly to the different radii (and hence cross\nsections) assumed for very massive stars in those two packages.\n\nThe solid and dashed curves in the figure show the runaway mass as a\nfunction of time for a Salpeter (1955) IMF with a lower limit of $1\nM_\\odot$ and $c \\approx 2.1$ ($W_0 = 9$) and $c \\approx 2.7$ ($W_0=12$).\nThe dash-dotted curves are for two models with $W_0=9$ with an upper\nlimit to the IMF of $50 M_\\odot$, instead of the standard $100\nM_\\odot$ used in the other calculations; these runs were terminated at\nthe moment the runaway star exploded as a supernova.  The\ndash-3-dotted curve shows the result for $W_0=12$ with a Salpeter IMF,\nwith 10\\% of the stars in primordial binaries---any tendency of these\nsystems to arrest core collapse is effectively offset by the larger\ncollision cross sections of the binary components.  Finally, the\ndotted curve shows results for $W_0=9$ and a Kroupa (2001) IMF with a\nminimum mass of $0.1 M_\\odot$, in a simulation of 585,000 stars.\n\n\n\\section{Summary and Discussion}\n\nRapid mass segregation in a dense star cluster leads to an effective\ncore collapse on a time scale $\\sim$$0.2 t_{rh}$ for typical initial\nmass functions.  This in turn can lead to a runaway series of\ncollisions in the cluster core and the possible formation of a\n$\\sim$1000 $M_\\odot$ IMBH there.  We therefore expect an association\nbetween ultraluminous X-ray sources and the cores of dense young star\nclusters.  A leading candidate for such an association is M82 X-1,\nwhich appears to lie in the massive young cluster MGG-11.  On the\nbasis of $N$-body simulations and elementary considerations of the\ntime scale on which massive stars sink to the cluster center, we can\nreadily explain why MGG-11 might host an IMBH while its more luminous\nneighbor MGG-9 does not.  High initial central concentrations are\nrequired in order for this process to operate even in MGG-11, but we\nnote that all of the ``local'' clusters listed in Table 1 of Portegies\nZwart et al.~(2004b) are in fact very centrally condensed.\n\nOf course, it must be conceded that next to nothing is known about the\ndetailed evolution and ultimate fate of stars hundreds or thousands of\ntimes more massive than the Sun, so we should perhaps not take too\nseriously the predictions of a $2000 M_\\odot$ ``star'' in some of our\nsimulations.  Nevertheless, the simulations described here do make it\nclear that the hearts of these dense stellar systems can easily\nproduce conditions suitable for repeated stellar collisions.  The\ncollision runaway at the center of such a system should be extremely\nluminous and eminently observable during its short lifetime.\nObservations of the cores of dense young star clusters in our Galaxy\nand beyond may thus shed light on the structure and lifetimes of such\nultramassive stellar objects.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\n\\begin{figure}[!ht]\n  \\centering\\includegraphics[width=\\linewidth]{img\/teasor_new.pdf}\n  \\vspace{-5.0mm}\n  \\caption{\\textbf{Illustration and challenges of cross-resolution person re-ID.} (\\emph{Top}) Existing methods for cross-resolution person re-ID typically leverage image super-resolution models with pre-defined up-sampling rates followed by person re-ID modules. Methods of this class, however, may not be applicable to query images of varying or unseen resolutions. (\\emph{Bottom}) In contrast, our method learns resolution-invariant representations, allowing our model to re-identify persons in images of varying and even unseen resolutions.}\n  \\vspace{-4.0mm}\n  \\label{fig:teaser}\n\\end{figure}\n\n\\IEEEPARstart{P}{\\revision{erson}} \\revision{re-identification (re-ID)~\\cite{zheng2016person,zhong2017re,wang2015zero,ye2020deep} aims at recognizing the same person across images taken by different cameras, and is an active research topic in computer vision and machine learning.} A variety of applications ranging from person tracking~\\cite{andriluka2008people}, video surveillance system~\\cite{khan2016person}, urban safety monitoring~\\cite{garcia2015person}, to computational forensics~\\cite{vezzani2013people} are highly correlated this research topic. Nevertheless, due to the presence of background clutter, occlusion, illumination or viewpoint changes, and even uncontrolled human pose variations, person re-ID remains a challenging task for practical applications.\n\nDriven by the recent success of convolutional neural networks (CNNs), several learning based methods~\\cite{lin2017improving,shen2018deep,hermans2017defense,zhong2017camera,si2018dual,chen2018group,zhang2019densely,hou2019interaction,zheng2019joint,zheng2019re} have been proposed to address the challenges in person re-ID. Despite promising performance, these methods are typically developed under the assumption that both query and gallery images are of \\emph{similar} or \\emph{sufficiently high} resolutions. This assumption, however, may not hold in practice since image resolutions would vary drastically due to the varying distances between cameras and persons of interest. For instance, query images captured by surveillance cameras are often of low resolution (LR) whereas those in the gallery set are carefully selected beforehand and are typically of high resolution (HR). As a result, direct matching of LR query images and HR gallery ones would lead to non-trivial \\emph{resolution mismatch} problems~\\cite{jing2015super,wang2016scale,jiao2018deep,wang2018cascaded}.\n\nTo address cross-resolution person re-ID, conventional methods typically learn a shared feature space for LR and HR images to mitigate the resolution mismatch problem~\\cite{li2015multi,jing2015super,wang2016scale}. These approaches, however, adopt hand-engineered descriptors which cannot adapt themselves to the task at hand. The lack of an end-to-end learning pipeline might lead to sub-optimal person re-ID performance. To alleviate this issue, a number of approaches~\\cite{wang2018cascaded,jiao2018deep} employing trainable descriptors are presented. These methods leverage image super-resolution (SR) models to convert LR input images into their HR versions, on which person re-ID is carried out. While performance improvements have been shown, these methods suffer from two limitations. First, each employed SR model is designed to upscale image resolutions by a particular factor. Therefore, these methods need to \\emph{pre-determine} the resolutions of LR query images so that the corresponding SR models can be applied. However, designing SR models for each possible resolution input makes these methods hard to scale. Second, in real-world scenarios, query images can be with \\emph{various} resolutions even with the resolutions that are \\emph{not seen} during training. As illustrated in the top of Figure~\\ref{fig:teaser}, query images with varying or unseen resolutions would restrict the applicability of the person re-ID methods that leverage SR models since one cannot assume the resolutions of the input images will be known in advance.\n\nIn this paper, we propose \\emph{Cross-resolution Adversarial Dual Network} (CAD-Net) for cross-resolution person re-ID. The key characteristics of CAD-Net are two-fold. First, to address the resolution variations, CAD-Net derives the \\emph{resolution-invariant representations} via adversarial learning. As shown in the bottom of Figure~\\ref{fig:teaser}, the learned resolution-invariant representations allow our model to handle images of \\emph{varying} and even \\emph{unseen} resolutions. Second, CAD-Net learns to recover the missing details in LR input images. Together with the resolution-invariant features, our model generates HR images that are \\emph{preferable for person re-ID}, achieving the state-of-the-art performance on cross-resolution person re-ID. It is worth noting that the above image resolution recovery and cross-resolution person re-ID are realized by a \\emph{single} model learned in an \\emph{end-to-end} fashion. \n\nMotivated by the multi-scale adversarial learning techniques in semantic segmentation~\\cite{tsai2018learning} and person re-ID~\\cite{chen2019learning}, which have been shown effective in deriving more robust feature representations, we employ multi-scale adversarial networks to align feature distributions between HR and LR images across different feature levels, resulting in consistent performance improvements over single-scale adversarial methods. On the other hand, since there are infinitely many HR images that reduce to the same LR image, it is difficult for a model to simultaneously handle the resolution variations and learn the mapping between LR and HR images. To alleviate this issue, we introduce a consistency loss in the HR feature space to enforce the consistency between the features of the recovered HR images and the corresponding HR ground-truth images, allowing our model to learn HR image representations that are more robust to the variations of HR image recovery. By jointly leveraging the above schemes, our method further improves the performance of cross-resolution person re-ID.\n\nIn addition to person re-ID, the resolution mismatch issue may occur in various applications such as vehicle re-ID~\\cite{kanaci2018vehicle}. To demonstrate the wide applicability of our method, we also evaluate our method on cross-resolution vehicle re-ID and show that our CAD-Net performs favorably against existing cross-resolution vehicle re-ID approaches. Furthermore, to manifest that our formulation is not limited to cross-resolution setting, we show that our proposed algorithm improves person re-ID performance even when no significant resolution variations are present, achieving competitive performance compared to existing person re-ID approaches. Finally, as image labeling process is often labor intensive, we extend our CAD-Net to semi-supervised settings. Experimental results further support the use and extension of our method for such practical yet challenging settings.\n\nThe contributions of this paper are highlighted as follows: \n\n\\begin{itemize}\n  \\item We propose an end-to-end trainable network which advances adversarial learning strategies for cross-resolution person re-ID.\n  \\item Our model learns resolution-invariant representations while being able to recover the missing details in LR input images, resulting in favorable performance in cross-resolution person re-ID.\n  \\item Our model is able to handle query images with varying or even unseen resolutions, without the need to pre-determine the input resolutions.\n  \\item Extensive experimental results on five person re-ID and two vehicle re-ID datasets show that our method achieves the state-of-the-art performance on both tasks in the cross-resolution setting, and further validate the effectiveness of our approach for real-world person re-ID applications in a semi-supervised manner.\n\\end{itemize}\n\nIn this work, we significantly extend our previous results~\\cite{CAD-Net} and summarize the main differences in the following.\n\n\\begin{itemize}\n  \\item \\textbf{Multi-scale adversarial learning for learning resolution-invariant representations.} Unlike our preliminary work that learns the resolution-invariant representations at a single scale, we adopt multi-scale adversarial network components in this work. The resultant model effectively aligns feature distributions in different levels and derives feature representations across image resolutions, achieving performance improvements over the single-scale model in cross-resolution person re-ID. We refer to our improved method as CAD-Net++.\n  \\item \\textbf{HR feature space consistency loss.} To allow our model to handle the variations of HR image recovery, we introduce a feature consistency loss that enforces the consistency between the features of the recovered HR images and the corresponding HR ground-truth images. This loss further consistently improves the performance of cross-resolution person re-ID on all five datasets.\n  \\item \\textbf{Applications.} In contrast to our preliminary work that focuses on a single task (i.e., cross-resolution person re-ID), we evaluate our proposed method under various settings with extensive ablation studies, including 1) cross-resolution person re-ID, 2) person re-ID with no significant resolution variations (we refer to this setting as standard person re-ID), 3) cross-resolution vehicle re-ID, and 4) semi-supervised cross-resolution person re-ID. Extensive experimental results confirm the effectiveness of our method in a wide range of scenarios.\n\\end{itemize}\n\\section{Related Work}\n\nPerson re-ID has been extensively studied in the literature. We review several topics relevant to our approach in this section.\n\n{\\flushleft {\\bf Person re-ID.}}\nMany existing methods, e.g., \\cite{lin2017improving,shen2018deep,shen2018person,kalayeh2018human,cheng2016person,chang2018multi,chen2018group,li2018adaptation,sun2018beyond,suh2018part}, are developed to address various challenges in person re-ID, such as background clutter, viewpoint changes, and pose variations. For instance, Yang~\\textit{et~al}\\mbox{.}~\\cite{zhong2017camera} learn a camera-invariant subspace to deal with the style variations caused by different cameras. Liu~\\textit{et~al}\\mbox{.}~\\cite{liu2018pose} develop a pose-transferable framework based on generative adversarial network (GAN)~\\cite{goodfellow2014generative} to yield pose-specific images for tackling pose variations. Several methods addressing background clutter leverage attention mechanisms to emphasize the discriminative parts~\\cite{li2018harmonious,song2018mask,si2018dual}. \\revision{In addition to these methods that learn global features, a few methods further utilize part-level information~\\cite{sun2018beyond} to learn more fine-grained features, adopt human semantic parsing for learning local features~\\cite{kalayeh2018human}, learn multi-scale re-ID representation~\\cite{li2017person}, or derive part-aligned representations~\\cite{suh2018part} for improving person re-ID.} \n\nAnother research trend focuses on domain adaptation~\\cite{ganin2015unsupervised,ganin2016domain,long2015learning,long2016unsupervised,chen2019crdoco,hoffman2017cycada} for person re-ID~\\cite{wei2018person,image-image18,ge2018fd,li2019cross}. These methods either employ image-to-image translation modules (e.g., CycleGAN~\\cite{zhu2017unpaired}) as a data augmentation technique to generate viewpoint specific images with labels~\\cite{wei2018person,image-image18}, or leverage pose information to learn identity related but pose unrelated representations~\\cite{ge2018fd} or pose-guided yet dataset-invariant representations~\\cite{li2019cross} for cross-dataset person re-ID.\n\nWhile promising performance has been demonstrated, the above approaches typically assume that both query and gallery images are of similar or sufficiently high resolutions, which might not be practical for real-world applications.\n\n{\\flushleft {\\bf Cross-resolution person re-ID.}}\n\\revision{A number of methods have been proposed to address the resolution mismatch issue in person re-ID. These methods can be categorized into two groups depending on the adopted feature descriptors: 1) hand-crafted descriptor based methods~\\cite{li2015multi,jing2015super,wang2016scale} and 2) trainable descriptor based methods~\\cite{jiao2018deep,wang2018cascaded,chen2019learning,mao2019resolution,li2018toward}.} Methods in the first group typically use an engineered descriptor such as HOG~\\cite{HoG} for feature extraction and then learn a shared feature space between HR and LR images. For instance, Li~\\textit{et~al}\\mbox{.}~\\cite{li2015multi} jointly perform multi-scale distance metric learning and cross-scale image domain alignment. Jing~\\textit{et~al}\\mbox{.}~\\cite{jing2015super} develop a semi-coupled low-rank dictionary learning framework to seek a mapping between HR and LR images. Wang~\\textit{et~al}\\mbox{.}~\\cite{wang2016scale} learn a discriminating scale-distance function space by varying the image scale of LR images when matching with the HR ones. Nevertheless, these methods adopt hand-crafted descriptors, which cannot easily adapt the developed models to the tasks of interest, and thus may lead to sub-optimal person re-ID performance.\n\n\\revision{To alleviate this issue, several trainable descriptor based approaches are presented for cross-resolution person re-ID~\\cite{jiao2018deep,wang2018cascaded,chen2019learning,mao2019resolution,li2018toward}.} The network of SING~\\cite{jiao2018deep} is composed of several SR sub-networks and a person re-ID module to carry out LR person re-ID. On the other hand, CSR-GAN~\\cite{wang2018cascaded} cascades multiple SR-GANs~\\cite{ledig2017photo} and progressively recovers the details of LR images to address the resolution mismatch problem. Mao~\\textit{et~al}\\mbox{.}~\\cite{mao2019resolution} develop a foreground-focus super-resolution model that learns to recover the resolution loss in LR input images followed by a resolution-invariant person re-ID module. In spite of their promising results, such methods rely on training pre-defined SR models~\\cite{jiao2018deep,wang2018cascaded} or annotating the foreground mask for each training image to guide the learning of image recovery~\\cite{mao2019resolution}. As mentioned earlier, the degree of resolution mismatch, i.e., the resolution difference between the query and gallery images, is typically \\emph{unknown beforehand}. If the resolution of the input LR query is unseen during training, the above methods cannot be easily applied or might not lead to satisfactory performance. On the other hand, the dependence on foreground masks would make such methods hard to scale for real-world applications. Different from the above methods that employ SR models, a recent method motivated by the domain-invariant representations in domain adaptation~\\cite{ganin2015unsupervised,ganin2016domain} is presented~\\cite{chen2019learning}. By advancing adversarial learning strategies in the feature space, the RAIN method~\\cite{chen2019learning} aligns the feature distributions of HR and LR images, allowing the model to be more robust to resolution variations.\n\nSimilar to RAIN~\\cite{chen2019learning}, our method also performs feature distribution alignment between HR and LR images. Our model differs from RAIN~\\cite{chen2019learning} in that our model learns to recover the missing details in LR input images and thus provides more discriminative evidence for person re-ID. By jointly observing features of both modalities in an end-to-end learning fashion, our model recovers HR images that are preferable for person re-ID, resulting in performance improvements on cross-resolution person re-ID. Experimental results demonstrate that our approach can be applied to input images of varying and even unseen resolutions using only a single model with favorable performance.\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{img\/model.pdf}\n  \\vspace{-6.0mm}\n  \\caption{\\textbf{Overview of the proposed Cross-resolution Adversarial Dual Network++ (CAD-Net++).} CAD-Net++ comprises cross-resolution GAN (CRGAN) (highlighted in blue) and cross-modal re-ID network (highlighted in green). The former learns resolution-invariant representations and recovers the missing details in LR input images, while the latter considers both feature modalities for cross-resolution person re-ID.}\n  \\vspace{-2.0mm}\n  \\label{fig:Model}\n\\end{figure*}\n\n{\\flushleft {\\bf Cross-resolution vision applications.}}\nThe issues regarding cross-resolution handling have been studied in the literature. For face recognition, existing approaches typically rely on face hallucination algorithms~\\cite{zhu2016deep,yu2017hallucinating} or SR mechanisms~\\cite{kim2016accurate,dahl2017pixel,dong2016image} to super-resolve the facial details. Unlike the aforementioned methods that focus on synthesizing the facial details, our model learns to recover re-ID oriented discriminative details. For vehicle re-ID, the resolution mismatch issue is also a challenging yet under studied problem~\\cite{kanaci2018vehicle}. While several efforts have been made~\\cite{zhou2018aware,wang2017orientation,shen2017learning} to address the challenges (e.g., viewpoint or appearance variations) in vehicle re-ID, these resultant methods are developed under the assumption that both query and gallery images are of similar or sufficiently high resolutions. To carry out cross-resolution vehicle re-ID, MSVF~\\cite{kanaci2018vehicle} designs a multi-branch network that learns a representation by fusing features from images of different scales. Our method differs from MSVF~\\cite{kanaci2018vehicle} in three aspects. First, MSVF~\\cite{kanaci2018vehicle} is tailored for cross-resolution vehicle re-ID while our method is developed to address cross-resolution person re-ID. Second, our model does not need to pre-determined the number of branches. Instead, our model carries out cross-resolution person re-ID using only a single model. Third, our model further learns to recover the missing details in LR input images. Through extensive experiments, we demonstrate that our algorithm performs favorably against existing cross-resolution vehicle re-ID approaches.\n\\section{Proposed Method}\n\nIn this section, we first provide an overview of our proposed approach. We then describe the details of each network component as well as the loss functions.\n\n\\subsection{Algorithmic Overview}\n\nWe first define the notations to be used in this paper. In the training stage, we assume we have access to a set of~$N$ HR images $X_H = \\{x_i^H\\}_{i=1}^N$ and its corresponding label set $Y_H = \\{y_i^H\\}_{i=1}^N$, where $x_i^H \\in {\\mathbb{R}}^{H \\times W \\times 3}$ and $y_i^H \\in {\\mathbb{R}}$ are the $i^\\mathrm{th}$ HR image and its label, respectively. To allow our model to handle images of different resolutions, we generate a \\emph{synthetic} LR image set $X_L = \\{x_i^L\\}_{i=1}^N$ by down-sampling each image in $X_H$, followed by resizing them back to the original image size via bilinear up-sampling (i.e., $x_i^L \\in {\\mathbb{R}}^{H \\times W \\times 3}$), where $x_i^L$ is the synthetic LR image of $x_i^H$. Obviously, the label set $Y_L$ for $X_L$ is identical to $Y_H$.\n\nAs shown in Figure~\\ref{fig:Model}, our network comprises two components: the Cross-Resolution Generative Adversarial Network (CRGAN) and the Cross-Modal Re-ID network. To achieve cross-resolution person re-ID, our CRGAN simultaneously learns a resolution-invariant representation $f \\in {\\mathbb{R}}^{h \\times w \\times d}$ ($h \\times w$ is the spatial size of $f$ whereas $d$ denotes the number of channels) from the input cross-resolution images, while producing the associated HR images as the decoder outputs. The recovered HR output image will be encoded as an HR representation $g \\in {\\mathbb{R}}^{h \\times w \\times d}$ by the HR encoder. For person re-ID, we first concatenate $f$ and $g$ to form a joint representation $\\mathbfit{v} = [f, g] \\in {\\mathbb{R}}^{h \\times w \\times 2d}$. The classifier then takes the joint representation $\\mathbfit{v}$ as input to perform person identity classification. The details of each component are elaborated in the following subsections.\n\nAs for testing, our network takes a query image resized to $H \\times W \\times 3$ as the input, and computes the joint representation $\\mathbfit{v} = [f, g] \\in {\\mathbb{R}}^{h \\times w \\times 2d}$. We then apply global average pooling ($\\mathrm{GAP}$) to $\\mathbfit{v}$ for deriving a joint feature vector $\\mathbfit{u} = \\mathrm{GAP}(\\mathbfit{v}) \\in {\\mathbb{R}}^{2d}$, which is applied to match the gallery images via nearest neighbor search with Euclidean distance. It is worth repeating that, the query image during testing can be with varying resolutions or with unseen ones during training (verified in experiments).\n\n\\subsection{Cross-Resolution GAN (CRGAN)}\n\nIn CRGAN, we have a cross-resolution encoder $\\mathcal{E}$ which converts input images across different resolutions into resolution-invariant representations, followed by a high-resolution decoder $\\mathcal{G}$ recovering the associated HR versions.\n\n{\\flushleft {\\bf Cross-resolution encoder $\\mathcal{E}$.}}\nSince our goal is to perform cross-resolution person re-ID, we encourage the cross-resolution encoder $\\mathcal{E}$ to extract resolution-invariant features for input images across resolutions (e.g., HR images in $X_H$ and LR ones in $X_L$). To achieve this, we advance adversarial learning strategies and deploy a resolution discriminator $\\mathcal{D}_{F}$ in the latent \\emph{feature space}. This discriminator $\\mathcal{D}_{F}$ takes the feature maps $f_H$ and $f_L$ as inputs to determine whether the input feature maps are from $X_H$ or $X_L$. \n\nTo be more precise, we define the feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$ as\n\\begin{equation}\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}} = &~ \\mathbb{E}_{x_H \\sim X_H}[\\log(\\mathcal{D}_{F}(f_H))]\\\\\n  + &~ \\mathbb{E}_{x_L \\sim X_L}[\\log(1 - \\mathcal{D}_{F}(f_L))],\n  \\end{aligned}\n  \\label{eq:adv_loss_feature}\n\\end{equation}\nwhere $f_H = \\mathcal{E}({x_H})$ and $f_L = \\mathcal{E}({x_L}) \\in {\\mathbb{R}}^{h \\times w \\times d}$ denote the encoded HR and LR image features, respectively.\\footnote{For simplicity, we omit the subscript $i$, denote HR and LR images as $x_H$ and $x_L$, and represent their corresponding labels as $y_H$ and $y_L$.}\n\nWhile aligning feature distributions between HR and LR images at a single feature level has been shown effective to some extent in our previous results~\\cite{CAD-Net}, similar to existing methods for semantic segmentation~\\cite{tsai2018learning} and person re-ID~\\cite{chen2019learning}, we adopt multi-scale adversarial networks and align feature distributions at multiple levels to learn more robust feature representations. In this work, we employ the ResNet-$50$~\\cite{he2016deep} as the cross-resolution encoder $\\mathcal{E}$, which has five residual blocks $\\{R_1, R_2, R_3, R_4, R_5\\}$. The feature maps extracted from the last activation layer of each residual block are denoted as $\\{f^1, f^2, f^3, f^4, f^5\\}$, where $f^j \\in {\\mathbb{R}}^{h_j \\times w_j \\times d_j}$ is of spatial size $h_j \\times w_j$ and with $d_j$ channels.\n\nAs shown in Figure~\\ref{fig:multi-discriminator}, our multi-scale discriminator $\\mathcal{D}_F^j$ takes the feature maps $f_H^j$ and $f_L^j$ extracted at the corresponding feature level as inputs, and determines whether the input feature map is from $X_H$ or $X_L$. Note that $j \\in \\{1, 2, 3, 4, 5\\}$ is the index of the feature levels, and $f_H^j$ and $f_L^j$ denote the feature maps of $x_H$ and $x_L$, respectively.\n\nTo train the cross-resolution encoder $\\mathcal{E}$ and the multi-scale resolution discriminators $\\{\\mathcal{D}_F^j\\}$ with cross-resolution input images $x_H$ and $x_L$, we extend the adversarial loss in Eq.~\\eqref{eq:adv_loss_feature} from single-scale to multi-scale adversarial learning and define the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$ as\n\\begin{equation}\n  \\begin{split}\n  \\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}} = \\sum_{j}^{}\\bigg(&~ \\mathbb{E}_{x_H \\sim X_H}[\\log(\\mathcal{D}_F^{j}(f_H^j))] \\\\\n  + &~ \\mathbb{E}_{x_L \\sim X_L}[\\log(1 - \\mathcal{D}_F^{j}(f_L^j))]\\bigg).\n  \\end{split}\n  \\label{eq:adv_multi}\n\\end{equation}\n\nWith the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$, our multi-scale resolution discriminators $\\{\\mathcal{D}_F^j\\}$ align the feature distributions across resolutions, carrying out the learning of resolution-invariant representations.\n\n{\\flushleft {\\bf High-resolution decoder $\\mathcal{G}$.}}\nIn addition to learning the resolution-invariant representation $f$, our CRGAN further synthesizes the associated HR images. This is to recover the missing details in LR inputs, together with the re-ID task to be performed later in the cross-modal re-ID network.\n\nTo achieve this goal, we have an HR decoder $\\mathcal{G}$ in our CRGAN which reconstructs (or recovers) the HR images as the outputs. To accomplish this, we apply an HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ between the reconstructed HR images and their corresponding HR ground-truth images. Specifically, the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ is defined as\n\\begin{equation}\n  \\label{eq:rec}\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{rec} = &~ \\mathbb{E}_{x_H \\sim X_H}[\\|\\mathcal{G}(f_H) - x_H\\|_1]\\\\\n  + &~ \\mathbb{E}_{x_L \\sim X_L}[\\|\\mathcal{G}(f_L) - x_{H}\\|_1],\n  \\end{aligned}\n\\end{equation}\nwhere the HR ground-truth image associated with $x_L$ is $x_H$. Following Huang~\\textit{et~al}\\mbox{.}~\\cite{huang2018munit}, we adopt the $\\ell_1$ norm in the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ as it preserves image sharpness. We note that both $X_H$ and $X_L$ will be shuffled during training. That is, images of the same identity but different resolutions will not necessarily be observed by the CRGAN at the same time.\n\nIt is worth noting that, while the aforementioned HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ could reduce information loss in the latent feature space, we follow Ledig~\\textit{et~al}\\mbox{.}~\\cite{ledig2017photo} and introduce skip connections between the cross-resolution encoder $\\mathcal{E}$ and the HR decoder $\\mathcal{G}$. This would facilitate the learning process of image reconstruction, as well as allowing more efficient gradient propagation.\n\nTo make the HR decoder $\\mathcal{G}$ produce more perceptually realistic HR outputs and associate with the task of person re-ID, we further adopt adversarial learning in the \\emph{image space} and introduce an HR image discriminator $\\mathcal{D}_{I}$ which takes the recovered HR images (i.e., $\\mathcal{G}(f_L)$ and $\\mathcal{G}(f_H)$) and their corresponding HR ground-truth images as inputs to distinguish whether the input images are real or fake~\\cite{ledig2017photo,wang2018cascaded}. Specifically, we define the image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ as\n\\begin{equation}\\scriptsize\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}} = &~ \\mathbb{E}_{x_H \\sim X_H}[\\log(\\mathcal{D}_{I}(x_H))] + \\mathbb{E}_{x_L \\sim X_L}[\\log(1 - \\mathcal{D}_{I}(\\mathcal{G}(f_L)))] \\\\\n  + &~ \\mathbb{E}_{x_H \\sim X_H}[\\log(\\mathcal{D}_{I}(x_H))] + \\mathbb{E}_{x_H \\sim X_H}[\\log(1 - \\mathcal{D}_{I}(\\mathcal{G}(f_H)))].\n  \\end{aligned}\n  \\label{eq:adv_loss_image}\n\\end{equation}\n\nIt is also worth repeating that the goal of this HR decoder $\\mathcal{G}$ is not simply to recover the missing details in LR inputs, but also to have such recovered HR images aligned with the learning task of interest (i.e., person re-ID). Namely, we encourage the HR decoder $\\mathcal{G}$ to perform \\textit{re-ID oriented} HR recovery, which is further realized by the following cross-modal re-ID network.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{img\/model_2.pdf}\n  \\caption{\\textbf{Multi-scale adversarial learning.} We adopt multiple discriminators to effectively align feature distributions between HR and LR images at different feature levels. This multi-scale adversarial learning strategy allows our model to learn resolution-invariant representations that are more robust to resolution variations.}\n  \\label{fig:multi-discriminator}\n  \\vspace{-4.0mm}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{img\/model_3.pdf}\n  \\caption{\\textbf{Illustration of the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$.} The HR encoder $\\mathcal{F}$ takes the recovered HR image $\\tilde{x}_H$ and the corresponding HR ground-truth image $x_H$ as inputs and derives their HR representations $\\tilde{g}$ and $g$, respectively. We then introduce the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ to enforce the consistency between $\\tilde{g}$ and $g$. This consistency loss allows our HR encoder $\\mathcal{F}$ to learn HR representations that are more robust to the variations of HR image recovery.}\n  \\label{fig:consist-loss}\n  \\vspace{-3.0mm}\n\\end{figure}\n\n\\subsection{Cross-Modal Re-ID}\n\nAs shown in Figure~\\ref{fig:Model}, the cross-modal re-ID network first applies an HR encoder $\\mathcal{F}$, which takes the reconstructed HR image from the CRGAN as input, to derive the HR feature representation $g \\in {\\mathbb{R}}^{h \\times w \\times d}$. Then, a classifier $\\mathcal{C}$ is learned to complete person re-ID.\n\nWhile enforcing the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ and the image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_I}$ allows our model to map LR input images of various resolutions into their HR versions to some extent~\\cite{CAD-Net}, it is still difficult for the model to simultaneously handle the resolution variations and learn the mapping between LR and HR images, especially when there are infinitely many mappings between LR and HR images. To address the variations of HR image recovery, we introduce a feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ that enforces the consistency between the features of the recovered HR images and the corresponding HR ground-truth images. As illustrated in Figure~\\ref{fig:consist-loss}, the HR encoder $\\mathcal{F}$ takes the recovered HR image $\\tilde{x}_H$ and its corresponding HR ground-truth image $x_H$ as inputs and derives the HR representations $\\tilde{g} = \\mathcal{F}(\\tilde{x}_H)$ and $g = \\mathcal{F}(x_H)$, respectively. We then enforce the consistency between $\\tilde{g}$ and $g$ using the $\\ell_1$ distance and define the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ as\n\\begin{equation}\n  \\label{eq:consist}\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{consist} = &~ \\mathbb{E}_{x_H \\sim X_H}[\\|\\mathcal{F}(\\tilde{x}_H) - \\mathcal{F}(x_H)\\|_1]\\\\\n  + &~ \\mathbb{E}_{x_L \\sim X_L}[\\|\\mathcal{F}(\\tilde{x}_H) - \\mathcal{F}(x_{H})\\|_1].\n  \\end{aligned}\n\\end{equation}\n\nEnforcing the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ allows the HR decoder $\\mathcal{G}$ to derive the HR representation $g$ which is more robust to the variations of HR image recovery.\n\nAs for the input to the classifier $\\mathcal{C}$, we jointly consider the feature representations of two different modalities for person identity classification, i.e., the resolution-invariant representation $f$ and the HR representation $g$. The former preserves content information, while the latter observes the recovered HR details for person re-ID. Thus, we have the classifier $\\mathcal{C}$ take the concatenated feature representation $\\mathbfit{v} = [f, g] \\in {\\mathbb{R}}^{h \\times w \\times 2d}$ as the input. In this work, the adopted classification loss $\\mathcal{L}_\\mathrm{cls}$ is the integration of the identity loss $\\mathcal{L}_\\mathrm{id}$ and the triplet loss $\\mathcal{L}_\\mathrm{tri}$~\\cite{hermans2017defense}, and is defined as\n\\begin{equation}\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{cls} = \\mathcal{L}_\\mathrm{id} + \\mathcal{L}_\\mathrm{tri},\n  \\end{aligned}\n  \\label{eq:cls}\n\\end{equation}\nwhere the identity loss $\\mathcal{L}_\\mathrm{id}$ computes the softmax cross entropy between the classification prediction and the corresponding ground-truth one hot vector, while the triplet loss $\\mathcal{L}_\\mathrm{tri}$ is introduced to enhance the discrimination ability during the re-ID process and is defined as\n\\begin{equation}\n  \\begin{aligned}\n  \\mathcal{L}_\\mathrm{tri}\n   = &~ \\mathbb{E}_{(x_H,y_H) \\sim (X_H,Y_H)}\\max(0, \\phi + d_\\mathrm{pos}^H - d_\\mathrm{neg}^H) \\\\\n  + &~ \\mathbb{E}_{(x_L,y_L) \\sim (X_L,Y_L)}\\max(0, \\phi + d_\\mathrm{pos}^L - d_\\mathrm{neg}^L),\n  \\end{aligned}\n  \\label{eq:tri}\n\\end{equation}\nwhere $d_\\mathrm{pos}$ and $d_\\mathrm{neg}$ are the distances between the positive (same label) and the negative (different labels) image pairs, respectively, and $\\phi > 0$ serves as the margin.\n\nIt can be seen that, the above cross-resolution person re-ID framework is very different from existing one like CSR-GAN~\\cite{wang2018cascaded}, which addresses image SR and person re-ID \\emph{separately}. More importantly, the aforementioned identity loss $\\mathcal{L}_\\mathrm{id}$ not only updates the classifier $\\mathcal{C}$, but also refines the HR decoder $\\mathcal{G}$ in our CRGAN. This is the reason why our CRGAN is able to produce \\emph{re-ID oriented} HR outputs, i.e., the recovered HR details preferable for person re-ID.\n\n\\input{exp\/psuedo}\n\n{\\flushleft {\\bf Full training objective.}}\nThe total loss $\\mathcal{L}$ for training our proposed CAD-Net++ is summarized as follows:\n\\begin{equation}\n  \\begin{split}\n  \\mathcal{L} & = \\mathcal{L}_\\mathrm{cls} + \\lambda_\\mathrm{adv}^{\\mathcal{D}_{F}}\\cdot\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}} + \\lambda_\\mathrm{rec}\\cdot\\mathcal{L}_\\mathrm{rec} \\\\ & + \\lambda_\\mathrm{adv}^{\\mathcal{D}_{I}}\\cdot\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}} + \\lambda_\\mathrm{consist}\\cdot\\mathcal{L}_\\mathrm{consist},\n  \\end{split}\n  \\label{eq:fullobj}\n\\end{equation}\nwhere $\\lambda_\\mathrm{adv}^{\\mathcal{D}_{F}}$, $\\lambda_\\mathrm{rec}$, $\\lambda_\\mathrm{adv}^{\\mathcal{D}_{I}}$, and $\\lambda_\\mathrm{consist}$ are the hyper-parameters used to control the relative importance of the corresponding losses. We note that $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$, $\\mathcal{L}_\\mathrm{rec}$, and $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ are developed to learn the CRGAN, $\\mathcal{L}_\\mathrm{consist}$ is introduced to update the cross-modal re-ID component, and $\\mathcal{L}_\\mathrm{cls}$ is designed to update the entire framework.\n\nTo learn our network with the HR training images and their down-sampled LR ones, we minimize the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ for updating our CRGAN, the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ for updating the HR encoder $\\mathcal{F}$, and the classification loss $\\mathcal{L}_\\mathrm{cls}$ for jointly updating the CRGAN and the cross-modal re-ID network. The image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_I}$ is computed for producing perceptually realistic HR images, while the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_F}$ is optimized for learning resolution-invariant representations.\n\n\\revision{The training details of CAD-Net++ are summarized in Algorithm~\\ref{alg:cadnet}. Specifically, we train our CAD-Net++ until all losses converge.}\n\\section{Experiments}\n\nIn this section, we first describe the implementation details, the adopted datasets for evaluation, and the experimental settings. We then present both quantitative and qualitative results, including ablation studies.\n\n\\subsection{Implementation Details}\n\nWe implement our model using PyTorch. The ResNet-$50$~\\cite{he2016deep} pretrained on ImageNet is used to build the cross-resolution encoder $\\mathcal{E}$ and the HR encoder $\\mathcal{F}$. Note that since $\\mathcal{E}$ and $\\mathcal{F}$ work for different tasks, these two components do not share weights. The classifier $\\mathcal{C}$ is composed of a global average pooling layer and a fully connected layer followed a softmax activation. The architecture of the resolution discriminator $\\mathcal{D}_F$ is the same as that adopted by Tsai~\\textit{et~al}\\mbox{.}~\\cite{tsai2018learning}. The structure of the HR image discriminator $\\mathcal{D}_I$ is similar to the ResNet-$18$~\\cite{he2016deep}. Our HR decoder $\\mathcal{G}$ is similar to that proposed by Miyato~\\textit{et~al}\\mbox{.}~\\cite{miyato2018cgans}. Components $\\mathcal{D}_F$, $\\mathcal{D}_I$, $\\mathcal{G}$, and $\\mathcal{C}$ are all randomly initialized. We use stochastic gradient descent to train the proposed model. For components $\\mathcal{E}$, $\\mathcal{G}$, $\\mathcal{F}$, and $\\mathcal{C}$, the learning rate, momentum, and weight decay are $1 \\times 10^{-3}$, $0.9$, and $5 \\times 10^{-4}$, respectively. For the two discriminators $\\mathcal{D}_F$ and $\\mathcal{D}_I$, the learning rate is set to $1 \\times 10^{-4}$. The batch size is $32$. The margin $\\phi$ in the triplet loss $\\mathcal{L}_\\mathrm{tri}$ is set to $2$. We set the hyper-parameters in all the experiments as follows: $\\lambda_\\mathrm{adv}^{\\mathcal{D}_{F}}$ = 1, $\\lambda_\\mathrm{rec}$ = 1, $\\lambda_\\mathrm{adv}^{\\mathcal{D}_{I}}$ = 1, and $\\lambda_\\mathrm{consist}$ = 1. All images of various resolutions are resized to $256 \\times 128 \\times 3$ in advance. We train our model on a single NVIDIA GeForce GTX $1080$ GPU with $12$ GB memory.\n\n\\subsection{Datasets}\n\nWe adopt five person re-ID datasets, including CUHK03~\\cite{li2014deepreid}, VIPeR~\\cite{gray2008viewpoint}, CAVIAR~\\cite{Cheng:BMVC11}, Market-$1501$~\\cite{zheng2015scalable}, and DukeMTMC-reID~\\cite{zheng2017unlabeled}, and two vehicle re-ID datasets, including VeRi-$776$~\\cite{liu2016deep} and VRIC~\\cite{kanaci2018vehicle} for evaluation. The details of each dataset are described as follows.\n\n{\\flushleft {\\bf CUHK03~\\cite{li2014deepreid}.}}\nThe CUHK03 dataset is composed of $14,097$ images of $1,467$ identities with $5$ different camera views. Following CSR-GAN~\\cite{wang2018cascaded}, we use the $1,367\/100$ training\/test identity split.\n\n{\\flushleft {\\bf VIPeR~\\cite{gray2008viewpoint}.}}\nThe VIPeR dataset contains $632$ person-image pairs captured by $2$ cameras. Following SING~\\cite{jiao2018deep}, we randomly divide this dataset into two non-overlapping halves based on the identity labels. That is, images of a subject belong to either the training set or the test set.\n\n{\\flushleft {\\bf CAVIAR~\\cite{Cheng:BMVC11}.}}\nThe CAVIAR dataset is composed of $1,220$ images of $72$ person identities captured by $2$ cameras. Following SING~\\cite{jiao2018deep}, we discard $22$ people who only appear in the closer camera, and split this dataset into two non-overlapping halves according to the identity labels.\n\n{\\flushleft {\\bf Market-$1501$~\\cite{zheng2015scalable}.}}\nThe Market-$1501$ dataset consists of $32,668$ images of $1,501$ identities with $6$ camera views. We use the widely adopted $751\/750$ training\/test identity split.\n\n{\\flushleft {\\bf DukeMTMC-reID~\\cite{zheng2017unlabeled}.}}\nThe DukeMTMC-reID dataset contains $36,411$ images of $1,404$ identities captured by $8$ cameras. We utilize the benchmarking $702\/702$ training\/test identity split.\n\n{\\flushleft {\\bf VeRi-$776$~\\cite{liu2016deep}.}}\nThe VeRi-$776$ dataset is divided into two subsets: a training set and a test set. The training set is composed of $37,781$ images of $576$ vehicles, and the test set has $11,579$ images of $200$ vehicles. Following the evaluation protocol in \\cite{liu2016deep}, the image-to-track cross-camera search is performed, where we treat one image of a vehicle from one camera as the query, and search for tracks of the same vehicle in other cameras.\n\n{\\flushleft {\\bf VRIC~\\cite{kanaci2018vehicle}.}}\nThe VRIC dataset is a newly collected dataset, which consists of $60,430$ images of $5,656$ vehicle IDs collected from $60$ different cameras in traffic scenes. VRIC differs significantly from existing datasets in that vehicles were captured with variations in image resolution, motion blur, weather condition, and occlusion. The training set has $54,808$ images of $2,811$ vehicles, while the rest $5,622$ images of $2,811$ identities are used for testing.\n\n\\subsection{Experimental Settings and Evaluation Metrics}\n\nWe evaluate the proposed method under three different settings: (1) \\emph{cross-resolution setting}~\\cite{jiao2018deep,kanaci2018vehicle}, (2) \\emph{standard setting}~\\cite{ge2018fd,zhong2017camera}, and (3) \\emph{semi-supervised setting}~\\cite{chen2019learning}. For cross-resolution setting, the test (query) set is composed of LR images while the gallery set contains HR images only. For standard setting (i.e., re-ID with no significant resolution variations), both query and gallery sets contain HR images. For semi-supervised setting (i.e., re-ID with partially labeled datasets), we follow cross-resolution setting where the test (query) set consists LR images while the gallery set comprises HR images only.\n\nIn all of the experiments, we adopt the standard single-shot re-ID setting~\\cite{jiao2018deep,liao2015person}. We note that the cross-resolution re-ID setting analyzes the \\emph{robustness against the resolution variations}, while the standard re-ID setting examines if our method still improves re-ID if no significant resolution variations are present. The semi-supervised re-ID setting aims to investigate whether our proposed algorithm still exhibits sufficient ability in re-identifying images with less supervision.\n\nWe adopt the multi-scale resolution discriminators $\\{\\mathcal{D}_F^j\\}$ which align feature distributions at different feature levels. To balance between learning efficiency and performance, we select the index of feature level with $j \\in \\{1, 2\\}$, and denote our method as ``Ours (multi-scale)'' and the variant of our method with single-scale resolution discriminator ($j = 1$) as ``Ours (single-scale)''.\n\nFor performance evaluation, we adopt the average cumulative match characteristic as the evaluation metric. We note that the performance of our method can be further improved by applying pre-\/post-processing methods, attention mechanisms, or re-ranking. For fair comparisons, no such techniques are used in all of our experiments.\n\n\\input{exp\/exp-ReID.tex}\n\n\\subsection{Evaluation of Cross-Resolution Setting}\n\nWe evaluate our proposed algorithm on both person re-ID~\\cite{jiao2018deep,CAD-Net} and vehicle re-ID~\\cite{kanaci2018vehicle} tasks.\n\n\\subsubsection{Cross-Resolution Person Re-ID}\n\nFollowing SING~\\cite{jiao2018deep}, we consider multiple low-resolution (MLR) person re-ID and evaluate the proposed method on \\emph{four synthetic} and \\emph{one real-world} benchmarks. To construct the synthetic MLR datasets (i.e., MLR-CUHK03, MLR-VIPeR, MLR-Market-$1501$, and MLR-DukeMTMC-reID), we follow SING~\\cite{jiao2018deep} and down-sample images taken by one camera by a randomly selected down-sampling rate $r \\in \\{2, 3, 4\\}$ (i.e., the size of the down-sampled image becomes $\\frac{H}{r} \\times \\frac{W}{r} \\times 3$), while the images taken by the other camera(s) remain unchanged. The CAVIAR dataset inherently contains realistic images of multiple resolutions, and is a \\emph{genuine} and more challenging dataset for evaluating MLR person re-ID.\n\n\\revision{We compare our proposed approach (CAD-Net++) with methods developed for cross-resolution person re-ID, including JUDEA~\\cite{li2015multi}, SLD$^2$L~\\cite{jing2015super}, SDF~\\cite{wang2016scale}, RAIN~\\cite{chen2019learning}, DenseNet-121~\\cite{huang2017densely}, SE-ResNet-50~\\cite{hu2018squeeze}, ResNet-50~\\cite{he2016deep}, FFSR~\\cite{mao2019resolution}, RIFE~\\cite{mao2019resolution}, FSRCNN-reID~\\cite{dong2016accelerating} (FSRCNN followed by the same representation learning method as SING~\\cite{jiao2018deep}), SING~\\cite{jiao2018deep}, and CSR-GAN~\\cite{wang2018cascaded}, and approaches developed for standard person re-ID, including PCB~\\cite{sun2018beyond}, SPreID~\\cite{kalayeh2018human}, Part Aligned~\\cite{suh2018part}, CamStyle~\\cite{zhong2017camera}, and FD-GAN~\\cite{ge2018fd}.} For methods developed for cross-resolution person re-ID, the training set contains HR images and LR ones with all three down-sampling rates $r \\in \\{2, 3, 4\\}$ for each person. For methods developed for standard person re-ID, the training set contains HR images for each identity only.\n\n\\revision{\n{\\flushleft {\\bf Results.}}\nTable~\\ref{table:exp-ReID} reports the quantitative results recorded at ranks $1$, $5$, and $10$ on all five adopted datasets. For CSR-GAN~\\cite{wang2018cascaded} on the MLR-CUHK03, CAVIAR, MLR-Market-$1501$, and MLR-DukeMTMC-reID datasets, and PCB~\\cite{sun2018beyond}, SPreID~\\cite{kalayeh2018human}, Part Aligned~\\cite{suh2018part}, CamStyle~\\cite{zhong2017camera}, FSRCNN-reID~\\cite{dong2016accelerating}, and FD-GAN~\\cite{ge2018fd} on all five datasets, their results are obtained by running the official code with the default implementation setup.} For SING~\\cite{jiao2018deep}, we reproduce their results on the MLR-Market-$1501$ and MLR-DukeMTMC-reID datasets. \n\nOur method adopting either single-scale or multi-scale resolution discriminators performs favorably against all competing methods on all five adopted datasets. The performance gains can be ascribed to three main factors. First, unlike most existing person re-ID methods, our model performs cross-resolution person re-ID in an end-to-end learning fashion. Second, our method learns the resolution-invariant representations, allowing our model to recognize persons in images of different resolutions. Third, our model learns to recover the missing details in LR images, thus providing additional discriminative evidence for person re-ID.\n\n\\input{exp\/vehicle_reid_mlr.tex}\n\n{\\flushleft {\\bf Effect of multi-scale adversarial learning.}}\nThe effect of adopting multi-scale adversarial learning strategy can be observed by comparing two of the variant methods, i.e., Ours (single-scale) and Ours (multi-scale). We observe that adopting multi-scale adversarial learning strategy consistently improves the performance over adopting single-scale adversarial learning strategy on all five datasets. \n\n{\\flushleft {\\bf Effect of deriving joint representation.}}\nThe advantage of deriving joint representation $\\mathbfit{v} = [f,g]$ can be assessed by comparing with two of our variant methods, i.e., Ours (multi-scale) ($f$ only) and Ours (multi-scale) ($g$ only). In the Ours (multi-scale) ($f$ only) method, the classifier $\\mathcal{C}$ only takes the resolution-invariant representation $f$ as input. In the Ours (multi-scale) ($g$ only) method, the classifier $\\mathcal{C}$ only takes the HR representation $g$ as input. We observe that deriving joint representation $\\mathbfit{v}$ consistently improves the performance over these two variant\/baseline methods. \n\n\\input{exp\/standard_reID.tex}\n\n\\subsubsection{Cross-Resolution Vehicle Re-ID}\n\nSimilar to cross-resolution person re-ID, we also consider multiple low-resolution (MLR) setting for vehicle re-ID and evaluate the proposed method on one \\emph{synthetic} and one \\emph{real-world} benchmarks. To construct the synthetic MLR-VeRi$776$ dataset, we down-sample images taken by one camera by a randomly selected down-sampling rate $r \\in \\{2, 3, 4\\}$, whereas the images taken by the other camera(s) remain unchanged. The VRIC dataset is a \\emph{genuine} and more challenging dataset for evaluating MLR vehicle re-ID and contains realistic images of multiple resolutions.\n\n\\revision{We compare our approach with cross-resolution re-ID methods, including FSRCNN-reID~\\cite{dong2016accelerating}, SING~\\cite{jiao2018deep}, CSR-GAN~\\cite{wang2018cascaded}, and MSVF~\\cite{kanaci2018vehicle}, and standard vehicle re-ID approaches, including Siamese-Visual~\\cite{shen2017learning}, OIFE~\\cite{wang2017orientation}, and VAMI~\\cite{zhou2018aware}.} For cross-resolution re-ID methods, the training set contains HR images and LR ones with all three down-sampling rates $r \\in \\{2, 3, 4\\}$ for each vehicle. For standard vehicle re-ID approaches, the training set comprises only HR images for each vehicle.\n\n{\\flushleft {\\bf Results.}}\nTable~\\ref{table:exp-vehicle-reid-mlr} presents the quantitative results recorded at ranks $1$ and $5$, and mAP on the two adopted datasets. We observe that our proposed algorithm achieves the state-of-the-art performance on both datasets. While our method is designed for cross-resolution person re-ID, the favorable performance (about $4\\%$ performance gains at rank 1 on both datasets) over all competing approaches (some of the competing methods are particularly tailored for cross-resolution vehicle re-ID task) demonstrates the generalization of our proposed algorithm.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{img\/Semi-super.png}\n  \\caption{\\textbf{Semi-supervised cross-resolution person re-ID on the MLR-CUHK03 dataset (\\%).} Even if the training data is only partially labeled, our method still exhibits sufficient ability in re-identifying person images of various resolutions.}\n  \\label{fig:exp-semi}\n  \\vspace{-4.0mm}\n\\end{figure}\n\n\\subsection{Evaluation of Standard Setting}\n\n\\revision{To examine if our method still improves re-ID performance when no significant resolution variations are present, we consider standard person re-ID and compare with existing approaches, including JLML~\\cite{Li-2017-IJCAI}, TriNet~\\cite{Hermans-2017-arXiv}, DML~\\cite{Zhang-2018-CVPR}, MGCAM~\\cite{Song-2018-CVPR}, DPFL~\\cite{Cheng-2016-CVPR}, PAN~\\cite{Zheng-2018-TCSVT}, PoseTransfer~\\cite{Zhong-2018-CVPR}, AlignedReID~\\cite{zhang2017alignedreid}, SVDNet~\\cite{sun2017svdnet}, CamStyle~\\cite{zhong2017camera}, PN-GAN~\\cite{qian2017pose}, SPreID~\\cite{kalayeh2018human}, FD-GAN~\\cite{ge2018fd}, Part Aligned~\\cite{suh2018part}, PCB~\\cite{sun2018beyond}, and DG-Net~\\cite{zheng2019joint}, in which the training set contains HR images for each identity only. As for our model, we take the same training set and \\emph{augment} it by down-sampling each image with three down-sampling rates $r \\in \\{2, 3, 4\\}$ (i.e., our training set contains HR images and LR ones with $r \\in \\{2, 3, 4\\}$) per person.}\n\n\\input{exp\/psnr.tex}\n\n\\input{exp\/GAN.tex}\n\n\\revision{To demonstrate that our method can help improve the state-of-the-art methods, we initialize our HR encoder $\\mathcal{F}$ with different pre-trained models as our method employs the same backbone (i.e., ResNet-$50$~\\cite{he2016deep}) as most of these methods. Specifically, we initialize our HR encoder $\\mathcal{F}$ with the pre-trained weights from FD-GAN~\\cite{ge2018fd}, Part Aligned~\\cite{suh2018part}, PCB~\\cite{sun2018beyond}, and DG-Net~\\cite{zheng2019joint}, and denote these variants of our method as Ours (FD-GAN), Ours (Part Aligned), Ours (PCB), and Ours (DG-Net), respectively.}\n\n{\\flushleft {\\bf Results.}}\n\\revision{Table~\\ref{table:exp-standard-reid} reports the quantitative results recorded at rank $1$ and the mAP on the two adopted datasets. We observe that our method results in further improvements over existing methods, achieving the state-of-the-art performance on the Market-$1501$~\\cite{zheng2015scalable} and DukeMTMC-reID~\\cite{zheng2017unlabeled} datasets in the standard person re-ID setting.}\n\nThe above quantitative results demonstrate that when significant resolution variations are present (i.e., cross-resolution setting), methods developed for standard re-ID suffer from the negative effect caused by the resolution mismatch issue. When considering the standard re-ID setting, the proposed method achieves further improvements over existing methods.\n\n\\subsection{Evaluation of Semi-Supervised Setting}\n\nIn the following, we conduct a series of semi-supervised experiments, and evaluate whether the proposed CAD-Net++ remains effective for cross-resolution person re-ID when only a subset of the labeled training data is available. Namely, less labeled training data can be used when computing the classification loss $\\mathcal{L}_\\mathrm{cls}$ (Eq. (\\ref{eq:cls})).\n\nWe evaluate the proposed method on the MLR-CUHK$03$ dataset. For performance evaluation, we choose $k\\%$ of the training data and keep their labels, while ignoring the labels of the rest, for $k\\in\\{0, 20, 40, 60, 80, 100\\}$. Note that the unlabeled data are still utilized in optimizing the multi-scale feature-level adversarial loss (Eq. (\\ref{eq:adv_multi})), the HR reconstruction loss (Eq. (\\ref{eq:rec})), the image-level adversarial loss (Eq. (\\ref{eq:adv_loss_image})), and the feature consistency loss (Eq. (\\ref{eq:consist})). We compare our proposed approach with RAIN~\\cite{chen2019learning} and two baseline methods: ``Baseline (train on HR)''~\\cite{chen2019learning} and ``Baseline (train on HR \\& LR)''~\\cite{chen2019learning}.\n\nFigure~\\ref{fig:exp-semi} presents the model performance at rank $1$. We observe that without any label information, our method achieves $8\\%$ at rank $1$. When the fraction of labeled data is increased to $20\\%$, our model reaches $69.2\\%$ at rank $1$, and is even better than SING~\\cite{jiao2018deep} ($67.7\\%$ at rank $1$) learned with $100\\%$ labeled data. When the fraction of labeled data is set to $40\\%$, our model achieves $77.1\\%$ at rank $1$ and compares favorably against most existing approaches reported in Table~\\ref{table:exp-ReID} that are learned with $100\\%$ labeled data.\n\nThe promising results demonstrate that sufficient re-ID ability is exhibited by our method even if only a small portion of labeled training data are available. This favorable property increases the applicability of the proposed CAD-Net++ in real-world re-ID applications. We attribute this property to the elaborately developed loss functions. Except for the classification loss in Eq. (\\ref{eq:cls}), all other loss functions can utilize unlabeled training data to regularize model training.\n\n\\subsection{Evaluation of the Recovered HR Images}\n\nTo demonstrate that our CRGAN is capable of recovering the missing details in LR input images of varying and even unseen resolutions, we evaluate the quality of the recovered HR images on the MLR-CUHK03 \\emph{test set} using SSIM, PSNR, and LPIPS~\\cite{zhang2018unreasonable} metrics. We employ the ImageNet-pretrained AlexNet~\\cite{krizhevsky2012imagenet} when computing LPIPS. We compare our CRGAN with CycleGAN~\\cite{zhu2017unpaired}, SING~\\cite{jiao2018deep}, and CSR-GAN~\\cite{wang2018cascaded}. For CycleGAN~\\cite{zhu2017unpaired}, we train its model to learn a mapping between LR and HR images. We report the quantitative results of the recovered image quality and person re-ID in Table~\\ref{table:image-comparison} with two different settings: (1) LR images of resolutions seen during training, i.e., $r \\in \\{2, 3, 4\\}$ and (2) LR images of unseen resolution, i.e., $r = 8$.\n\nFor seen resolutions (i.e., left block), we observe that our results using SSIM and PSNR metrics are slightly worse than the CSR-GAN~\\cite{wang2018cascaded} while compares favorably against SING~\\cite{jiao2018deep} and CycleGAN~\\cite{zhu2017unpaired}. However, our method performs favorably against these three methods using LPIPS metric and achieves the state-of-the-art performance when evaluating on cross-resolution person re-ID task. These results indicate that (1) SSIM and PSNR metrics are low-level pixel-wise metrics, which do not reflect high-level perceptual tasks and (2) the end-to-end learning of cross-resolution person re-ID would result in better re-ID performance and recover more perceptually realistic HR images as reflected by LPIPS. \n\nFor unseen resolution (i.e., right block), our method performs favorably against all three competing methods on all the adopted evaluation metrics. These results suggest that our method is capable of handling unseen resolution (i.e., $r = 8$) with favorable performance in terms of both image quality and person re-ID. Note that we only train our model with HR images and LR ones with $r \\in \\{2, 3, 4\\}$.\n\nFigure~\\ref{fig:HR-img} presents six examples. For each person, there are four different resolutions (i.e., $r \\in \\{1, 2, 4, 8\\}$). Note that images with down-sampling rate $r = 1$ indicate that the images remain their original sizes and are the corresponding HR images of the LR ones. We observe that when LR images with down-sampling rate $r = 8$ are given, our model recovers the HR details with the highest quality among all competing methods. In addition, we present four examples in Figure~\\ref{fig:HR-uni} to show that the proposed CRGAN is able to recover the missing details in LR images of \\emph{various} resolutions. In each example, we have eight input images with down-sampling rates $r \\in \\{1, 2, 3, 4, 5, 6, 7, 8\\}$, respectively, where $r \\in \\{1, 2, 3, 4\\}$ are seen during training while the rest are unseen. The corresponding recovered HR images are displayed in Figure~\\ref{fig:HR-uni}. Both quantitative and qualitative results above confirm that our model can handle \\emph{a range of} seen resolutions and generalize well to \\emph{unseen} resolutions using just one single model, i.e., CRGAN.\n\n\\input{exp\/recovery.tex}\n\n\\input{exp\/exp-ablation-hrgan.tex}\n\n\\input{exp\/exp-ablation-consis-loss.tex}\n\n\\input{exp\/exp-ablation-multi-scale.tex}\n\n\\subsection{Ablation Study}\n\n\\subsubsection{Loss Functions}\n\nTo analyze the importance of each developed loss function, we conduct an ablation study on the MLR-CUHK03 dataset. Table~\\ref{table:exp-abla} reports the quantitative results of the recovered HR images and the performance of cross-resolution person re-ID recorded at rank $1$.\n\n{\\flushleft {\\bf Multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$.}}\nWithout $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$, our model does not learn the resolution-invariant representations and thus suffers from the resolution mismatch issue. Significant performance drops in the recovered image quality and re-ID performance occur, indicating the importance of our method for learning the resolution-invariant representations to address the problem of resolution mismatch.\n\n{\\flushleft {\\bf HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$.}}\nOnce $\\mathcal{L}_\\mathrm{rec}$ is excluded, there is no explicit supervision to guide the CRGAN to perform image recovery, and the model suffers from information loss in compressing visual images into semantic feature maps. Severe performance drops in terms of the recovered image quality and re-ID performance are hence caused.\n\n{\\flushleft {\\bf Image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$.}}\nWhen $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ is turned off, our model is not encouraged to produce perceptually realistic HR images as reflected by LPIPS, resulting in the performance drop of $3.6\\%$ at rank $1$.\n\n{\\flushleft {\\bf Classification loss $\\mathcal{L}_\\mathrm{cls}$.}}\nAlthough our model is still able to perform image recovery without $\\mathcal{L}_\\mathrm{cls}$, our model cannot perform discriminative learning for person re-ID since data labels are not used during training. Thus, significant performance drop in person re-ID occurs.\n\n\\revision{\n{\\flushleft {\\bf Consistency loss $\\mathcal{L}_\\mathrm{consist}$.}}\nAs shown in Table~\\ref{table:exp-abla}, once $\\mathcal{L}_\\mathrm{consist}$ is disabled, the quality of the recovered HR images almost remains the same but a performance drop of $0.8\\%$ at rank $1$ is occurred. To further evaluate the contribution of the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$, we report the results of the Ours w\/o $\\mathcal{L}_\\mathrm{consist}$ method on all five cross-resolution person re-ID datasets (i.e., MLR-CUHK$03$, MLR-VIPeR, CAVIAR, MLR-Market-$1501$, and MLR-DukeMTMC-reID) and all two cross-resolution vehicle re-ID datasets (i.e., MLR-VeRi$776$ and VRIC) in Table~\\ref{table:ablation-consis-oss}. We observe that without the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$, our model suffers from performance drops on all seven datasets. While the performance improvement in person re-ID contributed by the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ is marginal on the MLR-CUHK$03$ dataset, our results show that incorporating such a simple loss function consistently improves the performance on all seven datasets of both tasks without increasing the model complexity and capacity (i.e., the number of model parameters remains the same). In particular, on the MLR-Market-$1501$ and MLR-DukeMTMC-reID datasets, introducing the feature consistency loss $\\mathcal{L}_\\mathrm{consist}$ results in $3.7\\%$ and $3.5\\%$ performance improvement at Rank $1$, respectively.\n}\n\nThe ablation study demonstrates that the losses $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$, $\\mathcal{L}_\\mathrm{rec}$, and $\\mathcal{L}_\\mathrm{cls}$ are crucial to our method, while the losses $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ and $\\mathcal{L}_\\mathrm{consist}$ are helpful for further improving the performance.\n\n\\vspace{-4.0mm}\n\\revision{\\subsubsection{Multi-scale adversarial learning}}\n\n\\revision{For learning the resolution-invariant representations $f$, we perform an ablation study on the MLR-CUHK$03$ dataset to analyze the effect of introducing feature-level discriminators $\\{\\mathcal{D}_F^j\\}$ at different feature levels (scales) to align feature distributions between HR and LR images. We report the results recorded at rank $1$ for cross-resolution person re-ID, evaluate the quality of the recovered HR images using SSIM, PSNR, and LPIPS~\\cite{zhang2018unreasonable} metrics, and report the numbers of parameters of the feature-level discriminators $\\{\\mathcal{D}_F^j\\}$. As shown in Table~\\ref{table:exp-multi-abla}, the performance in rank $1$ of cross-resolution person re-ID and the quality of the recovered HR images improve as the number of feature distribution alignments increases. We observe that when introducing five feature-level discriminators $\\{\\mathcal{D}_F^j\\}_{j=1}^{5}$ to align feature distributions at all five feature levels, our model reaches the best performance with the best quality of the recovered HR images, but the number of parameters of the feature-level discriminators $\\{\\mathcal{D}_F^j\\}_{j=1}^{5}$ increases accordingly. To balance the parameter number (efficiency) and performance, we select the setting with the first two levels (i.e., $j \\in \\{1, 2\\}$) for our method, and denote it as ``Ours (multi-level)'' for performance comparison.}\n\n\\begin{figure}[t]\n  \\vspace{-2.0mm}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{img\/quali_psnr_hr.pdf}\n  \\vspace{-6.0mm}\n  \\caption{\\revision{\\textbf{Ablation study of the recovered HR images.} We present the recovered HR images obtained from our method as well as the ablation methods on the MLR-CUHK03 \\emph{test set} (i.e., cross-resolution person re-ID setting).}}\n  \\label{fig:abl}\n  \\vspace{-5.0mm}\n\\end{figure}\n\n\\input{exp\/tsne.tex}\n\n\\input{exp\/tsne-compare.tex}\n\n\\input{exp\/rank-person.tex}\n\n\\input{exp\/rank-vehicle.tex}\n\n\\vspace{3.0mm}\n\n\\subsubsection{The Recovered HR Images}\n\nHere, we present the recovered HR images generated by our method and the ablation methods on the MLR-CUHK03 \\emph{test set} using the cross-resolution person re-ID setting.\n\nWe visualize two examples of the recovered HR images in Figure~\\ref{fig:abl}. In each example, the recovered HR images with different input down-sampling rates $r = \\{1, 2, 4, 8\\}$ are shown. We observe that without applying the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ (Eq. (\\ref{eq:rec})), our model is not able to recover high-quality HR images. Without the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$ (Eq. (\\ref{eq:adv_multi})), our model does not learn resolution-invariant representations. Thus, our model cannot recover the HR details of images of an unseen resolution, e.g., $r = 8$. While our model can still reconstruct the feature maps to their HR images without the image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ (Eq. (\\ref{eq:adv_loss_image})), the recovered HR images may not look perceptually realistic, especially for input images with the down-sampling rate $r = 8$.\n\nThe examples in Figure~\\ref{fig:abl} demonstrate that the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ is essential to image recovery. The multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{F}}$ enables our model to deal with images of unseen resolutions while the image-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_{I}}$ encourages our model to recover perceptually realistic HR images.\n\n\\input{exp\/paras.tex}\n\n\\subsection{Resolution-Invariant Representation}\n\n\\subsubsection{Effect of Feature-level Adversarial Loss}\n\nTo demonstrate the effectiveness of our model in deriving the resolution-invariant representations $f$, we first apply global average pooling to $f$ to obtain the resolution-invariant feature vector $\\mathbfit{w} = \\mathrm{GAP}(f) \\in {\\mathbb{R}}^d$. We then visualize $\\mathbfit{w}$ on the MLR-CUHK03 \\emph{test set} in Figure~\\ref{fig:tsne}.\n\nWe select $50$ different identities, each of which is indicated by a unique color, as shown in Figure~\\ref{fig:tsne-baseline} and Figure~\\ref{fig:tsne-identity}. In Figure~\\ref{fig:tsne-baseline}, we observe that without the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_F}$ (Eq. (\\ref{eq:adv_multi})), our model cannot establish a well-separated feature space. When loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_F}$ is imposed, the projected feature vectors are well separated as shown in Figure~\\ref{fig:tsne-identity}. These two figures indicate that without loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_F}$, our model does not learn resolution-invariant representations, thus implicitly suffering from the negative impact induced by the resolution mismatch issue.\n\nWe note that the projected feature vectors in Figure~\\ref{fig:tsne-identity} are well separated, suggesting that sufficient re-ID ability can be exhibited by our model. On the other hand, for Figure~\\ref{fig:tsne-resolution}, we colorize each image resolution with a unique color in each identity cluster (four different down-sampling rates $r \\in \\{1, 2, 4, 8\\}$). We observe that the projected feature vectors of the same identity but different down-sampling rates are all well clustered. We note that images with down-sampling rate $r = 8$ are not presented in the training set (i.e., unseen resolution).\n\nThe above visualizations demonstrate that our model learns resolution-invariant representations and generalizes well to unseen image resolution (e.g., $r = 8$) for cross-resolution person re-ID.\n\n\\subsubsection{Visual Comparisons of the Re-ID Feature Vector}\n\nWe visualize the feature vector for person re-ID on the MLR-CUHK$03$ \\emph{test set} via t-SNE and present visual comparisons with SING~\\cite{jiao2018deep} and CSR-GAN~\\cite{wang2018cascaded}. The comparisons are conducted on a subset of $50$ identities. Note that for our method, we use the joint feature vector $\\mathbfit{u} = \\mathrm{GAP}(\\mathbfit{v}) \\in {\\mathbb{R}}^{2d}$ for person re-ID.\n\nFigure~\\ref{fig:tsne-identity-comp} shows the visual results of the three competing methods, including SING~\\cite{jiao2018deep} in Figure~\\ref{fig:tsne-identity-naive}, CSR-GAN~\\cite{wang2018cascaded} in Figure~\\ref{fig:tsne-identity-better}, and our method in Figure~\\ref{fig:tsne-identity-ours}. In each figure, we plot the projected re-ID feature vectors of each identity with a specific color. We observe that both SING~\\cite{jiao2018deep} and CSR-GAN~\\cite{wang2018cascaded} do not separate instances of different identities very well. In contrast, as shown in Figure~\\ref{fig:tsne-identity-ours}, our method successfully recognizes most identities, implying the reliable re-ID ability of our proposed method.\n\nIn Figure~\\ref{fig:tsne-reso-comp}, we adopt resolution-specific coloring and display the visual results of SING~\\cite{jiao2018deep} in Figure~\\ref{fig:tsne-reso-naive}, CSR-GAN~\\cite{wang2018cascaded} in Figure~\\ref{fig:tsne-reso-better}, and our method in Figure~\\ref{fig:tsne-reso-ours}. As shown in Figure~\\ref{fig:tsne-reso-naive} and Figure~\\ref{fig:tsne-reso-better}, SING~\\cite{jiao2018deep} and CSR-GAN~\\cite{wang2018cascaded} tend to mix images yielded with the down-sampling rate $r = 8$, i.e., those in red, even if these images are of different categories. The visual results indicate that both SING~\\cite{jiao2018deep} and CSR-GAN~\\cite{wang2018cascaded} suffer from the resolution mismatch problem. Our method, on the other hand, learns resolution-invariant representations. As shown in Figure~\\ref{fig:tsne-reso-ours}, images even with \\emph{unseen} different down-sampling rates (e.g., $r = 8$) are well clustered with respect to the identities.\n\nThe above visual comparisons verify that through learning resolution-invariant representations, our method works well on images of diverse and even unseen resolutions.\n\n\\subsection{Top-Ranked Gallery Images}\n\n{\\flushleft {\\bf Person re-ID.}}\nAs shown in Figure~\\ref{fig:rank-person}, given an LR query image with down-sampling rate $r = 4$ or $r = 8$, we present the first $7$ top-ranked HR gallery images in Figure~\\ref{fig:rank-person-4} and Figure~\\ref{fig:rank-person-8}, respectively. We compare our method (bottom row) with two approaches developed for cross-resolution person re-ID SING~\\cite{jiao2018deep} (top row) and CSR-GAN~\\cite{wang2018cascaded} (middle row). The green and red boundaries indicate correct and incorrect matches, respectively. In Figure~\\ref{fig:rank-person-4}, all three approaches including ours achieves almost correct matches. However, from the results in the top row of Figure~\\ref{fig:rank-person-8}, we observe that SING~\\cite{jiao2018deep} does not have any correct matches, while CSR-GAN~\\cite{wang2018cascaded} achieves $1$ out of $7$ correct matches. Our method, on the contrary, achieves $6$ out of $7$ correct matches, which again verifies the effectiveness and robustness of our model. Note that the resolution ($r = 8$) of the query image is not seen during training.\n\n{\\flushleft {\\bf Vehicle re-ID.}}\nSimilarly, as shown in Figure~\\ref{fig:rank-vehicle}, given an LR query image with down-sampling rates $r = 4$ and $r = 8$, we present the first $7$ top-ranked HR gallery images in Figure~\\ref{fig:rank-vehicle-4} and Figure~\\ref{fig:rank-vehicle-8}, respectively. We also compare our method (bottom row) with two existing methods, i.e, SING~\\cite{jiao2018deep} (top row) and CSR-GAN~\\cite{wang2018cascaded} (middle row). In Figure~\\ref{fig:rank-vehicle-4}, all three approaches achieve satisfactory re-ID results. We then consider the case where the resolution ($r = 8$) of the query image is unseen during training. In Figure~\\ref{fig:rank-vehicle-8}, SING~\\cite{jiao2018deep} and CSR-GAN~\\cite{wang2018cascaded} only have $2$ and $3$ out of $7$ correct matches, respectively. In contrast, our method achieves $6$ out of $7$ correct matches. The comparison with existing methods also supports the applicability of our method to cross-resolution vehicle re-ID.\n\n\\subsection{Sensitivity Analysis of Hyper-parameters}\n\nWe further analyze the sensitivity of our model against the hyper-parameters introduced in the total loss $\\mathcal{L}$ (Eq. (\\ref{eq:fullobj})). We conduct sensitivity analysis by varying the value of each hyper-parameter and report the results on the MLR-CUHK$03$ \\emph{validation set} using the cross-resolution person re-ID setting. Figure~\\ref{fig:lambda} presents the experimental results.\n\nFor $\\lambda_\\mathrm{rec}$ and $\\lambda_\\mathrm{adv}^{\\mathcal{D}_F}$, if their values are set to $0$, our model suffers from performance drops, as shown Figure~\\ref{fig:lambda-rec} and Figure~\\ref{fig:lambda-adv-F}. When $\\lambda_\\mathrm{rec}$ and $\\lambda_\\mathrm{adv}^{\\mathcal{D}_F}$ lie in a certain range (near $1$), the performance of our method is improved and remains stable. However, once their values are too large, e.g., $100$, significant performance drop occurs since the corresponding losses (i.e., the HR reconstruction loss $\\mathcal{L}_\\mathrm{rec}$ and the multi-scale feature-level adversarial loss $\\mathcal{L}_\\mathrm{adv}^{\\mathcal{D}_F}$) dominate the total loss. In Figure~\\ref{fig:lambda-adv-I}, we observe that when the value of $\\lambda_\\mathrm{adv}^{\\mathcal{D}_I}$ is set to $0$, modest performance drop happens. If $\\lambda_\\mathrm{adv}^{\\mathcal{D}_I}$ varies within a reasonable region, e.g., $0.01\\sim2$, the performance remains stable. However, when the value of $\\lambda_\\mathrm{adv}^{\\mathcal{D}_I}$ is too large, e.g., larger than $10$, severe performance drop occurs since the image-level adversarial loss $\\lambda_\\mathrm{adv}^{\\mathcal{D}_I}$ dominates the total loss.\n\nIn sum, the performance of our model remains stable when the values of the hyper-parameters lie in a certain range (near $1$). As a result, we set $\\lambda_\\mathrm{rec} = 1$, \n$\\lambda_\\mathrm{adv}^{\\mathcal{D}_F} = 1$, and $\\lambda_\\mathrm{adv}^{\\mathcal{D}_I} = 1$ without further fine-tuning them, though applying grid search or random search for hyper-parameter optimization might lead to further performance improvement.\n\\section{Conclusions}\n\nWe have presented an \\emph{end-to-end trainable} generative adversarial network, CAD-Net++, for addressing the resolution mismatch issue in person re-ID. The core technical novelty lies in the unique design of the proposed CRGAN which learns the \\emph{resolution-invariant} representations while being able to recover \\emph{re-ID oriented} HR details preferable for person re-ID. Our cross-modal re-ID network jointly considers the information from two feature modalities, resulting in improved re-ID performance. \\revised{Extensive experimental results show that our approach performs favorably against existing cross-resolution person re-ID methods on five challenging benchmarks, achieves competitive performance against existing approaches even when no significant resolution variations are present, and produces perceptually higher quality HR images using only a \\emph{single} model. Visualization of the resolution-invariant representations further verifies our ability in handling query images with \\emph{varying} or even \\emph{unseen} resolutions. Furthermore, we demonstrate the applicability of our method through cross-resolution vehicle re-ID task. Experimental results confirm the generalization of our model on cross-resolution visual tasks. The extensions to semi-supervised settings also demonstrate the superiority of our method over existing approaches. Thus, the use of our model for practical re-ID applications can be strongly supported.}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCompressive sensing has triggered significant research activity in recent years. Its central motif is that sparse signals can be recovered from what was previously believed to be highly incomplete\ninformation \\cite{carota06,do06-2}.  In particular, it is now known \\cite{carota06,cata06,ruve08,ra07,ra08,ra09-1} that an $s$-sparse trigonometric polynomial of maximal degree $N$ can be recovered from $m \\asymp s \\log^4(N)$ sampling points.  These $m$ samples  can be chosen as a random subset from  the discrete set $\\{j\/N\\}_{j=0}^{N-1}$ \\cite{carota06,cata06,ruve08}, or independently from the uniform measure on $[0,1]$, see \\cite{ra07,ra08,ra09-1}.  \n\nUntil now, all sparse recovery results of this type required that the underlying basis be uniformly bounded like the trigonometric system, so as to be \\emph{incoherent} with point samples \\cite{caro07}.   As the main contribution of this paper, we show that this condition may be relaxed, obtaining comparable sparse recovery results for any basis that is bounded by a square-integrable envelope function.  As a special case, we focus on the Legendre system over the domain $[-1,1]$.    To account for the blow-up of the Legendre system near the endpoints of its domain, the random sampling points are drawn according to the Chebyshev probability measure.  This aligns with classical results on Lagrange interpolation which support the intuition that Chebyshev points are much better suited for the recovery of polynomials than uniform points are \\cite{br97-1}.\n\nIn order to deduce our main results we establish the \\emph{restricted isometry property} (RIP) for a preconditioned\nversion of the matrix whose entries are the Legendre polynomials evaluated at sample points chosen from the Chebyshev measure.  The concept of preconditioning seems to be new in the context\nof compressive sensing, although it has appeared within the larger scope of sparse approximation in a different context in \\cite{sv08}. It is likely that the idea of preconditioning can be exploited in other situations of interest as well.\n\n\nSparse expansions of multivariate polynomials in terms of tensor products\nof Legendre polynomials recently appeared in the problem of numerically \nsolving stochastic or parametric PDEs \\cite{codesc10,alho10}. Our results indeed extend easily to tensor\nproducts of Legendre polynomials, and the application of our techniques in this context \nof numerical solution of SPDEs seems very promising. \nOur results may also be transposed into the setting of function approximation.  In particular,  we show that the aforementioned sampling and reconstruction procedure is guaranteed to produce near-optimal approximations to functions\nin infinite-dimensional spaces of functions having $\\ell_p$-summable Fourier-Legendre\ncoefficients ($0<p < 1$), provided that\nthe maximal polynomial degree in the $\\ell_1$-reconstruction procedure is fixed appropriately in terms of the sparsity level. \n\n\nOur original motivation for this work was the recovery of sparse spherical \nharmonic expansions \\cite{anasro99} from randomly located samples on the sphere. While our preliminary results in this context\nseem to be only suboptimal \\cite{rw-prep}, \nthe results in the present paper apply at least to the recovery of functions on the\nsphere that are invariant under rotations of the sphere around a fixed axis. \nSparse spherical harmonic expansions were recently exploited with good numerical success\nin the spherical inpainting problem for the cosmic microwave background \\cite{abdefamongst08}, but \nso far this problem had lacked a theoretical understanding. \n\nWe note that the Legendre polynomial transform has fast \nalgorithms for matrix vector multiplication; see for instance \\cite{dadeja97,herokose96,da03-1,postta98-1,ty08}.\nThis fact is of crucial importance in numerical algorithms used for reconstructing the original function from its sample values -- especially when the dimension\nof the problem gets large.\n\nOur results extend to any polynomial system which is orthogonal with respect to a finitely-supported weight function satisfying a mild continuity condition; this includes the Jacobi polynomials, of which the Legendre polynomials are a special case.   It turns out that the Chebyshev measure is universal for this rich class of orthogonal polynomials, in the\nsense that our corresponding result requires the random sampling points to be drawn according\nto the Chebyshev measure, independent of the particular weight function. \n\n\\medskip\n\nOur paper is structured as follows: Section 2 contains the main results for recovery of\nLegendre-sparse polynomials. Section 3 illustrates these results with numerical experiments.\nIn Section 4 we recall known theorems on $\\ell_1$-minimization and in Section 5, we prove the results presented in Section 2. Section 6 extends the results to a rich class of orthogonal polynomial systems, including the Jacobi polynomials, while Section 7 contains our main result on the recovery of continuous\nfunctions that are well approximated by Legendre-sparse polynomials.\n\n\n\\paragraph{\\bf Notation.}\n\nLet us briefly introduce some helpful notation.  The $\\ell_p$-norm on ${\\mathbb{R}}^N$ is defined as \n$\n\\|z\\|_p = \\Big( \\sum_{j=1}^N |z_j|^p \\Big)^{1\/p}, 1 \\leq p < \\infty,\n$\nand $\\|z\\|_\\infty = \\max_{j=1,\\hdots,N} |z_j|$ as usual. \nThe ``$\\ell_0$-norm'', $\\|z\\|_0 = \\#\\{j: z_j \\neq 0\\}$,\ncounts the number of non-zero entries of $z$. A vector $z$ is called\n$s$-sparse if $\\|z\\|_0 \\leq s$, and the error of best $s$-term approximation\nof a vector $z\\in {\\mathbb{R}}^N$ in $\\ell_p$ is defined as\n\\[\n\\sigma_s(z)_p = \\inf_{y : \\|y\\|_0 \\leq s} \\|y-z\\|_p.\n\\]\nClearly, $\\sigma_s(z)_p = 0$ if $z$ is $s$-sparse. Informally, $z$ is called compressible\nif $\\sigma_s(z)_1$ decays quickly as $s$ increases. A result due to Stechkin, see e.g.\\ \\cite[Lemma 3.1]{fora10-1}, states that, for \n$q<p$, \n\\begin{equation}\\label{Stechkin}\n\\sigma_s(x)_p \\leq s^{1\/p-1\/q} \\|x\\|_q;\n\\end{equation}\nthus, vectors $x \\in B_q^N = \\{x \\in {\\mathbb{R}}^N, \\|x\\|_q \\leq 1\\}$ for $0<q\\leq 1$ \ncan be considered a good model for\ncompressible signals.\n\nFor $N \\in {\\mathbb{N}}$, we use the notation $[N] = \\{1,\\hdots,N\\}$.  In this article, $C>0$ will always denote a universal constant that\nmight be different in each occurence. \n\nThe Chebyshev probability measure (also referred to as arcsine distribution) on $[-1,1]$ is given by  $d\\nu(x) = \\pi^{-1} (1 - x^2)^{-1\/2}dx$. \nIf a random variable $X$ is uniformly distributed on $[0,\\pi]$, then the random variable $Y = \\cos{X}$ is distributed according to the Chebyshev measure.\n\n\\section{Recovery of Legendre-sparse polynomials from a few samples}\n\nConsider the problem of recovering a polynomial $g$ \nfrom $m$ sample values $g(x_1),\\hdots,g(x_m)$.  If the number of sampling points is less than or \nequal to the degree of $g$, such reconstruction is impossible in general due to \ndimension reasons. Therefore, as usual in the compressive sensing literature, \nwe make a sparsity assumption. \nIn order to introduce a suitable\nnotion of sparsity we consider the basis of Legendre polynomials $L_n$ on $[-1,1]$, normalized so as to be orthonormal with respect to the uniform measure on $[-1,1]$, i.e. $\\frac{1}{2} \\int_{-1}^{1} L_n(x) L_{\\ell}(x) dx = \\delta_{n,\\ell}$.  \n\n\n\n\n\nAn arbitrary real-valued \npolynomial $g$ of degree $N-1$ can be expanded in terms of Legendre polynomials\n\\begin{equation}\\label{gLegendre}\ng(x) = \\sum_{n=0}^{N-1} c_n L_n(x), \\quad x \\in [-1,1].\n\\end{equation}\nIf the coefficient vector $c \\in {\\mathbb{R}}^{N}$ is $s$-sparse, we call the corresponding polynomial\n\\emph{Legendre $s$-sparse}, or simply Legendre-sparse. If $\\sigma_s(c)_1$ decays \nquickly as $s$ increases, then $g$ is called Legendre--compressible.\n\nWe aim to reconstruct Legendre--sparse polynomials, and more generally Legendre--compressible polynomials, of maximum degree $N-1$ from $m$ samples $g(x_1),\\hdots,g(x_m)$, where $m$ \nis desired to be small -- at least smaller than $N$. Writing $g$ in the form \\eqref{gLegendre}\nthis task clearly amounts to reconstructing the coefficient vector $c \\in {\\mathbb{R}}^{N}$. \n\nTo the set of $m$ sample points $(x_1, \\hspace{1mm} \\hdots \\hspace{1mm}, x_m)$ we associate the $m \\times N$ \\emph{Legendre} matrix $\\Phi$ defined component-wise by \n\\begin{equation}\n\\label{legendre_matrix}\n\\Phi_{j,k} = \nL_{k-1}(x_j), \\quad j \\in [m],\\hspace{2mm}  k \\in [N].\n\\end{equation}\nNote that the samples $y_j = g(x_j)$ may be expressed concisely in terms of the coefficient \nvector $c \\in \\mathbb{R}^{N}$ according to\n$$\ny =\n\\Phi c.\n$$\nReconstructing $c$ from the vector $y$ amounts to solving this system of linear\nequations. As we are interested in the underdetermined case $m < N$, this system\ntypically has infinitely many solutions, and our task is to single out the original sparse $c$. The obvious but naive approach for doing this is by solving for the sparsest solution that agrees with the measurements, \n\\begin{equation}\\label{l0:prog}\n\\min_{ z \\in {\\mathbb{R}}^{N}}  \\|z\\|_0 \\quad \\mbox{subject to}\\quad \\Phi z = y.\n\\end{equation}\nUnfortunately, this problem is NP-hard in general \\cite{avdama97,aw10}. To overcome this computational\nbottleneck the compressive sensing\nliterature has suggested various tractable alternatives \\cite{gitr07,carota06,netr08}, \nmost notably $\\ell_1$-minimization (basis pursuit) \\cite{chdosa99,carota06,do06-2}, \non which we focus in this paper. Nevertheless, it follows from our findings that \ngreedy algorithms such as CoSaMP \\cite{netr08} or Iterative Hard Thresholding \\cite{blda09} may also be\nused for reconstruction.\n\n\\medskip\n\nOur main result is that any Legendre $s$-sparse polynomial may be recovered efficiently from a number of samples $m \\asymp s \\log^3(s)\\log(N)$.  Note that at least up to the logarithmic factors, this rate is optimal. Also the \ncondition on $m$ is implied by the simpler one $m \\asymp s \\log^4{N}$\nReconstruction is also robust: \\emph{any} polynomial may be recovered efficiently to within a factor of its best approximation by a Legendre $s$-sparse polynomial, and, if the measurements are corrupted by noise, $g(x_1) + \\eta_1, \\hdots, g(x_m) + \\eta_m$, to within an additional factor of the noise level $\\varepsilon = \\| \\eta \\|_{\\infty}$.  We have\n\n\\begin{theorem}\n\\label{uniform:noise}\nLet $N,m,s \\in {\\mathbb{N}}$ be given such that\n$$\nm \\geq C s \\log^3(s) \\log(N).\n$$\nSuppose that $m$ sampling points $(x_1, \\hdots, x_m)$ are drawn independently at random from the Chebyshev  \nmeasure, and consider the $m \\times N$ Legendre matrix $\\Phi$ with entries $\\Phi_{j,k} = L_{k-1}(x_j)$, and the $m \\times m$ diagonal matrix ${\\cal A}$ with entries $a_{j,j} = (\\pi\/2)^{1\/2}(1-x_j^2)^{1\/4}$.   Then with probability exceeding\n$1-N^{-\\gamma \\log^3(s)}$ the following \nholds for all polynomials $g(x) = \\sum_{k=0}^{N-1} c_k L_k(x)$.\nSuppose that noisy sample values $y = \\big( g(x_1) + \\eta_1, \\hdots, g(x_m) + \\eta_m \\big) = \\Phi c + \\eta$ are observed, and $\\|{\\cal A}\\eta\\|_{\\infty} \\leq \\varepsilon$.   Then the coefficient vector $c = (c_0, c_1, \\hdots, c_{N-1})$ is recoverable to within a factor of its best $s$-term approximation error \nand to a factor of the noise level by solving the inequality-constrained $\\ell_1$-minimization problem\n\\begin{align}\\label{relaxed}\nc^{\\#} = \\arg \\min_{z \\in {\\mathbb{R}}^N} \\| z \\|_1 \\quad \\mbox{ subject to } \\quad \\| {\\cal A} \\Phi z - {\\cal A} y \\|_2 \\leq \\sqrt{m}\\varepsilon.\n\\end{align}\nPrecisely, \n\\begin{equation}\\label{l1:approx2}\n\\| c -c^{\\#} \\|_{2} \\leq \\frac{C_1 \\sigma_s(c)_1}{\\sqrt{s}} + C_2\\varepsilon,\n\n\\end{equation}\n\nand\n\\begin{equation}\\label{l1:approx}\n\\|c-c^{\\#}\\|_{1} \\leq D_1 \\sigma_s(c)_1 + D_2 \\sqrt{s} \\varepsilon.\n\\end{equation}\nThe constants $C, C_1,C_2,D_1,D_2$, and $\\gamma$ are universal.\n\\end{theorem}\n\n\\begin{remark}\\label{rem22}\n\\begin{itemize}\n\\item[(a)] \\emph{In the noiseless ($\\varepsilon = 0$) and exactly $s$-sparse case ($\\sigma_s(x)_1 = 0$), the above theorem\nimplies exact recovery via} \n\\[\nc^{\\#} = \\arg \\min_{z \\in {\\mathbb{R}}^N} \\| z \\|_1 \\quad \\mbox{ \\emph{subject to} } \\quad \\Phi z = y.\n\\]\n\\item[(b)] \\emph{The condition $\\|{\\cal A}\\eta\\|_{\\infty} \\leq \\varepsilon$ is satisfied in particular if $\\|\\eta\\|_\\infty \\leq \\varepsilon$.}\n\n\\item[(c)]\n\\emph{The proposed recovery method \\eqref{relaxed} is \\emph{noise-aware}, in that it requires knowledge of the noise level $\\varepsilon$ a priori.\nOne may remove this drawback by using other reconstruction algorithms such as \nCoSaMP \\cite{netr08} or Iterative Hard Thresholding \\cite{blda09} which also achieve the reconstruction rates \\eqref{l1:approx2} and \\eqref{l1:approx} under the stated hypotheses, but do not require knowledge of $\\varepsilon$ \\cite{blda09,netr08}.  Actually, those algorithms always return $2s$-sparse vectors as approximations, in which case the $\\ell_1$-stability result \\eqref{l1:approx} follows immediately from \\eqref{l1:approx2}, see \\cite[p.\\ 87]{b09} for details.   }\n\n\n\n\\end{itemize}\n\\end{remark}\n\n\\section{Numerical Experiments}\n\nLet us illustrate the results of Theorem \\ref{uniform:noise}.  In Figure $1(a)$ we plot a polynomial $g$ that is $5$-sparse in Legendre basis and with maximal degree $N = 80$ along with $m = 20$ sampling points drawn independently from the Chebyshev measure.  This polynomial is reconstructed exactly from the illustrated sampling points as the solution to the $\\ell_1$-minimization problem \\eqref{relaxed} with $\\varepsilon = 0$.   In Figure $1(b)$ we plot the same \nLegendre-sparse polynomial in solid line, but the $20$ samples \nhave now been corrupted by zero-mean Gaussian noise $y_j = g(x_j) + \\eta_j$.  Specifically, we take ${\\mathbb{E}} \\hspace{1mm} (|\\eta_j|^2 )  = 0.025$, so that the expected noise level $\\varepsilon \\approx 0.16$.    In the same figure, we superimpose \nin dashed line the polynomial obtained from these noisy measurements as the solution \nof the inequality-constrained $\\ell_1$-minimization problem \\eqref{relaxed} with noise level \n$\\varepsilon  = 0.16$.  \n\n\\begin{figure}[h]\n\\label{fig:illustrate}\n\\subfigure{\n\\includegraphics[width=0.45\\textwidth]{P_exact_80_20_5}\n}\\hfill\n\\subfigure{\n\\includegraphics[width=0.45\\textwidth]{Pnoisy80_20_5_recon}\n}\n\\caption{(a) A Legendre-$5$-sparse polynomial of maximal degree $N = 80$, and its exact reconstruction from $20$ samples drawn independently from the Chebyshev distribution.  (b) The same polynomial (solid line), and its approximate reconstruction from $20$ samples corrupted with noise (dashed line).}\n\\end{figure}\n\n\nTo be more complete, we plot a phase diagram illustrating, for $N = 300$, and several values of $s\/m$ and $m\/N$ between $0$ and $.7$, the success rate of $\\ell_1$-minimization in exactly recovering Legendre $s$-sparse polynomials $g(x) = \\sum_{k=0}^{N-1} c_k L_k(x)$.  \nThe results, illustrated in Figure $2$, show a sharp transition between uniform\nrecovery (in black) and no recovery whatsoever (white).  This transition curve is similar to the phase transition curves obtained for other compressive sensing matrix ensembles, e.g.\\ the random partial discrete Fourier matrix or the Gaussian ensemble.  For more details, we refer the reader to \\cite{dt}.\n\n\n\\begin{SCfigure}\n\\centering\n{\\label{legendre:plot}\n    \\psfrag{ylabel}{$\\frac{s}{m}$}\n                \\psfrag{xlabel}{$\\frac{m}{N}$}}\n\\includegraphics[width=7cm, height = 6cm]{phase}\n\\caption{Phase diagram illustrating the transition between uniform recovery (black) and no recovery whatsoever (white) of Legendre-sparse polynomials of sparsity level $s$ and using $m$ measurements, as $s$ and $m$ vary over the range $s \\leq m \\leq N = 300$.  In particular, for each pair $(s\/m, m\/N)$, we record the rate of success out of $50$ trials of $\\ell_1$-minimization in recovering $s$-sparse coefficient vectors with random support over $[N]$ and with i.i.d. standard Gaussian coefficients from $m$ measurements distributed according to the Chebyshev measure.  }\n\\centering\n\\end{SCfigure}\n\n\n\n\n\\section{Sparse recovery via restricted isometry constants}\n We prove Theorem \\ref{uniform:noise} by showing that the preconditioned Legendre matrix ${\\cal A}\\Phi$ satisfies the  \\emph{restricted isometry property} (RIP) \\cite{cata06,carota06-1}.  To begin, let us recall the notion of restricted isometry constants for a matrix $\\Psi$.\n\n\\begin{definition}[Restricted isometry constants]\nLet $\\Psi \\in {\\mathbb{C}}^{m \\times N}$. For $s \\leq N$, the restricted isometry constant $\\delta_s$ \nassociated to $\\Psi$ is the smallest number $\\delta$ for which\n\\begin{equation}\\label{def:RIP}\n(1-\\delta) \\|c\\|_2^2 \\leq \\|\\Psi c\\|_2^2 \\leq (1+\\delta) \\|c\\|_2^2\n\\end{equation}\nfor all $s$-sparse vectors $c \\in {\\mathbb{C}}^N$.\n\\end{definition}\n\nInformally, the matrix $\\Psi$ is said to have the restricted isometry property if\n$\\delta_s$ is small for $s$ reasonably large compared to $m$.\nFor matrices satisfying the restricted isometry property, the following $\\ell_1$-recovery results can be shown\n\\cite{carota06-1,ca08,fola09,fo09}.\n\n\\begin{theorem}[Sparse recovery for RIP-matrices]\n\\label{thm:l1:stable} \nLet $\\Psi \\in {\\mathbb{C}}^{m \\times N}$. Assume\nthat its restricted isometry constant $\\delta_{2s}$ \nsatisfies\n\\begin{equation}\n\\label{RIP:const}\n\\delta_{2s} < 3\/(4 + \\sqrt{6}) \\approx 0.4652.\n\\end{equation}\nLet $x \\in {\\mathbb{C}}^N$ and assume noisy measurements $y = \\Psi x + \\eta$ are given with $\\|\\eta\\|_2 \\leq \\varepsilon$. Let $x^\\#$ be the minimizer of \n\\begin{align}\\label{l1eps:prog}\n\\arg \\min_{z \\in {\\mathbb{C}}^N} \\quad \\| z \\|_1 \\mbox{ subject to } \\quad \\|\\Psi z - y \\|_2 \\leq \\varepsilon.\n\\end{align}\nThen\n\\begin{align}\n\\label{l2noise}\n\\|x - x^\\#\\|_2 \\leq  C_1 \\frac{\\sigma_s(x)_1}{\\sqrt{s}} + C_2 \\varepsilon,\n\\end{align}\nand\n\\begin{align}\n\\label{l1noise}\n\\|x - x^\\#\\|_1 \\leq D_1 \\sigma_s(x)_1 + D_2 \\sqrt{s} \\varepsilon.\n\\end{align}\nThe constants $C_1, D_1, C_2, D_2 >0$ depend only on $\\delta_{2s}$.\nIn particular, if $x$ is $s$-sparse then reconstruction is exact, $x^\\# = x$.\n\\end{theorem}\nThe constant in \\eqref{RIP:const} is\nthe result of several refinements.\nCand{\\`e}s provided the value\n$\\sqrt{2}-1$ in \\cite{ca08}, Foucart and Lai the value $0.45$ in \\cite{fola09}, while\nthe version in \\eqref{RIP:const} was shown in \\cite{fo09}. \nThe proof of \\eqref{l2noise} can be found in \\cite{ca08}.  The $\\ell_1$-error bound \\eqref{l1noise} is straightforward from these calculations, but does not seem to appear explicitly in the literature.  \n\n\\medskip\n\nSo far, all good constructions of matrices with the restricted isometry property use randomness. \nThe RIP constant for a matrix whose entries are (properly normalized) independent and identically distributed Gaussian or Bernoulli random variables satisfies $\\delta_{s} \\leq \\delta$ with probability\nat least $1- e^{-c_1(\\delta) m}$ provided\n\\begin{equation}\nm \\geq  c_2(\\delta) s \\log(N\/s); \n\\label{s}\n\\end{equation}\nsee for example \\cite{badadewa08,cata06,rascva08,ra09-1}. To be more precise, it can be shown that \n$c_1(\\delta) = C_1 \\delta^{2}$ and $c_2(\\delta) = C_2 \\delta^{-2}$. \nLower bounds for Gelfand widths of $\\ell_1$-balls show that the bound \\eqref{s} is optimal \\cite{do06-2,codade09,foparaul10}. \n\n If one allows for slightly more measurements than the optimal number \\eqref{s}, the restricted isometry property also holds for a rich class of \\emph{structured} random matrices; the structure of these matrices allows for fast matrix-vector multiplication, which accelerates the speed of reconstruction procedures such as $\\ell_1$ minimization.\nA quite general class of structured random matrices are those associated to \\emph{bounded orthonormal systems}.  This concept is introduced in \\cite{ra09-1}, although it is already contained somewhat implicitly\nin \\cite{cata06,ruve08} for discrete systems.  Let ${\\cal{D}}$ be a measurable space -- for instance, a measurable subset of\n${\\mathbb{R}}^d$ -- endowed with a probability measure $\\nu$. Further, let $\\{ \\psi_j$, $j \\in [N] \\}$, \nbe an orthonormal system of (real or complex-valued) functions on ${\\cal D}$, i.e.,\n\\begin{equation}\n\\int_{\\cal D} \\psi_j(x) \\overline{\\psi_k(x)} d\\nu(x) = \\delta_{j,k}, \\quad k,j \\in [N].\n\\end{equation}\nIf this orthonormal system is uniformly bounded,\n\\begin{equation}\n\\label{Linf_bound}\n\\sup_{j \\in [N]} \\|\\psi_j\\|_\\infty = \\sup_{j \\in [N]} \\sup_{x \\in {\\cal D}} |\\psi_j(x)| \\leq K\n\\end{equation}\nfor some constant $K\\geq 1$, we call systems $\\{\\psi_j\\}$ satisfying\nthis condition \\emph{bounded orthonormal systems}.\n\n\n\\begin{theorem}[RIP for bounded orthonormal systems]\n\\label{thm:BOS:RIP} \nConsider \nthe matrix $\\Psi \\in {\\mathbb{C}}^{m \\times N}$ with entries\n\\begin{equation}\\label{def:Phi:matrix}\n\\Psi_{\\ell,k} = \\psi_k(x_\\ell), \\quad \\ell \\in [m], k \\in [N],\n\\end{equation}\nformed by i.i.d.\\ samples $x_\\ell$ drawn from the orthogonalization measure $\\nu$\nassociated to the bounded\northonormal system $\\{ \\psi_j$, $j \\in [N] \\}$ having uniform bound $K\\geq 1$ in \\eqref{Linf_bound}. \nIf \n\\begin{equation}\\label{BOS:RIP:cond}\nm \\geq C\\delta^{-2} K^2 s \\log^3(s) \\log(N),\n\\end{equation}\nthen with probability at least \n$1-N^{-\\gamma \\log^3(s)},$\nthe restricted isometry constant $\\delta_s$ of $\\frac{1}{\\sqrt{m}} \\Psi$ satisfies $\\delta_s \\leq \\delta$. The constants $C,\\gamma>0$ are universal.\n\\end{theorem}\n\nWe note that condition \\eqref{BOS:RIP:cond} is stated slightly different in \\cite{ra09-1}, namely as\n\\[\n\\frac{m}{\\log(m)} \\geq C\\delta^{-2} K^2 s \\log^2(s) \\log(N).\n\\]\nHowever, it is easily seen that \\eqref{BOS:RIP:cond} implies this condition (after possibly adjusting constants).\nNote also  that \\eqref{BOS:RIP:cond} is implied by the simpler condition\n\\[\nm \\geq C K^2 \\delta^{-2} s \\log^4(N).\n\\]\n\n\\medskip\n\nAn important special case of a bounded orthonormal system is the random partial Fourier matrix, which is formed by choosing a random subset of $m$ rows from the $N \\times N$ discrete Fourier matrix.   The continuous analog of this system is the matrix associated to the trigonometric polynomial basis \n$\\{ x \\mapsto e^{2\\pi i n x } , \\quad n = 0, \\hdots, N-1 \\}$ evaluated at $m$ sample points chosen \nindependently from the uniform measure on $[0,1]$.  Note that the trigonometric system has \ncorresponding optimal uniform bound $K=1$.  Another example is the matrix associated to the \nChebyshev polynomial system evaluated at sample points chosen independently \nfrom the corresponding orthogonalization measure, the Chebyshev measure. In this case, $K = \\sqrt{2}$.  \n\n\\section{Proof of Theorem \\ref{uniform:noise}}\n\nAs a first approach towards recovering Legendre-sparse polynomials from random samples, one may try to apply Theorem \\ref{thm:BOS:RIP} directly, selecting the sampling points $\\{ x_j, j \\in [m] \\}$, independently from the normalized Lebesgue measure on $[-1,1]$, the orthogonalization measure for the Legendre polynomials.  However, as shown in \\cite{szego}, the $L^{\\infty}$-norms of the Legendre polynomials grow according to $\\| L_n \\|_{\\infty} = | L_n (1) | = |L_n(-1) | = (2n + 1)^{1\/2}$.  \nApplying $K = \\| L_{N-1} \\|_{\\infty} = (2N-1)^{1\/2}$ in Theorem \\ref{thm:BOS:RIP} produces a required number of samples\n\\[\nm \\asymp N \\delta^{-2} s \\log^3(s) \\log(N). \n\\]\nOf course, this bound is completely useless, because the required number of samples\nis now larger than $N$ -- an almost trivial estimate. Therefore, in order to deduce\nsparse recovery results for the Legendre polynomials, we must take a different approach.\n\nDespite growing unboundedly with increasing degree at the endpoints $+1$ and $-1$,  an \nimportant characteristic of the Legendre polynomials is that they are \nall bounded by the same envelope function.\nThe following result \\cite[Theorem 7.3.3]{szego}, gives a precise estimate \nfor this bound.\n \n\\begin{lemma}\n\\label{thm:growth}\nFor all $n \\geq 1$ and for all $x \\in [-1,1]$,\n\\[\n\\label{uniform_bound}\n(1 - x^2)^{1\/4} | \\thinspace L_n(x) \\thinspace | < 2\\pi^{-1\/2}\\Big( 1 + \\frac{1}{2n} \\Big)^{1\/2}, \\hspace{5mm} -1 \\leq x \\leq 1;\n\\]\nhere, the constant $2 \\pi^{-1\/2}$ cannot be replaced by a smaller one. \n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{uniform:noise}]\n\nIn light of Lemma \\ref{thm:growth}, we apply a preconditioning\ntechnique to transform the Legendre polynomial system into a bounded orthonormal system.  Consider the functions \n\\begin{equation}\\label{reweight}\nQ_n(x) = (\\pi\/2)^{1\/2} (1 - x^2)^{1\/4} L_n(x).\n\\end{equation}\nThe matrix $\\Psi$ with entries $\\Psi_{j,n}=Q_{n-1}(x_j)$ may be written as $\\Psi = {\\cal A} \\Phi$ where ${\\cal A}$ is the diagonal matrix with entries $a_{j,j} =  (\\pi\/2)^{1\/2} (1 - x_j^2)^{1\/4}$ as in Theorem \\ref{thm:growth}, and $\\Phi \\in {\\mathbb{R}}^{m \\times N}$ is the Legendre matrix with entries $\\Phi_{j,n} = L_{n-1}(x_j)$.  By Lemma \\ref{thm:growth}, the system $\\{ Q_n \\}$ is uniformly bounded on $[-1,1]$ and satisfies the bound $\\| Q_n \\|_{\\infty} \\leq \\sqrt{2 + \\frac{1}{n}} \\leq \\sqrt{3}$.  Due to the orthonormality of the Legendre system with respect to the normalized Lebesgue measure on $[-1,1]$, the $Q_n$ are orthonormal with respect to the Chebyshev probability measure $d \\nu(x) = \\pi^{-1}(1 - x^2)^{-1\/2} dx$ on $[-1,1]$:\n\\begin{eqnarray}\n\\label{ortho}\n\\int_{-1}^1 \\pi^{-1}Q_n(x) Q_{k}(x) (1-x^2)^{-1\/2} dx &=& \\frac{1}{2}\\int_{-1}^1 L_n(x) L_{k}(x) dx  = \\delta_{n,k}. \\nonumber\n\\end{eqnarray}\n\nTherefore, the $\\{ Q_n \\}$ form a bounded orthonormal system in the sense of Theorem \\ref{thm:BOS:RIP} with uniform bound $K = \\sqrt{3}$.  By Theorem \\ref{thm:BOS:RIP}, the renormalized matrix $\\frac{1}{\\sqrt{m}} \\Psi$ has the restricted isometry property with constant $\\delta_s \\leq \\delta$ with high probability once $m \\geq C\\delta^{-2}s \\log^4(N)$.  We then apply Theorem \\ref{thm:l1:stable} to the noisy samples $\\frac{1}{\\sqrt{m}} {\\cal A}y$ where $y = \\big( g(x_1) + \\eta_1, ... , g(x_m) + \\eta_m \\big)$ and observe that $\\| {\\cal A} \\eta \\|_{\\infty} \\leq \\varepsilon$ implies $\\frac{1}{\\sqrt{m}} \\| {\\cal A} \\eta \\|_2 \\leq \\varepsilon$.  This gives Theorem \\ref{uniform:noise}.\n\n\\end{proof}\n\n\n\n\n\\section{Universality of the Chebyshev measure}\nThe Legendre polynomials are orthonormal with respect to the uniform measure on $[-1,1]$; we may instead consider an arbitrary weight function $v$ on $[-1,1]$, and the polynomials $\\{p_n\\}$ that are orthonormal with respect to $v$.  \nSubject to a mild continuity condition on $v$, a result similar to Lemma \\ref{thm:growth} concerning the uniform growth of ${p_n}$ still holds, and the sparse recovery results of Theorem \\ref{uniform:noise} extend to this more general scenario.  In all cases, the sampling points are chosen according to the Chebyshev measure. \n\nLet us recall the following general bound, see e.g.\\ Theorem 12.1.4 in Szeg{\\\"o} \\cite{szego}. \n\n\\begin{theorem} \n\\label{Lip:Dini}\nLet $v$ be a weight function on $[-1,1]$ and set\n$f_v(\\theta) = v(\\cos \\theta)|\\sin(\\theta)|$. Suppose that\n$f_v$ satisfies the Lipschitz-Dini condition, that is,\n\\begin{equation}\\label{LipschitzDini}\n|f_v(\\theta+\\delta) - f_v(\\theta)| \\leq L |\\log(1\/\\delta)|^{-1-\\lambda}, \\quad \\mbox{ for all } \\theta \\in [0,2\\pi), \\delta > 0,\n\\end{equation}\nfor some constants $L,\\lambda > 0$. \n Let $\\{p_n, n \\in {\\mathbb{N}}_0\\}$, be the associated orthonormal polynomial system. Then\n\\begin{equation}\\label{weight:estimate}\n(1-x^2)^{1\/4} v(x)^{1\/2} |p_n(x)| \\leq C_v \\quad \\mbox{ for all } n\\in {\\mathbb{N}}, x \\in [-1,1].\n\\end{equation}\nThe constant $C_v$ depends only on the weight function $v$.\n\\end{theorem}\nThe Lipschitz-Dini condition \\eqref{LipschitzDini} is satisfied for a range\nof Jacobi polynomials $p_n = p_n^{(\\alpha,\\beta)}$, $n \\geq 0,$ $\\alpha,\\beta \\geq -1\/2$, \nwhich are orthogonal\nwith respect to the weight function $v(x) = (1-x)^\\alpha(1+x)^\\beta$. \nThe Legendre polynomials are a special case of the Jacobi polynomials corresponding to $\\alpha = \\beta = 0$; more generally, the case $\\alpha = \\beta$ correspond to the ultraspherical polynomials. The Chebyshev polynomials are another important special case of ultraspherical polynomials, corresponding to parameters $\\alpha = \\beta = -1\/2$, and Chebyshev measure. \n\n\nFor any orthonormal polynomial system satisfying a bound of the form \\eqref{weight:estimate}, the following RIP-estimate applies.\n\\begin{theorem}\n\\label{RIP:precondition}\nConsider a positive weight function $v$ on $[-1,1]$ satisfying the conditions of Theorem \\ref{Lip:Dini}, and consider the orthonormal polynomial system $\\{ p_n \\}$ with respect to the probability measure $d\\nu(x) = c\\,v(x) dx$ on $[-1,1]$ where \n$c^{-1} = \\int_{-1}^1 v(x)dx$.  \n\nSuppose that $m$ sampling points $(x_1, \\hdots, x_m)$ are drawn independently at random from the Chebyshev  \nmeasure, and consider the $m \\times N$ composite matrix $\\Psi = {\\cal A} \\Phi$, where $\\Phi$ is the matrix with entries $\\Phi_{j,n} = p_{n-1}(x_j)$, and ${\\cal A}$ is the diagonal matrix with entries $a_{j,j} = (c\\pi)^{1\/2} (1-x_j^2)^{1\/4} v(x_j)^{1\/2}$.   Assume that \n\\begin{equation}\\label{RIP:prec:cond}\nm \\geq C \\delta^{-2} s \\log^3(s) \\log(N).       \n\\end{equation}\nThen with probability at least $1-N^{-\\gamma \\log^3(s)}$ the restricted isometry\nconstant of the composite matrix $\\frac{1}{\\sqrt{m}} \\Psi = \\frac{1}{\\sqrt{m}} {\\cal A} \\Phi$ satisfies $\\delta_s \\leq \\delta$. The constant $C$ depends only on $v$, and the constant $\\gamma > 0$ is universal.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem \\ref{RIP:precondition}]\nObserve that $\\Psi_{j,n} = Q_{n-1}(x_j)$, where \n$$Q_n(x) =(c \\pi)^{1\/2} (1-x^2)^{1\/4} v(x)^{1\/2} p_n(x).$$\nFollowing Theorem \\ref{Lip:Dini},  the system $\\{ Q_n \\}$ is uniformly bounded on $[-1,1]$ and satisfies the bound $\\| Q_n \\|_{\\infty} \\leq (c\\pi)^{-1\/2} C_v$; moreover, due to the orthonormality of the polynomials $\\{ p_n \\}$ with respect to the measure $d\\nu(x)$,  the $\\{Q_n\\}$ are orthonormal with respect to the Chebyshev measure:\n\\begin{eqnarray}\n\\label{ortho_nu}\n\\int_{-1}^1 \\pi^{-1}Q_n(x) Q_{k}(x) (1-x^2)^{-1\/2} dx &=& \\int_{-1}^1 c p_n(x) p_{k}(x) v(x)dx  = \\delta_{n,k}. \n\\end{eqnarray}\nTherefore, the $\\{ Q_n \\}$ form a bounded orthonormal system with associated matrix $\\Psi$ as in Theorem \\ref{RIP:precondition} formed from samples $\\{ x_j \\}$ drawn from the Chebyshev distribution.  Theorem \\ref{thm:BOS:RIP} implies that the renormalized composite matrix $\\frac{1}{\\sqrt{m}}\\Psi$ has the restricted isometry property as stated.\n\\end{proof}\n\n\\begin{corollary}\n\\label{poly:noise}\nConsider an orthonormal polynomial system $\\{ p_n \\}$ associated to a measure $v$ satisfying the conditions of Theorem \\ref{Lip:Dini}.\nLet $N,m,s \\in {\\mathbb{N}}$ satisfy the conditions of Theorem \\ref{RIP:precondition}, and \nconsider the matrix $\\Psi = {\\cal{A}} \\Phi$ as defined there.  \n\nThen with probability exceeding\n$1-N^{-\\gamma \\log^3(s)}$ the following \nholds for all polynomials $g(x) = \\sum_{k=0}^{N-1} c_k p_k(x)$.\nIf noisy sample values $y = \\big( g(x_1) + \\eta_1, \\hdots, g(x_m) + \\eta_m \\big) = \\Phi c + \\eta$ are observed, and $\\|\\eta\\|_{\\infty} \\leq \\varepsilon$, then the coefficient vector $c = (c_0, c_1, \\hdots, c_{N-1})$ is recoverable to within a factor of its best $s$-term approximation error \nand to a factor of the noise level by solving the inequality-constrained $\\ell_1$-minimization problem\n\\begin{align}\\label{relaxed2}\nc^{\\#} = \\arg \\min_{z \\in {\\mathbb{R}}^N} \\| z \\|_1 \\quad \\mbox{ subject to } \\quad \\| {\\cal A} \\Phi z - {\\cal A} y \\|_2 \\leq \\sqrt{m}\\varepsilon.\n\\end{align}\nPrecisely, \n$$\n\\| c -c^{\\#} \\|_{2} \\leq \\frac{C_1 \\sigma_s(c)_1}{\\sqrt{s}} + D_1\\varepsilon,\n\n$$\nand\n\\begin{equation}\\label{l1:approx3}\n\\|c-c^{\\#}\\|_{1} \\leq C_2 \\sigma_s(c)_1 + D_2 \\sqrt{s} \\varepsilon.\n\\end{equation}\nThe constants $C_1,C_2,D_1,D_2$ and $\\gamma$ are universal.\n\\end{corollary}\n\n\n\nAs a byproduct of Theorem \\ref{RIP:precondition}, we also obtain condition number estimates\nfor preconditioned orthogonal polynomial matrices that should be of interest on their own, and improve on the results in \\cite{grpora07}.   Theorem \\ref{RIP:precondition} implies that all submatrices of a preconditioned random orthogonal polynomial matrix \n$\\frac{1}{\\sqrt{m}}\\Psi = \\frac{1}{\\sqrt{m}} {\\cal A} \\Phi \\in {\\mathbb{R}}^{m \\times N}$ with at most $s$ columns are simultaneously well-conditioned, provided \\eqref{RIP:prec:cond} holds. If one is only interested in a particular subset of $s$ columns, i.e., a particular subset of $s$ orthogonal polynomials, the number of measurements in \\eqref{RIP:prec:cond} can be reduced to \n\\begin{equation}\\label{m:est}\nm \\geq C s \\log(s);\n\\end{equation}\nsee Theorem 7.3 in \\cite{ra09-1} for more details. \n\n\\medskip\n\n\\noindent\n{\\bf Stability with respect to the sampling measure.}\n\nThe requirement that sampling points $x_j$ are drawn from the Chebyshev measure in the previous theorems can be relaxed somewhat.    In particular, suppose that the sampling points $x_j$ are drawn  not from the Chebyshev measure, but from a more general probability measure $d\\nu(x) = \\rho(x) dx$ on $[-1,1]$ with \n$\\rho(x) \\geq c' (1-x^2)^{-1\/2}$ (and $\\int_{-1}^1 \\rho(x) dx = 1$).  Now assume a weight function $v$ satisfying the Lipschitz-Dini condition \\eqref{LipschitzDini} and the associated orthonormal polynomials $p_n(x)$ are given. Then, by Theorem \\ref{Lip:Dini} the functions\n\\begin{equation}\\label{Qn:def}\nQ_n(x) =(c \\pi)^{1\/2} \\rho(x)^{-1\/2} v(x)^{1\/2} p_n(x)\n\\end{equation}\nform a bounded orthonormal system with respect to the probability measure $\\tilde{c} \\rho(x) v(x) dx$. Therefore, all previous\narguments are again applicable. \nWe note, however, that taking $\\rho(x) dx$ to be the Chebyshev measure produces the smallest constant $K$\nin the boundedness condition \\eqref{Linf_bound} due to normalization reasons.\n\n\n\\section{Recovery in infinite-dimensional function spaces}\n\nWe can transform the previous results into approximation results on the level of continuous functions.  For simplicity, we restrict the scope of this section to the Legendre basis, although all of our results extend to any orthonormal polynomial system with a Lipschitz-Dini weight function, as well as to the trigonometric system, for which related results have not been\nworked out yet, either.\n\nWe introduce the following weighted norm on continuous functions in $[-1,1]$:\n\\[\n\\|f\\|_{\\infty,w} := \\sup_{x \\in [-1,1]} |f(x)| w(x), \\quad w(x) = \\sqrt{\\frac{\\pi}{2}} (1-x^2)^{1\/4}.\n\\]\nFurther, we define\n\\begin{equation}\n\\label{weight:norm}\n\\sigma_{N,s}(f)_{\\infty, w} := \\inf_{c \\in {\\mathbb{R}}^N} \\left\\{ \\sigma_s(c)_{1} + \\sqrt{s} \\|f- \\sum_{k=0}^{N-1} c_k L_k \\|_{\\infty, w}\\right\\}.\n\\end{equation}\nThe above quantity involves the best $s$-term approximation error of $c$, as well as the ability of Legendre coefficients \n$c \\in {\\mathbb{R}}^{N}$ to approximate the given function $f$ in the $L_\\infty$-norm. In some sense, it provides a mixed linear and\nnonlinear approximation error. The $c$ which ``balances'' both error terms determines $\\sigma_{N,s}(f)_\\infty$.\nThe factor $\\sqrt{s}$ scaling the ``linear approximation part'' may seem to lead to non-optimal estimates at first sight, but \nlater on, the strategy will actually be to choose $N$ in dependence of $s$ such that $\\sigma_{N,s}(f)_\\infty$ becomes\nof the same order as $\\sigma_s(c)_1$. In any case, we \nnote the (suboptimal) estimate\n\\[\n\\sigma_{N,s}(f)_{\\infty, w} \\leq \\sqrt{s}\\, \\rho_{N,s}(f)_{\\infty,w},\n\\]\nwhere\n\\[\n\\rho_{N,s}(f)_{\\infty,w} := \\inf_{c \\in {\\mathbb{R}}^N, \\|c\\|_0 \\leq s} \\|f- \\sum_{k=0}^{N-1} c_k L_k \\|_{\\infty, w}.\n\\]\n\nOur aim is to obtain\na good approximation to a continuous function $f$ from $m$ sample values, and to compare the approximation\nerror with $\\sigma_{N,s}(f)_{\\infty,w}$. We have\n\n\n\n\\begin{proposition}\\label{thm:function:approx} Let $N,m,s$ be given with\n\\[\nm \\geq C s \\log^3(s) \\log(N).\n\\]\nThen there exist sampling points $x_1,\\hdots,x_m$ (i.e., chosen i.i.d.\\ from the Chebyshev measure) and an efficient \nreconstruction procedure (i.e., $\\ell_1$-minimization), such that for\nany continuous function $f$ with associated error $\\sigma_{N,s}(f)_{\\infty, w}$, the\npolynomial $P$ of degree at most $N$ \nreconstructed from $f(x_1),\\hdots,f(x_m)$ satisfies\n\\[\n\\|f-P\\|_{\\infty,w} \\leq C' \\sigma_{N,s}(f)_{\\infty, w}.\n\\] \nThe constants $C, C'>0$ are universal.\n\\end{proposition}\n\nThe quantity $\\sigma_{N,s}(f)_{\\infty,w}$ involves \nthe two numbers $N$ and $s$.\nWe now describe how $N$ can be chosen in dependence on $s$, reducing the number of parameters to one. \nWe illustrate this strategy below in a more concrete\nsituation. To describe the setup we introduce analogues of the Wiener algebra in the Legendre polynomial setting. \nLet $c(f)$ with entries\n\\[\nc_k(f) = \\frac{1}{2} \\int_{-1}^1 f(x) L_k(x) dx,\\quad k \\in {\\mathbb{N}}_0, \n\\]\ndenote the vector of Fourier-Legendre coefficients of $f$. Then we define\n\\[\nA_p := \\{ f \\in C[-1,1], \\|c(f)\\|_p < \\infty \\},\\quad 0<p\\leq 1,\n\\]\nwith quasi-norm $\\|f\\|_{A_p} := \\|c(f)\\|_p$. The use of the $p$-norm is motivated by the \nStechkin estimate \\eqref{Stechkin} below, which\ntells us that elements in $\\ell_p$ can be considered compressible. \nSince $\\|L_k w\\|_{\\infty} \\leq \\sqrt{3}$ it follows that\n\\[\nf(x) w(x) = \\sum_{k \\in {\\mathbb{N}}_0} c_k(f) L_k(x) w(x)\n\\] \nconverges uniformly for  $f \\in A_1$, so that $fw \\in C[-1,1]$, and\n$\\|f\\|_{\\infty,w} \\leq \\sqrt{3}\\|f\\|_{A_1}$. Since $\\|f\\|_{A_1} \\leq \\|f\\|_{A_p}$ for $0 <p\\leq 1$ this holds also for \n$f \\in A_p$, $0<p\\leq 1$.  Now we introduce\n\\[\n\\sigma_s(f)_{A_1} := \\inf_{c \\in \\ell_2({\\mathbb{N}}_0), \\|c\\|_0 \\leq s} \\|f - \\sum_{k} c_k L_k\\|_{A_1} = \\sigma_s( c(f) )_1.\n\\]\nBy Stechkin's estimate \\eqref{Stechkin} (which is also valid in infinite dimensions) we\nhave, for $0< q < 1$,\n\\begin{equation}\\label{Stechkin:rate}\n\\sigma_s(f)_{A_1} \\leq s^{1-1\/q} \\|f\\|_{A_q}.\n\\end{equation}\nOur goal is to realize this approximation rate for $f \\in A_q$ when only\nsample values of $f$ are given. Additionally, the number of samples should be close (up to $\\log$-factors) to the number $s$ of degrees of freedom of the reconstructed function. Unfortunately, for this task we have to at least know roughly a finite set $[N]$ containing  the Fourier-Legendre coefficients of \na good $s$-sparse approximation\nof $f$. In order to deal with this problem, we introduce, for $\\alpha >0$, a weighted Wiener type space $A_{1,\\alpha}$, containing the functions $f \\in C[-1,1]$ with finite norm\n\\[\n\\|f\\|_{A_{1,\\alpha}} := \\sum_{k \\in {\\mathbb{N}}_0} (1+k)^\\alpha |c_k(f)|.\n\\]\nOne should imagine $\\alpha\\ll 1$ very small, so that $f \\in A_{1,\\alpha}$ \ndoes not impose a severe restriction on $f$, compared to $f \\in A_q$. Then instead of\n$f \\in A_q$ we make the slightly stronger requirement $f \\in A_q \\cap A_{1,\\alpha}$, $0<q<1$. The next theorem states that under such assumptions, the optimal \nrate \\eqref{Stechkin:rate} can be realized when only a small number of sample values of $f$ are available.\n\n\\begin{theorem}\\label{thm:Ap} Let $0<q<1$, $\\alpha >0$, and $m,s \\in {\\mathbb{N}}$ be given such that\n\\begin{equation}\\label{ms:rel}\nm \\geq C \\alpha^{-1}\\left(\\frac{1}{q}-\\frac{1}{2}\\right) s \\log^4(s).\n\\end{equation}\nThen there exist sampling points $x_1,\\hdots,x_m \\in [-1,1]$ (i.e., random Chebyshev points) such that for every \n$f \\in A_q \\cap A_{1,\\alpha}$ a polynomial $P$ of degree at most \n$N = \\lceil s^{(1\/q-1\/2)\/\\alpha}\\rceil$\ncan be reconstructed from the sample values $f(x_1),\\hdots,f(x_m)$ such that\n\\begin{equation}\\label{recovery:rate}\n\\frac{1}{\\sqrt{3}}\\|f - P\\|_{\\infty,w} \\leq \\|f-P\\|_{A_1} \\leq C(\\|f\\|_{A_q} + \\|f\\|_{A_{1,\\alpha}}) s^{1-1\/q}.\n\\end{equation}\n\\end{theorem}\nNote that up to $\\log$-factors the number of required samples is of the\norder of the number $s$ of degrees of freedom (the sparsity) \nallowed in the estimate \\eqref{Stechkin}, and the reconstruction \nerror \\eqref{recovery:rate} satisfies the same rate. Clearly $\\ell_1$-minimization or greedy alternatives can be used for reconstruction. This result \nmay be considered\nas an extension of the theory of compressive sensing to infinite dimensions (although all the\nkey tools are actually finite dimensional).\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Proposition \\ref{thm:function:approx}}\n\nLet $P_{opt} = \\sum_{k=0}^{N-1} c_{k,opt} L_k$ denote the polynomial of degree at most $N-1$ whose coefficient vector $c_{opt}$ realizes the approximation error $\\sigma_{N,s}(f)_{\\infty, w}$, as defined in \\eqref{weight:norm}. The samples $f(x_1), \\hdots, f(x_m)$ can be seen as noise corrupted samples of $P_{opt}$, that is, $f(x_{\\ell}) = P_{opt}(x_{\\ell}) + \\eta_{\\ell},$ and $| \\eta_{\\ell} | w(x_{\\ell}) \\leq \\| f - P_{opt} \\|_{\\infty, w} := \\varepsilon$.  The preconditioned system reads then $f(x_{\\ell}) w(x_{\\ell}) = \\sum_{k=0}^{N-1} c_{k,opt} L_k(x_{\\ell}) w(x_{\\ell}) + \\varepsilon_{\\ell}$, with $| \\varepsilon_{\\ell} | \\leq \\varepsilon$.  According to Theorem \\ref{thm:BOS:RIP} and Theorem \\ref{thm:growth}, the matrix $\\frac{1}{\\sqrt{m}}\\Psi$ consisting of entries $\\Psi_{\\ell,k} = w(x_{\\ell})L_{k-1}(x_\\ell)$ satisfies the RIP with high probability, provided the stated condition on the minimal number of samples holds.\nDue to Theorem \\ref{thm:l1:stable}, an application of noise-aware $\\ell_1$-minimization \\eqref{l1eps:prog} to $y = (f(x_\\ell)w(x_{\\ell}))_{\\ell=1}^m$ with $\\varepsilon$ replaced\nby $\\sqrt{m}\\varepsilon$ yields a coefficient vector $c$ satisfying $\\| c - c_{opt} \\|_1 \\leq C_1 \\sigma_s(c_{opt})_1 + C_2 \\sqrt{s} \\varepsilon$.   We denote the polynomial corresponding to this coefficient vector by $P(x) = \\sum_{k=0}^{N-1} c_k L_k(x)$. Then \n\\begin{align}\n\\| f - P \\|_{\\infty, w} &\\leq\n \\| f - P_{opt} \\|_{\\infty, w} + \\| P_{opt} - P \\|_{\\infty, w}\n\\leq \\frac{\\sigma_{N,s}(f)_{\\infty, w}}{\\sqrt{s}} + \\sqrt{3} \\| c - c_{opt} \\|_1 \\nonumber \\\\\n&\\leq \\frac{\\sigma_{N,s}(f)_{\\infty, w}}{\\sqrt{s}} + \\sqrt{3} \\Big[ C_1\\sigma_s(c_{opt})_1 +  C_2 \\sqrt{s}  \\| f - P_{opt} \\|_{\\infty, w}\\Big] \n\\leq C \\sigma_{N,s}(f)_{\\infty, w}.\\nonumber\n\\end{align}\nThis completes the proof.\n\nThe attentive reader may have noticed that our recovery method, noise-aware \n$\\ell_1$-minimization \\eqref{l1eps:prog}, requires knowledge of $\\sigma_{N,s}(f)$, see also\nRemark \\ref{rem22}(c). \nOne may remove this drawback by considering CoSaMP \\cite{netr08} or\nIterative Hard Thresholding \\cite{blda09} instead.  The required \nerror estimate in $\\ell_1$ follows from the $\\ell_2$-stability results for these algorithms\nin \\cite{blda09,netr08}, as both algorithms produce a $2s$-sparse vector, \nsee \\cite[p.\\ 87]{b09} for details.   \n\n\\subsection{Proof of Theorem \\ref{thm:Ap}}\n\nLet $f \\in A_q \\cap A_{1,\\alpha}$ with Fourier Legendre coefficients $c_k(f)$.\nLet $N > s$ be a number to be chosen later and introduce the truncated Legendre expansion\n\\[\nf_N(x) = \\sum_{k=0}^{N-1} c_k(f) L_k(x),\n\\]\nwhich has truncated Fourier-Legendre coefficient vector $c^{(N)}$ with entries\n$c^{(N)}_k =c_k(f)$ if $k<N$ and $c^{(N)}_k = 0$ otherwise. Clearly, \n$\n\\|c^{(N)}\\|_q \\leq \\|c(f)\\|_q = \\|f\\|_{A_q}.\n$\nFurther note that\n\\begin{align}\n\\frac{1}{\\sqrt{3}}\\|f-f_N\\|_{\\infty,w} & \\leq \\|f-f_N\\|_{A_1} = \\|c-c^{(N)}\\|_1 \n= \\sum_{k=N}^\\infty |c_k(f)| \\leq N^{-\\alpha} \\sum_{k=N}^{\\infty} (1+k)^{\\alpha} |c_k(f)|\\notag\\\\\n& \\leq N^{-\\alpha} \\|f\\|_{A_{1,\\alpha}}.\\notag\n\\end{align}\nNow we proceed similarly as in the proof of Theorem \\ref{thm:function:approx} and treat the samples of $f$ as perturbed\nsamples of $f_N$, that is $f_N(x_j) = f(x_j) + \\eta_j$ with \n$|\\eta_j| w(x_j) \\leq \\|f-f_N\\|_{\\infty,w} \\leq \\sqrt{3}N^{-\\alpha} \\| f \\|_{A_{1,\\alpha}}$. Then following the same\narguments as in the proof of Theorem \\ref{thm:function:approx}, if\n\\begin{equation}\\label{msN:choice}\nm \\geq C s \\log^3(s) \\log(N),\n\\end{equation}\nwe can reconstruct a \ncoefficient vector $\\widetilde{c}$ from samples $f(x_1),\\hdots,f(x_m)$ \nwith support contained in $\\{0,1,\\hdots,N-1\\}$ such that \n\\begin{align}\n\\|c^{(N)} - \\widetilde{c}\\|_1 \\leq C_1 \\sigma_{s}(c^{(N)})_1 + C_2 \\sqrt{s} \\|f-f_N\\|_{\\infty,w}\n\\leq C_1 s^{1-1\/q} \\|f\\|_{A_q} + C_2 \\sqrt{s} N^{-\\alpha} \\|f\\|_{A_{1,\\alpha}}.\\notag\n\\end{align}\nHere, we applied Stechkin's estimate \\eqref{Stechkin}. Therefore,\n\\begin{align}\n\\|c-\\widetilde{c}\\|_1 &\\leq \\|c-c^{(N)}\\|_1 + \\|c^{(N)} - \\widetilde{c}\\|_1\n\\leq N^{-\\alpha} \\|f\\|_{A_{1,\\alpha}} + C_1 s^{1-1\/q} \\|f\\|_{A_q} + C_2 \\sqrt{s} N^{-\\alpha} \\|f\\|_{A_{1,\\alpha}}\\notag\\\\\n&\\leq C_1 s^{1-1\/q} \\|f\\|_{A_q} + C_2' \\sqrt{s} N^{-\\alpha} \\|f\\|_{A_{1,\\alpha}}.\\notag\n\\end{align}\nNow we choose \n\\begin{align}\\label{N:choice}\nN = \\lceil s^{1\/\\alpha(1\/q-1\/2)} \\rceil\n\\end{align}\nwhich yields $\\sqrt{s} N^{-\\alpha} \\leq s^{1-1\/q}$. With this choice\n\\[\n\\frac{1}{\\sqrt{3}}\\|f-\\tilde{f}_N\\|_{\\infty,w} \\leq \\|f-\\tilde{f}_N\\|_{A_1} = \\|c-\\widetilde{c}\\|_1 \\leq C' (\\|f\\|_{A_q} + \\|f\\|_{A_{1,\\alpha}}) s^{1-1\/q}.\n\\]\nPlugging \\eqref{N:choice} into \\eqref{msN:choice} yields \\eqref{ms:rel}, \nand the proof is finished.\n\n\n\n\\begin{remark}\n\\emph{Analogous function approximation results can be derived from Theorem \\ref{RIP:precondition} for any orthogonal polynomial basis whose weight function satisfies the conditions of Theorem \\ref{Lip:Dini}.  The associated norm is $\\| f \\|_{v, \\infty} = \\| \\sqrt{3} v^{1\/2} f w \\|_{\\infty}.$    For the Chebyshev polynomials, $\\| f \\|_{v, \\infty} = \\| f \\|_{\\infty}$, and the corresponding function approximation results in this case are with respect to the unweighted uniform norm.  }\n \n\\end{remark}\n\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Albert Cohen, Simon Foucart, and Joseph Ward\nfor valuable discussions on this topic, and are also grateful to Laurent Gosse for helpful comments.   Rachel Ward gratefully awknowledges the partial support of National Science Foundation Postdoctoral Research Fellowship.  Holger Rauhut gratefully acknowledges support by \nthe Hausdorff Center for Mathematics and by the WWTF project SPORTS (MA 07-004).\nParts of this manuscript have been written during a stay of the first author at the Laboratoire\nJacques-Louis Lions of \nUniversit{\\'e} Pierre et Marie Curie in Paris. He greatly acknowledges the warm hospitality of\nthe institute and especially of his host Albert Cohen.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nClustering is a widely used technique in Computer Science with\napplications to unsupervised learning, classification, data mining\nand other fields. We study two variants of the clustering problem\nin the geometric setting. The \\emph{geometric $k$-median\nclustering} problem is the following: Given a set $P$ of points in\n$\\Re^d$, compute a set of $k$ points in $\\Re^d$ such that the sum\nof the distances of the points in $P$ to their respective nearest\nmedian is minimized. The $k$-means differs from the above in that\ninstead of the sum of distances, we minimize the sum of squares of\ndistances.  Interestingly the $1$-mean is the center of mass of\nthe points, while the $1$-median problem, also known as the\nFermat-Weber problem, has no such closed form. As such the problems\nhave usually been studied separately from each other even in the\napproximate setting. We propose techniques which can be used for\nfinding approximate $k$ centers in both variants.\n\nIn the data stream model of computation, the points are read\nin a sequence and we desire to compute a function,\nclustering in our case, on the set of points seen so far. In\ntypical applications, the total volume of data is very large\nand can not be stored in its entirety. Thus we usually\nrequire a data-structure to maintain an aggregate of the\npoints seen so far so as to facilitate computation of the\nobjective function. Thus the standard complexity measures in\nthe data stream model are the storage cost, the update cost\non seeing a new point and the time to compute the function\nfrom the aggregated data structure.\n\n\\begin{table}[t]\n    \\centerline{\n    }\n    \\begin{tabular}{|l||l|l|r|r|r|r|}\n      \\hline\n      Problem\n      & Previous Results\n      & Our Results\n      \\\\\n     \n      \\hline\n      \\hline\n      $k$-median\n      &        \n        \\begin{minipage}{2.5in}\n            \\smallskip%\n            $O(\\varrho}%{C_{\\mathrm{kr}} n(\\log{n})\\log {k})$ \\cite{kr-nltas-99} ~~{(*)}\\\\\n               $~~~~~$ $\\varrho}%{C_{\\mathrm{kr}} = \\CkrExp$\\\\\n           \\end{minipage}\n           &\n           \\begin{minipage}{2.5in}\n           $O \\pth{ n +\n              \\varrho}%{C_{\\mathrm{kr}} k^{O(1)} \\log^{O(1)} n }$\\\\\n           $[$\\thmref{fast:k:median}$]$\n           \\end{minipage}\\\\      \n      \\hline\n           \\begin{minipage}{0.8in}\n           discrete\\\\ $k$-median\n           \\end{minipage}&\n               $O(\\varrho}%{C_{\\mathrm{kr}} n\\log{n}\\log {k})$ \\cite{kr-nltas-99}\n           &\n             \\begin{minipage}{1.8in}\n                 \\smallskip%\n                 $O \\pth{ n +\n                    \\varrho}%{C_{\\mathrm{kr}} k^{O(1)} \\log^{O(1)} n }$\\\\\n                 \\smallskip%\n                 $[$\\thmref{approx:k:median:discrete}$]$\n             \\end{minipage}\\\\[0.1cm]\n           %\n           %\n           \\hline\n           \\begin{minipage}{0.8in}\n               $k$-means\n           \\end{minipage}&\n           $O_k(n (\\log{n})^k \\eps^{-2k^2 d})$\n           \\cite{m-agc-00} ~~{(**)}\n           &\n           \\begin{minipage}{2.4in}\n               \\smallskip%\n               $O (n +{k^{k+2}}{\\eps^{-(2d+1)k}} {\\log^{k+1}{n}}\\log^k\\frac{1}{\\eps})$\\\\\n               $[$\\thmref{k:means:approx}$]$\n               \\smallskip%\n           \\end{minipage}\\\\\n      \\hline\n           \\begin{minipage}{0.8in}\n               Streaming\n           \\end{minipage}\n           &\n           \\begin{minipage}{2.4in}\n              \n               $k$-median\\\\\n               \\si{Const} factor; Any metric space\\\\\n               $O( k \\polylog )$ space ~\\cite{cop-bsacp-03}\n           \\end{minipage}&\n           \\begin{minipage}{2in}\n               \\smallskip%\n               $k$-means and $k$-median\\\\\n               $(1+\\eps)$-approx; Points in $\\Re^d$. \\\\\n               $O( k \\eps^{-d} \\log^{2d+2}\n               n )$ space\\\\\n               $[$\\thmref{k:means:streaming}$]$\n\n               \\smallskip\n           \\end{minipage}\\\\\n           \\hline\n       \\end{tabular}%\n    \\caption{For our results, all the running time bounds are in\n       expectation and the algorithms succeed with high\n       probability. (*) Getting this running time\n       requires non-trivial modifications of the algorithm\n       of Kolliopoulos and Rao \\cite{kr-nltas-99}. (**) The\n       $O_k$ notation hides constants that depends solely on\n       $k$. }\n    \\vspace{-0.4cm}\n    \\tablab{results}\n\\end{table}\n\n\\smallskip\n\\noindent{\\bf $k$-median clustering.} The $k$-median problem turned\nout to be nontrivial even in low dimensions and achieving a good\napproximation proved to be a challenge. Motivated by the work of Arora\n\\cite{a-ptase-98}, which proposed a new technique for geometric\napproximation algorithms, Arora, Raghavan and Rao \\cite{arr-asekm-98}\npresented a $O\\pth{n^{O(1\/\\eps)+1}}$ time $(1+\\eps)$-approximation\nalgorithm for points in the plane. This was significantly improved by\nKolliopoulos and Rao \\cite{kr-nltas-99} who proposed an algorithm with\na running time of $O(\\varrho}%{C_{\\mathrm{kr}} n\\log{n}\\log {k})$ for the discrete version\nof the problem, where the medians must belong to the input set and\n$\\varrho}%{C_{\\mathrm{kr}} = \\CkrExp$. The $k$-median problem has been studied extensively\nfor arbitrary metric spaces and is closely related to the\nuncapacitated facility location problem.  Charikar \\textit{et~al.}\\xspace{}\n\\cite{cgts-cfaak-99} proposed the first constant factor approximation\nto the problem for an arbitrary metric space using a natural linear\nprogramming relaxation of the problem followed by rounding the\nfractional solution. The fastest known algorithm is due to Mettu and\nPlaxton \\cite{mp-otbap-02} who give an algorithm which runs in\n$O(n(k+\\log n))$ time for small enough $k$ given the distances are\nbound by $2^{O\\pth{n\/ \\log\\pth{n\/k}}}$. It was observed that if the\nconstraint of having exactly $k$-medians is relaxed, the problem\nbecomes considerably easier \\cite{cg-icaflkm-99,jv-pdaam-99}. In\nparticular, Indyk \\cite{i-stams-99} proposed a constant factor\napproximation algorithm which produces $O(k)$ medians in\n$\\tldO \\pth{n k}$ time. In the streaming context, Guha \\textit{et~al.}\\xspace {}\n\\cite{gmmo-cds-00} propose an algorithm which uses $O(n^\\eps)$ memory\nto compute $2^{1\/\\eps}$ approximate $k$-medians.  Charikar \\textit{et~al.}\\xspace {}\n\\cite {cop-bsacp-03} improve the algorithm by reducing the space\nrequirement to $O(k \\cdot \\polylog(n))$.\n\n\\smallskip\n\\noindent{\\bf $k$-means clustering.} Inaba \\textit{et~al.}\\xspace{} \\cite{iki-awvdr-94}\nobserve that the number of Voronoi partitions of $k$ points in $\\Re^d$\nis $n^{k d}$ and can be done exactly in time $O(n^{k d+1})$. They also\npropose approximation algorithms for the $2$-means clustering problem\nwith time complexity $O(n^{O(d)})$.  \\si{de} la Vega \\textit{et~al.}\\xspace{}\n\\cite{vkkr-ascp-03} proposes a $(1+\\eps)$-approximation algorithm, for\nhigh dimensions, with running time $O(g(k,\\eps) n \\log^k n)$, where\n$g(k,\\eps) = \\exp [ (k^3\/\\eps^8) (\\ln(k\/\\eps)) \\ln k]$ (they refer to\nit as $\\ell_2^2$ $k$-median clustering).  \\Matousek{} \\cite{m-agc-00}\nproposed a $(1+\\eps)$-approximation algorithm for the geometric\n$k$-means problem with running time $O\\pth{n\\eps^{-2k^2d}\\log^k{n} }$.\n\n\n\\paragraph{Our Results.}\nWe propose fast algorithms for the approximate $k$-means and\n$k$-medians problems.  The central idea behind our\nalgorithms is computing a weighted point set which we call a\n\\emph{$\\pth{k,\\eps}$-coreset}. For an optimization problem,\na coreset is a subset of input, such that we can get a good\napproximation to the original input by solving the\noptimization problem directly on the coreset. As such, to\nget good approximation, one needs to compute a coreset, as\nsmall as possible from the input, and then solve the problem\non the coreset using known techniques. Coresets have been\nused for geometric approximation mainly in low-dimension\n\\cite{ahv-aemp-04,h-cm-04,apv-aaklc-02}, although a similar\nbut weaker concept was also used in high dimensions\n\\cite{bhi-accs-02,bc-ocsb-03,hv-pchdu-02}. In low dimensions\ncoresets yield approximation algorithm with linear or near\nlinear running time with an additional term that depends\nonly on the size of the coreset.\n\nIn the present case, the property we desire of the\n$\\pth{k,\\eps}$-coreset is that the clustering cost of the coreset for\nany arbitrary set of $k$ centers is within $\\pth{1\\pm \\eps}$ of the\ncost of the clustering for the original input. To facilitate the\ncomputation of the coreset, we first show a linear time algorithm (for\n$k =O(n^{1\/4})$), that constructs a $O( k\\polylog )$ sized set of\ncenters such that the induced clustering gives a constant factor\napproximation to both the optimal $k$-means and the optimal\n$k$-medians clustering. We believe that the technique used for this\nfast algorithm is of independent interest. Note that it is faster than\nprevious published fast algorithms for this problem (see\n\\cite{mp-otbap-02} and references therein), since we are willing to\nuse more centers.  Next, we show how to construct a suitably small\ncoreset from the set of approximate centers.  We compute the $k$\nclusterings for the coresets using weighted variants of known\nclustering algorithms. Our results are summarized in \\tabref{results}.\n\nOne of the benefits of our new algorithms is that in the resulting\nbounds, on the running time, the term containing `$n$' is decoupled\nfrom the ``nasty'' exponential constants that depend on $k$ and\n$1\/\\eps$. Those exponential constants seems to be inherent to the\nclustering techniques currently known for those problems.\n\nOur techniques extend very naturally to the streaming model\nof computation. The aggregate data-structure is just a\n$(k,\\eps)$-coreset of the stream seen so far. The size of\nthe maintained coreset is $O( k\\eps^{-d} \\log{n})$, and the\noverall space used is $O((\\log^{2d+2} {n})\/\\eps^d )$. The\namortized time to update the data-structure on seeing a new\npoint is $O(k^5+\\log^2(k\/\\eps))$.\n\nAs a side note, our ability to get linear time algorithms for fixed\n$k$ and $\\eps$, relies on the fact that our algorithms need to solve a\nbatched version of the nearest neighbor problem. In our algorithms,\nthe number of queries is considerably larger than the number of sites,\nand the distances of interest arise from clustering. Thus, a small\nadditive error which is related to the total price of the clustering\nis acceptable. In particular, one can build a data-structure that\nanswers nearest neighbor queries in $O(1)$ time per query, see\n\\apndref{fast:n:n}. Although this is a very restricted case, this\nresult may nevertheless be of independent interest, as this is the\nfirst data-structure to offer nearest neighbor queries in constant\ntime, in a non-trivial settings.\n\nThe paper is organized as follows. In \\secref{coresets}, we\nprove the existence of coresets for $k$-median\/means\nclustering. In \\secref{fast:const:factor}, we describe the\nfast constant factor approximation algorithm which generates\nmore than $k$ means\/medians. In \\secref{eps:approx:k:median}\nand \\secref{eps:approx:k:mean}, we combine the results of the two preceding\nsections, and present an $(1+\\eps)$-approximation algorithm\nfor $k$-means and $k$-median respectively. In\n\\secref{streaming}, we show how to use coresets for space\nefficient streaming. We conclude in \\secref{conclusions}.\n\n\n\n\\section{Preliminaries}\n\n\\begin{defn}\n    For a point set $X$, and a point $p$, both in $\\Re^d$, let\n    $\\mathbf{d}\\hspace{-1pt}(p,X)= \\min_{x \\in X} \\dist{x p}$ denote the \\emph{distance\n       of $p$ from $X$}.\n\\end{defn}\n\n\\begin{defn}[Clustering]\n    For a weighted point set $P$ with points from $\\Re^d$, with an\n    associated weight function $w:P\\rightarrow \\mathbb{Z}^+$ and any\n    point set $C$, we define\n    $\\nu_C\\pth{P} = \\sum_{p\\in P} w(p)\\mathbf{d}\\hspace{-1pt}(p,C)$ as the\n    \\emph{price} of the $k$-median clustering provided by $C$. Further\n    let\n    $\\nu_{\\mathrm{opt}}(P,k) =\\min_{C\\subseteq \\Re^d, \\cardin{C} = k}\n    \\nu_C\\pth{P}$ denote the price of the \\emph{optimal $k$-median}\n    clustering for $P$.\n\n    Similarly, let $\\mu_C(P) = \\sum_{p \\in P}\n    w(p)\\pth{\\mathbf{d}\\hspace{-1pt}(p,C)}^2$ denote the price of the\n    $k$-means clustering of $P$ as provided by the set of\n    centers $C$.  Let $\\mu_{\\mathrm{opt}}(P,k) = \\min_{C \\subseteq\n       \\Re^d, \\cardin{C} = k} \\mu_C(P)$ denote the price\n    of the \\emph{optimal $k$-means clustering} of $P$.\n\\end{defn}\n\n\\begin{remark}\n    We only consider positive integer weights. A regular\n    point set $P$ may be considered as a weighted set with weight $1$\n    for each point, and total weight $\\cardin{P}$.\n\\end{remark}\n\n\\begin{defn}[Discrete Clustering]\n    In several cases, it is convenient to consider the\n    centers to be restricted to lie in the original point\n    set. In particular, let $\\nu^D_{\\mathrm{opt}}(P,k) =\\min_{C\\subseteq\n       P, \\cardin{C} = k} \\nu_C\\pth{P}$ denote the price\n    of the \\emph{optimal discrete $k$-median} clustering for\n    $P$ and let $\\mu^D_{\\mathrm{opt}}(P,k) = \\min_{C \\subseteq P,\n       \\cardin{C} = k} \\mu_C(P)$ denote the price of the\n    \\emph{optimal discrete $k$-means} clustering of $P$.\n\\end{defn}\n\n\\begin{observation}\n    For any point set $P$, we have $\\mu_{\\mathrm{opt}}(P,k) \\leq \\mu^D_{\\mathrm{opt}}(P,k) \\leq\n    4 \\mu_{\\mathrm{opt}}(P,k)$, and $\\nu_{\\mathrm{opt}}(P,k) \\leq \\nu^D_{\\mathrm{opt}}(P,k) \\leq 2\n    \\nu_{\\mathrm{opt}}(P,k)$.\n\\end{observation}\n\n\n\\section{Coresets from Approximate Clustering}\n\\seclab{coresets}\n\n\\begin{defn}[Coreset]\n    For a weighted point set $P\\subseteq \\Re^d$, a weighted\n    set ${\\mathcal{S}} \\subseteq \\Re^d$, is a\n    \\emph{$\\pth{k,\\eps}$-coreset} of $P$ for the $k$-median\n    clustering, if for any set $C$ of $k$ points in $\\Re^d$,\n    we have $(1-\\eps)\\nu_C(P) \\leq \\nu_C({\\mathcal{S}})\n    \\leq (1+\\eps)\\nu_C(P)$.\n\n    Similarly, ${\\mathcal{S}}$ is a \\emph{$\\pth{k,\\eps}$-coreset}\n    of $P$ for the $k$-means clustering, if for any set $C$\n    of $k$ points in $\\Re^d$, we have $(1-\\eps)\\mu_C(P)\n    \\leq \\mu_C({\\mathcal{S}}) \\leq (1+\\eps)\\mu_C(P)$.\n\\end{defn}\n\n\\subsection{Coreset for $k$-Median}\n\nLet $P$ be a set of $n$ points in $\\Re^d$, and\n$A = \\brc{x_1,\\ldots, x_m}$ be a point set, such that\n$\\nu_A(P) \\leq c \\nu_{\\mathrm{opt}}(P,k)$, where $c$ is a constant. We give\na construction for a $(k,\\eps)$-coreset using $A$. Note that we do\nnot have any restriction on the size of $A$, which in subsequent\nuses will be taken to be $O(k\\polylog)$.\n\n\\subsubsection{The construction}\n\\seclab{construction}\n\nLet $P_i$ be the points of $P$ having $x_i$ as their nearest\nneighbor in $A$, for $i=1,\\ldots, m$. Let $R =\n\\nu_A(P)\/\\pth{c n}$ be a lower bound estimate of the\naverage radius $\\mathrm{R^{\\nu}_{opt}}(P,k) = \\nu_{\\mathrm{opt}}(P,k)\/n$.  For any $p\n\\in P_i$, we have $\\dist{p x_i} \\leq c n R$, since $\\dist{p\n   x_i} \\leq \\nu_A(P)$, for $i=1,\\ldots, m$.\n\nNext, we construct an appropriate exponential grid around each $x_i$,\nand snap the points of $P$ to those grids. Let $Q_{i,j}$ be an\naxis-parallel square with side length $R 2^j$ centered at $x_i$, for\n$j=0,\\ldots, M$, where $M=\\ceil{2\\lg (c n)}$.  Next, let\n$V_{i,0} = Q_{i,0}$, and let\n$V_{i,j} = Q_{i,j} \\setminus Q_{i,j-1}$, for\n$j=1,\\ldots, M$.  Partition $V_{i,j}$ into a grid with side length\n$r_{j} = \\eps R 2^j\/(10 c d)$, and let $G_i$ denote the resulting\nexponential grid for $V_{i,0},\\ldots, V_{i,M}$.  Next, compute for\nevery point of $P_i$, the grid cell in $G_i$ that contains it.  For\nevery non empty grid cell, pick an arbitrary point of $P_i$ inside it\nas a representative point for the coreset, and assign it a weight\nequal to the number of points of $P_i$ in this grid cell. Let\n${\\mathcal{S}}_i$ denote the resulting weighted set, for $i=1,\\ldots, m$,\nand let ${\\mathcal{S}} = \\cup_i {\\mathcal{S}}_i$.\n\nNote that $\\cardin{{\\mathcal{S}}} = O\\pth{\\pth{\\cardin{A}\\log n}\/\n   \\eps^d }$.  As for computing ${\\mathcal{S}}$ efficiently. Observe that\nall we need is a constant factor approximation to $\\nu_A(P)$\n(i.e., we can assign a $p \\in P$ to $P_i$ if $\\dist{p, x_i} \\leq\n2\\mathbf{d}\\hspace{-1pt}(p,A)$).  This can be done in a naive way in $O(n m)$ time,\nwhich might be quite sufficient in practice. Alternatively, one can\nuse a data-structure that answers constant approximate\nnearest-neighbor queries in $O(\\log m)$ when used on $A$ after\n$O(m \\log{m})$ preprocessing \\cite{amnsw-oaann-98}.  Another option\nfor computing those distances between the points of $P$ and the set\n$A$ is using \\thmref{batch:n:n} that works in $O(n + m n^{1\/4}\n\\log{n})$ time.  Thus, for $i=1,\\ldots, m$, we compute a set $P_i'$\nwhich consists of the points of $P$ that $x_i$ (approximately) serves.\nNext, we compute the exponential grids, and compute for each point of\n$P_i'$ its grid cell.  This takes $O(1)$ time per point, with a\ncareful implementation, using hashing, the floor function and the\n$\\log$ function. Thus, if $m=O(\\sqrt{n})$ the overall running time is\n$O(n + m n^{1\/4} \\log{n}) = O(n)$ and $O( m\\log{m} + n \\log{m} + n) =\nO( n\\log{m})$ otherwise.\n\n\n\\subsubsection{Proof of Correctness}\n\n\\begin{lemma}\n    \\lemlab{correctness}%\n    %\n    The weighted set ${\\mathcal{S}}$ is a $\\pth{k,\\eps}$-coreset for $P$\n    and $\\cardin{{\\mathcal{S}}} = O\\pth{|A|\\eps^{-d}\\log{n}}$.\n\\end{lemma}\n\n\n\\begin{proof}\n    Let $Y$ be an arbitrary set of $k$ points in $\\Re^d$.\n    For any $p\\in P$, let $p'$ denote the image of $p$ in\n    ${\\mathcal{S}}$. The error is $\\E =\n    \\cardin{\\nu_Y\\pth{P} - \\nu_Y({\\mathcal{S}})} \\leq\n    \\sum_{p\\in P}\\cardin{\\mathbf{d}\\hspace{-1pt}(p, Y) - \\mathbf{d}\\hspace{-1pt}(p', Y)}$.\n\n    Observe that $\\mathbf{d}\\hspace{-1pt}(p,Y) \\leq \\dist{p p'} +\\mathbf{d}\\hspace{-1pt}(p',Y)$\n    and $\\mathbf{d}\\hspace{-1pt}(p',Y) \\leq \\dist{p p'} +\\mathbf{d}\\hspace{-1pt}(p,Y)$ by the\n    triangle inequality. Implying that $\\cardin{\\mathbf{d}\\hspace{-1pt}(p, Y)\n       - \\mathbf{d}\\hspace{-1pt}(p',Y)} \\leq \\dist{p p'}$.  It follows that\n    \\[\n    \\E \\leq \\sum_{p\\in P}\\dist{p p'} =\n    {\\!\\!\\!\\!\\!\\sum_{\\substack{p\\in P,\\\\\n             \\mathbf{d}\\hspace{-1pt}(p,A) \\leq R} } \\!\\!\\!\\!\\! \\dist{p p'}\n    } +\n    \\!\\!\\!\\!\\!{\\sum_{\\substack{p\\in P,\\\\\n             \\mathbf{d}\\hspace{-1pt}(p,A) > R} } \\!\\!\\!\\!\\!  \\dist{p p'} }\n    \\leq \\frac{\\eps}{10c} n R +\\frac{\\eps}{10c}\\sum_{p\\in P}\n    \\mathbf{d}\\hspace{-1pt}\\pth{p,A} \\leq \\frac{2\\eps}{10c}\n    \\nu_A(P) \\leq \\eps \\nu_{\\mathrm{opt}}\\pth{P,k},\n    \\]\n    since $\\dist{p p'} \\leq \\frac{\\eps}{10c}\\mathbf{d}\\hspace{-1pt}(p,A)$ if\n    $\\mathbf{d}\\hspace{-1pt}(p,A) \\geq R$, and $\\dist{p p'} \\leq\n    \\frac{\\eps}{10c} R$, if $\\mathbf{d}\\hspace{-1pt}(p,A) \\leq R$, by the\n    construction of the grid. This implies\n    $\\cardin{\\nu_Y\\pth{P} - \\nu_Y({\\mathcal{S}})} \\leq \\eps\n    \\nu_Y \\pth{ P}$, since $\\nu_{\\mathrm{opt}}(P,k) \\leq\n    \\nu_Y(P)$.\n\\end{proof}\n\nIt is easy to see that the above algorithm can be easily\nextended for weighted point sets.\n\\begin{theorem}\n    \\thmlab{coreset:fast:k:median}%\n   \n    Given a point set $P$ with $n$ points, and a point set $A$\n    with $m$ points, such that $\\nu_A(P) \\leq c \\nu_{\\mathrm{opt}}(P,k)$,\n    where $c$ is a constant. Then, one can compute a weighted set\n    ${\\mathcal{S}}$ which is a $(k,\\eps)$-coreset for $P$, and\n    $\\cardin{{\\mathcal{S}}} = O\\pth{ (\\cardin{A} \\log{n}) \/\\eps^d}$.\n    The running time is $O(n )$ if $m=O(\\sqrt{n})$ and $O( n\\log{m})$\n    otherwise.\n\n    If $P$ is weighted, with total weight $W$, then\n    $\\cardin{{\\mathcal{S}}} = O\\pth{ (\\cardin{A} \\log{W})\n       \/\\eps^d}$.\n\n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Coreset for $k$-Means}\n\\seclab{k:means:coreset}\n\nThe construction of the $k$-means coreset follows the $k$-median\nwith a few minor modifications.  Let $P$ be a set of $n$ points in\n$\\Re^d$, and a $A$ be a point set $A = \\brc{x_1,\\ldots,\nx_m}$, such that $\\mu_A(P) \\leq c \\mu_{\\mathrm{opt}}(P,k)$.  Let $R =\n\\sqrt{(\\mu_A(P)\/(c n))}$ be a lower bound estimate of the\naverage mean radius $\\mathrm{R^{\\mu}_{opt}}(P,k) = \\sqrt{ \\mu_{\\mathrm{opt}}(P,k)\/n}$.  For\nany $p \\in P_i$, we have $\\dist{p x_i} \\leq \\sqrt{c n} R$, since\n$\\dist{p\n   x_i}^2 \\leq \\mu_A(P)$, for $i=1,\\ldots, m$.\n\nNext, we construct an exponential grid around each point of $A$,\nas in the $k$-median case, and snap the points of $P$ to this grid,\nand we pick a representative point for such grid cell. See\n\\secref{construction} for details.  We claim that the resulting set of\nrepresentatives ${\\mathcal{S}}$ is the required coreset.\n\n\\begin{theorem}\n    \\thmlab{k:means:weighted:coreset}%\n    %\n    Given a set $P$ with $n$ points, and a point set $A$ with $m$\n    points, such that $\\mu_A(P) \\leq c \\mu_{\\mathrm{opt}}(P,k)$, where $c$\n    is a constant. Then, can compute a weighted set ${\\mathcal{S}}$ which\n    is a $(k,\\eps)$-coreset for $P$, and $\\cardin{{\\mathcal{S}}} = O\\pth{\n       (m \\log{n}) \/(c\\eps)^d}$.  The running time is $O(n )$ if\n    $m=O(n^{1\/4})$ and $O( n\\log{m})$ otherwise.\n    \n    If $P$ is a weighted set with total weight $W$, then the\n    size of the coreset is $O\\pth{ (m \\log{W})\/\\eps^d}$.\n\\end{theorem}\n\n\\begin{proof}\n    We prove the theorem for an unweighted point set.  The\n    construction is as in \\secref{k:means:coreset}.  As for\n    correctness, consider an arbitrary set $B$ of $k$ points in\n    $\\Re^d$.  The proof is somewhat more tedious than the median case,\n    and we give short description of it before plunging into the\n    details. We partition the points of $P$ into three sets: (i)\n    Points that are close (i.e., $\\leq R$) to both $B$ and\n    $A$. The error those points contribute is small because they\n    contribute small terms to the summation.  (ii) Points that are\n    closer to $B$ than to $A$ (i.e., $P_A$). The error\n    those points contribute can be charged to an $\\eps$ fraction of\n    the summation $\\mu_A(P)$. (iii) Points that are closer to\n    $A$ than to $B$ (i.e., $P_B$). The error is here\n    charged to the summation $\\mu_B(P)$.  Combining those\n    three error bounds, give us the required result.\n\n    For any $p\\in P$, let $p'$ the image of $p$ in ${\\mathcal{S}}$; namely,\n    $p'$ is the point in the coreset ${\\mathcal{S}}$ that represents $p$.\n    Now, we have\n    \\begin{equation*}\n        \\E%\n        =%\n        \\cardin{\\mu_B(P) - \\mu_B(S) } \\leq\n        \\sum_{p\\in P} \\cardin{ {\\mathbf{d}\\hspace{-1pt}(p,B) }^2\n           - \\mathbf{d}\\hspace{-1pt}(p', B)^2}%\n        \\leq\n        \\sum_{p\\in P} \\cardin{ \\pth{\\mathbf{d}\\hspace{-1pt}(p, B)\n              - \\mathbf{d}\\hspace{-1pt}(p', B)}\n           \\pth{\\Bigl.\\!\\mathbf{d}\\hspace{-1pt}(p, B) + \\mathbf{d}\\hspace{-1pt}(p',B)} }\n    \\end{equation*}\n    Let\n    $P_R = \\Set{ p \\in P }{ \\mathbf{d}\\hspace{-1pt}(p, B ) \\leq R \\text{ and }\n       \\mathbf{d}\\hspace{-1pt}(p , A)\\leq R}$,\n    $P_A = \\Set{ p \\in P \\setminus P_R }{ \\mathbf{d}\\hspace{-1pt}(p,B) \\leq\n       \\mathbf{d}\\hspace{-1pt}(p,A)}$, and let\n    $P_B = P \\setminus \\pth{ P_R \\cup P_A}$. By the triangle\n    inequality, for $p \\in P$, we have\n    $\\mathbf{d}\\hspace{-1pt}(p',B) + \\dist{p p'} \\geq \\mathbf{d}\\hspace{-1pt}(p, B)$ and\n    $\\mathbf{d}\\hspace{-1pt}(p, B) + \\dist{p p'} \\geq \\mathbf{d}\\hspace{-1pt}(p',B)$.  Thus,\n    $\\dist{p p'} \\geq \\cardin{ \\mathbf{d}\\hspace{-1pt}(p, B) - \\mathbf{d}\\hspace{-1pt}(p', B)}$.\n    \n    Also,\n    $\\mathbf{d}\\hspace{-1pt}(p, B) + \\mathbf{d}\\hspace{-1pt}(p', B) \\leq 2\\mathbf{d}\\hspace{-1pt}(p, B) +\n    \\dist{p p'}$, and thus\n    \\begin{align*}\n      \\E_R%\n      &=%\n        \\sum_{p\\in P_R} \\cardin{ \\pth{ \\mathbf{d}\\hspace{-1pt}(p, B) -\n        \\mathbf{d}\\hspace{-1pt}(p',B)} \\pth{\\mathbf{d}\\hspace{-1pt}(p, B) + \\mathbf{d}\\hspace{-1pt}(p',B)}\n        }%\n       \n      \\leq %\n      \\sum_{p\\in P_R} \\dist{p p'} \\pth{ 2 \\mathbf{d}\\hspace{-1pt}(p, B) + \\dist{p\n      {}p'}}%\n      \\\\&%\n      \\leq%\n      \\sum_{p\\in P_R} \\frac{\\eps}{10} R \\pth{ 2 R\n      + \\frac{\\eps}{10} R}%\n  \n      \\leq %\n      \\frac{\\eps}{3} \\sum_{p\\in P_R} R^2 \\leq \\frac{\\eps}{3}\n      \\mu_{\\mathrm{opt}}(P,k),\n    \\end{align*}\n    since by definition, for $p \\in P_R$, we have\n    $\\mathbf{d}\\hspace{-1pt}(p, A), \\mathbf{d}\\hspace{-1pt}(p,B) \\leq R$.\n    \n    By construction $\\dist{p p'} \\leq (\\eps\/10c)\\mathbf{d}\\hspace{-1pt}(p,A)$, for\n    all $p \\in P_A$, as $\\mathbf{d}\\hspace{-1pt}(p, A) \\geq R$. Thus,\n    \\begin{align*}\n      \\E_A\n      &=%\n        \\sum_{p\\in P_A} \\dist{p p'} \\pth{ 2\n        \\mathbf{d}\\hspace{-1pt}(p, B) + \\dist{p p'}}\n        \\leq \\sum_{p\\in P_A} \\frac{\\eps}{10c}\n        \\mathbf{d}\\hspace{-1pt}(p,A) \\pth{ 2 + \\frac{\\eps}{10c}}\n        \\mathbf{d}\\hspace{-1pt}(p, A) %\n      \\\\&%\n      \\leq%\n      \\frac{\\eps}{3c} \\sum_{p \\in\n      P_A} \\pth{ \\mathbf{d}\\hspace{-1pt}(p,A)}^2 \\leq\n      \\frac{\\eps}{3} \\mu_{\\mathrm{opt}}(P,k)\n      \\leq \\frac{\\eps}{3} \\mu_B(P).\n    \\end{align*}\n\n    As for $p \\in P_B$, we have $\\dist{p p'} \\leq\n    \\frac{\\eps}{10c}\\mathbf{d}\\hspace{-1pt}(p, B)$, since $\\mathbf{d}\\hspace{-1pt}(p, B) \\geq\n    R$, and $\\mathbf{d}\\hspace{-1pt}(p,A) \\leq \\mathbf{d}\\hspace{-1pt}(p, B)$. Implying\n    $\\dist{p p'} \\leq (\\eps\/10c)\\mathbf{d}\\hspace{-1pt}(p,B)$ and thus\n    \\begin{align*}\n        \\E_B\n      &=%\n        \\sum_{p\\in P_B} \\dist{p p'} \\pth{ 2\n           \\mathbf{d}\\hspace{-1pt}(p, B) + \\dist{p p'}}\n        \\leq\n        \\sum_{p \\in P_B}\n        \\frac{\\eps}{10c}\\mathbf{d}\\hspace{-1pt}(p,B)\n        \\pth{ 2\n        \\mathbf{d}\\hspace{-1pt}(p, B) + \\frac{\\eps}{10c}\\mathbf{d}\\hspace{-1pt}(p,B)}%\n      \\\\&%\n      \\leq%\n        \\sum_{p \\in P_B} \\frac{\\eps}{3} \\mathbf{d}\\hspace{-1pt}(p, B)^2\n        \\leq \\frac{\\eps}{3} \\mu_B(P).\n    \\end{align*}\n    We conclude that $\\E = \\cardin{\\mu_B(P) - \\mu_B(S)\n    } \\leq \\E_R + \\E_A + \\E_B \\leq \\frac{3\\eps}{3}\n    \\mu_B(P), $ which implies that $(1-\\eps)\\mu_B(P)\n    \\leq \\mu_B(S) \\leq (1+\\eps) \\mu_B(P)$, as\n    required. It is easy to see that we can extend the analysis\n    for the case when we have weighted points.\n\\end{proof}\n\n\n\n\n\n\\section{Fast Constant Factor Approximation Algorithm}\n\\seclab{fast:const:factor}\n\nLet $P$ be the given point set in $\\Re^d$. We want to quickly\ncompute a constant factor approximation to the $k$-means\nclustering of $P$, while using more than $k$ centers. The number\nof centers output by our algorithm is $O\\pth{k \\log^3 n}$.\nSurprisingly, the set of centers computed by the following\nalgorithm is a good approximation for both $k$-median and\n$k$-means. To be consistent, throughout this section, we refer to\n$k$-means, although everything holds nearly verbatim for\n$k$-median as well.\n\n\\begin{defn}[bad points]\n    For a point set $X$, define a point $p \\in P$ as\n    \\emph{bad} with respect to $X$, if the cost it pays in\n    using a center from $X$ is prohibitively larger than the\n    price $C_{\\mathrm{opt}}$ pays for it; more precisely $\\mathbf{d}\\hspace{-1pt}(p,X)\n    \\geq 2 \\mathbf{d}\\hspace{-1pt}(p,C_{\\mathrm{opt}})$. A point $p \\in P$ which is not\n    bad, is by necessity, if not by choice, \\emph{good}.\n    Here $C_{\\mathrm{opt}} =C_{\\mathrm{opt}}(P,k)$ is a set of optimal $k$-means\n    centers realizing $\\mu_{\\mathrm{opt}}(P,k)$.\n\\end{defn}\nWe first describe a procedure which given $P$, computes a\nsmall set of centers $X$ and a large $P'\\subseteq P$ such\nthat $X$ induces clusters $P'$ well. Intuitively we want a\nset $X$ and a large set of points $P'$ which are \\emph{good}\nfor $X$.\n\n\n\\subsection{Construction of the Set $X$ of Centers}\n\n\\seclab{good:subset:centers}\n\nFor $k=O(n^{1\/4})$, we can compute a $2$-approximate $k$-center\nclustering of $P$ in linear time \\cite{h-cm-04}, or alternatively, for\n$k=\\Omega(n^{1\/4})$, in $O(n\\log{k})$ time, using the algorithm of\nFeder and Greene \\cite{fg-oafac-88}.  This is the \\emph{min-max\n   clustering} where we cover $P$ by a set of $k$ balls such the\nradius of the largest ball is minimized. Let $V$ be the set of $k$\ncenters computed, together with the furthest point in $P$ from those\n$k$ centers.\n\nLet $L$ be the radius of this $2$-approximate clustering.  Since both\nthose algorithms are simulating the (slower) algorithm of Gonzalez\n\\cite{g-cmmid-85}, we have the property that the minimal distance\nbetween any points of $V$ is at least $L$.  Thus, any $k$-means\nclustering of $P$, must have price at least $(L\/2)^2$, and is at most\nof price $n L^2$, and as such $L$ is a rough estimate of\n$\\mu_{\\mathrm{opt}}(P,k)$. In fact, this holds even if we restrict out attention\nonly to $V$; explicitly $(L\/2)^2 \\leq \\mu_{\\mathrm{opt}}(V, k) \\leq \\mu_{\\mathrm{opt}}(P,k)\n\\leq n L^2$.\n\nNext, we pick a random sample $Y$ from $P$ of size $\\rho = \\gamma k\n\\log^2 n$, where $\\gamma$ is a large enough constant whose value\nwould follow from our analysis.  Let $X = Y \\cup V$ be the required\nset of cluster centers. In the extreme case where $\\rho > n$, we just\nset $X$ to be $P$.\n\n\n\n\\subsection{A Large Good Subset for $X$}\n\\seclab{good:subset:points}\n\n\\subsubsection{Bad points are few}\nConsider the set $C_{\\mathrm{opt}}$ of $k$ optimal centers for the $k$-means, and\nplace a ball ${b}_i$ around each point of $c_i\\inC_{\\mathrm{opt}}$, such that\n${b}_i$ contain $\\eta = n\/(20k\\log{n})$ points of $P$. If\n$\\gamma$ is large enough, it is easy to see that with high\nprobability, there is at least one point of $X$ inside every ball\n${b}_i$. Namely, $X \\cap {b}_i \\ne \\emptyset$, for $i=1,\\ldots,\nk$.\n\nLet $P_{\\mathrm{bad}}$ be the set of all bad points of $P$. Assume, that\nthere is a point $x_i \\in X$ inside ${b}_i$, for $i=1,\\ldots,\nk$.  Observe, that for any $p \\in P \\setminus {b}_i$, we have\n$\\dist{p x_i} \\leq 2\\dist{p c_i}$. In particular, if $c_i$ is the\nclosest center in $C_{\\mathrm{opt}}$ to $p$, we have that $p$ is good.  Thus,\nwith high probability, the only bad points in $P$ are the one that\nlie inside the balls ${b}_1,\\ldots, {b}_k$. But every one of\nthose balls, contain at most $\\eta$ points of $P$. It\nfollows, that with high probability, the number of bad points in\n$P$ with respect to $X$ is at most $\\beta= k \\cdot \\eta =\nn\/(20\\log n)$.\n\n\\subsubsection{Keeping Away from Bad Points}\n\nAlthough the number of bad points is small, there is no easy way to\ndetermine the set of bad points. We instead construct a set $P'$\nensuring that the clustering cost of the bad points in $P'$ does not\ndominate the total cost.  For every point in $P$, we compute its\napproximate nearest neighbor in $X$.  This can be easily done in\n$O(n\\log \\cardin{X} + \\cardin{X} \\log \\cardin{X})$ time using\nappropriate data structures \\cite{amnsw-oaann-98}, or in $O(n +\nn\\cardin{X}^{1\/4} \\log n)$ time using \\corref{batch:n:n:2} (with $D=n\nL$). This stage takes $O(n)$ time, if $k=O(n^{1\/4})$, else it takes\n$O(n \\log{\\cardin{X}} + \\cardin{X} \\log{\\cardin{X}}) = O(n \\log( k\\log\nn))$ time, as $\\cardin{X} \\leq n$.\n\nIn the following, to simplify the exposition, we assume that we\ncompute exactly the distance $r(p) = \\mathbf{d}\\hspace{-1pt}(p,X)$, for $p \\in P$.\n\nNext, we partition $P$ into classes in the following way.  Let\n$P[a,b] = \\Set{ p \\in P }{a\\leq r(p)< b}$. Let $P_0 = P[0, L\/(4n)]$,\n$P_\\infty = P[2Ln,\\infty]$ and $P_i = P\\pbrc{2^{i-1} L\/n, 2^{i} L\/n}$,\nfor $i=1,\\ldots, M$, where $M = 2\\ceil{\\lg n} + 3$. This partition of\n$P$ can be done in linear time using the $\\log$ and floor function.\n\nLet $P_\\alpha$ be the last class in this sequence that\ncontains more than $2\\beta = 2(n\/(20\\log{n}))$ points. Let\n$P' = V \\cup \\bigcup_{i \\leq \\alpha} P_i$. We claim that\n$P'$ is the required set.  Namely, $\\cardin{P'} \\geq n\/2$\nand $\\mu_X(P') = O(\\mu_{C_{\\mathrm{opt}}}(P'))$, where $C_{\\mathrm{opt}}\n=C_{\\mathrm{opt}}(P,k)$ is the optimal set of centers for $P$.\n\n\\subsubsection{Proof of Correctness}\n\nClearly the set $P'$ contains at least $\\pth{n - \\cardin{P_\\infty}\n- M \\cdot \\pth{2n\/20 \\log{n}}}$ points. Since $P_\\infty \\subseteq\nP_{\\mathrm{bad}}$ and $\\cardin{P_{\\mathrm{bad}}} \\leq \\beta$, hence $|P'| > n\/2$.\n\nIf $\\alpha > 0$, we have $\\cardin{P_\\alpha} \\geq 2\\beta =\n2(n\/(20\\log{n}))$. Since $P'$ is the union of all the classes with\ndistances smaller than the distances in $P_\\alpha$, it follows that\nthe worst case scenario is when all the bad points are in $P_\\alpha$.\nBut with high probability the number of bad points is at most $\\beta$,\nand since the price of all the points in $P_\\alpha$ is roughly the\nsame, it follows that we can charge the price of the bad points in\n$P'$ to the good points in $P_\\alpha$.\n\nFormally, let $Q'= P_\\alpha \\setminus P_{\\mathrm{bad}}$.  For any point $p\n\\in P' \\cap P_{\\mathrm{bad}}$ and $q \\in Q'$, we have $\\mathbf{d}\\hspace{-1pt}(p,X) \\leq\n2\\mathbf{d}\\hspace{-1pt}(q,X)$. Further $|Q'| > |P_{\\mathrm{bad}}|$. Thus, $\\mu_X(P' \\cap\nP_{\\mathrm{bad}}) \\leq 4\\mu_X(Q') \\leq 16 \\mu_{C_{\\mathrm{opt}}}(Q') \\leq\n16\\mu_{C_{\\mathrm{opt}}}(P')$. Thus, \n\\[\n\\mu_X(P') = \\mu_X(P' \\cap P_{\\mathrm{bad}}) + \\mu_X(P'\n\\setminus P_{\\mathrm{bad}} ) \\leq 16 \\mu_{C_{\\mathrm{opt}}}(P') +\n4\\mu_{C_{\\mathrm{opt}}}(P') = 20 \\mu_{C_{\\mathrm{opt}}}(P').\n\\]\n\nIf $\\alpha =0$ then for any point $p \\in P'$, we have $(\\mathbf{d}\\hspace{-1pt}(p,X))^2\n\\leq n(L\/4n)^2 \\leq L^2\/(4n)$. and thus $\\mu_X(P') \\leq L^2\/4 \\leq\n\\mu_{C_{\\mathrm{opt}}}(V) \\leq \\mu_{C_{\\mathrm{opt}}}(P')$, since $V \\subseteq P'$.\n\nIn the above analysis we assumed that the nearest neighbor data\nstructure returns the exact nearest neighbor. If we were to use an\napproximate nearest neighbor instead, the constants would slightly\ndeteriorate.\n\n\\begin{lemma}\n    \\lemlab{good:subset}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, and parameter $k$, one\n    can compute sets $P'$ and $X \\subseteq P$ such that, with high\n    probability, $\\cardin{P'} \\geq n\/2$, $\\cardin{X} = O(k \\log^2 n)$,\n    and $\\mu_{C_{\\mathrm{opt}}}(P') \\geq \\mu_X(P')\/32$, where $C_{\\mathrm{opt}}$ is\n    the optimal set of $k$-means centers for $P$. The running time of\n    the algorithm is $O(n)$ if $k = O(n^{1\/4})$, and $O(n \\log{(k\n       \\log{n})} )$ otherwise.\n\\end{lemma}\n\n\nNow, finding a constant factor $k$-median clustering is easy. Apply\n\\lemref{good:subset} to $P$, remove the subset found, and repeat on\nthe remaining points. Clearly, this would require $O(\\log{n})$\niterations. We can extend this algorithm to the weighted case, by\nsampling $O(k \\log^2 W)$ points at every stage, where $W$ is the total\nweight of the points. Note however, that the number of points no\nlonger shrink by a factor of two at every step, as such the running\ntime of the algorithm is slightly worse.\n\n\\begin{theorem}[Clustering with more centers]\n    \\thmlab{k:cluster:const:rough}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, and parameter $k$, one\n    can compute a set $X$, of size $O(k\\log^3 n)$, such that\n    $\\mu_X(P) \\leq 32 \\mu_{\\mathrm{opt}}(P,k)$. The running time of the\n    algorithm is $O(n)$ if $k = O(n^{1\/4})$, and $O(n \\log{(k \\log\n       n)})$ otherwise.\n    \n    Furthermore, the set $X$ is a good set of centers for\n    $k$-median.  Namely, we have that $\\nu_X(P) \\leq 32\n    \\nu_{\\mathrm{opt}}(P,k)$.\n    \n    If the point set $P$ is weighted, with total weight $W$, then the\n    size of $X$ becomes $O(k \\log^3 W)$, and the running time becomes\n    $O(n \\log^2 W )$.\n\\end{theorem}\n\n\n\\section{$(1+\\eps)$-Approximation for $k$-Median}\n\\seclab{eps:approx:k:median}\n\nWe now present the approximation algorithm using exactly $k$\ncenters. Assume that the input is a set of $n$ points. We use the set\nof centers computed in \\thmref{k:cluster:const:rough} to compute a\nconstant factor coreset using the algorithm of\n\\thmref{coreset:fast:k:median}.  The resulting coreset ${\\mathcal{S}}$, has\nsize $O(k \\log^4 n )$.  Next we compute a $O(n)$ approximation to the\n$k$-median for the coreset using the $k$-center (min-max) algorithm\n\\cite{g-cmmid-85}.  Let $C_0 \\subseteq {\\mathcal{S}}$ be the resulting set\nof centers.  Next we apply the local search algorithm, due to Arya\n\\textit{et~al.}\\xspace{} \\cite{agkmm-lshkm-04}, to $C_0$ and ${\\mathcal{S}}$, where the set\nof candidate points is ${\\mathcal{S}}$. This local search algorithm, at\nevery stage, picks a center $c$ from the current set of centers\n$C_{curr}$, and a candidate center $s \\in {\\mathcal{S}}$, and swaps $c$ out\nof the set of centers and $s$ into the set of centers. Next, if the\nnew set of centers\n$C_{curr}' = C_{curr} \\setminus \\brc{c} \\cup \\brc{s}$ provides a\nconsiderable improvement over the previous solution (i.e.,\n$\\nu_{C_{curr}}({\\mathcal{S}}) \\leq\n(1-\\eps\/k)\\nu_{C_{curr}'}({\\mathcal{S}})$ where $\\eps$ here is an\narbitrary small constant), then we set $C_{curr}$ to be $C_{curr}'$.\nArya \\textit{et~al.}\\xspace{} \\cite{agkmm-lshkm-04} showed that the algorithm\nterminates, and it provides a constant factor approximation to\n$\\nu^D_{\\mathrm{opt}}({\\mathcal{S}}, k)$, and as hence to $\\nu_{\\mathrm{opt}}(P,k)$.  It is easy to\nverify that it stops after $O(k \\log{n})$ such swaps. Every swap, in\nthe worst case, requires considering $\\cardin{{\\mathcal{S}}}k$\nsets. Computing the price of clustering for every such candidate set\nof centers takes $O\\pth{ \\cardin{{\\mathcal{S}}} k }$ time.  Thus, the\nrunning time of this algorithm is\n$O \\pth{ \\cardin{{\\mathcal{S}}}^2 k^{3} \\log n}=O \\pth{ k^5 \\log^9 n\n}$. Finally, we use the new set of centers with\n\\thmref{coreset:fast:k:median}, and get a $(k,\\eps)$-coreset for\n$P$. It is easy to see that the algorithm works for weighted\npoint-sets as well. Putting in the right bounds from\n\\thmref{k:cluster:const:rough} and \\thmref{coreset:fast:k:median} for\nweighted sets, we get the following.\n\n\\begin{lemma}[coreset]\n    \\lemlab{k:coreset:small:median}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, one can\n    compute a $k$-median $(k,\\eps)$-coreset ${\\mathcal{S}}$ of\n    $P$, of size $O\\pth{ (k\/\\eps^d)\\log{n} }$, in time $O\n    \\pth{ n + k^5 \\log^9 n }$.\n    \n    If $P$ is a weighted set, with total weight $W$, the\n    running time of the algorithm is $O( n \\log^2 W + k^5\n    \\log^9 W)$.\n\\end{lemma}\n\nWe would like to apply the algorithm of Kolliopoulos and Rao\n\\cite{kr-nltas-99} to the coreset, but unfortunately, their\nalgorithm only works for the discrete case, when the medians are\npart of the input points. Thus, the next step is to generate from\nthe coreset, a small set of candidate points in which we can\nassume all the medians lie, and use the (slightly modified)\nalgorithm of \\cite{kr-nltas-99} on this set.\n\n\\begin{defn}[Centroid Set]\n    Given a set $P$ of $n$ points in $\\Re^d$, a set $\\mathcal{D}\n    \\subseteq \\Re^d$ is an \\emph{$(k,\\eps)$-approximate\n       centroid set} for $P$, if there exists a subset\n    $C \\subseteq \\mathcal{D}$ of size $k$, such that\n    $\\nu_C(P) \\leq (1+\\eps)\\nu_{\\mathrm{opt}}(P,k)$.\n\\end{defn}\n\n\\begin{lemma}\n    \\lemlab{k:median:cen:set}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, one can\n    compute an $(k,\\eps)$-centroid set $\\mathcal{D}$ of size\n    $O(k^2\\eps^{-2d}\\log^2{n})$.  The running time of this\n    algorithm is $O \\pth{ n + k^5 \\log^9 n +\n       k^2\\eps^{-2d}\\log^2{n} }$.\n    \n    For the weighted case, the running time is\n    $O \\pth{ n \\log^2 W + k^5 \\log^9 W + k^2\\eps^{-2d}\\log^2{W} }$,\n    and the centroid set is of size $O(k^2\\eps^{-2d}\\log^2{W})$.\n\\end{lemma}\n\n\\begin{proof}\n    Compute a $(k,\\eps\/12)$-coreset ${\\mathcal{S}}$ using\n    \\lemref{k:coreset:small:median}. We retain the set\n    $B$ of $k$ centers, for which $\\nu_B( P )\n    =O( \\nu_{\\mathrm{opt}}(P,k) )$, which is computed during the\n    construction of ${\\mathcal{S}}$.  Further let $R =\n    \\nu_B(P) \/ n$.\n    \n    Next, compute around each point of ${\\mathcal{S}}$, an exponential grid\n    using $R$, as was done in \\secref{construction}.  This results in\n    a point set $\\mathcal{D}$ of size of $O(k^2\\eps^{-2d}\\log^2{n})$. We\n    claim that $\\mathcal{D}$ is the required centroid set.  The proof\n    proceeds on similar lines as the proof of\n    \\thmref{coreset:fast:k:median}.\n    \n    Indeed, let $C_{\\mathrm{opt}}$ be the optimal set of $k$ medians.\n    We snap each point of $C_{\\mathrm{opt}}$ to its nearest neighbor in\n    $\\mathcal{D}$, and let $X$ be the resulting set. Arguing as\n    in the proof of \\thmref{coreset:fast:k:median}, we have\n    that $\\cardin{\\nu_X({\\mathcal{S}})\n       -\\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}})}$ $\\leq\n    (\\eps\/12)\\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}})$. On the other hand,\n    by definition of a coreset, $\\cardin{\\nu_{C_{\\mathrm{opt}}}(P) -\n       \\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}})}\\leq (\\eps\/12)$\n    $\\nu_{C_{\\mathrm{opt}}}(P)$ and $\\cardin{\\nu_{X}(P) -\n       \\nu_{X}({\\mathcal{S}})}\\leq (\\eps\/12)$ $\\nu_{X}(P)$.\n    As such, $\\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}}) \\leq\n    (1+\\eps\/12)\\nu_{C_{\\mathrm{opt}}}(P)$ and it follows\n    \\begin{equation*}\n        \\cardin{\\nu_X({\\mathcal{S}}) -\\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}})}\n        \\leq (\\eps\/12)(1+\\eps\/12)\\nu_{C_{\\mathrm{opt}}}(P) \\leq\n        (\\eps\/6) \\nu_{C_{\\mathrm{opt}}}(P).        \n    \\end{equation*}\n    As such,\n    \\begin{align*}\n      \\nu_X(P) %\n      &\\leq%\n        \\frac{1}{1-\\eps\/12}\n        \\nu_X({\\mathcal{S}}) \\leq 2\n        \\nu_X({\\mathcal{S}})\n        \\leq 2 \\pth{ \\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}}) +\n        \\frac{\\eps}{6} \\nu_{C_{\\mathrm{opt}}}(P)}%\n      \\\\&%\n      \\leq%\n      2 \\pth{\\pth{1+ \\frac{\\eps}{12}} \\nu_{C_{\\mathrm{opt}}}(P) + \\frac{\\eps}{6}\n      \\nu_{C_{\\mathrm{opt}}}(P)}\n      \\leq%\n      3\\nu_{C_{\\mathrm{opt}}}(P),\n    \\end{align*}\n    for $\\eps < 1$. We conclude that $\\cardin{\\nu_X(P)\n       - \\nu_X({\\mathcal{S}}) } \\leq (\\eps\/12)\\nu_X(P) \\leq\n    (\\eps\/3) \\nu_{C_{\\mathrm{opt}}}(P)$.  Putting things together,\n    we have\n    \\begin{align*}\n      \\cardin{\\nu_{X}(P) - \\nu_{C_{\\mathrm{opt}}}(P) }\n      &\\leq%\n        \\cardin{\\nu_X(P) - \\nu_{X}({\\mathcal{S}}) }\n        +  \\cardin{\\nu_{X}({\\mathcal{S}}) -\n        \\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}}) }\n        +  \\cardin{\\nu_{C_{\\mathrm{opt}}}({\\mathcal{S}}) -\n        \\nu_{C_{\\mathrm{opt}}}(P) } \\\\\n      &\\leq%\n        \\pth{\\frac{\\eps}{3} + \\frac{\\eps}{6} +\n        \\frac{\\eps}{12}} \\nu_{C_{\\mathrm{opt}}}(P)\n        \\leq \\eps\\nu_{C_{\\mathrm{opt}}}(P).\n    \\end{align*}\n\\end{proof}\n\nWe are now in the position to get a fast approximation\nalgorithm.  We generate the centroid set, and then we modify\nthe algorithm of Kolliopoulos and Rao so that it considers\ncenters only from the centroid set in its dynamic\nprogramming stage.  For the weighted case, the depth of the\ntree constructed in \\cite{kr-nltas-99} is $O(\\log{W})$\ninstead of $O(\\log{n})$. Further since their algorithm works\nin expectation, we run it independently\n$O(\\log(1\/\\delta)\/\\eps)$ times to get a guarantee of\n$(1-\\delta)$.\n\n\\begin{theorem}[\\cite{kr-nltas-99}]\n    \\thmlab{kr:algo}%\n    %\n    Given a weighted point set $P$ with $n$ points in $\\Re^d$, with\n    total weight $W$, a centroid set $\\mathcal{D}$ of size at most $n$,\n    and a parameter $\\delta > 0$, one can compute\n    $(1+\\eps)$-approximate $k$-median clustering of $P$ using only\n    centers from $\\mathcal{D}$. The overall running time is\n    $O \\pth{ \\varrho}%{C_{\\mathrm{kr}} n (\\log k) (\\log W) \\log (1\/\\delta) }$, where\n    $\\varrho}%{C_{\\mathrm{kr}} = \\CkrExp$.  The algorithm succeeds with probability\n    $\\geq 1-\\delta$.\n\n\\end{theorem}\n\\remove{\n\\begin{proof}\n    We need to modify the algorithm of \\cite{kr-nltas-99} so that it\n    considers only centers from the centroid set.  This is a\n    straightforward modification of their dynamic programming stage.\n\n    We execute the algorithm of \\cite{kr-nltas-99} $M =\n    O(\\log(1\/\\delta)\/\\eps)$ times independently. Since the\n    algorithm of \\cite{kr-nltas-99} works in expectation, it\n    follows by the Markov inequality, in each such execution\n    the algorithm succeeds with probability larger $\\eps\/2$.\n    Thus, it fails with probability $\\leq (1-\\eps\/2)^M \\leq\n    \\exp(-M \\eps\/2) \\leq \\delta$.\n\n    As for the running time, the depth of the tree\n    constructed by \\cite{kr-nltas-99} is $O(\\log W)$ in this\n    case (instead of $O(\\log n)$ as in the original\n    settings). Thus, the bound on the running time follows.\n\\end{proof}\n}\n\nThe final algorithm is the following: Using the algorithms\nof \\lemref{k:coreset:small:median} and\n\\lemref{k:median:cen:set} we generate a $(k,\\eps)$-coreset\n${\\mathcal{S}}$ and an $\\eps$-centroid set $\\mathcal{D}$ of $P$,\nwhere $\\cardin{{\\mathcal{S}}} = O(k\\eps^{-d} \\log{n})$ and\n$\\cardin{\\mathcal{D}} = O(k^2\\eps^{-2d} \\log^2{n})$. Next, we\napply the algorithm of \\thmref{kr:algo} on ${\\mathcal{S}}$ and\n$\\mathcal{D}$.\n\\begin{theorem}[$(1+\\eps)$-approx $k$-median]\n    \\thmlab{fast:k:median}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, and parameter $k$, one\n    can compute a $(1+\\eps)$-approximate $k$-median clustering of $P$\n    (in the continuous sense) in\n    $O \\pth{ n + k^5 \\log^9 n + \\varrho}%{C_{\\mathrm{kr}} k^2 \\log^5 n }$ time, where\n    $\\varrho}%{C_{\\mathrm{kr}} = \\CkrExp$ and $c$ is a constant.  The algorithm outputs a\n    set $X$ of $k$ points, such that\n    $\\nu_X(P) \\leq (1+\\eps)\\nu_{\\mathrm{opt}}(P,k)$. If $P$ is a weighted set,\n    with total weight $W$, the running time of the algorithm is\n    $O ( n \\log^2 W + k^5 \\log^9 W + \\varrho}%{C_{\\mathrm{kr}} k^2 \\log^5 W )$.\n\\end{theorem}\n\nWe can extend our techniques to handle the discrete median\ncase efficiently as follows.\n\\begin{lemma}\n    \\lemlab{k:median:cen:set:discrete}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, one can compute a\n    discrete $(k,\\eps)$-centroid set $\\mathcal{D} \\subseteq P$ of size\n    $O(k^2\\eps^{-2d}\\log^2{n})$. The running time of this algorithm is\n    $ \\displaystyle O \\pth{ n + k^5 \\log^9 n + k^2\\eps^{-2d}\\log^2{n}\n    }$ if $k \\leq \\eps^d n^{1\/4}$ and $ \\displaystyle O \\pth{ n\n       \\log{n} + k^5 \\log^9 n + k^2\\eps^{-2d}\\log^2{n} }$ otherwise.\n\\end{lemma}\n\n\\begin{proof}\n    We compute a $\\eps\/4$-centroid set $\\mathcal{D}$, using\n    \\lemref{k:median:cen:set}, and let $m =\\cardin{\\mathcal{D}} =\n    O(k^2\\eps^{-2d}\\log^2{n})$. Observe that if $m > n$ then we set\n    $\\mathcal{D}$ to be $P$. Next, snap every point in $P$ to its\n    (approximate) nearest neighbor in $\\mathcal{D}$, using\n    \\corref{batch:n:n:2}.  This takes $O(n+ m n^{1\/10} \\log(n) )= O(n\n    + k^2 n^{1\/10} \\eps^{-2d} \\log^3 n) = O(n)$ time, if $k \\leq\n    \\eps^d n^{1\/4}$, and $O(n \\log n)$ otherwise (then we use the\n    data-structure of \\cite{amnsw-oaann-98} to perform the nearest\n    neighbor queries).  For every point $x\\in\\mathcal{D}$, let $P(x)$ be\n    the of points in $P$ mapped to $x$.  Pick from every set $P(x)$\n    one representative point, and let $U \\subseteq P$ be the resulting\n    set. Consider the optimal discrete center set $C_{\\mathrm{opt}}$, and\n    consider the set $X$ of representative points that corresponds to\n    the points of $C_{\\mathrm{opt}}$.  Using the same argumentation as in\n    \\lemref{k:median:cen:set} it is easy to show that $\\nu_X(P)\n    \\leq (1+\\eps)\\nu^D_{\\mathrm{opt}}(P,k)$.\n\\end{proof}\n\nCombining \\lemref{k:median:cen:set:discrete} and\n\\thmref{fast:k:median}, we get the following.\n\\begin{theorem}[Discrete k-medians]\n    \\thmlab{approx:k:median:discrete}%\n    %\n    One can compute an $(1+\\eps)$-approximate discrete $k$-median of a\n    set of $n$ points in time $\\displaystyle O \\pth{ n + k^5 \\log^9 n\n       + \\varrho}%{C_{\\mathrm{kr}} k^2 \\log^5 n }$, where $\\varrho}%{C_{\\mathrm{kr}}$ is the constant from\n    \\thmref{kr:algo}.\n\\end{theorem}\n\n\\begin{proof}\n    The proof follows from the above discussion. As for the running\n    time bound, it follows by considering separately the case when\n    $1\/\\eps^{2d} \\leq 1\/n^{1\/10}$, and the case when $1\/\\eps^{2d} \\geq\n    1\/n^{1\/10}$, and simplifying the resulting expressions. We omit\n    the easy but tedious computations.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{A $(1+\\eps)$-Approximation Algorithm for $k$-Means}\n\\seclab{eps:approx:k:mean}\n\n\n\\subsection{Constant Factor Approximation}\n\\seclab{exact:k:means:const}\n\nIn this section we reduce the number of centers to be exactly $k$.  We\nuse the set of centers computed by \\thmref{k:cluster:const:rough} to\ncompute a constant factor coreset using the algorithm of\n\\thmref{k:means:weighted:coreset}.  The resulting coreset ${\\mathcal{S}}$,\nhas size $O(k \\log^4 n)$. Next we compute a $O(n)$ approximation to\nthe $k$-means for the coreset using the $k$-center (min-max) algorithm\n\\cite{g-cmmid-85}.  Let $C_0 \\subseteq {\\mathcal{S}}$ be the resulting set\nof centers.  Next we apply the local search algorithm, due to Kanungo\n\\textit{et~al.}\\xspace{} \\cite{kmnpsw-lsaak-04}, to $C_0$ and ${\\mathcal{S}}$, where the set\nof candidate points is ${\\mathcal{S}}$. This local search algorithm, at\nevery stage, picks a center $c$ from the current set of centers\n$C_{curr}$, and a candidate center $s \\in {\\mathcal{S}}$, and swaps $c$ out\nof the set of centers and $c$ into the set of centers. Next, if the\nnew set of centers\n$C_{curr}' = C_{curr} \\setminus \\brc{c} \\cup \\brc{s}$ provides a\nconsiderable improvement over the previous solution (i.e.,\n$\\mu_{C_{curr}}({\\mathcal{S}}) \\leq\n(1-\\eps\/k)\\mu_{C_{curr}'}({\\mathcal{S}})$ where $\\eps$ here is an\narbitrary small constant), then we set $C_{curr}$ to be\n$C_{curr}'$. Extending the analysis of Arya \\textit{et~al.}\\xspace{}\n\\cite{agkmm-lshkm-04}, for the $k$-means algorithm, Kanungo \\textit{et~al.}\\xspace{}\n\\cite{kmnpsw-lsaak-04} showed that the algorithm terminates, and it\nprovides a constant factor approximation to $\\mu^D_{\\mathrm{opt}}({\\mathcal{S}}, k)$,\nand as hence to $\\mu_{\\mathrm{opt}}(P,k)$.  It is easy to verify that it stops\nafter $O(k \\log{n})$ such swaps. Every swap, in the worst case,\nrequires considering $\\cardin{{\\mathcal{S}}}k$ sets. Computing the price of\nclustering for every such candidate set of centers takes\n$O\\pth{ \\cardin{{\\mathcal{S}}} k }$ time.  Thus, the running time of this\nalgorithm is\n$O \\pth{ \\cardin{{\\mathcal{S}}}^2 k^{3} \\log n}=O \\pth{ k^5 \\log^9 n }$.\n\n\\begin{theorem}\n    \\thmlab{k:means:const:fast}%\n    %\n    Given a point set $P$ in $\\Re^d$ and parameter $k$, one can\n    compute a set $X \\subseteq P$ of size $k$, such that $\\mu_X(P)\n    =O(\\mu_{\\mathrm{opt}}(P,k))$.  The algorithm succeeds with high probability.\n    The running time is $O(n + k^5 \\log^9 n)$ time.\n\n    If $P$ is weighted, with total weight $W$, then the\n    algorithm runs in time $O ( n + k^5 \\log^4 n$ $\\log^5 W\n    )$.\n\\end{theorem}\n\n\\remove{\n\\begin{proof}\n    The unweighted case follows by the above discussion. As\n    for the weighted case, computing a set $C$ of $O(k\n    \\log^2 n \\log W)$ centers, which are constant factor\n    approximation to the $k$-means clustering of $P$, can be\n    done in $O(n \\log{n} + k \\log^3 n \\log W)$ time, using\n    \\thmref{k:cluster:const:rough}. Computing a constant\n    factor coreset from $C$, takes $O(n\\log{ \\cardin{C}}) =\n    O(n (\\log{n} + \\log\\log W))$ time, using\n    \\thmref{k:means:weighted:coreset}, and results in a\n    coreset ${\\mathcal{S}}$ of size $O(\\cardin{C} \\log W) = O(k\n    \\log^2 n \\log^2 W)$.\n\n    Finally, using the local search algorithm of\n    \\cite{kmnpsw-lsaak-04} takes\n    $O \\pth{ \\cardin{{\\mathcal{S}}}^2 k^{3} \\log W } = O \\pth{k^5 \\log^4 n\n       \\log^5 W }$ time.\n\\end{proof}\n}\n\n\n\\subsection{The $(1+\\eps)$-Approximation}\n\\seclab{exact:k:means:eps}\n\nCombining \\thmref{k:means:const:fast} and\n\\thmref{k:means:weighted:coreset}, we get the following result\nfor coresets.\n\\begin{theorem}[coreset]\n    \\thmlab{k:coreset:small:means}%\n    %\n    Given a set $P$ of $n$ points in $\\Re^d$, one can\n    compute a $k$-means $(k,\\eps)$-coreset ${\\mathcal{S}}$ of\n    $P$, of size $O\\pth{ (k\/\\eps^d)\\log{n} }$, in time $O\n    \\pth{ n + k^5 \\log^9 n }$.\n    \n    If $P$ is weighted, with total weight $W$, then the\n    coreset is of size $O\\pth{ (k\/\\eps^d)\\log{W} }$, and the\n    running time is $O(n \\log^2 W + k^5 \\log^9 W)$.\n\\end{theorem}\n\n\\begin{proof}\n    We first compute a set $A$ which provides a constant factor\n    approximation to the optimal $k$-means clustering of $P$, using\n    \\thmref{k:means:const:fast}. Next, we feed $A$ into the\n    algorithm \\thmref{k:means:weighted:coreset}, and get a\n    $(1+\\eps)$-coreset for $P$, of size $O((k\/\\eps^d) \\log{W})$.\n\\end{proof}\n\nWe now use techniques from \\Matousek{} \\cite{m-agc-00} to compute the\n$(1+\\eps)$-approximate $k$-means clustering on the coreset.\n\\begin{defn}[Centroid Set]\n    Given a set $P$ of $n$ points in $\\Re^d$, a set $T\n    \\subseteq \\Re^d$ is an \\emph{$\\eps$-approximate centroid\n       set} for $P$, if there exists a subset $C \\subseteq\n    T$ of size $k$, such that $\\mu_C(P) \\leq\n    (1+\\eps)\\mu_{\\mathrm{opt}}(P,k)$.\n\\end{defn}\n\n\\Matousek{} showed that there exists an $\\eps$-approximate centroid\nset of size $O(n\\eps^{-d} \\log(1\/\\eps))$.  Interestingly enough, his\nconstruction is weight insensitive. In particular, using an\n$(k,\\eps\/2)$-coreset ${\\mathcal{S}}$ in his construction, results in a\n$\\eps$-approximate centroid set of size $O\\pth{ \\cardin{{\\mathcal{S}}}\n   \\eps^{-d} \\log(1\/\\eps)}$.\n\n\\begin{lemma}\n    For a weighted point set $P$ in $\\Re^d$, with total\n    weight $W$, there exists an $\\eps$-approximate centroid\n    set of size $O(k\\eps^{-2d}\\log{W} \\log{(1\/\\eps)} )$.\n\\end{lemma}\n\nThe algorithm to compute the $(1+\\eps)$-approximation now follows\nnaturally. We first compute a coreset ${\\mathcal{S}}$ of $P$ of size\n$O\\pth{(k\/\\eps^d)\\log{W} }$ using the algorithm of\n\\thmref{k:coreset:small:means}. Next, we compute in\n$O\\pth{\\cardin{{\\mathcal{S}}} \\log \\cardin{{\\mathcal{S}}} + \\cardin{{\\mathcal{S}}}\n   e^{-d} \\log{\\frac{1}{\\eps}}}$ time a $\\eps$-approximate centroid\nset $U$ for ${\\mathcal{S}}$, using the algorithm from \\cite{m-agc-00}. We\nhave $\\cardin{U} = O(k\\eps^{-2d}\\log{W} \\log{(1\/\\eps)} )$. Next we\nenumerate all $k$-tuples in $U$, and compute the $k$-means clustering\nprice of each candidate center set (using ${\\mathcal{S}}$). This takes\n$O\\pth{ \\cardin{U}^k \\cdot k \\cardin{{\\mathcal{S}}}}$ time. And clearly,\nthe best tuple provides the required approximation.\n\n\\begin{theorem}[$k$-means clustering]\n    \\thmlab{k:means:approx}%\n    %\n    Given a point set $P$ in $\\Re^d$ with $n$ points, one can compute\n    $(1+\\eps)$-approximate $k$-means clustering of $P$ in time\n    \\[\n    O \\pth{n + k^5 \\log^9 n + {k^{k+2}}{\\eps^{-(2d+1)k}}\n       {\\log^{k+1}{n}} \\log^k(\\nfrac{1}{\\eps})}.\n    \\]\n    For a weighted set, with total weight $W$, the running\n    time is\n    \\[\n        O \\pth{ n \\log^2 W + k^5 \\log^4 n \\log^5 W +k^{k+2}\n           \\eps^{-(2d+1)k} \\log^{k+1} W \\log^{k} (1\/\\eps)}.\n    \\]\n\\end{theorem}\n\n\n\n\n\n\n\n\n\\section{Streaming}\n\\seclab{streaming}\n\nA consequence of our ability to compute quickly a\n$\\pth{k,\\eps}$-coreset for a point set, is that we can\nmaintain the coreset under insertions quickly.\n\n\\begin{observation}\n    (i) If $C_1$ and $C_2$ are the $(k,\\eps)$-coresets for\n    disjoint sets $P_1$ and $P_2$ respectively, then\n    $C_1\\cup C_2$ is a $(k,\\eps)$-coreset for $P_1\\cup P_2$.\n\n    (ii) If $C_1$ is $(k,\\eps)$-coreset for $C_2$, and $C_2$\n    is a $(k,\\delta)$-coreset for $C_3$, then $C_1$ is a\n    $(k,\\eps+\\delta)$-coreset for $C_3$.\n\\end{observation}\n\nThe above observation allows us to use Bentley and Saxe's technique\n\\cite{bs-dspsd-80} as follows. Let $P = \\pth{p_1,p_2,\\ldots,p_n}$ be\nthe sequence of points seen so far. We partition $P$ into sets $P_0,\nP_1,P_2,\\ldots,P_t$ such that each either $P_i$ empty or $|P_i| =\n2^{i}M$, for $i>0$ and $M = O(k\/\\eps^d)$. We refer to $i$ as the rank\nof $i$.\n\nDefine $\\rho_j = \\eps\/\\pth{c(j+1)^2}$ where c is a large\nenough constant, and $1+\\delta_j = \\prod_{l=0}^j(1+\\rho_l)$,\nfor $j=1,\\ldots, \\ceil{\\lg n}$. We store a $\\pth{k,\\delta_j\n}$-coreset $Q_{j}$ for each $P_j$.  It is easy to verify\nthat $1+\\delta_{j} \\leq 1+\\eps\/2$ for $j=1,\\ldots, \\ceil{\\lg\n   n}$ and sufficiently large $c$. Thus the union of the\n$Q_i$s is a $(k,\\eps\/2)$-coreset for $P$.\n\nOn encountering a new point $p_{u}$, the update is done in\nthe following way: We add $p_u$ to $P_0$. If $P_0$ has less\nthan $M$ elements, then we are done. Note that for $P_0$ its\ncorresponding coreset $Q_0$ is just itself. Otherwise, we\nset $Q_1' = P_0$, and we empty $Q_0$. If $Q_1$ is present,\nwe compute a $(k,\\rho_2)$ coreset to $Q_1\\cup Q'_1$ and call\nit $Q'_2$, and remove the sets $Q_1$ and $Q'_1$. We continue\nthe process until we reach a stage $r$ where $Q_r$ did not\nexist. We set $Q_r'$ to be $Q_r$. Namely, we repeatedly\nmerge sets of the same rank, reduce their size using the\ncoreset computation, and promote the resulting set to the\nnext rank. The construction ensures that $Q_r$ is a\n$(k,\\delta_r)$ coreset for a corresponding subset of $P$ of\nsize $2^r M$. It is now easy to verify, that $Q_r$ is a $(k,\n\\prod_{l=0}^j(1+\\rho_l) - 1)$-coreset for the corresponding\npoints of $P$.\n\nWe further modify the construction, by computing a\n$(k,\\eps\/6)$-coreset $R_i$ for $Q_i$, whenever we compute $Q_i$. The\ntime to do this is dominated by the time to compute $Q_i$. Clearly,\n$\\cup R_i$ is a $(k,\\eps)$-coreset for $P$ at any point in time, and\n$\\cardin {\\cup R_i} = O(k\\eps^{-d} \\log^2{n})$.\n\n\\paragraph{Streaming $k$-means}\nIn this case, the $Q_i$s are coresets for $k$-means clustering.\nSince $Q_i$ has a total weight equal to $2^i M$ (if it is not\nempty) and it is generated as a $(1+\\rho_i)$ approximation, by\n\\thmref{k:coreset:small:means}, we have that $|Q_{i}| = O\\pth{k\n\\eps^{-d} \\pth {i+1}^{2d}(i+\\log{M})}$. Thus the total storage\nrequirement is $O\\pth{\\pth{k\\log^{2d+2}{n}}\/\\eps^d}$.\n\nSpecifically, a $(k,\\rho_j)$ approximation of a subset $P_j$ of\nrank $j$ is constructed after every $2^j M$ insertions, therefore\nusing \\thmref{k:coreset:small:means} the amortized time spent for\nan update is\n\\begin{align*}\n  &\\hspace{-1cm}\\sum_{i=0}^{\\ceil{\\log{(n\/M)}}} \\frac{1}{2^i M}\n    O\\pth{\\cardin{Q_i} \\log^2 \\cardin{P_i} + k^5 \\log^9\n    \\cardin{P_i}}%\n  \\\\&%\n  =%\n  \\sum_{i=0}^{\\ceil{\\log{(n\/M)}}}  \\frac{1}{2^i M}\n  O\\pth{\\pth{ \\frac{k}{\\eps^d} i^{2d}\\pth{ i +\n  \\log M}^2 + k^5 \\pth{ i +\n  \\log M}^9 }}\n  = O \\pth{ \\log^2 (k\/\\eps) + k^5}.\n\\end{align*}\nFurther, we can generate an approximate $k$-means clustering\nfrom the $(k,\\eps)$-coresets, by using the algorithm of\n\\thmref{k:means:approx} on $\\cup_i R_i$, with $W=n$. The\nresulting running time is $O(k^5 \\log^9 n +\n{k^{k+2}}{\\eps^{-(2d+1)k}} {\\log^{k+1}{n}}\n\\log^k(\\nfrac{1}{\\eps}))$.\n\n\\paragraph{Streaming $k$-medians}\nWe use the algorithm of \\lemref{k:coreset:small:median} for\nthe coreset construction. Further we use\n\\thmref{fast:k:median} to compute an\n$(1+\\eps)$-approximation to the $k$-median from the current\ncoreset. The above discussion can be summarized as follows.\n\n\\begin{theorem}\n    \\thmlab{k:means:streaming}%\n    %\n    Given a stream $P$ of $n$ points in $\\Re^d$ and $\\eps > 0$, one\n    can maintain a $(k,\\eps)$-coresets for $k$-median and $k$-means\n    efficiently and use the coresets to compute a\n    $(1+\\eps)$-approximate $k$-means\/median for the stream seen so\n    far. The relevant complexities are:\n    \\begin{itemize}\n        \\item Space to store the information:\n        $O\\pth{k\\eps^{-d} \\log^{2d+2}{n}}$.\n\n        \\item Size and time to extract coreset of the current set:\n        $O(k\\eps^{-d} \\log^2 n)$.\n        \n        \\item Amortized update time: $O\\pth{ \\log^2 (k\/\\eps)\n           + k^5}$.\n\n        \\item Time to extract $(1+\\eps)$-approximate $k$-means clustering:\\\\\n        $O\\pth{k^5 \\log^9 n + {k^{k+2}}{\\eps^{-(2d+1)k}}\n           {\\log^{k+1}{n}} \\log^k(\\nfrac{1}{\\eps})}$.\n        \n        \\item Time to extract $(1+\\eps)$-approximate $k$-median\n        clustering:\\\\ $O\\pth{\\varrho}%{C_{\\mathrm{kr}} k \\log^7 n}$, where $\\varrho}%{C_{\\mathrm{kr}} =\n        \\CkrExp$.\n    \\end{itemize}\n\\end{theorem}\nInterestingly, once an optimization problem has a coreset, the\ncoreset can be maintained under both insertions and deletions,\nusing linear space. The following result follows in a plug and\nplay fashion from \\cite[Theorem 5.1]{ahv-aemp-04}, and we omit the\ndetails.\n\\begin{theorem}\n    Given a point set $P$ in $\\Re^d$, one can maintain a\n    $(k,\\eps)$-coreset of $P$ for $k$-median\/means, using\n    linear space, and in time $O(k \\eps^{-d} \\log^{d+2} n\n    \\log \\frac{k \\log{n}}{\\eps} + k^5 \\log^{10} n )$ per\n    insertion\/deletions.\n\\end{theorem}\n\n\n\\section{Conclusions}\n\\seclab{conclusions}\n\nIn this paper, we showed the existence of small coresets for\nthe $k$-means and $k$-median clustering.  At this point,\nthere are numerous problems for further research. In\nparticular:\n\\begin{enumerate}\n    \\item Can the running time of approximate $k$-means clustering be\n    improved to be similar to the $k$-median bounds? Can one do FPTAS\n    for $k$-median and $k$-means (in both $k$ and $1\/\\eps$)?\n    Currently, we can only compute the $(k,\\eps)$-coreset in fully\n    polynomial time, but not extracting the approximation itself from\n    it.\n\n    \\item Can the $\\log{n}$ in the bound on the size of the coreset be\n    removed?\n\n    \\item Does a coreset exist for the problem of $k$-median and\n    $k$-means in high dimensions? There are some partial relevant\n    results \\cite{bhi-accs-02}.\n\n    \\item Can one do efficiently $(1+\\eps)$-approximate streaming for\n    the discrete $k$-median case?\n    \n    \\item Recently, Piotr Indyk \\cite{i-eabca-04} showed how to\n    maintain a $(1+\\eps)$-approximation to $k$-median under insertion\n    and deletions (the number of centers he is using is roughly $O(k\n    \\log^2{\\Delta})$ where $\\Delta$ is the spread of the point set). \n    It would be interesting to see if one can extend our techniques to\n    maintain coresets also under deletions. It is clear that there is\n    a linear lower bound on the amount of space needed, if one assume\n    nothing. As such, it would be interesting to figure out what are\n    the minimal assumptions for which one can maintain\n    $(k,\\eps)$-coreset under insertions and deletions.\n\\end{enumerate}\n\n\\section*{Acknowledgments}\n\nThe authors would like to thank Piotr Indyk, Satish Rao and\nKasturi Varadarajan for useful discussions of problems\nstudied in this paper and related problems.\n\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n \\providecommand{\\CNFX}[1]{ {\\em{\\textrm{(#1)}}}}\n  \\providecommand{\\tildegen}{{\\protect\\raisebox{-0.1cm}{\\symbol{'176}\\hspace{-0.03cm}}}}\n  \\providecommand{\\SarielWWWPapersAddr}{http:\/\/sarielhp.org\/p\/}\n  \\providecommand{\\SarielWWWPapers}{http:\/\/sarielhp.org\/p\/}\n  \\providecommand{\\urlSarielPaper}[1]{\\href{\\SarielWWWPapersAddr\/#1}{\\SarielWWWPapers{}\/#1}}\n  \\providecommand{\\Badoiu}{B\\u{a}doiu}\n  \\providecommand{\\Barany}{B{\\'a}r{\\'a}ny}\n  \\providecommand{\\Bronimman}{Br{\\\"o}nnimann}  \\providecommand{\\Erdos}{Erd{\\H\n  o}s}  \\providecommand{\\Gartner}{G{\\\"a}rtner}\n  \\providecommand{\\Matousek}{Matou{\\v s}ek}\n  \\providecommand{\\Merigot}{M{\\'{}e}rigot}\n  \\providecommand{\\Hastad}{H\\r{a}stad\\xspace}\n  \\providecommand{\\CNFCCCG}{\\CNFX{CCCG}}\n  \\providecommand{\\CNFBROADNETS}{\\CNFX{BROADNETS}}\n  \\providecommand{\\CNFESA}{\\CNFX{ESA}}\n  \\providecommand{\\CNFFSTTCS}{\\CNFX{FSTTCS}}\n  \\providecommand{\\CNFIJCAI}{\\CNFX{IJCAI}}\n  \\providecommand{\\CNFINFOCOM}{\\CNFX{INFOCOM}}\n  \\providecommand{\\CNFIPCO}{\\CNFX{IPCO}}\n  \\providecommand{\\CNFISAAC}{\\CNFX{ISAAC}}\n  \\providecommand{\\CNFLICS}{\\CNFX{LICS}}\n  \\providecommand{\\CNFPODS}{\\CNFX{PODS}}\n  \\providecommand{\\CNFSWAT}{\\CNFX{SWAT}}\n  \\providecommand{\\CNFWADS}{\\CNFX{WADS}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPattern recognition is becoming more important in various fields due to the development of machine learning. Traditional pattern recognition focuses on single-task learning (STL), with multi-task learning (MTL) generally being disregarded. The MTL aims to use helpful information in several related tasks to improve the generalization performance of all tasks. Multi-task learning aims to enhance predictions by exchanging group knowledge amongst related training data sets known as ``tasks''.\nTherefore, multi-task learning is a significant area of research in machine learning. \nThe study of multi-task learning has been of interest in diverse fields, including multi-level analysis~\\cite{bakker2003task}, medical diagnosis~\\cite{bi2008improved}, semi-supervised learning~\\cite{ando2005framework},  web search ranking~\\cite{chapelle2010multi}, speech recognition~\\cite{birlutiu2010multi}, cell biology~\\cite{ren2016multicell}, person identification~\\cite{su2017multi}, drug interaction extraction~\\cite{zhou2018position},  object tracking~\\cite{cheng2015multi}, etc.\n\nSeveral MTL approaches have been proposed throughout the years, which can be classified into several groups. SVM-based methods are among such approaches. Because of the effectiveness of support vector machine (SVM) \\cite{burges1998tutorial} in multi-task learning, several researchers have concentrated on multi-task SVM \\cite{ji2013multitask,shiao2012implementation,xue2016multi,yang2010multi}. Evgeniou et al.~\\cite{evgeniou2004regularized} developed a multi-task learning strategy based on the minimization of regularization functions similar to those used in SVM.\n\nAs we know, Jeyadeva et al.~\\cite{khemchandani2007twin} proposed twin support vector machine (TSVM) for binary classification in 2007, based on the main notion of GEPSVM \\cite{mangasarian2005multisurface}. TSVM divides the positive and negative samples by producing two non-parallel hyperplanes via solving two smaller quadratic programming problems (QPP) rather than one large QPP considered in SVM. In contrast to the substantial research conducted on multi-task support vector machines, there have been few efforts to incorporate multi-task learning into twin support vector machines (TSVM). For instance, inspired by multi-task learning and TSVM, Xie and Sun proposed a multi-task twin support vector machine \\cite{xie2012multitask}. They used twin support vector machines for multi-task learning and referred to the resulting model as directed multi-task twin support vector machine (DMTSVM). Following that, multi-task centroid twin support vector machines (MCTSVM) \\cite{xie2015multitask} were suggested to cope with outlier samples in each task. In addition, motivated by least-squares twin support vector machine (LS-TWSVM) \\cite{kumar2009least}, Mei and Xu~\\cite{mei2019multi} proposed a multi-task least-squares twin support vector machine (MTLS-TWSVM). Instead of the two QPP problems addressed by DMTSVM, MTLS-TWSVM solves only two smaller linear equations, resulting in quick computation. \n\nUniversum data is defined as a set of unlabeled samples that do not belong to any class \\cite{chapelle2007analysis,vapnik200624,weston2006inference}. These data demonstrate the ability to encode past knowledge by providing meaningful information in the same domain as the problem. The Universum data have effectively improved learning performance in classification and clustering.  By incorporating the Universum data into SVM, Vapnik \\cite{vapnik2006estimation} proposed a novel model and referred to it as support vector machine with Universum ($\\mathfrak{U}$-SVM). Weston et al.~\\cite{weston2006inference} investigated this new framework (Universum data) and proved that the use of these data outperformed approaches that just used labeled samples. \n\nInspired by $\\mathfrak{U}$-SVM, Sinz et al.~\\cite{chapelle2007analysis} introduced  least-squares support vector machine with Universum data ($\\mathfrak{U}_{LS}$-SVM). Also, Zhang et al.~\\cite{zhang2008semi} proposed semi-supervised algorithms based on a graph for the learning of labeled samples, unlabeled samples, and Universum data. Various studies confirm the helpfulness of Universum data for supervised and semi-supervised learning. The training procedure incorporates Universum data, which increases the total number of samples and adds substantial computing complexity. As a result, the classical $\\mathfrak{U}$-SVM has disadvantages such as high computational complexity due to facing a larger QPP.\nFortunately, in 2012, Qi et al.~\\cite{qi2012twin} proposed a twin support vector machine using Universum data ($\\mathfrak {U}$-TSVM) that addressed this computational shortcoming. Instead of solving one large QPP, which is done in the standard $\\mathfrak {U}$-SVM algorithm, this approach solves two smaller QPPs. The authors demonstrated that the approach not only reduced the time of computation, but also outperformed the traditional $\\mathfrak{U}$-SVM in terms of classification accuracy.  Following that, Xu et al.~\\cite{xu2016least} developed  least-squares twin support vector machine using Universum data ($\\mathfrak{U}_{LS}$-TSVM) for classification. They described the way two nonparallel hyperplanes could be found by solving a pair of systems of linear equations. As a result, $\\mathfrak{U}_{LS}$-TSVM works faster than $\\mathfrak{U}$-TSVM. Inspired by $ \\nu$-TSVM and $\\mathfrak{U}$-TSVM, Xu et al.~\\cite{xu2016nu} presented a $ \\nu$-TSVM  with Universum data ($\\mathfrak{U}_{\\nu}$-TSVM).  It allows the incorporation of the prior knowledge embedded in the unlabeled samples into supervised learning to improve generalization performance. Xiao et al.~\\cite{xiao2021new} established Universum learning in 2021 to make non-target task data behave as previous knowledge, and suggested a novel multi-task support vector machine using Universum data (U-MTLSVM). In general, Universum models have been of interest to many researchers because of their simple structures and good generalization performance \\cite{moosaei2021DC,moosaei2021sparse,richhariya2018improved,richhariya2018eeg}.\n\nDespite the work done in MTL, there is still a need to create more efficient approaches in terms of accuracy and other field measures. Inspired by DMTSVM and Universum data, we present a significant multi-task twin support vector machine using Universum data ($ \\mathfrak{U} $MTSVM).\nThis paper presents two approaches to the solution of the proposed model.\nWe obtain the dual formulation of $\\mathfrak{U}$MTSVM, and try to solve the quadratic programming problems at the first approach. In addition, we propose a least-squares version of the multi-task twin support vector machine with Universum data (referred to as LS-$\\mathfrak{U}$MTSVM) to further increase the generalization performance and reduce the time of computation. The LS-$\\mathfrak{U}$MTSVM only deals with two smaller linear equations instead of the two dual quadratic programming problems which are used in $ \\mathfrak{U} $MTSVM. \n\nThe contributions of our research can be summarized as follows. \n\\begin{itemize}\n\t\\item  Using Universum data, we present a new multi-task twin support vector machine model. This model naturally extends DMTSVM by adding Universum data.\n\t\\item The proposed model has the same advantages as DMTSVM, and furthermore, improves its performance.\n\t\\item We propose two approaches to finding the solution of the proposed model, namely, solving the dual problem instead of the\n\tprimal problem, and introducing the least-squares version of $\\mathfrak{U}$MTSVM, which is called LS-$\\mathfrak{U}$MTSVM, for solving the primal problem.\n\\end{itemize}\n\nThe remainder of this paper is organized as follows. Following a quick review of\nDMTSVM and MTLS-TWSVM in Section~2, we describe the details of our proposed $\\mathfrak{U}$MTSVM and introduce its dual problem in Section~3. In Section~4, we present a new algorithm, namely, LS-$\\mathfrak{U}$MTSVM. Next, we provide some numerical experiments in Section~5. Finally, we summarize our findings in Section~6 within a brief conclusion.\n\n\\paragraph{Notation.}\nWe use $ \\mathbb{R}^n $ for  the $ n$-dimensional real vector space and $I$ for the identity matrix. The transpose and Euclidean norm of a matrix $ A $ are denoted by the symbols $A^T $ and $ \\|\\cdot\\| $, respectively.\n \n \n\n    The gradient of the function $f$ with respect to the variable $ x $ is denoted by $ \\nabla_{x} f(x)$ or simply $ \\nabla f(x)$.\n  Next, we use $\\langle x,y\\rangle =x^{T}y  $ to denote the inner product of two  $ n$-dimensional vectors $ x $ and $ y $. The symbol $blkdiag({{P}_{1}},\\ldots,{{P}_{T}})$ denotes the block-diagonal matrix created by ${{P}_{1}},\\ldots,{{P}_{T}}$ matrices.\n\n\n\\section{Related work}\nIn this section, we first give an overview of the multi-task problem. Then, we introduce the direct multi-task twin support vector machine (DMTSVM)  and the multi-task least-squares twin support vector machine (MTLS-TWSVM). It is preferable to define the fundamental foundations of these procedures since they serve as a solid basis for our suggested method.  For the multi-task problem, we have  $T$ training task, and we assume the set $S_{t}$, for $t=1,\\ldots, T$, stores  the labeled  samples for the $t$-th task. It is given by \n$$S_{t}=\\{(x_{1t}, y_{1t}),\\ldots,(x_{{n_{t}}t}, y_{{n_{t}}t}) \\}\n,$$\nwhere, $n_{t}$ is the number of samples in task $t$, $x_{it}\\in \\mathbb{R}^{n}$, $y_{it}\\in \\{\\pm 1\\},\\,\\, i=1,\\ldots,n_{t}$.\n\n\n\\subsection{Multi-task twin support vector machine}\\label{2.1} \n\nXie et al.~\\cite{xie2012multitask} introduced a new classification method that directly incorporates the regularized multi-task learning (RMTL) \\cite{evgeniou2004regularized}  concept into TSVM and called it direct multi-task twin support vector machine (DMTSVM). \n\nSuppose positive samples and positive samples in \n$t$-th task are presented by $X_{p}$ and $X_{pt}$, respectively, while $X_{n}$  represents the negative samples and negative samples in $t$-th task are presented by $X_{nt}$,  \n that is, $X_p^T=[X_{p1}\\,X_{p2}\\, \\dots X_{pT}]$.\nNow, for every task $t\\in \\{1, \\ldots, T\\}$ we define:\n\\[ A=[X_{p}\\,\\,\\, e_1],~ A_{t}=[X_{pt}\\,\\,\\, e_{1t}],~ B=[X_{n}\\,\\,\\, e_2],~ B_{t}=[X_{nt}\\,\\,\\, e_{2t}], \\]\nwhere $e_1,e_2,e_{1t}$ and $e_{2t}$ are one vectors of appropriate dimensions. Assume that all tasks have two mean hyperplanes in common, i.e., $u_{0}=[w_{1}, b_{1}]^{T}$ and $v_{0}=[w_{2}, b_{2}]^{T}$. The two hyperplanes in the $t$-th task for positive and negative classes are \n$(u_{0}+u_{t})=[w_{1t}, b_{1t}]^{T}$  and $(v_{0}+v_{t})=[w_{2t}, b_{2t}]^{T}$, respectively. The bias between task $t$  and the common mean vectors  $u_{0}$ and $v_{0}$ is represented by $u_{t}$ and $v_{t}$, respectively.\n\nThe DMTSVM optimization problems are expressed below:\n\\begin{align}\\label{1} \n\\underset{u_{0}, u_{t}, \\xi_{t}}\\min\\ \n &\\dfrac{1}{2}\\| Au_{0}\\|^{2}\n  +\\dfrac{\\mu_{1}}{2T}\\sum_{t=1}^{T} \\| A_{t}u_{t}\\|^{2}+c_{1}\\sum_{t=1}^{T}e_{2t}^{T}\\xi_{t},\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,& -B_{t}(u_{0}+u_{t})+\\xi_{t}\\geq e_{2t},\\nonumber\\\\\n&\\hspace*{2.7cm} \\xi_{t}\\geq 0,\n\\end{align}\nand\n\\begin{align}\\label{2} \n\\underset{v_{0}, v_{t}, \\eta_{t}}\\min\\ &\\dfrac{1}{2}\\| Bv_{0}\\|^{2}+\\dfrac{\\mu_{2}}{2T}\\sum_{t=1}^{T} \\| B_{t}v_{t}\\|^{2}+c_{2}\\sum_{t=1}^{T}e_{1t}^{T}\\eta_{t},\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,& A_{t}(v_{0}+v_{t})+\\eta_{t}\\geq e_{1t},\\nonumber\\\\\n&\\hspace*{2.2cm} \\eta_{t}\\geq 0.\n\\end{align}\n In problems \\eqref{1} and \\eqref{2},  $t\\in\\{1,\\ldots,T\\}$, $c_{1}$, $c_{2}$, and $e_{1t}$ and $e_{2t}$ are one vectors of appropriate dimensions.  Next, $\\mu_{1}$ and $\\mu_{2}$  are positive parameters used for correlation of all tasks.\n If  $\\mu_{1}$ and $\\mu_{2}$ give small penalty on vectors $ u_{t} $ and $ v_{t} $, then $ u_{t} $ and $ v_{t} $ tend to be larger. As a consequence, the models give less similarity.\n When $\\mu_{1}\\rightarrow \\infty$ and $\\mu_{2}\\rightarrow \\infty$, $ u_{t} $ and $ v_{t} $ tend to be smaller and make the $T$ models similar~\\cite{evgeniou2004regularized}.\n\nBy defining\n\\begin{align*}\n& Q =B{{({{A}^{T}}A)}^{-1}}{{B}^{T}}, \\quad  \n {{P}_{t}}={{B}_{t}}{{(A_{t}^{T}A_{t})}^{-1}}B_{t}^{T}, \\\\\n& \\alpha=[\\alpha_{1}^{{T}},\\ldots,\\alpha_{T}^{{T}}]^{T},\\quad\n P=blkdiag({{P}_{1}},\\ldots,{{P}_{T}}),  \n\\end{align*}\nthe dual problem of the problem \\eqref{1} may be expressed as follows:\n\\begin{align}\\label{13} \n\\underset{\\alpha}\\max&\\  -\\dfrac{1}{2}\\alpha^{T}(Q+\\textstyle\\frac{T}{\\mu_{1}}P)\\alpha\n +e_{2}^T\\alpha\\nonumber\\\\\n\\text{s.t.}\\,& \\ \\ 0\\leq\\alpha\\leq c_{1}e_{2}.\n\\end{align}\nBy resolving the aforementioned dual problem, we may discover:\n\\begin{align} \\nonumber\n& u_{0}= -(A^{T}A)^{-1}B^{T}\\alpha, \\\\  \n& u_{t}= -\\dfrac{T}{\\mu_{1}}(A^{T}_{t}A_{t})^{-1}B^{T}_{t}\\alpha_{t}. \\nonumber\n\\end{align}\n\nSimilarly, we may derive the dual problem of the problem \\eqref{2} as follows:\n\\begin{align}\\label{14} \n\\underset{\\alpha^{\\ast}}\\max&\\  -\\dfrac{1}{2}\\alpha^{\\ast^{T}}(R+\\textstyle\\frac{T}{\\mu_{2}}S)\\alpha^{\\ast}\n+e_{1}^T\\alpha^{\\ast}\\nonumber\\\\\n\\text{s.t.}\\,& \\ \\ 0\\leq\\alpha^{\\ast}\\leq c_{2}e_{1},\n\\end{align}\nwhere $\\alpha^{\\ast}=[\\alpha_{1}^{\\ast^{T}},\\ldots,\\alpha_{T}^{\\ast^{T}}]^{T}$,\n$R=A{{({{B}^{T}}B)}^{-1}}{{A}^{T}}$, and ${{S}_{t}}={{A}_{t}}{{(B_{t}^{T}{{B}_{t}})}^{-1}}A_{t}^{T}$ and $S=blkdiag({{S}_{1}},\\ldots ,{{S}_{T}})$.\nBy solving problem \\eqref{13} and \\eqref{14}, we can set the hyperplanes of every task $(u_{0}+u_{t})$ and \n$(v_{0}+v_{t})$. Meanwhile, a new data point $x$ in the $t$-th  task is determined to class $i\\in \\{+1, -1\\}$ by using the following decision function:\n\\begin{align}\\label{15} \nf(x)=\\arg \\underset{k=1,2}\\min\\, |x^{T}w_{kt}+b_{kt}|.\n\\end{align}\n\n\n\\subsection{Multi-task least squares twin support vector machine}\\label{2.2} \nInspired by DMTSVM and the least squares twin support vector machine (LSTWSVM), Mei et al.~\\cite{mei2019multi} proposed  a novel multi-task least squares twin support vector machine and called MTLS-TWSVM. \n\nThe notation of $X_{p}, X_{pt},X_{n},X_{nt}, A, A_{t}, B$ and $B_{t}$ is the same as that used in subsection \\eqref{2.1}. The MTLS-TWSVM problems are formulated as follows:\n\\begin{align}\\label{16} \n\\mathop {\\min }_{u_{0}, u_{t}, \\xi_{t}\\,}  \\, &\\dfrac{1}{2}\\| Au_{0}\\|^{2} +\\dfrac{\\mu_{1}}{2T}\\sum_{t=1}^{T} \\| A_{t}u_{t}\\|^{2}+\\dfrac{c_{1}}{2}\\sum_{t=1}^{T}\\| \\xi_{t}\\|^{2},\\nonumber\\\\\n\\text{ s.t.} ~~& -B_{t}(u_{0}+u_{t})+\\xi_{t}=e_{2t},\n\\end{align}\nand\n\\begin{align}\\label{17} \n\\mathop {\\min }_{v_{0}, v_{t}, \\eta_{t}\\,}  \\,&   \\dfrac{1}{2}\\| Bv_{0}\\|^{2} +\\dfrac{\\mu_{2}}{2T}\\sum_{t=1}^{T} \\| B_{t}v_{t}\\|^{2}+\\dfrac{c_{2}}{2}\\sum_{t=1}^{T}\\|\\eta_{t}\\|^{2},\\nonumber\\\\\n\\text{ s.t.} ~~&  A_{t}(v_{0}+v_{t})+\\eta_{t}= e_{1t},\n\\end{align}\nwhere $\\mu_{1}, c_{1},\\mu_{2}$ and $c_{2}$ are positive parameters. \nThe Lagrangian function associated with the problem \\eqref{16} is defined by\n\\begin{align}\\label{18} \nL_{1}= \\dfrac{1}{2}\\| Au_{0}\\|^{2}&+\\dfrac{\\mu_{1}}{2T}\\sum_{t=1}^{T} \\parallel A_{t}u_{t}\\parallel^{2}+\\dfrac{c_{1}}{2}\\sum_{t=1}^{T}\\| \\xi_{t}\\parallel^{2}\\nonumber\\\\\n& -\\sum_{t=1}^{T}\\alpha_{t}^{T}\\big(- B_{t}(u_{0}+u_{t})+\\xi_{t}-e_{2t}\\big),\n\\end{align}\nwhere  $\\alpha=[\\alpha_{1}^{T},\\ldots,\\alpha_{T}^{T}]^{T}$ are the Lagrangian multipliers. \nAfter writing the partial derivatives of Lagrangian function \\eqref{18} with respect to $ u_{0}$, $ u_{t} $, $ \\xi_{t} $, and $ \\alpha_{t} $,  we derive \n\\begin{equation}\\nonumber\n\\alpha = \\left(Q+\\frac{T}{\\mu_{1}}P+\\frac{1}{c_{1}}I\\right)^{-1}e_{2},\n\\end{equation}\nwhere $Q=B(A^{T}A)^{-1}B^{T}$, $P_{t}=B_{t}(A_{t}^{T}A_{t})^{-1}B^{T}_{t}$, and\n$P=blkdiag(P_{1},\\ldots,P_{T})$. \nThen we can compute  the solution of problem~\\eqref{16}:\n\\begin{align} \\nonumber\nu_{0}=-(A^{T}A)^{-1}B^{T}\\alpha,~~~\n u_{t}=-\\dfrac{T}{\\mu_{1}}(A^{T}_{t}A_{t})^{-1}B^{T}_{t}\\alpha_{t}.\n\\end{align}\n\n\n\n\n\nSimilarly, the following relations may be used to find the solution of \\eqref{17}:\n\\begin{align}\\nonumber \n\\alpha^{\\ast}=\\left(R+\\frac{T}{\\mu_{2}}S+\\frac{1}{c_{2}}I\\right)^{-1}e_{1},\n\\end{align}\nwhere $R=A(B^{T}B)^{-1}A^{T}$, $S_{t}=A_{t}(B_{t}^{T}B_{t})^{-1}A^{T}_{t}$, \n$S=blkdiag(S_{1},\\ldots,S_{T})$ and $\\alpha^{\\ast}=[\\alpha_{1}^{\\ast^{T}},\\ldots,\\alpha_{T}^{\\ast^{T}}]^{T}$.\nAs a result, the classifier parameters $u_{0}, u_{t},v_{0}$ and $v_{t}$ of the $t$-th task are determined.\n\n\n\\section{Multi-task twin support vector machine with Universum data}\nMotivated by $\\mathfrak{U}$-TSVM and DMTSVM, we would like to introduce a new multi-task model and name it as multi-task twin support vector machine with Universum data ($\\mathfrak{U}$MTSVM).\n\nFor a multi-task problem, we have $\\widetilde{T}$ training sets, and suppose that the training set $\\widetilde{T}_{t}$ for $t=1,\\ldots,T$ consists of two subsets and each task $t$ contains $n_{t}$ samples as follows\n\t$$\\widetilde{T}_{t}=S_{t}\\cup X_{\\mathfrak{U}t},$$\n\twhere\n\t\\begin{align*}\n\t& S_{t}=\\{(x_{1t}, y_{1t}),\\ldots,(x_{{n_{t}}t}, y_{{n_{t}}t}) \\},\n   \\\\\n\t& X_{\\mathfrak{U}t}=\\{x_{1t}^{\\ast},\\ldots,x_{u_{t}t}^{\\ast} \\},\n\t\\end{align*}\nwith $x_{it}\\in \\mathbb{R}^{n}$, $y_{i}\\in \\{\\pm 1\\}$ and $i=1,\\ldots,n_{t}$. \nHence, the set $S_{t}$ denotes the labeled\nsamples for the $t$-th task, and  $X_{\\mathfrak{U}t}$ contains the Universum data for the $t$-th task. For every task, we expect to build the classifier based on positive  and negative labeled samples as well as Universum data of this task.\n\n All tasks have two mean hyperplanes  $u_{0}=[w_{1},b_{1}]^{T}$  and $v_{0}=[w_{2},b_{2}]^{T}$.                  \nThe two  hyperplanes in the $t$-th task for positive and negative classes are $(u_{0}+u_{t})=[w_{1t},b_{1t}]^{T}$  and  $(v_{0}+v_{t})=[w_{2t},b_{2t}]^{T}$, respectively. \nWe employ the same notation of $X_{p},X_{pt}, X_{n},X_{nt}, A, A_{t}, B$ and  $B_{t}$  as we used in subsections~\\ref{2.1} and~\\ref{2.2}. \nIn addition, $X_{\\mathfrak{U}}$   denotes  the Universum samples, and Universum samples in $t$-th task is presented by matrix~$X_{\\mathfrak{U}t}$. Then, for every \ntask  $t\\in \\{1,\\ldots,T\\}$, we can define:\n\\[\\mathfrak{U}=[X_{\\mathfrak{U}}~e_{u}] ~\\mbox{ and } ~\\mathfrak{U}_t=[X_{\\mathfrak{U}t}~e_{ut}],\\]\nwhere   $e_{u}$  and   $e_{ut}$  are vector ones of appropriate dimensions.\n\n\\subsection{Linear case}\n In this part, we introduce the linear case of our new model ($\\mathfrak{U}$MTSVM).  Before define the $ \\delta-$insensitive loss function for Universum data, we define the hinge loss function as $H_{\\delta}[\\theta]=\\max \\lbrace 0,\\delta-\\theta \\rbrace $.\n\t We will define the $ \\delta-$insensitive loss $ U^{t}[\\theta]=H^{t}_{-\\delta}[\\theta]+H^{t}_{-\\delta}[-\\theta],~t=1,...,T $ for Universum data in each task.\n\t This loss measures the real-valued output of our classifier \n$ f_{w_{1t},b_{1t}}(x)=w_{1t}^{T}x+b_{1t} $ and $ f_{w_{2t},b_{2t}}(x)=w_{2t}^{T}x+b_{2t} $ on $X_{\\mathfrak{U}}$ and penalizes outputs that are far from zero \\cite{qi2012twin}. We then wish to minimize the total losses $ \\sum_{t=1}^{T} \\sum_{j=1}^{u_{t}} U^{t}[f_{w_{1t},b_{1t}}(x^{*}_{jt})] $ and $ \\sum_{t=1}^{T} \\sum_{j=1}^{u_{t}} U^{t}[f_{w_{2t},b_{2t}}(x^{*}_{jt})]$, and the classifiers have greater possibility when these values are less, and vice versa \\cite{zhang2008semi}. Therefore by adding the following  terms in the objective functions of DMTSVM, we introduce our new model ($\\mathfrak{U}$MTSVM):\n\\[c_{u} \\sum_{t=1}^{T} \\sum_{j=1}^{u_{t}} U^{t}[f_{w_{1t},b_{1t}}(x^{*}_{jt})], ~~\\mbox{and} ~~c^{*}_{u}\\sum_{t=1}^{T} \\sum_{j=1}^{u_{t}} U^{t}[f_{w_{2t},b_{2t}}(x^{*}_{jt})],\\]\nwhere $ c_{u}  $ and $ c^{*}_{u} $ controls the loss of  Universum data. Hence,  the optimization formulas of our proposed $\\mathfrak{U}$MTSVM  may be stated as follows:\n\n\n\n\\begin{align}\n\\mathop {\\min }_{u_{0},u_{t},\\xi_{t},\\psi _{t}\\,} \\,& \\frac{1}{2}\\|A u_{0}\\|^{2}+\\dfrac{\\mu_{1}}{2T} \\sum_{t=1}^{T}\\|A_{t}u_{t}\\|^{2}    +c_{1}\\sum_{t=1}^{T} e_{2t}^{T} \\xi _{t}+c_{u}\\sum_{t=1}^{T} e_{ut}^{T} \\psi _{t}\\nonumber \\\\\n\\text{ s.t.} ~~& -  B_{t} (u_{0}+u_{t})+{{\\xi }_{t}}\\geq e_{2t},\\nonumber \\\\ \n& \\mathfrak{U}_{t} (u_{0}+u_{t})+\\psi_{t}\\geq  (-1+\\varepsilon)e_{ut}, \\nonumber\\\\ \n& \\xi _{t} \\geq 0,  \\ \\ \n \\psi_{t}\\geq 0,\n\\label{26}\n\\end{align}\nand\n\\begin{align}\n\\mathop {\\min }_{v_{0},v_{t},\\eta_{t},\\psi^{*} _{t}\\,} \\,& \\frac{1}{2}\\|B v_{0}\\|^{2}+\\dfrac{\\mu_{2}}{2T} \\sum_{t=1}^{T}\\|B_{t}v_{t}\\|^{2}  +c_{2}\\sum_{t=1}^{T} e_{1t}^{T} \\eta_{t}+c^{*}_{u}\\sum_{t=1}^{T} e_{ut}^{T} \\psi _{t}^{*}\\nonumber  \\\\\n\\text{ s.t.}~~ &   A_{t} (v_{0}+v_{t})+{{\\eta }_{t}}\\geq e_{1t},\\nonumber \\\\ \n& -\\mathfrak{U}_{t} (v_{0}+v_{t})+\\psi^{*}_{t}\\geq  (-1+\\varepsilon)e_{ut}, \\nonumber\\\\ \n& \\eta _{t} \\geq 0,  \\ \\  \n \\psi^{*}_{t}\\geq 0,\n\\label{27}\n\\end{align}\nwhere  $c_{1}, c_{2}, c_{u}$ and  $c_{u}^{*}$  are penalty parameters.   $\\xi_{t}, \\eta_{t}$, $ \\psi_{t}=(\\psi_{1},...,\\psi_{ut})$  and $\\psi_{t}^{*}=(\\psi_{1}^{*},...,\\psi_{ut}^{*})$\nare the corresponding slack vectors.  $T$ denotes the number of task parameters, and $\\mu_{1}$ and  $\\mu_{2}$ are the positive\nparameters, which controls preference of the tasks.\n\n\n\n\n\nThe Lagrangian function associated with problem (\\ref{26}) is denoted by\n\\begin{align}\\label{28}\nL_{1}= & \\frac{1}{2}\\|A u_{0}\\|^{2}+\\dfrac{\\mu_{1}}{2T} \\sum_{t=1}^{T}\\|A_{t}u_{t}\\|^{2}    +c_{1}\\sum_{t=1}^{T} e_{2t}^{T} \\xi _{t}\n +c_{u}\\sum_{t=1}^{T} e_{ut}^{T} \\psi _{t}-\\sum_{t=1}^{T} \\alpha_{1t}^{T}(-B_{t}(u_{0}+u_{t})+\\xi_{t}\\nonumber \\\\\n& -e_{2t})- \\sum_{t=1}^{T} \\beta_{1t}^{T}\\xi_{t}-\\sum_{t=1}^{T} \\alpha_{2t}^{T}(\\mathfrak{U}_{t}(u_{0}+u_{t})+\\psi_{t}\n -(-1+\\varepsilon)e_{ut})- \\sum_{t=1}^{T} \\beta_{2t}^{T}\\psi_{t},\n\\end{align}\nwhere $\\alpha_{1t},\\alpha_{2t}, \\beta_{1t}$ and $\\beta_{2t}$ are the  Lagrange multipliers.\nThus, denoting $\\alpha_1=[\\alpha_{11}^{T},\\ldots, \\alpha_{1T}^{T}]^{T}$ and $\\alpha_2=[\\alpha_{21}^{T},\\ldots, \\alpha_{2T}^{T}]^{T}$, the KKT necessary and sufficient optimality conditions for \\eqref{26} are given by\n\n\n\\begin{align}\n\\dfrac{\\partial L_{1}}{\\partial u_{0}}& =A^{T}Au_{0}+B^{T}\\alpha_{1}-\\mathfrak{U}^{T}\\alpha_{2}=0,\\label{29} \\\\\n\\dfrac{\\partial L_{1}}{\\partial u_{t}}& =\\dfrac{\\mu_{1}}{T}A_{t}^{T}A_{t}u_{t}+B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t}=0,\\label{30} \\\\\n\\dfrac{\\partial L_{1}}{\\partial \\xi_{t}}& =c_{1}e_{2}-\\alpha_{1}-\\beta_{1}=0,\\label{31} \\\\\n\\dfrac{\\partial L_{1}}{\\partial \\psi_{t}}& =c_{u}e_{u}-\\alpha_{2}-\\beta_{2}=0.\\label{32} \n\\end{align}\nSince  $\\beta_{1}\\geq 0$  and  $\\beta_{2}\\geq 0$, from (\\ref{31}) and (\\ref{32}), we have \n\\begin{align} \\nonumber                                                     0\\leq \\alpha_{1}\\leq c_{1}e_{2}, ~~~\n0\\leq \\alpha_{2}\\leq c_{u}e_{u}. \\nonumber\n\\end{align}\nAlso, from the equations (\\ref{29}) and  (\\ref{30}), we have \n\\begin{align} \\nonumber                                                      u_{0}&=-(A^{T}A)^{-1} (B^{T}\\alpha_{1}-\\mathfrak{U}^{T}\\alpha_{2}), \\nonumber \\\\ \nu_{t}&=-\\dfrac{T}{\\mu_{1}}(A_{t}^{T}A_{t})^{-1} (B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t}). \\nonumber\n\\end{align}\nThen, substituting  $u_{0}$  and  $u_{t}$  into (\\ref{28}) :\n\\begin{align*}                                                           L_{1}={}&\\dfrac{1}{2}(\\alpha^{T}_{1}B -\\alpha_{2}^{T}\\mathfrak{U})(A^{T}A)^{-1} (B^{T}\\alpha_{1}-\\mathfrak{U}^{T}\\alpha_{2})  \\\\\n&+\\frac{T}{ 2\\mu_{1}}\\sum_{t=1}^{T}(\\alpha_{1t}^{T}B_{t}- \\alpha_{2t}^{T}\\mathfrak{U}_{t}) (A_{t}^{T}A_{t})^{-1}(B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t})\\\\\n& -\\frac{T}{ \\mu_{1}}\\sum_{t=1}^{T}\\alpha_{1t}^{T}B_{t}  (A_{t}^{T}A_{t})^{-1} (B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t})\\\\\n&-\\sum_{t=1}^{T}\\alpha_{2t}^{T}\\mathfrak{U}_{t}(A_{t}^{T}A_{t})^{-1} (B ^{T}\\alpha_{1}-\\mathfrak{U}^{T}\\alpha_{2})\\\\\n& -\\dfrac{T}{ \\mu_{1}}  \\sum_{t=1}^{T}\\alpha_{1t}^{T}\\mathfrak{U}_{t}(A_{t}^{T}A_{t})^{-1} (B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t})\\\\\n&=-\\dfrac{1}{2}(\\alpha^{T}_{1}B -\\alpha_{2}^{T}\\mathfrak{U})(A^{T}A)^{-1} (B^{T}\\alpha_{1}-\\mathfrak{U}^{T}\\alpha_{2})\\\\\n&-\\dfrac{T}{ 2\\mu_{1}}  \\sum_{t=1}^{T}(\\alpha_{1t}^{T}B_{t}- \\alpha_{2t}^{T}\\mathfrak{U}_{t}) (A_{t}^{T}A_{t})^{-1} (B_{t}^{T}\\alpha_{1t}-\\mathfrak{U}_{t}^{T}\\alpha_{2t}).\n\\end{align*}\nDefining\n\\begin{align*}\nQ& =\\left[ \\begin{matrix}\nB\\\\\n\\mathfrak{U}\n\\end{matrix}\\right] (A^{T}A)^{-1}\\left[ \\begin{matrix}\nB^{T}&\\mathfrak{U}^{T}\n\\end{matrix}\\right],\\\\\nP_{t}& =\\left[ \\begin{matrix}\nB_{t}\\\\\n\\mathfrak{U}_{t}\n\\end{matrix}\\right] (A^{T}A)^{-1}\\left[ \\begin{matrix}\nB_{t}^{T}&\\mathfrak{U}^{T}_{t}\n\\end{matrix}\\right],\\\\\nP&= blkdiag \\,(P_{1},\\ldots,P_{T}),\n\\end{align*}\nthe dual problem of (\\ref{26}) may be expressed as \n\\begin{align}\\label{37}\n\\mathop {\\max }_{\\alpha_{1}, \\alpha_{2}} \\,& -\\frac{1}{2}\\left[ \\alpha_{1}^{T}, \\alpha_{2}^{T}\\right] \\left( Q+\\dfrac{T}{\\mu_{1}}P\\right)     \\left[ \\begin{matrix}\n\\alpha_{1}\\\\\n\\alpha_{2}\n\\end{matrix}\\right]\n+\\left[ \\alpha_{1}^{T}, \\alpha_{2}^{T}\\right]  \\left[ \\begin{matrix}\ne_{2}\\\\\n(-1+\\varepsilon)e_{u}\n\\end{matrix}\\right]\\nonumber\\\\\n\\text{s.t.} \\,\\,\\,\\,\\,& 0\\leq\\alpha_{1}\\leq c_{1}e_{2},\\nonumber \\\\ \n& 0\\leq\\alpha_{2}\\leq c_{u}e_{u}.\n\\end{align}\nSimilarly, by introducing \n\\begin{align*}\n&R =\\left[ \\begin{matrix}\nA\\\\\n\\mathfrak{U}\n\\end{matrix}\\right] (B^{T}B)^{-1}\\left[ \\begin{matrix}\nA^{T}&\\mathfrak{U}^{T}\n\\end{matrix}\\right], ~~\nS_{t} =\\left[ \\begin{matrix}\nA_{t}\\\\\n\\mathfrak{U}_{t}\n\\end{matrix}\\right] (A_{t}^{T}A_{t})^{-1}\\left[ \\begin{matrix}\nA_{t}^{T}&\\mathfrak{U}^{T}_{t}\n\\end{matrix}\\right],\\\\\n&S= blkdiag\\,(S_{1},\\ldots, S_{T}),~~\n\\alpha^{*}_{1}=\\big[(\\alpha_{11}^{\\ast})^T,\\ldots,(\\alpha_{1T}^{\\ast})^T\\big]^{T},~~ \n\\alpha^{*}_{2}=\\big[(\\alpha_{21}^{\\ast})^T,\\ldots,(\\alpha_{2T}^{\\ast})^T\\big]^{T},\n\\end{align*}\nthe dual problem of (\\ref{27}) can be obtained as \n\\begin{align}\\label{38}\n\\mathop {\\max }_{\\alpha^{*}_{1}, \\alpha^{*}_{2}} \\,& -\\frac{1}{2}\\left[ \\alpha_{1}^{*T}, \\alpha_{2}^{*T}\\right] \n\\left( R+\\dfrac{T}{\\mu_{2}}S\\right)   \n\\left[ \\begin{matrix}\n\\alpha^{*}_{1}\\\\\n\\alpha^{*}_{2}\n\\end{matrix}\\right]\n+\\left[ \\alpha_{1}^{*T}, \\alpha_{2}^{*T}\\right]  \\left[ \\begin{matrix}\ne_{1}\\\\\n(-1+\\varepsilon)e_{u}\n\\end{matrix}\\right]\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,\\, & 0\\leq\\alpha^{*}_{1}\\leq c_{2}e_{1},\\nonumber \\\\ \n& 0\\leq\\alpha^{*}_{2}\\leq c^{*}_{u}e_{u}.\n\\end{align}\nBy solving problems (\\ref{37}) and (\\ref{38}), we find the  $\\alpha_{1}, \\alpha_{2}, \\alpha_{1}^{*}$ and  $\\alpha_{2}^{*}$, and then the classifiers parameters  $u_{0}, u_{t}, v_{0}$ and  $v_{t}$   of the  $t$-th  task can be obtained.\nThe label of a new sample   $x\\in \\mathbb{R}^{n}$  is  determined by (\\ref{15}).\nA linear $\\mathfrak{U}$MTSVM can be obtained by the steps Algorithm~\\ref{A1}.\n\\begin{algorithm} [t] \n\t\\caption{\\label{A1} A linear  multi-task twin support vector machine with Universum ($\\mathfrak{U}$MTSVM) }\n\n\t\\algsetup{linenosize=\\normalsize}\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\begin{algorithmic}[1]\n\t\t\\normalsize\n\t\t\\REQUIRE{\\mbox{}\\\\-- The training set $\\tilde{T}$ and Universum data $ X_{\\mathfrak{U}} $;\\\\\n\t\t\t-- Decide on the total number of tasks included in the data set and assign this value to T;\\\\\n\t\t\t-- Select classification task $ S_{t}~(t=1,\\ldots,T) $ in training  data set $\\tilde{T}$;\\\\\n\t\t\t-- Divide Universum data $X_{\\mathfrak{U}} $ by $t$-task and get $ X_{\\mathfrak{U}t}~ (t=1,\\ldots,T)$;\\\\\n\t\t\t-- Choose appropriate parameters\n\t\t\t$c_{1}, c_{2},c_{u}$, $c_{u}^{*}$, $ \\mu_{1} $, $ \\mu_{2} $, and  parameter $\\varepsilon \\in (0,1)$.}\\\\\n\t\t{\\textbf{The outputs:}}\n\t\t\\begin{list}{--}{}\n\t\t\t\\item $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\end{list}\n\t\t\n\t\t{\\textbf{The process:}}\n\t\t\n\t\t\\STATE\n\t\tSolve the optimization problems (16) and (17), and get $ \\alpha_{1},~\\alpha_{2},~\\alpha^{*}_{1}$, and $\\alpha^{*}_{2}. $\n\t\t\\STATE\n\t\tCalculate  $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\STATE\n\t\tBy utilizing the decision function (\\ref{15}), assign a new point $ x $ in the $t$-th task to class $ +1 $ or $ -1 $.\n\t\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Nonlinear case}\n\nIt is understandable that a linear classifier would not be appropriate for training data that are linearly inseparable. To deal with such issues, employ the kernel technique. To that end, we introduce the kernel function $K(\\cdot,\\cdot)$ and define\n $D=\\left[ A_{1}^{T}, B_{1}^{T}, A_{2}^{T}, B_{2}^{T},\\ldots, A_{T}^{T},B_{T}^{T}\\right] ^{T}$, $\\overline{A}=\\left[ K(A, D^{T}),e_{1}\\right] $, $\\overline{A}_{t}=\\left[ K(A_{t}, D^{T}),e_{1t}\\right] $, $\\overline{B}=[K(B, D^{T}),e_{2}]$, $\\overline{B}_{t}=\\left[ K(B_{t}, D^{T}),e_{2t}\\right] $, $\\overline{\\mathfrak{U}}=\\left[ K(X_{\\mathfrak{U}}, D^{T}),e_{u}\\right] $ and $\\overline{\\mathfrak{U}}_{t}=\\left[ K(X_{\\mathfrak{U}t}, D^{T}),e_{ut}\\right] $. Hence, the nonlinear formulations for   $\\mathfrak{U}$MTSVM  are defined as follow:\n\\begin{align}\n\\mathop {\\min }_{u_{0},u_{t},\\xi_{t},\\psi _{t}\\,} \\,& \\frac{1}{2}\\|\\bar{A} u_{0}\\|^{2}+\\frac{\\mu_{1}}{2T} \\sum_{t=1}^{T}\\|\\bar{A}_{t}u_{t}\\|^{2}    +c_{1}\\sum_{t=1}^{T} e_{2t}^{T} \\xi _{t}\n+c_{u}\\sum_{t=1}^{T} e_{ut}^{T} \\psi _{t}\\nonumber \\\\\n\\text{s.t.} ~~& -  \\overline{B}_{t} (u_{0}+u_{t})+{{\\xi }_{t}}\\geq e_{2t},\\nonumber \\\\ \n& \\bar{\\mathfrak{U}}_{t} (u_{0}+u_{t})+\\psi_{t}\\geq  (-1+\\varepsilon)e_{ut}, \\nonumber\\\\ \n& \\xi _{t} \\geq 0, \\ \\ \n  \\psi_{t}\\geq 0,\n\\label{40}\n\\end{align}\nand\n\\begin{align}\n\\mathop {\\min }_{v_{0},v_{t},\\eta_{t},\\psi^{*} _{t}\\,} \\,& \\frac{1}{2}\\|\\bar{B} v_{0}\\|^{2}+\\frac{\\mu_{2}}{2T} \\sum_{t=1}^{T}\\|\\bar{B}_{t}v_{t}\\|^{2}    +c_{2}\\sum_{t=1}^{T} e_{1t}^{T} \\eta _{t}\n+c^{*}_{u}\\sum_{t=1}^{T} e_{ut}^{T} \\psi _{t}\\nonumber  \\\\\n\\text{s.t.}  ~~&   \\bar{A}_{t} (v_{0}+v_{t})+{{\\eta }_{t}}\\geq e_{1t},\\nonumber \\\\ \n& -\\overline{\\mathfrak{U}}_{t} (v_{0}+v_{t})+\\psi^{*}_{t}\\geq  (-1+\\varepsilon)e_{ut}, \\nonumber\\\\ \n& \\eta _{t} \\geq 0, \\ \\ \n  \\psi^{*}_{t}\\geq 0,\n\\label{41}\n\\end{align}\nhere $c_{1}$, $c_{2}$, $c_{u}$ and  $c_{u}^{*}$   are penalty parameters, and\n$\\xi_{t}$, $\\eta_{t}$, $\\psi_{t}$  and $\\psi_{t}^{*}$\nare the corresponding slack vectors. By $T$ we denote the number of task parameters, and $\\mu_{1}$ and  $\\mu_{2}$ are the positive\nparameters, which control preference of the tasks.\nAfter using the Lagrange multipliers and KKT conditions, the duals of problems (\\ref{40}) and (\\ref{41}) read as follows:\n\\begin{align}\\label{42}\n\\mathop {\\max }_{\\alpha_{1}, \\alpha_{2}} \\,& -\\frac{1}{2}\\left[ \\alpha_{1}^{T}, \\alpha_{2}^{T}\\right]\n\\left( Q+\\dfrac{T}{\\mu_{1}}P\\right)     \\left[ \\begin{matrix}\n\\alpha_{1}\\\\\n\\alpha_{2}\n\\end{matrix}\\right]\n+\\left[ \\alpha_{1}^{T}, \\alpha_{2}^{T}\\right]  \\left[ \\begin{matrix}\ne_{2}\\\\\n(-1+\\varepsilon)e_{u}\n\\end{matrix}\\right]\\nonumber \\\\\n\\text{s.t.} \\,\\,\\,\\,\\, &  0\\leq\\alpha_{1}\\leq c_{1}e_{2},\\nonumber \\\\ \n& 0\\leq\\alpha_{2}\\leq c_{u}e_{u},\n\\end{align}\nand\n\\begin{align}\\label{43}\n\\mathop {\\max }_{\\alpha^{*}_{1}, \\alpha^{*}_{2}} \\,& -\\frac{1}{2}\\left[ \\alpha_{1}^{*T}, \\alpha_{2}^{*T}\\right] \n\\left( R+\\dfrac{T}{\\mu_{2}}S\\right)     \\left[ \\begin{matrix}\n\\alpha^{*}_{1}\\\\\n\\alpha^{*}_{2}\n\\end{matrix}\\right]\n+\\left[ \\alpha_{1}^{*T}, \\alpha_{2}^{*T}\\right]  \\left[ \\begin{matrix}\ne_{1}\\\\\n(-1+\\varepsilon)e_{u}\n\\end{matrix}\\right]\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,\\, & 0\\leq\\alpha^{*}_{1}\\leq c_{2}e_{1},\\nonumber \\\\ \n& 0\\leq\\alpha^{*}_{2}\\leq c^{*}_{u}e_{u},\n\\end{align}\nwhere\n\\begin{align*}\nQ& =\\left[ \\begin{matrix}\n\\bar{B}\\\\\n\\bar{ \\mathfrak{U}}\n\\end{matrix}\\right] ( \\bar{A}^{T} \\bar{A})^{-1}\\left[ \\begin{matrix}\n\\bar{B}^{T}& \\bar{ \\mathfrak{U}}^{T}\n\\end{matrix}\\right],~~\nP_{t} =\\left[ \\begin{matrix}\n\\bar{B}_{t}\\\\\n\\bar{ \\mathfrak{U}}_{t}\n\\end{matrix}\\right] ( \\bar{A}_{t}^{T} \\bar{A}_{t})^{-1}\\left[ \\begin{matrix}\n\\bar{B}_{t}^{T}& \\bar{ \\mathfrak{U}}^{T}_{t}\n\\end{matrix}\\right],\\\\\nP&=blkdiag \\,(P_{1},\\ldots, P_{T}),~~\nR =\\left[ \\begin{matrix}\n\\bar{A}\\\\\n\\bar{ \\mathfrak{U}}\n\\end{matrix}\\right] (\\bar{B}^{T}\\bar{B})^{-1}\\left[ \\begin{matrix}\n\\bar{A}^{T}& \\bar{ \\mathfrak{U}}^{T}\n\\end{matrix}\\right],\\\\\nS_{t}& =\\left[ \\begin{matrix}\n\\bar{A}_{t}\\\\\n\\bar{ \\mathfrak{U}}_{t}\n\\end{matrix}\\right] (\\bar{A}_{t}^{T}\\bar{A}_{t})^{-1}\\left[ \\begin{matrix}\n\\bar{A}_{t}^{T}& \\bar{ \\mathfrak{U}}^{T}_{t}\n\\end{matrix}\\right], ~~\nS= blkdiag \\,(S_{1},\\ldots, S_{T}).\n\\end{align*}\n\n\nA new data point $x$ in the $t$-th  task is determined to class $i\\in \\{+1, -1\\}$ by using the following decision function:\n\\begin{align}\\label{n15} \nf(x)=\\arg \\underset{k=1,2}\\min\\, \\big|K\\big(x,D^{T}\\big)w_{kt}+b_{kt}\\big|.\n\\end{align}\nThe nonlinear $\\mathfrak{U}$MTSVM is described in the steps of Algorithm~\\ref{A2}.\n\\begin{algorithm} [t] \n\t\\caption{\\label{A2} A nonlinear  multi-task twin support vector machine with Universum ($\\mathfrak{U}$MTSVM) }\n\n\t\\algsetup{linenosize=\\normalsize}\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\begin{algorithmic}[1]\n\t\t\\normalsize\n\t\t\\REQUIRE{\\mbox{}\\\\-- The training set $\\tilde{T}$ and Universum data $X_{\\mathfrak{U}} $;\\\\\n\t\t\t-- Decide on the total number of tasks included in the data set and assign this value to T;\\\\\n\t\t\t-- Select classification task $ S_{t}~(t=1,\\ldots,T) $ in training  data set $\\tilde{T}$;\\\\\n\t\t\t-- Divide Universum data $X_{\\mathfrak{U}} $ by $t$-task and get $ X_{\\mathfrak{U}t}~ (t=1,\\ldots,T)$;\\\\\n\t\t\t-- Choose appropriate parameters\n\t\t\t$c_{1}, c_{2},c_{u}$, $c_{u}^{*}$, $ \\mu_{1} $, $ \\mu_{2} $, and  parameter $\\varepsilon \\in (0,1)$.\\\\\n\t\t-- Select proper kernel function and kernel parameter.\n\t}\\\\\n\t\t{\\textbf{The outputs:}}\n\t\t\\begin{list}{--}{}\n\t\t\t\\item $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\end{list}\n\t\t\n\t\t{\\textbf{The process:}}\n\t\t\n\t\t\\STATE\n\t\tSolve the optimization problems (20) and (21), and get $ \\alpha_{1},~\\alpha_{2},~\\alpha^{*}_{1}$, and $\\alpha^{*}_{2}. $\n\t\t\\STATE\n\t\tCalculate  $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\STATE\n\t\tAssign a new point $ x $ in the $t$-th task to class $ +1 $ or $ -1 $ by using decision function~(\\ref{n15}).\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Least squares multi-task twin support vector machine with Universum data}\nIn this section, we introduce the least-squares version of $\\mathfrak{U}$MTSVM for the linear and nonlinear cases, to which we refer as  least-squares multi-task twin support vector machine with Universum data (LS-$\\mathfrak{U}$MTSVM). Our proposed method combines the advantages of DMTSVM, $ \\mathfrak{U}_{LS} $-TSVM, and MTLS-TWSVM. In terms of generalization performance, the proposed method is superior to MTLS-TWSVM, because it improves prediction accuracy by absorbing previously embedded knowledge embedded in the Universum data. In terms of the time of computation, LS-$\\mathfrak{U}$MTSVM works faster than DMTSVM by solving two systems of linear equations instead of two quadratic programming problems.\n\n\\subsection{Linear case}\nWe modify problems (\\ref{26}) and \n(\\ref{27}) in the least  squares  sense and  replace the inequality constraint with equality requirements as follows\n\\begin{align}\\label{44}\n\\mathop {\\min }_{u_{0},u_{t},\\xi_{t},\\psi _{t}\\,} \\,& \\frac{1}{2}\\|A u_{0}\\|^{2}+\\dfrac{\\mu_{1}}{2T} \\sum_{t=1}^{T}\\|A_{t}u_{t}\\|^{2}    +\\dfrac{c_{1}}{2}\\sum_{t=1}^{T} \\| \\xi _{t}\\|^{2}\n+\\dfrac{c_{u}}{2}\\sum_{t=1}^{T} \\|\\psi _{t}\\|^{2}\\nonumber \\\\\n\\text{s.t.} ~~& -  B_{t} (u_{0}+u_{t})+{{\\xi }_{t}}= e_{2t},\\nonumber \\\\ \n& \\mathfrak{U}_{t} (u_{0}+u_{t})+\\psi_{t}=  (-1+\\varepsilon)e_{ut}, \n\\end{align}\nand\n\\begin{align}\\label{45}\n\\mathop {\\min }_{v_{0},v_{t},\\eta_{t},\\psi^{*} _{t}\\,} \\,& \\frac{1}{2}\\|B v_{0}\\|^{2}+\\dfrac{\\mu_{2}}{2T} \\sum_{t=1}^{T}\\|B_{t}v_{t}\\|^{2}    +\\dfrac{c_{2}}{2}\\sum_{t=1}^{T} \\| \\eta_{t}\\|^{2}\n+\\dfrac{c^{*}_{u}}{2}\\sum_{t=1}^{T}\\|\\psi _{t}^{*}\\|^{2}\\nonumber  \\\\\n\\text{s.t.}~~ &   A_{t} (v_{0}+v_{t})+{{\\eta }_{t}}= e_{1t},\\nonumber \\\\ \n& -\\mathfrak{U}_{t} (v_{0}+v_{t})+\\psi^{*}_{t}=  (-1+\\varepsilon)e_{ut}.\n\\end{align}\nHere, $c_{1},c_{2},c_{u}$,  and   $c_{u}^{\\ast}$   are penalty parameters,   $\\xi_{t}, \\eta_{t}, \\psi_{t}$  and  $\\psi_{t}^{\\ast}$   are slack variables for $t$-th task and  $e_{1t}, e_{2t}$,  and  $e_{ut}$  are vectors of appropriate dimensions whose all components are equal  to $1$. \n\n\nIt is worth noting that the loss functions in \\eqref{44} and \\eqref{45} are the square of the 2-norm of the slack variables $\\psi$ and $\\psi^{*}$ rather than the 1-norm  in problems (\\ref{26}) and (\\ref{27}), which renders the constraints $\\psi_{t}\\geq 0$ and $\\psi^{*}_{t}\\geq 0$ superfluous.\n\nThe Lagrangian function for the problem \\eqref{44} can be written as follows: \n\\begin{align}\\label{46} \nL_{1} ={}& \\dfrac{1}{2}\\| Au_{0}\\|^{2}+\\frac{\\mu_{1}}{2T}\\sum_{t=1}^{T} \\| A_{t}u_{t}\\|^{2}+\\frac{c_{1}}{2}\\sum_{t=1}^{T}\\| \\xi_{t}\\|^{2}\\nonumber\\\\\n& -\\sum_{t=1}^{T}\\alpha_{t}^{T}(- B_{t}(u_{0}+u_{t})+\\xi_{t}-e_{2t})+\\dfrac{c_{u}}{2}\\sum_{t=1}^{T}\\| \\psi_{t}\\|^{2}\\nonumber\\\\\n& -\\sum_{t=1}^{T}\\beta_{t}^{T}(\\mathfrak{U}_{t}(u_{0}+u_{t})+\\psi_{t}-(-1+\\varepsilon)e_{ut}),\n\\end{align}\nwhere  $\\alpha_{t}$   and  $\\beta_{t}$   are the Lagrange multipliers.\nThe Lagrangian function \\eqref{46} is differentiable and the KKT optimally conditions can be obtained as follows:\n\\begin{align}\n&\\dfrac{\\partial L}{\\partial u_{0}}=A^{T}Au_{0}+B^{T}\\alpha -\\mathfrak{U}^{T} \\beta =0,\\label{47} \\\\\n&\\dfrac{\\partial L}{\\partial u_{t}}=\\dfrac{\\mu_{1}}{T}A^{T}_{t}A_{t}u_{t}+B^{T}_{t}\\alpha_{t} - \\mathfrak{U}^{T}_{t} \\beta_{t}=0,\\label{48} \\\\\n&\\dfrac{\\partial L}{\\partial \\xi_{t}}=c_{1}\\xi_{t}-\\alpha_{t} =0, \\label{49} \\\\\n&\\dfrac{\\partial L}{\\partial \\psi_{t}}=c_{u}\\psi_{t}-\\beta_{t} =0, \\label{50} \\\\\n&\\dfrac{\\partial L_{1}}{\\partial \\alpha_{t}}=B_{t}(u_{0}+u_{t})-\\xi_{t}+e_{2t}=0,\\label{51} \\\\\n&\\dfrac{\\partial L_{1}}{\\partial \\beta_{t}}=-\\mathfrak{U}_{t}(u_{0}+u_{t})-\\psi_{t}+(-1+\\varepsilon)e_{ut}=0.\\label{52} \n\\end{align}\nFrom equations \\eqref{47}--\\eqref{50}, we derive\n\\begin{align}\n& u_{0}= -(A^{T}A)^{-1}(B^{T}\\alpha -\\mathfrak{U}\\beta), \\label{53}\\\\ \n& u_{t}= -\\dfrac{T}{\\mu_{1}}(A^{T}_{t}A_{t})^{-1}(B^{T}_{t}\\alpha_{t}-\\mathfrak{U}_{t}^{T}\\beta_{t}), \\label{54}\\\\ \n& \\xi_{t}=\\dfrac{\\alpha_{t}}{c_{1}},\\label{55} \\\\\n& \\psi_{t}=\\dfrac{\\beta_{t}}{c_{u}}.\\label{56} \n\\end{align}\nBy substituting  $u_{0}$,  $u_{t}$,  $\\xi_{t}$  and  $\\psi_{t}$  into the equations  \\eqref{51} and \\eqref{52}, we have \n\\begin{align}\\label{57} \n{{B}_{t}}\\left[ -{{({{A}^{T}}A)}^{-1}}({B}^{T}\\alpha - \\mathfrak{U}^{T}\\beta)\n-\\frac{T}{{{\\mu }_{1}}}{{({{A}^{T}_{t}}A_{t})}^{-1}}(B_{t}^{T}{{\\alpha }_{t}}\n-\\mathfrak{U}_{t}^{T}\\beta_{t}) \\right]\n-\\frac{{{\\alpha }_{t}}}{{{c}_{1}}}\n&=  -{{e}_{2t}},\n\\\\%\\end{align}\n\\label{58} \n{ -\\mathfrak{U}_{t}}\\left[ -{{({{A}^{T}}A)}^{-1}}({B}^{T}\\alpha - \\mathfrak{U}^{T}\\beta)\n -\\frac{T}{{{\\mu }_{1}}}{{({{A}^{T}_{t}}A_{t})}^{-1}}(B_{t}^{T}{{\\alpha }_{t}}-\\mathfrak{U}_{t}^{T}\\beta_{t}) \\right]-\\frac{{{\\beta }_{t}}}{{{c}_{u}}}\n &=-(-1 + \\varepsilon){{e}_{ut}},\n\\end{align}\nwhere   $t\\in \\{1, \\ldots, T\\}$, $\\alpha=[\\alpha_{1}^{T},\\ldots, \\alpha_{T}^{T}]^{T}$  and $\\beta=[\\beta_{1}^{T},\\ldots, \\beta_{T}^{T}]^{T}$. Here, we define \n\\begin{align*}\n& Q_{1} =B{{({{A}^{T}}A)}^{-1}}{{B}^{T}}, ~~\n Q_{2} =B{{({{A}^{T}}A)}^{-1}}{{\\mathfrak{U}}^{T}}, ~~\nS_{1} =\\mathfrak{U}{{({{A}^{T}}A)}^{-1}}{{B}^{T}}, ~~ S_{2} =\\mathfrak{U}{{({{A}^{T}}A)}^{-1}}{{\\mathfrak{U}}^{T}}, \n\\\\\n& {{P}_{1t}}={{B}_{t}}{{(A_{t}^{T}A_{t})}^{-1}}B_{t}^{T}, ~~ {{P}_{2t}}={{B}_{t}}{{(A_{t}^{T}A_{t})}^{-1}}\\mathfrak{U}_{t}^{T},~~ P_{1}=blkdiag({{P}_{11}},\\ldots ,{{P}_{1T}}),\n\\\\\n& P_{2}=blkdiag({{P}_{21}},\\ldots ,{{P}_{2T}}),~~ R_{1t}=\\mathfrak{U}_{t}{{({{A}^{T}_{t}}A_{t})}^{-1}}{{B}^{T}_{t}},~~ R_{2t}=\\mathfrak{U}_{t}{{({{A}^{T}_{t}}A_{t})}^{-1}}{{\\mathfrak{U}}^{T}_{t}}, \n\\\\\n& R_{1}=blkdiag({{R}_{11}},\\ldots ,{{R}_{1T}}),~~ R_{2}=blkdiag({{R}_{21}},\\ldots ,{{R}_{2T}}).\n\\end{align*}\nThen \\eqref{57} and \\eqref{58} can be converted to the following equations:\n\\begin{align}\n& \\left( {{Q}_{1}}\\alpha -{{Q }_{2}}\\beta  \\right)+\\frac{T}{{{\\mu }_{1}}}\\left( {{P}_{1}}\\alpha -{{P}_{2}}\\beta  \\right)+\\frac{1}{{{c}_{1}}}{{I}_{1}}\\alpha  ={{e}_{2}}, \\label{59} \\\\ \n& \\left( -{{S}_{1}}\\alpha +{{S}_{2}}\\beta  \\right)-\\frac{T}{{{\\mu }_{2}}}\\left( {{R}_{1}}\\alpha -{{R}_{2}}\\beta  \\right)+\\frac{1}{{{c}_{u}}}{{I}_{2}}\\beta \n=\\left( -1+\\varepsilon  \\right){{e}_{u}}. \\label{60}\n\\end{align}\nCombining equations \\eqref{59} and\\eqref{60}, we obtain\n\\begin{align}\n\\left[ \\begin{matrix}\nQ_{1}&-Q_{2}\\\\\n-S_{1}&S_{2}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha\\\\\n\\beta\n\\end{matrix} \\right] &+ \\dfrac{T}{\\mu_{1}}\\left[ \\begin{matrix}\nP_{1}&-P_{2}\\\\\n-R_{1}&R_{2}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha\\\\\n\\beta\n\\end{matrix} \\right] \n+ \\left[ \\begin{matrix}\n\\dfrac{1}{c_{1}}I_{1}&0\\\\\n0&\\dfrac{1}{c_{u}}I_{2}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha\\\\\n\\\\\n\\beta\n\\end{matrix} \\right] \n= \\left[\\begin{matrix}\ne_{2}\\\\\n\\\\\n(-1+\\varepsilon)e_{u}\n\\end{matrix} \\right],\\label{61} \n\\end{align}\nthen, we can write\n\\begin{align}\n\\left[ \\begin{matrix}\n\\alpha   \\\\\n\\beta   \\\\\n\\end{matrix} \\right]&=\\left[ \\left[ \\begin{matrix}\n{{Q}_{1}} & -{{Q }_{2}}  \\\\\n-{{S}_{1}} & {{S}_{2}}  \\\\\n\\end{matrix} \\right]+\\frac{T}{{{\\mu }_{1}}}\\left[ \\begin{matrix}\n{{P}_{1}} & -{{P}_{2}}  \\\\\n-{{R}_{1}} & {{R}_{2}}  \\\\\n\\end{matrix} \\right] \n + \\left[ \\begin{matrix}\n\\dfrac{1}{{{c}_{1}}}{{I}_{1}} & 0  \\\\\n0 & \\dfrac{1}{{{c}_{u}}}{{I}_{2}}  \\\\\n\\end{matrix} \\right] \\right]^{-1}\\left[ \\begin{matrix}\n{{e}_{2}}  \\\\\n(-1+\\varepsilon ){{e}_{u}}  \n\\end{matrix} \\right]. \\label{62} \n\\end{align}\nSimilarly, the Lagrangian function for the problem \\eqref{45} can be written as follows:\n\\begin{align}\\label{63} \nL_{2} ={}& \\dfrac{1}{2}\\| Bv_{0}\\|^{2}+\\dfrac{\\mu_{2}}{2T}\\sum_{t=1}^{T} \\| B_{t}v_{t}\\|^{2}+\\dfrac{c_{2}}{2}\\sum_{t=1}^{T}\\| \\eta_{t}\\|^{2}\\nonumber\\\\\n& -\\sum_{t=1}^{T}\\alpha_{t}^{\\ast^{T}}(A_{t}(v_{0}+v_{t})+\\eta_{t}-e_{1t})+\\dfrac{c_{u}^{\\ast}}{2}\\sum_{t=1}^{T}\\| \\psi_{t}^{\\ast}\\|^{2}\\nonumber\\\\\n& -\\sum_{t=1}^{T}\\beta_{t}^{\\ast^{T}}(\\mathfrak{U}_{t}(v_{0}+v_{t})+\\psi_{t}^{\\ast}-(-1+\\varepsilon)e_{ut}).\n\\end{align}\nBy performing a similar process, the following equations are obtained:\n\\begin{align}\n\t(Q _{1}^{*}\\alpha _{1}^{*}-Q _{2}^{*}{{\\beta }^{*}})+\\frac{T}{{{\\mu }_{2}}}\\left( P_{1}^{*}{{\\alpha }^{*}}-P_{2}^{*}{{\\beta }^{*}} \\right)+\\frac{1}{{{c}_{2}}}{{I}_{1}}{{\\alpha }^{*}}\n&={{e}_{1}},\\label{64} \n\\\\%\t\\end{align}\n\\left( {{S}^{*}_{1}}\\alpha^{*}-S_{2}^{*}{{\\beta }^{*}} \\right)+\\frac{T}{{{\\mu }_{2}}}( R_{1}^{*}{{\\alpha }^{*}}\n-R_{2}^{*}{{\\beta }^{*}} )\n-\\frac{1}{{{c}_{u}^{\\ast}}}{{I}_{2}}{{\\beta }^{*}}\n&= -(-1+\\varepsilon)e_{u}, \\label{65} \n\\end{align}\nwhere\n\\begin{align*}\n& Q_{1}^{\\ast} =A{{({{B}^{T}}B)}^{-1}}{{A}^{T}}, ~~ Q_{2}^{\\ast} =A{{({{B}^{T}}B)}^{-1}}{{\\mathfrak{U}}^{T}}, ~~ S_{1}^{\\ast} =\\mathfrak{U}{{({{B}^{T}}B)}^{-1}}{{A}^{T}},  \n\\\\\n&S_{2}^{\\ast} =\\mathfrak{U}{{({{B}^{T}}B)}^{-1}}{{\\mathfrak{U}}^{T}},~~ {{P}_{1t}^{\\ast}}={{A}_{t}}{{(B_{t}^{T}B_{t})}^{-1}}A_{t}^{T}, ~~ {{P}_{2t}^{\\ast}}={{A}_{t}}{{(B_{t}^{T}B_{t})}^{-1}}\\mathfrak{U}_{t}^{T},\\\\\n& P_{1}^{\\ast}=blkdiag({{P}_{11}^{\\ast}},\\ldots ,{{P}_{1T}^{\\ast}}),~~ P_{2}^{\\ast}=blkdiag({{P}_{21}^{\\ast}},\\ldots ,{{P}_{2T}^{\\ast}}),~~ R_{1t}^{\\ast}=\\mathfrak{U}_{t}{{({{B}^{T}_{t}}B_{t})}^{-1}}{{A}^{T}_{t}}, \\\\\n& R_{2t}^{\\ast}=\\mathfrak{U}_{t}{{({{B}^{T}_{t}}B_{t})}^{-1}}{{\\mathfrak{U}}^{T}_{t}}, \n~~ R_{1}^{\\ast}=blkdiag({{R}_{11}^{\\ast}},\\ldots ,{{R}_{1T}^{\\ast}}),\n~~ R_{2}^{\\ast}=blkdiag({{R}_{21}^{\\ast}},\\ldots ,{{R}_{2T}^{\\ast}}).\n\\end{align*}\nCombining equations \\eqref{64} and \\eqref{65}, we have\n\\begin{align}\n\\left[ \\begin{matrix}\nQ_{1}^{\\ast}&-Q_{2}^{\\ast}\\\\\nS_{1}^{\\ast}&-S_{2}^{\\ast}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha^{\\ast}\\\\\n\\beta^{\\ast}\n\\end{matrix} \\right] &+ \\dfrac{T}{\\mu_{2}}\\left[ \\begin{matrix}\nP_{1}^{\\ast}&-P_{2}^{\\ast}\\\\\nR_{1}^{\\ast}&-R_{2}^{\\ast}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha^{\\ast}\\\\\n\\beta^{\\ast}\n\\end{matrix} \\right] \n+ \\left[ \\begin{matrix}\n\\dfrac{1}{c_{2}}I_{1}&0\\\\\n0&-\\dfrac{1}{c_{u}^{\\ast}}I_{2}\n\\end{matrix}\\right] \\left[\\begin{matrix}\n\\alpha^{\\ast}\\\\\n\\\\\n\\beta^{\\ast}\n\\end{matrix} \\right] \n= \\left[\\begin{matrix}\ne_{1}\\\\\n\\\\\n-(-1+\\varepsilon)e_{u}\n\\end{matrix} \\right],\n\\end{align}\nthen, we can write\n\\begin{align}\n\\left[ \\begin{matrix}\n\\alpha^{\\ast}   \\\\\n\\beta^{\\ast}   \\\\\n\\end{matrix} \\right]&=\\left[ \\left[ \\begin{matrix}\n{{Q}_{1}^{\\ast}} & -{{Q }_{2}^{\\ast}}  \\\\\n{{S}_{1}^{\\ast}} & {-{S}_{2}^{\\ast}}  \\\\\n\\end{matrix} \\right]+\\frac{T}{{{\\mu }_{2}}}\\left[ \\begin{matrix}\n{{P}_{1}^{\\ast}} & -{{P}_{2}^{\\ast}}  \\\\\n{{R}_{1}^{\\ast}} & -{{R}_{2}^{\\ast}}  \\\\\n\\end{matrix} \\right] + \\left[ \\begin{matrix}\n\\dfrac{1}{{{c}_{2}}}{{I}_{1}} & 0  \\\\\n0 & \\dfrac{1}{{{c}_{u}^{\\ast}}}{{I}_{2}}  \\\\\n\\end{matrix} \\right] \\right]^{-1}\\left[ \\begin{matrix}\n{{e}_{1}}  \\\\\n- (-1+\\varepsilon ){{e}_{u}}  \n\\end{matrix} \\right].\\label{66} \n\\end{align}\nFinally, by finding  solutions \\eqref{62} and \\eqref{66}, the classifier parameters  $u_{0}$,  $u_{t}$,  $v_{0}$   and   $v_{t}$  are obtained.\nThe decision function \\eqref{15}  can be used to assign a new data point  $x\\in \\mathbb{R}^{n}$   to its appropriate class.\nAccording to the discussion above, we illustrate the LS-$\\mathfrak{U}$MTSVM via Algorithm~\\ref{A3}.\n\\begin{algorithm} [t] \n\t\\caption{\\label{A3} A linear least squares multi-task twin support vector machine with Universum (LS-$\\mathfrak{U}$MTSVM) }\n\n\t\\algsetup{linenosize=\\normalsize}\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\begin{algorithmic}[1]\n\t\t\\normalsize\n\t\t\\REQUIRE{\\mbox{}\\\\-- The training set $\\tilde{T}$ and Universum data $ X_{\\mathfrak{U}} $;\\\\\n\t\t\t-- Decide on the total number of tasks included in the data set and assign this value to T;\\\\\n\t\t\t-- Select classification task $ S_{t}~(t=1,\\ldots,T) $ in training  data set $\\tilde{T}$;\\\\\n\t\t\t-- Divide Universum data $ X_{\\mathfrak{U}} $ by $t$-task and get $ X_{\\mathfrak{U}t}~ (t=1,\\ldots,T)$;\\\\\n\t\t\t-- Choose appropriate parameters\n\t\t\t$c_{1}, c_{2},c_{u}$, $c_{u}^{*}$, $ \\mu_{1} $, $ \\mu_{2} $, and  parameter $\\varepsilon \\in (0,1)$.}\\\\\n\t\t{\\textbf{The outputs:}}\n\t\t\\begin{list}{--}{}\n\t\t\t\\item $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\end{list}\n\t\t\n\t\t{\\textbf{The process:}}\n\t\t\n\t\t\\STATE\n\t\tSolve the two small systems of linear equations (41) and (46), and get $ \\alpha,~\\beta,~\\alpha^{*}$, and $\\beta^{*}. $\n\t\t\\STATE\n\t\tCalculate  $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\STATE\n\t\t\tBy utilizing the decision function (\\ref{15}), assign a new point $ x $ in the $t$-th task to class $ +1 $ or $ -1 $.\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\subsection{Nonlinear case}\n\nIn the following, we introduce a nonlinear version of  our proposed LS-$\\mathfrak{U}$MTSVM  because there are situations that are not linearly separable, in which case the kernel trick can be used.\nTherefore, we use the kernel function $K(\\cdot,\\cdot)$ and define\n\\begin{align*}\n& D={{\\left[ A_{1}^{T},B_{1}^{T},A_{2}^{T},B_{2}^{T},\\ldots ,A_{T}^{T},B_{T}^{T} \\right]}^{T}}, \\\\ \n& \\overline{A}=\\left[ K(A,{{D}^{T}}),e_{1} \\right],\\,\\,\\,\\,{{\\overline{A}}_{t}}=\\left[ K({{A}_{t}},{{D}^{T}}),{{e}_{1t}} \\right], \\\\ \n& \\overline{B}=\\left[ K(B,{{D}^{T}}),e_{2} \\right],\\,\\,\\,\\,{{\\overline{B}}_{t}}=\\left[ K({{B}_{t}},{{D}^{T}}),{{e}_{2t}} \\right], \\\\ \n& \\overline{\\mathfrak{U}}=\\left[ K(X_{\\mathfrak{U}},{{D}^{T}}),e_{u} \\right],\\,\\,\\,\\,{{\\overline{\\mathfrak{U}}}_{t}}=\\left[ K({X_{\\mathfrak{U}t}},{{D}^{T}}),{{e}_{ut}} \\right].\n\\end{align*}\nSo, the nonlinear formulations of the optimization problems \\eqref{44} and \\eqref{45} can be written as\n\\begin{align}\\label{67} \n\\underset{u_{0}, u_{t}, \\xi_{t},\\psi_{t}}\\min  ~& \\dfrac{1}{2}\\| \\overline{A}u_{0}\\|^{2} +\\dfrac{\\mu_{1}}{2T}\\sum_{t=1}^{T} \\| \\overline{A}_{t}u_{t}\\|^{2}+\\dfrac{c_{1}}{2}\\sum_{t=1}^{T}\\| \\xi_{t}\\|^{2}+\\dfrac{c_{u}}{2}\\sum_{t=1}^{T}\\| \\psi_{t}\\|^{2}\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,\\,\\,& -\\overline{B}_{t}(u_{0}+u_{t})+\\xi_{t}= e_{2t},\\\\\n&\\overline{\\mathfrak{U}}_{t}(u_{0}+u_{t})+\\psi_{t}=(-1+\\varepsilon)e_{ut},\\nonumber\n\\end{align}\nand\n\\begin{align}\\label{68} \n\\underset{v_{0}, v_{t}, \\eta_{t},\\psi_{t}^{\\ast}}\\min ~&\\dfrac{1}{2}\\| \\overline{B}v_{0}\\|^{2} +\\dfrac{\\mu_{2}}{2T}\\sum_{t=1}^{T} \\| \\overline{B}_{t}v_{t}\\|^{2}+\\dfrac{c_{2}}{2}\\sum_{t=1}^{T}\\| \\eta_{t}\\|^{2}+\\dfrac{c_{u}^{\\ast}}{2}\\sum_{t=1}^{T}\\| \\psi_{t}^{\\ast}\\|^{2}\\nonumber\\\\\n\\text{s.t.}\\,\\,\\,\\,\\,\\,& \\overline{A}_{t}(v_{0}+v_{t})+\\eta_{t}= e_{1t},\\\\\n&-\\overline{\\mathfrak{U}}_{t}(v_{0} +v_{t})+\\psi_{t}^{\\ast}=(-1+\\varepsilon)e_{ut},\\nonumber\n\\end{align}\nhere, parameters $ c_{1}$, $c_{2}$, $c_{u}$, $c_{u^*}$, $\\mu_{1}$, and $\\mu_{2} $ are as defined in section~4.1. In a similar way to the linear case, we can written the Lagrangian function problems (\\ref{67}) and (\\ref{68}) and KKT optimality conditions. After that, the optimal solutions of problems (\\ref{67}) and (\\ref{68}) take the form\n\\begin{align}\n\\left[ \\begin{matrix}\n\\alpha   \\\\\n\\beta   \\\\\n\\end{matrix} \\right]\n&=\\left[ \\left[ \\begin{matrix}\n{{Q}_{1}} & -{{Q }_{2}}  \\\\\n-{{S}_{1}} & {{S}_{2}}  \\\\\n\\end{matrix} \\right]+\\frac{T}{{{\\mu }_{1}}}\\left[ \\begin{matrix}\n{{P}_{1}} & -{{P}_{2}}  \\\\\n-{{R}_{1}} & {{R}_{2}}  \\\\\n\\end{matrix} \\right]   \n+ \\left[ \\begin{matrix}\n\\dfrac{1}{{{c}_{1}}}{{I}_{1}} & 0  \\\\\n0 & \\dfrac{1}{{{c}_{u}}}{{I}_{2}}  \\\\\n\\end{matrix} \\right] \\right]^{-1}\\left[ \\begin{matrix}\n{{e}_{2}}  \\\\\n(-1+\\varepsilon ){{e}_{u}}  \n\\end{matrix} \\right], \\label{69} \n\\end{align}\nand\n\\begin{align}\n\\left[ \\begin{matrix}\n\\alpha^{\\ast}   \\\\\n\\beta^{\\ast}   \\\\\n\\end{matrix} \\right]&=\\left[ \\left[ \\begin{matrix}\n{{Q}_{1}^{\\ast}} & -{{Q }_{2}^{\\ast}}  \\\\\n{{S}_{1}^{\\ast}} & {-{S}_{2}^{\\ast}}  \\\\\n\\end{matrix} \\right]+\\frac{T}{{{\\mu }_{2}}}\\left[ \\begin{matrix}\n{{P}_{1}^{\\ast}} & -{{P}_{2}^{\\ast}}  \\\\\n{{R}_{1}^{\\ast}} & -{{R}_{2}^{\\ast}}  \\\\\n\\end{matrix} \\right]  + \\left[ \\begin{matrix}\n\\dfrac{1}{{{c}_{2}}}{{I}_{1}} & 0  \\\\\n0 & \\dfrac{1}{{{c}_{u}^{\\ast}}}{{I}_{2}}  \\\\\n\\end{matrix} \\right] \\right]^{-1}\\left[ \\begin{matrix}\n{{e}_{1}}  \\\\\n- (-1+\\varepsilon ){{e}_{u}}  \n\\end{matrix} \\right],\\label{70} \n\\end{align}\nwhere\n\\begin{align*}\n& Q_{1} =\\overline{B}{{({{\\overline{A}}^{T}}\\overline{A})}^{-1}}{{\\overline{B}}^{T}}, ~~\nQ_{2} =\\overline{B}{{({{\\overline{A}}^{T}}\\overline{A})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}}, ~~\nS_{1} =\\overline{\\mathfrak{U}}{{({{\\overline{A}}^{T}}\\overline{A})}^{-1}}{{\\overline{B}}^{T}}, ~~ S_{2} =\\overline{\\mathfrak{U}}{{({{\\overline{A}}^{T}}\\overline{A})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}}, \n\\\\\n& {{P}_{1t}}={{\\overline{B}}_{t}}{{(\\overline{A}_{t}^{T}\\overline{A}_{t})}^{-1}}\\overline{B}_{t}^{T}, ~~ {{P}_{2t}}={{\\overline{B}}_{t}}{{(\\overline{A}_{t}^{T}\\overline{A}_{t})}^{-1}}\\overline{\\mathfrak{U}}_{t}^{T},~~ P_{1}=blkdiag({{P}_{11}},\\ldots ,{{P}_{1T}}),\n\\\\\n& P_{2}=blkdiag({{P}_{21}},\\ldots ,{{P}_{2T}}),~~ R_{1t}=\\overline{\\mathfrak{U}}_{t}{{({{\\overline{A}}^{T}_{t}}\\overline{A}_{t})}^{-1}}{{\\overline{B}}^{T}_{t}},~~ R_{2t}=\\overline{\\mathfrak{U}}_{t}{{({{\\overline{A}}^{T}_{t}}\\overline{A}_{t})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}_{t}}, \n\\\\\n& R_{1}=blkdiag({{R}_{11}},\\ldots ,{{R}_{1T}}),~~ R_{2}=blkdiag({{R}_{21}},\\ldots ,{{R}_{2T}}),\n\\end{align*}\nand\n\\begin{align*}\n& Q_{1}^{\\ast} =\\overline{A}{{({{\\overline{B}}^{T}}\\overline{B})}^{-1}}{{\\overline{A}}^{T}}, ~~ Q_{2}^{\\ast} =\\overline{A}{{({{\\overline{B}}^{T}}\\overline{B})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}}, ~~ S_{1}^{\\ast} =\\overline{\\mathfrak{U}}{{({{\\overline{B}}^{T}}\\overline{B})}^{-1}}{{\\overline{A}}^{T}}, \n\\\\\n&S_{2}^{\\ast} =\\overline{\\mathfrak{U}}{{({{\\overline{B}}^{T}}\\overline{B})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}},~~ {{P}_{1t}^{\\ast}}={{\\overline{A}}_{t}}{{(\\overline{B}_{t}^{T}\\overline{B}_{t})}^{-1}}\\overline{A}_{t}^{T}, ~~ {{P}_{2t}^{\\ast}}={{\\overline{A}}_{t}}{{(\\overline{B}_{t}^{T}\\overline{B}_{t})}^{-1}}\\overline{\\mathfrak{U}}_{t}^{T},\n\\\\\n& P_{1}^{\\ast}=blkdiag({{P}_{11}^{\\ast}},\\ldots ,{{P}_{1T}^{\\ast}}),~~ P_{2}^{\\ast}=blkdiag({{P}_{21}^{\\ast}},\\ldots ,{{P}_{2T}^{\\ast}}),~~ R_{1t}^{\\ast}=\\overline{\\mathfrak{U}}_{t}{{({{\\overline{B}}^{T}_{t}}\\overline{B}_{t})}^{-1}}{{\\overline{A}}^{T}_{t}}, \n\\\\\n& R_{2t}^{\\ast}=\\overline{\\mathfrak{U}}_{t}{{({{\\overline{B}}^{T}_{t}}\\overline{B}_{t})}^{-1}}{{\\overline{\\mathfrak{U}}}^{T}_{t}}, \n~~ R_{1}^{\\ast}=blkdiag({{R}_{11}^{\\ast}},\\ldots ,{{R}_{1T}^{\\ast}}),\n~~ R_{2}^{\\ast}=blkdiag({{R}_{21}^{\\ast}},\\ldots ,{{R}_{2T}^{\\ast}}).\n\\end{align*}\nThen the corresponding decision function of the $t$-th task can be computed by~(\\ref{n15}). \nAlgorithm~\\ref{A4} describes the process of nonlinear case.\n\n\\begin{algorithm} [t] \n\t\\caption{\\label{A4} A nonlinear least squares multi-task twin support vector machine with Universum (LS-$\\mathfrak{U}$MTSVM)}\n\n\t\\algsetup{linenosize=\\normalsize}\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\begin{algorithmic}[1]\n\t\t\\normalsize\n\t\t\\REQUIRE{\\mbox{}\\\\-- The training set $\\tilde{T}$ and Universum data $ X_{\\mathfrak{U}} $;\\\\\n\t\t\t-- Decide on the total number of tasks included in the data set and assign this value to T;\\\\\n\t\t\t-- Select classification task $ S_{t}~(t=1,\\ldots,T) $ in training  data set $\\tilde{T}$;\\\\\n\t\t\t-- Divide Universum data $ X_{\\mathfrak{U}} $ by $t$-task and get $ X_{\\mathfrak{U}t}~ (t=1,\\ldots,T)$;\\\\\n\t\t\t-- Choose appropriate parameters\n\t\t\t$c_{1}, c_{2},c_{u}$, $c_{u}^{*}$, $ \\mu_{1} $, $ \\mu_{2} $, and  parameter $\\varepsilon \\in (0,1)$.\\\\\n\t\t-- Select proper kernel function and kernel parameter.} \\\\\n\t\t{\\textbf{The outputs:}}\n\t\t\\begin{list}{--}{}\n\t\t\t\\item $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\end{list}\n\t\t\n\t\t{\\textbf{The process:}}\n\t\t\n\t\t\\STATE\n\t\tSolve the two small systems of linear equations (41) and (46), and get $ \\alpha,~\\beta,~\\alpha^{*}$, and $\\beta^{*}. $\n\t\t\\STATE\n\t\tCalculate  $ u_{0},~u_{t},~v_{0}$, and $v_{t}. $\n\t\t\\STATE\n\t\t\tBy utilizing the decision function (\\ref{15}), assign a new point $ x $ in the $t$-th task to class $ +1 $ or $ -1 $.\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical experiments}\nThis section presents the results of experiments on various single-task learning algorithms and multi-task learning algorithms. \n\n The single-task learning algorithms considered consist of TBSVM \\cite{shao2011improvements}, I$ \\nu $-TBSVM \\cite{wang2018improved} and $ \\mathfrak{U}_{LS} $-TSVM \\cite{xu2016least}, while the multi-task learning methods are DMTSVM \\cite{xie2012multitask}, MTLS-TWSVM \\cite{mei2019multi}, and our proposed methods, i.e., $\\mathfrak{U}$MTSVM and LS-$\\mathfrak{U}$MTSVM.\n All numerical\nexperiments for both linear and nonlinear models were performed in Matlab R2018b on a PC with 4 GB of RAM and Core(TM) i7 CPU @2.20 GHz under the Microsoft Windows 64-bit operating system. Moreover, to determine the classification accuracies and performance of the algorithms, we used a five-fold strategy for cross-validation. The following steps describe the cross-validation procedure.\n\n\\begin{itemize}\n\t\\item  Partition the data sets randomly into five separate subsets of equal size.\n\t\\item  Apply the model to four of the subsets selected as the training data.\n\t\\item  Consider the one remaining subsets as the test data and evaluate the model on it.\n\t\\item  Repeat the process until each of the five sets has been utilized as test data.\n\\end{itemize}\nThe accuracy is defined as the number of accurate predictions divided by the total number of forecasts. The value is then multiplied by 100 to give the percentage accuracy. We randomly selected an equal amount of data from each class for the benchmark and medical data sets, then used half of them to build the Universum data by averaging pairs of samples from different classes.\n\\subsection{Parameters selection}\n\nIt is obvious that the performance of classification algorithms depends on the selection of proper parameters \\cite{lu2018svm,xiao2021new}. Therefore, we will discuss selecting the optimal parameters of the single-task and multi-task learning methods. This subsection used five-fold cross-validation using the grid search approach to select parameters. The 3D surface plots in Figure~\\ref{fig1}  demonstrate the influence of changing in the various values of parameters $c_{1}$, $c_{2}$, $c_{u}$, $c_{u^{*}}$, $\\mu_{1}$ and $\\mu_{2}$ on accuracy for proposed LS-$\\mathfrak{U}$MTSVM method on Caesarian data set in linear state. From Figure~\\ref{fig1}(a--b), it can be inferred that the accuracy obtained is quite sensitive to the selected parameters. Of course, the accuracy may be stable at some values. Therefore, the choice of parameters depends on the distribution of points in a particular data set.\n\n\\begin{figure}[hbt!]\n\t\\centering\n\t\\includegraphics[width=12cm]{multiparameter.png}\n\t\\caption{The effect of different values of parameter on the Caesarian data sets}\n\t\\label{fig1}\n\\end{figure}\nThe performance of the particular algorithms is determined by the parameters  $ c_{1}$, $ c_{2}$, $ c_{3}$ and $c_{4}$ in TBSVM; $ c_{1}$, $ c_{2}$, $ c_{3}$, $ c_{4}$  and $ \\nu_{1},  \\nu_{2} $  in I$ \\nu $-TBSVM; $ c_{1}$, $ c_{2}$, $ c_{u}$, $c_{u}^{*}$ and $ \\varepsilon $  in $ \\mathfrak{U}_{LS} $-TSVM; $ c_{1}$, $c_{2}$, $\\mu_{1}$ and $\\mu_{2}$ in DMTSVM; $ c_{1},~c_{2},~\\mu_{1}$ and $\\mu_{2}$ in MTLS-TWSVM; $ c_{1}$, $c_{2}$, $c_{u}$, $c_{u}^{*}$, $\\mu_{1}$, $\\mu_{2}$  and $ \\varepsilon $ in $\\mathfrak{U}$MTSVM; and $ c_{1}$, $c_{2}$, $c_{u}$, $c_{u}^{*}$, $\\mu_{1}$, $\\mu_{2}$ and $ \\varepsilon $ in LS-$\\mathfrak{U}$MTSVM. Therefore, the following ranges are considered for selecting the optimal values of the parameters. In our experiments, parameters $ c_{1}$, $c_{2}, c_{3}$, $c_{4}$, $c_{u}$, $c_{u}^{*}$, $\\mu_{1}$ and $\\mu_{2} $ are all selected from the set $ \\lbrace 2^i \\mid i=-10,\\dots,10\\rbrace $; $  \\nu_{1}$,  $\\nu_{2}$ and $ \\varepsilon $ are selected from the set $ \\lbrace 0.1,\\dots,0.9\\rbrace $.\n\n\nIn our experiments, due to the better performance for inseparably data sets, we use the Gaussian kernel function (that is, $K(x,y)=\\exp(-\\gamma \\|x-y\\|^2)$, $\\gamma > 0$).\nWe choose a value for the kernel parameter $ \\gamma  $ from the range $ \\lbrace 2^i \\mid i=-10,\\dots,10\\rbrace$.\n\n\\subsection{Benchmark data sets}\nIn this subsection, we have compared multi-task learning algorithms on five benchmark data sets involving: Monk, Landmine, Isolet, Flags and Emotions. Table~\\ref{tab1} shows details of data sets. \n\n\\begin{table}[htp!]\n\t\\small\n\t\t\\centering\n\t\\caption{The information of benchmark data sets.}\\label{tab1}\n\t\\begin{tabular}{cccc}\n\t\t\\toprule\n\t\tData set   & $\\#$ Samples & $\\#$ Features & $\\#$ Tasks\\\\\n\t\t\\midrul\n\t\n\t\tMonk   &432 &6 & 3 \\\\\n\t\n\t\tLandmine  &9674 &9 & 4 \\\\\n\t\n\t\tIsolet& 7797&617 & 5 \\\\\n\t\n\t\tFlags   & 194& 19 & 7\\\\\n\t\n\t\tEmotions   &593 & 72 & 6 \\\\\n\t\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\nThe information of these data sets is as follows. \n\n\\begin{itemize}\n\n\t\\item \\textbf{Monk}: In July 1991, the monks of Corsendonk Priory were confronted with the 2nd European Summer School on Machine Learning, which was hosted at their convent. After listening to various learning algorithms for more than a week, they were confused: which algorithm would be the best? Which ones should you stay away from? As a consequence of this quandary, they devised the three MONK's challenges as a basic goal to which all learning algorithms should be compared.\n\t\\item \\textbf{Landmine}: This data collection was gathered from 29 landmine regions, each of which corresponded to a separate landmine field. The data set comprises nine characteristics, and each sample is labeled with a 1 for a landmine or a 0 for a cluster, reflecting positive and negative classifications. The first 15 areas correlate too strongly with foliated regions, whereas the latter 14 belong to bare ground or uninhabited places. We adopted the following procedures for the Landmine data set to get better and more equitable experimental outcomes. This data set has more negative labels than positive ones; as a result, we started by removing some negative samples to balance things out. In our experiment, we partition the data into four tasks.\nAs a result, we picked four densely foliated areas as a selection of positive data. We constructed an experimental data set using four places from bare ground or desert regions as a negative data subset. To produce the data set represented by Landmine in Table~\\ref{tab2}, we selected the four sites 1, 2, 3, and 4 from foliated regions and identified the four areas 16, 17, 18, and 19 from bare earth regions.\n\n\t\\item\\textbf{Isolet}: Isolet is a widely used data set in speech recognition that is collected as follows. One hundred fifty peoples speak 26 letters of the English alphabet twice. Therefore, 52 training samples are generated for each speaker.  Each speaker is classified into one of five categories. Consequently, we have five sets of data sets that can be considered five tasks.\n\tOn the one hand, these five tasks are closely related because they are gathered from the same utterances. On the other hand, the five tasks differ because speakers in different groups pronounce the English letters differently.\n\tIn this paper, we selected four pairs of similar-sounding letters, including  (O, U), (X, Y), (H, L) and (P, Q) for our experiments.\n\t\\item \\textbf{Flags}: The flags data set offers information about the flags of various countries. It contains 194 samples and 19 features. This data set is divided into seven tasks based on different colors. Each task is represented by a 19-dimensional feature vector derived from flag images of several nations. \n\tEach sample may have a maximum of seven labels, because the task of recognizing each label may be seen as connected. Hence, we consider it as a multi-task learning problem. Thus, this data set contains seven tasks. In Table~\\ref{tab2}, we compare the performance of the aforementioned multi-task learning methods on this data set.\n\t\\item\\textbf{Emotions}: Emotion recognition from text is one of the most difficult challenges in Natural Language Processing. The reason for this is the lack of labeled data sets and the problem's multi-class character. Humans experience a wide range of emotions, and it is difficult to gather enough information for each feeling, resulting in the issue of class imbalance. We have labeled data for emotion recognition here, and the goal is to construct an efficient model to identify the emotion.\nThe Emotions data collection comprises Twitter posts in English that depict six primary emotions: anger, contempt, fear, joy, sorrow, and surprise. All samples are labeled in six different ways. Each sample may contain more than one label (or emotion). Different emotion recognition tasks have similar characteristics and can be considered related tasks.  So it may be viewed as a multi-task classification issue, with each task requiring the identification of a single kind of emotion. We use 50 samples from this data set to test various multi-task learning algorithms in this experiment. The results of comparing the performance of multi-task learning algorithms on this data set are reported in Table~\\ref{tab2}.\n\\end{itemize}\nAs mentioned, we compare the performance of the proposed methods with DMTSVM and MTLS-TWSVM in this subsection. Table~\\ref{tab2} shows the average accuracies (``Acc''), standard deviations (``Std'') and the running time (``Time'')  on the five popular multi-task data sets. \nAs is seen in Table~\\ref{tab2}, the best performance is achieved by the proposed LS-$\\mathfrak{U}$MTSVM followed by $ \\mathfrak{U}$MTSVM. For example, on the Emotions data set, the accuracies for DMTSVM and MTLS-TWSVM  are $65.33\\%$, and $66\\%$, respectively. In comparison, the  $ \\mathfrak{U}$MTSVM and LS-$\\mathfrak{U}$MTSVM method achieved the accuracies $69.30\\%$, and  $75.73\\%$, which performs better than the other two multi-task learning algorithms. As a result, due to the nature of multi-task learning, it is advantageous to combine data with Universum data throughout the training phase.\nFurthermore, the results seem to match our intuition that Universum data play an essential role in the performance of $ \\mathfrak{U}$MTSVM and LS-$\\mathfrak{U}$MTSVM. When $ \\mathfrak{U}$MTSVM and LS-$\\mathfrak{U}$MTSVM compare with DMTSVM and MTLS-TWSVM, we find that our proposed algorithms indeed exploit Universum data to improve the prediction accuracy and stability.\nTherefore, $ \\mathfrak{U}$MTSVM and LS-$\\mathfrak{U}$MTSVM perform better than other  multi-task learning algorithms, i.e., DMTSVM and MTLS-TWSVM. \n\nIn terms of learning speed, although our proposed methods are not the fastest ones due incorporating Universum data,  they offer better accuracies at an acceptable time.\n\n\\begin{table}[htp!]\n\t\\small\n\t\\caption{Performance comparison of nonlinear  multi-task learning methods on benchmark data sets.}\\label{tab2}\n\t\\begin{tabular}{ccccc}\n\t\t\\toprul\n\t\t & DMTSVM& $ \\mathfrak{U} $MTSVM& MTLS-TWSVM & LS-$\\mathfrak{U}$MTSVM\\\\\n\t\tData set  & Acc ($\\%$)$ \\pm $Std &Acc ($\\%$) $ \\pm $Std& Acc ($\\%$)$ \\pm $Std & Acc ($\\%$)$ \\pm $Std \\\\\n\t\t& Time ($s$)&Time ($s$)& Time ($s$)& Time ($s$)\\\\\n\t\t\\midrul\n\t\tMonk&92.76$ \\pm $0.02&99.61$ \\pm $0.00&94.80$ \\pm $0.02&\\textbf{99.80$ \\pm $0.02}\\\\\n\t\t&27.82& 36.99&25.46&30.21\\\\\n\t\n\t\tLandmine&92.33$ \\pm $0.01&93$ \\pm $0.11 &94.05$ \\pm $0.05&\\textbf{94.50$ \\pm $0.00}\\\\\n\t\t&40.69&42.21 &36.69&39.88\\\\\n\t\n\t\tIsolet (O, U) &99.60$ \\pm $0.00&99.67$ \\pm $0.01 &99$ \\pm $0.02&\\textbf{99.77$ \\pm $0.01}\\\\\n\t\t&11.49&15.86&10.21&11.54\\\\\n\t\n\t\tIsolet (X, Y) &99.50$ \\pm $0.07&98.83$ \\pm $0.01 &99.60$ \\pm $0.00&\\textbf{100$ \\pm $0.00}\\\\\n\t\t&11.60&15.28 &10.39&11.99\\\\\n\t\n\t\tIsolet (H, L) &98.16$ \\pm $0.02&\\textbf{100$ \\pm $0.04} &99.83$ \\pm $0.01&\\textbf{100$ \\pm $0.00}\\\\\n\t\t&11.55&15.37&10.27&11.89\\\\\n\t\n\t\tIsolet (P, Q) &96.83$ \\pm $0.04&96.33$ \\pm $0.05 &97.83$ \\pm $0.03&\\textbf{100$ \\pm $0.00}\\\\\n\t\t&12.24&15.26 &10.32&11.88\\\\\n\t\n\t\tFlags &55.29$ \\pm $0.25&57.48$ \\pm $0.07 &57.13$ \\pm $0.26&\\textbf{59.57$ \\pm $0.28}\\\\\n\t\n\t\t&3.49& 3.88&2.52&3.12\\\\\n\t\n\t\tEmotions&65.33$ \\pm $0.19&69.30$ \\pm $0.21 &66$ \\pm $0.23&\\textbf{75.73$ \\pm $0.18}\\\\\n\t\n\t\t&2.25& 2.60&1.15&2.30\\\\\n\t\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\\subsection{Medical data sets}\nIn this subsection of our experiments, we focus on comparing our proposed methods and several classifier methods, including single-task and multi-task learning methods in linear and nonlinear states. Therefore, we select four popular medical data sets to test these algorithms, including  Immunotherapy,  Ljubljana Breast Cancer, \tBreast Cancer Coimbra,  and Caesarian data sets. \nA summary of the data sets information is provided in Table~\\ref{tab3}. The details of the data sets are as follows.\n\n\\begin{table}[htp!]\n\t\\small\n\t\\caption{The information of medical data sets.}\\label{tab3}\n\t\t\\centering\n\t\\begin{tabular}{cccc}\n\t\t\\toprul\n\t\tData set   & $\\#$ Samples & $\\#$ Features & $\\#$ Tasks\\\\\n\t\t\\midrul\n\t\n\t\tImmunotherapy    &90 & 8& 3 \\\\\n\t\n\t\tLjubljana Breast Cancer   &286 & 9& 5 \\\\\n\t\n\t\tBreast Cancer Coimbra &116 & 9 & 3 \\\\\n\t\n\t\tCaesarian   &80 & 5 & 2 \\\\ \n\t\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\n\\begin{itemize}\n\t\\item \\textbf{Immunotherapy}: This data collection provides information regarding wart treatment outcomes of 90 individuals utilizing Immunotherapy. The Immunotherapy data set includes 90 instances, and each instance has eight features. The features of this data include sex, age,  type, number of warts, induration diameter, area and the result of treatment.\n\tFor this data set, we partition the data into three tasks using the variable type: task 1 (type \\quo{1} = Common, 47 instances), task 2 (type \\quo{2} = Plantar, 22 instances), and task 3 (type \\quo{3} = Both, 21 instances). Since the kind of wart within each job are varied, this variable is also incorporated in our model.\n\t\n\t\\item \\textbf{Ljubljana Breast Cancer}: Nowadays, breast cancer is one of the most common malignancies in women, which has captured the attention of people all around the globe. The illness is the leading cause of mortality in women aged 40 to 50, accounting for around one-fifth of all fatalities in this age range. Every year, more than 14,000 individuals die, and the number is increasing \\cite{wang2021data}. Thus, there remains a need to remove the cancer early to reduce recurrence. Since recurrence within five years of diagnosis is correlated with the chance of death, understanding and predicting recurrence susceptibility is critical. The Ljubljana breast cancer data set provides 286 data points on the return of breast cancer five years following surgical removal of a tumor. After deleting nine instances where values are missing, we are left with 277.  Each data point has one class label (for recurrence or no-recurrence events) and nine attributes, including age, menopausal status, tumor size, invasive nodes, node caps, degree of malignancy, breast (left, right), breast quadrant (left-up, left-low, right-up, right-low, central), and irradiation (yes, no).\n\tHere, we divide the data into five tasks using the variable tumor size: task 1 (0 $ \\leq $tumor size$ \\leq $19), task 2 (20 $ \\leq $ tumor size $ \\leq $24), task 3 (25 $ \\leq $ tumor size $ \\leq $29), task 4 (30 $ \\leq $ tumor size $ \\leq $34), and task 5 (35 $ \\leq $tumor size $ \\leq $54).  \n\t\n\t\\item \\textbf{Breast Cancer Coimbra}: The Breast Cancer Coimbra data set is the second breast cancer data set we utilize for comparison. The Gynecology Department of the Coimbra Hospital and University Center (CHUC) in Portugal collected this data set between 2009 and 2013. The Breast Cancer Coimbra data set contains 116 instances, each with nine features. This data set consists of 9 quantitative attributes and a class label attribute indicating if the clinical result is positive for existing cancer or negative (patient or healthy). Clinical characteristics were observed or assessed in 64 patients with breast cancer and 52 healthy controls. Age, BMI, insulin, glucose, HOMA, leptin, resistin, adiponectin,  and MCP-1 are all quantitative characteristics. The features are anthropometric data and measurements acquired during standard blood analysis. The qualities have the potential to be employed as a biomarker for breast cancer.\n\tIn this experiment, we partition the data set into three tasks using the feature BMI. Based on tissue mass (muscle, fat, and bone) and height, the BMI is a simple rule of thumb to classify a person as underweight, normal weight, overweight, or obese. Underweight (less than 18.5 $kg\/m^2$), normal weight (18.5 $kg\/m^2$ to 24.9 $kg\/m^2$), overweight (25 $kg\/m^2$ to 29.9 $kg\/m^2$), and obese (30 $kg\/m^2$ or more) are the four major adult BMI categories.\n\tHence, we consider that the first task is underweight people, the second task is normal-weight people, and the third task is overweight and obese.\n\t\n\t\\item\\textbf{Caesarian}: This data set, which aims to deliver via cesarean section or natural birth, provides information on 80 pregnant women who have had the most extreme delivery complications  in the medical field. The Caesarian data set includes 80 instances, and each instance has\n\tfive features. The features of this data include age, delivery number, blood pressure, delivery time,  and heart problem. The heart problem\n\tfeature in Caesarian data sets has two forms. We separate the data into two tasks using the variable a heart problem; task 1: The patient has\n\ta heart problem, and task 2: The patient does not have a heart problem.\n\\end{itemize}\nTo analyze the performance of our proposed methods, we used medical data sets and compared our proposed algorithms to five (multi-task and single-task learning) algorithms, i.e.,  TBSVM, I$ \\nu $-TBSVM, $ \\mathfrak{U}_{LS}$-TSVM, DMTSVM, and MTLS-TWSVM.\n We can see from the results of Tables~\\ref{tab4} and~\\ref{tab5} that our algorithms outperform all algorithms in linear and nonlinear states.  This occurs because proposed methods add Universum data to the model learning process to modify the classification decision boundaries. The proposed methods train all tasks simultaneously, and they can take advantage of the underlying information among all tasks and improve their performance.\n\n\\begin{table}[htp!]\n\t\\small\n\t\\caption{Performance comparison of linear single-task and multi-task learning methods on medical data sets.}\\label{tab4}\n\t\\scalebox{0.85}{\n\t\\begin{tabular}{ccccc}\n\t\t\\toprul\n\t\t  & Immunotherapy& \tLjubljana Breast Cancer & \tBreast Cancer Coimbra &Caesarian \\\\\n\t\tMethods  & Acc ($\\%$)$ \\pm $Std &Acc ($\\%$) $ \\pm $Std& Acc ($\\%$)$ \\pm $Std & Acc ($\\%$)$ \\pm $Std \\\\\n\t\t& Time ($s$)&Time ($s$)& Time ($s$)& Time ($s$)\\\\\n\t\t\\midrul\n\t\tTBSVM&77.81$ \\pm $0.09 &75.11 $ \\pm $0.06&72.36$ \\pm $0.02&69.09$ \\pm $0.03 \\\\\n\t\t&1.41 &1.49 &1.40&1.42 \\\\\n\t\n\t\tI$\\nu$-TBSVM &79.97$ \\pm $0.15& 73.30$ \\pm $0.03&73.26$ \\pm $0.15&65.01$ \\pm $0.06 \\\\\n\t\t&1.63&3.22 &1.76& 1.58\\\\\n\t\n\t\t$ \\mathfrak{U}_{LS} $-TSVM&78.92$ \\pm $0.10&  74.72$ \\pm $0.08   &79.31$ \\pm $0.09&71.59$ \\pm $0.11\\\\\n\t\t&0.04&0.04&0.45&0.04\\\\\n\t\n\t\tDMTSVM &   81.29$ \\pm $0.06&71.14$ \\pm $0.09 &75.43$ \\pm $0.20&63.86$ \\pm $0.12\\\\\n\t\t&1.49& 1.60&1.50&1.48\\\\\n\t\n\t\t$ \\mathfrak{U} $MTSVM &84.63$ \\pm $0.07&73.26$ \\pm $0.08 &80.24$ \\pm $0.13&71.67$ \\pm $0.13\\\\\n\t\t&1.53&1.64 &1.55&1.48\\\\\n\t\n\t\tMTLS-TWSVM &86.11$ \\pm $0.08&75.09$ \\pm $0.19 &83.39$ \\pm $0.11&76.95$ \\pm $0.11\\\\\n\t\t&0.20&0.25 &0.19&0.16\\\\\n\t\n\t\tLS-$\\mathfrak{U}$MTSVM &\\textbf{88.11$ \\pm $0.11}& \\textbf{75.41$ \\pm $0.00}  &\\textbf{85.37$ \\pm $0.12}&\\textbf{78.52$ \\pm $0.12}\\\\\n\t\n\t\t&0.25& 0.33&0.25&0.20\\\\\n\t\n\t\t\\bottomrule\n\t\\end{tabular}}\n\\end{table}\n\n\\begin{table}[htp!]\n\t\\small\n\t\\caption{Performance comparison of nonlinear single-task and multi-task learning methods on medical data sets.}\\label{tab5}\n\\scalebox{0.85}{\n\t\\begin{tabular}{ccccc}\n\t\t\t\\toprul\n\t\t  & Immunotherapy& Ljubljana Breast Cancer & Breast Cancer Coimbra &Caesarian \\\\\n\t\tMethods  & Acc ($\\%$)$ \\pm $Std &Acc ($\\%$) $ \\pm $Std& Acc ($\\%$)$ \\pm $Std & Acc ($\\%$)$ \\pm $Std \\\\\n\t\t& Time ($s$)&Time ($s$)& Time ($s$)& Time ($s$)\\\\\n\t\t\t\\midrul\n\t\tTBSVM&78.88$ \\pm $0.02 &72.54$ \\pm $0.05&60.39$ \\pm $0.07&71.35$ \\pm $0.04 \\\\\n\t\t&1.50 &1.49 &1.46&1.47 \\\\\n\t\n\t\tI$\\nu$-TBSVM &78.92$ \\pm $0.01& 72.90$ \\pm $0.03&60.87$ \\pm $0.17&71.14$ \\pm $0.14 \\\\\n\t\t&1.43&1.59 &1.45&1.46 \\\\\n\t\n\t\t$ \\mathfrak{U}_{LS} $-TSVM&78.92$ \\pm $0.02&74.74$ \\pm $0.37&63.84$ \\pm $0.06&72.28$ \\pm $0.02\\\\\n\t\t&0.06&0.11&0.07&0.07\\\\\n\t\tDMTSVM &80.25$ \\pm $0.05&73.71$ \\pm $0.09 &63.74$ \\pm $0.09&72.56$ \\pm $0.13\\\\\n\t\t&1.58& 2.14&1.67&1.56\\\\\n\t\n\t\t$ \\mathfrak{U} $MTSVM &80.25$ \\pm $0.05&75.62$ \\pm $0.11 &65.71$ \\pm $0.19&74.14$ \\pm $0.1\\\\\n\t\t&1.53&2.15 &1.69&1.57\\\\\n\t\n\t\tMTLS-TWSVM &80.22$ \\pm $0.25&75.32$ \\pm $0.71 &65.33$ \\pm $0.11&73.15$ \\pm $0.09\\\\\n\t\t&0.25&0.71 &0.30&0.21\\\\\n\t\n\t\tLS-$\\mathfrak{U}$MTSVM &\\textbf{80.56$ \\pm $0.05}&\\textbf{77.15$ \\pm $0.09} &\\textbf{67.17$ \\pm $0.14}&\\textbf{76.74$ \\pm $0.11}\\\\\n\t\n\t\t&0.38& 1.55&0.46&0.27\\\\\n\t\n\t\t\\bottomrule\n\t\\end{tabular}\n}\n\\end{table}\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we introduced the twin support vector machine in the framework of multi-task learning with Universum data sets and proposed a new model called $\\mathfrak{U}$MTSVM. In addition, we suggested two approaches to solving our novel model. As the first approach, we solved the $\\mathfrak{U}$MTSVM by the dual problem, a quadratic programming problem. Also, we suggested the least-squares version of $\\mathfrak{U}$MTSVM and called it LS-$\\mathfrak{U}$MTSVM. The LS-$\\mathfrak{U}$MTSVM only dealt with two systems of linear equations. Hence, comprehensive experiments on several popular multi-task data sets and medical data sets demonstrated the effectiveness of our proposed methods in terms of classification performance. The experiments confirmed that our algorithms achieved better experimental results compared to three single-task learning algorithms and two multi-task learning algorithms. \n\n\\subsubsection*{Acknowledgments} \nH. Moosaei and M~Hlad\\'{\\i}k were supported by the Czech Science Foundation Grant P403-22-11117S. \n In addition, the work of H. Moosaei was supported by the Center for Foundations of Modern Computer Science (Charles Univ.\\ project UNCE\/SCI\/004). \n \n\\section*{Conflict of interest}\nThe authors state that they do not have any conflicts of interest.\n\n\\bibliographystyle{spmpsci}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNuclear mass is one of the most fundamental properties of nuclei.\nIt is of great importance not only in the nuclear physics but also in the astrophysics~\\cite{Lunney2003RMP, Blaum2006PR}.\nFor example, the masses of nuclei widely ranged from the valley of stability to the vicinity of neutron drip line are involved in simulating the rapid neutron capture (r-process) of stellar nucleosynthesis~\\cite{Arnould2007PR}.\nAlthough considerable achievements have been made in mass measurement~\\cite{Wang2017CPC}, vast amount of nuclei on the neutron-rich side away from valley of stability are still beyond the experimental capability in the foreseeable future.\nTherefore, reliable nuclear models for high-precision description of nuclear masses are strongly required.\n\nDuring the past decades, various global nuclear models have been proposed to describe the nuclear mass, including the finite-range droplet model (FRDM)~\\cite{Moller1995ADaNDT, Moeller2016ADNDT}, the semi-empirical Weizs$\\mathrm{\\ddot{a}}$cker-Skyrme (WS) model~\\cite{Wang2010PRC, Wang2014PLB}, the non-relativistic~\\cite{Goriely2009PRLa, Goriely2013PRC, Erler2012N, Goriely2016TheEuropeanPhysicalJournalA52202} and relativistic~\\cite{Geng2005PoTP, Afanasjev2013PLB, Agbemava2014PRC,Zhang2014FoP, Lu2015PRC} density functional theories (DFTs), etc.\nThe nuclear DFTs start from universal density functionals containing a few parameters determined by fitting to the properties of finite nuclei or nuclear matter.\nThey can describe the nuclear masses, ground and excited state properties in a unified way~\\cite{Ring1996PPNP, Bender2003RMP, Vretenar2005PR}.\nIn particular, due to the consideration of the Lorentz symmetry, the relativistic or covariant density functional theory (CDFT) naturally includes the nucleonic spin degree of freedom and the time-odd mean fields, which play an essential role in describing the moments of inertia for nuclear rotations~\\cite{Koenig1993PRL, Afanasjev2000NPA, Afanasjev2000PRC, Zhao2012PRC}.\nUp to now, the CDFT has received wide attentions because of its successful description of many nuclear phenomena~\\cite{Ring1996PPNP, Vretenar2005PR, Meng2006PiPaNP, Niksic2011PiPaNP, Meng2013FoP, meng2016relativistic, Zhao2018Int.J.Mod.Phys.E1830007}.\n\nIn the framework of CDFT, the masses for over 7000 nuclei with $8\\leq Z\\leq100$ up to the proton and neutron drip lines were investigated based on the axial relativistic mean field (RMF) theory~\\cite{Geng2005PoTP}.\nLater on, to explore the location of the proton and neutron drip lines, a systematic investigation has been performed for even-even nuclei within the axial relativistic Hartree-Bogoliubov (RHB) theory~\\cite{Afanasjev2013PLB, Agbemava2014PRC, Afanasjev2015PRC}.\nVery recently, the ground-state properties of nuclei with $8\\leq Z\\leq 120$ from the proton drip line to the neutron drip line have been calculated using the spherical relativistic continuum Hartree-Bogoliubov (RCHB) theory, in which the couplings between the bound states and the continuum can be considered properly~\\cite{Xia2018ADaNDT}.\nThe root-mean-square (rms) deviation with respect to the experimental nuclear masses in these pure CDFT calculation is typically around several MeV.\nTo achieve a higher precision, one needs to go beyond the mean-field approximation and consider the beyond-mean-field dynamical correlation energies (DCEs).\n\nIn Ref.~\\cite{Zhang2014FoP}, Zhang \\emph{et al.} have carried out a global calculation of the binding energies for 575 even-even nuclei ranging from $Z=8$ to $Z=108$ based on the axial RMF, and the Bardeen-Cooper-Schrieffer (BCS) approximation is adopted to consider the pairing correlations.\nIn this axial RMF+BCS calculation, the DCEs, namely the rotational and vibrational correlation energies were obtained by cranking prescription.\nAfter including the DCEs, the rms deviation for binding energies of the 575 even-even nulcei reduces from 2.58 MeV to 1.24 MeV.\nLater on, the DCEs of these nuclei were revisited in Ref.~\\cite{Lu2015PRC} using the five-dimensional collective Hamiltonian (5DCH) method with the collective parameters determined by the CDFT calculations~\\cite{Niksic2009PRC, Li2009PRC}.\nThe 5DCH method takes into account the DCEs in a more proper way, and the resulting rms deviation reduces from 2.52 MeV to 1.14 MeV~\\footnote{Note that in Ref.~\\cite{Zhang2014FoP}, the adopted experimental mass data are taken from Ref.~\\cite{Audi2003NuclearPhysicsA729337-676}, while in Ref.~\\cite{Lu2015PRC}, the experimental mass data are from Ref.~\\cite{Audi2012ChinesePhysicsC361287--1602}.\nMoreover, compared to the theoretical results shown in Ref.~\\cite{Zhang2014FoP}, the energies associated with triaxial deformation are further included in Ref.~\\cite{Lu2015PRC}.}\n\nThe studies shown in Refs.~\\cite{Zhang2014FoP, Lu2015PRC} demonstrate that the inclusion of the DCEs can significantly improve the description of the nuclear masses.\nSo far, the inclusion of DCEs in the CDFT is still confined to the nuclei with mass known and the DCEs of most neutron-rich nuclei crucial in simulating the r-process still remain uninvestigated.\nTherefore, it is necessary to extend the investigation from the nuclei with mass known to the boundary of nuclear landscape.\nMeanwhile, the pairing correlations are treated by BCS approximation in Refs.~\\cite{Zhang2014FoP, Lu2015PRC}.\nFor the description of nuclei around the neutron drip line, this approximation is questionable because the continuum effect can not be taken into account properly ~\\cite{Dobaczewski1984NPA}.\nNevertheless, the methods with the Bogoliubov transformation can provide a better description for the pairing correlations in weakly bound nuclei.\nTherefore, in present paper, nuclear masses of even-even nuclei from O to Sn isotopes ranging from the proton drip line to the neutron drip line are performed within the triaxial relativistic Hartree-Bogoliubov theory~\\cite{PhysRevC.81.054318}, and the beyond mean-field quadrupole DCEs are included by the 5DCH method.\n\n\\section{Numerical details}\nTo this end, the large-scale deformation constrained triaxial RHB calculation is carried out to generate the mean-filed states in the whole $(\\beta,\\gamma)$ plane.\nThe well-known density functional PC-PK1~\\cite{Zhao2010PRC} is adopted in the particle-hole channel.\nThis density functional particularly improves the description for isospin dependence of binding energies and it has been successfully used in describing the Coulomb displacement energies between mirror nuclei~\\cite{Sun2011Sci.ChinaSer.G-Phys.Mech.Astron.210}, nuclear masses~\\cite{Zhao2012Phys.Rev.C64324,Lu2015PRC}, quadrupole moments~\\cite{Zhao2014Phys.Rev.C11301,Yordanov2016Phys.Rev.Lett.32501,Haas2017EPL62001}, superheavy nuclei~\\cite{Zhang2013Phys.Rev.C54324,Lu2014Phys.Rev.C14323,Agbemava2015Phys.Rev.C54310,Li2015Front.Phys.268}, nuclear shape phase transitions~\\cite{Quan2018Phys.Rev.C31301, Quan2017PRC}, magnetic and antimagnetic rotations~\\cite{Zhao2011PRL,Zhao2011PLB,Meng2013FoP,Meng2016PhysicaScripta53008}, chiral rotations~\\cite{Zhao2017Phys.Lett.B1}, etc.\nA finite range separable pairing force with $G=-728$ MeV~\\cite{Tian2009PLB} is used in the particle-particle channel.\nThe triaxl RHB equation is solved by a three-dimensional harmonic oscillator basis expansion in Cartesian coordinate with 12 and 14 major shells for nuclei with $Z<20$ and $20\\leq Z\\leq50$, respectively.\nThe obtained quasiparticle energies and wave functions are used to calculated the mass parameters, moments of inertia, and collective potentials in the 5DCH, which are functions of the quadrupole deformation parameters $\\beta$ and $\\gamma$.\nThe DCE $E_{\\mathrm{corr}}$ is defined as the energy difference between the lowest mean-field state and the $0_1^+$ state of 5DCH.\n\n\\section{Results and discussion}\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=11cm]{Figure1.pdf}\\\\\n  \\caption{Even-even nuclei from O to Sn isotopes predicted by the triaxial RHB approach with (panel (b)) and without (panel (a)) dynamical correlation energies.\n  Discrepancies of the calculated binding energies with the data~\\cite{Wang2017CPC} are denoted by colors.\n  The proton and neutron drip lines predicted by spherical RCHB (PC-PK1)~\\cite{Xia2018ADaNDT}, axial RHB (DD-PC1)~\\cite{Agbemava2014PRC} and axial RMF+BCS (TMA)~\\cite{Geng2005PoTP} are also plotted for comparison.}\\label{1}\n\\end{figure}\nThe bound nuclear regions from O to Sn isotopes predicted by the triaxial RHB approach with and without DCEs are shown in Figure~\\ref{1}.\nThe discrepancies of the calculated binding energies with respect to the data are scaled by colors.\nThe binding energies calculated by triaxial RHB approach shown in panel (a) are given by the binding energies of the lowest mean-field states, while in panel (b), the DCEs are taken into account.\n\nIn the triaxial RHB calculations without DCEs, it is found that the binding energies are systematically underestimated.\nMost of the deviations are in the range of 0.5 MeV $\\sim$ 4.5 MeV, resulting in the rms deviation of 2.50 MeV.\nBy including the DCEs, the underestimation of binding energies are improved significantly, and the rms deviation is reduced from 2.50 MeV to 1.59 MeV.\nHowever, in the region $(N,Z)\\sim(24,12)$, large deviations exist even though the DCEs have been considered.\nThis might be associated with the complex shell evolution around this region.\nTo have a better description of binding energies in this region, the tensor interaction~\\cite{Long2007Phys.Rev.C34314} may need to be included in the adopted density functional, which is beyond the scope of the present investigations.\n\nIn order to estimate the number of bound nuclei from O to Sn isotopes, two-proton and two-neutron drip lines predicted by present triaxial RHB approach with and without DCEs are also plotted in Figure.~\\ref{1}.\nThe predicted number of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes without DCEs is 569.\nThe inclusion of DCEs has little impact on the proton and neutron drip lines and the corresponding number of bound nuclei is 564.\nFor comparison, the drip lines predicted by the spherical RCHB (PC-PK1)~\\cite{Xia2018ADaNDT}, axial RHB(DD-PC1)~\\cite{Agbemava2014PRC} and axial RMF+BCS(TMA)~\\cite{Geng2005PoTP} are also shown.\nIt is found that theoretical differences for proton drip lines are rather small.\nHowever, the neutron drip lines predicted by different approaches differ considerably and the differences increase with the mass number.\nThe neutron drip line predicted by the triaxial RHB approach locates in between of those by the axial RHB and spherical RCHB approaches.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=11cm]{Figure2.pdf}\\\\\n  \\caption{Contour map of the dynamical correlation energies $E_\\mathrm{corr}$ calculated by the 5DCH model based on triaxial RHB calculations as functions of neutron and proton numbers.}\\label{2}\n\\end{figure}\nFigure \\ref{2} displays the contour map of the dynamical correlation energies $E_\\mathrm{corr}$ calculated by the 5DCH model based on triaxial RHB calculations.\nThe calculated $E_\\mathrm{corr}$ ranges from 0 to 5 MeV, and varies mainly in the region of $2.0$---$4.0$ MeV.\nDue to the shape fluctuations, the dynamical correlation energies are pronounced for nuclei around $Z \\sim 32, 40$ and $N \\sim 34, 60$.\nSimilarly to the finding reported in Ref.~\\cite{Lu2015PRC}, the dynamical correlation energies for the semi-magic nuclei with $Z=28, 50$ and $N = 28, 82$ are nonzero or even rather large.\nThis is caused by the fact that the potential energy surfaces for these nuclei are either soft or with shape coexisting phenomena.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=10cm]{Figure3.pdf}\\\\\n  \\caption{The dynamical correlation energies calculated by 5DCH based on triaxial RHB (circles) in comparison with those based on triaxial RMF+BCS~\\cite{Lu2015PRC}  (triangles).}\\label{3}\n\\end{figure}\nIn Ref.~\\cite{Lu2015PRC}, the binding energies of 575 even-even nuclei in the region of $8\\leq Z\\leq108$ have been calculated using the 5DCH method in the framework of triaxial RMF+BCS.\nFor the 228 nuclei with $8\\leq Z\\leq50$ in Ref.~\\cite{Lu2015PRC}, the rms deviation with respect to data is 1.23 MeV, whereas the rms deviation in present calculations for these nuclei is 1.47 MeV.\nIt is found that the lowest mean-field binding energies given by these two calculations are of little difference, so the differences mainly come from the $E_{\\mathrm{corr}}$.\n\nThe dynamical correlation energies $E_{\\mathrm{corr}}$ calculated by 5DCH based on triaxial RHB and triaxial RMF+BCS are plotted in Figure \\ref{3} as functions of neutron number $N$.\nEven though the systematics of $E_{\\mathrm{corr}}$ are similar for both calculations, the triaxial RHB-based $E_{\\mathrm{corr}}$ are systematically larger than those based on triaxial RMF+BCS.\nThe rms deviation between these two results is 0.53 MeV and this leads to the overall difference in the binding energies.\nThe systematic difference of $E_{\\mathrm{corr}}$ might be originated from the different treatments of pairing correlations.\nThe pairing correlations in present calculations are considered by the Bogoliubov transformation, while in Ref.~\\cite{Lu2015PRC}, the pairing correlations are considered by the BCS approximation.\nDifferent pairing properties may lead to different zero point energies, and thus results in different dynamical correlation energies $E_{\\mathrm{corr}}$~\\cite{Lu2015PRC}.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=11cm]{Figure4.pdf}\\\\\n  \\caption{Contour map of the quadrupole deformation $\\beta$ calculated by the triaxial RHB approach as functions of the neutron and proton numbers. Nuclei with triaxial deformation are denoted by black triangles.}\n  \\label{4}\n\\end{figure}\nThe contour map of triaxial RHB calculated quadrupole deformation $\\beta$ are presented in Figure \\ref{4}.\nThe quadrupole deformation corresponds to the energy minima on the whole $(\\beta,\\gamma)$ plane.\nHere, $\\beta$ is defined as positive for $0^\\circ\\leq\\gamma<30^\\circ$ and negative for $30^\\circ<\\gamma\\leq60^\\circ$.\nIn general, the nuclei near magic numbers possess small or vanishing deformation.\nHowever, it is found that single magic numbers do not enforce sphericity, especially for neutron-rich nuclei.\nFor example, neutron-rich isotones with $N = 28, Z<20$ and $N = 50, Z<28$ show remarkable deformation.\nIn addition, the deformation develops when moving away from the magic numbers either isotopically or isotonically.\nThere are four large deformed regions located at $(N, Z)\\sim (24,14), (34,32), (60,40)$ and $(94,46)$.\nThese regions with large deformation correspond to the regions with large DCEs as shown in Figure~\\ref{2}.\n\nThe nuclei with triaxial deformation i.e. $\\gamma\\neq 0^\\circ, 60^\\circ$, are also shown in Figure~\\ref{4}.\nThere are 15 nuclei with triaxial deformation and most of them belong to Ge, Mo, and Ru isotopes.\nOur theoretical investigations provide good candidates for the experiment to study the possibility of triaxial deformations.\n\n\\section{Summary}\nIn summary, the nuclear masses of even-even nuclei with $8\\leq Z\\leq 50$ ranging from the proton drip line to neutron drip line are systematically investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1, and the quadrupole dynamical correlation energies are taken into account by solving the five-dimensional collective Hamiltonian.\nWith the inclusion of dynamical correlation energies, the prediction of triaxial relativistic Hartree-Bogoliubov theory for 252 nuclei masses is improved significantly with the rms deviation reducing from 2.50 MeV to 1.59 MeV.\nIt is found that the dynamical correlation energies have little influence on the positions of proton and neutron drip lines, and the predicted numbers of bound even-even nuclei between proton and neutron drip lines with and without dynamical correlation energies are 569 and 564 respectively.\nIn present calculations, the obtained dynamical correlation energies range from 0 to 5 MeV which are slightly larger than the results of previous work~\\cite{Lu2015PRC}.\nThe discrepancies might be caused by the different treatments of pairing correlations, which would lead to different zero point energies, and thus different dynamical correlation energies.\nThe contour map of quadrupole deformation $\\beta$ and $\\gamma$ associated with the dynamical correlation energies is also discussed in detail.\nThere are 15 nuclei predicted to have triaxial deformation, which provide good candidates for the experiment to study the possibility of triaxial deformations.\nThe final aim of this project is to build a whole nuclear mass table including both triaxial degrees of freedom and dynamical correlation energies. The works following this line are still in progress.\n\n\\ack\nThe authors are grateful to Prof. Zhi Pan Li and Prof. PengWei Zhao for providing\nthe numerical computation codes and the fruitful discussions as well as critical readings\nof our manuscript.\nThis work was partly supported by the National Key R\\&D Program of China (Contract No. 2018YFA0404400 and No. 2017YFE0116700) and\nthe National Natural Science Foundation of China (Grants No. 11621131001, No. 11875075, No. 11935003, and No. 11975031).\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe test set problem is NP-hard. The polynomial time approximation\nalgorithms using in practice includes \"greedy\" heuristics\nimplemented by set cover criterion or by information\ncriterion\\cite{ms}. Test set can not be approximated within\n$(1-\\varepsilon)\\ln n$ for any $\\varepsilon>0$ unless $NP\\subseteq\nDTIME(n^{\\log\\log n})$\\cite{bh,bdk}. Recently, the authors of\n\\cite{bdk} designed a new information type greedy algorithm,\ninformation content heuristic (ICH for short), and proved its\nperformance guarantee $\\ln n+1$, which almost matches the\ninapproximability results.\n\nThe setcover greedy algorithm (SGA for short) is a natural\napproximation algorithm for test set. In practice, its average\nperformance is virtually the same as information type greedy\nalgorithms\\cite{ms,dk}. The performance guarantee $2\\ln n$ of SGA is\nobtained by transforming the test set problem as a set cover\nproblem. The authors of \\cite{bh} give the tight performance\nguarantee $11\/8$ of SGA on instances with the size of tests no\ngreater than 2.\n\nOblivious rounding, a derandomization technique to obtain simple\ngreedy algorithm for set cover problems by conditional probabilities\nwas introduced in \\cite{y}. Young observed the number of elements\nuncovered is an \"potential function\" and the approximation algorithm\nonly need to drive down the potential function at each step, thus he\nshowed another proof of the well-known performance guarantee $\\ln\nn+1$.\n\nIn this paper, the author presents a tighter analysis of SGA. The author uses\nthe potential function technique of \\cite{y} to improve the performance\nguarantee $2\\ln n$ which derives from set cover problem to $1.1354\\ln n$, and\nconstruct instances to give a nontrivial lower bound $1.0004609\\ln n$ of the\nperformance guarantee. The latter result confirms the fact ICH is slightly\nbetter than SGA in worst case. In this analysis, the author refers to the tight\nanalysis of the greedy algorithm for set cover problem in \\cite{s}.\n\nIn Section 2, the author shows the two main theorems, and some definitions,\nnotations and facts are given. In Section 3, the author analyzes\ndifferentiation distribution of item pairs and uses the potential function\nmethod to prove the improved performance guarantee. In Section 4, the author\nshows the nontrivial lower bound by constructing instances of test set with\narbitrary large size. Section 5 is some discussions.\n\n\\section{Overview}\nThe input of test set problem consists of $S$, a set of items (called\nuniverse), and $\\mathcal T$, a collection of subsets (called tests) of $S$. An\nitem pair $\\{i,j\\}$ is a subset of $S$ containing two different items of $S$. A\ntest $T$ differentiates item pair $\\{i,j\\}$ if $|T\\cap \\{i,j\\}|=1$. $\\mathcal\nT$ is a test set of $S$, i.e. any item pair of $S$ is differentiated by one\ntest in $\\mathcal T$. The objective is to find $\\mathcal T'\\subseteq \\mathcal\nT$ with minimum cardinality which is also a test set of $S$. We use $\\mathcal\nT^*$ to represent the optimal test set. Denote $n=|S|$, and $m^*=|\\mathcal\nT^*|$. In this paper, we assume $m^*\\ge 2$.\n\nAmong an instance of test set problem, there are $n\\choose 2$ different item\npairs. Let $i,j$ be two different items, and $S_{1},S_{2}$ be two disjoint\nsubsets of $S$. If $i,j\\in S_{1}$, we say $\\{i,j\\}$ is an item pair inside\n$S_{1}$, and if $i\\in S_{1}$ and $i\\in S_{2}$, we say $\\{i,j\\}$ is an item pair\nbetween $S_{1}$ and $S_{2}$.\n\nLet $S'$ be a subset of $S$, $\\mathcal T$ be a collection of tests of $S$, we\nsay $\\mathcal T$ is a test set of $S'$ iff any item pair inside $S'$ is\ndifferentiated by one test in $\\mathcal T$. Notice tests in $\\mathcal T$ may\ncontains items that are not in $S'$. Clearly, if $\\mathcal T$ is a test set of\n$S$, $\\mathcal T$ is a test set of $S'$.\n\nWe use $\\{i,j\\}\\perp T$ to represent that $T$ differentiates $\\{i,j\\}$ and\n$\\{i,j\\}\\parallel T$ to represent that $T$ does not differentiate $\\{i,j\\}$. We\nuse $\\{i,j\\}\\perp \\mathcal T$ to represent that at least one test in $\\mathcal\nT$ differentiates $\\{i,j\\}$, $\\{i,j\\}\\parallel \\mathcal T$ to represent that\nany test in $\\mathcal T$ does not differentiate $\\{i,j\\}$, and $\\perp\n(\\{i,j\\},\\mathcal T)$ to represent the number of tests in $\\mathcal T$ that\ndifferentiate $\\{i,j\\}$.\n\n{\\bf Fact 1.} \\it For three different items $i$, $j$ and $k$, if\n$\\{i,j\\}\\parallel \\mathcal T$ and $\\{i,k\\}\\parallel \\mathcal T$,\nthen $\\{j,k\\}\\parallel \\mathcal T$.\\rm\n\n{\\bf Fact 2.} \\it For three different items $i$, $j$ and $k$, and a test $T$,\nif $\\{i,j\\}\\perp T$ and $\\{i,k\\}\\perp T$, then $\\{j,k\\}\\parallel T$.\\rm\n\nGiven $\\mathcal T'\\subseteq \\mathcal T$, we define a binary relation\n$\\backsim_{\\mathcal T'}$ on $S$: for two item $i,j$,\n$i\\backsim_{\\mathcal T'}j$ iff $\\{i,j\\}\\parallel \\mathcal T'$. By\nFact 1, $\\backsim_{\\mathcal T'}$ is an equivalent relation. The\nequivalent classes containing $i$ is denoted as $[i]$.\n\n{\\bf Fact 3.} \\it If $\\mathcal T$ is a minimal test set, then\n$|\\mathcal T|\\le n-1$.\\rm\n\n{\\bf Fact 4.} \\it If $\\mathcal T$ is a test set, then $|\\mathcal\nT|\\ge \\log_{2} n$.\\rm\n\nTest set $\\mathcal T$ with $|\\mathcal T|=\\log_{2} n$ is\ncalled a compact test set. If $\\mathcal T$ is a compact test set,\nthen $|S|=2^q, q\\in Z^+$.\n\nIn set cover problem, we are given $U$, the universe, and $\\mathcal C$, a\ncollection of subsets of $U$. $\\mathcal C$ is a set cover of $U$, i.e.\n$\\bigcup_{c\\in \\mathcal C}=U$. The objective is to find $\\mathcal C'\\subseteq\n\\mathcal C$ with minimum cardinality which is also a set cover of $S$.\n\nThe greedy algorithm for set cover runs like that. In each\niteration, simply select a subset covering most uncovered elements,\nrepeat until all elements are covered, and return the set of\nselected subsets. Let $N$ be the size of the universe, and $M^*$ be\nthe size of the optimal set cover. The greedy algorithm for set\ncover has performance guarantee $\\ln N-\\ln\\ln N+\\Theta(1)$ by\n\\cite{s}.\n\nWe give two lemmas about the greedy algorithm for set cover. Lemma 1\nis a corollary of Lemma 2 in \\cite{s} and Lemma 2 is a corollary of\nLemma 1 and Lemma 4 in \\cite{s}.\n\n{\\bf Lemma 1.} \\it The size of set cover\nreturned by the greedy algorithm is at most $M^*(\\ln N-\\ln M^*+1)$.\\rm\n\n{\\bf Lemma 2.} \\it Given $N$ and $M^*$, there are instance of set\ncover problem such that the size of set cover returned by the greedy\nalgorithm is at least $(M^*-1)(\\ln N-\\ln M^*)$.\\rm\n\nTest set problem can be transformed to set cover problem in a\nnatural way. Let $(S,\\mathcal T)$ be an instance of test set, we\nconstruct an instance $(U,\\mathcal C)$ of set cover, where\n$U=\\{\\{i,j\\}|i,j\\in S,i\\ne j\\}$, and $\\mathcal C=\\{c(T)|T\\in\n\\mathcal T\\},c(T)=\\{\\{i,j\\}|i\\in T,j\\in S-T\\}.$\n\nClearly,  $\\mathcal T'$ is a test set of $S$ iff $\\mathcal\nC'=\\{c(T)|T\\in \\mathcal T'\\}$ is a set cover of $U$.\n\nSGA can be described as:\\\\\n\n\\indent{\\bf Input:} $S$,$\\mathcal T$;\\\\\n\\indent{\\bf Output:} a test set of $S$;\\\\\n\\indent$\\bar\\mathcal T\\leftarrow\\varnothing$;\\\\\n\\indent {\\bf while} $\\#(\\bar\\mathcal T)>0$ {\\bf do}\\\\\n\\indent\\indent select $T$ in $\\mathcal T-\\bar\\mathcal T$\nminimizing $\\#(\\bar\\mathcal T\\cup\\{T\\})$;\\\\\n\\indent\\indent $\\bar\\mathcal T\\leftarrow\\bar\\mathcal T\\cup\\{T\\}$;\\\\\n\\indent {\\bf endwhile}\\\\\n\\indent {\\bf return} $\\bar\\mathcal T$;\\\\\n\\rm\n\nIn SGA, we call $\\bar\\mathcal T$ the partial test set. The\ndifferentiation measure of $\\bar\\mathcal T$, $\\#(\\bar\\mathcal T)$,\nis defined as the number of item pairs not differentiated by\n$\\bar\\mathcal T$. The differentiation measure of $T$ w.r.t.\n$\\bar\\mathcal T$ is defined as $\\#(T,\\bar\\mathcal T)=\\#(\\bar\\mathcal\nT)-\\#(\\bar\\mathcal T\\cup \\{T\\})$.\n\nSGA is isomorphic to the greedy algorithm for set cover under the\nnatural transformation. Thus we immediately obtain the performance\nguarantee $2\\ln n$ of SGA. This paper shows a better performance\nguarantee and a nontrivial lower bound of performance guarantee. The\ntwo main theorems are:\n\n{\\bf Theorem 1.} \\it The performance guarantee of SGA can be\n$1.1354\\ln n$. \\rm\n\n{\\bf Theorem 2.} \\it There are arbitrarily large instances of test set problem\nsuch that the performance ratio of SGA on these instances is at least\n$1.0004609\\ln n$. \\rm\n\nIn this paper, denote $[n]:=\\{1,2,\\cdots,n\\}$. Denote $\\phi(x):=\\frac{1}{x}(\\ln\nx-1)$. The harmonious number is defined as $H_n:=\\sum_{i=1}^n{\\frac{1}{i}}$.\n\nTwo inequalities are listed here for convenience of proof in Section 3.\n\n{\\bf Fact 5.} \\it For any $0<x<1$, $(1-x)^{1\/x}<1\/e$. \\rm\n\n{\\bf Fact 6.} \\it For any $x>1$, $\\phi(x)\\le 1\/e^2=0.135\\cdots$. \\rm\n\n\\section{Improved Performance Guarantee}\n\n\\subsection{Differentiation Distribution}\nIn this subsection, the author analyzes the distribution of times for\nwhich item pairs are differentiated in instances of test set,\nespecially the relationship between the differentiation distribution\nand the size of the optimal test set.\n\n{\\bf Lemma 3.} \\it Given two disjoint subsets $S_1,S_2\\subseteq S$, and\n$\\mathcal T$, a set of tests of $S$, suppose $\\mathcal T$ is a test set of\n$S_1$ and a test set of $S_2$ , then at most $\\min(|S_1|,|S_2|)$ item pairs\nbetween $S_1$ and $S_2$ are not\ndifferentiated by any test in $\\mathcal T$.\\\\\n\\indent Proof. \\rm Suppose $|S_1|\\le |S_2|$. We claim for any item $i\\in S_1$,\nthere is at most one item $j$ in $S_2$ satisfying $\\{i,j\\}\\parallel \\mathcal\nT$. Otherwise there are two different items $j,k$ in $S_2$ such that\n$\\{i,j\\}\\parallel \\mathcal T$ and $\\{i,k\\}\\parallel \\mathcal T$, then by Fact 1\n, $\\{j,k\\}\\parallel \\mathcal T$, which contradicts $\\mathcal T$ is a test set\nof $S_2$. $\\Box$\n\n{\\bf Lemma 4.} \\it At most $n\\log_{2}n$ item pairs are\ndifferentiated by exactly one test in $\\mathcal T^*$.\\\\\n\\indent Proof. \\rm Let $B$ be the set of item pairs that are\ndifferentiated by exactly one test in $\\mathcal T^*$. We prove\n$|B|\\le n\\log_{2}n$ by induction. When $n=1$, $|B|=n\\log_{2}n$.\nSuppose the lemma holds for any $n\\le h-1$, we prove the lemma holds\nfor $n=h$.\n\nSelect $T\\in \\mathcal T^*$ such that $T\\neq \\varnothing$ and $T\\neq\nS$, then $|T|\\le h-1$, $|S-T|\\le h-1$. Since $\\mathcal T^*$ is a\ntest set of $T$, by induction hypothesis, at most $|T|\\log_{2}|T|$\nitem pairs inside $T$ are differentiated by exactly one test in\n$\\mathcal T^*$. Similarly, at most $|S-T|\\log_{2}|S-T|$ item pairs\ninside $S-T$ are differentiated by exactly one test in $\\mathcal\nT^*$.\n\nBy Lemma 3, at most $\\min(|T|,|S-T|)$ item pairs between $T$ and\n$S-T$ are not differentiated by any test in $\\mathcal T^*-\\{T\\}$.\nTherefore at most $\\min(|T|,|S-T|)$ item pairs between $T$ and\n$S-T$ are differentiated by exactly one test in $\\mathcal T^*$.\n\nW.l.o.g, suppose $|T|\\le |S-T|$, then\n\\begin{eqnarray*}\n&|B|&\\le|T|\\log_{2}|T|+|S-T|\\log_{2}|S-T|+|T|\\\\\n&&=|T|\\log_{2}(2|T|)+|S-T|\\log_{2}|S-T|\\\\\n&&\\le|T|\\log_{2}|S|+|S-T|\\log_{2}|S|\\\\\n&&=|S|\\log_{2}|S|.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 5.} \\it Given $S''\\subseteq S'\\subseteq S$, and $\\mathcal T$, a set\nof tests of $S'$, suppose $\\mathcal T$ is a test set of $S''$ and a test set of\n$S'-S''$ , then at most $|S'|\\log_2{|S'|}$ item pairs between $S''$ and\n$S'-S''$ are differentiated by exactly one test in $\\mathcal T$.\\\\\n\n\\indent Proof. \\rm Let $B$ be the set of item pairs between $S''$ and $S'-S''$\nwhich are differentiated by exactly one test in $\\mathcal T$. We prove that\n$|B|\\le |S'|\\log_2{|S|'}$ by induction. When $|S|=1$ and $|S|=2$, the lemma\nholds. Suppose the lemma holds for any $|S|\\le h-1$, $h\\ge 3$, we prove the\nlemma holds for $|S|=h$.\n\nSelect $T\\in \\mathcal T$ such that $T\\neq \\varnothing$ and $T\\neq S'$,\n then $|T|\\le h-1$, $|S'-T|\\le h-1$ (see Figure 1).\nSince $\\mathcal T-\\{T\\}$ is a test set of $S''\\cap T$ and a test\nset of $(S'-S'')\\cap T$, by induction hypothesis, at most\n$|T|\\log_2{|T|}$ item pairs between $S''\\cap T$ and $(S'-S'')\\cap T$\nare differentiated by exactly one test in $\\mathcal T$. Similarly,\nat most $|S'-T|\\log_{2}|S'-T|$ item pairs between $S''\\cap (S'-T)$\nand $(S'-S'')\\cap (S'-T)$\n are differentiated by exactly one test in $\\mathcal T$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,bb=140 290 440 560]{lemma.eps}\n\\caption{illustration of Lemma 5}\n\\end{center}\n\\end{figure}\n\nSince $\\mathcal T-\\{T\\}$ is a test set of $S''\\cap T$ and a test set\nof $(S'-S'')\\cap (S'-T)$, by Lemma 3, at most $\\min(|S''\\cap\nT|,|(S'-S'')\\cap (S'-T)|)$ item pairs between $S''\\cap T$ and\n$(S'-S'')\\cap (S'-T)$ are not differentiated by any test in\n$\\mathcal T-\\{T\\}$. Hence at most $\\min(|S''\\cap T|,|(S'-S'')\\cap\n(S'-T)|)$ item pairs between $S''\\cap T$ and $(S'-S'')\\cap (S'-T)$\nare differentiated by exactly one test in $\\mathcal T$. Similarly,\nat most $\\min(|(S'-S'')\\cap T|,|S''\\cap (S'-T)|)$ item pairs between\n$(S'-S'')\\cap T$ and $S''\\cap (S'-T)$ are differentiated by exactly\none test in $\\mathcal T$\n\nClearly,\n\\begin{eqnarray*}\n&|T|\\ge&\\min(|S''\\cap T|,|(S'-S'')\\cap (S'-T)|)\\\\\n&&+\\min(|(S'-S'')\\cap T|,|S''\\cap (S'-T)|).\n\\end{eqnarray*}\n\nW.l.o.g, suppose $|T|\\le |S'-T|$, then\n\\begin{eqnarray*}\n&|B|&\\le|T|\\log_{2}|T|+|S'-T|\\log_{2}|S'-T|+|T|\\\\\n&&=|T|\\log_{2}(2|T|)+|S'-T|\\log_{2}|S'-T|\\\\\n&&\\le|T|\\log_{2}|S'|+|S'-T|\\log_{2}|S'|\\\\\n&&=|S'|\\log_{2}|S'|.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 6.} \\it At most $n\\log_2{n}{m^*}^{t-1}$ item pairs are\ndifferentiated by exactly $t$ test in $\\mathcal T^*$, where $t\\ge\n2$.\\\\\n\\indent Proof. \\rm Let $B_{t}$ be the set of item pairs that are\ndifferentiated by exactly $t$ test in $\\mathcal T^*$. For any\ncombination $\\pi$ of $t-1$ tests in $\\mathcal T^*$, let $B_{\\pi}$\nbe the subset of $B_{t}$ such that each item pair in $B_{\\pi}$ is\ndifferentiated by any test in $\\pi$.\n\nLet $\\backsim_{\\pi}$ be the equivalent relation induced by\n $\\pi$. For any equivalent class $[i]$, there is exactly one\n equivalent class $[j]$, such that each item pair between $[i]$\n and $[j]$ is differentiated by any test in $\\pi$ (Fact 2).\n\nSince $\\mathcal T^*-\\pi$ is a test set of $[i]$ and a test set of\n$[j]$, by Lemma 5, at most $(|[i]\\cup[j]|)\\log_2{|[i]\\cup[j]|}$ item\npairs between $[i]$ and $[j]$ are differentiated by exactly one test\nin $\\mathcal T^*-\\pi$. In another word, at most\n$(|[i]\\cup[j]|)\\log_2{|[i]\\cup[j]|}$ item pairs between $[i]$ and\n$[j]$ are differentiated by exactly $t$ tests in $\\mathcal T^*$.\nHence\n\\begin{eqnarray*}\n&|B_{\\pi}|&\\le\\sum_{[i],[j]}{|[i]\\cup[j]|\\log_2{|[i]\\cup[j]|}}\\le n\\log_2{n}.\n\\end{eqnarray*}\n\nTherefore,\n\\begin{eqnarray*}\n&|B_{t}|&\\le\\sum_{\\pi}{|B_{\\pi}|}\\le{m^*\\choose\n{t-1}}n\\log_2{n}\\le n\\log_2{n}{m^*}^{t-1}.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 7.} \\it At most $2n\\log_2{n}{m^*}^{t-1}$ item pairs\nare differentiated by at most $t$ test in $\\mathcal T^*$, where\n$t\\ge 2$.\\\\\n\\indent Proof. \\rm Let $B$ be the set of item pairs that are\ndifferentiated by at most $t$ test in $\\mathcal T^*$, and $B_{t}$ be\nthe set of item pairs that are differentiated by exactly $t$ test in\n$\\mathcal T^*$. By Lemma 6,\n\\begin{eqnarray*}\n&|B|&=|B_{1}|+|B_{2}|+\\cdots+|B_{t}|\\\\\n&&\\le n\\log_2{n}(1+m^*+\\cdots+{m^*}^{t-1})\\\\\n&&\\le 2n\\log_2{n}{m^*}^{t-1}.\n\\end{eqnarray*}\ngu $\\Box$\n\n\\subsection{Proof of Theorem 1}\nIn this subsection, the author uses the potential function technique\nto derive improved performance guarantee of SGA for test set. Our\nproof is based on the trick to \"balance\" the potential function by\nappending a negative term to the differentiation measure.\n\nLet $I=\\lceil \\ln\\frac{n-1}{4\\log_2{n}}\/\\ln m^*\\rceil$, then\n$2n\\log_2{n}{m^*}^{I-1}<{n\\choose 2}\\le 2n\\log_2{n}{m^*}^I$. Let\n$\\#_{0}=1$, $\\#_{1}=n\\log_2{n}$, $\\#_{t}=2n\\log_2{n}{m^*}^{t-1},2\\le\nt\\le I$, and $\\#_{I+1}=n(n-1)\/2$. Let $k_{t}=\\frac{m^*}{t}\\ln\n\\frac{t\\#_{t}}{\\#_{t-1}}$, $2\\le t\\le I+1$.\n\nDenote by $p$ the probability distribution on tests in $\\mathcal\nT^*$ drawing one test uniformly from $\\mathcal T^*$. For any $T\\in\n\\mathcal T^*$ , the probability of drawing $T$ is\n$p(T)=\\frac{1}{m^*}$.\n\nWe divide a run of the algorithm into $I+1$ phases. For $I+1\\ge t\\ge\n1$, Phase $t$ begins when $\\#(\\bar\\mathcal T)\\ge\\#_{t-1}$ and lasts\nuntil $\\#(\\bar\\mathcal T)<\\#_{t-1}$. Phase $t$ is blank if when\nPhase $t+1$ ends, $\\#(\\bar\\mathcal T)<\\#_{t-1}$.\n\nLet the set of selected tests in Phase $t$ is $\\mathcal T_{t}$, the\npartial test set when Phase $t$ ends is $\\bar\\mathcal T_{t}$, and\nthe returned test set is $\\mathcal T'$. Then $\\bar\\mathcal\nT_{t}=\\cup_{t\\le s\\le I+1}\\mathcal T_{s}$, $1\\le t\\le I+1$, and\n$\\mathcal T'=\\bar\\mathcal T_{2}\\cup\\mathcal T_{1}$. Set\n$\\bar\\mathcal T_{I+2}=\\varnothing$. If Phase $t$ is not blank, let\nthe last selected test in Phase $t$ is $T'_{t}$.\n\nIn Phase $t$, $I+1\\ge t\\ge 2$, define the potential function as\n$$f(\\bar\\mathcal T)=(\\#(\\bar\\mathcal T)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\n\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal T-\\bar\\mathcal T_{t+1}|}.$$\n\nBy the definition of $\\bar\\mathcal T_{t+1}$ and Fact 5,\n$$f(\\bar\\mathcal T_{t+1})<(\\#_{t}-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\\frac{t}{m^*})^{k_{t}}\n<\\frac{\\#_{t-1}}{t}.$$\n\nBy the definition of $f(\\bar\\mathcal T)$ and the facts $p(T)\\ge 0$\nand $\\sum_{T\\in \\mathcal T^*}{p(T)}=1$,\n\\begin{eqnarray*}\n&&\\min_{T\\in \\mathcal T}{f(\\bar\\mathcal T\\cup \\{T\\})}\\\\\n&&\\le\\min_{T\\in \\mathcal T^*}{f(\\bar\\mathcal T\\cup \\{T\\})}\\\\\n&&\\le\\sum_{T\\in \\mathcal T^*}({p(T)}f(\\bar\\mathcal T\\cup\n\\{T\\}))\\\\\n&&=(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1}-\\sum_{T\\in \\mathcal\nT^*}{(p(T)\\#(T,\\bar\\mathcal\nT))})(1-\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal T-\\bar\\mathcal\nT_{t+1}|-1}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&&\\sum_{T\\in \\mathcal\nT^*}{(p(T)\\#(T,\\bar\\mathcal T))}\\\\\n&&=\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T}{\\sum_{T\\in\n\\mathcal T^*:\\{i,j\\}\\perp T}{p(T)}}\\\\\n&&\\ge\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T}{\\frac{t}{m^*}}\n-\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T:\\perp(\\{i,j\\},\\mathcal T^*)\\le t-1}{\\frac{t-1}{m^*}}\\\\\n&&\\ge(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})\\frac{t}{m^*}\n\\end{eqnarray*}\nby Lemma 4 and Lemma 7.\n\nTherefore,\n$$\\min_{T\\in \\mathcal T}{f(\\bar\\mathcal T\\cup\\{T\\})}\\le(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal\nT-\\bar\\mathcal T_{t+1}|}=f(\\bar\\mathcal T).$$\n\nDuring Phase $t$, the algorithm selects $T$ in $\\mathcal T$ to\nminimize $f(\\bar\\mathcal T\\cup\\{T\\}$). Therefore, $f(\\bar\\mathcal\nT_{t}-\\{T'_{t}\\})\\le f(\\bar\\mathcal T_{t+1})<\\frac{\\#_{t-1}}{t}$.\n\nOn the other hand, $\\#(\\bar\\mathcal T_{t}-\\{T'_{t}\\})\\ge \\#_{t-1}$\nby definition of Phase $t$. Hence\n$$f(\\bar\\mathcal T_{t}-\\{T'_{t}\\})=\\frac{\\#_{t-1}}{t}(1-\\frac{t}{m^*})^{k_{t}-|\\mathcal\nT_{t}-\\{T'_{t}\\}|}.$$\n\nTherefore, $(1-\\frac{t}{m^*})^{k_{t}-|\\mathcal\nT_{t}-\\{T'_{t}\\}|}<1$, $|\\mathcal T_{t}-\\{T'_{t}\\}|<k_{t}$, and\n$|\\mathcal T_{t}|<k_{t}+1$.\n\nTo sum up,\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&<\\sum_{2\\le t\\le I+1}{k_{t}}+I\\\\\n&&=m^*(\\sum_{2\\le t\\le I+1}{\\frac{1}{t}\\ln\\frac\n{\\#_{t}}{\\#_{t-1}}}+\\sum_{2\\le t\\le I+1}{\\frac{\\ln t}{t}})+I.\n\\end{eqnarray*}\n\nWhen all Phase $t$, $I+1\\ge t\\ge 2$, end, we obtain an instance of\nset cover $(U,\\mathcal C)$, where $U=\\{\\{i,j\\}|\\{i,j\\}\\parallel\n\\bar\\mathcal T_{2}\\}$ and $\\mathcal C=\\{c(T)|c(T)\\cap U\\ne\n\\varnothing\\}$. Clearly, $|U|<\\#_{1}$. Let $M^*$ be the size of the\noptimal set cover of this instance. Then $|M^*|\\le m^*$.\n\nConsider the following two cases: (a)$|M^*|\\le \\frac{m^*}{2}$;\n(b)$|M^*|>\\frac{m^*}{2}$.\n\nIn case (a),\n$$|\\mathcal T_{1}|\\le M^*(\\ln\\#_{1}+1)\\le m^*(\\textstyle\\frac12\\displaystyle+o(1))\\ln n,$$\n\n\\noindent and\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&\\le m^*(\\sum_{2\\le t\\le I+1}{\\frac12\\ln\\frac\n{\\#_{t}}{\\#_{t-1}}}+\\frac12 \\ln^2 (I+2))+I\\\\\n&&=m^*(\\textstyle\\frac12\\displaystyle+o(1))\\ln n.\n\\end{eqnarray*}\n\nHence\n$$|\\mathcal T'|=m^*(1+o(1))\\ln n.$$\n\nIn case (b), by Lemma 1,\n\\begin{eqnarray*}\n&|\\mathcal T_{1}|&\\le M^*(\\ln\\#_{1}-\\ln M^*+1)=m^*((1+o(1))\\ln n-\\ln\nm^*),\n\\end{eqnarray*}\n\n\\noindent and\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&< m^*(H_{I+1}\\ln m^*+\\frac12\\ln 2+\\frac12 \\ln^2 (I+2))+I\\\\\n&&\\le m^*(\\ln \\frac{\\ln n}{\\ln m^*}\\ln m^*+o(1)\\ln n).\n\\end{eqnarray*}\n\nNotice $\\frac{\\ln n}{\\ln m^*}>1$, by Fact 3.\n\nBy Fact 6,\n$$|\\mathcal T'|\\le m^*(1+\\phi(\\frac{\\ln n}{\\ln m^*})+o(1))\\ln n\n\\le m^*(1.13533\\cdots+o(1))\\ln n.$$\n\n\\section{Lower Bound}\nIn this section, we discuss a variation of test set problem. Given disjoint\nsets $S^{1},\\cdots,S^{r}$ and $\\mathcal T$, set of subsets of the universe\n$S=S^{1}\\cup\\cdots\\cup S^{r}$, we seek $\\mathcal T'\\subseteq \\mathcal T$  with\nminimum cardinality which is a test set of any $S^{p}$ for $1\\le p \\le r$.\nDenote the instance by $(S^p;\\mathcal T)$.\n\nIn our construction, $r=2N-J!2^{q}$, let $2^{k-1}<N^2\\le 2^{k}$, we could use\nthe copy-split trick similar to that used in \\cite{bh} to make $2^{k}$ copies\nof $S^{1},\\cdots,S^{r}$, split all copies by $k$ tests and split all $S^{p}$\nfor each copy by $r-1$ tests. Thus the splitting overhead for each copy could\nbe ignored.\n\nSuppose $\\hat\\mathcal T$ is a compact test set of $[2^q]$. For example, we can\nlet $\\hat\\mathcal T=\\{T_{k}|k\\in[q]\\}$, where $T_{k}$ contains integer $x$\nbetween $1$ and $2^q$ such that the $k$-th bit of $x$'s binary representation\nis $1$.\n\n\\subsection{Atom Instances}\nFirstly, we give the level-$t$ atom instances $(S_t^{y};\\mathcal T)$. The\nuniverse $S_t$ includes integral points in $(t+1)$-dimension Euclid space.\\\\\n\n{\\bf Construction of Atom Instances.} $S_t=\\bigcup_{y}{S_t^{y}}$.\n$S_t^{y}=\\{(x_{1},\\cdots,x_{t},y)|x_{i}\\in [2^q]\\}$, $1\\le y\\le 2^{q-2}$.\n$\\mathcal T_t=\\mathcal T^*_t\\cup \\mathcal T'_t$. $\\mathcal T^*_t=\\mathcal\nT^*_{t,1}\\cup\\cdots\\cup\\mathcal T^*_{t,t}$. $\\mathcal\nT^*_{t,i}=\\{T^*_{t,i;j}|j\\in[2^{q}]\\}$, $1\\le i\\le t$. $\\mathcal T'_t=\\mathcal\nT'_{t,1}\\cup\\cdots\\cup\\mathcal T'_{t,t}$, $\\mathcal\nT'_{t,i}=\\{T'_{t,i;j,k}|j\\in[2^{q-2}],k\\in[q]\\}$, $1\\le i\\le t$.\n\n$|\\mathcal T^*_t|=t 2^{q}$, and $|\\mathcal T'_{t}|=\\frac{q}{4}t 2^{q}$.\n$T^*_{t,i;j}$ contains points in $S_t^{y}$ with $x_{i}=j$. $T'_{t,i;j,k}$\ncontains points in $S_t^{j}$ with $x_{i}$ in the $k$-th test in $\\hat\\mathcal\nT$. We assign an order to tests $T'_{t,i;j,k}$ in $\\mathcal T'_t$, called {\\it\nnatural order}, as the\nlexical order of $(i,j,k)$.\\\\\n\nAn atom instance with $q=3$ and $t=2$ is shown in Figure 2.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth,bb=70 330 480 520]{atom.eps}\n\\caption{atom instance with $q=3$ and $t=2$}\n\\end{center}\n\\end{figure}\n\nWe claim SGA could return $\\mathcal T'_t$ according to their natural order in\nthe atom instance.\n\nAt the beginning of the algorithm, the differentiation measure of tests in\n$\\mathcal T'_t$ is $2^{2qt-2}$, and the differentiation measure of tests in\n$\\mathcal T_t^*$ is\n$$2^{q(t-1)}(2^{qt}-2^{q(t-1)})2^{q-2}=2^{2qt-2}(1-2^{-q}).$$\n\nThe algorithm could first select tests in $\\mathcal T'_{t,1}$ according to\ntheir natural order. After that, the differentiation measure of tests in\n$\\mathcal T'_t-\\mathcal T'_{t,1}$ decreases by a factor $2$, and the\ndifferentiation measure of tests in $\\mathcal T_t^*$ decreases by a factor at\nleast $2$. Hence, the algorithm could subsequently select tests in $\\mathcal\nT'_{t,2},\\cdots,\\mathcal T'_{t,t}$ according to their natural order.\n\n\\subsection{Level-$t$ Instances}\nSecondly, we construct a series of level-$t$ instances $(S_{t}^{y,z,w};\\mathcal\nT_{t})$, $1\\le t\\le J$ based on the atom instances. Tests in the atom instances\nare {\\it stretched} in $w$ dimension and {\\it cloned} in $z$ dimension. Let\n$M^*=J!2^q$, and $N=J!2^{q(J+1)}$. The universe $S_t$ includes $N$ integral\npoints in $(t+3)$-dimension Euclid space.\\\\\n\n{\\bf Construction of Level-$t$ Instances.} $S_t=\\bigcup_{y,z,w}{S_t^{y,z,w}}$.\\\\\n$S_t^{y,z,w}=\\{(x_{1},\\cdots,x_{t},y,z,w)|x_{i}\\in[2^q]\\}$, $1\\le y\\le\n2^{q-2},1\\le z\\le\\frac{J!}{t},1\\le w\\le t2^{q(J-t)+2}$. $\\mathcal T_t=\\mathcal\nT^*_t\\cup \\mathcal T'_t$. $\\mathcal T^*_t=\\mathcal\nT^*_{t,1}\\cup\\cdots\\cup\\mathcal T^*_{t,t}$. $\\mathcal\nT^*_{t,i}=\\{T^*_{t,i;j,l}|j\\in[2^{q}],l\\in[\\frac{J!}{t}]\\}$, $1\\le i\\le t$.\n$\\mathcal T'_t=\\mathcal T'_{t,1}\\cup\\cdots\\cup\\mathcal T'_{t,t}$, $\\mathcal\nT'_{t,i}=\\{T'_{t,i;j,k,l}|j\\in[2^{q-2}],k\\in[q],l\\in[\\frac{J!}{t}]\\}$, $1\\le\ni\\le t$.\n\n$|\\mathcal T^*_t|=M^*$, and $|\\mathcal T'_{t}|=\\frac{qM^*}{4}$. $T^*_{t,i;j,l}$\ncontains points in $S_t^{y,l,w}$ with $x_{i}=j$ for any $w$. $T'_{t,i;j,k,l}$\ncontains points in $S_t^{j,l,w}$ with $x_{i}$ in the $k$-th test in\n$\\hat\\mathcal T$ for any $w$. We assign an order to tests $T'_{t,i;j,k,l}$ in\n$\\mathcal T'_t$,\ncalled {\\it natural order}, as the lexical order of $(i,j,k,l)$.\\\\\n\nWe claim SGA could select all tests in $\\mathcal T'_{t,1}$ according to their\nnatural order in the first phase of the algorithm.\n\nAt the beginning of the algorithm, it is easy to prove that the differentiation\nmeasure of tests in $\\mathcal T'_{t,1}$ is $\\#^{begin}_t=2^{q(t-1)}N$, and the\ndifferentiation measure of tests in $\\mathcal T_t^*$ is\n$2^{q(t-1)}(1-2^{-q})N$.\n\nThe algorithm could select tests in $\\mathcal T'_{t,1}$ according to their\nnatural order while the differentiation measure of the selected test is kept\nequal to the differentiation measures of tests in $\\mathcal T'_{t,i}$ for $2\\le\ni\\le t$ and no less than the differentiation measure of any test in $\\mathcal\nT_t^*$.\n\nWhen the algorithm select the last test in $\\mathcal T'_{t,1}$, its\ndifferentiation measure is $\\#^{end}_t=2\\cdot 2^{q(t-2)}N$.\n\nThe algorithm could subsequently select tests in $\\mathcal\nT'_{t,2},\\cdots,\\mathcal T'_{t,t}$ according to their natural order, and\nreturns $\\mathcal T'_t$.\n\nWe explicitly show the three claims for proof of Section 4.4. \\\\\n\n{\\bf Claim 1.} \\it In each step of SGA, the differentiation measure of the\nselected test is no less than the differentiation measure of a subsequent test\naccording to their natural order.\\rm \\\\\n\n{\\bf Claim 2.} \\it In each step of SGA, the differentiation measure of the\nselected test in every step is no less than any test in $\\mathcal T_t^*$.\\rm\\\\\n\n{\\bf Claim 3.} \\it In the first phase of SGA, the differentiation measure of\nthe selected test in every step is at most $\\#^{begin}_t$ and at least\n$\\#^{end}_t$.\\rm\n\n\\subsection{Complete Instances}\nLet $(U,\\mathcal C)$ be the instance in Lemma 2, $U=\\{e_{1},\\cdots,e_{N}\\}$,\n$\\mathcal C^*$ be the optimal set cover, and $\\mathcal C'$ be the set cover\nreturned by the greedy algorithm. Construct an instance of test set\n$(S_0^{p};\\mathcal T_{0})$. $S_0=\\bigcup_{p}{S_0^p}$.\n$S_0^{p}=\\{e_{p},f_{p}\\}$, $1\\le p\\le N$. $\\mathcal T_{0}=\\mathcal\nT'_{0}\\cup\\mathcal T^*_{0}$. $\\mathcal T'_{0}=\\mathcal C'$. $\\mathcal\nT^*_{0}=\\mathcal C^*$.\n\nOn $(S_0^{p};\\mathcal T_{0})$, the algorithm could select all the tests in\n$\\mathcal T'_{0}$, the differentiation measure of selected tests ranges from\n$\\#^{begin}_0=N\/M^*$ to $\\#^{end}_0=1$ by the proof of Lemma 1 in \\cite{s}.\n\nConsequently, we construct a series of level-$t$ instances\n$(S_{t}^{y,z,w};\\mathcal T_{t})$, $1\\le t\\le J$. We modify tests in $\\mathcal\nT'_{t,1}$ by two operations: Enlargement and Merging. In the Enlargement\noperation, tests in $\\mathcal T'_{t,1}$ are enlarged by a factor $2$. In the\nMerging operation, tests in $\\mathcal T'_{t,i}$ for $i\\ge 2$ are merged to\ntests in $\\mathcal T'_{s,1}$ for $t-1\\ge\ns\\ge 1$. \\\\\n\n{\\bf Enlargement.} Let $\\mathcal T'_{t,1}=\\{T'_{t,1;j,k,l}|j\\in[2^{q-3}],k\\in[\nq],l\\in[\\frac{J!}{t}]\\}$ for $J\\ge t\\ge 1$. $T_{t,1;j,k,l}$ contains points in\n$S_t^{2j-1,l,w}$ and $S_t^{2j,l,w}$ with $x_{1}$ in the $k$-th test in\n$\\hat\\mathcal T$ for any $w$. As a result, $|\\mathcal\nT'_{t,1}|=\\frac{qM^*}{8t}$.\n\n{\\bf Merging.} By the decreasing order of $t$ for $J\\ge t\\ge 2$, merge tests in\n$\\mathcal T'_{t}-\\mathcal T'_{t,1}$ by their natural order one-by-one to tests\nin $\\mathcal T'_{s,1}$ for $t-1\\ge s\\ge 1$ by the decreasing order of $s$\n(primarily) and their natural order in $\\mathcal T'_{s,1}$ until tests in\n$\\mathcal T'_{t}-\\mathcal T'_{t,1}$ are exhausted.\\\\\n\nThe illustration of the Merging operations with $J=4$ is shown in Figure 3.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth,bb=160 310 460 510]{merging.eps}\n\\caption{Merging operation with $J=4$}\n\\end{center}\n\\end{figure}\n\nFor any $J\\ge t\\ge 2$, tests in $\\mathcal T'_{s,1}$ suffice in the Merging\noperation, provided that\n$$|\\mathcal T'_{t}-\\mathcal T'_{t,1}|=\\frac14(1-\\frac{1}{t})qM^*\n\\le\\frac{1}{8}H_{t-1}qM^*=|\\bigcup_{s=1}^{t-1}\\mathcal T'_{s,1}|.$$\n\nLet $\\mathcal T'=\\mathcal T'_{0}\\cup\\bigcup_{t=1}^{J}\\mathcal T'_{t,1}$ after\nthe two operations are performed. Note that after the Merging operation, tests\nin $\\mathcal T'_{t,1}$ for $J-1\\ge t\\ge 1$ will contain items from $S_{s}$ for\nall $J\\ge s\\ge t$. Let $\\mathcal T=\\mathcal T^*\\cup\\mathcal T'$.\n\nThe complete instance is $(S_0^{p},S_{t}^{y,z,w};\\mathcal T^*\\cup \\mathcal\nT')$. Let $S=\\bigcup_{t=0}^{J}{S_t}$, then $n=|S|=(J+1)N$. Suppose $\\mathcal\nT^*_{t}=\\{T^*_{t,1},\\cdots,T^*_{t,M^*}\\}$, $0\\le t\\le J$, let $\\mathcal\nT^*=\\{T^*_{0,1}\\cup\\cdots\\cup T^*_{J,1},\\cdots,T^*_{0,M^*}\\cup\\cdots\\cup\nT^*_{J,M^*}\\}$. Since $\\mathcal T^*_{0}$ is an optimal test set of $S_0$,\n$\\mathcal T^*$ is an optimal solution of the complete instance, and\n$m^*=|\\mathcal T^*|=M^*$.\n\n\\subsection{Proof of Theorem 2}\nWe intend to prove the performance ratio of SGA on this instance is at least\n$(1+\\frac{1}{J+1}(\\frac{H_{J}}{8\\ln 2}-1)-o(1))\\ln n$, for fixed $J$. When\n$J=391$, this performance ratio is at least $1.0004609\\ln n$. To accomplish the\nproof, we analyze the behavior of SGA on the complete instance. Indeed, the\nsequence of the selected tests projected onto $S_t$ is almost the same as the\nselected tests of SGA on the original level-$t$ instance for $J\\ge t\\ge 1$,\nexcept that the tests in $\\mathcal T'_{t,1}$ are enlarged.\n\nBefore the algorithm selects a test, let $\\#_t$ be the maximum differentiation\nmeasure of tests in $\\mathcal T'_{t,1}$ for $J\\ge t\\ge 1$, and $\\#^*$ to the\nmaximum differentiation measure of tests in $\\mathcal T^*$.\n\nLet $\\#_{t,s}$ be the number of item pairs inside of $S_s$ contributing to\n$\\#_t$ for $t\\le s\\le J$, $\\#_{t-1,s}$ be the number of item pairs inside $S_s$\ncontributing to $\\#_{t-1}$ for $t-1\\le s\\le J$,  $\\#_s^*$ be the number of item\npairs inside $S_s$ contributing to $\\#^*$ for $1\\le s\\le J$, , and $\\#_0^*$ be\nthe number of item pairs inside $S_0^{p}$ contributing to $\\#^*$. Then\n$\\#_t=\\sum_{s=t}^{J}{\\#_{t,s}}$, $\\#_{t-1}=\\sum_{s=t-1}^{J}{\\#_{t-1,s}}$, and\n$\\#^*=\\sum_{s=0}^{J}{\\#_s^*}$.\n\nNote that $\\#_{t,t}$ for $J\\ge t\\ge 1$ is twice of the corresponding number in\nthe original level-$t$ instance. Remember $\\#^{begin}_s=2^{q(s-1)}N$,\n$\\#^{end}_s=2\\cdot 2^{q(s-2)}N$, for $J\\ge s\\ge 1$, and\n$\\#^{begin}_0=\\frac{1}{J!}2^{-q}N$.\n\nBy Claim 1, $\\#_{t,s}\\ge \\#_{t-1,s}$ for $s>t$ and by Claim 3,\n$$\\#_{t,t}=2\\#_{t-1,t}\\ge\\#_{t-1,t}+\\#^{end}_t=\\#_{t-1,t}\n+2\\#^{begin}_{t-1}\\ge\\#_{t-1,t}+\\#_{t-1,t-1},$$\n\n\\noindent it follows that $\\#_t\\ge\\#_{t-1}$. Hence $\\#_t\\ge\\#_{s}$, for any\n$1\\le s< t$.\n\nBy Claim 2, $\\#_{t,s}\\ge \\#_s^*$ for $s>t$ and by Claim 3,\n$$\\#_{t,t}\\ge \\#_t^*+\\#^{end}_{t}= \\#_t^*+2\\#^{begin}_{t-1}\\ge\n\\#_t^*+\\sum_{s=0}^{t-1}\\#^{begin}_{s}\\ge \\#_t^*+\\sum_{s=0}^{t-1}{\\#^*_{s}},$$\n\n\\noindent it follows that $\\#_t\\ge\\#^*$.\n\nWe conclude the algorithm could select all tests in $\\mathcal T'_{t,1}$ in\ntheir natural order, for $J\\ge t\\ge 1$, and select all tests in $\\mathcal\nT'_{0}$, finally return $\\mathcal T'$.\n\nRemember $M^*=J!2^q$, $N=J!2^{q(J+1)}$, $m^{*}=M^{*}$, and $n=(J+1)N$. In the\ncondition $J$ is fixed, the size of returned solution is\n\\begin{eqnarray*}\n&|\\mathcal T'|&\\ge(M^*-1)(\\ln N-\\ln M^*)+\\frac{qM^*}{8}H_{J}\\\\\n&&= m^*(1+\\frac{1}{J+1}(\\frac{H_{J}}{8\\ln 2}-1)-o(1))\\ln n.\n\\end{eqnarray*}\n\\section{Discussion}\nThe author notes this is the first time to distinguish precisely the\nworst case performance guarantees of two types of \"greedy\nalgorithms\" implemented by set cover criterion and by information\ncriterion. In fact, the author definitely shows the pattern of\ninstances on which ICH performs better than SGA.\n\nIn a preceding paper\\cite{cl}, we proved the performance guarantee\nof SGA can be $(1.5+o(1))\\ln n$, and the proof can be extended to\nweighted case, where each test is assigned a positive weight, and\nthe objective is modified as to find a test set with minimum total\nweight.\n\nIn the minimum cost probe set problem\\cite{bc} of bioinformatics,\ntests are replaced with partitions of items. The objective is to\nfind a set of partitions with smallest cardinality to differentiate\nall item pairs. It is easily observed that the improved\nperformance guarantee in this paper is still applicable to this generalized case.\\\\\n\n\\noindent{\\bf Acknowledgements.} The author would like to thank Tao\nJiang and Tian Liu for their helpful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec introduction}\n\\setcounter{equation}{0}\n\n\nLet $X$ be a proper non compact metric space and  $o$ be a point in $X$. Given a discrete group $\\Delta$ of isometries of $X$, consider the \\textit{orbital counting function}\n\n\n\n\\begin{center}\n$N_\\Delta(o,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_X(o,g\\cdot o)\\leq t \\rbrace$,\n\\end{center}\n\n\\noindent where $t\\geq 0$. The \\textit{orbital counting problem} consists on the study of the asymptotic behaviour of $N_\\Delta(o,t)$ as $t\\longrightarrow\\infty$.\n\n\n\nWhen $X=\\mathbb{R}^2$ and $\\Delta=\\mathbb{Z}^2$ this is known as the \\textit{Gauss circle problem} (see Phillips-Rudnick \\cite{PR}). For a negatively curved Hadamard manifold $X$ and $\\Delta$ co-compact, this problem was studied by Margulis in his PhD Thesis \\cite{Mar}: the author shows a purely exponential asymptotic for $N_\\Delta(o,t)$, the exponent being the topological entropy of the geodesic flow of the quotient space $\\Delta \\backslash X$. Many authors have generalized the work of Margulis to different contexts, see Roblin \\cite{Rob} and references therein for a fairly complete picture in the negatively curved setting. \n\n\n\nWhen $X$ is a (not necessarily Riemannian) symmetric space associated to a semisimple Lie group $G$ and $\\Delta <G$ is a lattice, these kind of problems were studied notably by Eskin-McMullen \\cite{EMcM} and Duke-Rudnick-Sarnak \\cite{DRS}. In the non-lattice case but restricted to Riemannian symmetric spaces, one also finds the work of Quint \\cite{Qui} and Sambarino \\cite{Sam2}. Quint deals with the case in which $\\Delta$ is a Schottky group (in the sense of Benoist \\cite{Ben4}). Sambarino treats more generally the case of Anosov subgroups (in the full flag variety of $G$) introduced by Labourie \\cite{Lab}.\n\n\n\n\n\nBefore stating precise results we discuss in an informal way the problems adressed by this paper. Fix $d:=p+q$ where $p\\geq 1$ and $q\\geq 2$, and let $\\langle\\cdot,\\cdot\\rangle_{p,q}$ be the bilinear symmetric form on $\\mathbb{R}^{d}$ defined by\n\n\\begin{center}\n$ \\langle (x_1,\\dots,x_d),(y_1,\\dots,y_d)\\rangle_{p,q}:= \\displaystyle\\sum_{i=1}^p x_iy_i - \\displaystyle\\sum_{i=p+1}^d x_{i}y_{i}$.\n\\end{center}\n\n\n\\noindent We denote by $G:=\\textnormal{PSO}(p,q)$ the group of projectivized matrices in $\\textnormal{SL}(d,\\mathbb{R})$ preserving $\\langle\\cdot,\\cdot\\rangle_{p,q}$ and by $X_G$ the \\textit{Riemannian symmetric space} of $G$, that is, the space of $q$-dimensional subspaces of $\\mathbb{R}^{d}$ on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ is negative definite. Let $d_{X_G}$ be the distance in $X_G$ induced by a $G$-invariant Riemannian metric. For closed subsets $A$ and $B$ of $X_G$, set\n\n\\begin{center}\n$d_{X_G}(A,B):=\\inf\\lbrace d_{X_G}(a,b):\\hspace{0,3cm} a\\in A, b\\in B\\rbrace$.\n\\end{center}\n\n\n\\noindent On the other hand, the \\textit{pseudo-Riemannian hyperbolic space of signature $(p,q-1)$} is the set\n\n\\begin{center}\n$\\mathbb{H}^{p,q-1}:=\\left\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}<0  \\right\\rbrace$,\n\\end{center}\n\n\n\\noindent endowed with a $G$-invariant pseudo-Riemannian metric coming from restriction of the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ to tangent spaces.\n\nLet $\\Delta$ be a discrete subgroup of $G$ and fix a point $o$ in $\\mathbb{H}^{p,q-1}$. In this paper we study counting problems in $X_G$ and in $\\mathbb{H}^{p,q-1}$.\n\n\n\\begin{itemize}\n\\item \\textbf{Counting in $X_G$:} Denote by\n\n\n\\begin{center}\n$S^o:=\\lbrace \\tau\\in X_G:\\hspace{0,3cm} o\\subset \\tau \\rbrace $.\n\\end{center}\n\n\n\n\n\\noindent It is a totally geodesic sub-manifold of $X_G$ isometric to the Riemannian symmetric space of $\\textnormal{PSO}(p,q-1)$. We define two counting functions in this setting. The first one is\n\n\n\\begin{center}\n$N_\\Delta(S^o,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{X_G}(S^o,g \\cdot S^o)\\leq t\\rbrace$.\n\\end{center}\n\n\n \n\n\\noindent For the second one we pick a point $\\tau\\in S^o$ and define\n\n\\begin{center}\n$N_\\Delta(S^o,\\tau,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{X_G}(\\tau,g\\cdot S^o)\\leq t\\rbrace$.\n\\end{center}\n\n\\item \\textbf{Counting in $\\mathbb{H}^{p,q-1}$:} We provide a geometric interpretation of the function $N_\\Delta(S^o,t)$ in $\\mathbb{H}^{p,q-1}$. It is the amount of \\textit{space-like} geodesic segments\\footnote{That is, geodesic segments which are tangent to positive vectors.} of length at most $t$, that connect $o$ with points of $\\Delta\\cdot o$. The function $N_\\Delta(S^o,\\tau,t)$ has a geometric interpretation in this setting as well, which is more involved, and that we postpone until Subsection \\ref{subsec counting hpq introd}.\n\n\\end{itemize}\n\n\n\\begin{rem}\nIf $q=1$ one has $\\mathbb{H}^{p}=X_G=\\mathbb{H}^{p,q-1}$ and $o=S^o=\\tau$. We have as well the equalities\n\\begin{center}\n$N_\\Delta(o,t)=N_\\Delta(S^o,t)=N_\\Delta(S^o,\\tau,t)$\n\\end{center} \n\n\\noindent and our results correspond to the classical and well-known counting theorems already quoted.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nIn contrast with the counting function $N_\\Delta(o,t)$ described at the beginning, the functions $N_\\Delta(S^o,t)$ and $N_\\Delta(S^o,\\tau,t)$ could in general be equal to infinity for large values of $t$. Part of the results that we present here concern the study of conditions for the choice of $o$ (and $\\tau$) that guarantee that the new counting functions are real-valued for every $t\\geq 0$. Once this is established, one may ask if the \\textit{exponential growth rate}\n\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log N_\\Delta(S^o,t)}{t}$\n\\end{center}\n\n\n\\noindent is positive, finite and independent on the choice of $o$ (and the analogue questions for $N_\\Delta(S^o,\\tau,t)$). A more subtle problem is to find an asymptotic for the functions $N_\\Delta(S^o,t)$ and $N_\\Delta(S^o,\\tau,t)$ as $t\\longrightarrow\\infty$. The main goal of this paper is to give an answer to this more subtle problem for an interesting class of subgroups $\\Delta$: images of word hyperbolic groups under \\textit{projective Anosov representations}. \n\n\n\n\n\n\n\n\\subsection{Main results in $X_G$} \\label{subsec counting XG introd}\n\n \nIn order to formally state our results we need to recall some basic facts concerning (projective) Anosov representations. Anosov representations are (a stable class of) faithful and discrete representations from word hyperbolic groups into semisimple Lie groups that share many geometrical and dynamical features with holonomies of convex co-compact hyperbolic manifolds. They were introduced by Labourie \\cite{Lab} in his study of the Hitchin component and further extended to arbitrary word hyperbolic groups by Guichard-Wienhard \\cite{GW}. After that, Anosov representations had been object of intensive research in the field of geometric structures on manifolds and their deformation spaces (see for instance the surveys of Kassel \\cite{Kas} or Wienhard \\cite{Wie} and references therein).\n\n\n\n\n\n\n\n\nLet $P_1^{p,q}$ be the stabilizer of an isotropic line in $\\mathbb{R}^d$, i.e. a line on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ equals zero. Then $P_1^{p,q}$ is a parabolic subgroup of $G$ and the quotient space $\\partial\\mathbb{H}^{p,q-1}:= G\/P_1^{p,q}$, called the \\textit{boundary} of $\\mathbb{H}^{p,q-1}$, identifies with the set of isotropic lines in $\\mathbb{R}^d$.\n\nFix a non elementary word hyperbolic group $\\Gamma$ and let $\\partial_\\infty\\Gamma$ be its Gromov boundary. Let $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation. By definition (see Section \\ref{sec anosov}) this means that there exists a continuous equivariant map\n\n\n\n\n\\begin{center}\n$\\xi:\\partial_\\infty\\Gamma\\longrightarrow \\partial\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\\noindent with the following properties:\n\n\\begin{itemize}\n\\item \\textbf{Transversality:} Let $\\cdot^{\\perp_{p,q}}$ denote the orthogonal complement with respect to the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$. Then the map $\\eta:=\\xi^{\\perp_{p,q}}$ satisfies $\\xi(x)\\oplus\\eta(y)=\\mathbb{R}^d$ for every $x\\neq y$ in $\\partial_\\infty\\Gamma$.\n\\item \\textbf{Uniform hyperbolicity:} Some flow associated to $\\rho$ satisfies a uniform contraction\/dilation property (see \\cite{Lab,GW}).\n\\end{itemize}\n\n\n\nWhen $\\rho$ is $P_1^{p,q}$-Anosov all infinite order elements in $\\rho(\\Gamma)$ are \\textit{proximal}. This means that they act on $\\mathbb{P}(\\mathbb{R}^d)$ with a unique attractive fixed line and a unique repelling hyperplane. The \\textit{limit set} of $\\rho$ is, by definition, the closure  of the set of attractive fixed lines of proximal elements in $\\rho(\\Gamma)$. It is denoted by $\\Lambda_{\\rho(\\Gamma)}$ and coincides with the image of $\\xi$.\n\n\n\nDefine\n\n\n\n\\begin{center}\n\n$\\pmb{\\Omega}_{\\rho}:=\\lbrace o=[\\hat{o}]\\in\\mathbb{H}^{p,q-1}: \\langle \\hat{o},\\hat{\\xi} \\rangle_{p,q}\\neq 0 \\textnormal{ for all }\\xi=[\\hat{\\xi}]\\in\\Lambda_{\\rho(\\Gamma)}\\rbrace$.\n\\end{center}\n\n\n\n\nIn the study of discrete groups of projective transformations, it is standard to consider sets similar to $\\pmb{\\Omega}_\\rho$ (see for instance Danciger-Gu\u00e9ritaud-Kassel \\cite{DGK1,DGK2} and references therein). Without any further assumption the set $\\pmb{\\Omega}_\\rho$ could be empty. An important class of Anosov representations for which $\\pmb{\\Omega}_\\rho$ is non empty is given by \\textit{$\\mathbb{H}^{p,q-1}$-convex co-compact subgroups} introduced in \\cite{DGK1}. However in our results we do not assume that $\\rho$ is $\\mathbb{H}^{p,q-1}$-convex co-compact, we only need that $\\pmb{\\Omega}_\\rho\\neq\\emptyset$ (see Example \\ref{ex omegarho no vacio}).\n\n\n\n\\begin{propsn}[Propositions \\ref{prop counting with nu is well defined} and \\ref{prop counting with lambdauno is well defined}]\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $\\tau\\in S^o$. Then for every $t\\geq 0$ one has\\footnote{Even though finiteness of $N_{\\rho(\\Gamma)}(S^o,\\tau,t)$ follows directly from finiteness of $N_{\\rho(\\Gamma)}(S^o,t)$, in our proof we first show $N_{\\rho(\\Gamma)}(S^o,\\tau,t)<\\infty$ and use it to prove $N_{\\rho(\\Gamma)}(S^o,t)<\\infty$.}\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,\\tau,t)<\\infty$ and $N_{\\rho(\\Gamma)}(S^o,t)<\\infty$.\n\\end{center}\n\n\n\n\n\\end{propsn}\n\n\nThe main results of this paper in the Riemannian context are Theorems \\ref{teorema A} and \\ref{teorema B}. The notation $f(t) \\sim g(t)$ stands for\n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\frac{f(t)}{g(t)}= 1$.\n\\end{center}\n\n\n\n\n\\begingroup\n\\setcounter{tmp}{\\value{teo}\n\\setcounter{teo}{0}\n\\renewcommand\\theteo{\\Alph{teo}\n\n\n\n\\begin{teo} \\label{teorema A}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. There exist positive constants $h=h_\\rho$ and $M=M_{\\rho,o}$ such that\n\n\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,t)\\sim \\dfrac{e^{ht}}{M}$.\n\\end{center}\n\n\n\\end{teo}\n\n\n\n\\begin{teo} \\label{teorema B}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $\\tau\\in S^o$. There exist positive constants $h=h_\\rho$ and $M'=M'_{\\rho,\\tau}$ such that\n\n\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,\\tau,t)\\sim \\dfrac{e^{ht}}{M'}$.\n\\end{center}\n\n\n\\end{teo}\n\n\n\n\\endgroup\n\n\n\n\nThe constant $h$ is the same in both Theorems \\ref{teorema A} and \\ref{teorema B} and it is independent on the choice of $o$ in $\\pmb{\\Omega}_\\rho$ (and $\\tau$ in $S^o$). It coincides with the topological entropy of the \\textit{geodesic flow} $\\phi^\\rho$ of $\\rho$, introduced by Bridgeman-Canary-Labourie-Sambarino \\cite{BCLS}, and can be computed as\n\n\\begin{center}\n$h=\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace[\\gamma]\\in[\\Gamma]:\\hspace{0,3cm}\\lambda_1(\\rho(\\gamma))\\leq t\\rbrace}{t}$.\n\\end{center}\n\n\\noindent Here $[\\gamma]$ denotes the conjugacy class of $\\gamma$ and $\\lambda_1(\\rho(\\gamma))$ denotes the logarithm of the spectral radius of $\\rho(\\gamma)$. The constants $M$ and $M'$ are related to the total mass of specific measures in the Bowen-Margulis measure class of $\\phi^\\rho$ (recall that the Bowen-Margulis measure class is the homothety class of measures maximizing entropy of $\\phi^\\rho$).\n\n\n\n\n\n\n\n\nSince the work of Margulis \\cite{Mar}, in order to obtain a counting result one usually studies the ergodic properties of a well chosen dynamical system. In order to find a dynamical system adapted to Theorem \\ref{teorema A} we introduce a decomposition of a specific subset of $G$, analogue to the Cartan Decomposition, but replacing \\textit{the} maximal compact subgroup of $G$ by $\\textnormal{PSO}(p,q-1)$ and \\textit{the} Cartan subspace by a smaller abelian subalgebra (Subsection \\ref{subsec HexpliebH}). For Theorem \\ref{teorema B} we use the more studied \\textit{polar decomposition} of $G$ (Subsection \\ref{subsec KexpliebH}).\n\n\n\\subsubsection{\\textnormal{\\textbf{Relation with the work of Oh-Shah}}} \\label{subsub ohshah}\n\n\n\n\n\n\nMotivated by the study of \\textit{Apollonian circle packings} on the Riemann sphere, Oh-Shah \\cite{OS} studied counting problems similar to ours. Indeed, let $p=1$ and $q=3$. Then $\\mathbb{H}^{1,2}$ identifies with the space of circles of the Riemann sphere or, equivalently, the space of totally geodesic isometric copies of $\\mathbb{H}^2$ inside $\\mathbb{H}^3$. In \\cite[Theorem 1.5]{OS} the cited authors prove that for a well-chosen $S^o\\cong \\mathbb{H}^2\\subset\\mathbb{H}^3$ and any point $\\tau\\in\\mathbb{H}^3$ one has\n\n\\begin{center}\n$\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{\\mathbb{H}^3}(\\tau,g\\cdot S^o)\\leq t  \\rbrace\\sim M^{-1}e^{ht}$.\n\\end{center}\n\n\n\\noindent Hence Theorem \\ref{teorema B} can be interpreted as a higher rank generalization of this result. We note however that, for $p=1$ and $q=3$, our results only concern convex co-compact groups, while Oh-Shah's Theorem applies to a wider class of geometrically finite Kleinian subgroups. A slightly different counting theorem in $\\mathbb{H}^{1,2}$ was obtained by the cited authors in \\cite{OS3}. Effective versions of Oh-Shah's results (i.e. with an error term) have been obtained by Lee-Oh \\cite{LO} and Mohammadi-Oh \\cite{MO}. Our Theorem \\ref{teorema A} seems to be new even in this setting.\n\n\n\nThe approach by Oh-Shah is similar to the one of Eskin-McMullen \\cite{EMcM}: they study the equidistribution, with respect to certain measures, of the orthogonal translates of $S^o$ under the geodesic flow of $\\Delta\\backslash\\mathbb{H}^3$ (see Oh-Shah \\cite{OS2} for precisions). Here we use different techniques. We follow the approach by Sambarino \\cite{Sam} and construct a dynamical system on a compact space that contains the required geometric information.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Interpretation in $\\mathbb{H}^{p,q-1}$} \\label{subsec counting hpq introd}\n\nAnother part of our contributions concern geometric interpretations of Theorems \\ref{teorema A} and \\ref{teorema B} in $\\mathbb{H}^{p,q-1}$. We now state these interpretations.\n\n\nGeodesics in $\\mathbb{H}^{p,q-1}$ are intersections of projectivized $2$-dimensional subspaces of $\\mathbb{R}^d$ with $\\mathbb{H}^{p,q-1}$ and they are classified in three types, depending on the sign of the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ on its tangent vectors (see Subsection \\ref{subsub geod hpq}). We are mainly interested in \\textit{space-like geodesics}, i.e. geodesics associated to planes on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ has signature $(1,1)$. Let $o,o'\\in\\mathbb{H}^{p,q-1}$ be two points joined by a space-like geodesic and let $\\ell_{o,o'}$ be the length of this geodesic segment (see Subsection \\ref{subsub lengths of spacelike geod}). We denote by $\\mathscr{C}^>_o$ the set of points of $\\mathbb{H}^{p,q-1}$ that can be joined to $o$ by a space-like geodesic and we set\n\n\\begin{center}\n$\\mathscr{C}^>_{o,G}:=\\lbrace g\\in G:\\hspace{0,3cm} g\\cdot o\\in\\mathscr{C}^>_o\\rbrace$.\n\\end{center}\n\n\n\n\n\n\n\n\n\\begin{propsn}[Proposition \\ref{prop ell dXG y vertboverto}]\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $g\\in \\mathscr{C}^>_{o,G}$. Then \n\n\\begin{center}\n$\\ell_{o,g\\cdot o}=d_{X_G}(S^o,g\\cdot S^o)$.\n\\end{center}\n\\end{propsn}\n\nIn Corollary \\ref{cor gammao in cowmayor} we prove that given a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$ and $o$ in $\\pmb{\\Omega}_\\rho$, then apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ one has $\\rho(\\gamma)\\in \\mathscr{C}^>_{o,G}$. By Proposition \\ref{prop counting with lambdauno is well defined} we have\n\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in \\mathscr{C}^>_{o,G} \\textnormal{ and } \\ell_{o,\\rho(\\gamma)\\cdot o}\\leq t\\rbrace<\\infty$\n\\end{center}\n\n\n\\noindent for every positive $t$. Moreover, Theorem \\ref{teorema A} implies that this function is asymptotic to $M^{-1}e^{ht}$ as $t\\longrightarrow\\infty$.\n\n\n\n\nIn order to state the corresponding geometric interpretation of Theorem \\ref{teorema B} we follow Kassel-Kobayashi \\cite[p.151]{KK}. Let $o\\in\\mathbb{H}^{p,q-1}$ and $\\tau\\in S^o$. Then\n\n\\begin{center}\n$\\mathbb{H}^p_\\tau:=(o\\oplus\\tau^{\\perp_{p,q}})\\cap\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\n\\noindent is a space-like totally geodesic copy of $\\mathbb{H}^{p}$ passing through $o$. Let $K^\\tau$ be the (maximal compact) subgroup of $G$ stabilizing $\\tau$. As we shall see, for every $g$ in $G$ the point $g\\cdot o$ lies in the $K^\\tau$-orbit of a point $o_g$ in $\\mathbb{H}^{p}_\\tau$. The counterpart of Theorem \\ref{teorema B} in $\\mathbb{H}^{p,q-1}$ is provided by the following proposition.\n\n\n\n\\begin{propsn}[Proposition \\ref{prop interpretation of btau in symg}]\nFor every $g$ in $G$ one has\n\n\\begin{center}\n$\\ell_{o,o_g}=d_{X_G}(\\tau,g\\cdot S^o)$.\n\\end{center}\n\n\\end{propsn}\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Relation with the work of Glorieux-Monclair and Kassel-Kobayashi}}}\\label{subsub GM and KK in introd}\n\n\nGlorieux-Monclair \\cite{GM} introduced an orbital counting function for $\\mathbb{H}^{p,q-1}$-convex co-compact representations that differs from\n\n\\begin{center}\n$t\\mapsto\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in \\mathscr{C}^>_{o,G} \\textnormal{ and } \\ell_{o,\\rho(\\gamma)\\cdot o}\\leq t\\rbrace$\n\\end{center}\n\n\n\\noindent by a constant. Indeed, they define an \\textit{$\\mathbb{H}^{p,q-1}$-distance}\n\n\\begin{center}\n$d_{\\mathbb{H}^{p,q-1}}(o,o'):=\\left\\{\\begin{array}{cc} \\ell_{o,o'} \\hspace{0,1cm}\\textrm{ if } o'\\in\\mathscr{C}^>_o \\\\ 0  \\hspace{0,6cm}\\textrm{ otherwise }\\end{array}\\right.$,\n\\end{center} \n\n\\noindent and show that it satisfies a version of the triangle inequality in the convex hull of the limit set of $\\rho$. This is used to prove that the exponential growth rate of the counting function\n\n\n\\begin{center}\n$t\\mapsto\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm}  d_{\\mathbb{H}^{p,q-1}}(o,\\rho(\\gamma)\\cdot o)\\leq t\\rbrace$\n\\end{center}\n\n\n\\noindent is independent on the choice of the basepoint $o$. The authors interpret this exponential rate as a \\textit{pseudo-Riemannian Hausdorff dimension} of the limit set of $\\rho$, with the purpose of finding upper bounds for this number (\\cite[Theorem 1.2]{GM}). A consequence of Theorem \\ref{teorema A} and Proposition \\ref{prop ell dXG y vertboverto} (see Remarks \\ref{rem crit exponent coindes with the one of GM} and \\ref{rem crit exponent coincides with the entropy}) is that this rate coincides with the topological entropy $h$ of $\\phi^\\rho$.\n\n\n\nOn the other hand, as we shall see in Section \\ref{sec generalized cartan} the number $\\ell_{o,o_g}$ is related to the \\textit{polar projection} of $g$ and therefore Theorem \\ref{teorema B} addresses the problems treated by Kassel-Kobayashi in \\cite[Section 4]{KK}. In \\cite{KK} the authors study the orbital counting function of Theorem \\ref{teorema B} for \\textit{sharp} subgroups of a real reductive symmetric space (see \\cite[Section 4]{KK}). Kassel-Kobayashi obtain some estimates on the growth of this function, but no precise asymptotic is established.\n\nThe method of \\cite{GM} is based on pseudo-Riemannian geometry: they construct analogues of Busemann functions, Gromov products and Patterson-Sullivan densities in $\\mathbb{H}^{p,q-1}$ using this viewpoint. Our approach is inspired by \\cite{KK} and has Lie-theoretic flavor: we study linear algebraic interpretations of the geometric quantities involved in the definition of the counting functions. This allows us to establish finiteness of these functions, to make a link between the different symmetric spaces and to apply Ledrappier's \\cite{Led} framework to our setting.\n\n\\subsection{Outline of the proof} \\label{subsec outline}\n\n\nThere are three major steps in the proof of Theorems \\ref{teorema A} and \\ref{teorema B}.\n\n\\subsubsection*{\\textnormal{\\textbf{First step}}} As we said, we interpret the geometric quantities involved in Theorems \\ref{teorema A} and \\ref{teorema B} as linear algebraic quantities. \n\n\nLet us be more precise. Fix $o \\in \\mathbb{H}^{p,q-1}$ and denote by $H^o$ the stabilizer in $G$ of this point. If we consider the symmetry of $\\mathbb{R}^d$ given by $J^o:=\\textnormal{id}_o\\oplus\\left(-\\textnormal{id}_{o^{\\perp_{p,q}}}\\right)$, we have that $H^o$ equals the fixed point set of the involution\n\\begin{center}\n$\\sigma^o:g\\mapsto J^ogJ^o$\n\\end{center}\n\n\n\\noindent of $G$ (see Subsection \\ref{subsubsec struc sym hpq}). This identifies the tangent space at $o$ of $\\mathbb{H}^{p,q-1}$ with the subspace of $\\mathfrak{so}(p,q)$ defined by $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1\\rbrace$. In Propositions \\ref{prop ell dXG y vertboverto} and \\ref{prop linear alg interpr of bo} we prove that for every $g\\in \\mathscr{C}^>_{o,G}$ one has\n\\begin{equation}\\label{eq igualddad distancia con vap}\nd_{X_G}(S^o,g\\cdot S^o)=\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1}).\n\\end{equation}\n\n\n\n\\noindent The main ingredient in the proof of equality (\\ref{eq igualddad distancia con vap}) is the following version of the classical Cartan Decomposition of $G$.\n\n\\begin{propsn}[Proposition \\ref{prop HBH}]\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $\\mathfrak{b}^+\\subset\\mathfrak{q}^o$ be a ray such that $\\exp(\\mathfrak{b}^+)\\cdot o$ is space-like. Given $g\\in \\mathscr{C}^>_{o,G}$ there exists $h,h'\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$ such that\n\n\n\\begin{center}\n$g=h\\exp(X)h'$.\n\\end{center}\n\n\\end{propsn} \n\nOn the other hand, the linear algebraic interpretation of the quantity $d_{X_G}(\\tau,g\\cdot S^o)$ is the following: the choice of $\\tau$ induces a norm $\\Vert\\cdot\\Vert_\\tau$ on $\\mathbb{R}^d$ invariant under the action of $K^\\tau$. We show in Propositions \\ref{prop interpretation of btau in symg} and \\ref{prop computing nu} that for every $g\\in G$ the following equality holds\n\\begin{equation}\\label{eq igualddad distancia con vasing}\nd_{X_G}(\\tau,g\\cdot S^o)=\\frac{1}{2}\\log\\Vert J^ogJ^og^{-1}\\Vert_\\tau.\n\\end{equation}\n\\noindent Once again the proof of this equality relies on a generalization of Cartan Decomposition (see Schlichtkrull \\cite[Chapter 7]{Sch}): every $g\\in G$ can be written as \n\n\\begin{center}\n$g=k\\exp(X)h$\n\\end{center}\n\n\\noindent for some $k\\in K^{\\tau}$, $h\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$.\n\n\n\\subsubsection*{\\textnormal{\\textbf{Second step}}} In order to simplify the exposition we assume that $\\Gamma$ is torsion free. In this case every $\\gamma\\neq 1$ in $\\Gamma$ has a unique attractive (resp. repelling) fixed point in $\\partial_\\infty\\Gamma$, denoted by $\\gamma_+$ (resp. $\\gamma_-$). Consider $\\rho:\\Gamma\\longrightarrow G$ a $P_1^{p,q}$-Anosov representation. The key feature of choosing $o$ in $\\pmb{\\Omega}_\\rho$ is that it guarantees some \\textit{transversality condition} for the proximal matrices $J^o\\rho(\\gamma)J^o$ and $\\rho(\\gamma^{-1})$ and this allows to estimate the quantities (\\ref{eq igualddad distancia con vap}) and (\\ref{eq igualddad distancia con vasing}) in terms of the spectral radius of $\\rho(\\gamma)$.\n\n\nMore precisely, we will see in Proposition \\ref{prop fijos de Jo en borde} that \n\\begin{equation} \\label{eq omegarho en introduccion}\n\\pmb{\\Omega}_\\rho=\\lbrace o\\in\\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot\\xi(x)\\notin\\eta(x) \\textnormal{ for all } x\\in\\partial_\\infty\\Gamma \\rbrace.\n\\end{equation}\n\n\n\n\\noindent Fix $o\\in\\pmb{\\Omega}_\\rho$ and a distance $d$ in $\\mathbb{P}(\\mathbb{R}^d)$ induced by the choice of an inner product in $\\mathbb{R}^d$. By compactness of $\\partial_\\infty\\Gamma$ there exists a positive constant $r$ such that\n\n\n\\begin{center}\n$d(J^o\\cdot\\xi(x),\\eta(x))\\geq r$\n\\end{center}\n\n\\noindent holds for every $x\\in\\partial_\\infty\\Gamma$ (here $d(J^o\\cdot\\xi(x),\\eta(x))$ is the minimal distance between $J^o\\cdot\\xi(x)$ and the lines included in $\\eta(x)$). Further, if $\\gamma_+$ is uniformly far from $\\gamma_-$, with respect to some visual distance in $\\partial_\\infty\\Gamma$, then $\\xi(\\gamma_+)$ (resp. $\\xi(\\gamma_-)$) is uniformly far from $\\eta(\\gamma_-)$ (resp. $\\eta(\\gamma_+)$). In Lemma \\ref{lema jrhojrho prox dos} we combine all these facts with Benoist's work \\cite{Ben1} to conclude that  the product $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is proximal. Moreover, we obtain a comparison between the quantity (\\ref{eq igualddad distancia con vap}) (resp. (\\ref{eq igualddad distancia con vasing})) and\n\n\\begin{center}\n$\\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\n\\noindent with very precise control on the error made in this comparison.\n\n\n\\subsubsection*{\\textnormal{\\textbf{Third step}}} \n\nWe apply Sambarino's outline \\cite{Sam} to our particular context\\footnote{The results in \\cite{Sam} are proved for fundamental groups of closed negatively curved manifolds. However, all the results obtained there remain valid when $\\Gamma$ is an arbitrary word hyperbolic group admitting an Anosov representation. This is explained in detail in Appendix \\ref{appendix distribution utaugamma y uogamma}.}. To a H\u00f6lder cocycle $c$ on $\\partial_\\infty\\Gamma$ the author associates a H\u00f6lder reparametrization $\\psi_t^c$ of the geodesic flow of $\\Gamma$. Recall that a \\textit{H\u00f6lder cocycle} is a map $c:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ satisfying\n\n\n\n\\begin{center}\n$c(\\gamma_0\\gamma_1,x)=c(\\gamma_0,\\gamma_1\\cdot x)+c(\\gamma_1,x)$\n\\end{center}\n\n\n\\noindent for every $\\gamma_0,\\gamma_1$ in $\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ and such that the map $c(\\gamma_0,\\cdot)$ is H\u00f6lder (with the same exponent for every $\\gamma_0$). The cocycle $c'$ is said to be \\textit{cohomologous} to $c$ if there exists a H\u00f6lder continuous function $U:\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ such that for every $\\gamma$ in $\\Gamma$ and $x$ in $\\partial_\\infty\\Gamma$ one has\n\n\n\n\\begin{center}\n$c(\\gamma,x)-c'(\\gamma,x)=U(\\gamma\\cdot x)-U(x)$.\n\\end{center}\n\n\\noindent In that case $\\psi_t^c$ is conjugate to $\\psi_t^{c'}$ (see \\cite[Section 3]{Sam}). By considering a Markov coding and applying Parry-Pollicott's Prime Orbit Theorem \\cite{PP}, Sambarino obtains an asymptotic for the number of periodic orbits of $\\psi_t^c$ of period less than or equal to $ t$ (see \\cite[Corollary 4.1]{Sam}). Obviously this is a purely dynamical result, i.e. changing $\\psi_t^c$ in its conjugacy class does not affect the asymptotics. \n\n\nHowever our problem is more subtle: one must find a particular cocycle, with some geometric meaning, and not just \\textit{any} cocycle in the given cohomology class. Indeed, the cocycles that we consider to prove Theorems \\ref{teorema A} and \\ref{teorema B} are cohomologous, but only the specific choices in such a cohomology class yield the respective results.\n\n\n\n\n\n\nLet us briefly sketch the proof of Theorem \\ref{teorema A} (Theorem \\ref{teorema B} is proved in a similar way). Fix $o\\in\\pmb{\\Omega}_\\rho$ and consider\n\n\\begin{center}\n$c_o:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_o(\\gamma,x):=\\dfrac{1}{2}\\log\\left\\vert\\dfrac{\\langle\\rho(\\gamma)\\cdot v_x,J^o\\rho(\\gamma)\\cdot v_x\\rangle_{p,q}}{\\langle v_x,J^o\\cdot v_x\\rangle_{p,q}}\\right\\vert$\n\\end{center}\n\n\\noindent where $v_{x}\\neq 0$ is any vector in $\\xi(x)$\\footnote{When $q=1$ this coincides with the \\textit{Busemann cocycle} of $\\mathbb{H}^{p}$, i.e. $c_o(\\gamma,x)=\\beta_{\\xi(x)}(\\rho(\\gamma^{-1})\\cdot o,o)$ where $\\beta_\\cdot(\\cdot,\\cdot):\\partial\\mathbb{H}^{p}\\times\\mathbb{H}^{p}\\times\\mathbb{H}^{p}\\longrightarrow\\mathbb{R}$ is the Busemann function.}. This is a well-defined function thanks to (\\ref{eq omegarho en introduccion}) and it is a H\u00f6lder cocycle.\n\n\n\nLet $\\partial_\\infty^{2}\\Gamma$ be the set of pairs of distinct points in $\\partial_\\infty\\Gamma$ and consider the action of $\\Gamma$ on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ given by\n\n\\begin{center}\n$\\gamma\\cdot (x,y,s):=(\\gamma \\cdot x,\\gamma\\cdot y, s-c_o(\\gamma,y))$.\n\\end{center}\n\n\n\n\n\n\n\\noindent We denote by $\\textnormal{U}_o\\Gamma$ the quotient space. The \\textit{translation flow} on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ given by\n\n\\begin{center}\n$\\psi_t(x,y,s):=(x,y,s-t)$\n\\end{center}\n\n\\noindent descends to a flow $\\psi_t=\\psi_t^o$ on $\\textnormal{U}_o\\Gamma$. As Sambarino shows in \\cite[Theorem 3.2(1)]{Sam} (see also Lemma \\ref{lema conj urhogamma y uogamma}) the flow $\\psi_t$ is conjugate to a H\u00f6lder reparametrization of the geodesic flow of $\\Gamma$ introduced by Gromov \\cite{Gro}. We will show (see Lemma \\ref{lema conj urhogamma y uogamma}) that periodic orbits of $\\psi_t$ are parametrized by conjugacy classes of \\textit{primitive} elements in $\\Gamma$, i.e. elements which cannot be written as a power of another element. If $\\gamma$ is primitive, the corresponding period is given by\n\n\\begin{center}\n$\\ell_{c_o}(\\gamma):=\\lambda_1(\\rho(\\gamma))$.\n\\end{center}\n\n\nWe show the following property concerning spectral radii in a projective Anosov representation.\n\n\n\n\\begin{propsn}[Proposition \\ref{prop geod flow is weak mixing}]\n\n\nLet $\\rho$ be a projective Anosov representation of $\\Gamma$. Then the set $\\lbrace\\lambda_1(\\rho(\\gamma))\\rbrace_{\\gamma\\in\\Gamma}$ spans a non discrete subgroup of $\\mathbb{R}$.\n\n\\end{propsn} \n\n\n\n\n\n\n\nDenote by $h$ the topological entropy of $\\psi_t$. The probability of maximal entropy of $\\psi_t$ can be constructed as follows: define the \\textit{Gromov product}\n\n\\begin{center}\n$[\\cdot,\\cdot]_o:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_o:=-\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\langle v_x,J^o\\cdot v_x\\rangle_{p,q}\\langle v_y,J^o\\cdot v_y\\rangle_{p,q}}{\\langle v_x,v_y\\rangle_{p,q}\\langle v_y,v_x\\rangle_{p,q}}\\right\\vert$.\n\\end{center}\n\n\n\\noindent This function is well-defined thanks to (\\ref{eq omegarho en introduccion}) and transversality of $\\xi$ and $\\eta$. One can prove that\n\n\n\\begin{center}\n$[\\gamma \\cdot x,\\gamma\\cdot y]_o - [x,y]_o=-(c_o(\\gamma,x)+c_o(\\gamma,y))$\n\\end{center}\n\n\\noindent holds for every $\\gamma$ in $\\Gamma$ and $(x,y)\\in\\partial_\\infty^{2}\\Gamma$. Let $\\mu_o$ be a \\textit{Patterson-Sullivan probability} associated to $c_{o}$, that is, $\\mu_o$ is a probability on $\\partial_\\infty\\Gamma$ that satisfies\n\n\\begin{center}\n$\\dfrac{d\\gamma_*\\mu_o}{d\\mu_o}(x)=e^{-hc_{o}(\\gamma^{-1},x)}$\n\\end{center}\n\n\n\\noindent for every $\\gamma\\in\\Gamma$\\footnote{Recall that if $f:X\\longrightarrow Y$ is a map and $m$ is a measure on $X$ then $f_*(m)$ denotes the measure on $Y$ defined by $A\\mapsto m(f^{-1}(A))$.}. For the existence of such a probability see Subsection \\ref{subsub PS}. The measure\n\n\\begin{center}\n$e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt$\n\\end{center}\n\n\n\\noindent on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ is $\\Gamma$-invariant. It induces on the quotient $\\textnormal{U}_o\\Gamma$ the measure of maximal entropy of $\\psi_t$, which is unique up to scaling (see \\cite[Theorem 3.2(2)]{Sam} or Proposition \\ref{prop product is of maximal entropy}). \n\n\nDenote by $C_c^*(\\partial_\\infty^{2}\\Gamma)$ the dual of the space of compactly supported real continuous functions on $\\partial_\\infty^{2}\\Gamma$ equipped with the weak-star topology. For $x$ in $\\partial_\\infty\\Gamma$ let $\\delta_x$ be the Dirac mass at $x$. Inspired by the work of Roblin \\cite{Rob}, Sambarino \\cite[Proposition 4.3]{Sam} shows\n \n\\begin{center}\n$Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, \\ell_{c_o}(\\gamma)\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\\noindent on $C_c^*(\\partial_\\infty^{2}\\Gamma)$ as $t\\longrightarrow\\infty$ (for a proof in our context see Proposition \\ref{prop distribution of periodic orbits}). The constant $M=M_{\\rho,o}>0$ equals the product of $h$ with the total mass of $e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt$ on the quotient space $\\textnormal{U}_o\\Gamma$.\n\n\nAs we show in Lemma \\ref{lema computing gromov on gammapm o}, the number $[\\gamma_-,\\gamma_+]_o$ is the precise error term in the comparison between $\\ell_{c_o}(\\gamma)$ and $\\frac{1}{2}\\lambda_1(J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1}))=d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o)$ provided by Benoist's Theorem \\ref{teo benoist}. This is the geometric step: we replace the period $\\ell_{c_o}(\\gamma)$ by the number $d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o)$ in the previous sum, using the Gromov product. \n\n\n\n\n\\begin{propsn}[Proposition \\ref{prop distribution on bg for length}]\nLet $\\Gamma$ be a torsion free word hyperbolic group, $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. Then\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o) \\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow \\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\\noindent on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$ as $t\\longrightarrow\\infty$.\n\\end{propsn}\n\n\nThe proof of Proposition \\ref{prop distribution on bg for length} follows line by line the proof of \\cite[Theorem 6.5]{Sam}, which is again inspired by Roblin's work \\cite{Rob}.\n\nIt turns out that the previous proposition can be used to deduce Theorem \\ref{teorema A} in the general case, that is, if we admit torsion elements in $\\Gamma$.\n\n\n\\begin{propsn}[Proposition \\ref{prop distribution on bg for length with torsion}]\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. Then\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o) \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\end{propsn}\n\n\n\n\n\n\n\n\n\\subsection{Organization of the paper}\n\n\n\nIn Section \\ref{sec symmetric spaces} we recall basic facts on the symmetric spaces $X_G$ and $\\mathbb{H}^{p,q-1}$. Of particular importance is Subsection \\ref{subsub endign of spacelike}, which is devoted to the study of end points of space-like geodesics passing through our preferred point $o\\in\\mathbb{H}^{p,q-1}$. We give several characterizations of this set that will allow us to understand $\\pmb{\\Omega}_\\rho$ in different ways, all of them used indistinctly in Sections \\ref{sec the set omegarho}, \\ref{section distrib wrt bo} and \\ref{section distrib wrt btau}. In Section \\ref{sec generalized cartan} we study the geometric quantities involved in Theorems \\ref{teorema A} and \\ref{teorema B}. Equalities (\\ref{eq igualddad distancia con vap}) and (\\ref{eq igualddad distancia con vasing}) are proven respectively in Subsections \\ref{subsec HexpliebH} and \\ref{subsec KexpliebH}. In Section \\ref{sec proximality} we recall Benoist's results on products of proximal matrices and Section \\ref{sec anosov} is devoted to reminders on Anosov representations. In Section \\ref{sec the set omegarho} we define the set $\\pmb{\\Omega}_\\rho$ and study the action of $\\Gamma$ on this set. We show in particular that the orbital counting functions involved in Theorems \\ref{teorema A} and \\ref{teorema B} are well-defined (Proposition \\ref{prop counting with lambdauno is well defined} and Proposition \\ref{prop counting with nu is well defined}). We also obtain some estimates for the spectral radius and operator norm of elements $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ which are of major importance (c.f. Lemma \\ref{lema jrhojrho prox dos}). In Section \\ref{section distrib wrt bo} (resp. Section \\ref{section distrib wrt btau}) we prove Theorem \\ref{teorema A} (resp. Theorem \\ref{teorema B}). Finally, in Appendix \\ref{appendix distribution utaugamma y uogamma} we explain how to adapt the results of \\cite{Sam} to the context of arbitrary word hyperbolic groups admitting an Anosov representation.\n\n\n\n\n\n\\subsection*{Acknowledgements}\n\nThese problems were proposed to me by Rafael Potrie and Andr\u00e9s Sambarino. Without their guidance, their support and the (many) helpful discussions this work would not have been possible. I am extremely grateful for this.\n\nThe author also acknowledges Olivier Glorieux, Tal Horesh and Fanny Kassel for several enlightening discussions and comments.\n\n\nFinally, I would like to thank the referee of this article for careful reading and useful suggestions.\n\n\n\\section{Two symmetric spaces associated to $\\textnormal{PSO}(p,q)$} \\label{sec symmetric spaces}\n\\setcounter{equation}{0}\n\n\n\n\n\nFix two integers $p,q\\geq 1$ and let $d:=p+q$. We assume $d>2$. Denote by $\\mathbb{R}^{p,q}$ the vector space $\\mathbb{R} ^d$ endowed with the quadratic form\n\n\\begin{center}\n$ \\langle (x_1,\\dots,x_d),(y_1,\\dots,y_d)\\rangle_{p,q}:= \\displaystyle\\sum_{i=1}^p x_iy_i - \\displaystyle\\sum_{i=p+1}^d x_{i}y_{i}$.\n\\end{center}\n\n\n\n\n\\noindent From now on we denote by $G:=\\textnormal{PSO}(p,q)$ the subgroup of $\\textnormal{PSL}(d,\\mathbb{R})$ consisting on elements whose lifts to $\\textnormal{SL}(d,\\mathbb{R})$ preserve the form $ \\langle \\cdot,\\cdot\\rangle_{p,q}$.\n\n\nFor a subspace $\\pi$ of $\\mathbb{R}^d$ we denote by $\\pi^{\\perp_{p,q}}$ its orthogonal complement with respect to  $\\langle \\cdot,\\cdot\\rangle_{p,q}$, i.e. \n\n\\begin{center}\n$\\pi^{\\perp_{p,q}}:=\\lbrace x\\in\\mathbb{R}^d: \\hspace{0,3cm} \\langle x,y\\rangle_{p,q}=0 \\textnormal{ for all } y\\in \\pi\\rbrace$.\n\\end{center}\n\n\n\n\n\n\nLet $\\mathfrak{g}:=\\mathfrak{so}(p,q)$ be the Lie algebra of $G$. If $\\cdot^t$ denotes the \\textit{usual} transpose operator one has that $\\mathfrak{g}$ equals the set of matrices of the form\n\n\\begin{center}\n$\\left(\\begin{matrix}\nX_1 & X_2\\\\\nX_2^t & X_3\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent where $X_1$ is of size $p\\times p$, $X_3$ is of size $q\\times q$ and both are skew-symmetric with respect to $\\cdot^t$. The \\textit{Killing form} of $G$ is the symmetric bilinear form $\\kappa$ on $\\mathfrak{g}$ defined by\n\n\\begin{center}\n$\\kappa(X,Y):=\\textnormal{tr}(\\textnormal{ad}_X\\circ\\textnormal{ad}_Y)$,\n\\end{center}\n\n\n\\noindent where $\\textnormal{ad}:\\mathfrak{g}\\longrightarrow\\textnormal{End}(\\mathfrak{g})$ is the adjoint representation.  It can be seen that the following equality holds:\n\n\n\n\n\n\n\n\\begin{center}\n$\\kappa(X,Y)=(d-2)\\textnormal{tr}(XY)$\n\\end{center}\n\n\n\n\n\n\\noindent  (see Helgason {\\cite[p.180 \\& p.189]{Hel}}).\n\n\n\n\n\\subsection{The Riemannian symmetric space $X_G$} \\label{subsec XG}\n\n\n\nA \\textit{Cartan involution} of $G$ is an involutive automorphism $\\tau:G\\longrightarrow G$ such that the bilinear form\n\n\\begin{center}\n$(X,Y)\\mapsto -\\kappa(X,d\\tau(Y))$\n\\end{center}\n\n\n\\noindent is positive definite. The fixed point set $K^\\tau$ of such an involution is a maximal compact subgroup of $G$ (see Knapp \\cite[Theorem 6.31]{Kna}). The \\textit{Riemannian symmetric space} of $G$ is the set consisting on Cartan involutions of $G$. It is denoted by $X_G$ and it is equipped with a natural action of $G$ which is transitive (c.f. \\cite[Corollary 6.19]{Kna}). The stabilizer of $\\tau$ is $K^\\tau$, thus\n\n\\begin{center}\n$G\/K^\\tau\\cong X_G$.\n\\end{center}\n\n\n\\begin{rem}\\label{rem Xg space of qplanes}\nThe space $X_G$ can be identified with the space of $q$-dimensional subspaces of $\\mathbb{R}^d$ on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ is negative definite. Explicitly, to a $q$-dimensional negative definite subspace $\\pi$ one associates the Cartan involution of $G$ determined by the inner product of $\\mathbb{R}^d$ which equals $-\\langle\\cdot,\\cdot\\rangle_{p,q}$ (resp. $\\langle\\cdot,\\cdot\\rangle_{p,q}$) on $\\pi$ (resp. $\\pi^{\\perp_{p,q}}$) and for which $\\pi$ and $\\pi^{\\perp_{p,q}}$ are orthogonal.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nThe choice of a point $\\tau$ in $X_G$ determines a \\textit{Cartan decomposition}\n\n\n\n\\begin{center}\n$\\mathfrak{g}=\\mathfrak{p}^\\tau\\oplus\\mathfrak{k}^\\tau$\n\\end{center}\n\n\n\\noindent where $\\mathfrak{p}^{\\tau}:=\\lbrace d\\tau=-1\\rbrace$ and $\\mathfrak{k}^{\\tau}:=\\lbrace d\\tau=1\\rbrace$. The group $K^\\tau$ is tangent to $\\mathfrak{k}^\\tau$ and one has a $G$-equivariant identification \n\\begin{equation} \\label{eq liep es el tangente}\n\\mathfrak{p}^\\tau\\cong T_\\tau X_G\n\\end{equation}\n\n\\noindent given by $X\\mapsto \\left. \\frac{d}{dt}\\right\\vert_0 \\exp{(tX)}\\cdot\\tau$ (see \\cite[Theorem 3.3 of Ch. IV]{Hel}).\n\n\n\n\n\\begin{ex} \\label{ex explicit cartan involution}\n\nConsider the involution of $G$ defined by $\\tau(g):=(g^{-1})^t$. One sees that $\\tau\\in X_G$ and $\\mathfrak{p}^{\\tau}$ (resp. $\\mathfrak{k}^{\\tau}$) is the set of symmetric matrices (resp. skew-symmetric matrices) in $\\mathfrak{so}(p,q)$. Moreover $K^{\\tau}$ is the subgroup $\\textnormal{PS}(\\textnormal{O}(p)\\times \\textnormal{O}(q))$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\nThe Killing form $\\kappa$ is positive definite (resp. negative definite) on $\\mathfrak{p}^\\tau$ (resp. $\\mathfrak{k}^\\tau$). Thanks to (\\ref{eq liep es el tangente}) any positive multiple of $\\kappa$ induces a $G$-invariant Riemannian metric on $X_G$. It is well-known (see \\cite[Theorem 4.2 of Ch. IV]{Hel}) that $X_G$ equipped with any of these metrics is a symmetric space which is non-positively curved. \n\n\nWe already mentioned that in this paper we study counting problems not only in $X_G$ but also in $\\mathbb{H}^{p,q-1}$. In the next section we construct $\\mathbb{H}^{p,q-1}$, whose metric is induced by the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$. However, we will see that the Killing form induces as well a $G$-invariant metric on $\\mathbb{H}^{p,q-1}$. These two metrics differ by the scaling factor $(2(d-2))^{-1}$ (see Remark \\ref{rem form on qo} for further precisions). Since we want a simultaneous treatment of the geometry of the spaces $X_G$ and $\\mathbb{H}^{p,q-1}$, we fix the following normalization for the metric on $X_G$:\n\\begin{equation} \\label{eq distance in XG and killing}\nd_{X_G}(\\tau,\\exp(X)\\cdot\\tau):=\\left(\\dfrac{1}{2(d-2)}\\kappa(X,X)\\right)^{\\frac{1}{2}}\n\\end{equation}\n\n\n\n\\noindent for all $\\tau\\in X_G$ and all $X\\in\\mathfrak{p}^\\tau$.\n\n\n\n\n\n\\subsection{The pseudo-Riemannian hyperbolic space $\\mathbb{H}^{p,q-1}$} \\label{subsec hpq}\n\n\n\n\n\n\n\nLet \n\n\n\\begin{center}\n$\\hat{\\mathbb{H}}^{p,q-1}:=\\lbrace \\hat{o}\\in\\mathbb{R}^{p,q}: \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}=-1 \\rbrace$\n\\end{center}\n\n\\noindent endowed with the restriction of the form $\\langle \\cdot,\\cdot\\rangle_{p,q}$ to tangent spaces. This metric induces on\n\n\\begin{center}\n$\\mathbb{H}^{p,q-1}:=\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^{p,q}): \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}<0 \\rbrace$\n\\end{center}\n\n\\noindent a pseudo-Riemannian structure invariant under the projective action of $G$. This space is called the \\textit{pseudo-Riemannian hyperbolic space of signature $(p,q-1)$}. The \\textit{boundary} of $\\mathbb{H}^{p,q-1}$ is the space of \\textit{isotropic lines} defined by\n\n\n\\begin{center}\n$\\partial\\mathbb{H}^{p,q-1}:=\\lbrace \\xi=[\\hat{\\xi}]\\in\\mathbb{P}(\\mathbb{R}^{p,q}): \\hspace{0,3cm} \\langle\\hat{\\xi},\\hat{\\xi}\\rangle_{p,q}=0 \\rbrace$.\n\\end{center}\n\n\n\\noindent It is also equipped with the natural (transitive) action of $G$. If we denote by $P_1^{p,q}$ the (parabolic) subgroup of $G$ stabilizing an isotropic line, then\n\n\\begin{center}\n$\\partial\\mathbb{H}^{p,q-1}\\cong G\/P_1^{p,q}$.\n\\end{center}\n\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Structure of symmetric space}}} \\label{subsubsec struc sym hpq}\n\n\n\nThe action of $G$ on $\\mathbb{H}^{p,q-1}$ is transitive, hence $\\mathbb{H}^{p,q-1}\\cong G\/H^o$ where $H^o$ is the stabilizer in $G$ of the point $o\\in\\mathbb{H}^{p,q-1}$. For instance, when $o=[0,\\dots,0,1]\\in\\mathbb{H}^{p,q-1}$ one has\n\n\n\\begin{center}\n$H^o=\\left\\lbrace \\left[\\begin{matrix}\n\\hat{g} & 0\\\\\n0 & 1\n\\end{matrix}\\right]\\in G: \\hspace{0,3cm} \\hat{g}\\in \\textnormal{O}(p,q-1)  \\right\\rbrace$.\n\\end{center} \n\n\n\nFix any $o\\in\\mathbb{H}^{p,q-1}$. Since $o$ and $o^{\\perp_{p,q}}$ are transverse we can consider the matrix \n\n\\begin{center}\n$J^o:=\\textnormal{id}_o\\oplus\\left(-\\textnormal{id}_{o^{\\perp_{p,q}}}\\right)$. \n\\end{center}\n\n\\noindent It follows that $H^o=\\textnormal{Fix}(\\sigma^o)$ where $\\sigma^o$ is the involution of $G$ defined by\n\\begin{equation} \\label{eq involution}\n\\sigma^o(g):=J^ogJ^o.\n\\end{equation}\n\n\n\\noindent Thus $\\mathbb{H}^{p,q-1}\\cong G\/H^o$ is a symmetric space of $G$.\n\n\n\n\\begin{rem} \\label{rem form on qo}\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1 \\rbrace$. There exists a $G$-equivariant identification\n\n\n\\begin{center}\n$\\mathfrak{q}^o\\cong  T_o\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\n\n\n\\noindent given by $X\\mapsto \\left. \\frac{d}{dt}\\right\\vert_0 \\exp{(tX)}\\cdot o$. We denote by $\\langle\\cdot,\\cdot\\rangle$ the pull-back of the $(p,q-1)$-form on $T_o\\mathbb{H}^{p,q-1}$ under this map and, for $X\\in\\mathfrak{q}^o$, we set $\\vert X\\vert:=\\langle X,X\\rangle$\\footnote{This number can be positive, negative or zero for $X\\neq 0$ in $\\mathfrak{q}^o$.}. \n\n\nRecall that $\\kappa$ is the Killing form of $\\mathfrak{so}(p,q)$. From explicit computations (that we omit) one can conclude that the equality\n\\begin{equation} \\label{eq form on qo and killing}\n\\vert X\\vert=\\dfrac{1}{2(d-2)}\\kappa(X,X)\n\\end{equation}\n\\noindent holds for every $X\\in\\mathfrak{q}^o$. This justifies the choice of normalization made in Subsection \\ref{subsec XG}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\begin{rem} \\label{rem action of Ho en el tangente en SOpqmenos1}\n\n\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then the action of the connected component of $H^o$ containing the identity is conjugate to the action of $\\textnormal{SO}(p,q-1)$ on $\\mathbb{R}^{p,q-1}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Geodesics of $\\mathbb{H}^{p,q-1}$}}}\\label{subsub geod hpq}\n\n\n\nGeodesics of $\\mathbb{H}^{p,q-1}$ are the intersections of straight lines of $\\mathbb{P}(\\mathbb{R}^{p,q})$ with $\\mathbb{H}^{p,q-1}$. They are divided in three types:\n\n\\begin{itemize}\n\\item \\textit{Space-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(1,1)$. They have positive speed and meet the boundary $\\partial\\mathbb{H}^{p,q-1}$ in two distinct points.\n\\item \\textit{Time-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(0,2)$. They have negative speed and do not meet the boundary (they are closed). \n\n\\item \\textit{Light-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(0,1)$, that is, is degenerate but has a negative eigenvalue. They have zero speed and meet the boundary in a single point.\n\\end{itemize}\n\n\n\n\n\\noindent For a point $o\\in\\mathbb{H}^{p,q-1}$ we denote by $\\mathscr{C}_o^{0}$ (resp. $\\mathscr{C}_o^{>}$) the set of points of $\\mathbb{H}^{p,q-1}$ that can be joined with $o$ by a light-like (resp. space-like) geodesic. Its closure in $\\mathbb{P}(\\mathbb{R}^{p,q})$ is denoted by $\\overline{\\mathscr{C}_o^0}$ (resp. $\\overline{\\mathscr{C}_o^>}$). \n\n\\subsubsection{\\textnormal{\\textbf{Light-cones}}}\\label{subsub light cones}\n\n\nThe following lemma is proved by Glorieux-Monclair in \\cite[Lemma 2.2]{GM}.\n\n\n\n\\begin{lema}\\label{lema geod between o and xi}\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}=o^{\\perp_{p,q}}\\cap\\partial\\mathbb{H}^{p,q-1}$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Lenghts of space-like geodesics}}}\\label{subsub lengths of spacelike geod}\n\n\n\nFor a point $o'$ in $\\mathscr{C}_o^>$ we denote by $\\ell_{o,o'}$ the length of the geodesic segment connecting $o$ with $o'$. For instance the geodesic\n\\begin{equation} \\label{eq geod alpha}\ns\\mapsto [\\sinh(s),0\\dots,0,\\cosh(s)]\\in\\mathbb{H}^{p,q-1}\n\\end{equation}\n\n\n\\noindent is parametrized by arc-length.\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Space-like copies of $\\mathbb{H}^p$}}} \\label{subsub space copies Hp}\nLet $\\pi$ be a $(p+1)$-dimensional subspace of $\\mathbb{R}^d$ of signature $(p,1)$. Then $\\mathbb{P}(\\pi)\\cap\\mathbb{H}^{p,q-1}$ identifies with\n\n\n\n\n\\begin{center}\n$\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^{p,1}) \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,1}<0 \\rbrace$.\n\\end{center}\n\n\\noindent It follows that $\\mathbb{P}(\\pi)\\cap\\mathbb{H}^{p,q-1}$ is a totally geodesic isometric copy of $\\mathbb{H}^p$ inside $\\mathbb{H}^{p,q-1}$. Moreover this sub-manifold is space-like, in the sense that any of its tangent vectors has positive norm.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{End points of space-like geodesics}}} \\label{subsub endign of spacelike}\n\n\n\nLet $o$ be a point in $\\mathbb{H}^{p,q-1}$. Note that $J^o$ preserves the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ and thus acts on $\\partial\\mathbb{H}^{p,q-1}$. Set\n\n\\begin{center}\n$\\mathcal{O}^o:=\\lbrace \\xi\\in\\partial\\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot \\xi\\neq \\xi\\rbrace$. \n\\end{center}\n\n\n\n\\begin{prop} \\label{prop fijos de Jo en borde}\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then the following equalities hold:\n\n\n\n\n\\begin{equation*}\n\\begin{split}\n\\mathcal{O}^o & = \\lbrace \\xi\\in\\partial\\mathbb{H}^{p,q-1}: \\hspace{0,3cm} J^o\\cdot \\xi\\notin\\xi^{\\perp_{p,q}}\\rbrace\\\\ & = \\partial\\mathbb{H}^{p,q-1}\\setminus o^{\\perp_{p,q}} \\\\ & =\\partial\\mathbb{H}^{p,q-1}\\setminus \\overline{\\mathscr{C}_o^0}.\n\\end{split}\n\\end{equation*}\n\n\n\n\n\\end{prop}\n\n\nWe conclude that, unless $q=1$, the set $\\mathcal{O}^o$ is not the whole boundary of $\\mathbb{H}^{p,q-1}$. \n\n\n\\begin{proof}[Proof of Proposition \\ref{prop fijos de Jo en borde}]\n\n\nThe equality $\\partial\\mathbb{H}^{p,q-1}\\setminus o^{\\perp_{p,q}}=\\partial\\mathbb{H}^{p,q-1}\\setminus \\overline{\\mathscr{C}_o^0}$ is a consequence of Lemma \\ref{lema geod between o and xi}. The other equalities follow from definitions.\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalized Cartan decompositions} \\label{sec generalized cartan}\n\\setcounter{equation}{0}\n\n\n\n\nThe goal of this section is to define two generalized Cartan projections and to provide a link between them and Theorems \\ref{teorema A} and \\ref{teorema B}. The first one (Subsection \\ref{subsec KexpliebH}) is called the \\textit{polar projection} of $G$ and it is well-known. The second one (Subsection \\ref{subsec HexpliebH}) is new and can only be defined for elements in $G$ that satisfy some special property with respect to the choice of the basepoint $o$.\n\n\n\n\n\\subsection{Notations} \\label{subsec notations generalized cartan}\n\n\n\n\nThrough this section we fix a point $o\\in\\mathbb{H}^{p,q-1}$ and let $H^o=\\textnormal{Fix}(\\sigma^o)$ be its stabilizer in $G$ (c.f. Subsection \\ref{subsubsec struc sym hpq}). Let $\\mathfrak{h}^o$ be the Lie algebra of fixed points of $d\\sigma^o$ and $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1\\rbrace$. One has the following decomposition of the Lie algebra $\\mathfrak{g}$ of $G$:\n\n\\begin{center}\n\n$\\mathfrak{g}=\\mathfrak{h}^o\\oplus\\mathfrak{q}^o$.\n\n\\end{center}\n\n\\noindent Moreover, this decomposition is orthogonal with respect to the Killing form of $\\mathfrak{g}$.\n\n\nLet $\\tau$ be a Cartan involution  commuting with $\\sigma^o$: such involutions always exist and two of them differ by conjugation by an element in $H^o$ (see Matsuki \\cite[Lemma 4]{Mat}). Let $K^{\\tau}:=\\textnormal{Fix}(\\tau)$, which is a maximal compact subgroup of $G$. Let $\\mathfrak{p}^\\tau$ and $\\mathfrak{k}^\\tau$ be the subspaces defined in Subsection \\ref{subsec XG}. As $\\sigma^o$ and $\\tau$ commute, the following holds:\n\n\n\\begin{center}\n$\\mathfrak{g}=(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o) \\oplus(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o)\\oplus(\\mathfrak{k}^{\\tau}\\cap\\mathfrak{q}^o)\\oplus(\\mathfrak{k}^{\\tau}\\cap\\mathfrak{h}^o)$.\n\\end{center}\n\n\n\n\\noindent Let $\\mathfrak{b}\\subset\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$ be a (necessarily abelian) maximal subalgebra: two of them differ by conjugation by an element in $K^{\\tau}\\cap H^o$. We will consider closed Weyl chambers in $\\mathfrak{b}$ corresponding to positive systems of restricted roots of $\\mathfrak{b}$ in $\\mathfrak{g}^{\\sigma^o\\tau}:= (\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o)\\oplus (\\mathfrak{k}^{\\tau}\\cap\\mathfrak{h}^o)$. These closed Weyl chambers will be denoted by $\\mathfrak{b}^+$.\n\n\n\n\n\n\n\n\\begin{ex} \\label{ex explicit example}\nLet $o=[0,\\dots,0,1]$. Then $H^o$ is the upper left corner embedding of $\\textnormal{O}(p,q-1)$ in $G$ and the involution $\\sigma^o$ is obtained by conjugation by $J^o=\\textnormal{diag}(-1,\\dots,-1,1)$. One sees that $\\mathfrak{h}^o$ equals the upper left corner embedding of $\\mathfrak{so}(p,q-1)$ in $\\mathfrak{so}(p,q)$ and that\n\n\\begin{center}\n\n$\\mathfrak{q}^o=\\left\\lbrace\\left(\\begin{matrix}\n0 & 0 & Y_1\\\\\n0 & 0 & Y_2\\\\\nY_1^t & -Y_2^t & 0\n\\end{matrix}\\right): \\hspace{0,3cm} Y_1\\in \\textnormal{M}(p\\times 1,\\mathbb{R}), \\hspace{0,3cm} Y_2\\in \\textnormal{M}((q-1)\\times 1,\\mathbb{R})\\right\\rbrace$.\n\\end{center}\n\n\nLet $\\tau$ be the Cartan involution of Example \\ref{ex explicit cartan involution}. One observes that $\\tau$ commutes with $\\sigma^o$ and\n\n\n\n\n\\begin{center}\n\n$\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o=\\left\\lbrace X\\in\\mathfrak{q}^o: \\hspace{0,3cm} Y_2=0\\right\\rbrace$ \\hspace{0,5cm} $\\mathfrak{k}^{\\tau}\\cap\\mathfrak{q}^o=\\left\\lbrace X\\in\\mathfrak{q}^o: \\hspace{0,3cm} Y_1=0\\right\\rbrace$ .\n\\end{center}\n\n\n\n\n\\noindent Pick $\\mathfrak{b}$ to be the subset of $\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$ of matrices with $Y_1$ of the form\n\n\\begin{center}\n$\\left(\\begin{matrix}\ns\\\\\n0\\\\\n\\vdots\\\\\n0\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\in\\mathbb{R}$: this is a maximal subalgebra of $\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$. A closed Weyl chamber $\\mathfrak{b}^+$ is defined by the inequality $ s\\geq 0$. \n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\n\n\n\n\nThe following remark will be used repeatedly in the sequel.\n\n\n\\begin{rem} \\label{rem J preserva norma}\nEven though $G$ does not act on $\\mathbb{R}^d$, it makes sense to ask if an element $g$ of $G$ preserves a norm on $\\mathbb{R}^d$ (this notion does not depend on the choice of a lift of $g$ to $\\textnormal{SL}(d,\\mathbb{R})$). Given a Cartan involution $\\tau$ commuting with $\\sigma^o$, let $\\Vert\\cdot\\Vert_\\tau$ be a norm on $\\mathbb{R}^d$ preserved by $K^\\tau$. We claim that this norm is preserved by $J^o$. Indeed, this is obvious for the choices of Example \\ref{ex explicit example} and follows in general by conjugating by an element $g$ in $G$ that takes $[0,\\dots,0,1]$ to the point $o$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsection{The sub-manifold $S^o$}\\label{subsec So} Define\n\n\\begin{center}\n$S^o:=\\lbrace\\tau\\in X_G:\\hspace{0,3cm} \\tau\\sigma^o=\\sigma^o\\tau\\rbrace$.\n\\end{center}\n\n\n\n\n\\begin{rem}\\label{rem So space of hp trough o}\nRecall from Remark \\ref{rem Xg space of qplanes} that $X_G$ can be identified with the space of $q$-dimensional negative definite subspaces of $\\mathbb{R}^d$. Under this identification $S^o$ corresponds to the set of subspaces that contain the line $o$. By considering the $\\langle\\cdot,\\cdot\\rangle_{p,q}$-orthogonal complement we see that $S^o$ parametrizes the space of totally geodesic space-like copies of $\\mathbb{H}^{p}$ inside $\\mathbb{H}^{p,q-1}$ passing through $o$ (c.f. Subsection \\ref{subsub space copies Hp}).\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nUsing the fact that two elements of $S^o$ differ by conjugation by an element in $H^o$ one observes that for any $\\tau\\in S^o$ the following holds\n\n\\begin{center}\n$S^o=H^o\\cdot\\tau$.\n\\end{center}  \n\n\\noindent Further, the group $H^o$ has several connected components but one can see that the connected component containing the identity acts transitively on $S^o$. Hence $S^o$ is connected and one can show that\n\n\\begin{center}\n$S^o=\\exp(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o)\\cdot\\tau$. \n\\end{center}\n\n\n\\noindent It follows that $S^o$ is a totally geodesic sub-manifold of $X_G$ and $T_\\tau S^o\\cong \\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o$ (see \\cite[Theorem 7.2 of Ch. IV]{Hel}). \n\n\n\n\n\n\n\n\\subsection{$K\\exp(\\mathfrak{b}^+)H$-decomposition}\\label{subsec KexpliebH}\n\n\n\nFor the rest of this section we fix a Cartan involution $\\tau\\in S^o$, a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+\\subset\\mathfrak{b}$. By Schlichtkrull \\cite[Proposition 7.1.3]{Sch} the following decomposition of $G $ holds:\n\\begin{equation}\\label{eq polar decomposition}\nG=K^{\\tau}\\exp(\\mathfrak{b}^+)H^o\n\\end{equation}\n\\noindent where the $\\exp(\\mathfrak{b}^+)$-component is uniquely determined and one can define\n\\begin{equation} \\label{eq def nu}\nb^\\tau:G\\longrightarrow\\mathfrak{b}^+\n\\end{equation}\n\n\n\\noindent by taking the $\\textnormal{log}$ of this component. This is a continuous map called the \\textit{polar projection} of $G$ associated to the choice of $\\tau$ and $\\mathfrak{b}^+$. It generalizes the usual Cartan projection of $G$.\n\n\\begin{rem} \\label{rem btau is proper}\n\nNote that $b^\\tau$ is not proper (unless $q=1$). However it descends to a map $\\mathbb{H}^{p,q-1}\\cong G\/H^o\\longrightarrow\\mathfrak{b}^+$ which, by definition, is proper.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\nWe now discuss geometric interpretations of the polar projection $b^\\tau$. The geometric interpretation in $\\mathbb{H}^{p,q-1}$ follows Kassel-Kobayashi \\cite[p.151]{KK}, while the geometric interpretation in $X_G$ is inspired by the work of Oh-Shah \\cite{OS} for the case $p=1$ and $q=3$.\n\nLet us begin with the interpretation in the pseudo-Riemannian setting. By Remark \\ref{rem So space of hp trough o}, the choice of $\\tau\\in S^o$ determines a totally geodesic space-like copy of the $p$-dimensional hyperbolic space, inside $\\mathbb{H}^{p,q-1}$ and passing through $o$. We denote this copy by $\\mathbb{H}^{p}_\\tau$. From explicit computations one can show that \n\n\\begin{center}\n$\\mathbb{H}^{p}_\\tau=\\exp(\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o)\\cdot o$.\n\\end{center}\n\n\n\\noindent In particular $\\mathbb{H}^{p}_\\tau$ contains the geodesic ray $\\exp(\\mathfrak{b}^+)\\cdot o$ starting from $o$. Equality (\\ref{eq polar decomposition}) tell us that for every $g$ in $G$ the point $g\\cdot o$ lies in the $K^\\tau$-orbit of $o_g:=\\exp(b^\\tau(g))\\cdot o$ (see Figure \\ref{fig interp geom polar}). The geometric interpretation of the polar projection is now clear: the number $\\vert b^\\tau(g)\\vert^{\\frac{1}{2}}$ equals the length of the geodesic segment connecting $o$ with $o_g$\\footnote{Recall that $\\vert\\cdot\\vert$ is the form on $\\mathfrak{q}^o$ defined in Remark \\ref{rem form on qo}.}.\n\n\n\n\n\n\n \n\n\n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\scalebox{0.8}{%\n\\begin{overpic}[scale=1, width=1\\textwidth, tics=5]{fig1.pdf}\n\n \\put (30,79) { \\Large$g\\cdot o$}\n  \\put (87,92) { \\Large\\textcolor{blue}{$K^\\tau \\cdot o_g$}}\n    \\put (78,50) { \\Large\\textcolor{red}{$\\exp(\\mathfrak{b}^+)\\cdot o$}}\n        \\put (65,53) { \\Large$o_g$}\n        \\put (48,52) { \\Large$o$}\n        \\put (45,35) { \\Large$\\mathbb{H}^{p,q-1}$}\n        \\put (29,51) { \\Large$\\mathbb{H}^{p}_\\tau$}\n        \\put (5,35) { \\Large$\\partial\\mathbb{H}^{p,q-1}$}\n \\end{overpic}}\n \n\\hspace{0,3cm}\n\n\\begin{changemargin}{3cm}{3cm}    \n    \\caption{Geometric interpretation of polar projection in $\\mathbb{H}^{p,q-1}$.}\n  \\label{fig interp geom polar}\n\\end{changemargin}\n\n  \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\nWe now turn our attention to the Riemannian symmetric space $X_G$.\n\n\n\\begin{prop} \\label{prop interpretation of btau in symg}\nFor every $g$ in $G$ one has \n\n\\begin{center}\n$\\vert b^\\tau(g)\\vert^{\\frac{1}{2}}=d_{X_G}(g^{-1}\\cdot \\tau,S^o)$.\n\\end{center}\n\\end{prop}\n\n\n\\begin{proof}\nThe function $g\\mapsto d_{X_G}(g^{-1}\\cdot\\tau,S^o)$ is $K^{\\tau}$-invariant on the left and $H^o$-invariant on the right, hence it suffices to check that the equality of the statement holds when $g=\\exp(X)$ for some $X\\in\\mathfrak{b}^+$.\n\n\nSince $X_G$ is non-positively curved, there exists a unique geodesic through $\\exp(-X)\\cdot \\tau$ which is orthogonal to $S^o=\\exp(\\mathfrak{p}^\\tau\\cap\\mathfrak{h}^o)\\cdot \\tau$. This geodesic is $\\exp(\\mathfrak{b})\\cdot \\tau$ and intersects $S^o$ in $\\tau$, hence\n\n\n\n\n\\begin{center}\n$d_{X_G}(\\exp(-X)\\cdot\\tau,S^o)=d_{X_G}(\\exp(-X)\\cdot\\tau,\\tau)$.\n\\end{center}\n\n\n\\noindent Thanks to Remark \\ref{rem form on qo} and (\\ref{eq distance in XG and killing}) the proof is complete.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\nWe finish this subsection with a linear algebraic interpretation of the polar projection. Let $\\Vert\\cdot\\Vert_{\\tau}$ be a norm on $\\mathbb{R}^d$ invariant under the action of $K^\\tau$.\n\n\\begin{prop} \\label{prop computing nu}\nFor every $g$ in $G$ one has\n\\begin{center}\n$ \\vert b^\\tau(g)\\vert^{\\frac{1}{2}}=\\frac{1}{2}\\log\\Vert J^o gJ^og^{-1}\\Vert_{\\tau} $.\n\\end{center}\n\n\n\\end{prop}\n\n\n\n\n\n\\begin{proof}\n\nWe prove the proposition for the particular choices of Example \\ref{ex explicit example}, the general case follows from this one by conjugating by appropriate elements of $G$.\n\n\nBy Remark \\ref{rem J preserva norma} the matrix $J^o$ preserves $\\Vert\\cdot\\Vert_\\tau$ thus  \n\n\\begin{center}\n$\\frac{1}{2}\\log\\Vert J^o gJ^og^{-1}\\Vert_{\\tau}=\\frac{1}{2}\\log\\Vert gJ^og^{-1}\\Vert_{\\tau}$.\n\\end{center}\n\n\n\\noindent The map $g\\mapsto\\frac{1}{2}\\log\\Vert  gJ^og^{-1}\\Vert_{\\tau}$ is $K^{\\tau}$-invariant on the left and $H^o$-invariant on the right, hence it remains to check that the equality of the statement holds on $\\exp(\\mathfrak{b}^+)$. Let $X\\in\\mathfrak{b}^+$, that is,\n\n\n\n\n\n\n\\begin{center}\n$X=\\left(\\begin{matrix}\n &   & &  & s\\\\\n  & &   & 0 & \\\\\n & & \\iddots &  & \\\\\n &  0  &  &  & \\\\\ns &  & & &   \n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\geq 0$. Since $X\\in\\mathfrak{q}^o$, one has $J^o\\exp(-X)=\\exp(X)J^o$ and thus\n\n\n\\begin{center}\n$\\vert X\\vert^\\frac{1}{2}=s=\\frac{1}{2}\\log\\Vert  \\exp(X)J^o\\exp(-X)\\Vert_{\\tau}$.\n\\end{center}\n\\end{proof}\n\n\n\n\n\n\n\\subsection{$H\\exp(\\mathfrak{b}^+)H$-decomposition}\\label{subsec HexpliebH}\n\n\nRecall from Subsection \\ref{subsub geod hpq} the definition of the set $\\mathscr{C}^>_o$ and define\n\n\n\\begin{center}\n$\\mathscr{C}_{o,G}^>:=\\lbrace g\\in G:\\hspace{0,3cm} g\\cdot o\\in \\mathscr{C}_{o}^>\\rbrace$. \n\\end{center}\n\n\n\n\n\\begin{prop}\\label{prop HBH}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one can write\n\n\\begin{center}\n$g=h\\exp (X)h' $\n\\end{center}\n\n\n\\noindent for some $h,h'\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$.\n\\end{prop}\n\n\nIt is clear that this decomposition of $g$ can only hold when $g\\in\\mathscr{C}_{o,G}^>$.\n\n\\begin{proof}[Proof of Proposition \\ref{prop HBH}]\n\nTake $h$ in $H^o$ such that $h^{-1}g\\cdot o\\in\\exp(\\mathfrak{b}^+)\\cdot o$. There exists then $X\\in\\mathfrak{b}^+$ and $h'\\in H^o$ such that $h^{-1}g=\\exp(X)h'$. Note that $X$ is unique since it is determined by the length of the geodesic segment connecting $o$ with $g\\cdot o$.\n\n\\end{proof}\n\n\nWe define the map\n\\begin{equation} \\label{eq def de bo}\nb^o:\\mathscr{C}_{o,G}^>\\longrightarrow\\mathfrak{b}^+: \\hspace{0,3cm} g\\mapsto b^o(g)\n\\end{equation}\n\n\\noindent where $g=h\\exp(b^o(g))h'$ for some $h,h'\\in H^o$. Note that $b^o$ descends to the quotient $\\mathscr{C}_{o}^>$ but this map is not proper (compare with Remark \\ref{rem btau is proper}).\n\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop ell dXG y vertboverto}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one has\n\n\\begin{center}\n$\\ell_{o,g\\cdot o}=\\vert b^o(g)\\vert^{\\frac{1}{2}}=d_{X_G}(S^o,g\\cdot S^o)$.\n\\end{center}\n\\end{prop}\n\n\n\\begin{proof}\n\n\nThe first equality was already discussed in the proof of Proposition \\ref{prop HBH}. For the second one write $g=h\\exp(b^o(g))h'$. Since $S^o=H^o\\cdot \\tau$ we have\n\n\\begin{center}\n$d_{X_G}(S^o,h\\exp(b^o(g))h'\\cdot S^o)=d_{X_G}(H^o\\cdot\\tau,\\exp(b^o(g))H^o\\cdot\\tau)$.\n\\end{center}\n\n\nSet $X:=b^o(g)$. If $X=0$ there is nothing to prove, so assume $X\\neq 0$. In that case $H^o\\cdot \\tau$ is disjoint from $\\exp(X)H^o\\cdot\\tau$: since the action of $\\mathfrak{b}$ on the geodesic $\\exp(\\mathfrak{b})\\cdot\\tau$ is free, this follows from the fact that $X_G$ is non-positively curved and the fact that $\\exp(\\mathfrak{b})\\cdot\\tau$ intersects orthogonally $H^o\\cdot\\tau$ (resp. $\\exp(X)H^o\\cdot\\tau$) in $\\tau$ (resp. $\\exp(X)\\cdot\\tau$).\n\n\n\\begin{cla} \\label{claim in prop interp geom of bo}\n\nTake $\\tau'\\in H^o\\cdot\\tau$ and $\\tau''\\in \\exp(X)H^o\\cdot\\tau$. Then the following holds:\n\\begin{center}\n$d_{X_G}(\\tau',\\tau'')\\geq d_{X_G}(\\tau,\\exp(X)\\cdot\\tau)$.\n\\end{center}\n\\end{cla}\n\n\n\\begin{proof}[Proof of Claim \\ref{claim in prop interp geom of bo}]\n\nLet $\\beta_1\\subset H^o\\cdot \\tau$ (resp. $\\beta_2\\subset \\exp(X)H^o\\cdot\\tau$) be the unit-speed geodesic connecting $\\beta_1(0)=\\tau$ (resp. $\\beta_2(0)=\\exp(X)\\cdot \\tau$) with $\\tau'$ (resp. $\\tau''$). Then $\\beta_1$ and $\\beta_2$ are disjoint and from the fact that $X_G$ is non-positively curved follows that the map\n\\begin{center}\n$(t,s)\\mapsto d_{X_G}(\\beta_1(t),\\beta_2(s))$\n\\end{center}\n\\noindent is smooth (see Petersen \\cite[p.129]{Pet}). Moreover, since $\\exp(\\mathfrak{b})\\cdot \\tau$ is orthogonal both to $H^o\\cdot\\tau$ and $\\exp(X)H^o\\cdot\\tau$ we conclude that the differential at $(0,0)$ of this map is zero. \n\nTake $t_0>0$ such that $\\beta_1(t_0)=\\tau'$ and a positive $a$ such that the geodesic $t\\mapsto\\beta_2(at)$ equals $\\tau''$ in $t_0$. By Busemann \\cite[Theorem 3.6]{Bus} the map\n\n\\begin{center}\n$t\\mapsto d_{X_G}(\\beta_1(t),\\beta_2(at))$\n\\end{center}\n\n\n\n\\noindent is convex. Since it has a critical point at $t=0$ the proof of the claim is finished.\n\n\n\\end{proof}\n\nThanks to Remark \\ref{rem form on qo} and (\\ref{eq distance in XG and killing}) the proof of Proposition \\ref{prop ell dXG y vertboverto} is now complete.\n\n\n\\end{proof}\n\n\n\n\n\n\n\nRecall that $\\lambda_1(g)$ denotes the logarithm of the spectral radius of $g\\in G$.\n\n\n\n\\begin{prop}\\label{prop linear alg interpr of bo}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one has\n\n\\begin{center}\n$\\vert b^o(g)\\vert^{\\frac{1}{2}}=\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1})$.\n\\end{center}\n\n\n\n\\end{prop}\n\n\\begin{proof}\n\nIt suffices to prove the proposition for the choices of $o$ and $\\mathfrak{b}^+$ of Example \\ref{ex explicit example}. Write $g=h\\exp(b^o(g))h' $ with\n\n\n\n\\begin{center}\n$b^o(g)=\\left(\\begin{matrix}\n &  & &   & s\\\\\n & & & 0 & \\\\\n & &  \\iddots &    & \\\\\n & 0 & & & \\\\\ns &  &   & &\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\geq 0$. We have $\\vert b^o(g)\\vert^\\frac{1}{2}=s$. On the other hand, $J^o$ commutes with elements of $H^o$ and thus the number $\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1})$ equals to\n\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1(J^oh\\exp(b^o(g))J^o\\exp(b^o(g))^{-1}h^{-1})=\\frac{1}{2}\\lambda_1(J^o\\exp(b^o(g))J^o\\exp(b^o(g))^{-1})$.\n\\end{center}\n\n\n\n\\noindent Since $b^o(g)\\in\\mathfrak{q}^o$ we have $J^o\\exp(b^o(g))^{-1}=\\exp(b^o(g))J^o$ and the proof is complete.\n\\end{proof}\n\n\n\n\\section{Proximality} \\label{sec proximality}\n\\setcounter{equation}{0}\n\n\n\nIn this section we recall basic facts on product of proximal matrices, the main one being Benoist's Theorem \\ref{teo benoist}. This results are well-known but we provide proofs for those which are not explicitly stated in the literature (the reader familiarized with these concepts may skip this section). Standard references are the works of Benoist \\cite{Ben3,Ben4,Ben1}. \n\n\n\\subsection{Notations and basic definitions}\\label{subsec notation definitions in sec proximality}\n\n\n\nA norm $\\Vert \\cdot \\Vert$ on $\\mathbb{R}^d$ will be fixed in the whole section. For $\\xi_{1},\\xi_2\\in\\mathbb{P}(\\mathbb{R}^d)$ define the distance\n\n\\begin{center}\n$d(\\xi_1,\\xi_2):=\\inf\\lbrace \\Vert v_{\\xi_1}-v_{\\xi_2}\\Vert: \\hspace{0,3cm} v_{\\xi_i}\\in \\xi_i \\textnormal{ and } \\Vert v_{\\xi_i}\\Vert =1 \\textnormal{ for all } i=1,2 \\rbrace$.\n\\end{center}\n\n\n\\noindent Let $\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$ be the Grassmannian of $(d-1)$-dimensional subspaces of $\\mathbb{R}^d$. There exists a $G$-equivariant identification $\\mathbb{P}((\\mathbb{R}^d)^*)\\longrightarrow \\mathsf{Gr}_{d-1}(\\mathbb{R}^d) $ given by \n\n\\begin{center}\n$\\theta\\mapsto\\ker\\theta$\n\\end{center}\n\n\n\\noindent where the action of $G$ on the left side is given by $g\\cdot\\theta:=\\theta\\circ g^{-1}$. This identification will be used from now on whenever convenient.\n\n\nFor $\\eta_1,\\eta_2\\in\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$ we let\n\n\\begin{center}\n$d(\\xi_1,\\eta_1):=\\min\\lbrace d(\\xi_1,\\xi):\\hspace{0,3cm} \\xi\\in\\mathbb{P}(\\eta_1)\\rbrace$\n\\end{center}\n\n\\noindent and we denote by $d^*(\\eta_1,\\eta_2)$ the distance on $\\mathbb{P}((\\mathbb{R}^d)^*)$ induced by the operator norm on $(\\mathbb{R}^d)^*$. Given a positive $\\varepsilon$ we set\n\n\\begin{center}\n\n$b_\\varepsilon(\\xi_1):=\\lbrace \\xi\\in\\mathbb{P}(\\mathbb{R}^d):\\hspace{0,3cm} d(\\xi_1,\\xi)<\\varepsilon\\rbrace$\n\\end{center}\n\n\n\\noindent and\n\n\n\\begin{center}\n\n$B_\\varepsilon(\\eta_1):=\\lbrace \\xi\\in\\mathbb{P}(\\mathbb{R}^d):\\hspace{0,3cm} d(\\xi,\\eta_1)\\geq\\varepsilon\\rbrace$.\n\\end{center}\n\n\nOn the other hand, let\n\n\\begin{center}\n$\\mathbb{P}^{(2)}:=\\lbrace (\\theta,v)\\in\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} v\\notin\\ker\\theta \\rbrace$\n\\end{center}\n\n\n\\noindent and\n\n\n\\begin{center}\n$\\mathbb{P}^{(4)}:=\\lbrace (\\theta,v,\\phi,u)\\in\\mathbb{P}^{(2)}\\times\\mathbb{P}^{(2)}: \\hspace{0,3cm} v\\notin\\ker\\phi \\textnormal{ and } u\\notin\\ker\\theta \\rbrace$.\n\\end{center}\n\n\n\\noindent Observe that \n\\begin{equation} \\label{eq defi gcursiva}\n\\mathscr{G}_{\\Vert\\cdot\\Vert}=\\mathscr{G}:\\mathbb{P}^{(2)}\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} \\mathscr{G}( \\theta,v):=\\log\\dfrac{\\left\\vert\\theta(v)\\right\\vert}{\\Vert \\theta\\Vert\\Vert v\\Vert}\t\n \\end{equation} \n\\noindent is well-defined. Similarly the following map is well-defined\n\\begin{equation}\\label{eqdef crossratio}\n\\mathbb{B}:\\mathbb{P}^{(4)}\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} \\mathbb{B}(\\theta,v,\\phi,u):= \\log\\left\\vert\\dfrac{\\theta(u)}{\\theta(v)}\t\\dfrac{\\phi(v)}{ \\phi(u)}\\right\\vert\t\n \\end{equation} \n\n\n\\noindent and is called de \\textit{cross-ratio} of $(\\theta,v,\\phi,u)$\\footnote{Sometimes $e^{\\mathbb{B}}$ is called the cross-ratio.}. Both $\\mathscr{G}$ and $\\mathbb{B}$ are continuous.\n\n\n\n\n\n\\subsection{Product of proximal matrices}\\label{subsec product of proximal}\n\n\nGiven $g$ in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ we denote by\n\n\\begin{center}\n$\\lambda_1(g)\\geq\\dots\\geq\\lambda_d(g)$\n\\end{center}\n\n\\noindent the logarithms of the moduli of the eigenvalues of $g$, repeated with multiplicity (we use the convention $\\log 0=-\\infty$). The matrix $g$ is said to be \\textit{proximal} in $\\mathbb{P}(\\mathbb{R}^d)$ if $\\lambda_1(g)$ is simple. In that case we let $g_+$ (resp. $g_-$) to be the attractive fixed line (resp. repelling fixed hyperplane) of $g$ in $\\mathbb{P}(\\mathbb{R}^d)$. Note that if $g$ is non invertible then $g_-$ contains the kernel of $g$.\n\n\nWe now define a quantified version of proximality. The definition that we propose is (slightly) weaker than the one given by Benoist in \\cite{Ben3,Ben4,Ben1}. We provide proofs of the basic facts established in those works when necessary.\n\n\n\\begin{dfn} \\label{dfn of repsilon proximality}\nLet $0<\\varepsilon \\leq r$ and $g\\in\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ be a proximal matrix. The matrix $g$ is called \\textit{$(r,\\varepsilon)$-proximal} if $d(g_+,g_-)\\geq 2r$ and $g\\cdot B_\\varepsilon(g_-)\\subset b_\\varepsilon(g_+)$. \n\\end{dfn}\n\n\n\n\\begin{lema}[Benoist {\\cite[Corollaire 6.3]{Ben3}}]\\label{lema benoist proximal comp vasing y vap}\nLet $0<\\varepsilon\\leq r$. There exists a constant $ c_{r,\\varepsilon}>0$ such that for every $(r,\\varepsilon)$-proximal matrix $g$ one has\n\n\\begin{center}\n$\\log\\Vert g \\Vert - c_{r,\\varepsilon}\\leq \\lambda_1(g)\\leq \\log\\Vert g \\Vert$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\nThe following criterion of $(r,\\varepsilon)$-proximality will be very useful in the sequel.\n\n\n\\begin{lema}[Benoist {\\cite[Lemme 6.2]{Ben3}}]\\label{lema benoist lemma 1.2}\nLet $g$ be an element in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$, $\\eta\\in\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$, $\\xi\\in\\mathbb{P}(\\mathbb{R}^d)$ and $0<\\varepsilon\\leq r$. If $d(\\xi,\\eta)\\geq 6r$ and $g\\cdot B_\\varepsilon(\\eta)\\subset b_\\varepsilon(\\xi)$ then $g$ is $(2r,2\\varepsilon)$-proximal with $d(g_+,\\xi)\\leq\\varepsilon$ and $d^*(g_-,\\eta)\\leq\\varepsilon$.\n\n\\end{lema}\n\n\n\\begin{proof}\nConsider the Hilbert distance on the convex set $B_\\varepsilon(\\eta)$ (see \\cite{Ben2}). The condition $g\\cdot B_\\varepsilon(\\eta)\\subset b_\\varepsilon(\\xi)$ implies that $g$ is contracting for this metric and thus has a unique fixed point in $B_\\varepsilon(\\eta)$, which belongs in fact to $b_\\varepsilon(\\xi)$. The proof now finishes as in \\cite[Lemme 6.2]{Ben3}.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{cor}[Benoist {\\cite[Lemme 1.4]{Ben1}}]\\label{cor product of proximal transverse is proximal}\nLet $0<\\varepsilon\\leq r$. If $g_1$ and $g_2$ are $(r,\\varepsilon)$-proximal and satisfy \n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\\noindent then $g_1g_2$ is $(2r,2\\varepsilon)$-proximal. \n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{cor}\n\n\n\n\n\nLet $g_1$ and $g_2$ be two matrices as in Corollary \\ref{cor product of proximal transverse is proximal}. The goal now is to state a theorem (Theorem \\ref{teo benoist}) which provides a comparison between the spectral radius and operator norm of $g_1g_2$ in terms of the spectral radii of $g_1$ and $g_2$ and the maps $\\mathscr{G}$ and $\\mathbb{B}$.\n\n\n\n\\begin{lema}\\label{lemma atracrep of product}\n\nFix $r>0$ and $\\delta> 0$. For every $\\varepsilon$ small enough, the following property is satisfied: for every pair of $(r,\\varepsilon)$-proximal elements $g_1$ and $g_2$ such that\n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\\noindent one has\n\n\\begin{center}\n$\\vert \\mathscr{G}(g_{2_-},g_{1_+})-\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+)\\vert<\\delta$.\n\\end{center}\n\n\n\\end{lema}\n\n\n\n\\begin{proof}\nFor every $0<\\varepsilon\\leq r$, consider the compact set $C_{r,\\varepsilon}$ of pairs $(g_1,g_2)$ of norm-one $(r,\\varepsilon)$-proximal matrices in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ satisfying \n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$.\n\\end{center}\n\n\\noindent The function\n\\begin{center}\n$(g_1,g_2)\\mapsto \\vert \\mathscr{G}(g_{2_-},g_{1_+})-\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+)\\vert$\n\\end{center}\n\n\n\\noindent is continuous and equals zero on $C_r:=\\displaystyle\\cap_{\\varepsilon >0}C_{r,\\varepsilon}\\subset \\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$.\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\\begin{teo}[Benoist {\\cite[Lemme 1.4]{Ben1}}] \\label{teo benoist}\n\n\nFix $r>0$ and $\\delta> 0$. Then for every $\\varepsilon$ small enough, the following properties are satisfied: for every pair of $(r,\\varepsilon)$-proximal elements $g_1$ and $g_2$ such that\n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\n\\noindent one has:\n\n\n\\begin{enumerate}\n\\item The number\n\n\n\\begin{center}\n\n\n$\\left\\vert \\lambda_1( g_1g_2)  - (\\lambda_1(g_1)+\\lambda_1(g_2))-\\mathbb{B}(g_{1_-},g_{1_+},g_{2_-},g_{2_+}) \\right\\vert$\n\n\\end{center}\n\n\n\\noindent is less than $\\delta$.\n\n\\item The number\n\n\\begin{center}\n$\\left\\vert \\log\\Vert g_1g_2 \\Vert - (\\lambda_1(g_1)+\\lambda_1(g_2))-\\mathbb{B}(g_{1_-},g_{1_+},g_{2_-},g_{2_+})+\\mathscr{G}(g_{2_-},g_{1_+}) \\right\\vert$\n\\end{center}\n\n\n\\noindent is less than $\\delta$.\n\n\\end{enumerate}\n\n\\end{teo}\n\n\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item[(1)] See \\cite[Lemme 1.4]{Ben1}.\n\n\\item[(2)] Let $\\varepsilon$ be as in (1). For every $g_1$ and $g_2$ as in the statement, Corollary \\ref{cor product of proximal transverse is proximal} implies that $g_1g_2$ is $(2r,2\\varepsilon)$-proximal. By \\cite[Lemma 5.6]{Sam} (and taking $\\varepsilon$ smaller if necessary) we have\n\n\n\\begin{center}\n$\\left\\vert \\log\\Vert g_1g_2 \\Vert - \\lambda_1(g_1g_2)+\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+) \\right\\vert<\\delta$.\n\\end{center}\n\n\n\n\\noindent Lemma \\ref{lemma atracrep of product} finishes the proof.\n\n\n\\end{enumerate}\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Projective Anosov representations} \\label{sec anosov}\n\\setcounter{equation}{0}\n\n\nAnosov representations were introduced by Labourie \\cite{Lab} for surface groups and extended by Guichard-Wienhard \\cite{GW} to word hyperbolic groups. In this section we recall the definition of (projective) Anosov representations and some well-known facts concerning $(r,\\varepsilon)$-proximality of matrices in the image of such a representation. \n\n\n\n\n\n\n\n\n\\subsection{Singular values}\\label{subsec cartan dec in sec anosov}\n\n\nThe most useful characterization of Anosov representations for our purposes is the one given in terms of \\textit{singular values}. We begin by recalling this notion and we fix also some notations that we will use in the rest of the paper.\n\n\n\n\n\n\n\nLet $\\tau$ be a $q$-dimensional subspace of $\\mathbb{R}^d$ which is negative definite for $\\langle\\cdot,\\cdot\\rangle_{p,q}$. Consider $\\langle \\cdot,\\cdot\\rangle_\\tau$ to be the inner product of $\\mathbb{R}^d$ that coincides with $-\\langle\\cdot,\\cdot\\rangle_{p,q}$ (resp. $\\langle\\cdot,\\cdot\\rangle_{p,q}$) on $\\tau$ (resp. $\\tau^{\\perp_{p,q}}$) and for which $\\tau$ and $\\tau^{\\perp_{p,q}}$ are orthogonal. Given $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$, we let $g^{*_\\tau}$ to be the adjoint operator with respect to $\\langle \\cdot,\\cdot\\rangle_\\tau$. Set\n\n\\begin{center}\n\n$a_1^\\tau(g)\\geq \\dots \\geq a_d^\\tau(g)$\n\\end{center}\n\n\\noindent to be the logarithms of the eigenvalues of $\\sqrt{g^{*_\\tau}g}$ repeated with multiplicity. These are called the $\\tau$-\\textit{singular values} of $g$. Geometrically, they represent the (logarithms of the) lengths of the semi axes of the ellipsoid which is the image by $g$ of the unit sphere\n\n\\begin{center}\n$\\mathbb{S}^{d-1}_\\tau:=\\lbrace x\\in\\mathbb{R}^d:\\hspace{0,3cm} \\langle x,x\\rangle_\\tau=1 \\rbrace$.\n\\end{center}\n\n\n\n\n\n\nLet $i=1,\\dots,d-1$. Given an element $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$ such that $a^\\tau_i(g)>a^\\tau_{i+1}(g)$ we denote by $U_i(g)$ the $i$-dimensional subspace of $\\mathbb{R}^d$ spanned by the $i$ biggest axes of $g\\cdot \\mathbb{S}^{d-1}_\\tau$. We also set\n\n\n\\begin{center}\n$S_{d-i}(g):=U_{d-i}(g^{-1})$.\n\\end{center} \n\n\n\\begin{rem} \\label{rem complemento de Sdmenosuno va en Uuno}\nLet $\\varepsilon>0$. It follows from Singular Value Decomposition (see Horn-Johnson \\cite[Section 7.3 of Chapter 7]{HJ}), that there exists $L>0$ such that for every $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$ satisfying $a^\\tau_1(g)-a^\\tau_2(g)>L$ one has\n\n\\begin{center}\n$g\\cdot B_\\varepsilon(S_{d-1}(g))\\subset b_\\varepsilon( U_1(g))$,\n\\end{center}\n\n\\noindent where $B_\\varepsilon(S_{d-1}(g))$ and $b_\\varepsilon( U_1(g))$ are defined as in Subsection \\ref{subsec notation definitions in sec proximality}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{The definition of projective Anosov representations}\\label{subsec deifnition anosov}\n\nA lot of work has been done in order to simplify the original definition of Anosov representations, here we follow mainly the work of Bochi-Potrie-Sambarino \\cite{BPS} (see also Guichard-Gu\u00e9ritaud-Kassel-Wienhard \\cite{GGKW} or Kapovich-Leeb-Porti \\cite{KLP1}).\n\n\n\nFix $\\tau$ as in the previous subsection and let $\\Gamma$ be a finitely generated group. Consider a finite symmetric generating set $S$ of $\\Gamma$ and take $\\vert\\cdot\\vert_\\Gamma$ to be the associated word length: for $\\gamma$ in $\\Gamma$, it is the minimum number required to write $\\gamma$ as a product of elements of $S$\\footnote{This number depends on the choice of $S$. However, the set $S$ will be fixed from now on hence we do not emphasize the dependence on this choice in the notation.}. Let $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}{(d,\\mathbb{R})}$ be a representation. We say that $\\rho$ is \\textit{projective Anosov} if there exist positive constants $C$ and $\\alpha$ such that for all $\\gamma\\in\\Gamma$ one has\n\\begin{equation}\\label{eq def anosov}\na^\\tau_1(\\rho(\\gamma))-a^\\tau_2(\\rho(\\gamma))\\geq \\alpha\\vert\\gamma\\vert_\\Gamma-C.\n\\end{equation}\nBy Kapovich-Leeb-Porti \\cite[Theorem 1.4]{KLP3} (see also \\cite[Section 3]{BPS}), condition (\\ref{eq def anosov}) implies that $\\Gamma$ is word hyperbolic\\footnote{We refer the reader to the book of Ghys-de la Harpe \\cite{GdlH} for definitions and standard facts on word hyperbolic groups.}. We assume in this paper that $\\Gamma$ is non elementary. Let $\\partial_\\infty\\Gamma$ be the Gromov boundary of $\\Gamma$ and $\\Gamma_{\\textnormal{H}}$ be the set of infinite order elements in $\\Gamma$. Every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ has exactly two fixed points in $\\partial_\\infty\\Gamma$: the attractive one denoted by $\\gamma_+$ and the repelling one denoted by $\\gamma_-$. The dynamics of $\\gamma$ on $\\partial_\\infty\\Gamma$ is of type \\textit{north-south}.\n\n\n\n\nFix $\\rho:\\Gamma\\longrightarrow\\textnormal{PSL}(d,\\mathbb{R})$ a projective Anosov representation. By \\cite{BPS,GGKW,KLP1} we know that there exist continuous equivariant maps\n\n\\begin{center}\n$\\xi:\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{P}(\\mathbb{R}^d)$ and $\\eta:\\partial_\\infty\\Gamma\\longrightarrow\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$\n\\end{center}\n\n\n\\noindent which are \\textit{transverse}, i.e. for every $x\\neq y$ in $\\partial_\\infty\\Gamma$ one has\n\\begin{equation} \\label{eq transv condition}\n\\xi(x)\\oplus\\eta(y)=\\mathbb{R}^d.\n\\end{equation}\n\n\n\n\n\\noindent One can see that condition (\\ref{eq def anosov}) implies that for every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ the matrix $\\rho(\\gamma)$ is proximal. Equivariance of $\\xi$ and $\\eta$ implies that\n\n\\begin{center}\n$\\xi(\\gamma_+)=\\rho(\\gamma)_+$ and $\\eta(\\gamma_+)=\\rho(\\gamma^{-1})_-$.\n\\end{center}\n\n\n\\noindent It follows that both $\\xi$ and $\\eta$ are homeomorphisms onto their images. In fact, these homeo\\-mor\\-phisms are H\u00f6lder (see Bridgeman-Canary-Labourie-Sambarino \\cite[Lemma 2.5]{BCLS}).\n\n\n\nWe denote by $\\Lambda_{\\rho(\\Gamma)}\\subset\\mathbb{P}(\\mathbb{R}^d)$ the image of $\\xi$, which is called the \\textit{limit set} of $\\rho(\\Gamma)$: it is the closure of the set of attractive fixed points in $\\mathbb{P}(\\mathbb{R}^d)$ of proximal elements in $\\rho(\\Gamma)$. The image of $\\eta$  is called the \\textit{dual limit set} of $\\rho(\\Gamma)$.\n\n\n\n\nHere is another characterization of the limit sets which is very useful. An explicit reference is \\cite[Theorem 5.3]{GGKW} (it can also be deduced from {\\cite[Subsection 3.4]{BPS}}). Let $d=d_\\tau$ (resp. $d^*=d^*_\\tau$) be the distance on $\\mathbb{P}(\\mathbb{R}^d)$ (resp. $\\mathbb{P}((\\mathbb{R}^d)^*)$) associated to $\\langle\\cdot,\\cdot\\rangle_\\tau$.\n\n\\begin{prop}\\label{prop limit with S and U and Uuno cerca gammamas}\nLet $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}(d,\\mathbb{R})$ be a projective Anosov representation. Then $\\xi(\\partial_\\infty\\Gamma)$ (resp. $\\eta(\\partial_\\infty\\Gamma)$) equals the set of accumulation points of sequences $\\lbrace U_1(\\rho(\\gamma_n))\\rbrace_n$ (resp. $\\lbrace S_{d-1}(\\rho(\\gamma_n))\\rbrace_n$) where $\\gamma_n\\longrightarrow\\infty$. Moreover, given a positive $\\varepsilon$ there exists $L>0$ such that for every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ with $\\vert\\gamma\\vert_\\Gamma>L$ one has\n\n\\begin{center}\n\n$d(U_1(\\rho(\\gamma)),\\rho(\\gamma)_+)<\\varepsilon$ and $d^*(S_{d-1}(\\rho(\\gamma)),\\rho(\\gamma)_-)<\\varepsilon$.\n\n\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n\n\n\nWe are interested in projective Anosov representations whose image is contained in $G=\\textnormal{PSO}(p,q)$. The following remark is then important for our purposes.\n\n\\begin{rem}\nLet $\\rho:\\Gamma\\longrightarrow\\textnormal{PSL}(d,\\mathbb{R})$ be a projective Anosov representation. If $\\rho(\\Gamma)$ is contained in $G$ we say that $\\rho$ is \\textit{$P_1^{p,q}$-Anosov} (recall that $P_1^{p,q}$ denotes the (parabolic) subgroup of $G$ stabilizing an isotropic line). In this case, the image of $\\xi$ is contained in $\\partial\\mathbb{H}^{p,q-1}$ and the dual map $\\eta$ equals $\\xi^{\\perp_{p,q}}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{Proximality properties}\\label{subsec proximality anosov}\nThe following lemma will be useful in the next section.\n\n\\begin{lema}[c.f. {\\cite[Lemma 5.7]{Sam}}]\\label{lema sambarino lemma 5.7}\nLet $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}(d,\\mathbb{R})$ be a  projective Anosov representation and $0<\\varepsilon\\leq r$. Then\n\n\\begin{center}\n$\\#\\lbrace\\gamma\\in\\Gamma_{\\textnormal{H}}: \\hspace{0,3cm} d(\\rho(\\gamma)_+,\\rho(\\gamma)_-)\\geq 2r \\textnormal{ and } \\rho(\\gamma) \\textnormal{ is not } (r,\\varepsilon)\\textnormal{-proximal}\\rbrace<\\infty$.\n\\end{center}\n\n\n\\end{lema}\n\n\n\\begin{proof}\n\nConsider a sequence $\\gamma_n\\longrightarrow\\infty$ in $\\Gamma_{\\textnormal{H}}$ such that $d(\\rho(\\gamma_n)_+,\\rho(\\gamma_n)_-)\\geq 2r$ for all $n$. By Proposition \\ref{prop limit with S and U and Uuno cerca gammamas} for every $n$ big enough the following holds\n\n\\begin{center}\n$b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma_n)))\\subset b_{\\varepsilon}(\\rho(\\gamma_n)_+)$\n\\end{center}\n\n\\noindent and\n\n\\begin{center}\n$B_{\\varepsilon}(\\rho(\\gamma_n)_-)\\subset B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma_n)))$.\n\\end{center}\n\n\n\n\\noindent By Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and (\\ref{eq def anosov}) the condition $\\rho(\\gamma_n)\\cdot B_{\\varepsilon}(\\rho(\\gamma_n)_-)\\subset b_{\\varepsilon}(\\rho(\\gamma_n)_+)$ is sa\\-tis\\-fied for sufficiently large $n$.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{The set $\\pmb{\\Omega}_\\rho$}\\label{sec the set omegarho}\n\\setcounter{equation}{0}\n\n\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and define\n\n\n\n\n\n\n\\begin{center}\n$\\pmb{\\Omega}_{\\rho}:=\\lbrace o\\in \\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot \\xi(x)\\notin\\eta(x) \\textnormal{ for all } x\\in\\partial_\\infty\\Gamma\\rbrace$.\n\\end{center}\n\n\n\n\n\\noindent This section is structured as follows. In Subsection \\ref{subsec dynam on omegarho} we prove that the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous. Moreover, we show that if $o$ is a point in $\\pmb{\\Omega}_\\rho$ then the geodesic connecting $o$ with $\\rho(\\gamma)\\cdot o$ is space-like (apart from possibly finitely many exceptions $\\gamma\\in\\Gamma$). In Subsection \\ref{subsec proxim of jrhojrho} we study the matrices $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ for a point $o$ in $\\pmb{\\Omega}_\\rho$: we apply to them Benoist's work on proximality. Finiteness of our counting functions is proved in Subsection \\ref{subsec orbital counting functions}. Finally, in Subsection \\ref{subsec weak triangle} we prove a proposition that will be needed in the proof of Proposition \\ref{prop distribution on bg for length}.\n\n\nBefore we start, let us discuss some examples for which $\\pmb{\\Omega}_\\rho$ is non empty. From Proposition \\ref{prop fijos de Jo en borde} we know that the following alternative description of $\\pmb{\\Omega}_\\rho$ holds\n\n\\begin{center}\n$\\pmb{\\Omega}_{\\rho}=\\lbrace o=[\\hat{o}]\\in \\mathbb{H}^{p,q-1}:\\hspace{0,3cm} \\langle \\hat{o},\\hat{\\xi}\\rangle_{p,q}\\neq 0 \\textnormal{ for all } \\xi=[\\hat{\\xi}]\\in\\Lambda_{\\rho(\\Gamma)}\\rbrace$.\n\\end{center}\n\n\n\\noindent We have the following important example.\n\n\n\n\\begin{ex}\\label{ex omegarho no vacio}\n\n\\item \n\n\\begin{itemize}\n\\item Let $\\Gamma$ be the fundamental group of a convex co-compact hyperbolic manifold of dimension $m\\geq 2$ and $\\iota_0:\\Gamma\\longrightarrow \\textnormal{SO}(m,1)$ be the holonomy representation. Fix $p\\geq m$ and $q\\geq 2$. Consider the embedding $\\mathbb{R}^{m,1}\\hookrightarrow \\mathbb{R}^{p,q}$ given by\n\n\\begin{center}\n$\\mathbb{R}^{m,1}\\cong \\textnormal{span}\\lbrace e_{p-m+1},\\dots,e_{p+1}\\rbrace$,\n\\end{center}\n\n\n\\noindent where $e_i$ is the vector of $\\mathbb{R}^d$ with all entries equal to zero except for the $i$-th entry which is equal to one. This induces a projection $j:\\textnormal{SO}(m,1)\\longrightarrow G$ and a representation $\\rho_0:\\Gamma\\longrightarrow G$ defined by\n\n\n\\begin{center}\n$\\rho_0:=j\\circ\\iota_0$.\n\\end{center}\n\n\n\\noindent Thus $\\rho_0$ is $P_1^{p,q}$-Anosov, because $\\iota_0$ is $P_1^{m,1}$-Anosov. The set $\\pmb{\\Omega}_{\\rho_0}$ is non empty: every point $o\\in\\mathbb{H}^{p,q-1}$ for which the subspace\n\n\\begin{center}\n$\\textnormal{span}\\lbrace o,e_{p+2},\\dots,e_d\\rbrace$\n\\end{center}\n\n\n\\noindent has signature $(0,q)$ belongs to $\\pmb{\\Omega}_{\\rho_0}$. Since the condition of being Anosov is open in the space of representations of $\\Gamma$ into $G$ and the limit map $\\xi$ varies continuously with the representation (see Guichard-Wienhard \\cite[Theorem 5.13]{GW}), we obtain that if $\\rho$ is a small deformation of $\\rho_0$ then $\\pmb{\\Omega}_\\rho$ is non empty.\n\n\n\n\\item The previous example generalizes to a large class of representations introduced by Danciger-Gu\u00e9ritaud-Kassel in \\cite{DGK1,DGK2} called \\textit{$\\mathbb{H}^{p,q-1}$-convex co-compact}\\footnote{These are inclusion representations induced by taking an infinite discrete subgroup $\\Gamma< G$ which preserves some properly convex non empty open set $\\Omega\\subset\\mathbb{P}(\\mathbb{R}^d)$ whose boundary is strictly convex and of class $C^1$. One requires that $\\Gamma$ preserves some \\textit{distinguished} non empty convex subset of $\\Omega$ on which the action is co-compact (see \\cite{DGK1,DGK2} for precisions).}. Let $\\Gamma <G$ be a $\\mathbb{H}^{p,q-1}$-convex co-compact group and $\\rho:\\Gamma\\longrightarrow G$ be the inclusion representation, which is $P_1^{p,q}$-Anosov as proved in \\cite[Theorem 1.25]{DGK2}. Let $\\Omega$ be a non empty $\\Gamma$-invariant properly convex open subset of $\\mathbb{H}^{p,q-1}$. By \\cite[Proposition 4.5]{DGK2}, $\\Omega$ is contained in $\\pmb{\\Omega}_\\rho$.\n\n\\item There exist examples of $P_1^{p,q}$-Anosov representations $\\rho$ whose image is not $\\mathbb{H}^{p,q-1}$-convex co-compact but satisfy $\\pmb{\\Omega}_\\rho\\neq\\emptyset$ (see \\cite[Examples 5.2 \\& 5.3]{DGK1}).\n\n\\end{itemize}\n\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\n\n\n\n\\subsection{Dynamics on $\\pmb{\\Omega}_\\rho$}\\label{subsec dynam on omegarho}\n\n\n\n\n\n\n\n\nObserve that $\\pmb{\\Omega}_\\rho$ is $\\Gamma$-invariant. The following proposition is well-known, we include a proof for completeness.\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop action on omegarho is prop discont and limit set is limit set}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation. Then the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous, that is, for every compact set $C\\subset\\pmb{\\Omega}_\\rho$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\cdot C\\cap C\\neq \\emptyset\\right\\rbrace<\\infty$.\n\\end{center}\n\n\\noindent Moreover, for any point $o$ in $\\pmb{\\Omega}_\\rho$ the set of accumulation points of $\\rho(\\Gamma)\\cdot o$ in $\\mathbb{H}^{p,q-1}\\cup\\partial\\mathbb{H}^{p,q-1}$ coincides with the limit set $\\Lambda_{\\rho(\\Gamma)}$.\n\\end{prop}\n\n\n\\begin{proof}\n\n\n\n\n\n\n\nLet $C\\subset\\pmb{\\Omega}_\\rho$ be a compact set and fix a norm on $\\mathbb{R}^d$. By definition of $\\pmb{\\Omega}_\\rho$ we can take a positive $\\varepsilon$ such that\n\n\n\\begin{center}\n$C\\cap\\displaystyle\\bigcup_{x\\in\\partial_\\infty\\Gamma}b_\\varepsilon(\\xi(x))=\\emptyset$ and $C\\subset\\displaystyle\\bigcap_{x\\in\\partial_\\infty\\Gamma}B_\\varepsilon(\\eta(x))$.\n\\end{center}\n\n\n\n\n\nBy Proposition \\ref{prop limit with S and U and Uuno cerca gammamas}, Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and (\\ref{eq def anosov}) we know that, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$, the following holds:\n\n\n\\begin{center}\n$b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma)))\\subset \\displaystyle\\bigcup_{x\\in\\partial_\\infty\\Gamma}b_\\varepsilon(\\xi(x))$,\n\n\\vspace{0,3cm}\n\n$\\displaystyle\\bigcap_{x\\in\\partial_\\infty\\Gamma}B_\\varepsilon(\\eta(x))\\subset B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma)))$\n\n\n\\end{center}\n\n\n\\noindent and\n\n\\begin{center}\n\n$\\rho(\\gamma)\\cdot B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma))) \\subset b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma)))$.\n\n\\end{center}\n\n\\noindent For these $\\gamma$ we have then that $\\rho(\\gamma)\\cdot C$ is contained in the $\\varepsilon$-neighbourhood of $\\Lambda_{\\rho(\\Gamma)}$ and thus is disjoint from $C$.\n \n\n\n\nWe have shown that the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous and that for any point $o$ in $\\pmb{\\Omega}_\\rho$ the accumulation points of $\\rho(\\Gamma)\\cdot o$ belong to $\\Lambda_{\\rho(\\Gamma)}$. Conversely, the $\\Gamma$-orbit of any point in $\\Lambda_{\\rho(\\Gamma)}$ is dense in the limit set and now the proof is complete.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\nLet $o\\in\\pmb{\\Omega}_\\rho$ and recall the notations introduced in Subsection \\ref{subsub geod hpq}. Given an open set $W\\subset\\partial\\mathbb{H}^{p,q-1}$ disjoint from $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}$ we denote by $\\mathscr{C}_{o}^{>_W}$ the subset of $\\mathscr{C}_o^{>}$ consisting of points $o'$ such that the (space-like) geodesic ray connecting $o$ with $o'$ has its end point in $W$.\n\n\nThe following corollary has been proved by Glorieux-Monclair \\cite{GM} for $\\mathbb{H}^{p,q-1}$-convex co-compact groups.\n\n\n\\begin{cor}\\label{cor gammao in cowmayor}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $W\\subset\\partial\\mathbb{H}^{p,q-1}$ an open set containing $\\Lambda_{\\rho(\\Gamma)}$ with closure disjoint from $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}$. Then apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ one has $\\rho(\\gamma)\\cdot o \\in\\mathscr{C}_{o}^{>_W}$. In particular the geodesic joining $o$ with $\\rho(\\gamma)\\cdot o$ is space-like.\n\\end{cor}\n\n\n\n\\begin{proof}\n\nLet $C$ be the closure of $\\mathbb{H}^{p,q-1}\\setminus\\mathscr{C}_{o}^{>_W}$ in $\\mathbb{H}^{p,q-1}\\cup\\partial\\mathbb{H}^{p,q-1}$. Note that $C$ is compact and by Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set} does not contain accumulation points of $\\rho(\\Gamma)\\cdot o$, hence $\\rho(\\Gamma)\\cdot o\\cap C$ is finite.  Since $\\gamma\\mapsto\\rho(\\gamma)\\cdot o$ is proper the proof is complete.\n\n\n\\end{proof}\n\n\\subsection{Proximality of $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$} \\label{subsec proxim of jrhojrho}\n\n\nFor the rest of the section we fix a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$, a point $o\\in\\pmb{\\Omega}_\\rho$ and a Cartan involution $\\tau\\in S^o$.\n\n\nThe next lemma is a direct consequence of Proposition \\ref{prop limit with S and U and Uuno cerca gammamas}, transversality condition (\\ref{eq transv condition}) and the definition of $\\pmb{\\Omega}_\\rho$. \n\n\n\n\\begin{lema}\\label{lema JUuno lejos de Sdmenos1}\nLet $d_\\tau$ be the distance on $\\mathbb{P}(\\mathbb{R}^d)$ induced by the norm $\\Vert\\cdot\\Vert_\\tau$. There exists a positive constant $D$ such that\n\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma:\\hspace{0,3cm} d_\\tau(J^o\\cdot U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma^{-1})))< D\\rbrace <\\infty$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\\begin{lema} \\label{lema jrhojrho prox uno}\nThere exist $0<\\varepsilon\\leq r$ such that, apart from possibly finitely many exceptions $\\gamma\\in\\Gamma$, the matrix $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal.\n\\end{lema}\n\n\n\n\\begin{proof}\n\n\n\nWe apply a \\textit{ping-pong} argument together with Lemma \\ref{lema benoist lemma 1.2}. By Lemma \\ref{lema JUuno lejos de Sdmenos1} we can take a positive constant $r$ and a finite subset $F\\subset\\Gamma$ such that for every $\\gamma\\in\\Gamma\\setminus F$ one has\n\\begin{equation}\\label{eq jrhojrho prox uno}\nd_\\tau (J^o\\cdot U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma^{-1})))\\geq 6r.\n\\end{equation}\n\n\n\n\\noindent Take $0<\\varepsilon\\leq r$ such that for every $\\gamma\\in\\Gamma\\setminus F$ one has\n\n\\begin{center}\n$b_\\varepsilon(J^o\\cdot U_1(\\rho(\\gamma)))\\subset B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))$.\n\\end{center}\n\n\n\\noindent By Remark \\ref{rem J preserva norma} the matrix $J^o$ preserves $d_\\tau$ thus\n\n\n\n\n\\begin{center}\n$J^o\\cdot b_\\varepsilon( U_1(\\rho(\\gamma)))\\subset B_\\varepsilon( S_{d-1}(\\rho(\\gamma^{-1})))$.\n\\end{center}\n\n\n\\noindent By taking $F$ larger if necessary we have that\n\n\\begin{center}\n$\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(U_1(\\rho(\\gamma^{-1})))$\n\\end{center}\n\n\\noindent holds for every $\\gamma$ in $\\Gamma\\setminus F$. It follows that \n\n\\begin{center}$J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset B_\\varepsilon(S_{d-1}(\\rho(\\gamma)))$\n\\end{center}\n\n\\noindent and applying $\\rho(\\gamma)$ we obtain\n\n\\begin{center}\n$\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(U_1(\\rho(\\gamma)))$.\n\\end{center}\n\n\\noindent Then\n\\begin{center}\n$J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(J^o\\cdot  U_1(\\rho(\\gamma)))$.\n\\end{center}\n\n\\noindent By (\\ref{eq jrhojrho prox uno}) and Lemma \\ref{lema benoist lemma 1.2} the proof is finished.\n\n\\end{proof}\n\n\nThe following is a strengthening of Lemma \\ref{lema jrhojrho prox uno}. It provides a link between the generalized Cartan projections $b^o$ and $b^\\tau$ and the spectral radii of proximal elements in $\\rho(\\Gamma)$. For the remainder of the section we fix a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+$. \n\n\n\n\n\n\\begin{lema}\\label{lema jrhojrho prox dos}\n\nFix any $\\delta>0$ and  $A$ and $B$ two compact disjoint sets in $\\partial_\\infty\\Gamma$. Then there exist $0< \\varepsilon\\leq r$ such that, apart from possibly finitely many exceptions $\\gamma\\in \\Gamma_{\\textnormal{H}}$ with $\\gamma_-\\in A$ and $\\gamma_+\\in B$, the following holds:\n\n\\begin{enumerate}\n\\item The matrices $J^o\\rho(\\gamma)J^o$ and $\\rho(\\gamma^{-1})$ are $(r,\\varepsilon)$-proximal.\n\\item $d_\\tau(J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-)\\geq 6r$ and $d_\\tau(\\rho(\\gamma^{-1})_+,J^o\\cdot \\rho(\\gamma)_-)\\geq 6r$.\n\\item $d_\\tau((J^o\\rho(\\gamma)J^o)_+,\\rho(\\gamma^{-1})_-)\\geq 6r$ and $d_\\tau(\\rho(\\gamma^{-1})_+,(J^o\\rho(\\gamma)J^o)_-)\\geq 6r$.\n\\item The matrix $\\rho(\\gamma)$ belongs to $\\mathscr{C}_{o,G}^>$ and the number \n\n\\begin{center}\n$\\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}   - \\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\\noindent is at distance at most $\\delta$ from\n\n\\begin{center}\n$\\frac{1}{2}\\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)$.\n\\end{center}\n\n\n\\item The number\n\n\\begin{center}\n\n$\\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} - \\lambda_1(\\rho(\\gamma)) $\n\\end{center}\n\n\\noindent is at distance at most $\\delta$ from\n\n\\begin{center}\n$\\frac{1}{2} \\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)-\\frac{1}{2}\\mathscr{G}_\\tau(\\rho(\\gamma^{-1})_-,J^o\\cdot \\rho(\\gamma)_+)$.\n\\end{center}\n\n\n\\end{enumerate}\n\n\\end{lema}\n\n\n\n\n\\begin{proof}\n\nBy transversality condition (\\ref{eq transv condition}) there exists $r>0$ such that\n\\begin{equation}\\label{eq transvversalidad A y B distribucion nu}\nd_{\\tau}(\\xi(x),\\eta(y))\\geq 2r \\textnormal{ and } d_{\\tau}(\\xi(y),\\eta(x))\\geq 2r\n\\end{equation}\n\\noindent for all $(x,y)\\in A\\times B$. Further, since $o\\in\\pmb{\\Omega}_{\\rho}$ we may assume\n\\begin{equation} \\label{eq seisr en teo distribucion nu}\nd_{\\tau}(J^o\\cdot\\xi(x),\\eta(x))\\geq 6r\n\\end{equation}\n\n\n\\noindent for all $x\\in\\partial_\\infty\\Gamma$. Given these $r>0$ and $2\\delta>0$, we consider $\\varepsilon>0$ as in Benoist's Theorem \\ref{teo benoist}. \n\nBy Lemma \\ref{lema sambarino lemma 5.7} there exists a finite subset $F$ of $\\Gamma_{\\textnormal{H}}$ outside of which elements satisfying $d_{\\tau}(\\rho(\\gamma)_+,\\rho(\\gamma)_-)\\geq 2r$ are $(r,\\varepsilon)$-proximal. Thanks to (\\ref{eq transvversalidad A y B distribucion nu}), for all $\\gamma\\in\\Gamma_{\\textnormal{H}}\\setminus F$ with $\\gamma_-\\in A$ and $\\gamma_+\\in B$ one has that $\\rho(\\gamma^{\\pm 1})$ is $(r,\\varepsilon)$-proximal. Moreover, since $J^o=(J^o)^{-1}$ preserves $\\Vert\\cdot\\Vert_{\\tau}$ we have that $J^o\\rho(\\gamma)J^o$ is $(r,\\varepsilon)$-proximal with $(J^o\\rho(\\gamma)J^o)_{\\pm}=J^o\\cdot\\rho(\\gamma)_{\\pm}$. In fact, by (\\ref{eq seisr en teo distribucion nu}) we have\n\n\\begin{center}\n$d_{\\tau}(J^o\\cdot\\rho(\\gamma)_{+},\\rho(\\gamma^{-1})_{-})\\geq 6r \\textnormal{ and } d_{\\tau}(\\rho(\\gamma^{-1})_{+},J^o\\cdot\\rho(\\gamma)_{-})\\geq 6r$.\n\\end{center}\n\n\n\n\\noindent Thanks to Proposition \\ref{prop linear alg interpr of bo} (and Corollary \\ref{cor gammao in cowmayor}), Proposition \\ref{prop computing nu}, Theorem \\ref{teo benoist} and the fact that $\\lambda_1(\\rho(\\gamma^{-1}))$ equals $\\lambda_1(\\rho(\\gamma))$ for all $\\gamma$, the proof is finished.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{The orbital counting functions of Theorems \\ref{teorema A} and \\ref{teorema B}}\\label{subsec orbital counting functions}\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop counting with nu is well defined}\n\nFor every $t\\geq 0$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t\\right\\rbrace<\\infty$.\n\\end{center}\n\\end{prop}\n\n\n\n\\begin{proof}\n\n\nBy Remark \\ref{rem btau is proper} the map $b^\\tau$ descends to a proper map in $\\mathbb{H}^{p,q-1}\\cong G\/H^o$, that we still denote by $b^\\tau$. Hence\n\n\\begin{center}\n$C:=\\lbrace o'\\in\\mathbb{H}^{p,q-1}:\\hspace{0,3cm}  \\vert b^\\tau(o')\\vert\\leq t^2 \\rbrace$\n\\end{center}\n\n\n\n\n\n\n\\noindent is compact. By Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set}, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$, we have that $\\rho(\\gamma)\\cdot o$ does not belong to $C$.\n\n\n\\end{proof}\n\n\n\nThe next proposition follows from a combination of Propositions \\ref{prop linear alg interpr of bo} and \\ref{prop computing nu}, Lemmas \\ref{lema jrhojrho prox uno} and \\ref{lema benoist proximal comp vasing y vap}, and the previous proposition.\n\n\n\n\n\\begin{prop}\\label{prop counting with lambdauno is well defined}\n\nFor every $t\\geq 0$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t\\right\\rbrace<\\infty$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\\begin{rem}\\label{rem crit exponent coindes with the one of GM}\nAssume that $\\rho$ is $\\mathbb{H}^{p,q-1}$-convex co-compact and the basepoint $o$ belongs to the convex hull of the limit set of $\\rho$. By Corollary \\ref{cor gammao in cowmayor} and Proposition \\ref{prop ell dXG y vertboverto} we have that\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t\\rbrace}{t}$\n\\end{center}\n\n\\noindent coincides with\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} d_{\\mathbb{H}^{p,q-1}}(o,\\rho(\\gamma)\\cdot o)\\leq t\\rbrace}{t}$,\n\\end{center}\n\n\\noindent where $d_{\\mathbb{H}^{p,q-1}}$ is the $\\mathbb{H}^{p,q-1}$-distance introduced in \\cite{GM}.\n\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\\subsection{Weak triangle inequality}\\label{subsec weak triangle}\n\n\n\nThe following proposition is inspired by \\cite[Theorem 3.5]{GM}. \n\n\n\n\n\n\\begin{prop} \\label{prop traingle inequality}\nThere exists a constant $L>0$ such that for every $f\\in\\Gamma$ there exists $D_f>0$ with the following property:  for every $\\gamma\\in\\Gamma$  with $\\vert\\gamma\\vert_\\Gamma>L$ one has \n\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1(J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1}))\\leq D_f+\\frac{1}{2}\\lambda_1(J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1}))$.\n\\end{center}\n\n\n\n\\end{prop}\n\n\nWe can think about the content of Proposition \\ref{prop traingle inequality} as follows. Fix $f\\in\\Gamma$ such that $\\rho(f)\\in\\mathscr{C}_{o,G}^>$. By Corollary \\ref{cor gammao in cowmayor} for every $\\gamma$ with $\\vert\\gamma\\vert_\\Gamma$ large enough one has $\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$ and $\\rho(f)\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$. Thanks to Proposition \\ref{prop ell dXG y vertboverto} and Proposition \\ref{prop linear alg interpr of bo}, the inequality established in Proposition \\ref{prop traingle inequality} can be stated as\n\n\\begin{center}\n$\\ell_{o,\\rho(f)\\rho(\\gamma)\\cdot o}\\leq D_f+\\ell_{\\rho(f)\\cdot o,\\rho(f)\\rho(\\gamma)\\cdot o}$,\n\\end{center}\n\n\\noindent where the constant $D_f$ depends on the choice of $o$ and $f$ (and $\\rho$) but not on the choice of $\\gamma$. Even though the function $\\ell_{\\cdot,\\cdot}$ is not a distance, we can heuristically think about $D_f$ as the term that replaces $\\ell_{o,\\rho(f)\\cdot o}$ in the usual triangle inequality for distances.\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop traingle inequality}]\n\n\n\nTake $0<\\varepsilon\\leq r$ as in Lemma \\ref{lema jrhojrho prox uno}. Let $L>0$ such that for every $\\gamma$ in $\\Gamma$ with $\\vert \\gamma\\vert_\\Gamma >L$ the matrix $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal. Fix $f\\in\\Gamma$ and let $\\gamma$ be a element in $\\Gamma$ with $\\vert \\gamma\\vert_\\Gamma >L$. We have\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1( J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1}))\\leq\\frac{1}{2}\\log\\Vert J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1})\\Vert_\\tau$.\n\\end{center}\n\n\\noindent By Remark \\ref{rem J preserva norma} the right side number equals $\\frac{1}{2}\\log\\Vert \\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1})\\Vert_\\tau$ which is less than or equal to\n\n\\begin{center}\n$  D_f'+\\frac{1}{2}\\log\\Vert J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\Vert_\\tau$\n\\end{center}\n\n\n\n\\noindent where $D_f':=\\frac{1}{2}\\log\\Vert \\rho(f)\\Vert_\\tau +\\frac{1}{2}\\log \\Vert\\rho(f^{-1})\\Vert_\\tau$. Since $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal, we conclude by applying Lemma \\ref{lema benoist proximal comp vasing y vap}.\n\n\\end{proof}\n\n\n\n\n\n\\section{Distribution of the orbit of $o$ with respect to $ b^o$}\\label{section distrib wrt bo}\n\\setcounter{equation}{0}\n\nIn this section we prove Theorem \\ref{teorema A}. The section is structured as follows: in Subsection \\ref{subsec cocycle co} we define a H\u00f6lder cocycle on $\\partial_\\infty\\Gamma$ and the corresponding flow. In Subsection \\ref{subsec dual and gromov co} we study the associated Gromov product. Theorem \\ref{teorema A} in the torsion free case (resp. general case) is proved in Subsection \\ref{subsec dist of fixed wrt bo} (resp. Subsection \\ref{subsec proof teo A}).\n\n\n\n\n\nFor the rest of the section we fix $\\rho:\\Gamma\\longrightarrow G$ a $P_1^{p,q}$-Anosov representation and a point $o$  in $\\pmb{\\Omega}_\\rho$.\n\n\n\\subsection{The cocycle $c_o$}\\label{subsec cocycle co}\n\nObserve that by definition of $\\pmb{\\Omega}_\\rho$ and equivariance of the curves $\\xi$ and $\\eta$ the following map is well-defined.\n\n\n\\begin{dfn} \\label{dfn cocycle o}\n\nLet\n\n\\begin{center}\n$c_o:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_o(\\gamma,x):=\\dfrac{1}{2}\\log\\left\\vert\\dfrac{\\theta_{x}\\left(\\rho(\\gamma^{-1})J^o\\rho(\\gamma)\\cdot v_x\\right)}{\\theta_{x}\\left(J^o\\cdot v_x\\right)}\\right\\vert$,\n\\end{center}\n\n\\noindent where $\\theta_{x}:\\mathbb{R}^d\\longrightarrow\\mathbb{R}$ is a non-zero linear functional whose kernel equals $\\eta(x)$ and $v_{x}\\neq 0$ belongs to $\\xi(x)$.\n\\end{dfn}\n\n\nA geometric interpretation of the map $c_o$ is provided by the following remark. This characterization will not be used in the sequel.\n\n\n\\begin{rem} \\label{rem co es la buseman de gm}\n\n\nOne can prove that for every $\\gamma\\in\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has\n\n\n\\begin{center}\n$c_{o}(\\gamma,x)=\\beta_{\\xi(x)}(\\rho(\\gamma^{-1})\\cdot o,o)$\n\\end{center}\n\n\n\\noindent where $\\beta_\\cdot(\\cdot,\\cdot)$ is the pseudo-Riemannian Busemann function defined by Glorieux-Monclair \\cite[Definition 3.8]{GM}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\nRecall that a \\textit{H\u00f6lder cocycle} is a function $c:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ satisfying that for every $\\gamma_0,\\gamma_1$ in $\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has \n\n\\begin{center}\n$c(\\gamma_0\\gamma_1,x)=c(\\gamma_0,\\gamma_1\\cdot x)+c(\\gamma_1,x)$\n\\end{center}\n\n\\noindent and such that the map $c(\\gamma_0,\\cdot)$ is H\u00f6lder (with the same exponent for every $\\gamma_0$). The \\textit{period} of (an infinite order element) $\\gamma\\in \\Gamma_{\\textnormal{H}}$ is defined by $\\ell_{c}(\\gamma):=c(\\gamma,\\gamma_+)$.\n\n\n\n\\begin{lema}\\label{lema cocycle o and periods}\nThe map $c_o$ is a H\u00f6lder cocycle. The period of $\\gamma\\in\\Gamma_{\\textnormal{H}}$ is given by\n\n\\begin{center}\n$\\ell_{c_o}(\\gamma)=\\lambda_1(\\rho(\\gamma))>0$.\n\\end{center}\n\n\n\n\\end{lema}\n\n\n\n\n\\begin{proof} \n\n\n\nA direct computation shows that $c_o$ is a H\u00f6lder cocycle.\n\n\nOn the other hand let $\\gamma\\in\\Gamma_{\\textnormal{H}}$ and fix a particular choice of a linear functional $\\theta_{\\gamma_+}$. Since $\\lambda_1(\\rho(\\gamma))=\\lambda_1(\\rho(\\gamma^{-1}))$ one sees that $\\theta_{\\gamma_+}\\circ(\\pm\\rho(\\gamma^{-1}))$ coincides with $ e^{\\lambda_1(\\rho(\\gamma))}\\theta_{\\gamma_+}$ up to a sign (here $\\pm\\rho(\\gamma^{-1})$ denotes some lift of $\\rho(\\gamma^{-1})$ to $\\textnormal{SO}(p,q)$). The proof is now complete.\n\n\n\n\\end{proof}\n\n\nSet $\\partial_\\infty^{2}\\Gamma:=\\lbrace(x,y)\\in\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma:\\hspace{0,3cm} x\\neq y\\rbrace$ and consider the \\textit{translation flow} on $\\partial_\\infty^{2}\\Gamma\\times \\mathbb{R}$ defined by\n\\begin{equation}\\label{eq translation flow}\n\\psi_t(x,y,s):=(x,y,s-t).\n\\end{equation}\n\\noindent The group $\\Gamma$ acts on $ \\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ by\n\\begin{equation} \\label{eq action via co}\n\\gamma\\cdot(x,y,s):=(\\gamma\\cdot x,\\gamma\\cdot y, s-c_o(\\gamma,y)).\n\\end{equation}\n\\noindent This action is proper and co-compact and we denote the quotient space by $\\textnormal{U}_o\\Gamma$. The flow $\\psi_t$ descends to a flow on $\\textnormal{U}_o\\Gamma$, still denoted $\\psi_t$, which is a H\u00f6lder reparametrization of the Gromov geodesic flow of $\\Gamma$ \\cite{Gro}. This is the analogue of Sambarino's Theorem \\cite[Theorem 3.2(1)]{Sam} (see also Lemma \\ref{lema conj urhogamma y uogamma}).\n\n\n\nWe say that an element $\\gamma$ in $\\Gamma$ is \\textit{primitive} if cannot be written as a positive power of another element in $\\Gamma$. Periodic orbits of $\\psi_t$ are in one-to-one correspondence with conjugacy classes of primitive elements in $\\Gamma$. If $[\\gamma]$ is such a conjugacy class, the period of the corresponding periodic orbit is \n\n\\begin{center}\n$\\ell_{c_o}(\\gamma)=\\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\\noindent (see Fact \\ref{fact urhogamma is anosov} and Lemma \\ref{lema conj urhogamma y uogamma}). The topological entropy of $\\psi_t$ coincides with the \\textit{entropy} of $\\rho$ defined by Bridgeman-Canary-Labourie-Sambarino \\cite{BCLS}:\n\n\n\n\\vspace{0,2cm}\n\n\n\\begin{center}\n$h_\\textnormal{top}(\\psi_t)=h_\\rho:=\\displaystyle\\limsup_{t\\longrightarrow\\infty} \\dfrac{\\log\\#\\lbrace [\\gamma]\\in [\\Gamma]:\\hspace{0,3cm} \\gamma \\textnormal{ is primitive and }\\lambda_1(\\rho(\\gamma))\\leq t\\rbrace}{t}$.\n\n\\end{center} \n\n\n\\vspace{0,2cm}\n\n\n\\noindent It is positive and finite (c.f. Fact \\ref{fact entropy, equidistribution and counting for urhogamma}) and will be denoted by $h$ from now on.\n\n\n\n\n\n\n\\begin{rem}\nOne can prove that if we \\textit{push} all this construction by the limit map $\\xi:\\partial_\\infty\\Gamma\\longrightarrow\\Lambda_{\\rho(\\Gamma)}$, we recover the geodesic flow defined in \\cite[Subsection 6.1]{GM} for $\\mathbb{H}^{p,q-1}$-convex co-compact groups. This remark will not be used in the sequel.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsection{Dual cocycle and Gromov product}\\label{subsec dual and gromov co}\n\n\nThanks to transversality condition (\\ref{eq transv condition}) and the fact that $o$ belongs to $\\pmb{\\Omega}_\\rho$ the following map is well-defined.\n\n\n\n\n\\begin{dfn}\nLet\n\n\n\\begin{center}\n$[\\cdot,\\cdot]_o:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_o:=-\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\theta_{x}\\left(J^o\\cdot v_x\\right)\\theta_{y}\\left(J^o\\cdot v_y\\right)}{\\theta_{x}\\left(v_y\\right)\\theta_{y}\\left(v_x\\right)}\\right\\vert$,\n\\end{center}\n\n\\noindent where $\\theta_x$ (resp. $\\theta_y$) is a non-zero linear functional whose kernel is $\\eta(x)$ (resp. $\\eta(y)$) and $v_x$ (resp. $v_y$) is a non-zero vector in $\\xi(x)$ (resp. $\\xi(y)$).\n\\end{dfn}\n\n\\begin{rem}\\label{rem gromov o is the one of GM}\nThe map $[\\cdot,\\cdot]_o$ coincides, up to a sign, with the Gromov product introduced in \\cite[Subsection 3.5]{GM}. The authors give geometric interpretations of this function using pseudo-Riemannian geometry.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\\begin{rem}\\label{rem coco are dual}\nThe cocycle $c_o$ is dual to itself, i.e. $\\ell_{c_o}(\\gamma)=\\ell_{c_{o}}(\\gamma^{-1})$ for every $\\gamma\\in\\Gamma_{\\textnormal{H}}$. Indeed, this follows from Lemma \\ref{lema cocycle o and periods} and the fact that $\\lambda_1(g)=\\lambda_1(g^{-1})$ for all $g$ in $ G$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nThe proof of the following lemma is a direct computation.\n\n\n\\begin{lema}\\label{lema gromov o}\nThe map $[\\cdot,\\cdot]_o$ is a Gromov product for the pair $\\lbrace c_o,c_o\\rbrace$, that is, for every $\\gamma\\in\\Gamma$ and every $(x,y)\\in\\partial_\\infty^{2}\\Gamma$ one has\n\n\n\\begin{center}\n$[\\gamma\\cdot x,\\gamma\\cdot y]_o-[x,y]_o=-(c_o(\\gamma,x)+c_o(\\gamma,y))$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\n\n\nThe following lemma will be very important in the proof of Theorem \\ref{teorema A}. It provides a geometric interpretation of the Gromov product different from the one given in Remark \\ref{rem gromov o is the one of GM}.\n\n\\begin{lema}\\label{lema computing gromov on gammapm o}\nLet $\\gamma$ be an element of $\\Gamma_{\\textnormal{H}}$. Then\n\n\\begin{center}\n$[\\gamma_-,\\gamma_+]_o=-\\frac{1}{2} \\mathbb{B}(J^o\\cdot\\rho(\\gamma)_-,J^o\\cdot\\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)$.\n\\end{center}\n\\end{lema}\n\n\n\\begin{proof}\n\n\nFrom Section \\ref{sec anosov} we know that $\\rho(\\gamma^{\\pm 1})$ is proximal and that the following holds:\n\n\\begin{center}\n$\\rho(\\gamma)_+=\\xi(\\gamma_{+})$, \\hspace{0,5cm} $\\rho(\\gamma^{-1})_+=\\xi(\\gamma_{-})$, \\hspace{0,5cm} $\\rho(\\gamma)_-=\\eta(\\gamma_-)$, \\hspace{0,5cm} $\\rho(\\gamma^{-1})_-=\\eta(\\gamma_{+})$.\n\n\\end{center}\n\n\n\n\n\n\\noindent Since $J^o=(J^o)^{-1}$, the matrix $J^o\\rho(\\gamma) J^o$ is proximal and one has the equalities\n\n\\begin{center}\n$(J^o\\rho(\\gamma) J^o)_{+}=J^o\\cdot \\xi(\\gamma_+)$ and $(J^o\\rho(\\gamma) J^o)_{-}=J^o\\cdot \\eta(\\gamma_-)$.\n\\end{center}\n\n\n\n\n\n\n\n\\noindent The proof finishes by a direct computation.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Distribution of attractors and repellors with respect to $ b^o$} \\label{subsec dist of fixed wrt bo}\n\n\nRecall that $h=h_{\\textnormal{top}}(\\psi_t)$ and let $\\mu_{o}$ be a \\textit{Patterson-Sullivan probability} on $\\partial_\\infty\\Gamma$ associated to $c_{o}$, i.e. $\\mu_o$ satisfies\n\n\\begin{center}\n$\\dfrac{d\\gamma_*\\mu_o}{d\\mu_o}(x)=e^{-hc_{o}(\\gamma^{-1},x)}$\n\\end{center}\n\n\n\\noindent for every $\\gamma\\in\\Gamma$. Such a probability exists (see Subsection \\ref{subsub PS}). By Lemma \\ref{lema gromov o} the measure\n\\begin{equation}\ne^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt\n\\end{equation}\n\\noindent on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ is $\\Gamma$-invariant and induces on the quotient $\\textnormal{U}_o\\Gamma$ a $\\psi_t$-invariant measure. By Sambarino \\cite[Theorem 3.2(2)]{Sam} this measure is, up to scaling, the probability of maximal entropy of $\\psi_t$ (see Proposition \\ref{prop product is of maximal entropy}).\n\n\n\nFor a metric space $X$ we denote by $C_c^*(X)$ the dual of the space of compactly supported continuous real functions on $X$ equipped with the weak-star topology. If $x$ is a point in $X$, let $\\delta_x\\in C_c^*(X)$ be the Dirac mass at $x$.\n\n\n\n\n\\begin{prop}[Sambarino {\\cite[Proposition 4.3]{Sam}}\\footnote{For a proof in our setting see Proposition \\ref{prop distribution of periodic orbits}.}]\\label{prop sambarino distribution wrt periods}\n\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},\\ell_{c_o}(\\gamma)\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C_c^*(\\partial_\\infty^{2}\\Gamma)$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n\n\nFrom Proposition \\ref{prop sambarino distribution wrt periods} we deduce Proposition \\ref{prop distribution on bg for length} which directly implies Theorem \\ref{teorema A} in the torsion free case. \n\n\nFix a point $\\tau\\in S^o$, a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+$ contained in $\\mathfrak{b}$.\n\n\\begin{prop} \\label{prop distribution on bg for length}\n\nThere exists a  constant $M=M_{\\rho,o}>0$ such that\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$.\n\n\\end{prop}\n\n\n\nRecall that the generalized Cartan projection $b^o$ is defined in the set $\\mathscr{C}_{o,G}^>$. The sum in Proposition \\ref{prop distribution on bg for length} is taken then over all elements $\\gamma\\in\\Gamma_{\\textnormal{H}}$ for which $\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$ and $ \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t$. To make the formula more readable we do not emphasize the fact that $\\rho(\\gamma)$ must belong to $\\mathscr{C}_{o,G}^>$. On the other hand, by Corollary \\ref{cor gammao in cowmayor} this condition holds apart from finitely many exceptions $\\gamma\\in\\Gamma$.\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop distribution on bg for length}]\n\n\n\nSet \n\n\\begin{center}\n\n$\\theta_t:=M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}$.\n\\end{center}\n\n\n\n\nWe first prove the statement outside the diagonal, that is, on subsets of $\\partial_\\infty^{2}\\Gamma$. Let $\\delta>0$ and $A,B\\subset\\partial_\\infty\\Gamma$ disjoint open sets. Consider an element $\\gamma\\in\\Gamma_{\\textnormal{H}}$ such that $\\gamma_-\\in A$ and $\\gamma_+\\in B$ and let $s:=[\\gamma_-,\\gamma_+]_{o}$. By taking $A$ and $B$ smaller we may assume \n\\begin{equation} \\label{eq aprox grom prod on dist thm nu}\n\\vert [x,y]_{o}-s\\vert<\\delta\n\\end{equation}\n\\noindent for all $(x,y)\\in A\\times B$.\n\n\n\n\nBy Lemma \\ref{lema jrhojrho prox dos}, apart from possibly finitely many exceptions $\\gamma\\in \\Gamma_{\\textnormal{H}}$ with $(\\gamma_-,\\gamma_+)\\in A\\times B$, the following holds:\n\n\n\n\n\\begin{center}\n$\\left\\vert  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}-\\lambda_1(\\rho(\\gamma))  -\\frac{1}{2}\\mathbb{B}(J^o\\cdot\\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)\\right\\vert <\\delta$.\n\\end{center}\n\n\n\n\n\n\n\n\\noindent Applying Lemma \\ref{lema cocycle o and periods} and Lemma \\ref{lema computing gromov on gammapm o} we conclude that\n\n\\begin{center}\n$\\left\\vert   \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}-\\ell_{c_o}(\\gamma)  +[\\gamma_-,\\gamma_+]_o\\right\\vert <\\delta$.\n\\end{center}\n\n\n\n\n\n\\noindent By (\\ref{eq aprox grom prod on dist thm nu}) it follows that\n \n\n\\begin{center}\n$ \\ell_{c_{o}}(\\gamma)-s-2 \\delta< \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}<\\ell_{c_{o}}(\\gamma) -s+2 \\delta$\n\\end{center}\n\n\n\\noindent holds apart from finitely many exceptions $\\gamma\\in\\Gamma_{\\textnormal{H}}$ such that $\\gamma_-\\in A$ and $\\gamma_+\\in B$. From now on, the proof of the convergence\n\n\n\n\n\\begin{center}\n$\\theta_t(A\\times B)\\longrightarrow \\mu_o(A)\\mu_o(B)$\n\\end{center}\n\n\\noindent follows line by line the proof of \\cite[Theorem 6.5]{Sam}. \n\n\n\nIt remains to prove the convergence in the diagonal $\\lbrace(x,x): \\hspace{0,3cm} x\\in\\partial_\\infty\\Gamma\\rbrace$, but once again, the proof is the same as the one given in \\cite[Theorem 6.5]{Sam}. For completeness we briefly sketch it.\n\n\n\nSince $\\mu_o$ has no atoms (see Lemma \\ref{lema PS non atomic}), for every $\\gamma$ in $\\Gamma$ the diagonal has $(\\mu_o\\otimes \\gamma_*\\mu_o)$-measure equal to zero. We fix two elements $\\gamma_0,\\gamma_1\\in\\Gamma_{\\textnormal{H}}$ with no common fixed point in $\\partial_\\infty\\Gamma$ and let $\\varepsilon_0>0$. There exists a finite open covering $\\mathscr{U}$ of $\\partial_\\infty\\Gamma$ such that for $i=0,1$ one has\n\n\\begin{center}\n$\\displaystyle\\sum_{U\\in\\mathscr{U}}  \\mu_o(U) \\mu_o(\\gamma_i^{-1}\\cdot U)<\\varepsilon_0$.\n\\end{center}\n\n\n\\noindent We can assume that for every $U\\in\\mathscr{U}$ there exists $i\\in\\lbrace 0,1\\rbrace$ such that $\\gamma_i^{-1}\\cdot\\overline{U}$ is disjoint from $\\overline{U}$. There exists an open covering $\\mathscr{V}$ of $\\partial_\\infty\\Gamma$ with the following properties:\n\n\\begin{enumerate}\n\n\\item $\\displaystyle\\sum_{V\\in\\mathscr{V}}  \\mu_o(V) \\mu_o(\\gamma_i^{-1} \\cdot V)<\\varepsilon_0$ for $i=0,1$.\n\\item The closure of every element in $\\mathscr{U}$ is contained in a unique element of $\\mathscr{V}$ and if $\\gamma_i^{-1}\\cdot\\overline{U}$ is disjoint from $\\overline{U}$ the same holds for this element in $\\mathscr{V}$.\n\\item Suppose that $\\gamma_i^{-1}\\cdot\\overline{U}\\cap \\overline{U}=\\emptyset$ and let $V\\in\\mathscr{V}$ be the unique element such that $\\overline{U}\\subset V$. Then apart from finitely many exceptions $\\gamma$ such that $\\gamma_{\\pm}\\in U$ one has $(\\gamma_i^{-1}\\gamma)_-\\in V$ and $(\\gamma_i^{-1}\\gamma)_+\\in \\gamma_i^{-1}\\cdot V$.\n\n\\end{enumerate}\n\nSet $D:=\\displaystyle\\max_{i=0,1}\\lbrace D_{\\gamma_i^{-1}}\\rbrace$ where $D_{\\gamma_i^{-1}}$ is the constant given by Proposition \\ref{prop traingle inequality} and take $U\\in\\mathscr{U}$ as in (3). By Proposition \\ref{prop traingle inequality} we have\n\n\n\n\n\n\\begin{equation*}\n\\begin{split}\n\\theta_t(U\\times U) & \\leq Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t+D} \\delta_{\\gamma_-}(V)\\delta_{\\gamma_+}(\\gamma_i^{-1}\\cdot V)\\\\ & +Me^{-ht}\\# F\n\\end{split}\n\\end{equation*}\n\n\n\n\n\\noindent where $F$ is a finite set independent of $t$. Since $V\\times\\gamma_i^{-1}\\cdot V$ is far from the diagonal the right side converges to\n\n\n\n\\begin{center}\n$e^D\\mu_o(V)\\mu_o(\\gamma_i^{-1}\\cdot V)$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow \\infty$. Adding up in $U\\in\\mathscr{U}$ we conclude\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\displaystyle\\sum_{U\\in\\mathscr{U}}\\theta_t(U\\times U)\\leq 2e^D\\varepsilon_0$.\n\\end{center}\n\n\n\\noindent Hence $\\theta_t(\\lbrace(x,x):\\hspace{0,3cm} x\\in\\partial_\\infty\\Gamma\\rbrace)$ converges to zero and since the diagonal has measure zero for $\\mu_o\\otimes\\mu_o$ the proof is finished.\n\n\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{teorema A}} \\label{subsec proof teo A}\n\n\n\nThe following is a corollary of Proposition \\ref{prop distribution on bg for length}.\n\n\n\n\\begin{cor}\\label{cor distr orbit o in gammah for bo}\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\n\n\\end{cor}\n\n\n\\begin{proof}\n\nSet\n\n\\begin{center}\n$\\nu_t^{\\textnormal{H}}:= M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}$\n\\end{center}\n\n\n\\noindent and take $\\theta_t$ the measure defined in the proof of Proposition \\ref{prop distribution on bg for length}. We know that \n\n\\begin{center}\n$(\\eta,\\xi)_*(\\theta_t)\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$.\n\\end{center}\n\n\n\\noindent Hence we only have to show the following convergence\n\\begin{equation}\\label{eq en cor orbit o para hiperbolicos}\n\\nu_t^{\\textnormal{H}}-(\\eta,\\xi)_*(\\theta_t)\\longrightarrow 0.\n\\end{equation}\n\n\n\nTake a small positive $\\delta$. By Proposition \\ref{prop limit with S and U and Uuno cerca gammamas} and the proof of Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set} we know that, apart from finitely many exceptions $\\gamma$ in $\\Gamma_{\\textnormal{H}}$, one has\n\n\\begin{center}\n$d(\\rho(\\gamma)\\cdot o,\\rho(\\gamma)_+)<\\delta$ and $d(\\rho(\\gamma^{-1})\\cdot o,\\rho(\\gamma^{-1})_+)<\\delta$.\n\\end{center}\n\n\nBy taking $\\cdot^{\\perp_{p,q}}$ we can assume further that $d^*(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)_-)<\\delta$. Now the proof of (\\ref{eq en cor orbit o para hiperbolicos}) follows from evaluation on continuous functions of $\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d)$.\n\n\n\\end{proof}\n\n\nWe now include torsion elements to the previous statement and finish the proof of Theorem \\ref{teorema A}.\n\n\n\\begin{prop}\\label{prop distribution on bg for length with torsion}\n\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma,  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\end{prop}\n\n\n\n\n\\begin{proof}\n\n\nThe structure of the proof is the same as that of Proposition \\ref{prop distribution on bg for length}, that is, we first prove the statement outside the diagonal and deduce from that the statement on the diagonal. Here by \\textit{diagonal} we mean the set\n\n\\begin{center}\n$\\Delta:=\\lbrace (\\theta,v)\\in \\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} \\theta(v)=0 \\rbrace$.\n\\end{center}\n\n\nLet \n\n\\begin{center}\n$\\nu_t:= M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma,  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}$\n\\end{center}\n\n\n\\noindent and take $\\nu_t^{\\textnormal{H}}$ as in the proof of Corollary \\ref{cor distr orbit o in gammah for bo}.\n\n\nConsider first a continuous function $f$ on $\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d)$ whose support $\\textnormal{supp}(f)$ is disjoint from $\\Delta$.\n\n\n\\begin{cla} \\label{claim f suppor far from diagonal implies g in gh}\n\nThe following holds\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} (\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in \\textnormal{supp}(f) \\textnormal{ and } \\gamma\\notin\\Gamma_{\\textnormal{H}}\\rbrace<\\infty$.\n\\end{center}\n\n\\end{cla} \n\n\n\n\n\\begin{proof}[Proof of Claim \\ref{claim f suppor far from diagonal implies g in gh}]\n\n\n\n\nFix a positive $D$ such that for every $(\\theta,v)\\in \\textnormal{supp}(f)$ one has $d(\\theta,v)> D$. As we saw in the proof of Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set}, the distances\n\n\\begin{center}\n$d(\\rho(\\gamma)\\cdot o,U_1(\\rho(\\gamma)))$ and $d^*(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},S_{d-1}(\\rho(\\gamma)))$\n\\end{center}\n\n\\noindent converge to zero as $\\gamma\\longrightarrow\\infty$. We conclude that, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ with $(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in \\textnormal{supp}(f)$, one has\n\n\\begin{center}\n$d(U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma)))>D$.\n\\end{center}\n\n\n\nNow apply (\\ref{eq def anosov}), Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and Benoist's Lemma \\ref{lema benoist lemma 1.2} to conclude that for $\\vert\\gamma\\vert_\\Gamma$ large enough the matrix $\\rho(\\gamma)$ is proximal.\n\n\\end{proof}\n\n\n\nFrom Claim \\ref{claim f suppor far from diagonal implies g in gh} we conclude that \n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\nu_t(f)=\\displaystyle\\lim_{t\\longrightarrow\\infty}\\nu_t^{\\textnormal{H}}(f)$\n\\end{center}\n\n\n\n\\noindent which by Corollary \\ref{cor distr orbit o in gammah for bo} equals $(\\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o))(f)$.\n\n\nIt remains to prove the convergence on the diagonal. It suffices to prove that for every positive $\\varepsilon_0$ there exists an open covering $\\lbrace U^*\\times U\\rbrace$ of $\\Delta$ such that\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\nu_t\\left(\\displaystyle\\bigcup (U^*\\times U)\\right)\\leq \\varepsilon_0$.\n\\end{center}\n\n\n\n\nThe proof is the same as in Proposition \\ref{prop distribution on bg for length}. Namely, take two elements $\\gamma_0,\\gamma_1$ in $\\Gamma_{\\textnormal{H}}$ with no common fixed point in $\\partial_\\infty\\Gamma$ and a coverings $\\mathscr{U}=\\lbrace U^*\\times U\\rbrace$ and $\\mathscr{V}=\\lbrace V^*\\times V\\rbrace$ of $\\Delta$ by open sets with the following properties:\n\n\n\n\\begin{enumerate}\n\\item For every $U^*\\times U$ in $\\mathscr{U}$ there exists $i=0,1$ such that $\\rho(\\gamma_i^{-1})\\cdot\\overline{U}$ is transverse to $\\overline{U^*}$.\n\\item $\\displaystyle\\sum_{V^*\\times V\\in\\mathscr{V}} (\\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o))(V^*\\times\\rho(\\gamma_i^{-1})\\cdot V)<\\varepsilon_0$ for $i=0,1$.\n\\item The closure of every element in $\\mathscr{U}$ is contained in a unique element of $\\mathscr{V}$ and if $\\rho(\\gamma_i^{-1})\\cdot \\overline{U}$ is transverse to $\\overline{U^*}$ the same holds for this element in $\\mathscr{V}$.\n\\item Suppose that $\\rho(\\gamma_i^{-1})\\cdot \\overline{U}$ is transverse to $\\overline{U^*}$ and let $V^*\\times V\\in\\mathscr{V}$ be the unique element such that $\\overline{U}\\subset V$ and $\\overline{U^*}\\subset V^*$. Then, apart from possibly finitely many exceptions $\\gamma$ such that $(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in U^*\\times U$, one has\n\n\\begin{center}\n$(\\rho((\\gamma_i^{-1}\\gamma)^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma_i^{-1}\\gamma)\\cdot o)\\in V^*\\times \\rho(\\gamma_i^{-1})\\cdot V$.\n\\end{center}\n\\end{enumerate}\n\n\n\n\nProvided with this construction, the proof finishes in the same way as that of Proposition \\ref{prop distribution on bg for length}.\n\n\n\n\\end{proof}\n\n\n\\begin{rem}\\label{rem crit exponent coincides with the entropy}\nFrom Proposition \\ref{prop distribution on bg for length with torsion} we deduce that\n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t\\rbrace}{t}$\n\\end{center}\n\n\\noindent coincides with the entropy $h=h_\\rho$ of $\\rho$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\\section{Distribution of the orbit of $o$ with respect to $ b^\\tau$}\\label{section distrib wrt btau}\n\\setcounter{equation}{0}\n\n\nThe proof of Theorem \\ref{teorema B} follows the same lines of the proof of Theorem \\ref{teorema A}, we just have to pick a (slightly) different flow $\\psi_t$.\n\n\nFix a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$, a point $o$ in $\\pmb{\\Omega}_\\rho$ and $\\tau\\in S^o$.\n\n\n\n\\subsection{The cocycle $c_\\tau$}\\label{subsec cocycle ctau}\n\n\n\nLet $\\Vert\\cdot\\Vert_\\tau$ be the norm introduced in Subsection \\ref{subsec cartan dec in sec anosov}.\n\n\n\\begin{dfn} \\label{dfn cocycle tauo}\n\nLet\n\n\\begin{center}\n$c_{\\tau}:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_{\\tau}(\\gamma,x):=\\dfrac{1}{2}\\log\\left(\\dfrac{\\Vert\\rho(\\gamma)\\cdot\\theta_{x}\\Vert_{\\tau}\\Vert\\rho(\\gamma)\\cdot v_{x}\\Vert_{\\tau}}{\\Vert\\theta_{x}\\Vert_{\\tau}\\Vert v_{x}\\Vert_{\\tau}}\\right)$\n\\end{center}\n\n\n\\noindent where $\\theta_{x}:\\mathbb{R}^d\\longrightarrow\\mathbb{R}$ is a non-zero linear functional whose kernel equals $\\eta(x)$ and $v_{x}\\neq 0$ belongs to $\\xi(x)$.\n\n\\end{dfn}\n\n\\begin{rem} \\label{rem ctau es el betauno}\nOne can prove that for every $\\gamma\\in\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has\n\n\n\\begin{center}\n$c_{\\tau}(\\gamma,x)=\\log\\dfrac{\\Vert\\rho(\\gamma)\\cdot v_{x}\\Vert_{\\tau}}{\\Vert v_{x}\\Vert_{\\tau}}$,\n\\end{center}\n\n\\noindent that is, $c_\\tau$ coincides with the map $\\beta_1(\\cdot,\\cdot)$ of \\cite[Section 5]{Sam}. This remark will not be used in the sequel.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\nThe following lemma holds by straightforward computations.\n\n\n\\begin{lema}\\label{lema cocycle tau and periods}\n\nThe function $c_{\\tau}$ is a H\u00f6lder cocycle. The period of $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ is given by\n\n\\begin{center}  \n$\\ell_{c_{\\tau}}(\\gamma)=\\lambda_1(\\rho(\\gamma))>0$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\nThe quotient space of $\\partial_\\infty^{2}\\Gamma\\times \\mathbb{R}$ by the action of $\\Gamma$ induced by $c_\\tau$ will be denoted by $\\textnormal{U}_\\tau\\Gamma$. It is equipped with a flow that lifts to the translation flow (\\ref{eq translation flow}) on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$.\n\n\n\n\n\n\n\\subsection{Dual cocycle and Gromov product}\\label{subsec dual and gromov ctau}\n\n\n\n\n\n\n\\begin{dfn}\n\nLet\n\\begin{center}\n$[\\cdot,\\cdot]_{\\tau}:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_{\\tau}:=\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\theta_{y}\\left(v_x\\right)\\theta_{x}\\left(v_y\\right)}{\\theta_{x}\\left(J^o\\cdot v_x\\right)\\Vert\\theta_{y}\\Vert_{\\tau}\\Vert v_y\\Vert_{\\tau}} \\right\\vert$.\n\\end{center}\n\\end{dfn}\n\n\n\\begin{rem} \\label{rem co dual to ctauo} \nRecall that $c_o$ is the cocycle defined in Section \\ref{section distrib wrt bo}. The cocycle $c_{\\tau}$ is dual to $c_o$, i.e. $\\ell_{c_o}(\\gamma)=\\ell_{c_{\\tau}}(\\gamma^{-1})$ for every $\\gamma\\in\\Gamma_{\\textnormal{H}}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\nThe proof of the following lemma is a direct computation.\n\n\\begin{lema}\nFor every $\\gamma\\in\\Gamma$ and every $(x,y)\\in\\partial_\\infty^{2}\\Gamma$ one has\n\n\n\\begin{center}\n$[\\gamma\\cdot x,\\gamma\\cdot y]_{\\tau}-[x,y]_{\\tau}=-(c_o(\\gamma,x)+c_{\\tau}(\\gamma,y))$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\\begin{lema}\\label{lema computing gromov tauo on gammapm}\nLet $\\gamma$ be an element of $\\Gamma_{\\textnormal{H}}$. Then\n\n\\begin{center}\n$[\\gamma_-,\\gamma_+]_{\\tau}=-\\frac{1}{2}\\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot\\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)+\\frac{1}{2}\\mathscr{G}_{\\tau}(\\rho(\\gamma^{-1})_-,J^o\\cdot \\rho(\\gamma)_+)$.\n\\end{center}\n\\end{lema}\n\n\n\\begin{proof}\n\nRecall the definition of $[\\cdot,\\cdot]_o$ from Subsection \\ref{subsec dual and gromov co}. One has \n\n\n\\begin{center}\n\n$[\\gamma_-,\\gamma_+]_\\tau=[\\gamma_-,\\gamma_+]_o+\\dfrac{1}{2}\\log\\dfrac{\\left\\vert \\theta_{\\gamma_+}(J^o\\cdot v_{\\gamma_+}) \\right\\vert}{\\Vert \\theta_{\\gamma_+} \\Vert_\\tau\\Vert  v_{\\gamma_+}\\Vert_\\tau} $.\n\n\\end{center}\n\\noindent The proof then follows from Lemma \\ref{lema computing gromov on gammapm o} and Remark \\ref{rem J preserva norma}.\n\n\\end{proof}\n\n\n\n\n\\subsection{Distribution of attractors and repellors with respect to $ b^\\tau$} \\label{subsec dist of fixed wrt btau}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet $\\mu_{\\tau}$ be a Patterson-Sullivan probability on $\\partial_\\infty\\Gamma$ associated to $c_{\\tau}$ and recall that $\\mu_o$ is the one associated to $c_o$. The analogue of Proposition \\ref{prop sambarino distribution wrt periods} is available for the flow on $\\textnormal{U}_\\tau\\Gamma$. The limit measure can be written in this case as\\footnote{For a proof, see Remark \\ref{rem BM for ctau and distribution}.}\n\n\\begin{center}\n$e^{-h[\\cdot,\\cdot]_\\tau}\\mu_o\\otimes\\mu_\\tau $.\n\\end{center}\n\n\n\nLet $\\mathfrak{b}^+$ be a closed Weyl chamber of a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$.\n\n\n\\begin{prop} \\label{prop distribution on bg nu}\nThere exists a constant $M'=M'_{\\rho,\\tau}>0$ such that\n\n\n\\begin{center}\n$M' e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow \\mu_o\\otimes\\mu_\\tau$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$.\n\n\\end{prop}\n\n\n\n\n\n\\begin{proof}\n\n\nThe proof is the same that the one given in Proposition \\ref{prop distribution on bg for length} adapted to the pair $\\lbrace c_o,c_\\tau \\rbrace$ and the Gromov product $[\\cdot,\\cdot]_\\tau$: apply item (5) of Lemma \\ref{lema jrhojrho prox dos} and Lemma \\ref{lema computing gromov tauo on gammapm}.\n\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{teorema B}} \\label{subsec proof teo B}\n\n\nThe following proposition, which implies Theorem \\ref{teorema B}, can be proved in the same way as Proposition \\ref{prop distribution on bg for length with torsion}.\n\n\n\\begin{prop}\\label{prop distribution on bg for btau with torsion}\nThere exists a constant $M'=M'_{\\rho,\\tau}>0$ such that\n\n\n\\begin{center}\n$M' e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_\\tau)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectromagnetic metamaterials have been used to realize many exotic scattering responses over the last two decades. Effects including negative refractive index and cloaking have generated significant interest and have served to drive the community \\cite{smith2000composite, pendry2000negative, shelby2001experimental, smith2004metamaterials, schurig2006metamaterial}. A more applied, but still relevant metamaterials achievement, is that of graded designs \\cite{greegor2005simulation, smith2005gradient}. It was shown early on that unit-cells which form metamaterials may be designed with a spatial dependence across a surface and \/ or volume, and various lensing effects were shown that utilize spatial degrees of freedom. A principle of gradient metasurfaces is that their scattering properties change slowly as a function of spatial coordinate. Broadband metamaterial absorbers \\cite{landy2008perfect, avitzour2009wide}, and metamaterial spatial light modulators \\cite{shrekenhamer2013four}, also make use of dissimilar neighboring unit-cells, however there is no such requirement to make neighbors as alike as possible, and designs are simply cobbled together to achieve the desired response. This fact highlights a design feature of sub-wavelength metal-based metamaterials, i.e. their scattering response is primarily connected to their geometry and -- due to minimal neighbor interaction -- the unit-cell largely governs the electromagnetic properties of the array. Conventional optimization techniques built-in to modern day electromagnetic mode solvers are sufficient to achieve a desired response, and designs of differing units may be assembled with little change to the overall response.\n\nIn principle, all-dielectric metasurface (ADM) unit-cells may also be used to tesselate a surface in an arbitrary fashion similar to that achieved with metal-based metamaterials. However, ADMs resonators are only marginally sub-wavelength, and modes utilized are often not confined within their physical bound -- with an evanescent tail lying just outside their surface -- resulting in significant neighbor interaction. Further, surface modes related to the periodicity are not restricted to nearest neighbors, and may persist over several spatial periods. Despite the success of ADMs \\cite{sautter2015active, headland2015terahertz}, it is conceivable that still-richer electromagnetic scattering can be achieved if more complex geometries are employed. However, physical understanding for such metasurfaces is poor -- simple functional relationships, or even heuristic guidance regarding super unit-cell geometry and final electromagnetic properties is unavailable. The only contemporary means to estimate such metasurface properties, given a candidate geometry, is simulation or fabrication.  However, given the vast space of potential designs and the speed of conventional simulation and fabrication, it is completely infeasible to iterate over designs in order to achieve a desired response. \n\nA viable design alternative to numerical simulation for structured materials, including metamaterials \\cite{liu2018generative, ma2018deep, jiang2019global, nadell2019deep, campbell2019review}, photonic band gap \\cite{peurifoy2018nanophotonic, liu2018training, kiarashinejad2020knowledge, huang2020deep}, and plasmonics studies \\cite{inampudi2018neural, jiang2019free, wiecha2019deep}, is deep learning.  Deep neural networks (DNNs) have successfully learned a forward mapping $s=f(g)$ between a metasurface geometry $g$ and the resulting frequency dependent electromagnetic scattering $s$, where $f$ is an unknown complex (e.g., highly nonlinear) function. \\cite{gareth2013introduction}  A DNN -- once trained -- can effectively act as a high-speed simulator that may be used to find the electromagnetic scattering of a candidate geometry substantially faster (e.g., by six orders of magnitude \\cite{nadell2019deep}) than conventional simulation.\n\n\n\\section{Neural-Adjoint Method}\nWhile deep learning enables substantially faster evaluation of candidates, given the vast number of possibilities for many problems (e.g., approx. $10^{12}$ in our case), it is nonetheless difficult or impossible to iterate over all or most candidate designs. From a design perspective, what would be of greatest utility would be to instead specify a desired frequency dependent electromagnetic response $s$, and have a model or solver compute a specific approximate geometry $\\hat{g}$, which yields $s$ (here $\\hat{}$ denotes approximate). The solution used to search for such an ideal geometry may be cast as an inverse problem written as $\\hat{g}=\\hat{f}^{-1}(s)$. Hadamard described a solution method as well-posed if it met three criteria -- $H_1$: existence (at least one solution), $H_2$: uniqueness (only one solution), and $H_3$: stability (solution must depend continuously on $s$). A solution approach is then ill-posed if one of the $H_1-H_3$ conditions fails \\cite{mueller2012linear}. Past work has shown that finding a specific ADM consisting of distinct neighboring resonators in a super unit-cell geometry for a desired scattering state is an ill-posed non-linear inverse problem and, in particular, conditions $H_1$ and $H_2$ are not met \\cite{liu2018generative}.\n\n\\begin{figure}[t!]\n    \\centering\\includegraphics[width=0.5\\textwidth]{images\/fig1.pdf}\n    \\caption{(a) 3D illustration of an all-dielectric metasurface absorber. A schematic of the all-dielectric metasurfaces super-cell is shown -- top right. (b) Training phase of the neural-adjoint method which uses fixed geometry inputs and corresponding simulated spectrum. (c) The inference phase of the neural-adjoint method, in which the target spectrum drives the search of the optimal geometry. The geometry is shown in a green box here to indicate it is the only trainable parameter.}\n\\label{1}\n\\end{figure}\n\nHere we explore a recently-proposed deep inverse modeling approach, called the neural-adjoint (NA) method \\cite{ren2020benchmarking}, which outperformed many other methods for solving ill-posed inverse problems.  Here we adapt the NA method to solve a challenging problem involving the design of ADMs geometries with 14 free geometrical parameters; a much larger space than most recent work (e.g., $5-10$ free parameters in  \\cite{peurifoy2018nanophotonic, ma2018deep, nadell2019deep}).  Furthermore, existing inverse methods often involve complex training procedures, and ultimately produce sub-optimal solutions. In contrast, the NA only requires that we train a single conventional feed-forward neural network, and -- as we show -- appears to find close approximations to the globally optimal solution within just one minute, even for our complex ADM design problem.  Furthermore, we demonstrate how the NA method can be utilized to expand the design search space, essentially providing a form of active learning that is specifically tailored to solve inverse problems. We explore an example inverse problem, where a frequency dependent infrared absorptivity ($A(\\omega)=s$) is desired, which we would like to achieve using an ADM consisting of a square array of 2$\\times$2 resonators, each with an elliptical generatrix lying perpendicular to its directrix (line of length $h$) -- depicted in Fig. \\ref{1}. Although past experience and single unit-cell simulations guide us to choose approximate metasurface dimensions which yield resonances in the desired spectral range, our inverse problem is ill-posed, since we do not know if a solution exists, i.e. condition $H_1$ is not met. Further, we do not meet the condition of uniqueness, $H_2$, since many metasurface solutions (disparate geometries) may yield the same spectra -- at least within the accuracy of our chosen loss metric, and numerical precision. The proposed ADMs consists of a geometry space of fourteen parameters: [$h$, $p$, $r_{x_1}$--$r_{x_4}$, $r_{y_1}$--$r_{y_4}$, $\\theta_1$--$\\theta_4$]. As shown in Fig. \\ref{1}, $h$ is the height of all four elliptical resonators, $p$ is the periodicity of the super cell, $r_{x_1}$ -- $r_{x_4}$ and $r_{y_1}$ -- $r_{y_4}$ are the x-axis and y-axis radii of four elliptical resonators respectively, and $\\theta_1$ -- $\\theta_4$ are the rotational angles with respect to the center axis of each elliptical resonator. All geometry values are randomly sampled from the data grid shown in Table S1, which is included in the Supplemental. The numerical simulations use SiC for the proposed ADMs, and the Supplemental contains more details.\n\n\\begin{figure}[t!]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig2.pdf}\n    \\caption{Forward model mean-squared-error (MSE) between DNN predicted spectra and numerical simulations. The DNN result with (a) the best, (b) average, and (c) below average performance. (d) Histogram detailing the distribution of MSE for the entire validation set.}\n    \\label{2}\n\\end{figure}\n\n\n\\section{Results}\nThe DNN forward model is trained by pair of 14 geometrical parameters and 2000 frequency points from 100-500 THz. Results of the forward model trained on 60k $\\{g,s\\}$ pairs (approximately $5.75\\times10^{-8}$ of the entire geometry space) are shown in Fig. \\ref{2} (a)-(c), where red curves are numerical simulations, and blue curves are the DNN predictions. The absolute error between simulation and prediction is plotted as the gray curve on the right vertical axis of Fig. \\ref{2} (a)-(c). In Fig. \\ref{2} (d) we show the histogram of the mean-squared-error (MSE) for the entire validation set, and find that 95\\% have an MSE$\\leq2.49\\times10^{-3}$ (dashed gray line), and an average MSE of $1.2\\times10^{-3}$. Having an accurate trained forward model we next turn toward the inverse solution. The NA method \\cite{ren2020benchmarking} finds the optimal inverse solution by fixing all the weights and biases of $\\hat{f}$, and computing the forward model's gradient solely with respect to the input to the network (i.e., the geometry), starting from randomly chosen values, denoted $\\hat{g}_0$. It is important to highlight that $\\hat{f}$ is a closed-form differentiable expression, and thus calculation of $\\partial \\hat{f}\/\\partial g$ is trivial. Further, we can estimate the gradient of the input geometry with respect to a loss function $\\mathcal{L}$ that we are free to specify (e.g., mean squared error). Therefore, if $s$ is our desired spectrum, and $\\hat{g}_i$ is our current best estimate of the metasurface geometry, we can iteratively move along the loss surface to find a better solution using, \n\n\\begin{equation}\n    \\left. \\hat{g}_{i+1} = \\hat{g}_{i}+\\alpha\\frac{\\partial \\mathcal{L}(\\hat{f}(\\hat{g}_{i}),s)}{\\partial g} \\right\\vert_{g = \\hat{g}_{i}}\\\\\n    \\label{eq1}\n\\end{equation}\n\n\\noindent where $\\alpha$ is the learning rate. The $\\hat{g}_i$ can then be evaluated iteratively until some convergence criteria is satisfied (e.g., $\\mathcal{L}$ changes very little after each iteration).  Because this is a gradient-based procedure, it will only converge to a locally optimal solution.  As a result, the NA method prescribes that this search process be repeated $T$ times, each time starting from a different randomly chosen value of $\\hat{g}_{0}$. In practice we find that we may run greater than $10^4$ $T$ initializations in parallel with no cost in speed -- only limited by available memory. Therefore the NA method produces $T$ candidate designs, and we can choose the best design (or several designs, if desired) by passing each design back into $\\hat{f}$ and evaluating their similarity to the target scattering properties.\n\n\\begin{figure}[t]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig3.pdf}\n    \\caption{Neural adjoint inverse results for prediction of target spectra that exist within the geometry space. Example spectra from geometry predictions that have (a) the best, (b) average, and (c)-(d) below average performance compared to the target spectrum.}\n\\label{3}\n\\end{figure}\n\nNotably, while the user can specify many different choices for $\\mathcal{L}$, the NA method prescribes that a so-called \"boundary loss\", $\\mathcal{L}_{bnd}$ should be added to any user-chosen loss \\cite{ren2020benchmarking} and is given by: $\\mathcal{L}_{bnd} = ReLU(|\\hat{g}-\\mu_g|-\\frac{1}{2}R_g)$, where rectified linear unit (ReLU) is the activation function, $\\mu_g$ is the mean of the geometry training data, and $R_g$ is its range.  This boundary loss punishes the inference process with increasing loss if the geometry search process steps out of the space of the training data, where the forward model may produce inaccurate estimates of scattering parameters.  In our experiments we use the following total loss: $\\mathcal{L}= (s-\\hat{f}(\\hat{g}))^2 + \\mathcal{L}_{bnd}$. As an initial test of the NA inverse method, we feed in frequency dependent absorptivities $A(\\omega)$ where we know apriori that a solution $g$ exists, i.e. $s$ is a numerical simulation from which $\\hat{f}$ originates from. In Fig. \\ref{3} we show results of the NA inverse method and each sub-plot shows characteristic results of (a) the best results, (b) average results, and (c) and (d) below average results, all based on MSE.  In all of the these examples, the NA method identifies a close approximation to the correct solution.  This is impressive given the complexity of the spectra present in Fig. \\ref{3}. We suspect the small remaining errors in the predicted design are due largely to the limited precision of gradient descent as it nears solutions (i.e., minima) in the error space; due to the non-zero learning rate in Eq. \\ref{eq1}, it cannot converge to the exact minimum point.  We note however that learning rate can be gradually reduced during the search process, at the cost of additional computation time, until a solution of desired precision is obtained.\n\nWe next turn to the significantly more challenging task of applying the NA method to a spectra where we are unaware if a solution exists within the chosen geometrical parameter space, i.e. criterion $H_1$ for inverse problems is violated. We chose the frequency dependent external quantum efficiency (EQE) of gallium antimonide (GaSb) as $s$, shown as the gray curves of Fig. \\ref{4}. The metasurface will operate at elevated temperature, and thus we consider the so-called graybody spectra --  also termed the spectral exitance $M_{e,\\nu}(T)$ -- which is given by the blackbody radiation curve times the absorptivity. We keep the top 16,000 neural adjoint solutions (spectra) and determine $M_{e,\\nu}(T)$ for each of these at 100 temperatures between 1500 and 2500 k --  a total of $1.6\\times 10^6$ candidates. The shape of the EQE curve differs significantly from typical spectra we see in our geometry space (Fig. \\ref{3}). None-the-less we find a best solution resulting from the NA method at a temperature of $T=2100k$ that consists of a geometry of [$h=0.566$, $p=1.440$, $r_{x_1}=0.180$, $r_{x_2}=0.155$, $r_{x_3}=0.214$, $r_{x_4}=0.278$, $r_{y_1}=0.285$, $r_{y_2}=0.253$, $r_{y_3}=0.146$, $r_{y_4}=0.256$, $\\theta_1=-0.901^{\\circ}$, $\\theta_2=-20.677^{\\circ}$, $\\theta_3=-37.982^{\\circ}$, and $\\theta_4=39.046^{\\circ}$]. The spectral exitance resulting from this geometry -- calculated from $\\hat{f}$ -- is shown as the blue curve of Fig. \\ref{4} (b), and we find an MSE, compared to the EQE of GaSb, of 1.06$\\times$10$^{-2}$. We also apply a weighting function $W(\\nu)=1\\chi_{[100,275]}+0\\chi_{(275,300]}$ on the MSE forcing the NA method to focus on the region of interest for energy harvesting purposes. To verify our neural adjoint results, we numerically simulate the predicted geometry and plot the resulting $M_{e,\\nu}(T)$ in Fig. \\ref{5} (a) as the red curve -- again compared to the EQE of GaSb (gray curve). As can be seen, the simulated curve has many relatively sharp peaks that are not present in Fig. \\ref{4}(b). This is because, as noted earlier, $\\hat{f}$ does not perfectly match the numerical simulator, and this will introduce errors in the design process. Thus since the NA method relies on $\\hat{f}$ to search for designs, it is also limited by the accuracy of the forward model estimate. We also found that since $\\hat{f}$ is trained from geometries constrained to a grid, the discrepancy between NA solutions and numerical simulation arises because NA solutions are not confined to the grid, where our model is most accurate. None-the-less we find that our simulated $M_{e,\\nu}(T)$ achieves an MSE of  1.65$\\times10^{-2}$, as shown in Fig. \\ref{5} (a).\n\n\\begin{figure}[t]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig4.pdf}\n    \\caption{Neural adjoint inverse results for matching the EQE of GaSb. The target spectrum (gray), DNN prediction for $M_{e,\\nu}(T=2100K)$ (blue), for the expanded geometry space (a), and  the original geometry space (b) explored. Neural adjoint predictions versus $p$ and $h$ for (c) the expanded geometry space, and (d) the original space. Symbol colors indicate MSE, with values given in the colorbar.}\n\\label{4}\n\\end{figure}\n\nAnother major obstacle is that our design search space does not contain a geometry that can realize our target spectrum (i.e., Hadamard's criteria $H_1$).  This is suggested by the fact that our best NA solution, shown in Fig. \\ref{4}(a), still does not match our target spectrum.  However, we can use the NA output to identify where we should expand our search space so that it will include better designs. We can do this by visualizing the error of all inverse solutions returned by NA, and looking for trends e.g., we may find that all the best solutions are bunched up against some edge of our initial search space, suggesting that expanding along that dimension may yield better results. However, since we have a 14 dimensional design space, we are unable to easily visualize these data. To address this problem, we use the Uniform Manifold Approximation and Projection (UMAP) \\cite{mcinnes2018umap}, which is a type of dimensionality reduction method permitting us to visualize the distribution of our inverse solutions performance in 2D, so that we may more easily identify patterns. From this initial investigation with UMAP in Fig. S1 -- shown in Supplemental -- we find that our best NA inverse solutions are limited by height. Shown in Fig. \\ref{4} (d) are NA solutions as a function of height and periodicity color mapped by corresponding MSE values. It is evident that not only are our best solutions grouped at the maximum height allowed in our geometrical space, $h=0.6$ $\\mu$m, but also that the solutions improve as a function of height. Encouraged by these results, we expanded our original geometry space to include increased height values from 0.6 to 0.75 $\\mu$m, by simulating an additional 24k $\\{g,s\\}$ pairs. The NA model now trained on the expanded geometrical space indeed finds an improved solution, shown as the blue curve in Fig. \\ref{4} (a), where we realize an MSE that is reduced by a factor of 2.7. The simulated red curve in Fig. \\ref{5} (b) further validates the result that the MSE of numerical simulations  is also reduced -- here by a factor of 4.8. A plot of the 1000 best NA solutions in the expanded geometrical space shown in Fig. \\ref{4} (c), however, indicate that we may be able to make continued improvements, since we still have a gradient pushing for greater heights -- although the periodicity seems to be honing in on a value of 1.2 $\\mu$m.\n\n\\begin{figure}[t]\n   \\centering\\includegraphics[width=\\textwidth]{images\/fig5.pdf}\n    \\caption{Numerically simulated $M_{e,\\nu}(T=2100K)$ (red) of the optimal geometry predicted by the neural-adjoint method to match the EQE of GaSb (gray), for the original geometry space (a), and the expanded geometry space (b).}\n\\label{5}\n\\end{figure}\n\n\\section{Conclusion}\nWe have adapted the neural-adjoint inverse design method \\cite{ren2020benchmarking} to accurately predict the high-dimensional all-dielectric metasurface geometry needed to produce a targeted infrared absorptivity spectra. When the geometry needed to produce a desired spectrum lies outside of the bounds, the NA method appears to find the best possible solution within the permissible search space. Unlike other adjoint inverse approaches \\cite{lalau2013adjoint}, the NA method does not require any domain knowledge specific to the problem. In the event that the inverse solution does not exist in the parameter space explored, NA may be used to guide one to a better solution, through expansion of the design parameter search space. This may help to reduce the initial required number of numerical simulations, and to instead use NA guided simulation exploration. With its exceptional computational speed, high accuracy, and potential use in active learning that is explored here, the neural-adjoint method has an impressive future in not only ADMs thermal emitter but also any ADMs inverse problems. The NA method is not restricted to the case presented, but may be applied to many other systems including photonic band gap and plasmonics.\n\n\n\n\\section{Numerical Simulation}\nThe cylindrical resonators' geometry was previously demonstrated in the THz regime by\\cite{liu2017experimental, fan2017all}. To prove deep neural work capability with high-dimensional inputs, we increased the geometrical dimensionality by introducing the elliptical structures to previous cylindrical resonators, and each elliptical resonator undergoes a rotation angle ranging from -45 to 45 degrees. Furthermore, governing the fabrication practicality, we fixed all elliptical resonators to have the same height. To migrate from the THz regime to the infrared, materials and geometry sizes are scaled accordingly. We sized down our unit cell volume refers to the ratio of THz frequency and infrared frequency used in the legacy design and current design. Then an optimization on an adequate scale to finalize our geometry boundary listed in Table 1. We chose SiC for our simulations, considering its high melting point, high oxidation resistance, and reasonable absorption coefficient in near-infrared. We used experimentally measured relative permittivity data of SiC from 0 to 300 THz to fit the dispersive materials property of SiC in our simulations. To implement the unit cell boundary conditions in CST Microwave Studio we used for our simulation, we used finite frequency solvers to perform the numerical simulations, which also take considerations of the coupling effect between the four resonators within the unit cell. To minimize the time cost of the simulations, we lightly comprise the simulation accuracy to simulation speed. We tuned the Floquent mode to have one mode at both ports. The simulation mesh is tetrahedral, and we used a second-order solver with accuracy at $1e^-6$. The resulting spectrum of the simulation has 2001 data points within the range of 100-500 THz.\n\n\\begin{table}[htbp!]\n\\caption{Grid definition for the 14-dimensional input geometry parameters. h, p, and r are in units of microns. $\\theta$ is in unit of degrees.}\n\\centering\\begin{tabular}{ccccc}\n \\\\\n Step&h&p&$r_{x_n}$\/$r_{y_n}$&$\\theta_n$\\tnote{*}\\\\ \\hline\n 1&0.3&1&0.1&-45 \\\\\n 2&0.375&1.125&0.1125&-22.5 \\\\\n 3&0.45&1.25&0.125&0 \\\\\n 4&0.525&1.375&0.1375&22.5 \\\\\n 5&0.6&1.5&0.15&45 \\\\\n 6&-&-&0.1625&- \\\\\n 7&-&-&0.175&- \\\\\n 8&-&-&0.1875&- \\\\\n 9&-&-&0.25&- \\\\\n \\hline\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\centering\\item[*] $n$ corresponds to the first to the fourth elliptical resonator in one super cell\n\\end{tablenotes}\n\\end{table}\n\nThe total possible number of geometrical combinations of our 2$\\times$2 metasurface is $8^9*6^5=1.04\\times 10^{12}$. It is impossible to use a conventional numerical simulation approach to exploit the entire geometry space to achieve the targeted spectrum. We find that the average simulation time per geometrical configuration per CPU is approximately three minutes. Thus it would take about 600 million years to finish exploring the entire geometry space with one CPU. The fast forward dictionary search (FFDS) inverse method was shown feasible for THz ADMs absorbers, where all 812 million possible geometries can be computed in a day \\cite{nadell2019deep}. To compute our entire geometry space with a size of over a trillion parameters would take FFDS over three and a half years. Thus the NA method is a good choice when the parameter space become too large for a FFDS approach. \n\n\\section{Deep Neural Network Architecture}\nWe built the entire network and neural-adjoint method using the PyTorch platform. The DNN used for the neural adjoint method consists of twelve fully connected linear layers, four 1D transpose convolutional layers for upsampling, and one final 1D convolutional layer for spectrum smoothing. The linear fully connected layers have the following structure[14, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 500\n], and all hidden layers except the last linear layer are batch normalized, activated by Leaky\\_Relu, followed by dropout layers with p = 0.05. The transpose convolutional layers have kernel size [16, 16, 33, 33] and filter size [4, 4, 4, 4]. The final convolutional layer has kernel size 1 and stride at 1. The data loader takes geometry inputs and the first 2000 data points of the absorptivity spectrum to generate the training and test datasets.  The post-processing truncates the predicted spectra's first and last fifty points to drop the convolutional layers bump at the edges.  We use L2 regularization, batch normalization, and the ADAM optimizer.\n\n\\section{NA Inverse Method}\nBecause NA a gradient-based procedure, it will only converge to a locally optimal solution. Because the NA method is amenable to parallelization on graphics processing units (standard hardware for deep learning), this entire process can be computed in very little time.  In our experiments, we can run the NA method with $T=1000$ on our desktop computer with an Nvidia 2080ti GPU and complete processing in under 1 minute.Since the NA method finds the globally optimal solution even in its worst-performing cases, our results suggest that the NA always (or nearly always) finds the globally optimal solutions, even for highly complex problems like ours.  Interestingly, this suggests that the main obstacle of custom design is no longer the inverse model, but rather the space over which we choose to search for designs i.e., the shapes we consider (cylinder, crosses, etc) and their parameter settings (e.g., radii, height).  Although powerful, to use deep learning methods we must necessarily define a range of these settings so that we can collect simulations to train our models, and this space limits where we can search for designs.  However, the design needed to realize our targeted scattering parameters may not exist in this initial search space.   As we show subsequently, the NA method can also be used to identify where this initial search space can be expanded so that it is most likely to contain the desired solution, providing a solution to this emergent obstacle to complex material design.    \n\n\\section{Data Augmentation}\nOur simulations' unit cell boundary conditions allow us to do four times data augmentation on our dataset because the infinite plane of unit cells consists of four different resonators' combinations that give almost identical spectra with fluctuations from CST software. However, we learned that the DNN could quickly learn the correlation between four different resonators' combinations. The forward model will know which input geometries share the same spectrum in high fidelity if the entire dataset is augmented before splitting into the training and validation dataset. Therefore, the forward model will give a false mean square error much lower than actual loss performance on an independent validation set. After applying data augmentation to the training and validation sets after the splits, we observe that the four times augmentation did not significantly improve accuracy. We believe that, with our 60000 simulations (24000 simulations after augmentation), the augmented data points are still too sparse to cover the entire geometry space defined by our geometry boundaries. \n\n\\section{Geometry Space Exploration Through UMAP}\nWe use Uniform Manifold Approximation and Projection (UMAP) to explore our solution geometry space and realize that angles have more random impacts on distributing the best NA solutions.  Thus, we plotted the UMAP with ten parameters, excluding rotational angles. The plotted UMAP demonstrates a clear trend that the MSE is decreasing in one direction. To confirm that the decreasing trend matches the increasing of resonators' height, we further marked the points from maximum and minimum height boundaries, respectively. The clustering of points towards the best MSE performance suggests that the NA method is finding the best local minima. \n\n\\begin{figure}[h!]\n    \\centering\\includegraphics[width=3.0in]{images\/figs1_umap.pdf}\n    \\caption{Uniform Manifold Approximation and Projection plotted with 10-dimensional geometry inputs indicates a strong correlation between the MSE performance and the increasing height.}\n\\label{5}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{I. Introduction}\n\nCorrelation functions in disordered systems usually decay nonexponentially in time \\cite{angell}. The microscopic origin of this behavior is of importance for a detailed understanding of the dynamics in these systems. In principle, the stretched exponential form of the correlation functions can be explained by two extreme scenarios. Either different Debye-type relaxation rates superimpose to the total response, or the broadened form is an intrinsic property of the dynamics in glassy systems. These different scenarios have been denoted as dynamic heterogeneous and homogeneous \\cite{hethom}. Several, but not numerous, experimental techniques have been developed that allow the investigation of dynamic heterogeneity, like four-time NMR measurements \\cite{4timenmr}, a deep bleach experiment \\cite{deepbleach} and nonresonant hole burning (NHB) \\cite{NHB}. \n\nIn a theoretical investigation of NHB \\cite{gregor} it has been shown that indeed heterogeneous and homogeneous dynamics can be distinguished. Experimental realizations of NHB have been conducted on numerous systems like supercooled liquids \\cite{NHB,duvvuri,blochi}, relaxor materials \\cite{oli} and spin glasses \\cite{chamberlin}. In all these studies indications for dynamic heterogeneity surpass those for homogeneity, but also homogeneous behavior was found, e.g., in an amorphous ion conductor \\cite{richert}. Recently a mechanical variant of NHB has been developed \\cite{mckenna}.\n\n\nMainly accountable for the distinction between heterogeneous and homogeneous dynamics is the possibility to adress certain dynamics in the sample, i.e., the selection of specific dynamic subensembles. Frequency-selective behavior is obtained only if the measured response function is nonlinear in the applied field, which holds for the dynamic Kerr effect \\cite{kerreffect} as well as for NHB. In Kerr effect relaxation, frequency-selective behavior can be achieved if a driving AC field with varying frequency is applied.  We demonstrated frequency-selectivity in a Kerr effect experiment in the terahertz range in a recent publication \\cite{mykerr}. Treating the vibrational dynamics around the boson peak in a Brownian oscillator model we concluded that the frequency dependence of the vibrations' damping can be extracted experimentally. \n\nThe Kerr effect has to our knowledge not yet been used for an investigation of dynamic heterogeneities. In this article, we study the Kerr effect response in the range of slow reorientational motions in supercooled liquids. We propose the Kerr effect for a distinction between dynamic heterogeneous and homogeneous relaxation. The advantage of the Kerr effect compared to NHB lies in the fact that the Kerr effect response is purely nonlinear in the applied field. Therefore no separation of nonlinear contributions from the linear response background is required, and field strengths remarkably weaker than those usually applied in NHB ($100\\ kV\/cm$ \\cite{NHB,duvvuri,blochi}) may suffice. It should also be possible to study dynamics at different temperatures ranging from the glass transition temperature to high temperatures. Here, possible changes in a heterogeneous distribution with temperature \\cite{breitetemp,gammaverteilung} might be investigated.  \n\nThe paper is organized as follows. Section II outlines the principles of the Kerr effect, explains our suggested experiment and defines heterogeneous and homogeneous models. In Section III we give expressions for the Kerr effect response in the different approaches and we discuss the results. It turns out that indeed the distinction between dynamic homogeneity and heterogeneity is possible. The conclusions are given in Section IV.\n \n\n\\section*{II. Theory}\n\\subsection*{1. Dynamic Kerr effect}\nIf an anisotropically polarizable sample is exposed to a time-dependent electric field it becomes birefringent. This phenomenon is known as dynamic Kerr effect. In a theoretical description the coupling of external electric fields $E(t)$ to matter via the sample's permanent dipole moment $\\mu$ has to be taken into account as well as the coupling via the polarizability $\\alpha$. The structure of the Hamiltonian describing the interaction is thus of the form \\cite{hamop}\n\\be \\label{exthamilton}\n\\mathcal{H}_{ext}=-\\mu E(t)P_1(\\cos \\theta)-\\alpha E^2(t) P_2(\\cos \\theta)\n\\ee\nThe permanent dipole moment interaction is linear in $E$, while the polarizability interaction is of order $E^2$ since the induced dipole moment itself is $\\propto E$. The appearance of the Legendre polynomials $P_L(\\cos \\theta)$ is due to the tensorial nature of the dipole moments. Since the permanent dipole moment is a vector and the polarizability is a matrix, they transform like first and second rank Legendre polynomials, respectively. The scalar $\\alpha$ in Eq.(\\ref{exthamilton}) is to be identified with the difference between the polarizabilities parallel and perpendicular to the internal symmetry axis \\cite{deschardaeng}, and $\\theta$ is the angle between this axis and the applied field. \n\nThe time-dependent polarizability is the quantity of interest for a description of the dynamic Kerr effect. We therefore have to calculate the expectation value of the second Legendre polynomial\n\\be\\label{alpha}\n\\langle \\alpha P_2(\\cos \\theta (t) )\\rangle  =\\alpha \\langle P_2(t) \\rangle .\n\\ee\nThe brackets denote an expectation value over the whole sample. In this work we focus on reorientational motions. Here, the time dependence of the orientation (described by the angle $\\theta$) determines the time dependence of the polarizability. In the following, we will use the short hand notation on the right hand side of Eq.(\\ref{alpha}).\n\nThe expectation value $\\langle P_2(t) \\rangle$ is calculated in some dynamic model, where the applied external field that determines the time dependence is treated in perturbation theory. Even without specifying the model yet, it is clear that the linear response of order $E$ must vanish as long as isotropic systems are considered. This is because the linear response is always proportional to the first rank Legendre polynomial, see Eq.(\\ref{exthamilton}). Because of the orthogonality of the Legendre polynomials the linear response vanishes if the expectation value of the second rank Legendre polynomial, Eq.(\\ref{alpha}), is calculated. The Kerr effect response is thus of order $E^2$.\n \nIn some more detail we have two contributions to the expectation value $\\langle P_2(t) \\rangle$. The first one is proportional to the polarizability $\\alpha$ (formally calculated with the second term in the external Hamiltonian (\\ref{exthamilton}) in first order perturbation theory), the second one is quadratic in the permanent dipole moment $\\mu$ (second order perturbation theory with the first term in (\\ref{exthamilton})). We will denote these contributions in the following as $\\langle P_2^\\alpha(t) \\rangle$ and $\\langle P_2^{\\mu \\mu}(t) \\rangle$. \n\n\\subsection*{2. Experiment}\n\nWe propose an experiment as follows. First, a sinusoidal electric field is applied for an arbitrary number $N$ of (full) cycles to a sample in equilibrium.\n\\be\\label{feld}\nE(\\tau )=E \\sin(\\Omega_p \\tau) \\qquad \\tau <t_p=\\frac{2\\pi N}{\\Omega_p}\n\\ee\nThe most important experimental parameter is the pump frequency $\\Omega_p$. After the 'pump time' $t_p$ the field is switched off and the relaxation of the sample's polarizability back to thermal equilibrium is measured. In our terminology the time variable $t$ starts with $0$ at $t_p$. \n\nThe experiment is based on the same idea as NHB. If the dynamics is assumed to be heterogeneous (different relaxation rates), then it should be possible to address 'slow' or 'fast' dynamics separately by varying the pump frequency from 'low' to 'high'. The so-selected ensembles will then show 'slow' or 'fast' relaxation behavior afterwards. In a Fourier transformed representation of the relaxation, one would therefore expect an extremum position proportional to the applied field frequency. On the other hand, in a homogeneous system one would expect relaxation on a single time scale only, and different pump frequencies should not alter the time-dependence or extremum position in the Fourier transform. Thus, shifts in the extremum position can be considered as indications for dynamic heterogeneous relaxation while no shifts would support the homogeneous picture.\n\nThe aim of this paper is to demonstrate that the concept outlined above indeed holds in heterogeneous and homogeneous models. Because of the nonlinearity of the Kerr effect the contributions $\\langle P_2^\\alpha(t) \\rangle$ and $\\langle P_2^{\\mu \\mu}(t) \\rangle$ are not trivially related to the correlation function, and have thus to be explicitly calculated for a given field sequence like Eq.(\\ref{feld}). We specify the dynamic models used in the calculation of the response functions in the following section. \n\n\n\\subsection*{3. Dynamics in the rotational diffusion model}\n\nFor a description of the dynamics we use the well-known rotational diffusion model \\cite{kubo}. The rotational diffusion equation for the conditional probability $P_i(\\Omega,t|\\Omega ',t')$ to find the orientation $\\Omega$ at time $t$, assuming it was $\\Omega '$ at time $t'<t$ is\n\\be\\label{simplefopl}\n\\frac{\\partial}{\\partial t} P_i(\\Omega,t|\\Omega ',t')=\\hat \\Pi_i(\\Omega ,t) P_i(\\Omega,t|\\Omega ',t')\n\\ee\nwith the Fokker-Planck operator\n\\be\\label{rotdiff}\n\\hat \\Pi_i(\\Omega ,t) =D_i(t)\\left[ \\frac{1}{\\sin\\theta} \\partial_\\theta \\sin\\theta \\left( \\partial_\\theta +\\beta \\frac{\\partial {\\mathcal H}_{ext}}{\\partial \\theta}\\right) + \\frac{1}{\\sin^2\\theta} \\partial^2_\\phi \\right] .\n\\ee\nHere, $\\theta$ and $\\phi$ denote polar and azimuth angles with respect to the molecules internal symmetry axis. These angles constitute the orientation $\\Omega$. $D_i(t)$ is a (possibly time-dependent) rotational diffusion constant, $\\beta=(k_B T)^{-1}$ and ${\\mathcal H}_{ext}$ is the Hamiltonian for the interaction with the applied external field in Eq.(\\ref{exthamilton}). \n\nThe solution of Eq.(\\ref{rotdiff}) without driving field (${\\mathcal H}_{ext}=0$) is achieved via an expansion in Wigner rotation matrices $D^{L}_{MN}$ \\cite{wigner}. The ansatz\n\\begin{equation}\\label{ansatz}\nP_i(\\Omega,t|\\Omega ',t')=\\sum_{LMN} \\frac{2L+1}{8\\pi^2} D^{L}_{MN}(\\Omega) \\left[ D^{L}_{MN}(\\Omega ')\\right]^\\ast G^{(L)}_i(t,t')\n\\end{equation}\ninserted in Eq.(\\ref{simplefopl}) yields\n\\begin{equation}\\label{glvont}\nG^{(L)}_i(t,t')=\\exp\\left[ -L(L+1)\\int_{t'}^t d\\tau \\ D_i(\\tau)\\right]\n\\end{equation}\nand rotational correlation functions of the $L$-th Legendre polynomial are given by\n\\be\\label{corr}\n C^L_i(t,t') = \\langle  P_L(\\cos\\theta(t)) P_L(\\cos\\theta(t')) \\rangle_i =\\frac{1}{2L+1} G^{(L)}_i(t,t'). \n\\ee\nIn the presence of external fields perturbation theory is used to calculate the response functions. The corresponding perturbation operators result from the Fokker-Planck operator (\\ref{rotdiff}) including the external Hamiltonian in Eq.(\\ref{exthamilton}). Linear response functions obtained this way are related to the correlation function via the  fluctuation dissipation theorem \\cite{calabrese}.\n\n\n\n\n\n\n\n\\subsection*{Heterogeneous model}\nIn a heterogeneous scenario one assumes the coexistence of different, 'slow' and 'fast' environments related to small and large rotational diffusion constants $D_i$. If $D_i$ is time-independent, the propagator $P_i(\\Omega,t|\\Omega ',t')$ is time-translational invariant and the correlation function in case of a single rotational diffusion constant is given by\n\\be\\label{}\nC_i^L(t,t')=C_i^L(t-t')=\\frac{1}{2L+1} \\exp\\left[-L(L+1) D_i(t-t')\\right].\n\\ee\nDynamic heterogeneities are known to have finite life times of similar magnitude as the $\\alpha$-relaxation time \\cite{otptauschtaus,austauschsim}. Finite life times can be modelled via exchange models. Various approaches exist to include exchange in a model \\cite{xeigenwerte}. In a master equation ansatz \\cite{sillescu} one assumes that slow environments become fast and vice versa. The environments are related to rotational diffusion constants, and exchange between the environments leads to effectively time-dependent rotational diffusion constants. Time-translational invariance is preserved within this framework. We will denote the environments by $\\epsilon_i$ in the following, and we add a second index to quantities like the propagator. $P_{ij}(\\Omega, t | \\Omega ',t')=P_{ij}(\\Omega, t-t' |\\Omega ',0)$ thus is the probability to find the orientation $\\Omega$ in environment $\\epsilon_i$ at time $t$, assuming it was $\\Omega '$ in environment $\\epsilon_j$ at time $t'$. Transition rates $\\gamma_{ij}$ determine transitions from environment $\\epsilon_j$ to $\\epsilon_i$, meaning that the rotational diffusion constant changes its value from $D_j=D(\\epsilon_j)$ to $D_i=D(\\epsilon_i)$. Furthermore, the behavior of the orientation during a transition has to be specified. If we assume that each transition is associated with a rotational random jump, then an exchange model can be defined by \\cite{sillescu} \n\\be\\label{beckpfeif}\n\\frac{\\partial}{\\partial t} P_{ki}(\\Omega,t|\\Omega ',t')=\\left( \\hat \\Pi_k(\\Omega)-\\gamma_k \\right)  P_{ki}(\\Omega,t|\\Omega ',t') +  \\frac{1}{8\\pi^2} \\sum_{j \\not= k} \\gamma_{kj} \\int d\\Omega_0 P_{ji}(\\Omega_0,t|\\Omega ',t').\n\\ee\nHere, $\\hat \\Pi_k(\\Omega)$ is the Fokker-Planck operator (\\ref{rotdiff}) with $D_i(t)$ replaced by the time-independent $D_k$ and $\\gamma_k=\\sum_{j\\not= k}\\gamma_{jk}$ is the total rate out of state $\\epsilon_k$. \n\nThe solution of Eq.(\\ref{beckpfeif}) without external fields is obtained in a similar manner as the solution of Eq.(\\ref{simplefopl}). Here, the function $G^{(L)}_i(t,t')$ in Eq.(\\ref{ansatz}) is replaced $G^{(L)}_{ki}(t,t')$. For $L\\not= 0$ the elements of the matrix ${\\bf G}^{(L)}(t,t')$ are given by\n\\be\\label{c0gdiag}\nG^{(L\\not= 0)}_{ki}(t,t')=\\delta_{ki} \\exp\\left[ \\Lambda^{(L)}_i (t-t')\\right]\n\\ee\nwith the eigenvalues\n\\be\\label{xeigen}\n\\Lambda^{(L)}_i=-L(L+1)D_i-\\gamma_i.\n\\ee\nThe equilibrium probabilities $P_i^{eq}$ to find the system in environment $\\epsilon_i$ in the presence of exchange (finite $\\gamma_{ij}$) are obtained from the long-time limit of the propagator, $P_i^{eq}=\\lim_{(t-t') \\to \\infty} \\int d\\Omega  P_{ik}(\\Omega,t|\\Omega ',t')=G^{(0)}_{ik}(\\infty ,0)$, which guarantees detailed balance. Correlation functions in the heterogeneous model are then calculated as\n\\be\\label{hetcorr}\nC^L(t)=\\sum_i C^L_i(t) P^{eq}_i=\\frac{1}{2L+1}\\sum_i \\exp\\left[\\Lambda_i^{(L)}t\\right] P^{eq}_i .\n\\ee\nThe larger the exchange rates are, the more they dominate the eigenvalues in comparison to the $L$-dependent term in Eq.(\\ref{xeigen}). This also holds in other exchange models \\cite{xeigenwerte}. Large exchange rates thus lead to a weaker $L$-dependence of the eigenvalues, and the exponents in correlation functions (\\ref{hetcorr}) for different values of $L$ become more and more similar. The inclusion of exchange in the rotational diffusion model thus leads to very similar correlation times for different $L$, as usually found experimentally \\cite{willi}. The calculation of response functions is straightforward.\n\nExchange effects are neglected if $\\gamma_{ij}=0$ is chosen. Then the eigenvalues in Eq.(\\ref{hetcorr}) are given by $\\Lambda_i^{(L)}=-L(L+1)D_i$, and a distribution of rotational diffusion constants for a heterogeneous model can be arbitrarily chosen. If the relaxation times $\\tau_i \\propto D_i^{-1}$ are distributed according to the generalized Gamma distribution \\cite{gammaverteilung}, a stretched exponential form of the correlation function is obtained.\n\n\n \n\n\n\\subsection*{Homogeneous model}\nIn a homogeneous scenario one assumes that the relaxation rate (or rotational diffusion constant) depends on time. A decaying power law function\n\\be\\label{homd}\nD_i(t)=\\frac{\\beta_K}{2\\tau}  \\left( t\/\\tau \\right)^{\\beta_K-1} \n\\ee\ninserted into the correlation function (\\ref{corr}) using Eq.(\\ref{glvont}) yields \n\\be\\label{homcorr}\n\\langle P_1(t)P_1(t')\\rangle = \\frac{1}{3} \\exp\\left[ -(t'\/\\tau)^{\\beta_K}  \\right] \\exp\\left[ -(t\/\\tau)^{\\beta_K}  \\right] .\n\\ee\nThus, the special choice in Eq.(\\ref{homd}) reproduces a stretched exponential for the correlation function. Note that Eq.(\\ref{homcorr}) is not time-translational invariant, which is due to the time dependence of $D_i(t)$ \\cite{calabrese}. Due to the lack of time-translational invariance the system has an 'internal clock' that is hard to justify from a physical point of view. The consequences in our case will be discussed below. An alternative for modelling homogeneous dynamics are the so-called fractional Fokker-Planck approaches \\cite{fractional}. Here, time-translational invariance is maintained \\cite{fracttti}. \n\n\n\n\\section*{III. Results}\nIn this section we discuss the Kerr effect response to the alternating field in the various approaches considered in the previous section. After the orientational integrations have been carried out we find the expressions \n\\be\\label{alal}\n\\langle P_2^{\\alpha}(t) \\rangle=\\frac{6}{5} \\beta \\alpha E^2 \\sum_{i} \\int_0^{t_p} dt' \\sin^2(\\Omega_p t') D_i(t') G_{i}^{(2)}(t+t_p,t') P_i^{eq}\n\\ee\nfor the $\\alpha \\alpha$- contribution to the Kerr effect response and\n\\Be \\nonumber  \n\\langle P_2^{\\mu \\mu}(t) \\rangle=\\frac{4}{5}\\beta^2 \\mu^2 E^2 \\times \\\\ \\label{almumu}\n\\times \\sum_{i} \\int_0^{t_p} dt' \\int_0^{t'} dt'' \\sin(\\Omega_p t') \\sin(\\Omega_p t'') D_i(t') D_i(t'') G_{i}^{(2)}(t+t_p,t') G_{i}^{(1)}(t',t'') P_i^{eq} \n\\Ee\nfor the $\\alpha \\mu \\mu$-contribution. Eqns.(\\ref{alal},\\ref{almumu}) are valid for heterogeneous and homogeneous models. In the heterogeneous case the $D_i$ are time-independent and in the homogeneous case no summation over $i$ is required. In the presence of exchange $G_i^{(L)}(t_1,t_2)$ is to be identified with the diagonal elements of the matrix ${\\mathbf G}^{(L)}(t_1,t_2)$, see Eq.(\\ref{c0gdiag}).\n\nNote that Eq.(\\ref{alal}) is formally identical to the expression for the linear (dielectric) response, except for the fact that the external field of sine form enters quadratically and $G^{(2)}_i$ appears instead of $G^{(1)}_i$. As a consequence of this quasi-linearity the $\\alpha \\alpha$-contribution is -like the dielectric linear response- unable to distinguish between homogeneous and heterogeneous scenarios.\n\n\nLet us briefly discuss the simplest case where the system is described by a single, time-independent rotational diffusion constant. Here, $G^{(L)}_i(t_1,t_2)$ are exponential functions according to Eq.(\\ref{glvont}). We furthermore have to set $D_i=const$ and $\\sum_i P^{eq}_i=1$ in Eqns.(\\ref{alal},\\ref{almumu}). The time integrals can then be carried out analytically, leading to response functions of the form\n\\be\\label{respeinerallg}\n\\langle P_2^{\\Theta}(t) \\rangle=A^{\\Theta}(\\Omega_p, D_i,t_p)\\exp(-6 D_i t)\n\\ee\nwhere $\\Theta$ stands for $\\alpha$ or for $\\mu\\mu$. In the following discussion of the response functions we will show the imaginary part of Fourier-transformed response functions, denoted as $\\langle P_2^\\Theta(\\omega)\\rangle$. The symbol $\\omega_{max}$ will be used for the extremum positions in $\\langle P_2^\\Theta(\\omega)\\rangle$.\n\nBoth contributions (\\ref{respeinerallg}) with $\\Theta=\\alpha$ and $\\Theta=\\mu\\mu$ show the same time dependence, a monoexponential decay with the decay rate of the second Legendre polynomial. This is analogous to the step field-off-response discussed in Ref.\\cite{cole}. The maximum position in $\\langle P_2^\\Theta(\\omega)\\rangle$ therefore is located at frequency $\\omega_{max}=6D_i$ for both terms.\n\n\nThe amplitude functions $A^\\Theta(\\Omega_p,D_i,t_p)$ are given by \\cite{meinsenf}\n\\Be\\label{respeineralal}\nA^{\\alpha}(\\Omega_p,D_i,t_p) & = & \\frac{1}{30} \\beta \\alpha E^2   \\frac{\\Omega_p}{D_i} \\varepsilon ''(6D_i,2\\Omega_p) \\left[1-e^{-6 D_i t_p} \\right]   \\\\ \\nonumber\nA^{\\mu \\mu}(\\Omega_p,D_i,t_p) & = & \\frac{1}{30} \\beta^2 \\mu^2 E^2  \\varepsilon ''(2 D_i,\\Omega_p) \\times  \\\\\n\\label{respeineralmumu} & \\hspace{-0.6cm} \\times & \\hspace{-0.6cm} \\left( 3\\varepsilon ''(4D_i,\\Omega_p) \\left[e^{-6 D_i t_p}-e^{-2 D_i t_p} \\right] +\\frac{5}{3} \\varepsilon ''(6D_i,2\\Omega_p) \\left[ 1-e^{-6 D_i t_p} \\right] \\right)  \n\\Ee\nwith the dielectric loss defined as $\\varepsilon ''(x,y)=xy\/(x^2+y^2)$. In the context of frequency-selectivity the behavior of the amplitudes as a function of pump frequency is important. $A^{\\mu \\mu}(\\Omega_p,D_i,t_p)$ plotted versus pump frequency shows a peak because the amplitudes become large if the arguments in the dielectric loss functions $\\varepsilon ''(x,y)$ are equal, that means for $\\Omega_p\\approx 2D_i,3D_i,4D_i$. If for a moment we neglect the fact that the arguments of the dielectric loss functions have varying prefactors, then $E^2 \\varepsilon ''(D_i,\\Omega_p)^2$ represents the pump frequency-dependent energy absorbed by the system.\n\nNo frequency-selective behavior can be expected from the quasilinear $\\alpha \\alpha$-term, where the additional factor $\\Omega_p$ appearing in the nominator of $A^{\\alpha}(\\Omega_p,D_i,t_p)$ does not allow for a 'resonant' energy absorption. This is because the product $\\Omega_p \\varepsilon ''(6D_i,2\\Omega_p)$ is a strictly increasing function of $\\Omega_p$ in contrast to $\\varepsilon ''(6D_i,2\\Omega_p)$ which exhibits a peak at $\\Omega_p=3D_i$.\n\n\\subsection*{Heterogeneous model}\nIn a first discussion of Kerr effect responses in the heterogeneous model we neglect exchange effects. We choose the rotational diffusion constants distributed according to a generalized Gamma distribution with parameters such that the resulting (normalized) $(L=1)$-correlation has the form $\\exp\\left[ -(t\/\\tau)^{\\beta_K} \\right]$ with the KWW-exponent $\\beta_K=0.5$ and $\\tau =1$. The distribution of rotational diffusion constants is shown in the upper panel of Figure $1$. The lower panel shows the Kerr effect signal $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ for different pump frequencies $\\Omega_p=0.01,0.1,1,10,100$ (indicated by letters A,B,C,D,E in the figure). One cycle has been applied to the system for all pump frequencies. \n\nIn order to understand the curves in Figure $1$ we refer to the previous section where we discussed the response of a single $D_i$. Here, the energy absorption was shown to have a maximum for $\\Omega_p \\approx 2D_i, 3D_i, 4D_i$. This means that in a distribution of $D_i$ most energy is absorbed if $D_i \\approx \\Omega_p\/2$ holds. Thus, a pump frequency dependent selection is possible. The $D_i$ corresponding to $\\Omega_p=10^{-2}\\dots 10^2$ are marked by the vertical arrows in the distribution function (upper panel of Fig.$1$). Under the assumption that the response is mostly governed by the selected $D_i$ we obtain a maximum position at $\\omega_{max}=6D_i$, thus at $\\omega_{max}\\approx 3 \\Omega_p$. Although the prefactor $3$ is not exactly recovered in the extremum positions in the lower panel of Fig.$1$, we indeed observe a proportionality of extremum position (indicated by dotted lines) and pump frequency for $\\Omega_p=100\\dots 0.1$ (E,D,C,B). These shifts of the extremum position are the hallmark of dynamic heterogeneity. For $\\Omega_p=0.01$ (A) the relative shift becomes weaker. Here, a selection of $D_i\\approx 0.01\/2$ does not determine the total response because these small $D_i$ are underrepresented in the distribution function, see the arrow (A) in the upper panel. In that case, rotational diffusion constants around $D_i \\approx 0.05$ (B in the upper panel) determine the extremum position. This also holds for still smaller pump frequencies. Here, the extremum position becomes independent of $\\Omega_p$ while the amplitudes of the signal strongly decrease. A similar behavior is observed at the high-frequency end of the distribution, which we do not show explicitly. The maximum position as a function of $\\Omega_p$ will be discussed in detail later compared to results in the homogeneous limit.\n\n\nIn order to discuss the width of the responses, we added a Debye function (dotted line near C in the lower panel of Figure $1$). The heterogeneous response (C) is somewhat broadened in comparison. The curve for $\\Omega_p=100$ (E) has a negative onset on the low-frequency side of the peak. Two remarks concerning this finding of a partially negative response function are appropriate. First, the negative sign of the response function is due to the first term in Eq.(\\ref{respeineralmumu}), which is proportional to the difference $\\left[\\exp(-6 D_i t_p)-\\exp(-2 D_i t_p) \\right]$. This is negative for finite $t_p$ in contrast to the other term $\\propto \\left[1-\\exp(-6 D_i t_p) \\right]$. A negative response function therefore is a switch-on effect, since the former contribution vanishes for large $t_p$. Indeed, if several cycles are applied, all response functions become positive in the whole frequency range. Second, even for finite $t_p$, the negative term can be considered as an artefact of the rotational diffusion model. This is because the rates $6D_i$, $2D_i$ in the exponents are the decay rates of the second and the first Legendre polynomial, which are known to be of similar magnitude in real systems as discussed in Section II. In any model where these rates are very similar (e.g. the exchange model discussed in Section II) or equal (e.g. isotropic rotational jumps instead of rotational diffusion) no negative sign of the response occurs for arbitrary $t_p$. \n\nFigure $2$ shows the contributions of the quasilinear term $\\langle P_2^\\alpha(\\omega)\\rangle$  with the same parameters as in Figure $1$. The extremum position is not proportional to pump frequency although it weakly depends on $\\Omega_p$. While the pump frequency spreads over five decades, the maximum position shifts only for less than one decade. The shift of the extremum positions is further diminished if several cycles are applied. This demonstrates the expected result that the quasilinear term is not able to detect dynamic heterogeneities. In the following discussion, we will therefore focus on the $\\alpha \\mu\\mu$-contribution. The quasilinear term can be viewed as a background contribution to the response function. As it is of quadratic order in the applied field like the $\\alpha \\mu \\mu$-contribution, no phase-cycling procedure allows the extraction of either of the contributions. However, the terms should be easily distinguishable since the quasilinear signal is at a more or less constant position. Furthermore, the problem with the quasilinear term is trivially minimized if systems with large permanent dipole moment $\\mu$ (compared to their polarizability $\\alpha$) are studied. In this case, $\\langle P_2^{\\alpha} (\\omega)\\rangle$ is much smaller than $\\langle P_2^{\\mu \\mu} (\\omega)\\rangle$.   \n\n\nWe now discuss the influence of exchange on the Kerr effect response in the framework of the exchange model outlined in Section II. For our calculations we assume that the exchange rates factor into terms that depend on initial and final environment only, $\\gamma_{ij}=\\rho(\\epsilon_i) X \\exp(\\beta \\epsilon_j)$. Here, $\\rho(\\epsilon_i)$ (probability to end in final state $\\epsilon_i$) is some normalized density of states and $X \\exp(\\beta \\epsilon_j)$ (probability to leave initial state $\\epsilon_j$) can be interpreted as thermally activated jumps out of environment $\\epsilon_j$. Here, we have identified the so-far not specified $\\epsilon_j$ with energies. The prefactor $X$ allows to adjust the 'strength' of exchange. We furthermore relate the rotational diffusion constants to $\\epsilon_i$ as $D_i=Y \\exp(\\beta \\epsilon_i)$.  The model is designed such that the total rate out of environment $\\epsilon_i$ is proportional to the appropriate rotational diffusion constant, $\\gamma_i\\approx (X\/Y) D_i$, meaning that slow environments exchange slowly and fast environments exchange fast. The ratio of $\\gamma_i$ and $D_i$, given by $(X\/Y)$, determines the influence of exchange on the decay of rotational correlation functions.\n\nFor the curves shown in Figure $3$ we chose $\\rho(\\epsilon_i)$ to be Gaussian with width $\\sigma=3$ and we have set $\\beta=1$. In order to obtain correlation times around unity we set $Y=100$. The upper panel of Figure $3$ shows normalized correlation functions for $L=1,2$ without exchange ($X\/Y=0$, dashed lines) and in the presence of exchange ($X\/Y=10$, solid lines). In the latter case the decay is strongly affected by exchange. All correlation functions shown can be well fitted with a function of KWW-type with a stretching exponent $\\beta_K \\approx 0.5\\ $. The ratios of correlation times $\\tau_L$ obtained from the fit are $\\tau_1\/\\tau_2 \\approx 3.0$ without exchange and $\\tau_1\/\\tau_2\\approx 1.3$ with exchange.\n\n\nThe nonlinear Kerr effect response $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ in the presence of exchange is shown in the lower panel of Figure $3$. All parameters describing exchange are chosen as for the solid lines in the upper panel. In the limit without exchange, $(X\/Y)=0$, the model discussed in Figure $1$ is analytically recovered with a modified distribution of rotational diffusion constants. The resulting curves in that case are very similar to the responses in Fig.$1$. For that reason we show only the curves with exchange ($X\/Y=10$). The pump frequencies are chosen as $\\Omega_p=10^{-2} \\dots 10^{2}$ with one applied cycle for each frequency. The shifts of the extremum position obtained are very similar as in the heterogeneous case without exchange, cf. Fig.$1$. The peaks become slightly broader in the presence of exchange. No negative sign of the response occurs in any of the curves. \n\nWe therefore conclude that exchange does not affect the possibility to detect dynamic heterogeneities. It is important to stress that the Kerr effect experiment does not allow to conduct information regarding the lifetime of dynamic heterogeneities.\n\n\n\n\\subsection*{Homogeneous model}\nWe now discuss the responses in a homogeneous scenario, where the distribution of rotational diffusion constants is replaced by a single, but time-dependent rotational diffusion constant. The time dependence is chosen in a manner that the correlation function for $L=1$ is proportional to $\\exp\\left[ -(t\/\\tau)^{\\beta_K} \\right]$ with $\\beta_K=0.5$ and $\\tau=1$. The correlation function is thus the same as in the situation we discussed in Figure $1$. Inserting Eqns.(\\ref{glvont}),(\\ref{homd}) into (\\ref{almumu}) we find\n\\Be \n\\nonumber & & \\hspace{-1.6cm} \\langle P_2^{\\mu \\mu}(t) \\rangle=\\frac{1}{5} \\beta^2 \\mu^2 E^2 \\left(\\frac{\\beta_K}{\\tau}\\right)^2  \\exp\\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right] \\times  \\\\ \\label{homoalmumu}\n& & \\hspace{-1.6cm} \\times \\int_0^{t_p} dt' \\int_0^{t'} dt'' \\sin(\\Omega_p t') \\sin(\\Omega_p t'') \\left(\\frac{t'}{\\tau} \\right)^{\\beta_K-1} \\left(\\frac{t''}{\\tau} \\right)^{\\beta_K-1}  \\exp\\left[ 2 \\left(\\frac{t'}{\\tau} \\right)^{\\beta_K}\\right]  \\exp\\left[ \\left(\\frac{t''}{\\tau} \\right)^{\\beta_K}\\right] .  \n\\Ee\nThe double integral does not depend on time $t$ and therefore represents only a prefactor to the response. The time-dependence is solely governed by the function $ \\exp \\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right]$. Thus, the maximum position in $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ is determined by the Fourier-transform of this function.\n\nFigure $4$ shows the homogeneous responses for the same pump frequencies as in Figure $1$. One cycle has been applied for each pump frequency, $t_p=2\\pi\/\\Omega_p$. It is this pump frequency-dependence of $t_p$ which leads to the shifts in the extremum positions in Figure $4$ (see the function above). Although the extrema are not at a constant position as usually expected in a homogeneous scenario, the shifts are less pronounced than in the heterogenous limit. The dashed curves corresponding to $\\Omega_p=1$ and $\\Omega_p=10$ are plotted negative. This is an example for the fact that signs of nonlinear responses are in general not easy to predict. The change of sign is not a systematic feature in our calculations.\n\n\n\n\\subsection*{Comparison of heterogeneous and homogeneous models}\n\nIn Figure $5$ we compare the maximum position in heterogeneous (upper straight line) and homogeneous (lower straight line) models, plotted versus pump frequency $\\Omega_p$ for one applied cycle. The distribution in the heterogeneous case is as in Fig.$1$, and the time-dependent rotational diffusion constant of the homogeneous model is as in Fig.$4$. In the homogeneous case a maximum position independent of pump frequency is obtained only for sufficiently large $\\Omega_p$. Clear shifts of the extremum position are observed in the heterogeneous limit. The shifting becomes weaker for $\\Omega_p < 0.1$. This is due to the shape of the chosen distribution function as elucidated in the discussion of Fig.$1$. \n \nFor a distinction between homogeneity and heterogeneity, the straight lines in Figure $5$ do not look too promising at first glance because the homogeneous extrema are not at a constant position. However, this finding should not be overemphasized. The observation of shifts in the homogeneous extremum positions relies directly on the lack of time-translational invariance of the specific homogeneous model used in the calculation. This leads to the shifted time $(t+t_p)$ in the function $\\exp \\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right] $. The fact that the extrema are not at a constant position is only due to the appearance of the pump time $t_p$ in this function, which is related to the pump frequency via $t_p=2\\pi\/\\Omega_p$. It is furthermore easy to eliminate this effect. If $t_p$ is chosen independent of the pump frequency, then the homogeneous model obviously yields extremum positions completely independent of the pump frequency. This can be achieved by applying several cycles $N(\\Omega_p)\\propto \\Omega_p$ because of $t_p =2\\pi N(\\Omega_p)\/\\Omega_p$.\n  \nWe added some extremum positions for that case in Fig.$5$, denoted by symbols. We chose $t_p=2\\pi\/0.01$, meaning that one cycle is applied for the smallest pump frequency considered, $\\Omega_p=0.01$, and $10^4$ cycles are applied for the largest $\\Omega_p=100$. Then the homogeneous model extrema (open symbols) do not depend on $\\Omega_p$. The influence of several applied cycles on the extremum positions in the heterogeneous case is small. The extrema are located at somewhat smaller frequencies in comparison to the curve for one applied cycle. This is due to the pump time dependence of the function $A^{\\mu \\mu}(\\Omega_p,D_i,t_p)$. \n\nThe theoretical finding that the homogeneous ansatz leads to strictly constant extremum positions only if the pump time is constant for all applied frequencies also holds for NHB. If NHB is carried out as outlined in Ref.\\cite{mynhb} and the dynamics are treated in a rotational diffusion ansatz as in this work, we find in the homogeneous limit the same expression as Eq.(\\ref{homoalmumu}) up to prefactors and a modified time dependence. It is again the lack of time-translational invariance that leads to terms like $(t+t_p)^{\\beta_K}$ in the appearing exponential functions. We therefore suggest also in NHB to apply several cycles for higher pump frequencies. Also the heterogeneous model for NHB leads to very similar results as presented in the previous sections. Shifts of the extremum position are observed if the pump frequencies are varied in the range defined by the distribution of rotational diffusion constants. \n\n\\subsection*{Distribution width in the heterogeneous model}\n\nFinally, we discuss the influence of the width of a distribution of $D_i$ in a heterogeneous model. A linear behavior of the extremum position as a function of pump frequencies, $\\omega_{max} \\propto \\Omega_p^\\eta$ with $\\eta =1$, is obtained only if the distribution of rotational diffusion constants is sufficiently flat in the studied range. If the distribution is narrow, then usually exponents $\\eta <1$ are found.\n\nIn Figure $6$, extremum positions obtained from the heterogeneous model with a logarithmic Gaussian distribution of rotational diffusion constants $g(D_i)\\propto \\exp\\left(-[\\ln D_i - \\ln D_{\\infty}]^2\/[2\\sigma^2] \\right)$ are shown for three different distribution widths $\\sigma=1.8$ (diamonds), $\\sigma=1.2$ (squares) and $\\sigma=0.6$ (circles). Exchange effects are neglected. In each case, $D_{\\infty}$ was chosen such that the correlation time of the $(L=1)$-correlation function in a KWW-fit is $\\tau_1=1$. The corresponding KWW-exponents obtained from the fits of the correlation function are $\\beta_K=0.51$ (diamonds), $\\beta_K=0.67$ (squares) and $\\beta_K=0.88$ (circles). We applied the field for fixed pump time $t_p=2\\pi\/0.001$ independent of the pump frequency.\n\nHeterogeneities are observed via shifts of the extremum position. Constant behavior of $\\omega_{max}(\\Omega_p)$ is observed for small and large $\\Omega_p$. This is because, similar as in Figure $5$, the 'selected' $D_i$ in these cases are located beyond the main contributions to the distribution functions. In Fig.$5$ a constant behavior for large $\\Omega_p$ is not reached in the frequency range considered due to the tailing of the corresponding distribution of rotational diffusion constants, see the upper panel of Fig.$1$. We do not expect that a constant extremum position for large $\\Omega_p$ would be obtained experimentally. Here, other dynamics than isotropic reorientations will occur in real systems.\n\n\nOne can see that the rise of $\\omega_{max}$ with $\\Omega_p$ is the steeper the broader the distribution is, approaching the limit $\\eta =1$ for a very flat distribution. Acceptable power law fits for the shown curves are obtained in the window $\\Omega_p=0.1 \\dots 10$. Here, we find the exponents $\\eta \\approx 0.84$ (diamonds), $\\eta \\approx 0.63$ (squares) and $\\eta \\approx 0.2$ (circles). \n\n\\section*{IV. Conclusions}\n\nWe have calculated the Kerr after-effect response to an external oscillating electric field for reorientational dynamics in an isotropic system. Such an experiment is very similar to NHB, thus aiming at the question of the nature of the dynamics. The different assumptions of heterogeneous and homogeneous dynamics have been treated in a rotational diffusion model. We find that homogeneous and heterogeneous dynamics show different behaviour in the Kerr effect response function. \n\nWe propose the Kerr effect as an alternative to NHB. NHB and Kerr effect have in common that they are both nonlinear experiments. The advantage of the Kerr effect is that the signal has no linear response background. In an experiment, one would measure the reequilibration of the polarizability after the sample has been excited by a sinusoidal electric field. The frequency of the latter is then varied. If the time-dependent functions obtained in this way are plotted in a Fourier-transformed representation (imaginary part), then extremum positions as a function of the pump frequency shift under the assumption of heterogeneous dynamics, and they are constant in the homogeneous limit. The Kerr effect signal is thus able to distinguish between dynamic heterogeneity and homogeneity.\n\nThe interesting relation between dynamic and spatial heterogeneities is not resolved yet. We therefore emphasize that the Kerr effect experiment outlined in this article does not allow to obtain any information about spatial aspects of the heterogeneities.\n\nThe influence of exchange, and thus a finite lifetime of heterogeneities on the responses is small. Exchange in our model does not affect the possibility of frequency selective modification.\n\nIt turned out that the general trends of homogeneous and heterogeneous limits are obtained more clearly if several cycles of the pump field are applied. This is because switch-on effects are minimized for longer pump times. In addition, the extremum positions in our homogeneous model depend on the pump time $t_p=2\\pi N\/\\Omega_p$, and therefore on the pump frequency. In order to suppress this artificial dependence, we suggest to keep the pump time constant rather than the number of cycles applied if measurements with different $\\Omega_p$ are compared. If, e.g., $N$ cycles (where ideally $N>1$) are applied for pump frequency $\\Omega_p$, then $2N$ cycles should be applied if the pump frequency is $2\\Omega_p$.\n\n\nIf the width of a relaxation time distribution decreases then the shifts of the extremum position with pump frequency become weaker. This point might be of interest in the investigation of a sample at different temperatures. A steeper rise of the extremum position with pump frequency for lower sample temperatures would indicate an increasing width of the corresponding distribution function. A point we have not discussed explicitly is that of course also combinations of heterogeneous and homogeneous behavior may occur in real systems. If the weight of homogeneous and heterogenous character of the dynamics changes with temperature this should be observable in more or less pronounced shifts of the extrema.\n\n\n\n\n\\section*{Acknowledgement}\nThis work has been supported by the DFG under contract No. Di693\/1-2. \n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is widely assumed that our understanding of gravity according to Einstein, together with effective local quantum field theories give an accurate description of IR physics. On the other-hand, many recent insights within the fields of holography and entropic gravity continue to corroborate the notion that spacetime and the laws of gravity should emerge from a UV-complete theory of quantum gravity\\cite{J95, M01, VR2010, susskind13, pad15, susskind17, felix17}\\nocite{smc} \\nocite{vl18}\n\\nocite{Heem}. It has become clear, particularly in the context of AdS\/CFT, that the emergence of the known laws of gravity is intimately connected with an entanglement area-law \\cite{ryu06, hubeny, myers13, swingle14, C16, C17}. Conversely, deviations from the area-law entanglement could lead to violations of gravitational physics. \n\nThat there could be novel physical ramifications of high-energy physics upon the laws that govern arbitrarily-large scales might seem very surprising, but just such a connection was recently proposed by E. Verlinde in the recent work \\cite{verl2016}. This recent work offers a novel hypothesis for the relation between the de Sitter entropy associated with the cosmological horizon, in terms of the dark energy, and its response due to the addition of matter. It is argued that the entropy carried by the dark energy obeys a volume-law, and that adding matter to a region of space generates competition between the volume-law and the area-law entropy associated with the matter. It is shown that there is a length-scale, which is related to the mass and the curvature scale, where the volume-law for the dark energy overwhelms the area-law scaling of the matter entropy. This leads to a violation of the holographic principle above this length-scale. It is this key role played by the violation of holography, at sufficiently large length-scales that is highlighted in the present work. \n\nIn \\cite{verl2016}, it was shown that the interplay between the de Sitter entropy, carried by the dark energy, and its local reduction due to the addition of a mass $M$, pertains to the following criteria,\n\n\\begin{equation}\n\\label{criteria}\n\\frac{M}{A(R)} \\lessgtr \\frac{a_0}{8 \\pi G},\n\\end{equation}\nwhere $a_0 = \\frac{ c^2 }{ L }$ is the MOND acceleration scale, $L$ is the Hubble length and $A(R)$ is the area of a sphere with radius $R$. This entropic condition coincides exactly with the empirical criteria for the scale upon which the phenomena attributed to dark matter become manifest in galactic rotation curves. This empirical observation is then given physical motivation by Verlinde's new hypothesis. Furthermore, it is shown that one can reproduce many key empirical findings in cosmology, such as Milgrom's fitting formula of MOND and the baryonic Tully-Fisher relation within this formalism \\cite{milgrom, btf}. There is also strong agreement between this framework and weak-lensing observations \\cite{brouwer16}.\n\nThe paper \\cite{verl2016} has been criticised on the basis of apparent inconsistencies in terms of associating an effective elastic description to the response governed by the removal (by the addition of matter) of a certain volume of the dark energy medium. Several of these features have since been addressed in the recent paper \\cite{sabine}, where a covariant proposal for the effective elastic response of the dark energy was considered, and the roles of the identifications between gravitational and elastic variables were clarified. Since the elastic description is intended to give an effective description, its negation would not necessarily rule out the underlying postulates of the emergent gravity framework. In the present work we make contact with the key results of \\cite{verl2016}, without using an effective elastic description.\n\nThe main proposal of the present paper is to use a straightforward modification of the setup described in \\cite{verl2010}, where Newtonian and Einstein gravity were derived from arguments involving so-called holographic screens. The modification entails a violation of holography above a critical length-scale, which thereby emulates a key feature of Verlinde's new theory. In the present paper, it is demonstrated that this modification is sufficient to describe the emergence of a dark gravity force, as well as a version of the baryonic Tully-Fisher relation. This work does not claim to go further than Verlinde's proposals, but instead the present goal is to provide a framework for the underlying ideas that is mathematically simpler, and which clarifies the key role played by the breakdown of holography in the emergence of dark gravity.\n\nThe rest of this paper is organised as follows. Section \\ref{review} contains a review of the recent proposal \\cite{verl2016} which identifies the core components that will be utilised in the derivation of emergent dark gravity, presented in section \\ref{main}. In section \\ref{main}, the main result is presented, where a modified thermodynamic setup based on arguments involving a holographic screen is constructed. It is demonstrated that the emergent entropic force associated to this system receives an additional contribution from the information in the bulk which is no-longer encoded holographically. This additional force has a $r^{-1}$ scaling-law required to describe flattening galactic rotation curves. In this setup it is also shown that the bulk and boundary energies obey a relation which is analogous to the baryonic Tully-Fisher relation, and that the bulk energy manifests as an apparent mass. By inputting the critical scale $r_c(M,L)$ identified in Verlinde's new theory to the setup, we find that the entropic force associated to the bulk then takes exactly the form of the force required to describe flat galactic rotation curves. It is also found that this value for the critical scale reproduces the baryonic Tully-Fisher relation up to a numerical factor. The key assumptions that are used in the present work are then reviewed. Finally, in section \\ref{relation} the present work is compared to the work of several recent papers with related goals.\n\n\n\\section{Emergent Gravity in de Sitter Space}\n\\label{review}\n\nIn \\cite{verl2016} a radical new explanation for the phenomena attributed to dark matter was offered in terms of emergent gravity in de Sitter space. In this section, we briefly review the key features of this work.\n\nIn \\cite{verl2016}, the static patch of de Sitter space was considered, with the metric,\n\\begin{equation}\n\\label{dsstatic}\nds^2 = -f(r) dt^2 + \\frac{1}{f(r)} dr^2 + d\\Omega_{d-2}^2,\n\\end{equation}\nwith $f(r) = 1 - \\frac{ r^2 }{ L^2 }$, where the cosmological horizon is at $r=L$. The Bekenstein-Hawking formula associates an entropy to de Sitter space which is determined by the area $A(L)$ of the cosmological horizon, as follows,\n\\begin{equation}\n\\label{dSent}\nS_{DE} = \\frac{ A(L) c^3 }{ 4 \\hbar G },\n\\end{equation}\nwhere, for reasons that will shortly be explained, the subscript ``DE\" denotes the dark energy. In \\cite{verl2016}, an interpretation for the entropy \\eqref{dSent} is proposed wherein the total entropy of de Sitter is associated to the dark energy which is distributed throughout the volume of de Sitter, leading to constant entropy density that obeys a volume-law. Accordingly, if we consider a spherical region of size $r$, the entropy associated to the dark energy within this region is proportional to the volume $V(r)$ of the sphere, so that we have\n\\begin{equation}\n\\label{SV}\nS_{DE}(r) = \\frac{ V(r) }{ V_0 },\n\\end{equation}\nwhere $V_0$ is the volume per unit of entropy of the dark energy. The condition that the entropy \\eqref{SV} coincides with the de Sitter entropy \\eqref{dSent} when $r=L$ then leads to the following formula for $V_0$,\n\\begin{equation}\n\\label{V0}\nV_0 = \\frac{ 4 G \\hbar L}{ d-1 }.\n\\end{equation}\nIt is then straightforward to show that formula \\eqref{dSent} can be re-written in the following way,\n\\begin{equation}\n\\label{SVA}\nS_{DE}(r) = \\frac{r}{L} \\frac{ A(r) c^3}{ 4 G \\hbar},\n\\end{equation}\nwhich makes it clear to see that when $r=L$, the above formula reproduces the formula for the total de Sitter entropy \\eqref{dSent}. In \\cite{verl2016}, the emergence of dark gravity is traced to the interplay that results from the local removal of a portion of the dark energy degrees of freedom due to the addition of matter. To ascertain this effect, one can turn on a matter source for a point mass M at the origin of the static patch \\eqref{dsstatic} by introducing the Newtonian potential with the following replacement,\n\\begin{equation}\n\\label{withpot}\nf(r) \\rightarrow 1 - \\frac{ r^2 }{ L^2 } + 2 \\phi(r),\n\\end{equation}\nwith,\n\\begin{equation}\n\\label{phi}\n\\phi(r) = \\frac{- GM }{r }.\n\\end{equation}\nIt is shown that the negative sign of $\\phi(r)$ leads to a reduction in the total de Sitter entropy when we add the mass $M$ to the origin of the static patch. This total reduction of the de Sitter entropy in fact corresponds to a local reduction of the entropy associated with the dark energy in the region surrounding the point mass. Accordingly one can consider the change in the growth of the area of a spherical region as a function of geodesic distance in the case with and without the matter to arrive at the following formula for reduction of the de Sitter entropy due to the addition of the mass $M$,\n\\begin{equation}\n\\label{SM}\nS_M(r) = \\frac{ 2 \\pi M c}{\\hbar} r.\n\\end{equation}\nA key result of \\cite{verl2016} is that, given a point mass M, there is a length-scale, depending on the mass and the curvature scale, where the de Sitter entropy \\eqref{SVA} exactly equals the amount of entropy \\eqref{SM} which is removed by the addition of a mass $M$. Using the formulas \\eqref{SVA} and \\eqref{SM} it is easy to calculate this scale to be the following,\n\\begin{equation}\n\\label{rcrit}\nr_c(M) = \\sqrt{\\frac{ G M L}{c^2}} = \\sqrt{\\frac{ G M}{ a_0}},\n\\end{equation}\nwhere the MOND acceleration scale $a_0 = c^2 \/ L$ \\cite{milgrom} has been identified. Below this critical length-scale, all of the de Sitter entropy is removed by the mass $M$; in this case there are only matter degrees of freedom in the bulk which are encoded in the degrees of freedom at the boundary. This corresponds to what is called the Newtonian regime. Conversely, for regions that are larger than this scale, the mass $M$ does not remove all of the de Sitter entropy. In this sub-Newtonian or ``dark gravity\" regime, there is therefore information associated to the dark energy in the bulk which is entangled with the bulk mass. This volume-law entanglement, which contains information about the bulk mass, then spoils the holographic encoding of the bulk mass. This scenario is depicted in figure \\ref{bigone}. As we will see, this is a key feature of these proposals that this work seeks to emulate in the thermodynamic setup presented in section \\ref{main}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]\n{EGDS.jpg}\n\\caption{(Top) Cartoon illustrating the encoding of the information (shaded blue) pertaining to the bulk mass $M$ in de Sitter space. For regions much smaller than $r_c$, the bulk information pertaining to $M$ is redundantly encoded in the boundary bits $N_\\partial$ which obey an area-law. Conversely, for regions that are large compared to $r_c$, there are bulk degrees of freedom that scale extensively with the bulk and which are not encoded in the bits $N_\\partial$. (Bottom) Cartoon of a portion of a tensor network representation for the entanglement structure of de Sitter space. The bulk legs correspond to the green indices. The entanglement structure that builds the emergent geometry is encoded in the short-range correlations described by the grey, internal legs of the network. The state is endowed with a constant density of long-range entanglement by the single, large red tensor. The bulk indices participate in the long-range entanglement via their connection to the large-red tensor via the dark red legs. (Bottom Right) adding matter locally removes some of the long-range entanglement. We imagine that the long-range indices in the center (white) have been deleted. In this case, the bulk state in the yellow legs can be approximately reconstructed from the state on the portion of network bounded by the cut which passes through the blue legs of the network. For larger portions of the network, such as for the cut which passes through the purple legs, the state in the bulk legs within this cut cannot be reconstructed from the legs passing through the cut, leading to a breakdown of holography for regions that are sufficiently large.}\n\\label{bigone}\n\\end{figure}\n\nIn \\cite{verl2016}, a remarkable feature is observed when we consider the following criteria, associated to the length-scale above which the volume-law associated with the dark energy overwhelms the matter entropy \\eqref{SM}, that is where we have the following:\n\\begin{equation}\n\\label{entcrit}\nS_M(r) < S_{DE}(r).\n\\end{equation}\nUsing equations \\eqref{SVA} and \\eqref{SM}, we observe that the above criteria are equivalent to the criteria \\eqref{criteria}, as claimed in the introduction. \n\nSince the volume-law of the dark energy overwhelms the area-law scaling of the matter entropy in the dark gravity regime, then accordingly what emerges are not the known laws of gravity. Instead, in \\cite{verl2016}, this regime is understood by considering the effective, elastic response of the dark energy medium to its removal, from local inclusion regions, by the addition of matter. It is shown that this leads to the fitting formula of MOND and also the baryonic Tully-Fisher relation \\cite{milgrom, btf}. It is thereby argued that this framework provides an alternative explanation for the phenomena that are currently attributed to dark matter, which does not require dark matter to exist. This work will make contact with these two key results, but the effective elastic description described in \\cite{verl2016} will not be required.\n\n\\section{Emergent Dark Gravity from a (Non)Holographic Screen}\n\\label{main}\n\nThis section describes how to obtain the emergence of a dark gravitational force, which scales like $\\frac{1}{r}$ above a critical length-scale $r_c$, in a purely thermodynamic setting which is very similar to the setup considered in \\cite{verl2010}. The modification presented here is inspired by the arguments in \\cite{verl2016}, described in section \\ref{review}, where the interplay between the entropy associated to the dark energy and the entropy which is removed by the addition of matter leads to a violation of the holographic encoding of bulk beyond the critical length-scale $r_c$ as in \\eqref{rcrit}. Accordingly, the modification proposed here is the introduction of an arbitrary critical length-scale that controls the scale at which the holographic encoding of the bulk fails. \n\n\\subsection{A (Non)Holographic Screen}\n\\label{nhs}\n\nAt first we will assume precisely the setup described in section 3.2 of \\cite{verl2010}. Namely one is to imagine that there is a spherical region of space, known as a holographic screen, which separates the interior ``unemerged\" part of space from the exterior ``emerged\" part of space. A particular thermodynamic system is ascribed to this setup in which the ``unemerged\" part of space emerges. The emergence of space is imagined to arise due to a series of coarse-graining steps that push the holographic screen into the ``unemerged\" part of space, which leads to an overall reduction in the microscopic degrees of freedom associated with the holographic screen. In the emerged part of space, we imagine that there is a massive test particle of mass $m$ which is located at a small displacement $\\delta x$ from the holographic screen. The setup is contrived so that a change in the test particle's position contributes a change to the entropy of the screen, which leads to an entropic force acting on the test particle. The setup so-far described is depicted in figure \\ref{EDG} a). \n\nThe role of the holographic screen, within the earlier work, is to encode the information within the unemerged space. To make a clear connection between this setup and the AdS\/CFT literature, the unemerged part of space will be referred to, hereafter, as the bulk. The holographic screen then corresponds to the boundary of the bulk, but since this is an arbitrary boundary in space, and not a geometric boundary at spatial infinity as in AdS\/CFT, the boundary of the bulk (unemerged part of space) will be referred to as a screen. In the present setup, holography will be explicitly violated, so the term ``holographic screen\" will not be appropriate. The term non-holographic screen may then seem more appropriate, but for convenience this artificial boundary will simply be referred to as a screen.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]\n{edgfinal.jpg}\n\\caption{ a) the test particle of mass $m$ is located in the emerged part of space, and is displaced from the spherical screen of radius $r$ by $\\delta x \\ll 1$. b) for regions $r \\ll r_c$, the test particle contributes to a change in the entropy associated to the bits on the screen. c) for regions $r \\sim r_c$, the test particle contributes to a significant change in the entropy associated to the bulk.}\n\\label{EDG}\n\\end{figure}\n\nIn \\cite{verl2010} it is assumed that the information pertaining to an emergent mass $M$ in the bulk is encoded holographically in a quantity of bits $N_\\partial$ which is proportional to the area of the screen, in the following way,\n\\begin{equation}\n\\label{nbound}\nN_\\partial = \\frac{ A(r) c^3}{ 4 G \\hbar}.\n\\end{equation}\nThe main modification of \\cite{verl2010} presented in this paper is to assert that the holographic principle is violated in a way that depends on a critical length-scale $r_c$. We suppose that, in addition to the bits $N_\\partial$, there are a set of bits $N_\\Sigma$ that one also needs to have in order to be able to completely reconstruct the emergent mass $M$ in the bulk. We assert that the bits $N_\\Sigma$ pertaining to $M$ are not encoded holographically at the screen. Motivated by \\cite{verl2016} we assert that $N_\\Sigma$ should scale extensively with the \\emph{volume} of the bulk. We then associate the critical scale $r_c$ with the ratio of the numbers of bulk and screen bits as follows,\n\\begin{equation}\n\\label{ratio}\n\\frac{N_\\Sigma}{N_\\partial} = \\frac{r}{r_c},\n\\end{equation}\nwhich leads to the following form for $N_\\Sigma$,\n\\begin{equation}\n\\label{BV}\nN_\\Sigma = \\frac{r}{r_c} \\frac{ A(r) c^3}{ 4 G \\hbar}.\n\\end{equation}\nBased on the above observations we conclude that the role of the critical scale $r_c$ is to determine when the holographic, area-law scaling of the information associated to $M$ in the bulk is overwhelmed by the volume-law contribution of the bulk bits $N_\\Sigma$ that are not encoded holographically. Thus the critical scale $r_c$ that we have described plays a very similar role to the critical scale $r_c$ as in \\eqref{rcrit} identified in \\cite{verl2016}. In our setup, one could suppose that the scale $r_c$ that we are considering, should depend on the amount of information associated to the mass $M$ and the curvature scale $L$. This is indeed the case for the critical scale \\eqref{rcrit}. For now, this dependence will not be explicitly assumed.\n\nIf one imagines inserting a mass into the bulk, we can call the number of bits that characterise this matter $\\tilde{N}_\\Sigma$. In general this is different from $N_\\Sigma$, because when the holographic principle holds, we anticipate that the former are encoded redundantly at a screen, of size $r$, as a quantum secret-sharing scheme among the bits $N_\\partial(r)$ \\cite{harlow14, harlow15, harlow16}. In contrast, the bits $N_\\Sigma$ pertain to information about the bulk mass that is not encoded holographically, which do not therefore form a subset of the bits $N_\\partial$. The latter can be compared to the formula \\eqref{SVA}, so that the bits $N_\\Sigma$ here play the same role as the dark energy in \\cite{verl2016}. \n\n\\subsection{Thermodynamic Setup}\n\nNow we describe how to associate a thermodynamic system with the setup described in \\ref{nhs}. As per \\cite{verl2010}, we begin with the assumption that displacing the test particle which is outside the screen, but which is close to it, leads to a change to the entropy of the screen and the bulk. The change in the entropy of the total system will take the following form,\n\\begin{equation}\n\\label{composite}\n\\delta S = \\delta S_\\partial + \\delta S_\\Sigma ,\n\\end{equation}\nwhere $\\delta S_\\partial$ coincides with $\\delta S$ defined in section 3.2 of \\cite{verl2010}, which we identify as the change in the entropy of the screen. The quantity $\\delta S_\\Sigma$ is similarly defined as the contribution to the entropy of the bulk, which, as we have explained, is not encoded holographically. \n\nA change to the entropy as per \\eqref{composite} will lead to a change to the energy of the system $\\delta E = \\delta E_\\partial + \\delta E_\\Sigma$, where we have identified the contributions $\\delta E_\\partial$ and $\\delta E_\\Sigma$ which correspond to changes to the energy associated with the screen and bulk degrees of freedom respectively. Following \\cite{verl2010}, we assume that when the displacement $\\delta x$ of the test particle is small, the corresponding changes to the energies of the bulk and screen systems should be evenly divided over their respective degrees of freedom, so that the equipartition theorem holds for each system.\n\n\\begin{equation}\n\\label{equipartitions}\n\\delta E_\\partial = \\frac{1}{2} N_\\partial k_B T_\\partial \\hspace{1cm} \\delta E_\\Sigma = \\frac{1}{2} N_\\Sigma k_B T_\\Sigma ,\n\\end{equation}\nwhere $T_\\partial$ and $T_\\Sigma$ are the temperatures associated to the bulk and boundary systems. To proceed we further make the assumption that the bulk and boundary systems are in thermal equilibrium. In general this is something that we may expect to hold for situations where the emergent bulk is static. For now we will assume that this holds, so that we can identify $T_\\partial = T_\\Sigma = T$, where $T$ is the equilibrium temperature associated to the combined ensemble of the bulk and the screen. \n\nIf the first-law of thermodynamics holds for this system, then for near-equilibrium configurations the following relation must hold for the bulk and boundary subsystems,\n\\begin{equation}\n\\label{1stlaw}\n\\delta E_\\partial =T \\delta S_\\partial \\hspace{1cm} \\delta E_\\Sigma = T \\delta S_\\Sigma .\n\\end{equation}\nUsing equations \\eqref{equipartitions} and \\eqref{1stlaw}, we find that the changes to the entropy of the bulk and the screen obey the following relation,\n\\begin{equation}\n\\label{heidi}\n\\delta S_\\Sigma = \\frac{N_\\Sigma}{N_\\partial} \\delta S_\\partial = \\frac{r}{r_c} \\delta S_\\partial .\n\\end{equation}\nThe above relation has the intuitive interpretation that for scales $r \\ll r_c$, we have $\\delta S_\\Sigma \\sim 0$, and the significant contribution to the entropy of the system comes from the contribution to the screen, as depicted in figure \\ref{EDG} b). Conversely, at $r=r_c$, the test particle contributes the same amount of information to the bulk and boundary subsystems at $r=r_c$. Hence $r_c$ naturally described a scale where the contribution of a test particle to the bulk first becomes comparable to its corresponding contribution to the screen. For scales $r>r_c$, the corresponding contribution to the bulk overwhelms the contribution to the screen, as depicted in figure \\ref{EDG} c). Putting together equations \\eqref{heidi} and \\eqref{equipartitions} gives the following relation between the changes to the energies of the bulk and boundary subsystems,\n\\begin{equation}\n\\label{heidi2}\n\\delta E_\\Sigma = \\frac{N_\\Sigma}{N_\\partial} \\delta E_\\partial = \\frac{r}{r_c} \\delta E_\\partial .\n\\end{equation}\nTo obtain an emergent entropic force associated to this system, we express the change in energy given by the first law \\eqref{1stlaw} in terms of the work done by the force $F$ that displaces the test particle a distance $\\delta x$ \\emph{towards} the screen. In this way the entropic force must take the following form in terms of the change to the entropy of the system.\n\\begin{equation}\n\\label{ef}\nF = T \\Bigg( \\frac{\\partial S_\\Sigma}{\\partial x}+\\frac{\\partial S_\\partial}{\\partial x} \\Bigg) = T \\frac{\\partial S_\\partial}{\\partial x}\\Bigg( 1 + \\frac{r}{r_c} \\Bigg),\n\\end{equation}\nwhere equation \\eqref{heidi} was used in the last step. The term $\\frac{r}{r_c}$ includes a modification to the setup described in \\cite{verl2010} which is associated with the breakdown of holography that we have described. In particular, when $r \\ll r_c$ we can effectively set this term to zero, and in this case we would obtain the same result as \\cite{verl2010}, where the holographic principle is assumed to hold exactly.\n\n\\subsection{Emergent Dark Gravity}\n\\label{mainmain}\n\nIn \\cite{verl2010}, the form for the change $\\delta S_\\partial$ to the entropy of the screen can be motivated as follows. Following Bekenstein's derivation of the black hole entropy, one can propose that when the test particle is one Compton-wavelength from the screen, it should contribute a single bit of information to it \\cite{bek}. This contribution is assumed to be linear in $\\delta x$, at least approximately when $\\delta x$ is sufficiently small. This leads to the following form for the change of the boundary entropy,\n\\begin{equation}\n\\label{ds1}\n\\delta S_\\partial = \\frac{\\pi k_B}{2} \\frac{ m c }{\\hbar } \\delta x .\n\\end{equation}\nTo relate the quantities in this thermodynamic system to an emergent mass we suppose that the following relations hold,\n\\begin{equation}\n\\label{firstlaw}\n\\delta E_\\partial =M_B c^2 \\hspace{1cm} \\delta E_\\Sigma = M_D c^2 .\n\\end{equation}\nFor now $M_B$ and $M_D$ are just arbitrary constants with the units of mass. The relation \\eqref{heidi2} implies that these quantities obey the relation,\n\\begin{equation}\n\\label{BTF}\n\\frac{M_D}{M_B} = \\frac{r}{r_c} .\n\\end{equation}\nWe can now use either of the equations \\eqref{equipartitions} to determine the equilibrium temperature $T$ which is found to be the following,\n\\begin{equation}\n\\label{combtemp}\nT = \\frac{ 2 M_B G \\hbar}{c k_B \\pi r^2} .\n\\end{equation}\nNow we have all of the pieces to obtain the form for the entropic force that emerges from this thermodynamic system. In the case where $r \\ll r_c$, the bulk contribution to the entropy change can be neglected. In this limit we can neglect our modification to the setup described in \\cite{verl2010} and we accordingly obtain the same result contained therein, which is the emergence of Newton's gravitational force-law,\n\\begin{equation}\n\\label{Enewton}\nF = \\frac{ G m M_B }{ r^2 } .\n\\end{equation} \nEquation \\eqref{Enewton} justifies the interpretation of $M_B$ as describing an emergent mass in the bulk. For $r \\sim r_c$, the bulk contribution to the change in entropy of the screen becomes non-trivial, and using the formula \\eqref{combtemp} for the equilibrium temperature, together with the formulae \\eqref{heidi} and \\eqref{ds1} for the changes in the bulk and boundary entropy (respectively) and plugging these results into \\eqref{ef} for the entropic force, the entropic force takes the form,\n\\begin{equation}\n\\label{darkgravity}\nF = \\frac{ G m M_B }{ r^2 } + \\frac{G M_B m}{ r \\cdot r_c} .\n\\end{equation}\nThis force has the correct $\\frac{1}{r}$ scaling required to describe flat galactic rotation curves, where $M_B$ is the baryonic point mass located at the origin of the bulk. It is worth emphasising that this result essentially derived from the violation of holography as per equation \\eqref{ratio}, which eventually led to equation \\eqref{heidi}. In view of the derivation of Newtonian gravity in section 3.2 of \\cite{verl2010}, the additional factor of $r$ contained in \\eqref{heidi}, in this work, is what led to a force that scales like $r^{-1}$.\n\nPresently, the critical scale appearing in \\eqref{darkgravity} is arbitrary. On the other-hand, the critical length-scale \\eqref{rcrit} plays essentially the same role in \\cite{verl2016}. If we identify our critical scale with the scale \\eqref{rcrit}, we obtain the following emergent force,\n\\begin{equation}\n\\label{DEG}\nF= \\frac{G M_B m}{r^2} + \\sqrt{M_B G a_0} \\frac{m}{r},\n\\end{equation}\nwhich has precisely the form of Newtonian gravity with an additional dark gravity force which is observed at galactic scales above $r_c$ as per equation \\eqref{rcrit} \\cite{milgrom}. \n By inserting the relation \\eqref{BTF} into \\eqref{darkgravity} we see that the entropic force can be written in the following form,\n\\begin{equation}\n\\label{darkgravity2}\nF = \\frac{ G m }{ r^2 }  \\big( M_B + M_D(r) \\big) .\n\\end{equation}\nThis result serves to clarify the role of the quantity $M_D$, which describes the change in the bulk energy according to \\eqref{firstlaw}. In view of \\eqref{darkgravity2}, we conclude that the failure of holography for scales $r \\sim r_c$ leads to a contribution to the bulk energy (of the test particle to the screen) that manifests as an apparent, additional mass $M_D$ in the bulk. Notice that the scale $r_c$ drops out of equation \\eqref{darkgravity2}, but to ascertain the relation between the apparent mass $M_D$  and the mass $M_B$, we need to use the relation \\eqref{heidi2}, which does implicate $r_c$. This setup can therefore mimic the dark matter hypothesis, independently of the choice of $r_c$. We see that the identification between the mass $M_B$ and the apparent mass $M_D$ given by equation \\eqref{heidi2} has a similar role to the baryonic Tully-Fisher relation, which relates the baryonic mass distribution to the apparent distribution of dark matter \\cite{btf}. \n\nIf we consider the emergence of a spherically-symmetric mass distribution in the bulk which is entirely contained in a screen of size $r$. That is, suppose that there is an emergent mass profile $M_B(r)$ in the bulk, contained inside a screen of size $r$ and which corresponds to the change in energy $\\delta E_\\partial$ of the screen. Then if we then take the formula \\eqref{BTF}, with the critical scale $r_c$ identified by \\eqref{rcrit}, we recover a relation between the emergent mass profile $M(r)$ and the apparent mass profile $M_D(r)$, which, up to a numerical factor, has been shown to be equivalent to the baryonic Tully-Fisher relation \\cite{btf},\n\\begin{equation}\n\\label{BTFfinal}\nM_D(r)^2 = \\frac{a_0}{2 \\pi G} M_B(r),\n\\end{equation}\nwhich was shown to determine the apparent dark matter profile $M_D(r)$, given a profile $M_B(r)$ of (observed) baryonic matter in galaxies. Using this framework, we can therefore make contact with the main results obtained in \\cite{verl2016}. \n\n\\subsection{Review of Key Assumptions}\nThe key features of our main result, presented in \\ref{mainmain}, follow from the application of the following assumptions, whose role should be clarified,\n\\begin{enumerate}\n\\item{The bulk is not encoded holographically at the screen, and the non-holographic bulk degrees of freedom obey a volume-law, such that the failure to encode the bulk holographically is controlled by a length-scale $r_c$.}\n\\item{Changing the displacement of the test particle changes the entropy associated with the bulk and boundary subsystems.}\n\\item{The change in energy of the bulk and boundary subsystems obeys the equipartition theorem.}\n\\item{The bulk and boundary systems are in thermal equilibrium.}\n\\item{The first law of thermodynamics holds.}\n\\end{enumerate}\nAssumption 1 is our main assumption in this work, and it is the main modification to the setup described in \\cite{verl2010}. This is the modification that makes a connection to the new theory \\cite{verl2016}, where a violation of the holographic principle occurs for length-scales which are sufficiently large so that the inequality \\eqref{entcrit} holds. Assumption 2 is also made in \\cite{verl2010}, and this is the key feature that is contrived to produce an entropic force acting on the test particle. Assumption 3 is also made in \\cite{verl2010} in relation to what we have called the boundary system. That we should imagine this to hold for the bulk system seems like a natural extension, since when the particle is sufficiently close to the screen, we can imagine that the change in the energy of the bulk is evenly divided over the bulk bits. As in \\cite{verl2010}, we may not expect this to hold for displacements that are large compared to the Compton wavelength of the test particle, and we refer the reader to the former work for a justification of this. Assumption 4 is new here and it seems like a natural assumption to make when the emergent bulk and matter distribution is static, as was the case for the static patch of de Sitter that was considered in \\cite{verl2016}. In more general situations, where there are non-trivial dynamics, we may not expect assumption 4 to hold exactly. Assumption 5 was assumed in \\cite{verl2010}; the first law should evidently hold for any equilibrium thermodynamic system, such as the one that has been described here. The role of the first law is, as in the earlier work, that it allows one to relate the an entropic force, to the entropy gradient and the equilibrium temperature as per \\eqref{ef}.\n\nIn many cases, it may appear that the fundamental constants have been introduced in an ad-hoc fashion. This objection could also be raised about the work \\cite{verl2010}. Here, as in that work, the role of these contants is essentially to give quantities with the correct dimensions. Nevertheless, we see that $\\hbar$ drops out in the calculations of the dark gravity force \\eqref{DEG}, as it must in order to match with the idea that we are considering a Newtonian limit (that does not albeit lead to Newtonian gravity). So, as with the previous work, the constant $\\hbar$ remains arbitrary in this work.\n\n\\section{Comparison with Previous Work}\n\\label{relation}\n\nThe setup described in section \\ref{main} is strongly based on the setup described in \\cite{verl2010}. However, a key modification is included which is based on a central observation of the recent proposals \\cite{verl2016}, that the emergence of dark gravity is due to the breakdown of the holographic principle above a certain length-scale \\eqref{rcrit}, where the inequality \\eqref{entcrit} holds, as described in section \\ref{review}. Indeed, one of the aims of this work is to make a clear connection between the relatively simpler framework described in \\cite{verl2010} and the recent work \\cite{verl2016}, by extracting this key feature and implementing it as we have described in section \\ref{main}. The main results \\eqref{DEG} and \\eqref{BTFfinal} that we have derived for the emergent dark gravity force and the baryonic Tully-Fisher relation (respectively), rely on the identification for our critical scale as per \\eqref{rcrit}. We cannot motivate the use of this scale based on the efficacy of the empirical observations because in the present work, the role of the critical scale is to control the breakdown of holography, which does not immediately follow from the empirically observed behaviour above this scale. This effect is indeed currently widely attributed to the dark matter hypothesis, which is an entirely different proposal. Thus, the motivation for us to consider the scale \\eqref{rcrit}, in the present work, is tied to the proposals of \\cite{verl2016} which initially make contact with the observed criteria. In that work, this length-scale is motivated by according a particular interpretation to the de Sitter entropy, which it not considered here; it is instead shown that the ensuing violation of holography, with a volume-law scaling of the emergent matter degrees of freedom, essentially leads to the key features that match the observed galaxy-scale phenomena. To this end the effective elastic description that was described in \\cite{verl2016} was not required.\n\nThere have been several recent papers with related goals, which have considered how to derive the MOND fitting formula in terms of an entropic force which emerges from a thermodynamic setup involving holographic screens. These works include \\cite{Zhang}, where an argument analogous to \\cite{verl2010} is considered, but with the introduction of a modified inertia relation which accordingly produces a modified gravity force. In this work we do not assume that such a modified inertial relation holds. Rather, as we have said, the present setup is inspired by the observation of Verlinde in \\cite{verl2016} that the emergence of a dark gravity force is attributed to a breakdown of holography at a certain scale, which has been implemented explicitly in setion \\ref{main} of the present paper. There are also a pair of more recent papers \\cite{Abreu1,Abreu2} which have considered how to derive the MOND relation, again inspired by the holographic screen arguments \\cite{verl2010}, but the authors have made use of the Tsallis entropy to derive a modified gravity relation. Again, this differs significantly from the present approach, where Tsallis statistic have not been used; the key results presented here derive from a breakdown of holography which is not assumed in the previous papers \\cite{Abreu1,Abreu2}.\n\n\\section{Discussion \\& Outlook}\n\nIn this work, a clear connection has been established between the earlier work \\cite{verl2010} and the recent proposals \\cite{verl2016}, by implementing a breakdown of holography which is controlled by a critical length-scale via a straightforward modification to the earlier framework. Furthermore, when the value for the critical scale \\eqref{rcrit}, identified in the recent proposals, is adopted, the exact form of the dark gravity force, in addition to (up to a numerical factor) the baryonic Tully-Fisher relation, which are observed to produce flattening galactic rotation curves above precisely this scale, are obtained. This work then essentially clarifies the key role played by the breakdown of holography for sufficiently large length-scales (given an emergent mass $M$) in E. Verlinde's new hypothesis, whilst providing a mathematically simple framework with which one could explore this, and related ideas. \n\nIn figure \\ref{bigone}, a tensor network has been depicted which provides an analogy for the entanglement structure of an emergent de Sitter geometry as described in \\cite{verl2016}. The particular network represented here does not accurately depict the tensor network one might use to describe an emergent de Sitter geometry, however one could in principle obtain this via a discretisation of a constant time slice \\cite{B16, ev}, sewn with tensors that, as well as bulk and internal indices, each carry a small additional index which is contracted with the tensor that thereby endows the state with a constant density of long-range entanglement, whose role can therefore mimic the dark energy as per \\cite{verl2016}. Furthermore, this work offers the interpretation that the addition of a mass $M$ can be framed, in these terms, as a deletion of a portion of the long-range legs attached to a closed subregion of the network. This would be consistent with the idea that mass is associated with relative entropy in emergent gravity\\cite{verl2016}. This question is left open for future work.\n\n A particularly interesting open question for this work relates to the possibility of a finding covariant formulation, which could make contact with the recent work \\cite{sabine, wang}. Most of the components needed for this have been identified in the present work, but the formulation of a covariant model is left for future work. Another interesting and partially-related question  concerns a possible relation between the framework presented here and the effective elastic description offered in \\cite{verl2016} to describe the dark gravity regime.\n\nDespite the many successes of the dark matter paradigm, we have presently yet to observe the dark matter particle, and Verlinde's new theory offers the exciting possibility that understanding this regime may require a radical revision of our widely-held belief in the efficacy of GR and EFT on cosmological scales. In this work we hope to have provided a framework that will help to further our understanding of this new proposal and its underlying microscopic description.\n\n\n\\section{Acknowledgements}\nThe author would like to thank Henry Maxfield and Vaios Ziogas for helpful discussion and comments.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nBeing able to replicate, and therefore investigate, the structure and function of real-world complex networks is a profoundly difficult problem. However, the pervasiveness of systems that could be more accurately interpreted as a result cannot be overstated: social networks  \\cite{girvan2002community}, the spread of disease \\cite{newman2003structure}, artificial intelligence \\cite{hopfield1982neural}, language structure \\cite{ronen2014links}, and transportation networks \\cite{santi2014quantifying}. Accordingly, a number of network models and network generating algorithms have been proposed~\\cite{watts1998collective, newman2001random, kim2004performance, volz2004random, serrano2005tuning, bansal2009exploring, ritchie2014higher, ritchie2014beyond, overbury2015using}. Many of these network models seek to reproduce a specific network property or characteristic: the degree distribution \\cite{barabasi1999emergence, newman2001random}, the small worldness \\cite{watts1998collective}, degree-degree correlations \\cite{newman2002assortative, newman2003structure} or clustering, the propensity of 3-cycles in a network \\cite{milo2002network, newman2003properties}. However, investigations of \\emph {higher-order} structure, subgraphs and arrangements of subgraphs not specified by standard network metrics, have been limited by a lack of accurate and versatile network models. Some progress has been made using the \\emph{configuration model}~\\cite{ karrer2010random,ritchie2014higher, ritchie2014beyond}, and it is this work we seek to build upon.  \n\nIn the standard configuration model, triangle subgraphs appear infrequently as a by-product of working with finite size networks~\\cite{bollobas1980probabilistic}. But what if one \\emph{wants} triangle subgraphs to appear in a network, in particular, if one wants to model a complex network with clustering? An extension of the configuration model to this case exists~\\cite{miller2009percolation, newman2009random}. In this extension a node is allocated a number of stubs, that may go on to form standard edges, as well as a number of triangle `corners' or \\emph{hyperstubs}, pairs of stubs that will form triangles. While edges are formed in the usual way, triangles are formed by selecting three triangle hyperstubs at random and connecting their pairs of constituent stubs. As for edges the number of all stubs must be divisible by two, the total number of triangle hyperstubs must be divisible by three for the triangle hyperstub sequence to be graphical. Another similarity this model shares with the standard configuration model is that the probability that any two triangles will share an edge, thus forming a $G_\\boxslash$ subgraph (see Figure~\\ref{fig:examples}), vanishes in the limit of large network size~\\cite{karrer2010random}. Just as a network composed of lines only is limited in recreating real-world networks, so is a model that can only include edges and triangles. Obviously, this may depend on properties and structure of the real networks, but in many cases edges and triangles are not enough to produce an accurate enough artificial replica of the real network. \n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\begin{tikzpicture}\n\\node [place] (2) at (0, 0) {};\n\\node [place] (1) at (0.001, 1) {};\n \\draw [ultra thick](1) to (2);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n  \\newdimen\\R\n  \\R=0.58cm\n  \\draw (0:\\R)\n     \\foreach \\x in {120,240,360} { (\\x:\\R) }\n               (360:\\R) node [place] (3) {}\n               (240:\\R) node [place] (2) {}\n               (120:\\R) node [place] (1) {};\n \n     \\draw [ultra thick](2) to (3);\n     \\draw [ultra thick](3) to (1);\n\\end{tikzpicture} \n&\n\\begin{tikzpicture}\n  \\newdimen\\R\n  \\R=0.58cm\n  \\draw (0:\\R)\n     \\foreach \\x in {120,240,360} { (\\x:\\R) }\n               (360:\\R) node [place] (3) {}\n               (240:\\R) node [place] (2) {}\n               (120:\\R) node [place] (1) {};\n  \n     \\draw [ultra thick](1) to (2);\n     \\draw [ultra thick](2) to (3);\n     \\draw [ultra thick](3) to (1);\n\\end{tikzpicture} \n&\n\\begin{tikzpicture}\n\\node [place] (1) at (0.001, 0) {};\n\\node [place] (2) at (0.001, 1.001) {};\n\\node [place] (3) at (1, 1) {};\n\\node [place] (4) at (1, 0.001) {};\n\\draw [ultra thick](2) to (3);\n\\draw [ultra thick](3) to (4);\n\\draw [ultra thick](4) to (1);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n  \\newdimen\\R\n  \\R=0.65cm\n  \\draw (0:\\R)\n     \\foreach \\x in {90,210,330} { (\\x:\\R) }\n               (330:\\R) node [place] (3) {}\n               (210:\\R) node [place] (2) {}\n               (90:\\R) node [place] (1) {};\n     \\node [place] (4) at (0, 0) {};\n     \\draw [ultra thick](1) to (4);\n     \\draw [ultra thick](2) to (4);\n     \\draw [ultra thick](3) to (4);\n\\end{tikzpicture} \n\\\\\n$G_0$ &    \n$u3$ &\n$G_\\triangle~\/~t3$ &\n$u4$ &\n$s4$ \\\\ \n\\begin{tikzpicture}\n\\node [place] (1) at (0.001, 0) {};\n\\node [place] (2) at (0.001, 1.001) {};\n\\node [place] (3) at (1, 1) {};\n\\node [place] (4) at (1, 0.001) {};\n\\draw [ultra thick](1) to (2);\n\\draw [ultra thick](2) to (3);\n\\draw [ultra thick](4) to (1);\n\\draw [ultra thick](1) to (3);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n\\node [place] (1) at (0.001, 0) {};\n\\node [place] (2) at (0.001, 1.001) {};\n\\node [place] (3) at (1, 1) {};\n\\node [place] (4) at (1, 0.001) {};\n\\draw [ultra thick](1) to (2);\n\\draw [ultra thick](2) to (3);\n\\draw [ultra thick](3) to (4);\n\\draw [ultra thick](4) to (1);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n\\node [place] (1) at (0.001, 0) {};\n\\node [place] (2) at (0.001, 1.001) {};\n\\node [place] (3) at (1, 1) {};\n\\node [place] (4) at (1, 0.001) {};\n\\draw [ultra thick](1) to (2);\n\\draw [ultra thick](2) to (3);\n\\draw [ultra thick](3) to (4);\n\\draw [ultra thick](4) to (1);\n\\draw [ultra thick](1) to (3);\n\\end{tikzpicture} \n&\n\\begin{tikzpicture}\n\\node [place] (1) at (0.001, 0) {};\n\\node [place] (2) at (0.001, 1.001) {};\n\\node [place] (3) at (1, 1) {};\n\\node [place] (4) at (1, 0.001) {};\n\\draw [ultra thick](1) to (2);\n\\draw [ultra thick](2) to (3);\n\\draw [ultra thick](3) to (4);\n\\draw [ultra thick](4) to (1);\n\\draw [ultra thick](1) to (3);\n\\draw [ultra thick](2) to (4);\n\\end{tikzpicture} \n&\n~\n\\\\\n$i4$ &    \n$G_\\boxempty~\/~e4$ &\n$G_\\boxslash~\/~d4$ &\n$G_\\boxtimes~\/~c4$ &\n\t\t\t \\\\ \n\\begin{tikzpicture}\n  \\newdimen\\R\n  \\R=0.75cm\n  \\draw (0:\\R)\n     \\foreach \\x in {1,71,...,281} { (\\x:\\R) }\n               (1:\\R)   node [place] (1) {}\n               (71:\\R)  node [place] (2) {}\n               (141:\\R) node [place] (3) {}\n               (211:\\R) node [place] (4) {}\n               (281:\\R) node [place] (5) {};\n     \\draw [ultra thick](1) to (2);\n     \\draw [ultra thick](2) to (3);\n     \\draw [ultra thick](3) to (4);\n     \\draw [ultra thick](4) to (5);\n     \\draw [ultra thick](5) to (1);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n  \\newdimen\\R\n  \\R=0.75cm\n  \\draw (0:\\R)\n     \\foreach \\x in {1,61,...,301} { (\\x:\\R) }\n               (1:\\R)   node [place] (1) {}\n               (61:\\R)  node [place] (2) {}\n               (121:\\R) node [place] (3) {}\n               (181:\\R) node [place] (4) {}\n               (241:\\R) node [place] (5) {}\n               (301:\\R) node [place] (6) {};\n     \\draw [ultra thick](1) to (2);\n     \\draw [ultra thick](2) to (3);\n     \\draw [ultra thick](3) to (4);\n     \\draw [ultra thick](4) to (5);\n     \\draw [ultra thick](5) to (6);\n     \\draw [ultra thick](6) to (1);\n\\end{tikzpicture}\n&\n~\n&\n~\n&\n\\\\\n$G_{\\pentagon}$&    \n$G_{\\hexagon}$&\n~&\n~& \n\\end{tabular}\n\\caption{The set of subgraphs that have been used in this paper. The subgraphs denoted by: $\\{G_0, G_\\triangle, G_\\boxempty, G_\\boxslash, G_\\boxtimes, G_{\\pentagon}, G_{\\hexagon}\\}$, are those that have been used as input for the proposed network construction algorithms. We use: $\\{u3, t3, u4, s4, i4, e4, d4, c4\\}$, to denote the total number of uniquely counted subgraphs given by the subgraph counting algorithm~\\cite{ritchie2014higher}.}\\label{fig:examples}\n\\end{center}\n\\end{figure}\n\nThe configuration model has since received further attention to address this~\\cite{karrer2010random}. Building on the edge-triangle model a more general subgraph-based approach is taken where one may specify distributions of edges alongside distributions of arbitrary subgraphs. In the case of complete subgraphs it is obvious how to do this. For example, $G_\\boxtimes$ subgraphs can be formed by allocating to nodes hyperstubs composed of three stubs. Then, four of these hyperstubs can be selected at random to form a $G_\\boxtimes$ subgraph. However, it is not clear how this may work for subgraphs that are not symmetric. For example, in a $G_\\boxslash$, there are two different types of hyperstubs and it is necessary for any network model or construction algorithm to be able to make this distinction. Karrer and Newman proposed that it is possible to identify a node's \\emph{role} within a subgraph using \\emph{orbits}. To find the orbits of a subgraph one must first list all possible automorphisms of the subgraph, that is, permutations of nodes that do not create or destroy edges. The orbit of a node is a set of other nodes with which it may be permuted so that no edges are created or destroyed. Of course, computing the automorphism group of subgraphs is a computationally hard problem but so long as subgraphs with few nodes are used, this is not a problem~\\cite{karrer2010random}. \n\nNetwork models are rarely used independently of other processes. Instead, they typically provide the substrate for dynamical processes to operate upon. For example, the compartmental Susceptible-Infected-Recovered ($SIR$) model of contagion is often embedded into a network to help better understand how the network and its properties affect the epidemic. Previous work~\\cite{ritchie2014beyond} successfully incorporated the Karrer and Newman approach into an approximate ODE or mean-field model for $SIR$ epidemics on networks displaying higher-order structure, and this mean-field model showed excellent agreement with simulation results. In order to achieve this, Ritchie et~al. bypassed the need to classify a node's role in a subgraph via the automorphism group. Instead, nodes within asymmetric subgraphs were uniquely enumerated, even if they were topologically equivalent to one another, and this enumeration defined their role. The motivation for this adaptation was to simplify the derivation of the ODE model. Using the orbit approach or the full enumeration are different ways of satisfying different modelling needs, and these are not the only possible approaches. In fact, when modelling networks and nodes within subgraphs one can instead classify nodes by the stub cardinality of their hyperstubs. \n\nA common method across all of the above models, i.e., edge-triangle, the more general Karrer-Newman model, and that proposed by Ritchie et al., is that sequences of hyperstubs must be specified for each and every subgraph that is to be included. From these sequences it is possible to recover the network's degree sequence by multiplying them by the stub cardinality of the hyperstub which they represent, and then summing the resulting sequences. Therefore the degree sequence of the network is a result of the construction of the network rather than a quantity that is controlled for. However, given that the degree sequence of the network is probably the single most important characteristic of a network, there is a need for methods that can generate networks with a particular subgraph family and distribution yet preserve a given degree sequence. In~\\cite{ritchie2014beyond}, we recently showed that it is possible to constrain the  hyperstub sequences so that the $1^{st}$ and $2^{nd}$ moments of the resulting degree sequence are controlled. In this paper, we go beyond this work and propose two generation algorithms that provide full control over the degree sequence and clustering. \n \nThe paper is organised as follows. In Section~\\ref{sec:MM}, we describe in detail the two generation algorithms, including tuning of clustering. In Section~\\ref{sec:results}, we validate our algorithms and we explore the diversity of the generated networks by comparing them to the widely used Big-V rewiring scheme. We further analyse networks generated by using different subgraph families or distributions. Epidemic and complex contagion models are simulated on these networks and we show that degree distribution and global clustering alone are not sufficient to predict the outcome of these processes. Finally, we discuss extensions and further research questions relating to our work.\n\n\n\\section{Materials and methods}\\label{sec:MM}\n\nIn this section we propose two new algorithms, both of which are parametrised by a degree sequence and a set of subgraphs. The algorithms  construct hyperstub degree sequences (from which the input degree sequence may be recovered exactly) that can be used in a modified configuration model style connection procedure to realise a network. \n\nThere are some caveats regarding the preservation of the input degree sequence that are common to all configuration-like models. Firstly a degree sequence must sum to a an even number to be graphical. If it does not, a stub must be created or destroyed to satisfy this constraint. In general, hyperstub degree sequences must sum with multiplicity equal to the number of times they appear in their parent subgraphs, i.e., a triangle hyperstub sequence must be divisible by 3. When selecting stubs or hyperstubs at random to form subgraphs it is possible that self or multi-edges may form. The number of these events happening depends only on the average degree $\\langle k \\rangle$ and thus remains constant with network size. It is possible to simply delete self-edges or collapse multi-edges down to a single edge. If this approach is taken then the guiding degree sequence will be violated. Instead we disallow such connections by reselecting nodes in the connection procedure until no self or multi-edges will be created by forming the subgraph. This is known as the \\emph{matching algorithm}~\\cite{milo2003uniform}. Finally, it is possible for the process to be left with no option other than to add subgraphs over existing links or selecting multiple instances of the same node. In this case we completely reset the algorithm, regenerating hyperstub sequences and forming subsequent connections until a network is formed. \n\n\\subsection{The underdetermined sampling algorithm -- UDA}\\label{sec:UDA}\n\nThe concept underpinning this algorithm is that for each node there are combinations of hyperstubs that will satisfy its degree. For example, a node with $k=3$ classical edges could form 3 single $G_0$ edges or 1 $G_0$ edge and one $G_\\triangle$ hyperstub. The number of possible arrangements will depend on the degree of the node and number of input subgraphs. From these arrangements a single one is selected at random. For a given degree $k$ this problem is equivalent to solving an underdetermined linear Diophantine equation equal to $k$ subject to positivity constraints. The coefficients are given by the edge counts of the hyperstubs, that are induced by the input subgraphs, and the solution will give the number of each hyperstub so that the degree of the node is matched exactly.  \n\nTo generate a network using this algorithm, let us assume that a degree sequence, $D=\\{d_1,d_2, \\dots, d_{N}\\}\\in \\mathbb{N}_0^{1 \\times N}$, and the set of subgraphs to be included in the network's construction, $G = \\{G_1, G_2, \\dots, G_l\\}$, is given. Then, for each subgraph we classify its hyperstubs by their edge cardinality. It is now possible to form a vector that has elements specifying the number of edges in each hyperstub. From this vector we take the unique elements. For example, the $G_\\boxslash$ subgraph will have a corresponding hyperstub vector of $\\alpha = (2,3)$. For a given degree $k$ we must consider all possible hyperstubs and hyperstub combinations that yield a classical degree equal to $k$. To systematically list all such combinations, we first concatenate all the hyperstub vectors into a single vector, $\\boldsymbol{\\alpha}$, to be used as coefficients for the following linear underdetermined Diophantine equation \n\\begin{eqnarray}\\label{eq:dio}\nk = \\alpha_1 x_1 + \\alpha_2 x_2 + \\dots \\alpha_{r}x_{r},\n\\end{eqnarray}\nwhere $k = k_{min},k_{min}+1,\\dots,k_{max}$ and $r$ denotes the number of eligible hyperstubs -- a node with degree $k=3$ can only go on to form subgraphs where the hyperstubs contain no more than three edges --, for the given degree $k$ and which is solved subject to the constraint $\\boldsymbol{x} \\in \\mathbb{N}_0^{r}$. A solution $\\boldsymbol{x}$ of this equation corresponds to the number of each type of hyperstubs required to result in a node of degree $k$. For example, if $\\alpha_1$ and $\\alpha_2$ take values $1$ and $2$ corresponding to hyperstubs of $G_0$ and $G_\\triangle$ respectively and the degree of the node is $k=5$, the Diophantine equation would take the form $5 = x_1 + 2x_2$ and the solution space of this equation is given by the pairs $(x_1,x_2) =\\{(5,0),~(3,1),~(1,2)\\}$. In general these equations may be solved recursively by fixing a trial value $x_i = j$ and reducing the dimensionality of the equation by absorbing this term. This is repeated until the equation becomes of the standard form: $k' = \\alpha_1x_1 + \\alpha_2x_2$, which can be solved explicitly. A solution obtained this way will form a single solution of the original equation. This process is then repeated for a different starting trial solution, and since we seek only positive solutions and $k$ is finite, the corresponding solution space has a finite number of elements. Matlab code for this process is available at \\url{https:\/\/github.com\/martinritchie\/Network-generation-algorithms}.\n\nOnce the entire solution space for each degree has been found it is possible to start forming the hyperstub degree sequences. To proceed, the algorithm works sequentially through the degree sequence $D =\\{d_1,d_2, \\dots, d_N\\}$ of the $N$ nodes, where $d_i \\in \\{k_{min}, k_{min}+1, \\dots, k_{max}\\}$. By selecting at random a solution from the solution space that corresponds to $k=d_i$, that specifies the hyperstub configuration, and by concatenating all the selected solutions for all the nodes a hyperstub degree sequence of dimension $h \\times N$, where $h$ denotes the total number of hyperstubs induced by the input subgraphs, is formed.\n\nFor incomplete subgraphs it is not possible to select solutions of the Diophantine equations' solution spaces at random. The reason for this is two-fold: (1) not all asymmetric subgraphs are composed of equal quantities of each of their constituent hyperstubs, and (2) hyperstubs with lower stub cardinality will appear more frequently than hyperstubs of higher stub cardinality because hyperstubs with fewer edges can be more readily accommodated into the degree of a node. Problem (1) may be addressed by representing every hyperstub induced by a subgraph in the vector of coefficients opposed to grouping hyperstubs by their stub cardinality. Problem (2) may be addressed by decomposing hyperstubs generated in excess into simple\/classical edges. This is the approach we take in our implementation. This choice is motivated by its wider applicability. For example, when using $G_\\boxslash$ as an input subgraph there will be more degree-2 corners generated than degree-3 corners. However, once all degree-3 corners are allocated to $G_\\boxslash$ subgraphs any leftover degree-2 corners may be decomposed back into stubs that can form edges, thus preserving the degree sequence. Finally, it should be noted that if the input subgraphs do not admit hyperstub combinations that can sum to a  particular degree in the network then the proposed method will fail. For example, it is almost always necessary to include $G_0$ (edge) as input.  \n\nPseudo-code for the UDA algorithm is given in Appendix~\\ref{UDA}, and the Matlab code is available from \\url{https:\/\/github.com\/martinritchie\/Network-generation-algorithms}.\n\n\\subsubsection{A priori clustering calculation}\\label{sec:UDA_c}\n\nThe global clustering coefficient is defined as the ratio between the total number of triangles and the total number of connected triples of nodes $\\triangle + \\vee$, since each triangle contains 3 triples of nodes: $C = \\frac{\\triangle}{\\triangle + \\vee}$. It should be noted that each unique triangle is counted 6 times and each unique triple is counted twice. The number of triples incident to a node of degree $k$ is given by $\\triangle + \\vee = k(k-1)$ since a node will form a triple with every pair of its neighbours and each triple is counted twice. The expected number of triples for a node of degree $k$ is therefore given by $P(K=k)\\times k(k-1)$, where $P(K=k)$ is the probability of finding a node of degree $k$. The expected number of triangles incident to a node of degree $k$, $\\langle \\triangle_k \\rangle$, may be obtained from the Diophantine equations' solution space associated with that degree. To do this, one needs to sum all occurrences of triangle corners, regardless of which subgraph they belong to, from that solution space and divide by the number of solutions in that particular solution space, since solutions are selected uniformly at random. Finally we are in a position to compute the global clustering coefficient as\n\\begin{eqnarray}\nC = \\sum_{k=2}^{k_{max}} \\frac{\\langle \\triangle_k \\rangle}{P(K=k) \\times k(k-1).}\n\\end{eqnarray}\nFor example, let us consider the homogeneous network with $k=5$ and the input subgraphs $G_0$ and $G_\\boxslash$. These subgraphs induce the vector of coefficients $\\boldsymbol{\\alpha} = (1,2,3)$ that, for $k=5$, has the following solution space\n\\[\\begin{array}{rccccc}\n         G_0:   & 5 & 3 & 2 & 1 & 0,\\\\\n         g_2:   & 0 & 1 & 0 & 2 & 1, \\\\\n         g_3:   & 0 & 0 & 1 & 0 & 1,\n\\end{array}\\]\nwhere the rows give the number of each hyperstub, the columns give an individual solution and $g_2$ and $g_3$ denote the double and triple hyperstub of $G_\\boxslash$ respectively. From this we may calculate the expected number of triangles $\\langle \\triangle_5 \\rangle$. In this example we can see that on average for every $g_3$ corner the UDA algorithm will generate two $g_2$ corners. Since the excess $g_2$ corners will be decomposed into edges, one observes that $g_2$ and $g_3$ will be generated in equal quantities. So the expected number of $g_2$ is given by the expected number of $g_3$, e.g., $2\/5$ per node. Since $g_2$ denotes a triangle corner, the number of $g_2$ corners also gives the total \nnumber of triangles, that is uniquely counted and per node. So the expected number of triangle per node is $12\/5$, each triangle being counted 6 times, and this network will have a theoretical global \nclustering of $C=0.12$. Computationally, we verify this by generating such networks with $N=5000$, and find that the number of open triples and triangles is  exactly $|\\vee| = 100000$ and $|\\triangle| = \n12120$, resulting in a global clustering of $0.1212$, as expected. \n\n\\subsection{Cardinality matching -- CMA}\\label{sec:CMA}\n\nThe cardinality matching algorithm (CMA) requires as input a degree sequence, a set of subgraphs and corresponding \\emph{subgraph sequences}, i.e., multiple sequences specifying to which and how many subgraphs nodes belong to. Note that these sequences are not yet allocated to nodes. The algorithm proceeds to allocate hyperstubs of subgraphs to nodes that have a sufficient number of stubs to accommodate the hyperstub degree. The algorithm outputs hyperstub degree sequences, from which the input degree sequence may be recovered exactly. This then can be used to realise a network based on a modification of the configuration model.\n\nTo generate a CMA network one needs to first decide on a degree sequence $D$, a subgraph set $G = \\{G_1, G_2, \\dots, G_l\\}$ and a set of subgraph sequences $S = \\{S_1,S_2,\\dots, S_l\\}$, where $S_j(k)$, with $j=1, 2, \\dots, l$ and $k=1, 2,\\dots, N$, gives the number of times a node will be part of a $G_j$ subgraph without specifying the precise hyperstubs that connect the node to a $G_j$ subgraph. Our goal is to map the subgraph sequences into hyperstub sequences which can then be allocated to nodes that can accommodate them. From the hyperstub sequence, it is possible to work out the lower bound on the degree of nodes that can accommodate a specific hyperstub sequence. To complete this mapping one needs to differentiate between complete and incomplete subgraphs.\n\nFor complete subgraphs the subgraph sequence is identical to its hyperstub sequence since there is only one way or hyperstub by which a node can connect to such a subgraph. Thus, multiplying the hyperstub degree by the number of edges in the hyperstub will give us the lower bound on the degree of nodes that can accommodate the hyperstub sequence. For incomplete subgraphs the subgraph sequence does not specify how the node connects to the subgraph. Hence, we need to determine how the various hyperstubs are allocated to nodes. To see how to do this let us consider an arbitrary subgraph $G$ with subgraph sequence $S$. Given that the subgraph has $m$ distinct hyperstubs, let $p=(p_1, p_2, \\dots, p_m)$ be the vector of probabilities of picking different hyperstubs. We note that the values of $p$ reflects the proportion of each hyperstub found in the subgraph. For example, $G_\\boxslash$ has two distinct hyperstubs that both appear with multiplicity two, in this case $p=(1\/2,1\/2)$. This will ensure that their numbers are balanced and subgraphs can be formed.\n\nNext, using the multinomial distribution corresponding to subgraph $G$, $M^{G}(s^{G}_i,P)$ where $s^{G}_i$ denotes the subgraph sequence of index $i$ (this is not yet a node label), we pick hyperstub types to transform the subgraph sequence into hyperstub degree. For each $s^{G}_i$ this will result in a vector of length $m$ specifying the exact number of each hyperstub. It is possible to concatenate all the resulting choices from all multinomial distributions $M^{G}(s^{G}_i,p)$, where $i=1,2, \\dots, N$ to form the following matrix\n$$\n\\bordermatrix{\n       & s_1^{G}    & s_2^{G} & \\dots & s_N^{G} \\cr\nh_1^{G}    & h_1^{G}(1) & h_1^{G}(2) & \\dots & h_1^{G}(N) \\cr\nh_2^{G}    & h_2^{G}(1) & h_2^{G}(2) & \\dots & h_2^{G}(N) \\cr\n\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\cr\nh_m^{G}    & h_m^{G}(1) & h_m^{G}(2) & \\dots & h_m^{G}(N)\n\\cr}=H^{G},\n$$\nwhere $h_i^{G}(j)$ denotes the number of $h_i$ hyperstubs contributing to the subgraph degree $s_j^{G}$. We now need to compute the total number of edges specified by each column of the above matrix or by the hyperstub degree. This is given by $H^{G}(i)=\\sum_{j=1}^{m}|h_j^{G}|h_j^{G}(i)$ that denotes the total number of edges required by the subgraph degree $s_i^{G}$, and where $|h_j^{G}|$ represents the number of edges needed to form hyperstub $j$ in subgraph $G$ and $i=(1,2,\\dots,N)$. This process needs to be repeated for each subgraph to be included in the networks construction, i.e., for each subgraph $G_i$ with subgraph sequence $S^{G_i}=(s_1^{G_i},s_2^{G_i},\\dots, s_N^{G_i})$ there is a corresponding $H^{G_i}$ with elements that the algorithm will use as the lower bound on the degree of the nodes that can accept such a selection of hyperstubs.\n\nThe algorithm then proceeds by choosing the largest values, $H_{\\max}$, from all $H^{G_i}$ matrices, and this is used as the lower bound on the degree of nodes that can accept the hyperstub configuration associated with $H_{\\max}$, i.e., have enough edges of the classical type. From this list of all nodes with degree equal to or larger than $H_{\\max}$, a node is selected uniformly at random. The degree of the selected node is reduced accordingly, and the index of the node is now associated with the hyperstub degree to $H_{\\max}$. This node is then removed from the pool of eligible nodes for that particular subgraph, as otherwise it may be selected twice for the same subgraph thus violating the subgraph degree sequence. Similarly, the element $H_{\\max}$ is also removed from the pool of subgraph degree sequences that have yet to be allocated to nodes. This needs to be repeated until all elements of each subgraph degree sequence are allocated to nodes. Any edges that are not allocated to a particular hyperstub or subgraph are left to form edges.\n\nIn some cases it may be necessary to impose some cardinality constraints on the subgraph sequences. Obviously, if the network is homogeneous with $k=3$ we cannot include complete pentagon subgraphs or allocate two $G_\\triangle$ subgraphs to each node. More generally, it may be necessary to constrain the moments of the subgraph sequences. Let $\\langle k \\rangle$ denote the mean degree of the given degree sequence and let $G_i$ be a subgraph composed of a single hyperstub with cardinality $\\alpha$ and having subgraph degree sequence with mean $\\langle s \\rangle$ then: $ \\langle \\alpha s \\rangle = \\alpha \\langle  s \\rangle \\leq \\langle k \\rangle$ is a necessary condition for the two sequences to be graphical. In the case of more than one hyperstub, this is extended to $\\sum_{i = 1}^m\\alpha_i \\langle s_i \\rangle \\leq \\langle k \\rangle$, where $m$, $\\alpha_i$ and $s_i$  denote the number of hyperstubs, hyperstub cardinality and associated subgraph sequence respectively. For the networks generated in this paper, the degree sequence and subgraph sequences were measured from networks previously generated by the UDA such that prior knowledge about the sequences being graphical was available without the need to impose any such constraints.  \n\nClustering calculations for this algorithm are trivial since the subgraph degree sequences are known. One simply sums a sequence and then multiplies this figure by the number of triangles induced by that subgraph, being careful not to double count across multiple sequences for the same subgraph. The number of triples of connected nodes can be calculated following the method given for the UDA given in Section~\\ref{sec:UDA_c}. Pseudo-code for the CMA is given in Appendix~\\ref{CMA}, with the corresponding Matlab code available from \\url{https:\/\/github.com\/martinritchie\/Network-generation-algorithms}.\n \n\\subsection{Connection process}\\label{sec:CP}\n\nWe describe this process for a single incomplete subgraph. The case of the complete subgraph is trivial and has already been described (see Section~\\ref{sec:intro}). This process was first presented by Karrer \\& Newman \\cite{karrer2010random}. Consider a subgraph composed of three different hyperstub types, $h_1$, $h_2$ and $h_3$ that occur with a multiplicity of 1, 2 and 3 respectively, i.e., the subgraph is composed of 6 nodes. For these hyperstub sequences to be graphical we require\n\\begin{eqnarray}\n\\sum_{i=1}^N |h_1|_i = \\frac{1}{2}\\sum_{i=1}^N |h_2|_i = \\frac{1}{3}\\sum_{i=1}^N |h_3|_i,\n\\end{eqnarray}\nwhere $|h_i|_j$ specifies the $h_i$ hyperstub degree of node $j$. If these conditions are not met, one needs to decompose any surplus hyperstubs into stubs that may form classical edges in order to preserve the degree sequence. \n\nUsing the hyperstub sequences, one can create three dynamic lists for the three hyperstub types where a node appears with multiplicity equal to its hyperstub degree. Once the dynamic lists are fully populated, the connections process can start. This is done by selecting the following: 1 node from the $h_1$ bin, 2 from the $h_2$ bin and 3 from the $h_3$ bin, and all the selection processes done  uniformly at random and without replacement. Before forming the connections between these 6 nodes, one must ensure  that: (1) the selection contains no duplicates (that will form self-edges) and (2) that no single pair of nodes are already connected. If a connection already exists, a multi-edge may form and\/or subgraphs will share edges. If neither of these conditions are violated then the connections may be formed. Otherwise all nodes are returned to their bins and a new selection is made. As previously discussed, it is possible to delete self and multi-edges, however, this will destroy the degree sequence. The method of reselecting nodes has been previously introduced and is known as the \\emph{matching algorithm} \\cite{milo2003uniform}. It is possible that after many selections no valid combinations of nodes remain. For example, all bins may contain the same node. In this and other non-viable cases, all bins are re-populated and the connection process is started anew. It should be noted that, as none of the construction constraints discussed above involve the neighbours of the nodes being connected, it is possible for previously created subgraphs to become connected into a set of subgraphs with overlap, see Figure~\\ref{fig:byproductexample} for an illustration. Evidence of such by-products will be shown in Section~\\ref{sec:UDA_CMA}.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tikzpicture}[scale=.5]\n           \\node[place, minimum size = 8mm] (A) at ( 2.5, -4.33) {\\scalebox{1}{$A$}};\n           \\node[place, minimum size = 8mm] (B) at ( 0,0) {\\scalebox{1}{$B$}};\n           \\node[place, minimum size = 8mm] (D) at ( 5,0) {\\scalebox{1}{$D$}};            \n           \\node[place, minimum size = 8mm] (E) at ( 10,0) {\\scalebox{1}{$E$}};\n           \\node[place, minimum size = 8mm] (C) at ( 2.5,4.33) {\\scalebox{1}{$C$}};\n           \\node[place, minimum size = 8mm] (F) at ( 7.5, 4.33) {\\scalebox{1}{$F$}};            \n           \\node[place, minimum size = 8mm] (G) at ( 5, 8.65) {\\scalebox{1}{$G$}};\n\n           \\path[fill=red!20,opacity=.5] (B.center) to (C.center) to (F.center) to (D.center) to (B.center);\n           \\path[fill=green!20,opacity=.5] (C.center) to (F.center) to (E.center) to (D.center) to (C.center);\n           \\path[fill=blue!20,opacity=.5] (C.center) to (G.center) to (F.center) to (D.center) to (C.center);\n\n           \\node[place, minimum size = 8mm] (A) at ( 2.5, -4.33) {\\scalebox{1}{$A$}};\n           \\node[place, minimum size = 8mm] (B) at ( 0,0) {\\scalebox{1}{$B$}};\n           \\node[place, minimum size = 8mm] (D) at ( 5,0) {\\scalebox{1}{$D$}};            \n           \\node[place, minimum size = 8mm] (E) at ( 10,0) {\\scalebox{1}{$E$}};\n           \\node[place, minimum size = 8mm] (C) at ( 2.5,4.33) {\\scalebox{1}{$C$}};\n           \\node[place, minimum size = 8mm] (F) at ( 7.5, 4.33) {\\scalebox{1}{$F$}};            \n           \\node[place, minimum size = 8mm] (G) at ( 5, 8.65) {\\scalebox{1}{$G$}};\n\n         \n           \\draw [line width=1pt, dashed] (C) -- (G);\n           \\draw [line width=1pt, dashed] (G) -- (F);\n           \\draw [line width=1pt, dashed] (F) -- (C);\n\n           \\draw [line width=1pt] (E) -- (F);\n           \\draw [line width=1pt] (E) -- (D);\n           \\draw [line width=1pt] (D) -- (F);\n\n           \\draw [line width=1pt] (C) -- (B);\n           \\draw [line width=1pt] (D) -- (C);\n           \\draw [line width=1pt] (B) -- (D);\n\n           \\draw [line width=1pt] (A) -- (B);\n           \\draw [line width=1pt] (A) -- (D);\n           \\draw [line width=1pt] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n\\caption{Unintended generation of subgraphs with overlap. Despite satisfying the generation constraints given in Section~\\ref{sec:CP}, the addition of triangle (C,G,F) to toast (A,B,C,D) and triangle (D,F,E) results in 3 unintended distinct toasts \\{(B,C,F,D) in red, (D,C,F,E) in green, and (D,C,G,F) in blue\\} overlapping on one unintended triangle (C,F,D), in gray.}\n\\label{fig:byproductexample}\n\\end{figure}\n\n\\subsection{The Big-V algorithm}\\label{sec:bigv}\nThe Big-V algorithm does not generate networks as such, but is a widely-used, see~\\cite{house2010impact,house2011insights,ritchie2014higher,green2010large} for example, degree-preserving rewiring algorithm making it possible to control clustering. At each iteration, the algorithm selects a linear chain of 5 nodes at random, e.g., $\\{a,b,c,d,e\\}$ with 4 edges $\\{(a,b),(b,c),(c,d),(d,e)\\}$. It then delete edges $(a,b)$ and $(d,e)$ to form $(a,e)$ and $(b,d)$. When starting from an unclustered network, this process will lead to at least one extra $G_\\triangle$ being created~\\cite{bansal2009exploring}. This is repeated until the desired level of clustering is achieved. It is possible to include a Metropolis-style augmentation whereby at each step the local clustering coefficient is computed for the five nodes before and after rewiring, and the rewired configuration is only accepted if it results in an increase in average local clustering. It is worth noting that this algorithm leads to a positive degree-degree correlation which was not necessarily present in the original network.\n\nIn this paper, we use the Big-V algorithm to demonstrate that our newly proposed algorithms are able to sample from a larger part of the state space of all possible networks with a given degree sequence and global clustering coefficient.\n\n\\subsection{Models of contagion}\\label{sec:contagion}\n\nIn order to illustrate the impact of network structure -- and higher-order structure particularly -- different epidemic dynamics were simulated on the generated networks. Three different models were chosen: Susceptible-Infected-Susceptible ($SIS$), Susceptible-Infected-Recovered ($SIS$) \nand complex contagion \\cite{miller2015complex, osullivan2015mathematical}. To simulate $SIS$ and $SIR$ dynamics, the fully susceptible network of nodes is perturbed by infecting a small number of nodes. Infected nodes spread the infection to susceptible neighbours at a per-link rate of infection $\\tau$. Infected individuals recover independently of the network at rate $\\gamma$ and become  susceptible again (for $SIS$ dynamics) or become removed (for $SIR$ epidemics). In contrast to the infection process in the previous two dynamics, the complex contagion process requires that susceptible nodes are exposed to multiple infectious events before becoming infected. These events must be from different infectious neighbours as only the first infection attempt from an infectious node counts.  This critical infection threshold for each node is set in advance and is usually bounded from above by the degree of the node. To simulate the complex contagion dynamics, nodes are allocated infection thresholds $r_i\\in \\mathbb{N}$, where $i=1,2\\dots, N$, and the fully susceptible population of nodes is perturbed by infecting an initial number of nodes chosen at random. In this model a susceptible node $i$ becomes infected as soon as it has received at least $r_{i}$ infectious contacts from $r_{i}$ distinct infectious neighbours. There is no recovery in this model and infected individuals remain infected for the duration of the epidemic. \n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Algorithm validation}\\label{sec:validation}\n\nTo validate our algorithms, we generated a number of networks with pre-specified degree distribution and subgraph set, as well as a multinomial distribution of subgraph corners or hyperstubs around nodes. We verified that the networks generated were as expected given the input. \n\nAs described in Section~\\ref{sec:MM} the algorithms preserve the degree sequence, permitting at most a single edge to be deleted if the degree sequence sums to an odd number. The ability to exercise control over the networks' subgraph topology is illustrated by Figure~\\ref{fig:example_nets}. Note that Figure~\\ref{fig:small_random} shows a \\emph{random} network that includes $G_\\triangle$ subgraphs. When constructing networks using the configuration model it is possible to create $G_\\triangle$ subgraphs with non-zero probability and this is to be expected~\\cite{newman2009networks}. However, this is a function of mean degree not network size, and this probability goes to zero with network size going to infinity. \n\n\\begin{table}[!htbp]\n\\centering\n\\begin{tabular}{ccccccccc}\\rowcolor{LightBlue} \n       & c4 & d4 & e4 & i4 & s4 & t3 & u3 & u4 \\\\ \\hline\nRandom & 0 & 0 & 42 & 17 & 446 & 6 & 482 & 1706 \\\\\\rowcolor{LightBlue} \nBig-V  & 1 & 23 & 10 & 10 & 212 & 7 & 386 & 1220 \\\\\nUDA    & 7 & 10 & 22 & 5 & 243 & 1 & 389 & 1239 \\\\\\rowcolor{LightBlue} \nCMA   & 0 & 9  & 10 & 40 & 185 & 24 & 389 & 1201\n\\end{tabular}\n\\caption{Subgraph counts for the networks of Figure~\\ref{fig:example_nets}. Note: if one adds a single $G_\\triangle$ so that it shares a single edge with a $G_\\boxslash$ and this edge is not the diagonal edge of $G_\\boxslash$, then $d4$ increases by one but $t3$ will have only increased by one, not two. We note that $2\\cdot d3$ yields the maximum number of possible $G_\\triangle$ induced by $G_\\boxslash$. In general, calculating the number of $G_\\triangle$ in this way will always yield the maximum possible count but not necessarily the true count because a single $G_\\triangle$ could be shared by more than one $G_\\boxslash$.}\n\\label{tab:example_counts}\n\\end{table}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{small_random}\n  \\caption{Random}\n  \\label{fig:small_random}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{small_bigv}\n  \\caption{Big-V, C=0.22}\n  \\label{fig:small_big_v}\n\\end{subfigure}\n\\vskip\\baselineskip\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{small_c4}\n  \\caption{UDA, C=0.22}\n  \\label{fig:small_uda}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{small_card}\n  \\caption{CMA, C=0.22}\n  \\label{fig:small_card}\n\\end{subfigure}\n\\caption{Small networks generated by the Big-V, UDA and CMA algorithms. All networks have the same homogeneous degree sequence with $k=5$. The Big-V algorithm re-wired the random network, Figure~\\ref{fig:small_random}. The UDA was parametrised with subgraphs $G_0$ $G_\\boxslash$ and $G_\\boxtimes$. The CMA was parametrised so that every node was incident to 2 $G_\\triangle$. The Big-V, UDA, and CMA networks all have a global clustering coefficient of $C=0.22$. The network nodes are coloured so that green\/orange\/pink denotes nodes of low\/medium\/high clustering, respectively.}\n\\label{fig:example_nets}\n\\end{figure}\n\nTo properly demonstrate the proposed algorithms' control over the building blocks in the network, we used a recently described subgraph counting algorithm~\\cite{ritchie2014higher} to count the number of subgraphs \\emph{a posteriori}. In our implementation we counted subgraphs composed of 4 nodes or less -- see the top two rows of Figure~\\ref{fig:examples}, as well as 5- and 6-cycles. Table~\\ref{tab:example_counts} provides the subgraph counts for the networks displayed in Figure~\\ref{fig:example_nets}. It confirms that the random network given in Figure~\\ref{fig:small_random} contains 6 $G_\\triangle$, counted uniquely, as observed above. The table also reveals that, through increasing the frequency of $G_\\triangle$, the Big-V algorithm also introduced $G_\\boxslash$ and $G_\\boxtimes$ subgraphs. The UDA was parametrised with $\\{G_0, G_\\boxslash, G_\\boxtimes\\}$ and the table confirms a significant presence of these subgraphs when compared to the random network. Although the CMA was parametrised solely with $G_\\triangle$ subgraphs distributed so that each node was incident to 2 $G_\\triangle$ subgraphs, the subgraph counts reveal that this network contains 9 $G_\\boxslash$ subgraphs. This is a consequence of attempting to generate \\emph{small} networks with such a high prevalence of triangles: it is highly likely that the algorithms will select nodes that already share one other common neighbour later in the connection process. One expects the proportion of these events to become increasingly negligible with greater network size.   \n\nNext, we used the above motif counting algorithm to evaluate the extent to which the proposed algorithms can exert control over the prevalence of subgraphs in the generated networks. Figure~\\ref{fig:homo_k5_counting} compares \\emph{measured} counts of subgraphs in UDA and CMA networks with \\emph{expected} counts. Here, an important observation must be made at the outset. Even in random networks, cycles ($G_\\boxempty$, $G_{\\pentagon}$ and $G_{\\hexagon}$) appear in significant quantities: 33, 100 and 333 times respectively, and regardless of network size. They are a natural consequence of the fact that the probability of selecting two nodes in different branches of a finite tree-like network is non-zero. Therefore, our \\emph{expected} counts are the sum of the counts expected \\emph{by construction} and those \\emph{measured} in the random networks. For example, since the CMA networks were generated with each node being incident to a single $G_{\\hexagon}$ subgraph, a total of 833 uniquely counted $G_{\\hexagon}$ subgraphs were expected \\emph{by construction} in networks of size $N=5000$. However, because an average of 344 $G_{\\hexagon}$ subgraphs were counted in random networks of size $N=5000$, our \\emph{expected} count was $833+344=1177$. The \\emph{measured} count was found to be 1165. More generally, we found the \\emph{expected} counts to match well with the \\emph{measured} counts, indicating that the generating algorithms did not create by-products in addition to those observed at random\\footnote{Although we will show in Section~\\ref{sec:UDA_CMA} that for specific parameterisations of CMA, by-products are possible.}. However, these results also suggest that the level of control exerted by the algorithms over subgraph prevalence depends on how often those subgraphs appear naturally as by-products. Control is strongest for subgraphs that do not appear naturally as by-products. When considering subgraphs that appear naturally with high frequency, e.g., $G_{\\pentagon}$, real control over their prevalence can only be achieved if an even higher frequency is imposed, which may not always be possible for a given degree sequence and global clustering. \n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.49\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{rand_and_uda}\n  \\caption{UDA}\n  \\label{fig:rand_and_uda}\n\\end{subfigure}%\n\\begin{subfigure}{0.49\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{rand_and_card}\n  \\caption{CMA}\n  \\label{fig:rand_and_card}\n\\end{subfigure}\n\\caption{A comparison of subgraphs found in the UDA and CMA networks to their random network analogues and expected counts plotted with thick lines, thin lines and discrete markers respectively. $p5$ and $h6$ denote the counts of $G_{\\pentagon}$ and $G_{\\hexagon}$ respectively. All networks have the same homogeneous degree sequence with $k=5$ but with increasing size: $N=250,500,1000,2500,5000$, where 100 of each size was generated. (a) The UDA algorithm was parametrised with subgraphs $\\{G_\\triangle, G_\\boxempty, G_{\\pentagon}, G_{\\hexagon}\\}$, and the resulting average subgraph counts are shown on the left. (b) The CMA algorithm was parametrised so that each node was incident to a single $G_{\\pentagon}$ and $G_{\\hexagon}$ subgraph, and the resulting average subgraph counts are shown on the right. The expected values were calculated by summing the total counts from the subgraph sequences, dividing them by the subgraphs' node cardinality, and adding these figures to the number of subgraphs found as by-products in the random networks.}\n\\label{fig:homo_k5_counting}\n\\end{figure}\n\nIn what follows, we set out to highlight differences between the new algorithms compared to classic ones and also to emphasise the diversity within networks generated by the same algorithms.\n\n\\subsection{Sampling from a different area of the network state space}\\label{sec:UDA_Big_V}\n\nIn this section, we seek to highlight the versatility of the proposed generation mechanisms by showing that, given a degree distribution and a global clustering, they sample different areas of the network state space than existing methods such as Big-V. We begin by reminding the reader that the Big-V algorithm searches for paths of 5 nodes and rewires such paths so that additional triangles are created. In other words, the principal building block of this algorithm is the $G_\\triangle$ subgraph and subgraphs that may be constructed by overlapping $G_\\triangle$ subgraphs. It follows that this algorithm is unlikely to give rise to a higher than expected at random number of $G_\\boxempty$ or other `empty' cycles. The UDA algorithm was therefore parametrised with  subgraph family $\\{G_0, G_\\triangle, G_\\boxempty, G_{\\pentagon}, G_{\\hexagon} \\}$. In order to eliminate the effect of degree heterogeneity, a homogeneous degree sequence with $k=5$ was used. The resulting networks had a global clustering coefficient of $C=0.04$, induced by 666 (uniquely counted) $G_\\triangle$ subgraphs. We then used the Big-V algorithm to rewire random networks constructed using the same degree sequence until the desired level of clustering, $C=0.04$, was achieved. Significant differences between generated networks would confirm that the Big-V and UDA generated networks are sampled from different areas of the state space of networks satisfying that degree sequence and global clustering. As a further point of reference, data taken from a random network realisation of the degree sequence was included in all of our analyses. Henceforth we shall refer to these three types of networks as network family \\textbf{A}.\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_avpl}\n  \\caption{Average path length}\n  \\label{fig:homo_k5_cycles_avpl}\n\\end{subfigure}%\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_betw_avg}\n  \\caption{Average betweenness}\n  \\label{fig:homo_k5_cycles_avg_betw}\n\\end{subfigure}\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_betw_max}\n  \\caption{Maximum betweenness}\n  \\label{fig:homo_k5_cycles_max_betw}\n\\end{subfigure}\n\\caption{Plots of the average path length and diameter for homogeneous networks ($N=5000$ and $k=5$) for network family \\textbf{A}. The Big-V algorithm was parametrised solely by clustering, in this case $C=0.04$, to best suit the networks produced by the UDA. The differences in average path length, average betweenness centrality and maximum betweenness centrality between the random network and its Big-V analogue were of similar magnitude as the differences between the Big-V network and the cycle-based UDA networks, and these were significant.}\n\\label{fig:homo_k5_cycles_betw}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[scale=1]{homo_k5_cycles_sg_count}\n\\caption{Distributions of total number of subgraphs in network family \\textbf{A} ($N=5000$, $k=5$). The Big-V and UDA networks have a global clustering coefficient of $C=0.04$. All given counts are unique. The $t3$ counts denote the number of $G_\\triangle$ subgraphs that are not involved in any subgraphs of four nodes (i.e., $G_\\boxslash$ and $G_\\boxtimes$). However, the $c4$ and $d4$ counts may include $G_\\triangle$ subgraphs shared by $G_\\boxslash$ and $G_\\boxtimes$. The number of $G_\\boxempty$ subgraphs generated by the Big-V algorithm is very close to the counts found in random networks.}   \n\\label{fig:homo_k5_counts}\n\\end{figure}\n\nIn Figure~\\ref{fig:homo_k5_cycles_betw}, the distributions of the average path length, average betweenness centrality and maximum betweenness centrality for the above networks are given. In general, an increase in clustering results in a higher value of the average path length -- see the average path length of random and Big-V networks in Figure~\\ref{fig:homo_k5_cycles_avpl}. This is a known result ~\\cite{bansal2009exploring}. Surprisingly, a similar magnitude of difference in average path length and average and maximum  betweenness centrality is observed between the Big-V and UDA networks despite them having the same global clustering, see Figure~\\ref{fig:homo_k5_cycles_avpl}, ~\\ref{fig:homo_k5_cycles_avg_betw} and ~\\ref{fig:homo_k5_cycles_max_betw}, respectively. Output from the subgraph counting algorithm (Figure~\\ref{fig:homo_k5_counts}) confirms that, as expected, the Big-V algorithm does not generate more $G_\\boxempty$ subgraphs than are observed in the random network. More generally, the results show that the Big-V and UDA networks exhibit markedly different subgraph topologies with the Big-V networks relying heavily on $G_\\boxslash$ to cluster the networks unlike UDA networks that rely almost exclusively on $G_\\triangle$ not appearing as part of any other subgraph. It may be that such variation was facilitated by the low level of clustering considered, and that with higher clustering, eliciting such differences might be more challenging. However, these results provide evidence that the UDA algorithm can sample from a different part of the state space than the Big-V algorithm.\n\n\\subsection{Diversity within the newly proposed algorithms}\\label{sec:UDA_CMA}\n\nIn this section, we illustrate the diversity of networks generated with UDA and CMA by exploring the impact of subgraph distribution over nodes (for identical degree distribution and global clustering) and how it may change network characteristics. \n\nTo do this we first parametrised the UDA with subgraph family $\\{G_0, G_\\triangle, G_\\boxempty, G_\\boxslash, G_\\boxtimes\\}$ (chosen due to its frequent use in the literature, e.g., ~\\cite{ritchie2014beyond, ritchie2014higher, house2010generalised, house2009motif, bansal2009exploring, karrer2010random}), and a heterogeneous degree sequence generated using the Poisson distribution with $\\lambda = 5$. Since it is difficult to control for the number of subgraphs that appear in a network generated using the UDA we counted the total number of each subgraph, from UDA-produced subgraph sequences, and used these counts to create alternative subgraph sequences as input to the CMA, see Section~\\ref{sec:CMA}, rather than drawing such sequences from a theoretical distribution. The resulting networks were therefore  expected to have identical degree sequence, global clustering of 0.13 and subgraph counts. Since the CMA allows us to choose arbitrary sequences of subgraphs, we opted to push the clustered subgraphs, $\\{ G_\\triangle, G_\\boxslash, G_\\boxtimes\\}$, onto the higher-degree nodes to accentuate the effect of clustering. We did this by specifying that these subgraphs had to appear with multiplicity greater than one. For example, a degree-three $G_\\boxtimes$ hyperstub required a minimum $k=9$-degree node. As previously, we included a random network realisation of the heterogeneous degree sequence for comparison. Henceforth, we shall refer to these three types of networks as network family \\textbf{B}. \n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_ass}\n  \\caption{Assortativity}\n  \\label{fig:pois_k5_ass}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{k5_pois_semilog}\n  \\caption{Degree dependent clustering}\n  \\label{fig:pois_k5_ddc}\n\\end{subfigure}\n\\caption{Plots of assortativity and degree-dependent clustering for network family \\textbf{B} with $k \\sim Pois(5)$. The UDA and CMA networks have a global clustering coefficient of $C=0.13$. The distribution of subgraphs in CMA networks was manipulated so that the clustered subgraphs $\\{G_\\triangle, G_\\boxslash, G_\\boxtimes\\}$ appeared around nodes with multiplicity greater than one. In order to preserve the subgraph degree sequence these aggregated subgraphs were allocated to the higher degree nodes, resulting in higher assortativity and a more positively skewed distribution of degree-dependent clustering. The dash-dotted line corresponds to $c(k)=k^{-1}$.}\n\\label{fig:pois_k5_hetm}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_avpl}\n  \\caption{Average path length}\n  \\label{fig:pois_k5_avpl}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_diam}\n  \\caption{Diameter}\n  \\label{fig:pois_k5_diam}\n\\end{subfigure}\n\\caption{Plots of average path length and diameter for network family \\textbf{B} with $k \\sim Pois(5)$. The UDA and CMA networks have a global clustering coefficient of $C=0.13$. The increased average path length and diameter between the UDA and random networks is attributable to the higher clustering. The similar increase between UDA and CMA networks is a reflection of the higher assortativity of the CMA networks.}\n\\label{fig:pois_k5_pl}\n\\end{figure}\n\nThe heterogeneity in degree distribution allows us to use additional degree-dependent metrics: degree-degree correlations and degree-dependent clustering~\\cite{newman2002assortative, serrano2005tuning}. These have been plotted in Figure~\\ref{fig:pois_k5_hetm}. The plot for the degree-degree correlation coefficient shows that by aggregating clustered subgraphs around high-degree nodes, the CMA-constructed networks yield a higher assortativity than that of UDA and random networks, see Figure~\\ref{fig:pois_k5_ass}. This is an important property of the methodology since the clustering potential of a network is bounded by the degree-degree correlation coefficient~\\cite{serrano2005tuning}. Moreover, if one wishes to maximise clustering in heterogeneous networks, it is necessary for nodes of similar degree to mix preferentially. Figure~\\ref{fig:pois_k5_ddc} shows that the CMA networks yield a negatively skewed distribution of degree-dependent clustering, with nodes of degree $k\\geq 9$ contributing most to clustering. The ability to manipulate the degree and clustering relationship as well as assortativity clearly demonstrates the broader scope of the CMA when sampling from the ensemble of networks with same degree distribution and global clustering.   \n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_min_betw}\n  \\caption{Minimum}\n  \\label{fig:pois_k5_min_betw}\n\\end{subfigure}%\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_avg_betw}\n  \\caption{Average}\n  \\label{fig:pois_k5_avg_betw}\n\\end{subfigure}\n\\begin{subfigure}{0.31\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_max_betw}\n  \\caption{Maximum}\n  \\label{fig:pois_k5_max_betw}\n\\end{subfigure}\n\\caption{Plots of betweenness centrality for network family \\textbf{B} with $k \\sim Pois(5)$. The UDA and CMA networks have a global clustering coefficient of $C=0.13$. A trend of increasing average and maximum betweenness centrality is observed between random, UDA and CMA networks, respectively.}\n\\label{fig:pois_k5_betw}\n\\end{figure}\n\nAs with network family \\textbf{A}, an increase in average path length, diameter, average and maximum betweenness centrality of UDA and CMA networks over random networks will be attributable to the increased global clustering coefficient, $C=0.13$, see Figure~\\ref{fig:pois_k5_pl} and ~\\ref{fig:pois_k5_betw}. However, since UDA and CMA networks share the same degree sequence and global clustering coefficient differences in these metrics between UDA and CMA can only be due to increased degree-degree correlation and negatively skewed distribution of degree-dependent clustering. It has previously been noted that increased assortativity corresponds to an increase in average path length \\cite{xulvi2004reshuffling} and this will be compounded by the higher-degree nodes (which inevitably serve as central hubs) being more clustered. Similarly, an increase in diameter (a function of path length) will be due to these highly clustered high-degree nodes. Finally, Figures~\\ref{fig:pois_k5_avg_betw} and~\\ref{fig:pois_k5_max_betw} show a significant increase in average and maximum betweenness centrality between UDA and CMA networks. This is yet another manifestation of the presence of these highly-clustered high-degree nodes.\n\nTable~\\ref{tab:countsB} presents a comparison between \\emph{measured} and \\emph{expected} average subgraph counts for the networks in family \\textbf{B}.\nWhereas there is good agreement for UDA networks, it is observed that CMA networks have produced by-products other than what was expected at random, e.g., an additional 50\\% $G_\\boxslash$ have appeared as by-products. The effects of finite size have been exacerbated by aggregating clustered subgraphs around higher degree nodes, effectively excluding lower to medium degree nodes during this part of the connection process. Within this densely connected component it is easy to envisage a situation where adding only a single edge may create additional (unwanted) subgraphs. This highlights the fact that whilst the total number of $G_\\triangle$ is preserved (as evidenced by identical global clustering), the way these subgraphs contribute to higher-order structure can vary significantly.\n\n\\begin{table}[!htbp]\n\\centering\n\\begin{tabular}{ccccc}\\rowcolor{LightBlue} \n   & c4 & d4 & e4 & t3 \\\\ \\hline \nRandom & 0   & 0 & 79 & 21 \\\\ \\rowcolor{LightBlue}\nUDA & 243   & 504 & 587 & 718 \\\\\nCMA & 232  & 743 & 772 & 691\\\\ \\rowcolor{LightBlue}\nExpected & 243  & 504 & 619 & 741\n\\end{tabular}\n\\caption{Subgraph counts for network  \\textbf{B} ($N=5000$, $k \\sim Pois(5)$ and $C=0.13$). The counts are unique. The expected counts are computed by summing the total counts from the subgraph sequences, dividing them by the subgraphs' node cardinality, and adding these figures to the number of subgraphs found as by-products in the random network. The counts for $t3$ are for $G_\\triangle$ subgraphs that do not appear in any other subgraphs.}\n\\label{tab:countsB}\n\\end{table}\n\nThis Section has highlighted that control over the choice of subgraph families and their distributions makes it possible to flexibly explore the solution space of networks with the same degree distribution and global clustering. This in turn provides us with the means to investigate specific areas of this solution space as well as further our understanding of how network metrics deal with such diversity.\n\n\\subsection{Does higher-order structure matter?}\\label{sec:SIS}\n\nIn order to answer this question we make use of the network families \\textbf{A} and \\textbf{B} detailed above and test the impact of higher-order structure by considering the outcome and evolution of widely used dynamics on networks, namely, $SIS$, $SIR$ and the complex contagion model.  \n\nFor each network type in families {\\textbf A} and {\\textbf B} a series of networks were generated. For each network we performed a single Gillespie realisation of the $SIS$, $SIR$ and complex contagion epidemics. The mean time evolution of infectious prevalence was then calculated, plotted and compared between network types. Complex contagion dynamics was simulated in a similar way but without recovery and remembering that a single infectious contact was usually not sufficient to result in an infected node. Different thresholds of infection and infectious seeds were used and these are specified in figure captions. Matlab code for the $SIS$ and $SIR$ Gillespie algorithms is available from \\url{https:\/\/github.com\/martinritchie\/Dynamics}.\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_SIS}\n  \\caption{SIS}\n  \\label{fig:k5_homo_cycles_SIS}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_SIR}\n  \\caption{SIR}\n  \\label{fig:k5_homo_cycles_SIR}\n\\end{subfigure}\n\\vskip\\baselineskip\n\\begin{subfigure}{0.9\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{homo_k5_cycles_cc}\n  \\caption{Complex contagion}\n  \\label{fig:homo_k5_cycles_cc}\n\\end{subfigure}\n\\caption{Epidemic dynamics for network family \\textbf{A} with $k=5$. The Big-V and CMA networks have a global clustering coefficient of $C=0.04$. In (a) and (b) the orange line, blue circles and green vertical markers correspond to the random, Big-V and UDA networks respectively. In (c) and (d) the same colour scheme is used but with bars. The $SIS$ and $SIR$ epidemics represent the average of single Gillespie simulations on each of the 1000 network realisations from each network generation algorithm. The $SIS$ and $SIR$ epidemics were seeded with an initial infectious seed of $I_0=10$ and had a per link rate of infection of $\\tau=1$ and recovered independently at rate $\\gamma=1$. The complex contagion epidemics had an initial infectious seed of $I_0=250$ and a fixed threshold of infection of $r=2$.}\n\\label{fig:k5_homo_cycles_epi}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_SIS}\n  \\caption{SIS}\n  \\label{fig:pois_k5_sis}\n\\end{subfigure}%\n\\begin{subfigure}{0.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_SIR}\n  \\caption{SIR}\n  \\label{fig:pois_k5_sir}\n\\end{subfigure}\n\\vskip\\baselineskip\n\\begin{subfigure}{0.9\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{pois_k5_classical_cc}\n  \\caption{Complex contagion}\n  \\label{fig:pois_k5_classical_cc}\n\\end{subfigure}\n\\caption{Epidemic dynamics for network family \\textbf{B} with $k \\sim Pois(5)$. The UDA and CMA networks have a global clustering coefficient of $C=0.13$. In (a) and (b) the orange line, blue circles and green vertical markers correspond to the random, UDA and CMA networks respectively. In (c) and (d) the same colour scheme is used for the bars. The $SIS$ and $SIR$ epidemics represent the average of single Gillespie simulations on each of the 1000 network realisations from each network generation algorithm. The $SIS$ and $SIR$ epidemics were seeded with an initial infectious seed of $I_0=10$ and had a per link rate of infection of $\\tau=1$ and recovered independently at rate $\\gamma=1$. The complex contagion epidemics had an initial infectious seed of $I_0=1000$ and a fixed threshold of infection of $r=3$. In the $SIS$ and $SIR$ dynamics the effect of clustering on the high degree nodes, inhibiting disease propagation, dominates the epidemiological encouraging effect of increased assortativity.}\n\\label{fig:pois_k5_epi}\n\\end{figure}\n\nWe know by construction that members of network family \\textbf{A} were generated using different subgraphs and Section~\\ref{sec:UDA_CMA} has shown that observable differences were found between networks in terms of average path length, betweenness centrality and subgraph composition. Despite this, Figures~\\ref{fig:k5_homo_cycles_SIS} and~\\ref{fig:k5_homo_cycles_SIR}, which show the time evolution for $SIS$ and $SIR$ dynamics respectively, illustrate that these dynamics can display a certain degree of insensitivity to these differences in structure. In this case, it is the $SIR$ dynamics that show the greatest difference, in peak infectious prevalence (Figure~\\ref{fig:k5_homo_cycles_SIR}) albeit quite marginal. In contrast, complex contagion dynamics do show sensitivity to structural differences found between Big-V and UDA networks. Figure~\\ref{fig:homo_k5_cycles_cc} reveals that for Big-V networks the epidemic fully percolates in almost 100\\% of the simulations instead of only 80\\% of the cases for UDA networks. This indicates that whilst Big-V networks operate in the super critical regime, UDA networks are closer to the transition point. Locating this transition is possible but is beyond the scope of this paper. \n\nWhen network family \\textbf{A} is used, the networks' degree distribution and clustering appear to be the main determinants of the time evolution and outcome of the $SIS$ and $SIR$ epidemics. In contrast, when network family \\textbf{B} is used, Figure~\\ref{fig:pois_k5_epi} shows that all dynamics considered are impacted by differences in network topology. For Figures~\\ref{fig:pois_k5_sis} and~\\ref{fig:pois_k5_sir}, a trend of inhibited spread of infection is observed from the random to UDA to CMA networks. It has already been shown that clustering slows the spread of infection~ \\cite{keeling1999effects,green2010large}, and we see that this effect dominates over higher assortativity, which usually leads to faster initial spread of the epidemic \\cite{kiss2008effect}. Similarly, Figure~\\ref{fig:pois_k5_classical_cc} which shows the distribution of the final epidemic size for the complex contagion dynamics reveals that: (a) the higher clustering observed in the UDA networks fails to have a significant impact when compared to the random network equivalent and (b) the CMA networks significantly slow the pace of the epidemic as well as reduce its final size compared to both random and UDA networks. Hence, for the UDA and CMA networks where both degree distribution and global clustering are identical the observed differences are explained by the combined effect of varying distributions of subgraph around nodes and varying prevalence of subgraphs (both of which are related to one another to some extent) as shown by\n    Table~\\ref{tab:countsB}.\n\nTaken together, our simulation data shows that even though the proposed algorithms construct networks with identical degree sequence and global clustering, these networks can give rise to measurable differences in resulting epidemics, be it it time evolution or final outcome. With the exception of $SIS$ and $SIR$ epidemics on network family \\textbf{A} (still with some small differences) we found significant differences in all other instances. A more systematic investigation of more network models and wider parameter range for the dynamics is needed but left to future work.\n\n\\section{Discussion}\\label{sec:discussion}\n\nIn this paper, we have described two novel network generating algorithms that strictly preserve a given degree sequence whilst permitting control over the building blocks of the network and enabling tuning of global clustering. We have compared these algorithms to one another as well as to the widely used Big-V rewiring algorithm. Using our algorithms we have empirically demonstrated that it is possible to create networks that are identical with respect to degree sequence and global clustering, yet elicit significant differences in network metrics and in the outcome of dynamical processes unfolding on them. We have presented evidence to suggest that the methods sample from different areas of the network state space and that these sampling variations do matter. \n\nOf the two algorithms proposed, UDA is the simplest to use. It requires less input and is conceptually elegant. We believe that this algorithm, when parametrised with complete subgraphs, would more likely yield analytical results due to its combinatorial nature. Note that whilst varying levels of clustering can be achieved and estimated before network construction it is not possible to target a specific level of clustering, due to the emergent nature of the distribution of subgraphs around nodes. A second potential limitation of this algorithm is its dependence on solving the underdetermined Diophantine equations that reside in high dimensional spaces. Computationally, it may become difficult to include large families of subgraphs. However, we did not encounter such problem in our experiments.\n\nThe CMA algorithm is more complex but also more versatile. Being able to specify distributions of subgraphs alongside a given degree sequence, and preserve both, is highly novel. Fixing the degree sequence allows for some interesting ways to construct the subgraph sequences. Knowing the number of nodes in each degree class $k$ allows us to combine such nodes to form complete subgraph with $k$ nodes, as this requires ($k-1$) links. The remaining single edges can then be used to connect to the rest of the network. We used this for the heterogeneous degree sequence presented in the results section, yielding a network with a global clustering coefficient of $C=0.67$ and a giant component of $N\\approx 4800$ out of $N=5000$ nodes. We were unable to achieve such high clustering with either Big-V or UDA algorithms. It must be noted that in our application of this algorithm, we had the luxury of using hyperstubs sequences we knew to be graphical -- them being output from the UDA -- to guide how we parametrised the CMA. In general, the stub sequences induced by the hyperstub sequences would have to be constrained to ensure that they are graphical. This is possible but must be taken into account when considering applying this algorithm. \n\nWe have shown that despite identical degree distribution and global clustering, significant diversity in networks can still be elicited. This has occurred in two ways: (1) by construction, by redistributing the same number of subgraphs and (2) unexpectedly, through the emergence of by-products. We conjecture that any controlled -- or believed to be controlled -- network generation algorithm will yield by-products, unless heuristic constraints are introduced to reduce the likelihood of subgraphs sharing lower-order subgraph components for example. As witnessed in our results, even configuration model networks lead to a large number of loops with 4, 5, and 6 nodes (longer cycles were not measured). This problem can only be exacerbated when control of more sophisticated structures is implemented. As such, care has to be taken when parametrising algorithms. For example, one would need to specify a relatively large number $G_{\\hexagon}$ subgraphs in a network's construction to impact the subgraph count beyond what one would observe by chance in a random network. More surprisingly, as we witnessed with $G_\\boxslash$ subgraphs in the CMA networks from network family \\textbf{B}, significant numbers of subgraph by-products can appear in addition to what was observed in the random networks depending on how one wishes to place the subgraphs around nodes.  \n\nWe have seen that by using a very modest selection of subgraphs, we have been able to substantially influence dynamics running on the network, particularly complex contagion dynamics. All results relating to this model indicate that constraining a network by degree sequence and clustering is not sufficient to accurately predict the course of the epidemic. More importantly, the results appear to suggest that the location of the critical regime depends on the higher-order structure of the network (above and beyond clustering). \n\nBeing able to generate networks with different structural properties or higher-order structure is a key feature of any network construction algorithm. However, if such structural details do not impact on dynamics unfolding on the network, then models for such dynamics can rely with high confidence on a limited set of network descriptors. Although degree sequence, degree-degree correlations and global clustering coefficient were observed to be the main drivers of disease transmission in models such as $SIS$ and $SIR$, we found it not to be true in general. This is an important finding because one should remember that the dynamics simulated here are modest in complexity, when compared to models of neuronal dynamics for example, and yet, we were able to elicit significant differences by simply tuning the network structure above and beyond triangles. This implies that determining the role and impact of higher-order structure may yet hold many important and surprising answers.  \\\\\n\n\n\\noindent{\\textbf{Acknowledgements:}} Martin Ritchie gratefully acknowledges EPSRC (Engineering and Physical Sciences Research Council) and the University of Sussex for funding for his PhD. We would also like to thank Dr J.C. Miller for useful discussions on the complex contagion model~\\cite{miller2015complex}, and for sharing his code for simulating the complex contagion model on networks~\\cite{miller2015private}.\n\n\n\n\\clearpage\n\n\\section{Appendix}\\label{sec:app}\n\n\\subsection{Pseudocode for UDA}\\label{UDA}\n\\begin{algorithm}[!htbp]\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{yellow!30}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\Input{$D=(d_1,d_2,\\dots,d_N)$, $G=\\{G_1,G_2,\\dots,G_l\\}$}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{yellow!30}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\Output{$H \\in \\mathbb{N}_0^{l \\times N}$.}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{blue!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\KwSty{Variables} \\\\\n$D$: degree sequence, $N$: number of nodes, \\\\\n$G$: set of subgraphs, $l$: number of subgraphs, \\\\\n$g_i$: subgraph adjacency matrix, $X_k$: solution space for degree $k$, \\\\\n$H$: hyperstub degree sequence \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{orange!50}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\KwSty{Procedure} \\\\\n\\For{Each subgraph, $G_i$}{\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Identify the degree sequences of the subgraphs.} \\\\\n$s_i = \\sum g_i$ \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Take the unique elements.} \\\\\n$s_i = unique(s_i)$} \n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Concatenate into a single vector.} \\\\\n $S = (s_1, s_2, \\dots s_l)$ \\\\ \n \\For{$k=1,2,\\dots k_{max}$}{\n \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% $X_k(i,:)$ denotes a hyperstub arrangement for a degree $k$ node.} \\\\\n $X_k = diorecur(S,k)$ \\\\\n }\n \\For{n = 1,2,\\dots, N}{\n \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Take random element from the solution space.} \\\\\n $r = rand$; $h_n =  X_{D(n)}(r,\\cdot)$}\n \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Concatenate into a single matrix.} \\\\\n $H = (h_1, h_2, \\dots, h_l)$ \\\\ \n \\nl \\KwRet\n\\caption{Pseudocode for the underdetermined network generation algorithm (UDA). This pseudocode focuses on the salient points of the UDA, namely, how the algorithm draws solutions from the solution space of an underdetermined Diophantine equation to determine the arrangement of hyperstubs around a particular node. Other steps, such as ensuring the handshake lemma is satisfied for both lines and subgraphs, are detailed in Section~\\ref{sec:UDA} and can be viewed in the source code. The output hyperstub degree sequence $H$ must be used as input for a modified configuration model connection process to realise a network, see Section~\\ref{sec:CP}.}\\label{algo:UDA}\n\\end{algorithm}\n\n\\clearpage\n\n\\subsection{Pseudocode for CMA}\\label{CMA}\n\\IncMargin{1em}\n\\begin{algorithm}[!htbp]\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\\SetKwFunction{FRecurs}{FnRecursive}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{yellow!30}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\Input{$D=(d_1,d_2,\\dots,d_N)$, $G=\\{G_1,G_2,\\dots,G_l\\}$, $S=\\{S_1,S_2,\\dots,S_l\\}$.}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{yellow!30}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\Output{$H \\in \\mathbb{N}_0^{|s| \\times N}$.}\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{blue!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\KwSty{Variables} \\\\\n$D$: degree sequence, $N$: number of nodes, \\\\\n$G$: set of subgraphs, $l$: number of subgraphs, \\\\\n$S$: subgraph sequence, $g_i$: subgraph adjacency matrix, \\\\\n$|s|$: number of unique corners in a subgraph, $H$: hyperstub degree sequence \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{orange!50}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}\\KwSty{Procedure} \\\\\n\\For{Each subgraph, $G_i$}{\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Identify the degree sequence, $s$, of the subgraph.} \\\\\n$s_i = \\sum g_i$, $s_i = unique(s_i)$, $m = length(s_i)$ \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% $p$ reflects the proportions of hyperstubs} \\\\\n$p_i = (p_1,p_2,\\dots,p_m)$ \\\\\n\t\\For{$j = 1,2,\\dots,N$}{\n    \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% The subgraph sequence is decomposed into a hyperstub} \\\\\n    \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% sequence using the multinomial distribution, $M$, } \\\\\n    \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% so that $H_i \\in \\mathbb{N}_0^{m \\times N}$  } \\\\\n    $H_i(j) = M(S_i(j),p_i)$, \\\\}\n    \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% $H'_i$ is a sequence of the true stub count } \\\\\n    $H'_i=H_i \\cdot s_i$\\\\\n    \\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Sum so that $H'_i \\in \\mathbb{N}_0^{1 \\times N}$ } \\\\\n    $H'_i(j) = \\sum_{\\alpha=1}^{m}H'_i(\\alpha,j)$ \\\\\n}\n\\While{elements of each $H_i$ are non-zero}{\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Find the largest subgraph degree, } \\\\\n$h_i(j) = \\max \\{\\max \\{H'_1\\}, \\max \\{H'_2 \\}, \\cdots, \\max \\{H'_l\\} \\}$ \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% i.e., the $j^{th}$ element of $H_i$. } \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% Find all elements of the degree sequence at least this large and } \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% select an element from $d'$ at random} \\\\\n$d' = \\{d \\in D: d \\geq m \\}$, $\\delta = d'(random)$ \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% pair $H_i(j)$ to $\\delta$ and update} \\\\\n\\leavevmode\\rlap{\\hbox to \\hsize{\\color{green!20}\\leaders\\hrule height .8\\baselineskip depth .5ex\\hfill}}{\\% $\\delta$'s available degree and $H_i$} \\\\\n$\\delta = \\delta - H_i(j)$,~ $H_i(j) = 0$ \\\\\n}\n\\caption{Pseudocode for the cardinality matching algorithm (CMA). Other steps, such as ensuring the handshake lemma is satisfied for both lines and subgraphs, are identical to what is used for the UDA and are detailed in Section~\\ref{sec:UDA} and can be viewed in the Matlab source code. The output hyperstub degree sequence $H$ must be used as input for a modified configuration model connection process to realise a network, see Section~\\ref{sec:CP}.}\n\\end{algorithm}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPrincipal Component Analysis (PCA) is a celebrated dimension reduction method. It was first described by \\cite{pearson1901}; and it was developed further by several authors \\citep[see \\emph{e.g.}][and references therein]{jolliffe1986}. In a nutshell, PCA summarises high-dimensional data $(x_1,...,x_m)\\in\\mathbb{R}^d$, $m\\in\\mathbb{N}^*$, into a smaller space, which is designed to be `meaningful' and more easily interpretable. By `meaningful' we mean that this new subspace still captures efficiently the correlations between data points, while at the same time reducing drastically the dimension of the space. A popular tool to design this meaningful subspace is the \\emph{Gram Matrix} of the data, defined as $(\\langle x_i,x_j\\rangle)_{i,j}$. PCA then considers the eigenvectors of this matrix. Note that this is a linear operation, in the sense that PCA consists of an orthogonal transformation of the coordinate system in which we describe our data, followed by a projection onto the first $k$ directions in the new system, corresponding to the largest $k$ eigenvalues of the Gram matrix.\n\nOver the past two decades, PCA has been studied and enriched \\citep[\\emph{e.g.}, principal curves as a nonlinear extension of PCA, as done by][]{guedj2018sequential}. The particular extension of PCA that we focus on is 'kernel PCA' \\citep[which may be traced back to][]{scholkopf1998}. Using a kernel, we map our data \ninto a reproducing kernel Hilbert space\\footnote{We refer the reader to  \\cite{hein2004kernels} or  \\cite{hofmann2005tutorial} for an introduction to RKHS and their uses in machine learning.} (RKHS).\nThe linear PCA then operates in this Hilbert space to yield a finite-dimensional subspace onto which we project new data points. The final step is to assess how close from the original data is this projection. Kernel PCA is widely used in the machine learning literature \\citep[\\emph{e.g.},][to name but a few recent works]{kim2020,xu2019,vo2016} which makes the need of a better theoretical understanding even more pressing.\n\nA first theoretical study has been made in \\cite{shawe-taylor2005eigenspectrum} who derived PAC (Probably Approximately Correct) guarantees for kernel PCA. The PAC bounds proposed by \\cite{shawe-taylor2005eigenspectrum} were set up to control the averaged projection of new data points onto a finite-dimensional subspace of the RKHS into which data is embedded.\n\n\nBounds in \\cite{shawe-taylor2005eigenspectrum} include a Rademacher complexity. Rademacher complexity terms are known to be challenging to compute in many settings as they typically blow up combinatorially. To provide more numerically friendly results, we investigate a different route than \\cite{shawe-taylor2005eigenspectrum} and introduce the first PAC-Bayesian study of kernel PCA which, as a byproduct, allows to replace the Rademacher term by a Kullback-Leibler divergence (which has a closed form when distributions are Gaussian, and can be approximated by Monte Carlo in other cases). PAC-Bayes theory is a powerful framework to study generalisation properties of randomised predictors, and was introduced in the seminal works of \\cite{shawe1997pac,McAllester1998,McAllester1999}. PAC-Bayes theory has then been developed further by \\cite{seeger2002,McAllester2003,maurer2004note,catoni2007} among others. PAC-Bayes has emerged in the past few years as one of the promising leads to study generalisation properties of deep neural networks \\citep{conf\/uai\/Dziugaite2017,letarte2019}. A surge of recent papers in a variety of different settings illustrates the flexibility and relevance of PAC-Bayes as a principled tool  \\citep{conf\/icml\/Amit2018,conf\/icml\/Dziugaite2018,conf\/nips\/Dziugaite2018,conf\/nips\/Rivasplata2018,rivasplata2020, holland2019,haddouche2020,nozawa2020,mhammedi19pac,mhammedi2020pacbayesian,cantelobre2020pacbayesian}.\nWe refer to the recent survey \\cite{guedj2019primer} and tutorial \\cite{GueSTICML}, and the paper \\cite{rivasplata2020}, for details on PAC-Bayes theory.\n\n\n\\paragraph{Our contributions.} We aim at PAC and PAC-Bayesian bounds on the performance of kernel PCA.\nWe provide empirical PAC bounds which improve on those from \\cite{shawe-taylor2005eigenspectrum}. We introduce the first PAC-Bayes lower and upper bounds for kernel PCA, which clarify the merits and limitations of the overall method. These results are unprecedented, to the best of our knowledge.\n\n\n\\paragraph{Outline.} We introduce our notation and recall existing theoretical results on kernel PCA in \\cref{sec:notation}. \\cref{sec:pac} contains two new PAC bounds for kernel PCA, and \\cref{sec:pacb} is devoted to two new PAC-Bayes bounds, along with our proofs. The paper closes with a brief illustration of the numerical value of our bounds (\\cref{sec: experiments}) and concluding remarks (\\cref{sec:end}). We gather proofs of technical results in \\cref{sec:proofs}.\n\n\n\\section{Notation and preliminaries}\n\\label{sec:notation}\n\nWe let $\\mathbb{R}^{m \\times n}$ denote the space of matrices of shape $m \\times n$ with real entries. The data space is $\\mathcal{X} \\subseteq \\mathbb{R}^d$. We assume to have access to $s = (x_1, \\ldots, x_m) \\in \\mathcal{X}^m$, a realisation of the size-$m$ random vector $S = (X_1, \\ldots, X_m) \\in \\mathcal{X}^m$.\n\nWe let $\\mathcal{M}_1(\\mathcal{X})$ denote the space of probability distributions over $\\mathcal{X}$ and $\\mu \\in \\mathcal{M}_1(\\mathcal{X})$ stands for the distribution that generates one random example $X \\in \\mathcal{X}$. Its empirical counterpart is given by $\\hat{\\mu} = \\frac{1}{m}\\sum_{i=1}^{m} \\delta_{X_i}$, \\emph{i.e.}, the empirical distribution defined by the random sample. We assume the collected sample to be independent and identically distributed (iid): $S \\sim \\mu^m$, where $\\mu^m = \\mu\\otimes\\cdots\\otimes\\mu$ ($m$ copies).\n\n$\\mathbb{E}_{\\nu}[f] = \\mathbb{E}_{X \\sim\\nu}[f(X)] = \\int_{\\mathcal{X}} f(x) \\nu(dx)$ denotes the expectation under $\\nu \\in \\mathcal{M}_{1}(\\mathcal{X})$, for $f : \\mathcal{X}\\to\\mathbb{R}$.\nWe denote by $\\mathcal{H}$ a (separable) Hilbert space, equipped with an inner product $\\langle \\cdot,\\cdot \\rangle$. We let $\\| u \\| = \\langle u,u \\rangle^{1\/2}$ be the norm of $u \\in \\mathcal{H}$.\nThe operator $P_V : \\mathcal{H} \\to \\mathcal{H}$ is the orthogonal projection onto a subspace $V$, and \n$P_{v} = P_{\\operatorname{span}\\{ v \\}}$. In what follows, $\\mathcal{F}$ is a set of predictors, and $\\pi,\\pi^0 \\in \\mathcal{M}_1(\\mathcal{F})$ represent probability distributions over $\\mathcal{F}$. Finally, $\\mathbb{E}_{\\pi}[L] = \\mathbb{E}_{f \\sim\\pi}[L(f)] = \\int_{\\mathcal{F}} L(f) \\pi(df)$ is the expectation under $\\pi \\in \\mathcal{M}_1(\\mathcal{F})$, for $L : \\mathcal{F}\\to\\mathbb{R}$.\n\n\n\\paragraph{On Reproducing Kernel Hilbert Spaces (RKHS).} We recall results from \\cite{hein2004kernels} on the links between RKHS and different mathematical structures.\n\nLet us start by a primer on kernels.\nThe key idea is that while data belongs to a data space $\\mathcal{X}$, a kernel function $\\kappa : \\mathcal{X}\\times\\mathcal{X} \\to \\mathbb{R}$ implicitly embeds data into a Hilbert space (of real-valued functions), where there is an abundance of structure to exploit. Such a function $\\kappa$ is required to be \\emph{symmetric} in the sense that $\\kappa(x_1,x_2) = \\kappa(x_2,x_1)$ for all $x_1,x_2 \\in \\mathcal{X}$.\n\n\\begin{definition}[PSD kernels]\n    A symmetric real-valued function $\\kappa:\\mathcal{X}\\times\\mathcal{X}\\rightarrow \\mathbb{R}$ is said to be a \\emph{positive semi definite} (PSD) kernel if \n    $\\forall n\\geq 1$, $\\forall x_1,...,x_n \\in\\mathcal{X}$, $\\forall c_1,...,c_n\\in\\mathbb{R}$:\n    \\begin{align*}\n        \\sum_{i,j=1}^n c_i c_j \\kappa(x_i,x_j)\\geq 0.\n    \\end{align*}\nIf the inequality is strict (for non-zero coefficients $c_1,\\ldots,c_n$), then the kernel is said to be \\emph{positive definite} (PD).\n\\end{definition}\n\nFor instance, polynomial kernels $\\kappa(x,y)= (x^{T}y+r)^n$, and Gaussian kernels  $\\kappa(x,y)=\\exp(-||x-y||^2\/2\\sigma^2)$ (for $n\\geq 1,(x,y)\\in(\\mathbb{R}^d)^2, r\\geq 0,\\sigma>0$) are PD kernels.\n\n\\begin{definition}[RKHS]\n    A \\emph{reproducing kernel Hilbert space} (RKHS) on $\\mathcal{X}$ is a Hilbert space $\\mathcal{H} \\subset \\mathbb{R}^\\mathcal{X}$ (functions from $\\mathcal{X}$ to $\\mathbb{R}$) where all evaluation functionals $\\delta_x : \\mathcal{H} \\rightarrow \\mathbb{R}$, defined by $\\delta_x(f) = f(x)$, are continuous.\n\\end{definition}\nNote that, since the evaluation functionals are linear, an equivalent condition to continuity (of all the $\\delta_x$'s) is that for every $x\\in\\mathcal{X}$, there\nexists $M_x < +\\infty$ such that\n\\[ \n\\forall f \\in \\mathcal{H}, \\hspace{3mm} |f(x)|\\leq M_x ||f||.  \n\\]\nThis condition is the so-called \\emph{reproducing property}.\nWhen $\\mathcal{H}$ is an RKHS over $\\mathcal{X}$, there is a kernel $\\kappa:\\mathcal{X}\\times\\mathcal{X}\\rightarrow \\mathbb{R}$ and a mapping $\\psi:\\mathcal{X}\\to\\mathcal{H}$ such that $\\kappa(x_1,x_2) = \\langle \\psi(x_1),\\psi(x_2) \\rangle_{\\mathcal{H}}$. Intuitively: the `feature mapping' $\\psi$ maps data points $x \\in \\mathcal{X}$ to `feature vectors' $\\psi(x) \\in \\mathcal{H}$, while the kernel computes the inner products between those feature vectors without needing explicit knowledge of $\\psi$.\n\nThe following key theorem from \\cite{aronszajn1950} links PD kernels and RKHS.\n\n\\begin{theorem}[Moore-Aronszajn, 1950]\nIf $\\kappa$ is a positive definite kernel, then there exists a unique reproducing kernel Hilbert space $\\mathcal{H}$ whose kernel is $\\kappa$.\n\\end{theorem}\n\nIn \\cite{hein2004kernels}, a sketch of the proof is provided: from a PD kernel $\\kappa$, we build an RKHS from the pre-Hilbert space \n$$\nV=\\operatorname{span}\\left\\{\\kappa(x,.)\\mid x\\in\\mathcal{X} \\right\\}.    \n$$\nWe endow V with the following inner product:\n\\begin{equation}\\label{remark_RKHS_finite_data_space_X}\n\\left\\langle \\sum_{i}a_i\\kappa(x_i,.), \\sum_j b_j \\kappa(x_j,.)  \\right\\rangle_V = \\sum_{i,j} a_i b_j \\kappa(x_i,x_j) .\n\\end{equation}\nIt can be shown that this is indeed a well-defined inner product.\nThus, the rest of the proof consists in the completion of $V$ into an Hilbert space (of functions) verifying the reproducing property, and the verification of the uniqueness of such an Hilbert space.\n\nA important special case is when $|\\mathcal{X}|< +\\infty$, then $V$ is a finite-dimensional vector space. Thus if we endow it with an inner product, $V$ is already an Hilbert space (it already contains every pointwise limits of all Cauchy sequences of elements of $V$). As a consequence, the associated RKHS is finite-dimensional in this case.\n\n\\begin{definition}[Aronszajn mapping]\n    \\label{aronszajn_mapping}\n    For a fixed PD kernel $\\kappa$, we define the \\emph{Aronszajn mapping} \n   \n    $\\psi: \\mathcal{X}\\rightarrow \\left(\\mathcal{H},\\langle \\cdot,\\cdot\\rangle\\right)$ such that\n    \\[\n    \\forall x \\in\\mathcal{X}, \\hspace{3mm} \\psi(x)= \\kappa(x,.)    ,\n    \\]\n    where we denote by $\\mathcal{H}$ the RKHS given by the Moore-Aronszajn theorem and $\\langle\\cdot,\\cdot\\rangle$ is the inner product given in  \\eqref{remark_RKHS_finite_data_space_X}.\n    In the sequel, we refer to the Aronszajn mapping as the pair $(\\psi,\\mathcal{H})$ when it is important to highlight the space $\\mathcal{H}$ into which $\\psi$ embeds the data.\n\\end{definition}\n\nWhen it comes to embedding points of $\\mathcal{X}$ into a Hilbert space through a feature mapping $\\psi$, several approaches have been considered \\citep[see][Section 3.1]{hein2004kernels}. The Aronszajn mapping is one choice among many.\n\n\n\\paragraph{On kernel PCA.} We present here the results from \\cite{shawe-taylor2005eigenspectrum}. Fix a PD kernel $\\kappa$. We denote by $(\\psi,\\mathcal{H})$ the Aronszajn mapping of $\\mathcal{X}$ into $\\mathcal{H}$.\n\n\\begin{definition}\n\\label{d:gram_matrix}\nThe kernel Gram matrix of a data set $s=(x_1,\\ldots,x_m) \\in \\mathcal{X}^m$ is the element $K(s)$ of $\\mathbb{R}^{m \\times m}$ defined as\n\\[ \nK(s)= (\\kappa( x_i,x_j))_{i,j}.\n\\]\nWhen the data set is clear from the context, we will shorten this notation to $K = (\\kappa( x_i,x_j))_{i,j}$.\n\\end{definition}\nNote that for a random sample $S = (X_1, \\ldots, X_m)$, the corresponding $K(S)$ is a random matrix. \n\nThe goal of kernel PCA is to analyse $K$ by putting the intricate data sample of size $m$ from the set $\\mathcal{X}$ into $\\mathcal{H}$, where data are properly separated, and then find a small (in terms of dimension) subspace $V$ of $\\mathcal{H}$ which catches the major part of the information contained in the data. \nWe define $\\mu \\in \\mathcal{M}_1(\\mathcal{X})$, a probability measure over $\\mathcal{X}$, as the distribution representing the way data are spread out over $\\mathcal{X}$.\n\n\nIn other words, we want to find a subspace $V \\subseteq \\mathcal{H}$ such as\n\\[\n\\forall x\\in \\mathcal{X}, \\hspace{3mm}\n\\left| ||P_V(\\psi(x)||^2 - ||\\psi(x)||^2 \\right| \\approx 0\n\\]\nwhere $P_V$ is the orthogonal projection over the subspace $V$.\n\nThe notation $[m] = \\{ 1,\\ldots,m \\}$ is convenient.\nRecall that $\\psi$ and $\\mathcal{H}$ are defined such that we can express the elements of $K$ as a scalar product in $\\mathcal{H}$:\n\\[\n\\forall i,j \\in [m], \\hspace{3mm}\nK_{i,j}= \\kappa(x_i,x_j) = \\langle \\psi(x_i),\\psi(x_j)\\rangle_\\mathcal{H}.\n\\]\n\n\n\\begin{definition}\nFor any probability distribution $\\nu$ over $\\mathcal{X}$,\nwe define the self-adjoint operator on $L^2(\\mathcal{X},\\nu)$ associated to the kernel function $\\kappa$ as:\n\\[\n\\mathcal{K}_\\nu (f)(x) = \\int_{\\mathcal{X}} f(x') \\kappa(x,x')\\nu(dx').\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{d:e_values}\nWe use the following conventions:\n\\begin{itemize}[leftmargin=15pt,itemsep=1pt]\n    \\item If $\\mu$ is the data-generating distribution, then we rename $\\mathcal{K} := \\mathcal{K}_{\\mu}$.\n    \\item If $\\hat{\\mu}$ is the empirical distribution of our $m$-sample $(x_i)_i$, then we name $\\hat{\\mathcal{K}} := \\mathcal{K}_{\\hat{\\mu}}$.\n    \\item $\\lambda_1\\geq \\lambda_2\\geq\\cdots$ are the eigenvalues of the operator $\\mathcal{K}$.\n    \\item $\\hat{\\lambda}_1\\geq\\cdots\\geq \\hat{\\lambda}_m\\geq 0$ are the eigenvalues of the kernel matrix $K$.\n\\end{itemize}\n\\end{definition}\n\nMore generally, let $\\lambda_1(A) \\geq \\lambda_2(A) \\geq \\cdots$ be the eigenvalues of a matrix $A$, or a linear operator $A$. Then in Definition \\ref{d:e_values}, we use the shortcuts $\\lambda_i = \\lambda_i(\\mathcal{K})$ and $\\hat{\\lambda}_i = \\lambda_i(K)$. \nNotice that $\\forall i \\in\\{1,\\dots,m\\}$, we have $\\lambda_i(\\hat{\\mathcal{K}})= \\frac{\\hat{\\lambda}_i}{m}$.\n\n\n\\begin{definition}\n\\label{d:sums}\nFor a given sequence of real scalars $(a_i)_{i \\ge 1}$ of length $m$, which may be infinity, we define for any $k$ the initial sum and the tail sum as\n\\[\na^{\\leq k}:= \\sum_{i= 1}^k a_i\n\\hspace{3mm}\\text{ and}\\hspace{3mm}\na^{>k}:= \\sum_{i= k+1}^m a_i.\n\\]\n\\end{definition}\n\n\n\\begin{definition}\n\\label{d:corr_matrix}\nThe \\emph{sample covariance matrix} of a random data set $S=(X_1,\\ldots,X_m)$ is the element $C(S)$ of $\\mathbb{R}^{m \\times m}$ defined by\n\\[ \nC(S)= \\frac{1}{m}\\sum_{i=1}^m \\psi(X_i)\\psi(X_i)',\n\\]\nwhere $\\psi(x)'$ denotes the transpose of $\\psi(x)$. \nNotice that this is the sample covariance in feature space.\nWhen $S$ is clear from the context, we will shorten $C(S)$ to $C$.\n\\end{definition}\n\n\nOne could object that $C$ may not be finite-dimensional, because $\\mathcal{H}$ is not (in general). However, notice that the subspace of $\\mathcal{H}$ spanned by  $\\psi(x_1),...,\\psi(x_m)$ is always finite-dimensional, hence by choosing a basis of this subspace, $C$ becomes effectively a finite-dimensional square matrix (of size no larger than $m \\times m$).\n\n\n\n\\begin{definition}\n\\label{def: subspaces}\nFor any probability distribution $\\nu$ over $\\mathcal{X}$, we define $\\mathcal{C}_\\nu : \\mathcal{H} \\to \\mathcal{H}$ as the mapping $\\alpha \\mapsto \\mathcal{C}_\\nu(\\alpha)$ where:\n\\[ \n\\mathcal{C}_\\nu(\\alpha) = \\int_{\\mathcal{X}}\\langle \\psi(x),\\alpha\\rangle \\psi(x)\\nu(dx).\n\\]\nIf $\\nu$ has density $v(x)$, then we write \n$\\mathcal{C}_v (\\alpha) = \\int_{\\mathcal{X}}\\langle \\psi(x),\\alpha\\rangle \\psi(x)v(x)dx$.\nNotice that the eigenvalues of $\\mathcal{K}_\\nu$ and $\\mathcal{C}_\\nu$ are the same for any $\\nu$, the proof of this fact may be found in \\cite{shawe-taylor2005eigenspectrum}.\nWe similarly define the simplified notations $\\mathcal{C} := \\mathcal{C}_{\\mu}$ (when $\\mu$ is the population distribution) \nand $\\hat{\\mathcal{C}} = \\mathcal{C}_{\\hat{\\mu}}$ (when $\\hat{\\mu}$ is the empirical distribution).\nWe then define for any $k\\in\\{1,\\dots,m\\}$ \n\\begin{itemize\n    \\item $V_k$ the subspace spanned by the $k$-first eigenvectors of $\\mathcal{C}$,\n    \\item $\\hat{V_k}$ the subspace spanned by the $k$-first eigenvectors of $\\hat{\\mathcal{C}}$.\n\\end{itemize}\n\\end{definition}\nNotice that $\\hat{\\mathcal{C}}$ coincides with the sample covariance matrix $C$, \\emph{i.e.} we have\n\\[\n\\hat{\\mathcal{C}}(\\alpha)=  C\\alpha,\n\\hspace{5mm} \\forall \\alpha \\in \\mathcal{H}.\n\\]\nSo for any $k>0$, $\\hat{V_k}$ is the subspace spanned by the first $k$ eigenvectors of the matrix $C$.\n\n\n\\begin{proposition}[Courant-Fischer's corollary]\n\\label{p:courant-fischer}\nIf $(u_i)_i$ are the eigenvectors associated to $(\\lambda_i(\\mathcal{K}_\\nu))_i$ and $V_k$ is the space spanned by the $k$ first eigenvectors:\n\\[ \n\\lambda_k(\\mathcal{K}_\\nu) \n= \\mathbb{E}_\\nu[||P_{u_k}(\\psi(x))||^2] \n= \\max_{\\dim(V)=k} \\min_{0\\neq v\\in V}\\mathbb{E}_\\nu[||P_{v}(\\psi(x))||^2],\n\\]\n\\[\n\\lambda^{\\leq k}(\\mathcal{K}_\\nu) = \\max_{\\dim(V)=k} \\mathbb{E}_\\nu[||P_{V}(\\psi(x))||^2],\n\\]\n\\[\n\\lambda^{>k}(\\mathcal{K}_\\nu) = \\min_{\\dim(V)=k} \\mathbb{E}_\\nu[||P_{V}^\\bot(\\psi(x))||^2].\n\\]\n\\end{proposition}\n\nNow we will denote by $\\mathbb{E_{\\mu}}$ the expectation under the true data-generating distribution $\\mu$ and by $\\mathbb{E}_{\\hat{\\mu}}$ the expectation under the empirical distribution of an $m$-sample $S$.\nCombining the last properties gives us the following equalities.\n\n\\begin{proposition}\n\\label{p: characterization_eigenvalues}\n We have\n \\begin{align*}\n  \\mathbb{E}_{\\hat{\\mu}} [ ||P_{\\hat{V_k}}(\\psi(x))||^2] \n  &= \\frac 1 m \\sum_{i=1}^m ||P_{\\hat{V _k}}(\\psi(x_i))||^2  = \\frac 1 m \\sum_{i=1}^k \\hat{\\lambda}_i ,\\\\\n  \\mathbb{E}_{\\mu} [ ||P_{V_k}(\\psi(x))||^2] \n  &= \\sum_{i=1}^k \\lambda_i .\n \\end{align*} \n\\end{proposition}\n\nWe now recall the main results from \\cite{shawe-taylor2005eigenspectrum} before introducing our own results.\n\nWith the notation introduced in Definition \\ref{d:sums}, when projecting onto the subspace $\\hat{V_k}$ spanned by the first $k$ eigenvectors of $\\hat{\\mathcal{C}} = \\hat{\\mathcal{K}}$,\nthe tail sum $\\lambda^{>k} = \\sum_{i>k} \\lambda_i$ lower-bounds the expected squared residual.\n\n\n\\begin{theorem}[\\citealp{shawe-taylor2005eigenspectrum}, Theorem 1]\n\\label{th : st_2005_residual}\nIf we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability of at least $1-\\delta$ over random $m$-samples $S$ we have for all $k \\in [m]$ if we project new data $x$ onto the space $\\hat{V}_k$, \nthe expected square residual satisfies:\n\\begin{align*}\n\\lambda^{>k}  \\leq \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}^\\bot(\\psi(x))||^2]\n  &\\leq \\min_{1\\leq \\ell \\leq k} \\left[\\frac 1 m\\hat{\\lambda}^{>\\ell}(S) + \\frac{1 + \\sqrt{\\ell}} {\\sqrt{m}} \\sqrt{\\frac 2 m \\sum_{i=1}^m \\kappa(x_i,x_i)^2}\\right] \\\\\n  & + R^2 \\sqrt{\\frac {18} {m} \\log\\left(\\frac {2m} {\\delta}\\right)} .\n\\end{align*}\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\n\\begin{theorem}[\\citealp{shawe-taylor2005eigenspectrum}, Theorem 2]\n\\label{th : st_2005_proj}\nIf we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability of at least $1-\\delta$ over random $m$-samples $S$ we have for all $k \\in [m]$ if we project new data $x$ onto the space $\\hat{V}_k$, the expected square projection satisfies:\n\\begin{align*}\n\\lambda^{\\leq k}  \\geq \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}(\\psi(x))||^2]\n  &\\geq \\max_{1\\leq \\ell\\leq k} \\left[\\frac 1 m\\hat{\\lambda}^{\\leq \\ell}(S) - \\frac{1 + \\sqrt{\\ell}} {\\sqrt{m}} \\sqrt{\\frac 2 m \\sum_{i=1}^m \\kappa(x_i,x_i)^2}\\right] \\\\\n  & - R^2 \\sqrt{\\frac {19} {m} \\log\\left(\\frac {2(m+1)} {\\delta}\\right)} .\n\\end{align*}\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\n\n\nNotice that the purpose of those two theorems is to control, by upper and lower bounds, the theoretical averaged squared norm projections of a new data point $x$ into the empirical subspaces $\\hat{V}_k$ and $\\hat{V}_k^\\bot$:  $ \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}(\\psi(x))||^2] $ and $\\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}^\\bot(\\psi(x))||^2]$. Let us note that for each theorem, only one side of the inequality is empirical (while the other one consists in an unknown quantity, $\\lambda\n^{\\leq k}$ or $\\lambda^{>k}$, respectively). \n\nOur contribution is twofold: \n\\begin{itemize}\n    \\item We first propose two empirical PAC bounds improving (in some cases) the results of \\cite{shawe-taylor2005eigenspectrum}. These are collected in \\cref{sec:pac}.\n    \\item Casting this onto the PAC-Bayes framework, we then move on to two more sophisticated empirical bounds which are replacing the theoretical quantities  $\\lambda^{>k}$ and $\\lambda\n^{\\leq k}$ in Theorems \\ref{th : st_2005_residual} and \\ref{th : st_2005_proj}. This is the core of the paper (\\cref{sec:pacb}).\n\\end{itemize}\n\n\n\n\\section{A theoretical analysis of kernel PCA: PAC bounds}\\label{sec:pac}\n\nWe present in this section two PAC bounds, which are directly comparable to those of \\citet{shawe-taylor2005eigenspectrum} which were recalled in \\cref{th : st_2005_residual} and \\cref{th : st_2005_proj} (see also \\cref{sec: experiments} for a brief numerical comparison). These bounds exploit the classical idea of splitting a data set in half, one being used as a training set and the other as a test set.\n\\begin{theorem}\n\\label{th : pac_inequality_1}\nIf we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability at least  $1-\\delta$ over random $2m$-samples $S=S_1\\cup S_2$ (where $S_1=\\{x_1,...,x_m\\}$, $S_2=\\{x_{m+1},...,x_{2m}\\}$ are two disjoints $m$-samples, we have for all $k \\in [m]$, if we project new data $x$ onto the space $\\hat{V}_k(S_1)$, the expected square projection is bounded by :\n\\begin{align*}\n      \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right] & \\geq \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i}))||^2 -R^2\\sqrt{\\frac{2}{m}\\log\\left(\\frac{1}{\\delta}\\right)} .\n\\end{align*}\nWhere $\\hat{V}_k(S_1)$ is the subspace spanned by the $k$ eigenvectors of the covariance matrix $C(S_1)$ corresponding to the $k$ largest eigenvalues of $C(S_1)$.\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\nThis result provides an empirical lower bound for the theoretical expected squared projection. In other words, it quantifies how accurate the projection on a new data point onto our empirical subspace is.\n\n\\begin{theorem}\n\\label{th : pac_inequality_2}\nIf we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability at least  $1-\\delta$ over random $2m$-samples $S=S_1\\cup S_2$ (where $S_1=\\{x_1,...,x_m\\}$, $S_2=\\{x_{m+1},...,x_{2m}\\}$ are two disjoints $m$-samples, we have for all $k \\in [m]$, the expected square residual is bounded by:\n\\begin{align*}\n      \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k^{\\bot}(S_1)}(\\psi(x))||^2\\right] & \\leq \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k^{\\bot}(S_1)}(\\psi(x_{m+i}))||^2 +R^2\\sqrt{\\frac{2}{m}\\log\\left(\\frac{1}{\\delta}\\right)} .\n\\end{align*}\nWhere $\\hat{V}_k^{\\bot}(S_1) $ is the orthogonal complement of $\\hat{V}_k(S_1)$, the subspace spanned by the $k$ eigenvectors of the covariance matrix $C(S_1)$ corresponding to the $k$ largest eigenvalues of $C(S_1)$.\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\n\n\\cref{th : pac_inequality_2} provides an upper bound on the residual squared projection. It therefore measures how much information is lost by the projection of a new data point onto our empirical subspace.\n\nThe rest of this section is devoted to the proofs of \\cref{th : pac_inequality_1} and \\cref{th : pac_inequality_2}. Numerical implementations of both theorems are gathered in \\cref{sec: experiments}.\n\n\nWe first recall a classical concentration inequality of \\citet{mcdiarmid1989}.\n\n\\begin{theorem}[Bounded differences, McDiarmid]\n\\label{th: mcdiarmid}\nLet $X_1,...,X_n$ be independent random variables taking values in $\\mathcal{X}$ and $f: \\mathcal{X}^n \\rightarrow \\mathbb{R}$. Assume that for all $1\\leq i\\leq n$ and all $x_1,\\ldots,x_{i-1},x_{i+1},\\ldots,x_n \\in \\mathcal{X}$ we have:\n\\[\n\\sup_{x_i,\\hat{x}_i} \n|f(x_1,\\ldots,x_{i-1},x_{i},x_{i+1},\\ldots,x_n) - f(x_1,\\ldots,x_{i-1},\\hat{x}_{i},x_{i+1},\\ldots,x_n)| \\leq c_i .\n\\]\nThen for all $\\delta >0$:\n\\[\n\\mathbb{P}\\left( f(X_1,...,X_n) - \\mathbb{E}[f(X_1,...,X_n)] > \\delta \\right) \\leq \\exp\\left(- \\frac{2\\delta^2}{\\sum_{i=1}^n c_i^2}\\right) .\n\\]\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem \\ref{th : pac_inequality_1}]\n\nLet $S=S_1\\cup S_2$ a size-$2m$ sample. We recall that our data are iid.\nWe first apply Proposition \\ref{p:courant-fischer} and Proposition \\ref{p: characterization_eigenvalues}\nto $S_1$.\n\nWe define $\\hat{V}_k(S_1)$ the subspace spanned by the $k$ eigenvectors of the covariance matrix $C(S_1)$ corresponding to the top $k$ eigenvalues of $C(S_1)$.\nWe need now to study \n$\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right]$.\n\nNote that\n\\begin{align*}\n\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right] \n&= \\left(\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right]- \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i}))||^2\\right) \\\\\n&\\hspace*{17mm}+ \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i}))||^2  .\n\\end{align*}\nBecause our data are iid, following the distribution $\\mu$, we know that $S_1$ and $S_2$ are independent, hence \n\\[\n\\mathbb{E}_{S_2}\\left[ \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i})||^2 \\right]  = \\mathbb{E}_{\\mu}\\left[||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right]  .\n\\]\nWe can now apply McDiarmid's inequality: with probability $1-\\delta$,\n\\[\n\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S_1)}(\\psi(x))||^2\\right] - \\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i}))||^2 \\geq - R^2 \\sqrt{\\frac{2}{m}\\log\\left(\\frac{1}{\\delta}\\right)} .\n\\]\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{th : pac_inequality_2}]\nThe proof is similar to the previous one, just replace $P_{\\hat{V}_k(S_1)}$ by $P_{\\hat{V}_k^{\\bot}(S_1)}$ and use McDiarmid's inequality.\n\\end{proof}\n\n\n\\section{A theoretical analysis of kernel PCA: PAC-Bayes bounds}\\label{sec:pacb}\n\nThis section contains our main results which harness the power and flexibility of PAC-Bayes. We bring bounds of a new kind for kernel PCA: while our PAC bounds (in \\cref{sec:pac}) were assessing that kernel PCA was efficient with a certain amount of confidence, the two next theorems, on the contrary, explicit the limitations we face when projecting onto an empirical subspace, therefore contributing to a better overall theoretical understanding.\n\n\\begin{theorem}\n\\label{th: pac_bayes_thm_proj}\nFor a finite data space $\\mathcal{X}$, for $\\alpha\\in\\mathbb{R}$, $\\delta\\in]0,1]$, if we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability of at least  $1-\\delta$ over random $m$-samples $S$, we have for all $k \\in [m]$, the expected square projection is bounded by:\n\\begin{align*}\n    \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}(\\psi(x))||^2]\n  &\\leq \\frac{1}{m}\\hat{\\lambda}^{\\leq k} + \\frac{\\log(1\/\\delta)}{m^{\\alpha}} + \\frac{R^4}{2m^{1-\\alpha}} .\\end{align*}\nand the optimal value for $\\alpha$ is $\\alpha_0= \\frac{1}{2}+\\frac{1}{2\\log(m)}\\log\\left( \\frac{2\\log(1\/\\delta)}{R^{4}} \\right)$.\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\nThis theorem shows that the expected squared projection cannot improve on a quantity close to the partial sum of empirical eigenvalues. This demonstrates that measuring the efficiency of our projection through this empirical sum is therefore relevant.\n\nThe next theorem is intended in the same spirit, but holds for empirical squared residuals.\n\n\n\\begin{theorem}\n\\label{th: pac_bayes_thm_residual}\nFor a finite data space $\\mathcal{X}$, for $\\alpha\\in\\mathbb{R}$, $\\delta\\in]0,1]$, if we perform PCA in the feature space defined by the Aronszjan mapping $(\\psi,\\mathcal{H})$ of a kernel $\\kappa$, then with probability at least  $1-\\delta$ over random $m$-samples $S$, we have for all $k \\in [m]$, the expected square residual is bounded by:\n\\begin{align*}\n    \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k^{\\bot}}(\\psi(x))||^2]\n  &\\geq \\frac{1}{m}\\hat{\\lambda}^{>k} - \\frac{\\log(1\/\\delta)}{m^{\\alpha}} - \\frac{R^4}{2m^{1-\\alpha}}.\n\\end{align*}\nand the optimal value for $\\alpha$ is $\\alpha_0= \\frac{1}{2}+\\frac{1}{2\\log(m)}\\log\\left( \\frac{2\\log(1\/\\delta)}{R^{4}} \\right)$.\nThis holds under the assumption that $\\| \\psi(x) \\| \\leq R$, for any $x \\in \\mathcal{X}$.\n\\end{theorem}\n\n\\begin{remark}\n The assumption $|\\mathcal{X}|< +\\infty$ may appear to be restrictive at first glance. As a matter of fact, it covers the case of $\\mathcal{X}\\subseteq\\mathbb{R}^d$ bounded if one decides to discretise $\\mathcal{X}$ into a finite grid $\\mathcal{G}$. With a large number of points on the grid, one can approximate $\\mathcal{X}$ efficiently and also apply Theorems \\ref{th: pac_bayes_thm_proj} and \\ref{th: pac_bayes_thm_residual} and those theorems provide solid bounds independent of the number of points inside $\\mathcal{G}$.\n\\end{remark}\n\nNumerical implementations of those two bounds are gathered in \\cref{sec: experiments}.\n\n\n\nWe now move to the proofs of \\cref{th: pac_bayes_thm_proj} and \\cref{th: pac_bayes_thm_residual}. We start with additional technical background.\n\n\\paragraph{Self-bounding functions.}\n\nWe use the notion of \\emph{self-bounding function}   \\citep[presented for instance in][Definition 2]{boucheron2009} which allows to deal with a certain type of exponential moment. This tool is at the core of the recent work from \\cite{haddouche2020}, to obtain PAC-Bayesian generalisation bounds when the loss is unbounded \\citep[as typically assumed in most of the PAC-Bayes literature, see the discussion of][and references therein]{haddouche2020}.\n\n \n\n\\begin{definition}[\\citealp{boucheron2009}, Definition 2]\n\\label{d: self-bounding}\n    A function $f:\\mathcal{X}^m\\rightarrow\\mathbb{R}$ is said to be $(a,b)$-self-bounding with $(a,b)\\in\\left(\\mathbb{R}^{+}\\right)^2\\backslash\\{(0,0)\\}$, if there exists $f_i :\\mathcal{X}^{m-1}\\rightarrow \\mathbb{R}$ for every $i\\in[m]$ such that for all $i\\in[m]$ and $x\\in\\mathcal{X}$:\n    \\[\n    0\\leq f(x) - f_i(x^{(i)}) \\leq 1  \n    \\]\n and\n \\[\n \\sum_{i=1}^m f(x)-f_i(x^{(i)}) \\leq af(x) +b  .\n \\]\n Where for all $1\\leq i \\leq m$, the removal of the $i$th entry is $x^{(i)}= (x_1,...,x_{i-1},x_{i+1},...,x_m)$.\n We denote by $\\texttt{SB}(a,b)$ the class of\n functions that satisfy this definition.\n\\end{definition}\n\n\nIn \\cite{boucheron2009}, the following bound has been presented to deal with the exponential moment of a self-bounding function.\nLet $c_{+}:=\\max(c,0)$ denote the positive part of $c \\in \\mathbb{R}$. We define $c_{+}^{-1}:= +\\infty$ when $c_{+}=0$.\n\n\\begin{theorem}[\\citealp{boucheron2009}, Theorem 3.1]\n\\label{th: exp_inequality2009 }\n    Let $Z=g(X_1,...,X_m)$ where $X_1,...,X_m$ are independent (not necessarily identically distributed) $\\mathcal{X}$-valued random variables.\n    We assume that $\\mathbb{E}[Z]<+\\infty$. \n   \n   \n    If $g\\in\\texttt{SB}(a,b)$, then defining $c=(3a-1)\/6$, \n    for any $s\\in[0;c_{+}^{-1})$ we have:\n    \\[ \n    \\log\\left(\\mathbb{E}\\left[e^{s(Z-\\mathbb{E}[Z])}  \\right] \\right) \\leq \\frac{\\left(a\\mathbb{E}[Z] +b \\right)s^2}{2(1-c_{+}s)}.   \n    \\]\n   \n\\end{theorem}\n\n\n\n\\paragraph{PAC-Bayes.}\n\n We will consider a fixed learning problem with data space $\\mathcal{Z}$, set of predictors $\\mathcal{F}$, and loss function $\\ell : \\mathcal{F}\\times \\mathcal{Z} \\rightarrow \\mathbb{R}^{+} $.\n We denote by $\\mu$ a probability distribution over $\\mathcal{Z}$, \n  $s = (z_1,\\ldots,z_m)$ is a size-$m$ sample.\n We denote as $\\Sigma_{\\mathcal{F}}$ the considered $\\sigma$-algebra on $\\mathcal{F}$. \n Finally,  we define for any $f\\in\\mathcal{F}$, $L(f)= \\mathbb{E}_{z\\sim \\mu}[\\ell(f,z)]$ and $\\hat{L}_s(f)=\\frac{1}{m}\\sum_{i=1}^m \\ell(f,z_i)$.\n\nLet us highlight that in supervised learning, for instance, $\\mathcal{Z} = \\mathcal{X}\\times\\mathcal{Y}$\nwhere $\\mathcal{X} \\subset \\mathbb{R}^d$ is a space of inputs, and $\\mathcal{Y}$ a space of labels. In this case predictors are functions $f:\\mathcal{X}\\to\\mathcal{Y}$. One may be interested in applying PCA over the input space $\\mathcal{X}$ to reduce the input dimension.\n\n\nWe use the PAC-Bayesian bound given by \\cite{rivasplata2020}, in which the authors developed a PAC-Bayesian bound allowing the use of a data-dependant prior (we refer to the discussion and references therein for an introduction to data-dependent prior distributions in PAC-Bayes).\n\n\\begin{definition}[Stochastic kernels]\n    A \\emph{stochastic kernel} from $\\mathcal{Z}^m$ to $\\mathcal{F}$ is defined as a mapping $Q: \\mathcal{Z}^m\\times \\Sigma_{\\mathcal{F}} \\rightarrow [0;1]$ where\n    \\begin{itemize}\n        \\item For any $B\\in \\Sigma_{\\mathcal{F}}$, the function  $s=(z_1,...,z_m)\\mapsto Q(s,B)$ is measurable,\n        \\item For any $s\\in\\mathcal{Z}^m$, the function $B\\mapsto Q(s,B)$ is a probability measure over $\\mathcal{F}$.\n    \\end{itemize}\n    We will denote by $\\texttt{Stoch}(\\mathcal{Z}^m,\\mathcal{F})$ the set of all stochastic kernels from $\\mathcal{Z}^m$ to $\\mathcal{F}$.\n\\end{definition}\n\n\\begin{definition}\n    For any $Q\\in \\texttt{Stoch}(\\mathcal{Z}^m,\\mathcal{F}) $ and $s\\in\\mathcal{Z}^m$, we define $Q_s[L]:=\\mathbb{E}_{f\\sim Q_s}[L(f)]$ and $Q_s[\\hat{L}_s]:=\\mathbb{E}_{f\\sim Q_s}[\\hat{L}_s(f)]$. We generalise this definition to the case where we consider $S=(Z_1,...,Z_m)$ a random sample. Then we consider the random quantities $Q_S[L]$ and $Q_S[\\hat{L}_S]$.\n\\end{definition}\n\nFor the sake of clarity, we now present a slightly less general version of one of the main theorems of \\cite{rivasplata2020}. We define $\\mu^m := \\mu \\otimes\\cdots \\otimes \\mu$ ($m$ times).\n\n\\begin{theorem}[\\citealp{rivasplata2020}]\n\\label{th: rivasplata2020}\n   For any $F:\\mathbb{R}^2\\rightarrow \\mathbb{R}$ convex, for any $Q^{0}\\in\\texttt{Stoch}(\\mathcal{Z}^m,\\mathcal{F})$, we define\n   \\[ \\xi:= \\mathbb{E}_{S\\sim \\mu^m} \\mathbb{E}_{f\\sim Q^{0}_S}\\left[\\exp\\left(F(\\hat{L}_S(f),L(f))\\right)\\right] . \\]\n   Then for any $Q\\in\\texttt{Stoch}(\\mathcal{Z}^m,\\mathcal{F})$, any $\\delta\\in ]0;1]$, with probability at least $1-\\delta$ over the random draw of the sample $S\\sim \\mu^m$\n   \\[ F(Q_S[(\\hat{L}_S,L]) \\leq \\operatorname{KL}(Q_S,Q_S^{0}) + \\log(\\xi\/\\delta)  . \\]\n\\end{theorem}\n\nFor a fixed PD kernel $\\kappa$, $(\\psi,\\mathcal{H})$ is the associated Aronszajn mapping.\nLet us now consider a finite data space $\\mathcal{X}$. We therefore have\n\\[N_\\mathcal{H}:=\\dim(\\mathcal{H})< +\\infty.\\] \nFor the sake of clarity, we will assume that $\\mathcal{H}=\\mathbb{R}^{N_\\mathcal{H}}$ endowed with the Euclidean inner product. The space is provided with the Borelian $\\sigma$-algebra $\\mathcal{B}(\\mathbb{R}^{N_\\mathcal{H}})$.\n\n\nWe assume that our data distribution $\\psi(\\mu)$  over $\\mathcal{H}$ is bounded in $\\mathcal{H}$ by a scalar $R$:\n\\begin{equation*}\n\\forall x\\in\\mathcal{X}, \\hspace{3mm}\n||\\psi(x)||\\leq R .\n\\end{equation*}\n\\begin{remark}\n Note that this assumption is not especially restrictive. Indeed, if $\\kappa$ is bounded by $C$, then for all $x\\in\\mathcal{X}, ||\\psi(x)||^2=\\kappa(x,x)\\leq C$.\n\\end{remark}\n\n\nFirst, note that the function $||P_{V_k}(\\psi(x))||^2$ is expressed with a quadratic dependency over the coordinates of $\\psi(x)$. However, it would be far more convenient better to consider linear functions.\n\n\\begin{proposition}[\\citealp{shawe-taylor2005eigenspectrum}, Prop. 11]\n\\label{p:shawetaylor_prop_11}\nLet $\\hat{V}$ be the subspace spanned by some subset $I$ of the set of eigenvectors of the correlation matrix $C(S)$ associated to the kernel matrix $K(S)$. There exists an Hilbert space mapping $\\hat{\\psi}:\\mathcal{X}\\rightarrow \\mathbb{R}^{N_\\mathcal{H}^2}$ such that the projection norm $||P_{\\hat{V}}(\\psi(x))||^2$ is a linear function $\\hat{f}$ in $\\mathbb{R}^{N_\\mathcal{H}^2}$ such that, for all  $(x,z)\\in\\mathcal{X}^2$\n\\[\n\\langle \\hat{\\psi}(x),\\hat{\\psi}(z)\\rangle = \\kappa(x,z)^2.\n\\]\nFurthermore, if $|I| = k$, then the 2-norm of the function $\\hat{f}$ is $\\sqrt{k}$.\n\\end{proposition}  \n\n\\begin{proof}\nLet $X=U\\Sigma V'$ be the SVD of the sample matrix $X$ (where each column represents a data point).\nThe projection norm is then given by:\n\\[|| P_{\\hat{V}}(\\psi(x)||^2 = \\psi(x)' U(I)U(I)' \\psi(x)    \\]\nwhere $U(I)\\in\\mathbb{R}^{N_\\mathcal{H}\\times k}$ containing the $k$ columns of $U$ in the set $I$ (or equivalently $U(I)\\in\\mathbb{R}^{N_\\mathcal{H}\\times{N_\\mathcal{H}}}$ filled with zeros). If we define $w:= U(I)U(I)'\\in\\mathbb{R}^{N_\\mathcal{H}^2}$ then we have:\n\\[ \n|| P_{\\hat{V}}(\\psi(x)||^2 \n= \\sum_{i,j=1}^{N_\\mathcal{H}} w_{i,j}\\psi(x)_i \\psi(x)_j \n= \\sum_{i,j=1}^{N_\\mathcal{H}} w_{i,j} \\hat{\\psi}(x)_{i,j} \n= \\langle w, \\hat{\\psi}(x)\\rangle ,\n\\]\nwhere for all $i$, $\\psi(x)_i$ represents the $i$-th coordinate of $\\psi(x)$ and for all $x\\in\\mathcal{X},\\hat{\\psi}(x)= (\\psi(x)_i\\psi(x)_j)_{i,j=1..N_\\mathcal{H}}$.\n\nThus, the projection norm is a linear function $\\hat{f}$ in $\\mathbb{R}^{N_\\mathcal{H}^2}$ with the mapping $\\hat{\\psi}$. Note that considering the 2-norm of $\\hat{f}$ is equivalent to consider the 2-norm of $w$.\nThen, if we denote by $(u_i)_i$ the columns of $U$, we have:\n\\begin{align*}\n    ||\\hat{f}||^2 \n    & = \\sum_{i,j=1}^{N_\\mathcal{H}} w_{i,j}^2= ||U(I)U'(I)||^2 \\\\\n    & = \\left\\langle\\sum_{i\\in I}u_i  u_i', \\sum_{j\\in I}u_j u_j'      \\right\\rangle \\\\\n    &= \\sum_{i,j\\in I} (u'_i u_j)^2 \\\\ &= k.\n\\end{align*}\nHence the 2-norm of $\\hat{f}$ is $\\sqrt{k}$. Finally, for $(x,z)\\in\\mathcal{X}^2$:\n\\begin{align*}\n    \\kappa(x,z)^2\n    &= \\langle \\psi(x),\\psi(z) \\rangle ^2 \n        = \\left( \\sum_{i=1}^{N_\\mathcal{H}}\\psi(x)_i \\psi(z)_i \\right)^2 \\\\\n    &= \\sum_{i,j=1}^{N_\\mathcal{H}} \\psi(x)_i \\psi(z)_i \\psi(x)_j \\psi(z)_j \\\\\n    &= \\sum_{i,j=1}^{N_\\mathcal{H}} (\\psi(x)_i \\psi(x)_j) (\\psi(z)_i \\psi(z)_j)\\\\\n    &=\\langle \\hat{\\psi}(x), \\hat{\\psi}(z) \\rangle.\n\\end{align*}\n\\end{proof}\nNow, recall that for a fixed $k$, $P_{V_k}$ minimises the shortfall between the squared norm projection of $\\psi(x)$ and $||\\psi(x)||^2$ for any $x$ over the space of projection functions over a subspace of dimension at most $k$. We therefore introduce the following learning framework.\n\nThe data space is $\\mathcal{X}$ with the probability distribution $\\mu$. \nThe space $\\mathcal{X}$ is endowed with a $\\sigma$-algebra $\\Sigma_\\mathcal{X}$.\n\nThe goal is to minimise the loss over the set of linear predictors in $\\mathbb{R}^{N_\\mathcal{H}^2}$ corresponding to projections into a $k$-dimensional subspace of $\\mathbb{R}^{N_\\mathcal{H}}$ \n\\begin{align*}\n    \\mathcal{F}_{k} &:= \\left\\{ f_{w}: x \\mapsto \\langle w , \\hat{\\psi}(x)\\rangle \\mid \\exists V\\subseteq \\mathbb{R}^{N_\\mathcal{H}}, \\text{dim}(V)=k, \\; \\forall x\\in \\mathcal{X},  f_{w}(x)= ||P_V(\\psi(x)||^2 \\right\\}.\n\\end{align*}\n\nBecause $k$ may variate between $1$ and $N_{\\mathcal{H}}$, we also define the space $\\mathcal{F}$ such that for all $k$, $\\mathcal{F}_k\\subseteq\\mathcal{F}$:\n\\[ \\mathcal{F} := \\left\\{ f_{w}: x \\mapsto \\langle w , \\hat{\\psi}(x)\\rangle \\mid \\exists V\\subseteq \\mathbb{R}^{N_\\mathcal{H}}, \\forall x\\in \\mathcal{X},  f_{w}(x)= ||P_V(\\psi(x)||^2 \\right\\}.  \\] \nWhen needed, we will assimilate $f_w$ to $w$.\n\nThe loss function is $\\ell : \\mathcal{F} \\times \\mathcal{X} \\to [0,R^2]$ such that for $f_w \\in \\mathcal{F}$ and $ x\\in\\mathcal{X}$:\n\\[ \n\\ell(f,x)=   ||\\psi(x)||^2 - f_w(x) .\n\\]\n\n\nThis loss is indeed non-negative because all the vectors of $\\mathcal{F}$ are representing the squared norm of projectors in $\\mathbb{R}^{N_\\mathcal{H}}$ so we have for all $f_w\\in\\mathcal{F},$ $x\\in\\mathcal{X}, f_w(x)\\leq ||\\psi(x)||^2$.\n\n\n\\begin{remark}\nIn other contexts and tasks other loss functions are used. The reason that the loss function just defined above makes sense in our task is given by Proposition \\ref{p:courant-fischer}. Indeed, we know that if $f$ is a projector over a space $V$ of dimension $k$, then we have $\\ell(f,x)=||P_{V}^{\\bot}(\\psi(x)||^{2} $ and moreover the expected squared residual $\\lambda^{>k}$ satisfies:\n\\[  \\lambda^{>k} = \\min_{\\dim(V)=k} \\mathbb{E}_{\\mu}[||P_{V}^\\bot(\\psi(x))||^2] = \\mathbb{E}_{\\mu} [ ||P_{V_k}^{\\bot}(\\psi(x))||^2]. \\]\n\\end{remark}\nWe assume $|\\mathcal{X}|< +\\infty$.\nTo apply our PAC-Bayesian theorem (\\cref{th: rivasplata2020}) we need to introduce an adapted stochastic kernel. For a fixed $k$, we set $Q^k, P^k$ as follows, $\\forall s\\in\\mathcal{X}^m$, $\\forall B\\in \\Sigma_{\\mathcal{F}_{k}}$:\n\\[  Q^k(s,B)= \\mathds{1}\\left\\{  f_{\\hat{w}_k(s)}\\in B   \\right\\},   \\]\n\\[  P^k(s,B)= \\mathds{1}\\left\\{  f_{\\hat{w}^{\\bot}_k(s)}\\in B   \\right\\},   \\]\nwhere $\\Sigma_{\\mathcal{F}_k}$ is the $\\sigma$-algebra on $\\mathcal{F}_k$, and: \n\\begin{itemize}\n    \\item For all $k$, the vector $\\hat{w}_k(s)$  of $\\mathbb{R}^{N_{\\mathcal{H}^2}}$ is such that $f_{\\hat{w}_k}(x)= ||P_{\\hat{V}_k(s)}(.)||^2 $ where $\\hat{V_k}(s)$ is the $k-$dimensional subspace defined in Definition \\ref{def: subspaces} obtained from the sample $s$;\n    \\item For all $k$, the vector $\\hat{w}^{\\bot}_k(s)$ of $\\mathbb{R}^{N_{\\mathcal{H}^2}}$ is such that $f_{\\hat{w}^{\\bot}_k}(x)= ||P_{\\hat{V}^{\\bot}_k(s)}(.)||^2 $ where $\\hat{V_k}^{\\bot}(s)$ is the orthogonal of $\\hat{V}_k(s)$.\n\\end{itemize}\nWe need the following technical results.\n\\begin{theorem}\n\\label{th: measurability_kernel_Q}\nFor all $k$, $Q^k$ is a stochastic kernel. \n\\end{theorem}\n\n\\begin{proof}\nThe proof is postponed to \\cref{sec:proofs}.\n\\end{proof}\n\n \\begin{theorem}\n \\label{th: measurability_kernel_P}\n For all $k$, $P^k$ is a stochastic kernel.\n \\end{theorem}\n\n\\begin{proof}\nThe proof is postponed to Section \\ref{sec:proofs}.\n\\end{proof}\n\n\\begin{remark}\nThe proof of Theorem \\ref{th: measurability_kernel_Q} exploits the hypothesis $|\\mathcal{X}|< +\\infty$. Indeed, the fact that $\\mathcal{H}$ is finite-dimensional allows to consider well-defined symmetric matrices and to exploit a result from \\cite{wilcox1972}.\n\\end{remark}\n\n\n\n\nNow we prove \\cref{th: pac_bayes_thm_proj}, which we recall here for convenience: for a finite data space $\\mathcal{X}$, for $\\alpha\\in\\mathbb{R}$, $\\delta\\in]0,1]$, for any $1\\leq k\\leq m$, we have with probability $1-\\delta$ over the random $m$-sample $S$\n\\begin{align*}\n    \\mathbb{E}_{\\mu}[||P_{\\hat{V}_k}(\\psi(x))||^2]\n  &\\leq \\frac{1}{m}\\sum_{i=1}^k \\hat{\\lambda}_i + \\frac{\\log(1\/\\delta)}{m^{\\alpha}} + \\frac{R^4}{2m^{1-\\alpha}}.\n\\end{align*}\nand the optimal value for $\\alpha$ is $\\alpha_0= \\frac{1}{2}+\\frac{1}{2\\log(m)}\\log\\left( \\frac{2\\log(1\/\\delta)}{R^{4}} \\right)$.\n\n\n\\begin{proof}[Proof of \\cref{th: pac_bayes_thm_proj}]\n\nLet $k\\in[m]$.\nWe first apply \\cref{th: rivasplata2020} with probability $1-\\delta$, $F:(x,y)\\mapsto m^{\\alpha}(y-x)$ and the stochastic kernel $P^k$ (thanks to \\cref{th: measurability_kernel_P}) as prior and posterior which nullify the KL-divergence term. \nWe then have, with probability $1-\\delta$,\n\\[ m^{\\alpha}( P^k_S(L(f))- P^k_S(\\hat{L}_S(f)) \\leq \\operatorname{KL}(P^k_S,P_S^{k}) + \\log(\\xi\/\\delta)   \\]\nwhere $\\xi:= \\mathbb{E}_{S\\sim \\mu^m} \\mathbb{E}_{f\\sim P^{k}_S}\\left[\\exp\\left(m^{\\alpha}(\\hat{L}_S(f)-(L(f))\\right)\\right]$.\n\nNotice that for any sample $S$, $P^k_S$ is the Dirac measure in $f_{\\hat{w}^{\\bot}_k(S)}$ \\emph{i.e.} the function representing the squared norm projection $||P_{\\hat{V}^{\\bot}_k(S)}(.)||^2$. Hence $P^k_S(L(f))= L(f_{\\hat{w}^{\\bot}_k(S)})$, $Q^k_S(\\hat{L}_S(f))= \\hat{L}_S(f_{\\hat{w}^{\\bot}_k(S)})$.\n\nFinally we have\n \\[ m^{\\alpha}\\left(L(f_{\\hat{w}^{\\bot}_k(S)})- \\hat{L}_S(f_{\\hat{w}^{\\bot}_k(S)})\\right) \\leq  \\log(\\xi\/\\delta)   \\]\nhence:\n\\[ m^{\\alpha}\\left(\\mathbb{E}_{\\mu}\\left[ ||\\psi(x)||^2- ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]  -  \\mathbb{E}_{\\hat{\\mu}}\\left[ ||\\psi(x)||^2- ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]\\right) \\leq  \\log(\\xi\/\\delta)   \\] \nwhere $\\hat{\\mu}$ is the empirical distribution over the $m$-sample $S$.\n\nFurthermore, because we are considering orthogonal projections, we can see that for any $x\\in\\mathcal{X}$, we have \n\\[ \n||\\psi(x)||^2- ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2 =  ||P_{\\hat{V}_k(S)}(\\psi(x))||^2.\n\\]\nSo\n\\begin{align*}\n\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] &\\leq  \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]  +\\frac{\\log(\\xi\/\\delta)}{m^{\\alpha}}.\n\\end{align*}\n\nThanks to Proposition \\ref{p: characterization_eigenvalues}, we know that $\\mathbb{E}_{ \\hat{\\mu}}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]= \\frac{1}{m}\\sum_{i=1}^k \\hat{\\lambda}_i$, which yields\n\\begin{align*}\n\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] &\\leq  \\frac{1}{m}\\sum_{i=1}^k \\hat{\\lambda}_i  +\\frac{\\log(\\xi\/\\delta)}{m^{\\alpha}}.\n\\end{align*}\nFinally, we need to control $\\log(\\xi)$. To do so, we use \\cref{th: exp_inequality2009 }. We first recall that\n\\begin{align*}\n\\xi &:= \\mathbb{E}_{S\\sim \\mu^m} \\mathbb{E}_{f\\sim P^{k}_S}\\left[\\exp\\left(m^{\\alpha}(L(f)-\\hat{L}_S(f))\\right)\\right] \\\\\n& = \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}(L(f_{\\hat{w}^{\\bot}_k})-\\hat{L}_S(f_{\\hat{w}^{\\bot}_k}))\\right)\\right] \\\\\n&= \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}(\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] )\\right)\\right].\n\\end{align*}\nYet, thanks to Propositions \\ref{p:courant-fischer} and \\ref{p: characterization_eigenvalues}, we know that:\n\\begin{align*}\n    \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] \\leq \\max_{\\dim(V)=k} \\mathbb{E}_\\mu[||P_{V}(\\psi(x))||^2] = \\mathbb{E}_{\\mu}\\left[ ||P_{V_k(S)}(\\psi(x))||^2  \\right]\n\\end{align*}\nand \n\\begin{align*}\n    \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]]& = \\max_{\\dim(V)=k} \\mathbb{E}_{\\hat{\\mu}}[||P_{V}(\\psi(x))||^2]\\\\\n    & \\geq \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V_k}(\\psi(x))||^2  \\right].\n\\end{align*}\nThus we have \n\\[ \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]  \\leq \\mathbb{E}_{\\mu}\\left[ ||P_{V_k}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V_k}(\\psi(x))||^2  \\right]\\]\n\\begin{remark}\n \n The interest of this maneuver is to replace the projector $P_{\\hat{V}_k(S)}$, which is data-dependent, by $P_{V_k}$, which is not. Doing so, if we set $Y= \\frac{1}{m}\\sum_{i=1}^m Y_i  $ and for all $i$, $Y_i=||P_{V_k}(\\psi(x_i))||^2 $ (where the $Y_i$ are iid), we can write properly: \n\\[\\mathbb{E}_{\\mu}\\left[ ||P_{V_k}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V_k}(\\psi(x))||^2  \\right]= \\mathbb{E}_{\\mathcal{S}}[Y] -Y \\]\nwhile $\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right] \\neq \\mathbb{E}_{\\mathcal{S}}\\left[\\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]\\right]$.\n\\end{remark}\nSo, finally we have\n\\[\\xi \\leq  \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}(\\mathbb{E}_{\\mathcal{S}}[Y]- Y )\\right)\\right].  \\] \nWe define the function $f : \\mathcal{X}^{m} \\to \\mathbb{R}$ as\n    \\[\n    f:z \\mapsto \\frac{1}{R^2}\\hspace{1mm}\\sum_{i=1}^{m}\\left(R^2-||P_{V_k}(\\psi(z_i))||^2\\right)\n    \\hspace{7mm}\\text{for}\\hspace{2mm} z=(z_1,...,z_m)\\in\\mathcal{X}^{m}.\n    \\]\n    We define $Z=f(X_1,...,X_m)$ and notice that $\\mathbb{E}_{\\mathcal{S}}[Y]- Y= \\frac{R^2}{m}\\left( Z- \\mathbb{E}_{\\mathcal{S}}[Z] \\right)$. \n    We first prove that $f\\in\\texttt{SB}(\\beta,1-\\beta)$ (cf. Def \\ref{d: self-bounding}) for any $\\beta\\in[0,1]$.\n   \n    Indeed, for all $1\\leq i\\leq m$, we define\n    \\[\n    f_i(z^{(i)})= \\frac{1}{R^2}\\sum_{j\\neq i} \\left(R^2-||P_{V_k}(\\psi(z_i))||^2\\right)\n    \\]\n    where $z^{(i)}=(z_1,...,z_{i-1},z_{i+1},...,z_{m})\\in\\mathcal{X}^{m-1}$ for any $z\\in\\mathcal{X}^{m}$ and for any $i$.\n   \n    Then, since $0\\leq ||P_{V_k}(\\psi(z_i))||^2 \\leq R^2$ for all $i$, we have \n    \\[\n    0\\leq f(z)-f_i(z^{(i)})= \\frac{R^2-||P_{V_k}(\\psi(z_i))||^2}{R^2} \\leq 1 .\n    \\]\nMoreover, because $f(z)\\leq m$ for any $z\\in\\mathcal{X}^m$, we have\n\\begin{align*}\n    \\sum_{i=1}^m f(z)-f_i(z^{(i)}) \n    & =  \\sum_{i=1}^m \\frac{R^2-||P_{V_k}(\\psi(z_i))||^2}{R^2} \\\\\n    & = f(z) =\\beta f(z) + (1-\\beta)f(z)\n    \\leq \\beta f(z) + (1-\\beta)m.\n\\end{align*}\nSince this holds for any $x\\in\\mathcal{X}^{m}$, this proves that $f$ is $(\\beta,1-\\beta)$-self-bounding.\n\nNow, to complete the proof, we will use \\cref{th: exp_inequality2009 }. Because $Z$ is $(1\/3,(2\/3)m)$-self-bounding, we have for all $s\\in\\mathbb{R}^{+}$\n\\[ \n\\log\\left(\\mathbb{E}_{\\mathcal{S}}\\left[e^{s(Z-\\mathbb{E}_{\\mathcal{S}}[Z])}  \\right] \\right) \\leq \\frac{\\left(\\frac{1}{3}\\mathbb{E}_{\\mathcal{S}}[Z] +\\frac{2m}{3} \\right)s^2}{2}.  \n\\]\nAnd since $Z\\leq m$:\n\\begin{align*}\n    \\mathbb{E}_{\\mathcal{S}}\\left[ e^{m^\\alpha(\\mathbb{E}_{\\mathcal{S}}[Y]-Y)}\\right] \n    &= \\mathbb{E}_{\\mathcal{S}}\\left[ e^{\\frac{R^2}{m^{1-\\alpha}}(Z-\\mathbb{E}_{\\mathcal{S}}[Z])}\\right] \n        & \\\\\n    &\\leq \\exp\\left( \\frac{\\left(\\frac{1}{3}\\mathbb{E}_{\\mathcal{S}}[Z] +\\frac{2m}{3} \\right)R^4}{2m^{2-2\\alpha}}   \\right)\n        & (\\emph{\\cref{th: exp_inequality2009 }})\\\\\n    & \\leq \\exp\\left( \\frac{R^4}{2m^{1-2\\alpha}} \\right).\n        & (\\textrm{since } \\mathbb{E}_{\\mathcal{S}}[Z]\\leq m).\n\\end{align*}\nSo, finally, we have\n\\[ \\frac{\\log(\\xi)}{2m^{\\alpha}} \\leq \\frac{R^4}{m^{1-\\alpha}},  \\]\nhence the final result.\n\nTo obtain the optimal value of $\\alpha$, we simply study the derivative of the univariate function $f_{R,\\delta}(\\alpha):= \\frac{\\log(1\/\\delta)}{m^{\\alpha}} + \\frac{R^{4}}{m^{1-\\alpha}}$.\n\\end{proof}\n\nWe now prove \\cref{th: pac_bayes_thm_residual} which deals with the expected square residuals. We recall the theorem: for a finite data space $\\mathcal{X}$, for $\\alpha\\in\\mathbb{R}$, $\\delta\\in]0,1]$, for any $1\\leq k\\leq m$, we have with probability $1-\\delta$ over the random $m$-sample $S$:\n\\begin{align*}\n    \\mathbb{E}_{\\mu}[||P_{\\hat{V}^{\\bot}_k}(\\psi(x))||^2]\n  &\\geq \\frac{1}{m}\\sum_{i=k+1}^m \\hat{\\lambda}_i - \\frac{\\log(1\/\\delta)}{m^{\\alpha}} - \\frac{R^4}{2m^{1-\\alpha}}\n\\end{align*}\nand the optimal value for $\\alpha$ is $\\alpha_0= \\frac{1}{2}+\\frac{1}{2\\log(m)}\\log\\left( \\frac{2\\log(1\/\\delta)}{R^{4}} \\right)$.\n\n\n\\begin{proof}[Proof of \\cref{th: pac_bayes_thm_residual}]\n    The proof is similar to the one of \\cref{th: pac_bayes_thm_proj} but  it rather involves the stochastic kernel $Q^k$. Let $k\\in[m]$. We first apply \\cref{th: rivasplata2020} with probability $1-\\delta$, $F:(x,y)\\mapsto m^{\\alpha}(x-y)$ and the stochastic kernel $Q^k$ (cf. Theorem \\ref{th: measurability_kernel_Q}) as prior and posterior which nullify the KL-divergence term. We then have with probability $1-\\delta$:\n\\[ m^{\\alpha}( Q^k_S(\\hat{L}_S(f))-Q^k_S(L(f))) \\leq \\operatorname{KL}(Q^k_S,Q_S^{k}) + \\log(\\xi\/\\delta)   \\] \nwhere $\\xi:= \\mathbb{E}_{S\\sim \\mu^m} \\mathbb{E}_{f\\sim Q^{k}_S}\\left[\\exp\\left(m^{\\alpha}(\\hat{L}_S(f)-(L(f))\\right)\\right]$.\n\nWe notice that for any sample $S$, $Q^k_S$ is the Dirac measure in $f_{\\hat{w}_k(S)}$ i.e. the function representing the squared norm projection $||P_{\\hat{V}_k(S)}(.)||^2$. Hence $Q^k_S(L(f))= L(f_{\\hat{w}_k(S)})$, $Q^k_S(\\hat{L}_S(f))= \\hat{L}_S(f_{\\hat{w}_k(S)}) $.\nFinally we have:\n \\[ m^{\\alpha}\\left( \\hat{L}_S(f_{\\hat{w}_k(S)})-L(f_{\\hat{w}_k(S)})\\right)) \\leq  \\log(\\xi\/\\delta)   \\]\nhence\n\\[ m^{\\alpha}\\left(\\mathbb{E}_{\\hat{\\mu}}\\left[ ||\\psi(x)||^2- ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]  -  \\mathbb{E}_{\\mu}\\left[ ||\\psi(x)||^2- ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]\\right) \\leq  \\log(\\xi\/\\delta),   \\] \nwhere $\\hat{\\mu}$ is the empirical distribution over the $m$-sample $S$.\n\nFurthermore, because we are considering orthogonal projections, we remark that for any $x\\in\\mathcal{X}$, we have: \n\\[ ||\\psi(x)||^2- ||P_{\\hat{V}_k(S)}(\\psi(x))||^2 =  ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2. \\]\nSo we have by multiplying by $-1$:\n\\begin{align*}\n\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] &\\geq  \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]  -\\frac{\\log(\\xi\/\\delta)}{m^{\\alpha}}.\n\\end{align*}\nThanks to Proposition \\ref{p: characterization_eigenvalues}, we know that $\\mathbb{E}_{ \\hat{\\mu}}\\left[ ||P_{\\hat{V}_k(S)}(\\psi(x))||^2  \\right]= \\frac{1}{m}\\sum_{i=k+1}^m \\hat{\\lambda}_i$ and then we have: \n\\begin{align*}\n\\mathbb{E}_{x\\sim \\mu}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] &\\geq  \\frac{1}{m}\\sum_{i=k+1}^m \\hat{\\lambda}_i  -\\frac{\\log(\\xi\/\\delta)}{m^{\\alpha}}.\n\\end{align*}\nFinally we need to control $\\log(\\xi)$. To do so, we use \\cref{th: exp_inequality2009 }. We first recall that:\n\\begin{align*}\n\\xi &:= \\mathbb{E}_{S\\sim \\mu^m} \\mathbb{E}_{f\\sim Q^{k}_S}\\left[\\exp\\left(m^{\\alpha}(\\hat{L}_S(f)-L(f))\\right)\\right] \\\\\n& = \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}(\\hat{L}_S(f_{\\hat{w}_k})-L(f_{\\hat{w}_k}))\\right)\\right] \\\\\n&= \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}\\left(\\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] \\right)\\right)\\right].\n\\end{align*}\nYet, thanks to Propositions \\ref{p:courant-fischer} and \\ref{p: characterization_eigenvalues}, we know that:\n\\begin{align*}\n    \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] \\geq \\min_{\\dim(V)=k} \\mathbb{E}_\\mu[||P_{V^{\\bot}}(\\psi(x))||^2] = \\mathbb{E}_{\\mu}\\left[ ||P_{V^{\\bot}_k(S)}(\\psi(x))||^2  \\right]\n\\end{align*}\nand \n\\begin{align*}\n    \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]]& =  \\min_{\\dim(V)=k} \\mathbb{E}_{\\hat{\\mu}}[||P_{V^\\bot}(\\psi(x))||^2]\\\\\n    & \\leq \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V^{\\bot}_k}(\\psi(x))||^2  \\right].\n\\end{align*}\nThus we have \n\\[ \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V^{\\bot}}_k(S)}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V^{\\bot}}_k(S)}(\\psi(x))||^2  \\right]  \\leq \\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V^{\\bot}_k}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\mu}\\left[ ||P_{V^{\\bot}_k}(\\psi(x))||^2  \\right].\\]\n\n\\begin{remark}\n The interest of this maneuver is to replace the projector $P_{\\hat{V}_k(S)}$, which is data-dependent, by $P_{V_k}$, which is not. Doing so, if we set $Y= \\frac{1}{m}\\sum_{i=1}^m Y_i  $ and for all $i$, $Y_i=||P_{V^{\\bot}_k}(\\psi(x_i))||^2 $ (where the $Y_i$ are iid), we can write properly: \n\\[\\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{V^{\\bot}_k}(\\psi(x))||^2  \\right] - \\mathbb{E}_{\\mu}\\left[ ||P_{V^{\\bot}_k}(\\psi(x))||^2  \\right]= Y-\\mathbb{E}_{\\mathcal{S}}[Y]  \\]\nwhile $\\mathbb{E}_{\\mu}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right] \\neq \\mathbb{E}_{\\mathcal{S}}\\left[\\mathbb{E}_{\\hat{\\mu}}\\left[ ||P_{\\hat{V}^{\\bot}_k(S)}(\\psi(x))||^2  \\right]\\right]$.\n\\end{remark}\nSo, finally, we have:\n\\[\\xi \\leq  \\mathbb{E}_{S\\sim \\mu^m} \\left[\\exp\\left(m^{\\alpha}(Y-\\mathbb{E}_{\\mathcal{S}}[Y] )\\right)\\right] . \\] \nWe define the function $f : \\mathcal{X}^{m} \\to \\mathbb{R}$ as\n    \\[\n    f:z \\mapsto \\frac{1}{R^2}\\hspace{1mm}\\sum_{i=1}^{m}||P_{V^{\\bot}_k}(\\psi(z_i))||^2\\\n    \\hspace{7mm}\\text{for}\\hspace{2mm} z=(z_1,...,z_m)\\in\\mathcal{X}^{m}.\n    \\]\n\n    We define $Z=f(X_1,...,X_m)$ and notice that $\\mathbb{E}_{Y-\\mathcal{S}}[Y]= \\frac{R^2}{m}\\left( Z- \\mathbb{E}_{\\mathcal{S}}[Z] \\right)$. \n    We first prove that $f\\in\\texttt{SB}(\\beta,1-\\beta)$ (cf. Def \\ref{d: self-bounding}) for any $\\beta\\in[0,1]$.\n    \n    Indeed, for all $1\\leq i\\leq m$, we define:\n    \\[\n    f_i(z^{(i)})= \\frac{1}{R^2}\\sum_{j\\neq i} ||P_{V_k}(\\psi(z_i))||^2\n    \\]\n    where $z^{(i)}=(z_1,...,z_{i-1},z_{i+1},...,z_{m})\\in\\mathcal{X}^{m-1}$ for any $z\\in\\mathcal{X}^{m}$ and for any $i$.\n   \n    Then, since $0\\leq ||P_{V^{\\bot}_k}(\\psi(z_i))||^2 \\leq R^2$ for all $i$, we have \n    \\[\n    0\\leq f(z)-f_i(z^{(i)})= \\frac{||P_{V^{\\bot}_k}(\\psi(z_i))||^2}{R^2} \\leq 1 .\n    \\]\nMoreover, because $f(z)\\leq m$ for any $z\\in\\mathcal{X}^m$, we have:\n\\begin{align*}\n    \\sum_{i=1}^m f(z)-f_i(z^{(i)}) \n    & =  \\sum_{i=1}^m \\frac{||P_{V^{\\bot}_k}(\\psi(z_i))||^2}{R^2} \\\\\n    & = f(z) =\\beta f(z) + (1-\\beta)f(z)\n    \\leq \\beta f(z) + (1-\\beta)m.\n\\end{align*}\nSince this holds for any $x\\in\\mathcal{X}^{m}$, this proves that $f$ is $(\\beta,1-\\beta)$-self-bounding.\n\nNow, to complete the proof, we use \\cref{th: exp_inequality2009 }. Because $Z$ is $(1\/3,(2\/3)m)$-self-bounding, we have for all $s\\in\\mathbb{R}^{+}$:\n\\[ \n\\log\\left(\\mathbb{E}_{\\mathcal{S}}\\left[e^{s(Z-\\mathbb{E}_{\\mathcal{S}}[Z])}  \\right] \\right) \\leq \\frac{\\left(\\frac{1}{3}\\mathbb{E}_{\\mathcal{S}}[Z] +\\frac{2m}{3} \\right)s^2}{2}.  \n\\]\nAnd since $Z\\leq m$:\n\\begin{align*}\n    \\mathbb{E}_{\\mathcal{S}}\\left[ e^{m^\\alpha(\\mathbb{E}_{\\mathcal{S}}[Y]-Y)}\\right] \n    &= \\mathbb{E}_{\\mathcal{S}}\\left[ e^{\\frac{R^2}{m^{1-\\alpha}}(Z-\\mathbb{E}_{\\mathcal{S}}[Z])}\\right] \n        & \\\\\n    &\\leq \\exp\\left( \\frac{\\left(\\frac{1}{3}\\mathbb{E}_{\\mathcal{S}}[Z] +\\frac{2m}{3} \\right)R^4}{2m^{2-2\\alpha}}   \\right)\n        & (\\cref{th: exp_inequality2009 })\\\\\n    & \\leq \\exp\\left( \\frac{R^4}{2m^{1-2\\alpha}} \\right).\n        & (\\textrm{since } \\mathbb{E}_{\\mathcal{S}}[Z]\\leq m).\n\\end{align*}\nSo, finally, we have\n\\[ \\frac{\\log(\\xi)}{m^{\\alpha}} \\leq \\frac{R^4}{2m^{1-\\alpha}}.  \\]\nHence the final result.\n\nTo obtain the optimal value of $\\alpha$, we simply study the derivative of the univariate function $f_{R,\\delta}(\\alpha):= \\frac{\\log(1\/\\delta)}{m^{\\alpha}} + \\frac{R^{4}}{m^{1-\\alpha}}$.\n\\end{proof}\n    \n\n\\section{A brief numerical illustration}\n\\label{sec: experiments}\n\nIn this section we briefly illustrate the numerical behaviour of our lower and upper bounds, with respect to \\cite{shawe-taylor2005eigenspectrum}. We conduct two experiments below.\n\n\\paragraph{Experiment 1}\nWe exploit the dataset used in  \n\\citet{shawe-taylor2005eigenspectrum} to compare our bound in \\cref{th : pac_inequality_1}, which we recall here:\n\\[ \n\\frac{1}{m}\\sum_{i=1}^m ||P_{\\hat{V}_k(S_1)}(\\psi(x_{m+i}))||^2 -R^2\\sqrt{\\frac{2}{m}\\log\\left(\\frac{1}{\\delta}\\right)},\n\\]\nwith the one from \\citet{shawe-taylor2005eigenspectrum}, given by\n\\[ \n\\max_{1\\leq \\ell\\leq k} \\left[\\frac 1 m\\hat{\\lambda}^{\\leq \\ell}(S) - \\frac{1 + \\sqrt{\\ell}} {\\sqrt{m}} \\sqrt{\\frac 2 m \\sum_{i=1}^m \\kappa(x_i,x_i)^2}\\right] - R^2 \\sqrt{\\frac {19} {m} \\log\\left(\\frac {2(m+1)} {\\delta}\\right)}.\n\\]\nWe choose $\\delta=0.05$, and the dataset size is $N= 696$. We take $m=\\frac N 4 = 174$, and $k\\in\\{1,\\dots,100\\}$.\n\nWe generate a $2m$-sized dataset $S=S_1\\cup S_2$, we apply our kernel PCA method over $S_1$ (to obtain the projection space $\\hat{V}_{\\mathcal{S_1}}$), then we compute our bound by using $S_2= \\{x_{m+1},\\cdots, x_{2m}\\}$. This is the blue curve in \\autoref{experiment1_pac_inequality}.\n\nWe then compute the bound from \\citet{shawe-taylor2005eigenspectrum} by using the eigenvalues of $K(S)$ (the orange curve in \\autoref{experiment1_pac_inequality}).\nFinally, we draw $\\lambda^{\\leq k}$ (green curve).\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{experiment_1.png}\n    \\caption{Evaluation of \\cref{th : pac_inequality_1}. The x-axis is the number $k$ of considered eigenvalues to compute the bound, the y-axis is the amount of information contained in the projection, from 0 to 1.} \n    \\label{experiment1_pac_inequality}\n\\end{figure}\nClearly, on this specific instance of the kernel PCA problem, our \\cref{th : pac_inequality_1} leads to a much tighter bound than the one of \\cite{shawe-taylor2005eigenspectrum}. Let us stress here that this is merely a safety check on a specific example.\n\n\n\\paragraph{Experiment 2}\n\nWe are using the same experimental framework as in Experiment 1. We now compute four curves: the theoretical eigenvalues, the bound from \\cref{th: pac_bayes_thm_proj} with the `naive' choice of $\\alpha=1\/2$  and also with the optimised $\\alpha_0$. We also compute the bound from \\cref{th : pac_inequality_1}. Results are shown in \\autoref{experiment_2}. Clearly, the choice of $\\alpha$ significantly influences the tightness of the bound.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{experiment_3.png}\n    \\caption{The x-axis is the number $k$ of considered eigenvalues to compute the bound, the y-axis is the amount of information contained in the projection, from 0 to 1.}\n    \\label{experiment_2}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec:end}\n\nWe provided empirical bounds for two quantities: the expected squared norm and the expected residual of a new data point projected onto the empirical (small) subspaces given by the kernel PCA method. This outperforms (as illustrated on an example) the existing bounds given by \\cite{shawe-taylor2005eigenspectrum}. Another improvement on the seminal work of \\cite{shawe-taylor2005eigenspectrum} is that we provide both lower and upper empirical bounds for each studied quantity. Doing so, we contribute a better theoretical understanding of kernel PCA, which we hope will translate into practical insights on strengths and limitations of kernel PCA for machine learning theoreticians and practitioners.\n\n\n\\section{Proofs -- technical results}\n\\label{sec:proofs}\n\n\\subsection{Proof of \\cref{th: measurability_kernel_Q}}\n\n\n\n\nFirst, for all $k$ and $s\\in\\mathcal{X}^m$, the function $B\\mapsto Q^k_s(B)$ is the Dirac in $||P_{\\hat{V}_k(s)}(.)||^2 \\in \\mathcal{F}_k$ hence it is a well-defined probability law.\nNow, we fix $k\\in\\{1.. N_{\\mathcal{H}}\\}$ and $B\\in \\Sigma_{\\mathcal{F}_k}$. We need to prove that the function $A:s\\mapsto Q^k(s,B)$ is measurable. We first decompose $A$ into several functions:\n\n\\begin{align*} \ns & \\stackrel{\\psi}{\\longmapsto} \\hspace{1mm} (\\psi(x_1)\\cdots \\psi(x_m)) \\hspace{1mm} \\stackrel{A_1}{\\longmapsto} \\hspace{1mm} C(s) \\hspace{1mm}\\stackrel{A_2}{\\longmapsto}\\hspace{1mm} \\text{eigenvectors of } C(s)  \\\\\n& \\stackrel{A_3}{\\longmapsto} \\hspace{1mm} f_{\\hat{w}_k} \\hspace{1mm} \\stackrel{A_4}{\\longmapsto} \\hspace{1mm} \\mathds{1}\\left\\{  f_{\\hat{w}_k}\\in B   \\right\\}\n\\end{align*}\nFor all the intermediate spaces (which are all finite-dimensional vector spaces, or subsets of them), we will consider them with their Lebesgue $\\sigma-$algebra (or the one induced on the corresponding subset), which will allow us to consider the usual notion of continuity onto those spaces. Then for every $s$, we have $A(s)=  A_4 \\circ A_3 \\circ A_2 \\circ A_1 \\circ \\psi (s)$.\\\\\nNow our goal is to prove that all those functions are measurable, doing so, $A$ will be measurable as composition of measurable functions.\n\nFirst, because the $\\sigma$-algebra on $\\mathcal{X}$ is $\\mathcal{P}(\\mathcal{X})$, we know that $\\psi$ is measurable. \n\n\\paragraph{Measurablity of $A_1$.} Thanks to the definition of $C(S)$, we know that every coordinate of $C(s)$ consists in a linear combination of the coordinates of $(\\psi(x_1),\\cdots, \\psi(x_m)$. Thus, $A_1$ is continuous therefore measurable.\n\\paragraph{Measurablity of $A_2$.} To prove that $A_2$ is measurable, we need to show that the eigenvectors from a symmetric matrix ($C(s)$ is indeed symmetric) are a measurable function of this matrix. This result is true: to prove it, we will detail the problem treated in \\cite{wilcox1972}. Let us consider a polynomial \n\\[M(p)= \\sum_{|\\alpha|\\leq q} M_\\alpha p^\\alpha    \\]\n where $ q\\in\\mathbb{N}$ $\\alpha= (\\alpha_1,\\cdots,\\alpha_n)\\in\\mathbb{N}^n, |\\alpha|= \\alpha_1+\\cdots+\\alpha_n $. Also, $p=(p_1,...,p_n)\\in\\mathbb{R}^n, p^{\\alpha}= p_1^{\\alpha1}\\cdots p_n^{\\alpha_n}$.\n Finally, for any $\\alpha$, $M_\\alpha$ is an Hermitian matrix of size $d\\times d$. Let us consider the following eigenvalue problem for $E$ an Hermitian positive definite matrix: \n\\begin{align*}\n\\label{eq: eigenvalue_problem}\n    M(p)x= \\lambda E x\\hspace{5mm} x\\in\\mathbb{R}^d.\n\\end{align*}\nIf we denote by $\\lambda_1(p)\\geq\\cdots\\geq \\lambda_d(p)$ the $d$ eigenvalues of $M(p)$ in descending order, we can use the following:\n\\begin{theorem}[\\citealp{wilcox1972}, Theorem 2]\n\\label{th: wilcox1972}\n    There exists functions\n    $v_i : \\mathbb{R}^n \\to \\mathbb{R}^d$, $i \\in [d]$, \n    such that \n    \\begin{itemize}\n        \\item For all $i\\in [d]$ and $p\\in\\mathbb{R}^n$, it holds that $M(p)v_i(p)= \\lambda_i(p)v_i(p)$;\n        \\item For all $1\\in [d]$, the function $v_i(p)$ is Lebesgue measurable.\n    \\end{itemize}\n\\end{theorem}\nWe now prove the following lemma:\n\\begin{lemma}\n    Let $M = (m_{i,j})_{i,j}\\in\\mathbb{R}^{d\\times d}$ be a symmetric matrix, then the eigenvectors of $M$ are measurable functions on the coordinates of $M$.\n\\end{lemma}\n\\begin{proof}\nWe set for $(i,j)\\in[d]\\times[d]$, $E_{i,j}$ the matrix with value $1$ in coordinate $(i,j)$ and $0$ everywhere else. Then we take $n= d(d+1)\/2$ and we define $p=(m_{i,j})_{i\\leq j}\\in\\mathbb{R}^{n}$. We also define for $i<j$, $M_{i,j}:= E_{i,j}+ E_{j,i}$ and if i=j $M_{i,i}= E_{i,i}$. In any case, $M_{i,j}$ is an Hermitian matrix. \n\nWe set for $i\\leq j$, $\\alpha_{i,j}$ the vector of $\\mathbb{R}^n$ whom all the coordinates are null except the $(i,j)$-th which is equal to 1.\nDoing so, we have for all $(i,j)$, $p^{\\alpha_{i,j}}= m_{i,j}$. \nFinally we prove that we have the equality \n\\[M= M(p)= \\sum_{i\\leq j} M_{i,j}p^{\\alpha_{i,j}}   \\]\nbecause $M$ is symmetric. Now, we take $E=I_d$, then the considered eigenvalue problem is for any $x\\in\\mathbb{R}^d$:\n\\[M(p)x= \\lambda x.\\]\n\nNow we can apply \\cref{th: wilcox1972}, we obtain that the eigenvectors of $M$ are measurable on $p$, \\emph{i.e.} the coordinates of $M$.\n\\end{proof}\n\nFinally we just have to apply that last lemma to $M=C(s)\\in\\mathbb{R}^{N_{\\mathcal{H}}\\times N_{\\mathcal{H}}} $ to conclude that $A_2$ is measurable in $C(s)$.\n\nHence $A_2$ is measurable.\n\n\\paragraph{Measurability of $A_3$.} Thanks to the proof of Proposition \\ref{p:shawetaylor_prop_11}, we know that if $U$ is the matrix whom the $i$-th column correspond to $v_i(s)$ the $i$-th eigenvector of $C(s)$, then we have, if $I_k=\\{1..k\\}$:\n\n\\[\\forall x\\in\\mathcal{X}, f_{\\hat{w}_k(s)}(x)= ||P_{\\hat{V}_k(s)}(\\psi(x))||^2= \\psi(x)'U(I_k)U(I_k)' \\psi(x)   \\]\nwhere $U(I_k)\\in\\mathbb{R}^{N_{\\mathcal{H}\\times N_{\\mathcal{H}}}}$  consists in the $k$ first columns of U filled with $\\mathcal{N_{\\mathcal{H}}}-k$ columns of 0. Hence, if we reordinate the coordinate one can affirm that $\\hat{w}_k= U(I_k)U(I_k)' $.\n\nThen the coordinates of $\\hat{w}_k(s)$ consists in a linear combination of the coordinates of the $k-$first eigenvectors of $C(s)$. \nSo the function from $(\\mathbb{R}^{N_{\\mathcal{H}}})\\times{N_{\\mathcal{H}}}$ to $\\mathbb{R}^{N_{\\mathcal{H}}^2}$ which transform the eigenvectors into $\\hat{w}_k$ is continuous therefore measurable. Furthermore, $w \\mapsto f_w$ is an isomorphism between $\\mathbb{R}^{N_{\\mathcal{H}}^2} $ and its dual which are two finite-dimensional spaces, it is then continuous, so measurable.\n\nFinally we can conclude that $A_3$ is measurable.\n\\paragraph{Measurability of $A_4$.}\nBecause we chose on $\\mathcal{F}_k$ the $\\sigma$-algebra $\\mathcal{P}(\\mathcal{F}_k)$, we know that $B$ is measurable on $\\mathcal{F}_k$ and so is the indicator function $\\mathds{1}\\{\\cdot\\in B\\}$. So $A_4$ is measurable, hence the final result.\n\n\n\\subsection{Proof of \\cref{th: measurability_kernel_P}}\n\n\nThe proof is similar to the one of Theorem \\ref{th: measurability_kernel_Q}. Indeed; for some fixed $k$ and $B\\in\\Sigma_{\\mathcal{F}_k}$, we have the following decomposition of $A':s\\mapsto P^k(s,B):$\n\n\\begin{align*} \ns & \\stackrel{\\psi}{\\longmapsto} \\hspace{1mm} (\\psi(x_1)\\cdots \\psi(x_m)) \\hspace{1mm} \\stackrel{A_1}{\\longmapsto} \\hspace{1mm} C(s) \\hspace{1mm}\\stackrel{A_2}{\\longmapsto}\\hspace{1mm} \\text{eigenvectors of } C(s)  \\\\\n& \\stackrel{A'_3}{\\longmapsto} \\hspace{1mm} f_{\\hat{w}^{\\bot}_k} \\hspace{1mm} \\stackrel{A_4}{\\longmapsto} \\hspace{1mm} \\mathds{1}\\left\\{  f_{\\hat{w}^{\\bot}_k}\\in B   \\right\\}\n\\end{align*}\nWhere the functions $\\psi,A_1,A_2,A_4$ are the same than those in the proof of \\cref{th: measurability_kernel_Q} and we already proved their measurability. Thus, because $A'=A_4 \\circ A'_3 \\circ A_2 \\circ A_1 \\circ \\psi$, we just have to prove that $A'3$ is measurable. For that we suppose that we have $(v_1,...,v_m)$ the eigenvectors of $C(S)$ (ordinate according to the value of their asociated eigenvalue), if we take as in Proposition \\ref{p:shawetaylor_prop_11}, $U$ the matrix of the eigenvectors then we have, if $J_k=\\{k+1,...,m\\}$:\n\\[\\forall x\\in\\mathcal{X}, f_{\\hat{w}^{\\bot}_k(s)}(x)= ||P_{\\hat{V}^{\\bot}_k(s)}(\\psi(x))||^2= \\psi(x)'U(J_k)U(J_k)' \\psi(x)   \\]\nwhere $U(J_k)\\in\\mathbb{R}^{N_{\\mathcal{H}\\times N_{\\mathcal{H}}}}$  consists in the $N_\\mathcal{H}-k$ last columns of U filled with $k$ columns of 0. Hence, if we reordinate the coordinate one can affirm that $\\hat{w}_k= U(J_k)U(J_k)' $.\n\nThen the coordinates of $\\hat{w}^{\\bot}_k(s)$ consists in a linear combination of the coordinates of the $k-$first eigenvectors of $C(s)$. \nSo the function from $(\\mathbb{R}^{N_{\\mathcal{H}}})\\times{N_{\\mathcal{H}}}$ to $\\mathbb{R}^{N_{\\mathcal{H}}^2}$ which transform the eigenvectors into $\\hat{w}_k$ is continuous therefore measurable. Furthermore, $w \\mapsto f_w$ is an isomorphism between $\\mathbb{R}^{N_{\\mathcal{H}}^2} $ and its dual, which are two finite-dimensional spaces, it is then continuous, so measurable.\n\nFinally we can claim that $A'_3$ is measurable.\nHence $A'$ is measurable, which concludes the proof and this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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De Angelis, N. Merkulov, J. Palczewski]{Tiziano De Angelis, Nikita Merkulov, Jan Palczewski}\n\\address{T.\\ De Angelis: School of Management \\& Economics, Dept.\\ ESOMAS, University of Turin, C.so Unione Sovietica 218bis, 10134, Torino, Italy}\n\\email{tiziano.deangelis@unito.it (T. De Angelis)} \n\\address{N.\\ Merkulov and J.\\ Palczewski: School of Mathematics, University of Leeds, LS2 9JT, Leeds, UK}\n\\email{mmnme@leeds.ac.uk (N. Merkulov)}\n\\email{j.palczewski@leeds.ac.uk (J. Palczewski)}\n\\thanks{2020 {\\em Mathematics Subject Classification}: 91A27, 91A55, 91A15, 60G07, 60G40}\n\\keywords{non-Markovian Dynkin games, partial information, asymmetric information, optimal stopping, randomised stopping times, regular processes, predictable-jump processes.}\n\\thanks{{\\em Acknowledgments}: T.~De Angelis gratefully acknowledges support by the EPSRC grant EP\/R021201\/1. We are grateful to an anonymous referee who pointed us to the existence of optimal strategies and suggested the equivalence of our topology with the one used by Baxter and Chacon \\cite{BaxterChacon} and Meyer \\cite{Meyer} (see our Lemma \\ref{lem:top})}\n\n\\begin{abstract}\nWe prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous time with partial\/asymmetric information. Instead, we focus on a probabilistic and functional analytic approach based on the general theory of stochastic processes and Sion's min-max theorem (M.\\ Sion, Pacific J. Math., {\\bf 8}, 1958, pp.\\ 171-176). Our framework encompasses examples found in the literature on continuous time Dynkin games with asymmetric information and we provide counterexamples to show that our assumptions cannot be further relaxed.\n\\medskip\n\n \n\\end{abstract}\n\n\\maketitle\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\nIn this paper we develop a framework for the study of the existence of a value (also known as {\\em Stackelberg equilibrium}) in zero-sum Dynkin games with partial\/asymmetric information  in a non-Marko\\-vian setting, when the payoffs are general c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} measurable  processes and players are allowed to use randomised stopping times. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The games are considered on both finite and infinite-time horizon and the horizon is denoted by $T$.  The payoff processes can be decomposed into the sum of a {\\em regular} process (in the sense of Meyer \\cite{Meyer}) and a pure jump process with mild restrictions on the direction of predictable jumps for one of the two players. Regular processes form a very broad class encompassing, for example, all c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} processes that are also quasi left-continuous (i.e., left-continuous over stopping times). \n\nWe allow for a very general structure of the information available to the players. All processes are adapted to an overarching filtration $(\\mathcal{F}_t)$ whereas each player makes decisions based on her own filtration, representing her access to information. Letting $(\\mathcal{F}^{\\,i}_t)$ be the filtration of the $i$-th player, with $i=1,2$, we only need to assume that $\\mathcal{F}^{\\,i}_t\\subseteq\\mathcal{F}_t$ for all $t\\in[0,T]$. In particular, we cover the case in which players are equally (partially) informed, i.e., $\\mathcal{F}^1_t=\\mathcal{F}^2_t$, and, more importantly, the case in which they have asymmetric (partial) information, i.e., $\\mathcal{F}^1_t\\neq\\mathcal{F}^2_t$.\n\nUnder this generality we prove that Dynkin games with second-mover advantage admit a value in mixed strategies (which in this context are represented by randomised stopping times) and optimal strategies exist for both players. \n\nOur framework encompasses most (virtually all) examples of zero-sum Dynkin games (in continuous time) with partial\/asymmetric information that we could find in the literature (see, e.g., De Angelis et al. \\cite{DGV2017}, \\cite{DEG2020}, Gensbittel and Gr\\\"un \\cite{GenGrun2019}, Gr\\\"un  \\cite{Grun2013} and Lempa and Matom\\\"aki \\cite{lempa2013}) and we give a detailed account of this fact in Section \\ref{sec:examples} (notice that \\cite{DGV2017} and \\cite{lempa2013} obtain a saddle point for the game in pure strategies, i.e., using stopping times, but in very special examples). Broadly speaking, all those papers' solution methods hinge on variational inequalities and PDEs and share two key features: (i) a specific structure of the information flow in the game and (ii) the Markovianity assumption. In our work instead we are able to analyse games at a more abstract level that allows us to drop the Markovianity assumption and to avoid specifying an information structure. Of course the cost to pay for such greater generality is a lack of explicit results concerning the value and the optimal strategies (beyond their existence), which instead may be obtained in some problems satisfying (i) and (ii) above. We also show by several counterexamples that our main assumptions cannot be further relaxed as otherwise a value for the game may no longer exist.\n\nOur methodology draws on the idea presented in Touzi and Vieille \\cite{TouziVieille2002} of using Sion's min-max theorem (Sion \\cite{Sion1958}). In \\cite{TouziVieille2002} the authors are interested in non-Markovian zero-sum Dynkin games with full information in which first- and second-mover advantage may occur at different points in time, depending on the stochastic dynamics of the underlying payoff processes. In that context randomisation is essentially used by the players to attain a value in the game and avoid stopping simultaneously (another general study on such class of problems, but without using Sion's theorem, is contained in Laraki and Solan \\cite{LarakiSolan2005}). Since our set-up is different, due to the partial\/asymmetric information features and relaxed assumptions on the payoff processes, we encounter some non-trivial technical difficulties in repeating arguments from \\cite{TouziVieille2002}; for example, our class of randomised stopping times is not closed with respect to the topology used in \\cite{TouziVieille2002} (see Remark \\ref{rem-TV-norm-doesnt-work}). For this reason we develop an alternative approach based on the general theory of stochastic processes combined with ideas from functional analysis.\n\n\\subsection{Literature review}\nExistence of a value (Stackelberg equilibrium) in zero-sum Dynkin games is a research question that goes back to the 70's in the classical set-up where players have full and symmetric information. A comprehensive and informative review of the main results since Dynkin's inception of stopping games \\cite{dynkin1969} is contained in the survey paper by Kifer \\cite{kifer2013}. Here we recall that early results on the existence of a value in a diffusive set-up were obtained by Bensoussan and Friedman \\cite{bensoussan1974} via PDE methods, and by Bismut \\cite{bismut1977} via probabilistic methods (and allowing for processes with jumps). Those results were later extended to right-continuous Markov processes by Stettner \\cite{stettner1982} (see also Stettner \\cite{stettner1982b} and \\cite{stettner1984}). In the non-Markovian setting the early results are due to Lepeltier and Maingueneau \\cite{lepeltier1984}. Around the year 2000 zero-sum Dynkin games gained popularity thanks to their applications in mathematical finance suggested by Kifer \\cite{kifer2000} (see also Kyprianou \\cite{kyprianou2004} for an early contribution). Numerous other papers have addressed related questions, including the existence of value and optimal strategies, with various methods. The interested reader may consult Ekstr\\\"om and Peskir \\cite{ekstrom2008} for modern results in a general Markovian setting or Ekstr\\\"om and Villeneuve \\cite{ekstrom2006} for the special case of one-dimensional linear diffusions. It is also worth mentioning that nonzero-sum Dynkin games have been studied by many authors including, e.g., Hamadene and Zhang \\cite{hamadene2010} for non-Markovian games,  Attard \\cite{attard2018} for general Markovian games and De Angelis et al. \\cite{de2018nash} for games on one-dimensional linear diffusions. All the papers mentioned in this (largely incomplete) literature review deal with players holding full and symmetric information and the value is found in {\\em pure strategies}, i.e., in stopping times. \n\nYasuda \\cite{yasuda1985} was probably the first to study the existence of a value for Dynkin games with {\\em randomised stopping times}, in the case of discrete-time Markov processes. In that context, randomisation is specified by assigning a probability of stopping at each time $n=1,2,\\ldots$. A similar type of games for Markov chains with an absorbing state was also studied by Domansky \\cite{domansky2002}. Rosenberg, Solan and Vieille \\cite{rosenberg2001} developed more general results by removing the Markovianity assumption and the assumption of an ordering of payoffs (i.e., they did not require that there be a first- or second-mover advantage). Randomised stopping times are also used in mathematical finance, in the context of pricing and hedging game options in discrete time with transaction costs (see \\cite[Sec.\\ 5]{kifer2013}), and in mathematical economics, to construct subgame-perfect equilibria in (nonzero-sum) Dynkin games (see, e.g., Riedel and Steg \\cite{riedel2017}). In continuous time, the most recent results on the existence of a value for non-Markovian zero-sum Dynkin games with randomised stopping times are contained in Touzi and Vieille \\cite{TouziVieille2002} and in Laraki and Solan \\cite{LarakiSolan2005}.\n\nWe emphasise that in all the papers mentioned above players are equally informed and the need for randomisation stems mainly from a specific structure of the game's payoff (often due to the lack of an ordering of the payoff processes that induces alternating first- and second-mover advantage). Our work instead is inspired by a more recent strand of the literature on continuous-time Dynkin games that addresses the role of information across players and its impact on their strategies (see also Section \\ref{sec:examples} for a fuller account). We believe this strand was initiated with work by Gr\\\"un \\cite{Grun2013}, where one of the players knows the payoff process in the game while the other one has access only to an initial distribution of possible payoff processes. Gr\\\"un proves existence of a value in randomised strategies and the existence of an optimal strategy for the informed player. In Gr\\\"un and Gensbittel \\cite{GenGrun2019} players observe two different stochastic processes and the payoffs in the game depend on both processes. The authors prove existence of a value and, under some additional conditions, of optimal strategies for both players. Both these papers attack the problem via a characterisation of the value as the viscosity solution of a certain variational inequality (of a type which is rather new in the literature and is inspired by similar results in the context of differential games; see, e.g., Cardaliaguet and Rainer \\cite{cardaliaguet2009}). Free-boundary methods in connection with randomised stopping times are instead used in De Angelis et al. \\cite{DEG2020}, where players have asymmetric information regarding the drift of a linear diffusion underlying the game, and in Ekstr\\\"om et al. \\cite{ekstrom2017}, where the two players estimate the drift parameter according to two different models. The methods used in those papers cannot be extended to the non-Markovian framework of our paper.\n\nAlthough not directly related to Dynkin games, we notice that methods from functional analysis and the general theory of stochastic processes have been used recently to study optimal stopping problems by Pennanen and Perkk\\\"io \\cite{pennanen2018}. By relaxing the problem to include randomised stopping times the authors reduce the optimal stopping problem to an optimisation of a linear functional over a convex set of randomised stopping times and find that the solution exists as an extreme point, i.e., a pure stopping time. Closely related contributions date back to Baxter and Chacon \\cite{BaxterChacon} and Meyer \\cite{Meyer} who establish compactness of the set of randomised stopping times in weak topologies defined by functionals which can be interpreted as stopping of quasi left-continuous processes, in \\cite{BaxterChacon}, and regular processes, in \\cite{Meyer}. In our game framework we need to rely on min-max arguments instead of convex optimisation as in the optimal stopping case, so compactness arguments are not immediately applicable. However, \\cite{BaxterChacon, Meyer} inspired our approach and some of our convergence results in Section \\ref{sec:tech}. Furthermore, the topology of \\cite{Meyer} on the set of randomised stopping times turns out to be equivalent to the topology obtained via different routes in our paper (Lemma \\ref{lem:top}).\n\n\n\n\\subsection{Structure of the paper}\nThe paper is organised as follows. The problem is set in Section \\ref{sec:setting} where we also state our main result on the existence of a value in full generality (Theorem \\ref{thm:main2}). For the ease of readability we also state a version of the result under slightly stronger conditions on the underlying processes (Theorem \\ref{thm:main}), which allows a more linear approach to the proof. An extension to the case of a game with conditioning on some initial information is also stated as Theorem \\ref{thm:ef_0_value}. Before turning to proofs of our results we use Section \\ref{sec:examples} to illustrate how our framework encompasses Dynkin games in continuous time with partial and asymmetric information that have appeared in the literature to date. Section \\ref{sec:reform} is used to reformulate the Dynkin game in terms of a game of increasing (singular) controls. Section \\ref{sec-Sion-existence-of-value} begins with a statement of Sion's min-max theorem which is followed by a (short) proof of our Theorem \\ref{thm:main}. The latter is based on several technical results which are addressed in detail in Sections \\ref{sec:tech}, \\ref{sec:verif} and \\ref{sec:approx}. The proof of Theorem \\ref{thm:main2} is then given in Section \\ref{sec:relax} and the one of Theorem \\ref{thm:ef_0_value} is finally given in Section \\ref{sec:ef_functional}. We close the paper with a few counterexamples in Section \\ref{sec:Nikita-examples} showing that our conditions cannot be relaxed.\n\n\n\\section{Problem setting and main results}\\label{sec:setting}\nFix a complete probability space $(\\Omega,\\mathcal{F}, \\mathbb{P})$ equipped with a filtration $(\\mathcal{F}_t)_{t\\in[0,T]}$, where $T\\in(0,\\infty]$ is the time horizon of our problem. All random variables, processes and stopping times are considered on this filtered probability space unless specified otherwise. We write $\\mathbb{E}$ for the expectation with respect to measure $\\mathbb{P}$. By a \\emph{measurable process} we mean a stochastic process which is $\\mathcal{B}([0,T])\\times\\mathcal F$-measurable. We denote by $\\mcalL$ a Banach space of c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} measurable processes with the norm\n\\[\n\\| X \\|_{\\mcalL} := \\mathbb{E} \\Big[\\sup_{t\\in[0,T]} |X_t|\\Big] < \\infty.\n\\]\nA process $(X_t)_{t\\in [0,T]} \\in \\mcalL$ is called \\emph{regular} if \n\\begin{align}\\label{eq:cond-reg}\n\\mathbb{E} [X_\\eta - X_{\\eta-}|\\mathcal{F}_{\\eta-}] = 0\\quad \\mathbb{P}\\as \\text{ for all predictable $(\\mathcal{F}_t)$-stopping times $\\eta$.} \n\\end{align}\nNotice that if $T=\\infty$, then $\\infty$ is a one-point compactification of $[0, \\infty)$, so that c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} and regular processes are understood as follows (c.f. \\cite[Remark VI.53e]{DellacherieMeyer}): a process $(X_t)_{t \\in [0, \\infty]}$ is c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} if it is c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} on $[0, \\infty)$ and the limit $X_{\\infty-}:=\\lim_{t \\to \\infty} X_t$ exists; the random variable $X_\\infty$ if $\\mathcal{F}_\\infty$-measurable and $\\mathcal{F}_\\infty$ is potentially different from $\\mathcal{F}_{\\infty-} = \\sigma\\big(\\cup_{t \\in [0, \\infty)} \\mathcal{F}_t\\big)$. Furthermore, $(X_t)_{t\\in[0,\\infty]}$ is regular, if it is regular on $[0, \\infty)$ and \n\\[\n\\mathbb{E}[X_\\infty - \\lim_{t \\to \\infty} X_t | \\mathcal{F}_{\\infty-} ] = 0.\n\\]\nThroughout the paper we consider several filtrations and stochastic processes, so in order to keep the notation simple we will often use $(\\mathcal{F}_t)$ instead of $(\\mathcal{F}_t)_{t\\in[0,T]}$ and similarly $(X_t)$ (or simply $X$) for a process $(X_t)_{t\\in[0,T]}$.\n \n\nWe consider two-player zero-sum Dynkin games on the (possibly infinite) horizon $T$. Actions of the first player are based on the information contained in a filtration $(\\mathcal{F}^1_t) \\subseteq (\\mathcal{F}_t)$. Actions of the second player are based on the information contained in a filtration $(\\mathcal{F}^2_t) \\subseteq (\\mathcal{F}_t)$. Each player selects a random time (taking values in $[0,T]$) based on the information she acquires via her filtration: the first player's random time is denoted by $\\tau$ while the second player's random time is $\\sigma$. The game terminates at time $\\tau \\wedge \\sigma \\in[0, T]$ with the first player delivering to the second player the payoff\n\\begin{equation}\\label{eqn:payoff}\n\\mathcal{P} (\\tau, \\sigma) = f_{\\tau} \\ind{\\{\\tau<\\sigma\\}} \n+\ng_{\\sigma} \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh_{\\tau} \\ind{\\{\\tau=\\sigma\\}}.\n\\end{equation}\nThe first player (or $\\tau$-player) is the {\\em minimiser} in the game whereas the second player (or $\\sigma$-player) is the {\\em maximiser}. That means that the former will try to minimise the expected payoff (see \\eqref{eq-uninf-payoff} below) while the latter will try to maximise it.\n\nThe {\\em payoff processes} $f$, $g$ and $h$ satisfy the following conditions:\n\\begin{assumption}\n\\item[(A1)]\\label{eq-integrability-cond} $f, g \\in\\mcalL$, \n\\item[(A2)]\\label{ass:regular} $f, g$ are $(\\mathcal{F}_t)$-adapted regular processes,\n\\item[(A3)]\\label{eq-order-cond} $f_t\\ge h_t \\ge g_t$  for all $t\\in[0,T]$, $\\mathbb{P}$\\as,\n\\item[(A4)]\\label{eq-terminal-time-order-cond} $h$ is an $(\\mathcal{F}_t)$-adapted, measurable process.\n\\end{assumption}\nIn particular, we do not assume that $h$ is c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}.\n\nIn the context of zero-sum games that we will address below, our assumption \\ref{eq-order-cond} corresponds to the so-called games with the {\\em second-mover advantage}, i.e., games in which both players have an incentive to wait for the opponent to end the game.\n\n\nAssumption \\ref{eq-integrability-cond} is natural in the framework of optimal stopping problems (see, e.g., \\cite[Section 2.2]{Peskir2006}, \\cite[Eq. (D.29)]{Karatzas1998}) and Dynkin games (\\cite{TouziVieille2002}). With \\ref{ass:regular} we replace semimartingale assumptions on $f$ and $g$ from \\cite{TouziVieille2002} while imposing the regularity condition dating back to \\cite{Meyer} in the optimal stopping framework. Regular processes encompass a large family of stochastic processes encountered in applications. It is straightforward to see that quasi left-continuous processes \\cite[Section III.11]{RogersWilliams} are regular. In the Markovian framework all standard processes \\cite[Def. I.9.2]{Blumenthal1968} and, in particular, weak Feller processes (\\cite[Def. III.6.5 and Thm. III.11.1]{RogersWilliams} or \\cite[Thm. I.9.4]{Blumenthal1968}) are regular. Hence, strong and weak solutions to stochastic differential equations (SDEs) driven by multi-dimensional Brownian motion (or L\\`evy process and bounded coefficients) \\cite[Section 6.7]{Appelbaum2009} and solutions to jump-diffusion SDEs are also regular processes. More generally, a regular process is allowed to jump at predictable times $\\eta$, provided that such jumps have zero mean conditional on $\\mathcal{F}_{\\eta-}$. \n\nWe subsequently relax Assumption \\ref{ass:regular} by allowing payoff processes with predictable jumps of nonzero mean. That is, we replace \\ref{ass:regular} with the following:\n\\begin{assumption}\n \\item[(A2')] \\label{ass:regular_gen} Processes $f$ and $g$ have the decomposition $f = \\tilde f + \\hat f$, $g = \\tilde g + \\hat g$ with \n \\begin{enumerate}\n  \\item $\\tilde f, \\tilde g \\in\\mcalL$,\n  \\item $\\tilde f, \\tilde g $ are $(\\mathcal{F}_t)$-adapted regular processes,\n  \\item $\\hat f, \\hat g$ are $(\\mathcal{F}_t)$-adapted (right-continuous) piecewise-constant processes of integrable variation with $\\hat f_{0} = \\hat g_0 = 0$, $\\Delta \\hat f_T = \\hat f_{T} - \\hat f_{T-} = 0$ and $\\Delta \\hat g_T =\\hat g_T- \\hat g_{T-}= 0$,\n  \\item either $\\hat f$ is non-increasing or $\\hat g$ is non-decreasing.\n \\end{enumerate}\n\\end{assumption}\nNotice that there are non-decreasing processes $(\\hat f^+_t), (\\hat f^-_t), (\\hat g^+_t), (\\hat g^-_t) \\in \\mcalL$ starting from $0$ such that $\\hat f = \\hat f^+-\\hat f^-$ and $\\hat g = \\hat g^+ - \\hat g^-$ \\cite[p. 115]{DellacherieMeyer}.\n\nUnder Assumption \\ref{ass:regular_gen}, we allow jumps of $\\hat f$ in any direction and only upward jumps of $\\hat g$, or, viceversa, jumps of $\\hat g$ in any direction and downward jumps of $\\hat f$. This ensures a certain closedness property (see Section \\ref{sec:relax}).\nIt is worth emphasising that further relaxation of condition \\ref{ass:regular_gen} is not possible in the generality of our setting as demonstrated in Remark \\ref{rem:contrad} and in Section \\ref{subsec:example_jumps}. While regular processes have no restrictions on non-predictable jumps, \\ref{ass:regular_gen} relaxes condition in Eq. \\eqref{eq:cond-reg} by allowing predictable jumps with non-zero (conditional) mean. The necessity to restrict the direction of predictable jumps of one of the payoff processes is a new feature introduced by the asymmetry of information. In classical Dynkin games it is not necessary, see \\cite{ekstrom2008, lepeltier1984, stettner1982b}.\n\nWe further require a technical assumption\n\\begin{assumption}\n\\item[(A5)]\\label{ass:filtration} The filtrations $(\\mathcal{F}_t)$ and $(\\mathcal{F}^i_t)$, $i=1,2$, satisfy the usual conditions, i.e., they are right-continuous and $\\mathcal{F}^i_0$, $i=1,2$, contain all sets of $\\mathbb{P}$-measure zero.\n\\end{assumption}\n\nPlayers assess the game by looking at the \\emph{expected payoff}\n\\begin{equation} \nN(\\tau,\\sigma)= \\mathbb{E} \\big[ \\mathcal{P} (\\tau, \\sigma) \\big].\n\\label{eq-uninf-payoff}\n\\end{equation}\nThe game is said to have a \\emph{value} if\n\\[\n\\sup_\\sigma \\operatornamewithlimits{inf\\vphantom{p}}_\\tau N(\\tau, \\sigma) = \\operatornamewithlimits{inf\\vphantom{p}}_\\tau \\sup_\\sigma  N(\\tau, \\sigma),\n\\]\nwhere, for now, we do not specify the nature of the admissible random times $(\\tau,\\sigma)$. The mathematical difficulty with establishing existence of a value lies in the possibility to swap the order of `inf' and `sup' and this is closely linked to the choice of admissible random times. Furthermore, an admissible pair $(\\tau_*,\\sigma_*)$ is said to be a {\\em saddle point} (or a pair of \\emph{optimal strategies}) if\n\\[\nN(\\tau_*,\\sigma)\\le N(\\tau_*,\\sigma_*)\\le N(\\tau,\\sigma_*),\n\\]\nfor all other admissible pairs $(\\tau,\\sigma)$.\n\n\\begin{remark}\\label{rem:ineq}\nOur problem formulation enjoys a symmetry which will be later used in proofs. Since we do not make assumptions on the sign of $f$, $g$, $h$, if the value exists for the game with payoff $\\mathcal{P}(\\tau,\\sigma)$, then it also exists for the game with payoff $\\mathcal{P}'(\\tau,\\sigma):=-\\mathcal{P}(\\tau,\\sigma)$. However, in the latter game the $\\tau$-player is a maximiser and the $\\sigma$-player is a minimiser, by the simple fact\n\\begin{align}\\label{eq:swap}\n\\sup_\\sigma\\operatornamewithlimits{inf\\vphantom{p}}_\\tau \\mathbb{E}[\\mathcal{P}(\\tau,\\sigma)]=-\\operatornamewithlimits{inf\\vphantom{p}}_\\sigma\\sup_\\tau \\mathbb{E}[\\mathcal{P}'(\\tau,\\sigma)],\n\\end{align}\nwhere, obviously, in $\\mathcal{P}'$ we have the payoff processes $f'_t:=-f_t$, $g'_t:=-g_t$ and $h'_t:=-h_t$.\n\\end{remark}\n\nIt has been indicated in the literature that games with asymmetric information may not have a value if players' strategies are stopping times for their respective filtrations, see \\cite[Section 2.1]{Grun2013}. Indeed, in Section \\ref{sec:Nikita-examples} we demonstrate that the game studied in this paper may not, in general, have a value if the  first player (resp. the second player) uses $(\\mathcal{F}^1_t)$-stopping times (resp. $(\\mathcal{F}^2_t)$-stopping times). It has been proven in certain Markovian set-ups that the relaxation of player controls to randomised stopping times may be sufficient for the existence of the value (see, e.g., \\cite{Grun2013}, \\cite{GenGrun2019}). The goal of this paper is to show that this is indeed true in the generality of our non-Markovian set-up for the game with payoff \\eqref{eq-uninf-payoff}.\n\nThe framework of this paper encompasses all two-player zero-sum Dynkin games in continuous time that we found in the literature. Indeed, when $(\\mathcal{F}_t) = (\\mathcal{F}^1_t) = (\\mathcal{F}^2_t)$, the game \\eqref{eq-uninf-payoff} is the classical Dynkin game with full information for both players. The case of $(\\mathcal{F}^1_t) = (\\mathcal{F}^2_t)$ but $(\\mathcal{F}^1_t) \\ne (\\mathcal{F}_t)$ corresponds to a game with partial but symmetric information about the payoff processes (e.g., \\cite{DGV2017}), whereas $(\\mathcal{F}^1_t) \\ne (\\mathcal{F}^2_t)$ is the game with asymmetric information. One can have $(\\mathcal{F}^1_t)=(\\mathcal{F}_t)$, i.e., only the second player is uninformed (e.g., \\cite{Grun2013}), or $(\\mathcal{F}^1_t)\\ne(\\mathcal{F}_t)$ and $(\\mathcal{F}^2_t) \\ne (\\mathcal{F}_t)$, i.e., both players access different information flows and neither of them has full knowledge of the underlying world (e.g., \\cite{GenGrun2019}). In Section \\ref{sec:examples}, we present in full detail how games with asymmetric information studied in the literature fit into our framework.\n\nAs mentioned above the concept of randomised stopping time is central in our work, so we introduce it here. For that we need to consider increasing processes: given a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$ let\n\\begin{align*}\n{\\mathcal{A}^\\circ} (\\mathcal{G}_t):=&\\,\\{\\rho\\,:\\,\\text{$\\rho$ is $(\\mathcal{G}_t)$-adapted with $t\\mapsto\\rho_t(\\omega)$ c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}},}\\\\\n&\\qquad\\,\\text{non-decreasing, $\\rho_{0-}(\\omega)=0$ and $\\rho_T(\\omega)=1$ for all $\\omega\\in\\Omega$}\\}.\n\\end{align*}\nIn the definition of ${\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ we take the opportunity to require that the stated properties hold for all $\\omega \\in \\Omega$. This leads to no loss of generality if $\\mathcal{G}_0$ contains all $\\mathbb{P}$-null sets of $\\Omega$. Hence for any $\\omega\\in \\mathcal{N} \\subset \\Omega$ with $\\mathbb{P}(\\mathcal{N})=0$ we can simply set $\\rho_t(\\omega)=0$ for $t\\in[0,T)$ and $\\rho_T(\\omega)=1$. Recall that in the infinite-time horizon case, $T=\\infty$, we understand $\\rho_T$ as an $\\mathcal{F}_{\\infty}$-measurable random variable while $\\rho_{T-}:=\\lim_{t\\to\\infty}\\rho_t$ (which exists by the assumption that $(\\rho_t)$ is a c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} process).\nRandomised stopping times can be defined as follows.\n\\begin{definition}\\label{def-rand-st}\nGiven a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, a random variable $\\eta$ is called a \\emph{$(\\mathcal{G}_t)$-randomised stopping time} if there exists a random variable $Z$ with uniform distribution $U([0,1])$, independent of $\\mathcal{F}_T$, and a process $\\rho\\in{\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ \nsuch that \n\\begin{equation}\n\\eta=\\eta(\\rho,Z)=\\operatornamewithlimits{inf\\vphantom{p}}\\{t\\in [0,T]: \\rho_t > Z\\}, \\quad \\mathbb{P}\\as\n\\label{eq-def-rand-st}\n\\end{equation}\nThe variable $Z$ is called a \\emph{randomisation device} for the randomised stopping time $\\eta$, and the process $\\rho$ is called the \\emph{generating process}. The set of $(\\mathcal{G}_t)$-randomised stopping times is denoted by $\\mathcal{T}^R(\\mathcal{G}_t)$. It is assumed that randomisation devices of different stopping times are independent.\n\\end{definition}\nWe refer to \\cite{Solanetal2012}, \\cite{TouziVieille2002} for an extensive discussion on various definitions of randomised stopping times and conditions that are necessary for their equivalence. To avoid unnecessary complication of notation, we assume that the probability space $(\\Omega, \\mathcal{F}, \\mathbb{P})$ supports two independent random variables $Z_\\tau$ and $Z_\\sigma$ which are also independent of $\\mathcal{F}_T$ and are the randomisation devices for the randomised stopping times $\\tau$ and $\\sigma$ of the two players.  \n\n\\begin{definition}\\label{def-value-rand-strat}\nDefine\n\\begin{equation*}\nV_*:=\\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N(\\tau,\\sigma)\\quad\\text{and}\\quad V^*:= \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)}\\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)}  N(\\tau,\\sigma).\n\\end{equation*}\nThe \\emph{lower value} and {\\em upper value of the game in randomised strategies} are given by $V_*$ and $V^*$, respectively.  If they coincide, the game is said to have a \\emph{value in randomised strategies} $V=V_*=V^*$. \n\\end{definition}\nThe following theorem states the main result of this paper.\n\\begin{theorem}\n\\label{thm:main2}\nUnder assumptions \\ref{eq-integrability-cond}, \\ref{ass:regular_gen}, \\ref{eq-order-cond}-\\ref{ass:filtration}, the game has a value in randomised strategies. Moreover, if $\\hat f$ and $\\hat g$ in \\ref{ass:regular_gen} are non-increasing and non-decreasing, respectively, there exists a pair $(\\tau_*,\\sigma_*)$ of optimal strategies.\n\\end{theorem}\nFor the clarity of presentation of our methodology, we first prove a theorem with more restrictive regularity properties of payoff processes and then show how to extend the proof to the general case of Theorem \\ref{thm:main2}.\n\\begin{theorem}\n\\label{thm:main}\nUnder assumptions \\ref{eq-integrability-cond}-\\ref{ass:filtration}, the game has a value in randomised strategies and there exists a pair $(\\tau_*,\\sigma_*)$ of optimal strategies.\n\\end{theorem}\nProofs of the above theorems are given in Section \\ref{sec:sions}. They rely on two key results: an approximation procedure (Propositions \\ref{thm:conv_lipsch} and \\ref{thm:conv_lipsch_gen}) and an auxiliary game with `nice' regularity properties (Theorem \\ref{th-value-cont-strat} and \\ref{th-value-cont-strat_gen}) which enables the use of a known min-max theorem (Theorem \\ref{th-the-Sion}).\n\n\nThe $\\sigma$-algebra $\\mathcal{F}_0$ is not assumed to be trivial. It is therefore natural to consider a game in which players assess their strategies ex-post, i.e., after observing the information available to them at time $0$ when their first action may take place. Allowing for more generality, let $\\mathcal{G}$ be a $\\sigma$-algebra contained in $\\mathcal{F}^1_0$ and in $\\mathcal{F}^2_0$, i.e., containing only information available to both players at time $0$. The expected payoff of the game in this case takes the form (recall that $\\tau,\\sigma\\in[0,T]$):\n\\begin{equation}\\label{eqn:cond_func}\n\\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big]\n=\n\\mathbb{E}\\big[ f_{\\tau} \\ind{\\{\\tau<\\sigma\\}} \n+\ng_{\\sigma} \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh_{\\tau} \\ind{\\{\\tau=\\sigma\\}}\\big| \\mathcal{G} \\big].\n\\end{equation}\n\nThe proof of the following theorem is in Section \\ref{sec:ef_functional}.\n\\begin{theorem}\\label{thm:ef_0_value}\nUnder the assumptions of Theorem \\ref{thm:main2} and for any $\\sigma$-algebra $\\mathcal{G} \\subseteq \\mathcal{F}^1_0 \\cap \\mathcal{F}^2_0$, the $\\mathcal{G}$-conditioned game has a value, i.e.\n\\begin{equation}\\label{eqn:value_ef0}\n\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)} \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big] = \\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)}\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)}  \\mathbb{E}\\big[\\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big], \\qquad \\mathbb{P}\\as\n\\end{equation}\nMoreover, if $\\hat f$ and $\\hat g$ in \\ref{ass:regular_gen} are non-increasing and non-decreasing, respectively, there exists a pair $(\\tau_*,\\sigma_*)$ of optimal strategies in the sense that\n\\begin{equation}\\label{eqn:saddleG}\n\\mathbb{E}\\big[ \\mathcal{P}(\\tau_*, \\sigma) \\big| \\mathcal{G} \\big]\n\\le\n\\mathbb{E}\\big[ \\mathcal{P}(\\tau_*, \\sigma_*) \\big| \\mathcal{G} \\big]\n\\le\n\\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma_*) \\big| \\mathcal{G} \\big], \\qquad \\mathbb{P}\\as\n\\end{equation}\nfor all other admissible pairs $(\\tau,\\sigma)$.\n\n\\end{theorem}\n\n\\section{Examples}\\label{sec:examples}\nBefore moving on to prove the theorems stated above, in this section we illustrate some of the specific games for which our general results apply. We draw form the existing literature on two-player zero-sum Dynkin games in continuous time and show that a broad class of these (all those we are aware of) fits within our framework. Since our contribution is mainly to the theory of games with partial\/asymmetric information, we exclude the well-known case of games with full information which has been extensively studied (see our literature review in the introduction). \n\n\\subsection{Game with partially observed scenarios}\\label{subsec:game_1}\nOur first example extends the setting of \\cite{Grun2013} and it reduces to that case if $J=1$ and the {\\em payoff processes} $f$, $g$ and $h$ are deterministic functions of an It\\^o diffusion $(X_t)$ on $\\mathbb{R}^d$, i.e., $f_t=f(t,X_t)$, $g_t=g(t,X_t)$ and $h_t=h(t, X_t)$. On a discrete probability space $(\\Omega^s, \\mathcal{F}^s, \\mathbb{P}^s)$, consider two random variables $\\mcalI$ and $\\mcalJ$ taking values in $\\{1,\\ldots,\\I\\}$ and in $\\{1,\\ldots,\\J\\}$, respectively. Denote their joint distribution by $(\\pi_{i,j})_{i=1, \\ldots, \\I,j=1,\\ldots,\\J}$ so that $\\pi_{i,j} = \\mathbb{P}^s(\\mcalI = i,\\mathcal J=j)$. The indices $(i,j)$ are used to identify the {\\em scenario} in which the game is played and are the key ingredient to model the asymmetric information feature. Consider another probability space $(\\Omega^p, \\mathcal{F}^p, \\mathbb{P}^p)$ with a filtration $(\\mathcal{F}^p_t)$ satisfying the usual conditions, and $(\\mathcal{F}^p_t)$-adapted payoff processes $f^{i,j}$, $g^{i,j}$, $h^{i,j}$, with $(i,j)$ taking values in $\\{1,\\ldots,\\I\\} \\times \\{1,\\ldots,\\J\\}$. For all $i,j$, we assume that $f^{i,j}$, $g^{i,j}$, $h^{i,j}$ satisfy conditions \\ref{eq-integrability-cond}-\\ref{eq-terminal-time-order-cond}.\n\nThe game is set on the probability space $(\\Omega, \\mathcal{F}, \\mathbb{P}) := (\\Omega^p \\times \\Omega^s, \\mathcal{F}^p \\vee \\mathcal{F}^s, \\mathbb{P}^p \\otimes \\mathbb{P}^s)$. The first player is informed about the outcome of $\\mcalI$ before the game starts but never directly observes $\\mcalJ$. Hence, her actions are adapted to the filtration $\\mathcal{F}^1_t = \\mathcal{F}^p_t \\vee \\sigma(\\mcalI)$. Conversely, the second player knows $\\mcalJ$ but not $\\mcalI$, so her actions are adapted to the filtration $\\mathcal{F}^2_t = \\mathcal{F}^p_t \\vee \\sigma(\\mcalJ)$. Given a choice of random times $\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)$ and $\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)$ for the first and the second player, the payoff is\n\\begin{equation*}\n\\mathcal{P} (\\tau, \\sigma) = f^{\\mcalI,\\mcalJ}_{\\tau} \\ind{\\{\\tau<\\sigma\\}} \n+\ng^{\\mcalI,\\mcalJ}_{\\sigma} \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh^{\\mcalI,\\mcalJ}_\\tau \\ind{\\{\\tau = \\sigma\\}}.\n\\end{equation*}\nPlayers assess the game by looking at the expected payoff as in \\eqref{eq-uninf-payoff}. It is worth noticing that this corresponds to the so-called `{\\em ex-ante}' expected payoff, i.e., the expected payoff before the players acquire the additional information about the values of $\\mcalI$ and $\\mcalJ$. The structure of the game is common knowledge, i.e., both players know all processes $f^{i,j}$, $g^{i,j}$ and $h^{i,j}$ involved; however, they have partial and asymmetric knowledge on the couple $(i,j)$ which is drawn at the start of the game from the distribution of $(\\mcalI,\\mcalJ)$.\n\nDrawing a precise parallel with the framework of Section \\ref{sec:setting}, the above setting corresponds to $f_t = f^{\\mcalI,\\mcalJ}_t$, $g_t = g^{\\mcalI,\\mcalJ}_t$, and $h_t = h_t^{\\mcalI,\\mcalJ}$  with the filtration $\\mathcal{F}_t = \\mathcal{F}^p_t \\vee \\sigma(\\mcalI, \\mcalJ)$. The observation flows for the players are given by $(\\mathcal{F}^1_t)$ and $(\\mathcal{F}^2_t)$, respectively. \n\nThe particular structure of players' filtrations $(\\mathcal{F}^1_t)$ and $(\\mathcal{F}^2_t)$ allows for the following decomposition of randomised stopping times, see \\cite[Proposition 3.3]{esmaeeli2018} (recall the radomisation devices $Z_\\tau\\sim U([0,1])$ and $Z_\\sigma\\sim U([0,1])$, which are mutually independent and independent of $\\mathcal{F}_T$). \n\\begin{Lemma}\\label{lem:tau_decomposition}\nAny $\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)$ has a representation\n\\begin{equation}\\label{eqn:tau_decomposition}\n\\tau = \\sum_{i=1}^\\I \\ind{\\{\\mcalI = i\\}} \\tau_i, \n\\end{equation}\nwhere $\\tau_1,\\ldots,\\tau_\\I \\in \\mathcal{T}^R(\\mathcal{F}^p_t)$, with generating processes $\\xi^1,\\ldots,\\xi^\\I \\in {\\mathcal{A}^\\circ} (\\mathcal{F}^p_t)$ and a common randomisation device $Z_\\tau$.\nAn analogous representation holds for $\\sigma$ with $\\sigma_1, \\ldots, \\sigma_\\J \\in \\mathcal{T}^R(\\mathcal{F}^p_t)$, generating processes $\\zeta^1_t, \\ldots, \\zeta^\\J_t \\in {\\mathcal{A}^\\circ} (\\mathcal{F}^p_t)$, and a common randomisation device $Z_\\sigma$. \n\\end{Lemma}\n\\begin{cor}\nAny $(\\mathcal{F}^1_t)$-stopping time $\\tau$ has a decomposition \\eqref{eqn:tau_decomposition} with $\\tau_1,\\ldots,\\tau_\\I$ being $(\\mathcal{F}^p_t)$-stopping times (and analogously for $(\\mathcal{F}^2_t)$-stopping times).\n\\end{cor}\nHence, given a realisation of the idiosyncratic scenario variable $\\mcalI$ (resp.\\ $\\mcalJ$), the first (second) player chooses a randomised stopping time whose generating process is adapted to the common filtration $(\\mathcal{F}^p_t)$. The resulting expected payoff can be written as\n\\begin{equation*}\nN(\\tau, \\sigma) = \\sum_{i=1}^\\I \\sum_{j=1}^\\J \\pi_{i,j} \\mathbb{E} \\Big[ f^{i,j}_{\\tau_i} \\ind{\\{\\tau_i<\\sigma_j\\}}+\ng^{i,j}_{\\sigma_j} \\ind{\\{{\\sigma_j}<{\\tau_i}\\}}+ h^{i,j}_{\\tau_i} \\ind{\\{\\tau_i = \\sigma_j\\}} \\Big].\n\\end{equation*}\n\n\\subsection{Game with a single partially observed dynamics} \\label{subsec:game_2}\nOur second example generalises the set-ups of \\cite{DGV2017} and \\cite{DEG2020} and reduces to those cases when $J=2$, the time horizon is infinite and the payoff processes are (particular) time-homogeneous functions of a (particular) one-dimensional diffusion. Here the underlying dynamics of the game is a diffusion, whose drift depends on the realisation of an independent random variable $\\mcalJ\\in\\{1,\\ldots, J\\}$. Formally, on a probability space $(\\Omega, \\mathcal{F}, \\mathbb{P})$ we have a Brownian motion $(W_t)$ on $\\mathbb{R}^d$, an independent random variable $\\mcalJ\\in\\{1,\\ldots, J\\}$ with distribution $\\pi_j=\\mathbb{P}(\\mcalJ=j)$, and a process $(X_t)$ on $\\mathbb{R}^d$ with the dynamics\n\\[\ndX_t=\\sum_{j=1}^J \\ind{\\{\\mcalJ=j\\}} \\mu_j(X_t)dt+\\sigma(X_t)dW_t,\\quad X_0=x,\n\\]\nwhere $\\sigma$, $(\\mu_j)_{j=1,\\ldots J}$ are given functions (known to both players) that guarantee existence of a unique strong solution of the SDE for each $j=1,\\ldots J$. The payoff processes are deterministic functions of the underlying process, i.e., $f_t=f(t,X_t)$, $g_t=g(t,X_t)$ and $h_t=h(t,X_t)$, and they are known to both players. We assume that the payoff processes satisfy conditions \\ref{eq-integrability-cond}-\\ref{eq-terminal-time-order-cond}. It is worth to remark that in the specific setting of \\cite{DGV2017} the norms $\\| f \\|_{\\mcalL}$ and $\\| g \\|_{\\mcalL}$ are not finite so that our results cannot be directly applied. However, the overall structure of the game in \\cite{DGV2017} is easier than ours so that some other special features of the payoff processes can be used to determine existence of the value therein.\n\nTo draw a precise parallel with the notation from Section \\ref{sec:setting}, here we take $\\mathcal{F}_t=\\mathcal{F}^W_t\\vee\\sigma(\\mcalJ)$, where $(\\mathcal{F}^W_t)$ is the filtration generated by the Brownian sample paths and augmented with $\\mathbb{P}$-null sets. Both players observe the dynamics of $X$, however they have partial\/asymmetric information on the value of $\\mcalJ$. In \\cite{DGV2017} neither of the two players knows the true value of $\\mcalJ$, so we have $(\\mathcal{F}^1_t)=(\\mathcal{F}^2_t)=(\\mathcal{F}^X_t)$, where $(\\mathcal{F}^X_t)$ is generated by the sample paths of the process $X$ and it is augmented by the $\\mathbb{P}$-null sets (notice that $\\mathcal{F}^X_t\\subsetneq \\mathcal{F}_t$). In \\cite{DEG2020} instead, the first player (minimiser) observes the true value of $\\mcalJ$. In that case $(\\mathcal{F}^1_t)=(\\mathcal{F}_t)$ and $(\\mathcal{F}^2_t)=(\\mathcal{F}^X_t)$, so that $\\mathcal{F}^2_t\\subsetneq \\mathcal{F}^1_t$. Using the notation $X^\\mcalJ$ to emphasise the dependence of the underlying dynamics on $\\mcalJ$, and given a choice of random times $\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)$ and $\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)$ for the first and the second player, the game's payoff reads\n\\begin{equation*}\n\\mathcal{P} (\\tau, \\sigma) = f(\\tau,X^\\mcalJ_\\tau) \\ind{\\{\\tau<\\sigma\\}} \n+\ng (\\sigma,X^\\mcalJ_\\sigma) \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh (\\tau, X^\\mcalJ_\\tau) \\ind{\\{\\tau = \\sigma\\}}.\n\\end{equation*}\nPlayers assess the game by looking at the expected payoff as in \\eqref{eq-uninf-payoff}. Finally, we remark that under a number of (restrictive) technical assumptions and with infinite horizon \\cite{DGV2017} and \\cite{DEG2020} show the existence of a value and of a saddle point in a smaller class of strategies. In \\cite{DGV2017} both players use $(\\mathcal{F}^X_t)$-stopping times, with no need for additional randomisation. In \\cite{DEG2020} the uninformed player uses $(\\mathcal{F}^X_t)$-stopping times but the informed player uses $(\\mathcal{F}_t)$-randomised stopping times.\n\n\\subsection{Game with two partially observed dynamics\nHere we show how the setting of \\cite{GenGrun2019} also fits in our framework. This example is conceptually different from the previous two because the players observe two different stochastic processes. On a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ two processes $(X_t)$ and $(Y_t)$ are defined (in \\cite{GenGrun2019} these are finite-state continuous-time Markov chains). The first player only observes the process $(X_t)$ while the second player only observes the process $(Y_t)$. In the notation of Section \\ref{sec:setting}, we have $(\\mathcal{F}^1_t)=(\\mathcal{F}^X_t)$, $(\\mathcal{F}^2_t)=(\\mathcal{F}^Y_t)$ and $(\\mathcal{F}_t)=(\\mathcal{F}^X_t\\vee\\mathcal{F}^Y_t)$, where the filtration $(\\mathcal{F}^X_t)$ is generated by the sample paths of $(X_t)$ and $(\\mathcal{F}^Y_t)$ by those of $(Y_t)$ (both filtrations are augmented with $\\mathbb{P}$-null sets). The payoff processes are deterministic functions of the underlying dynamics, i.e., $f_t=f(t,X_t,Y_t)$, $g_t=g(t,X_t,Y_t)$ and $h_t=h(t, X_t,Y_t)$, and they satisfy conditions \\ref{eq-integrability-cond}-\\ref{eq-terminal-time-order-cond}. Given a choice of random times $\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)$ and $\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)$ for the first and the second player, the game's payoff reads\n\\begin{equation*}\n\\mathcal{P} (\\tau, \\sigma) = f(\\tau,X_\\tau,Y_\\tau) \\ind{\\{\\tau<\\sigma\\}} \n+\ng (\\sigma,X_\\sigma,Y_\\sigma) \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh (\\tau, X_\\tau,,Y_\\tau) \\ind{\\{\\tau = \\sigma\\}}.\n\\end{equation*}\nPlayers assess the game by looking at the expected payoff as in \\eqref{eq-uninf-payoff}. We remark that the proof of existence of the value in \\cite{GenGrun2019} is based on variational inequalities and relies on the finiteness of the state spaces of both underlying processes, and therefore cannot be extended to our general non-Markovian framework.\n\n\\subsection{Game with a random horizon\nHere we consider a non-Markovian extension of the framework of \\cite{lempa2013}, where the time horizon of the game is exponentially distributed and independent of the payoff processes. On a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ we have a filtration $(\\mathcal{G}_t)_{t\\in[0,T]}$, augmented with $\\mathbb{P}$-null sets, and a positive random variable $\\theta$ which is independent of $\\mathcal{G}_T$ and has a continuous distribution. Let $\\Lambda_t:=\\ind{\\{t\\ge \\theta\\}}$ and take $\\mathcal{F}_t=\\mathcal{G}_t\\vee\\sigma(\\Lambda_s,\\,0\\le s\\le t)$.\n\n\nThe players have asymmetric knowledge of the random variable $\\theta$. The first player observes the occurrence of $\\theta$, whereas the second player does not. We have $(\\mathcal{F}^1_t)=(\\mathcal{F}_t)$ and $(\\mathcal{F}^2_t)=(\\mathcal{G}_t)\\subsetneq (\\mathcal{F}^1_t)$.\nGiven a choice of random times $\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)$ and $\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)$ for the first and the second player, the game's payoff reads\n\\begin{align}\\label{eq:PLM}\n\\mathcal{P} (\\tau, \\sigma) \n&= \n\\indd{\\tau \\wedge \\sigma \\le \\theta} \\big(f^0_\\tau \\ind{\\{\\tau<\\sigma\\}} \n+\ng^0_\\sigma \\ind{\\{{\\sigma}<{\\tau}\\}}\n+\nh^0_\\tau \\ind{\\{\\tau = \\sigma\\}} \\big),\n\\end{align}\nwhere $f^0$, $g^0$ and $h^0$ are $(\\mathcal{G}_t)$-adapted processes that satisfy conditions \\ref{eq-integrability-cond}-\\ref{eq-terminal-time-order-cond} and $f^0 \\ge 0$.\n\nNotice that the problem above does not fit directly into the framework of Section \\ref{sec:setting}: Assumption \\ref{eq-integrability-cond} is indeed violated, because the processes $(\\indd{t \\le \\theta} f^0_t),(\\indd{t \\le \\theta}g^0_t)$ are not c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}. However, we now show that the game can be equivalently formulated as a game satisfying conditions of our framework. The expected payoff can be rewritten as follows\n\\begin{align*\nN^0(\\tau, \\sigma) := \\mathbb{E}\\big[\\mathcal{P} (\\tau, \\sigma) \\big]\n&= \n\\mathbb{E}\\big[\\indd{\\tau \\le \\theta} \\ind{\\{\\tau<\\sigma\\}} f^0_\\tau \n+\n\\indd{\\sigma \\le \\theta} \\ind{\\{{\\sigma}<{\\tau}\\}} g^0_\\sigma \n+\n\\indd{\\sigma \\le \\theta} \\ind{\\{\\tau = \\sigma\\}} h^0_\\tau\\big]\\\\\n&= \n\\mathbb{E}\\big[\\indd{\\tau \\le \\theta} \\ind{\\{\\tau<\\sigma\\}} f^0_\\tau \n+\n\\indd{\\sigma < \\theta} \\ind{\\{{\\sigma}<{\\tau}\\}} g^0_\\sigma \n+\n\\indd{\\sigma < \\theta} \\ind{\\{\\tau = \\sigma\\}} h^0_\\tau\\big],\\notag\n\\end{align*}\nwhere the second equality holds because $\\theta$ is continuously distributed and independent of $\\mathcal{F}^2_T$, so $\\mathbb{P}(\\sigma=\\theta) = 0$ for any $\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)$. Fix $\\varepsilon > 0$ and set\n\\begin{align*\nf^\\varepsilon_t:=f^0_{t}\\ind{\\{t<\\theta+\\varepsilon\\}},\\quad g_t:=g^0_t\\ind{\\{t < \\theta\\}}, \\quad h_t:=h^0_t\\ind{\\{t < \\theta\\}}, \\qquad t \\in [0, T].\n\\end{align*}\nWe see that conditions \\ref{eq-integrability-cond}, \\ref{eq-order-cond}, \\ref{eq-terminal-time-order-cond} hold for the processes $(f^\\varepsilon_t)$, $(g_t)$, $(h_t)$ (for condition \\ref{eq-order-cond} we use that $f^0 \\ge 0$). Condition \\ref{ass:regular} (regularity of payoffs $f^\\varepsilon$ and $g$) is satisfied, because $\\theta$ has a continuous distribution, so it is a totally inaccessible stopping time for the filtration $(\\mathcal{F}_t)$ by \\cite[Example VI.14.4]{RogersWilliams}. Therefore, by Theorem \\ref{thm:main}, the game with expected payoff\n\\[\nN^\\varepsilon(\\tau, \\sigma) =\\mathbb{E}\\big[\\mathcal{P}^\\varepsilon(\\tau,\\sigma)\\big]:= \\mathbb{E} \\big[\\ind{\\{\\tau<\\sigma\\}} f^\\varepsilon_\\tau \n+\n\\ind{\\{{\\sigma}<{\\tau}\\}} g_\\sigma \n+\n\\ind{\\{\\tau = \\sigma\\}} h_\\tau\\big]\n\\]\nhas a value and a pair of optimal strategies exists.\n\nWe now show that the game with expected payoff $N^0$ has the same value as the one with expected payoff $N^\\varepsilon$, for any $\\varepsilon > 0$. First observe that\n\\begin{align*\nN^\\varepsilon(\\tau, \\sigma) - N^0(\\tau, \\sigma) = \\mathbb{E}\\big[\\indd{\\tau < \\sigma} \\indd{\\theta < \\tau < \\theta + \\varepsilon} f^0_\\tau\\big] \\ge 0\n\\end{align*}\nby the assumption that $f^0 \\ge 0$. Hence, \n\\begin{equation}\\label{eqn:N_eps_upper}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} \\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} N^\\varepsilon (\\tau, \\sigma)\n\\ge\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} \\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} N^0 (\\tau, \\sigma).\n\\end{equation}\nTo derive an opposite inequality for the lower values, fix $\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)$. For $\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)$, define\n\\[\n\\hat \\tau =\n\\begin{cases}\n\\tau, & \\tau \\le \\theta,\\\\\nT, & \\tau > \\theta.\n\\end{cases}\n\\]\nThen, using that $\\mathcal{P}^\\varepsilon(\\tau,\\sigma)=\\mathcal{P}(\\tau,\\sigma)$ on $\\{\\tau\\le \\theta\\}$ and $\\mathcal{P}^\\varepsilon(T,\\sigma)=g^0_\\sigma\\ind{\\{\\sigma<\\theta\\}}=\\mathcal{P}(\\tau,\\sigma)$ on $\\{\\tau>\\theta\\}$, we have\n$N^\\varepsilon(\\hat \\tau, \\sigma) = N^0(\\tau, \\sigma)$. It then follows that\n\\[\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N^\\varepsilon (\\tau, \\sigma) \\le \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N^0 (\\tau, \\sigma),\n\\]\nwhich implies\n\\begin{equation}\\label{eqn:N_eps_lower}\n\\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N^\\varepsilon (\\tau, \\sigma) \\le \n\\sup_{\\sigma \\in \\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N^0 (\\tau, \\sigma).\n\\end{equation}\n\nSince the value of the game with expected payoff $N^\\varepsilon$ exists, combining \\eqref{eqn:N_eps_upper} and \\eqref{eqn:N_eps_lower} we see that the value of the game with expected payoff $N^0$ also exists. It should be noted, though, that this does not imply that an optimal pair of strategies for $N^\\varepsilon$ is optimal for $N^0$. \n\nIt is worth noticing that in \\cite{lempa2013} the setting is Markovian with $T=\\infty$, $f^0_t=h^0_t=e^{-rt} \\bar f(X_t)$, $g^0_t=e^{-rt} \\bar g(X_t)$, $\\bar f$, $\\bar g$ deterministic functions, $r\\ge 0$, $\\theta$ exponentially distributed and $(X_t)$ a one-dimensional linear diffusion. Under specific technical requirements on the functions $\\bar f$ and $\\bar g$ the authors find that a pair of optimal strategies for the game \\eqref{eq:PLM} exists when the first player uses $(\\mathcal{F}^1_t)$-stopping times and the second player uses $(\\mathcal{F}^2_t)$-stopping times (in the form of hitting times to thresholds), with no need for randomisation. Their methods rely on the theory of one-dimensional linear diffusions (using scale function and speed measure) and free-boundary problems, hence do not admit an extension to a non-Markovian case.\n\n \n\\section{Reformulation as a game of (singular) controls} \\label{sec:reform}\n\nIn order to integrate out the randomisation devices for $\\tau$ and $\\sigma$ and obtain a reformulation of the payoff functional $N(\\tau, \\sigma)$ in terms of generating processes for randomised stopping times $\\tau$ and $\\sigma$, we need the following two auxiliary lemmata. We remark that if $\\eta$ is a $(\\mathcal{G}_t)$-randomised stopping time for $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, then $\\eta$ is also an $(\\mathcal{F}_t)$-randomised stopping time. Therefore, the results below are formulated for $(\\mathcal{F}_t)$-randomises stopping times.\n\n\\begin{Lemma}\\label{lem-eta-xi}\nLet $\\eta\\in\\mathcal{T}^R(\\mathcal{F}_t)$ with the generating process $(\\rho_t)$. Then, for any $\\mathcal{F}_T$-measurable random variable $\\kappa$ with values in $[0,T]$, \n\\begin{alignat}{3}\n&\\mathbb{E}[\\ind{\\{\\eta\\le \\kappa\\}}|\\mathcal{F}_T]=\\rho_\\kappa, \\qquad &&\\mathbb{E}[\\ind{\\{\\eta>\\kappa\\}}|\\mathcal{F}_T]=1-\\rho_\\kappa, \\label{eq-xi-eta-1}\\\\  \n&\\mathbb{E}[\\ind{\\{\\eta<\\kappa\\}}|\\mathcal{F}_T]=\\rho_{\\kappa_-},\\qquad &&\\mathbb{E}[\\ind{\\{\\eta\\ge \\kappa\\}}|\\mathcal{F}_T]=1-\\rho_{\\kappa_-}. \\label{eq-xi-eta-3} \n\\end{alignat}\n\\end{Lemma}\n\\begin{proof}\nThe proof of \\eqref{eq-xi-eta-1} follows the lines of \\cite[Proposition 3.1]{DEG2020}. Let $Z$ be the randomisation device for $\\eta$. Since $\\rho$ is right-continuous, non-decreasing and (\\ref{eq-def-rand-st}) holds, we have\n\\begin{equation*}\n\\{\\rho_\\kappa > Z\\}\\subseteq \\{\\eta\\le \\kappa\\}\\subseteq\\{\\rho_\\kappa\\ge Z\\}.\n\\end{equation*}\nUsing that $\\rho_\\kappa$ is $\\mathcal{F}_T$-measurable, and $Z$ is uniformly distributed and independent of $\\mathcal{F}_T$, we compute\n\\begin{equation*}\n\\mathbb{E}[\\ind{\\{\\eta\\le \\kappa\\}}|\\mathcal{F}_T]\\ge \\mathbb{E}[\\ind{\\{\\rho_\\kappa> Z\\}}|\\mathcal{F}_T] = \\int_0^1 \\ind{\\{\\rho_\\kappa> y\\}} dy = \\rho_\\kappa,\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathbb{E}[\\ind{\\{\\eta\\le \\kappa\\}}|\\mathcal{F}_T]\\le \\mathbb{E}[\\ind{\\{\\rho_\\kappa\\ge Z\\}}|\\mathcal{F}_T] = \\int_0^1 \\ind{\\{\\rho_\\kappa\\ge y\\}} dy = \\rho_\\kappa.\n\\end{equation*}\nThis completes the proof of the first equality in \\eqref{eq-xi-eta-1}. The other one is a direct consequence.\n\nTo prove $(\\ref{eq-xi-eta-3})$, we observe that, by (\\ref{eq-xi-eta-1}), for any $\\varepsilon>0$ we have\n\\[\n\\ind{\\{\\kappa>0\\}}\\mathbb{E}[\\ind{\\{\\eta\\le (\\kappa-\\varepsilon) \\vee (\\kappa\/2)\\}}|\\mathcal{F}_T]=\\ind{\\{\\kappa>0\\}} \\rho_{(\\kappa-\\varepsilon) \\vee (\\kappa\/2)}.\n\\]\nDominated convergence theorem implies \n\\begin{align*}\n\\mathbb{E}[\\ind{\\{\\eta< \\kappa\\}}|\\mathcal{F}_T] &= \\ind{\\{\\kappa > 0\\}}\\, \\mathbb{E}[\\ind{\\{\\eta< \\kappa\\}}|\\mathcal{F}_T] \n= \\lim_{\\varepsilon\\downarrow 0} \\ind{\\{\\kappa>0\\}}\\,\\mathbb{E}[\\ind{\\{\\eta\\le (\\kappa-\\varepsilon) \\vee (\\kappa\/2)\\}}|\\mathcal{F}_T]\\\\\n&= \\lim_{\\varepsilon\\downarrow 0} \\ind{\\{\\kappa>0\\}}\\, \\rho_{(\\kappa-\\varepsilon) \\vee (\\kappa\/2)} = \\ind{\\{\\kappa>0\\}}\\, \\rho_{\\kappa-} = \\rho_{\\kappa-},\n\\end{align*}\nwhere in the last equality we used that $\\rho_{0-}=0$. This proves the first equality in \\eqref{eq-xi-eta-3}. The other one is a direct consequence.\n\\end{proof}\n\n\n\\begin{Lemma}\\label{lem:integ_out}\nLet $\\eta,\\theta\\in\\mathcal{T}^R(\\mathcal{F}_t)$ with generating processes $(\\rho_t)$, $(\\chi_t)$ and independent randomisation devices $Z_\\eta$, $Z_\\theta$. For $(X_t)$ measurable, adapted and such that $\\|X\\|_{\\mcalL}<\\infty$ (but not necessarily {c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}\\!\\!}),\nwe have\n\\begin{align*}\n&\\mathbb{E}\\left[X_\\eta \\ind{\\{\\eta\\le\\theta\\}\\cap\\{\\eta<T\\}}\\right]=\\mathbb{E}\\left[\\int_{[0, T)} X_t(1-\\chi_{t-})d\\rho_t\\right],\\\\\n&\\mathbb{E}\\left[X_\\eta \\ind{\\{\\eta<\\theta\\}}\\right]=\\mathbb{E}\\left[\\int_{[0, T)} X_t(1-\\chi_{t})d\\rho_t\\right],\n\\end{align*}\nwhere we use the notation $\\int_{[0, T)}$ for the (pathwise) Lebesgue-Stieltjes integral.\n\\label{lem-alt-repr-both-rand}\n\\end{Lemma}\n\\begin{proof}\n\nFor $y\\in[0,1)$, define a family of random variables\n\\begin{equation*}\nq(y)=\\operatornamewithlimits{inf\\vphantom{p}}\\{t\\in [0, T]: \\rho_t > y\\}.\n\\end{equation*}\nThen, $\\eta=q(Z_\\eta)$. Using that $Z_\\eta \\sim U(0,1)$ and Fubini's theorem, we see that\n\\begin{align*}\n\\mathbb{E}\\left[X_\\eta \\ind{\\{\\eta\\le\\theta\\}\\cap\\{\\eta<T\\}}\\right]&=\\mathbb{E}\\left[\\int_0^1 X_{q(y)} \\ind{\\{q(y)\\le\\theta\\}\\cap\\{q(y)<T\\}} dy\\right]\\\\\n&=\\int_0^1 \\mathbb{E}\\left[\\mathbb{E}\\left[X_{q(y)} \\ind{\\{q(y)\\le\\theta\\}\\cap\\{q(y)<T\\}}|\\mathcal{F}_T\\right]\\right]dy.\n\\end{align*}\nSince $X_{q(y)} \\ind{\\{q(y)<T\\}}$ is $\\mathcal{F}_T$-measurable and the randomization device $Z_\\theta$ is independent of $\\mathcal{F}_T$, we continue as follows:\n\\begin{align*}\n\\int_0^1 \\mathbb{E}\\left[\\mathbb{E}\\left[X_{q(y)} \\ind{\\{q(y)\\le\\theta\\}\\cap\\{q(y)<T\\}}|\\mathcal{F}_T\\right]\\right]dy&=\\int_0^1 \\mathbb{E}\\left[X_{q(y)} \\ind{\\{q(y)<T\\}} \\mathbb{E}[\\ind{\\{q(y)\\le\\theta\\}} | \\mathcal{F}_T]\\right] dy\\\\\n&=\\mathbb{E}\\left[\\int_0^1 X_{q(y)} \\ind{\\{q(y)<T\\}} (1-\\chi_{q(y)-}) dy\\right]\\\\\n&=\\mathbb{E}\\left[\\int_{[0, T)} X_t(1-\\chi_{t-})d\\rho_t\\right],\n\\end{align*}\nwhere in the second equality we apply Lemma \\ref{lem-eta-xi} with $\\kappa = q(y)$ and in the third equality we change the variable of integration applying \\cite[Proposition 0.4.9]{revuzyor} $\\omega$-wise and using the fact that the function $y \\mapsto q(y)(\\omega)$ is the generalized inverse of $t\\mapsto \\rho_t(\\omega)$. The first statement of the lemma is now proved.\n\nFor the second statement, we adapt the arguments above to write\n\\begin{align*}\n\\mathbb{E}\\left[X_\\eta \\ind{\\{\\eta<\\theta\\}}\\right] &= \\int_0^1 \\mathbb{E}\\left[X_{q(y)} \\mathbb{E}[\\ind{\\{q(y)<\\theta\\}} | \\mathcal{F}_T]\\right] dy\n=\\mathbb{E}\\left[\\int_0^1 X_{q(y)} (1-\\chi_{q(y)}) dy\\right]\\\\\n&=\\mathbb{E}\\left[\\int_{[0, T]} X_t(1-\\chi_{t})d\\rho_t\\right]\n=\\mathbb{E}\\left[\\int_{[0, T)} X_t(1-\\chi_{t})d\\rho_t\\right],\n\\end{align*}\nwhere in the last equality we used that $\\chi_T = 1$. \n\\end{proof}\n\n\\begin{cor}\\label{cor:j}\nUnder the assumptions of Lemma \\ref{lem:integ_out}, we have\n\\[\n\\mathbb{E}[X_\\eta\\ind{\\{\\eta=\\theta\\}}]=\\mathbb{E}\\bigg[\\sum_{t\\in[0,T]}X_t\\Delta\\rho_t\\Delta\\chi_t\\bigg],\n\\]\nwhere $\\Delta \\rho_t = \\rho_t - \\rho_{t-}$ and $\\Delta \\chi_t = \\chi_t - \\chi_{t-}$.\n\\end{cor}\n\\begin{proof}\nFrom Lemma \\ref{lem-alt-repr-both-rand} we have\n\\begin{align*}\n\\mathbb{E}[X_\\eta\\ind{\\{\\eta=\\theta\\}\\cap\\{\\eta<T\\}}]=&\\mathbb{E}\\big[X_{\\eta}\\big(\\ind{\\{\\eta\\le\\theta\\}\\cap\\{\\eta<T\\}}-\\ind{\\{\\eta<\\theta\\}}\\big)\\big]\\\\\n=&\\mathbb{E}\\bigg[\\int_{[0,T)}X_t\\Delta\\chi_t d\\rho_t\\bigg]=\\mathbb{E}\\bigg[\\sum_{t\\in[0,T)} X_t\\Delta\\chi_t \\Delta\\rho_t\\bigg],\n\\end{align*}\nwhere the final equality is due to the fact that $t\\mapsto\\chi_t(\\omega)$ has countably many jumps for each $\\omega \\in \\Omega$ and the continuous part of the measure $d\\rho_t(\\omega)$ puts no mass there. Further, we notice that \n\\begin{align*}\n\\mathbb{E}\\big[\\ind{\\{\\eta=\\theta=T\\}}|\\mathcal{F}_T\\big]\n&=\n\\lim_{n\\to\\infty}\\mathbb{E}\\big[\\ind{\\{\\eta>T-1\/n\\}}\\ind{\\{\\theta>T-1\/n\\}}|\\mathcal{F}_T\\big]\n=\n\\lim_{n\\to\\infty}\\mathbb{E}\\big[\\ind{\\{\\rho_{T-1\/n}\\le Z_\\eta\\}}\\ind{\\{\\chi_{T-1\/n}\\le Z_\\theta\\}}|\\mathcal{F}_T\\big]\\\\\n&=\n\\lim_{n\\to\\infty}(1-\\rho_{T-1\/n})(1-\\chi_{T-1\/n})\n=\n\\Delta\\rho_T\\Delta\\chi_T,\n\\end{align*}\nwhere the second equality is by \n\\[\n\\{\\rho_{T-1\/n}<Z_\\eta\\}\\subseteq\\{\\eta>T-\\tfrac{1}{n}\\}\\subseteq\\{\\rho_{T-1\/n}\\le Z_\\eta\\},\n\\]\nand analogous inclusions for $\\{\\theta\\!>\\!T\\!-\\!\\frac{1}{n}\\}$. The third equality uses that $\\rho_{T-1\/n}$ and $\\chi_{T-1\/n}$ are $\\mathcal{F}_T$-measurable, and $Z_\\eta$, $Z_\\theta$ are independent of $\\mathcal{F}_T$. The final equality follows since $\\rho_T=\\chi_T=1$. Combining the above gives\nthe desired result.\n\\end{proof}\n\nApplying Lemma \\ref{lem-alt-repr-both-rand} and Corollary \\ref{cor:j} to \\eqref{eqn:payoff} and \\eqref{eq-uninf-payoff}, we obtain the following reformulation of the game.\n\n\\begin{Proposition}\\label{prop-functionals-equal}\nFor $\\tau\\in \\mathcal{T}^R (\\mathcal{F}^1_t)$, $\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)$, \n\\begin{equation}\nN(\\tau,\\sigma)= \\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi_t + \\int_{[0, T)} g_t(1-\\xi_t)d\\zeta_t + \\sum_{t \\in [0, T]} h_t \\Delta\\xi_t \\Delta\\zeta_t\\bigg],\n\\label{eq-functional-in-terms-of-controls}\n\\end{equation}\nwhere $(\\xi_t)$ and $(\\zeta_t)$ are the generating processes for $\\tau$ and $\\sigma$, respectively.\n\\end{Proposition}\n\nWith a slight abuse of notation, we will denote the right-hand side of \\eqref{eq-functional-in-terms-of-controls} by $N(\\xi,\\zeta)$. \n\n\\begin{remark}\\label{rem-Laraki-Solan}\nIn the Definition \\ref{def-value-rand-strat} of the lower value, the infimum can always be replaced by infimum over \\emph{pure} stopping times (cf. \\cite{LarakiSolan2005}). Same holds for the supremum in the definition of the upper value.\n\nLet us look at the upper value: take arbitrary $\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)$, $\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)$, and define the family of stopping times\n\\begin{equation*}\nq(y)=\\operatornamewithlimits{inf\\vphantom{p}}\\{t\\in [0,T]: \\zeta_t > y\\}, \\qquad y \\in [0,1),\n\\end{equation*}\nsimilarly to the proof of Lemma \\ref{lem-alt-repr-both-rand} and with $(\\zeta_t)$ the generating process of $\\sigma$. Then,\n\\begin{equation*}\nN(\\tau,\\sigma)=\\int_0^1 N(\\tau,q(y)) dy \\le \\sup_{y\\in[0,1)} N(\\tau,q(y))\\le \\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)} N(\\tau,\\sigma),\n\\end{equation*}\nwhere $\\mathcal{T}(\\mathcal{F}^2_t)$ denotes the set of pure $(\\mathcal{F}^2_t)$-stopping times. Since $\\mathcal{T}(\\mathcal{F}^2_t) \\subset \\mathcal{T}^R(\\mathcal{F}^2_t)$, we have\n\\begin{equation*}\n\\sup_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)}N(\\tau,\\sigma)= \\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)} N(\\tau,\\sigma),\n\\end{equation*}\nand, consequently, the `{\\em inner}' optimisation can be done over pure stopping times:\n\\begin{equation*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)}\\sup_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)} N(\\tau,\\sigma)= \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)}\\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)} N(\\tau,\\sigma).\n\\end{equation*}\nBy the same argument one can show that\n\\begin{equation*}\n\\sup_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)} N(\\tau,\\sigma)= \\sup_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}(\\mathcal{F}^1_t)} N(\\tau,\\sigma).\n\\end{equation*}\nHowever, in general an analogue result for the `{\\em outer}' optimisation does not hold, i.e.,\n\\begin{equation*}\n\\sup_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N(\\tau,\\sigma)\\neq \\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)} \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)} N(\\tau,\\sigma)\n\\end{equation*}\nas shown by an example in Section \\ref{sec:Nikita-examples}.\n\\end{remark}\n\n\n\\section{Sion's theorem and existence of value}\\label{sec-Sion-existence-of-value}\\label{sec:sions}\nThe proofs of Theorems \\ref{thm:main2} and \\ref{thm:main}, i.e., that the game with payoff \\eqref{eq-uninf-payoff} has a value in randomised strategies, utilises Sion's min-max theorem \\cite{Sion1958} (see also \\cite{Komiya1988} for a simple proof). The idea of relying on Sion's theorem comes from \\cite{TouziVieille2002} where the authors study zero-sum Dynkin games with full and symmetric information. Here, however, we need different key technical arguments as explained in, e.g., Remark \\ref{rem-TV-norm-doesnt-work} below.\n\nLet us start by recalling Sion's theorem.\n\\begin{theorem}[Sion's theorem]\\label{th-the-Sion}\n\\cite[Corollary 3.3]{Sion1958}\nLet $A$ and $B$ be convex subsets of a linear topological space one of which is compact. Let $\\varphi(\\mu,\\nu)$ be a function $A\\times B \\mapsto \\mathbb{R}$ that is quasi-concave and upper semi-continuous in $\\mu$ for each $\\nu\\in B$, and quasi-convex and lower semi-continuous in $\\nu$ for each $\\mu\\in A$. Then,\n\\begin{equation*}\n\\sup_{\\mu\\in A}\\operatornamewithlimits{inf\\vphantom{p}}_{\\nu\\in B} \\varphi(\\mu,\\nu)=\\operatornamewithlimits{inf\\vphantom{p}}_{\\nu\\in B}\\sup_{\\mu\\in A} \\varphi(\\mu,\\nu).\n\\end{equation*}\n\\end{theorem}\n\nThe key step in applying Sion's theorem is to find a topology on the set of randomised stopping times, or, equivalently, on the set of corresponding generating processes so that the functional $N(\\cdot, \\cdot)$ satisfies the assumptions. We will use the weak topology of \n\\[\n\\mathcal{S} := L^2 \\big([0, T] \\times \\Omega, \\mathcal{B}([0, T]) \\times \\mathcal{F}, \\lambda \\times \\mathbb{P}\\big),\n\\]\nwhere $\\lambda$ denotes the Lebesgue measure on $[0, T]$. Given a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, in addition to the class of increasing processes ${\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ introduced in Section \\ref{sec:setting}, here we also need\n\\begin{align*}\n{\\mathcal{A}^\\circ_{ac}} (\\mathcal{G}_t) :=&\\,\\{\\rho\\in {\\mathcal{A}^\\circ}(\\mathcal{G}):\\,\\text{$t\\mapsto\\rho_t(\\omega)$ is absolutely continuous on $[0,T)$ for all $\\omega\\in\\Omega$}\\}.\n\\end{align*}\nIt is important to notice that $\\rho\\in{\\mathcal{A}^\\circ_{ac}}(\\mathcal{G}_t)$ may have a jump at time $T$ if\n\\[\n\\rho_{T-}(\\omega):=\\lim_{t\\uparrow T}\\int_0^t\\big(\\tfrac{d}{d t}\\rho_s\\big)(\\omega)d s<1=\\rho_T(\\omega).\n\\] \nAs with ${\\mathcal{A}^\\circ}(\\mathcal{G}_t)$, in the definition of ${\\mathcal{A}^\\circ_{ac}}(\\mathcal{G}_t)$ we require that the stated properties hold for all $\\omega \\in \\Omega$, which causes no loss of generality if $\\mathcal{G}_0$ contains all $\\mathbb{P}$-null sets of $\\Omega$. It is clear that ${\\mathcal{A}^\\circ_{ac}} (\\mathcal{G}_t) \\subset{\\mathcal{A}^\\circ}(\\mathcal{G}_t) \\subset\\mathcal{S}$.\n\nFor reasons that will become clear later (e.g., see Lemma \\ref{lem-strat-set-compact}), we prefer to work with slightly more general processes than those in ${\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ and ${\\mathcal{A}^\\circ_{ac}}(\\mathcal{G}_t)$. Let us denote\n\\begin{align*}\n\\mathcal{A}(\\mathcal{G}_t) :=&\\, \\{ \\rho \\in \\mathcal S : \\,\\exists\\; \\hat\\rho\\in{\\mathcal{A}^\\circ} (\\mathcal{G}_t) \\,\\text{such that $\\rho = \\hat \\rho$ for $(\\lambda \\times \\mathbb{P})$\\ae $(t,\\omega)\\in[0,T]\\times\\Omega$}\\},\\\\\n\\mathcal{A}_{ac}(\\mathcal{G}_t) := &\\,\\{ \\rho \\in \\mathcal S : \\,\\exists\\; \\hat\\rho\\in{\\mathcal{A}^\\circ_{ac}} (\\mathcal{G}_t) \\,\\text{such that $\\rho = \\hat \\rho$ for $(\\lambda \\times \\mathbb{P})$\\ae $(t,\\omega)\\in[0,T]\\times\\Omega$}\\}.\n\\end{align*}\nWe will call $\\hat \\rho$ in the definition of the set $\\mathcal{A}$ (and $\\mathcal{A}_{ac}$) the \\emph{c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}} (and {\\em absolutely continuous}) {\\em representative} of $\\rho$. Although it is not unique, all c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives are indistinguishable (Lemma \\ref{lem:cadlag_indis}). Hence, all c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives $\\hat\\rho$ of $\\rho\\in\\mathcal{A}$ define the same positive measure on $[0,T]$ for $\\mathbb{P}$\\ae $\\omega\\in\\Omega$ via a non-decreasing mapping $t\\mapsto\\hat\\rho_t(\\omega)$.\nThen, given any bounded measurable process $(X_t)$ the stochastic process (Lebesgue-Stieltjes integral)\n\\begin{equation*}\nt \\mapsto \\int_{[0, t]} X_s\\, d\\hat \\rho_s, \\qquad t \\in [0, T],  \n\\end{equation*}\ndoes not depend on the choice of the c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representative $\\hat \\rho$ in the sense that it is defined up to indistinguishability.\n\nThe next definition connects the randomised stopping times that we use in the construction of the game's payoff (Proposition \\ref{prop-functionals-equal}) with processes from the classes $\\mathcal{A}(\\mathcal{F}^1_t)$ and $\\mathcal{A}(\\mathcal{F}^2_t)$. Note that $\\mathcal{A}(\\mathcal{G}_t) \\subseteq \\mathcal{A}(\\mathcal{F}_t)$ whenever $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, so the definition can be stated for $\\mathcal{A}(\\mathcal{F}_t)$ without any loss of generality.\n\\begin{definition}\\label{def:integral}\nLet $(X_t)$ be measurable and such that $\\|X\\|_{\\mcalL}\\!<\\!\\infty$ (not necessarily {c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}\\!\\!}). For $\\chi,\\rho \\in \\mathcal{A}(\\mathcal{F}_t)$, we define the Lebesgue-Stieltjes integral processes\n\\[\nt \\mapsto \\int_{[0, t]} X_s\\, d\\rho_s,\\quad t\\mapsto\\int_{[0, t]} X_s\\,(1-\\chi_{s}) d\\rho_s\\quad\\text{and}\\quad t\\mapsto\\int_{[0, t]} X_s\\,(1-\\chi_{s-}) d\\rho_s \\qquad t \\in [0, T],   \n\\]\nby \n\\[\nt \\mapsto \\int_{[0, t]} X_s\\, d\\hat{\\rho}_s,\\quad t\\mapsto\\int_{[0, t]} X_s\\,(1-\\hat{\\chi}_{s}) d\\hat{\\rho}_s\\quad\\text{and}\\quad t\\mapsto\\int_{[0, t]} X_s\\,(1-\\hat{\\chi}_{s-}) d\\hat{\\rho}_s \\qquad t \\in [0, T],   \n\\]\nfor any choice of the c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives $\\hat \\rho$ and $\\hat \\chi$, uniquely up to indistinguishability.\n\\end{definition}\n\nWith a slight abuse of notation we define a functional $N: \\mathcal{A}(\\mathcal{F}^1_t) \\times \\mathcal{A}(\\mathcal{F}^2_t) \\to \\mathbb{R}$ by the right-hand side of \\eqref{eq-functional-in-terms-of-controls}. It is immediate to verify using Definition \\ref{def-value-rand-strat} and Proposition \\ref{prop-functionals-equal} that the lower and the upper value of our game satisfy\n\\begin{align}\\label{eq:VV}\nV_{*}=\\sup_{\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)}\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)} N(\\xi,\\zeta), \\qquad V^*=\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)} \\sup_{\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)} N(\\xi,\\zeta).\n\\end{align}\nNotice that even though according to Definition \\ref{def-rand-st} the couple $(\\xi,\\zeta)$ should be taken in ${\\mathcal{A}^\\circ}(\\mathcal{F}^1_t)\\times{\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$, in \\eqref{eq:VV} we consider $(\\xi,\\zeta)\\in \\mathcal{A}(\\mathcal{F}^1_t)\\times\\mathcal{A}(\\mathcal{F}^2_t)$. This causes no inconsistency thanks to the discussion above and Definition \\ref{def:integral} for integrals.\n\n\n\\begin{remark} \nThe mapping $\\mathcal{A}(\\mathcal{F}^1_t) \\times \\mathcal{A}(\\mathcal{F}^2_t) \\ni (\\xi, \\zeta) \\mapsto N(\\xi, \\zeta)$ does not satisfy the conditions of Sion's theorem under the strong or the weak topology of $\\mathcal{S}$. Indeed, taking $\\xi^n_t = \\ind{\\{t \\ge T\/2 + 1\/n\\}}$, we have $\\xi^n_t \\to \\ind{\\{t \\ge T\/2\\}}=:\\xi_t$ for $\\lambda$\\ae $t \\in [0, T]$, so that by the dominated convergence theorem $(\\xi^n)$ also converges to $\\xi$ in $\\mathcal{S}$. Then, fixing $\\zeta_t = \\ind{\\{t \\ge T\/2\\}}$ in $\\mathcal{A}(\\mathcal{F}^2_t)$ we have $N(\\xi^n, \\zeta) = \\mathbb{E}[g_{T\/2}]$ for all $n\\ge 1$ whereas $N(\\xi, \\zeta) =\\mathbb{E}[ h_{T\/2} ]$. So the lower semicontinuity of $\\xi \\mapsto N(\\xi, \\zeta)$ cannot be ensured if, for example, $\\mathbb{P}(h_{T\/2}>g_{T\/2})>0$.\n\\end{remark}\n\nDue to issues indicated in the above remark, as in \\cite{TouziVieille2002}, we `smoothen' the control strategy of one player in order to introduce additional regularity in the payoff. We will show that this procedure does not change the value of the game (Proposition \\ref{thm:conv_lipsch}). We choose (arbitrarily and with no loss of generality, thanks to Remark \\ref{rem:ineq}) to consider an auxiliary game in which the  first player can only use controls from $\\mathcal{A}_{ac} (\\mathcal{F}^1_t)$. Let us define the associated upper\/lower values:\n\\begin{equation}\\label{eq-value-cont-restriction}\nW_{*}=\\sup_{\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)}\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)} N(\\xi,\\zeta)\\quad\\text{and}\\quad W^*=\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)} \\sup_{\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)} N(\\xi,\\zeta).\n\\end{equation}\n\nHere, we work under the regularity assumption on the payoff processes \\ref{ass:regular}. Relaxation of this assumption is conducted in Section \\ref{sec:relax}.\nThe main results can be distilled into the following theorems:\n\n\\begin{theorem}\\label{th-value-cont-strat}\nUnder assumptions \\ref{eq-integrability-cond}-\\ref{ass:filtration}, the game (\\ref{eq-value-cont-restriction}) has a value, i.e.\n\\begin{equation*}\nW_{*}=W^{*}:=W.\n\\end{equation*}\nMoreover, the $\\zeta$-player (maximiser) has an optimal strategy, i.e. there exists $\\zeta^*\\in\\mathcal{A}(\\mathcal{F}^2_t)$ such that\n\\begin{equation*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)} N(\\xi,\\zeta^*)=W.\n\\end{equation*}\n\\end{theorem}\n\\begin{Proposition}\\label{thm:conv_lipsch}\nUnder assumptions \\ref{eq-integrability-cond}-\\ref{ass:filtration}, for any $\\zeta \\in \\mathcal{A}(\\mathcal{F}^2_t)$ and $\\xi \\in \\mathcal{A}(\\mathcal{F}^1_t)$, there is a sequence $\\xi^n \\in \\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ such that\n\\[\n\\mathop{\\lim\\sup}_{n \\to \\infty} N(\\xi^n, \\zeta) \\le N(\\xi, \\zeta).\n\\]\n\\end{Proposition}\n\nThe proofs of the above theorems will be conducted in the following subsections: Section \\ref{sec:tech} contains a series of technical results which we then use to prove Theorem \\ref{th-value-cont-strat} (in Section \\ref{sec:verif}) and Proposition \\ref{thm:conv_lipsch} (in Section \\ref{sec:approx}). With the results from Theorem \\ref{th-value-cont-strat} and Proposition \\ref{thm:conv_lipsch} in place we can provide a (simple) proof of Theorem \\ref{thm:main}.\n\\begin{proof}[{\\bf Proof of Theorem \\ref{thm:main}}]\nObviously, $V_* \\le W_*$ and $V^* \\le W^*$. However, Proposition \\ref{thm:conv_lipsch} implies that \n\\begin{align}\\label{eq:W*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)} N(\\xi,\\zeta) = \\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)} N(\\xi,\\zeta)\\quad\\text{for any $\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)$},\n\\end{align} \nso $V_* \\ge W_*$ and therefore $V_* = W_*$. Then, thanks to Theorem \\ref{th-value-cont-strat}, we have a sequence of inequalities which completes the proof of existence of the value\n\\[\nW = W_* = V_* \\le V^* \\le W^* = W.\n\\]\nIn \\eqref{eq:W*} we can choose $\\zeta^*$ which is optimal for $W$ (its existence is guaranteed by Theorem \\ref{th-value-cont-strat}). Then,\n\\[\nV=V_*=\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)} N(\\xi,\\zeta^*).\n\\]\nThanks to Remark \\ref{rem:ineq}, we can repeat the same arguments above with the roles of the two players swapped as in \\eqref{eq:swap}, i.e., the $\\tau$-player ($\\xi$-player) is the maximiser and the $\\sigma$-player ($\\zeta$-player) is the minimiser. Thus, applying again Theorem \\ref{th-value-cont-strat} and Proposition \\ref{thm:conv_lipsch} (with $\\mathcal{P}'$ as in Remark \\ref{rem:ineq} in place of $\\mathcal{P}$) we arrive at \n\\[\n-V=:V'=\\operatornamewithlimits{inf\\vphantom{p}}_{\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)} \\mathbb{E}\\big[\\mathcal{P}'(\\xi^*,\\zeta)\\big],\n\\]\nwhere $\\xi^*\\in\\mathcal{A}(\\mathcal{F}^1_t)$ is optimal for the maximiser in the game with value $W'=-W$. Hence $\\xi^*$ is optimal for the minimiser in the original game with value $V$ and the couple $(\\xi^*,\\zeta^*)\\in\\mathcal{A}(\\mathcal{F}^1_t)\\times\\mathcal{A}(\\mathcal{F}^2_t)$ is a saddle point. The corresponding randomised stopping times, denoted $(\\tau_*,\\sigma_*)$, are an optimal pair for the players. \n\\end{proof}\n\n\n\n\n\\subsection{Technical results}\\label{sec:tech}\n\nIn this section we give a series of results concerning the convergence of integrals when either the integrand or the integrator converge in a suitable sense. We start by stating a technical lemma whose easy proof is omitted. \n\\begin{Lemma}\\label{lem:cadlag_indis}\nLet $(X_t)$ and $(Y_t)$ be c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} measurable processes such that $X_t = Y_t$, $\\mathbb{P}$\\as for $t \\in D \\subset [0, T)$ countable and dense, $X_{0-} = Y_{0-}$ and $X_T = Y_T$, $\\mathbb{P}$\\as Then $(X_t)$ is indistinguishable from $(Y_t)$.\n\\end{Lemma}\n\n\n\\begin{definition}\\label{def:def_C}\nGiven a c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} measurable process $(X_t)$, for each $\\omega\\in\\Omega$ we denote\n\\[\nC_X(\\omega):= \\{ t\\in[0,T]: X_{t-}(\\omega)=X_t(\\omega) \\}.\n\\]\n\\end{definition}\nOur next result tells us that the convergence $(\\lambda\\times\\mathbb{P})$\\ae of processes in $\\mathcal{A} (\\mathcal{G}_t)$ can be lifted to $\\mathbb{P}$\\as convergence at all points of continuity of the corresponding c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives.\n\\begin{Lemma}\\label{lem:cadlag_convergence}\nFor a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, let $(\\rho^n)_{n\\ge 1}\\subset\\mathcal{A}(\\mathcal{G}_t)$ and $\\rho \\in \\mathcal{A}(\\mathcal{G}_t)$ with $\\rho^n \\to \\rho$ $(\\lambda \\times \\mathbb{P})$\\ae as $n\\to\\infty$. \nThen for any c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives $\\hat \\rho^n$ and $\\hat \\rho$ we have\n\\begin{equation}\\label{eqn:cadlag_convergence}\n\\mathbb{P}\\Big(\\big\\{\\omega \\in \\Omega:\\ \\lim_{n\\to\\infty}\\hat \\rho^n_t(\\omega)= \\hat \\rho_t(\\omega) \\:\\:\\text{for all $t\\in C_{\\hat \\rho}(\\omega)$}\\big\\}\\Big) = 1.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nThe $(\\lambda \\times \\mathbb{P})$\\ae convergence of $\\rho^n$ to $\\rho$ means that the c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives $\\hat \\rho_n$ converge to $\\hat \\rho$ also $(\\lambda \\times \\mathbb{P})$\\ae. Hence, there is a set $D \\subset [0, T]$ with $\\lambda([0,T]\\setminus D) = 0$ such that $\\hat \\rho^n_t \\to \\hat \\rho_t$ $\\mathbb{P}$\\as for $t \\in D$. Since $\\lambda([0,T]\\setminus D) = 0$, there is a countable subset $D_0 \\subset D$ that is dense in $[0, T]$. Define \n\\[\n\\Omega_0 := \\{ \\omega \\in \\Omega:\\ \\hat \\rho^n_t (\\omega) \\to \\hat \\rho_t (\\omega)\\:\\: \\text{for all $t \\in D_0$}\\}. \n\\]\nThen $\\mathbb{P}(\\Omega_0) = 1$.\n\nNow, fix $\\omega\\in\\Omega_0$ and let $t\\in C_{\\hat \\rho}(\\omega) \\cap (0, T)$. Take an increasing sequence $(t^1_k)_{k\\ge 1}\\subset D_0$ and a decreasing one $(t^2_k)_{k\\ge 1}\\subset D_0$, both converging to $t$ as $k\\to\\infty$. For each $k\\ge 1$ we have\n\\begin{equation}\\label{eqn:upward_conv}\n\\hat \\rho_t(\\omega)=\\lim_{k\\to\\infty}\\hat \\rho_{t^2_k}(\\omega)=\\lim_{k\\to\\infty}\\lim_{n\\to\\infty}\\hat \\rho^n_{t^2_k}(\\omega)\\ge \\mathop{\\lim\\sup}_{n\\to\\infty}\\hat \\rho^n_t(\\omega), \n\\end{equation}\nwhere in the final inequality we use that $\\hat \\rho^n_{t^2_k}(\\omega)\\ge \\hat\\rho^n_t(\\omega)$ by monotonicity. By analogous arguments we also obtain\n\\[\n\\hat \\rho_t(\\omega)=\\lim_{k\\to\\infty}\\hat \\rho_{t^1_k}(\\omega)=\\lim_{k\\to\\infty}\\lim_{n\\to\\infty}\\hat \\rho^n_{t^1_k}(\\omega)\\le \\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n\\to\\infty}\\hat \\rho^n_t(\\omega), \n\\]\nwhere the first equality holds because $t\\in C_{\\hat \\rho}(\\omega)$. Combining the above we get \\eqref{eqn:cadlag_convergence} (apart from $t\\in\\{0,T\\}$) by recalling that $\\omega\\in\\Omega_0$ and $\\mathbb{P}(\\Omega_0)=1$. The convergence at $t = T$, irrespective of whether it belongs to $C_{\\hat\\rho}(\\omega)$, is trivial as $\\hat \\rho^n_T(\\omega) = \\hat \\rho_T(\\omega) = 1$. If $0 \\in C_{\\hat\\rho}(\\omega)$, then $\\hat \\rho_0 (\\omega) = \\hat \\rho_{0-}(\\omega) = 0$. Inequality \\eqref{eqn:upward_conv} reads $0 = \\hat \\rho_0 (\\omega) \\ge \\mathop{\\lim\\sup}_{n \\to \\infty} \\hat \\rho^n_0(\\omega)$. Since $\\hat \\rho^n_0(\\omega) \\ge 0$, this proves that $\\hat\\rho^n_0(\\omega) \\to \\hat \\rho_0(\\omega)=0$.\n\\end{proof}\n\n\\begin{Lemma}\\label{prop-terminal-time-jump-limit}\nFor a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, let $(\\rho^n)_{n\\ge 1}\\subset{\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ and $\\rho\\in{\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ with $\\rho^n\\to \\rho$ $(\\lambda\\times\\mathbb{P})$\\ae as $n\\to\\infty$. For any $t \\in [0, T]$ and any random variable $X \\ge 0$ with $\\mathbb{E}[X]<\\infty$, we have\n\\begin{equation*}\n\\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}[X\\Delta \\rho^n_t]\\le \\mathbb{E}[X\\Delta \\rho_t].\n\\end{equation*}\n\\end{Lemma}\n\\begin{proof}\nFix $t \\in (0, T)$. Using $(\\lambda\\times\\mathbb{P})$\\ae convergence of $\\rho^{n}$ to $\\rho$, i.e., that $\\int_0^T \\mathbb{P}\\big(\\lim_{n\\to\\infty}\\rho^{n}_t=\\rho_t\\big) d t=T$, there is a decreasing sequence $\\delta_m \\to 0$ such that\n\\begin{equation*}\n\\lim_{n\\to\\infty} \\rho^{n}_{t-\\delta_m} = \\rho_{t-\\delta_m},\n\\qquad\n\\lim_{n\\to\\infty} \\rho^{n}_{t+\\delta_m} = \\rho_{t+\\delta_m},\\qquad \\mathbb{P}\\as\n\\end{equation*}\nThen, by the dominated convergence theorem,\n\\begin{align*}\n\\mathbb{E}[X\\Delta \\rho_t]&=\\lim_{m \\to \\infty} \\mathbb{E}[ X (\\rho_{t + \\delta_m} - \\rho_{t - \\delta_m})]\\\\\n&=\n\\lim_{m\\to\\infty}\\lim_{n\\to\\infty} \\mathbb{E}[X(\\rho^{n}_{t+\\delta_m}-\\rho^{n}_{t-\\delta_m})]\\\\\n&=\n\\lim_{m\\to\\infty}\\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}[X(\\rho^{n}_{t+\\delta_m}-\\rho^{n}_{t-\\delta_m})]\\\\\n&=\n\\lim_{m\\to\\infty}\\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}[X(\\rho^{n}_{t+\\delta_m}-\\rho^{n}_{t} + \\rho^{n}_{t-} - \\rho^{n}_{t-\\delta_m} + \\Delta \\rho^{n}_{t})] \\ge \\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}[X \\Delta \\rho^{n}_t],\n\\end{align*}\nwhere the last inequality is due to $t\\mapsto\\rho^{n}_t$ being non-decreasing. This finishes the proof for $t\\in(0,T)$.\nThe proof for $t \\in\\{ 0, T\\}$ is a simplified version of the argument above, since $\\rho^n_T=\\rho_T=1$ and $\\rho^n_{0-}=\\rho_{0-}=0$, $\\mathbb{P}$\\as\n\\end{proof}\n\nWe need to consider a slightly larger class of processes ${\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t) \\supset {\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ defined by\n\\begin{align*}\n{\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t):=&\\,\\{\\rho\\,:\\,\\text{$\\rho$ is $(\\mathcal{G}_t)$-adapted with $t\\mapsto\\rho_t(\\omega)$ c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}},}\\\\\n&\\qquad\\,\\text{non-decreasing, $\\rho_{0-}(\\omega)=0$ and $\\rho_T(\\omega)\\le 1$ for all $\\omega\\in\\Omega$}\\}.\n\\end{align*}\n\n\\begin{Proposition}\\label{prop:r-convergence}\nFor a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, let $(\\rho^n)_{n\\ge 1}\\subset{\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t)$ and $\\rho\\in{\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t)$. Assume\n\\[\n\\mathbb{P}\\Big(\\big\\{\\omega \\in \\Omega:\\ \\lim_{n\\to\\infty}\\rho^n_t(\\omega)=\\rho_t(\\omega)\\:\\:\\text{for all $t\\in C_\\rho(\\omega)\\cup\\{T\\}$} \\big\\} \\Big) = 1.\n\\]\nThen for any $X\\in\\mcalL$ that is also $(\\mathcal F_t)$-adapted and regular, we have\n\\begin{equation}\\label{eqn:b_t}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T]} X_t d\\rho^n_t\\bigg] = \\mathbb{E}\\bigg[\\int_{[0, T]} X_t d\\rho_t\\bigg].\n\\end{equation}\n\\end{Proposition}\n\\begin{proof}\nLet us first assume that $(X_t) \\in \\mcalL$ has continuous trajectories (but is not necessarily adapted). If we prove that \n\\begin{equation}\\label{eqn:b_t_omega}\n\\lim_{n\\to\\infty}\\int_{[0, T]} X_t(\\omega) d \\rho^n_t(\\omega)= \\int_{[0, T]} X_t(\\omega) d \\rho_t (\\omega),\\quad\\text{for $\\mathbb{P}$\\ae $\\omega\\in\\Omega$,}\n\\end{equation}\nthen the result in \\eqref{eqn:b_t} will follow by the dominated convergence theorem. By assumption there is $\\Omega_0\\subset \\Omega$ with $\\mathbb{P}(\\Omega_0)=1$ and such that $\\rho^n_t(\\omega)\\to\\rho_t(\\omega)$ at all points of continuity of $t\\mapsto \\rho_t(\\omega)$ and at the terminal time $T$ for all $\\omega \\in \\Omega_0$. Since $d\\rho^n_t(\\omega)$ and $d\\rho_t(\\omega)$ define positive measures on $[0,T]$ for each $\\omega\\in\\Omega_0$, the convergence of integrals in \\eqref{eqn:b_t_omega} can be deduced from the weak convergence of finite measures, see \\cite[Remark III.1.2]{Shiryaev}. Indeed, if $\\omega\\in\\Omega_0$ is such that $\\rho_T(\\omega)=0$, the right-hand side of \\eqref{eqn:b_t_omega} is zero and we have\n\\begin{equation*}\n\\mathop{\\lim\\sup}_{n\\to\\infty}\\left|\\int_{[0,T]} X_t(\\omega) d \\rho^n_t(\\omega)\\right| \\le \\mathop{\\lim\\sup}_{n\\to\\infty}\\sup_{t \\in [0, T]} |X_t(\\omega)| \\rho^n_T(\\omega)= 0,\n\\end{equation*}\nwhere we use $X\\in\\mcalL$ to ensure that $\\sup_{t \\in [0, T]} |X_t(\\omega)|<\\infty$. If instead, $\\omega\\in\\Omega_0$ is such that $\\rho_T(\\omega)>0$, then for all sufficiently large $n$'s, we have $\\rho^n_T(\\omega) > 0$ and $t \\mapsto \\rho^n_t(\\omega) \/ \\rho^n_T(\\omega)$ define cumulative distribution functions (cdfs) converging pointwise to $\\rho_t(\\omega) \/ \\rho_T(\\omega)$ at the points of continuity of $\\rho_t(\\omega)$. Since $t \\mapsto X_t(\\omega)$ is continuous, \\cite[Thm III.1.1]{Shiryaev} justifies\n\\begin{align*}\n\\lim_{n\\to\\infty}\\int_{[0, T]} X_t (\\omega) d \\rho^{n}_t(\\omega) =&\\,\\lim_{n\\to\\infty} \\rho^{n}_T (\\omega) \\int_{[0, T]} X_t(\\omega) d\\left(\\frac{\\rho^{n}_t(\\omega)}{\\rho^{n}_T(\\omega)}\\right) \\\\\n=&\\,\\rho_T(\\omega) \\int_{[0, T]} X_t(\\omega) d\\left(\\frac{\\rho_t(\\omega)}{\\rho_T(\\omega)}\\right) = \\int_{[0, T]} X_t(\\omega) d \\rho_t(\\omega).\n\\end{align*}\n\nNow we drop the continuity assumption on $X$. We turn our attention to c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}}, $(\\mathcal{F}_t)$-adapted and regular $(X_t) \\in \\mcalL$. By \\cite[Theorem 3]{Bismut1978} there is $(\\tilde X_t) \\in \\mcalL$ with continuous trajectories (not necessarily adapted) such that $(X_t)$ is an $(\\mathcal{F}_t)$-optional projection of $(\\tilde X_t)$. From the first part of the proof we know that \\eqref{eqn:b_t} holds for $(\\tilde X_t)$. To show that it holds for $(X_t)$ it is sufficient to notice that $(\\rho^n_t)$ and $(\\rho_t)$ are $(\\mathcal{F}_t)$-optional processes, and apply \\cite[Thm VI.57]{DellacherieMeyer} to obtain\n\\[\n \\mathbb{E}\\bigg[\\int_{[0, T]} X_t d \\rho^n_t\\bigg] = \\mathbb{E}\\bigg[\\int_{[0, T]} \\tilde X_t d \\rho^n_t\\bigg]\\qquad \\text{and} \\qquad \n \\mathbb{E}\\bigg[\\int_{[0, T]} X_t d \\rho_t\\bigg] = \\mathbb{E}\\bigg[\\int_{[0, T]} \\tilde X_t d \\rho_t\\bigg].\n\\]\n\\end{proof}\n\n\\begin{remark}\nThe statement of Proposition \\ref{prop:r-convergence} can be strengthened to include all processes in $\\mcalL$ which are regular but not necessarily $(\\mathcal{F}_t)$-adapted. One can prove it by adapting arguments of the proof of \\cite[Thm.\\ 3]{Meyer}. \n\\end{remark}\n\n\n\\begin{Proposition}\\label{prop-specific-convergence-2}\nFor a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, let $\\chi\\in {\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ and $\\rho\\in\\mathcal{A}_{ac}(\\mathcal{G}_t)$ and consider $X\\in\\mcalL$ which is $(\\mathcal{F}_t)$-adapted and regular. If $(\\rho^n)_{n\\ge 1}\\subset\\mathcal{A}_{ac}(\\mathcal{G}_t)$ converges $(\\lambda \\times \\mathbb{P})$\\ae to $\\rho$ as $n\\to\\infty$, then\n\\begin{equation}\\label{eq:lim00}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho^n_t\\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho_t\\bigg].\n\\end{equation}\n\\end{Proposition}\n\\begin{proof}\nDefine absolutely continuous adapted processes\n\\begin{equation*}\nR^n_t = \\int_{[0, t]} (1-\\chi_{s-})d\\rho^n_s\\quad\\text{and}\\quad R_t = \\int_{[0, t]} (1-\\chi_{s-})d\\rho_s,\n\\end{equation*}\nso that\n\\begin{equation}\\label{eq:intdR}\n\\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho^n_t=\\int_{[0, T]} X_tdR^n_t\\quad\\text{and}\\quad \\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho_t=\\int_{[0, T]} X_tdR_t.\n\\end{equation}\nWith no loss of generality we can consider the absolutely continuous representatives of $\\rho$ and $\\rho^n$ from the class ${\\mathcal{A}^\\circ_{ac}}(\\mathcal{G}_t)$ in the definition of all the integrals above (which we still denote by $\\rho$ and $\\rho^n$ for simplicity). In light of this observation it is clear that $(R^n)_{n\\ge 1}\\subset{\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t)$ and $R\\in {\\tilde{\\mathcal{A}}^\\circ}(\\mathcal{G}_t)$. The idea is then to apply Proposition \\ref{prop:r-convergence} to the integrals with $R^n$ and $R$ in \\eqref{eq:intdR}.\n\nThanks to Lemma \\ref{lem:cadlag_convergence} and recalling that $\\rho^n_T = \\rho_T = 1$, the set\n\\[\n\\Omega_0 = \\big\\{\\omega\\in\\Omega:\\lim_{n\\to\\infty}\\rho^n_t(\\omega)= \\rho_t(\\omega) \\text{ for all $t \\in [0, T]$}\\big\\}\n\\]\nhas full measure, i.e., $\\mathbb{P}(\\Omega_0)=1$. For any $\\omega \\in \\Omega_0$ and $t\\in[0,T]$, integrating by parts (see, e.g., \\cite[Prop. 4.5, Chapter 0]{revuzyor}), using the dominated convergence theorem and then again integrating by parts give\n\\begin{equation}\\label{eqn:conv_R}\n\\lim_{n \\to \\infty} R^n_t= \\lim_{n \\to \\infty} \\bigg[(1-\\chi_{t})\\rho^n_t - \\int_{[0,t]} \\rho^n_sd(1-\\chi_{s}) \\bigg]\n= (1-\\chi_{t})\\rho_t - \\int_{[0,t]} \\rho_sd(1-\\chi_{s})=R_t.\n\\end{equation}\nHence $R^n$ and $R$ satisfy the assumptions of Proposition \\ref{prop:r-convergence} and we can conclude that \\eqref{eq:lim00} holds.\n\\end{proof}\n\n\\begin{cor}\\label{cor-specific-convergence-2}\nUnder the assumptions of Proposition \\ref{prop-specific-convergence-2}, we have \n\\begin{equation*\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} X_t(1-\\chi_{t})d\\rho^n_t + X_T \\Delta \\chi_T \\Delta \\rho^n_T\\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T)} X_t(1-\\chi_{t})d\\rho_t + X_T \\Delta \\chi_T \\Delta \\rho_T\\bigg].\n\\end{equation*}\n\\end{cor}\n\\begin{proof}\nRecall that $\\rho^n$ and $\\rho$ are continuous everywhere apart from $T$. Hence, we can rewrite the left- and right-hand side of \\eqref{eq:lim00} as\n\\[\n\\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho^n_t = \\int_{[0, T]} X_t(1-\\chi_{t})d\\rho^n_t + X_T \\Delta \\chi_T \\Delta \\rho^n_T\n\\]\nand\n\\[\n\\int_{[0, T]} X_t(1-\\chi_{t-})d\\rho_t = \\int_{[0, T]} X_t(1-\\chi_{t})d\\rho_t + X_T \\Delta \\chi_T \\Delta \\rho_T\\,,\n\\]\nrespectively. It remains to note that $\\int_{[0, T]} X_t(1-\\chi_{t})d\\rho^n_t = \\int_{[0, T)} X_t(1-\\chi_{t})d\\rho^n_t$ because $\\chi_T = 1$.\n\\end{proof}\n\nWe close this technical section with a similar result to the above but for approximations which are needed for the proof of Proposition \\ref{thm:conv_lipsch}. The next proposition is tailored for our specific type of regularisation of processes in $\\mathcal{A}(\\mathcal{F}^1_t)$. Notice that the left hand side of \\eqref{eq:lim-in-A} features $\\chi_{t-}$ while the right hand side has $\\chi_t$.\n\n\\begin{Proposition}\\label{prop-specific-convergence-3}\nFor a filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$, let $\\chi,\\rho \\in{\\mathcal{A}^\\circ} (\\mathcal{G}_t)$, $(\\rho^n)_{n\\ge 1}\\subset{\\mathcal{A}^\\circ} (\\mathcal{G}_t)$ and consider $X\\in\\mcalL$ which is $(\\mathcal{F}_t)$-adapted and regular. Assume the sequence $(\\rho^n)_{n\\ge 1}$ is non-decreasing and for $\\mathbb{P}$\\ae $\\omega\\in\\Omega$\n\\begin{align}\\label{eq:conv-rho}\n\\lim_{n\\to\\infty}\\rho^{n}_t(\\omega)=\\rho_{t-}(\\omega)\\:\\: \\text{for all $t\\in[0,T)$}.\n\\end{align}\nThen\n\\begin{equation}\\label{eq:lim-in-A}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} X_t(1-\\chi_{t-})d\\rho^n_t\\bigg]= \\mathbb{E}\\bigg[\\int_{[0, T)} X_t(1-\\chi_{t})d\\rho_t\\bigg],\n\\end{equation}\nand for $\\mathbb{P}$\\ae $\\omega \\in \\Omega$\n\\begin{equation}\\label{eqn:lim-in-t-}\n\\lim_{n\\to\\infty}\\rho^n_{t-}(\\omega)=\\rho_{t-}(\\omega)\\quad \\text{for all $t \\in [0, T]$}.\n\\end{equation}\n\\end{Proposition}\n\\begin{proof}\nDenote by $\\Omega_0$ the set on which the convergence \\eqref{eq:conv-rho} holds. The first observation is that for all $\\omega\\in\\Omega_0$ and $t \\in (0, T]$\n\\begin{align}\\label{eq:lim-nt}\n\\lim_{n\\to\\infty}\\rho^n_{t-}(\\omega)=\\lim_{n\\to\\infty}\\lim_{u\\uparrow t}\\rho^n_{u}(\\omega)=\\lim_{u\\uparrow t}\\lim_{n\\to\\infty}\\rho^n_{u}(\\omega)=\\lim_{u\\uparrow t}\\rho_{u-}(\\omega)=\\rho_{t-}(\\omega),\n\\end{align}\nwhere the order of limits can be swapped by monotonicity of the process and of the sequence. The convergence at $t=0$ is obvious as $\\rho^n_{0-} = \\rho_{0-} = 0$. This proves \\eqref{eqn:lim-in-t-}.\n\nDefine for $t \\in [0, T)$,\n\\begin{equation}\\label{eqn:def_Rn}\nR^n_t=\\int_{[0, t]} (1-\\chi_{s-}) d\\rho^n_s, \\qquad R_t=\\int_{[0, t]} (1-\\chi_{s}) d\\rho_s,\n\\end{equation}\nand extend both processes to $t=T$ in a continuous way by taking $R^n_{T}:=R^n_{T-}$ and $R_{T}:=R_{T-}$. By construction we have $(R^n)_{n\\ge 1}\\subset\\tilde{{\\mathcal{A}^\\circ}} (\\mathcal{G}_t)$ and $R\\in \\tilde{{\\mathcal{A}^\\circ}}(\\mathcal{G}_t)$ and the idea is to apply Proposition \\ref{prop:r-convergence}. First we notice that for all $\\omega\\in\\Omega$ and any $t\\in[0,T)$ we have\n\\[\n\\Delta R_t(\\omega)=(1-\\chi_t(\\omega))\\Delta\\rho_t(\\omega),\n\\] \nso that we can write the set of points of continuity of $R$ as (recall Definition \\ref{def:def_C})\n\\[\nC_R(\\omega)=C_{\\rho}(\\omega)\\cup \\{t\\in[0,T]:\\chi_t(\\omega)=1\\}.\n\\] \n\nFor any $t\\in[0, T)$ and all $\\omega\\in\\Omega_0$, integrating $R^n_t(\\omega)$ by parts (\\citep[Prop. 4.5, Chapter 0]{revuzyor}) and then taking limits as $n\\to\\infty$ we get\n\\begin{align}\\label{eq:Rcon}\n\\lim_{n\\to\\infty}R^n_t(\\omega)=&\\,\\lim_{n\\to\\infty} \\Big[(1-\\chi_{t}(\\omega))\\rho^n_t(\\omega) - \\int_{[0, t]} \\rho^n_s(\\omega)d (1-\\chi_{s}(\\omega))\\Big]\\\\\n=&\\, (1-\\chi_{t}(\\omega))\\rho_{t-}(\\omega) - \\int_{[0, t]} \\rho_{s-}(\\omega)d (1-\\chi_{s}(\\omega)) \\notag\\\\\n=&\\, R_t(\\omega) - (1-\\chi_t(\\omega)) \\Delta \\rho_t(\\omega)=R_{t-}(\\omega),\\notag \n\\end{align}\nwhere the second equality uses dominated convergence and \\eqref{eq:conv-rho}, and the third equality is integration by parts. We can therefore conclude that \n\\begin{align*}\n\\lim_{n\\to\\infty}R^n_t(\\omega)=R_t(\\omega),\\quad\\text{for all $t\\in C_R(\\omega) \\cap[0,T)$ and all $\\omega\\in\\Omega_0$}.\n\\end{align*}\n\nIt remains to show the convergence at $T$ which is in $C_R(\\omega)$ by our construction of $R$. Since the function $t \\mapsto \\rho_t(\\omega)$ is non-decreasing and the sequence $(\\rho^n(\\omega))_n$ is non-decreasing, the sequence $(R^n(\\omega))_n$ is non-decreasing too (an easy proof of this fact involves integration by parts and observing that $t\\mapsto d (1-\\chi_t(\\omega))$ defines a negative measure; notice also the link to the first-order stochastic dominance). As in \\eqref{eq:lim-nt}, we show that $\\lim_{n \\to \\infty} R^n_{T-}(\\omega) = R_{T-} (\\omega)$ for $\\omega \\in \\Omega_0$. By construction of $R^n$ and $R$, this proves convergence of $R^n_T$ to $R_T$.\n\nThen, the processes $R^n$ and $R$ fulfil all the assumptions of Proposition \\ref{prop:r-convergence} whose application allows us to obtain \\eqref{eq:lim-in-A}.\n\\end{proof}\n\nFrom the convergence \\eqref{eq:Rcon}, an identical argument as in \\eqref{eq:lim-nt} proves convergence of left-limits of processes $(R^n)$ at any $t \\in [0, T]$. The following corollary formalises this observation. It will be used in Section \\ref{sec:relax}.\n\\begin{cor}\\label{cor:lim_R_in-t-}\nConsider the processes $(R^n)$ and $R$ defined in \\eqref{eqn:def_Rn}. For $\\mathbb{P}$\\ae $\\omega \\in \\Omega$ we have\n\\begin{equation*}\n\\lim_{n\\to\\infty} R^n_{t-}(\\omega)=R_{t-}(\\omega)\\quad \\text{for all $t \\in [0, T]$}.\n\\end{equation*}\n\\end{cor}\n\n\n\\subsection{Verification of the conditions of Sion's theorem}\\label{sec:verif}\n\nFor the application of Sion's theorem, we will consider a weak topology on $\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ and $\\mathcal{A}(\\mathcal{F}^2_t)$ inherited from the space $\\mathcal{S}$. In our arguments, we will often use that for convex sets the weak and strong closedness are equivalent \\cite[Theorem 3.7]{Brezis2010} (although weak and strong convergence are not equivalent, c.f. \\cite[Corollary 3.8]{Brezis2010}).\n\n\\begin{Lemma} \\label{lem-strat-set-compact}\nFor any filtration $(\\mathcal{G}_t) \\subseteq (\\mathcal{F}_t)$ satisfying the usual conditions, the set $\\mathcal{A}(\\mathcal{G}_t)$ is weakly compact in $\\mathcal{S}$.\n\\end{Lemma}\n\\begin{proof}\nWe write $\\mathcal{A}$ for $\\mathcal{A}(\\mathcal{G}_t)$ and ${\\mathcal{A}^\\circ}$ for ${\\mathcal{A}^\\circ}(\\mathcal{G}_t)$. The set $\\mathcal{A}$ is a subset of a ball in $\\mathcal{S}$. Since $\\mathcal{S}$ is a reflexive Banach space, this ball is weakly compact (Kakutani's theorem, \\cite[Theorem 3.17]{Brezis2010}). Therefore, we only need to show that $\\mathcal{A}$ is weakly closed. Since $\\mathcal{A}$ is convex, it is enough to show that $\\mathcal{A}$ is strongly closed \\cite[Theorem 3.7]{Brezis2010}.\n\nTake a sequence $(\\rho^n)_{n\\ge 1}\\subset\\mathcal{A}$ that converges strongly in $\\mathcal{S}$ to $\\rho$. We will prove that $\\rho\\in\\mathcal{A}$ by constructing a c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} non-decreasing adapted process $(\\hat \\rho_t)$ such that $\\hat\\rho_{0-} = 0$, $\\hat \\rho_T = 1$, and $\\hat \\rho = \\rho$ $(\\lambda\\times\\mathbb{P})$\\ae With no loss of generality we can pass to the c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives $(\\hat \\rho^n)_{n\\ge 1}\\subset{\\mathcal{A}^\\circ}$ which also converge to $\\rho$ in $\\mathcal S$. Then, there is a subsequence $(n_k)_{k\\ge 1}$ such that $\\hat\\rho^{n_k}\\to \\rho$ $(\\lambda \\times \\mathbb{P})$\\ae \\cite[Theorem 4.9]{Brezis2010}.\n\nSince \n\\[\n\\int_0^t \\mathbb{P}\\big(\\lim_{k\\to\\infty}\\hat\\rho^{n_k}_s=\\rho_s\\big) d s=t,\\quad\\text{for all $t\\in[0,T]$,}\n\\]\nwe can find $\\hat D\\subset [0,T]$ with $\\lambda([0,T]\\setminus\\hat D)=0$ such that $\\mathbb{P}(\\Omega_t)=1$ for all $t\\in \\hat D$, where \n\\[\n\\Omega_t:=\\{\\omega\\in\\Omega: \\lim_{k\\to\\infty}\\hat \\rho^{n_k}_t(\\omega)=\\rho_t(\\omega)\\}.\n\\]\nThen we can take a dense countable subset $D\\subset \\hat D$ and define $\\Omega_0:=\\cap_{t\\in D}\\Omega_t$ so that $\\mathbb{P}(\\Omega_0)=1$ and \\[\n\\lim_{k\\to\\infty}\\hat \\rho^{n_k}_t(\\omega)=\\rho_t(\\omega),\\qquad\\text{for all $(t,\\omega)\\in D\\times\\Omega_0$.}\n\\]\nSince $\\hat \\rho^{n_k}$ are non-decreasing, so is the mapping $D\\ni t\\mapsto \\rho_t(\\omega)$ for all $\\omega\\in\\Omega_0$. Let us extend this mapping to $[0,T]$ by defining $\\hat \\rho_t(\\omega):=\\rho_t(\\omega)$ for $t\\in D$ and\n\\[\n\\hat{\\rho}_t(\\omega):=\\lim_{s\\in D:s\\downarrow t} \\rho_s(\\omega),\\quad\\hat{\\rho}_{0-}(\\omega):=0,\\quad \\hat{\\rho}_{T}(\\omega):=1,\\quad\\text{for all $\\omega\\in \\Omega_0$,}\n\\]\nwhere the limit exists due to monotonicity. For $\\omega\\in \\mathcal{N}:=\\Omega\\setminus\\Omega_0$, we set $\\hat \\rho_t(\\omega) = 0$ for $t < T$ and $\\hat \\rho_T(\\omega)=1$.  Notice that $\\mathcal{N}\\in\\mathcal{G}_0$ since $\\mathbb{P}(\\mathcal{N})=0$ so that $\\hat{\\rho}_t$ is $\\mathcal{G}_t$-measurable for $t\\in D$. Moreover, $\\hat{\\rho}$ is c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} by construction and $\\hat{\\rho}_t$ is measurable with respect to $\\cap_{s\\in D, s > t\\,}\\mathcal{G}_s=\\mathcal{G}_{t+}=\\mathcal{G}_t$ for each $t\\in[0,T]$ by the right-continuity of the filtration.  Hence, $\\hat \\rho$ is $(\\mathcal{G}_t)$-adapted and $\\hat \\rho \\in {\\mathcal{A}^\\circ}$.\n\nIt remains to show that $\\hat \\rho^{n_k}\\to \\hat{\\rho}$ in $\\mathcal{S}$ so that $\\hat \\rho=\\rho$ $(\\lambda\\times\\mathbb{P})$\\ae and therefore $\\rho\\in\\mathcal{A}$. It suffices to show that $\\hat \\rho^{n_k}\\to \\hat{\\rho}$ $(\\lambda\\times\\mathbb{P})$\\ae and then conclude by dominated convergence that $\\hat \\rho^{n_k}\\to \\hat{\\rho}$ in $\\mathcal{S}$. For each $\\omega\\in\\Omega_0$ the process $t\\mapsto\\hat \\rho(\\omega)$ has at most countably many jumps (on any bounded interval) by monotonicity, i.e., $\\lambda([0,T]\\setminus C_{\\hat \\rho}(\\omega))=0$ (recall Definition \\ref{def:def_C}). Moreover, arguing as in the proof of Lemma \\ref{lem:cadlag_convergence}, we conclude\n\\[\n\\lim_{k\\to\\infty}\\hat \\rho^{n_k}_t(\\omega)=\\hat \\rho_t(\\omega),\\quad\\text{for all $t\\in C_{\\hat \\rho}(\\omega)$ and all $\\omega\\in\\Omega_0$}.\n\\]\nSince $(\\lambda\\!\\times\\!\\mathbb{P})(\\{(t,\\omega)\\!:\\!t\\in C_{\\hat\\rho}(\\omega)\\cap B,\\omega\\in\\Omega_0\\})\\!=\\!\\lambda(B)$ for any bounded interval $B\\subseteq[0,T]$ then $\\hat \\rho^{n_k}\\!\\to\\! \\hat{\\rho}$ in $\\mathcal{S}$ and $\\mathcal{A}$ is strongly closed in $\\mathcal{S}$.\n\\end{proof}\n \n\\begin{remark}\\label{rem-TV-norm-doesnt-work}\nOur space $\\mathcal{A}(\\mathcal{G}_t)$ is the space of processes that generate randomised stopping times and for any $\\rho\\in\\mathcal{A}(\\mathcal{G}_t)$ we require that $\\rho_T(\\omega)=1$, for all $\\omega\\in\\Omega$. In the finite horizon problem, i.e., $T<\\infty$, such specification imposes a constraint that prevents a  direct use of the topology induced by the norm considered in \\cite{TouziVieille2002}. Indeed, in \\cite{TouziVieille2002} the space $\\mathcal S$ is that of $(\\mathcal{G}_t)$-adapted processes $\\rho$ with\n\\begin{equation*}\n\\|\\rho\\|^2:=\\mathbb{E}\\bigg[\\int_{0}^T (\\rho_t)^2 d t + (\\Delta \\rho_T)^2\\bigg] < \\infty,\\quad \\Delta\\rho_T:=\\rho_T-\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{t\\uparrow T}\\rho_t.\n\\end{equation*}\nThe space of generating processes $\\mathcal{A}(\\mathcal{G}_t)$ is not closed in the topology induced by $\\|\\cdot\\|$ above: define a sequence $(\\rho^n)_{n\\ge 1}\\subset\\mathcal{A}(\\mathcal{G}_t)$ by\n\\begin{equation*}\n\\rho^n_t = n \\bigg(t - T + \\frac{1}{n}\\bigg)^+, \\qquad t \\in [0, T].\n\\end{equation*}\nThen $\\|\\rho^n\\|\\to 0$ as $n\\to\\infty$ but $\\rho\\equiv 0\\notin\\mathcal{A}(\\mathcal{G}_t)$ since it fails to be equal to one at $T$ (and it is not possible to select a representative from $\\mathcal{A}(\\mathcal{G}_t)$ with the equivalence relation induced by $\\|\\,\\cdot\\,\\|$). \n\\end{remark}\n\nIt is of interest to explore the relationship between the topology on $\\mathcal{A}(\\mathcal{G}_t)$ implied by the weak topology on $\\mathcal{S}$ (denote it by $\\mcalO_2$) and the topology introduced in \\cite{BaxterChacon, Meyer} (denote it by $\\mcalO_1$). The topology $\\mcalO_1$ is the coarsest topology in which all functionals of the form\n\\begin{equation}\\label{eqn:top_O2}\n\\mathcal{A}(\\mathcal{G}_t) \\ni \\rho \\mapsto \\mathbb{E} \\Big[\\int_{[0, T]} X_t\\, d \\rho_t\\Big]\n\\end{equation}\nare continuous for any $X \\in \\mcalL$ with continuous trajectories. Our topology $\\mcalO_2$, instead, is the restriction to $\\mathcal{A}(\\mathcal{G}_t)$ of the weak topology on $\\mathcal{S}$. That is, $\\mcalO_2$ is the coarsest topology for which all functionals of the form\n\\begin{equation*\n\\mathcal{A}(\\mathcal{G}_t) \\ni \\rho \\mapsto \\mathbb{E} \\Big[\\int_{[0, T]} \\rho_t\\, Y_t\\, d t\\Big]\n\\end{equation*}\nare continuous for all $Y \\in \\mathcal{S}$. \n\\begin{Lemma}\\label{lem:top}\nTopologies $\\mcalO_1$ and $\\mcalO_2$ are identical.\n\\end{Lemma}\n\\begin{proof}\nDenoting \n\\begin{align}\\label{eq:Xint}\nX_t = \\int_{[0, t]} Y_t\\, d t\n\\end{align} \nand integrating by parts, we obtain for $\\rho \\in \\mathcal{A} (\\mathcal{G}_t)$\n\\[\n\\mathbb{E} \\Big[\\int_{[0, T]} \\rho_t\\, Y_t\\, d t\\Big]\n=\n\\mathbb{E} \\Big[X_T \\rho_T - X_0 \\rho_{0-} - \\int_{[0, T]} X_t\\, d \\rho_t \\Big]\n=\n\\mathbb{E} \\Big[X_T - \\int_{[0, T]} X_t\\, d \\rho_t \\Big],\n\\]\nwhere we used that $\\rho_T = 1$ and $\\rho_{0-} = 0$, $\\mathbb{P}$\\as Hence, $\\mcalO_2$ is the coarsest topology on $\\mathcal{A} (\\mathcal{G}_t)$ for which functionals \\eqref{eqn:top_O2} are continuous for all processes $X$ defined as in \\eqref{eq:Xint}. Since these processes $X$ are continuous, we conclude that $\\mcalO_2 \\subset \\mcalO_1$.\n\nThe set $\\mathcal{A} (\\mathcal{G}_t)$ is compact in the topologies $\\mcalO_1$ \\citep[Theorem 3]{Meyer} and $\\mcalO_2$ (see Lemma \\ref{lem-strat-set-compact} above). The compact Hausdorff topology is the coarsest among Hausdorff topologies \\cite[Cor.~3.1.14, p. 126]{Engelking}. Since $\\mcalO_2$ is Hausdorff by \\cite[Prop~3.3]{Brezis2010}, so is $\\mcalO_1$ and we have $\\mcalO_1 = \\mcalO_2$.\n\n\\end{proof}\n\n\\begin{remark}\nMeyer \\cite[Thm.~4]{Meyer} shows that if $\\mathcal{F}$ is separable (i.e., countably generated) then the topology $\\mcalO_1$ (hence $\\mcalO_2$) is metrizable. This could also be seen directly for the topology $\\mcalO_2$ by \\cite[Thm.~3.29]{Brezis2010}, because $\\mathcal{A}(\\mathcal{G}_t)$ is bounded in $\\mathcal{S}$ and $\\mcalO_2$ is the restriction to $\\mathcal{A}(\\mathcal{G}_t)$ of the weak topology on $\\mathcal{S}$. Indeed, it emerges from this argument for the metrizability of $\\mcalO_2$ that it is sufficient to require that only $\\mathcal{G}_T$ be separable.\n\\end{remark}\n\n\n\\begin{Lemma}\\label{lem:semi-cont}\nGiven any $(\\xi,\\zeta)\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t) \\times\\mathcal{A} (\\mathcal{F}^2_t)$, the functionals $N(\\xi,\\cdot):\\mathcal{A}(\\mathcal{F}^2_t)\\to\\mathbb{R}$ and $N(\\cdot,\\zeta):\\mathcal{A}_{ac}(\\mathcal{F}^1_t)\\to\\mathbb{R}$ are, respectively, upper semicontinuous and lower semicontinuous in the strong topology of $\\mathcal{S}$.\n\\end{Lemma}\n\\begin{proof}\nSince $\\xi \\in \\mathcal{A}_{ac} (\\mathcal{F}^1_t)$, we have from \\eqref{eq-functional-in-terms-of-controls} that the contribution of simultaneous jumps reduces to a single term:\n\\begin{equation}\\label{eqn:N_cont}\nN(\\xi,\\zeta)= \\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi_t + \\int_{[0, T)} g_t(1-\\xi_t)d\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg].\n\\end{equation}\n\n\\emph{Upper semicontinuity of $N(\\xi,\\cdot)$}. Fix $\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ and consider a sequence $(\\zeta^{n})_{n \\ge 1}\\subset\\mathcal{A}(\\mathcal{F}^2_t)$ converging to $\\zeta \\in \\mathcal{A}(\\mathcal{F}^2_t)$ strongly in $\\mathcal{S}$. We have to show that\n\\begin{equation*}\n\\mathop{\\lim\\sup}_{n\\to\\infty} N(\\xi,\\zeta^{n})\\le N(\\xi,\\zeta).\n\\end{equation*}\nAssume, by contradiction, that $\\mathop{\\lim\\sup}_{n\\to\\infty} N(\\xi,\\zeta^{n}) > N(\\xi,\\zeta)$. There is a subsequence $(n_k)$ over which the limit on the left-hand side is attained. Along a further subsequence we have $(\\mathbb{P}\\times\\lambda)\\ae$ convergence of $\\zeta^n$ to $\\zeta$ \\citep[Theorem 4.9]{Brezis2010}. With an abuse of notation we will assume that the original sequence possesses those two properties, i.e., the limit $\\lim_{n\\to\\infty} N(\\xi,\\zeta^{n})$ exists, it strictly dominates $N(\\xi,\\zeta)$, and there is $(\\mathbb{P} \\times \\lambda)\\ae$ convergence of $\\zeta^{n}$ to $\\zeta$.\n\nSince $\\xi$ is absolutely continuous on $[0, T)$, \n\\begin{equation*}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta^{n}_{t})d\\xi_t\\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi_t \\bigg]\n\\end{equation*}\nby the dominated convergence theorem. For the last two terms of $N(\\xi, \\zeta^n)$ in \\eqref{eqn:N_cont} we have\n\\begin{align*}\n\\mathbb{E}\\bigg[\\int_{[0, T)} g_t(1-\\xi_t)d\\zeta^n_t + h_T\\Delta\\xi_T\\Delta\\zeta^n_T\\bigg]\n&=\n\\mathbb{E}\\bigg[\\int_{[0, T)} g_t(1-\\xi_{t-})d\\zeta^n_t + h_T\\Delta\\xi_T\\Delta\\zeta^n_T\\bigg]\\\\\n&=\n\\mathbb{E}\\bigg[\\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta^n_t + (h_T - g_T) \\Delta\\xi_T\\Delta\\zeta^n_T\\bigg],\n\\end{align*}\nwhere the first equality is by the continuity of $\\xi$ and for the second one we use that $1-\\xi_{T-} = \\Delta \\xi_T$. From Lemma \\ref{lem:cadlag_convergence} and the boundedness and continuity of $(\\xi_t)$ we verify the assumptions of Proposition \\ref{prop:r-convergence} (with $X_t=g_t(1-\\xi_{t-})$ therein since $\\xi_{t-}$ is continuous on $[0,T]$), so\n\\begin{equation*}\n\\lim_{n \\to \\infty} \\mathbb{E}\\bigg[\\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta^n_t \\bigg] = \\mathbb{E}\\bigg[\\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta_t \\bigg].\n\\end{equation*}\nRecalling that $g_T \\le h_T$, we obtain from Lemma \\ref{prop-terminal-time-jump-limit}\n\\begin{equation*}\n\\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}\\big[(h_T-g_T)\\Delta\\xi_T\\Delta\\zeta^n_T\\big] \\le \\mathbb{E}\\big[(h_T-g_T)\\Delta\\xi_T\\Delta\\zeta_T\\big].\n\\end{equation*}\nCombining above convergence results contradicts $\\lim_{n\\to\\infty} N(\\xi,\\zeta^n) >N(\\xi,\\zeta)$, hence, proves the upper semicontinuity.\n\n\\emph{Lower semicontinuity of $N(\\cdot,\\zeta)$}. Fix $\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)$ and consider a sequence $(\\xi^{n})_{n\\ge 1}\\subset\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ converging to $\\xi \\in \\mathcal{A}_{ac} (\\mathcal{F}^1_t)$ strongly in $\\mathcal{S}$. Arguing by contradiction as above, we assume that there is a subsequence of $\\xi^{n}$, which we denote the same, such that $\\xi^{n} \\to \\xi$ $(\\mathbb{P}\\times\\lambda)$\\ae and \n\\begin{equation}\\label{eqn:two_terms2}\n\\lim_{n\\to\\infty} N(\\xi^{n},\\zeta) < N(\\xi,\\zeta). \n\\end{equation}\nBy Lemma \\ref{lem:cadlag_convergence} and the continuity of $(\\xi_t)$ we have for $\\mathbb{P}$\\ae $\\omega\\in\\Omega$\n\\[\n\\lim_{n\\to\\infty} \\xi^{n}_t(\\omega) = \\xi_t(\\omega)\\quad\\text{for all $t \\in [0,T)$}.\n\\] \nThen by dominated convergence for the second term of $N(\\xi^n,\\zeta)$ in \\eqref{eqn:N_cont} we get\n\\[\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} g_t(1-\\xi^{n}_t)d\\zeta_t\\bigg]= \\mathbb{E}\\bigg[\\int_{[0, T)} g_t(1-\\xi_t)d\\zeta_t\\bigg].\n\\]\nFor the remaining terms of $N(\\xi^{n}, \\zeta)$, we have \n\\[\n\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi^n_t + h_T\\Delta\\xi^n_T\\Delta\\zeta_T\\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi^n_t + f_T \\Delta\\xi^n_T\\Delta\\zeta_T + (h_T-f_T)\\Delta\\xi^n_T\\Delta\\zeta_T\\bigg]. \n\\]\nObserve that, by Lemma \\ref{prop-terminal-time-jump-limit},\n\\begin{equation*}\n\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n\\to\\infty}\\mathbb{E}\\big[(h_T-f_T)\\Delta\\xi^n_T\\Delta\\zeta_T\\big]\\ge \\mathbb{E}\\big[(h_T-f_T)\\Delta\\xi_T\\Delta\\zeta_T\\big],\n\\end{equation*}\nbecause $h_T-f_T\\le 0$. Further,\n\\begin{equation*}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi^n_t + f_T \\Delta\\xi^n_T\\Delta\\zeta_T \\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi_t + f_T \\Delta\\xi_T\\Delta\\zeta_T\\bigg]\n\\end{equation*}\nby Corollary \\ref{cor-specific-convergence-2}. The above results contradict \\eqref{eqn:two_terms2}, therefore, proving the lower semicontinuity.\n\\end{proof}\n\nWe are now ready to prove that the game with continuous randomisation for the first player ($\\tau$-player) has a value.\n\\begin{proof}[{\\bf Proof of Theorem \\ref{th-value-cont-strat}}]\nWe will show that the conditions of Sion's theorem hold (recall the notation in Theorem \\ref{th-the-Sion}) with $(A,B)=(\\mathcal{A}(\\mathcal{F}^2_t),\\mathcal{A}_{ac}(\\mathcal{F}^1_t))$ on the space $\\mathcal{S} \\times \\mathcal{S}$ equipped with its weak topology. For the sake of compactness of notation, we will write $\\mathcal{A}$ for $\\mathcal{A}(\\mathcal{F}^2_t)$ and $\\mathcal{A}_{ac}$ for $\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$. It is straightforward to verify that the sets $\\mathcal{A}$ and $\\mathcal{A}_{ac}$ are convex. Compactness of $\\mathcal{A}$ in the weak topology of $\\mathcal{S}$ follows from Lemma \\ref{lem-strat-set-compact}. It remains to prove the convexity and semi-continuity properties of $N$ with respect to the weak topology of $\\mathcal{S}$. This is equivalent to showing that for any $a\\in\\mathbb{R}$, $\\hat{\\xi}\\in\\mathcal{A}_{ac}$ and $\\hat{\\zeta}\\in\\mathcal{A}$ the level sets \n\\[\n\\mathcal{K}(\\hat{\\zeta},a)=\\{\\xi\\in\\mathcal{A}_{ac}:N(\\xi,\\hat{\\zeta})\\le a\\} \\qquad \\text{and}\\qquad  \\mathcal{Z}(\\hat{\\xi},a)=\\{\\zeta\\in\\mathcal{A}:N(\\hat{\\xi},\\zeta)\\ge a\\}\n\\]\nare convex and closed in $\\mathcal{A}_{ac}$ and $\\mathcal{A}$, respectively, with respect to the weak topology of $\\mathcal{S}$. For any $\\lambda \\in [0,1]$ and $\\xi^{1}, \\xi^{2} \\in \\mathcal{A}_{ac}$, $\\zeta^{1}, \\zeta^{2} \\in \\mathcal{A}$, using the expression in \\eqref{eq-functional-in-terms-of-controls} it is immediate (by linearity) that \n\\begin{align*}\nN(\\lambda \\xi^{1} + (1-\\lambda) \\xi^{2}, \\hat \\zeta) &= \\lambda N(\\xi^{1}, \\hat \\zeta) + (1-\\lambda) N(\\xi^{2}, \\hat\\zeta),\\\\\nN(\\hat\\xi, \\lambda \\zeta^{1} + (1-\\lambda)\\zeta^{2}) &= \\lambda N(\\hat\\xi, \\zeta^{1}) + (1-\\lambda) N(\\hat\\xi, \\zeta^{2}).\n\\end{align*}\nThis proves the convexity of the level sets. Their closedness in the strong topology of $\\mathcal{S}$ is established in Lemma \\ref{lem:semi-cont}. The latter two properties imply, by \\cite[Theorem 3.7]{Brezis2010}, that the level sets are closed in the weak topology of $\\mathcal{S}$. Sion's theorem (Theorem \\ref{th-the-Sion}) yields the existence of the value of the game: $W_* = W^*$.\n\nThe second part of the statement results from using a version of Sion's theorem proved in \\cite{Komiya1988} which allows to write $\\max$ instead of $\\sup$ in \\eqref{eq-value-cont-restriction}, i.e.,\n\\begin{equation*}\n\\sup_{\\zeta\\in\\mathcal{A}}\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}} N(\\xi,\\zeta)=\\max_{\\zeta\\in\\mathcal{A}}\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}} N(\\xi,\\zeta)=\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}} N(\\xi,\\zeta^*),\n\\end{equation*}\nwhere $\\zeta^*\\in\\mathcal{A}$ delivers the maximum.\n\\end{proof}\n\n\n\\subsection{Approximation with continuous controls}\\label{sec:approx}\nWe now prove Proposition \\ref{thm:conv_lipsch} by constructing a sequence $(\\xi^{n})$ of Lipschitz continuous processes with the Lipschitz constant for each process bounded by $n$ for all $\\omega$. This uniform bound on the Lipschitz constant is not used in this paper as we only need that each of the processes $(\\xi^{n}_t)$ has absolutely continuous trajectories with respect to the Lebesgue measure on $[0,T)$ so that it belongs to $\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$.\n\n\n\\begin{proof}[Proof of Proposition \\ref{thm:conv_lipsch}]\n\nFix $\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)$. We need to show that for any $\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)$, there exists a sequence $(\\xi^{n})_{n \\ge 1} \\subset \\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ such that\n\\begin{equation}\n\\mathop{\\lim\\sup}_{n\\to\\infty} N(\\xi^{n},\\zeta)\\le N(\\xi,\\zeta).\n\\label{eq-liminf-M}\n\\end{equation}\n\nWe will explicitly construct absolutely continuous $\\xi^{n}$ that approximate $\\xi$ in a suitable sense. As $N(\\xi, \\zeta)$ does not depend on the choice of  c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} representatives, by Definition \\ref{def:integral}, without loss of generality we assume that $\\xi \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^1_t)$ and $\\zeta \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$. Define a function $\\phi^n_t = (nt)\\wedge 1\\vee 0$. Let $\\xi^{n}_t = \\int_{[0, t]} \\phi^n_{t-s} d\\xi_s$ for $t\\in[0,T)$, and $\\xi^{n}_T = 1$. We shall show that $(\\xi^{n}_t)$ is $n$-Lipschitz, hence absolutely continuous on $[0, T)$.\nNote that $\\phi^n_t\\equiv 0$ for $t\\le 0$, and therefore $\\xi^{n}_t = \\int_{[0, T]} \\phi^n_{t-s} d\\xi_s$ for $t \\in [0, T)$. For arbitrary $t_1,t_2\\in[0,T)$ we have\n\\begin{align*}\n|\\xi^{n}_{t_1}-\\xi^{n}_{t_2}| \n&= \n\\left|\\int_{[0, T]} (\\phi^n_{t_1-s}-\\phi^n_{t_2-s}) d\\xi_s\\right| \n\\le\n\\int_{[0, T]} |\\phi^n_{t_1-s}-\\phi^n_{t_2-s}| d\\xi_s\\\\\n&\\le \n\\int_{[0, T]} n|(t_1-s)-(t_2-s)| d\\xi_s \n=\n\\int_{[0, T]} n|t_1-t_2| d\\xi_s=n|t_1-t_2|,\n\\end{align*}\nwhere the first inequality is Jensen's inequality (which is applicable since $\\xi(\\omega)$ is a cumulative distribution function on $[0, T]$ for each $\\omega$), and the second inequality follows by the definition of $\\phi^n$.\n\nWe will verify the assumptions of Proposition \\ref{prop-specific-convergence-3}. Clearly the sequence $(\\xi^n)$ is non-decreasing in $n$, as the measure $d \\xi(\\omega)$ is positive for each $\\omega \\in \\Omega$ and the sequence $\\phi^n$ is non-decreasing. By the construction of $\\xi^{n}$ we have $\\xi^{n}_0 = 0 \\to \\xi_{0-}$ as $n\\to\\infty$. Moreover, for any $t \\in (0, T)$ and $n > 1\/t$\n\\begin{align*}\n\\xi^{n}_t = \\int_{[0, t)}\\phi^n_{t-s}d\\xi_s=\\xi_{t-\\tfrac{1}{n}}+\\int_{(t-\\tfrac{1}{n}, t)}n(t-s)d\\xi_s,\n\\end{align*}\nwhere the first equality uses that $\\phi^n_{0}=0$, so that jumps of $\\xi$ at time $t$ give zero contribution, and the second one uses the definition of $\\phi^n$. Letting $n\\to\\infty$ we obtain $\\xi^{n}_t\\to \\xi_{t-}$ as the second term above vanishes since\n\\begin{align*}\n0\\le \\int_{(t-\\tfrac{1}{n}, t)}n(t-s)d\\xi_s\\le \\xi_{t-} - \\xi_{t-\\tfrac{1}{n}}\\to 0. \n\\end{align*}\nThe continuity of $\\xi^n$ on $[0, T)$ and Proposition \\ref{prop-specific-convergence-3} imply that\n\\begin{equation*}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)}f_t(1-\\zeta_{t})d\\xi^{n}_t\\bigg]\n=\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)}f_t(1-\\zeta_{t-})d\\xi^{n}_t\\bigg]\n= \n\\mathbb{E}\\bigg[\\int_{[0, T)}f_t(1-\\zeta_{t})d\\xi_t\\bigg],\n\\end{equation*}\nand $\\lim_{n \\to \\infty} \\xi^{n}_{T-} = \\xi_{T-}$ so that\n\\[\n\\lim_{n\\to\\infty}\\Delta\\xi^{n}_T =\\Delta\\xi_T,\n\\]\nsince $\\xi^{n}_T=1$ for all $n\\ge 1$.\nThe dominated convergence theorem (applied to the second integral below) also yields\n\\begin{equation}\\label{eqn:M_ij_conv}\n\\begin{aligned}\n\\lim_{n\\to\\infty} N(\\xi^{n},\\zeta)\n&=\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)}f_{t}(1-\\zeta_{t})d\\xi^{n}_{t}+ \\int_{[0, T)} g_t(1-\\xi^{n}_{t})d\\zeta_t + h_T\\Delta\\xi^{n}_T\\Delta\\zeta_T\\bigg]\\\\\n&= \\mathbb{E}\\bigg[\\int_{[0, T)}f_{t}(1-\\zeta_{t})d\\xi_{t} + \\int_{[0, T)} g_t(1-\\xi_{t-})d\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg].\n\\end{aligned}\n\\end{equation}\nNote that\n\\begin{align}\\label{eq-remove-common-jumps}\nN(\\xi,\\zeta)&=\\mathbb{E}\\bigg[\\!\\int_{[0, T)}\\! f_t(1-\\zeta_{t})d\\xi_t +\\! \\int_{[0, T)}\\! g_t(1\\!-\\xi_t)d\\zeta_t + \\sum_{t \\in [0, T]} h_t \\Delta\\xi_t\\Delta\\zeta_t\\bigg]\\notag\\\\\n&= \\mathbb{E}\\bigg[\\!\\int_{[0, T)}\\! f_t(1-\\zeta_{t})d\\xi_t +\\! \\int_{[0, T)}\\! g_t(1-\\xi_{t-})d\\zeta_t\n+\\! \\sum_{t \\in [0, T)}\\!\\! (h_t-g_t)\\Delta\\xi_t\\Delta\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg]\\\\\n&\\ge \\mathbb{E}\\bigg[\\!\\int_{[0, T)}\\! f_t(1-\\zeta_{t})d\\xi_t +\\! \\int_{[0, T)}\\! g_t(1-\\xi_{t-})d\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg],\\notag\n\\end{align}\nwhere the last inequality is due to Assumption \\ref{eq-order-cond}. Combining this with \\eqref{eqn:M_ij_conv} completes the proof of \\eqref{eq-liminf-M}.\n\\end{proof}\n\n\\subsection{Relaxation of Assumption \\ref{ass:regular}}\\label{sec:relax}\n\nAssumption \\ref{ass:regular} which requires that the payoff processes be regular can be relaxed to allow for a class of jumps including predictable ones with nonzero conditional mean (i.e., violating regularity, see Eq. \\eqref{eq:cond-reg}). In this section we extend Theorem \\ref{th-value-cont-strat} and Proposition \\ref{thm:conv_lipsch} to the case of Assumption \\ref{ass:regular_gen} with $(\\hat g_t)$ from the decomposition of the payoff process $g$ being non-decreasing. In this case we must `smoothen' the generating process $\\xi$ of the minimiser in order to guarantee the desired semi-continuity properties of the game's expected payoff (see Remark \\ref{rem:contrad}). Arguments when $(\\hat f_t)$ from the decomposition of $f$ in Assumption \\ref{ass:regular_gen} is non-increasing are analogous thanks to the symmetry of the set-up pointed out in Remark \\ref{rem:ineq}. However, in that case we restrict strategies of the maximiser to absolutely continuous generating processes $\\zeta\\in\\mathcal{A}_{ac}(\\mathcal{F}^2_t)$ and the first player (minimiser) picks $\\xi\\in\\mathcal{A}(\\mathcal{F}^1_t)$.\n\n\\begin{theorem}\\label{th-value-cont-strat_gen}\nUnder assumptions \\ref{eq-integrability-cond}, \\ref{ass:regular_gen}, \\ref{eq-order-cond}-\\ref{ass:filtration}, (with $\\hat g$ non-decreasing) the game \\eqref{eq-value-cont-restriction} has a value, i.e.\n\\begin{equation*}\nW_{*}=W^{*}:=W.\n\\end{equation*}\nMoreover, the $\\zeta$-player (maximiser) has an optimal strategy, i.e. there exists $\\zeta^*\\in\\mathcal{A}(\\mathcal{F}^2_t)$ such that\n\\begin{equation*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)} N(\\xi,\\zeta^*)=W.\n\\end{equation*}\n\\end{theorem}\n\\begin{Proposition}\\label{thm:conv_lipsch_gen}\nUnder assumptions \\ref{eq-integrability-cond}, \\ref{ass:regular_gen}, \\ref{eq-order-cond}-\\ref{ass:filtration}, (with $\\hat g$ non-decreasing) for any $\\zeta \\in \\mathcal{A}(\\mathcal{F}^2_t)$ and $\\xi \\in \\mathcal{A}(\\mathcal{F}^1_t)$, there is a sequence $\\xi^n \\in \\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ such that\n\\[\n\\mathop{\\lim\\sup}_{n \\to \\infty} N(\\xi^n, \\zeta) \\le N(\\xi, \\zeta).\n\\]\n\\end{Proposition}\n\\begin{proof}[{\\bf Proof of Theorem \\ref{thm:main2}}]\nThe proof of the existence of the value is identical to the proof Theorem \\ref{thm:main} but with references to Theorem \\ref{th-value-cont-strat} and Proposition \\ref{thm:conv_lipsch} replaced by the above results. \n\nFor the existence of the saddle point, the additional requirement that $\\hat g$ be non-decreasing {\\em and} $\\hat f$ be non-increasing guarantees the complete symmetry of the problem when swapping the roles of the two players as in Remark \\ref{rem:ineq}. Thus, the same proof as in Theorem \\ref{thm:main} can be repeated verbatim.\n\\end{proof}\n \n\nIn the rest of the section we prove Theorem \\ref{th-value-cont-strat_gen} and Proposition \\ref{thm:conv_lipsch_gen}. Processes $\\hat f, \\hat g$ have the following decomposition according to Theorem VI.52 in \\cite{DellacherieMeyer} and remarks thereafter: there are $(\\mathcal{F}_t)$-stopping times $(\\eta^f_k)_{k \\ge 1}$ and $(\\eta^g_k)_{k \\ge 1}$, non-negative $\\mathcal{F}_{\\eta^f_k}$-measurable random variables $X^f_k$, $k \\ge 1$, and non-negative $\\mathcal{F}_{\\eta^g_k}$-measurable random variables $X^g_k$, $k \\ge 1$, such that \n\\begin{equation}\\label{eqn:decomposition_piecewise}\n\\hat f_t = \\sum_{k=1}^\\infty (-1)^k X^f_k \\ind{\\{t \\ge \\eta^f_k\\}}, \\qquad \\hat g_t = \\sum_{k=1}^\\infty X^g_k \\ind{\\{t \\ge \\eta^g_k\\}}.\n\\end{equation}\nThe alternating terms in the sum for $(\\hat f_t)$ come from interweaving sequences for the two non-decreasing processes $(\\hat f^+_t)$ and $(\\hat f^-_t)$ from $\\mcalL$ arising from the decomposition of the integrable variation process $(\\hat f_t)$ (recall $\\hat f_t=\\hat f^+_t-\\hat f^-_t$). This is for notational convenience and resulting in no mathematical complications as the infinite sum is absolutely convergent. Recall that $\\hat g$ is assumed non-decreasing.\n\nThe condition that $\\hat f_0 = \\hat g_0 = 0$ means that $\\eta^f_k, \\eta^g_k > 0$ for all $k \\ge 1$. Since $\\hat f, \\hat g$ have integrable variation (in the sense of \\cite[p. 115]{DellacherieMeyer}), the infinite sequences in \\eqref{eqn:decomposition_piecewise} are dominated by integrable random variables $X^f$ and $X^g$: for any $t \\in [0, T]$\n\\begin{align}\\label{eq:Xfg}\n|\\hat f_t| \\le X^f := \\sum_{k=1}^\\infty X^f_k, \\qquad \\text{and}\\qquad \\hat g_t \\le X^g := \\sum_{k=1}^\\infty X^g_k.\n\\end{align}\n\nTo handle convergence of integrals with piecewise-constant processes, we need to extend the results of Proposition \\ref{prop:r-convergence}.\n\\begin{Proposition}\\label{prop:A}\nFor a filtration $(\\mathcal{G}_t)\\subseteq (\\mathcal{F}_t)$, consider $(\\rho^n)_{n \\ge 1} \\subset \\tilde{\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ and $\\rho \\in \\tilde{\\mathcal{A}^\\circ}(\\mathcal{G}_t)$ with\n\\[\n\\mathbb{P}\\Big(\\Big\\{\\omega \\in \\Omega:\\ \\lim_{n\\to\\infty}\\rho^n_t(\\omega)=\\rho_t(\\omega),\\quad\\text{for all $t\\in C_\\rho(\\omega)\\cup\\{T\\}$} \\Big\\} \\Big) = 1.\n\\]\nThen for any $\\mathcal{F}$-measurable random variables $\\theta\\in(0,T]$ and $X\\in[0,\\infty)$  with $\\mathbb{E}[X] < \\infty$ we have\n\\begin{equation}\\label{eqn:theta_t}\n\\mathop{\\lim\\sup}_{n\\to\\infty}\\mathbb{E}\\Big[\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho^n_t\\Big] \\le \\mathbb{E}\\Big[\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho_t\\Big].\n\\end{equation}\nFurthermore, if $\\mathbb{P} (\\{\\omega:\\ \\theta(\\omega) \\in C_\\rho(\\omega) \\text{ or } X(\\omega) = 0\\}) = 1$, then\n\\begin{equation}\\label{eqn:theta_t_eq}\n\\lim_{n\\to\\infty}\\mathbb{E}\\Big[\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho^n_t\\Big] = \\mathbb{E}\\Big[\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho_t\\Big].\n\\end{equation}\n\\end{Proposition}\n\\begin{proof}\nLet $\\Omega_0$ be the set of $\\omega \\in \\Omega$ for which $\\rho^n_t(\\omega) \\to \\rho_t(\\omega)$ for all $t \\in C_\\rho(\\omega) \\cup \\{T\\}$. \nFix $\\omega \\in \\Omega_0$. For any $t$ such that $t \\in C_\\rho(\\omega)$ and $t < \\theta(\\omega)$ (such $t$ always exists as $\\theta(\\omega) > 0$ and $\\rho$ has at most countably many jumps on any bounded interval) we have $\\rho^n_t(\\omega) \\le \\rho^n_{\\theta(\\omega)-}(\\omega)$ so that by assumption \n\\[\n\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n \\to \\infty} \\rho^n_{\\theta(\\omega)-} (\\omega) \\ge \\rho_{t} (\\omega).\n\\]\nSince $C_{\\rho}(\\omega)$ is dense in $(0, T)$, by arbitrariness of $t<\\theta(\\omega)$ we have\n\\begin{equation}\\label{eqn:hash}\n\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n \\to \\infty} \\rho^n_{\\theta(\\omega)-} (\\omega) \\ge \\rho_{\\theta(\\omega)-} (\\omega).\n\\end{equation}\nWe rewrite the integral as follows: $\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho^n_t = X (\\rho^n_T - \\rho^n_{\\theta-})$. Therefore,\n\\begin{align*}\n\\mathop{\\lim\\sup}_{n\\to\\infty}\\mathbb{E}\\Big[\\int_{[0, T]} \\ind{\\{t \\ge \\theta\\}} X d\\rho^n_t\\Big]\n=\n\\mathop{\\lim\\sup}_{n\\to\\infty}\\mathbb{E}\\big[X (\\rho^n_T - \\rho^n_{\\theta-}) \\big]\n=\n\\mathop{\\lim\\sup}_{n \\to \\infty} \\mathbb{E} [X \\rho^n_T] - \\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n \\to \\infty} \\mathbb{E}[X \\rho^n_{\\theta-}].\n\\end{align*}\nThe dominated convergence theorem yields that $\\lim_{n \\to \\infty} \\mathbb{E} [X \\rho^n_T] = \\mathbb{E} [X \\rho_T]$, while applying Fubini's theorem gives\n\\[\n\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n \\to \\infty} \\mathbb{E}[X \\rho^n_{\\theta-}] \\ge  \\mathbb{E}[\\mathop{\\lim\\operatornamewithlimits{inf\\vphantom{p}}}_{n \\to \\infty} X \\rho^n_{\\theta-}] \\ge \\mathbb{E}[ X \\rho_{\\theta-}],\n\\]\nwhere the last inequality is by \\eqref{eqn:hash}. Combining the above estimates completes the proof of \\eqref{eqn:theta_t}.\n\nAssume now that $\\theta(\\omega) \\in C_\\rho(\\omega)$ or $X(\\omega) = 0$ for $\\mathbb{P}$\\ae $\\omega \\in \\Omega_0$. This and the dominated convergence theorem yield\n\\[\n\\mathbb{E}[X(\\rho_T - \\rho_{\\theta-})] = \\mathbb{E}[X(\\rho_T - \\rho_{\\theta})] = \\lim_{n \\to \\infty} \\mathbb{E}[X(\\rho^n_T - \\rho^n_{\\theta})] \\le \\mathop{\\lim\\sup}_{n \\to \\infty} \\mathbb{E}[X(\\rho^n_T - \\rho^n_{\\theta-})],\n\\]\nwhere the last inequality follows from the monotonicity of $\\rho^n$. This estimate and \\eqref{eqn:theta_t} prove \\eqref{eqn:theta_t_eq}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:contrad0}\nThe inequality \\eqref{eqn:theta_t} in Proposition \\ref{prop:A} can be strict even if $\\rho^n_t \\to \\rho_t$ for all $t \\in [0, T]$ because this condition does not imply that $\\rho^n_{t-} \\to \\rho_{t-}$. One needs further continuity assumptions on $(\\rho_t)$ to establish equality \\eqref{eqn:theta_t_eq}.\n\\end{remark}\n\n\\begin{proof}[{\\bf Proof of Theorem \\ref{th-value-cont-strat_gen}}]\nCompared to the proof of the analogue result under the more stringent condition \\ref{ass:regular} (i.e., Theorem \\ref{th-value-cont-strat}), we only need to establish lower and upper semicontinuity of the functional $N$, while all other remaining arguments stay valid. For the semicontinuity, we extend arguments of Lemma \\ref{lem:semi-cont}. \n\n\\emph{Upper semicontinuity of $N(\\xi,\\cdot)$}. Fix $\\xi\\in\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ and consider a sequence $(\\zeta^{n})_{n \\ge 1}\\in\\mathcal{A}(\\mathcal{F}^2_t)$ converging to $\\zeta \\in \\mathcal{A}(\\mathcal{F}^2_t)$ strongly in $\\mathcal{S}$. Arguing by contradiction, we assume that there is a subsequence of $(\\zeta^n)_{n \\ge 1}$ denoted the same with an abuse of notation, that converges $(\\mathbb{P} \\times \\lambda)\\ae$ to $\\zeta$ and such that\n\\begin{equation*}\n\\lim_{n\\to\\infty} N(\\xi,\\zeta^{n}) > N(\\xi,\\zeta).\n\\end{equation*}\nWithout loss of generality, we can further require that $(\\zeta^n)_{n \\ge 1} \\subset {\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$ and $\\zeta \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$. \nSince $\\xi$ is absolutely continuous on $[0, T)$, \n\\begin{equation}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta^{n}_{t})d\\xi_t\\bigg]=\\mathbb{E}\\bigg[\\int_{[0, T)} f_t(1-\\zeta_{t})d\\xi_t \\bigg]\n\\label{eq-int-conv-1a}\n\\end{equation}\nby the dominated convergence theorem. For the last two terms of $N(\\xi, \\zeta^n)$ (recall \\eqref{eqn:N_cont}) we have\n\\begin{align*}\n\\mathbb{E}\\bigg[\\int_{[0, T)} g_t(1-\\xi_t)d\\zeta^n_t + h_T\\Delta\\xi_T\\Delta\\zeta^n_T\\bigg]\n&=\n\\mathbb{E}\\bigg[\\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta^n_t + (h_T - g_T) \\Delta\\xi_T\\Delta\\zeta^n_T\\bigg].\n\\end{align*}\nAs in the proof of Lemma \\ref{lem:semi-cont}, for the regular part $\\tilde g$ of the process $g$ we have\n\\begin{equation}\\label{eqn:tl_g_conv}\n\\lim_{n \\to \\infty} \\mathbb{E}\\bigg[\\int_{[0, T]} \\tilde g_t(1-\\xi_{t-})d\\zeta^n_t \\bigg] = \\mathbb{E}\\bigg[\\int_{[0, T]} \\tilde g_t(1-\\xi_{t-})d\\zeta_t \\bigg].\n\\end{equation}\n\nFor the pure jump part $\\hat g$ of the process $g$, we will prove that \n\\begin{equation}\\label{eqn:hat_g_conv}\n\\mathop{\\lim\\sup}_{n \\to \\infty} \\mathbb{E}\\bigg[\\int_{[0, T]} \\hat g_t(1-\\xi_{t-})d\\zeta^n_t \\bigg] \\le \\mathbb{E}\\bigg[\\int_{[0, T]} \\hat g_t(1-\\xi_{t-})d\\zeta_t \\bigg].\n\\end{equation}\nTo this end, let us define\n\\[\nR^n_t = \\int_{[0, t]} (1-\\xi_{s-})d\\zeta^n_s, \\qquad R_t = \\int_{[0, t]} (1-\\xi_{s-})d\\zeta_s, \\qquad \\text{for $t\\in[0,T]$,}\n\\]\nwith $R^n_{0-} = R_{0-} = 0$ and then we are going to apply Proposition \\ref{prop:A} with $R^n$ and $R$ instead of $\\rho^n$ and $\\rho$. We need $R^n_t (\\omega) \\to R_t(\\omega)$ as $n\\to\\infty$ for $t \\in C_R(\\omega)=C_{\\zeta}(\\omega)\\cup \\{t\\in[0,T]:\\xi_t(\\omega)=1\\}$, for $\\mathbb{P}$\\ae $\\omega \\in \\Omega$. \nThe latter is indeed true. Setting $\\Omega_0 = \\{\\omega \\in \\Omega:\\ \\lim_{n \\to \\infty} \\zeta^n_t(\\omega) = \\zeta_t(\\omega)\\ \\forall\\, t \\in C_\\zeta(\\omega) \\}$, we have $\\mathbb{P}(\\Omega_0) = 1$ by Lemma \\ref{lem:cadlag_convergence}. For any $\\omega \\in \\Omega_0$ and $t \\in C_\\zeta(\\omega)$, invoking the absolute continuity of $(\\xi_t)$, we obtain (omitting the dependence on $\\omega$)\n\\begin{equation*}\n\\lim_{n \\to \\infty} R^n_t \n=\n\\lim_{n \\to \\infty} \\Big[ (1-\\xi_t) \\zeta^n_t + \\int_{[0, t]} \\zeta^n_s d \\xi_s \\Big]\n=\n(1-\\xi_t) \\zeta_t + \\int_{[0, t]} \\zeta_s d \\xi_s = R_t,\n\\end{equation*}\nwhere the convergence of the second term is the consequence of the dominated convergence theorem and the fact that $\\lambda ([0,T]\\setminus C_\\zeta(\\omega)) = 0$ and $\\zeta^n_T = \\zeta_T = 1$. \n\n\nFor any $k \\ge 1$, since $X^g_k \\ge 0$, Proposition \\ref{prop:A} gives (recall \\eqref{eqn:decomposition_piecewise})\n\\begin{equation}\\label{eqn:limsup01}\n\\mathop{\\lim\\sup}_{n \\to \\infty} \\mathbb{E} \\bigg[ \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg] \\le \\mathbb{E} \\bigg[\\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R_t \\bigg].\n\\end{equation}\nWe apply the decomposition of $\\hat g$ and then the monotone convergence theorem\n\\[\n\\mathbb{E}\\bigg[\\int_{[0, T]} \\hat g_t(1-\\xi_{t-})d\\zeta^n_t \\bigg] = \\mathbb{E} \\bigg[ \\sum_{k=1}^\\infty \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg]\n= \\sum_{k=1}^\\infty \\mathbb{E} \\bigg[ \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg].\n\\]\nSince $\\hat g \\in \\mcalL$ we have the bound (recall \\eqref{eq:Xfg})\n\\[\n\\sum_{k=1}^\\infty \\sup_n \\mathbb{E} \\bigg[ \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg] \\le \\sum_{k=1}^\\infty \\mathbb{E} [ X^g_k ] \n< \\infty .\n\\]\nThen we can apply (reverse) Fatou's lemma (with respect to the counting measure on $\\mathbb{N}$)\n\\begin{align*}\n\\mathop{\\lim\\sup}_{n \\to \\infty} \\sum_{k=1}^\\infty \\mathbb{E} \\bigg[ \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg]\n&\\le\n\\sum_{k=1}^\\infty \\mathop{\\lim\\sup}_{n \\to \\infty} \\mathbb{E} \\bigg[ \\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R^n_t \\bigg]\\\\\n&\\le\n\\sum_{k=1}^\\infty \\mathbb{E} \\bigg[\\int_{[0, T]} X^g_k \\ind{\\{t \\ge \\eta^g_k\\}} d R_t \\bigg]\n=\n\\mathbb{E}\\bigg[\\int_{[0, T]} \\hat g_t(1-\\xi_{t-})d\\zeta_t \\bigg],\n\\end{align*}\nwhere the last inequality is due to \\eqref{eqn:limsup01} and the final equality follows by monotone convergence and the decomposition of $\\hat g$. This completes the proof of \\eqref{eqn:hat_g_conv}.\n\nRecalling that $g_T \\le h_T$, we obtain from Lemma \\ref{prop-terminal-time-jump-limit}\n\\begin{equation*}\n\\mathop{\\lim\\sup}_{n\\to\\infty} \\mathbb{E}\\big[(h_T-g_T)\\Delta\\xi_T\\Delta\\zeta^n_T\\big] \\le \\mathbb{E}\\big[(h_T-g_T)\\Delta\\xi_T\\Delta\\zeta_T\\big],\n\\end{equation*}\nand combining the latter with \\eqref{eqn:tl_g_conv}, \\eqref{eqn:hat_g_conv} and \\eqref{eq-int-conv-1a} shows that\n\\begin{align}\\label{eq:usc_gen}\n\\mathop{\\lim\\sup}_{n \\to \\infty} N(\\xi, \\zeta^n) \\le N(\\xi, \\zeta).\n\\end{align}\nHence we have a contradiction with $\\lim_{n\\to\\infty} N(\\xi,\\zeta^n) >N(\\xi,\\zeta)$, which proves the upper semicontinuity.\n\n\\emph{Lower semicontinuity of $N(\\cdot,\\zeta)$}. The proof follows closely the argument of the proof of Lemma \\ref{lem:semi-cont}: we fix $\\zeta\\in\\mathcal{A}(\\mathcal{F}^2_t)$, consider a sequence $(\\xi^{n})_{n\\ge 1}\\subset\\mathcal{A}_{ac}(\\mathcal{F}^1_t)$ converging to $\\xi \\in \\mathcal{A}_{ac} (\\mathcal{F}^1_t)$ strongly in $\\mathcal{S}$, assume that \\eqref{eqn:two_terms2} holds and reach a contradiction. We only show how to handle the convergence for $(\\hat f_t)$ as all other terms are handled by the proof of Lemma \\ref{lem:semi-cont}.\n\nBy Lemma \\ref{lem:cadlag_convergence} and the continuity of $(\\xi_t)$ we have $\\mathbb{P} \\big( \\lim_{n\\to\\infty} \\xi^{n}_t(\\omega) = \\xi_t(\\omega)\\ \\forall\\,t \\in [0,T)\\big) = 1$. \nLet\n\\[\nR^n_t = \\int_{[0,t]} (1-\\zeta_{t-})d\\xi^n_t, \\qquad R_t = \\int_{[0,t]} (1-\\zeta_{t-})d\\xi_t,\n\\]\nwith $R^n_{0-} = R_{0-} = 0$. Due to the continuity of $(\\xi^n_t)$ and $(\\xi_t)$ for $t \\in [0, T)$, processes $(R^n_t)$ and $(R_t)$ are continuous on $[0, T)$ with a possible jump at $T$. From \\eqref{eqn:conv_R} in the proof of Proposition \\ref{prop-specific-convergence-2} we conclude that for $\\mathbb{P}$\\ae $\\omega \\in \\Omega$\n\\[\n\\lim_{n \\to \\infty} R^n_t(\\omega) = R_t(\\omega) \\quad \\text{for all $t \\in [0, T]$}.\n\\]\nSince $\\Delta \\hat f_T = 0$ (see Assumption \\ref{ass:regular_gen}), there is a decomposition such that $X^f_k \\ind{\\{\\eta^f_k = T\\}} = 0$ $\\mathbb{P}$\\as for all $k$. Recalling that $(R_t)$ is continuous on $[0, T)$, we can apply \\eqref{eqn:theta_t_eq} in Proposition \\ref{prop:A}: for any $k \\ge 1$\n\\[\n\\lim_{n \\to \\infty} \\mathbb{E} \\bigg[ \\int_{[0, T]} X^f_k \\ind{\\{t \\ge \\eta^f_k\\}} d R^n_t \\bigg] = \\mathbb{E} \\bigg[\\int_{[0, T]} X^f_k \\ind{\\{t \\ge \\eta^f_k\\}} d R_t \\bigg].\n\\]\nCombining the latter with decomposition \\eqref{eqn:decomposition_piecewise} and the dominated convergence theorem (with the bound $X^f$) we obtain\n\\[\n\\lim_{n \\to \\infty} \\mathbb{E} \\bigg[ \\int_{[0, T]} \\hat f_t d R^n_t \\bigg] = \\mathbb{E} \\bigg[\\int_{[0, T]} \\hat f_t d R_t \\bigg].\n\\]\nArguing as in the proof of Corollary \\ref{cor-specific-convergence-2}, we have\n\\begin{equation*\n\\lim_{n \\to \\infty} \\mathbb{E} \\bigg[ \\int_{[0, T)} \\hat f_t (1-\\zeta_t) d \\xi^n_t + \\hat f_T \\Delta \\zeta_T \\Delta \\xi^n_T\\bigg] = \\mathbb{E} \\bigg[\\int_{[0, T)} \\hat f_t (1-\\zeta_t) d \\xi_t + \\hat f_T \\Delta \\zeta_T \\Delta \\xi_T \\bigg].\n\\end{equation*}\nCorollary \\ref{cor-specific-convergence-2} implies an analogous convergence for $(\\tilde f_t)$ and the rest of the proof of lower semicontinuity from Lemma \\ref{lem:semi-cont} applies.\n\\end{proof}\n\n\\begin{remark}\\label{rem:contrad}\nIn the arguments above, item (4) in Assumption \\ref{ass:regular_gen} implies in particular that the payoff process $(g_t)$ does not have predictable jumps that are $\\mathbb{P}$\\as negative. This assumption cannot be further relaxed as this may cause the proof of the upper semicontinuity in Theorem \\ref{th-value-cont-strat_gen} to fail. Recall that the process $(g_t)$ corresponds to the payoff of the second player and her strategy $(\\zeta_t)$ is not required to be absolutely continuous. For example, fix $t_0\\in(0,T)$ and take $g_t=1-\\ind{\\{t\\ge t_0\\}}$, $\\zeta_t = \\ind{\\{t\\ge t_0\\}}$ and $\\xi_t = \\ind{\\{t = T\\}}$. Let us consider the sequence $\\zeta^n_t =\\ind{\\{t\\ge t_0-\\frac{1}{n}\\}}$, which converges to $\\zeta$ pointwise and also strongly in $\\mathcal{S}$. We have\n\\begin{equation*}\n\\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta^n_t\\equiv 1, \\:\\:\\: \\text{for all $n$'s, but}\\:\\:\\: \\int_{[0, T]} g_t(1-\\xi_{t-})d\\zeta_t\\equiv 0,\n\\end{equation*}\nhence \\eqref{eqn:hat_g_conv} fails and so does \\eqref{eq:usc_gen}.\n\\end{remark}\n\n\n\n\\begin{proof}[{\\bf Proof of Proposition \\ref{thm:conv_lipsch_gen}}]\nHere, we also only show how to extend the proof of Proposition \\ref{thm:conv_lipsch} to the more general setting. Fix $\\zeta \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$ and $\\xi \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^1_t)$. Construct a sequence $(\\xi^n) \\subset {\\mathcal{A}^\\circ_{ac}}(\\mathcal{F}^1_t)$ as in the proof of Proposition \\ref{thm:conv_lipsch}. It is sufficient to show that\n\\begin{equation}\\label{eqn:lipch_conv_liminf}\n\\mathop{\\lim\\sup}_{n \\to \\infty} N(\\xi^n, \\zeta) \\le N(\\xi, \\zeta).\n\\end{equation}\n\nFrom the proof of Proposition \\ref{thm:conv_lipsch} we have that\n\\begin{equation}\\label{eqn:M_ij_conv_gen}\n\\begin{aligned}\n&\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)}\\tilde f_{t}(1-\\zeta_{t})d\\xi^{n}_{t}+ \\int_{[0, T)} \\tilde g_t(1-\\xi^{n}_{t})d\\zeta_t + h_T\\Delta\\xi^{n}_T\\Delta\\zeta_T\\bigg]\\\\\n&= \\mathbb{E}\\bigg[\\int_{[0, T)}\\tilde f_{t}(1-\\zeta_{t})d\\xi_{t} + \\int_{[0, T)} \\tilde g_t(1-\\xi_{t-})d\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg].\n\\end{aligned}\n\\end{equation}\nFor $t \\in [0, T]$, define\n\\[\nR^n_t = \\int_{[0, t]} (1-\\zeta_{s-})d\\xi^n_s, \\qquad R_t = \\int_{[0, t]} (1-\\zeta_{s})d\\xi_s\n\\]\nwith $R^n_{0-} = R_{0-} = 0$. Corollary \\ref{cor:lim_R_in-t-} implies that for $\\mathbb{P}$\\ae $\\omega \\in \\Omega$\n\\begin{align}\\label{eq:convR}\n\\lim_{n \\to \\infty} R^n_{t-}(\\omega) = R_{t-}(\\omega) \\quad \\text{for all $t \\in [0, T]$}.\n\\end{align}\nBy the decomposition of $(\\hat f_t)$ in \\eqref{eqn:decomposition_piecewise} and the dominated convergence theorem for the infinite sum (recalling \\eqref{eq:Xfg}) we obtain\n\\begin{align*}\n\\mathbb{E}\\bigg[\\int_{[0, T)}\\hat f_{t}(1-\\zeta_{t})d\\xi^{n}_{t}\\bigg] \n&= \n\\mathbb{E}\\bigg[\\int_{[0, T)}\\hat f_{t}(1-\\zeta_{t-})d\\xi^{n}_{t}\\bigg]\n=\n\\sum_{k=1}^\\infty \\mathbb{E}\\bigg[(-1)^k \\int_{[0, T)} X^f_k \\ind{\\{t \\ge \\eta^f_k\\}} d R^n_t \\bigg]\\\\\n&=\n\\sum_{k=1}^\\infty \\mathbb{E} \\big[ (-1)^k X^f_k (R^n_{T-} - R^n_{\\eta^f_k-})\\big],\n\\end{align*}\nwhere the first equality follows from the continuity of $(\\xi^n_t)$ on $[0, T)$.\nWe further apply dominated convergence (with respect to the product of the counting measure on $\\mathbb N$ and to the measure $\\mathbb{P}$) to obtain\n\\begin{equation}\\label{eqn:part2}\n\\begin{aligned}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\!\\int_{[0, T)}\\!\\hat f_{t}(1-\\zeta_{t})d\\xi^{n}_{t}\\bigg] \n&=\\sum_{k=1}^\\infty  \\mathbb{E} \\big[(-1)^k \\lim_{n\\to\\infty}  X^f_k (R^n_{T-} - R^n_{\\eta^f_k-})\\big]\\\\\n&=\n\\sum_{k=1}^\\infty \\mathbb{E} \\big[ (-1)^k X^f_k (R_{T-} - R_{\\eta^f_k-})\\big]=\n\\mathbb{E}\\bigg[\\int_{[0, T)}\\hat f_{t}(1-\\zeta_{t})d\\xi_{t}\\bigg],\n\\end{aligned}\n\\end{equation}\nwhere the second equality uses \\eqref{eq:convR} and the final one the decomposition of $\\hat f$. \nRecalling that $\\xi^n_t\\to\\xi_{t-}$ as $n\\to\\infty$ by construction, dominated convergence gives\n\\begin{equation}\\label{eqn:part3}\n\\lim_{n\\to\\infty}\\mathbb{E}\\bigg[\\int_{[0, T)}\\hat g_{t}(1-\\xi^n_{t})d\\zeta_{t}\\bigg]\n=\n\\mathbb{E}\\bigg[\\int_{[0, T)}\\hat g_{t}(1-\\xi_{t-})d\\zeta_{t}\\bigg].\n\\end{equation}\nPutting together \\eqref{eqn:M_ij_conv_gen}, \\eqref{eqn:part2} and \\eqref{eqn:part3} shows\n\\[\n\\lim_{n \\to \\infty} N(\\xi^n, \\zeta) = \\mathbb{E}\\bigg[\\int_{[0, T)} f_{t}(1-\\zeta_{t})d\\xi_{t} + \\int_{[0, T)} g_t(1-\\xi_{t-})d\\zeta_t + h_T\\Delta\\xi_T\\Delta\\zeta_T\\bigg].\n\\]\nIt remains to notice that by \\eqref{eq-remove-common-jumps} the right hand side is dominated by $N(\\xi, \\zeta)$, which completes the proof of \\eqref{eqn:lipch_conv_liminf}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:ef_0_value}} \\label{sec:ef_functional}\nRandomisation devices $Z_\\tau$ and $Z_\\sigma$ associated to a pair $(\\tau,\\sigma)\\in\\mathcal{T}^R(\\mathcal{F}^1_t)\\times\\mathcal{T}^R(\\mathcal{F}^2_t)$ are independent of $\\mathcal{G}$. Denoting by $(\\xi_t) \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^1_t)$ and $(\\zeta_t) \\in {\\mathcal{A}^\\circ}(\\mathcal{F}^2_t)$ the generating processes for $\\tau$ and $\\sigma$, respectively, the statement of Proposition \\ref{prop-functionals-equal} can be extended to encompass the conditional functional \\eqref{eqn:cond_func}:\n\\begin{equation}\\label{eqn:cond_reform}\n\\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big] = \\mathbb{E}\\bigg[\\int_{[0, T)} f_{t}(1-\\zeta_{t})d \\xi_t +  \\int_{[0, T)} g_{t}(1-\\xi_t) d \\zeta_t + \\sum_{t \\in [0, T]} h_t \\Delta \\xi_t \\Delta \\zeta_t \\bigg|\\mathcal{G}\\bigg].\n\\end{equation}\nWe can also repeat the same argument as in Remark \\ref{rem-Laraki-Solan} to obtain that\n\\[\n\\underline V:=\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)}\\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)}  \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big]=\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)}\\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau \\in \\mathcal{T}(\\mathcal{F}^1_t)}  \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big]\n\\]\nand \n\\[\n\\overline V:=\\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} \\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)} \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big]=\\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)}\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)}  \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big].\n\\]\nNotice that $\\overline V\\ge \\underline V$, $\\mathbb{P}$\\as We will show that\n\\begin{align}\\label{eq:EV}\n\\mathbb{E}[\\,\\underline V\\,]=\\mathbb{E}[\\,\\overline V\\,],\n\\end{align}\nso that $\\overline V= \\underline V$\\,, $\\mathbb{P}$\\as as needed.\n\nIn order to prove \\eqref{eq:EV}, let us define\n\\begin{equation*}\n\\overline{M}(\\tau):=\\operatornamewithlimits{\\mathrm{ess\\,sup}}_{\\sigma \\in \\mathcal{T}(\\mathcal{F}^2_t)}  \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big],\\quad\\text{for $\\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)$},\n\\end{equation*}\nand \n\\begin{align*}\n\\underline{M}(\\sigma):=\\operatornamewithlimits{\\mathrm{ess\\,inf\\vphantom{p}}}_{\\tau \\in \\mathcal{T}(\\mathcal{F}^1_t)}  \\mathbb{E}\\big[ \\mathcal{P}(\\tau, \\sigma) \\big| \\mathcal{G} \\big],\\quad\\text{for $\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)$}.\n\\end{align*}\nThese are two standard optimal stopping problems and the theory of Snell envelope applies (see, e.g., \\cite[Appendix D]{Karatzas1998} and \\cite{elkaroui1981}). We adapt some results from that theory to suit our needs in the game setting.\n\\begin{Lemma}\\label{lem:directed}\nThe family $\\{\\overline{M}(\\tau),\\,\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)\\}$ is downward directed and the family $\\{\\underline{M}(\\sigma),\\,\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)\\}$ is upward directed. \n\\end{Lemma}\n\\begin{proof}\nLet $\\tau^{(1)},\\tau^{(2)}\\in\\mathcal{T}^R(\\mathcal{F}^1_t)$ and let $\\xi^{(1)},\\xi^{(2)}\\in{\\mathcal{A}^\\circ}(\\mathcal{F}^1_t)$ be the corresponding generating processes. Fix the $\\mathcal{G}$-measurable event $B=\\{\\overline{M}(\\tau^{(1)})\\le\\overline{M}(\\tau^{(2)})\\}$ and define another $(\\mathcal{F}^1_t)$-randomised stopping time as $\\hat{\\tau}=\\tau^{(1)} \\ind{B} +\\tau^{(2)} \\ind{B^c}$. We use $\\mathcal{G}\\subset\\mathcal{F}^1_0$ to ensure that $\\hat \\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)$. The generating process of $\\hat \\tau$ reads $\\hat \\xi_t=\\xi^{(1)}_t \\ind{B} +\\xi^{(2)}_t \\ind{B^c}$ for $t\\in[0,T]$. Using the linear structure of $\\hat \\xi$ and recalling \\eqref{eqn:cond_reform}, for any $\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)$, we have\n\\begin{align*}\n\\mathbb{E}\\big[\\mathcal{P}(\\hat{\\tau}, \\sigma)|\\mathcal{G}\\big]\n&=\n\\ind{B}\\mathbb{E}\\bigg[\\int_{[0, \\sigma)} f_{u}d \\xi^{(1)}_u +  g_{\\sigma}(1-\\xi^{(1)}_\\sigma) + h_\\sigma \\Delta \\xi^{(1)}_\\sigma \\bigg|\\mathcal{G}\\bigg]\\\\\n&\\hspace{12pt}+\\ind{B^c}\\mathbb{E}\\bigg[\\int_{[0, \\sigma)} f_{u}d \\xi^{(2)}_u +  g_{\\sigma}(1-\\xi^{(2)}_\\sigma) + h_\\sigma \\Delta \\xi^{(2)}_\\sigma\\bigg|\\mathcal{G}\\bigg]\\\\\n&=\\ind{B}\\mathbb{E}\\big[\\mathcal{P}(\\tau^{(1)} ,\\sigma)|\\mathcal{G}\\big]+\\ind{B^c}\\mathbb{E}\\big[\\mathcal{P}(\\tau^{(2)} ,\\sigma)|\\mathcal{G}\\big]\\\\\n&\\le\\ind{B}\\overline{M}(\\tau^{(1)})+\\ind{B^c}\\overline{M}(\\tau^{(2)})=\\overline{M}(\\tau^{(1)})\\wedge\\overline{M}(\\tau^{(2)}),\n\\end{align*} \nwhere the inequality is by definition of essential supremum and the final equality by definition of the event $B$.\nThus, taking the supremum over $\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)$ we get\n\\[\n\\overline{M}(\\hat \\tau)\\le \\overline{M}(\\tau^{(1)})\\wedge\\overline{M}(\\tau^{(2)}),\n\\]\nhence the family $\\{\\overline{M}(\\tau),\\,\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)\\}$ is downward directed. A symmetric argument proves that \nthe family $\\{\\underline{M}(\\sigma),\\,\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)\\}$ is upward directed.\n\\end{proof}\nAn immediate consequence of the lemma  and of the definition of essential supremum\/infimum is that (see, e.g., \\cite[Lemma I.1.3]{Peskir2006}) we can find sequences $(\\sigma_n)_{n\\ge 1}\\subset \\mathcal{T}^R(\\mathcal{F}^2_t)$ and $(\\tau_n)_{n\\ge 1}\\subset\\mathcal{T}^R(\\mathcal{F}^1_t)$ such that $\\mathbb{P}$\\as\n\\begin{align}\\label{eq:limV}\n\\overline V=\\lim_{n\\to\\infty}\\overline{M}(\\tau_n)\\quad\\text{and}\\quad\\underline V=\\lim_{n\\to\\infty}\\underline{M}(\\sigma_n),\n\\end{align}\nwhere the convergence is monotone in both cases. \n\nAnalogous results hold for the optimisation problems defining $\\overline M(\\tau)$ and $\\underline M(\\sigma)$. The proof of the following lemma is similar to that of Lemma \\ref{lem:directed} and omitted.\n\\begin{Lemma}\\label{lem:directed2}\nThe family $\\{\\mathbb{E}\\big[\\mathcal{P}(\\tau,\\sigma)|\\mathcal{G}\\big],\\,\\sigma\\in \\mathcal{T}(\\mathcal{F}^2_t)\\}$ is upward directed for each $\\tau\\in \\mathcal{T}^R(\\mathcal{F}^1_t)$. The family $\\{\\mathbb{E}\\big[\\mathcal{P}(\\tau,\\sigma)|\\mathcal{G}\\big],\\,\\tau\\in \\mathcal{T}(\\mathcal{F}^1_t)\\}$ is downward directed for each $\\sigma\\in \\mathcal{T}^R(\\mathcal{F}^2_t)$. \n\\end{Lemma}\nIt follows that for each $\\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)$ and $\\sigma\\in\\mathcal{T}^R(\\mathcal{F}^2_t)$, there are sequences $(\\sigma^\\tau_n)_{n\\ge 1}\\subset\\mathcal{T}(\\mathcal{F}^2_t)$ and $(\\tau^\\sigma_n)_{n\\ge 1}\\subset\\mathcal{T}(\\mathcal{F}^1_t)$ such that \n\\begin{align}\\label{eq:limM}\n\\overline M(\\tau)=\\lim_{n\\to\\infty}\\mathbb{E}\\big[\\mathcal{P}(\\tau,\\sigma^\\tau_n)|\\mathcal{G}\\big]\\quad\\text{and}\\quad\\underline M(\\sigma)=\\lim_{n\\to\\infty}\\mathbb{E}\\big[\\mathcal{P}(\\tau^\\sigma_n,\\sigma)|\\mathcal{G}\\big],\n\\end{align}\nwhere the convergence is monotone in both cases. Equipped with these results we can prove the following lemma which will quickly lead to \\eqref{eq:EV}.\n\n\\begin{Lemma}\\label{cor-exp-for-values}\nRecall $V_*$ and $V^*$ as in Definition \\ref{def-value-rand-strat}. We have\n\\begin{equation}\\label{eq-cor-exp}\n\\mathbb{E}[\\overline{V}] = V^*, \\qquad \\text{and}\\qquad \\mathbb{E}[\\underline{V}] = V_*.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nFix $\\tau\\in\\mathcal{T}^R(\\mathcal{F}^1_t)$. By \\eqref{eq:limM} and the monotone convergence theorem\n\\[\n\\mathbb{E}[ \\overline{M}(\\tau) ] = \\lim_{n\\to\\infty}\\mathbb{E}[\\mathcal{P}(\\tau,\\sigma^\\tau_n)]\\le \\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)}\\mathbb{E}[\\mathcal{P}(\\tau,\\sigma)].\n\\]\nThe opposite inequality follows from the fact that $\\overline{M}(\\tau) \\ge \\mathbb{E}[\\mathcal{P}(\\tau,\\sigma)|\\mathcal{G}]$ for any $\\sigma \\in \\mathcal{T}(\\mathcal{F}^2_t)$ by the definition of the essential supremum. Therefore, we have\n\\begin{equation}\\label{eqn:M1}\n\\mathbb{E}[ \\overline{M}(\\tau) ] = \\sup_{\\sigma\\in\\mathcal{T}(\\mathcal{F}^2_t)}\\mathbb{E}[\\mathcal{P}(\\tau,\\sigma)]. \n\\end{equation}\nFrom \\eqref{eq:limV}, similar arguments as above prove that\n\\begin{equation}\\label{eqn:M2}\n\\mathbb{E}[\\overline{V}] = \\operatornamewithlimits{inf\\vphantom{p}}_{\\tau \\in \\mathcal{T}^R(\\mathcal{F}^1_t)} \\mathbb{E}[ \\overline{M}(\\tau) ].\n\\end{equation}\nCombining \\eqref{eqn:M1} and \\eqref{eqn:M2} completes the proof that $\\mathbb{E}[\\overline{V}] = V^*$. The second part of the statement requires analogous arguments.\n\\end{proof}\n\nFinally, \\eqref{eq-cor-exp} and Theorem \\ref{thm:main2} imply \\eqref{eq:EV}, which concludes the proof of Theorem \\ref{thm:ef_0_value}.\n\n\n\n\n\n\n\\section{Counterexamples}\n\\label{sec:Nikita-examples}\n\nIn the three subsections below we show that: (a) relaxing condition \\ref{eq-order-cond} may lead to a game without a value, (b) in situations where one player has all the informational advantage, the use of randomised stopping times may still be beneficial also for the uninformed player, and (c) Assumption \\ref{ass:regular_gen} is tight in requiring that either $(\\hat f_t)$ is non-increasing or $(\\hat g_t)$ is non-decreasing. \n\nIn order to keep the exposition simple we consider the framework of Section \\ref{subsec:game_1} with $\\I = 2$, $\\J = 1$, and impose that $(\\mathcal{F}^p_t)$ be the trivial filtration (hence all payoff processes are deterministic, since they are $(\\mathcal{F}^p_t)$-adapted). Furthermore we restrict our attention to the case in which $f^{1,1} = f^{2,1} = f$, $g^{1,1} = g^{2,1} = g$ and $h^{1,1}_t\\ind{\\{t<T\\}}=h^{2,1}_t\\ind{\\{t<T\\}}=f_t\\ind{\\{t<T\\}}$. Only the terminal payoff depends on the scenario, i.e., $h^{1,1}_T\\neq h^{2,1}_T$ (both deterministic). For notational simplicity we set $h^{1}:=h^{1,1}_T$ and $h^{2}:=h^{2,1}_T$.\n\nNotice that only the first player (minimiser) observes the true value of $\\mcalI$, so she has a strict informational advantage over the second player (maximiser).\nThe second player will be referred to as the \\emph{uninformed player} while the first player as the \\emph{informed player}.\n\nWe denote by $\\mathcal{T}^R$ the set of $(\\mathcal{F}^p_t)$-randomised stopping times. The informed player chooses two randomised stopping times $\\tau_1, \\tau_2$ (one for each scenario, recall Lemma \\ref{lem:tau_decomposition}) with the generating processes $\\xi^1,\\xi^2$ which, due to the triviality of the filtration $(\\mathcal{F}_t^p)$, are deterministic functions. Pure stopping times are constants in $[0,T]$. Similarly, the uninformed player's randomised stopping time $\\sigma$ has the generating process $\\zeta$ that is a deterministic function. \n\n\n\\subsection{A game without a value when \\ref{eq-order-cond} fails}\nLet us consider specific payoff functions \n\\begin{equation*}\nf \\equiv 1,\\quad g_t=\\frac{1}{2}t, \\quad h^1= 2,\\quad h^2=0,\n\\end{equation*}\nand let us also set $T=1$, $\\pi_1 = \\pi_2 =\\frac{1}{2}$.\n\n\\begin{Proposition}\nIn the example of this subsection we have\n\\begin{equation*}\nV_{*}\\le\\frac{1}{2} \\qquad \\text{and}\\qquad V^* > \\frac12,\n\\end{equation*}\nso the game does not have a value.\n\\end{Proposition}\n\\begin{proof}\nFirst we show that $V_{*}\\le\\frac{1}{2}$. Recall that (c.f. Remark \\ref{rem-Laraki-Solan})\n\\begin{equation*}\nV_*=\\sup_{\\sigma\\in\\mathcal{T}^R}\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2\\in\\mathcal{T}^R}N((\\tau_1,\\tau_2),\\sigma)=\\sup_{\\sigma\\in\\mathcal{T}^R}\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2\\in[0,1]}N((\\tau_1,\\tau_2),\\sigma),\n\\end{equation*} \nso we can take $\\tau_1, \\tau_2 \\in [0, 1]$ deterministic in the arguments below. Take any $\\sigma \\in \\mathcal{T}^R$ and the corresponding generating process $(\\zeta_t)$ which is, due to the triviality of the filtration $(\\mathcal{F}^p_t)$,\na deterministic function.  For $\\tau_1\\in[0,1)$, $\\tau_2=1$ we obtain\n\\begin{align*}\nN((\\tau_1,\\tau_2),\\sigma)&= \\mathbb{E}\\big[(\\frac{1}{2}\\sigma \\indd{\\sigma<\\tau_1}+1\\cdot \\ind{\\{\\sigma\\ge\\tau_1\\}})\\indd{\\mcalI = 1} + (\\frac{1}{2}\\sigma \\indd{\\sigma<1}+0 \\cdot \\indd{\\sigma=1}) \\indd{\\mcalI = 2}\\big]\\\\\n&\\le \\frac{1}{2}(\\frac{1}{2}\\zeta_{\\tau_1-}+(1-\\zeta_{\\tau_1-}))+ \\frac{1}{4}\\zeta_{1-}\n= \\frac{1}{2}-\\frac{1}{4}\\zeta_{\\tau_1-}+\\frac{1}{4}\\zeta_{1-},\n\\end{align*}\nwhere we used that $\\sigma$ is bounded above by $1$ and that $\\mcalI$ is independent of $\\sigma$ with $\\mathbb{P}(\\mcalI=1)=\\mathbb{P}(\\mcalI=2)=\\tfrac{1}{2}$.\nIn particular,\n\\begin{equation*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2 \\in [0, 1]} N((\\tau_1,\\tau_2),\\sigma) \\le \\lim_{\\tau_1\\to 1-} N((\\tau_1,1),\\sigma) = \\frac{1}{2}.\n\\end{equation*}\nThis proves that $V_* \\le \\frac12$.\n\nNow we turn our attention to demonstrating that $V^{*}>\\frac{1}{2}$. Noting again that\n\\begin{equation*}\nV^*=\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2\\in\\mathcal{T}^R}\\sup_{\\sigma\\in\\mathcal{T}^R}N(\\tau_1,\\tau_2,\\sigma)=\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2\\in\\mathcal{T}^R}\\sup_{\\sigma\\in[0,1]}N(\\tau_1,\\tau_2,\\sigma),\n\\end{equation*} \nwe can restrict our attention to constant $\\sigma \\in [0, 1]$. Take any $\\tau_1, \\tau_2 \\in \\mathcal{T}^R$ and the corresponding generating processes $(\\xi^1_t), (\\xi^2_t)$ which are also deterministic functions. \n\nTake any $\\delta \\in (0, 1\/2)$. If $\\xi^1_{1-} > \\delta$, then for any $\\sigma<1$ we have\n\\begin{align*}\nN((\\tau_1,\\tau_2),\\sigma)\n&\\ge \n\\mathbb{E}\\big[\\big(1 \\cdot \\indd{\\tau_1\\le\\sigma}+ \\frac{1}{2}\\sigma \\indd{\\sigma<\\tau_1}\\big) \\indd{\\mcalI=1}+\\frac{1}{2}\\sigma \\indd{\\mcalI=2}\\big]\\\\\n&=\n\\mathbb{E}\\big[\\big(\\xi^1_\\sigma + \\frac{1}{2}\\sigma(1-\\xi^1_\\sigma)\\big) \\ind{\\{\\mcalI=1\\}}+\\frac{1}{2}\\sigma \\ind{\\{\\mcalI=2\\}}\\big]\\\\\n&=\n\\frac{1}{2}\\xi^1_\\sigma - \\frac{1}{4}\\sigma\\xi^1_\\sigma+\\frac{1}{2}\\sigma\n=\n\\frac{1}{2}\\xi^1_\\sigma(1 - \\frac{1}{2}\\sigma)+\\frac{1}{2}\\sigma,\n\\end{align*}\nand, in particular,\n\\begin{equation*}\n\\sup_{\\sigma \\in [0, 1]} N((\\tau_1,\\tau_2),\\sigma) \\ge \\lim_{\\sigma\\to 1-} N((\\tau_1,\\tau_2),\\sigma) \\ge \\frac{1}{4}\\xi^1_{1-} + \\frac{1}{2}\\ge \\frac12 + \\frac14\\delta>\\frac12.\n\\end{equation*}\nOn the other hand, if $\\xi^1_{1-} \\le \\delta$, taking $\\sigma=1$ yields\n\\begin{equation*}\n\\sup_{\\sigma \\in [0, 1]} N((\\tau_1,\\tau_2),\\sigma) \\ge N((\\tau_1,\\tau_2),1) \\ge \\mathbb{E}[2\\cdot \\ind{\\{\\tau_1= 1\\}} \\ind{\\{\\mcalI=1\\}}] = 1-\\xi^1_{1-}  \\ge 1 - \\delta > \\frac{1}{2}.\n\\end{equation*}\nThis completes the proof that $V^* > \\frac12$.\n\\end{proof}\n\n\n\\subsection{Necessity of randomization}\\label{sec-example-necessity-of-rand}\n\nHere we argue that randomisation is not only sufficient in order to find the value in Dynkin games with asymmetric information but in many cases it is also necessary. In \\cite{DEG2020} there is a rare example of explicit construction of optimal strategies for a zero-sum Dynkin game with asymmetric information in a diffusive set-up (see Section \\ref{subsec:game_2} above for details). The peculiarity of the solution in \\cite{DEG2020} lies in the fact that the informed player uses a randomised stopping time whereas the uninformed player sticks to a pure stopping time. An interpretation of that result suggests\nthat the informed player uses randomisation to `gradually reveal' information about the scenario in which the game is being played, in order to induce the uninformed player to act in a certain desirable way. Since the uninformed player has `nothing to reveal' one may be tempted to draw a general conclusion that she should never use randomised stopping rules. However, Proposition \\ref{prop-uninf-benefits-from-rand} below shows that such conclusion would be wrong in general and even the \\emph{uninformed} player may benefit from randomisation of stopping times.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{example_graph.png}\n\\end{center}\n\\caption{Payoff functions $f$ in blue, $g$ in orange.}\n\\label{fig:1}\n\\end{figure}\n\nWe consider specific payoff functions $f$ and $g$ plotted on Figure \\ref{fig:1}. Their analytic formulae read\n\\[\nf_t = (10t+4)\\ind{\\{t\\in[0,\\frac{1}{10})\\}} + 5\\ind{\\{t\\in[\\frac{1}{10},1]\\}}, \\qquad \ng_t = (15t - 6) \\ind{\\{t\\in[\\frac{2}{5},\\frac{1}{2})\\}}+(9-15 t)\\ind{\\{t\\in[\\frac{1}{2},\\frac{3}{5})\\}}\n\\]\nwith\n\\begin{equation*}\nh^1 = 0 = g_{1-},\\quad h^2=5=f_{1-}.\n\\end{equation*}\nWe also set $T=1$, $\\pi_1 = \\pi_2 =\\frac{1}{2}$.\nAs above, we identify randomized strategies with their generating processes. In particular, we denote by $\\zeta$ the generating process for $\\sigma \\in \\mathcal{T}^R$. \n\nBy Theorem \\ref{thm:main},  the game has a value in randomised strategies, i.e., $V^* = V_*$. Restriction of the uninformed player's (player 2) strategies to pure stopping times affects only the lower value, see Remark \\ref{rem-Laraki-Solan}. The lower value of the game in which player 2 is restricted to using pure stopping times reads\n\\begin{equation*}\n\\widehat{V}_*:=\\sup_{\\sigma\\in[0,1]}\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1, \\tau_2 \\in \\mathcal{T}^R} N((\\tau_1, \\tau_2),\\sigma)=\\sup_{\\sigma\\in[0,1]}\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1, \\tau_2 \\in [0,1]}  N((\\tau_1, \\tau_2),\\sigma),\n\\end{equation*}\nwhere the equality is again due to Remark \\ref{rem-Laraki-Solan} (notice that here all pure stopping times are $(\\mathcal{F}^p_t)$-stopping times hence deterministic, because $(\\mathcal{F}^p_t)$ is trivial). As the following proposition shows, $\\widehat{V}_*<V_*$, so the game in which the uninformed player does not randomise does not have a value. This confirms that the randomisation can play a strategic role beyond manipulating information.\n\\begin{Proposition}\\label{prop-uninf-benefits-from-rand}\nIn the example of this subsection, we have\n\\begin{equation*}\nV_*>\\widehat{V}_*.\n\\end{equation*}\n\\end{Proposition}\n\\begin{proof}\nFirst, notice that \n\\begin{equation*}\n\\widehat{V}_*\\le\\sup_{\\sigma\\in [0,1]} N(\\hat{\\tau}(\\sigma),\\sigma),\n\\end{equation*}\nwhere we take\n\\[\n\\hat \\tau (\\sigma)= (\\tau_1(\\sigma), \\tau_2(\\sigma)) = \\begin{cases}\n(1,1),& \\text{for } \\sigma\\in[0,1),\\\\\n(1,0),& \\text{for } \\sigma=1.\n            \\end{cases}\n\\]\nIt is easy to verify that $\\sup_{\\sigma\\in[0,1]} N(\\hat{\\tau}(\\sigma),\\sigma)=2$. \n\nWe will show that the $\\sigma$-player can ensure a strictly larger payoff by using a randomised strategy. Define $\\zeta_t=a \\ind{\\{t\\ge\\frac{1}{2}\\}}+(1-a)\\ind{\\{t=1\\}}$, i.e., the corresponding $\\sigma\\in\\mathcal{T}^R$ prescribes to `stop at time $\\frac{1}{2}$ with probability $a$ and at time $1$ with probability $1-a$'. The value of the parameter $a \\in [0,1]$ will be determined below. We claim that\n\\begin{equation}\\label{eqn:zb}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1, \\tau_2 \\in[0,1]} N((\\tau_1, \\tau_2),\\zeta) = N((1,0),{\\zeta})\\wedge N((1,1),{\\zeta}).\n\\end{equation}\nAssuming that the above is true, we calculate\n\\[\nN((1,0),{\\zeta})=2+\\frac{3}{4}a, \\qquad N((1,1),{\\zeta}) = \\frac{5}{2}-a.\n\\]\nPicking $a = \\frac27$ the above quantities are equal to $\\frac{31}{14}$. Hence $V_* \\ge \\frac{31}{14}>2$.\n\nIt remains to prove \\eqref{eqn:zb}. Recall that $\\zeta_t=a \\ind{\\{t\\ge\\frac{1}{2}\\}}+(1-a)\\ind{\\{t=1\\}}$ is the generating process of $\\sigma$ and the expected payoff reads\n\\begin{equation*}\nN((\\tau_1, \\tau_2), \\zeta) = \\sum_{i=1}^2 \\mathbb{E} \\big[ \\ind{\\{\\mcalI=i\\}}\\left(f_{\\tau_i} \\ind{\\{\\tau_i \\le \\sigma\\} \\cap \\{\\tau_i < 1\\}} + g_{\\sigma} \\ind{\\{\\sigma<\\tau_i\\} \\cap \\{\\sigma < 1\\}} + h^i \\ind{\\{\\tau_i = \\sigma = 1\\}}\\right) \\big].\n\\end{equation*}\nIt is clear that on the event $\\{\\mcalI=1\\}$ the infimum is attained for $\\tau_1=1$, irrespective of the choice of $\\zeta$. On the event $\\{\\mcalI=2\\}$ the informed player would only stop either at time zero, where the function $f$ attains the minimum cost $f_0=4$, or at time $t>\\frac12$ since  $\\zeta$ only puts mass at $t=\\frac12$ and at $t=1$ (the informed player knows her opponent may stop at $t=\\frac12$ with probability $a$). The latter strategy corresponds to a payoff $5-\\frac72 a$ and can also be achieved by picking $\\tau_2=1$. Then the informed player needs only to consider the expected payoff associated to the strategies $(\\tau_1,\\tau_2)=(1,0)$ and $(\\tau_1,\\tau_2)=(1,1)$, so that \\eqref{eqn:zb} holds.\n\\end{proof}\n\n\\subsection{Necessity of Assumption \\ref{ass:regular_gen}} \\label{subsec:example_jumps}\n\nOur final counter-example shows that violating Assumption \\ref{ass:regular_gen} by allowing both predictable upward jumps of $f$ \\emph{and} predictable downward jumps of $g$ may also lead to a game without a value.\n\nConsider the payoffs\n\\[\nf_t=1+2\\ind{\\{t\\ge \\frac{1}{2}\\}},\\quad g_t=-\\ind{\\{t\\ge \\frac{1}{2}\\}},\\quad h^1=3,\\quad h^2=-1,\n\\]\nso that $h^1=f_{1-}$ and $h^2=g_{1-}$ and let us also set $T=1$, $\\pi_1=\\pi_2=\\tfrac{1}{2}$. Assumption \\ref{ass:regular_gen} is violated as $g$ has a predictable downward jump and $f$ has a predictable upward jump at time $t=\\frac{1}{2}$.\n\n\\begin{Proposition}\nIn the example of this subsection we have\n\\begin{equation*}\nV_{*}\\le 0,\\quad\\text{and}\\quad V^{*}>0,\n\\end{equation*}\nso the game does not have a value.\n\\end{Proposition}\n\\begin{proof}\nFirst we show that $V_{*}\\le 0$. For this step, it is sufficient to restrict our attention to pure stopping times $\\tau_1,\\tau_2\\in[0,1]$ for the informed player (c.f. Remark \\ref{rem-Laraki-Solan}). Let $\\sigma\\in\\mathcal{T}^R$ with a (deterministic) generating process $(\\zeta_t)$ and fix $\\varepsilon\\in (0, \\frac12)$. For $\\tau_1=\\frac{1}{2}-\\varepsilon$ and $\\tau_2=1$ we obtain\n\\begin{align*}\nN((\\tau_1,\\tau_2),\\sigma)&\n=\\mathbb{E}\\big[\\indd{\\mcalI=1}(0\\cdot\\indd{\\sigma<\\tau_1}+1\\cdot\\indd{\\sigma\\ge\\tau_1}) + \\indd{\\mcalI=2}(0\\cdot\\indd{\\sigma<\\frac{1}{2}}-1\\cdot\\indd{\\sigma\\ge \\frac{1}{2}})\\big]\\\\\n&=\\frac12\\big(1-\\zeta_{(\\frac{1}{2}-\\varepsilon)-}\\big)-\\frac12\\big(1-\\zeta_{\\frac{1}{2}-}\\big).\n\\end{align*}\nTherefore, using that $(\\zeta_t)$ has c\\`adl\\`ag\\@ifnextchar.{}{\\@ifnextchar,{}{\\@ifnextchar;{}{ }}} trajectories we have\n\\begin{equation*}\n\\operatornamewithlimits{inf\\vphantom{p}}_{\\tau_1,\\tau_2\\in[0,1]} N((\\tau_1,\\tau_2),\\sigma) \\le \\lim_{\\varepsilon\\to 0} \\frac12\\cdot(\\zeta_{\\frac{1}{2}-}-\\zeta_{(\\frac{1}{2}-\\varepsilon)-}) =0.\n\\end{equation*}\nSince the result holds for all $\\sigma\\in\\mathcal{T}^R$ we have $V_*\\le 0$.\n\nNext, we demonstrate that $V^{*}>0$. For this step it is sufficient to consider pure stopping times $\\sigma\\in[0,1]$ for the uninformed player (Remark \\ref{rem-Laraki-Solan}). Let $\\tau_1,\\tau_2\\in\\mathcal{T}^R$ and let $\\xi^1,\\xi^2$ be the associated (deterministic) generating processes. Consider first the case in which $\\xi^1_{\\frac{1}{2}-}+\\xi^2_{\\frac{1}{2}-}>\\delta$ for some $\\delta\\in(0,1)$ and fix $\\varepsilon \\in (0, \\frac12)$. For $\\sigma=\\frac{1}{2}-\\varepsilon$ we have\n\\begin{align*}\nN((\\tau_1,\\tau_2),\\sigma)\n&=\n\\mathbb{E}\\big[\\indd{\\mcalI=1}(1\\cdot\\indd{\\tau_1\\le\\sigma}+0\\cdot\\indd{\\sigma<\\tau_1}) + \\indd{\\mcalI=2}(1\\cdot\\indd{\\tau_2\\le\\sigma}+0\\cdot\\indd{\\sigma<\\tau_2})\\big]\\\\\n&=  \\frac12\\big(\\xi^1_{\\frac{1}{2}-\\varepsilon} + \\xi^2_{\\frac{1}{2}-\\varepsilon}\\big),\n\\end{align*}\nthus implying\n\\begin{equation}\\label{eq:last0}\n\\sup_{\\sigma\\in[0,1]} N((\\tau_1,\\tau_2),\\sigma) \\ge \\lim_{\\sigma\\to \\frac{1}{2}-} N((\\tau_1,\\tau_2),\\sigma)=  \\frac12(\\xi^1_{\\frac{1}{2}-}+\\xi^2_{\\frac{1}{2}-})>\\frac{\\delta}{2}>0.\n\\end{equation}\nIf, instead, $\\xi^1_{\\frac{1}{2}-}+\\xi^2_{\\frac{1}{2}-}\\le\\delta$ so that, in particular, $\\xi^1_{\\frac{1}{2}-}\\vee\\xi^2_{\\frac{1}{2}-}\\le\\delta$, then\n\\begin{equation}\\label{eq:last1}\n\\begin{aligned}\n\\sup_{\\sigma\\in[0,1]} N((\\tau_1,\\tau_2),\\sigma)\n&\\ge \nN((\\tau_1,\\tau_2),1)\\\\\n&\\ge \n\\mathbb{E}\\big[\\indd{\\mcalI=1}(1\\cdot\\indd{\\tau_1<\\frac{1}{2}}+3\\cdot\\indd{\\tau_1\\ge\\frac{1}{2}}) + \\indd{\\mcalI=2}(-1)\\big]\\\\\n&\\ge \\frac{1}{2}\\left(\\xi^1_{\\frac{1}{2}-}+3\\big(1-\\xi^1_{\\frac{1}{2}-}\\big)\\right) - \\frac{1}{2}\n=1-\\xi^1_{\\frac{1}{2}-}\\ge 1-\\delta>0.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{eq:last0} and \\eqref{eq:last1} we have $V^*>0$.\n\\end{proof}\n\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe AdS\/CFT correspondence \\cite{Maldacena:1997re, Witten:1998qj, Gubser:1998bc}, the celebrated conjecture relating type IIB strings on $AdS_5\\times S^5$ to $\\mathcal{N}=4$ super Yang-Mills theory, has received a lot of attention given the possibilities of extracting information from the strongly coupled gauge theory, by means of performing perturbative computations in the gravitational dual. However, this same property has made it difficult to find a way to prove the conjecture in all generality, and one needs to rely in tests restricted to the BPS sector.\n\nIn particular, evaluation of four-point correlation functions of BPS operators in tree level supergravity has allowed to check  the correspondence in the limit $N\\rightarrow \\infty$, large $\\lambda$.  Four-point functions are very interesting objects as they are not completely fixed by conformal symmetry, and they can be given an Operator Product Expansion (OPE) interpretation, which is known to encode all the dynamical information of the theory. Moreover, their quantum behaviour is severely restricted due to the existence of a lagragian formulation of $d=4$ SYM, so the predictions on the dynamical piece can be verified by direct computation.\n\nThe present availability of the spectrum has limited the calculations to fields arising in the compactification of IIB supergravity on $AdS_5\\times S^5$. The standard AdS\/CFT dictionary relates the infinite tower of KK scalar excitations originating from the trace of the graviton and the five-form on $S^5$ to $1\/2$-BPS operators of $\\mathcal{N}=4$ SYM theory. These operators are known to have protected conformal dimensions, two- and three-point functions  \\cite{Lee:1998bxa,D'Hoker:1998tz}.  Four-point functions are then the simplest objects which exhibit non-trivial dynamics when going to the strongly coupled regime. Therefore, comparison of results obtained from supergravity with those obtained either from free or perturbative YM often reveal new insights into the behaviour of the theory, while also constituting a probing test for the duality.\n\nGiven the technical difficulty associated with evaluating diagrams for generic operators, supergravity induced four-point functions have been studied only for specific examples\\footnote{Other known examples involving superconformal descendents can be found in \\cite{Liu:1998ty,Uruchurtu:2007kq}.}. The first example in the literature, in which the basic techniques for evaluating amplitudes were developed, was the four-point function of dilaton-axion fields \\cite{D'Hoker:1999pj}, whose dual operators belong to the (ultrashort) current multiplet of $\\mathcal{N}=4$ SYM. Four-point functions of superconformal primaries followed later since the cubic and quartic couplings are difficult to evaluate \\cite{Arutyunov:1999en,Arutyunov:1999fb}. The examples have been restricted to those involving four identical operators with weight $\\Delta=2,3,4$ \\cite{Arutyunov:2000py,Arutyunov:2002fh,Arutyunov:2003ae}, and the results have shown to have the dynamical structure predicted by the gauge theory and superconformal symmetry. The first example that explored the dynamics in the $t$-channel between massless fields and Kaluza-Klein (KK) excitations was presented in \\cite{Berdichevsky:2007xd}, and so far, there are not known computations from supergravity that address fields transforming in generic representations, this is, of the form $[0,n,0]$. \n\nIn this paper we then continue the programme of evaluating new examples of four-point functions involving BPS operators. In this case we will consider two operators of lowest conformal dimension $\\Delta=2$, and two operators of generic conformal dimension $\\Delta=n$. This example generalises the result in \\cite{Berdichevsky:2007xd} and is the first one involving operators transforming in generic representations of the $R$-symmetry group. This constitutes a first step towards computing the four-point function of 1\/2-BPS primaries of arbitrary weight, while also allowing the emergence of interactions between the massless graviton multiplet and the infinite tower of KK excitations. We will start by establishing the general structure of the amplitude by restricting the functional dependence using superconformal symmetry and the dynamical procedure known as the insertion procedure. We then evaluate the amplitude in AdS supergravity and compare this result against the predictions made in the gauge theory side. \n\nTo this end, one needs to obtain the on-shell value of the five-dimensional effective action for type IIB supergravity on $AdS_5\\times S^5$ relevant for the calculation. These terms can be found in \\cite{Lee:1998bxa,Arutyunov:1999en,Arutyunov:1999fb, Arutyunov:1998hf}. To calculate the on-shell action, we use the techniques in \\cite{Arutyunov:2002fh, Berdichevsky:2007xd, D'Hoker:1999ni} for evaluating the AdS $z$-integrals. However, for the evaluation of the effective vertices coming from the integrals over the $S^5$, we introduce a new method, as the direct evaluation of sums of products of $SO(6)$ $C$-tensors cannot be evaluated in a closed form when including representations depending on generic values\\footnote{And even in cases in which $n>4$ it becomes very involved and one requires the use of a computer algebra program.}\\cite{Arutyunov:2002fh}. We then show that as in the previous cases in the literature, the four derivative terms in the effective lagrangian can be re-expressed in terms of two and zero derivative terms, so the lagrangian is of $\\sigma$-model type. We also show how the resulting quartic lagrangian has a rather simple form, after the dramatic simplification coming from adding the different contributions. Finally, we will verify that the result for the strongly coupled four-point amplitude splits into a free and an interacting piece, which has the structure predicted by the insertion procedure \\cite{Arutyunov:2002fh,Intriligator:1998ig}. This phenomena has also been observed in all other four-point functions involving superconformal primary operators, and is a highly non--trivial result as there is no argument supporting this splitting in the gravitational theory. This result serves then as further evidence for the AdS\/CFT correspondence\n\nThe plan of this paper is as follows. In section \\ref{sec:structure} we consider the general structure of the four-point amplitude of $1\/2$-BPS operators using the different symmetries (i.e. conformal, crossing and $R$-symmetry) and we see that the dependence is contained in four functions of conformal ratios. In section \\ref{sec:insertion}, we introduce further constraints on the interacting piece from the insertion procedure, that reduces the number of independent functions from four to one. Section \\ref{sec:chiralpsdiffweight} is devoted to the evaluation of the four-point function of interest in the supergravity approximation. Some technical details are postponed to the appendices, including the derivation of the quartic lagrangian and the novel method for computing the effective interaction vertices coming from integrals on $S^5$. In section \\ref{sec:verifying} we analyse the supergravity result in the light of the predictions obtained from the CFT side, and verify that indeed, the supergravity-induced amplitude splits into a free and an interacting piece. We also reveal a puzzling result pertaining to one of the coefficient functions entering the amplitude. Finally, section \\ref{sec:conclusions} summarises our results and presents some interesting problems that could be addressed in the future.\n\\section{General Structure of the Four-Point Function}\n\\label{sec:structure}\nThe general structure of the process we are considering is constrained by $R$ and crossing summetry. In this paper we are concerned with four-point functions of $1\/2$-BPS superconformal primaries of $\\mathcal{N}=4$ supersymmetric Yang-Mills theory. The canonically normalised operators \\cite{Lee:1998bxa}  with conformal dimension $\\Delta=k$ are given by\n\\begin{equation}\n\\mathcal{O}_k^{I}(\\vec{x})=\\frac{(2\\pi)^k}{\\sqrt{k\\lambda^k}}C_{i_1\\cdots i_k}^{I}\\mathrm{tr}(\\varphi^{i_1}(\\vec{x})\\cdots \\varphi^{i_k}(\\vec{x}))\n\\label{normCPOk}\n\\end{equation}\nwhere $C_{i_1\\cdots i_k}^I$ are totally symmetric traceless $SO(6)$ tensors of rank $k$ and the index $I$ runs over a basis of a representation of $SO(6)$ specified by $k$. The four-point function we wish to study has the form\n\\begin{equation}\n\\langle \\mathcal{O}^{I_1}_{2}(\\vec{x}_1) \\mathcal{O}^{I_2}_{2}(\\vec{x}_2) \\mathcal{O}^{I_3}_{n}(\\vec{x}_3) \\mathcal{O}^{I_4}_{n}(\\vec{x}_4)\\rangle\n\\label{diffweightprocess}\n\\end{equation}\nThe content of the OPE's is given by operators in the representations arising in the tensor of the $SU(4)$ representations $[0,2,0]$ and $[0,n,0]$.  This is\n\\begin{equation}\n\\label{4pdiffweightreps}\n\\langle \\mathcal{O}_{2}(\\vec{x}_1)\\mathcal{O}_{2}(\\vec{x}_2)\\mathcal{O}_{n}(\\vec{x}_3)\\mathcal{O}_{n}(\\vec{x}_4)\\rangle\n\\in [0,2,0] \\otimes [0,2,0] \\otimes [0,n,0] \\otimes [0,n,0]\n\\end{equation}\nwhere\n\\begin{equation}\n[0,n,0]\\otimes[0,n,0]=\\sum_{k=0}^{n}\\sum_{l=0}^{n-k}[l,2n-2l-2k,l] \n\\end{equation}\nAll the OPE channels with $l=0,1$ contain only short and semishort operators. We now follow the ideas and methods in \\cite{Arutyunov:2002fh}. An appropriate basis to study the content of a four-point function is given by the \\emph{propagator basis} arising in free field theory. Recall that the propagator for scalar fields is given by\n\\begin{equation}\n\\label{scalarpropYM}\n\\langle \\varphi^{i}(\\vec{x}_1)\\varphi^j(\\vec{x}_2)\\rangle=\\frac{\\delta^{ij}}{|\\vec{x}_{12}|^2}\n\\end{equation}\nLet us introduce the harmonic (complex) variables $u^i$ satisfying the following constraints\n\\begin{equation}\nu_iu_i=0 \\qquad \\qquad u_i\\bar{u}_i=1\n\\end{equation}\nThese variables parametrise the coset $SO(6)\/SO(2)\\times SO(4)$ so that under an $SO(6)$ transformation, the highest weight vector representation transforms as $u^{i_1}\\cdots u^{i_n}$, so projections onto representations $[0,n,0]$ can be achieved by writing\n\\begin{equation}\n\\mathcal{O}^{(n)}=u_{i_1}\\cdots u_{i_n}\\mathrm{tr}(\\varphi^{i_1}\\cdots \\varphi^{i_n})\n\\end{equation}\nwith $(n)$ denoting the highest weight of the representation $[0,n,0]$. Scalar fields can also be projected\n\\begin{equation}\n\\varphi^{i_1}(\\vec{x}_1)=\\varphi(1)\\bar{u}_1^{i_1}\n\\end{equation} \nso (\\ref{scalarpropYM}) can be rewritten as\n\\begin{equation}\n\\langle \\varphi(1)\\varphi(2) \\rangle =\\frac{ {u_1}^{i_1} {u_2}^{i_2}\\delta^{i_1 i_2}}{|\\vec{x}_{12}|^2}\n=\\frac{ ({u_1}^{i_1} {u_2}^{i_2})}{|\\vec{x}_{12}|^2}\n\\end{equation}\nWe can now construct four-point functions by connecting pairs of points by propagators. For the case in hand, the amplitude will have $n+2$ contractions, so the propagator basis for (\\ref{4pdiffweightreps}) is determined from six graphs belonging to four equivalence classes, as depicted in figure \\ref{colourbasis}. \n\\begin{figure}[ht]\n\\begin{center}\n\\resizebox{60mm}{100mm}{\\input{colourd.pdftex_t}}\n\\end{center}\n\\caption{Propagator basis for the process $\\langle \\mathcal{O}_{2}(\\vec{x}_1) \\mathcal{O}_{2}(\\vec{x}_2) \\mathcal{O}_{n}(\\vec{x}_3)\\mathcal{O}_{n}(\\vec{x}_4)\\rangle$. The graphs are arranged in four equivalence classes. The symbol $n$ stands\nfor the $n$ propagators coming out from the corresponding vertices.}\n\\label{colourbasis}\n\\end{figure}\nEach of the propagator structures can be multiplied by an arbitrary function of the conformally invariant ratios $u$ and $v$\n\\begin{equation}\nu=\\frac{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^2}{|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2} \\qquad \\qquad\nv=\\frac{|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2}{|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2} \n\\label{crossradii}\n\\end{equation}\nHence, the most general four-point amplitude with the required transformation properties is given by\n\\begin{eqnarray}\n&&\\langle \\mathcal{O}_{2}(\\vec{x}_1) \\mathcal{O}_{2}(\\vec{x}_2) \\mathcal{O}_{n}(\\vec{x}_3)\\mathcal{O}_{n}(\\vec{x}_4)\\rangle=\na(u,v)\\frac{({u_1}^{i_1}{u_2}^{i_2})^2({u_3}^{i_3}{u_4}^{i_4})^n}{|\\vec{x}_{12}|^4|\\vec{x}_{34}|^{2n}}\n\\nonumber \\\\\n&+&b_1(u,v)\\frac{({u_1}^{i_1}{u_2}^{i_2})({u_3}^{i_3}{u_4}^{i_4})^{n-1}({u_1}^{i_1}{u_3}^{i_3})({u_2}^{i_2}{u_4}^{i_4})}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2}\n\\nonumber \\\\\n&+&b_2(u,v)\\frac{({u_1}^{i_1}{u_2}^{i_2})({u_3}^{i_3}{u_4}^{i_4})^{n-1}({u_1}^{i_1}{u_4}^{i_4})({u_2}^{i_2}{u_3}^{i_3})}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2}\n\\nonumber \\\\\n&+&c_1(u,v)\\frac{({u_1}^{i_1}{u_3}^{i_3})^2({u_2}^{i_2}{u_4}^{i_4})^2({u_3}^{i_3}{u_4}^{i_4})^{n-2}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{13}|^{4}|\\vec{x}_{24}|^{4}}\n+c_2(u,v)\\frac{({u_1}^{i_1}{u_4}^{i_4})^2({u_2}^{i_2}{u_3}^{i_3})^2({u_3}^{i_3}{u_4}^{i_4})^{n-2}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{14}|^{4}|\\vec{x}_{23}|^{4}}\n\\nonumber \\\\\n&+&d(u,v)\\frac{({u_1}^{i_1}{u_3}^{i_3})({u_2}^{i_2}{u_4}^{i_4})({u_2}^{i_2}{u_3}^{i_3})({u_1}^{i_1}{u_4}^{i_4})({u_3}^{i_3}{u_4}^{i_4})^{n-2}}{|\\vec{x}_{13}|^2|\\vec{x}_{24}|^{2}|\\vec{x}_{23}|^{2}|\\vec{x}_{14}|^2|\\vec{x}_{34}|^{2(n-2)}}\n\\label{structure4p}\n\\end{eqnarray}\nPermutation symmetries under exchange of $1\\leftrightarrow 2$ and $3 \\leftrightarrow 4$ reduce the number of coefficient functions to four since\n\\begin{eqnarray}\na(u,v)&=&a(u\/v,1\/v) \\nonumber \\\\\nb_2(u,v)&=&b_1(u\/v,1\/v) \\nonumber \\\\\nc_2(u,v)&=&c_1(u\/v,1\/v) \\nonumber \\\\\nd(u,v)&=&d(u\/v,1\/v)\n\\end{eqnarray}\nThe harmonic variables in  (\\ref{structure4p}) can be re-expressed in terms of $SO(6)$ $C$-tensors (Appendix \\ref{sec:SphereInts}) as\n\\begin{eqnarray}\n&&\\langle \\mathcal{O}_{2}^{I_1}(\\vec{x}_1) \\mathcal{O}_{2}^{I_2}(\\vec{x}_2) \\mathcal{O}_{n}^{I_3}(\\vec{x}_3)\\mathcal{O}_{n}^{I_4}(\\vec{x}_4)\\rangle=\na(u,v)\\frac{\\delta^{I_1 I_2}_2\\delta^{I_3 I_4}_n}{|\\vec{x}_{12}|^4|\\vec{x}_{34}|^{2n}}\n+b_1(u,v)\\frac{C^{I_1I_2I_3I_4}}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2}\n\\nonumber \\\\\n&+&b_2(u,v)\\frac{C^{I_1I_2I_4I_3}}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2}\n+c_1(u,v)\\frac{\\Upsilon^{I_1I_2I_3I_4}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{13}|^{4}|\\vec{x}_{24}|^{4}}\n\\nonumber \\\\\n&+&c_2(u,v)\\frac{\\Upsilon^{I_1I_2I_4I_3}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{14}|^{4}|\\vec{x}_{23}|^{4}}\n+d(u,v)\\frac{S^{I_1I_2I_3I_4}}{|\\vec{x}_{13}|^2|\\vec{x}_{24}|^{2}|\\vec{x}_{23}|^{2}|\\vec{x}_{14}|^2|\\vec{x}_{34}|^{2(n-2)}}\n\\label{structure4pCten}\n\\end{eqnarray}\nIt is possible to compute the value of the coefficient functions using free field theory in the large $N$ limit (\\emph{e.g.} contribution form planar diagrams only). This was done in \\cite{Rayson:2007th} and the results are reproduced here\\footnote{The coefficient of the disconnected piece is set to be one as a consequence of the normalisation choice for the two-point functions.}\n\\begin{equation}\na=1 \\qquad b_i=\\frac{2n}{N^2} \\qquad c_i=\\frac{n(n-1)}{2N^2}\\left(\\frac{X_{i_1\\cdots i_{n-2}kk}X_{j_1\\cdots j_{n-2}ll}}{X_{m_1\\cdots m_n}X_{m_1\\cdots m_n}}\\right) \\qquad d=\\frac{2n(n-1)}{N^2}\n\\label{largeNfreecoeff}\n\\end{equation}\nwhere $X_{i_1\\cdots i_n}$ is a totally symmetric rank $n$ colour tensor, so that the value of $c_i$ is dependent on a non-trivial tensor calculation\\footnote{For $n=2$, $c_i=1$ and for $n=3$, $c_i=0$. For $n\\geq 4$ it was shown in \\cite{Rayson:2007th} that it the value of $c_i$ can be approximated as\n\\begin{equation}\nc_i\\simeq \\frac{2n(n-2)}{N^2}\\simeq (n-2)b_i \n\\end{equation}\n}. Notice also that $d=(n-1)b_i$ for any value of $n$ and $N$.  \n\\section{The Insertion Formula}\n\\label{sec:insertion}\nWe now follow the ideas developed in \\cite{Arutyunov:2002fh} to restrict the dynamical piece of the four-point function. The derivative with respect to the coupling $g_{YM}^2$ of the amplitude (\\ref{diffweightprocess}) can be expressed as (see also \\cite{Intriligator:1998ig})\n\\begin{equation}\n\\frac{\\partial}{\\partial g_{YM}^2}\\langle \\mathcal{O}_{2}\\mathcal{O}_{2}\\mathcal{O}_{n}\\mathcal{O}_{n}\\rangle \n\\propto \\int d^{4}\\vec{x_0} d^{4}\\theta_0\n\\langle \\mathcal{O}_\\tau(\\vec{x_0})\\mathcal{O}_{2}\\mathcal{O}_{2}\\mathcal{O}_{n}\\mathcal{O}_{n}\\rangle \n\\end{equation}\nThe integration above is consistent with supersymmetry as the $\\theta$-expansion for the case $\\mathcal{O}_2$ terminates at four $\\theta$'s, and one can show that the five-point function in the right side of the previous expression, gives rise to a nilpotent superconformal covariant. By following this procedure in which we insert and additional \\emph{ultrashort} operator, it is possible to extract more information about the four-point function we have been studying. As the construction of nilpotents covariants if of technical nature, we refer to \\cite{Arutyunov:2002fh} for references and the derivation of the results reproduced below.\n\nThe key idea is to assume that the nilpotent covariant must have the following form\n\\begin{equation}\n\\label{inserting}\n\\langle \\mathcal{O}_\\tau(\\vec{x_0})\\mathcal{O}_{2}\\mathcal{O}_{2}\\mathcal{O}_{n}\\mathcal{O}_{n}\\rangle=R^{2222}(\\theta_0)^4F^{00n-2n-2}(\\vec{x_0},\\cdots,\\vec{x}_4,u_1,\\cdots,u_4)\n\\end{equation}\nso the five-point function is factorised into a kernel with weight $2$ and an additional factor carrying the remaining $SO(6)$ quantum numbers, so at each point the weight is $k_i'=k_i-2$.  Note here that the Grassmann factor $(\\theta_0)^4$ carries the full harmonic dependence at the insertion point. The relevant expressions are given by\n\\begin{eqnarray}\n\\label{Rkernel}\nR^{2222}&=&u \\frac{({u_1}^{i_1}{u_2}^{i_2})^2({u_3}^{i_3}{u_4}^{i_4})^2}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^2}+(v-u-1)\\frac{({u_1}^{i_1}{u_2}^{i_2})({u_3}^{i_3}{u_4}^{i_4})({u_1}^{i_1}{u_3}^{i_3})({u_2}^{i_2}{u_4}^{i_4})}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^2|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2}\n\\nonumber \\\\\n&+&(1-u-v)\n\\frac{({u_1}^{i_1}{u_2}^{i_2})({u_3}^{i_3}{u_4}^{i_4})({u_1}^{i_1}{u_4}^{i_4})({u_2}^{i_2}{u_3}^{i_3})}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^2|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2}+\n \\frac{({u_1}^{i_1}{u_3}^{i_3})^2({u_2}^{i_2}{u_4}^{i_4})^2}{|\\vec{x}_{13}|^4|\\vec{x}_{24}|^4}\n \\nonumber \\\\\n&+& \\frac{({u_1}^{i_1}{u_4}^{i_4})^2({u_2}^{i_2}{u_3}^{i_3})^2}{|\\vec{x}_{14}|^4|\\vec{x}_{23}|^4}+(u-v-1)\\frac{({u_1}^{i_1}{u_3}^{i_3})({u_1}^{i_1}{u_4}^{i_4})({u_2}^{i_2}{u_4}^{i_4})({u_2}^{i_2}{u_3}^{i_3})}{|\\vec{x}_{13}|^2|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2|\\vec{x}_{24}|^2}\n\\end{eqnarray}\nand\n\\begin{equation}\nF^{0 k_1'k_2'k_3'}=\n\\left(\\frac{{u_2}^{i_2}{u_3}^{i_3}}{|\\vec{x}_{23}|^2}\\right)^{\\frac{1}{2}(k_1+k_2-k_3-2)}\n\\left(\\frac{{u_2}^{i_2}{u_4}^{i_4}}{|\\vec{x}_{24}|^2}\\right)^{\\frac{1}{2}(k_1+k_3-k_2-2)}\n\\left(\\frac{{u_3}^{i_3}{u_4}^{i_4}}{|\\vec{x}_{34}|^2}\\right)^{\\frac{1}{2}(k_2+k_3-k_1-2)}f(\\vec{x}_0,\\cdots, \\vec{x}_4)\n\\end{equation}\nSubstitution of these expressions into (\\ref{inserting}) and integration over the Grassman variable $\\theta_0$  lead to the following dependence on the coupling of the four-point function (\\ref{diffweightprocess})\n\\begin{eqnarray}\n\\label{insertionres}\n&&\\frac{\\partial}{\\partial g_{YM}^2}\\langle \\mathcal{O}_{2}^{I_1}\\mathcal{O}_{2}^{I_2}\\mathcal{O}_{n}^{I_3}\\mathcal{O}_{n}^{I_4}\\rangle=u G(u,v)\\frac{\\delta_{2}^{I_1I_2}\\delta_{n}^{I_3I_4}}{|\\vec{x}_{12}|^{4}|\\vec{x}_{34}|^{2n}}+\n(v-u-1)G(u,v)\\frac{C^{I_1I_2I_3I_4}}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{13}|^2|\\vec{x}_{24}|^2}\n\\nonumber \\\\\n&&+(1-u-v)G(u,v)\\frac{C^{I_1I_2I_4I_3}}{|\\vec{x}_{12}|^2|\\vec{x}_{34}|^{2(n-1)}|\\vec{x}_{14}|^2|\\vec{x}_{23}|^2}+G(u,v)\\frac{\\Upsilon^{I_1I_2I_3I_4}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{13}|^{4}|\\vec{x}_{24}|^{4}}\n\\nonumber \\\\\n&&+vG(u,v)\\frac{\\Upsilon^{I_1I_2I_4I_3}}{|\\vec{x}_{34}|^{2(n-2)}|\\vec{x}_{14}|^{4}|\\vec{x}_{23}|^{4}}+(u-v-1)G(u,v)\\frac{S^{I_1I_2I_3I_4}}{|\\vec{x}_{13}|^2|\\vec{x}_{24}|^{2}|\\vec{x}_{23}|^{2}|\\vec{x}_{14}|^2|\\vec{x}_{34}|^{2(n-2)}}\\nonumber \\\\\n\\end{eqnarray}\nwith\n\\begin{equation}\nG(u,v)=\\int d^4 \\vec{x}_0 f(\\vec{x}_0,\\cdots, \\vec{x}_4)\n\\end{equation}\nSo comparing (\\ref{insertionres}) with (\\ref{structure4pCten}) one realises that the amplitude depends on a single function $\\mathcal{F}(u,v)$, satisfying\n\\begin{eqnarray}\na(u,v)&=&u\\mathcal{F}(u,v) \\nonumber \\\\\nb_1(u,v)&=&(v-u-1)\\mathcal{F}(u,v)\\nonumber \\\\\nb_2(u,v)&=&(1-u-v)\\mathcal{F}(u,v)\\nonumber \\\\\nc_1(u,v)&=&\\mathcal{F}(u,v)\\nonumber \\\\\nc_2(u,v)&=&v\\mathcal{F}(u,v)\\nonumber \\\\\nd(u,v)&=&(u-v-1)\\mathcal{F}(u,v)\n\\label{insertrels}\n\\end{eqnarray}\nThis is a (partial) non-renormalisation theorem for the structure of the amplitude (i.e. a dynamical constraint), so verification of this result from the supergravity calculation constitutes an indirect test for the AdS\/CFT correspondence.\n\\section{Supergravity Calculation}\n\\label{sec:chiralpsdiffweight}\nThe precise relation between the operators in the gauge theory and the fields in the bulk was established in \\cite{Witten:1998qj,Gubser:1998bc} and refined in \\cite{Mueck:1999kk,Mueck:1999gi,Bena:1999jv}. The proposition is \n \\begin{equation}\n\\langle \\exp\\{ \\int d^{4}x \\phi_{0}(\\vec{x})\\mathcal{O}(\\vec{x})\\}\\rangle_{CFT}=\\exp \\{ -S_{IIB}[\\phi_{0}(\\vec{x})] \\}\n\\label{prescriptionadscft}\n\\end{equation}\nOn the left hand side of (\\ref{prescriptionadscft}) the field $\\phi_0(\\vec{x})$, which stands for the boundary value of the bulk field $\\phi(z_0,\\vec{x})$, is a source for the operator $\\mathcal{O}(\\vec{x})$, and the expectation value is computed by expanding the exponential and evaluating the correlation functions in the field theory. On the right hand side, one has the generating functional encompassing all dynamical processes of IIB strings on $AdS_5\\times S^{5}$. In the supergravity approximation, $S_{IIB}$ is just the type IIB supergravity action on $AdS_5\\times S^5$, and it is assumed here that all the bulk fields $\\phi(z_0,\\vec{x})$ have appropriate boundary behaviour so they source the YM operators on the left hand side. Hence in practice, one first finds the boundary data for the corresponding gravitational fields and then computes correlation functions as a function of these values (on-shell), by functional differentiation.\n\nGiven that we are interested in computing correlation functions of superconformal primaries, we first need to identify the bulk fields whose value in the boundary serve as sources.  From looking at the representations, we see that the fields dual to superconformal primaries are obtained from mixtures of modes from the graviton and the five form on the $S^5$ \\cite{Kim:1985ez} and are denoted as $s_k^I$, with $I$ running over the basis of the corresponding $SO(6)$ irrep. with Dynkin labels $[0,k,0]$. The four-point function can then be determined from the expression\n\\begin{equation}\n\\langle \\mathcal{O}^{I_1}_{k_1}(\\vec{x}_{1})\\mathcal{O}^{I_2}_{k_2}(\\vec{x}_{2})\\mathcal{O}^{I_3}_{k_3}(\\vec{x}_{3})\\mathcal{O}^{I_4}_{k_4}(\\vec{x}_{4})\\rangle=\n\\frac{\\delta}{\\delta s_{k_1}^{I_{1}}(\\vec{x}_{1})}\\frac{\\delta}{\\delta s_{k_2}^{I_{2}}(\\vec{x}_{2})}\n\\frac{\\delta}{\\delta s_{k_3}^{I_{3}}(\\vec{x}_{3})}\\frac{\\delta}{\\delta s_{k_4}^{I_{4}}(\\vec{x}_{4})}(-S_{IIB})\n\\end{equation}\n\\subsection{On-Shell Lagrangian}\n\\label{subsec:onshelldiffCPO}\nWe are interested in computing (\\ref{diffweightprocess}) in strongly coupled $\\mathcal{N}=4$ SYM theory, using the supergravity approximation. The prescription (\\ref{prescriptionadscft}) indicates that we need to evaluate the on-shell value of the five-dimensional effective action of compactified type IIB supergravity on $AdS_5\\times S^5$. We write this action as\n\\begin{equation}\nS=\\frac{N}{8\\pi^2}\\int [dz] \\left(\\mathcal{L}_2+\\mathcal{L}_3+\\mathcal{L}_4\\right)\n\\end{equation}\nwhich involves the sum of quadratic, cubic and quartic terms. The normalisation of the action can be derived from expressing the ten dimensional gravitational coupling as $2\\kappa_{10}^2=(2\\pi)^7g_s^2\\alpha'^4$ and using the volume of $S^5$ to get the five dimensional gravitational coupling\n\\begin{equation}\n\\label{normaction5d}\n\\frac{1}{2\\kappa_{5}^2}=\\frac{\\mathrm{Vol}(S^5)}{2\\kappa_{10}^2}=\\frac{N^2}{8\\pi^2l^3}\n\\end{equation}\nwith $l$ being the $AdS_5$ radius, which will be set to one. The quadratic terms \\cite{Kim:1985ez,Arutyunov:1998hf} read\n\\begin{eqnarray}\n\\mathcal{L}_{2}&=&\\frac{1}{4}(D_{\\mu}{s_{2}}^{1}D^{\\mu}{s_{2}}^{1}-4{s_{2}}^{1}{s_{2}}^{1})\n               +\\frac{1}{4}(D_{\\mu}{s_{n}}^{1}D^{\\mu}{s_{n}}^{1}+n(n-4){s_{n}}^{1}{s_{n}}^{1})\n               \\nonumber \\\\\n               &+&\\frac{1}{2}({F_{\\mu \\nu,1}}^{1})^{2} +\\frac{1}{2}(({F_{\\mu \\nu,n-1}}^{1})^{2}+2n(n-2)(A^{1}_{\\mu,n-1})^{2})\n               \\nonumber \\\\\n               &+&\\frac{1}{4}D_{\\mu}\\phi_{\\nu \\rho,0}D^{\\mu}\\phi_{0}^{\\nu \\rho}-\\frac{1}{2}D_{\\mu}\\phi^{\\mu \\nu,0}D^{\\rho}\\phi_{\\rho \\nu,0}\n               +\\frac{1}{2}D_{\\mu}\\phi^{\\nu}_{\\nu,0}D_{\\rho}\\phi^{\\mu \\rho}_{0}-\\frac{1}{4}D_{\\mu}\\phi^{\\nu}_{\\nu,0}D^{\\mu}\\phi^{\\rho}_{\\rho,0}\n               \\nonumber \\\\\n               &-&\\frac{1}{2}\\phi_{\\mu \\nu,0}\\phi^{\\mu \\nu}_{0}+\\frac{1}{2}(\\phi^{\\mu}_{\\mu,0})^{2}\n               \\nonumber \\\\\n               &+&\\frac{1}{4}D_{\\mu}\\phi_{\\nu \\rho,n-2}D^{\\mu}\\phi_{n-2}^{\\nu \\rho}-\\frac{1}{2}D_{\\mu}\\phi^{\\mu \\nu,n-2}D^{\\rho}\\phi_{\\rho \\nu,n-2}\n               +\\frac{1}{2}D_{\\mu}\\phi^{\\nu}_{\\nu,n-2}D_{\\rho}\\phi^{\\mu \\rho}_{n-2}\n               \\nonumber \\\\\n               &-&\\frac{1}{4}D_{\\mu}\\phi^{\\nu}_{\\nu,n-2}D^{\\mu}\\phi^{\\rho}_{\\rho,n-2}\n               +\\frac{(n^2-6)}{4}\\phi_{\\mu \\nu,n-2}\\phi^{\\mu \\nu}_{n-2}-\\frac{(n^2-2)}{4}(\\phi^{\\mu}_{\\mu,n-2})^{2}\n\\end{eqnarray}\nwhere $F_{\\mu \\nu,k}=\\partial_{\\mu}A_{\\nu,k}-\\partial_{\\nu}A_{\\mu,k}$, and summation over upper indices is assumed, running over the basis of the irreducible representation corresponding to the field\\footnote{We often use the notation $s_k^{I_m}\\equiv s_{k}^{m}$.}. We should point out that the fields have been rescaled in order to simplify the action. In this case, the corresponding rescaling factors are given by\n\\begin{equation}\ns_{n} \\rightarrow \\sqrt{ \\frac{(n+1)}{2^{6}n(n-1)(n+2)}} s_{n} \\qquad A_{\\mu,n-1} \\rightarrow\n2\\sqrt{\\frac{n+1}{n}} A_{\\mu,n-1}\n\\end{equation}\nand all symmetric tensors are left unscaled. The cubic couplings \\cite{Arutyunov:1999en, Lee:1998bxa, Lee:1999pj} are given by\n\\begin{eqnarray}\n\\mathcal{L}_{3}&=&-\\frac{1}{3}\\langle C^{1}_{2}C^{2}_{2}C^{3}_{[0,2,0]}\\rangle s_{2}^{1}s_{2}^{2}s_{2}^{3}\n                -\\frac{n(n-1)}{2}\\langle C^{1}_{n}C^{2}_{n}C^{3}_{[0,2,0]} \\rangle s_{n}^{1}s_{n}^{2}s_{2}^{3}\n                \\nonumber \\\\\n               &-&\\frac{1}{4}\\left(D^{\\mu}s_{2}^{1}D^{\\nu}s_{2}^{1}\\phi_{\\mu \\nu,0}-\\frac{1}{2}(D^{\\mu}s_{2}^{1}D_{\\mu}s_{2}^{1}\n               -4s_{2}^{1}s_{2}^{1})\\phi^{\\nu}_{\\nu,0} \\right)\n               \\nonumber \\\\\n               &-&\\frac{1}{4}\\left(D^{\\mu}s_{n}^{1}D^{\\nu}s_{n}^{1}\\phi_{\\mu \\nu,0}-\\frac{1}{2}(D^{\\mu}s_{n}^{1}D_{\\mu}s_{n}^{1}\n               +n(n-4)s_{n}^{1}s_{n}^{1})\\phi^{\\nu}_{\\nu,0} \\right)\n               \\nonumber \\\\\n               &-&\\frac{1}{2}\\langle C_{2}^{1}C_{n}^{1}C^{3}_{[0,n-2,0]}\\rangle \\left(D^{\\mu}s_{2}^{1}D^{\\nu}s_{n}^{1}\\phi_{\\mu \\nu,n-2}\n               -\\frac{1}{2}(D^{\\mu}s_{2}^{1}D_{\\mu}s_{n}^{1}-2n s_{2}^{1}s_{n}^{1})\\phi^{\\nu}_{\\nu,n-2} \\right)\n               \\nonumber \\\\\n               &-&\\langle C^{1}_{2}C^{2}_{2}C^{3}_{[1,0,1]}\\rangle s_{2}^{1}D^{\\mu}s_{2}^{2}A_{\\mu,1}^{3}\n               -\\frac{n}{2}\\langle C^{1}_{n}C^{2}_{n}C^{3}_{[1,0,1]}\\rangle s_{n}^{1}D^{\\mu}s_{n}^{2}A_{\\mu,1}^{3}\n               \\nonumber \\\\\n               &-&\\sqrt{\\frac{n(n-1)}{2}}\\langle C^{1}_{2}C^{2}_{n}C^{3}_{[1,n-2,1]}\\rangle s_{2}^{1}D^{\\mu}s_{n}^{2}A_{\\mu,n-1}^{3}\n               -\\sqrt{\\frac{n(n-1)}{2}}\\langle C^{1}_{n}C^{2}_{2}C^{3}_{[1,n-2,1]}\\rangle s_{n}^{1}D^{\\mu}s_{2}^{2}A_{\\mu,n-1}^{3}\n               \\nonumber\n\\end{eqnarray}\nAs one can see, there are different contributions to the $s$ and $t$-channels. Finally, the quartic couplings are given by\n\\begin{equation}\n\\mathcal{L}_{4}=\\mathcal{L}_{4}^{(0)}+\\mathcal{L}_{4}^{(2)}+\\mathcal{L}_{4}^{(4)}\n\\end{equation}\nwhere the supraindex indicates contributions coming from zero, two and four-derivative terms, which are given by\n\\begin{eqnarray}\n\\mathcal{L}_{4}&=&\\mathcal{L}_{k_{1}k_{2}k_{3}k_{4}}^{(0)I_{1}I_{2}I_{3}I_{4}}s_{k_{1}}^{I_{1}}s_{k_{2}}^{I_{2}}s_{k_{3}}^{I_{3}}s_{k_{4}}^{I_{4}}+\n\\mathcal{L}_{k_{1}k_{2}k_{3}k_{4}}^{(2)I_{1}I_{2}I_{3}I_{4}}s_{k_{1}}^{I_{1}}D_{\\mu}s_{k_{2}}^{I_{2}}s_{k_{3}}^{I_{3}}D^{\\mu}s_{k_{4}}^{I_{4}}\n\\nonumber \\\\\n&+&\\mathcal{L}_{k_{1}k_{2}k_{3}k_{4}}^{(4)I_{1}I_{2}I_{3}I_{4}}s_{k_{1}}^{I_{1}}D_{\\mu}s_{k_{2}}^{I_{2}}D^{\\nu}D_{\\nu}(s_{k_{3}}^{I_{3}}D^{\\mu}s_{k_{4}}^{I_{4}})\n\\end{eqnarray}\nThe explicit form of these terms has been computed in \\cite{Arutyunov:1999fb}. \nFor our case, two of the $k_{i}$'s are equal to 2 and the other two are equal to $n$. This allows for six possible permutations, where the indices $I_{i}$ run over the basis of the representation $[0,k_{i},0]$ which is being summed over. The less trivial part of the calculation is to compute the explicit coefficients of these terms. It can be shown, however, that the relevant interactions can be reduced to a simple expression, as it occurs in all the examples that have been computed previously. We refer to appendix \\ref{sec:QuarticSimp} for the details, and reproduce the final expression here\n\\begin{eqnarray}\n\\mathcal{L}_{4}&=&-\\frac{1}{4}(C^{1234}+S^{1234})s_{2}^{1}D_{\\mu}s_{2}^{2}s_{n}^{3}D^{\\mu}s_{n}^{4}\n\\nonumber \\\\\n&+&\\frac{1}{8}n(-\\delta^{12}_{2}\\delta^{34}_{n}+(6+n)C^{1234}+(3n-4)S^{1234}-n\\Upsilon^{1234})s_{2}^{1}s_{2}^{1}s_{n}^{3}s_{n}^{4}\n\\end{eqnarray}\nwhich can be shown to reproduce the $n=3$ case in \\cite{Berdichevsky:2007xd}. The quantities in this expression will be defined later. It should be noted that all four derivative terms disappear, which is consistent with the fact that this is a sub-subextremal process, i.e. $k_{1}=k_{2}+k_{3}+k_{4}-4$, as indicated in \\cite{Arutyunov:2000ima,D'Hoker:2000dm}.\n\n\nNow that the relevant terms in the lagrangian have been specified, it remains to compute its on-shell value. From the couplings, one can determine the diagrams that need to be computed. In the $s$-channel, one has a scalar exchange of $s_{2}^{I}$, a vector exchange $A^{I}_{a,[1,0,1]}$ and a graviton exchange, $\\phi_{ab,[0,0,0]}$. In the $t$-channel, one has a scalar exchange of $s_{n}^{I}$, a vector exchange $A^{I}_{a,[1,n-1,1]}$ and a massive symmetric tensor $\\phi_{ab,[0,n-2,0]}$. Finally one has contact diagrams contributing to the process. The Witten diagrams for the $s$-channel are shown on Fig. \\ref{schanneldiffw}. The corresponding diagrams for the $t$-channel and the contact diagram are shown on Fig. \\ref{tchanneldiffw}.\n\\begin{figure}[ht]\n\\begin{center}\n\\resizebox{120mm}{40mm}{\\input{diffweights2.pdftex_t}}\n\\end{center}\n\\caption{Witten Diagrams for the $s$-channel process. \\emph{(a)} exchange by a scalar with $m^2=-4$ \\emph{(b)} exchange by a massless vector \\emph{(c)} graviton exchange}\n\\label{schanneldiffw}\n\\end{figure}\n\\begin{figure}[ht]\n\\begin{center}\n\\resizebox{160mm}{40mm}{\\input{diffweights.pdftex_t}}\n\\end{center}\n\\caption{Witten Diagrams for the $t$-channel process. \\emph{(a)} exchange by a scalar of mass $m^2=\\Delta(\\Delta-4)$ \\emph{(b)} exchange by a vector of mass $m_{k}^2=k^2-1$ \\emph{(c)} exchange by a tensor field of mass $f_k=k(k+4)$ \\emph{(d)} Contact diagram. \\ }\n\\label{tchanneldiffw}\n\\end{figure}\n\nIt is convenient to introduce the currents\n\\begin{eqnarray}\nT_{\\mu\\nu}&=&D_{(\\mu}s_{k_{1}}D_{\\nu)}s_{k_{1}}-\\frac{1}{2}g_{\\mu\\nu}\\left(D^{\\rho}s_{k_{1}}D_{\\rho}s_{k_{2}}+\\frac{1}{2}(m^2_{k_{1}}+m^2_{k_{2}}-k_{3}(k_{3}+4))s_{k_{1}}s_{k_{2}}\\right) \\nonumber \\\\\nJ_{\\mu}&=&s_{k_{1}}D_{\\mu}s_{k_{2}}-s_{k_{2}}D_{\\mu}s_{k_{1}}\n\\end{eqnarray}\nwhere $k_{1},k_{2},k_{3}$ are the conformal weights of the corresponding scalar operators and the primaries here have the appropriate weight depending of the channel one is considering. One then represents the solution to the equations of motion in the form\n\\begin{equation}\ns_{k}=s_{k}^{0}+\\tilde{s}_{k} \\qquad A_{\\mu}=A_{\\mu}^{0}+\\tilde{A}_{\\mu} \\qquad \\phi_{\\mu\\nu}=\\phi_{\\mu\\nu}^{0}+\\tilde{\\phi}_{\\mu\\nu}\n\\end{equation}\nwhere $s^{0}_{k}$, $A_{\\mu}^{0}$ and $\\phi_{\\mu\\nu}^{0}$ are solutions to the linearised equations with fixed boundary conditions and $\\tilde{s}_{k}$, $\\tilde{A}_{\\mu}$ and $\\tilde{\\phi}_{\\mu\\nu}$ represent the fields in the AdS bulk with vanishing boundary conditions. It is then possible to express these fields in terms of an integral on the bulk, involving the corresponding Green function. For the $s$-channel process one needs\n\\begin{eqnarray}\n\\tilde{s}_{2}^{5}(w)&=&2\\langle C^{1}_{2}C^{2}_{2}C^{5}_{[0,2,0]}\\rangle \\int [dz] G_{2}(z,w) s^{1}_{2}(z)s^{2}_{2}(z) +n(n-1)\\langle C^{1}_{2}C^{2}_{n}C^{5}_{[0,n,0]}\\rangle\\int [dz] G_{n}(z,w)s^{1}_{2}(z)s^{2}_{n}(z) \n\\nonumber \\\\\n\\tilde{A}_{\\mu,1}^{5}(w)&=&\\frac{1}{4}\\langle C^{1}_{2}C^{2}_{2}C^{5}_{[1,0,1]} \\rangle \\int [dz] {G_{\\mu}}^{\\nu}(z,w)J_{\\nu}(z) \n\\nonumber \\\\\n\\tilde{\\phi}^{5}_{\\mu\\nu,0}(w)&=&\n\\frac{1}{4}\\langle C^{1}_{2}C^{3}_{2}C^{5}_{[0,0,0]} \\rangle\\int [dz]G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}(z)\n\\end{eqnarray}\nwhere the $z$-integral is being done on the vertex involving the $\\mathcal{O}_{2}$'s. For the $t$-channel process, the bulk fields couple to a $\\Delta=2$ primary and a $\\Delta=n$ primary, so the $z$-integrals read\n\\begin{eqnarray}\n\\tilde{s}_{n}^{5}(w)&=&2n(n-1)\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[0,n,0]}\\rangle \\int [dz] G_{n}(z,w) s^{1}_{2}(z)s^{3}_{n}(z) \n\\nonumber \\\\\n\\tilde{A}_{\\mu,n-1}^{5}(w)&=&\\frac{1}{2}\\sqrt{\\frac{n(n-1)}{2}}\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[1,n-2,1]} \\rangle \\int [dz] {G_{\\mu}}^{\\nu}(z,w)J_{\\nu}(z) \n\\nonumber \\\\\n\\tilde{\\phi}^{5}_{\\mu\\nu,n-2}(w)&=&\\frac{1}{2}\n\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[0,n-2,0]} \\rangle \\int [dz]G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}(z)\n\\end{eqnarray}\nand the currents are defined with the appropriate weights. We will drop the tilde in the following. Using the expressions above, we arrive at the following expression for the on-shell value of the action for each of the channels we are considering. For the $s$-channel, the amplitude is determined by\n\\begin{eqnarray}\n\\mathcal{L}_{s-channel}&=&-n(n-1)\n\\langle C_{2}^{1}C_{2}^{2}C_{2}^{5}\\rangle \\langle C_{n}^{3}C_{n}^{4}C_{2}^{5}\\rangle\n\\int [dz] s^{1}_{2}(z)s^{2}_{2}(z)G(z,w)s^{3}_{n}(w)s^{4}_{n}(w) \\nonumber \\\\\n&-&\\frac{n}{2^4}\\langle C^{1}_{2}C^{2}_{2}C^{5}_{[1,0,1]} \\rangle\\langle C^{3}_{2}C^{2}_{4}C^{5}_{[1,0,1]} \\rangle  \\int [dz]J^{\\mu}(z)G_{\\mu\\nu}(z,w)J^{\\nu}(w) \\nonumber \\\\\n&-&\\frac{1}{2^4}\\langle C_{2}^{1}C_{2}^{2}C_{[0,0,0]}^{5}\\rangle \\langle C_{n}^{3}C_{n}^{4}C_{[0,0,0]}^{5}\\rangle\n\\int [dz] T^{\\mu\\nu}_{22}(z)G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}_{nn}(w) \\label{s-channel}\n\\end{eqnarray}\nand for the $t$-channel one has\n\\begin{eqnarray}\n\\mathcal{L}_{t-channel}&=&-n^2(n-1)^2\n\\langle C_{2}^{1}C_{n}^{3}C_{n}^{5}\\rangle \\langle C_{2}^{2}C_{n}^{4}C_{n}^{5}\\rangle\n\\int [dz] s^{1}_{2}(z)s^{3}_{n}(z)G(z,w)s^{2}_{2}(w)s^{4}_{n}(w) \n\\nonumber \\\\\n&-&\\frac{n(n-1)}{2^3}\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[1,n-2,1]} \\rangle\\langle C^{2}_{2}C^{4}_{n}C^{5}_{[1,n-2,1]} \\rangle \\int [dz]J^{\\mu}(z)G_{\\mu\\nu}(z,w)J^{\\nu}(w) \n\\nonumber \\\\\n&-&\\frac{1}{2^3}\n\\langle C_{2}^{1}C_{n}^{3}C_{[0,n-2,0]}^{5}\\rangle \\langle C_{2}^{2}C_{n}^{4}C_{[0,n-2,0]}^{5}\\rangle\n\\int [dz] T^{\\mu\\nu}_{2n}(z)G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}_{2n}(w) \\label{t-channel}\n\\end{eqnarray}\nThe expressions in brackets arise from the integrals over $S^5$ and are defined in appendix \\ref{sec:SphereInts}. We will worry about contact interactions later. So far, we see that we need to compute three Witten Diagrams for each channel, involving exchanges of scalars, massless and massive gauge bosons and massless and massive gravitons. In order to do so, we extend the methods developed in \\cite{D'Hoker:1999pj,D'Hoker:1999ni,Berdichevsky:2007xd} to perform the computations.\n\\subsection{Results for Exchange Integrals}\nWe now carry out the integrals and write the results in terms of $\\bar{D}$-functions, which are functions of $u$ and $v$ and are related to the more familiar $D$-functions \\cite{D'Hoker:1999pj} which are defined as\n\\begin{equation}\nD_{\\Delta_1\\Delta_2\\Delta_3,\\Delta_4}(\\vec{x}_1,\\vec{x}_2,\\vec{x}_3,\\vec{x}_4)=\\int [dw] \\tilde{K}_{\\Delta_1}(w,\\vec{x}_1)\\tilde{K}_{\\Delta_2}(w,\\vec{x}_2)\\tilde{K}_{\\Delta_3}(w,\\vec{x}_3)\\tilde{K}_{\\Delta_4}(w,\\vec{x}_4) \n\\end{equation}\nwhere $\\tilde{K}_{\\Delta}(w,\\vec{x})$ is the unit normalised bulk-to-boundary propagator for a scalar of conformal dimension $\\Delta$\n\\begin{equation}\n\\tilde{K}_{\\Delta}(z,\\vec{x})=\\left(\\frac{z_{0}}{z_{0}^2+(\\vec{z}-\\vec{x})^2}\\right)^{\\Delta}\n\\end{equation}\n$D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ can be identified as a quartic scalar interactions (see Fig. \\ref{wittendfunc}). The relation between the $D$-functions and the $\\bar{D}$-functions, and their properties can be found in appendix \\ref{sec:Dfunc}. \n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{60mm}{38mm}{\\input{dfunction.pdftex_t}}\n\\end{center}\n\\caption{Graphic representation of a $D$-function.}\n\\label{wittendfunc}\n\\end{figure}\nLet us first introduce the following notation for the various exchange integrals that contribute to the amplitude. \n\\begin{eqnarray}\nS_{\\Delta_{1}\\Delta_{2}\\Delta_{3}\\Delta_{4}}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&\\int [dw] [dz]\\tilde{K}_{\\Delta_{1}}(z,\\vec{x}_{1})\\tilde{K}_{\\Delta_{2}}(z,\\vec{x}_{2})G(z,w)\\tilde{K}_{\\Delta_{3}}(w,\\vec{x}_{3})\\tilde{K}_{\\Delta_{4}}(w,\\vec{x}_{4})\n\\nonumber \\\\\nV_{\\Delta_{1}\\Delta_{2}\\Delta_{3}\\Delta_{4}}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&\\int [dw] [dz]\\tilde{K}_{\\Delta_{1}}(z,\\vec{x}_{1})\\buildrel\\leftrightarrow\\over{D^{\\mu}}\\tilde{K}_{\\Delta_{2}}z,\\vec{x}_{2})\nG_{\\mu\\nu}(z,w)\\tilde{K}_{\\Delta_{3}}(w,\\vec{x}_{3})\\buildrel\\leftrightarrow\\over{D^{\\nu}}\\tilde{K}_{\\Delta_{4}}(w,\\vec{x}_{4}) \n\\nonumber \\\\\nT_{\\Delta_{1}\\Delta_{2}\\Delta_{3}\\Delta_{4}}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&\\int [dz] [dw] T^{\\mu\\nu}_{\\Delta_{1}\\Delta_{2}}(z,\\vec{x}_{1},\\vec{x}_{2})G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}_{\\Delta_{3},\\Delta_{4}}(w,\\vec{x}_{3},\\vec{x}_{4}) \n\\end{eqnarray}\nwith the bulk-to-bulk propagators appropriately chosen, depending on the particle that is being exchanged.  For our case, the $s$-channel integrals yield\n\\begin{eqnarray}\nS_{22nn}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&\\frac{\\pi^2}{8}\\frac{1}{(n-1)\\Gamma(n)} \n\\frac{u}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\\bar{D}_{11nn}\n\\nonumber \\\\\nV_{22nn}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&-\\frac{\\pi^2}{4\\Gamma(n)}\\frac{u}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\n\\left\\{-2\\bar{D}_{21nn+1}+\\bar{D}_{21n+1n}+\\bar{D}_{12nn+1}\\right\\}\n\\nonumber \\\\\nT_{22nn}(\\vec{x}_{1},\\vec{x}_{2},\\vec{x}_{3},\\vec{x}_{4})&=&-\\frac{\\pi^2}{2\\Gamma(n)}\\frac{u}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\n\\left\\{\\frac{1}{3}n\\bar{D}_{11nn}-n(n-1)u\\bar{D}_{22nn}\\right. \n\\nonumber \\\\\n&-&\\left.n(1+v-u)\\bar{D}_{22n+1n+1}\\right\\}\n\\label{schannresults}\n\\end{eqnarray}\nand the $t$-channel amplitudes are given by\n\\begin{eqnarray}\nS_{2n2n}(\\vec{x}_{1},\\vec{x}_{3},\\vec{x}_{2},\\vec{x}_{4})&=&\\frac{\\pi^2}{8}\\frac{1}{(n-1)\\Gamma(n)}\\frac{u^2}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\\bar{D}_{12n-1n}\n\\nonumber \\\\\nV_{2n2n}(\\vec{x}_{1},\\vec{x}_{3},\\vec{x}_{2},\\vec{x}_{4})&=&-\\frac{\\pi^2}{2n\\Gamma(n)}\\frac{u^2}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\n\\left\\{-\\bar{D}_{31nn}+\\bar{D}_{12nn+1}-(n-1)\\bar{D}_{22n-1n+1}\\right.\n\\nonumber \\\\\n&+&\\left.(n-1)u\\bar{D}_{23n-1n}\\right\\}\n\\nonumber \\\\\nT_{2n2n}(\\vec{x}_{1},\\vec{x}_{3},\\vec{x}_{2},\\vec{x}_{4})&=&-\\frac{\\pi^2}{\\Gamma(n)}\\left[\\frac{n}{(n+1)(n+2)}\\right]\\frac{u^2}{{|\\vec{x}_{12}|}^{4}{|\\vec{x}_{34}|}^{2n}}\n\\left\\{2\\bar{D}_{31n+1n+1}\\right.\n\\nonumber \\\\\n&+&+n(n-1)u\\bar{D}_{33n-1n+1}+\\left.2n(1-v-u)\\bar{D}_{23nn+1}\\right\\}\n\\label{rchannresults}\n\\end{eqnarray}\nwhere $u$ and $v$ were introduced in (\\ref{crossradii}).\nThese expressions are to be substituted in the action, including an overall factor of $C(n)^2C(2)^2$ where\n\\begin{equation}\nC(n)=\\begin{cases}\n              \\frac{\\Gamma(n)}{\\pi^{2}\\Gamma(n-2)}, \\qquad n> 2 \\\\\n              \\frac{1}{\\pi^2}, \\hspace{16mm} n=2\n\\end{cases}\n\\end{equation}\n\\subsection{Contact Diagrams}\nOne starts from the quartic lagrangian\n\\begin{eqnarray}\n\\mathcal{L}_{4}&=&-\\frac{1}{4}(C^{1234}-S^{1234})s_{2}^{1}\\nabla_{\\mu}s_{2}^{2}s_{n}^{3}\\nabla^{\\mu}s_{n}^{4}\n\\nonumber \\\\\n&+&\\frac{1}{8}n(-\\delta_{2}^{12}\\delta_{n}^{34}+(6+n)C^{1234}+(3n-4)S^{1234}\n-n\\Upsilon^{1234})s_{2}^{1}s_{2}^{1}s_{n}^{3}s_{n}^{4}\n\\label{quartic}\n\\end{eqnarray}\nWe record the useful identity\n\\begin{equation}\nD_{\\mu}K_{\\Delta_{1}}(z,\\vec{x}_{1})D^{\\mu}K_{\\Delta_{2}}(z,\\vec{x}_{2}) =\\Delta_{1}\\Delta_{2}\n(K_{\\Delta_{1}}(z,\\vec{x}_{1})K_{\\Delta_{2}}(z,\\vec{x}_{2})-2|\\vec{x}_{12}|^{2}K_{\\Delta_{1}+1}(z,\\vec{x}_{1})K_{\\Delta_{2}+1}(z,\\vec{x}_{2}))\n\\end{equation}\nUsing this expression and the definition of the $D$-functions, we see that the contribution to the amplitude from\nthe quartic lagrangian is given by\n\\begin{eqnarray}\n\\mathcal{L}_{4}&=&-\\frac{1}{4}(C^{1234}-S^{1234})(2n D_{22nn}-4n |\\vec{x}_{24}|^{2}D_{23nn+1})\n\\nonumber \\\\\n&+&\\frac{1}{8}n(-\\delta^{12}_{2}\\delta^{34}_{n}+(6+n)C^{1234}+(3n-4)S^{1234}-n\\Upsilon^{1234})D_{22nn}\n\\end{eqnarray}\nwhere again an overall factor of $C(n)^{2}C(2)^{2}$ was omitted, but should be included. We can rewrite this\nexpression in terms of the $\\bar{D}$-functions\n\\begin{eqnarray}\n\\mathcal{L}_{4}&=&\\pi^2\\frac{(C(2)C(n))^{2}}{\\Gamma(n)}\\frac{u^2}{|\\vec{x}_{12}|^{4}|\\vec{x}_{34}|^{2n}}\\left[-\\frac{n}{4}(C^{1234}-S^{1234})(\n\\bar{D}_{22nn}-\\bar{D}_{23nn+1})\\right.\n\\nonumber \\\\\n&+&\\left.\\frac{1}{2^4}n(-\\delta^{12}_{2}\\delta^{34}_{n}+(6+n)C^{1234}+(3n-4)S^{1234}-n\\Upsilon^{1234})\\bar{D}_{22nn}\\right]\n\\label{quarticfinal}\n\\end{eqnarray}\nThe final result for the on-shell action is then given by substituting the expressions for the exchange amplitudes on equations (\\ref{s-channel}) and (\\ref{t-channel}) and by equation (\\ref{quarticfinal}).\n\\subsection{Results for the Four-Point Function}\nWe collect the results for the relevant on-shell action. First we write down the part of the lagrangian that contributes to the four-point function of interest\n\\begin{eqnarray}\n\\mathcal{L}_{on-shell}&=&-n(n-1)\\langle C_{2}^{1}C_{2}^{2}C_{2}^{5}\\rangle \\langle C_{n}^{3}C_{n}^{4}C_{2}^{5}\\rangle\n\\int [dz] s^{1}_{2}(z)s^{2}_{2}(z)G(z,w)s^{3}_{2}(w)s^{4}_{2}(w) \n\\nonumber \\\\\n&-&n^2(n-1)^2\\langle C_{2}^{1}C_{n}^{3}C_{n}^{5}\\rangle \\langle C_{2}^{2}C_{n}^{4}C_{n}^{5}\\rangle\n\\int [dz] s^{1}_{2}(z)s^{3}_{n}(z)G(z,w)s^{2}_{2}(w)s^{4}_{n}(w) \n\\nonumber \\\\\n&-&\\frac{n}{2^4}\\langle C^{1}_{2}C^{2}_{2}C^{5}_{[1,0,1]} \\rangle \\langle C^{3}_{n}C^{4}_{n}C^{5}_{[1,0,1]} \\rangle\n\\int [dz]s^{1}_{2}(z)\\buildrel\\leftrightarrow\\over\\nabla^{\\mu}s^{2}_{2}(z)\nG_{\\mu\\nu}(z,w)s_{n}^{3}(w)\\buildrel\\leftrightarrow\\over\\nabla^{\\nu }s^{4}_{n}(w) \n\\nonumber \\\\ \n&-&\\frac{n(n-1)}{2^3}\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[1,n-1,1]} \\rangle \\langle C^{2}_{2}C^{4}_{n}C^{5}_{[1,n-1,1]} \\rangle\n\\int [dz]s_{2}^{1}(z)\\buildrel\\leftrightarrow\\over\\nabla^{\\mu}s_{n}^{3}(z)\nG_{\\mu\\nu}(z,w)s^{2}_{2}(w)\\buildrel\\leftrightarrow\\over\\nabla^{\\nu}s_{n}^{4}(w) \n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n&-&\\frac{1}{2^4} \\langle C^{1}_{2}C^{2}_{2}C^{5}_{[0,0,0]} \\rangle \\langle C^{3}_{3}C^{4}_{3}C^{5}_{[0,0,0]} \\rangle\n\\int [dz] T^{\\mu\\nu}_{22}(z)G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}_{nn}(w) \n\\nonumber \\\\\n&-&\\frac{1}{2^3}\\langle C^{1}_{2}C^{3}_{n}C^{5}_{[0,n-2,0]} \\rangle \\langle C^{2}_{2}C^{4}_{n}C^{5}_{[0,n-2,0]} \\rangle\n\\int [dz] T^{\\mu\\nu}_{2n}(z)G_{\\mu\\nu\\mu'\\nu'}(z,w)T^{\\mu'\\nu'}_{2n}(w) \n\\nonumber \\\\\n&-&\\frac{1}{2^2}(C^{1234}-S^{1234})s_{2}^{1}(w)\\nabla_{\\mu}s_{2}^{2}(w)s_{n}^{3}(w)\\nabla^{\\mu}s_{n}^{4}(w)\n\\nonumber \\\\\n&+&\\frac{1}{2^3}n\\left(-\\delta^{12}_{2}\\delta^{34}_{n}+(6+n)C^{1234}+(3n-4)S^{1234}-n\\Upsilon^{1234}\\right)s_{2}^{1}(w)s_{2}^{1}(w)s_{n}^{3}(w)s_{n}^{4}(w)\n\\end{eqnarray}\nWe now substitute the summation of overlapping $SO(6)$ tensors (see appendix \\ref{sec:SphereInts}) and use the results for the exchange integrals. After relabelling the indices, one finally gets the on-shell value of the action that determines the four-point function \n\\begin{eqnarray}\n\\mathcal{S}&=&\\frac{N^{2}}{8\\pi^{2}}\\frac{(n-1)^2(n-2)^2}{4\\pi^{6}\\Gamma(n)}\n\\int d^{4}\\vec{x}_{1}d^{4}\\vec{x}_{2}d^{4}\\vec{x}_{3}d^{4}\\vec{x}_{4}\ns_{2}^{1}(\\vec{x}_{1})s_{2}^{2}(\\vec{x}_{2})s_{n}^{3}(\\vec{x}_{3})s_{n}^{4}(\\vec{x}_{4})\n\\frac{u}{|\\vec{x}_{12}|^{4}|\\vec{x}_{34}|^{2n}}\\left\\{\\right.\n\\nonumber \\\\\n&+&\\delta^{12}_{2}\\delta^{34}_{n}\\frac{n}{2^{5}}\\left[\\bar{D}_{11nn}-(n+1)u\\bar{D}_{22nn}-(1+v-u)\\bar{D}_{22n+1n+1}\\right]\n\\nonumber \\\\\n&+&C^{1234}\\frac{n}{2^4}\\left[-2\\bar{D}_{11nn}-2(n-1)u\\bar{D}_{12n-1n}+(n+6)u\\bar{D}_{22nn}-2\\bar{D}_{21nn+1}+2\\bar{D}_{12nn+1}\\right.\n\\nonumber \\\\\n&-&\\left.(u\\bar{D}_{31nn}-(n-1)u^2\\bar{D}_{23n-1n})-u((n-1)\\bar{D}_{22n-1n+1}-\\bar{D}_{12nn+1})\\right]\n\\nonumber \\\\\n&+&C^{1243}\\frac{n}{2^2}\\left[u\\bar{D}_{23nn+1}-u\\bar{D}_{22nn}\\right]\n\\nonumber  \\\\\n&+&\\Upsilon^{1234}\\frac{n}{2^4(n+2)}\\left[\\frac{2(n-1)^{2}(n+2)}{(n+1)}u\\bar{D}_{12n-1n}+(n-2)(u\\bar{D}_{31nn}-(n-1)u^2\\bar{D}_{23n-1n})\\right.\n\\nonumber \\\\\n&+&(n-2)u((n-1)\\bar{D}_{22n-1n+1}-\\bar{D}_{12nn+1})-n(n+2)u\\bar{D}_{22nn}\n\\nonumber \\\\\n&+&\\left.\\frac{2}{n+1}(n(n-1)u^2\\bar{D}_{33n-1n+1}+2u\\bar{D}_{31n+1n+1}+2n(1-u-v)u\\bar{D}_{23nn+1})\\right]\n\\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\n&+&S^{1234}\\frac{n}{2^4}\\left[-2(n-1)^2u\\bar{D}_{12n-1n}+3nu\\bar{D}_{22nn}\n-4u\\bar{D}_{23nn+1}\\right.\n\\nonumber \\\\\n&+&\\left.\\left.(u\\bar{D}_{31nn}-(n-1)u^2\\bar{D}_{23n-1n})+((n-1)u\\bar{D}_{22n-1n+1}-u\\bar{D}_{12nn+1})\\right]\n\\right\\}\n\\end{eqnarray}\nHere we have made use of some identities relating $\\bar{D}$-functions (appendix \\ref{sec:Dfunc}) to simplify the expressions. Notice that here we are abusing of the notation, as the scalar fields now refer to the boundary sources, and so depend on the $\\vec{x}_i$ coordinates. We are now ready to compute the four-point function (\\ref{diffweightprocess}) using the AdS\/CFT prescription given in (\\ref{prescriptionadscft}). Of course, we need first to canonically normalise the corresponding 1\/2-BPS operators, taking into account the rescaling we did to the action at the beginning of this computation\n\\begin{equation}\n\\tilde{s}^{I}_{n}=\\frac{N}{4\\pi^{2}}(n-2)^{1\/2}(n-1)s^{I}_{n}\n\\qquad \\qquad\n\\tilde{s}_{2}^{I}=\\frac{N}{4\\sqrt{2}\\pi^2}s^{I}_{2}\n\\end{equation}\nThis implies that the connected piece of the four-point function is of order $\\mathcal{O}(1\/N^2)$. The explicit form can be determined from\n\\begin{equation}\n\\langle \\mathcal{O}_{2}(\\vec{x}_{1})\\mathcal{O}_{2}(\\vec{x}_{2})\\mathcal{O}_{n}(\\vec{x}_{3})\\mathcal{O}_{n}(\\vec{x}_{4})\\rangle=\n\\frac{2^{9}\\pi^{8}}{N^4}\\frac{1}{(n-2)(n-1)^2}\n\\frac{\\delta}{\\delta s_{2}^{I_{1}}(\\vec{x}_{1})}\\frac{\\delta}{\\delta s_{2}^{I_{2}}(\\vec{x}_{2})}\n\\frac{\\delta}{\\delta s_{n}^{I_{3}}(\\vec{x}_{3})}\\frac{\\delta}{\\delta s_{n}^{I_{4}}(\\vec{x}_{4})}(-S)\n\\end{equation}\nUpon functional differentiation, the contribution to the amplitude from each of the tensor structures will be given by the corresponding orbit, this is, the $s$, $t$ and $u$ channels obtained by independent permutations of the points $1 \\leftrightarrow 2$, $3 \\leftrightarrow 4$. Here we make use of the symmetries of the $SO(6)$ tensors, so the final result reads as follows\n\\begin{eqnarray}\n&&\\langle \\mathcal{O}_{2}^{I_{1}}(\\vec{x}_{1})\\mathcal{O}_{2}^{I_{2}}(\\vec{x}_{2})\\mathcal{O}_{n}^{I_{3}}(\\vec{x}_{3})\\mathcal{O}_{n}^{I_{4}}(\\vec{x}_{4})\\rangle\n=\\frac{1}{{\\vec{x}_{12}}^4{\\vec{x}_{34}^{2n}}}\\left\\{A(u,v)\\delta^{I_1I_2}_{2}\\delta^{I_3I_4}_{n}+B_{1}(u,v)C^{I_{1}I_{2}I_{3}I_{4}}\n\\right.\n\\nonumber \\\\\n&&\\left.+B_{2}(u,v)C^{I_{1}I_{2}I_{4}I_{3}}+c_{1}(u,v)\\Upsilon^{I_{1}I_{2}I_{3}I_{4}}+C_{2}(u,v)\\Upsilon^{I_{1}I_{2}I_{4}I_{3}}\n+D(u,v)S^{I_{1}I_{2}I_{3}I_{4}}\\right\\}\n\\label{resultdiffweightsugra}\n\\end{eqnarray}\nwhere the functions $(A, B_{1}, B_{2}, C_{1}, C_{2}, D)$ are given by \n\\begin{equation}\n(A, B_{1}, B_{2}, C_{1}, C_{2}, D)=\\frac{2^{4}(n-2)}{\\Gamma(n)}\\frac{1}{N^2}(\\tilde{A},\\tilde{B}_{1},\\tilde{B}_{2},\\tilde{C}_1,\\tilde{C}_2, \\tilde{D})\n\\end{equation}\nand\n\\begin{eqnarray}\n\\tilde{A}(u,v)&=&\n-\\frac{n}{2^3}u\\left\\{\\bar{D}_{11nn}-(n+1)u\\bar{D}_{22nn}-(1+v-u)\\bar{D}_{22n+1n+1}\\right\\}\n\\nonumber \\\\\n\\tilde{B}_1(u,v)&=&\n-\\frac{n}{2^3}u \\left\\{-2\\bar{D}_{11nn}-2(n-1)u\\bar{D}_{12n-1n}\n-2(\\bar{D}_{21nn+1}-\\bar{D}_{21n+1n})+(n+6)u\\bar{D}_{22nn}\\right.\n\\nonumber \\\\\n&-&(u\\bar{D}_{31nn}-(n-1)u^2\\bar{D}_{23n-1n})-((n-1)u\\bar{D}_{22n-1n+1}-u\\bar{D}_{12nn+1})\n\\nonumber \\\\\n&-&\\left.4u(\\bar{D}_{22nn}-\\bar{D}_{32nn+1})\\right\\}\n\\nonumber \\\\\n\\tilde{B}_2(u,v)&=&\n-\\frac{n}{2^3}u\\left\\{-2\\bar{D}_{11nn}-2(n-1)u\\bar{D}_{12nn-1}\n-2(\\bar{D}_{12nn+1}-\\bar{D}_{21nn+1})+(n+6)u\\bar{D}_{22nn}\\right.\n\\nonumber \\\\\n&-&(u\\bar{D}_{13nn}-(n-1)u^2\\bar{D}_{23nn-1})-((n-1)u\\bar{D}_{22n+1n-1}-u\\bar{D}_{12n+1n})\n\\nonumber \\\\\n&-&\\left.4u(\\bar{D}_{22nn}-\\bar{D}_{23nn+1})\\right\\}\n\\nonumber \\\\\n\\tilde{C}_1(u,v)&=&\n-\\frac{n}{2^3(n+2)}u^2\\left\\{\\frac{2(n-1)^2(n+2)}{(n+1)}\\bar{D}_{12n-1n}\n-n(n+2)\\bar{D}_{22nn} \\right.\n\\nonumber \\\\\n&+&\\frac{2n(n-1)}{n+1}u\\bar{D}_{33n-1n+1}\n+\\frac{4}{n+1}\\bar{D}_{31n+1n+1}+\\frac{4n}{n+1}(1-u-v)\\bar{D}_{23nn+1}\n\\nonumber \\\\\n&+&\\left.(n-2)((n-1)\\bar{D}_{22n-1n+1}-\\bar{D}_{12nn+1})+(n-2)(\\bar{D}_{31nn}-(n-1)u\\bar{D}_{23n-1n})\\right\\}\n\\nonumber \\\\\n\\tilde{C}_2(u,v)&=&\n-\\frac{n}{2^3(n+2)}u^2\\left\\{\\frac{2(n-1)^2(n+2)}{(n+1)}\\bar{D}_{12nn-1}\n-n(n+2)\\bar{D}_{22nn}\\right.\n\\nonumber \\\\\n&+&\\frac{2n(n-1)}{n+1}u\\bar{D}_{33n+1n-1}\n+\\frac{4}{n+1}\\bar{D}_{13n+1n+1}+\\frac{4n}{n+1}(v-u-1)\\bar{D}_{23n+1n}\n\\nonumber \\\\\n&+&\\left.(n-2)((n-1)\\bar{D}_{22n+1n-1}-\\bar{D}_{12n+1n})+(n-2)(\\bar{D}_{13nn}-\n(n-1)u\\bar{D}_{23nn-1}\\right\\}\n\\nonumber \\\\\n\\tilde{D}(u,v)&=&\n-\\frac{n}{2^3}u^2\\left\\{-2(n-1)^2(\\bar{D}_{12n-1n}+\\bar{D}_{12nn-1})+6n\\bar{D}_{22nn}-4(\\bar{D}_{23nn+1}+\\bar{D}_{32nn+1})\\right.\n\\nonumber \\\\\n&+&(n-1)(\\bar{D}_{22n-1n+1}+\\bar{D}_{22n+1n-1})-(\\bar{D}_{12nn+1}+\\bar{D}_{12n+1n})\n\\nonumber \\\\\n&+&\\left.(\\bar{D}_{31nn}+\\bar{D}_{13nn})-(n-1)u(\\bar{D}_{23n-1n}+\\bar{D}_{23nn-1})\\right\\}\n\\label{orbits}\n\\end{eqnarray}\nFrom (\\ref{orbits}) it is possible to see that the crossing symmetries are respected and that the overall form of the four-point amplitude is consistent with conformal symmetry.\n\\section{Verifying the CFT Predictions}\n\\label{sec:verifying}\nWe now try to verify the dynamical constraints imposed on the amplitude by the insertion procedure, on the supergravity result. To do this, we need to rewrite the result (\\ref{resultdiffweightsugra}) in a simpler way. We will follow the notation in \\cite{Rayson:2007th}, which is based on ideas developed in \\cite{Nirschl:2004pa, Dolan:2003hv} and introduce the conformal invariants\n\\begin{equation}\n\\sigma=\\frac{u_1\\cdot u_3 u_2\\cdot u_4}{u_1\\cdot u_2 u_3 \\cdot u_4} \\qquad \\qquad \\tau=\\frac{u_1 \\cdot u_4 u_2 \\cdot u_3}{u_1 \\cdot u_2 u_3 \\cdot u_4} \n\\label{ucrossrad1}\n\\end{equation}\nso the four-point function  (\\ref{diffweightprocess}) is given by\n\\begin{equation}\n \\langle \\mathcal{O}_{2}(\\vec{x}_1,u_1)\\mathcal{O}_{2}(\\vec{x}_2,u_2)\\mathcal{O}_{n}(\\vec{x}_3,u_3)\\mathcal{O}_{n}(\\vec{x}_4,u_4)\\rangle =\\left(\\frac{u_1.u_2}{|\\vec{x}_{12}|^2}\\right)^{2}\\left(\\frac{u_3.u_4}{|\\vec{x}_{34}|^2}\\right)^{n}\\mathcal{G}^{(2,2,n,n)}(u,v;\\sigma,\\tau)\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathcal{G}^{(2,2,n,n)}(u,v;\\sigma,\\tau)=\\mathcal{G}_{0}(u,v;\\sigma,\\tau)+s(u,v;\\sigma,\\tau)\\mathcal{H}_{I}(u,v;\\sigma,\\tau)\n\\label{generalstrucG}\n\\end{equation}\n$\\mathcal{H}_I$ contains all the non-trivial dynamic contributions and $\\mathcal{G}_{0}$ is the free field part, which has the following structure\n\\begin{equation}\n\\mathcal{G}_{0}(u,v;\\sigma,\\tau)=k+G_{f}(u,v;\\sigma,\\tau)+s(u,v;\\sigma,\\tau)\\mathcal{H}_{0}(u,v,\\sigma,\\tau)\n\\end{equation}\nIn these expressions\n\\begin{equation}\ns(u,v;\\sigma,\\tau)=v+\\sigma^2 uv +\\tau^2 u +\\sigma v (v-1-u)+\\tau (1-u-v) + \\sigma \\tau u (u-1-v)\n\\end{equation}\nThe free field term in the $22\\rightarrow nn$ channel is given by the expression \\cite{Rayson:2007th,Nirschl:2004pa, Dolan:2003hv}\n\\begin{equation}\n\\label{freeg}\n\\mathcal{G}_0(u,v;\\sigma,\\tau)=1+b_1\\left(\\sigma u  +\\tau \\frac{u}{v}\\right)+c_1\\left(\\sigma^2 u^2 +\\tau^2 \\frac{u^2}{v^2}\\right)+d \\sigma \\tau \\frac{u}{v}\n\\end{equation}\nwith $b_1, c_1$ and $d$ are given in (\\ref{largeNfreecoeff}). The $2n \\rightarrow 2n$ channel can be obtained using crossing symmetry. From (\\ref{resultdiffweightsugra}), one can read the expression in the interacting theory \n\\begin{equation}\n\\mathcal{G}(u,v;\\sigma,\\tau)=a(u,v)+\\left(\\sigma u b_{1}(u,v)+\\tau\\frac{u}{v}b_{2}(u,v) \\right)+\\left(\\sigma^2 u^2 c_{1}(u,v)+\\tau^2\\frac{u^2}{v^2}c_{2}(u,v)\\right)+\\sigma\\tau \\frac{u^2}{v}d(u,v)\n\\label{intg}\n\\end{equation}\nwhere $a(u,v)=A(u,v)$, $b_1(u,v)=\\frac{B_{1}(u,v)}{u}$, $c_1(u,v)=\\frac{C_{1}(u,v)}{u^2}$ and $d(u,v)=\\frac{v}{u^2}D(u,v)$. $b_2(u,v)$ and $c_2(u,v)$ can be obtained from crossing symmetry, as the supergravity result (\\ref{largeNfreecoeff}) satisfies this property. Notice also that the cross-ratios $\\sigma$ and $\\tau$ defined in (\\ref{ucrossrad1}) arise naturally from expressing the products of $C$-tensors in terms of harmonic polynomials (see appendix \\ref{sec:HarmPoly}).\n\nIt is possible to rewrite (\\ref{intg}) by simplifying the result (\\ref{largeNfreecoeff}), using identities between $\\bar{D}$-functions (see appendix \\ref{sec:Dfunc}). The simplification was done in \\cite{Rayson:2007th} and we reproduce it here. One gets\n\\begin{equation}\n\\mathcal{G}(u,v;\\sigma,\\tau)=1+\\frac{2n}{N^2}\\left(\\sigma u +\\tau \\frac{u}{v}+(n-1)\\sigma\\tau \\frac{u^2}{v}-\\frac{1}{(n-2)!}s(u,v;\\sigma,\\tau)u^n\\bar{D}_{nn+222}(u,v) \\right)\n\\end{equation}\nwhere the disconnected piece has been normalised to 1. In the free field limit, $\\mathcal{G}\\rightarrow \\mathcal{G}_0$, so comparing (\\ref{freeg}) with (\\ref{intg}) one has\n\\begin{equation}\na(u,v)\\rightarrow 1 \\qquad b_i(u,v) \\rightarrow b_i \\qquad c_i(u,v) \\rightarrow c_i \\qquad d(u,v) \\rightarrow d\n\\end{equation}\nfrom where we can identify $k=1+(n+1)b_i+2c_i$ and from (\\ref{generalstrucG}) one sees that\\footnote{This can be read of from $a(u,v)$ as its connected piece has no free field contributions.}\n\\begin{equation}\n\\mathcal{H}_I(u,v)=-\\frac{2n}{N^2}\\frac{1}{(n-2)!}u^n \\bar{D}_{nn+222}(u,v)\n\\label{Hint}\n\\end{equation}\nIn the $2n\\rightarrow 2n$ channel the previous expression reads\n\\begin{equation}\n\\hat{\\mathcal{H}}_I(u,v)=-\\frac{2n}{N^2}\\frac{1}{(n-2)!}u^2 \\bar{D}_{2n+22n}(u,v)\n\\end{equation}\nIt is now clear that one can write\n\\begin{eqnarray}\na(u,v)&=&1+v \\mathcal{H}_I(u,v) \\hspace{28.mm} d(u,v)=d+\\frac{v}{u}(u-v-1)\\mathcal{H}_I(u,v)\n\\nonumber\\\\\nb_{1}(u,v)&=&b_1+\\frac{v}{u}(v-u-1)\\mathcal{H}_I(u,v) \\qquad b_{2}(u,v)=b_2+\\frac{v}{u}(1-u-v)\\mathcal{H}_I(u,v)\n\\nonumber \\\\\nc_{1}(u,v)&=&c_1+\\frac{v}{u}\\mathcal{H}_I(u,v)  \\hspace{25.mm} c_{2}(u,v)=c_2+\\frac{v^2}{u}\\mathcal{H}_I(u,v)\n\\label{splitcoeff}\n\\end{eqnarray}\nso the supergravity result also splits into a free and a quantum part, as it was predicted by superconformal symmetry. Defining \n\\begin{equation}\n\\mathcal{H}_I(u,v)=\\frac{u}{v}\\mathcal{F}(u,v)\n\\end{equation}\nit becomes clear that the relations (\\ref{insertrels}) are satisfied. We consider this fact as a strong evidence in favour of the AdS\/CFT correspondence. \n\nWe can also read off the values of the coefficients $b_i, c_i$ and $d$ from the free part of the function $\\mathcal{G}(u,v;\\sigma,\\tau)$. The results are\n\\begin{equation}\nb_i=\\frac{2n}{N^2} \\qquad c_i=0 \\qquad d = \\frac{2n(n-1)}{N^2} \n\\label{cizero}\n\\end{equation}\nNotice that the values of $b_i$ and $d$ agree with those computed using free field theory. This is a highly non-trivial result. However, $c_i$ vanish, which is apparently at odds with what was obtained using free fields, but recall that $c_i$ was dependent on the colour structure of the operators. This might suggest that this quantity receives quantum corrections. It should also be noticed that in the case $n=3$, one has $c_i=0$ so there is agreement \\cite{Berdichevsky:2007xd}. \n\n\\section{Conclusions and Outlook}\n\\label{sec:conclusions}\nIn this paper, we have investigated four-point functions of different weight operators in the context of the AdS\/CFT correspondence. We have looked at a specific computations in the supergravity approximation (large $\\lambda$, large $N$), of a process involving fields dual to primaries of conformal dimension $2$ and primaries of conformal dimension $n$. The results have been analysed using results from free field Yang-Mills theory and superconformal symmetry. Some of our key results are summarised below:\n\\begin{itemize}\n\\item The connected piece of the four-point function of 1\/2-BPS superconformal primaries of conformal weights 2 and $n$, was shown to have a structure that is consistent with superconformal symmetry. Moreover, we have seen it naturally separates into a free and an interacting (quantum) piece, which involves all the non-trivial dynamics and satisfies the restrictions imposed by the insertion procedure. \n\n\\item A new method was used for evaluating effective couplings in the lagrangian arising from integrals over $S^5$. This allowed the determination of the on-shell lagrangian for KK scalars dual to superconformal primaries in the YM side. \n\n\\item We provided further evidence for the possibility that the quartic four-derivative Lagrangian of \\cite{Arutyunov:1999fb} vanishes, as now we have extended the computation of the lagrangian to include primaries with different conformal weights, with two of them being generic (i.e. no specification of the representation content). As it has been argued before in \\cite{Arutyunov:2002fh, Arutyunov:2003ae}, this would imply the existence of a $\\sigma$-model action describing the extension of  $d=5$ $\\mathcal{N}=8$ supergravity to include massive KK modes of the IIB compactification.\n\\end{itemize}\nWith the techniques developed in appendix \\ref{sec:HarmPoly} to compute the interaction couplings arising from the products of $C$-tensors, it seems likely that the computation of the correlation function \n\\begin{equation*}\n\\langle \\mathcal{O}_{n_1}(\\vec{x}_1)\\mathcal{O}_{n_1}(\\vec{x}_2)\\mathcal{O}_{n_2}(\\vec{x}_3)\\mathcal{O}_{n_2}(\\vec{x}_4)\\rangle\n\\end{equation*} \nin AdS supergravity could be evaluated.  This would give us further information on the dynamics of KK scalars, and would provide additional evidence for the vanishing of the quartic four-derivative lagrangian in the five-dimensional effective theory.\n\nAnother problem one could explore is the effect of $\\mathcal{R}^4$ corrections to four-point functions of superconformal primary operators. Recalling that the dual fields are built from the trace of the graviton in the $S^5$ and the RR four-form on $S^5$ and given that all the terms at order $\\alpha'^3$ involving the metric and the four-form are known from \\cite{Paulos:2008tn}, it is conceivable that the corrections to the five-dimensional effective lagrangian can be obtained. This indeed would be a difficult task, but a first step would be to consider the case of lowest scale dimension primaries($\\Delta=2$). In this way, it should be possible to compute the order $(g_{YM}^2N)^{-3\/2}$ correction to the four-point function of lowest weight primaries.  \n\nA puzzle that remains to be addressed is the mismatch of the $c_i$ coefficient function from the supergravity computation, eq. (\\ref{cizero}), and the free-field theory one,  eq. (\\ref{largeNfreecoeff}). Given that the supergravity result gives $c_i=0$, one might imagine that there should be stringy corrections to this quantity. Corrections in $\\alpha'$ could be considered once the higher order corrections to the five-dimensional effective action are known. Another interesting avenue would be to consider the potential contribution coming from non-perturbative effects \\cite{Green:2002vf}.\n\nFinally it should be mentioned that the supergravity result obtained here can be used to ana\\-lyse the structure of the OPE of the primaries at strong coupling and to evaluate anomalous dimensions. Some results in this matter can be found in \\cite{Rayson:2007th}.\n\\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\\section{Introduction}\n\nIn a recent paper  \\cite{EHIIM11}, Eto et al. investigated electromagnetic\nproperties of baryons under the influence of external electromagnetic field,\nbased on the Skyrme model \\cite{Skyrme61} with Wess-Zumino-Witten term\nincluding electromagnetism \\cite{WZ71},\\cite{Witten83}, thereby\nconcluding that a nucleon in external electromagnetic\nfields has anomalous charge distribution due to the chiral anomaly.\nFurthermore, the Gell-Mann-Nishijima relation, $Q = I_3 + N_B \/ 2$\n($Q$ : electric charge, $I_3$ : the third component of isospin,\n$N_B$ : baryon number), acquires an additional term due to the quantum\nanomaly. As a consequence, non-zero net charge, which is generally\nnon-integer, is induced even for a neutron. This astounding conclusion\nstems from the gauged Wess-Zumino-Witten action with two flavors,\ngiven in the form \\cite{KRS84},\\cite{PR85} :\n\\begin{equation}\n S_{WZW} [U, A_\\mu] \\ = \\ - \\ e \\,\\int \\,d^4 x \\,A_\\mu \\,\n \\left(\\, \\frac{N_c}{6} \\,j^\\mu_B \\ + \\ \\frac{1}{2} \\,j^\\mu_{anm} \\,\\right),\n \\label{WZW1} \n\\end{equation}\nwhere\n\\begin{eqnarray}\n j^\\mu_B &=& - \\,\\frac{1}{24 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\n \\mbox{\\rm tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta) , \\label{current_B} \\\\\n j^\\mu_{anm} &=& \\frac{i \\,e \\,N_c}{96 \\,\\pi^2} \\,\n \\epsilon^{\\mu \\nu \\alpha \\beta} \\,F_{\\nu \\alpha} \\,\n \\mbox{\\rm tr} \\,\\tau_3 \\,(L_\\beta + R_\\beta) , \\label{current_anm}\n\\end{eqnarray}\nwith\n\\begin{equation}\n L_\\mu \\ \\equiv \\ U \\,\\partial_\\mu U^\\dagger, \\ \\ \\  \n R_\\mu \\ \\equiv \\ \\partial_\\mu U^\\dagger \\,U . \n\\end{equation}\n(We point out that our definition of $L_\\mu$ and $R_\\mu$ is different\nfrom that in \\cite{EHIIM11}.)\nHere, $j^\\mu_B$ is the well-known baryon current giving an integer baryon\nnumber \\cite{Witten83}. According to \\cite{EHIIM11}, in the presence of\nbackground electromagnetic fields, not only\nthe first term but also the second term of Eq.~(\\ref{WZW1}) is important.\nThe electric charge $Q$ with the contribution of anomaly is then written as\n\\begin{equation}\n Q \\ = \\ I_3 \\ + \\ \\frac{N_B}{2} \\ + \\ \\frac{Q_{anm}}{2} ,\n\\end{equation}\nwhere $N_B = \\int d^3 x \\,j^0_B$ and $Q_{anm} = \\int \\,d^3 x \\,j^0_{anm}$.\nThis means that the Gell-Mann-Nishijima relation receives\na remarkable modification under the background electromagnetic fields.\n\nIt appears to us, however, that the above-mentioned anomalous induction\nof non-zero net charge for a nucleon (or a Skyrmion) is not in good harmony\nwith the schematic physical picture illustrated in Fig.1 of their paper.\nThis schematic diagram represents electric charge generation of a nucleon\nthrough the anomalous coupling between one pion and two photons\n(or electromagnetic fields). Since the electromagnetic fields (as abelian\ngauge fields) carries no electric charge, the exchanged pion in this\nfigure must be neutral. In fact, this lowest order diagram results\nfrom the same vertex as describing the famous decay process\n$\\pi^0 \\rightarrow 2 \\,\\gamma$ due to the triangle\nanomaly \\cite{Adler69},\\cite{BJ69},\nwhich is legitimately contained in the\ngauged Wess-Zumino-Witten action \\cite{Witten83}\\nocite{KRS84}-\\cite{PR85}.\nNaturally, the gauged Wess-Zumino-Witten\naction also contains higher-power terms in the pion fields.\nHowever, even if one considers diagrams in which more pions are exchanged\nbetween the nucleon and the electromagnetic fields, the exchanged pions\nmust be electrically neutral as a whole, since the electromagnetic fields\ncarry no electric charge. \nWhat we are worrying about here is a conflict between this intuitive thought\nand the principle conclusion of the paper \\cite{EHIIM11}, i.e. the anomalous\ninduction of non-zero net charge for a nucleon.\n\nThe purpose of the present paper is to unravel the origin of this contradiction.\nHere, we unavoidably encounter the problem of how to define electromagnetic\nhadron current in an unambiguous manner by starting with the gauged\nWess-Zumino action.\nA subtlety arises from the fact that the gauged Wess-Zumino Witten action\ncontains nonlinear terms in the electromagnetic fields. In fact, if it contains\nonly linear terms in the electromagnetic fields, it is clear that one can easily\nread off the electromagnetic hadron current as a coefficient of the\nelectromagnetic field. For handling this delicate point, \nfirst in sect.II, we briefly analyze the familiar lagrangian of\nscalar electrodynamics containing couplings between photons and\ncomplex scalar fields, which is nonlinear in the photon fields.\nA particular emphasis here is put on how to define electromagnetic matter\ncurrent based on a solid guiding principle. It will be shown there that the two\nforms of current, i.e. the one defined on the basis of the Noether theorem\nand the other defined as a source current of the Maxwell equation through\nthe equations of motion, perfectly coincides with each other.\nIt is also shown that this current is gauge-invariant and conserved,\nthereby ensuring the consistency of scalar electrodynamics as a quantum\ngauge theory. In section III, we shall carry out a similar analysis for the\nnonlinear meson action with the Wess-Zumino-Witten action including\nelectromagnetism to find something unexpected, which is thought to be\nthe origin of the discrepancy pointed out above. Finally, in sect. IV,\nwe briefly summarize what we have found in the present paper. \n\n\n\\section{A lesson learned from scalar electrodynamics}\n\nLet us start with the familiar lagrangian of scalar electrodynamics given by\n\\begin{equation}\n {\\cal L} \\ = \\ - \\,\\frac{1}{4} \\,F_{\\mu \\nu} \\,F^{\\mu \\nu} \\ + \\ \n D_\\mu \\phi^* \\,D^\\mu \\phi \\ - \\ V (\\phi^* \\,\\phi) ,\n\\end{equation}\nwith\n\\begin{eqnarray}\n F_{\\mu \\nu} &\\equiv& \\partial_\\mu \\,A_\\nu \\ - \\ \n \\partial_\\nu A_\\mu , \\\\\n D^\\mu \\,\\phi (x) &\\equiv& [\\,\\partial^\\mu \\ + \\ i \\,e \\,A^\\mu (x) \\,] \\,\n \\phi(x) .\n\\end{eqnarray}\nThis lagrangian is manifestly gauge-invariant under the following gauge\ntransformation : \n\\begin{eqnarray}\n \\phi (x) &\\rightarrow& e^{- \\,i \\,e \\,\\alpha (x)} \\,\\phi (x), \\ \\ \\ \n A^\\mu (x) \\ \\rightarrow \\ A^\\mu (x) \\ + \\ \\partial^\\mu \\,\\alpha (x).\n \\label{SED_GT}\n\\end{eqnarray}\nThe equations of motion derived from the above lagrangian are given by\n\\begin{eqnarray}\n \\partial_\\mu \\,F^{\\mu \\nu} &=& j^\\nu , \\label{SED_Maxwell_eq}\\\\\n D_\\mu \\,D^\\mu \\,\\phi &=& - \\,\\partial V \\,\/ \\,\\partial \\phi^*, \\\\\n \\left( D_\\mu \\,D^\\mu \\,\\phi \\right)^* &=& - \\,\\partial V \\,\/ \\,\\partial \\phi .\n\\end{eqnarray}\nHere, the source current $j^\\nu$ of the Maxwell\nequation (\\ref{SED_Maxwell_eq}) is given by\n\\begin{equation}\n j^\\nu \\ = \\ i \\,e \\,\\left[\\, \\phi^* \\,D^\\mu \\,\\phi \\ - \\ \n (D^\\mu \\,\\phi)^* \\,\\phi \\,\\right] \\ = \\ \n i \\,e \\,\\phi^* \\,\\overleftrightarrow{\\partial^\\nu} \\,\\phi \\ - \\ \n 2 \\,e^2 \\,\\phi^* \\,\\phi \\,A^\\nu , \n\\end{equation}\nwith $\\overleftrightarrow{\\partial^\\nu} = \\overrightarrow{\\partial^\\nu} - \\overleftarrow{\\partial^\\nu}$.\nBy using the equations of motion, it can be shown that this matter current\n$j^\\nu$ is conserved, i.e.\n\\begin{equation}\n \\partial_\\nu \\,j^\\nu \\ = \\ 0 .\n\\end{equation}\nOne can also convince that this matter (or source) current is\ninvariant under the full gauge transformation (\\ref{SED_GT}).\nOne should recognize that the conservation of source current is crucially\nimportant. In fact, if it were broken, one encounters a serious contradiction\nwith the Maxwell equation (\\ref{SED_Maxwell_eq}) in such a way that\n\\begin{equation}\n 0 \\ = \\ \\partial_\\nu \\,\\partial_\\mu \\,F^{\\mu \\nu} \\ = \\ \n \\partial_\\nu \\,j^\\nu \\ \\neq \\ 0.\n\\end{equation}\nFor later discussion, it is useful to remember the fact that the matter current\nabove can also be obtained by using the standard Noether prescription.\nTo confirm it, we first consider the infinitesimal version of the gauge\ntransformation given by\n\\begin{eqnarray}\n \\delta \\phi \\ = \\ - \\,i \\,e \\,\\epsilon (x) \\, \\phi , \\ \\ \\ \n \\delta \\phi^* \\ = \\ i \\,e \\,\\epsilon (x) \\,\\phi^*, \\ \\ \\ \n \\delta A^\\mu \\ = \\ \\partial^\\mu \\epsilon (x).\n\\end{eqnarray}\nNaturally, the lagrangian of the scalar electrodynamics is invariant under\nthis gauge transformation.\nThe Noether current is obtained by considering another variation\n\\begin{eqnarray}\n \\delta^\\prime \\phi \\ = \\ - \\,i \\,e \\,\\epsilon (x) \\, \\phi , \\ \\ \\ \n \\delta^\\prime \\phi^* \\ = \\ i \\,e \\,\\epsilon (x) \\,\\phi^*, \\ \\ \\ \n \\delta^\\prime  A^\\mu \\ = \\ 0.\n\\end{eqnarray}\nThe variation of the whole lagrangian under this transformation\nis reduced to the form\n\\begin{equation}\n \\delta^\\prime {\\cal L} \\ = \\ \\epsilon (x) \\,\\partial_\\mu \\,J^\\mu \\ + \\ \n \\partial_\\mu \\epsilon (x) \\,J^\\mu,\n\\end{equation} \nwhich defines the current $J^\\mu$ such a way that \n\\begin{eqnarray}\n J^\\mu \\ &=& \\frac{\\partial (\\delta^\\prime {\\cal L})}{\\partial (\\partial_\\mu \\epsilon (x))}, \n \\label{SED_Noether_current} \\\\\n \\partial_\\mu \\,J^\\mu &=& \\ \\frac{\\partial (\\delta^\\prime {\\cal L})}{\\partial \\epsilon (x)}.\n\\end{eqnarray} \nIf the lagrangian ${\\cal L}$ is invariant under a space-time independent\n(global) transformation\n\\begin{eqnarray}\n \\delta^\\prime \\phi \\ = \\ - \\,i \\,e \\,\\epsilon \\, \\phi , \\ \\ \\ \n \\delta^\\prime \\phi^* \\ = \\ i \\,e \\,\\epsilon \\,\\phi^*, \n\\end{eqnarray}\nwhich is indeed the case with our lagrangian (1), we conclude that\n\\begin{equation}\n 0 \\ = \\ \\delta^\\prime {\\cal L} \\ = \\ \\epsilon \\,\\,\\partial_\\mu \\,J^\\mu.\n\\end{equation}\nThis means that the current $J^\\mu$ defined by the equation\n(\\ref{SED_Noether_current}) is in fact\na conserved Noether current. The above-explained method of obtaining\nthe Noether current is known as the Gell-Mann-Levy method.\n\nNow, for the lagrangian of the scalar electrodynamics, we have\n\\begin{eqnarray}\n \\delta^\\prime [\\,\\partial_\\mu \\,\\phi^* \\,\\partial^\\mu \\phi \\,]\n &=& i \\,e \\,\\partial_\\mu \\,\\epsilon (x) \\,\n [\\,\\phi^* \\,\\partial^\\mu \\,\\phi \\ - \\ \\partial^\\mu \\,\\phi^* \\,\\phi \\,], \\\\\n \\delta^\\prime \\,[\\,i \\,e \\,(\\,\\partial_\\mu \\,\\phi^* \\,\\phi \\ - \\ \n \\phi^* \\,\\partial_\\mu \\,\\phi \\,) \\,A^\\mu \\,] &=&\n \\partial_\\mu \\epsilon (x) \\,( - \\,2 \\,e^2 ) \\,\\phi^* \\,\\phi \\,A^\\mu , \\\\\n \\delta^\\prime \\,[\\,e^2 \\,A_\\mu \\,A^\\mu \\,\\phi^* \\,\\phi \\,] &=& 0 ,\n\\end{eqnarray}\nthereby being led to\n\\begin{equation}\n \\delta^\\prime \\,{\\cal L} \\ = \\ \\partial_\\mu \\,\\epsilon (x) \\,\n \\left\\{\\,i \\,e \\,[\\,\\phi^* \\,\\partial^\\mu \\,\\phi \\ - \\ \n (\\partial^\\mu \\,\\phi)^* \\,\\phi \\,]\n \\ - \\ 2 \\,e^2 \\,\\phi^* \\,\\phi \\,A^\\mu \\,\\right\\} .\n\\end{equation}\nThe resultant Noether current is therefore given by\n\\begin{eqnarray}\n J^\\mu &=& i \\,e \\,\\,\\phi^* \\,\\overleftrightarrow{\\partial^\\mu} \\,\\phi \n \\ - \\ 2 \\,e^2 \\,\\phi^* \\,\\phi \\,A^\\mu\n \\ = \\ \n i \\,e \\,[\\,\\phi^* \\,D^\\mu \\,\\phi \\ - \\ (D^\\mu \\,\\phi)^* \\,\\phi \\,].\n\\end{eqnarray}\nOne confirms that this conserved Noether current precisely coincides\nwith the source current appearing in the Maxwell\nequation (\\ref{SED_Maxwell_eq}) for the photon\nfield. This ensures the consistency of the scalar electrodynamics as a\nclassical and a quantum field theory. Somewhat embarrassingly, we shall\nsee below that the familiar gauged Wess-Zumino-Witten action does not\nsatisfy the same sense of consistency.\n\n\n\\section{electromagnetic hadron current resulting from the gauged\nWess-Zumino-Witten action}\n\n\\subsection{Matter current derived from Noether principle}\n\nHere, we start with the gauged Wess-Zumino-Witten action with two flavor\nexpressed in the following form :\n\\begin{eqnarray}\n S_{WZW} [\\,U, A_\\mu \\,] &=& S_{WZ} [U] \\ - \\ \n \\frac{1}{2} \\,e \\,\\int \\,d^4 x \\,A_\\mu\n \\nonumber \\\\\n &\\times& \\left\\{\\,- \\,\\frac{1}{24 \\,\\pi^2} \\,\n \\epsilon^{\\mu \\nu \\alpha \\,\\beta} \\,\n \\mbox{tr} \\,(L_\\nu \\, L_\\alpha \\,L_\\beta ) \\right. \\nonumber \\\\\n &\\,& \\ \\ \\left. + \\ \\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\n \\epsilon^{\\mu \\nu \\alpha \\beta} \\,\n F_{\\nu \\alpha} \\,\\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta) \\,\\right\\} ,\n \\label{WZW2}\n\\end{eqnarray}\nwhere $Q$ is the SU(2) charge matrix given as\n\\begin{equation}\n Q \\ = \\ \\left( \\begin{array}{cc}\n \\frac{2}{3} & 0 \\\\\n 0 & - \\,\\frac{1}{3} \\\\\n \\end{array} \\right) .\n\\end{equation}\n(As is well-known, although $S_{WZ} [U]$ vanishes in the SU(2) case,\nits gauge variation does not. We therefore retain it here.) \nBy construction, i.e. as a consequence of the\n``trial and error'' gauging {\\it a la} Witten \\cite{Witten83}, the gauged\nWess-Zumino-Witten action is invariant under the following\ninfinitesimal gauge transformation : \n\\begin{equation}\n \\delta U \\ = \\ i \\,\\epsilon (x) \\,[Q, U] , \\ \\ \n \\delta U^\\dagger \\ = \\ i \\,\\epsilon (x) \\,[Q, U^\\dagger], \\ \\ \n \\delta \\,A_\\mu \\ = \\ - \\,\\frac{1}{e} \\,\\,\\partial_\\mu \\,\\epsilon (x) .\n \\label{fullGT}\n\\end{equation}\nLet us first try to see what answer we shall obtain for the Noether current,\nif we apply the Gell-Mann-Levy method to the above lagrangian (\\ref{WZW2}).\nThe transformation, which we consider to this end, is given by\n\\begin{equation}\n \\delta^\\prime U \\ = \\ i \\,\\epsilon (x) \\,[Q, U] , \\ \\ \n \\delta^\\prime U^\\dagger \\ = \\ i \\,\\epsilon (x) \\,[Q, U^\\dagger], \\ \\ \n \\delta^\\prime \\,A_\\mu \\ = \\ 0 . \n\\end{equation}\nMaking use of the relation\n\\begin{eqnarray}\n \\delta^\\prime \\,S_{WZW} &=& - \\,\\int \\,d^4 x \\,\\,\\partial_\\mu \\epsilon (x) \\,\n \\frac{1}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\mbox{tr} \\,\n (L_\\nu \\,L_\\alpha \\,L_\\beta ) \\nonumber \\\\\n &\\,& - \\ \\int \\,d^4 x \\,\\,\\partial_\\nu \\,\\epsilon (x) \\,\\frac{3 \\,i \\,e}{48 \\,\\pi^2}\n \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,A_\\mu \\,\\partial_\\alpha \\,\n \\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta),\n\\end{eqnarray}\nwe readily find that the corresponding Noether current is given by\n\\begin{eqnarray}\n J^\\mu_{\\rm I} \\ \\equiv \\ \\frac{\\delta \\,(\\delta^\\prime \\,S_{WZW})}\n {\\delta (\\partial_\\mu \\,\\epsilon (x))} &=&\n - \\,\\frac{1}{48 \\,\\pi^2} \\,\\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n \\mbox{tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta) \\nonumber \\\\\n &\\,& + \\,\\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n A_\\nu \\,\\partial_\\alpha \\,\n \\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta) .\n\\end{eqnarray}\nOne can also verify that this current is invariant under the full\ngauge transformation (\\ref{fullGT}). Unfortunately, this current is not conserved.\nIn fact, we find that\n\\begin{eqnarray}\n \\partial_\\mu \\,J^\\mu_{\\rm I} &=& \\frac{3 \\,i \\,e}{24 \\,\\pi^2} \\,\\,\n \\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\partial_\\mu A_\\nu \\,\\,\n \\partial_\\alpha \\,\\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta) \\ \\neq \\ 0. \\label{div_current}\n\\end{eqnarray}\nHowever, one can verify that the r.h.s. of (\\ref{div_current}) is a total\nderivative of another four-vector as\n\\begin{equation}\n \\partial_\\mu \\,J^\\mu_{\\rm I} \\ = \\ \\partial_\\mu \\,X^\\mu,\n\\end{equation}\nwith\n\\begin{equation}\n X^\\mu \\ \\equiv \\ \\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\n \\epsilon^{\\mu \\nu \\alpha \\beta} \\,A_\\nu \\,\\partial_\\alpha \\,\n \\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta) .\n\\end{equation}\nThis means that, if we define another current $j^\\mu_{\\rm II}$ by\n\\begin{equation}\n J^\\mu_{\\rm II} \\ \\equiv \\ J^\\mu_{\\rm I} \\ - \\ X^\\mu \\ = \\ \n - \\,\\frac{1}{24 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\n \\mbox{tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta ),\n\\end{equation}\nthen, $J^\\mu_{\\rm II}$ is conserved. The price to pay is that the new current\n$J^\\mu_{\\rm II}$ is no longer gauge-invariant.\n\nIncidentally, in the case of Poincare symmetry not of internal symmetry,\nthe ambiguous nature of the Noether current is widely known.\nFor example, in the case of quantum\nchromodynamics (QCD), the 2nd-rank energy momentum tensor obtained from\na naive Noether procedure does not satisfy the desired symmetry property\nunder the exchange of two Lorentz indices \\cite{JM90}. However, there exists\na well-known procedure for ``improving'' the Noether current by adding\na superpotential - divergence of anti-symmetric tensor - which does not\nspoil the current conservation. The symmetric energy momentum tensor\nof QCD obtained in such a procedure is sometimes called Belinfante\nsymmetrized energy-momentum tensor. \n\nSummarizing the analysis in this subsection, we have applied the familiar\nGell-Mann-Levy method to the gauged Wess-Zumino-Witten action for\nobtaining a Noether current as a candidate of electromagnetic hadron\ncurrent. However, we have ended up\nwith two different forms of currents, i.e. $J^\\mu_{\\rm I}$ and $J^\\mu_{\\rm II}$.\nThe current $J^\\mu_{\\rm I}$ is gauge-invariant but not conserved, while\nthe current $J^\\mu_{\\rm II}$ is conserved but not gauge-invariant.\nAs pointed out in the paper by Son and Stephanov \\cite{SS08}, one can\nconstruct the 3rd current, which satisfies both of gauge-invariance and current\nconservation, by using the ``trial and error'' gauging method as proposed by\nWitten. It is given by\n\\begin{eqnarray}\n J^\\mu_{\\rm III} &=& - \\,\\frac{1}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\n \\mbox{tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta) \\nonumber \\\\\n &\\,& - \\,\\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\n \\partial_\\nu \\,[\\,A_\\alpha \\,\\mbox{tr} \\,Q \\,(L_\\beta + R_\\beta) \\,].\n\\end{eqnarray}\nUnfortunately, it is not a current derived from the gauged Wess-Zumino-Witten\naction on the basis of a definite prescription as guidelined by the Noether principle.\n\nIn this way, we must conclude that, quite different from the case of\nscalar electrodynamics, the standard Noether method does not do a\ndesired good job to derive the electromagnetic hadron current\ncorresponding to the gauged Wess-Zumino-Witten action, in the sense \nthat it fails to give a candidate of electromagnetic hadron current,\nsatisfying both of gauge-invariance and conservation.\nIn the next subsection, we shall investigate the nature of another candidate\nof electromagnetic hadron current, i.e. the source current, which is defined\nthrough the equation of motion for the electromagnetic field. \n\n\n\\subsection{Matter current as a source of Maxwell equation}\n\nThe full action of the two-flavor Skyrme model coupled to the\nelectromagnetic field $A_\\mu$ is given by\n\\begin{equation}\n S \\ = \\ S_\\gamma [A_\\mu] \\ + \\ S_{Skyrme} [U, A_\\mu] \\ + \\ \n S_{WZW} [U, A_\\mu] .\n\\end{equation}\nHere, the 1st term\n\\begin{equation}\n S_\\gamma \\ = \\ - \\,\\frac{1}{4} \\,\\int \\,d^4 x \\,\\,F_{\\mu \\nu} \\,F^{\\mu \\nu}\n\\end{equation}\nis the kinetic part of the electromagnetic field, while the 2nd term,\n$S_{Skyrme} [U,A_\\mu]$, stands for the non-anomalous part of action\nfor the two-flavor Skyrme model minimally coupled to the electromagnetic\nfield. The 3rd term, i.e. $S_{WZW} [U, A_\\mu]$, gives the gauged\nWess-Zumino-Witten action given by (\\ref{WZW2}).\nIn the following, we shall discard\nthe part $S_{Skyrme} [U, A_\\mu]$ for simplicity, since it plays no essential role\nin our discussion below. The Euler-Lagrange equation of motion for the\nelectromagnetic field is therefore written down from\n\\begin{equation}\n \\frac{\\delta}{\\delta A_\\nu} \\,\\left\\{\\,S_\\gamma [A_\\mu] \\ + \\ \n S_{WZW} [U, A_\\mu] \\,\\right\\} \\ = \\ 0 ,\n\\end{equation}\nwhich gives the Maxwell equation\n\\begin{equation}\n \\partial_\\mu \\,F^{\\mu \\nu} \\ = \\ j^\\nu ,\n\\end{equation}\nwith the definition of the source current $j^\\nu$ as\n\\begin{equation}\n e \\,j^\\nu \\ \\equiv \\ - \\,\\frac{\\delta}{\\delta A_\\nu} \\,S_{WZW} [U, A_\\mu] .\n\\end{equation}\n(Naturally, if we had included the part $S_{Skyrme} [U, A_\\mu]$, it would also\ncontribute to the source current of Maxwell equation. However, this part\nof current is conserved itself and does not cause any trouble as discussed\nbelow.) An immediate question is whether the above definition, given as a\nfunctional derivative of the gauged Wess-Zumino-Witten action with respect\nto the electromagnetic fields, offers us the same answer as obtained with\nthe Noether prescription. The answer is no.\nWe find that the source current is given by\n\\begin{eqnarray}\n j^\\mu &=& - \\,\\frac{1}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n \\mbox{\\rm tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta) \\nonumber \\\\\n &\\,& + \\,\\frac{3 \\,i \\,e}{96 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n F_{\\nu \\alpha} \\,\\mbox{\\rm tr} \\,Q \\,(L_\\beta + R_\\beta) \\nonumber \\\\\n &\\,& + \\,\\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n \\partial_\\nu \\,\\left[\\,\n A_\\alpha \\,\\mbox{\\rm tr} \\,Q \\,(L_\\beta + R_\\beta) \\,\\right] . \n \\label{source_current}\n\\end{eqnarray}\nwhich does not coincide with any of the currents\n$J^\\mu_{\\rm I}$, $J^\\mu_{\\rm II}$, and $J^\\mu_{\\rm III}$ discussed in the\nprevious subsection. Somewhat unexpectedly, it turns out that this current\n$j^\\mu$ is not gauge-invariant. More serious problem is that it\nis not conserved, owing to the presence of the 2nd term\nof (\\ref{source_current}). In fact, we find that\n\\begin{equation}\n \\partial_\\mu \\,j^\\mu \\ = \\ \\partial_\\mu \\,X^\\mu \\ \\neq \\ 0,\n\\end{equation}\nwith\n\\begin{equation}\n X^\\mu \\ = \\ \\frac{3 \\,i \\,e}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n A_\\nu \\,\\partial_\\alpha \\,\\mbox{\\rm tr} \\,Q \\,(L_\\beta + R_\\beta) .\n\\end{equation}\nAs emphasized in the example of scalar electrodynamics, non-conservation\nof source current is not permissible, since it causes an incompatibility\nwith the fundamental equation of electromagnetism, i.e.\nthe Maxwell equaion \\cite{Jackiw85}.\nHow can we make a compromise with this trouble. One possible attitude would\nbe to follow the argument as given by Kaymakcalan, Rajeev and Schechter many\nyears ago \\cite{KRS84}.\nThey argue that the low-energy effective action for QCD involves many more\nnew fields and interactions so one should not worry too much about the\ncomplete consistency of equation of motion. The effective action is, after all,\nbeing used as a handy mnemonic to read off the relevant vertices. The\ngauged Wess-Zumino-Witten action certainly describes the typical\nanomalous processes containing the photons like $\\pi^0 \\rightarrow 2 \\,\\gamma$\nand\/or $\\gamma \\rightarrow 3 \\,\\pi$ consistently with the low energy theorem,\ni.e. the anomalous Ward identities.\n\nNow we are in a position to pinpoint the origin of somewhat astounding conclusion\nobtained in the paper \\cite{EHIIM11}, i.e. the anomalous induction of net electric\ncharge for a nucleon in the magnetic fields. This conclusion follows from\nthe electromagnetic hadron current given as a half of the sum\nof $j^\\mu_B$ in (\\ref{current_B}) and $j^\\mu_{anm}$ in (\\ref{current_anm}).\nSetting $N_c = 3$, this reduces to\n\\begin{eqnarray}\n j^\\mu &=& - \\,\\frac{1}{48 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n \\mbox{\\rm tr} \\,(L_\\nu \\,L_\\alpha \\,L_\\beta) \\nonumber \\\\\n &\\,& + \\,\\frac{i \\,e \\,N_c}{192 \\,\\pi^2} \\,\\epsilon^{\\mu \\nu \\alpha \\beta} \\,\\,\n F_{\\nu \\alpha} \\,\\,\\mbox{\\rm tr} \\,\\,\\tau_3 \\,(L_\\beta + R_\\beta) .\n \\label{current_B+anm}\n\\end{eqnarray}\nIn consideration of the fact that $Q = \\frac{1}{6} + \\frac{\\tau_3}{2}$,\nthis current just coincides with the sum of the 1st and 2nd terms\nin the source current (47), which we have derived above.\nSince the 3rd term of the current (\\ref{source_current}) is of a\ntotal derivative form, it does not contribute to the net charge of a\nnucleon. We thus find that the 2nd term of the current (\\ref{source_current})\nor of the current (\\ref{current_B+anm}) is\nthe cause of trouble, which prevents\nthe conservation of source current of the Maxwell equation.\nIn any case, what we can say definitely from the analysis above is that\nthe anomalous induction of non-zero net charge for a nucleon (or a Skyrmion)\nclaimed in the paper \\cite{EHIIM11} is inseparably connected with\nthis unfavorable feature of the gauged Wess-Zumino-Witten action.\nStill, what is lacking in our understanding is a deep explanation of why the\ngauged Wess-Zumino-Witten action, which was constructed so as to fulfill the\nelectromagnetic gauge-invariance with use of the ``trial and error'' method,\ndoes not satisfy the consistency with the Maxwell equation. \n \nA final comment is on a related work by Kharzeev, Yee, and Zahed \\cite{KEZ11},\nwhich was done motivated by the paper \\cite{EHIIM11}.\nStarting with a simple effective lagrangian of QCD (it corresponds to\nthe lowest power term in the pion field in the gauged Wess-Zumino-Witten\naction), they investigated the effect of anomaly induced charge distribution\nin the nucleon. Under a certain kinematical approximation concerning the\nclassical equation of motion for the pion field in a nucleon, they\nconclude that the abelian anomaly of QCD induces a quadrupole moment\nfor a neutron but it does not induce net electric charge for it.\nThe last statement, i.e. no induction of net electric charge for a neutron\nappears to be consistent with the nature of their effective lagrangian\nand also with the intuitive consideration given in the introduction of\nthe present paper. \n\n\n\\section{Summary and conclusion}\n\nTo conclude, motivated by the recent claim that, under the external magnetic\nfields, the anomalous couplings between mesons and electromagnetic fields\ncontained in the gauged Wess-Zumino-Witten action induces non-zero\nnet electric-charge for a nucleon, we have carefully re-investigated the\nproblem of how to define the electromagnetic hadron current from this\nwidely-known action. To this end, we first compare the two methods of\nobtaining electromagnetic hadron current for the familiar lagrangian\nof the scalar electrodynamics. The one is the Gell-Mann-Levy\nmethod to obtain the Noether current, while the other is the method of\nusing equations of motion of actions to define source current.\nFor this standardly-known lagrangian, we confirm that these\ntwo methods give precisely the same form of the electromagnetic\nhadron current. It can also be verified that this current is gauge-invariant\nand conserved. Unfortunately, this is not the case with\nthe gauged Wess-Zumino-Witten action.\nThat is, the currents obtained by these two methods do not coincide with\neach other. Particularly troublesome here is the fact that the source current\nof Maxwell equation is not conserved. This means that\nthe gauged Wess-Zumino-Witten action, which was constructed so as to\nfulfill the electromagnetic gauge-invariance by using the ``trial and error''\nmethod, does not satisfy the consistency with the the fundamental equation\nof electromagnetism. Although mysterious, it seems at the least\nclear that the recently claimed anomalous induction of net electric charge\nfor a nucleon in the magnetic fields is inseparably connected with this\nunwelcome feature of the gauged Wess-Zumino-Witten action. \n\n\\begin{acknowledgments}\nThe author would like to thank Prof. T.~Kubota for useful discussion.\nThis work is supported in part by a Grant-in-Aid for Scientific\nResearch for Ministry of Education, Culture, Sports, Science\nand Technology, Japan (No.~C-21540268)\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLow magnetic field neutron star X-ray binaries (NSXBs) are often divided in two classes,\naccording to their correlated spectral and timing properties: the ``atoll'' and the ``Z''\nsources \\cite{13}, named after the shape of the track they draw in\nthe X-ray color-color diagrams (CDs) and the hardness-intensity diagrams (HIDs).\n\nZ sources are brighter than atoll sources and are believed to accrete at\nnear-Eddington luminosities (0.5-1.0 $L_{Edd}$, \\cite{34}). To date seven Galactic\nNSXBs have been classified as Z sources: six of them are persistent,\none of them is transient (XTE J1701-462, \\cite{16}; as the source faded to quiescence it\nswitched to atoll-type on the way down, \\cite{17}). They are\ncharacterized by three-branched tracks in their CDs and HIDs that in some cases resemble the\ncharacter ``Z''. The three branches are called horizontal branch (HB),\nnormal branch (NB) and flaring branch (FB).\nThe mass accretion rate, $\\dot{m}$, is assumed to drive the transitions between the branches,\nincreasing monotonically from the HB to the FB, affecting both the spectral and\nthe timing properties.\n\nPower density spectra (PDS) of the Z sources show several types of quasi periodic\noscillations (QPOs) and noise components \\cite{34}, whose presence and\nproperties are strongly correlated with the position of the source along the Z track\n\\cite{13}. Three types of low frequency QPOs ($<$100 Hz) are seen in the\nZ sources, each of them usually detected in just one of the three branches (but see \\cite{35},\n\\cite{36}); twin kHz QPOs have also been detected in all the Z sources (only marginally detected in\nXTE J1701-462, \\cite{16}).\nOther characteristic noise components of Z sources are the very low frequency noise (VLFN)\nand the low frequency noise (LFN).\n\nAll the Z-type NS sources are detected in the radio band, showing large and rapid variability, optically\nthin and optically thick emission. In 1998 it has been found for the first time (in GX 17+2)\nthat the radio emission varies as a function of the position in the X-ray CD \\cite{27}, decreasing with\nincreasing mass accretion rate from the HB (strongest radio emission) to the FB (weakest radio\nemission). Recently it has been suggested suggested that this behaviour could be universal \\cite{21}\n(but see \\cite{31} for GX 5-1). Extended radio \njets have been spatially resolved for Sco X-1 and have also\nbeen associated to ultra-relativistic ejections \\cite{11}.\n\n\\subsection{Cir X-1}\nCir X-1 was discovered in 1971 \\cite{20} and has been showing flares with a period of\n16.55 days, observed first in the X-ray band \\cite{19} and then in the infrared \\cite{12}, radio\n\\cite{14} and optical bands \\cite{22}: this fact is interpreted as enhanced accretion\nclose to the periastron passage of a highly eccentric binary orbit ($e \\, \\sim \\, 0.8$, \\cite{23},\n\\cite{24}). The source is located in the galactic plane at a\ndistance that has been reported to lie in the range 4-12 kpc (see \\cite{18} for\na recent discussion).\nMany properties of Cir~X-1 would suggest that this is a black hole candidate (BHC): it has\nstrong radio emission (e.g. \\cite{14}),\nultra-relativistic radio jets (the most relativistic detected so far within our galaxy, \\cite{9}),\nhard X-ray emission \\cite{6} and very strong X-ray variability \\cite{26}.\nThe first strong indication that the binary system harbours a neutron\nstar has been reported in 1986 \\cite{32}, when type-I X-ray bursts have been detected in\nEXOSAT data. After \\cite{32} no type-I X-ray bursts have been reported.\nShirey et al, in an extensive analysis of RXTE\/PCA data, identified typical low-frequency Z-source\nfeatures in Cir~X-1 PDS \\cite{29} and then a complete  Z track in its high-luminosity\norbital phases \\cite{30}. In 1999 it has been noted that the\ncharacteristic timing frequencies of Cir X-1 lie in between those typically associated with NS and BH\nsystems \\cite{28}. Recently twin kHz QPOs have been reported for the first time\nin Cir X-1 PDS \\cite{3}, a further indication of the nature of the compact object.\n\nCir X-1 lies within a radio nebula \\cite{33} and shows radio jets \nthat have been spatially resolved \\cite{8}; the radio nebula is \nproduced by synchrotron emission likely originating in the interaction \nbetween the jet and the interstellar medium. Recently \nevidence of an extended arcmin-scale X-ray jet around the source has been reported \\cite{15},\nin the same direction as the receding radio jet, making Cir X-1 the first secure neutron star\nsystem for which an extended X-ray jet has been resolved.\n\n\\section{Observations and data analysis}\n\\subsection{X-ray data} We analysed 22 RXTE\/PCA observations made between 2000\nOctober 1 and 2000 October 26 and 23 RXTE\/PCA observations performed between 2002 December 3 and\n2002 December 10.\n\nBackground subtracted light curves with a time resolution of 16 seconds were obtained from\nthe ``Standard 2''-mode data, covering the energy range 2-18 keV and dead-time corrections\nwere applied. We defined two X-ray colors, a  hard color (HC) and a broad color (BC),\nin the following energy bands: (8.5-13)\/(13-18) keV (HC) and (2-6.3)\/(6.3-13) keV (BC).\\\\\nIn addition, for each 128-second time interval, we accumulated power density spectra in the\n$\\sim$2-33 keV and $\\sim$2-10 keV energy ranges (respectively for 2000 October and 2002 December),\nwith a Nyquist frequency of 8192 Hz. Further details on the data analysis will be given in a\nfuture paper (Soleri et al., in preparation).\\\\\nNew radio ephemeris (determined in 2007) was used to calculate the orbital phases \\cite{25}.\n\n\\subsection{Radio data}\nWe have observed Cir X-1 in radio (simultaneously with X-ray) over multiple epochs on 2000 October \nand 2002 December at 4.8 and 8.6 GHz using the Australia Telescope Compact Array (ATCA). We used \nPKS J1939-6342 (PKS B1934-638) as primary calibrator and PMN J1524-5903 as secondary calibrator\n(B1520-58).\n\n\\section{Results}\nFigure \\ref{fig:licu_PCA} shows the RXTE\/PCA light curves of Cir X-1 for our data set: in 2000\nOctober more than an entire orbit was covered, albeit sparsely, while in 2002 December\nthe data were focused around periastron passage (phase 0.0 of the orbital period).\n\\begin{figure}\n\\begin{tabular}{c}\n\\hspace{0.75cm}\n\\includegraphics[width=12cm]{licu_PCA_proc_new2.ps}\n\\end{tabular}\n\\caption{2-18 keV RXTE\/PCA light curves for the two data sets examined here. The bin size is 16\nseconds. Only counts from PCU2 were used. Horizontal lines in the bottom part of each light curve\ncorrespond to ATCA radio observations, vertical dashed lines correspond to\nphase 0.0 of the orbital period, vertical dotted lines correspond to phase 0.5.}\n\\label{fig:licu_PCA}\n\\end{figure}\nSince a detailed discussion about the evolution of the X-ray\/radio properties in the whole analysed\ndata set is beyond the scope of this paper, for now we will focus our attention on two particular\norbital phases where we noticed remarkable behaviour. An extensive discussion of the X-ray and\nthe correlated X-ray\/radio properties of this data set will be presented in a future paper\n(Soleri et al., in preparation).\n\n\\subsection{2002 December - Phase 0.0 of the orbital period}  \\label{par:phase0.0}\nFigure \\ref{fig:Dec02_flare} shows the radio light curve, the X-ray light curve and the X-ray hardness\ncurve for the orbital phase interval 0.99 - 1.15, in 2002 December. After the passage through\nphase 0.0 (but not immediately) a multiple-flare event was clearly detected in radio (both at\n4.8 and 8.6 GHz), simultaneous with a sudden increase of the X-ray flux and variations in the X-ray\nhardness.  \n\\begin{figure}\n\\begin{tabular}{c}\n\\hspace{1.7cm}\n\\resizebox{10cm}{!}{\\includegraphics[angle=-90]{licu_X_radio_around_05Dec02_new2.ps}}\n\\end{tabular}\n\\caption{Radio and X-ray light curves (top and middle panels) and X-ray hardness curve\n(bottom panel) for 2002 December 03-06, between phase 0.99 and 1.15 of the orbital period.\nMultiple radio flares are clearly visible after the passage through the periastron}\n\\label{fig:Dec02_flare}\n\\end{figure}\nIn Figure \\ref{fig:hid_Dec2002} (left panel) we present a X-ray HID where points corresponding to\ndifferent days are plotted in different colours: the track drawn by the source in the HID\nchanges its position and morphology after the passage through phase 0.0 of the orbital phase,\npassing from a ``cloud'' located in the left-side of the diagram to a series of horizontal ``strips''\nin the bottom side. The transition happens on December 5 (third HID in Figure\n\\ref{fig:hid_Dec2002} left panel, red points), although it is not sharp: a transition cloud-strips\noccurs, followed after approximately 5 hours by a rapid transition (about 200 s) strips-cloud-strips.\\\\\n\\begin{figure}\n\\begin{tabular}{c}\n\\includegraphics[width=14cm]{figura_Dec02_HID_PDS_new2.ps}\n\\end{tabular}\n\\caption{{\\it Left panel:} Hardness-intensity diagram for 2002 December. The bin size is 16 seconds.\nIn each panel, points corresponding to specific orbital intervals are plotted in different colours.\n``Cloud'' and ``horizontal strips\" refer to specific zones of the HID (see the text).\n{\\it Right panel:} power density spectra averaged before (top spectrum) and after (bottom spectrum) the\nspectral transition around phase 0.0. Spectra before\/after the transition have not been averaged on the whole\ncloud\/strips region respectively, but on a smaller selection of points (details\nwill be given in Soleri et al., in preparation). The continuous line represents the best\nfit to the data. Spectra are plotted in the $\\nu P_{\\nu}$ representation \\cite{1}.}\n\\label{fig:hid_Dec2002}\n\\end{figure}\nPower density spectra averaged in the HID before and right after the X-ray spectral transition are shown\nin Figure \\ref{fig:hid_Dec2002} (right panel). Their properties change considerably: the spectrum\nbefore the transition is characterized by a weak VLFN ({\\it rms} = 3.15$\\pm$\n0.18~\\% integrated in the range 0.01-100 Hz) while after the transition the {\\it rms} of the VLFN\nincreases (6.06$\\pm$0.58~\\% in the same range) and a normal branch oscillation appears ($\\nu$ = 5.87$\\pm$\n0.62 Hz, {\\it rms} = 4.48$\\pm$0.28~\\%).\n\n\\subsection{2000 October - Phase 0.5 of the orbital period}  \nFigure \\ref{fig:Oct00_flare} shows the radio light curve, the X-ray light curve and the X-ray hardness\ncurve for the orbital phase interval 0.40 - 0.55, in 2000 October.\n\\begin{figure}\n\\begin{tabular}{c}\n\\hspace{1.7cm}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{licu_X_radio_around_09Oct00_new2.ps}}\n\\end{tabular}\n\\caption{Radio and X-ray light curves (top and middle panels) and X-ray hardness curve\n(bottom panel) for 2000 October 07-10, between phase 0.40 and 0.55 of the orbital period.\nMultiple radio flares are clearly visible close to phase 0.5}\n\\label{fig:Oct00_flare}\n\\end{figure}\nFrom the radio light curve, a sequence of radio flares nearby the passage\nthrough the apastron is evident.\nRadio flares close to this phase are peculiar and unexpected with respect to what can be found in the\nliterature where significant radio flux density enhancements are associated\njust to phase 0.0 (Tudose et al., in preparation). To date there is only one claim of a\nradio flare associated with phase 0.5 \\cite{7}.\\\\\n\\begin{figure}\n\\begin{tabular}{c}\n\\includegraphics[width=14cm]{Oct_200_flare_new2.ps}\n\\end{tabular}\n\\caption{{\\it Left panel:} zoom of Figure 4 around 2000 October 09-10; {\\it Right\npanel:} HID for 2000 October. Different colours in the HID correspond to different sides\n(in respect to the vertical line) of the X-ray light curve}\n\\label{fig:hid_Oct2002}\n\\end{figure}\nTo test whether this sequence is associated with any spectral change in the X-rays, in Figure\n\\ref{fig:hid_Oct2002} (left panel) we show a ``zoom'' of Figure \\ref{fig:Oct00_flare} corresponding to\nOctober 09-10 and the hardness-intensity diagram for all 2000 October (right panel). Purple and blue points\nin the HID correspond to the two portions of the light curve in the left panel and are used to mark two\ndifferent spectral branches. Since a classification of the spectral branches in the hardness-intensity\ndiagram is beyond the scope of this paper, for now we will assert without demonstration that the purple\npoints track a HB and the blue points a NB. A discussion will be presented in a future paper\n(Soleri et al., in preparation). On 2000 October 09-10 RXTE clearly caught a spectral transition\nbetween the HB and the NB, however this clearly occurred {\\em after} the radio flare. \n\n\\section{Conclusions}\nWe have analysed simultaneous X-ray\/radio observations of Cir X-1 obtained with RXTE and ATCA respectively.\n\n{\\bf Phase 0.0:} On 2002 December 05, near periastron passage, we\ndetected a multiple radio flare event, corresponding to a fast change of the X-ray\nspectral\/timing properties (in \\S \\ref{par:phase0.0} we already remarked that in the HID the transition is\nnot sharp). Our detection could be the first direct observation of a radio flare associated with a spectral \ntransition in the X-ray band in a NS system, since no clear association between X-ray spectral\/timing\nchanges and the emission of a radio flare has been reported yet.\nRecently it has been suggested that in NS-Z sources the spectral transitions from the HB to the NB \nmight be the moment of the launch of transient radio jets \\cite{21}, by analogy with what is\nsuggested for BHCs, where\nthe hard intermediate state to soft intermediate state transition might drive the ejection of transient\nradio plasmons \\cite{4}, \\cite{10}. In BHCs these state transitions have\nbeen in turn associated with the presence, in the PDS, of a transient low-frequency QPO, the so called type-B\n\\cite{2}. An association between the type-B QPO detected in BHCs and the normal branch oscillation\nobserved in Z sources has been suggested in \\cite{5}, since these QPOs present\nsimilar properties: on 2002 December we could associate the radio flare to a X-ray spectral\ntransition that is in turn associated with the presence of a normal branch oscillation, although we do\nnot see any obvious HB to NB passage.   \n\n{\\bf Phase 0.5:} On 2000 October 09 we detected a radio flare near the passage through\nthe apastron. In the literature there is only circumstantial evidence for  radio flares at phase 0.5 \\cite{7},\nmaybe partly because the observing strategies concentrated around phase 0.0. This is the first clear\nevidence of a radio flare near phase 0.5 (Tudose et al., in preparation). In the RXTE data we also\ndetected a spectral transition on 2000 October 09 about 2.5 hours after the end of the\nsecond radio flare, with the source moving from the HB to the NB. As it has been suggested in \\cite{21},\nthis transition might be the moment of the launch of transient radio jets, but\nin our specific case the time difference between the two events does not allow us to associate them.\nUnluckily we do not have RXTE data right before the beginning of the radio flaring activity (see Figure\n\\ref{fig:Oct00_flare}).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nAdvances in homoepitaxial chemical vapor deposition (CVD) of high-purity single-crystal diamond (SCD) have made the exceptional material properties of SCD available for a variety of new and exciting applications \\cite{Balmer_JPCM_09,Gracio_JPD_10,E6_handbook_17,Tokuda_NAD_15}. In particular, the wide bandgap, high carrier mobility, large thermal conductivity, corrosion resistance, and biocompatibility of SCD have enabled new devices for high-power electronics \\cite{Achard_DRM_11}, ultraviolet light sources \\cite{Makino_APL_11} and detectors \\cite{Balducci_APL_05}, nonlinear optics \\cite{Mildren_OpEx_08, Hausmann_NP_14, Latawiec_Optica_15}, quantum information processing \\cite{Toyli2013}, biomedical applications \\cite{Balasubramanian_COBC_14, Narayan_MT_11}, magnetometry \\cite{Rondin_RPP_14}, and integrated photonics \\cite{Aharonovich_NP_11, Aharonovich_AOM_14, Schroder_JOSAB_16,Grote_APLP_16,Gao_JLT_16}. \n\n\\figureOne\n\\cleaveProcess\n\nMany of these diverse applications benefit from the advantageous mechanical, thermal, and bond properties along $\\langle 111\\rangle$. For example, the (111) crystal plane is the hardest face \\cite{Wilks1972,Field_RPP_12}, making it more resilient to damage and desirable for industrial scale diamond cutting tools \\cite{Balmer_JPCM_09}. Similarly, the Raman gain coefficient is strongest when the optical polarization axis is aligned along $\\langle 111\\rangle$ \\cite{Mildren_OED_13,Sabella_OL_10}. For this reason, integrating on-chip Raman lasers requires precise crystallographic orientation of the diamond layer to maximize the laser efficiency~\\cite{Latawiec_Optica_15}.  High-power electronic devices also benefit from (111) faces, which provide improved donor incorporation efficiency and correspondingly higher mobility \\cite{Katagiri_APL_04,Kato_APL_05,Kato_DRM_11,Kato_APL_16,Achard_DRM_11,Tallaire_DRM_14}. Finally, point defects in SCD that are used as single-photon sources and spin qubits, such as the nitrogen-vacancy and silicon-vacancy centers, have a symmetry axis aligned along $\\langle 111\\rangle$ directions \\cite{Awschalom2013,Hepp_PRL_14,Bhaskar_arXiv_16,Neu_APL_14}. Thus, a (111) face maximizes the interaction with normally-incident light fields and optimizes the photon collection efficiency from these emitters, which is critically important for light-matter quantum interfaces \\cite{Bernien2013} and high-sensitivity magnetometry \\cite{Pham_PRB_12,Ozawa_APE_17}.\n\n\nWhile (100)-faced SCD plates are commonly produced by homoepitaxial CVD, growth of (111)-faced SCD plates has proven challenging due to twinning and stacking faults that form along $\\langle 111\\rangle$ directions \\cite{Friel_DRM_09, Gracio_JPD_10,Tallaire_DRM_14}. Despite some recent advances \\cite{Tokuda_NAD_15,Widmann_DRM_16}, inherent challenges still remain in growing (111)-oriented layers thicker than $\\approx$\\SI{100}{\\nano\\meter} with low surface roughness. For this reason, macroscopic (111)-faced plates are typically cut from larger (100)-faced starting plates using micro-machining laser ablation tools \\cite{Stoupin2016} and subsequent polishing steps are used to reduce the surface roughness \\cite{Hird_OED_13,Konov_OED_13}.  This process results in substantial loss of costly SCD material (kerf loss) along with misalignment of the resulting face relative to [111], since it is nearly impossible to polish the (111) face using traditional methods \\cite{Wilks1972}.  Furthermore, the mechanical polishing step induces sub-surface damage that can only be removed by further chemical etching \\cite{Friel_DRM_09}. \n\nAn alternate approach, traditionally used to shape diamond gemstones, is mechanical cleaving, where a small scribe, or kerf, is created and struck by hand with a knife to initiate a cleave. It is not feasible, however, to perform this process by hand with the micron-scale accuracy needed to create large, thin, parallel plates for electronic and optical applications from small samples of SCD.  Instead, it would be valuable to establish a process similar to silicon stealth dicing \\cite{Kumagai_TSM_07}, whereby a laser is used to nucleate and propagate a mechanical cleave with high spatial precision and control of the cleaving dynamics.  \n\nIn this paper, we present a laser-nucleated cleaving (LNC) process for SCD.   Rather than machining the entire surface of a desired face, the LNC process uses a pulsed, focused laser with sub-bandgap photon energy (\\SI{532}{\\nano\\meter}) to locally graphetize a thin ($\\approx$\\SI{30}{\\micro\\meter} diameter) line that propagates a mechanical cleave along a desired (111) plane.  As shown in Fig.~\\ref{optical_image}, (111)-faced plates are fabricated by generating two parallel (111) cleave planes, forming a parallelepiped shape. This process results in large, atomically-flat (111)-faced SCD plates while eliminating kerf loss and removing the need for mechanical polishing.  \n\n\nThe paper is organized as follows: first, a step-by-step description of the LNC process is provided, followed by atomic force microscopy (AFM) and X-ray diffraction measurements that determine the surface roughness and confirm the crystal orientation, respectively. Then, the potential physical mechanisms that contribute to the LNC process are discussed. Finally, avenues for improving the process to optimize the size and flatness of (111)-faced SCD plates are presented.\n\n\n\\section{Materials and methods} \nThe LNC process is illustrated in Fig.~\\ref{cleave_process} and the individual steps are described in the section below. We begin by discussing the types of SCD plates used in this study followed by the micro-machining laser specifications. Then, we outline the LNC process which is grouped into four sub-processes consisting of nucleation site creation, initiation of the cleave, propagating the cleave front, and backside cleaving to produce a free standing (111)-faced SCD plate. Finally, the details concerning the characterization of the SCD plates' physical properties are covered. Additional details are included in the supporting information \\cite{Supporting}.\n\n\\subsection{Materials}\nWe perform LNC on commercially available  ``standard grade'' (SG, $<1$~ppm nitrogen and boron) and high-purity ``electronics grade'' (EG, $<5$~ppb nitrogen and boron) type-IIa, CVD single-crystal diamond from Element6 \\cite{E6_website}.  The edge and face orientations of these samples are shown in Fig.~\\ref{optical_image}(c)-(d). \n\n\\subsection{Nucleation site creation}\nThe LNC process begins with the creation of a nucleation site on the bottom surface of the starting SCD plate. The nucleation site acts as a seed for producing the initial cleave.  For samples with two polished (100) faces, textured regions [Fig.~\\ref{cleave_process}(a), inset] are laser-machined onto the backside using an excimer laser \\cite{Supporting}. Only one textured site per desired (111) face is required. For single-side polished pieces, the unpolished side of as-received samples with a surface roughness of $R_{q} = ~\\SI{250}{\\nano\\meter}$ \\cite{E6_website} was found to be sufficient to nucleate cleaves without any additional patterning.\n\n\\subsection{Micromachining laser beam preparation}\nWe perform LNC using a nanosecond pulsed, frequency-doubled Yb-doped fiber laser operating at a wavelength of \\SI{532}{\\nano\\meter} with a repetition rate of \\SI{150}{\\kilo\\hertz}, integrated in a micromachining laser system (IPG Photonics, IX-280-ML). The primary focusing optic is a \\SI{100}{\\milli\\meter}-focal-length objective that produces a beam waist of diameter $<$\\SI{17}{\\micro\\meter}, a beam quality factor of $M^2 = 1.5$, and a depth of field $\\approx${\\SI{800}{\\micro\\meter}. In this system, the laser focus and course lateral position are controlled using a motorized stage, and cuts are made by steering the beam with a galvo scanner. The starting SCD plate is fixed to a \\SI{1.0}{\\milli\\meter}-thick glass microscope slide with double-sided tape so that the micromaching laser can be focused through the air\/diamond interface (referred to as the top surface) onto the tape-diamond interface (referred to as the bottom surface).  \n\nThe beam focus is positioned such that the bottom surface of the SCD plate ablates while the top surface does not. SCD ablation by sub-bandgap light is a nonlinear optical process, and the ablation rate depends sensitively on the pulse fluence \\cite{Konov_LPR_12}.  Positioning the laser focus close to the bottom surface establishes a fluence gradient such that the top surface is  unaffected by the micromaching laser beam, whereas the bottom surface is above the threshold for ablation and graphetization. The presence of a nucleation site or unpolished bottom surface further lowers the threshold for ablation. Figure~\\ref{fluence} depicts the experimentally measured ablation rate of a top polished SCD surface as a function of pulse energy for our micromaching laser, exhibiting an abrupt threshold around \\SI{140}{\\micro\\joule}.\n\n\n\\fluence\n\n\\subsection{Initiating the cleave}\nTo initiate a cleave, the micromachining laser beam is aligned over a textured nucleation site, or a target region of an unpolished bottom surface using the sample stage. The laser beam is then switched on and laterally scanned in $x$ at a speed of \\SI{8}{\\centi\\meter\\per\\second} by a distance $w$, as illustrated in Fig.~\\ref{cleave_process}(b). The scan distance is calculated according to $w=h\/\\tan\\SI{55}{\\degree}$, where $h$ is the starting SCD plate thickness, and \\SI{55}{\\degree} is the angle between the [111] and [100] lattice vectors. This translation distance ensures that the cleave extends through the full thickness of the SCD ($x$-$z$ plane). After this step, a darkened line appears and a new reflective surface can be seen within the diamond [Fig.~\\ref{cleave_process}(b), inset]. The combination of these two observations heralds the presence of a cleave along a (111) face. \n\n\n\n\n\n\n\\subsection{Propagating the cleave}\nAs seen in Fig.~\\ref{cleave_process}(b), a typical cleave only propagates a few hundred microns away from the laser scan line in the $y$-direction. The (111) crystal face is extended by repeating the lateral laser scans used in the previous step, but with the beam translated along $y$ near the edge of the newly-created (111) face. Scanning the laser again along $x$ extends the cleave further along $y$, as shown in Fig.~\\ref{cleave_process}(c).  As such, a textured nucleation site is only required at the position of the first laser scan line to initiate a cleavage plane. After each subsequent extension of the (111) face, a new boundary is identified in the microscope and the laser scan process is repeated until the cleave has been extended across the entire width of the starting SCD plate, creating a fully exposed (111) face [Fig.~\\ref{cleave_process}(d)].  \n\n\n\n\n\n\\subsection{Backside cleaving}\nTo complete the (111)-faced SCD plate fabrication, a second (111) crystal face is generated that is parallel to the initial exposed face. The distance between the two nucleation sites sets the resulting plate thickness $t$ as can be seen in Fig.~\\ref{cleave_process}(f). For reliable performance, we find that the second nucleation site should be offset by a distance greater than the beam translation distance, ($d>w$) [Fig.~\\ref{cleave_process}(e)], hence the minimum (111)-faced plate thickness is limited by the starting plate height, $h$. \n\n\\subsection{Characterization}\nMeasurements of the cleaved-face lattice orientation were made at room temperature using X-Ray diffraction with a $\\lambda = \\SI{0.1544}{\\nano\\meter}$ source, a \\SI{1}{\\degree} divergence slit, \\SI{1}{\\degree} scatter slit, \\SI{0.6}{\\milli\\meter} receiving slit, and a scintillator detector.  Plates were mounted on a glass slide using crystal bond.\nSurface roughness was characterized using an Asylum MFP-3D atomic-force scanning probe with $z$-axis noise $<$\\SI{0.06}{\\nano\\meter} (average deviation) in a \\SI{0.1}{\\hertz}-\\SI{1}{\\kilo\\hertz} bandwidth.  \n\n\\section{Results}\n\\samples\n\n\\subsection{Plate dimensions}\nRepresentative (111)-faced plates produced by the LNC process are shown in Fig.~\\ref{optical_image}(a)-(b), Fig.~\\ref{cleave_process}(f), and Fig.~\\ref{samples}. The dark lines on the top and bottom surfaces of the plates are the locations of the micromachining laser scan lines. A laser-scan-line separation in $y$ of $L=$~\\SI{300}{\\micro\\meter} was found to be suitable for reliably propagating the cleave using an automated process [Fig.~\\ref{samples}, bottom surface]. However, by manually identifying the cleavage plane boundary in the microscope, much larger spacings can be realized [Fig.~\\ref{samples}, top surface], resulting in laser-scan-line-free areas ranging from \\SI{0.18}{\\milli\\meter}$^2$ [Fig.~\\ref{cleave_process}(f) inset] to \\SI{1.2}{\\milli\\meter}$^2$ [Fig.~\\ref{samples}(b)]. \n\nFabricated plate thicknesses vary from \\SI{50}{\\micro\\meter} [Fig.~\\ref{cleave_process}(f) inset] to \\SI{520}{\\micro\\meter} [Fig.~\\ref{samples}(b)]. Minimum thicknesses of \\SI{170}{\\micro\\meter} and \\SI{290}{\\micro\\meter} are reliably produced for starting plate thicknesses of \\SI{300}{\\micro\\meter} and \\SI{500}{\\micro\\meter}, respectively. A single $\\SI{3}{\\milli\\meter} \\times \\SI{3}{\\milli\\meter}$ (100) SCD plate can typically yield up to 10 (111)-faced plates. The number is limited due to the increasing number of stochastically-generated cracks along undesired planes as the original SCD plate is diced into multiple (111)-faced plates. \n\n\\diamondsurface\n\n\\subsection{Surface Morphology}\nLines running along the (111) face in the direction of the cleave propagation can be observed in the differential-interference-contrast optical microscope image shown in Fig.~\\ref{diamond_surface}(a). These nanometer-scale steps or ridges are known as Wallner lines, which appear ubiquitously on cleaved surfaces of brittle materials as a result of the interaction between the crack front and transverse acoustic waves \\cite{Rabinovitch_JAP_06,Hird_OED_13}. Using atomic force microscopy (AFM), we find that the Wallner lines typically exhibit step heights of 10--\\SI{50}{\\nano\\meter} [Fig~\\ref{diamond_surface}(b)], with separations that vary from $\\approx$\\SI{10}{\\micro\\meter} near the initial laser scan line to $>$\\SI{100}{\\micro\\meter} in regions farther away. \n\nAs shown in Fig.~\\ref{diamond_surface}(c), the areas between Wallner lines exhibit atomically-flat (111) surfaces characterized by micron-scale, triangular terraces with step heights consistent with the expected \\SI{0.206}{\\nano\\meter} spacing of (111) atomic planes \\cite{Tokuda_NAD_15}. The terraces are occasionally intersected by linear features as observed in the bottom-left of Fig.~\\ref{diamond_surface}(c). For comparison, the polished (100) surface of an EG-SCD plate, as-received from the manufacturer \\cite{E6_website} following cleaning using an acid bath and O$_2$ plasma treatment, is displayed in Fig.~\\ref{diamond_surface}(d). \n\nClearly, the morphologies of the cleaved and polished surfaces are very different.  Figure~\\ref{diamond_surface}(e) shows the power spectral density (PSD) of the AFM scans in Figs.~\\ref{diamond_surface}(c,d), as well as the PSD for the region of a single atomic plateau  indicated by a red box in Fig.~\\ref{diamond_surface}(c). Over the entire $\\SI{12}{\\micro\\meter}\\times\\SI{12}{\\micro\\meter}$ AFM scan windows, we calculate root-mean-square (RMS) surface roughness values of $R_{q}=\\SI{0.22}{\\nano\\meter}$ and \\SI{0.46}{\\nano\\meter} for the cleaved (111) and polished (100) surfaces, respectively.  The LNC surface roughness represents an order-of-magnitude improvement compared to the manufacturer's specification for the polished surface (average roughness $<$\\SI{5}{\\nano\\meter}) and a smoothness comparable to the best values reported from advanced polishing techniques  of $<\\SI{0.1}{\\nano\\meter}$ \\cite{DDK}. Furthermore, the majority of the cleaved-surface roughness occurs in the form of atomic steps with micron-scale spacings, whereas the polished surface exhibits roughness over a broad range of spatial frequencies.  The calculated RMS roughness of the \\SI{25}{\\micro\\meter\\squared} region on an atomic terrace is $R_{q} = \\SI{0.05}{\\nano\\meter}$, corresponding to the noise floor of the AFM tool. Further details the AFM surface roughness analysis are provided in the supporting information \\cite{Supporting}.\n\n\n\\subsection{Face Orientation}\nCrystallographic faces were indexed using X-ray diffraction [Fig.~\\ref{diamond_surface}(f)], showing a peak at $2\\theta=\\SI{44}{\\degree}$ that corresponds to first order Bragg diffraction from \\{111\\} planes.  Since the spacing between individual (111) planes is $d = a\/\\sqrt{3}=\\SI{0.206}{\\nano\\meter}$, where $a = \\SI{0.357}{\\nano\\meter}$ is the lattice constant of diamond, the first (111) diffraction peak should appear at $2\\theta = 2\\sin^{-1}\\left(\\lambda\/(2d)\\right) = \\SI{44.0}{\\degree}$. These measurements have insufficient accuracy to determine the miscut angle, but we can use the features in AFM scans like Fig.~\\ref{diamond_surface}(b)-(c) to estimate the miscut based on the heights and spacings of atomically-flat plateaus.  In between Wallner lines, we typically observe atomic terraces with $\\approx\\SI{2}{\\micro\\meter}$ spacings, corresponding to a miscut angle of $\\sim\\SI{0.006}{\\degree}$. Over larger length scales, Wallner lines of height $\\approx$\\SI{50}{\\nano\\meter} are typically separated by $\\approx$\\SI{10}{\\micro\\meter}, corresponding to an overall miscut angle $\\approx\\SI{0.3}{\\degree}$ \\cite{Supporting}.\n\n\n\n\n\\section{Discussion} \n\n\nWhile further studies are required to fully elucidate the physics underlying the LNC process, the observation of dark laser-induced lines in Figs.~\\ref{optical_image}(a), \\ref{cleave_process}, and~\\ref{samples} suggest that the LNC process initiates and propagates a graphitization wave with a sufficiently large diameter to locally stress the SCD beyond its fracture threshold. In effect, LNC is a mechanical cleaving process in which the laser acts as a sub-micron knife striking a kerf, with the potential for creating large-area, atomically flat faces in SCD plates and gemstones for a variety of applications.  \n\n\n\\Wallners\nPulsed-laser graphitization and laser-guided graphitization wave propagation in SCD have been studied extensively in the context of 3D-laser writing \\cite{Konov_OED_13}. In LNC, graphitization waves are propagated by focusing the laser through the starting SCD plate such that the beam fluence is sufficiently large at the bottom surface nucleation site to initiate graphitization, while the defocused beam throughout the bulk of the SCD does not create any material damage, as illustrated in Fig.~\\ref{fluence}. Once the graphitization wave is initiated at the nucleation site, the ablation threshold drops abruptly such that the fluence required to propagate the wave is much lower than the bulk graphitization threshold \\cite{Konov_OED_13}, allowing for the wavefront to be guided along a (111) plane by laterally scanning the laser in the $x$-direction as indicated in Fig.~\\ref{cleave_process}. Using the appropriate laser fluence, spot size, and scan speed, we are able to create a graphite wave with a sufficiently large transverse diameter to fracture SCD as it propagates in a (111) plane.\n\nLocal crack formation from laser-guided graphitization waves has been previously observed in diamond \\cite{Kononenko_DRM_11}, and has been attributed to the substantially greater specific volume (lower density) of graphite as compared to SCD, resulting in strong tensile stresses in the vicinity of the graphitized region. Assuming that the laser can provide adequate fluence, the requirement for producing sufficiently large internal stresses to cause cracking is determined by the area and speed of graphite generation. Romano \\textit{et al.} developed an empirical threshold for crack propagation of $V\\pi d^2\/4R < \\SI{20}{\\micro\\meter^3}$ per laser shot, where $V$ is the laser scan speed, $d$ is the laser spot size, and $R$ is the pulse repetition rate \\cite{Konov_OED_13}. The laser micromachining system used for our LNC process has a minimum spot size $d \\lesssim\\SI{17}{\\micro\\meter}$ and a scan speed of $V = \\SI{8}{\\centi\\meter\\per\\second}$ with a laser pulse repetition rate of \\SI{150}{\\kilo\\hertz}, which gives a value of $V\\pi d^2\/4R \\approx \\SI{120}{\\micro\\meter^3}$ per laser shot. The fact that our processing parameters are well above the threshold for exceeding the tensile strength of SCD provides further evidence that laser-guided graphitization waves are the dominant physcial mechanism behind LNC.  Since SCD preferentially cleaves along \\{111\\} planes \\cite{Hird_OED_13} due to the tensile strength being lowest along $\\langle 111\\rangle$ directions \\cite{Telling_PRL_00}, the graphitization wave and resultant cleave are self guiding along the chosen (111) surface, even though the laser focus is fixed below the sample throughout the entire scan.\n\n\nIn contrast to traditional mechanical cleaving of diamond, LNC provides the opportunity for precise control of the cleave propagation by tuning the scan speed of the laser. Similar to what has been done with silicon stealth dicing \\cite{Ohmura_JAMDSM_08}, where the laser power and stage translation speed have been tuned to achieve Wallner-line-free cleaves \\cite{Casquel_ASS_11}, further analysis of the mechanisms involved in LNC should allow for drastic reduction of the Wallner lines that form during crack propagation~\\cite{Hird_OED_13}. \nFurthermore, by simultaneously scanning the laser position and focus, it may be possible to propagate cleaves along \\{100\\} or \\{110\\} planes in SCD. \n\nThe precision and flexibility of the LNC process will make it useful for many applications. Since kerf loss is minimized, LNC may be desirable for faceting diamond gems or cutting tools. The large, perfectly flat (111) atomic planes are believed to be ideal for stabilizing the spin and electronic properties of sub-surface quantum sensors for nanoscale magnetic resonance imaging \\cite{Chou2017}.\nAdditionally, the LNC process has potential for reducing or completely eliminating plate wedge, which adversely affects diamond photonic devices \\cite{Latawiec_Optica_15,Hausmann_NP_14,Hausmann_NL_13,Gao_JLT_16}. In addition to generating (111)-faced SCD substrates for diamond electronics, photonics, and quantum technologies, LNC will also prove useful for fabricating seed crystals required for improving CVD growth of customized [111]-oriented samples \\cite{Friel_DRM_09, Gracio_JPD_10,Tallaire_DRM_14,Tokuda_NAD_15,Widmann_DRM_16,Ohno2012}.\n\n\n\n\\section{Conclusion}\nWe have demonstrated a new process for fabricating (111)-faced SCD plates by laser-nucleated cleaving.  Multiple plates with thicknesses down to \\SI{170}{\\micro\\meter} and face areas up to \\SI{3.0}{\\milli\\meter}$\\times$\\SI{0.4}{\\milli\\meter} are reliably produced from (100) SG crystals with starting dimensions \\SI{3.0}{\\milli\\meter}$\\times$\\SI{3.0}{\\milli\\meter}$\\times$\\SI{0.3}{\\milli\\meter}.  The plates exhibit micron-scale atomically-flat terraces, with large areas (up to \\SI{1.0}{\\milli\\meter}$\\times$\\SI{0.4}{\\milli\\meter}) unmarred by laser damage. With the ability to produce large (111) faces out of commercially available CVD diamond plates comes the opportunity for further exploration of, and access to, SCD as an integral component in high-power electronics, architectures for quantum computing and quantum communication, and ultra-sensitive magnetometers. \n\n\\section*{Acknowledgements}\nWe thank S. Szewczyk for assistance with X-ray diffraction measurements; B. Peterson and E. Johnston for assistance with micromaching lasers; and A. L. Exharos, T.-Y. Huang, and J. Tabeling for their insightful discussions and comments. This work was supported by an NSF CAREER grant (ECCS-1553511) and the Singh Center for Nanotechnology at the University of Pennsylvania.\n\n\\section*{References}\n\n\n\\section{Sample details}\nDetails regarding the single-crystal diamond (SCD) starting plates as provided by the manufacturer \\cite{E6_website} are provided in Table~\\ref{sample_table}.  The edge and face orientations of both standard grade (SG) and electronics grade (EG) starting plates are depicted in Fig.~1(c,d) of the main text.  \n\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n&Electronics grade & Standard grade\\\\\n\\hline\nN content&$<$~5~ppb&$<$~1~ppm\\\\\n\\hline\nB content&$<$~1~ppb&$<$~0.05~ppm\\\\\n\\hline\nLateral Dims. & $\\SI{2.0}{\\milli\\meter}\\times\\SI{2.0}{\\milli\\meter}$ & $\\SI{3.0}{\\milli\\meter}~\\times~\\SI{3.0}{\\milli\\meter}$ \\\\\n\\hline\nThickness & \\SI{0.5}{\\milli\\meter} & \\SI{0.3}{\\milli\\meter} \\\\\n\\hline\nPart no.&145-500-0385&  145-500-0253 \\\\\n\\hline\n\\end{tabular}\\caption{\\label{sample_table}Type IIa CVD-grown SCD starting plate specifications from Element6 \\cite{E6_website}.}\n\\end{table}\n\n\n\\section{Micromachining laser details}\nSpecifications for the laser micromachining tools used for LNC are given in Table~\\ref{laser_params}.  The \\SI{198}{\\nano\\meter} excimer laser tool was used to pattern cleave nucleation sites on the bottom surface of double-side polished samples, while the \\SI{532}{\\nano\\meter} Yb-doped fiber laser tool was used for initiating and propagating cleaves.  The spot size of the excimer laser is defined by a variable aperture, which is set here to a $\\SI{10}{\\micro\\meter}\\times\\SI{10}{\\micro\\meter}$ square mask.\n\n\n\\begin{table}[!h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n & Excimer & Yb-doped fiber\\\\\n \\hline\nModel & IX-255 & IX-280-ML \\\\\n \\hline\nWavelength & \\SI{193}{\\nano\\meter} & \\SI{532}{\\nano\\meter} \\\\\n\\hline \nMax Pulse energy  & \\SI{800}{\\micro\\joule} & \\SI{200}{\\micro\\joule} \\\\\n\\hline\nMax Rep. rate  & \\SI{120}{\\hertz}  & \\SI{300}{\\kilo\\hertz} \\\\\n\\hline\nPulse Length & \\SI{10}{\\nano\\second} & \\SI{2}{\\nano\\second} \\\\\n\\hline\nMin Spot diam. & \\SI{10}{\\micro\\meter} (apertured) & $<$\\SI{17}{\\micro\\meter} \\\\\n\\hline\nMax Fluence & \\SI{25}{\\joule\\per\\centi\\meter^2} & \\SI{90}{\\joule\\per\\centi\\meter^2} \\\\\n\\hline\n\\end{tabular}\\caption{\\label{laser_params}Specifications for the micromachining lasers used (IPG Photonics).}\n\\end{table}\n\n\\subsection{Finding and setting focus}\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[scale=0.65] {Fig_S1_v2.png}}\n\\caption{\\label{focus}\nAblation spot diameter on the top polished diamond surface for varying stage heights. Increasing stage height corresponds to the sample surface being translated towards the laser focusing lens.}\n\\end{figure}\n\nThe focus of the \\SI{532}{\\nano\\meter} laser is found by setting the laser power to 100\\% with a reprate of \\SI{150}{\\kilo\\hertz} and attempting to ablate the surface at different stage heights. The results of this calibration can be seen in Fig.~\\ref{focus}. The minimum ablated spot diameter gives us an upper bound on the beam waist of \\SI{17}{\\micro\\meter}, however the exact value if difficult to measure due to the nonlinear nature of the ablation process. The laser focus is determined to be the stage height producing the minimum ablated region width. In our coordinate system, increasing stage heights imply that the sample is being translated closer to the focusing lens. We found that the inspection camera focus and the laser focus differed by $\\approx$\\SI{500}{\\micro\\meter} and that care had to be taken to position the diamond sample in the proper location such that the top face was below the ablation threshold as is seen in Figure~3 of the main text. This calibration is performed before every cleave process to ensure that the sample is positioned properly.\n\n\n\n\\subsection{Ablation rate}\nThe ablation rate as a function of pulse energy in Fig.~3 of the main text was measured by setting the top diamond surface at the point of laser focus, fixing the laser rep rate at \\SI{150}{\\kilo\\hertz}, and linearly varying the laser pulse energy between \\SI{10}{\\micro\\joule} and \\SI{200}{\\micro\\joule} in 10 steps with a variable attenuator to produce the series of ablation sites shown in Fig.~\\ref{nuc_site}(a). Five exposures at each pulse energy were performed and averaged to arrive at the measured depth of the resulting ablation site, which were measured by AFM.\n\n\n\n\\subsection{Creating the nucleation site}\nThe textured nucleation site shown in the main text [Fig.~2(a)] is fabricated using an Excimer laser [Tab. \\ref{laser_params}]. The nucleation site is fabricated on the bottom surface of the SCD plate (see main text for surface name convention). A \\SI{50}{\\micro\\meter}x\\SI{50}{\\micro\\meter} area of ablated diamond [Fig.~\\ref{nuc_site}(b)] is created by rastering the \\SI{10}{\\micro\\meter}x\\SI{10}{\\micro\\meter} aperture mask in steps of \\SI{5}{\\micro\\meter} in two perpendicular directions. The roughened region produced is then suitable for nucleating a cleave.\n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[scale=1] {Figure2V7_SI.png}}\n\\caption{\\label{nuc_site} \n(a) Region used for measuring ablation rate in figure 3 of the main text.\n(b) Nucleation site fabricated using the UV Excimer laser [Tab. \\ref{laser_params}].}\n\\end{figure}\n\n\\section{Processing details}\n\n\\subsection{Cleaving Single-Side Polished SG-SCD}\nThe unpolished-surface of single-side polished (SSP) starting plates allowed for the formation (111)-oriented SG-SCD plates without the need for a nucleation site. We found that a scribe line could be cut into the rough backside using selective ablation that allows for a cleave to propagate the length of the sample unhindered,\\---\\,producing (111) faces unmarred by cleave lines for $>\\SI{2}{\\milli\\meter}$. This procedure, outlined in Figure~\\ref{cleave_process}, requires that the roughened surface be placed on the bottom. A scribe line is then created on the bottom surface, oriented along the desired crystal plane, using the selective ablation procedure with the laser reprate set at \\SI{75}{\\kilo\\hertz} (half the normal operating reprate of \\SI{150}{\\kilo\\hertz}). Following creation of the scribe line, the laser is rastered across 3-5 lines separated by $=\\SI{212}{\\micro\\meter}$ at 100$\\%$ power [Fig.~\\ref{cleave_process}(a)]. The first raster initiates the cleave, and the subsequent lines propagate the cleave across the entire scribe line to the sample edge.\n\n\n\\begin{figure}[t!]\n\\includegraphics[scale=0.98] {Figure1V2_SI.png}\n\\caption{\\label{cleave_process} Illustration of the cleaving process (a) Orientation of the scribe line cut with respect to the standard orientation of SG-SCD (b) Side-on profile view of the process for cleaving a \\{111\\} plate with parallel faces. $w_{\\textrm{max}}$ denotes the maximum lateral separation between nucleation sites, $t_{\\textrm{max}}$ denotes the maximum plate thickness, $w_{\\textrm{min}}$ denotes the minimum lateral separation between nucleation sites, and $t$ denotes the minimum plate thickness. (c) Expanded image of the set of cleaves required for the scribe-assisted SG-SCD cleaving process.}\n\\end{figure}\n\n\\subsection{Controlling plate thickness}\nThe (111)-faced SCD plate thickness is chosen by considering the distance between the two desired crystal faces, $t$, and then selecting the appropriate distance between nucleation sites, $w$, as shown in Fig.~\\ref{cleave_process}(b). We generally observe that the laser cannot generate or propagate a cleave when intersecting a previously-created crystal face.  Hence the minimum plate thickness we can reliably produce is given by $t_{\\textrm{min}}=h\\cos(\\SI{55}{\\degree})$, where $h$ is the sample thickness. We occasionally generate thinner samples by accident, and we anticipate that by performing the cleaves on alternating sides of the diamond, thinner plates could be obtained by this method. The maximum thickness that results in two parallel, overlapping (111) faces is also shown in Fig.~\\ref{cleave_process}(b).\n\n\\section{Surface topography analysis} \n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[scale=1] {Figure6v1_SI_corrected.png}}\n\\caption{\\label{miscut}\n(a) Atomic force micrograph of cleaved (111) face. A portion of the ablated region after a cleave propagation line is seen in the bottom left corner, and a line trace is specified in yellow near the top, seen in (b). (b) Line trace of a cleaved (111) face.}\n\\end{figure}\n\n\\subsection{Power spectral density} Power spectral density (PSD) analysis of the surfaces of LNC fabricated (111) plates is used to characterize the surface roughness at distinct length scales of topographical features relevant for the various uses of LNC SCD (111) plates. The PSD can be integrated in the spatial frequency domain to infer roughness characteristics for a given length scale \\cite{jacobs2017}. Peaks in a given PSD curve correspond to distinct frequencies of roughness, and their heights are related to their contribution to the RMS roughness at that spatial frequency.\n\nWe start with the discrete inverse and forward discrete Fourier transforms describing the surface topography, $h_x$:\n\\begin{eqnarray}\nh_{x} = \\frac{1}{L_{x}} \\sum_{q}{\\tilde{h}_{q}e^{iqx}} \\\\\n\\tilde{h}_{q} = l_{x} \\sum_{x}{h_{x}e^{-iqx}},\n\\end{eqnarray}\n\\noindent where $L_{x}$ is the length of the AFM scan in the x-direction, $q$ is the vector of spatial frequencies, and $l_{x}$ is the pixel size of the AFM scan, $l_{x} = N_{x}\/L_{x}$ where $N_{x}$ is the number of points across a single horizontal line scan. The PSD of the topography above is then defined as \n\\begin{equation}\\label{PSD}\nC_{q}^{1D} = L_{x}^{-1} |\\tilde{h_{q}}|^2\n\\end{equation}\nIt follows that Eq. \\ref{PSD} is related to the RMS roughness ($R_q$) by\n\\begin{equation}\nR_q = \\sqrt{L_{x}^{-1} \\sum_{q}{C_{q}^{1D}}}\n\\end{equation}\nTo calculate the PSD, for a given AFM scan, we use the fast Fourier transform (FFT) algorithm to calculate the Fourier components $h_q$ and corresponding PSD for each line scan of the image, and then average them together to calculate the PSD for the full image.  No image adjustments or corrections have been applied to the AFM data prior to this analysis. Note that, as calculated, the PSD only reflects spatial frequencies in the (horizontal) $x$-direction of each AFM image, so the specific shape of the curve can depend on the orientation of the image relative to specific sample features.  However, the integrated value of the RMS roughness is independent of sample orientation.\n\n\n\\subsection{Miscut measurement}\nWe estimate the miscut of LNC-fabricated (111) plates using large-scale AFM scans. \nFigure~\\ref{miscut} is a typical example of a region on cleaved (111) surfaces exhibiting Wallner lines. The miscut angle is calculated by counting the number and height of steps in line scans such as in Fig. \\ref{miscut}(b). In those data, a distance of \\SI{67}{\\micro\\meter} was scanned with a total vertical offset of \\SI{0.31}{\\micro\\meter}, corresponding to a miscut angle of $\\tan^{-1}(0.31\/67) = \\SI{0.27}{\\degree}$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\nIn General Relativity (GR), just as in Maxwell's electrodynamics,\none finds that\ngiven a spherically symmetric system, there are no dynamical\ndegrees of freedom\nin the theory.\nThis is Birkhoff's theorem, and implies that a time dependent\nsource will not\nexcite modes in the gravitational system, so that outside this\nsource the system\nmust be physically equivalent to the Schwarzschild solution (the\nnon-trivial\nstatic spherically symmetric spacetime).\nIt has been further established\n\\cite{Regge+Wheeler:1957,Vishveshwara:1970} that\nthe solution is stable when perturbed, so that small deviations\nfrom spherical\nsymmetry do not alter the large scale features of the spacetime,\nand systems\nthat are only approximately spherically symmetric are therefore\nstill very well\nmodeled by the Schwarzschild solution.\nThis establishes that the phenomenology of the Schwarzschild\nsolution is\nphysically relevant.\n\nTo see that Birkhoff's theorem is not a generic feature of\nphysical theories one\nneed look no further than a scalar field.\nHowever the example that is important for this work is the\n(massive) Kalb-Ramond\naction \\cite{Kalb+Ramond:1974} which (as will be demonstrated in\nSection\n\\ref{section:Schw pert}) has a single mode that is time dependent\nin general.\nAs will be shown in Section \\ref{section:Massive NGT}, the\nmassive Nonsymmetric\nGravitational Theory \\cite{Moffat:1994,Moffat:1995b} (mNGT as\nopposed to the\nolder versions of the theory, referred to as NGT, or massless\nNGT) becomes\nidentically a massive Kalb-Ramond field with an additional\ncurvature coupling\nterm when considered as a perturbation about a Ricci-flat GR\nbackground, so the\nresult that NGT has a monopole mode is not surprising.\nThe mode considered here is short ranged, so that far enough away\nfrom the\nsource one finds that the solution will be dominated by\nSchwarzschild behavior,\nand there is no energy flux.\nHowever, after demonstrating that spherically symmetric fields in\nthe skew\nsector are not static in general, it will be shown in Section\n\\ref{section:Wyman\npert} that the symmetric sector will also no longer be static,\nthrough\nexamination of a similar perturbation about the mNGT background\ndiscussed in\nSection \\ref{section:Wyman}.\nThis means that no static solutions can be considered rigorously\nas an exterior\nsolution unless the solution is globally static (i.e. the\ninterior is static as\nwell).\n\nThe results in this paper are obtained through an examination of\nlinearized\nperturbations, although the conclusions must hold in general as\nBirkhoff's\ntheorem would imply that these fields must be static as well.\nWhat cannot be examined in this fashion is whether in the full\nnonlinear theory\nmore modes become excited.\nIn particular, one will see in the case of a perturbation about a\nGR background\n(in Section \\ref{section:Massive NGT}) that there are three\npropagating modes,\neven though the absence of gauge invariant kinetic terms in the\nfull theory\nwould suggest that all six modes in the antisymmetric metric\ncould be\nindependent degrees of freedom (as yet, the number of degrees of\nfreedom in mNGT\nhas not been rigorously established).\nAlthough this issue is not addressed directly in this work, the\nability to\nrecast the theory in a general basis given in Section\n\\ref{section:GenBasis\nintro} sets the stage for a complete analysis of the spherically\nsymmetric\nsystem in NGT.\nGiven that there is an additional mode in the general spherically\nsymmetric\nsystem, how the fields may or may not approach an asymptotically\nflat spacetime\nshould then be addressed, and also whether evolution\nsingularities of the type\ndiscussed in \\cite{Isenberg+Nestor:1977} are encountered.\n\n\\section{Massive NGT}\n\\label{section:Massive NGT}\n\nThe original version of NGT \\cite{Moffat:1979} grew out of a\nre-interpretation\nof the Einstein-Straus \\cite{Einstein:1945,Einstein+Straus:1946}\nunified field\ntheory as a purely gravitational system.\nThe antisymmetric part of the metric and connection operationally\nproduce\ndifferent modes of parallel transport and index contraction\n\\cite{Moffat:1979,Moffat:1990,Einstein:1956}, where the algebra\nis consistent\nwith an enlargement of the tangent vector space to its hyperbolic\ncomplex\nextension\n\\cite{Kunstatter+Yates:1981,Mann:1989,Kunstatter+Moffat+Malzan:1983}.\nIt is important to note that the action cannot support the\nadditional Bianchi\nidentities and gauge invariance related to the extension of the\ntangent bundle,\nsimply because the base manifold is locally diffeomorphic to\n${\\cal R}^4$,\nand the variational principle is based on an integration over\nthis real\nmanifold.\nAny change of gauge that mixes real and hyperbolic complex\ncovectors will cause\nthe volume element to pick up a hyperbolic complex piece, and the\naction will no\nlonger be real.\n\nThe hyperbolic complex structure is unnecessary for an\noperational discussion of\nthe theory (although it may be relevant for a more fundamental\ndiscussion of its\nphysical interpretation), and in this paper all quantities will\nbe considered\nreal, allowing antisymmetric contributions to the metric and\nconnection\ncoefficients.\nThe dynamics of the theory will be determined from the first\norder action\n($G=c=1$):\n\\begin{equation}\\label{mNGTAct}\nS=\\int d^4x\\{-{\\bf\ng}^{\\mu\\nu}R_{\\mu\\nu}^{\\text{\\tiny{NS}}}[\\Gamma]\n-{\\bf g}^{\\mu\\nu}\\partial_{[\\mu}W_{\\nu]}\n+{\\bf l}^\\mu\\Gamma_\\mu\n+\\case{1}{2}\\alpha{\\bf g}^{(\\mu\\nu)}W_\\mu W_\\nu\n+\\case{1}{4}m^2{\\bf g}^{[\\mu\\nu]}g_{[\\mu\\nu]}\\}+S_M,\n\\end{equation}\nwhere:\n\\begin{equation}\\label{mMAct}\n\\frac{\\delta S_M}{\\delta g^{\\mu\\nu}}={\\bf T}_{\\mu\\nu},\n\\end{equation}\nis the matter stress energy tensor that acts as a source in the\ngravitational\nfield equations.\nThe Ricci-like tensor for NGT appearing in (\\ref{mNGTAct}) is\nwritten as:\n\\begin{equation}\\label{NGTRicci}\nR_{\\mu\\nu}^{\\text{\\tiny{NS}}}=\n\\partial_\\alpha\\Gamma^\\alpha_{\\mu\\nu}-\\case{1}{2}\n(\\partial_\\nu\\Gamma^\\alpha_{\\mu\\alpha}+\\partial_\\mu\\Gamma^\\alpha_\n{\\alpha\\nu})\n+\\Gamma^\\alpha_{\\mu\\nu}\\Gamma^\\beta_{(\\alpha\\beta)}\n-\\Gamma^\\alpha_{\\beta\\nu}\\Gamma^\\beta_{\\mu\\alpha},\n\\end{equation}\nand a mass term for $g_{[\\;\\;]}$ has been included along with a\nterm quadratic\nin $W$ as new features of the action.\nAs will become clear shortly, $\\alpha$ may be fixed uniquely by\nrequiring good\nasymptotic behavior of perturbations about GR backgrounds.\nThe new parameter in the massive action ($m$) is an inverse\nlength scale that\nmust be constrained by experiment, and $l$ is a Lagrange\nmultiplier employed to\nenforce the vanishing of the trace of the antisymmetric part of\nthe connection\ncoefficients: $\\Gamma_\\mu=\\Gamma^\\alpha_{[\\mu\\alpha]}$.\n\nThe field equations related to the metric compatibility\nconditions are derived\nthrough the variations:\n\\begin{mathletters}\n\\label{mVars}\n\\begin{eqnarray}\n\\label{mVars:a}\n\\frac{\\delta S}{\\delta {\\bf l}^\\mu}&=&\\Gamma_\\mu=0,  \\\\\n\\label{mVars:b}\n\\frac{\\delta S}{\\delta W_\\mu}&=&\\partial_\\nu{\\bf\ng}^{[\\nu\\mu]}+\\alpha{\\bf\ng}^{(\\mu\\nu)}W_\\nu=0,  \\\\\n\\label{mVars:c}\n\\frac{\\delta S}{\\delta \\Gamma^\\gamma_{\\sigma\\omega}}&=&\n\\partial_\\gamma{\\bf g}^{\\sigma\\omega}\n-{\\bf g}^{\\sigma\\omega}\\Gamma^\\alpha_{(\\gamma\\alpha)}\n+{\\bf g}^{\\alpha\\omega}\\Gamma^\\sigma_{\\alpha\\gamma}\n+{\\bf g}^{\\sigma\\alpha}\\Gamma^\\omega_{\\gamma\\alpha}\\nonumber  \\\\\n&&\\quad -\\case{1}{2}\\delta^\\omega_\\gamma\n(\\partial_\\alpha{\\bf g}^{\\sigma\\alpha}\n+{\\bf g}^{\\alpha\\beta}\\Gamma^\\sigma_{\\alpha\\beta}\n-{\\bf l}^\\sigma)\n-\\case{1}{2}\\delta^\\sigma_\\gamma\n(\\partial_\\alpha{\\bf g}^{\\alpha\\omega}\n+{\\bf g}^{\\alpha\\beta}\\Gamma^\\omega_{\\alpha\\beta}\n+{\\bf l}^\\omega).\n\\end{eqnarray}\n\\end{mathletters}\nContracting on either index of (\\ref{mVars:c}) and solving for\nthe\n(anti-)symmetric parts of the divergence of the densitized\ninverse metric\nresults in the determination of the Lagrange multiplier using\n(\\ref{mVars:b}):\n${\\bf l}^\\sigma=\\frac{\\alpha}{3}{\\bf\ng}^{(\\sigma\\omega)}W_\\omega$.\nThis also allows one to simplify the Kronecker-$\\delta$ terms,\nand determine the\ncompatibility conditions in undensitized form as:\n\\begin{equation}\\label{Comp2m}\n\\partial_\\gamma g_{\\mu\\nu}\n-g_{\\mu\\alpha}\\Gamma^\\alpha_{\\nu\\gamma}\n-g_{\\alpha\\nu}\\Gamma^\\alpha_{\\gamma\\mu}\n=\\case{2}{3}\\alpha\n(g_{\\mu[\\gamma}g_{\\alpha]\\nu}\n+\\case{1}{2}g_{\\mu\\nu}g_{[\\gamma\\alpha]})g^{(\\alpha\n\\beta)}W_\\beta,\n\\end{equation}\nwhere the inverse of the metric has been defined by\n$g_{\\mu\\alpha}g^{\\alpha\\nu}=\\delta^\\nu_\\mu$, which has been used\nin order to\nrewrite the compatibility conditions in terms of the components\nof the metric\n$g_{\\mu\\nu}$.\n\nThe remaining field equations derived from the variation of the\naction with\nrespect to $g^{\\mu\\nu}$ may be written as:\n\\begin{equation}\\label{mNGTFeq}\n{\\cal R}_{\\mu\\nu}:=R_{\\mu\\nu}^{\\text{\\tiny{NS}}}\n+\\partial_{[\\mu}W_{\\nu]}-\\case{1}{2}\\alpha W_{\\mu}W_{\\nu}\n-\\case{1}{4}m^2(g_{[\\mu\\nu]}\n-g_{\\alpha\\mu}g_{\\nu\\beta}g^{[\\alpha\\beta]}\n+\\case{1}{2}g_{\\nu\\mu}g^{[\\alpha\\beta]}g_{[\\alpha\\beta]})\n=T_{\\mu\\nu}-\\case{1}{2}g_{\\nu\\mu}T,\n\\end{equation}\nwhere $T=g^{\\mu\\nu}T_{\\mu\\nu}$, and the tensor ${\\cal R}$ has\nbeen\nintroduced in order to simplify the discussion of the field\nequations.\nOne may translate the conventions used here to those in\n\\cite{Moffat:1994} by\ntaking $W\\rightarrow-\\frac{2}{3}W, T_{\\mu\\nu}\\rightarrow -8\\pi\nT_{\\mu\\nu}$,\n$\\alpha\\rightarrow -\\frac{9}{4}\\sigma$, and adjusting the\ndefinitions of the\ninverse metric: $g^{\\mu\\nu}\\rightarrow g^{\\nu\\mu}$.\nTo see the equivalence of the action, one further needs to\nrewrite $\\Gamma$ in\nterms of the unconstrained $W$ connection, and drop the\ncontribution from the\nLagrange multiplier $l$.\n\nThe action for massless NGT is given by (\\ref{mNGTAct}) with\n$m=\\alpha=0$.\nAs will be demonstrated, the new terms have been introduced in\norder to make all\nskew modes short ranged when considering perturbations about GR\nbackgrounds.\nOne performs this expansion about a symmetric, Ricci-flat\nbackground, where one\nassumes that all background curvatures fall off at worst as 1\/r\nas\n$r\\rightarrow\\infty$.\nThis allows one to talk sensibly of energy-momentum and decompose\nfields via a\nspin projection, so that higher order poles and negative energy\n(ghost) modes\nmay be identified, as well as avoiding the full nonsymmetric\nstructure of a more\ngeneral background that would make the analysis far more\ncomplicated.\nOne considers a perturbation of all quantities about a symmetric\nGR background\nas in \\cite{Kelly:1991}:\n\\begin{eqnarray}\\label{foexp}\ng_{\\mu\\nu}&&\\rightarrow g_{(\\mu\\nu)}+h_{\\mu\\nu}, \\nonumber \\\\\n\\Gamma^\\alpha_{\\mu\\nu}&&\\rightarrow \\{\\stackrel{{\\scriptstyle\n\\alpha}}{{\\scriptstyle \\mu\\nu}}\\}\n+\\gamma^\\alpha_{\\mu\\nu},\n\\end{eqnarray}\nwhere $W$, $l$ and $T$ are are considered to be first order in\nthe perturbation\n(as the background is assumed to be Ricci-flat, there one has\n$T_{\\mu\\nu}=0$).\nAs usual, indices will be `raised' and `lowered' by the symmetric\nbackground\nmetric, and the covariant derivative $\\nabla$ is associated with\nthe background\nChristoffel symbols $\\{\\;\\}$ determined from the background\nmetric in the usual\nmanner.\nCorrections to the background curvatures, field equations and\nLagrangian at each\norder in the perturbation will be indicated by a superscript as:\n${}^0\\! R, {}^1\\! R\\cdots$.\n\nThe first order correction to the compatibility equation\n(\\ref{Comp2m}) can be\nsolved explicitly for $\\gamma$ to yield:\n\\begin{equation}\\label{lcomp}\n\\gamma^\\alpha_{\\mu\\nu}=\\case{1}{2}g^{\\alpha\\beta}\n(\\nabla_\\nu h_{\\beta\\mu}+\\nabla_\\mu h_{\\nu\\beta}-\\nabla_\\beta\nh_{\\mu\\nu})\n+\\case{2}{3}\\alpha\\delta^\\alpha_{[\\mu}W_{\\nu]},\n\\end{equation}\nand $\\gamma_\\mu=\\gamma^\\alpha_{[\\mu\\alpha]}$ is seen to vanish by\nthe\nlinearization of the skew divergence equation:\n\\begin{equation}\\label{ldiv}\n-\\nabla_\\nu h^{[\\mu\\nu]}=\\alpha W^\\mu.\n\\end{equation}\n\nIn massless NGT \\cite{Moffat:1990}, one had\n(\\ref{lcomp},\\ref{ldiv}) with\n$\\alpha=0$, and hence there was no relation between the metric\ndegrees of\nfreedom and those of $W$.\nThe skew part of the linearization of equation (\\ref{mNGTFeq})\nwith $m=0$ as\nwell as $\\alpha=0$ became:\n\\begin{equation}\\label{NGTlFeq}\n{}^1\\! {\\cal R}_{[\\mu\\nu]}={}^1\\!\nR_{[\\mu\\nu]}+\\partial_{[\\mu}W_{\\nu]}\n=\\nabla_\\alpha\\gamma^\\alpha_{[\\mu\\nu]}+\\partial_{[\\mu}W_{\\nu]}\n=T_{[\\mu\\nu]}.\n\\end{equation}\nThe symmetric contribution, are the equations for a metric\nperturbation in GR\n\\cite{Wald:1984}, and will be ignored in the remainder of this\nsection.\nUsing (\\ref{ldiv}) with $\\alpha=0$, the assumption that the\nbackground is\nRicci-flat (${}^0\\! R_{\\mu\\nu}=0$), and the commutation\nrelation (for\nan arbitrary tensor $B$):\n\\begin{equation}\\label{commcovder}\n\\nabla_{[\\alpha}\\nabla_{\\beta]}B_{\\mu\\nu}\n=-\\case{1}{2}(B_{\\omega\\nu}{}^0\\! R^\\omega_{\\;\\;\\mu\\alpha\\beta}\n+B_{\\mu\\omega}{}^0\\! R^\\omega_{\\;\\;\\nu\\alpha\\beta}),\n\\end{equation}\n(\\ref{NGTlFeq}) simplifies to:\n\\begin{equation}\\label{lNGTf}\n\\nabla^\\alpha\\nabla_\\alpha h_{[\\mu\\nu]}\n-2\\partial_{[\\mu}W_{\\nu]}\n-4{}^0\\! R^{\\alpha\\;\\;\\beta}_{\\;\\;\\mu\\;\\;\\nu}h_{[\\alpha\\beta]}\n=\\nabla^\\alpha F_{\\mu\\nu\\alpha}\n-2\\partial_{[\\mu}W_{\\nu]}-8{}^0\\!\nR^{\\alpha\\;\\;\\beta}_{\\;\\;\\mu\\;\\;\\nu}h_{[\\alpha\\beta]}\n=-2T_{[\\mu\\nu]}.\n\\end{equation}\nThe second form is given in terms of the curl of the skew metric\n($F_{\\gamma\\mu\\nu}=\\partial_\\gamma h_{\\mu\\nu}+\\partial_\\nu\nh_{\\gamma\\mu}+\\partial_\\mu h_{\\nu\\gamma}$), in order to more\neasily demonstrate\nthe result found previously by Damour, Deser, and McCarthy\n\\cite{Damour+Deser+McCarthy:1992,Damour+Deser+McCarthy:1993}\n(using the fact\nthat $\\nabla^\\gamma\\nabla^\\nu F_{\\mu\\nu\\gamma}=0$ about a\nRicci-flat\nbackground), that these antisymmetric perturbations will in\ngeneral have bad\nasymptotic behavior.\nAlthough written as if the skew metric were a gauge field, the\npresence of the\ncurvature coupling term implies that the associated gauge\ninvariance is not\npresent \\cite{Kelly:1992}, and one is not allowed to make any\nchoice of gauge in\norder to simplify this sector.\nOne proceeds by taking the divergence of (\\ref{lNGTf}) and\nchoosing the gauge\n$\\nabla^\\alpha W_\\alpha=0$ (the theory had a $U(1)$ invariance as\n$W$ only\nappeared in the action in a curl) to find:\n\\begin{equation}\\label{lNGTdiv}\n\\nabla^\\alpha\\nabla_\\alpha W_\\mu\n-8\\nabla^\\nu[{}^0\\!\nR^{\\alpha\\;\\;\\beta}_{\\;\\;\\mu\\;\\;\\nu}h_{[\\alpha\\beta]}]\n=-2\\nabla^\\nu T_{[\\mu\\nu]}.\n\\end{equation}\nNotice that the background curvature acts as a source here, so\nthat even if one\npostulates that the matter source is conserved, this curvature\ncoupling (and in\ngeneral other nonlinear terms) will still exist as a source,\ncausing $W$ to\npropagate and have asymptotic behavior consistent with a massless\nfield ($\\sim\n1\/r$ along the forward light cone).\nUsing this asymptotic behavior to determine $h$ from\n(\\ref{lNGTf}) results in a\nsource with $\\sim 1\/r$ behavior, causing the field $h$ not to\nfall off as\n$r\\rightarrow\\infty$ along the forward light cone.\nThis analysis is correct since one has assumed that the\nbackground curvature\nfalls off fast enough, and hence the potential term in\n(\\ref{lNGTdiv}) can be\ntreated as a source, without changing the asymptotic behavior of\nthe fields.\nIt must be stressed that one is assuming that the background {\\em\nand} the\nradiative fields fall off at least as fast as $\\sim 1\/r$, and\nwhat has actually\nbeen derived here is a contradiction of this, since $h$ is driven\nto a constant\nand can no longer be considered as a perturbative mode.\n\nAny analysis of this sort also supposes that a solution of the\nlinearized field\nequations does in fact correspond to an exact solution of the\nfull nonlinear\nfield equations.\nThis is the case in GR \\cite{Wald:1984,Fischer+Marsden:1979} at\nleast for\nsource free equations, but no such result exists yet for any of\nthe models\nconsidered here.\nIt is possible to take the stance that NGT is not\nlinearization-stable, so that\nthis sort of analysis necessarily produces spurious results that\ndo not\ncorrespond to global solutions, but then one is denying the\nability to do any\nsort of perturbative analysis without the existence of an exact\nsolution to back\nit up.\nDue to the scarcity of solutions, and the apparent existence of\nweak-field\nperturbative situations, this would seem an unreasonable position\nto adopt.\n\nThis result is not confined to curved backgrounds, and in fact\nthe analysis\nabout Minkowski space will serve to explicitly demonstrate the\nhigher order pole\nleading to bad fall-off.\nSince this curvature coupling, and any nonlinear effects in\ngeneral, will act as\na nonconserved source term in the skew sector, the linearization\nthat correctly\nrepresents the full nonlinear field equations in the asymptotic\nregion will have\na nonconserved source term.\nThis is no more than the observation that once again, the full\nNGT action does\nnot possess any form of additional gauge invariance in the skew\nsector.\nA gauge field coupled to a source (or matter) in a non-gauge\ninvariant manner\nmay have drastically different behavior than the empty space and\napparently\ngauge invariant field equations, if indeed the action is\nconsistent at all.\nAny analysis that attempts to determine the propagator or\nasymptotic behavior of\nthe field must take the form of the source (or coupling to other\nfields) into\naccount.\n\nA trivial example of this is given by considering the Maxwell\naction.\nCoupling the usual gauge invariant kinetic terms to a\nnonconserved source gives\nan inconsistent set of field equations,\nand adding some sort of gauge fixing term will give a consistent\nset of\nequations, but the scalar ghost mode will be excited and\ndepending on the gauge\nthere may be higher order poles in the solution.\nThe linearized field equations considered outside the source\nresemble those of\nthe gauge invariant theory in a particular gauge, but treating\nthem as such will\nnot give asymptotic behavior that follows from coupling to the\nnonconserved\nsource.\nThe situation in NGT is more akin to enforcing the gauge\ncondition in the action\nthrough the use of an auxiliary field as: $b\\partial_\\mu A^\\mu$\n\\cite{Nakanishi:1967,Goto+Obara:1967}.\nSource conservation and absence of ghosts relies on whether or\nnot the scalar\nLagrange multiplier field $b$ has a source or not in the wave\nequation that\ndetermines it, and is thus a global question.\nGiven that the source for $A$ is not conserved, then $b$\npropagates and there\nare higher order poles in the solution for $A$, leading to fields\nthat do not\nfall off as $r\\rightarrow\\infty$ along the forward light cone.\n\nConsidering the field equations for massless NGT linearized from\n(\\ref{lNGTf})\nabout Minkowski space:\n\\begin{mathletters}\n\\label{lNGTflat}\n\\begin{eqnarray}\n\\label{lNGTflat:a}\n\\square h_{[\\mu\\nu]}-2\\partial_{[\\mu}W_{\\nu]}\n&=&-2T_{[\\mu\\nu]}, \\\\\n\\label{lNGTflat:b}\n\\partial_\\nu h^{[\\mu\\nu]}&=&0,\n\\end{eqnarray}\n\\end{mathletters}\n($\\partial^\\nu T_{[\\nu\\mu]}\\neq 0$),\none may take a divergence (to find a wave equation for $W$) or a\ncurl (to remove\n$W$) of the first of these, resulting in the unique consistent\nsolution:\n\\begin{eqnarray}\\label{LCh}\nh_{[\\mu\\nu]}&=&-2\\square^{-1}\\left[T_{[\\mu\\nu]}\n+2\\square^{-1}\\partial^\\alpha\\partial_{[\\mu}T_{[\\alpha\\nu]]}\n\\right], \\nonumber \\\\\nW_\\mu&=&-2\\square^{-1}\\partial^\\nu T_{[\\mu\\nu]}.\n\\end{eqnarray}\nThe presence of the higher order pole (and consequent bad\nfall-off) is now\nobvious from the presence of the $\\square^{-2}$ term in the Green\nfunction\nsolution, and is no more than the result of vanNieuwenhuizen\n\\cite{vanNieuw:1973} who showed that the only healthy quadratic\nactions built of\nantisymmetric tensor fields are the so-called Kalb-Ramond\n\\cite{Kalb+Ramond:1974} (massless or massive) actions.\nOne also notes that there are 5 modes here: 3 in $h$, since 3 are\ndetermined\nalgebraically by the second equation in (\\ref{lNGTflat}), and 2\nin $W$ due to\nthe previously mentioned $U(1)$ gauge invariance\n\\cite{Kunstatter+Leivo+Savaria:1984}.\nIf it is assumed that the (matter) source is conserved, the\nhigher order poles\nare removed at linear order, but show up in the second order\ncorrection to the\nfields, again causing a breakdown of the perturbative analysis.\n\nThis analysis correctly represents the asymptotic behavior of the\nfields\n($W,h$), and is equivalent to equation (18) of\n\\cite{Mann+Moffat:1982}, where\nthe higher order pole resides in the projection operator:\n$P(1^+)$.\nOne also sees the true propagating nature of $W$, and this is\nborne out by the\nanalysis in \\cite{Moffat:1980,McDow+Moffat:1982} where there are\nfive degrees of\nfreedom evolving from each Cauchy surface, the extra two of which\nare associated\nwith the field $W$.\nThat a Lagrange multiplier is propagating merely signifies that\nit is a\ndetermined multiplier, with its evolution derived from the field\nequations\n\\cite{Kelly:1991} and not freely fixable as was done in\n\\cite{Moffat:1981,Mann:1986} and in the next to last section of\n\\cite{Kunstatter+Leivo+Savaria:1984} where {\\it ad hoc}\nconstraints were imposed\non the linearized theory in order to obtain the dynamics of a\nKalb-Ramond\ntheory.\nThat these constraints cannot exist is clear from the lack of\ngauge invariance\nin the full NGT action.\n\nThe result of vanNieuwenhuizen does however motivate a potential\nsolution to\nthis problem, since the massive Kalb-Ramond theory does not\nrequire a conserved\ncurrent and yet has no ghost modes, higher order poles or\ntachyons.\nThe additional terms in the action for mNGT (\\ref{mNGTAct}) are\nintroduced in\norder to allow the linearized field equations of NGT to take on\nthis form in the\nantisymmetric sector.\nThese two terms play slightly different roles: the $W^2$ term\ncauses $W$ to be\ndetermined in terms of metric functions directly ($\\alpha$ is\nfixed in order to\nfind the correct form of the kinetic energy terms), and the mass\nterm for\n$g_{[\\;]}$ makes the skew sector short-ranged, and ensures that\nthe linearized\nfield equations remain consistent when expanding about a flat\nbackground.\n\nThus mNGT should  have a linearization about Minkowski space of\nthe form:\n\\begin{equation}\\label{lmNGTflat}\n\\partial^\\alpha F_{\\mu\\nu\\alpha}\n+m^2h_{[\\mu\\nu]}\n=J_{[\\mu\\nu]}.\n\\end{equation}\nThe solution to (\\ref{lmNGTflat}) can be found by taking a\ndivergence and\nsubstituting back in to find:\n\\begin{eqnarray}\\label{MKRs}\nh_{[\\mu\\nu]}&=&\\square^{-1}\\left[J_{[\\mu\\nu]}\n+\\case{2}{m^2}\\partial^\\alpha\\partial_{[\\mu}J_{[\\alpha\\nu]]}\n\\right],\\nonumber  \\\\\n\\partial^\\nu h_{[\\mu\\nu]}&=&\\case{1}{m^2}\\partial^\\nu\nJ_{[\\mu\\nu]}.\n\\end{eqnarray}\nThe higher order poles have disappeared, and it can be shown that\nthe linearized\nHamiltonian is weakly positive definite and that ghost modes are\nremoved through\nthe algebraic conditions that couple them locally to the source\nin (\\ref{MKRs}).\nAbout a more general background one can allow a curvature\ncoupling term, since\nit will not affect the behavior of the fields asymptotically once\nthe background\nis assumed to fall off appropriately.\nChoosing the theory that results in this behavior in the\nlinearized theory will\nfix $\\alpha$ uniquely.\n\nReturning now to the field equations of mNGT expanded about a GR\nbackground\nfollowing from (\\ref{mNGTFeq}), one finds:\n\\begin{equation}\\label{minit}\n{}^1\\! {\\cal R}_{\\mu\\nu}={}^1\\! R_{\\mu\\nu}\n+\\case{1}{\\alpha}\\nabla_{[\\mu}\\nabla^\\alpha h_{[\\alpha\\nu]]}\n-\\case{1}{2}m^2h_{[\\mu\\nu]}=T_{\\mu\\nu}-\\case{1}{2}g_{\\nu\\mu}T,\n\\end{equation}\nwhere the first order correction to the `Ricci' tensor is given\nby:\n\\begin{equation}\\label{1Ricci}\n{}^1\\! R_{\\mu\\nu}=\n\\nabla_\\alpha\\gamma^\\alpha_{\\mu\\nu}\n-\\nabla_{(\\nu}\\gamma^\\alpha_{\\alpha\\mu)}.\n\\end{equation}\nAgain ignoring the symmetric GR perturbations, the antisymmetric\npart of\n(\\ref{minit}) is:\n\\begin{eqnarray}\\label{Aspt}\n&&\\nabla_\\alpha\\gamma^\\alpha_{[\\mu\\nu]}\n+\\case{1}{\\alpha}\\nabla_{[\\mu}\\nabla^\\alpha h_{[\\alpha\\mu]]}\n-\\case{1}{2}m^2h_{[\\mu\\nu]}  \\nonumber \\\\\n\\quad\\quad &=&-\\case{1}{2}(\\nabla^\\alpha\nF_{\\mu\\nu\\alpha}+m^2h_{[\\mu\\nu]})\n-2\\nabla^\\alpha\\nabla_{[\\mu}h_{[\\alpha\\nu]]}\n+\\case{1}{\\alpha}\n[1+\\case{2}{3}\\alpha]\n\\nabla_{[\\mu}\\nabla^\\alpha h_{[\\alpha\\nu]]}=T_{[\\mu\\nu]}.\n\\end{eqnarray}\nRequiring that this reduce to the massive Kalb-Ramond field\nequations\n(\\ref{lmNGTflat}) determines the (previously arbitrary) coupling:\n$\\alpha=3\/4$.\nThe last two terms can be reduced to a curvature term to give:\n\\begin{equation}\\label{equation:KR curved}\n\\nabla^\\alpha F_{\\mu\\nu\\alpha}\n+m^2h_{[\\mu\\nu]}\n-4{}^0\\!\nR_{\\;\\;[\\mu\\;\\;\\nu]}^{\\alpha\\;\\;\\beta}h_{[\\alpha\\beta]}\n=-2T_{[\\mu\\nu]},\n\\end{equation}\nso the skew sector perturbations are well-behaved when perturbing\nabout any\nasymptotically flat GR background.\n\nExpanding the action (\\ref{mNGTAct}) to second order (ignoring\nsurface terms)\ngives:\n\\begin{equation}\\label{FL2}\n{}^2\\! {\\cal L}=-{}^2\\! R-\\case{1}{2}h{}^1\\! R\n+h^{\\mu\\nu}{}^1\\! R_{\\mu\\nu}\n+h^{\\mu\\nu}\\partial_{[\\mu}W_{\\nu]}+\\case{1}{2}\\alpha W^\\mu W_\\mu\n+l^\\mu\\Gamma_\\mu\n-\\case{1}{4}m^2h^{[\\mu\\nu]}h_{[\\mu\\nu]}\n\\end{equation}\nand once compatibility is imposed, followed by the removal of\n$W$, this becomes:\n\\begin{equation}\\label{mLNGT2}\n{}^2\\! {\\cal L}=\\case{1}{12}F^{\\mu\\nu\\gamma}F_{\\mu\\nu\\gamma}\n-\\case{1}{4}m^2h^{[\\mu\\nu]}h_{[\\mu\\nu]}\n-\\nabla^\\gamma h^{[\\mu\\nu]}\\nabla_\\nu h_{[\\gamma\\mu]}\n-(\\case{1}{2\\alpha}+\\case{1}{3})\n\\nabla_\\nu h^{[\\mu\\nu]}\\nabla^\\gamma h_{[\\mu\\gamma]}.\n\\end{equation}\nChoosing $\\alpha=3\/4$ results in kinetic terms identical to those\nof Kalb-Ramond\ntheory on a GR background, giving the skew sector action:\n\\begin{equation}\\label{mNGTGRB}\n{}^2\\! {\\cal L}=\\case{1}{12}F^{\\mu\\nu\\gamma}F_{\\mu\\nu\\gamma}\n-\\case{1}{4}m^2h^{[\\mu\\nu]}h_{[\\mu\\nu]}\n-h^{[\\mu\\nu]}h^{[\\alpha\\beta]} {}^0\\! R_{\\alpha\\mu\\beta\\nu},\n\\end{equation}\nwhich reproduces the linearized field equations (\\ref{equation:KR\ncurved}).\nThus the massive NGT action will be (\\ref{mNGTAct}) with\n$\\alpha=3\/4$\n\\cite{Moffat:1994}, giving the action (\\ref{mNGTGRB}) for\nperturbations about a\nGR background and guaranteeing good asymptotic behavior for these\nfields.\n\nAlthough it has been established that the perturbation equations\nabout a GR\nbackground are a consistent system resulting in good fall-off for\nthe skew\nsector, it is not clear whether an asymptotic perturbation\nactually corresponds\nto a global solution (linearization stability).\nThe (seemingly contrived) asymptotic limit of gauge invariant\nkinetic terms\ncannot be reflected in the full action, since there is no room\nfor the\nadditional gauge invariance in theories constructed from\nantisymmetric fields in\nthis manner.\nThis means that in general that one expects more (perhaps all 6)\ndegrees of\nfreedom in the skew sector evolving as degrees of freedom in a\nCauchy analysis,\nwhereas in any spacetime that has an asymptotically flat region\nonly three will\nsurvive.\nThis situation could be similar to that found in\n\\cite{Isenberg+Nestor:1977},\nwhere vector fields were seen to increase their degrees of\nfreedom when\ngravitational effects are taken into account.\nIn order to obtain an asymptotically flat spacetime (with the\nreduced degrees of\nfreedom of the vector fields) from physically reasonable initial\ndata, the\nevolution equations were seen to have to encounter singularities.\nThis is generally considered to be a sign of instability, and\ncertainly not a\ndesirable feature in any theory.\nPerturbations about NGT backgrounds should also be considered,\nsince the\nphysically interesting NGT solutions are most likely those that\nare not `close'\nto a GR solution \\cite{Cornish+Moffat:1994}.\nIt is hoped that a more complete analysis of the general\nspherically symmetric\nsystem should be able to say something about this issue, since it\nwill certainly\ntell one how many degrees of freedom survive and how they couple\nto external\nfields, and hopefully something about how the system may or may\nnot approach an\nasymptotically flat spacetime.\n\nIt is also true that the form of the action (\\ref{mNGTAct}) is\nfar from unique.\nIn particular, one could replace the $W^2$ term with some\ncombination of $W^2$\nand $\\frac{1}{\\sqrt{-g}}g_{(\\mu\\nu)}\\partial_\\alpha[{\\bf\ng}^{[\\mu\\alpha]}]\\partial_\\gamma[{\\bf g}^{[\\nu\\gamma]}]$, giving\nthe same\nperturbation equations (\\ref{equation:KR curved}), and resulting\nin an arbitrary\ncoupling constant in the action.\nFurther, since there is nothing preventing one from adding\n$\\Gamma_{[\\;]}$ terms\n(they are tensors) or even infinite strings of terms of the form:\n$g_{(\\alpha\\beta)}g^{(\\beta\\gamma)}\\cdots$ or\n$g_{[\\alpha\\beta]}g^{[\\beta\\gamma]}\\cdots$ (which conveniently\ndisappear in the\nasymptotic expansion), there is clearly an infinite number of\nactions that do\nthis.\nThese examples seem extremely unnatural and will not be\nconsidered further here,\nalthough the results of this paper would not change significantly\nfor these more\ngeneral actions.\n\n\\section{Spherically Symmetric Perturbation of the Schwarzschild\nsolution in a\nCoordinate basis}\n\\label{section:Schw pert}\n\nThe absence of a Birkhoff theorem may be derived from the\nperturbation equations\n(\\ref{equation:KR curved}) developed in Section\n\\ref{section:Massive NGT}.\nIn general the spherically symmetric fields in the skew sector\nwill not be\nstatic, although the symmetric sector will remain static in the\nperturbation\nabout the GR solution considered here.\nThe background metric is Schwarzschild with (coordinate basis)\nmetric written\nas: $g={\\rm diag}(A(r),-1\/A(r),-r^2,-r^2\\sin^2(\\theta))$, where\n$A(r)=1-2M_s\/r$\nand $M_s$ is the Schwarzschild mass parameter.\nThe perturbation considered will be one that is spherically\nsymmetric but not\nnecessarily static.\nA killing vector analysis yields the general form of the\nspherically symmetric\nperturbation:\n\\begin{equation}\\label{equation:perturbation metric}\n| h_{\\mu\\nu} |=\\left[\n\\begin{array}{cccc}\nh_{00}(t,r)&h_{(01)}(t,r)+h_{[01]}(t,r)&0&0\\\\\nh_{(01)}(t,r)-h_{[01]}(t,r)&h_{11}(t,r)&0&0\\\\\n0&0&h_{22}(t,r)&h_{[23]}(t,r)\\sin(\\theta)\\\\\n0&0&-h_{[23]}(t,r)\\sin(\\theta)&h_{22}(t,r)\\sin^2(\\theta)\n\\end{array}\\right].\n\\end{equation}\nMaking a change of coordinates of the background geometry is\nequivalent to\nmaking a change of gauge on the perturbation: $\\delta\nh=\\pounds_\\varepsilon[g]$,\nwhere\n$\\varepsilon$ is the spherically symmetric vector gauge parameter\ngenerating\ndiffeomorphisms between spherically symmetric spacetimes.\nThis allows one to simplify the form of the perturbation by a\nsuitable choice of\ngauge.\nChoosing the gauge parameter as:\n\\begin{mathletters}\n\\begin{eqnarray}\n\\varepsilon^0&=&-\\int\\left(\\frac{h_{(01)}}{A(r)}-\\frac{\\partial_t\n[h_{22}]}{2rA(r)\n^2}\\right)dr,\n \\\\\n\\varepsilon^1&=&\\frac{h_{22}}{2r},\n\\end{eqnarray}\n\\end{mathletters}\nremoves the $\\theta-\\theta$, $\\phi-\\phi$, and symmetric $t-r$\nperturbations altogether, and a remaining gauge transformation\n$\\varepsilon^0=\\epsilon(t)$ allows one to remove an\narbitrary function from the $t-t$ component of the form: $\\delta\nh_{00}=2A(r)\\epsilon(t)$.\n\nThe field equations will be written without the source terms for\nsimplicity\nalthough it is straightforward to include them and relate the\nconstants of\nintegration to properties of the source.\nFirst reviewing how the symmetric (in this case identically GR)\nperturbations\nbecome static, it is simplest to begin with the field equation:\n${}^1\\! {\\cal R}_{(01)}=0$, which implies:\n\\begin{equation}\n\\partial_t[h_{11}(t,r)]=0,\n\\end{equation}\nimmediately showing that $h_{11}$ must be static.\nBy considering ${}^1\\! {\\cal R}_{22}=0$, it is determined to\nbe:\n\\begin{equation}\nh_{11}(t,r)=-\\frac{2\\delta M_s}{rA^2},\n\\end{equation}\nwhere the integration constant has been combined with $M_s$ and\ninterpreted as a\nperturbation of\nthe Schwarzschild mass parameter: $\\delta M_s$.\nThen one considers: $g^{00}\\,{}^1\\! {\\cal\nR}_{00}-g^{11}\\,{}^1\\! {\\cal R}_{11}=0$, leading to:\n\\begin{equation}\nh_{00}(t,r)=B(t)A-\\frac{2\\delta M_s}{r},\n\\end{equation}\nalso giving a contribution arising from the perturbed mass\nparameter, as well as\nan arbitrary function of time as an integration constant,\nremovable by the\nremaining choice of gauge noted above with:\n$\\epsilon(t)=-B(t)\/2$.\nThus one has that the symmetric perturbations are static and\ninterpretable as\nbeing due to a small change in the total energy of the system:\n$\\delta M_s$.\n\nIn the skew sector, the $t-r$ field equation gives:\n\\begin{equation}\\label{equation:h01}\n{}^1\\! {\\cal\nR}_{[01]}=\\case{1}{2}(2A^{\\prime\\prime}-m^2)h_{[01]}=0,\n\\end{equation}\nfrom which one must conclude that $h_{[01]}$ vanishes outside the\nsource.\n(Primes will denote the derivative of a function of one variable\nwhere\nconvenient.)\nIn massless Kalb-Ramond theory, this is the surviving spherically\nsymmetric\nghost mode which in that case is pure gauge.\nWhen a mass term is added, although these modes are now no longer\npure gauge,\nthey do not propagate since they are locally coupled to the\nsource.\nIt is these modes that one eventually must worry about, since in\nthe full theory\nthey may play a nontrivial dynamical role.\nIn the $\\theta-\\phi$ sector:\n\\begin{equation}\n{}^1\\! {\\cal\nR}_{[23]}=-\\frac{1}{2}\\left\\{\\frac{1}{A}\\partial_t^2[h_{[23]}]\n-A\\partial_r^2[h_{[23]}]\n-(A^\\prime-\\frac{2A}{r})\\partial_r[h_{[23]}]\n+(\\frac{4}{r^2}(1-A)+m^2)h_{[23]}\\right\\}\\sin(\\theta)=0.\n\\end{equation}\nIn order to derive the asymptotic form of this perturbation, it\nis convenient to\ndefine $h_{[23]}=rf(t,r)$,  leading to:\n\\begin{equation}\\label{equation:Schw f}\n\\frac{1}{A}\\partial_t^2[f]-A\\partial_r^2[f]\n-A^\\prime\\partial_r[f]\n+(\\frac{3A^\\prime}{r}+\\frac{2A}{r^2}+m^2)f=0.\n\\end{equation}\nIntroducing the coordinate:\n\\begin{equation}\nr^*=\\int\\frac{dr}{A(r)}=r+2M_s\\ln(\\frac{r}{2M_s}-1),\n\\end{equation}\none obtains (after multiplying by $A$) the partial differential\nequation for $f$\nin normal form:\n\\begin{equation}\\label{equation:Wave 1}\n\\partial_t^2[f]-\\partial_{r^*}^2[f]\n+A(m^2+\\frac{2A}{r^2}+\\frac{3A^\\prime}{r})f=0\n\\end{equation}\nwhere $r$ is considered as a function of $r^*$ as are $A(r)$ and\n$\\partial_rA(r)$, and the perturbation $f=f(t,r^*)$.\nIn this form it is obvious that (\\ref{equation:Wave 1}) is a\nhyperbolic wave\nequation, and that the field $f$ is therefore nonlocally related\nto the source.\n\nUsing the fact that $1\/r-1\/r^*\\sim o(1\/(r^*)^2)$ as\n$r^*\\rightarrow\\infty$, one\nkeeps only the constant mass term asymptotically, as all other\npotential terms\nwill be dominated by it.\nThis leaves the massive scalar wave equation to determine the\nasymptotic form of\nthe perturbation:\n\\begin{equation}\\label{equation:Wave equation}\n\\partial_t^2[f]-\\partial_{r^*}^2[f]+m^2f\\sim 0.\n\\end{equation}\nThe static solution of this is easily seen to have the asymptotic\nform:\n\\begin{equation}\nh_{[23]}(r)\\sim F_0\\frac{r}{m}e^{-mr^*}\n\\sim F_0\\frac{r}{m}e^{-mr} (\\frac{r}{2M_s})^{-2mM_s},\n\\end{equation}\nwhere a factor of $m$ has been introduced in order to make the\nconstant $F_0$\ndimensionless.\nThe general time dependent case may be handled by noting that the\nretarded and\nadvanced Green functions for the massive scalar wave equation\n\\cite{Bogoliubov+Shirkov:1959} ($x^2=t^2-\\vec{x}^2$):\n\\begin{equation}\\label{equation:Green functions}\nD^{{\\rm ret, adv}}(x)= D^{{\\rm ret, adv}}(t,r)=\n\\frac{1}{2\\pi}\\theta(\\pm\nx^0)\\left[\\delta(x^2)-\\frac{m\\theta(x^2)}{2\\sqrt{x^2}}J_1(m\\sqrt{\nx^2})\\right],\n\\end{equation}\ndepends only on $(t,r)$, and $r^*D^{{\\rm ret, adv}}(t,r^*)$ will\nsolve\n(\\ref{equation:Wave equation}).\nThe asymptotic behavior of $h_{[23]}$ is then determined from:\n\\begin{equation}\\label{equation:h23 wave solution}\nh_{[23]}\\sim rr^* D^{{\\rm ret, adv}}(t,r^*).\n\\end{equation}\nNote that the behavior on the light cone is determined from just\nthe massless\nGreen function $\\delta(x^2)$ \\cite{Itzykson+Zuber:1980}, and so\nit would appear\nthat $h_{[23]}$ will behave as $r$ as $r\\rightarrow\\infty$ along\nthe forward\nlight cone.\nThis is misleading, as it can be demonstrated explicitly\n\\cite{Reed+Simon:1979}\nthat for $C^\\infty$ initial data with compact spatial support, a\nmassive\nKlein-Gordon field is bounded everywhere by: $\\phi\\leq d(1+\\mid\nt\\mid)^{-3\/2}$,\nfor some constant $d$, and therefore cannot radiate energy.\nThis can also be understood by noting that because the field is\nmassive, the\neffects propagating on the light cone must be fields of infinite\nenergy, and\ngiven some physically reasonable source distribution, these\ninfinite energy\nmodes will not be excited.\n\nThe existence of time dependent solutions thus proves that\nBirkhoff theorem is\nnot valid in mNGT, although the short-ranged nature of the skew\nsector implies\nthat monopole radiation will not exist.\nThe symmetric sector has remained static in this system, but as\nwill be shown in\nSection \\ref{section:Wyman pert}, through a perturbation about an\napproximated\nmNGT solution, this will not be the case in general.\nThe perturbation equations about a mNGT background have not been\ngiven in\ncovariant form, primarily due to the complication involved\n(although it is\npossible in principle using a generalization of the inversion of\nthe\ncompatibility equation given in \\cite{Tonnelat:1982}).\nInstead the system may be developed in each case separately, and\nthe analysis\nsimplified  by considering the field equations in a vierbein\nframe given in the\nnext Section.\n\n\\section{Nonsymmetric Theories in a General Frame}\n\\label{section:GenBasis intro}\n\nThe structure of the compatibility relations and field equations\nin nonsymmetric\ntheories can be formulated in terms of components in a general\nmoving frame (in\nthe sense of global section of the general linear frame bundle\n${\\it\nGL}{\\cal M}$ of all linear frames over ${\\cal M}$).\nThe formalism given here is essentially a more systematic\ndevelopment of the\napproach in \\cite{Vlachynsky:1988}, and differs slightly from\nthat of Hlavaty\n\\cite{Hlavaty:1958} in that the (in general nonsymmetric in a\ncoordinate basis)\nconnection coefficients have been split up into a connection that\nis torsion\nfree, and another that is purely antisymmetric, instead of\ndefining two types of\ncovariant derivative, one associated with the NGT Christoffel\nsymbols, and\nanother that is in general non-symmetric and not in general\ntorsion-free.\nThe construction here has the advantage of only defining one\ncovariant\nderivative, and the fact that it is torsion-free implies that the\nantisymmetric\ncomponents in a general (non-coordinate) frame are related in the\nstandard way\nto the structure constants.\nIn a coordinate basis this is the usual split between the\nsymmetric and\nantisymmetric components, however it is easily generalized to any\nbasis by\nconsidering the antisymmetric components as a separate\nantisymmetric tensor, and\nthe symmetric components as a torsion-free but generally\nnon-compatible\nconnection.\n\nThis provides a simple way to split the GR and NGT contribution\nin weak field\nsituations, as well as generating computationally simpler systems\nto solve when\ninverting the compatibility relations.\nNote that although the formalism is developed for a general\nbasis, the\nspecialization to a vierbein basis (the reduction of ${\\it\nGL}{\\cal M}$ to\n${\\it L}{\\cal M}$, the Lorentz frame bundle consisting of all\nLorentz frames\nabove ${\\cal M}$) which will be utilized in the rest of this\npaper, is\naccomplished through the choice of the fiber metric as\n$g_{(\\;)}\\rightarrow\\eta$\nabove all points of the manifold.\nThis is possible in NGT for the same reason that it is possible\nin GR:\nmathematically formulating a physical theory in a diffeomorphism\ninvariant\nmanner will always allow the introduction of these general linear\nframes.\nThe reduction to Lorentz frames is also possible as one is\nassuming that the\nsymmetric part of the metric that one is attempting to\ndiagonalize is\nnondegenerate, allowing the reduction of the frame bundle.\nThis construction will be of importance when considering the\ncanonical analysis\nof NGT, as one would like to work in a surface compatible\n(generally\nnon-coordinate) basis in order to avoid specialization to a\nparticular choice of\ntime parameter fixed by the foliation of the manifold, and is\neasily applied to\nother systems with a nonsymmetric metric and connection\n\\cite{Mann:1986}.\n\n\\subsection{Metric, Compatibility and Curvature}\n\\label{section:GenBasis compat}\nThe compatibility conditions in a coordinate basis (\\ref{Comp2m})\nwill be\nwritten for convenience as:\n\\begin{equation}\\label{coordcomp}\n\\partial_\\gamma [g_{\\mu\\nu}]\n-g_{\\mu\\alpha}\\Gamma^\\alpha_{\\nu\\gamma}\n-g_{\\alpha\\nu}\\Gamma^\\alpha_{\\gamma\\mu}\n=-\\Delta^0_{\\gamma\\mu\\nu}\n\\end{equation}\nwhere $\\Delta^0$ depends only on the metric or quantities\ndirectly derivable\nfrom it (and possibly other quantities, but for the purposes of\nthis\nconstruction it does not depend on the connection coefficients).\nParallel transport (and the related covariant derivative) will\nthen be defined\nusing just the symmetric part of the coordinate basis connection,\nand its action\non the (coordinate) basis vectors is:\n\\begin{equation}\\label{parallaltransp}\n\\nabla_{e_\\alpha}[e]_\\beta=\\Gamma^\\gamma_{(\\alpha\\beta)}e_\\gamma,\n\\quad\n\\nabla_{e_\\alpha}[\\theta]^\\gamma=-\\Gamma^\\gamma_{(\\alpha\\beta)}\n\\theta^\\beta,\n\\end{equation}\nand the connection is split into a symmetric connection and an\nantisymmetric\ntensor:\n\\begin{equation}\\label{Connectionsplit}\n\\Gamma^\\gamma_{\\mu\\nu}\\rightarrow\\Gamma^\\gamma_{(\\mu\\nu)}\n+\\Lambda^\\gamma_{[\\mu\\nu]}.\n\\end{equation}\nThus $\\Gamma$ will refer from this point onwards to the\ntorsion-free (symmetric\nin a coordinate basis) part of the connection, and $\\Lambda$ to\nthe remaining\ntensor contribution.\nIn this way, $\\Gamma$ is a torsion-free (but non-compatible)\ncovariant\nderivative since:\n\\begin{equation}\\label{coordtorsion}\nT^\\gamma_{\\mu\\nu}\n=\\theta^\\gamma\\left[\\nabla_{e_\\mu}e_\\nu-\\nabla_{e_\\nu}e_\\mu\n-[e_\\mu,e_\\nu]\\right]\n=2\\Gamma^\\gamma_{[\\mu\\nu]}=0.\n\\end{equation}\nThe compatibility equation (\\ref{coordcomp}) then becomes:\n\\begin{equation}\\label{compat}\n\\nabla_{e_\\gamma}[g]_{\\mu\\nu}\n=e_\\gamma[g_{\\mu\\nu}]\n-g_{\\mu\\alpha}\\Gamma^\\alpha_{\\gamma\\nu}\n-g_{\\alpha\\nu}\\Gamma^\\alpha_{\\gamma\\mu}\n=g_{\\mu\\alpha}\\Lambda^\\alpha_{\\nu\\gamma}\n+g_{\\alpha\\nu}\\Lambda^\\alpha_{\\gamma\\mu}\n-\\Delta^0_{\\gamma\\mu\\nu},\n\\end{equation}\nwhere the basis vectors are just directional derivatives along\nthe coordinates:\n$e_\\gamma[\\;]=\\partial_\\gamma[\\;]$.\n\nWith this definition of the covariant derivative and related\nconnection\ncoefficients, the geometric curvature is found as usual from:\n\\begin{eqnarray}\\label{Riemann}\nR^\\alpha_{\\;\\beta\\mu\\nu}\n&=&\\theta^\\alpha[(\\nabla_{e_\\mu}\\nabla_{e_\\nu}\n-\\nabla_{e_\\nu}\\nabla_{e_\\mu}-\n\\nabla_{[e_\\mu,e_\\nu]})e_\\beta] \\nonumber \\\\\n&=&e_\\mu[\\Gamma^\\alpha_{\\nu\\beta}]\n-e_\\nu[\\Gamma^\\alpha_{\\mu\\beta}]+\n\\Gamma^\\gamma_{\\nu\\beta}\\Gamma^\\alpha_{\\mu\\gamma}\n-\\Gamma^\\gamma_{\\mu\\beta}\\Gamma^\\alpha_{\\nu\\gamma},\n\\end{eqnarray}\nand defining the two independent contractions:\n\\begin{mathletters}\n\\begin{eqnarray}\\label{Riccis}\nR^1_{\\mu\\nu}&=&R^\\alpha_{\\;\\mu\\alpha\\nu}\n=e_\\alpha[\\Gamma^\\alpha_{\\nu\\mu}]\n-e_\\nu[\\Gamma^\\alpha_{\\alpha\\mu}]+\n\\Gamma^\\gamma_{\\nu\\mu}\\Gamma^\\alpha_{\\alpha\\gamma}\n-\\Gamma^\\gamma_{\\alpha\\mu}\\Gamma^\\alpha_{\\nu\\gamma},  \\\\\nR^2_{\\mu\\nu}&=&R^\\alpha_{\\;\\alpha\\mu\\nu}\n=e_\\mu[\\Gamma^\\alpha_{\\nu\\alpha}]\n-e_\\nu[\\Gamma^\\alpha_{\\mu\\alpha}],\n\\end{eqnarray}\n\\end{mathletters}\nThe Ricci tensor will be defined as:\n\\begin{eqnarray}\\label{Ricci}\nR_{\\mu\\nu}&=&\nR^1_{\\mu\\nu}-\\case{1}{2}R^2_{\\mu\\nu}\\nonumber   \\\\\n&=&e_\\alpha[\\Gamma^\\alpha_{\\nu\\mu}]-\\case{1}{2}\n(e_\\nu[\\Gamma^\\alpha_{\\alpha\\mu}]\n+e_\\mu[\\Gamma^\\alpha_{\\nu\\alpha}])\n+\\Gamma^\\gamma_{\\nu\\mu}\\Gamma^\\alpha_{\\alpha\\gamma}\n-\\Gamma^\\gamma_{\\alpha\\mu}\\Gamma^\\alpha_{\\nu\\gamma}.\n\\end{eqnarray}\nThis particular combination is symmetric, and obviously reduces\nto the GR Ricci\ntensor when the NGT antisymmetric terms vanish.\nDecomposition of (\\ref{NGTRicci}) into $R_{\\mu\\nu}$ and another\nthat depends on\n$\\Lambda$ as:\n$R^{\\text{\\tiny{NS}}}_{\\mu\\nu}=R_{\\mu\\nu}+R^\\Lambda_{\\mu\\nu}$\nwhere ($\\Lambda_\\mu=\\Lambda^\\alpha_{\\mu\\alpha}$) gives:\n\\begin{equation}\nR^\\Lambda_{\\mu\\nu}=\\nabla_{e_\\alpha}[\\Lambda]^\\alpha_{\\mu\\nu}\n+\\nabla_{e_{[\\mu}}[\\Lambda]_{\\nu]}\n+\\Lambda^\\alpha_{\\mu\\beta}\\Lambda^\\beta_{\\nu\\alpha}.\n\\end{equation}\n\nA more general basis is introduced at each point on the manifold\nthrough\n$e_A=E_A^{\\;\\;\\mu}e_\\mu$, where $E$ is locally an element of\n$Gl(4,{\\cal\nR})$, and these bases are smoothly joined up to form sections of\nthe tangent\nbundle $T({\\cal M})$ \\cite{Choquet-Bruhat+:1989,Nakahara:1990}.\nThe general basis vectors are then given in terms of a coordinate\nbasis through\nthe vierbein-like quantities, which can be used to translate\ntensors from one\nchoice of basis to the other:\n\\begin{equation}\\label{Evbns}\ne_A=E_A^{\\;\\;\\mu}e_\\mu, \\quad\nE_A^{\\;\\;\\mu}E_B^{\\;\\;\\nu}g_{\\mu\\nu}=g_{AB}, \\quad {\\rm etc.}.\n\\end{equation}\n(In the usual orthonormal basis, one transforms the symmetric\npart of the metric\nto the Minkowski space metric, and the $E$'s provide the\nisomorphism between\ncoordinate basis tensors and locally Lorentzian tensors.)\nThe dual basis of $T^*({\\cal M})$ is introduced through the usual\nrelation:\n$\\theta^A[e_B]=\\delta^A_B$, and the inverse of the vierbeins is\ndefined through:\n$E_A^{\\;\\;\\mu}E^A_{\\;\\;\\nu}=\\delta^\\mu_\\nu$.\nIn this paper capital letters from the beginning of the alphabet:\n$A,B,C,\\cdots$\nwill refer to components of the object decomposed in the general\nbasis.\n\nParallel transport of the basis vectors now defines the\ngeneralized connection\ncoefficients:\n\\begin{equation}\\label{paralltptncb}\n\\nabla_{e_A}[e]_B=\\Gamma^C_{AB}e_C,\\quad\n\\nabla_{e_A}[\\theta]^C=-\\Gamma^C_{AB}\\theta^B.\n\\end{equation}\nThe definition of the basis in (\\ref{Evbns}) implies that it is\nno longer a\ncoordinate basis in general, and hence the directional\nderivatives no longer\nnecessarily commute, giving rise to the structure constants:\n\\begin{mathletters}\n\\begin{equation}\\label{ncbStructure}\n[e_A,e_B]={C_{AB}}^Ce_C,\n\\end{equation}\ngiven by:\n\\begin{equation}\\label{Stcnst}\n{C_{AB}}^C=E^C_{\\;\\;\\nu}(\nE_ A^{\\;\\;\\mu}\\partial_\\mu[E_ B^{\\;\\;\\nu}]\n-E_ B^{\\;\\;\\mu}\\partial_\\mu[E_ A^{\\;\\;\\nu}]),\n\\end{equation}\n\\end{mathletters}\ncalculated by noting that $e_A[\\;]={E_A}^\\mu\\partial_\\mu[\\;]$.\nThis also implies that a torsion-free connection will no longer\nbe symmetric,\nand vanishing torsion now gives:\n\\begin{equation}\\label{ncbtorsion}\nT^A_{BC}\n=\\theta^A\\left[\\nabla_{e_B}e_C-\\nabla_{e_C}e_B-[e_B,e_C]\\right]\n=2\\Gamma^A_{[BC]}-{C_{BC}}^A=0,\n\\end{equation}\nallowing one to determine the antisymmetric part as usual from\nthe structure\nconstants.\nThis is the motivation for splitting up the connection in this\nway.\nGiven some alternate split where $\\Gamma$ is not torsion free,\none would have to\ndistinguish between the effects of the general basis on the skew\npart of the\nconnection coefficients, and that of the NGT effects (themselves\ntensors).\n\nThe compatibility condition (\\ref{coordcomp}) can now be written\nas:\n\\begin{equation}\\label{ncbcompat}\n\\nabla_{e_C}[g]_{AB}\n=g_{AD}\\Lambda^D_{BC}\n+g_{DB}\\Lambda^D_{CA}\n-\\Delta^0_{CAB},\n\\end{equation}\nwhere since $\\Delta^0$ and $\\Lambda$ are tensors, they are just\nredefined by\nmultiplication by the appropriate combination of vierbeins.\nThe symmetric part of this can now be solved for the symmetric\npart of $\\Gamma$\nin terms of the antisymmetric part, the structure constants, and\n$\\Lambda$, to\ngive:\n\\begin{equation}\\label{symmconn}\n\\Gamma_{C(AB)}=\\case{1}{2}\\Delta_{C(AB)}-\\Gamma_{A[BC]}+\\Gamma_{B\n[CA]}\n-A^D_{\\;A}\\Lambda_{DBC}+A^D_{\\;B}\\Lambda_{DCA},\n\\end{equation}\nwhere the quantities:\n\\begin{equation}\\label{defns}\n\\Gamma_{ABC}=g_{(AD)}\\Gamma^D_{BC},\\quad\n\\Lambda_{ABC}=g_{(AD)}\\Lambda^D_{BC},\\quad\nA^A_{\\;B}=S^{(AC)}g_{[CB]},\n\\end{equation}\nhave been defined for convenience, and $S$ is the inverse of the\nsymmetric part\nof the metric defined by: $S^{(AB)}g_{(BC)}=\\delta^A_C$.\nAlso appearing is the symmetric part of:\n\\begin{equation}\\label{delta}\n\\Delta_{CAB}=e_B[g_{CA}]+e_A[g_{BC}]-e_C[g_{AB}]\n+\\Delta^0_{BCA}+\\Delta^0_{ABC}-\\Delta^0_{CAB}.\n\\end{equation}\nThe antisymmetric part of the compatibility conditions can now be\nrecast (using\n(\\ref{symmconn})) as 24 algebraic equations for $\\Lambda$:\n\\begin{equation}\\label{compantisymm}\n\\Lambda_{CAB}-A^D_{\\;A}A^E_{\\;B}\\Lambda_{ECD}\n-A^D_{\\;A}A^E_{\\;C}\\Lambda_{EBD}\n+A^D_{\\;B}A^E_{\\;A}\\Lambda_{ECD}\n+A^D_{\\;B}A^E_{\\;C}\\Lambda_{EAD}\n=\\Omega_{C[AB]},\n\\end{equation}\nwhere:\n\\begin{eqnarray}\\label{omega}\n\\Omega_{C[AB]}&=&\\case{1}{2}(\\Delta_{C[AB]}\n+A^D_{\\;B}\\Delta_{D(CA)}\n-A^D_{\\;A}\\Delta_{D(BC)}) \\nonumber \\\\\n\\quad\\quad &+&A^D_{\\;A}(\\Gamma_{B[CD]}+\\Gamma_{C[BD]})\n-A^D_{\\;B}(\\Gamma_{A[CD]}+\\Gamma_{C[AD]})\n-A^D_{\\;C}\\Gamma_{D[AB]}.\n\\end{eqnarray}\n\nThe method for solving the compatibility conditions is to first\ndetermine the\nauxiliary quantities appearing in this relation:\n($A,\\Gamma_{[\\;]},\\Delta,\\Omega$) in terms of the vierbeins and\nmetric\nquantities, then solve for $\\Lambda$ through\n(\\ref{compantisymm}), determine\n$\\Gamma_{(\\;)}$ from (\\ref{symmconn}), and then use $S$ with\n$\\Gamma_{[\\;]}$ and\n$\\Gamma_{(\\;)}$ to form $\\Gamma^A_{BC}$ and $\\Lambda^A_{BC}$.\nThis may not seem like much of a simplification, but when\nspecialized to a\nLorentz frame, many of these quantities simplify considerably (as\nis the case in\nthe Wyman sector in Section \\ref{section:Wyman}).\n\nThe curvature tensor (\\ref{Riemann}) becomes:\n\\begin{eqnarray}\\label{ncbRiemann}\nR^A_{\\;BCD}\n&=&\\theta^A[(\\nabla_{e_C}\\nabla_{e_D}\n-\\nabla_{e_D}\\nabla_{e_C}-\n\\nabla_{[e_C,e_D]})e_B] \\nonumber \\\\\n&=&e_C[\\Gamma^A_{DB}]\n-e_D[\\Gamma^A_{CB}]+\n\\Gamma^E_{DB}\\Gamma^A_{CE}\n-\\Gamma^E_{CB}\\Gamma^A_{DE}\n-{C_{CD}}^E\\Gamma^A_{EB},\n\\end{eqnarray}\nand the contractions:\n\\begin{mathletters}\n\\begin{eqnarray}\nR^1_{AB}&=&R^C_{\\;ACB}=e_C[\\Gamma^C_{BA}]-e_B[\\Gamma^C_{CA}]\n+\\Gamma^D_{BA}\\Gamma^C_{CD}-\\Gamma^D_{CA}\\Gamma^C_{BD}-{C_{CB}}^D\n\\Gamma^C_{DA},\n\\\\\nR^2_{AB}&=&R^C_{\\;CAB}=e_A[\\Gamma^C_{BC}]-e_B[\\Gamma^C_{AC}]-{C_{\nAB}}^D\\Gamma^C_{\nDC},\n\\end{eqnarray}\n\\end{mathletters}\ncombine to give the Ricci tensor:\n\\begin{eqnarray}\\label{ncbRicci}\nR_{AB}&=&\nR^1_{AB}-\\case{1}{2}R^2_{AB} \\nonumber \\\\\n&=&e_C[\\Gamma^C_{BA}]\n-e_B[\\Gamma^C_{CA}]\n-\\case{1}{2}e_A[\\Gamma^C_{BC}]\n+\\case{1}{2}e_B[\\Gamma^C_{AC}]\\nonumber  \\\\\n&&\\quad+\\Gamma^D_{BA}\\Gamma^C_{CD}\n-\\Gamma^D_{CA}\\Gamma^C_{BD}\n-{C_{CB}}^D\\Gamma^C_{DA}\n+\\case{1}{2}{C_{AB}}^D\\Gamma^C_{DC}\\nonumber  \\\\\n&=&e_C[\\Gamma^C_{BA}]\n-e_B[\\Gamma^C_{CA}]\n-\\case{1}{2}e_A[\\Gamma^C_{BC}]\n+\\case{1}{2}e_B[\\Gamma^C_{AC}] \\nonumber \\\\\n&&\\quad+\\Gamma^D_{BA}\\Gamma^C_{CD}\n-\\Gamma^D_{CB}\\Gamma^C_{DA}\n+\\Gamma^D_{[AB]}\\Gamma^C_{DC}.\n\\end{eqnarray}\nIn the split $R_{AB}^{\\text{\\tiny{NS}}}=R_{AB}+R^\\Lambda_{AB}$:\n\\begin{equation}\\label{ncbRNgt}\nR^\\Lambda_{AB}=\n\\nabla_{e_C}[\\Lambda]^C_{AB}\n+\\nabla_{e_{[A}}[\\Lambda]_{B]}\n+\\Lambda^C_{AD}\\Lambda^D_{BC}\n\\end{equation}\nas expected.\n\nSince by construction $\\Gamma$ is a torsion free connection, and\n(\\ref{ncbRiemann}) is the standard curvature tensor constructed\nfrom it, one\nobtains the usual Bianchi identities \\cite{Nakahara:1990} on the\ncurvature\ntensor.\nOne should note though that the connection is not compatible, and\nso the\nrotation coefficients are not antisymmetric.\nThe relevant Ricci tensors are also not constructed in the same\nmanner as in GR,\nso the implications of these identities are somewhat different.\nThe first Bianchi identity gives the usual cyclic identity on the\nlast three\nindices of (\\ref{ncbRiemann}), and leads to the result:\n\\begin{equation}\nR^1_{[AB]}=\\case{1}{2}R^2_{AB}\n\\end{equation}\nwhen one contracts on any lowered index.\n(This can also be proven directly using the Jacobi identity.)\nThis tells us that the NGT Ricci tensor is symmetric\n($R_{[AB]}=0$) in general,\nnot just in a coordinate basis.\n\nA detailed study of the contractions of the second Bianchi\nidentity (the cyclic\ncovariant derivative):\n\\begin{equation}\\label{equation:2nd Bianchi}\n\\nabla_{e_C}[R]^A_{\\;BDE}+\\nabla_{e_D}[R]^A_{\\;BEC}+\\nabla_{e_E}[\nR]^A_{\\;BCD}=0,\n\\end{equation}\nshould result in a derivation of the equations of motion for\nmatter fields\n\\cite{Moffat:1987,Legare+Moffat:1995} from the field equations.\n\n\\subsection{The NGT Action and Field Equations in a General\nBasis}\n\\label{FEQ}\n\nThe translation of the field equations\n(\\ref{mVars:b},\\ref{Comp2m},\\ref{mNGTFeq}) is accomplished\nthrough an almost\nstraightforward substitution:\n\\begin{mathletters}\n\\label{mNGTF}\n\\begin{eqnarray}\n\\label{mNGTF:b}\n\\Lambda_A&=&0, \\\\\n\\label{mNGTF:c}\n\\nabla_{e_B}[{\\bf g}]^{[AB]}&=&\\alpha{\\bf g}^{(AB)}W_B,\\\\\n\\label{mNGTF:a}\n{\\cal\nR}_{AB}:=R^{\\text{\\tiny{NS}}}_{AB}+\\nabla_{e_{[A}}[W]_{_{B]}}\n&-&\\case{1}{2}\\alpha\nW_{A}W_{B}\n-\\case{1}{4}m^2M_{AB}\n=0,\n\\end{eqnarray}\n\\end{mathletters}\nwhere the density is $\\sqrt{-g}=\\sqrt{-{\\rm det}(g_{AB})}$, the\nmass tensor:\n\\begin{equation}\\label{equation:mass tensor}\nM_{AB}=g_{[AB]}-g_{CA}g_{BD}g^{[CD]}\n+\\case{1}{2}g_{BA}g^{[CD]}g_{[CD]}\n\\end{equation}\nhas been defined, and the tensor appearing in the compatibility\nequations is:\n\\begin{equation}\\label{deltas}\n\\Delta^0_{CAB}=-\\case{2}{3}\\alpha\n(g_{A[C}g_{D]B}+\\case{1}{2}g_{AB}g_{[CD]})g^{(DE)}W_E.\n\\end{equation}\nOne must be careful to treat totally antisymmetric derivatives\nproperly (the\nstructure constants now come into the curl of a vector), and\ntranslate the\nmetric density properly.\n\nIn order to define the action, one should note that the inverse\nof the metric is\nnow: $g_{AB}g^{BC}= g^{CB}g_{BA}=\\delta^C_A$, and the direct\ntranslation of the\ndensity results in:\n$\\sqrt{-g}\\rightarrow\\sqrt{-EgE^t}=E\\sqrt{-g}$ where $g={\\rm\ndet}(g_{AB})$ and $E={\\rm det}(E^A_{\\;\\;\\mu})$.\nThen (\\ref{mNGTAct}) is rewritten:\n\\begin{equation}\\label{mNGTActncb}\nS=\\int_{\\cal M}d^4x\\,E\\left\\{-{\\bf\ng}^{AB}R^{\\text{\\tiny{NS}}}_{AB}\n-{\\bf g}^{AB}\\nabla_{e_{[A}}[W]_{_{B]}}+{\\bf l}^A\\Lambda_A\n+\\case{1}{2}\\alpha{\\bf g}^{(AB)}W_AW_B\n+\\case{1}{4}m^2{\\bf g}^{[AB]}g_{[AB]}\\right\\}.\n\\end{equation}\n(Note that in a Lorentz basis, the inverse of the metric is not\n$\\eta$.)\n\nDeriving the equations of motion from this action should be\napproached with\ncare.\nAs it stands there are too many fields (the metric and the\nvierbeins share\ndegrees of freedom) and one typically must choose either a\ncoordinate basis (as\nin Section \\ref{section:Massive NGT}), a Lorentz basis (so that\nall symmetric\nmetric degrees of freedom are contained in the vierbeins), or a\nwell-defined\ncombination of the two.\nOne must also realize that (\\ref{mNGTActncb}) as it stands\nassumes that the\nconnection $\\Gamma$ is torsion-free {\\it a priori}, so that when\nvarying the\nvierbein, $\\Gamma_{[\\;]}$ must be varied as well.\nAs an alternative, one may impose the torsion-free condition\nthrough additional\nLagrange multiplier terms: $L^{BC}_AT^A_{BC}$ in the action,\nvarying the full\nconnection coefficients and vierbeins separately.\n\n\\section{Approximation of the Wyman Sector Solution in a Vierbein\nBasis}\n\\label{section:Wyman}\n\nIn general, the spherically symmetric Killing vector analysis for\na $(0,2)$\ntensor gives both $t-r$ and $\\theta-\\phi$ skew components.\nHowever it is possible to show from the general spherically\nsymmetric field\nequations that it is consistent to put either (or both) of these\nskew components\nto zero separately, since in either case one loses the\ncorresponding field\nequation, and the system of equations remains consistent.\nWhether it is physically reasonable to do this or not depends on\nthe details of\nthe matter coupling in the theory, and how it alters the global\nbehavior of the\nskew sector.\nHere will be considered the field equations for what will be\nreferred to as the\nWyman sector \\cite{Wyman:1950} (keeping just the $\\theta-\\phi$\nsector), although\nthe asymptotics of the $t-r$ sector will be discussed briefly at\nthe end of this\nsection, where it will be argued that there are no static\nsolutions with\nasymptotic behavior that is dominated by Schwarzschild (or\nequivalently,\nNewtonian) effects.\nThis will allow an analysis of the perturbation equations for the\nspherically\nsymmetric modes, in order to see the effects of the antisymmetric\nbackground.\n\nIn a coordinate basis, the Wyman metric looks like:\n\\begin{equation}\n| g_{\\mu\\nu}| ={\\rm\ndiag}\\{\\gamma(r),-\\alpha(r),-r^2,-r^2\\sin^2(\\theta)\\},\\quad\ng_{[23]}=f(r)\\sin(\\theta).\n\\end{equation}\n(An appropriate coordinate system has been chosen in order to\nremove\nthe symmetric $t-r$ metric component, and fix the $\\theta-\\theta$\ncomponent.)\nIntroducing the usual choice of vierbein (using the functions\ndefined by:\n$F=f(r)\/r^2, E_0=1\/\\sqrt{\\gamma(r)}, E_1=1\/\\sqrt{\\alpha(r)}$):\n\\begin{equation}\\label{equation:vierbeins}\nE_A^{\\;\\;\\mu}={\\rm diag}\\{E_0^{\\;\\;0}=E_0,\\;\nE_1^{\\;\\;1}=E_1,\\;\nE_2^{\\;\\;2}=\\frac{1}{r},\\;\nE_3^{\\;\\;3}=\\frac{1}{r\\sin(\\theta)}\\},\n\\end{equation}\nthe metric becomes:\n\\begin{equation}\\label{equation:Wyman metric}\n| g_{(AB)} |=\\eta_{AB},\\quad\ng_{[23]}=F,\n\\end{equation}\nand the density $\\sqrt{-g}=\\sqrt{1+F^2}$.\n\nAt this point one can invert the compatibility conditions and\ncompute the field\nequations using the method of Section \\ref{section:GenBasis\nintro}, given in\nsome detail in Appendix \\ref{appendix:Wyman}.\nNo attempt will be made here to solve the field equations\nexactly, although\nnumerical evidence for the existence of an exact solution with\nasymptotic\nbehavior that matches that given here has been found\n\\cite{Cornish:1994},\nensuring that the approximations given come from a global\nsolution.\nInstead, an approximation will be given that describes the\nasymptotic behavior\nof the exact solution.\nThe idea will be to consider the skew sector as a small\ncorrection (of order\nsome small dimensionless parameter $\\kappa$, to be explicitly\ndefined later) to\nthe Schwarzschild solution far enough away from the source.\nThis should be reasonable since one expects from the results of\nthe perturbation\nin Section \\ref{section:Schw pert} that the skew sector will\nbehave\nasymptotically as a decaying exponential, while the symmetric\nsector should\nbehave as $\\sim 1\/r$, so that far enough away from the\ngravitational source the\nskew sector should be completely dominated by GR effects.\n\nTo lowest order in $\\kappa$ (the skew sector) the work is already\ndone, as the\nfield equation for the skew function will be essentially the same\nas the static\nperturbation about a Schwarzschild background already considered\nin Section\n\\ref{section:Schw pert}.\nIn the vierbein basis, this is derived as before from ${\\cal\nR}_{[23]}$\n(\\ref{equation:Wyman field equations}), and gives:\n\\begin{equation}\\label{equation:Wyman F}\nA\\partial_r^2[F]+(A^\\prime+\\frac{2A}{r})\\partial_r[F]-\\frac{2}{r}\n(A^\\prime+\\frac\n{A}{r})F-m^2F=0,\n\\end{equation}\nand it is trivial to see that when using: $F=f\/r$, this reduces\nto the static\nlimit of (\\ref{equation:Schw f}), giving the asymptotic form for\n$F$:\n\\begin{equation}\\label{equation:Wyman asymptF}\nF\\sim F_0\\frac{e^{-mr^*}}{mr}.\n\\end{equation}\n\nOne now must consider how the presence of the skew sector affects\nthe symmetric\nsector, particularly whether it really is a higher order effect.\nThe asymptotic form of these corrections due to $F$ may be\ncalculated by\nconsidering order $\\kappa^2$ corrections to the vierbeins (order\n$\\kappa$ terms\nwill not depend on $F$, and so will be solely $\\delta M_s$\ncorrections),\ncalculated from the symmetric field equations with $F$ from\n(\\ref{equation:Wyman\nasymptF}) acting as a source.\nWriting the corrections to the vierbeins as: $E_{0,1}\\rightarrow\nE_{0,1}+E_{0,1}^{(2)}$, and the corrections to the field\nequations as ${\\cal\nR}^{(2)}$, one calculates:\n\\begin{equation}\\label{equation:Wyman appx12}\n{\\cal R}_{00}^{(2)}+{\\cal\nR}_{11}^{(2)}=-\\frac{2A}{r}\\partial_r[\\sqrt{A}E^{(2)}_0+\\frac{E^{\n(2)}_1}{\\sqrt{A\n}}]\n-AFF^{\\prime\\prime}-\\frac{3}{2}A(F^\\prime)^2\n-\\frac{2A}{r}FF^\\prime=0,\n\\end{equation}\nwhich, after translating it into a differential equation in $r^*$\nand keeping\nonly the asymptotically dominant terms, results in:\n\\begin{equation}\\label{equation:Wyman asympt12}\n\\partial_{r^*}[\\sqrt{A}E^{(2)}_0+\\frac{E^{(2)}_1}{\\sqrt{A}}]\n\\sim -\\frac{5}{4}(F_0)^2\\frac{e^{-2mr^*}}{r^*}.\n\\end{equation}\nthis integrates to give (the constant of integration is ignored\nas one could\neliminate it through an appropriate choice of gauge as in Section\n\\ref{section:Schw pert}):\n\\begin{equation}\\label{equation:Wyman asymptsum}\n\\sqrt{A}E^{(2)}_0+\\frac{E^{(2)}_1}{\\sqrt{A}}\\sim\n\\frac{5}{4}(F_0)^2\\frac{e^{-2mr^*}}{2mr^*}.\n\\end{equation}\nConsidering next ${\\cal R}_{33}^{(2)}$ and using\n(\\ref{equation:Wyman\nappx12}) leads to the asymptotic equation:\n\\begin{equation}\\label{equation:Wyman asympt1}\n\\partial_{r^*}[r\\sqrt{A}E^{(2)}_1]\\sim-\\case{5}{4}(F_0)^2e^{-2mr^\n*}.\n\\end{equation}\nThe solution of this combined with the results of\n(\\ref{equation:Wyman\nasymptsum}) gives (once again the constant of integration is\nignored, this time\nas it would be interpretable as a perturbation of the mass\nparameter and not due\nto the effects of the skew sector):\n\\begin{equation}\\label{equation:Wymna d1}\nE^{(2)}_1\\sim \\frac{5}{4}(F_0)^2\\frac{e^{-2mr^*}}{2mr^*},\\quad\nE^{(2)}_0\\sim o(\\frac{e^{-2mr^*}}{(r^*)^2}),\n\\end{equation}\nwhere the dominant correction to the symmetric sector is\n$E^{(2)}_1$, and\n$E^{(2)}_0$ is down by $o(1\/r^*)$.\nIt is not hard to see that these corrections are indeed of an\norder higher than\nthe effects in the skew sector.\nClearly one may define a small parameter\n$\\kappa=F_0\\exp(-mr^*_0)$, where\n$r^*_0$ is chosen such that $F(r^*_0)\\ll 1$, to define the small\nsize of the\nskew sector when $r^*>r^*_0$.\nThe corrections to the symmetric sector are seen to be of order\n$\\kappa^2$, and\nwill therefore be neglected in the approximation of the\nbackground required in\nSection \\ref{section:Wyman pert}.\n\nOne may attempt to do the same sort of analysis keeping the\n$g_{[01]}$\ncomponent, however the linearized field equation implies\nimmediately that the\nfield must vanish (it is identical to (\\ref{equation:h01})).\nConsidering higher orders in the field in an attempt to generate\na solution\nother than this trivial result, the third order correction gives\n(writing\n$W(r)=\\sqrt{X(r)}$):\n\\begin{equation}\\label{equation:Wyman W3}\n{\\cal\nR}_{[01]}=-\\frac{\\sqrt{X}}{6}\\left[A\\partial_r^2[X]\n+(A^\\prime+\\frac{2A}{r})\\partial_r[X]\n+(\\frac{12A}{r^2}+\\frac{4A^\\prime}{r}+\\frac{3}{2}m^2)X+\\frac{12A^\n\\prime}{r}+3m^2\n\\right]=0.\n\\end{equation}\nWriting $X=Y\/r$ and transforming to the $r^*$ coordinate as\nbefore gives in\ncanonical form:\n\\begin{equation}\\label{equation:W3 canonical}\n\\partial^2_{r^*}[Y]+A(\\frac{12A}{r^2}+\\frac{3A^\\prime}{r}+\\frac{3\n}{2}m^2)Y+A(12A\n^\\prime+3m^2r)=0,\n\\end{equation}\nand keeping the dominant terms:\n\\begin{equation}\\label{equation:W3 canonical asyptotic}\n\\partial^2_{r^*}[Y]+\\case{3}{2}m^2Y+3m^2r^*=0,\n\\end{equation}\neasily giving the asymptotic form of the solution:\n\\begin{equation}\\label{equation:W3 soln}\nW^2(r^*)=-2+\\frac{a}{r^*}\\cos(\\sqrt{\\case{3}{2}}mr^*)\n+\\frac{b}{r^*}\\sin(\\sqrt{\\case{3}{2}}mr^*),\n\\end{equation}\n(where $(a,b)$ are arbitrary constants).\nThe dominant part of this solution implies that $W$ is imaginary,\nand must be\ndiscarded.\nHowever one also sees that this solution is not in fact a small\ncorrection to\nthe Schwarzschild metric asymptotically, and would have to be\ndiscarded for that\nreason alone.\nThis is not surprising as one is trying to match a function that\nis small\nasymptotically (by hypothesis) to one that is constant, so\nkeeping higher orders\nin $W$ will not change this.\nThis implies that nontrivial static solutions that include this\nsector fail to\nbe dominantly Schwarzschild for large $r$.\nThis of course does not exclude solutions with asymptotic\nbehavior that is of\nsome other form, nor can one exclude the possibility that $W$ is\nnonvanishing\nonly inside some finite radius.\n\n\\section{Spherically Symmetric Perturbation about a Wyman\nBackground}\n\\label{section:Wyman pert}\n\nIn an attempt to consider the perturbation equations for NGT\nabout a general\nnon-symmetric background, one finds that the compatibility\nconditions prevent\none from formulating the inversion in a useful form.\nThis means that a fairly straightforward covariant formulation\n(like that given\nin Section \\ref{section:Massive NGT}) is not feasible, and\ninstead one must\ntreat each situation separately, in this case a spherically\nsymmetric\nperturbation about the approximated mNGT Wyman solution given in\nthe previous\nsection.\nHere it is demonstrated that despite the remaining gauge freedom\nin the\nsymmetric sector, both symmetric functions will in general pick\nup time\ndependence from the skew sector.\nAlthough this cross coupling is demonstrated explicitly in a\nperturbative\nscenario, it will certainly persist in a more general sense.\nThe results here will show that the perturbations in the\nsymmetric sector pick\nup time dependence that is algebraically determined by the skew\nfunction $F$,\nwithout themselves becoming independent degrees of freedom.\nThe canonical analysis of the general spherically symmetric\nsystem will address\nrigorously how many degrees of freedom exist in each sector.\nIf there are more in the nonperturbative theory, one can examine\nthe dynamical\napproach to an asymptotically flat spacetime\nlooking for possible singular behavior similar to that found in\n\\cite{Isenberg+Nestor:1977}.\n\nThe perturbation of the Wyman metric (\\ref{equation:Wyman\nmetric}) in a\ncoordinate basis will look identical to\n(\\ref{equation:perturbation metric})\n(using the gauge choice to simplify it as before).\nThe background vierbeins will be the same as those in the Wyman\nsolution\n(\\ref{equation:vierbeins}), where now the perturbations of the\nvierbeins and\nskew metric functions are related to perturbations in the\ncoordinate basis by:\n\\begin{equation}\n\\delta W=\\frac{h_{[01]}(t,r)}{\\sqrt{\\alpha(r)\\gamma(r)}},\\quad\n\\delta F=\\frac{\\delta h_{[23]}(t,r)}{r^2},\\quad\n\\delta\nE_0=-\\frac{1}{2}\\frac{h_{00}(t,r)}{\\gamma(r)^{\\frac{3}{2}}},\\quad\n\\delta\nE_1=-\\frac{1}{2}\\frac{h_{11}(t,r)}{\\alpha(r)^{\\frac{3}{2}}}.\n\\end{equation}\nIn the vierbein basis the metric perturbation has nonvanishing\ncomponents:\n\\begin{equation}\nh_{[01]}=\\delta W,\\quad h_{[23]}=\\delta F.\n\\end{equation}\nThe approximation of the background Wyman solution given in\nSection\n\\ref{section:Wyman} greatly simplifies the algebra necessary to\ndevelop the\nperturbation given in Appendix \\ref{appendix:Wyman perturbation}.\nApproximating the symmetric sector by the Schwarzschild solution\nand the\nantisymmetric sector by (\\ref{equation:Wyman asymptF}), first\norder in this\nstatic antisymmetric background is kept, as is the first order in\nthe\nperturbations.\nAs one shall see, this will be a reasonable approximation since\nit will be\npossible to keep the perturbations small compared to the\nbackground by an\nappropriate choice of integration constants (similarly to $\\delta\nM_s$ in the\nSchwarzschild case).\n\nThe field equation: ${}^1\\! {\\cal R}_{[01]}=0$, yields\nprecisely\nthe same field equation as in the Schwarzschild case\n(\\ref{equation:h01}),\nallowing one to immediately set $\\delta W=0$.\nThe symmetric part: ${}^1\\! {\\cal R}_{(01)}=0$, can be written\nas\na total time derivative:\n\\begin{equation}\n{}^1\\! {\\cal\nR}_{(01)}=\\partial_t\\left[\\frac{2}{r\\sqrt{A}}\\delta\nE_1\n+\\frac{3}{2}F^\\prime\\delta F\n+(\\frac{1}{r}-\\frac{A^\\prime}{2A})F\\delta F\n+F\\partial_r[\\delta F]\\right]=0.\n\\end{equation}\nThis last field equation is then integrated, introducing an\narbitrary static\nfunction $\\delta E(r)$:\n\\begin{equation}\\label{equation:Wyman deltaE1}\n\\frac{2}{r\\sqrt{A}}(\\delta E_1-\\delta E)\n=-\\frac{3}{2}F^\\prime\\delta F\n-(\\frac{1}{r}-\\frac{A^\\prime}{2A})F\\delta F\n-F\\partial_r[\\delta F].\n\\end{equation}\nNow computing\n(${\\rm Tr}[{\\cal R}_{AB}]:={\\cal R}_{00}+{\\cal R}_{11}+2{\\cal R}_{22}$):\n\\begin{eqnarray}\n{\\rm Tr}[{}^1\\! {\\cal\nR}_{AB}]&=&\\frac{4}{r^2}\\partial_r[r\\sqrt{A}\\delta E_1]\n-\\frac{F}{A}\\partial_t^2[\\delta F]\n+3AF\\partial_r^2[\\delta F] \\nonumber \\\\\n&+&(\\frac{8A}{r}F+2A^\\prime F+5A F^\\prime)\\partial_r[\\delta F]\n+(3AF^{\\prime\\prime}+\\frac{8A}{r}F^\\prime+2A^\\prime\nF^\\prime-m^2F)\\delta F=0,\n\\end{eqnarray}\nand inserting (\\ref{equation:Wyman deltaE1}) gives:\n\\begin{eqnarray}\\label{equation:Wyman TrRsimp}\n{\\rm Tr}[{}^1\\! {\\cal\nR}_{AB}]&=&\\frac{4}{r^2}\\partial_r[r\\sqrt{A}\\delta E] \\nonumber\n\\\\\n&-&F\\left( \\frac{1}{A}\\partial_t^2[\\delta F]\n-A\\partial_r^2[\\delta F]\n-(A^\\prime+\\frac{2A}{r})\\partial_r[\\delta F]\n+(\\frac{2A^\\prime}{r}+\\frac{2A}{r^2}+m^2)F\\delta F\\right)=0.\n\\end{eqnarray}\nAlso,\n\\begin{eqnarray}\\label{equation:Wyman R00plusR11}\n{}^1\\! {\\cal R}_{00}+{}^1\\! {\\cal R}_{11}\n&=&\\frac{2A}{r}\\partial_r[\\sqrt{A}\\delta E_0+\\frac{\\delta\nE_1}{\\sqrt{A}}]\\nonumber  \\\\\n&+&\\frac{F}{A}\\partial_t^2[\\delta F]\n+AF\\partial_r^2[\\delta F]\n+A(\\frac{2F}{r}+3F^\\prime)\\partial_r[\\delta F]\n+A(\\frac{2F^\\prime}{r}+F^{\\prime\\prime})\\delta F=0,\n\\end{eqnarray}\ngives the equality of spatial derivatives of $\\sqrt{A}\\delta E_0$\nand $\\delta\nE_1\/\\sqrt{A}$ up to order $F$.\nThis will be useful when considering:\n\\begin{eqnarray}\\label{equation:R23 one}\n{}^1\\! {\\cal R}_{[23]}\n&=&(\\frac{F}{r}-\\frac{F^\\prime}{2})A\\partial_r\\left[\\frac{\\delta\nE_1}{\\sqrt{A}}-\\sqrt{A}\\delta E_0\\right]\n-\\left[AF^{\\prime\\prime}+(A^\\prime+\\frac{2A}{r})F^\\prime\n-\\frac{2}{r}(A^\\prime+\\frac{A}{r})F\\right]\\frac{\\delta\nE_1}{\\sqrt{A}}\\nonumber \\\\\n&&\\quad+\\frac{1}{2}\\left(\\frac{1}{A}\\partial_t^2[\\delta F]\n-A\\partial_r^2[\\delta F]\n-(A^\\prime+\\frac{2A}{r})\\partial_r[\\delta F]\n+(\\frac{2A^\\prime}{r}+\\frac{2A}{r^2}+m^2)\\delta F\\right)\n\\nonumber \\\\\n&=&(\\frac{F}{r}-\\frac{F^\\prime}{2})A\\partial_r\\left[\\frac{\\delta\nE_1}{\\sqrt{A}}-\\sqrt{A}\\delta E_0\\right]\n-m^2F\\frac{\\delta E_1}{\\sqrt{A}}\\nonumber \\\\\n&&\\quad+\\frac{1}{2}\\left(\\frac{1}{A}\\partial_t^2[\\delta F]\n-A\\partial_r^2[\\delta F]\n-(A^\\prime+\\frac{2A}{r})\\partial_r[\\delta F]\n+(\\frac{2A^\\prime}{r}+\\frac{2A}{r^2}+m^2)\\delta F\\right)=0,\n\\end{eqnarray}\nwhere use has been made of (\\ref{equation:Wyman F}).\nOne derives a simple field equation by inserting\n(\\ref{equation:R23 one}) in\n(\\ref{equation:Wyman TrRsimp}) and dropping the resulting terms\nthat are of\nsecond order in the background skew field $F$:\n\\begin{equation}\\label{equation:Wyman deltaE}\n\\partial_r[r\\sqrt{A}\\delta E]=0\n\\rightarrow \\delta E=\\frac{\\delta M_s}{r\\sqrt{A}},\n\\end{equation}\nwhere the constant of integration has been identified with the\nGR-like\nperturbation of the Schwarzschild mass parameter.\n\nNow (\\ref{equation:Wyman R00plusR11}) can be used to replace\n$\\delta E_0$ with\n$\\delta E_1$ at this order, and (\\ref{equation:Wyman deltaE1}) to\nreplace\n$\\delta E_1$ with $\\delta E$ to find:\n\\begin{eqnarray}\\label{equation:Wyman R23}\n{}^1\\! {\\cal R}_{[23]}&=&\\frac{1}{2}\\left\\{\n\\frac{1}{A}\\partial_t^2[\\delta F]\n-A\\partial_r^2[\\delta F]\n-(A^\\prime+\\frac{2A}{r})\\partial_r[\\delta F]\n+(\\frac{2A^\\prime}{r}+\\frac{2A}{r^2}+m^2)\\delta\nF\\right\\}\\nonumber  \\\\\n&&\\quad-A(F^\\prime-\\frac{2F}{r})\\partial_r[\\frac{\\delta E}{\\sqrt{A}}]\n-m^2F\\frac{\\delta E}{\\sqrt{A}}=0,\n\\end{eqnarray}\nand using (\\ref{equation:Wyman deltaE}) in this yields the wave\nequation for\n$\\delta F$:\n\\begin{eqnarray}\\label{equation:Wyman deltaF}\n\\frac{1}{A}\\partial_t^2[\\delta F]\n-A\\partial_r^2[\\delta F]\n&-&(A^\\prime+\\frac{2A}{r})\\partial_r[\\delta F]\n+(m^2+\\frac{2A}{r^2}+\\frac{2A^\\prime}{r})\\delta F\\nonumber  \\\\\n&=&\\frac{\\delta\nM_s}{rA}[m^2F+4(\\frac{F}{r}-\\frac{F^\\prime}{2})(\\frac{A}{r}\n+A^\\prime)].\n\\end{eqnarray}\nNote that this is a static source and so will not in itself\ninduce any wave\nsolutions, but as before the effects of a matter source will show\nup\nasymptotically.\nThe static part of the solution may be derived using the methods\nin Section\n\\ref{section:Schw pert}:\n\\begin{equation}\\label{equation:Wyman static dF}\n\\delta F=-2F_0\\frac{\\delta\nM_s}{r}e^{-2mr^*}\\ln(\\frac{r^*}{2M_s}),\n\\end{equation}\nand is consistent with the static solution (\\ref{equation:Wyman\nasymptF})\nderived about a Schwarzschild background with mass parameter\n$M_s+\\delta M_s$.\nTime dependent solutions are identical to those found from\n(\\ref{equation:Wyman\nF}), and will induce time dependence in the symmetric sector\nthrough\n(\\ref{equation:Wyman deltaE1}).\nSince $\\delta E_1$ is related to $\\delta F$ locally, it is not an\nindependent\ndegree of freedom, and since the skew field is short-ranged, it\nwill not radiate\nenergy at infinity.\n\nUsing (\\ref{equation:Wyman deltaF}) and (\\ref{equation:Wyman F}),\none reduces\n(\\ref{equation:Wyman R00plusR11}) to an algebraic relation for\n$\\delta E_0$:\n\\begin{eqnarray}\\label{equation:Wyman deltaE0}\n&&\\frac{2A}{r}\\partial_r[\\sqrt{A}\\delta E_0+\\frac{\\delta\nE}{\\sqrt{A}}]\n+AF\\partial_r^2[\\delta F]\n+(\\frac{3A^\\prime}{2}F+\\frac{2A}{r}F\n+\\frac{A}{2}F^\\prime)\\partial_r[\\delta F]\\nonumber \\\\\n&&\\quad +[(A^\\prime+\\frac{A}{2r})F^\\prime-\n(\\frac{(A^\\prime)^2}{2A}+\\frac{7A^\\prime}{2r}+\\frac{3A}{r^2}\n+\\frac{3}{4}m^2)F]\\delta F=0.\n\\end{eqnarray}\nThe solution for $\\delta E_0$ can be written as:\n\\begin{equation}\\label{equation:Wyamn deltaE0soln}\n\\sqrt{A}\\delta E_0=\\sqrt{A}\\delta\\tilde{E}-\\frac{\\delta\nE}{\\sqrt{A}}+B(t)\n\\end{equation}\nwhere $B(t)$ is an arbitrary function of time (removable in the\nusual way using\nthe remaining gauge transformation), the second term corresponds\nto the static\nSchwarzschild perturbation from Section \\ref{section:Schw pert},\nand\n$\\delta\\tilde{E}$ solves the remainder of (\\ref{equation:Wyman\ndeltaE0}).\nNote that although not independent degrees of freedom, neither\n$\\delta E_0$ nor\n$\\delta E_1$ is static.\nThis is in fact what one would expect when considering the effect\nof a\nspherically symmetric matter field to which Birkhoff's theorem\ndoes not apply,\non the GR background.\nThe presence of the non-static field will induce time dependence\nin the\ngravitational fields, without exciting any independent modes.\nThis is expected to continue to be the case in NGT: the general\nspherically\nsymmetric system should only have degrees of freedom in the skew\nsector.\n\n\\section{Conclusions}\n\nThe asymptotic behavior of the antisymmetric sector for the case\nof a static\nWyman-type metric has been determined, and the corrections to the\nsymmetric\nsector shown to be negligible provided one considers regions of\nspacetime far\nenough away from the gravitational source.\nIt has also been determined that if one keeps the antisymmetric\n$t-r$ component,\nthen one cannot have asymptotic behavior that is dominated by the\nSchwarzschild\nmetric, and so it must be discarded.\nThis analysis was facilitated by the introduction of a vierbein\nbasis, although\nthe formalism has been given for a general basis for\ncompleteness.\n\nBy considering a spherically symmetric perturbation of the\nSchwarzschild metric,\nit has been shown that NGT does not have a rigorous Birkhoff\ntheorem as the\nantisymmetric sector will not remain static in general.\n(This has also been noted previously in a Unified Field Theory\nbased on Lyra\ngeometries \\cite{Dunn:1971}.)\nPerturbing an approximate Wyman background in a vierbein basis\nhas shown that\nthe symmetric sector is also not static in general, although no\nadditional modes\nbecome excited.\nThis is important phenomenologically since one cannot consider\nthe static\nsolutions (Schwarzschild and Wyman) as the only spherically\nsymmetric exterior\nsolutions to the field equations, and one must therefore match an\ninterior\nsolution to a non-static exterior in general.\n\nPerturbations of GR backgrounds have been shown to have good\nasymptotic behavior\nin general, since the ghost modes do not become excited and the\nremaining\ndegrees of freedom are short ranged by construction.\nHowever this is not good enough since one expects that the\nphysically\ninteresting solutions to mNGT will not be the purely GR\nsolutions, and one would\ntherefore like to examine the behavior of perturbations on\ngeneric,\nasymptotically-flat, mNGT backgrounds.\nA covariant perturbative scheme, although possible in principle,\nwould seem to\nbe too complex to be of any practical value.\nInstead one may treat each case separately and consider the\nbehavior of (perhaps\nseveral) modes about a particular background, as was done here\nfor the\nspherically symmetric perturbation about a Wyman background.\n\nHowever this also may not be adequate to fully understand the\ndynamics of the\nskew sector in mNGT.\nThe lack of additional gauge invariance in the skew sector may\nmean that there\nare more modes in the rigorous theory that will be seen in any\nsort of weak\nfield, perturbative analysis.\nTo determine whether or not this is the case will require a\ncanonical analysis\nof the full theory.\nPartial information may be obtained by considering the full set\nof fields in a\nspherically symmetric system, and looking for global information\nabout the\nbehavior of the skew modes given a general coupling to external\nsources.\nThis is not likely to be a tractable problem in a coordinate\nbasis, and even in\na Lorentz frame the field equations are not expected to be\nparticularly\nenlightening, due to their complication alone.\nHowever the canonical analysis of this system will show which\nfields propagate\nin the general case, and allow one to get at the dynamics of the\napproach to an\nasymptotically well behaved spacetime.\n\n\n\\acknowledgments\n\nThe author would like to thank the Natural Sciences and\nEngineering Research\nCouncil of Canada and the University of Toronto for funding\nduring part of this\nwork, the hospitality of the Department of Physics, Cave Hill\nCampus, University\nof West Indies, Barbados, and the Inter America Development Bank\nfor supporting\nthe stay in Barbados.\nThanks also go to L. Demopoulos for suggestions, and J. W.\nMoffat, J.\nL\\'{e}gar\\'{e}, P. Savaria, and N. Cornish for discussions\nrelated to this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe SuperWASP (Wide Angle Search for Planets) project \\citep{pollacco}\nhas conducted wide-field time-domain photometric surveys since 2003 in\nthe northern hemisphere, and since 2005 in the southern hemisphere.\nEmploying 30~s exposures and 11~cm telescope apertures, its emphasis\nhas been on obtaining high-cadence (6--40~minute) observations of\nbright ($V\\sim8$--15~mag) stars over almost the whole sky, with a\nprimary goal of identifying transiting exoplanet candidates.  Although\nit has been highly successful in this (69 exoplanets announced by\nearly October 2012, out of\n287\\footnote{http:\/\/exoplanet.eu\/catalog.php}), it is also well-suited\nto the detection and study of variable stars of many types, and in\nparticular eclipsing binaries.\n\nHere we have used the SuperWASP archive of $\\sim$30 million objects to\nsearch for and analyse main sequence eclipsing binaries with very\nshort orbital periods ($<$20\\,000~s or $\\sim$0.2315~d).  This should\nyield an interesting sample around the observed short-period limit for\nsuch binary systems of $\\sim$0.2~d \\citep{ruc92, ruc07}, potentially\nilluminating the causes of this cut-off point in the period\ndistribution.  (We should note, however, that systems in this period\nrange must be expected to be relatively low in mass and hence\nintrinsically faint, such that SuperWASP will not detect them with the\nsame efficiency as longer-period eclipsing binaries.)  In earlier\nwork, \\citet{norton} presented 53 candidate eclipsing systems in this\nperiod range, using SuperWASP archived data; here, with a more\nthorough search, we have sought to detect further such objects which\nmight have been missed.  Also, \\citet{lohr} presented the results of a\nsearch for period changes in these 53 objects, finding three which\nexhibited statistically significant period decrease; here again, we\nhave used an improved period change detection method to search for\nperiod changes in the eclipsing systems found with periods below\n20\\,000~s.  This paper, then, is primarily intended to update\n\\citet{norton} and \\citet{lohr}.  However, in the process of analysing\nour findings, we believe we have discovered a new doubly eclipsing\nquadruple system, which is reported in Subsect.~\\ref{sub:J093010}\n\n\n\\section{Method}\n\\label{method}\n\nAn initial list of 36\\,758 SuperWASP identifiers was obtained from the\ncatalogue, with associated possible periods in the range\n8000--10\\,000~s.  This range would correspond to potential binary\norbital periods below 20\\,000~s, since there are two eclipses per\ncycle.  (The catalogue periods are the result of a uniform period\nsearch applied to the majority of data as part of the initial\nprocessing pipeline; the code used is described in \\citet{norton07}\nand is run separately for data from different seasons and cameras.  As\na consequence, several different periods can be listed in the\ncatalogue for a single object.)  A frequency plot of the periods\nrevealed substantial excesses of objects in the ranges 8610--8625~s\nand 9565--9586~s i.e. in the neighbourhood of 1\/10 and 1\/9 of a\nsidereal day respectively.  Since the vast majority of these harmonic\nperiodicities are expected to be spurious, objects in these ranges\nwere excluded from further consideration.  It is likely that only\nabout three genuine eclipsing binaries with periods below 20\\,000~s\nwill have been missed as a consequence.  Repeated identifiers\n(occurring when a single identifier had multiple possible\nperiodicities listed in the catalogue, in the ranges of interest) were\nalso removed at this stage.\n\nThis left 5743 distinct identifiers, of which around 5190 probably\nrepresented distinct astrophysical sources: since SuperWASP uses the\nUSNO-B1 input catalogue to label objects, it is possible for a single\nbright source in a field of view containing many faint sources to be\ncatalogued under multiple identifiers.  Such `duplicates' can usually\nbe easily recognised by their near-identical periods and coordinates;\nhowever, they were not excluded at this stage so that the brightest or\nclearest lightcurve for each source could be preferentially selected\nfor more detailed analysis later.  Lightcurves with fluxes corrected\nby the Sys-Rem algorithm \\citep{tamuz,mazeh} were obtained from the\narchive for these objects.\n\nA custom-written IDL program was then used to check and refine the\ncatalogue periods in a two-step process which improved on that used in\n\\citet{lohr}.  First, up to 50 short sections (depending on file\nlength) of each lightcurve were used for trial fitting with a\nsinusoidal function, using the Levenberg-Marquardt algorithm\n\\citep{lev,marq}.  If a frequency plot of the resulting periods\nyielded a single dominant approximate period, this was used as the\nperiod estimate for the second step; otherwise the object was\nclassified as probably non-periodic or possessing a period outside the\n8000--10\\,000~s range, and not considered further.  4434 objects of\ninterest remained after this step.\n\nThe approximate periods found for these objects were then refined to\nthe nearest 0.001~s by a form of phase dispersion minimization\ni.e. recursively folding the lightcurve on trial periods separated by\nsmall intervals, and selecting the period which minimized the standard\ndeviation of flux values in each of 100 phase bins.  This step was\nrepeated with the initial trial period being doubled, and objects were\ngiven a preliminary classification as possible eclipsing binaries if\nthe minimum phase dispersion was lower with the doubled period than\nwith the single period; the doubled period was then retained as the\nbinary's orbital period.  If the single period yielded a lower minimum\nphase dispersion, the object was tentatively classified as a periodic\nvariable of a different type (probably a pulsating or rotating\nvariable, in this period range, or a non-eclipsing contact binary),\nand the single period was retained as the star's pulsational or\nrotational period.\n\nSince this method will not always separate pulsators and eclipsing\nbinaries reliably (e.g. in cases where eclipsing systems show primary\nand secondary eclipses of equal depth), a final visual check was made\nof the folded lightcurves of the more distinctive objects with periods\nbelow 20\\,000~s.  This included all those where the amplitude of the\nmean lightcurve exceeded the amplitude of data scatter about the mean\nlightcurve, facilitating a clear identification of variable type.  It\nalso included some where the amplitude of variation was between 50\\%\nand 100\\% of the scatter, but where the objects might be expected to\nhave distinctive lightcurves on other grounds e.g. very bright\nobjects; objects with high numbers of observations; objects with very\nwell-defined periods found in step one of the program.\n\nApproximately 1000 identifiers were checked, and 201 were selected as\nprobable eclipsing binaries of W~UMa type, corresponding to 143\ndistinct astrophysical objects.  Fainter duplicates were rejected at\nthis stage.  The selected objects had nearly all been automatically\nclassified as eclipsing binaries, and many showed a clear difference\nin depths of primary and secondary eclipses; some had different\nheights of maxima, presumably due to the O'Connell effect\n\\citep{oconnell}; others were chosen for their relatively broad,\nsymmetric maxima and narrow eclipses.  Purely sinusoidal lightcurves\nwere excluded; although some of them were probably generated by\ngenuine eclipsing binaries, our photometric data was insufficient to\ndistinguish them from other periodic variable types with reasonable\nprobability.  Radial velocities would need to be determined for these\nambiguous objects to establish their variability type.\n\nEvidence of period change was then searched for in these 143 objects\nby means of automated construction of O$-$C (observed minus\ncalculated) diagrams, using an improvement of the method described in\n\\citet{lohr}.  The main difference was in the method for determining\naccurate times of observed (primary) minima.  Rather than trying to\nfit local regions of the lightcurve with quadratic, Gaussian,\nsinusoidal or other analytic functions, the binned mean lightcurve for\neach object, found during phase dispersion minimization, was used as a\nfitting `function'.  This shape, being derived from the combined\nobservations of hundreds of cycles, might be expected to provide an\nexcellent fit for each individual observed cycle, since it represents\nthe true underlying shape of the object's lightcurve.  A similar\napproach was used by \\citet{pribulla08} for finding eclipse minima in\na close quadruple system.\n\nThis approach has the advantages that it does not require the eclipses\nto be remotely symmetrical, or the same phase range to be used for\neach fit, as is preferred with the method of \\citet{kwee}.  The method\nmight be expected to break down, however, in objects whose lightcurve\nvaries in shape over time, such as RR~Lyrae pulsators exhibiting the\nBlazhko effect \\citep{blazhko}.  The number of phase bins to be used\nfor the fitting curve needs to be chosen carefully: if too many bins\nare used, the fitting curve will appear spiky, with features which are\nnot part of the underlying lightcurve; if too few bins are used, the\nfitting curve will blur distinctive features of the underlying\nlightcurve which may be necessary for optimal fits e.g. a difference\nin depth or shape between primary and secondary eclipses.  For this\nstudy, the program automatically picked a bin number based on the\nnumber of data points in the object file and the brightness of the\nobject, but this was optimized by hand in some cases.  An optimal\nnight for fitting the zero epoch (essential for determining reliable\ncalculated times of eclipses for the O$-$C diagram) was also selected\nmanually from a range near the middle night of each object file.\n\nAfter construction of each O$-$C diagram, a small number of\nlocally-outlying values were stripped out automatically; manual\nchecking of the corresponding nights of observation indicated that\nthese were typically caused by irregular features of the data\n(instrumental or astrophysical) rather than failures of the fitting\nmethod.  Linear and quadratic fits of the O$-$C values were then\nattempted; where a quadratic function gave a superior fit, and the\nrate of change was significantly different from zero ($\\ge 1\\sigma$),\nthe object was counted as exhibiting secular period change.\n\n\n\\section{Results}\n\nDuring the checking process described above, periodic variables of\nseveral different types were observed.  A large number of pulsators\n(probably $\\delta$~Scuti and RR~Lyrae variables in this short-period\nrange) were evident, characterized by narrow minima and broad maxima,\nand\/or asymmetric lightcurve shapes.  As well as eclipsing binaries in\nthe period range of interest, many eclipsing binaries were seen with\nperiods between 32\\,000 and 40\\,000~s, whose quarter-periods had\nevidently been picked up by the period detection code used in the\nSuperWASP catalogue.  Also, the subdwarf B (sdB) eclipsing binaries\n\\object{NY~Vir}, \\object{HS~2231+2441} and \\object{NSVS~14256825},\nwhich have orbital periods in the range 8000--10\\,000~s, were detected\nand phase-folded appropriately in spite of their strongly\nnon-sinusoidal lightcurves.  These detections suggest that the period\ndetermination method used here, despite using sinusoidal fitting\nduring the initial stage, is effective with periodic variables\nexhibiting a wide variety of lightcurve shapes.\n\n\\subsection{Periods}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{pcumul.eps}}\n\\caption{Cumulative period distribution of 143 candidate eclipsing binaries.\nTypical values for the short-period limit are shown.}\n\\label{perioddist}\n\\end{figure}\n\nPeriod and magnitude statistics for the 143 candidate (main sequence)\neclipsing binaries found with periods under 20\\,000~s are presented in\nTable~\\ref{143data} (online only).  They include 44 of the 53\nshort-period candidate eclipsing binaries presented in \\citet{norton}.\n(The other nine, as discussed in \\citet{lohr}, have periods slightly\nlonger than 20\\,000~s.)  Three of the 99 additional objects presented\nhere are known periodic variables: \\object{CC~Com}, listed as a\nW~UMa-type eclipsing binary in the GCVS, with the same period as found\nhere; \\object{LL~Eri}, listed as a rotating ellipsoidal binary in the\nGCVS, again with the period found here; and\n\\object{ROTSE1~J164349.58+325637.8}, listed in \\citet{akerlof} as a\n$\\delta$~Scuti variable with period half that found here.  Since the\nSuperWASP lightcurve for the latter object is particularly\nwell-observed, and its folded lightcurve exhibits clear (though small)\ndifferences in the depths of primary and secondary minima, we suggest\nthat the object is more likely to be an eclipsing binary.  Therefore,\n97 of the objects presented here are new candidate eclipsing binaries\nnear the short-period limit.  Figure~\\ref{perioddist} gives their\ncumulative period distribution, and Fig.~\\ref{appfig1} (online only)\nshows their individual folded lightcurves.\n\n\n\\subsection{Period changes}\n\\label{pdots}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{Periodchangedistribution.eps}}\n\\caption{Period change distribution of 74 candidate eclipsing binaries\n  showing significant change.  Period changes found with at least\n  $3\\sigma$ confidence are indicated with solid lines (38 objects);\n  dashed lines indicate changes found at a $1\\sigma$ level or more.}\n\\label{periodchangedist}\n\\end{figure}\n\n12 objects were excluded from the period change search, since only a\nsingle year of data was available for them.  Of the remaining 131, 74\nshowed evidence of significant secular change i.e. the uncertainty\nranges on their $\\dot{P}$ values did not include zero.  (Note that\nhighly significant period change may be small in magnitude: J121206\nshows a change of just 0.0265~s~yr\\textsuperscript{-1}, but its\nremarkably bright and well-defined lightcurve makes this value\nsignificant at $23\\sigma$.)  The remainder have not necessarily been\ndemonstrated to have unchanging periods: although the O$-$C diagrams\nof some were indeed better fitted by a linear function than a\nquadratic one, others exhibited apparent secular change at a\nnon-significant level, and may be regarded as having indeterminate\nstatus.  Period changes and significance levels are listed for these\n74 objects in Table~\\ref{143data} (online only), and the distribution\nof significant changes is plotted in Fig.~\\ref{periodchangedist}.\nFigure~\\ref{appfig1} (online only) shows the O$-$C diagrams of all\nobjects.\n\nSince a few of the changes significant at 1 or $2\\sigma$ may be\nexpected to be spurious, resulting by chance alone in a data set of\nthis size, the distribution of changes significant at $\\ge3\\sigma$ is\nalso shown in Fig.~\\ref{periodchangedist} (38 objects); we would not\nexpect to find even one instance of period change this significant out\nof 131 objects by chance.  Two of the objects found to exhibit highly\nsignificant period change in \\citet{lohr} are among this set\n(J133105 and J234401); the third (1SWASP~J174310.98+432709.6)\nhas a period slightly greater than 20\\,000~s and so was not included\nin the present study.  The reproduction of our earlier findings\nsupports the validity of the modified period change detection\napproach, and the increased number of highly-significant period\nchanges detected implies an improvement in sensitivity.\n\n\n\\section{Discussion}\n\n\\subsection{Periods}\n\nIt may be seen that the period distribution found here\n(Fig.~\\ref{perioddist}) straddles two frequently-quoted values for the\nperiod cut-off (0.20 and 0.22~d), and follows a smooth tail-off\ntowards shorter periods.  It tallies well with similar distributions\nfound from other wide-field time-domain surveys, covering wider period\nranges, such as those in \\citet{szymanski}, Fig.~9, and\n\\citet{paczynski}, Fig.~6, which show contact binary periods tailing\noff towards lower periods from a maximum around 0.38~d, with none\nobserved below 0.20~d.\n\nOnly one object (J201816) is included here with a period shorter\nthan that of \\object{BX~Tri} (=GSC~02314-0530, here J022050),\ngenerally regarded as the main-sequence eclipsing binary with the\nshortest-known period \\citep{dimitrov}.  However, since this object is\nvery poorly-observed by SuperWASP, with only a couple of thousand data\npoints mostly from a single year, we are reluctant to make too strong\na claim for it.  Although it apparently exhibits substantial\ndifferences in the heights of its maxima, as well as small differences\nin primary and secondary minima depths, it is possible that these are\nartefacts of the limited observations, and that the object is really a\npulsating variable.\n\nThus our search supports the existence of a short-period limit at\naround 0.20~d, and perhaps favours particular types of explanation for\nit.  \\citet{stepien} argued that the current age of the Universe\nindirectly explained the limit, on the grounds that lower-mass\ndetached binaries lose angular momentum more slowly than high-mass\nsystems, and so take longer to evolve into (stable) contact\nconfigurations; 0.20~d would then simply be the current minimum period\nthat a system would have had time to reach.  Such a model might imply\na much sharper, more cliff-like cut-off point for the binary period\ndistribution than seen here (and indeed, a cut-off period which would\nbecome shorter as the Universe aged).  Also, \\citet{jiang} have\nindicated a number of known short-period binaries with measured masses\nlower than \\citeauthor{stepien}'s formula should allow, and\n\\citet{nefs} have claimed the discovery of four M-dwarf eclipsing\nbinaries with periods below 0.18~d (one as short as 0.11~d), which\nwould certainly conflict with \\citeauthor{stepien}'s model.\n\nHowever, other models suggest that objects are leaving the\nshort-period end of the distribution through rapid merger, in addition\nto entering it from above through evolution from detached into contact\nconfigurations.  \\citet{jiang} argued that binary systems with\nparticular combinations of low primary mass and low mass ratio would\nevolve into unstable states and merge rapidly; 0.20~d would then be\nthe shortest period corresponding to a possible stable configuration.\n\\citet{stepien12} also proposes a new series of binary models\nincluding evolution towards coalescence within the contact stage; the\nlowest period obtained for any of these models, at the time of\ncoalescence, is 0.201~d.  Such explanations might fit better with the\nobserved distribution of binary periods, with some objects reaching\nunstable states and heading rapidly towards merger at periods somewhat\nabove 0.20~d, and others able to remain stable even at the cut-off\npoint.\n\n\\subsection{Period changes}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{Pvpdot.eps}}\n\\caption{Plot of significant period changes against periods, for 74\nobjects significant at $\\ge1\\sigma$ (small crosses) and 38 objects\nsignificant at $\\ge3\\sigma$ (larger diagonal crosses).}\n\\label{pvpdot}\n\\end{figure}\n\nThe distribution of significant period changes observed here\n(Fig.~\\ref{periodchangedist}) also tallies broadly with one found by\n\\citet{kubiak}, Fig.~5, for 134 OGLE contact binaries with periods\nbelow 1~d.  It is symmetrical around zero (a Kolmogorov-Smirnov test\nsupports symmetry at $P=0.91$ for the 74 objects with period change\nsignificant at $\\ge1\\sigma$ and at $P=0.69$ for the 38 objects\nsignificant at $\\ge3\\sigma$), and approximately normal, with a\nhalf-width at half-maximum around 0.1~s~yr\\textsuperscript{-1}, where\n\\citeauthor{kubiak} found a rather lower value around\n0.03~s~yr\\textsuperscript{-1}.  The extremely short-period objects\nconsidered here, then, appear equally likely to show increases and\ndecreases in period.  Figure~\\ref{pvpdot} plots significant period\nchanges against periods.  No particular relation between either\ndirection or magnitude of period change and period length seems\napparent.\n\nIt might be argued that the period change distribution found here has\nan effective hole at zero, and a deficiency near zero, since we do not\ninclude objects where no significant change was detected.  As\nindicated in Subsect.~\\ref{pdots}, these objects are not of a single\ntype, and cannot be confidently claimed to have unchanging periods.\nTo include all or some of them in our distribution would distort it\nand make it hard to compare with other results obtained using\ndifferent methods.  \\citeauthor{kubiak}'s distribution of objects\n(with $\\mathcal{P}$-statistic $>63.3$, described as ``statistically\nconfirmed'' period changes) also has a gap near zero, for similar\nreasons to ours: it is more difficult to detect and quantify small\nperiod changes since their uncertainties must also be small.\n\nAnother potential confounding factor would be the presence of\nwandering spots on the surface of a star.  These could in theory\nchange the lightcurve shape in such a way as to move the detected\ntimes of minima and so create a spurious curvature in the O-C diagram,\nleading us to conclude erroneously that the object's orbital period is\nchanging.  In practice, visual checks of individual nights suggest the\nlightcurves do not vary substantially in shape for the objects showing\nthe most significant period change, though it is possible that spot\nmovement is contributing to data scatter in some of the fainter\nand less well-observed lightcurves.\n\nSome of the O$-$C diagrams suggest a periodic, sinusoidal variation in\nperiod, in addition to or instead of a secular trend (e.g. J142312,\nJ172717, J210423).  Such cyclical variations are often interpreted\nas indicating the presence of a third body in the binary system\n(e.g. \\citet{lee}), through the light-time effect.  However, the\nApplegate mechanism \\citep{applegate}, involving the magnetic activity\ncycle of a star in a close binary system, may often provide a more\nplausible explanation \\citep{hilditch,christopoulou}; this mechanism\nexplains orbital period changes as gravitational quadrupole responses\nto a cyclic redistribution of angular momentum within the layers of an\nactive, convective star, associated with varying levels of\ndifferential rotation at different times in the star's magnetic cycle.\nIn such a case, longer-term luminosity variations would be expected to\nshow the same period as the orbital period modulation, since they\narise from a common cause, and any other variability associated with\nmagnetic activity e.g. coronal X-ray luminosity, should also show this\nperiod.\n\n\\subsection{J234401}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J234401lc.eps}}\n\\caption{Lightcurve of object J234401 folded at period 18\\,461.639~s.}\n\\label{J234401lc}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J234401oc.eps}}\n\\caption{O$-$C diagram for object J234401 (uncertainties not plotted\nfor clarity).  Dashed line shows best linear fit ($\\chi^2=22.53$);\nsolid line shows best quadratic fit ($\\chi^2=17.83$), corresponding to\na secular period change of $-0.1422\\pm\n0.0041$~s~yr\\textsuperscript{-1}.}\n\\label{J234401oc}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J234401oc2.eps}}\n\\caption{O$-$C diagram for object J234401 with final year times of\nminima adjusted by half a cycle, to test the hypothesis that secondary\nminima are closer to the calculated times than primary minima, owing\nto rapidity of period decrease.  Dashed line shows best linear fit\n($\\chi^2=124.67$); solid line shows best quadratic fit\n($\\chi^2=10.86$), corresponding to a secular period change of\n$-0.6902\\pm 0.0040$~s~yr\\textsuperscript{-1}.}\n\\label{J234401oc2}\n\\end{figure}\n\nJ234401 (Fig.~\\ref{J234401lc}) may provide an example of such cyclical\nvariation.  In \\citet{lohr}, we advanced this system as illustrating\nvery rapid period decrease ($-0.313\\pm0.019$~s~yr\\textsuperscript{-1},\nsignificant at $16\\sigma$) on the basis of the first four years of\nSuperWASP observations.  However, with the addition of a further\npartial year, following a year-long gap in observations of the object,\nthe O$-$C diagram now presents a rather different picture\n(Fig.~\\ref{J234401oc}).  Highly significant period change is still\nindicated (at $35\\sigma$ now), but its magnitude is weakened by the\naddition of the further data, and the quadratic fit is evidently a\npoor match.  One possible explanation is that the object is in fact\nvarying in period sinusoidally, with a period of around 4--4.5 years.\nSuch cyclical period variation could itself be superimposed on a\nsecular change, the direction of which would require a longer\ntime-base of observations to establish.\n\nIf the period variation is sinusoidal, and due to the Applegate\nmechanism, the object would have a (semi-amplitude) $\\Delta P\/P$ of\n$\\sim3.3\\times$10\\textsuperscript{-5}, according to\n\\citeauthor{applegate}'s equation (38).  This is at the upper end of\nperiod modulation amplitudes considered by \\citeauthor{applegate} as\nexplicable by his mechanism, but of the same order of magnitude.\nThere is also a possible suggestion of average flux variation from\nyear to year, which could support the Applegate mechanism as the\nexplanation here.  However, if the sinusoidal variation were due to a\nthird body, this object would have to be of comparable mass to the\nbinary system itself (using equations 10 and 11 in \\citet{pribulla},\nand estimates for binary parameters described in \\citet{lohr}).  With\na separation of $\\sim$3~AU between third body and binary, we would not\nexpect to be able to resolve its light separately from the contact\nsystem, but whether such a triple system would be stable in the long\nterm is unclear, and would require modelling to assess.\n\nAlternatively, J234401's period could have continued to decrease\nafter the first four years, with the O$-$C values for the final year\nbeing so large and negative that they have been `wrapped round' by\nhalf a cycle i.e. the secondary eclipses now occur near to where the\nprimary eclipses would be calculated to occur on the basis of an\nunchanging period.  Since the primary and secondary eclipses are of\nsimilar depth and shape, this would not create a problem for the\nfitting algorithm.  Figure~\\ref{J234401oc2} indicates the O$-$C\ndiagram that would result if this had in fact occurred.  A quadratic\nfunction now provides a far better fit, though an acceleration of\nperiod decrease would be indicated\n($-0.6902\\pm0.0040$~s~yr\\textsuperscript{-1}) and the data for year 4\nnow stands out as discrepant.  Period determinations for the\nindividual years of data also do not support a continuing rapid\ndecrease, though the uncertainties in period are substantial for the\nfinal year, which contains relatively few nights of observations.\n\nWhat is clear, however, is that J234401 is undergoing highly\nsignificant and dramatic period changes of some sort.  We hope in the\nnear future to use newly-obtained spectroscopic data to learn more\nabout this interesting object.\n\n\\subsection{J102328}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J102328lc.eps}}\n\\caption{Lightcurve of object J102328 folded at period 18\\,125.146~s.}\n\\label{J102328lc}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J102328oc.eps}}\n\\caption{O$-$C diagram for object J102328 (uncertainties not plotted\nfor clarity).  Dashed line shows best linear fit ($\\chi^2=1.74$);\nsolid line shows best quadratic fit ($\\chi^2=1.46$), corresponding to\na secular period change of $-0.254\\pm\n0.037$~s~yr\\textsuperscript{-1}.}\n\\label{J102328oc}\n\\end{figure}\n\nAnother object of note is J102328 (Figs.~\\ref{J102328lc} and\n\\ref{J102328oc}), which shows period decrease nearly as rapid as that\napparently seen in J234401 on the basis of its first four years of\ndata: $-0.254\\pm0.037$~s~yr\\textsuperscript{-1}, significant at\n$6\\sigma$.  If this decrease continued it would imply a merger\ntimescale ($P\/\\dot{P}$) of at most 71\\,500 years; however, only three\nyears of SuperWASP observations are available for this object, so\ncaution is warranted: future years of data might support a sinusoidal\nperiod variation instead.  We may also note that this magnitude of\nperiod decrease falls within the symmetrical distribution found for\nthe 38 highly significant objects taken as a group.\n\n\\subsection{J093010}\n\\label{sub:J093010}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[trim = 7mm 0mm 0mm 0mm,\nclip,angle=270]{temp_pspec.eps}}\n\\caption{Power spectrum for J093010, showing strongest signal at $f_1$\n(associated with the period of the contact binary) and a weaker signal\nat $f_2$ and its harmonics (associated with the period of the\nAlgol-type binary).}\n\\label{powerspec}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J093010.eps}}\n\\caption{Top: lightcurve of object J093010 (combined with duplicate)\nfolded at dominant contact binary period 19\\,674.574~s, with median\nbinned lightcurve overplotted.  Middle: lightcurve of Algol-type\neclipsing binary folded at period 112\\,799.109~s, after subtraction of\nmedian binned lightcurve.  Bottom: lightcurve of W~UMa-type eclipsing\nbinary after subtraction of Algol median binned lightcurve.}\n\\label{J093010}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{J093010ind.eps}}\n\\caption{Individual nights of observations of object J093010, with\nmedian binned lightcurves of Algol and contact binaries overplotted\n(flux levels adjusted to allow direct comparison).  Top: the Algol's\nprimary eclipse coincides with the contact system's secondary eclipse.\nMiddle: the Algol's secondary eclipse nearly coincides with a maximum\nof the contact system.  Bottom: the Algol's primary eclipse falls\nbetween the contact system's minima.}\n\\label{J093010ind}\n\\end{figure}\n\nOf particular interest is object J093010.  This does not exhibit\nsignificant period change, but its O$-$C diagram showed a surprising\namount of scatter.  In \\citet{lohr}, the same object's apparently\nsignificant period change was rejected as the result of contamination\nby a nearby star, but here, a fuller explanation was pursued.  Prior\nto analysis, the data for J093010 was combined with that from a\n`duplicate' object with a similar identifier (see Sect.~\\ref{method})\nto maximize the available observations.  J093010's lightcurve, folded\nat 19\\,674.574~s (Fig.~\\ref{J093010}, top) then showed a typical contact\nbinary shape, but with numerous non-random data points below the main\ncurve.  A visual examination of the object's full lightcurve suggested\nthe cause was additional deep eclipses on certain individual nights,\nimplying an additional eclipsing body in the field of view.  A\nfrequency power spectrum also supported an additional periodic signal\nnear 1.3~d (Fig.~\\ref{powerspec}).  Stripping out the median binned\nlightcurve (corresponding to the contact binary) from the data yielded\nthe lightcurve of an Algol-type eclipsing binary with period\n112\\,799.109~s (Fig.~\\ref{J093010}, middle).  Stripping the median\nbinned lightcurve of the latter object out of the combined data also\nyielded a clean lightcurve for the contact eclipsing binary\n(Fig.~\\ref{J093010}, bottom).  Figure~\\ref{J093010ind} illustrates\nhow the observations during three representative nights of J093010 are\nwell-reproduced as the sum of these two eclipsing binary lightcurves.\n\nThe question is then, are these two systems physically related, or is\ntheir juxtaposition on the sky coincidental?  Two sources were\nobserved at this location by Hipparcos as \\object{TYC~3807-759-1} and\n\\object{TYC~3807-759-2}, with equivalent Johnson $V$ magnitudes 9.851 and\n10.990 respectively (corresponding to SuperWASP fluxes around 110 and\n42), a separation of 1.88$\\arcsec$, and a common proper motion (pmRA:\n-8.0~mas~yr\\textsuperscript{-1}, pmDE:\n-9.4~mas~yr\\textsuperscript{-1}).  This would seem to strongly favour\nan interpretation of the two eclipsing binary systems as being\ngravitationally bound in a rare quadruple doubly-eclipsing system.\nThe distance to TYC~3807-759-1 has been calculated as 35.17~pc,\nyielding a separation of 66.1~AU.  Assuming that this separation\ncorresponds to apastron, and that the total mass for both systems is\n$\\sim2$ solar masses, a meta-orbital period of $\\sim380$ years is\nindicated.\n\nOnly five other doubly eclipsing quadruple systems have been proposed:\n\\object{BV~Dra}+\\object{BW~Dra}, a contact+contact system\n\\citep{batten}; \\object{V994~Her}, an Algol+Algol system\n\\citep{lee08}; \\object{OGLE-LMC-ECL-16545}, Algol+contact\n\\citep{graczyk}; \\object{KIC~4247791}, Algol+Algol\n\\citep{lehmann}; and \\object{Cze~V343}, another Algol+contact system\n\\citep{cagas}.  The contact binary in this new quadruple has a shorter\nperiod than any system in the other five quadruples, making it\nparticularly amenable to further observations.\n\n\\subsection{Other systems of note}\n\nSeveral lightcurves in Fig.~\\ref{appfig1} (online only) resemble\n$\\beta$ Lyrae-type variables (EB), showing notable differences in\nprimary and secondary eclipse depths: J011732, J022050, J070953,\nJ093443, J102328, J215826 and J222302 are perhaps the clearest\nexamples.  Surprisingly for such short-period objects, these may\nrepresent detached or semi-detached systems not in thermal contact.\nIndeed, J022050 (=BX~Tri), which shows substantial differences in\ndepths of minima, as well as a large O'Connell effect\n\\citep{oconnell}, has been studied in detail by \\citet{dimitrov}, who\nmodelled it as not quite in contact, but with one star nearly filling\nits Roche lobe.  Other objects showing pronounced O'Connell effects\n(maxima of different heights) are J033242, J084408, J115326, J201816\n(if a genuine eclipsing binary), and J221117, of which the last is\nparticularly striking and unusual in shape.  Finally, we observe\nsignificant variations in average flux in several objects, including\nJ121206 (which shows distinct well-defined curves at different\nlevels), J134430, J150957, J151146, J152022, J172717 and J173003\n(which show broader banding or continuous variation in average flux\nlevel).  In some cases this variation seems intrinsic to the system,\nbeing observable within a single night's observations with a single\ncamera; in other cases it probably has instrumental causes e.g. the\nobject was observed over a long time-base by cameras in both the\nnorthern and southern hemispheres.\n\n\n\\section{Conclusions}\n\nOur new search of the SuperWASP archive yielded 143 plausible\ncandidate eclipsing binaries with orbital periods $<$20\\,000~s, of which\n97 are new discoveries.  This updates the findings of \\citet{norton},\nand provides a useful new sample of extremely short period eclipsing\nsystems near the cut-off point.  Their period distribution fits neatly\nat one end of previous period distributions found for eclipsing\nbinaries seen by other comparable surveys.  The shape of the\ndistribution may also inform understanding of the reasons for the\nshort-period limit for main sequence binaries.\n\nAn improved period change detection method was employed with the new\nsample, which measured observed times of minima by fitting each night\nof observations with the object's own binned mean lightcurve, obtained\nfrom its full data set during period determination.  This approach\nconfirmed some earlier results in \\citet{lohr}, and found significant\nevidence for period change in 74 of the candidate binaries, of which\n38 are significant at $3\\sigma$ or more.  The distribution of\nsignificant period changes found agrees substantially with a previous\ncomparable distribution, showing equal numbers of systems increasing\nand decreasing in period.  Of the highly significant period changes,\nnone exceed 0.3~s~yr\\textsuperscript{-1} in magnitude, and the\nhalf-width at half-maximum of the distribution is\n$\\sim0.1$~s~yr\\textsuperscript{-1}, slightly larger than that found in\na previous study of binaries with longer periods.\n\nThree of the systems found were of particular interest.  J234401,\ndiscussed in \\citet{lohr}, continues to show highly significant and\ndramatic period changes, though it is currently unclear whether it is\nvarying in period sinusoidally, or steadily decreasing in period with\nunusual rapidity.  J102328 currently shows the most rapid period\ndecrease of the whole sample: $-0.254\\pm\n0.037$~s~yr\\textsuperscript{-1}, significant at $6\\sigma$, which would\nindicate a merger timescale $\\le$70\\,000 years if the decrease\ncontinues.  J093010 appears to be the sixth reported\ndoubly eclipsing quadruple system, consisting of a contact binary with\na 19\\,674.575~s period and an Algol-type binary with a 112\\,799.109~s\nperiod, separated by 66.1~AU, and plausibly orbiting each other with a\nperiod of $\\sim 400$ years.\n\n\\begin{acknowledgements}\nThe WASP project is funded and operated by Queen's University Belfast,\nthe Universities of Keele, St. Andrews and Leicester, the Open\nUniversity, the Isaac Newton Group, the Instituto de Astrofisica de\nCanarias, the South African Astronomical Observatory and by STFC.\nThis work was supported by the Science and Technology Funding Council\nand the Open University.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nA long-standing problem in the study of topological dynamical systems is the conjugacy problem, i.e., the problem of determining whether two dynamical systems which appear different actually exhibit the same dynamical behavior. A related problem is to determine when a topological dynamical system factors onto another one, i.e., when there is a surjective continuous map from the first to the second which intertwines their actions. Such maps are called (topological) factor maps, and they have been widely studied. We focus on these problems in the context of symbolic dynamical systems, also called subshifts.\n\nFor any natural number $d$ and finite set $\\mathcal{A}$ (given the discrete topology), a $\\mathbb{Z}^d$ subshift is any closed subset (with respect to the product topology) of $\\mathcal{A}^{\\mathbb{Z}^d}$ which is invariant under every translation $\\sigma_t$ by a vector $t \\in \\mathbb{Z}^d$.\nWe often refer to a subshift by the set $X$, with the understanding that the dynamics are always provided by the restriction of $\\sigma$ to $X$. Examples of easily defined subshifts are the so-called $\\mathbb{Z}^d$ shifts of finite type (or $\\mathbb{Z}^d$ SFTs): for any finite set $\\mathcal{F}$ of finite patterns, $X(\\mathcal{F})$ is defined as the set of all elements of $\\mathcal{A}^{\\mathbb{Z}^d}$ which do not contain any pattern in $\\mathcal{F}$. A special case is $X(\\mathcal{\\varnothing}) = \\mathcal{A}^{\\mathbb{Z}^d}$, called the full shift. \n\nThere are two well-known necessary conditions for the existence of a factor map $\\phi$ from $X$ onto $Y$. First, note that if $\\sigma_t(x) = x$ for some $x \\in X$ and $t \\in \\mathbb{Z}^d$, then $\\sigma_t(\\phi(x)) = \\phi(x)$.   \nThus, $X$ and $Y$ must satisfy Condition (P): for every $x \\in X$, there exists $y \\in Y$ such that if $\\sigma_t(x) = x$, then $\\sigma_t(y) = y$.\nNote that this condition is always satisfied when $Y$ contains a fixed point, i.e. $y \\in Y$ where $\\sigma_t(y) = y$ for all $t \\in \\mathbb{Z}^d$.\nSecond, the topological entropy of a $\\mathbb{Z}^d$ subshift $X$ \n(denoted by $h(X)$; see Section~\\ref{defs} for the definition) cannot increase under a factor map, and so $h(X) \\geq h(Y)$ must hold. Surprisingly, for restricted classes of subshifts, these necessary conditions also seem to be nearly sufficient. (A stronger form of the following theorem appears in \\cite{boyle}.)\n\n\\begin{theorem}[\\cite{boyle}] \nFor mixing $\\mathbb{Z}$ SFTs $X$ and $Y$ with $h(X) > h(Y)$, there exists a factor map from $X$ onto $Y$ if and only if $X$ and $Y$ satisfy Condition (P). \n\\end{theorem}\n\nWhen $d=1$ and $Y$ is a full shift, even the equal entropy case ({i.e., $h(X) = h(Y)$) has been solved. In this case, $Y$ automatically contains a fixed point, and so no additional periodic point hypothesis is necessary.\n\n\\begin{theorem}[\\cite{boyle}, \\cite{marcus}] \nFor a $\\mathbb{Z}$ SFT $X$ and a full shift $Y$ with $h(X) \\geq h(Y)$, there exists a factor map from $X$ onto $Y$.\n\\end{theorem}\n\nUnfortunately, the situation is much more complicated for $d > 1$. In particular, there are several different candidates for a proper extension of ``mixing'' to the multidimensional case. One commonly used condition is the block gluing condition defined in \\cite{BPS}, and a much stronger one is the existence of a so-called safe symbol (definitions are given in Section~\\ref{defs}). We do not attempt to summarize the entire literature on this topic, but here are a few representative results. First, the theorems for $\\mathbb{Z}$ subshifts do not directly extend to $\\mathbb{Z}^d$ subshifts when $d > 1$.\n\n\\begin{theorem}[\\cite{BPS}] \n\\label{theoremBPS1}\nFor every $d > 1$, there exist topologically mixing $\\mathbb{Z}^d$ SFTs with arbitrarily high entropy which do not factor onto any nontrivial full shift.\n\\end{theorem}\n\n\\begin{theorem}[\\cite{PS}]\n\\label{theoremPS}\nFor every $d \\geq 3$ and every nontrivial $\\mathbb{Z}^d$ full shift $Y$, there exists a block gluing $\\mathbb{Z}^d$ SFT $X$ with $h(X) = h(Y)$ such that there is no factor map from $X$ onto $Y$.\n\\end{theorem}\n\n\nUnder a strict entropy inequality, the block gluing hypothesis, which allowed for the negative examples of Theorem~\\ref{theoremPS}, implies a positive result for $d > 1$ even for general subshifts.\n\n\n\n\\begin{theorem}[\\cite{BPS}]\n\\label{theoremBPS2}\nIf $X$ is a block gluing $\\mathbb{Z}^d$ subshift, $Y$ is a $\\mathbb{Z}^d$ SFT with a safe symbol, and $h(X) > h(Y)$, then there exists a factor map from $X$ onto $Y$.\n\\end{theorem}\n\nWe also note that the safe symbol hypothesis in Theorem \\ref{theoremBPS2} is very restrictive, and is not at all invariant under topological conjugacy.\n\n\n\nIn this work, we define a new condition called the finite extension property, which is significantly weaker than the existence of a safe symbol. We prove that this condition is conjugacy-invariant, and then we prove the following main result\n\n\\begin{theorem}\\label{mainthm}\nIf $X$ is a block gluing $\\mathbb{Z}^d$ subshift, $Y$ is a $\\mathbb{Z}^d$ SFT with a fixed point and the finite extension property, and $h(X) > h(Y)$, then there exists a factor map from $X$ onto $Y$.\n\\end{theorem}\n\n\nFor a $\\mathbb{Z}^d$ SFT defined by a set of forbidden pairs of adjacent letters, an easily verified (but not conjugacy invariant) condition is single-site fillability or SSF (\\cite{MP}). For $d=2$, SSF means \nthat for any choice of letters $a,b,c,d \\in \\mathcal{A}$, there exists $e \\in \\mathcal{A}$ for which the pattern $\\begin{smallmatrix} & a & \\\\ b & e & c\\\\ & d & \\end{smallmatrix}$ contains none of the forbidden adjacent pairs. Using the forbidden adjacencies as the set of forbidden patterns, it is straightforward to check that SSF implies the finite extension property. The following corollary is immediate.\n\n\\begin{corollary}\\label{maincor}\nIf $X$ is a block gluing $\\mathbb{Z}^2$ subshift, $Y$ is a $\\mathbb{Z}^2$ SFT that satisfies single-site fillability and has a fixed point, and $h(X) > h(Y)$, then there exists a factor map from $X$ onto $Y$.\n\\end{corollary}\n\nCorollary~\\ref{maincor} can be used to create explicit examples of new subshifts to which our results apply, since there are many nearest-neighbor $\\mathbb{Z}^d$ SFTs which have fixed points and satisfy SSF without having a safe symbol. For instance, one can take any alphabet $\\mathcal{A}$ with $|\\mathcal{A}| \\geq 2d+1$, take any non-identity involution $f$ on $\\mathcal{A}$, and define $Y$ by the rule that no pair of letters $\\{a,f(a)\\}$ ($a \\in \\mathcal{A}$) can be adjacent.\n\n\\section*{acknowledgements} \nThe authors would like to thank the anonymous referee for making many useful comments, which significantly improved the clarity and presentation of this work.\n\n\n\n\\section{Definitions}\\label{defs}\n\nWe begin with some basic geometric definitions for $\\mathbb{Z}^d$. Anytime we refer to distance in $\\mathbb{Z}^d$, it is with respect to the $\\ell_{\\infty}$ distance given by $d((v_i)_{i=1}^d, (w_i)_{i=1}^d) = \\max_i(|v_i - w_i|)$.\nFor sets $A, B \\subset \\mathbb{Z}^d$, we define $d(A,B) = \\min_{a \\in A, b \\in B} d(a,b)$. For every $k$, we use $C_k$ and $Q_k$ to denote the hypercubes $[0,k-1]^d$ and $[-k,k]^d$ respectively. For any set $S \\subset \\mathbb{Z}^d$, we define its \\textbf{inner $k$-boundary} $\\partial_k S$ to be the set of all $t \\in S$ within distance $k$ from some $t' \\in S^c$.\n\n\n\\begin{definition}\nA \\textbf{pattern} over a finite alphabet $\\mathcal{A}$ is a member of $\\mathcal{A}^S$ for some $S \\subset \\mathbb{Z}^d$, which is said to have \\textbf{shape} $S$. We may refer to any pattern with finite shape as a \\textbf{finite pattern}.\n\\end{definition}\n\nWe consider patterns to be defined up to translation: if $u \\in \\mathcal{A}^S$ for a finite $S \\subset \\mathbb{Z}^d$ and $v \\in \\mathcal{A}^T$, where $T = S+t$ for some $t \\in \\mathbb{Z}^d$, then we write $u = v$ to mean that $u(s) = v(s+t)$ for each $s$ in $S$. \n\nFor any patterns $v \\in \\mathcal{A}^S$ and $w \\in \\mathcal{A}^T$ with $S \\cap T = \\varnothing$, we define the concatenation $vw$ to be the pattern in $\\mathcal{A}^{S \\cup T}$ defined by $(vw)(S) = v$ and $(vw)(T) = w$.\n\n\\begin{definition}\nFor any finite alphabet $\\mathcal{A}$, the \\textbf{$\\mathbb{Z}^d$-shift action} on $\\mathcal{A}^{\\mathbb{Z}^d}$, denoted by $\\{\\sigma_t\\}_{t \\in \\mathbb{Z}^d}$, is defined by $(\\sigma_t x)(s) = x(s+t)$ for $s,t \\in \\mathbb{Z}^d$. \n\\end{definition}\n\nWe always think of $\\mathcal{A}^{\\mathbb{Z}^d}$ as being endowed with the product discrete topology, with respect to which it is compact. \n\n\\begin{definition}\nA \\textbf{$\\mathbb{Z}^d$ subshift} is a closed subset of $\\mathcal{A}^{\\mathbb{Z}^d}$ that is invariant under the $\\mathbb{Z}^d$-shift action.\n\\end{definition}\n\nAny $\\mathbb{Z}^d$ subshift inherits a topology from $\\mathcal{A}^{\\mathbb{Z}^d}$, with respect to which it is compact. Each $\\sigma_t$ is a homeomorphism on any $\\mathbb{Z}^d$ subshift, and so any $\\mathbb{Z}^d$ subshift, when paired with the $\\mathbb{Z}^d$-shift action, is a topological dynamical system. \n\nAny $\\mathbb{Z}^d$ subshift can also be defined in terms of forbidden patterns: for any set $\\mathcal{F}$ of finite patterns over $\\mathcal{A}$, one can define the set\n$$X(\\mathcal{F}) := \\{x \\in \\mathcal{A}^{\\mathbb{Z}^d} \\ : \\ x(S) \\notin \\mathcal{F} \\ \\text{ for all finite } S \\subset \\mathbb{Z}^d\\}.$$It is well known that any set of the form $X(\\mathcal{F})$ is a $\\mathbb{Z}^d$ subshift, and all $\\mathbb{Z}^d$ subshifts may be presented in this way. \n\n\\begin{definition}\nA \\textbf{$\\mathbb{Z}^d$ shift of finite type (SFT)} is a $\\mathbb{Z}^d$ subshift equal to $X(\\mathcal{F})$ for some finite set $\\mathcal{F}$ of forbidden finite patterns.\n\\end{definition}\n\n\n\\begin{definition}\nThe \\textbf{language} of a $\\mathbb{Z}^d$ subshift $X$, denoted by $L(X)$, is the set of all patterns that appear in elements of $X$. For any $S \\subset \\mathbb{Z}^d$, let $L_S(X) := L(X) \\cap \\mathcal{A}^S$, the set of patterns in the language of $X$ with shape $S$. A finite pattern $w$ will be called a \\textbf{first offender} for $X$ if it is not in $L(X)$ but every proper subpattern of $w$ belongs to $L(X)$.\n\\end{definition}\n\n\\begin{remark}\nWe have defined the language of a subshift to include both the finite and infinite patterns that appear in elements of $X$. We adopt this convention for convenience of presentation, despite the fact that many authors do not include infinite patterns in the language.\n\\end{remark}\n\n\\begin{definition}\nSuppose $X$ and $Y$ are compact, metrizable spaces. Further suppose that $\\mathbb{Z}^d$ acts on each of these spaces by homeomorphisms, with actions denoted by $\\sigma$ and $\\tau$, respectively.\nA (topological) \\textbf{factor map} is any continuous surjection $\\phi : X \\to Y$ such that $\\phi \\circ \\sigma_t = \\tau_t \\circ \\phi$ for each $t \\in \\mathbb{Z}^d$. In this case, the pair $(Y,\\tau)$ is called a \\textbf{factor} of $(X,\\sigma)$, and we say that $X$ factors onto $Y$. A bijective factor map is called a \\textbf{topological conjugacy}.\n\\end{definition}\n\nFor the purposes of this work, we restrict attention to factor maps between subshifts. It is well-known that any factor map $\\phi$ between $\\mathbb{Z}^d$ subshifts is a so-called sliding block code, i.e., there exists $n \\in \\mathbb{N}$ so that $x(t + [-n,n]^d)$ uniquely determines $(\\phi(x))(t)$ for any $x \\in X$ and $t \\in \\mathbb{Z}^d$; such $n$ is usually called a radius for the sliding block code. (See \\cite{LM} for a proof for $d = 1$, which extends to $d > 1$ without changes.) When convenient, for a pattern $w$ with shape $S$, we may use $\\phi(w)$ to denote its image under a sliding block code $\\phi$ with radius $n$, with shape $S \\setminus \\partial_n S$.\n\n\n\n\\begin{definition}\\label{entdef}\nThe \\textbf{topological entropy} of a $\\mathbb{Z}^d$ subshift $X$ is\n$$h(X) := \\lim_{n \\rightarrow \\infty} \\frac{1}{n^d} \\log | L_{C_n}(X) |.$$\nThis limit exists by a standard subadditivity argument.\n\\end{definition}\n\n\nFinally, let us define the mixing properties for $\\mathbb{Z}^d$ subshifts which we will need\n\n\n\n\\begin{definition}\\label{block}\nA $\\mathbb{Z}^d$ subshift $X$ is {\\bf block gluing} if there exists $g \\geq 0$ so that for any hyperrectangles $R, R' \\subset \\mathbb{Z}^d$ with $d(R, R') > g$ and any $w \\in L_R(X)$ and $w' \\in L_{R'}(X)$, there exists $x \\in X$ with $x(R) = w$ and $x(R') = w'$.\n\\end{definition}\n\n\\begin{definition}\\label{safe}\nA letter $* \\in \\mathcal{A}$ is a \\textbf{safe symbol} for a $\\mathbb{Z}^d$ subshift $X$ if for any point $x \\in X$ and any $S \\subseteq \\mathbb{Z}^d$, changing each letter of $x$ on $S$ to $*$ yields a point in $X$.\n\\end{definition}\n\n\n\\begin{definition}\\label{extprop}\nFor $g \\in \\mathbb{N}$, a $\\mathbb{Z}^d$ SFT $X$ has the \\textbf{$g$-extension property} if there exists a finite set $\\mathcal{F}$ of forbidden finite patterns inducing $X$ with the following property: if a pattern $w$ with shape $S$ can be extended to a pattern on $S + Q_g$ which does not contain any patterns from $\\mathcal{F}$, then $w \\in L(X)$, i.e., it can be extended to a point on all of $\\mathbb{Z}^d$ which does not contain any patterns from $\\mathcal{F}$. We say that $X$ has the \\textbf{finite extension property} if it has the $g$-extension property for some $g$.\n\\end{definition}\n\n(The reader may check that any $X$ with the $g$-extension property is block gluing at distance $2g$ plus the maximum diameter over $w \\in \\mathcal{F}$.)\n\n\n\n\n\n\n\nThe \\textbf{topological strong spatial mixing (TSSM) property} for $\\mathbb{Z}^d$ SFTs was introduced in \\cite{briceno}, where it was also shown to be equivalent to the existence of only finitely many first offenders for $X$.\n\n\\begin{proposition}\nA $\\mathbb{Z}^d$ SFT $X$ has the TSSM property if and only if it has the $0$-extension property.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that $X$ has the TSSM property and therefore has only finitely many first offenders. Let $\\mathcal{F}$ denote the list of first offenders. We claim that $X$ has the $0$-extension property for $\\mathcal{F}$. In fact, $X = X(\\mathcal{F})$ and if $w$ is a pattern not in $L(X)$, then $w$ must contain a minimal subpattern not in $L(X)$, which by definition is a first offender.\n\nFor the reverse implication, suppose that $X$ has the $0$-extension property for a finite set $\\mathcal{F}'$ of forbidden finite patterns of diameter at most $g$. Assume, for the sake of contradiction, that $w$ is a first offender of diameter greater than $g$. Then, $w \\notin \\mathcal{F}'$ and, by definition of first offender, every proper subpattern of $w$ is in $L(X)$ and so not in $\\mathcal{F}'$. Therefore, by the $0$-extension property, $w$ is in $L(X)$, contradicting the assumption that $w$ is a first offender. We conclude that first offenders have bounded diameter, so there must be finitely many of them.\n\\end{proof}\n\nIt is known that the existence of a safe symbol implies TSSM (see \\cite{briceno}). Thus we have the following corollary.\n\n\\begin{corollary}\nIf $X$ is a $\\mathbb{Z}^d$ SFT with a safe symbol, then $X$ has the $0$-extension property.\n\\end{corollary}\n\n\n\nAs noted in the introduction, the finite extension property is also invariant under topological conjugacy.\n\n\\begin{theorem}\\label{conjinv}\nIf $X$ and $Y$ are conjugate $\\mathbb{Z}^d$ SFTs and $X$ has the finite extension property, then $Y$ has the finite extension property.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $X$ has the $g$-extension property (for forbidden list $\\mathcal{F}$) and that $\\phi: X \\rightarrow Y$ is a conjugacy. Denote by $r$ the radius of $\\phi$ and by $s$ the radius of $\\phi^{-1}$. Define a list of patterns on $\\mathcal{A}_Y$ as follows:\n\\[\n\\mathcal{F}' := \\{w \\in \\mathcal{A}_Y^{S + Q_s} \\ : \\ v \\in \\mathcal{F}, \\textrm{ $v$ has shape $S$, } \\phi^{-1}(w) \\textrm{ contains } v\\}.\n\\]\n\nClearly $\\mathcal{F}'$ is a finite list of finite patterns, and we claim that it induces the shift of finite type $Y$. \nIndeed, by definition, if $y \\in Y$, then $\\phi^{-1}(y) \\in X$, and therefore $y$ contains no pattern in $\\mathcal{F}'$. On the other hand, if $y \\in \\mathcal{A}_Y^{\\mathbb{Z}^d}$ contains no pattern in $\\mathcal{F}'$, then the point $x$ defined by $x(t) = \\phi^{-1}(y(t+Q_s))$ contains no pattern in $\\mathcal{F}$, so is in $X$, and therefore $y = \\phi(x)$ is in $Y$.\n \nNow assume that a pattern $w \\in \\mathcal{A}_Y^S$ can be extended to a pattern $v \\in \\mathcal{A}_Y^{S + Q_{g+r+s}}$ containing no patterns from $\\mathcal{F'}$. Then, by definition, $\\phi^{-1}(v)$ contains no patterns from $\\mathcal{F}$; say that $\\phi^{-1}(v)$ has shape $T$, and note that $T \\supseteq S + Q_{g+r}$. Then by $g$-extension of $X$, the pattern $(\\phi^{-1}(v))(T \\setminus \\partial_g(T))$ is in $L(X)$. Then obviously \n$\\phi((\\phi^{-1}(v))(T \\setminus \\partial_g(T))) \\in L(Y)$, and we note that its shape contains $S$. Finally, by definitions of $r$ and $s$, we have $\\phi((\\phi^{-1}(v))(T \\setminus \\partial_g(T)))(S) = w$, and so $w \\in L(Y)$, completing the proof.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{mainthm}}\\label{proof}\n\nThe overall structure of our proof is similar to previous proofs which used\nmixing properties to construct factor maps onto various shifts (see \\cite{BPS} and \\cite{desai}). By this, we mean that the proof involves using marker patterns to define ``surrounded patterns'' in points of the domain, which will be used to assign patterns on ``determined zones'' after application of the map.\nVery roughly speaking, given $x \\in X$, its image $\\phi(x)$ will have patterns on determined zones that depend on corresponding surrounded patterns in $x$, and $\\phi(x)$ will look like the fixed point of $Y$ at all sites not near a determined zone. Then we will fill the area between determined zones and the fixed point ``background'' in stages using the $g$-extension property of $Y$. First we give the proof for $d = 2$ in order to present a streamlined argument with illustrations, and then we describe the changes that need to be made for $d > 2$.\n\nTo begin the formal proof, choose any $X$ and $Y$ as in the theorem, with alphabets $\\mathcal{A}_X$ and $\\mathcal{A}_Y$, respectively. We assume without loss of generality that $g \\geq 0$ is a gap distance for the block gluing of $X$, that $Y$ has the $g$-extension property for a finite list $\\mathcal{F}$ of forbidden finite patterns with diameters less than or equal to $g$,\nand that the fixed point $*^{\\mathbb{Z}^2}$ is in $Y$. \n\nWe now construct markers in $X$ following \\cite{BPS}, but we repeat some details here to set notation. Let $p > 5g$, and choose a pattern $P \\in L_{C_p}(X)$ so that $h(X_P) > h(Y)$, where $X_P$ is the subshift consisting of points of $X$ which do not contain the pattern $P$ (see \\cite{QT}). Then define a pattern $Q \\in L_{C_q}(X_P)$ (for some $q \\in \\mathbb{N}$ perhaps much larger than $p$) for which $Q$ cannot overlap itself at any nonzero vector in $Q_{g+p} = [-g-p,g+p]^2$, i.e., for every such vector $t$, there does not exist $x \\in \\mathcal{A}^{\\mathbb{Z}^2}$ for which $x(C_q) = x(C_q + t) = Q$ (see \\cite{BPS,desai}). Then use block gluing to create a marker pattern $M \\in L_{C_m}(X)$ ($m = 2p + 2g + q$) with $P$ at each corner, $Q$ in the center, and patterns $G_i \\in L(X_P)$, $1 \\leq i \\leq 4$, along each edge, as in the left half of Figure~\\ref{TSSMpic1}. Any pattern as in the right half of Figure~\\ref{TSSMpic1}, where $W \\in L_{C_k}(X_P)$ and each $H_i \\in L(X_P)$, $1 \\leq i \\leq 4$, is called a \\textbf{surrounding frame}, whose central occurrence of $W$ is called a \\textbf{surrounded pattern}. \nThe side length $k$ of the shape of $W$ is for now arbitrary, and will be fixed later. For any surrounding frame $x(t+C_{k+2g+2m})$ in $x$, we refer to the region $t + (g+m) \\vec{1} + C_{k+g+m}$ as a \\textbf{determined zone} in $\\phi(x)$. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.6125]{Fig1b-eps-converted-to.pdf}\n\\caption{A marker pattern (left) and a surrounding frame (right)}\n\\label{TSSMpic1}\n\\end{figure}\n\nWe need a few simple facts about the locations of determined zones. Firstly, shown exactly as in \\cite{BPS} and \\cite{desai}, by the marker properties defining $M$, any two determined zones have distance more than $g$ from each other. In fact, for any two determined zones $t_1 + C_{k+g+m}$ and $t_2 + C_{k+g+m}$ with distance exactly $g+1$, the surrounding frames $x(t_1 - (g+m) \\vec{1} + C_{k+2g+2m})$ and $x(t_2 - (g+m) \\vec{1} + C_{k+2g+2m})$ have overlap consisting of either exactly one occurrence of $M$ or a rectangle with dimensions $m$ and $k + 2g + 2m$ with occurrences of $M$ at the extreme ends. (See Figure~\\ref{TSSMpic2}.) In either case, we say that those determined zones are \\textbf{adjacent}. We use the term \\textbf{component of determined zones} to refer to a maximal connected component with respect to this notion of adjacency. Finally, we claim that if two determined zones $Z_1$ and $Z_2$ are not adjacent, then\n\\begin{equation} \\label{Eqn:Banff}\nd(Z_1,Z_2) > 2g+p > 7g.\n\\end{equation} \n\nTo see this, suppose for a contradiction that two determined zones are separated by distance more than $g$ and less than or equal to $2g + p$. This means that $x$ contains two surrounding frames separated by a vector $t = (t_1, t_2)$ where $k + 2g + m < \\max(|t_1|, |t_2|) \\leq k + 3g + m + p$, which without loss of generality we can take to be $x(C_{k+3g+m})$ and $x(t + C_{k + 3g + m})$. Our argument will rely only on the general structure of surrounded frames (and not the specific values of $W$ or the $H_i$), and so is unaffected by reflections about horizontal, vertical, or diagonal lines. Therefore, we may assume without loss of generality that $k + 2g + m < t_1 \\leq k + 3g + m + p$ and $0 \\leq t_2 \\leq k + 3g + m + p$. \n\nWe note that if $t_2 \\in [0, g+p]$, then $d(t, (k+2g+m,0)) \\leq g + p$, meaning that the lower-right copy of $M$ within $x(C_{k+2g+2m})$ and the lower-left copy of $M$ within $x(t + C_{k+3g+m})$ would have separation by a nonzero vector in $Q_{g+p}$. This contradicts the definition of $Q$ and so is impossible. The case $t_2 \\in [k+2g+m, k+3g+m+p]$ is also not possible, by a similar argument using the upper-right copy of $M$ within $x(C_{k+2g+2m})$ and the lower-left copy of $M$ within $x(t + C_{k+3g+m})$. Therefore, $t_2 \\in (g+p, k+2g+m)$. However, this implies that the lower-left copy of $M$ within $(t + C_{k + 2g + 2m})$ overlaps the pattern $H$ along the right side of $x(C_{k+2g+2m})$ in a rectangle with height at least $p$ and width at least $m - g - p = p + g + q$. This yields a contradiction since $M$ has a copy of $P$ in each corner and $H$ was assumed in $L(X_P)$. We have thus established (\\ref{Eqn:Banff}), a fact which will be useful later. \\\\\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.6125]{Fig2b-eps-converted-to.pdf}\n\\caption{Surrounding frames and component of determined zones induce by them (left). Reduced determined zones after Stage 3 in a $*$-background (right).}\n\\label{TSSMpic2}\n\\end{figure}\nNow let $x$ be in $X$. Informally speaking, $\\phi(x)$ will be defined in six alternating stages, determined completely by the surrounded patterns in $x$. After each odd-indexed stage $2i-1$ ($i = 1,2$), $\\phi(x)$ will be defined on a set $U_{2i-1}$ as a pattern $u_{2i-1} \\in L(Y)$. Then, the following (even-indexed) stage $2i$ will define $\\phi(x)$ on a set $S_{2i}$, where $\\phi(x)(S_{2i})$ is a pattern $s_{2i}$ for which $v_{2i} = u_{2i-1} s_{2i}$ on $V_{2i} := U_{2i-1} \\sqcup S_{2i}$ contains no patterns from $\\mathcal{F}$. The following (odd-indexed) stage $2i+1$ will remove all letters on $\\partial_g V_{2i}$, yielding a pattern $u_{2i+1}$ on $U_{2i+1} := V_{2i} \\setminus \\partial_g V_{2i}$. Then $u_{2i+1} \\in L(Y)$ by the $g$-extension property, allowing the process to continue. The patterns placed during even-indexed stages are dependent only on nearby surrounded patterns in $x$; to describe this dependency, we require the following auxiliary function.\n\nSince $h(X_P) > h(Y)$, for sufficiently large $k$ it is the case that\n\\begin{equation}\\label{d2bound}\n|L_{C_k}(X_P)| > |L_{C_{k+g+m}}(Y)| \\cdot |\\mathcal{A}_Y|^{12g(k-3g+m) + 196g^2}.\n\\end{equation}\nFix any such $k$ (which does not depend on $x$), and then define a surjection $\\psi$ from $L_{C_k}(X_P)$ to the set of all tuples of the form \n$(i_j)_{1 \\leq j \\leq 9}$, where $1 \\leq i_1 \\leq |L_{C_{k+g+m}}(Y)|$, $1 \\leq i_j \\leq |\\mathcal{A}_Y|^{3g(k-3g+m)}$ for $2 \\leq j \\leq 5$, and $1 \\leq i_j \\leq |\\mathcal{A}_Y|^{49g^2}$ for $6 \\leq j \\leq 9$. We are now ready to describe the stages of defining the factor map $\\phi$.\\\\\n\n\\textbf{Stage 1:} Define $U_1$ to be the set of all $t \\in \\mathbb{Z}^2$ at a distance of more than $g$ from all determined zones, and define $u_1 = *^{U_1}$. Clearly $u_1 \\in L(Y)$ since $*^{\\mathbb{Z}^2} \\in Y$. We note that after Stage 1, the undefined portion of $\\phi(x)$ consists of components of determined zones, along with all sites within distance $g$ of them; we use the term ``island'' to denote the set of sites within distance $g$ of such a component. By (\\ref{Eqn:Banff}), any two nonequal islands have distance more than $5g$.\nFor any island $I$, and for $i = 1,2$, define $T_i(I)$ to be the sets of $e_1$- and $e_2$-coordinates (respectively) which appear in some determined zone in $I$.\\\\\n\n\\textbf{Stage 2:} For each island $I$, the set $I \\cap (T_1(I) \\times T_2(I))$ is the disjoint union of the determined zones in the component inducing $I$. Let $S_2 = \\bigcup_I (I \\cap (T_1(I) \\times T_2(I)))$. We define a pattern $s_2$ on $S_2$ as follows. For any determined zone $t + C_{k+g+m}$, by definition $x(t + C_k)$ is a surrounded pattern in $x$. Let the tuple $(i_j)_{1 \\leq j \\leq 9}$ be defined by $\\psi(x(t+C_k)) = (i_j)_{1 \\leq j \\leq 9}$, and then let $s_2(t+C_{k+g+m})$ be the $i_1$th pattern in $L_{C_{k+g+m}}(Y)$ according to the lexicographic ordering. Then $s_2$ is just the concatenation of these patterns.\n\nWe define $V_2 = U_1 \\sqcup S_2$ and $v_2 := u_1 s_2$. Each pattern placed on a determined zone was assumed to be in $L(Y)$, and so contained no patterns from $\\mathcal{F}$. As noted above, the same is true for the $*$-pattern $u_1$ placed on $U_1$.} Since patterns in $\\mathcal{F}$ have diameters less than $g$ and since determined zones have distance greater than $g$ from each other and from $U_1$, $v_2$ contains no patterns from $\\mathcal{F}$.\\\\\n\n\\textbf{Stage 3:} Define $U_3 = V_2 \\setminus \\partial_g V_2$, and $u_3 := v_2(U_3)$. By the $g$-extension property, $u_3 \\in L(Y)$. \n\nTo more easily describe future stages, we describe the structure of the set $U_3$. Namely, $U_3$ consists of two types of sites: those at distance more than $2g$ from all determined zones, and those within a determined zone in an island $I$ and for which both coordinates are at distance more than $g$ from the corresponding $T_i(I)^c$. For each island $I$, $U_3 \\cap I$ consists of a disjoint union of squares obtained from removing the inner $g$-boundary from each determined zone; we call these squares ``reduced determined zones.'' (See Figure~\\ref{TSSMpic2}.) \\\\\n\n\\textbf{Stage 4:} Define $S_4$ to be the set of all sites which are within distance $2g$ of some determined zone in an island $I$, \nhave one coordinate which is within distance $g$ of the corresponding $T_i(I)^c$, and one coordinate which has a distance of more than $2g$ from the corresponding $T_i(I)^c$. Informally, $S_4$ is the (disjoint) union of all rectangles with dimensions $3g$ and $k-3g+m$ that share (at least one of) their longest side(s) with a reduced determined zone and are centered along the corresponding side of that reduced determined zone. \nAny two such rectangles are separated by distance greater than $g$; if they're part of the same island then this is true since reduced determined zones have side length greater than $m - g > 2p - g > g$, and if they are part of different islands then this follows from (\\ref{Eqn:Banff}).\n\nWe now define a pattern $s_4$ on $S_4$. Choose any of the rectangles $R$ comprising $S_4$. First, we need a way to associate a determined zone to $R$; to this end, choose the first direction in the ordering $\\{$up, left, down, right$\\}$ for which there is a reduced determined zone adjacent to $R$ in that direction, which came from some determined zone. Since $u_3 \\in L(Y)$, there exists a pattern on $R$ which yields a pattern in $L(Y)$ when concatenated with $u_3$. However, we need to choose such a pattern on $R$ using only the portion of $x$ which lies within a uniformly bounded distance of $R$ to ensure that $\\phi$ is a sliding block code, and if the island $I$ is quite large, then there is no obvious way to do so. Instead, we settle for choosing a pattern on $R$ which creates no patterns from $\\mathcal{F}$ when concatenated with $u_3$. That is, consider the collection of patterns $\\{w \\in (\\mathcal{A}_Y)^R :  u_3 w \\textrm{ contains no patterns from } \\mathcal{F}\\}$; note that this collection depends only on the portion of $u_3$ within distance $g$ of $R$. Since $u_3$ was in $L(Y)$, this collection is nonempty, and trivially, it has cardinality bounded from above by $|\\mathcal{A}_Y|^{|R|} \\leq |\\mathcal{A}_Y|^{3g(k-3g+m)}$. \n\nWe then define $s_4(R)$ to be the $i_j$th pattern in this collection according to the lexicographic ordering, where $t + C_{k+g+m}$ was the determined zone associated to $R$ above, $\\psi(x(t + C_k)) = (i_j)_{1 \\leq j \\leq 9}$, and $j$ is taken to be $2$, $3$, $4$, or $5$ based on whether $t + C_{k+g+m}$ is reached by moving up, left, down, or right from $R$. (We adopt the convention, here and later, that for a totally ordered set $S$ and $n > |S|$, the $n$th element of $S$ is just taken to be the maximal element.) We note for future reference that no $i_j$ determines patterns on two different rectangles $R$. Now, $s_4$ is just the concatenation of these patterns.\n\n\n\nDefine $V_4 = U_3 \\sqcup S_4$ and $v_4 = u_3 s_4$. (See Figure~\\ref{TSSMpic8}.) No forbidden pattern in $\\mathcal{F}$ can intersect two rectangles $R$ since distinct rectangles $R$ are separated by distance more than $g$. No forbidden pattern in $\\mathcal{F}$ can intersect exactly one rectangle $R$ since $u_3 s_4(R)$ was assumed not to contain such patterns. Finally, no forbidden pattern in $\\mathcal{F}$ can occur disjointly from all rectangles $R$ since $u_3 \\in L(Y)$. Therefore, $v_4$ contains no patterns from $\\mathcal{F}$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.6125]{Fig3b-eps-converted-to.pdf}\n\\caption{Sites assigned during Stages 1,2, and 4 are in dark gray, light gray, and medium gray, respectively (left). Sites assigned during Stage 6 are in white and doubly reduced determined zones in light gray (right).}\n\\label{TSSMpic8}\n\\end{figure}\n\n\\textbf{Stage 5:} Define $U_5 = V_4 \\setminus \\partial_g V_4$, and $u_5 := v_4(U_5)$. By the $g$-extension property, $u_5 \\in L(Y)$. \nAgain we explicitly describe the structure of $U_5$. \nNow, $U_5$ consists of three types of sites. The first are those which are at a distance of more than $3g$ from all determined zones. The second are those which are within distance $3g$ from a determined zone in an island $I$, and for which both coordinates have distance more than $2g$ from the corresponding $T_i(I)^c$. Such sites form a disjoint union of squares obtained by removing the inner $2g$-boundary from all determined zones; we call these ``doubly reduced determined zones.'' \nThe third type are those which are within distance $3g$ from a determined zone in an island $I$, have one coordinate within distance $2g$ from the corresponding $T_i(I)^c$, and one coordinate with distance more than $3g$ from the corresponding $T_i(I)^c$.\\\\\n\n\n\\textbf{Stage 6:} We define $S_6 = U_5^c$. From the description above, it should be clear that the sites in $S_6$ have the following properties: they are within distance $3g$ from a determined zone in an island $I$, have one coordinate within distance $2g$ from the corresponding $T_i(I)^c$, and the other coordinate within distance $3g$ from the corresponding $T_i(I)^c$. By (\\ref{Eqn:Banff}), sites in $S_6$ associated to different islands have distance at least $g$. Since doubly reduced determined zones have side length greater than \n$m - 3g > 2p - 3g > 3g$, we see that $S_6$ consists of a disjoint union of connected components with diameters at most $7g$ separated by distance more than $g$, which we call holes. \n\nWe fill the holes with patterns in much the same way as in Stage 4. We again associate a determined zone to each hole $H$; to this end, choose the first direction in the ordering $\\{$up-left, up-right, down-left, down-right$\\}$ for which there is a doubly reduced determined zone adjacent to $H$ in that direction, which came from some determined zone.\n\nFor each hole $H$, consider the collection of patterns \n$\\{w \\in (\\mathcal{A}_Y)^H \\ : \\ u_5 w \\textrm{ contains no patterns from } \\mathcal{F}\\}$. Since $u_5 \\in L(Y)$, this collection is nonempty, and its cardinality is at most $|\\mathcal{A}_Y|^{|H|} \\leq |\\mathcal{A}_Y|^{49g^2}$. \n\nWe define $s_6(H)$ to be the $i_j$th pattern in this collection according to the lexicographic ordering, where \n$t + C_{k+g+m}$ was the determined zone associated to $H$ above, $\\psi(x(t + C_k)) = (i_j)_{1 \\leq j \\leq 9}$, and $j$ is taken to be $6$, $7$, $8$, or $9$ based on whether $t + C_{k+g+m}$ is reached by moving up-left, up-right, down-left, or down-right from $H$. As in Stage 4, no $i_j$ determines patterns on two different holes $H$.\n\n\nNow, $s_6$ is just the concatenation of these patterns on holes. Define $V_6 = U_5 \\sqcup S_6 = \\mathbb{Z}^2$ and \n$v_6 = u_5 s_6$. Exactly as in Stage 4, $v_6$ contains no patterns from $\\mathcal{F}$, since $u_5$ was in $L(Y)$ and holes are separated by distances of at least $g$. Then $v_6 \\in Y$, and so we define $\\phi(x) = v_6$.\\\\\n\nFinally, we must show that $\\phi$ is shift-commuting, continuous, and surjective. For shift-commuting and continuity, we claim that $\\phi$ is a sliding block code. To see this, we first note that the status of any site $t$ (meaning either its assigned symbol or the fact that no symbol has been assigned) after Stage 1 clearly depends only on whether $t$ is within distance $g$ from a determined zone, which is determined by knowledge of $x$ on sites within distance $k+3g+2m$ from $t$. For any subsequent stage $i$, the status of any site $t$ depends only on the status of sites after stage $i - 1$ within distance $k+3g+2m$ of $t$. Therefore,  $\\phi$ is a sliding block code with radius $6(k+3g+2m)$.\n\nThe proof that $\\phi$ is surjective is quite similar to the ones from \\cite{BPS} and \\cite{desai}, and so we only outline some slight differences here. Firstly, we only consider $x \\in X$ consisting of a lattice of aligned overlapping surrounding frames as in the left-hand side of \nFigure~\\ref{TSSMpic7} showing that their $\\phi$-images already cover all of $Y$. In that figure, the right-hand side displays the regions of $\\phi(x)$, partitioned (by color) by the stage which determined their values. However, since $\\psi$ was a surjection and each $i_j$ from any $\\psi(x(t + C_k))$ is used at most once, it's clear that for any $y \\in Y$, the surrounded patterns $W_i$ on the left can be chosen to yield the desired subpatterns of $y$ on the right, and so $\\phi$ is a surjective factor map. This completes the proof of Theorem~\\ref{mainthm} for $d=2$. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.65]{Fig4b-eps-converted-to.pdf}\n\\caption{An element of $X$ and its image under $\\phi$}\n\\label{TSSMpic7}\n\\end{figure}\n\nIt remains only to describe necessary changes in the proof for $d > 2$. Markers are constructed exactly as before, with shapes which are \n$d$-dimensional hypercubes rather than squares. We choose $p > (2d+1)g$, yielding a version of (\\ref{Eqn:Banff}) guaranteeing distance more than \n$(2d+3)g$ between all non-adjacent determined zones. The surjection $\\psi$ for $d = 2$ had nine coordinates; one for the determined zones themselves, four for the rectangles placed in Stage 4 along edges, and four for the holes placed in Stage 6 near corners. For $d > 2$, $\\psi$ has $3^d$ coordinates, again corresponding to the main bulk of a determined zone plus all its lower-dimensional ``faces.'' This requires a version of (\\ref{d2bound}) in which $12g(k-3g+m) + 196g^2$ is replaced by a more complicated polynomial expression $f_d(k,g,m)$ dependent on sizes of the sets $S_i$ (defined below), and bounded from above by $d((k+3g+m+2dg)^d-(k+g+m-2dg)^d)$ ($d$ times the volume difference of two $d$-dimensional hypercubes). This polynomial has degree $d-1$ in $k$, thus the desired inequality still holds for large enough $k$ by definition of entropy.\n\n\nThe definition of $\\phi$ proceeds in alternating stages exactly as before; for arbitrary $d$ there will be $2(d+1)$ stages. Again $U_1$ consists of sites which are at distance more than $g$ from all determined zones, and $u_1 = *^{U_1}$. Similarly, $S_2$ consists of the union of all determined zones, and $s_2$ is determined on each determined zone by knowledge of the corresponding surrounded pattern in $x$. Then, for each $j \\geq 1$, $V_{2j} = U_{2j-1} \\sqcup S_{2j}$ and $U_{2j+1} = V_{2j} \\setminus \\partial_g V_{2j}$, and so we must only describe the sets $S_{2j}$. For $1 < j \\leq d+1$, $S_{2j}$ consists of all sites $t$ with the following properties:\n\\begin{itemize}\n\\item $t$ is within distance $jg$ of some determined zone in an island $I$,\n\\item for all $i < j$, $i$ coordinates of $t$ are within distance $(j-2+i)g$ of the corresponding $T_i(I)^c$, and\n\\item $d - j + 1$ coordinates of $t$ have distance more than $(2j-2)g$ from the corresponding $T_i(I)^c$.\\\\\n\\end{itemize}\n\nWe leave it to the reader to check that with this definition, each $S_{2j}$ is disjoint from $U_{2j-1}$, and $V_{2(d+1)} = \\mathbb{Z}^d$. The proof that $\\phi$ is a  factor map is analogous to the $d = 2$ proof, and the proof that $\\phi$ is surjective simply uses $d$-dimensional versions of the points in Figure~\\ref{TSSMpic7} (see \\cite{BPS} and \\cite{desai}); we again leave the details to the reader. \n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nErgodicity of many-body systems and its breaking is one of the central research areas in modern statistical mechanics.\nThe basic definition is well understood: the physical system can be called ergodic, if during its time evolution all accessible microstates are visited.\nHowever, the detailed division whether the ergodicity is broken or not, especially, when applied to a large variety of quantum systems is yet to be established~\\cite{DAlessio2016}.\nA few general examples of ergodicity breaking in these are quantum scars \\cite{Bernien2017, Serbyn2021}, Bethe ansatz integrable systems~\\cite{QuantumCradle}, lattice gauge theories \\cite{LatticeGaugeTheory}, fractons and confinement~\\cite{Fractons_confinement}, and Hilbert space fragmentation \\cite{Sala2020, PhysRevB.101.174204}.\n\nAll these examples of ergodicity breaking and lack of thermalization in closed systems provoke a number of questions on the microscopic characterization of ergodicity in quantum systems. \nIn particular, there are interesting connections between thermalization and properties of eigenstates of microscopic Hamiltonian, which are nicely summarized in the form of the eigenstate thermalization hypothesis \\cite{SrednickiETH, DeutschETH}, and also with the quasiclassical limit, where these notions overlap with classical chaos \\cite{haake1991quantum}.\nRather generally, the notions of quantum chaos and thermalization in closed systems can be used interchangeably, and we thereafter use different criteria of quantum chaoticity, as, in particular, spectral statistics, to detect absence of thermalization. \n\nAs for experimental verification of the mentioned model studies and uncovering new related phenomena,\nover the past two decades, the range of accessible quantum many-body systems has been sufficiently extended. \nThis progress is largely due to an impressive development of experimental techniques for cooling and loading atoms into optical lattices \\cite{Bloch2008RMP, Esslinger2010ARCMP}.\nIn these artificial systems, many relevant parameters can be controlled and tuned with a high degree of freedom: the external potential (with additional disorder, linear, or any specific), the interaction (both the amplitude and the range), the initial lattice filling, particle statistics, symmetries, etc.\nAs a natural consequence of this freedom, in particular, the celebrated Anderson localization phenomenon is now viewed as only a member of the wider class of many-body localization (MBL) transitions in interacting systems \\cite{Nandkishore2015, Abanin2019RMP}.\n\nFrom the success of  the Anderson localization, the natural platforms for MBL were initially the systems with disorder.\nThese platforms were successfully realized in experiments, see, e.g., Ref.~\\cite{ExperimentMBL}, where one required artificial disorder produced by quasiperiodic potentials or other means.\nRecently, it was shown that analogous systems with only short-range interactions and disorder-free (linear or harmonic) potentials can exhibit localized behavior in a wide range of Hamiltonian parameters, which was named as the Stark (or Bloch) localization  \\cite{Nieuwenburg2019, StarkLocalization, Yao2020} and was also observed experimentally \\cite{Scherg2021}.\n\nThe disorder-free potentials with a linear tilt are common in the field of cold atoms in optical lattices \\cite{Raizen1997}.\nTypically, the interactions between cold atoms are short-range, however, there are many cases, where these become sufficiently nonlocal, as in gases of atomic isotopes possessing the dipole moment in the ground state~\\cite{DipolarGasesOpticalLattice} and atoms in the metastable excited Rydberg states~\\cite{Rydberg_atoms_review}. The latter are especially attractive in the context of ergodicity breaking due to quantum scar effects \\cite{Bernien2017}.\nFurthermore, there is a number of both experimental and theoretical studies on many-body regimes in atomic gases with cavity-mediated interactions \\cite{Dogra2016,Landig2016,Sierant2019} (see also the review \\cite{CavityQED}). Coupling to the cavity modes in these systems sufficiently extends the effective range of interactions between atoms.\n\nFrom the theoretical point of view, the combination of the above realizations, namely, the tilted lattice systems, where atoms or quantum spins interact nonlocally, was not studied in detail. \nThis motivates us to focus on a wide class of model Hamiltonians, with various types of experimentally available long-range interactions or long-range hopping processes and analyze the fate of many-body localization in these systems.  As we show below, long-range interactions also impact the spectrum in the same regularizing way as an additional disordered or harmonic potential for short-range interacting systems. It turns out that it is sufficient to employ the disorder-free linear external potential with moderate-range interactions between particles to observe and study MBL transitions. \n\n\n\nIt should be noted that aspects of Stark localization in similar context attracted much interest recently. In particular, there are studies of  the tilted Heisenberg spin chain with the next-nearest couplings \\cite{vernek2021robustness}, tilted lattice systems with long-range hopping \\cite{PhysRevB.102.085133} and with cavity-mediated interactions \\cite{Chanda2021manybody}.\n\n\n\\section{Models and methods}\n\\subsection{Lattice models}\nIn this section, we introduce one-dimensional theoretical models in the order of increasing complexity. Starting from the noninteracting limit with only (long-range) hopping and external linear potential, we discuss the localized wave functions and influence of hopping. \nWe extend further our description by including power-law interactions, various representations and symmetries of these models. Furthermore, we study a model of localization in all-to-all potentials, which can be realized in cavities \\cite{Landig2016}.\n\n\n\\subsubsection{Noninteracting model}\nThe Hamiltonian of noninteracting lattice model consists of the external linear potential and the hopping term, which describes long-range tunneling processes. Here, we start from an infinite system, i.e., neglect the boundary effects for simplicity,\n\\begin{equation}\\label{eq:Hnonint}\n    \\hat{H} = - \\sum_{j=1}^{m}J_{j}\\sum_{k} \n    (\\hat{a}^{\\dagger}_{k} \\hat{a}_{k+j} +{\\rm H.c.})  \n    + F \\sum_{k} k \\hat{a}^{\\dagger}_{k} \\hat{a}_{k},\n\\end{equation}\nwhere $\\hat{a}^{\\dagger}_{k}$ and $\\hat{a}_{k}$ are bosonic or fermionic creation and annihilation operators on site $k$, respectively. The quantity $F$ characterizes the amplitude of external linear potential and $J_{j}$ are the hopping amplitudes, which depend on the distance~$j$ between the lattice sites. The upper limit~$m$ in the sum denotes the maximal range of hopping. This maximal range can be both finite or infinite in the case of power-law hopping $J_{j} \\propto 1\/j^\\beta$.\n\nThe introduced model is quadratic in creation and annihilation operators, thus it is sufficient to solve its one-particle sector. \nHence, the wave function can be written in the form\n\\begin{equation}\n    |\\psi\\rangle = \\sum_{k} c_{k} \\hat{a}^{\\dagger}_{k} |0\\rangle,\n\\end{equation}\nwhere $c_{k}$ are the coefficients and $|0\\rangle$ is the vacuum state.\nWe can map the Hilbert space built on the basis states $\\hat{a}^{\\dagger}_{k}|0\\rangle$ onto the Hilbert space of functions on the circle according to the rule \\cite{Hartmann_2004}\n\\begin{equation}\n    \\hat{a}^{\\dagger}_{k} |0\\rangle \\rightarrow \n    \\frac{\\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}},\n\\end{equation}\nwhere $\\phi$ is the polar angle.  This results in the mapping\n\\begin{equation}\n    |\\psi\\rangle \n    \\rightarrow \n    \\psi(\\phi)=\\sum_{k} \\frac{c_{k} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}}.\n\\end{equation}\n\nWithin the introduced procedure, it is possible to map operators entering the Hamiltonian~\\eqref{eq:Hnonint} to differential operators on the circle. For the linear potential term, the corresponding mapping can be written as follows:\n\\begin{multline}\\label{eq:map1}\n    \\sum_{m} m a^{\\dagger}_{m} a_{m} \\sum_{k} c_{k} a^{\\dagger}_{k} |0\\rangle = \\sum_{k} k c_{k} a^{\\dagger}_{k} |0\\rangle \\rightarrow \n    \\\\ \n    \\rightarrow \\sum_{k} \\frac{ k c_{k} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}} \n    =\n    -i \\frac{d}{d\\phi} \\psi(\\phi).\n\\end{multline}\nWe see that the external linear potential is mapped to the derivative. \nFinally, let us perform analogous mapping for the hopping terms,\n\\begin{multline}\\label{eq:map2}\n    \\sum_{m} \\hat{a}^{\\dagger}_{m} \\hat{a}_{m+j} \n    \\sum_{k} c_{k} \\hat{a}^{\\dagger}_{k} |0\\rangle \n    = \\sum_{k} c_{k+j} \\hat{a}^{\\dagger}_{k} |0\\rangle \\rightarrow \n    \\\\ \n    \\rightarrow \\sum_{k} \\frac{ c_{k+j} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}} \n    = \n    \\exp{\\left(-i j \\phi \\right)} \\psi(\\phi).\n\\end{multline}\nThese terms are mapped to the basis functions multiplied by the range-dependent phase factors. \n\nBy means of the obtained mapping rules \\eqref{eq:map1} and \\eqref{eq:map2}, the Hamiltonian~\\eqref{eq:Hnonint} can be expressed as \n\\begin{equation}\n    {H} = -2\\sum_{j=1}^{m} J_{j} \\cos{\\left(j\\phi\\right)} -i F \\frac{d}{d\\phi}.\n\\end{equation}\nThe eigenstates of this Hamiltonian can be determined by solving the first-order differential equation, while the eigenvalues are obtained by the condition that the eigenstates must be periodic functions on the circle.\nAs a result, we obtain the eigenstates,\n\\begin{equation}\\label{eq:psi_n}\n    \\psi_{n}(\\phi) = \\frac{\\exp{\\left(i n \\phi + 2 i \\sum_{j=1}^{m} \\frac{J_{j} \\sin{(j \\phi)}}{j F}\\right)}}{\\sqrt{2 \\pi}} ,\n\\end{equation}\nand the eigenvalues\n\\begin{equation}\n    E_{n} = F n, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\nThe spectrum of the introduced model is independent of the hopping amplitudes and it is the same as of the Hamiltonian with only a potential term. \nIt is natural to suggest that the wave functions in the presence of hopping are continuously connected to the wave functions in the atomic limit (the latter are completely localized on one site). The nonzero hopping processes lead to broadening of the wave functions around that center site with a corresponding exponential decay of the density distribution. \nBelow, we also show it more directly by expressing the coefficients~$c_k$ that determine the wave function in the initial basis~$|\\psi\\rangle$. \n\nFrom the form of the wave function~\\eqref{eq:psi_n} it is clear that eigenfunctions with $n\\neq0$ can be obtained from the eigenstate with $n=0$ simply by translation. \nIt can also be deduced from the fact that a commutator of the shift operator with the Hamiltonian results in the shift operator itself. \nHence, for the models considered below, the shift operator can be viewed as a raising operator: all eigenstates can be obtained by repeated action of the shift operator on a particular eigenstate. For $n=0$, we obtain the following coefficients $c_{k}$ in the initial basis:\n\\begin{equation}\n    c_{k} = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\exp{\\left(-i k \\phi + 2 i \\sum_{j=1}^{m} \\frac{J_{j} \\sin{(j \\phi)}}{j F}\\right)} d\\phi\n\\end{equation}\n\nIn the simplest case of only the nearest-neighbor hopping, these coefficients are determined in terms of the Bessel functions~${\\cal J}_k(x)$ as $c_{k} = {\\cal J}_{k}\\left({2J_{1}}\/{F}\\right)$. \nIn a similar case of hopping only between the next-nearest neighbors ($J_{2}\\neq0$, while $J_j=0$ for $j\\neq2$), the wave functions vanish for odd $k$, while for even $k$ they are given by $c_{2k} = {\\cal J}_{k}\\left({J_{2}}\/{F}\\right)$.\nNote that in the case of nearest-neighbor hopping, the localization is generally stable to interactions if $F>2J_{1}$, or if the argument of the Bessel functions is smaller than one.\nNote that in this case, the exponential vanishing of the wave functions is clear from the expansion of the Bessel functions into series over small argument $x=2J_1\/F$, which gives ${\\cal J}_{k}(x) \\propto x^{k}(1+O(x^2))$. \nIn case of the next-nearest-neighbor hopping, we can conjecture analogously that the localization is stable to interactions if the argument of the Bessel function for the noninteracting wave function is smaller than one, or if $J_{2} < F$. \nMore generally, we can conclude that the single particle Stark localization is stable if $2J_{m}<mF$. \nTherefore, to suppress localization, the hopping amplitude between the farther neighbors must be larger than the one between the nearest neighbors, which is rarely the case in cold-atom realizations. \nTo check this conjecture, we numerically study localization with nearest and next-nearest hoppings and show that $J_{2}$ starts to influence localization only if it is larger than the amplitude~$J_{1}$. The stability of Stark localization towards long-range hopping was also studied in Ref.~\\cite{PhysRevB.102.085133}. \n\nSo far, we have discussed the behavior of noninteracting wave functions, if the argument of the Bessel function is smaller than one ($x<1$). In the noninteracting case, the wave function is localized for all values of couplings $J_m$.\nThe only change at large $x$ concerns the number of maxima of the wave function and their positions, since with a decrease of $F$ the wave functions exhibit several oscillations before their exponential tails. \nIn more general case, the wave functions can be obtained by expansions in Bessel series as discussed in Ref.~\\cite{PhysRevA.91.023606}. \n\nIt is also possible to check the case of long-range hoppings given by $J_{m} = J\/m^{\\beta}$. \nIn the numerical exact diagonalization approach for finite systems, we observe that eigenfunctions are localized even at $\\beta = 0$ (the case of all-to-all hopping) that contrasts to the Anderson localization, which is stable only for $\\beta>1$.\n\n\n\\subsubsection{Models with power-law long-range interactions}\nAs one can see from the preceding results, the many-body states of the noninteracting model are localized for all values of parameters.\nBut the localization may not be stable with respect to interactions between particles. \nNow, we introduce the long-range many-body interactions and study the possibility of localization in this system.\nThe issue of MBL in the presence of long-range interactions is also of conceptual value, since for a long time it was accepted that systems with long-range interactions described by the power-law dependence cannot demonstrate localization features.\n\nFor definiteness, let us introduce the interacting one-dimensional system consisting of spinless fermions on the finite lattice with $L$ sites [see also Fig.~\\subfigref{models}{a}].\nIt is described by the Hamiltonian\n\\begin{equation}{\\label{H_fermionic}}\n    \\hat{H} \n    = -\\sum_{k=1}^{m} J_{k} \\sum_{i=1}^{L-k} \n    (\\hat{f}^{\\dagger}_{i} \\hat{f}_{i+k} +{\\rm H.c.}) \n    +F\\sum_{i=1}^{L} i \\hat{n}_{i} + U \\sum_{1\\leq i < j}^L\n    \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}},\n\\end{equation}\nwhere $\\hat{f}^{\\dagger}_{i}$ and $\\hat{f}_{i}$ are the fermionic creation and annihilation operators on site $i$, respectively, and $\\hat{n}_{i}=\\hat{f}^{\\dagger}_{i}\\hat{f}^{\\phantom{\\dagger}}_{i}$ is the corresponding number operator on site $i$. $F$ determines the strength of the external linear potential, as before, $U$ corresponds to the magnitude of interactions between particles, and $\\alpha$ is the exponent characterizing the power-law decay of interactions. \n$J_{k}$ are the hopping amplitudes, while $m$ determines the maximal range of hopping as in the noninteracting model. \n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig1_models.pdf}\n  \\caption{\\label{fig:models}%\n     Schematic illustration of many-body systems under study and relevant couplings.\n     }\n\\end{figure}\nBelow, we focus on small values of the exponent $\\alpha$, in particular, $\\alpha \\in [0.5,3]$, since at larger values of $\\alpha$ the system behaves as the one with short-range interactions. \nThe case $\\alpha = 3$ is especially relevant, as it can be realized experimentally with dipolar ultracold gases \\cite{DipolarGasesOpticalLattice}. \n\nIn case of bosonic system, the Hamiltonian is analogous to Eq.~\\eqref{H_fermionic}, except of the additional possibility of the on-site interaction, controlled by the parameter $V$ [see also Fig.~\\subfigref{models}{b}]. For the completeness, we specify the explicit form as follows:\n\\begin{multline}{\\label{H_bosonic}}\n    \\hat{H} = -\\sum_{k=1}^{m} J_{k} \\sum_{i=1}^{L-k} \n    (\\hat{a}^{\\dagger}_{i} \\hat{a}_{i+k} + {\\rm H.c.}) \n    \\\\ \n    +V\\sum_{i=1}^{L} \\hat{n}_{i}(\\hat{n}_{i}-1)\n    +F\\sum_{i=1}^{L}  i \\hat{n}_{i} + U \\sum_{1\\leq i < j \\leq L} \n    \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}},\n\\end{multline}\nwhere $\\hat{a}^{\\dagger}_{i}$ and $\\hat{a}_{i}$ are the bosonic creation and annihilation operators on site $i$, respectively. \nAll other quantities have the same meaning, as in the fermionic case. In numerical calculations, the dimension of the local bosonic Hilbert space has to be restricted to a finite value. We set the maximal number of bosons on the same site equal to three, which is sufficient at moderate values of $V$.\n\nNote that for the purpose of succeeding analysis in the framework of the time-dependent variational principle (TDVP) and the Shrieffer-Wolff transformation, it is necessary to reformulate the fermionic Hamiltonian in the bosonic language. \nFor this purpose, we employ the Jordan-Wigner transformation to map the fermionic system onto the spin-1\/2 chain. \nIn this procedure, the creation and annihilation operators are mapped onto the Jordan-Wigner chains according to the rules: $\\hat{f}^{\\dagger}_{i} \\to \\prod_{j=1}^{i-1} (-\\hat{\\sigma}^{z}_{j}) \\hat{S}^{+}_{i}$ \nand $\\hat{f}_{i} \\to \\prod_{j=1}^{i-1} (-\\hat{\\sigma}^{z}_{j}) \\hat{S}^{-}_{i}$, where $\\hat{\\sigma}^{z}$ is the Pauli matrix with a conventional correspondence to the spin projection operator to the $z$ axis, $\\hat{S}^z=\\hat{\\sigma}^z\/2$, while $\\hat{S}^{+}=(\\hat{\\sigma}^x+i\\hat{\\sigma}^y)\/2$ and $\\hat{S}^{-}=(\\hat{\\sigma}^x-i\\hat{\\sigma}^y)\/2$ are the spin-raising and spin-lowering operators, respectively. \nThe particle number operator $\\hat{n}_i$ is mapped to the local projection operator as $\\hat{n}_i \\to 1\/2 + \\hat{S}^{z}_{i}$.\nUsing these rules, it is possible to map the fermionic Hamiltonian with only the nearest-neighbor hopping ($m=1$) to the following spin-chain Hamiltonian:\n\\begin{multline}{\\label{H_spin}}\n    \\hat{H} = - J_{1}\\sum_{i=1}^{L-1} (\\hat{S}^{+}_{i}\\hat{S}^{-}_{i+1} + \\hat{S}^{-}_{i}\\hat{S}^{+}_{i+1})\n    + U \\sum_{1\\leq i < j \\leq L} \n    \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}}\n    \\\\\n    + \\sum_{i=1}^{L} (F i +w_{i}) \\hat{S}^{z}_{i}\n     , \n    \\quad\n    w_{i} = \\frac{U}{2}\\sum_{j=1, j\\neq i}^{L} \\frac{1}{|i-j|^{\\alpha}}.\n\\end{multline}\nIn the given form, this model describes the $XXZ$ spin chain in the external inhomogeneous magnetic field $h_i^z\\equiv(Fi+w_i)$, which has additional Ising-type couplings between the spins located farther from each other than nearest neighbors [see also Fig.~\\subfigref{models}{c}].\nThe additional on-site potential term $w_{i}$ becomes constant in the limit of infinite $L$. On the finite lattice, this term is almost constant in the bulk and decreases only at the boundaries. \n\n\n\\subsubsection{Model in a cavity}\\label{subsec:cavity}\nIn addition to the model with power-law long-range interactions, let us also introduce the model of localization in the all-to-all potential. \nThis kind of interatomic potential can be realized with a system in a cavity, where presence of the cavity modes, strongly interacting with particles, can induce completely nonlocal interaction patterns. Here, we study a simple model, which captures some basic characteristics of real cavities. In this model [see also Fig.~\\subfigref{models}{d}], we introduce an additional term \\cite{Dogra2016, Landig2016, Sierant2019, CavityQED, BoseHubbardQuenches}, which is added to the above-specified Hubbard-type Hamiltonians \\eqref{H_fermionic} or \\eqref{H_bosonic},\n\\begin{equation}{\\label{H_cavity}}\n    \\hat{H}_{\\rm cav} = - \\frac{U_{\\rm cav}}{L} \\left( \\sum_{i=1}^{L} (-1)^{i+1} \\hat{n}_{i} \\right)^{2}.\n\\end{equation}\n\nThis cavity term leads to all-to-all interactions with the same strength between particles irrespectively of the separating distance. \nThe interaction amplitude $U_{\\rm cav}$ is normalized by the lattice size $L$ to be meaningful in the infinite-lattice limit. \nIt is interesting to investigate whether the localization induced by a linear potential is stable to these long-range interactions. \n\n\n\\subsection{Methods}\nTo distinguish between chaotic and MBL phases, we employ several methods, which are commonly used in the literature on many-body localization. For small system sizes (up to $L = 18$ for spinless fermionic or spin-1\/2 systems), it is possible to obtain full spectrum using exact diagonalization. \nSince chaotic and localized systems have different level statistics \\cite{Alet2018ManybodyLA}, characteristics of the spectrum can be used as probes of localization.\nAs the system size grows, exact diagonalization quickly becomes infeasible due to exponential growth of the Hilbert space with the number of the latticce sites~$L$. \nTherefore, below we also employ methods based on the matrix product states to access dynamics of much larger bosonic and fermionic systems after quenches. \nIn these simulations, MBL manifests itself as a lack of thermalization of local observables and slow logarithmic growth of the entanglement entropy, similar to observations given, e.g., in Refs.~\\cite{Alet2018ManybodyLA, EntanglementGrowth1,EntanglementGrowth2, EntanglementGrowth3, EntanglementGrowth4}. \nFor a large linear tilt $F$, we apply the Schrieffer-Wolff transformation to obtain effective Hamiltonians. Within the effective models, we also analyze limiting cases of spectral characteristics and evolution of relevant physical observables.\n\n\n\\subsubsection{Exact diagonalization and level statistics}\\label{subsec:ED}\nFor a small system size and moderate local Hilbert space dimensions (or in a dilute limit, not studied in this work), it is possible to determine full spectrum of the system Hamiltonian in the fixed symmetry sector. The exponential growth of the Hilbert space with a number of lattice sites~$L$ limits these calculations to $L \\approx 18$ for spinless fermions, spins or hard-core bosons, or even smaller numbers for bosons with moderate on-site interactions or spinful fermions. \n\nChaotic and MBL spectra have different level statistics: Poissonian for MBL phase and Wigner-Dyson for chaotic systems \\cite{Alet2018ManybodyLA}. It is connected to the fact that the MBL phase has an extensive set of quasilocal integrals of motion. \nThe eigenstates with different eigenvalues of these integrals have uncorrelated energy eigenvalues that leads to the Poisson distribution. However, Hamiltonians of chaotic systems can be represented as random matrices. Eigenvalues of random matrices are distributed according to the Dyson-Wigner ensembles \\cite{Mehta2004}. \nDue to the fact that the full spectrum statistics contains an immense amount of information, it is more reasonable to employ a simple quantity, which distinguishes chaotic and MBL systems. \nSuch a commonly-used criterion is the gap ratio, usually denoted by $r$. \nTo evaluate it, we sort the energy spectrum and calculate the quantity\n\\begin{equation}{\\label{r}}\n    {r_i} = \\frac{\\min(E_{i}-E_{i-1}, E_{i+1} - E_{i})}{\\max(E_{i}-E_{i-1}, E_{i+1} - E_{i})}.\n\\end{equation}\nNext, this ratio is averaged over all neighbor triples $\\{E_{i-1}, E_{i}, E_{i+1}\\}$ in the sorted spectrum, $r=\\langle r_i\\rangle$.\nIt is established that $r\\approx0.38$ for the MBL systems and $r\\approx0.53$ for the chaotic ones \\cite{PhysRevLett.110.084101,PhysRevB.75.155111}. \nBy analyzing this criterion as a function of system parameters, i.e., by constructing an effective ``phase diagram'', we can determine boundaries between chaotic and localized behavior \\cite{PhysRevA.92.041601, PhysRevB.82.174411}.\nIn this study, we perform the corresponding numerical analysis by means of the \\textsc{QuSpin} open-source package \\cite{SciPostPhys.2.1.003, 10.21468\/SciPostPhys.7.2.020}.\n\nThe exact diagonalization (ED) technique can also be employed for studying time dynamics after quenches. \nWithin this method, we directly obtain all Hamiltonian eigenstates and project the initial wave function onto them.\nAlthough ED is feasible for small system sizes, its substantial benefit is that the evolution of physical observables can be analyzed on exponentially large timescales (in contrast to the TDVP approach, where the complexity scales linearly with time). \nAccess to quantities in this regime allows us to study asymptotic behavior of relevant observables and their fluctuations. \n\n\n\\subsubsection{Ensemble-based analysis}\\label{subsec:ensembles}\nThe physical observables obtained within the ED approach can also be compared with those from the diagonal and microcanonical ensembles \\cite{DAlessio2016}. \nThis comparison is an explicit test for thermalization or its absence, since for the thermalized system the local observables must stay in agreement with those provided by the microcanonical ensemble. \nThe observables from the microcanonical or diagonal ensembles can be evaluated from the available data obtained in the simulation of system dynamics.\n\nFor the microcanonical ensemble, we calculate observables in the following way. First, we calculate the expectation value~$E^{(0)}=\\langle\\psi^{(0)}| \\hat{H} |\\psi^{(0)} \\rangle$ of the Hamiltonian in the initial state~$|\\psi^{(0)} \\rangle$ before the quench.\nNext, we specify the range of energies $\\Delta E$ and determine all eigenstates ${\\psi_i}$ with the energies $E_i\\in[E^{(0)} - \\Delta E, E^{(0)} + \\Delta E]$.\nWe choose $\\Delta E$ in the way that the number $N_{\\rm st}$ of the available eigenstates in the interval is about $N_{\\rm st}=50$. \nFinally, we evaluate the expectation values of the operator $\\hat{\\cal O}$ in the microcanonical ensemble (ME) according to the standard formula, \n\\begin{equation}\\label{eq:obs_ME}\n    \\langle \\hat{\\cal O}\\rangle_{\\rm ME}\n    = {N_{\\rm st}}^{-1}\\sum_{i} \n    {\\langle \\psi_{i}|\\hat{\\cal O}|\\psi_{i} \\rangle}, \n\\end{equation}\nwhere the summation is performed over all eigenstates in the specified energy range. \n\nThe diagonal ensemble describes long-time asymptotics of expectation values \\cite{DAlessio2016}. To access it, we calculate all eigenstates $|v_{j}\\rangle$ in the symmetry sector (e.g., the block with the fixed total number of particles) of the initial state $|\\psi^{(0)}\\rangle$.\nNext, we calculate the projection coefficients of the initial state onto the eigenstates $\\langle v_{j}|\\psi^{(0)}\\rangle$.\nThe average values in the diagonal ensemble (DE) are calculated according to the formula\n\\begin{equation}\\label{eq:obs_DE}\n    \\langle \\hat{\\cal O}\\rangle_{\\rm DE}\n    =\\sum_{j} |\\langle v_{j}|\\psi^{(0)}\\rangle|^{2} \\langle v_{j}|\\hat{\\cal O}|v_{j} \\rangle,\n\\end{equation}\nwhere the summation is performed over all energy eigenstates~$|v_{j}\\rangle$. \nSince the long-time asymptotes of physical observables after quench are equal to expectation values in the diagonal ensemble, we compare $\\langle \\hat{\\cal O}\\rangle_{\\rm DE}$ with $\\langle \\hat{\\cal O}\\rangle_{\\rm ME}$ to determine whether the system is thermalized. \n\nAs an additional important observable, we also calculate the entanglement entropy. To this end, we consider a state~$|\\psi\\rangle \\equiv|\\psi\\rangle_{AB}$ and a bipartition of the system $AB$ into two parts: $A$ and $B$ with the respective sizes $L_{A}$ and $L_{B}$.\nThen, we can define the entanglement entropy of the subsystem $A$ as the von Neumann entropy of the reduced density matrix $\\rho_{A}$ characterizing the subsystem $A$,\n\\begin{equation}\\label{eq:ent_entropy}\n    S=-\\operatorname{Tr}(\\rho_{A}\\ln{\\rho_{A}}).\n\\end{equation}\nWe calculate the reduced density matrix according to the standard formula $\\rho_{A} = \\operatorname{Tr}_{B}\\rho$, where $\\rho = |\\psi\\rangle \\langle \\psi|$ is the density matrix of the full system under study and the trace is taken over degrees of freedom in the subsystem $B$. \n\n\\subsubsection{Matrix-product state approaches}\nAs we mentioned above, for large systems the ED approach is not feasible.\nHowever, it is still possible to employ algorithms based on matrix product states (MPS). \nIn these methods, it is assumed that the targeted state can be represented as an MPS of a relatively small bond dimension~$D$ (typically, $D\\lesssim 100$). There are several classes of such algorithms applicable to MBL systems. \nIn particular, the density-matrix renormalization group (DMRG) approach with the corresponding generalization (DMRG-X) can be used to determine eigenstates in the middle of the spectrum of MBL systems \\cite{PhysRevLett.116.247204}. \nUnitary matrix-product operator algorithm \\cite{PhysRevB.94.041116} finds the full unitary matrix that diagonalizes the Hamiltonian of the localized system.\nTDVP \\cite{PhysRevB.94.165116} and time evolving block decimation (TEBD) \\cite{PhysRevLett.93.040502} can be used to determine dynamics of wave functions after quenches or time dynamics of operators in the Heisenberg picture. \nBelow, we restrain ourselves to studying only time dynamics of wave functions. \nAlthough both TEBD and TDVP approaches can be employed for this purpose, TEBD is restricted to Hamiltonians with short-range interactions. \nAs the Hamiltonians of our models contain the long-range terms, TDVP becomes more beneficial for the simulation of quenches. \nIn this study, we perform the corresponding tensor-network calculations by means of the \\textsc{ITensor} numerical package~\\cite{itensor}.\n\nMPS-based approaches are powerful in representing the states with low entanglement entropy. They have a control parameter, the bond dimension $D$, with the maximal entanglement of the representable states, which scales as $\\log(D)$. \nOne can use unentangled states as initial wave functions, which can be exactly represented as MPS. \nThen, we propagate this state in time within the TDVP approach. \nNaturally, the entanglement entropy increases during the time evolution.\nAs soon as the entropy reaches approximately the same value as the maximal entanglement entropy for the given bond dimension, results from TDVP become unreliable \\footnote{In fact, as results for the system with short-range interactions show, TDVP may become unreliable significantly earlier, see, e.g., the analysis of TDVP convergence in Ref.~\\cite{Sierant2021}}. \nFor this reason, TDVP is effective only on finite time intervals. However, it can be effectively used for a detection of the MBL regime in which the dynamics significantly slows down. It is much more difficult to unambiguously confirm MBL phase without investigating much longer timescales \\cite{Sierant2021}, and there are clear differences between disordered systems and systems with quasiperiodic potentials. The question of how Stark localized systems fit in this scheme needs further investigation, which is beyond the scope of the current study.\n\nFor chaotic systems, the entropy increases linearly in time after quenches. Due to this fact, TDVP is applicable on relatively small timescales. In contrast, for MBL systems, the entropy grows logarithmically in time, thus numerical simulations can cover significantly larger time intervals at moderate bond dimensions. \nMoreover, the entropy evolution can be used by itself as a criterion of localization in numerical algorithms. \nThereafter, the growth of the entropy is used both as an indicator of reliability of the obtained results and as one of representative quantities, which are sensitive to transition between chaotic and localized behavior.\n\nIn the subsequent analysis, we use the following quench protocol: we initialize the wave function in the product state, where all even sites of the lattice are filled with one particle and all odd sites are empty (in the case of spin chain, even and odd sites are occupied by spin-up and spin-down particles, respectively); then, the time dynamics of this state for the given model Hamiltonian is studied. In case of fermionic system, we perform the Jordan-Wigner transformation to map the system to the spin chain [see also Eqs.~\\eqref{H_fermionic} and \\eqref{H_spin}], where one can apply the TDVP approach in a straightforward manner.\n\n\nWhile analyzing time evolution of the system, we measure several quantities characterizing the many-body wave function. \nOne of them is the above-mentioned entanglement entropy \\eqref{eq:ent_entropy}. \nIt is also possible to compute expectation values of operators, which can characterize ergodicity breaking. \nSuch an experimentally-relevant observable (see, e.g., Ref.~\\cite{Scherg2021}), which is especially convenient for our quench protocol, is the even-odd site occupation imbalance~$I$ (or the so-called amplitude of the charge-density wave). It is defined as a difference between the number of particles on even and odd sites of the lattice ($N_{e}$ and $N_{o}$, respectively), normalized by the total number of particles $N$,\n\\begin{equation}\\label{imbalance}\n    I(t)\n    = \\frac{1}{N} \\sum_{i=1}^{L} (-1)^{i} \n    \\langle\\psi(t)|\\hat{n}_{i}|\\psi(t)\\rangle\n    = ({N_{e} - N_{o}})\/{N}.\n\\end{equation}\n\nFor the above-specified initialization of the wave function, the initial state yields $I(0)=1$ and this is its maximal value, i.e., $|I(t)|\\leq1$. \nDuring the time evolution, this observable typically decreases to a certain constant value and then oscillates around this value with a small amplitude. \nIn the chaotic phase, this constant value is close to zero. In contrast, in the MBL phase this value remains relatively large. \nThis shows that MBL phase contains some memory of the initial state and its inhomogeneities, which are partly measured by the parameter~$I$. \nTherefore, the asymptotic behavior of imbalance at long times can be used as reliable indicator of localization.\n\n\n\\subsubsection{Schrieffer-Wolff transformation}\\label{subsec:SWT}\nLet us briefly discuss the case of large external potential in the Hamiltonian.\nNote that the corresponding amplitude~$F$ is proportional to the dipole moment of particles or spins. \nThe spectrum of the dipole operator entering the Hamiltonian is highly degenerate. Therefore, it seems natural to employ the degenerate perturbation theory based on the Schrieffer-Wolff transformation to effectively describe the system under study. \n\nThe traditional Schrieffer-Wolff transformation (SWT) \\cite{sw66} relies on the following procedure: we divide the Hamiltonian into the leading term $\\hat{H}_{0}$, which determines the largest energy scale of the full Hamiltonian, and the residual part. \nThe latter can be additionally divided into parts $\\hat{T}$ and $\\hat{V}$ containing operators that do not commute and commute with $\\hat{H}_{0}$, respectively.\nFor definiteness, the spin Hamiltonian~\\eqref{H_spin} can be written as\n\\begin{equation}\\label{SW-decomposition}\n    \\hat{H} = \\hat{H}_{0} + \\hat{T} +\\hat{V},\n    \\qquad\n    \\hat{H}_{0} = F\\sum_{i=1}^{L}  i \\hat{S}^{z}_{i},\n\\end{equation}\nwhere the part commuting with $\\hat{H}_0$ is given by\n\\begin{equation}\n    \\hat{V} = \\sum_{i=1}^{L} w_{i} \\hat{S}^{z}_{i} + U \\sum_{1\\leq i < j \\leq L} \n    \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}}\n\\end{equation}\nand the noncommuting perturbation has the form\n\\begin{equation}\n    \\hat{T} = - J_{1}\\sum_{i=1}^{L-1} (\\hat{S}^{+}_{i}\\hat{S}^{-}_{i+1} + \\hat{S}^{-}_{i}\\hat{S}^{+}_{i+1}).\n\\end{equation}\n\nUpon this (or a similar) division, we apply the unitary transformation $\\hat{\\cal U}$ to the Hamiltonians \\eqref{H_fermionic}--\\eqref{H_spin}. This unitary transformation is represented in the form $\\hat{\\cal U} = \\exp{\\hat{\\cal S}}$,  where $\\hat{\\cal S}$ is anti-hermitian operator. \nThe operator $\\hat{\\cal S}$ is expressed in terms of a series in the expansion parameter (${1}\/{F}$ in our case) in the way to cancel terms in the Hamiltonian that do not commute with $\\hat{H}_{0}$.\nThis transformation yields an effective Hamiltonian, which is block-diagonal (up to small higher-order corrections in the expansion parameter), with the size of blocks determined by the degeneracy of $\\hat{H}_{0}$. \nFor the models with the linear potential we obtain the dipole-conserving Hamiltonians. Note that in the limit of infinite system, the resulting effective Hamiltonian is translationally invariant. In this sense, the systems in linear or quadratic external potentials are close to translational invariance. \n\n\n\nLet us now discuss the form of the effective Hamiltonians for the above-specified models. For the spin Hamiltonian~\\eqref{H_spin}, we obtain\n\\begin{multline}\\label{H_effective_spin}\n    \\hat{H}_{\\rm eff} = F\\sum_{i=1}^{L}  i \\hat{S}^{z}_{i} + \n    + \\sum_{i=1}^{L} w_{i} \\hat{S}^{z}_{i} + U \\sum_{1\\leq i < j \\leq L} \n    \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}} \n    \\\\\n    + \\frac{J_{1}^{2}}{F} \\left( \\hat{S}_{L}^{z} - \\hat{S}_{1}^{z} \\right) \n    + \\hat{H}_{\\rm eff}^{(2)},\n\\end{multline}\nwhere the explicit form of the second-order terms $\\hat{H}_{\\rm eff}^{(2)}$ is given in Appendix~\\ref{App1} for the sake of compactness.\nAll terms in the effective model commute with the dipole operator $\\sum_{i=1}^{L} i \\hat{S}^{z}_{i}$. If only short-range interactions are present, this Hamiltonian is additionally fragmented into noninteracting sectors, as described in Ref.~\\cite{Sala2020}.\n\nEither from the effective Hamiltonian \\eqref{H_effective_spin} with the inverse Jordan-Wigner transformation, or directly from the Fermi-Hubbard model \\eqref{H_fermionic}, the effective Hamiltonian can be written as\n\\begin{multline}\\label{H_effective_fermionic}\n    \\hat{H}_{\\rm eff} \n    = F \\sum_{i=1}^{L} i \\hat{n}_{i} \n    + U \\sum_{1\\leq i < j \\leq L} \n    \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}} \n    \\\\\n    + \\frac{J_{1}^{2}}{F} \\left( \\hat{n}_{L} - \\hat{n}_{1} \\right)\n    + \\hat{H}_{\\rm eff}^{(2)}.\n\\end{multline}\n\nUp to quadratic terms in the expansion series over ${1}\/{F}$, the generator $\\hat{\\cal S}$ for the spin model~\\eqref{H_spin} has the following form:\n\\begin{equation}\\label{SW_spin}\n    \\hat{\\cal S} \n    = -\\frac{J_{1}}{F}  \\sum_{i=1}^{L-1} (\\hat{S}_{i}^{-} \\hat{S}_{i+1}^{+} - \\hat{S}_{i}^{+} \\hat{S}_{i+1}^{-}) \n    + \\hat{\\cal S}^{(2)},\n\\end{equation}\nsee also Appendix~\\ref{App1} for the explicit form of $\\hat{\\cal S}^{(2)}$.\nNote that in the fermionic system, the transformation has a similar form except for the absence of terms with $w_{i}$ in the operator $\\hat{\\cal S}$. All other terms can be obtained from Eq.~\\eqref{SW_spin} by applying the Jordan-Wigner fermionization rules. \n\nThe bosonic model \\eqref{H_bosonic} contains an additional on-site interaction term with the coupling $V$, thus the effective Hamiltonian up to linear terms in $1\/F$ differs from Eq.~\\eqref{H_effective_fermionic} only by the term $V\\sum_{i=1}^{L} \\hat{n}_{i}(\\hat{n}_{i}-1)$. \nAt the same time, the explicit forms of the quadratic corrections $\\hat{H}_{\\rm eff}^{(2)}$ and $\\hat{\\cal S}^{(2)}$ are substantially different for cases of fermions and bosons; these are given separately in Appendix~\\ref{App1}.\n\n\n\\section{Results}\\label{sec:results}\n\\subsection{Spectral characteristics}\nIn this section, we discuss results for the above-introduced ergodicity criterion $r$ [see Eq.~\\eqref{r}], which we evaluate by means of the exact diagonalization of various Hamiltonians with long-range deformations of the Hubbard model in the presence of a linear potential (see Subsec.~\\ref{subsec:ED}).\nWe begin our analysis from the Fermi-Hubbard model~\\eqref{H_fermionic} with the next-nearest hopping processes ($m=1$).\nIn Fig.~\\ref{fig:fermionic_spectrum} we show the characteristic diagrams of the parameter $r$ in the fermionic system with long-range interactions described by different values of exponents $\\alpha$ ranging from $\\alpha=0.5$ to $\\alpha = 3.0$.\nLet us emphasize that the former ($\\alpha=0.5$) is far beyond the predicted boundary values of $\\alpha$, where localization can occur according to the perturbation theory \\cite{Yao2014, Burin2015a, Burin2015b}. Note that there are also ED studies  of the MBL persistence in the presence of similar long-range interactions and aperiodic potentials revealing similar behavior \\cite{Nag2019, Prasad2021}.\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig2_fermions.pdf}\n  \\caption{\\label{fig:fermionic_spectrum}\n    Dependencies of the parameter $r$ on the strength of the external linear tilt $F$ and the interaction strength $U$ at different $\\alpha=\\{0.5,1,2,3\\}$ for the fermionic long-range interacting model~\\eqref{H_fermionic} with $L=16$, $N=8$, $J_1=1$, and $m=1$. \n    }\n\\end{figure}\n\n\nOne of central observations of our study is that the systems with small but nonzero long-range interaction typically remain localized at $F>2$.\nThis holds in a wide range of the employed parameters $\\alpha$ and $U$. \nWe attribute it to the fact that long-range interactions completely lift all degeneracies in the spectrum yielding completely regular spectrum statistics with no need for further introduction of the on-site disorder or harmonic potential. \n\nAt large amplitudes of the interaction potential $U$, the systems under study are localized for almost every value of $F$, but this effect is caused rather by conventional Mott-like localization, than by the external linear tilt. \nSince these systems are spinless and do not have internal degrees of freedom, their dynamics is trivial in the strong-coupling limit (in contrast to the effective Heisenberg chains for systems with internal degrees of freedom).\n\nAs one can see from Fig.~\\ref{fig:fermionic_spectrum}, the chaotic phase is the most pronounced in the interval $U \\in (2 ,5)$. To further clarify the influence of the exponent $\\alpha$, we fix $U=3.5$ and study the dependence of the gap ratio $r(F, \\alpha)$ as shown in Fig.~\\ref{fig:Alpha_dependency}. At large values of $\\alpha$, the system is effectively short-range and the localization boundary only moderately depends on the exponent~$\\alpha$. \nAt small $\\alpha$, the system becomes additionally localized due to long-range interactions. We use finite-size scaling analysis as described in Appendix~\\ref{App2} to extract the critical value $F_{c}$ at different $\\alpha$ ~\\cite{StarkMobilityEdge, StarkSuperconductingCircuits}. The obtained critical values are indicated by circles in Fig.~\\ref{fig:Alpha_dependency}.\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig3_alphadep.pdf}\n \n  \\caption{\\label{fig:Alpha_dependency}\n    Dependencies of the parameter $r$ in the fermionic system on the strength of the external linear tilt $F$ and the exponent $\\alpha$ at $L=16$, $N=8$, and $U=3.5$. Red points correspond to the critical amplitude~$F_{c}$ obtained from the finite-size scaling analysis.\n    }\n\\end{figure}\n\n\n\nUp to this moment, we analyzed stability of localization with respect to introduction of long-range interactions with different power-law dependencies. \nLet us also discuss how long-range hopping can influence MBL. \nTo this end, we introduce the next-nearest neighbor (nnn) hopping term with the amplitude $J_{2}$ [see also Eqs.~\\eqref{H_fermionic} and \\eqref{H_bosonic}] and study its influence on the many-body localization. Note, that the influence of long-range hopping was studied in Ref.~\\cite{vernek2021robustness} for the $J_{1}$-$J_{2}$ spin chain in external linear field. Our results agree well with the observations of that study. \n\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig4_fermions_lr-L16.pdf}\n  \\caption{\\label{fig:lr_hopping}%\n    Dependencies of the parameter $r$ on the strength of the external linear tilt $F$ and the next-nearest-neighbor hopping amplitude $J_{2}$ at $\\alpha=1$ and two different $U=\\{3,6\\}$ for the fermionic long-range interacting model~\\eqref{H_fermionic} with $L=16$, $N=8$, $J_1=1$, and $m=2$. \n    }\n\\end{figure}\nFigure~\\ref{fig:lr_hopping} shows the dependence of $r$ on the external tilt $F$ and the hopping amplitude $J_{2}$ at two values of long-range interaction strength: $U=3$ and $U=6$. \nIn the regime of small $J_{2}$ ($J_{2} \\lesssim J_{1}$), the nnn hopping does not impact the localization transition in a visible way. \nOnly at $J_{2} > J_{1}$ the transition becomes substantially determined by the amplitude~$J_{2}$. \nIn particular, the transition curve exhibits approximately linear dependence of the critical tilt $F_{\\rm loc}$ on $J_{2}$, as one can also conclude from the noninteracting model. \nSince in experimental realizations the longer-range hopping terms are usually smaller than the nearest-neighbor terms, the former become largely irrelevant to the issue of stability of Stark localization. We should note that at large interaction strength $U$ the dependence of the gap ratio $r$ on $J_2$ can be more complex, as the system shows localized value of $r$ for all $F$ at $J_{2} =0$. In this case, the nonzero $J_{2}$ can drive the system into the chaotic phase. This behavior is partly shown in Fig.~\\ref{fig:lr_hopping} at $U=6$, where the system has intermediate values of $r$ in the limit $J_{2} = 0$.\n\nWe have also checked the case of more general power-law hopping $J_{m} = {J_1}\/{m^{\\beta}}$. The hopping processes parameterized in this way do not destroy localization even at small values of the parameter~$\\beta$ ($1< \\beta < 2$). \nNote that the stability of MBL was theoretically studied for the lattice model with long-range interactions and the same parametrization of long-range hopping in Ref.~\\cite{Nag2019}, but with aperiodic potentials. The given results agree with our observations.\n\n\nThe observed robustness of MBL even upon inclusion of the long-range hopping can be partly explained by arguments based on resonances, which were used to predict breaking of MBL in the case of disordered potentials \\cite{Yao2014,Burin2015a,Burin2015b}. \nResonances are generally present if the difference of energies $|\\tilde E_i-\\tilde E_{j}|$\nbetween the two eigenstates $|\\tilde{\\psi}_{i}\\rangle$ and $|\\tilde{\\psi}_{j}\\rangle$ of the Hamiltonian without hopping (which includes both many-body interactions and external potential) are smaller than the hopping matrix element between these two respective states. \nIn case of an external disorder potential with a randomly distributed amplitude $\\epsilon_{n} \\in [-W,W]$, there is a nonzero probability that  $|\\tilde E_i-\\tilde E_{j}|$ is very small, and resonances are present. \nIf the number of such resonances diverges with distance between resonating sites, MBL is not stable.\nFrom this, one can derive that for stability of MBL in one-dimensional case with random external potential and local interactions, the condition $\\beta > 1$ must be fulfilled.\nIf more general interactions between the resonances are considered, even more strong restrictions on $\\alpha$ and $\\beta$ can be obtained. \nFor Stark localization, generally, if states $|\\psi_{i}\\rangle$ and $|\\psi_{j}\\rangle$ are coupled by a single hopping process between, for example, the sites $m$ and $n$, then the difference between the respective energies $| E_i- E_{j}|$ will be mainly determined by the external tilt, $| E_i- E_{j}| \\approx F|m-n|$. \nResonance will be present only if ${J_1}\/{|m-n|^{\\beta}} > F|m-n|$, which is generally not the case for large enough $F$ and $|m-n|$ with $\\beta > 0$. \nHence, the natural mechanism of MBL destabilization by long-range hopping is significantly suppressed by the nature of potential, which largely inhibits the possibility of resonances between the states coupled by a single hopping process. \n\nThe above analysis reveals stability of the Stark many-body localization upon inclusion of long-range interactions and long-range hopping processes. The natural question arises on experimentally realistic interaction terms that are able to make Stark MBL unstable or, at least, to shift the localization boundary to the larger values of $F$.\nThese interactions must contain even the longer-range coupling than in the power-law dependence.\nThe obvious type of interactions to examine are the cavity-mediated interactions (see Subsec.~\\ref{subsec:cavity}), which have already been studied in the context of MBL \\cite{Sierant2019,Chanda2021manybody}.\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig5_fermions_cav.pdf}\n  \\caption{\\label{fig:cavity_spectrum}%\n    Dependence of the parameter $r$ on the amplitude of the cavity-mediated interaction $U_{\\rm cav}$ and the external linear tilt $F$. The fixed parameters are $L=16$, $N=8$, $U=3$, and $\\alpha = 3$. \n    }\n\\end{figure}\nIn Fig.~\\ref{fig:cavity_spectrum} we analyze the ergodicity parameter~$r$ as a function of $F$ and $U_{\\rm cav}$, while the amplitudes $U$ and $\\alpha$ are kept fixed. \nWe observe that the localization boundary shifts to the larger values of $F$ compared to the case of $U_{\\rm cav}=0$. \nAt larger values of the external tilt $F$ (in particular, $F \\gtrsim 5$ for the chosen set of parameters), this system remains localized.\n\n\n\nNote that in the limit of large tilt $F$, we also verified the obtained ED results for spectral characteristics by means of the SWT-based calculations [see Subsec.~\\ref{subsec:SWT}].\nThe corresponding analysis confirms, in particular, that the spectral characteristics of the effective Hamiltonians \\eqref{H_effective_spin} and \\eqref{H_effective_fermionic} are the same as of the full models \\eqref{H_fermionic}--\\eqref{H_spin} up to corrections proportional to $1\/F^{3}$.\n\n\n\\subsection{Dynamics: Imbalance and entropy}\nThe localized and chaotic regimes can be identified by clear signatures in the dynamics of physical observables, such as the particle imbalance $I$ [see Eq.~\\eqref{imbalance}].\nCharacteristic examples of this dynamics are shown in Fig.~\\ref{fig:Imbalance_dyn} for the fermionic system in the chaotic ($F=0.5$) and localized ($F=3.0$) regimes. \n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig6_dynamics-ED_corr2.pdf}\n \n  \\caption{\\label{fig:Imbalance_dyn}%\n    Time evolution of the imbalance $I$ and entanglement entropy~$S$ after quench in the fermionic model at two different values of the external tilt, $F=0.5$ and $F=3.0$. \n    For sake of visibility, the entropy growth is shown on two timescales: linear (c) and logarithmic (d).\n    Other parameters are $L=18$, $U=2$, and $\\alpha = 1$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n    }\n\\end{figure}\nNote the difference in timescales used in the corresponding subfigures. \nIn the chaotic regime [see Fig.~\\subfigref{Imbalance_dyn}{a}], the expectation values in microcanonical and diagonal ensembles are nearly the same and close to zero. \nThe initial imbalance relaxes to this expectation value on the timescale of the order of $1\/J$ (here and below $\\hbar=1$, $J\\equiv J_1$, {$N=L\/2$}, and $m=1$). After this relaxation, the fluctuations of the imbalance become negligibly small. \nIn contrast, in the localized regime with much larger $F$ [see Fig.~\\subfigref{Imbalance_dyn}{b}], the diagonal and microcanonical ensembles provide us with different expectation values of the imbalance. \nThis observable oscillates around the respective expectation value in the diagonal ensemble for a significantly larger period of time. \n\nClear indications of localization can also be observed in the dynamics of the entanglement entropy $S$ [see Eq.~\\eqref{eq:ent_entropy}], as we show in Figs.~\\subfigref{Imbalance_dyn}{c} and \\subfigref{Imbalance_dyn}{d}. \nIn the chaotic regime with $F=0.5$, the entanglement entropy grows linearly for a short period of time and then saturates to a constant value. \nThe period of the linear growth is approximately the same as a period of relaxation of the imbalance~$I$, see also Fig.~\\subfigref{Imbalance_dyn}{a}. \nIn contrast, in the localized regime with $F=3$, the entropy~$S$ grows only logarithmically in time and demonstrates characteristic oscillations. \nAt much longer times it also saturates, but to a smaller value than in the chaotic regime. A more general discussion of the entanglement growth in long-range interacting localized systems can be found in Ref.~\\cite{NonAlgebraicEntanglementGrowth}.\n\nWe can use these observations on the behavior of the imbalance~$I(t)$ to study localization transition in more detail. \nBelow, we obtain full spectrum of the Hamiltonians under study and calculate the imbalance both from the diagonal and microcanonical ensembles, as discussed in Subsec.~\\ref{subsec:ensembles}.\nFor the purpose of quantifying the observed differences in system dynamics, we introduce an auxiliary ergodicity parameter~$\\rho$,\n\\begin{equation}\n    \\rho=-\\log|\\langle I \\rangle_{\\rm DE}-\\langle I \\rangle_{\\rm ME}|.\n\\end{equation}\n\nIn terms of $\\rho$, first, we compare predictions given by these two ensembles for the fermionic model~\\eqref{H_fermionic} in Fig.~\\ref{fig:Dynamics_fermions}.\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig7_all.pdf}\n  \\caption{\\label{fig:Dynamics_fermions}%\n     Dependence of the auxiliary ergodicity parameter $\\rho$ on the strength of linear tilt $F$ and strength of long-range interactions $U$ for the fermionic (left, $L=16$, $\\alpha = 1$) and bosonic (right, $L=12$, $\\alpha=2$, $V=4$) systems.\n    }\n\\end{figure}\nHere, the boundary between the chaotic and localized behavior at $F\\approx2$ and moderate $U$ can be seen much more clearly. \nThis boundary starts to shift to larger values of $F$ at higher interaction strength $U$, but it is necessary to note that in this regime both microcanonical and diagonal ensembles predict large values of the final imbalance. \nThis shift of the localization boundary to larger interaction strengths is also confirmed by calculations of the entanglement entropy, which exhibits a linear growth to large values even at $F=2.4$ and $U=9$. \n\nFor small values of the interaction strength~$U$, the results from the dynamics and level statistics show some discrepancies (cf. Figs.~\\ref{fig:fermionic_spectrum} and \\ref{fig:Dynamics_fermions}). \nWe further checked the behavior of the entanglement entropy~$S$ in the region of parameter space, where dynamics and level statistics suggest different results. \nThe entanglement entropy shows a logarithmic growth to the values typical for chaotic systems, while the imbalance~$I$ fluctuates as in the localized system, but, at the same time, predictions for the mean values $\\langle I\\rangle_{\\rm ME}$ and $\\langle I\\rangle_{\\rm DE}$ agree. \nFurthermore, the microcanonical ensemble predictions become sensitive to the energy range~$\\Delta E$ (equivalently, to the number $N_{\\rm st}$) used in the definition~\\eqref{eq:obs_ME} of the corresponding observables more strongly than in the case of large interaction strength.\nIn Appendix~\\ref{App3} we discuss how the impact of the mentioned discrepancy can be further reduced by analyzing temporal fluctuations of main observables.\n\n\nNext, for the bosonic model~\\eqref{H_bosonic}, we obtain results for the imbalance dynamics. \nLet us note that, according to additional analysis, the results for the spectrum statistics are different at larger values of $V$, as there are states in the spectrum with double or triple occupancies on some sites and these states have energies uncorrelated with other states. \nFor bosonic system we restrict ourselves to half-filling to compare with the fermionic case. For larger densities chaotic behavior can survive to higher values of $F$, as effective hopping is enhanced by bosonic statistics.\nTherefore, the results from the dynamics become more relevant.\nWe show the characteristic phase diagram in Fig.~\\ref{fig:Dynamics_fermions}. At small and intermediate interaction strength, $\\langle I\\rangle_{\\rm ME}$ and $\\langle I\\rangle_{\\rm DE}$ agree at $F<2$. At larger interactions, as in the fermionic case, microcanonical and diagonal ensembles show similar results only at relatively large tilts $F$. \n\nFor larger systems ($L\\geq20$), we employ the TDVP approach to study the imbalance and entanglement entropy behavior. \nIn Fig.~\\ref{fig:ImbEntTDVP}, we show the results for imbalance dynamics in the bosonic and fermionic systems ($L=50$) at different values of the tilt $F$.  In Fig.~\\ref{fig:EntTDVPLogScale}, we additionally analyze the dynamics of entanglement entropy on logarithmic timescale to ensure logarithmic growth of entropy in the localized phase. \n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig8_tdvp_corr.pdf}\n  \\caption{\\label{fig:ImbEntTDVP}%\n    Time dynamics of the imbalance $I$ and entanglement entropy~$S$ for fermionic (left) and bosonic (right)  systems obtained by TDVP. The parameters are $\\alpha = 1$, $U = 3$, $V = 5$ (for bosons), $L = 50$, and $D = 100$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n    }\n\\end{figure}\n\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{fig9_entr_logscale.pdf}\n \n  \\caption{\\label{fig:EntTDVPLogScale}%\n  Time dynamics of the entanglement entropy on logarithmic timescale in fermionic system for different values of the tilt $F$ and system sizes $L$. The parameters are $\\alpha = 1$, $U=3$, and $D=50$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n    }\n\\end{figure}\n\nIt is clear that the dynamics of imbalance changes for both statistics at $F\\approx2$. \nAt larger $F$, the imbalance~$I$ exhibits oscillations and relaxes to a nonzero value, while at $F\\approx1$ it quickly approaches zero. Behavior is qualitatively the same as was observed in small systems with the exact diagonalization. \nIn the TDVP analysis, we employ a relatively small value of the bond dimension ($D=100$), which restricts our calculations at small tilts $F$ to short times, since for larger times the employed MPS approach is not able to accurately represent the amount of entanglement in the wave function. \nThis effect can also be seen in Fig.~\\ref{fig:ImbEntTDVP}, which shows the growth of the entanglement entropy with time. At small $F$, the entropy reaches the maximum value allowed by the bond dimension~$D$ at several $Jt$. \nThis invalidates our results at larger values of $t$,  but also heralds the chaoticity of the system.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe theoretically studied the many-body localization in the case of one-dimensional lattice systems with long-range interaction between particles and linear external potential.\nThe obtained results reveal that the systems with small but nonzero long-range interaction typically remain localized at moderate and large amplitude of the external linear potential.\nThese observations hold in a wide range of parameters characterizing long-range interaction potential including the cases of cavity-mediated interactions and long-range tunneling. This makes inclusion of the additional on-site disorder or harmonic potential unnecessary. \n\n\nIn addition to quantitative analysis of spectral characteristics of systems in wide ranges of parameters, we analyzed the dynamical evolution of relevant physical observables: even-odd site occupation imbalance and entanglement entropy.\nThe dynamics of both quantities clearly indicates differences between the chaotic and localized many-body regimes in lattice systems with the external linear tilt.\n\nUpon calculation of the imbalance within the microcanonical and diagonal ensembles, we introduced an auxiliary (ensemble-based) ergodicity parameter.\nFor the fermionic systems, we observe qualitative agreement in structures of phase diagrams constructed by means of the ergodicity paramenters from different (spectrum- and ensemble-based) approaches, whereas for the bosonic system, the ensemble-based ergodicity parameter becomes more accurate in certain regimes of the on-site interaction strengths.\nDepending on the system size, we applied both ED and TDVP approaches, which agree well in determining localization transitions.\nWe also confirmed the obtained numerical results in the limit of the applicability of the effective models, where we derived the effective Hamiltonians for the systems under study.\n\nIn general, our findings significantly extend the class of systems, where the transitions between the localized and chaotic many-body regimes can be studied in detail by accessing relevant observables in cold-atom experiments \\cite{Scherg2021}.\nThe systems under study are completely disorder free and quasi translationally invariant in the sense that the shift operator commutes with the Hamiltonians up to a constant.\nThis makes the system identical at different spatial positions and allows one to study it in a kind of thermodynamic limit.\nThus, the approach becomes efficient for the Wegner-flow and Schrieffer-Wolff studies relying on translational invariance \\cite{Pekker2017, LongRangeWegnerFlow}.\n \n\n\n\\begin{acknowledgments}\nThe authors acknowledge support from \nthe National Research Foundation of Ukraine, Grant No.~0120U104963,\nthe Ministry of Education and Science of Ukraine, Research Grant No.~0120U102252, and the National Academy of Sciences of Ukraine, Project No. 0121U108722.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nThe \\gls{iot} makes it possible to remotely monitor and control a wide set of\nheterogeneous objects through an Internet connection. This paradigm foresees\nmultiple applications in a large variety of scenarios: from fleet tracking and\nprocess monitoring in industrial scenarios to smarter garbage collection and\nintelligent light control in cities; from monitoring of soil moisture in\nagriculture to home temperature control and personal health monitoring\n\\cite{zanella2014internet, yuehong2016internet, dlodlo2015internet,\n  chiariotti2018symbiocity}. Also, the \\gls{iot} paradigm can be applied to\nsurveillance-related applications~\\cite{bovenzi2018iot}, as event detectors and\nalarms~\\cite{dos2020performance}.\n\nThe presence of several use cases spawned an ample market, and encouraged the\ndevelopment of multiple technologies meeting the need for low-cost ubiquitous\nconnectivity. A large part of \\gls{iot} nodes will consist in sensors that\ngenerate sporadic traffic, without strict constraints in terms of latency and\nthroughput. This calls for new wireless solutions able to support a massive\nnumber of devices, with an affordable cost for both user equipment and network\ninfrastructure. Therefore, high energy efficiency, extended coverage, and\ninfrastructure simplicity are aspects of primary importance.\n\nSuch requirements motivated the creation of a new family of wireless\ntechnologies collectively called \\glspl{lpwan}, characterized by long\ncoverage range and low power consumption. A prominent \\gls{lpwan}\ntechnology is LoRaWAN, which claims up to 10~years of battery lifetime for\ndevices, and a transmission range between 1.5~km in urban scenarios and\n30~km in rural areas~\\cite{centenaro2016long}.\n\n\n\nSince the deployment of a dense \\gls{iot} network is expensive and time\nconsuming, performance assessments using simulations and mathematical models\nbecome essential to gauge the effect of network parameters and estimate the\nperformance at a reduced cost. In this work, we propose an analytical model of\nthe performance of a LoRaWAN network, accounting for the most relevant features\nof the LoRaWAN standard. This model considers network-layer performance,\nassuming perfect orthogonality between signals modulated with different\n\\glspl{sf}. However, compared to previous models in the literature (discussed in\nSec. III), our model takes into account a wider range of aspects of the PHY and\nMAC layers, such as the possibility of transmitting multiple times both\nconfirmed and unconfirmed packets, the limitations on the channel occupancy time\nimposed by the different national regulations, the interference produced by\nmultiple overlapping transmissions, the capture effect, and the limited number\nof demodulators available at the \\gls{gw}. Furthermore, the model formulation\noffers great flexibility in setting some system parameters, thus making it\npossible to analyze the system performance under different conditions and to\nshed light on possible trade-offs. We consider as performance\nmetrics the packet success probability, average delays, and fairness, from which\nit is possible to derive other measures of interest, such as energy consumption,\nsystem's reliability and the achievable \\gls{qos} in multiple scenario.\nThe proposed model is validated by comparing the results with those obtained\nthrough detailed ns-3 simulations. The analysis shows how the model can be used\nto maximize different performance metrics, proving a very powerful and\nconvenient tool to determine the best network configuration.\n\n\nThe rest of this work is structured as follows. To make the paper\nself-contained, in Sec.~\\ref{sec:technology} we present the main features of the\nLoRaWAN standard, while in Sec.~\\ref{sec:soa} we give an overview of the current\nstate of the art in the performance modeling of this technology.\nSec.~\\ref{sec:model} introduces the proposed model and describes how some of its\nparameters can be tuned to explore different behaviors of the network, while\nSec.~\\ref{sec:simulation} briefly describes the simulation framework used for\nvalidation. Sec.~\\ref{sec:results}, then, compares the output of the analytical\nand simulation models, also showing how they can be used to provide different\ninsights of the network behavior. Finally, Sec.~\\ref{sec:conclusion} draws the\nconclusions and discusses possible future developments.\n\n\n\n\n\n\\section{Technology overview}\n\\label{sec:technology}\n\nThis section describes the key LoRaWAN features, dwelling upon the elements and\nproperties that have a significant impact on the system-level performance, which\nwill then be considered in the model formulation.\n\n\\subsection{The LoRa modulation}\n\\glsreset{sf}\n\nLoRa is a modulation technique based on \\gls{css}, patented by Semtech. Bitrate\nand coverage range depend on the \\gls{sf} parameter that can vary from 7 to 12. Lower \\gls{sf}\nvalues achieve higher data rates and shorter transmission times, but require\nhigher signal powers at the receiver for correct decoding, which implies shorter\ncoverage ranges. On the other hand, signals transmitted using higher \\gls{sf}\nvalues are more robust to channel impairments and can thus achieve longer\ntransmission distances, at the price of an increased transmission time due to\ntheir lower data rates. Furthermore, signals modulated with different \\glspl{sf}\nare almost orthogonal: even if overlapping in time and frequency, two or more\nsignals transmitted with different \\glspl{sf} can be simultaneously decoded,\nprovided that their received powers satisfy some\nconditions~\\cite{croce2018impact}.\n\n\nWhen multiple packets transmitted with the same \\gls{sf} overlap in time and\nfrequency, instead, they may generate destructive mutual interference,\ndisrupting each other's reception and resulting in what is called a\n\\textit{packets collision} event. However, if one signal is significantly\nstronger than the others, by a power margin greater than the so-called\n``co-channel rejection parameter'' $CR_{\\rm dB}$, then it can be received\ncorrectly despite the interference, giving rise to a \\emph{capture} phenomenon.\n\nThe value of $CR_{dB}$ has been estimated to be around 6~dB\nin~\\cite{goursaud2015dedicated}. In order to take advantage of these features,\nthe SX1301 LoRa PHY chipset, typically employed in \\glspl{gw}~\\cite{sx1301},\nprovides 8 parallel demodulation chains, which allow the chip to demodulate up\nto 8 different signals simultaneously, irrespective of their \\glspl{sf} and\nfrequency. We also remark that the \\glspl{gw} do not support full-duplex\ntransmission and reception: in order to send a \\gls{dl} packet they have to\ninterrupt any ongoing reception, regardless of the frequency channels in which\ntransmission and reception occur.\n\n\n\n\\subsection{The LoRaWAN standard}\n\n\\glsreset{ns} \\glsreset{ed} \\glsreset{gw} The LoRaWAN standard~\\cite{lorawan}\ndefines a star-of-stars topology, as represented in Fig.~\\ref{fig:infrastruc},\nwith three kinds of devices: the \\textit{\\gls{ns}}, which is the central network\ncontroller and can be located anywhere in the Internet; the \\textit{\\glspl{ed}},\nperipheral nodes (usually sensors or actuators) that transmit using the LoRa\nmodulation; and the \\textit{\\glspl{gw}}, relay nodes that collect messages from\nthe \\glspl{ed} through the LoRa interface and forward them to the \\gls{ns} using\na reliable IP connection, and \\textit{vice versa}.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\figurescaling\\linewidth]{figures\/lorawanInfrastructureBN.png}\n  \\caption{LoRaWAN network infrastructure. Dotted lines represent LoRa links,\n    while solid lines represent IP connections.}\n  \\label{fig:infrastruc}\n  \\vspace{-1em}\n\\end{figure}\nThe standard also defines three classes of \\glspl{ed}, which differ for the time\nthey spend in reception mode. This article considers \\textit{Class A} devices,\nwhich have the strictest requirements in terms of energy consumption. In order\nto save battery, these devices stay in sleep mode most of the time, opening two\nreception windows only 1 and 2 seconds after the end of an \\gls{ul} packet\ntransmission. Fig.~\\ref{fig:cycle} depicts the different operational phases of a\nClass-A device when transmitting a \\gls{ul} packet and, then, receiving a\n\\gls{dl} packet in the second receive window.\n\nThe \\glspl{ed} have the possibility of transmitting \\textit{unconfirmed} or\n\\textit{confirmed} packets. In the latter case, an \\gls{ack} is expected in one\nof the two reception opportunities after the transmission to confirm the correct\nreception of the packet by the \\gls{ns}.\\footnote{Although in this paper we\n  focus on ACK transmissions, the model and the analysis equally hold for any\n  \\gls{dl} packet returned by the \\gls{ns} to the ED after the reception of a\n  \\gls{ul} packet by the NS.} If the \\gls{ack} is not received in either of the\ntwo reception windows, a re-transmission can be performed at least\nRETRANSMIT\\_TIMEOUT seconds after the end of the second receive window. The\nstandard recommends to randomly pick the value for RETRANSMIT\\_TIMEOUT\nuniformly between 1 and 3 seconds~\\cite{regional}.\nIf the \\gls{ack} is not received, the same confirmed message can be\nre-transmitted up to $m-1$ times, after which the packet is dropped. The value\nof $m$ can be configured by the \\gls{ns}.\\footnote{This behavior holds for the\n  LoRaWAN 1.1 standard~\\cite{lorawan, regional}: other versions of the standard\n  may differ.} To increase the robustness of unconfirmed transmissions, instead,\nthe \\gls{ed} can transmit each packet $h$ times. Once again, the value of $h$\ncan be set by the \\gls{ns}. It is worth noting that the reception windows are\nopened after every UL transmission, irrespective of whether or not an \\gls{ack}\nis expected, in order to give the \\gls{ns} the opportunity to send a \\gls{dl}\npacket to the \\glspl{ed}, if needed. The \\gls{ul} messages transmitted by an\n\\gls{ed} are collected by all the \\glspl{gw} in the coverage range of the\ntransmitter, and forwarded to the \\gls{ns}. If the \\gls{ed} requires a reply,\nthe \\gls{ns} can pick any of these \\glspl{gw} to transmit the \\gls{dl} message.\n\n\\begin{figure}[t]\n  \\centering\n      \\input{figures\/cycle.tex}\n      \\caption{Example of operational phases for a Class-A \\gls{ed}. The device,\n        initially in sleep mode, wakes up to transmit from time\n        $t_0$ to $t_1$. Then, the node remains in the idle state for 1 second, and\n        at time $t_2=t_1+1~s$ opens the \\acrfull{rx1}. If no packets are\n        received, the device remains idle until the \\acrfull{rx2} is opened at\n        time $t_3=t_1+2~s$. }\n  \\label{fig:cycle}\n  \\vspace{-1em}\n\\end{figure}\n\nLoRaWAN operates in the ISM unlicensed spectrum, the use of which is subject to\nnational regulations that define the maximum transmit power, and\nthe \\gls{dc} limit, i.e., the maximum percentage of time a node can actively\ntransmit on a certain frequency band. The frequency bands, power and \\gls{dc}\nrestrictions that apply to different regions are reported in the\nstandard~\\cite{regional}. In particular, Table~\\ref{tab:channels}\nshows the configuration mandated for the European region, which entails three\nbidirectional channels and a fourth channel reserved to \\gls{dl} transmissions\nonly. The 868.1, 868.3, 868.5~MHz channels belong to the same regulatory\nsub-band (\\gls{sb1}), and have to share a \\gls{dc} limitation of 1\\%, while the\nchannel reserved for \\gls{dl}, located in the 869~MHz sub-band (\\gls{sb2}), can\nbenefit from a more lenient \\gls{dc} of 10\\% and a higher transmission power.\n\n\\begin{table}[h]\n  \\footnotesize\n  \\centering\n  \\caption{Available LoRaWAN channels in the two sub-bands.}\n  \\label{tab:channels}\n  \\begin{tabular}{llrr}\n    \\toprule\n    Sub-band & Frequency [MHz] & Use & Duty Cycle \\\\\n    \\midrule\n    \\multirow{3}{*}{SB1} & 868.1 & UL\/DL & 1\\%, shared \\\\\n    & 868.3 & UL\/DL & 1\\%, shared \\\\\n    & 868.5 & UL\/DL & 1\\%, shared \\\\\n    \\arrayrulecolor{black!70}\\midrule\n    SB2 & 869.525 & DL & 10\\%, dedicated \\\\\n    \\arrayrulecolor{black}\\bottomrule\n  \\end{tabular}\n\\end{table}\n\nThe \\gls{sf} used for a device's transmission is configured by the device itself\nor set by the \\gls{ns} according to some network management policies. By\ndefault, \\glspl{ed} open the \\gls{rx1} on the same frequency channel of the\n\\gls{ul} transmission, and expect a signal modulated with the same \\gls{sf}. The\n\\gls{rx2}, instead, is opened on the 869.525~MHz channel and the incoming signal is\nassumed to use \\gls{sf} 12, to maximize the coverage rate. The standard allows\nthe \\gls{ns} to modify this pre-defined configuration by communicating the new\nsettings to the \\gls{ed} through appropriate \\gls{mac} commands, allowing for\nthe use of any \\gls{sf} in the second window.\n\n\n\\subsection{Packet life cycle}\n\\label{sec:lifecycle}\n\nMessages transmitted by \\glspl{ed} to the \\gls{gw} are subject to multiple\ncauses of losses:\n\n\\begin{itemize}\n\\item \\textit{Interference}: packets sent in the same frequency channel and\n  with the same \\gls{sf} collide. A transmission can survive a collision\n  event if its received power is sufficiently higher than that of the other\n  overlapping signals (capture effect).\n\\item \\textit{\\gls{gw} already in transmission}: the \\gls{gw} can not lock\n  on a \\gls{ul} packet while performing a \\gls{dl} transmission.\n\\item \\textit{\\gls{gw} starting a transmission}: an ongoing packet\n  reception may be interrupted if the \\gls{gw} needs to send a \\gls{dl}\n  packet.\n\\item \\textit{No available demodulation chains at the \\gls{gw}}: all demodulators\n  are already busy decoding incoming signals.\n\\end{itemize}\n\nMoreover, confirmed \\gls{ul} messages cause the \\gls{ns} to generate\n\\glspl{ack} that need to be transmitted by the \\gls{gw}. Such DL transmissions\nmay as well be impaired by a number of events:\n\\begin{itemize}\n\\item \\textit{Unavailability of receive windows}: this event occurs when\n  all available \\glspl{gw} are prevented from transmitting in both the\n  receive windows because of the \\gls{dc} constraint or other ongoing\n  transmissions.\n\\item \\textit{Interference}: \\gls{dl} packets transmitted in \\gls{rx1} can\n  collide with \\gls{ul} packets transmitted by other \\glspl{ed} in the same\n  channel and with the same \\gls{sf}.\n\\end{itemize}\n\nIn this work, we provide a network model that accounts for all these\nevents.\n\n\n\n\\section{State of the art in LoRaWAN modeling}\n\\label{sec:soa}\n\nIn the last years,\nmathematical modeling has been applied to assess the network\nperformance with respect to various metrics. In~\\cite{adelantado2017under}, the\nauthors address high-level questions about LoRaWAN's suitability for a range of\nsmart city applications, from metering to video surveillance, by modeling the\nsystem as a superposition of different Aloha networks. They conclude that, even if\nthe long coverage range of a single \\gls{gw} makes the infrastructure able to\nserve several devices, the network must be carefully dimensioned to meet the\napplication requirements. The work presented in~\\cite{georgiou2017low} is one of\nthe first to address the issue of scalability, using stochastic geometry to\nmodel interference in a LoRaWAN network. However, the study considers scenarios\nwith only \\gls{ul} traffic.\nIn~\\cite{sorensen2017analy} instead, queueing theory is applied to model latency\nand throughput of an \\gls{ed} subject to \\gls{dc} constraints, again focusing on\n\\gls{ul} communication only. The authors of~\\cite{bankov2017mathem,\n  bankov2019lorawan} provide a model based on Poisson arrival processes which\ntakes \\gls{dl} communications, re-transmissions and capture effect into account.\nHowever, the analysis holds only in limited-size networks, where nodes can\nemploy any transmission rate and their received powers are similar. The authors\nof~\\cite{croce2019lora} consider the features of the technology at the \\gls{phy}\nlayer, by focusing on the capture effect and imperfect orthogonality between\n\\glspl{sf}: after performing empirical measurements, they model these effects\nand derive the throughput achieved by the network for different cell\nconfigurations and number of \\glspl{gw}. In~\\cite{heusse2020capacity}, the\nproblem of network scalability is faced through mathematical modeling and\nPython-based simulations, taking into account also the capture effect, and\nevaluating the impact of \\gls{sf} allocation and power control. In all these\nworks, however, the main focus is on the \\gls{phy} layer, and downlink traffic\nand re-transmissions are not considered. Finally, the work presented\nin~\\cite{khan2019model} proposes a model to calculate energy consumption and\ndelay for reliable \\gls{ul} traffic in a LoRaWAN network. The results for a\nlimited number of devices are compared to real test-bed measurements and to the\noutcome of ns-3 simulations. The analysis, based on Markov-chain theory,\nneglects the \\gls{dc} constraints in the different sub-bands, and assumes that\n\\glspl{ack} are always sent in one specific receive window (either \\gls{rx1} or\n\\gls{rx2}). Markov chains are also proposed in~\\cite{delgado2021batteryless} to\ncharacterize the performance of a LoRaWAN battery-less device; however, the\nstudy considers a single device, and the network analysis is left for future\nwork.\n\nThe work presented in this paper is an extension of our previous conference\npaper~\\cite{capuzzo2018mathematical}, where we modeled a wide network with\nPoisson packet arrivals, considering the \\gls{dc} limitations and a set of\nnetwork parameters. Here we revise the model by developing a novel approach to\naccurately consider the limited availability of reception chains at the \\gls{gw},\nthe peculiarities of the two receive windows, and the \\gls{dc} constraints.\nAdditionally, we include packet re-transmissions and the capture\neffect. Compared to the state of the art, our model includes the ability to take\ninto account the coexistence of unconfirmed and confirmed traffic and, at the\nsame time, maintains the possibility of estimating the network behavior under\nseveral network configurations with minimal effort. The results obtained through\nthis model are compared with those given by a state-of-the-art and open source\nLoRaWAN simulator, presented in~\\cite{magrin2017performance}, further attesting\nthe accuracy of the proposed approach and exploring the impact of common\nassumptions. Finally, we also show some possible usages of the model to evaluate\na wide variety of network configurations with limited effort.\n\n\n\n\n\\section{Model}\n\\label{sec:model}\n\nThe aim of the model proposed in this paper is to characterize the behavior of a\nLoRaWAN network with a single \\gls{gw}, which receives packets from a set of\n\\glspl{ed} and needs to reply in one of the two receive windows when an \\gls{ed}\nrequires confirmation. The system performance is assessed in terms of packet\nsuccess probability, following the approach used\nin~\\cite{capuzzo2018mathematical} and extending it with a more accurate\ncharacterization of the \\gls{gw} behavior. This performance metric is proxy to\nother fundamental metrics, such as throughput and network capacity, which can be\nstraightforwardly derived from it. The following sub-sections are structured as\nfollows. The reference scenario, model assumptions, system parameters and their\neffects are described in Sec.~\\ref{sec:scenario}, together with a brief\npresentation of the structure of the model and its underlying rationale;\nSec.~\\ref{sec:quantities}, then, describes some relevant quantities and\nparameters of the proposed model. We then delve into the analytical formulation\nby decoupling the analysis of the \\gls{ul} traffic\n(Sec.~\\ref{sec:ultrafficrates} and~\\ref{sec:phy-probs}) and \\gls{dl} messages\n(Sec.~\\ref{sec:acktransmission}), and derive the formulas for \\gls{dl} success\nprobabilities in Sec.~\\ref{sec:succprobs}. Finally, Sec.~\\ref{sec:metrics},\ndescribes different performance metrics and their computation. Note that,\nbecause of the mutual dependency of some values, some terms may be described and\nintroduced before the corresponding equation can be derived, in which case\nreferences are provided in the text.\n\n\\subsection{Scenario and assumptions}\n\\label{sec:scenario}\n\nWe consider a scenario where the \\glspl{ed} are randomly and uniformly\ndistributed around a single \\gls{gw}. Application-layer packets are generated\naccording to a Poisson Process with aggregate packet generation rate $\\lambda$\n[pck\/s], and can be either confirmed or unconfirmed.\n\nFor tractability, we assume perfect orthogonality between different \\glspl{sf},\ni.e., only packets employing the same \\gls{sf} can collide. In this case, one of\nthe two packets can survive if its received power is sufficiently higher than\nthat of the colliding packet (collisions with more than two packets happen with\nnegligible probability and are not considered). While the orthogonality\nassumption has been shown to have an impact on the \\gls{phy}-layer performance\nof \\gls{ul} only traffic~\\cite{croce2018impact}, the results discussed in\nSec.~\\ref{sec:results} show that the effect is much more limited in the presence\nof confirmed traffic, where the performance is severely limited by other\nfactors.\n\n\\begin{figure}[t]\n  \\centering \\input{figures\/diagram}\n  \\caption{Representation of the model's packet filtering structure. $R^{phy}$ is\n    the rate of \\gls{ul} traffic (see~\\eqref{eq:rphytot}), while $r^1$ and\n    $r^2$ represent the rate of \\glspl{ack} sent in \\gls{sb1} and \\gls{sb2},\n    respectively (see~\\eqref{eq:r1},~\\eqref{eq:r2}).}\n  \\label{fig:diagram}\n  \\vspace{-1em}\n\\end{figure}\n\nFig.~\\ref{fig:diagram} shows the structure of the packet reception model,\nconsisting in successive filtering of Poisson processes. At the base of the\nfigure, arrows are used to represent the \\gls{ul} traffic generated by the\n\\glspl{ed}, including both new packet transmissions and re-transmissions of\nfailed packets. This process is assumed to be Poisson for tractability,\nignoring the fact that re-transmissions of a certain packet are correlated\nin time because of \\gls{dc} limitations. An initial filtering of this\nprocess excludes some arrivals, modeling packet losses due to interference\nfrom other \\glspl{ed}, unavailability of \\gls{gw} demodulators, or ongoing\n\\gls{dl} transmissions from the \\gls{gw}. This yields a process with a\nreduced rate, which now represents the packets that are correctly received\nby the \\gls{gw}.\n\nWhen the received \\gls{ul} message requires confirmation, an \\gls{ack} must be\nsent by the \\gls{gw} during one of the two receive windows of the target\n\\gls{ed}. The ability of the \\gls{gw} to perform such a transmission is modeled\nthrough two independent alternating renewal processes, in which the system\nalternates between the ON and OFF states. The two processes represent the\nopportunity of sending the \\gls{ack} in \\gls{rx1} or \\gls{rx2}, respectively,\nwhich are opened on \\gls{sb1} or \\gls{sb2}, i.e., on the shared or dedicated\nsub-band. If a confirmed packet finds a process in the ON state, it means that\nthe \\gls{gw} will be able to send an \\gls{ack} in that sub-band. In this case,\nthe process will switch to the OFF state to model the unavailability of that\nsub-channel for a certain period of time following the ACK transmission, due to\nthe DC restrictions.\n\nSince the sub-bands are disjoint, we assume that the two processes are\nuncorrelated, neglecting the fact that the very packets that need to be served\nin \\gls{sb2} are those that found \\gls{sb1} in the OFF state. If\nthe \\gls{dl} packet finds at least one of the two processes in the ON state, an\n\\gls{ack} is sent. If the \\gls{ack} is sent on \\gls{sb1} (hence, using frequencies\nshared by \\gls{ul} and \\gls{dl} traffic), it can be destroyed by the interference created by other \\glspl{ed}.\nIf the \\gls{ack} is sent on \\gls{sb2}, instead, it is assumed to be\nalways successful.\n\nFor the sake of clarity, the following list describes some examples of the\nlife cycle of the packets in Fig.~\\ref{fig:diagram}:\n\\begin{enumerate}[(A)]\n\\item This packet is lost because of interference or \\gls{gw} transmission\n  or unavailability of demodulators. Hence, it does not pass the first\n  filter.\n\\item This is an unconfirmed \\gls{ul} packet, which is successfully\n  received by the \\gls{gw}. It does not generate any \\gls{ack}.\n\\item This is a confirmed packet successfully received by the \\gls{gw}. It\n  generates an \\gls{ack}, which finds the \\gls{sb1} process in the ON state. The\n  \\gls{ack} is successfully sent, and the \\gls{sb1} process switches to the\n  OFF state.\n\\item This is another confirmed packet which is successfully received by\n  the \\gls{gw}. Since the \\gls{gw} has just sent an \\gls{ack} for packet\n  (C), it cannot reply in \\gls{sb1} due to \\gls{dc} constraints; \\gls{sb2}\n  is however in the ON state, and the \\gls{gw} can thus reply to the\n  \\gls{ed}, making the second process switch to the OFF state.\n\\item This is another confirmed packet, which gets a treatment similar to\n  that of packet (D). However, since the \\gls{gw} has transmitted the\n  \\gls{ack} for packet (D) and is still under the \\gls{dc} constraints, it\n  cannot reply to packet (E) in either of the two receive windows (both\n  \\gls{sb1} and \\gls{sb2} processes are in the OFF state). The \\gls{dl}\n  packet is hence discarded, and the \\gls{ed} will re-transmit the \\gls{ul}\n  message at a later time.\n\\end{enumerate}\n\n\\subsection{Model Quantities}\\label{sec:quantities}\n\nOur model offers some tunable parameters to increase its flexibility,\nenabling the evaluation of the network performance in various\nconfigurations with minimal effort. The model makes it possible to specify\nthe following values:\n\\begin{itemize}\n  \\item $\\mathcal{SF}=\\{7,\\ldots,12\\}$ indicates the set of all SFs.\n  \\item $\\alpha$: fraction of application-layer traffic requiring confirmation;\n  \\item $p^u_{i}, p^c_{i}$: fraction of devices generating unconfirmed and\n    confirmed traffic with a specific SF $i \\in \\mathcal{SF}$, respectively.\n    Note that $\\sum_{i \\in \\mathcal{SF}} p_i^u = \\sum_{i \\in \\mathcal{SF}} p_i^c\n    = 1$;\n  \\item $h$: number of times an application-layer unconfirmed packet is\n        transmitted;\n      \\item $m$: maximum number of transmission attempts for confirmed packets;\n      \\item $\\delta_{SB1}$ and $\\delta_{SB2}$: ratio between silent time and\n        transmission time in SB$k$, corresponding to the \\gls{dc} constraint.\n        For instance, in Europe, we have $\\delta_{SB1}=99$ and $\\delta_{SB2}=9$\n        corresponding to a \\gls{dc} of 1\\% in SB1 and 10\\% in SB2. In general,\n        when $\\delta_{SBk}>0$ the \\gls{dc} constraint applies to all devices\n        transmitting in subchannel SB$k$. Instead, the setting $\\delta_{SB_k}=0$\n        corresponds to a DC constraint of 100\\%, which means that there is no\n        limitation on the transmission time\\footnote{This setting is not allowed\n          by current RF recommendations but is considered in this study to gain\n          insights on the impact of \\gls{dc} limitations in the considered\n          scenarios.};\n      \\item $\\tau_1$ and $\\tau_2$: prioritization flags. If $\\tau_k=0$, the\n        \\gls{gw} prioritizes reception operations over transmission during the\n        $k$-th receive window, with $k=1, 2$. In this case, the \\gls{gw} will\n        drop any DL message that needs to be transmitted while a \\gls{ul}\n        reception is ongoing. Instead, if TX is prioritized ($\\tau_k = 1$), the\n        reception of any incoming packet will be interrupted in order to send\n        the \\gls{ack};\n  \\item $C$: number of \\gls{ul} frequency channels. Note that each \\gls{ul}\n        channel can also be used for \\gls{dl} transmissions. Instead, the\n        channel in \\gls{sb2} is \\gls{dl} only;\n  \\item $T_{i}^{ack_2}$: duration of the transmission of the \\gls{ack} in\n        \\gls{rx2} when using \\gls{sf} $i$. (The standard requires the use of SF\n        12 in \\gls{rx2} as a pre-configured setting, corresponding to\n        $T_{12}^{ack_2}$. Note that this default setting can be changed by the\n        NS, and accordingly in our model.)\n      \\item $T_i^{data}$ and $T_i^{ack_1}$ indicate the time durations of a data\n        packet and of an \\gls{ack} transmitted in \\gls{sb1} with \\gls{sf} $i$,\n        respectively. If \\glspl{ack} transmitted in \\gls{sb2} use \\gls{sf}12,\n        irrespective of the \\gls{sf} employed in the \\gls{ul} transmission, then\n        $T_i^{ack_2} = T_{12}^{ack_1}, \\: \\forall i \\in \\mathcal{SF}$.\n \n \n \n \n \n \n\\end{itemize}\n\n\nIn the formulas, the notation generally respects the following scheme. The\nprobability is indicated with $S$ or $F$ if it corresponds to a ``success''\nor ``failure'' event, respectively; if this rule does not apply, the probability\nis denoted simply as $P$. The superscript indicates the considered event, while\nthe subscript the \\gls{sf}. For example, in~\\eqref{eq:Sint}, the symbol\n$S_i^{INT}$ represents the probability of successfully surviving interference\nwhen using \\gls{sf}~$i$. Different uses of the notation are specified in the\ntext. The following sections provide a mathematical formulation for some\nrelevant quantities in this model.\n\n\\subsection{Uplink traffic rates}\n\\label{sec:ultrafficrates}\n\nThe assumption of perfect orthogonality between different \\glspl{sf} makes it\npossible to split the network traffic in different logical channels that do not\ninterfere with each other. The traffic load for each \\gls{sf} $i$ is split\nuniformly over the given $C$ frequency channels (since \\glspl{ed} pick a random\n\\gls{ul} frequency for each transmission attempt). Thus, the traffic generated\nat the application layer by the \\glspl{ed} using confirmed and unconfirmed\nmessages is, respectively, given by:\n\\begin{align}\n  \\label{eq:RateApp}\n  R_i^{c, app} &= \\frac{p_i^c \\cdot \\lambda}{C} \\cdot \\alpha, \\\\\n  R_i^{u, app} &= \\frac{p_i^u \\cdot \\lambda}{C} \\cdot (1 - \\alpha).\n\\end{align}\n\nSince \\glspl{ed} using unconfirmed traffic will perform $h$ transmissions of\neach application-layer packet, the PHY rate of these devices can be computed as\n$R_i^{u, phy} = R_i^{u, app} \\cdot h$. For \\glspl{ed} transmitting confirmed\nmessages, instead, the number of re-transmitted packets depends on the success\nof both the \\gls{ul} transmission and the corresponding \\gls{ack}. We indicate\nas $P_{i,j}^{DL}$ the probability that a confirmed \\gls{ul} packet sent with\n\\gls{sf} $i$ is successfully received and acknowledged at the $j$-th\ntransmission attempt, which will be derived in~\\eqref{eq:psucc_dl}.\nTherefore, we have that the rate of confirmed packets transmitted at \\gls{sf}\n$i$, $R_i^{c,phy}$, is given by the product of the application-level rate,\n$R_i^{c,app}$, and the average number of times a confirmed packet is transmitted\nat the \\gls{phy} layer.\n\n\\begin{align}\n  \\label{eq:Rc}\n  \\begin{split}\n    R_i^{c, phy} = R_i^{c, app} \\Bigg[ &\\sum_{j = 1}^{m-1} j \\cdot\n    P_{i,j}^{DL}\n    + m \\left( 1 - \\sum_{j = 1}^{m-1}P_{i,j}^{DL} \\right) \\Bigg].\n  \\end{split}\n\\end{align}\nThe first summation in the square brackets of~\\eqref{eq:Rc} takes into account\ntransmissions that are successfully received before the $m$th attempt, while the\nsecond term considers the case when the packet is transmitted $m$ times\n(irrespective of whether the last transmission is successful or not).\n\nThe total traffic for a single frequency channel and for \\gls{sf}~$i$ is\ntherefore given by\n\\begin{equation}\n  \\label{eq:rphytot}\n  R_i^{phy} = R_i^{u, phy} + R_i^{c, phy}.\n\\end{equation}\n\nIn general, the distribution of the \\glspl{sf} for the transmitted packets at the\n\\gls{phy} layer will differ from the native distribution of \\glspl{sf} among the\ndevices, \\{$p_i^u, p_i^c$\\}, because of re-transmissions. Thus, we\ndefine\n\\begin{equation}\n  \\label{eq:d}\n  d_i = \\frac{R_i^{phy}}{\\sum_j R_j^{phy}},\n\\end{equation}\nas the ratio of \\gls{phy} layer packets that are transmitted at \\gls{sf} $i \\in\n\\mathcal{SF}$.\n\n\n\n\n\\subsection{PHY layer probabilities}\n\\label{sec:phy-probs}\n\nA \\gls{ul} packet is successfully received by the \\gls{gw} if all the following\nconditions are met: (i) it does not overlap with another \\gls{ul} transmission\nusing the same \\gls{sf} on the same frequency, or it overlaps with another\n\\gls{ul} packet, but the received power is sufficiently large to allow for\ncorrect decoding despite the interference (capture), (ii) it does not overlap\nwith a \\gls{gw} \\gls{dl} transmission in any channel, and (iii) it finds an\navailable demodulator. These conditions are represented by the first filter in\nFig.~\\ref{fig:diagram}.\n\nSince packets are generated following a Poisson process, the probability of\nevent (i) is given by two components. The first is the probability that there\nare no other arrivals during the $2T_i^{data}$ vulnerability period across the\npacket arrival instant. The second, considers a collision with one packet, and\nthe fact that the receiver successfully captures the frame. For the \\gls{ul}, we\nconsider the capture probability $ \\mathbb{W}^{GW}$ as computed\nin~\\cite{bankov2017mathem}. Since these two events are disjoint, the probability\nof surviving interference (event (i)) is given by the sum of the two components,\nwhich results in\n\\begin{equation}\n  \\label{eq:Sint}\n  S_i^{INT} = e^{-2T_i^{data}R_i^{phy}} +\n  2T_i^{data}R_i^{phy}e^{-2T_i^{data}R_i^{phy}} \\cdot \\mathbb{W}^{GW},\n\\end{equation}\nwhere, in the right-most term, we computed the probability that either of the\ntwo colliding packets is captured (collision events with more than two packets\nare neglected).\n\n\n\n\n\n\nTo compute the probability of event (ii), we observe that a \\gls{ul} message is\nalways lost when it arrives at the \\gls{gw} during the transmission of an\n\\gls{ack}. Otherwise, the \\gls{gw} will start the reception of the \\gls{ul}\nmessage, which will take a time $T_i^{data}$. If reception on SB$k$ is\nprioritized (i.e., $\\tau_k=0$), this process cannot be interrupted, and the\n\\gls{ul} message will be successfully delivered to the \\gls{ns}. Conversely, if\n$\\tau_k=1$, i.e., we prioritize transmission on SB$k$, the reception of the\n\\gls{ul} packet may be aborted at any time during the period $T_i^{data}$, in\norder to give priority on the \\gls{ack} transmission. Therefore, the\nvulnerability period is given by the \\gls{ack} transmission time\n$T_{s}^{ack_k}$, to which we need to add the interval $T_i^{data}$ only if\n$\\tau_k=1$. Denoting by $b_s^k$ the probability that an \\gls{ack} is transmitted\non SB$k$ with \\gls{sf} $s\\in \\mathcal{SF}$ (which will be derived later\nin~\\eqref{eq:bk}), the average vulnerability period is then given by\n$\\overline{T_k} = \\sum_{s\\in \\mathcal{SF}} b_s^k T_s^{ack_k} + T_i^{data}\\cdot\n\\tau_k$. Now, according to the Poisson Arrivals See Time Averages (PASTA)\nproperty, the probability that a \\gls{ul} packet arrival falls in the\nvulnerability period of channel SB$k$, with $k=1,2$, can be expressed as\n\\begin{equation}\n  \\label{eq:f}\n  F_{i}^{TXk}=\\frac{\\sum_{s\\in \\mathcal{SF}}{b_s^k T_s^{ack_k}}+ T_i^{data} \\cdot \\tau_k}{E_{ON}^k+E_{OFF}^k},\n\\end{equation}\nwhere the denominator is the mean renewal time of the SB$k$ process, given\nby the sum of $E_{ON}^k$ and $E_{OFF}^k$, i.e., the expected times the SB$k$\nprocess spends in the ON and OFF states during a renewal period (ON-OFF cycle),\nwhich will be computed in~\\eqref{eq:eon} and~\\eqref{eq:eoff}. Then, assuming\n(for ease of analysis) that events in \\gls{sb1} and \\gls{sb2} are independent,\nthe probability that a \\gls{ul} packet is successfully received (event (ii)) is\ngiven by\n\\begin{equation}\n  \\label{eq:Sitx}\n  S_i^{TX} =(1-F_i^{TX1})(1-F_i^{TX2}).\n\\end{equation}\n\nNext, we compute the probability of event (iii), i.e., that at least one\ndemodulator out of 8 is available. Each demodulator chain is modeled through an\nalternating renewal process, where the demodulator can be in an ``available''\nstate $A$, when idle or in a ``locked'' state $L$, when occupied with the\nreception of another signal. We assume that the different demodulators are\nactivated in succession: if all are available, an incoming signal will be\nreceived by the first demodulator; if the first demodulator is in the $L$ state,\nthe packet will be handled by the second demodulator, and so on. Let $E^L$ be\nthe expected time a demodulator will be locked on a incoming signal. Since the\noccupation will last for the duration of \\gls{ul} LoRa packets at the \\gls{phy}\nlayer, we have:\n\\begin{equation}\n  \\label{eq:el}\n  E^L = \\sum_{i \\in \\mathcal{SF}}d_i \\cdot T_i^{data}.\n\\end{equation}\nThe average time the first demodulator is in the $A$ state, instead, is\ncomputed as the average inter-arrival time of \\gls{ul} packets, regardless\nof their \\gls{sf} and selected frequency:\n\\begin{equation}\n  E^{A, 1} = \\frac{1}{C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}}.\n\\end{equation}\nThen, the process of packets that require the second demodulator is filtered by\nthe probability of finding the first demodulator occupied. Thus, the expected\ntime the second demodulator is available is given by\n\\begin{equation}\n  E^{A, 2} = \\frac{E^{A,1}}{P^{L,1}} = \\frac{1}{P^{L, 1} \\cdot C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}},\n\\end{equation}\nwhere $P^{L,1}$ is the probability that the first modulator is in the $L$ state\n(see~\\eqref{eq:pl1}).\nWith a similar reasoning, we compute the expected time for which the $j$-th\ndemodulator is available as\n\\begin{equation}\n  E^{A, j} = \\frac{E^{A,j-1}}{P^{L,j-1}} = \\frac{1}{\\prod_{\\ell=1}^{j-1}P^{L, \\ell} \\cdot C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}}.\n\\end{equation}\nThe probability $P^{L,\\ell}$ of finding the $\\ell$-th demodulator in the $L$ state, in turn, can be expressed as\n\\begin{equation}\n  \\label{eq:pl1}\n  P^{L, \\ell} = \\frac{E^{L}}{E^{A, \\ell} + E^{L}}.\n\\end{equation}\n\nThen, a packet finds an available demodulator (event (iii)) with probability:\n\\begin{equation}\n  \\label{eq:Sdemod}\n  S^{demod} = 1 - \\prod_{j=1}^8 P^{L, j}.\n\\end{equation}\n\nThe overall \\gls{ul} packet success probability, considering events (i), (ii)\nand (iii) described above, is finally expressed as\n\\begin{equation}\n  \\label{eq:Sul} S_i^{UL} = S_i^{INT} \\cdot S_i^{TX} \\cdot S^{demod} .\n\\end{equation}\n\n\\subsection{\\gls{ack} transmission}\n\\label{sec:acktransmission}\nOnce a confirmed packet is correctly received by the \\gls{gw}, an \\gls{ack}\nneeds to be transmitted back to the \\gls{ed}. Eq.~\\eqref{eq:Sul} gives the\nprobability of successful packet reception at the \\gls{gw}. Therefore, the rate\nof \\gls{ack} messages that the \\gls{gw} will try to send in \\gls{sb1} is:\n\\begin{equation}\n  \\label{eq:r1}\n  r_i^1 = R_i^{c, phy} \\cdot S_i^{UL}.\n\\end{equation}\n\\begin{figure}[t]\n  \\centering\n  \\input{figures\/dldiagram.tex}\n  \\caption{Diagram for successful \\gls{ack} reception.}\n  \\label{fig:dldiagram}\n  \\vspace{-1em}\n\\end{figure}\nA visual representation of the possible \\gls{ack} life cycles considered in the\nmodel is shown in Fig.~\\ref{fig:dldiagram}. Labels refer to the probabilities of\nthe different events, which we derive next. In general, an \\gls{ack} is\ntransmitted in SB$k$ if both the following conditions hold: (i) $\\tau_k=1$ (TX\nis prioritized) or $\\tau_k=0$ and the GW is idle; (ii) SB$k$ is available (i.e.,\nnot blocked by DC constraints). If either condition is not satisfied, the\n\\gls{ack} is dropped.\n\nLet $T$ denote the event ``the \\gls{gw} \\textit{may} transmit,''\nwhich depends on the TX\/RX prioritization policy. If $\\tau_k=1$, the\n\\gls{gw} can transmit the \\gls{dl} packet whenever it needs to; otherwise,\nif $\\tau_k=0$, the \\gls{gw} can transmit in SB$k$ only if no reception is\nongoing. We denote by $P^{T,k}$ the probability of $T$, which can be computed as\n\\begin{equation}\n  \\label{eq:Pnorx}\n  P^{T, k} =\n  \\begin{cases}\n    1, & \\textrm{if $\\tau_k$ = 1}; \\\\\n    e^{-\\sum_{i \\in \\mathcal{SF}} C \\cdot R_i^{phy} T_i^{data}}, &\n    \\textrm{if $\\tau_k$ = 0};\n  \\end{cases}\n\\end{equation}\nwhere the second expression is the probability that no \\gls{ul} packet was\ngenerated in the last $T^{data}_i$ seconds.\n\nIf \\gls{sb1} is not available, the \\gls{gw} will try to process the\n\\gls{ack} in \\gls{sb2}. Such packets form a process with rate\n\\begin{equation}\n  \\label{eq:r2}\n  r_i^2= r_i^1 [P^{OFF, 1} + P^{ON, 1}(1 - P^{T, 1})],\n\\end{equation}\nwhere $P^{ON, 1}$ and $P^{OFF, 1}$ are the probabilities of finding \\gls{sb1}\nin the ON and OFF state, respectively, and $(1 - P^{T, 1})$ is the\nprobability that the \\gls{gw} is not available for \\gls{dl} transmission.\nThe ON and OFF probabilities for the SB$k$ process, with $k=1,2$, are given\nby\n\\begin{align}\n  \\label{eq:Ponoff}\n  P^{ON, k}  &= \\frac{E^{ON, k}}{E^{ON, k} + E^{OFF ,k}}, \\\\\n  P^{OFF, k} &= \\frac{E^{OFF, k}}{E^{ON, k} + E^{OFF, k}},\n\\end{align}\nwhere $E^{ON,k}$ and $E^{OFF,k}$ are the mean sojourn times in ON and OFF\nstates, respectively, which are computed as follows.\nBy considering the arrival rate of successful \\gls{ul} packets in the\n$k$-th sub-band, we have:\n\\begin{align}\n  \\label{eq:eon}\n  \\begin{split}\n    E^{\\mathrm{ON}k} &= \\frac{1}{\\sum_{i \\in \\mathcal{SF}} C \\cdot r_i^k}.\n  \\end{split}\n\\end{align}\nNote that the switch from the ON to the OFF state will be caused by a packet\nsent in any of the $C$ \\gls{ul} channels: therefore, we need to multiply the\nrates $r_i^k$ of arrivals to SB$k$ with SF $i$ by the number of available\nchannels.\n\nIn order to compute the expected duration of the OFF periods, we first need\nto derive the probability distribution $b_i^k$ of the \\glspl{sf} used for\n\\gls{ack} transmissions, which is given by\n\\begin{equation}\n  \\label{eq:bk}\n  b_i^k = \\frac{r_i^k}{\\sum\\limits_{s \\in \\mathcal{SF}}r_s^k}.\n\\end{equation}\nIn our model, the OFF period accounts for the time the \\gls{gw} is\nprevented from transmitting a new data packet, which includes the time to\nsend the ACK using the given \\gls{sf}, plus the waiting time imposed by the\n\\gls{dc} limitations. We hence have\n\\begin{align}\n  \\label{eq:eoff}\n  \\begin{split}\n    E^{\\rm OFF, 1} &= \\sum_{s \\in \\mathcal{SF}} b_s^1 (T_s^{\\rm ack_1} + \\delta_{SB1} \\cdot T_s^{\\rm ack_1}), \\\\\n    E^{\\rm OFF, 2} &= \\sum_{s \\in \\mathcal{SF}} b_s^2 (T_s^{\\rm ack_2} + \\delta_{SB2} \\cdot\n    T_s^{\\rm ack_2}).\n  \\end{split}\n\\end{align}\n(Note that, by including the parameter $\\delta_{SBk}$ as defined in\nSec.~\\ref{sec:scenario}, we can change the \\gls{dc} limitations in the $k$-th\nsub-band, thus making it possible to analyze its impact.)\n\n\n\n\nFinally, we remark that \\gls{dl} packets sent by the \\gls{gw} in \\gls{sb1} also\nhave to avoid interference from other \\glspl{ed}. In the absence of collisions,\nthe vulnerability period is given by the sum of two terms. The first term\ncorresponds to the case of no \\gls{ul} transmissions starting while the \\gls{dl}\npacket is being sent ($T^{ack_1}$); the second term represents the event where\nno \\gls{ul} transmissions started before the \\gls{ack} is sent. Note that if\n$\\tau_1=0$ the second term is not present, since in that case the \\gls{ack}\nwould not be generated at all. Furthermore, an \\gls{ack} can survive an\ninterfering packet sent by another \\gls{ed} in case of capture, which happens\nwith probability $\\mathbb{W}^{ED}$ (equivalent to the $\\mathbb{W}^{Mote}$ as\nderived in~\\cite{bankov2017mathem}). Therefore, the probability that the\n\\gls{ack} does not collide with a \\gls{ul} packet in \\gls{sb1}, or is captured\ndespite the collision, is equal to\n\\begin{equation}\n  \\label{eq:SintAck}\n  S_i^{INT, ack_1} =\n  e^{-R_i^{phy} (T_i^{ack_1} + \\tau_1 \\cdot T_i^{data})} +\n  R_i^{phy} (T_i^{ack_1} + T_i^{data}) \\cdot e^{-R_i^{phy} (T_i^{ack_1} + T_i^{data})} \\cdot \\mathbb{W}^{ED}.\n\\end{equation}\nFor packets sent in \\gls{sb2}, instead, the reception is assumed to\nbe always successful, since the 869.525~MHz channel is dedicated to \\gls{dl}\ncommunication and the \\gls{gw} only transmits one packet at a time (note\nthat this assumption does not hold in the case of multiple \\glspl{gw}).\n\n\\subsection{\\gls{dl} success probability}\n\\label{sec:succprobs}\nGiven that a confirmed \\gls{ul} packet sent with \\gls{sf} $i$ has been\nsuccessfully received by the \\gls{gw}, the probability that the corresponding\n\\gls{ack} is also successfully returned to the \\gls{ed} is expressed as\n\\begin{equation}\n  \\label{eq:Sdl1ack} S_i^{\\rm DL} = S_{i}^{\\textrm{SB1}} +  S^{\\textrm{SB2}},\n\\end{equation}\nwhere $S_{i}^{\\textrm{SB1}} $ describes the probability of a successful\n\\gls{ack} transmission in \\gls{sb1} with \\gls{sf} $i$, while $S^{\\textrm{SB2}}$\naccounts for the probability that \\gls{sb1} is not available, and the \\gls{ack}\nis successfully sent in \\gls{sb2}. These probabilities, in turn, can be\nexpressed as follows:\n\\begin{align}\n  \\label{eq:SB1B2}\n  S_{i}^{\\textrm{SB1}} &=\\; P^{ON, 1} \\cdot P^{T, 1} \\cdot S_i^{INT, ack_1}, \\\\\n  S^{\\textrm{SB2}} &=\\;[P^{OFF, 1} + P^{ON, 1} \\cdot (1 - P^{T, 1}) ]\\cdot P^{ON, 2} \\cdot P^{T, 2}.\n\\end{align}\n\nFig.~\\ref{fig:dldiagram} can be used as a reference for the computation of\nthis quantity.\n\n\nFinally, we can compute the success probabilities over $m$ transmissions. We\nrecall that, for the sake of simplicity, we neglect the time correlation of\npacket re-transmissions due to \\gls{dc} constraints, (the impact of this\napproximation will be analyzed by simulation). We recall that $P_{i,j}^{UL}$\nindicates the probability that a \\gls{ul} packet with \\gls{sf} $i$ is\nsuccessfully received at the \\gls{gw} at exactly the $j$-th transmission\nattempt, which can be computed as:\n\\begin{equation}\n  \\label{eq:psucc_ul}\n  P_{i,j}^{UL} = S_i^{UL} \\left(1 - S_i^{UL}\\right)^{j - 1} .\n\\end{equation}\nThen, the \\gls{ed} successfully receives the \\gls{ack} at exactly the $j$-th\nattempt if both the \\gls{ul} and the \\gls{dl} transmissions succeed. The\nprobability $P_{i,j}^{DL}$ of this event is hence given by:\n\\begin{equation}\n  \\label{eq:psucc_dl}\n  P_{i,j}^{DL} = \\left[1 - (S_i^{UL}S_i^{DL})\\right]^{j - 1} \\cdot (S_i^{UL}S_i^{DL}).\n\\end{equation}\n\nOnce all intermediate quantities are computed, the model can be summarized\nby two inter-dependent equations:\n\\begin{equation*}\n  \\begin{cases*}\n    S^{UL} = f(S^{UL}, S^{DL}),\\\\\n    S^{DL} = g(S^{UL}, S^{DL}).\n  \\end{cases*}\n\\end{equation*}\nwhere $S^{UL}=[S^{UL}_7,\\ldots,S^{UL}_{12}]$ and\n$S^{DL}=[S^{DL}_7,\\ldots,S^{DL}_{12}]$, while $f()$ and $g()$ are implicit\nfunctions given by the chaining of the sequence of operations that\nyield~\\eqref{eq:Sul} and~\\eqref{eq:Sdl1ack}, respectively.\n\nThis system admits a fixed-point solution, which can be found through\nfixed-point iteration. From a practical perspective, when initialized with the\nstates $S^{UL} = S^{DL} = [1, 1, 1, 1, 1, 1]$, the iterative process has always\nreached convergence to the stable fixed point after a few iterations (order of\nfew units) for all the parameter combinations considered in this work. The proof\nof the system's convergence is provided in~\\cite{magrin2021proof}. An\nimplementation of the model, allowing the interested readers to easily replicate\nthe results shown in this paper, is publicly available\nat~\\cite{publishedmodelcode}.\n\n\n\n\n\n\\subsection{Performance metrics}\n\\label{sec:metrics}\nTo evaluate the system performance, we consider three classes of key performance\nSndicators, namely: reliability, delay, and fairness metrics which are better\ndetailed in the remainder of this section together with the methodology to\ndetermine their value using the proposed model. Once a set of parameters is\nfixed, the model can be solved and the performance metrics can be estimated\nstarting from $S^{UL}$ and $S^{DL}$. Conversely, it is possible to employ the\nmodel to optimize a given performance metric, finding the parameter setting that\nmaximizes it, as shown in Sec.~\\ref{sec:results}.\n\n\\subsubsection{Reliability Metrics}\n\nWe consider three \\gls{pdr} indexes, namely:\n\\begin{itemize}\n\\item \\textit{\\gls{uu}}: fraction of (application-layer) unconfirmed\n  packets that are successfully received by the \\gls{gw};\n\\item \\textit{\\gls{cu}}: fraction of (application-layer) confirmed packets that\n  are successfully received by the \\gls{gw}, irrespective of whether or not the\n  corresponding \\gls{ack} is successfully returned to the \\gls{ed};\n\\item \\textit{\\gls{cd}}: fraction of (application-layer) confirmed packets\n  that are successfully acknowledged by the NS.\n\\end{itemize}\nClearly, CD $\\leq$ CU, since a packet needs to be successfully received by\nthe \\gls{gw} in order to be acknowledged. Note that the \\gls{cu} metric\ncaptures the performance of applications for which it is important to\ndeliver packets to the \\gls{ns} and \\glspl{ack} are only used to stop\nre-transmissions (and thus avoid a useless increase in traffic), while\n\\gls{cd} is more interesting for applications that require the \\glspl{ed}\nto get explicit feedback from the \\gls{ns}, for instance containing control\ninformation addressed to the \\gls{ed}.\n\n\n\nWe obtain the \\gls{uu} and \\gls{cu} values by averaging the \\gls{ul} success\nprobability ($UU_i$ and $CU_i$ for unconfirmed and confirmed packets,\nrespectively) for each  \\gls{sf} $i$ over the \\gls{sf} distribution, i.e.,\n\\begin{equation}\n  \\label{eq:uu}\n  {\\rm UU} = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^u \\cdot \\mathrm{UU}_i\\right) = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^u \\cdot \\sum\\limits_{j=1}^h P_{i, j}^{UL}\\right),\n\\end{equation}\n\\begin{equation}\n  \\label{eq:cu}\n  {\\rm CU} = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^c \\cdot \\mathrm{CU}_i \\right) = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^c \\cdot \\sum\\limits_{j=1}^mP_{i, j}^{UL}\\right).\n\\end{equation}\n\nSimilarly, \\gls{cd} is computed as the probability of success for a\nconfirmed packet within the available re-transmission attempts\n\\begin{equation}\n  \\label{eq:cd}\n  {\\rm CD} = \\sum\\limits_{i \\in \\mathcal{SF}} \\left( p_i^c \\cdot \\sum\\limits_{j=1}^mP^{DL}_{i,j}\\right).\n\\end{equation}\n\n\\subsubsection{Delay Metrics}\nWe define two delay metrics, considering confirmed traffic only: $\\Delta^{\\rm\n  UL}$ measures the time from the first transmission attempt to the successful\ndelivery to the \\gls{gw} of an \\gls{ul} confirmed packet, while $\\Delta^{\\rm\n  DL}$ accounts for the time from the first transmission of a confirmed packet\nto the successful reception of the corresponding reply. Delays are computed for\nsuccessful packets only, and the propagation delay is assumed to be negligible.\nTo compute these metrics with our model, we assume the RETRANSMIT\\_TIMEOUT value\nto be a uniformly distributed random variable with mean $\\mu$, and consider that\n\\glspl{ed} employ the shared sub-band with $\\delta_{SB1}$ \\gls{dc} limitations.\nTherefore, the average time between two transmissions of the same MAC-layer\npacket by a device is given by:\n\\begin{equation}\n  \\label{eq:intertranmissionTime}\n  \\gamma_i = (\\delta_{SB1} + 1) \\cdot T_i^{data} + \\mu.\n\\end{equation}\n\nThe average delay from the successful reception of a packet at the \\gls{gw}\nto the transmission of the \\gls{ack} is given by:\n\\begin{equation}\n  \\label{eq:avgAckTransmissionTime}\n  \\phi_i = S_i^{\\rm SB1} \\cdot (1 + T_i^{ack_1}) + S^{\\rm SB2} \\cdot (2 + T_i^{ack_2}),\n\\end{equation}\nwhere we take into account that the \\gls{ack} will be served in SB1 (opened\nafter 1 second) with probability $S_i^{\\rm SB1}$, and in SB2 (opened after 2\nseconds) with probability $S^{\\rm SB2}$.\n\nIf a packet is re-transmitted $m$ times, each re-transmission $j$ is\nassociated with a certain \\gls{ul} success probability $P_{i,j}^{\\rm UL}$.\nThe average delay at each \\gls{sf} $i \\in \\mathcal{SF}$ can be computed as:\n\\begin{equation}\n  \\label{eq:uldelay}\n  \\Delta^{\\rm UL} = \\sum_{i\\in\\mathcal{SF}} p_i^c \\cdot \\left(  \\sum_{j=1}^m \\bar{P}_{i,j}^{\\rm UL} \\left(T_i^{data} + (j-1) \\cdot \\gamma_i\\right)\\right),\n\\end{equation}\nwhere we define $\\bar{P}_{i,j}^{\\rm UL} = P_{i,j}^{\\rm\n  UL}\/\\sum_jP_{i,j}^{\\rm UL}$ to obtain the distribution of successful\n\\gls{ul} packet transmissions.\n\nSimilarly, we can compute the average \\gls{ack} delay:\n\\begin{equation}\n  \\label{eq:dldelay}\n  \\Delta^{\\rm DL} = \\sum_{i\\in\\mathcal{SF}} p_i^c \\cdot \\left( \\sum_{j=1}^m \\bar{P}_{i,j}^{\\rm DL} \\left(T_i^{data} + (j-1) \\cdot \\gamma_i + j \\cdot \\phi_i\\right) \\right),\n\\end{equation}\nwhere, in addition to the inter-transmission time between two packets, we\nalso account for the time to perform the \\gls{ack} transmission.\n\n\n\n\n\n\\subsubsection{Fairness}\nFinally, we consider the fairness of the system in different scenarios. Indeed,\n\\glspl{ed} employing confirmed traffic or higher \\glspl{sf} will use more\nsystem resources (e.g., channel occupancy), possibly affecting the application\nperformance of devices that employ different settings.\nTo this aim, we use Jain's fairness index, defined as\n\\begin{equation}\n  \\label{eq:j}\n  J(\\mathbf{x}) = \\frac{\\Big(\\sum_{i=1}^{n}x_i\\Big)^2}{n \\cdot \\sum_{i=1}^{n}x_i^2},\n\\end{equation}\nwhere $n$ is the total number of user categories, each with throughput\n$x_i$. Note that $1\/n \\leq J(\\mathbf{x}) \\leq 1$, and the system is perfectly fair if\n$J(\\mathbf{x}) = 1$. In particular, in the following section, we will consider\nthe fairness among devices employing different \\glspl{sf}. Furthermore, since\nall the devices have equal packet generation rate, and transmit packets with the same length, instead of the throughput\nwe can simply consider the \\gls{ul} success probability, i.e., \\gls{uu} for\nnodes employing unconfirmed traffic and \\gls{cu} for devices transmitting confirmed\nmessages. Therefore, the fairness is computed by taking\n$\\mathbf{x} = [\\mathbf{x^u}, \\mathbf{x^c}]$, where the elements correspond to\n$x_i^u = UU_i$, and $x_i^c = CU_i$, as defined in~\\eqref{eq:uu},~\\eqref{eq:cu}.\n\n\\section{Network Simulations}\n\\label{sec:simulation}\n\nIn order to validate our model, we compared the performance estimates obtained\nfrom the model with those observed in more realistic simulations, in which most\nof the simplifying assumptions of the model are removed.\n\nThis section describes how we employ the LoRaWAN ns-3 module described\nin~\\cite{magrin2020thorough} to perform such a validation. To be noted that the\nmore accurate modeling of the LoRaWAN standard considered in the simulator comes\nat the cost of a much larger computational time to assess the system\nperformance. Indeed, for the same parameter set, the performance evaluation is\nbasically instantaneous when employing the theoretical model, while each ns-3\nsimulation run takes in the order of tens of seconds, with execution times\nrapidly increasing when the traffic load, the number of devices and the number\nof required randomized runs grow.\n\nThe merit of the simulator is that it strives to be as realistic as possible,\nalso taking into account some factors that are overlooked by the model for\ntractability reasons. For instance, the assumption of perfect orthogonality\nbetween transmissions employing different \\glspl{sf} is removed, and the\nsimulator relies on the link-level model provided\nin~\\cite{goursaud2015dedicated} to determine the actual reception probability in\ncase of overlapping transmissions, which also accounts for the capture effect.\n\n\\begin{table}[t]\n  \\footnotesize\n  \\centering\n  \\caption{Values of $T^{data}$, $T^{ack}$ and SF distributions $p$.\n     Payload of data packets is 10 bytes; \\glspl{ack} have no\n    payload.}\n  \\begin{tabular}[c]{ccccc}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\toprule\n    SF & $T^{data}$ [s] & $T^{ack}$ [s] & $p_{\\rm equal}$ & $p_{\\rm EXPLoRa}$\\\\\n    \\midrule\n    7  & 0.051 & 0.041 & 0.166 & 0.487 \\\\\n    8  & 0.102 & 0.072 & 0.166 & 0.243 \\\\\n    9  & 0.185 & 0.144 & 0.166 & 0.135 \\\\\n    10 & 0.329 & 0.247 & 0.166 & 0.076 \\\\\n    11 & 0.659 & 0.495 & 0.166 & 0.038 \\\\\n    12 & 1.318 & 0.991 & 0.166 & 0.019 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\label{tab:resparams}\n\\end{table}\n\nThe simulation setting is as follows.\n\\begin{itemize}\n  %\n\\item \\textit{Traffic load} -- The number of \\glspl{ed} is fixed to 1200, and the \\glspl{ed}' application layer is set to periodically generate packets to be transmitted by the MAC layer. The traffic load in the network is modified by varying the packet generation\n  period. It is to be noted that this periodic traffic generation pattern is likely more realistic than the Poisson traffic assumed in the model. Nonetheless, the good match of simulation and analytical results confirms that the Poisson assumption is valid when the number of nodes is sufficiently large.\n\n  %\n\\item \\textit{Channel allocation} -- We consider the typical frequency\n  allocation scheme for Europe, as reported in Tab.~\\ref{tab:channels}.\n  Therefore, the number of different frequency channels for \\gls{ul} is $C\n  = 3$.\n  %\n\\item \\textit{Duty cycle} -- The simulator considers the \\gls{dc} limitations applied in the\n  European region~\\cite{regional}, which corresponds to setting $\\delta_{SB1} = 99$ and $\\delta_{SB2} = 9$ in the model.\n\\item \\textit{Channel model} -- Differently from the model, simulated LoRaWAN\n  nodes experience a log-distance propagation path loss,\n  as for an open-air scenario. Thus, farther devices will suffer increased\n  loss, and their performance will be penalized with respect to \\glspl{ed} that\n  are close to the \\gls{gw}. Note that we do not include fast-fading components,\n  which are supposed to be averaged out by the LoRa modulation, nor\n  time-dependent variations in the channel, which remains constant throughout the\n  entire simulation. Also, the channel is assumed to be symmetric, and\n  \\gls{dl} transmissions will suffer the same impairments as in the \\gls{ul}.\n  %\n  \\item \\textit{\\gls{sf} distribution} -- \\glspl{ed} are located around the\n        single \\gls{gw} in a circular area of radius 2500~m, which allows for\n        communications with any SFs with negligible channel error probability\n        (in the absence of interference). Instead, the positions of the nodes\n        are randomly picked at each simulation run. \\glspl{sf} are assigned\n        uniformly (see Tab.~\\ref{tab:resparams}, $p_{\\rm equal}$). A different\n        \\gls{sf} distribution ($p_{\\rm EXPLoRa}$) is considered in some\n        scenarios, to evaluate the impact of this parameter on the different\n        metrics.\n  %\n  \\item \\textit{Interference and capture effect} -- To model interference, in\n        the simulator we consider the collision matrix provided\n        in~\\cite{goursaud2015dedicated} and the overlapping time between\n        packets, as described in~\\cite{magrin2017performance}.\\footnote{Note\n        that, in the simulator, the capture event is determined also considering\n        the partial overlapping of the colliding packets.} A packet survives\n        interference from a signal modulated with the same \\gls{sf} if its power\n        is at least $CR_{dB} = 6$~dB higher than the colliding one. In order to\n        provide a comparison with this scenario, in the analytical model we\n        leverage the assumption of uniformly distributed \\glspl{ed} around the\n        \\gls{gw} to compute the capture probabilities as\n        in~\\cite{bankov2017mathem}, which results in $\\mathbb{W}^{GW} = 0.1796$,\n        and $\\mathbb{W}^{ED} = 0.5682 $. We remark that different distributions\n        of \\glspl{ed} around the \\gls{gw} can be modeled by adapting this\n        derivation.\n\\end{itemize}\n\n\nSince the \\gls{gw} implementation in the simulator attempts to emulate the\nbehavior of a real device, a \\gls{ul} packet is successfully received when all the\nfollowing conditions are satisfied:\n\\begin{enumerate}\n  %\n\\item The packet finds an available demodulator;\n  %\n\\item The packet's reception is not interrupted by \\gls{dl} transmissions;\n  %\n\\item Once the reception is finished, the packet was not corrupted by\n  interference.\n  %\n\\end{enumerate}\n\nTo count packets at the \\gls{phy} layer coherently with the simulator\nimplementation, the model's packet loss probabilities due to lack of\ndemodulators ($F_{NMD}$), \\gls{gw} transmission ($F_{GWTX}$) and interference\n($F_{INT}$) are plotted in the following section using, respectively, the\nfollowing expressions:\n\\begin{enumerate}\n\\item $F_{\\rm NMD}= 1 - S^{demod}$;\n\\item $F_{\\rm GWTX} = E_{i}\\left[S^{demod} \\cdot (1 - S_i^{TX})\\right]$;\n\\item $F_{\\rm INT} = E_{i}\\left[S^{demod} \\cdot S_i^{TX} \\cdot (1 - S_i^{INT})\\right]$;\n\\end{enumerate}\nby exploiting~\\eqref{eq:Sint},~\\eqref{eq:Sitx}, and~\\eqref{eq:Sdemod}, and where\n$E_i\\left[\\cdot\\right]$ indicates the expectation over the distribution of\n\\glspl{sf} and $S^{demod}$ the probability that, in the simulations, a packet\ncan lock on an available demodulator.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nThis section provides a comparison between the performance estimated with the\nproposed model and by the ns-3 simulator. Results are presented for both\n\\gls{phy} and \\gls{mac} layer, and the impact of the model's assumptions is\nshown to be mostly negligible, or at least acceptable. Finally, some results\nwill showcase how the model can be used to gain insight on the behavior of the\nLoRaWAN technology in a quick and effortless way, analyzing the effects of\nvarious parameters on the performance of the network. In the plots of this\nsection the analytical results are represented by lines, while markers\ncorrespond to simulation outcomes.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/phy-confirmed.pdf}\n\t\\caption{PHY-level performance with $m=8$, $\\alpha=1$.}\n\t\\label{fig:phy}\n  \\vspace{-1em}\n\\end{figure}\nFig.~\\ref{fig:phy} shows the packet outcome probabilities at the \\gls{phy}\nlayer in a network employing confirmed traffic. Although obtained with\ndifferent approaches, such probabilities are overall consistent, proving\nthe effectiveness of the model.\n\n\\begin{figure}[t]\n\t\\centering\n  \\begin{subfigure}[t]{\\figurescaling\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{modelFigures\/cu-confirmed.pdf}\n    \\vspace{-0.5cm}\n    \\caption{CU for different values of $m$, $\\alpha=1$}\n    \\label{fig:cu}\n  \\end{subfigure}%\n  \\begin{subfigure}[t]{\\figurescaling\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{modelFigures\/cd-confirmed.pdf}\n    \\vspace{-0.5cm}\n    \\caption{CD for different values of $m$, $\\alpha=1$}\n    \\label{fig:cd}\n  \\end{subfigure}%\n\t\\caption{Comparison of model and simulation results in terms of CU and\n    CD.}\n\t\\label{fig:cucd}\n  \\vspace{-1em}\n\\end{figure}\n\nThe good match between model and simulation is also reflected in\nFig.~\\ref{fig:cucd}, which shows the \\gls{cu} and \\gls{cd} metrics for a network\nin which all \\glspl{ed} generate confirmed traffic ($\\alpha=1$), and for\ndifferent values of $m$. Also in this case, the model results are quite close to\nthose given by the simulations. Fig.~\\ref{fig:cu} shows that the number of\navailable transmissions helps the correct delivery of the message at the\n\\gls{mac} layer, providing performance above 0.9 also for relatively high\ntraffic levels, when an average of one packet per second is generated by the\nnetwork at the application layer. The \\gls{cd} performance shown in\nFig.~\\ref{fig:cd} exhibits a similar behavior, but reaches much lower values\nmostly because the rate of \\gls{dl} messages that the \\gls{gw} can generate is\nlimited by the \\gls{dc} restrictions. The fact that this loss in performance is\ncaused by the \\gls{gw}'s \\gls{dc} is confirmed by the lilac dash-dotted line in\nFig.~\\ref{fig:cd}: to obtain these results, the \\gls{dc} restrictions were\nlifted by setting $\\delta_{SB1} = \\delta_{SB2} = 0$ in the model, producing\nmarkedly better results when compared to the corresponding green curve, where\n\\gls{dc} is enabled. Another example of the model's flexibility in considering\nalso non-standard settings is given by the densely dash-dotted brown line, which\nrepresents the \\gls{cd} metric when $\\delta_{SB1} = \\delta_{SB2} = 9$, i.e.,\nwhen transmissions in both sub-bands are subject to a \\gls{dc} of 10\\%. Although\nbeing an ideal setting, this case shows that even a small increase in the\n\\gls{dc} allowance in SB1 can yield considerable performance gains.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/alpha.pdf}\n\t\\caption{Performance when varying the fraction of confirmed traffic, with\n    $\\lambda=1, m=8, h=1$.}\n\t\\label{fig:alpha}\n  \\vspace{-1em}\n\\end{figure}\nFig.~\\ref{fig:alpha} compares simulation and theoretical results, in terms of\n\\gls{uu}, \\gls{cu} and \\gls{cd}, when different fractions of confirmed traffic\nare employed in the network. For this comparison, we set the network application\nlayer packet arrival rate to $\\lambda=1$ pck\/s, the maximum number of\ntransmissions for confirmed traffic to $m=8$, and the number of repetitions for\nunconfirmed traffic to $h=1$. As the fraction of \\glspl{ed} employing confirmed\ntraffic increases, the data delivery performance decreases for all the\n\\glspl{ed}, in particular for nodes employing unconfirmed traffic which do not\nhave the chance of re-transmitting their packets. The match between the\nsimulator and the model is confirmed to be excellent for all values of $\\alpha$.\n\n\\begin{figure}[t]\n\t\\centering\n  \\begin{floatrow}\n  \\ffigbox{\\includegraphics[width=0.95\\linewidth]{modelFigures\/delays-confirmed.pdf}}{\\caption{Delays for a confirmed traffic network, $m=8$.}\\label{fig:delays}}\n  \\ffigbox{\\includegraphics[width=0.95\\linewidth]{modelFigures\/fairness.pdf}}{\\caption{Fairness for different SF distributions when $m=8, h=8, \\tau=1, \\alpha=0.3$.}   \\label{fig:fairness}}\n  \\end{floatrow}\n\\end{figure}\nThe final metric that we evaluate through both model and simulation is the\ndelay, as described in Sec.~\\ref{sec:metrics}. Fig.~\\ref{fig:delays} shows\nhow delays generally increase with the traffic load, since more\nre-transmissions are needed to successfully deliver a packet. Note that for\nhigh values of $\\lambda$ the average \\gls{ack} delay $\\Delta^{\\textrm{DL}}$\ndecreases: this is explained by the fact that devices employing higher\n\\glspl{sf}, (which may increase the average delay due to their longer\ninter-packet transmission times) heavily suffer from interference and are\noften dropped (unsuccessful packets are not considered in the delay\ncomputation). Although not shown here, it is worth noting that the model\nformulation makes it easy to extract per-\\gls{sf} metrics that can help\ntroubleshoot the network configuration under study.\n\nWe now analyze how the fairness varies with the traffic load for different\nconfigurations of $\\alpha$, $p^u$ and $p^c$. We consider the \\gls{sf}\ndistributions $p_{\\rm equal}$ and $p_{\\rm EXPLoRa}$ as defined in\nTab.~\\ref{tab:config}. The $p_{\\rm EXPLoRa}$ distribution, first presented\nin~\\cite{cuomo2017explora}, aims at equalizing the aggregate time on air of each\ngroup of devices employing the same \\gls{sf} to minimize the collision\nprobability. In Fig.~\\ref{fig:fairness} we can observe that, when the \\glspl{sf}\nare uniformly allocated independently of the traffic type (i.e.,\n$p^u = p^c = p_{\\rm equal}$), the fairness decreases for an increasing traffic\nintensity. Indeed, as the traffic grows, nodes employing lower \\glspl{sf} will\nsuffer less from interference because of the shorter transmission times. The\nfairness grows when $\\alpha=0.3$ and $p^c = p_{\\rm EXPLoRa}$, since with this\nconfiguration 30\\% of the generated packets will use lower \\glspl{sf} with\nhigher probability, diminishing the channel and \\gls{gw} occupancy. However,\nsince the traffic load is high and the fairness is measured on the uplink\nperformance (\\gls{uu} and \\gls{cu}), the beneficial effect of allocating\n\\glspl{sf} according to the $p_{\\rm EXPLoRa}$ distribution are more evident when\nit is used for most of the devices, i.e., the 70\\% of nodes employing\nunconfirmed traffic. Finally, the maximum fairness is achieved when the\n\\glspl{sf} are allocated using $p_{\\rm EXPLoRa}$ both for $p^u$ and $p^c$\n(dotted line in\nFig.~\\ref{fig:fairness}).\nNote that, when $\\lambda \\leq 1$, the load in the network is low enough to have\n$J=1$ for every $p^u, p^c$, since the collision probability is low and the \\gls{gw} is not busy with\n\\gls{ack} transmissions.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/retxdistribution.pdf}\n  \\caption{Distribution of re-transmissions, $m=4$, $\\alpha=1$.}\n  \\label{fig:retxdistribution}\n  \\vspace{-1em}\n\\end{figure}\n\nAn example of insight that the analytical model can offer is presented in\nFig.~\\ref{fig:retxdistribution}, which shows the fraction of traffic that\nachieves success after a certain number of re-transmission attempt for different\ntraffic loads, derived from $P^{DL}_{i,j}$. This data, for instance, can be used\nto estimate the power consumption at the nodes: for low traffic loads the vast\nmajority of \\gls{mac} layer packet transmissions succed with just one \\gls{phy}\nlayer transmission attempt. As the traffic load increases, the fraction of\ndevices needing multiple re-transmissions to correctly receive an \\gls{ack}\ncorrespondingly increases. After a certain point, packet reception fails with\nsuch a high rate that most \\glspl{ed} need to employ the maximum number of\ntransmissions and, despite the high energy expenditure, still fail to receive an\n\\gls{ack} from the \\gls{gw}.\n\n\\begin{table}\n  \\footnotesize\n  \\centering\n  \\caption{Configurations employed in Fig.~\\ref{fig:improvements}}\n  \\label{tab:config}\n  \\begin{tabular}{lccccc}\n    \\toprule\n    Configuration & $\\tau_1$ & $\\tau_2$ & $m$ & $h$ & $p^u = p^c$ \\\\\n    \\midrule\n    C1 & 1 & 1 & 1 & 1 & $p_{\\rm equal}$   \\\\\n    C2 & 0 & 1 & 1 & 4 & $p_{\\rm EXPLoRa}$ \\\\\n    C3 & 0 & 1 & 4 & 4 & $p_{\\rm EXPLoRa}$ \\\\\n   \n   \n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/improvements.pdf}\n\t\\caption{UU and CD performance for different network configurations,\n    $\\alpha=0.3$.}\n\t\\label{fig:improvements}\n  \\vspace{-1em}\n\\end{figure}\nFinally, we show how the model can be applied to investigate the impact of\ndifferent network parameters on the performance. In the example of\nFig.~\\ref{fig:improvements}, 30\\% of the \\glspl{ed} employ confirmed traffic,\nand we show results obtained with the proposed mathematical model. The parameter\nconfigurations are summarized in Table~\\ref{tab:config}. Configuration C1\nprovides a baseline: priority is given to \\gls{dl} transmission in both windows,\ndevices employ a single transmission attempt for both confirmed and unconfirmed\ntraffic, and \\glspl{sf} are uniformly distributed. In this case the curves have\na shape similar to those shown in Fig.~\\ref{fig:cucd} for $m=1$, but, since\nfewer devices require \\glspl{ack}, the \\gls{gw} is able to receive more packets\nand profitably send replies, leading to better performance. To improve \\gls{uu}\na second configuration (C2) considers the prioritization of \\gls{ack}\ntransmissions in \\gls{rx2}, where their reception suffers less interference.\nMoreover, unconfirmed packets are sent multiple times and we use $p^u = p^c =\np_{\\rm EXPLoRa}$. This configuration provides a considerable\nimprovement with respect to the \\gls{uu} metric, and some gains are also\nachieved in the \\gls{cu} performance. To improve also the results for confirmed\ntraffic, a further step (configuration C3) is to set $m=4$. This provides a\nsignificant improvement of \\gls{cu}, at the cost of a (minimal) decrease in\n\\gls{uu} performance. As a final step, we fully leverage the analytical model to\nidentify the optimal parameter configuration (i.e., $m$, $h$, $p_{u}$ and\n$p_{c}$) for each plotted traffic load, with the objective of maximizing the\naverage of \\gls{uu} and \\gls{cu}. The red curves of this setting (C4) show how\nthis optimization process enabled by the model can significantly improve the\nglobal performance of the network, significantly improving the \\gls{cd}\nperformance at the price of a very small reduction in packet success rate for\nunconfirmed devices.\n\nThe optimization problem that is solved to obtain configuration C4 is defined\nas:\n\\begin{equation}\n\\begin{aligned}\n\\max_{p_{u}, p_{c}} \\quad & \\textrm{UU} + \\textrm{CD} \\\\\n\\textrm{s.t.} \\quad     & 0 \\le p_{i}^{u} \\le 1 \\\\\n                        & 0 \\le p_{i}^{c} \\le 1 \\\\\n                        & \\sum_{i} p_{i}^{u} = 1 \\\\\n                        & \\sum_{i} p_{i}^{c} = 1 \\\\\n\\end{aligned}\n\\label{eq:optimization}\n\\end{equation}\nwhere we explore the entire space defined by $m$, $h$ and $\\lambda$, by solving~\\eqref{eq:optimization}\nto find the best $p_{u}$ and $p_{c}$, and finally pick the best solution for\neach $\\lambda$. The search is performed using the trust region method as\nimplemented by the \\texttt{scipy} library, and we always set $p_{i}^{u} =\np_{i}^{c} = 1\/6$ as the initial parameter value for the algorithm.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.6\\linewidth]{modelFigures\/distributions.pdf}\n\t\\caption{Optimal values of $p_{u}$, $p_{c}$, $m$ and $h$ as computed through model-driven optimization, for various values of $\\lambda$.}\n\t\\label{fig:optimal-parameters}\n  \\vspace{-1em}\n\\end{figure}\nFigure~\\ref{fig:optimal-parameters} displays the parameters of configuration C4\nfor some representative values of $\\lambda$, showing $p_{u}$ in the first row,\n$p_{c}$ in the second row, and a combination of the two weighed on $\\alpha$ on\nthe third row. For a low value of generated traffic ($\\lambda = 0.1$, first\ncolumn), we see that the optimization stops almost immediately, yielding a\ndistribution that is very similar to the initial value of $p_{u}$ and $p_{c}$.\nIn this case, as can also be seen in Figure~\\ref{fig:improvements}, since the\ntraffic load is low the performance is indeed very good for high values of $m$\nand $h$, and needs little optimization of the \\gls{sf} distributions. For\n$\\lambda = 1$, instead, the optimization process yields a more distinctive value\nof $p_{c}$, setting almost all devices to use \\gls{sf}7. This is motivated by\nthe fact that, \\gls{rx1} is set to employ the same \\gls{sf} used in the\n\\gls{ul}. Therefore, having most of the confirmed devices employ an \\gls{sf} as\nlow as possible is advantageous, since it guarantees faster \\gls{ack}\ntransmissions in the \\gls{dl} and, as a consequence, shorter silent times\nimposed by the \\gls{dc}, and a larger set of devices can thus be served. Devices\nemploying unconfirmed traffic, instead, are set to use a variety of \\gls{sf}\nvalues. Notably, the selected values are such that the aggregated distribution\nconsidering both unconfirmed and confirmed traffic (visible in the third row)\ntakes a shape that is very similar to that of $p_{\\rm EXPLoRa}$. This behavior\nis even more marked when $\\lambda = 10$, with the notable difference that higher\n\\gls{sf} values are not used in the optimized network: this is because of the\nlimited number of demodulators at the \\gls{gw} (a factor which is accounted for\nin our model). Indeed, although using all \\gls{sf} values would bring an\nadditional gain, a packet with high \\gls{sf} value occupies a demodulator for\nquite a long time, increasing the probability that other incoming packets are\ndropped because of unavailability of reception chains at the \\gls{gw}. Finally,\nwe note that $m$ and $h$ are consistently set to their maximum values (8 here)\nup to $\\lambda = 1$. After this value, instead, it pays off to reduce the number\nof repetitions employed by both unconfirmed and confirmed \\glspl{ed}.\n\nAlthough this analysis showcases the potential of the mathematical model to\nidentify the optimal settings, an evaluation of the trade-offs associated to\nparameter configurations and their effect on other metrics of interest, such as\ndelays and energy consumption, needs a deeper investigation, which we leave for\nfuture work.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this work, we presented a model for the performance evaluation of a\nLoRaWAN network in the presence of both confirmed and unconfirmed traffic,\ntaking into account the influence of different settings of multiple network\nconfiguration parameters.\n\nThe model is able to capture both the \\gls{phy} layer and \\gls{mac} layer\nperformance, and describes the multiple events that affect both \\gls{ul} packet\nreception and \\gls{dl} transmission: interference, capture effect, availability\nof demodulator, \\gls{dc} constraints, ongoing transmissions and receptions.\nWe validated the model results with ns-3 simulations, showing the consistency\namong the two sets of results. Finally, we presented some examples of how the\nmodel can be employed to analyze the effects of possible changes to the standard\nparameter settings, and to identify optimal configurations with minimum effort.\n\nSeveral extensions of this work are possible. A first improvement to the model\nis the inclusion of multi-\\gls{gw} scenarios, where \\gls{ul} packets are\npotentially received by several \\glspl{gw}, and the network \\gls{dl} capacity is\nincreased. A second aspect of interest is to leverage the proposed model to\nbetter investigate trade-offs among different network parameters in various\nscenarios, or when specific performance requirements are provided. A third\npossible improvement would involve characterizing the capture effect for\nnon-uniform spatial distribution of the devices. Finally, a fourth direction is\nto employ the proposed model to identify optimal network settings when different\nmetrics of interest are used as optimization functions, as we showed in the\nresults section with some simple cases. We point out that the target of the\nmodel was to explore the capabilities of LoRaWAN networks, thus, in this work,\nwe neglected some features of LoRa, such as the interference between overlapping\npackets modulated with different \\glspl{sf}. The model can be extended by\nincluding this, as well as other specific features of the LoRa technology. Such\nextensions are left for future work.\n\nWe remark that all figures contained in this paper, covering both\nmodel evaluations and simulation results, can be easily reproduced using\nthe tool available at~\\cite{publishedmodelcode}.\n\n\\section*{Acknowledgment}\n\nPart of this work was supported by MIUR (Italian Ministry for Education and\nResearch) under the initiative \"Departments of Excellence\" (Law 232\/2016).\n\n\n\\bibliographystyle{IEEEtran} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\\label{Sec:intro}\n\nThe conduction electron Wannier orbitals in transition-metal compounds are generally  \nfairly localized in space so that electronic correlations, i.e. all effects that deviate from the \nindependent-particle picture,  are sometimes strong enough  to give rise to metal-insulator \ntransitions in particular temperature and pressure conditions. The correlation-driven \nmetal-insulator transition, known as Mott transition,\\cite{Mott_original, Mott} is \noften accompanied by rather spectacular phenomena that appear in its proximity, high-temperature \nsuperconductivity being the most popular example. This makes $3d$ metal \nelements and \ncompounds a natural laboratory for intriguing many-body physics, which despite \na rich history and many studies is worth exploring further. \n\nElectronic structure methods (sometimes referred to as \"first principles\" methods) that rely on independent \nparticle descriptions, such as \\ac{hf} or \\ac{dft} within \\ac{lda}, are by construction \nincapable of capturing the Mott transition, \nwhich has no counterpart in a one electron picture.\nFor this reason, \\ac{hf} and \\ac{lda},  while \ngenerally quite successful for many materials,\nmay sometimes fail in the description of solids involving transition metals.  In fact, most of our knowledge \nabout Mott electron localization has been attained by means of simplified lattice models, the best known \nbeing the Hubbard model,\\cite{Hubbard} which are accessible by methods better suited to deal with correlations, \nsuch as quantum Monte Carlo,\\cite{QMC_review} density-matrix renormalization group\\cite{Schwollock_DMRG} \nand dynamical mean-field theory.\\cite{DMFT} \n\nClearly, for  the purpose of a quantitative understanding of real materials, it is of key importance to \nsew the two worlds together, bringing in particular the many body\nexpertise gained on lattice models over to realistic, off-lattice, first-principles calculations of solids.\n\nThis has historically been attempted through ad-hoc improvements of \\ac{dft}. \nFor instance, the inclusion (in fact, the addition and subtraction) of an intra-site Coulomb  repulsion $U$ (the \"Hubbard $U$\") \nin the Kohn-Sham Hamiltonian permits a decrease of the so-called self-interaction error,  \na severe flaw of \\ac{lda} for partially or fully occupied  localized orbitals -- just the case of transition metals.  \nWhen added to \\ac{lda}, this procedure, the so called \n\\ac{ldau},\\cite{LDAU_Anisimov_Andersen, LDAU_jphys} often improves results, and can for example stabilize magnetic phases which \nstraight \\ac{lda} would miss. Yet, \\ac{ldau} remains basically a mean-field, independent particle approach that cannot \ndescribe  Mott localization. The problem can be overcome if, for instance, the Kohn-Sham Hamiltonian of \\ac{lda} \nsupplemented by $U$ is solved through \\ac{dmft},  by the so-called \\ac{ldadmft}.\\cite{LDADMFT_theory} \nAlternatively, variational Quantum Monte Carlo \\ac{vqmc} approaches\\cite{Ceperley_VMC,Sorella_SR} have been successfully \napplied to  the electronic \nproperties of atoms and simple molecules,\\cite{Sorella_dimers} and its development appears to be promising \nfor more ambitious applications.\n\nAt present, both \\ac{ldadmft} and \\ac{vqmc} are numerically much more involved and far more demanding \nthan conventional \\ac{lda} or even \\ac{ldau}, which owe much of their success  \nto simplicity. The desirability of approaches joining together the simplicity of \\ac{lda} and the description of\ncorrelations typical of many body methods is therefore still very high. \nIn the context of lattice models, a simple approach to \nstrong correlations was proposed long ago by Martin C. Gutzwiller.\\cite{Gutzwiller1,Gutzwiller2} This method,\nprojecting out of a trial Slater determinant an adjustable proportion of costly configurations and \nevaluating average values by approximate formulas, is strictly variational in the limit of \ninfinite lattice-coordination\\cite{Gebhard} -- the same limit where \\ac{dmft} is exact -- providing much more accurate \nresults than \\ac{hf}.  That success invites the use of the Gutzwiller method even when \nthe lattice space dimension, and thus the site coordination, is finite, as people do with \\ac{dmft}. \n\\ac{ga}  electronic structure calculations \nhave the great advantage \nto couple extreme LDA-level simplicity with qualitatively, often quantitatively, increased accuracy in the\ndescription of correlations. For example, \\ac{ga} has been able to describe conducting materials that are \ninsulators ``in disguise'',\\cite{Fazekas} i.e. whose properties depend on correlations that are already present \nin their Mott insulating phase, and that continue to play an important role even in the nearby metallic phases. \nA well known example is the RVB scenario for high-temperature superconductors,\\cite{Anderson_RVB_Science} \nwhere Cooper pairing is explained as a byproduct of doping a parent state of resonating valence bonds, \nwhich is the remnant of antiferromagnetism when N\\`eel long range order disappears. Another famous \nresult of the \\ac{ga} is the Brinkman-Rice description of the Mott transition in vanadium sesquioxide, originally \nderived by the \\ac{ga} solution to the Hubbard model.\\cite{brinkman&rice}. \n\nBecause of its simplicity, a great deal of effort has therefore been \ndevoted in recent years to extend \\ac{ga} from simple lattice models to more realistic off-lattice \ncases.\\cite{Ho_LDAG_condmat, ZhongFang_LDAG, Andersen_Gebhard_Gutzwiller, ZhongFang_LDAG_app1, Lanata_efficient, Ho,Lanata-Kotliar} \nHere we  implement  a density self-consistent algorithm that exploits the Gutzwiller variational wave \nfunction together with the conventional \\ac{lda} for the density functional. The Levy-Lieb constrained-search \nformulation of \\ac{dft} provides a solid theoretical framework for the introduction of Gutzwiller variational \nparameters in the density functional, while a localized atomic basis set (we use in particular the Siesta electronic structure code) makes \nthe definition of the Gutzwiller-projected states straightforward.\n\nWe test the power of the \\ac{ldag} functional by calculating the electronic structure of nonmagnetic\nand ferromagnetic $bcc$ Fe, motivated  by long standing basic \nquestions about the electronic origin of magnetic order,  \n\\cite{vollhardt2001} including \na recent \\ac{ldadmft} study \nby Anisimov and coworkers\\cite{anisimov} \nsuggesting that $bcc$ iron might be an orbital-selective Mott insulator. According to that picture,  the \npoorly dispersive $e_{g}$-type electrons of metallic Fe may be fully localized due to interactions, \nso that conduction phenomena are restricted within the $t_{2g}$ manifold (besides of course \nthe $s$ electrons). In that picture\\cite{stollhoff_ironloc, Goodenough_ironloc, stearns_ironloc} \nferromagnetic alignment \nwould not be due to inter-site Coulomb exchange, as is \nordinarily assumed, but rather to double-exchange, as in colossal magnetoresistance manganites.\\cite{manganites}\nThe Mott localized $e_g$ electrons \nform spin-1 moments that  couple \nferromagnetically via intra-atomic Hund's exchange to the electrons in the  \nnearly-full itinerant $t_{2g}$ bands. In order to preserve coherent $t_{2g}$ hole motion, \nthe local $e_g$ moments order ferromagnetically. As in the manganites, ferromagnetism is \nthus driven by a kinetic energy gain rather than a potential energy one.   \nEven though our \\ac{ldag} approach is still mean-field and thus cannot address dynamical  \nphenomena such as orbital selective Mott transitions --\nespecially so in a delicate case where the two sets of orbitals, $e_{g}$ and $t_{2g}$,  hybridize with each other \nin the Brillouin zone --  we find that calculation of the total energy and a detailed analysis of its \nseparate kinetic and potential energy contributions actually supports  double-exchange \nas the driving mechanism of  \nferromagnetism in iron, rather than the conventional Stoner instability. On the whole, this work \nmay also be of general use as a very detailed\nexample of {\\sl ab-initio} application of Gutzwiller correlations to a realistic electronic structure problem.\n\nThe plan of this article is as follows: in \\sect{Sect1} we introduce the formalism of \\ac{ldag} starting \nfrom the constrained-search formulation of Density Functional Theory,\ndemonstrating\nhow the Gutzwiller \nwavefunction can be used to generalize \\ac{ldau} by allowing the expectation value of the atomic \nHamiltonian to be computed on a multi-determinant wavefunction. In \\sect{Sect:Gutz_expect} and\n\\sect{Sec:Gutz_practice} we then show how the different terms of the \\ac{ldag} density functional can be computed \nby means of \\ac{ga}, and how the total energy of a correlated electronic system can be minimized by a three-step \niterative procedure. In \\sect{Sec:results} we finally present and comment on the physical results for paramagnetic and \nferromagnetic $bcc$ Fe, and connect back to the basics questions about the origin of ferromagnetic order.\n\n\n\\section{Constrained-search formulation of a Gutzwiller Density Functional Theory}\\label{Sect1}\n\nA convenient way to introduce a Gutzwiller density functional is through the formalism \nindependently proposed by Levy~\\cite{Levy1,Levy2} and Lieb~\\cite{Lieb1}.\nStarting from the Rayleigh-Ritz definition for the ground state energy $E_{\\rm GS}$ of a system\n\\begin{align}\\label{Eq_min_GSenergy}\nE_{\\rm GS} = \\min_{\\Psi} \\expect{\\Psi}{\\hat{H}}\\,,\n\\end{align}\nwhere the electron Hamiltonian $\\hat{H}$ includes the kinetic energy $\\hat{T}$, the electron-electron interaction $\\hat{V}_{\\rm ee}$, \nand a local external potential $\\hat{V}_{\\rm ext}$, Levy and Lieb converted the variational principle for \nthe ground state wavefunction into a variational principle for the ground state density through a constrained minimization at fixed density $n({\\bm r})$\n\\begin{eqnarray}\n\\label{Eq_LL_DF}\nE_{\\rm GS}[V_{\\rm ext}({\\bm r})] &=& \\min_{n({\\bm r})} \\Bigg\\{ \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}+\\hat{V}_{\\rm ee}} \\nonumber \\\\\n&& ~~~~~~~~~~+ \\int V_{\\rm ext}({\\bm r}) n({\\bm r})\\Bigg\\}\\,.\n\\end{eqnarray}\nThe first term on the right-hand side of (\\ref{Eq_LL_DF}) is nothing but the constrained-search definition of \nthe Hohenberg-Kohn functional\\cite{DFT_HKtheorem}, i.e.\n\\begin{align}\\label{Eq_LL_HKfunc_def}\nF_{\\rm HK}[n({\\bm r})] = \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}+\\hat{V}_{\\rm ee}}\\,,\n\\end{align}\nwhich is independent of the external potential $V_{\\rm ext}$.\nThe wavefunction $\\Psi$ in the definition~\\eqn{Eq_LL_HKfunc_def} should span the whole many-body Hilbert space, \ngenerally too large \nto allow a \nstraightforward numerical evaluation of $F_{\\rm HK}[n({\\bm r})]$.\nWithin the Kohn-Sham scheme, the generality of \\eqn{Eq_LL_HKfunc_def} is abandoned in \nfavor of a more practical definition of the Hohenberg and Kohn functional, in which the latter is split \ninto kinetic, Hartree, and exchange-correlation terms, namely\n\\begin{align}\\label{Eq_HK_semilocal}\nF_{\\rm HK}[n({\\bm r})] = T_{s}[n({\\bm r})] +E_{\\rm H}[n({\\bm r})] + E_{\\rm xc}[n({\\bm r})],\n\\end{align}\nwhere $E_{\\rm H}[n({\\bm r})]$ is simply the electrostatic energy of the electron density regarded as a classical charge distribution.\nA constrained search is then retained only for the kinetic contribution\n\\begin{align}\\label{Eq_HK_nonint}\nT_{s}[n({\\bm r})] = \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}}\\,,\n\\end{align}\nwhich, because $\\hat{T}$ is a one-body operator, has a solution within the class of Slater determinants, \na relatively simple task to accomplish through \nauxiliary non-interacting electron Hamiltonians whose ground state local density $n({\\bm r})$ coincides \nwith that of the physical interacting model. The insurmountable difficulties of the original many-body problem \nhave thus been hidden in the unknown exchange-correlation functional $E_{\\rm xc}[n({\\bm r})]$. All DFT approximation \nschemes correspond just to  different guesses of a physically sensible functional form of $E_{\\rm xc}[n({\\bm r})]$ \nin terms of the local density. \n\nThe main problem that arises from the density-dependent parametrization \\eqn{Eq_HK_nonint} \nis that $E_{\\rm H}[n({\\bm r})]$ contains a spurious \\ac{si} \nterm -- finite even when $n({\\bm r})$ is the density of a single electron! -- a term which should be identically \ncancelled in the exact $E_{\\rm xc}[n({\\bm r})]$. Unfortunately, all semi-local approximations to $E_{\\rm xc}[n({\\bm r})]$, \nsuch as \\ac{lda} and  \\ac{gga}, fail to fully subtract such a \\ac{si} term from the density functional, which \nbrings about results that by construction contain a certain level of \nself-interaction \nerror. \n\nThe spurious SI one-electron energy is larger for spatially localized electronic wavefunctions. \nFor instance, a single electron with a simple gaussian wavefunction feels an \\ac{si} that is\ninversely proportional to the standard deviation of the gaussian, only $70\\%$ of which is \nsubtracted by the \\ac{lda} exchange functional. The improvements  attained by \nbetter functionals do not seem major.\\cite{korzdorfer_SIC}\nAll density-functional calculations are affected to some extent by the \\ac{si} error, more important \nwhen the real-space density matrix is more localized. That is especially \nthe case for most transition metals \nand transition-metal oxides. In a density functional calculation with semi-local functionals, the spurious \\ac{si} term acts effectively as a penalty term \npreventing electronic localization,  thus often spoiling \nagreement with experimental data for band gaps, \nmagnetization, and other physical observables such as lattice constant and bulk modulus.\n\n\\subsection{LDA+U}\nA popular way to reduce the \\ac{si} \nwhile \nstill remaining in the context of local or semi-local density functionals \nis by including in the kinetic functional \\eqn{Eq_HK_nonint} also part of the electron-electron interaction, \nspecifically the projection $\\hat{H}_{\\rm at}$ of $\\hat{V}_{\\rm ee}$ on atomic-like orbital (se below).\nThe common choice is \nto consider only orbitals that are partially occupied within standard LDA, hence which suffer more from the \\ac{si}. \nThe non-interacting kinetic functional $T_{s}[n({\\bm r})]$ is thus turned into a modified kinetic functional $T_i[n({\\bm r})]$:\n\\begin{align}\\label{Eq_HK_modified}\nT_{s}[n({\\bm r})] \\rightarrow T_i[n({\\bm r})] = \\min_{\\Psi_{0} \\rightarrow n({\\bm r})} \\expect{\\Psi_{0}}{\\hat{T}+\\hat{H}_{\\rm at}}\\,,\n\\end{align}\nand the Hohenberg and Kohn functional changes into \n\\begin{align}\nF_{\\rm HK}[n({\\bm r})] &= T_i[n({\\bm r})] +E_{\\rm H}[n({\\bm r})] \\nonumber\\\\\n& + E_{\\rm xc}[n({\\bm r})] - {E_{\\rm dc}}[n({\\bm r})]\\,,\\label{Eq_HK_plusU}\n\\end{align}\nwhere ${E_{\\rm dc}}[n({\\bm r})]$ is a double-counting energy which must cancel\nthe contribution of $\\hat{H}_{\\rm at}$ already included within LDA.\n\nIn \\eqn{Eq_HK_modified} the constrained-search is still restricted to the space of Slater-determinants \n$\\Psi_{0}$, so that the modified kinetic functional can be dealt with within an independent-particle picture, \nand therefore included in the Kohn-Sham scheme. Essentially, the interaction $\\hat{H}_{\\rm at}$ is treated by Hartree-Fock, \nwhich is devoid of SI -- while still unable to capture the Mott localization phenomenon, \na correlation effect. \nIn section \\ref{Sect:gutz} we shall discuss how to improve the functional $T_i$ so as to make Mott physics \naccessible. Here in addition we  \nbriefly discuss how to define properly $\\hat{H}_{\\rm at}$.\nTypically $\\hat{H}_{\\rm at}=\\sum_{{\\bm R}} \\hat{H}_{\\rm at}^{({\\bm R})}$, with $\\hat{H}_{\\rm at}^{({\\bm R})}$ accounting for the leading order \nmultipolar expansion of the Coulomb interaction projected onto a selected set of \natomic-like orbitals $\\ket{\\phi^{(l)}_{{\\bm R},m}}$ with angular momentum $l$  \nat atomic site ${\\bm R}$ in the lattice, \n\\begin{eqnarray}\n\\hat{H}^{({\\bm R})}_{\\text{at}} &=& \\frac{F_0}{2}\\,\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)\n+ \\frac{1}{2}\\,\\sum_{L>0}^{2l}F_L\\,\\left(C^{l0}_{l0\\,L0}\\right)^2\\nonumber\\\\\n&& ~~~~~~\\sum_{M=-L}^L\\,(-1)^M C^{lm}_{lm'\\,LM}\nC^{lm_1}_{lm'_1\\,L-M}\\nonumber\\\\\n&& ~~~~~~~~~~~~~~c^\\dagger_{{\\bm R},m\\sigma}c^\\dagger_{{\\bm R},m_1\\sigma_1}\nc^{\\phantom{\\dagger}}_{{\\bm R},m'_1\\sigma_1}c^{\\phantom{\\dagger}}_{{\\bm R},m'\\sigma}\n\\label{Eq:hHat_exact}\n\\end{eqnarray}\nwhere $\\hat{N}_{\\bm R}$ is the total electron number operator at site ${\\bm R}$ projected onto \nthe selected set of atomic orbitals, $L = 2n$ with $n=1,\\dots,l$, and $C^{lm}_{lm'\\,LM}$ are \nthe Clebsch-Gordan coefficients. The parameters $F_L$ are commonly \nknown as Slater integrals. The first term on the right-hand side of \\eqn{Eq:hHat_exact}, \nwhich we shall denote hereafter as $\\hat{H}_{\\rm Hub}^{({\\bm R})}$, is a pure charge repulsion usually referred \nto as the Hubbard term, its coupling constant $F_0$ generally called the \"Hubbard $U$\". The remaining terms instead \nenforce Hund's first and second rules, hence they may be referred to as \nthe Hund's rule exchange ($\\hat{H}_{\\rm Hund}$).\nIn fact, in the case  of $p$ orbitals ($l=1$), the exact multipolar expansion can be rewritten solely in terms \nof the number operator $\\hat{N}_{\\bm R}$, the total spin $\\mathbf{S}_{\\bm R}$ and total angular momentum  $\n\\mathbf{L}_{\\bm R}$ operators projected on the set $\\ket{\\phi^{(1)}_{{\\bm R},m}}$:\n\\begin{align}\n\\label{Eq_Coulomb_on_p}\n\\hat{H}_{\\rm at} &= \\frac{F_0}{2} \\left[\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-\\mathds{1}\\right)\\right] \\\\\n&+ \\frac{F_2}{2}\\left[\\frac{4}{5}\\hat{N}_{\\bm R}-\\frac{\\hat{N}_{\\bm R}^2}{5}-\n\\frac{3}{25}\\left(4 \\hat{\\mathbf{S}}_{\\bm R}\\cdot\\hat{\\mathbf{S}}_{\\bm R}  +\\hat{\\mathbf{L}}_{\\bm R}\\cdot\\hat{\\mathbf{L}}_{\\bm R}\\right)\\right]\\,,\\nonumber\n\\end{align}\nexplicitly \nshowing the content of the first two Hund rules. For $l>1$, it is no longer\npossible to  \nrewrite \\eqn{Eq:hHat_exact} in terms of simple operators like spin and angular momentum.\n\nThe well-known \\ac{ldau} method truncates the multipolar expansion of the Coulomb operator, \n\\eqn{Eq:hHat_exact}, to the zeroth-order term, therefore setting $\\hat{H}_{\\rm at}=\\hat{H}_{\\rm Hub}$.\nWith this recipe, the density dependence of the expectation value $\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}$ can be written \nin terms of the matrix elements \n$n^{(0)}_{lm\\sigma{\\bm R},lm'\\sigma'{\\bm R}} = \\expect{\\Psi_{0}}{c^\\dagger_{{\\bm R}, lm\\sigma}\nc^{\\phantom{\\dagger}}_{{\\bm R},lm'\\sigma'}}$ \nof the local single-particle density matrix $\\hat{n}^{(0)}_{\\bm R}$ , which is an implicit function of the density $n({\\bm r})$. \nIf  lattice periodicity is unbroken and the set of correlated orbitals is characterized by a single \nvalue of the angular momentum, we can drop both indices $l$ and ${\\bm R}$ in any local operator, \nand write $\\hat{n}^{(0)}_{\\bm R} = \\hat{n}^{(0)}$,  $\\forall {\\bm R}$.\nThe double-counting correction ${E_{\\rm dc}}[n({\\bm r})]$ in \\ac{ldau} is commonly chosen \nso as to cancel   $\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}$ in the limiting case \nof an idempotent single-particle density matrix $\\hat{n}^{(0)}$,\\cite{mazin} which corresponds to assuming \nthat, within straight \\ac{lda}, $\\langle  \\hat{n}^{(0)}_{\\bm R} \\hat{n}^{(0)}_{\\bm R} \\rangle = \n\\langle  \\hat{n}^{(0)}_{\\bm R}  \\rangle\\langle  \\hat{n}^{(0)}_{\\bm R} \\rangle $.\nWith this assumption, the $U$-dependent part of Kohn-Sham Hamiltonian is equal to \nthe positive definite contribution\n\\begin{align}\n\\label{Eq:LDAU_total_ham}\n\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}-{E_{\\rm dc}}[n({\\bm r})] = \\frac{U}{2} \\text{Tr}\\Big[\\hat{n}^{(0)}\\big(\n1-\\hat{n}^{(0)}\\big)\\Big].\n\\end{align}\nAn optimal value of $U$ can be estimated by linear response calculations \\cite{Cococcioni_DeGironcoli_LinLDAU, \nAnisimov_Zaanen}, or \nempirically determined by agreement with  experimental data. \n\nThe advantage of using \\eqn{Eq:LDAU_total_ham} to improve the description of systems with \nstrongly localized electrons is both its simplicity, involving no further computational effort than \nthat needed to solve the Kohn-Sham equations, and its success in removing the self-interaction \nwhenever $U$ is sensibly chosen.\nHowever, there are of course situations in which the empirical \\ac{ldau} functional will \nnot be adequate.\nWe previously mentioned that Mott localization \nbecause of its genuinely many-body, collective nature,  \nis not accessible by \\ac{ldau} \nnor by any other technique that relies on a single-particle description.\nMoreover, it is well known that only the spherically-averaged strength of the exchange-correlation \nhole is correctly accounted for by the LDA functional, but not its angular dependence. For these reasons \none cannot expect that\n\\ac{ldau} will be apt to describe systems that display strongly orbital-dependent correlations, as was shown \nto be the case of body-centered cubic iron.\\cite{anisimov} Indeed recent studies\non iron pnictides and chalcogenides\\cite{Kotliar_coherence_incoherence, Kotliar_ironcalco, magnetism_chargedyn_pnictides}  \nsuggest that the orbital selectivity displayed by these iron compounds crucially depends on atomic Hund's rules. These \nobservations indicate that a way to further improve LDA beyond \\ac{ldau} will not only be the inclusion \nof correlations in the modified kinetic functional so as to make Mott localization accessible, \nbut also \nthe introduction of an appropriate expression for Hund's interaction $\\hat{H}_{\\text{Hund}}$ in \nthe atomic Hamiltonian $\\hat{H}_{\\text{at}}$, so as to account for orbital selectivity. However, \nwhen Hund's rule exchange, the second term in \nthe r.h.s. of \\eqn{Eq:hHat_exact}, is taken into account, one faces the problem of finding a proper expression \nfor electron double counting. The latter should by definition be equivalent to the LDA approximation \nto the atomic interaction energy, \\eqn{Eq:hHat_exact}. However, that average depends in principle on the \nspecific point symmetry of the system, and one cannot find a general expression valid for every case. \nThe conventional way to proceed is to dismiss the hope of including within \\ac{ldau} the whole \natomic interaction \\eqn{Eq:hHat_exact}, and instead be content with only terms that depend \non angular-averaged local operators, specifically the total number operator $\\hat{N}_{\\bm R}$ and total spin \n$\\hat{\\mathbf{S}}_{\\bm R}$. These terms are identified by noting that, using the \nre-coupling formula\n\\begin{eqnarray*}\n&&\\sum_M\\,  (-1)^M\\,  C^{lm}_{lm'\\, LM} C^{lm_1}_{lm'_1\\,L-M} = \n\\sum_{\\Lambda\\lambda}\\, (2\\Lambda+1)\\, \n\\begin{Bmatrix}\nL & l & l\\\\\n\\Lambda & l & l\n\\end{Bmatrix} \\\\\n&&  ~~~~~~~~~~~~~~~~(-1)^{L+\\Lambda}\\, (-1)^{\\lambda}\\,C^{lm}_{lm'_1\\,\\Lambda\\lambda} \n\\, C^{lm_1}_{lm'\\,\\Lambda-\\lambda}\\, ,\n\\end{eqnarray*}\nthe $L>0$ contribution of \\eqn{Eq:hHat_exact} can be also written as \n\\begin{eqnarray*}\n\\hat{H}^{({\\bm R})}_{\\text{at}\\, L>0} &=& \n- \\frac{1}{2}\\sum_{mm_1m'm'_1}\\sum_{\\sigma\\sigma_1}\\sum_{L>0}^{2l}F_L\\,\\left(C^{l0}_{l0\\,L0}\\right)^2\\nonumber\\\\\n&& ~~\\sum_{\\Lambda}\\sum_{\\lambda=-\\Lambda}^\\Lambda \n(-1)^{\\Lambda+\\lambda} (2\\Lambda+1) \\begin{Bmatrix}\nL & l & l\\\\\n\\Lambda & l &l \n\\end{Bmatrix}\\nonumber\\\\\n&& C^{lm}_{lm_1'\\,\\Lambda\\lambda}C^{lm_1}_{lm'\\,\\Lambda-\\lambda}\\,\nc^\\dagger_{lm\\sigma}c^{\\phantom{\\dagger}}_{lm'_1\\sigma_1}c^\\dagger_{lm_1\\sigma_1}c^{\\phantom{\\dagger}}_{lm'\\sigma}\\,,\n\\end{eqnarray*}\nwhere $\\{\\dots\\}$ denote the Wigner $6j$-symbols. We can then select out the term with $\\Lambda=0$, \nwhich depends on rotationally invariant densities, \nre-couple back $m$ with $m'$ and $m_1$ with $m'_1$ in the remaining terms, and iterate the procedure.\nAt the end, we obtain a term that involves rotationally invariant densities, plus another interaction that \ncannot be expressed by any means in terms of those densities. The former \ntogether with the Hubbard $U$ define the part of the atomic interaction \\eqn{Eq:hHat_exact} \neasier to implement within \\ac{ldau}, namely  \n\\begin{align}\n\\hat{H}^{({\\bm R})}_{\\text{at}} &\\simeq& \\frac{U}{2} \\hat{N}_{\\bm R}\\Big(\\hat{N}_{\\bm R}-1\\Big)\n- \\frac{2l+1}{2l+2}\\,J \\,\\Bigg[\\hat{\\mathbf{S}}_{{\\bm R}}\\cdot \\hat{\\mathbf{S}}_{{\\bm R}}\n-\\frac{3}{4}\\,\\hat{N}_{\\bm R}  \\nonumber\\\\\n&& + \\fract{\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)}{4} \n+\\fract{\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)}{2(2l+1)} \n\\Bigg],\\label{Eq:hHat_approssimata}\n\\end{align}\nwhere $J$ is conventionally defined as \\cite{Cococcioni_DeGironcoli_LinLDAU, Anisimov_Zaanen} \n\\begin{equation}\nJ = \\frac{1}{2l}\\sum_{L>0}^{2l} \\, \\left(C_{l0\\, L0}^{l0}\\right)^2\\, F_L,\\label{def:J}\n\\end{equation}\nwhich, for $d$-orbitals, i.e. $l=2$, is $J=(F_0+F_4)\/14$. The double counting term associated with \n\\eqn{Eq:hHat_approssimata} is obtained  analogously as before and reads in the general case of a \nspin-polarized calculation \n\\begin{align}\nE_{\\text{dc}} &=& \\frac{U}{2} N\\big(N-1\\big) - \\frac{2l+1}{4l+4}\\,J \\,\\bigg[ N_\\uparrow\\left(N_\\uparrow-1\\right) \n\\nonumber\\\\\n&& ~~~~~~~~~~+ \n N_\\downarrow\\left(N_\\downarrow-1\\right) + \\fract{N(N-1)}{2l+2}\\bigg].\\label{double-counting-Hund}\n\\end{align}\nThe expression \\eqref{Eq:hHat_approssimata} can be further simplified to get rid of the $l$ dependence, \nby readsorbing the $l$ in the definition of $J$, and by adopting a simplified version\nof the last term in square brackets, leading to the following results\n\\begin{align}\\label{Eq:recast_J}\n\\hat{H}_{\\rm Hund}=-J \\left\\{ \\hat{S}^2 - \\frac{3}{4} \\hat{N} + \\frac{\\hat{N} (\\hat{N}-1)}{4}  +\\sum_{m} \\hat{n}_{m\\uparrow}\\hat{n}_{m{\\downarrow}} \\right\\}\\,.\n\\end{align}\nfor which we choose a double-counting \nenergy \nof the type\n\\begin{align}\\label{Eq:Edc_ours}\n&{E_{\\rm dc}}^{\\rm Hund}[n({\\bm r})] = \\nonumber \\\\\n&= -J \\sqbra{\\frac{N_{\\up} (N_{\\up}-1)}{2} +\\frac{N_{\\dw}(N_{\\dw}-1)}{2} + \\frac{N_{\\up}N_{\\dw}}{2l+1} }\\,.\n\\end{align}\n\\subsection{Extending LDA+U to LDA+Gutzwiller}\\label{Sect:gutz}\nThe key difference between \\ac{ldag} and \\ac{ldau} resides in the definition of the modified kinetic \nfunctional $T_{\\rm i}$. Within \\ac{ldag}, the definition \\eqn{Eq_HK_modified} changes to\n\\begin{align}\\label{Eq_GW_kinetic}\nT_{\\rm i}[n({\\bm r})] \\rightarrow T_{\\rm G}[n({\\bm r})] = \\min_{{\\Psi_{\\rm G}} \\rightarrow n({\\bm r})} \\expect{{\\Psi_{\\rm G}}}{\\hat{T}+\\hat{H}_{\\rm at}}\\,,\n\\end{align}\nwhere the  wavefunction $\\ket{\\Psi_{\\rm G}}$ is defined as\n\\begin{align}\\label{Eq_Gutzwav_def}\n{\\Psi_{\\rm G}} = \\Gpg{} \\ket{\\Psi_{0}} = \\prod_{{\\bm R}} \\Gpl{{\\bm R}} \\ket{\\Psi_{0}}\\,.\n\\end{align}\nIn the above equation, $\\ket{\\Psi_{0}}$ is still a Slater determinant, and the elements of novelty are \nthe operators $\\Gpl{{\\bm R}}$, which are linear transformations acting on the configurational space of \na chosen set of local orbitals at lattice site ${\\bm R}$. As in \\ac{ldau}, this set of orbitals $\\phi_{m,{\\bm R}}$ \nretain well defined atomic angular momentum $l$, $m$ being its projection on a given quantization axis.  \nThe operator $\\Gpl{{\\bm R}}$ can be generally written as\n\\begin{align}\\label{Eq_multiband_projector}\n\\Gpl{{\\bm R}} = \\sum_{\\Gamma \\Gamma'} \\Lambda_{\\Gamma\\Gamma',{\\bm R}}\\,\n\\ket{\\Gamma,{\\bm R}}\\bra{\\Gamma',{\\bm R}}\\,,\n\\end{align}\nwhere $\\ket{\\Gamma,{\\bm R}}$ denote many-body configurations of electrons occupying the \norbitals $\\phi_{m,{\\bm R}}$. Differently from \\ac{ldau}, the expectation value of the kinetic plus atomic \ninteraction operators will not depend solely on the Slater determinant $\\ket{\\Psi_{0}}$,  but also on the variational \nparameters $\\Lambda_{\\Gamma\\Gamma',{\\bm R}}$ that define $\\Gpl{{\\bm R}}$. \n\nComputing exact expectation values on the Gutzwiller wavefunction for lattices of finite coordination \nis a task that can be accomplished only numerically, e.g. through Variational Quantum Monte Carlo.\\cite{Sorella_SR,Sorella-VMC} \nFor infinite-coordination lattices, an exact expression can be instead computed analytically. There is in fact \na close connection between the Gutzwiller variational approach in the limit of infinite lattice coordination  \nand dynamical mean field theory.\\cite{DMFT} In that limit, the single particle self-energy matrix \n$\\Sigma(\\epsilon,{\\bm k}) = \\Sigma(\\epsilon)$ becomes purely local, hence momentum independent. \nDMFT allows to evaluate exactly $\\Sigma(\\epsilon)$ by solving an auxiliary Anderson impurity model \nconstructed in such a way as to have the same self-energy. The Gutzwiller variational approach is instead \na consistent approximation to the exact solution, which assumes a Fermi-liquid expression \n$\\Sigma(\\epsilon) \\simeq \\Sigma(0) + \\left(1-Z^{-1}\\right)\\epsilon$, where $Z$ is commonly refereed \nto as the quasiparticle weight. Because of this assumption, the Gutzwiller wavefunction can describe only states \nwhose elementary excitations are quasiparticles,\nsuch as \nLandau-Fermi liquids and insulators that can be \nrepresented through a Slater determinant. However, the additional freedom brought by the parameter $Z$, \nwhose value is strictly \n$Z = 1$ within Hartree-Fock \nand in \\ac{ldau}, opens the possibility to access  \nstrongly correlated metals, $Z\\ll 1$,  \nand thus the approach to a  \nMott transition, where $Z\\to 0$. \nAlthough DMFT is exact only in the limit of infinite coordination, it is currently used as an approximation \nin realistic finite-coordination lattices, under the hypothesis that (strong) correlation effects beyond Hartree-Fock (HF) are well represented by \n$\\Sigma(\\epsilon,{\\bm k}) \\simeq \\Sigma_{\\text{HF}}({\\bm k}) + \\Sigma(\\epsilon)$, where \n$\\Sigma_{\\text{HF}}({\\bm k}) $ is the HF self-energy, eventually including frequency-dependent \nrandom-phase-like contributions,\\cite{LDA+cRPA+DMFT} and the correction $\\Sigma(\\epsilon)$ is momentum independent and can be obtained by DMFT. Under the same assumptions, one can keep using the \nformal results of the Gutzwiller variational approach, that are strictly valid only in infinite-coordination lattices, also in finite-coordination ones, an approximation refereed to as the {\\sl Gutzwiller approximation} (\\acs{ga}). \nIn other words, the GA should be better regarded as an approximation to DMFT, when either of them are used in finite-coordination lattices, rather than an approximation to the exact evaluation of average values on the Gutzwiller wavefunction, \n\\eqn{Eq_Gutzwav_def}. This viewpoint, which we underwrite, is our motivation for adopting the Gutzwiller \napproximation in combination with \\ac{ldau} as an alternative to \\ac{ldadmft}, \nat the cost of less rigor, but as we shall show with gain in simplicity and flexibility. \n\\subsubsection{Expectation values in the Gutzwiller Approximation}\\label{Sect:Gutz_expect}\nIn order to  determine \nthe functional $T_{\\rm G}[n({\\bm r})]$, one should be able to compute expectation values \nof both many-body on-site operators such as those contained in $\\hat{H}_{\\rm at}$, and off-site single-particle \noperators, which are present in the definition of the kinetic operator $\\hat{T}$. In all what follows, \nwe shall use the formalism presented in Ref.~\\onlinecite{BaroneLanata}. \n\nFirst of all, the Slater determinant $\\mid\\Psi_0\\rangle$ defines the uncorrelated one-body local \ndensity-matrix $\\hat{n}^{(0)}_{\\bm R}$ (the same matrix that enters the \\ac{ldau} energy \ncorrection term \\eqn{Eq:LDAU_total_ham}), with elements \n\\begin{equation}\nn^{(0)}_{{\\bm R} m\\sigma,{\\bm R} m'\\sigma'} = \\langle\\Psi_0\\mid \nc^\\dagger_{{\\bm R},m\\sigma} c^{\\phantom{\\dagger}}_{{\\bm R},m'\\sigma'}\\mid\\Psi_0\\rangle, \\label{II.B-n0}\n\\end{equation}\nwhere \n$c^\\dagger_{{\\bm R},m\\sigma}$ \ncreates a spin-$\\sigma$ electron in orbital $\\phi_{m,{\\bm R}}$. \n$\\hat{n}^{(0)}_{\\bm R}$ is diagonalized by a unitary transformation that turns the \noriginal basis of operators \n$c^\\dagger_{{\\bm R}, m\\sigma}$ \ninto the natural basis of operators \n$c^\\dagger_{{\\bm R}, \\gamma\\sigma}$\n, assuming \ninvariance with respect to spin rotations around the $z$-axis. In the natural basis, the one-body \ndensity matrix is therefore diagonal, with eigenvalues $n^{(0)}_{{\\bm R},\\gamma\\sigma}$. \nIn the natural-orbital Fock basis, with states \n\\[\n\\mid \\{n_{{\\bm R},\\gamma\\sigma}\\}\\rangle \n\\equiv \\prod_{\\gamma\\sigma}\\,\\left(c^\\dagger_{{\\bm R},\\gamma\\sigma}\\right)^{n_{{\\bm R},\\gamma\\sigma}}\\,\n\\mid 0\\rangle,\n\\] \nit follows that the probability matrix \n\\begin{eqnarray}\n&&P^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}\\{m_{{\\bm R},\\gamma\\sigma}\\}} \\equiv \n\\langle \\Psi_0\\mid \\, \n\\mid \\{m_{{\\bm R},\\gamma\\sigma}\\}\\rangle\\langle \\{n_{{\\bm R},\\gamma\\sigma}\\}\\mid\\, \n\\mid \\Psi_0\\rangle \\nonumber\\\\\n&&~~~~~~~~~=\nP^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}}\\,\\delta_{\\{n_{{\\bm R},\\gamma\\sigma}\\}\n\\{m_{{\\bm R},\\gamma\\sigma}\\}} \\nonumber\\\\\n&& ~~~~~~~~=\\prod_{\\gamma\\sigma} \\left(n^{(0)}_{{\\bm R},\\gamma\\sigma}\\right)^{n_{{\\bm R},\\gamma\\sigma}}\\,\n\\left(1-n^{(0)}_{{\\bm R},\\gamma\\sigma}\\right)^{1-n_{{\\bm R},\\gamma\\sigma}}\\,, \\label{Eq_pizero_explicit}\n\\end{eqnarray}\nis diagonal, too. It is actually convenient\\cite{BaroneLanata} to rewrite the operator \\eqn{Eq_multiband_projector} in a mixed basis representation\nas \n\\begin{align}\\label{IIB.Eq_multiband_projector}\n\\Gpl{{\\bm R}} = \\sum_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\}} \\left(\n\\fract{\\Phi_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\},{\\bm R}} }{P^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}}}\\right)\n\\ket{\\Gamma,{\\bm R}}\\bra{\\{n_{{\\bm R},\\gamma\\sigma}\\}}\\,,\n\\end{align}\nwhere $\\mid \\Gamma,{\\bm R}\\rangle$ is a state, e.g. a Fock state, \nin the original basis, whereas $\\mid \\{n_{{\\bm R},\\gamma\\sigma}\\}\\rangle$ is a Fock state in the natural basis. This mixed representation \nsimplifies considerably the calculations. \nIn order to use the Gutzwiller approximation, we  \nneed to impose the two following \nconstraints on the matrix $\\hat{\\Phi}_{\\bm R}$ with elements $\\Phi_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\},{\\bm R}}$:\\cite{BaroneLanata}\n\\begin{align}\n\\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}} &= 1\\,, \\label{Eq_Gw2_constr1}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma\\sigma}\\oc_{{\\bm R}, \\gamma'\\sigma'}} &= \nn^{(0)}_{{\\bm R},\\gamma\\sigma}\\delta_{\\gamma\\gamma'}\\,\\delta_{\\sigma\\sigma'}\\,,\\label{Eq_Gw2_constr2}\n\\end{align}\nwhere $\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma\\sigma}$ is the matrix representation of the Fermi operator in its Fock basis. \nIf these constraints are fulfilled, then within the Gutzwiller approximation, which we recall is exact for infinite-coordination lattices,\nwe have\n\\begin{align}\\label{Eq:renorm_onsite}\n\\expect{{\\Psi_{\\rm G}}}{\\hat{O}_{{\\bm R}}}= \\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{O}_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}}\\,,\n\\end{align}\nwhere $\\hat{O}_{{\\bm R}}$ is the matrix representation of any local operator. \nThe inter-site density matrix can be computed from\n\\begin{align}\\label{Eq:renorm_offsite}\n\\expect{{\\Psi_{\\rm G}}}{\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\\oc_{{\\bm R},m'\\sigma}} = \n&\\sum_{\\gamma\\gamma'}\\, R_{\\gamma m;\\sigma,{\\bm R}}^\\dagger \\,R_{m'\\gamma';\\sigma,{\\bm R'}}^{\\phantom{\\dagger}}\\nonumber\\\\\n& \\expect{\\Psi_{0}}{c^\\dagger_{{\\bm R},\\gamma\\sigma'}c^{\\phantom{\\dagger}}_{{\\bm R'},\\gamma'\\sigma'}} \\,,\n\\end{align}\nwhere\n\\begin{align}\\label{Eq_Rparam2} \nR^{\\dagger}_{\\gamma m,\\sigma,{\\bm R}} = \\fract{\\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\,\n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\oc_{{\\bm R}, \\gamma \\sigma}}}{\\sqrt{n^{(0)}_{{\\bm R},\\gamma\\sigma} (1-n^{(0)}_{{\\bm R},\\gamma\\sigma})}} \\,,\n\\end{align}\ncan be regarded as a wavefunction renormalization matrix. \nHere $\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}$ is the matrix representation of the original operators in the basis of states \n$\\mid \\Gamma,{\\bm R}\\rangle$. When this is the Fock basis constructed by the same original operators, \ntheir matrix representation is actually independent of the basis of single-particle wavefunctions which \nthey refer to, hence it is the same as for the  $\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma \\sigma}$ operators of the natural basis. \nIn reality, in most cases that are relevant for real materials the natural basis that diagonalizes the local density \nmatrix is determined fully by the lattice symmetry, hence it is possible and convenient to write the Hamiltonian \ndirectly in that basis. In the above formulas, this corresponds to identifying the set of labels \n$\\{m\\}$ with $\\{\\gamma\\}$.  Since the natural basis is such both for the uncorrelated on-site density matrix \n\\begin{align}\nn^{(0)}_{{\\bm R} m\\sigma,m'\\sigma'} &= \\langle\\Psi_0\\mid \n\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma} \\oc_{{\\bm R},m'\\sigma'}\\mid\\Psi_0\\rangle \\nonumber\\\\\n&=  \\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\, \\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\oc_{{\\bm R}, m' \\sigma}\n} \\nonumber\\\\\n&= \\delta_{mm'}\\, n^{(0)}_{{\\bm R}, m\\sigma},\\label{II.B-density-matrix-var}\n\\end{align}\nand for the correlated one\n\\begin{align}\nn_{{\\bm R} m\\sigma,m'\\sigma'} &= \\langle\\Psi_{\\text{G}}\\mid \n\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma} \\oc_{{\\bm R},m'\\sigma'}\\mid{\\Psi_{\\rm G}}\\rangle \\nonumber \\\\\n&= \\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\, \\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\oc_{{\\bm R}, m' \\sigma}\\, \n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}} \\nonumber\\\\\n&=\n\\delta_{mm'}\\, n_{{\\bm R}, m\\sigma},\\label{II.B-density-matrix-true}\n\\end{align}\ngenerally with different eigenvalues, it is not difficult to realize that the wavefunction renormalization \nmatrix \\eqn{Eq_Rparam2} becomes diagonal, i.e. \n\\begin{align}\\label{Eq_Rparam2-1} \nR^{\\dagger}_{m' m,\\sigma,{\\bm R}} = \\fract{\\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\,\n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\oc_{{\\bm R}, m' \\sigma}}}{\\sqrt{n^{(0)}_{{\\bm R},m'\\sigma} (1-n^{(0)}_{{\\bm R},m'\\sigma})}} \n= \\delta_{mm'}\\,R_{m\\sigma,{\\bm R}}^\\dagger\\,.\n\\end{align} \n\nThe Eqs.~(\\ref{Eq:renorm_onsite})--(\\ref{Eq_Rparam2-1}) are the basic formulas that allow to \nevaluate the average value of the Hamiltonian as a functional of the Slater determinant and of the matrices \n$\\hat{\\Phi}_{\\bm R}$, hence to solve the variational problem.\n\\section{The Gutzwiller functional in practice}\\label{Sec:Gutz_practice}\nIn this section we show how to perform a density-self-consistent \\ac{ldag} calculation on a realistic system, \nnamely\n$bcc$ Fe which,\nas mentioned in the Introduction, although a\nbasic and supposedly simple system, still exhibits controversial aspects. \n\nWe first have to select the {\\sl correlated} orbitals to be treated by the Gutzwiller operator. In the present \ncase the choice is simple: the $3d$ orbitals of Fe.\nThis case \nis one of those mentioned earlier \nin which the natural basis is determined by symmetry and corresponds to the cubic crystal field split \n$d$ orbitals, namely the $e_g$ doublet and the $t_{2g}$ triplet. In this representation the \nformulas Eqs.~(\\ref{II.B-density-matrix-var})--(\\ref{Eq_Rparam2-1}) hold, which is a great simplification. \nFurthermore, since $bcc$ is a Bravais lattice, the positions ${\\bm R}$ of Fe atoms also label unit cells, hence \nby translational symmetry we can safely assume that the variational matrix parameters $\\hat{\\Phi}_{\\bm R} \n= \\hat{\\Phi}$ are independent of ${\\bm R}$. So are therefore the eigenvalues of the local density matrices, \n$n^{(0)}_{{\\bm R},m\\sigma} = n^{(0)}_{m\\sigma}$ and $n_{{\\bm R},m\\sigma}=n_{m\\sigma}$, as well as the wavefunction \nrenormalization $R_{m\\sigma,{\\bm R}}=R_{m\\sigma}$. To lighten notations, in what follows the orbital labels \n$m$ will refer both to the correlated set and to the uncorrelated ones, unaffected by the action of the Gutzwiller operator. In the last paragraph of this section we shall come back to this point.   \n\nWe define the Gutzwiller density functional as\n\\begin{align}\\label{Eq:F_functional-1}\n&{\\cal F}[n({\\bm r})] = \\min_{{\\Psi_{\\rm G}} \\rightarrow n({\\bm r})} {\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})]\\,.\n\\end{align}\nwhere the quantity ${\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})]$ undergoing constrained minimization is\n\\begin{align}\\label{Eq:E_functional}\n&{\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})] = \\expect{{\\Psi_{\\rm G}}}{\\hat{T}+\\hat{H}_{\\rm int}}+\\nonumber\\\\&+\\int V_{\\rm ext}({\\bm r}) n({\\bm r}) \\de{{\\bm r}}+\\tilde{E}_{\\rm H}[n({\\bm r})] + \\tilde{E}_{\\rm xc}[n({\\bm r})] - {E_{\\rm dc}}[n({\\bm r})]\\,.\n\\end{align}\nFor our purposes, it is convenient to rewrite \\eqn{Eq:F_functional-1} as a minimization constrained \nwith respect to the ``uncorrelated'' density $n^{(0)}({\\bm r})$,\n\\begin{align}\\label{Eq:F_functional}\n&{\\cal F}[n^{(0)}({\\bm r})] = \\min_{\\Gpl{},\\Psi_{0} \\rightarrow n^{(0)}({\\bm r})} {\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})]\\,,\n\\end{align}\nwhere ${\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})] = {\\cal E}[{\\Psi_{\\rm G}}(\\Psi_{0},\\Gpl{}),n(\\Psi_{0},\\Gpl{})]$.\nThe dependence of the ``correlated'' density $n({\\bm r})$ upon the ``uncorrelated'' density $n^{(0)}({\\bm r})$ can \nbe made explicit once one writes them in terms of the one-body ``correlated'' density-matrix of the periodic system\n\\begin{align}\\label{Eq:1b_corr_densmat}\n{D}_{mm',\\sigma,{\\bm R}} = \\langle \\Psi_{\\text{G}}\\mid \\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\n\\oc_{\\bm 0,m'\\sigma}\\mid\\Psi_{\\text{G}}\\rangle,\n\\end{align}\nand of the ``uncorrelated'' density-matrix \n\\begin{align}\\label{Eq:1b_uncorr_densmat}\nD^{(0)}_{mm',\\sigma,{\\bm R}}= \\langle \\Psi_{0}\\mid \\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\n\\oc_{\\bm 0,m'\\sigma}\\mid\\Psi_{0}\\rangle,\n\\end{align}\nnamely\n\\begin{align}\nn^{(0)}({\\bm r}) &= \\sum_\\sigma \\, n^{(0)}_\\sigma({\\bm r}) \\nonumber\\\\\n&= \\sum_{m,m',\\sigma, {\\bm R}} D^{(0)}_{mm',\\sigma,{\\bm R}}\\, \n\\phi^\\ast_{m,{\\bm R}}({\\bm r}) \\,\n\\phi_{m',\\bm 0}(\\bm{r})\\, , \\label{Eq_dens_uncorr}\\\\\nn(\\bm{r}) &= \\sum_\\sigma \\, n_\\sigma({\\bm r}) \\nonumber\\\\\n&= \n\\sum_{m,m',\\sigma, \\bm R} {D}_{mm',\\sigma,{\\bm R}} \\, \\phi^\\ast_{m,{\\bm R}}({\\bm r}) \\,\n\\phi_{m',\\bm 0}(\\bm{r})\\, .\n\\label{Eq_dens_corr}\n\\end{align}\nIndeed, ${D}_{mm',\\sigma,{\\bm R}}$ can be obtained by  $D^{(0)}_{mm',\\sigma,{\\bm R}}$ using the recipe of the Gutzwiller Approximation:\n\\begin{align}\\label{Eq_rendensmat}\n{D}_{mm',\\sigma,{\\bm R}} = \\begin{cases}\nR_{m\\sigma}^\\dagger\\, \nD^{(0)}_{mm',\\sigma,{\\bm R}} \\,R_{m'\\sigma}\\,, & \\hspace{-0.3cm}{\\bm R}\\neq \\bm 0\\,,\\vspace{0.3cm}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger \\,\\hat{n}_{mm',\\sigma}\\,\\hat{\\Phi}^{\\phantom{\\dagger}}\\hspace{-0.2cm}}= \n\\delta_{mm'} n_{m\\sigma}\\,,\n& \\hspace{-0.3cm}{\\bm R} = {\\bm 0}\\,,\n\\end{cases}\n\\end{align}\nwhere $\\hat{n}_{mm',\\sigma}$ is the matrix representation on the local Fock space at site ${\\bm R}$ of \n$\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\\hat{c}^{\\dagger}_{{\\bm R},m'\\sigma}$, which is independent of ${\\bm R}$ for a periodic system, \nand where $n_{m\\sigma}$ is equal to $n_{{\\bm R}=0,m\\sigma}$ defined in \\eqn{II.B-density-matrix-true}.\n\nIn order to write ${\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})]$ explicitly in terms of the new variables, we start from \nthe first and second terms of \\eqn{Eq:E_functional}.\nWe can now treat the kinetic and the external potential terms on the same footing through \n\\begin{align}\\label{Eq_kinextgutz_trace}\n& \\expect{{\\Psi_{\\rm G}}}{\\hat{T}} + \\int n({\\bm r}) V_{\\rm ext}({\\bm r}) d{\\bm r} = \\nonumber\\\\\n& \\sum_{m,m',\\sigma,{\\bm R}} \\Big(T_{mm',{\\bm R}} + V^{(\\rm ext)}_{mm',{\\bm R}}\\Big)\n\\,{D}_{mm',\\sigma,{\\bm R}}\\,,\n\\end{align}\nwhere values of $T_{mm',{\\bm R}}$ and $V^{(\\rm ext)}_{mm',{\\bm R}}$ are the  spin-independent matrix elements of \nthe kinetic and external potential operators computed between our basis orbitals at sites ${\\bm R}$ and $\\bm 0$, \ni.e. \n\\begin{align}\nV^{(\\rm ext)}_{mm',{\\bm R}} = \\int \\phi^\\ast_{m,{\\bm R}}({\\bm r}) V_{\\rm ext}({\\bm r}) \n\\phi_{m',\\bm 0}({\\bm r})d {\\bm r}\\,,\\\\\nT_{mm',{\\bm R}} = -\\frac{\\hbar^2}{2m} \\int \\phi^\\ast_{m,{\\bm R}}({\\bm r}) \n\\Big[\\nabla^2 \\phi_{m',\\bm 0}({\\bm r})\\Big] d {\\bm r}\\,\n\\end{align}\nand compute the value of the atomic interaction energy $\\expect{{\\Psi_{\\rm G}}}{\\hat{H}_{\\rm at}}$ using the Gutzwiller Approximation recipe\n\\begin{align}\\label{Eq:eat}\nE_{\\rm at}[\\Psi_{0},\\Gpl{}] = \\expect{{\\Psi_{\\rm G}}}{\\hat{H}_{\\rm at}} = \\Tr{\\hat{\\Phi}^\\dagger \\hat{H}_{\\rm at} \\hat{\\Phi}^{\\phantom{\\dagger}}}\n\\end{align}\nIn order to simplify the density self-consistent \\ac{ldag} minimization we decided to use the\nHartree $\\tilde{E}_{\\text{H}}[n({\\bm r})]$ and exchange-correlation $\\tilde{E}_{\\text{xc}}[n({\\bm r})]$ \nfunctionals as the \\ac{lda} functionals linearized around the uncorrelated density $n^{(0)}({\\bm r})$. \nWe checked {\\sl a posteriori} the accuracy of such a linearization. \nThe modified Hartree functional then reads\n\\begin{align}\\label{Eq:Hartree1}\n\\tilde{E}_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &\\simeq \\frac{e^2}{2}\\int d{\\bm r} d{\\bm r'}\\; \\fract{n^{(0)}({\\bm r}) n^{(0)}({\\bm r'})}{|{\\bm r} - {\\bm r'}|}\n\\nonumber \\\\\n&~~~+ \\int d{\\bm r} \\,\\delta n({\\bm r})\\,v_{\\rm H}[n^{(0)}({\\bm r})]\\,,\n\\end{align}\nwhere $\\delta n({\\bm r}) = \\sum_\\sigma \\delta n_\\sigma({\\bm r})=\\sum_\\sigma n_\\sigma({\\bm r})-n^{(0)}_\\sigma({\\bm r})$ \nand $v_{\\rm H}[n^{(0)}({\\bm r})]$ is the conventional Hartree potential, whereas the exchange-correlation functional is \n\\begin{align}\\label{Eq:exc1}\n\\tilde{E}_{\\rm xc}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &=  \\sum_\\sigma\\, \\int \\de{{\\bm r}} n^{(0)}_\\sigma({\\bm r})\\, \n\\epsilon_{\\text{xc},\\sigma}[n^{(0)}({\\bm r})] \n\\nonumber\\\\\n&~~+  \\int d{\\bm r} \\, v_{\\text{xc},\\sigma}[n^{(0)}({\\bm r})] \\, \\delta n_\\sigma({\\bm r})\\,,\n\\end{align}\n$v_{\\rm xc}[n^{(0)}({\\bm r})]$ being the \\ac{lda} exchange-correlation potential.\nNote that the choice of $\\tilde{E}_{\\rm H}$ involves neglecting a term\n\\begin{align}\n\\DeltaE_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &= \\tilde{E}_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right]-E_{\\rm H}[n({\\bm r})] =\\nonumber\\\\\n&=\\frac{e^2}{2}\\int d{\\bm r} d{\\bm r'}\\, \\fract{\\delta n({\\bm r}) \\delta n({\\bm r'})}{|{\\bm r} - {\\bm r'}|}\n\\end{align}\nwhich can be interpreted as the energy of correlation-induced charge fluctuations. This term, together with \nthe corresponding one neglected for the exchange-correlation functional, $\\DeltaE_{\\rm xc}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right]$, \ncan be computed at the end of the \\ac{ldag} calculation in order to provide an estimate of the error due to \napproximations~\\eqref{Eq:Hartree1} and~\\eqref{Eq:exc1} (see \\rtab{Tab:delta_energies}).\nIt is worth mentioning that the linearization~\\eqref{Eq:exc1} of exchange-correlation energy around \nthe ``uncorrelated'' density does not spoil the sum rule for the \\ac{lda} exchange-correlation hole.\nAs for the double-counting term, similarly to what is done within \\ac{ldau}, it is chosen as a function \nof the local ``uncorrelated'' density-matrix $n^{(0)}$ only, ${E_{\\rm dc}}[n({\\bm r})] = {E_{\\rm dc}}[n^{(0)}]$. In \\sect{Sec:results} we take\nas its explicit form the one of \\eqn{double-counting-Hund}, having chosen our atomic interaction Hamiltonian $\\hat{H}_{\\rm at}$\nto be the expression of \\eqn{Eq:hHat_approssimata}.\n\n\\subsection{Three-step minimization of the LDA+Gutzwiller functional}\\label{Sec:threestep_mini}\n\nThe two densities $n({\\bm r})$ and $n^{(0)}({\\bm r})$ must be such that Gutzwiller constraints are fulfilled. \nIn our case where original and natural basis coincide, the constraints on the density matrix can be written as\n\\begin{align}\nD^{(0)}_{mm', \\sigma,{\\bm R}= \\bm 0} &= n^{(0)}_{m\\sigma}\\,\\delta_{mm'}\\,,\\label{Eq_df2_gwconst1}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}} \\hat{n}_{mm',\\sigma}} &= n^{(0)}_{m\\sigma}\\,\\delta_{mm'}\\,,\\label{Eq_df2_gwconst2}\n\\end{align}\nwhere we regard $n^{(0)}_{m\\sigma}$ as an additional independent variational parameter of the density \nfunctional. These constraints can be enforced with Lagrange multipliers, together with the first Gutzwiller constraint \n\\begin{align}\n\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}}} &= 1\\,.\\label{Eq_df_gwconst}\n\\end{align}\n\nSumming up all contributions and adding the electrostatic ion-ion interaction $E_{\\rm ion}$, \nwe find that the overall functional we need to minimize has the form\n\\begin{widetext} \n\\begin{align}\\label{Eq_Gwdensfunc_full}\n{\\cal F}\\Big[n({\\bm r}),n^{(0)}({\\bm r}),n^{(0)}_{m\\sigma}\\Big] &= \\max_{\\lambda\\lambda'\\lambda_0} \n\\Bigg[{\\cal K}[n({\\bm r})] +E_{\\rm at}[n({\\bm r})] -{E_{\\rm dc}}[n^{(0)}_{m\\sigma}]+E^{(0)}_{\\rm H}[n^{(0)}({\\bm r})] + E^{(0)}_{\\rm xc}[n^{(0)}({\\bm r})]\n- \\lambda_0 \\left(\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}}} - 1\\right)\\nonumber \\\\\n& -\\sum_{mm'\\sigma}\\, \n\\lambda'_{mm',\\sigma}\\left(D^{(0)}_{mm',\\sigma, {\\bm R}=\\bm 0}-n^{(0)}_{m\\sigma}\\delta_{mm'}\\right) \n- \\lambda_{mm',\\sigma} \\left(\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}} \\hat{n}_{mm',\\sigma}}\n-n^{(0)}_{m\\sigma}\\delta_{mm'}\\right)\n\\Bigg] +E_{\\rm ion},\n\\end{align}\n\\end{widetext}\nwhere the functional ${\\cal K}[n({\\bm r})]$ contains all terms which depend on $n({\\bm r})$ linearly through \nthe renormalized density matrix ${D}$, namely\n\\begin{align}\n{\\cal K}({D}) &= \\sum_{mm',\\sigma,{\\bm R}} \\bigg[T_{mm',{\\bm R}}+V^{(\\rm H)}_{mm',{\\bm R}}\n+\nV^{(\\rm xc)}_{mm',\\sigma,{\\bm R}}\\nonumber\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~ +V^{(\\rm ext)}_{mm',{\\bm R}}\\bigg]\\, {D}_{mm',\\sigma,{\\bm R}}\\nonumber\\\\\n&\\equiv \\sum_{mm',\\sigma,{\\bm R}} \\, {\\cal K}_{mm',\\sigma,{\\bm R}}\\, {D}_{mm',\\sigma,{\\bm R}}\\,,\n\\end{align}\nwhere $V^{(\\rm H)}_{mm',{\\bm R}}$ and $V^{(\\rm xc)}_{mm',\\sigma,{\\bm R}}$ are the matrix elements of $v_{\\rm H}$ and $v_{\\rm xc}$ \nbetween the basis orbitals. \nFor every fixed value of $n^{(0)}_{m\\sigma}$, we can optimize ${\\cal F}$ with respect to the two \ndensities $n^{(0)}({\\bm r})$ and $n({\\bm r})$. In practice, by inspection of equations~\\eqref{Eq_dens_uncorr}, \n\\eqref{Eq_dens_corr} and \\eqref{Eq_rendensmat} one can see that this is equivalent to a minimization \nwith respect to the Slater determinant $\\ket{\\Psi_{0}}$ and the Gutzwiller parameters\\ contained in the operator $\\hat{\\Phi}^{\\phantom{\\dagger}}$.\nThis minimization can be carried out in two separate steps:\n\\begin{enumerate}\n\\item first carry out a Siesta self-consistent calculation to find the Slater determinant $\\Psi_{0}$ \nthat optimizes ${\\cal F}[n({\\bm r}),n^{(0)}({\\bm r}), n^{(0)}_{m\\sigma}]$ with respect to $n^{(0)}({\\bm r})$, enforcing the \nconstraint~\\eqref{Eq_df2_gwconst1} through an Augmented Lagrangian Method~\\cite{Fletcher}.\nThe Gutzwiller parameters, and therefore the hopping renormalization parameters $R_{m\\sigma}$, are kept fixed throughout \nthis optimization. The atomic energy $E_{\\rm at}[n({\\bm r})]$ does not change, nor does the double-counting energy \n${E_{\\rm dc}}[n^{(0)}({\\bm r})]$, which is a function of $n^{(0)}({\\bm r})$ only through $n^{(0)}_{m\\sigma}$.\nThe self-consistent single-particle Kohn-Sham equations allowing the minimization with respect to $\\ket{\\Psi_{0}}$ are\n\\begin{align}\n\\sum_{m'{\\bm R}}\\,\\mathcal{H}_{mm',\\sigma,{\\bm R}}\\,\\psi_{m'\\sigma,{\\bm R}} &= \\varepsilon\\, \\psi_{m,\\sigma,\\bm0}\\,,\n\\end{align}\nwhere\n\\[\n\\mathcal{H}_{mm',\\sigma,{\\bm R}} = {\\cal K}_{mm',\\sigma,{\\bm R}}+V^{(\\rm 0)}_{mm',\\sigma,{\\bm R}} - \\lambda'_{mm',\\sigma}\n\\delta_{{\\bm R}\\bm 0},\n\\]\nand \n\\begin{align}\nV^{(\\rm 0)}_{mm',\\sigma,{\\bm R}} &= \\int \\de{{\\bm r}} \\phi^\\ast_{m,{\\bm R}}({\\bm r})\\Big\\{v_{\\rm H}[n^{(0)}({\\bm r})] \\nonumber\\\\\n&~~~~~~~~~+ v_{\\rm xc}[n^{(0)}({\\bm r})]\\Big\\}\\phi_{m',\\bm 0}({\\bm r})\\,.\n\\end{align}\n\\item next, optimize ${\\cal F}$ with respect to Gutzwiller parameters\\ by a Lanczos-improved \\ac{lm} algorithm (see ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection*{Introduction}\n\nVERITAS, located at the Fred Lawrence Whipple Observatory in southern\nArizona, is one of three major imaging atmospheric Cherenkov telescope\nfacilities in operation worldwide. It consists of an array of four,\n12-meter diameter telescopes, providing a $<1\\%$ Crab-flux sensitivity in\nthe energy range between $100\\U{GeV}$ and $50\\U{TeV}$. The array\nhas been operating since 2007 and has detected 42 objects from\n$\\sim10$ different source classes, including many new discoveries\n\\cite{veritas_status}.\n\nIn 2012, the VERITAS collaboration established a long-term plan\ndescribing a scientific strategy for operations. Many of the goals\noutlined address problems relevant to the charge of the CF6\nsubgroup. An additional white paper describing the VERITAS indirect\ndark matter detection program has been submitted to subgroup CF2\n\\cite{andyWP}. A prerequisite for achieving these goals was the\nsuccess of a major upgrade to the array which was completed, with no\ndisruption to the array operations, in the summer of 2012. The upgrade\ninvolved the installation of a new trigger system, and the replacement\nof all of the photodetectors with super-bialkali photomultiplier\ntubes. This has resulted in an increase of at least 35\\% in photon\ncollection efficiency. VERITAS will remain the premier VHE facility in\nthe Northern Hemisphere for some time, while the next-generation\nCherenkov Telescope Array (CTA) project is under development. The\ncontemporaneous overlap of VERITAS operations with Fermi-LAT, HAWC,\nIceCube and Auger will be of critical importance to many of the\nscience goals described here.\n\n\\subsubsection*{Cosmic Particle Acceleration as Signal and Background}\n\nVERITAS has made significant contributions to the study of Galactic\nparticle accelerators, including pulsars and their nebulae, gamma-ray\nbinary systems and supernova remnants. Highlight results include the\ndetection of $>100\\U{GeV}$ emission from the Crab pulsar\n\\cite{crabpulsar}, and the first TeV detection of Tycho's SNR\n\\cite{tycho}. The Crab pulsar result can only be easily explained by a\nnew emission mechanism, or an additional component at high energies,\nThe Tycho detection, combined with results from the Fermi-LAT,\nprovides compelling evidence for hadronic particle acceleration in\nSNR. Complementary evidence for a link between cosmic ray production\nand star formation activity was provided by the discovery of\ngamma-ray emission from the starburst galaxy M82 \\cite{m82}. \n\nHAWC \\cite{HAWC} will soon begin observations, and will provide a complete\nTeV map of the northern sky. Follow-up observations with high\nsensitivity and better angular and energy resolution will be performed\nby VERITAS. In particular, this is key to determining the nature of\nunidentified 'dark accelerators', as already demonstrated by VERITAS\nobservations of Milagro sources. The resolution of gamma-ray emission\nin the region of MGRO~J2019+37 and the Cygnus OB1 association into at\nleast two distinct sources, one clearly associated with the pulsar\nwind nebula CTB~87, demonstrates the importance of the excellent\nangular resolution provided by the imaging technique. Contemporaneous\noperation of VERITAS and HAWC will also allow a rapid response to\ntransient events, such as blazar flares and gamma-ray bursts.\n\nThe study of VHE gamma-ray emission from particle accelerators is, of\ncourse, interesting from a purely astrophysical perspective. It is\nalso critical to understand the nature and properties of astrophysical\nbackgrounds in searches for new physical effects (this issue is\naddressed in detail in a separate CF6 white paper\n\\cite{amandaWP}). The interpretation of indirect dark matter searches,\nLorentz Invariance Violation (LIV) tests, studies of gamma-ray and\nantimatter backgrounds and searches for axion-like particles all rely\non an accurate knowledge of the potential astrophysical backgrounds\nand their spectral, morphological and temporal properties. A classic\nexample of this is the case of the Galactic Center, which has the\nhighest local concentration of dark matter, but also hosts multiple\nknown and potential astrophysical TeV sources, both point-like and\nextended \\cite{andyWP}. VERITAS observations of the Galactic Center\nare ongoing as part of our long-term observing plan. For this southern\nsource, the observations take place at low elevation angles, resulting\nin a high energy threshold but providing an increase in the effective\ncollection area at high energies. This allows us to probe the end\npoint of the spectrum of the Galactic Center gamma-ray emission, which\nmay hold the key to resolving the nature of the source.\n\n\n\\subsubsection*{Probing Fundamental Physics}\n\nVERITAS is a mature experiment, and has moved beyond the initial\nsource discovery phase. Fundamental physics topics now play an\nincreasingly important role in the observing plan. The success of\nthese studies, which often require long and technically challenging\nexposures, relies on stable operation and a thorough knowledge of\nthe detector performance, calibration and associated Monte Carlo\nsimulations. After five years of operations, all of these aspects of\nVERITAS are very well understood. Indirect dark matter searches are\ndescribed elsewhere \\cite{andyWP}. Other topics which we plan to\ninvestigate with VERITAS in the coming years include:\n\n\\begin{itemize}\n\\item{\\bf Antimatter studies:} The rising positron fraction identified\n  by PAMELA \\cite{PAMELA} and confirmed by Fermi \\cite{fermipositron,\n    fermipositron2} up to a few hundred GeV is an intriguing result. It may be explained by a contribution from local astrophysical\n  sources, or possibly by annihilating dark matter. First results from\n  AMS-02 confirm that the positron fraction continues to rise up to at\n  least $250\\U{GeV}$, at which point it appears to flatten \\cite{AMS}.  A\n  measurement of the positron fraction at higher energies would\n  provide a key discriminant between the competing\n  explanations. VERITAS is attempting to make such a measurement by\n  observing the shadow of the Moon in both electrons and positrons, as\n  proposed by Colin \\cite{colin}. This is technically challenging, due\n  to the optical sky brightness close to the Moon, and the limited\n  amount of observing time available at high elevations. We have\n  developed short-wavelength optical filter plates for the telescope cameras\n  to enable us to observe close to the Moon, and the results of\n  preliminary test observations are encouraging. Observations over the\n  next few years should allow us to build up the necessary exposure\n  required for this unique measurement.\n\n\\item{\\bf Primordial Black Holes:} In addition, or as an alternative\n  to, particle dark matter, primordial black holes (PBHs) formed\n  during the early universe can serve as a viable candidate for\n  cosmological dark matter (see \\cite{PBH}). PBHs can evaporate\n  through Hawking radiation, where the evaporation rate is directly\n  coupled to their mass. Consequently, during the final seconds of\n  their lifetime, PBHs can release a large flux of gamma rays within\n  the sensitivity range of VERITAS. Dedicated searches for these PBH\n  signals have already commenced with VERITAS \\cite{gordana}, and an\n  evaporation rate limit of\n  $\\rho_{PBH}<1.29\\times10^5\\UU{pc}{-3}\\UU{yr}{-1}$ has been placed\n  using only 700 hours of VERITAS observations. This limit is already\n  an order of magnitude below previous limits. VERITAS accrues\n  approximately 800 hours of Moonless observations each year, so a\n  significant refinement of the result can be expected.\n\n\\item{\\bf Cosmological measurements using the EBL and IGMF:} The\n  gamma-ray spectra of blazars are modified by interactions with\n  intergalactic radiation fields through pair-production and\n  subsequent cascade processes. As a result, these spectra contain an\n  imprint of the extragalactic background light (EBL) and the\n  intergalactic magnetic field (IGMF). The EBL comprises the combined\n  flux of all extragalactic sources integrated over the history of the\n  Universe, and carries unique information regarding the epoch of\n  galaxy formation and the history of galaxy evolution. This topic is\n  discussed in detail in a related white paper \\cite{frankWP}. The\n  IGMF strength is only weakly constrained, and impossible to measure\n  directly. VERITAS observations of the spectra, angular distribution\n  and arrival times of gamma-rays from distant blazars will provide\n  constraints to, or a measurement of, the IGMF strength which is not\n  accesible to other techniques. A positive measurement would be\n  important, possibly implying the existence of a primordial\n  field produced in the early Universe. Both EBL and IGMF measurements\n  require deep, multi-year exposures of numerous blazars over a range\n  of redshifts out to $z\\sim0.5$, as envisaged in our long-term\n  observing plan.\n\n\\item{\\bf Tests of Lorentz Invariance Violation (LIV):} Blazar observations\n  provide the most stringent tests of LIV for VERITAS, thanks to their\n  large distance and rapid timescale of variability. Four bright,\n  high-energy peaked BL Lac objects have been identified for deep\n  monitoring exposures of $\\geq100\\U{hours}$ in our long-term plan. An\n  additional target-of-opportunity program allows us to respond\n  rapidly to alerts of enhanced emission from instruments at other\n  wavelengths. The detection of VHE emission from the Crab pulsar also\n  raises the possibility of using pulsar time profiles to constrain LIV\n  \\cite{crabLIV}, and we plan to substantially augment our already\n  extensive Crab pulsar dataset over the coming years, as well as to\n  search for pulsed emission from other candidate sources. The energy\n  threshold reduction provided by the 2012 upgrade will be\n  particularly important in this regard.\n\n\\end{itemize}\n\n\\subsubsection*{UHECRs and Neutrino Astrophysics}\n\nVERITAS observations impact the related fields of ultra-high energy\ncosmic rays (UHECRs) and neutrino astrophysics. The UHECRs are most\nlikely extragalactic in origin, with active galactic nuclei (AGN)\namong the best candidates for the accelerators. Gamma-ray observations\nin the GeV-TeV range are essential to constrain models of particle\nacceleration and gamma-ray\/ neutrino emission in these sources (see\n\\cite{dummWP} for more details). Our long-term plan calls for regular\nmonitoring of most of the northern hemisphere VHE blazar population\nover the next five years, allowing us to accumulate deep exposures of\nthe sources in various emission states, and maximizing our chances of\ndetecting bright VHE flares. Observations of the nearby radio galaxy\nM87 will also continue, and will be complemented by high resolution\nX-ray and radio observations in the event of a flare. A clear\ncorrelation between morphological changes in the jet structure and the\nVHE emission state could help to finally pin down the particle\nacceleration and photon emission region in AGN jets.\n\nVERITAS can also act as a flare alert system for the UHECR and\nneutrino observatories, and provide rapid, high sensitivity follow-up\nobservations. In response to the early Auger reports of a correlation\nbetween ultra-high energy cosmic rays and AGN, VERITAS was the first\ninstrument to provide follow-up TeV gamma-ray observations\n\\cite{UHECRs}. No gamma-ray emission was seen, and the evidence for a\ncorrelation has diminished over time, but the observations demonstrate\nthe substantial overlap between the two instruments, despite their\nlocations in different hemispheres. IceCube, conversely, can easily\nview the northern sky, and VERITAS and IceCube are very well-matched\nin energy range (IceCube has a minimum neutrino energy threshold of\n50-100GeV, and an optimal response above $1\\U{TeV}$\n\\cite{IceCube}). Numerous predictions of measurable neutrino fluxes\nassociated with astrophysical particle accelerators exist in the\nliterature, including both Galactic (SNRs, binary systems,\nunidentified TeV sources and pulsar wind nebulae \\cite{neutrinos_gal1,\n  neutrinos_gal2}) and extragalactic (GRBs, active and starburst\ngalaxies \\cite{neutrinos_xgal}) objects. VERITAS is the best instrument\nto search for and characterize the electromagnetic signatures of\nparticle interactions in these objects, which will be necessary to\nassess the relative contributions of leptonic and hadronic particle\npopulations. We will perform follow-up gamma-ray observations of any\nreported neutrino sources, and have established a memorandum of\nunderstanding between VERITAS and IceCube which allows us to rapidly\ntrigger observations of any transient neutrino excess. IceCube and\nAuger will be at their most productive over the coming five years, and\nVERITAS observations will both complement and augment their results.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Motivation}\n\\section{Motivation}\\label{s:motivation}\n\nEvidence is accumulating that star formation follows rapidly upon molecular\ncloud formation (e.g. \\citealp{2001ApJ...562..852H} and \\citealp{2007RMxAA..43..123B}\nfor the solar neighborhood; \\citealp{2003ApJS..149..343E} for M33;\n\\citealp{2007ApJ...668.1064E} in the context of M51). This rapid onset\nsuggests that the clouds need to acquire high, non-linear density enhancements\nduring their formation, since massive, finite clouds are highly susceptible\nto global gravitational collapse which could overwhelm small-scale fragmentation\nnecessary for (local) star formation \\citep{2004ApJ...616..288B}. \nThus to understand the initial conditions for star formation, we need to \nunderstand the formation of the parental clouds.\n \n\\citet{1999ApJ...527..285B} and \\citet{2001ApJ...562..852H} proposed that the build-up\nof clouds in large-scale, converging flows of diffuse atomic gas\ncould explain the crossing time problem, i.e. the observation that the stellar age \nspreads in a large number of local star forming regions are substantially smaller \nthan the lateral crossing time \\citep{2001ApJ...562..852H,2007RMxAA..43..123B}. \nIn this picture, there need not be a causal connection between star formation events \nin the plane perpendicular to the large-scale flows (see also \\citealp{2000ApJ...530..277E}). \nRapid star formation is a necessary requirement for this scenario to work.\n\nNumerical models of flow-driven cloud formation (we give only an early and the most\nrecent numerical work of each group, namely \n\\citealp{1999A&A...351..309H} and \\citealp{2008A&A...486L..43H};\n\\citealp{2000ApJ...532..980K} and \\citealp{2008ApJ...687..303I};\n\\citealp{2005ApJ...633L.113H} and \\citealp{2008ApJ...674..316H}; \n\\citealp{2006ApJ...643..245V} and \\citealp{2007ApJ...657..870V})\nhave identified the thermal and dynamical instabilities that are responsible for the \nrapid fragmentation of the  nascent cloud (see \\citealp{2008ApJ...683..786H} \nfor an assessment of the roles of the physical processes). Despite these promising successes, \nmany questions about the physics at play remain unanswered, among one of the most pressing is  \nthe role of magnetic fields during the cloud formation process. \n\nThe role of magnetic fields in the flow-driven cloud formation scenario has been largely\nenvisaged as one of ``guiding the flows'' to assemble the clouds, whether in\nform of the Parker instability along galactic spiral arms \n\\citep{1966ApJ...145..811P,1967ApJ...149..517P,1974A&A....33...73M},\nin a generally turbulent interstellar medium (ISM) \n\\citep{1995ApJ...455..536P,2001ApJ...562..852H}, \nor during the sweep-up of gas in spiral shocks \\citep{2006ApJ...646..213K,2008MNRAS.383..497D}.\nBased on the models of Passot et al., Hartmann et al. suggested that the field orientation\nwith respect to the flows selects the locations of cloud formation, namely that\nclouds will only form if the fields are aligned with the flows. A perpendicular\nfield will reduce the compression of the post-shock gas and thus will limit the strong cooling and \nthe thermal instability (TI, \\citealp{1965ApJ...142..531F}) necessary for the rapid \nflow fragmentation and the build-up of high-density contrasts \n\\citep{2004ApJ...616..288B,2008ApJ...674..316H,2008ApJ...683..786H}. \n\nGiven sufficiently high strengths,\nfields aligned with the inflows can suppress the dynamical instabilities responsible\nfor the generation of turbulence, namely the non-linear thin shell instability (NTSI, \n\\citealp{1994ApJ...428..186V}; for a magnetic version see \\citealp{2007ApJ...665..445H}) \nand the Kelvin-Helmholtz instability (KHI, e.g. \\citealp{1961hhs..book.....C}, \nmore recently \\citealp{2008MNRAS.385.1494K}, and for numerical studies \\citealp{2008ApJ...678..234P}).\nYet magnetic fields are intrinsically three-dimensional, and already two-dimensional\nmodels by \\citet{2008ApJ...687..303I} show that even for fields perpendicular to the inflow,\ncold (albeit diffuse) clouds can form. \n\nThus, three-dimensional models of flow-driven cloud formation including magnetic fields \nare needed. \\citet{2008A&A...486L..43H} present a first approach to the problem, \nmodeling the formation of a cloud in converging, perturbed flows, including fields and\nself-gravity. Here, we focus on the \neffects of magnetic field strength and orientation on the early stages\nof flow-driven cloud formation. We work in the ideal MHD limit (i.e. we do not\nexplicitly consider ambipolar drift or resistivity), and we do not include gravity in the models.\n\nAll our models start out with field strengths below equipartition with the kinetic energy\nof the inflows. At a factor of $4.3$ below equipartition -- corresponding to an\nabsolute field strength of $5\\mu$G at flow densities and velocities of $1$~cm$^{-3}$ \nand $16$~km~s$^{-1}$ -- , the fields already suppress the dynamical instabilities \n(and thus the generation of turbulence) leading to slab-like molecular clouds, while weaker \nfields -- at $2.5\\mu$G, corresponding to a factor of $17$ below equipartition -- lead to clouds \nclosely resembling the hydrodynamical case, albeit with more coherent filaments. \nFields at $0.5\\mu$G perpendicular to the inflows suppress the build-up of massive clouds in the collision\nplane, while they lead to the formation of diffuse, cold filaments perpendicular\nto the mean field, reminiscent of the cold HI clouds discussed by \\citet{2003ApJ...586.1067H}.\nA tangled field allows the assembly of substantial column densities\nin regions where the lateral field component is small or vanishing.\nOur results are consistent with the notion \nthat magnetic fields select the location of cloud formation \\citep{2001ApJ...562..852H}.\n\n\\section{Technical Details \\& Parameters}\n\n\\subsection{Athena}\nCalculations were performed with Athena \\citep{2005JCoPh.205..509G,2008JCoPh.227.4123G}, \nan unsplit, second-order accurate Godunov scheme, using the corner transport upwind method\n\\citep{2008JCoPh.227.4123G} and a linearized Roe solver \\citep{1981JCoPh..43..357R}. \nThe divergence of the magnetic field is kept zero by using constrained transport\n\\citep{1988ApJ...332..659E}. Dissipative terms (viscosity, heat conduction and resistivity) \nare not explicitly included.\nFor a detailed description and test results, the reader is referred to \n\\citet{2005JCoPh.205..509G,2008JCoPh.227.4123G} and \\citet{2008ApJS..178..137S}.\n\nWe implemented heating and cooling as an additional energy source term at 2nd order in time.\nA tabulated cooling function provides the energy change rate as function of density and temperature\nat each grid cell. We decided to keep the iterative approach we had used in our earlier studies\nof cloud formation \\citep{2005ApJ...633L.113H,2006ApJ...648.1052H,2008ApJ...674..316H}, \nwith a slight modification. Instead of advancing the fluid evolution at the usual time step\ngiven by the Courant-Friedrichs-Levy (CFL) condition and subcycling on the energy equation in case\nthe cooling timescale is shorter than the CFL timestep, we lower the CFL timestep according to \n\\begin{equation}\n  \\Delta t = \\Delta t_{CFL} \\min(1,(\\tau_c\/\\Delta t_{CFL})^p),\n\\end{equation}\nwith $0\\leq p\\leq 1$. For increasing $p$, small cooling timesteps will control the overall CFL\ntimestep. The earlier version (see references above) would be equivalent\nto $p\\equiv 0$. Yet this choice can lead to inconsistencies in the hydrodynamical evolution \nonce the cooling timesteps get substantially shorter than the fluid timesteps, leading to a \nnumerical overemphasis of the acoustic mode of the TI, since regions can cool substantially \nwithout accounting for the resulting pressure drop in the dynamics. While these inconsistencies \nmay not affect the overall results, they turn out to affect the stability of the solution. \nFor the models presented here, $0.5<p<0.7$ yields a stable and accurate solution. \nWhile the iterative approach is more time-consuming than a direct integration as suggested by \ne.g. \\citet{2007ApJ...657..870V}, it allows us to take into account the increasing thermal \ntimescale of a fluid parcel cooling down towards the thermal equilibrium line \n(see e.g. Fig~3 of \\citealp{2008ApJ...683..786H}). This effectively keeps \na fraction of $40$--$50$\\% of the gas mass in the thermally unstable warm neutral medium (WNM), \nconsistent with observed fractions (\\citealp{2003ApJ...586.1067H}; also \\S\\ref{ss:thermals}).\n\n\\subsection{Setup and Parameters}\nThe initial conditions and flow parameter are similar to those used in our earlier\nwork \\citep{2008ApJ...674..316H}.\nTwo identical uniform flows collide head-on at an interface whose position is perturbed\nusing an amplitude that is chosen randomly, but with the constraint that only the long\nwavelengths (down to a quarter of the box size) are non-zero. The maximum perturbation\namplitude measures $10$\\% of the box size. The flows are otherwise in thermal equilibrium\nand do not have substructure. Thus we can test the most unfavorable conditions for \nsubstructure and turbulence generation in the resulting clouds. \n\nThe simulation domain has a resolution of  $256^3$, and \nthe linear box size measures $44$~pc in all cases, resulting in a nominal resolution of $0.18$~pc. \nThe boundaries in $y$ (and $z$) are periodic, while those in $x$ are set to a constant inflow speed \nof $v_0=16$~km~s$^{-1}$, a density of $n_0=1$~cm$^{-3}$, a temperature of $T_0=8\\times 10^3$~K and a \nmagnetic field (if applicable). The adiabatic exponent $\\gamma=5\/3$.\nThe flow speed $v_0$ is consistent with shock speeds along streamlines passing through Galactic\nspiral shocks at the solar circle \\citep{1972ApJ...173..557S}.\n Our thermal pressure is a factor $2-3$ higher than the pressures estimated for the CNM\/WNM \n\\citep{2001ApJS..137..297J,2003ApJ...586.1067H}, increasing the ratio of thermal over \nmagnetic pressure $\\beta_{th}$ above observed values (\\citealp{2005ApJ...624..773H}, hereafter HT05). \nYet the measure relevant for the dynamical importance of magnetic fields is the ratio of the \n{\\em ram} pressure over the magnetic pressure,\n$\\beta_{ram}$, since the instabilities responsible for turbulence generation and fragmentation\ndepend on the ratio of the flow velocities over the Alfv\\'{e}n speed\n(\\citealp{1961hhs..book.....C}; \\citealp{2007ApJ...665..445H}). In the diffuse, local \nISM, kinetic and magnetic energy are in approximate equipartition ($\\beta_{ram}\\approx 1$; HT05), \nwhile in regions of ordered, large-scale flows -- such as Galactic spiral shocks or expanding\nsupernova shells -- with $v_0\\gtrsim 10$~km~s$^{-1}$, $\\beta_{ram}>1$ should be expected. \n\nThere are four classes of models: hydrodynamical (series H), field aligned with flow (series X),\nfield perpendicular to the flow (series Y), and (series XR) a uniform field component aligned with \nthe flow plus a random field component of similar size, consistent with (although a little smaller than) \nobserved magnetic field \nstrength estimates (e.g. \\citealp{1996ASPC...97..457H}; \\citealp{2004Ap&SS.289..293B};\n\\citealp{2006ChJAS...6b.211H}). \nIn the latter series, we do not perturb the \ncollision interface but rely on the tangled field component to trigger the dynamical instabilities.  \nTable~\\ref{t:param} summarizes the model parameters. Self-gravity is not included in the models.\n\nTo initialize the random field component, we set the amplitudes and phases of e.g. the $x$-component\nof the (edge-centered) vector potential to \n\\begin{equation}\n  A_x(x,y,z) = \\sum_{i,j,k=1}^{max} |k|^{-p}\\sin(k_xx+k_yy+k_zz+\\phi_{i,j,k}^x),\\label{e:vecpot}\n\\end{equation}\nwhere $|k|\\equiv k_x^2+k_y^2+k_z^2$ and e.g. $k_x\\equiv 2\\pi i\/L_x$ with the box length $L_x$. \nWe set $p\\equiv 4$, mimicking a (steep) turbulent energy spectrum as observed in detailed\nnumerical simulations of magneto-hydrodynamic turbulence (e.g. \\citealp{2003MNRAS.345..325C}).\nThe wavenumbers $k_{x,y,z}$ are chosen such that $1\\leq |k| \\leq 4$, i.e. all\ncombinations of $(i,j,k)$ in the sum over $k$-space are used that satisfy the constraint on $|k|$. \nThe phases $\\phi_{i,j,k}^x$ in $k$-space are chosen from a uniform random distribution. Each vector\npotential component $A_{x,y,z}$ requires a separate phase array $\\phi^{x,y,z}$. \n\nThis formulation in real space instead of in Fourier space (see e.g. \n\\citealp{1998PhRvL..80.2754M}; \\citealp{1998ApJ...508L..99S}; \\citealp{2008ApJ...682L..97L} for\nvelocity fields) allows us to easily regenerate \nthe vector potential (and the field) at the inflow boundaries by\n\\begin{equation}\n  A_x(\\pm L_x\/2,y,z,t) = A_x(\\pm (L_x\/2+v_0t),y,z),\\label{e:bc}\n\\end{equation}\nwhere the negative value refers to the lower $x$-boundary, and the positive to the upper one. \nThe face-centered fields are then computed from the vector potential by \n$\\mathbf{B}=\\nabla\\times\\mathbf{A}$. \n\nThe choice of the wave-number range $1 \\leq |k| \\leq 4$ does not constitute a restriction\nin terms of generality of our simulations, since the energy distribution over spatial scales\nis determined by the (steep) power law index $p$. This is fortunate in a sense, since the \ngeneration of the boundary conditions (eq.~[\\ref{e:bc}]) would consume substantially more time\nif we had to sum over all available $|k|$ in equation~(\\ref{e:vecpot}).\n\n\\begin{deluxetable}{lcccccc}\n\\tablewidth{0pt}\n\\tablecaption{Model Parameters\\label{t:param}}\n\\tablehead{\\colhead{Name}&\\colhead{$B_{x0}$ [$\\mu$G]}\n                         &\\colhead{$B_{y0}$ [$\\mu$G]}\n                         &\\colhead{$B_{rms}$ [$\\mu$G]}\n                         &\\colhead{$\\beta_{th}$}\n                         &\\colhead{$\\beta_{ram}$}}\n\\startdata\nH      & $0.0$ & $0.0$ & $0.0$ & $\\infty$ & $\\infty$ \\\\\nX25    & $2.5$ & $0.0$ & $0.0$ & $4.5$    & $17$     \\\\\nX50    & $5.0$ & $0.0$ & $0.0$ & $1.1$    & $4.3$    \\\\\nY05    & $0.0$ & $0.5$ & $0.0$ & $110$    & $430$    \\\\\nXR25   & $2.5$ & $0.0$ & $2.5$ & $2.2$    & $8.5$      \n\\enddata\n\\tablecomments{1st column: model name, \n2nd: magnetic field strength $B_x$; 3rd: $B_y$, 4th: random field $B_{rms}$, \n5th: thermal plasma $\\beta$, 6th: ram plasma $\\beta$.}\n\\end{deluxetable}\n\n\\subsection{Physical Interpretation of the Initial Conditions}\n\nObviously, our initial conditions are somewhat idealized, e.g. \ngenerally, the flows would be expected to have substructure, \nthe flows might not be expected to collide always head-on, and the\nmagnetic fields will have parallel and perpendicular components with respect\nto the inflows. Yet the initial conditions can be seen as idealized versions \nof different physical environments.\n\nThe case of uniform fields aligned with the inflows (models X25, X50) \ncould be identified with the sweep-up of material by an expanding supernova \nshell along an ordered background field, or with the collision of two expanding \nshells in such a field. The initial field strength of $5\\mu$G (model X50) \nis close to the local median (total) field strength in the CNM (e.g. \nHT05; \\citealp{2005ASPC..343...64T}). \nUsing Nakano \\& Nakamura's (\\citeyear{1978PASJ...30..671N})\nexpression for the critical surface density $N_c\\equiv B\/\\sqrt{4\\pi^2 G}$ \nabove which gravitational collapse is possible under flux-freezing conditions, \nthe swept-up  clouds would reach approximately $0.5N_c$ after $12$~Myr, while model\nX25 ($B_{x0}=2.5\\mu$G) would be marginally critical at the same time. We defer\nthe discussion of the mass-to-flux ratio in the clouds to a subsequent paper\nincluding gravity. \n\nAn ordered field aligned with the flows plus a large-scale random\ncomponent of similar amplitude (model XR25) introduces a large-scale shear and \nmight be considered a general situation for sweep-up of gas in spiral shocks, \nwhile the perpendicular field case (Y05) would address the (probably common) \nsituation of an oblique field whose lateral component is amplified by flow compressions.\n\nWe emphasize that while we attempt to address the extreme situations\nof field orientations, the finite size of our simulation domains cannot fully capture the \neffects of the magnetic field's boundary conditions. These will be set on larger \nscales than our local simulations can cover. In that sense, our results should be \nviewed as providing insight into magnetized cloud formation under idealized conditions \nrather than under physically realistic ones.\n\n\\subsection{A Comment on Resolution}\n\nWe decided to keep the resolution of our models constant, foregoing a resolution\nstudy in favor of a parameter study. Resolution effects have been discussed \nby \\citet{2007A&A...465..431H}. In addition, we have performed a systematic\nresolution study for two-dimensional cloud formation models (unpublished -- the\nmodels are similar to the ones discussed by \n\\citet{2005ApJ...633L.113H,2006ApJ...648.1052H}), covering a factor of $32$ in\nspatial resolution (from $256^2$ to $8192^2$ cells). As has been pointed out,\nthe critical length scale to resolve is the cooling \nlength of the thermal instability. If not resolved, the thermal instability will \nbe partially suppressed. At parameters of the WNM, the cooling\nlength is on the order of a parsec, while for the cold neutral medium (CNM), \nit drops to a fraction of a parsec. Thus, while more substructures should form with\nincreasing resolution, we expect our models to follow the general evolution\nof the thermal and dynamical instabilities sufficiently accurately for our purposes.\n\n\\section{Results}\n\n\\subsection{Morphologies}\n\nFigure~\\ref{f:polmap} summarizes the morphological effects of magnetic fields during\nthe build-up of a cloud. From top to bottom, it shows logarithmic column density maps of \nthe hydrodynamical model H, and the four MHD models X25 through XR25. The three columns stand \nfor projections along each coordinate axis, namely along the inflow ($x$-axis, {\\em left}), \nand perpendicular to the inflow (along $y$ and $z$-axes, {\\em center} and {\\em right}). The \nmaps of the MHD-models show polarization vectors which have been determined by integrating \nthe density-weighted Stokes $Q$ and $U$ parameters along the respective line-of-sight \n(see \\citealp{1996ASPC...97..486Z}; \\citealp{2001ApJ...561..800H}). \n\n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[width=0.7\\textwidth]{f1.eps}\n  \\end{center}\n  \\caption{\\label{f:polmap}Logarithmic column density projections (in cm$^{-2}$) \n           along the three \n           grid axes ({\\em left}: along inflow, {\\em center} and {\\em right}:\n           perpendicular to the inflow) for models H, X25, X50, Y05 and XR25 as indicated,\n           at $t=9.5$~Myr after flow collision. The mean field direction is denoted by \n           the symbols in the panel labels ($\\rightarrow$,$\\uparrow$,$\\times$).}\n\\end{figure*}\n\n\\subsubsection{Field parallel to inflow}\n\nModel H shows the strong fragmentation due to thermal and dynamical instabilities\nsimilar to the models discussed by \\citet{2008ApJ...674..316H}. \nSpecifically, the large-scale initial perturbation\ntriggers the NTSI, due to whose rapid growth some of the dense material has already reached\nthe inflow boundaries. Viewed along the inflow (top left panel in Fig.~\\ref{f:polmap}), the\ncold dense fragments appear clumpy rather than filamentary.\n\nIntroducing a magnetic field {\\em aligned} with the flow (model X25, second row)\nsuppresses fragmentation compared to model H.\nThe face-on view ({\\em left}) shows several large-scale coherent filaments with\ndenser cores. The edge-on views ({\\em center} and {\\em right}) \ndemonstrate that the magnetic field is not dynamically\ndominant. The polarization vectors are aligned with local structures.\n\nIncreasing the magnetic field (model X50, third row from top) \nseems to suppress much of the fragmentation. Specifically, the NTSI is only very weakly \n(if at all) present, since the magnetic field is\nstrong enough to suppress the lateral momentum transport necessary for triggering the NTSI\n\\citep{2007ApJ...665..445H}. Nonetheless, the flows still fragment, albeit into a tight \nnetwork of filaments\n(X50, left) instead of a few large, more clumpy and fuzzy structures \n(models H, X25). The suppression of the NTSI leads to the formation of a more or less coherent \nfilament in the lateral projection (center and right column for model X50). Local shear modes \nlead to strong distortions of the field from its initial alignment with the inflow, as \nindicated by the polarization vectors which mostly trace out the mean background field.\n\n\\subsubsection{Field perpendicular to inflow}\n\nThe introduction of a field {\\em perpendicular} to the inflow changes the\nmorphology completely (4th row of Fig.~\\ref{f:polmap}, model Y05), despite the by a \nfactor of $10$ \nweaker field (see Table~\\ref{t:param}). The perpendicular field breaks the symmetry in the plane of\nthe flow collision, leading to filaments perpendicular to the mean field direction (note that\nthe mean field in the left panel of the 4th row of Fig.~\\ref{f:polmap} is oriented horizontally). \nThese filaments form due to motions along the field lines, but perpendicular to the incoming flows\n(see also \\citealp{2007ApJ...665..445H} and \\citealp{2008ApJ...687..303I} \nfor two-dimensional models). The magnetic field suppresses \none degree of freedom in the gas motions, also leading to lower column density contrasts\nthan in models H and X50. The two lateral views of model Y05 exhibit another effect of \nthe perpendicular field. Seen along the mean field direction, a large scale NTSI-driven mode \nis discernible, while\nthe projection perpendicular to the inflow and to the mean field (Y05 right) just shows a \nslab (albeit with substructure). In the former, the field lines are just shuffled around and\ncontribute to the dynamics only via the pressure term in the Lorentz force, thus lowering\nthe column densities and broadening the slab (compare to center panel of model X50).\nIn the latter, the tension term of the Lorentz force prevents the growth of the NTSI.\nThis is evidence for the presence of interchange modes in the NTSI, similar to e.g. \nthe Rayleigh-Taylor instability \\citep{2007ApJ...671.1726S}.\nStill, material is free to move along the field lines (and thus perpendicular to the inflows), \nleading to the formation of the filaments {\\em parallel} to the inflows. \n\nThe magnetic field perpendicular to the inflow resists compression, leading\nto a suppression of the thermal instability, which is also mirrored in the total mass budget\nof all models (Fig.~\\ref{f:masses}). Model X50 has the highest fraction of cold gas, due to \nthe strong guide field leading to a strong compression of the gas, while model Y05 shows the\nsmallest cold mass fraction, because the lateral field resists compression by the flows, and thus\nreduces the cooling rates. \n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{f2.eps}\n  \\caption{\\label{f:masses}Total masses against time, below and above $T=100$~K. \n           The perpendicular field\n           reduces the compression and thus lowers the cold gas mass, while the \n           field aligned with the flow leads to higher compressibility.}\n\\end{figure}\n\n\\subsubsection{Tangled field}\n\nThe bottom row of Figure~\\ref{f:polmap} shows the maps for model XR25, which starts out\nwith a uniform field aligned with the flow at $2.5\\mu$G and a random field component of \nequal magnitude. Although the (varying) lateral field components contain $10$ times\nas much energy as the perpendicular field in model Y05, the fields do not suppress\nthe formation of clouds with column densities in excess of $10^{22}$~cm$^{-2}$; a tangled\nfield is substantially less efficient in preventing compression. Since there are regions\nwhere  the field will be aligned with the flow, it leads \nto a selection effect in the sense that the clouds form at positions where\nthe lateral random components of the fields are weakest over time and\/or where bends\nin the fields determine the position of cloud formation (see \\citealp{2001ApJ...562..852H}).\nThe resulting clouds \nare more isolated, with larger voids between them (bottom left panel of Fig.~\\ref{f:polmap}).\nThe side view (bottom center and right) exhibits a diffuse halo of thermally unstable gas, \nmaterial which is caught in the tangled field between the bounding shocks and the dense cold\ngas. \n\n\\subsection{Dynamics}\n\nThe cold mass fractions of models H and X25 (Fig.~\\ref{f:masses}) are slightly lower than  that\nof X50, indicating that the developing turbulence due to the flow fragmentation is also broadening\nthe slab. While this notion is already suggested by Figure~\\ref{f:polmap}, it is confirmed by comparing\nthe $rms$ velocity dispersion in the cold ($T<300$~K) gas (Fig.~\\ref{f:vrms}), and it can also be \ngleaned from a more detailed look at the laterally averaged pressure profiles (Fig.~\\ref{f:prssprof}). \n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{f3.eps}\n  \\caption{\\label{f:vrms}Density-weighted $rms$ velocity dispersion against time for\n           all models. Fields parallel to the inflows seem to suppress turbulence in the \n           cold gas.}\n\\end{figure}\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{f4.eps}\n  \\caption{\\label{f:prssprof}Pressure profiles along the inflow direction $(x)$, averaged over\n           the perpendicular directions $(y,z)$, for three times as indicated, and for all models.\n           Shown are total (solid line), kinetic (dashed),\n           internal (dash-dot) and -- if applicable -- magnetic (dash-3-dot) pressures. Note \n           that we show the pressures, not the logarithm of the pressures.}\n\\end{figure*}\n\nShown is a time sequence of the pressure profiles along the $x$-axis (i.e. along the inflows) for \nall models. In the absence of gravity, the slabs are all overpressured by the ram pressure of\nthe colliding flows (solid lines). At early times, all five models show a drop in kinetic pressure\nand an increase in thermal pressure in the collision region. The flows have not fully fragmented\nyet, and the cooling is not in full strength yet because of the still low densities. With evolving\ntime, the thermal pressure peak for model H drops due to increasing cooling. \n\nThis is markedly different for model X50, where the thermal pressure continues to be enhanced\nby a factor of more than 2 above that of the inflow. Also, the kinetic pressure\ndrops, reaching an approximate equipartition with the thermal pressure. This is due\nto the strong magnetic guide field, which ``splits'' the cloud into a network of dense filaments \nwith low-density, high-temperature voids in between (see top panel of 2nd column of \nFig.~\\ref{f:polmap}, model X50). Figure~\\ref{f:prssdens} offers a different view of the same\nphenomenon, showing the pressure-density distributions for all four models, at $t=9.5$~Myr. \n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{f5.eps}\n  \\caption{\\label{f:prssdens}Greyscale-coded mass fraction of thermal pressure against volume density \n           for all models as indicated in the plots, at $t=9.5$~Myr. The solid line indicates the\n           thermal equilibrium curve, while the vertical and horizontal dashed lines denote the \n           initial conditions in density and pressure. Diagonal dashed lines stand for isotherms\n           at $T=10^4$, $300$ and $10$~K as indicated.}\n\\end{figure*}\n\nThe high-temperature voids of model X50 show up at $\\log n \\approx 0.5$ and $\\log P \\approx 4.3$, \nwhile the high-density filaments sit all on the stable low-temperature branch of the thermal\nequilibrium curve at $T\\approx 40$~K. Note that a substantial amount of the gas mass is actually\nthermally over-pressured, in contrast to model H. \n\nReducing the field aligned with the flow (model X25) leads to pressure profiles \n(Fig.~\\ref{f:prssprof}) and thermal states (Fig.~\\ref{f:prssdens}) similar to the hydrodynamical \nmodel H. In other words, while the field in model X25 is non-negligible in the sense that its \npresence still makes a morphological difference (see Fig.~\\ref{f:polmap}), it does not noticeably \naffect the overall dynamics of the cloud.\n\nThe field is obviously dynamically important in model Y05. Because of the strong flow \ncompression perpendicular to the field lines, the magnetic pressure takes over the role of the\nthermal pressure, which leads to a substantial amount of thermally underpressured gas \nin model Y05 (Fig.~\\ref{f:prssprof} and \\ref{f:prssdens}). There is only a small amount of \nmaterial at high ($\\log n > 2$) densities.\n\nIntroducing the tangled field component on top of a field aligned with the inflows leads to an\nover-pressurization of the slab by a factor of more than $2$ (Fig.~\\ref{f:prssprof} right, model XR25).\nThis is mainly due to a combined increase in magnetic and kinetic pressure, i.e. the tangled field\nleads to more turbulence than all other field geometries. The increase in kinetic pressure cannot\nbe solely due to enhanced densities -- model X50 should show a similar increase then. \nAlthough the tangled field in the diffuse gas phase is not force-free, it does not contribute\nperceptibly to turbulent motions in the inflows, as can be seen by comparing the kinetic pressure\nlevels in the inflows between models XR25 and e.g. X50. Also, the kinetic pressure of model\nXR25 does not increase when moving closer towards the midplane, until one enters the post-shock region.\nThe thermal pressure peaks in the diffuse\nenvelopes due to warm gas being unable to cool down (see bottom right panel of Fig.~\\ref{f:prssdens}),\nbut it drops back to the ambient {\\em thermal} pressure at the cloud midplane ($x=0$). Obviously,\nthe averaged thermal pressure alone is not a very accurate indicator of the cloud's physical state. \n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{f6.eps}\n  \\caption{\\label{f:prsstime}Pressures against time for all models, sorted according to temperature\n          regimes (rows; total, $T>3000$K, $300<T<3000$K and $T<300$K) and \n          components (columns; total, $x$, $y$ and $z$). Black lines indicate thermal pressure, red\n          lines kinetic and blue lines magnetic pressure. Note that these are again linear pressures in\n          units of $10^4$~K~cm$^{-3}$.}\n\\end{figure*}\n\nFigure~\\ref{f:prsstime} summarizes the pressure evolution over time for all models. The pressures\nare sorted according to temperature regimes (rows; total, stable WNM, unstable WNM, and CNM) \nand vector components (columns; total, $x$ parallel to\ninflow, $y$ perpendicular to inflow, and $z$ again perpendicular to inflow). \nThe total pressures ({\\em top} row) stay constant with time except for the magnetic components of\nmodels XR25 and Y05 due to the compression of the transverse field component (see $y$ column), \nand the $y$-component of the kinetic energy of model XR25. \nNot surprisingly, the stable warm neutral phase does not show any strong pressure variations with\ntime. In the thermally unstable regime, $300<T<3000$~K, the\nmodels with transverse field components (XR25, Y05) show an increase in the $y$ and $z$ component\nof the magnetic pressure. A large fraction of the magnetic energy in model XR25 is thus stored in\nthe diffuse cloud halo (see Fig.~\\ref{f:polmap}). \nThe magnetic fields also lead to a ``redistribution'' of kinetic energy between the vector components:\nthe bulk of the kinetic energy of the cold gas in model H is found in the $x$-component, parallel\nto the inflow, indicating that the dynamics are dominated by the NTSI. The magnetic field models\nhave at least as much kinetic energy in the transverse ($y$ and $z$) components. For model\nXR25, the transverse components even dominate. The substantial overpressure at the midplane\nof model XR25 (see Fig.~\\ref{f:prssprof}) arises mainly from the transverse components of the\nmagnetic and kinetic energy in the cold gas. While for $T<300$~K, the magnetic energies are larger\nthan the kinetic ones, the kinetic energy content dominates the total energy budget (top left\npanel of Fig.~\\ref{f:prsstime}). \n\nFigure~\\ref{f:fieldtime} offers a simpler view of the field amplification depending on mean field\ndirection. It shows the ordered (mean) field component and the random (turbulent) component \nagainst the time. Only the random components are growing with time, indicating that the fields\nare mainly amplified by fieldline stretching. Only models Y05 and XR25 have field components\ninitially perpendicular to the inflows; these are growing linearly with time due to the compression\nby the flows. The random (or turbulent) field components of models X25 and X50 evolve in a similar\nway, indicating that a similar amount of energy is stored in the random component. For models\nX25 and Y05, the ordered and random components reach similar levels (see \\S\\ref{ss:components}).\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{f7.eps}\n  \\caption{\\label{f:fieldtime}Ordered (thick lines) and turbulent (thin lines) magnetic\n           field components against time for models X25, X50, Y05 and XR25. For models\n           X25 and XR25, the ordered and turbulent components are approximately equal.}\n\\end{figure}\n\n\\subsection{Turbulence and Thermal States}\\label{ss:thermals}\n\nThe ratio of gas mass in the thermally unstable regime to the mass in the WNM \ncan be compared to observational constraints. \n\\citet{2003ApJ...586.1067H} find a fraction of $48$\\%. Figure~\\ref{f:vrmsunm}\nshows the mass fractions for\nall models against the velocity dispersion in the CNM, averaged\nover a time interval $7.5<t<9.5$~Myr. Higher fractions of thermally unstable\ngas are found at higher velocity dispersions, a correlation already pointed out earlier\n(e.g. \\citealp{2001ApJ...557L.121G,2005A&A...433....1A,2006ApJ...648.1052H,2007A&A...465..431H}). \nHowever, the magnetic field\nintroduces a second dependency: of the three models with the highest velocity dispersion\n(H, Y05 and XR25), model Y05 has the highest fraction of thermally unstable gas.\nAs discussed above, in this model the lateral field components reduce the density contrasts,\nkeeping a substantial fraction of the gas in the thermally unstable regime. The large\nvelocity dispersion of model Y05 arises from the large-scale NTSI interchange mode\n(see Fig.~\\ref{f:polmap}). \n\nIt is maybe not surprising that the structures which bear closest resemblance to \nthe CNM clouds studied by \\citet{2003ApJ...586.1067H,2005ApJ...624..773H}\nbelong to the model with a thermally unstable mass fraction closest to the observed \none (Y05). \n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{f8.eps}\n  \\caption{\\label{f:vrmsunm}Mass fraction of thermally unstable gas ($300<T<3000$~K) \n           with respect to all WNM ($T>300$~K) against the velocity\n           dispersion in the CNM (Fig.~\\ref{f:vrms}), averaged between\n           $7.5$ and $9.5$~Myr. The error bars show errors on the mean.}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{The Role of  Magnetic Fields for Cloud Formation}\n\nThe field strength, and the orientation of the mean magnetic field with respect to the \nflows sweeping up the gas play a crucial role for the flow-driven formation of molecular clouds\n(Fig.~\\ref{f:polmap}). If the fields are dynamically dominant, the only chance to \nbuild up substantial clouds is by channeling the flows along the fields. \nThis is the situation shown in model X50, and it is also \nborne out by simulations of molecular cloud formation in galactic spiral arms\n\\citep{2006ApJ...646..213K}, where the clouds tend to be oriented perpendicularly to \nthe large-scale field (also possibly visible in the models by \n\\citealp{2008MNRAS.383..497D}), until sufficient material has been accumulated that they\ndecouple dynamically from the large-scale field. Similarly, for the sweep-up\nof material by e.g. H{\\small{II}}-regions or supernova shells, one would expect the densest clouds \nto appear at the locations where the field is perpendicular to the shell\n(see Fig.~2 of \\citealp{2006ApJ...641..905H}, although the effect might\nbe less clear in a highly turbulent environment, see Fig.~10 of \n\\citealp{2004ApJ...617..339B}). \n\n\\citet{2001ApJ...562..852H} point out -- based on simulations by\n\\citet{1995ApJ...455..536P} -- that dynamically weak (but not necessarily\nordered) fields would lead to a general selection effect for the formation of molecular\nclouds. Since $\\beta_{ram}>1$, the flows stretch out the\nfieldlines, leading to a natural alignment. In this picture, clouds form in the bends\nof large-scale fields (see Figs. 4 \\& 5 of \\citealp{2001ApJ...562..852H}).\nSuch a scenario is to some extent\naddressed by model XR25, where  varying field orientations entail \na local selection effect, picking out the formation sites of molecular clouds \nover e.g. a broad shock front. Note that while the field is dynamically weak \n($\\beta_{ram}=8.5$, $\\beta_{th}=4.3$) in the initial conditions (and in the inflows) of model XR25, \n$\\beta_{th} < 1$ within the cloud (Fig.~\\ref{f:prssprof}).\n\nThis selection effect comes about because already a small oblique component can\nbe amplified sufficiently to withstand the compression, preventing the high densities\nneeded for cloud formation \\citep{2001ApJ...562..852H}. Such a situation is addressed in\nthe extreme by model Y05. \nA field perpendicular to the sweeping-up flow can suppress the formation of massive clouds,\nalthough the three-dimensional situation is much less clear-cut than its one-dimensional\ncounterpart (see e.g. \\citealp{1980ARA&A..18..219M}; \\citealp{2004ApJ...612..921B}). \nIn one dimension, a density increase by a factor of $100$ from e.g. $n=1$~cm$^{-3}$ to\n$100$~cm$^{-3}$ would entail the same factor for the magnetic field strength since $B\\propto n$. \nFigure~\\ref{f:babsdens} shows this is not the case in three dimensions.\nThe weak perpendicular field (model Y05) has been amplified by a peak factor of $\\approx 30$, \nwhile the density has increased by up to a factor of $300$. Generally, our models \nshow a weak correlation of field strength with density over the whole thermal range, from \nthe WNM to the CNM, consistent with observations of the field-density relation \nin the WNM and CNM (\\citealp{1986ApJ...301..339T}; HT05), and with\nnumerical results (e.g. \\citealp{2005A&A...436..585D,2008A&A...486L..43H}).\nFor models X50 and X25 a weak correlation between field and density is not overly surprising.\nFor model Y05, the decorrelation\\footnote{The seemingly strongly correlated B(n) for $n<1$~cm$^{-3}$\nin model Y05 does not affect the argument. These are a few regions (low mass fraction) at the edges\nof the expanding slab, subjected to numerical reconnection.} is a consequence of the fact that material is still free to \nmove along the field lines perpendicular to the original inflow \\citep{2007ApJ...665..445H}, \nthus leading to the build-up of \nhigher-density filaments perpendicular to the field (but aligned with the inflow), see\nFigure~\\ref{f:polmap}. Also, other effects, such as the acceleration of magnetic field\ntransport by turbulence (\\citealp{2002ApJ...567..962Z,2002ApJ...570..210F,2004ApJ...603..165H}\nfor ion-neutral drift, \\citealp{1999ApJ...517..700L} for reconnection),\nor a decorrelation due to MHD waves \\citep{2003A&A...398..845P}\ncould explain the observed weak correlation.\n\nMagnetic fields will rarely be completely uniform. Model XR25 tests the more\ngeneral case of a uniform field at $2.5\\mu$G and a random component of equal size, consistent\nwith (although slightly lower than) observational estimates for magnetic field strengths in the diffuse\ngas \\citep{1996ASPC...97..457H,2004Ap&SS.289..293B,2006ChJAS...6b.211H}. \n\\citet{2007ApJ...663L..41G} \nshowed in a two-dimensional numerical experiment that pre-existing  perturbations in the \ninflows can lead to substantial magnetic field amplification due to a rippling of the \nshockfront and subsequent fieldline stretching. We observe a similar effect in model XR25, \nalthough our Mach numbers are substantially lower (their study addressed the propagation of a \nsupernova shock front). \\citet{2008A&A...486L..43H} perturb the velocities of the inflows and \nfind only a modest increase of the field strength. Clearly, the initially tangled field leads \nto rather different dynamics in the forming cloud (Figs~\\ref{f:polmap}, \\ref{f:prsstime}). \n\nModels X25 and X50 demonstrate that not only\nthe field orientation will play a role during cloud formation (see model Y05), but also\nthe field strength, since all the instabilities involved have threshold limits for the\nfield strength -- at least in two dimensions. It might well be that the stronger field\nin model X50 suppressing the formation of filaments could be offset by higher inflow speeds\nor substructures in the flows. This remains to be studied. \n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{f9.eps}\n  \\caption{\\label{f:babsdens}Magnetic field strength against volume density for models\n           as indicated in the plots. The colors denote temperatures, and the intensity\n           the mass fraction. Generally, there is no clear correlation between field strength\n           and density. The steeper of the two dashed lines denotes $B\\propto n$, the \n           flatter one $B\\propto n^{1\/2}$. Dotted lines denote the initial conditions.}\n\\end{figure*}\n\n\\subsection{Turbulence and Thermal States}\\label{ss:turbtherm}\nFields aligned with the inflows tend to suppress the NTSI, and thus lead to an approximate \nequipartition between the spatial components of the kinetic energy in the cold and\nin the thermally unstable gas \n(Fig.~\\ref{f:prssprof}, bottom couple of rows). For comparison, the hydrodynamical model \nH has the bulk of the kinetic energy in the (flow-aligned) $x$-component. Magnetic fields may play\nan important role to isotropize highly directional flows. Thus, searching for observational\nsignatures of flow-driven cloud formation should focus on the warm, diffuse gas phase, \nsince the inflow signature will be erased in the cold dense gas.\n\nFractions of thermally unstable gas (Fig.~\\ref{f:vrmsunm}) for flow-aligned\nfields (models X50, X25) are lower than values observed for diffuse CNM \nclouds \\citep{2003ApJ...586.1067H}. A lateral field component results in\na thermally unstable gas fraction of $\\approx 50$\\%, consistent with observations.\nBased on these findings, one could feel tempted to extend the above argument\nabout the selection effect introduced by magnetic fields: not only could\nmagnetic fields control the locations of molecular cloud formation, but they\nalso could lead to ``failed'' molecular clouds, i.e. diffuse atomic hydrogen\nclouds, if there is a non-negligible field component perpendicular to the\nsweeping-up flow (see also \\citet{2008ApJ...687..303I}\nfor a similar argument based on two-dimensional simulations). \n\n\\subsection{Ordered vs. Random Component}\\label{ss:components}\n\nAnother observational constraint is given by the ratio of the ordered \nover the unordered (or turbulent) field component. The observational\nevidence points to the components being of similar magnitude \n(e.g. \\citealp{1996ASPC...97..457H}; \\citealp{2004Ap&SS.289..293B};\n\\citealp{2004ASSL..315..277B}; \\citealp{2006ChJAS...6b.211H}; see also\ndiscussion in HT05). A direct comparison to\nour models is hampered by the fact that in order to see the varying \ncomponent, sufficiently large scales need to be addressed, which is why\nHT05 argue that their observed median field strength of $6\\mu$G should\nbe identified with the {\\em total} magnetic field strength. Likewise,\nit is not obvious that the components should be of equal magnitude locally \neverywhere. Bearing this limitation in mind, it is clear from \nFigure~\\ref{f:fieldtime} that only for models X25 and Y05 the components are \ncomparable. \n\n\\subsection{Gravity}\n\nWe deliberately left out self-gravity in our simulations, in order to get a clearer \nview of the role of magnetic fields during the early cloud formation phase. Thus, our\nclouds are only confined by the ram pressure of the inflows, and at later stages, \nthe dynamics of the clouds are probably underestimated since gravity as a source\nof turbulence is missing (e.g. \\citealp{2008MNRAS.385..181F}). \nAs a result of the restricted physics, a comparison of our models with observations\nis only meaningful for models where gravity is not expected to play a role, i.e. \nfor model Y05 addressing the formation of diffuse HI clouds. For all other models, \nwe expect gravity to be relevant during the cloud formation process \\citep{2008ApJ...689..290H}.\n\n\\section{Summary}\n\nExtending our previous work and complementing a model by \nHennebelle et al. (\\citeyear{2008A&A...486L..43H}; see also \\citealp{2008arXiv0808.0986B}),\nwe study the role of magnetic field strength and orientation on the process\nof flow-driven cloud formation. Our models include the usual heating and cooling effects,\nallowing rapid fragmentation of the flows, they use uniform inflows to study the most unfavorable\nconditions for structure formation, and they envisage the formation of clouds in two head-on\ncolliding flows, i.e. the extreme case for building up massive clouds. We do not include\nself-gravity, focusing on the early stages of cloud formation, during which gravity\nmight be less important. \n\nUnder these conditions, we find that the effects of magnetic fields on the morphology and\non the thermal state of the resulting clouds depend very strongly not only on the field\norientation with respect to the inflow, but also on the field strength. Initial field energies\nare below equipartition with the kinetic energies (by a factor of $4.3$, corresponding\nto a field strength of $5\\mu$G for our flow parameters) even for the strongest field\ncase in our study (model X50), yet they result in significantly different cloud properties\nthan those for a field weaker by a factor of $2$ (model X25, $2.5\\mu$G). \nMagnetic fields also lead to a redistribution of the inflow energy to the transverse spatial \ndirections (Fig.~\\ref{f:prsstime}). \nHence searching for signatures of colliding flows should focus on the diffuse\ngas phase, since the cold gas will have no memory of the original flow direction.\n\nNot surprisingly, weak magnetic fields ($0.5\\mu$G) perpendicular to the inflows can suppress the build-up\nof massive clouds (model Y05). Yet substructure still can arise in the post shock gas, in \nthe form of diffuse filaments perpendicular to the field, and of wave-like patterns \n(possibly magnetosonic waves). \nThe filaments are a consequence of lateral gas transport \n(see also \\citealp{2007ApJ...665..445H}; \\citealp{2008ApJ...687..303I}). \nThe straight-forward correlation $B\\propto n$ is not obeyed (Fig.~\\ref{f:babsdens}).\nMass fractions of thermally unstable gas for the model with a lateral field component (Y05)\nare consistent with observed values for diffuse HI clouds \\citep{2003ApJ...586.1067H}. For\nall other models, the fractions are lower (Fig.~\\ref{f:vrmsunm}). The ratio of ordered\nvs. random field component is consistent with observations only for the weak-field model\nX25, and for the diffuse HI cloud model Y05 (Fig.~\\ref{f:fieldtime}).\n\nA weak ($2.5\\mu$G) uniform field together with a random component of equal size \nleads to a strong over-pressurization of the cloud due to a combined \nincrease of magnetic and kinetic pressure (Fig.~\\ref{f:prssprof}), with the magnetic\npressure dominating the thermal pressure within the cloud.\nHigh column densities are assembled at locations\nwhere the perpendicular field component is weakest over time. Thus, a tangled field can lead\nto a selection effect for cloud formation while not preventing it globally. \n\nOur numerical models address the ideal MHD limit, i.e. we do not take into account ion-neutral\ndecoupling or resistive dissipation. It remains to be seen how non-ideal MHD processes affect\nthe structure formation during the build-up of the clouds (e.g. \\citealp{2008ApJ...687..303I}).\n\n\n\\acknowledgements\nWe thank the referee for a critical and very helpful report.\nComputations were\nperformed at the National Center for Supercomputing Applications\n(AST 060034) and on the local PC cluster Star, perfectly maintained and\nadministered by J.~Hallum \\& R. Bonser. FH is supported by the University\nof Michigan and NSF grant AST 0807305.\nThis work has made use of NASA's Astrophysics Data System.\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe dynamics of wall-bounded turbulent flows are linked closely to processes that dominate the flow close to the wall. A prominent feature of such flows is the presence of slow moving wavy `streaks' of fluid, which intermittently and abruptly lift-up away from the wall, and eject slow moving fluid towards the faster core~\\citep{Kline1967,Offen1975}. These `bursts' of slow moving streaks have been identified in a number of experiments using hydrogen bubble visualization~\\citep{Kline1967}, dye visualization~\\citep{Kim1971}, and observation of neutrally buoyant colloidal particles~\\citep{Corino1969}. The ejections are usually followed by `sweeps' of faster moving fluid towards the wall~\\citep{Corino1969}, completing the cycle of momentum exchange between the low speed near-wall layers and the high speed core. Several studies note that these intermittent bursts are important sources for the generation and dissipation of turbulent kinetic energy within boundary layers, control of transport phenomena, and are also responsible for the majority of turbulent drag acting on the wall~\\citep{Kline1967, Corino1969, Kim1971, Wallace1972, Lumley1998, Jimenez2012}. \n\nAlthough the existence of intermittent bursts in wall-bounded turbulence is widely accepted, there has been some ambivalence regarding their role in near-wall dynamics. \\citet{Robinson1991}, \\citet{Moin1998}, and \\citet{Schoppa2002} have suggested that bursts may not play as crucial a role in turbulence generation as previously thought. The main argument in favour of this viewpoint is that the intermittent events observed by \\citet{Kline1967} were caused by the passage of streamwise vortices over static measurement locations. However, certain studies have remarked that these strong intermittent events are not merely artefacts of vortices passing by, but should instead be viewed as intrinsic components of the near-wall dynamics~\\citep{Jimenez2012,Jimenez2013}. \\citet{Lumley1998} considered bursts to be integral to the formation and evolution of coherent structures, and proposed that the inhibition of bursts should be a crucial element of potential control strategies. \\citet{Jimenez2013} notes that frictional drag on the wall increases abruptly and substantially during bursting events. Several studies have proposed that coherent hairpin vortices may be consequences of instabilities and ejections associated with low-speed streaks~\\citep{LozanoDuran2014,Hack2018}. \\citet{Schlatter2014} suggested that hairpin vortices may be artefacts of the relatively moderate Reynolds numbers that prior DNSs had been restricted to owing to computational limitations. It is evident that there have been differences of opinion regarding the exact nature of near-wall dynamics, which highlights the need for novel analytical tools that can help interpret nonlinear turbulent flow data more effectively.\n\nWhile we have witnessed steady progress in both experimental diagnostics and simulation capabilities since some of the seminal studies discussed above, a comprehensive understanding of fundamental processes in near-wall turbulence, and more importantly, effective means of influencing them are still being sought~\\citep{Jimenez2018}. Disentangling the non-linear spatial and temporal correlations inherent in turbulent flows has proved to be the principal obstacle, and has been particularly challenging for reduced order modelling approaches such as Principal Component Analysis (also known as Proper Orthogonal Decomposition - POD), and Dynamic Mode Decomposition (DMD)~\\citep{Schmid2010}. Recently, novel techniques that have undergone rapid development owing to advances by the machine learning and computer vision community, have seen increased adoption for prediction and analysis tasks in fluid mechanics. Very early uses of Artificial Neural Networks (ANNs) for this purpose include studies by~\\citet{Fan1993}, and~\\citet{Lee1997}. \\citet{Milano2002} compared the prediction and reconstruction capabilities of nonlinear autoencoders to those of Principal Component Analysis in a turbulent channel flow simulation. \\citet{Hack2016} used ANNs to predict the transition to turbulence in a spatially developing boundary layer, by identifying near-wall streaks that were most likely to breakdown and induce the formation of turbulent spots. \\citet{Maulik2017} trained a single layer feedforward ANN to deconvolve low-pass filtered turbulent datasets, in order to reconstruct the subfilter length scales. \\citet{Fukami2019} and~\\citet{Liu2020} have also explored deconvolution to reconstruct subfilter scales, albeit using Convolutional Neural Networks (CNNs)~\\citep{Fukushima1980}, which preserve spatial correlations inherent in the data. CNNs have  proved to be effective for predicting both steady~\\citep{Guo2016,Sekar2019} and unsteady~\\citep{Lee2019} laminar flows around bluff bodies, airfoils, and cylinders. CNNs have also been used in low Reynolds number flows to predict unsteady force coefficients for bluff bodies~\\citep{Miyanawala2018}, pressure distribution on a cylinder~\\citep{Ye2020}, and drag for arbitrary 2D shapes in laminar flows~\\citep{Viquerat2019}.\n\nGiven the integral role of bursting events in the turbulence generation cycle, and the innate ability of Neural Networks to identify nonlinear correlations, we train a 3D CNN to predict the intensity of strong and intermittent ejection events that occur in the near-wall region. This is done by first `labelling' 3D velocity fields extracted from a turbulent channel flow DNS with their corresponding ejection intensities, and then using the velocity fields as input, and ejection intensities (labels) as output for training. Once the CNN is able to correctly predict ejection intensities for out-of-sample velocity data, we visualize localized regions of the flow that the trained CNN focuses on in order to make accurate predictions. This allows us to look beyond the black-box nature of the neural network, to reveal physical processes that such networks are capable of identifying in extremely complex flow fields. Details regarding the numerical methods and training procedure for the CNN are provided in \\S\\ref{sec:methods}. Results demonstrating the identification capabilities of the CNN are presented in \\S\\ref{sec:results}, followed by concluding remarks in \\S\\ref{sec:conclusion}.\n\n\\section{Methods}\\label{sec:methods}\n\\subsection{Direct Numerical Simulation}\nThe data used for training the CNN was generated using a DNS of a periodic turbulent channel flow. The simulation is based on the incompressible Navier-Stokes equations, which are solved using a high order conservative finite difference scheme~\\citep{Desjardins2008}. The flow is driven by imposing a pressure gradient in the streamwise direction, which changes in time to maintain a constant mass flow rate. The simulation domain and its dimensions are shown in Figure~\\ref{fig:diagram}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{domain-para6.pdf}\n\\caption{A snapshot of the flow field from a turbulent channel flow simulation at $Re_\\tau = 300$. The horizontal plane shows an isocontour of the horizontal velocity component $u$, coloured using the vertical velocity $v$. Low-speed streaks manifest as sinuous ridges, and bright spots mark regions where the flow is being ejected away from the wall. The pink blobs denote high intensity ejection parcels where positive fluctuations for $v$ exceed 2 standard deviations, i.e.,  $v > \\mean{v} + 2\\sigma_v$. The grid cell sizes were kept uniform in the streamwise and spanwise directions ($\\Delta x=\\Delta z = 3.5\\delta^+$), whereas the cell heights were stretched from the wall to the channel center in a sinusoidal manner ($0.03\\delta^+ \\leq \\Delta y \\leq 2.4\\delta^+$). The white box in the bottom left corner depicts MFU-sized sections that the snapshots were divided into for training the CNN.}\n\\label{fig:diagram}\n\\end{figure}\nThe channel uses periodic boundaries in the streamwise and spanwise directions, and the no-slip boundary condition at the top and bottom walls. The friction Reynolds number for the data used for training the CNN is approximately $Re_\\tau = u_{\\tau} (L_y\/2)\/\\nu = 300$. Here, $u_\\tau = \\sqrt{\\tau\/\\rho}$ is the friction velocity, $\\tau = \\mu \\partial u\/\\partial y$ is the surface shear stress, $\\nu = \\mu\/\\rho$ is the kinematic viscosity, and $\\rho$ is the fluid density. The mean velocity and rms velocity profiles for two distinct simulations at $Re_\\tau = 300$ and $670$ are shown in Figure~\\ref{fig:loglaw}.\n\\begin{figure}\n\\centering\n\\subfloat[\\label{sfig:loglaw1}]{%\n\\includegraphics[width=0.5\\linewidth]{loglaws.pdf}\n}\n\\subfloat[\\label{sfig:loglaw2}]{%\n\\includegraphics[width=0.49\\linewidth]{plot_urms.pdf}\n}\n\\caption{\\protect\\subref{sfig:loglaw1} Mean horizontal velocity profile shown in wall units for $Re_\\tau=300$ (blue) and $Re_\\tau=670$ (red). \\protect\\subref{sfig:loglaw2} The corresponding rms velocity profiles shown in wall units. The symbols correspond to data from \\citet{Moser1999} for $Re_\\tau=395$ and $590$.}\n\\label{fig:loglaw}\n\\end{figure}\nOnce the flow is statistically stationary, several snapshots are recorded at intervals of approximately $40t^+$, which allows the individual snapshots to be temporally decorrelated. Here, $t^+ = \\delta^+\/u_\\tau$ is the viscous time scale and $\\delta^+ = \\nu\/u_{\\tau}$ is the viscous length scale. Each full-channel snapshot is divided up into Minimal Flow Unit-sized sections~\\citep{Jimenez1991}, as depicted by the white box in Figure~\\ref{fig:diagram}. Similarly, MFU-sized samples are extracted from the upper wall after flipping the wall-normal and spanwise velocities appropriately, so as to maintain the same orientation as the lower wall. This procedure yields 450 three-dimensional sections (velocity samples) per wall for each snapshot, and a total of 10,800 velocity samples from 12 independent full-channel snapshots.\n\n\\subsection{Labelling the burst intensity in 3D velocity samples}\n\nThe quadrant method introduced by \\citet{Wallace1972} has been used widely to classify bursts and sweeps ($u'<0, \\ v'>0$ for bursts, and $u'>0,  \\ v'<0$ for sweeps). However, \\citet{Luchik1987} note that this technique experiences difficulties with detecting entire burst or ejection events. Moreover, the quadrant criteria do not require intense bursting activity, as they are merely associated with fluctuation signs with respect to the mean values. In the present work, we associate bursts with strong intermittent events as described by \\citet{Kline1967}, and consider ejections to be associated with large deviations in the vertical velocity. To determine the intensity of these ejection events, we compute the percentage of cells where positive fluctuations in $v$ exceed 2 standard deviations, i.e., $v > \\mean{v} + 2\\sigma_v$. This provides a useful indication of activity within each velocity sample, without having to rely on adjustable parameters. Each velocity sample is then interpolated onto a grid of size $64\\times40\\times64$ with uniformly spaced cells in the wall-normal direction, and reduced to half-precision floating point numbers to conserve memory during training.\n\n\\subsection{Training procedure and saliency maps}\n\\label{subsec:training}\nAfter labelling, the 10,800 velocity samples are split randomly into $85\\%$ training, $7.5\\%$ validation, and $7.5\\%$ test sets. The training samples are fed in batches of five to the CNN as input, along with the corresponding labels as output (Figure~\\ref{fig:arch}). We note that only the vertical velocity component $v$ is used for training, since it is most closely related to ejection events.\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\textwidth]{architecture5-color.pdf}\n\\caption{The Convolutional Neural Network takes a 3D velocity field (only the $v$ component) as input, and predicts the ejection intensity as output. The architecture consists of 4 convolution + pooling layers, which learn to identify and extract the most important flow features from the data. The 3D data is then flattened out, followed by two fully-connected layers terminating in the output node marked `prediction'. The number of distinct filtering kernels  used at each convolution layer are shown as $\\times 32$, $\\times 64, \\cdots$, and the layer sizes are shown as $(64,40,64)$, and so on. Altogether, there are 2.2 million unknown parameters (weights and biases) that must be learned during training.}\n\\label{fig:arch}\n\\end{figure}\nThe max-pooling layers downsample the data by retaining a single cell out of every $2\\times2\\times2$ block of cells. This reduces the dimensionality of the data by one eighth after every pooling operation. The function of the convolution and pooling layers is to extract 3D features from the flow, whereas the fully connected layers towards the end learn to associate the assortment of feature maps with the appropriate ejection intensity value. The training was implemented using Keras and TensorFlow, which are open-source machine learning libraries. The loss-function was defined as the percentage error between the predicted value and the actual label for each sample, and the weights were updated using the Adam optimizer to minimize this loss. The network architecture and training procedure were optimized through a series of hyperparameter sweeps, and the optimal combination that yields the highest accuracy is shown in Table~\\ref{tab:kd}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccc}\n     \\emph{Architecture and Training} & \\emph{Parameters}  \\\\[3pt]\n       Kernel Size   & 3x3x3  \\\\\n       Pooling Size   & 2x2x2  \\\\\n       Weight Initialization  & he uniform \\\\\n       Bias Initialization   & Zeros \\\\\n       Loss Function & Percent Error \\\\\n       Optimization & Adam \\\\\n\\end{tabular}\n\\quad\n\\vline\n\\begin{tabular}{ccc}\n      \\emph{Hyperparameters}  & \\emph{Value}  \\\\[3pt]\n       Batch Size   & 5 \\\\\n       Epochs   & 57  \\\\\n       Dropout & 0.5 \\\\\n       Learning Rate   & 0.0001 \\\\\n       Decay  & 0.0001 \\\\\n      Activation Function & ReLU \\\\\n\\end{tabular}\n\\caption{Parameters and hyperparameters related to the architecture and training.}\n\\label{tab:kd}\n\\end{center}\n\\end{table}\n\nOnce the network weights have been trained, there are several methods that can be used to understand what the network has `learned' to be important. We may plot the filter kernels, the feature maps (the output at each layer), or saliency maps as described by \\citet{Simonyan2013}. Saliency maps provide a visual representation of sensitivity analysis, and are generated by perturbing each point of the input data and measuring the resulting change in the output. An improved version which adjusts data points at convolutional layers throughout the architecture, but with respect to the final convolutional layer's feature maps, was developed by \\citet{Selvaraju2016}. This technique is referred to as Gradient-weighted Class Activation Mapping (Grad-CAM), and is used in the present work to identify salient regions in the near-wall flow.\n\n\\section{Results}\n\\label{sec:results}\nAfter successful training, the CNN was used for predicting ejection intensities for velocity samples extracted from a time-decorrelated snapshot. The results shown in Figure~\\ref{fig:preds} indicate that the CNN's predictions match the ground truth very well, regardless of whether the samples contain high intensity ejections or minimal activity.\n\\newsavebox{\\measurebox}\n\\begin{figure}\n\\sbox{\\measurebox}{%\n \\begin{minipage}[b]{0.4\\textwidth}\n  {\\includegraphics[width=\\textwidth]{mini1.png}%\n}\n\\vfill\n\\subfloat\n  []\n  {\\label{fig:predictA}\\includegraphics[width=\\textwidth]{mini2.png}}\n\\end{minipage}\n\n \\begin{minipage}[b]{.6\\textwidth\n  \\subfloat\n    []\n    {\\label{fig:predictB}\\includegraphics[width=\\textwidth]{prediction017-fin.pdf}}\n \\end{minipage}}\n\\usebox{\\measurebox}\\qquad\n\\caption{\\protect\\subref{fig:predictA} Two test samples that were not seen by the CNN during training or validation. The actual labels for the two datasets are $0.284\\% $ (top) and $4.751\\%$ (bottom), whereas the values predicted by the CNN are $0.282\\%$ and $4.749\\%$, respectively. \\protect\\subref{fig:predictB} Comparison of the labels (blue dots) and the predicted ejection intensities (red) for an out-of-sample snapshot which is time-decorrelated from the training dataset. The mean absolute percentage error in the predicted values is $9.7\\%$.}\n\\label{fig:preds}\n\\end{figure}\nWe examine the crucial flow features that the CNN has learned to focus on, by highlighting the salient regions using the Grad-CAM technique discussed in \\S\\ref{subsec:training}. For an intuitive explanation of the Grad-CAM technique, Figures~\\ref{fig:gradcamA} and~\\ref{fig:gradcamB} show how an image-classification network focuses on a dog's floppy ears, its eyes and the collar in order to make its determination that the picture is that of a dog. \n\\begin{figure}\n\\centering\n\\begin{minipage}[b]{0.38\\textwidth}\n\\centering\n\\subfloat\n[]\n{\\label{fig:gradcamA}\\includegraphics[width=0.4\\textwidth]{thor6reshape.png}}\n\\quad\n\\subfloat\n[]\n{\\label{fig:gradcamB}\\includegraphics[width=0.4\\textwidth]{gradcam-jet_trim.jpg}}\n\\end{minipage\n\\usebox{\\measurebox\n\\begin{minipage}[b]{0.58\\textwidth}\n\\centering\n\\subfloat\n  []\n  {\\label{fig:gradcamC}\\includegraphics[width=0.5\\textwidth]{fig3snap.png}}\n\\subfloat\n  []\n  {\\label{fig:gradcamD}\\includegraphics[width=0.5\\textwidth]{fig3grad.png}} \n\\end{minipage}\n\\caption{\\protect\\subref{fig:gradcamA} Input image and \\protect\\subref{fig:gradcamB} the corresponding Grad-CAM output from a CNN trained to discern between cats and dogs. The red and yellow areas depict the salient regions which most influence the CNN's prediction, namely, the ears, the eyes and the collar. \\protect\\subref{fig:gradcamC} Post-processed image for an input velocity sample, and \\protect\\subref{fig:gradcamD} the corresponding Grad-CAM resulting from the trained 3D CNN. The golden structures in \\protect\\subref{fig:gradcamD} indicate localized regions of the flow that are most influential for making the correct prediction. These salient regions correlate very well with the high-intensity ejection parcels and the bursting streak.}\n\\label{fig:grads}\n\\end{figure}\nSimilarly, figures~\\ref{fig:gradcamC} and~\\ref{fig:gradcamD} show the post-processed visualization of a velocity sample, and the corresponding Grad-CAM output when it is processed by the trained CNN. The pink fluid parcels in Figure~\\ref{fig:gradcamC} indicate regions of high ejection intensity, similar to Figure~\\ref{fig:diagram}. We also observe a bursting streak towards the back of the image, denoted by a brightly coloured ridgeline in the horizontal plane. From the Grad-CAM image in Figure~\\ref{fig:gradcamD}, we note that the CNN focuses on both the ejection parcels as well as the bursting streak, as is evident from the golden structures occupying the same spatial regions as the pink parcels, as well as engulfing the bursting streak in the back. This is a notable result, especially since the CNN was provided with no a priori knowledge of the flow patterns that it should focus on; rather, this ability was gained by the CNN through training on velocity samples that were assigned a single metric, i.e., the ejection intensity.\n\nWe now examine the ability of the CNN to track salient regions as the flow evolves in time. Figure~\\ref{fig:series} shows successive snapshots taken at a particular spatial location, with post-processed input velocity data overlayed with the Grad-CAM output. \n\\begin{figure\n\\centering\n\\subfloat[\\label{sfig:series1}]{%\n\\label{fig:seriesA}\\includegraphics[width=0.4\\linewidth]{series2Ann1.pdf\n}\n\\subfloat[\\label{sfig:series2}]{%\n\\label{fig:seriesB}\\includegraphics[width=0.4\\linewidth]{series2Ann2.pdf}\n}\n\\\\ \\centering\n\\subfloat[\\label{sfig:series3}]{%\n\\label{fig:seriesC} \\includegraphics[width=0.4\\linewidth]{series2Ann3.pdf}\n}\n\\subfloat[\\label{sfig:series4}]{%\n\\label{fig:seriesD}\\includegraphics[width=0.4\\linewidth]{series2Ann4.pdf}}\n\\caption{Four successive time instances showing an overlay of the Grad-CAM output over the corresponding flow field (animation available in Supplementary Movie 1). In \\protect\\subref{fig:seriesA} the CNN focuses its attention on ejection parcels that are already well formed, as well as on the streak that is undergoing bursting in the lower right corner. \\protect\\subref{fig:seriesB} As a new ejection parcel enters the field of view from the left, the CNN includes it as part of the salient regions. \\protect\\subref{fig:seriesC}, \\protect\\subref{fig:seriesD} As the bursting streak and ejection parcels move out of the field of view, the CNN switches its attention to the strong ejection parcel developing on the left.}\n\\label{fig:series}\n\\end{figure}\nAt $t_0$, the salient regions identify three distinct ejection packets, as well two bursting streaks towards the left and right edges. One viscous time unit later (i.e., at $t_0+t^+$), the CNN considers the larger ejection parcel entering the field of view from the left to be more important to its prediction, and focuses less on the parcel that has started dissipating near the lower right edge. At this instant, the ejection parcel towards the back and the bursting streak at the right edge are still influential in the CNN's prediction.  At $t_0+2 t^+$ and $t_0+3 t^+$, the large ejection parcel that has entered the field of view is considered to be most significant for predicting the ejection intensity.\n\nTo determine how well the CNN trained at $Re_\\tau=300$ generalizes to different flow conditions, we test its prediction ability at a higher $Re_\\tau = 670$ in Figure~\\ref{fig:highRE}. The dimensions of the new velocity samples were identical to the $Re_\\tau=300$ samples in wall units, and the velocity was rescaled by multiplying with $u_{\\tau300}\/u_{\\tau670}$.  \n\\begin{figure}\n\\centering\n\\subfloat[\\label{fig:predRe}]{%\n\\includegraphics[width=0.5\\linewidth]{predictionReshape1.pdf}\n}\n\\subfloat[\\label{fig:gradRe}]{%\n\\includegraphics[width=0.4\\linewidth]{re670_gradcam_hires.png}\n}\n\\caption{\\protect\\subref{fig:predRe} Prediction for $Re_{\\tau}= 670$ data using a CNN trained on the $Re_{\\tau}=300$ database. \\protect\\subref{fig:gradRe} The salient regions for the high $Re_{\\tau}=670$ samples.}\n\\label{fig:highRE}\n\\end{figure}\nDespite a notable difference in the Reynolds number, the network is able to make predictions with a mean absolute percentage error of less than $24\\%$. Moreover, the CNN is still able to discern the most relevant ejection parcels and bursting streaks. This highlights the capability of CNNs to reveal important physical processes that persist across diverse flow conditions.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, we have trained a three dimensional Convolutional Neural Network (CNN) to predict the intensity of strong intermittent ejection events that occur in the near-wall layer of a turbulent channel flow simulation. The CNN is able to accurately predict ejection intensities in velocity samples taken from a snapshot that was not part of the training dataset, and was sufficiently removed so as to be temporally decorrelated. To understand which part of the data most influences the network's ability to make accurate predictions, we visualize salient regions in the flow where the CNN focuses its attention, using the Gradient-weighted Class Activation Mapping (Grad-CAM) technique. We observe that the resulting salient regions correlate well with high intensity ejection parcels as well as with low-speed streaks undergoing bursting. This indicates that the CNN is able to reveal dynamically crucial regions within the turbulent flow field, without a-priori knowledge of the intrinsic dynamics. Finally, we demonstrate that the CNN trained on data at $Re_\\tau=300$ is able to predict ejection intensities for samples at $Re_\\tau=670$. This suggests that the trained CNN is generalizable in its prediction ability, especially with regard to physical processes that persist across varying flow conditions. The results indicate that Convolutional Neural Networks, which were originally developed for image recognition and classification, have immense potential for uncovering non-linear correlations and spatial features in turbulent flow fields.\n\n\\section*{Acknowledgements}\nThis work was supported by the Department of Ocean and Mechanical Engineering at the Florida Atlantic University, as part of the start-up package of Siddhartha Verma. The authors thank Prof. Petros Koumoutsakos for helpful discussions, and for providing access to computational resources. Computational resources were provided by the Swiss National Supercomputing Centre (CSCS) under project ID s929, and by the National Science Foundation under grant CNS-1828181.\n\n\\bibliographystyle{jfm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\mathcal{A}$ be an associative algebra over a field $F$, and let $f\\in F\\langle X\\rangle$ be a multilinear polynomial from the free associative algebra $F\\langle X\\rangle$. Lvov posed the question to determine whether the image of a multilinear $f$ when evaluated on $\\mathcal{A}=M_n(F)$, is always a vector subspace of $M_n(F)$, see \\cite[Problem 1.93]{dnestr}. The original question is attributed to Kaplansky and asks the determination of the image of a polynomial $f$ on $\\mathcal{A}$. It is well known that the above question is equivalent to that of determining whether the image of a multilinear $f$ on $M_n(F)$ is 0, the scalar matrices, $sl_n(F)$ or $M_n(F)$. Clearly the first possibility corresponds to $f$ being a polynomial identity on $M_n(F)$, and the second gives the central polynomials. \n\nRecall here that the description of all polynomial identities on $M_n(F)$ is known only for $n\\le 2$, see \\cite{razmal, dral} for the case when $F$ is of characteristic 0, and \\cite{pkm2} for the case of $F$ infinite of characteristic $p>2$. The same holds for the central polynomials  \\cite{okhitin, jcpk}. The theorem of Amitsur and Levitzki gives the least degree polynomial identity for $M_n(F)$, the standard polynomial $s_{2n}$, see \\cite{am_lev}. Recall also that one of the major breakthroughs in PI theory was achieved by Formanek and by Razmyslov \\cite{for, razm} who proved the existence of nontrivial (that is not identities) central polynomials for the matrix algebras. As for $sl_n(F)$, a theorem of Shoda \\cite{shoda} gives that every $n\\times n$ matrix over a field of zero characteristic is the commutator of two matrices; later on Albert and Muckenhoupt \\cite{albert} generalized this to arbitrary fields. Hence all four conjectured possibilities for the image of a multilinear polynomial on $M_n(F)$ can be achieved. \n\nThe study of images of polynomials on the full matrix algebra is of considerable interest for obvious reasons, among these the relation to polynomial identities. In \\cite{belov1} the authors settled the conjecture due to Lvov in the case of $2\\times 2$ matrices over a quadratically closed field $F$ (that is if $f$ is a given polynomial in several variables then every polynomial in one variable of degree $\\le 2\\deg f$ over $F$ has a root in $F$). They proved that for every multilinear polynomial $f$ and for every field $F$ that is quadratically closed with respect to $f$, the image of $f$ on $M_2(F)$ is 0, $F$, $sl_2(F)$ or $M_2(F)$. It should be noted that the authors in \\cite{belov1} proved a stronger result. Namely they considered a so-called semi-homogeneous polynomial $f$: a polynomial in $m$ variables $x_1$, \\dots, $x_m$ of weights $d_1$, \\dots, $d_m$ respectively such that every monomial of $f$ is of (weighted) degree $d$ for a fixed $d$. They proved that the image of such a polynomial on $M_2(F)$ must be 0, $F$, $sl_2(F)$, the set of all non-nilpotent traceless matrices, or a dense subset of $M_2(F)$. Here the density is according to the Zariski topology. Later on in \\cite{malev} the author gave the solution to the problem for $2\\times 2$ matrices for the case when $F$ is the field of the real numbers. In the case of $3\\times 3$ matrices the known results can be found in \\cite{belov2}. The images of polynomials on $n\\times n$ matrices for $n>3$ are hard to describe, and there are only partial results, see for example \\cite{belov3}. Hence in the case of $n\\times n$ matrices one is led to study images of polynomials of low degree. Interesting results in  this direction are due to \\v Spenko \\cite{spenko}, she proved the conjecture raised by Lvov in case $F$ is an algebraically closed field of characteristic 0, and $f$ is a multilinear Lie polynomial of degree at most 4. Further advances in the field were made in  \\cite{br_kl1, br_kl2, br}. In \\cite{br} the author proved that if $A$ is an algebra over an infinite field $F$ and $A=[A,A]$ then the image of an arbitrary polynomial which is neither an identity nor a central polynomial, equals $A$. Recently Malev \\cite{malev_quat} described completely the images of multilinear polynomials on the real quaternion algebra; he also described the images of semi-homogeneous polynomials on the same algebra.\n\nIf the base field $F$ is finite, a theorem of Chuang \\cite{chuang} states that the image of a polynomial without constant term can be every subset of $M_n(F)$ that contains $0$ and is closed under conjugation by invertible matrices. In the same paper it was also shown that such a statement fails when $F$ is infinite. \n\nTherefore it seems likely it should be very difficult to describe satisfactory the images of multilinear polynomials on $M_n(F)$. That is why people started studying images of polynomials on ``easier\" algebras, and also on algebras with an additional structure. The upper triangular matrix algebras are quite important in Linear Algebra because of their applications to different branches of Mathematics and Physics. They are also very important in PI theory: they describe, in a sense, the subvarieties of the variety of algebras generated by $M_2(F)$ in characteristic 0. Block-triangular matrices appear in the description of the so-called minimal varieties of algebras. The images of polynomials on the upper triangular matrices have been studied rather extensively. In \\cite{wang} the author described the images of multilinear polynomials on $UT_2(F)$, the $2\\times 2$ upper triangular matrices over a field $F$. The images of multilinear polynomials of degree up to 4 on $UT_n=UT_n(F)$ for every $n$ were classified in \\cite{mello_fag}, and in \\cite{Fag} the first named author of the present paper described the images of arbitrary multilinear polynomials on the strictly upper triangular matrices. It turned out that if $f$ is a multilinear polynomial of degree $m$ then its image on the strictly upper triangular matrix algebra $J$ is either 0 or $J^{m}$. The following conjecture was raised in \\cite{mello_fag}: Is the image of a multilinear polynomial on $UT_n(F)$ always a vector subspace of $UT_n(F)$? This conjecture was solved independently in \\cite{LWa}, for infinite fields (or finite fields with sufficiently many elements), and in \\cite{GMe}. Further results concerning images of polynomials on the upper triangular matrix algebra can be found in \\cite{tcm, zw, wzl}.\n\nGradings on algebras appeared long ago; the polynomial ring in one or several variables is naturally graded by the infinite cyclic group $\\mathbb{Z}$ by the degree. Gradings on algebras by finite groups are important in Linear Algebra and also in Theoretical Physics: the Grassmann (or exterior) algebra is naturally graded by the cyclic group of order 2, $\\mathbb{Z}_2$. In fact the Grassmann algebra is the most well-known example of a superalgebra. It should be noted that while in the associative case, the term ``superalgebra\" is synonymous to ``$\\textbf{Z}_2$-graded algebra\", if one considers nonassociative algebras these notions are very different: a Lie or a Jordan superalgebra seldom is a Lie or a Jordan algebra. We are not going to discuss further such topics because these are not relevant for our exposition.\n\nIn \\cite{wall}, Wall classified the finite dimensional $\\mathbb{Z}_2$-graded algebras that are graded simple. Later on the description of all gradings on matrix algebras was obtained as well as on simple Lie and Jordan algebras. We refer the readers to the monograph \\cite{ek} for the state-of-art and for further references. In PI theory gradings appeared in the works of Kemer, see \\cite{kemer}, and constituted one of the main tools in the classification of the ideals of identities of associative algebras, which in turn led him to the positive solution of the long-standing Specht Problem. It turns out that the graded identities are easier to describe than the ordinary ones; still they provide a lot of information on the latter. It is somewhat surprising that the images of polynomials have not been studied extensively in the graded setting. \nIn \\cite{Kul}, Kulyamin described the images of graded polynomials on matrix algebras over the group algebra of a finite group over a finite field. \n\nThe upper triangular matrix algebra admits various gradings, these were shown to be isomorphic to elementary ones, see \\cite{VZa}. A grading on a subalgebra $A$ of $M_n(F)$ is elementary if all matrix units $e_{ij}\\in A$ are homogeneous in the grading. All elementary gradings on $UT_n$ were classified in \\cite{VKV}; in the same paper the authors described the graded identities for all these gradings. In this paper we fix an arbitrary field $F$ and the upper triangular matrix algebra $UT_n$. \n\nIn Section \\ref{sect3} we prove that for an arbitrary group grading on $UT_n$, $n>1$, there are no nontrivial graded central polynomials. (Hence the image of a graded polynomial on $UT_n$ cannot be equal to the scalar matrices whenever $n>1$.) In Section \\ref{sect4} we consider a specific grading on $UT_n$, and describe all possible images of multilinear graded polynomials for that grading. It turns out that the images are always homogeneous vector subspaces. We impose a mild restriction on the cardinality of the base field. As a by-product of the proof we obtain a precise description of the graded identities for this grading.\n\nIn Section \\ref{sect5} we give a sufficient condition for the traceless matrices to be contained in the image of a multilinear graded polynomial. Once again we require a mild condition on the cardinality of the field. Section \\ref{sect6} studies the graded algebras $UT_2$ and $UT_3$. We prove that the image of a multilinear graded  polynomial on $UT_2$, for every group grading, is a homogeneous subspace. In the case of $UT_3$, the image of such a polynomial is also a homogeneous subspace provided that the grading is nontrivial. In the case of the trivial grading, if the field contains at least 3 elements then the image of every multilinear polynomial is a vector subspace. Then we proceed with the Jordan algebra structure $UJ_n$ obtained from $UT_n$ by the Jordan (symmetric) product $a\\circ b= ab+ba$ provided that the characteristic of the base field is different from 2. The description of all group gradings on $UJ_n$ is more complicated than that on $UT_n$, see \\cite{KYa}, there appear gradings that are not isomorphic to elementary ones. The gradings on $UJ_2$ were described in \\cite{KMa}. We consider each one of these gradings, and prove that the image of a multilinear graded  polynomial is always a homogeneous subspace. No restrictions on the base field are imposed (apart from the characteristic being different from 2). An analogous result is deduced for the Lie algebra $UT_2^{(-)}$ obtained from $UT_n$ by substituting the associative product by the Lie bracket $[a,b]=ab-ba$. Finally we consider $UJ_3$ equipped with the natural $\\mathbb{Z}_3$-grading: $\\deg e_{ij} = j-i\\pmod{3}$ for every $i\\le j$, assuming the base field infinite and of characteristic different from 2. We prove that the image of a multilinear graded polynomial is always a homogeneous subspace. \n\nWe hope that this paper will initiate a more detailed study of images of polynomials on algebras with additional structures. \n\n\n\n\\section{Preliminaries}\n\nUnless otherwise stated, we denote by $F$ an arbitrary field and $\\mathcal{A}$ an associative algebra over $F$. Given a group $G$, a $G$-grading on $\\mathcal{A}$ is a decomposition of $\\mathcal{A}$ in a direct sum of subspaces $\\mathcal{A}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ such that $\\mathcal{A}_{g}\\mathcal{A}_{h}\\subset \\mathcal{A}_{gh}$, for all $g$, $h \\in G$. We define the support of a $G$-grading on $\\mathcal{A}$ as the subset $supp(\\mathcal{A})=\\{g\\in G\\mid \\mathcal{A}_{g}\\neq0\\}$. A subspace $U$ of $\\mathcal{A}$ is called homogeneous if $U=\\bigoplus_{g\\in G}(U\\cap \\mathcal{A}_{g})$. A graded homomorphism between two graded algebras $\\mathcal{A}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ and $\\mathcal{B}=\\bigoplus_{g\\in G}\\mathcal{B}_{g}$ is defined as an algebra homomorphism $\\varphi\\colon \\mathcal{A}\\rightarrow \\mathcal{B}$ such that $\\varphi(\\mathcal{A}_{g})\\subset \\mathcal{B}_{g}$ for every $g\\in G$. We denote by $F\\langle X \\rangle^{gr}$ the free $G$-graded associative algebra generated by a set of noncommuting variables $X=\\{x_{i}^{(g)}\\mid i\\in\\mathbb{N},g\\in G\\}$. We also denote the neutral (that is of degree $1\\in G$) variables by $y$ and call them even variables, and the non neutral ones by $z$ and  we call them odd variables. We draw the reader's attention that odd variables may have different degrees in the $G$-grading. \n\nWe define the image of a graded polynomial on an algebra as in \\cite{Kul}.\n\n\\begin{defi}\nLet $f\\in F\\langle X \\rangle^{gr}$ be a $G$-graded polynomial. The image of $f$ on the $G$-graded algebra $\\mathcal{A}$ is the set\n\\[\nIm(f)=\\{a\\in\\mathcal{A}\\mid a=\\varphi(f) \\ \\mbox{for some graded homomorphism} \\ \\varphi\\colon F\\langle X \\rangle^{gr}\\rightarrow \\mathcal{A}\\}\n\\]\n\\end{defi}\n\nEquivalently, if $f(x_{1}^{(g_{1})},\\dots,x_{n}^{(g_{n})})\\in F\\langle X \\rangle^{gr}$, then the image of $f$ on the algebra $\\mathcal{A}$ is the set $Im(f)=\\{f(a_{1}^{(g_{1})},\\dots,a_{n}^{(g_{n})})\\mid a_{i}^{(g_{i})}\\in \\mathcal{A}_{g_{i}}\\}$. We will also denote the image of $f$ on $\\mathcal{A}$ by $f(\\mathcal{A})$.\n\nWe now recall some basic properties of images of graded polynomials on algebras that will be used throughout the paper. \n\n\\begin{prp}\\label{basicprop}\nLet $f\\in F\\langle X \\rangle^{gr}$ be a polynomial and $\\mathcal{A}$ a $G$-graded algebra. \n\\begin{enumerate}\n    \\item Let $U$ be one-dimensional subspace of $\\mathcal{A}$ such that $Im(f)\\subset U$ and assume that $\\lambda Im(f)\\subset Im(f)$ for every $\\lambda \\in F$. Then either $Im(f)=\\{0\\}$ or $Im(f)=U$;\n    \\item If $1\\in \\mathcal{A}$ and $f\\in F\\langle X \\rangle^{gr}$ is a multilinear polynomial in neutral variables such that the sum of its coefficients is nonzero, then $Im(f)=\\mathcal{A}_{1}$;\n    \\item $Im(f)$ is invariant under graded endomorphisms of $F\\langle X \\rangle^{gr}$;\n    \\item If $supp(\\mathcal{A})$ is abelian and $f\\in F\\langle X \\rangle^{gr}$ is multilinear, then $Im(f)$ is a homogeneous subset.\n    \\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nThe proofs of the first and third items are straightforward. For the second item it is enough to recall that if $\\mathcal{A}$ is a graded algebra with $1$, then $1\\in\\mathcal{A}_{1}$. Hence, given $a\\in\\mathcal{A}_{1}$ we have $a=f(\\alpha^{-1} a,1,\\dots,1)$, where $\\alpha\\neq0$ is the sum of the coefficients of $f$, and then $Im(f)=\\mathcal{A}_{1}$. For the last item, let $g_{1}$, \\dots, $g_{m}$ be the homogeneous degree of the variables that occur in $f$. If some $g_{i}\\notin supp(\\mathcal{A})$, then $Im(f)=\\{0\\}$ is a homogeneous subspace. Otherwise, since $supp(\\mathcal{A})$ is abelian, we have that each monomial of $f$ is of homogeneous degree $g_{1}\\cdots g_{m}$, and hence the same holds for $f$.  \n\\end{proof}\n\nWe say that a nonzero polynomial $f\\in F\\langle X \\rangle^{gr}$ is a graded polynomial identity for a $G$-graded algebra $\\mathcal{A}$ if its image on $\\mathcal{A}$ is zero. The set of all graded polynomial identities of $\\mathcal{A}$ will be denoted by $Id^{gr}(\\mathcal{A})$. It is easy to check that $Id^{gr}(\\mathcal{A})$ is actually an ideal of $F\\langle X \\rangle^{gr}$ invariant under graded endomorphisms of $F\\langle X \\rangle^{gr}$. It is called the $T_{G}$-ideal of $\\mathcal{A}$. Given a nonempty subset $S$ of $F\\langle X \\rangle^{gr}$, we denote by $\\langle S \\rangle^{T_{G}}$ the $T_{G}$-ideal generated by $S$, that is the least $T_{G}$-ideal that contains the set $S$. The linearisation process also holds for graded polynomials, and as in the ordinary case we have the following statement.\n\n\\begin{prp}\\label{multiidentity}\nIf $\\mathcal{A}$ satisfies a graded polynomial identity, then $\\mathcal{A}$ also satisfies a multilinear one. Moreover, if $char(F)=0$, then $Id^{gr}(\\mathcal{A})$ is generated by its multilinear polynomials.\n\\end{prp}\n\n\nLet now $\\mathcal{A}=UT_{n}$ be the algebra of $n\\times n$ upper triangular matrices over the field $F$. A $G$-grading on $\\mathcal{A}$ is said to be elementary if all elementary matrices are homogeneous in this grading, or equivalently, if there exists an $n$-tuple $(g_{1},\\dots,g_{n})\\in G^{n}$ such that $\\deg(e_{ij})=g_{i}^{-1}g_{j}$. A theorem of Valenti and Zaicev states that every grading on $UT_{n}$ is essentially elementary.\n\n\\begin{thm}[\\cite{VZa}]\\label{gradingsupper}\nLet $G$ be a group and let $F$ be a field. Assume that $UT_{n}=\\mathcal{A}=\\bigoplus_{g\\in G} \\mathcal{A}_{g}$ is $G$-graded. Then $\\mathcal{A}$ is $G$-graded isomorphic to $UT_{n}$ endowed with some elementary $G$-grading.\n\\end{thm}\n\nBy Proposition 1.6 from \\cite{VKV} we have that an elementary grading on $UT_{n}$ is completely determined by the sequence $(\\deg(e_{12}),\\deg(e_{23}),\\ldots,\\deg(e_{n-1,n}))\\in G^{n-1}$.\n\n\nWe recall now some recent results about the description of images of multilinear polynomials on the algebra of upper triangular matrices. We start with the definition of the so-called commutator degree of an associative polynomial.\n\n\\begin{defi}\nLet $f\\in F\\langle X \\rangle$ be a polynomial. We say that $f$ has commutator degree $r$ if \n\\[\nf\\in\\langle [x_{1},x_{2}]\\cdots [x_{2r-1},x_{2r}]\\rangle^{T} \\ \\mbox{and} \\ f\\notin\\langle [x_{1},x_{2}]\\cdots [x_{2r+1},x_{2r+2}]\\rangle^{T}.\n\\]\n\\end{defi}\nIn \\cite{GMe}, Gargate and de Mello used the above definition to give a complete description of images of multilinear polynomials on $UT_{n}$ over infinite fields. Denoting by $J$ the Jacobson radical of $UT_{n}$ and  $J^{0}=UT_{n}$, they proved the following theorem.\n\n\\begin{thm}\\label{tGargateThiago}\nLet $F$ be an infinite field and let $f\\in F\\langle X \\rangle$ be a multilinear polynomial. Then $Im(f)$ on $UT_{n}$ is $J^{r}$ if and only if $f$ has commutator degree $r$.\n\\end{thm}\n\nOne of the main steps in the proof of Theorem \\ref{tGargateThiago} was the characterization the polynomials of commutator degree $r$ in terms of their coefficients. An instance of such characterization has already been known, see \\cite{GMe} Lemma 3.3(2). \n\n\\begin{lem}[\\cite{GMe}]\\label{sumofcoeffi}\nLet $F$ be an arbitrary field and let $f\\in F\\langle X \\rangle$ be a multilinear polynomial. Then $f\\in \\langle [x_{1},x_{2}]\\rangle^{T}$ if and only if the sum of its coefficients is zero.\n\\end{lem}\n\nIt is worth mentioning that the above theorem has been extended for a larger class of fields by Luo and Wang in \\cite{LWa}.\n\n\\begin{thm}[\\cite{LWa}]\\label{TLuoWang}\nLet $n\\geq 2 $ be an integer, let $F$ be a field with at least $n(n-1)\/2$ elements and let $f\\in F\\langle X \\rangle$ be a multilinear polynomial. If $f$ has commutator degree $r$, then $Im(f)$ on $UT_{n}$ is $J^{r}$.\n\\end{thm}\n\nIn the next corollary we denote by $UT_{n}^{(-)}$ the Lie algebra defined on $UT_{n}$ by means of the Lie bracket $[a,b]=ab-ba$.\n\n\\begin{cor}\nLet $F$ be a field with at least $n(n-1)\/2$ elements and let $f\\in L(X)$ be a multilinear Lie polynomial. Then $Im(f)$ on $UT_{n}^{(-)}$ is $J^{r}$, for some $0\\leq r \\leq n$.\n\\end{cor}\n\n\\begin{proof}\nWe use the Poincar\u00e9-Birkhoff-Witt Theorem (and more precisely the Witt Theorem) to consider the free Lie algebra $L(X)$ as the subalgebra of $F\\langle X \\rangle^{(-)}$ generated by the set $X$. Since $F\\langle X\\rangle$ is the universal enveloping algebra of $L(X)$, given a multilinear Lie polynomial $f\\in L(X)$ there exists an associative polynomial $\\tilde{f}\\in F\\langle X \\rangle$ such that $Im(f)$ on $UT_{n}^{(-)}$ is equal to $Im(\\tilde{f})$ on $UT_{n}$. Now it is enough to apply Theorem \\ref{TLuoWang}.\n\\end{proof}\n\n\nLet $UT_{n}(d_{1},\\ldots,d_{k})$ be the upper block-triangular matrix algebra, that is, the subalgebra of $M_{n}(F)$ consisting of all block-triangular matrices of the form\n\\[\n\\begin{pmatrix}\n     A_{1}& & *  \\\\\n     & \\ddots & \\\\\n     0 & & A_{k} \n\\end{pmatrix}\n\\]\nwhere $n=d_{1}+\\cdots+d_{k}$ and  $A_{i}$ is a $d_{i}\\times d_{i}$ matrix. We will denote by $T$ the subalgebra of $UT_{n}(d_{1},\\ldots,d_{k})$ which consists of only triangular blocks of sizes $d_{i}$ on the main diagonal and zero elsewhere. That is, \n\\[\nT=\\begin{pmatrix}\nUT_{d_{1}} & &0 \\\\\n & \\ddots & \\\\\n 0& & UT_{d_{k}}\n\\end{pmatrix}\n\\]\nAs a consequence of the above theorem we obtain the following lemma.\n\n\\begin{lem}\\label{lblock}\nLet $F$ be a field with at least $n(n-1)\/2$ elements and let $f\\in F\\langle X \\rangle$ be a multilinear polynomial of commutator degree $r$. Then the image $Im(f)$ on $T$ is $J^{r}$, where $J=Jac(T)$ is the Jacobson radical of $T$.\n\\end{lem}\n\n\\begin{proof}\nWe note that $T\\cong UT_{d_{1}}\\times\\cdots\\times UT_{d_{k}}$. Hence, by \\cite[Proposition 5.60]{Bre},\n\\[\nJ=\\begin{pmatrix}\n    J_{d_{1}} & & \\\\\n     & \\ddots & \\\\\n     & & J_{d_{k}}\n\\end{pmatrix}\n\\]\nwhere $J_{d_{i}}=Jac(UT_{d_{i}})$. Therefore by Theorem \\ref{TLuoWang}, we have\n\\[\nf(T)=\\begin{pmatrix}\n    f(UT_{d_{1}}) & &  \\\\\n     &\\ddots &\\\\\n     & & f(UT_{d_{k}})\n\\end{pmatrix} =\\begin{pmatrix}\n    J_{d_{1}}^{r} & &  \\\\\n     &\\ddots &\\\\\n     & & J_{d_{k}}^{r}\n\\end{pmatrix}=J^{r}.\n\\qedhere\n\\]\n\\end{proof}\n\nThroughout this paper we use the letters $w_{i}$ and $w_{i}^{(j)}$ to denote commuting variables. We recall the following well known result about commutative polynomials.\n\n\\begin{lem}\\label{lcomutpoly}\nLet $F$ be an infinite field and let $f_{1}(w_{1},\\dots,w_{m})$, \\dots,  $f_{n}(w_{1},\\dots,w_{m})$ be commutative polynomials. Then there exist $a_{1}$, \\dots,  $a_{m}\\in F$ such that \n\\[\nf_{1}(a_{1},\\dots,a_{m})\\neq0,\\quad \\dots, \\quad f_{n}(a_{1},\\dots,a_{m})\\neq0.\n\\]\n\\end{lem}\n\nA similar result also holds for finite fields, as long as some boundedness  on the degrees of the variables is given (see \\cite[Proposition 4.2.3]{Dre}).\n\n\\begin{lem}\\label{lcomutpolyfinite}\nLet $F$ be a finite field with $n$ elements and let $f=f(w_{1},\\dots,w_{m})$ be a nonzero polynomial. If $\\deg_{w_{i}}(f)\\leq n-1$ for every $i=1$, \\dots, $n$, then there exist $a_{1}$, \\dots, $a_{m}\\in F$ such that $f(a_{1},\\dots,a_{m})\\neq0$.\n\\end{lem}\n\n\\begin{cor}\\label{ccomutpolyfinite}\nLet $F$ be a finite field with $n$ elements and let $f_{1}(w_{1}\\dots,w_{m})$, \\dots, $f_{n-1}(w_{1},\\dots,w_{m})$ be nonzero polynomials in commuting variables. If $\\deg_{w_{i}}(f_{j})\\leq 1$ for all $i$ and $j$, then there exist $a_{1}$,  \\dots, $a_{m}\\in F$ such that\n\\[\nf_{1}(a_{1},\\dots,a_{m})\\neq0,\\quad \\dots,\\quad f_{n-1}(a_{1},\\dots,a_{m})\\neq0.\n\\]\n\\end{cor}\n\n\n\\section{Graded central polynomials for $UT_{n}$}\n\\label{sect3}\n\nOur goal in this section is to prove the non existence of graded central polynomials for the graded algebra of upper triangular matrices with entries in an arbitrary field. It is well known that the algebra of upper block triangular matrices has no central polynomials, see \\cite[Lemma 1]{gz_ijm}.\n\nWe will denote by $Z(\\mathcal{A})$ the centre of the algebra $\\mathcal{A}$.  \n\n\\begin{defi}\nLet $f\\in F\\langle X \\rangle^{gr}$. We say that $f$ is a graded central polynomial for the algebra $\\mathcal{A}$ if $Im(f)\\subset Z(\\mathcal{A})$ and $f\\notin Id^{gr}(\\mathcal{A})$.\n\\end{defi}\n\nWe recall the following fact from \\cite[Lemma 1.4]{VKV}.\n\n\\begin{lem}\nLet $UT_{n}$ be endowed with some elementary grading. Then the subspace of all diagonal matrices is homogeneous of neutral degree.\n\\end{lem}\n\n\\begin{thm}\nLet $UT_{n}=\\mathcal{A}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ be a $G$-grading on the algebra of upper triangular matrices over an arbitrary field. If $n>1$ then there exist no graded central polynomials for $\\mathcal{A}$.\n\\end{thm}\n\n\\begin{proof}\nBy Theorem \\ref{gradingsupper} we have that $\\mathcal{A}$ is graded isomorphic to some elementary grading on $UT_{n}$. Hence we may reduce our problem to elementary gradings. Now we assume that $f\\in F\\langle X \\rangle^{gr}$ is a polynomial with zero constant term, such that $Im(f)$ on $\\mathcal{A}$ is contained in $F=Z(\\mathcal{A})$. We write $f$ as $f=f_{1}+f_{2}$ where $f_{1}$ contains neutral variables only and $f_{2}$ has at least one non neutral variable in each of its monomials. Consider $a_{1}$, \\dots, $a_{m}\\in\\mathcal{A}_{1}$, and $b_{1}$, \\dots, $b_{l}$ non neutral variables (of homogeneous degree $\\ne 1$) that occur in $f$. Hence $f(a_{1},\\dots,a_{m},b_{1},\\dots,b_{l})=f_{1}(\\overline{a}_{1},\\dots,\\overline{a}_{m})+j_{1}+j_{2}$ where $j_{1}$, $j_{2}\\in J$, the Jacobson radical of $\\mathcal{A}$, and $\\overline{a}_{i}$ is the diagonal part of $a_{i}$. Since $Im(f)\\subset F$, then $j_{1}+j_{2}=0$ and hence $Im(f)=Im(f_{1})$, where the image of $f_{1}$ is taken on diagonal matrices only. Now, note that if $\\lambda_{1}$, \\dots, $\\lambda_{m}\\in F$ are arbitrary, then \n\\[\nf_{1}(\\lambda_{1}e_{11},\\dots,\\lambda_{m}e_{11})=f_{1}(\\lambda_{1},\\dots,\\lambda_{m})e_{11}.\n\\]\nSince $Im(f_{1})\\subset F$, we must have $f_{1}(\\lambda_{1},\\dots,\\lambda_{m})=0$. Hence, for diagonal matrices $D_{i}=\\displaystyle\\sum_{k=1}^{n}\\lambda_{k}^{(i)}e_{kk}$ we have \n\\[\nf_{1}(D_{1},\\dots,D_{m})=\\sum_{k=1}^{n}f_{1}(\\lambda_{1}^{(k)},\\dots,\\lambda_{k}^{(m)})e_{kk}=0,\n\\]\nand thus $Im(f)=\\{0\\}$. We conclude the non existence of graded central polynomials for $UT_{n}$.\n\\end{proof}\n\n\\section{Certain ${\\ensuremath{\\mathbb{Z}}}_{q}$-gradings on $UT_{n}$}\n\\label{sect4}\n\nThroughout this section we denote $UT_{n}=\\mathcal{A}$, endowed with the elementary $\\mathbb{Z}_{q}$-grading given by the following sequence in $\\mathbb{Z}_{q}^{n}$\n\\[\n(\\overline{0},\\overline{1},\\dots,\\overline{q-2},\\underbrace{\\overline{q-1},\\overline{q-1},\\dots,\\overline{q-1}}_{\\text{$n-q+1$ times}}). \\]\nGiven $q\\leq n$ an integer, we study the images of multilinear graded polynomials on $\\mathcal{A}$.\n\nOne can see that for $q=n$ we have the natural ${\\ensuremath{\\mathbb{Z}}}_{n}-$grading on $UT_{n}$ given by $\\deg e_{ij}=j-i\\pmod{n}$ for every $i\\le j$.\n\nWe note that the neutral component of $UT_{n}$ is given by a block triangular matrix with $q-1$ triangular blocks of size one each and a triangular block of size $n-q+1$ in the bottom right corner\n\\[\n\\mathcal{A}_{0}=\\begin{pmatrix}\n    * &   &   & 0  \\\\\n      & \\ddots & & \\\\\n      & & * & \\\\\n      0& & & UT_{n-q+1}\n\\end{pmatrix}\n\\]\nFor $l\\in\\{1,\\dots,q-1\\}$ we have that the homogeneous component of degree $\\overline{l}$ is given by\n\\[\n\\mathcal{A}_{\\overline{l}}=span\\{e_{i,i+l}, e_{q-l,j} \\mid i=1,\\dots,q-l, j=q+1,\\dots,n\\}.\n\\]\nFor $1\\leq r \\leq n-q$ we also define the following homogeneous subspaces of $A_{\\overline{l}}$\n\\[\n\\mathcal{B}_{\\overline{l},r}=span\\{e_{q-l,j} \\mid j=q+r,\\ldots,n\\}.\n\\]\nAn immediate computation shows that the following are graded identities for $\\mathcal{A}$\n\\begin{align}\n&[y_{1},y_{2}]z\\equiv 0 \\label{identity1} \\\\\n&z_{1}z_{2}\\equiv 0  \\label{identity2} \\\\\n&{[y_{1},y_{2}]}\\cdots [y_{2(n-q+1)-1},y_{2(n-q+1)}]\\equiv 0 \\label{identity3}\n\\end{align}\nwhere the variables $y_i$ are neutral ones, $z$, $z_{1}$, $z_{2}$ are non neutral variables and $\\deg(z_{1})+\\deg(z_{2})=\\overline{0}$. A complete description of the graded polynomial identities for elementary gradings on $UT_{n}$ was given in \\cite{VKV} for infinite fields and in \\cite{GRi} for finite fields.\n\nWe state several lemmas concerning the description of some graded polynomials on $\\mathcal{A}$. In the upcoming lemmas, unless otherwise stated, we assume that the field $F$ has at least $n(n-1)\/2$ elements and $f\\in F\\langle X \\rangle^{gr}$ is a multilinear polynomial.\n\n\\begin{lem}\\label{l1Zq}\nIf $f=f(y_{1},\\dots,y_{m})$, then $Im(f)$ on $\\mathcal{A}$ is a homogeneous vector subspace.\n\\end{lem}\n\n\\begin{proof}\nIt is enough to apply Lemma \\ref{lblock}.\n\\end{proof}\n\n\nIn the next two lemmas we will assume that $f=f(z_{1},\\dots,z_{l},y_{l+1},\\dots,y_{m})$ where $\\deg(z_{i})=\\overline{1}$, $1\\leq i \\leq l$. It is obvious that in this case one must have $Im(f)$ on $\\mathcal{A}$ as a subset of $\\mathcal{A}_{\\overline{l}}$. Modulo the identity $(1)$ we rewrite the polynomial $f$ as\n\\[\nf=\\sum_{\\bm{i_{1}},\\dots, \\bm{i_{l}}}y_{\\bm{i_{1}}}z_{1}y_{\\bm{i_{2}}}z_{2}\\cdots y_{\\bm{i_{l}}}z_{l}g_{\\bm{i_{1}},\\dots,\\bm{i_{l}}}+h\n\\]\nwhere $y_{\\bm{i_{j}}}=y_{i_{j_{1}}}\\cdots y_{i_{j_{k_{j}}}}$ is such that $i_{j_{1}}<\\cdots <i_{j_{k_{j}}}$. Moreover $g_{\\bm{i_{1}},\\dots,\\bm{i_{l}}}$ is the polynomial obtained by permuting the neutral variables whose indices are different from either of $\\bm{i_{1}}$, \\dots, $\\bm{i_{l}}$, and forming a linear combination of such monomials. Furthermore, $h$ is the sum of  polynomial that differ from the first summand of $f$ by nontrivial permutations of the odd variables.\n\nAmong all polynomials $g_{\\bm{i_{1}},\\dots,\\bm{i_{l}}}$ (including those in $h$), we choose one of least commutator degree, say $g$, of commutator degree $r$. Up to permuting the odd variables, we can assume that the polynomial $g$ occurs in the first summand of $f$. \n\nHence, in case $1\\leq r \\leq n-q$, we can improve the inclusion $Im(f)\\subset \\mathcal{A}_{\\overline{l}}$ to $Im(f)\\subset \\mathcal{B}_{\\overline{l},r}$. Our goal is to prove that $Im(f)=\\mathcal{A}_{\\overline{l}}$ in case $r=0$ and $Im(f)=\\mathcal{B}_{\\overline{l},r}$ otherwise.\n\n\\begin{lem}\\label{l2Zq}\nIf $1\\leq r \\leq n-q$, then $Im(f)=\\mathcal{B}_{\\overline{l},r}$.\n\\end{lem}\n\n\\begin{proof}\nLet $\\overline{g}=y_{\\bm{i_{1}}}z_{1}y_{\\bm{i_{2}}}z_{2}\\cdots y_{\\bm{i_{l}}}z_{l}g$ be a nonzero summand of $f$ written as above, where the commutator degree of $g$ is $r$. We consider the following evaluations: the variables in $y_{\\bm{i_{1}}}$ by $e_{q-l,q-l}$, the ones in $y_{\\bm{i_{2}}}$ by $e_{q-l+1,q-l+1}$,\\dots, and all variables in $y_{\\bm{i_{l}}}$ by $e_{q-1,q-1}$. We also put $z_{1}=e_{q-l,q-l+1}$, $z_{2}=e_{q-l+1,q-l+2}$, \\dots, $z_{l-1}=e_{q-2,q-1}$, and $z_{l}=\\sum_{k=q}^{n}w_{k}e_{q-1,k}$. Since $g$ is of commutator degree $r$, Theorem \\ref{TLuoWang} enables us to evaluate the even variables in $g$ by matrices from \n\\[\n\\begin{pmatrix}\n0&0\\\\\n0&UT_{n-q+1}\n\\end{pmatrix}\n\\]\nin order to obtain the matrix $e_{q,q+r}+e_{q+1,q+r+1}+\\cdots+ e_{n-r,n}$.\n\nNote that the evaluations that we have considered allow us to reduce the study of the image of $f$ to the polynomial $\\overline{g}$. Under these evaluations we have \n\\begin{align*}\n\\overline{g}&=(w_{q}e_{q-l,q}+w_{q+1}e_{q-l,q+1}+\\cdots+w_{n}e_{q-l,n})(e_{q,q+r}+e_{q+1,q+r+1}+\\cdots+e_{n-r,n})\\\\\n&=w_{q}e_{q-l,q+r}+w_{q+1}e_{q-l,q+r+1}+\\cdots+w_{n-r}e_{q-l,n}.\n\\end{align*}\n\nTaking a matrix $B\\in \\mathcal{B}_{\\overline{l},r}$, say $B=b_{q}e_{q-l,q+r}+\\cdots+b_{n-r}e_{q-l,n}$ we can easily realize $B$ as image of $\\overline{g}$ by choosing $w_{j}=b_{j}$, $j=q$, \\dots, $n-r$. \n\nHence $f(\\mathcal{A})=\\mathcal{B}_{\\overline{l},r}$.\n\\end{proof}\n\n\\begin{lem}\\label{l3Zq}\nIf $F$ is a field with at least $n$ elements and $r=0$, then  $Im(f)=\\mathcal{A}_{\\overline{l}}$.\n\\end{lem}\n\n\\begin{proof}\nDenoting by $\\mathcal{D}$ the homogeneous subspace of diagonal matrices of $\\mathcal{A}$, we consider the following homogeneous subalgebra of $\\mathcal{A}$:\n\\[\n\\mathcal{S}=\\mathcal{D}\\oplus \\bigoplus_{1\\leq l\\leq q-1}\\mathcal{A}_{\\overline{l}}.\n\\]\nWe will show that $Im(f)$ on $\\mathcal{S}$ is $\\mathcal{A}_{\\overline{l}}$ which is enough to conclude the lemma. Note that $\\mathcal{S}$ still satisfies the identity (\\ref{identity2}) and it also satisfies $[y_{1},y_{2}]\\equiv0$.\n\nBy the identity $[y_{1},y_{2}]\\equiv0$, we may write the polynomial $\\overline{g}$ as \n\\[\n\\beta y_{\\bm{i_{1}}}z_{1}y_{\\bm{i_{2}}}z_{2}\\cdots y_{\\bm{i_{l}}}z_{l}y_{\\bm{i_{l+1}}}\n\\]\nwhere $\\beta$ is the sum of all coefficients of the polynomial $g$ and $y_{\\bm{i_{l+1}}}$ is the product of the variables of $g$ in increasing order of the indices. We claim that $\\beta\\neq 0$. To this end it is enough to check that $\\overline{g}$ is not a consequence of a commutator. In fact,  if $\\overline{g}$ is a consequence of a commutator it must be in the variables of $g$ since the remaining variables of $\\overline{g}$ are given in a fixed order in all of its monomials. However $g$ has commutator degree zero, which proves our claim.  \n\nNow we write $f=f(z_{1},\\dots,z_{l},y_{l+1},\\dots,y_{m})$ as \n\\[\nf=\\sum_{j=1}^{l}f_{j}\n\\]\nwhere $f_{j}$ is the sum of all monomials of $f$ such that the variable $z_{l}$ is in the $j$-th position in relation to the odd variables.\n\nFor each $j=1$, \\dots, $l$, we write\n\\[\nf_{j}=\\sum_{\\sigma\\in S_{l}^{(j)}}f_{j,\\sigma}\n\\]\nwhere $S_{l}^{(j)}=\\{\\sigma\\in S_{l}|\\sigma(l)=j\\}$, $f_{j,\\sigma}$ is the sum of all monomials of $f_{j}$ where the order of the odd variables is given by the permutation $\\sigma$. \n\nTaking $\\displaystyle z_{i}=\\sum_{k=1}^{q-1}w_{k}^{(i)}e_{k,k+1}+w_{q}^{(i)}e_{q-1,q+1}+\\cdots+w_{n-1}^{(i)}e_{q-1,n}$ and $\\displaystyle y_{j}=\\sum_{k=1}^{n}w_{k}^{(j)}e_{kk}$ we have\n\\begin{align*}\n&f_{l,id}(z_{1},\\dots,z_{l},y_{l+1},\\dots,y_{m})=\\sum_{k=1}^{q-l}p_{k}w_{k}^{(1)}w_{k+1}^{(2)}\\cdots w_{k+l-1}^{(l)}e_{k,k+l}\\\\\n&+ p_{q-l+1}w_{q-l}^{(1)}\\cdots w_{q-2}^{(l-1)}w_{q}^{(l)}e_{q-l,q+1}+\\cdots+ p_{n-l}w_{q-l}^{(1)}\\cdots w_{q-2}^{(l-1)}w_{n-1}^{(l)}e_{q-l,n}\n\\end{align*}\nwhere $p_{k}$, $k=1$, \\dots, $n-l$, are polynomials in the variables $w^{(l+1)}$, \\dots, $w^{(m)}$.\nWe note that all polynomials $p_{k}$, $k=1$, \\dots, $n-l$, are nonzero ones. Indeed, we just have to check that different monomials in $f_{l,id}$ give  different monomials in $p_{k}$. To this end, note that if $m_{1}$ and $m_{2}$ are different monomials in $f_{l,id}$, then there exists some even variable $y_{j}$ such that the quantity of preceding odd variables in relation to $y_{j}$ is distinct in $m_{1}$ and $m_{2}$. This gives us variables $w^{(j)}$ with different lower indices in the two monomials in $p_{k}$ given by $m_{1}$ and $m_{2}$, which proves our claim. Moreover, we note that every variable in each monomial of the polynomial $p_{k}$ appears exactly once.  \n\nSince we have at most $n-1$ polynomials $p_{k}$, by Corollary \\ref{ccomutpolyfinite} there exist evaluations of the even variables $y_{j}$ by diagonal matrices $D_{j}$ such that $p_{k}$ take nonzero values simultaneously for all $k$. In case $F$ is infinite the same conclusion holds by applying Lemma \\ref{lcomutpoly}.\nHence \n\\begin{align*}\nf_{l}&=\\sum_{k=1}^{q-l}\\bigg(\\sum_{\\sigma\\in S_{l}^{(l)}}\\alpha_{\\sigma}w_{k}^{(\\sigma(1))}\\cdots w_{k+l-2}^{(\\sigma(l-1))}\\bigg)w_{k+l-1}^{(l)}e_{k,k+l}\\\\ \n&+ \\bigg(\\sum_{\\sigma\\in S_{l}^{(l)}}\\alpha_{\\sigma}w_{q-l}^{(\\sigma(1))}\\cdots w_{q-2}^{(\\sigma(l-1))}\\bigg)w_{q}^{(l)}e_{q-l,q+1}\\\\\n&+\\cdots+ \\bigg(\\sum_{\\sigma\\in S_{l}^{(l)}}\\alpha_{\\sigma}w_{q-l}^{(\\sigma(1))}\\cdots w_{q-2}^{(\\sigma(l-1))}\\bigg)w_{n-1}^{(l)}e_{q-l,n}\n\\end{align*}\nwith $\\alpha_{id}\\neq0$. So the polynomials inside the brackets above are nonzero ones and each of their monomials have variables of degree one. Applying Corollary \\ref{ccomutpolyfinite} once again we may evaluate the variables $z_{1}$, \\dots, $z_{l-1}$ by matrices in $C_{1}$, \\dots, $C_{l-1}\\in A_{\\overline{1}}$ such that all these polynomials take nonzero values on $F$ (in case $F$ is infinite we apply Lemma \\ref{lcomutpoly}). Denote by $\\alpha_{l,k}\\in F\\setminus\\{0\\}$, $k =1$, \\dots, $q-l$, the values of the polynomials inside the brackets after such evaluations. \n\nTherefore\n\\begin{align*}\n&f(C_{1},\\dots,C_{l-1},z_{l},D_{l+1},\\dots,D_{m})\\\\\n&=\\sum_{k=1}^{q-l}\\bigg(\\alpha_{1,k}w_{k}^{(l)}+\\cdots +\\alpha_{l-1,k}w_{k+l-2}^{(l)} +\\alpha_{l,k}w_{k+l-1}^{(l)}\\bigg)e_{k,k+l}\\\\\n&+\\bigg(\\alpha_{1,q}w_{q-l}^{(l)}+\\cdots + \\alpha_{l-1,q}w_{q-2}^{(l)}+\\alpha_{l,q-l}w_{q}^{(l)}\\bigg)e_{q-l,q+1}\\\\\n&+\\cdots+ \\bigg(\\alpha_{1,n-1}w_{q-l}^{(l)}+\\cdots + \\alpha_{l-1,n-1}w_{q-2}^{(l)}+\\alpha_{l,q-l}w_{n-1}^{(l)}\\bigg)e_{q-l,n}\n\\end{align*}\nwhere $\\alpha_{l,k}\\neq0$ for every $k=1$, \\dots, $q-l$.\n\nThen given a matrix $B=\\displaystyle\\sum_{k=1}^{q-l}b_{k}e_{k,k+l}+b_{q-l+1}e_{q-l,q+1}+\\cdots+ b_{n-l}e_{q-l,n}\\in \\mathcal{A}_{\\overline{l}}$ we take \n\\[\nf(C_{1},\\dots,C_{l-1},z_{l},D_{l+1},\\dots,D_{m})=B\n\\]\nand we obtain a linear system in the variables $w^{(l)}$ whose solution (not necessarily unique) can be found recursively. \n\\end{proof}\n\n\n\\begin{thm}\\label{maintheorem}\nLet $F$ be a field with at least $n(n-1)\/2$ elements, let $UT_{n}=\\bigoplus_{k\\in\\mathbb{Z}_{q}}\\mathcal{A}_{k}$ be endowed with the elementary $\\mathbb{Z}_{q}$-grading given by the sequence $(\\overline{0},\\overline{1},\\dots,\\overline{q-2},\\overline{q-1},\\dots,\\overline{q-1})$ and let $f\\in F\\langle X \\rangle^{gr}$ be a multilinear polynomial. Then $Im(f)$ on $UT_{n}$ is $\\{0\\}$, $J^{r}$, $\\mathcal{B}_{\\overline{l},r}$ or $\\mathcal{A}_{\\overline{l}}$, where $J=Jac(\\mathcal{A}_{0})$. In particular, the image is always a homogeneous vector subspace.\n\\end{thm}\n\n\\begin{proof}\nBy Lemmas \\ref{l1Zq}, \\ref{l2Zq} and \\ref{l3Zq} we only need to analyse the case where $f$ is a polynomial in neutral variables and non neutral ones (that is of degree different from $\\overline{1}$). However we can reduce this case to the aforementioned lemmas. Indeed, modulo the graded identities (\\ref{identity1}), (\\ref{identity2}), (\\ref{identity3}), let $f$ and $g$ be as in the comments before Lemma \\ref{l2Zq}  and let $r$ be the commutator degree of $g$. Hence $Im(f)\\subset \\mathcal{B}_{\\overline{l},r}$ if $r\\neq0$ and $Im(f)\\subset \\mathcal{A}_{\\overline{l}}$ otherwise. By Proposition \\ref{basicprop}(3), the image of the polynomial $\\tilde{f}$ obtained from $f$ by evaluating every non neutral variable $z_{i}$ of homogeneous degree $\\overline{k}$ by a product of $k$ variables of homogeneous degree $\\overline{1}$, is contained in $Im(f)$. But the polynomial $g$ defined for $f$ (see the comments before Lemma \\ref{l2Zq}) is the same as the one defined for $\\tilde{f}$. This allows us to get $\\mathcal{B}_{\\overline{l}}^{r}\\subset Im(\\tilde{f})$ in case $r\\neq0$ and $\\mathcal{A}_{\\overline{l}}\\subset Im(\\tilde{f})$ otherwise.\n\\end{proof}\n\n\\begin{rem}\nConsidering similar computations one can easily see that the result is also valid for the elementary grading defined by the identities $z[y_{1},y_{2}]\\equiv0$, (\\ref{identity2}), and (\\ref{identity3}). \n\\end{rem}\n\nIn the next corollary we are assuming that $F$ is a field of characteristic zero and $\\mathcal{A}=UT_{n}$ is endowed with the elementary ${\\ensuremath{\\mathbb{Z}}}_{q}$-grading given by the sequence $(\\overline{0},\\overline{1},\\dots,\\overline{q-2},\\overline{q-1},\\dots,\\overline{q-1})$.  \n\n\\begin{cor}\nThe $T_{G}$-ideal $Id^{gr}(\\mathcal{A})$ is generated by the graded identities (\\ref{identity1}), (\\ref{identity2}), and (\\ref{identity3}). \n\\end{cor}\n\n\\begin{proof}\nLet $f\\in Id^{gr}(\\mathcal{A})$. By Proposition \\ref{multiidentity} we may assume that $f$ is multilinear. Note that in the proof of Theorem \\ref{maintheorem} and in the lemmas that precede it, we have shown that if $f$ is not a consequence of the identities (\\ref{identity1}), (\\ref{identity2}), and (\\ref{identity3}), then $Im(f)\\neq\\{0\\}$. In other words, if $f\\in Id^{gr}(\\mathcal{A})$, then $f$ is a consequence of the aforementioned identities.  \n\\end{proof}\n\nWe recall that in case $q=n$ we have the natural ${\\ensuremath{\\mathbb{Z}}}_{n}$-grading on $UT_{n}$. \n\n\\begin{cor}\\label{cnaturalUTn}\nLet $F$ be a field with at least $n$ elements, let $UT_{n}$ be endowed with the natural $\\mathbb{Z}_{n}$-grading and let $f\\in F\\langle X \\rangle^{gr}$ be a multilinear polynomial. Then the image of $f$ on $UT_{n}$ is either zero or some homogeneous component.\n\\end{cor}\n\n\\begin{proof}\nIt follows from Proposition \\ref{basicprop}(2), Lemma \\ref{sumofcoeffi}, and from the proof of Lemma \\ref{l3Zq}.\n\\end{proof}\n\n\n\n\\begin{rem}\nNote that the same result also holds if we consider the natural ${\\ensuremath{\\mathbb{Z}}}$-grading on $UT_{n}$. Analogous results hold for the lower triangular matrix algebra $LT_{n}$ as well (we will use this remark in the next section).\n\\end{rem}\n\n\n\\section{Graded identities and traceless matrices}\n\\label{sect5}\n\nIn this section we give a sufficient condition for the subspace of the traceless matrices to be contained in the image of a multilinear polynomial on the full matrix algebra.\n\nWe start by recalling the following result from \\cite{ARo}.\n\n\\begin{thm}[\\cite{ARo}]\\label{tAR}\nLet $D$ be a division ring, $n \\geq 2$ an integer, and $A \\in M_n(D)$ a non central matrix. Then $A$ is similar (conjugate) to a matrix in $M_n(D)$ with at most one non-zero entry on the main diagonal. In particular, if $A$ has trace zero, then it is similar to a matrix in $M_n(D)$ with only zeros on the main diagonal.\n\\end{thm}\n\nConsider the natural $\\mathbb{Z}$-grading on $M_{n}(F)=\\bigoplus_{r\\in\\mathbb{Z}}(M_{n}(F))_{r}$ given by\n\\[\n(M_{n}(F))_{r}=\\left\\{\\begin{array}{cl}\n    span\\{e_{k,k+r}\\mid k=1,\\dots,n-r\\}, &  \\mbox{if} \\ 0\\leq r \\leq n-1 \\\\\n    span\\{e_{k-r,k}\\mid k=1,\\dots,n+r\\}, &  \\mbox{if} \\ -n+1\\leq r\\leq -1\n    \\\\\n    \\{0\\}, & \\mbox{elsewhere}\n\\end{array}\\right.\n\\]\n\nIn this section we denote $F\\langle X \\rangle^{gr}$ the free $\\mathbb{Z}$-graded algebra. We also keep the notation of using variables $y$'s for neutral variables and $z$'s for non neutral ones. \n\n\\begin{thm}\nLet $n\\geq2$ be an integer, let $F$ be a field with at least $(n-1)n+1$ elements where $char(F)$ does not divide $n$, and let $f\\in F\\langle X \\rangle$ be a multilinear polynomial. If  $f(y_{1},\\dots,y_{m-1},z)\\notin \\langle [y_{1},y_{2}] \\rangle^{T_{\\mathbb{Z}}}$ for every non neutral variable $z$, then $Im(f)$ on $M_{n}(F)$ contains $sl_{n}(F)$.\n\\end{thm}\n\n\\begin{proof}\nSince $Im(f)$ is invariant under automorphisms, by Theorem \\ref{tAR} it is enough to show that $Im(f)$ contains all matrices with zero diagonal. Let $A$ be a zero diagonal matrix and write $A$ as the sum of its homogeneous components\n\\[\nA=\\sum_{i=-n+1}^{-1}A_{i}+\\sum_{i=1}^{n-1}A_{i}\n\\]\nwhere $A_{i}=\\displaystyle\\sum_{k=1}^{n+i}a_{k-i,k}e_{k-i,k}$ for $i=-n+1$, \\dots, $-1$ and $A_{i}=\\displaystyle\\sum_{k=1}^{n-i}a_{k,k+i}e_{k,k+i}$ for $i=1$, \\dots, $n-1$.\n\nBy hypothesis $f(y_{1},\\dots,y_{m-1},z^{(i)})$ is not a graded polynomial identity for $UT_{n}$ with the natural $\\mathbb{Z}$-grading, for every variable $z^{(i)}$ of homogeneous degree $i$ where $1\\leq i\\leq n-1$. \n\nWe now consider the following evaluations on generic matrices: $y_{j}=\\displaystyle\\sum_{k=1}^{n}w_{k}^{(j)}e_{kk}$ for all $j=1$, \\dots, $m-1$ and $z^{(i)}=\\displaystyle\\sum_{k=1}^{n-i}w_{k}^{(m,i)}e_{k,k+i}$. \n\nHence\n\\[\nf(y_{1},\\dots,y_{m-1},z^{(i)})=\\sum_{k=1}^{n-i}p_{k,i}w_{k}^{(m,i)}e_{k,k+i}\n\\]\nwhere $p_{k,i}$ is a polynomial in the variables $w_{k}^{(j)}$. Since $f\\notin Id^{gr}(UT_{n})$, Corollary \\ref{cnaturalUTn} gives us that the image of $f(y_{1},\\dots,y_{m-1},z^{(i)})$ on $UT_{n}$ is exactly $(UT_{n})_{i}$. Hence all $p_{k,i}$ are nonzero polynomials. Moreover note that $p_{k,i}$ is such that all its monomials are multilinear ones. \n\nAnalogously, we also have that $f(y_{1},\\dots,y_{m-1},z^{(i)})$ is not a graded polynomial identity for the lower triangular matrix algebra $LT_{n}$ endowed with the natural $\\mathbb{Z}$-grading, for $i=-n+1$, \\dots, $-1$. Therefore \n\\[\nf(y_{1},\\dots,y_{m-1},z^{(i)})=\\sum_{k=1}^{n-i}q_{k,i}w_{k}^{(m,-i)}e_{k+i,k}\n\\]\nwhere $z^{(i)}=\\displaystyle\\sum_{k=1}^{n+i}w_{k}^{(m,-i)}e_{k-i,k}$ and $q_{k,i}$ are nonzero commutative polynomials with multilinear monomials.\n\nThe number of polynomials $p_{k,i}$ and $q_{k,i}$ is exactly $(n-1)n$. We now apply Corollary \\ref{ccomutpolyfinite} to get an evaluation of all variables $w_{k}^{(j)}$ such that the polynomials $p_{k,i}$ and $q_{k,i}$ assume simultaneously nonzero values in $F$. Such evaluations give us diagonal matrices $D_{1}$, \\dots, $D_{m}$ such that  \n\\[\nf(D_{1},\\dots,D_{m-1},z^{(i)})=\\sum_{k=1}^{n-i}\\alpha_{i,k}w_{k}^{(m,i)}e_{k,k+i}\n\\]\nwhere $\\alpha_{i,k}$ are nonzero scalars. Therefore each matrix $A_{i}$ can be realized as $f(D_{1},\\dots,D_{m-1},B_{i})$ for a suitable matrix $B_{i}\\in (UT_{n})_{i}$, for every $i=1$, \\dots, $n-1$. Similarly we also have that each matrix $A_{i}$ can be realized as $f(D_{1},\\dots,D_{m-1},C_{i})$ for a suitable matrix $C_{i}\\in (LT_{n})_{i}$, for all $i=-n+1$, \\dots, $-1$. Hence\n\\[\nA=\\sum_{i=-n+1}^{-1}A_{i}+\\sum_{i=1}^{n-1}A_{i}=\\sum_{i=-n+1}^{-1}f(D_{1},\\ldots,D_{m-1},C_{i})+\\sum_{i=1}^{n-1}f(D_{1},\\dots,D_{m-1},B_{i})\n\\]\nand it is enough to use the linearity of $f$ in one variable to get $A\\in Im(f)$.\n\\end{proof}\n\n\n\\section{The low dimension cases}\n\\label{sect6}\n\n\n\\subsection{Arbitrary gradings on $UT_{2}$ and $UT_{3}$}\n\nWe start this section with the following proposition.\n\n\\begin{prp}\\label{lowprop}\nLet $G$ be a group and let $\\mathcal{A}$ and $\\mathcal{B}$ be two $G$-graded algebras such that $\\mathcal{B}$ is a graded homomorphic image of $\\mathcal{A}$. Let $f\\in F\\langle X \\rangle^{gr}$ be a graded  polynomial and assume that $f(\\mathcal{A})$ is a homogeneous subspace of $\\mathcal{A}$. Then $f(\\mathcal{B})$ is also a homogeneous subspace of $\\mathcal{B}$. \n\\end{prp} \n\n\\begin{proof}\nLet $\\phi\\colon \\mathcal{A}\\rightarrow \\mathcal{B}$ be a graded epimorphism. We start by noting that $\\phi(f(\\mathcal{A}))=f(\\phi(\\mathcal{A}))$. Then taking $b_{i}^{(1)}$, $b_{i}^{(2)}\\in \\mathcal{B}_{g_{i}}$, $i=1$, \\dots, $m$, we have $b_{i}^{(j)}=\\phi(a_{i}^{(j)})$ for some $a_{i}^{(j)}\\in \\mathcal{A}_{g_{i}}$, since $\\phi$ is surjective. This leads us to\n\\begin{align*}\n&\\alpha f(b_{1}^{(1)},\\dots,b_{m}^{(1)})+f(b_{1}^{(2)},\\dots,b_{m}^{(2)})\\\\\n&=\\alpha f(\\phi(a_{1}^{(1)}),\\dots,\\phi(a_{m}^{(1)}))+f(\\phi(a_{1}^{(2)}),\\dots,\\phi(a_{m}^{(2)}))\\\\\n&=\\phi(\\alpha f(a_{1}^{(1)},\\dots,a_{m}^{(1)})+f(a_{1}^{(2)},\\dots,a_{m}^{(2)}))\n\\end{align*}\nwhich is an element from $\\phi(f(\\mathcal{A}))=f(\\phi(\\mathcal{A}))$ since we are assuming that $f(\\mathcal{A})$ is a subspace. Then $f(\\mathcal{B})$ is also a subspace.\n\nNow we assume that $f(\\mathcal{A})$ is a homogeneous subspace and let $b=b_{h_{1}}+\\cdots + b_{h_{k}}$ be an element in $f(\\mathcal{B})$ written as the sum of its homogeneous components. Since $b\\in f(\\mathcal{B})$ let \n\\[\nb=f(b_{1},\\dots,b_{m})=f(\\phi(a_{1}),\\dots,\\phi(a_{m}))=\\phi(f(a_{1},\\dots,a_{m}))\n\\]\nfor some $b_{i}=\\phi(a_{i})$, and let \n\\[\nf(a_{1},\\dots,a_{m})=a_{g_{1}}+\\cdots + a_{g_{l}}\n\\]\nbe the sum of its homogeneous components. It follows that \n\\[\nb=\\phi(a_{g_{1}})+\\cdots +\\phi(a_{g_{l}})\n\\]\nand since $\\phi$ is a graded homomorphism we must have $k=l$ and every $g_{t}$ must be equal to some $h_{s}$. Without loss of generality, we assume $b_{g_{t}}=\\phi(a_{g_{t}})$. Now it is enough to use that $f(\\mathcal{A})$ is homogeneous and $\\phi(f(\\mathcal{A}))=f(\\phi(\\mathcal{A}))$.\n\\end{proof}\n\n\\begin{rem}\\label{remarkprop}\nIn the proof of Proposition \\ref{lowprop} we have not used the associativity of $\\mathcal{A}$. Therefore it also holds for arbitrary algebras, in particular an analogous proposition holds for graded Jordan algebras. \n\\end{rem}\n\nAs a consequence of Proposition \\ref{lowprop} we have the following theorem.\n\n\\begin{thm}\\label{theoremUT2}\nLet $UT_{2}=\\mathcal{A}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ be some grading on $\\mathcal{A}$ and let $f\\in F\\langle X \\rangle^{gr}$ be a multilinear graded polynomial. Then $f(\\mathcal{A})$ is a homogeneous subspace of $\\mathcal{A}$.\n\\end{thm}\n\n\\begin{proof}\nBy Theorem \\ref{gradingsupper} and Proposition \\ref{lowprop}, it is enough to consider images of multilinear graded polynomials on elementary gradings only. We note that just two elementary $G$-gradings can be defined on $\\mathcal{A}=UT_{2}$. Indeed, an elementary grading on $UT_{2}$ is completely determined by the homogeneous degree of $e_{12}$. If $\\deg(e_{12})=1$, then we have the trivial grading, and we apply Theorem \\ref{TLuoWang}. Hence we assume  $\\mathcal{A}_{1}=span\\{e_{11},e_{22}\\}$ and $\\mathcal{A}_{g}=span\\{e_{12}\\}$, where $g\\neq1$. In this grading the images of multilinear polynomials in neutral variables are handled by Lemma \\ref{sumofcoeffi} and Proposition \\ref{basicprop}(2). Since $\\mathcal{A}_{g}^{2}=\\{0\\}$, it is enough to consider multilinear polynomials in one variable of homogeneous degree $g$ and all remaining variables of neutral degree. In this case the image is contained in $\\mathcal{A}_{g}$ and by Proposition \\ref{basicprop}(1) we are done.\n\\end{proof}\n\nNow we prove an analogous fact to Theorem \\ref{theoremUT2} with $\\mathcal{A}=UT_{3}$ instead of $UT_{2}$. From now on in this subsection we assume that $\\mathcal{A}$ is endowed with some elementary $G$-grading given by a tuple $(g_{1},g_{2})\\in G^{2}$. Hence $g_{1}=\\deg(e_{12})$, $g_{2}=\\deg(e_{23})$, and $g_{3}:=g_{1}g_{2}=\\deg(e_{13})$.\n\nHence the elementary gradings on $UT_{3}$ are exactly the ones given by following relations.\n\n\\begin{itemize}\n\\item[(I)] $\\{1\\}\\cap \\{g_{1},g_{2},g_{3}\\}\\neq \\emptyset$.\n\\begin{itemize}\n\\item[(a)] $g_{1}=g_{2}=1$, which implies $g_{3}=1$;\n\\item[(b)] $g_{1}=1$, which implies $g_{2}=g_{3}$;\n\\item[(c)] $g_{2}=1$, which implies $g_{1}=g_{3}$;\n\\item[(d)] $g_{3}=1$ and $g_{1}=g_{2}$;\n\\item[(e)] $g_{3}=1$ and $g_{1}\\neq g_{2}$.\n\\end{itemize}\n\\item[(II)] $\\{1\\}\\cap \\{g_{1},g_{2},g_{3}\\}=\\emptyset$.\n\\begin{itemize}\n\\item[(a)] $1,g_{1},g_{2},g_{3}$ are pairwise distinct elements;\n\\item[(b)] $g_{1}=g_{2}\\neq g_{3}$.\n\\end{itemize}\n\\end{itemize}\n\nIn the following lemmas we discuss the grading on $UT_{3}$ determined by each relation above and the respective image of a multilinear graded polynomial on such a graded algebra. \n\n\\begin{lem}\\label{l1UT3}\nLet $UT_{3}$ be endowed with the grading (I)(b). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nWe denote $g_{2}=g$, then we have $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33},e_{12}\\}$ and $\\mathcal{A}_{g}=span\\{e_{13},e_{23}\\}$. Note that $\\mathcal{A}_{g}^{2}=\\{0\\}$ and hence we only need to analyse multilinear polynomials in at most one variable of homogeneous degree $g$. \n\nThe case when $f$ is a multilinear polynomial in neutral variables is settled by Lemma \\ref{sumofcoeffi} and Proposition \\ref{basicprop}(2).\n\nNow we consider $f$ as a multilinear polynomial in one non neutral variable and $m-1$ neutral ones. Since $\\mathcal{A}$ satisfies the graded identity $z[y_{1},y_{2}]\\equiv 0$, then modulo this identity we write $f$ as\n\\[\n\\sum_{1\\leq i_{1}<\\cdots< i_{k}\\leq m-1}h_{i_{1},\\dots,i_{k}}z_{m}y_{i_{1}}\\cdots y_{i_{k}}.\n\\]\nIf all polynomials $h_{i_{1}, \\dots, i_{k}}$ have commutator degree different from $0$, then $Im(f)\\subset span\\{e_{13}\\}$ and then we apply Proposition \\ref{basicprop}(2). Otherwise we may assume, without loss of generality, that $h_{1,\\dots,k}$ has commutator degree $0$. Then we perform the following evaluations: $y_{1}=\\cdots=y_{k}=e_{33}$, $y_{j}=e_{11}+e_{22}$ for every $j\\notin\\{1,\\dots,k\\}$, and $z_{m}=\\alpha^{-1}(a_{1}e_{13}+a_{2}e_{23})$, where $\\alpha$ is the sum of the coefficients of $h_{1,\\dots,k}$. Note that under such an evaluation we have $a_{1}e_{13}+a_{2}e_{23}\\in Im(f)$ which proves that $Im(f)=\\mathcal{A}_{g}$.\n\\end{proof}\n\n\\begin{lem}\nLet $UT_{3}$ be endowed with the grading (I)(c). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nNote that $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33},e_{23}\\}$, $\\mathcal{A}_{g_{1}}=span\\{e_{12},e_{13}\\}$ and also that $\\mathcal{A}$ satisfies the identities $[y_{1},y_{2}]z\\equiv0$ and $z_{1}z_{2}\\equiv0$. Thus, the proof is similar to the one for (I)(b).\n\\end{proof}\n\n\\begin{lem}\nLet $UT_{3}$ be endowed with the grading (I)(e). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nHere we must have $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33},e_{13}\\}$,  $\\mathcal{A}_{g_{1}}=span\\{e_{12}\\}$, $\\mathcal{A}_{g_{2}}=span\\{e_{23}\\}$. Note that $\\mathcal{A}_{g_{1}}^{2}=\\mathcal{A}_{g_{2}}^{2}=\\{0\\}$,  $\\mathcal{A}_{g_{2}}\\mathcal{A}_{g_{1}}=\\{0\\}$, and $\\mathcal{A}_{g_{1}}\\mathcal{A}_{g_{2}}\\subset span\\{e_{13}\\}$.  \n\nThe case when $f$ is a multilinear polynomial in neutral variables can be treated as in the grading (I)(b). \nHence we may consider $f$ is a multilinear polynomial in: one variable of degree $g_{1}$ (respectively $g_{2}$) and $m-1$ neutral variables, or in one variable of degree $g_{1}$, one of degree $g_{2}$ and $m-2$ neutral ones. In each of these situations we have that $Im(f)$ is contained in a \none-dimensional space and we apply Proposition \\ref{basicprop}(2). \n\\end{proof}\n\n\n\n\\begin{lem}\nLet $UT_{3}$ be endowed with the grading (II)(a). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nWe have $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33}\\}$, $\\mathcal{A}_{g_{1}}=span\\{e_{12}\\}$, $\\mathcal{A}_{g_{2}}=span\\{e_{23}\\}$, and $\\mathcal{A}_{g_{3}}=span\\{e_{13}\\}$. The only non trivial relation among the non neutral homogeneous components is given by $\\mathcal{A}_{g_{1}}\\mathcal{A}_{g_{3}}=\\mathcal{A}_{g_{2}}$. \n\nThe case of $f$ in neutral variables is the same as for the grading (I)(b).\n\nSince the non neutral components are one-dimensional, then the image of a multilinear polynomial in one non neutral variable and $m-1$ neutral ones is always zero or the respective homogeneous component. \n\nIn case $f$ has one variable of homogeneous degree $g_{1}$, one of degree $g_{3}$ and $m-2$ neutral ones then the image is contained in $\\mathcal{A}_{g_{2}}$, and we are done.\n\\end{proof}\n\n\\begin{lem}\nLet $UT_{3}$ be endowed with the grading (II)(b). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nNote that $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33}\\}$, $\\mathcal{A}_{g_{1}}=span\\{e_{12},e_{23}\\}$ and $\\mathcal{A}_{g_{3}}=span\\{e_{13}\\}$. We   only need to consider the case when $f$ is a multilinear polynomial in $m-1$ neutral variables and one of homogeneous degree $g_{1}$, since the remaining cases can be treated as above. We write $f=\\displaystyle\\sum_{j=1}^{m}f_{j}$ where $f_{j}$ is the sum of all monomials from $f$ which contain the variable $z_{m}$ in the $j$-th position. Hence, modulo $[y_{1},y_{2}]\\equiv 0$ we have\n\\[\nf_{j}=\\sum_{1\\leq i_{1}<\\cdots<i_{j-1}\\leq m-1}\\alpha_{i_{1},\\dots,i_{j-1}}y_{i_{1}}\\cdots y_{i_{j-1}}z_{m}y_{k_{1}}\\cdots y_{k_{m-j}}\n\\]\nwhere $k_{1}<\\cdots<k_{m-j}$. We evaluate $y_{i}=w_{1}^{(i)}e_{11}+e_{22}+w_{3}^{(i)}e_{33}$ and $z_{m}=w_{1}^{(m)}e_{12}+w_{2}^{(m)}e_{23}$. Thus $f(y_{1},\\dots,y_{m-1},z_{m})$ is given by\n\\[\n\\begin{pmatrix}\n     0&p_{1}(w_{1}^{(1)},\\dots,w_{1}^{(m-1)})w_{1}^{(m)} & 0\\\\\n     &0 &p_{2}(w_{3}^{(1)},\\dots,w_{3}^{(m-1)})w_{2}^{(m)} \\\\\n     & &0\n\\end{pmatrix}\n\\]\nwhere $p_{1}(w_{1}^{(1)},\\dots,w_{1}^{(m-1)})=\\displaystyle\\sum_{j=1}^{m}\\sum_{1\\leq i_{1}<\\cdots<i_{j-1}\\leq m-1}\\alpha_{i_{1},\\dots,i_{j-1}}w_{1}^{(i_{1})}\\cdots w_{1}^{(i_{j-1})}$ and $p_{2}$ is given analogously.  \n\nWe claim that $p_{1}$ takes nonzero values on $F$. Indeed, assume that $p_{1}$ is a polynomial identity for $F$ and denote $e_{j}=\\displaystyle\\sum_{1\\leq i_{1}<\\cdots< i_{j-1}\\leq m-1}\\alpha_{i_{1},\\dots,i_{j-1}}w_{1}^{(i_{1})}\\cdots w_{1}^{(i_{j-1})}$. Note that $e_{1}\\in F$ and taking $w_{1}^{(1)}=\\cdots=w_{1}^{(m-1)}=0$ we have $e_{1}=0$. Taking $w_{1}^{(l)}=1$ and zero for the remaining values of $w_{1}$ we have $\\alpha_{l}=0$ for all $l\\in\\{1,\\dots,m-1\\}$ and hence $e_{2}=0$. Now assume $e_{l}=0$ for all $l<k$, we shall prove that $e_{k}=0$. For each chosen $i_{1}$, \\dots, $i_{k-1}$ we take $w_{1}^{(r)}=0$ for all $r\\notin\\{i_{1},\\dots,i_{k-1}\\}$, then  $e_{l}=0$ for all $l>k$ and $e_{k}=\\alpha_{i_{1},\\dots,i_{k-1}}w_{1}^{(i_{1})}\\cdots w_{1}^{(i_{k-1})}$. Then we take $w_{1}^{(i_{1})}=\\cdots=w_{1}^{(i_{k-1})}=1$ and we conclude that $\\alpha_{i_{1},\\dots,i_{k-1}}=0$. Hence $p_{1}=0$, which is a contradiction. Analogous claim holds for $p_{2}$. Therefore it is enough to use the variables $w_{1}^{(m)}$ and $w_{2}^{(m)}$ to realize any matrix in $\\mathcal{A}_{g_{1}}$ in the image of $f$. \n\\end{proof}\n\n\\begin{lem}\nLet $UT_{3}$ be endowed with the grading (I)(d). Then $Im(f)$ on $UT_{3}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nWe denote $g_{1}=g$ and note that  $\\mathcal{A}_{1}=span\\{e_{11},e_{22},e_{33},e_{13}\\}$ and $\\mathcal{A}_{g}=span\\{e_{12},e_{23}\\}$. Then $\\mathcal{A}_{g}^{2}\\subset span\\{e_{13}\\}$ and $\\mathcal{A}$ satisfies the identities $z[y_{1},y_{2}]\\equiv 0$ and $[y_{1},y_{2}]z\\equiv 0$. The case when $f$ has one variable of homogeneous degree $g$ and $m-1$ neutral variables can be treated as in the previous lemma. The remaining cases are considered as above.  \n\\end{proof}\n\nHence we have the following theorem.\n\n\\begin{thm}\nLet $F$ be an arbitrary field, let $UT_{3}=\\mathcal{A}=\\bigoplus_{g\\in G}A_{g}$ be some non trivial grading on $\\mathcal{A}$, and let $f\\in F\\langle X \\rangle^{gr}$ be a multilinear graded  polynomial. Then $Im(f)$ on $\\mathcal{A}$ is a homogeneous subspace of $\\mathcal{A}$. If $|F|\\geq 3$ and $\\mathcal{A}$ is equipped with the trivial grading, then the image is also a subspace. \n\\end{thm}\n\n\\begin{proof}\nThe proof is clear from the previous lemmas and Proposition \\ref{lowprop}.\n\\end{proof}\n\n\\subsection{The graded Jordan algebra $UJ_{2}$}\n\nThroughout this subsection we assume that $F$ is a field of characteristic different from $2$ and we denote by $UJ_{n}$ the Jordan algebra of the upper triangular matrices with product $a\\circ b=ab+ba$. Unlike the associative setting, gradings on $UJ_{n}$ are not only elementary ones. Actually, a second kind of gradings also occurs on $UJ_{n}$, the so-called mirror type gradings, and we define these below. First of all let us introduce the following notation. \n\nLet $i$, $m$ be non negative integers and set \n\\[\nE_{i:m}^{+}=e_{i,i+m}+e_{n-i-m+1,n-i+1} \\ \\mbox{and} \\ E_{i:m}^{-}=e_{i,i+m}-e_{n-i-m+1,n-i+1}.\n\\]\n\\begin{defi}\nA $G$-grading on $UJ_{n}$ is called of mirror type if the matrices $E_{i:m}^{+}$ and $E_{i:m}^{-}$ are homogeneous, and $\\deg(E_{i:m}^{+})\\neq \\deg(E_{i:m}^{-})$.\n\\end{defi}\n\nWe recall the following theorem from \\cite{KYa}.\n\n\\begin{thm}[\\cite{KYa}]\nThe $G$-gradings on the Jordan algebra $UJ_{n}$ are, up to a graded isomorphism, elementary or of mirror type.\n\\end{thm}\n\n\nIn particular we have the following classification of the gradings on $UJ_{2}$.\n\n\n\\begin{prp}\nUp to a graded isomorphism, the gradings on $UJ_{2}$ are given by $UJ_{2}=\\mathcal{A}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ where\n\n\\begin{itemize}\n\\item[(I)] elementary ones\n\\begin{itemize}\n\\item[(a)] trivial grading;\n\\item[(b)]  $\\mathcal{A}_{1}=\\begin{pmatrix}\n    a & 0 \\\\\n     & b\n\\end{pmatrix}$, $\\mathcal{A}_{g}=\\begin{pmatrix}\n    0 & c \\\\\n     & 0\n\\end{pmatrix}$\n\\end{itemize}\n\\item[(II)] mirror type ones;\n\\begin{itemize}\n\\item[(a)] $\\mathcal{A}_{1}=\\begin{pmatrix}\n    a & 0 \\\\\n     & a\n\\end{pmatrix}$, $\\mathcal{A}_{g}=\\begin{pmatrix}\n    b & c \\\\\n     & -b\n\\end{pmatrix}$\n\\item[(b)] $\\mathcal{A}_{1}=\\begin{pmatrix}\n    a & b \\\\\n     & a\n\\end{pmatrix}$, $\\mathcal{A}_{g}=\\begin{pmatrix}\n    c & 0 \\\\\n     & -c\n\\end{pmatrix}$\n\\item[(c)] $\\mathcal{A}_{1}=\\begin{pmatrix}\n    a & 0 \\\\\n     & a\n\\end{pmatrix}$, $\\mathcal{A}_{g}=\\begin{pmatrix}\n    b & 0 \\\\\n     & -b\n\\end{pmatrix}$, $\\mathcal{A}_{h}=\\begin{pmatrix}\n    0 & c \\\\\n     & 0\n\\end{pmatrix}$\n\\end{itemize}\n\\end{itemize}\nwhere $g$, $h\\in G$ are elements of order $2$.\n\\end{prp}\n\n\n\nIn \\cite{KYa} it was also proved that the support of a grading on $UJ_{n}$ is always abelian (see \\cite{KYa} Theorem 24). Hence by Proposition \\ref{basicprop} (4) we have that $Im(f)$ on $UJ_{n}$ is a homogeneous subset for any multilinear graded polynomial $f\\in J(X)$.\n\nNext we analyse the images of a multilinear graded Jordan polynomial $f$ on the gradings considered above.\n\n\\begin{lem}\\label{l1jordan}\nLet $UJ_{2}$ be endowed with the grading (I)(b). Then $Im(f)$ on $UJ_{2}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nWe start with a multilinear polynomial $f$ in $m$ neutral variables. We evaluate each variable $y_{i}$ to an arbitrary diagonal matrix $D_{i}$. Therefore each monomial $\\mathbf{m}$ in $f$ is evaluated to $2^{m-1}\\beta D_{1}\\cdots D_{m}$, where $\\beta\\in F$ is the coefficient of $\\mathbf{m}$. Hence \n\\[\nf(D_{1},\\dots,D_{m})=2^{m-1}\\alpha D_{1}\\cdots D_{m}\n\\]\nwhere $\\alpha\\in F$ is the sum of all coefficients of $f$. In case $\\alpha=0$, then $f=0$ is a graded polynomial identity for $UJ_{2}$, otherwise we can take $D_{2}=\\cdots=D_{m}=I_{2}$ and use $D_{1}$ in order to obtain every diagonal matrix in the image of $f$.\n\nSince $UJ_{2}$ satisfies the graded identity $z_{1}\\circ z_{2}=0$ such that $\\deg(z_{1})=\\deg(z_{2})=g$, then we only need to analyse the case where $f$ is a multilinear polynomial in $m-1$ neutral variables and one of homogeneous degree $g$. Obviously we must have $Im(f)\\subset \\mathcal{A}_{g}$ and this homogeneous component is one-dimensional, then we are done.\n\\end{proof}\n\nFor the grading (II)(a) we recall a lemma from \\cite{GSa} applied to multilinear polynomials. In order to make the notation more compact we omit the symbol $\\circ$ for the Jordan product, and we write $ab$ instead of $a\\circ b$. If no brackets are given in a product, we assume these left-normed, that is $abc=(ab)c$.\n\n\\begin{lem}[\\cite{GSa}]\\label{l1DimasSalomao}\nLet $UJ_{2}$ be endowed with the grading (II)(a) and let $f\\in J(X)_{g}$ be a multilinear $\\mathbb{Z}_{2}$-graded polynomial. Then, modulo the graded identities of $UJ_{2}$, we can write $f$ as a linear combination of monomials of the type\n\\[\ny_{1}\\cdots y_{l}z_{i_{0}}(z_{i_{1}}z_{i_{2}})\\cdots (z_{i_{2m-1}}z_{i_{2m}}), 1<\\cdots<l, i_{1}<i_{2}<i_{3}<\\cdots<i_{m}<i_{m+1}, i_{0}>0.\n\\]\n\\end{lem}\n\n\\begin{lem}\\label{l2jordan}\nLet $UJ_{2}$ be endowed with the grading (II)(a). Then $Im(f)$ on $UJ_{2}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nSince $\\dim\\mathcal{A}_{1}=1$ it follows that if the image of a multilinear polynomial on $UJ_{2}$ is contained in $\\mathcal{A}_{1}$ then it must be either $\\{0\\}$ or $\\mathcal{A}_{1}$.\n\nNow we consider a multilinear polynomial $f$ in homogeneous variables of degree $1$ and $g$ such that $\\deg f=g$. Let $\\mathbf{m}=y_{1}\\cdots y_{l}z_{i_{0}}(z_{i_{1}}z_{i_{2}})\\cdots (z_{i_{2m-1}}z_{i_{2m}})$ be a monomial as in Lemma \\ref{l1DimasSalomao}. We note that the main diagonal of a matrix in $m(UJ_{2})$ is such that the entry $(k,k)$ is given by $(-1)^{k+1}2^{m+l+1}a$ where $a$ is the product of the entries at position $(1,1)$ of all matrices $y$ and $z$. Hence every matrix in $Im(f)$ is of the form\n\\[\n\\begin{pmatrix}\n    2^{m+l+1}\\alpha\\cdot a & * \\\\\n     & -2^{m+l+1}\\alpha\\cdot a\n\\end{pmatrix}\n\\]\nwhere $\\alpha$ is the sum of all coefficients of $f$.\n\nIn case $\\alpha=0$, then $Im(f)\\subset span\\{e_{12}\\}$ and then the image is completely determined. \n\nWe consider now $\\alpha\\neq0$. Without loss of generality, we assume that the nonzero scalar occurs in the monomial $y_{1}\\cdots y_{l}z_{0}(z_{1}z_{2})\\cdots (z_{2m-1}z_{2m})$. Then we make the following evaluation: $y_{1}=\\cdots=y_{l}=I_{2}$, $z_{0}=w_{1}(e_{11}-e_{22})+w_{2}e_{12}$ and $z_{i}=e_{11}-e_{22}$ for every $i=1$, \\dots, $2m$, where $w_{1}$, $w_{2}$ are commutative variables. Therefore\n\\[\nf(y_{1},\\dots,y_{l},z_{0},\\dots,z_{2m})=\\begin{pmatrix}\n    2^{m+l+1}\\alpha w_{1} & 2^{m+l+1}w_{2} \\\\\n     & -2^{m+l+1}\\alpha w_{1}\n\\end{pmatrix}.\n\\]\nSince $char(F)\\neq 2$ and $\\alpha\\neq0$, it follows that $Im(f)=\\mathcal{A}_{g}$. \n\\end{proof}\n\nNow we consider the grading (II)(b) and we recall another lemma from \\cite{GSa}.\n\n\\begin{lem}[\\cite{GSa}]\\label{l2GSa}\nLet $f\\in J(X)_{1}$ be a multilinear polynomial. Then, modulo the graded identities of $UJ_{2}$, $f$ can be written as a linear combination of monomials of the form\n\\begin{enumerate}\n    \\item $(y_{i_{1}}\\cdots y_{i_{r}})(z_{j_{1}}\\cdots z_{j_{l}})$;\n    \\item $(((y_{i}z_{j_{1}})z_{j_{2}})y_{i_{1}}\\cdots y_{i_{r}})z_{j_{3}}\\cdots z_{j_{l}}$,\n\\end{enumerate}\nwhere $l\\geq 0$ is even, $r\\geq 0$, $i_{1}<\\cdots <i_{r}$, and $z_{j_{1}}<z_{j_{2}}<z_{j_{3}}<\\cdots <z_{j_{l}}$.\n\\end{lem}\n\n\\begin{lem}\\label{l3jordan}\nLet $UJ_{2}$ be endowed with the grading (II)(b). Then $Im(f)$ on $UJ_{2}$ is a homogeneous subspace.\n\\end{lem}\n\n\\begin{proof}\nWe start with a multilinear polynomial $f$ in $m$ neutral variables. Note that a multilinear monomial of degree $m$ evaluated on $\\mathcal{A}_{1}$ is a matrix whose main diagonal is given by $2^{m-1}aI_{2}$ where $a$ is the product of the entries on the main diagonal of the matrices used in the evaluation. Hence a matrix in $Im(f)$ must be of the form\n\\[\n\\begin{pmatrix}\n    2^{m-1}\\alpha\\cdot a & * \\\\\n     & 2^{m-1}\\alpha\\cdot a\n\\end{pmatrix}\n\\]\nwhere $\\alpha$ is the sum of all coefficients of $f$. In case $\\alpha=0$, we have $Im(f)\\subset span\\{e_{12}\\}$ and the image is completely determined. From now on we assume $\\alpha\\neq0$, and we evaluate $m-1$ matrices by $I_{2}$ and one matrix, say $y_{1}$, by $w_{1}(e_{11}+e_{22})+w_{2}e_{12}$ where $w_{1}$, $w_{2}$ are commuting variables. Therefore\n\\[\nf(y_{1},\\dots,y_{m})=\\begin{pmatrix}\n    2^{m-1}\\alpha w_{1} & 2^{m-1}\\alpha w_{2} \\\\\n     & 2^{m-1}\\alpha w_{1}\n\\end{pmatrix},\n\\]\nand since $char(F)\\neq2$ and $\\alpha\\neq0$, we have that $Im(f)=\\mathcal{A}_{1}$.\n\nNow we consider a multilinear polynomial $f$ which has at least one variable of homogeneous degree $g$. In case $\\deg f=g$, the image $Im(f)$ is completely determined, since $\\dim\\mathcal{A}_{g}=1$. So we assume $\\deg f=1$. In case $f$ is a multilinear polynomial in variables of homogeneous degree $g$, then $Im(f)$ is contained in the vector space of the scalars matrices, and therefore the image is completely determined. Hence we assume further that $f$ has at least one variable of neutral degree and let $f$ be  a multilinear polynomial in $l$ neutral variables $y_{1}$, \\dots, $y_{l}$, and $m-l$ variables $z_{l+1}$, \\dots, $z_{m}$ of homogeneous degree $g$. Then by Lemma \\ref{l2GSa}, we write $f$ as \n\\[\nf=\\alpha_{1}(y_{1}\\cdots y_{l})(z_{l+1}\\cdots z_{m})+\\sum_{i=1}^{l}\\alpha_{i+1}(((y_{i}z_{l+1})z_{l+2})y_{1}\\cdots \\widehat{y}_{i}\\cdots  y_{l})z_{l+3}\\cdots z_{m}.\n\\]\nHere $\\widehat{y}_{i}$ means that the variable $y_{i}$ does not appear in the product $y_{1}\\cdots \\widehat{y}_{i}\\cdots  y_{l}$.\n\nWe replace $y_{i}=w_{1}^{(i)}(e_{11}+e_{22})+w_{2}^{(i)}e_{12}$ and $z_{j}=w_{1}^{(j)}(e_{11}-e_{22})$, where the $w$'s are commuting variables. Note that the Jordan product of two matrices $y_{1}$ and $y_{2}$ is given by $2y_{1}\\cdot y_{2}$ where the dot $\\cdot$ stands for the usual product of matrices. On the other hand, the usual product of $n$ matrices $y_{1}$,  \\dots,  $y_{n}$ is given by \n\\[\n\\begin{pmatrix}\n    w_{1}^{(1)}\\cdots w_{n}^{(n)} & w  \\\\\n     & w_{1}^{(1)}\\cdots w_{n}^{(n)} \n\\end{pmatrix}.\n\\]\nHere $w=\\displaystyle\\sum_{\\substack{1\\leq i_{1}<\\cdots<i_{n-1}\\leq n \\\\\ni_{n}\\in\\{1,\\dots,n\\}\\setminus\\{i_{1},\\dots,i_{n-1}\\}}}w_{1}^{(i_{1})}\\cdots w_{1}^{(i_{n-1})}w_{2}^{(i_{n})}$, as one can see by induction on $n$. Hence the image of the monomial $\\alpha_{i+1}(((y_{i}z_{l+1})z_{l+2})y_{1}\\cdots \\widehat{y}_{i}\\cdots  y_{l})z_{l+3}\\cdots z_{m}$ is equal to\n\\[\n2^{m-1}\\alpha_{i+1}\\begin{pmatrix}\n   w_{1}^{(l)}\\cdots w_{1}^{(l)}w_{1}^{(l+1)}\\cdots w_{1}^{(m)}  & w_{i}w_{1}^{(l+1)}\\cdots w_{1}^{(m)} \\\\\n     & w_{1}^{(l)}\\cdots w_{1}^{(l)}w_{1}^{(l+1)}\\cdots w_{1}^{(m)}\n\\end{pmatrix}\n\\]\nwhere $w_{i}$ is given as $w$ above but $i_{n}\\neq i$. \n \nTherefore the main diagonal of $f(y_{1},\\dots,y_{l},z_{l+1},\\dots,z_{m})$ is given by\n\\[\n2^{m-1}\\alpha w_{1}^{(1)}\\cdots w_{1}^{(m)}\n\\]\nwhere $\\alpha$ is the sum of all coefficients in $f$. The entry at position $(1,2)$ is \n \\[\n \\displaystyle\\sum_{\\substack{1\\leq i_{1}<\\cdots<i_{l-1}\\leq l \\\\\nk\\in\\{1,\\dots,l\\}\\setminus\\{i_{1},\\dots,i_{l-1}\\}}}\\beta_{k}w_{1}^{(i_{1})}\\cdots w_{1}^{(i_{l-1})}w_{2}^{(k)}w_{1}^{(l+1)}\\cdots w_{1}^{(m)}\n \\]\n where $\\beta_{k}=\\alpha_{1}+\\displaystyle\\sum_{\\substack{j=1 \\\\ j\\neq k}}^{l}\\alpha_{j+1}$.\n\nIf all $\\beta_{k}$ are equal to zero, then $Im(f)$ is contained in the space of the scalar matrices and we are done. So we may assume that some of the $\\beta_{k}$ is nonzero, and without loss of generality we suppose $\\beta_{l}\\neq0$. In this case we claim that the image will be the whole neutral component. Indeed, take $A=a_{1}(e_{11}+e_{22})+a_{2}e_{12}\\in \\mathcal{A}_{1}$. We evaluate the matrices of degree $g$ by $e_{11}-e_{22}$, that is, we take $w_{1}^{(l+1)}=\\cdots=w_{1}^{(m)}=1$. We also evaluate the neutral matrices $y_{1}$, \\dots, $y_{l-1}$ by the identity matrix, that is, we take $w_{1}^{(1)}=\\cdots = w_{1}^{(l-1)}=1$ and $w_{2}^{(1)}=\\cdots =w_{2}^{(l-1)}=0$. Hence the equality\n\\[\nf(I_{2},\\dots,I_{2},y_{l},e_{11}-e_{22},\\dots,e_{11}-e_{22})=A\n\\]\n leads us to the following linear system\n \\[\n \\left\\{\\begin{array}{c}\n     2^{m-1}\\alpha w_{1}^{(l)}=a_{1}  \\\\\n     2^{m-1}\\beta_{l}w_{2}^{(l)}=a_{2} \n \\end{array}\\right.\n \\]\n which has $w_{1}^{(1)}=(2^{m-1}\\alpha)^{-1}a_{1}$ and $w_{2}^{(2)}=(2^{m-1}\\beta_{1})^{-1}a_{2}$ as the solution.\n\\end{proof} \n \n\\begin{thm}\nLet $UJ_{2}=\\bigoplus_{g\\in G}\\mathcal{A}_{g}$ be a non trivial $G$-grading and let $f\\in J(X)$ be a multilinear graded  Jordan polynomial. Then $Im(f)$ on $UJ_{2}$ is a homogeneous subspace. The same conclusion also holds for the trivial grading on $UJ_{2}$ for arbitrary fields.\n\\end{thm}\n \n\\begin{proof}\nWe first consider a non trivial grading on $UJ_{2}$. By Remark \\ref{remarkprop} we may reduce the defined grading on $UJ_{2}$ to one of those described above. We note that the case of the grading (II)(c) follows from the fact that $Im(f)$ on $UJ_{2}$ is a homogeneous subset and all homogeneous components in this grading are one dimensional. We use Lemmas \\ref{l1jordan},\\ref{l2jordan} and \\ref{l3jordan} for the remaining non trivial gradings. Now we consider the trivial grading on $UJ_{2}$. Let $f\\in J(X)$ be a multilinear polynomial. We may assume that $f\\notin Id(UJ_{2})$. By \\cite{Sli}, the algebra $J(X)\/Id(UJ_{2})$ is a special Jordan algebra,  and hence we may assume $f$ is an element in the free special Jordan algebra. Therefore, the image $Im(f)$ on $UJ_{2}$ is equal to the image of some associative polynomial on $UT_{2}$. Hence $Im(f)\\in\\{J,UJ_{2}\\}$.\n\\end{proof}\n\n\\begin{rem}\nConsider the Lie algebra $UT_{n}^{(-)}$ with product given by the Lie bracket. Given a grading on $UT_{n}^{(-)}$, note that $J=[UT_{n}^{(-)},UT_{n}^{(-)}]$ is always a homogeneous ideal. We also note that if $f\\in L(X)^{gr}$ is a multilinear polynomial of degree $\\ge 2$, then $Im(f)$ on $UT_{n}^{(-)}$ is contained in $J$. In particular, for $n=2$ we must have that $Im(f)$ is contained in $span\\{e_{12}\\}$ which is a homogeneous subspace. Since the image of multilinear polynomials of degree $1$ is trivial, we have that $Im(f)$ on the graded algebra $UT_{2}^{(-)}$ is always a homogeneous subspace, regardless of the grading defined on $UT_{2}^{(-)}$.\n\\end{rem}\n\n\\subsection{The natural elementary ${\\ensuremath{\\mathbb{Z}}}_{3}$-grading in the Jordan algebra $UJ_{3}$}\n\nIn this section we study images of multilinear polynomials on the Jordan algebra $\\mathcal{A}=UJ_{3}$ endowed with the elementary ${\\ensuremath{\\mathbb{Z}}}_{3}$-grading given by the sequence $(\\overline{0},\\overline{1},\\overline{2})$, that is, $\\mathcal{A}_{\\overline{0}}=span\\{e_{11},e_{22},e_{33}\\}$, $\\mathcal{A}_{\\overline{1}}=span\\{e_{12},e_{23}\\}$, and $\\mathcal{A}_{\\overline{2}}=span\\{e_{13}\\}$. \n\nWe denote by $(x_{1},x_{2},x_{3})=(x_{1}x_{2})x_{3}-x_{1}(x_{2}x_{3})$ the associator of the elements $x_{1}$, $x_{2}$, $x_{3}$.\n\nWe recall the following identity which holds in any Jordan algebra.\n\n\\begin{lem}\\label{lJacobson}\nLet $\\mathcal{J}$ be a Jordan algebra. Then\n\\[\nabcd+adcb+bdca=(ab)(cd)+(ac)(bd)+(ad)(bc)\n\\]\nfor all $a$, $b$, $c$, $d\\in \\mathcal{J}$.\n\\end{lem}\n\n\\begin{proof}\nSee for example \\cite[Page 34]{Jac}.\n\\end{proof}\n\nAs an easy consequence of Lemma \\ref{lJacobson} we have \n\\begin{equation}\\label{iJacobson}\nabcd+adcb+bdca=abdc+acdb+bcda\n\\end{equation}\nfor every $a$, $b$, $c$, $d\\in \\mathcal{J}$.\n\n\nThe next lemma point out some graded identities for the algebra $UJ_{3}$.\n\n\\begin{lem}\\label{lidentitiesjordan}\nThe identities \n\\[\n(y_{1},y_{2},y_{3})\\equiv 0, (y_{1},z,y_{2})\\equiv 0 \\ \\mbox{and} \\ z_{1}z_{2}\\equiv 0\n\\]\nhold for $UJ_{3}$, where $z$ is an odd variable and $\\deg(z_{1})+\\deg(z_{2})=\\overline{0}$.\n\\end{lem}\n\n\\begin{proof}\nA straightforward computation, hence omitted.\n\\end{proof}\n\n\n\nThe next lemma has the same proof as \\cite[Lemma 5.3]{GSa}. However we will consider its proof here for the sake of completeness.\n\n\\begin{lem}\nThe polynomial \n\\[\ng=y_{1}(y_{2}(y_{3}z))-\\frac{1}{2}\\bigg(y_{1}(z(y_{2}y_{3})) +y_{2}(z(y_{1}y_{3}))+y_{3}(z(y_{1}y_{2}))-z(y_{1}(y_{2}y_{3}))\\bigg)\n\\]\nis a consequence of $(y_{1},z,y_{2})$, where $\\deg(z)\\in\\{\\overline{1},\\overline{2}\\}$.\n\\end{lem}\n\n\\begin{proof}\nBy identity (\\ref{iJacobson}) we have\n\\[\n-((y_{2}y_{3})z)y_{1}-((y_{1}y_{3})z)y_{2}-((y_{1}y_{2})z)y_{3}+((y_{2}y_{3})y_{1})z=-((y_{2}z)y_{1})y_{3}-((y_{3}z)y_{1})y_{2}.\n\\]\nHence, we can write $h=2g$ as\n\\begin{align*}\n h= & \\ 2(y_{1}(y_{2}(y_{3}z))-((y_{2}z)y_{1})y_{3}-((y_{3}z)y_{1})y_{2}\\\\\n = & \\ (y_{3},z,y_{2})y_{1}+(y_{2},zy_{3},y_{1})+(y_{3},zy_{2},y_{1})\n\\end{align*}\nwhich implies that $g$ is a consequence of $(y_{1},z,y_{2})$.\n\\end{proof}\n\nGiven two even variables $y_{i}$ and $y_{j}$ we set $y_{i}<y_{j}$ if $i<j$. Hence we define an order on words in even variables $Y_{1}<Y_{2}$ considering the left lexicographic order in case $Y_{1}$ and $Y_{2}$ have the same length, and $Y_{1}<Y_{2}$ in case $Y_{2}$ is longer than $Y_1$.  For the next lemma we use ideas from \\cite[Lemma 5.6]{GSa}. We denote by $T$ the $T$-ideal generated by the identities from Lemma \\ref{lidentitiesjordan}.\n\n\\begin{lem}\\label{ljordanUJ3}\nLet $f=f(y_{1},\\dots,y_{m-1},z_{m})\\in J(X)$ be a multilinear polynomial, where $\\deg(z_{m})\\in \\{\\overline{1},\\overline{2}\\}$. Then modulo $T$, $f$ is a linear combination of monomials of the form $Y_{1}(zY_{2})$, where each $Y_{i}$ is an increasingly ordered product of even variables and $Y_{1}<Y_{2}$. \n\\end{lem}\n\n\\begin{proof}\nIt is enough to consider $f=f(y_{1},\\dots,y_{m-1},z_{m})$ as a monomial. We apply induction on $m$. If $m=1$ or $m=2$, then the conclusion is obvious. So we assume $m\\geq 3$ and we write $f=gh$ where $g$, $h\\in J(X)$. Without loss of generality we may assume that the odd variable $z_{m}$ occurs in $g$. Hence $h=Y_{1}$ and by the induction hypothesis we must have $g$ as a linear combination of monomials of the form $Y_{2}(zY_{3})$. On the other hand, we have\n\\[\n(Y_{2}(zY_{3}))Y_{1}=\\frac{1}{2}\\bigg(Y_{1}(z(Y_{2}Y_{3}))+Y_{2}(z(Y_{1}Y_{3})+Y_{3}(z(Y_{1}Y_{2}))-z(Y_{1}(Y_{2}Y_{3}))\\bigg).\n\\]\nNow it is enough to use the identities $(y_{1},y_{2},y_{3})\\equiv0$,  $(y_{1},z,y_{2})\\equiv0$ and the commutativity of the Jordan product to get the desired conclusion. \n\\end{proof}\n\nNow we are ready to prove the main theorem of this subsection.\n\n\\begin{thm}\nLet $F$ be an infinite field of characteristic different from 2 and let $f\\in J(X)$ be a multilinear graded  polynomial. Then the image of $f$ on the graded Jordan algebra $UJ_{3}$ endowed with the natural elementary ${\\ensuremath{\\mathbb{Z}}}_{3}$-grading is either $\\{0\\}$ or some homogeneous component.\n\\end{thm}\n\n\\begin{proof}\nSince $f$ is a homogeneous polynomial in the graded algebra $J(X)$ and $z_{1}z_{2}\\equiv0$ holds on $UJ_{3}$, for $\\deg z_{1}+\\deg z_{2}=\\overline{0}$, we will consider the following three cases in our proof.\n\n{\\it Case 1:} $\\deg f=\\overline{0}$. Here we must have $f=f(y_{1},\\dots,y_{m})$ and the proof is the same as the first paragraph of the proof of Lemma \\ref{l1jordan}. \n\n{\\it Case 2:} $\\deg f=\\overline{1}$. Let $f=f(y_{1},\\dots,y_{m-1},z_{m})$ be such that $\\deg z_{m}=\\overline{1}$. By Lemma \\ref{ljordanUJ3}, modulo $T$, we may write $f$ as a linear combination of monomials of the form $Y_{1}(z_{m}Y_{2})$, where $Y_{1}<Y_{2}$. On the other hand, given \n\\begin{equation}\\label{J3evaluation}\ny_{i}=\\sum_{k=1}^{3}w_{k}^{(i)}e_{kk} \\ \\mbox{and} \\ z_{m}=w_{1}^{(m)}e_{12}+w_{2}^{(m)}e_{23},\n\\end{equation}\nnote that $y_{i}z_{m}=\\begin{pmatrix}\n     0&(w_{1}^{(i)}+w_{2}^{(i)})w_{1}^{(m)} &0  \\\\\n     &0 &(w_{2}^{(i)}+w_{3}^{(i)})w_{2}^{(m)}  \\\\ \n     & &0\n\\end{pmatrix}$ and then\n\\[\nf(y_{1},\\dots,y_{m-1},z_{m})=\\begin{pmatrix}\n     0&p_{1}w_{1}^{(m)} &0  \\\\\n     &0 &p_{2}w_{2}^{(m)}  \\\\ \n     & &0\n\\end{pmatrix}\n\\]\nwhere $p_{1}$ and $p_{2}$ are polynomials in the variables $w^{(i)}$, $i=1$, \\dots, $m-1$. We claim that if $f\\neq0$ modulo $T$, then $p_{1}\\neq0$ and $p_{2}\\neq0$. Indeed, consider the monomial $\\mathbf{m}=\\alpha Y_{1}(z_{m}Y_{2})$, where   $Y_{1}=y_{j_{1}}\\cdots y_{j_{r}}$, $Y_{2}= y_{l_{1}}\\cdots y_{l_{s}}$ and $Y_{1}<Y_{2}$. Note that the $(1,2)$ entry of the image of $\\mathbf{m}$ under the evaluation (\\ref{J3evaluation}) is given by\n\\[\n2^{a}\\alpha (w_{1}^{(j_{1})}\\cdots w_{1}^{(j_{r})}+w_{2}^{(j_{1})}\\cdots w_{2}^{(j_{r})})w_{1}^{(m)}(w_{1}^{(l_{1})}\\cdots w_{1}^{(l_{s})}+w_{2}^{(l_{1})}\\cdots w_{2}^{(l_{s})}) \n\\]\nfor some power of $2$. Hence $p_{1}$ contains the following monomials \n\\[\n2^{a}\\alpha w_{1}^{(j_{1})}\\cdots w_{1}^{(j_{r})}w_{2}^{(l_{1})}\\cdots w_{2}^{(l_{s})} \\ \\mbox{and} \\ 2^{a}\\alpha w_{2}^{(j_{1})}\\cdots w_{2}^{(j_{r})}w_{1}^{(l_{1})}\\cdots w_{1}^{(l_{s})}.\n\\]\nSince $Y_{1}<Y_{2}$, the two monomials above can only be obtained from the monomial $\\mathbf{m}$. Hence, if $f\\neq 0$ modulo $T$, then $f$ contains some monomial $\\mathbf{m}$ as above for some nonzero $\\alpha$, which will imply in nonzero monomials in $p_{1}$ that are not multiple of any other one that comes from the remaining monomials of $f$. The same ideas also prove that $p_{2}\\neq0$.\n\nNow we use the fact that $F$ is infinite to get evaluations of the even variables for diagonal matrices such that $p_{1}$ and $p_{2}$ assume nonzero values on $F$, simultaneously. We finally use the variables $w_{1}^{(m)}$ and $w_{2}^{(m)}$ to get arbitrary odd matrices in $Im(f)$, that is, $Im(f)=(UJ_{3})_{\\overline{1}}$.\n\n{\\it Case 3:} $\\deg f=\\overline{2}$. This last case follows from the fact that the homogeneous component of degree $\\overline{2}$ is one dimensional. \n\\end{proof}\n\n\\section*{Funding}\n\nP. Fagundes was supported by S\\~ao Paulo Research Foundation (FAPESP), grant \\# 2019\/16994-1. P. Koshlukov was partially supported by S\\~ao Paulo Research Foundation (FAPESP), grant \\# 2018\/23690-6 and by CNPq grant\n\\# 302238\/2019-0.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Lov\\'{a}sz Local Lemma, due to Erd\\H{o}s and Lov\\'{a}sz \\nolinebreak\\latexcite{ErdosLovasz},\nhas become a very useful tool for showing the existence of objects with special properties.\nIt is particularly important in combinatorics, theoretical computer science,\nand also in number theory.\n\nThe statement of the LLL involves a probability space on $\\Omega$ with events $E_1,\\ldots,E_n$.\nThe desired conclusion is that $\\Pr[ \\cap_i \\overline{E_i} ] > 0$.\nThe hypothesis is that there exists a graph on vertex set $[n] = \\set{1,\\ldots,n}$ such that\n\\begin{enumerate}\n\\item[\\rm (i)] non-neighbors in the graph have, in a certain sense, limited dependence, and\n\\item[\\rm(ii)] the probabilities of the events must satisfy a certain upper bound.\n\\end{enumerate}\nIn the original formulation of the LLL \\nolinebreak\\latexcite{ErdosLovasz}, \ncondition (i) is that each event must be independent from its non-neigbors,\nand condition (ii) is that each event must have probability at most $1\/4d$, where $d$ \nis the maximum degree in the graph.\n\nOver the years, new formulations of condition (i) were discovered,\nof which a very general one is stated below as inequality \\eqref{eq:Dep}.\nInstead of requiring independence between non-neighbors,\nit allows arbitrary dependencies, as long as one can establish a useful upper bound on the\nprobability of $E_i$ conditioned on any set of its non-neighboring events not occurring.\nWe believe this condition first appeared in a paper by Albert, Frieze and Reed \\nolinebreak\\latexcite{Albert},\nand is sometimes referred to as the ``lopsided\" version of the LLL.\n(This is more general than the condition used by Erdos and Spencer~\\nolinebreak\\latexcite{ErdosSpencer}.)\n\nSeveral new formulations of condition (ii) have been proposed over the years,\nnotably by Spencer \\nolinebreak\\latexcite{Spencer75,Spencer77} and by Shearer \\nolinebreak\\latexcite{Shearer}.\nShearer's condition is actually optimal, assuming that the graph is undirected.\nUnfortunately Shearer's condition is difficult to use in applications, so\nresearchers have also studied weaker conditions that are easier to use.\nOne of the most useful of those is the ``cluster expansion'' condition, due to Bissacot\net~al.~\\nolinebreak\\latexcite{Bissacot}.\n\nIn this note, we present short, self-contained proofs of the LLL \nin which condition (i) is formalized using \\eqref{eq:Dep}, as in Albert et~al.~\\nolinebreak\\latexcite{Albert},\nand condition (ii) is formalized using either Shearer's condition \\nolinebreak\\latexcite{Shearer}\nor the cluster expansion condition \\nolinebreak\\latexcite{Bissacot}.\n\\Section{ShearerShort} gives a short proof for Shearer's condition.\nOur proof follows the line of Shearer's original argument,\nalthough we believe our exposition is simpler and more direct.\n\\Section{cluster} gives a short proof for the cluster expansion condition.\nWhereas Bissacot et al.\\ used analytic methods inspired \nby statistical physics, we found a short combinatorial inductive argument. \nThis combinatorial proof originally appeared in Section 5.7 of \\nolinebreak\\latexcite{HV-arxiv}, \nbut since that may be somewhat difficult to find, we reproduce it here.\nTo conclude, we show that the cluster expansion condition implies the\nnear-optimal $p \\leq \\frac{1}{ed}$ condition for the symmetric LLL.\n\n\n\n\n\\section{Shearer's Lemma}\n\\SectionName{ShearerShort}\n\nThe following result is the ``lopsided Shearer's Lemma\", a generalization of the LLL\ncombining conditions from Albert et~al.~\\nolinebreak\\latexcite{Albert} and Shearer \\nolinebreak\\latexcite{Shearer}.\nThis formulation also appears in \\nolinebreak\\latexcite{Knuth}.\nLet $\\Gamma(i)$ denote the neighbors of vertex $i$ and let $\\Gamma^+(i) = \\Gamma(i) \\cup \\set{i}$.\nLet $\\operatorname{Ind}=\\operatorname{Ind}(G)$ denote the collection of all independent sets in the graph $G$.\n\n\\begin{lemma}[lopsided Shearer's Lemma]\n\\LemmaName{extShearer}\nSuppose that $G$ is a graph and $E_1,\\ldots,E_n$ events such that\n\\begin{equation}\n\\EquationName{Dep}\n\\Pr[E_i \\mid {\\textstyle \\bigcap}_{j \\in J} \\overline{E_j}] ~\\leq~ p_i\n\\qquad\\forall i \\in [n] ,\\, J \\subseteq [n] \\setminus \\Gamma^+(i).\n\\end{equation}\nFor each $S \\subseteq [n]$, define\n$$\\breve{q}_S ~=~ \\breve{q}_S(p) ~=~ \\sumstack{I \\subseteq S \\\\ I \\in \\operatorname{Ind}(G)} (-1)^{|I|} \\prod_{i \\in I} p_i.$$\nIf $\\breve{q}_S \\geq 0$ for all $S \\subseteq [n]$, then for each $A \\subseteq [n]$, we have\n$$ \\Pr[{\\textstyle \\bigcap}_{j \\in A} \\overline{E_j}] \\geq \\breve{q}_A.$$\n\\end{lemma}\n\nWe present an inductive proof of this lemma. First, we state the following recursive identity for $\\breve{q}_A$.\n\n\\begin{claim}[The ``fundamental identity'' for $\\breve{q}$.]\n\\ClaimName{fundamental-q}\nFor any $a \\in A$, we have\n$$\\breve{q}_A ~=~ \\breve{q}_{A \\setminus \\set{a}} \\,-\\, p_a \\cdot \\breve{q}_{A \\setminus \\Gamma^+(a)}.$$\n\\end{claim}\n\n\\begin{proof}\nEvery independent set $I \\subseteq A$ either contains $a$ or does not. In addition, if $a \\in I$\nthen $I$ is independent iff $I \\setminus \\{a\\}$ is an independent subset of $A \\setminus \\Gamma^+(a)$. Thus the terms in $\\breve{q}_A$ correspond one-to-one to terms on the right-hand side.\n\\end{proof}\n\nNext. define $\\breve{P}_A = \\Pr[\\bigcap_{i \\in A} \\overline{E_i}]$.\nThe following claim analogous to \\Claim{fundamental-q} is the key inequality in the original proof\nof the LLL \\nolinebreak\\latexcite{ErdosLovasz,Spencer75,AlonSpencer} although typical expositions do not\ncall attention to it.\n\n\\begin{claim}[The ``fundamental inequality\" for $\\breve{P}$]\n\\ClaimName{fundamentalP}\nAssume that \\eqref{eq:Dep} holds.\nThen for each $a \\in A$,\n$$\\breve{P}_A \\geq \\breve{P}_{A - a} - p_a \\breve{P}_{A \\setminus \\Gamma^+(a)}.$$\n\\end{claim}\n\n\\begin{proof}\nThe claim is derived as follows.\n$$\n\\breve{P}_A \n ~=~ \\breve{P}_{A-a} - \\Pr\\Bigg[ E_a \\cap \\bigcap_{i \\in A-a} \\overline{E_i} \\Bigg]\n ~\\geq~ \\breve{P}_{A-a} - \\Pr\\Bigg[ E_a \\cap \\bigcap_{i \\in A \\setminus \\Gamma^+(a)} \\overline{E_i} \\Bigg]\n ~\\geq~ \\breve{P}_{A-a} - p_a \\breve{P}_{A \\setminus \\Gamma^+(a)}\n$$\nThe first inequality is trivial (by monotonicity of measure with respect to taking subsets) and the\nsecond inequality is our assumption with $J = A \\setminus \\Gamma^+(a)$.\n\\end{proof}\n\nGiven these two claims, Shearer's Lemma follows by induction.  \\vspace{3pt}\n\n\\begin{proofof}{\\Lemma{extShearer}}\nWe claim by induction on $|A|$ that for all $a \\in A$, \n\\begin{equation}\n\\EquationName{ShearerInduction}\n\\frac{\\breve{P}_{A}}{\\breve{P}_{A-a}} \\geq \\frac{\\breve{q}_{A}}{\\breve{q}_{A-a}}.\n\\end{equation}\nThe base case, $A = \\{a\\}$, holds because $\\breve{P}_{\\{a\\}} = \\breve{q}_{\\{a\\}} = 1 - p_a$\nand $\\breve{P}_\\emptyset = \\breve{q}_\\emptyset = 1$.\n\nFor $|A|>1$, the inductive hypothesis applied successively to the elements of $A \\cap \\Gamma(a)$ yields\n$$ \\frac{\\breve{P}_{A-a}}{\\breve{P}_{A \\setminus \\Gamma^+(a)}} \\geq\n\\frac{\\breve{q}_{A-a}}{\\breve{q}_{A \\setminus \\Gamma^+(a)}}.$$\nUsing this inequality, \\Claim{fundamental-q} and \\Claim{fundamentalP}, we obtain\n$$ \\frac{\\breve{P}_{A}}{\\breve{P}_{A-a}} \\geq 1 - p_a \\frac{\\breve{P}_{A \\setminus\n\\Gamma^+(a)}}{\\breve{P}_{A-a}} \\geq 1 - p_a \\frac{\\breve{q}_{A \\setminus \\Gamma^+(a)}}{\\breve{q}_{A-a}} \n= \\frac{\\breve{q}_A}{\\breve{q}_{A-a}}.$$\nThis proves the inductive claim.\n\nCombining \\eqref{eq:ShearerInduction} with the fact $\\breve{P}_\\emptyset = \\breve{q}_\\emptyset = 1$\nshows that $\\breve{P}_A \\geq \\breve{q}_A$ for all $A$.\n\\end{proofof}\n\n\\paragraph{Comparison to Shearer's original lemma.}\nShearer's lemma was originally stated as follows \\nolinebreak\\latexcite{Shearer}.\n\n\\begin{lemma}\n\\LemmaName{origShearer}\nSuppose that\n\\begin{equation}\n\\EquationName{Dep2}\n\\Pr[E_i \\mid {\\textstyle \\bigcap}_{j \\in J} \\overline{E_j}] ~=~ \\Pr[E_i]\n\\qquad\\forall i \\in [n] ,\\, J \\subseteq [n] \\setminus \\Gamma^+(i).\n\\end{equation}\nLet $p_i = \\Pr[E_i]$ and for each $S \\subseteq [n]$, define\n$$q_S = \\sumstack{I \\in \\operatorname{Ind}(G) \\\\ S \\subseteq I} (-1)^{|I \\setminus S|} \\prod_{i \\in I} p_i.$$\nIf $q_S \\geq 0$ for all $S \\subseteq [n]$, then\n$$ \\Pr[{\\textstyle \\bigcap}_{i=1}^{n} \\overline{E_i}] \\geq q_\\emptyset.$$\n\\end{lemma}\n\nThere are two differences between \\Lemma{extShearer} and \\Lemma{origShearer}.\nRegarding condition (i), \\Lemma{extShearer} uses \\eqref{eq:Dep}\nwhereas \\Lemma{origShearer} uses \\eqref{eq:Dep2};\nas discussed above, the former condition is more general.\nThe other main difference is the use of coefficients $q_S$ in \\Lemma{origShearer} as opposed to $\\breve{q}_S$ in \\Lemma{extShearer}. \nThe condition $q_S \\geq 0 \\:\\forall S$ turns out to be equivalent to $\\breve{q}_S \\geq 0 \\:\\forall S$,\nso \\Lemma{extShearer} and \\Lemma{origShearer} have equivalent formulations of condition (ii) (see \\nolinebreak\\latexcite{HV-arxiv} for more details).\nWe chose to state \\Lemma{extShearer} using $\\breve{q}_S$ because those are the coefficients that\nnaturally arise in the proof.\n\nShearer gives an interpretation of these coefficients as follows:\nthere is a unique probability space called the ``tight instance\" that minimizes the\nprobability of $\\Pr[ {\\textstyle \\bigcap}_i \\overline{E_i} ]$.\nIn that probability space, $q_S$ is exactly the probability that the events $\\setst{ E_i }{ i \\in S }$\noccur and the events $\\setst{ E_j }{ j \\notin S }$ do not occur.\nIn contrast, the coefficient $\\breve{q}_S$ is the probability that the events $\\setst{ E_i }{ i \\in S }$\ndo not occur. \nIn general, the coefficients are related by the identity $\\breve{q}_S = \\sum_{T \\subseteq [n] \\setminus\nS} q_T$, which can be proved by inclusion-exclusion.\nThe conclusion of \\Lemma{extShearer} is that $\\Pr[\\bigcap_{i=1}^{n} \\overline{E_i}] \\geq \\breve{q}_{[n]}$ and it is easy to see that $\\breve{q}_{[n]} = q_\\emptyset$. \nHence we recover \\Lemma{origShearer} from \\Lemma{extShearer}.\nThe tight instance also shows that the conclusion of Shearer's lemma is tight. \n\n\n\n\n\n\n\n\n\\section{Cluster Expansion}\n\\label{sec:cluster}\n\nNext we turn to a variant of the LLL that is stronger than the early formulations\n\\nolinebreak\\latexcite{ErdosLovasz,Spencer75,Spencer77} but weaker than Shearer's Lemma.\nThis lemma has been referred to as the {\\em cluster expansion} variant of the LLL; it was proved by Bissacot et al.~\\nolinebreak\\latexcite{Bissacot} using analytic techniques inspired by statistical physics. Although it is subsumed by Shearer's Lemma, it is typically easier to use in applications and\nprovides stronger quantitative results than the original LLL.\n\nFor variables $y_1,\\ldots,y_n$, we define\n$$\nY_S ~=~ \\sumstack{I \\in \\operatorname{Ind} \\\\ I \\subseteq S} y^I,\n$$\nwhere $y^I$ denotes $\\prod_{i \\in I} y_i$.\nThis is similar to the quantity $\\breve{q}_S$, but without the alternating sign.\n\n\\begin{lemma}[the cluster expansion lemma]\n\\LemmaName{cluster}\nSuppose that\n\\eqref{eq:Dep} holds \nand there exist $y_1,\\ldots,y_n>0$ such that for each $i \\in [n]$,\n\\begin{equation}\n\\EquationName{CLL}\np_i ~\\leq~ \\frac{y_i}{Y_{\\Gamma^+(i)}}.\n\\end{equation}\nThen\n$$ \\Pr[{\\textstyle \\bigcap}_{i=1}^{n} \\overline{E_i}] ~\\geq~ \\frac{1}{Y_{[n]}} ~>~ 0.$$\n\\end{lemma}\n\nHere we present an inductive combinatorial proof of \\Lemma{cluster}. \nFirst, some preliminary facts.\n\n\\begin{claim}[The ``Fundamental Identity'' for $Y$]\n\\ClaimName{fundamentalY}\nFor any $a \\in A$, we have\n$$ Y_{A} = Y_{A - a} + y_a Y_{A \\setminus \\Gamma^+(a)}.$$\n\\end{claim}\n\n\\begin{proof}\nThis follows from \\Claim{fundamental-q} since we can write $Y_S = \\breve{q}_S(-y)$.\nOr directly, every summand $y^J$ on the left-hand side either appears in $Y_{A - a}$\nif $a \\not\\in J$, or as a summand in $y_a Y_{A \\setminus \\Gamma^+(a)}$ if $a \\in J$.\n\\end{proof}\n\n\\begin{claim}[Log-subadditivity of $Y$]\n\\ClaimName{submult}\nIf $A, B$ are disjoint then $Y_{A \\union B} \\leq Y_A \\cdot Y_B$.\n\\end{claim}\n\\begin{proof}\nEvery summand $y^J$ of $Y_{A \\cup B}$ appears in the expansion of the product\n$$\nY_A \\cdot Y_B = \n \\sumstack{J' \\subseteq A \\\\ J' \\in \\operatorname{Ind}} \n \\sumstack{J'' \\subseteq B \\\\ J'' \\in \\operatorname{Ind}} y^{J'} y^{J''}\n$$\nby taking $J' = J \\intersect A$ and $J'' = J \\intersect B$.\nAll other terms on the right-hand side are non-negative.\n\\end{proof}\n\nThe following is the key inductive inequality, analogous to \\Equation{ShearerInduction}\nin the proof of Shearer's Lemma. Note that here the induction runs\nin the opposite direction for the $Y$ coefficients, which are indexed by complementary sets;\nthe reason for this lies in the fundamental identity for $Y$ (\\Claim{fundamentalY}) which\nhas the opposite sign compared to \\Claim{fundamental-q}.\nFor a set $S \\subseteq [n]$,  we will use the notation $S^c = [n] \\setminus S$.\n\n\\begin{lemma}\n\\LemmaName{cluster-induction}\nSuppose that $p$ satisfies \\eqref{eq:CLL}.\nThen for every $a \\in S \\subseteq [n]$, $\\breve{P}_S > 0$ and\n$$\n\\frac{\\breve{P}_{S}}{\\breve{P}_{S-a}} ~\\geq~ \\frac{Y_{S^c}}{Y_{(S-a)^c}}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nFirst, note that \\eqref{eq:CLL} implies that $p_i < 1$ for all $i$.\nWe proceed by induction on $|S|$. The base case is $S = \\{a\\}$. In that case we have\n$ \\frac{\\breve{P}_{\\{a\\}}}{\\breve{P}_\\emptyset} = \\Pr[\\overline{E_a}] \\geq 1 - p_a > 0$.\nOn the other hand, by the two claims above and \\eqref{eq:CLL}, we have \n$$ Y_{[n]} ~=~ Y_{[n]-a} + y_a Y_{[n] \\setminus \\Gamma^+(a)}\n ~\\geq~ Y_{[n] - a} + p_a Y_{\\Gamma^+(a)} Y_{[n] \\setminus \\Gamma^+(a)}\n ~\\geq~ Y_{[n] - a} + p_a Y_{[n]}.$$\nTherefore, $\\frac{Y_{[n]-a}}{Y_{[n]}} \\leq 1 - p_a$ which proves the base case.\n\nWe prove the inductive step by similar manipulations.  Let $a \\in S$.\nWe can assume that $\\breve{P}_{S-a} > 0$ by the inductive hypothesis.\nBy \\Claim{fundamentalP}, we have\n$$ \\frac{\\breve{P}_S}{\\breve{P}_{S-a}} ~\\ge~ \n1 - p_a \\frac{\\breve{P}_{S \\setminus \\Gamma^+(a)}}{\\breve{P}_{S-a}}.$$\nThe inductive hypothesis applied repeatedly to the elements of $S \\cap \\Gamma(a)$ yields\n\\begin{equation*}\n1 - p_a \\frac{\\breve{P}_{S \\setminus \\Gamma^+(a)}}{\\breve{P}_{S-a}}\n ~\\geq~ 1 - p_a \\frac{Y_{(S \\setminus \\Gamma^+(a))^c}}{Y_{(S-a)^c}}\n ~=~ 1 - p_a \\frac{Y_{S^c \\cup \\Gamma^+(a)}}{Y_{S^c + a}}. \n\\end{equation*}\nBy the two claims above and \\eqref{eq:CLL}, we have\n$$ Y_{S^c+a} ~=~ Y_{S^c} + y_a Y_{S^c \\setminus \\Gamma^+(a)}\n ~\\geq~ Y_{S^c} + p_a Y_{\\Gamma^+(a)} Y_{S^c \\setminus \\Gamma^+(a)}\n ~\\geq~ Y_{S^c} + p_a Y_{S^c \\cup \\Gamma^+(a)}.$$\nWe conclude that\n$$ \\frac{\\breve{P}_{S}}{\\breve{P}_{S-a}}\n ~\\geq~ 1 - p_a \\frac{Y_{S^c \\cup \\Gamma^+(a)}}{Y_{S^c+a}}\n ~\\geq~ 1 - \\frac{Y_{S^c+a} - Y_{S^c}}{Y_{S^c+a}} = \\frac{Y_{S^c}}{Y_{(S-a)^c}} $$\nwhich also implies $\\breve{P}_S > 0$.\n\\end{proof}\n\nNow we can complete the proof of \\Lemma{cluster}.\n\n\\begin{proofof}{\\Lemma{cluster}}\nBy \\Lemma{cluster-induction}, we have $\\frac{\\breve{P}_{S}}{\\breve{P}_{S-a}} ~\\geq~ \\frac{Y_{S^c}}{Y_{(S-a)^c}}$ for all $a \\in S$.\nHence,\n$$ \\Pr[\\bigcap_{i=1}^{n} \\overline{E_i}] ~=~ \\breve{P}_{[n]}\n ~=~ \\prod_{i=1}^{n} \\frac{\\breve{P}_{[i]}}{\\breve{P}_{[i-1]}}\n ~\\geq~ \\prod_{i=1}^n \\frac{Y_{[i]^c}}{Y_{[i-1]^c}}\n ~=~ \\frac{1}{Y_{[n]}}.\n$$\n\\end{proofof}\n\nBy a similar proof, it can be proved that $\\frac{\\breve{q}_{S}}{\\breve{q}_{S-a}} ~\\geq~ \\frac{Y_{S^c}}{Y_{(S-a)^c}}$ for all $a \\in S$,\nwhich relates the cluster expansion lemma to Shearer's Lemma. We refer the reader to \\nolinebreak\\latexcite{HV-arxiv}.\n\n\n\\section{The Symmetric LLL}\n\nThe ``symmetric'' LLL does not use a different upper bound $p_i$\nfor each event $E_i$, and instead assigns $p_i = p$ for all $i$.\nThe question then becomes, given a dependency graph,\nwhat is the maximum $p$ such that the conclusion of the LLL holds?\nAs mentioned above, Erd\\H{o}s and Lov\\'{a}sz \\nolinebreak\\latexcite{ErdosLovasz} show that one may take $p=1\/4d$\nif the graph has maximum degree $d$.\nSpencer \\nolinebreak\\latexcite{Spencer77} showed the improved result $p = d^d\/(d+1)^{d+1} > \\frac{1}{e(d+1)}$,\nand Shearer \\nolinebreak\\latexcite{Shearer} improved that to the value\n$p = (d-1)^{d-1}\/d^d > \\frac{1}{ed}$, which is optimal as $n \\rightarrow \\infty$.\n\nWe now show that the cluster expansion lemma (\\Lemma{cluster}) gives a short proof\nof the $\\frac{1}{ed}$ bound, which is just slightly suboptimal.\nAlternative proofs of the $(d-1)^{d-1}\/d^d$ and $\\frac{1}{ed}$ bounds may be found in\nKnuth's exercises 323 and 325 \\nolinebreak\\latexcite{Knuth}.\n\n\\begin{lemma}[Near-optimal symmetric LLL]\nSuppose that $G$ has maximum degree $d \\geq 2$ and let\n$p = \\max_{i \\in [n]} \\, \\max_{J \\subseteq [n] \\setminus \\Gamma^+(i)} \\,\n\\Pr[E_i \\mid {\\textstyle \\bigcap}_{j \\in J} \\overline{E_j}]$.\nIf $$p \\leq \\frac{1}{ed}$$\nthen $$\\Pr[ {\\textstyle \\bigcap}_i \\overline{E_i} ] > 0.$$\n\\end{lemma}\n\\begin{proof}\nWe set $p_i=p$ and $y_i=y=\\frac{1}{d-1}$ for all $i$,\nthen apply \\Lemma{cluster}.\nTo do so, we must check that \\eqref{eq:CLL} is satisfied.\nNote that $Y_{\\Gamma^+(i)} = y + Y_{\\Gamma(i)} \\leq y + (1+y)^d$.\nThen\n$$\n\\frac{y_i}{Y_{\\Gamma^+(i)}}\n    \\geq \\frac{y}{y+(1+y)^d}\n   \n       = \\frac{1}{1+\\frac{d^d}{(d-1)^{d-1}}}.\n$$\nThe claim is that this is at least $\\frac{1}{ed}$.\nBy simple manipulations, this claim is equivalent to\n\\begin{equation}\n\\EquationName{SymmIneq}\ne ~\\geq~ \\frac{1}{d} + \\Big( \\frac{d}{d-1} \\Big)^{d-1},\n\\end{equation}\nwhich we prove by a short calculus argument.\nFirst we derive the bound\n\\begin{align*}\n\\ln \\Big( \\frac{d}{d-1} \\Big)^{d-1}\n\\,=\\, \\!- (d-1) \\ln \\Big( 1 - \\frac{1}{d} \\Big)\n\\,=\\, (d-1) \\sum_{k=1}^{\\infty} \\frac{1}{k d^k}\n\\,=\\, 1 - \\sum_{k=1}^{\\infty} \\Big(\\frac{1}{k}-\\frac{1}{k+1}\\Big) \\frac{1}{d^k}\n\\,<\\, 1 - \\frac{1}{2d}.\n\\end{align*}\nFrom here, we obtain\n$$\n\\Big( \\frac{d}{d-1} \\Big)^{d-1}\n~<~ \\exp\\Big( 1 - \\frac{1}{2d} \\Big)\n~<~ e \\cdot \\Big( 1 - \\frac{1}{2d} \\Big)\n~<~ e - \\frac{1}{d}\n$$\nwhich establishes \\eqref{eq:SymmIneq}.\n\\end{proof}\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDecentralization is back in the spotlight. While peer-to-peer (P2P) systems were popular in the 2000s, they subsequently lost their appeal to centralized social networks and streaming services. The tide has recently turned for two reasons. The first is the fear over excessive centralization of user data, sparked by pivotal incidents, such as the 2013 leaking of the PRISM surveillance program by Edward Snowden \\cite{greenwald2013nsa_prism} and \nthe 2016 Cambridge Analytica scandal \\cite{cadwalladr2017great_cambridgeanalytica}. The second is the increasing mainstream appeal of cryptocurrencies and blockchain-based applications \\cite{chen2018survey_blockchainapps}.\n\nAs a result, influential web technologists have called for a decentralized web with actions like the Decentralized Web Summits by the Internet Archives\\footnote{\\url{https:\/\/www.decentralizedweb.net\/}} and the Solid project by Tim Berners-Lee\n\\cite{mansour2016demonstration_solid}. From their part, policy-makers have answered this call, with the European Commission supporting decentralized technologies in its flagship Next Generation Internet initiative\\footnote{\\url{https:\/\/digital-strategy.ec.europa.eu\/en\/policies\/next-generation-internet-initiative}}. \n\nCurrently, the most popular decentralized technologies are distributed hash tables (DHTs) and blockchain. Of these, DHTs come from the previous wave of P2P research and enable document retrieval via unique textual identifiers, with strong guarantees on retrieval delay. For example, the Kademlia DHT powers the Interplanetary File System (IPFS), a decentralized file storage solution that aspires to become a pillar of the decentralized Web\\footnote{\\url{https:\/\/ipfs.io\/}}.\nOn the other hand, blockchain allows decentralized nodes to maintain common states via consensus mechanisms like proof-of-work and broadcasting. Instead of data, blockchain is typically used to broadcast monetary transactions and reward nodes for executing decentralized operations. For example, Filecoin builds on IPFS and uses blockchain to reward nodes for offering file storage. \n\nBut how can one \\textit{find} documents in decentralized systems? DHTs require previous knowledge of document identifiers, which must be acquired externally. Alternatively, they can implement distributed inverted indexes by storing relevant document identifiers for search keywords \\cite{reynolds2003efficient_dhtsearchengine}, as the  YaCy search engine does\\footnote{\\url{https:\/\/yacy.net\/}}.\n\nHowever, this practice carries fundamental bandwidth and storage constraints\\cite{li2003feasibility_dhtwebindexing} and exact keyword matching is dated compared to the semantic awareness of modern search engines. On the other hand, unstructured search techniques, such as flooding, random walks, index sharing, and query caching\\cite{khatibi2021resource_unstructuredsurvey} often suffer from high communication overhead and unpredictable delays. Finally, blockchain has been used to reward nodes for executing indexing and retrieval operations in decentralized search engines, such as Presearch\\footnote{\\url{https:\/\/presearch.org\/}}, but broadcasting indexes to all nodes is prohibitive in terms of bandwidth and storage.\n\nWhile research on decentralized search has stagnated on the above bottlenecks, centralized search engines have evolved to better understand query semantics. This evolution has been driven by advancements in \\textit{embeddings}, latent representations of text and other types of content \\cite{lin2021pretrained_transformersforretrieval}. Retrieval with embeddings often follows a vector space model, which extracts vector representations for documents and queries and compares their relevance with a simple similarity metric, such as the dot product or cosine similarity. This way, the retrieval can be cast as a \\textit{nearest-neighbor} problem, which tries to find the nearest documents to a query according to the selected similarity metric. In contrast to term-frequency vectors, embeddings are lower-dimensional and enable semantic rather than exact term matching, giving rise to \\textit{dense retrieval}.\n\nHere, we argue that decentralized search can benefit from modern techniques employed by centralized search engines. To this end, we revisit the decentralized search problem from an embedding-based standpoint. We further employ a graph signal processing technique to implement similarity-based P2P query routing. We propose composing node embeddings from local node documents and diffusing them through P2P networks with decentralized implementations of graph filters, such as Personalized PageRank (PPR). We then use the diffused embeddings to guide decentralized search towards nodes with relevant documents. We experiment with a simulation of a real-world P2P network and investigate how our solution scales with the number of documents in the network. Our approach successfully locates relevant documents in nearby nodes but accuracy sharply declines as the number of documents increases, highlighting the need for further research.\n\n\\section{Background and Related Work}\nThis section explores related work on decentralized search (Subsection~\\ref{sec:decentralized search}) and then presents dense information retrieval (Subsection~\\ref{sec:dense retrieval}) and graph signal processing (Subsection~\\ref{sec:gsp}) background to contextualize later analysis.\n\n\\label{sec:background}\n\\subsection{Decentralized search} \\label{sec:decentralized search}\nDecentralized search received attention in the early 2000s for P2P file sharing systems, such as Gnutella and Freenet \\cite{aberer2002overview_overviewgnutellaetal}. Gnutella introduced \\textit{flooding}, the simplest technique for search, which forwards search queries to all nodes within a specified number of hops. As P2P platforms grew in size, flooding was soon found to not scale in terms of bandwidth consumption \\cite{ritter2001gnutella_gnutellascalability}, giving rise to alternatives, such as random walks, index sharing, and super-peer architectures \\cite{lua2005survey_earlysurveyonp2p}. Of these, \\textit{informed} methods exploit hints about possible document locations and outperform \\textit{blind} methods, like flooding and random walks in terms of delay and communication cost. This comes at the expense of costly state maintenance at nodes \\cite{tsoumakos2006analysis_blindinformeddistinction}. \n\\par\nInformed search methods rely on query routing and can be further categorized into \\textit{document-} and \\textit{query-oriented} ones \\cite{arour2015learning_querycontentoriented}. In document-oriented methods, P2P nodes exchange information about their stored documents \\cite{crespo2002routing_firstdocumentrouting,  kumar2005efficient_bloomfilters}. As the storage cost increases with the number of documents in the network, the advertisement radius is limited and summarization is employed to compress the advertisements, for instance with Bloom filters \\cite{kumar2005efficient_bloomfilters}. Both techniques introduce routing errors. In query-oriented methods, nodes store information of passing queries and their results \\cite{kalogeraki2002local_firstqueryrouting, li2006improve_queryroutingwithrl} and, when a new query arrives, it is forwarded to the most successful route travelled by similar past queries. These methods are attractive because they avoid storing information about unpopular documents. On the other hand, they are blind to unseen queries, especially at the beginning of the network's operation when no information is available (cold-start problem).\n\nWhile informed search identifies the locations of relevant documents through routing, DHTs decouple these two operations with a clever application of hashing \\cite{lua2005survey_earlysurveyonp2p}. In particular, DHT nodes agree to store documents whose hash values are the closest to their own address, according to a distance function. As a result, when nodes search for a document, they can resolve its location and reach it through routing. For efficiency, most DHT systems, such as Chord, Pastry, and Kademlia, structure P2P networks so that all locations are reachable within a maximum number of hops \\cite{lua2005survey_earlysurveyonp2p}, although this structuring is not strictly required\\footnote{Efficient addressing can be enforced on networks with arbitrary structure, for example with greedy embeddings \\cite{hofer2013greedy_greedyembeddings}.}.\n\nThe theoretical properties and practicality of DHTs have made them attractive for modern decentralized systems, such as IPFS, but they are best suited for key-based retrieval. For other types of search, such as range and nearest neighbor queries, adaptations or other distributed data structures are needed, such as skip-lists and skip-graphs  \\cite{reynolds2003efficient_dhtsearchengine, bongers2015survey_multidimensionalrangequeries, gao2007efficient_dhtssimilaritysearch}. These solutions carry their own limitations, including security concerns and poor load balancing of traffic.\n\n\\subsection{Dense retrieval}\\label{sec:dense retrieval}\nInformation retrieval is often based on vector space models that represent documents and queries as vectors and estimate document relevance to queries via a similarity metric. Text vector representations are traditionally derived from bag-of-words models based on word frequencies, predominantly the TF-IDF and BM25 models\\cite{manning2010introduction}. Those yield high-dimensional sparse vectors that can be efficiently stored in inverted index tables but do not capture the underlying semantics, such as implied contexts, synonyms, or word co-usage patterns. To address this issue, research has moved towards lower-dimensional dense representations, which encode latent semantics and enable soft matches. Dense retrieval has recently demonstrated definite improvement over sparse retrieval (represented by the BM25 model) \\cite{lin2019neural_neuralhype}, owing to the successful transfer of deep learning advances \\cite{lin2021neural_neuralhyperecant}. \n\nKey steps in this process have been the development of efficient vector representations for words with the Word2Vec and Glove frameworks \\cite{pennington2014glove_glove}, which were later extended to sentences. While sentence embeddings are less understood, they were shown to capture linguistic information \\cite{conneau2018you_sentenceembeddings} and are useful to retrieval \\cite{yang2019simple_sbertforadhoc}. \n\nCurrently, the state of the art for dense retrieval focuses on pre-trained transformer models, commonly based on BERT \\cite{devlin2018bert}, \nwhich are subsequently fine-tuned on downstream retrieval tasks \\cite{lin2021pretrained_transformersforretrieval}. There are two extreme approaches in using BERT for retrieval, \\textit{cross-encoders} and \\textit{bi-encoders}. Cross-encoders consider all interactions among query and document words, which yields the best accuracy but with high processing and energy costs. For instance, cross-encoders need to process all documents and queries at query-time, which incurs unreasonable delays. In contrast, bi-encoders conform to the vector space model in that documents and queries are transformed separately to vectors and interact via simple operations, such as the dot product or cosine similarity. While bi-encoders are less accurate than cross-encoders, they outperform BM25, enable proactive document indexing, and their inference is quick and cheap with approximate nearest-neighbor algorithms \\cite{aumuller2020ann}. Therefore, the vector space model and nearest-neighbor algorithms remain relevant for modern search applications.\n\n\\subsection{Graph signal processing}\\label{sec:gsp}\nGraph signal processing is a recently popularized field that generalizes traditional signal processing principles to graphs \\cite{ortega2018graph_gspsurvey,huang2018graph}. With this approach, graph signals are defined as collections of node values, e.g., scalars, vectors, and graph filters study their propagation through graphs. In particular, a graph convolution operation is defined, which performs one-hop propagation of node values through matrix multiplication, and graph filters are defined by weighted aggregation of multihop propagations. Popular graph filters, such as PPR and heat kernels perform the equivalent of low-pass filtering by placing higher importance to node values that are propagated fewer hops away.\n\\par\nWhen node values are vectors, graph filters operate independently on each vector dimension. This type of propagation is useful by itself for downstream predictive tasks, such as prediction propagation in graph neural networks \\cite{klicpera2018predict,dong2021equivalence}. In this work, we consider low-pass graph filters as a type of smoothing that concentrates around a small area around nodes. This area can be tuned by a single parameter of the PPR filter.\n\n\\section{Problem Setting}\n\\label{sec:problem_seting}\nThis section first presents dense retrieval operations, as they would be applied by modern centralized search engines (Subsection \\ref{subsec:centralized_setting}), and then re-formulates them in a decentralized setup (Subsection \\ref{subsec:decentralized_setting}).\n\n\\subsection{Centralized Setting}\n\\label{subsec:centralized_setting}\n\nIn the centralized setting, we consider search engines that are responsible for answering queries over collections of stored documents $\\mathcal{D}$. When engines receive queries $q$, they compute relevance scores $s(d,q)$ for all documents $d \\in \\mathcal{D}$. They then estimate the top-$k$ most relevant documents per\n\\begin{equation}\n    \\underset{d \\in \\mathcal{D}}{\\text{arg top-}k} ~ s(d, q).\n\\end{equation}\n\nIn this paper, we consider the bi-encoder model of dense retrieval, which splits the score computation in two parts: i) an \\textit{encoding} part that transforms queries $q$ and documents $d$ to $\\nu$-dimensional embedding vectors $\\mathbf{e}_q, \\mathbf{e}_d$ respectively ($\\mathbf{e}_q, \\mathbf{e}_d \\in \\mathbb{R}^\\nu$), and ii) a \\textit{comparison} part that derives the score $s$ from the embeddings. This is formalized as \n\\begin{equation}\n\\label{candidate_document}\n    s = \\phi(\\mathbf{e}_q, \\mathbf{e}_d) = \\phi\\left(\\eta_q (q), \\eta_d (d)\\right)\n\\end{equation}\nwhere $\\eta_q, \\eta_d$ are encoding functions for queries and documents respectively, and $\\phi$ is a comparison mechanism \\cite{deepretrievalframework}. The above formulation is attractive because it contains the computational complexity to the encoding function $\\eta$, which can be pre-computed during indexing. In contrast, the comparison function $\\phi$ is executed at query time and is therefore chosen to be computationally lightweight; usually, the dot product or cosine similarity is chosen\\footnote{These are equivalent when the embeddings are L2-normalized.}. These choices cast the retrieval as a $k$ nearest-neighbor problem, which can be computed efficiently with popular approximation algorithms, e.g., based on locality sensitive hashing or hierarchical navigable small world graphs \\cite{aumuller2020ann}. \n\n\\subsection{Decentralized Setting}\n\\label{subsec:decentralized_setting}\n\nTo move to the decentralized setting, we consider a P2P network whose nodes maintain their own private document collections. The network is modeled as an \\textit{undirected} graph $\\mathcal{G}=(\\mathcal{V}, \\mathcal{E})$, where $\\mathcal{V}$ is the set of nodes and $\\mathcal{E}\\subseteq \\mathcal{V}\\times \\mathcal{V}$ their communication edges, while  $\\mathcal{D}_u\\subseteq\\mathcal{D}$ represents the local documents of node $u$.\n\\par\nWhen nodes initiate queries, they first execute the retrieval operations of subsection \\ref{subsec:centralized_setting} over their local document collections, and then forward queries to their one-hop neighbors to retrieve more results. Farther nodes can be contacted by relaying the queries along nodes. Since contacting all nodes would induce non-scalable communication costs and delays, we allow the search to fail to find relevant documents, even if these could have been retrieved by centralized search engines. The goal of our analysis is to make clever forwarding decisions to achieve high search hit accuracy of relevant documents.\n\n\\section{Diffusion-based decentralized search}\n\\label{sec:proposal}\nOur decentralized scheme for search is a document-oriented solution where nodes maintain a summary of documents available from their neighbors. These summaries take the form of \\textit{node embedding} vectors, denoted by $\\mathbf{e}_u$, which are composed from the embeddings of both local and nearby documents. To generate the node embeddings, when new nodes enter the network or update their document collections, they compute \\textit{personalization vectors}, denoted by $\\mathbf{e}_u^{(0)}$, which characterize their local document collections (Subsection \\ref{subsec:personalization}). Subsequently, the nodes diffuse their personalization vectors to the network with an iterative and asynchronous diffusion algorithm based on PPR (Subsection \\ref{subsec:diffusion}). This algorithm converges to the node embedding vectors and also keeps track the embeddings of the one-hop neighbors for each node. At query-time, the nodes can use their stored neighbor embeddings to forward queries towards promising next hops (Subsection \\ref{subsec:forwarding}).\n\n\\subsection{Node personalization}\n\\label{subsec:personalization}\n Ideally, for each node $u$, we would like to estimate the maximum score of all neighbors $v$, as in \\eqref{candidate_document}, without knowing their documents $\\mathcal{D}_v$. A simple way is to represent each node with the personalization vector $\\mathbf{e}_u^{(0)}$ that is the sum of the node's document embeddings. This has the attractive property that,  due to the linearity of the interaction function, the dot product of the query with the neighbor embedding yields the total relevance of the neighbor's documents:\n\\begin{equation}\n    \\mathbf{e}_q \\cdot \\mathbf{e}_v^{(0)} = \\mathbf{e}_q \\cdot \\sum_{d \\in \\mathcal{D}_v} \\mathbf{e}_d= \\sum_{d \\in \\mathcal{D}_v} \\mathbf{e}_q \\cdot \\mathbf{e}_d.\n\\end{equation}\nThis approach tends to score higher nodes with a larger number of documents. This is desirable in general although it runs the risk of prioritizing nodes with many irrelevant documents over nodes with a few but relevant documents. \n\n\\subsection{Diffusion of embeddings}\n\\label{subsec:diffusion}\nAfter computing their personalization vectors, the nodes transmit them to their neighbors. Instead of traditional $n$-hop advertising, we consider a diffusion scheme based on graph signal processing. A typical diffusion has the form:\n\\begin{equation}\n\\label{diffusion}\n\\mathbf{E} = \\mathbf{H} \\mathbf{E}^{(0)} ~ \\Rightarrow ~\n    \\mathbf{e}_u = \\sum_{v \\in V} h_{u v} \\mathbf{e}_v^{(0)}\n\\end{equation}\nwhere $\\mathbf{E}^{(0)}$, $\\mathbf{E}$ are the initial and diffused embeddings in matrix form, $\\mathbf{H}$ is the weight matrix or impulse response of diffusion, whose elements $h_{u v}$ represent the impact of node $v$ to $u$. While the diffusion weights $\\mathbf{H}$ could be learned with a machine learning algorithm, the complexity of learning would scale with $\\mathcal{O}(N^2)$, which would be intractable for large graphs. Therefore, we have chosen the PPR algorithm for calculating the weights, which is a popular approach in the literature \\cite{klicpera2018predict}, and can be implemented in a decentralized and asynchronous way \\cite{krasanakis2021p2pgnn_asynchronousppr}, which is a highly desirable feature. \n\nIn PPR, we associate $h_{u v}$ with the probability to reach $v$ via a random walk that starts from $u$. If the random walk were allowed to progress, as in the traditional PageRank, it would forget its origin $u$ and converge to a probability characterizing only $v$. To avoid this, in PPR, we force the walker to teleport back to node $u$ with probability $a$. Thus,  $h_{u v}$ is associated with the probability to reach node $v$ from $u$ with a short walk of average length $1\/a$.\n\nFormally, denoting by $\\boldsymbol{\\pi} [v]$ the probability of arriving at node $v$, and by $\\boldsymbol{\\delta}_u[v]$ the one-hot vector at node $u$, i.e., $\\boldsymbol{\\delta}_u[u]=1$ and $\\boldsymbol{\\delta}_u[v]=0$ for $v \\neq u$, we have\n\\begin{equation}\n\\label{ppr_recursive}\n    \\boldsymbol{\\pi} [v] = (1-a) \\mathbf{A} \\boldsymbol{\\pi} [v] + a \\boldsymbol{\\delta}_u[v] \\Rightarrow \\boldsymbol{\\pi} [v] = a (\\mathbf{I}-(1-a)\\mathbf{A})^{-1} \\boldsymbol{\\delta}_u[v]\n\\end{equation}\nwhere $\\mathbf{I}$ is the identity matrix and $\\mathbf{A}$ the transition matrix of the Markov chain, based on a suitable normalization of the adjacency matrix of $\\mathcal{G}$ or external weights. Considering the definition of $\\boldsymbol{\\delta}_u[v]$, it is clear that the columns of $a (\\mathbf{I}-(1-a)\\mathbf{A})^{-1}$ correspond to the desired probabilities for different origins $u$. The diffused embeddings of \\eqref{diffusion} are thus given by:\n\\begin{equation}\n\\label{ppr_solution}\n\\mathbf{E} = a (\\mathbf{I}-(1-a)\\mathbf{A})^{-1} \\mathbf{E}^{(0)}\n\\end{equation}\nWhile the embeddings are propagated to the whole graph, the \\textit{effective} range of the diffusion is tuned by the parameter $a$. \n\nFor the decentralized and asynchronous implementation, we first express \\eqref{ppr_solution} iteratively as:\n\\begin{equation}\n\\label{ppr_iterative}\n    \\mathbf{E}^{(t)}  = (1-a) \\mathbf{A} \\mathbf{E}^{(t-1)} + a \\mathbf{E}^{(0)},\n\\end{equation}\nwhich converges to \\eqref{ppr_solution} but is synchronous. Subsequently, we make the iteration \\textit{asynchronous} by letting node pairs exchange and update embeddings. As proven in \\cite{krasanakis2021p2pgnn_asynchronousppr}, if the update intervals are not arbitrarily long, the embeddings converge to \\eqref{ppr_recursive} in distribution, which is a good approximation of the centralized scheme.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=2.5in]{node.pdf}\n\\caption{Node operations when a query is received.}\n\\label{img:node_operations}\n\\end{figure}\n\n\\subsection{Forwarding operations}\n\\label{subsec:forwarding}\nNode embeddings are used at query-time to guide search towards promising nodes, essentially performing a biased random walk. Queries keep track of the $k$ most relevant documents they have encountered along with their relevance score\\footnote{If documents are too large, the message can track the IP addresses of the source nodes or content identifiers if available, e.g., IPFS content IDs.}. Since visiting all nodes in the network is impractical, we impose a maximum number of hops with a time-to-live (TTL) field in the query message, which helps prevent queries from circulating in the network indefinitely. Due to the TTL limitation, we prioritize unvisited nodes for forwarding. To this end, the nodes keep track of the neighbors from which they have received and to which they have sent messages. We purposefully reject the alternative (and slightly more efficient) solution of recording the visited nodes in the query message in order to protect the privacy of node connections. In our solution, nodes relay the queries recursively, i.e., from node to node, and when their TTL expires, a response message is returned to the querying nodes via backtracking.\n\nFig. \\ref{img:node_operations} illustrates the node operations when a new query arrives. As described in subsection \\ref{subsec:decentralized_setting}, nodes first evaluate the query on their local documents according to the retrieval operations of subsection \\ref{subsec:centralized_setting}. Afterwards, they decrement the TTL field of the query message by 1 and check if the message is still alive. If the TTL has expired, the nodes discard the query and send a query response message to the reverse path, otherwise, they commence the forwarding procedure: nodes first determine a set of candidate next hops from their neighbors, which excludes previously visited nodes remembered by the nodes\\footnote{If no neighbors remain after this step, nodes consider all their neighbors as candidates as we do not want to waste opportunities for forwarding considering the TTL limitation.}. Nodes then match via dot product the embeddings of the candidate next hops with the query embedding, and select a few neighbors with the highest score. When a single neighbor is selected, the outcome is a simple random walk, otherwise, multiple walks are executed in parallel.\n\n\\section{Experimental Evaluation}\\label{sec:experiments}\n\\label{sec:setup}\nWe evaluate a retrieval operation in a social P2P network based on two datasets: a social network graph and a corpus of pre-trained embeddings (Subsection~\\ref{sec:datasets}). Through simulation (Subsection~\\ref{sec:simulation}), we investigate the scalability of our scheme with the number of stored documents in the network, $M$, in terms of the hit accuracy (Subsection~\\ref{sec:acc}) and the average number of hops of successful queries (Subsection~\\ref{sec:hop}).\n\n\\subsection{Datasets}\n\\label{sec:datasets}\nExperiments are conducted on the Facebook social circles graph\\cite{leskovec2012learning_fbdataset} hosted by the SNAP project\\footnote{\\url{http:\/\/snap.stanford.edu}}. This is an undirected graph of 4,039 Facebook users (nodes) and their 88,234 friend relations (edges). We consider this graph representative of P2P networks built on top of social relations, which are expected to resemble friend relations of centralized social networks.\n\nDocuments and queries are represented using 300-d word embeddings, trained by the Glove model on Wikipedia articles \\cite{pennington2014glove_glove} and distributed by the GenSim library\\footnote{\\url{https:\/\/radimrehurek.com\/gensim\/}}. While Glove embeddings are not ideal for retrieval, they are good predictors of similarity with the cosine similarity metric. As mentioned in Section \\ref{sec:setup}, the nearest-neighbor search mechanism is independent from the embedding method, which allows us to study search in isolation. In fact, queries and documents can refer to any type of content, even multimedia, provided relevance is a linear function of their embeddings.\n\n\\begin{figure}[!t]\n   \\centering\n\\fbox{\\parbox{0.8\\linewidth}{\\begin{algorithmic}[1]\n    \\STATE Generate documents and queries from Glove\n    \\STATE Distribute $N$ documents uniformly over $\\mathcal{G}$\n    \\STATE Compute node embeddings\n    \\REPEAT\n    \\STATE Diffuse node embeddings asynchronously\n    \\UNTIL embeddings converge\n    \\STATE Distribute queries\n    \\REPEAT\n    \\STATE Forward queries\n    \\UNTIL all queries expire\n\\end{algorithmic}}}\n\\caption{Pseudo-code for the simulation of the decentralized search setting.}\n\\label{fig:pseudocode}\n\\end{figure}\n\n\\subsection{Simulation setup}\n\\label{sec:simulation}\nFig. \\ref{fig:pseudocode} presents our simulation in pseudo-code. We first generate queries and documents from the Glove dataset using 1000 random words as queries and their nearest neighbors as gold documents, provided that their cosine similarity is over 0.6 and the two sets do not overlap. The remaining words are treated as a pool of irrelevant documents. We further distribute the documents over the graph's nodes uniformly and compute the node embeddings. This is followed by a warm up period, in which we diffuse the node embeddings over the network with the asynchronous PPR algorithm. The algorithm runs until the embeddings converge.\n\nWe then proceed with evaluating the top-$1$ document retrieval performance over sampled queries, whose number depends on the simulation scenario. In each iteration, queries are distributed over the network and are forwarded independently. For simplicity, each query performs a simple random walk, which is the most challenging case and can be easily extended to parallel walks. In the future, we plan to investigate parallel walks more thoroughly along with time-evolving conditions and the top-$k$ performance. More realistic document distributions are also worthwhile; in fact, they are expected to aid diffusion, since they naturally exhibit spatial correlation.\n\n\\subsection{Hit Accuracy}\n\\label{sec:acc}\nIn this series of experiments, we evaluate the accuracy of our algorithm over the number of stored documents in the network, $M$, and the teleport probability of PPR, $\\alpha$, which determines the average diffusion radius. For $M$, we select $10$, $100$, $1000$, and $10000$ documents to investigate 4 orders of magnitude. In each iteration, we store one gold and $M$-1 irrelevant documents in the network, and sample multiple querying nodes, one from each radius away from the location of the gold document. At the end of simulation, the accuracy is computed as the percentage of queries that retrieved the gold document within a TTL of 50 hops. The simulation is repeated for three different values of $\\alpha$, $0.1$, $0.5$, and $0.9$, as examples of heavy, moderate, and light diffusion respectively. The results are depicted in Fig. \\ref{fig:acc_analysis}.\n\nFigs. \\ref{fig:10docs} and \\ref{fig:100docs} show that our algorithm excels at finding documents within 2 hops away, provided that there are few documents in the network. In contrast, the accuracy starts to decline at 3 hops and deteriorates significantly farther away. Surprisingly, heavy diffusion does not aid accuracy, as more documents are discovered when the teleport probability is 0.9. The results change radically with more stored documents. In Figs. \\ref{fig:1000docs} and \\ref{fig:10000docs}, we see that the accuracy remains high mainly for documents in neighboring nodes and the impact of $\\alpha$ is more varied. In this case, heavier diffusion is better at small distances although $a=0.9$ appears beneficial at 3 and 4 hops when the stored documents are 1000. With 10000 documents, the performance deteriorates considerably.\n\nThe above show that the PPR diffusion is useful for local neighborhood search but its accuracy declines with the number of stored documents. This is attributed to the loss of information for individual documents when many embeddings are summed, either during summarization or diffusion. The behavior with $\\alpha$ can also be explained by the following trade-off: heavy diffusion (low $\\alpha$) announces documents within a wider range but adds more noise due to the summation of the embeddings. In contrast, light diffusion (high $\\alpha$) adds less noise but may fail to notify nearby nodes. Considering this trade-off, when few documents are stored in the network (Figs. \\ref{fig:10docs} and \\ref{fig:100docs}), it is preferable to leave fewer and cleaner hints as the random walk will eventually find the correct document. In contrast, with more documents (Figs. \\ref{fig:10docs} and \\ref{fig:100docs}), there is already noise in the network and light diffusion may hinder the random walk from finding documents even 1 hop away. \n\n\\subsection{Hop Count Analysis}\n\\label{sec:hop}\nIn this experiment, we compute the average hop count for successful queries until the gold document is found. As in Section \\ref{sec:acc}, the queries are considered successful when they retrieve the correct document within 50 hops. We note that, since the queries do not know when they find the gold document and must complete their TTL, the average hop count does not indicate bandwidth consumption but can guide the choice of TTL. For the setup, we execute 500 iterations in each of which we distribute 10 queries uniformly in the network, for a total of 5000 samples. We also choose the value 0.5 for the teleport probability $\\alpha$, scale the number of documents for 10 to 10000, and randomize the document distribution at each iteration, as in the accuracy experiment. Our results are summarized in Table \\ref{hop_analysis}.\n\nTable \\ref{hop_analysis} shows that less queries are successful when the stored documents increase, consistently with the accuracy results of Section \\ref{sec:acc}. Furthermore, with more documents, longer walks are required as both the median and the mean hops to reach the gold documents increase. The discrepancy between the median and the mean hops implies a skewed distribution, i.e., a few walks succeed after a large number of hops and drive the mean higher, which is corroborated by the high standard deviation. Combined with the results of the accuracy experiment, the above show that, even though documents are found predominantly by nearby nodes, some queries need to circulate for additional hops until they succeed. It is encouraging though that success is still possible with a high number of documents, such as 10000.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Average Hop Count}\n\\label{hop_analysis}\n\\centering\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n$M$ documents & success rate & median hops & mean hops & std hops \\\\\n\\hline\n10 & 1905 \/ 5000 & 3 & 7.62 & 10.83 \\\\\n100 & 1265 \/ 5000& 4 & 11.21 & 13.37 \\\\\n1000 & 1054 \/ 5000 & 9 & 15.26 & 14.55 \\\\\n10000 & 877 \/ 5000 & 9 & 14.31 & 13.36 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[!t]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.41\\linewidth]{fb_glove_10docs.pdf}\n\\label{fig:10docs}}\n\\subfloat[]{\\includegraphics[width=0.41\\linewidth]{fb_glove_100docs.pdf}\n\\label{fig:100docs}}\n\n\\subfloat[]{\\includegraphics[width=0.41\\linewidth]{fb_glove_1000docs.pdf}\n\\label{fig:1000docs}}\n\\subfloat[]{\\includegraphics[width=0.41\\linewidth]{fb_glove_10000docs.pdf}\n\\label{fig:10000docs}}\n\\caption{Accuracy analysis for a) 10, b) 100, c) 1000, and d) 10000 documents in the network.}\n\\label{fig:acc_analysis}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nAs decentralization is becoming an increasingly important feature of the future Internet, new algorithms are needed for effective decentralized search. In this paper, we revisit this long-standing problem from a combined embedding and graph diffusion perspective. Specifically, considering a P2P network with nodes of only local knowledge over their document collections, we apply the PPR algorithm to diffuse summarized information about the documents in the network. Our results show that this diffusion can be beneficial for local neighborhood search but further enhancements are needed to improve the performance for global search. Our current line of research is to exploit correlations in the document distribution and derive more sophisticated aggregation methods that encode more information about the grouped documents. \n\n\\section*{Acknowledgment}\nThis research was supported by the EU H2020 projects AI4Media (Grant Agreement 951911), MediaVerse (GA 957252) and HELIOS (GA 825585). \nThe authors want to thank Dr. Ioannis Sarafis for his productive feedback on the decentralized search scheme.\n\n\\balance\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Result}\n The enumeration of maps has a long history, in which the techniques and tools became more and more efficient and the classes of maps more and more sophisticated: In his \\textit{Census of Planar Maps}, William Tutte achieved groundbreaking progress in the 1960's \\cite{Tuttbij}.  Bender and Canfield then left the realm of planar maps in the 1980's and also took an embedding into higher-genus surfaces into consideration \\cite{Bender}. In the 2000's, the branch of mathematical physics established a powerful and efficient universal procedure to reach all topological sectors in a recursive way: Topological recursion (TR) of Chekhov, Eynard and Orantin \\cite{Eynard:2007kz,Chekhov:2006vd} built a bridge between enumerative and complex geometry (and, based on the work \\cite{Kontsevich:1992ti}, bridges to intersection theory and integrable hierachies, which we will neglect here) and thus covered numerous, seemingly disconnected areas of mathematical fields, by one universal recursion procedure.\n\nTopological recursion possesses the initial data $(\\Sigma,x,y,B)$, where $x:\\Sigma\\to \\Sigma_0$ is a ramified covering of Riemann surfaces, $\\omega_{0,1}=y\\, dx$ is a meromorphic differential 1-form on $\\Sigma$ regular at the ramification points and $\\omega_{0,2}=B$ a symmetric bilinear differential form on $\\Sigma\\times \\Sigma$ with double pole on the diagonal and no residue. From this initial data, TR computes recursively in the negative Euler characteristic $-\\chi=2g+n-2$ an infinite sequence of symmetric meromorphic $n$-forms $\\omega_{g,n}$ on $\\Sigma^n$ with poles only at the ramification points for $-\\chi>0$. The precise formula and more details are given in Ch. \\ref{ch:proof}. For specific choices of the initial data $(\\Sigma,x,y,B)$, the meromorphic $n$-forms are encoding some enumerative problems.\n\nThe prime example of this framework was the recursive computation of generating functions counting objects known in the literature as \\textit{ordinary maps}  (a very readable derivation can be found in \\cite{Eynard:2016yaa}):\n\\begin{theorem}[\\cite{Eynard:2016yaa}]\n\\label{th:eyn}\nThe spectral curve $(\\mathbb{P}^1,x_{ord},y_{ord}, \\frac{dz_1\\, dz_2}{(z_1-z_2)^2})$ with\n\\begin{align*}\n\tx_{ord}(z)=\\gamma \\bigg ( z+ \\frac{1}{z} \\biggl ) \\qquad y_{ord}(z)= \\sum_{k=0}^{d-1} u_{2k+1}z^{2k+1}\n\\end{align*}\nwhere\n\\begin{align*}\n\t\\gamma^2=1+\\sum_{k\\geq 1} t_{2k} \\binom{2k-1}{k} \\gamma^{2k}, \\qquad u_{2k+1}=\\gamma \\bigg(\\delta_{k,0}-\\sum_{j\\geq k+1}t_{2j}\\binom{2j-1}{j+k}\\gamma^{2j-2}\\bigg)\n\\end{align*}\ncomputes via TR (see formula \\eqref{BTR-intro}) generating functions for the enumeration of ordinary maps with $n$ marked faces of even boundary lengths. The faces have even degrees up to $2d$, where a face of degree $2k$ is weighted by $t_{2k}$.\n\\end{theorem}\\noindent\nThe theorem includes in general also faces of odd degree, but for later purposes, we want to state it in this form.\n\nSeveral more classes of maps, e.g. subsets of the ordinary maps, were then discovered to be governed by TR, as \\textit{ciliated} and \\textit{fully simple maps} \\cite{Borot:2017agy,Borot:2021eif}.\n\n  In this letter, we will focus on another subset of maps, the \\textit{bipartite maps} containing only those ordinary maps of even face degrees, for which the corresponding maps have vertices in black and white such that no monochromatic edge occurs. A bipartite map is called \\textit{rooted}, if one edge is distinguished and oriented. This \\textit{rooted edge} (also called marked edge) conventionally  has its origin in a white vertex (the \\textit{root vertex}). Rooting an edge creates a boundary of a certain even length $2l_k$ following the face to the right of the rooted edge. Several edges can be rooted such that the roots do not correspond to the same boundary. Bipartite maps already showed up in the context of TR, namely in \\cite{Chapuy2016} in which the authors were motivated by TR and established a recursive formula sharing many characteristics with the TR \\footnote{A more formal, but less illustrative definition for bipartite maps can be found in \\cite{Chapuy2016}}. However, the aim of their work is rather to prove rationality statements about bipartite maps and is thus written more in a combinatorist's language. All these statements are a direct consequence of TR. Their recursion and its proof are mainly built on ideas of TR, but no spectral curve was provided. Thus, the relation of their work to complex geometry will be established in the following Chapter \\ref{ch:proof}. We will deduce:\n\\begin{theorem}\\label{th:main}\nThe spectral curve $(\\mathbb{P}^1,x_{bip},y_{bip},\\frac{dz_1\\,dz_2}{(z_1-z_2)^2})$ with \n\\begin{align*}\n\tx_{bip}(z)=\\gamma^2\\bigg ( z+ \\frac{1}{z} \\bigg ) + 2\\gamma^2 \\qquad y_{bip}(z)=\\frac{\\sum_{k=0}^{d-1} u_{2k+1}z^{k+1}}{\\gamma (1+z)}\n\\end{align*}\ncomputes via TR generating functions for the enumeration of bipartite maps with $n$ marked faces (or rooted edges) of even boundary lengths. The faces have even degrees up to $2d$, where a face of degree $2k$ is weighted by $t_{2k}$. We have the following relation to Thm. \\ref{th:eyn}:\n\\begin{align*}\nx_{bip}(z^2)=x_{ord}(z)^2 \\qquad \\quad y_{bip}(z^2) = \\frac{y_{ord}(z)}{ x_{ord}(z)}\n\\end{align*}\n\\end{theorem}\nIn order to avoid misunderstandings, we would like to mention that an unconventional definition of bipartite maps, deviating from the one in this paper, is given in \\cite{Eynard:2016yaa} and coincides just for genus zero and one boundary. \n\nGiven these two spectral curves for ordinary and bipartite maps, the machinery of TR gives rise to generating functions as follows: Let $\\tilde{\\mathcal{T}}^{(g)}_{2l_1,...,2l_n}$ denote the generating function of bipartite maps with a natural embedding into a genus-$g$ surface with $n$ boundaries of length $2l_1,...,2l_n$ ($n$-fold rooted bipartite maps) and in the same manner $\\mathcal{T}^{(g)}_{2l_1,...,2l_n}$ for ordinary maps with faces of even degree. Note that in particular $2^{n-1}\\tilde{\\mathcal{T}}^{(0)}_{2l_1,...,2l_n}=\\mathcal{T}^{(0)}_{2l_1,...,2l_n}$ holds for genus $g=0$, however not for $g>0$, where only a small subset of ordinary maps are still bipartite. The prefactor $2^{n-1}$ has an easy combinatorial explanation: As described earlier, the \\textit{root vertex} is by convention a white one, the black-white coloring of the vertices is completely determined by fixing a root. Ignoring  the colouring, as is it is done for ordinary maps, an other boundary can have twice the number of labellings. Inductively, this gives rise to $2^{n-1}$ distinct graphs, if $n$ faces are marked, as ordinary maps.\n\n Define the \\textit{correlators} $W$ and $\\tilde W$  as \n  \\begin{align}\n\\label{resolv}\n W_n^{(g)}(x_{ord,1},...,x_{ord,n}) =   \\sum_{l_1,...,l_n=1}^\\infty \\frac{\\mathcal{T}^{(g)}_{2l_1,...,2l_n}}{x_{ord,1}^{2l_1+1}...x_{ord.n}^{2l_n+1}} \\\\\n \\tilde {W}_n^{(g)}(x_{bip,1},...,x_{bip,n}) =   \\sum_{l_1,...,l_n=1}^\\infty \\frac{\\tilde{\\mathcal{T}}^{(g)}_{2l_1,...,2l_n}}{x_{bip,1}^{l_1+1}...x_{bip,n}^{l_n+1}}\n\\end{align}\nfrom which the generating functions can be read off as a simple residue operation, e.g. for bipartite maps:\n\\begin{align*}\n\\tilde{\\mathcal{T}}^{(g)}_{2l_1,...,2l_n} = (-1)^n \\Res_{x_{bip,1}...x_{bip,n}\\to \\infty}  x_{bip}^{l_1}\\cdot ...\\cdot x_{bip}^{l_n} \\tilde {W}_n^{(g)}(x_{bip,1},...,x_{bip,n})dx_{bip,1}...dx_{bip,n}\n\\end{align*}\nThe crucial connection to the infinite sequence of meromorphic $n$-forms $\\omega_{g,n}$ generated by TR is the following identification for $2g+n-2>0$\n\\begin{align}\n\t\\omega_{g,n}(z_1,...,z_n)=\\tilde{W}^{(g)}_n(x_{bip}(z_1),...,x_{bip}(z_n))dx_{bip}(z_1)...dx_{bip}(z_n).\n\\end{align}\nFor the stable topologies $2g+n-2\\leq 0$, the situation is a bit subtle.\n\nFrom Thm. \\ref{th:main}, we deduce the following equivalent representation of the generating functions of bipartite maps, building the bridge to TR:\n\\begin{corollary}\n\\label{cor1}\nLet $\\omega_{g,n}$ be the correlators of TR  generated by \\eqref{BTR-intro} with $(x_{bip},y_{bip})$ given by Theorem \\ref{th:main}. Then  $\\tilde{\\mathcal{T}}^{(g)}_{2l_1,...,2l_n}$ can be achieved as follows:\n\\begin{align*}\n\\tilde{\\mathcal{T}}^{(g)}_{2l_1,...,2l_n} = (-1)^n \\Res_{z_1,...,z_n \\to \\infty} x_{bip}(z_1)^{l_1} \\cdot ... \\cdot x_{bip}(z_n)^{l_n}  \\omega_{g,n}(z_1,...,z_n)\n\\end{align*}\n\\end{corollary}\nAnalogously, generating functions for ordinary maps are obtained from the spectral curve of Theorem \\ref{th:eyn} (see \\cite{Eynard:2016yaa} for more details). This spectral curve, together with the recursion formula of \\cite[Thm. 3.9]{Chapuy2016}, will be the basis for the proof of Theorem \\ref{th:main} by direct identification.\n\n\\section*{ Acknowledgements}\nWe thank Guillaume Chapuy and Wenjie Fang for helpful discussions. JB is supported\\footnote{``Funded by\n  the Deutsche Forschungsgemeinschaft (DFG, German Research\n  Foundation) -- Project-ID 427320536 -- SFB 1442, as well as under\n  Germany's Excellence Strategy EXC 2044 390685587, Mathematics\n  M\\\"unster: Dynamics -- Geometry -- Structure.\"} by the Cluster of\nExcellence \\emph{Mathematics M\\\"unster}. He would like to thank the University of Oxford for its hospitality. AH is supported by\nthe Walter-Benjamin fellowship\\footnote{``Funded by\n  the Deutsche Forschungsgemeinschaft (DFG, German Research\n  Foundation) -- Project-ID 465029630}.\n \n\\section{Proof and Discussion}\n\\label{ch:proof}\n\\subsection{Reminder of previous results}\nFirst, we briefly recapitulate the procedure of topological recursion. Starting with the initial data, the spectral curve  $(\\Sigma,x,y,B)$, TR constructs recursively in $2g+n-2$ an infinite sequence of meromorphic $n$-forms  $\\omega_{g,n}$, starting with\n  \\begin{align*}\n\\omega_{0,1}(z) = y(z)\\, dx (z) \\qquad \\omega_{0,2}(z_1,z_2) = B(z_1,z_2),\n\\end{align*}\n via the following residue formula:\n\\begin{align}\n&  \\omega_{g,n+1}(I,z)\n  \\label{BTR-intro}\n  \\\\\n  & =\\sum_{\\beta_i}\n  \\Res\\displaylimits_{q\\to \\beta_i}\n  K_i(z,q)\\bigg(\n  \\omega_{g-1,n+2}(I, q,\\sigma_i(q))\n  +\\hspace*{-1cm} \\sum_{\\substack{g_1+g_2=g\\\\ I_1\\uplus I_2=I\\\\\n            (g_1,I_1)\\neq (0,\\emptyset)\\neq (g_2,I_2)}}\n  \\hspace*{-1.1cm} \\omega_{g_1,|I_1|+1}(I_1,q)\n  \\omega_{g_2,|I_2|+1}(I_2,\\sigma_i(q))\\!\\bigg).\n\\end{align}\nHere $I=\\{z_1,\\dots,z_n\\}$ is a collection of $n$ variables $z_j$, the sum is over the ramification points\n$\\beta_i$ of $x$ defined by $dx(\\beta_i)=0$. The kernel $K_i(z,q)$ is\ndefined in the vicinity of $\\beta_i$ by\n$K_i(z,q)=\\frac{\\frac{1}{2}\\int^{q}_{\\sigma_i(q)}\n  B(z,q')}{\\omega_{0,1}(q)-\\omega_{0,1}(\\sigma_i(q))}$, where\n$\\sigma_i\\neq \\mathrm{id}$ is the local Galois involution\n$x(q)=x(\\sigma_i(q))$ near $\\beta_i$, and $\\beta_i$ as a fixed point.\n\nA TR-like formula to recursively generate correlators for bipartite maps was found in the aforementioned paper \\footnote{We adapt the notation of \\cite{Chapuy2016} to the TR literature by $p_k \\mapsto t_{2k}$, $z \\mapsto \\gamma^2$, $u \\mapsto z$, $F_g\\mapsto U_g$}:\n\\begin{theorem}[\\cite{Chapuy2016}]\n\\label{th:chap}\nLet  $x(z)= \\frac{z}{(1+z\\gamma^2)^2}$. A correlators function $U_g(x(z))= \\sum_{l=1}^\\infty \\tilde{\\mathcal{T}}^{(g)}_{2l} x^l$, $g>1$, can be recursively obtained in the following way:\n\\begin{align*}\nU_g(x(z)) = \\frac{1}{P(z)} \\Res_{q \\to \\pm \\frac{1}{\\gamma^2}} \\frac{P(q)}{z-q} \\frac{x(q)}{Y(q)} \\bigg (U_{g-1}^{(2)}(q)+ \\sum_{\\substack{g_1+g_2=g\\\\g_i>0}} U_{g_1}(q)U_{g_2}(q) \\bigg ) \n\\end{align*}\nwith $P(q)=\\frac{1-\\gamma^2 q}{1+ \\gamma^2 q}$, $Y(q)$ see below. $U^{(2)}_g=\\sum_{l_1,l_2=1}^\\infty \\tilde{\\mathcal{T}}^{(g)}_{2l_1,2l_2} x^{l_1+l_2}$.\n\\end{theorem}\n\n\\subsection{Proof of the spectral curve}\nThe heuristic deduction of $(x_{bip},y_{bip})$ that finally turns Thm. \\ref{th:chap} into TR works as follows: \n\\begin{itemize}\n \\item $x_{bip}$: The work of Chapuy and Fang mainly relies on two important variable transformations. The first is the definition of $\\gamma^2$, arising already for ordinary maps and earlier works of Bender and Canfield \\cite{Bender}. The second, $x(z)= \\frac{z}{(1+z\\gamma^2)^2}$ will determine $x_{bip}$. Thm. \\ref{th:chap} creates generating functions as a series in positive powers of $x$. Sending $z \\to \\frac{z}{\\gamma^2}$ and then taking the reciprocal of $x$ gives the correct curve ramified covering.  We confirm this with the relation $x_{bip}(z^2)=x_{ord}^2(z)$ together with a comparison of the correlators $W$ and $\\tilde W$, up to a global factor of $\\frac{1}{x_{bip}}$ on which we comment later - this factor becomes decisive for the geometry of the spectral curve.\n\\item $y_{bip}$: Analogously to ordinary maps, the expression $y_{bip}(z)-y_{bip}(\\sigma(z))$ can be directly read off from the kernel representation of the Tutte equation for the disk. This kernel $Y(z)=y_{bip}(1\/z)-y_{bip}(z)$ is already given in Prop. 3.3 in \\cite{Chapuy2016} and shows up in the recursion formula Thm. \\ref{th:chap} as well. After changing the variables as for $x_{bip}$, we can extract from \\cite[Chap. 5.1]{Chapuy2016} a suitable expression for $Y(z)\\cdot x_{bip}(z)$:  \n\\begin{align*}\n&\\qquad \\quad Y(z)\\cdot x_{bip}(z) =\\\\\n& \\qquad \\quad  \\gamma^2\\frac{(1+z)^2}{z} - (1+z)\\bigg [2- \\sum_{k=1}^d t_{2k} \\gamma^{2k} \\bigg (\\sum_{l=1}^{k-1}z^l \\binom{2k-1}{k+l} - \\sum_{l=-k}^0 z^l \\binom{2k-1}{k+l} \\bigg ) \\bigg ].\n\\end{align*}\nInserting the implicit equation of $\\gamma^2$ from Theorem \\ref{th:eyn} in the first term cancels partially the terms for $l=-1,0$. After some further lengthy but trivial algebra, the expression can be ordered in positive and negative powers of $z$, where the positive powers give  \n\\begin{align}\\label{yeq}\n\ty_{bip}(z)\\cdot x_{bip}(z)= 1+z-(1+z)\\sum_{k=1}^{d-1} \\sum_{j\\geq k+1} t_{2j} \\binom{2j-1}{j+k} \\gamma^{2j} z^k.\n\\end{align}\nFinally, the definition of $u_{2k+1}$ in terms of $t_{2j}$ yields the desired identifications shown in Theorem \\ref{th:main}. \n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\subsection{Discussion of the result}\n\\label{ch:disc}\nOf particular interest is the somewhat unusual geometry of the spectral curve for bipartite maps. Its branch cut goes from $x_{bip}(1)=a=4\\gamma^2$ to $x_{bip}(-1)=b=0$. We naturally have the same Zhukovsky parametrisation as for $x_{ord}(z)$:\n\\begin{align}\\label{zhu}\n\\frac{a+b}{2}+\\frac{a-b}{4} \\bigg (z + \\frac{1}{z} \\bigg ) \\qquad  \\mathrm{and} \\qquad \\sqrt{(x-a)(x-b)} =\\gamma^2\\bigg (z - \\frac{1}{z} \\bigg )\n\\end{align}\nHowever, the branch point at $0=x(\\beta_2)$ corresponding to the ramification point $\\beta_2=-1$ affects the pole structure of all $\\omega_{g,n}$ - the highest degree of the poles is different for the two ramification points. Due to the fact that $y_{bip}$ is irregular at the ramification point $\\beta_2=-1$ (whereas as required $\\omega_{0,1}=y \\,dx$ is still regular), the maximum order of poles $\\frac{1}{(z+1)^k}$ is reduced by two in comparison to the poles $\\frac{1}{(z-1)^l}$. However, this does not change the fact that one can generate symmetric $n$-forms from $(x_{bip},y_{bip})$.  This is an interesting deviation from the most spectral curves. Despite the uncommon pole distribution at the ramification points, the universal symmetry under the Galois involution naturally holds: \n \\begin{align*} \n\\frac{\\omega_{g,n}(z,z_I)}{dx(z)} + \\frac{\\omega_{g,n}(\\frac{1}{z},z_I)}{dx(\\frac{1}{z})}=0 \\quad \\forall 2g+n-2>0 \n\\end{align*}\n  For illustrative purposes, we give $\\omega_{1,1}$ as an example and set $\\tilde{y}_{bip}(z) = \\frac{1}{\\gamma} \\sum_{k=0}^{d-1}u_{2k+1} z^{k+1}$\n \\begin{align*} \n\\omega_{1,1}(z) =& \\frac{1}{16 \\gamma^2 (1+z)^2 \\tilde{y}'_{bip}(-1)}- \\frac{1}{16 \\gamma^2 (z-1)^4 y'_{bip}(1)}\\\\\n&-\\frac{1}{16 \\gamma^2 (z-1)^3 y'_{bip}(1)} + \\frac{3 y'_{bip}(1)+3 y''_{bip}(1)+ y'''_{bip}(1)}{96 \\gamma^2 (z-1)^2 y'^2_{bip}(1)}\n\\end{align*}\n\n \nFinally, we want to collect some interesting  open questions: It is known \\cite{Borot:2017agy,Borot:2021eif} that the exchange of $x_{ord}$ and $y_{ord}$ gives rise to generating functions of fully simple maps. Does any sort of exchange of $x_{bip}$ and $y_{bip}$ have a comparable strong implication? Another question arises from the matrix models as realisations of those various types of maps. As known from the classical literature, bipartite maps arise from the complex matrix model, having a structural equivalence to the Hermitian 2-matrix model \\cite{Eynard:2005}. This model is (for certain boundary structures) already solved by TR. What is the relation between the two distinct spectral curves? A final question is dedicated only to quadrangulations. In \\cite{Branahl:2020yru} the quartic Kontsevich model (QKM) was shown to be solvable in terms of correlators $\\omega_{g,n}$ that follow an extension of TR. In this so-called blobbed topological recursion (BTR; general framework developed in \\cite{Borot:2015hna}), the $\\omega_{g,n}$ split into parts with poles at the ramification points (polar part) and with poles somewhere else (holomorphic part). In \\cite{Branahl:2021} it was stated that the pure (normalised) TR results contributing to  $\\omega_{g,n}$ of the QKM generate bipartite (rooted) quadrangulations, whereas ordinary quadrangulations are generated when taking the complete BTR into account. Understanding this different approach to the partition functions of the complex and hermitian 1-matrix model from the beginning will be an interesting challenge for the future.\n\n\\subsection{Example of quadrangulations}\\label{a}\n In order to underpin the correctness of our spectral curve, let us only allow for $t_4 \\neq 0$ and $n=l=1$, yielding (with $u_1=\\frac{1}{\\gamma}$ and $u_3=-t_4\\gamma^3$):\n \\begin{align*} \nx_{bip}(z) = \\gamma^2 \\bigg( z+\\frac{1}{z} \\bigg) + 2\\gamma^2, \\qquad y_{bip}(z)= \\frac{\\frac{z}{\\gamma^2}-t_4 \\gamma^2 z^2}{1+z}, \\qquad \\gamma^2 = \\frac{1-\\sqrt{1-12t_4}}{6t_4}.\n\\end{align*}\n The expansions in $t_4$ by computer algebra can be found in Tab. \\ref{tab1}. Bipartite rooted quadrangulations are in particular interesting, since Tutte's famous bijection \\cite{Tuttbij} relates them to rooted ordinary maps for faces of any (not only even) degree.\n\\begin{table}[h]\n\\centerline{\\begin{tabular}[h!b]{|c|c|c|c||c|c|c|}\n\\hline\nOrder &$\\tilde{\\mathcal{T}}_2^{(0)}$&$\\tilde{\\mathcal{T}}_2^{(1)}$&$\\tilde{\\mathcal{T}}_2^{(2)}$ &$  \\mathcal{T}_2^{(0)}$&$ \\mathcal{T}_2^{(1)}$&$ \\mathcal{T}_2^{(2)}$ \\\\\n\\hline\n$(t_4)^0$ & 1 &0 &0  & 1 & 0 & 0  \\\\\n\\hline\n$(t_4)^1$ & 2 & 0 & 0 & 2 & 1 & 0  \\\\\n\\hline\n$(t_4)^2$ & 9 & 1 & 0 & 9 & 15 & 45  \\\\\n\\hline\n$(t_4)^3$ & 54 & 20 & 0 & 54 & 198 & 2007 \\\\\n\\hline\n$(t_4)^4$ & 378 & 307 & 21 & 378  & 2511 & 56646 \\\\\n\\hline\n$(t_4)^5$ & 2916  & 4280 & 966  & 2916  & 31266 & 1290087  \\\\\n\\hline\n\\end{tabular}}\n\\hspace*{1ex}\n\\caption{These numbers are generated by Thm \\ref{th:main} and Thm. \\ref{th:eyn} together with Cor. \\ref{cor1} and coincide with \\cite{Bender} and with OEIS no. A006300 $(g=1)$ and no. A006301 ($g=2$) for $\\tilde{\\mathcal{T}}_{2}^{(g)}$.}\n\\label{tab1}\n\\end{table}\n \n \n\\bibliographystyle{halpha-abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\nThe discovery of quantum vacuum nonlinearities\n\\cite{Heisenberg:1935qt,Weisskopf,Schwinger:1951nm} under controlled\nlaboratory conditions using real photons or macroscopic\nelectromagnetic fields is a major goal of contemporary strong-field\nphysics. Many proposals rely on a pump-probe scheme, where a\nwell-controlled, say optical, photon beam probes a region of space\nthat is exposed to a strong field (``pump''). A typical example is given\nby schemes intended to verify vacuum birefringence\n\\cite{Toll:1952,Baier,BialynickaBirula:1970vy,Adler:1971wn} that can\nbe searched for using macroscopic magnetic fields\n\\cite{Cantatore:2008zz,Berceau:2011zz} or with the aid of\nhigh-intensity lasers \\cite{Heinzl:2006xc}, see e.g.,\n\\cite{Dittrich:2000zu,Marklund:2008gj,Dunne:2008kc,DiPiazza:2011tq} for reviews.\n\nAs these setups require techniques such as high-purity ellipsometry\n\\cite{Cantatore:2008zz,Marx:2011} to separate the (small) signal from\na typically huge background, a recent proposal has focused on a\nquantum-reflection scheme that facilitates a built-in noise\nsuppression \\cite{Gies:2013yxa}. In this scheme, incident\nprobe photons propagate towards a spatially localized field\ninhomogeneity (``pump''), as, e.g., generated in the focal spots of a\nhigh-intensity laser system.  Even though the inhomogeneity acts similar to\nan attractive potential, probe photons can be scattered backwards due to\nquantum reflection.  Looking for reflected photons in the field free\nregion, this scenario inherently allows for a clear geometric\nseparation between signal and background.  First estimates of the\nnumber of reflected photons attainable in present and near future\nlaser facilities look promising. Figure~\\ref{fig:QRef} depicts a\ntypical Feynman diagram contributing to the effect.\n\nAs quantum reflection crucially relies on the presence of an\ninhomogeneous pump field, it belongs to a general class of\nquantum-induced interference effects\n\\cite{King:2013am,Tommasini:2010fb,Hatsagortsyan:2011} with the\nparticular property of optimizing the signal-to-noise ratio.\n\nThe pump-probe scheme is typically also reflected by the theoretical\ndescription, in which the nonlinearities are kept for the pump-probe\ninteraction, but the equations are linearized with respect to the\nprobe propagation. In the present work, we rely again on an optical\npump-probe setup which however requires a nonlinear treatment of the\nprobe-field. The idea is to look for laser photon merging in the\npresence of an electromagnetic field inhomogeneity.  This effect\nresembles the standard nonlinear optical process of {\\it second\n  harmonic generation} (SHG) -- or in general {\\it high harmonic\n  generation} -- with the nonlinear crystal replaced by the quantum\nvacuum subject to strong electromagnetic fields. Higher harmonic\ngeneration in an electromagnetized vacuum has been discussed on the\nlevel of the Heisenberg-Euler action in\n\\cite{BialynickaBirula:1981,Kaplan:2000aa,Valluri:2003sp,Fedotov:2006ii},\nsee also the discussion in \\cite{Marklund:2006my}, or using the\nconstant-field polarization tensor in \\cite{DiPiazza:2005jc}. Laser\nphoton merging in proton-laser collisions have been investigated in\ndetail in \\cite{DiPiazza:2007cu,DiPiazza:2009cq}, where a promising\nscenario has been proposed for a discovery of the merging phenomenon\nthat involves a nowadays conventional optical high-intensity laser at\na high-energy proton collider. A related effect is called four-wave\nmixing for which also a concrete experimental proposal has been\nexplored in \\cite{Lundin:2006aa,King:2012aw}.  The same underlying\nquantum vacuum nonlinearity could even be used to radiate photons from\nthe focal spot of a single focused laser beam (``vacuum emission'') as\nproposed in \\cite{Monden:2011}. More generally, frequency mixing induced\nby quantum vacuum effects has even been suggested as a sensitive probe\nto search for new hypothetical particles \\cite{Dobrich:2010hi}.\n\n\\begin{figure}\n\\includegraphics[width=0.57\\textwidth]{QRef} \n\\caption{Typical Feynman diagram contributing to the effect of quantum reflection \\cite{Gies:2013yxa}. For field strengths of the inhomogeneity well below the critical field strength (cf. main text), the leading contribution arises from diagrams with two couplings to the field inhomogeneity. As there is no energy transfer from static fields, the frequencies of the incident and outgoing photons match.}\n\\label{fig:QRef}\n\\end{figure}\n\nIn the present work, we concentrate on an ``all-optical'' parameter\nregime realizable with high-intensity lasers. As the signal is\nexpected to be very small, we again consider specifically the\nkinematics of the reflection process for an appropriate\nsignal-to-noise reduction.  As in \\cite{Gies:2013yxa}, we limit\nourselves to the study of time-independent field inhomogeneities, such\nthat there is no energy transfer from the field inhomogeneity. Depending on the spatial field\ninhomogeneity, the propagation direction of the merged photons can\ndiffer from that of the incident probe photons.  For the specific\nreflecting kinematic situation, the merged photons can even\npropagate -- somewhat counter-intuitively -- into the backward\ndirection.  For a straightforward comparison of the signals resulting\nfrom quantum reflection \\cite{Gies:2013yxa} and the photon merging\nscenario of this work, we focus on a one-dimensional magnetic field\ninhomogeneity. As is shown by an explicit calculation below, our\nfindings confirm the expectation that the merging process for the\nreflective scenario is dominated by the quantum reflection process for\nthe all-optical parameter regime. Nevertheless, due to a different\npolarization and frequency dependence, filtering techniques might\nallow for a discovery of the merging process in this set up as well.\n\nLet us briefly outline the theoretical framework of our study,\ntailored to an all-optical scenario.\nOptical lasers operate at frequencies $\\omega\\sim{\\cal O}({\\rm eV})$\nmuch smaller than the electron mass $m\\approx511\\,{\\rm keV}$,\nconstituting a typical scale associated with quantum effects in\nquantum electrodynamics (QED), such that $\\frac{\\omega}{m}\\ll1$.\nMoreover, the maximum field strengths attainable with present and near\nfuture laser facilities are small in comparison to the {\\it critical\n  field strength} $E_{\\rm cr}\\equiv\\frac{m^2}{e}$\n\\cite{Heisenberg:1935qt}, i.e., $\\{\\frac{e{\\mathfrak\n    E}}{m^2},\\frac{eB}{m^2}\\}\\ll1$, with $\\mathfrak{E}$ denoting the\nelectric field strength of the probe laser and $B$ the peak magnetic field\nstrength of the spatially localized inhomogeneity.\nHence, for a given number $2n$, $n\\in\\mathbb{N}$, of probe laser\nphotons of frequency $\\omega$ (wavelength\n$\\lambda=\\frac{2\\pi}{\\omega}$), the dominant merging process into a\nsingle photon of frequency $2n\\omega$ is expected to arise from an\ninteraction of the type depicted in Fig.~\\ref{fig:merging_cartoon},\nexhibiting a single coupling to the (magnetic) field\ninhomogeneity. Higher order couplings to the field inhomogeneity are\nstrongly suppressed due to the fact that $\\frac{eB}{m^2}\\ll1$.\nFurry's theorem (charge conjugation symmetry of QED) dictates the\ninteraction to vanish for any odd number of couplings to the\nelectron-positron loop, which justifies that we have \ntailored the merging process to $2n$ laser photons. \nThe dominant contribution in the weak-field limit is expected to arise from\nthe merging of two laser photons, described by Feynman diagrams with\nfour legs (cf. Fig.~\\ref{fig:merging_cartoon}).\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{merging_cartoon} \n\\caption{Cartoon of the photon merging process. In the presence of a stationary but\n  spatially inhomogeneous electromagnetic field $2n$ laser photons of frequency\n  $\\omega$ can merge into a single photon of frequency $2n\\omega$.\n  Depending on the spatial field inhomogeneity, the propagation\n  direction of the merged photons can differ from that of the incident\n  probe photons. In curly braces we introduce our notation for the corresponding fields\/polarizations and four-momenta; cf. also Eqs.~\\eqref{eq:background}, \\eqref{eq:Ak} and \\eqref{eq:M4}, as well as Fig.~\\ref{fig:perspective}.}\n\\label{fig:merging_cartoon}\n\\end{figure}\n\nA sketch of the geometry of the reflective scenario\nof the merging process\nto be investigated in this paper can be found in  Fig.~\\ref{fig:perspective}.\nHere we already summarize the notation to be introduced and discussed below.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{QS-beschr-crop} \n\\caption{Schematic depiction of the two-photon merging process. Incident probe photons (wave vector $\\vec{\\kappa}$, energy $\\kappa^0=|\\vec{\\kappa}|=\\omega$) hit a one-dimensional field inhomogeneity $\\vec{B}({\\rm x})=B({\\rm x})\\vec{e}_{\\rm z}$ of width $w$ under an angle of $\\theta$. Due to nonlinear effective couplings between electromagnetic fields mediated by virtual charged particle fluctuations, the field inhomogeneity can impact incident probe photons to merge and form an outgoing photon (wave-vector $\\vec{k}_f$) of twice the energy of the incident probe photons, i.e., $k_f^0=|\\vec{k}_f|=2\\omega$. \nMost notably, the inhomogeneity can affect the outgoing merged photons to reverse their momentum component along $\\vec{e}_{\\rm x}$ with respect to the incident probe photons.\nThe vectors $\\vec{a}_1$, $\\vec{a}_2$ and $\\vec{\\epsilon}^{\\,(1)}$, $\\vec{\\epsilon}^{\\,(2)}$ span the polarization degrees of freedom of the incident and outgoing photons, respectively. For the depiction we specialized to $\\varphi=\\varphi'=0$ (cf. the main text).}\n\\label{fig:perspective}\n\\end{figure}\n\nThe leading quantum reflection process in the perturbative regime also\narises from four leg diagrams (cf. Fig.~\\ref{fig:QRef}).  While\nquantum reflection necessitates at least two couplings to the field\ninhomogeneity, photon merging just needs a single coupling to the\ninhomogeneity. Conversely, quantum reflection can be considered as a\ntwo-photon (one incident, one outgoing) process, whereas photon\nmerging involves at least three photons (two incident, one\noutgoing). From this observation, it can already be anticipated\nthat the dependence of the observables on the various parameters\nwill differ between the two processes.\n\nNotably, the merging process in Fig.~\\ref{fig:merging_cartoon} can be\nevaluated straightforwardly, owing to the fact that the photon\npolarization tensor is explicitly known for generic monochromatic\nplane wave backgrounds \\cite{Baier:1975ff,Becker:1974en}.\nInterpreting the plane wave background in terms of incident probe\nphotons of frequency $\\omega$, the two open legs of the polarization\ntensor can be identified with the field inhomogeneity and the outgoing\nmerged photon, respectively.  The polarization of the incident photons\ncan be controlled by adjusting the polarization of the monochromatic\nplane wave background.\n\nOur paper is organized as follows: In Sec.~\\ref{sec:calculation} we\nexplain in detail the various steps needed to evaluate the photon\nmerging process.  A crucial technical step is to find a\ncontrolled approximation to the photon polarization tensor in a plane\nwave background, facilitating an analytical treatment of the photon\nmerging process.  Such an approximation, especially suited to the\nparameters of an all-optical experimental scenario, is derived in\nSec.~\\ref{subsec:Pi}.  Section~\\ref{seq:Ex+Res} is devoted to the\ndiscussion of explicit examples and results. It contains a thorough\ncomparison of the effects of laser photon merging and quantum reflection. \nWe end with conclusions and an outlook in Sec.~\\ref{seq:Con+Out}.\n\n\n\\section{Calculation} \\label{sec:calculation}\n\n\\subsection{Photon polarization tensor in plane wave field} \\label{subsec:Pi}\n\nWe briefly recall and summarize the basic structure of the photon\npolarization tensor in a generic, elliptically polarized monochromatic\nplane wave background \\cite{Baier:1975ff,Becker:1974en}. The latter is parametrized by the following gauge potential in Coulomb-Weyl gauge\n\\begin{equation}\n{\\cal A}_\\mu(x)={\\mathfrak a}_{1}a_{1\\mu}\\cos(\\kappa x)+{\\mathfrak a}_{2}a_{2\\mu}\\sin(\\kappa x), \\label{eq:background}\n\\end{equation}\nwith $\\mathcal{A}_0=0$, $\\kappa^2=0$ and\n$a_1\\kappa=a_2\\kappa=a_1a_2=0$. Moreover, we will use\nthe frequency $\\omega\\equiv\\kappa^0$.  The four-vectors $a_{i\\mu}$ with\n$i\\in\\{1,2\\}$ are normalized to unity, i.e., $a_i^2=1$, and the field\namplitude is encoded in the coefficients ${\\mathfrak a}_i\\geq0$.  Our\nmetric convention is $g_{\\mu \\nu}=\\mathrm{diag}(-1,+1,+1,+1)$, and we use $c=\\hbar=1$.\n\nFor the normalized plane wave field strength in momentum space,\nwe introduce $f^{\\mu\\nu}_i=\\kappa^\\mu a_i^\\nu-\\kappa^\\nu a_i^\\mu$. \nIn the following, we will frequently use the shorthand notation $(kf_i)^\\mu=k_\\nu f^{\\nu\\mu}_i$. \n\nIn momentum space the photon polarization tensor mediates between two\nfour-momenta $k_1$ and $k_2$.  Since the wave~\\eqref{eq:background} is\ncharacterized by the single four-momentum $\\kappa$ and a change in the\nincident momentum is determined by an interaction with the wave, the\nkinematics are such that $k_2=k_1+C\\kappa$, with scalar constant $C$\n\\cite{Baier:1975ff}.  Correspondingly, $\\kappa k_2=\\kappa\nk_1\\equiv\\kappa k$ and also $(k_1f_i)^\\mu=(k_2f_i)^\\mu=(kf_i)^\\mu$.\n\nFollowing \\cite{Baier:1975ff}, the associated photon\npolarization tensor can then be compactly represented as\n\\begin{equation}\n \\Pi^{\\mu\\nu}(k_1,k_2)=c_1\\Lambda_1^\\mu\\Lambda_2^\\nu+ c_2\\Lambda_2^\\mu\\Lambda_1^\\nu+c_3\\Lambda_1^\\mu\\Lambda_1^\\nu+c_4\\Lambda_2^\\mu\\Lambda_2^\\nu+c_5\\Lambda_3^\\mu\\Lambda_4^\\nu, \\label{eq:PIstructure}\n\\end{equation}\nwith scalar coefficients $c_j(k_1,k_2)$, $j\\in\\{1,\\ldots,5\\}$.\nThe tensor structure is encoded in products of the normalized four vectors\n\\begin{multline}\n \\Lambda_i^\\mu=\\frac{(kf_i)^\\mu}{(\\kappa k)}=a_i^\\mu-\\frac{(ka_i)}{(\\kappa k)}\\kappa^\\mu \\quad \\text{for} \\quad i\\in\\{1,2\\},\\\\\n \\Lambda_3^\\mu=\\frac{\\kappa^\\mu k_1^2-k_1^\\mu(\\kappa k)}{(\\kappa k)\\sqrt{-k_1^2}}, \\quad \\Lambda_4^\\mu=\\frac{\\kappa^\\mu k_2^2-k_2^\\mu(\\kappa k)}{(\\kappa k)\\sqrt{-k_2^2}}, \\label{eq:Lambdas}\n\\end{multline}\nfulfilling\n$\\Lambda_1^2=\\Lambda_2^2=\\Lambda_3^2=\\Lambda_4^2=1$. This tensor\nstructure guarantees that $\\Pi^{\\mu\\nu} (k_1,k_2)$ satisfies the\nWard identities $k_{1,\\mu}\\Pi^{\\mu\\nu} (k_1,k_2)=\\Pi^{\\mu\\nu}(k_1,k_2)k_{2,\\nu}=0$.\n\nApart from a trivial overall factor of $\\alpha=e^2\/(4\\pi)$,\nthe coefficients $c_j$ depend on the kinematic variables $k_1$, $k_2$\nand $\\kappa$ as well as the electron mass $m$, and account for\nthe entire field strength dependence. The latter dependence is most\nconveniently expressed in terms of the two invariant intensity\nparameters $\\xi_i=\\frac{e\\mathfrak{a}_i}{m}$ with $i\\in\\{1,2\\}$.\nIn Coulomb-Weyl gauge, the amplitude $\\mathfrak{a}_i$ is intimately related to the amplitude\nof the associated electric field $\\mathfrak{E}_i$ via\n$\\mathfrak{a}_i=\\frac{\\mathfrak{E}_i}{\\omega}$, such that -- in terms\nof parameters directly accessible in the lab -- we have\n$\\xi_i=\\frac{e\\mathfrak{E}_i}{m\\omega}$.\n\nIn consequence of Furry's theorem, the field dependence can be encoded in\n$\\xi_1^2$, $\\xi_2^2$ and $\\xi_1\\xi_2$, i.e., combinations even in the\ncharge $e$, only.  It is moreover helpful to introduce the\ndimensionless parameter $\\lambda=-\\frac{\\kappa k}{2m^2}$,\nparametrizing the relative momenta of the involved photons. \nIn summary, the relevant dimensionless parameters for the\noff-shell  polarization tensor in a plane wave field are given by\n\\begin{equation}\n\\xi_i=\\frac{e\\mathfrak{E}_i}{m\\omega}, \\quad \\lambda=-\\frac{\\kappa k}{2m^2}, \\quad \\frac{k_1 k_2}{4m^2}, \n\\end{equation}\nwhere the last parameter characterizes the relative momenta of the in- and outgoing photon legs.\n\nIn the following, we are only interested in a situation with actual interactions\nwith the plane wave field~\\eqref{eq:background} and thus omit the zero\nfield contribution in \\Eqref{eq:PIstructure}.\\footnote{More precisely,\n  the coefficients $c_j$ provided in the following correspond to the\n  quantity $\\Pi^{\\mu\\nu}(A)-\\Pi^{\\mu\\nu}(A=0)$.  The zero field term\n  can be included straightforwardly, noting that\n  $g^{\\mu\\nu}-\\frac{k_1^\\mu\n    k_1^\\nu}{k_1^2}=\\Lambda_1^\\mu\\Lambda_1^\\nu+\\Lambda_2^\\mu\\Lambda_2^\\nu+\\Lambda_3^\\mu\\Lambda_3^\\nu$. \\label{ftnt}}\n\nThe coefficients $c_j$ generically decompose into an {\\it elastic}\npart characterized by zero momentum exchange with the wave and an {\\it\n  inelastic} part with finite momentum exchange.  The latter part is\nmade up of an infinite number $l\\in\\mathbb{Z}\\setminus\\{0\\}$ of\ncontributions with momentum transfer $2\\kappa l$ to be associated with\nthe absorption\/release of $2l$ laser photons.  Correspondingly, we\nwrite\n\\begin{equation}\n c_j=i(2\\pi)^4m^2\\frac{\\alpha}{\\pi}\\Bigl[\\delta(k_1-k_2)G_j^0+\\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}}\\delta(k_1-k_2-2l\\kappa)G_j^l\\Bigr], \\label{eq:c_j}\n\\end{equation}\nwhere the dimensionless coefficients $G_j^l(k_1,k_2)$, with\n$l\\in\\mathbb{Z}$, are most conveniently represented in terms of double\nparameter integrals that cannot be tackled analytically in a\nstraightforward way.  One of the integrals is over a proper-time type\nparameter $\\rho\\in [0,\\infty[$, and the other one over an additional parameter $\\nu\\in [-1,1]$\nrelated to the momentum routing in the loop.\n\nIn order to state them most compactly, it is convenient to define\n\\begin{multline}\n A=\\frac{1}{2}\\Bigl(1-\\frac{\\sin^2\\rho}{\\rho^2}\\Bigr),\\quad A_0=\\frac{1}{2}\\rho(\\partial_\\rho A),\\quad A_1=A+2A_0, \\\\\n z=\\frac{2(\\xi_1^2-\\xi_2^2)}{|\\lambda|(1-\\nu^2)}\\rho A_0,\\quad  y=\\frac{2(\\xi_1^2+\\xi_2^2)}{|\\lambda|(1-\\nu^2)}\\rho A. \\label{eq:defs}\n\\end{multline}\nTaking these definitions into account, the explicit expressions for $G_j^l$ read\n\\begin{equation}\n G_j^l=\\int_{-1}^1{\\rm d}\\nu\\int_0^\\infty\\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\,g_j^l\\,{\\rm e}^{-iy}, \\label{eq:Gs}\n\\end{equation}\nwhere\n\\begin{equation}\n \\phi_0=\\frac{2}{|\\lambda|(1-\\nu^2)}\\Bigl[1-i\\epsilon+\\frac{k_1k_2}{4m^2}(1-\\nu^2)\\Bigr],\n\\end{equation}\nwith $\\epsilon\\to0^+$, and\n\\begin{align}\n g_1^l&=\\xi_1\\xi_2\\Bigl(2\\,{\\rm sign}(\\lambda)\\frac{1+\\nu^2}{1-\\nu^2}\\,\\rho A_0-A_1\\frac{l}{z}\\Bigr)i^lJ_l(z), \\nonumber\\\\\n g_2^l&=g_1^l\\left(A_0\\to-A_0,z\\to z,A_1\\to A_1\\right), \\nonumber\\\\\n g_3^l&=\\Bigl(\\xi_1^2A_1-\\frac{\\xi_1^2-\\xi_2^2}{1-\\nu^2}\\sin^2\\rho\\Bigr)i^{l}\\bigl(J_l(z)-iJ_l'(z)\\bigr)+\\xi_1^2\\frac{1+\\nu^2}{1-\\nu^2}\\sin^2\\rho\\, i^lJ_l(z) \\nonumber\\\\\n      &\\quad +\\frac{1}{4}\\Bigl(\\frac{k_1k_2}{m^2}-\\frac{i|\\lambda|(1-\\nu^2)}{\\rho}\\Bigr)i^l\\bigl(J_l(z)-\\delta_{l0}\\,{\\rm e}^{iy} \\bigr), \\nonumber\\\\\n g_4^l&=g_3^l\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right)(-1)^l, \\nonumber\\\\\n g_5^l&=-\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(1-\\nu^2)i^l\\bigl(J_l(z)-\\delta_{l0}\\,{\\rm e}^{iy} \\bigr), \\label{eq:gs}\n\\end{align}\nfor $l\\in\\mathbb{Z}$. Here, $J_l(z)$ denotes the Bessel function of\nthe first kind, and $\\delta_{ll'}$ is the Kronecker delta.\nEquations~\\eqref{eq:PIstructure}-\\eqref{eq:gs} constitute the full\nexpression of the photon polarization tensor in a generic plane wave\nbackground of type~\\eqref{eq:background}\n\\cite{Baier:1975ff,Becker:1974en}; see \\cite{Meuren:2013oya} for\na more recent derivation and an alternative representation.\nNoteworthily, whenever one of the momenta $k_1$ and $k_2$ is on the\nlight cone, i.e., either $k_1^2=0$ or $k_2^2=0$, the coefficients\n$G_5^l$ vanish for all $l\\in\\mathbb{Z}$, such that $c_5=0$. Except for\nthe zero field contribution (cf. footnote~\\ref{ftnt}), the tensor\nstructure of the photon polarization tensor {under these conditions\n  can be written entirely in terms of the four-vectors $\\Lambda_1^\\mu$\n  and $\\Lambda_2^\\mu$.\n\nFor completeness, note that for a circularly polarized plane wave\nbackground, corresponding to the choice of $\\xi_1=\\xi_2$, we have\n$z=0$.  Hence, taking into account that $J_l(z)\\sim z^{|l|}$\n[cf. \\Eqref{eq:Jseries} below], the only nonvanishing\ncontributions~\\eqref{eq:gs} are those with $l\\in\\{0,\\pm1\\}$,\ncorresponding to the possibility of an elastic interaction and an\ninteraction involving the emission\/absorption of just two photons from\nthe circularly polarized wave.  The physical reason for this is that a\ncircularly polarized wave has definite chirality, such that\ntransitions are only possible without a change in the chirality of the\nincident photon $(l=0)$ or with a reversal of its chirality ($l=\\pm1$)\n\\cite{Baier:1975ff,Becker:1974en}.\n\nAs the expressions are rather cumbersome, we subsequently aim at an\napproximation particularly suited for all-optical experiments.  Our\nstrategy to achieve this relies on series expansions of the expression\n$g_j^l\\,{\\rm e}^{-iy}$ in the integrand of \\Eqref{eq:Gs}, such that\nboth integrals can be performed explicitly and handy approximations\nfor the polarization tensor are obtained. Similar expansion\nstrategies have recently also led to new analytical insights into\nthe well-known polarization tensor for constant fields \\cite{Karbstein:2013ufa}.\n\nFor this purpose it is particularly helpful to note that $A$ and $A_0$\nhave the following infinite series representations\n[cf. \\Eqref{eq:defs}],\n\\begin{equation}\n A=\\frac{\\rho^2}{6}\\sum_{n=0}^\\infty A^{(2n)}\\rho^{2n},\\quad  A_0=\\frac{\\rho^2}{6}\\sum_{n=0}^\\infty (1+n)A^{(2n)}\\rho^{2n}, \\label{eq:As_series}\n\\end{equation}\nwith $A^{(2n)}=\\frac{3}{2}\\frac{(2i)^{2n+4}}{(2n+4)!}$; our definitions are such that $A^{(0)}=1$.\n\nThe above series representations suggest to define\n\\begin{equation}\n \\zeta^\\pm\\equiv\\frac{(\\xi_1^2\\pm\\xi_2^2)\\rho^3}{3|\\lambda|(1-\\nu^2)}\n\\end{equation}\nand to rewrite the quantities $y$ and $z$ as follows\n\\begin{equation}\n  y=\\zeta^+\\sum_{n=0}^\\infty A^{(2n)}\\rho^{2n},\\quad z=\\zeta^-\\sum_{n=0}^\\infty (1+n)A^{(2n)}\\rho^{2n}. \\label{eq:yzseries}\n\\end{equation}\n\nAnother important ingredient in our approach is the series\nrepresentation of $J_l(z)$, which, for $l\\in\\mathbb{Z}$, reads\n(cf. formulae 8.404 and 8.440 of \\cite{Gradshteyn})\n\\begin{equation}\n J_{l}(z)=\\sum_{j=0}^{\\infty}\\frac{(-1)^j[{\\rm sign}(l)]^l}{j!\\,(|l|+j)!}\\left(\\frac{z}{2}\\right)^{|l|+2j}\\quad {\\rm for}\\quad |{\\rm arg}(z)|<\\pi\\,, \\label{eq:Jseries}\n\\end{equation}\nwhere $[{\\rm sign}(l)]^l=1$ for $l=0$ is implicitly understood.\nInserting \\Eqref{eq:yzseries} into \\Eqref{eq:Jseries}, all the Bessel\nfunctions occurring in \\Eqref{eq:gs} can be expanded in powers of $\\zeta^-$ and $\\rho^2$.\nAnalogously, factors of ${\\rm e}^{-iy}$ can be expanded in powers of\n$\\zeta^+$ and $\\rho^2$.\n\nIn the following, let us\nassume that $|\\frac{k_1k_2}{4m^2}|<1$, which is well compatible with\nan all-optical experimental scenario.  Building on this assumption,\nand resorting to the identity $\\int_0^\\infty \\frac{{\\rm\n    d}\\rho}{\\rho}\\,{\\rm\n  e}^{-i\\phi_0\\rho}\\,\\rho^{l+1}=l!\\bigl(\\frac{-i}{\\phi_0}\\bigr)^{l+1}$\nfor $l\\in\\mathbb{N}_0$, we obtain\n\\begin{equation}\n \\int_0^\\infty \\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\,\\rho^{l+1}=l!\\left(-\\frac{i}{2}|\\lambda|(1-\\nu^2)\\right)^{l+1}\\sum_{n=0}^\\infty\\binom{n+l}{n}\\left(-\\frac{k_1k_2}{4m^2}(1-\\nu^2)\\right)^n  .\n\\end{equation}\nHaving implemented the above expansions, the polarization tensor can formally be written in terms of multiple infinite sums.\nNoteworthily, all $\\nu$ integrals are of the following type\n\\begin{align}\n \\int_{-1}^1{\\rm d}\\nu\\,(1-\\nu^2)^n&=\\frac{2^{2n+1}(n!)^2}{(2n+1)!}, \\nonumber\\\\\n \\int_{-1}^1{\\rm d}\\nu\\,(1+\\nu^2)(1-\\nu^2)^n&=\\left(1+\\frac{1}{2n+3}\\right)\\int_{-1}^1{\\rm d}\\nu\\,(1-\\nu^2)^n,\n\\end{align}\nwith $n\\in\\mathbb{N}_0$, and can straightforwardly be performed explicitly for each contribution.\n\nThus, with the collective notation $\\xi^2\\in\\{\\xi_1^2,\\xi_2^2,\\xi_1\\xi_2\\}$ a generic\ncontribution to the photon polarization tensor reads\n\\begin{multline}\n \\int_{-1}^1{\\rm d}\\nu\\int_0^\\infty \\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\Bigl(\\frac{\\xi^2\\rho^{2}}{6}\\Bigr)^s\\rho^{l}(\\zeta^+)^n(\\zeta^-)^j\n \\left\\{\n \\begin{array}{c}\n  1 \\\\\n  1-\\nu^2 \\\\\n  \\frac{1}{1-\\nu^2}\\\\\n  \\frac{1+\\nu^2}{1-\\nu^2}\n \\end{array}\n \\right\\} \\\\\n=\\Bigl(-\\frac{2\\xi^2\\lambda^2}{3}\\Bigr)^{s}\\Bigl(i\\frac{2(\\xi_1^2+\\xi_2^2)\\lambda^2}{3}\\Bigr)^n\\Bigl(i\\frac{2(\\xi_1^2-\\xi_2^2)\\lambda^2}{3}\\Bigr)^j\\bigl(-2i|\\lambda|\\bigr)^{l}\\\\\n\\times c(n,j,s,l)\n \\left\\{\n \\begin{array}{c}\n  1 \\\\\n  1-\\frac{1}{4(n+j+s)+2l+3} \\\\\n  1+\\frac{1}{2}\\frac{1}{2(n+j+s)+l}\\\\\n  1+\\frac{1}{2(n+j+s)+l}\n \\end{array}\n \\right\\} \n\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^2)\\Bigr) , \\label{eq:genblock}\n\\end{multline}\nwith integers $\\{l,n,j\\}\\in\\mathbb{N}_0$ and $s\\in\\{0,1\\}$, fulfilling\n$l+n+j+s>0$. The components in the columns in braces exhaust all possible\ntypes of occurring $\\nu$ integrands. The explicit\nexpression for the numeric coefficient in \\Eqref{eq:genblock} is\n\\begin{equation}\n c(n,j,s,l)=\\frac{2[3(n+j)+2s+l-1]!\\{[2(n+j+s)+l]!\\}^2}{[4(n+j+s)+2l+1]!}.\n\\end{equation}\nBoth integrations can be carried out, and \\Eqref{eq:genblock} provides\nus with the full numeric prefactor for given integers $l$, $n$, $j$\nand $s$ at leading order in a double expansion in\n$|\\frac{k_1k_2}{4m^2}|\\ll1$ and $|\\lambda|\\ll1$, both\ncorresponding to a soft-photon limit.\nMost importantly, the parameters} $\\xi_i$ never come alone but\nalways appear in combination with a factor of $\\lambda$. This\nimplies that any perturbative expansion of the photon polarization\ntensor in plane wave backgrounds which is superficially in powers of\n$\\xi^2$ in fact amounts to an expansion in the combined parameter\n$\\xi^2\\lambda^2$. This is of substantial practical relevance, as\noptical high-intensity lasers are entering the regime $\\xi\\gg1$.\nStill the present expansion remains valid as long as\n$\\xi^2\\lambda^2\\ll 1$ which is typically well satisfied for\ncontemporary optical high-intensity lasers.\nFirst indications of a larger validity regime of the naive ``small-$\\xi$'' expansion had\nalready been observed in \\cite{DiPiazza:2009cq}. Our all-order series expansion of the\npolarization tensor now clarifies the systematics of the underlying\nphysical parameter regimes.\n\nCorrespondingly, the photon polarization tensor can be organized in\nterms of an expansion in the dimensionless quantities\n$\\frac{k_1k_2}{4m^2}$, $\\lambda$ and $\\xi^2\\lambda^2$.  In particular,\nthe leading contributions to \\Eqref{eq:c_j} are of ${\\cal\n  O}(\\xi^2\\lambda^2)$ and read\n\\begin{align}\n G_1^0&=-G_2^0=\\frac{32}{315}\\xi_1\\xi_2\\lambda^2i\\lambda\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\nonumber\\\\\n G_3^0&=-\\frac{2}{45}\\left(4\\xi_1^2\\lambda^2+7\\xi_2^2\\lambda^2\\right)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\nonumber\\\\\n G_4^0&=G_3^0\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right), \\nonumber\\\\\n G_5^0&=-\\frac{8}{105}\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(\\xi_1^2\\lambda^2+\\xi_2^2\\lambda^2)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\label{eq:G_j^0}\\\\\n\\intertext{and}\n G_1^{\\pm 1}&=G_2^{\\pm 1}=\\pm\\frac{i}{15}\\,\\xi_1\\xi_2\\lambda^2\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\nonumber\\\\\n G_3^{\\pm 1}&=\\frac{1}{45}\\left(4\\xi_1^2\\lambda^2-7\\xi_2^2\\lambda^2\\right)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\nonumber\\\\\n G_4^{\\pm 1}&=-G_3^{\\pm 1}\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right), \\nonumber\\\\\n G_5^{\\pm 1}&=\\frac{4}{105}\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(\\xi_1^2\\lambda^2-\\xi_2^2\\lambda^2)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\label{eq:G_j^1}\n\\end{align}\nwhereas the leading contributions to $G^l_j$ with $|l|\\geq2$ scale as\n$\\sim(\\xi^2\\lambda^2)^{|l|}$ and thus are at least of ${\\cal\n  O}((\\xi^2\\lambda^2)^2)$.\nPlugging these terms into Eqs.~\\eqref{eq:PIstructure}-\\eqref{eq:c_j},\nwe obtain a compact approximation to the photon polarization tensor\nfor a generic, elliptically polarized plane wave background in the\nparameter regime where\n$\\{\\xi^2\\lambda^2,|\\lambda|,|\\frac{k_1k_2}{4m^2}|\\}\\ll1$.  The above\nfindings imply that the infinite sum in \\Eqref{eq:c_j} at ${\\cal\n  O}(\\xi^2\\lambda^2)$ receives contributions only for $l=\\pm1$.\nHence, the persistent inelastic interactions can be associated with\nthe absorption\/release of just two laser photons.\n\nAs a particular example, we consider\nthe special case of an incoming on-shell photon with $k_1^\\mu=\\omega_1(1,\\vec{k}_1\/|\\vec{k}_1|)$, fulfilling\n$k_1^2=0$. In this case, the parameter $\\lambda$ can be written as\n$\\lambda\\to\\frac{\\omega\\omega_1}{2m^2}\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)$,\nsuch that\n\\begin{equation}\n \\lambda^2\\xi^2 \\quad\\to\\quad \\Bigl(\\frac{e\\mathfrak{E}}{m^2}\\Bigr)^2\\frac{\\omega_1^2}{4m^2}\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)^2,\n\\end{equation}\nwhere we employed the shorthand notation\n$\\mathfrak{E}^2\\in\\{\\mathfrak{E}^2_1,\\mathfrak{E}^2_2,\\mathfrak{E}_1\\mathfrak{E}_2\\}$. Obviously,\nthe dependence on the frequency $\\omega$ of the plane wave background\ndrops out and the combination $\\lambda^2\\xi^2$ becomes $\\omega$\nindependent.  Correspondingly, the photon polarization tensor at ${\\cal O}(\\xi^2\\lambda^2)$ in the\nlimit $\\omega\\to0$ is obtained straightforwardly in this case:\nIt is given by \\Eqref{eq:PIstructure} with $c_5=0$ [see the remarks\n  below \\Eqref{eq:gs}], and the projectors~\\eqref{eq:Lambdas} and\nother coefficients~\\eqref{eq:c_j} specialized to $\\omega=0$.\nObviously, it only features an elastic contribution and its\ncoefficients [cf. \\Eqref{eq:c_j}] are given by\n\\begin{equation}\n c_j\\quad\\to\\quad i(2\\pi)^4m^2\\frac{\\alpha}{\\pi}\\delta(k_1-k_2)\\tilde G_j, \\label{eq:c_jCrossed}\n\\end{equation}\nwith $\\tilde G_j\\equiv\\bigl[G_j^0+G_j^{+1}+G_j^{-1}\\bigr]\\big|_{\\omega=0}$ and $j\\in\\{1,\\ldots,4\\}$. \nInserting the explicit expressions from Eqs.~\\eqref{eq:G_j^0} and \\eqref{eq:G_j^1} into \\Eqref{eq:c_jCrossed}, we obtain $\\tilde G_1=\\tilde G_2=0$ as well as $\\tilde G_3=-\\frac{28}{45}\\xi_2^2\\lambda^2$ and $\\tilde G_4=-\\frac{16}{45}\\xi_2^2\\lambda^2$.\nAs expected the dependence on $\\xi_1$ completely drops out and the polarization tensor in this limit eventually depends only on the single field strength $\\mathfrak{E}_2$. Recall that the electromagnetic field components follow by differentiations of the four-vector potential~\\eqref{eq:background}, which explains why the electric field $\\mathfrak{E}_2$, persists even though it comes along with a factor of $\\sin(\\kappa x)$ in \\Eqref{eq:background}.\nPutting everything together, we finally obtain\n\\begin{equation}\n \\Pi^{\\mu\\nu}(k_1,k_2) \\ \\ \\to\\ \\ -i(2\\pi)^4\\delta(k_1-k_2)\\frac{\\alpha}{\\pi}\\,\\omega_1^2\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)^2\\Bigl(\\frac{e\\mathfrak{E}_2}{m^2}\\Bigr)^2\\biggl[ \\frac{7}{45}\\Lambda_1^\\mu\\Lambda_1^\\nu+\\frac{4}{45}\\Lambda_2^\\mu\\Lambda_2^\\nu\\biggr]. \\label{eq:PIstructureCrossed}\n\\end{equation}\nThis reproduces the photon polarization tensor for constant crossed fields at ${\\cal\n  O}\\bigl((\\frac{e\\mathfrak{E}}{m^2})^2\\bigr)$ and on-the-light-cone\ndynamics \\cite{narozhnyi:1968,ritus:1972}.\n\n\n\\subsection{Laser photon merging} \\label{subsec:photonmerging}\n\nFor a given laser photon polarization, i.e., a particular\nchoice of the monochromatic plane wave\nbackground~\\eqref{eq:background}, the photon merging amplitude\ndepends on both the explicit expression for the field inhomogeneity\nand the polarization state $\\epsilon_\\mu^{*(p)}(k)$ of the outgoing\nphoton, with $p$ labeling the two transverse photon polarizations.  It\nis given by \\cite{Yakovlev:1967}\n\\begin{equation}\n {\\cal M}^{(p)}(k)=\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\\int\\frac{{\\rm d}^4q}{(2\\pi)^4}\\,\\Pi^{\\mu\\nu}(k,q)A_\\nu(q)\\,, \\label{eq:M}\n\\end{equation}\nwhere $A_\\nu(q)=\\int_{x}\\,{\\rm e}^{-ixq}A_\\nu(x)$ is the Fourier\ntransform of the gauge field representing the inhomogeneous electromagnetic field \nin position space; the star symbol $^*$ denotes complex\nconjugation.  The explicit expression for $k^\\mu=(k^0,\\vec{k})$\ndepends of course on the specific merging process to be\nconsidered. For the merging of $2n$ laser photons of frequency\n$\\omega$ in a static field, momentum conservation and the fact that\nthe outgoing photon is real and propagates on the light cone imply\nthat $k^0=|\\vec{k}|=2n\\omega$.  Moreover, given this condition, the\ncoefficient $c_5$ in \\Eqref{eq:PIstructure} vanishes [cf. below\n  \\Eqref{eq:gs}], such that the tensor structure of\n$\\Pi^{\\mu\\nu}(q,k)$ can be expressed solely in terms of\n$\\Lambda_1^\\mu$ and $\\Lambda_2^\\mu$.\n\nAs outlined in detail above, in this article we limit ourselves to the\nstudy of the merging process in a static magnetic field.  We consider\nfield inhomogeneities of the form $\\vec{B}(x)=B(x)\\vec{e}_B$, such\nthat the direction of the magnetic field $\\vec{e}_B$ is fixed globally\nand only its amplitude is varied.  More specifically, we set\n$\\vec{e}_B=\\vec{e}_{\\rm z}$ and focus on a one dimensional spatial\ninhomogeneity in $\\rm x$ direction, i.e., $B(x)\\to B({\\rm x})$, such\nthat $\\vec{\\nabla}B({\\rm x})\\sim\\vec{e}_{\\rm x}$.  The wave vector of\nthe laser photons is assumed to be $\\vec{\\kappa}={\\kappa}_{\\rm\n  x}\\vec{e}_{\\rm x}+{\\kappa}_{\\rm y}\\vec{e}_{\\rm y}$, i.e., the\nincident laser photons do not have a momentum component parallel to\nthe magnetic field (cf. Fig~\\ref{fig:perspective}). Even if they\nhad, such a component would not be affected due to translational\ninvariance along the ${\\rm z}$ direction.\n\nUtilizing $\\kappa^2=0$ it is convenient to introduce the angle parameter\n$\\theta\\in[0\\ldots\\frac{\\pi}{2}]$ and write\n$\\kappa^\\mu=\\omega(1,\\cos\\theta,\\sin\\theta,0)$ with $\\omega>0$.\nCorrespondingly, the orthogonality relations\n$a_1\\kappa=a_2\\kappa=a_1a_2=0$ imply that the parametrization of the\northonormal vectors $a_1^\\mu$ and $a_2^\\mu$ just requires one\nadditional angle parameter which we denote by\n$\\varphi\\in[0\\ldots2\\pi)$.  We write\n\\begin{align}\n a_1^\\mu&=(0,-\\sin\\theta\\cos\\varphi,\\cos\\theta\\cos\\varphi,-\\sin\\varphi), \\nonumber\\\\\n a_2^\\mu&=(0,-\\sin\\theta\\sin\\varphi,\\cos\\theta\\sin\\varphi,\\cos\\varphi), \\label{eq:a_12}\n\\end{align}\ni.e., our conventions are such that the spatial components of\n$\\kappa^\\mu$, $a_1^\\mu$ and $a_2^\\mu$ form a right-handed trihedron\n(cf. Fig~\\ref{fig:perspective}).  The choice of $\\theta$ fixes the\npropagation direction $\\vec{\\kappa}$ of the incident photons relative\nto the inhomogeneity, while $\\varphi$ controls the orientation of the\nvectors $\\vec a_1$ and $\\vec a_2$ spanning the spatial subspace\ntransverse to $\\vec{\\kappa}$.\n\nA convenient choice for the four-vector potential giving rise to a\nmagnetic field of the desired type is\n\\begin{equation}\n A^\\mu(x)=A({\\rm x})e^\\mu_{\\rm y}, \\quad\\text{with}\\quad A({\\rm x})=\\int^{\\rm x}{\\rm d}{\\rm x}'\\,B({\\rm x}'), \\label{eq:Ax}\n\\end{equation}\nwhere we have defined $e^\\mu_{\\rm y}\\equiv(0,\\vec{e}_{\\rm y})$.\nThe lower limit of the integral is left unspecified as it does\nnot have any observable consequences and thus can be chosen\narbitrarily.  Finally, a Fourier transform of \\Eqref{eq:Ax} yields the\nmomentum space representation of the four-vector potential as needed\nin \\Eqref{eq:M},\n\\begin{equation}\n A^\\mu(q)=(2\\pi)^3\\delta(q_0)\\delta(q_{\\rm y})\\delta(q_{\\rm z})A(q_{\\rm x})e^\\mu_{\\rm y}, \\quad\\text{with}\\quad A(q_{\\rm x})=\\int_{-\\infty}^\\infty{\\rm d}{\\rm x}\\,{\\rm e}^{-i{\\rm x}q_{\\rm x}}\\,A({\\rm x}) . \\label{eq:Ak}\n\\end{equation}\n\nPlugging this expression into \\Eqref{eq:M} and introducing $\\bar q^\\mu\\equiv(0,q_{\\rm x}\\vec{e}_{\\rm x})$, the photon merging amplitude can be simplified significantly and reads\n\\begin{equation}\n {\\cal M}^{(p)}(k)=\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\\int\\frac{{\\rm d}q_{\\rm x}}{2\\pi}\\,\\Pi^{\\mu2}(k,\\bar q)\\,A(q_{\\rm x})\\,. \\label{eq:M1}\n\\end{equation}\nSubstituting $k_2\\to\\bar q$ into the expressions for $\\Lambda_1^\\mu$ and $\\Lambda_2^\\mu$ in\n\\Eqref{eq:Lambdas} we obtain together with \\Eqref{eq:a_12}\n\\begin{align}\n \\Lambda_1^\\mu&=\\bigl(\\tan\\theta\\cos\\varphi,0,\\tfrac{\\cos\\varphi}{\\cos\\theta},-\\sin\\varphi\\bigr), \\nonumber\\\\\n \\Lambda_2^\\mu&=\\bigl(\\tan\\theta\\sin\\varphi,0,\\tfrac{\\sin\\varphi}{\\cos\\theta},\\cos\\varphi\\bigr). \\label{eq:lambda12}\n\\end{align}\nAnalogously to \\Eqref{eq:c_j}, we write \n\\begin{equation}\n \\Pi^{\\mu2}(k,\\bar q)=(2\\pi)^4\\sum_{l\\in\\mathbb{Z}}\\delta(k-\\bar q-2l\\kappa)\\Pi^{\\mu2}_l(k,\\bar q), \\label{eq:Pi2nu}\n\\end{equation}\nwhere the explicit representation\n\\begin{equation}\n \\Pi^{\\mu2}_l= im^2\\frac{\\alpha}{\\pi}\\frac{1}{\\cos\\theta}\\Bigl[\\Lambda_1^\\mu\\,(G_1^l\\sin\\varphi+G_3^l\\cos\\varphi)+\\Lambda_2^\\mu\\,(G_2^l\\cos\\varphi+G_4^l\\sin\\varphi)\\Bigr]\n\\end{equation}\nmakes use of \\Eqref{eq:lambda12}.  Using \\Eqref{eq:Pi2nu} in\n\\Eqref{eq:M1}, the residual integration over $q_{\\rm x}$ can be\nperformed and we obtain\n\\begin{equation}\n {\\cal M}^{(p)}(k)=(2\\pi)^3 \\delta(k_{\\rm z})\\sum_{l\\in\\mathbb{Z}}\\delta(k^0-2l\\omega)\\delta(k_{\\rm y}-2l\\omega\\sin\\theta)\\,\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\n\\Pi^{\\mu2}_l(k,\\tilde k)A(\\tilde k_{\\rm x})\\,, \\label{eq:M2}\n\\end{equation}\nwith $\\tilde k^\\mu\\equiv(0,(k_{\\rm x}-2l\\omega\\cos\\theta)\\vec{e}_{\\rm x})$.\n\nTaking into account the fact that the outgoing photon has positive\nenergy ($k^0>0$) and propagates on the light cone ($k_\\mu k^\\mu=0$),\nand also because of the $\\delta$ functions for the ${\\rm y}$ and ${\\rm z}$\nmomentum components, we identify $k^0\\equiv2l\\omega$ and\nrewrite the $\\delta$ function implementing energy conservation in\n\\Eqref{eq:M2} as follows,\n\\begin{equation}\n \\delta(k^0-2l\\omega)\\ \\to\\ \\delta_{l0}\\,\\delta(k_{\\rm x})+ \\Theta(l+0^+)\\,\\frac{1}{\\cos\\theta}\\Bigl[\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)+\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)\\Bigr]\\,, \\label{eq:delta}\n\\end{equation}\nwhere $\\Theta(.)$ is the Heaviside function.\nCorrespondingly, we have\n\\begin{multline}\n {\\cal M}^{(p)}(k)=(2\\pi)^3 \\delta(k_{\\rm z})\\sum_{l=1}^{\\infty}\\frac{1}{\\cos\\theta}\\Bigl[\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)+\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)\\Bigr] \\\\ \\times\\delta(k_{\\rm y}-2l\\omega\\sin\\theta)\\,\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{4l\\omega}}\\,\\Pi^{\\mu2}_l(k,\\tilde k)A(\\tilde k_{\\rm x})\\,, \\label{eq:M3}\n\\end{multline}\nwith $k^\\mu=(2l\\omega,k_{\\rm x},k_{\\rm y},0)$, where we have\nmade use of the fact that the $l=0$ contribution vanishes: it scales\n$\\sim\\delta(\\vec{k})\\,\\frac{\\Pi^{\\mu2}_l(k,\\tilde\n  k)}{\\sqrt{4l\\omega}}\\sim\\delta(\\vec{k})\\,l^{3\/2}\\to0$ [cf. also\n  \\Eqref{eq:expansionparameters->} below].\n\nWhen adapted to the particular kinematics in \\Eqref{eq:M3} (cf. the arguments of the photon polarization tensor), the dimensionless parameters $\\frac{k_1k_2}{4m^2}$, $\\lambda$ and $\\xi^2\\lambda^2$\ngoverning the expansion of the photon polarization tensor performed in Sec.~\\ref{subsec:Pi} all vanish for the contribution $\\sim\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)$.\nFor the contribution $\\sim\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)$ they are non-zero and read\n\\begin{align}\n \\frac{k_1k_2}{4m^2}\\equiv \\frac{k \\tilde{k}}{4m^2} &\\quad\\to\\quad\\frac{1}{2}\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)^2, \\nonumber\\\\\n \\lambda&\\quad\\to\\quad\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)\\frac{\\omega\\cos\\theta}{m}, \\nonumber\\\\\n \\xi^2\\lambda^2&\\quad\\to\\quad\\left(\\frac{e\\mathfrak{E}}{m^2}\\right)^2\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)^2\\cos^2\\theta. \\label{eq:expansionparameters->}\n\\end{align}\nNeglecting higher-order contributions of ${\\cal O}(\\frac{k_1k_2}{4m^2})\\sim{\\cal O}(\\lambda)\\sim{\\cal O}(\\frac{\\omega^2}{m^2})$, our result will of course be fully governed by the remaining parameters $\\xi_1^2\\lambda^2$, $\\xi_2^2\\lambda^2$ and $\\xi_1\\xi_2\\lambda^2$.\n\nAs a result, the number of merged photons with four wave-vector $k_f^\\mu$\nand polarization $p$ according to Fermi's golden rule is given by \n\\begin{equation}\n {\\cal N}^{(p)}(k_f)=\\int\\frac{d^3k}{(2\\pi)^3}\\,\\bigl|{\\cal M}^{(p)}(k)\\bigr|^2= TL_{\\rm y}L_{\\rm z}\\sum_{l=1}^{\\infty}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_l(k_f,\\tilde k_f)A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4l\\omega\\cos\\theta}\\,, \\label{eq:M4}\n\\end{equation}\nwith $k^\\mu_f=2l\\omega(1,-\\cos\\theta,\\sin\\theta,0)$, i.e., the\noutgoing photon of energy $2l\\omega$ propagates in\n$(-\\cos\\theta,\\sin\\theta,0)$ direction.  Moreover, $\\tilde\nk^\\mu_f=-4l\\omega\\cos\\theta(0,\\vec{e}_{\\rm x})$ encodes the momentum\ntransfer from the field inhomogeneity, $T$ is the interaction time and\n$L_{\\rm y}L_{\\rm z}$ is the interaction area transverse to the\ninhomogeneity.  The total number of merged photons is\n\\begin{equation}\n {\\cal N}(k_f)=\\sum_{p}{\\cal N}^{(p)}(k_f).  \\label{eq:N}\n\\end{equation}\nObviously the dominant contribution is due to the merging of just two\nlaser photons, $l=1$, as higher photon processes are suppressed\nby at least a factor of $\\xi^2\\lambda^2$.  Correspondingly,\n\\Eqref{eq:M3} can be written as\n\\begin{equation}\n {\\cal N}^{(p)}(k_f)= TL_{\\rm y}L_{\\rm z}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1(k_f,\\tilde k_f)A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4\\omega\\cos\\theta}\\bigl(1+{\\cal O}(\\tfrac{e^2{\\mathfrak E}^2}{m^4}\\tfrac{\\omega^2}{m^2})\\bigr)\\,. \\label{eq:M4a}\n\\end{equation}\nWe emphasize that the terms written out explicitly in \\Eqref{eq:M4a} account for\nthe entire two-photon merging process.  We approximate the infinite\nsum in \\Eqref{eq:M4} by its contribution for $l=1$, and thereby\nneglect merging processes of $2l$ laser photons with $l>1$.\n\nEmploying the substitutions $\\varphi\\to\\varphi'$ and $\\theta\\to\\pi-\\theta$ in \\Eqref{eq:a_12}, we introduce the following two vectors\n\\begin{align}\n \\epsilon^{(1)\\mu}(k_f)&=(0,-\\sin\\theta\\cos\\varphi',-\\cos\\theta\\cos\\varphi',-\\sin\\varphi'), \\nonumber\\\\\n \\epsilon^{(2)\\mu}(k_f)&=(0,-\\sin\\theta\\sin\\varphi',-\\cos\\theta\\sin\\varphi',\\cos\\varphi'), \\label{eq:epsilons}\n\\end{align}\nwith $\\varphi'\\in[0\\ldots2\\pi)$ fixed, to span the subspace transverse to the wave-vector $\\vec{k}_f$ of the merged photon.\nThe two polarization degrees of freedom of the outgoing photon are then conveniently expressed in terms of the vectors $\\epsilon^{(p)\\mu}(k_f)$, with $p\\in\\{1,2\\}$, representing linear polarization states in the particular basis characterized by a particular choice of $\\varphi'$.\nPolarizations other than linear can be obtained through linear combinations of the vectors~\\eqref{eq:epsilons}.\n\nWe are now in a position to provide the explicit expressions of\nthe polarization tensor in \\Eqref{eq:M4} contracted with a given\npolarization vector of the outgoing photon, which read\n\\begin{multline}\n \\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_l\n= im^2\\frac{\\alpha}{\\pi}\\frac{1}{2\\cos\\theta}\\Bigl[\\sin\\varphi'\\,(G_1^l-G_2^l)-\\sin(\\varphi'+2\\varphi)(G_1^l+G_2^l)  \\\\\n-\\cos(\\varphi'+2\\varphi)\\,(G_3^l-G_4^l) -\\cos\\varphi'\\,(G_3^l+G_4^l)\\Bigr], \\label{eq:epsilonPi}\n\\end{multline}\nand\n\\begin{equation}\n \\epsilon_\\mu^{*(2)}(k_f)\\Pi^{\\mu2}_l=\\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_l\\big|_{\\varphi'\\,\\to\\,\\varphi'-\\frac{\\pi}{2}}\\,. \\label{eq:epsilonPi2}\n\\end{equation}\nIntroducing the dimensionless field strengths $\\varepsilon_i\\equiv\\frac{e\\mathfrak{E}_i}{m^2}$ with $i\\in\\{1,2\\}$, \nin particular the $l=1$ contribution to \\Eqref{eq:epsilonPi} can be written as\n\\begin{multline}\n \\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_{1}\n=i(\\omega \\cos\\theta)^2\\frac{\\alpha}{\\pi}\\frac{2}{15}\\cos\\theta\\Bigl[-2i\\sin(\\varphi'+2\\varphi)\\,\\varepsilon_1\\varepsilon_2  \\\\\n+\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^2+\\varepsilon_2^2) -\\frac{11}{3}\\cos\\varphi'\\,(\\varepsilon_1^2-\\varepsilon_2^2)\\Bigr]\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr), \\label{eq:epsilonPi_l=1}\n\\end{multline}\nwhere we have made use of Eqs.~\\eqref{eq:G_j^1} and\n\\eqref{eq:expansionparameters->}.\n\nIf $A(\\tilde k_{f,{\\rm x}})$ is either purely real or imaginary\nvalued, which is true for the field inhomogeneities symmetric in\n  $\\rm x$ to be considered below, the modulus squared can be split\nand \\Eqref{eq:M4a} be represented as follows,\n\\begin{equation}\n {\\cal N}^{(p)}(k_f)= TL_{\\rm y}L_{\\rm\n   z}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1(k_f,\\tilde\n   k_f)|^2\\,|A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4\\omega\\cos\\theta}\\bigl(1+{\\cal \n   O}(\\tfrac{e^2{\\mathfrak\n     E}^2}{m^4}\\tfrac{\\omega^2}{m^2})\\bigr)\\,. \\label{eq:M4b}\n\\end{equation}\n\nThe modulus squared of \\Eqref{eq:epsilonPi_l=1} is obtained straightforwardly and reads\n\\begin{multline}\n \\bigl|\\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_1\\bigr|^2\n=(\\omega \\cos\\theta)^4\\frac{\\alpha^2}{\\pi^2}\\frac{4}{225}\\cos^2\\theta\\Bigl\\{4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{22}{3}\\cos\\varphi'\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4) \\\\\n+\\Bigl[\\frac{121}{9}\\cos^2\\varphi'+\\cos^2(\\varphi'+2\\varphi)\\Bigr](\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr\\}\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n\\label{eq:modpisquared_1}\n\\end{multline}\nwhile the analogous expression for the other polarization mode ($p=2$) follows from \\Eqref{eq:epsilonPi2}.\n\nAiming at the total number of merged photons in the polarization basis characterized by a particular choice of $\\varphi'$, we have to add the moduli squared corresponding to the two different polarization states [cf. Eqs.~\\eqref{eq:N} and \\eqref{eq:M4b}].\nThis results in\n\\begin{multline}\n \\sum_{p=1}^2\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1\\bigr|^2\n=(\\omega \\cos\\theta)^4\\frac{\\alpha^2}{\\pi^2}\\frac{8}{225}\\cos^2\\theta \\\\\n\\times\\Bigl[4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{11}{3}\\cos(2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4)\n+\\frac{65}{9}(\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr]\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n\\label{eq:modpisquared}\n\\end{multline}\nwhich is completely independent of the choice of $\\varphi'$, as it should.\nNoteworthily, in case of circularly polarized incident\nlaser photons for which $\\xi_1=\\xi_2$ and thus\n$\\varepsilon_1=\\varepsilon_2$, the contributions for both polarization\nmodes individually become independent of $\\varphi$ and $\\varphi'$;\ncf. \\Eqref{eq:modpisquared_1}. Equation \\eqref{eq:modpisquared}\nupon insertion into \\Eqref{eq:N} and accounting for the prefactors\ndisplayed in \\Eqref{eq:M4a} represent a central result of this work.\n\nSubsequently, we assume the probe laser to deliver incident laser\npulses of duration $\\tau$, entering under an angle $\\theta$ and\nfeaturing a circular transverse beam profile.  The longitudinal\nevolution of the probe laser pulses follows the envelope of a Gaussian\nbeam, with beam waist right at the intersection with the field\ninhomogeneity.  We denote the transverse cross-section area at the\nbeam waist by $\\sigma$.  Correspondingly, the transversal area $L_{\\rm\n  y}L_{\\rm z}$ can be identified with the intersection area of such a\nbeam profile with the ${\\rm y}$--${\\rm z}$ plane, i.e., $L_{\\rm y}L_{\\rm\n  z}=\\frac{\\sigma}{\\cos\\theta}$ (cf. Fig.~\\ref{fig:yzSchnitt}).\nAssuming that the magnetic field inhomogeneity is long-lived as compared to the\npulse duration $\\tau$ of the probe laser, it is reasonable to consider\n$\\tau$ as a measure of the interaction time $T$, and set $T=\\tau$.\nHence, we can make use of the following substitution,\n\\begin{equation}\n TL_{\\rm y}L_{\\rm z} \\quad \\to \\quad \\frac{\\sigma\\tau}{\\cos\\theta}\\,. \\label{eq:subst}\n\\end{equation}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.67\\textwidth]{Gaussbeam-beschr-crop} \n\\caption{Sketch of the envelope of a Gaussian beam intersecting the ${\\rm y}$--${\\rm z}$ plane in\n the vicinity of its waist under an angle of $\\theta$ (cf. also\n Fig.~\\ref{fig:perspective}).  Given that the transverse cross-section\n area of the Gaussian beam at the beam waist is a circle of area\n $\\sigma$, the intersection area is an ellipse with area $\\frac{\\sigma}{\\cos\\theta}$.}\n\\label{fig:yzSchnitt}\n\\end{figure}\n\n\n\\section{Results and Discussion} \\label{seq:Ex+Res}\n\nLet us now consider explicit examples of localized magnetic field inhomogeneities which can be tackled analytically.\nWe limit ourselves to two elementary shapes, characterized by just two parameters, namely an amplitude $B$ and a typical extension $w$.\nFor a Lorentz profile characterized by its full width at half maximum (FWHM),\n\\begin{equation}\n B({\\rm x})=\\frac{B}{1+(\\frac{2{\\rm x}}{w})^2}, \\label{eq:Lorentz}\n\\end{equation}\nthe associated gauge field in position space can be determined by \\Eqref{eq:Ax}. We obtain\n\\begin{equation}\n A({\\rm x})=\\frac{Bw}{2}\\arctan\\!\\left(\\frac{2{\\rm x}}{w}\\right),\n\\end{equation}\nand Fourier transforming to momentum space via \\Eqref{eq:Ak},\n\\begin{equation}\n A(q_{\\rm x})=-i\\frac{\\pi Bw}{2q_{\\rm x}}\\,{\\rm e}^{-\\frac{|q_{\\rm x}|w}{2}}. \\label{eq:inh1}\n\\end{equation}\n\n\nAnalogously, for a Gaussian type inhomogeneity characterized by its full width at $1\/{\\rm e}$ of its maximum,\n\\begin{equation}\n B({\\rm x})=B\\,{\\rm e}^{-\\left(\\frac{2{\\rm x}}{w}\\right)^2}, \\label{eq:Gauss}\n\\end{equation}\nwe obtain\n\\begin{equation}\n A({\\rm x})=\\frac{\\sqrt{\\pi}Bw}{4}\\,{\\rm erf}\\!\\left(\\frac{2{\\rm x}}{w}\\right),\n\\end{equation}\nwhere ${\\rm erf}(.)$ denotes the error function, and finally\n\\begin{equation}\n A(q_{\\rm x})=-i\\frac{\\sqrt{\\pi} Bw}{2q_{\\rm x}}\\,{\\rm e}^{-\\left(\\frac{q_{\\rm x}w}{4}\\right)^2}. \\label{eq:inh2}\n\\end{equation}\nEquations~\\eqref{eq:inh1} and \\eqref{eq:inh2} share an overall prefactor $\\sim(-i\\frac{\\sqrt{\\pi} Bw}{2q_{\\rm x}})$, but differ in the exponential decay.\nFor the Lorentz profile the decay is linear in $|q_{\\rm x}|w$, while for the Gaussian inhomogeneity it is quadratic in this dimensionless parameter.\n\nIt is now straightforward to derive the number of merged laser photons, \\Eqref{eq:M4b}, for these inhomogeneities.\nThe number of outgoing merged laser photons with polarization\n$p=1$ and energy $2\\omega$ reads [cf. Eqs.~\\eqref{eq:M4b} and\n  \\eqref{eq:subst}]\n\\begin{multline}\n {\\cal N}^{(1)}(k_f)\n= w\\sigma\\tau(eB)^2(\\omega w)\\frac{\\alpha\\cos^2\\theta}{57600\\,\\pi}\\left\\{\\begin{array}{c}\n            {\\rm e}^{-(4\\omega\\cos\\theta) w} \\\\\n            \\frac{1}{\\pi}\\,{\\rm e}^{-\\frac{1}{8}(4\\omega\\cos\\theta)^2w^2}\n        \\end{array}\\right\\} \\\\\n        \\times\\Bigl\\{4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{22}{3}\\cos\\varphi'\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4) \\\\\n+\\Bigl[\\frac{121}{9}\\cos^2\\varphi'+\\cos^2(\\varphi'+2\\varphi)\\Bigr](\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr\\}\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n \\label{eq:Np1}\n\\end{multline}\nwhere the upper line in braces is the result for the Lorentz~\\eqref{eq:Lorentz}\nand the lower line that for the Gaussian~\\eqref{eq:Gauss} profile, and\n\\begin{equation}\n {\\cal N}^{(2)}(k_f)={\\cal N}^{(1)}(k_f)\\big|_{\\varphi'\\,\\to\\,\\varphi'-\\frac{\\pi}{2}}\\,. \\label{eq:Np2}\n\\end{equation}\nAs the results for the Lorentz and Gaussian inhomogeneities -- apart\nfrom the different exponential behavior -- are of very similar\nstructure, we find it convenient to adopt the two-component notation\nemployed in \\Eqref{eq:Np1} in the remainder of this paper.\nEquation \\eqref{eq:Np1} exhibits several characteristic\ndependencies on the involved parameters: as to be expected from the\nunderlying Feynman diagram, the leading order effect is proportional\nto the square of the plane wave intensity, i.e., $\\sim\n\\mathfrak{E}^4$, and to the square of the magnetic field $\\sim B^2$.\nIn particular the latter dependence represents a comparatively\nstrong increase of the effect with an enhancement of the peak\nmagnetic background field. Other typical nonlinear phenomena such\nas photon scattering off a magnetic field $\\sim B^4$ or photon splitting\n$\\sim B^6$ are more strongly suppressed since the $B$ field scale is\nmeasured in terms of the electron mass scale. On the other hand, the\ninhomogeneous field has to provide the necessary momentum transfer\n$\\sim 4\\omega\\cos\\theta$, and the effect is exponentially damped with $\\sim(4\\omega\\cos\\theta)w$.\n\nIn this respect, it is instructive to compare these expressions\nwith the number of photons experiencing quantum reflection\n\\cite{Gies:2013yxa} for the very same conditions, i.e., for incident\nphotons of the same energy, angle of incidence and polarization, and\nexactly the same field inhomogeneities as in Eqs.~\\eqref{eq:Lorentz}\nand \\eqref{eq:Gauss}.\n\nFor completeness, we note that in Ref.~\\cite{Gies:2013yxa}, the field inhomogeneity was not accounted for exactly in the sense that the photon polarization tensor was evaluated {\\it a priori} in the presence of the magnetic field inhomogeneity,\nbut rather the inhomogeneity was built in {\\it a posteriori} by resorting to the result for a constant magnetic background field and using the constant-field expressions locally.\nAs argued in detail in~\\cite{Gies:2013yxa}, such an approach is\njustifiable for inhomogeneities whose typical scale of variation $w$\nis much larger than the Compton wavelength $\\lambda_c$ of the charged\nvirtual particles, i.e., $w\\gg \\lambda_c$.  Particularly in quantum\nelectrodynamics (QED), where the virtual particles are electrons,\n$\\lambda_c\\approx2\\cdot10^{-6}{\\rm eV}^{-1}\\approx3.9\\cdot10^{-13}{\\rm\n  m}$, many field inhomogeneities available in the laboratory can be\ndealt with along these lines.\n\nReference~\\cite{Gies:2013yxa} identifies two situations for which the\ncalculations become particularly simple, corresponding to\nspecial alignments of the incident photons' wave vector $\\vec{k}$ and\npolarization plane, the magnetic field $\\vec{B}$, and the direction of\nthe inhomogeneity $\\vec{\\nabla}B$.\nThe one reconcilable with incident photons of four wave-vector\n$\\kappa^\\mu=\\omega(1,\\cos\\theta,\\sin\\theta,0)$ and\n$\\vec{B}\\sim\\vec{e}_{\\rm z}$ is that with polarization vector in the\nplane spanned by $\\vec{\\kappa}$ and $\\vec{B}$, labeled by $\\parallel$\nin~\\cite{Gies:2013yxa}.  To bring the $\\parallel$ case\nof~\\cite{Gies:2013yxa} and the merging scenario discussed here into\nfull kinematic agreement, we specialize the quantum reflection formulae to\n$\\varangle(\\vec{\\kappa},\\vec{B})=\\frac{\\pi}{2}$ and set\n$\\varphi=\\varphi'=0$, $\\varepsilon_1=0$ and\n$\\varepsilon_2=\\frac{e\\mathfrak{E}}{m^2}$ in Eqs.~\\eqref{eq:Np1} and\n\\eqref{eq:Np2}, i.e., we specialize to incident laser photons\npolarized linearly along ${\\rm z}$, and look for induced outgoing\nphotons in the same polarization basis.  Incidentally, it can be shown\nstraightforwardly that the polarization direction is conserved under\nthese circumstances for quantum reflection (cf. \\cite{Gies:2013yxa}),\ni.e., the quantum reflected photons are still polarized along ${\\rm z}$,\nwhile for laser photon merging the induced outgoing photons are\npolarized differently, namely their polarization vector lies in the\n${\\rm x}$--${\\rm y}$ plane [cf. \\Eqref{eq:Np-linpol1} below].\n\nThe number of quantum reflected photons ${\\cal N}_{\\rm Qref}$ is\nobtained by multiplying the number of incident probe photons $N_{\\rm\n  probe}$ with the adequate reflection coefficient, given in Eqs.~(27)\nand (29) of \\cite{Gies:2013yxa}.\nIn order to allow for a more direct comparison with the merging result, we first rewrite $N_{\\rm probe}$: The number of incident photons per pulse amounts to the ratio of the pulse energy of the probe laser $\\cal E$ and its frequency $\\omega$, i.e., $N_{\\rm probe}=\\frac{\\cal E}{\\omega}$.\nThe intensity $I_{\\rm probe}$ at the focal spot, which is related to the electric field strength in the focus via $I_{\\rm probe}={\\mathfrak E}^2$, is determined by $I_{\\rm probe}=\\frac{\\cal E}{\\sigma\\tau}$.\nHence, the number of probe photons can be expressed as $N_{\\rm probe}=\\frac{{\\mathfrak E}^2\\sigma\\tau}{\\omega}$, and -- neglecting corrections of ${\\cal O}\\bigl((\\tfrac{eB}{m^2})^6\\bigr)$ -- we finally obtain\n\\begin{equation}\n{\\cal N}_{\\rm Qref}\n= w\\sigma\\tau \\frac{49\\,\\alpha}{129600\\pi}\n\\biggl(\\frac{eB}{m^2}\\biggr)^4(e{\\mathfrak E})^2(\\omega w)\\frac{1}{\\cos^2\\theta}\n\\left\\{\\begin{array}{c}\n        \\frac{1}{4}(1+\\omega w\\cos\\theta)^2\\,{\\rm e}^{-2\\omega w\\cos\\theta}\\\\\n        \\frac{1}{2\\pi}\\,{\\rm e}^{-\\frac{1}{4}(\\omega w\\cos\\theta)^2} \n       \\end{array}\\right\\} . \\label{eq:Np-Qref}\n\\end{equation}\nFor the merging process, Eqs.~\\eqref{eq:Np1} and \\eqref{eq:Np2}, the same choice of parameters results in\n\\begin{equation}\n{\\mathcal N}^{(1)}\n= w\\sigma\\tau\\frac{49\\,\\alpha}{129600\\,\\pi}\\biggl(\\frac{e\\mathfrak{E}}{m^2}\\biggr)^4(eB)^2(\\omega w)  \\cos^2\\theta \\left\\{\\begin{array}{c}\n            {\\rm e}^{-4\\omega w\\cos\\theta} \\\\\n            \\frac{1}{\\pi}\\,{\\rm e}^{-2(\\omega w\\cos\\theta)^2}\n        \\end{array}\\right\\}\n \\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr) , \\label{eq:Np-linpol1}\n\\end{equation}\nwhile ${\\mathcal N}^{(2)}=0$, such that ${\\mathcal N}_{\\rm\n  merg}\\equiv{\\mathcal N}^{(1)}$. Both results exhibit an exponential\nsuppression with exponent $\\sim w\\omega\\cos\\theta=w\\kappa_{\\rm\n  x}$, with $\\kappa_{\\rm x}$ being the momentum component of the\nincident probe photons in the direction of the inhomogeneity\n[cf. above \\Eqref{eq:a_12}]. The suppression is more pronounced for the merging\nprocess.  This can also be understood intuitively by recalling that\nthe momentum transfer from the inhomogeneity is $|2\\kappa_{\\rm x}|$\nfor the process of quantum reflection (cf. \\cite{Gies:2013yxa}), while\nit is twice as large, namely $|4\\kappa_{\\rm x}|$, for the merging of\ntwo laser photons.\n\nAnother important point to notice is that in \\Eqref{eq:Np-Qref} the\ntransition to large incidence angles $\\theta\\lesssim\\pi\/2$ provides a\nconvenient handle to damp the exponential suppression while at the\nsame time increasing the overall prefactor, which scales inversely\nwith $\\cos^2\\theta$.  Conversely, in \\Eqref{eq:Np-linpol1} an\nanalogous increase of the angle of incidence to $\\theta\\lesssim\\pi\/2$\ndiminishes the overall prefactor $\\sim\\cos^2\\theta$.\nThe ratio of Eqs.~\\eqref{eq:Np-linpol1} and \\eqref{eq:Np-Qref} can be derived straightforwardly, and reads\n\\begin{equation}\n\\frac{\\mathcal{N}_{\\rm merg}}{{\\cal N}_{\\rm Qref}}\n\\approx 4\\,\n\\biggl(\\frac{\\mathfrak{E}}{B}\\,\\cos^2\\theta\\biggr)^2 \n\\left\\{\\begin{array}{c}\n        \\frac{1}{(1+\\omega w\\cos\\theta)^{2}}\\,{\\rm e}^{-2\\omega w\\cos\\theta} \\\\\n        \\frac{1}{2}\\,{\\rm e}^{-\\frac{7}{4}(\\omega w\\cos\\theta)^2}\n       \\end{array}\\right\\} . \\label{eq:ratio}\n\\end{equation}\nIt is governed by just two dimensionless quantities, namely the product $\\omega w \\cos\\theta$, measuring the width $w$ of the inhomogeneity in units of the inverse of the momentum component of the incident photons in $\\vec{\\nabla}B$ direction,\nand $\\mathfrak{E}\/B\\,\\cos^2\\theta$, i.e., the ratio of the field strength of the probe relative to that of the pump, augmented by an extra factor of $\\cos^2\\theta$.\n\nIt is now natural to ask for the conditions which have to be met such that photon merging dominates quantum reflection, i.e., $\\mathcal{N}_{\\rm merg}\\geq{\\cal N}_{\\rm Qref}$.\nInserting this condition into \\Eqref{eq:ratio}, we obtain\n\\begin{equation}\n\\frac{\\mathfrak{E}}{B}\\,\\cos^2\\theta\n\\geq \\frac{1}{2}\n\\left\\{\\begin{array}{c}\n        |1+\\omega w\\cos\\theta|\\,{\\rm e}^{\\omega w\\cos\\theta} \\\\\n        \\sqrt{2} \\,{\\rm e}^{\\frac{7}{8}(\\omega w\\cos\\theta)^2}\n       \\end{array}\\right\\}\\geq \\frac{1}{2}\n\\left\\{\\begin{array}{c}\n        1 \\\\\n        \\sqrt{2}\n       \\end{array}\\right\\} , \\label{eq:ratio2}\n\\end{equation}\nwhere we made use of the fact that the expression on the right-hand\nside of the first inequality is bounded from below by its value for\n$\\omega w\\cos\\theta=0$. \nThe latter condition tells us that for the particular set-up\nconsidered here, the yields for photon merging can dominate those for\nquantum reflection only if the quantity $(\\mathfrak{E}\/B)\\cos^2\\theta$ is larger\nthan the numerical bounds given on the rightmost side of\n\\Eqref{eq:ratio2}.\n\nIn Fig.~\\ref{fig:ratio}, we exemplarily set $\\mathfrak{E}=B$\nwhich is a natural choice if all fields are provided by a\nhigh-intensity laser system. We\ninvestigate the implications of the first inequality in\n\\Eqref{eq:ratio2} as a function of $\\theta$ and $\\omega w$.\nObviously, for this choice of the field strengths laser photon merging\ncan only dominate quantum reflection if\n$\\cos\\theta\\geq\\frac{1}{\\sqrt{2}}$ $\\leftrightarrow$\n$\\theta\\leq45^\\circ$ ($\\cos\\theta\\geq 2^{-1\/4}$ $\\leftrightarrow$\n$\\theta\\leq32.7^\\circ$) for a Gaussian (Lorentzian) inhomogeneity.\nQualitatively speaking, the merging process tends to dominate for\nsmall angles of incidence $\\theta$ and small values of $\\omega w$.\nEquation~\\eqref{eq:ratio} implies that this region (in the\n$\\theta$--$\\omega w$ plane) can be enlarged by increasing the ratio of\n$\\mathfrak{E}\/B$.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{140514merging_ratios_E=B_omegawofc} \n\\caption{Choosing $\\mathfrak{E}=B$ as an example,\nwe depict the regimes where photon merging dominates quantum\nreflection and vice versa based on \\Eqref{eq:ratio2}.  Photon\nmerging dominates quantum reflection in the regime in the lower left\nbounded by the blue (solid) and red (dotted) lines for Gauss and\nLorentz type inhomogeneities, respectively.}\n\\label{fig:ratio}\n\\end{figure}\n\nSo far we only focused on the relative\nimportance of the two effects, but did not\nprovide absolute quantitative estimates.  Most obviously, as\nboth effects are suppressed by powers of $\\frac{eB}{m^2}$ and\n$\\frac{e\\mathfrak{E}}{m^2}$ [cf. Eqs.~\\eqref{eq:Np-Qref} and\n  \\eqref{eq:Np-linpol1}], in order to increase them it is preferable\nto enlarge the field strengths as much as possible.\n\nBefore providing some explicit quantitative estimates, let us briefly\ndiscuss the generic features of \\Eqref{eq:Np-linpol1} and confront it\nwith \\Eqref{eq:Np-Qref}.\nConsider the first derivative of the number of merged photons~\\eqref{eq:Np-linpol1} with respect to $w\\cos\\theta$,\n\\begin{equation}\n\\frac{d{\\mathcal N}_{\\rm merg}}{d(w\\cos\\theta)}\n\\approx  \\frac{2\\,{\\mathcal N}_{\\rm merg}}{w\\cos\\theta}\n\\left\\{\\begin{array}{c}\n            1-2\\omega w\\cos\\theta \\\\\n            1-2(\\omega w\\cos\\theta)^2\n        \\end{array}\\right\\}\\stackrel{!}{=}0 \\quad \\to\\quad\n w\\cos\\theta = \\frac{1}{2\\omega}\\left\\{\\begin{array}{c}\n        1\\\\\n        \\sqrt{2}\n       \\end{array}\\right\\}.\n \\label{eq:N_diff}\n\\end{equation}\nTaking into account the sign of the second derivative, we find that \nthe number of outgoing merged photons has a\nmaximum as a function of $w\\cos\\theta$ for the above values and reads \n\\begin{equation}\n{\\mathcal N}_{\\rm merg}\\big|_{\\rm max}\n\\approx \\frac{\\sigma\\tau}{\\omega}\\frac{49\\,\\alpha}{129600\\,\\pi}\\biggl(\\frac{e\\mathfrak{E}}{m^2}\\biggr)^4(eB)^2\\,\\frac{1}{4}\\left\\{\\begin{array}{c}\n            {\\rm e}^{-2} \\\\\n            \\frac{2}{\\pi}\\,{\\rm e}^{-1}\n        \\end{array}\\right\\} . \\label{eq:Nmax}\n\\end{equation}\n\nHence, keeping $w$ fixed, \nthe number of merged photons increases monotonically as a\nfunction of $\\theta$ from its value for $\\theta=0$ until it reaches\na maximum at $\\theta = \\arccos(\\frac{1}{2\\omega w})$ in case of the\nGaussian, and $\\theta = \\arccos(\\frac{1}{\\sqrt{2}\\omega w})$ for the\nLorentz type inhomogeneity. Increasing $\\theta$ even further, it\ndecreases rapidly until it reaches ${\\mathcal N}_{\\rm merg}=0$ at\n$\\theta=90^\\circ$.\n\nConversely, for fixed $\\omega$ the number of quantum reflected photons~\\eqref{eq:Np-Qref} exhibits a monotonic increase throughout the interval from $\\theta=0$ to $\\theta=90^\\circ$. \nActually, ${\\cal N}_{\\rm Qref}$ even diverges for $\\theta\\to90^\\circ$ due to the cosine squared term in its denominator, an unphysical feature which can be attributed to the unphysical limit of an infinitely long interaction of the probe photons and the inhomogeneity at ``grazing incidence'' $\\theta\\to90^\\circ$.\n\nFinally, we provide some rough estimates on the numbers of merged and\nquantum reflected photons attainable in an all optical pump--probe\nexperiment based on high-intensity lasers.  Even though we have just\nfocused on a one-dimensional field inhomogeneity, as in\n\\cite{Gies:2013yxa} we exemplarily adopt the design parameters of the\ntwo high-intensity laser systems to become available in Jena \\cite{Jena}:\nJETI~200 \\cite{JETI200} ($\\lambda=800{\\rm nm}\\approx4.06{\\rm\n  eV}^{-1}$, ${\\cal E}=4{\\rm J}\\approx2.50\\cdot10^{19}{\\rm eV}$,\n$\\tau=20{\\rm fs}\\approx30.4{\\rm eV}^{-1}$) as probe, and POLARIS\n\\cite{POLARIS} ($\\lambda_{\\rm pump}=1030{\\rm nm}\\approx5.22{\\rm\n  eV}^{-1}$, ${\\cal E}_{\\rm pump}=150{\\rm\n  J}\\approx9.36\\cdot10^{20}{\\rm eV}$, $\\tau_{\\rm pump}=150{\\rm\n  fs}\\approx228{\\rm eV}^{-1}$) as pump.\nThis is meant to give a first order of magnitude estimate of the\nnumber of induced outgoing photons.  Let us \nemphasize that it is certainly a rather crude approximation to adopt\nthe formula derived for a stationary, one-dimensional magnetic field\ninhomogeneity of Gaussian type~\\eqref{eq:Gauss} to mimic the field\ninhomogeneity as generated in the focal spot of a high-intensity\nlaser.  Such an approximation ignores the {\\it longitudinal modulation\n  and evolution} of the pump laser pulse.  A more rigorous and refined\ntreatment in the context of an all optical pump--probe experiment\nwould require us to account also for the temporal structure and\nevolution of field inhomogeneities. Fully accounting for pulse shape\ndependencies has become a subject of increasing importance in\nstrong-field phenomenology with high-intensity lasers. Progress has\nalready been made, for instance, for the case of vacuum birefringence\n\\cite{DiPiazza:2006pr,Dinu:2014tsa}.\n\nIn generic high-intensity laser experiments the focal spot area cannot\nbe chosen at will, but is limited by diffraction.\nAssuming Gaussian beams, the effective focus area is conventionally\ndefined to contain $86\\%$ of the beam energy ($1\/e^2$ criterion for\nthe intensity).  The minimum value of the beam diameter in the focus\nis given by twice the laser wavelength multiplied with $f^\\#$, the\nso-called $f$-number, defined as the ratio of the focal length and the\ndiameter of the focusing aperture \\cite{Siegman}; $f$-numbers as low\nas $f^\\#=1$ can be realized experimentally.  Thus, assuming both probe\nand pump lasers to be focused down to the diffraction limit, the\nattainable field strengths are of the order of\n\\begin{equation}\n \\mathfrak{E}^2=I_{\\rm probe}\\approx \\frac{0.86\\,{\\cal E}}{\\tau\\,\\sigma}\\,, \\quad B^2=2I_{\\rm pump}\\approx2\\,\\frac{0.86\\,{\\cal E}_{\\rm pump}}{\\tau_{\\rm pump}\\,\\sigma_{\\rm pump}}\\,,\n\\label{eq:EBpump}\n\\end{equation}\nwith $\\sigma\\approx\\pi\\lambda^2$ and $\\sigma_{\\rm pump}\\approx\\pi\\lambda_{\\rm pump}^2$. The additional factor of two in the definition of $B$ accounts for the fact that, focusing on a purely magnetic field inhomogeneity, the entire laser intensity is considered to be available in terms of a magnetic field, as could, e.g., be realized by superimposing two counter propagating laser beams.\n\nIn the most straightforward experimental setting to imagine, the pump laser beam propagates along the ${\\rm y}$ axis, while its transversal profile,\nparametrized by the coordinate ${\\rm x}$, evolves along the well-defined envelope of a Gaussian beam, and in the vicinity of the beam waist is to be understood as constituting the Gaussian field inhomogeneity~\\eqref{eq:Gauss} of width $w\\approx2\\lambda_{\\rm pump}$.\n\nFor beams focused down to the diffraction limit, the Rayleigh length \nis given by the wavelength of the beam multiplied with a factor of\n$\\pi$ \\cite{Siegman}, i.e., for the pump, $z_{\\rm R}=\\pi\\lambda_{\\rm\n  pump}$.  Hence, over distances of the order of several wave lengths\n$\\lambda_{\\rm pump}$ about the beam waist, the beam diameter remains\napproximately constant along $\\vec{e}_{\\rm y}$ and an experimental\nsetting resembling Fig.~\\ref{fig:perspective} is conceivable.\n\nIn Fig.~\\ref{fig:qualitative}, we plot the number of induced outgoing photons for both effects as a\nfunction of $\\theta$. The respective results are obtained\nstraightforwardly by plugging the design parameters of the Jena\nhigh-intensity laser systems JETI~200 and Polaris given above into\n\\Eqref{eq:EBpump} and the lower components of Eqs. \\eqref{eq:Np-Qref},\n\\eqref{eq:Np-linpol1} and \\eqref{eq:Nmax}. \n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{140514merging_ratios_E=B_omegawofc_PolarisJETI} \n\\caption{Number of induced outgoing photons per shot ${\\cal N}_{\\rm\n    merg}$ due to the effects of laser photon merging and quantum\n  reflection as a function of $\\theta$, adopting the design parameters of the\n  Jena high-intensity laser systems, JETI~200 and Polaris (cf. main\n  text).  The horizontal dashed line shows where the number of induced\n  outgoing photons per shot becomes one.  For quantum reflection this\n  is the case for $\\theta\\geq78^\\circ$ \\cite{Gies:2013yxa}.\n  Conversely, the number of outgoing merged photons reaches a maximum\n  at $\\theta\\approx87^\\circ$ and stays below ${\\cal N}_{\\rm\n    merg}\\big|_{\\rm max}\\approx1.3\\cdot10^{-4}$ throughout the\n  interval $0\\leq\\theta\\leq90^\\circ$; cf. \\Eqref{eq:Nmax} for the\n  Gaussian inhomogeneity and the discussion below.  For completeness,\n  we note that ${\\cal N}_{\\rm\n    Qref}\\bigr|_{\\theta=0}\\approx3\\cdot10^{-29}$ while ${\\cal N}_{\\rm\n    merg}\\bigr|_{\\theta=0}\\approx3\\cdot10^{-228}$ .}\n\\label{fig:qualitative}\n\\end{figure}\n\nObviously, for this particular all-optical experimental setup the\nphoton merging process is substantially suppressed in comparison with\nquantum reflection.  As detailed below \\Eqref{eq:Np-linpol1}, the\ndifferences observed in Fig.~\\ref{fig:qualitative} can be attributed\nto the different scaling of Eqs. \\eqref{eq:Np-Qref} and\n\\eqref{eq:Np-linpol1} with $\\cos^2\\theta$.  While quantum reflection\nreceives an overall enhancement with $\\sim\\frac{1}{\\cos^2\\theta}$ for\nlarge angles of incidence $\\theta\\lesssim90^\\circ$, photon merging\nbecomes maximal if the condition~\\eqref{eq:Nmax} is met (for the\nJETI~200 -- Polaris setup this is the case for an angle of\n$\\theta\\approx 87^\\circ$, wherefore ${\\cal N}_{\\rm merg}\\big|_{\\rm\n  max}\\approx1.3\\cdot10^{-3}$) and dies off to zero for\n$\\theta\\to90^\\circ$.\n\nIn practice, an all-optical setup designed to benefit from the\ngeometric noise reduction will work at a reflection angle near or\nsomewhat above $\\theta \\simeq 80^\\circ$. For parameters similar to\nthe ones studied here, photon merging then is clearly a negligible\nbackground to the quantum reflection signal. Nevertheless, because\nof its different polarization and frequency dependence, appropriate\nfiltering techniques could still render photon merging detectable in the long run.\n\n\n\n\\section{Conclusions and Outlook} \\label{seq:Con+Out}\n\nIn this paper we have studied laser photon merging in the presence of\na one dimensional, stationary magnetic field inhomogeneity.  We have\nin particular confronted the number of outgoing merged photons with\nthe number of quantum reflected photons for the same conditions and\ndiscussed in detail the similarities and differences of the two\neffects.  Sticking to the design parameters of the high-intensity\nlaser facilities to be available in Jena, consisting of a petawatt and\na terawatt class laser system, we have provided a first rough estimate\nof the number of merged photons to be potentially attainable in an\nall-optical experiment. Our results confirm that the quantum\nreflection signal is a most promising candidate for the discovery of\nquantum vacuum nonlinearities under controlled laboratory conditions\nwith high-intensity lasers. In particular, it dominates photon\nmerging in a wide parameter range.\n\nThe expression for the photon merging number is determined most\nstraightforwardly from\nthe photon polarization tensor in a plane wave background.  Actually,\nthe main difficulty in determining the number of outgoing merged laser\nphotons is the problem of finding a convenient and controllable\nexpansion of the photon polarization tensor, allowing us to represent\nour results in concise expressions.  This has led us to adopt a novel\nexpansion strategy to obtain analytical insights into the photon\npolarization in plane wave backgrounds. We believe that this\nrepresentation will also be useful in many other strong field physics\nquestions beyond the merging process.\n\nOf course, a natural extension of our present study in the future\nwould be the investigation of the photon merging process in more\ngeneric, time-dependent inhomogeneities.  Such a study is necessary to\nallow for definitive answers about the the numbers of outgoing merged\nphotons attainable in the focal spot of high-intensity lasers, taking\ninto account the full longitudinal evolution of the pump laser pulse.\n\n\n\\section*{Acknowledgments}\n\nWe are particularly indebted to Maria~Reuter for creating\nFigs.~\\ref{fig:perspective} and \\ref{fig:yzSchnitt}.  FK is\ngrateful to Matt~Zepf for many interesting and enlightening discussions.\nHG acknowledges support by the DFG under grants Gi 328\/5-2 (Heisenberg\nprogram) and SFB-TR18. RS acknowledges support by the Ministry of Education\nand Science of the Republic of Kazakhstan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nControl charts such as the Shewhart chart \\citep{Shewhart1931ECo} and\nthe cumulative sum (CUSUM) chart \\citep{Page1954CIS} have been\nvaluable tools in many areas, including reliability\n\\citep{OConnor2002Pre,Xie2002Sec}, medicine\n\\citep{Carey2003Ihw,Lawson2005Sas,Woodall2006Tuo} and finance\n\\citep{Frisen2008FS}.  See \\cite{Stoumbos2000SoS} and the special\nissues of ``Sequential Analysis'' (2007, Volume 26, Issues 2,3) for an\noverview.  Often, heterogeneity between observations is accounted for\nby using risk-adjusted charts based on fitted regression models\n\\citep{Grigg2004oor,Horvath2004Mci,Gandy2010ram}.\n\nA common convention in monitoring based on control charts is to assume\nthe probability distribution of in-control data to be known.  In\npractice this usually means that the distribution is estimated based\non a sample of in-control data and the estimation error is ignored.\nExamples of this are\n\\cite{Steiner2000Msp,Grigg2004oor,Bottle2008Iin,Biswas2008rCi,Fouladirad2008Otu,Sego2009Rmo,Gandy2010ram}.\n\nHowever, the estimation error has a profound effect on the performance\nof control charts. This has been mentioned at several places in the\nliterature, e.g.\\ \\cite{jones2004rld,Albers2004Esc,jensen2006epe,Stoumbos2000SoS,Champ2007PoM}.\n\nTo illustrate the effect of estimation, we consider a CUSUM chart\n\\citep{Page1954CIS} with normal observations and estimated in-control\nmean. We observe a stream of independent random variables\n$X_1,X_2,\\ldots$ which in control have an $ N(\\mu,1)$ distribution and\nout of control have an $N(\\mu+\\Delta,1)$ distribution, where $\\Delta>0$\nis the shift in the mean.  The chart\nswitches from the in-control state to the out-of-control state at an\nunknown time $\\kappa$.  The unknown in-control mean $\\mu$ is estimated\nby the average $\\hat\\mu$ of $n$ past in-control observations\n$X_{-n},\\dots,X_{-1}$ (this is often called phase 1 of the monitoring;\nthe running of the chart is called phase 2). We consider the  CUSUM chart\n$$S_t=\\max(0, S_{t-1}+X_t-\\hat \\mu - \\Delta\/2), \\quad S_0=0 $$\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\n\n\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{simpaper\/estimerr_guaranteed_CUSUM.pdf}\n  \\caption{In-control distribution of ARL=$\\E(\\tau|\\hat\\mu)$ for\n    CUSUMs for standard normally distributed data.  The mean $\\hat\n    \\mu$ used in the monitoring is estimated based on $n$ past\n    observations.  The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%,\n    75\\%, 90\\% and 97.5\\% quantiles.The top part of the plot shows\n    the situation when estimation error is ignored. In the middle part  the\n    threshold has been chosen to give an unconditional ARL of 100\n    (averaging out the parameter estimation). In the bottom part the threshold is\n    adjusted to guarantee  with 90\\% probability an in-control ARL of at least 100.  }\n  \\label{fig:estimerr}\n\\end{figure}\n\nThe in-control average run length,\n$\\ARL=\\E(\\tau|\\hat\\mu,\\kappa=\\infty)$, depends on $\\hat \\mu$ and is\nthus a random quantity. The top part of the plot in\nFigure~\\ref{fig:estimerr} shows boxplots of its distributions with\nthreshold $c=2.84$, $\\Delta=1$ and various numbers of past\nobservations. If  $\\hat \\mu=\\mu$, i.e.\\ $\\mu$ was know,  this\nwould give an in-control $\\ARL$ of 100. The estimation error is having a substantial effect on\nthe attained $\\ARL$ even for  large samples such as\n$n=1000$.\nFor further illustrations of the impact of estimation error see\n\\cite{jones2004rld} for CUSUM charts and  \\cite{Albers2004Esc} for\nShewhart charts.\n\n\nSo far, no general approach for taking the estimation error into\naccount has been developed, but there are many special constructions\nfor specific situations.  For instance, for some charts so called\nself-starting charts\n\\citep{Hawkins1987SCC,Hawkins1998csc,Sullivan2002SCC}, maximum\nlikelihood surveillance statistics\n to eliminate parameters\n\\cite[e.g.][]{Frisen2009}, correction\nfactors for thresholds\n\\citep{Albers2004Esc,jones2002statistical},\nmodified thresholds \\citep{Zhang2011TSX} and threshold\nfunctions \\citep{Horvath2004Mci,Aue2006Cpm} have been\ndeveloped. Various bootstrap schemes for specific situations have also\nbeen suggested, see for instance\n\\cite{Kirch2008BSC,Chatterjee2009Dcs,Capizzi2009Bdo,Huskova10Bsc}.\nFurther, some nonparametric charts which account for the estimation\nerror in past data have been proposed, see \\cite{Chakraborti2007Ncc}\nand references therein.   Recently some modified charts for monitoring\nvariance in the normal distribution with estimated parameters have\nbeen suggested by \\cite{Maravelakis2009AEC}  and\n\\cite{Castagliola2011ACC}.\n\nWhen addressing estimation error, the above methods mainly  focus\non the performance of the charts averaged over both the estimation of\nthe in-control state as well as  running  the chart once.\nIn the middle part of Figure~\\ref{fig:estimerr}, the threshold has\nbeen chosen such that,  averaged over both the estimation of\nthe in-control state as well as  running  the chart once, the average\nrun length is $100$ (this results in a different threshold\nfor each $n$). It turns out that only a small change in the threshold\nis needed and that the distribution of the conditional $\\ARL=\\E(\\tau|\\hat\n\\mu)$ is only changed slightly. This bias correction\nfor the $\\ARL$ actually goes in the wrong direction in the sense that\nit implies more short $\\ARL$s. This is due to the $\\ARL$ being\nsubstantially influenced by the right tail of the run length\ndistribution, see the discussion in Section 2 of \\cite{Albers2006SAC}.\n\n\n\nHowever, usually, after the chart parameters are estimated, the\nchart is run for some time without any reestimation of the in-control\nstate even if the chart signals. Moreover, in some situations, several\ncharts are run based on the same estimated parameters.\nIn these situations the ARL conditional on the estimated in-control\nstate is more relevant than the unconditional ARL. In the middle and\nupper part of Figure \\ref{fig:estimerr}, one sees that the conditional\nARL can be much lower than 100, meaning that both the unadjusted\nthreshold and the\n threshold adjusted for bias in the unconditional ARL\nlead, with a substantial probability, to  charts\nthat have a considerably decreased time until false alarms.\n\n\nTo overcome these problems we will look at the performance of the\nchart conditional on the estimated in-control distribution, averaging\nonly over different runs of the chart. This will lead to the\nconstruction of charts that with high probability have an in-control\ndistribution with desired properties conditional on the observed past\ndata, thus reducing the situations in which there are many false\nalarms due to estimation error.\n\n\nThe bottom part of Figure \\ref{fig:estimerr}  shows the distribution of the in control\nARL when   the threshold for each set of past data is adjusted to\nguarantee an in-control ARL of at least 100 with probability 90\\%. The adjustment is\ncalculated using a bootstrap procedure explained later in the paper.\nThe adjustment succeeds to avoid the too low ARLs with the\nprescribed probability, and we will see later that the cost in a\nhigher out-of-control ARL is modest.  Using hitting probabilities instead of ARL as\ncriterion leads to similar results.\n\n\\xx{talk about adjustments}\n\n\nOur approach is similar in spirit to the exceedance probability\nconcept developed by Albers and Kallenberg for various types of\nShewhart \\citep{Albers2004AEC,Albers2005Ncf,Albers2005EPF} and\nnegative binomial charts \\citep{Albers2009CUM,Albers2010Toc}. They\ncalculate approximate adjusted thresholds such that there is only a\nsmall prescribed probability that some performance measure, for\ninstance an ARL, will be a certain amount below or above a specified\ntarget.\n\n\nThe main difference between their approach and what we present is that\nour approach applies far more widely, to many different types of\ncharts and without having to derive specific approximation formulas in\neach setting. If we apply a nonparametric bootstrap, the\nproposed procedure will be robust against model misspecification.  In\naddition to that, our approach allows not only to adjust the threshold\nbut also to give a confidence interval for the in-control performance\nof a chart for a fixed threshold.  Lastly, even though not strongly\nadvocated in this paper, the bootstrap procedure we propose can also\nbe used to do a bias correction for the unconditional performance of\nthe chart, as in the middle part of Figure \\ref{fig:estimerr}.\n\n\n\nNext, we describe our approach more formally.\nSuppose we want to use a monitoring scheme and that the in-control\ndistribution $P$ of the observations is unknown, but that\nbased on past in-control behaviour we have an estimate $\\hat P$ of the\nin-control distribution.  Let $q$ denote the in-control property of\nthe chart we want to compute, such as the $\\ARL$, the false alarm\nprobability or the threshold needed for a certain $\\ARL$ or false alarm\nprobability. In the above example we were interested to find a\nthreshold such that the in-control ARL is 100.\n\nGenerally, $q$ may depend on both the true in-control distribution $P$\nand on estimated parameters of this distribution which for many charts\nare needed to run the chart. We denote these parameters by $\\hat\n\\xi=\\xi(\\hat P)$.\nIn the above CUSUM chart example $\\hat \\xi = \\hat\n\\mu$.  We are interested in $q(P;\\hat\\xi)$, that\nis the in-control performance of the chart conditional on the\nestimated parameter. In the above CUSUM example,  $q(P;\\hat\\xi)$ is the threshold\nneeded to give an $\\ARL$ of 100 if the observations are from the true\nin-control distribution  $P$ and the\nestimated parameter $\\hat \\mu$ is used. As $P$ is not observed $q(P;\\hat\n\\xi)$ is not observable.  As mentioned above, many papers pretend\nthat the estimated in-control distribution $\\hat P$ equals the true\nin-control distribution $P$ and thus use $ q(\\hat P;\\hat \\xi)$.\nOur suggestion is to use bootstrapping of past data to construct an\napproximate one-sided confidence intervals for $q(P;\\hat \\xi)$. From\nthis we get a  guaranteed conditional performance of the control\nscheme.\n\n\n\n\nIn Section~\\ref{sec:monitorhomobs} we present the general idea in the\nsetting with homogeneous observations, and discuss this for Shewhart\nand CUSUM charts.  The main theoretical results are presented in\nSection~\\ref{sec:gentheor}, with most of the proofs given in the\nAppendix.  Section \\ref{sec:simulsingle} contains simulations\nillustrating the performance of charts for homogeneous observations.\nIn Section~\\ref{sec:regmod} extensions to charts based on regression\nand survival analysis models are presented. Some concluding comments\nare given in Section~\\ref{sec:conclusion}.  The suggested methods are\nimplemented in a flexible R-package, that will be made available on the\nComprehensive R Archive Network (CRAN).\n\n\n\n\\section{Monitoring homogeneous observations}\n\\label{sec:monitorhomobs}\n\\subsection{General idea}\n\\label{subsec:generalidea}\n\n\nSuppose that in control we have independent observations\n$X_1,X_2,\\dots$ following an unknown distribution $P$.  We want to use\nsome monitoring scheme\/control chart that detects when $X_{i}$ is no\nlonger coming from $P$. The particular examples we discuss in\nthis paper are Shewhart and CUSUM charts, but the methodology we\nsuggest applies more widely.\n\nTo run the charts, one often needs certain parameters $\\xi$. For\nexample, in the CUSUM control chart of the introduction, we\nneeded $\n\\xi= \\mu$, the assumed in-control mean.  These parameters will usually\nbe estimated.\n\nLet $\\tau$ denote the time at which the chart signals a change. As $\\tau$ may\ndepend on $\\xi$, we sometimes write $\\tau(\\xi)$.  The\ncharts we consider use a threshold $c$, which determines how quickly\nthe chart signals (larger $c$ lead to a later signal).\n\nThe performance of such a control chart with the in-control\ndistribution $P$ and the parameters $ \\xi$ can, for example, be\nexpressed as one of the following.\n\\begin{itemize}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item $\\ARL(P;\\xi)=\\E(\\tau( \\xi))$, where $\\E$ is the expectation with respect to $P$.\n\\item ${\\hit}(P;\\xi)=\\Prob(\\tau( \\xi)\\leq T)$ for\n  some finite $T>0$, where $\\Prob$ is the probability measure under which $X_1,X_2,\\dots\\sim P$. This is the false alarm probability in $T$ time\n  units.\\xx{do we want to distinguish between $\\Prob$ and $P$?}\n\\item $c_{\\ARL}(P;\\xi)=\\inf\\{c>0:\\ARL(P;\\xi)\\geq\\gamma\\}$ for some\n  $\\gamma>0$. Assuming appropriate continuity, this is the threshold\n  needed to give an in-control average run length of $\\gamma$.\n\\item $c_{\\hit}(P;\\xi)=\\inf\\{c>0:\\hit(P;\\xi)\\leq\\beta\\}$ for\n  some $0<\\beta<1$. This is the threshold needed\n  to give a false alarm probability of $\\beta$.\n\\end{itemize}\nThe latter two quantities are very important in practice, as they are\nneeded to decide which threshold to use to run a chart.  In the\nnotation we have suppressed the dependence of the quantities on $c$,\n$T$, $\\gamma$, $\\beta$ and $\\Delta$.\n\nIn the following, $q$ will denote one of $\\ARL$, $\\hit$, $c_{\\ARL}$\nor $c_{\\hit}$, or simple transformations such as $\\log(\\ARL)$,\n$\\logit(\\hit)$, $\\log(c_{\\ARL})$ and $\\log(c_{\\hit})$, where\n$\\logit(x)=\\log\\left(\\frac{x}{1-x}\\right)$.\n\nThe true in-control distribution $P$ and the parameters $\\xi=\\xi(P)$\nneeded to run the chart are usually estimated.  We assume that we have\npast in-control observations $X_{-n},\\dots,X_{-1}$ (independent of $X_1,X_2,\\dots$), which\nwe use to  estimate the in-control distribution $P$ parametrically or\nnon-parametrically. We denote this estimate by $\\hat P$.  The estimate\nof $\\xi$ will be denoted by $\\hat \\xi=\\xi(\\hat P)$. For example, in\nthe CUSUM control chart of the introduction, $\\hat \\xi=\\hat \\mu$ is\nthe estimated in-control mean.\n\nThe observed performance of the chart will depend on the true\nin-control distribution $P$ as well as on the estimated parameters\n$\\hat \\xi$ that are used to run the chart.  Thus we are interested in\n$q(P;\\hat\\xi)$, the performance of the control chart\n\\emph{conditional} on $\\hat \\xi$.  This is an unknown quantity as $P$\nis not known.  Based on the estimator $q(\\hat P;\\hat\\xi)$, we\nconstruct a one-sided confidence interval for this quantity to\nguarantee, with high probability, a certain performance for the\nchart. We choose to call the interval a  confidence interval,\neven though the quantity $q(P;\\hat\\xi)$ is random.\n\nWe suggest the following for guaranteeing an upper bound on $q$ (which\nis relevant for  $q=\\hit$, $q=c_{\\ARL}$ or $q=c_{\\hit}$).  For $\\alpha\\in (0,1)$,\nlet $p_\\alpha$ be a constant such that\n$$\n\\Prob(q(\\hat P;\\hat\\xi) - q(P;\\hat\\xi) >p_\\alpha)=1-\\alpha,\n$$\nassuming that such a $p_{\\alpha}$ exists.\nHence,\n$$\n\\Prob(q(P;\\hat\\xi)< q(\\hat P;\\hat\\xi)-p_{\\alpha})= 1-\\alpha.\n$$\nThus $(-\\infty,q(\\hat P;\\hat\\xi)-p_{\\alpha})$ could be considered an\nexact lower one-sided confidence interval of $q(P;\\hat\\xi)$.\n\\xx{Or,\n tolerance interval?}\n\\cc{This is not really a confidence interval in the classical sense -\n $q( P;\\hat\\xi)$ is an\n  unobserved random quantity... and not just a fixed parameter.\nThe article  Weerahandi (1993, JASA) ``Generalized Confidence Intervals'' might be a useful reference. }\n\nOf course, $p_{\\alpha}$ is unknown.  We suggest to obtain an\napproximation of $p_{\\alpha}$ via bootstrapping.  In the following,\n$\\hat P^\\ast$ denotes a parametric or non-parametric bootstrap\nreplicate of the estimated in-control distribution $\\hat P$.\nWe can approximate $p_\\alpha$ by $p^\\ast_\\alpha$ such that\n$$\\Prob(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)> p^\\ast_\\alpha|\\hat P)=1-\\alpha.$$\n\\cc{alternatively we could use $\\hat P(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)> p^\\ast_\\alpha)=1-\\alpha.$\n}\nThus\n\\begin{equation}\n  \\label{eq:onesidedapproxconfint}\n (-\\infty,q(\\hat P;\\hat\\xi)-p^\\ast_\\alpha)\n\\end{equation}\nis a one-sided (approximate) confidence interval for $q(P;\\hat\\xi)$.\nIn this paper, we will use the following generic algorithm to\nimplement the bootstrap.\n\\begin{algorithm}[Bootstrap]\n  \\label{alg:Bootstrap}\n\\hspace*{2mm}\\\\[-7mm]\n\\begin{enumerate}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item From the past data $X_{-n},\\dots,X_{-1}$, estimate\n  $\\hat P$ and  $\\hat\\xi$.\n\\item Generate bootstrap samples $X^{\\ast}_{-n},\\dots,X^{\\ast}_{-1}$\n  from $\\hat P$.  Compute the corresponding estimate $\\hat P^{\\ast}$\n  and $\\hat \\xi^{\\ast}$. Repeat $B$ times to get $\\hat\n  P^{\\ast}_1,\\dots,\\hat P^{\\ast}_B$ and $\\hat \\xi^{\\ast}_1,\\dots,\\hat\n  \\xi^{\\ast}_B$.\n\\item Let $p_{\\alpha}^{\\ast}$ be the $1-\\alpha$ empirical quantile of\n  $q(\\hat P^{\\ast}_b;\\hat\\xi^{\\ast}_b)-q(\\hat P;\\hat\\xi^{\\ast}_b)$, $b=1,\\dots,B$.\n\\end{enumerate}\n\\end{algorithm}\n\n\n\n\nFor guaranteeing a lower bound on $q$, which is for example relevant\nfor $q=\\ARL$, a similar upper one-sided confidence interval can be\nconstructed.\n\n\n\n\nIn a practical situation, the focus would be on deciding which\nthreshold to use for the control chart to obtain desired in-control\nproperties. We suggest to use either $q=c_{\\ARL}$ or $q=c_{\\hit}$, or\nlog transforms of these, and\nthen run the chart with the adjusted threshold\n\\begin{equation}\n  \\label{eq:adjThreshold}\nq(\\hat P;\\hat \\xi)-p^{\\ast}_{\\alpha}.\n\\end{equation}\nThis will guarantee that in (approximately) $1-\\alpha$ of the\napplications of this method, the control chart actually has the\ndesired in-control properties.\n\n\n\\cc{The following are some comments which are probably not quite relevant to practice.\nIn some application of control charts the chart parameters $\\xi$ are\nnot estimated but determined according to certain specifications the\nprocess should meet. A typical example would be industrial\napplications like monitoring of properties of mass produced units\nwhere there are precise specification of physical properties of the\nunits which should be monitored. Then $\\xi$ may be determined\naccording to these specifications, and the point of the monitoring is\nto detect deviations from the specifications.  However, the full\nin-control distribution $P$ would usually still be unknown and our\napproach would still apply for constructing confidence intervals for\n$q(P;\\xi_s)$ where $\\xi_s$ denotes a specified $\\xi$.\n\n\nWould this in practice be relevant? Or would one also specify $P$,\n  or at least parts of $P$ like the  mean?  Could using our approach\n  here e.g.\\ lead to picking a far too   large $c$ to get a guaranteed\n  ARL if $P$ actually is far off from   where it ``should be''?\n}\n\n\n\n\\subsection{Specific charts}\n\n\\subsubsection{Shewhart charts}\n\\label{subsubsec:Shewhart}\n\nThe one-sided Shewhart chart \\citep{Shewhart1931ECo} signals at\n$$\n\\tau=\\inf\\{t \\in \\{1,2,\\dots\\}: f(X_t,\\xi)>c\\}\n$$\nfor some threshold $c$, where  $f$ is some function, $X_t$ is the\nobservation at time $t$ and $\\xi$ are\nsome parameters.\n $X_t$ can be a single\nmeasurement or e.g.\\ the average, range or standard deviation of a\nspecified number of measurements, or some other statistic like a\nproportion.\nIt is common to use a Shewhart chart with a threshold of\nthe mean plus 3 times the standard deviation, in\nthis case one would use $c=3$ and   $f(x,\\xi)=\\frac{x-\\xi_1}{\\xi_2}$\nwith $\\xi_1$ being the mean and $\\xi_2$ being the standard deviation.\n For two-sided charts one could just use  $f(x,\\xi)=\\frac{|x-\\xi_1|}{\\xi_2}$.\n\nConditionally on fixed parameters  $ \\xi$, the stopping time $\\tau$ follows a\ngeometric distribution with parameter\n$p=p(c;P,\\xi)=\\Prob(f(X_t,\\xi)>c)$.\nThen the performance measures mentioned in the previous section simplify to\n\\begin{align*}\n\\ARL(P;\\xi)=&\\frac{1}{p(c;P,\\xi)},& \\hit(P;\\xi)=&1-(1-p(c;P,\\xi))^T,\\\\\n c_{\\ARL}(P;\\xi)=&p^{-1}\\left( \\frac{1}{\\gamma}\n   ;P,\\xi\\right)\n\\;\\;\\;\\;\\;\\;\n\\text{ and}&\n c_{\\hit}(P;\\xi)=&p^{-1}\\left( 1-(1-\n    \\beta)^{\\frac{1}{T}};P,\\xi\\right),\n\\end{align*}\n where $p^{-1}(\\cdot;P,\\xi)$ is the inverse of $p(\\cdot;P,\\xi)$.\n\\cc{To get the formula for $c_{\\hit}$ set\n    $\\beta=\\hit$ in the second item and solve for $c$.}\n\n\nSuppose that the in-control distribution comes from a parametric\nfamily $P_{\\theta}, \\theta\\in \\Theta$.  Furthermore, suppose that we\nhave some way of computing an estimate $\\hat\\theta$ of $\\theta$ based on the\nsample.\nThen we can use  Algorithm \\ref{alg:Bootstrap} with $\\hat P=P_{\\hat \\theta}$ to compute a\nconfidence interval as given by (\\ref{eq:onesidedapproxconfint}).\n\nShewhart charts depend heavily on the tail behaviour of the\ndistribution of the observations. This is particularly problematic\nwhen the sample size is small and we use non-parametric methods or a\nsimple non-parametric bootstrap. We thus primarily suggest to use a\nparametric bootstrap for Shewhart charts.\n\n\n\\begin{remark}\nIn certain cases the parametric bootstrap will actually be exact when\n$B \\to \\infty$. This happens when the distribution of\n$q(P_{\\hat\\theta};\\hat\\xi) - q(P_{\\theta};\\hat\\xi)$ under $P_{\\theta}$\ndoes not depend on $\\theta$. In particular, this implies that\n$q(P_{\\hat\\theta^\\ast};\\hat\\xi^\\ast) - q(P_{\\hat\\theta};\\hat\\xi^\\ast)$\nhas the same distribution and $p^\\ast_\\alpha\\to p_\\alpha$ as $B\\to\n\\infty$.\n\n As an example, consider the case when\n $f(x,\\xi)=\\frac{x-\\xi_1}{\\xi_2}$ and $X_t$ follows an\n $N(\\xi_1,\\xi_2^2)$ distribution and $q$ is any of the performance\n measures described above. We use $\\theta=\\xi$ and as estimator $\\hat\n \\xi_1$ we use the sample mean and as estimator $\\hat\\xi_2$ we use the\n sample standard deviation. Then\n\\begin{align*}\np(c;P_{\\!\\xi},\\hat\\xi)\n=\\Prob_{\\!\\xi}\\!\\left(\\frac{X_t-\\hat\\xi_1}{\\hat\\xi_2}> c\\right)\n=1-\\Phi\\left(\\frac{c\\hat\\xi_2+\\hat\\xi_1-\\xi_1}{\\xi_2}\\right),\n\\end{align*}\nwhere $\\Phi$ is the cdf of the standard normal distribution, and\nunder $P_\\xi$,\n$$\n\\frac{c\\hat\\xi_2+\\hat\\xi_1-\\xi_1}{\\xi_2}=c\\frac{\\hat\\xi_2}{\\xi_2}+\\frac{\\hat\\xi_1-\\xi_1}{\\xi_2}\\sim\\frac{c}{\\sqrt{n-1}}\\sqrt{W}+\\frac{1}{\\sqrt{n}}Z,\n$$\nwhere $W\\sim\\chi_{n-1}^2$ and $Z\\sim N(0,1)$ are independent. Thus the\ndistribution of $p(c;P_\\xi,\\hat\\xi)$, and hence $q(P_{\\xi};\\hat\\xi)$,\nis completely known.  As $\np(c;P_{\\!\\hat\\xi},\\hat\\xi)=\\Prob_{\\!\\hat\\xi}\\left(\\frac{X_t-\\hat\\xi_1}{\\hat\\xi_2}>\n  c\\right) =1-\\Phi(c)$, and thus $q(P_{\\!\\hat\\xi};\\hat\\xi)$, is not\nrandom, the distribution of $q(P_{\\!\\hat\\xi};\\hat\\xi) -\nq(P_{\\xi};\\hat\\xi)$ also does not depend on any unknown\nparameters. Thus the parametric bootstrap is exact in this example.\n\\end{remark}\n\n\n\\subsubsection{CUSUM charts}\n\\label{subsubsec:CUSUM}\nThis section considers the one-sided CUSUM chart \\citep{Page1954CIS}.\nThe classical CUSUM chart was designed to detect a shift of size\n$\\Delta>0$ in the mean of normally distributed observations. Let $\\mu$ and $\\sigma$\ndenote, respectively, the in-control mean and standard\ndeviation. A CUSUM chart can be defined by\n\\begin{equation}\n  \\label{eq:discrCUSUM_meanshift}\nS_t=\\max(0, S_{t-1}+(X_t-\\mu - \\Delta\/2)\/\\sigma), \\quad S_0=0\n\\end{equation}\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\n\nAlternatively, we could drop the scaling and not divide by the\n standard deviation $\\sigma$ in\n(\\ref{eq:discrCUSUM_meanshift}). See Chapter 1.4 in\n\\cite{Hawkins1998csc} for a discussion on scaled versus unscaled\nCUSUMs.\n\n\nMore generally, to accommodate observations with general in-control distribution with\ndensity $f_0$ and general out-of-control distribution with density $f_{1}$, it\nis optimal in a certain sense \\citep{Moustakides1986OST} to modify the\nCUSUM chart by replacing $(X_t-\\mu - \\Delta\/2)\/\\sigma$ by the log\nlikelihood ratio $\\log(f_1(X_t,\\theta)\/f_0(X_t, \\theta))$ such\nthat the CUSUM chart is\n\\begin{equation}\n  \\label{eq:discrCUSUM_loglikelihood}\nS_t=\\max(0, S_{t-1}+\\log(f_1(X_t,\\theta)\/f_0(X_t, \\theta))), \\quad S_0=0.\n\\end{equation}\n\nLet $\\xi$ denote either $(\\mu,\\sigma)$ in\n(\\ref{eq:discrCUSUM_meanshift}) or $\\theta$ in\n(\\ref{eq:discrCUSUM_loglikelihood}).  Usually, $\\xi$ needs to be\nestimated from past data, and we can then use Algorithm\n\\ref{alg:Bootstrap} to compute a confidence interval\n(\\ref{eq:onesidedapproxconfint}) for the performance measure\n$q(P;\\hat\\xi)$. For (\\ref{eq:discrCUSUM_loglikelihood}) it is most\nnatural to use a parametric bootstrap with $\\hat P=P_{\\hat \\theta}$,\nwhile for (\\ref{eq:discrCUSUM_meanshift}) we can use either a\nparametric or a nonparametric bootstrap. In the latter case we let\n$\\hat P$ be the empirical distribution of $X_{-n},\\dots,X_{-1}$, i.e.\nin Algorithm \\ref{alg:Bootstrap}, $X^{\\ast}_{-n},\\dots,X^{\\ast}_{-1}$\nare sampled with replacement from $X_{-n},\\dots,X_{-1}$.\n\n\n\\begin{remark}\nSimilar as for Shewhart charts, this parametric bootstrap is exact\nwhen the distribution of\n$q(P_{\\hat\\theta};\\hat\\xi)-q(P_{\\theta};\\hat\\xi)$ does not have any\nunknown parameters.  This is, for instance, the case if we use\n(\\ref{eq:discrCUSUM_loglikelihood}) for an exponential distribution\nwith the out-of-control distribution specified as an exponential\ndistribution with mean $\\Delta\\lambda$, where $\\lambda$ is the\nin-control mean. Another example of this is when we have normally\ndistributed data and use a CUSUM with the increments\n$(X_t-\\hat\\mu)\/\\hat\\sigma-\\Delta\/2$.\n\\end{remark}\n\n\n\n\n\n\\section{General theory}\n\\label{sec:gentheor}\n\nIn this section, we show that asymptotically, as the number of past\nobservations $n$ increases, our procedure works.  An established way\nof showing asymptotic properties of bootstrap procedures is via a\nfunctional delta method \\citep{Vaart1996WCa,Kosorok2008ItE}. Whilst we\nwill follow a similar route, our problem does not fit directly into\nthe standard framework, because the quantity of interest, $q(P,\\hat\n\\xi)$, contains the random variable $\\hat \\xi$.\nWe present the setup and\nthe main result in Section \\ref{sec:th:main}, followed by examples\n(Section \\ref{sec:th:examples}).\n\\cc{The asymptotic development in this section only show that things\n  do not go badly wrong as $n\\to \\infty$. They only establish\n  consistency of the correction\/confidence intervals.  However, the need to use\n  these confidence intervals disappears as $n$ increases.}\n\n\n\\subsection{Main theorem}\n\\label{sec:th:main}\nLet $D_q$ be the set in which  $P$ and its\nestimator $\\hat P$ lie, i.e.\\ a set describing the potential\nprobability distribution of our observations. This could be a subset\nof $\\mathbb{R}^d$ for parametric distributions, the set of cumulative\ndistribution functions for non-parametric situations, or the set of\njoint distributions of covariates and observations.  We assume that\n$D_q$ is a subset of a complete normed vector space $D$. \\cc{Do we\n  want to \/need to assume that $D$ is complete (every Cauchy sequence\n  converges)?  This should not be a problem as $R^k$ and\n  $l_{\\infty}(\\mathbb{R})$ are complete metric spaces. } Let $\\Xi$ be a\nnon-empty topological space containing the potential parameters $\\xi$\nused for running the chart. In our examples, we will let $\\Xi\\subset\n\\mathbb{R}^d$ be an open set.\n\nWe assume that $\\hat P^{\\ast}=\\hat P^{\\ast}(\\hat P, W_n)$ is a\nbootstrapped version of $\\hat P$ based both on the observed data $\\hat P$ and\non an independent random vector $W_n$.  For example, when resampling\nwith replacement then $W_n$ is a weight vector of length $n$,\nmultinomially distributed, that determines how often a given\nobservation is resampled.  In a parametric bootstrap, $W_n$ is the\nvector of random variables needed to generate observations from the\nestimated parametric distribution.\n\n\nIn the main theorem we will need that the mapping $q:D_q\\times\n\\Xi\\to\\mathbb{R}$, which returns the property of the chart we are interested\nin, satisfies the following extension of Hadamard differentiability.\nFor the usual definition of Hadamard differentiability see e.g.\\\n\\citep[Section 20.2]{Vaart1998AS}. The extension essentially consists in\nrequiring Hadamard differentiability in the first component when the second\ncomponent is converging.\n\\begin{definition}\n\\label{def:haddiffamily}\nLet $D,E$ be metric spaces, let $D_f\\subset D$ and let $\\Xi$ be a\nnon-empty topological space. \\cc{We need at least to be able to speak\n  about convergence in $\\Xi$.}  The family of functions\n$\\{f(\\cdot;\\xi):D_f \\to E: \\xi\\in \\Xi\\}$ is called \\emph{Hadamard\n  differentiable at $\\theta\\in D_f$ around $\\xi \\in \\Xi$ tangentially\n  to $D_0\\subset D$}  if there exists a continuous linear map \\cc{this is a requirement that also appears in the original definition and which we may be using in our proofs; however, we never prove for our derivatives that they are continuous and linear}\n$f'(\\theta;\\xi):D_0\\to E$ such that\n$$\n\\frac{f(\\theta+t_nh_n;\\xi_n)-f(\\theta;\\xi_n)}{t_n}\\to\nf'(\\theta;\\xi)(h)\\quad(n\\to\\infty)\n$$\nfor all sequences $(\\xi_n)\\subset \\Xi$, $(t_n)\\subset \\mathbb{R}$, $(h_n)\\subset D$\nthat satisfy $\\theta+t_nh_n\\in D_f \\,\\forall n$ and $\\xi_n\\to \\xi$, $t_n\\to 0$, $h_n\\to h\\in D_0$ as $n\\to \\infty$.\n\\end{definition}\n\n\n\nIn the following theorem we understand convergence in distribution, denoted by $\\leadsto$, as defined\nin \\citet[Def 1.3.3]{Vaart1996WCa} or in \\citet[p.108]{Kosorok2008ItE}.\n\\cc{ Let\n  $(\\Omega_n, {\\cal A}_n, P_n)$ be a sequence of probability spaces,\n  let $(\\Omega, \\cal A, P)$ be a further probability space, let $D$ be\n  a metric space and let $X_n:\\Omega_n\\to D$ be a sequence of maps and\n  let $X:\\Omega\\to D$ be a Borel measurable map. Then $X_n\\leadsto X$\n  if $\\E^{\\ast}f(X_n) \\to \\E f(X)$ for all continuous, bounded $f:D\\to\n  \\mathbb{R}$.  }\n\\cc{Outer expectation is defined in \\cite{Vaart1996WCa} and\n  in \\cite{Kosorok2008ItE}, essentially $\\E^{\\ast}X = \\inf \\{\\E Y:\n  Y\\geq X, Y \\text{ measurable}\\}$.}\n\\begin{theorem}\n\\label{th:main}\n  Let $q:D_q\\times \\Xi\\to \\mathbb{R}$ be a mapping, let $P\\in D_q$ and let\n  $\\xi:D_q\\to \\Xi$ be a continuous function.\n  Suppose that the following conditions are satisfied.\n\\begin{itemize}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item[a)] $q$ is Hadamard differentiable at $P$ around $\\xi$ tangentially to $D_0$ for some $D_0\\subset D$.\n\\item[b)] $\\hat P$ is a sequence of random elements in $D_q$ such that\n$\n  \\sqrt{n}(\\hat P-P)\\leadsto Z\n$ as $n\\to \\infty$\nwhere $Z$ is some tight random element in $D_0$.\n\\item[c)]\n$\\sqrt{n}(\\hat P^{\\ast}-\\hat P)\\condweakconv{\\hat P} Z$ as $n\\to\\infty$\nwhere $\\condweakconv{\\hat P}$ denotes weak convergence conditionally on $\\hat\nP$ in probability as defined in \\citet[p.19]{Kosorok2008ItE}.  \\cc{\n  i.e. $\\sup_{h\\in \\text{BL}_1}|E_Wh(\\hat X_n) - E\n  h(X)|\\stackrel{P}{\\to}0$ and $E_Wh(\\hat X_n)^{\\ast}-E_Wh(\\hat\n  X_n)_{\\ast}\\stackrel{P}{\\to}0$ for all $f\\in \\text{BL}_1$ where the\n  subscript $W$ denotes conditional expectation over the weights given\n  the remaining data.  }\n\\item[d)] The cumulative distribution function  of  $q'(P;\\xi)Z$ is continuous.\n\\item[e)] Outer-almost surely, the map $W_n\\mapsto h(\\hat P^{\\ast}(\\hat P, W_n))$ is measurable for each $n$ and for every continuous bounded function $h:D_q\\to \\mathbb{R}$.\n\\item[f)] $q(\\hat P; \\hat \\xi)-q(P;\\hat \\xi)$ and  $p_\\alpha^{\\ast}$ are  random variables, i.e.\\ measurable,\nwhere $\\hat \\xi =\\xi(\\hat P)$ and $p^{\\ast}_{\\alpha}=\\inf\\{t\\in \\mathbb{R}:\n\\hat \\Prob(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)\\leq t)\\geq\n\\alpha\\}$.\n\\end{itemize}\nThen\n\\begin{equation*}\n\\Prob(q(P;\\hat\\xi)\\in (-\\infty,  q(\\hat P;\\hat\\xi)-p^{\\ast}_{\\alpha}))\\to  1-\\alpha \\quad (n\\to \\infty).\n\\end{equation*}\n\\end{theorem}\nA similar result  holds for upper confidence intervals.\n\nThe proof is in Appendix \\ref{sec:proof}.  The theorem essentially is\nan extension of the delta-method.  Condition a) ensures the necessary\ndifferentiability.  Conditions b) and c) are standard assumptions for\nthe functional delta method; b) for the ordinary delta method and c)\nfor the bootstrap version of it. Condition d) ensures that, after\nusing an extension of the delta-method, the resulting confidence\ninterval will have the correct asymptotic coverage probability.\nCondition e) is a technical measurability condition, which will be\nsatisfied in our examples. Condition f) is a measurability condition,\nwhich should usually be satisfied.\n\n\n\\subsection{Examples}\n\\label{sec:th:examples}\nThe following sections give examples in which Theorem \\ref{th:main}\napplies.  We consider hitting probabilities ($q=\\hit$) and thresholds to\nobtain certain hitting probabilities ($q=c_{\\hit}$).\n\nThese examples are\nmeant to be illustrative rather than exhaustive. For example, other\nparametric setups could be considered along similar lines to Section\n\\ref{sec:cusum-charts-with}. Furthermore, other performance measures such as\n$\\log(c_{\\hit})$ or $\\logit(\\hit)$ would essentially require application of chain rules\nto show differentiability.\n\n\\subsubsection{Simple nonparametric setup for CUSUM charts}\n\\label{sec:theor:ex:CUSUM:nonpar}\nWe show how the above theorem applies to the CUSUM chart described in\n(\\ref{eq:discrCUSUM_meanshift}) when using a non-parametric bootstrap\nversion of Algorithm \\ref{alg:Bootstrap}.\n\nLet $D=l_{\\infty}(\\mathbb{R})$ be the set of bounded functions $\\mathbb{R}\\to\\mathbb{R}$\nequipped with the sup-norm $\\|x\\|=\\sup_{t\\in \\mathbb{R}}|x_t|$. \\cc{This is a\n  Banach space, i.e. a complete normed vector space}\nLet $D_q\\subset D$ be the set of cumulative distribution functions on\n$\\mathbb{R}$ with finite second moment.\nThe parameters needed to run the chart are the mean and the standard deviation of the in-control observations, thus we may choose\n $\\Xi=\\mathbb{R}\\times(0,\\infty)$ and  $\\xi:D_q\\to \\Xi, P\\mapsto (\\int x P(dx),\n\\int x^2 P(dx)-(\\int x P(dx))^2)$.\n\n\nAs quantities $q$ of interest we are considering hitting probabilities\n($q=\\hit$) and thresholds ($q=c_{\\hit}$) needed to achieve a certain hitting\nprobability.  The probability $\\hit:D_q\\times \\Xi\\to \\mathbb{R}$ of hitting a\nthreshold $c>0$ up to step $T>0$ can be written as\n$\\hit(P;\\xi)=\\Prob(m(Y) \\geq c)$, where\n$m(Y)=\\max_{i=1,\\dots,T}R_i(Y)$ is the maximum value of the chart up\nto time $T$,\n$R_i(Y)=\\sum_{j=1}^iY_j-\\min_{0\\leq k\\leq i}\\sum_{j=1}^kY_j$ is the\nvalue of the CUSUM chart at time $i$, $Y=(Y_1,\\dots,Y_T)$,\n$Y_t=\\frac{X_t-\\xi_1-\\Delta\/2}{\\xi_2}$ and $X_1,\\dots,X_T \\sim P$ are the\nindependent observations.  The threshold needed to achieve a certain hitting\nprobability $\\beta \\in (0,1)$ is $c_{\\hit}: D_q\\times \\Xi\\to \\mathbb{R}$,\n$c_{\\hit}(P;\\xi)=\\inf\\{c>0:\\hit(P;\\xi)\\leq \\beta\\}$.\n\nThe setup for the nonparametric bootstrap is as follows. $W_{n}$ is an\n$n$-variate multinomially distributed random vector with probabilities\n$1\/n$ and $n$ trials. The resampled distribution is $\\hat\nP^{\\ast}=\\frac{1}{n}\\sum_{j=1}^nW_{nj}\\delta_{X_{-j}}$, where $\\delta_x$\ndenotes the Dirac measure at $x$.\n\n\nThe following lemma shows\ncondition a) of  Theorem \\ref{th:main},\nthe Hadamard differentiability of $\\hit$ and $c_{\\hit}$.\n\\begin{lemma}\n\\label{le:HaddiffCUSUMhit}\nFor every $P\\in D_q$, and every $\\xi \\in \\mathbb{R}\\times(0,\\infty)$, the\nfunction $\\hit$ is Hadamard differentiable at $P$ around $\\xi$\ntangentially to $D_0=\\{H:\\mathbb{R}\\to \\mathbb{R}: H\\text{ continuous}, \\lim_{t\\to\n  \\infty}H(t)=\\lim_{t\\to-\\infty}H(t)=0\\}$.  If, in addition, $P$ has a\ncontinuous bounded positive derivative $f$ with $f(x)\\to 0$ as $x\\to\n\\pm \\infty$, then $c_{\\hit}$ is also Hadamard differentiable at $P$\naround $\\xi$ tangentially to $D_0$.\n\\end{lemma}\nThe proof  is in Appendix \\ref{sec:haddifhitprobex}, with preparatory results in\nAppendix \\ref{sec:chain-rule} - \\ref{sec:diff-hitt-prob}.\n\nConditions b) and c) of Theorem \\ref{th:main} follow directly from empirical process theory,\nsee e.g.\\  \\cite[p.17,Theorems 2.6 and 2.7]{Kosorok2008ItE}.\n\\cc{To see conditions b) and c) of  Theorem \\ref{th:main}, we can argue as follows.\n  In the language of empirical process theory, consider ${\\cal\n    F}=\\{\\mathbb{R}\\to \\mathbb{R}, x\\mapsto 1_{(-\\infty,a]}(x):a\\in \\mathbb{R}\\}$ and let\n  $l_{\\infty}({\\cal F})$ be the set of all bounded function ${\\cal\n    F}\\to\\mathbb{R}$.  As $\\cal F$ can be identified with $\\mathbb{R}$, we can\n  idenfity $l_{\\infty}({\\cal F})$ with $l_{\\infty}(\\mathbb{R})$, the set of\n  bounded functions $\\mathbb{R}\\to \\mathbb{R}$.  By \\cite[p.17]{Kosorok2008ItE}, $\\cal F$ is\n  Donsker, i.e.\\ if $X_1,\\dots,X_n\\sim P$ independently, and letting\n  $P_n=\\frac{1}{n}\\sum_{i=1}^n\\delta_{X_i}$ be the corresponding\n  empirical measure, then $G_n=\\sqrt{n}(P_n-P)\\leadsto G$ in\n  $l_{\\infty}(\\cal F)$ (or equivalently in $l_{\\infty}(\\mathbb{R})$) for some\n  $G$.\n\n  thus $G_n$ is considered a random element in $l_{\\infty}({\\cal\n      F})$ (or equivalently $l_{\\infty}(\\mathbb{R})$), via\n    $\\sqrt{n}(P_n-P)(1_{(-\\infty,a]})=\\sqrt{n}(P_n((-\\infty,a])-P((-\\infty,a]))$\n\n  Now, Theorems 2.6 and 2.7 of \\cite{Kosorok2008ItE} give conditional convergence\n  results for the nonparametric bootstrap, i.e. they show that\n$\\hat G_n\\condweakconv{\\hat P} G$ in $l_{\\infty}({\\cal F})$ and that the sequence $\\hat G_n$ is asymptotically measurable.\nSufficient conditions for c)  are e.g.\\ given in Theorems 3.6.1 and 3.6.2 on p.347 of \\cite{Vaart1996WCa}\n}\nCondition e)  is satisfied as well, see  bottom of p.189  and after Theorem 10.4 (p.184) of \\cite{Kosorok2008ItE}.\n\nVerifying condition d) in full is  outside the scope of the present paper.\nA starting point could be the fact that by\nthe Donsker theorem, $Z\\sim G\\circ P$, where $G$ is a Brownian bridge.\n\\cc{We would need to consider the derivative in Lemma\n  \\ref{le:diffhitprob} and in Lemma \\ref{le:Haddiffinversemap}.}\n\n\n\n\\subsubsection{CUSUM charts with normally distributed observations}\n\\label{sec:cusum-charts-with}\nIn this section, we consider a similar setup to the monitoring based\non (\\ref{eq:discrCUSUM_meanshift}) considered in the previous\nsubsection with the difference that we now use parametric assumptions.\nMore specifically,  the observations $X_i$ follow a normal\ndistribution with unknown mean $\\mu$ and variance $\\sigma^2$. We will\nuse this both for computing the properties of the chart as well as in\nthe bootstrap, which will be a parametric bootstrap version of\nAlgorithm \\ref{alg:Bootstrap}.\n\n\nThe distribution of the observations can be identified with its\nparameters which we estimate by $\\hat P = (\\hat \\mu, \\hat \\sigma^2)$,\nwhere $\\hat \\mu=\\frac{1}{n}\\sum_{i=1}^nX_{-i}$ and $\\hat\n\\sigma^2=\\frac{1}{n-1}\\sum_{i=1}^n(X_{-i}-\\hat \\mu)^2$.  The set of\npotential parameters is $D_q=\\mathbb{R}\\times(0,\\infty)$ which is a subset of\nthe Euclidean space $D=\\mathbb{R}^2$. The parameters needed to run the chart\n(\\ref{eq:discrCUSUM_meanshift}) are just the same as the one needed to\nupdate the distribution, thus $\\Xi=D_q$ and $\\xi:D_q\\to \\Xi,\n(\\mu,\\sigma)\\mapsto (\\mu,\\sigma)$ is just the identity.\n\nAs before, we are interested in hitting probabilities within the first\n$T$ steps.  Using the function $\\hit$ defined in the previous\nsubsection, we can write the hitting probability in this parametric\nsetup as $\\hit^N:D_q\\times \\Xi\\to\\mathbb{R}$, $(\\mu,\\sigma;\\xi)\\mapsto\\hit(\n\\Phi_{\\mu,\\sigma^2};\\xi)$, where $\\Phi_{\\mu,\\sigma^2}$ is the cdf of\nthe normal distribution with mean $\\mu$ and variance $\\sigma^{2}$ and\nthe superscript $N$ stands for normal distribution.  Furthermore,\nusing $c_{\\hit}$ from the previous subsection, the threshold needed to\nachieve a given hitting probability is $c_{\\hit}^N:D_q\\times\\Xi\\to\\mathbb{R}$,\n$(\\mu,\\sigma;\\xi)\\mapsto c_{\\hit}(\\Phi_{\\mu,\\sigma^2};\\xi)$.\n\nThe resampling is a parametric resampling. To put this in the framework of the main theorem, we let $W_n=(W_{n1},\\dots,W_{nn})$, where\n$W_{n1},\\dots,W_{nn}\\sim N(0,1)$ are independent. The\nresampled parameters are then  $\\hat\n\\mu^{\\ast}_n=\\frac{1}{n}\\sum_{i=1}^nX_{ni}^{\\ast}$ and $\\hat \\sigma^{\\ast\n  2}_n=\\frac{1}{n-1}\\sum_{i=1}^n(X^{\\ast}_{ni}-\\hat \\mu^{\\ast}_n)^2$\nwhere $ X^{\\ast}_{ni}=\\hat P_2W_{ni}+\\hat P_1$.\n\nThe following lemma shows that condition a) of  Theorem \\ref{th:main} is  satisfied.\n\\begin{lemma}\n\\label{le:HaddiffCUSUMhitNORMAL}\nFor every $\\theta\\in \\mathbb{R}\\times (0,\\infty)$ and every  $\\xi \\in \\mathbb{R}\\times(0,\\infty)$,\nthe functions $\\hit^N$ and $c_{\\hit}^{N}$ are Hadamard differentiable at $\\theta$ around $\\xi$.\n\\end{lemma}\nThe proof can be found in Appendix \\ref{sec:haddifhitprobex}, using again the preparatory results of\nAppendix \\ref{sec:chain-rule} - \\ref{sec:diff-hitt-prob}.\n\nConcerning the other conditions of Theorem \\ref{th:main}: Condition b)\ncan be shown using standard asymptotic theory, e.g.\\ maximum likelihood\ntheory, which will yield that $Z$ is normally distributed. \\cc{could\n  argue via the $(\\hat \\mu, (n-1)\/n\\hat \\sigma^2)$ being the MLE}\nCondition c) is essentially the requirement that the parametric\nbootstrap of normally distributed data is working.  \\cc{This should be\n  easy to shown by arguing conditionally on the estimators. There may\n  be something in vdVaart, asymptotic statistics - but he is just\n  using nonparametric resampling.}  As $Z$ is a normally distributed\nvector, condition d) holds unless $q'$ equals 0.  Condition e) is\nsatisfied, as the mapping $W_n\\mapsto\\hat P^{\\ast}(\\hat P, W_n) =\n(\\hat\\mu_n^{\\ast}, \\hat\\sigma_n^{\\ast 2})$ is continuous and hence\nmeasurable.\n\n\n\n\n\n\n\\subsubsection{Setup for Shewhart charts}\nFor Shewhart charts, the same setup as in the previous two sections\ncan be used, the only difference is the choice of $q$. Conditions b),\nc) and e) are as in the previous two sections.  We conjecture that it is possible to show the  Hadamard\ndifferentiability  more directly, as the properties are\navailable in closed form, see Section \\ref{subsubsec:Shewhart}.\n\n\n\\cc{\nWith $G=1-p$,\n $\\hit(G)=(c\\mapsto 1-G(c)^T)$,  $\\hit'(G)(H)=(c\\mapsto -G(c)^{T-1}H(c))$,\n\n\n $\\frac{\\partial}{\\partial c} \\hit(G)(c)=-T G(c)^{T-1}g(c)$\n\n $\\ARL(G)=(c\\mapsto\\frac{1}{1-G(c)})$, $\\ARL'(G)(H)=(c\\mapsto\\frac{1}{(1-G(c))^2}H(c))$ \\cc{see \\cite[Lemma 3.9.25]{Vaart1996WCa}}\n $\\frac{\\partial}{\\partial c}\\ARL(G)(c)=\\frac{1}{(1-G(c))^2}g(c)$\n}\n\n\n\n\\section{Simulations for homogeneous observations}\n\\label{sec:simulsingle}\n\n\nWe now illustrate our approach by some simulations using\nCUSUM charts. The simulations were done in R \\citep{R}.\n\n\nWe use two past sample sizes, $n=50$ and\n$n=500$. The in-control distribution of $X_t$ is $N(0,1)$ and\nwe use 1000 replications and $B=1000$ bootstrap\nreplications. We employ both the parametric bootstrap and the\nnonparametric bootstrap mentioned in the previous sections.  For the\nparametric bootstrap we used the sample mean and sample standard\ndeviation of $X_{-n},\\dots,X_{-1}$ as estimates for the mean and the standard deviation of\nthe observations.\n\nFor the performance measures $\\ARL$, $\\log(\\ARL)$, $\\hit$ and\n$\\logit(\\hit)$ we use a threshold\nof $c=3$. For $c_{\\ARL}$ we calibrate to  an $\\ARL$ of $100$\nin control and for $c_{\\hit}$ we calibrate to a false alarm probability of\n$5\\%$ in 100 steps.\n\n\n\nWe use the CUSUM chart (\\ref{eq:discrCUSUM_meanshift}) with $\\Delta=1$\nand $\\mu$ and $\\sigma$ estimated from the past data. To compute\nproperties such as $\\ARL$ or hitting probabilities, we  use a\nMarkov chain approximation (with 75 grid points), similar to the one\nsuggested in \\cite{BROOK1972atp}\\cc{there is a precise description of\n  a grid that is being used in that paper - I think we are using\n  something similar but most likely not completely identical}. \\cc{We\nchecked that this gave very good  approximations.}\n\n\n\n\n\n\\subsection{Coverage probabilities}\n\\label{subsec:simulcoverage}\n\n\nTable \\ref{tab:covprob_simnormal} contains coverage probabilities of\nnominal 90\\% confidence intervals. These are the one-sided lower confidence\nintervals given by (\\ref{eq:onesidedapproxconfint}), except for\n$q=\\ARL$ and $\\log(ARL)$ where the corresponding upper interval is used.\n\n\n\\begin{table}\n    \\caption{Coverage probabilities of nominal 90\\% confidence intervals for CUSUM charts.\n  \\label{tab:covprob_simnormal}}\n\\begin{center}\n\\parbox{0.58\\textwidth}{\n    \\input{simpaper\/tablepaper_stdNormal_CenterScale.tex}\\\\\n      The standard deviation of the results is roughly 0.01.\n }\n    \\end{center}\n\\end{table}\n\n\nIn the parametric case, for $n=50$, the coverage probabilities are\nsomewhat off for untransformed versions, in particular for $q=\\ARL$.\nUsing $\\log$ or $\\logit$ transformations seems to improve the coverage\nprobabilities considerably.  In the parametric case, for $n=500$, all\ncoverage probabilities seem to be fine, except for $q=\\ARL$, which\nalthough  shows some marked improvement compare to $n=50$.  In the\nnonparametric case, a similar picture emerges, but the coverage\nprobabilities are a bit worse than in the parametric case.\n\n\n\\begin{remark}\n\\label{rem:scalingdoesnotmatter}\n  For $q= \\log(c_{\\ARL})$ and $q= \\log(c_{\\hit})$ the division by\n  $\\hat\\sigma$ in\n  (\\ref{eq:discrCUSUM_meanshift}) could be skipped without making a\n  difference to the coverage probabilities.  Indeed, the division by\n  $\\hat\\sigma$ just scales the chart (and the resulting threshold) by\n  a multiplicative factor, which is turned into an additive factor by\n  $\\log$ and which then cancels out in our adjustment.\n\\end{remark}\n\n\n\n\n\\subsection{The benefit of an adjusted threshold}\nIn this section, we consider both the in- and out-of-control\nperformance of CUSUM charts when adjusting the threshold $c$ to give a\nguaranteed in-control $\\ARL$ of 100.  Setting the threshold is, in our\nopinion, the most important practical application of our method.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/adjusted_unadjusted_ARL_CenterScale_boxplot.pdf}\n  \\caption{Distribution of the conditional $\\ARL$ for CUSUMs in a\n    normal distribution setup. Thresholds are calibrated to an\n    in-control $\\ARL$ of 100.  The adjusted thresholds have a\n    guarantee of 90\\%.  A log transform is used in the calibration.\n    The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%, 90\\% and\n    97.5\\% quantiles.  The white boxplots are in-control, the gray\n    boxplots out-of-control.}\n  \\label{fig:adjusted_unadjusted_ARL}\n\\end{figure}\n\n\nFigure \\ref{fig:adjusted_unadjusted_ARL} shows average run lengths\nfor both  the unadjusted threshold $c(\\hat P;\\hat \\mu, \\hat \\sigma)$ and the\nadjusted threshold $\\exp(\\log( c(\\hat P;\\hat \\mu, \\hat \\sigma))-p^{\\ast}_{0.1})$, where $p^{\\ast}_{0.1}$ is computed via the parametric\nbootstrap using $q=\\log(c_{\\ARL})$.  Thus, with  90\\%  probability, the  adjusted threshold should\nlead to an $\\ARL$ that is above 100. In this and in all  following\nsimulations,\nthe out-of-control ARL refers\nto the  situation where the chart is  out-of-control from the\nbeginning, i.e.\\ from\ntime 0 onwards.\n\nFor the unadjusted threshold, the desired in-control average run\nlength is only reached in roughly half the cases.  More importantly,\nfor $n=50$, the probability of having an in-control $\\ARL$ of below $50$\nis greater than 20\\%.\n\n\nWith the adjusted threshold we should get an average run length of at\nleast 100 in 90\\% of the cases.  This is achieved.  The\nout-of-control $\\ARL$ using the adjusted thresholds increases only\nslightly compared to the unadjusted version.\n\n\nSimilarly to Remark \\ref{rem:scalingdoesnotmatter}, removing the\nscaling by $\\hat \\sigma$ in (\\ref{eq:discrCUSUM_meanshift}) would not\nchange the results of this section.\n\n\n\\subsection{Nonparametric bootstrap - advantages and disadvantages}\n\nIn this section, we compare the parametric and the non-parametric\nbootstrap.  We consider CUSUM charts that are calibrated to an\nin-control average run length of 100 assuming a normal distribution.\nWe use the adjusted threshold $\\exp(\\log( c_{\\ARL}(\\hat P;\\hat\n\\mu, \\hat \\sigma))-p^{\\ast}_{0.1})$.\n\nFigure \\ref{fig:par_nonpar_ARL} shows the distribution of $\\ARL$ for\n$n=50$ and $n=500$ for both the parametric bootstrap that assumes a normal\ndistribution of the updates and the nonparametric bootstrap. We consider both a\ncorrectly specified model where $X_t\\sim N(0,1)$ as well as two\nmisspecified models where $X_t\\sim \\text{Exponential}(1)$ and $\\sqrt{20}X_t\\sim\n\\chi^2_{10}$ (all of the $X_t$ have variance 1).  We show both the in- as well as the\nout-of-control performance of the charts.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/par_nonpar_ARL_CenterScale_boxplot.pdf}\n  \\caption{Effects of misspecification. Thresholds are calibrated to\n    an in-control ARL of 100 and adjusted to the estimation error with\n    a guarantee of 90\\%.  A log transform is used in the\n    calibration. The white boxplots are in-control, the gray\n    boxplots are  out-of-control. The boxplots show\n    the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%, 90\\% and 97.5\\% quantiles.}\n  \\label{fig:par_nonpar_ARL}\n\\end{figure}\n\nIn the correctly specified model ($X_t\\sim N(0,1)$), the performance\nof the parametric and the non-parametric chart seems to be almost\nidentical. The only difference is a slightly worse in-control\nperformance for the non-parametric chart for $n=50$.\n\nIn the misspecified model with $X_t\\sim \\text{Exponential}(1)$, the\nparametric chart does not have the desired in-control\nprobabilities. The non-parametric chart seems to be doing well, in\nparticular for $n=500$.  We have a similar results in the other\nmisspecified model, with $\\sqrt{20}X_t\\sim \\chi^2_{10}$.\n\n\n\n\n\n\\section{Regression models}\n\\label{sec:regmod}\n\nIn many monitoring situations, the units being monitored are heterogeneous,\nfor instance when monitoring patients at hospitals or bank customers.\nTo make sensible monitoring systems in such situations, the explainable\npart of the heterogeneity should be accounted for by relevant\nregression models. The resulting charts are often called risk\nadjusted, and an overview of some such charts can be  found in \\cite{Grigg2004oor}.\n\nTo run risk adjusted charts, the regression model needs to be estimated\nbased on past data, and this estimation needs to be accounted for.  Our\napproach for setting up charts with a guaranteed performance applies\nalso to risk adjusted charts,\nand we will in particular look at linear, logistic and survival\nmodels.\n\n\\subsection{Linear models}\n\\label{subsec:linmod}\n\nSuppose we have independent observations $(Y_1,X_1),$  $(Y_2,X_2)$, $\\ldots$,\nwhere $Y_i$ is a response of interest and $X_i$ is a corresponding\nvector of covariates, with the first component usually equal to 1.\nLet $P$ denote the joint distribution of $(Y_i,X_i)$ and suppose that\nin control $\\E(Y_i|X_i)=X_i\\xi$.  From some observation\n$\\kappa$ there is a shift in the mean response to\n$\\E(Y_i|X_i)=\\Delta+X_i\\xi$ for $i=\\kappa,\\kappa+1,\\dots$.\n\nMonitoring schemes for detecting changes in regression models can\nnaturally be based on residuals of the model, see for instance\n\\cite{Brown1975TfT} and \\cite{Horvath2004Mci}.\n We can, for instance,\ndefine a CUSUM to monitor changes in the conditional mean of $Y$ by\n$$S_t=\\max(0, S_{t-1}+Y_t-X_t \\xi - \\Delta\/2), \\quad S_0=0, $$\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\nIn a similiar manner we could also set up charts for\nmonitoring changes in other components of  $\\xi$.\n\nThe parameter vector $\\xi$ is estimated from past in\ncontrol data, e.g.\\ by the standard least squares estimator.  We\nsuggest to use a nonparametric version of the general Algorithm\n\\ref{alg:Bootstrap} with $\\hat P$ being the\nempirical distribution putting weight $1\/n$ on each of the past\nobservations $(Y_{-n},X_{-n}),\\dots,(Y_{-1},X_{-1})$. Resampling is\nthen equivalent to resampling\n$(Y^{\\ast}_{-n},X^{\\ast}_{-n}),\\dots,(Y^{\\ast}_{-1},X^{\\ast}_{-1})$ by\ndrawing with replacement from $\\hat P$.\n\nThe suggested method should  work  even if the linear model is misspecified,\ni.e.\\  $\\E(Y_i|X_i)=X_i\\xi$ does not necessarily hold. The nonparametric\nbootstrap should take  this into account.\n\n\nAn analogous approach can be used for Shewhart charts. In settings\nwhere it is reasonable to consider the covariate vector to be\nnon-random one could alternatively use bootstrapping of residuals, see\nfor example \\cite{Freedman1981BRM}.\n\n\n\\subsubsection{Theoretical considerations}\n\nObtaining precise results is more demanding than in the examples\nwithout covariates in Section~\\ref{sec:th:examples}. We only\ngive an idea of the setup that might be used.\n\n\nThe set of distributions of the observations $D_q$ can be chosen as the\nset of cdfs on $\\mathbb{R}^{d+1}$ with finite second moments, where $d$ is the dimension of the covariate. The first cdf corresponds to the responses, the others to the covariates. $D_q$ is contained\nin the vector space $D=l_{\\infty}(\\mathbb{R}^{d+1})$, the set of bounded functions $\\mathbb{R}^{d+1}\\to \\mathbb{R}$.\nThe parameters needed to run the chart are the regression coefficients contained in the set  $\\Xi=\\mathbb{R}^d$.\nThese parameters are obtained from the distribution of the observations via  $\\xi:D_q\\to \\Xi$, $F\\mapsto\n(E(X^TX))^{-1}E(X Y)$ where $(Y,X)\\sim F$ where $X$ is considered to\nbe a row vector.\n\n\nWe conjecture that the conditions of Theorem \\ref{th:main} are broadly\nsatisfied if the cdf  of $Y-X\\xi$ is differentiable and if for\nthe property $q$ we use  hitting probabilities or thresholds to\nachieve a given hitting probability.  In particular, it should be\npossible to show Hadamard differentiability similarly to Lemma\n\\ref{le:HaddiffCUSUMhit}: write $q$ as concatenation of two functions\nand use the chain rule in Lemma \\ref{le:chainrule}.  The first mapping\nreturns the distribution of the updates of the chart depending on\n$F\\in D_q$ and $\\xi\\in \\Xi$ via $(F;\\xi)\\mapsto {\\cal L}(Y-X\\xi-\n\\Delta)$, where ${\\cal L}$ denotes the law of a random variable.  The\nsecond takes the distribution of the updates and returns the property of\ninterests. The differentiability of the second map has been shown in\nLemmas \\ref{le:Haddiffinversemap} and  \\ref{le:diffhitprob}.\n\n\n\\subsubsection{Simulations}\n\\label{example:CUSUMLinReg}\n\nWe illustrate the performance of the bootstrapping scheme using a\nCUSUM and the linear in-control model\n$Y=X_{1}+X_{2}+X_3+\\epsilon$. Let $\\epsilon\\sim N(0,1)$, $X_{1}\\sim\n\\text{Bernoulli}(0.4)$, $X_2\\sim U(0,1)$ and $X_3\\sim N(0,1)$, where\n$X_1,X_2,X_3$ and $\\epsilon$ are all independent. The out-of-control\nmodel is $Y=1+X_{1}+X_{2}+X_3+\\epsilon$,\ni.e. $\\Delta=1$. Figure~\\ref{fig:regression_ARL} shows the distribution\nof the attained ARL for CUSUMs with thresholds calibrated to give an\nin control ARL of 100. We see that the behaviour of the adjusted\nversus unadjusted thresholds are very similar to what we observed for\nthe simpler model in Figure~\\ref{fig:adjusted_unadjusted_ARL}. The\ncoverage probabilities obtained for this regression model, not\nreported here, are also very similar to the covarage probabilities\nreported in Table~\\ref{tab:covprob_simnormal}, though with a tendency\nto be slightly worse.\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/regression_ARL_boxplot.pdf}\n  \\caption{Distribution of the conditional $\\ARL$ for CUSUMs in a\n    linear regression setup. Thresholds are calibrated to an\n    in-control $\\ARL$ of 100. A log transform is used in the\n    calibration.  The adjusted thresholds have a guarantee of\n    90\\%. The white boxplots are in control, the gray\n    out-of-control. The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%,\n    75\\%, 90\\% and 97.5\\% quantiles.}\n  \\label{fig:regression_ARL}\n\\end{figure}\n\n\n\n\n\n\\subsection{Logistic regression}\n\\label{subsec:logreg}\n\nControl charts, in particular CUSUM charts, based on logistic\nregression models are popular for modelling of binary\noutcomes in medical contexts. See e.g.\\ \\cite{Lie1993nsp}, \\cite{Steiner2000Msp},\n\\cite{Grigg2004oor} and \\cite{Woodall2006Tuo}.\n\nSuppose we have independent observations $(Y_1,X_1),(Y_2,X_2),\\ldots,$\nwhere $Y_i$ is a binary response variable and $X_i$ is a corresponding\nvector of covariates. Further, suppose that in control the log odds\nratio is $\\logit(\\Prob(Y_i=1|X_i))=X_i\\xi$, and that from some observation\n$\\kappa$ there is a shift in the log odds ratio to\n$\\logit(\\Prob(Y_i=1|X_i))=\\Delta+X_i\\xi$ for $i=\\kappa,\\kappa+1,\\dots$\n\nA CUSUM to monitor changes in the odds ratio can be defined by \\citep{Steiner2000Msp}\n$$S_t=\\max(0, S_{t-1}+R_t), \\quad S_0=0, $$\nwhere $R_t$ is the log likelihood ratio between the in-control and out-of-control model for observation $t$. More precisely\n$$\n\\exp(R_t)=\\frac{\\exp(\\Delta+X_t\\xi)^{Y_t}\/(1+\\exp(\\Delta+X_t\\xi))}{\\exp(X_t\\xi)^{Y_t}\/(1+\\exp(X_t\\xi))}\n=\\exp(Y_t\\Delta)\\frac{1+\\exp(X_t\\xi)}{1+\\exp(\\Delta+X_t\\xi)}.\n$$\n\nThe parameter vector $\\xi$ is estimated from past in-control\ndata by e.g.\\  the standard maximum likelihood estimator. The same\nnonparametric bootstrap approach as described for the linear model in\nSection~\\ref{subsec:linmod} can now be applied to this CUSUM based on\nthis logistic regression model. Moreover, this approach would also\napply to control charts based on other generalized linear models, for\ninstance Poisson regression models for monitoring count data. The only\namendment needed is to replace $R_t$ by the relevant log likelihood\nratio.\n\n\n\nWe have run simulations, not reported here, based on the same covariate\nspecifications as in Section~\\ref{example:CUSUMLinReg}.  The results\nare similar to the results for the linear model of\nSection~\\ref{example:CUSUMLinReg}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Survival analysis models}\n\nRecently, risk adjusted control charts based on survival models have\nstarted to appear, see\n\\cite{Biswas2008rCi,Sego2009Rmo,Steiner2009ras,Gandy2010ram}. In none\nof these papers any adjustment for estimation error is done, but\n\\cite{Sego2009Rmo} are  illustrating, by simulations, the impact of\nestimation error on the attained average run length  for the\naccelerated failure time model based CUSUM studied in their paper.\n\n\nIn the following, we provide a brief simulation example of our\nadjustment in a survival setup where we use the methods described in\n\\cite{Gandy2010ram}.\n\nWe observe the survival of individuals over a fixed time interval of\nlength $n$ (we will use $n=100$ and $n=500$).  Individuals arrive at\ntimes $B_i$ (in our simulation according to a Poisson process with\nrate $1$), and survive for $T_i$ time units. Individuals may arrive\nbefore the observation interval, as long as $B_i+T_i$ is after the\nstart of the observation interval.  Right-censoring, at $C_i$ time\nunits after arrival, is taking place after a maximum follow-up time of\n$t=60$ time units or after the individuals leave the observation\ninterval.  In the simulation, the true hazard rate of $T_{i}$ is\n$h_i(t)= 0.1\\exp( X_{1i}+X_{2i})$, where $X_{1i}\\sim\n\\text{Bernoulli}(0.4)$ and $X_{2i}\\sim N(0,1)$ are covariates.\n\nBased on the observed data we fit a Cox proportional hazard model\nwith $X_{1i}$ and $X_{2i}$ as covariates and nonparametric baseline,\ngiving estimates $\\hat \\beta$ for the covariate effects and $\\hat\n\\Lambda_0(t)$ for the the integrated baseline.\n\nWe use the CUSUM chart described in\n\\cite{Gandy2010ram} against a proportional alternative with\n$\\rho=1.25$.  The parameters\nneeded to run the chart are $\\xi=(\\beta, \\Lambda_0)$ estimated by\n$\\hat\\xi=(\\hat \\beta, \\hat \\Lambda_0)$.\nTo be precise, the chart signals at time\n$\\tau=\\inf\\{t>0:S(t)\\geq c\\}$, where\n$S(t)=R(t)-\\inf_{s\\leq t}R(s)$,\n$\n                              R (t ) = \\log(\\rho ) N (t ) - (\\rho - 1)\n\\Lambda(t),\n$\n $N (t )$ is the number of events until time $t$ and $\\Lambda(t ) =\n\\sum_{i} \\exp( \\beta_1X_{i1}+ \\beta_2X_{2i})\n\\Lambda_0(\\min((t-B_i)^{+},T_i,C_i))$.\n\n\nWe are interested in finding a threshold that gives a desired\nhitting probability, i.e. we use $q=c_{\\hit}$.  We compute\n$c_{\\hit}(P,\\xi)$ via simulations (simulate new data from $P$ and run\nthe chart with $\\xi$). We estimate the threshold needed to get a 10\\%\nfalse alarm probability in $n$ time units in control, by the 90\\% quantile of 500\nsimulations of the maximum of the chart.\n\nTo resample, we resample individuals with replacement. We use 500\nbootstrap samples.  Figure \\ref{fig:hitprobsurvanal} shows the\ndistribution of the resulting hitting probabilities based on 500\nsimulated observation intervals.\n\n\n\n\\begin{figure}\n \\centering\n\\includegraphics[width=\\linewidth]{simpaper\/adjusted_unadjusted_coxhitprob_boxplot.pdf}\n\\caption{Distribution of the conditional hitting probability for\n  survival analysis CUSUMs. Thresholds are calibrated to an in-control\n  hitting probability of 0.1.  The adjusted thresholds have a\n  guarantee of 90\\%.  The white boxplots are in control, the gray\n  out-of-control. The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%,\n  90\\% and 97.5\\% quantiles. \\label{fig:hitprobsurvanal}}\n\\end{figure}\n\n\nIn control, without the adjustment, the desired false\nalarm probability of 0.1 is only reached in roughly 60\\% of the cases.  The\nbootstrap correction seems to work fine, leading to a false alarm probability\nof at most 10\\% in roughly 90\\% of the cases.  As expected, increasing the\nlength of the fitting period and the length of time the chart is run\nfrom $n=100$ to $n=500$ results in higher out-of-control hitting probabilities.\n\n\n\nIf the length of the fitting period and the deployment period of the chart differ then\na somewhat more complicated resampling procedure needs to be used.\nFor example, one could\n resample arrival times and survival times\/covariates separately.\nThe former could be done by assuming a Poisson process as arrival time and the\nlatter either by resampling with replacement or by sampling from an estimated\nCox model and an estimated censoring distribution.\n\n\n\n\n\n\\cc{\n              In the survival analysis case with a proportional alternative, the chart is  based on\n              $$\n              R(t) = \\log(\\rho ) N (t )- (\\rho - 1)\\hat \\Lambda (t ),\n              $$\n              After the time transformation of $N$ to the standard Poisson process $\\tilde N$ this becomes\n              \\begin{align*}\n                \\tilde R(t) = R(\\Lambda^{-1}(t))=\\log(\\rho)\\tilde\n                N(t)-(\\rho-1)\\hat\\Lambda(\\Lambda^{-1}(t))\n              \\end{align*}\n              Thus the nice Markov-approximation will not work $\\hat\n              \\Lambda$ and $\\Lambda$ will not have independent\n              increments. Therefore we needed to simulate.\n}\n\n\n\n\n\n\n\n\n\n\\section{Conclusions and discussion}\n\\label{sec:conclusion}\n\n\nWe have presented a general approach for handling estimation error in\ncontrol charts with estimated parameters and unknown in-control\ndistributions. Our suggestion is, by bootstrap methods, to tune the\nmonitoring scheme to guarantee, with high probability, a certain\nconditional in-control performance (conditional on the estimated\nin-control distribution). If we apply a nonparametric bootstrap, the\napproach is  robust against model specification error.\n\nIn our opinion, focusing on a guaranteed conditional in-control  performance is\ngenerally more relevant than focusing on some average\nperformance, as an estimated chart usually is run for some time without\nindependent reestimation. Our approach can also easily be adapted to\nmake for instance bias adjustments. Bias adjustments, in\ncontrast to guaranteed performance, tend to\nbe substantially influenced by tail behaviour for heavy\ntailed distributions which for instance the average run length has.\nThis implies that the bias adjustments need not be useful in the\nmajority of cases as the main effect of the adjustment is to adjust\nthe tail behaviour.\n\n\nWe have in particular demonstrated our approach for various variants\nof Shewhart and CUSUM charts, but the general approach will\napply to other charts as well. The method is generally\nrelevant when the in-control distribution is unknown and the conditions of\nTheorem~\\ref{th:main} hold. We conjecture that this will be the case\nfor many of the most commonly used control charts.\n\\cc{for instance be the case for charts like  EWMA charts \\citep{Roberts1959CCT}, general\nlikelihood ratio based charts \\cite{Frisen1991Ops,Frisen2003SSO},  the\nSets method \\citep{Chen1978SSC,Grigg2004ARA}}\nNumerous extensions of control charts to other settings exist, for example\nto other  regression\nmodels, to  autocorrelated data, to multivariate data.\nWe do  conjecture that our approach will also apply  in\nmany of these settings.\n\n\n\n\n\n\n\n\\small\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFew-shot learning is a research challenge that assesses a model's ability to quickly adapt to new tasks or new environments.\nThis has been the leading area where researchers apply meta-learning algorithms - where a strategy that learns to learn quickly is likely to be the most promising.\nHowever, it was recently shown by Tian et al. \\cite{Tian2020} that a model with a good embedding is able to match and beat many modern sophisticated meta-learning algorithms on a number of few-shot learning benchmarks.\nIn addition, there seems to be growing evidence that this is a real phenomena \\cite{Chen2019, Chen, Dhillon2019, Huang2019}.\nFurthermore, analysis of the representations learned by Model Agnostic Meta-Learning (MAML) \\cite{maml} (on few-shot learning tasks) revealed that MAML mainly works by learning a feature that is re-usable for many tasks \\cite{Raghu} -- what we are calling a good embedding in this paper.\n\nThese discoveries reveal a lack of understanding on when and why meta-learning algorithms work and are the main motivation for this work. \nIn particular, our contributions are:\n\n\\begin{enumerate}\n    \\item We show that it is possible to define a synthetic task that results in lower degree of feature re-use, thus suggesting that \n   current few-shot learning benchmarks might not have the properties needed for the success of meta-learning algorithms;\n   \\item Meta-overfitting occurs when the number of classes (or concepts) are finite, and the issue disappears once the tasks have an unbounded number of concepts;\n   \\item More adaptation for MAML does not necessarily result in representations that change significantly or even perform better at meta-test time.\n\\end{enumerate}\n\n\\section{Unified Framework for Studying Meta-Learning and Absolute Performance}\\label{metric_ml}\n\nWe propose that future work on meta-learning should not only report absolute performance, but also quantify and report the degree of meta-learning.\nWe hypothesize this is important because previous work \\cite{Tian2020} has observed that supervised learning (while only fine-tuning the final layer) is sufficient to solve current meta-learning benchmarks.\nThis might give the potentially false impression that current trends in meta-learning are irrelevant.\nTo avoid that, we hypothesize that measuring the degree of meta-learning we provide in this section will provide a step forward in explaining those important observations.  \n\nIn this work, we make an important first step by emphasizing the analysis done by \\cite{Raghu}, by defining the degree of meta-learning as the normalized degree of change in the representation of a neural network $nn_{\\theta}$ after using meta-adaptation $A$:\n\\begin{equation}\\label{eq_ml}\n    ML(nn_{\\theta}) = \\mathrm{Diff}( nn_{\\theta},  A(nn_{\\theta}) ).\n\\end{equation}\nIn this work we set $ML(nn_{\\theta})$ to be distance based Canonical Correlation Analysis (dCCA) \\cite{Morcos}.\nNote that dCCA is simply 1 minus CCA to switch the similarity based metric to a difference based metric between 0 and 1.\n\n\\section{Benchmarks that Require Meta-Learning}\n\n\\subsection{Background}\n\n\\textbf{Model-Agnostic Meta-Learning (MAML).}\nThe MAML algorithm \\cite{maml} attempts to meta-learn an initialization of parameters for a neural network that is primed for quick gradient descent adaptation. \nIt consists of two main optimization loops: 1) an outer loop used to prime the parameters for fast adaptation, and 2) an inner loop that does the fast adaptation.\nDuring meta-testing, only the inner loop is used to adapt the representation learned by the outer loop.\n\n\\textbf{Feature re-use.} \nIn the context of MAML, this term usually means that the inner loop provides little adaptation during meta-testing, when solving an unseen task.\nIn particular, Raghu et al. \\cite{Raghu} showed that MAML has little representation change as measured with CCA and CKA after adaptation, during meta-testing on the MiniImageNet few-shot learning benchmark.\n\n\\subsection{Motivation for Our Work}\n\nThe analysis by Raghu et al. \\cite{Raghu} showing that MAML works mainly by feature re-use is the main motivation for our work.\nHowever, we argue that their conclusion is highly dependent on the data set (or benchmark) used.\nThis motivates us to construct a different benchmark and show that by {\\em only} constructing a different benchmark, we can exhibit lower degrees of feature re-use in a statistically significant way.\nTherefore, our goal will be to show a lower degree of feature re-use than them.\nIn particular, their work \\cite{Raghu} showed that the representation layer of a neural network trained with MAML had a dCCA of $0.1 \\pm 0.02$ \\cite{Raghu}.\n{\\em Therefore, our concrete goal will be to show that the dCCA on our task is greater than $0.12$}.\nIf this is achieved, it is good evidence that this new benchmark benefits from meta-learning and can be detected at a higher degree than previous work \\cite{Raghu} in a statistically significant way.\nThis is our main result of this section and is discussed in detail in Section \\ref{main_result}.\n\n\\subsection{Synthetic Task that Requires Meta-learning}\n\n\\subsubsection{Overview and Goal}\\label{goals}\nThe main idea is to sample functions to be approximated, such that the final layer needs little or no adaptation, but the feature layers require a large amount of adaptation.\nThis type of task would forcibly require that the meta-learner learns a representation that requires the feature layers to change to achieve good meta-test performance (i.e., it cannot rely solely on feature re-use).\nTherefore, to perform well, not only would it be good to adapt the representation layers, but additionally performance is likely to be obtained from a (meta-learned) initialization that is primed to change flexibly.\nIn summary, our goal will be to construct synthetic tasks such achieving high meta-test performance and detectable meta-learning - as discussed in section \\ref{metric_ml} - one needs to go beyond feature sharing.\n\n\\subsubsection{Definition}\\label{def}\n\nIn this section, we describe a family of benchmarks that exhibits detectable meta-learning and requires more than a re-usable representation layer to be solved.\nWe propose a set of regression functions specified as a fully connected neural network (FCNN), such that the magnitude of parameters of the representation are larger than the head.\nIn particular, we sample the parameters of the representation layer from a Gaussian with a larger standard deviation, compared to the parameter sampling of the head.\nWe define the representation layer to be the first $L-1$ layers, and the head to be the final layer.\n\nNext, we describe the process to sample one function (regression task\n) from a Gaussian distribution.\nWe have two pairs of benchmark parameters $[(\\mu^{(1)}, \\sigma^{(1)}), (\\mu^{(2)}, \\sigma^{(2)})]$:\n$(\\mu^{(1)}, \\sigma^{(1)})$ to sample the parameters for the representation layer, \nand $(\\mu^{(2)}, \\sigma^{(2)})$ to sample the parameters for the final layer.\nThen each regression task $f^{(t)}$ (with index $t$) is sampled as follows:\n\\begin{itemize}\n    \\item Sample the representation parameters $w^{(l)} \\sim N(\\mu^{(1)}, \\sigma^{(1)})$ for each layer $l \\in [L-1]$ in the representation layers\n    \\item Sample the final layer parameters $w^{(L)} \\sim N(\\mu^{(2)}, \\sigma^{(2)})$\n\\end{itemize}\n\nThe idea is that for some $c \\in \\mathbb R$ we have $\\sigma^{(1)} > c \\cdot  \\sigma^{(2)}$ such that the variance in tasks is due to the representation layers, and therefore adapting the representation layers is necessary.\nFor all our experiments $\\sigma^{(2)}=1.0$.\nAn example task can be seen in Figure \\ref{fun_reg}.\nDuring meta-training, points are uniformly sampled from $[-1,1]$, and the standard support set and query set are constructed by computing $f^{(t)}_{w}(x)$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.46\\linewidth]{figs\/fun_reg.png}\n\\caption{An example regression task constructed as described in Section \\ref{def}. Addressing such tasks requires high degree of meta-learning.}\n\\label{fun_reg}\n\\end{figure}\n\n\\subsubsection{Results on Benchmarks that Require Meta-Learning}\\label{main_result}\n\nIn this section, we show a higher degree of meta-learning and a lower degree of feature re-use from an initialization trained with MAML on the benchmarks described in Section \\ref{def}. \nIn particular, we show this in Figure \\ref{best_relu_vs_std} because the dCCA value exhibited is much larger than $0.12$ of previous work \\cite{Raghu}.\n{\\em Most importantly, the results are statistically significant, because the error bars do not intersect with the red dotted line with (worst case) dCCA value of $0.12$. }\nThe red dotted line is the top error band of previous work - i.e. the mean plus the standard deviation.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/best_relu_vs_std.png}\n\\caption{\nShows the of lack of feature re-use and a higher degree of meta-learning, as the standard deviation of the representation layer $\\sigma^{(1)}$  for generating regression.\nThe x-axis is the standard deviation (std) of the parameter $\\sigma^{(1)}$ for generating the tasks for the data sets. \nThe models used for each point in the plot are models selected from early stopping (using the meta-validation MSE loss) when meta-trained with MAML.\nThe models are the same architecture as the target function (4 layers fully connected neural network) with ReLU activation function. \nWe also show the meta-validation loss vs the standard deviation of the task.\nThe dCCA was computed by from the average and standard deviation over the representation layers, in this case the first three layers.\nThe average is across different runs using the same meta-learned initialization.\nThe red dotted line shows the value of $0.12$ that our models have to be statistically significant.\nThe only difference of this figure with respect to figure \\ref{relu_metaoverfitted} is that we selected a model with the best validation here and in the figure \\ref{relu_metaoverfitted} we selected the model in last step.\n}\n\\label{best_relu_vs_std}\n\\end{figure}\n\nNote that a dCCA higher than $0.12$ was observed across all of our experiments in over sixteen different benchmarks. \nIn particular, this happened even in models that had meta-overfitted, e.g., see Figure \\ref{relu_metaoverfitted}.\nThis is strongly suggestive that the benchmarks we defined in Section \\ref{def} require meta-learning, since they do not solely rely on feature re-use to be solved.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]\n                   {figs\/relu_metaoverfitted.png}\n\\caption{\nThis figure supports the main result of the paper because a higher degree of meta-learning and a lack of feature re-use are present -- even in models that are meta-overfitted.\nA meta-overfitted model can be easily obtained in our experiments by selecting a model at the final iteration.\nThe x-axis is the standard deviation (std) of the parameter $\\sigma^{(1)}$ for generating the tasks for the data sets.\nThe red dotted line shows the value of $0.12$ that our models have to be above for statistically significant results that support our claims.\nThe only difference of this figure with respect to figure \\ref{best_relu_vs_std} is that we selected a model in last step (after trough, and it had meta-overfitted) while in figure \\ref{best_relu_vs_std} we select the model with lowest meta-validation loss.\n}\n\\label{relu_metaoverfitted}\n\\end{figure}\n\n\\section{Meta-Overfitting} \\label{meta_overfitting}\n\nIn this section, we show how being armed with the additional metric discussed in Section \\ref{metric_ml}, we are able to identify an increasing gap between the meta-test and meta-train losses\/accuracy -- a term we refer to as \\textit{meta-overfitting}.\nIn particular, this phenomenon is observed when we meta-train models with MAML, and becomes more pronounced as the number of iterations increases.\nWe attribute this to the adaptation, because this increase in the meta-generalization gap is observed in conjunction with the low degree of feature re-use (as discussed in Section \\ref{main_result}), and is most noticeable in our synthetic benchmarks compared to MiniImagent \\cite{Raghu}. \nNote that the dCCA of the models was much larger in our synthetic benchmarks than in MiniImagent.\nIn addition, we show that if the number of regression tasks (in this case functions) is not fixed, then the meta-overfitting issue is no longer observed\n\n\\subsection{Finite Number of Tasks}\\label{finite_metaoverfit}\n\nWhen the number of regression tasks (functions) is finite ($200$ in our experiments), we consistently observe meta-overfitting.\nWe show this in Figure \\ref{meta_overfit1} by increasing the meta-generalization gap (i.e. an increase in the difference between the meta-train and the meta-validation losses).\nThis is consistently observed in over $30$ experiments with a finite number of regression tasks.\n\nFurthermore, meta-overfitting is also observed in a few-shot image recognition benchmark.\nThis is shown in Figure \\ref{overfit_mini} with MiniImagent.\nWith a PyTorch ResNet-18 model, one can observe a meta-generalization gap of about $30\\%$.\nWith a state-of-the-art ResNet-12 \\cite{Tian2020}, the meta-generalization gap is instead about $20\\%$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.55\\linewidth]{figs\/meta_overfit1.png}\n\\caption{\nShows meta-overfitting when the number of tasks (functions) is finite at $200$ regression tasks because the meta-validation loss increases as the meta-train loss decreases.\nIn particular, the dCCA for these models was $0.36 \\pm 0.12$ corresponding to $\\sigma^{(1)}=1.0$. \nThe plot is the learning curve for a 4-layered fully connected neural network trained with MAML \\cite{maml} using episodic meta-learning.\nNote that we use a (large) meta-batch size of $75$ to decrease the noise during training in the figure.\nThe main difference of this figure with figure \\ref{no_overfit} is that in this one has a finite set of tasks using our synthetic benchmark while the other has an infinite set of tasks using the sinusoidal tasks suggested in \\cite{maml}.\n}\n\\label{meta_overfit1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.67\\linewidth]{figs\/resnet18_metaoverfitting_horizontal.png}\n\\caption{ \nShows that meta-overfitting is a real phenomenon in MiniImagent.\nWe interpret this due to the peak in the meta-validation accuracy followed by a decline as the number of iterations increases.\nImportantly, the meta-train accuracy continues to increase as it converges.\nThe model trained is an out-of-the-box PyTorch ResNet-18.\nNote that the higher noise of the meta-validation accuracy is due to having a meta-batch size of $2$ to speed up experiments.\nWe smoothed the meta-validation curve with a TensorBoard smoothing weight of $0.8$.\nWe consistently saw that increases in meta-batch size lead to decreases in noise in the learning curves, but we didn't re-run these experiments since it can take up to a week to reproduce an episodic meta-learning run - even on a Quadro RTX 6000.\n}\n\\label{overfit_mini}\n\\end{figure}\n\n\\subsection{Infinite Number of Tasks}\n\nIt is interesting to highlight that meta-overfitting was not observed when the number of regression tasks is unbounded, as shown in Figure \\ref{no_overfit}.\nThis evidence suggests that, when the number of tasks is unbounded but sampled from a related set of tasks, meta-learning algorithms can leverage their power to adapt without meta-overfitting.\n\nTo measure the amount of meta-learning and the lack of feature re-use, we compute the dCCA value of the model as in Section \\ref{main_result} and observe a value of  $0.44 \\pm 0.11$.\nThis also implies that the degree of meta-learning is higher when the number of tasks is unbounded. \n\nThe main contribution is that this evidence suggests {\\it we need to rethink how we define the few-shot learning benchmarks for meta-learning}. \nWe hypothesize this is true because changing the property - like the number of concepts available to the learning - changes the behaviors of meta-learning algorithms. \nIn particular, MAML stops meta-overfitting.\nThis suggests to practitioners that MAML is a good algorithm for online or lifelong learning benchmarks - rather than deploying it to benchmarks with a fixed number of concepts.\nOverall, our evidence suggests that, as a research community, we are applying meta-learning algorithms to the wrong data sets and benchmarks.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.55\\linewidth]{figs\/no_overfit.png}\n\\caption{\nShows that meta-overfitting does not occur and perfect meta-generalization occurs when the number of tasks (functions) is unbounded when training with MAML.\nIn other words, the meta-train and meta-validation error are indistinguishable and decrease together as the meta-iterations increases.\nThe main difference of this figure with figure \\ref{meta_overfit1} is that in this one has a finite set of tasks using our synthetic benchmark while the other has an infinite set of tasks using the sinusoidal tasks suggested in \\cite{maml}.\n}\n\\label{no_overfit}\n\\end{figure}\n\n\\section{Effects of More Meta-Adaptation}\n\nIn this section, we show that increasing the number of inner steps for MAML during adaptation does not necessarily change the representation further as measured with dCCA (as in Equation \\ref{eq_ml}).\nIn addition, the meta-validation performance also does not change.\n\nTo show this, we obtain a single neural network meta-trained with MAML using a dataset as described in Section \\ref{def}.\nThen we plot how the representation changes and how the meta-validation error changes as a function of the inner steps.\nWe show this in Figures \\ref{ml_loss_vs_inner_steps_sigmoid_best} and \\ref{ml_loss_vs_inner_steps_relu_best}.\nWe observe that the MAML neural networks are robust to meta-overfitting with respect to the inner steps of its inner adaptation rule.\n\nNote that this is different from what was observed in Section \\ref{finite_metaoverfit}, because that section shows it as a function of the meta iterations (what is sometimes called outer iterations).\nIn addition, it is important to emphasize that the representation change in the plots is above the $0.12$ compared to previous work \\cite{Raghu}, supporting the main results of section \\ref{main_result}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/ml_loss_vs_inner_steps_sigmoid_best.png}\n\\caption{\nShows 1) the lack of representation change and b) meta-validation change as the number of inner steps increases.\n1 is shown by the relative flatness of the blue and orange lines in the upper plot.\nSimilarly, 2 is shown by the flatness of the green line in the lower plot.\nIn particular, notice that we exponentially increase the inner steps from 1 to 2 to 32.\nThe models used are 4 layered FCNN trained with MAML with 1 inner step and 0.1 inner learning rate, selected using early stopping using the meta-validation set with the Sigmoid activation function.\nThe only difference of this figure with figure \\ref{ml_loss_vs_inner_steps_relu_best} is that this figure uses a sigmoid activation and the other one uses a ReLU.\nNote that this is the model used for figure \\ref{meta_overfit1}.\nNote the dCCA value remains above 0.12 suggesting lower degree of feature re-use.\n}\n\\label{ml_loss_vs_inner_steps_sigmoid_best}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/ml_loss_vs_inner_steps_relu_best.png}\n\\caption{\nShows 1) the lack of representation change and b) meta-validation change as the number of inner steps increases.\n1 is shown by the relative flatness of the blue and orange lines in the upper plot.\nSimilarly, 2 is shown by the flatness of the green line in the lower figure.\nWe want to emphasize that we exponentially increase the inner steps from 1 to 2 to 32.\nThe models used are 4 layered FCNN trained with MAML with 1 inner step and 0.1 inner learning rate, selected using early stopping using the meta-validation set with the ReLU activation function.\nThe only difference of this figure with figure \\ref{ml_loss_vs_inner_steps_sigmoid_best} is that this figure uses a ReLU activation and the other one uses a sigmoid.\nNote the dCCA value remains above 0.12 suggesting lower degree of feature re-use.\n}\n\\label{ml_loss_vs_inner_steps_relu_best}\n\\end{figure}\n\n\\section{Related Work}\\label{related_section}\n\nOh et al. \\cite{boil} show that one can encourage models to use less feature re-use purely algorithmically by setting the inner learning rate to zero for the final layer. \nThey showed BOIL outperforms MAML in both traditional few-shot learning (e.g. meta-trained on MiniImagent then meta-tested on MiniImagent) and cross-domain few-shot learning (meta-trained on MiniImagent then meta-tested on tiered ImageNet). \nIn particular, their cross-domain few-shot learning is similar in spirit to the synthetic task we propose in section \\ref{def}.\nHowever, note that we show that even MAML - an algorithm that has been shown to work by feature re-use \\cite{Raghu, boil} - can exhibit large representation changes if it is trained solely on a task that requires large feature changes.\nConcisely, we encourage rapid learning by only changing the task, while Oh et al. \\cite{boil} encourage it by changing the algorithm itself.\n\nGuo et al.'s \\cite{bscd_fsl} work is similar to ours in that they focus on defining a benchmark more appropriate for meta-learning and transfer learning. \nThey propose that meta-learning should be done in a fashion where the distribution of tasks sampled changes considerably when moving from meta-training to meta-evaluation.\nOur work is different in that we emphasize that the meta-training tasks themselves need to have diversity to be able to encourage meta-learning.\nAlthough Guo et al.'s \\cite{bscd_fsl} meta-evaluation procedure is excellent, we hypothesize - based on our results - that their benchmark won't have enough diversity to encourage large representation changes during meta-training.\nHowever, we conjecture that combining our ideas and theirs is a promising step for creating a better benchmark for meta-learning.\n\n\nSimilar work by Triantafillou et al. \\cite{Triantafillou2019} attempt to improve benchmarks by merging more data sets, but we hypothesize their data sets are not diverse enough to achieve this.\nIn terms of methods, our work is most similar to Raghu et al. \\cite{Raghu}, but they lack an analysis of the role of the tasks in explaining their observations.\nThere is also other work \\cite{Chen2019, Chen, Dhillon2019, Huang2019} that shows that a good representation is sufficient to achieve a high meta-accuracy on modern few-shot learning tasks e.g. MiniImagent, tiered-Imagenet, Cifar FS, FC100, Omniglot, \\cite{Tian2020}, which we hope to analyze in the future.\nWe conjecture in is imperative that a definition of meta-learning is developed and explicitly connected to the general intelligence.\nChollet \\cite{Chollet2019} takes this direction, but to our understanding the proposed definition is mostly focused on program synthesis. \nWe also hope that in the future a metric for AI safety is ubiquitously reported as suggested in Miranda et al. \\cite{foundationsmetalearning}.\n\n\\section{Discussion}\n\nIt is exciting evidence that by only changing the few shot learning benchmark, one can consistently show higher degrees of representation changes as measured by two different metrics. \nWe hypothesize this is the case because the meta-learning system has to be trained explicitly with a task that demands it to learn to adapt.\n\nAn important  discussion point is the lack of an authoritative definition for measuring meta-learning in our work and in the general literature.\nIn particular, in our work, we decided to not report any results with CKA.\nWe decided this because Ding et al. \\cite{Ding} showed that it's possible to remove up to 97\\% of the principal components of the weights of a layer until CKA starts to detect it.\nThus, we used dCCA which doesn't have the problem.\nIt instead has a higher variance, but it's easier to address this with experiment repetition sand error bars (which we did).\nHowever, we hypothesize it would be interesting to use and extend Orthogonal Procrustes as suggest by \\cite{Ding} in future work.\n\nThe most obvious gap in our work is a thorough analysis with a real world vision data set.\nWe hope to repeat our work with the hinted extension in section \\ref{related_section} benchmarks as suggested in \\cite{bscd_fsl, Triantafillou2019}. \n\nIn addition, Figures \\ref{ml_loss_vs_inner_steps_sigmoid_best}, \\ref{ml_loss_vs_inner_steps_relu_best} shows that as the number of inner steps increases, the dCCA does not increase.\nThis is somewhat surprising given the meta-overfitting results observed in section \\ref{meta_overfitting} and further experiments would be valuable.\n\n\n\n\\begin{ack}\nWe'd like to thank Intel for providing our team with access to their Academic Cluster Environment (ACE). \nTheir computational resources and support from their staff were essential to the successful completion of our project.\nIn addition, this work utilized resources supported by the National Science Foundation's Major Research Instrumentation program, grant 1725729, as well as the University of Illinois at Urbana-Champaign \\cite{Kindratenko2020}.\nWe'd like to acknowledge the work and authors of Anatome, TorchMeta and higher \\cite{anatome, torchmeta, higher} for making their code available and answering ours questions in their project's GitHub repository.\nWe'd like to acknowledge the weights and biases (wandb) framework for powerful tracking of experiments \\cite{wandb}.\nWe acknowledge the feedback from Open Philanthropy on the proposal on the foundations of meta-learning \\cite{foundationsmetalearning} that inspired this work.\nWe acknowledge the anonymous reviewers from NeurIPS for the valuable feedback for this work.\n\\end{ack}\n\n\n\n\\medskip\n\n\n\\printbibliography\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nChange detection obtains ground feature change information by comparing images from different periods.\nRemote sensing images have become common data for detecting changes in the surface due to their high spatial coverage and high time resolution \\cite{Hussain.2013}. At the same time, the increased spatial resolution of remote sensing images can provide more details of ground objects.\nIn-depth study of urban change is essential to promote sustainable urbanization \\cite{Huang.2017}. Therefore, UCD has become a research hotspot. \nUrban areas often have a wide variety of objects and strong regional heterogeneity. Ground objects, even in the same class, may have very different geometric shapes, and local features. In order to better analyze urbanization, different requirements are also put forward for the usage data.\n\\textbf{(1) Higher spatial resolution.}\n Higher spatial resolution images can provide more information to distinguish features between different images to obtain a clear boundary of change.\n\\textbf{(2) Richer prior information on land cover.} Knowing the prior information about land cover can detect the direction of change and analyze land cover changes.\n\\textbf{(3) Longer time series images.} \nThe changes in ground objects are time-dependent, and a more continuous sequence of images can realize time series analysis to monitor urban changes.\n\nWe collected public UCD datasets (Table~\\ref{tab:dataset}) and found that they have some limitations: \n\\textbf{(1) Lack of high spatial resolution images.} The image spatial resolutions of OSCD \\cite{Daudt.2018}, ZY3 \\cite{Zhang.2020}, and SZTAKI AirChange \\cite{Benedek.2009} are 10 m, 5.8 m, and 1.5 m respectively. Although the resolution is gradually increasing, it still cannot meet the requirements of UCD, especially for buildings.\n\\textbf{(2) Lack of semantic annotation.} ABCD \\cite{Fujita.2017}, LEVIR-CD \\cite{Chen.2020}, WHU Building \\cite{Ji.2019b} only label building-related changes, and Season-varying \\cite{Lebedev.2018} directly labeled land cover related changes, all of them are lack of semantic changes. It is difficult to perform multi-class change detection to obtain fine change directions. Although HRSCD \\cite{CayeDaudt.2019} provided the direction of changes, its labeling accuracy is only\n80\\% to 85\\%. In addition, the ground objects of the urban area are classified into five categories,\nwhich is relatively rough and difficult to reflect the changes of typical objects in urban areas.\n\\textbf{(3) Lack of long-range multi-temporal images.} The above-mentioned public datasets only contain bi-temporal images of the same area. Therefore, it is difficult to obtain satisfactory refined detection results for UCD.\n\n\\begin{table*}[!t]\n  \\centering\n  \\caption{ Open datasets in remote sensing change detection } \\label{tab:dataset}\n   \n   \n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{cccccccc}\n      \\toprule\n      {Dataset}& Resolution (m) & \\#Images & Image size (Pixel)  & Years & Change &Interesting object& Classes \\\\\n      \\midrule\n\n      OSCD \\cite{Daudt.2018}  & 10 & 24 & $600\\times600$  & 2 & Binary change & Land cover&- \\\\\n    \n      ZY3 \\cite{Zhang.2020}  & 5.8 & 1 & $1154\\times740$  & 2  & Binary change & Land cover&- \\\\\n\n       SZTAKI AirChange \\cite{Benedek.2009}  & 1.5 & 13 & $952\\times640$  & 2 & Binary change & Land cover&-\\\\\n  \n     \n       AICD \\cite{Bourdis.2011}  & 0.5 & 1000 & $800\\times600$  & 2 & Binary change & Land cover&-\\\\\n      \n       {ABCD\\cite{Fujita.2017}}  & {0.4} & 8506\/8444 & $160\\times160$\/$120\\times120$  &{2} &Binary change & {Building} &-\\\\\n\n        LEVIR-CD \\cite{Chen.2020}  & 0.5 & 637 & $1024\\times1024$  & 2 & Binary change & Building&-\\\\\n\n       WHU Building \\cite{Ji.2019b}& 0.2 & 1 & $32207\\times15354$  & 2  & Binary change & Building&-\\\\\n\n      Season-varying \\cite{Lebedev.2018}&0.03-1&16000& $256\\times256$&2&Binary change&Land cover&-\\\\\n      HRSCD \\cite{CayeDaudt.2019}  & 0.5 & 291 & $10000\\times10000$  & 2 & Semantic change & Land cover&5 \\\\\n      Hi-UCD (ours)  & 0.1 & 1293 & $1024\\times1024$  & 3 & Semantic change & Land cover & 9 \\\\\n      \\bottomrule\n    \\end{tabular}}\n\n\\end{table*}\n\nTo solve these problems, we introduce a large-scale semantic annotated ultra-high resolution UCD dataset named Hi-UCD.\nOur dataset uses aerial images with a spatial resolution of 0.1m to clearly show the spatial details of ground objects and capture small changes in them.\nHi-UCD obtains fine semantic changes of objects by labeling the land cover classes of images in different periods. We have selected 9 land cover classes including natural and artificial objects to achieve full coverage of urban ground objects.\nIn addition, Hi-UCD contains images of three time phases in the same area, which is conducive to studying the temporal correlation of changes in ground features.\nOverall, Hi-UCD is a large-scale, multi-temporal, ultra-high resolution urban semantic change detection data set, which can realize comprehensive detection and analysis of urban changes.\nTo verify the validity of Hi-UCD, we selecte the classic method in the binary and multi-class change detection task to conduct the experiments, finally provide a benchmark.\n\n\n\n\n\\section{Hi-UCD Dataset}\n\nHi-UCD focuses on urban changes and uses ultra-high resolution images to construct multi-temporal semantic changes to achieve refined change detection.\nThe study area of Hi-UCD is a part of Tallinn, the capital of Estonia, with an area of 30 $km^2$. The Estonian Land Board provides aerial images\\footnote{ Orthophoto, Land Board 2020.} taken by Leica ADS100-SH100 in 2017, 2018, and 2019, with topographic database\\footnote{Estonian Topographic Database, Land Board 2020.} for the area. Hi-UCD obtained semantic changes by annotating the land cover classes in different periods. We have considered topographic documents and changes in ground objects to\nselect 9 land cover classes to achieve complete coverage of ground objects in Estonia.\n Finally, we cut each year's images into patches with a size of $1024\\times1024$, and filter out patches with change pixels more than 200 to form the Hi-UCD dataset. There are 359 image pairs in 2017-2018, 386 pairs in 2018-2019, and 548 pairs in 2017-2019, including images, semantic maps and change maps at different times.\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[scale=0.4,trim=0 140 150 0,clip]{figure\/classess.pdf}\n  \\caption{Examples of Hi-UCD dataset for images, semantic maps and change maps.}\\label{fig:example}\n\\end{figure}\n In Figure \\ref*{fig:example}, examples of the Hi-UCD dataset are given. Compared with other public datasets, its characteristics are as follows:\n\n \\begin{enumerate}\n   \\item \\textbf{Ultra-high spatial resolution.}  \n   Hi-UCD uses aerial images with a spatial resolution of 0.1m, which is the highest resolution in public data. In these images, the geometric shape of the ground objects is clear, and the boundary is obvious, which provides rich spatial texture information. Therefore, it is conducive to detecting local changes of ground objects and realizing refined change detection.\n\n  \\item \\textbf{Multi-temporal images.} Hi-UCD contains the images of the three years from 2017 to 2019, and gives the semantic annotation and change mask every two years (2017-2018, 2018-2019, 2017-2019). Changes are highly time-dependent, and multi-temporal data can provide temporal features, which helps researchers to conduct long-term serial studies and improve the temporal precision of UCD. In addition, the images of different years have undergone orthorectification without registration errors. At the same time, they were taken in the same season, which greatly reduced the influence of seasonal changes in vegetation.\n\n  \\item \\textbf{Semantic annotation.} Considering typical urban objects and change-related objects, we developed Hi-UCD semantic annotation categories.\n   There are 9 types of objects, including natural objects (water, grassland, woodland, bare land), artificial objects (Building, greenhouse, road, bridge), and others (change-related), basically include all types of urban land cover in Estonia. The above categories are mapped with the shapefile layer in the Estonian Topographic Database (ETD). Due to the inconsistency of the vector boundaries between different years in ETD, the buildings are based on the vector of each year, and the other classes are based on the vector of 2019. Through visual interpretation, we check the topographic shapefiles and modify them. Meanwhile, we compare images of different years to determine the relevant objects of the change and add the category \"other\". Finally, the binary and multi-classes change masks generated through the semantic annotation results.\n  \\end{enumerate}\nBecause of these characteristics, Hi-UCD is full of challenges: (1) the increase in spatial resolution has aggravated the shadows and occlusions in the image.\n(2) The changes in uninteresting ground objects such as cars will cause serious background noise during change detection. (3) High-rise buildings are tilted and geometrically mismatched due to different shooting angles at different times. \n(4) The number of category transitions caused by changes is much greater than the number of semantic categories, which increases the difficulty of multi-classes change detection task. In summary, Hi-UCD is far more diverse, comprehensive, and challenging.  \n\n\\section{Benchmark}\n\n\nIn order to establish a fair benchmark,  we evaluated the classic methods of binary and multi-class change detection\nunder a unified experimental setting and data division conditions.\n\n\\textbf{Methods} After decades of development, change detection methods have evolved from pixel-based direct comparison to data-driven deep learning methods \\cite{Hussain.2013,Tewkesbury.2015,Shi.2020}. \nWe chose different methods according to the different detection task. \nFor binary change detection, these methods are the commonly used as comparison methods, including traditional methods (change vector analysis (CVA) \\cite{Malila.1980}, multivariate alteration detection (MAD) \\cite{Nielsen.1998}, the regularized iteratively reweighted MAD (IRMAD) \\cite{Nielsen.2007}), and deep learning methods (FC\\_EF \\cite{CayeDaudt.2018}, FC\\_Siam\\_diff \\cite{CayeDaudt.2018}, FC\\_Siam\\_diff \\cite{CayeDaudt.2018}, FC\\_Res\\_EF \\cite{CayeDaudt.2019}).\nFor multi-class change detection, the commonly used method is the post-classification comparison. After classifying images of different time phases through a classifier, like support vector machines [18], random forest [19], convolutional neural network [20] are compared to obtain change information. Considering the complexity of the objects in Hi-UCD, we only chose the classic semantic segmentation deep learning networks in computer vision ( Deeplab v3 \\cite{chen2017rethinking.2017}, Deeplab v3+ \\cite{Chen_2018_ECCV}, PSPNet \\cite{Zhao_2017_CVPR} ) and  remote sensing ( FarSeg \\cite{Zheng.2020b} ) for classification to obtain multi-class changes.\n\n\\textbf{Settings} We used 300 pairs of images in 2017 and 2018 for training, the remaining 59 pairs as the validation set, and 386 pairs of images in 2018 and 2019 for testing.\nIn the traditional method, the clustering method proposed in \\cite{Celik.2009} was used to obtain the change mask. For all deep learning methods, the learning rate was 0.01 and use a polynomial decay with a decay factor of 0.9. The batch size was 4 and trained on a single GPU.  The stochastic gradient descent (SGD) was used for optimization with weight decay of 0.0001 and a momentum of 0.9. For data augmentation, horizontal and vertical flip, rotation of 90 degrees and random cut ( $size=(512,512)$) were adopted during training. \nIn binary change detection, the number of iterations is 10k, and the loss function is the binary cross-entropy and dice loss. While in classification,  we used the cross-entropy loss function with 20k iterations. The backbone used for classification methods was ResNet-50, which was pre-trained on ImageNet \\cite{Deng.2009}. \n\n\\textbf{Metrics}\n We used overall accuracy (OA), kappa coefficient to evaluate the overall performance of the change detection results. For binary change detection methods, we used intersection over union (IoU) to only evaluate the ability to detect changes. In addition, We added mean intersection over union (mIoU) to evaluate algorithm performance in classification and multi-class change detection. The parameters and number of operations measured\n by multiply-adds (MAdd) calculated by a tensor with a size of $1\\times C\\times256\\times256 (C=3,6)$ are given to show deep learning model complexity.\n The accuracy evaluation results of different methods are shown in Table~\\ref{tab:result}.\n\n \\textbf{Analysis} In Table~\\ref{tab:firsttable},  most binary change detection methods can effectively detect unchanged ground objects, the IoU of change does not exceed 50\\%.\n IRMAD performed the best with kappa 8\\% higher than the other traditional methods. Deep learning methods are significantly higher than traditional methods in all metrics, which fully reflects the potential of deep learning in change detection. Traditional methods cannot distinguish false changes caused by shadows, occlusions, and uninteresting objects, while deep learning methods rely on their powerful learning capabilities to effectively remove background noise.\n Among them, the FC\\_Siam\\_diff method is slightly better than FC\\_Siam\\_conc in all metrics. FC\\_EF improves IoU of change by nearly 7\\%. After adding the residual module, FC\\_EF\\_Res increased by nearly 5\\% and has the smallest parameters.  \n In Table~\\ref{tab:secondtable}, the metrics of all methods for land cover classification in 2018 are higher than in 2019. Because the change of the ground objects leads to differences in the distribution of ground features at different times. \n Through post-classification to get the results of multi-class change detection, there are many false alarms at the boundary of the ground objects in multi-class change results. In Table~\\ref{tab:firsttable}, although Deeplab v3 obtains the best accuracy in multi-class change detection, it also has the highest computational complexity. \n Besides, the accuracy of the change highly depends on the accuracy of the classification,\nHow to obtain reliable multi-class change detection results in urban areas is still a problem that needs research. \n\n\n\n\\begin{table} [tb]\n  \\caption{The quantitative evaluation of the baseline methods for Hi-UCD}\n  \\label{tab:result}\n  \\centering\n  \\subtable[ Change detection accuracy ]{\n    \\resizebox{0.48\\linewidth}{!}{\n    \\begin{tabular}{ccccccccc}\n      \\toprule\n      {Binary change detection} &\n      \\#Params (M)&\n      MAdds (B)&\n       OA (\\%) &\n       Kappa (\\%) & \n       IoU (\\%) &\n     \n     \n     \n        \\\\\n       \\midrule\n      CVA \\cite{Malila.1980} &-&-&40.79\t&\t3.98\t&\t11.51\t\t\\\\\n       MAD \\cite{Nielsen.1998}&-&-\t&\t88.95\t&\t3.64\t&\t3.41\t\\\\\n       IRMAD \\cite{Nielsen.2007}&-&-\t&\t88.08\t&\t11.78\t&\t9.38\t\t\\\\  \n       FC\\_Siam\\_conc \\cite{CayeDaudt.2018}  &1.546&5.8& 91.25 \t&\t44.16\t&\t32.26 \t \t \t\\\\\n       FC\\_Siam\\_diff \\cite{CayeDaudt.2018}&1.35&4.67 &91.74 \t&\t47.67  \t&\t35.19 \t \t \t\\\\\n       FC\\_Siam\\_EF \\cite{CayeDaudt.2018}&1.35&3.54 &91.50 \t&\t54.92 \t\t&\t42.51 \t \t \t \t\\\\\n        Siam\\_Res\\_EF \\cite{CayeDaudt.2019}& 1.104&1.98&93.05 \t&\t60.67 \t&\t47.62  \t \t \\\\\n       \\midrule\n       { Multi-class change detection} &\n       \\#Params (M)&\n       MAdds (B)&\n         OA (\\%) &\n         Kappa (\\%) & \n         mIoU (\\%) \\\\\n         \\midrule\n   \n       Deeplab v3 \\cite{chen2017rethinking.2017}&39.046&80.58& 76.82\t&\t29.48\t&\t17.51\\\\\n       Deeplab v3+ \\cite{Chen_2018_ECCV}&39.897&26.34&75.83\t&\t28.31\t&\t15.65\\\\\n       PSPNet \\cite{Zhao_2017_CVPR}&46.588&88.60\t& 76.17\t&\t28.49\t&\t14.29\\\\\n       FarSeg \\cite{Zheng.2020b}&33.881&28.68& 73.58\t&\t25.68\t&\t13.34\\\\\n        \\bottomrule\t\n         \\end{tabular}}\n         \\label{tab:firsttable}\n  }\n  \\hfill\n  \\subtable[Land cover accuracy ]{  \n    \\resizebox{0.48\\linewidth}{!}{      \n    \\begin{tabular}{ccccccc}\n      \\toprule\n       Methods&\n       \\#Params (M)&\n       MAdds (B)&\n       Year&\n        OA (\\%) &\n        Kappa (\\%) & \n        mIoU (\\%) \n        \\\\\n        \\midrule\n        \\multirow{2}{*}{Deeplab v3 \\cite{chen2017rethinking.2017}}\n        &\\multirow{2}{*}{39.046}&\\multirow{2}{*}{40.29}&2018\t&\t87.59\t& 83.94 &\t72.39 \\\\\n        &&&2019&77.19&69.74&42.55\\\\\n        \\midrule\n        \\multirow{2}{*}{Deeplab v3+ \\cite{Chen_2018_ECCV}} &\\multirow{2}{*}{39.897 }&\\multirow{2}{*}{13.17}& 2018\t&\t86.28\t& 82.22&\t67.24 \\\\\n        &&&2019&76.23&68.45&37.98\\\\\n        \\midrule\n        \\multirow{2}{*}{PSPNet \\cite{Zhao_2017_CVPR}} &\\multirow{2}{*}{46.588 }&\\multirow{2}{*}{44.30} &2018\t&\t86.50\t&82.50&69.88 \\\\\n        &&&2019&74.50&65.79&37.37\\\\\n        \\midrule\n        \\multirow{2}{*}{FarSeg \\cite{Zheng.2020b}} &\\multirow{2}{*}{33.881 }&\\multirow{2}{*}{14.34 }& 2018\t&\t86.78\t&82.88 &69.98 \\\\\n        &&&2019&75.58&64.83&36.02\\\\\n        \\bottomrule\n         \\end{tabular}}\n         \\label{tab:secondtable}\n  }\n  \\end{table}\n\n\n\n\n\\section{Conclusion}\nIn this article, we introduce a new multi-temporal ultra-high-resolution aerial image UCD dataset, which has rich semantic annotations to detect more details of urban change.\nAt the same time, we have established a benchmark for UCD in binary and multi-class change detection tasks.\nIn the next work, we will continue to expand the scale of the dataset and provide different large-area test sets to verify the generalization and migration of the algorithm better. We hope the release of Hi-UCD will promote the development of UCD.\n\n\n\\medskip\n\n\\ack\nThis work was supported by National Natural Science Foundation of China under Grant Nos.41771385, 41801267 and 42071350. The authors would like to thank the Estonian Land Board for acquiring and providing the data used in this study.\n\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction to the problem}\n\\noindent \nConsider $u(x,t)$ as a function describing the temperature at the time $t$ of a point $x$ in an infinite isolated rod, being hence a solution of the heat equation. As usual, it is assumed that heat has spread from hotter zones to colder ones. Now, if one considers an ancient non-negative solution, the diffusive process has already gone on for an infinite amount of time, and it is reasonable to question if $u(x,t)$ has become constant. This fact, stated in this way, is generally false, as shown by the following examples:\n\\begin{equation}\\label{examples}\n    u_1(x,t)= e^{x_N+t}, \\quad u_2(x,t)=e^{-t} \\sin(x_1), \\quad x \\in {\\mathbb R}^N.\n\\end{equation}\nThe two functions above are {\\it eternal} solutions of the heat equation, i.e. solutions in ${\\mathbb R}^N \\times {\\mathbb R}$. We call {\\it ancient} solutions those solutions that solve the parabolic equation in $ {\\mathbb R}^N \\times (-\\infty, T)$ for some time $T \\in {\\mathbb R}$. In line with the literature, we call {\\it Liouville property} any rigidity condition that ensures the triviality of solutions. It is clear from $u_1$ that a sign condition is not enough to confirm our suspect, while the sign-changing solution $u_2$ shows that boundedness at a fixed time is not enough. Although Appel \\cite{Appel} already proved in 1892 that an ancient solution to the heat equation which is two-sided bounded (as for instance $0 \\leq u(x,t) \\leq M$) is constant, the first optimal parabolic Liouville theorem for ancient solutions was found in 1952 by Hirschman (see \\cite{Hirschman}, Bear \\cite{Bear} and Widder \\cite{Widder}, \\cite{WidderBook} for the case $N=1$), stating that a non-negative ancient solution to the heat equation is constant if one adds the assumption that, for a time $t_o<T$,\n\\begin{equation}\\label{log}\n    \\lim_{r \\uparrow + \\infty} \\frac{\\log(\\sup_{|x|<r}u(x,t_o))}{r} \\leq 0, \\qquad \\text{that is,} \\qquad u(x,t_o) \\leq e^{o(|x|)} \\quad \\text{as} \\quad |x|\\rightarrow +\\infty.\n\\end{equation} This result was sharp in the sense that any function of the kind\n\\begin{equation*}\\label{count-Hirsch}\n    u(x,t)= e^{a^2t} \\cosh{ax}\n\\end{equation*} shows that if \\eqref{log} above is violated then $u(x,t)$ is not necessarily constant; but \\eqref{log} is just a condition on the space variables. Sub-exponential optimal growth conditions have been generalized to different metric contexts; see for instance \\cite{Sunra} and references therein. Not much later, in 1958, Friedman gave a condition on the behavior of non-negative ancient solutions to more general second order parabolic equations as\n\\begin{equation}\\label{seond-order}\n    \\partial_t u(x,t) = \\sum_{i,j=1}^N a_{i,j}\\frac{\\partial^2 u(x,t)}{\\partial x_i \\partial x_j} + \\sum_{i=1}^N b_i \\frac{\\partial_i u(x,t)}{x_i} +c u(x,t),  \\quad \\text{in} \\quad {\\mathbb R}^N \\times {\\mathbb R}_+,\n\\end{equation} being $b_i,c$ real numbers and $\\{a_{i,j}\\}_{i,j}$ a positive matrix. Now the assumption concerns infinite past times as\n\\begin{equation} \\label{fried}\n\\lim_{t\\downarrow - \\infty} \\frac{\\log(u(0,t))}{t}= c+ \\gamma, \\qquad \\gamma>0,\\quad  c+\\gamma \\ge 0. \\end{equation} \\noindent See \\cite{Friedman} for the result and \\cite{Eidelman} for the earlier case of systems. Furthermore, conditions guaranteeing the stabilization of the solution to a constant were studied for a fixed space variable (see \\cite{Eidelman-Kamin-Tedeev} and its references for an account). This short preamble is just to highlight that different assumptions, mainly on the second bound, may be requested to solutions of these parabolic equations in order to ensure Liouville property; it is therefore an incomplete list. The literature on these rigidity results is wide, so we refer the reader to the book \\cite{QS-libro} and the survey \\cite{Kogoj} for a more complete account.\\vskip0.1cm\n\\noindent \nThe heat equation can be regarded as a special case of the anisotropic $p$-Laplacian equation\n\n\\noindent \\begin{equation} \\label{EQ}\n    \\partial_t u= \\sum_{i=1}^N \\partial_i (|\\partial_i u|^{p_i-2} \\partial_i u),  \n   \n\\end{equation} \\noindent\nwhen $p_i\\equiv 2$ for all $i=1,\\dots,N$. When $2<p_i< \\bar{p}(1+1\/N)$, this equation describes the effect of competing diffusions along coordinate axes in finite speed of propagation (see \\cite{Ant-Sh}, \\cite{Mosconi}, \\cite{Ruzicka} for an introduction to the parabolic problem). In one spatial dimension, the equation \\eqref{EQ} is a one-dimensional $p$-Laplacian, which has a very interesting change of behavior from the degenerate case ($p>2$) to the singular one ($1<p<2$). Roughly speaking, the solutions to $p$-Laplacian singular equations behave more like solutions to elliptic equations and only one bound is enough to infer that they are constant (see \\cite{DGV-Liouville} for more details). On the other hand, in the degenerate case two bounds are required to infer a Liouville property; see Section \\ref{LiouvilleSection} below for a counterexample. In this paper we show some Liouville properties for non-negative solutions to \\eqref{EQ} for a range of $p_i$s which is degenerate and allows a finite speed of propagation. In many physical circumstances, this is a more reasonable assumption than the sudden infinite expansion of the support of solutions to the heat equation.\\vskip0.2cm \n\\noindent The theory of regularity for solutions to \\eqref{EQ}, even if much investigated, is still incomplete and fragmented (see, e.g., \\cite{MingRadu} and \\cite{Mar-survey} for an account on the elliptic case). The Liouville properties that we are about to describe are entailed by recent Harnack estimates, obtained with an approach of expansion of positivity. This has been shown relying on the behavior of abstract fundamental solutions in \\cite{Ciani-Mosconi-Vespri}. Here we start from the aforementioned Harnack inequality (see Section \\ref{Preliminaries}, estimate \\eqref{Harnack}), which is formulated in an intrinsic geometry (\\ref{anisocubes}) reflecting the natural scaling of the equation, and study some rigidity connections between local and global behavior of solutions.\\newline\nSimilarly to the Liouville property inferred by Hirschman, we will prove that it is sufficient to have a one-sided bound (say, from below) and an estimate from the other side (say, from above), just for a fixed time. If these conditions are met, solutions are forced to be constant (Theorem \\ref{Liouville1}). This clearly implies that a solution which is bounded both from above and below is bounded; on the other hand, it is unreasonable to expect that just a one-sided bound suffices (see the example in Section \\ref{LiouvilleSection}) for our range of $p_i$s. As a known fact, we comment that a precise decay on oscillation given by H\\\"older continuity estimates is enough as Liouville property. This decay is usually easier to show than a complete Harnack inequality, and as such deserves its own attention: already in the range $\\bar{p}(1+1\/N)\\leq p_i<\\bar{p}(1+2\/N)$, although continuity is expected by the regularizing properties of diffusion, no Harnack estimate may be available, because the competition among the diffusions is too strong (see for instance \\cite{AS}). It would be of interest to compare this behavior to the one in porous materials by the sole control on the oscillation, for example as in \\cite{Eurica}.\\newline\nOn a similar track to Friedman's result, we prove that, fixed any spatial point and assuming that the solution is bounded at infinity in time, again it is forced to be constant (Theorem \\ref{Liouville2}). Finally we state a Harnack inequality that frees the time variable to be intrinsic (Theorem \\ref{WeakHarnack}, see \\cite{DB} for the isotropic counterpart): being the Harnack estimate not anymore intrinsic in time, the estimate is more suitable for an application to rigidity. Moreover, this turns out to be useful to determine the optimal growth on the initial data when $|x|\\rightarrow \\infty $ for the solvability of the Cauchy problem for \\eqref{EQ} (see for instance \\cite{DB-He}). Clearly, this implies that the domain where the equation is solved must be, in turn,\n`compatible' with the anisotropy of the diffusion: this is certainly the case for ancient solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Structure of the paper}\nSection \\ref{Preliminaries} is devoted to set up the functional framework and to recall some known properties of the solutions, as the existence of fundamental solutions, comparison principles and the Harnack inequality. In Section \\ref{LiouvilleSection} we prove the Liouville-type results, while Section \\ref{WHSection} concerns an alternative formulation of the Harnack inequality, which turns out to be locally equivalent to the known one. Finally we gather in the Appendix a detailed proof of the H\\\"older continuity of solutions.\n\n\\section*{Notations}\n\\begin{itemize}\n\\item[-] We define the following function of $p_i$s, called {\\it harmonic mean}: $\\,\n\\bar{p}=N(\\sum_{i=1}^N1\/p_i)^{-1}$.\\newline We suppose that $ 1<p_1\\leq p_2 \\leq \\ldots \\leq p_N$ are ordered, as well as $\\bar{p}<N$.\n\\vskip0.1cm \\noindent \n    \n\\item[-] For any $\\rho, \\theta>0$ and $x\\in{\\mathbb R}^N$, we denote by $K_{\\rho}(x) \\subset {\\mathbb R}^N$ the cube of side $2\\rho$ centered at $x$.\n\\noindent Let $x_o+\\mathcal{K}_{\\rho}(\\theta)$ stand for the anisotropic cube of radius $\\rho$, ``magnitude'' $\\theta$, and center $x_o$, i.e.,\n\\begin{equation*}\\label{anisocubes}\nx_o+\\mathcal{K}_{\\rho}(\\theta)= \\prod_{i=1}^N\\bigg{\\{}|x-x_{o,i}|<\\theta^{{(p_i-\\bar{p})}\/{p_i}}\\rho^{{\\bar{p}}\/{p_i}}\\bigg{\\}}.\n\\end{equation*}\nIf either $\\theta=\\rho$ or $p_i=p$ for all $i=1,\\ldots,N$, then $x_o+\\mathcal{K}_{\\rho}(\\theta)=K_{\\rho}(x_o)$.\n\n\\item[-] For any $\\rho, \\theta,C >0$ and $(x_o,t_o) \\in {\\mathbb R}^{N+1}$, we consider the following anisotropic cylinders:\n\\begin{equation*}\\label{cylinders}\n\\begin{cases}\n\\text{centered: }(x_o,t_o)+\\mathcal{Q}_{\\rho}(\\theta,C)=\n(x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times (t_o-\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}},t_o+\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}});\\\\\n\\text{forward: }(x_o,t_o)+\\mathcal{Q}^+_{\\rho}(\\theta,C)= (x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times [t_o,t_o+\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}});\\\\\n\\text{backward: }(x_o,t_o)+\\mathcal{Q}^-_{\\rho}(\\theta,C)=\n(x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times (t_o-\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}},t_o].\n\\end{cases}\n\\end{equation*} \\noindent We omit the index $C$ when the constant is clear from the context. \\vskip0.1cm \\noindent \n\\item[-] For $\\Omega \\subset \\subset {\\mathbb R}^N$, i.e., $\\Omega$ open and bounded set in ${\\mathbb R}^N$, we denote with $\\Omega_T= \\Omega \\times [-T,T]$, $T>0$, the parabolic domain, and for $s\\in {\\mathbb R}$ with $S_s= {\\mathbb R}^N \\times (-\\infty, s)$ the space strip.\\vskip0.1cm \\noindent \n\\item[-] We adopt the nowadays classic convention that constants may change from line to line.\n\\end{itemize}\n\n\n\n\n\n\\section*{Acknowledgements}\n\\noindent \nWe are grateful to S.A. Marano and V. Vespri for encouraging us toward this project. We wish to thank professor S. Mosconi for his precious suggestions and Emanuele Macca for a numerical insight about Barenblatt-type solutions. Moreover we are indebted with Eurica Henriques for the careful reading of the manuscript and for pointing out an early mistake about H\\\"older continuity of solutions. Finally, S. Ciani is supported by the department of Mathematics of the Technical University of Darmstadt, and U. Guarnotta is supported by: (i) PRIN 2017 `Nonlinear Differential \nProblems via Variational, Topological and Set-valued Methods' (Grant \nNo. 2017AYM8XW) of MIUR; (ii) GNAMPA-INdAM Project \nCUP$\\underline{\\phantom{x}}$E55F22000270001; (iii) grant `PIACERI \n20-22 Linea 3' of the University of Catania. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries and Tools of the Trade} \\label{Preliminaries}\n\\noindent \nWe begin with the definition of local weak solution. For $\\Omega \\subseteq {\\mathbb R}^N$ open rectangular domain and $T>0$, the Banach spaces \n\\[W^{1,{\\bf{p}}}_{loc}(\\Omega):= \\{ u \\in W^{1,1}_{loc}(\\Omega) |\\,  \\partial_i u \\in L^{p_i}_{loc}(\\Omega) \\}, \\]\n\\[ L^{{\\bf{p}}}_{loc}(0,T;W^{1,{\\bf{p}}}_{loc}(\\Omega)):= \\{u \\in W^{1,1}_{loc}(0,T;L^1_{loc}(\\Omega))|\\, \\partial_i u \\in L^{p_i}_{loc}(0,T;L^{p_i}_{loc}(\\Omega))   \\}, \\]\nare called anisotropic spaces (see, for instance, \\cite{Ant-Sh}). When $\\bar{p}>N$ and $\\partial \\Omega$ is regular enough, the space $W^{1,{\\bf{p}}}(\\Omega)$ is embedded in the space of H\\\"older continuous functions \\cite{VenTuan}. A function \\[ u \\in C^0_{loc}(0,T; L^2_{loc}({\\mathbb R}^N)) \\cap L^{\\bf{p}}_{loc}(0,T;W^{1,{\\bf{p}}}_{loc}({\\mathbb R}^N))\\] is called a {\\it local weak solution} of \\eqref{EQ} in $S_T$ if, for all $0<t_1<t_2<T$ and any $\\varphi \\in C^{\\infty}_{loc}(0,T;C_o^{\\infty}({\\mathbb R}^N))$, it holds\n\\begin{equation} \\label{anisotropic-localweaksolution}\n\\int u \\varphi \\, dx \\bigg|_{t_1}^{t_2}+ \\int_{t_1}^{t_2} \\int (-u \\, \\varphi_t + \\sum_{i=1}^N \\, |\\partial_i u|^{p_i-2} \\partial_i u \\, \\partial_i \\varphi) \\, dx dt=0,\n\\end{equation} \\noindent where the integral is assumed to be taken in ${\\mathbb R}^N$ when no otherwise specified. When $\\Omega\\subset {\\mathbb R}^N$ is bounded we still call local weak solution to \\eqref{EQ} in $\\Omega_T$ a function satisfying the above requirements with ${\\mathbb R}^N$ replaced by $\\Omega$.\nHereafter we suppose that\n\\begin{equation}\\label{pi}\n    2<p_1\\leq p_N<\\bar{p}(1+1\/N),\\qquad \\bar{p}<N.\n\\end{equation}\nMoreover, throughout the paper, the constants $C_i$, $i=1,2,3$, are the ones of Theorem \\ref{Harnack-Inequality} below.\\newline\nNow we recall the main tools for our proofs, namely, the Harnack inequality, the existence of an abstract Barenblatt-type solution, and the comparison principle.\n\\begin{theorem} \\label{Harnack-Inequality}\nLet $u\\ge 0$ be a  local weak solution to \\eqref{EQ} in $\\Omega \\times [-T,  T]$ and let \\eqref{pi} be valid. Suppose that $u(x_o,t_o)>0$ for a Lebesgue point $(x_o,t_o) \\in \\Omega_T$ for $u$. Then there exist $C_{1}\\ge 0, C_3\\ge C_2\\ge 1$, depending only on $N$ and the $p_{i}$s, such that, letting $\\theta=u(x_o,t_o)\/C_1$, it holds\n\\begin{equation}\\label{Harnack}\n  \\frac{1}{C_{3}}\\sup_{x_o+\\mathcal{K}_{\\rho}(\\theta)}u(\\,\\cdot\\, , t_o - \\theta^{2-\\bar p}\\, (C_{2}\\, \\rho)^{\\bar p} )\\le  u(x_o,t_o) \\le C_{3} \\inf_{x_o+\\mathcal{K}_{\\rho}(\\theta)} u(\\,\\cdot\\, ,   t_o + \\theta^{2-\\bar{p}}\\, (C_{2}\\, \\rho)^{\\bar{p}})\n    \\end{equation}\n    whenever \n    \\begin{equation} \\label{side-condition}\n    \\theta^{2-\\bar p}\\, (C_{3}\\,  \\rho)^{\\bar p}<T-|t_o| \\qquad \\text{and} \\qquad x_o+\\mathcal{K}_{C_{3}\\, \\rho}(\\theta)\\subseteq \\Omega.\\end{equation}\n\\end{theorem}\n\n\n\\noindent The assumption $u(x_o,t_o)>0$ is defined by a suitable limit process, as customary. Semi-continuity clarifies this definition, as long as a theoretical maximum principle is in force (see \\cite{CianiGuarnotta}, \\cite{Mosconi}, \\cite{Liao} for an account). \nWe stress this definition because Theorem \\ref{Harnack-Inequality} has been proven in \\cite{Ciani-Mosconi-Vespri} without the assumption of H\\\"older continuity of solutions, that can be shown (see Section \\ref{Appendix}) to be a sole consequence of \\eqref{Harnack}. This important property has been faced several times in the past, with imprecise proofs or an unclear geometric setting. For this reason, and in order to explain the main adversities that anisotropic diffusion obliges us to face, we include in Section \\ref{Appendix} the proof of local H\\\"older continuity of solutions to \\eqref{EQ}, which follows the Moser's ideas \\cite{Moser} through an appropriate anisotropic intrinsic geometry. Taking for granted the continuity of solutions, in what follows we will refer directly to the pointwise values of solutions.\\vskip0.1cm \\noindent Secondly, let us comment Theorem \\ref{Harnack-Inequality} from a global point of view: if we pick a point $(x_o,t_o) \\in \\Omega_T$ where $u$ is positive, it is possible to `detect' the sets where the pointwise controls \\eqref{Harnack} hold true. This is the core of the next proposition.\n\n\\begin{proposition}\\label{paraboloids}\nSuppose the assumptions of Theorem \\ref{Harnack-Inequality} to be satisfied for $(x_0,t_0) \\in \\Omega\\times[-T,T]$. Then\n\\begin{equation} \\label{estimate-paraboloid}\n\\inf_{\\mathcal{P}_\\theta^+(x_0,t_0)} u \\ge u(x_0,t_0)\/C_3 \\qquad \\text{and}\\qquad \\sup_{\\mathcal{P}_\\theta^-(x_0,t_0)}u \\leq C_3 u(x_0,t_0),\\end{equation}\nwhere, setting $\\theta= u(x_o,t_o)\/C_1$, the paraboloids $\\mathcal{P}^+_{\\theta}(x_o,t_o)$ and $\\mathcal{P}^-_{\\theta}(x_o,t_o)$ are defined by\n\\[\n\\mathcal{P}_\\theta^+(x_0,t_0)= \\bigg{\\{}(x,t) \\in \\Omega_T:\\, \\,  C_2^{\\bar{p}} |x_i-x_{0,i}|^{p_i}\\theta^{2-p_i}\\leq  (t-t_0)\\leq C_2^{\\bar{p}}\\varrho^{\\bar{p}}\\theta^{2-\\bar{p}}, \\, \\,   \\forall i=1,..N \\bigg{\\}},\n\\]\n\\[\n\\mathcal{P}_\\theta^-(x_0,t_0)= \\bigg{\\{}(x,t) \\in \\Omega_T:\\,\\,    -C_2^{\\bar{p}}\\varrho^{\\bar{p}}\\theta^{2-\\bar{p}}\\leq (t-t_0) \\leq -C_2^{\\bar{p}}|x_i-x_{0,i}|^{p_i} {\\theta}^{2-p_i},  \\, \\,   \\forall i=1,..N \\bigg{\\}},\n\\] \n\\noindent with $\\varrho$ depending on $u$, $\\Omega_T$, and $(x_0,t_0)$ according to the following expression (see Figure \\ref{FigA}): \n\\begin{equation}\\label{rho+}\n    \\varrho^{\\bar{p}}= C_3^{-\\bar{p}} \\bigg( \\frac{u(x_0,t_0)}{C_1} \\bigg)^{\\bar{p}-2} \\min_{i=1,\\dots,N} \\bigg{\\{}(T-|t_0|), \\, \\bigg( \\frac{\\mathrm{dist}(x_0, \\partial \\Omega)}{2}\\bigg)^{p_i} \\bigg( \\frac{u(x_0,t_0)}{C_1} \\bigg)^{2-p_i} \\bigg{\\}}.\n\\end{equation}\n\\end{proposition}\n\\noindent It is remarkable that estimate \\eqref{Harnack} is prescribed on a {\\it{space}} configuration depending on the solution, in contrast to what happens with $p$-Laplacian type equations. This is due to the natural scaling of the equation (see \\cite{CianiGuarnotta} and the end of Section \\ref{WHSection}), because the expansion of positivity of solutions is readily checked via comparison with the following family of Barenblatt-type solutions.\n\n\n\\begin{theorem}\n\\label{Barenblatt}\nSet $\\lambda=N(\\bar{p}-2)+\\bar{p}$ and let \\eqref{pi} be satisfied. For each $\\sigma >0$ there exists $\\tilde{\\eta}>0$ and a local weak solution $\\mathcal{B}_{\\sigma}(x, t)$ to \\eqref{EQ} \nwith the following properties, valid for any $t\\in(0,T)$:\n\\begin{enumerate}\n\\item $\\displaystyle{\\|\\mathcal{B}_{\\sigma}(\\cdot, t)\\|_{\\infty}=\\sigma \\,  t^{-\\alpha}}$,\n\\item\n$\\displaystyle{{\\rm supp}(\\mathcal{B}_{\\sigma}(\\cdot, t))\\subseteq \\prod_{i=1}^N \\big{\\{} |x_i|\\le    \\sigma^{(p_i-2)\/p_i}\\, t^{\\alpha_i} \\big{\\}}}$, $\\qquad \\qquad$  $\\alpha=N\/\\lambda$, $\\alpha_i=(1+2\\alpha)\/p_i-\\alpha$, \n\\item\n$\\displaystyle{\\{\\mathcal{B}_{\\sigma}(\\cdot, t)\\ge \\eta\\,  \\sigma \\, t^{-\\alpha}\\}\\supseteq \\prod_{i=1}^N \\big{\\{} |x_i|\\le \\eta\\, \\sigma^{(p_{i}-2)\/p_{i}}\\, t^{\\alpha_i} \\big{\\}}=:\\mathcal{P}_t}$.\n\\end{enumerate}\n\\end{theorem}\n\\noindent The existence of a Barenblatt Fundamental solution $\\mathcal{B}$ is a consequence of the finite speed of propagation of solutions to \\eqref{EQ} combined with a particular correspondence of the Cauchy problems associated to \\eqref{EQ} and to an anisotropic Fokker-Planck equation. On the other hand, the properties of $\\mathcal{B}$ stated above stem from comparison techniques and the invariance of the equation \\eqref{EQ} under scaling, which entitles $\\mathcal{B}$ to be a self-similar solution. We refer to \\cite{Ciani-Mosconi-Vespri} for the proofs of these facts; see also \\cite{CSV}, \\cite{Vazquez} for the singular case.\n\\begin{proposition} \\label{local-comparison}\nLet $\\Omega \\subset {\\mathbb R}^N$ be a bounded open set and $u,v$ be weak local solutions to the equation \\eqref{EQ} in $\\Omega_T$, satisfying $u(x,t) \\ge v(x,t)$ in the parabolic boundary of $\\Omega_T$. Then $u \\ge v$ in $\\Omega_T$.\n\\end{proposition} \\noindent \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Liouville-type results}\\label{LiouvilleSection}\n\\noindent In their origins, Liouville properties were discovered for harmonic functions. Indeed, for solutions to $\\Delta u =0$ in ${\\mathbb R}^N$, Liouville properties occur when $u$ is just one-sided bounded, or also when it grows sublinearly at infinity. \nThe two classic examples in \\eqref{examples} follow respectively from an application of Harnack's inequality and from gradient estimates. Here we observe that gradient bounds of logarithmic type are unknown for solutions to the stationary counterpart of \\eqref{EQ} and seem very difficult to obtain, chiefly because of the lack of homogeneity of the operator. On the other hand, for parabolic equations a one-side bound is not sufficient to imply that solutions are constant, as we remarked. The Liouville property is false, stated as it is, also for non-negative solutions to degenerate $p$-Laplacian equations (i.e., for $p>2$). Indeed, the one-parameter family of non-negative functions \n\\[\n{\\mathbb R} \\times {\\mathbb R}  \\ni (x,t)\\rightarrow u(x,t;c)= c^{{1}\/({p-2})} \\bigg(\\frac{p-2}{p-1}  \\bigg)^{{(p-1)}\/{(p-2)}} (1-x+ct)_+^{{(p-1)}\/{(p-2)}}\n\\] is a family of non-negative, non-constant weak solutions to $u_t=\\Delta_{p}u$ in ${\\mathbb R}^2$. This naturally provides a counterexample also in case of equation \\eqref{EQ} in one spatial dimension. Similarly, the anisotropic driving example we have in mind is\n\\[{\\mathbb R}^N \\times {\\mathbb R} \\ni (x,t) \\rightarrow u(x,t;c)= \\bigg(1-ct + \\sum_{i=1}^N {(\\alpha_i\/p_i') |x_i|^{p_i'}}\\bigg)_+,\\]\nfor $\\alpha_i >0$ such that $\\sum_{i=1}^N |\\alpha_i|^{p_i-1}\\alpha_i=c$ and being $p_i'$ the H\\\"older conjugate of $p_i$ for all $i=1,\\dots, N$. On the other hand, if a lower bound is coupled with a specific upper bound at some time level, Liouville property is true, as the following result shows.\n\n\\begin{theorem}\\label{Liouville1}\nLet $T\\in {\\mathbb R}$, $S_T={\\mathbb R}^N \\times (-\\infty, T)$ and $u$ be a solution to \\eqref{EQ} which is bounded below in $S_T$. Assume moreover that, for some $s<T$, one has \n\\begin{equation}\\label{rs-bound}\n    \\sup_{{\\mathbb R}^N} u(\\cdot,s)=M_s <\\infty.\n\\end{equation} Then $u$ is constant in $S_s= {\\mathbb R}^N \\times (-\\infty, s)$.\n\\end{theorem} \n\\begin{corollary}\n\\label{LiouvilleCor}\nLet $T\\in {\\mathbb R}$, $S_T={\\mathbb R}^N \\times (-\\infty, T)$, and $u$ be a solution to \\eqref{EQ}. If $u$ is bounded from above and below in $S_T$, then it is constant.\n\\end{corollary}\n\n\\begin{proof}[Proof of Theorem \\ref{Liouville1}]\nLet $u$ be a solution to \\eqref{EQ} bounded from below in $S_T$. We define\n\\[\nm:=\\inf_{S_T} u.\n\\]\nWe prove the following fact, which is interesting in its own:\n\\begin{equation}\\label{perse}\n    \\lim_{t\\rightarrow -\\infty} u(x,t)= \\inf_{S_T} u \\quad \\mbox{for any} \\quad x \\in {\\mathbb R}^N.\n\\end{equation}\\noindent To this aim, fix any $x\\in{\\mathbb R}^N$ and $\\varepsilon>0$. Notice that there exists a point $(y_\\varepsilon, s_\\varepsilon) \\in S_T$ such that $u(y_{\\varepsilon},s_{\\varepsilon})-m\\leq \\varepsilon\/C_3$. Set $\\theta_{\\varepsilon}= (u(y_{\\varepsilon},s_{\\varepsilon})-m)\/C_1$. Exploiting the left-hand side of \\eqref{Harnack}, written for the solution $u-m$, we obtain the estimate\n\\begin{equation}\\label{infimum}\nm \\leq u(y,s) \\leq m+\\varepsilon, \\quad \\text{for all} \\quad (y,s) \\in \\mathcal{P}_{\\theta_\\varepsilon}^-(y_{\\varepsilon},s_{\\varepsilon}).\n\\end{equation}\nConsider the half line $R:=\\{x\\}\\times(-\\infty,T)$. Observe that\n\\[\nR \\cap \\mathcal{P}_{\\theta_\\varepsilon}^-(y_{\\varepsilon},s_{\\varepsilon}) = \\{x\\} \\times (-\\infty,t_{\\varepsilon,x}), \\quad \\mbox{being} \\quad t_{\\varepsilon,x} := s_\\varepsilon-C_2^{\\bar{p}}(2-|x_i-y_{\\varepsilon,i}|)^{p_i}\\theta^{2-p_i}.\n\\]\nAccording to \\eqref{infimum}, this shows that\n\\begin{equation*}\nm \\leq u(x,s) \\leq m+\\varepsilon, \\quad \\text{for all} \\quad s < t_{\\varepsilon,x}.\n\\end{equation*}\nAccordingly, \\eqref{perse} is proved, by arbitrariness of $x$ and $\\varepsilon$. \nA similar argument shows that\n\\begin{equation}\\label{perse2}\n    \\sup_{S_T} u <\\infty \\quad \\Rightarrow \\quad \\lim_{t\\rightarrow -\\infty} u(x,t) = \\sup_{S_T} u \\quad \\forall x \\in {\\mathbb R}^N.\n\\end{equation}\nEventually this implies that any $u$ solution to \\eqref{EQ} which is bounded from both above and below in the whole $S_T$ is necessarily constant. Indeed, by \\eqref{perse} and \\eqref{perse2} we have $\\sup_{S_T} u= \\inf_{S_T} u$. This argument proves Corollary \\ref{LiouvilleCor}.\n\\vskip0.2cm \\noindent \nIn order to conclude the proof of Theorem \\ref{Liouville1}, we use the assumption that there exists $\\bar{s} \\in (-\\infty,\\, T)$ such that $u(\\cdot, \\bar{s})$ is bounded from above in the whole ${\\mathbb R}^N$ by a suitable $M_s\\in {\\mathbb R}$. Indeed, letting $\\theta_x=(u(x,\\bar{s})-m)\/C_1$ for any $x\\in{\\mathbb R}^N$ and using the intrinsic backward Harnack inequality for $u-m$ again, we get the uniform bound \n\\[\nu(y,s) \\leq C_3 u(x,\\bar{s}) \\leq C_3 M_{\\bar{s}}, \\quad \\text{for all} \\quad x \\in {\\mathbb R}^N \\quad \\mbox{and} \\quad (y,s) \\in \\mathcal{P}_{\\theta_{x}}^-(x,\\bar{s}).\n\\] Reasoning as above, with $\\mathcal{P}_{\\theta_x}^-(x,\\bar{s})$ instead of $\\mathcal{P}_{\\theta_{\\varepsilon}}^- (y_{\\varepsilon},\\, s_{\\varepsilon})$, besides recalling that $u$ bounded from both above and below in $\\mathcal{P}_{\\theta_x}^-(x,\\bar{s})$ uniformly in $x \\in {\\mathbb R}^N$, we conclude that $u$ is constant in $S_{\\bar{s}}$.\n\\end{proof} \n\n\\noindent As a general principle, the bigger the set where the equation is solved the stronger the rigidity: for solutions of \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$, it suffices to check their asymptotic (in time) two-side boundedness at a single point $y \\in {\\mathbb R}^N$ to infer that they are constant, as shown by the next theorem. \n\\begin{theorem}\\label{Liouville2}\nSuppose \\eqref{pi} to be satisfied. Let $u$ be a local weak solution to \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$ which is bounded from below. If, in addition, there exists $y \\in {\\mathbb R}^N$ and a sequence $\\{ s_n\\} \\subset {\\mathbb R}$, $s_n\\to+\\infty$, such that $\\{u(y, s_n)\\}$ is bounded, then $u$ is constant.\n\\end{theorem}\\noindent \n\\begin{remark}\nWe explicitly point out the following straightforward consequence of Theorem \\ref{Liouville2}. Let $u$ be a local weak solution to \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$ which is bounded from below. Suppose that, for some $y \\in {\\mathbb R}^N$, one has\n\\begin{equation}\\label{Liouville2HP}\n    \\liminf_{t\\rightarrow +\\infty} u(y,t)=\\alpha \\in {\\mathbb R}.\n\\end{equation}\\noindent Then $u$ is constant.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{Liouville2}]\nLet $m:=\\inf u$ and consider $\\tilde{u}:=u+m+C_1$, which is a solution to \\eqref{EQ}. By assumption, there exist $M \\in {\\mathbb R}$ and $\\{s_n\\}\\subset{\\mathbb R}$ such that $s_n \\to +\\infty$ and\n\\[\n\\tilde{u}(y,s_n)<M \\quad  \\quad \\forall n \\in {\\mathbb N}.\n\\]\nLet us fix arbitrarily $\\bar{s}\\in {\\mathbb R}$ and let $\\bar{n} \\in {\\mathbb N}$ be big enough such that $s_n >\\bar{s}$ for all $n \\ge \\bar{n}$. Then, for all $n\\ge\\bar{n}$, we set $\\theta_n:=\\tilde{u}(y,s_n)\/C_1$ and define a sequence of radii $\\{\\rho_n\\}$ through\n\\[\ns_n-\\theta_n^{2-\\bar{p}} (C_2 \\rho_n)^{\\bar{p}} =\\bar{s}, \\quad \\quad \\mbox{that is,} \\quad \\quad \\rho_n= \\bigg[\\theta_n^{\\bar{p}-2}\\frac{(s_n-\\bar{s})}{C_2^{\\bar{p}}} \\bigg]^{1\/\\bar{p}}.\n\\] We want to apply the Harnack inequality to deduce an upper bound for $\\tilde{u}(\\cdot,\\bar{s})$ in the whole ${\\mathbb R}^N$; so we need to check that the intrinsic anisotropic cubes $\\mathcal{K}_{\\rho_n}(\\theta_n)$ expand as $s_n\\to+\\infty$. An explicit computation yields\n\\[\n\\mathcal{K}_{\\rho_n}(\\theta_n)= \\prod_{i=1}^N \\bigg{\\{}|x_i|<\\theta_n^{{(p_i-2)}\/{p_i}}  \\left(\\frac{s_n-\\bar{s}}{C_2^{\\bar{p}}}\\right)^{{1}\/{p_i}} \\bigg{\\}}\\quad \\xrightarrow[n \\to \\infty]{} \\quad {\\mathbb R}^N,\n\\] since $1\\le\\theta_n\\le M\/C_1$ and $\\{s_n\\}$ diverges. By the intrinsic backward Harnack inequality \\eqref{Harnack} we have\n\\[\n\\sup_{y+\\mathcal{K}_{\\rho_n}(\\theta_n)} \\tilde{u} \\bigg(\\, \\,  \\cdot\\, \\, ,\\,  s_n-\\theta_n^{2-\\bar{p}} (C_2\\rho_n)^{\\bar{p}} \\bigg)\\leq C_3 \\,\\tilde{u}(y,s_n) \\leq C_3 M \\quad \\forall n\\ge \\bar{n}.\n\\] Thus, recalling the definition of $\\{\\rho_n\\}$, we get the uniform estimate \n\\[\n\\sup_{y+\\mathcal{K}_{\\rho_n}(\\theta_n)} \\tilde{u}(\\cdot, \\bar{s} ) \\leq C_3 M \\quad \\forall n\\ge \\bar{n},\n\\]\nwhence, letting $n\\to\\infty$,\n\\[\n\\sup_{{\\mathbb R}^N} \\tilde{u}(\\cdot, \\bar{s}) \\leq C_3 M.\n\\]\nNow we are in the position to apply Theorem \\ref{Liouville1}. The proof is concluded by arbitrariness of $\\bar{s}\\in{\\mathbb R}$, after transforming $\\tilde{u}$ back to $u$.\n\\end{proof} \n\n\n\\noindent Finally, we show that the oscillation estimates \\eqref{control} constitute a Liouville property for non-negative ancient solutions. This allows us to get rid of the range of $p_i$s of finite speed of propagation in Theorem \\ref{Harnack-Inequality}, at the price of assuming a suitable oscillation decay. \n\n\n\\begin{theorem} \\label{Cutilisci}\nLet $u$ be a bounded function in $S_T$ satisfying the oscillation estimates \\eqref{control}. Then $u$ is constant.\n\\end{theorem}\n\n\n\\begin{proof} The proof is an adaptation of an early idea already present in \\cite{Glagoleva1} (see also \\cite{Landis}); here the adversity is the intrinsic geometry, which in the case of global boundedness turns out to be simpler. The natural geometry will be dictated by $\\omega_o= 2 ||u||_{\\infty, S_T}$. Indeed, let $A,B \\in S_T$ be two points such that $u(A) \\ne u(B)$ and call $Z_T= \\{T\\} \\times {\\mathbb R}^N$. Setting\n\n\\[d= \\max \\{\\mathrm{dist}(A, Z_T),\\,  \\mathrm{dist}(B, Z_T),\\, \\mathrm{dist}(A,B) \\},\\]\nwe choose a radius $R_o>0$ big enough to enclose $A$ and $B$ inside an intrinsic backward cylinder $\\tilde{\\mathcal{Q}}_0:=\\mathcal{Q}_{R_o}^-(\\omega_o\/C_1,C_2)$,\nso that $R_o$ satisfies \n\\begin{equation*}\n    \\begin{cases}\n  (\\omega_o\/C_1)^{p_i-\\bar{p}} R_o^{{\\bar{p}}} >d^{p_i},\\quad i =1,..,N,\\\\\n({\\omega_o}\/{C_1})^{2-\\bar{p}} (C_2 R_o)^{\\bar{p}} >d.\n    \\end{cases\n\\end{equation*}\nLet us set $\\delta=4C_3\/(1+4C_3) \\in (0,1)$ and define $\\varepsilon= \\delta^{({\\bar{p}-2})\/{\\bar{p}}}\/A \\in (0,1)$, being $A=4^{p_N}$ as in Proposition \\ref{birra} below. Suppose, without loss of generality, that $T=0$, and construct the sequence of expanding backward cylinders $\\tilde{\\mathcal{Q}}_n$ centered in $(y,s)=(0,0)$\nas\n\\[\n\\tilde{\\mathcal{Q}}_n= \\prod_{i=1}^N \\bigg{\\{}|x_i|<[\\omega_o\/(C_1\\delta^n)]^{{(p_i-\\bar{p})}\/{p_i}} (R_o\/\\varepsilon)^{{\\bar{p}}\/{p_i}}  \\bigg{\\}} \\times [(\\omega_o\/(C_1 \\delta^{n})]^{2-\\bar{p}} [(C_2 R_o)\/\\varepsilon) ]^{\\bar{p}} ,\\, 0\\bigg].\n\\] Using the oscillation inequalities \\eqref{control}, we arrive at the contradiction \n\\[\n    2 ||u||_{\\infty, S_T} \\ge \\operatornamewithlimits{osc}_{\\tilde{\\mathcal{Q}}_0} u \\ge (\\operatornamewithlimits{osc}_{\\tilde{\\mathcal{Q}}_n} u)\/ \\delta^n \\ge |u(A)-u(B)|\/ \\delta^{n} \n\\] for large $n \\in {\\mathbb N}$.\n\\end{proof}\n\\noindent Theorem \\ref{Cutilisci} can be formulated without the assumption that $u$ is a solution of any equation. Indeed, this general principle goes far beyond equation \\eqref{EQ} and is a key argument to prove rigidity results in a very general class of equations (see, e.g., \\cite[Prop. 18.4]{DBGV-mono}). Its importance enters into play when a Harnack inequality ceases to hold true.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Time-extrinsic Harnack inequality} \\label{WHSection}\n In this section we show how it is possible to free the Harnack inequality from its intrinsic geometry in time. More specifically, we give a formulation of the Harnack inequality allowing the solution to be evaluated at any time level, independently of the anisotropic geometry, provided there is enough room for the anisotropic evolution inside $\\Omega_T$. Unlike the isotropic case, here it looks more difficult to get rid of the intrinsic geometry along the space variables. The proof of the next theorem exploits a comparison with the abstract Barenblatt fundamental solution $\\mathcal{B}$ of Theorem \\ref{Barenblatt} to control the positivity.\n\n\n\\begin{theorem}\n\\label{WeakHarnack} Let $u \\ge 0$ be a local weak solution to \\eqref{EQ} in $\\Omega_T$, and assume \\eqref{pi}. Then there exist $ \\tilde{\\eta} >0$ and $\\gamma>1$, depending only on $N$ and $p_i$s, such that for all $(x_o,t_o)\\in \\Omega_T$ and $\\rho, \\tilde{\\theta}>0$ fulfilling the condition\n\\begin{equation}\\label{domain}\n(x_o, t_o+ \\tilde{\\theta})+\\mathcal{Q}_{C_3 \\rho}(u(x_o,t_o)\/C_1,C_2) \\subseteq \\Omega_T\n\\end{equation} we have\n\\begin{equation}\\label{WH}\n    u(x_o,t_o)\\leq \\gamma \\bigg{\\{} \\bigg(\\frac{\\rho^{\\bar{p}}}{\\tilde{\\theta}}  \\bigg)^{{1}\/{(\\bar{p}-2)}}+ \\bigg( \\frac{\\tilde{\\theta}}{\\rho^{\\bar{p}}} \\bigg)^{N\/\\bar{p}} \\bigg[\\inf_{x_o+K_{\\tilde{\\eta}\\rho}(\\tilde{\\eta} u(x_o,t_o)\/C_1)} u( \\cdot,\\, t_o+\\tilde{\\theta})  \\bigg]^{\\lambda\/\\bar{p}}\\bigg{\\}},\n\\end{equation} \\noindent where $C_1,C_3>1$ come from Theorem \\ref{Harnack-Inequality} while $\\lambda,\\tilde{\\eta}>0$ stem from Theorem \\ref{Barenblatt}. \\end{theorem}\n\\begin{proof}\nLet $\\rho, \\tilde{\\theta} >0$ be such that \\eqref{domain} holds true. Set\n\\begin{equation} \\label{t*}\nt^*:= \\bigg( \\frac{C_1}{u(x_o,t_o)} \\bigg)^{\\bar{p}-2} (C_2 {\\rho})^{\\bar{p}}.\n\\end{equation} We can suppose $t^*<\\tilde{\\theta}\/2$; otherwise we get $u(x_o,t_o) \\leq \\gamma ({\\rho}^{\\bar{p}}\/\\tilde{\\theta})^{1\/(\\bar{p}-2)}$ for a suitable $\\gamma= \\gamma (C_1, C_2,\\bar{p})$, and \\eqref{WH} is valid. Observe that $t^*<\\tilde{\\theta}\/2$ and \\eqref{domain} imply\n\\[\nt_0+\\bigg( \\frac{C_1}{u(x_o,t_o)}\\bigg)^{\\bar{p}-2} (C_2 {\\rho})^{\\bar{p}} <t_0+ \\tilde{\\theta}\/2 < T \\quad \\mbox{and} \\quad x_o + \\mathcal{K}_{C_3 {\\rho}}(u(x_o,t_o)\/C_1) \\subset \\Omega.\n\\] Hence the forward Harnack inequality \\eqref{Harnack} furnishes\n\\[\nu(x_o,t_o) \\leq C_3 u(x,\\, t_o+t^*) \\quad \\quad \\forall \\, x \\in x_o + \\mathcal{K}_{{\\rho}}(u(x_o,t_o)\/C_1).\n\\] This initial value can be considered for a comparison with the Barenblatt solution  $\\mathcal{B}_{\\sigma}(x-x_o,t-s)$ centered at $(x_o,s)$, being $s,\\sigma>0$ to be chosen such that $\\mathcal{B}_{\\sigma}(x-x_o,t_o+t^*-s)$ lies below $u$ in $x_0+\\mathcal{K}_{\\rho}(u(x_o,t_o)\/C_1)$. These requirements can be written as\n\\begin{equation}\\label{supportami}\n\\begin{cases}\n\\operatorname{supp}{\\mathcal{B}_{\\sigma}}(\\cdot-x_o, t_o+t^*-s) \\subseteq x_o+ \\mathcal{K}_{\\rho}(u(x_o,t_o)\/C_1),\\\\\n\\|\\mathcal{B}_{\\sigma}(\\cdot-x_o, t_o+t^*-s)\\|_\\infty \\leq u(x_o,t_o)\/C_3.\n\\end{cases}\n\\end{equation}\nAccording to Theorem \\ref{Barenblatt}, conditions in \\eqref{supportami} are fulfilled as long as\n\\begin{equation}\\label{supportami2}\n\\begin{cases}\n \\sigma^{(p_i-2)\/p_i} (t_o+t^*-s)^{\\alpha_i} \\leq \\rho^{\\bar{p}\/p_i} (u(x_o,t_o)\/C_1)^{(p_i-\\bar{p})\/p_i},\\\\\n \\sigma (t_o+t^*-s)^{-\\alpha} \\leq u(x_o,t_o)\/C_3.\n\\end{cases}\n\\end{equation}\n\\noindent Inequalities in \\eqref{supportami2} are in turn ensured by choosing\n\\[\n\\sigma= (t_o+t^*-s)^{N\/\\lambda} u(x_o,t_o)\/C_3 \\quad \\mbox{and} \\quad s= t_o+t^*- \\bigg( \\frac{\\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}} \\bigg) \\gamma_1,\\] \nwhere $\\gamma_1=\\min \\{(C_3^{p_i-2})\/(C_1^{p_i-\\bar{p}})\\, |\\, i=1,\\dots,N\\}$. Therefore the comparison principle, applied at the time $t_o+\\tilde{\\theta}>t_o+t^*$, gives\n\\begin{equation}\\label{comparison} \\begin{aligned}\nu(x, t_o+\\tilde{\\theta}) &\\ge \\tilde{\\eta} \\sigma |t_o+t^*-(t_o+\\tilde{\\theta})|^{-\\alpha} = \\tilde{\\eta} \\bigg( \\frac{u(x_o,t_o)}{C_3} \\bigg) (t_o+t^*-s)^{N\/\\lambda} (\\tilde{\\theta}-t^*)^{-N\/\\lambda}\\\\\n&\\ge \\tilde{\\eta} \\bigg( \\frac{u(x_o,t_o)}{C_3} \\bigg) \\bigg(  \\frac{\\gamma_1  \\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}} \\bigg)^{N\/\\lambda} \\tilde{\\theta}^{-N\/\\lambda} \\ge \\gamma u(x_o,t_o)^{\\bar{p}\/\\lambda} \\bigg( \\frac{\\rho^{\\bar{p}}}{\\tilde{\\theta}} \\bigg)^{N\/\\lambda},\n\\end{aligned} \\end{equation} being $\\gamma= \\gamma(\\gamma_1,\\tilde{\\eta})$, for every $x$ in the set of positivity\n\\begin{equation*}\n\\begin{aligned}\n\\mathcal{P}_{t_o+\\tilde{\\theta}-s}(x_o)\\supseteq \\mathcal{P}_{t_o+t^*-s}(x_o) &= \\prod_{i=1}^N\\{|x_i-x_{o,i}|\\leq\\tilde{\\eta} \\rho^{\\bar{p}\/p_i} (u(x_o,t_o)\/C_1)^{(p_i-\\bar{p})\/p_i} \\}\\\\\n&=x_o+ \\mathcal{K}_{\\tilde{\\eta}\\rho}(\\tilde{\\eta}u(x_o,t_o)\/C_1),\n\\end{aligned}\n\\end{equation*}\nwith $\\tilde{\\eta}$ depending only on the data $N, p_i$. Passing \\eqref{comparison} to the infimum on $x_o+ \\mathcal{K}_{\\tilde{\\eta}\\rho}(\\tilde{\\eta}u(x_o,t_o)\/C_1)$ concludes the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\nIn Theorem \\ref{WeakHarnack} the lower bound $u(x_o,t_o)>0$ is not required; moreover, $\\tilde{\\theta}>0$ is arbitrarily chosen between those numbers that preserve the inclusion of the intrinsic cylinder translated to time $\\tilde{\\theta}$ into $\\Omega_T$. Henceforth, when the equation is solved in ${\\mathbb R}^{N+1}$, the proof furnishes inequality \\eqref{WH} without the first term on the right, and one can infer similar Liouville properties as Theorem \\ref{Liouville2}. Actually, Theorems \\ref{Harnack-Inequality} and \\ref{WeakHarnack} are equivalent for small radii.\\vskip0.2cm \n\n\\noindent Indeed, we proved that Theorem \\ref{Harnack-Inequality} implies Theorem \\ref{WeakHarnack}. Now we show that the converse statement can be obtained by a simple choice of $\\tilde{\\theta}$. Indeed, let us pick\n\\[ \\tilde{\\theta}= \\frac{(2\\gamma)^{\\bar{p}-2}\\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}},\n\\] and suppose that $(x_o,t_o+\\tilde{\\theta}) + \\mathcal{Q}_{C_3\\rho} (u(x_o,t_o)\/C_1)\\subset \\Omega_T$. The weak Harnack inequality \\eqref{WH} leads to\n\\[\nu(x_o,t_o) \\leq \\gamma \\bigg{\\{} \\frac{u(x_o,t_o)}{2\\gamma}+ \\bigg( \\frac{2\\gamma}{u(x_o,t_o)} \\bigg)^{{N(\\bar{p}-2)}\/{\\bar{p}}}\\bigg[ \\inf_{x_o+ \\mathcal{K}_{\\tilde{\\eta} \\rho}(\\tilde{\\eta} u(x_o,t_o)\/C_1)} u(\\cdot, \\, t_o+ \\bigg( \\frac{u(x_o,t_o)}{2\\gamma} \\bigg)^{2-\\bar{p}} \\rho^{\\bar{p}}) \\bigg]^{{\\lambda}\/{\\bar{p}}}  \\bigg{\\}},\n\\]\nwhence\n\\[\nu(x_o,t_o) \\leq \\tilde{C_3} \\inf_{x_o+ \\mathcal{K}_{\\tilde{\\rho}}(M)} u(\\cdot, \\, t_o+ \\tilde{C_2} M^{2-\\bar{p}} \\tilde{\\rho}^{\\bar{p}}), \\quad M= u(x_o,t_o)\/\\tilde{C}_1, \\] for all $\\tilde{\\rho} \\leq \\tilde{\\eta} \\rho$ and with positive constants \n\\[\\tilde{C_1}= C_1\/\\tilde{\\eta}, \\quad \\quad  \\tilde{C_2}=\\frac{(2\\gamma\/\\tilde{C}_1)^{\\bar{p}-2}}{\\tilde{\\eta}^2}, \\quad \\quad \\tilde{C_3}= 2 \\gamma.\n\\] \\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix: H\\\"older Continuity of solutions}\n\\label{Appendix}\n\n\n\\begin{theorem}\\label{HC}\nUnder condition \\eqref{pi}, any local weak solution $u$ to \\eqref{EQ} is locally H\\\"older continuous. More precisely, there exist $\\gamma>1$ and $\\chi \\in (0,1)$, depending only upon $p_i,N$, with the following property: for each compact set $K \\subset \\subset \\Omega_T$ there exist a set $\\Lambda$ and $\\omega_o=\\omega_o(K, \\|u\\|_{\\infty,K})$ such that $K \\subset \\Lambda \\subseteq \\Omega_T$ and, for every $(x,t)$, $(y,s) \\in K$,\n\\begin{equation} \\label{HContinuity}\n    |u(x,t)-u(y,s)| \\leq \\gamma \\omega_o \\bigg(\\frac{\\sum_{i=1}^N |x_i-y_i|^{{p_i}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-p_i)}\/{\\bar{p}}}+ |t-s|^{1\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-2)}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda) } \\bigg)^{\\chi},\n\\end{equation}\\noindent with\n\\begin{equation} \\label{pi-dist}\n\\begin{aligned}\n&{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda):=\\inf \\{ {\\bf{p}}_x, {\\bf{p}}_t \\}, \\quad \\text{being}\\\\\n& {\\bf{p}}_x=\\inf \\bigg{\\{} \n|x_i-y_i|^{{p_i}\/{\\bar{p}}}(\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}\\, : \\, (x,t) \\in K, (y,s) \\in \\partial \\Lambda,\\, i=1,..,N\\bigg{\\}},\\\\\n& {\\bf{p}}_t=\\inf \\bigg{\\{} \n|t-s|^{{1}\/{\\bar{p}}}(\\omega_o\/C_1)^{{(\\bar{p}-2)}\/{\\bar{p}}}\\, : \\, (x,t) \\in K, (y,s) \\in \\partial \\Lambda\\bigg{\\}}.\n\\end{aligned}\n\\end{equation} \\noindent Furthermore, if $u$ is bounded in $\\Omega_T$ then \\eqref{HContinuity} holds with $\\Lambda= \\Omega_T$.\n\\end{theorem} \\noindent We prove Theorem \\ref{HC} in four steps, without assuming that $u$ is globally bounded.\n\\begin{proof}\nLet us fix a compact set $K \\subset \\subset \\Omega_T$ and two points $(y,s), (x,t) \\in K$. \\vskip0.2cm \n\\noindent {\\small{STEP 1-{\\it A global bound for the solution in $K$.}}}\n\\vskip0.2cm \\noindent Set $g(k)=\\sum_{i=1}^{N}k^{ p_{i}-2}$ and $h(k)=\\left(\\sum_{i=1}^{N}k^{p_{i}-\\bar p_{2}}\\right)^{-1}$. We use the estimates in \\cite[Lemma 4.2]{Mosconi}: under the exponent range \\eqref{pi}, subsolutions to \\eqref{EQ} satisfy the estimate \n\\begin{equation}\n\\label{supest}\n\\|u_{+}\\|_{L^{\\infty}(Q_{\\lambda\/2, M})}\\leq g^{-1}(1\/M)+ h^{-1}\\left(C\\Big(M\\, \\bint\\kern-0.15cm\\bint_{Q_{\\lambda, M}} u_{+}^{\\bar p_{2}}\\, dx\\Big)^{{\\bar p}\/{(N+\\bar p)}}\\right)\n\\end{equation} in the anisotropic cylinders \n\\begin{equation} \\label{anisocylinder}\nQ_{\\lambda, M}= \\prod_{i=1}^{N}\\left[-\\lambda^{{1}\/{p_{i}}}, \\lambda^{{1}\/{p_{i}}}\\right]\\times [-M\\, \\lambda, 0],\\quad \\quad  M, \\lambda>0.\n\\end{equation}\n\\vskip0.2cm \\noindent By compactness of $K$, we find $(x_i,t_i) \\in K$ and $\\lambda_i, M_i \\in \\mathbb{R}_+$, $i=1,\\dots,m$, for $m\\in{\\mathbb N}$, such that \n\\begin{equation*}\n    K \\subset \\Lambda:=\\bigcup_{j=1}^m \\{(x_j,t_j)+Q_{\\lambda_j,M_j}\\}\\subseteq \\bigcup_{j=1}^m \\{(x_j,t_j)+Q_{2\\lambda_j,M_j} \\}\\subseteq \\Omega_T,\n\\end{equation*} \\noindent being $Q_{\\lambda,M}$ as in \\eqref{anisocylinder}. \n\\noindent According to \\eqref{supest}, for each anisotropic cylinder  $\\hat{Q}_{\\lambda_j,M_j}=(x_j,t_j)+Q_{\\lambda_j,M_j}$, $j=1,\\dots,m$, we deduce the estimate \n\\begin{equation*} \\begin{aligned}\\label{A}\n\\| u\\|_{L^{\\infty}(\\hat{Q}_{\\lambda_j, M_j})} &\n\\leq g^{-1} (1\/\\min_{j} M_j)+ h^{-1} \\bigg( C \\max_{j=1,\\dots,m}  \\bigg( M_j \\Xint{\\raise4pt\\hbox to7pt{\\hrulefill}} \\Xint{\\raise4pt\\hbox to7pt{\\hrulefill}}_{\\hat{Q}_{2\\lambda_j,M_j}} |u|^{\\bar{p}_2}\\,  dxdt\\bigg)^{{\\bar{p}}\/{(N+\\bar{p})}} \\bigg)=: \\mathcal{I},\n\\end{aligned}\n\\end{equation*} \\noindent because $h$,$g$, are monotone increasing.\n\\noindent Finally, we define $\\omega_o=\\omega_o(K)$ as \\begin{equation} \\label{0}\n\\omega_o:= 2 \\mathcal{I}. \\end{equation} \\noindent  Accordingly,\n\\[\nK \\subset \\bigcup_{j=1}^m \\hat{Q}_{\\lambda_j, M_j}(x_j,t_j)= \\Lambda \\qquad \\mbox{and} \\qquad 2 \\|u\\|_{L^{\\infty}(K)} \\leq \\omega_o.\n\\]\n\\vskip0.2cm \\noindent {\\small{STEP 2-{\\it Accommodation of degeneracy and alternatives.}}}\n\\vskip0.2cm \\noindent\nRecalling \\eqref{pi-dist}\nwe define $R:= [{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)]\/(2C_3)$. Now, by definition of $R$, the intrinsic cylinder centered at $(y,s)\\in K$ and constructed with $R$ and $\\omega_o$ is contained inside  $\\Lambda$, that is,\n\\[\n(y,s)+ \\mathcal{Q}_{R}(\\omega_o\/C_1,C_2) \\subseteq \\Lambda.\n\\]\n\n\n\\noindent Now consider any other point $(x,t) \\in K$. We reduce the study of the oscillation only in $(y,s) + Q_R(\\omega_o\/C_1)$, having elsewhere the H\\\"older continuity of $u$. Indeed, if $|s-t| \\ge (\\omega_o\/C_1)^{2-\\bar{p}}(C_2 R)^{\\bar{p}}$, we have\n\\[\n|u(y,s)-u(x,t)|\\leq |u(y,s)|+|u(x,t)|\\leq \\omega_o \\leq 2C_3 \\omega_o \\bigg( \\frac{(\\omega_o\/C_1)^{{(\\bar{p}-2)}\/{\\bar{p}}}|s-t|^{{1}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)} \\bigg) \\] by definition of $R$.\nSimilarly, if $|y_i-x_i| \\ge (\\omega_o\/C_1)^{{(p_i-\\bar{p})}\/{\\bar{p}}} R^{{\\bar{p}}\/{p_i}}$ for some $i \\in \\{1,\\dots,N\\}$, the same conclusion follows from\n\\[\n|u(y,s)-u(x,t)|\\leq |u(y,s)|+|u(x,t)|\\leq \\omega_o \\leq 2C_3 \\omega_o \\bigg( \\frac{(\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}|y_i-x_i|^{{p_i}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)} \\bigg).\n\\]\nHence we can assume that\n\\begin{equation}\\label{exclusion}\n   |s-t|<(\\omega_0\/C_1)^{2-\\bar{p}} (C_2 R)^{\\bar{p}} \\quad \\text{and}  \\quad  |y_i-x_i|< (\\omega_o\/C_1)^{{(p_i-\\bar{p})}\/{p_i}} R^{{\\bar{p}}\/{p_i}}  \\quad \\forall i=1,\\dots, N, \n\\end{equation}\nthat is, \n\\[(x,t) \\in (y,s)+ \\mathcal{Q}_R(\\omega_o\/C_1,C_2). \\]\nWe take the cylinder $\\mathcal{Q}_0:=(y,s)+\\mathcal{Q}_R^-(\\omega_o\/C_1,C_2)$ as the first element of a net $\\{\\mathcal{Q}_n\\}_n$ of cylinders shrinking to the center $(y,s)$. This net will be constructed to control uniformly the oscillation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vskip0.2cm \\noindent \n\n\n\\noindent {\\small{STEP 3- { \\it Controlled reduction of oscillation }}} \n\\begin{proposition}[Reduction of oscillation in shrinking cylinders] \\label{birra} Let the hypothesis of Theorem \\ref{HC} be satisfied and assume also \\eqref{exclusion}. Then, setting \n\\begin{equation*}\n    \\begin{cases}\n    \\omega_0= \\omega_o(K),\\\\\n    \\omega_n=\\delta \\omega_{n-1}, \\, n\\ge 1,\n    \\end{cases} \n    \\begin{cases}\n    \\theta_n= \\omega_n\/C_1, \\, n\\ge 0,\\\\\n    \\rho_0=R,\\\\\n    \\rho_n= \\varepsilon \\rho_{n-1}, \\, n\\ge 1,\\\\\n    \\end{cases}\n    \\begin{cases}\n    \\delta=4C_3\/(1+4C_3),\\\\\n    \\varepsilon=\\delta^{{(\\bar{p}-2)}\/{\\bar{p}}}\/A, \\\\\n    A=4^{p_N}, \\end{cases}\n\\end{equation*} \\noindent we have both the inclusions\n\\[\n\\mathcal{Q}_{n}\\subset \\mathcal{Q}_{n-1}, \\quad \\text{with} \\quad \\mathcal{Q}_n= (y,s)+ \\mathcal{Q}_{\\rho_n}^-(\\theta_n)= \\prod_{i=1}^N \\bigg{\\{}|y_i-x_i|<\\theta_n^{{(p_i-\\bar{p})}\/{p_i}}\\rho_n^{{\\bar{p}}\/{p_i}}  \\bigg{\\}} \\times \\bigg(s-\\theta_n^{2-\\bar{p}} (C_2\\rho_n)^{\\bar{p}} ,\\, s\\bigg],\n\\] and the inequalities\n\\begin{equation}\\label{control}\n    \\operatornamewithlimits{osc}_{\\mathcal{Q}_n} u \\leq \\omega_n = \\delta^n \\omega_o.\n\\end{equation}\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{birra}]\\noindent First of all, we prove that $\\mathcal{Q}_{n} \\subset \\mathcal{Q}_{n-1}$ for all $n\\in{\\mathbb N}$. By direct computation,\n\\begin{equation*} \\begin{aligned}\n\\theta_{n}^{2-\\bar{p}} (C_2\\rho_{n})^{\\bar{p}} = \\bigg(\\frac{\\delta \\omega_{n-1}}{C_1}\\bigg)^{2-\\bar{p}}\\bigg((C_2\\rho_{n-1}\/A)^{\\bar{p}}\\delta^{\\bar{p}-2}\\bigg)= \\theta_{n-1}^{2-\\bar{p}} (C_2\\rho_{n-1}\/A)^{\\bar{p}}.\n\\end{aligned} \\end{equation*}  For each $i\\in \\{1,..,N\\}$, since $p_i>2$ and $\\delta \\in (0,1)$, it holds\n\\[\n\\theta_{n}^{p_i-\\bar{p}} \\rho_{n}^{{\\bar{p}}} = \\delta^{p_i-2} \\theta_{n-1}^{{p_i-\\bar{p}}} ( {\\rho_{n-1}}\/{A} )^{{\\bar{p}}} \\leq \\theta_{n-1}^{{p_i-\\bar{p}}} ( {\\rho_{n-1}}\/{A} )^{{\\bar{p}}}.\n\\] This computation shows a little more, by allowing indeed $\\mathcal{Q}_{n}\\subset (y,s)+\\mathcal{Q}_{\\rho_{n-1}\/A}^-(\\theta_{n-1})\\subset \\mathcal{Q}_{n-1}$.\n\\noindent Now we prove \\eqref{control} by induction. The base step holds true: indeed, the accommodation of degeneracy (see Step 2 above) entails $\\mathcal{Q}_0\\subset\\Lambda$, so that the bound produced in Step 1 yields\n\\[\n\\operatornamewithlimits{osc}_{\\mathcal{Q}_0} u \\leq \\operatornamewithlimits{osc}_{\\Lambda} u \\leq 2\\|u\\|_{L^\\infty(\\Lambda)} \\leq \\omega_o.\n\\]\nWe assume now that the statement \\eqref{control} is true until step $n$ and we show it for $n+1$. This will determine the number $A$. More precisely, we assume that $\\operatornamewithlimits{osc}_{\\mathcal{Q}_n}u \\leq \\omega_n$ and, by contradiction, that $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}} u > \\omega_{n+1}$. We set\n\\[\nM_{n}= \\sup_{\\mathcal{Q}_{n}} u, \\qquad m_{n}= \\inf_{\\mathcal{Q}_{n}}u, \\qquad P_{n}=(y,\\, s-\\theta_{n}^{2-\\bar{p}}(C_2\\rho_{n})^{\\bar{p}}).\n\\] Now we observe that one of the following two inequalities must hold:\n\\[\nM_{n}-u(P_{n}) > \\omega_{n+1}\/4 \\quad \\text{or} \\qquad u(P_{n})-m_{n} > \\omega_{n+1}\/4.\n\\] Indeed, if both alternatives are violated, then by adding the opposite inequalities we obtain $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n}} u  \\leq \\omega_{n+1}\/2< \\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}$, generating an absurd because $\\mathcal{Q}_{n+1} \\subseteq \\mathcal{Q}_n$. Let us suppose $M_{n}-u(P_{n}) \\ge  \\omega_{n+1}\/4$, the other case being similar. In particular we have the double bound\n\\begin{equation}\n    \\label{doublebound}\n    \\omega_{n+1}\/4\\leq M_n - u(P_n) \\leq \\omega_n.\n\\end{equation}\nLet us set $\\hat{\\theta}_n=(M_n-u(P_n))\/C_1$. We work in the half-paraboloid $\\mathcal{P}^+_n=\\mathcal{P}^+_{\\hat{\\theta}_n}(P_n)$ for times restricted to the ones of $\\mathcal{Q}_n$.\n\\noindent\nWe notice that the starting time of $P_n^+$ is the same of $\\mathcal{Q}_n$ (see \\ref{FigA}). To show that $\\mathcal{P}_n^+\\subset\\mathcal{Q}_n\\subset\\Omega_T$, we control the space variables. By definition of $\\omega_n$, we obtain the following estimate:\n\\begin{equation}\\label{spatial-est}\n|x_i-y_i|^{p_i}<  \\bigg(\\frac{M_n-u(P_n)}{C_1}\\bigg)^{p_i-2}\\rho_n^{\\bar{p}}\\bigg(\\frac{\\omega_n}{C_1}\\bigg)^{2-\\bar{p}}=\\bigg(\\frac{M_n-u(P_n)}{C_1}\\bigg)^{p_i-2} \\bigg(\\frac{R}{A^n}\\bigg)^{\\bar{p}} \\bigg(\\frac{\\omega_o}{C_1} \\bigg)^{2-\\bar{p}},\n\\end{equation}\nfor all $x\\in\\pi_x(\\mathcal{P}_n^+)$, being $\\pi_x$ is the projection on the space variables. \\\\\n\\noindent\nNow we show that, after a certain time $\\bar{t}$, the whole cylinder $\\mathcal{Q}_{n+1}$ is contained in the paraboloid $ \\mathcal{P}^+_n$; see Figure \\ref{FigA} for a representation. For times $t >s-(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}}$, we denote by $\\mathcal{P}^+_n(t)$ the time-section of $\\mathcal{P}^+_n$ at time $t$:\n\\[ \\mathcal{P}^+_n(t)= \\bigg{\\{} x \\in {\\mathbb R}^N: \\, \\,    |x_i-y_i|^{p_i}< C_2^{-\\bar{p}} [(M_n-u(P_n))\/C_1]^{p_i-2}(t-s+ (\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}})  \\bigg{\\}}.\\]\n\\noindent Let us set \\begin{equation} \\label{t} \\bar{t}=s-(\\omega_{n+1}\/C_1)^{2-\\bar{p}}(C_2 \\rho_{n+1})^{\\bar{p}},\\end{equation}\nand let us prove that at time $\\bar{t}$ we have the inclusion $\\pi_{x}(\\mathcal{Q}_{n+1})\\subset \\mathcal{P}^+_n(\\bar{t})$. This reduces to show that\n\\[\\rho_{n+1}^{\\bar{p}} (\\omega_{n+1}\/C_1)^{p_i-\\bar{p}} \\leq (A^{\\bar{p}}-1)  [(M_n-u(P_n)\/C_1)]^{p_i-2} \\rho_{n+1}^{\\bar{p}} \n (\\omega_{n+1}\/C_1)^{2-\\bar{p}},\\]\n that is,\n\\[\\omega_{n+1}^{p_i-2} \\leq (A^{\\bar{p}}-1) (M_n-u(P_n))^{p_i-2}.\\]\nAccording to \\eqref{doublebound}, this inequality holds true if we choose $A$ such that $4^{p_N-2}<A^{\\bar{p}}-1$, as for instance $A= 4^{p_N}$. \\vskip0.1cm \\noindent Therefore, by the Harnack inequality \\eqref{Harnack} and \\eqref{doublebound}, we can estimate the infimum of $M_{n}-u$ in $\\mathcal{Q}_{n+1}$ as\n\\begin{equation} \\label{bound}\n \\inf_{\\mathcal{Q}_{n+1}} (M_{n}-u) \\ge \\frac{M_{n}-u(P_{n})}{C_3} \\ge\\, \\,  \\omega_{n+1}\/(4C_3),\n\\end{equation}\\noindent again referring to Figure \\ref{FigA}. Thus\n\\[\nM_{n} \\ge \\sup_{\\mathcal{Q}_{n+1}} u+ \\omega_{n+1}\/(4C_3).  \n\\]\nAdding $-\\inf_{\\mathcal{Q}_n} u \\ge - \\inf_{\\mathcal{Q}_{n+1}}u$ to both sides, besides using $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}u >\\omega_{n+1}$, we get \n\\[\\omega_n \\ge M_n - \\inf_{\\mathcal{Q}_n} u \\ge  \\sup_{\\mathcal{Q}_{n+1}} u+ \\omega_{n+1}\/(4C_3)  - \\inf_{\\mathcal{Q}_{n+1}} u= \\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}u+\\omega_{n+1}\/(4C_3)> \\bigg(1+\\frac{1}{4C_3}  \\bigg) \\omega_{n+1}\\, .\n\\]This leads to a contradiction by definition of $\\delta$, since\n\\[\n\\omega_n > \\bigg(1+\\frac{1}{4C_3}  \\bigg) \\delta \\omega_{n}=  \\bigg(\\frac{4C_3}{1+4C_3}\\bigg) \\bigg(1+\\frac{1}{4C_3} \\bigg) \\omega_n = \\omega_n.\n\\] \n\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.25]\n\n\\draw[thick,->] (25,0) -- (25,6) node[anchor=north west] {\\small{$x \\in {\\mathbb R}^N$}};\n\\draw[thick,->] (25,0) -- (31,0) node[anchor=south west] {\\small{$t \\in {\\mathbb R}$}};\n\\draw (20,0) rectangle (-6,6);\n\n\n\\draw (20,0)  rectangle (6,4);\n\n\n\n\\draw (20,0) rectangle (-6,-6);\n\n\n\\draw (20,0) rectangle (6,-4);\n\n\\draw (18, 2) node{$\\mathcal{Q}_{n+1}$};\n\\draw (-4, 4) node{$\\mathcal{Q}_{n}$};\n\\draw (4, 4.6) node{\\textcolor{red}{$\\mathcal{P}^+_n$}};\n\\draw (-7, 0) node{$P_n$};\n\\draw (21, 0) node{$s$};\n\n\n\\draw (5.6,-0.65) node{$\\bar{t}$};\n\n\n\n\n\\draw[red]  (20,5.6) parabola (-6,0);\n\n\n\n\n\\draw[red]  (20,-5.6) parabola (-6,0);\n\n\n\n\n\\end{tikzpicture}\n\\caption{{\\small Scheme of the proof of \\eqref{bound}. The anisotropic paraboloid $\\mathcal{P}^+_n$ (in red), that is centered in $P_n=(\\,y,\\, s-(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}})$, evolves in a time $(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}}$ to cover completely $\\mathcal{Q}_{n+1}$.}}\n \\label{FigA}\n\\end{figure}\n    \n\n\n\n\\end{proof}\n\\noindent \n{\\small STEP 4.{\\it Conclusion of the proof of Theorem \\ref{HC}}}\n\\vskip0.2cm \n\n\\noindent If we consider a point $(x,t) \\in (y,s) +\\mathcal{Q}_R^-(\\omega_o\/C_1)$, let $n \\in \\mathbb{N}$ be the last number such that we have $(x,t)\\in \\mathcal{Q}_n$, so that $(x,t) \\not\\in \\mathcal{Q}_{n+1}$. From the first condition and \\eqref{control} we have \n\n\\[|u(x,t)-u(y,s)| \\leq \\operatornamewithlimits{osc}_{\\mathcal{Q}_n} u\\leq \\delta^n \\omega_o.\\]\nThe rest of the job is standard and consists in determining from condition $(x,t) \\not\\in \\mathcal{Q}_{n+1}$ an upper bound for $\\delta^n$. For the sake of simplicity, we just show the case $x \\not\\in y+\\mathcal{K}_{\\rho_{n+1}}$.\n\n\\noindent \nLet $\\beta>0$ be such that $\\delta^{{(\\bar{p}-2)}\/{\\bar{p}}}\/A=\\delta^{\\beta}$. By assumption, there must be an index $i \\in \\{1,\\dots, N\\}$ such that\n\\begin{equation*}\\label{i}\n\\begin{aligned}\n|x_i-y_i|^{p_i}> \\rho_{n+1}^{{\\bar{p}}} (\\omega_{n+1}\/C_1)^{{p_i-\\bar{p}}} = \\gamma(A)(\\delta^n)^{p_i-2}R^{\\bar{p}}(\\omega_o\/C_1)^{p_i-\\bar{p}} \\geq \\gamma(A)(\\delta^n)^{[{\\bar{p}(\\beta-1)+p_i}]}R^{\\bar{p}}(\\omega_o\/C_1)^{p_i-\\bar{p}}\n\\end{aligned}\n\\end{equation*}\n\\noindent that gives us, for $\\chi_i=\\bar{p}\/(\\bar{p}(\\beta-1)+p_i)$, the following estimate of $\\delta^n$:\n \\begin{equation*} \\begin{aligned}\n \\delta^n \\leq&\n \\gamma  \\bigg( \\frac{ |x_i-y_i|^{{p_i}\/{\\bar{p}}}  (\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}}{R} \\bigg)^{{\\bar{p}}\/[{{\\bar{p}(\\beta-1)+p_i}}]} \\\\\n &\\leq \\gamma \\bigg(\\frac{\\sum_{i=1}^N |x_i-y_i|^{{p_i}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-p_i)}\/{\\bar{p}}}+ |t-s|^{{1}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-2)}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda) } \\bigg)^{\\chi_i}.\\end{aligned} \\end{equation*}\n \\noindent\n From $A>4>\\delta^{-1-2\/\\bar{p}}$ we infer $\\beta>2$, whence $\\chi_i\\in(0,1)$. A similar estimate follows from the case where times are not contained, with $\\chi_t= \\bar{p}\/(\\bar{p}(\\beta-1)+2)$. Therefore, recalling that $p_N>2$, we choose the H\\\"older exponent\n\\begin{equation} \\label{alfa}\n    \\chi = \\min \\{\\chi_i, \\chi_t, \\quad i=1,\\dots,N \\}=\\frac{\\bar{p}}{\\bar{p}(\\beta-1)+p_N}.\n\\end{equation}\n \\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\small\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\\label{Sec-Introduction}\n\nComputational science and engineering (CSE) involves the integration\nof a number of different techniques, requiring expertise in data\nstructures, algorithms, numerical analysis, programming methodologies,\nsimulation, visualization, data analysis, and performance\noptimization.  The CSE community has embraced Python as a platform for\nattacking a wide variety of research problems, in part because of\nPython's support for easily gluing together tools from different\ndomains to solve complex problems.  Teaching the theory and practice\nof CSE requires touching on all the subjects mentioned above, in the\ncontext of important and interesting scientific problems.  Many of the\nsame advantages that Python brings to CSE research make it useful for\nteaching too.  Traditionally, courses have tended to focus more\nnarrowly on particular aspects of CSE, such as numerical analysis,\nalgorithms, or high-performance computing.  In developing a new,\nbroadly focused laboratory course in computational science,\nengineering and biology, we have sought to introduce students to the\nwide swath of techniques necessary to do effective research in CSE.\nPython and its many batteries serve remarkably well in this endeavor.\n\n\\emph{Computational methods for nonlinear systems} is a graduate\ncomputational science laboratory course jointly developed and taught\nby us.  We initiated course development in the summer of 2004 to\nsupport the curricular needs of the Cornell IGERT program in nonlinear\nsystems, a broad and interdisciplinary graduate fellowship program\naimed at introducing theoretical and computational techniques\ndeveloped in the study of nonlinear and complex systems to a range of\nfields.  The focal themes of the IGERT program span a number of areas\n- including complex networks, biological locomotion and manipulation,\npattern formation, and gene regulation - broadly interpreted in the\ncontext of complex systems and nonlinear dynamics.  These themes form\nthe core of our course curriculum, augmented with other problems of\ninterest arising in the fields of statistical mechanics, applied\nmathematics, and computer science.\n\nThe format of the course is somewhat unusual.  As a computational\nlaboratory course, it provides relatively little in the way of\nlectures: we prefer to have students learn by doing, rather than\nhaving us tell them how to do things.  The course is autonomous,\nmodular, and self-paced: students choose computational modules to work\non from a large (and hopefully growing) suite of those available, and\nthen proceed to implement relevant simulations and analyses as laid\nout in the exercise.  We provide \\emph{Hints} files to help the\nstudents along: these consist of documented skeletal code that the\nstudents are meant to flesh out.  (In practice, we develop a module\nourselves, document each of the relevant pieces using Python's\ndocstrings, and then replace all the code bodies with the Python\nkeyword \\verb+pass+ so that the students can repopulate those code\nbodies themselves.)  We have written several different visualization tools\nto provide visual feedback.  We find these help to \nengage the students in new problems and are useful in code debugging.\n\nPython is a useful language for teaching for several reasons (even \nthough most of our incoming students have had no previous\nexperience with Python).  Its clean syntax enables students to learn\nthe language quickly, and allows us to provide concise \nprogramming hints in our documented code fragments.  Python's dynamic\ntyping and high-level, built-in datatypes enables students to get\nprograms working quickly, without having to struggle with type\ndeclarations and compile-link-run loops.  Since Python is interpreted,\nstudents can learn the language by executing and analyzing individual\ncommands, and we can help them debug their programs by working with\nthem in the interpreter.\n\nOne of the other key advantages that Python brings to scientific\ncomputing is the availability of many packages supporting numerical\nalgorithms and visualization.  While some of our exercises require\ndevelopment of algorithms from scratch, others rely on established\nnumerical routines implemented in third-party libraries.  It is of\ncourse important to understand the fundamentals of algorithms, error\nanalysis, and algorithmic complexity, but it is also useful to know\nwhen and how to use existing solutions that have been developed by\nothers.  We make heavy use of the {numpy}\\cite{Numpy} and\n{scipy}\\cite{Scipy} packages, for construction of efficient arrays and\nfor access to routines for generation of random numbers, integration\nof ordinary differential equations, root-finding, computation of\neigenvalues, etc.  We use {matplotlib}\\cite{Matplotlib} for x-y\nplotting and histograms.  We have written several visualization\nmodules that we provide to students, based on the {Python Imaging\nLibrary (PIL)}\\cite{PIL},  using PIL's ImageDraw\nmodule to place graphics primitives within\nan image, and the ImageTk module to paste an image into a Tk\nwindow for real-time animation.  We recommend the use of the {ipython}\ninterpreter, which facilitates exploration by students\\cite{IPython}.\nAnd we have used {VPython}\\cite{VPython} to generate three-dimensional\nanimations to accompany some of our modules.\n\n\\section*{Course modules}\n\\label{Sec_Course_modules}\n\nThere are too many course modules to describe in detail here, and we\nrefer interested readers to our course website \\cite{CM4NS} for\ninformation on all the modules, as well as access to problems, hints,\nand answers.  (Many of the exercises have also been incorporated into\na new textbook written by one of us.\\cite{Sethna2006})  Here, we\nhighlight a few of the modules, in order to illustrate both the\nbreadth of science that can be usefully taught with Python and variety\nof tools and techniques that Python can bring to bear on such\nproblems.\n\n\\subsection*{Small world networks}\n\nThe study of complex networks has flourished over the last several\nyears as researchers have discovered commonalities among networked\nstructures that arise in diverse fields such as biology, ecology,\nsociology, and computer science\\cite{Barabasi2002}.  An interesting\nproperty found in many complex networks is exemplified in the popular\nnotion of ``six degrees of separation'', which suggests that any two\npeople on earth are connected through at most roughly five\nintermediate acquaintances.  Duncan Watts and Steve\nStrogatz\\cite{Watts1998} developed a simple model of random networks\nthat demonstrate this ``small-world'' property.  Our course module\nenables students to construct small-world networks and to examine how\nthe average path length connecting two nodes decreases rapidly as\nrandom, long-range bonds are introduced into a network consisting\ninitially of only short-ranged bonds (Figure 1).\n\nComputationally, this module introduces students to data structures\nfor the representation of undirected graphs, object-oriented\nencapsulation of those data structures, and graph traversal\nalgorithms.  Python makes the development of an undirected graph data\nstructure exceedingly simple, a point made long ago by Python creator\nGuido van Rossum in one of his early essays on\nPython\\cite{VanRossum1998}.  In an undirected graph, nodes are\nconnected to other nodes by edges.  A simple way to implement this is\nto combine the two cornerstones of container-based programming in\nPython: lists and dictionaries.  In our \\verb+UndirectedGraph+ class,\na dictionary of network neighbor connections (a neighbor dictionary)\nmaps a node identifier to a list of other nodes to which the reference\nnode is connected.  Because the graph edges are undirected, we\nduplicate the connection information for each node: if an edge is\nadded connecting node 1 and node 2, the neighbor dictionary must be\nupdated so that node 2 is added to node 1's list of neighbors, and\nvice versa.\n\n\\begin{figure}\n\\includegraphics[height=2in]{betweenness.ps}\n\\caption{\\label{SmallWorldFig}\nNode and edge betweenness in a model of small-world networks.  Nodes\n(red dots) are connected by undirected edges (black lines).\nBetweenness measures how central each node and edge is to the shortest\nnetwork paths connecting any two nodes.  In this plot, node diameter\nand edge thickness are propotional to node and edge betweenness,\nrespectively.  (Our simple graph visualization tool uses the Python Imaging\nLibrary.)\n}\n\\end{figure}\n\nWe can of course hide the details of adding edges \ninside an \\verb+AddEdge+ method defined on an \\verb+UndirectedGraph+ class:\n\n\\begin{verbatim}\nclass UndirectedGraph:\n    # ...\n    def AddEdge(self, node1, node2):\n        self.AddNode(node1)\n        self.AddNode(node2)\n        if node2 not in self.neighbor_dict[node1]:\n            self.neighbor_dict[node1].append(node2)\n        if node1 not in self.neighbor_dict[node2]:\n            self.neighbor_dict[node2].append(node1)\n\\end{verbatim}\n\nIn the small-world networks exercise, we choose to label nodes simply\nby integers, but Python's dynamic typing does not require this.  If we\nwere playing the ``Kevin Bacon game'' of searching for\nshortest paths in actor collaboration networks, we could use our code\nabove to build a graph connecting names of actors (encoded as\nstrings).  This dynamic typing allows for significant code reuse (as\ndescribed below in the section on Percolation).  And it is worth\nmentioning that, while our \\verb+UndirectedGraph+ class is exceedingly simple\nand built to support only the analyses relevant to our course module,\nthe same basic principles are at work in a much more comprehensive,\nPython-based, graph construction and analysis package - named NetworkX\n- that has been developed at Los Alamos National Labs.\\cite{NetworkX}\n\n\\subsection*{Percolation}\n\nPercolation is the study of how objects become connected (or\ndisconnected) as they are randomly wired together (or cut apart).\nPercolation is an important and classic problem in the study of phase\ntransitions, and has practical relevance as well: considerable\ninterest over the years in percolation phenomena has come from the oil\nand gas industry, for example, where one is interested in extracting a\nfluid through a network of pores in rock.\n\nAlthough percolation is traditionally studied on regular lattices, it\nis a problem more generally applicable to arbitrary networks, and in\nfact, we are able to reuse some of the code developed in the\nsmall-world networks module to support the study of percolation.  As\nnoted above, Python's dynamic typing makes our definition of a node in\na graph very flexible; in a percolation problem on a lattice, we can\nreuse our \\verb+UndirectedGraph+ class described previously by making node\nidentifiers be lattice index tuples $(i,j)$.  We can thus easily make an\ninstance of bond percolation on a 2D square lattice of size $L$ (with\nperiodic boundary conditions) and bond fraction $p$:\n\n\\begin{verbatim}\ndef MakeSquareBondPercolation(L, p):\n    g = UndirectedGraph()\n    for i in range(L):\n        for j in range(L):\n            g.AddNode((i,j))\n            if random.random() < p:\n                g.AddEdge((i,j), ((i+1\n            if random.random() < p:\n                g.AddEdge((i,j), (i, (j+1\n    return g\n\\end{verbatim}\n\n\\begin{figure}\n\\includegraphics[height=2in]{BondPercolation_10_0.4_1.ps}\n\\includegraphics[height=2in]{BondPercolation_1024_0.5_2.ps}\n\\includegraphics[height=2in]{SitePercolation_20_0.5_4.ps}\n\\caption{\\label{PercolationFig}\nTwo instances of bond percolation on a 2D square lattice, \nand an instance of site percolation on a hexagonal lattice.\nIn bond percolation, neighboring lattice points are connected with\nprobability $p$, and connected clusters in the resulting network are\nidentified via breadth-first search.  Separately clusters are colored\ndistinctly, for a 10x10 grid (left) and a 1024x1024 grid (middle).  In\nsite percolation (right), lattice sites are filled with probability\n$p$, and clusters connect neighboring sites that are filled.  We study\nboth bond and site percolation to to introduce the concept of\nuniversality of phase transitions.}\n\\end{figure}\n\nInstances of percolation networks generated by this procedure are\nillustrated in Figure 2.  Students use breadth-first search to identify\nall connected clusters in such a network, and our PIL-based\nvisualization tool colors each separate cluster distinctly, taking as\ninput a list of all nodes in each cluster.  \n\nThe concept of universality of phase transitions is also introduced:\ndespite their microscopic differences, site-percolation on a 2D\nhexagonal lattice and bond-percolation on a 2D square lattice are\nindistinguishable from each other on long length scales, exhibiting\nthe same critical behavior (e.g., scaling exponents).  Scaling\ncollapses are a useful construct for revealing the universality of\nphase transitions, and typically involve transforming the $x$ and $y$\naxes in specified ways to get disparate data sets to ``collapse'' onto\none universal scaling form.  With Python, we can support \nsuch scaling collapses very flexibly by using the built-in\n\\verb+eval()+ function that evaluates expressions encoded as strings.\nRather than hard-coding particular functional forms for scaling collapses,\narbitrary mathematical expressions can be simply encoded and evaluated.\n\n\\subsection*{Biomechanics: The Walker}\n\nResearch in biomechanics aims to understand how living beings move,\nand robotics and prosthetics are two important technological areas\nthat can benefits from advances in the field.  While much research in\nrobotics is focused on active sensing and control, Andy Ruina and\ncollaborators have been interested in passive biolocomotive systems,\nwhich are more properly understood as dynamical systems than as\ncontrol systems.  The ``simplest walking model'' of Garcia et\nal.\\cite{Garcia1998} provides the basis of our Walker module.  This\nmodel consists of a pair of legs connected at the hip (a double\npendulum), walking down an inclined ramp under the influence of\ngravity, with a heelstrike that imparts angular momemtum to the Walker\nas the swing leg strikes the floor and becomes the stance leg.  As a\nwarmup, students integrate the equation of motion for a single\npendulum under gravity, and compute the period of the motion as a\nfunction of the initial pendulum angle.\n\nThe Pendulum and Walker modules introduce several important scientific\nand computational aspects.  Ordinary differential equations (ODEs) \ndescribing the time evolution of the Pendulum and Walker \nneed to be integrated forward in time.  In \nthe context of the simpler Pendulum, we highlight the properties of \naccuracy, fidelity, and stability in numerical integration, having \nstudents explore errors introduced by a finite time step $\\Delta t$.\nWe also highlight the need for event detection in many numerical \nintegration problems.  In the Walker, for example, \na heelstrike occurs when the swing leg hits the floor.\nAccurately solving for the heelstrike collision involves transforming\nto a new set of integration variables, where an appropriate\ncombination of the pendulum angles becomes the independent variable,\nand time a dependent variable.  We then integrate backwards in angle\nto find the time at which the heelstrike occurred.  We use the\nscipy.integrate.odeint function to execute these integrations,\nproviding a function \\verb+dydt+ that evaluates the instantaneous time\nderivative of the Walker state vector field $\\vec y$ and a function\n\\verb+dzdc+ that evaluates the instantaneous time derivative of the\ntransformed system for heelstrike detection (where the independent variable\nis the ``collision variable'' $c = \\phi-2\\theta$).\n\n\\begin{verbatim}\ndef dydt(self, y,t):\n    theta,thetaDot,phi,phiDot = y\n    thetaDotdot = scipy.sin(theta-self.gamma)\n    phiDotdot = thetaDotdot + \\\n                (thetaDot**2)*sin(phi)-cos(theta-self.gamma)*sin(phi)\n    return [thetaDot,thetaDotdot,phiDot,phiDotdot]\n\nself.trajectory = scipy.integrate.odeint(self.dydt, self.GetStateVector(),\\\n                  timepoints)\n\ndef dzdc(self, z, c):\n    theta,thetaDot,phi,phiDot,t = z\n    y = array([theta, thetaDot, phi, phiDot])\n    thetaDot, thetaDotdot, phiDot, phiDotdot = self.dydt(y, t)\n    cDot = phiDot - 2.*thetaDot\n    return [thetaDot\/cDot,thetaDotdot\/cDot,phiDot\/cDot,phiDotdot\/cDot,\n            1.\/cDot]\n\nz = scipy.integrate.odeint(self.dzdc, [y[0],y[1],y[2],y[3],t], \n    scipy.array([self.CollisionCondition(), 0.])\n\n\\end{verbatim}\n\n\\begin{figure}\n\\includegraphics[height=2in]{Walker23.ps}\n\\caption{\\label{WalkerFig}\nSnapshot in the perambulation of the Walker.  The model consists of a\npair of coupled pendula (legs) walking down a ramp.  The stance leg (red)\nremains fixed with respect to the floor, while the swing leg (orange)\nswings forward.  Once the swing leg hits the floor ahead of the stance leg\n(heelstrike), the two legs switch roles.  \nReal-time animation of the Walker is accomplished using VPython.\n}\n\\end{figure}\n\nThe Walker exhibits an interesting period-doubling route to chaos as\nthe slope of the inclined ramp is increased.  Simple periodic walking\nis stable for small ramp angles, but becomes unstable to a period-two\ngait at a critical angle.  The period-two orbit bifurcates to a\nperiod-four gait, etc., with increasing angle, culminating in a\nchaotic walk.  (The chaos is, however, remarkably subtle, as is also\ntrue in systems like dripping faucets.) A snapshot of the Walker is\nshown in Figure 3.  Other modules in our course\nenable students to study in considerable more detail these sorts of\nperiod-doubling bifurcations and chaotic dynamics in iterated\none-dimensional maps.\n\n\\subsection*{Pattern formation in cardiac dynamics}\n\nPattern formation is ubiquitous in spatially-extended nonequilibrium\nsystems.  Many patterns involve regular, periodic phenomena in space\nand time, but equally important are localized coherent structures that\nbreak or otherwise interrupt these periodic structures.  Patterns lie\nat the root of much activity in living tissues: the regular beating of\nthe human heart is perhaps our most familiar reminder of the\nspatiotemporal rhymicity of biological patterns.  Cardiac tissue is an\nexcitable medium: rhythmic voltage pulses, initiated by the heart's\npacemaker cells (in the sinoatrial node), spread as a wave through the\nrest of the heart inducing the heart muscle to contract, thereby\npumping blood in a coherent fashion.  In some situations, however,\nthis regular beating can become interrupted by the presence of spiral\nwaves in the heart's electrical activity (see Figure 4).  These\nspiral waves generate voltage pulses on their own, disrupting the\ncoordinated rhythm of the normal heart, leading to cardiac arrythmia.\nA simple model of cardiac dynamics - the two-dimensional\nFitzHugh-Nagumo equations\\cite{FitzHugh1961, Nagumo1962} - \nis introduced in this course module, which \nwe developed in conjunction with Niels Otani.  The FitzHugh-Nagumo model\ndescribes the coupled time evolution of two fields, the transmembrane\npotential $V$ and the recovery variable $W$ (given parameters\n$\\epsilon$, $\\gamma$ and $\\beta$):\n$$\n\\frac{\\partial V}{\\partial t} = \\nabla^2 V + \\frac{1}{\\epsilon}\n (V - V^3\/3 - W)\\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\frac{\\partial W}{\\partial t} = \\epsilon (V - \\gamma W + \\beta)\n$$\n\nFixed point solutions to the FitzHugh-Nagumo equations are found by \nroot-finding, which we accomplish using the \\verb+brentq+ function in scipy:\n\n\\begin{verbatim}\ndef FindFixedPoint(gamma, beta):\n    f = lambda v, gamma, beta: (v-(v**3)\/3.)-((1.\/gamma)*(v+beta))\n    vstar = scipy.optimize.brentq(f, -2., 2., args=(gamma, beta))\n    wstar = ((1.\/gamma)*(vstar+beta))\n    return vstar, wstar\n\\end{verbatim}\n\nWe also introduce students to finite-difference techniques for\ncomputing spatial derivatives in the solution of partial differential\nequations (PDEs).  Numpy arrays are used to represent the $V$ and $W$\nfields of the FitzHugh-Nagumo model, and an important operation is the\ncomputation of the laplacian of the voltage field, $\\nabla^2 V(x,y)$.\nWe introduce the stencil notation for characterizing finite-difference\napproximations to $\\nabla^2 V$, and use a combination of array\narithmetic and array slicing to compactly and efficiently compute the\nderivative on the interior (non-boundary) cells of the simulation\ndomain.  Students are asked to implement two different approximations\nto the laplacian operator (a five-point and nine-point stencil), \nand compare their effects on the detailed form of propagating electrical waves.\nThe computation of the five-point stencil is shown here:\n\n\\begin{figure}\n\\includegraphics[height=2in]{Cardiac.ps}\n\\includegraphics[height=2in]{CardiacShock.ps}\n\\caption{\\label{CardiacFig}\nSnapshots in the time evolution of the FitzHugh-Nagumo model of \ncardiac dynamics.  The transmembrane voltage $V$ is depicted via a \ngrayscale map (higher voltages in lighter grays).  Spiral waves in \nthe voltage field can lead to cardiac arrythmias by disrupting the \nnormal periodic rhythm generated by the sinoatrial node.  (Right) \nUsers can administer local voltage pulses (white rectangle) to trigger\nspiral wave formation or to shock the arrhythmic heart back to a \nnormal beating state.\n}\n\\end{figure}\n\n\\begin{verbatim}\ndef del2_5(a, dx):\n    \"\"\"del2_5(a, dx) returns a finite-difference approximation of the\n    laplacian of the array a, with lattice spacing dx, using the five-point\n    stencil:\n        0   1   0\n        1  -4   1\n        0   1   0\n    \"\"\"\n    del2 = scipy.zeros(a.shape, float)\n    del2[1:-1, 1:-1] = (a[1:-1,2:] + a[1:-1,:-2] + \\\n                        a[2:,1:-1] + a[:-2,1:-1] - 4.*a[1:-1,1:-1])\/(dx*dx)\n    return del2     \n\\end{verbatim}\n\nWe provide an animation tool that we have written, based on PIL and\nTkinter, that enables students to update the display of the voltage\nfield V at every time step, and to use the mouse to introduce local\n``shocks'' to the system.  (See Figure 4.) These shocks are both\nuseful in initiating spiral waves and in resetting the global\nelectrical state of the system as a defribillator might do.  Optional\nextensions to the module, developed by our collaborator Otani,\nallow for simulations of spontaneous pacemakers, dead regions of\ntissue, and more complex heart-chamber geometries, by letting the\nvarious parameters of the model become spatially-varying fields\nthemselves (again implemented via numpy arrays).\n\n\\subsection*{Gene regulation and the Repressilator}\n\nGene regulation describes a set of processes by which the expression\nof genes within a living cell - i.e., their transcription to messenger\nRNA and ultimately their translation to protein - is controlled.\nWhile modern genome sequencing has provided great insights into the\nconstituent parts (genes and proteins) of many organisms, much less is\nknown about how those parts of turned on and off and mixed and matched\nin different contexts: how is that a brain cell and a hair cell, for\nexample, can derive from the same genomic blueprint but have such\ndifferent properties? \n\nThe Repressilator is a relatively simple synthetic gene regulatory\nnetwork developed by Michael Elowitz and Stan\nLeibler\\cite{Elowitz2000}. Its name derives from its use of three\nrepressor proteins arranged to form a biological oscillator: these\nthree repressors act in a manner akin to the ``rock-paper-scissors''\ngame where TetR inhibits $\\lambda$ cI, which in turn inhibits LacI,\nwhich in turn inhibits TetR.  A snapshot in the time evolution of the\nRepressilator is shown in Figure 5.\n\n\\begin{figure}\n\\includegraphics[height=2in]{Repressilator47.ps}\n\\caption{\\label{RepressilatorFig}\nSnapshot in the stochastic time evolution of the Repressilator.  The \nstate for this model consists of 15 components: 3 protein concentrations \n(back row), 3 mRNA concentrations (middle row), and 3 sets of promoter-binding\nstates (front row): each promoter can be either unbound, singly-bound, \nor doubly-bound.  At this instant, TetR (red) concentrations are high, \nleading to suppression of $\\lambda$ cI (yellow).  Since $\\lambda$ cI is \nlow, however, LacI (green) concentrations are allowed to grow.  This will\nlead to the eventual suppression of TetR.\n}\n\\end{figure}\n\nOne of the important scientific and computational features that we\nemphasize in this module are the differences between stochastic and\ndeterministic representations of chemical reaction networks.  (We\nfirst introduce these concepts in a warmup exercise, Stochastic Cells,\nin which students simulate a much simpler biochemical network: one\nrepresentation the binding and unbinding of two monomer molecules $M$\nto form a single dimer $D$: $M + M \\leftrightarrow D$.)  We introduce\nstudents to Petri nets as a graphical notation for encoding such\nnetworks, and then have them, from the underlying Petri net\nrepresentation, (a) synthesize differential equations describing the\ndeterministic time evolution of the system, and (b) implement the\nGillespie algorithm (a form of continuous time Monte Carlo) for\nstochastic simulation.\\cite{Gillespie1977}  \nGillespie's ``direct method'' involves\nchoosing a particular reaction and reaction time based on the\ninstantaneous reaction rates.  For the Repressilator, \nthis can be done quite compactly using\narray operations within numpy\/scipy:\n\n\\begin{verbatim}\nclass StochasticRepressilator (Repressilator):\n    # ...\n    def Step(self, dtmax):\n        self.ComputeReactionRates()\n        total_rate = sum(self.rates)\n        # get exponentially distributed time\n        ran_time = -scipy.log(1.-random.random())\/total_rate\n        if ran_time > dtmax:\n            return dtmax\n        # get uniformly drawn rate in interval defined by total_rate\n        ran_rate = total_rate*random.random()\n        # find interval corresponding to random rate\n        reac_index = len(self.rates) - sum(scipy.cumsum(self.rates) > ran_rate)\n        reaction = self.reactions[reac_index]\n        # execute specified reaction\n        for chem, dchem in reaction.stoichiometry.items():\n            chem.amount += dchem\n        # return time at which reaction takes place\n        return ran_time\n\\end{verbatim}\n\n\\subsection*{Other modules}\n\nOur course consists of a number of other modules which we can only\nmention in passing here.  As noted, there is a suite of problems\nintroducing various aspects of chaos and bifurcations in iterated\nmaps.  There is also a suite of small modules exploring properties of\nrandom walks and extremal statistics.  We have two exercises examining\nconnections between statistical mechanics and computational\ncomplexity, by probing the nature of phase transitions in NP-complete\nproblems such as 3SAT.  A random matrix theory module examines the\nnature of universality of eigenvalue distributions, and two other\nmodules explore the thermodynamics of large collective systems (the\nIsing model of simple magnets, and the molecular dynamics of large\nnumbers of atoms).\n\nWe continue to look for new problems to add to this collection, and for \ncollaborators interested in contributing their scientific and computational \nexpertise to this endeavor.  (Please contact us if you have ideas for\ninteresting modules.)  Our goal is to provide a hands-on introduction\nin scientific computing, and it is our hope that this course can help \nserve a number of educational objectives in the part of a larger curriculum \nin computational science and engineering.\n\n\\section*{Acknowledgments}\n\nWe thank our colleagues who have helped us develop computational\nmodules and have given us useful feedback: Steve Strogatz, Andy Ruina,\nNiels Otani, Bart Selman, Carla Gomes, and John Guckenheimer.  We also\nthank all the students who have taken our course and have helped us\nwork the bugs out of exercises and solutions.  Funding from the NSF\nIGERT program (award NSF DGE-0333366) and NSF DMR-0218475 helped\nsupport some initial development of course modules.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nMultimessenger observations of neutron-star (NS) mergers have the potential to revolutionize nuclear astrophysics much in the same way as observations of the cosmic microwave background (CMB) radiation revolutionized particle astrophysics. Neutron-star merger events simultaneously emit gravitational waves (GWs) and electromagnetic (EM) signals, from gamma-rays, X-rays, optical, infrared, to radio waves, and neutrinos. The first observation of a NS merger, GW170817 in the GW spectrum, GRB 170817A in the gamma-ray spectrum, and AT~2017gfo in the electromagnetic (EM) spectrum, was made on August 17, 2017, and in the weeks thereafter~\\cite{TheLIGOScientific:2017qsa,GBM:2017lvd,Monitor:2017mdv,Abbott:2018wiz}. Triggered by the Fermi and Integral telescopes~\\cite{Monitor:2017mdv,Savchenko:2017ffs}, this observation provided detailed spectral and temporal features both in GWs and EM radiation. Theoretical efforts to interpret this data has provided insights into the production of heavy r-process elements in NS mergers~\\cite{Drout:2017ijr}, and constraints on the EOS of dense matter~\\cite{Annala:2017llu,Fattoyev:2017jql,Most:2018hfd,Lim:2018bkq,Tews:2018iwm}. NS mergers have the potential to provide detailed information on the properties of the merging compact stars, such as their masses and radii~\\cite{Bauswein:2017vtn}, as well as on the properties of the densest baryonic matter to be observed in the universe. Future detections of NS mergers, anticipated during the next observing run of the Advanced LIGO and VIRGO detectors, could provide even stronger constraints on the EOS of strongly-interacting matter and the r-process. \n\nWe are pleased to contribute to this topical issue on \"First joint gravitational wave and electromagnetic observations:  Implications for nuclear physics\", which contains several articles devoted to the theory and computing needed to improve the description of dense matter and to model neutron-star mergers - efforts that will play a key role in extracting insights from GW170817 and future detections. Here, we elaborate on earlier work in Ref.~\\cite{Tews:2018iwm}, where we analyzed GW170817 constraints on the dense matter EOS, to provide additional details, discussions, and new results. \n\nOur contribution is structured as follows. In Sec.~\\ref{sec:models} we describe the NS equation-of-state models employed in our analysis. In particular, we use two models: the minimal model or meta-model (MM), see Sec.~\\ref{sec:minmod} and the maximal or speed-of-sound model (CSM), see Sec.~\\ref{sec:maxmod}. Both models are constrained at low densities by state-of-the-art calculations of neutron-rich matter from chiral effective field theory (EFT). We discuss these models in the context of GW170817 in great detail in Sec.~\\ref{sec:results} and  analyze the impact of phase transitions or future GW detections. Finally, we summarize our results and provide an outlook in Sec.~\\ref{sec:summary}.\n\n\\section{Models}\n\\label{sec:models}\n\nIn this section, we discuss the dense-matter models we use in our analysis. Calculations of the EOS of neutron matter based on Hamiltonians derived from chiral EFT provide a reliable method to estimate the uncertainties associated with poorly constrained aspects of two- and many-body nuclear forces at short-distance~\\cite{Lynn:2015jua,Tews:2018kmu}. Chiral EFT is a systematic expansion for nuclear forces in powers of momenta, and provides an efficient way to estimate theoretical uncertainties. It is however limited to momenta up to the so-called breakdown scale, $\\Lambda_b$, which signals the breakdown of the effective theory due to additional high-momentum physics, e.g. the onset of new degrees of freedom. Since $\\Lambda_b$ is expected to be of the order of $\\simeq 500-600$~MeV~\\cite{Melendez:2017phj}, chiral EFT is not applicable at all densities encountered in neutron stars and chiral EFT interactions have typically been used to describe neutron matter only up to saturation density, $n_{sat}$.  Here, using insights obtained in Ref.~\\cite{Tews:2018iwm}, we will analyze to which extent chiral EFT predictions up to 2$n_{sat}$ with conservative error estimates provide useful constrains for the nuclear equation of state, even though uncertainties grow fast with density.\n\nTo describe the EOS at higher densities, we will consider two extrapolation schemes rooted in low-density microscopic predictions and widely covering our present uncertainties at higher density. These two schemes are the minimal model or meta-model (MM), based on a smooth extrapolation of chiral EFT results, and the maximal model or speed-of-sound model (CSM), which explores the widest possible domain for the EOS and contains also more drastic behavior with density; see Ref.~\\cite{Tews:2018iwm} for the first analysis of GWs with these models. These two models show some overlap for properties of dense neutron-star matter, as suggested from the masquerade phenomenon~\\cite{Alford:2004pf}, but also highlight differences: The confrontation of these models with each other and with observations sheds light on the impact of the presence of strong phase transitions at high density, as is detailed hereafter.\n\n\\subsection{Pure neutron matter from chiral EFT}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 2.0cm 1.5cm, clip=,width=0.9\\columnwidth]{NMEOS-band.pdf}\\hspace{0.4cm}\n\\includegraphics[trim= 0.0cm 0 2.0cm 1.5cm,\nclip=,width=0.9\\columnwidth]{NMEOS-Pband.pdf}\n\\end{center}\n\\caption{\\label{fig:chiralPNM} The energy per particle and pressure of pure neutron matter as functions of baryon density up to $2n_{\\rm sat}$. We show the constraints from Ref.~\\cite{Tews:2018kmu} based on AFDMC calculations with local chiral potentials at N$^2$LO (red bands). As a comparison, we show results at LO (black dashed lines), NLO (black dashed-dotted lines), as well as calculations using phenomenological $NN$ interactions only (black dotted lines) and including also phenomenological $3N$ forces (black solid lines). We also indicate the unitary-gas bound of Ref.~\\cite{Kolomeitsev:2016sjl} (blue dashed-dotted lines) and the part of the uncertainty band that we use for our NS modeling (red dotted lines); see text for more details.}\n\\end{figure*} \n\nNeutron stars are ideal laboratories to test theories of the strong interaction at finite chemical potential: the structure of neutron stars is governed by the knowledge of the EOS of neutron-star matter, relating energy density, pressure, and temperature. Additional uncertainties may come from rotation and magnetic field distribution in the star, but the dense-matter EOS is the key input. Since neutron stars explore densities from a few gram per cubic centimeter up to 10 times the nuclear saturation density, $n_{\\rm sat}=0.16 \\,\\mathrm{fm}^{-3} = 2.7\\!\\cdot\\! 10^{14} \\rm{g\\, cm}^{-3}$, the knowledge of the EOS is required for densities covering several orders of magnitude. Though young proto-neutron stars or neutron-star remnants also explore the EOS at high temperatures up to several tens of MeV, older neutron stars can typically be considered as cold objects at $T=0$. This is especially true for two binary NS during the inspiral phase of a neutron-star merger, whose properties can be analyzed from the premerger GW signal.  \n\nWhile the EOS of the neutron-star crust, reaching up to $n_{\\rm sat}\/2$, is rather well constrained, the uncertainty of the EOS increases fast with density and the composition of the inner core of NS is still unknown. Nevertheless, in the density range from $n_{\\rm sat}\/2$ up to about $2n_{\\rm sat}$, the neutron-star EOS can be constrained by state-of-the-art nuclear-theory models. The starting point for these constraints are calculations of pure neutron matter (PNM). PNM is an idealized, infinite system consisting solely of neutrons, but it is much easier to compute than systems containing also protons. The reason is that certain parts of the nuclear interaction, e.g., tensor interactions, are weaker or do not contribute at all among neutrons. In contrast to symmetric nuclear matter, PNM is also not unstable with respect to density fluctuations below $n_\\mathrm{sat}$, and uniform matter remains the true ground state of PNM at all densities, simplifying its calculation.\n\nTo reliably describe neutron matter, one needs precise and accurate quantum many-body methods in combination with a reliable model for the nuclear interaction. Neutron matter has been extensively studied in the last decade, using a multitude of nuclear interactions and advanced \\textsl{ab initio}  many-body methods. Among these are, e.g., many-body perturbation theory~\\cite{Hebeler:2009iv,Drischler:2016djf,Holt:2016pjb}, the coupled-cluster method~\\cite{Hagen:2013yba}, quantum Monte Carlo methods~\\cite{Gandolfi:2011xu}, or the self-consistent Green's function method~\\cite{Carbone:2014mja}. A comparison of these different studies, see e.g., Refs.~\\cite{Gandolfi:2015jma,Hebeler:2015hla}, shows that neutron matter is rather well constrained by these multiple \\textsl{ab initio} approaches using diverse nuclear Hamiltonians.  In this paper, we will use calculations of neutron matter obtained with the auxiliary-field diffusion Monte Carlo (AFDMC) method~\\cite{Carlson:2014vla} together with modern nuclear Hamiltonians from chiral EFT. \n\nQuantum Monte Carlo methods are among the most precise many-body methods for strongly interacting systems~\\cite{Carlson:2014vla}. They provide the ground state of a many-body system, governed by a non-relativistic nuclear Hamiltonian defining the Schr\\\"odinger equation, by evolving a trial wave function $\\Psi_T$ in imaginary time, \n\\begin{equation}\n\\Psi_{GS}=\\lim_{\\tau \\to \\infty} e^{- \\mathcal{H}\\tau}\\Psi_T\\,,\n\\end{equation}\nwhere $\\Psi_T$ is constructed so that it has a non-vanishing overlap with the ground state $\\Psi_{GS}$. Expanding $\\Psi_T$ in eigenfunctions of the Hamiltonian, one can easily see that contributions of excited states decay with time, and only the ground-state component of the trial wave function remains. Quantum Monte Carlo methods have been used to successfully describe nuclei up to \\isotope[16]{O}~\\cite{Carlson:2014vla,Piarulli:2017dwd,Lonardoni:2017hgs} and neutron matter~\\cite{Gandolfi:2011xu,Lynn:2015jua}. At very low densities, where neutron matter is close to the unitary limit and interactions are dominated by large scattering-length physics, these methods~\\cite{Carlson:2008zza} have been successfully confronted to experimental measurements of cold atomic gases~\\cite{Nascimbene2010,Navon2010,Zwierlein:2015}. Due to its great success to study strongly-interacting matter and nuclei~\\cite{Gandolfi:2011xu,Lonardoni:2014bwa,Lynn:2015jua,Gandolfi:2016bth,Lonardoni:2017hgs}, we employ in this work the AFDMC method to determine PNM properties. For more details on Quantum Monte Carlo methods we refer the reader to Ref.~\\cite{Carlson:2014vla}. \n\nOn the interaction side, chiral EFT~\\cite{Epelbaum2009,Machleidt:2011zz} is a modern theory for nuclear forces that is consistent with the symmetries of Quantum Chromodynamics and systematically describes the nucleon-nucleon interaction in terms of explicitly resolved longer-range pion exchanges as well as short-range nucleon contact interactions.  Chiral EFT is based on a momentum expansion in terms of $p\/\\Lambda_b$, where $p$ is the typical momentum of the nuclear system at hand, and $\\Lambda_b$ is the breakdown scale already discussed. The short-range interaction terms parametrize all unresolved and unknown high-energy physics beyond the breakdown scale, and depend on a set of low-energy couplings (LECs), which are typically fitted to nucleon-nucleon ($NN$) scattering data and properties of light nuclei. Chiral EFT does not only describe $NN$ interactions but also consistent three-body ($3N$) and higher many-body forces. It has been successfully applied to calculate properties of ground and excited states of nuclei, nuclear matter, as well as electroweak processes; see, e.g, Ref.~\\cite{Hebeler:2015hla} for a review. Most importantly, the systematic chiral EFT expansion enables the estimation of theoretical uncertainties for these physical systems.\n\nIn our analysis in this work, we use local chiral EFT interactions that have been constructed especially for the use in QMC methods in Refs.~\\cite{Lynn:2015jua,Gezerlis:2013ipa,Gezerlis:2014zia,Tews:2015ufa}. These interactions have been successfully tested in light- to medium-mass nuclei and in n-$\\alpha$ scattering~\\cite{Lynn:2015jua,Lonardoni:2017hgs} and agree with our current knowledge of the empirical parameters of nuclear matter~\\cite{Kolomeitsev:2016sjl,Margueron:2017eqc}. In Ref.~\\cite{Tews:2018kmu}, these interactions have been used to study neutron matter up to $2n_{\\rm sat}$ with theoretical uncertainty estimates using the AFDMC method.  \nFor more details on QMC calculations with local chiral interactions we refer the reader to Ref.~\\cite{Lynn:2019rdt}. \n\nIn particular, in this work we use local chiral interactions at a cutoff scale $R_0=1.0$ fm with its systematic uncertainty estimates. In Fig.~\\ref{fig:chiralPNM} we show the results for the energy per particle and pressure of neutron matter at leading order (LO), next-to-leading order (NLO), and at next-to-next-to-leading order (N$^2$LO) with its uncertainty band for densities ranging from 0.04~fm$^{-3}$  up to $2n_{\\rm sat}$. We find that the uncertainty bands increase fast with density and are quite sizable at $2 n_{\\rm sat}$. In addition to the results for chiral interactions, we also show in Fig.~\\ref{fig:chiralPNM} AFDMC results employing the phenomenological AV8' $NN$ and AV8' $NN$ plus UIX $3N$ interactions as a comparison. It is interesting to note that the AV8' and NLO $NN$ interactions agree very well with each other, which highlights the fact that many-body forces are a considerable source of uncertainty. Finally, we also compare all calculations with the unitary-gas limit of Ref.~\\cite{Kolomeitsev:2016sjl}.\n\n\\subsection{Discussion of uncertainties}\n\nThe uncertainty bands shown in Fig.~\\ref{fig:chiralPNM} include the following sources of uncertainty: i) the truncation of the nuclear Hamiltonian within the chiral expansion, ii) the regularization scheme and scale, which are needed to implement nuclear Hamiltonians in many-body methods, iii) the uncertainties in the determination of low-energy couplings from data, and iv) the many-body uncertainty that originates in approximations made when solving the Schr\\\"odinger equation for the nuclear many-body system. The first three sources, which originate in the nuclear Hamiltonian, dominate over the many-body uncertainty from QMC methods. Among these three, the truncation uncertainty is the dominant source of uncertainty and we will discuss it in the following.\n\nThe truncation uncertainty can be expressed in the following way. \nIntroducing the dimensionless expansion parameter $Q=p\/\\Lambda_b$ and following Ref.~\\cite{Furnstahl:2015rha}, under the prerequisite that chiral EFT is a converging theory, one can define the order-by-order contributions to an observable $X$ using the following infinite summation,\n\\begin{equation}\\label{eq:chiralExp}\nX=X_0\\sum_{i=0}^{\\infty} c_i Q^{i}\\,.\n\\end{equation}\nHere, $X_0$ sets the natural scale expected for the observable $X$, e.g., the leading-order result, $X_0=X_{\\rm{LO}}$ ($c_0=1$), and the $c_{i\\ge 1}$ denote the expansion coefficients. In calculations of nuclear systems, due to practical reasons this sum has to be truncated at a certain order $n$, inducing the so-called truncation uncertainty. This uncertainty is intrinsic to \\emph{all} nuclear Hamiltonians but can be specified for chiral EFT Hamiltonians by  \n\\begin{equation}\n\\Delta X=X- X_0\\sum_{i=0}^{n}c_i Q^{i}\\,.\n\\end{equation}\n\nIt has been shown in Ref.~\\cite{Furnstahl:2015rha} that for practical purposes an estimate of the magnitude of the first truncated term in Eq.~\\eqref{eq:chiralExp}, given by $i=n+1$, is a sufficient uncertainty estimate. To obtain this estimate, both the size of the unknown expansion coefficient $c_{n+1}$ and of the expansion parameter $Q$ are required. A conservative choice for the coefficient $c_{n+1}$ is the maximum of all previously found coefficients, \n\\begin{equation}\nc_{n+1}=\\max_{i=0}^n{c_i}\\,,\n\\end{equation} \nwhile $Q$ has to be estimated from the typical momentum scale for the system at hand. This uncertainty prescription is similar to the one presented by Epelbaum, Krebs, and Mei{\\ss}ner (EKM)~\\cite{Epelbaum:2014efa}, and the truncation uncertainty, e.g., at N$^2$LO, can be obtained from an order-by-order calculation as \n\\begin{align}\n\\Delta X^{\\nxlo{2}}=\\max &\n\\left(\\vphantom{X^{\\nxlo{2}}}Q^{4} \\left|X^{\\nxlo{0}}-X^{\\rm free}\\right|,Q^2 \\left|X^{\\nxlo{1}}-X^{\\nxlo{0}}\\right|,\\right. \\nonumber \\\\\n&\\quad \\left. Q\\left|X^{\\nxlo{2}}-X^{\\nxlo{1}}\\right|\n\\right)\\nonumber\\\\\n&= Q^4 X_0 \\max_{i=0}^n{c_i}\\,. \\label{eq:uncertainty}\n\\end{align}\nWe have used this uncertainty estimate to compute the truncation uncertainty, using $Q=\\sqrt{3\/5}k_F\/\\Lambda_b$, with the Fermi momentum $k_F$ and $\\Lambda_b=500 \\,\\mathrm{MeV}$. \n\n\\begin{table*}[t]\n\\centering\n\\setlength{\\tabcolsep}{10pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccccccccc}\n\\hline\n$P_{\\alpha}$ & $E_{sat}$ & $E_{sym}$ & $n_{sat}$ & $L_{sym}$ & $K_{sat}$ & $K_{sym}$ & $Q_{sat}$ & $Q_{sym}$ & $Z_{sat}$ & $Z_{sym}$ & $b$\\\\\n                    & MeV           & MeV           & fm$^{-3}$           & MeV           & MeV           & MeV           & MeV           & MeV           & MeV           & MeV & \\\\\n\\hline\nMax             & -15 & 38 & 0.17 & 90  &  270  &  200  &   1000 &   2000 &   3000 &    3000 & 14 \\\\\nMin              & -17 & 26 & 0.15 & 20  &  190  & -400  &  -1000 &   -2000 &  -3000 & -3000 &1 \\\\\n\\hline\n\\end{tabular}\n\\caption{Empirical parameters and their domain of variation entering into the definition of the MM~(\\ref{eq:MM:energy}). The parameters $\\kappa_{sat}$ and $\\kappa_{sym}$  are fixed such that $m_{sat}^*\/m=0.75$ in symmetric matter and $m_n^*\/m-m_p^*\/m=-0.1$ in neutron matter.}\n\\label{tab:epbound}\n\\end{table*}\n\nThe total uncertainty bands in Fig.~\\ref{fig:chiralPNM} additionally include the other three sources of uncertainty. The regularization scheme dependence has been explored by explicitly including regulator artifacts for local regulators. Specifically, in Fig.~\\ref{fig:chiralPNM}, the neutron-matter uncertainty bands include three different local chiral Hamiltonians which explore short-range $3N$ regulator artifacts; see Ref.~\\cite{Lynn:2015jua} for details on the Hamiltonians and Ref.~\\cite{Huth:2017wzw} for details on the regulator artifacts. These two sources of uncertainties dominate the total uncertainty band, while the many-body uncertainty is negligible.\n\nTo estimate the convergence of the chiral expansion at different densities, the series of expansion coefficients of Eq.~\\eqref{eq:chiralExp} can provide insights. In Ref.~\\cite{Tews:2018kmu}, we have studied the convergence of the chiral series in pure neutron matter and found it to be reasonable up to a density of $2 n_{\\rm sat}$. Beyond that, we expect the chiral expansion to break down even though the expansion parameter only increases by approximately 25\\% from $n_{\\rm sat}$ to $2 n_{\\rm sat}$. Therefore, we restrict the chiral EFT input to densities up to $2 n_{\\rm sat}$. In addition, we exclude one chiral Hamiltonian from further consideration because its regulator artifacts lead to a spurious and unphysical attractive $3N$ contribution in neutron matter, as discussed in Ref.~\\cite{Tews:2018kmu}. This Hamiltonian represents the lower, soft part of the uncertainty band and  is also in conflict with the unitary-gas bound of Ref.~\\cite{Kolomeitsev:2016sjl}, shown in Fig.~\\ref{fig:chiralPNM} as a blue dashed line. Excluding this Hamiltonian changes the lower bound of the uncertainty band to the red-dotted line in Fig.~\\ref{fig:chiralPNM}, in good agreement with the unitary-gas constraint.\n\nIn the following, we use this chiral EFT band up to a density $n_{\\text{tr}}$ to constrain two different modelings for the high density equation of state. By varying $n_{\\text{tr}}$ from $n_{\\text{sat}}$ to $2n_{\\text{sat}}$, we will show that, despite the rapid increase of the uncertainty of the neutron-matter EOS with density, chiral EFT constraints remain extremely useful up to $2 n_{\\rm sat}$.  \n\n\\subsection{The minimal model}\n\\label{sec:minmod}\n\nThe first model that we consider in this analysis, the minimal model or meta-model (MM), assumes the EOS to be smooth enough to be describable in terms of a density expansion about $n_{sat}$. Here, we briefly describe the MM, but see also Refs.~\\cite{Margueron:2017eqc,Margueron:2017lup} for more details. \n\nThe MM is described in terms of the empirical parameters of nuclear matter, which are defined as the Taylor coefficients of the density expansion of the energy per particle of symmetric nuclear matter $e_{sat}(n)$ and the symmetry energy $s_{sym}(n)$, \n\\begin{align}\ne_{sat}(n) &= E_{\\text{sat}} + \\frac 1 2 K_{\\text{sat}} x^2 + \\frac 1 6 Q_{\\text{sat}} x^3 + \\frac 1 {24} Z_{\\text{sat}} x^4 + ... \\label{eq:esat}\\\\\ns_{sym}(n) &= E_{\\text{sym}} + L_{\\text{sym}} x+ \\frac{1}{2} K_{\\text{sym}} x^2 + \\frac{1}{6} Q_{\\text{sym}} x^3 \\nonumber  \\label{eq:esym}\\\\ \n& +\\frac{1}{24} Z_{\\text{sym}} x^4 + ... \\,,\n\\end{align}\nwhere the expansion parameter $x$ is defined as $x=(n-n_{\\text{sat}})\/(3n_{\\text{sat}})$ and $n=n_n+n_p$ is the baryon density, $n_{n\/p}$ are the neutron and proton densities.\nA good representation of the energy per particle around $n_{sat}$ and for small isospin asymmetries $\\delta=(n_n-n_p)\/n$ can be obtained from the following quadratic approximation,\n\\begin{equation}\ne(n,\\delta)=e_{sat}(n)+s_{sym}(n)\\, \\delta^2\\, .\n\\end{equation}\nThe lowest order empirical parameters can be extracted from nuclear experiments~\\cite{Margueron:2017eqc}, but typically carry uncertainties. Especially the symmetry-energy parameters are of great interest to the nuclear physics community and considerable effort is invested into a better estimation of their size.\n\nThe MM constructs the energy per nucleon as,\n\\begin{eqnarray}\ne^N(n,\\delta)=t^{FG*}(n,\\delta)+v^N(n,\\delta),\n\\label{eq:MM:energy}\n\\end{eqnarray}\nwhere the kinetic energy is expressed as \n\\begin{eqnarray}\nt^{FG^*}(n,\\delta)&=&\\frac{t_{sat}^{FG}}{2}\\left(\\frac{n}{n_{sat}}\\right)^{2\/3} \n\\bigg[ \\left( 1+\\kappa_{sat}\\frac{n}{n_{sat}} \\right) f_1(\\delta) \\nonumber \\\\\n&& \\hspace{2.5cm} + \\kappa_{sym}\\frac{n}{n_{sat}}f_2(\\delta)\\bigg] ,\n\\label{eq:MM:kin}\n\\end{eqnarray}\nand the functions $f_1$ and $f_2$ are defined as\n\\begin{eqnarray}\nf_1(\\delta) &=& (1+\\delta)^{5\/3}+(1-\\delta)^{5\/3} \\, , \\\\\nf_2(\\delta) &=& \\delta \\left( (1+\\delta)^{5\/3}-(1-\\delta)^{5\/3} \\right) .\n\\end{eqnarray}\nThe parameters $\\kappa_{sat}$ and $\\kappa_{sym}$ control the density and asymmetry dependence of the Landau effective mass as ($q$=n or p),\n\\begin{equation}\n\\frac{m}{m^*_q(n,\\delta)} = 1 + \\left( \\kappa_{sat} + \\tau_3 \\kappa_{sym} \\delta \\right) \\frac{n}{n_{sat}} ,\n\\label{eq:effmass}\n\\end{equation}\nwhere $\\tau_3=1$ for neutrons and -1 for protons.\nTaking the limit $\\kappa_{sat}=\\kappa_{sym}=0$, Eq.~(\\ref{eq:MM:kin}) provides the free Fermi gas energy.\n\nThe potential energy in Eq.~(\\ref{eq:MM:energy}) is expressed as a series expansion in the parameter $x$ and is quadratic in the asymmety parameter $\\delta$,\n\\begin{eqnarray}\nv^N(n,\\delta)=\\sum_{\\alpha\\geq0}^N \\frac{1}{\\alpha!}( v_{\\alpha}^{sat}+ v_{\\alpha}^{sym} \\delta^2) x^\\alpha u^N_{\\alpha}(x) ,\n\\label{eq:MM:pot}\n\\end{eqnarray}\nwhere the function $u^N_{\\alpha}(x)=1-(-3x)^{N+1-\\alpha}\\exp(-b n\/n_{sat})$ ensures the limit $e^N(n=0,\\delta)=0$.\nThe parameter $b$ is taken large enough for the function $u^N_{\\alpha}$ to fall sufficiently fast with density and to not contribute at densities above $n_{sat}$. A typical value is $b=10\\ln2\\approx 6.93$ such that the exponential function is $1\/2$ for $n=n_{sat}\/10$.\nThe MM parameters $v_{\\alpha}^{sat}$ and $v_{\\alpha}^{sym}$ are simply expressed in terms of the empirical parameters. The MM as expressed in Eqs.(\\ref{eq:MM:energy}), (\\ref{eq:MM:kin}), and (\\ref{eq:MM:pot}) coincides with the meta-model ELFc described in Ref.~\\cite{Margueron:2017eqc}, where detailed relations can be found.\nTo obtain the neutron-star EOS, we extend our models to $\\beta$-equilibrium and include a crust as described in Ref.~\\cite{Margueron:2017lup}. By varying the empirical parameters within their known or estimated uncertainties, it was shown that the MM can reproduce many existing neutron-star EOS that are based on the assumption that a nuclear description is valid at all densities probed in neutron stars. Therefore, this model is a reliable representation for EOS without exotic phases of matter separated from the nucleonic phase through strong phase transitions.\n\nIn the following, the parameter space for the MM will be explored within a Markov-Chain Monte-Carlo algorithm, where the MM parameters are allowed to freely evolve inside the boundaries given in Table.~\\ref{tab:epbound}. The resulting models satisfy the chiral EFT predictions in neutron matter for the energy per particle and the pressure up to $n_{\\rm tr}$, causality, stability, positiveness of the symmetry energy ($s_{sym}(n)>0$), and also reach the maximum observed neutron-star mass $M_{\\rm max}^{\\rm obs}$, see the discussion in Sec.~\\ref{sec:MMandCSM}. The maximum density associated with each EOS within the MM is given either by the break-down of causality, stability, or positiveness of the symmetry energy condition, or by the end point of the stable neutron-star branch.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{EpsPcomp.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{EpsPcomp800.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{EpsPcomp_032.pdf}\n\\caption{\\label{fig:EpsPcomp}\nComparison of the allowed EOS envelopes for the MM (black bands) and the CSM (red bands). We show three cases: a) the most general case, where $n_{\\text{tr}}=n_{\\text{sat}}$ and only $M_{\\rm{max}}\\geq 1.9 M_{\\odot}$ is enforced, b) for $n_{\\text{tr}}=n_{\\text{sat}}$ when enforcing $70\\leq \\tilde{\\Lambda} \\leq 720$ and c) for $n_{\\text{tr}}=2 n_{\\text{sat}}$. When additionally enforcing $R_{1.6}\\geq 10.68$ km~\\cite{Bauswein:2017vtn}, the hatched regions are excluded.\n}\n\\end{center}\n\\end{figure*}\n\n\\subsection{The maximal model}\n\\label{sec:maxmod}\n\nThe second model that we consider in this analysis, the maximal model (CSM), is based on an extension of the speed of sound in neutron-star matter. Starting from the pure neutron matter calculations, we construct the neutron-star EOS up to $n_{\\rm tr}$ by constructing a crust as described in Ref.~\\cite{Tews:2016ofv} and extending the neutron-matter results to $\\beta$ equilibrium above the crust-core transition. Having constructed the EOS up to $n_{\\rm tr}$ we compute the speed of sound, \n\\begin{equation}\nc_S^2 = \\frac{\\partial p(\\epsilon)}{\\partial \\epsilon}\\,,\n\\end{equation}\nwhere $p$ is the pressure and $\\epsilon$ is the energy density. Above $n_{\\rm tr}$, we parametrize the speed of sound in a very general way: we randomly sample a set of points $c_S^2(n)$, where the values for $c_S$ have to be positive and are limited by the speed of light (stability and causality), and interpolate between the different sampling points using linear segments. The individual points are randomly distributed in the interval $n_{\\rm tr}-12 n_{\\rm sat}$.  From the resulting speed-of-sound curve, we reconstruct the EOS step-by-step starting at $n_{\\text{tr}}$, where $\\epsilon(n_{\\text{tr}})$,  \n$p(n_{\\text{tr}})$, and $\\epsilon'(n_{\\text{tr}})$ are known:\n\\begin{align}\nn_{i+1}&= n_i + \\Delta n \\\\\n\\epsilon_{i+1} &= \\epsilon_i +\\Delta\\epsilon= \\epsilon_i + \\Delta n \\cdot \\left(\\frac{\\epsilon_i+p_i}{n_i}\\right) \\\\\np_{i+1} &= p_i + c_S^2 (n_i) \\cdot \\Delta \\epsilon\\,,\n\\end{align}\nwhere $i=0$ defines the transition density $n_{\\text{tr}}$. In the second line we have used the thermodynamic relation $p=n \\partial \\epsilon\/\\partial n -\\epsilon$, which is valid at zero temperature. \n\nIn that way, we iteratively obtain the high-density EOS. We have explored extensions for a varying number of $c_S^2(n)$ points, i.e., for 5-10 points, and found that the differences between these extensions are marginal. We, therefore, choose 6 sampling points. For each sampled EOS, we generate a second version which includes a strong first-order phase transition with a random onset density and width, to explicitly explore such extreme density behavior.\n\nThe CSM for neutron-star applications was introduced in Ref.~\\cite{Tews:2018kmu}, and represents and extension of the model of Ref.~\\cite{Alford:2013aca}. A similar model was used in Ref.~\\cite{Greif:2018njt}. However, in contrast to Ref.~\\cite{Tews:2018kmu} we have extended this model to explore the complete allowed parameter space for the speed of sound, by abandoning the specific functional form of Ref.~\\cite{Tews:2018kmu} in favor of an extension using linear segments. This more conservative choice leads to slightly larger uncertainty bands, but allows us to make more definitive statements about neutron-star properties. The resulting EOS parameterizations represent possible neutron-star EOS and may include drastic density dependences, e.g., strong phase transitions which lead to intervals with a drastic softening or stiffening of the EOS. \nThis represents a stark contrast to the MM, which does not include such behavior, and might give insights into the constituents of neutron-star matter at high-densities. The predictions of the CSM represent the widest possible domain for the respective neutron-star observables consistent with the low density input from chiral EFT. If observations outside of this domain were to be made, this would imply a breakdown of nuclear EFTs at densities below the corresponding $n_{\\rm tr}$. \n\nSince the CSM represents very general EOSs only governed by the density dependence of the speed-of-sound, it does not allow any statements about possible degrees of freedom. In this sense, it is very similar to extensions using piecewise polytropes which were introduced in Ref.~\\cite{Read:2008iy} and have been used extensively to determine neutron-star properties; see, e.g., Ref.~\\cite{Hebeler:2013nza,Raithel:2016bux,Annala:2017llu}. However, in contrast to polytropic extensions, in the CSM the speed of sound is continuous except when first-order phase transition are explicitly accounted for. Discontinuities in the speed of sound affect the study of tidal polarizabilities, where $c_S^{-1}$ enters, by introducing features whose source is solely the choice of parametrization.\n\n\\subsection{Comparison of MM and CSM}\n\\label{sec:MMandCSM}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{MRcomp.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{MRcomp_800.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.65\\columnwidth]{MRcomp_032.pdf}\n\\caption{\\label{fig:MRcomp}\nComparison of the allowed MR envelopes for the MM (black bands) and the CSM (red bands). We show three cases: a) the most general case, where $n_{\\text{tr}}=n_{\\text{sat}}$ and only $M_{\\rm{max}}\\geq 1.9 M_{\\odot}$ is enforced, b) for $n_{\\text{tr}}=n_{\\text{sat}}$ when enforcing $70\\leq \\tilde{\\Lambda} \\leq 720$, and c) for $n_{\\text{tr}}=2 n_{\\text{sat}}$. When additionally enforcing $R_{1.6}\\geq 10.68$ km~\\cite{Bauswein:2017vtn}, the hatched regions are excluded.\n}\n\\end{center}\n\\end{figure*} \n\nFor both the MM and CSM we generate thousands of EOSs that are consistent with low-density constraints from chiral EFT. In addition, the observations of heavy two-solar-mass pulsars in recent years~\\cite{Demorest2010,Antoniadis2013,Fonseca2016} place important additional constraints on these EOSs, which we enforce by requiring  $M_{\\text{max}}>M_{\\rm max}^{\\rm obs}$ for all our EOSs. \nTo be conservative, as the limit for $M_{\\rm max}^{\\rm obs}$ we choose the centroid of the maximum observed mass minus twice the error-bar on the observation. For the two heaviest neutron stars observed up to now~\\cite{Demorest2010,Antoniadis2013,Fonseca2016}, this gives $M_{\\rm max}^{\\rm obs}\\approx 1.9 M_\\odot$. \n\nWe now compare the predictions of both the MM (black bands with solid contour) and CSM (red bands with dotted contour) for the EOS of neutron-star matter, see Fig.~\\ref{fig:EpsPcomp}, and the mass-radius (MR) relation, see Fig.~\\ref{fig:MRcomp}. In the respective figures, we show the EOS and MR envelopes for $n_{\\rm tr}=n_{\\rm sat}$ [panels (a)] and for $n_{\\rm tr}=2 n_{\\rm sat}$ [panels (c)], where ragged edges are due to the limited number of models. In all cases, the MM is a subset of the CSM, as expected. Also, the two models, which treat the neutron-star crust with different prescriptions, show excellent agreement at low densities. For $n_{\\rm tr}=n_{\\rm sat}$, the MM and CSM EOSs agree very well up to  $n_{\\rm tr}$, while for $n_{\\rm tr}=2 n_{\\rm sat}$ the MM only samples a subset of the chiral EFT input, because the $M_{\\rm max}^{\\rm obs}$ constraint forces the EOS to be sufficiently stiff which excludes the softest low-density neutron-matter EOS. This is a consequence of the smooth density expansion around $n_{\\rm sat}$ in the MM. In the CSM, instead, a non-smooth stiffening of these softest EOS at higher densities can help stabilize heavy neutron stars, which is why the complete low-density band from chiral EFT is sampled.  We also find that going from $n_{\\rm tr}=n_{\\rm sat}$ to $n_{\\rm tr}=2 n_{\\rm sat}$ allows to considerable reduce the EOS uncertainty for the CSM. The MM uncertainty is also slightly reduced and the MM band gets narrower. These results show that even though the theoretical uncertainties in the neutron-matter EOS increase fast in the density range $1-2 n_{\\text{sat}}$, the additional information provided allows to substantially reduce uncertainties in the CSM EOS:   essentially, the chiral EFT constraint excludes the possibility of phase transitions in the region going from 1 to $2n_{sat}$. The impact of phase transitions above $2n_{sat}$ on the EOS is very much reduced compared to the case where they are allowed to appear at lower densities, because we impose the $M_{\\rm max}^{\\rm obs}$ constraint. A large domain of soft CSM EOSs is, thus, excluded. The stiff MM and CSM EOS are very close up to $2n_{sat}$, as expected.\n\nThese observations are also reflected in the MR predictions of both models. For $n_{\\rm tr}=n_{\\rm sat}$ [panel (a)], the CSM (MM) leads to a radius range of a typical neutron star of $1.4 M_{\\odot}$ of $8.4-15.2$ km ($10.9-13.5$ km). This range reduces dramatically for $n_{\\rm tr}=2 n_{\\rm sat}$ [panel (c)], where we find $8.7-12.6$ km ($10.9-12.0$ km) for the CSM (MM). \n\nIn the last case, the radius uncertainty for a typical neutron star is only about 1 km in the MM, compatible with the expected uncertainty of the NICER mission~\\cite{NICER1}. This allows for a possible tight confrontation between the MM and the NICER results. If such an observation should be made in the near future, we will be able to better constrain dense-matter phase transitions. In contrast, the CSM, which includes EOS with sudden softening or stiffening at higher densities, dramatically extends the allowed envelopes for the EOS and the MR relation as compared with the MM. These differences in the predictions of the MM and CSM can be used to identify regions for the neutron-star observables, for which statements about the constituents of matter might be possible. For example, the observation of a typical neutron star with a radius of 10 km would imply the existence of a softening phase transition, that would hint on new phases of matter appearing in the core of neutron stars. Instead, in regions were both the MM and CSM agree, the masquerade problem does not allow statements about the constituents of neutron-star matter at high densities~\\cite{Alford:2004pf}.\n\nIn Fig.~\\ref{fig:MRcomp}, the maximum mass for $n_\\mathrm{tr}=n_\\mathrm{sat}$ is almost $4M_\\odot$ while it is only $2.9M_\\odot$ if $n_\\mathrm{tr}=2n_\\mathrm{sat}$.\nIt is interesting to compare these findings with previous predictions for the maximum mass of neutron stars. Connecting a nucleonic EOS to the stiffest possible EOS at $n_\\mathrm{tr}=2n_\\mathrm{sat}$, the maximum mass was predicted to be  $2.9 M_\\odot$~\\cite{Rhoades1974}, as in our case. With a similar approach but defining $n_\\mathrm{tr}$ to lie between 1 and $2n_\\mathrm{sat}$, Ref.~\\cite{Kalogera1996} predicted the maximum mass to be  $3.2 M_\\odot$. Note, however, that by lowering $n_\\mathrm{tr}$, the authors found $3.9 M_\\odot$ as the maximum mass, again very close to our prediction. The maximum mass of neutron stars is therefore tightly correlated with $n_\\mathrm{tr}$ for both the MM and CSM models, as shown in Fig.~\\ref{fig:MRcomp}.\n\nFinally, due to the rather soft density dependence of chiral EFT constraints in the density range $1-2 n_{\\rm sat}$, $n_{\\rm tr}=2 n_{\\rm sat}$ together with the constraint $M_{\\text{max}}>M_{\\rm max}^{\\rm obs}$ seems to strongly disfavor EOS that lead to the appearance of disconnected compact-star branches, as suggested in Ref.~\\cite{Paschalidis:2017qmb}. Such EOS need very strong first-order phase transitions, which would soften the EOS so much that heavy two-solar-mass neutron stars cannot be supported, in accordance  with the findings in Ref.~\\cite{Alford:2015dpa}. Instead, chiral EFT calculations up to $n_{\\rm tr}=2 n_{\\rm sat}$ imply that EOSs with first-order phase transitions lead to neutron stars of the classification \"A\" or \"C\" of Ref.~\\cite{Alford:2013aca}. \n\n\\section{Results for GW170817}\\label{sec:results}\n\nIn this section, we confront the recent neutron-star merger observation GW170817 by the LIGO-Virgo (LV) collaboration with our two classes of EOS models. \n\n\\subsection{Posterior of the LIGO-Virgo analysis}\n\\label{sec:posterior}\n\nThe LV collaboration observed the GW signal of GW170817 for about $100 s$ (several 1000 revolutions, starting from 25 Hz) and performed detailed analyses of the wave front~\\cite{Abbott:2018wiz}. Because the chirp mass $M_{\\text{chirp}}$, defined as \n\\begin{equation}\nM_{\\text{chirp}}=\\frac{(m_1 m_2)^{3\/5}}{(m_1+m_2)^{1\/5}}\\,,\n\\end{equation} \ncan be extracted from the entire signal, this observation allowed to put tight constraints on it. For GW170817, the LV collaboration precisely determined $M_{\\text{chirp}}= 1.186\\pm 0.001 M_{\\odot}$.\n\n\\begin{figure}[t]\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{M1M2histo.pdf}\\\\\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{Lamhisto.pdf}\n\\caption{\\label{fig:posteriors}\nPosteriors for the LV observation of GW170817. Upper panel: The mass distributions for $m_1$ and $m_2$ from Ref.~\\cite{Abbott:2018wiz} (histograms) and the distributions used in this work (solid lines), see Eq.~\\eqref{eq:massdist}. Lower panel: Marginalized and normalized posterior probability for the distribution $p(\\tilde{\\Lambda})$ as defined in this work. We also show the corresponding distributions for the analysis of the LV collaboration (LVC), and the reanalysis of Ref.~\\cite{De:2018uhw} for the two extreme cases [uniform mass prior (u) and mass prior informed by double neutron stars (d)].\n}\n\\end{figure} \n\n\n\\begin{table*}\n\\caption{\\label{tab:LVCLtilde}\nFit parameters of the Gaussians of Eq.~\\eqref{eq:Gaussians}}\n\\centering\n\\begin{tabularx}{\\textwidth}{XXXXXXXXXX}\n\\hline\n\\hline\nN & $a_1$ & $\\Lambda_1$ & $\\sigma_1$ & $a_2$ &$\\Lambda_2$ &$\\sigma_2$ & $a_3$ &$\\Lambda_3$ & $\\sigma_3$\\\\\n\\hline\n2  & 281.6 & 212.6 & 76.2 & 106.5\n& 547.5 & 171.0 &  & &\\\\\n3 & 266.6 & 212.4 & 74.2 & 85.0 & 523.6 & 219.2 & 38.6\n& 560.8 & 49.5\\\\\n\\hline\n\\hline\n\\end{tabularx}\n\\end{table*}\n\nThe extraction of higher-order GW parameters from the wavefront is complicated for several reasons. First, higher-order parameters are sensitive to the GW signal at later times and, thus, only a smaller part of the signal is suitable for their extraction. Second, there exist ambiguities between different higher-order parameters, e.g., between the individual neutron-star spins and the tidal polarizability. Because of this, the LV collaboration provided results for both a low-spin and a high-spin scenario. In this work, we only investigate the low-spin scenario for two reasons. First, large spins are not expected from the observed galactic binary NS population. Second, because neutron stars spin down over time, low spins are also expected from the extremely long merger time of GW170817 of the order of gigayears. Therefore, the low spin scenario is expected to be the more realistic scenario for binary neutron-star mergers such as GW170817.\n\nThe above mentioned problems in the extraction of higher-order parameters lead to weaker constraints on the individual masses of the two component neutron stars in GW170817. With $m_1$ being the mass of the heavier and $m_2$ being the mass of the lighter neutron star in the binary, the mass distribution of the individual stars is typically described in terms of the parameter $q=m_2\/m_1$. The observed mass distributions for $m_1$ and $m_2$ are presented as histograms in the upper panel of Fig.~\\ref{fig:posteriors}. To use this information in our calculations, we describe the posterior of the LV collaboration for $M_{\\text{chirp}}$ and $q$ by the analytical probability distribution~\\cite{Margalit:2017}\n\\begin{equation}\np(q,M_{\\text{chirp}}) = p(q) p(M_{\\text{chirp}})\\,,\n\\end{equation}\nwhere\n\\begin{equation}\np(M_{\\text{chirp}}) \\propto \\exp [- (M_{\\text{chirp}}-\\bar{M}_{\\text{chirp}})^2\/2\\sigma_{M}^2]\\,,\n\\end{equation}\nwith $\\bar{M}_{\\text{chirp}}=1.186M_{\\odot}$ and $\\sigma_{M}= 10^{-3}M_{\\odot}$~\\cite{Abbott:2018wiz}. For the mass asymmetry $q$, we have fitted the function \n\\begin{equation}\np(q)=\\exp \\left(-\\frac12 v(q)^2 -\\frac{c}{2} v(q)^4 \\right)\\,,\\label{eq:massdist}\n\\end{equation}\nto the LV posterior for the component masses. We find $c=1.83$ and $v(q)=(q-0.89)\/0.20$, and compare the resulting normalized analytic distributions with the observed data in the upper panel of Fig.~\\ref{fig:posteriors}.\n\nSince in this work we will confront the gravitational-wave observations of the LV collaboration with nuclear physics constraints, i.e., use our set of EOSs together with the source properties of GW170817 to postdict the distribution of $\\tilde{\\Lambda}$, we do not make use of the observed probability distribution for $\\tilde{\\Lambda}$. However, for reasons of completeness, we have fitted functions consisting of two and three Gaussians of the form\n\\begin{eqnarray}\np(\\tilde{\\Lambda}) = \\sum_{i=1}^N a_i e^{-\\frac 1 2 \\left( \\frac{\\tilde{\\Lambda}-\\Lambda_i}{\\sigma_{i}}\\right)^2}\\label{eq:Gaussians}\n\\end{eqnarray}\nto the observed LV posterior for $\\tilde{\\Lambda}$. The resulting parameters $a_i$, $q_i$ and $\\sigma_{qi}$ are reported in Table~\\ref{tab:LVCLtilde}, and the resulting functions as well as the LV result are plotted in the lower panel of Fig.~\\ref{fig:posteriors}, where the horizontal black line represents the 90\\% LV confidence level for $\\tilde{\\Lambda}$. We also show the posteriors for the reanalysis of Ref.~\\cite{De:2018uhw} for the two extreme cases [uniform mass prior (u) and mass prior informed by double neutron stars (d)]. The main difference between the two analyses lies in the appearance of a second peak in the posterior probability distribution around $\\tilde{\\Lambda}\\sim 600$ for the LV result. The origin of this second peak is not well understood: the peak may be washed out considering a wider domain of frequencies, starting from 23~Hz as in Ref.~\\cite{De:2018uhw}. The presence of the second peak is indeed an important issue for the prediction of $\\tilde{\\Lambda}$: including the second peak, the upper boundary for the 90\\%-CL is 720, while it drops if the second peak is absent.\n\nTherefore, in the following, we consider a structureless flat probability distribution in $\\tilde{\\Lambda}$, and sample the mass distributions for $m_1$ and $m_2$ in GW170817 from the analytic function $p(q,M_{\\text{chirp}})$.\n\n\\subsection{Areas of constant $\\Lambda$}\n\n\\begin{figure}[t]\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{MRLam.pdf}\n\\caption{\\label{fig:MRLam}\nMass-radius envelopes for $n_{\\rm tr}=n_{\\rm sat}$ of Fig.~\\ref{fig:MRcomp}(a) and areas of constant $\\Lambda$ for all CSM EOS parametrizations. We show areas for $\\Lambda=200$ (red), $\\Lambda=400$ (green), $\\Lambda=800$ (blue), and for $\\Lambda=1600$ (brown). For a typical $1.4 M_{\\odot}$ neutron star (horizontal dashed line), a constraint on $\\Lambda$ is equivalent to a radius constraint. The corresponding values for the MM (not shown) always lie withing the areas for the CSM. \n}\n\\end{figure} \n\nBefore addressing GW170817, we focus on the tidal polarizability $\\Lambda$ of individual neutron stars. The tidal polarizability describes how a neutron star deforms under an external gravitational field, and depends on neutron-star properties as\n\\begin{align}\n\\Lambda &=\\frac23 k_2 \\left(\\frac{c^2}{G} \\frac{R}{M}\\right)^5\\,.\n\\end{align}\nHere, $k_2$ is the tidal love number, that is computed together with the Tolman-Oppenheimer-Volkoff equations; see, for example, Refs.~\\cite{Flanagan2008,Damour2009,Moustakidis:2016sab} for more details.\n\n\\begin{figure*}[t]\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.67\\columnwidth]{MchirpLam016.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.67\\columnwidth]{MchirpLam016_800.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.67\\columnwidth]{MchirpLam032.pdf}\\\\\n\\null\\hfill\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.67\\columnwidth]{MchirpLam016_800_fic.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.67\\columnwidth]{MchirpLam032_fic.pdf}\n\\caption{\\label{fig:MchirpLam}\nEnvelopes for the CSM (red) and the MM (black) for the predicted tidal polarizability parameter $\\tilde{\\Lambda}$ as a function of chirp mass for neutron-star binaries with component masses in the range $1.0-1.9 M_{\\odot}$. We show: panel (a) the results for $n_{\\text{tr}}=n_{\\text{sat}}$, panel (b) for $n_{\\text{tr}}=n_{\\text{sat}}$ when additionally enforcing the LV constraint from GW170817, and panel (c) for $n_{\\text{tr}}=2 n_{\\text{sat}}$. \nIn panels (d) and (e), we show how this band reduces under a fictitious observation of a merger of two $1.6 M_{\\odot}$ neutron stars when $\\tilde{\\Lambda}$ would be measured to be $200-300$. We indicate GW170817 and the fictitious measurement (blue error bars) and the corresponding chirp masses (dotted vertical lines). In panel (e), the GW observations together with nuclear physics constraints would rule out the MM.}\n\\end{figure*} \n\n\nIt is interesting to look at areas of constant $\\Lambda$ within the MR plane. In this case, the relation of neutron-star mass and radius is given by \n\\begin{align}\nM&=\\left(\\frac32 \\frac{\\Lambda}{k_2}\\right)^{-\\frac15} \\frac{c^2}{G} R\\,,\n\\end{align}\nleading to the following scaling relation,\n\\begin{align}\n\\left(\\frac{M}{M_{\\odot}}\\right)&=0.6243 \\left(\\frac{\\Lambda}{k_2}\\right)^{-\\frac15} \\left(\\frac{R}{1 \\,\\mathrm{km}}\\right)\\,.\n\\label{eq:scaling}\n\\end{align}\nFor constant $\\Lambda$, this implies an almost linear relationship between M and R, because the love number $k_2$ does not vary strongly in that case. In addition, for different values of $\\Lambda$, the slopes are rather similar due to the small exponent $-1\/5$. In Fig.~\\ref{fig:MRLam}, we plot the mass-radius relation for $n_{\\text tr}=n_{\\rm sat}$ for the CSM, together with areas of constant $\\Lambda$. In particular, we show areas for $\\Lambda=200, 400, 800$, and $1600$.\n\nWhile there is a tight correlation between radii and tidal polarizabilities, from Fig.~\\ref{fig:MRLam} one can see that both quantities still provide complementary information. For example, an exact observation of the tidal polarizability of a neutron star, i.e., with vanishing uncertainty, would still lead to a remaining uncertainty for the radius of a typical $1.4 M_{\\odot}$ neutron star. To be specific, for $\\Lambda=200$, the remaining radius uncertainty is still $\\approx 1$ km, compatible with the expected uncertainty of NICER~\\cite{NICER1}. For larger values of $\\Lambda$ this uncertainty decreases and for $\\Lambda=800$ it is only $\\approx 0.5$ km. However, based on GW170817 values larger than $720$ are ruled out for typical neutron stars. Hence, both tidal deformabilities and radii offer complementary information on neutron-star global structure. \n\nFinally, from Eq.~(\\ref{eq:scaling}), one can infer the following fit,\n\\begin{align}\n\\left(\\frac{M}{M_{\\odot}}\\right)&= \\frac{a}{(b+\\Lambda)^{1\/5}} \\left(\\frac{R}{1 \\,\\mathrm{km}}\\right)\\,,\n\\label{eq:scalingFit}\n\\end{align}\nwhere we find $a=0.406435$ and $b= 68.5$.\n\n\\subsection{Tidal polarizabilities of GW170817}\n\nFor neutron-star mergers, the GW signal allows the extraction of the binary tidal polarizability parameter $\\tilde{\\Lambda}$. This parameter is defined as a mass-weighted average of the individual tidal polarizabilities, \n\\begin{equation}\n\\tilde{\\Lambda}~=~\\frac{16}{13} \\left[\\frac{(m_1+12m_2)m_1^4\\Lambda_1 }{m_{\\text{tot}}^5}+ \\frac{(m_2+12m_1)m_2^4\\Lambda_2 }{m_{\\text{tot}}^5}\\right]\\,.\n\\end{equation}\nAs discussed in Sec.~\\ref{sec:posterior}, the extraction of the binary tidal polarizability suffers from increased uncertainties, due to its importance only during the last few orbits~\\cite{Flanagan2008,Damour2009} and correlations among the parameters. In the initial publication of the LV collaboration~\\cite{Abbott:2017}, the constraint on $\\tilde{\\Lambda}\\leq 800$ was reported with 90\\% confidence (corrected to be $\\tilde{\\Lambda}\\leq 900$ in Ref.~\\cite{Abbott:2018wiz}). This analysis, however, was very general and did not assume both objects in the binary system to have the same EOS. Several reanalyses have since improved this constraint. Assuming that both compact objects were neutron stars governed by the same EOS, Ref.~\\cite{De:2018uhw} used polytropic EOS models and a Bayesian parameter estimation with additional information on the source location from EM observations to derive limits on $\\tilde{\\Lambda}$ for different prior choices for the component masses: for uniform priors the reported 90\\% confidence interval was $\\tilde{\\Lambda}=84-642$, for a component mass prior\ninformed by radio observations of Galactic double neutron stars the result was $\\tilde{\\Lambda}=94-698$, and for a component mass\nprior informed by radio pulsars the reported result was $\\tilde{\\Lambda}=89-681$. A reanalysis by the LV collaboration found a new 90\\% confidence of $70 \\leq\\tilde{\\Lambda}\\leq 720$~\\cite{Abbott:2018wiz}; see Fig.~\\ref{fig:posteriors}. Finally, the LV collaboration reported an additional result, assuming that both merging objects were neutron stars governed by the same EOS~\\cite{Abbott:2018exr}. This EOS was based on the Lindblom parametrization~\\cite{Lindblom:2010bb}\nstitched to the SLy EOS for the crust, and resulted in $\\tilde{\\Lambda}=70-580$ with 90\\% confidence. For the different extractions, the lower limit is rather stable, but the upper limit varies from 580-800. \n\nIn general, the uncertainty range for all extractions is sizable. In the following, we will investigate the resulting $\\tilde{\\Lambda}$ obtained from state-of-the-art nuclear-physics models at low densities. To obtain these results, for all our EOS models we compute the combined tidal polarizability $\\tilde{\\Lambda}$ for thousands of NS-NS binaries where the sample the mass $m_1$ of the heavier neutron star in the range $1.0-1.9 M_{\\odot}$ and the mass of the lighter neutron star $m_2$ in the range $1.0 M_{\\odot}-m_1$ (implying $q\\leq 1$). This allows us to explore a wide range of mass asymmetries and chirp masses ranging from $0.871M_{\\odot}$ to $1.654 M_{\\odot}$, which naturally includes the chirp masses for several known neutron-star binaries as well as GW170817.  We show the resulting envelopes for $\\tilde{\\Lambda}$ as a function of $M_{\\rm{chirp}}$ in Fig.~\\ref{fig:MchirpLam}. We also indicate the chirp mass for GW170817, $M_{\\rm chirp}^{\\rm GW170817}=1.186 M_{\\odot}$~\\cite{Abbott:2018wiz}  (blue dashed vertical lines) that allows to extract nuclear-physics constraints on $\\tilde{\\Lambda}$ of GW170817. \n\nUsing nuclear-physics constraints from chiral EFT up to $n_\\text{sat}$ [panel (a)] leads to the widest allowed range for $\\tilde{\\Lambda}$ for a given chirp mass. This is true for both the MM and the CSM, but the CSM envelope is much larger due to the wider flexibility of the EOS at higher densities. For GW170817 ($M_{\\rm chirp}^{\\rm GW170817}=1.186 M_{\\odot}$), we find $\\tilde{\\Lambda}_{\\text{CSM}}=60-2180$ and $\\tilde{\\Lambda}_{\\text{MM}}=230-950$; for the CSM, the uncertainty in $\\tilde{\\Lambda}$ is much larger than the LV constraint for GW170817. For this transition density, both the MM and the CSM can be constrained by the LV constraint on GW170817 and, as a result, GW170817 adds information on the mass-radius relation of neutron stars. \n\nTo explore the impact of the LV constraint of Ref.~\\cite{Abbott:2018wiz}, we make use of $p(q,M_{\\text{chirp}})$ and, using a uniform prior, select only EOS-$m_{1,2}$ combinations with $70\\leq\\tilde{\\Lambda}\\leq 720$. In panel (b) of Fig.~\\ref{fig:MchirpLam} we show the resulting envelope for $\\tilde{\\Lambda}(M_{\\rm{chirp}})$ for the MM and CSM. In addition, we also show the resulting envelopes for the EOS and the MR relation in panels (b) of Figs.~\\ref{fig:EpsPcomp} and~\\ref{fig:MRcomp}, respectively. Please note that the resulting range of tidal polarizabilities for $M_{\\rm{chirp}}=1.186$ of $\\tilde{\\Lambda}=70-1020$ in Fig.~\\ref{fig:MchirpLam}(b) is larger than the LV constraint. The reason is that we accept all EOS that fulfill the LV constraint for any value of $q$ allowed according to $p(q)$. The range in Fig.~\\ref{fig:MchirpLam}(b), however, is computed for many more values of $q$. For example, if an EOS passes the constraint $\\tilde{\\Lambda}\\leq 720$ for $q=0.7$ than the resulting $\\tilde{\\Lambda}$ for $q=1$ will be larger. \n\nNaturally, enforcing this constraint rules out a considerable part of EOSs that lie both on the high-pressure and low-pressure side at high energy densities. This, again, is reflected in the mass-radius relation, where neutron stars with large radii are excluded by the LV constraint. For our analysis and the CSM, we find that the radius of a $1.4 M_{\\odot}$ neutron star, $R_{1.4}$, can be constrained to be $9.0\\, \\text{km}< R_{1.4}<13.6$ km. This was also found in Ref.~\\cite{Annala:2017llu}, where a polytropic EOS expansion was used to constrain the radius of neutron stars by enforcing the constraint $\\Lambda_{1.4}<800$ (the initial LV constraint of Ref.~\\cite{Abbott:2017}). Ref.~\\cite{Annala:2017llu} found that $R_{1.4}<13.6$ km, and both analyses are in excellent agreement. \n\nFinally, we assume the chiral EFT constraint to be valid up to $2n_\\text{sat}$ [panel (c)]. Even though the uncertainties are still sizable, the predicted total range for $\\tilde{\\Lambda}$ reduces dramatically. For GW170817, we find $\\tilde{\\Lambda}_{\\text{CSM}}=80-580$ and $\\tilde{\\Lambda}_{\\text{MM}}=280-480$. Our constraint, which is solely guided by nuclear-EFT input, is much tighter than the observational LV constraint and in excellent agreement with the recent detailed reanalysis by the LV collaboration~\\cite{Abbott:2018exr}. We emphasize, though, that our analysis is more constraining than the LV reanalysis: our 100\\% envelopes are compatible with the 90\\% contour of Ref.~\\cite{Abbott:2018exr}. Therefore, the sentiment that the neutron-star merger GW170817 revolutionized our understanding of the EOS, is a bit of an exaggeration. GW170817, however, represents a new hope for obtaining different constraints on the EOS that might also offer the possibility to investigate new phases of dense matter. In this sense, GW170817 and the expected future detections will surely contribute to answering the long standing question of the nature of the NS core.\n\nWe explicitly stress that our results imply that current nuclear physics knowledge in the relevant density range of $1-2 n_{\\rm sat}$, as obtained by ab inito calculations using modern nuclear Hamiltonians and state-of-the art many-body methods, is compatible with the recent neutron-star merger observation but more constraining for neutron-star observables and the EOS. In addition, efforts in the nuclear-theory community to improve nuclear interactions might allow to considerably reduce the theoretical uncertainty for the neutron-star-matter EOS between $1-2 n_{\\rm sat}$, which will tighten our constraints even more. In general, this very interesting density range provides an excellent laboratory to probe nuclear-theory predictions against astrophysical observations and heavy-ion collision experiments.\n\n\n\\subsection{Impact of varying ${\\bf n_{\\text{tr}}}$ and the validity of chiral EFT predictions}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{ntrRminRmax-band.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{ntrLamminmax-band.pdf}\n\\end{center}\n\\caption{\\label{fig:ntrRminRmax}\nRadius of a typical $1.4 M_{\\odot}$ neutron star, $R_{1.4}$ (left), and $\\tilde{\\Lambda}$ for $M_{\\rm chirp}=1.186 M_{\\odot}$ (right) as functions of $n_{\\rm{tr}}$. We show the envelopes for the CSM in red and for the MM in black. For the CSM, when requiring $c_S^2\\leq 0.5$ instead of $c_S^2\\leq1.0$, the hatched areas are excluded. We also indicate the constraints from GW170817 and the values of $n_{\\rm{tr}}$, above which nuclear-theory input alone becomes more constraining than observations.}\n\\end{figure*} \n\nThese present studies as well as the one of Ref.~\\cite{Tews:2018kmu} are the first to use chiral EFT calculations of the neutron matter EOS up to twice nuclear saturation density with reliable error estimates to compute tidal polarizabilities for GW170817. Reliable uncertainty estimates are critical for understanding the impact that GW detections will have on elucidating the properties of dense matter inside neutron stars, and theoretical calculations of the dense-matter EOS without uncertainty estimates are of limited value for a meaningful analysis of GW data. Uncertainty estimates have shown that chiral EFT input remains useful up to $2 n_{\\rm sat}$, and we find, in contrast to other recent publications~\\cite{Annala:2017llu,Fattoyev:2017jql,Most:2018hfd}, that GW170817 does \\emph{not} provide new insight about the EOS that cannot be obtained from current nuclear physics knowledge. This message tempers claims made in these recent publications which state that the upper limit on the tidal polarizability derived from GW data rules out stiff nuclear EOS. While this inference is correct, such stiff EOSs are already ruled out based on state-of-the-art nuclear Hamiltonians. In other words, models of dense matter excluded by the upper limit on the tidal deformability from GW170817 are already incompatible with the current microscopic EOSs at densities where error estimates can still be justified. \n\nNevertheless, the reliability of chiral interactions at these densities has been questioned. Although the convergence of the chiral expansion cannot be strictly proven in this density range, we present arguments to show that the order-by-order convergence of the chiral expansion for the EOS up to $2n_{\\rm sat}$ is still reasonable. First, the expansion parameter increases by only about 25\\% over the density interval $1-2 n_{\\rm sat}$. Second, Ref.~\\cite{Tews:2018kmu} analyzed the order-by-order convergence of the employed Hamiltonians at $2 n_{\\rm sat}$, and showed that, even though the reliability naturally decreases with increasing density, the order-by-order convergence remains reasonable and consistent with simple power counting arguments within the theoretical uncertainty estimates. Nevertheless, densities around $2 n_{\\rm sat}$ seem to provide an upper limit to the applicability of the chiral Hamiltonians we use in this work.\n\nTo support our main statement - namely that the constraints from GW170817 are compatible with but less restrictive than predictions of the EOS based on realistic nuclear potentials and do not yield specific new information about nuclear Hamiltonians or about possible phase transitions at supra-nuclear density - in this context, we investigate which density range for chiral EFT input is sufficient to justify our statement. We present the total uncertainty ranges for $R_{1.4}$ (left panel) and $\\tilde{\\Lambda}$ for $M_{\\rm chirp}=1.186 M_{\\odot}$(right panel) as functions of the density $n_{\\rm tr}$ in Fig.~\\ref{fig:ntrRminRmax}. For $R_{1.4}$, we indicate the upper limit on the radii of Ref.~\\cite{Annala:2017llu}, $R_{1.4}\\leq 13.6$ km, which was obtained using $n_{\\rm tr}=n_{\\rm sat}$ and the LV constraint (horizontal dotted line). We find that the CSM alone constrains the radii to be smaller than this bound for $n_{\\rm tr}>0.23 \\,\\mathrm{fm}^{-3} \\approx 1.44 n_{\\rm sat}$ (an 11\\% increase of the expansion parameter compared to $n_{\\rm sat}$). For the tidal polarizability, we indicate the LV constraint as a horizontal blue band and find that the CSM leads to $\\tilde{\\Lambda}\\leq 720$ as soon as $n_{\\rm tr}> 0.285 \\,\\mathrm{fm}^{-3} \\approx 1.78 n_{\\rm sat}$ (a 20\\% increase of the expansion parameter compared to $n_{\\rm sat}$). We would like to emphasize that these crucial values for $n_{\\rm tr}$ for both observables do not necessarily have to agree, as seen in Fig.~\\ref{fig:ntrRminRmax}. The reason is that the upper limit on $\\tilde{\\Lambda}$ depends on $q$ while $R_{1.4}$ does not. In Fig.~\\ref{fig:MchirpLam}(b) we have seen that when varying $q$ in the range allowed by GW170817, $\\tilde{\\Lambda}$ can increase to values $\\sim 1000$ for the EOS that pass the LV constraint from GW170817. Chiral EFT input becomes compatible with this value at $n_{\\rm tr}\\sim 0.23 \\,\\mathrm{fm}^{-3}$, in agreement with the value for $R_{1.4}$. At these values for $n_{\\rm tr}$, in particular at $1.44 n_{\\rm sat}$, arguments for the validity of chiral interactions remain even stronger,  which strengthens the validity of our main statement.\n\nFinally, the value of $n_{\\rm tr}$ also affects the speed of sound inside neutron stars. The speed of sound is expected to approach the conformal limit of $c_S^2=1\/3$ at very high densities~\\cite{Kurkela:2010}. In neutron stars, though, it is not clear if this conformal limit remains valid or not. As discussed in detail in Ref.~\\cite{Tews:2018kmu}, the neutron-matter EOS up to $n_{\\rm tr}=2 n_{\\rm sat}$ requires the speed of sound to pass the conformal limit to be sufficiently stiff to stabilize the observed two-solar-mass neutron stars. In fact, for chiral models the speed of sound has to increase beyond the conformal limit for $n_{\\rm tr}>0.28 \\,\\mathrm{fm}^{-3}$ and even for phenomenological nuclear Hamiltonians, which lead to stiffer neutron-matter EOS, this statement remains valid for $n_{\\rm tr}>0.31 \\,\\mathrm{fm}^{-3}$. While there might be EOS that are much stiffer below $2 n_{\\rm sat}$ and, hence, stabilize the heaviest neutron stars while still obeying the conformal limit, such EOS are ruled out by modern nuclear Hamiltonians. \n\nTherefore, the neutron-matter EOS up to $2 n_{\\rm sat}$ for state-of-the-art nuclear Hamiltonians requires the speed of sound in neutron stars to experience a non-monotonous behavior, i.e, increasing beyond $c_S^2=1\/3$ but decreasing at higher densities to approach this limit.\nFor example, for chiral EFT interactions and $n_{\\rm tr}=2 n_{\\rm sat}$, the speed of sound has to reach values $c_S^2\\geq 0.4$. \nThe question remains, though, which forms of strongly-interacting matter lead to such a behavior for the speed of sound. \nIn order to estimate the impact of the speed-of-sound behavior on $R_{1.4}$ and $\\tilde{\\Lambda}$, we present hatched areas in Fig.~\\ref{fig:ntrRminRmax} which are excluded for $c_s^2\\leq0.5$. We choose this limiting value solely for illustrative purposes.\nThis constraint slightly reduces the upper bound on neutron-star radii but it would mostly rule out low-radius neutron stars.  \nThe reason is that neutron stars can have very small radii only for strong first-order phase transitions with low onset densities. To simultaneously support $2M_{\\odot}$ neutron stars, the EOSs has to experience a sudden subsequent stiffening, i.e., the speed of sound has to increase dramatically. For a larger possible speed of sound, stronger phase transitions are allowed, which leads to stars with smaller radii. Limits on $c_S^2$, on the other hand, rule out the strongest phase transitions, and increase the smallest possible radius. For $c_S^2\\leq 0.5$, the lower limit on the radius of a $1.4M_{\\odot}$ neutron star is approximately 10 km, of the order of the constraint of Ref.~\\cite{Bauswein:2017vtn}.\n\n\\subsection{Impact of additional constraints}\n\n\\begin{figure}[t]\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{RLam.pdf}\n\\caption{\\label{fig:RLam}\nEnvelopes for the correlation between $\\tilde{\\Lambda}$ of GW170817 and the radius of a $1.4 M_{\\odot}$ (red) and the radius of a $1.6 M_{\\odot}$ (blue) neutron star for $n_{\\text{tr}}=2 n_{\\text{sat}}$ and the CSM. The corresponding values for the MM (not shown) lie within the CSM envelopes. \nWe also show the lower limit of the LV constraint on the tidal polarizability of GW170817~\\cite{Abbott:2018wiz}, the proposed constraint of Ref.~\\cite{Radice:2017lry} and its update of Ref.~\\cite{Radice:2018ozg}, and the radius constraint for a $1.6 M_{\\odot}$ neutron star from Ref.~\\cite{Bauswein:2017vtn}. \n}\n\\end{figure} \n\nEven though the tidal polarizabilities extracted from GW170817 alone may not revolutionize our understanding of the EOS, several additional constraints based on the EM counterpart were proposed. These additional constraints were mostly based on the fact that the EM signal of GW170817 does not seem to imply a prompt collapse of the hypermassive merger remnant to a black hole. Instead, it is argued that the merger remnant survived for several 100 milliseconds before collapse. Based on this assumption, several groups independently suggested the maximum mass of neutron stars to be less than $\\approx 2.2-2.3 M_{\\odot}$~\\cite{Margalit:2017,Shibata:2017xdx,Rezzolla:2017aly}. While this constraint is powerful for smooth EOS models, which exhibit a strong correlation between $M_{\\rm max}$ and radii of typical neutron stars, the appearance of strong first-order phase transitions in general EOS models implies that the maximum mass is not very constraining for the structure of typical neutron stars; see also Ref.~\\cite{Tews:2018iwm}. \n\nAdditional constraints for radii and tidal polarizabilities were proposed based on the same assumptions. Ref.~\\cite{Bauswein:2017vtn} suggested that the EM observation can be used to argue that $R_{1.6}\\geq 10.68_{-0.04}^{+0.15}$ km. In contrast to the $M_{\\rm max}$ constraint, a radius constraint has a sizable impact on the CSM: In Figs.~\\ref{fig:EpsPcomp}(b) and (c) as well as Figs.~\\ref{fig:MRcomp}(b) and (c) we indicate parts of the envelopes which are excluded by $R_{1.6}\\geq 10.68_{-0.04}^{+0.15}$ km by hatched areas. In addition,  Ref.~\\cite{Radice:2017lry} suggested that the amount of ejecta determined from the EM observations implies $\\tilde{\\Lambda}>400$. This constraint was later updated to $\\tilde{\\Lambda}>300$~\\cite{Radice:2018ozg}.  In Fig.~\\ref{fig:RLam}, we show the correlation between $\\tilde{\\Lambda}$ and the radii of a $1.4 M_{\\odot}$ neutron star, $R_{1.4}$, and a $1.6 M_{\\odot}$ neutron star, $R_{1.6}$, for $n_\\text{tr}=2n_\\text{sat}$ and the CSM. While in general radius and tidal polarizabilities are correlated, the appearance of phase transitions washes this correlation out. Fig.~\\ref{fig:RLam} again highlights the fact that even an exact determination of $\\tilde{\\Lambda}$ leaves a considerable radius uncertainty. Therefore, independent observations of radii and tidal polarizabilities are crucial to pin down the high-density EOS of nuclear matter.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{Diffq0710.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{Diffq1007.pdf}\n\\end{center}\n\\caption{\\label{fig:q0710diff}\nEquations of state for $n_{\\rm tr}=n_{\\rm sat}$ which pass the LV constraint $70\\leq \\tilde{\\Lambda}\\leq 720$ for $q=0.7$ but not for $q=1.0$ [panel (a)] and vice versa [panel (b)].\n}\n\\end{figure*} \n\nIn Fig.~\\ref{fig:RLam}, we also show the constraints of Refs.~\\cite{Bauswein:2017vtn,Radice:2017lry,Radice:2018ozg}. The radius constraint implies that $\\tilde{\\Lambda}\\geq 180$ while the constraint of Ref.~\\cite{Radice:2017lry} (Ref.~\\cite{Radice:2018ozg}) implies $R_{1.6} \\sim R_{1.4}\\geq 11.5$ km ($10.5$ km). All of these constraints are based on empirical formulas extracted from simulations for a limited set of model EOSs. Especially for the constraints of Refs.~\\cite{Radice:2017lry,Radice:2018ozg}, this set contains only four nucleonic EOS and, therefore, is likely overestimated~\\cite{Tews:2018iwm}. \nIn the case of the first constraint, a similar argument might be made. Nevertheless, in that case the authors try to explore the full EOS dependence which results in a more conservative constraint.\nIn both cases, however, future numerical simulations with additional EOSs, including, e.g., phase transitions, can be used to refine these constraints and improve their robustness.\n\nIn addition to inferences from GW170817, additional future observations might dramatically improve our understanding of the EOS.  \nThe NICER~\\cite{NICER1} and eXTP~\\cite{Watts:2018iom} missions will provide neutron-star radii with a few percent uncertainty: the NICER mission is expected to provide first results within this year. As we have seen above, these future radius observations might considerably reduce the ambiguity of the allowed EOS models. A measurement of $R_{1.4}$ with a 5\\% accuracy will add valuable information and might help distinguish EOSs with and without phase transitions; see also Ref.~\\cite{Tews:2018kmu}.\n\nIn addition, in the next years additional neutron-star merger observations by the LV collaboration are expected. While the uncertainty for the tidal polarizability associated with GW170817 is not sufficient to constrain the EOS, this might change for future observations. For example, mergers with better signal-to-noise ratios could be observed, or sufficiently many mergers are observed so that accurate information can be extracted. In addition, third generation GW detectors might provide tidal-polarizability measurements with 10\\% uncertainty. To illustrate the possibilities offered by such new GW events, we inject in Fig.~\\ref{fig:MchirpLam}(d) and (e) a fictituous measurement of $M_{chirp}=1.385$ and $\\tilde{\\Lambda}$ to be measured in the range $200-300$. Such an observation would dramatically reduce the uncertainties in the EOS: \nit would reduce the allowed radius range for a typical neutron star to \n11.7-13.4 km for $n_{\\rm tr}=n_{\\rm sat}$ and to only 11.7-12.5 km for $n_{\\rm tr}=2 n_{\\rm sat}$. Also, it is interesting to note that in this case the MM cannot reproduce the two events, GW170817 and the fictitious one. There is, therefore, a great potential to combine future detections as a filter for EOS models and the accumulation of GW tidal deformabilities may offer the possibility to make statements about the existence of phase transitions in dense matter.\n\n \\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{RtilLamtil016.pdf}\n\\includegraphics[trim= 0.0cm 0 0 0, clip=,width=0.9\\columnwidth]{RtilLamtil032.pdf}\n\\end{center}\n\\caption{\\label{fig:EmpRel}\nRelation connecting the common radius $\\hat{R}$  and the binary tidal polarizability $\\tilde{\\Lambda}$ for $0.7<q<1.0$ and for $n_{\\rm tr}=n_{\\rm sat}$ (left panel) or $n_{\\rm tr}=2 n_{\\rm sat}$ (right panel). As a comparison, we show the relation Eq.~(5) of Ref.~\\cite{De:2018uhw}\nwith its uncertainty (black dotted lines) and our fits (blue dashed line).}\n\\end{figure*} \n\n\\subsection{Impact of phase transitions on tidal polarizability}\n\\label{sec:PTimpact}\n\nIn the previous sections, we have seen that ranges for all neutron-star observables are larger for the CSM than the MM because the CSM permits regions of drastic stiffening or softenting of the EOS. In this section, we briefly discuss the impact that strong phase transitions have on neutron-star tidal polarizabilities. \n\nOf special interest for the interpretation of merger observations is the behavior of the EOS for stars in the mass range of the two component masses: For GW170817 this range is around $M=1.4M_{\\odot}$. EOS with strong first-order phase transitions appearing in stars of this mass range might be probed by future merger observations. For instance, the CSM, which includes such phase transitions, permits small values for $\\tilde{\\Lambda}$ due to strong softening and subsequent stiffening of the EOS, but the MM prevents $\\tilde{\\Lambda}$ to be below $\\approx 250$. These observable differences among the two models allow us to identify ranges of tidal deformabilities (and neutron-star observables in general) for which a strong first-order phase transition is preferred or even necessary, providing a means to probe new states of matter inside neutron stars. In the above example, an observation of $\\tilde{\\Lambda}\\leq 250$ would indicate a softening of the EOS that smooth (nucleonic) EOS cannot provide.\n\nWe have also seen before that strong phase transitions weaken the correlation between $R$ and $\\tilde{\\Lambda}$. For EOSs with phase transitions in the relevant mass range, which produce lighter stars with larger radii and heavier stars with smaller radii, a significant mass asymmetry of a merging binary keeps the EOS compatible with a constraint on $\\tilde{\\Lambda}$ but permits larger radii for typical neutron stars and, therefore, washes out this correlation.\n\nIn Fig.~\\ref{fig:q0710diff}, we illustrate this behavior for $n_{\\rm tr}=n_{\\rm sat}$ for two interesting cases: EOSs which pass the LV constraint for $q=0.7$ but are excluded for $q=1.0$ and vice versa. We show the EOSs belonging to the first class of models in Fig.~\\ref{fig:q0710diff}(a) and the EOSs belonging to the second class of models in Fig.~\\ref{fig:q0710diff}(b). In general, for a given EOS, heavier neutron stars have smaller tidal polarizabilities, and increasing the mass asymmetry in the binary, i.e., lowering $q$, results in slightly smaller values for $\\tilde{\\Lambda}$ for a given chirp mass. Therefore, several smooth EOSs, i.e., without phase transitions, pass the LV constraint for $q=0.7$ but not for $q=1.0$, which can be seen in Fig.~\\ref{fig:q0710diff}(a).\n\nThe more interesting case are EOS models with a strong phase transition occurring around $1.4 M_{\\odot}$ and leading to a kink in the MR curve. Below the kink, radii and tidal polarizabilities are larger but drastically decrease beyond the phase transition. Two cases can be distinguished: the phase transition appears at masses above $1.4 M_{\\odot}$ or below $1.4 M_{\\odot}$. For the first case, $q=1$ for GW170817 implies that both stars have the same mass $\\sim 1.4 M_{\\odot}$ and, therefore, larger radii and tidal polarizabilities $\\tilde{\\Lambda}=\\Lambda_1=\\Lambda_2$. Lowering $q$, so that the heavier star probes the phase transition, suddenly decreases $\\tilde{\\Lambda}$ by a fair amount. Therefore, some EOS will be rejected for $q=1$ but accepted for lower $q$, e.g., $q=0.7$. We show these models in Fig.~\\ref{fig:q0710diff}(a).  In contrast to the smooth models, though, these models permit much larger radii for typical neutron stars, which can also be seen in Fig.~\\ref{fig:MRcomp}(b).\n\n\nIf the phase transition appears below $1.4 M_{\\odot}$, the inverted situation can appear: EOSs are ruled out for $q=0.7$ but allowed for $q=1.0$. We show these cases in the right panel of Fig.~\\ref{fig:q0710diff}. If the phase transition happens in very low-mass stars at densities close to saturation density, then the EOS produces neutron stars with very small radii of the order of $R_{1.4}\\sim 9$ km. In this case, $\\tilde{\\Lambda}$ is reduced for smaller values of $q$ and the EOS is ruled out due to the lower constraint the tidal polarizability, $70\\leq \\tilde{\\Lambda}$. However, this is an extremely rare situation and we find only one such EOS among tens of thousands of samples, see Fig.~\\ref{fig:q0710diff}(b). \nIf the phase transition appears in stars slightly below $1.4 M_{\\odot}$, for $q=1$ both stars in GW170817 would have been hybrid stars and the $\\tilde{\\Lambda}$ would have been small enough for these models to pass the constraint. Increasing the mass asymmetry, $\\Lambda_1$ decreases but $\\Lambda_2$ increases rapidly, leading to the EOS being rejected by the upper constraint on $\\tilde{\\Lambda}$. We found a few such models, see Fig.~\\ref{fig:q0710diff}(b). \n\nIn any case, information on possible strong first-order phase transitions might be obtained by neutron-star merger observations. The observation of two mergers with similar chirp mass but different mass asymmetries and dramatically different binary tidal polarizabilities might shed light on the location of a strong first-order phase transition. In addition, future observations accessing regions allowed by the CSM but forbidden by the MM might also provide information on such a phase transition. For these extractions, however, higher-order GW parameters need be constrained much more precisely in future observations.  \n\n\\subsection{Empirical relations for $\\bf \\tilde{\\Lambda}$} \n\\label{sec:Relations}\n\nFinally, we use our EOS models to investigate the empirical relation between the tidal polarizability and the radius of neutron stars.  Such a relation was reported in Eq. (5) of Ref.~\\cite{De:2018uhw}, that related the binary tidal polarizability $\\tilde{\\Lambda}$ to the common radius of a neutron-star binary:\n\\begin{equation}\n\\tilde{\\Lambda}=0.0042(4) \\left(\\frac{\\hat{R}c^2}{G M_{\\rm{chirp}}} \\right)^6= 0.000146(13) \\left(\\frac{\\hat{R}}{km} \\right)^6\\,.\n\\end{equation}\nSimilarly, a relation between the tidal polarizability and radius of a typical $1.4 M_{\\odot}$ neutron star was reported in Ref.~\\cite{Annala:2017llu}:\n\\begin{equation}\n\\Lambda_{1.4}=2.88\\cdot 10^{-6}\\left(\\frac{R_{1.4}}{km} \\right)^{7.5}\\,.\n\\end{equation}\nInterestingly, even though both approaches are based on a piecewise polytropic expansion for the EOS, the resulting relations and especially exponents are rather different (for $q=1$, $\\tilde{\\Lambda}\\sim \\Lambda_{1.4}$ and $\\hat{R} \\sim R_{1.4}$).\n\nWe constructed similar relations between $\\tilde{\\Lambda}$ and the average radius of the two binary neutron stars in GW170817 for the CSM\nand $n_{\\rm tr}=n_{\\rm sat}$ and $n_{\\rm tr}=2 n_{\\rm sat}$. We show density plots for our data points and the resulting fit functions in Fig.~\\ref{fig:EmpRel}, together with the result of Ref.~\\cite{De:2018uhw}. For $n_{\\rm tr}=n_{\\rm sat}$ (left panel), we find the relation \n \\begin{equation}\n\\tilde{\\Lambda}=0.00057(6) \\left(\\frac{\\hat{R}c^2}{G M_{\\rm{chirp}}} \\right)^{7.05}\\,.\n \\end{equation}\nIn this case, the exponent lies in between the other two determinations but is closer to the result of Ref.~\\cite{Annala:2017llu}.  For $n_{\\rm tr}=2 n_{\\rm sat}$, we find instead\n \\begin{align}\n\\tilde{\\Lambda}&=0.0047(8) \\left(\\frac{\\hat{R}c^2}{G M_{\\rm{chirp}}} \\right)^{5.94}\\,,\n\\end{align}\nin very good agreement to the relation of Ref.~\\cite{De:2018uhw}. Comparing the findings, we see that these relations are not universal but depend on the EOS input used. \n\n\\subsection{Comparisons to other recent works}\n\nThere is general consensus that the upper bound on the tidal deformability $\\Lambda_\\text{1.4}<800$ derived by the initial analysis by the LIGO-Virgo scientific collaboration in Ref.\\cite{Abbott:2017} implies that the radius $R_\\text{1.4}\\lesssim 13.6$ km. \nMaking the reasonable assumption that both compact objects were NSs, and that they are both described by the same EOS, other authors have discussed how the bound on the tidal deformability impacts our understanding of NSs and dense matter.  In what follows we compare our analysis to some of these studies. \n\nIn Ref.~\\cite{Annala:2017llu} the authors construct a model for the EOS based on the predictions of  \nchiral EFT up to a baryon number density $n_{sat}$ and use a set of four polytropes to describe matter at higher densities encountered in the core. They claim that perturbative calculations of QCD (pQCD) valid at very high density, far exceeding those encountered inside the NS core, can constrain the allowed parameter space of the polytropic EOSs.  This is then combined with the upper limit on the tidal deformability to constrain the relationship between mass and radius of all NSs and the EOS of matter encountered in their cores.  The maximal model we employ addresses the question of how improved constraints on the EOS from theory between  $n_{sat}$ and $2n_{sat}$ will alter the situation. We find no evidence for the usefulness of constraints from pQCD. The pressure in NS cores is much smaller than those encountered at the densities where pQCD is valid. Our maximal model is thermodynamically consistent and has adequate freedom to satisfy constraints from pQCD, but is uninformed by it. \n\nIn Ref. \\cite{Fattoyev:2017jql} the authors use a model EOS for neutron-rich matter that describes matter at sub-nuclear density encountered inside nuclei and at higher densities encountered inside neutron stars.  They find a strong correlation between  the neutron-skin thickness of neutron-rich nuclei and the neutron star tidal deformability, similar to the correlation between the skin-thickness and neutron-star radii found earlier \\cite{Horowitz:2001}.  Such a correlation is expected because the NS radius and the tidal deformability are tightly correlated in models that do not contain phase transitions. For their models they report a tight correlation given by $\\Lambda \\simeq 7.76 \\times 10^{-4}~ (R\/\\text{km})^{5.3}$. Using the correlation between neutron skin thickness and NS radius they show that the experimental lower bound on the neutron-skin thickness of $^{208}$Pb implies $R_\\text{1.4}>12.55$ km. This, combined with the correlation between $\\Lambda$ and $R$, is used to deduce that  $\\Lambda_\\text{1.4}>490$. As discussed earlier, both these correlations are \\emph{model dependent}.  It is useful to compare these inferences to the predictions of our minimal model shown in Fig.~\\ref{fig:ntrRminRmax} which assumes a smooth EOS without phase transitions, does not violate experimental data for the neutron-skin thickness of $^{208}$Pb, but can accommodate smaller values for $R_\\text{1.4}$ and $\\Lambda_\\text{1.4}$. \n\nIn Ref. \\cite{Most:2018hfd}, the authors impose an additional constraint requiring that $M_\\text{max} < 2.16~M_\\odot$ and employ EOSs with and without strong first-order phase transitions to determine limits on the neutron star radius and deformability. In the absence of phase transitions they find that $12~\\text{km} < R_{1.4} < 13.45~ \\text{km}$ and require $\\Lambda_{1.4}> 375$. This range is deduced as the $2\\sigma$ interval by exploring a large suite of hadronic models. Our analysis based on the minimal model finds that smaller radii are possible. Further, we caution against using a probabilistic interpretation of the allowed ranges for  $R_{1.4}$ and  $\\Lambda_{1.4}$ because it is difficult to assign likelihoods to a specific realization of the EOS. The inclusion of strong phase transitions in \\cite{Most:2018hfd} allows for the existence of \"twin star\" solutions containing two separate stable branches of NSs. In this case, smaller values for  $R_{1.4}$ and  $\\Lambda_{1.4}$ are allowed and the constraints weaken to $R_{1.4}>8.53~\\text{km}$ and  $\\Lambda_{1.4}>35.5$. The results obtained using the maximal model (CSM) are in good agreement with these limits. \n\n\\section{Summary}\n\\label{sec:summary}\n\nTo summarize, we confronted the recent GW observation with modern nuclear-physics constraints from chiral EFT. We elaborated on our results of Ref.~\\cite{Tews:2018iwm} and provided many additional results.  \n\nIn particular, we have used two different classes of models to extend QMC results with chiral EFT interactions to higher densities encountered in the core of neutron stars. We have used a minimal model, that is based on a density expansion around saturation density, and a maximal model based on a very general expansion in the speed of sound, that explores all EOSs consistent with the low-density input from chiral EFT. We used these models to study the uncertainties for the EOS and neutron-star observables for chiral EFT input up to either $n_{\\rm sat}$ or $2 n_{\\rm sat}$.\n\nWe used these models with input from nuclear physics up to nuclear saturation density and data from GW170817 to deduce that the radius of a typical neutron star has to be $R_{1.4}\\leq 13.6$ km. If instead EFT predictions for the EOS are used up to twice nuclear saturation density we find that $\\tilde{\\Lambda}<580$ and $R_{1.4}\\leq 12.6$ km. These smaller ranges suggest that future observations need to provide much more precise constraints to enable conclusions about the EOS or provide evidence for novel phases of matter in neutron stars.  We compared our results to other recent works, which arrived at the opposite conclusion, and discussed the robustness of our main statement. \n\nWe studied the impact of additional constraints on our findings. Most of these additional constraints are derived from interpretations of the EM counterpart of GW170817, and provide limits on radii, tidal polarizabilities, or the maximum mass. We showed that constraints on the maximum mass do not reduce the EOS uncertainty for typical neutron stars, in contrast to radius information, which is rather valuable. We also investigated how an upper limit on the speed of sound in neutron stars affects our findings.\n\nWe finally investigated the impact of strong first-order phase transitions on our predictions. Contrasting the predictions of the MM and the CSM may provide useful insights on how future measurements of $\\tilde{\\Lambda}$ from neutron-star mergers can help to identify new forms of matter at densities beyond nuclear saturation. \n\nTo conclude, we pose the question if and when the accuracy of gravitational-wave observations will be sufficiently small to provide constraints on the EOS that are tighter than the ones from nuclear theory. From our results, we estimate that the uncertainty $\\tilde{\\Lambda}$ needs to be of the order of $\\Delta\\tilde{\\Lambda}<300$ to test the chiral EFT prediction in the density range $n_{\\rm sat}-2n_{\\rm sat}$. Based on the contrast between MM and CSM, we expect that $\\Delta\\tilde{\\Lambda}<100$ is needed to shed light on the possible existence of phase transitions in dense matter.\n\n\\begin{acknowledgement}\nThis work was supported in part by the U.S.~DOE under Grants \nNo.~DE-AC52-06NA25396 and DE-FG02-00ER41132, by the LANL LDRD program, and by the National Science Foundation Grant No.~PHY-1430152 (JINA Center for the Evolution of the Elements). \nJ.M. was partially supported by the IN2P3 Master Project MAC, \"NewCompStar\" COST Action MP1304, PHAROS COST Action MP16214.\nThis research used resources provided by the Los Alamos National\nLaboratory Institutional Computing Program, which is supported by the\nU.S. Department of Energy National Nuclear Security Administration under Contract No. 89233218CNA000001. Computational resources have been provided by the National Energy Research Scientific Computing Center (NERSC), which is supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-05CH11231. Computational resources have also been provided by the J\\\"ulich\nSupercomputing Center.\n\\end{acknowledgement}\n\n\\bibliographystyle{epj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the past few years, GAN \\cite{5} has been widely used in image generation, image inpainting, style transfer and super-resolution reconstruction, etc. However, with the great progresses in GAN, an essential problem has always been with us, that's mode collapse. This phenomenon heavily harms the diversity and quality of images generated by generator. In this paper, we mainly focus on mode collapse alleviation and aim to generate data in high diversity based on the available, and further, apply the augmentated data into downstream tasks for better performance.\n\n\\begin{figure}[t]\n\\begin{center}\n   \\includegraphics[width=1.0\\linewidth]{.\/figure\/top_figure_color.png}\n\\end{center}\n\\caption{\\textbf{Domain translation with PDPM.} Both StarGAN and PDPM are trained 10 epochs. PDPM can transfer the facial expressions of attribute images to other faces while the vanilla StarGAN can not capture the change of eyes, this indicates PDPM converges much faster. Besides, it is clear to see that PDPM generates accurate attention masks of changes compared with the first column while StarGAN captures much more background.}\n\\label{fig:animation}\n\\end{figure}\n\nIn general, mode collapse usually manifests as that the trained generator can only generate images in some specific classes which really harms the data diversity. Currently, to the best of our knowledge, there are two main ways to alleviate mode collapse, modifying the architecture (or training method) of GANs or refining the loss function. The main drawback of the former is its poor generalization performance since it is effective just for some specific networks, for example, in Unrolled GAN \\cite{25}, the generator has to consider both its current state and the state of discriminator after $K$ iterations which is hard to apply to other models. By contrast, the latter method usually has better generalization ability, but it is difficult to design an universal module, for example, in DRAGAN \\cite{31} and MSGAN \\cite{24}, new penalty terms are introduced for improving data diversity, but in our experiments (see Figure \\ref{fig:men-women}), we notice these methods may generate some noisy pixels which harms the image quality. Besides, using multiple GANs can alleviate this problem to some extent, but due to its high cost, this method is rarely adopted in practice. Up to this day, most approaches of mode collapse alleviation start with the original data space while few methods deal with this problem via features of the fake images.\n\nMoreover, in our experiments (see Figure~\\ref{fig:dp-collapse-sample}), we notice an abnormal phenomenon that sometimes very different latent vectors may be mapped to similar images which is the essential characteristic of mode collapse. Besides, in traditional GANs, the images generated by generator are more like the combination of several images, and this usually leads to low image resolution and quality. In brief, the observations stated above are appearances of mode collapse, and  they indicatie the necessity of addressing this issue.\n\n\\begin{figure}[t]\n\\begin{center}\n   \\includegraphics[width=1.0\\linewidth]{.\/figure\/framework}\n\\end{center}\n\\caption{\\textbf{The proposed PDPM.} In the framework above, $f_1$ and $f_2$ are feature maps extracted from the discriminator, $z_1$ and $z_1$ are latent vectors while $g_1$ and $g_2$ are their corresponding fake images. $S(\\cdot)$ indicates the similarity measurement function. The key idea of PDPM is that the similarity relationship of fake images' features should be consistent with their corresponding latent vectors.}\n\\label{fig:proposed-framework}\n\\end{figure}\n\n\nTo alleviate the effects of mode collapse while avoiding the drawbacks of previous methods, a novel pluggable diversity penalty module is proposed in this paper, hereinafter, \\textbf{PDPM}. Figure~\\ref{fig:proposed-framework} shows the pipeline of our framework. Concretely, the more difference between latent vectors the more different their corresponding fake images should be, i.e., if two latent vectors are different, then PDPM enforces generator to generate two images with different features. Unlike current mainstream methods, PDPM performs constraints in feature space which is more robust than that in data space. In latent space, the similarity among latent vectors is given using Gram matrix. However, in data space, each image usually has a great amount of pixels which are uncessary for distinguishing, and in fact, ~\\cite{11} find that the feature representations can better describe an image than pixels. Thus, PDPM calculates the similarity of images via their corresponding feature maps. Besides, nonlinear mapping is performed for normalizing the similarity values. The key idea of PDPM  is that the similarity of feature pairs should be consistent with the similarity of their corresponding latent vector pairs. This paper has the following contributions: \n\\begin{itemize}\n\\item A novel block named PDPM is proposed to alleviate mode collapse in GAN. PDPM has better generalization ability compared with most current methods, it can be used in almost all GANs as a pluggable attachment. Besides, PDPM performs constraints in feature space which is more robust and has better pixel value stability;\n\n\\item PDPM has great transfer ability and low computation cost. It can be used in image generation, image data augmentation, domain translation and so on, this indicates PDPM is not sensitive to tasks. Besides, PDPM is almost parameters-free which has only one balance coefficient;\n\n\\item Compared with other complex methods, PDPM is effective yet easy to perform. The results in Figure~\\ref{fig:synthetic} on 2D synthetic dataset show that PDPM can help GAN capture much more modes effectively, and Figure~\\ref{fig:animation} also suggests PDPM has good performance in domain translation. In image data augmentation, PDPM introduces a markable accuracy improvement on ResNet. In image generation, PDPM outperforms MSGAN, WGAN\\_GP, WGAN\\_GP\\_MS and some other SOTA architectures both visually and quantitatively (IS and FID).\n\\end{itemize}\n\n\\section{Related Work}\n\n\\noindent\n\\textbf{Mode Collapse Reduction}\\quad For improving data diversity and stable training, researchers have done a lot of work. In Unrolled GAN~\\cite{25}, Metz \\emph{et al.} define the generator objective with respect to an unrolled optimization of the discriminator. The generator has to consider both its current state and the state of discriminator after $K$ iterations. This can lead to a better solution, but it is hard to apply to other models. In Energy-based GAN \\cite{32}, Zhao \\emph{et al.} use entropy to measure the diversity of images generated by generator while maintaining low energy state. In VEEGAN~\\cite{26}, Srivastava \\emph{et al.} introduce a variational principle for estimating implicit probability distributions which can help avoide mode collapse. Further, in PacGAN~\\cite{28}, Lin \\emph{et al.} let the discriminator make decisions based on multiple samples from the same class which can penalize generator with mode collapse. In BourGAN~\\cite{29}, Xiao \\emph{et al.} treat modes as a geometric structure of data distribution in a metric space which also leads to a better genertor. Recently, in MSGAN~\\cite{24}, Mao \\emph{et al.} modify the objective function for encouraging the generators to explore more minor modes in data space. By contrast, our proposed PDPM captures modes in feature space which is more robust than MSGAN.\\\\\n\n\\noindent\n\\textbf{Data Augmentation Learning}\\quad Image data augmentation has been proven to be effective in practice. In~\\cite{8}, data augmentation is used to reduce overfitting. Also in~\\cite{4}, Shorten \\emph{et al.} find that even simple techniques such as cropping, rotating and flipping can have markable effects on reducing overfitting. Currently, as \\cite{9}, image data augmentation mainly has three branches which are traditional transformations, generative methods and learning the augmentation. The former method has been well studied while the latter has very high computation cost like NAS in \\cite{33}. In generative models, GAN is the representative, but it is rarely used due to the limited diversity of generated data caused by mode collapse. That's part of PDPM's motivation.\\\\\n\n\\noindent\n\\textbf{Convergence and Stability of GANs}\\quad The stability of training process and convergence speed are vital for GANs. In~\\cite{5}, the vanilla GAN is proposed for generating high-quality images, but at that time, it is not an easy task to train GAN stably due to the imbalance between generator and discriminator. Further, in~\\cite{6}, wasserstein distance is used to measure the similarity between distributions instead of KL-divergence, this reduces the difficulty of GAN training greatly, and then, in~\\cite{7}, the gradient penalty term is proposed to enforce the Lipschiz constraint instead of using weight clipping as WGAN. These difficulties in training GANs suggest refining loss function is not a trivial task, and indicate the necessity of evaluating convergence and stability of GANs. \\\\\n\n\\noindent\n\\textbf{Feature Representations of CNN}\\quad A deep convolutional layer can extract the feature of an input image accurately. In~\\cite{10}, deconvnet is used to visualize the features that a fully trained model has learned. Furthermore, in~\\cite{11}, Zhou \\emph{et al.} demonstrate that the convolutional neural networks are able to localize the discriminative regions of image. Based on this finding, we use the features extracted from discriminator to represent the images instead of using images directly. And in~\\cite{1}, the proposed Grad-CAM method also supports the results in ~\\cite{11}. Figure~\\ref{fig:grad-cam} in Appendix shows some Grad-CAM results on CelebA~\\cite{23} with our trained discriminator, and these results show PDPM can capture image features accurately.\\\\\n\n\n\\section{Motivation}\nTo alleviate mode collapse, generator must capture much more modes of the available data. As in Figure \\ref{fig:mode_collapse}, $z$ represents latent vectors while $f(z)$ represents the features of corresponding fake image. In feature space, mode collapse manifests as that only a part of modes can be captured by the generator like {\\tt\\small{the vanilla GAN}} group in Figure \\ref{fig:mode_collapse}. And as a result, the data generated by generator will gather in some specific classes or some typical features. Inspired by this, PDPM lets the generator to generate images with more discrete features first, and then, with the development of discriminator, PDPM's penalty term will make the latent vectors distribute around centers of feature modes mainly. The case that different latent vectors clustered around the similar feature by generator will result in higher penalty loss. In brief, PDPM makes the features of fake images much more discrete first ({\\tt\\small{PDPM starts}} in Figure \\ref{fig:mode_collapse}), and then with the help of discriminator and regularization term, to assign these latent vectors to different mode centers ({\\tt\\small{PDPM ends}} in Figure \\ref{fig:mode_collapse}). In next section, the pipeline of PDPM  and its mechanism will be given in detail. \n\n\\begin{figure}[htpb]\n\\begin{center}\n   \\includegraphics[width=1.0\\linewidth]{.\/figure\/mode_collapse}\n\\end{center}\n\\caption{\\textbf{Illustration of mode collapse in feature space.} The {\\tt\\small{feature modes distribution}} is the implicit real distribution of data, and {\\tt\\small{the vanilla GAN}} indicates the modes captured by generator without PDPM.}\n\\label{fig:mode_collapse}\n\\end{figure}\n\n\\section{Pluggable Diversity Penalty Module}\nAs introduced in previous sections, PDPM penalizes the generator if two different latent vectors are mapped to images with similar features. In this section, $G(\\cdot)$ and $D(\\cdot)$ are used to indicate generator and discriminator, $p_z$ and $p_r$ are distributions of latent vectors and the real data. Besides, fake images indicate the images generated by generator, and if not specified, $f$ is used to represent the features of fake images extracted from discriminator.\n\n\\subsection{Measurement of similarity}\nSuppose $p_{z}(z)$ is the distribution of latent vectors which follows a standard normal distribution, a batch vectors $\\{z_1, z_2, ..., z_m\\}$ are randomly sampled from $p_{z}(z)$, and then, the normalized Gram matrix can be shown as:\n\\begin{equation}\nG_z^*(i, j)=\\frac{z_i^Tz_j}{||z_i||_2\\cdot ||z_j||_2} \\label{eq:original-latent-sim}\n\\end{equation}\nwhere $\\|\\cdot\\|_2$ represents l2-norm. It is reasonable to suppose that $z_i$ and $z_j$ are independent identically distributed (\\emph{i.i.d.}), and in fact, $G_{z}^*$ still follows a gaussian distribution which can be derived from the following claim:\\\\[6pt]\n\\emph{$f(x)$ and $g(x)$ are Gaussian PDFs with means $\\mu_f$ and $\\mu_g$ and standard deviations $\\sigma_f$ and $\\sigma_g$, then the product of $p(x)$ and $g(x)$ follows a scaled Gaussian distribution with $\\mu=\\frac{\\mu_f\\sigma_g^2+\\mu_g\\sigma_f^2}{\\sigma_f^2+\\sigma_g^2}$ and $\\sigma=\\sqrt{\\frac{\\sigma_f^2\\sigma_g^2}{\\sigma_f^2+\\sigma_g^2}}$. The scale factor is \\rm $s=\\frac{\\sigma_f^2\\sigma_g^2}{\\sqrt{\\sigma_f^2+\\sigma_g^2}}\\mathbf{exp}\\left [ -\\frac{(x-\\mu)^2}{2\\sigma^2} \\right ]$.}\\\\\n\nLikewise, the similarity of feature pairs can be got as follows:\n\\begin{equation}\nG_f^*(i, j)=\\frac{f_i^Tf_j}{||f_i||_2\\cdot ||f_j||_2} \\label{eq:original-feature-sim}\n\\end{equation}\nwhere $f_i$ represents flattened feature map of the \\emph{i-th} fake image extracted from discriminator. Since the values in Eq (\\ref{eq:original-latent-sim}) and Eq (\\ref{eq:original-feature-sim}) can be zero or negative, performing division directly doesn't make sense, and thus, the \\emph{sigmoid} function is used to scale them.\nThe scale factor is denoted by $s$ and Eqs (\\ref{eq:original-latent-sim}) and (\\ref{eq:original-feature-sim}) can be revised as:\n\\begin{equation}\nG_z(i, j)=\\sigma(s\\frac{z_i^Tz_j}{||z_i||_2\\cdot ||z_j||_2}) \\label{eq:revised-latent-sim}\n\\end{equation}\n\n\\begin{equation}\nG_f(i, j)=\\sigma(s\\frac{f_i^Tf_j}{||f_i||_2\\cdot ||f_j||_2}) \\label{eq:revised-feature-sim}\n\\end{equation}\n\n\\subsection{Loss function}\nFor alleviating mode collapse, the diversity penalty module should have the following attributes:\n\n\\begin{itemize}\n\\item if two latent vectors are similar, their corresponding fake images are unlikely to be very different.\n\\item if two latent vectors are different, their corresponding images have to be different likewise, which means the corresponding feature maps exist much difference.\n\\end{itemize}\n\n\\noindent Obviously, the diversity penalty module should pay much attention to the second situation which often results in mode collapse. Through these observations, the diversity penalty module is designed as follows:\n\\begin{equation}\nDP(z)=\\frac{1}{m^2}\\sum_{i=1}^{m}\\sum_{j=1}^{m}\\frac{G_f(i,j)}{G_z(i,j)}\\label{eq:dp-z}\n\\end{equation}\nwhere \\emph{m} represents the batch size, and $DP(z)$ has to be minimized when training GANs. Taking the vanilla GAN for example, $D(\\cdot)$ is trained to maximize the probability of assigning the correct label to both real images and fake images, also, $G(\\cdot)$ is trained simultaneously to get high score from $D(\\cdot)$. Thus, the basic loss function of GAN can be formulated as follows:\n\\begin{align}\n&\\max_{G}  L_G(z)= \\mathbb{E}_{z\\sim p_{z}}D(G(z)) \\\\\n&\\min_{D} L_D(z, x)=\\mathbb{E}_{z\\sim p_{z}}D(G(z))-\\mathbb{E}_{x\\sim p_r}D(x) \\\\\n&\\max_{G}\\min_{D}\\,\\mathbb{E}_{z\\sim p_{z}}D(G(z))-\\mathbb{E}_{x\\sim p_r}D(x) \n\\end{align}\nTo perform diversity penalty, we just need to add diversity penalty loss to generator. The loss function of GAN with PDPM can be formulated as follows: \n\\begin{align}\n&\\max_{G}  L_G(z)= \\mathbb{E}_{z\\sim p_{z}}D(G(z)) - \\lambda \\mathbb{E}_{z\\sim p_{z}}DP(z)\\label{eq:gloss}\\\\\n&\\min_{D} L_D(z, x)=\\mathbb{E}_{z\\sim p_{z}}D(G(z))-\\mathbb{E}_{x\\sim p_r}D(x) \\label{eq:dloss}\\\\\n&\\max_{G}\\min_{D}\\,\\mathbb{E}_{z\\sim p_{z}}D(G(z))-\\mathbb{E}_{x\\sim p_r}D(x) - \\lambda \\mathbb{E}_{z\\sim p_{z}}DP(z) \\label{eq:total-loss}\n\\end{align}\nwhere $\\lambda$ is a balance coefficient of diversity penalty term. The loss function of discriminator remains unchanged.\nAccording to Eqs (\\ref{eq:gloss}), (\\ref{eq:dloss}) and (\\ref{eq:total-loss}), the training pipeline can be summarized in \\textbf{Algorithm 1}.\n\n\\subsection{Mechanism Explanations}\nWhen training GANs, the discriminator is usually trained $k$ times while generator is trained only once, and that means discriminator usually converges better than generator. At the begining of training, PDPM enforces generator to generate fake images with discrete features, and this makes it possible for generator to capture more feature modes  like {\\tt\\small{PDPM starts}} in Figure \\ref{fig:mode_collapse}. At that time, the discriminator is not well trained, and it dose not penalize the generator severely. Then, as the discriminator is trained better and better, it will enfoece the generator to map the latent vectors around peaks of the feature distribution like {\\tt\\small{PDPM ends}} in Figure \\ref{fig:mode_collapse}, and the case that latent vectors are mapped to the saddle of feature distribution will get low score from discriminator. Thus, when PDPM converges, most of the latent vectors will be mapped to the surroundings of feature modes while little vectors scatter around untypical feature centers.\n\n\\begin{algorithm}[h] \n\\label{algorithm-1}\n\\caption{\\small{GAN} with \\small{PDPM} training via mini-batch \\small{Adam}} \n\\begin{algorithmic}[1] \n\\FOR{total training \\emph{epochs}}\n\\FOR{\\emph{k} times} \n\\STATE  Sample a batch data from $p_z$ : $\\{z_1, z_2, ...,z_m\\}$; \n\\STATE  Sample a batch data from $p_r$ : $\\{x_1, x_2, ...,x_m\\}$; \n\\STATE  Update discriminator :\n\\STATE  \\qquad $\\theta_d\\leftarrow \\;\\theta_d-\\nabla_{\\theta_d}\\frac{1}{m}\\sum_{i=1}^{m}L_D(z_i, x_i)$\n\\ENDFOR \n\\STATE  Sample a batch data from $p_z$ : $\\{z_1, z_2, ...,z_m\\}$;\n\\STATE  Update generator :\n\\STATE  \\quad \\;\\; \\qquad $\\theta_g\\leftarrow \\;\\theta_g-\\nabla_{\\theta_g}\\frac{1}{m}\\sum_{i=1}^{m}L_G(z_i)$\n\\ENDFOR \n\\end{algorithmic} \n\\end{algorithm}\n\n\n\\section{Experiments}\n\n\\begin{comment}\n\\begin{itemize}\n\\item Whether the similarity measurement in feature space is valid or not ? How dose the convergence performance and training stability of PDPM ?\n\\item Whether PDPM outperforms other SOTA architectures both visually and quantitatively or not ?\n\\item Whether PDPM alleviates mode collapse effectively compared with other GANs specially designed or not?\n\\item How well dose PDPM perform in other tasks such as image generation, image data augmentation, domain translation and so on ?  \n\\end{itemize}\n\\end{comment}\nIn this section, PDPM is evaluated from several different views. First, in {\\tt\\small{Basic Attribute Evaluation of PDPM}} part, the feasibility analysis of similarity measurement and the convergence performance of PDPM will be talked in detail, and next, in {\\tt\\small{Ablation Study}} part, both visual and quantitative comparisons between PDPM and other typical architectures such as ALI~\\cite{27}, Unrolled GAN~\\cite{25}, VEEGAN~\\cite{26}, PacGAN~\\cite{28} and BourGAN~\\cite{29} on 2D Synthetic Datasets will be given, which indicates the efficiency of PDPM. Further, in {\\tt\\small{Multi Task Applications}}, PDPM is applied into domain translation, image generation, image data augmentation and other tasks, and the results show PDPM outperforms most mainstream GANs such as DCGAN \\cite{15}, WGAN\\_GP \\cite{7} and MSGAN \\cite{24}.\n\n\\subsection{Datasets}\nThe datasets used in our experiments are MNIST, Fashion-MNIST, CIFAR-10, CelebA and 2D Synthetic Datasets. For the first three datasets, only training set is used while no changes are made in testing set. In some tables, M, F-M and C10 are used to represent MNIST, Fashion-MNIST and CIFAR-10 respectively. \n\n\\subsection{Training Details}\nUnless specified, $\\mathbf{Adam}$ optimizer with $\\beta_1$=0.5 and $\\beta_2$=0.9 is used for training GANs, and $\\mathbf{SGD}$ optimizer with weight decay(1e-4) and momentum(0.9) is used for training ResNet. The initial learning rates are 1e-4 and 1e-3 for GANs and ResNet respectively. The traditional data augmentation methods contain {\\tt RandomHorizontalFlip, RandomVerticalFlip, RandomResizedCrop, RandomRotation} and {\\tt RandomColorJitter}.\n\n\\subsection{Basic Attribute Evaluation of PDPM}\nSimilarity measurement must has two basic characteristics:\n\\begin{itemize}\n\\item The similarity value should be higher within classes than between classes.\n\\item Visually similar images should be close in feature space.\n\\end{itemize}\nPicking Fashion-MNIST images as samples, the similarity values among different categories are calculated via their corresponding feature maps extracted from discriminator. Results shown in Figure~\\ref{fig:similarity-fm} tell that the similarity value is higher within classes than between classes.\n\\begin{figure}[htpb]\n\\begin{center}\n   \\includegraphics[width=0.71\\linewidth]{.\/figure\/similarity_fm_no_title}\n\\end{center}\n   \\caption{\\textbf{Similarity Analysis on Fashion-MNIST.} \nFor avoiding occasionality, 5k images are sampled per class. The similarity is computed via Eq (\\ref{eq:revised-feature-sim}).} \n\\label{fig:similarity-fm}\n\\end{figure}\n\nFurther, similar operation is performed within one specific class on Fashion-MNIST to verify the second character stated above. Results are attached in Appendix Figure~\\ref{fig:fm-one-class}, they confirm that visually similar images are also similar in feature space and vice versa. These statistical results verify the reasonability of our similarity measurement.\n\nBesides, for GANs, whether it can converge stably or not is vital, and thus, the evaluation of PDPM on MNIST, Fashion-MNIST, CIFAR-10 and CelebA is made, respectively. Architectures of GAN are contained in Appendix Table~\\ref{architecture}. Figure \\ref{fig:gloss} gives the convergence results of domain translation on CelebA, the detailed results of other datasets are attached in Appendix Figure \\ref{fig:convergence}. In domain translation, StarGAN \\cite{30} is set as baseline, two groups with PDPM are set for comparison. From the results shown in Figure \\ref{fig:gloss}, it is clear to see that PDPM can accelerate the convergence of generator significantly. \n\n\\begin{figure}[htpb]\n\\begin{center}\n    \\includegraphics[width=0.45\\textwidth]{.\/figure\/gloss}\n\\end{center}\n    \\caption{\\textbf{Loss of generator.} The balance coefficient $\\lambda$ in PDPM is set to 1e-3 in StarGAN\\_PDPM 1 and 1e-4 in StarGAN\\_PDPM 2.}\n    \\label{fig:gloss}\n\\end{figure}\n\nThis acceleration is achieved because PDPM can capture accurate feature representations which are vital in facial expression transfer. Besides, in Figure \\ref{fig:animation}, the first column indicates the original images and their corresponding facial masks, the following columns are results of facial expression transfer and their corresponding attention masks. These attention masks should capture the changes between the image after transformation and the original. It can be seen that PDPM can generate much clearer facial attention masks with less background which can bring better and smoother detail changes compared with the vanilla StarGAN. Besides, StarGAN in Figure \\ref{fig:animation} can not transfer the changes of eyes to new face image, this indicates that when training with same epochs, the vanilla StarGAN converges much worse than PDPM.\n\n\n\\subsection{Ablation Study}\nIn this part, the effects of PDPM are stated in detail. First, in {\\tt\\small{Evaluation on Basic Datasets}}, both visual and quantitative results of mode collapse and mode collapse alleviation are given on MNIST, Fashion-MNSIT and CIFAR-10, further, in {\\tt\\small{Evaluation on 2D Synthetic Datasets}}, the comparison between some typical GANs with and without PDPM are given for precise comparison.\n\n\\subsubsection{Evaluation on Basic Datasets}\n\\begin{figure*}[t]\n\t\\subfigure[WGAN\\_GP without PDPM on MNIST]{\n        \\includegraphics[width=0.49\\textwidth]{.\/figure\/mode_collapse_mnist_samples}\n    }\n    \\subfigure[WGAN\\_GP with PDPM on MNIST]{\n        \\includegraphics[width=0.49\\textwidth]{.\/figure\/mode_collapse_dp}\n    }\n    \\caption{\\textbf{Alleviation of mode collapse via PDPM.} (a) WGAN\\_GP without PDPM. (b) WGAN\\_GP with PDPM $\\lambda$=5. The value above each image pair indicates the similarity value of their latent vectors. It can be found that in GAN without PDPM, latent vectors with low similarity value can be mapped to similar images while PDPM not.}\n    \\label{fig:dp-collapse-sample}\n\\end{figure*}\n\nIn vanilla GANs,the latent vectors even with very low similarity may be mapped to similar images, but with PDPM, this phenomenon is alleviated since this situation will result in higher loss. That is, PDPM makes similar fake images have corresponding latent vectors with higher similarity. Using the method shown in Appendix Figure \\ref{fig:mode-collapse-sample}, the similar images and their corresponding latent vectors can be got simultaneously with our trained generator. With these fake images and their latent vectors, the similarity value can be calculated via Eqs (\\ref{eq:revised-latent-sim}) and (\\ref{eq:revised-feature-sim}). Part of these results are shown in Figure \\ref{fig:dp-collapse-sample}, others are attached in Appendix. These results indicate that PDPM prevents the generator from mapping latent vectors with low similarity to similar fake images. \n\n\\begin{table}[htpb]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\caption{\\textbf{Statistic results of diversity penalty module.}}\n\\begin{tabular}{c|l|lll}\n\\hline\n\\multicolumn{2}{l|}{\\small{Dataset}}   & \\small{WGAN\\_GP}         & \\small{PDPM $\\lambda$=5}        & \\small{PDPM $\\lambda$=10}           \\\\ \\hline\n\\multirow{10}{*}{\\small{M \/\/ FM}} & 1  & 0.68 \/\/ 0.65 & 0.78 \/\/ 0.82 & 0.82 \/\/ 0.84 \\\\\n                          & 2  & 0.67 \/\/ 0.63 & 0.66 \/\/ 0.77 & 0.65 \/\/ 0.85 \\\\\n                          & 3  & 0.64 \/\/ 0.63 & 0.77 \/\/ 0.82 & 0.77 \/\/ 0.89 \\\\\n                          & 4  & 0.69 \/\/ 0.61 & 0.80 \/\/ 0.83 & 0.78 \/\/ 0.81 \\\\\n                          & 5  & 0.64 \/\/ 0.66 & 0.78 \/\/ 0.84 & 0.77 \/\/ 0.81 \\\\\n                          & 6  & 0.68 \/\/ 0.64 & 0.78 \/\/ 0.83 & 0.78 \/\/ 0.81 \\\\\n                          & 7  & 0.64 \/\/ 0.63 & 0.75 \/\/ 0.84 & 0.77 \/\/ 0.86 \\\\\n                          & 8  & 0.61 \/\/ 0.66 & 0.74 \/\/ 0.86 & 0.75 \/\/ 0.87 \\\\\n                          & 9  & 0.67 \/\/ 0.64 & 0.80 \/\/ 0.79 & 0.80 \/\/ 0.83 \\\\\n                          & 10 & 0.64 \/\/ 0.65 & 0.76 \/\/ 0.84 & 0.75 \/\/ 0.82 \\\\ \\hline\n\\end{tabular}\n\\label{fig:tab-statistics}\n\\end{table}\n\nFurther, to avoid occasionality, 5k similar fake image pairs per class are generated by the generator with and without PDPM, and Eq (\\ref{eq:revised-latent-sim}) is used for calculating similarity between latent vectors. In Table \\ref{fig:tab-statistics}, the value indicates the similarity of latent vector pairs whose corresponding fake images are similar under MSE metrics. It can be seen that PDPM reduces the chance of two different latent vectors mapped to similar fake images.\n\n\n\n\\begin{comment}\n\\begin{figure}[htpb]\n\\begin{center}\n    \\includegraphics[width=0.45\\textwidth]{.\/figure\/sta}\n\\end{center}\n    \\caption{\\textbf{Statistic result of diversity penalty.} The diversity penalty significantly improves the similarity value of similar images' corresponding latent vectors, and that means, it reduces the chance of two very different latent vectors mapped to similar images.$\\lambda_2$ indicates the coefficient of diversity penalty term.}\n    \\label{fig:dp-statistics}\n\\end{figure}\n\\end{comment}\n\nIn GANs, IS~\\cite{18,21} and FID~\\cite{19} are commonly accepted metrics used for evaluationg the quality and diversity of fake images. On the datasets stated above, 5k fake images per class are generated using the generator with and without PDPM for calculating IS and FID. The parameter $n_{splits}$ of IS is set to 10. Table \\ref{tab:score} shows the details. Greater IS value and lower FID value are signs of high quality and diversity of generated data.\n\n\\begin{table}[htpb]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\begin{spacing}{1.2}\n\\caption{\\textbf{Quantitative results of IS and FID.}}\n\\label{tab:score}\n\\begin{tabular}{l|l|lll}\n\\hline\n\\multicolumn{2}{l|}{\\small{Dataset}}         & \\small{WGAN\\_GP} & \\small{PDPM $\\lambda$=5} & \\small{PDPM $\\lambda$=10} \\\\ \\hline\n\\multirow{2}{*}{\\small{M}}         & $\\uparrow$IS  & 2.18$\\pm$.003  & 2.19$\\pm$.005 & \\textbf{2.31$\\pm$.005}       \\\\\n                               & $\\downarrow$FID & 7.36$\\pm$.012      & 6.43$\\pm$.009      & \\textbf{5.88$\\pm$.011}       \\\\ \\hline\n\\multirow{2}{*}{\\small{FM}} & $\\uparrow$IS  & 4.28$\\pm$.004        & \\textbf{4.38$\\pm$.006}       & 4.36$\\pm$.005       \\\\\n                     & $\\downarrow$FID & \\textbf{15.68$\\pm$.007}  & 15.97$\\pm$.013   & 15.72$\\pm$.011      \\\\ \\hline\n\\multirow{2}{*}{\\small{C10}}      & $\\uparrow$IS  & 7.35$\\pm$.007      & 7.52$\\pm$.005       & \\textbf{7.83$\\pm$.007}       \\\\\n                               & $\\downarrow$FID & 29.84$\\pm$.017       & \\textbf{28.45$\\pm$.015} & 29.03$\\pm$.013 \\\\ \\hline\n\\multirow{2}{*}{\\small{CelebA}}      & $\\uparrow$IS  & 2.78$\\pm$.002      & 2.91$\\pm$.005       & \\textbf{2.94$\\pm$.002}       \\\\\n                               & $\\downarrow$FID & 33.48$\\pm$.002       & 25.45$\\pm$.015 &  \\textbf{24.86$\\pm$.002} \\\\ \\hline\n\\end{tabular}\n\\end{spacing}\n\\end{table}\n\n\\subsubsection{Evaluation on 2D Synthetic Datasets}\nOn synthetic dataset, the quantitative evaluation results of mode collapse can be got accurately, because the distribution of data and its modes are known. As prior works, GANs with and without PDPM are evaluated on \\textbf{2D Ring} and \\textbf{2D Grid}. 2D Ring dataset contains eight 2D Gaussian distributions whose centers locate on a ring equally. 2D Grid contains twenty-five 2D Gaussian distributions whose centers locate on the meshgrid of a square. For comparison, PDPM is applied into the vanilla GAN, Unrolled GAN and BourGAN. The number of modes captured by generator and the percentage of points generated by generator in high-quality (h-q) are used as metrics. As in \\cite{26}, a sample is counted as high quality, if it is within three standard deviations of the nearest mode, and the number of modes captured by generator is the number of Gaussian centers which are nearest to at least one high quality sample.\n\n\\begin{table}[htpb]\n\\begin{spacing}{1.15}\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\caption{\\textbf{Quantitative results on 2D Synthetic Dataset.}}\n\\label{tab:synthetic_table}\n\\begin{tabular}{lllll}\n\\hline\n                 & \\multicolumn{2}{l}{2D Ring} & \\multicolumn{2}{l}{2D Grid} \\\\\n                 & modes        & h-q          & modes        & h-q          \\\\ \\hline\nGAN              & 1.0          & $\\times$        & 17.7         & 82.3         \\\\\nALI        & 2.8        & 0.13            & 12.8        & 1.6          \\\\\nUnrolled GAN     & 7.6          & 87.97        & 14.9         & 4.89         \\\\\nVEEGAN          & 8.0          & 86.77        & 24.4         & 77.16        \\\\\nPacGAN           & 7.8          & 98.21        & 24.3         & 79.46        \\\\\nBourGAN          & 8.0          & 99.76        & 25.0         & 95.91        \\\\ \\bottomrul\nGAN\\_PDPM          & 2.0          & $\\times$        & 21.3         & 80.8         \\\\\nUnrolled\\_PDPM & 8.0          & 99.36        & 21.7         & 75.21        \\\\\nBourGAN\\_PDPM      & 8.0          & 99.89        & 25.0         & 95.99        \\\\ \\bottomrul\n\\end{tabular}\n\\end{spacing}\n\\end{table}\n\n\\begin{figure*}[htpb]\n\\begin{center}\n   \\includegraphics[width=.98\\linewidth]{.\/figure\/synthetic.png}\n\\end{center}\n   \\caption{\\textbf{Visual results on Synthetic Dataset.} From the first two columns, it can be seen that PDPM help the vanilla GAN capture more modes, especially in 2D Grid, the GAN with PDPM captures four more modes than its vanilla counterpart.}\n\\label{fig:synthetic}\n\\end{figure*}\n\nFrom visual results in Figure~\\ref{fig:synthetic} and quantitative results in Table~\\ref{tab:synthetic_table}, it can be seen that GAN with PDPM captures more modes of the data distribution, and the vanilla GAN group with PDPM outperforms the ALI and Unrolled GAN on 2D Grid Dataset while closer to VEEGAN and PacGAN. Besides, from BourGAN and BourGAN\\_PDPM in Figure \\ref{fig:synthetic}, it is clear to see that the group with PDPM converges to the mode centers better than its vanilla counterpart. The results of Unrolled GAN, VEEGAN and PacGAN are from \\cite{29}, no official codes of VEEGAN and PacGAN are found until we finish this part, thus, PDPM is not applied into these GANs. \n\n\n\\subsection{Multi Task Applications}\nIn this part, PDPM is applied into image data augmentation, image generation and domain translation, results in these tasks all suggest that GANs with PDPM outperform their vanilla counterpart.\n\n\\subsubsection{Image Generation on CelebA}\nThe GANs are split into two groups which are DCGAN series with \\{DCGAN, DCGAN\\_MS, DCGAN\\_PDPM\\} and WGAN\\_GP series with \\{WGAN\\_GP, WGAN\\_GP\\_MS, WGAN\\_GP\\_PDPM\\}. Here MS represents the mode seeking regularization propsed in MSGAN. The coefficient $\\lambda_{ms}$ is set to 1, and the penalty coefficient $\\lambda$ of PDPM shown in Eq (\\ref{eq:total-loss}) is set to 10. $\\mathbf{Adam}$ with $\\beta_1$=0.5 and $\\beta_2$=0.9 is used as optimizer, and the learning rate is set to 1e-4. All GANs are trained with a batch size of 128 and 100 epochs in total. The details of architectures are attached in Appendix Table \\ref{gans-celeba}. Figure \\ref{fig:men-women} shows the results of linear interpolation in latent space, and Table \\ref{is-fid-celeba} gives the quantitative results of IS and FID. In Figure \\ref{fig:men-women}, MS group generates many noisy pixels, and the transition between images is not smooth since the man with glasses only appears in last two images. By contrast, PDPM can interpolate between two images without noisy pixels, and the transition is much more smoother.\n\n\\begin{figure}[htpb]\n\\begin{center}\n    \\includegraphics[width=0.45\\textwidth]{.\/figure\/man_mark}\n\\end{center}\n    \\caption{\\textbf{Linear interpolation in latent space.} From (1) to (6) are WGAN\\_GP, WGAN\\_GP\\_MS, WGAN\\_GP\\_PDPM, DCGAN, DCGAN\\_MS and DCGAN\\_PDPM. The MS group in blue box generates some noisy pixels while PDPM group in purple box not.}\n    \\label{fig:men-women}\n\\end{figure}\n\nIt can be seen that PDPM also gets higher IS value and lower FID value compared with DCGAN, WGAN\\_GP and MSGAN from quantitative results shown in Table \\ref{is-fid-celeba}.\n\n\\begin{table}[htpb]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\begin{spacing}{1.2}\n\\caption{\\textbf{IS and FID results on CelebA.}} \n\\label{is-fid-celeba}\n\\begin{tabular}{lllll}\n\\hline\n\\multicolumn{2}{l}{} &\\small{DCGAN}    &\\small{DCGAN\\_MS}    &\\small{DCGAN\\_PDPM}    \\\\ \\cline{3-5} \n\\multicolumn{2}{l}{$\\uparrow$\\small{IS}}      &2.113$\\pm$ 0.014  &2.360$\\pm$0.006  & \\textbf{2.379$\\pm$0.013}              \\\\\n\\multicolumn{2}{l}{$\\downarrow$\\small{FID}}     &24.23$\\pm$0.150 &23.51$\\pm$0.090 & \\textbf{21.76$\\pm$0.110}             \\\\ \\hline\n\\multicolumn{2}{l}{}        &\\small{WGAN\\_GP} & \\small{WGAN\\_GP\\_MS} & \\small{WGAN\\_GP\\_PDPM} \\\\ \\cline{3-5} \n\\multicolumn{2}{l}{$\\uparrow$\\small{IS}}      &2.775$\\pm$0.018    & 2.927$\\pm$0.016 & \\textbf{2.941$\\pm$0.021} \\\\\n\\multicolumn{2}{l}{$\\downarrow$\\small{FID}}     &33.48$\\pm$0.011 & 24.86$\\pm$0.020 & \\textbf{24.18$\\pm$0.031}\\\\ \\hline\n\\end{tabular}\n\\end{spacing}\n\\end{table}\n\n\n\\subsubsection{Image Data Augmentation with PDPM}\nGANs with PDPM are used for augmentating data on MNIST, Fashion-MNIST and CIFAR-10. The fake images are served as auxiliary training set. ResNet20 proposed in \\cite{20} is adopted as classification net. $\\mathbf{SGD}$ optimizer is used with learning rate decay.\nResults of accuracy on testing set are shown in Table~\\ref{acc-table}, training details are attached in Appendix Figure~\\ref{fig:resnet-acc}.\n\n\\begin{table}[htpb]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\begin{spacing}{1.2}\n\\caption{\\textbf{Testing Accuracy on Several Datasets.}}\n\\label{acc-table}\n\\begin{tabular}{l|lll}\n\\hline\nTesting Acc       & MNIST & Fashion-MNIST & CIFAR-10 \\\\ \\hline\nBaseline          & 0.9897    & 0.9257           & $\\times$         \\\\\nDA & $\\times$    & $\\times$               & 0.9172         \\\\\nWGAN\\_GP          & 0.9951     & 0.9394            & 0.9184        \\\\\nWGAN\\_GP\\_MS          & 0.9961     & 0.9430            & 0.9200        \\\\\nPDPM\\_1 $\\lambda=5$          & \\textbf{0.9975}     &0.9465             & \\textbf{0.9239}        \\\\\nPDPM\\_2    $\\lambda=10$       & 0.9969     & \\textbf{0.9527}            & 0.9212        \\\\ \\hline\n\\end{tabular}\nDA : Traditional Data Augmentation\n\\end{spacing}\n\\end{table}\n\nCompared with WGAN\\_GP, PDPM gains improvements of 0.24\\%, 1.33\\% and 0.55\\% on MNIST, Fashion-MNIST and CIFAR-10 respectively. More details about the training process refer to Appendix Figure~\\ref{fig:resnet-acc}.\n\n\n\\section{Conclusion}\nIn this paper, a pluggable block called diversity penalty module (PDPM) has been proposed to alleviate mode collapse in GAN. This penalty term is used to enforce the similarity between feature pairs to be consistent with that between latent vector pairs. The advantage of our proposed method is its generalization ability, it almost can be combined with all GANs in different architectures and vision tasks.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIf $X$ and $Y$ are finite $2$-complexes with $\\pi_1(X) \\cong \\pi_1(Y)$, then it is well-known that $X \\vee aS^2 \\simeq Y \\vee bS^2$ for some $a,b \\ge 0$ \\cite{Wh39}.\nThe set of homotopy types of finite $2$-complexes $X$ with fixed $\\pi_1(X) \\cong G$ can therefore be viewed as a tree (an acyclic graph) with edges between each $X$ and $X \\vee S^2$. This tree is graded by $\\chi(X)$, which has a minimum value $\\chi_{\\min}(G)$ and satisfies $\\chi(X \\vee S^2) = \\chi(X)+1$. \nEach $X$ is also homotopy equivalent to the presentation complex $X_{\\mathcal{P}}$ of some presentation $\\mathcal{P}$ of $G$.\n\nIn the 1960s-70s, the structure of this tree was investigated in a series of papers by Cockroft-Swan \\cite{CS61}, Dyer-Sieradski \\cite{DS73,DS75,SD79} and Dyer (see, for example, \\cite{Dy78,Dy79a,Dy79b}).\nHowever, it was not until 1976 that Dunwoody \\cite{Du76} and Metzler \\cite{Me76} independently found examples of finite $2$-complexes $X$, $Y$ such that $X \\vee S^2 \\simeq Y \\vee S^2$ but $X \\not \\simeq Y$.\nConversely, Browning \\cite{Br78} showed that, if $G$ is finite, then $\\chi(X) = \\chi(Y) >\\chi_{\\min}(G)$ implies $X \\simeq Y$ (see also \\cite{HK93}). \nThe tree of finite $2$-complexes $X$ with $\\pi_1(X) \\cong G$ finite is therefore of the form given in \\cref{figure:diagrams}a.\nThis raises the following question:\n\n\\begin{problem} \\label{problem:2-complexes}\nFor which $k$ do there exist homotopically distinct finite $2$-complexes $X_1$, $X_2$ such that $\\pi_1(X_i) \\cong G$ and $\\chi(X_i) = k + \\chi_{\\min}(G)$? {\\normalfont (see \\cref{figure:diagrams}b)}\n\\setcounter{thm}{\\value{thm}-1}\n\\end{problem}\n\nThere are two main approaches to this problem. The first is in the 1979 Problems List edited by Wall \\cite[Problem D5]{Wa79} and the second is from Dyer \\cite[Problem C]{Dy79a}.\n\\begin{clist}{(1)}\n\\item\nFind $X_1$ such that $\\chi(X_1) = k + \\chi_{\\min}(G)$ and $X_1 \\not \\simeq Y \\vee S^2$ for all finite $2$-complexes $Y$. We can then take $X_2 = X \\vee kS^2$ where $\\chi(X) = \\chi_{\\min}(G)$. \n\\item\nFind $X_1$, $X_2$ such that $X_1 \\vee (k+1)S^2 \\simeq X_2 \\vee (k+1)S^2$ but $X_1 \\vee kS^2 \\not \\simeq X_2 \\vee kS^2$.\n\\end{clist}\n\nThe examples of Metzler and Dunwoody showed that $k=0,1$ are possible respectively, though no examples were found for $k \\ge 2$.\nThis is, in part, owing to difficulties in related problems in algebra. For example, Dunwoody's construction involves realising non-free stably free $\\mathbb{Z} T$-modules of rank $k = 1$ as $\\pi_2(X)$ for $X$ a finite $2$-complex with $\\pi_1(X) \\cong T$  the trefoil group. However, there has previously been no known example of a stably free $\\mathbb{Z} G$-module of rank at least two \\cite[p623]{Jo12b}. \n\nOur main algebraic result is the construction of non-free stably free $\\mathbb{Z} G$-modules of arbitrary rank $k$. \nIn fact, we show that there are infinitely many stably free $\\mathbb{Z} G$-modules which are distinct even up to $\\Aut(G)$-isomorphism (see \\cref{subsection:Aut(G)}). Let $\\cd(G)$ denote the cohomological dimension of $G$.\n\n\\begingroup\n\\renewcommand\\thethm{\\Alph{thm}}\n\\begin{thm} \\label{thm:main-SF}\nFor all $k \\ge 1$, there exists a finitely presented group $G$ and infinitely many stably free $\\mathbb{Z} G$-modules of rank $k$ which are distinct up to $\\Aut(G)$-isomorphism. Furthermore, for all $d \\ge 2$, we can assume that $\\cd(G) = d$.\n\\end{thm}\n\\endgroup\n\nOur simplest example is when $G = \\ast_{i=1}^k T$ is a free product of trefoil groups $T$, which has $\\cd(G) = 2$. \nHere the case $k=1$ was shown by Berridge-Dunwoody \\cite{BD79}. For $k \\ge 2$, the main idea of our proof will be to use Bergman's theorem on modules over coproducts of rings \\cite{Be74} in the case of $\\mathbb{F}[\\ast_{i=1}^k T]$-modules for $\\mathbb{F}$ a field.\nThis strategy was proposed by Evans in \\cite{Ev99}, though an example was never given.\n\n\\begin{figure}[t] \\vspace{-4mm} \n\\begin{center}\n\\begin{tabular}{ccccc}\t\n\\begin{tabular}{l}\n\\begin{tikzpicture}\n\\draw[fill=black] (2,0) circle (2pt);\n\\draw[fill=black] (3,0) circle (2pt);\n\\draw[fill=black] (2,1) circle (2pt);\n\\draw[fill=black] (2,2) circle (2pt);\n\\draw[fill=black] (2,3) circle (2pt);\n\\node at (2,3.6) {$\\vdots$};\n\\node at (1.2,3) {(a)};\n\n\\end{tabular}\n&&&&\t\n\\begin{tabular}{l}\n\\begin{tikzpicture}\n\\draw[fill=black] (2,-1) circle (2pt);\n\\draw[fill=black] (2,0) circle (2pt);\n\\draw[fill=black] (1,1) circle (2pt);\n\\draw[fill=black] (2,1) circle (2pt);\n\\draw[fill=black] (3,1) circle (2pt);\n\\draw[fill=black] (2,2) circle (2pt);\n\n\\node at (2,2.6) {$\\vdots$};\n\\draw[black] (2,0) node[right]{\n$\\left.\n    \\begin{array}{ll}\n           \\\\ \\\\ \\\\\n    \\end{array}\n\\right \\} k$};\n\\node at (1.2,2) {(b)};\n\n\\end{tabular}\n\\end{tabular}\n\\end{center}\n\\caption{Branching phenomena in the tree of finite $2$-complexes $X$ with $\\pi_1(X) \\cong G$. The vertical height is the Euler characteristic.} \\label{figure:diagrams}\n\\vspace{-2mm}\n\\end{figure}\n\nNow recall that \\cref{problem:2-complexes} has a natural analogue in higher dimensions.\nFor $n \\ge 2$, a \\textit{$(G,n)$-complex} is an $n$-complex $X$ with $\\pi_1(X) \\cong G$ and such that $\\widetilde X$ is $(n-1)$-connected.\nIf $X$, $Y$ are finite $(G,n)$-complexes, then $X \\vee aS^n \\simeq Y \\vee bS^n$ for some $a,b \\ge 0$, and $\\chi(X)$ has a minimal value $\\chi_{\\min}(G,n)$ among finite $(G,n)$-complexes.\nThe natural extension of \\cref{problem:2-complexes} to $(G,n)$-complexes was considered by Dyer in \\cite{Dy79a,Dy79b}.\nHowever, there were still no known examples found for $k \\ge 2$.\n\nOur main topological result is the following, which gives a complete resolution of \\cref{problem:2-complexes} and its generalisation to $(G,n)$-complexes. \nThis corresponds to the first approach to \\cref{problem:2-complexes} and so answers both \\cite[Problem D5]{Wa79} and the more general question of Dyer \\cite[p378]{Dy79b} in the affirmative.  \nIn fact, we will show that infinitely many homotopically distinct $X_i$ exist for each $n$ and $k$.\n\n\\begingroup\n\\renewcommand\\thethm{\\Alph{thm}}\n\\begin{thm} \\label{thm:main}\nFor all $n \\ge 2$ and $k \\ge 0$, there exists a finitely presented group $G$ and  infinitely many homotopically distinct finite $(G,n)$-complexes $X_i$ such that $\\chi(X_i) = k + \\chi_{\\min}(G,n)$ and $X_i \\not \\simeq Y_i \\vee S^n$ for any finite $(G,n)$-complex $Y_i$.\n\\end{thm}\n\\endgroup\n\nFor $n = 2$, our simplest example is when $k \\ge 1$ and $G = \\ast_{i=1}^k T$. \nThe $X_i$ will be some infinite collection of finite $2$-complexes of the form $k X_{\\mathcal{P}_i} = \\bigvee_{j=1}^k X_{\\mathcal{P}_i}$ where\n\\[ \\mathcal{P}_i = \\langle x,y,a,b \\mid x^2=y^3, a^2=b^3, x^{2i+1}=a^{2i+1}, y^{3i+1}=b^{3i+1} \\rangle \\]\nare the presentations of Harlander-Jensen \\cite{HJ06a}.\nIf $k=0$, we will instead use the finite $2$-complexes $X_i$ constructed by Lustig in \\cite{Lu93}.\n\nWe will also use \\cref{thm:main} to show that syzygies $\\Omega_n^G(\\mathbb{Z})$ can have branching at all level $k \\ge 0$ (\\cref{cor:syzygies}). This gives some response to the remark made by Johnson \\cite[p.xiii]{Jo12a} that very little is known about the branching of $\\Omega_n^G(\\mathbb{Z})$ outside of the finite case where branching occurs only at the minimal level and level one.\n\nNow recall that, if $\\mathbb{Z} G$ is Noetherian of Krull dimension $d_G$ and $d = d_G +1$, then stably free $\\mathbb{Z} G$-modules of rank $\\ge d$ are free and, if $X_1$, $X_2$ are finite $(G,n)$-complexes with $\\chi(X_1) = \\chi(X_2) \\ge d + \\chi_{\\min}(G)$, then $X_1 \\simeq X_2$ (see \\cite{Ha19} for the case $n=2$). \nIf $G$ is polycyclic-by-finite, then $\\mathbb{Z} G$ is Noetherian and it is conjectured that these are the only such groups (see \\cite[p328]{KL19} for recent work). \nIf $\\mathbb{Z} G$ is not Noetherian, then such a bound $d$ can often still be found; for example, if $G$ is a free group, then we can take $d=0$ by results of Bass \\cite{Ba64} and Wall \\cite{Wa65}.\nHowever, there has been no known example of a group $G$ for which no such bound $d$ exists.\n\nOur next result will be to give an example of a group $G$ for which there exists non-free stably free $\\mathbb{Z} G$-modules of arbitrary rank.\nOur simplest example is $G = \\ast_{i=1}^\\infty T$.\n\n\\begingroup\n\\renewcommand\\thethm{\\Alph{thm}}\n\\begin{thm} \\label{thm:main-SF-further}\nThere exists a group $G$ such that, for all $k \\ge 1$, there are infinitely many stably free $\\mathbb{Z} G$-modules of rank $k$ which are distinct up to $\\Aut(G)$-isomorphism.\nFurthermore, for all $d \\ge 2$, we can assume that $\\cd(G) = d$.\n\\end{thm}\n\\endgroup\n\nIn all our examples, $G$ is not finitely generated and so there does not exist a \\textit{finite} $(G,n)$-complex and $\\chi(X)$ is not well-defined. \nWe nonetheless show the following.\n\n\\begingroup\n\\renewcommand\\thethm{\\Alph{thm}}\n\\begin{thm} \\label{thm:main-further}\nFor all $n \\ge 2$, there exists a group $G$ and an aspherical $(G,n)$-complex $Y$ such that, for all $k \\ge 1$, there are infinitely many homotopically distinct $(G,n)$-complexes $X_i$ with $X_i \\vee S^n \\simeq Y \\vee (k+1)S^n$.\n\\end{thm}\n\\endgroup\n\nFinally, we consider the question of whether or not the techniques used to prove Theorems \\ref{thm:main-SF} and \\ref{thm:main} can be applied to all groups of the form $G = \\ast_{i=1}^k G_i$ and finite $2$-complexes $X$ with $\\pi_1(X) \\cong G$.\nFor a field $\\mathbb{F}$, we show that $\\pi_2(X) \\otimes \\mathbb{F}$ is uniquely a direct sum of induced $\\mathbb{F} G_i$-modules provided it has no direct summand of the form $\\mathbb{F} G$ (Propositions \\ref{prop:existence}, \\ref{prop:uniqueness}). This makes it possible to distinguish finite $2$-complexes $X_1$, $X_2$ by distinguishing the component $\\mathbb{F} G_i$-modules. This works in the case $G_i = T$ but cannot work in many other cases such as if the $G_i$ are finite.\n\nIn contrast, we give examples to show that $\\pi_2(X)$ need not be a direct sum of induced $\\mathbb{Z} G_i$-modules (\\cref{thm:non-existence}) and that, if so, this decomposition need not be unique (\\cref{thm:non-uniqueness}). \nThis limits the potential to use the methods presented in this article to find a general cancellation theorem for finite $2$-complexes.\n\nWe will conclude this article with a list of five open problems. \n\n\\section{Preliminaries on $R G$-modules}\n\\label{section:RG-modules}\n\nLet $G$ be a group, let $R$ be a ring and let $R G$ denote the group ring of $G$ with coefficients in $R$. We will now develop the necessary preliminaries on $R G$-modules.\n\n\\subsection{Stably free $R G$-modules}\n\\label{subsection:stably-finite}\n\nFor a ring $R$, a finitely generated (left) $R$-module $S$ is \\textit{stably free} if there exists $n$, $m$ such that $S \\oplus R^n \\cong R^m$.\nIn order to have a well-defined notion of rank, certain conditions on $R$ must be imposed:\n\\begin{clist}{(I)}\n\\item For all $n, m$, $R^n \\cong R^m$ implies $n=m$ (\\textit{invariant basis number property})\n\\item For all $n, m$, $S \\oplus R^n \\cong R^m$ implies $n \\le m$ (\\textit{surjective rank property})\n\\item For all $n$, $S \\oplus R^n \\cong R^n$ implies $S=0$ (\\textit{stable finiteness property})\n\\end{clist}\nSuppose $R$ satisfies (I). If $S$ is a stably free $R$-module, then we can define the \\textit{rank} of $S$ to be $\\rank (S) = m-n$ for any $n$, $m$ such that $S \\oplus R^n \\cong R^m$. If $R$ satisfies (II), then $\\rank(S) \\ge 0$ for all $S$. If $R$ satisfies (III), then $S \\ne 0$ implies that $\\rank(S) \\ge 1$.\n\nIt is straightforward to see that (III) $\\Rightarrow$ (II) $\\Rightarrow$ (I). Conversely, examples were given by Cohn \\cite{Co66} to show that $(R \\ne 0)$ $\\not\\Rightarrow$ (I) $\\not\\Rightarrow$ (II) $\\not\\Rightarrow$ (III). Rings which satisfy (III) are also known as weakly finite and satisfy the equivalent condition that, for all $n$, one-sided inverses in $M_n(R)$ are two-sided, i.e. $uv=1$ if and only if $vu=1$.\n\nWe would now like to determine when conditions (I)-(III) hold for $R G$.\nThe following is a consequence of \\cite[Proposition 2.4, Theorem 2.6]{Co66}.\n\n\\begin{prop}\nLet $R$ be a commutative ring and let $G$ be a group. Then $R G$ has the surjective rank property, and hence also the invariant basis number property.\n\\end{prop}\n\nIt remains to determine when $R G$ is stably finite.\nIt was shown by Kaplansky \\cite{Ka72} that, if $\\mathbb{F}$ is a field of characteristic $0$, then $\\mathbb{F} G$ is stably finite for all groups $G$. This implies that $\\mathbb{Z} G$ is stably finite since $\\mathbb{Z} G \\subseteq \\mathbb{Q} G$.\nKaplansky conjectured that this holds for all fields $\\mathbb{F}$, but this remains open.\n\nThe best result for general fields $\\mathbb{F}$ is the following theorem of Elek-Szab\\'{o} \\cite{ES04}, which built upon earlier work of Ara, O'Meara and Perera \\cite[Theorem 3.4]{AOP02}. \n\n\\begin{thm} \\label{thm:ES}\nLet $\\mathbb{F}$ be a field and let $G$ be a sofic group. Then $\\mathbb{F} G$ is stably finite.\n\\end{thm}\n\nFor a definition of sofic, see \\cite[p430]{ES04}. For our purposes, it suffices to note that $G=1$ is sofic and that sofic groups are closed under direct\/free products, direct\/inverse limits, subgroups, and that the extension of an amenable group (see \\cite[p227]{AOP02}) by a sofic group is sofic. There is no known example of a non-sofic group.\n\nAll groups which will be considered in this article are sofic. We can therefore assume, when needed, that non-trivial stably free $\\mathbb{F} G$-modules have rank $\\ge 1$.\n\n\\subsection{$R G$-modules over free products}\n\\label{subsection:Bergman}\n\nFix groups $G_1, \\cdots, G_n$, let $G = \\ast_{k=1}^n G_k$ denote the free product and let $\\iota_k : G_k \\hookrightarrow G$ denote the inclusion map for each $k$.\n\nLet $R$ be a ring. If $M_k$ is an $R G_k$-module, then ${\\iota_k}_\\#(M_k) = R G \\otimes_{R G_k}  M_k$ is an $R G$-module. We say that an $R G$-module $M$ is \\textit{induced} if there exists $R G_k$-modules $M_k$ and an $R G$-module isomorphism\n\\[ M \\cong {\\iota_1}_\\#(M_1) \\oplus \\cdots \\oplus {\\iota_n}_\\#(M_n).\\]\n\nWe now define two special types of map between induced $R G$-modules.\nFirstly, if $M = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k)$ and $M' = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k')$ are induced $R G$-modules, then an $R G$-module homomorphism $f : M \\to M'$ is called an \\textit{induced homomorphism} if there exists $R G_k$-module homomorphisms $f_k : M_k \\to M_k'$ such that $f = \\oplus_{k=1}^n \\iota_*(f_k)$.\n\nNow, let $M = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k)$ be an induced $R G$-module and suppose there exists $a$ for which $M_a \\cong M_a' \\oplus R G_a$ for some $R G_a$-module $M_a'$. Then, for any $b \\ne a$, there is an isomorphism\n\\[f_{a,b} : {\\iota_a}_\\#(M_a' \\oplus R G_a) \\oplus {\\iota_b}_\\#(M_b) \\to {\\iota_a}_\\#(M_a') \\oplus {\\iota_b}_\\#(M_b \\oplus R G_b)\\]\ninduced by ${\\iota_a}_\\#(R G_a) \\cong R G \\cong {\\iota_b}_\\#(R G_b)$. \nWe define a \\textit{free transfer isomorphism} on $a,b$ to be the isomorphism $F_{a,b} : M \\to M'$ which extends $f_{a,b}$ by the identity map on the other components and where\n\\[ M' =  {\\iota_1}_\\#(M_1) \\oplus \\cdots \\oplus {\\iota_a}_\\#(M_a') \\oplus \\cdots \\oplus {\\iota_b}_\\#(M_b \\oplus R G_b) \\oplus \\cdots \\oplus  {\\iota_n}_\\#(M_n).\\]\n\nThe following can be viewed as a special case of Bergman's theorem on modules over coproducts of rings \\cite{Be74}. We now restrict to the case where $R = \\mathbb{F}$ is a field.\n\n\\begin{thm}[Bergman]\n\\label{thm:Bergman}\nLet $M$ be a finitely generated induced $\\mathbb{F} G$-module. Then:\n\\begin{clist}{(i)}\n\\item If $M' \\subseteq M$ is a submodule, then $M'$ is an induced $\\mathbb{F} G$-module.\n\\item If $M'$ is an induced $\\mathbb{F} G$-module, then $M \\cong M'$\nif and only if they are connected by a sequence of induced isomorphisms and free transfer isomorphisms.\n\\end{clist}\n\\end{thm}\n\nFor the convenience of the reader, we will briefly outline how this can be deduced from Bergman's results. Here will will use the terminology from \\cite[p1-4]{Be74}.\n\n\\begin{proof}[Proof \\normalfont (outline)]\nFirstly, note that $\\mathbb{F} G$ is a the coproduct of the $\\mathbb{F}$-rings $\\mathbb{F} G_k$ which are faithful since they come equipped with natural injections $\\iota_k : \\mathbb{F} G_k \\hookrightarrow \\mathbb{F} G$. \n\nPart (i) follows immediately from \\cite[Theorem 2.2]{Be74}. For part (ii), suppose $f: M \\to M'$ is an isomorphism of $\\mathbb{F} G$-modules. By \\cite[Theorem 2.3]{Be74}, and the remark on \\cite[p3]{Be74}, $f$ is the composition of induced isomorphisms, free transfer isomorphisms and transvections. Since transvections are module automorphisms, omitting them from the composition still leaves an isomorphism of $\\mathbb{F} G$-modules. \\end{proof}\n\n\\begin{corollary} \\label{cor:Bergman}\nLet $M = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k)$ be a finitely generated induced $\\mathbb{F} G$-module and suppose each $M_k$ has no direct summand of the form $\\mathbb{F} G_k$. Then:\n\\begin{clist}{(i)}\n\\item If $M' = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k')$ is an induced $\\mathbb{F} G$-module, then $M \\cong M'$ as $\\mathbb{F} G$-modules if and only if $M_k \\cong M_k'$ as $\\mathbb{F} G_k$-modules for all $k$.\n\\item $M$ has no direct summand of the form $\\mathbb{F} G$.\n\\end{clist}\n\\end{corollary}\n\n\\begin{proof}\nPart (i) follows from \\cref{thm:Bergman} (ii) since, if the $M_k$ have no direct summands of the form $\\mathbb{F} G_k$, then there are no free transfer isomorphisms by definition. \n\nTo see part (ii) note that, if $M \\cong M' \\oplus \\mathbb{F} G$, then $M' \\subseteq M$ is a submodule and so is an induced $\\mathbb{F} G$-module by \\cref{thm:Bergman} (i). If $M' = \\bigoplus_{k=1}^n {\\iota_k}_\\#(M_k')$, then $M \\cong {\\iota_1}_\\# (M_1' \\oplus \\mathbb{F} G_1) \\oplus \\bigoplus_{k=2}^n {\\iota_k}_\\#(M_k')$ which contradicts the result from (i).\n\\end{proof}\n\n\\subsection{$R G$-modules up to $\\Aut(G)$-isomorphism}\n\\label{subsection:Aut(G)}\n\nIf $M$ is an $R G$-module and $\\theta \\in \\Aut(G)$, then we can define $M_\\theta$ to be the $R G$-module with the same underlying abelian group as $M$ but with $G$-action given by $g \\cdot_{M_{\\theta}} m = \\theta(g) \\cdot_{M} m$ for $g \\in G$ and $m \\in M$. We say that $R G$-modules $M$ and $M'$ are \\textit{$\\Aut(G)$-isomorphic} if $M \\cong (M')_\\theta$ are isomorphic as $R G$-modules for some $\\theta \\in \\Aut(G)$.\nThis has a number of basic properties. In particular, if $M$ and $M'$ are $\\mathbb{Z} G$-modules and $\\theta \\in \\Aut(G)$, then $(M \\oplus M')_\\theta \\cong M_\\theta \\oplus (M')_\\theta$, and $(R G)_\\theta \\cong R G$ for all $\\theta \\in \\Aut(G)$. \n\nRecall that a subgroup $N \\subseteq G$ is \\textit{characteristic} if $\\theta(N) = N$ for all $\\theta \\in \\Aut(G)$. We also say that a surjective map $f: G \\twoheadrightarrow H$ is characteristic if $\\Ker(f) \\subseteq G$ is characteristic and, if so, then there is an induced map $\\bar{\\cdot} : \\Aut(G) \\to \\Aut(H)$.\n\nThe following is straightforward (see, for example, \\cite[Corollary 7.4]{Ni20a}).\n\n\\begin{prop} \\label{prop:modules-over-quotients}\nLet $G$ be a group, let $f: G \\twoheadrightarrow H$ be characteristic and let $\\bar{\\cdot} : \\Aut(G) \\to \\Aut(H)$ be the map induced by $f$. If $M$ is an $R G$ module and $\\theta \\in \\Aut(G)$, then\n$f_\\#(M_\\theta) \\cong (f_\\#(M))_{\\bar{\\theta}}$\nare isomorphic as $R H$-modules. \n\\end{prop}\n\nThe following will be of use in applying \\cref{prop:modules-over-quotients} to the case where $G$ is a free product. \nWe say that a group $G$ is \\textit{indecomposable} if it is non-trivial and $G \\cong G_1 \\ast G_2$ implies $G_1$ or $G_2$ is trivial.\n\n\\begin{prop} \\label{prop:free-product-char}\nLet $G = G_1 \\ast \\cdots \\ast G_n$ where each $G_k$ is indecomposable and not infinite cyclic. For each $k$, let $f_k : G_k \\twoheadrightarrow H_k$ be characteristic and such that, if $G_i \\cong G_j$, then $H_i \\cong H_j$ and $f_i$, $f_j$ differ by automorphisms of $G_i$, $H_i$.\n\nIf $f : G \\twoheadrightarrow H_1 \\ast \\cdots \\ast H_n$ is the map with $f \\mid_{G_k} = f_k$, then $f$ is characteristic.\n\\end{prop}\n\nOur proof will be a routine application of the following version of the Kurosh subgroup theorem \\cite[Theorem 5.1]{Ma77}.\n\n\\begin{thm}[Kurosh subgroup theorem]\nLet $G = G_1 \\ast \\cdots \\ast G_n$. If $H \\subseteq G$ is a subgroup, then\n\\[ H = F(X) \\ast (\\ast_{k=1}^n g_k H_k g_k^{-1})\\]\nwhere $F(X)$ is the free group on a set $X$, $H_k \\subseteq G_k$ is a subgroup and $g_k \\in G$.\n\\end{thm}\n\n\\begin{proof}[Proof of \\cref{prop:free-product-char}]\nLet $\\varphi \\in \\Aut(G)$. Then $\\varphi(G_k) \\subseteq G$ is indecomposable and not infinite cyclic and so, by the Kurosh subgroup theorem, we have $\\varphi(G_k) = g_{i_k} H_{i_k} g_{i_k}^{-1}$ for some subgroup $H_{i_k} \\subseteq G_{i_k}$. Since $\\varphi$ is an automorphism, we have:\n\\[ G = \\ast_{k=1}^n (g_{i_k} H_{i_k} g_{i_k}^{-1}) \\subseteq \\ast_{k=1}^n (g_{i_k} G_{i_k} g_{i_k}^{-1}) \\subseteq \\ast_{k=1}^n (g_k G_k g_k^{-1}) = G  \\]\nwhich implies that $H_{i_k} = G_{i_k}$ and that the $i_k$ are distinct.\n\nLet $N_k = \\Ker(f_k) \\subseteq G_k$ and note that $N = \\Ker(f)$ is generated by the subgroups $g N_k g^{-1}$ for $g \\in G$. If $\\varphi \\in \\Aut(G)$, then the above implies that $\\varphi \\mid_{G_k} = c_{g_{i_k}} \\circ \\varphi_{i,i_k}$ where $\\varphi_{i,i_k} : G_i \\to G_{i_k}$ is an isomorphism and $c_{g_{i_k}} : G_{i_k} \\to G$ is conjugation by $g_{i_k}$. Since $f_{i}, f_{i_k}$ differ by automorphisms of $G_i, G_{i_k}$, we have $\\varphi_{i,i_k}(N_i) = \\varphi_{i_k}(N_{i_k})$ for some $\\varphi_{i_k} \\in \\Aut(G_{i_k})$ and so $\\varphi_{i,i_k}(N_i) = N_{i_k}$ since $N_{i_k}$ is characteristic. Hence $\\varphi(g N_k g^{-1}) = (g g_{i_k}) N_{i_k} (g g_{i_k})^{-1} \\subseteq N$ and so $N$ is characteristic.\n\\end{proof}\n\n\\section{Groups of finite cohomological dimension}\n\\label{section:cd(G)}\n\nWe will now recall some basic facts about groups with finite cohomological dimension which are due to Serre \\cite{Se71}.\nA standard reference is the notes of Bieri \\cite{Bi81}.\n\nA group $G$ has \\textit{cohomological dimension $n$}, written $\\cd(G) = n$, if $n$ is the smallest integer for which there exists a projective resolution of $\\mathbb{Z} G$-modules of the form:\n\\[ 0 \\to P_n \\to \\cdots \\to P_1 \\to P_0 \\to \\mathbb{Z} \\to 0.\\]\nThis is equivalent to asking that $H^i(G;M)=0$ for all $i > n$ and all $\\mathbb{Z} G$-modules $M$ \\cite[Proposition 5.1(a)]{Bi81}.\nIf no such $n$ exists, then we take $\\cd(G) = \\infty$.\n\nA group $G$ is said to be \\textit{of type $\\FL$} if, for some $n \\ge 0$, there exists a resolution of finitely generated free $\\mathbb{Z} G$-modules of the form:\n\\[ 0 \\to F_n \\to \\cdots \\to F_1 \\to F_0 \\to \\mathbb{Z} \\to 0\\]\nThe following is \\cite[Propositions 1.5, 4.1(b)]{Bi81}.\n\n\\begin{prop} \\label{prop:FP+cd}\nLet $G$ be a group with $\\cd(G) = n$. If $G$ is of type $\\FL$, then there exists a resolution of finitely generated free $\\mathbb{Z} G$-modules of the form:\n\\[ 0 \\to F_n \\to \\cdots \\to F_1 \\to F_0 \\to \\mathbb{Z} \\to 0.\\]\n\\end{prop}\n\nWe now recall how these conditions are related under amalgamated free products and direct products. The following is \\cite[Proposition 2.13(a), Proposition 6.1]{Bi81}.\n\n\\begin{lemma} \\label{lemma:cd-amalg}\nLet $G = G_1 \\ast_H G_2$ for groups $G_1$, $G_2$ with a common subgroup $H$.\n\\begin{clist}{(i)}\n\\item If $G_1$, $G_2$ are of type $\\FL$ and $H$ is of type $\\FL$, then $G$ is of type $\\FL$\n\\item If $n = \\max\\{\\cd(G_1),\\cd(G_2)\\} < \\infty$ and  $\\cd(H) < n$, then $\\cd(G) = n$. \n\\end{clist}\n\\end{lemma}\n\nThe following is a consequence of more general results on group extensions which can be found in \\cite[Proposition 2.7, Theorem 5.5]{Bi81}.\n\n\\begin{lemma} \\label{lemma:cd-direct}\nLet $G = G_1 \\times G_2$ for groups $G_1$, $G_2$.\n\\begin{clist}{(i)}\n\\item If $G_1$, $G_2$ are of type $\\FL$, then $G$ is of type $\\FL$\n\\item If $\\cd(G_1), \\cd(G_2) < \\infty$, $G_1$ is of type $\\FL$ and $H^n(G_1;\\mathbb{Z} G_1)$ is $\\mathbb{Z}$-free for $n = \\cd(G_1)$, then $\\cd(G) = \\cd(G_1) + \\cd(G_2)$.\n\\end{clist}\n\\end{lemma}\n\nWe will now give a construction of groups which will be the basis for our examples in \\cref{thm:main-SF} in the case $d \\ge 3$. This is inspired by a construction of Lustig \\cite{Lu93}.\n\nLet $G$ be a group and let $m \\ge 2$ be an integer. Then define\n\\[ G_+ = ( G \\ast \\langle r \\mid \\hspace{-.8mm}-\\rangle )\/ [r^m,G], \\]\nwhich is isomorphic to $(G \\times \\langle q \\mid \\hspace{-.8mm}-\\rangle) \\ast_{\\langle q = r^{m} \\rangle} \\langle r \\mid \\hspace{-.8mm}-\\rangle$. For integers $m_1, \\cdots, m_{n-1} \\ge 2$, we can define $G_{(n)}$ inductively by letting $G_{(1)} = G$ and $G_{(i+1)} = (G_{(i)})_+$ for $i \\ge 1$. We will label the new generator by $r_i$.\nThe choice of $m_i \\ge 2$ will not matter for the purposes of this article; it suffices to consider the case $m_i=2$.\n\nLet $\\iota : G \\to G_{(n)}$ be the composition of the natural maps $G_{(i)} \\to G_{(i+1)}$ and let $f: G_{(n)} \\to G$ be the map which sends $r_i \\mapsto 1$ for each $i$. We have that $f \\circ \\iota = \\id_{G}$ and so $\\iota$ is injective, $f$ is surjective and $G$ is a retract of $G_{(n)}$.\n\n\\begin{prop} \\label{prop:group-construction}\nLet $n \\ge 1$ and let $G$ be a finitely presented group of type $\\FL$ with $\\cd(G) = d$. Then:\n\\begin{clist}{(i)}\n\\item $G_{(n)}$ is a finitely presented group of type $\\FL$ with $\\cd(G_{(n)})=n+d-1$\n\\item The map $f: G_{(n)} \\twoheadrightarrow G$ is characteristic.\n\\end{clist}\n\\end{prop}\n\nIn order to prove this, we will first need the following lemma. The proof is identical to the one given in \\cite[p174]{Lu93}.\n\n\\begin{lemma} \\label{lemma:group-construction}\nLet $G$ be a torsion free group and let $G_+ = (G \\times \\langle q \\mid \\hspace{-.8mm}-\\rangle) \\ast_{\\langle q = r^m \\rangle} \\langle r \\mid \\hspace{-.8mm}-\\rangle$ for some $m \\ge 2$. Then the map $f : G_+ \\twoheadrightarrow G$ which sends $r \\mapsto 1$ is characteristic.\n\\end{lemma}\n\n\\begin{proof}[Proof of \\cref{prop:group-construction}]\nIt is clear that $G_{(n)}$ is finitely presented. We now prove (i) by induction, noting that it is trivial in the case $n=1$.\n\nSuppose (i) holds for $n$ and note that $G_{(n+1)} \\cong (G_{(n)} \\times \\mathbb{Z}) \\ast_{\\mathbb{Z}} \\mathbb{Z}$. It is well known that $K(\\mathbb{Z},1) \\simeq S^1$ and so $\\mathbb{Z}$ is of type $\\FL$, $\\cd(\\mathbb{Z})=1$ and $H^1(\\mathbb{Z};\\mathbb{Z}[\\mathbb{Z}])=0$. By \\cref{lemma:cd-direct}, $G_{(n)} \\times \\mathbb{Z}$ is of type $\\FL$ and $\\cd(G_{(n)} \\times \\mathbb{Z}) = n+d$.  By \\cref{lemma:cd-amalg}, this implies that $G_{(n+1)}$ is of type $\\FL$ and $\\cd(G_{(n+1)})=n+d$ as required.\n\nSince $\\cd(G_{(n)}) < \\infty$, $G_{(n)}$ is torsion free for all $n$ \\cite[Proposition 4.11]{Bi81}. By \\cref{lemma:group-construction}, this implies that the map $f_{i+1}: G_{(i+1)} \\twoheadrightarrow G_{(i)}$, $r_{i+1} \\mapsto 1$ is characteristic for all $i \\ge 1$. Hence $f = f_n \\circ f_{n-1} \\circ \\cdots \\circ f_{2}$ is characteristic by composition.\n\\end{proof}\n\n\\section{Proof of \\cref{thm:main-SF}}\n\\label{section:proof-main-algebra}\n\nRecall that the trefoil group $T$ is defined as $\\pi_1(S^3 \\, \\setminus \\, N(K))$ where $N(K)$ is the knot exterior of the trefoil knot $K \\subseteq S^3$. It has presentation $\\mathcal{P} = \\langle x,y \\mid x^2 = y^3 \\rangle$.\n\nLet $T''$ denote the second derived subgroup of $T$, i.e. $T'' = (T')'$, and let $f: T \\twoheadrightarrow T\/T''$ be the quotient map.\nNote that $T\/T''$ is polycyclic since $(T\/T'')' \\cong \\mathbb{Z}^2$ and $(T\/T'')\/(T\/T'')' \\cong \\mathbb{Z}$.\nThe following was shown by P. H. Berridge and M. J. Dunwoody \\cite{BD79}, building upon previously work of Dunwoody \\cite{Du76}.\n\n\\begin{thm}[Berridge-Dunwoody] \\label{thm:BD}\nThere exists infinitely many rank one stably free $\\mathbb{Z} T$-modules $S_i$ for $i \\ge 1$ such that: \n\\begin{clist}{(i)}\n\\item $S_i \\oplus \\mathbb{Z} T \\cong \\mathbb{Z} T^2$\t.\n\\item There exists distinct primes $p_i$ for which $\\mathbb{F}_{p_i} \\otimes f_\\#(S_j) \\cong \\mathbb{F}_{p_i} [T\/T'']$ are isomorphic as $\\mathbb{F}_{p_i} [T\/T'']$-modules if and only if $i = j$.\n\\end{clist}\nIn particular, the $S_i$ are distinct up to $\\mathbb{Z} G$-module isomorphism.\n\\end{thm}\n\n\\begin{remark} \\label{remark:relation-module}\nFor $i \\ge 0$, let $M_i = \\Ker(\\cdot\\left(\\begin{smallmatrix} x^{2i+1}-1 \\\\ y^{3i+1}-1 \\end{smallmatrix}\\right) : \\mathbb{Z} T^2 \\twoheadrightarrow \\mathbb{Z} T)$ be the relation module for the generating set $\\{x^{2i+1},y^{3i+1}\\}$, which is a stably free $\\mathbb{Z} T$-module of rank one. It was shown in \\cite{BD79} that $\\mathbb{F}_p \\otimes f_\\#(M_i) \\cong \\mathbb{F}_p[T\/T'']$ as $\\mathbb{F}_p[T\/T'']$-modules if and only if $p \\mid i(i+1)$. \nThere exists integers $\\ell_i$ for $i \\ge 1$ and primes $p_i$ such that $p_i \\mid \\ell_j(\\ell_j+1)$ if and only if $i=j$, and so we can take $S_i = M_{\\ell_i}$ in \\cref{thm:BD}. It is not known whether or not the $M_i$ are all distinct up to $\\mathbb{Z} G$-module isomorphism.\n\\end{remark}\n\nFor the rest of this section, fix $k \\ge 1$ and $n \\ge 1$. Let $G = T_1 \\ast \\cdots \\ast T_k$ where $T_j \\cong T$ is the trefoil group and let $G_{(n)}$ be as defined in \\cref{section:cd(G)}.\nSince $T$ is a knot group, $T$ has type $\\FL$ and $\\cd(T) =2$ \\cite[p212]{Br82}.\nBy \\cref{lemma:cd-amalg} and \\cref{prop:group-construction}, this implies that $G_{(n)}$ has type $\\FL$ and $\\cd(G_{(n)}) = n+1$. The aim of the rest of this section will be to prove the following theorem which implies \\cref{thm:main-SF}.\n\n\\begin{thm} \\label{thm:main-SF-detailed}\nFor each $n \\ge 1$ and $1 \\le m \\le k$, there exists infinitely many stably free $\\mathbb{Z} G_{(n)}$-modules $\\widehat S_i$ for $i \\ge 1$ such that:\n\n\\begin{clist}{(i)}\n\\item $\\widehat S_i \\oplus \\mathbb{Z} G_{(n)} \\cong \\mathbb{Z} G_{(n)}^{m+1}$.\n\\item $\\widehat S_i$ has no direct summand of the form $\\mathbb{Z} G_{(n)}$.\n\\item The $\\widehat S_i$ for $i \\ge 1$ are distinct up to $\\Aut(G_{(n)})$-isomorphism of $\\mathbb{Z} G_{(n)}$-modules.\t\n\\end{clist}\n\\end{thm}\n\nNote that the case $m = k$ is sufficient to establish \\cref{thm:main-SF}. This result shows that the tree of stably free $\\mathbb{Z} G_{(n)}$-modules has branching at all ranks $1 \\le m \\le k$.\nWe do not know whether branching occurs at ranks $\\ge k+1$, even in the case $G = T$.\n\nIn order to prove \\cref{thm:main-SF-detailed}, we will begin with the following lemma.\n\n\\begin{lemma} \\label{lemma:f=char}\nLet $f_j : T_j \\twoheadrightarrow T_j\/(T_j)''$ be the quotient maps aand let\n\\[ f: G \\twoheadrightarrow (T_1\/T_1'') \\ast \\cdots \\ast (T_k\/T_k'') \\]\nbe the map induced by the $f_j$. Then $f$ is characteristic. \n\\end{lemma}\n\n\\begin{proof}\nFor any group $G$, it is well known that $G' \\subseteq G$ is characteristic and so $G'' \\subseteq G$ is characteristic also. Hence $f_j$ is characteristic for each $j$. Since $T$ is indecomposable and not infinite cyclic, $f$ is characteristic by \\cref{prop:free-product-char}.\n\\end{proof}\n\nFor simplicity, we will begin by proving \\cref{thm:main-SF-detailed} in the case $n=1$, i.e. where $G_{(n)}=G$.\nFrom now on, fix $1 \\le m \\le k$. For integers $i_1, \\cdots, i_m$, define\n\\[S_{i_1, \\cdots, i_m} = {\\iota_1}_\\#(S_{i_1}) \\oplus \\cdots \\oplus {\\iota_m}_\\#(S_{i_m})    \\]\nwhere $\\iota_j : T_j \\hookrightarrow G$ is the inclusion map.\nWe will now prove the following as a consequence of Bergman's theorem, which we will apply by using \\cref{cor:Bergman}.\n\n\\begin{prop} \\label{prop:SF-trefoil}\nFor integers $i_1, \\cdots, i_m$, we have:\n\\begin{clist}{(i)}\n\\item $S_{i_1, \\cdots, i_m} \\oplus \\mathbb{Z} G \\cong \\mathbb{Z} G^{m+1}$.\n\\item $S_{i_1,\\cdots, i_m}$ has no direct summand of the form $\\mathbb{Z} G$.\n\\item If $S_{i_1, \\cdots, i_m} \\cong S_{i_1',\\cdots, i_m'}$ are $\\Aut(G)$-isomorphic as $\\mathbb{Z} G$-modules then, as sets, we have $\\{i_1, \\cdots, i_m\\} = \\{i_1',\\cdots,i_m'\\}$.\n\\end{clist}\n\\end{prop}\n\n\\begin{proof}\nPart (i) is a straightforward consequence of \\cref{thm:BD} (i). \n\nLet $\\bar{G} = \\ast_{j=1}^n T_j\/T_j''$ and let $\\text{$\\bar{\\iota}_j$} : T_j\/T_j'' \\hookrightarrow \\bar{G}$ be inclusion. By \\cref{thm:BD} (ii), there exists $p$ such that $\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j}) \\not \\cong \\mathbb{F}_p[T_j\/T_j'']$ for all $j$. Fix $p$ and note that:\n\\[ \\mathbb{F}_p \\otimes f_\\#(S_{i_1,\\cdots, i_m}) \\cong \\textstyle \\bigoplus_{j=1}^m \\mathbb{F}_p \\otimes (f \\circ \\iota_j)_\\#(S_{i_j}) \\cong \\bigoplus_{j=1}^m \\text{$\\bar{\\iota}_j$}_\\#(\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j})) \\]\nis an induced $\\mathbb{F}_p \\bar{G}$ module. \nIn order to show that \\cref{cor:Bergman} applies, it remains to show that $\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j})$ has no direct summand of the form $\\mathbb{F}_p[T_j\/T_j'']$.\n\nIf $\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j}) \\cong S \\oplus \\mathbb{F}_p[T_j\/T_j'']$, then $S \\oplus \\mathbb{F}_p[T_j\/T_j'']^2 \\cong \\mathbb{F}_p[T_j\/T_j'']^2$.\nSince $T_j\/T_j''$ is polycyclic, it is amenable and so sofic. By \\cref{thm:ES}, $\\mathbb{F}_p[T_j\/T_j'']$ is stably finite and so $S= 0$. Hence $\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j}) \\cong \\mathbb{F}_p[T_j\/T_j'']$, which is a contradiction.\n\nTo show (ii) note that, if $S_{i_1,\\cdots, i_m}$ has a direct summand $\\mathbb{Z} G$, then $\\mathbb{F}_p \\otimes f_\\#(S_{i_1,\\cdots, i_m})$ has a direct summand $\\mathbb{F}_p \\bar{G}$. This contradicts \\cref{cor:Bergman} (ii).\n\nTo show (iii), suppose that $\\{i_1, \\cdots, i_m\\} \\ne \\{i_1',\\cdots,i_m'\\}$ as sets. By symmetry, we can assume that there exists $i_r' \\not \\in \\{i_1, \\cdots, i_m\\}$. Let $p = p_{i_r'}$ in the notation of \\cref{thm:BD}. By the argument above, $\\mathbb{F}_p \\otimes f_\\#(S_{i_1,\\cdots, i_m})$ has no direct summand of the form $\\mathbb{F}_p \\bar{G}$. On the other hand, $\\mathbb{F}_p \\otimes {f_r}_\\#(S_{i_r'}) \\cong \\mathbb{F}_p[T_r\/T_r'']$ which implies that\n\\begin{align*} \\mathbb{F}_p \\otimes f_\\#(S_{i_1',\\cdots, i_m'}) & \\cong \\textstyle \\bigoplus_{j=1, j \\ne r}^m \\text{$\\bar{\\iota}_j$}_\\#(\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j})) \\oplus \\mathbb{F}_p \\bar{G} \\\\\n& \\cong \\textstyle \\bigoplus_{j=1, j \\ne r}^{m-1} \\text{$\\bar{\\iota}_j$}_\\#(\\mathbb{F}_p \\otimes {f_j}_\\#(S_{i_j})) \\oplus \\mathbb{F}_p \\bar{G}^2 \\cong \\cdots \\cong \\mathbb{F}_p \\bar{G}^m.\t\n\\end{align*}\nIf $S_{i_1,\\cdots, i_m} \\cong S_{i_1',\\cdots, i_m'}$ are $\\Aut(G)$-isomorphic, then $S_{i_1,\\cdots, i_m} \\cong (S_{i_1',\\cdots, i_m'})_\\theta$ for some $\\theta \\in \\Aut(G)$. By \\cref{lemma:f=char}, $f$ is characteristic and so, by \\cref{prop:modules-over-quotients}, $f_\\#((S_{i_1',\\cdots, i_m'})_\\theta) \\cong (f_\\#(S_{i_1',\\cdots, i_m'}))_{\\bar{\\theta}}$ for some $\\bar{\\theta} \\in \\Aut(\\bar{G})$. In particular, we have:\n\\[ \\mathbb{F}_p \\otimes f_\\#(S_{i_1,\\cdots, i_m}) \\cong (\\mathbb{F}_p \\otimes f_\\#(S_{i_1',\\cdots, i_m'}))_{\\bar{\\theta}} \\cong (\\mathbb{F}_p \\bar G^m)_{\\bar{\\theta}} \\cong \\mathbb{F}_p \\bar{G}^m\\]\nwhich is a contradiction.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:main-SF-detailed}]\nLet $\\iota : G \\hookrightarrow G_{(n)}$ and $f: G_{(n)} \\twoheadrightarrow G$ be as defined in \\cref{section:cd(G)}. This satisfies $f \\circ \\iota = \\id_G$ and, by \\cref{prop:group-construction}, $f$ is characteristic. Define $\\widehat S_i = \\iota_\\#(\\widehat S_{i_1, \\cdots, i_m})$, where $i_j = i$ for all $j$. By \\cref{prop:SF-trefoil}, it is now straightforward to check that the $\\widehat S_i$ has the required properties.\n\\end{proof}\n\nWe conclude this section with extended remarks on \\cref{thm:main-SF} and  \\cref{thm:main-SF-detailed}.\n\n\\subsubsection{Relation modules}\n\nBy \\cref{remark:relation-module}, $S_i$ is the relation module for the generating set $\\{x^{2\\ell_i+1},y^{3\\ell_i+1}\\}$ of $T$. It follows that $S_{i_1, \\cdots, i_m}$ is the relation module for the generating set $\\{x_i^{2\\ell_i+1},y_i^{3\\ell_i+1}\\}_{i=1}^k$ of $G = T_1 \\ast \\cdots \\ast T_k$ where $T_i = \\langle x_i, y_i \\mid x_i^2 = y_i^3 \\rangle$.\n\n\\subsubsection{Change of field}\n\nIn the proof of \\cref{prop:SF-trefoil}, the $\\mathbb{Z} G$-modules were distinguished by passing to $\\mathbb{F}_p G$ for various $p$. An alternate approach is to instead pass to $\\mathbb{Q} G$ and use the results of Lewin \\cite{Le82}.\nThis has the advantage that $\\mathbb{Q} G$ is stably finite by results of Kaplansky, and so we need not rely on \\cref{thm:ES}. However, whilst non-free stably free $\\mathbb{Z} G$-modules can be detected on $\\mathbb{Q} G$ using \\cite{Le82}, it is not clear how one would detect infinitely many distinct stably free $\\mathbb{Z} G$-modules on $\\mathbb{Q} G$.\n\n\\subsubsection{Alternate constructions}\n\nThere are more ways to deduce \\cref{thm:main-SF} in the case $d \\ge 3$ from the case $d=2$.\nBy \\cref{prop:SF-trefoil} and the proof of \\cref{thm:main-SF-detailed}, it suffices to find a finitely presented group $G$ with $\\cd(G) = d$ and a characteristic quotient $f : G \\twoheadrightarrow \\ast_{i=1}^N T$ for some $N \\ge k$.\nTwo such constructions are as follows.\n\\begin{clist}{(1)}\n\\item \nLet $G = \\ast_{i=1}^r (\\ast_{j=1}^{n_i} T)_{(d-1)}$ where $1 \\le n_1 \\le \\cdots \\le n_r$ and $N = \\sum_{i=1}^r n_i$. Then $\\cd(G) = d$ and there is a characteristic quotient $f : G \\twoheadrightarrow \\ast_{i=1}^N T$.\nFor example, we can take $G = (\\ast_{i=1}^k T)_{(d-1)}$ as above, or $G = \\ast_{i=1}^k T_{(d-1)}$ (see \\cref{thm:main-SF-further-detailed}).\n\n\\item\nLet $G = (\\ast_{i=1}^N T) \\times \\Gamma$ where $\\Gamma$ is a finitely presented group with $\\cd(\\Gamma) = d-2$, $Z(\\Gamma)=1$ and which does not contain $\\ast_{i=1}^N T$ as a direct factor. By \\cref{lemma:cd-direct}, we have $\\cd(G) = d$. If $N \\ge 2$, then $Z(\\ast_{i=1}^N T)=1$ and it can be deduced from \\cite[Corollary 2.2]{Jo83} that $f: G \\twoheadrightarrow \\ast_{i=1}^N T$ is characteristic. \n\nFor example: If $d=3$, let $\\Gamma$ be a free group of rank $\\ge 2$.\nIf $d = 4$, let $\\Gamma$ be a surface group of genus $\\ge 2$.\nIf $d \\ge 5$, let $\\Gamma \\subseteq L$ be a cocompact torsion free lattice in a non-compact simple Lie group $L$ with dimension $d-2$ over its maximal compact subgroup. \nNote that there are infinitely many such $\\Gamma$ up to commensurability.\nI am indebted to F. E. A. Johnson for this observation.\n\\end{clist}\n\n\\section{Module invariants of CW-complexes}\n\\label{section:module-invariants}\n\nLet $X$ be a CW-complex and recall that its cellular chain complex $C_*(\\widetilde X)$ is a chain complex of free $\\mathbb{Z}[\\pi_1(X)]$-modules under the monodromy action. The chain homotopy type of $C_*(\\widetilde X)$ is a homotopy invariant for $X$ and so, for all $n$, the $\\mathbb{Z}[\\pi_1(X)]$-module $H_n(C_*(\\widetilde X))$ is also a homotopy invariant.\n\nIf $G$ is a group and $\\rho: \\pi_1(X) \\cong G$, then every $\\mathbb{Z}[\\pi_1(X)]$-module $M$ can be converted to a $\\mathbb{Z} G$-module with action $g \\cdot_{\\mathbb{Z} G} m := \\rho^{-1}(g) \\cdot_{\\mathbb{Z}[\\pi_1(X)]} m$ for $g \\in G$ and $m \\in M$. In this notation, $H_n(C_*(\\widetilde X))_\\rho$ is a $\\mathbb{Z} G$-module. We will denote this by $H_n(X;\\mathbb{Z} G)$ when $\\rho$ is understood. If $\\rho' : \\pi_1(X) \\cong G$ and $\\theta = \\rho \\circ (\\rho')^{-1} \\in \\Aut(G)$, then $H_n(C_*(\\widetilde X))_{\\rho'} \\cong (H_n(C_*(\\widetilde X))_\\rho)_\\theta$. In particular, the $\\Aut(G)$-isomorphism class of $H_n(X;\\mathbb{Z} G)$ is a homotopy invariant and is independent of the choice of $\\rho$.\n\nThe aim of this section will be to consider how $H_n(X;\\mathbb{Z} G)$ changes under wedge product. We will also give a mild variation of this invariant under group quotients.\n\n\\subsection{Homology of a wedge product}\n\\label{subsection:wedge}\n\nThe following is presumably well-known. However, we were not able to locate a suitable reference in the literature.\n\n\\begin{prop} \\label{prop:CW-of-a-wedge}\nLet $X_1$, $X_2$ be CW-complexes with a single 0-cell such that $\\pi_1(X_k) \\cong G_k$. Let $X = X_1 \\vee X_2$ which has $\\pi_1(X) \\cong G$ where $G= G_1 \\ast G_2$. Then:\n\\[ \nC_i(\\widetilde X) = \n\\begin{cases}\n{\\iota_1}_\\#(C_i(\\widetilde X_1)) \\oplus {\\iota_2}_\\#(C_i(\\widetilde X_2)), & \\text{if $i \\ge 1$} \\\\\n\\mathbb{Z} G , & \\text{if $i=0$}\n\\end{cases}\n\\]\nwhere $\\partial_i = {\\iota_1}_\\#(\\partial^{X_1}_{i}) \\oplus {\\iota_2}_\\#(\\partial^{X_2}_{i})$ for $i \\ge 2$, $\\partial_1 = ({\\iota_1}_\\#(\\partial^{X_1}_{1}), {\\iota_1}_\\#(\\partial^{X_2}_{1}))$ and $\\partial_0 = \\varepsilon_G$.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to compute an explicit model for $\\widetilde X$ in terms of $\\widetilde X_1$ and $\\widetilde X_2$. Such a model, which is often attributed to Scott-Wall \\cite{SW79}, is provided by taking the graph of spaces structure on $X = X_1 \\vee X_2$ and lifting it to $\\widetilde X$. \n\nDefine a graph $(V,E)$ with vertex set $V = V(X_1) \\sqcup V(X_2)$ where $V(X_1)$ is the set of elements in $G_1 \\ast G_2$ with final term in $G_2$, i.e. the identity $e$ as well as the elements of the form $g_n \\cdots g_1 g_1$ for $n \\ge 1$ where $g_i \\in G_2 \\setminus \\{ 1 \\}$ when $i$ is odd and $g_i \\in G_1 \\setminus \\{ 1 \\}$ otherwise. Define $V(X_2)$ similarly. Note that, whilst $V(X_1) \\cap V(X_2) = \\{1\\}$ as subsets of $G_1 \\ast G_2$, the elements $1 \\in V(X_i)$ are not identified in $V$.\n\nDefine $E = \\bigsqcup_{v \\in V(X_1)} (G_1 \\, \\setminus \\, \\{1\\})_v \\sqcup \\bigsqcup_{v \\in V(X_2)} (G_2 \\, \\setminus \\, \\{1\\})_v \\sqcup \\{e_{1,1} \\}$ where, for each $v \\in V(X_1)$ and $g \\in G_1 \\, \\setminus \\, \\{1\\}$, we have a directed edge $e_{v,vg} = (g)_v$ from $v$ to $vg$ which is labeled by $g \\in G$. Similarly for $V(X_2)$ and $G_2$. The edge $e_{1,1}$ from $1 \\in V(X_1)$ to $1 \\in V(X_2)$ is labeled by $1 \\in G$.\n\nLet $\\ast \\in X_i$ denote the $0$-cell and, for each $g \\in G_i$, let $\\ast_g \\in \\widetilde X_i$ denote its corresponding lift.\nOur model is the CW-complex\n\\[ X_{(V,E)} = \\left(\\bigsqcup_{v \\in V(X_1)} (\\widetilde X_1)_v \\sqcup \\bigsqcup_{v \\in V(X_2)} (\\widetilde X_2)_v \\right)\/\\sim \\]\nwhere, if we have a directed edge $e_{v_1,v_2} \\in E$ with label $g \\in G$, then $(\\ast_g)_{v_1} \\sim (\\ast_1)_{v_2}$ where, if $v_1 \\in V(X_i)$, then $(\\ast_g)_{v_1} \\in (\\widetilde X_i)_{v_1}$ and similarly for $(\\ast_1)_{v_2}$. \nBy comparing with the construction in \\cite{SW79}, we have $\\widetilde X \\simeq X_{(V,E)}$.\n\nWe now determine the induced action of $G = G_1 \\ast G_2$ on $X_{(V,E)}$. Note that $G_1$ acts $(\\widetilde X_1)_1$ by monodromy and freely permutes the $\\ast_g \\in (\\widetilde X_1)_1$. This action extends to all of $X_{(V,E)}$ inductively, and similarly for the action of $G_2$ on $(\\widetilde X_2)_2$. Since $G = \\langle G_1, G_2 \\rangle$, this determines the full action of $G$ on $X_{(V,E)}$. \n\nIt now remains to read off the cell structure of $X_{(V,E)}$ under this $G$-action. For $i \\ge 1$, the $i$-cells lie in the interior of the copies of $\\widetilde X_1$, $\\widetilde X_2$ and so are unaffected by the relation $\\sim$. This implies that:\n\\[C_i(X_{(V,E)}) = \\bigoplus_{v \\in V(X_1)} v \\cdot C_i(\\widetilde X_1) \\oplus \\bigoplus_{v \\in V(X_2)} v \\cdot C_i(\\widetilde X_2) \\]\nas an abelian group. Since $G$ acts on the $V(X_j)$ in the natural way, and the elements of $V(X_j)$ are coset representatives for $G\/G_j$, we have that: \n\\[ \\bigoplus_{v \\in V(X_1)} v \\cdot C_i(\\widetilde X_1) \\cong \\mathbb{Z} G \\otimes_{\\mathbb{Z} G_j} C_i(\\widetilde X_j) \\cong {\\iota_j}_\\#(C_i(\\widetilde X_j)) \\]\nas $\\mathbb{Z} G$-modules. We can determine $C_0(X_{(V,E)})$ and the $\\partial_i$ similarly.\n\\end{proof}\n\n\\begin{corollary} \\label{cor:pi_2-of-wedge}\nLet $X_1$ and $X_2$ be CW-complexes with a single $0$-cell such that $\\pi_1(X_i) \\cong G_i$. Let $X = X_1 \\vee X_2$ which has $\\pi_1(X) \\cong G$ where $G= G_1 \\ast G_2$. Then:\n\\[ H_n(X;\\mathbb{Z} G) \\cong {\\iota_1}_\\#(H_n(X; \\mathbb{Z} G_1)) \\oplus {\\iota_2}_\\#(H_n(X;\\mathbb{Z} G_2)). \\]\n\\end{corollary}\n\n\\begin{remark}\nThis could be deduced from the Mayer-Vietoris sequence for homology with local coefficients \\cite[Theorem 2.4]{Wh78}, though the above argument is more direct.\n\\end{remark}\n\n\\subsection{Homology under group quotients} \n\\label{subsection:change-of-group}\n\nLet $X$ be a CW-complex with $\\rho : \\pi_1(X) \\cong G$ and let  $C_*(\\widetilde X)_\\rho$ be the corresponding chain complex of $\\mathbb{Z} G$-modules. \nIf $f: G \\twoheadrightarrow H$ is a quotient of groups, then $f_\\#(C_*(\\widetilde X)_\\rho)$ is a chain complex of free $\\mathbb{Z} H$-modules with boundary maps $\\id_{\\mathbb{Z} H} \\otimes \\partial_i$, and $H_n(f_\\#(C_*(\\widetilde X)_\\rho))$ is a $\\mathbb{Z} H$-module. \nWe will denote this by $H_n(X;\\mathbb{Z} H)$ when $f$ and $\\rho$ are understood.\n\nSubject to conditions on $f$, this give an additional homotopy invariant for $X$.\n\n\\begin{prop} \\label{prop:ZH-homology}\nIf $f$ is characteristic, then the $\\Aut(H)$-isomorphism class of $H_n(X;\\mathbb{Z} H)$ is a homotopy invariant and is independent of the choice of $\\rho$.\t\n\\end{prop}\n\n\\begin{proof}\nIf $C_*(\\widetilde X)_\\rho \\simeq C_*(\\widetilde Y)_{\\rho'}$ are chain homotopic as chain complexes of $\\mathbb{Z} G$-modules, then $f_\\#(C_*(\\widetilde X)_\\rho) \\simeq f_\\#(C_*(\\widetilde Y)_{\\rho'})$ are chain homotopic as chain complexes of $\\mathbb{Z} H$-modules.\nLet $\\theta \\in \\Aut(G)$. Since $f$ is characteristic, \\cref{prop:modules-over-quotients} implies that $f_\\#((C_*(\\widetilde X)_\\rho)_\\theta) \\cong (f_\\#(C_*(\\widetilde X)_\\rho))_{\\bar{\\theta}}$\nfor some $\\bar{\\theta} \\in \\Aut(H)$. The result now follows.\n\\end{proof}\n\n\\section{Algebraic classification of finite $(G,n)$-complexes}\n\\label{section:Gn-complexes}\n\nA \\textit{$(G,n)$-complex} is an $n$-dimensional CW-complex $X$ such that $\\pi_1(X) \\cong G$ and the universal cover $\\widetilde X$ is $(n-1)$-connected. By contracting a maximal spanning tree, $X$ is homotopy equivalent to a $(G,n)$-complex with a single $0$-cell. For convenience, we will now assume that a $(G,n)$-complex has a single $0$-cell which is the basepoint.\n\nIf $i \\ge 2$, then $\\pi_i(X) \\cong \\pi_i(\\widetilde X)$ as abelian group. In this way, we can view $\\pi_i(X)$ as a $\\mathbb{Z} G$-module under the monodromy action.\nIf $2 \\le i < n$, then $\\pi_i(X) = 0$ since $\\widetilde X$ is $(n-1)$-connected. If $i = n$, then the Hurewicz theorem implies that:\n\\[ \\pi_n(X) \\cong H_n(\\widetilde X ;\\mathbb{Z}) \\cong H_n(X;\\mathbb{Z} G) \\]\nas $\\mathbb{Z} G$-modules. In particular, \\cref{cor:pi_2-of-wedge} applies to $\\pi_n(X)$.\n\n\\subsection{Algebraic $n$-complexes and the D2 problem}\n\\label{subsection:algebraic-n-complexes}\n\nLet $G$ be a group. \nAn \\textit{algebraic $n$-complex over $\\mathbb{Z} G$} is an exact chain complex:\n\\[ E = (F_n \\xrightarrow[]{\\partial_n} \\cdots \\xrightarrow[]{\\partial_2} F_1 \\xrightarrow[]{\\partial_1} F_0 \\xrightarrow[]{\\partial_0} \\mathbb{Z} \\to 0)\\]\nwhere the $F_i$ are finitely generated stably free $\\mathbb{Z} G$-modules. \n\nLet $\\text{\\normalfont{Alg}}(G,n)$ denote the equivalence classes of algebraic $n$-complexes over $\\mathbb{Z} G$ up to chain homotopy equivalences of the unaugmented complex $(F_i,\\partial_i)_{i=1}^n$.\nThe \\textit{$n$th homotopy group} of $E$ is the $\\mathbb{Z} G$ module $\\pi_n(E) = \\Ker(\\partial_n)$ and is an invariant of the chain homotopy class of $E$. \nIf $n \\ge 2$, we can assume the $F_i$ are free since every algebraic $n$-complex is chain homotopy equivalent to such a complex.\n\n Let $\\PHT(G,n)$ denote the polarised homotopy types of finite $(G,n)$-complexes, i.e. the homotopy types of pairs $(X,\\rho)$ where $\\rho: \\pi_1(X) \\cong G$.\nIf $(X,\\rho) \\in \\PHT(G,n)$, then $C_*(\\widetilde X)_{\\rho}$ is a chain complex of $\\mathbb{Z} G$-modules such that $H_0(C_*(\\widetilde X)_{\\rho}) \\cong \\mathbb{Z}$ and $H_i(C_*(\\widetilde X)_{\\rho}) = 0$ for $1 \\le i < n$. In particular, there is a map:\n\\[ \\Psi: \\PHT(G,n) \\to \\text{\\normalfont{Alg}}(G,n). \\]\n\nRecall that a finitely presented group $G$ has the \\textit{D2 property} if every finite CW-complex $X$ such that $\\pi_1(X) \\cong G$, $H_i(\\widetilde X;\\mathbb{Z}) = 0$ for $i > 2$ and $H^{n+1}(X;M)=0$ for all finitely generated $\\mathbb{Z} G$-modules $M$ is homotopy equivalent to a finite 2-complex.\nThe following is a mild improvement of Wall's results on finiteness conditions for CW-complexes due to Johnson \\cite{Jo03a} and Mannan \\cite{Ma09}. \nThis precise version follows from \\cite[Corollary 8.27]{Jo12a} in the case $n \\ge 3$ and \\cite[Theorem 2.1]{Ni19} in the case $n=2$.\n\n\\begin{prop} \\label{prop:realisation-thm}\nLet $G$ be a finitely presented group. \nIf $n \\ge 3$, then $\\Psi$ is bijective.\nIf $n = 2$, then $\\Psi$ is injective and is bijective if and only if $G$ has the {\\normalfont D2} property.\n\\end{prop}\n\n\\begin{remark}\nPart (i) is often vacuous since there are finitely presented groups $G$ for which no algebraic $n$-complex over $\\mathbb{Z} G$ exists for all $n \\ge 3$. The first example was found by Stallings in \\cite{St63} (see also \\cite[Proposition 2.14]{Bi81}) and was later generalised to a class of right-angled Artin groups by Bestvina-Brady \\cite[Main Theorem]{BB97}.\n\\end{remark}\n\n\\subsection{Realising $\\mathbb{Z} G$-modules by algebraic $n$-complexes}\n\\label{subsection:pi_n-realisation}\n\nThe $n$th \\textit{stable syzygy} $\\Omega_{n}^G(\\mathbb{Z})$ is the set of $\\mathbb{Z} G$-modules $M$ for which $M \\oplus \\mathbb{Z} G^i \\cong \\pi_{n-1}(E) \\oplus \\mathbb{Z} G^j$ for some $i, j \\ge 0$ and some algebraic $(n-1)$-complex $E$ over $\\mathbb{Z} G$. We will denote this by $\\Omega_n(\\mathbb{Z})$ when the choice of $G$ is clear from the context. This is well-defined and does not depend on the choice of $E$ \\cite[Theorem 8.9]{Jo12a}.\nIt also comes with a map:\n\\[ \\pi_n : \\text{\\normalfont{Alg}}(G,n) \\to \\Omega_{n+1}(\\mathbb{Z}). \\]\n\nThe following can be found in \\cite[Proposition 8.18]{Jo12a}.\n\n\\begin{prop} \\label{prop:realisation-of-syzygies}\nLet $n \\ge 2$ and let $G$ be an infinite finitely presented group of type $\\FL$ such that $H^{n+1}(G;\\mathbb{Z} G)=0$. Then $\\pi_n$ is bijective.\n\\end{prop}\n\nThe following is a straightforward consequence of Propositions \\ref{prop:FP+cd} and \\ref{prop:realisation-of-syzygies}.\n\n\\begin{prop} \\label{prop:syzygies-cd-finite}\nLet $G$ be a finitely presented group of type $\\FL$ with $\\cd(G) = d$.\n\\begin{clist}{(i)}\n\\item If $n \\ge d$, then $\\Omega_{n}(\\mathbb{Z})$ is the set of stably free $\\mathbb{Z} G$-modules\n\\item If $n \\ge d$, then $\\pi_n: \\text{\\normalfont{Alg}}(G,n) \\to \\Omega_{n+1}(\\mathbb{Z})$ is bijective\n\\item If $n = d-1$, then $0 \\not \\in \\IM(\\pi_{n} : \\text{\\normalfont{Alg}}(G,n) \\to \\Omega_{n+1}(\\mathbb{Z}))$.\n\\end{clist}\n\\end{prop}\n\n\\begin{remark}\nThis implies that, for $n \\ge 2$, $\\pi_n : \\text{\\normalfont{Alg}}(G,n) \\to \\Omega_{n+1}(\\mathbb{Z})$ is not surjective whenever $\\cd(G) = n+1$ (for example, $G = \\mathbb{Z}^{n+1}$). This was noted in \\cite[p107]{Jo12a}.\n\\end{remark}\n\nIt is possible to see that $\\cd(G) \\le n$ implies that $\\pi_n: \\text{\\normalfont{Alg}}(G,n) \\to \\Omega_{n+1}(\\mathbb{Z})$ is surjective directly (see, for example, \\cite[Theorem 4]{HJ06b}).\nThe following is now clear.\n\n\\begin{corollary} \\label{cor:homotopy-cd-finite}\nLet $n \\ge 3$ and let $G$ be a finitely presented group of type $\\FL$ with $\\cd(G) = n$. Then $\\pi_n$ gives a one-to-one correspondence between homotopy types of finite $(G,n)$-complexes and $\\Aut(G)$-isomorphism classes of stably free $\\mathbb{Z} G$-modules.\n\\end{corollary}\n\nFinally, we note the following where $\\rank(P)$ denotes the stably free rank of $P$.\n\n\\begin{prop} \\label{prop:rank-computation}\nLet $G$ be a finitely presented group of type $\\FL$ with $\\cd(G) = d$ and let $n \\ge d-1$. Then $\\chi(X) = k + \\chi_{\\min}(G,n)$ if and only if:\n\\[ \\rank(\\pi_n(X)) = k + \\min\\{ \\rank(\\pi_n(X_0)) : \\text{$X_0$ a finite $(G,n)$-complex} \\}. \\]\nIn particular, if $n \\ge \\max\\{3, d\\}$, then $k = \\rank(\\pi_n(X))$.\n\\end{prop}\n\n\\section{Proof of \\cref{thm:main}}\n\\label{section:proof-main-topological}\n\nWe will now prove \\cref{thm:main} separately in the two cases of non-minimal Euler characteristic ($k \\ge 1$) and minimal Euler characteristic ($k =0$). \nThroughout, $T_i \\cong T$ will denote the trefoil group and $G_{(n)}$ will be as defined in \\cref{section:cd(G)}.\n\n\\subsection{Finite $(G,n)$-complexes with non-minimal Euler characteristic}\n\\label{subsection:non-min-EC}\n\nThe aim of this section will be to prove the following. Note that, in the case $n \\ge 3$, we could also take $G$ to be one of the other groups listed at the end of \\cref{section:proof-main-algebra}.\t\n\n\n\\begin{thm} \\label{thm:main-non-min-EC}\nLet $n \\ge 2$, let $k \\ge 1$ and let $G = (T_1 \\ast \\cdots \\ast T_k)_{(n-1)}$. Then, for all $1 \\le m \\le k$, there exists infinitely many finite $(G,n)$-complexes $\\widehat X_i$ such that:\n\\begin{clist}{(i)}\n\\item $\\pi_n(\\widehat X_i) \\cong \\widehat S_i$ as $\\mathbb{Z} G$-modules (where $\\widehat S_i$ is as defined in \\cref{thm:main-SF-detailed})\n\\item $\\chi(\\widehat X_i) = m + \\chi_{\\min}(G,n)$\n\\item $\\widehat X_i \\not \\simeq Y_i \\vee S^2$ for any finite $(G,n)$-complex $Y_i$.\n\\end{clist}\n\\end{thm}\n\nSince the $\\Aut(G)$-isomorphism class of $\\pi_n(\\widehat X_i)$ is a homotopy invariant, it follows that the $\\widehat X_i$ are homotopically distinct by \\cref{thm:main-SF-detailed}. By restricting to the case $m=k$, this implies \\cref{thm:main} for $k \\ge 1$. \n\nWe will begin with the case $n=2$, where $G = T_1 \\ast \\cdots \\ast T_k$. Let $S_i$ be the stably free $\\mathbb{Z} T$-modules from \\cref{thm:BD} and, for $1 \\le m \\le k$, recall that:\n\\[ S_{i_1, \\cdots, i_m} = {\\iota_1}_\\#(S_{i_1}) \\oplus \\cdots \\oplus {\\iota_m}_\\#(S_{i_m}).\\]\nThe case of interest will be $\\widehat S_i = S_{i_1, \\cdots, i_m}$ where $i_j = i$ for all $j$.\n\nThe main result which we will use is the following, which is \\cite[Theorem 4.5]{HJ06a}.\n\n\\begin{thm}[Harlander-Jensen] \\label{thm:HJ}\nThe trefoil group $T$ has presentations\n\\[ \\mathcal{P}_i = \\langle x,y,a,b \\mid x^2=y^3, a^2=b^3, x^{2i+1}=a^{2i+1}, y^{3i+1}=b^{3i+1} \\rangle \\]\nfor $i \\ge 0$. For each $i$, there exists $\\ell_i$ such that $S_i \\cong \\pi_2(X_{\\mathcal{P}_{\\ell_i}})$ as $\\mathbb{Z} T$-modules.\n\\end{thm}\n\n\\begin{remark}\nNote that $\\mathcal{P}_0 \\simeq \\langle x, y \\mid x^2=y^3, 1 \\rangle$ and $\\mathcal{P}_1$ is homotopy equivalent to the presentation found by Dunwoody in \\cite{Du76}.\t\n\\end{remark}\n\nLet $X_i = X_{\\mathcal{P}_{\\ell_i}}$ for each $i \\ge 1$. For integers $i_j \\ge 1$, define:\n\\[ X_{i_1, \\cdots, i_n} = X_{i_1} \\vee \\cdots \\vee X_{i_n}\\]\nwhich is a finite $2$-complex with $\\pi_1(X_{i_1, \\cdots, i_n}) \\cong T_1 \\ast \\cdots \\ast T_k$. Let $\\widehat X_i = X_{i_1, \\cdots, i_n}$ where $i_j = i$ for all $j$.\nBy repeated application of \\cref{cor:pi_2-of-wedge}, we have that $\\pi_2(X_{i_1,\\cdots,i_n}) \\cong S_{i_1, \\cdots, i_n}$ and so $\\pi_2(\\widehat X_i) \\cong \\widehat S_i$. \nSince $\\rank(\\widehat S_i) = m$, we have that $\\chi(\\widehat X_i) = m + \\chi_{\\min}(G)$ by \\cref{prop:rank-computation}. Finally, if $\\widehat X_i \\simeq Y_i \\vee S^2$, then:\n\\[ \\widehat S_i \\cong \\pi_2(\\widehat X_i) \\cong \\pi_2(Y_i) \\oplus (\\mathbb{Z} G \\otimes_{\\mathbb{Z}} \\pi_2(S^2)) \\cong \\pi_2(Y_i) \\oplus \\mathbb{Z} G \\]\nwhich is a contradiction since $\\widehat S_i$ has no summand of the form $\\mathbb{Z} G$ by \\cref{thm:main-SF-detailed}.\t\nThis completes the proof of \\cref{thm:main-non-min-EC} in the case $n=2$.\n\nWe will now consider the case $n \\ge 3$. where $G = (T_1 \\ast \\cdots \\ast T_k)_{(n-1)}$. By \\cref{thm:main-SF-detailed}, there exists stably free $\\mathbb{Z} G$-modules $\\widehat S_i$ of rank $m$ and which have no summand of the form $\\mathbb{Z} G$. By \\cref{prop:group-construction}, we have that $\\cd(G) = n$ and so, by \\cref{cor:homotopy-cd-finite}, there exists finite $(G,n)$-complexes $\\widehat X_i$ such that $\\pi_n(\\widehat X_i) \\cong \\widehat S_i$. We can now argue similarly to the case $n=2$. This completes the proof of \\cref{thm:main-non-min-EC}.\n\n\\subsubsection{Application to Syzygies}\n\nWe now discuss consequences of \\cref{thm:main-non-min-EC} for syzygies.\nRecall that a $\\mathbb{Z} G$-module $M_0 \\in \\Omega_n(\\mathbb{Z})$ is \\textit{minimal} if $M \\in \\Omega_n(\\mathbb{Z})$ implies that $M \\oplus \\mathbb{Z} G^i \\cong M_0 \\oplus \\mathbb{Z} G^j$ for some $i \\le j$. For $k \\ge 0$, we say that $M \\in \\Omega_n(\\mathbb{Z})$ has \\textit{level $k$} if $M \\oplus \\mathbb{Z} G^i \\cong M_0 \\oplus \\mathbb{Z} G^j$ where $j-i = k$ and $M_0$ is minimal.\nIf $X$ is a finite $(G,n)$-complex, then $\\pi_n(X) \\in \\Omega_{n+1}(\\mathbb{Z})$. If $\\cd(G) = n$ and $\\pi_n(X)$ is stably free $\\mathbb{Z} G$-module of rank $k$, then $\\pi_n(X)$ has level $k$. \nHence, by \\cref{thm:main-non-min-EC}, we have: \n\n\\begin{corollary} \\label{cor:syzygies}\nFor all $n \\ge 3$ and $k \\ge 1$, there exists a finitely presented group $G$ such that $\\Omega_n(\\mathbb{Z})$ has branching at level $k$. Furthermore, there exists infinitely many $\\mathbb{Z} G$-modules $M_i \\in \\Omega_n(\\mathbb{Z})$ at level $k$ which are distinct up to $\\Aut(G)$-isomorphism.\n\\end{corollary}\n\n\\subsection{Finite $(G,n)$-complexes with minimal Euler characteristic}\n\\label{subsection:min-EC}\n\nThe following is the main result of \\cite{Lu93}.\n\n\\begin{thm}[Lustig] \\label{thm:lustig}\nLet $G = T_{(2)}$. Then there exists infinitely many homotopically distinct finite $2$-complexes $X_i$ for $i \\ge 1$ such that $\\pi_1(X_i) \\cong G$ and $\\chi(X_i) = 1$.\n\\end{thm}\n\nThe aim of this section will be to give the following generalisation of Lustig's result, which includes an identification of $\\chi_{\\min}(T_{(2)})$.\n\n\\begin{thm} \\label{thm:main-min-EC}\nLet $n \\ge 2$ and let $G = T_{(n)}$. Then there exists infinitely many finite $(G,n)$-complexes $\\widehat X_i$ for $i \\ge 1$ such that:\n\\begin{clist}{(i)}\n\\item $H_n(\\widehat X_i; \\mathbb{Z} T) \\cong S_i$ as $\\mathbb{Z} T$-modules (where $S_i$ is as defined in \\cref{thm:BD})\n\\item $\\chi(\\widehat X_i) = \\chi_{\\min}(G,n)$\n\\end{clist}\n\\end{thm}\n\n\\begin{remark}\nThis corrects a statement made in \\cite[Section 5]{HJ06b} where it was suggested that $\\chi(X_i) = 1 + \\chi_{\\min}(T_{(2)})$. \nIn fact, we have $\\chi(\\widehat X_i) = \\chi_{\\min}(T_{(n)},n) = 1-n$.\n\\end{remark}\n\nBy \\cref{prop:ZH-homology}, the $\\Aut(T)$-isomorphism class of $H_n(\\widehat X_i; \\mathbb{Z} T)$ is a homotopy invariant and so the $\\widehat X_i$ are homotopically distinct by \\cref{thm:BD}. Hence this implies \\cref{thm:main} in the case $k = 0$.\n\nWe will begin with the following lemma, which can be verified directly.\n\n\\begin{lemma} \\label{lemma:alg(n)-alg(n+1)}\nLet $n \\ge 2$, let $G$ be a group and let $E = (\\mathbb{Z} G^{d_i}, \\partial_i)_{i=1}^n \\in \\text{\\normalfont{Alg}}(G,n)$.\nIf $G_{+} = (G \\times \\langle q \\mid \\hspace{-.8mm}-\\rangle) \\ast_{\\langle q = r^2 \\rangle} \\langle r \\mid \\hspace{-.8mm}-\\rangle$, then:\n\\[ E_+ = (\\mathbb{Z} G_+^{d_n} \\xrightarrow[]{\\partial_{n+1}^+} \\cdots \\xrightarrow[]{\\partial_2^+} \\mathbb{Z} G_+^{d_1+1} \\xrightarrow[]{\\partial_1^+} \\mathbb{Z} G_+ \\xrightarrow[]{\\varepsilon_{G_+}} \\mathbb{Z} \\to 0) \\in \\text{\\normalfont{Alg}}(G_+,n+1)\\]\nwhere $\\partial_1^+ = \\cdot\\left(\\begin{smallmatrix}\\partial_1 \\\\ r-1 \\end{smallmatrix}\\right)$, $\\partial_2^+ = \\cdot\\left(\\begin{smallmatrix}\\partial_2 & 0 \\\\ r^2-1 & -\\partial_1 \\cdot (r+1) \\end{smallmatrix}\\right)$ and $\\partial_i^+ = \\cdot\\left(\\begin{smallmatrix}\\partial_i & 0 \\\\ r^2-1 & -\\partial_{i-1} \\end{smallmatrix}\\right)$ for $i \\ge 3$. The $\\partial_*$ are the induced maps and we take $d_{n+1}=0$, $\\partial_{n+1}=0$.\n\\end{lemma}\n\n\\begin{remark}\nThis also works when $G_{+} = (G \\times \\langle q \\mid \\hspace{-.8mm}-\\rangle) \\ast_{\\langle q = r^m \\rangle} \\langle r \\mid \\hspace{-.8mm}-\\rangle$ for $m \\ge 2$.\n\\end{remark}\n\nLet $\\mathcal{P} = \\langle x, y \\mid x^2 = y^3 \\rangle$ be the standard presentation for $T$ and note that:\n\\[ C_*(\\widetilde X_{\\mathcal{P}}) \\cong (\\mathbb{Z} T \\xrightarrow[]{\\partial_2} \\mathbb{Z} T^2 \\xrightarrow[]{\\partial_1} \\mathbb{Z} T \\xrightarrow[]{\\varepsilon_T} \\mathbb{Z} \\to 0) \\in \\text{\\normalfont{Alg}}(T,2)\\]\nwhere $\\partial_1 = \\cdot\\left(\\begin{smallmatrix} x-1 \\\\ y-1 \\end{smallmatrix}\\right)$ and $\\partial_2 = \\cdot\\left(\\begin{smallmatrix} x+1 & -(y^2+y+1) \\end{smallmatrix}\\right)$.\nThis has $\\pi_2(C_*(\\widetilde X_{\\mathcal{P}})) = 0$.\n\nFor each $n \\ge 1$, define $\\widetilde E_n \\in \\text{\\normalfont{Alg}}(T_{(n)},n+1)$ by $\\widetilde E_1 = C_*(\\widetilde X_{\\mathcal{P}})$ and $\\widetilde E_{n} = (\\widetilde E_{n-1})_+$ for $n \\ge 2$ using  \\cref{lemma:alg(n)-alg(n+1)}.\nLet $E_n \\in \\text{\\normalfont{Alg}}(T_{(n)},n)$ denote the restriction to the first $n+1$ terms in $\\widetilde E_n$. Since $\\pi_2(\\widetilde E_1) = 0$, we have that $\\pi_{n+1}(\\widetilde E_n) = 0$ and this implies that\n$\\pi_{n}(E_n) = \\IM(\\partial_{n+1}^{\\widetilde E_n}) \\cong \\mathbb{Z} T_{(n)}$.\n\nFor $n \\ge 2$, let $\\Delta_n = \\partial_{n}^{\\widetilde E_n}$ denote the final boundary map in $E_n$, so that:\n\\[ \n\\Delta_1 = \\partial_1 \\cdot (r_1+1), \\quad \n\\Delta_{n} = \\cdot\\left(\\begin{smallmatrix} v_{n}  & 0  \\\\ r_{n-1}^2-1 & -\\Delta_{n-1} \\end{smallmatrix}\\right) : \\mathbb{Z} T_{(n)}^{n+1} \\to \\mathbb{Z} T_{(n)}^{\\frac{n(n+1)}{2}}\n\\]\nwhere $v_{n} = (r_{n-2}^2-1, (-1)(r_{n-3}^2-1), \\cdots, (-1)^{n-3}(r_1^2-1), (-1)^{n-2}\\partial_2)$. Here $\\Delta_1$ is defined for the purposes of this definition and \ndoes not coincide with $\\partial_1^{E_1} = \\partial_1$.\n\nLet $\\alpha_n, \\beta_n$ denote the last two row vectors in $\\Delta_n$, which are defined by: \n\\[ \\alpha_1 = (x-1)(r_1+1), \\quad \\beta_1 = (y-1)(r_1+1)\\]\n\\[ \\alpha_n = (\\underbrace{0, \\cdots, 0}_{n-2} , r_{n-1}^2-1, 0, -\\alpha_{n-1}), \\quad \\beta_n = (\\underbrace{0, \\cdots, 0}_{n-1} , r_{n-1}^2-1, -\\beta_{n-1}).\\]\nFor each $i \\ge 0$, let $\\alpha_n^{(i)} = \\Sigma_x \\alpha_n$, $\\beta_n^{(i)} = \\Sigma_y \\beta_n$ where $\\Sigma_x = \\sum_{j=0}^{2i} x^j$, $\\Sigma_y = \\sum_{j=0}^{3i} y^j$. \n\nWe will now show that following, where we adopt the notation of \\cref{subsection:change-of-group}.\n\n\\begin{prop} \\label{prop:alg-alterations}\nFor $n \\ge 2$, let $\\Delta_n^{(i)}$ be the matrix $\\Delta_n$ but with $\\alpha_n, \\beta_n$ replaced by $\\alpha_n^{(i)}, \\beta_n^{(i)}$, and let $E_{n}^{(i)}$ to be the resolution $E_n$ but with $\\Delta_n$ replaced by $\\Delta_n^{(i)}$. Then:\n\\begin{clist}{(i)}\n\\item $E_n^{(i)} \\in \\text{\\normalfont{Alg}}(T_{(n)},n)$\n\\item If $f : T_{(n)} \\twoheadrightarrow T$, then $H_n(E_n^{(i)};\\mathbb{Z} T) \\cong \\Ker(\\cdot\\left(\\begin{smallmatrix}x^{2i+1}-1 \\\\ y^{3i+1}-1 \\end{smallmatrix}\\right))$ as $\\mathbb{Z} T$-modules.\n\\end{clist}\n\\end{prop}\n\nFor the convenience of the reader, we will write this explicitly in the case $n=2$:\n\n\\newcommand{\\scriptsize{\\cdot\\left(\\begin{smallmatrix}{\\scriptsize{\\cdot\\left(\\begin{smallmatrix}\nx+1 & -(y^2+y+1) & 0 \\\\ \n(r_1^2-1)\\Sigma_x & 0 & (1-x^{2i+1})(r_1+1) \\\\ \n0 & (r_1^2-1)\\Sigma_y & (1-y^{3i+1})(r_1+1)  \n\\end{smallmatrix}\\right)}}\n\n\\newcommand{\\cdot\\left(\\begin{smallmatrix}x-1 \\\\ y-1 \\\\ r_1 -1 \\end{smallmatrix}\\right)}{\\cdot\\left(\\begin{smallmatrix}x-1 \\\\ y-1 \\\\ r_1 -1 \\end{smallmatrix}\\right)}\n\n\\vspace{-2mm}\n\\[ E_2^{(i)} = (\\mathbb{Z} T_{(2)}^3 \\xrightarrow[]{\\scriptsize{\\cdot\\left(\\begin{smallmatrix} \\mathbb{Z} T_{(2)}^3 \\xrightarrow[]{\\cdot\\left(\\begin{smallmatrix}x-1 \\\\ y-1 \\\\ r_1 -1 \\end{smallmatrix}\\right)} \\mathbb{Z} T_{(2)} \\xrightarrow[]{\\varepsilon} \\mathbb{Z} \\to 0). \\]\n\\vspace{-2mm}\n\nIn order to prove this, we will first need the following technical lemma.\n\n\\begin{lemma} \\label{lemma:identity}\nLet $G$ be a group with $T \\subseteq G$. For $i = 1, 2, 3$, there exists $\\lambda_i, \\mu_i \\in \\mathbb{Z} T \\subseteq \\mathbb{Z} G$ such that, for all $r \\in G$, we have:\n\n\\vspace{2mm}\n\\begin{adjustbox}{width=\\textwidth+3mm}\n$(r-1,0,1-x) = \\lambda_1 \\cdot (\\Sigma_x(r-1),0,1-x^{2i+1}) + \\lambda_2 \\cdot (0,\\Sigma_y(r-1),1-y^{3i+1}) + \\lambda_3 \\cdot (\\partial_2,0) \\cdot (r-1)$\n\\end{adjustbox}\n\n\\vspace{2mm}\n\\begin{adjustbox}{width=\\textwidth+3mm}\n$(0,r-1,1-y) = \\mu_1 \\cdot (\\Sigma_x(r-1),0,1-x^{2i+1})\n+ \\mu_2 \\cdot (0,\\Sigma_y(r-1),1-y^{3i+1}) + \\mu_3 \\cdot (\\partial_2,0) \\cdot (r-1)$\n\\end{adjustbox}\n\\end{lemma}\n\n\\begin{proof}[Proof of \\cref{prop:alg-alterations}]\nTo prove (i), it suffices to show that $\\IM(\\cdot\\Delta_n^{(i)}) = \\IM(\\cdot\\Delta_n)$ for $i \\ge 1$. We have $\\IM(\\cdot\\Delta_n^{(i)}) \\subseteq \\IM(\\cdot\\Delta_n)$, so it remains to show $\\alpha_n, \\beta_n \\in \\IM(\\cdot\\Delta_n^{(i)})$.\n\nBy the proof of \\cref{lemma:identity}, we have $\\mathbb{Z} T \\cdot \\{ x-1, y-1 \\} = \\mathbb{Z} T \\cdot \\{ x^{2i+1}-1,y^{3i+1}-1 \\}$. It follows that $\\mathbb{Z} T \\cdot \\{ \\alpha_1, \\alpha_2\\} = \\mathbb{Z} T \\cdot \\{\\alpha_1^{(i)}, \\beta_1^{(i)}\\}$ which implies that $\\alpha_1, \\beta_1 \\in \\IM(\\cdot\\Delta_1^{(i)})$.\nThe case $n = 2$ is done in \\cref{lemma:identity}, which provides $\\lambda_i$ such that:\n\\[ \\alpha_2 = \\lambda_1 \\cdot \\alpha_2^{(i)} + \\lambda_2 \\cdot \\beta_2^{(i)} + \\lambda_3(r_1^2-1) \\cdot (\\partial_2,0)  \\]\nand similarly for $\\mu_i$ and $\\beta_2$.\nLet $\\gamma_1, \\cdots , \\gamma_{n-1}$ denote the first $n-1$ rows of $\\Delta_n$, the remaining two rows being $\\alpha_n, \\beta_n$. It is now straightforward to see that:\n\\[ \\alpha_n = \\lambda_1 \\cdot \\alpha_n^{(i)} + \\lambda_2 \\cdot \\beta_n^{(i)} + \\lambda_3(-1)^n((r_{n-1}^2-1) \\cdot \\gamma_1 + \\sum_{i=2}^{n-1} (-1)^{i}(r_{n-i}^2-1) \\cdot \\gamma_{i}) \\]\nfor $n \\ge 2$, and similarly for $\\beta_n$. Hence $\\alpha_n, \\beta_n \\in \\IM(\\cdot\\Delta_n^{(i)})$ for all $n \\ge 2$.\n\nTo prove (ii), note that $H_n(E_n^{(i)}; \\mathbb{Z} T) = \\Ker(f_\\#(\\Delta_n^{(i)}))$. For each $n \\ge 2$, we have:\n\\[ f_\\#(\\Delta_n^{(i)}) = \\cdot \\left(\\begin{smallmatrix} f_\\#(v_n) & 0 \\\\ 0 & - f_\\#(\\Delta_{n-1}^{(i)})  \\end{smallmatrix}\\right). \\]\nSince $f_\\#(v_n) = (0, \\cdots, 0, (-1)^{n-2}\\partial_2)$ is injective, this implies that $\\Ker(f_\\#(\\Delta_n^{(i)})) = \\Ker(-f_\\#(\\Delta_{n-1}^{(i)}))$ and so, by induction:\n\\[ \\Ker(f_\\#(\\Delta_n^{(i)})) \\cong \\Ker(f_\\#(\\Delta_1^{(i)})) = \\Ker(\\cdot\\left(\\begin{smallmatrix} 2(x^{2i+1}-1) \\\\ 2(y^{3i+1}-1) \\end{smallmatrix}\\right)) = \\Ker(\\cdot\\left(\\begin{smallmatrix} x^{2i+1}-1 \\\\ y^{3i+1}-1 \\end{smallmatrix}\\right)). \\qedhere \\]\n\\end{proof}\n\\vspace{-1mm}\n\nLet $G = T_{(n)}$. For each $i \\ge 1$, there exists $\\ell_i$ such that $\\Ker(\\cdot\\left(\\begin{smallmatrix} x^{2\\ell_i+1}-1 \\\\ y^{3\\ell_i+1}-1 \\end{smallmatrix}\\right)) \\cong S_i$ where the $S_i$ are as defined in the discussion following \\cref{thm:BD}\n\nIf $n \\ge 3$, then \\cref{prop:realisation-thm} implies that there exists finite $(G,n)$-complexes $\\widehat X_i$ such that $C_*(X) \\simeq E_n^{(\\ell_i)}$ are chain homotopy equivalent where $X$ is the universal cover of $\\widehat X_i$. This is also true when $n=2$ by taking $\\widehat X_i = X_i = \\mathcal{P}_{\\ell_i}$ where:\n\\[ \\mathcal{P}_i = \\langle a, b, c \\mid a^2=b^3, [a^2,b^{2i+1}], [a^2,c^{3i+1}] \\rangle \\]\nare the presentations given by Lustig in \\cite{Lu93}.\n\nBy \\cref{prop:alg-alterations}, $H_n(\\widehat X_i; \\mathbb{Z} T) \\cong S_i$ as $\\mathbb{Z} T$-modules. It is straightforward to see that $\\rank(\\pi_n(E_n^{(\\ell_i)})) = \\rank(\\pi_n(E_n)) = 1$. By \\cref{prop:modules-over-quotients}, $\\cd(G) = n+1$ and so $0 \\not \\in \\IM(\\pi_n : \\PHT(G,n) \\to \\Omega_{n+1}(\\mathbb{Z}))$ by \\cref{prop:syzygies-cd-finite}. Hence, by \\cref{prop:rank-computation}, we have $\\chi(X_i) = \\chi_{\\min}(G,n)$. This completes the proof of \\cref{thm:main-min-EC}. By combining with \\cref{thm:main-non-min-EC}, this completes the proof of \\cref{thm:main}.\n\n\\section{Proofs of Theorems \\ref{thm:main-SF-further} and \\ref{thm:main-further}}\n\\label{section:proof-main-further}\n\nThe aim of this section will be to prove the following two theorems which imply Theorems \\ref{thm:main-SF-further} and \\ref{thm:main-further} respectively. The proofs are similar to that of Theorems \\ref{thm:main-SF} and \\ref{thm:main} and so many of the details will be omitted.\nWe will let $T$ denote the trefoil group.\n\n\\begin{thm} \\label{thm:main-SF-further-detailed}\nLet $d \\ge 2$ and let $G = \\ast_{i=1}^\\infty T_{(d-1)}$. Then $\\cd(G) = d$ and, for all $k \\ge 1$, there exists infinitely many stably free $\\mathbb{Z} G$-modules of rank $k$ which are distinct up to $\\Aut(G)$-isomorphism.\n\\end{thm}\n\nLet $S_i$ denote the stably free $\\mathbb{Z} T$-modules of \\cref{thm:BD} and let $\\iota_j : T_j \\hookrightarrow G$.\n\n\\begin{proof}\nLet $k \\ge 1$ and let $\\widehat S_i^{(k)} = \\bigoplus_{j=1}^k {\\iota_j}_\\#(S_i)$ for $i \\ge 1$. Since $\\widehat S_i^{(k)} \\oplus \\mathbb{Z} G \\cong \\mathbb{Z} G^{k+1}$, the $\\widehat S_i^{(k)}$ are stably free $\\mathbb{Z} G$-modules of rank $k$.\nLet $f: G \\twoheadrightarrow \\ast_{j=1}^\\infty T_j\/T_j''$ be induced by the characteristic quotients $f_j : (T_j)_{(d-1)} \\twoheadrightarrow T_j\/T_j''$.\nThis is characteristic by a mild generalisation of \\cref{prop:free-product-char} which applies since $T_j$ is finitely generated.\n\nFor $p$ prime, we have that $\\mathbb{F}_p \\otimes f_\\#(\\widehat S_i^{(k)}) \\cong \\oplus_{j=1}^k \\text{$\\bar{\\iota}_j$}_\\#(\\mathbb{F}_p \\otimes {f_j}_\\#(S_i))$ where $\\text{$\\bar{\\iota}_j$} : T_j\/T_j'' \\hookrightarrow \\ast_{j=1}^\\infty T_j\/T_j''$ is the inclusion map.\nSimilarly to the proof of \\cref{thm:main-SF-detailed}, there exists primes $p_i$ for $i \\ge 1$ such that $\\mathbb{F}_{p_i} \\otimes {f_j}_\\#(S_i) \\cong \\mathbb{F}_{p_i} [T_j\/T_j'']$ if and only if $i = j$. Since \\cref{thm:Bergman} and \\cref{cor:Bergman} also holds for infinite free products (see \\cite{Be74}), we get the $\\mathbb{F}_p \\otimes f_\\#(\\widehat S_i^{(k)})$ are distinct up to $\\Aut(G)$-isomorphism. Since $f$ is characteristic, the $\\widehat S_i^{(k)}$ are distinct up to $\\Aut(G)$-isomorphism also.\n\\end{proof}\n\n\\begin{thm} \\label{thm:main-further-detailed}\nLet $n \\ge 2$ and let $G = \\ast_{i=1}^\\infty T_{(n-1)}$.\nThen there exists an aspherical $(G,n)$-complex $Y$ such that, for all $k \\ge 1$, there are infinitely many homotopically distinct $(G,n)$-complexes $X_i$ with $X_i \\vee S^n \\simeq Y \\vee (k+1)S^n$.\n\\end{thm}\n\n\\begin{proof}\nBy \\cref{lemma:alg(n)-alg(n+1)}, there exists $\\widetilde E_{n-1} \\in \\text{\\normalfont{Alg}}(T_{(n-1)},n)$ with $\\pi_n(\\widetilde E_{n-1}) = 0$.  If $n \\ge 3$, then \\cref{prop:realisation-thm} implies that there exists a finite $(G,n)$-complex $Y_0$ such that $C_*(\\widetilde Y_0) \\simeq \\widetilde E_{n-1}$ are chain homotopy equivalent. This is also true when $n=2$ by taking $Y_0 = X_{\\mathcal{P}}$ where $\\mathcal{P} = \\langle x, y \\mid x^2=y^3 \\rangle$ is the standard presentation for $T$. Hence, for all $n \\ge 2$, $Y = \\vee_{i=1}^\\infty Y_0$ is an aspherical $(G,n)$-complex.\n\nFor all $i \\ge 1$, let $X_i = \\bigvee_{j=1}^k \\widehat X_i \\vee \\bigvee_{j=k+1}^\\infty Y_0$ where the $\\widehat X_i$ are the finite $(T_{(n-1)},n)$-complexes such that $\\pi_n(\\widehat X_i) \\cong S_i$ which were constructed in \\cref{thm:main-non-min-EC}.\nThen $X_i$ is a $(G,n)$-complex such that:\n\\[ \\pi_n(X_i) \\cong \\bigoplus_{j=1}^k {\\iota_j}_\\#(\\pi_n(\\widehat X_i)) \\oplus \\bigoplus_{j=k+1}^\\infty {\\iota_j}_\\#(\\pi_n(Y)) \\cong \\bigoplus_{j=1}^k {\\iota_j}_\\#(S_i) = \\widehat S_i^{(k)}.  \\]\nSince the $\\widehat S_i^{(k)}$ are distinct up to $\\Aut(G)$-isomorphism, this implies that the $X_i$ are homotopically distinct.\nBy \\cref{thm:BD} and \\cref{prop:syzygies-cd-finite}, we have that $\\widehat X_i \\vee S^n \\simeq Y_0 \\vee 2S^n$. It follows that $X_i \\vee S^n \\simeq Y \\vee (k+1)S^n$, as required.\n\\end{proof}\n\n\\section{Some remarks on induced module decompositions}\n\\label{subsection:induced}\n\nRecall that Theorems \\ref{thm:main-SF} and \\ref{thm:main} concerned stably free $\\mathbb{Z} G$-modules and finite $2$-complexes $X$ with $\\pi_1(X) \\cong G$ where $G = \\ast_{i=1}^k G_i$.\nIn our example, $\\pi_2(X) \\otimes \\mathbb{F}_p$ was an induced $\\mathbb{F}_p G$-module whose component $\\mathbb{F}_p T$-modules $M_i$ were unique up to $\\mathbb{F}_p T$-isomorphism where $G_i = T$ is the trefoil group.\n\nThe aim of this section will be to investigate the extent to which this applies to all groups of the form $G = \\ast_{i=1}^k G_i$ and to $\\pi_2(X)$ rather than just $\\pi_2(X) \\otimes \\mathbb{F}$.\nFor simplicity, we will restrict to the case of $2$-complexes. However, all results have analogues for $(G,n)$-complexes for $n \\ge 3$.\n\n\\subsection{Existence of induced module decompositions}\n\\label{subsection:existence}\n\nWe will begin by considering the question of existence. From now on, we will take $\\mathbb{F}$ to be a field.\n\n\\begin{prop}[Existence over \\text{$\\mathbb{F}[G_1 \\ast \\cdots \\ast G_k]$}] \\label{prop:existence}\nLet $X$ be a finite $2$-complex with $\\pi_1(X) \\cong G_1 \\ast \\cdots \\ast G_k$. Then $\\pi_2(X) \\otimes \\mathbb{F}$ is an induced $\\mathbb{F}[G_1 \\ast \\cdots \\ast G_k]$-module.\n\\end{prop}\n\n\\begin{proof}\nLet $X_i$ be a finite $2$-complex with $\\pi_1(X_i) \\cong G_i$. Then $\\pi_1(\\vee_{i=1}^k X_i) \\cong \\ast_{i=1}^k G_i$ and so there exists $a, b \\ge 0$ such that $X \\vee aS^2 \\simeq \\vee_{i=1}^k X_i \\vee bS^2$. This implies that\n\\[ (\\pi_2(X) \\otimes \\mathbb{F}) \\oplus \\mathbb{F} G^a \\cong {\\iota_1}_\\#((\\pi_2(X_1) \\otimes \\mathbb{F}) \\oplus \\mathbb{F} G_1^b) \\oplus  \\bigoplus_{j = 2}^k {\\iota_j}_\\#(\\pi_2(X_j) \\otimes \\mathbb{F}) \\]\nand so $\\pi_2(X) \\oplus \\mathbb{F}$ is a submodule of an induced $\\mathbb{F}[G_1 \\ast \\cdots \\ast G_k]$-module. Hence, by \\cref{thm:Bergman}, $\\pi_2(X) \\oplus \\mathbb{F}$ is an induced $\\mathbb{F}[G_1 \\ast \\cdots \\ast G_k]$-module.\n\\end{proof}\n\n\\begin{thm}[Non-existence over \\text{$\\mathbb{Z}[G_1 \\ast \\cdots \\ast G_k]$}] \\label{thm:non-existence}\nFor all $k \\ge 2$, there exists a finite $2$-complex $X$ with $\\pi_1(X) \\cong G_1 \\ast \\cdots \\ast G_k$ such that $\\pi_2(X)$ is not an induced $\\mathbb{Z}[G_1 \\ast \\cdots \\ast G_k]$-module.\n\\end{thm}\n\nIn order to prove this, we will need the following method of proving that presentation complexes are homotopy equivalent.\nIf $\\mathcal{P} = \\langle x_1, \\dots, x_n \\mid r_1, \\dots, r_m \\rangle$, then an \\textit{elementary transformation} on $\\mathcal{P}$ is an operation that replaces a relator $r_i$ with:\n\\begin{clist}{(i)}\n\\item $\\omega r_i \\omega^{-1}$ for a word $\\omega \\in F(x_1 \\cdots, x_n)$ (\\textit{conjugation})\n\\item $r_i^{-1}$ (\\textit{inversion})\n\\item $r_i r_j$ or $r_j r_i$ for some $j \\ne i$ (\\textit{left or right multiplication}).\n\\end{clist}\nWe say that two group presentations $\\mathcal{P}$ and $\\mathcal{Q}$ are \\textit{$Q$-equivalent} if they are related by a sequence of elementary transformations. If $\\mathcal{P}$ and $\\mathcal{Q}$ are $Q$-equivalent, then $X_{\\mathcal{P}}$ and $X_{\\mathcal{Q}}$ are (simple) homotopy equivalent \\cite[p20-29]{HMS93}.\n\nWe begin by noting the following, which is a generalisation of \\cite[Theorem 3]{HLM85}.\n\n\\begin{prop} \\label{prop:def(G*H)}\nLet $k \\ge 1$ and let $m_i, n_i \\ge 1$ for $i =1, \\cdots, k$. Suppose there exists integers $r_i$, $q_i$ such that $(q_i,q_j)=1$ for all $i \\ne j$ and, for all $i$, we have: \n\\[ r_i^{m_i}-1=n_iq_i, \\qquad r_i \\equiv 1 \\mod n_i , \\qquad (m_i,n_i) \\ne 1. \\]\nThen $G = \\ast_{i=1}^k (\\mathbb{Z}\/m_i \\times \\mathbb{Z}\/n_i)$ has a presentation\n\\[ \\mathcal{P} = \\langle a_1, b_1, \\dots, a_k, b_k \\mid a_1^{m_1}, \\dots, a_k^{m_k}, a_1b_1a_1^{-1}b_1^{-r_1}, \\dots, a_kb_ka_k^{-1}b_k^{-r_k}, b_1^{n_1} \\cdots b_k^{n_k} \\rangle  \\]\nof deficiency $-1$. Furthermore, if $\\mathcal{P}_i = \\langle a,b \\mid a^{n_i}, b^{m_i}, [a,b] \\rangle$ is the standard presentation for $\\mathbb{Z}\/m_i \\times \\mathbb{Z}\/n_i$, then $X_{\\mathcal{P}} \\vee (k-1)S^2 \\simeq X_{\\mathcal{P}_1} \\vee \\cdots \\vee X_{\\mathcal{P}_k}$.\n\\end{prop}\n\nThe conditions on $m_i, n_i$ are satisfied in the case where $m_i = n_i = p_i$ for distinct primes $p_i$. In particular, this applies to all groups of the form $G = \\ast_{i=1}^k (\\mathbb{Z}\/p_i)^2$.\n\n\\begin{proof}\nThat proof that $\\mathcal{P}$ presents $G$ is similar to the case $k=2$ (see \\cite[Theorem 3]{HLM85}), as so will be omitted.\nLet $\\mathcal{P}_+$ denote the presentation $\\mathcal{P}$ with additional relations $b_1^{n_1}, \\cdots, b_{k-1}^{n_{k-1}}$, so that $X_{\\mathcal{P}_+} \\simeq X_{\\mathcal{P}} \\vee (k-1)S^2$. In order to show that $X_{\\mathcal{P}} \\vee (k-1)S^2 \\simeq X_{\\mathcal{P}_1} \\vee \\cdots \\vee X_{\\mathcal{P}_k}$, it therefore suffices to show that $\\mathcal{P}_+$ and $\\mathcal{P}_1 \\ast \\cdots \\ast \\mathcal{P}_k$ are $Q$-equivalent. To see this, note that we can replace $b_1^{n_1} \\cdots b_k^{n_k} \\leadsto b_k^{n_k}$ by left-multiplying by the $b_i^{-n_k}$ for $1 \\le i < k$. Since $r_i \\equiv 1 \\mod n_i$, we can then replace $a_ib_ia_i^{-1}b_i^{-r_i} \\leadsto [a_i,b_i]$ by successively right-multiplying by $b_i^{n_i}$.\n\\end{proof}\n\nWe say that two $\\mathbb{Z} G$-modules $M$ and $M'$ are \\textit{stably isomorphic}, written $M \\cong_s M'$, if there exists $a,b \\ge 0$ such that $M \\oplus \\mathbb{Z} G^a \\cong M' \\oplus \\mathbb{Z} G^b$.\n\n\\begin{lemma} \\label{lemma:stable-uniqueness}\nFor $1 \\le i \\le k$, let $M_i$, $M_i'$ be finitely generated $\\mathbb{Z} G_i$-lattices such that \n\\[ {\\iota_1}_\\#(M_1) \\oplus \\cdots \\oplus {\\iota_k}_\\#(M_k) \\cong {\\iota_1}_\\#(M_1') \\oplus \\cdots \\oplus {\\iota_k}_\\#(M_k')\\] \nas $\\mathbb{Z}[G_1 \\ast \\cdots \\ast G_k]$-modules. Then $M_i \\cong_s M_i'$ for all $1 \\le i \\le k$.\n\\end{lemma}\n\n\\begin{proof}\nFor $1 \\le i \\le k$, let $q_i : G_1 \\ast \\cdots \\ast G_k \\twoheadrightarrow G_i$ be the projection map. By applying $(q_1)_\\#$ to the given isomorphism of $\\mathbb{Z}[G_1 \\ast G_2]$-modules, we get that\n\\[ M_1 \\oplus \\bigoplus_{j=2}^k {(q_1 \\circ \\iota_j)}_\\#(M_j) \\cong M_1' \\oplus \\bigoplus_{j=2}^k {(q_1 \\circ \\iota_j)}_\\#(M_j') \\]\nas $\\mathbb{Z} G_1$-modules.\nIf $j \\ne 1$, then $q_1 \\circ \\iota_j : G_j \\to G_1$, $g \\mapsto 1$. If $M$ is a finitely generated $\\mathbb{Z} G_j$-module, then ${(q_1 \\circ \\iota_j)}_\\#(M) \\cong \\mathbb{Z} G_1 \\otimes_{\\mathbb{Z}} (\\mathbb{Z} \\otimes_{\\mathbb{Z} G_j} M)$. If $\\mathbb{Z} \\otimes_{\\mathbb{Z} G_j} M \\cong \\mathbb{Z}^{r_M} \\oplus F_M$ for $F_M$ a finite abelian group and $r_M \\ge 0$, then ${(q_1 \\circ \\iota_j)}_\\#(M) \\cong \\mathbb{Z} G_1^{r_M} \\oplus F_M G_1$.\n\nIn particular, for some finite abelian groups $F, F'$ and some $r,r' \\ge 0$, we have $M_1 \\oplus \\mathbb{Z} G_1^{r} \\oplus F G_1 \\cong M_1' \\oplus \\mathbb{Z} G_1^{r'} \\oplus F' G_1$.\nSince $M_1, M_1'$ are $\\mathbb{Z} G_1$-lattices, this $\\mathbb{Z} G_1$-isomorphism must induce isomorphisms $F G_1 \\cong F' G_2$ and $M_1 \\oplus \\mathbb{Z} G_1^{r} \\cong M_1' \\oplus \\mathbb{Z} G_1^{r'}$. Hence $M_1 \\cong_s M_1'$ and, by symmetry, we have that $M_i \\cong_s M_i'$ for all $1 \\le i \\le k$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:non-existence}]\nLet $p_1, \\cdots, p_k$ be distinct primes and let $G = \\ast_{i=1}^k (\\mathbb{Z}\/p_i)^2$. By \t\\cref{prop:def(G*H)}, $G$ has a presentation $\\mathcal{P}$ of deficiency $-1$. \nWe claim that $\\pi_2(X_{\\mathcal{P}})$ is not an induced $\\mathbb{Z}[G_1 \\ast \\cdots \\ast G_k]$-module, where $G_i = (\\mathbb{Z}\/p_i)^2$ for all $i$.\n\nSuppose that $\\pi_2(X_{\\mathcal{P}}) = {\\iota_1}_\\#(M_1) \\oplus \\cdots \\oplus {\\iota_k}_\\#(M_k)$ for $\\mathbb{Z} G_i$-modules $M_i$. \nAgain by \\cref{prop:def(G*H)}, we have that $X_{\\mathcal{P}} \\vee (k-1)S^2 \\simeq X_{\\mathcal{P}_1} \\vee \\cdots \\vee X_{\\mathcal{P}_k}$ where the $\\mathcal{P}_i = \\langle a,b \\mid a^{p_i}, b^{p_i}, [a,b] \\rangle$ are the standard presentations for $G_i$.\nHence, we have:\n\\[ {\\iota_1}_\\#(M_1 \\oplus \\mathbb{Z} G_1^{k-1}) \\oplus  \\bigoplus_{j = 2}^k {\\iota_j}_\\#(M_j) \\cong \\bigoplus_{j = 1}^k {\\iota_j}_\\#(\\pi_2(X_{\\mathcal{P}_j})). \\]\nBy \\cref{lemma:stable-uniqueness}, this implies that $M_i \\cong_s \\pi_2(X_{\\mathcal{P}_i})$ for all $i$ and so $M_i \\in \\Omega_3^{G_i}(\\mathbb{Z})$.\n\t\nIt follows from \\cite[Proposition 2.1]{Sw65}\t that $\\pi_2(X_{\\mathcal{P}_i}) \\in \\Omega_3^{G_i}(\\mathbb{Z})$ is minimal and so $M_i \\oplus \\mathbb{Z} G_i^{r_i} \\cong \\pi_2(X_{\\mathcal{P}_i}) \\oplus \\mathbb{Z} G_i^{s_i}$ for some integers $r_i \\le s_i$. This gives that:\n\\[ \\pi_2(X_{\\mathcal{P}}) \\oplus \\mathbb{Z} G^{s_1 + \\cdots + s_k + k -1} \\cong \\pi_2(X_{\\mathcal{P}}) \\oplus \\mathbb{Z} G^{r_1 + \\cdots + r_k}. \\]\nBy \\cite[Proposition 2.1]{Jo12a}, $\\sum s_i + k -1 = \\sum r_i \\le \\sum s_i$ which is a contradiction.\t\n\\end{proof}\n\n\\subsection{Uniqueness of induced module decompositions}\n\\label{subsection:uniqueness}\n\nWe will now turn to the question of uniqueness. The following is an immediate consequence of \\cref{cor:Bergman}.\t\n\n\\begin{prop}[Uniqueness over \\text{$\\mathbb{F}[G_1 \\ast \\cdots \\ast G_k]$}] \\label{prop:uniqueness}\nLet $X$ be a finite $2$-complex with $\\pi_1(X) \\cong G_1 \\ast \\cdots \\ast G_k$.\nIf $\\pi_2(X) \\otimes \\mathbb{F} \\cong {\\iota_1}_\\#(M_1) \\oplus \\cdots \\oplus {\\iota_k}_\\#(M_k)$ for $\\mathbb{F} G_i$-modules $M_i$ such that $\\mathbb{F} G_i \\nmid M_i$\n, then the $M_i$ are unique up to $\\mathbb{F} G_i$-module isomorphism.\n\\end{prop}\n\n\\begin{thm}[Non-uniqueness over \\text{$\\mathbb{Z}[G_1 \\ast \\cdots \\ast G_k]$}] \\label{thm:non-uniqueness}\nFor all $k \\ge 2$, there exists finite $2$-complexes $X_i$, $Y_i$ with $\\pi_1(X_i) \\cong \\pi_1(Y_i) \\cong G_i$ for $1 \\le i \\le k$ such that \n\\[ \\pi_2(X_1 \\vee \\cdots \\vee X_k) \\cong \\pi_2(Y_1 \\vee \\cdots \\vee Y_k) \\] \nbut, for all $i$, $\\mathbb{Z} G_i \\nmid \\pi_2(X_i), \\, \\pi_2(Y_i)$ and $\\pi_2(X_i) \\not \\cong \\pi_2(Y_i)$ are not $\\Aut(G_i)$-isomorphic.\n\\end{thm}\n\nIn order to prove this, we will begin by proving the following.\nWe note that this holds for a larger class of abelian groups than elementary abelian $p$-groups.\n\n\\begin{prop} \\label{prop:abelian-collapse}\nLet $k \\ge 2$ and let $p_i$ be distinct primes and $n_i \\ge 1$ for $i =1, \\cdots, k$. If $\\mathcal{P}_i$, $\\mathcal{P}_i'$ are two presentations for $G_i = (\\mathbb{Z}\/p_i)^{n_i}$ with the same deficiency, then $X_{\\mathcal{P}_1} \\vee \\cdots \\vee X_{\\mathcal{P}_k} \\simeq X_{\\mathcal{P}_1'} \\vee \\cdots \\vee X_{\\mathcal{P}_k'}$.\n\\end{prop}\n\n\\begin{proof}\nFor ease of notation, we will let $k=2$. The general case is analogous. Let:\n\\[ \\mathcal{P}_r^{(i)} = \\langle a_1, \\cdots, a_{n_i} \\mid a_1^{p_i}, \\cdots, a_{n_i}^{p_i}, [a_1^r,a_2], \\{ [a_i,a_j] : i < j, (i,j) \\ne (1,2) \\} \\rangle\\]\nfor $r \\in \\mathbb{Z}$ with $(r,p_i)=1$. This is a presentation for $G_i$ and, since the homotopy type of $\\mathcal{P}_r^{(i)}$ can be shown to depend only on $r \\mod p_i$, we can take $r \\in (\\mathbb{Z}\/p_i)^\\times$.\n\nIt was shown by Browning \\cite{Br79} (see also \\cite[Proposition 9.2]{GL91}) that, if $\\mathcal{P}$ is a presentation for $(\\mathbb{Z}\/p_i)^{n_i}$, then $X_{\\mathcal{P}} \\simeq X_{\\mathcal{P}_r^{(i)}} \\vee \\ell S^2$ for some $r \\in (\\mathbb{Z}\/p_i)^\\times$, $\\ell \\ge 0$.\nIt suffices to show that $X_{\\mathcal{P}_{r}^{(1)}} \\vee X_{\\mathcal{P}_{s}^{(2)}} \\simeq X_{\\mathcal{P}_{1}^{(1)}} \\vee X_{\\mathcal{P}_{1}^{(2)}}$ for all $r \\in (\\mathbb{Z}\/p_1)^\\times$, $s \\in (\\mathbb{Z}\/p_2)^\\times$.\n\nAs in \\cref{prop:def(G*H)}, there exists integers $r_i$, $q_i$ such that $(q_i,q_j)=1$ for all $i \\ne j$ and such that $r_i^{p_i}-1=p_iq_i$ and $r_i \\equiv 1 \\mod p_i$ for all $i$.\nLet $r,s$ be integers such that $(r,p_1)=1$ and $(s,p_2)=1$. If $(rq_1,sq_2)=1$ then, by the same argument as given in \\cref{prop:def(G*H)}, $G = G_1 \\ast G_2$ has a presentation:\n\\begin{align*} \\mathcal{P}_{r,s} = &\\, \\langle a_1, \\dots, a_{n_1},  b_1, \\cdots, b_{n_2} \\mid \\{a_i^{p_1}\\}_{i=2}^{n_1}, \\{b_i^{p_2}\\}_{i=2}^{n_2}, a_1^{p_1} \\cdot b_1^{p_2}, \\\\ \n& a_2(a_1^r)a_2^{-1}(a_1^r)^{-r_1},  b_2(b_1^s)b_2^{-1}(b_1^s)^{-r_2}, \\{[a_i,a_j], [b_i,b_j] : i < j, (i,j) \\ne (1,2) \\} \\rangle.\n\\end{align*}\nThis form is general for all $r, s$ since, by Dirichlet's theorem on arithmetic progressions, there exists $r',s'$ such that $r' \\equiv r \\mod p_1$, $s '\\equiv s \\mod p_2$ and $(r'q_1,s'q_2)=1$.\n\nLet $(\\mathcal{P}_{r,s})_+$ denote the presentation $\\mathcal{P}_{r,s}$ with the additional relation $a_1^{p_1}$.\nIn $(\\mathcal{P}_{r,s})_+$, we can replace $a_1^{p_1} \\cdot b_1^{p_2} \\leadsto b_1^{p_2}$ by left multiplying with $a_1^{-p_1}$, then replace $a_2(a_1^r)a_2^{-1}(a_1^r)^{-r_1} \\leadsto [a_2, a_1^r]$ by right multiplying with $a_1^{r_1-1}$ (which works since $r_i \\equiv 1 \\mod p_i$), and similarly $b_2(b_1^s)b_2^{-1}(b_1^s)^{-r_2} \\leadsto [b_2, b_1^s]$. This implies that $(\\mathcal{P}_{r,s})_+$ and $\\mathcal{P}_{r}^{(1)} \\ast \\mathcal{P}_{s}^{(2)}$ are $Q$-equivalent and so\n$X_{\\mathcal{P}_{r,s}} \\vee S^2 \\simeq X_{(\\mathcal{P}_{r,s})_+} \\simeq X_{\\mathcal{P}_{r}^{(1)}} \\vee X_{\\mathcal{P}_{s}^{(2)}}$.\n\nNote that $\\mathcal{P}_{1,s}$ differs from $\\mathcal{P}_{r,s}$ by a changing $a_2a_1a_2^{-1}a_1^{-r_1} \\leadsto a_2(a_1^r)a_2^{-1}(a_1^r)^{-r_1}$. Since both relations hold in $G$, we can add $a_2(a_1^r)a_2^{-1}(a_1^r)^{-r_1}$ to $\\mathcal{P}_{1,s}$ and add $a_2a_1a_2^{-1}a_1^{-r_1}$ to $\\mathcal{P}_{r,s}$ to get that $X_{\\mathcal{P}_{1,s}} \\vee S^2 \\simeq X_{\\mathcal{P}_{r,s}} \\vee S^2$. By symmetry, we also have that $X_{\\mathcal{P}_{r,1}} \\vee S^2 \\simeq X_{\\mathcal{P}_{r,s}} \\vee S^2$ and so $X_{\\mathcal{P}_{r}^{(1)}} \\vee X_{\\mathcal{P}_{s}^{(2)}} \\simeq X_{\\mathcal{P}_{1}^{(1)}} \\vee X_{\\mathcal{P}_{1}^{(2)}}$.\n\\end{proof}\n\nThe following can be found in \\cite[Theorem 1.2 (3)(iv)]{Li93}. This can also be deduced by combining the earlier work \\cite[Proposition 9]{SD79} with \\cite[Theorem 1.7]{Br79}.\n\n\\begin{lemma} \\label{lemma:abelian-non-cancellation}\nLet $G = (\\mathbb{Z}\/p)^n$ for $p$ prime and $n \\ge 1$. Let $\\delta(G)$ denote the number of $\\Aut(G)$-isomorphism classes of modules $\\pi_2(X_{\\mathcal{P}})$ for $\\mathcal{P}$ a presentation with $\\Def(\\mathcal{P}) = \\Def(G)$. If $p=2$, then $\\delta(G) = 1$ and, if $p$ is odd, then:\n\\[ \\delta(G) = \\begin{cases} (\\frac{p-1}{2},n-1), & \\text{if $n$ is even} \\\\\n (\\frac{p-1}{2}, \\frac{n-1}{2}), & \\text{if $n$ is odd}.\n \\end{cases}\n \\]\n\\end{lemma}\n\n\\begin{proof}[Proof of \\cref{thm:non-uniqueness}]\nLet $k \\ge 2$ and, for $i = 1, \\cdots, k$, let $p_i$ be distinct primes with $p_i \\equiv 1 \\mod 4$ and let $G_i = (\\mathbb{Z}\/p_i)^3$. By \\cref{lemma:abelian-non-cancellation}, we have that $\\delta(G_i) = 2$ and so there exists presentations $\\mathcal{P}^{(i)}$, $\\mathcal{Q}^{(i)}$ for $G_i$ such that $\\Def(\\mathcal{P}^{(i)}) = \\Def(\\mathcal{Q}^{(i)}) = \\Def(G)$ and $\\pi_2(X_{\\mathcal{P}^{(i)}}) \\not \\cong \\pi_2(X_{\\mathcal{Q}^{(i)}})$ are not $\\Aut(G_i)$-isomorphic.\n\t\nSimilarly to the proof of \\cref{thm:non-existence}, $\\pi_2(X_{\\mathcal{P}^{(i)}}), \\pi_2(X_{\\mathcal{Q}^{(i)}}) \\in \\Omega_3^{G_i}(\\mathbb{Z})$ are minimal by \\cite[Proposition 2.1]{Sw65}. This implies that $\\mathbb{Z} G_i \\nmid \\pi_2(X_{\\mathcal{P}^{(i)}}), \\, \\pi_2(X_{\\mathcal{Q}^{(i)}})$ for all $i$.\nBy \\cref{prop:abelian-collapse}, we have that $X_{\\mathcal{P}^{(1)}} \\vee \\cdots \\vee X_{\\mathcal{P}^{(k)}} \\simeq X_{\\mathcal{Q}^{(1)}} \\vee \\cdots \\vee X_{\\mathcal{Q}^{(k)}}$ and so $\\pi_2(X_{\\mathcal{P}^{(1)}} \\vee \\cdots \\vee X_{\\mathcal{P}^{(k)}}) \\cong \\pi_2(X_{\\mathcal{Q}^{(1)}} \\vee \\cdots \\vee X_{\\mathcal{Q}^{(k)}})$ as required.\t\n\\end{proof}\n\n\\section{List of open problems}\n\\label{subsection:problems}\n\nWe now collect together a list of open problems on stably free $\\mathbb{Z} G$-modules and the homotopy type of finite $2$-complexes. Problems \\ref{problem:2}, \\ref{problem:4} and \\ref{problem:5} have analogues for $(G,n)$-complexes and are open for all $n \\ge 2$.\n\n\\subsection{Branching behaviour under multiple stabilisations}\n\nAs in the introduction, it is possible to resolve \\cref{problem:2-complexes} by exhibiting two types of branching behaviour.\nThe first type is exhibited in Theorems \\ref{thm:main-SF} and \\ref{thm:main}. It therefore remains to determine whether examples of the second type also exist.\n\n\\begin{problemlist} \\label{problem:1}\nFor which $k \\ge 1$ does there exist a group $G$ and stably free $\\mathbb{Z} G$-modules $S_1$, $S_2$ such that $S_1 \\oplus \\mathbb{Z} G^k \\cong S_2 \\oplus \\mathbb{Z} G^k$ and $S_1 \\oplus \\mathbb{Z} G^{k-1} \\not \\cong S_2 \\oplus \\mathbb{Z} G^{k-1}$?\n\\end{problemlist}\n\n\\begin{problemlist} \\label{problem:2}\nFor which $k \\ge 1$ do there exist finite $2$-complexes $X_1$, $X_2$ with $\\pi_1(X_1) \\cong \\pi_1(X_2)$ such that $X_1 \\vee kS^2 \\simeq X_2 \\vee kS^2$ and $X_1 \\vee (k-1)S^2 \\not \\simeq X_2 \\vee (k-1)S^2$?\n\\end{problemlist}\n\nTo the best of our knowledge, both problems are open for all $k \\ge 2$. \\cref{problem:2} in the case $k=2$ can be found in \\cite[Problem C]{Dy79a} and later appeared in \\cite[p124]{HMS93}.\n\n\\begin{figure}[h] \\vspace{-4mm} \n\\begin{center}\n\\begin{tabular}{ccccc}\n\\begin{tabular}{l}\n\\begin{tikzpicture}\n\\draw[fill=black] (1.5,1) circle (2pt);\n\\draw[fill=black] (3,0) circle (2pt);\n\\draw[fill=black] (2.5,1) circle (2pt);\n\\draw[fill=black] (2,2) circle (2pt);\n\\draw[fill=black] (2,3) circle (2pt);\n\\node at (2,3.6) {$\\vdots$};\n\\draw[black] (2.75,1) node[right]{\n$\\left.\n    \\begin{array}{ll}\n           \\\\ \\\\ \\\\\n    \\end{array}\n\\right \\} k$};\n\\node at (1,3) {(a)};\n\n\\end{tabular}\n&&&&\n\\begin{tabular}{l}\n\\begin{tikzpicture}\n\\draw[fill=black] (1,1) circle (2pt);\n\\draw[fill=black] (2,1) circle (2pt);\n\\draw[fill=black] (3,1) circle (2pt);\n\n\\draw[fill=black] (2,2) circle (2pt);\n\n\\draw[fill=black] (1,3) circle (2pt);\n\\draw[fill=black] (2,3) circle (2pt);\n\\draw[fill=black] (3,3) circle (2pt);\n\n\\draw[fill=black] (2,4) circle (2pt);\n\n\\node at (2,2.6) {$\\vdots$};\n\\node at (2,4.6) {$\\vdots$};\n\n\\node at (1,4) {(b)};\n\n(1,1) -- (2,2) (2,1) -- (2,2) (3,1) -- (2,2)  (1,3) -- (2,4) (2,3) -- (2,4) (3,3) -- (2,4);\n\\end{tikzpicture}\n\\end{tabular}\n\\end{tabular}\n\\end{center}\n\\caption{Further branching phenomena in the tree of stably free $\\mathbb{Z} G$-modules or the tree of finite $2$-complexes $X$ with $\\pi_1(X) \\cong G$.} \\label{figure:further-branching}\n\\vspace{-2mm}\n\\end{figure}\n\n\\subsection{Finite $2$-complexes with arbitrary non-minimal Euler characteristic}\n\nWe can also ask whether or not the behaviour exhibited in \\cref{thm:main-SF-further} actually holds for finitely presented groups and finite $2$-complexes (see \\cref{figure:further-branching}b). As explained in the introduction, both problems have a negative answer when $\\mathbb{Z} G$ is Noetherian.\n\n\\begin{problemlist} \\label{problem:3}\nDoes there exist a finitely presented group $G$ such that, for infinitely many $k \\ge 1$, there exists a non-free stably free $\\mathbb{Z} G$-module of rank $k$?\n\\end{problemlist}\n\n\\begin{problemlist} \\label{problem:4}\nDoes there exist a finitely presented group $G$ such that, for infinitely many $k \\ge 0$, there exists homotopically distinct finite $2$-complexes $X_1$, $X_2$ with $\\pi_1(X_i) \\cong G$ and $\\chi(X_i) = k + \\chi_{\\min}(G)$?\n\\end{problemlist}\n\n\\subsection{Towards a general cancellation theorem for $2$-complexes}\n\nThe following is motivated by the results in \\cref{subsection:induced}.\nRecall that a CW-complex $X$ is irreducible if $X \\simeq Y \\vee Z$ for CW-complexes $Y$, $Z$ implies $Y$ or $Z$ is contractible.\n\n\\begin{problemlist} \\label{problem:5}\nLet $X_i$, $Y_i$ be irreducible non-simply connected finite $2$-complexes. When does $X_1 \\vee \\cdots \\vee X_k \\simeq Y_1 \\vee \\cdots \\vee Y_k$ imply that $X_i \\simeq Y_{\\sigma(i)}$ for some $\\sigma \\in S_k$?\n\\end{problemlist}\n\nHere irreducibility is necessary since it rules out the following two situations:\n\n\\begin{clist}{(a)}\n\\item \\textit{Exchange of subfactors}: If $X \\not \\simeq Z$, $Y \\not \\simeq \\ast$, then $(X \\vee Y) \\vee Z \\simeq X \\vee (Y \\vee Z)$.\n\\item \\textit{Non-cancellation}: If $X \\vee S^2 \\simeq Y \\vee S^2$, $X \\not \\simeq Y$, then $X \\vee (Z \\vee S^2) \\simeq Y \\vee (Z \\vee S^2)$. \n\\end{clist}\n\nThe finite $2$-complexes given in the proof of \\cref{thm:non-uniqueness} are irreducible and so show that some further conditions must be imposed.\nThis was shown to be true by Jajodia \\cite[Corollary 4]{Ja79} when the $X_i$, $Y_i$ have a single $2$-cell.\n\nFor a field $\\mathbb{F}$, it is possible to use Propositions \\ref{prop:existence} and \\ref{prop:uniqueness} to show that, subject to certain conditions, $\\pi_2(X_i) \\otimes \\mathbb{F} \\cong \\pi_2(Y_{\\sigma(i)}) \\otimes \\mathbb{F}$ are $\\Aut(G_i)$-isomorphic for some $\\sigma \\in S_k$. Results of this type are a first approximation to \\cref{problem:5} but are only useful in cases such as $G_i = T$ where information about $\\pi_2(X_i)$ is not lost by passing to $\\mathbb{F} G_i$.\nIn contrast, if $G_i$ is finite, then $\\pi_2(X_i) \\otimes \\mathbb{F}$ is determined by $\\chi(X_i)$ ssince $\\pi_2(X_i) \\otimes \\mathbb{F} \\cong I_{\\mathbb{F}}(G) \\oplus \\mathbb{F} G^{\\chi(X_i)-1}$ where $I_{\\mathbb{F}}(G) = \\Ker(\\varepsilon: \\mathbb{F} G \\twoheadrightarrow \\mathbb{F})$ \\cite[p120]{Jo03a}.\n\n\\section*{Acknowledgements}\nI would like to thank Martin Dunwoody, Jens Harlander and Francis E. A. Johnson for useful correspondence and a number of helpful comments.\nThis work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP\/N509577\/1.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nLarge language models (LMs) can perform unseen tasks by conditioning on a few labeled examples, effectively inferring the underlying tasks through a process known as \\textit{in-context learning} \\citep{gpt3}.\nHowever, task inference is implicit, and the ability of models to \\textit{explicitly} reason about it remains unexplored.\nIn this work, we show that LMs can explicitly describe an underlying task, in natural language, given a few labeled examples.\n\nWe introduce the \\textit{instruction induction} challenge, in which a model is provided with a few input-output demonstrations, and is requested to generate a natural language instruction describing the connection between the input-output pairs.\nIn our experiments, inducing instructions is done in a zero-shot manner by simply prompting the models to explain a small set of given demonstrations, as shown in Figure \\ref{fig:induction_example};\nwe do not perform fine-tuning or use any labeled instruction induction data.\n\nWe examine instruction induction on 24 tasks, ranging from morphosyntactic tasks (e.g., pluralization) to style transfer (e.g., formality) and sentiment analysis.\nAs a basic evaluation protocol, we collect human annotations and use them as gold-standard references; the generated instructions are then compared to these references using BERTScore \\citep{bertscore}.\nMoreover, we suggest a novel evaluation metric for instruction induction: \\textit{execution accuracy}. The execution accuracy of a generated instruction is measured by testing whether LMs can correctly perform the task in a zero-shot manner by using the generated instruction alone, without any demonstrations.\n\nOur experiments reveal a surprising ability at generating correct instructions.\nThe best-performing model, InstructGPT \\citep{instruct-gpt}, achieves an average BERTScore of 44.4, compared to human performance of 60.0; when measuring execution accuracy, the model reaches 43.6, with human-written instructions reaching 66.4.\nFor some tasks, the model's performance is on par or even better than human performance.\nWhen qualitatively examining the generated instructions, we often observe accurate instructions, even for some of the more challenging tasks.\nFor instance, in the task of formality style transfer, generated instructions include ``Translate the inputs into more formal language'' and ``Use formal language''.\nFor semantic text similarity, the generated instructions include ``For each input, rate the similarity of the two sentences on a scale of 0 to 5, with 5 being a perfect match'' and ``Determine whether the two sentences are about the same thing''.\n\nDespite these impressive results, we find that this ability is currently unique to InstructGPT \\citep{instruct-gpt}, which is both very large (175B parameters) and was especially fine-tuned to follow instructions.\nAblations on smaller versions of InstructGPT as well as the original 175B-parameter GPT-3 \\citep{gpt3} yield dramatically weaker performance.\nThese findings are in line with recent work showing that increasing model size unlocks new capabilities \\citep{palm,predictability-and-surprise}, and serves as additional evidence for the strength of instruction tuning \\citep{sanh2022multitask,wei2022finetuned,instruct-gpt}, perhaps even pointing to the necessity of complementing standard next-word prediction with additional objectives.\n\n\nThe fact that models can induce natural language instructions suggests that instruction-induction may serve as a learning paradigm of its own, where the goal is to find the best description in the natural language hypothesis space. While we currently provide a proof-of-concept for that idea, extending it by grounding models in natural language has the immediate benefit of human interpretability, and might also help alleviate overfitting and other issues associated with spurious correlations.\n\n\n\\begin{figure}[t!]\n\\centering\n \\includegraphics[width=1.0\\textwidth]{images\/fig1_office.pdf}\n \\caption{An example of instruction induction for the task of formality style transfer. \\textit{Left:} the standard in-context learning setting; given five demonstrations, complete the sixth. \\textit{Right:} instruction induction; the language model is prompted to generate a natural language instruction that describes the demonstrations. Model completions are in \\textcolor{fig1blue}{blue}, prompt templates are in \\textcolor{fig1pink}{pink}.}\n \\label{fig:induction_example}\n\\end{figure}\n\n\n\\section{Instruction Induction}\n\n\n\nWe begin by formulating the task of instruction induction. Given a sequence of $n$ demonstrations $\\{x_k, y_k\\}_{k \\in \\{1,\\ldots,n\\}}$, the goal is to generate a \\textit{single} natural language instruction, such that for each $x_k$, following the instruction results in $y_k$.\nThis format is similar to in-context learning \\citep{gpt3}, only here the desired output is an \\textit{instruction} describing the relation between the inputs and outputs of the demonstrations. We require models to perform this in a zero-shot setting, without any fine-tuning on labeled data.\nFigure~\\ref{fig:induction_example} illustrates the difference between standard in-context prompting and instruction-induction prompting.\n\nTo elicit models to generate instructions, we consider prompts that would elicit humans to do so. We design a meta-prompt presenting instruction induction as a challenge puzzle and verify its clarity in a human study (\\S\\ref{sec:verification}).\nThe prompt is presented in Figure~\\ref{fig:induction_example} (right side, in pink).\\footnote{We found this prompt informative for both humans and models in preliminary experiments.}\n\nWhile prior work already shows that large LMs are often able to infer a latent task from a given set of demonstrations, this has been largely based on their ability to \\textit{execute} the task on a held-out example.\nInstruction induction requires that the model \\textit{describe} the underlying task in natural language.\n\n\\section{Data}\n\\label{sec:data}\n\nWe evaluate on 24 tasks, listed in Table \\ref{tab:tasks}.\nWe select these tasks as they vary in difficulty and represent different aspects of language understanding, ranging from surface-level spelling to sentence similarity and causality detection.\\footnote{See Appendix~\\ref{sec:data_construction} for the full details of each task.}\nWe review the dataset's format, the annotation and verification processes we conducted to ensure that the tasks are viable, and finally discuss a theoretical limitation of this setup.\n\n\\input{03a_data_tab}\n\n\n\\subsection{Format}\n\nIn every task, each single \\textit{demonstration} $(x_k, y_k)$ is formatted as follows:\n\\begin{center}\nInput: $x_k$ ~~\\\\\nOutput: $y_k$\n\\end{center}\nFor instance, one demonstration in the pluralization task is ``Input: cat'' followed by ``Output: cats'' in a new line.\nWe split each task's demonstrations into two sets: an \\textit{induce} set, which we use for generating instructions, and an \\textit{execute} set, which is held out for the execution accuracy evaluation metric (see \\S\\ref{sec:execution}).\nEach \\textit{instruction induction example} is composed of 5 demonstrations sampled randomly without replacement from the induce set, concatenated with new-line separators; we create 100 examples for each task.\nWhen generating instructions, each example is placed inside the instruction induction prompt, and fed to the model (Figure~\\ref{fig:induction_example}, right).\n\n\n\\subsection{Annotating Reference Instructions}\n\\label{sec:annotations}\n\nWe collect 10 gold-reference human-annotated instructions via college-graduate English-speaking annotators.\nFor each task, we provide the annotators with the exact same input we intend to provide a model: 5 input-output demonstrations wrapped by the instruction-induction prompt (Figure \\ref{fig:induction_example}).\nWe manually verify each annotation and discard ones that do not correctly describe the task. \nWe refer to this set of annotations as the \\textit{gold} annotations, and use them for reference-based evaluation (see \\S\\ref{sec:eval}).\n\n\n\n\\subsection{Verification}\n\\label{sec:verification}\n\nPrior to the instruction induction experiments, we conduct two tests to ensure that either models or humans can infer the underlying task given 5 demonstrations.\nWe first verify that models can indeed execute our tasks given 5 demonstrations using in-context learning.\nSecondly, we conduct a human study to confirm that 5 demonstrations are enough for humans to describe the latent tasks.\n\n\\paragraph{In-Context Learning}\nWe prompt models with 5 input-output demonstrations and concatenate an additional test input $x_{k+1}$,\nand verify that the models are able to correctly predict $y_{k+1}$ (Figure~\\ref{fig:induction_example}, left).\nFor each task, we repeat this experiment 100 times, each with a different set of demonstrations and test inputs.\nWe do not provide the model with any instruction beyond the ``Input: $x_k$ Output: $y_k$'' format. We evaluate each task using its predefined evaluation metric.\\footnote{All metrics are variants of simple string matching, with some task-specific heuristics, for example, to allow for multiple correct answers. See Appendix~\\ref{sec:data_construction} for exact details.}\nThe in-context results for GPT-3 \\citep{gpt3} and InstructGPT \\citep{instruct-gpt} (see model details in \\S\\ref{sec:results}) are reported in Table \\ref{tab:verification_tab} in Appendix~\\ref{sec:data_verification}, which shows that in-context learning can reach 80\\% accuracy and above on most tasks.\n\n\\paragraph{Human Study}\nTo assess the human ability to induce instructions, we collect human-written instructions, using annotators that \\textit{did not} participate in the gold references collection. As in the gold-reference annotation process, we provide annotators with the same input we intend to provide to models. We refer to this set of annotations as the \\textit{control} annotations. We then manually count, for each task, the number of annotators that provided a correct instruction, and report the correct instructions percentage in Table \\ref{tab:verification_tab} (Appendix~\\ref{sec:data_verification}). \nIn all but one task (\\textit{Larger Animal}), at least 4 out of 5 annotators were able to produce correct task descriptions.\n\nWe also use the control group's annotations to establish a human baseline for automatic evaluation metrics.\nFor reference-based evaluation (\\S\\ref{sec:bertscore}), we treat the control annotations as generated instructions and compare them against the gold annotations, while for execution accuracy (\\S\\ref{sec:execution}), we use the control annotations to measure human performance, and the gold references as a ceiling metric.\n\n\n\\subsection{Ambiguity}\n\\label{sec:ambiguity}\n\nA theoretical challenge in inducing instructions is ambiguity.\nFor example, when given the single demonstration ``Input: The coffee is too hot. Output: The, too, hot'', one could infer that the underlying task is either ``write all the words containing the letter T'' or ``write all the three-lettered words'', both valid interpretations.\nAmbiguity might confuse models tasked with instruction induction while also making evaluation less reliable.\nIn practice, providing 5 demonstrations typically resolves the ambiguity in our set of tasks.\nAs evident from the data verification process, our tasks can typically be inferred by models and\/or humans.\n\nInducing more complex task descriptions, such as predicting detailed annotation guidelines, may pose a greater challenge in terms of ambiguity. We hypothesize that providing more than 5 demonstrations could mitigate some of that challenge, and leave further exploration of this avenue to future work.\n\n\n\n\\section{Evaluating Generated Instructions}\n\\label{sec:eval}\nAs a standard text generation metric, we report BERTScore \\citep{bertscore}. However, the instruction induction challenge has a unique property, which does not usually hold for other text generation tasks: the instructions are \\textit{executable}. Their correctness can therefore be measured directly by utilizing them as prompts.\n\n\n\n\\subsection{Reference-Based Evaluation}\n\\label{sec:bertscore}\n\nWe use BERTScore \\citep{bertscore} to compare the model-generated instructions against the collected gold annotations.\nAs mentioned in \\S\\ref{sec:annotations}, we use only the correct, verified annotations as references. \nWe take the maximal BERTScore-F1 over all gold-reference annotations to account for natural variations in instruction formulation.\\footnote{We use BERTScore version 0.3.11 with the DeBERTa-xl-MNLI model \\citep{he2021deberta, nangia-etal-2017-repeval}.}\nWe also establish a human baseline for each task using the \\textit{control} annotations, which were collected from a separate control group of annotators (\\S\\ref{sec:data_verification}), which we compare against the \\textit{gold} annotations in exactly the same way as model-generated instructions.\n\n\n\\subsection{Execution Accuracy}\n\\label{sec:execution}\n\nWe introduce \\textit{execution accuracy}, a new metric unique to the instruction induction task.\nTo measure the execution accuracy of a predicted instruction $I$ (e.g., ``Write the plural form of the given word.'') for a task $T$ (pluralization), we prompt a model with $I$ and an input $x$ (``cat'').\nWe then test, given $I$ and $x$, whether the model can correctly predict $y$, the output of performing $T$ on the input $x$ (\\textit{cats}).\n\nTo obtain meaningful results, we measure execution accuracy on the 100 held-out \\textit{execute} examples for each task.\nThe execution accuracy of an instruction $I$ is therefore computed by taking the average over $Score_{T}(I(x_n),y_n)$ for all $x_n$ in the \\textit{execute} set, where $Score_{T}$ denotes the task's corresponding metric (see Appendix~\\ref{sec:data_construction}), and $I(x_n)$ is the result of prompting a predefined language model with the instruction $I$ and the input $x_n$.\nAs recent models are trained to follow instructions \\citep{sanh2022multitask,wei2022finetuned,instruct-gpt}, and due to the relative clarity of our tasks, we expect correct instructions to yield high execution accuracy when using a sufficiently powerful execution model.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\paragraph{Baseline Models}\nWe experiment with eight versions of GPT-3 \\citep{gpt3}, a Transformer decoder language model.\nFirst, we experiment with the most current version available in the OpenAI API, for each of the four available model sizes.\nThough not stated explicitly in the API, we assume these models are those reported by \\citet{instruct-gpt}, and we therefore refer to them as \\textit{Instruct} models.\\footnote{Concretely, we use: text-davinci-002, text-curie-001, text-babbage-001, text-ada-001.}\nWe also experiment with the four originally published GPT-3 versions.\\footnote{davinci, curie, babbage, ada.} \nBy default, we refer to the largest Instruct model as \\textit{InstructGPT}, and the original 175B-parameter model as \\textit{GPT-3}.\nAll model generations were produced using the greedy decoding algorithm.\n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{@{}lrr@{}}\n\\toprule\n\\textbf{Model} & \\textbf{BERTScore} & \\textbf{Execution} \\\\\n\\midrule\n\\textit{GPT-3} & & \\\\\n~~~~Ada & -7.7 & 4.0 \\\\\n~~~~Babbage & 4.1 & 3.2 \\\\\n~~~~Curie & 13.9 & 7.9 \\\\\n~~~~DaVinci & 14.6 & 6.5 \\\\\n\\midrule\n\\textit{InstructGPT} & & \\\\\n~~~~Ada & 5.9 & 4.4 \\\\\n~~~~Babbage & -0.5 & 3.8 \\\\\n~~~~Curie & 10.7 & 8.8 \\\\\n~~~~DaVinci & 44.4 & 43.6 \\\\\n\\midrule\n\\textit{Human (Control)}  & 60.0 & 66.4 \\\\\n\\bottomrule\n\\\\\n\\end{tabular}\n\\caption{Average BERTScore and execution accuracy across tasks. BERTScore is measured against the gold references. The execution accuracy for all generated instructions is measured using InstructGPT as the execution model. Human performance is measured using the human control group's instructions.}\n\\label{tab:averages}\n\\end{table}\n\\label{sec:results_bertscore}\n\n\n\\subsection{Comparing to Gold Annotations}\n\nFigure~\\ref{fig:bertscore} presents the average BERTScore per task (see \\S\\ref{sec:bertscore}).\nResults show that the InstructGPT model has, to some extent, the ability to induce instructions from a few demonstrations; in 13 out of 24 tasks it achieves at least 75\\% of human performance.\nGPT-3, on the other hand, is quite far from human performance across the board.\n\nTable~\\ref{tab:averages} shows the average scores across all tasks. We observe the same trend; while InstructGPT's BERTScore is 15.6 points lower than human performance, the gap between GPT-3 and humans is 45.4 points.\nMoreover, we observe that smaller models -- even those fine-tuned to follow instructions -- do not exhibit any instruction-induction abilities. Scores are slightly higher for larger models of the same family (except for the InstructGPT-Babbage outlier), but are overall low.\nExcluding the largest models, there does not appear to be a significant advantage for Instruct models over the originals when controlling for model size.\n\n\n\\subsection{Execution Accuracy}\\label{sec:results_execution}\n\nWe compute the execution accuracy as detailed in \\S\\ref{sec:execution}, and report the average over 100 generated instructions for each task.\nAs an execution model, we use the largest InstructGPT model.\nWe also use this model to induce instructions, and while using it as an execution model might bias results towards its own generations, preliminary experiments show that no other model is as good at following instructions as InstructGPT.\nAs a point of reference, we apply the execution accuracy evaluation protocol to human-written instructions.\nFirst, to compare models with human performance, we measure the execution accuracy of the \\textit{control} annotation set.\nSecond, to account for limitations in the execution model, we measure execution accuracy of the correct (manually verified) \\textit{gold} annotations, which acts as an approximated ceiling metric.\n\nFigure~\\ref{fig:execution_acc} presents the execution accuracy per task.\nIn 12 out of 24 tasks, InstructGPT achieves at least 75\\% of the execution accuracy measured for the human-written instructions.\nGPT-3 shows much weaker execution accuracy, scoring less than 10\\% on 20 of the 24 tasks.\nIn fact, only in the cases of formality, passivization, and cause selection does it approach human performance, and that is largely an artifact of a more lenient evaluation metric in the case of formality and cause selection, or due to the execution model being right for the wrong reasons in the case of passivization (see \\S\\ref{sec:analysis}).\nIn some tasks, the control annotations are of high quality and reach a higher score than the verified gold annotations, likely due to variance of the execution model in such cases.\n\nTable~\\ref{tab:averages} shows the same trends. On average, InstructGPT achieves 65.7\\% of human performance, while GPT-3 reaches only 9.8\\% of human performance. When considering different model families or sizes, we do not see any substantial improvements when increasing model size or adding instruction tuning, with the exception of the largest InstructGPT model. The ability to generate instructions seems to only emerge when a model is both large enough and aligned to follow instructions.\nOverall, even the best-performing model still does not reach human performance, leaving room for future improvement.\n\n\n\n\n\n\n\n\\section{Analysis}\n\\label{sec:analysis}\n\n\n\n\\begin{table}[t]\n    \\small\n    \\centering\n    \\begin{tabular}{@{}p{0.18\\textwidth}p{0.35\\textwidth}p{0.41\\textwidth}@{}}\n    \\toprule\n    \\textbf{Task} & \\textbf{GPT-3} & \\textbf{InstructGPT} \\\\\n    \\midrule\n    First letter & The friend's output was: & Write the first letter of each word. \\\\\n    \\midrule\n    Sentence Similarity & The friend wrote the following output: & For each input, rate the similarity of the two sentences on a scale of 0 to 5, with 5 being a perfect match. \\\\\n    \\midrule\n    Pluralization & The friend's output was: & Add `s' to the end of each word. \\\\\n    \\midrule\n    Passivization & The friend wrote the following output: & Reverse the order of the subject and the object in the sentence. \\\\\n    \\midrule\n    Antonyms & The friend's output was: & Reverse the input. \\\\\n    \\bottomrule\n    \\\\\n    \\end{tabular}\n    \\caption{Examples of the instructions generated by GPT-3 and InstructGPT for five of our tasks.}\n    \\label{tab:analysis}\n\\end{table}\n\nTo gain further insight into the successes and failures of instruction induction prompting, we manually analyze the model-generated instructions of 5 tasks.\nTable \\ref{tab:analysis} shows the most common predictions of GPT-3 and InstructGPT for each of these tasks.\n\nInstructGPT obtains high, or close to human execution accuracy scores for three of these tasks (\\textit{First Letter}, \\textit{Sentence Similarity}, \\textit{Pluralization}).\nIndeed, the instructions for both \\textit{First Letter} and \\textit{Sentence Similarity} accurately describe the task.\nHowever, the instruction generated for \\textit{Pluralization} is not entirely precise, since it dismisses other forms of pluralization such as -es, -ies, and irregulars.\nAlthough the instruction only asks to add an ``s'', the execution model often ignores the specifics and produces the correct plural form; in one case, the input word was ``life'' and the output was ``lives''.\nWhile this particular instruction accounts for 24\\% of the induced instructions in the pluralization task, some predictions do explicitly mention pluralization, though not always accurately, e.g., ``Add -s to the end of each word to make it plural''.\n\nFor some tasks, InstructGPT fails to produce accurate instructions, even if it is able to solve via in-context learning (see Table~\\ref{tab:verification_tab}).\nIn \\textit{Passivization}, 98\\% of the predicted instructions were to simply ``reverse the order of the subject and object'', while ignoring additional surface-form manipulations needed to convert the given sentence into passive form; e.g., for the input ``The authors supported the scientist'', following the instructions produces the output ``The scientist supported the authors'', while the correct passive form is ``The scientist was supported by the authors''.\nSurprisingly, the instructions generated by GPT-3 obtained higher execution accuracy than the InstructGPT, even though they were entirely unrelated.\nIn 24\\% of the cases, GPT-3 predicted ``The friend wrote the following output:'' -- an instruction that apparently prompts the execution model to often rephrase the input in passive form.\nLastly, in \\textit{Antonyms}, 60\\% of InstructGPT's predictions were ``Reverse the input'', and another 11\\% were ``Reverse the word''.\nWhile one could imagine an interpretation of these instructions that reflects the task (reversing the \\textit{meaning} of the word), the execution model interprets them literally, and reverses the input words' letters.\n\nOverall, GPT-3 did not exhibit any instruction induction abilities, although it did often phrase outputs in imperative language.\nOne relatively common prediction was the generic instruction ``Write an output for every input''.\nBecause these empty instructions are in the right format, they tend to have some overlap with the reference instructions, which inflates their BERTScore.\nExecution accuracy, on the other hand, is robust to this phenomenon, and typically assigns GPT-3's outputs very low scores.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\\paragraph{In-Context Learning}\n\\citet{gpt3} suggest that models can learn a task by conditioning on few input-output demonstration pairs, without any fine-tuning or gradient updates. This paradigm, known as \\textit{in-context learning} or \\textit{prompt-based learning} \\citep{prompt-based}, has been the focus of many research efforts lately:\n\\citet{glam} suggest methods for more efficient in-context learning,\n\\citet{zhao-2021-calibrate} study methods for improving the stability and accuracy of prompt-based models, \\citet{chen-meta-incontext} and \\citet{min2022metaicl} conduct meta-training with an in-context learning objective,\nwhile other work studies the effect of the provided prompts \\citep{reynolds-2021-prompt-programming, webson-pavlick-2021, min2022rethinking}, or suggests prompt reframing techniques \\citep{reframing-prompts} and prompt retrieval methods \\citep{prompt-retrieval}.\nTo the best of our knowledge, all previous work study in-context learning through the lens of \\textit{executing} a latent task, while we focus on the ability to explicitly \\textit{describe} it.\n\n\\paragraph{The Instruction Paradigm}\n\\citet{turking-test} propose to learn new tasks from natural language instructions.\n\\citet{naturalinstructionsv1} and \\citet{naturalinstructionsv2} collect crowdsourcing instructions used to create NLP datasets into a benchmark for measuring the ability to solve tasks by reading instructions.\nRecent work shows that fine-tuning on task instructions (\\textit{instruction tuning}) improves the zero-shot learning abilities of LMs \\citep{sanh2022multitask,wei2022finetuned,instruct-gpt}. This work focuses on models' ability to \\textit{generate} instructions, rather than their ability to \\textit{execute} instructions written by humans.\n\n\\paragraph{Intermediate Reasoning Steps}\n\\citet{nye2022show} show that LMs can perform complex computations by writing intermediate steps on a ``scratchpad''.\nIn \\textit{chain of thought prompting} \\citep{wei2022chain}, input-output demonstrations are enriched with sentences elaborating intermediate task reasoning steps, improving the performance of LMs on tasks requiring reasoning skills.\nSubsequent work further improves the performance on such tasks using a \\textit{self-consistency} ensemble \\citep{self-consistency-cot}, which samples a set of diverse chain-of-thought reasoning paths, taking the majority vote over all generated answers.\n\\citet{star-cot} utilize a small set of examples labeled with chain-of-thought rationales and a large set of unlabeled data to iteratively bootstrap automatic rationale generation, thus creating a large dataset labeled with such rationales to enable fine-tuning.\nIn contrast, we study the ability of LMs to generate a description of the task, rather than generating intermediate reasoning steps as a means of executing complex tasks.\n\n\n\n\n\n\n\n\\section{Discussion}\n\nThis work demonstrates that large LMs can not only infer new tasks based on a handful of demonstrations, but also describe them in natural language.\nWe provide evidence of this ability on a diverse set of language tasks, and show that while instruction induction abilities are limited to a single state-of-the-art model, this model does indeed approach human performance on about half the tasks.\n\nIt is not unreasonable to assume that models in the near future will be even better at processing human-generated instructions, and it is therefore interesting to discuss the potential applications of instruction induction.\nIn particular, we envision a use case in which instruction induction serves as a machine learning approach; instead of converting a dataset into a set of continuous parameters, we could produce a natural language instruction that best describes the data. Grounding the model in concise natural language has the advantage of interpretability, and has the potential to solve fundamental issues pertaining to spurious correlations. While it is still too early to determine whether this approach is viable, we view it as an intriguing direction for future research.\n\n\n\n\n\\section{Dataset Details}\n\\label{sec:data_construction}\n\nThis appendix details each task's dataset (\\S\\ref{sec:data_construction_details}).\nSome datasets rely on a set of common English nouns (CEN), described at \\S\\ref{sec:common_nouns}.\n\n\n\\subsection{Tasks}\n\\label{sec:data_construction_details}\n\nWe elaborate on each task's data source, preprocessing protocol, and evaluation metric used in the in-context learning and execution accuracy experiments.\nAs mentioned in \\S\\ref{sec:data}, each task has \\textit{induce} and \\textit{execute} sets; unless stated otherwise, we sample 100 examples as the execute set for each task.\nWhen evaluating outputs, the generated text is first normalized; we take only the first generated sentence and  lowercase it.\nWe apply exact string match as the evaluation metric where applicable, elaborating only where alternative metrics are used.\n\n\\paragraph{First Letter}\nIn each demonstration, $x_k$ is a noun, and $y_k$ is the first letter of that noun. We construct the demonstrations by extracting the first letter of each word in CEN.\n\n\\paragraph{Second Letter}\nIdentical to the \\textit{First Letter} task, only here $y_k$ is the second letter of $x_k$.\n\n\\paragraph{List Letters}\n$x_k$ is a noun from CEN, and $y_k$ is a list of $x_k$'s letters, separated by spaces.\n\n\\paragraph{Starting With}\n$x_k$ contains a sentence and a letter in brackets, and $y_k$ lists the words in $x_k$ that start with the given letter. We avoid cases in which $y_k$ is empty, i.e., there is always at least one word in the input sentence starting with the given letter.\nSentences are taken from the CoLA dataset \\citep{warstadt2018neural}.\nFor the induce set, we create all (sentence, letter) pairs using CoLA's train set, and then sample 3,000 pairs.\nFor the \\textit{execute} set, we create all (sentence, letter) pairs from CoLA's in-domain and out-of-domain dev sets, and then sample 50 in-domain and 50 out-of-domain examples.\nWe evaluate using exact \\textit{set} match, by treating the output (and $y_k$) as a set of strings.\n\n\\paragraph{Pluralization}\nGiven a singular noun $x_k$, produce the plural form $y_k$.\nWe take noun inputs from the CEN set, filtering out mass nouns using a predefined list.\\footnote{\\url{https:\/\/gist.github.com\/sudodoki\/b5408fa4ba752cc22597250fc58a5970}}\nTo create the plural forms, we apply an automatic pluralization engine\\footnote{\\url{https:\/\/pypi.org\/project\/inflect\/}} and exclude nouns for which the engine's output did not appear at least 50 times in the Wikitext-103 corpus.\nThis results in 2,043 singular-plural noun pairs.\n\n\\paragraph{Passivization}\nGiven a simple active sentence $x_k$, rephrase the sentence in passive voice $y_k$.\nWe use the 1,000 HANS \\citep{mccoy-etal-2019-right} evaluation set active-passive entailed sentence pairs.\n\n\\paragraph{Negation}\n$y_k$ is the negation of the input sentence $x_k$.\nWe use the negated LAMA dataset \\citep{petroni-etal-2019-language,kassner-schutze-2020-negated}, taking the 304 negated SQuAD \\citep{rajpurkar-etal-2016-squad} sentences, 300 ConceptNet \\citep{speer-havasi-2012-representing} sentences, 200 T-REx \\citep{elsahar-etal-2018-rex} sentences and 200 Google-RE\\footnote{\\url{https:\/\/code.google.com\/archive\/p\/relation-extraction-corpus\/}} sentences. For ConceptNet and T-REx, we manually select these sentences to ensure their quality. For Google-RE, we automatically sample 100 sentences from the \\textit{place of birth} relation, and 100 from the \\textit{place of death} relation.\n\n\\paragraph{Antonyms}\n$y_k$ is the antonym of the input word $x_k$.\nWe use the antonym pairs from oLMpics \\citep{talmor-etal-2020-olmpics}, which were extracted from ConceptNet \\citep{speer-havasi-2012-representing} and WordNet \\citep{fellbaum1998wordnet}.\nFor uniformity, we verify that all pairs are indeed antonyms according to WordNet.\n\n\\paragraph{Synonyms}\n$x_k$ is a word and $y_k$ is its synonym.\nAs in the antonyms task, we use the synonym pairs of \\citet{talmor-etal-2020-olmpics}.\nSince there can be multiple synonyms for each input word, the task's in-context and execution accuracy are evaluated by testing whether the gold answer (a single word) is contained in the predicted answer (which may be a list of words).\n\n\\paragraph{Membership}\n$x_k$ is a list of words, where some of the words represent animals, and $y_k$ lists the animals from $x_k$.\nTo construct the task's data, we first select 6 word categories: animals, clothing, colors, food, vehicles, and professions.\nWe then take 10-50 words from each category, using only words that are categorized at the A1 or A2 levels according to the Common European Framework of Reference for Languages (CEFR).\\footnote{https:\/\/languageresearch.cambridge.org\/american-english}\nUsing these words, we create random lists containing between 5 to 7 words, where 3 or 4 are animals and the rest belong to one of the other 5 categories. The induce split is constructed by sampling 3,000 such combinations, using 80\\% of each category's words. The execute split is constructed by sampling 100 such combinations, using the remaining 20\\% of each category's words.\nThe task's in-context and execution accuracy are evaluated using an exact \\textit{set} match, by treating the output (and $y_k$) as a set of strings.\n\n\\paragraph{Rhymes}\n$y_k$ is a rhyme of the input word $x_k$.\nThe data was constructed by taking words categorized at the A1, A2, or B1 levels according to CEFR.\nWe then use CMU's pronouncing dictionary\\footnote{\\url{https:\/\/github.com\/cmusphinx\/cmudict}} to find rhyming groups for these words.\nThe execute split is constructed by sampling 30 rhyming groups, each containing two or more words, and sampling 100 unique words. The induce split is constructed using the rest of the rhyming groups.\nWe evaluate this task by checking whether the predicted word is contained in the rhyming group of $x_k$.\n\n\\paragraph{Larger Animal}\n$x_k$ is two animals, and $y_k$ is the (physically) larger one.\nWe use the object comparison data from oLMpics \\citep{talmor-etal-2020-olmpics}, taking the train split, which only contains animals.\nWe construct the induce set using a sample of 80\\% of the animals and the execute set by sampling 100 pairs out of the remaining 20\\% animals.\n\n\\paragraph{Cause Selection}\n$x_k$ contains two sentences describing related events, where one event caused the other; $y_k$ contains the cause sentence.\nAs data source, we use the 50 examples from the BIG-bench \\citep{bigbench} \\textit{Cause and Effect} task, randomly splitting them to equally-sized induce and execute sets.\nIn each of the induce demonstrations, we randomly sample the position of the cause sentence (either the first or the second sentence in $x_k$).\nFor examples in the execute set, we take both options for each cause and effect pair, doubling the data.\n\n\\paragraph{Common Concept}\n$x_k$ contains a few entities that share a non-trivial common underlying concept, while $y_k$ describes that common concept.\nWe use the 32 examples from \\textit{Novel Concepts} in BIG-bench \\citep{bigbench}, using half for induce and half for execute.\nAs the BIG-bench answers usually contain clear ``task markers'' (e.g., answers that start with ``They all have...'', indicating that the task was to find a common concept), we remove them from our demonstrations.\nThe task's in-context and execution accuracy are evaluated using unigram overlap (F1).\n\n\\paragraph{Formality}\n$x_k$ is a sentence in informal English, and $y_k$ is its paraphrase in more formal language.\nWe write 30 sentence pairs ourselves, following existing guidelines for converting informal sentences into formal ones.\\footnote{\\url{https:\/\/www.niu.edu\/writingtutorial\/style\/formal-and-informal-style.shtml}, \\url{https:\/\/www.uts.edu.au\/current-students\/support\/helps\/self-help-resources\/grammar\/formal-and-informal-language}}\nThe task's in-context and execution accuracy are evaluated using unigram overlap (F1).\n\n\\paragraph{Sum}\n$x_k$ contains two numbers separated by a space, and $y_k$ is their sum.\nFor each number in the range $[0,99]$, we enumerate over all pairs.\n\n\\paragraph{Difference}\n$x_k$ contains two numbers separated by a space, and $y_k$ is the difference between them.\nWe use all number pairs such that both input numbers are in the range $[0,198]$, and always subtract the smaller number from the bigger number.\n\n\\paragraph{Number to Word}\n$x_k$ is a number written in digits (e.g., 28), and $y_k$ is the same number written in words (e.g, twenty-eight).\nWe use all numbers in range [0,9999].\n\n\\paragraph{Translation}\n$x_k$ is an English word and $y_k$ is its translation to some target language -- either German, Spanish, or French. We use CEN as input words, and obtain their translations via Wiktionary.\\footnote{\\url{https:\/\/github.com\/open-dsl-dict\/wiktionary-dict}}\nFor evaluation, we check whether the predicted answer is contained in the set of the possible gold answers.\n\n\\paragraph{Sentiment Analysis}\n$x_k$ is a movie review and $y_k$ is a binary label, either ``positive'' or ``negative'', marking the review's sentiment.\nWe use the Stanford Sentiment Treebank dataset \\citep{socher-etal-2013-recursive} from GLUE \\citep{wang-etal-2018-glue}, taking the train split as our induce set and the dev split as the execute set.\nWe consider only full sentences, discarding sentence constituents and sentences containing more than 10 words.\nThis leaves us with an induce set of 1,167 examples. To create label-balanced instruction induction examples, we sample each sequence of 5 demonstrations such that there are at least 2 demonstrations for each label.\n\n\\paragraph{Sentence Similarity}\n$x_k$ contains two sentences, and $y_k$ reflects the semantic similarity of the two input sentences.\nThe similarity is measured on a scale of 0 to 5, and the labels contain an additional short textual description of the numerical label, e.g., ``5 - perfectly''.\nWe use the Semantic Textual Similarity Benchmark dataset \\citep{cer-etal-2017-semeval} from GLUE, rounding the similarity scores and taking the train split as the induce set and the dev split as the execute set.\nWe discard examples in which at least one of the sentences contains more than 10 words, which leaves us with an induce set of 3,716 examples.\nIn each instruction induction example, we sample at least one pair with a score of 0 and one with a score of 5, so that models will be exposed to the minimal and maximal scores when generating an instruction.\nWe evaluate whether the predicted answer matches one of three valid outputs for each label: the numerical label (``5''), the verbal label (``perfectly''), or the combined label (``5 - perfectly'').\n\n\\paragraph{Word in Context}\n$x_k$ contains a target word and two contexts (sentences) for that word, and $y_k$ is a binary label reflecting whether the word has the same meaning in both contexts.\nWe use the Word in Context dataset \\citep{pilehvar-camacho-collados-2019-wic} from SuperGLUE \\citep{superglue}, taking the train split as the induce set and the dev split as the execute set.\nWe discard examples in which at least one of the sentences contains more than 10 words, which leaves us with an induce set of 4,084 examples.\nTo create label-balanced instruction induction examples, we sample each sequence of 5 demonstrations such that there are at least 2 demonstrations for each label.\nWe evaluate whether the predicted label matches one of several possible outputs: ``same'', ``yes'', or ``true'' for an identical meaning, and ``not the same'', ``no'', or ``false'' for a different meaning.\n\n\n\\subsection{Common English Nouns}\n\\label{sec:common_nouns}\n\nWe create a dataset of common English nouns (CEN) by filtering high-frequency nouns from the Wikitext-103 corpus \\citep{merity2016pointer}.\nWe first create a vocabulary of the 10,000 most frequent words in the corpus, from which we will later select the nouns.\nWe then process the corpus with SpaCy's part-of-speech tagger and lemmatizer,\\footnote{\\url{https:\/\/spacy.io\/}}\nand retain only nouns that appear in their singular form by verifying that their part-of-speech tag is ``NN'' and testing whether the word's lemma is identical to the word itself.\nWe additionally filter nouns that have less than 3 letters. Overall, this leaves us with a set of 3,406 nouns.\n\n\n\n\\section{Data Verification}\n\\label{sec:data_verification}\n\n\nTable~\\ref{tab:verification_tab} shows the results for the data verification experiments (\\S\\ref{sec:verification}). As evident by these results, most of our tasks can be inferred in-context by models. Moreover, all tasks but one can be accurately described by at least 4 out 5 human annotators.\n\n\\input{03b_in_context_tab}\n\n\n\n\n\n\n\\section*{Checklist}\n\n\n\n\n\\begin{enumerate}\n\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes{}\n  \\item Did you describe the limitations of your work?\n    \\answerYes{See \\S\\ref{sec:ambiguity} and the comment regarding the execution model in \\S\\ref{sec:results_execution}.}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNA{}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes{}\n\\end{enumerate}\n\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results?\n    \\answerNA{}\n        \\item Did you include complete proofs of all theoretical results?\n     \\answerNA{}\n\\end{enumerate}\n\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{We include our data as well as model predictions. Our code will be made publicly available upon acceptance.}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{We didn't perform any training, only inference, for which we reported the inference settings, see \\S\\ref{sec:results}.}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerNA{}\n        \\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes{}\n\\end{enumerate}\n\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n    \\answerYes{}\n  \\item Did you mention the license of the assets?\n    \\answerNo{We cited the assets and provided necessary links, license details can be found there. All datasets used are open and publicly available.}\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n    \\answerYes{}\n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n    \\answerNo{}\n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNo{No such danger in our data.}\n\\end{enumerate}\n\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerYes{See \\S\\ref{sec:annotations} and \\S\\ref{sec:verification}.}\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA{}\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA{The annotations were very limited in scope (a few minutes of work from each annotator), which we got via personal friends who were not involved in the project.}\n\\end{enumerate}\n\n\n\\end{enumerate}\n\n\\section{Data Construction}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe classical definition of martingales is extended to a more general\ncase in the space of Banach lattices by V.~Troitsky \\cite\n{TroitskyMartingales:05}. In the Banach lattice framework,\nmartingales are defined without a probability space and the famous\nDoob's convergence theorem was reproduced. Moreover, under certain\nconditions on the Banach lattice, it was shown that the set of bounded\nmartingales forms a Banach lattice with respect to the point-wise order.\nIn 2011, H.~Gessesse and V.~Troitsky \\cite{GessesseMartingales:11}\nproduced several\\vadjust{\\goodbreak} sufficient conditions for the space of bounded\nmartingales on a Banach lattice to be a Banach lattice itself. They\nalso provided examples showing that the space of bounded martingales is\nnot necessarily a vector lattice. Several other works have been done by\nother authors with regard to martingales in vector lattices, such as\n\\cite{Watson:13, Grobler:15}.\n\nIn the theory of random processes,\nnot just the study of martingale convergence is important,\nbut the study of convergence of martingale-like stochastic sequences and processes, and the determination of interrelation between them are also crucial.\nSo it is natural to ask if martingale-like\nsequences can be defined in a vector lattice or Banach lattice\nframework. In this article, we define and study martingale-like\nsequences in Banach lattices along the same lines as martingales are\ndefined and studied in \\cite{TroitskyMartingales:05}.\n\nClassically, a martingale-like sequence is defined as follows (for\ninstance, see a paper by A.~Melnikov \\cite{Melnikov:82}). Consider a\nprobability space $(\\varOmega,\\mathcal{F},P)$ and a filtration\n$(\\mathcal{F}_n)_{n=1}^{\\infty}$, i.e., an increasing sequence of\ncomplete sub-sigma-algebras of $\\mathcal{F}$. An integrable stochastic\nsequence $x=(x_n,\\mathcal{F}_n)$ is an {\\bf$L^1$-martingale} if\n\\[\n\\lim_{n\\rightarrow\\infty} \\sup_{m\\geq n}E\\big|E(x_m |\n\\mathcal {F}_n) - x_n\\big| = 0.\n\\]\nAn integrable stochastic sequence $x=(x_n,\\mathcal{F}_n)$ is an {\\bf\n$E$-martingale} if\n\\[\nP\\bigl\\{ \\omega: E(x_{n+1} | \\mathcal{F}_n) \\neq\nx_n \\text{ infinitely often } \\bigr\\}=0.\n\\]\n\nHere we extend the definition of $L^1$-martingales and $E$-martingales\nin a general Banach lattice $X$ following the same lines as the\ndefinition of martingales in Banach lattices in \\cite\n{TroitskyMartingales:05}. First we mention some terminology and\ndefinitions from the theory of Banach lattices for the reader\nconvenience. For more detailed exploration, we refer the reader to\n\\cite{Aliprantis:85}. A {\\bf vector lattice} is a vector space\nequipped with a lattice order relation, which is compatible with the\nlinear structure. A {\\bf Banach lattice} is a vector lattice with a\nBanach norm which is monotone, i.e., $0\\leq x \\leq y$ implies $\\norm\n{x}\\leq\\norm{y}$, and satisfies $\\norm{x}=\\querymark{Q1} \\lVert\\abs{x}\n\\rVert$ for any two vectors $x$ and $y$. A vector lattice is said to\nbe {\\bf order complete} if every nonempty subset that is bounded above\nhas a supremum. We say that a Banach lattice has {\\bf order continuous\nnorm} if $\\norm{x_{\\alpha}}\\rightarrow0$ for every decreasing net\n$(x_{\\alpha})$ with $\\inf x_{\\alpha}=0$. A Banach lattice with order\ncontinuous norm is order complete. A sublattice $Y$ of a vector lattice\nis called an (order) {\\bf ideal} if $y\\in Y$ and $|x|\\leq|y|$ imply\n$x\\in Y$. An ideal $Y$ is called a {\\bf band} if\n$x = \\sup_{ \\alpha} x_{\\alpha}$ implies $x\\in Y$ for every positive\nincreasing net $(x_{\\alpha})$ in $Y$. Two elements $x$ and $y$ in a\nvector lattice are said to be {\\bf disjoint} whenever $|x|\\wedge|y| =\n0$ holds. If $J$ is a nonempty subset of a vector lattice, then its\n{\\bf disjoint complement} $J^d$ is the set of all elements of the\nlattice, disjoint to every element of $J$. A band $Y$ in a vector\nlattice $X$ that satisfies $X = Y\\otimes Y^d$ is refered to as a {\\bf\nprojection band}. Every band in an order complete vector lattice is a\nprojection band. An operator $T$ on a vector lattice X is positive if\n$Tx\\geq0$ for every $x\\geq0$. A sequence of positive projections\n$(E_n)$ on a vector lattice $X$ is called a {\\bf filtration} if $E_nE_m\n= E_{n\\wedge m}$. A sequence of positive contractive projections\n$(E_n)$ on a normed lattice $X$ is called a {\\bf contractive\nfiltration} if $E_nE_m = E_{n\\wedge m}$. A~filtration $(E_n)$ in a\nnormed lattice $X$ is called \\term{dense} if $E_nx\\rightarrow x$ for\neach $x$ in $X$. In many articles such as in \\cite\n{TroitskyMartingales:05}, a \\term{martingale} with respect to a\nfiltration $(E_n)$ in a vector lattice $X$ is defined as a sequence\n$(x_n)$ in $X$ such that $E_nx_m=x_n$ whenever $m\\ge n$.\n\n\\section{Main definitions}\n\n\\begin{definition}\\label{def1}\nA sequence $ (x_n)$ of elements of a normed lattice $X$ is called an\n{\\bf $X$-martingale} relative to a contractive filtration $(E_n)$ if\n\\[\n\\lim\\limits\n_{n\\rightarrow\\infty} \\sup_{m\\geq n}\\norm{E_nx_m\n- x_n} = 0.\n\\]\n\\end{definition}\n\n\\begin{definition}\\label{def2}\nA sequence $ (x_n)$ of elements of a vector lattice $X$ is called an\n$\\mathcal{E}$-\\textbf{martingale} relative to a filtration $(E_n)$ if\nthere exists $n\\geq1$ such that $E_m x_{m+1}=x_m$ for all $m\\geq n.$\n\\end{definition}\n\nNote that Definition~\\ref{def2} is equivalent to saying a sequence\n$(x_n)$ is an $\\mathcal{E}$-martingale if there exists $l\\geq1$ such\nthat $E_n x_{m}=x_n$ whenever $m\\geq n \\ge l$. The symbol ``$\\mathcal\n{E}$'' stands for eventual so when we say $(x_n)$ is an $\\mathcal\n{E}$-martingale, we are saying that after a first few finite elements\nof the sequence, the sequence becomes a martingale.\n\nSequences defined by Definition~\\ref{def1} and Definition~\\ref{def2}\nare collectively called {\\bf martin\\-gale-like sequences}.\nNotice that every martingale $(x_n)$ in a vector lattice $X$ with\nrespect to a filtration $(E_n)$ is obviously an $\\mathcal\n{E}$-martingale with respect to the filtration\n$(E_n)$. Moreover, every $\\mathcal{E}$-martingale $(x_n)$ in a Banach\nlattice $X$ with respect to a contractive filtration $(E_n)$ is an\n$X$-martingale with respect to the contrative filtration\n$(E_n)$.\nNote that for every $x$ in a vector lattice $X$ and a filtration\n$(E_n)$ in $X$, the sequence $(E_n x)$ is an $\\mathcal{E}$-martingale\nwith respect to the filtration $(E_n)$. If $x$ is in a normed space $X$\nand $(E_n)$ is a contractive filtration, then the sequence $(E_n x)$ is\nan $X$-martingale with respect to the contractive filtration $(E_n)$.\n\nBy considering any nonzero martingale $(x_n)$ in a Banach lattice $X$\nwith respect to filtration $(E_n)$ where $x_1$ is nonzero without loss\nof generality, we can define a sequence $(y_n)$ such that $y_1=2x_1$\nand $y_n=x_{n}$ for all $n\\geq2$. Then one can see that $(y_n)$ is an\n$\\mathcal{E}$-martingale with respect to the filtration $(E_n)$.\nHowever, $(y_n)$ is not a martingale.\n\nNote that every sequence which converges to zero is an $X$-martingale\nwith respect to any contractive filtration $(E_n)$ because if\n$x_n\\rightarrow0$ and $m> n$ then\n$\\norm{E_n x_m - x_n}\\leq\\norm{x_m} +\\norm{x_n} \\rightarrow0$ as\n$n\\rightarrow\\infty$. So one can easily create an $X$-martingale\n$(x_n)$ which is not $\\mathcal{E}$-martingale by setting $x_n=\\frac\n{1}{n}x$ where $x$ is a nonzero vector in $X$.\n\nA martingale-like sequence $A=(x_n)$ with respect to a contractive\nfiltration $(E_n)$ on a normed lattice $X$ is said to be {\\bf bounded}\nif its norm defined by $\\norm{A}=\\sup_n\\norm{x_n}$ is finite. Given\na contractive filtration $(E_n)$ on a normed lattice $X$, we denote the\nset of all bounded $X$-martingales with respect to the contractive\nfiltration $(E_n)$ by $M_X=M_X(X,(E_n))$ and the set of all bounded\n$\\mathcal{E}$-martingales with respect to the contractive filtration\n$(E_n)$ by $M_E=M_E(X,(E_n))$. With the introduction of the sup norm in\nthese spaces, one can show that $M_X$ and $M_E$ are normed spaces.\nKeeping the notation $M$ of \\cite{TroitskyMartingales:05} for all\nbounded martingales with respect to the contractive filtration $(E_n)$\nand from the preceding arguments, these spaces form a nested increasing\nsequence of linear subspaces $M \\subset M_E \\subset M_X \\subset\\ell\n_\\infty(X)$, with the norm being exactly the $\\ell_\\infty(X)$ norm.\\looseness=-1\n\n\\begin{theorem}\\label{Mx-BS}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$, then\nthe collection of $X$-martingales $M_X$ is a closed subspace of $\\ell\n_\\infty(X)$, hence a Banach space.\\looseness=-1\n\\end{theorem}\n\\begin{proof}\nSuppose a sequence $(A^m)=(x^m_n)$ of $X$-martingales converges to $A$\nin $\\ell_\\infty(X)$. We show $A$ is also an $X$-martingale. Indeed,\nfrom $\\norm{A^m-A}=\\sup_n\\norm{x^m_n-x_n}\\rightarrow0$ as\n$m\\rightarrow\\infty$, we have that for each $n\\geq1$, $\\norm\n{x^m_n-x_n}\\rightarrow0$ as $m\\rightarrow\\infty$.\nNote that for $l\\ge n$,\n\\begin{align*}\n\\norm{E_nx_l-x_n}&=\\norm{E_nx_l-E_nx^m_l+E_nx^m_l-x^m_n+x^m_n-x_n}\n\\\\\n&\\leq\\norm{E_nx_l-E_nx^m_l}\n+\\norm{E_nx^m_l-x^m_n}\n+\\norm{x^m_n-x_n}.\n\\end{align*}\nFrom these inequalities and the contractive property of the filtration,\nwe have\n\\[\n\\lim_{n\\rightarrow\\infty} \\sup_{l\\geq n}\\norm{E_nx_l\n- x_n} = 0.\\qedhere\n\\]\n\\end{proof}\n\n\\begin{corollary}\\label{inclusion}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$, then\n$\\overline{M_E} \\subset M_X.$\n\\end{corollary}\n\n\\begin{lemma}\\label{conv-mx}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$ and\n$A=(x_n)$ be in $M_X$ where $x_n\\rightarrow x$. Then\n\\[\n\\lim\\limits\n_{n\\rightarrow\\infty} \\sup_{m\\ge n}\\norm{E_mx -\nx_m} = 0.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $A=(x_n)$ be in $M_X$ where $x_n\\rightarrow x$. Thus, for $m\\geq n$\n\\[\n\\norm{E_nx - x_n}= \\norm{E_nx -\nE_nx_m+E_nx_m-x_n}\n\\leq\\norm{x-x_m} +\\norm{E_nx_m-x_n}.\n\\]\nTaking $\\lim\\limits_{n\\rightarrow\\infty} \\sup_{m\\geq n}$ on both\nsides of the inequality completes the proof.\n\\end{proof}\n\nThe following proposition confirms that for any convergent element\\querymark{Q2}\n$A=(x_n)$ of $M_X$ we can find a sequence in $M_E$ that converges to $A$.\n\\begin{proposition}\\label{halfdense}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$ and\n$A=(x_n)$ be a sequence in $M_X$ such that $x_n\\rightarrow x$. Then\nthere exists a sequence $A^m$ in $M_E$ such that $A^m \\rightarrow A$ in\n$\\ell_{\\infty}(X)$.\n\\end{proposition}\n\\begin{proof}\nSuppose $x_n \\rightarrow x$ as $n\\rightarrow\\infty$. First note that\nthe sequence $(E_nx)$ is in $M$. Now define $A^m=(a^m_n)$ such that\n\\[\na^m_n=\n\\begin{cases}\n x_n,&  \\text{for } n\\le m,\\\\\n E_nx,& \\text{for } n > m.\n\\end{cases}\n\\]\nThen $A^m\\in M_E$ and $A^m \\rightarrow A$ in $\\ell_{\\infty}(X)$,\nhence $A\\in\\overline{M_E}$. Indeed, by Lemma~\\ref{conv-mx},\n\\[\n\\lim_{m\\rightarrow\\infty}\\norm{A^m-A}=\\lim_{m\\rightarrow\\infty\n}\n\\sup_j\\norm{E_{m+j}x-x_{m+j}}=0.\\qedhere\n\\]\n\\end{proof}\n\nIn \\cite{TroitskyMartingales:05} and \\cite{GessesseMartingales:11}\nseveral sufficient conditions are es\\querymark{Q3}tablished where the set of bounded\nmartingales $M$ is a Banach lattice. In \\cite{GessesseMartingales:11},\ncounter examples are provided where $M$ is not a Banach lattice. So,\none may similarly ask when are $M_X$ and $M_E$ Banach spaces and Banach\nlattices? We start by showing a counter example that illustrates that\n$M_E$ is not necessarily a Banach space.\n\\begin{example}\nLet $c_0$ be the set of sequences converging to zero. Consider the\nfiltration $(E_n)$ where $E_n \\sum_{i=1}^{\\infty} \\alpha_i e_i =\n\\sum_{i=1}^{n} \\alpha_i e_i$. Thus the sequence $(y_n)$ where $y_n=\n\\sum_{i=1}^{n} \\frac{1}{i} e_i$ is an $E$-martingale with respect to\nthis filtration. We define a sequence of $E$-martingales $A^m$ as\n$A^m=(x_n^m)$ where\n\\[\nx_n^m=\n\\begin{cases}\n \\sum_{i=n}^{\\infty} \\frac{1}{i} e_i  ,&  \\text{for } n\\leq m,\\\\\n y_n\/m,& \\text{for } n>m.\n\\end{cases}\n\\]\nDefine a sequence $A=(x_n)$ where $x_n=\\sum_{i=n}^{\\infty} \\frac\n{1}{i} e_i $. We can see that $A$ is not an $E$-martingale. But one can\nshow that $A^m$ converges to $A$. Indeed,\n\\[\n\\bigl\\lVert A^m - A \\bigr\\rVert=\\sup_{n}\n\\big\\|x^m_n-x\\big\\|=\\sup_{n \\in\n\\{m+1,m+2, \\ldots\\}}\n\\Bigg\\|y_n\/m-\\sum_{i=n}^{\\infty}\n\\frac{1}{i} e_i \\Bigg\\| \\rightarrow0\n\\]\nas $m\\rightarrow\\infty.$\n\\end{example}\n\n\\section{When is $M_E$ a vector lattice?}\n\nGiven a vector (Banach) lattice $X$ and a filtration (respectively\ncontractive) $(E_n)$ on $X$, we can introduce order structure on the\nspaces $M_E$ and $M_X$ as follows. For two bounded $\\mathcal\n{E}$-martingales (respectively $X$-martingales) $A=(x_n)$ and\n$B=(y_n)$, we write $A\\geq B$ if $x_n\\geq y_n$ for each $n$. With this\norder $M_E$ and $M_X$ are ordered vector spaces and the monotonicity of\nthe norm follows from the monotonicity of the norm of $X$, i.e. for two\n$\\mathcal{E}$-martingales (respectively $X$-martingales) with $0\\leq A\n\\leq B$, we have $\\norm{A}\\leq\\norm{B}$. For two $\\mathcal\n{E}$-martingales (respectively $X$-martingales) $A=(x_n)$ and\n$B=(y_n)$, one may guess that $A\\lor B$ (or $A\\wedge B$) can be\ncomputed by the formulas $A\\lor B =(x_n\\lor y_n)$ (or $A\\wedge\nB=(x_n\\wedge y_n)$). We show in the following theorem that this is in\nfact the case in order for $M_E$ to be a vector lattice. However, this\nis not obvious to show in the case of $M_X$.\n\n\\begin{theorem}\\label{vl-equivalence}\nLet $X$ be a vector lattice. Then the following statements are equivalent.\n\\begin{enumerate}\n\\item[(i)]$M_E$ is a vector lattice.\n\\item[(ii)] For each $A=(x_n)$ in $M_E$, the sequence $(|x_n|)$ is an\n$\\mathcal{E}$-martingale and $\\abs{A}=(\\abs{x_n})$.\n\\item[(iii)] $M_E$ is a sublattice of $\\ell_{\\infty}(X)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFirst we show (\\textit{i})~$\\implies$~(\\textit{ii}). Suppose $M_E$ is a vector lattice and\n$A=(x_n)$ is in $M_E$. Since $M_E$ is a vector lattice, $\\abs{A}$\nexists in $M_E$, say $|A|=B:=(y_n)$. Since $\\pm A\\le B$, for each $n$,\n$\\pm x_n\\le y_n$. So, $\\abs{x_n}\\le y_n$ for each $n$. Since $B$ is in\n$M_E$, there exists $l$ such that $E_ny_m=y_n$ whenever $m\\ge n\\ge l$.\nNow we claim that $y_n=|x_n|$ for each $n$. Fix $k> l$. We show\n$y_n=|x_n|$ for each $n\\le k$.\n\nIndeed, define an $\\mathcal{E}$-martingale $C=(z_n)$ where\n\\[\nz_n=\n\\begin{cases}\n \\abs{x_n},& \\text{for } n\\le k,\\\\\n y_n, &\\text{for } n> k.\n\\end{cases}\n\\]\nSince $k>l$ we can easily see that $C$ is an $\\mathcal{E}$-martingale.\nMoreover, $C\\ge0$ and $\\pm A \\le C \\le B$. Since $\\abs{A}=B$, $C=B$.\nThus, for every $n\\le k$, $y_n=|x_n|$. This establishes (\\textit{ii}).\n\n(\\textit{ii})~$\\implies$~(\\textit{iii})~$\\implies$~(\\textit{i}) is straightforward.\n\\end{proof}\n\nUsing the equivalence in Theorem~\\ref{vl-equivalence}, the following\nexamples illustrate\nthat $M_E$ is not always a vector lattice.\n\n\\begin{example}\\label{ReviewerExample}\nConsider the classical martingale $(x_n)$ in $L_1[0,1]$ where\n$x_n=\\break2^n\\mathbf{1}_{[0,2^{-n}]} -\\mathbf{1}$ with the filtration\n$(\\mathcal{F}_n)$ where $\\mathcal{F}_n$ is the smallest sigma algebra\ngenerated by the set\n\\[\n\\bigl\\{ \\bigl[0,2^{-n}\\bigr], (2^{-n}, 2^{-n+1}],\n\\dots, (1-2^{-n}, 1] \\bigr\\}.\n\\]\nOne can easily show that\n\\[\nE_n\\abs{x_{n+1}}=E \\bigl[|x_{n+1}| |x_n \\bigr]\n\\ne|x_n|\n\\]\nfor\nevery $n$ and the sequence $(|x_n|)$ fails to be an $\\mathcal\n{E}$-martingale. Hence, Theorem~\\ref{vl-equivalence} implies that\n$M_E$ is not a vector lattice.\n\\end{example}\n\n\\begin{example}\\label{cexample}\nConsider the filtration $(E_n)$ defined on $c_0$ as follows. For each\n$n=0, 1, 2, \\ldots$\n\\[\nE_n =\n\\begin{bmatrix}\n1 & & & & & & & \\\\\n& \\ddots & & & & & &\\\\\n& & 1 & & & & &\\\\\n& & &1\/2 &1\/2 & & &\\\\\n& & &1\/2 &1\/2 & & &\\\\\n& & & & &1\/2 &1\/2 &\\\\\n& & & & &1\/2 &1\/2 &\\\\\n& & & & & & &\\ddots\\\\\n\\end{bmatrix}\n\\]\nwith $2n$ ones in the upper left corner. For each $e_i=(0,\\ldots, 0,\n\\underbrace{1}_{i^{\\text{th}}}, 0, \\ldots)$,\n$E_ne_i=e_i$ if $i\\leq2n$ and $E_ne_{2k-1}=E_ne_{2k}=\\frac\n{1}{2}(e_{2k-1}+e_{2k})$ if $n<k$. Now if we define a sequence\n$A=(x_n)$ where for each $n=0,1,2,\\ldots$,\n\\[\nx_n=(\\underbrace{-1, 1, \\dots, -1, 1}_{2n\\text{-tuple}}, 0, \\dots),\n\\]\none can show thi\\querymark{Q4}s is a martingale as a result an $\\mathcal\n{E}$-martingale. However, $|A|=(|x_n|)$ where\n\\[\n|x_n|=(\\underbrace{1, \\dots, 1}_{2n\\text{-tuple}}, 0,\\dots)\n\\]\nis not an $\\mathcal{E}$-martingale. So, Theorem~\\ref{vl-equivalence}\nimplies that $M_E$ is not a vector lattice.\n\\end{example}\n\n\\begin{proposition}\nIf a filtration $(E_n)$ is a sequence of band projections, then $M_E$\nis a vector lattice with coordinate-wise lattice operations.\n\\end{proposition}\n\n\\begin{proof}\nIf $A=(x_n)\\in M_E$, then there exists $l$ such that $E_nx_m=x_n$\nwhenever $m\\ge n \\ge l$. Thus, $E_n\\abs{x_m}=\\abs{E_nx_m}=\\abs\n{x_n}$. So, $|A|=(\\abs{x_n})$ and thus $M_E$ is a vector lattice.\n\\end{proof}\n\n\\begin{theorem}\\label{filt-char}\nIf $M_E$ is a normed lattice and the filtration $(E_n)$ is dense in\n$X$, then for each $x$ in $X$, there exists $l$ such that $\\abs\n{E_nx}=E_n\\abs{x}$ whenever $n\\ge l$.\n\\end{theorem}\n\n\\begin{proof}\nLet $x$ be in $X$. Then $(E_n)$ is dense means $E_n x\\rightarrow x$.\nMoreover, $(E_nx)$ is a martingale. Since $M_E$ is a vector lattice, by\nTheorem~\\ref{vl-equivalence}, $(\\abs{E_n x})$\nis an $E$-martingale. Thus there exists $l$ such that for any $m$ and\n$n$ with $m\\ge n\\ge l$, $\\abs{E_n E_m x}=\\abs{E_n x}$ and\n$E_n\\abs{ E_m x}=\\abs{E_n x}$. So, $\\abs{E_n E_m x}=E_n\\abs{ E_m\nx}$ and letting $m\\rightarrow\\infty$, we have $\\abs{E_n x}=E_n\\abs{ x}$.\n\\end{proof}\n\n\\section{When is $M_X$ a Banach lattice?}\n\nUnder the pointwise order structure on $M_X$, for an $X$-martingale\n$A=(x_n)$, we can refer to Example~\\ref{cexample} to show that the sequence\n$(|x_n|)$ is not necessarily an $X$-martingale. However, under certain\nassumptions, we can show that $(\\abs{x_n})$ is \\xch{an}{an an} $X$-martingale\nfor every $X$-martingale $A=(x_n)$ making $M_X$ a Banach lattice.\n\n\\begin{proposition}\nIf $(E_n)$ is a contractive filtration where $E_n$ is a band projection\nfor every $n$ then $M_X$ is a Banach lattice with coordinate-wise\nlattice operations.\n\\end{proposition}\n\n\\begin{proof}\nLet $A=(x_n)$ be an $X$-martingale. For each $n$ and $m$, $E_n$ is a\nband projection implies $E_n|x_m|=|E_nx_m|$.\nThus, by the fact that $\\Bigabs{ |x|-|y|}\\leq\\abs{x-y}$,\nfor $m\\geq n$,\n\\[\n\\big\\|E_n|x_m|-|x_n|\\big\\|=\n\\big\\||E_nx_m|-|x_n|\\big\\|\\leq\n\\norm{E_nx_m-x_n}.\n\\]\nThis implies\n\\[\n\\lim_{n\\rightarrow\\infty} \\sup_{m\\geq n}\\big\\|E_n|x_m|\n- |x_n|\\big\\| = 0\n\\]\nwhich implies $|A|=(|x_n|)$ is also an $X$-martingale.\n\\end{proof}\n\n\\begin{question}\nFrom Theorem~\\ref{vl-equivalence}, $M_E$ is a vector lattice if and\nonly if for each $\\mathcal{E}$-martingale $(x_n)$, the sequence $(\\abs\n{x_n})$ is also an $\\mathcal{E}$-martingale. This is the case when the\nfiltration is a sequence of band projections. Can one give a\ncharacterization of the filtrations for which $M_E$ is a vector\nlattice? Or, can one give an example of a filtration which\nis not a sequence of projections\nand the corresponding $M_E$ is a vector lattice?\n\\end{question}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn order to adapt to the development of the Internet, recommendation system is widely used in e-commerce platform and entertainment application. \nThe data scale of these platform is also expanding with the use of users and the items listed in them, more worse the computing pressure of the traditional centralized recommendation system increases, and the performance requirements of hardware are gradually improved. \nOn this basis, the distributed recommendation system is gradually rising. Its main idea is to store the data in each node separately, such as the personal data of each user stored on personal devices, and the recommendation system can be implemented through joint modeling. \nAt the same time, the conflict between the use of big data and data privacy protection inevitably breaks out. The EU's general data protection regulation (GDPR) is a new challenge to the traditional data processing mode of artificial intelligence.\n\nTo solve the privacy protection problem in the distributed model training, federal learning technology was proposed by Google in 2016\\cite{konevcny2016federated}.\nThe framework assumes that the data parties involved in the model training are not trustworthy, and the data can not be exposed to any party. \nThrough a series of cryptographic proofs, the model can still complete the joint training with high performance and accuracy without privacy disclosure. \nIn the context of distributed recommendation system, it is very important to use federated learning technology to protect privacy. \nWe must ensure that information such as user's age, property status and so on cannot be obtained by others in data use. \nMatrix factorization is the most commonly used recommendation system model in industry. It has high computational cost and high accuracy, and is easy to be applied in distributed recommendation environment. \n\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/fedml.png}}\n    \\caption{Federated learning in recommendation system}\n    \\label{fedml}\n\\end{figure}\n\nIn this paper, a distributed matrix factorization recommendation system based on secret sharing is proposed. We assume that user data is distributed across different data sources, such as Taobao and Amazon, which have different customers. \nEach data sources will calculate its own user vectors locally, and calculate the item vectors in the central server. The problem of privacy disclosure while updating the parameters can be solved by secret sharing. \nThe experimental results show that our algorithm has higher efficiency while preserving privacy. Our main contributions are as follows:\n\\begin{itemize}\n    \\item We propose a privacy protection recommendation system model, which can not only complete the distributed recommendation task, but avoid privacy disclosure.\n    \\item Our model is based on secret sharing technology. Compared with the existing schemes based on homomorphic encryption, the running speed of our model is greatly improved.\n    \\item Our proposal has been verified on real data sets, which fully proves its superiority.\n\\end{itemize}\n\\section{Related Work}\nIn this section, We first review the privacy protection techniques commonly used in federated learning, and then discuss some of their existing applications in recommendation systems. \n\\subsection{Federated learning}\nThe idea of federated learning is to complete the joint modeling by combining the partial data shared by each data source. Its application in the recommendation system is shown in the figure\\ref{fedml}. \nThe two recommendation platforms have some of the same products, but the customers are not the same. Therefore, the performance of the model can be improved by combining different user data. \nIn this process, privacy protection technology is needed to ensure security. Common methods are as follows.\n\\paragraph{Anonymization}\nAnonymization is to hide or generalize the part of data features with privacy exposure. Among them, k-anonymity \\cite{samarati1998generalizing} is the most widely used privacy protection method for anonymous data in the data publishing scenario. \nIt was proposed by samarati et al. In 1998, it specifies the maximum acceptable information disclosure degree through the parameter K, and requires at least k on the recognizable data field The data is relatively fuzzy, so the attacker can't lock a specific user by obtaining the information of this field.\n\\paragraph{Homomorphic encryption}\nHomomorphic encryption\\cite{rivest1978data} can ensure that the results obtained by decrypting the encrypted data directly are consistent with the results of the same operation on the plaintext data. \nIn the process of model learning parameter exchange, homomorphic encryption can protect the privacy of user data. The Encryption ensures that the data and the model itself will not be transmitted, thus reducing the privacy leakage at the data level.\n\\paragraph{Differential privacy}\nDifferential privacy\\cite{dwork2008differential} protection is the problem of privacy leakage caused by small changes in the data source. \nIt is difficult for the observer to detect the subtle changes in the data set by observing the output results, so as to achieve the purpose of privacy protection. The common method is to add random noise to the input or output, such as Laplacian noise, in order to cover up the real data\n\\subsection{Privacy-preserving recommender system}\nThe related technologies mentioned above are applied in privacy protection recommendation system. in \\cite{wei2018improving} the author used a k-anonymity  method in the collaborative filtering and used a potential factor model to reduce the sparsity of the matrix, which protects the data privacy.\nThere are many privacy protection strategies for recommendation models. Next, we will focus on the privacy protection algorithms related to the matrix factorization model in this paper.\n\nRaghavendran Balu and Teddy Furon\\cite{balu2016differentially} use sketching techniques to implicitly provide the differential privacy while training matrix factorization model.Its model scales well with data and is suitable for large scale application.\nSungwook Kim et al\\cite{kim2016efficient} design a efficient data structure to use fully homomorphic encryption in matrix factorization, its inputs and outputs are both encrypted so there is no privacy problem while training recommend system.\nQiang Tang and Husen Wang\\cite{tang2017privacy} proposed a hybrid recommender model by using incremental matrix factorization which makes the system privacy-preserving and more efficient than other model.\nNonnegative matrix factorization (NMF) has been successfully applied and in\\cite{qian2020fast} the author proposed a distributed sketched alternating nonnegative least squares (DSANLS) framework for NMF and show the framework can be adapted to the security setting.\nSecure matrix factorization\\cite{chai2019secure} used a user-level distributed framework which makes user update the gradient locally and  uploads the encrypted gradient to tune parameters in server.\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/SMF.png}}\n    \\caption{Shared MF: two data sources example}\n    \\label{smf}\n\\end{figure}\n\\section{Matrix factorization with secret sharing}\nFirstly, we propose the background knowledge of matrix factorization. Then, according to the characteristics of multi-source data, we design a distributed matrix factorization recommendation system model, and introduce secret sharing technology to solve the privacy protection problem. \nWe name this model as shared matrix factorization and figure\\ref{smf} shows its structure. Compared with the existing solutions based on homomorphic encryption, our model's privacy protection strategy will be better for its high efficiency and practicability.\n\\subsection{Matrix factorization}\nSuppose we have n users, m items and each user rated a subset of m items. Then we can get a n*m rating matrix M. let $r_{ij}\\in M$ denote the score of item j rating by user i. Because the items have a large amount of numbers, the rating matrix is inevitably sparse, and what we want to do is find a solution to get the unknow rate in M.\nMatrix factorization formulates this problem as fitting a bilinear model on the rating matrix. It believes that users and items each have a profile matrix, and the rating matrix is the product of them. In particular, denoting user profile matrix as $U\\in R^{n*d}$ and item profile matrix as $V\\in R^{d*m}$.\nThen the rating $r_{ij}$ can be computed by $\\left<u_i,v_j\\right>$ where $u_i$ denotes the $i$th row of U and $v_j$ denotes the $j$th column of V.\nThe solution of U,V can be found by solving the following equation \\eqref{lsm}:\n\\begin{equation}\n    \\min_{U,V}=\\frac{1}{M}\\sum_{i=1}^{n}\\sum_{j=1}^{m}(r_{ij}-\\left<u_i,v_j\\right>)^2+\\lambda||U||^2_2+\\mu||V||^2_2\n    \\label{lsm}\n\\end{equation}\n\nwhere $\\lambda$ and $\\mu$ are the regularization parameters. And we can solve the above problem by stochastic gradient descent with the following equations\\cite{koren2009matrix}:\n\\begin{equation}\n    u_i^{new} = u_i^{old}-\\alpha\\Delta_{u_i}F(U^{old},V^{old})\n    \\label{sgd1}\n\\end{equation}\n\\begin{equation}\n    v_j^{new} = v_j^{old}-\\alpha\\Delta_{v_j}F(U^{old},V^{old})\n    \\label{sgd2}\n\\end{equation}\nwhere\n\\begin{equation}\n    \\Delta_{u_i}F(U,V) = -2\\sum_{j=1}^{m}v_j(r_{ij}-\\left<u_i,v_j\\right>)+2\\lambda u_i\n    \\label{sgd3}\n\\end{equation}\n\\begin{equation}\n    \\Delta_{v_j}F(U,V) = -2\\sum_{i=1}^{n}u_i(r_{ij}-\\left<u_i,v_j\\right>)+2\\lambda v_j\n    \\label{sgd4}\n\\end{equation}\nThe parameters are updated iteratively until the model loss function is less than a fixed threshold or the gradient difference between the two iterations is small, then the model can be considered as convergent.\n\\subsection{Distributed recommendation system}\nWe suppose that in a distributed environment, items are shared but different users belong to different data sources. For example, for users who buy iPhones, some users may purchase through Taobao, while others  will buy on the official website for quality reasons. \nDifferent users will rate the same item on different data sources. We assume that there are T data sources, the distributed matrix factorization recommendation system can be represented by Algorithm\\ref{alg:DMF}.\n\n\\begin{algorithm}\n    \\caption{Distributed Matrix Factorization}\n    \\label{alg:DMF}\n    \\begin{algorithmic}\n    \\REQUIRE $U_1,U_2,..,U_T,V,\\delta$\n    \\STATE data sources init their user profile matrix $U_t$\n    \\STATE server init item profile matrix $V,\\delta$\n    \\REPEAT \n    \\STATE \\textbf{data sources update:}\n    \\FOR {$t=1;t<=T;t++$}\n    \\STATE $\\Delta_{U_t}F(U,V)=-2(R_{U_t}-U_tV)V^T+2\\mu U_t$\n    \\STATE $U_t^{new} = U_t^{old}-\\alpha\\Delta_{U_t}F(U,V)$\n    \\STATE $Graident_t= -2U^T(R_{U_t}-U_tV)+2\\mu V$\n    \\STATE send $Graident_t$ to server\n    \\ENDFOR\n    \\STATE \\textbf{server update:}\n    \\STATE receive $Graidents$ from data sources\n    \\STATE $G = \\sum_{t=1}^{T}Graident_t$\n    \\STATE $V^{new} = V^{old} - G$\n    \\UNTIL{$G<\\delta$}\n    \\end{algorithmic}\n\\end{algorithm}\n\nUnder this framework, each data source holds its user profile matrix and keeps it secret to the outside. \nthe pubilc item profile matrix is stored in the central server. Each party uses the local rating matrix to update the user parameters, and only exposes the gradient of the item matrix to the server. \nThe server updates the item matrix after summarizing the gradient. This method only involves the transmission of gradients, therefore the security of local data is protected. \nHowever,transmit gradient can also exposes privacy. knowing the gradients of a data source uploaded in two continuous steps, it can infer the rating information by the equations \\eqref{leakage1}\\eqref{leakage2}. And see more detail in the paper\\cite{chai2019secure}. Therefore this article introduces secret sharing method in the transmission of graident, which make gradient transmission more efficient and safe.\n\n\\begin{equation}\n    u_i^t=(r_{ij}-\\left< u_i^t,v_j^t\\right>)=G_j^t\n    \\label{leakage1}\n\\end{equation}\n\n\\begin{equation}\n    r_{ij}=\\frac{G_{jk}^t}{u_{ik}^t}+\\sum_{m=1}^{D}u_{im}^t v_{jm}^t\n    \\label{leakage2}\n\\end{equation}\n\\subsection{Secret sharing}\nThe idea of secret sharing is to split the secret in an appropriate way, and each share after splitting is managed by different participants. A single participant cannot recover the secret information, and only several participants can cooperate to recover the secret message.\n\nThe figure\\ref{ss} gives a simple example of how to use sercet sharing. Two data sources own the number $X$ and $Y$ respectively, the server want to know the sum $X+Y$ but it will know nothing about X and Y. The process can be described as follows: \nfirstly, the original data is decomposed into two sub parts, and one sub part is exchanged between the two sides, and then the sum of the remaining sub parts with the part from other side is calculated. Finally, the solution of the original problem is obtained by summarizing the calculated sum. \nIn the process, the original data will not be exposed, so the sum operation can be completed under the premise of protecting data privacy. In addition, the multiplication can be realized by setting additional triples. In\\cite{zheng2020industrial}, the author uses secret sharing technology to implement multi-source federated neural network.\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/secret.png}}\n    \\caption{An example of Secret sharing}\n    \\label{ss}\n\\end{figure}\n\\subsection{Put all together}\nIn order to solve the privacy problem that may be caused by the exposure gradient, we propose a shared matrix factorization (SMF) method based on secret sharing. \nAs shown in the algorithm\\ref{alg:SMF}, the data source calculates the local user profile matrix  parameters and the item matrix gradients are encrypted by secret sharing technology before transmitting to the server, and finally the encrypted gradients are summarized on the server to update the item profile matrix parameters.\n\\begin{algorithm}\n    \\caption{Shared Matrix Factorization}\n    \\label{alg:SMF}\n    \\begin{algorithmic}\n    \\REQUIRE $U_1,U_2,..,U_T,V,\\delta$\n    \\STATE all parties initialize related parameters\n    \\REPEAT \n    \\STATE \\textbf{data sources update:}\n    \\FOR {$t=1;t<=T;t++$}\n    \\STATE update user profile matrix $U_t$\n    \\STATE compute item matrix gradient $g_t^{plain}$\n    \\STATE generate random number that meets $g_t^{plain}=g_{t}^{sub_1}+g_{t}^{sub_2}+..+g_{t}^{sub_T}$\n    \\STATE keep $g_t^{sub_t}$ and send the rest to other data \n    \\STATE receive $g^{sub_t}$ from others \n    \\STATE compute hybrid gradient $g_t^{hybrid}=\\sum_{i=1}^T g_{i}^{sub_t}$\n    \\STATE send hybrid gradient to server\n    \\ENDFOR\n    \\STATE \\textbf{server update:}\n    \\STATE receive $g^{hybrid}$ from data sources\n    \\STATE $G = \\sum_{t=1}^{T}g_t^{hybrid}$\n    \\STATE $V^{new} = V^{old} - G$\n    \\UNTIL{$G<\\delta$}\n    \\end{algorithmic}\n\\end{algorithm}\n\n\\section{Evaluation}\n\\subsection{Dataset}\nTo make the recommendation algorithm be better applied to the actual scene, we choose the real world dataset Movielens, which has been applied in many recommendation systems, such as caser\\cite{tang2018personalized}, h4mf\\cite{wang2018modeling}. \nWe disorganize the rating matrix and randomly sampled the train\/test set according to the ratio of 7:3.\n\\subsection{Parameters}\nThrough training experience and super parameter adjustment, we choose a group of better parameter combinations, in which the profile matrix dimension $k = 100$,\nthe regularization parameters $reg_u=10^{-3}, reg_v=10^{-3}$, and the learning rate is $lr=10^{-2}$\n\\subsection{Environment}\nAll experiments are performed on a server with 2.5GHz 16-core CPU and 64GB RAM, where the operation system is Linux and the program language is Python. \nWe use multithreading to simulate multi-source data holder. And they communicate and exchange data through grpc. Each source will start a rpc server client to receive data from other clients \n\\subsection{Performance}\n\\paragraph{\\textbf{local and distributed comparison}}\nFirst, we tested the improvement that the distributed recommendation system can bring. We used the data provided by only local data, three data sources and five data sources. \nFor each additional data source, the number of rating users increased by 200, and the total number of movies remained at 500. The experimental results are shown in the figure\\ref{ld}. With the increase of data sources, the loss of the model decreases. \nThis is due to the increase in the number of users, the rating matrix is more perfect, which makes the item vector fitting better.\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/local_distribute.jpg}}\n    \\caption{local and distributed recommender system comparison}\n    \\label{ld}\n\\end{figure}\n\n\\paragraph{\\textbf{Horizontal comparison}}\nWe have tested the improvement brought by distributed recommendation. In federated learning, the main reason that affects the performance of distributed algorithms is the overhead of encryption methods. \nTherefore, we test the different performance between our algorithm and that without encryption. Since the main cost of secret sharing lies in the communication and exchange of sub secrets between nodes, we set different number of data sources for horizontal comparison. \nThe result is as shown in the figure\\ref{hc}. Compared with matrix factorization, the communication cost caused by secret sharing is less than the computation cost by matrix factorization. Therefore, the performance of shared MF is basically the same as that of common distributed recommendation system, which means our algorithm has strong practicability.\n\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.40\\textwidth]{mat\/raw_ss.jpg}}\n    \\caption{time consumption with different data source numbers}\n    \\label{hc}\n\\end{figure}\n\\paragraph{\\textbf{Vertical comparison}}\nIn the previous horizontal comparison, we studied the communication overhead caused by increasing data sources. In the process of secret sharing of each data source, the amount of data transmitted is determined by the size of the item profile matrix. \nTherefore, we select the appropriate number of data sources and set different number of items to test the algorithm performance. The experimental results are shown in the figure\\ref{vc}. There are three data sources on the left and five data sources on the right. \nIt is obvious that with the increase of the number of objects, the communication overhead does not increase significantly, which proves  our algorithm is also very adaptable to large-scale items.\n\n\\begin{figure}[htbp]\n    \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/items.jpg}}\n    \\caption{time consumption with different item numbers}\n    \\label{vc}\n\\end{figure}\n\\paragraph{\\textbf{why not homomorphic encryption}}\nFrom the perspective of cryptography, homomorphic encryption can guarantee zero leakage of data privacy. Therefore, the distributed recommendation system using this method has the best security in theory. \n\nHowever, the disadvantage of homomorphic encryption is very obvious. The computational cost of data encryption and decryption process is very high. We compared our algorithm with FedML which uses an addition Encryption Paillier and tested the time cost under the same condition. \n\nFrom the table\\ref{tab1}, we can see that homomorphic encryption scheme can work when the amount of data is small, but with the increase of data volume, the encryption time is obviously too high, which can not adapt to the actual large-scale recommendation scenarios.\n\n\\begin{table}[htbp]\n    \\caption{SharedMF vs FedML}\n    \\begin{center}\n    \\begin{tabular}{|c|c|c|c|}\n    \\hline\n    train time(sec)&items50&items200 &items500  \\\\\n    \\hline\n    FedML& 223.49 & 843.21 & 2064.62 \\\\\n    \\hline\n    SharedMF& 100.58 & 284.13 & 583.37 \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab1}\n    \\end{center}\n\\end{table}\n\n\n\\section{Concluson and futrue work}\nIn this paper, we propose a secure distributed matrix factorization recommendation system framework, called SharedMF. Specifically, we first construct a distributed recommendation scenario, and store user data and item information separately in the clients and a server. \nThe model is fitting by exchanging gradients between them, and the secret sharing technology is used to ensure the data privacy and security in the training process.\n\nIn the experimental stage, we first prove the usefulness of the distributed system to improve the accuracy of recommendation scenarios, and then compare the performance differences between our algorithm and the non-encrypted distributed recommendation to verify the practicability of the algorithm. \nMoreover, we test the existing solutions based on homomorphic encryption, which proves that our scheme is more robust to the increase in the number of users and items, and is more suitable for large-scale recommendation scenarios.\n\nWith the  importance of privacy protection in recommendation system and machine learning increasing, federated learning technology based on cryptography is bound to be widely used. \nThe secret sharing technology used in this paper skilfully avoids the high computational complexity of traditional homomorphic encryption algorithm, and effectively improves the performance of privacy protection algorithm. However, it is worth mentioning that in this paper, secret sharing is only used to solve the privacy problem in the traditional algorithm matrix factorization. \nHow to apply it in the current popular deep neural network will be our further research topic.\n\n\n\\bibliographystyle{mat\/IEEEtran.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe main notions in nonrelativistic quantum mechanics are the \\Sh\\\nand wave function $|\\psi\\rangle$. The density matrix is an\nartificial construction, which, as will be shown bellow, can be\ncontradictory. We will consider the simplest case of the density\nmatrix, describing a monochromatic nonpolarized neutron beam.\n\nA monochromatic non polarized neutron beam is characterized by the\ndensity matrix\n\\begin{equation}\\label{dm}\n\\rho=\\fr1{2}\\Big(|u\\rangle\\langle u|+|d\\rangle\\langle d|\\Big),\n\\end{equation}\nwhich is one half of the unit matrix. The states $|u,d\\rangle$\ncorrespond to wave functions for neutrons polarized along and\nopposite some direction, which is known as quantization axis. The\nchoice of the quantization axis, however, is not important,\nbecause the density matrix \\eref{dm} is invariant with respect to\nsuch a choice. Indeed, if one chooses the quantization axis along\nsome unit vector $\\av$, then the matrix \\eref{dm} becomes\n\\begin{equation}\\label{2}\n\\rho=\\fr1{2}\\Big(|\\av\\rangle\\langle \\av|+|-\\av\\rangle\\langle\n-\\av|\\Big).\n\\end{equation}\nIf one chooses another axis $\\bb$, then, since\n\\begin{equation}\\label{3}\n|\\av\\rangle=\\alpha|\\bb\\rangle+\\beta|-\\bb\\rangle,\\qquad\n|-\\av\\rangle=\\alpha^*|-\\bb\\rangle-\\beta^*|\\bb\\rangle,\n\\end{equation}\nwhere $|\\alpha|^2+|\\beta|^2=1$, one obtains\n\\begin{equation}\\label{4}\n\\rho=\\fr1{2}\\Big(\\lt[\\alpha|\\bb\\rangle+\\beta|-\\bb\\rangle\\rt]\\lt[\\alpha^*\\langle\\bb|+\\beta^*\\langle-\\bb|\\rt]\n+$$\n$$+\\lt[\\beta^*|\\bb\\rangle-\\alpha^*|-\\bb\\rangle\\rt]\\lt[\\beta\\langle\\bb|-\\alpha\\langle-\\bb|\\rt]\\Big)=$$\n$$=\n\\fr1{2}\\Big(|\\bb\\rangle\\langle \\bb|+|-\\bb\\rangle\\langle\n-\\bb|\\Big).\n\\end{equation}\nFor instance, if $\\av$ is along $y$ axis, and $\\bb$ is along\n$z$-axis, one has\n\\begin{equation}\\label{5}\n|y\\rangle=\\fr1{\\sqrt2}{1\\choose\ni}=\\fr1{\\sqrt2}\\lt(|z\\rangle+i|-z\\rangle\\rt),\\qquad\n|-y\\rangle=\\fr1{\\sqrt2}{i\\choose1}=\\fr1{\\sqrt2}\\lt[|-z\\rangle+i|z\\rangle\\rt],\n\\end{equation}\nand\n\\begin{equation}\\label{6}\n\\rho=\\fr1{2}\\Big(|+z\\rangle\\langle+z|+|-z\\rangle\\langle-z|\\Big)=\\fr1{2}\\Big(|+y\\rangle\\langle+y|+|-y\\rangle\\langle-y|\\Big).\n\\end{equation}\nSo two axes are equivalent for the density matrix.  However these\naxes can be discriminated by an experimental equipment, and our\ngoal is to show how it is possible. To achieve it let's first show\nhow one can find polarization direction of a polarized beam.\n\n\\section{A method for polarization direction measurement}\n\nThe principle is based on an effect known in neutron\noptics~~\\cite{vf,ga,uig}, and is related to spin flip with the\nhelp of a resonant radio frequency (rf) spin-flipper. Such a\nspin-flipper is a coil with a permanent magnetic field $\\B_0$ and\nperpendicular to it rotating counterclockwise rf-field\n\\begin{equation}\\label{rf}\n\\B_{rf}=b\\Big(\\cos(\\omega t),\\sin(\\omega t).0\\Big),\n\\end{equation}\nwhere $\\omega=2\\mu B_0\/\\hbar$, and $\\mu$ is magnetic moment of the\nneutron, which is aligned oppositely to the neutron spin $\\sbb$.\nDirection of $\\B_0$ can be accepted as the quantization z-axis.\nInteraction of neutrons with such a flipper can be solved exactly\nand analytically, and the solution can be explained as\nfollows~\\cite{uig}.\n\nThe neutron interaction with magnetic field is described by the\npotential $-\\mb\\cdot\\B_0$. Therefore neutrons in the state\n$|z\\rangle$ entering the field $\\B_0$ are decelerated because the\nfield in this case creates a potential barrier of height $\\mu\nB_0$.\n\nInside the flipper the rf-field turns the spin down, i.e.\ntransforms the state $|z\\rangle$ into $|-z\\rangle$. In this state\nthe interaction $-\\mb\\cdot\\B_0$ becomes negative, so the potential\nbarrier transforms into potential well of depth $\\mu B_0$.\nTherefore after exit from the flipper and its magnetic field\n$\\B_0$ the neutron decelerates once again. In total the neutron\nenergy after transmission through the spin flipper decreases by\namount $2\\mu B_0$, which means emission of an rf quantum:\n$\\hbar\\omega=2\\mu B_0$. The wave functions before and after spin\nflipper are\n\\begin{equation}\\label{rf2}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)|z\\rangle,\n\\end{equation}\n\\begin{equation}\\label{rf3}\n|\\psi_{out}(x,t)\\rangle=\\exp(ik_-(x-D)-i(\\Omega-\\omega)\nt)|-z\\rangle,\n\\end{equation}\nrespectively. Here $x$ is the axis of propagation, $D$ is\nthickness of the spin-flipper, $k$ is initial wave number,\n$\\Omega=\\hbar k^2\/2m$, $m$ is the neutron mass, and\n$k_-=\\sqrt{k^2-2m\\omega\/\\hbar}$. If the incident neutron has the\nstate $|-z\\rangle$ it accelerates, and after spin-flipper has\nenergy larger than original one by the amount $2\\mu B_0$, which\nmeans absorbtion of an rf quantum: $\\hbar\\omega=2\\mu B_0$. The\nwave functions before and after spin flipper in this case are\nrespectively\n\\begin{equation}\\label{rf4}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)|-z\\rangle,\n\\end{equation}\n\\begin{equation}\\label{rf5}\n|\\psi_{out}(x,t)\\rangle=\\exp(ik_+(x-D)-i(\\Omega+\\omega)\nt)|z\\rangle,\n\\end{equation}\nwhere $k_+=\\sqrt{k^2+2m\\omega\/\\hbar}$.\n\nIf the incident neutron has a polarization\n$|\\xi\\rangle=\\alpha|z\\rangle+\\beta|-z\\rangle$, its wave function\nbefore and after spin flipper are respectively\n\\begin{equation}\\label{rf6}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)(\\alpha|z\\rangle+\\beta|-z\\rangle),\n\\end{equation}\n\\begin{equation}\\label{rf7}\n|\\psi_{out}(x,t)\\rangle=\\alpha\\exp(ik_-(x-D)-i(\\Omega-\\omega)\nt)|-z\\rangle+$$\n$$+\\beta\n\\exp(ik_+(x-D)-i(\\Omega+\\omega) t)|z\\rangle.\n\\end{equation}\nThe spin arrow of this\nstate represents a rotating spin wave propagating along $x$-axis.\n\nLet's put at some position $x=x_0$ an analyzer, which transmits\nonly neutrons polarized along $y$-axis. Since\n\\begin{equation}\\label{5}\n|+z\\rangle=\\fr1{\\sqrt2}(|+y\\rangle-i|-y\\rangle),\\qquad\n|-z\\rangle=\\fr 1{i\\sqrt2}(|+y\\rangle+i|-y\\rangle),\n\\end{equation}\nwhere $|\\pm y\\rangle$ denote states with polarization along and\nopposite $y$ axis, the neutron state \\eref{rf7} after the analyzer\nis\n\\begin{equation}\\label{6}\n|\\psi_{+y}(x_0,t)\\rangle=$$ $$\\fr {|+y\\rangle}{i\\sqrt2}\\lt(\\alpha\ne^{ ik_-(x_0-D)-i(\\Omega-\\omega)t}+i\\beta e^{\nik_+(x_0-D)-i(\\Omega+\\omega)t}\\rt),\n\\end{equation}\nand intensity of the neutron beam after the analyzer at some\nposition $x_0$ is\n\\begin{equation}\\label{7}\nI_{+y}(x_0,t)=\\fr12\\lt[|\\alpha|^2+|\\beta|^2+2|\\alpha\\beta|\\cos(\\varphi+2\\omega\nt)\\rt],\\end{equation}\n where $\\varphi$ is some phase. We see that\n the beam has density modulation with time, and visibility of the modulation\n \\begin{equation}\\label{8}\n V=\\fr{2|\\alpha\\beta|}{|\\alpha|^2+|\\beta|^2}=\\fr{2|\\alpha\/\\beta|}{1+|\\alpha|^2\/|\\beta|^2}\n\\end{equation}\ndetermines ratio $|\\alpha\/\\beta|$ and, therefore, the polar angle\nof the incident neutron spin arrow with respect to $z$-axis. If\n$\\alpha$ or $\\beta$ are zero, i.e. incident neutron is polarized\nalong or opposite spin-flipper axis, oscillations are absent.\n\n\\section{An experimental possibility for discrimination between $z$ and $y$ quantization axes}\n\nNow let's suppose that quantization axis is directed along\n$y$-axis. It means that the number $N_+$ of particles in the state\n$|+y\\rangle$ is the same as the number $N_-$ in the state\n$|-y\\rangle$. Since $|\\pm y\\rangle=(|\\pm z\\rangle+ i|\\mp\nz\\rangle)\/\\sqrt2$, we have according to \\eref{6} the intensities\nafter $y$-analyzer for two incident components $|\\pm y\\rangle$\nmeasured by a detector at some position $x_0$ to be\n\\begin{equation}\\label{7a}\nI_{+y}^\\pm(x_0,t)=\\fr{N_\\pm}2\\lt[1\\pm\\cos(2\\omega\nt)\\rt],\\end{equation} where upper index points out what was the\nincident component, and for simplicity we put the phase $\\varphi$\nin \\eref{7a} to zero, because it is the same for all the\nparticles.\n\nThe sum of averaged over time two intensities is a constant\n\\begin{equation}\\label{7a1}\n\\langle I_{+y}(t)\\rangle=\\langle I^+_{+y}(t)\\rangle+\\langle\nI^-_{+y}(t)\\rangle=$$ $$=\\fr{\\langle\nN_+\\rangle}2\\lt[1+\\cos(2\\omega t)\\rt]+\\fr{\\langle\nN_-\\rangle}2\\lt[1-\\cos(2\\omega t)\\rt]=N_0,\\end{equation} where\n$N_0=\\langle N_+\\rangle=\\langle N_-\\rangle$.\n\nHowever besides the average value there are also fluctuations of\nneutron count rate. We can naturally suppose that the fluctuations\nof two incident spin components are independent, and obey the\nPoisson statistics. Then fluctuations of neutron flux density\nafter $y$-analyzer will be\n\n\\begin{equation}\\label{7a2}\n\\langle|\\delta I_{+y}(t)|^2\\rangle=\\langle|\\delta\nI^+_{+y}(t)|^2\\rangle+\\langle|\\delta I^-_{+y}(t)|^2\\rangle=$$\n$$=\\Big\\langle\\fr{\\delta N_+}2\\lt[1+\\cos(2\\omega\nt)\\rt]\\Big\\rangle^2+\\Big\\langle\\fr{\\delta N_-}2\\lt[1-\\cos(2\\omega\nt)\\rt]\\Big\\rangle^2=\\fr{N_0}{2}(1+\\cos^2(2\\omega\nt)).\\end{equation}\n\nTo see these oscillations one should divide the period\n$T=\\pi\/2\\omega$ over $N$ small intervals $\\Delta T=T\/N$ and sum\nthe value\n\\begin{equation}\\label{7aa2}\n\\fr{\\langle|\\delta\nI_{+y}(t_n)|^2\\rangle}{N_0}=\\fr{1}2\\lt[1+\\cos^2(t_n\/\nT)\\rt],\\end{equation} at $t_n=n\\Delta T$ over many periods $T$.\n\nThis way one can discriminate between two quantization axes $z$,\nand $y$. Therefore these quantization axes are not equivalent,\nwhereas according to density matrix expression they are absolutely\nequivalent. This is the contradiction we wanted to point to.\n\n\\section{Conclusion}\n\nThe main element of \\qm\\ is a wave function, and corresponding to\nit a pure state. If one has an ensemble of particles with\ndifferent pure states, and the distribution of different states is\ncharacterized by probabilities, one must calculate a process with\npure states and then average over probabilities. This is the way\nneutron scattering cross sections are calculated. First they are\ncalculated for a pure state of an incident plain wave, and then\nthe obtained cross section is averaged over probability\ndistribution of the incident plain waves. Of course the density\nmatrix also can be useful, but because of discovered\ncontradiction, one must be very careful with it.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nState-based models of concurrent systems are standardly considered\nunder a wide range of system equivalences, typically located between\ntwo extremes respectively representing \\emph{linear time} and\n\\emph{branching time} views of system evolution. Over labelled\ntransition systems, one specifically has the well-known \\emph{linear\n  time -- branching time spectrum} of system equivalences between\ntrace equivalence and\nbisimilarity~\\cite{vanglabbeek2001linear}. Similarly, e.g.\\\nprobabilistic automata have been equipped with various semantics\nincluding strong bisimilarity~\\cite{LarsenSkou91}, probabilistic\n(convex) bisimilarity~\\cite{SegalaLynch94}, and distribution\nbisimilarity (e.g.~\\cite{DengEA08,DoyenEA08}). New equivalences keep\nappearing in the literature, e.g.~for non-deterministic probabilistic\nsystems~\\cite{BonchiEA19,vanHeerdtEA18}.\n\nThis motivates the search for unifying principles that allow for a\ngeneric treatment of process equivalences of varying degrees of\ngranularity and for systems of different branching types\n(non-deterministic, probabilistic etc.). As regards the variation of\nthe branching type, universal coalgebra~\\cite{Rutten00} has emerged as\na widely-used uniform framework for state-based systems covering a\nbroad range of branching types including besides non-deterministic and\nprobabilistic, or more generally weighted, branching also, e.g.,\nalternating, neighbourhood-based, or game-based systems. It is based\non modelling the system type as an endofunctor on some base category,\noften the category of sets.\n\nUnified treatments of system equivalences, on the other hand, are so\nfar less well-established, and their applicability is often more\nrestricted. Existing approaches include coalgebraic trace semantics in\nKleisli~\\cite{HasuoEA07} and Eilenberg-Moore\ncategories~\\cite{KissigKurz10,JacobsEA12,sbbr13,bms13,BonchiEA19,vanHeerdtEA18},\nrespectively.  Both semantics are based on decomposing the coalgebraic\ntype functor into a monad, the \\emph{branching type}, and a functor,\nthe \\emph{transition type} (in different orders), and require suitable\ndistributive laws between these parts; correspondingly, they grow\nnaturally out of the functor but on the other hand apply only to\nfunctors that admit the respective form of decomposition. In the\npresent work, we build on a more general approach introduced by\nPattinson and two of us, based on mapping the coalgebraic type functor\ninto a \\emph{graded monad}~\\cite{MiliusEA15}. Graded monads correspond\nto algebraic theories where operations come with an explicit notion of\n\\emph{depth}, and allow for a stepwise evaluation of process\nsemantics. Maybe most notably, graded monads systematically support a\nreasonable notion of \\emph{graded logic} where modalities are\ninterpreted as \\emph{graded algebras} for the given graded monad. This\napproach applies to all cases covered in the mentioned previous\nframeworks, and additional cases that do not fit any of the earlier\nsetups. We emphasize that graded monads are geared towards\n\\emph{inductively} defined equivalences such as finite trace semantics\nand finite-depth bisimilarity; we leave a similarly general treatment\nof infinite-depth equivalences such as infinite trace equivalence and\nunbounded-depth bisimilarity to future work. To avoid excessive\nverbosity, we restrict to models with finite branching\nthroughout. Under finite branching, finite-depth equivalences\ntypically coincide with their infinite-depth counterparts, e.g.\\\nstates of finitely branching labelled transition systems are bisimilar\niff they are finite-depth bisimilar, and infinite-trace equivalent iff\nthey are finite-trace equivalent.\n\nOur goal in the present work is to illustrate the level of generality\nachievable by means of graded monads in the dimension of system\nequivalences. We thus pick a fixed coalgebraic type, that of labelled\ntransition systems, and elaborate how a number of equivalences from\nthe linear time -- branching time spectrum are cast as graded\nmonads. In the process, we demonstrate how to extract logical\ncharacterizations of the respective equivalences from most of the\ngiven graded monads. For the time being, none of the logics we find\nare sensationally new, and in fact van Glabbeek already provides\nlogical characterizations in his exposition of the linear time --\nbranching time spectrum~\\cite{vanglabbeek2001linear}; an overview of\ncharacteristic logics for non-deterministic and probabilistic\nequivalences is given by Bernardo and\nBotta~\\cite{bernardo-botta:characterising-logics}. The emphasis in the\nexamples is mainly on showing how the respective logics are developed\nuniformly from general principles.\n\nUsing these examples as a backdrop, we develop the theory of graded\nmonads and graded logics further. In particular,\n\\begin{itemize}\n\\item we give a more economical characterization of depth-$1$ graded\n  monads involving only two functors (rather than an infinite sequence\n  of functors);\n\\item we extend the logical framework by a treatment of propositional\n  operators -- previously regarded as integrated into the modalities\n  -- as first class citizens; \n\\item we prove, as our main technical result, a generic expressiveness\n  criterion for graded logics guaranteeing that inequivalent states are\n  separated by a trace formula. \n\\end{itemize}\nOur expressiveness criterion subsumes, at the branching-time end of\nthe spectrum, the classical Hennessy-Milner\ntheorem~\\cite{HennessyMilner85} and its coalgebraic\ngeneralization~\\cite{Pattinson04,Schroder08} as well as expressiveness\nof probabilistic modal logic with only\nconjunction~\\cite{DesharnaisEA98}; we show that it also covers\nexpressiveness of the respective graded logics for more coarse-grained\nequivalences along the linear time -- branching time spectrum. To\nillustrate generality also in the branching type, we moreover provide\nan example in a probabilistic setting, specifically we apply our\nexpressiveness criterion to show expressiveness of a quantitative\nmodal logic for probabilistic trace equivalence.\n\n\\myparagraph{Related Work} Fahrenberg and\nLegay~\\cite{FahrenbergLegay17} characterize equivalences on the linear\ntime -- branching time spectrum by suitable classes of modal\ntransition systems. We have already mentioned previous work on\ncoalgebraic trace semantics in Kleisli and Eilenberg-Moore\ncategories~\\cite{HasuoEA07,KissigKurz10,JacobsEA12,sbbr13,bms13,BonchiEA19,vanHeerdtEA18}. A\ncommon feature of these approaches is that, more precisely speaking,\nthey model \\emph{language} semantics rather than trace semantics --\ni.e.\\ they work in settings with explicit successful termination, and\nconsider only successfully terminating traces. When we say that graded\nmonads apply to all scenarios covered by these approaches, we mean\nmore specifically that the respective language semantics are obtained\nby a further canonical quotienting of our trace\nsemantics~\\cite{MiliusEA15}. Having said that graded monads are\nstrictly more general than Kleisli and Eilenberg-Moore style trace\nsemantics, we hasten to add that the more specific setups have their\nown specific benefits including final coalgebra characterizations and,\nin the Eilenberg-Moore setting, generic determinization procedures. A\nfurther important piece of related work is Klin and Rot's method of\ndefining trace semantics via the choice of a particular flavour of\ntrace logic~\\cite{KlinRot15}. In a sense, this approach is opposite to\nours: A trace logic is posited, and then two states are declared\nequivalent if they satisfy the same trace formulae. In our approach\nvia graded monads, we instead pursue the ambition of first fixing a\nsemantic notion of equivalence, and then designing a logic that\ncharacterizes this equivalence. Like Klin and Rot, we view trace\nequivalence as an inductive notion, and in particular limit attention\nto finite traces; coalgebraic approaches to infinite traces exist, and\nmostly work within the Kleisli-style\nsetup~\\cite{Jacobs04,Cirstea11,KerstanKonig13,Cirstea14,Cirstea15,UrabeHasuo15,Cirstea17}. Jacobs,\nLevy and Rot~\\cite{JacobsEA18} use corecursive algebras to provide a\nunifying categorical view on the above-mentioned approaches to traces\nvia Kleisli- and Eilenberg-Moore categories and trace logics,\nrespectively. This framework does not appear to subsume the approach\nvia graded monads, and like for the previous approaches we are not\naware that it covers semantics from the linear time -- branching time\nspectrum other than the end points (bisimilarity and trace\nequivalence).\n\n\\section{Preliminaries: Coalgebra}\\label{sec:prelim}\n\nWe recall basic definitions and results in \\emph{(universal)\n  coalgebra}~\\cite{Rutten00}, a framework for the unified treatment of\na wide range of reactive systems. We write~$1=\\{\\star\\}$ for a fixed\none-element set, and $!\\colon X\\to 1$ for the unique map from a set~$X$\ninto~$1$. We write $f\\cdot g$ for the composite of maps $g\\colon X\\to Y$,\n$f\\colon Y\\to Z$, and $\\langle f,g\\rangle\\colon X\\to Y\\times Z$ for the pair map\n$x\\mapsto(f(x),g(x))$ formed from maps $f\\colon X\\to Y$, $g\\colon X\\to Z$.\n\nCoalgebra encapsulates the branching type of a given species of\nsystems as a \\emph{functor}, for purposes of the present paper on the\ncategory of sets. Such a functor $G\\colon \\Set\\to\\Set$ assigns to each\nset~$X$ a set~$GX$, whose elements we think of as structured\ncollections over~$X$, and to each map $f\\colon X\\to Y$ a map $Gf\\colon\nGX\\to GY$,\npreserving identities and composition. E.g.\\ the \\emph{(covariant)\n  powerset functor}~$\\mathcal{P}$ assigns to each set~$X$ the powerset\n$\\mathcal{P} X$ of~$X$, and to each map $f\\colon X\\to Y$ the map\n$\\mathcal{P} f\\colon \\mathcal{P} X\\to\\mathcal{P} Y$ that takes direct images. (We mostly omit\nthe description of the action of functors on maps in the sequel.)\nSystems with branching type described by~$G$ are then abstracted as\n\\emph{$G$-coalgebras}, i.e.\\ pairs $(X,\\gamma)$ consisting of a\nset~$X$ of \\emph{states} and a map $\\gamma\\colon X\\to GX$, the\n\\emph{transition map}, which assigns to each state $x\\in X$ a\nstructured collection $\\gamma(x)$ of successors. For instance, a\n$\\mathcal{P}$-coalgebra assigns to each state a set of successors, and thus\nis the same as a transition system.\n\\begin{example}\\label{expl:coalg}\n  \\begin{longitemslist\n  \\item Fix a set~$\\mathcal{A}$ of \\emph{actions}. The functor\n    $\\mathcal{A}\\times(-)$ assigns to each set $X$ the set $\\mathcal{A}\\times X$;\n    composing this functor with the powerset functor, we obtain the\n    functor $G=\\mathcal{P}(\\mathcal{A}\\times(-))$ whose coalgebras are precisely\n    labelled transition systems (LTS): A $G$-coalgebra assigns to each\n    state~$x$ a set of pairs $(\\sigma,y)$, indicating that~$y$ is a\n    successor of~$x$ under the action~$\\sigma$.\n  \\item The \\emph{(finite) distribution functor}~$\\mathcal{D}$ maps a\n    set~$X$ to the set of finitely supported discrete probability\n    distributions on~$X$. These can be represented as probability mass\n    functions $\\mu\\colon X\\to[0,1]$, with $\\sum_{x\\in X}\\mu(x)=1$ and with\n    the \\emph{support} $\\{x\\in X\\mid \\mu(x)>0\\}$ being\n    finite. Coalgebras for~$\\mathcal{D}$ are precisely Markov\n    chains. Composing with $\\mathcal{A}\\times(-)$ as above, we obtain the\n    functor $\\mathcal{D}(\\mathcal{A}\\times(-))$, whose coalgebras are\n    \\emph{generative probabilistic transition systems}, i.e.~assign\n    to each state a distribution over pairs consisting of an action\n    and a successor state. \n  \\end{longitemslist}\n\\end{example}\nAs indicated in the introduction, we restrict attention to\n\\emph{finitary} functors~$G$, in which every element $t\\in GX$ is\nrepresented using only finitely many elements of~$X$; formally, each\nset~$GX$ is the union of all sets $Gi_Y[GY]$ where $Y$ ranges over\nfinite subsets of~$X$ and $i_Y$ denotes the injection $i_Y\\colon Y\\hookrightarrow X$.\nConcretely, this means that we restrict the set~$\\mathcal{A}$ of actions to\nbe finite, and work with the \\emph{finite powerset functor}~$\\Pow_\\omega$\n(which maps a set~$X$ to the set of its finite subsets) in lieu\nof~$\\mathcal{P}$. ($\\mathcal{D}$ as defined above is already finitary.)\n\nCoalgebra comes with a natural notion of \\emph{behavioural\n  equivalence} of states. A \\emph{morphism}\n$f\\colon (X,\\gamma)\\to(Y,\\delta)$ of $G$-coalgebras is a map $f\\colon X\\to Y$ that\ncommutes with the transition maps, i.e.\\\n$\\delta\\cdot f=Gf\\cdot\\gamma$. Such a morphism is seen as preserving\nthe behaviour of states (that is, behaviour is defined as being\nwhatever is preserved under morphisms), and consequently states\n$x\\in X$, $z\\in Z$ in coalgebras $(X,\\gamma)$, $(Z,\\zeta)$ are\n\\emph{behaviourally equivalent} if there exist coalgebra morphisms\n$f\\colon (X,\\gamma)\\to(Y,\\delta)$, $g\\colon (Z,\\zeta)\\to(Y,\\delta)$ such that\n$f(x)=g(z)$. For instance, states in LTSs are\nbehaviourally equivalent iff they are bisimilar in the standard sense,\nand similarly, behavioural equivalence on generative probabilistic\ntransition systems coincides with the standard notion of probabilistic\nbisimilarity~\\cite{Klin09}. We have an alternative notion of\nfinite-depth behavioural equivalence: Given a $G$-coalgebra\n$(X,\\gamma)$, we define a series of maps $\\gamma_n\\colon X\\to G^n1$\ninductively by taking $\\gamma_0$ to be the unique map $X\\to 1$, and\n$\\gamma_{n+1} = G\\gamma_n \\cdot\\gamma$. (These are the first $\\omega$\nsteps of the \\emph{canonical cone} from~$X$ into the \\emph{final\n  sequence} of~$G$~\\cite{AdamekKoubek77}.) Then states $x,y$ in\ncoalgebras $(X,\\gamma)$, $(Z,\\zeta)$ are \\emph{finite-depth\n  behaviourally equivalent} if $\\gamma_n(x)=\\zeta_n(y)$ for all $n$;\nin the case where~$G$ is finitary, finite-depth behavioural equivalence\ncoincides with behavioural equivalence~\\cite{worrell}.\n\n\n\\section{Graded Monads and Graded Theories}\n\n\\noindent We proceed to recall background on system semantics via\ngraded monads introduced in our previous work~\\cite{MiliusEA15}. We\nformulate some of our results over general base categories~$\\mathbf{C}$,\nusing basic notions from category theory~\\cite{MacLane98,Pierce91};\nfor the understanding of the examples, it will suffice to think of\n$\\mathbf{C}=\\Set$. Graded monads were originally introduced by\nSmirnov~\\cite{smirnov08} (with grades in a commutative monoid, which\nwe instantiate to the natural numbers):\n\\begin{defn}[Graded Monads]\n  A \\emph{graded monad}~$M$ on a category $\\cat C$ consists of a\n  family of functors $(M_n\\colon \\cat C \\to \\cat C)_{n<\\omega}$, a natural\n  transformation $\\ensuremath{\\eta}\\xspace\\colon \\Id \\to M_0$ (the \\emph{unit}) and a family\n  of natural transformations \n  \\iffull\n  \\[\n    \\ensuremath{\\mu}\\xspace^{nk}\\colon M_n M_k \\to M_{n+k}\\quad (n,k<\\omega)\n  \\]\n  \\else\\\/$\\ensuremath{\\mu}\\xspace^{nk}\\colon M_n M_k \\to M_{n+k}$ for $n,k<\\omega$, \\fi\n  (the \\emph{multiplication}), satisfying the \\emph{unit laws},\n  $\\ensuremath{\\mu}\\xspace^{0n}\\cdot\\ensuremath{\\eta}\\xspace M_n = \\id_{M_n} = \\ensuremath{\\mu}\\xspace^{n0}\\cdot M_n\\ensuremath{\\eta}\\xspace$,\n  for all $n<\\omega$, and the \\emph{associative law}\n  \\iffull\n  \\[\n    \\begin{tikzcd}\n    M_nM_kM_m\\ar{d}{\\ensuremath{\\mu}\\xspace^{nk}M_m}\\ar{r}{M_n\\ensuremath{\\mu}\\xspace^{km}} & M_nM_{k+m}\\ar{d}{\\ensuremath{\\mu}\\xspace^{n,k+m}} \\\\\n      M_{n+k}M_m\\ar{r}{\\ensuremath{\\mu}\\xspace^{n+k,m}} & M_{n+k+m}\n    \\end{tikzcd}\n    \\qquad\\text{for all $k,n,m<\\omega$.}\n  \\]\n  \\else\\\/$\\ensuremath{\\mu}\\xspace^{n,k+m} \\cdot M_n \\ensuremath{\\mu}\\xspace^{km} = \\ensuremath{\\mu}\\xspace^{n+k,m} \\cdot\n  \\ensuremath{\\mu}\\xspace^{nk}M_m$ for all $k,n,m<\\omega$.\\fi\n\\end{defn}\nNote that it follows that $(M_0, \\eta, \\mu^{00})$ is a (plain)\nmonad. For $\\cat C = \\Set$, the standard equivalent presentation of\nmonads as algebraic theories carries over to graded monads. Whereas\nfor a monad $T$, the set $TX$ consists of terms over $X$ modulo\nequations of the corresponding algebraic theory, for a graded monad\n$(M_n)_{n<\\omega}$, $M_nX$ consists of terms of uniform depth $n$\nmodulo equations:g\n\n\\begin{defn}[Graded Theories~\\cite{MiliusEA15}]\n  A \\emph{graded theory} $(\\Sigma,E,d)$ consists of an algebraic\n  theory, i.e.\\ a (possibly class-sized and infinitary) algebraic\n  signature $\\Sigma$ and a class $E$ of equations, and an assignment\n  $d$ of a \\emph{depth} $d(f)<\\omega$ to every operation symbol\n  $f\\in\\Sigma$. This induces a notion of a term \\emph{having uniform\n    depth $n$}: all variables have uniform depth $0$, and\n  $f(t_1,\\dots,t_n)$ with $d(f)=k$ has uniform depth $n+k$ if all\n  $t_i$ have uniform depth $n$. (In particular, a constant $c$ has\n  uniform depth $n$ for all $n\\ge d(c)$). We require that all\n  equations $t=s$ in $E$ have uniform depth, i.e.\\ that both $t$ and\n  $s$ have uniform depth~$n$ for some~$n$. Moreover, we require that\n  for every set $X$ and every $k<\\omega$, the class of terms of\n  uniform depth $k$ over variables from $X$ modulo provable equality\n  is small (i.e.\\ in bijection with a set).\n\\end{defn}\n\\noindent Graded theories and graded monads on $\\Set$ are essentially\nequivalent concepts~\\cite{smirnov08,MiliusEA15}. In particular, a\ngraded theory $(\\Sigma,E,d)$ induces a graded monad~$M$ by taking\n$M_nX$ to be the set of $\\Sigma$-terms over $X$ of uniform depth~$n$,\nmodulo equality derivable under $E$.\n\n\\begin{example}\\label{E:graded-monad}\n  We recall some examples of graded monads and theories~\\cite{MiliusEA15}.\n  \\begin{longitemslist\n  \\item\\label{E:graded-monad:bisim} For every endofunctor $F$ on\n    $\\cat C$, the $n$-fold composition $M_n = F^n$ yields a graded\n    monad with unit $\\eta = \\id_{\\Id}$ and $\\mu^{nk} = \\id_{F^{n+k}}$.\n\n  \\item\\label{E:graded-monad:kleisli} As indicated in the\n    introduction, distributive laws yield graded monads: Suppose that\n    we are given a monad $(T,\\ensuremath{\\eta}\\xspace,\\ensuremath{\\mu}\\xspace)$, an endofunctor $F$ on\n    $\\cat C$ and a distributive law of $F$ over $T$ (a so-called\n    \\emph{Kleisli law}), i.e.\\ a natural transformation\n    $\\lambda\\colon FT \\to TF$ such that $\\lambda \\cdot F\\eta = \\eta F$\n    and $\\lambda \\cdot F\\mu = \\mu F \\cdot T\\lambda \\cdot \\lambda T$.\n    Define natural transformations $\\lambda^n\\colon F^nT \\to TF^n$\n    inductively by $\\lambda^0 = \\id_T$ and\n    $\\lambda^{n+1} = \\lambda^{n}F \\cdot F^{n}\\lambda$. Then we obtain\n    a graded monad with $M_n = TF^n$, unit $\\eta$, and multiplication\n    $\\mu^{nk} = \\mu F^{n+1} \\cdot T\\lambda^n F^k$. The situation is\n    similar for distributive laws of~$T$ over~$F$ (so-called\n    \\emph{Eilenberg-Moore laws}).\n  \\item\\label{E:graded-monad:tr} As a special case of\n    \\ref{E:graded-monad:kleisli}., for every monad $(T, \\eta, \\mu)$ on\n    $\\Set$ and every set $\\mathcal{A}$, we obtain a graded monad with\n    $M_nX = T(\\mathcal{A}^n \\times X)$.  Of particular interest to us will be\n    the case where $T = \\Pow_\\omega$, which is generated by the algebraic\n    theory of join semilattices (with bottom). The arising graded monad\n    $M_n=\\Pow_\\omega(\\mathcal{A}^n\\times X)$, which is\n    associated with trace equivalence, is generated by the graded\n    theory consisting, at depth~$0$, of the operations and equations\n    of join semilattices, and additionally a unary operation of\n    depth~$1$ for each $\\sigma \\in \\mathcal{A}$, subject to (depth-$1$)\n    equations expressing that these unary operations distribute over\n    joins.\n  \\end{longitemslist}\n\\end{example}\n\n\\myparagraph{Depth-1 Graded Monads and Theories}\nwhere operations and equations have depth at most~$1$ are a particularly convenient case for\npurposes of building algebras of graded monads; in the following, we elaborate on this\ncondition.\n\\begin{defn}[Depth-1 Graded\n  Theory~\\cite{MiliusEA15}]\\label{D:d1}\n  A graded theory is called \\emph{depth-$1$} if all its operations\n  and equations have depth at most~$1$. A graded monad on $\\Set$ is\n  \\emph{depth-1} if it can be generated by a depth-1 graded theory.\n\\end{defn}\n\\begin{proposition}[Depth-1 Graded Monads~\\cite{MiliusEA15}]\\label{P:d1}\n  A graded monad $((M_n),\\eta,(\\mu^{nk}))$ on $\\Set$ is depth-$1$\n  iff the diagram below is objectwise a coequalizer diagram in\n  $\\Set^{M_0}$ for all $n<\\omega$:\n  \\begin{equation}\\label{eq:mu1n}\n    \\xymatrix@1{M_1M_0M_n \\ar@<3pt>[rr]^{M_1\\mu^{0n}} \n      \\ar@<-3pt>[rr]_{\\mu^{10}M_n} && M_1 M_n \\ar[r]^{\\mu^{1n}} & M_{1+n}}.\n  \\end{equation}\n\\end{proposition}\n\n\\begin{example}\\label{E:d1}\n  All graded monads in \\autoref{E:graded-monad} are depth $1$:\n  for~\\ref{E:graded-monad:bisim}., this is easy to see,\n  for~\\ref{E:graded-monad:tr}., it follows from the presentation as a\n  graded theory, and for~\\ref{E:graded-monad:kleisli}.,\n  \\iffull\\\/see~\\hyperref[S:d1]{Appendix~\\ref{S:d1}}.\\else\\\/see~\\cite{DorschEA19}.\\fi\n\\end{example}\n\n\\noindent One may use the equivalent property of \\autoref{P:d1} to\ndefine depth-1 graded monads over arbitrary base\ncategories~\\cite{MiliusEA15}. We show next that depth-1 graded monads\nmay be specified by giving only $M_0$, $M_1$, the unit~$\\eta$, and $\\mu^{nk}$\nfor $n+k \\leq 1$.\n\n\\begin{theorem}\n\\label{thm:depth-1-graded-monads-M0}\nDepth-$1$ graded monads are in bijective correspondence with\n$6$-tuples $(M_0,M_1,\\ensuremath{\\eta}\\xspace,\\ensuremath{\\mu}\\xspace^{00},\\ensuremath{\\mu}\\xspace^{10},\\ensuremath{\\mu}\\xspace^{01})$ such\nthat the given data satisfy all applicable instances of the graded monad\nlaws.\n\\end{theorem}\n\n\\myparagraph{Semantics via Graded Monads} We next recall how graded\nmonads define \\emph{graded semantics}:\n\\begin{defn}[Graded\n  semantics~\\cite{MiliusEA15}]\\label{def:alpha-trace-semantics}\n  Given a set functor~$G$, a \\emph{graded semantics} for\n  $G$-coalgebras consists of a graded monad\n  $((M_n),\\ensuremath{\\eta}\\xspace,(\\ensuremath{\\mu}\\xspace^{nk}))$ and a natural transformation\n  $\\alpha\\colon G\\to M_1$.  The $\\alpha$-\\emph{pretrace sequence}\n  $( \\gamma^{(n)}\\colon X\\to M_nX )_{n<\\omega}$ for a\n  $G$-coalgebra $\\gamma\\colon X\\to GX$ is defined by\n  \\[\n    \\gamma^{(0)} = (X \\xrightarrow{\\ensuremath{\\eta}\\xspace_X} M_0 X)\n    \\quad\\text{and}\\quad\n    \\gamma^{(n+1)} = (X\n    \\xrightarrow{\\alpha_X\\cdot\\gamma} M_1 X \\xrightarrow{M_1\\gamma^{(n)}} M_1 M_n X\n    \\xrightarrow{\\ensuremath{\\mu}\\xspace_X^{1n}} M_{n+1}X).\n  \\]\n  The $\\alpha$-\\emph{trace sequence} $T^\\alpha_\\gamma$ is the sequence\n  $( M_n!\\cdot\\gamma^{(n)}\\colon X\\to M_n1)_{n<\\omega}$.\n  \n  In \\Set, two states $x\\in X$, $y\\in Y$ of coalgebras\n  $\\gamma\\colon X\\to GX$ and $\\delta\\colon Y\\to GY$ are $\\alpha$-\\emph{trace} (or\n  \\emph{graded}) \\emph{equivalent} if\n  $M_n!\\cdot\\gamma^{(n)}(x) = M_n!\\cdot\\delta^{(n)}(y)$ for all\n  $n<\\omega$.\n\\end{defn}\nIntuitively, $M_nX$ consists of all length-$n$ \\emph{pretraces}, i.e.\\\ntraces paired with a poststate, and $M_n1$ consists of all length-$n$\ntraces, obtained by erasing the poststate. Thus, a graded semantics\nextracts length-$1$ pretraces from successor structures. In the\nfollowing two examples we have $M_1 = G$; however, in general $M_1$\nand $G$ can differ (\\autoref{sec:ltbt}).\n\n\\begin{example}\\label{expl:monads}\n  Recall from \\autoref{sec:prelim} that a $G$-coalgebra for the\n  functor $G = \\Pow_\\omega(\\mathcal{A} \\times -)$ is just a finitely branching LTS.\n  We recall two graded semantics that model the extreme ends of the\n  linear time -- branching time spectrum~\\cite{MiliusEA15}; more\n  examples will be given in the next section\n\n  \\begin{longitemslist\n  \\item \\emph{Trace equivalence.} For $x,y \\in X$ and $w\\in \\mathcal{A}^*$,\n    we write $x \\xrightarrow{w} y$ if $y$ can be reached from $x$ on a\n    path whose labels yield the word $w$, and\n    $\\ensuremath{\\mathcal T}\\xspace(x) = \\{w \\in \\mathcal{A}^* \\mid \\exists y \\in X .\\ x \\xrightarrow{w}\n    y\\}$\n    denotes the set of \\emph{traces} of $x \\in X$. States $x,y$ are\n    \\emph{trace equivalent} if $\\ensuremath{\\mathcal T}\\xspace(x)=\\ensuremath{\\mathcal T}\\xspace(y)$. To capture trace\n    semantics of labelled transition systems we consider the graded\n    monad with $M_nX = \\mathcal{P}(\\mathcal{A}^n \\times X)$\n   \n   \n    (see \\autoref{E:graded-monad}.\\ref{E:graded-monad:tr}). The\n    natural transformation $\\alpha$ is the identity. For a\n    $G$-coalgebra $(X, \\gamma)$ and $x \\in X$ we have that\n    $\\gamma^{(n)}(x)$ is the set of pairs $( w, y)$ with\n    $w \\in \\mathcal{A}^n$ and $x \\xrightarrow{w} y$, i.e.\\ pairs of\n    length-$n$ traces and their corresponding poststate. Consequently,\n    the~$n$-th component $M_n!  \\cdot \\gamma^{(n)}$ of the\n    $\\alpha$-trace sequence maps $x$ to the set of its length-$n$\n    traces. Thus, $\\alpha$-trace equivalence is standard trace\n    equivalence~\\cite{vanglabbeek2001linear}.\n\n      Note that the equations presenting the graded monad $M_n$ in\n      \\autoref{E:graded-monad}.\\ref{E:graded-monad:tr} bear a striking\n      resemblance to the ones given by van Glabbeek to axiomatize\n      trace equivalence of processes, with the difference that in his\n      axiomatization actions do not distribute over the empty join. In\n      fact, $a.0 = 0$ is clearly not valid for processes under trace\n      equivalence. In the graded setting, this equation just expresses\n      the fact that a trace which ends in a deadlock after $n$ steps\n      cannot be extended to a trace of length $n+1$.\n\n    \\item\\label{expl:monads:2} \\emph{Bisimilarity.} By the discussion\n      of the final sequence of a functor~$G$ (\\autoref{sec:prelim}),\n      the graded monad with $M_nX = G^nX$\n      (\\autoref{E:graded-monad}.\\ref{E:graded-monad:bisim}), with\n      $\\alpha$ being the identity again, captures finite-depth\n      behavioural equivalence, and hence behavioural equivalence\n      when~$G$ is finitary. In particular, on finitely branching LTS,\n      $\\alpha$-trace equivalence is bisimilarity in this case.\n  \\end{longitemslist}\n\\end{example}\n\n\\section{A Spectrum of Graded Monads}\\label{sec:ltbt}\n\nWe present graded monads for a range of equivalences on the linear\ntime -- branching time spectrum as well as probabilistic trace\nequivalence for generative probabilistic systems (GPS), giving in each\ncase a graded theory and a description of the arising graded\nmonads. Some of our equations bear some similarity to van Glabbeek's\naxioms for equality of process terms. There are also important\ndifferences, however.  In particular, some of van Glabbeek's axioms\nare implications, while ours are purely equational; moreover, van\nGlabbeek's axioms sometimes nest actions, while we employ only\ndepth-$1$ equations (which precludes nesting of actions) in order to\nenable the extraction of characteristic logics later. All graded\ntheories we introduce contain the theory of join semilattices, or in\nthe case of GPS convex algebras, whose operations are assigned\ndepth~$0$; we mention only the additional operations needed. We use\nterminology introduced in \\autoref{expl:monads}.\n\n\\myparagraph{Completed Trace\n  Semantics}\\label{ssec:completed-trace-semntics} refines trace\nsemantics by distinguishing whether traces can end in a deadlock.  We\ndefine a depth-$1$ graded theory by extending the graded theory for\ntrace semantics (\\autoref{E:graded-monad}) with a constant depth-$1$\noperation~$\\star$ denoting deadlock. The induced graded monad has\n$M_0 X= \\Pow_\\omega(X)$, $M_1 = \\Pow_\\omega(\\mathcal{A} \\times X + 1)$ (and\n$M_nX=\\Pow_\\omega(\\mathcal{A}^n\\times X+\\mathcal{A}^{<n})$ where $\\mathcal{A}^{<n}$ denotes the\nset of words over~$\\mathcal{A}$ of length less than~$n$).  The natural\ntransformation $\\alpha_X\\colon \\Pow_\\omega(\\mathcal{A}\\times X) \\to M_1 X$ is given\nby $\\alpha(\\emptyset)=\\{\\star\\}$ and\n$\\alpha(S)=S\\subseteq \\mathcal{A} \\times X + 1$ for\n$\\emptyset\\neq S\\subseteq \\mathcal{A}\\times X$.\n\n\\myparagraph{Readiness and Failures\n  Semantics}\\label{ssec:readiness_failures} refine completed trace\nsemantics by distinguishing which actions are available (readiness) or\nunavailable (failures) after executing a trace.  Formally, given an\nLTS, seen as a coalgebra $\\gamma\\colon X\\to\\Pow_\\omega(\\mathcal{A}\\times X)$, we write\n$I(x) = \\Pow_\\omega\\pi_1 \\cdot \\gamma(x) = \\pi_1[ \\gamma(x) ]$ ($\\pi_1$ being\nthe first projection) for the set of actions available at~$x$, the\n\\emph{ready set} of~$x$.  A \\emph{ready pair} of a state $x$ is a pair\n$(w,A)\\in\\mathcal{A}^*\\times\\Pow_\\omega(\\mathcal{A})$ such that there exists~$z$ with\n$x\\overset w\\to z$ and $A = I(z)$; a \\emph{failure pair} is defined in\nthe same way except that $A\\cap I(z)=\\emptyset$.  Two states are\n\\emph{readiness (failures) equivalent} if they have the same ready\n(failure) pairs.\n\nWe define a depth-$1$ graded theory by extending the graded theory for\ntrace semantics (\\autoref{E:graded-monad}) with constant depth-$1$\noperations $A$ for ready (failure) sets $A\\subseteq\\mathcal{A}$. In case of\nfailures we add a monotonicity condition $A + A\\cup B = A\\cup B$ on\nthe constant operations for the failure sets. The resulting graded\nmonads both have $M_0X = \\Pow_\\omega X$, and moreover\n\\( M_1X = \\Pow_\\omega(\\mathcal{A}\\times X+\\Pow_\\omega\\mathcal{A}) \\)\nfor readiness and \\( M_1X = \\Pow_\\omega^\\downarrow(\\mathcal{A}\\times X+\\Pow_\\omega\\mathcal{A}) \\)\nfor failures, where $\\Pow_\\omega^\\downarrow$ is down-closed finite powerset, w.r.t.\\\nthe discrete order on $\\mathcal{A}\\times X$ and set inclusion on $\\Pow_\\omega\\mathcal{A}$.\nThe natural transformation\n$\\alpha_X\\colon \\Pow_\\omega(\\mathcal{A}\\times X)\\to M_1X$ is defined by\n\\( \\alpha_X(S) = S \\cup \\{ \\pi_1[S] \\} \\)\nfor readiness and\n\\( \\alpha_X(S) = S \\cup \\{ A \\subseteq\\mathcal{A}\\mid A\\cap\\pi_1[S]=\\emptyset\n\\} \\) for failures semantics.\n\n\\myparagraph{Ready Trace and Failure Trace Semantics}\\label{ssec:ready-failure-trace}\nrefine readiness and failures semantics, respectively, by distinguishing\nwhich actions are available (ready trace) or unavailable (failure trace) at each step\nof the trace. Formally, a \\emph{ready trace} of a state~$x$ is a sequence\n$A_0a_1A_1\\ldots a_n A_n\\in(\\Pow_\\omega\\mathcal{A}\\times\\mathcal{A})^\\ast\\times\\Pow_\\omega\\mathcal{A}$\nsuch that there exist transitions\n$x=x_0\\overset{a_1}\\to x_1\\ldots \\overset{a_n}\\to x_n$ where each\n$x_i$ has ready set $I(x_i)=A_i$. A \\emph{failure trace} has the same\nshape but we require that each $A_i$ is a \\emph{failure set} of $x_i$,\ni.e.\\ $I(x_i)\\cap A_i=\\emptyset$. States are \\emph{ready (failure) trace equivalent} if they have the\nsame ready (failure) traces.\n\nFor ready traces, we define a depth-$1$ graded theory with depth-$1$\noperations $\\langle A,\\sigma \\rangle$ for $\\sigma\\in\\mathcal{A}$,\n$A\\subseteq\\mathcal{A}$ and a depth-$1$ constant $\\star$, denoting deadlock,\nand equations\n$ \\langle A,\\sigma \\rangle(\\sum_{j\\in J} x_j) = \\sum_{j\\in J}\\langle\nA,\\sigma \\rangle (x_j)$.\nThe resulting graded monad is simply the graded monad capturing\ncompleted trace semantics for labelled transition systems where the\nset of actions is changed from $\\mathcal{A}$ to $\\Pow_\\omega\\mathcal{A}\\times\\mathcal{A}$. For\nfailure traces, we additionally impose the equation\n$ \\langle A,\\sigma \\rangle(x)+\\langle A\\cup B,\\sigma\n\\rangle(x)=\\langle A\\cup B,\\sigma \\rangle(x)$,\nwhich in the set-based description of the graded monad corresponds to\ndownward closure of failure sets.\n\nThe resulting graded monads both have $M_0X =\\Pow_\\omega X$; for ready\ntraces, \\( M_1X = \\Pow_\\omega(( \\Pow_\\omega\\mathcal{A}\\times\\mathcal{A} ) \\times X + 1) \\)\nand for failure traces,\n\\( M_1X = \\Pow_\\omega^\\downarrow(( \\Pow_\\omega\\mathcal{A}\\times\\mathcal{A} ) \\times X + 1) \\),\nwhere $\\Pow_\\omega^\\downarrow$ is down-closed finite powerset, w.r.t.\\ the order\nimposed by the above equation.\n\nFor ready trace semantics we define the natural transformation\n$\\alpha_X\\colon \\Pow_\\omega(\\mathcal{A}\\times X) \\to M_1X$ by\n$\\alpha_X(\\emptyset) = \\{\\star\\}$ and\n$\\alpha_X(S) = \\{ (( \\pi_1[S], \\sigma),x) \\mid ( \\sigma ,x\n) \\in S \\})$ for $S \\neq\\emptyset$.\nFor failure traces we define\n$\\alpha_X(\\emptyset) = \\{\\star\\}$ and\n$\\alpha(S) = \\{(( A, \\sigma),x) \\mid\n(\\sigma,x)\\in S, A\\cap \\pi_1[S]=\\emptyset\\}$\nfor $S\\neq\\emptyset$; note that in the latter case, $\\alpha(S)$ is\nclosed under decreasing failure sets.\n\n\n\\myparagraph{Simulation Equivalence}\\label{ssec:simulation\ndeclares\ntwo states to be equivalent if they simulate each other in the\nstandard sense.  We define a depth-$1$ graded theory with the same\nsignature as for trace equivalence but instead of join preservation\nrequire only that each $\\sigma$ is monotone, i.e.\\\n$\\sigma(x + y) + \\sigma(x) = \\sigma(x + y)$. The arising graded\nmonad~$M_n$ is equivalently described as follows. We define the sets\n$M_nX$ inductively, along with an ordering on $M_nX$. We take\n$M_0X=\\Pow_\\omega X$, ordered by set inclusion. We then order the elements\nof $\\mathcal{A}\\times M_nX$ by the product ordering of the discrete order\non~$\\mathcal{A}$ and the given ordering on~$M_nX$, and take $M_{n+1}X$ to be\nthe set of downclosed subsets of $\\mathcal{A}\\times M_nX$, denoted\n$\\Pow_\\omega^\\downarrow(\\mathcal{A}\\times M_nX)$, ordered by set inclusion.  The natural\ntransformation\n$\\alpha_X\\colon \\mathcal{P}(\\mathcal{A}\\times X) \\to \\Pow_\\omega^\\downarrow(\\mathcal{A}\\times\\Pow_\\omega(X))$ is defined\nby $\\alpha_X(S)={\\downarrow}\\{(s,\\{x\\})\\mid (s,x)\\in S\\}$, where\n$\\downarrow$ denotes downclosure.\n\n\\myparagraph{Ready Simulation Equivalence\nrefines simulation equivalence by requiring\nadditionally that related states have the same ready set.\nStates $x$ and $y$ are \\emph{ready similar} if they are related by\nsome ready simulation, and ready simulation equivalent if there are\nmutually ready similar.\nThe depth-$1$ graded theory combines the signature for ready traces\nwith the equations for simulation, i.e.\\ only requires the operations\n$\\langle A,\\sigma \\rangle$ to be monotone.\n\n\n\\myparagraph{Probabilistic Trace Equivalence}\\label{ssec:GPS} is the\nstandard trace semantics for generative probabilistic systems (GPS),\nequivalently, coalgebras for the functor \\( \\ensuremath{\\mathcal D}\\xspace(\\mathcal{A}\\times\\Id) \\)\nwhere $\\ensuremath{\\mathcal D}\\xspace$ is the monad of finitary distributions\n(\\autoref{expl:coalg}).\n \n \nProbabilistic trace equivalence is captured by the graded monad\n$M_nX = \\ensuremath{\\mathcal D}\\xspace (\\mathcal{A}^n\\times X)$, as described in\n\\autoref{E:graded-monad}.\\ref{E:graded-monad:kleisli}. The\ncorresponding graded theory arises by replacing the join-semilattice\nstructure featuring in the above graded theory for trace equivalence\nby the one of \\emph{convex algebras}, i.e.\\ the algebras for the\nmonad~$\\ensuremath{\\mathcal D}\\xspace$. Recall~\\cite{Doberkat06,Doberkat08} that a convex\nalgebra is a set $X$ equipped with finite convex sum operations: For\nevery $p \\in [0,1]$ there is a binary operation $\\boxplus_p$ on $X$,\nand these operations satisfy the equations\n\\(\nx \\boxplus_p x = x,\nx \\boxplus_p y = y \\boxplus_{1-p} x,\nx \\boxplus_0 y = y,\nx \\boxplus_p (y \\boxplus_q z) = (x\n\\boxplus_{p\/r} y) \\boxplus_r z,\n\\)\nwhere $p, q \\in [0,1]$, $x, y, z \\in X$, and $r = (p + (1-p)q) > 0$\n(i.e.\\ $p+q>0$) in the last equation~\\cite{Jacobs10}.  Again, we have\ndepth-$1$ operations $\\sigma$ for action $\\sigma\\in\\mathcal{A}$, now\nsatisfying the equations\n\\begin{math}\n  \\sigma(x \\boxplus_p y) = \\sigma(x) \\boxplus_p \\sigma(y).\n\\end{math}\n\n\n\n\n\n\\section{Graded Logics}\\label{sec:logics}\n\n\n\n\\noindent Our next goal is to extract \\emph{characteristic logics}\nfrom graded monads in a systematic way, with \\emph{characterizing}\nmeaning that states are logically indistinguishable iff they are\nequivalent under the semantics at hand. We will refer to these logics\nas \\emph{graded logics}; the implication from graded equivalence to\nlogical indistinguishability is called \\emph{invariance}, and the\nconverse implication \\emph{expressiveness}. E.g.\\ standard modal logic\nwith the full set of Boolean connectives is invariant under\nbisimilarity, and the corresponding expressiveness result is known as\nthe \\emph{Hennessy-Milner theorem}. This result has been lifted to\ncoalgebraic generality early on, giving rise to the \\emph{coalgebraic\n  Hennessy-Milner theorem}~\\cite{Pattinson04,Schroder08}. In previous\nwork~\\cite{MiliusEA15}, we have related graded semantics to modal\nlogics extracted from the graded monad in the envisaged fashion. These\nlogics are invariant by construction; the main new result we\npresent here is a generic \\emph{expressiveness} criterion, to be\ndiscussed in \\autoref{sec:expr}. The key ingredient in this criterion\nare \\emph{canonical} graded algebras, which we newly introduce here,\nproviding a recursive-evaluation style reformulation of the semantics\nof graded logics.\n\nA further key issue in characteristic modal logics is the choice of\npropositional operators; e.g.\\ notice that when $\\trdiamond{\\sigma}$\ndenotes the usual Hennessy-Milner style diamond operator for an\naction~$\\sigma$, the formula\n$\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top$ is invariant under\ntrace equivalence (i.e.~the corresponding property is closed under\nunder trace equivalence) but the formula\n$\\trdiamond{\\sigma}(\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top)$,\nbuilt from the former by simply prefixing with~$\\trdiamond{\\sigma}$,\nis not, the problem being precisely the use of conjunction. While in\nour original setup, propositional operators were kept implicit, that\nis, incorporated into the set of modalities, we provide an explicit\ntreatment of propositional operators in the present paper. Besides\nadding transparency to the syntax and semantics, having first-class\npropositional operators will be a prerequisite for the formulation of\nthe expressiveness theorem.\n\n\\myparagraph{Coalgebraic Modal Logic} To provide context, we briefly\nrecall the setup of \\emph{coalgebraic modal\n  logic}~\\cite{Pattinson04,Schroder08}. Let~$2$ denote the\nset~$\\{\\bot,\\top\\}$ of Boolean truth values; we think of the set~$2^X$\nof maps $X\\to 2$ as the set of predicates on~$X$. Coalgebraic logic in\ngeneral abstracts systems as coalgebras for a functor~$G$, like we do\nhere; fixes a set~$\\Lambda$ of \\emph{modalities} (unary for the sake\nof readability); and then interprets a modality $L\\in\\Lambda$ by the\nchoice of a \\emph{predicate lifting}, i.e.\\ a natural transformation\n\\begin{equation*}\n  \\Sem{L}_X\\colon 2^X\\to 2^{GX}.\n\\end{equation*}\nBy the Yoneda lemma, such natural transformations are in bijective\ncorrespondence with maps $G2\\to 2$~\\cite{Schroder08}, which we shall\nalso denote as $\\Sem{L}$. In the latter formulation, the recursive\nclause defining the interpretation $\\Sem{L\\phi}\\colon X\\to 2$, for a\nmodal formula~$\\phi$, as a state predicate in a $G$-coalgebra\n$\\gamma \\colon X\\to GX$ is then\n\\begin{equation}\\label{eq:coalg-modality}\n  \\Sem{L\\phi}= (X\\xrightarrow{\\gamma}GX\\xrightarrow{ G\\Sem{\\phi}} G2\\xrightarrow{\\Sem{L}}2).\n\\end{equation}\nE.g.\\ taking $G=\\Pow_\\omega(\\mathcal{A}\\times-)$ (for labelled transition systems),\nwe obtain the standard semantics of the Hennessy-Milner diamond\nmodality $\\trdiamond{\\sigma}$ for~$\\sigma\\in\\mathcal{A}$ via the predicate\nlifting\n\\begin{equation*}\n  \\Sem{\\trdiamond{\\sigma}}_X(f)=\\{B\\in\\Pow_\\omega(\\mathcal{A}\\times X)\\mid\n  \\exists x.\\,(\\sigma,x)\\in B\\land f(x)=\\top\\}\\qquad(\\text{for\n    $f\\colon  X\\to 2$}).\n\\end{equation*}\nIt is easy to see that \\emph{coalgebraic modal logic}, which combines\ncoalgebraic modalities with the full set of Boolean connectives, is\ninvariant under finite-depth behavioural equivalence\n(\\autoref{sec:prelim}). Generalizing the classical Hennessy-Milner\ntheorem~\\cite{HennessyMilner85}, the \\emph{coalgebraic Hennessy-Milner\n  theorem}~\\cite{Pattinson04,Schroder08} shows that conversely,\ncoalgebraic modal logic \\emph{characterizes} behavioural equivalence,\ni.e.\\ logical indistinguishability implies behavioural equivalence,\nprovided that~$G$ is finitary (implying coincidence of behavioural\nequivalence and finite-depth behavioural equivalence) and~$\\Lambda$ is\n\\emph{separating}, i.e.\\ for every finite set~$X$, the set\n\\begin{equation*}\n  \\Lambda(2^X)=\\{\\Sem{L}(f)\\mid f\\in 2^X\\}\n\\end{equation*}\nof maps $GX\\to 2$ is jointly injective.\n\nWe proceed to introduce the syntax and semantics of graded logics.\n\\myparagraph{Syntax} We parametrize the syntax of \\emph{graded logics}\nover\n\\begin{itemize}\n\\item a set~$\\Theta$ of \\emph{truth constants},\n\\item a set~$\\mathcal{O}$ of \\emph{propositional operators} with assigned\n  finite arities, and\n\\item a set~$\\Lambda$ of \\emph{modalities} with assigned arities.\n\\end{itemize}\nFor readability, we will restrict the technical exposition to unary\nmodalities; the treatment of higher arities requires no more than\nadditional indexing (and we will use $0$-ary modalities in the\nexamples). E.g.\\ standard Hennessy-Milner logic is given by\n$\\Lambda=\\{\\trdiamond{\\sigma}\\mid \\sigma\\in\\mathcal{A}\\}$ and~$\\mathcal{O}$\ncontaining all Boolean connectives. Other logics will be determined by\nadditional or different modalities, and often by fewer propositional\noperators. Formulae of the logic are restricted to have uniform depth,\nwhere propositional operators have depth~$0$ and modalities have\ndepth~$1$; a somewhat particular feature is that truth constants can\nhave top-level occurrences only in depth-$0$ formulae.  That is,\nformulae~$\\phi,\\phi_1,\\dots$ of depth~$0$ are given by the grammar\n\\begin{equation*}\n  \\phi\\Coloneqq p(\\phi_1,\\dots,\\phi_k) \\mid c\n    \\qquad (p\\in\\mathcal{O}\\text{ $k$-ary}, c\\in\\Theta),\n\\end{equation*}\nand formulae~$\\phi$ of depth $n+1$ by\n\\begin{equation*}\n  \\phi\\Coloneqq p(\\phi_1,\\dots,\\phi_k) \\mid L\\psi\n    \\qquad (p\\in\\mathcal{O}\\text{ $k$-ary}, L\\in\\Lambda)\n\\end{equation*}\nwhere $\\phi_1,\\dots,\\phi_n$ range over formulae of depth $n+1$ and\n$\\psi$ over formulae of depth~$n$. \n\n\\myparagraph{Semantics} The semantics of graded logics is parametrized\nover the choice of \\emph{a functor~$G$, a depth-$1$ graded monad\n  $M=((M_n)_{n<\\omega},\\eta,$ $(\\mu^{nk})_{n,k<\\omega})$, and a\n  graded semantics~$\\alpha\\colon G\\to M_1$, which we fix for the\n  remainder of the paper}. It was originally given by translating\nformulae into \\emph{graded algebras} and then defining formula\nevaluation by the universal property of $(M_n1)$ as a free graded\nalgebra~\\cite{MiliusEA15}; here, we reformulate the semantics in a\nmore standard style by recursive clauses, using canonical graded\nalgebras. In general, the notion of graded algebra is defined as\nfollows~\\cite{MiliusEA15}.\n\\begin{defn}[Graded algebras]\n  Let $n<\\omega$. A \\emph{(graded) $M_n$-algebra}\n  $A=((A_k)_{k\\le n},(a^{mk})_{m+k\\le n})$ consists of carrier\n  sets~$A_k$ and structure maps\n  \\begin{equation*}\n    a^{mk}\\colon M_mA_k\\to A_{m+k}\n  \\end{equation*}\n  satisfying the laws \n  \\begin{equation}\\label{diag:alg}\n    \\begin{tikzcd}\n      A_k \\arrow{r}{\\eta_{A_k}} \\arrow[equal]{dr\n       & M_0 A_k\\arrow{d}{a^{0k}} & \n      M_m M_r A_k \\arrow{r}{M_m a^{rk}}\n        \\arrow{d}[left]{\\mu^{mr}_{A_k}} &\n        M_m A_{r+k} \\arrow{d}{a^{m,r+k}} \\\\\n        & A_k &  M_{m+r}A_k \\arrow{r}{a^{m+r,k}} & A_{m+r+k}\n    \\end{tikzcd}\n  \\end{equation}\n  for all $k\\le n$ (left) and all $m,r,k$ such that $m+r+k\\le n$\n  (right), respectively. An \\emph{$M_n$-morphism}~$f$ from~$A$ to an\n  $M_n$-algebra $B=((B_k)_{k\\le n},(b^{mk})_{m+k\\le n})$ consists of\n  maps $f_k\\colon A_k\\to B_k$, $k\\le n$, such that\n  $f_{m+k}\\cdot a^{mk}=b^{mk}\\cdot M_mf_k$\n \n \n \n \n \n \n \n \n  for all $m,k$ such that $m+k\\le n$.\n\\end{defn}\n\\noindent \nWe view the carrier~$A_k$ of an~$M_n$-algebra as the set of algebra\nelements that have already absorbed operations up to depth~$k$. As in\nthe case of plain monads, we can equivalently describe graded algebras\nin terms of graded theories: If $M$ is generated by a graded theory\n$\\mathbb{T}=(\\Sigma,E,d)$, then an $M_n$-algebra interprets each operation\n$f\\in\\Sigma$ of arity~$r$ and depth~$d(f)=m$ by maps\n$f^A_k\\colon A_k^r\\to A_{m+k}$ for all $k$ such that $m+k\\le n$; this\ngives rise to an inductively defined interpretation of terms\n(specifically, given a valuation of variables in~$A_m$, terms of\nuniform depth~$k$ receive values in~$A_{k+m}$, for $k+m\\le n$), and\nsubsequently to the expected notion of satisfaction of equations.\n\n\nWhile in general, graded algebras are monolithic objects, for\n\\mbox{depth-$1$} graded monads we can construct them in a modular\nfashion from $M_1$-algebras~\\cite{MiliusEA15}; we thus restrict\nattention to $M_0$- and $M_1$-algebras in the following. We note that\nan $M_0$-algebra is just an Eilenberg-Moore algebra for the\nmonad~$M_0$. An $M_1$-Algebra~$A$ consists of $M_0$-algebras\n$(A_0,a^{00}\\colon M_0A_0\\to A_0)$ and $(A_1,a^{01}\\colon M_0A_1\\to A_1)$, and a\n\\emph{main structure map} $a^{10}\\colon M_1A_0\\to A_1$ satisfying two\ninstances of the right-hand diagram in~\\eqref{diag:alg}, one of which\nsays that $a^{10}$ is a morphism of $M_0$-algebras\n(\\emph{homomorphy}), and the other that the diagram\n\\begin{equation}\n  \\label{diag:algebra-coeq}\n  \\begin{tikzcd}[column sep=large]\n    M_1M_0A_0 \\arrow[shift left]{r}[above]{\\mu^{10}}\n    \\arrow[shift right]{r}[below]{M_1a^{00}}& M_1A_0 \\arrow{r}{a^{10}} & A_1,\n  \\end{tikzcd}\n\\end{equation}\nwhich by the laws of graded monads consists of $M_0$-algebra\nmorphisms, commutes (\\emph{coequalization}). We will often refer to an\n$M_1$-algebra by just its main structure map.\n\nWe will use $M_1$-algebras as interpretations of the modalities in\ngraded logics, generalizing the previously recalled interpretation of\nmodalities as maps $G2\\to 2$ in branching-time coalgebraic modal\nlogic. We fix an $M_0$-algebra $\\Omega$ of \\emph{truth values}, with\nstructure map $o\\colon M_0\\Omega\\to\\Omega$ (e.g.\\\nfor~$G=\\Pow_\\omega$, $\\Omega$ is a join semilattice).  Powers~$\\Omega^n$\nof~$\\Omega$ are again\n$M_0$-algebras.\nA modality $L\\in\\Lambda$ is interpreted as an $M_1$-algebra\n$A=\\Sem{L}$ with carriers $A_0=A_1=\\Omega$ and\n$a^{01}=a^{00}=o$. Such an $M_1$-algebra is thus specified\nby its main structure map $a^{10}\\colon M_1\\Omega\\to\\Omega$ alone, so\nfollowing the convention indicated above we often write $\\Sem{L}$ for\njust this map.\nThe evaluation of modalities is defined using canonical\n$M_1$-algebras:\n\\begin{defn}[Canonical algebras]\n  The \\emph{$0$-part} of an~$M_1$-algebra~$A$ is the $M_0$-algebra\n  $(A_0,a^{00})$. Taking $0$-parts defines a functor $U_0$ from\n  $M_1$-algebras to $M_0$-algebras. An $M_1$-algebra is\n  \\emph{canonical} if it is free, w.r.t.\\ $U_0$, over its\n  $0$-part. For~$A$ canonical and a modality $L\\in\\Lambda$, we\n  denote the unique morphism $A_1\\to\\Omega$ extending an\n  $M_0$-morphism $f\\colon A_0\\to\\Omega$ to an $M_1$-morphism $A\\to\\Sem{L}$\n  by~$\\Sem{L}(f)$, i.e.\\ $\\Sem{L}(f)$ is the unique $M_0$-morphism\n  such that\n  \\iffull\n  the square below commutes:\n  \\begin{equation}\\label{diag:L(f)}\n    \\begin{tikzcd}\n      M_1 A_0 \\arrow{r}{M_1f} \\arrow{d}[left]{a^{10}} \n      & M_1\\Omega\\arrow{d}{\\Sem{L}}\\\\\n      A_1 \\arrow{r}[below]{\\Sem{L}(f)} & \\Omega\n    \\end{tikzcd}\n  \\end{equation}\n  \\else\n  the following equation holds:\n  \\begin{equation}\\label{diag:L(f)}\n   (M_1 A_0 \\xrightarrow{M_1 f} M_1\\Omega \\xrightarrow{\\Sem{L}} \\Omega)\n   =\n   (M_1 A_0 \\xrightarrow{a^{10}} A_1 \\xrightarrow{\\Sem{L}(f)} \\Omega).\n \\end{equation}\n \\fi\n\\end{defn}\n\\begin{lemma}\\label{lem:canonical}\n  An $M_1$-algebra~$A$ is canonical iff \\eqref{diag:algebra-coeq} is a\n  (reflexive) coequalizer diagram in the category of $M_0$-algebras.\n\\end{lemma}\n\\noindent By the above lemma, we obtain a key example of canonical\n$M_1$-algebras:\n\\begin{corollary}\n  If $M$ is a depth-$1$ graded monad, then for every~$n$ and every\n  set~$X$, the $M_1$-algebra with carriers $M_nX,M_{n+1}X$ and\n  multiplication as algebra structure is canonical.\n\\end{corollary}\n  \n\n\n\\noindent Further, we interpret truth constants $c\\in\\Theta$ as\nelements of~$\\Omega$, understood as maps $\\hat{c}\\colon 1\\to\\Omega$,\nand $k$-ary propositional operators $p\\in\\mathcal{O}$ as $M_0$-homomorphisms\n$\n\\Sem{p}\\colon\\Omega^k\\to\\Omega.\n$\nIn our examples on the linear time -- branching time spectrum,~$M_0$\nis either the identity or, most of the time, the finite powerset\nmonad. In the former case, all truth functions are $M_0$-morphisms. In\nthe latter case, the $M_0$-morphisms $\\Omega^k\\to \\Omega$ are the\njoin-continuous functions; in the standard case where $\\Omega=2$ is\nthe set of Boolean truth values, such functions~$f$ have the form\n$f(x_1,\\dots,x_k)=x_{i_1}\\lor\\dots\\lor x_{i_l}$, where\n$i_1,\\dots,i_l\\in\\{1,\\dots,k\\}$. We will see one case where $M_0$ is\nthe distribution monad; then $M_0$-morphisms are affine\nmaps.\n\nThe semantics of a formula~$\\phi$ in graded logic is defined recursively\nas an $M_0$-morphism\n$\\Sem{\\phi}\\colon (M_n1, \\mu^{0n}_1) \\to (\\Omega, o)$ by\n\\begin{equation*}\n  \\Sem{c}  = (M_01\\xrightarrow{M_0\\hat c}M_0\\Omega \\xrightarrow{o}\\Omega)\\quad\n  \\Sem{p(\\phi_1,\\dots,\\phi_k)}  =\\Sem{p}\\cdot\\langle\\Sem{\\phi_1},\\dots,\n                                 \\Sem{\\phi_k}\\rangle\\quad\n  \\Sem{L\\phi}  = \\Sem{L}(\\Sem{\\phi}).\n\\end{equation*}\nThe evaluation of~$\\phi$ in a coalgebra $\\gamma\\colon X\\to GX$ is then given\nby composing with the trace sequence, i.e.\\ as\n\\begin{equation}\\label{eq:formula-eval}\n  X\\xrightarrow{M_n!\\cdot\\gamma^{(n)}} M_n1\\xrightarrow{\\Sem\\phi}\\Omega.\n\\end{equation}\nIn particular, graded logics are, by construction, invariant under the\ngraded semantics. \n\n    \n\n\\begin{example}[Graded logics]\\label{expl:logics}\n  We recall the two most basic examples, fixing $\\Omega=2$ in both\n  cases, and $\\top$ as the only truth constant:\n  \\begin{longitemslist\n  \\item \\emph{Finite-depth behavioural equivalence:} Recall that the\n    graded monad $M_nX=G^nX$ captures finite-depth behavioural\n    equivalence on $G$-coalgebras. Since~$M_0$ is the identity monad,\n    $M_0$-algebras are just sets. Thus, every function $2^k\\to 2$ is\n    an $M_0$-morphism,\n   \n    so we can use all Boolean operators as propositional\n    operators. Moreover, $M_1$-algebras are just maps\n    $a^{10}\\colon GA_0\\to A_1$. Such an $M_1$-algebra is canonical iff\n    $a^{10}$ is an isomorphism, and modalities are interpreted as\n    $M_1$-algebras $G2\\to 2$, with the evaluation according\n    to~\\eqref{diag:L(f)} and~\\eqref{eq:formula-eval} corresponding\n    precisely to the semantics of modalities in coalgebraic\n    logic~\\eqref{eq:coalg-modality}. Summing up, we obtain precisely\n    coalgebraic modal logic as summarized above in this case. In our\n    running example $G=\\Pow_\\omega(\\mathcal{A}\\times(-))$, we take modalities\n    $\\Diamond_\\sigma$ as above, with\n    $\\Sem{\\Diamond_\\sigma}\\colon\\Pow_\\omega(\\mathcal{A}\\times 2)\\to 2$ defined by\n    $\\Sem{\\Diamond_\\sigma}(S)=\\top$ iff $(\\sigma,\\top)\\in S$,\n    obtaining precisely classical Hennessy-Milner\n    logic~\\cite{HennessyMilner85}.\n  \\item \\emph{Trace equivalence:} Recall that the trace semantics of\n    labelled transition systems with actions in~$\\mathcal{A}$ is modelled by\n    the graded monad $M_nX=\\Pow_\\omega(\\mathcal{A}^n\\times X)$. As indicated above,\n    in this case we can use disjunction as a propositional operator\n    since $M_0=\\Pow_\\omega$. Since the graded theory for $M_n$ specifies for\n    each $\\sigma\\in\\mathcal{A}$ a unary depth-$1$ operation that distributes\n    over joins, we find that the maps $\\Sem{\\Diamond_\\sigma}$ from the\n    previous example (unlike their duals $\\Box_\\sigma$) induce\n    $M_1$-algebras also in this case, so we obtain a graded trace\n    logic featuring precisely diamonds and disjunction, as expected.\n  \\end{longitemslist}\n  We defer the discussion of further examples, including ones where\n  $\\Omega=[0,1]$, to the next section, where we will simultaneously\n  illustrate the generic expressiveness result\n  (\\autoref{expl:ltbt-logics}). \n\\end{example}\n\n\\begin{remark}\n  One important class of examples where the above approach to\n  characteristic logics will \\emph{not} work without substantial\n  further development are simulation-like equivalences, whose\n  characteristic logics need\n  conjunction~\\cite{vanglabbeek2001linear}. Conjunction is not an\n  $M_0$-morphism for the corresponding graded monads identified in\n  \\autoref{sec:ltbt}, which both have $M_0=\\Pow_\\omega$. A related and maybe\n  more fundamental observation is that formula evaluation is not\n  $M_0$-morphic in the presence of conjunction; e.g.\\ over simulation\n  equivalence, the evaluation map\n  $M_11=\\Pow_\\omega^\\downarrow(\\mathcal{A}\\times\\Pow_\\omega(1))\\to 2$ of the formula\n  $\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top$ fails to be\n  join-continuous for distinct $\\sigma,\\tau\\in\\mathcal{A}$. We leave the\n  extension of our logical framework to such cases to future work,\n  expecting a solution in elaborating the theory of graded monads,\n  theories, and algebras over the category of partially ordered sets,\n  where simulations live more naturally (e.g.~\\cite{KapulkinEA12}).\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Expressiveness}\\label{sec:expr}\n\nWe now present our main result, an expressiveness criterion for graded\nlogics, which states that a graded logic characterizes the given\ngraded semantics if it has enough modalities propositional operators,\nand truth constants. Both the criterion and its proof now fall into\nplace naturally and easily, owing to the groundwork laid in the\nprevious section, in particular the reformulation of the semantics in\nterms of canonical algebras:\n\\begin{defn}\\label{def:separation}\n  We say that a graded logic with set~$\\Omega$ of truth values and\n  sets~$\\Theta$,~$\\mathcal{O}$,~$\\Lambda$ of truth constants,\n  propositional operators, and modalities, respectively, is\n  \\begin{longitemslist}\n  \\item \\emph{depth-$0$ separating} if the family of maps\n    $\\Sem{c}\\colon M_01\\to\\Omega$, for truth constants\n    $c\\in\\Theta$, is jointly injective; and\n  \\item \\emph{depth-$1$ separating} if, whenever $A$ is a canonical\n    $M_1$-algebra and $\\mathfrak{A}$ is a jointly injective set of\n    $M_0$-homomorphisms $A_0\\to\\Omega$ that is closed under the\n    propositional operators in~$\\mathcal{O}$ (in the sense that\n    $\\Sem{p}\\cdot \\langle f_1,\\dots,f_k\\rangle\\in\\mathfrak{A}$ for\n    $f_1,\\dots,f_k\\in\\mathfrak{A}$ and $k$-ary $p\\in\\mathcal{O}$), then the\n    set \n    \\[\n      \\Lambda(\\mathfrak{A})\\coloneqq\\{\\Sem{L}(f)\\colon A_1\\to\\Omega\\mid L\\in\\Lambda,f\\in\\mathfrak{A}\\}.\n    \\]\n    of maps is jointly injective.\n  \\end{longitemslist}\n\\end{defn}\n\\begin{theorem}[Expressiveness]\\label{thm:expr}\n  If a graded logic is both depth-$0$ separating and depth-$1$\n  separating, then it is expressive.\n\\end{theorem}\n\n\\begin{example}[Logics for bisimilarity]\\label{expl:bisim-logics}\n  We note first that the existing coalgebraic Hennessy-Milner theorem,\n  for branching time equivalences and coalgebraic modal logic with\n  full Boolean base over a finitary\n  functor~$G$~\\cite{Pattinson04,Schroder08}, as recalled in\n  Section~\\ref{sec:logics}, is a special case of \\autoref{thm:expr}:\n  We have already seen in \\autoref{expl:logics} that coalgebraic modal\n  logic in the above sense is an instance of our framework for the\n  graded monad $M_nX=G^nX$. Since $M_0=\\id$ in this case, depth-$0$\n  separation is vacuous. As indicated in \\autoref{expl:logics},\n  canonical $M_1$-algebras are w.l.o.g.\\ of the form $\\id\\colon GX\\to GX$,\n  where for purposes of proving depth-$1$ separation, we can restrict\n  to finite~$X$ since~$G$ is finitary. Then, a set~$\\mathfrak{A}$ as in\n  \\autoref{def:separation} is already the whole powerset $2^X$, so\n  depth-$1$ separation is exactly the previous notion of separation.\n \n  A well-known particular case is probabilistic bisimilarity on Markov\n  chains, for which an expressive logic needs only probabilistic\n  modalities $\\Diamond_p$ `with probability at least~$p$' and\n  conjunction~\\cite{DesharnaisEA98}. This result (later extended to\n  more complex composite functors~\\cite{MossViglizzo06}) is also\n  easily recovered as an instance of \\autoref{thm:expr}, using the\n  same standard lemma from measure theory as in \\emph{op.~cit.},\n  which states that measures are uniquely determined by their values\n  on a generating set of the underlying $\\sigma$-algebra that is\n  closed under finite intersections (corresponding to the set~$\\mathfrak{A}$\n  from \\autoref{def:separation} being closed under conjunction).\n\\end{example}\n\n\\begin{remark}\n  For behavioural equivalence, i.e.\\ $M_nX=G^nX$ as in the above\n  example, the inductive proof of our expressiveness theorem\n  essentially instantiates to Pattinson's proof of the coalgebraic\n  Hennessy-Milner theorem by induction over the terminal\n  sequence~\\cite{Pattinson04}. One should note that although the\n  coalgebraic Hennessy-Milner theorem can be shown to hold for larger\n  cardinal bounds on the branching by means of a direct quotienting\n  construction~\\cite{Schroder08}, the terminal sequence argument goes\n  beyond finite branching only in corner cases.\n\\end{remark}\n\n\\begin{example}[Expressive graded logics on the linear time -- branching time spectrum]\\label{expl:ltbt-logics}\n  We next extract graded logics from some of the graded monads\n  for the linear time -- branching time spectrum introduced in\n  \\autoref{sec:ltbt}, and show how in each case, expressiveness is an\n  instance of \\autoref{thm:expr}. Bisimilarity is already covered by\n  the previous example. Depth-$0$ separation is almost always trivial\n  and not mentioned further. Unless mentioned otherwise, all logics\n  have disjunction, enabled by $M_0$ being powerset as discussed in\n  the previous section. Most of the time, the logics are essentially\n  already given by van Glabbeek (with the exception that we show that\n  one can add disjunction)~\\cite{vanglabbeek2001linear}; the emphasis\n  is entirely on uniformization.\n  \\begin{longitemslist\n  \\item \\emph{Trace equivalence:} As seen in \\autoref{expl:logics},\n    the graded logic for trace equivalence features (disjunction and)\n    diamond modalities $\\Diamond_\\sigma$ indexed over actions\n    $\\sigma\\in\\mathcal{A}$. The ensuing proof of depth-$1$ separation uses\n    canonicity of a given $M_1$-algebra~$A$ only to obtain that the\n    structure map $a^{10}$ is surjective. The other key point is that\n    a jointly injective collection~$\\mathfrak{A}$ of $M_0$-homomorphisms\n    $A_0\\to 2$, i.e.\\ join preserving maps, has the stronger\n    separation property that whenever $x\\not\\le y$ then there exists\n    $f\\in\\mathfrak{A}$ such that $f(x)=\\top$ and $f(y)=\\bot$.\n  \\item Graded logics for completed traces, readiness, failures, ready\n    traces, and failure traces are developed from the above by adding\n    constants or additionally indexing modalities over sets of\n    actions, with only little change to the proofs of depth-$1$\n    separation. For completed trace equivalence, we just add a $0$-ary\n    modality $\\star$ indicating deadlock. For ready trace equivalence,\n    we index the diamond modalities $\\Diamond_\\sigma$ with sets\n    $I\\subseteq\\mathcal{A}$; formulae $\\Diamond_{\\sigma,I}\\phi$ are then read\n    `the current ready set is~$I$, and there is a $\\sigma$-successor\n    satisfying~$\\phi$'. For failure trace equivalence we proceed in\n    the same way but read the index~$I$ as `$I$ is a failure set at\n    the current state'. For readiness equivalence and failures\n    equivalence, we keep the modalities~$\\Diamond_\\sigma$ unchanged\n    from trace equivalence and instead introduce $0$-ary\n    modalities~$r_I$ indicating that~$I$ is the ready set or a failure\n    set, respectively, at the current state, thus ensuring that\n    formulae do not continue after postulating a ready set.\n  \\end{longitemslist}\n\\end{example}\n\n\\begin{example}[Probabilistic traces]\\label{expl:prob-trace}\n  We have recalled in \\autoref{sec:ltbt} that probabilistic trace\n  equivalence of generative probabilistic transition systems can be\n  captured as a graded semantics using the graded\n  monad~$M_nX=\\mathcal{D}(\\mathcal{A}^n\\times X)$, with $M_0$-algebras being convex\n  algebras. In earlier work~\\cite{MiliusEA15} we have noted that a\n  logic over the set $\\Omega=[0,1]$ of truth values (with the usual\n  convex algebra structure) featuring rational truth constants, affine\n  combinations as propositional operators (as indicated in\n  \\autoref{sec:logics}), and modal operators $\\langle\\sigma\\rangle$,\n  interpreted by $M_1$-algebras\n  $\\Sem{\\langle\\sigma\\rangle}\\colon M_1[0,1]\\to[0,1]$ defined by\n  \\iffull\n  \\begin{equation*}\\textstyle\n    \\Sem{\\langle\\sigma\\rangle}(\\mu)=\\sum_{r\\in [0,1]}r\\mu(\\sigma,r)\n  \\end{equation*}\n  \\else\\\/$\\Sem{\\langle\\sigma\\rangle}(\\mu)=\\sum_{r\\in\n    [0,1]}r\\mu(\\sigma,r)$ \\fi\n  is invariant under probabilistic trace equivalence. By our\n  expressiveness criterion, we recover the result that this logic\n  is expressive for probabilistic trace semantics\n  (see e.g.~\\cite{bernardo-botta:characterising-logics}).\n\\end{example}\n\\section{Conclusion and Future Work}\n\nWe have provided graded monads modelling a range of process\nequivalences on the linear time -- branching time spectrum, presented\nin terms of carefully designed graded algebraic theories. From these\ngraded monads, we have extracted characteristic modal logics for the\nrespective equivalences systematically, following a paradigm of graded\nlogics that grows out of a natural notion of graded algebra. Our main\ntechnical results concern the further development of the general\nframework for graded logics; in particular, we have introduced a\nfirst-class notion of propositional operator, and we have established\na criterion for \\emph{expressiveness} of graded logics that\nsimultaneously takes into account the expressive power of the\nmodalities and that of the propositional base. (An open question that\nremains is whether an expressive logic always exists, as it does in\nthe branching-time setting~\\cite{Schroder08}.) Instances of this\nresult include, for instance, the coalgebraic Hennessy-Milner\ntheorem~\\cite{Pattinson04,Schroder08}, Desharnais et al.'s\nexpressiveness result for probabilistic modal logic with only\nconjunction~\\cite{DesharnaisEA98}, and expressiveness for various\nlogics for trace-like equivalences on non-deterministic and\nprobabilistic systems. The emphasis in the examples has been on\nwell-researched equivalences and logics for the basic case of labelled\ntransition systems, aimed at demonstrating the versatility of graded\nmonads and graded logics along the axis of granularity of system\nequivalence. The framework as a whole is however parametric also over\nthe branching type of systems and in fact over the base category\ndetermining the structure of state spaces; an important direction for\nfuture research is therefore to capture (possibly new) equivalences\nand extract expressive logics on other system types such as\nprobabilistic systems (we have already seen probabilistic trace\nequivalence as an instance; see~\\cite{BonchiEA17} for a comparison of\nsome equivalences on probabilistic automata, which combine\nprobabilities and non-determinism) and nominal systems, e.g.\\ nominal\nautomata~\\cite{BojanczykEA14,SchroderEA17}. Moreover, we plan to\nextend the framework of graded logics to cover also temporal logics,\nusing graded algebras of unbounded depth.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n \nThe purpose of this paper is to prove some saturation bounds for the ideals of non-singular complex projective varieties and their powers.\n\n\n\nWe begin with some background. Consider the polynomial ring  $S = \\mathbf{C}[x_0, \\ldots, x_r]$  in $r+1$ variables, and fix homogeneous polynomials\n\\[f_0\\, , \\,  f_1 \\, , \\, \\ldots \\, , \\, f_p\\, \\in \\, S \\ \\ \\text{with\\ \\  $\\deg(f_i) = d_i$}.\\] We assume that\n$d_0  \\ge  d_1  \\ge   \\ldots  \\ge  d_p$, \nand we denote by \n\\[ J \\ = \\ \\big ( \\, f_0 \\, , \\, f_1 \\, , \\, \\ldots \\, ,\\, f_p \\, \\big) \\ \\subseteq \\ S \\]\nthe ideal that the polynomials span. Suppose now that $J$ is primary for the irrelevant maximal ideal $\\mathfrak{m} = (x_0,  \\ldots, x_r)$, or equivalently that $\\dim_\\mathbf{C} S\/J  < \\infty$. In this case $J$ contains all monomials of sufficiently large degree, and it is a classical theorem of Macaulay \\cite[Theorem 7.4.1]{CM-S.P} that \n\\begin{equation} \\label{Macaulay.Eqn.1}\nJ_t \\ = \\ S_t \\ \\ \\text{ for } \\ \\ t\\,  \\ge \\, d_0 + \\ldots + d_r - r. \n\\end{equation}\nMoreover this bound is (always) sharp when $p = r$. \nAlthough less well known,  a similar statement holds for powers of $J$:\n\\begin{equation} \\label{Macaulay.Eqn.2}\n(J^a)_t \\ = \\ S_t \\ \\ \\text{ for } \\ \\ t\\,  \\ge \\, ad_0 +d_1 + \\ldots + d_r - r.\\end{equation}\nThis again is always sharp when $ p = r$. \n\nIt is natural to ask whether there are analogous results for more general homogeneous ideals $J$, in particular when \n\\[    X \\ =_{\\text{def}} \\ \\Zero{J} \\ \\subseteq \\ \\mathbf{P}^r \\]\nis a smooth complex projective variety. Of course if $J$ has non-trivial zeroes, then it does not contain any power of the maximal ideal. However if one interprets \\eqref{Macaulay.Eqn.1} and \\eqref{Macaulay.Eqn.2} as saturation bounds, then the question makes sense more generally. Specifically, recall that the \\textit{saturation} of a homogeneous ideal $J$ is defined by\n\\begin{equation}  \\satt{J}  \\ = \\ \\big \\{ \\, f \\in S \\mid \\mathfrak{m}^k \\cdot f \\subseteq J     \\text{ for some $k \\ge 0$} \\, \\big \\}. \\notag \\end{equation}\nThe quotient $ \\satt{J} \/ J$ has finite length, and in particular\n\\[ ( \\satt{J} )_t \\ = \\ J_t \\ \\ \\text{ for } \\ t \\gg 0. \\]\nThe least such integer $t$ is called the \\textit{saturation degree} $\\textnormal{sat. \\!deg}(J)$ of $J$. Observing that $\\satt{J} = S$ if and only if $J$ is $\\mathfrak{m}$-primary,  statements \\eqref{Macaulay.Eqn.1} and \\eqref{Macaulay.Eqn.2} are equivalent to estimates for the saturation degrees of $J$ and $J^a$. So the problem becomes to   bound the saturation degree of an ideal in terms of the degrees of its generators.\n\nIt is instructive to consider some examples. Let $X \\subseteq \\mathbf{P}^r$ be a hyperplane defined by a linear form $\\ell \\in S$, and set\n\\begin{equation} \\label{Hyperplane.Example}  f_i \\, = \\, x_i^{d-1}\\cdot \\ell \\ \\  \\ , \\  \\ \\ J = (f_0, \\ldots, f_r)\\, \\subseteq \\, S. \\end{equation}\nThen $\\satt{J} = (\\ell)$, and it follows from Macaulay's theorem that\n\\[  \\textnormal{sat. \\!deg}(J) \\ = \\ (r+1)(d-1) - r + 1 \\ = \\ (r+1)d - 2r, \\]\nwhich is very close to the bound \\eqref{Macaulay.Eqn.1}. On the other hand, it is not the case that the saturation degree  of an arbitrary ideal is bounded linearly in the degrees of its generators. For instance, \n the ideals \\[  J \\ = \\ \\big( x^d, y^d, xz^{d-1} - yw^{d-1} \\big) \\ \\subseteq \\ \\mathbf{C}[x,y,z,w] \\]\nconsidered by Caviglia \\cite[Example 4.2.1]{Caviglia} have  $\\textnormal{sat. \\!deg}(J) \\approx d^2$.\n\nOur first main result asserts that for ideals defining smooth varieties, the Macaulay bounds remain true without modification.\n\\begin{theoremalpha} \\label{Intro.Sat.Deg.Thm}\nAs above, suppose that \n\\[ J \\ = \\ \\big ( \\, f_0 \\, , \\, f_1 \\, , \\, \\ldots \\, ,\\, f_p \\, \\big) \\ \\subseteq \\ S \\]\nis generated by forms of degrees $d_0 \\ge \\ldots \\ge d_p$, and assume that the projective scheme\n\\[    X \\ =_{\\text{def}} \\ \\Zero{J} \\ \\subseteq \\ \\mathbf{P}^r \\]\ncut out by the $f_i$ is a non-singular complex variety. Then \n$ \\textnormal{sat. \\!deg}(J)   \\le  d_0 + \\ldots + d_r - r$,  \nand more generally\n\\begin{equation} \\label{Intro.Thm.Equation} \\textnormal{sat. \\!deg}(J^a) \\ \\le \\ ad_0 + d_1 + \\ldots + d_r - r. \\end{equation}\n\\end{theoremalpha}\n\\noindent (If $p < r$, one takes $d_{p+1} = \\ldots = d_r = 0$.) We do not know whether the stated bound is best possible, but in any event it is asymptotically sharp. Indeed, if $J$ is the ideal considered in \\eqref{Hyperplane.Example}, then the Theorem predicts that $\\textnormal{sat. \\!deg}(J^a)\\le (a+r)d -r$, whereas in fact $\\textnormal{sat. \\!deg}(J^a) = (a+r)d - 2r$. \n\n\n\n\nGiven a reduced algebraic set $X \\subseteq \\mathbf{P}^r$ denote by $I_X \\subseteq S$ the saturated homogeneous ideal of $X$. Recall that the \\textit{symbolic powers} of $I_X$ are\n\\[\nI_X^{(a)} \\ = \\ \\big \\{ f \\in S \\mid \\textnormal{ord}_x(f) \\ge a  \\text{ for general (or every) }  x\\in X \\, \\big \\}. \n\\]\nEvidently $I_X^a \\subseteq I_X^{(a)}$, and there has been a huge amount of interest in recent years in understanding the connections between actual and symbolic powers (cf \\cite{ELS}, \\cite{Hochster.Huneke}, \\cite{BocciHarbourne}, \\cite{Dao.ea}). If $X$  is non-singular, then $I_X^{(a)} = \\satt{(I_X^a)}$. Therefore Theorem \\ref{Intro.Sat.Deg.Thm} implies\n\\begin{corollaryalpha}\nAssume that $X \\subseteq \\mathbf{P}^r$ is smooth, and that $I_X$ is generated in degrees $d_0 \\ge d_1 \\ge \\ldots \\ge d_p$. Then\n\\[   \\big ( I_X^{(a)} \\big)_t \\ = \\ ( I_X^a)_t  \\ \\ \\text{ for } \\, t \\, \\ge \\, ad_0 +d_1 + \\ldots + d_r -r. \\]\n\\end{corollaryalpha}\n\\noindent \nFor example, suppose that $X \\subseteq \\mathbf{P}^2$ consists of the three coordinate points, so that $I_X = (xy, yz, zx) \\subseteq \\mathbf{C}[x,y,z]$.  The Corollary guarantees that $I_X^a$ and $I_X^{(a)}$ agree in degrees $\\ge 2a + 2$, whereas in reality $\\textnormal{sat. \\!deg} (I_X^a) = 2a$. So here again the statement is asymptotically but not precisely sharp.\n\n\nIn the case of finite sets,  results of Geramita-Gimigliano-Pitteloud \\cite{GGP}, Chandler \\cite{Chandler} and Sidman \\cite{Sidman}   provide an alternative   bound that is often best-possible. Recall that a scheme $X \\subseteq \\mathbf{P}^r$ is said to be $m$-regular in the sense of Castelnuovo--Mumford if its ideal sheaf $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$ satisfies the vanishings:\n\\[\n\\HH{i}{\\mathbf{P}^r}{\\mathcal{I}_X(m-i)} \\  = \\ 0 \\ \\ \\text{ for \\ } i > 0.  \n\\]\nThis is equivalent to asking that $I_X$ be generated in degrees $\\le m$, that the first syzygies among minimal generators of $I_X$ appear in degrees $\\le m+1$, the second syzygies in degrees $\\le m+2$, and so on.\\footnote{For saturated ideals, Castelnuovo--Mumford regularity of $I_X$ agrees with an algebraic notion of regularity introduced by Eisenbud and Goto \\cite{Eisenbud.Goto} that we propose to call \\textit{arithmetic regularity}. An arbitrary ideal $J \\subseteq S$ is arithmetically $m$-regular if and only if $\\satt{J}$ is $m$-regular and $\\textnormal{sat. \\!deg}(J) \\le m$. Given that we are interested in establishing bounds on saturation degree, unless otherwise stated we always refer to regularity in the geometric sense.}\nThe authors just cited show that if $X \\subseteq \\mathbf{P}^r$ is an $m$-regular finite set, then\n\\[\n\\textnormal{sat. \\!deg} (I_X^a) \\ \\le \\ am. \n\\]\n This is optimal for the example of the three coordinate points in $\\mathbf{P}^2$. \n \n Our second main result asserts  that the same statement holds when $\\dim X = 1$. \n\\begin{theoremalpha} \\label{Regularity.Saturation.Bound.Curves}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth $m$-regular curve. Then\n\\[\n\\big( I_X^a\\big)_t \\ = \\ \\big( I_X^{(a)}\\big)_t \\ \\ \\text{for }   \\ t \\, \\ge \\, a m. \n\\]\n\\end{theoremalpha}\n\\noindent In fact, for the saturation bound  it suffices that the curve $X$ be reduced. The statement is optimal (for all $a$) for instance when $X \\subseteq \\mathbf{P}^4$ is a rational normal curve. \nWe also show that if $X \\subseteq \\mathbf{P}^r$ is a reduced surface, then $\\textnormal{reg}(\\mathcal{I}_X^a) \\le a \\cdot \\textnormal{reg}(\\mathcal{I}_X)$. \nWe do not know any examples where the analogous statements fail for smooth varieties of higher dimension. \n\nReturning to the setting of Theorem \\ref{Intro.Sat.Deg.Thm}, the first and third authors showed with Bertram some years ago \\cite{BEL}  that if   $X \\subseteq \\mathbf{P}^r$ is a smooth complex projective variety of codimension $e$ cut out as a scheme by homogeneous polynomials of degrees $d_0 \\ge \\ldots \\ge d_p$, then $\\mathcal{I}_X^a$ is $(ad_0 + d_1 +  \\ldots + d_{e-1} -e)$-regular in the sense of Castelnuovo-Mumford. Note however that this  does not address the questions of saturation required to control the arithmetic (Eisenbud--Goto) regularity of $I_X^a$.\\footnote{In particular, the proof of Proposition 2.2 in \\cite{AV} seems to be erroneous.}  In fact, one can view Theorem \\ref{Intro.Sat.Deg.Thm} as promoting the results of \\cite{BEL} to statements about arithmetic regularity:\n \\begin{corollaryalpha} Assume that $J \\subseteq S$ satisfies the hypotheses of Theorem \\ref{Intro.Sat.Deg.Thm}. Then\n \\[ \\textnormal{arith. \\!reg}(J^a) \\ \\le \\ ad_0 + (d_1 + \\ldots + d_r -r).\\]\n \\end{corollaryalpha}\n\\noindent \nIt is known (\\cite{Kod}, \\cite{CHK}) that if $J \\subseteq S$ is an arbitrary homogeneous ideal then \\[  \\textnormal{arith. \\!reg}(J^a) \\ =  \\ ad  + b  \\ \\text{ \\ when  } \\ a \\gg 0, \\] where $d$ is the maximal degree needed to generate a reduction of $J$ -- which coincides with  the generating degree of $J$ when it is equigenerated -- and $b$ is some constant. However computing the constant term $b$ has proven elusive, and  the Corollary gives a bound in the case at hand. \n\nThe proofs of these results revolve around using  complexes of sheaves to study the image in $\\HHHH{*}{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a} = \\satt{(I_X^{a})}$ of the powers of the ideal spanned by generators of $I_X$ or $J$: this approach was inspired in part by geometrizing the arguments of Cooper and coauthors for codimenson two subvarieties in \\cite{Cooper+}. Specifically, suppose that\n\\[    \\varepsilon : U_0 \\, =_{\\text{def}} \\, \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r}(-d_i) \\longrightarrow \\mathcal{I}_X  \\]\nis the surjective map of sheaves determined by generators of $I_X$ or $J$. If $X$ is $m$-regular, then this sits in an exact complex $U_\\bullet$ of bundles:\n\\[\n0 \\longrightarrow U_{r-1} \\longrightarrow U_{r-2} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon}  \\longrightarrow \\mathcal{I}_X \\longrightarrow 0\n\\] where $\\textnormal{reg}(U_i) \\le m + i$. Weyman \\cite{Weyman} (see also \\cite{Tchernev}) constructs a new complex $L_\\bullet = \\textnormal{Sym}^a(U_\\bullet)$  that takes the form\n\\[\n\\ldots \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow S^a(U_0) \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0\n\\]\nwhere $\\textnormal{reg}(L_i) \\le am + i$. This complex is exact only off $X$, but as in \\cite{GLP} when $\\dim X = 1$ one can still read off the surjectivity of \n\\[\n\\HH{0}{\\mathbf{P}^r}{S^a (U_0)(t)} \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)}  \n\\]\nfor $t \\ge am$. This gives Theorem \\ref{Regularity.Saturation.Bound.Curves}. \n\nTurning to Theorem  \\ref{Intro.Sat.Deg.Thm}, a natural idea is to start with the Koszul complex\n\\[ \\ldots \\longrightarrow \\Lambda^3 U_0 \\longrightarrow \\Lambda^2 U_0 \\longrightarrow U_0 \\longrightarrow \\mathcal{I}_X \\longrightarrow 0.\\]\nAs established by Buchsbaum--Eisenbud \\cite{Buchsbaum.Eisenbud}, this determines a new complex\n\\[\n\\ldots \\longrightarrow S^{a, 1^2}(U_0) \\longrightarrow S^{a,1}(U_0) \\longrightarrow S^a(U_0) \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0, \\tag{*}\n\\]\nwhere $S^{a, 1^k}(U_0)$ denotes the Schur power of $U_0$ corresponding to the Young diagram $(a, 1^{k})$. We observe that \n\\[   \\textnormal{reg} \\big(S^{a, 1^{i}}(U_0) \\big) \\ \\le  \\ ad_0 + d_1 + \\ldots + d_i, \\]\nso if (*) were exact then the statement of the Theorem would follow immediately. Unfortunately (*) is exact only if $X$ is a complete intersection, but by blowing up $X$ this construction yields an exact complex whose cohomology groups one can control with some effort. At the end of the day, the computation boils down to using Kodaira--Nakano vanishing on $X$ to prove \na vanishing statement for  symmetric powers of the normal bundle to $X$ in $\\mathbf{P}^r$:\n\\begin{propositionalpha} \\label{NB.Vanishing.Prop}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth complex projective variety, and denote by $N = N_{X\/\\mathbf{P}^r}$ the normal bundle to $X$ in $\\mathbf{P}^r$. Then\n\\[ \\HHH{i}{X}{S^k N \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell) } \\ = \\ 0  \\ \\ \\text{ for } i > 0\n\\]\nand every $k \\ge 0$, $\\ell \\ge -r$. \n\\end{propositionalpha}\n\\noindent (Similar but slightly different vanishings were established by Schneider and Zintl in \\cite{Schneider.Zintl}.)\n We hope that some of these ideas may find other applications in the future.\\footnote{We remark that some of the auxiliary results appearing here -- for example the Proposition just stated -- were known to the first and third authors some years ago in connection with their work on \\cite{BEL}. However they were put aside in favor of the simpler arguments  with vanishing theorems that eventually appeared in that paper. }\n  \n The paper is organized as follows. The first section is devoted to Theorem \\ref{Regularity.Saturation.Bound.Curves}. We collect in \\S 2 some preliminary results towards the Macaulay-type bounds. Specifically, we discuss the Buchsbaum--Eisenbud powers of Koszul complexes, the computation of some push-forwards from a blowing-up, and Proposition \\ref{NB.Vanishing.Prop}. The proof of Theorem \\ref{Intro.Sat.Deg.Thm} occupies \\S 3. We work throughout over the complex numbers.\n \n We are grateful to  Sankhaneel Bisui, David Eisenbud, Elo\\'isa Grifo and Claudia Miller for valuable remarks and correspondence.\n \n \n\n\n\n\\numberwithin{equation}{section}\n\\section{Saturation and regularity}\n\nThe present section is devoted to the proof of Theorem \\ref{Regularity.Saturation.Bound.Curves} from the Introduction.\n\nWe start with some general remarks. Let $X \\subseteq \\mathbf{P}^r$ be a  complex projective variety or scheme, with ideal sheaf $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$  and homogeneous ideal $I_X \\subseteq S$.  \nDenote by $U_\\bullet$  the locally free resolution of $\\mathcal{I}_X$ obtained by sheafifying a minimal graded free resolution of $I_X$:\n\\begin{equation} \\label{m-reg.Resoln.I}\n0 \\longrightarrow U_{r} \\longrightarrow U_{r-1} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon}  \\longrightarrow \\mathcal{I}_X \\longrightarrow 0.  \\end{equation}\n  Thus \neach $U_i$ is a direct sum of line bundles, and we recover the original resolution as the the complex $\\HHHH{*}{0}{\\mathbf{P}^r}{U_\\bullet}$ obtained from $U_\\bullet$ by taking global sections of all twists. \n\nConsider now the surjective homomorphism of sheaves\n\\[ S^a(\\varepsilon) \\, : \\, S^a U_0 \\longrightarrow \\mathcal{I}_X^a . \\]\nFor any   $t \\ge 0$  one has\n\\[ \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \\ = \\ \\left( \\satt{\\left( I_X^a \\right)} \\right)_t. \\]\nOn the other hand, the fact that  $U_0$ is constructed from minimal generators of $I_X$ implies that \n\\[\n\\textnormal{Im} \\Big( \\HH{0}{\\mathbf{P}^r}{S^a (U_0) (t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \\Big) \\ = \\  \\big( I_X^a)_t .\n\\]\nTherefore\n\\begin{lemma} \\label{Surjectivity.Suffices.Lemma} The degree $t$ pieces of $I_X^a$ and $\\satt{(I_X^a)}$ coincide if and only if the homomorphism\n\\[\n\\HH{0}{\\mathbf{P}^r}{S^a (U_0) (t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \n\\]\ndetermined by $S^a(\\varepsilon)$ is surjective.  \\qed\n\\end{lemma}\n\\noindent The plan is to study $S^a(\\varepsilon)$ by realizing it as the last map of a complex $S^a( U_\\bullet)$.\n\nSpecifically, consider a smooth variety $M$, a subvariety $X \\subseteq M$, and a locally free resolution $U_\\bullet$ of $\\mathcal{I}_X \\subseteq \\mathcal{O}_M$ as above:\n\\begin{equation} \\label{m-reg.Resoln.II}\n0 \\longrightarrow U_{r} \\longrightarrow U_{r-1} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon}  \\longrightarrow \\mathcal{I}_X \\longrightarrow 0.  \\end{equation}\nAs explained by Weyman \\cite{Weyman} and Tchernev \\cite{Tchernev}, $U_\\bullet$ determines  for fixed $a \\ge 1$ a new complex $L_\\bullet = S^a(U_\\bullet)$ having the shape\n\\begin{equation} \\label{Weyman.Complex.1}\n\\xymatrix @C=18pt{\n\\ldots \\ar[r] & L_4 \\ar[r] &L_3 \\ar[r] & { \\begin{matrix} S^{a-2}U_0 \\otimes \\Lambda^2 U_1 \\\\ \\oplus \\\\ S^{a-1}U_0 \\otimes U_2 \\end{matrix}} \\ar[r] &S^{a-1}U_0 \\otimes U_1 \\ar[r] & S^aU_0\\ar[r]& \\mathcal{I}_X^a  \\ar[r] & 0.\n}\n\\end{equation}\nThe last map on the right is $S^a(\\varepsilon)$, and the homomorphism  $S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^a U_0$ is the natural one arising as the composition\n \\[\n S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^{a-1}U_0 \\otimes U_0 \\longrightarrow S^a U_0.\n \\] The  $L_i$ are determined by setting\n \\begin{equation} \\label{Weyman.Complex.2}\n C^k (U_j) \\ = \\ \\begin{cases} \\ S^k U_j \\ & \\text{if $j$ is even} \\\\ \\ \\Lambda^k U_j \\ & \\text{if $j$ is odd} \\end{cases},\n \\end{equation}\nand then taking\n\\begin{equation} \\label{Weyman.Complex.3}\nL_i\\ = \\ \\bigoplus_{\n\\substack{k_0 + \\ldots + k_r = a \\\\ k_1 + 2k_2 + \\ldots + rk_r = i}} C^{k_0}(U_0) \\otimes C^{k_1}(U_1) \\otimes \\ldots \\otimes C^{k_r}(U_r).\n\\end{equation}\n\nIt follows from \\cite[Theorem 1]{Weyman} or \\cite[Theorem 2.1]{Tchernev} that:\n\\begin{equation} \\label{Exact.Away.From.X.Equation}\n\\text{The  complex } \\eqref{Weyman.Complex.1}\n  \\text{ is exact away from $X$}. \n\\end{equation}\nIn general one does not expect exactness at  points of $X$, but when $X$ is  smooth  the right-most terms at least  are well-behaved:\n\\begin{lemma} \\label{Right.Hand.Exactness.Weyman.Complex}\nAssume that $X$ is non-singular. Then the sequence\n\\[ S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^a U_0 \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0\\]\nis exact. \n\\end{lemma}\n\\begin{proof} The question being local, we can work over the local ring $\\mathcal{O} = \\mathcal{O}_{M,x}$ of $M$ at a point $x \\in X$. Since $X$ is smooth,  $\\mathcal{I} = \\mathcal{I}_{X, x} \\subseteq \\mathcal{O}$ is  generated by a regular sequence of length $e = \\textnormal{codim} \\, X$. Thus $\\mathcal{I}$ has a minimal presentation \n\\[ \\Lambda^2 \\mathcal{U} \\longrightarrow \\mathcal{U} \\longrightarrow \\mathcal{I} \\longrightarrow 0 \n\\]\ngiven by the beginning of a Koszul complex, where $\\mathcal U = \\mathcal{O}^e$ is a free module of rank $e$. Here  one checks by hand the exactness of \n\\[\nS^{a-1}\\mathcal{U} \\otimes \\Lambda^2 \\mathcal{U} \\longrightarrow S^a \\mathcal{U} \\longrightarrow \\mathcal{I}^a \\longrightarrow 0.\n\\]  \n(Compare Proposition  \\ref{Koszul.Complex.Power}\n below.) An arbitrary free presentation of $\\mathcal{I}$   then has the form\n\\[    \\Lambda^2 \\mathcal{U} \\oplus \\mathcal{A} \\oplus \\mathcal{B} \\longrightarrow \\mathcal{U} \\oplus \\mathcal{A} \\longrightarrow \\mathcal{I} \\longrightarrow 0,\\]\nwhere $\\mathcal{A}$ is a free module mapping to zero in $\\mathcal{I}$,  $\\mathcal{B}$ is a free module mapping to zero in $\\mathcal{U} \\oplus \\mathcal{A}$, and the left-hand map is the identity on $\\mathcal{A}$.  It suffices to verify the exactness of\n\\[\nS^{a-1}\\big( \\mathcal{U} \\oplus \\mathcal{A} \\big)  \\otimes \\big( \\Lambda^2 \\mathcal{U} \\oplus \\mathcal{A} \\big) \\longrightarrow S^a \\big( \\mathcal{U} \\oplus \\mathcal{A}\\big) \\longrightarrow \\mathcal{I}^a \\longrightarrow 0, \n\\]\nand this is clear upon writing  $S^a \\big( \\mathcal{U} \\oplus \\mathcal{A}\\big)=S^a \\mathcal{U} \\, \\oplus \\, \\mathcal{A} \\otimes \nS^{a-1}\\big( \\mathcal{U} \\oplus \\mathcal{A} \\big)$. \\end{proof}\n\n\nWith these preliminaries out of the way, we now prove (a slight strengthening of) Theorem \\ref{Regularity.Saturation.Bound.Curves} from the Introduction.\n\\begin{theorem} \\label{Reduced.Curve.Theorem} Let $X \\subseteq \\mathbf{P}^r$ be a reduced $($but possibly singular$)$ curve, and assume that $X$ is $m$-regular in the sense of Castelnuovo--Mumford. Denote by $I_X \\subseteq S$ the homogeneous ideal of $X$. Then\n\\[\n\\textnormal{sat. \\!deg}( I_X^a) \\ \\le \\ a m.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe $m$-regularity of $X$ means that we can take a resolution $U_\\bullet$ of $\\mathcal{I}_X$ as in \\eqref{m-reg.Resoln.I} where $U_i$ is a direct sum of line bundles of degrees $\\ge -m -i$, ie $\\textnormal{reg}(U_i) \\le m + i$. Consider the resulting Weyman complex $L_\\bullet = S^a(U_\\bullet)$:\n\\[\n\\longrightarrow L_3 \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow L_0 \\longrightarrow \\mathcal{I}_X^a  \\longrightarrow 0,  \\tag{*} \\]\nthe last map being the surjection $S^a(\\varepsilon) : L_0 = S^a U_0 \\longrightarrow \\mathcal{I}_X $. In view of Lemma \\ref{Surjectivity.Suffices.Lemma}, the issue is to establish the surjectivity of the homomorphism\n\\[  \\HH{0}{\\mathbf{P}^r}{L_0(t)} \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)}  \\tag{**} \\]\nfor $t \\ge am$. To this end, observe first from \\eqref{Weyman.Complex.2} and \n\\eqref{Weyman.Complex.3} that\n\\[  \\textnormal{reg}(L_i) \\ \\le \\ am + i. \\]\nConsider next the homology sheaves $\\mathcal{H}_i = \\mathcal{H}_i(L_\\bullet \\longrightarrow \\mathcal{I}_X^a)$ of the augmented complex (*).  (So for $i = 0$ we understand  $\\mathcal{H}_0 = \\ker ( L_0 \\longrightarrow \\mathcal{I}_X^a) \/ \\textnormal{Im}(L_1 \\longrightarrow L_0).)$   Thanks to \\eqref{Exact.Away.From.X.Equation}, these are all supported on the one-dimensional set $X$. Moreover  it follows from Lemma \\ref{Right.Hand.Exactness.Weyman.Complex} that $\\mathcal{H}_0$ is supported on the finitely many singular points of $X$. Therefore the required surjectivity (**) is a consequence of the first statement of the following Lemma.\n \\end{proof}\n \n \\begin{lemma} Consider a complex $L_\\bullet$ of coherent sheaves on $\\mathbf{P}^r$ sitting in a diagram\n \\begin{equation}\\label{Chopping.Lemma.Eqn} \\ldots \\longrightarrow L_3 \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow L_0\\overset{\\varepsilon} \\longrightarrow \\mathcal{F}  \\longrightarrow 0,   \\end{equation}\n and denote by $\\mathcal{H}_i = \\mathcal{H}_i(L_\\bullet \\longrightarrow \\mathcal{F})$ the $i^{\\text{th}}$ homology sheaf of the augmented complex \\eqref{Chopping.Lemma.Eqn}.\\footnote{So as above,  the group of zero-cycles used to compute  $\\mathcal{H}_0$ is $\\ker (\\varepsilon)$.}\nAssume that $\\varepsilon$ is surjective, and let $p$ be an integer with the property that  $L_i$ is $(p + i)$-regular  for every $i$.  \n\\begin{enumerate}\n\\item [$(i)$]  If each  $\\mathcal{H}_i$  is supported on a set of dimension $\\le i$, then the homomorphism\n\\[  \\HH{0}{\\mathbf{P}^r}{L_0(t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{F}(t)} \\]\nis surjective for $t \\ge p$. \n\\vskip 5pt\n\\item[$(ii)$] If each $\\mathcal{H}_i$ is supported on a set of dimension $\\le i + 1$, then $\\mathcal{F}$ is $p$-regular. \n\\end{enumerate}\n \\end{lemma}\n\\begin{proof}\n This is established by chopping $L_\\bullet$ into short exact sequences in the usual way and chasing through the resulting diagram. (Compare \\cite[B.1.2, B.1.3]{PAG}, but note that the sheaf $\\mathcal{H}_0$ there should refer to the augmented complex, as above.)  \\end{proof}\n\nWe conclude this section by observing that the same argument proves that  Castelnuovo--Mumford regularity of surfaces behaves submultiplicatively in powers. For curves, this has been known for some time \\cite{Chandler}, \\cite{Sidman}.\n\\begin{proposition}\nLet $X \\subseteq \\mathbf{P}^r$ be a reduced $($but possibly singular$)$ surface, and denote by $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$ the ideal sheaf of $X$. If $\\mathcal{I}_X$ is $m$-regular, then $\\mathcal{I}_X^a$ is $am$-regular.\n\\end{proposition}\n\\begin{proof}[Sketch of Proof.] One argues just as in the  proof of Theorem \\ref{Reduced.Curve.Theorem}, reducing to  statement (ii) of the previous Lemma. \n\\end{proof}\n\n\n\n\n\n\n\\newcommand{\\Schur}[2]{S^{{#1},1^{#2}}}\n\n\\section{Macaulay-type bounds: preliminaries}\n\nThis section is devoted to some preliminary results that will be used in the proof of Theorem \\ref{Intro.Sat.Deg.Thm} from the Introduction. In the first subsection, we discuss symmetric powers of a Koszul complex. The second is devoted to the computation of some direct images from a blow-up. Finally \\S  \\ref{Vanishing.Theorem.Normal.Bundles.Subsection} gives the proof of Proposition \\ref{NB.Vanishing.Prop} form the Introduction. \n\n\n\\subsection{Powers of Koszul complexes} \\label{Powers.of.Koszul.Subsection}\n\n\n\nIn this subsection we review the construction of symmetric powers of a Koszul complex. In the local setting this (and much more) appears in the paper \\cite{Buchsbaum.Eisenbud} of Buchsbaum and Eisenbud, and it was revisited by Srinivasan in \\cite{Srinivasan}. However for the convenience of the reader we give here a quick sketch of the particular facts we require. We continue to  work  over the complex numbers.\n\nLet $M$ be a smooth algebraic variety, and let $V$ be a vector bundle of rank $e$ on $M$. Fix integers $a, k \\ge 1$. We denote by $\\Schur{a}{k-1}(V)$ the Schur power of $V$ corresponding to the partition $(a, 1, \\ldots, 1)$ ($k-1$ repetitions of $1$). It follows from Pieri's rule that\n\\begin{equation} \\label{Schur.Equation}\n\\begin{aligned}\\Schur{a}{k-1}(V) \\ &= \\ \\ker \\Big( \\Lambda^{k-1}V \\otimes S^a V \\longrightarrow \\Lambda^{k-2}V \\otimes S^{a+1}V \\Big) \\\\ &= \\ \\textnormal{im} \\Big( \\Lambda^k V \\otimes S^{a-1}V \\longrightarrow  \\Lambda^{k-1} V \\otimes  S^a V \\Big). \n\\end{aligned}\n\\end{equation}\n\n\\begin{remark} [Properties of $\\Schur{a}{k-1}(V)$] \\label{Properties.of.Schur.Power} We collect some useful observations concerning this Schur power.\n\\begin{enumerate}\n\\item [(i).] If $ k = 1$ then $\\Schur{a}{k-1}(V)  = S^aV$, while if $a = 1$ then $\\Schur{a}{k-1}(V)  = \\Lambda^k V$. Moreover\n\\[   \\Schur{a}{k-1}(V) \\, = \\, 0 \\ \\ \\text{ when } k > \\rk V. \\]\n\\vskip 5pt\n\\item[(ii).] The bundle $\\Schur{a}{k-1}(V)$ is actually a summand of $S^{a-1}V \\otimes \\Lambda^k V$. In fact,  Pieri shows that\n\\[\nS^{a-1}V \\otimes \\Lambda^k V \\ = \\ \\Schur{a}{k-1}(V)\\,  \\oplus \\, \\Schur{a-1}{k}(V).\n\\]\n\\vskip 5pt\n\\item[(iii).] If $L$ is a line bundle on $M$, then it follows from \\eqref{Schur.Equation} or (ii) that \n\\[ \\Schur{a}{k-1} ( V \\otimes L ) \\ = \\  \\Schur{a}{k-1} ( V) \\, \\otimes \\, L^{\\otimes a + k -1}.\\]\n\\item[(iv).] Suppose that $M = \\mathbf{P}^r$ and \n\\[    V \\ = \\  \\mathcal{O}_{\\mathbf{P}^r}(-d_0 )  \\oplus \\, \\ldots \\, \\oplus \\,  \\mathcal{O}_{\\mathbf{P}^r}(-d_p) \\]\nwith $\\ d_0 \\ge \\ldots \\ge d_p.$ Then it follows from (ii) that  $\\Schur{a}{k-1}(V)$ is a direct sum of line bundles of degrees $\\ge \\, -(ad_0 + d_1 + \\ldots + d_{k-1})$, and moreover a summand of this degree appears. In other words,\n\\[   \\textnormal{reg} \\big( \\, \\Schur{a}{k-1}(V) \\, ) \\ = \\ ad_0 + d_1 + \\ldots + d_{k-1}.\\]\n\\end{enumerate}\n\\end{remark}\n\nOne can also realize $\\Schur{a}{k-1}(V)$ geometrically, \\`a la Kempf \\cite{Kempf}.\n\\begin{lemma} \\label{Kempf.Type.Lemma} \nLet $ \\pi : \\mathbf{P}(V) \\longrightarrow M $  be the projective bundle of one-dimensional quotients of $V$, and denote by $F$ the kernel of the canonical quotient $\\pi^* V \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(1)$, so that $F$ sits in the short exact sequence\n\\[\n0 \\longrightarrow F \\longrightarrow \\pi^* V \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(1) \\longrightarrow 0 \\tag{*}\n\\]\nof bundles on $\\mathbf{P}(V)$. Then\n\\[ \\Schur{a}{k-1}(V) \\ = \\ \\pi_* \\Big( \\, \\Lambda^{k-1}F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\, \\Big). \\]\n\\end{lemma} \n\\begin{proof} In fact, (*) gives rise to a long exact sequence\n\\small\n\\[\n0 \\longrightarrow \\Lambda^{k-1} F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\Lambda^{k-1} (\\pi^* V) \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\Lambda^{k-2} (\\pi^* V) \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a+1) \\longrightarrow \\ldots \\ . \n\\]\n\\normalsize\nThe assertion follows from \\eqref{Schur.Equation} upon taking direct images.\n\\end{proof}\n \n Now suppose given a map of bundles\n \\begin{equation} \\label{cosection}\n   \\varepsilon : V \\longrightarrow \\mathcal{O}_M  \\end{equation}\nwhose image is the ideal sheaf $\\mathcal{I} \\subseteq \\mathcal{O}_M$ of a subscheme $Z \\subseteq X$:  equivalently, $\\varepsilon$ is dual to a section $\\mathcal{O}_M \\longrightarrow V^*$ whose zero-scheme is $Z$.  We allow the possibility that $\\varepsilon$ is surjective, in which case $\\mathcal{I} = \\mathcal{O}_M$ and $Z = \\varnothing$. \n\n\nIf $Z$ has the expected codimension $e = \\rk(V)$, then $\\mathcal{I}$ is resolved by the Koszul complex associated to $\\varepsilon$. The following result of Buchsbaum and Eisenbud gives the  resolution of powers of  $\\mathcal{I}$.\n\\begin{proposition} [{\\cite[Theorem 3.1]{Buchsbaum.Eisenbud}, \\cite[Theorem 2.1]{Srinivasan}}] \\label{Power.Koszul.Complex.Proposition}\nFix $a \\ge 1$. Then $\\varepsilon$ determines a complex \n\\begin{equation} \\label{Koszul.Complex.Power}\n\\xymatrix@C=30pt{\n\\ldots  \\ar[r]  &\\Schur{a}{2}(V) \\ar[r] &S^{a,1}(V)  \\ar[r]     &S^a V     \\ar[r]^{S^a(\\varepsilon)}  & \\mathcal{I}^a \\ar[r]  & 0  \n}\n\\end{equation}\nof vector bundles on $M$.\nThis complex is exact provided that either $\\varepsilon$ is surjective, or that $Z$ has codimension $= \\rk(V)$. \n\\end{proposition}\n\\noindent Observe from \\ref{Properties.of.Schur.Power} (i) that this complex has the same length as the Koszul complex of $\\varepsilon$. \n\n\\begin{proof} Returning to the setting of Lemma \\ref{Kempf.Type.Lemma}, denote by $\\tilde{\\varepsilon} : F \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}$ the composition of the inclusion $F \\hookrightarrow \\pi^*V$ with $\\pi^*\\varepsilon : \\pi^* V \\longrightarrow \\pi^* \\mathcal{O}_M$. The zero-locus of $\\tilde{\\varepsilon}$ defines the natural embedding of $\\mathbf{P}(\\mathcal{I})$ in $\\mathbf{P}(V)$. Now consider the Koszul complex of $\\tilde \\varepsilon$. After twisting by $\\mathcal{O}_{\\mathbf{P}(V)}(a)$ this has the form:\n\\[\n\\ldots \\longrightarrow \\Lambda^2 F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\mathcal{O}_{\\mathbf{P}(\\mathcal{I})}(a) \\longrightarrow 0.  \\tag{*}\n\\]\nIn view of Lemma  \\ref{Kempf.Type.Lemma}, \\eqref{Koszul.Complex.Power} arises by taking direct images. If $\\varepsilon$ is surjective, or defines a regular section of $V^*$, then the Koszul complex (*) is exact. Since the higher direct images of all the terms vanish, (*) pushes down to an exact complex. Furthermore, in this case $\\pi_* \\mathcal{O}_{\\mathbf{P}(\\mathcal{I})}(a) = \\mathcal{I}^a$ (cf \\cite[Theorem IV.2.2]{Fulton.Lang}), and the exactness of \\eqref{Koszul.Complex.Power} follows. \\end{proof}\n\n\n\\begin{example} [Macaulay's Theorem]\nSuppose as in the Introduction that $f_0, \\ldots, f_p \\in    \\mathbf{C}[x_0, \\ldots, x_r]$ are homogeneous polynomials of degrees $d_0 \\ge \\ldots \\ge d_p$ that generate a finite colength ideal $J$. This gives rise to a surjective map\n\\[   V \\ = \\ \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r} (-d_i) \\longrightarrow \\mathcal{O}_{\\mathbf{P}^r} \\longrightarrow 0\\]\nof bundles on projective space. Keeping in mind Remark \\ref{Properties.of.Schur.Power} (iv), Macaulay's statements \\eqref{Macaulay.Eqn.1}\n and \\eqref{Macaulay.Eqn.2}\nfollow by looking at the cohomology of the resulting complex \\eqref{Koszul.Complex.Power}. When $p = r$ this complex has length $r+1$, so one can also read off the non-surjectivity of \n\\[  \\HH{0}{\\mathbf{P}^r}{S^a V (t)} \\longrightarrow  \\HH{0}{\\mathbf{P}^r}{\\mathcal{O}_{\\mathbf{P}^r}(t)} \\]\nwhen $t < ad_0+ d_1 + \\ldots + d_r -r$. \n\\end{example}\n\n\\begin{example} [Complete intersection ideals] Suppose that $Z\\subseteq \\mathbf{P}^r$ is a complete intersection of dimension $\\ge 0$. Applying Theorem \\ref{Koszul.Complex.Power}\nto the Koszul resolution of its homogeneous ideal $I_Z$, one sees that $I_Z^a$ is saturated for every $a\\ge 1$. This is a result of Zariski.\n\\end{example}\n\n\\subsection{Push-forwards from a blowing up}\n\\label{Pushfowards.from.Blowup.Subsection}\n\n \nWe compute here the direct images of multiples of the exceptional divisor under the blowing-up of a smooth subvariety.\n\nConsider then a smooth  variety $M$ and a non--singular subvariety $X \\subseteq M$ having codimension $e \\ge 2$ and ideal sheaf $\\mathcal{I} = \\mathcal{I}_X \\subseteq \\mathcal{O}_M$. We consider the blowing-up \n\\[  \\mu : M^\\prime = \\text{Bl}_X(M) \\longrightarrow  M \\]\nof $M$ along $X$.  Write $\\mathbf{E} \\subseteq  M^\\prime$ for the exceptional divisor of $M^\\prime$, so that $\\mathcal{I} \\cdot \\mathcal{O}_{M^\\prime} = \\mathcal{O}_{M^\\prime}(-\\mathbf{E})$. \nWe recall that if $a>0$ then\n\\begin{equation} \\label{BU.Eqn.1}\n\\mu_* \\mathcal{O}_{M^\\prime}(-a\\mathbf{E}) \\ = \\ \\mathcal{I}^a \\ \\ \\text{and } \\ \\ R^j \\mu_* \\mathcal{O}_{M^\\prime}(-a\\mathbf{E}) \\, = \\, 0 \\ \\text{for } j > 0.\n\\end{equation}\n\nThe following Proposition gives the analogous computation for positive multiples of $\\mathbf{E}$.\n\\begin{proposition} \\label{Blowup.Pushforward.Proposition}\nFix $a > 0$. Then\n\\begin{equation} \\label{Pushforward.Ext.Equation}\nR^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ \\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}\\Big( \\mathcal{I}^{a-e+1} \\, , \\, \\mathcal{O}_M \\Big).\\footnote{When $0 < a < e-1$ we take $\\mathcal{I}^{a-e+1} = \\mathcal{O}_M$.}\n\\end{equation}\nIn particular, $\\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) = \\mathcal{O}_M$, $R^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) = 0$ if $j \\ne 0, e-1$, and\n\\[\nR^{e-1}\\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\  \\mathcal{E}\\mathit{xt}^{e-1}_{\\mathcal{O}_M}\\big( \\mathcal{I}^{a-e+1} \\, , \\, \\mathcal{O}_M \\big). \\]\n\\end{proposition} \n\n\\begin{proof}[Proof of Proposition \\ref{Blowup.Pushforward.Proposition}]\nThis is a consequence of duality for $\\mu$, which asserts that\n\\[  \nR\\mu_* \\, R\\,\\mathcal{H}\\mathit{om}_{\\mathcal{O}_{M^\\prime}} \\big( \\mathcal{F} \\, , \\, \\omega_\\mu \\, \\big) \\ = \\  R\\, \\mathcal{H}\\mathit{om}_{\\mathcal{O}_M} \\big ( \\, R\\mu_* \\mathcal{F} \\, , \\mathcal{O}_M \\, \\big) \\tag{*}\n\\]\nfor any sheaf $\\mathcal{F}$ on $M^\\prime$, where $\\omega_\\mu$ denotes the relative dualizing sheaf for $\\mu$ (\\cite[(3.19) on page 86]{Huybrechts}).   We apply this with \n\\[  \\mathcal{F} \\ = \\ \\mathcal{O}_{M^\\prime}\\big( \\, (e-1-a)\\mathbf{E}\\, ). \\]\nThen \n$R \\mu_* \\mathcal{F} =  \\mathcal{I}^{a-e+1}$\nthanks to  \\eqref{BU.Eqn.1} (and a direct computation when  $0 < a < e-1$), and $\\omega_\\mu = \\mathcal{O}_{M^\\prime}\\big( (e-1)\\mathbf{E} \\big)$. Therefore the first assertion of the Proposition follows from (*). The vanishing of $\\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}(\\mathcal{I}^{a-e+1}, \\mathcal{O}_M)$ for $j \\ne 0, e-1$ follows from the perfection of powers of the ideal of a smooth variety (which in turn is a consequence eg of  Proposition \\ref{Power.Koszul.Complex.Proposition}). \n\\end{proof}\n\n\n\\begin{remark} [Generalization to multiplier ideal sheaves] Let $\\mathfrak{b} \\subseteq \\mathcal{O}_M$ be an arbitrary ideal sheaf, and let $\\mu : M^\\prime \\longrightarrow M$ be a log resolution of $\\mathfrak{b}$, with $\\mathfrak{b} \\cdot \\mathcal{O}_{M^\\prime} = \\mathcal{O}_{M^\\prime}(-\\mathbf{E})$. A completely parallel argument shows that for $a > 0$:\n\\[\nR^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ \\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}\\big( \\MI{\\mathfrak{b}^a} \\, , \\, \\mathcal{O}_M \\big),\\]\nwhere $\\MI{\\mathfrak{b}^a}$ is the multiplier ideal of $\\mathfrak{b}^a$. The formula \\eqref{Pushforward.Ext.Equation} is a special case of this. \n\\end{remark}\n\n\n\n\\begin{corollary} \\label{Filtration.of.Push.Forwards}\nContinuing to work in characteristic zero, fix  $a \\ge 1$ and   denote by $N = N_{X\/M}$ the normal bundle to $X$ in $M$.  If $a \\le e-1$, then \n\\[ R^{e-1} \\mu_* \\, \\mathcal{O}_{M^\\prime}(a\\mathbf{E})  \\ = \\  0. \\]    If $a \\ge e$, then\n$\nR^{e-1} \\mu_* \\, \\mathcal{O}_{M^\\prime}(a\\mathbf{E})  \n$\nhas a filtration with successive quotients\n\\[\nS^k N \\otimes \\det N \\ \\ \\text{ for }   \\ 0 \\, \\le \\, k \\, \\le \\, a-e.\n\\]\n\\end{corollary}\n\\begin{proof} The first statement follows directly from the previous Proposition. For the second, \nrecall first that if $E$ is any locally free $\\mathcal{O}_X$-module, then  -- $X$ being non-singular of codimension $e$ in $M$ -- \n\\[   \\mathcal{E}\\mathit{xt}_{\\mathcal{O}_M}^{e} \\big ( \\, E \\, , \\, \\mathcal{O}_M \\, \\big) \\ = \\ E^* \\otimes \\det  N, \\]\nwhile all the other $\\mathcal{E}\\mathit{xt}^j$ vanish. The claim then follows from Proposition \\ref{Pushforward.Ext.Equation}\n using the exact sequences\n\\[ 0 \\longrightarrow \\mathcal{I}^{k+1} \\longrightarrow \\mathcal{I}^k \\longrightarrow S^k N^* \\longrightarrow 0 \\]\ntogether with the isomorphism $\\big (S^k(N^*)\\big)^* = S^k N$ valid in characteristic zero.\n \\end{proof}\n\n\n\\begin{remark}\nRecalling that $\\mathbf{E} = \\mathbf{P}(N^*)$, one can inductively prove the Corollary directly, circumventing Proposition \\ref{Pushforward.Ext.Equation}, by pushing forward the exact sequences\n\\[   0 \\longrightarrow  \\mathcal{O}_{M^\\prime}\\big((k-1)\\mathbf{E}\\big) \\longrightarrow  \\mathcal{O}_{M^\\prime}\\big(k \n\\mathbf{E}\\big) \\longrightarrow \\mathcal{O}_{\\mathbf{E}}(k\\mathbf{E})\\longrightarrow 0. \\]\nHowever it seemed to us that the Proposition may be of independent interest. \n\\end{remark}\n\n\n\\subsection{A vanishing theorem for normal bundles}\n\\label{Vanishing.Theorem.Normal.Bundles.Subsection}\n\nThis final subsection is devoted to the proof of \n\\begin{proposition} \\label{Van.Thm.NB.Subsection.Statement}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth complex projective variety of dimension $n$, and denote by $N = N_{X\/\\mathbf{P}^r}$ the normal bundle to $X$. Then\n\\[ \\HHH{i}{X}{S^kN \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell)} \\ = \\ 0 \n\\]\nfor all $i > 0$,  $k\\ge 0$ and $\\ell \\ge -r$. \n\\end{proposition}\n\\noindent Here $\\mathcal{O}_X(k)$ denotes $\\mathcal{O}_{\\mathbf{P}^r}(k)|X$. We remark that similar statements were established by Schneider and Zintl in \\cite{Schneider.Zintl}, but this particular vanishing does not seem to appear  there. Other vanishings for normal bundles played a central role in \\cite{SAD}. \n\n\\begin{proof} [Proof of Proposition \\ref{Van.Thm.NB.Subsection.Statement}]\nWe use the abbreviation $\\mathbf{P} = \\mathbf{P}^r$. Starting from the  exact sequence $ 0 \\longrightarrow TX \\longrightarrow T\\mathbf{P}|X \\longrightarrow N \\longrightarrow 0$, we get a long exact sequence\n\\[ \\ldots \\longrightarrow S^{k-2}T\\mathbf{P}|X \\otimes \\Lambda^2 TX \\longrightarrow S^{k-1}T\\mathbf{P}|X \\otimes TX  \\longrightarrow S^k T\\mathbf{P}|X \\longrightarrow S^k N \\longrightarrow 0. \\tag{*}  \\]\nBy adjunction, $\\det N \\otimes \\mathcal{O}_X(\\ell) = \\omega_X \\otimes \\mathcal{O}_X(\\ell+r+1)$.  So after twisting through by $\\det N \\otimes \\mathcal{O}_X(\\ell)$ in (*), we see that the Proposition will follow if we prove:\n\\[\n\\HHH{i}{X}{S^{k-j}T\\mathbf{P}|X \\otimes  \\Lambda^j TX \\otimes \\omega_X \\otimes \\mathcal{O}_X(\\ell+ r + 1))} \\ = \\ 0 \\  \\ \\ \\text{for } \\ i \\, \\ge \\, j + 1 \\tag{**}\n\\]\nwhen $0 \\le j \\le k$ and  $\\ell \\ge -r$. It follows from the Euler sequence that $S^m T\\mathbf{P}|X$ has a presentation of the form\n\\[  0 \\longrightarrow \\oplus \\, \\mathcal{O}_X(m-1) \\longrightarrow \\oplus \\, \\mathcal{O}_X(m) \\longrightarrow S^m T\\mathbf{P} |X \\longrightarrow 0, \\]\nso for (**) it suffices in turn to verify that\n\\[ \\HH{i}{X}{\\Lambda^j TX \\otimes \\omega_X \\otimes \\mathcal{O}_X(\\ell_1)} \\ = \\ 0 \\] for $i \\ge j + 1$ and $\\ell_1 > 0$. But \n$ \\Lambda^j TX \\otimes \\omega_X = \\Omega^{n-j}_X$,\nso finally we're asking that\n\\[ \\HH{i}{X}{\\Omega^{n-j}_X \\otimes \\mathcal{O}_X(\\ell_1)} \\ = \\ 0 \\ \\ \\text{for } \\ i \\ge j + 1  \\ \\text{and }   \\ell _1 >0,\\]\nand this follows from  Nakano vanishing.   \\end{proof}\n\n\\section{Proof of Theorem \\ref{Intro.Sat.Deg.Thm} }\n\nWe now turn to the proof of Theorem \\ref{Intro.Sat.Deg.Thm}\n from the Introduction.\n \nConsider then a non-singular variety $X \\subseteq  \\mathbf{P}^r$ that is cut out  as a scheme by hypersurfaces of degrees $d_0 \\ge \\ldots \\ge d_p$. Equivalently, we are given a surjective homomorphism of sheaves:\n\\[ \\varepsilon : U \\longrightarrow \\mathcal{I}_X \\ \\ \\text{,} \\ \\ U \\ = \\ \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r}(-d_i). \\]\nLet \n$\\mu : \\mathbf{P}^\\prime = \\text{Bl}_X(\\mathbf{P}^r) \\longrightarrow \\mathbf{P}^r  $ be the blowing up of $X$, with exceptional divisor $\\mathbf{E} \\subseteq \\mathbf{P}^\\prime$, so that $\\mathcal{I}_X \\cdot \\mathcal{O}_{\\mathbf{P}^\\prime} = \\mathcal{O}_{\\mathbf{P}^\\prime} (- \\mathbf{E})$. Write $H$ for the pull-back to $\\mathbf{P}^\\prime$ of the hyperplane class on $\\mathbf{P}^r$, and set $U^\\prime = \\mu^* U$. Thus on $\\mathbf{P}^\\prime$ we have a surjective map of bundles:\n\\begin{equation} \\label{Map.of.Bundles.on.Blowup}\n\\varepsilon^\\prime : U^\\prime \\longrightarrow \\mathcal{O}_{\\mathbf{P}^\\prime}(-\\mathbf{E}).\n\\end{equation}\nNoting that\n\\[  \\HHH{0}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}( tH - a \\mathbf{E})} \\ = \\  \\HHH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a \\otimes\\mathcal{O}_{\\mathbf{P}^r}(t)},\n\\]\none sees as in Lemma \\ref{Surjectivity.Suffices.Lemma} that the question is to prove the surjectivity of \n\\begin{equation} \\label{Surjectivity.Required.for.Thm.A}\n\\HHH{0}{\\mathbf{P}^\\prime}{S^aU^\\prime\\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(tH)} \\longrightarrow \\HHH{0}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}(tH- a \\mathbf{E})} \n\\end{equation}\nfor $t \\ge ad_0 + d_1 + \\ldots + d_r - r$. \n\nTo this end, we pass to the Buchsbaum--Eisenbud complex \\eqref{Koszul.Complex.Power} constructed from \\[ U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(\\mathbf{E})   \\overset{\\varepsilon^\\prime} \\longrightarrow \\mathcal{O}_{\\mathbf{P}^\\prime} \\longrightarrow 0. \\]\nTwisting through by $\\mathcal{O}_{\\mathbf{P}^\\prime}(t H - a\\mathbf{E})$, we arrive at a long exact sequence of vector bundles on $\\mathbf{P}^\\prime$ having the form:\n\n\\vskip -10pt\n\\small\n\\begin{equation} \\label{Big.Complex.on.Blowup}\n\\xymatrix@C=9.5pt@R=12pt{\n\\ldots \\ar[r] &\\Schur{a}{2} U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H + 2\\mathbf{E}) \\ar[r] \\ar@{=}[d]&S^{a,1} U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H+ \\mathbf{E}) \\ar[r]\\ar@{=}[d] & S^aU^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H) \\ar[r] \\ar@{=}[d] &\\mathcal{O}_{\\mathbf{P}^\\prime}(t H - a \\mathbf{E}) \\ar[r] &0. \\\\ & C_2 & C_1 & C_0\n}\n\\end{equation}\n\\normalsize\nWith indexing as indicated, the $i^{\\text{th}}$ term of this sequence is given by\n\\[\nC_i \\ = \\ \\Schur{a}{i}(U^\\prime)   \\otimes  \\mathcal{O}_{\\mathbf{P}^\\prime}(tH + i \\mathbf{E}). \n\\]\n\n\nIn  order to establish the surjectivity \\eqref{Surjectivity.Required.for.Thm.A} it suffices upon chasing through \\eqref{Big.Complex.on.Blowup} to prove that\n\\begin{equation} \\label{Vanishing.Required.for.Thm.A}\n\\HH{i}{\\mathbf{P}^\\prime}{C_i} \\ = \\ 0 \\ \\ \\text{ for } \\ 1 \\le i \\le r\n\\end{equation}\nprovided that $t \\ge ad_0 + d_1 + \\ldots + d_r -r$. But now recall (Remark \\ref{Properties.of.Schur.Power}) that if $i \\le r$ then $\\Schur{a}{i}(U^\\prime)$ is a sum of line bundles $\\mathcal{O}_{\\mathbf{P}^\\prime}(mH)$ with \n\\[\nm  \\, \\ge \\, -ad_0 - d_i - \\ldots - d_i \\ge -ad_0 - d_1 - \\ldots - d_r.\n\\]\nHence when $t \\ge ad_0 + d_1 + \\ldots + d_r -r$, $C_i$ is a sum terms of the form\n\\[  \n\\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E})  \\ \\ \\text{with } \\ell \\ge -r.\n\\]\nTherefore \\eqref{Vanishing.Required.for.Thm.A}\n -- and with it Theorem \\ref{Intro.Sat.Deg.Thm} -- is a consequence of\n\\begin{proposition}\nIf $\\ell \\ge -r$, then\n\\[  \\HH{i}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}{(\\ell H + i \\mathbf{E})} } \\ = \\ 0 \\ \\text{  for \\ } i > 0.\\]\n\\end{proposition}\n\\begin{proof}\nThanks to the Leray spectral sequence, it suffices to show:\n\\[\n\\HH{j}{\\mathbf{P}^r}{R^k \\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E})} \\ = \\ 0 \\ \\ \\text{when  } j + k = i > 0. \\tag{*}\n\\]\nFor $ k = 0$, observe that $\\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E}) = \\mathcal{O}_{\\mathbf{P}^r}(\\ell)$, and these sheaves have no higher cohomology when $\\ell \\ge -r$. On the other hand, by Proposition \\ref{Pushforward.Ext.Equation} the only non-vanishing higher direct images are the $R^{e-1}\\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E})$, which do not appear when $i \\le e-1$. So (*) holds when $j = 0, k = e-1$. It remains to consider the case  $k = e-1$ and $i \\ge e$, so $j = i - (e-1) > 0$.  \nHere Corollary \\ref{Filtration.of.Push.Forwards}\n implies that the $R^{e-1}$ have a filtration with quotients\n \\[\n S^\\alpha N \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell),\n \\]\n where as above $N = N_{X\/\\mathbf{P}^r}$ is the normal bundle to $X$ in $\\mathbf{P}^r$. But since we are assuming $\\ell \\ge -r$,  Proposition \\ref{Van.Thm.NB.Subsection.Statement} guarantees that these sheaves have vanishing higher cohomology. This completes the proof. \\end{proof}\n\n\\begin{remark}\\label{Few.Equations}\nObserve that if $X$ is defined by $p < r$ equations, then the argument just completed goes through taking\n  $  d_{p+1} = \\ldots =  d_r = 0.$\n  \\end{remark}\n\n\n\n\n %\n %\n\n %\n %\n\n \n \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn the present paper, we study the obstacle problem related to the following nonlocal and nonlinear operator, defined formally as\n\\begin{equation}\\label{operatore}\n\\mathcal{L}u(x)=p.~\\!v. \\int_{{\\mathds R}^n} K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\,{\\rm d}y, \\qquad x\\in {\\mathds R}^n;\n\\end{equation}\nwe take differentiability of order $s\\in (0,1)$ and growth $p>1$. The kernel $K$ is of order $(s,p)$ (see~\\eqref{hp_k}) with merely measurable coefficients.  \nThe integral in~\\eqref{operatore} may be singular at the origin and must be interpreted in an appropriate sense. Since we assume that coefficients are merely measurable, the involved equation has to have a suitable weak formulation; see Section~\\ref{sec_preliminaries} below for the precise assumptions on the involved quantities.\n \n\\smallskip\n \nThe  obstacle problem involving fractional powers of the Laplacian operator naturally appears in many contexts, such as in the analysis of anomalous diffusion (\\cite{BG90}), in the so called quasi-geostrophic flow problem (\\cite{CV10}), and in pricing of American options regulated by assets evolving in relation to jump processes (\\cite{CT04}). \nIn particular, the last application made the obstacle problem very relevant in recent times in all its forms; the obstacle problem can be indeed stated in several ways.\nRoughly speaking, a solution $u$ to the fractional obstacle problem is a minimal weak supersolution to the equation\n\\begin{equation}\\label{equazione}\n\\mathcal{L}u=0\n\\end{equation} \nabove an obstacle function $h$.\n\n\\smallskip\n\nIn the linear case when $p=2$ and when the kernel~$K$ reduces to the Gagliardo kernel~$K(x,y)=|x-y|^{-n-2s}$ without coefficients, a large treatment of the fractional obstacle problem can be found for instance in the fundamental papers by Caffarelli, Figalli, Salsa, and Silvestre (see, e.~\\!g.,~\\cite{Sil07,CSS08,CF13} and the references therein). See also~\\cite{Foc10,Foc09} for the analysis of families of bilateral obstacle problems involving fractional type energies in aperiodic settings; \nthe paper~\\cite{PP15b} for the fractional obstacle problems with drift; and the recent papers~\\cite{Gua15,MN15} for related estimates and approximations results.\nThis topic, despite its relatively short history, has already evolved into quite an elaborate theory, with connections to numerous branches of Analysis. It is impossible to provide here a complete list of references. We refer the interested reader to the exhaustive recent lecture notes by Salsa (\\cite{Sal12}), for the obstacle problem in the pure fractional Laplacian case, with the natural connection to the thin obstacle problem in low dimensions (for which we refer to~\\cite{ACS08}).\n\n\\smallskip\n\nHowever, in the more general framework considered here, the panorama seems rather incomplete, if not completely deficient in results. Clearly, the main difficulty into the treatment of the operators~$\\mathcal{L}$ in~\\eqref{operatore} lies in their very definition, which combines the typical issues given by its nonlocal feature together with the ones given by its nonlinear growth behavior; also, further efforts are needed due to the presence of merely measurable coefficients in the kernel~$K$. For this, some very important tools recently introduced in the nonlocal theory, as the by-now classic $s$-harmonic extension (\\cite{CS07}), the strong three-term commutators estimates (\\cite{DLR11}), and other  successful tricks as e.~\\!g. the pseudo-differential commutator approach in~\\cite{PP14,PP15}, cannot be plainly applied and seem difficult to adapt to the nonlinear framework considered here (mainly due to the non-Hilbertian nature of the involved fractional Sobolev spaces $W^{s,p}$).\n\n\\smallskip\n\nNevertheless, some related regularity results have been very recently achieved in this context, in~\\cite{BL15,DKP14,DKKP15,DKP15,KMS15,KMS15b,IMS15,IS14,Sch15} and many others, where often a fundamental role to understand the nonlocality of the nonlinear operators~$\\mathcal{L}$ has been played by a special quantity,\n\\begin{equation}\\label{coda}\n{\\rm Tail}(u;x_0,r) := \\bigg(r^{sp} \\int_{{\\mathds R}^n \\setminus B_r(x_0)} |u(x)|^{p-1} |x-x_0|^{-n-sp} \\,{\\rm d}x \\bigg)^{\\frac{1}{p-1}};\n\\end{equation}\nthat is, {\\it the nonlocal tail} of a function $u$ in the ball of radius $r>0$ centered in $x_0 \\in {\\mathds R}^n$. This quantity, introduced by two of the authors with A. Di Castro in~\\cite{DKP15}, have been subsequently became a relevant factor in many instances when one requires a fine quantitative control of the long-range interactions, which naturally arise when dealing with nonlocal operators (see Section~\\ref{sec_preliminaries} below).\n\n\\smallskip\n\nFor what concerns the main topic in the present paper, i.~\\!e., the nonlinear fractional obstacle problem with coefficients, we will prove a series of both qualitative and quantitative results. Amongst them, \nwe will formulate the natural variational framework for the obstacle problem, and we will prove both the existence and uniqueness of the solution~$u$ to this variational formulation (Theorem~\\ref{obst prob sol}). We will show that such a solution is a weak supersolution and that it is the smallest supersolution above the obstacle in a suitable sense (Proposition~\\ref{smallest super}).\nWe will also demonstrate that the solution~$u$ inherits the regularity of the obstacle, namely the boundedness~(Theorem~\\ref{thm:obs bnd}), continuity (Theorem~\\ref{thm:obs cont}), and H\\\"older continuity (Theorem~\\ref{thm:obs H cont}). As a consequence,  assuming that the obstacle function~$h$ is continuous,~$u$ is a weak solution to~\\eqref{equazione} in the open set $\\{u>h\\}$ (Corollary~\\ref{obst prob free}). These results are in clear accordance with the aforementioned results for the obstacle problems in the pure fractional Laplacian $(-\\Delta)^s$ case. However, our approach here is different and, though we are dealing with a wider class of nonlinear integro-differential operators with coefficients, the proofs are even somehow simpler, since we can make effort of a new nonlocal set-up together with the recent quantitative estimates obtained in~\\cite{DKP14,DKP15,DKKP15}, by also extending to the fractional framework some important tools in the classical Nonlinear Potential Theory.\n\n\\smallskip\n\nFinally, we will deal with the regularity up to the boundary (Theorems~\\ref{thm:H cont bdry}-\\ref{thm:cont bdry}). As well known, in the contrary with respect to the interior case, boundary regularity for nonlocal equations driven by singular, possibly degenerate, operators as in~\\eqref{operatore} seems to be a difficult problem in a general {\\it nonlinear} framework under natural assumptions on the involved quantities (while we refer to the recent paper~\\cite{Ros14} and to the forthcoming survey~\\cite{Ros16} for the case $p=2$). In this respect, a first (and possibly the solely currently present in the literature) result of global H\\\"older regularity has been obtained very recently in the  interesting paper~\\cite{IMS15}, where the authors deal with the equation in~\\eqref{equazione}, in the special case when the operator $\\mathcal{L}$ in~\\eqref{operatore} coincides with the nonlinear fractional Laplacian~$(-\\Delta)^{s}_p$, by considering exclusively zero Dirichlet boundary data, and by assuming a strong $C^{1,1}$-regularity up to the boundary for the domain~$\\Omega$. Indeed, their proof is strongly based on the construction of suitable barriers near $\\partial \\Omega$, starting from the fact that, under their restrictive assumptions, the function $x\\mapsto x^s_+$ is an explicit solution in the half-space. Clearly, one cannot expect to plainly extend such a strategy in the general framework considered here, in view of the presence of merely measurable coefficients in~\\eqref{operatore}. Also, we will allow nonzero boundary Dirichlet data to be chosen, and we will assume the domain~$\\Omega$ only to satisfy a natural {\\it measure density condition} (precisely, just on the complement of~$\\Omega$; see Formula~\\eqref{eq:dens cond} on Page~\\pageref{eq:dens cond}); the latter being as expected in accordance with the classical Nonlinear Potential Theory (that is, when $s=1$).\nFor this, we will need a new proof, that will extend up to the boundary part of the results in~\\cite{DKP14,DKP15} together with a careful handling of the tail-type contributions (see Section~\\ref{sec_boundary}). Once again, it is worth stressing that all these results are new even in the pure fractional $p$-Laplacian case when the operator~$\\mathcal{L}$ does coincide with $(-\\Delta)^s_p$, and  in the case of the (linear) fractional Laplacian with coefficients.\n\n\\smallskip\n\nAll in all, let us summarize  \n the contributions of the present paper:  \n\\noindent\n\\\\ - We prove new regularity results in terms of boundedness, continuity, and H\\\"older continuity for the solutions to a very general class of nonlocal obstacle problems, by extending previous results in the literature valid only for the pure linear fractional Laplacian case $(-\\Delta)^s$ without coefficients, also giving new proofs even in that case;\n\\noindent\n\\\\ - We obtain new regularity results up to the boundary for nonlocal operators, and, since we allow the obstacle function~$h$ to be an extended real-valued function, the degenerate case when $h\\equiv -\\infty$ (i.~\\!e., no obstacle is present) does reduce the problem to the standard Dirichlet boundary value problem, so that the results proven here are new even when $\\mathcal{L}$ does coincide with the fractional $p$-Laplacian $(-\\Delta)^s_p$. Also, since we assume that the boundary data merely belong to an appropriate tail space~$L^{p-1}_{sp}({\\mathds R}^n)$, all the (inner and boundary) results here reveal to be an improvement with respect to the previous ones in the literature when the data are usually given in the whole fractional Sobolev space~$W^{s,p}({\\mathds R}^n)$;\n\\noindent\n\\\\ - By solving the fractional obstacle problem together with some of the expected basic results proven here, we provide an important tool for several further investigations and applications.\nIndeed, as well known, the obstacle problem is deeply related to the study of minimal surfaces and the capacity of a set in Nonlinear Potential Theory. Thus, by means of our framework, we possibly give the basis for the development of a {\\it nonlocal} Nonlinear Potential Theory. This can be already seen in some subsequent forthcoming papers, as, e.~\\!g., in ~\\cite{KKL16} where part of the results here are the key for the viscosity approach for nonlocal integro-differential operators, and in~\\cite{DKKP15} where the whole nonlocal obstacle set-up is needed to extend the classical Perron method to a nonlocal nonlinear setting.\n\n\\smallskip\n\nFinally, let us comment about some immediate open problems naturally arising in this framework. Firstly, one can argue about the optimal regularity for the solutions to the nonlocal obstacle problem. We recall that for the classical obstacle problem, when $\\mathcal{L}$ coincides with the Laplacian operator $-\\Delta$,  the solutions are known to belong to~$C^{1,1}$. \nThe intuition behind this regularity result goes as follows:  in the contact set one has $-\\Delta u = -\\Delta h$, while where $u >h$ one has $-\\Delta u=0$; since the Laplacian jumps from $-\\Delta h$ to $0$ across the free boundary, the second derivatives of $u$ must have a discontinuity, so that $C^{1,1}$ is the maximum regularity class that can be expected. In the contrary, when $\\mathcal{L}\\equiv(-\\Delta)^s$, despite\nthe previous local argument does suggest that the solutions $u$ belong to $C^{2s}$,  the optimal regularity is $C^{1,s}$, and this is quite surprising since the regularity exponent is higher than the order of the equation.  \nIn the general nonlocal framework, starting from the H\\\"older regularity proven here, we still expect higher regularity results as for the linear case; nevertheless, in view of the interplay between local and nonlocal contributions and without having the possibility to rely on the $s$-harmonic extension, it is not completely evident what the optimal exponent could be as the nonlinear growth does take its part.\\footnote{For preliminary results in this direction, it is worth mentioning the very recent paper~\\cite{CRS16}, where optimal regularity results of the solution to the obstacle problem, and of the free boundary near regular points, have been achieved for linear integro-differential operators as in~\\eqref{operatore} in the case when~$p=2$.}\n\n Secondly, it could be interesting to investigate the regularity in a generic point of the free boundary (known to be analytic in the case of the Laplacian, except on a well defined set of singular points, and smooth in the case of the fractional Laplacian). \n\n Thirdly, a natural goal is to investigate the parabolic version of the nonlocal obstacle problem, as it is inspired in the so-called optimal stopping problem with deadline, by corresponding to the American option pricing problem with expiration at some given time.  \nAn extension in the setting presented here could be of relevant interest as it could describe a situation which also takes into account  the interactions coming from far together with a natural inhomogeneity. Accordingly with the optimal stopping problem model, a starting point in such an investigation could be the special case when the obstacle~$h$ coincides with the boundary value~$g$.\n\n \n\n\\medskip\nThe paper is organized as follows. In Section~\\ref{sec_preliminaries} below, we fix the notation by also stating some general assumptions on the quantities we will deal with throughout the whole paper.\nIn Section~\\ref{sec_obstacle}, we introduce the nonlinear fractional obstacle problem, and state and prove the existence and uniqueness of the related solutions.\nThe last two sections are devoted to the proofs of all the aforementioned boundedness and continuity results (Section~\\ref{sec_regularity}), and up to the boundary (Section~\\ref{sec_boundary}).\n \n\n\\medskip\n\n\\section{Preliminaries}\\label{sec_preliminaries}\n\nIn this section, we state the general assumptions on the quantities we are dealing with. We keep these assumptions throughout the paper.\n\\smallskip\n\nFirst of all, we recall that the class of integro-differential equations in which we are interested is the following\n\\begin{equation}\\label{problema}\n\\mathcal{L}u(x)=\\int_{{\\mathds R}^n} K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\,{\\rm d}y = 0, \\quad x\\in {\\mathds R}^n.\n\\end{equation}\nThe nonlocal operator~$\\mathcal{L}$ in the display above (being read a priori in the principal value sense) is driven by its {\\it kernel} $K:{\\mathds R}^n\\times {\\mathds R}^n \\to [0,\\infty)$, which is a measurable function  satisfying the following property:\n\\begin{equation}\\label{hp_k}\n\\Lambda^{-1} \\leq K(x,y)|x-y|^{n+sp} \\leq \\Lambda \\quad \\text{for a.~\\!e. } x, y \\in {\\mathds R}^n,\n\\end{equation}\nfor some $s\\in (0,1)$, $p>1$, $\\Lambda \\geq1$. We immediately notice that in the special case when $p=2$ and $\\Lambda=1$, we recover the well-known fractional Laplacian operator~$(-\\Delta)^s$.\nAlso, notice that the assumption on $K$ can be weakened,\nand in \\eqref{problema} the dependence of $u(x)-u(y)$, in turn, can be weakened from $t \\mapsto |t|^{p-2}t$ (see, for instance,~\\cite{KMS15}).\nHowever, for the sake of simplicity, we will take \\eqref{problema} and we will\nwork under the assumption in~\\eqref{hp_k}, since the weaker assumptions would bring no relevant differences in all the forthcoming proofs. \n\n\\medskip\n\nWe now  recall the definition of {\\it the nonlocal  tail \\,{\\rm{Tail}$(f; z, r)$} of a function $f$ in the ball of radius $r>0$ centered in $z\\in {\\mathds R}^n$}. We have\n\\begin{equation} \\label{def_tail} \n{\\rm Tail}(f;z,r) := \\bigg(r^{sp} \\int_{{\\mathds R}^n \\setminus B_r(z)} |f(x)|^{p-1} |x-z|^{-n-sp} \\,{\\rm d}x \\bigg)^{\\frac{1}{p-1}},\n\\end{equation}\nfor any function $f$ initially defined in $L^{p-1}_{\\textrm{loc}}({\\mathds R}^n)$. As mentioned in the introduction, this quantity will play an important role in the rest of the paper. The nonlocal tail has been introduced in~\\cite{DKP15}, and used subsequently in several recent papers (see e.~\\!g.,~\\cite{BL15,DKP14,HRS15,KMS15,KMS15b,IMS15,IS14} and many others\\footnote{\nWhen needed, our definition of Tail can also be given in a more general way by replacing the ball~$B_r$ and the corresponding~$r^{sp}$ term by an open bounded set~$E\\subset{\\mathds R}^n$ and its rescaled measure~$|E|^{sp\/n}$, respectively. This is not the case in the present paper.\n}), where it has been crucial to control in a quantifiable way the long-range interactions which naturally appear when dealing with nonlocal operators of the type considered here in~\\eqref{problema}.\nWhen having to control the positive and negative interactions separately, we denote the positive part and the negative part of a function $u$ by $u_+:=\\max\\{u,0\\}$ and $u_-:=\\max\\{-u,0\\}$, respectively. \nIn the following, when the center point $z$ will be clear from the context, we shall use the shorter notation \\, Tail$(f; r)\\equiv$ Tail$(f; z, r)$. \n Now, in clear accordance with the definition in~\\eqref{def_tail}, for any $p>1$ and any $s\\in (0,1)$, one can consider the corresponding  {\\it tail space} $L^{p-1}_{sp}({\\mathds R}^n)$ given by\n\\begin{equation*} \nL^{p-1}_{sp}({\\mathds R}^n) := \\Big\\{ f \\in L_{\\rm loc}^{p-1}({\\mathds R}^n) \\; : \\; {\\rm Tail}(f;z,r)< \\infty \\quad \\forall z \\in {\\mathds R}^n, \\forall r \\in (0,\\infty)\\Big\\}.\n\\end{equation*}\nNotice that \n\\begin{equation*}\nL^{p-1}_{sp}({\\mathds R}^n) = \\Big\\{ f \\in L_{\\rm loc}^{p-1}({\\mathds R}^n) \\; : \\;   \\int_{{\\mathds R}^n} |f(x)|^{p-1} (1+|x|)^{-n-sp} \\,{\\rm d}x < \\infty \\Big\\}.\n\\end{equation*}\nAs expected, one can check that $W^{s,p}({\\mathds R}^n) \\subset L^{p-1}_{sp}({\\mathds R}^n)$, where we denoted by $W^{s,p}({\\mathds R}^n)$ the usual fractional Sobolev space of order $(s,p)$, \ndefined by the norm\n\\begin{align}\\label{def_seminorm}\n\\|v\\|_{W^{s,p}({\\mathds R}^n)} & :=  \\|v\\|_{L^p({\\mathds R}^n)}\n+ [v]_{{W}^{s,p}({\\mathds R}^n)}\n \\nonumber \\\\\n&\\,\\, = \\left(\\int_{{\\mathds R}^n} |v|^p\\,{\\rm d}x \\right)^{\\frac1p} + \\left(\\int_{{\\mathds R}^n}\\int_{{\\mathds R}^n}\\frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1p}, \n\\end{align}\nwhere $s\\in (0,1)$ and $p\\geq1$. The local fractional Sobolev space $W^{s,p}(\\Omega)$ for $\\Omega \\subset {\\mathds R}^{n}$ is defined similarly.\nBy $W_0^{s,p}(\\Omega)$ we denote the closure of $C_0^\\infty(\\Omega)$ in $W^{s,p}({\\mathds R}^n)$. Conversely, if $v \\in W^{s,p}(\\Omega')$ with $\\Omega \\Subset \\Omega'$ and $v=0$ outside of $\\Omega$ almost everywhere, then $v$ has a representative in $W_0^{s,p}(\\Omega)$ as well (see, for instance, \\cite{DPV12}). \n\\medskip\n\n\nWe  now recall the definitions of sub and supersolutions $u$ to the class of \nintegro-differential problems we are interested in.\nA function $u \\in W^{s,p}_{\\rm{loc}}(\\Omega)\\cap L^{p-1}_{sp}({\\mathds R}^n)$ is a {\\it  fractional weak $p$-supersolution} of~$\\eqref{problema}$ if\n\\begin{align} \\label{supersolution}\n\\langle \\mathcal{L}u,\\eta\\rangle & \\equiv  \\int_{{\\mathds R}^n} \\int_{{\\mathds R}^n}K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\big(\\eta(x)-\\eta(y)\\big)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\*\n& \\ge 0\n\\end{align} \nfor every nonnegative $\\eta \\in C^\\infty_0(\\Omega)$. Here $\\eta \\in C^\\infty_0(\\Omega)$ can be replaced by $\\eta \\in W^{s,p}_0(\\Omega')$ with\nevery $\\Omega' \\Subset \\Omega$. \nIt is worth noticing that the summability assumption of $u$ belonging to the tail space $L^{p-1}_{sp}({\\mathds R}^n)$ is what one expects in the  nonlocal framework considered here (see~\\cite{DKKP15}). \n\\\\ A function $u \\in W^{s,p}_{\\rm{loc}}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^n)$ is a {\\it  fractional weak \n$p$-subsolution} if $-u$ is a  fractional weak $p$-supersolution. Finally, a function $u$ is a {\\it  fractional weak $p$-solution} if it is both  fractional weak $p$-sub and supersolution. In the following, we simply refer to those $u$ as (weak) supersolutions, subsolutions and solutions. \n\nMoreover, let us remark that we will assume that the kernel~$K$ is symmetric, and once again this is not restrictive,  in view of the weak formulation presented above, since one may always define the corresponding symmetric kernel $K_{\\textrm{\\tiny sym}}$ given by\n$$\nK_{\\textrm{\\tiny sym}}(x,y):=\\frac1{2}\\Big(K(x,y)+K(y,x)\\Big).\n$$ \n\n\n\\medskip\nWe conclude this section by presenting some basic estimates which will be useful in the course of the forthcoming proofs. \nAs customary when dealing with nonlinear operators, we will often have to treat in a different way the superquadratic case when $p>2$ and the subquadratic case $1<p<2$.\nIn order to simplify the notation in the weak formulation in~\\eqref{supersolution}, from now on we denote by\n\\begin{equation}\\label{def_l}\nL(a,b):=|a-b|^{p-2}(a-b), \\quad a,b \\in {\\mathds R}.\n\\end{equation}\nNotice that $L(a,b)$ is increasing with respect to $a$ and decreasing with respect to $b$.\n\n\\begin{lemma} \\label{lemma:1<p<2}\nLet $1<p \\le 2$ and $a,b,a',b' \\in {\\mathds R}$. Then\n\\begin{align}\\label{lem_2.2}\n|L(a,b)-L(a',b')| \\le 4|a-a'-b+b'|^{p-1}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nDenoting by \n\\[\nf(t):=L\\big(ta+(1-t)a', tb+(1-t)b'\\big),\n\\]\nwe obtain by the chain rule\n\\begin{align} \\label{Lab-La'b'}\n|L(a,b)-L(a',b')| &= \\Big| \\int_0^1 f'(t) \\,{\\rm d}t \\Big| \\nonumber \\\\\n& = \\Big| \\int_0^1 (a-a')\\partial_a L + (b-b')\\partial_b L \\,{\\rm d}t \\Big| \\nonumber \\\\\n&= (p-1)|a-a'-b+b'|\\int_0^1 |t(a-b)+(1-t)(a'-b')|^{p-2} \\,{\\rm d}t,\n\\end{align}\nwhere we also used that\n\\[\n\\partial_a L(a,b)=(p-1)|a-b|^{p-2} \\quad\n\\text{and} \\quad \\partial_b L(a,b)=-(p-1)|a-b|^{p-2}.\n\\]\n\nNow, for $\\alpha \\in [-1,1]$, define \n\\[\ng(\\alpha) := \\int_0^{1} |t \\alpha + 1-t|^{p-2}\\,{\\rm d}t.\n\\]\nNote that $g(1) = 1$. If $\\alpha<1$, then changing variables as $\\tau=t\\alpha+ 1-t$ yields\n\\[\ng(\\alpha) \\, = \\, \\frac{1}{1-\\alpha} \\int_{\\alpha}^{1}  |\\tau|^{p-2}\\,{\\rm d}\\tau\n\\,  \\leq \\,  2\\int_{0}^{1}  \\tau^{p-2}\\,{\\rm d}\\tau \n\\, = \\,  \\frac2{p-1}.\n\\]\nBy using the estimate above for $\\alpha := \\beta\/\\gamma$ with $|\\beta|\\leq |\\gamma|$, we obtain\n\\begin{align}\\label{210b}\n|\\gamma-\\beta| \\int_0^{1} |t \\beta + (1-t)\\gamma|^{p-2}\\,{\\rm d}t & = |\\gamma-\\beta| |\\gamma|^{p-2} g(\\alpha) \\nonumber \\\\\n& \\leq \\frac{4}{p-1} |\\gamma-\\beta|^{p-1}.\n\\end{align}\nFinally, by combining \\eqref{210b} with \\eqref{Lab-La'b'}, letting $\\beta=a-b$ and $\\gamma=a'-b'$, we obtain the desired result.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:p>2}\nLet $p \\ge 2$ and $a,b,a',b' \\in {\\mathds R}$. Then \n\\begin{align} \\label{p>2aba'b}\n|L(a,b)-L(a',b)| \\le c\\,|a-a'|^{p-1}+c\\,|a-a'||a-b|^{p-2}\n\\end{align}\nand\n\\begin{align} \\label{p>2abab'}\n|L(a,b)-L(a,b')| \\le c\\,|b-b'|^{p-1}+c\\,|b-b'||a-b|^{p-2},\n\\end{align}\nwhere $c$ depends only on $p$.\n\\end{lemma}\n\\begin{proof}\nDenoting by $f(t):=L\\big(ta'+(1-t)a,\\,  b\\big)$, we obtain by the chain rule\n\\begin{align*}\n|L(a,b)-L(a',b)| &= \\Big| \\int_0^1 f'(t) \\,{\\rm d}t \\Big| = \\Big| \\int_0^1 (a'-a)\\partial_a L(ta'+(1-t)a,b) \\,{\\rm d}t \\Big| \\\\[1ex]\n&= (p-1)|a-a'| \\int_0^1 |ta'+(1-t)a-b|^{p-2} \\,{\\rm d}t \\\\[1ex]\n&\\le c\\,|a-a'|^{p-1}+c\\,|a-a'||a-b|^{p-2},\n\\end{align*}\nwhere we also used that\n\\[\n\\partial_a L(a,b)=(p-1)|a-b|^{p-2}.\n\\]\nThus, the inequality in~\\eqref{p>2aba'b} does hold. Similarly, one can prove the inequality in~\\eqref{p>2abab'}.\n\\end{proof}\n\nFinally, we would like to make the following observation. In the rest of the paper, we often use the fact that there is a constant $c>0$ depending only on $p$ such that\n\\begin{align} \\label{a-b bounds}\n\\frac1{c} \\le \\frac{\\big(|a|^{p-2}a-|b|^{p-2}b\\big)(a-b)}{(|a|+|b|)^{p-2}(a-b)^{2}} \\le c,\n\\end{align}  \nwhen $a,b \\in {\\mathds R}$, $a \\neq b$. In particular,\n\\begin{align} \\label{a-b positive}\n\\big(|a|^{p-2}a-|b|^{p-2}b\\big)(a-b) \\geq 0, \\quad a,b \\in {\\mathds R}.\n\\end{align}\n\n\n\n\\medskip\n\n\n\\section{The obstacle problem}\\label{sec_obstacle}\nAs mentioned in the introduction, by solving the fractional obstacle problem we will provide an important tool in the development of the fractional Nonlinear Potential Theory, and, in order to present such a topological approach, \nwe start by introducing a necessary set of notation. \nLet $ \\Omega \\Subset \\Omega'$ be open bounded subsets of ${\\mathds R}^n$. Let  \n$h \\colon {\\mathds R}^n \\to [-\\infty,\\infty)$ be an extended real-valued function, which is considered to be the obstacle,\nand let $g \\in W^{s,p}(\\Omega') \\cap L^{p-1}_{sp}({\\mathds R}^n)$ be the boundary values. We define\n$$\n\\mathcal K_{g,h}(\\Omega,\\Omega') := \\Big\\{u \\in W^{s,p}(\\Omega') \\,:\\, u \\geq h\\, \\text{ a.~\\!e. in } \\Omega, \\, u = g\\, \\text{ a.~\\!e. on } {\\mathds R}^n \\setminus \\Omega \\Big\\}.\n$$\nThe interpretation for the case $h \\equiv -\\infty$ is that \n$$\n\\mathcal K_{g}(\\Omega,\\Omega')  \\equiv \\mathcal K_{g,-\\infty}(\\Omega,\\Omega') := \\Big\\{u \\in W^{s,p}(\\Omega') \\,:\\, u = g\\, \\text{ a.~\\!e. on } {\\mathds R}^n \\setminus \\Omega \\Big\\},\n$$ \ni.~\\!e., the class where we are seeking solutions to the Dirichlet boundary value problem. A few observations are in order. First, a natural assumption for any existence theory is that $\\mathcal K_{g,h}(\\Omega,\\Omega')$ is a non-empty set. This is a property of functions $g$ and $h$. Second, we are assuming that $g$ has bounded fractional Sobolev norm in a set~$\\Omega'$ which is strictly containing the set $\\Omega$, and not necessarily in the whole~${\\mathds R}^n$ as previously in the literature.\n\n\\vspace{0.9mm}\n\\subsection{Existence of solutions}\nThe obstacle problem can be reformulated as a standard problem in the theory of variational inequalities on Banach spaces, by seeking the energy minimizers in the set of suitable functions defined above. For this, by taking into account the nonlocality of the involved operators here,\nit is convenient to define a functional $\\mathcal A \\colon \\mathcal K_{g,h}(\\Omega,\\Omega') \\to \\left[W^{s,p}(\\Omega')\\right]'$ given by\n\\begin{equation}\\label{def_a}\n\\mathcal Au(v)  := \\mathcal A_1u(v) +  \\mathcal A_2 u(v)\n\\end{equation}\nfor every $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ and $v \\in W^{s,p}(\\Omega')$, where\n\\[\n \\mathcal A_1 u(v) := \\int_{\\Omega'}\\int_{\\Omega'} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\]\nand\n\\[\n \\mathcal A_2 u(v) := 2 \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} L(u(x),g(y))v(x)K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\]\nThe motivation for the definitions above is as follows. Assuming that $v \\in W_{0}^{s,p}(\\Omega)$, and $u \\in W^{s,p}(\\Omega')$ is such that $u = g$ on ${\\mathds R}^n \\setminus \\Omega'$, we have that \n\\begin{align} \\label{eq:weak sol vs A} \n\\notag \n& \\int_{{\\mathds R}^n}\\int_{{\\mathds R}^n} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\\\* \\notag & \\qquad = \\int_{\\Omega'}\\int_{\\Omega'} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\\\* \\notag & \\qquad \\quad + 2\\int_{{\\mathds R}^n \\setminus \\Omega'} \\int_{\\Omega} L(u(x),g(y)) v(x) K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\\\*  & \\qquad \\equiv  \\mathcal A_1 u(v) +  \\mathcal A_2 u(v).\n\\end{align}\nIn the following we will use the usual brackets, as e.~\\!g.~$\\langle \\mathcal{A}_1(u)-\\mathcal{A}_1(w), v\\rangle$ to denote $\\mathcal{A}_1u(v) -\\mathcal{A}_1w(v)$, and so on.\n\n\\begin{remark} \nThe functional $\\mathcal A u$ really belongs to the dual of $W^{s,p}(\\Omega')$. Indeed, we have\n\\begin{align} \\label{A1uv}\n| \\mathcal A_1 u(v) | &\\le \\int_{\\Omega'}\\int_{\\Omega'} |u(x)-u(y)|^{p-1}|v(x)-v(y)|K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\le c \\left(\\int_{\\Omega'}\\int_{\\Omega'} |u(x)-u(y)|^p \\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}}\\right)^\\frac{p-1}{p} \\nonumber\\\\\n&\\qquad \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^p \\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}}\\right)^\\frac{1}{p} \\nonumber\\\\[1ex]\n&\\le c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1} \\|v\\|_{W^{s,p}(\\Omega')}\n\\end{align}\nby H\\\"older's Inequality. Also,\n\\begin{align} \\label{A2uv}\n| \\mathcal A_2 u(v)| &\\le 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u(x)-g(y)|^{p-1}|v(x)|K(x,y)\\,{\\rm d}x{\\rm d}y  \\nonumber\\\\[1ex] \\nonumber\n&\\le c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u(x)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y  \\nonumber\\\\\n& \\quad + c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |g(y)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\le c\\,r^{-sp} \\left(\\int_{\\Omega} |u(x)|^p \\,{\\rm d}x\\right)^\\frac{p-1}{p} \\left(\\int_{\\Omega} |v(x)|^p \\,{\\rm d}x\\right)^\\frac{1}{p} \\nonumber \\\\\n& \\quad + c\\,\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-1}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)\\int_{\\Omega}|v(x)|\\,{\\rm d}x \\nonumber \\\\[1ex]\n&\\le c\\,r^{-sp}\\Big(\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}+{\\rm Tail}(g;z,r)^{p-1} \\Big)\\|v\\|_{W^{s,p}(\\Omega')}\n\\end{align}\nholds, where $z \\in \\Omega$ and $r:=\\dist(\\Omega,\\partial \\Omega')>0$, and $c$ depends on $n,p,s,\\Lambda,\\Omega,\\Omega'$. \n\\end{remark}\n\n\\begin{remark} \\label{remark:A2}\nIn the definition \\eqref{def_a}, we could replace $A_2 u(v)$ by\n\\begin{equation} \\label{A2 alternative}\n2 \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega''} L(u(x),g(y))v(x)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\end{equation}\nfor $\\Omega''$ satisfying $\\Omega \\subset \\Omega'' \\Subset \\Omega'$.\nAnyway, we need a strictly positive distance between $\\partial \\Omega''$ and $\\partial \\Omega'$\nto deduce $\\mathcal A u \\in [W^{s,p}(\\Omega')]'$, as seen in the calculations for \\eqref{A2uv} above.\n\\end{remark}\n\nNow, we are ready  to provide the natural definition of solutions to the obstacle problem in the general nonlocal framework considered here. \n\\begin{definition}\nWe say that $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is {\\it a solution to the obstacle problem} in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ if\n\\[\n \\mathcal Au(v-u)  \\geq 0\n\\]\nwhenever $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$.\n\\end{definition}\nBelow, we state and prove the uniqueness of the solution to the obstacle problem and the fact that such a solution is a weak supersolution to~\\eqref{problema}. Also, under natural assumptions on the obstacle $h$, one can prove that the solution to the obstacle problem is (fractional) harmonic away from the contact set, in clear accordance with the classical results when $s=1$.\n We have\n\n\\begin{theorem} \\label{obst prob sol}\nThere exists a unique solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. Moreover, the solution to the obstacle problem is a weak supersolution to \\eqref{problema} in $\\Omega$.\n\\end{theorem}\n\n\\begin{corollary} \\label{obst prob free}\nLet $u$ be the solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. If $B_r \\subset \\Omega$ is such that\n\\[\n\\essinf_{B_r} (u - h) >0,\n\\]\nthen $u$ is a weak solution to \\eqref{problema} in $B_r$. In particular, if $u$ is lower semicontinuous and $h$ is upper semicontinuous in $\\Omega$, then $u$ is a weak solution to \\eqref{problema}\nin $\\Omega_+:=\\big\\{x \\in \\Omega : u(x)>h(x)\\big\\}$.\n\\end{corollary}\n\n\\begin{remark}\nWhen solving the obstacle problem in $\\mathcal K_{g,-\\infty}(\\Omega,\\Omega')$,\nwe obtain a unique weak solution to \\eqref{problema} in $\\Omega$ having the boundary values\n$g \\in W^{s,p}(\\Omega') \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ in ${\\mathds R}^{n} \\setminus \\Omega$.\nThis is a generalization of the existence results stated in \\cite{DKP15}, where $g \\in W^{s,p}({\\mathds R}^{n})$, and -- as already mentioned in the introduction -- in general of all the analyses of fractional obstacle problems in the previous literature when $\\Omega'$ does coincide with the whole~${\\mathds R}^n$.\n\\end{remark}\n\n\nBefore going into the related proofs, we need to state and prove some properties of the operator $\\mathcal{A}$ defined in \\eqref{def_a}. We have the following\n\\begin{lemma} \\label{Amcwc}\nThe operator $\\mathcal A$ is monotone, coercive and weakly continuous on the set~$\\mathcal K_{g,h}(\\Omega,\\Omega')$.\n\\end{lemma}\n\\begin{proof}\nWe start with the monotonicity, that is, we show that $\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle \\ge 0$ holds for every $u,v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$.\nTo this end, let $u,v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. We have\n\\begin{align*}\n& \\langle \\mathcal A_1 u - \\mathcal A_1 v, u-v \\rangle \\\\\n&\\qquad = \\int_{\\Omega'}\\int_{\\Omega'} \\big(|u(x)-u(y)|^{p-2}(u(x)-u(y))-|v(x)-v(y)|^{p-2}(v(x)-v(y))\\big) \\\\\n&\\qquad \\qquad\\qquad \\times \\big(u(x)-u(y)-v(x)+v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\end{align*}\nand this quantity is nonnegative in view of~\\eqref{a-b positive}.\nSimilarly, for $\\mathcal A_2$,\n\\begin{align*}\n&\\langle \\mathcal A_2 u - \\mathcal A_2 v, u-v \\rangle \\\\\n&\\qquad = 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\Big(|u(x)-g(y)|^{p-2}\\big(u(x)-g(y))-|v(x)-g(y)|^{p-2} \\\\\n&\\qquad \\qquad\\qquad \\times (v(x)-g(y)\\big)\\Big)\\big(u(x)-g(y)-v(x)+g(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\\n&\\qquad \\ge 0.\n\\end{align*}\nHence $\\mathcal A$ is monotone.\n\n\\medskip\nNext, we prove the weak continuity. Let $\\{u_j\\}$ be a sequence in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ converging to $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ in $W^{s,p}(\\Omega')$. The weak continuity condition is that $\\langle \\mathcal A u_j - \\mathcal Au,v \\rangle \\to  0$ for all $v \\in W^{s,p}(\\Omega')$. Thus, let $v \\in W^{s,p}(\\Omega')$.\nThen for $\\mathcal A_1$ and $1<p\\leq 2$, applying \\eqref{lem_2.2} and H\\\"older's Inequality, we obtain\n\\begin{align*}\n& |\\langle \\mathcal A_1 u_j - \\mathcal A_1 u, v \\rangle| \\\\[1ex]\n&\\qquad \\le \\int_{\\Omega'}\\int_{\\Omega'} \\big|L(u_j(x),u_j(y))-L(u(x),u(y))\\big||v(x)-v(y)|K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\qquad \\le c\\int_{\\Omega'}\\int_{\\Omega'} |u_j(x)-u_j(y)-u(x)+u(y)|^{p-1}|v(x)-v(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\\\[1ex]\n&\\qquad \\le c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}^{p-1} \\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich vanishes as $j \\to \\infty$.\nOn the other hand, when $p>2$, by \nusing \\eqref{a-b bounds},\nwe have, again by H\\\"older's Inequality, that\n\\begin{align*}\n& |\\langle \\mathcal A_1 u_j - \\mathcal A_1 u, v \\rangle| \\\\*[1ex]\n&\\qquad \\le \\int_{\\Omega'}\\int_{\\Omega'} \\big|L(u_j(x),u_j(y))-L(u(x),u(y))\\big||v(x)-v(y)|K(x,y)\\,{\\rm d}x{\\rm d}y \\\\*[1ex]\n&\\qquad \\le c\\int_{\\Omega'}\\int_{\\Omega'} \\big(|u_j(x)-u_j(y)|+|u(x)-u(y)|\\big)^{p-2} \\\\*\n&\\qquad \\qquad\\qquad \\times |u_j(x)-u_j(y)-u(x)+u(y)||v(x)-v(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\\\[1ex]\n&\\qquad \\le c\\int_{\\Omega'}\\int_{\\Omega'} \\bigg(\\frac{|u_j(x)-u_j(y)|^{p-2}}{|x-y|^{s(p-2)}}+\\frac{|u(x)-u(y)|^{p-2}}{|x-y|^{s(p-2)}}\\bigg) \\\\\n&\\qquad \\qquad\\qquad \\times \\frac{|u_j(x)-u_j(y)-u(x)+u(y)|}{|x-y|^{s}}\\,\\frac{|v(x)-v(y)|}{|x-y|^{s}}\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n}} \\\\[1ex]\n&\\qquad \\le c\\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u_j(x)-u_j(y)|^{p}}{|x-y|^{n+sp}}\n+ \\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac{p-2}{p}} \\\\\n&\\qquad \\qquad \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u_j(x)-u(x)-u_j(y)+u(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1{p}} \\\\\n&\\qquad \\qquad  \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1{p}} \\\\[1ex]\n&\\qquad \\le c\\,\\big(\\|u_j\\|_{W^{s,p}(\\Omega')}+\\|u\\|_{W^{s,p}(\\Omega')}\\big)^{p-2} \\|u_j-u\\|_{W^{s,p}(\\Omega')} \\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich vanishes as $j \\to \\infty$.\nSimilarly, for $\\mathcal A_2$ when $1<p \\le 2$, by using again \\eqref{lem_2.2}, we have\n\\begin{align*}\n& |\\langle \\mathcal A_2 u_j - \\mathcal A_2 u, v \\rangle| \\\\\n&\\qquad \\le 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big|L(u_j(x),g(y))-L(u(x),g(y))\\big||v(x)|K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\qquad \\le c\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u_j(x)-u(x)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\qquad \\le c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}^{p-1}\\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich tends to $0$ as $j \\to \\infty$. In the case when $p>2$, by using~\\eqref{p>2aba'b}--\\eqref{p>2abab'} we get\n\\begin{align*}\n& |\\langle \\mathcal A_2 u_j - \\mathcal A_2 u, v \\rangle| \\\\\n&\\qquad \\le c\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u_j(x)-u(x)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\\\\n&\\qquad \\quad + c\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u_j(x)-u(x)||u(x)-g(y)|^{p-2}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\qquad \\le c\\int_{\\Omega} |u_j(x)-u(x)|^{p-1}|v(x)|\\,{\\rm d}x + c\\int_{\\Omega} |u_j(x)-u(x)||u(x)|^{p-2}|v(x)|\\,{\\rm d}x \\\\\n&\\qquad \\quad + c\\,\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-2}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg) \\int_{\\Omega} |u_j(x)-u(x)||v(x)|\\,{\\rm d}x \\\\[1ex]\n&\\qquad \\le c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}^{p-1}\\|v\\|_{W^{s,p}(\\Omega')} \\\\\n&\\qquad \\quad + c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}\\|u\\|_{W^{s,p}(\\Omega')}^{p-2}\\|v\\|_{W^{s,p}(\\Omega')} \\\\\n&\\qquad \\quad + c\\,r^{-sp}\\,{\\rm Tail}(g;z,r)^{p-2}\\|u_j-u\\|_{W^{s,p}(\\Omega')}\\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich again tends to $0$ as $j \\to \\infty$. \nNotice that in the display above the nonlocal integral has been estimated as follows\n\\begin{align*}\n& \\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-2}|z-y|^{-n-sp}\\,{\\rm d}y \\\\\n&\\qquad \\le \\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-1}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)^\\frac{p-2}{p-1}\n\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)^\\frac{1}{p-1} \\\\[1.5ex]\n&\\qquad \\le c\\,r^{-sp}\\,{\\rm Tail}(g;z,r)^{p-2},\n\\end{align*}\nwhere $z \\in \\Omega$ and $r:=\\dist(\\Omega,\\partial \\Omega')>0$. Thus, $\\langle \\mathcal A u_j,v \\rangle \\to \\langle \\mathcal A u, v \\rangle$ for every $v \\in W^{s,p}(\\Omega')$ as $j \\to \\infty$, i.~\\!e., the weak continuity holds.\n\n\\medskip\nFinally, we prove the coercivity, which means that there exists a function $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ such that\n\\[\n\\frac{\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle}{\\|u-v\\|_{W^{s,p}(\\Omega')}} \\to \\infty \\quad \\text{as} \\quad \\|u\\|_{W^{s,p}(\\Omega')} \\to \\infty.\n\\]\nSince we are assuming that $\\mathcal K_{g,h}(\\Omega,\\Omega')$ is non-empty, there is at least one $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let this be fixed. By \\eqref{A1uv} and \\eqref{A2uv} we see that\n\\begin{align} \\label{AuAvv}\n|\\langle \\mathcal A u - \\mathcal A v, v \\rangle| \\le c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}+c,\n\\end{align}\nwhere the constant $c$ is independent of $u$. We now show that the contribution from $\\langle \\mathcal A u - \\mathcal A v, u \\rangle$ dominates when $\\|u\\|_{W^{s,p}(\\Omega')}$ is large.\nFor $\\mathcal A_1$, we obtain\n\\begin{align}\\label{A1uA1vu}\n&\\langle \\mathcal A_1 u - \\mathcal A_1 v, u \\rangle \\nonumber\\\\\n&\\qquad= \\int_{\\Omega'}\\int_{\\Omega'} \\big(L(u(x),u(y))-L(v(x),v(y))\\big)\\big(u(x)-u(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\qquad\\ge \\frac1{c} \\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}} \\,{\\rm d}x{\\rm d}y \\nonumber\\\\\n&\\qquad\\quad -c \\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^{p-1}|u(x)-u(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad\\ge  \n\\frac 1c \\left[ u-g \\right]_{W^{s,p}(\\Omega')}^{p} - c \\left[ g \\right]_{W^{s,p}(\\Omega')}^{p} \n \\nonumber\\\\\n&\\qquad\\quad -c \\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^{p-1}|u(x)-u(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad\\ge \\frac1{c} \\|u-g\\|_{W^{s,p}(\\Omega')}^{p}-c\\,\\|g\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|v\\|_{W^{s,p}(\\Omega')}^{p-1}\\|u\\|_{W^{s,p}(\\Omega')} \\nonumber \\\\\n&\\qquad\\ge \\frac1{c} \\|u\\|_{W^{s,p}(\\Omega')}^{p}-c\\,\\|g\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|v\\|_{W^{s,p}(\\Omega')}^{p-1}\\|u\\|_{W^{s,p}(\\Omega')},\n\\end{align}\nby using in particular H\\\"older's Inequality and the fractional Sobolev embeddings (see for instance~\\cite[Section~6]{DPV12}, and also \\cite[Appendix 6.3]{PSV13} for a simple proof). \nFor~$\\mathcal A_2$, in turn, we obtain\n\\begin{align}\\label{A2uA2vu}\n&\\langle \\mathcal A_2 u - \\mathcal A_2 v, u \\rangle \\nonumber\\\\\n&\\qquad = 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big(L(u(x),g(y))-L(v(x),g(y))\\big)\\big(u(x)-v(x)\\big){K}(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\ \n&\\qquad \\quad +\\, 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big(L(u(x),g(y))-L(v(x),g(y))\\big)v(x){K}(x,y)\\,{\\rm d}x{\\rm d}y\\nonumber \\\\[1ex]\n&\\qquad \\geq   -2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big|L(u(x),g(y))-L(v(x),g(y))\\big||v(x)|{K}(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\qquad \\geq  -c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\Big(|u(x)|^{p-1}|v(x)|+|g(y)|^{p-1}|v(x)|+|v(x)|^{p}\\Big)\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad \\geq  -c\\,\\|u\\|_{L^p(\\Omega')}^{p-1}\\|v\\|_{L^p(\\Omega')} - c\\,r^{-sp}{\\rm Tail}(g;z,r)^{p-1}\\|v\\|_{L^p(\\Omega')} - \\|v\\|_{L^p(\\Omega')}^p \n\\end{align}\nwith $z\\in\\Omega$ and $r:=\\dist(\\Omega,\\partial\\Omega')$, where we also used that the term on the second line is nonnegative by the monotonicity.\nBy combining the estimates \\eqref{AuAvv}, \\eqref{A1uA1vu} with \\eqref{A2uA2vu}, it yields\n\\begin{align*}\n\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle \\ge \\frac1{c} \\|u\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}-c\\,\\|u\\|_{W^{s,p}(\\Omega')}-c,\n\\end{align*}\nfor a constant $c$ independent of $u$, and thus\n\\begin{align*}\n\\frac{\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle}{\\|u-v\\|_{W^{s,p}(\\Omega')}}\n\\to \\infty \\quad \\text{as} \\quad \\|u\\|_{W^{s,p}(\\Omega')} \\to \\infty.\n\\end{align*}\nThis finishes the proof.\n\\end{proof}\n\nNow, we are ready to prove the existence of a unique solution to the obstacle problem.\n\n\\begin{proof}[\\bf Proof of Theorem~\\ref{obst prob sol}]\nWe first notice that $\\mathcal K_{g,h}(\\Omega,\\Omega') \\subset W^{s,p}(\\Omega')$ is nonempty, closed and convex. Also, in view of Lemma~\\ref{Amcwc} the operator $\\mathcal A$ is monotone, coercive and weakly continuous on $\\mathcal K_{g,h}(\\Omega,\\Omega')$. This will permit us to apply the standard theory of monotone operators (see, for instance, Corollary III.1.8 in~\\cite{KS80}, or~\\cite{HKM06}) in order to deduce the existence of a function~$u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ such that\n\\[\n\\langle \\mathcal Au,v-u \\rangle \\ge 0,\n\\]\nwhenever $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. In order to show  the uniqueness, suppose that there are two functions $u_1$ and $u_2$ solving the obstacle problem. As a consequence, \n\\[\n\\langle \\mathcal Au_1,u_2-u_1 \\rangle \\ge 0 \\quad \\text{and} \\quad \\langle \\mathcal Au_2,u_1-u_2 \\rangle \\ge 0,\n\\]\nand then, by summing the preceding inequalities, we obtain\n\\[\n\\langle \\mathcal Au_1-\\mathcal Au_2,u_1-u_2 \\rangle \\le 0.\n\\]\nThis is possible only if $u_1=u_2$ almost everywhere. Thus, the solution $u$ is unique.\n\nNow we show that the function $u$ is a weak supersolution to \\eqref{problema} in $\\Omega$.\nFirst, clearly $u \\in W^{s,p}_{\\rm loc}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$.\nThen, notice that for any given nonnegative function $\\phi \\in C_0^{\\infty}(\\Omega )$, the function $v:=u+\\phi$ belongs to $\\mathcal K_{g,h}(\\Omega,\\Omega')$. Consequently, we have\nas in~\\eqref{eq:weak sol vs A}  that \n\\begin{align*}\n0 \\le \\langle \\mathcal Au, \\phi \\rangle  \n&= \\int_{{\\mathds R}^{n}} \\int_{{\\mathds R}^{n}}L(u(x),u(y))\\big(\\phi(x)-\\phi(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\end{align*}\nThus, $u$ is a weak supersolution in $\\Omega$.\n\\end{proof}\n\n\n\\begin{proof}[\\rm \\bf Proof of Corollary \\ref{obst prob free}]\nBy Theorem \\ref{obst prob sol} $u$ is a weak supersolution  \nin $B_r \\subset \\Omega$. To show that $u$ is also a weak subsolution in $B_r$, let $\\eta \\in C_0^{\\infty}(B_r)$ be a nonnegative test function that is not identically $0$. Set \n$\\varepsilon := \\|\\eta\\|_\\infty^{-1} \\essinf_{B_r}(u-h)>0$. Then $v = u - \\varepsilon \\eta \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ and $\\langle \\mathcal Au,v-u \\rangle \\geq 0$ yields $\\langle \\mathcal Au,\\eta \\rangle \\leq 0$. Therefore, by~\\eqref{eq:weak sol vs A} we obtain that $u$ is a weak subsolution in $B_r$, and thus a weak solution there. \n\\end{proof}\n\nThe solution to the obstacle problem is the smallest supersolution above the obstacle in the following sense.\n\\begin{proposition} \\label{smallest super}\nLet $\\Omega \\Subset \\Omega'' \\subset \\Omega'$. Let $u$ be the solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and let $v$ be a weak supersolution in $\\Omega''$ such that $\\min\\{u,v\\} \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Then $u \\leq v$ almost everywhere.\n\\end{proposition}\n\\begin{proof}\nSince $u$ is the solution to the obstacle problem and $\\min\\{u,v\\}=u$ in ${\\mathds R}^{n}\\setminus\\Omega$,\n\\begin{align} \\label{uminuv}\n0 &\\leq \\langle \\mathcal Au, \\min\\{u,v\\}-u \\rangle \\\\*\n&= \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} L(u(x),u(y))\\Big(\\min\\{u,v\\}(x)-u(x)-\\min\\{u,v\\}(y)+u(y)\\Big) \\nonumber \\\\*\n& \\qquad \\qquad \\quad \\times K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber.\n\\end{align}\nSince $v$ is a weak supersolution in $\\Omega''$ and $u-\\min\\{u,v\\} \\in W^{s,p}_0(\\Omega)$ is nonnegative, we have\n\\begin{align} \\label{vminuv}\n0 &\\leq \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} L(v(x),v(y))\\Big(u(x)-\\min\\{u,v\\}(x)-u(y)+\\min\\{u,v\\}(y)\\Big) \\nonumber \\\\\n& \\qquad \\qquad \\quad \\times K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\end{align}\nSumming the preceding inequalities \\eqref{uminuv} and \\eqref{vminuv}, we obtain\n\\begin{align*}\n0 &\\leq \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} \\big(L(v(x),v(y))-L(u(x),u(y))\\big) \\\\*\n&\\qquad\\qquad\\qquad \\times \\big(u(x)-\\min\\{u,v\\}(x)-u(y)+\\min\\{u,v\\}(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&= \\int_{\\{u > v\\}} \\int_{\\{u>v\\}}\\big(L(v(x),v(y))-L(u(x),u(y))\\big) \\\\\n&\\qquad\\qquad\\qquad \\times \\big(u(x)-v(x)-u(y)+v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\\n&\\quad + 2\\int_{\\{u \\leq v\\}} \\int_{\\{u>v\\}}\\big(L(v(x),v(y))-L(u(x),u(y))\\big)\n\\big(u(x)-v(x)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\*[1ex]\n& \\leq 0\n\\end{align*}\nsince the first term is nonpositive by \\eqref{a-b positive},\nwhereas in the second term, $L(v(x),v(y))<L(u(x),u(y))$ and $u(x)>v(x)$.\nConsequently, $|\\{u>v\\}|=0$.\n\\end{proof}\n\n\\medskip\n\n\\section{Interior regularity}\\label{sec_regularity}\nIn this section, we state and prove that the regularity of the solution to the obstacle problem inherits the regularity of the obstacle, both in the case of boundedness and (H\\\"older) continuity. This is in clear accordance with the by-now classical results for the obstacle problems in the pure fractional Laplacian $(-\\Delta)^s$ case, as seen e.~\\!g. in~\\cite{ACS08,CSS08,Sil07}, via the Dirichlet-to-Neumann extension.\nHowever, our approach here is different and, though we are dealing with a wider class of nonlinear integro-differential operators with coefficients, the proofs are  new and even simpler, since we can make effort of the recent quantitative estimates presented in the previous sections and in~\\cite{DKP14,DKP15}, by taking care of the nonlocal tail quantities.\n\n\\begin{theorem} \\label{thm:obs bnd}\nLet $u$  be the solution to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$. Assume that $B_r(x_0) \\subset \\Omega'$ and set \n\\[\nM :=\\max\\bigg\\{ \\esssup_{B_r(x_0) \\cap \\Omega}h ,  \\esssup_{B_r(x_0) \\setminus \\Omega} g \\bigg\\}.\n\\]\nHere the interpretation is that $ \\esssup_{A} f = -\\infty$ if $A = \\emptyset$. \nIf $M$ is finite, then $u$ is essentially bounded from above in $B_{r\/2}(x_0)$  and \n\\begin{equation} \\label{eq_obs_bnd}\n\\esssup_{B_{r\/2}(x_0)}(u-m)_+ \\leq  \\delta\\, {\\rm Tail}((u-m)_+;{x_0},r\/2)+c\\, \\delta^{-\\gamma} \\left(\\mean{B_r(x_0)} (u-m)_+^t\\,{\\rm d}x\\right)^{\\frac 1t}\n\\end{equation}\nholds for all $m\\geq M$, $t \\in (0,p)$ and $\\delta \\in (0,1]$ with constants $\\gamma \\equiv \\gamma(n,p,s,t)$ and $c \\equiv c(n,p,s,t,\\Lambda)$. \n\\end{theorem}\n\\begin{proof}\nSuppose that $M<\\infty$. Letting $k\\geq 0$, $m\\geq M$,  and $\\phi \\in C_0^\\infty(B_r(x_0))$, $0 \\leq \\phi \\leq 1$, we see that $v = u-m - (u-m-k)_+ \\phi^p$ belongs to $\\mathcal{K}_{g-m,h-m}(\\Omega,\\Omega')$ and that $u_m := u-m$ solves the corresponding obstacle problem. Thus we have that \n\\begin{align*}\n&\\int_{{\\mathds R}^n} \\int_{{\\mathds R}^n} L(u_m(x),u_m(y))  \\big( (u_m(x) {-}k)_+ \\phi^p(x) {-} (u_m(y){-}k)_+ \\phi^p(y)\\big)\\nonumber \\\\\n& \\qquad \\quad \\, \\ \\times K(x,y) \\,{\\rm d}x{\\rm d}y\\  \\leq\\ 0.\n\\end{align*}\nAs observed in the proof of~\\cite[Theorem 1.4]{DKP15}, this is enough to prove first a Caccioppoli-type estimate with tail, and subsequently a local boundedness result (see~\\cite[Theorem 1.1]{DKP15}) which yields\n\\begin{equation} \\label{sup obs 1}\n\\esssup_{B_{\\rho\/2}(y)}(u_m)_+ \\leq \\tilde \\delta\\, {\\rm Tail}((u_m)_+;{y},\\rho\/2)+c\\, \\tilde \\delta^{-\\tilde\\gamma} \\left(\\mean{B_\\rho(y)} (u_m)_+^p\\,{\\rm d}x\\right)^{\\frac 1p},\n\\end{equation}\nwhenever $B_\\rho(y) \\subset B_r(x_0)$, \nfor any  $\\tilde \\delta \\in (0,1]$, and with positive $\\tilde\\gamma$ depending only on $n,p,s$ and $c$ only on $n,p,s,\\Lambda$. Now, a covering argument, which goes back to the one  in the proof of~\\cite[Theorem 1.1]{DKP14}, will allow us to prove that~\\eqref{sup obs 1} actually implies~\\eqref{eq_obs_bnd}. For this, set $\\rho=(\\sigma-\\sigma')r$ with $1\/2\\leq \\sigma'<\\sigma\\leq 1$, and take $y \\in \\sigma'B\\equiv B_{\\sigma' r}(x_0)$. We can estimate the nonlocal contribution in~\\eqref{sup obs 1} as follows\n\\begin{align}\\label{tailm}\n& {\\rm Tail}( (u_m)_+; y, \\rho\/2)^{p-1} \\nonumber \\\\[1ex]\n& \\qquad  \\leq \\left(\\frac{\\rho}{2}\\right)^{sp} \\sup_{\\sigma B} (u_m)_+^{p-1} \\int_{\\sigma B\\setminus B_{\\rho\/2}(y)}|x-y|^{-n-sp}\\,{\\rm d}x  \n\\nonumber\\\\\n&  \\qquad  \\quad  + \\left(\n\\frac{\\rho}{2}\\right)^{sp}\n \\sup_{x\\in {\\mathds R}^{\\mathds n}\\setminus\\sigma B}\\left(\\frac{|x-x_0|}{|x-y|}\\right)^{n+sp}\n\\int_{{\\mathds R}^n\\setminus \\sigma B} (u_m)_+^{p-1}|x-x_0|^{-n-sp}\\,{\\rm d}x \\nonumber \\\\[1ex]\n& \\qquad   \\leq \\ c\\,\\sup_{\\sigma B} (u_m)_+^{p-1}\n+ c\\,(\\sigma-\\sigma')^{-{n}}{\\rm Tail}((u_m)_+; x_0, r\/2)^{p-1}.\n\\end{align}\nFor what concerns the local contribution in~\\eqref{sup obs 1}, we can  apply Young's Inequality (with exponents $p\/t$ and $p\/(p-t)$) to get\n\\begin{align*}\n\\tilde \\delta^{-\\tilde\\gamma}\\left( \\mean{B_{\\rho}(y)} (u_m)_+^{p}\\,{\\rm d}x \\right)^{\\frac{1}{p}}\n&\\leq \\tilde \\delta^{-\\tilde\\gamma} \\sup_{B_{\\rho}(y)}(u_m)_+^{\\frac{p-t}{p}} \\left( \\mean{B_{\\rho}(y)} (u_m)_+^{t}\\,{\\rm d}x \\right)^{\\frac{1}{p}} \\\\[1ex]\n&\\leq \\frac{1}{4}\\sup_{\\sigma B} (u_m)_+ \n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}}\\left( \\mean{B_{\\rho}(y)} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}} \\\\[1ex]\n&\\leq \\frac{1}{4}\\sup_{\\sigma B} (u_m)_+\n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}}\n(\\sigma-\\sigma')^{-\\frac{n}{t}}\n\\left( \\mean{B_r} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}}.\n\\end{align*}\nThus, by reabsorbing with $\\tilde \\delta\\leq 1\/4c$ we deduce by three last displays that\n\\begin{align*}\n\\sup_{\\sigma' B}(u_m)_+ &\\leq\n\\frac{1}{2}\\sup_{\\sigma B} (u_m)_+\n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}} (\\sigma-\\sigma')^{-\\frac{n}{t}}\\left( \\mean{B_r} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}} \\\\\n&\\quad +\\, c\\,\\tilde \\delta (\\sigma-\\sigma')^{-\\frac{n}{p-1}}{\\rm Tail}((u_m)_+; x_0, r\/2),\n\\end{align*}\nso that finally a standard iteration argument, see for instance~\\cite[Lemma 3.38]{HKM06}, and choosing $\\tilde \\delta = \\delta\/c$ will give the desired result~\\eqref{eq_obs_bnd}.\n\\end{proof}\n\nThe solution to the obstacle problem inherits the continuity of the obstacle. \n\n\\begin{theorem} \\label{thm:obs H cont}\nSuppose that $h$ is locally H\\\"older continuous in $\\Omega$. Then the solution $u$ to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$ is locally H\\\"older continuous in $\\Omega$ as well.\nMoreover, for every $x_0 \\in \\Omega$ there is $r_0>0$ such that\n\\begin{align} \\label{eq:obs cont} \n\\osc_{B_\\rho(x_0)} u & \\leq c \\left(\\frac{\\rho}{r}\\right)^{\\alpha} \\left[ {\\rm Tail}(u -h(x_0);{x_0},r) + \\bigg(\\mean{B_{r}(x_0)} |u-h(x_0)|^p \\,{\\rm d}x \\bigg)^{\\frac 1p}\\right]  \n\\\\ & \\quad \\nonumber + c \\int_{\\rho}^r \\left(\\frac{\\rho}{t}\\right)^{\\alpha} \\omega_h\\left(  \\frac{t}{\\sigma} \\right) \\, \\frac{dt}{t}\n\\end{align}\nfor every $r\\in (0,r_0)$ and $\\rho \\in (0,r\/4]$, where $\\omega_h(\\rho) \\equiv \\omega_h(\\rho,x_0) := \\osc_{B_{\\rho}(x_0)} h$,\nand $\\alpha$, $c$ and $\\sigma$ depend only on $n$, $p$, $s$, and $\\Lambda$.\n\\end{theorem}\n\\begin{proof}\nLet us first analyze the contact set, by which we mean that $x_0$ belongs to the contact set if and only if for every $r \\in (0,R)$, $R:= \\dist(x_0,\\partial \\Omega)$, we have\n\\[\n\\inf_{B_{r}(x_0)} (u-h) = 0.\n\\]\nOur first goal is to show that for any such point $x_0$ and for any\n$r \\in (0,R)$ we find $\\sigma \\in (0,1)$ and $c$, both depending only on $n,p,s,\\Lambda$, such that  \n\\begin{align} \\label{eq:osc decay 000}\n& \\nonumber \\osc_{B_{\\sigma r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma r) \n\\\\ & \\qquad \\leq \\frac12  \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right)  + c\\,\\omega_h(r).\n\\end{align}\nTo this end, observe first that $u \\geq d: = h(x_0) - \\omega_h(r)$ almost everywhere in $B_r(x_0)$. Set $u_d := u-d$, which is then a nonnegative weak supersolution in $B_r(x_0)$ by Theorem~\\ref{obst prob sol}. Now apply Theorem~\\ref{thm:obs bnd} and the weak Harnack estimate in \\cite[Theorem 1.2]{DKP14}. We obtain by \\eqref{eq_obs_bnd} (applied with $m= d + 2\\omega_h(r) \\geq \\sup_{B_{2\\rho}(x_0)}h$) that \n\\begin{equation} \\label{eq:sup u_k 0}\n\\sup_{B_\\rho(x_0)} u_d  \\leq  2\\omega_h(r) + \\delta\\,{\\rm Tail}((u_d)_+ ;{x_0},\\rho)+c\\, \\delta^{-\\gamma} \\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x \\right)^{\\frac 1t}\n\\end{equation}\nfor $\\rho \\in(0,r]$, $t\\in(0,p)$ and $\\delta\\in(0,1]$, and the weak Harnack gives\n\\[\n\\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x\\right)^{\\frac 1t} \\leq c \\inf_{B_{4\\rho}(x_0)} u_d + c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}}{\\rm Tail}( (u_d)_-;{x_0},r)\n\\]\nwhenever $\\rho \\in(0,r\/4]$.\nSince $\\inf_{B_{\\rho}(x_0)}u_d \\leq \\omega_h(r)$ due to $\\essinf_{B_{\\rho}(x_0)} (u-h) = 0$, we obtain from the previous display that \n\\[\n\\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x\\right)^{\\frac 1t} \\leq c\\,\\omega_h(r) +c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r).\n\\]\nThus, recalling that $u_d \\geq 0$ in $B_r(x_0)$, we arrive at\n\\[\n\\osc_{B_\\rho(x_0)} u \\leq c\\,\\delta^{-\\gamma} \\omega_h(r) + c\\,\\delta\\,{\\rm Tail}(u_d;{x_0},\\rho)\n+  c\\,\\delta^{-\\gamma} \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r).\n\\]\nNow we observe that\n\\begin{equation} \\label{eq:tail u_k}\n{\\rm Tail}(u_d;{x_0},\\rho) \\leq c \\sup_{B_{r}(x_0)} |u_d|  + c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r),\n\\end{equation}\nwhere we can estimate \n\\begin{equation} \\label{eq:sup u_k}\n\\sup_{B_r(x_0)} |u_d|=\\sup_{B_r(x_0)} |u-h(x_0)+\\omega_h(r)| \\leq \\osc_{B_{r}(x_0)} u+2 \\omega_h(r).\n\\end{equation}\nNow, for any $\\varepsilon \\in (0,1)$, we can first choose $\\delta$  small and then $\\widetilde \\sigma \\in (0,1)$, correspondingly, so that we have\n\\[\nc\\, \\delta \\leq \\frac{\\varepsilon}{2} \\qquad \\mbox{and} \\qquad c\\,\\delta^{-\\gamma} \\widetilde \\sigma^{\\frac{sp}{p-1}} \\leq \\frac{\\varepsilon}{2},\n\\]\nand therefore, for $\\rho = \\widetilde \\sigma r$,\n\\begin{equation} \\label{eq:osc tildesigmarho}\n\\osc_{B_{\\widetilde \\sigma r}(x_0)} u \\leq \\varepsilon \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right)  + c_\\varepsilon\\, \\omega_h(r)\n\\end{equation}\nholds. Using this together with \\eqref{eq:tail u_k}, we further have that for any $\\sigma \\in (0,\\widetilde \\sigma)$\n\\begin{align*} \n{\\rm Tail}(u - h(x_0);{x_0},\\sigma r) & \\leq c  \\osc_{B_{\\tilde \\sigma r}(x_0)} u +c \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u - h(x_0);{x_0},\\widetilde \\sigma r)\n\\\\* & \\leq c\\, \\varepsilon \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right)  + c\\, c_\\varepsilon\\, \\omega_h(r) \\\\*\n&\\quad + c \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} \\left(\\osc_{B_r(x_0)}u + {\\rm Tail}(u - h(x_0);{x_0},r)\\right).\n\\end{align*}\nTherefore, adding \\eqref{eq:osc tildesigmarho} and taking $\\sigma$ and $\\varepsilon$ so small that\n\\[\nc \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} \\leq \\varepsilon \\qquad \\text{and} \\qquad (c+2)\\,\\varepsilon \\leq \\frac12,\n\\]\nyields \\eqref{eq:osc decay 000}.\n\nNext, iterating \\eqref{eq:osc decay 000} we obtain\n\\begin{align} \\label{eq:osc decay 001}\n& \\nonumber \\osc_{B_{\\sigma^k r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma^k r) \n\\\\ & \\qquad  \\leq 2^{1-k} \\left(\\osc_{B_{\\sigma r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma r) \\right)  + c \\sum_{j=0}^{k-2} 2^{-j} \\omega_h(\\sigma^{k-j-1} r)\n\\end{align}\nfor any $k \\in \\mathbb N$.\nUsing finally the fact $\\osc_{B_r} u = \\osc_{B_r} u_d \\leq \\sup_{B_r} u_d$ and the supremum estimate \\eqref{eq:sup u_k 0},  \nwe conclude the contact set analysis with\n\\begin{align} \\label{eq:osc decay 002}\n& \\nonumber \\osc_{B_{\\sigma^k r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma^k r) \n\\\\ & \\qquad  \\leq c\\, 2^{1-k} \\left( {\\rm Tail}(u -h(x_0);{x_0},r) + \\bigg(\\mean{B_{r}(x_0)} |u-h(x_0)|^t\\,{\\rm d}x\\bigg)^{\\frac 1t}\\right) \\\\\n& \\qquad\\quad + c \\sum_{j=0}^{k-1} 2^{-j} \\omega_h(\\sigma^{k-j-1} r). \\nonumber\n\\end{align}\nNotice here that if $h$ is continuous and uniformly bounded in $B_r$, then\n\\begin{equation*} \\label{eq:omega_h sum}\n\\lim_{k\\to \\infty} \\sum_{j=0}^{k-1} 2^{-j} \\omega_h(\\sigma^{k-j-1} r) = 0,\n\\end{equation*}\nimplying that $\\lim_{r \\to 0} \\osc_{B_{r}(x_0)} u = 0$ in this case. \n\nWe then analyze the continuity properties outside of the contact set. In this case we find $r_0 \\in (0,R)$ such that \n$$\n\\inf_{B_{r_0}(x_0)} (u-h) > 0.\n$$\nThen Corollary \\ref{obst prob free} says that $u$ is a weak solution in $B_{r_0}(x_0)$, and consequently we can use the results in~\\cite{DKP15}, by also noticing that in the proofs there it makes no difference to assume\n$u \\in W^{s,p}_{\\rm loc}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ instead of $u \\in W^{s,p}({\\mathds R}^{n})$.\nIn particular, \\cite[Theorem 1.2]{DKP15} implies that\n\\begin{equation*}  \\label{eq:osc decay 003}\n\\osc_{B_{\\rho}(x_0)} u \\leq c \\left(\\frac{\\rho}{r}\\right)^{\\alpha} \\left({\\rm Tail}(u-h(x_0);{x_0},r) +  \\bigg(\\mean{B_{r}(x_0)} | u -h(x_0)|^p\\,{\\rm d}x\\bigg)^{\\frac 1p} \\right).\n\\end{equation*}\nfor every $r \\in (0,r_0)$ and $\\rho \\in (0,r\/2]$.\nThe claim  \nfollows from this and \\eqref{eq:osc decay 002} (with $\\alpha \\leq -\\log 2 \/ \\log \\sigma$) after straightforward manipulations. \n\\end{proof}\n\nSlightly modifying the proof above, we easily obtain the following.\n\\begin{theorem} \\label{thm:obs cont}\nSuppose that $h$ is continuous in $\\Omega$. Then the solution to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$ is continuous in $\\Omega$ as well.\n\\end{theorem}\n\\begin{proof}\nThis plainly follows from the previous theorem, since if $\\omega_h(t) \\to 0$ as $t\\to 0$ and $\\omega_h$ is locally uniformly bounded, then it is easy to check that \n$$\n\\int_{\\rho}^r \\left(\\frac{\\rho}{t}\\right)^{\\alpha} \\omega_h\\left(  \\frac{t}{\\sigma} \\right) \\, \\frac{dt}{t} \\to 0 \n$$\nas $\\rho \\to 0$ for small enough $r$. \n\\end{proof}\n\n\n\\medskip\n\n\n\n\\section{Boundary regularity}\\label{sec_boundary}\n\nWe continue our investigation by considering the regularity of the solution to the obstacle problem on the boundary of $\\Omega$. \nIn what follows, we assume $x_0 \\in \\partial \\Omega$.\nFirstly, we would need a Caccioppoli-type estimate with tail, whose proof is\na verbatim repetition of the proof of~\\cite[Theorem~1.4]{DKP15} after noticing that $v = u \\mp w_\\pm \\phi^p$, $\\phi \\in C_0^\\infty(B_r(x_0))$, $0\\leq \\phi \\leq 1$, belongs to $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$  for all indicated $k_+$ and $k_-$. For other fractional Caccioppoli-type inequalities, though not taking into account the tail contribution, see~\\cite{Min07,Min11,FP14}. We have the following\n\\begin{lemma} \\label{lemma:cacc bnd}\nSuppose that $u \\in \\mathcal{K}_{g,h}(\\Omega,\\Omega')$ solves the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$.\nLet $x_0 \\in \\partial\\Omega$, let $r \\in (0,r_0)$ with $r_0 := \\dist(x_0,\\partial \\Omega')$, and suppose that \n\\[\nk_+ \\geq \\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h  \\bigg\\} \\quad \\text{and} \\quad\nk_- \\leq \\essinf_{B_r(x_0)} g.\n\\]\nThen, for $w_+ := (u - k_+)_+$ and $w_- := (k_--u)_+$, we have \n\\begin{eqnarray}\\label{cacio1}\n\\nonumber && \\int_{B_r(x_0)}\\int_{B_r(x_0)} |w_{\\pm}(x)\\phi(x)-w_{\\pm}(y)\\phi(y)|^p K(x,y) \\,{\\rm d}x{\\rm d}y \\\\[1ex]\n &&\\qquad   \\leq c\\int_{B_r(x_0)}\\int_{B_r(x_0)} w_{\\pm}^p(x) |\\phi(x)-\\phi(y)|^p K(x,y) \\,{\\rm d}x{\\rm d}y \\\\\n&&\\qquad  \\quad+\\,c \\int_{B_r(x_0)}w_{\\pm}(x)\\phi^p(x)\\,{\\rm d}x \\left(\\sup_{y\\,\\in\\,{\\rm supp}\\,\\phi}\\int_{\\mathds{R}^n\\setminus B_r(x_0)} w_{\\pm}^{p-1}(x)K(x,y)\\,{\\rm d}x \\right), \n\\nonumber\n\\end{eqnarray}\nwhenever $\\phi \\in C_0^\\infty(B_r(x_0))$ and $0\\leq \\phi \\leq 1$.\n\\end{lemma}\n\\begin{remark} \\label{remark:cacc contact}\nIf the maximum $\\max\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h \\}$ is infinite,\nor $\\essinf_{B_r(x_0)} g = -\\infty$,  \nthen the interpretation is that there is no test function of the type $w_+$ or $w_-$, respectively. \n\\end{remark}\n\n\nWhen the obstacle and the boundary values are bounded on the boundary, so is the solution to the obstacle problem.\n\n\\begin{theorem} \\label{thm:boundednessx0}\nSuppose that $u \\in \\mathcal{K}_{g,h}(\\Omega,\\Omega')$ solves the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$.\nLet $x_0 \\in \\partial\\Omega$ and suppose that \n\\[\n\\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h  \\bigg\\}  < \\infty \\quad \\text{and} \\quad \\essinf_{B_r(x_0)} g > -\\infty\n\\]\nfor $r \\in (0,r_0)$ with $r_0 := \\dist(x_0,\\partial \\Omega')$. Then $u$ is essentially bounded close to $x_0$. \n\\end{theorem}\n\\begin{proof}\nChoose \n\\[\nk_+ \\geq \\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h  \\bigg\\}  \\quad \\text{and} \\quad\nk_- \\leq  \\essinf_{B_r(x_0)} g.\n\\]\nThen, repeating the proof of \\cite[Theorem 1.1]{DKP15} using the estimate \\eqref{cacio1} in Lemma \\ref{lemma:cacc bnd} with $w_+ := (u-k_+)_+$ and $w_- := (k_--u)_+$, we get\n\\begin{equation*} \n\\esssup_{B_{r\/2}(x_0)} w_\\pm \\leq \n\\delta \\, {\\rm Tail}(w_\\pm ; x_0, r\/2)+ c\\,\\delta^{-\\gamma} \\left( \\mean{B_r(x_0)} w_\\pm^p \\,{\\rm d}x \\right)^{1\/p}.\n\\end{equation*}\nfor any $\\delta \\in (0,1]$ with $\\gamma \\equiv \\gamma(n,p,s)$ and $c\\equiv c(n,p,s,\\Lambda)$. Consequently, $u$ is essentially bounded in $B_{r\/2}(x_0)$.\n\\end{proof}\n\nTo prove the H\\\"older continuity of the solution to the obstacle problem on the boundary, we also need the following logarithmic estimate. \n\n\\begin{lemma} \\label{log lemma w}\nLet $B_r \\subset B_{R\/2}$ be concentric balls and let $w \\in W^{s,p}(B_R) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ satisfy\n\\[\n\\esssup_{B_R} w \\leq M < \\infty \\quad \\text{and} \\quad \\essinf_{B_R} w \\geq \\varepsilon > 0.\n\\]\nSuppose that\n\\begin{equation*}\n\\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}}L(w(x),w(y))\\left(\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)-\\frac{M-w(y)}{w(y)^{p-1}}\\phi^{p}(y)\\right)K(x,y)\\,{\\rm d}x{\\rm d}y \\geq 0,\n\\end{equation*}\nwhere $\\phi \\in C^{\\infty}_0(B_{3r\/2})$ satisfies $0\\leq\\phi\\leq 1$, $\\phi = 1$ in $B_{r}$ and $|\\nabla\\phi|<c\/r$.\nThen\n\\begin{align} \\label{log lemma claim w}\n&\\int_{B_{r}}\\int_{B_{r}}\\left|\\log\\frac{w(x)}{w(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\\n&\\qquad \\leq c\\,r^{n-sp}\\left(1+\\varepsilon^{1-p}\\left(\\frac rR\\right)^{sp}{\\rm Tail}(w_-,x_0,R)^{p-1}\\right).\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nBy splitting the integral, we obtain\n\\begin{align} \\label{log lemma I1I2}\n0 &\\leq \\int_{B_{2r}}\\int_{B_{2r}}L(w(x),w(y))\\left(\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)-\\frac{M-w(y)}{w(y)^{p-1}}\\phi^{p}(y)\\right)K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\*\n&\\quad + 2\\int_{{\\mathds R}^{n}\\setminus B_{2r}}\\int_{B_{2r}}L(w(x),w(y))\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)K(x,y)\\,{\\rm d}x{\\rm d}y  \\nonumber \\\\*[1ex]\n&=: I_1 + I_2.\n\\end{align}\nWe estimate the integrand of $I_1$ in the case $w(x)>w(y)$. By \\cite[Lemma 1.3]{DKP15} we have\n\\[\n\\phi^{p}(x) \\leq \\phi^{p}(y)+c\\,\\delta\\phi^{p}(y)+c\\,\\delta^{1-p}|\\phi(x)-\\phi(y)|^{p}\n\\]\nwhenever $\\delta \\in (0,1)$. Choosing\n\\[\n\\delta=\\sigma\\frac{w(x)-w(y)}{w(x)} \\in (0,1), \\quad \\sigma \\in (0,1),\n\\]\nin the display above, implies\n\\begin{align*}\n\\Psi(x,y)&:=(w(x)-w(y))^{p-1}\\left(\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)-\\frac{M-w(y)}{w(y)^{p-1}}\\phi^{p}(y)\\right) \\\\[1ex]\n&\\leq (w(x)-w(y))^{p-1}\\left(\\frac{M-w(x)}{w(x)^{p-1}}-\\frac{M-w(y)}{w(y)^{p-1}}+c\\,\\delta\\frac{M-w(x)}{w(x)^{p-1}}\\right)\\phi^{p}(y) \\\\\n&\\quad + c\\,\\delta^{1-p}(w(x)-w(y))^{p-1}\\frac{M-w(x)}{w(x)^{p-1}}|\\phi(x)-\\phi(y)|^{p} \\\\[1ex]\n&= \\left(\\frac{M-w(x)}{w(x)^{p-1}}-\\frac{M-w(y)}{w(y)^{p-1}}+c\\,\\sigma\\frac{(w(x)-w(y))(M-w(x))}{w(x)^{p}}\\right) \\\\\n&\\qquad \\times (w(x)-w(y))^{p-1}\\phi^{p}(y) + c\\,\\sigma^{1-p}(M-w(x))|\\phi(x)-\\phi(y)|^{p} \\\\[1ex]\n&=: \\Psi_1(x,y) + \\Psi_2(x,y).\n\\end{align*}\n\nWe estimate $\\Psi_1$ separately in the cases $w(x)>2w(y)$ and $w(x) \\leq 2w(y)$.\nWhen $w(x)>2w(y)$, we obtain\n\\begin{align*}\n\\Psi_1(x,y) &\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\left(2^{1-p}(M-w(x))-(M-w(y))+c\\,\\sigma M\\right)\\phi^{p}(y) \\\\[1ex]\n&\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\left((2^{-1}-2^{1-p})w(x)-(1-2^{1-p})M+c\\,\\sigma M\\right)\\phi^{p}(y).\n\\end{align*}\nIf $p \\geq 2$, then\n\\begin{align*}\n(2^{-1}-2^{1-p})w(x)-(1-2^{1-p})M \\leq (2^{-1}-2^{1-p})M-(1-2^{1-p})M = -\\frac12 M.\n\\end{align*}\nIf $1<p<2$, in turn, then\n\\begin{align*}\n(2^{-1}-2^{1-p})w(x)-(1-2^{1-p})M \\leq -(1-2^{1-p})M.\n\\end{align*}\nThus, choosing $\\sigma$ small enough in $\\Psi_1$, we obtain\n\\begin{align} \\label{psi1wx>>wy}\n\\Psi_1(x,y) &\\leq -\\frac1c M\\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\phi^{p}(y).\n\\end{align}\n\nWhen $w(x) \\leq 2w(y)$, we can estimate\n\\begin{align*}\n\\Psi_1(x,y) &\\leq \\left(\\frac{w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(x)^{p-1}\\right)}{w(y)^{p-1}(w(x)-w(y))}+c\\,\\sigma M\\right) \\\\*\n&\\qquad \\times \\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\\phi^{p}(y).\n\\end{align*}\nIf $w(x)<M\/2$, then\n\\begin{align*}\n&(M-w(x))w(y)^{p-1}-(M-w(y))w(x)^{p-1} \\\\[1ex]\n&\\qquad \\qquad\\qquad\\qquad \\leq (M-w(x))w(y)^{p-1}-(M-w(x))w(x)^{p-1} \\\\[1ex]\n&\\qquad\\qquad\\qquad\\qquad \\leq -\\frac1c (M-w(x))(w(x)-w(y))w(x)^{p-2} \\\\[1ex]\n&\\qquad \\qquad\\qquad\\qquad\\leq -\\frac1c M(w(x)-w(y))w(x)^{p-2},\n\\end{align*}\nand we obtain\n\\begin{align} \\label{psi1wx>wy}\n\\Psi_1(x,y) &\\leq -\\frac1c M \\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\\phi^{p}(y)\n\\end{align}\nwhen choosing $\\sigma$ small enough. If $w(x) \\geq M\/2$, in turn, then\n\\begin{align*}\n&w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(x)^{p-1}\\right) \\\\[1ex]\n&\\qquad\\qquad\\qquad \\leq w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(y)^{p-1}\\right) \\\\[1ex]\n&\\qquad\\qquad\\qquad \\leq -\\frac12 M(w(x)-w(y))w(y)^{p-1},\n\\end{align*}\nand again we obtain \\eqref{psi1wx>wy} when choosing $\\sigma$ small enough.\n\nLet us then estimate further to get logarithms visible. In the case $w(x)>2w(y)$, it holds\n\\begin{align} \\label{logwx>>wy}\n\\left(\\log\\frac{w(x)}{w(y)}\\right)^{p} \\leq c\\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\n\\end{align}\nsince $(\\log t)^{p} \\leq c\\,(t-1)^{p-1}$ when $t>2$. In the case $w(x) \\leq 2w(y)$, in turn, it holds\n\\begin{align} \\label{logwx>wy}\n\\nonumber \\left(\\log\\frac{w(x)}{w(y)}\\right)^{p} &= \\left(\\log\\left(1+\\frac{w(x)-w(y)}{w(y)}\\right)\\right)^{p} \\\\*[1ex]\n&\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p}\n\\ \\leq \\ 2^{p}\\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\n\\end{align}\nsince $\\log(1+t) \\leq t$ when $t \\geq 0$. Thus, combining \\eqref{psi1wx>>wy} with \\eqref{logwx>>wy} and \\eqref{psi1wx>wy} with \\eqref{logwx>wy}, we obtain\n\\begin{align}\n\\Psi_1(x,y) &\\leq -\\frac1c M \\left(\\log\\frac{w(x)}{w(y)}\\right)^{p}\\phi^{p}(y).\n\\end{align}\n\nFor $\\Psi_2$ we easily get\n\\begin{align*}\n\\Psi_2(x,y) &\\leq c\\,M |\\phi(x)-\\phi(y)|^{p} \\leq c\\,M r^{-p}|x-y|^{p}.\n\\end{align*}\nIn the case $w(x)<w(y)$ we can interchange the roles of $x$ and $y$, whereas the contribution from the case $w(x)=w(y)$ is zero.\nAfter integrating we have\n\\begin{align} \\label{log lemma I1}\nI_1 \\leq -\\frac1c M \\int_{B_{r}}\\int_{B_{r}} \\left|\\log\\frac{w(x)}{w(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y + c\\,M r^{n-sp}.\n\\end{align}\n\nIn the integrand of $I_2$ we can estimate\n\\begin{align*}\n&|w(x)-w(y)|^{p-2}(w(x)-w(y))\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x) \\\\[1ex]\n&\\qquad \\leq c\\,\\big(w(x)^{p-1}+w_-(y)^{p-1}\\big)\\frac{M}{w(x)^{p-1}}\\chi_{B_{3r\/2}}(x) \\\\[1ex]\n&\\qquad \\leq c\\,M\\left(1+\\varepsilon^{1-p}w_-(y)^{p-1} \\right)\\chi_{B_{3r\/2}}(x),\n\\end{align*}\nand integrating yields\n\\begin{align} \\label{log lemma I2}\nI_2 &\\leq c\\,M \\int_{{\\mathds R}^{n}\\setminus B_{2r}} \\int_{B_{3r\/2}}\\left(1+\\varepsilon^{1-p}w_-(y)^{p-1}\\right)K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\leq c\\,M \\int_{{\\mathds R}^{n}\\setminus B_{2r}} \\int_{B_{3r\/2}}\\left(1+\\varepsilon^{1-p}w_-(y)^{p-1}\\right)|y-x_0|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\leq c\\,M r^{n-sp}\\left(1+\\varepsilon^{1-p}\\,{\\rm Tail}(w_-,x_0,r)^{p-1}\\right) \\nonumber \\\\[1ex]\n&= c\\,M r^{n-sp}\\left(1+\\varepsilon^{1-p}\\left(\\frac rR\\right)^{sp}{\\rm Tail}(w_-,x_0,R)^{p-1}\\right)\n\\end{align}\nsince $w_-=0$ in $B_R$.\nFinally, the claim \\eqref{log lemma claim w} follows by combining \\eqref{log lemma I1I2}, \\eqref{log lemma I1} and \\eqref{log lemma I2}.\n\\end{proof}\n\nWe employ the previous lemma for particular truncations of the solutions to the obstacle problem. \n\n\\begin{lemma} \\label{log lemma u}\nSuppose that $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ solves the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. \nLet $B_R \\Subset \\Omega'$, let $B_r \\subset B_{R\/2}$ be concentric balls and let\n\\[\n\\infty > k_+ \\geq \\max\\bigg\\{\\esssup_{B_R} g, \\esssup_{B_R \\cap \\Omega} h\\bigg\\} \\quad \\text{and} \\quad -\\infty < k_- \\leq \\essinf_{B_R} g.\n\\]\nThen the functions\n\\[\nw_\\pm:=\\esssup_{B_R}(u-k_\\pm)_\\pm-(u-k_\\pm)_\\pm+\\varepsilon\n\\]\nsatisfy the following estimate\n\\begin{align} \\label{log lemma claim u}\n&\\int_{B_{r}}\\int_{B_{r}}\\left|\\log\\frac{w_\\pm(x)}{w_\\pm(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\qquad \\leq c\\,r^{n-sp}\\left(1+\\varepsilon^{1-p}\\left(\\frac rR\\right)^{sp}{\\rm Tail}((w_\\pm)_-,x_0,R)^{p-1}\\right)\n\\end{align}\nfor every $\\varepsilon>0$.\n\\end{lemma}\n\\begin{proof}\nLet $\\varepsilon>0$ and denote $H_\\pm := \\esssup_{B_R}(u-k_\\pm)_\\pm+\\varepsilon$. Notice that $H_\\pm$ is finite by Theorem \\ref{thm:boundednessx0}.\nLet $\\phi \\in C^{\\infty}_0(B_{3r\/2})$ be such that $0\\leq\\phi\\leq 1$, $\\phi \\equiv 1$ in $B_{r}$ and $|D\\phi|<c\/r$.\nDenoting by\n\\[\nv_\\pm = u \\mp \\varepsilon^{p-1} \\frac{(u-k_\\pm)_\\pm}{(H_\\pm - (u-k_\\pm)_\\pm)^{p-1}}\\phi^{p}\n= u \\mp \\varepsilon^{p-1} \\frac{H_\\pm-w_\\pm}{w_\\pm^{p-1}}\\phi^{p},\n\\]\nwe clearly have $v_\\pm \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ since, in particular,\n\\[\nv_+ =  u - \\varepsilon^{p-1} \\frac{(u-k_+)_+}{w_+^{p-1}}\\phi^{p} \\geq u - (u-k_+)_+ \\chi_{B_R} \\geq h\n\\]\nalmost everywhere in $\\Omega$ because $k_+ \\geq  \\esssup_{B_R \\cap \\Omega} h$.\n\nSince $u$ solves the obstacle problem, we have $\\langle \\mathcal Au, v_\\pm-u \\rangle \\geq 0$, and thus\n\\begin{align*}\n&\\mp\\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}}L(u(x),u(y))\\left(\\frac{H_\\pm-w_\\pm(x)}{w_\\pm(x)^{p-1}}\\phi^{p}(x)-\\frac{H_\\pm-w_\\pm(y)}{w_\\pm(y)^{p-1}}\\phi^{p}(y)\\right) \\\\\n& \\qquad \\qquad \\ \\ \\times K(x,y)\\,{\\rm d}x{\\rm d}y \\ \\geq \\ 0.\n\\end{align*}\nLet us estimate the integrand above first for $w_+$. If $u(x),u(y)>k_+$, we simply have $-L(u(x),u(y))=L(w_+(x),w_+(y))$, and consequently\n\\begin{align} \\label{-Luxuy<Lwxwy}\n&-L(u(x),u(y))\\left(\\frac{H_+-w_+(x)}{w_+(x)^{p-1}}\\phi^{p}(x)-\\frac{H_+-w_+(y)}{w_+(y)^{p-1}}\\phi^{p}(y)\\right) \\nonumber \\\\[1ex]\n&\\qquad \\leq L(w_+(x),w_+(y))\\left(\\frac{H_+-w_+(x)}{w_+(x)^{p-1}}\\phi^{p}(x)-\\frac{H_+-w_+(y)}{w_+(y)^{p-1}}\\phi^{p}(y)\\right)\n\\end{align}\nholds. If $u(x)>k_+\\geq u(y)$, then $w_+(y)=H_+$ and\n\\[\n-(u(x)-u(y))=-(H_+ -w_+(x) +k_+-u(y)) \\leq w_+(x)-w_+(y),\n\\]\nand \\eqref{-Luxuy<Lwxwy} follows. If, in turn, $u(y)>k_+\\geq u(x)$, we can just exchange the roles of $x$ and $y$ to obtain \\eqref{-Luxuy<Lwxwy}, whereas in the case $u(x),u(y) \\leq k_+$ \\eqref{-Luxuy<Lwxwy} is trivial since $w_+(x)=w_+(y)=H_+$.\nSimilarly, we obtain\n\\begin{align*} \\label{Luxuy<Lwxwy}\n&L(u(x),u(y))\\left(\\frac{H_- -w_-(x)}{w_-(x)^{p-1}}\\phi^{p}(x)-\\frac{H_- -w_-(y)}{w_-(y)^{p-1}}\\phi^{p}(y)\\right) \\nonumber \\\\*[1ex]\n&\\qquad \\leq L(w_-(x),w_-(y))\\left(\\frac{H_- -w_-(x)}{w_-(x)^{p-1}}\\phi^{p}(x)-\\frac{H_- -w_-(y)}{w_-(y)^{p-1}}\\phi^{p}(y)\\right)\n\\end{align*}\nfor $w_-$, and thus\n\\begin{align*}\n& \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}}L(w_\\pm(x),w_\\pm(y))\\left(\\frac{H_\\pm-w_\\pm(x)}{w_\\pm(x)^{p-1}}\\phi^{p}(x)-\\frac{H_\\pm-w_\\pm(y)}{w_\\pm(y)^{p-1}}\\phi^{p}(y)\\right) \\\\\n& \\qquad \\qquad \\ \\ \\times K(x,y)\\,{\\rm d}x{\\rm d}y \\  \\geq\\  0.\n\\end{align*}\nNow the claim \\eqref{log lemma claim u} follows from Lemma \\ref{log lemma w}.\n\\end{proof}\n\n\n\\subsection{H\\\"older continuity up to the boundary}\nBefore starting to prove the H\\\"older continuity up to the boundary, it is worth noticing that we have to assume $g \\in \\mathcal{K}_{g,h}(\\Omega,\\Omega')$\nsince otherwise the solution may be discontinuous on every boundary point.\n\\begin{example}\nSuppose that $sp<1$ and let $\\Omega=B_{1}(0)$ and $\\Omega'=B_{2}(0)$. Then the characteristic function $\\chi_\\Omega$ solves the obstacle\nproblem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ with constant functions $g\\equiv 0$ and $h \\equiv 1$.\nIndeed, $\\chi_\\Omega \\in W^{s,p}(\\Omega')$ when $sp<1$ and it is easy to see that it is a weak supersolution. Consequently, it is the solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ in view of Proposition \\ref{smallest super}.\n\\end{example}\n\nIn the following we will assume that the complement of $\\Omega$ satisfies the following measure density condition.\nThere exist $\\delta_\\Omega \\in (0,1)$ and $r_0>0$ such that for every $x_0 \\in \\partial\\Omega$\n\\begin{equation} \\label{eq:dens cond}\n\\inf_{0<r<r_0} \\frac{| ({\\mathds R}^n \\setminus \\Omega) \\cap B_r(x_0)|}{|B_r(x_0)|} \\geq \\delta_\\Omega.\n\\end{equation}\nAs mentioned in the introduction, this requirement is in the same spirit of the standard Nonlinear Potential Theory, rearranged as an information given only on the complement of the set in accordance with the nonlocality of the involved equations; and this is also an improvement with respect to the previous boundary regularity results in the fractional literature when strong smooth or Lipschitz regularity of the set is required.\n\n\\begin{lemma} \\label{lemma:dens cond}\nLet $\\Omega$ satisfy \\eqref{eq:dens cond} for $r_0>0$ and $\\delta_\\Omega>0$,\nand let $B\\equiv B_r(x_0)$ with $x_0 \\in \\partial\\Omega$ and $r \\in (0,r_0)$.\nSuppose that $f \\in W^{s,p}(B)$ and $f=0$ in $B \\setminus \\Omega$. Then\n\\begin{equation}\n\\mean{B}|f|^{p}\\,{\\rm d}x \\leq c\\left(1-(1-\\delta_\\Omega)^{1-1\/p}\\right)^{-p}r^{sp}\\int_B\\mean{B}\\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince $f=0$ in $B\\setminus\\Omega$,\n\\begin{align*}\n|f_B| &\\leq \\frac{|B\\cap\\Omega|}{|B|}\\mean{B\\cap\\Omega}|f|\\,{\\rm d}x \\leq \\frac{|B\\cap\\Omega|}{|B|}\\left(\\mean{B\\cap\\Omega}|f|^{p}\\,{\\rm d}x\\right)^{1\/p} \\\\[1ex]\n&\\leq\\left(\\frac{|B\\cap\\Omega|}{|B|}\\right)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p}\n\\ =  \\ (1-\\delta_\\Omega)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p},\n\\end{align*}\nand we can estimate\n\\begin{align*}\n\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p} &\\leq \\left(\\mean{B}|f-f_B|^{p}\\,{\\rm d}x\\right)^{1\/p} + |f_B| \\\\[1ex]\n&\\leq \\left(\\mean{B}|f-f_B|^{p}\\,{\\rm d}x\\right)^{1\/p} + (1-\\delta_\\Omega)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p}.\n\\end{align*}\nAbsorbing the last term yields\n\\begin{align*}\n\\mean{B}|f|^{p}\\,{\\rm d}x \\leq \\left(1-(1-\\delta_\\Omega)^{1-1\/p}\\right)^{-p}\\mean{B}|f-f_B|^{p}\\,{\\rm d}x,\n\\end{align*}\nand the claim follows from the fractional Poincar\\'e inequality.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma:osc reduction}\nAssume that $x_0 = 0 \\in \\partial \\Omega$ and $g(0)=0$, where $\\Omega$ satisfies \\eqref{eq:dens cond} for  all $r\\leq R$.  Let $\\omega > 0$. There exist $\\tau_0 \\in (0,1)$, $\\sigma \\in (0,1)$ and $\\theta \\in (0,1)$, all depending only on $n$, $p$, $s$ and $\\delta_\\Omega$, such that\nif\n\\begin{equation} \\label{oscB}\n\\osc_{B_R(0)}u + \\sigma{\\rm Tail}(u;0,R) \\leq \\omega \\quad \\text{and} \\quad \\osc_{B_R(0)}g \\leq \\frac\\omega 8\n\\end{equation}\nhold, then the decay estimate \n\\begin{equation} \\label{osctauB}\n\\osc_{B_{\\tau R}(0)}u + \\sigma{\\rm Tail}(u;0,\\tau R) \\leq (1-\\theta)\\omega\n\\end{equation}\nholds as well for every $\\tau \\in (0,\\tau_0]$.\n\\end{lemma}\n\\begin{proof}\nDenote $H=\\theta\/\\sigma$ and $B\\equiv B_R(0)$. We begin by estimating the tail term to obtain\n\\begin{align*}\n\\sigma^{p-1}{\\rm Tail}(u;0,\\tau R)^{p-1} &= \\sigma^{p-1}(\\tau R)^{sp}\\int_{B \\setminus \\tau B}\\frac{|u(x)|^{p-1}}{|x|^{n+sp}}\\,{\\rm d}x \\\\\n& \\quad  + \\sigma^{p-1}\\tau^{sp}{\\rm Tail}(u;0,R)^{p-1} \\\\[1ex]\n&\\leq c\\,\\sigma^{p-1}\\omega^{p-1}+\\tau^{sp}\\omega^{p-1}\n\\end{align*}\nby \\eqref{oscB}. Consequently,\n\\begin{equation} \\label{eq:Tail bndr osc 000}  \n\\sigma{\\rm Tail}(u;0,\\tau R) \\leq \\tilde c \\left(\\frac\\theta H + \\tau^{sp\/(p-1)}\\right)\\omega\n \\leq \\frac{2\\tilde c\\,\\theta} H \\omega = \\theta\\omega\n\\end{equation}\nwhen restricting $\\tau_0 \\leq \\sigma^{(p-1)\/(sp)}$ and choosing $H = 2\\tilde c \\geq 1$, where $\\tilde c \\equiv \\tilde c(n,p,s)$.\nThus, it suffices to prove that\n\\begin{equation} \\label{osctauB2}\n\\osc_{\\tau B}u \\leq (1-2\\theta)\\omega\n\\end{equation}\nfor all $\\tau \\leq \\tau_0$. To this end, let\n\\[\nk_+ := \\sup_{B} u- \\frac{\\omega}{4}, \\quad k_- := \\inf_{B}u + \\frac{\\omega}{4}  , \\quad \\varepsilon:=\\theta\\omega\n\\]\nand\n\\[\nw_\\pm:=\\sup_{B}(u-k_\\pm)_\\pm-(u-k_\\pm)_\\pm+\\varepsilon, \\quad \\tilde w_\\pm:=\\frac{w_\\pm}{\\sup_{B} w_\\pm}.\n\\] \n\nWe may assume $\\sup_{B}u \\geq \\frac38\\omega$ or $\\inf_{B}u \\leq -\\frac38\\omega$ since otherwise\n$\\osc_{\\tau B}u \\leq \\osc_{B}u \\leq \\frac34\\omega$ and there is nothing to prove if we assume that $\\theta \\leq 1\/8$.\nWe consider the case $\\sup_{B}u \\geq \\frac38\\omega$; the case $\\inf_{B}u \\leq -\\frac38\\omega$ is symmetric.\nNotice that we have $\\tilde w_+ = 1$ in $B\\setminus \\Omega$ due to the condition $u=g \\leq \\omega\/8$ in $B\\setminus \\Omega$. First, we estimate, by Lemmas \\ref{lemma:dens cond} and \\ref{log lemma u} with $r\\equiv 2\\tau R$ and \\eqref{oscB} when restricting $\\tau_0 \\leq 1\/4$ and $\\tau_0 \\leq \\sigma^{2(p-1)\/(sp)}$, to obtain \n\\begin{align*}\n\\mean{2\\tau B}\\left|\\log \\tilde w_+\\right|^{p}\\,{\\rm d}x &\\leq c\\,(\\tau R)^{sp}\\int_{2\\tau B}\\mean{2\\tau B}\\left|\\log\\frac{\\tilde w_+(x)}{\\tilde w_+(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\leq c\\,(\\tau R)^{sp}\\int_{2\\tau B}\\mean{2\\tau B}\\left|\\log\\frac{w_+(x)}{w_+(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\leq c\\,\\Big(1+(\\theta\\omega)^{1-p}\\tau^{sp}{\\rm Tail}((w_+)_-;0,R)^{p-1}\\Big) \\\\[1ex]\n&\\leq c\\left(1+(\\theta\\omega)^{1-p}\\sigma^{2(p-1)}{\\rm Tail}(u;0,R)^{p-1}\\right) \\\\[1ex]\n&\\leq c\\left(1+(\\theta\\omega)^{1-p}\\left(\\frac\\theta H\\right)^{p-1}\\omega^{p-1}\\right) \\\\[1ex]\n&\\leq c.\n\\end{align*}\nConsequently, by Chebyshev's Inequality we have\n\\begin{align} \\label{logtildew>M} \\nonumber\n\\frac{\\left|2\\tau B \\cap \\{|\\log \\tilde w_+| \\geq  \\left|\\log(20\\,\\theta)\\right| \\}\\right|}{|2\\tau B|} &\\leq  |\\log (20\\,\\theta)|^{-p}\\mean{2\\tau B}|\\log\\tilde w_+|^{p}\\,{\\rm d}x \n\\\\*[1ex]\n &\\leq c\\,|\\log (20\\,\\theta)|^{-p}.\n\\end{align}\n\nLet us estimate the left-hand side of \\eqref{logtildew>M}. Since, by definitions, $0 < \\tilde w_+ \\leq 1$ and $\\sup_{B} (u-k_+)_+ = \\omega\/4$, we have that\n\\begin{align*}\n\\left\\{|\\log \\tilde w_+| \\geq|\\log (20\\,\\theta)| \\right\\} &= \\left\\{\\tilde w_+ \\leq 20\\,\\theta\\right\\} \n\\\\*[1ex]\n & = \\Big\\{\\frac{\\omega}{4} + \\varepsilon - (u-k_+)_+ \\leq 20\\,\\theta \\Big(\\frac{\\omega}{4} + \\varepsilon\\Big) \\Big\\}\n\\\\*[1ex]\n & = \\Big\\{\\frac{\\omega}{4} + \\theta\\omega - u + \\sup_B u - \\frac{\\omega}{4} \\leq 5 \\theta \\omega + 20\\,\\theta^2 \\omega  \\Big\\}\n\\\\*[1ex]\n & \\supset \\Big\\{u  \\geq \\sup_B u - 4 \\theta \\omega \\Big\\}\n\\end{align*}\nprovided that $\\theta < 1\/20$. Consequently, by defining $\\tilde k \\equiv \\tilde k_+ := \\sup_B u - 4 \\theta \\omega$ and using the above two displays, we get\n\\begin{align*}\n\\left( \\mean{2\\tau B} (u - \\tilde k )_+^p \\,{\\rm d}x \\right)^{1\/p} & \\leq 4 \\theta \\omega  \\left( \\frac{ |2\\tau B \\cap \\{ u  \\geq \\sup_B u - 4 \\theta \\omega  \\}| }{|2\\tau B| }\\right)^{1\/p} \\\\[1ex]\n&\\leq 4 \\theta \\omega \\left( \\frac{ |2\\tau B \\cap \\{ |\\log \\tilde w_+|  \\geq|\\log (20\\,\\theta)|  \\}| }{|2\\tau B| }\\right)^{1\/p} \\\\[1ex]\n&\\leq \\frac{c\\,\\theta\\omega}{|\\log(20\\,\\theta)|}.\n\\end{align*}\n\nSince $\\tilde k \\geq \\sup_B g$, we have by Theorem \\ref{thm:boundednessx0} that\n\\begin{equation*} \n\\sup_{\\tau B} (u - \\tilde k )_+ \\leq \n\\delta \\, {\\rm Tail}((u - \\tilde k )_+ ; 0, \\tau R)+ c\\,\\delta^{-\\gamma} \\left( \\mean{2\\tau B} (u - \\tilde k )_+^p \\,{\\rm d}x \\right)^{1\/p}\n\\end{equation*}\nfor any $\\delta \\in (0,1]$, and hence\n\\begin{equation} \\label{suputauB0}\n\\sup_{\\tau B} u  \\leq \\sup_B u - 4 \\theta\\omega + \\delta \\, {\\rm Tail}((u - \\tilde k )_+ ; 0, \\tau R) + \\frac{c\\,\\delta^{-\\gamma }}{|\\log(20\\,\\theta)|} \\theta\\omega.\n\\end{equation}\nTo estimate the tail term, we proceed similarly as in~\\eqref{eq:Tail bndr osc 000} and obtain\n\\begin{align*}\n{\\rm Tail}((u - \\tilde k )_+; 0,\\tau R)^{p-1}  &\\leq (\\tau R)^{sp} \\int_{B \\setminus \\tau B} (u(x) - \\tilde k )_+^{p-1} |x|^{-n-sp}\\,{\\rm d}x \\\\\n&\\quad + \\tau^{sp} {\\rm Tail}(u;0,R)^{p-1} \\\\[1ex]\n&\\leq c\\,(\\theta\\omega)^{p-1} \\left( 1 + \\frac{\\tau^{sp}}{\\theta^{p-1}\\sigma^{p-1}}  \\right) \n \\\\*[1ex]\n  &\\leq c\\,(\\theta\\omega)^{p-1},\n\\end{align*}\nwhere we also used the facts $(u-\\tilde k)_+ \\leq 4\\theta\\omega$ in $B$, ${\\rm Tail}(u;0,R) \\leq \\omega\/\\sigma$ by \\eqref{oscB}, and  $\\tau^{sp} \\leq \\tau_0^{sp} \\leq \\theta^{p-1}\\sigma^{p-1}$. Thus, by choosing first $\\delta$ small and then $\\theta$ small accordingly, we deduce from \\eqref{suputauB0} that \n\\[\n\\sup_{\\tau B} u  \\leq \\sup_B u - 2 \\theta\\omega,\n\\]\nand \\eqref{osctauB2} follows, as desired. This finishes the proof.\n\\end{proof}\n \nNow, we have finally collected all the machinery to plainly deduce the H\\\"older continuity up the boundary. We have the following\n\\begin{theorem} \\label{thm:H cont bdry}\nSuppose that $u$ solves the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and assume $x_0\\in \\partial\\Omega$ and $B_{2R}(x_0) \\subset \\Omega'$.\nIf $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is H\\\"older continuous in $B_R(x_0)$ and $\\Omega$ satisfies \\eqref{eq:dens cond} for all $r\\leq R$, then $u$ is H\\\"older continuous in $B_R(x_0)$ as well.\n\\end{theorem}\n\\begin{proof}\nWe may assume $x_0=0$ and $g(0)=0$.\nMoreover, we may choose $R_0$ such that $\\osc_{B_0}g \\leq \\osc_{B_0}u$ for $B_0 \\equiv B_{R_0}(0)$ since otherwise we have nothing to prove, and define\n\\begin{equation} \\label{omega0}\n\\omega_0 := 8\\left(\\osc_{B_0}u + {\\rm Tail}(u;0,R_0)\\right).\n\\end{equation}\nBy Lemma \\ref{lemma:osc reduction} there exist $\\tau_0$, $\\sigma$ and $\\theta$ depending only on $n$, $p$, $s$ and $\\delta_\\Omega$ such that if\n\\begin{equation} \\label{oscB0}\n\\osc_{B_r(0)}u + \\sigma{\\rm Tail}(u;0,r) \\leq \\omega \\quad \\text{and} \\quad \\osc_{B_r(0)}g \\leq \\frac\\omega 8\n\\end{equation}\nhold for a ball $B_r(0)$ and for $\\omega>0$, then\n\\begin{equation} \\label{osctauB0}\n\\osc_{B_{\\tau r}(0)}u + \\sigma{\\rm Tail}(u;0,\\tau r) \\leq (1-\\theta)\\omega\n\\end{equation}\nholds for every $\\tau \\in (0,\\tau_0]$.  As we can take $\\tau \\leq \\tau_0$ such that\n\\begin{equation} \\label{osctauBg}\n\\osc_{\\tau^{j}B_0}g \\leq (1-\\theta)^{j}\\frac{\\omega_0}{8} \\qquad \\text{for every } j=0,1,\\dots.\n\\end{equation}\nNow, iterating \\eqref{osctauB0} with \\eqref{oscB0} and \\eqref{osctauBg} noticing also that the initial condition is satisfied by \\eqref{omega0}, we obtain\n\\begin{equation*} \\label{osctauBu}\n\\osc_{\\tau^{j}B_0}u \\leq (1-\\theta)^{j}\\omega_0 \\qquad \\text{for every } j=0,1,\\dots.\n\\end{equation*}\nConsequently, $u \\in C^{0,\\alpha}(B_0)$ with the exponent $\\alpha=\\log(1-\\theta)\/\\log \\tau \\in (0,1)$.\n\\end{proof}\n\nSlightly modifying the proof above, we easily obtain the following.\n\\begin{theorem} \\label{thm:cont bdry}\nSuppose that $u$ solves the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and assume $x_0\\in \\partial\\Omega$ and $B_{2R}(x_0) \\subset \\Omega'$.\nIf $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is continuous in $B_R(x_0)$ and $\\Omega$ satisfies \\eqref{eq:dens cond} for all $r\\leq R$, then $u$ is continuous in $B_R(x_0)$ as well.\n\\end{theorem}\n\nFor the sake of completeness, we gather our continuity results into two global theorems. The first one follows by combining Theorems \\ref{thm:obs H cont} and \\ref{thm:H cont bdry} and the second one by combining Theorems \\ref{thm:obs cont} and \\ref{thm:cont bdry}.\n\\begin{theorem} \\label{thm:H cont up to bdry}\nSuppose that $\\Omega$ satisfies \\eqref{eq:dens cond}  \nand $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let $u$ solve the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. \nIf $g$ is locally H\\\"older continuous in $\\Omega'$ and $h$ is locally H\\\"older continuous in $\\Omega$, then $u$ is locally H\\\"older continuous in $\\Omega'$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:cont up to bdry}\nSuppose that $\\Omega$ satisfies \\eqref{eq:dens cond}  \nand $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let $u$ solve the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$.\nIf $g$ is continuous in $\\Omega'$ and $h$ is continuous in $\\Omega$, then $u$ is continuous in $\\Omega'$.\n\\end{theorem}\n\n\n\n\\smallskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHelioseismology has probed the interior of the Sun over the last three decades. Combining the information provided by \nseveral hundred pressure-driven modes (p modes), it has been possible to put  constraints on our knowledge of the \nstructure and the dynamics of the solar interior \\citep{JCD2002,ThoJCD2003}. Unfortunately, only a small fraction of \nthese modes reaches the solar core. Moreover, due to the increase in  sound-speed velocity with depth, these modes \ngive little information on the deeper layers as they spend less time there than in the convection zone. Let us take as an \nexample the internal rotation rate of the Sun: the rotation profile is very well known in the convective zone \n\\citep{ThoTOO1996,1998ApJ...505..390S,HowJCD2000,2000ApJ...541..442A}, while the uncertainties grow in the radiative \nzone and towards the core of the Sun \\citep{1994ApJ...435..874J,ElsHow1995,CouGar2003,ChaSek2004,GarCor2004}. In order \nto put new constraints inside the solar core other kinds of modes are needed: the gravity (g) modes. For example, by \nmeasuring just a few of such modes, information on the core rotation rate can undoubtably be obtained \n\\citep{2008A&A...484..517M}, whereas   the deepest layers that could be probed using only p-modes are around \n0.2 $R_{\\odot}$ \\citep{2008SoPh..tmp...43G}.\n \nGravity (g) modes are buoyancy-driven modes that have the advantage of propagating inside the entire radiative \nregion. However, these waves become evanescent in the convective zone and reach the solar surface with tiny \namplitudes preventing us to detect them easily (see for example \\citet{Belkacem08} and references therein). Indeed, \nseveral claims for g-mode detections have been made in the past \\citep{1983Natur.306..651D,1988IAUS..123...79P,1995Natur.376..139T}; \nhowever, modern and better data sets cannot confirm them. \n\nIn 1995 was launched the Solar and Heliospheric Observatory (SoHO),  one of whose scientific objectives \nwas the detection and characterization of gravity modes \\citep{DomFle1995}. Recently, using data from the Global \nOscillation and Low Frequency (GOLF) instrument \\citep{GabGre1995}, the signature of the asymptotic properties of \n  $\\ell$=1 dipole g modes has been measured with a high confidence level \\citep{2007Sci...316.1591G,2008AN....329..476G}. \nThis signal was also found \\citep{2006ESASP.624E..23G} using photometric data from the Variability of solar \nIRradiance and Gravity Oscillations (VIRGO) experiment \\citep{1995SoPh..162..101F}. Even if certain constraints \ncan be imposed on the structure \\citep{2008SoPh..tmp...55G} and dynamics \\citep{2007Sci...316.1591G} of the \nsolar core thanks to the study of these asymptotic properties, it is extremely important to detect individual \ng modes.  After 10 years of observations, the level of noise at 200 $\\mu$Hz has been established at \n$\\sim$ 4.5 mm\/s when individual peaks are looked for and at 1.5 mm\/s when this research is done for multiplets \n\\citep{2006soho...18E..22E}. Indeed some peaks and patterns could be identified as potential gravity modes or\n mixed modes above  the  noise level \\citep{GabBau2002,STC04,2007ApJ...668..594M} but it has been impossible \n to  tag them unambiguously with the correct $\\ell$, $m$ and $n$.\n\nIn this paper we analyse a peak around 220.7 $\\mu$Hz that has been studied several times as part of a g-mode \ncandidate using different instruments on board SoHO (see for example \\citet{Gab99,Fin01,STC04,2007ApJ...668..594M}) \nbut also from the theoretical side (see for example \\citet{CoxGuz2004} and references therein). To do so, we \nstart in section 2 with a brief description of the helioseismic instruments used in this work and we analyse in \ndetail the data of the VIRGO\/SPM instruments (section 3). In section 4, we look for an instrumental origin\n for this peak without success by analysing all the housekeeping parameters of the VIRGO package as well as \n the SoHO pointing. Once it is stablished that this peak seems to have a solar origin we check for its\n  presence \n in all the other instruments of the VIRGO package (section 5) and in the velocity instruments GOLF, \n MDI and GONG (section 6). We then finish by discussing its possible nature.\n\n\\section{Instrumentation and data analysis}\n\nThe data of VIRGO (Variability of IRradiance and Gravity Oscillation), GOLF (Gravity Oscillation at Low \nFrequencies) and MDI (Michelson Doppler Imager) on board SoHO (Solar and Heliospheric Observatory) satellite have \nbeen used in this research together with the GONG (Global Oscillation Network Group) ground-based netwok.\n \n\\subsection{SoHO\/VIRGO}\n\nThe VIRGO package was designed to study the characteristics of pressure and internal gravity modes by \nobserving irradiance and radiance variations, to measure the solar total and spectral irradiance and to quantify \ntheir variability (Fr\\\"{o}hlich et al.\\ 1995, 1997). It is composed of three different types of sensors:\n\\begin{itemize}\n\\item Two types of absolute radiometers (one VIRGO\/DIARAD and two VIRGO\/PMO6-V) for the measurements of solar \ntotal irradiance and its variations with high accuracy and precision. The cadences of VIRGO\/DIARAD and VIRGO\/PMO6-V are 180s and 60s respectively.\n\\item Two 3-channel sunphotometers (SPM), one  permanently exposed to sun lght and another for backup, set at at 402 nm (blue), 500 nm (green), \nand 862 nm (red), looking at the Sun as a star with a 60 s cadence. \nThe bandwidth of the filters is 5~nm.\n\\item One Luminosity Oscillation Imager (VIRGO\/LOI) for the measurements of the radiance in 12 pixels over the \nsolar disc. The filter is at 500 nm with a bandwidth of 5 nm.The cadence is 60s.\n\\end{itemize}\n\nIn  1998 June, {\\itshape SoHO} was lost\n for several months, but, after a search campaign, was\nfinally found and resumed  operations around  1998 October. The VIRGO data after  {\\itshape SoHO}'s \n``vacations'' show  the same high quality as before the temporary loss of the probe.\n\n\\subsection{SoHO\/GOLF, SoHO\/SOI\/MDI and GONG}\n\n\\begin{itemize}\n\\item\nSoHO\/GOLF is a resonance scattering spectrophotometer (Gabriel et al.\\ 1995, 1997)\n that measures the line-of-sight velocity using the sodium\ndoublet, similar to the IRIS and BiSON ground-based networks. The\nGOLF window was  opened in 1996 January and became fully\noperative by the end of that month. Over the following months,\noccasional malfunctions in its rotating polarizing elements were\nnoticed that led to the decision to stop them in a predetermined\nposition; truly non-stop observations began by 1996 mid-April.\nSince then, GOLF has been continuously and satisfactorily operating\nin a  mode unforeseen before launch, showing fewer\nlimitations than anticipated. The signal, then, consists of two close\nmonochromatic photometric measurements in a very narrow band (25\nm\\AA) on a single wing of the sodium doublet.\nThis signal has been calibrated into velocity \\citep{UlrGar2000,garcia05} and  is indeed  similar in nature\nto other known velocity measurements, such as those of IRIS and BiSON \n(Pall\\' e et al.\\ 1999). The sampling of the\nGOLF data used in this paper is 60 s. Before {\\itshape SoHO}'s vacations (1998 June),\nGOLF data were obtained in the blue wing of the sodium line; thus, after the {\\itshape \nSoHO}\nvacations the GOLF team decided change to the red wing of the sodium line (see Garc\\'\\i a et al. 2005, for the latest report on the GOLF instrument).\n\n\\item The Solar Oscillations Investigation (SOI) uses a Michelson Doppler Imager (MDI) type of instrument \n\\citep{1995SoPh..162..129S}. MDI  consists of a pair of tunable Michelson interferometers,\nwhich image the Sun onto a 1024 pixel x 1024 pixel\nCCD camera in five wavelengths across the Ni i 676.8 nm line. These resolved data can be processed by\nforming a weighted combination of the pixel signals to yield a\nproxy for a Sun-as-a-star response (see \\cite{Henney99}).\n\n\\item  The ground-based Global Oscillation Network Group (GONG;\nHarvey et al. 1996) consists of six sites, with instruments that use\nthe Fourier tachometer approach to observe the Doppler shift, in\nthe Ni i line, with 1024 pixel resolution. Here, a\nSun-as-a-star proxy was formed from a simple integration over\nall pixels with a cadence of 60s.\n\\end{itemize}\n\n\\section{The 220.7 $\\mu$Hz peak seen in VIRGO\/SPM}\nA long time series of 4098 days of VIRGO\/SPM data has been used in this work starting on 1996 April 11.  \nAs our purpose is to study the time evolution of signals at low frequency, a total of five independent subseries \nof 800 days have been computed. Data were available to allow a 50 day shift up to June 2007. These 66 \noverlapped series were used only for plots while non-overlapping data were used as input for the statistical tests we have carried on. \n\nThe slow trends in the time series, due to the degradation in the instruments and long-term solar variability,\n have been removed by applying a running mean filter of one day. To check whether this filter could affect the detected \n signal, we have also used a backwards difference filter, in which every measured point is substituted by the \n difference of two consecutive points  $\\delta f_{n}=f_{n+1}-f_{n}$. To recover the correct amplitudes in the \n power spectrum, this latter should be divided by the transfer function of the filter $Q(\\nu)$, defined as follows \\citep{GarBal08}: \n\\begin{equation}\nQ(\\nu)=[2sin(\\pi\\nu\\Delta t)]^2\n\\end{equation}\n $\\nu$ being the frequency and $\\Delta t$ the sampling of the data.\n\nBoth filters gave the same results and, in the rest of this paper, we work only with the time series filtered by the 1-day running mean.\n\nWe therefore computed several power density spectra using a Fast Fourier Transform (FFT) algorithm and\n built the time-evolution power diagrams used in this work. Each of them has been computed from the \n time series which have been extended by four equal time intervals of zero signal . This oversampling makes it easier to detect the bins in which the \n power is concentrated \\citep{GabBau2002}. We  also verified that a sine wave fit (SWF), computed \n in steps of 0.0001 $\\mu$Hz between 220.5 and 221 $muHz$, yields the same results. Therefore, we have used the \n normal zero-padded FFT as it is much faster than the SWF. \n\nThe time-evolution power diagrams are built as follows: The 66 power spectra of the overlapped time series are \ncomputed and plotted vertically using  a colour scale for the power. The vertical axis is the frequency of the \npower spectra with the colour equivalent to the power as indicated on the right-hand side of the diagram and the \nhorizontal axis the number of the time series from 0 to 65, i.e.\\  the time span corresponding to the \ntime series. Looking these time-evolution power diagrams we  know  at which frequency and for how long \n a signal can have enough power to be observed above the noise level.\n   \n In figure~\\ref{spms} the time-evolution of the three VIRGO\/SPM channels are shown for the frequency range\n  220.5--221.0 $\\mu$Hz.  The x-axis spans 11 years of the SoHO mission.  A clear signal is observed \n  in the blue channel (top) at around 220.7 $\\mu$Hz, which is stable in time with power that goes from 6--7  to \n  16--17 $ppm^2\/\\mu$Hz. Around time series 60 this signal seems to  change its frequency slightly by around 0.3 \n  $\\mu$Hz. In the green and red channels the same continuous signal is visible as in the blue one but with \n  the expected decrease in power with wavelength. It is also important to note that in all these VIRGO\/SPM channels \n  a second high-amplitude signal is visible at $\\sim$ 220.64 $\\mu$Hz, parallel to the previous one, from time \n  series 20 until the last one but with a small gap between time series 44 and 48.\n\n \n \n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=13pc,angle =90]{f1.eps} \\\\\n  \t\\includegraphics[width=13pc,angle =90]{f2.eps} \\\\\n\t\\includegraphics[width=13pc,angle =90]{f3.eps} \\\\\n\\end{tabular}\n\\caption{\\label{spms}VIRGO\/SPM time-evolution power diagram of channels blue, green and red (top to bottom \nrespectively) from April 1996 to June 2007.  A stable signal at  220.7 $\\mu$Hz  is clearly observed. }\n\\end{figure} \n\n\n \n\\subsection{Confidence levels and Monte-Carlo simulations}\n\nIn the previous section we saw that there is a persistent signal around the target frequency of\n 220.7 $\\mu$Hz. Indeed, the VIRGO\/SPM blue channel power density spectrum of the full length time series \n (see Figure~\\ref{full}) shows the presence of a peak at a precise frequency of 220.667 $\\mu$Hz above the \n 90$\\%$ confidence level computed in a   10 $\\mu$Hz window following \\cite{App00}.\n\n\\begin{figure}[!htb]\n\\includegraphics[width=0.36\\textwidth,angle =90]{f4.eps}\n\\caption{\\label{full}VIRGO\/SPM blue channel power spectrum density computed with 4098-day time \nseries starting on 1996 April 11. The horizontal dotted line corresponds to the 90\\% confidence level \nthat a peak above this line would not be due to noise.  }\n\\end{figure} \n\nUsing subseries of 800 days and a frequency window of 10 $\\mu$Hz, the power level above which an observed\n peak has a 90\\% probability not due to noise is 8.87$\\sigma$ (e.g. see \\cite{App00}). In the case of \n zero-padded data, the points are no longer independent and  are correlated. Therefore, Monte-Carlo \n simulations should be used to derive a correction for the above-mentioned confidence level.  In our case, \n for a padding factor of 5 we have added a correction of $\\mathrm{ln}(2.8)=1.03$ derived by \\cite{GabBau2002} \n to the threshold computed using non-zero-padded data. In this conditions the 90\\% confidence level at around \n 9.9$\\sigma$. Using the VIRGO\/SPM blue channel, we found that the peak we are studying has a maximum power in a\n  range between 8.74 and 10.4$\\sigma$ considering only five independent realizations of 800 days. In Figure~\\ref{spms} \n  the 90\\% limit is obtained at around 14 $ppm^2\/\\mu$Hz (orange colour in Figure~\\ref{spms}). This means that, for example,\n   most subseries between the 28th and the 58th have the peak above the 90\\% confidence level, as well as other subseries \n   such as those at the very beginning of the time-span. It is important to notice that in this case the $\\sigma$ has \n   been averaged over the 66 time series and the value of 14 $ppm^2\/\\mu$Hz is an averaged magnitude. \n\nWe are interested in knowing  the probability of having a signal with the same properties to those that we have found in \nthe VIRGO\/SPM blue channel; i.e.\\ a peak that is above the 90$\\%$ level in the full power density spectrum of more \nthan 4098 days, and that is also present in the five independent subseries of 800 days with similar levels to what we have with \nthis instrument (i.e.\\ not necessary all above a 90\\% confidence level in these individual small subseries but around that \nlevel). This latest condition would be much more restrictive because  it means that the peak should maintain a certain \ncoherence during the full time-span. A Monte Carlo simulation of 1 million iterations has been done by simulating Gaussian \nnoise time series of 4000 days that have been cut into five intervals of 800 days. To speed up the procedure we have not computed \nthe full spectrum of the 4000 days but only the average of the power density spectrum of the five independent realizations of 800 \ndays (which have an SNR of $\\sim$ 9.2 $\\sigma$ in the VIRGO\/SPM blue channel). Thus, the algorithm looks first for a signal \nin the average spectrum with $\\sim$0.9 times the level found in the VIRGO\/SPM blue channel (8.3$\\sigma$) and, if it is \nfound, it looks for the presence of that signal in the five subseries (again with levels of 0.9 times those of VIRGO).\n Any signal with these properties found in the 10 $\\mu$Hz window will be flagged as a positive identification. The results \n show that, in a window of 10 $\\mu$Hz, the 220.7 $\\mu$Hz signal has a likelihood of 99.8$\\%$ (which is reduced to 91.3\\% if \n we only consider the constraint on the averaged spectrum). We have also checked how the likelihood is degraded when a bigger \n window is considered. Thus, for the 20 and 30 $\\mu$Hz windows we obtain  99.6 and 99.4\\% respectively.\n\nWe can conclude that it is extremely difficult to find a pure noise signal above the 90\\% confidence level after $\\sim$4000 days \nand with a coherence with time as  found in the VIRGO\/SPM instruments.\n\n \\section{Possible instrumental origin of the signal inside VIRGO and SoHO}\n \nOnce this interesting signal has been detected in VIRGO\/SPM the main question is to investigate its origin; in other words, determine \nif it is of solar or instrumental origin. In this section, we study all the possible non-solar \norigins of this signal, from the orbital and pointing corrections of the spacecraft to the housekeeping parameters \n(hereafter HK) of the VIRGO package. \n\nPeriodic manoeuvring of the SoHO probe at this frequency (220.7 $\\mu$Hz, i.e. a period around 1.25 hours) due to orbital \nadjustments or pointing corrections could modulate the signal of the instruments on board as a tracking system can produce guided frequencies.\nOn the other hand, a temperature variation at this frequency could also modulate the observed signal. These temperature \nvariations could originate in the sensor itself or in other instrument subsystems.\n\nFor all these parameters we follow the same analysis we as for the VIRGO data; thus, we build the corresponding \ntime-evolution power diagrams and we compare them with the VIRGO\/SPM ones. If the 220.7 $\\mu$Hz signal is produced by the \ntemporal variations of some of these parameters, the time-evolution power diagrams of both VIRGO\/SPM and the parameter must \nbe highly correlated.\n\nFor this purpose we analyse in the following subsections orbital and pointing corrections of the SoHO spacecraft and the different \nHK parameters of the VIRGO package that might modulate the signal. It is important to note that some of the HK data \nare in the scientific telemetry of VIRGO and have a cadence of 60s, while others are in the HK telemetry and have a cadence of 180s. \n\n\n\\subsection {Orbital corrections}\n\nThe radial distance is reduced to 1AU by the usual quadratic law $S_{0}=S \\cdot r^2$,  $r$ being the spacecraft-to-Sun distance in \nastronomical units. This correction normalizes the spectral irradiance to the solar constant definition and removes signal modulations \ndue to movements of the Earth, Moon and planets in their orbits.\n\nThe observed radiation $S$ of a moving blackbody source is \n\\begin{equation}\nS=S_{0}  \\frac{(1-v)^2}{(1-v^2)} \n\\end{equation}\nwhere $S_{0}$ is the radiation in motionless conditions and\n v is the speed in units of the light speed, c. With SoHO velocity being a few $10^{-6}$ of the speed-of-light one can safely omit terms in \n $v^2$ and thus approximate the reciprocal formula\n\\begin{equation}\nS=S_{0} \\frac{(1-v)^2}{(1-v^2)}\\sim \\frac{S_{0}}{(1-2v)} \\sim S_{0} (1+2*v)\n\\end{equation}\nThis Doppler correction removes a tiny ($10^{-5}$), slow (Halo orbit period is 6 months) modulation of the measured irradiance.\n\nIn this way, the orbital correction applied to the three channels of VIRGO\/SPM is:\n\\begin{equation}\nSPM_{channel}=SPM_{channel} \\cdot radius^2 \\cdot (1+2 \\cdot vel)\n\\end{equation}\nwhere ``radius\" is the spacecraft-to-Sun distance in astronomical units and ``vel\" is the radial velocity in units of the speed-of-light.\n\n\\begin{figure}[!htb]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=13pc,angle =90]{f5.eps}\n\\end{tabular}}\n\\caption{\\label{orbit}Time-evolution power diagram of the orbital corrections applied to the VIRGO data. In addition to the order of\n magnitude which is 6 orders of magnitude smaller than VIRGO\/SPM no correlation is found.}\n\\end{figure} \n\nThe orbital parameters (radius and vel) are provided by NASA in a 10 minute  cadence and  are linearly interpolated to get the \nsame 60s as VIRGO\/SPM. The time series of the orbital correction applied, i.e.\\ $radius^2(1+2\\cdot vel)$, has been analysed in the same \nway as VIRGO\/SPM and the resulting time-evolution power diagram diagram is shown in Figure~\\ref{orbit}.  The orbital correction \nsignal is around 6 orders of magnitude smaller than the VIRGO\/SPM one and no correlation has been found.\n\n\n\\subsection{Spacecraft Pointing}\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f6.eps} \\\\\n  \t\\includegraphics[width=12pc,angle =90]{f7.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f8.eps} \\\\\n\\end{tabular}\n\\caption{\\label{pointing}Power time-evolution diagrams of the pitch, yaw and roll angles (respectively from top to bottom) \nof the SoHO spacecraft. Only data up to September 2002 (time series 32) are available. No correlation with the first 32 time series of VIRGO\/SPM is found.}\n\\end{figure} \n\nThe three critical flight dynamics parameters are rotations in three dimensions around the vehicle's coordinate-system origin, the \ncentre of mass. These angles are pitch, roll and yaw. Pitch is the rotation around the lateral or transverse axis. Therefore, \nmovements of the spacecraft to the north or south of the Sun. Yaw is the rotation about the vertical axis; thus, movements of \nthe spacecraft to the west or east of the Sun and, finally, roll is a rotation around the longitudinal axis i.e.\\ movements of the \nspacecraft from the north or south to the west or east of the Sun.\n\nFor an instrument that looks at the Sun as a star (integrated light) the most plain pointing correction would be divided \nby $cos(\\sqrt(yaw^2 + pitch^2)$, i.e.\\ the cosine of the angle between instrument optical axis and the line-of-sight\ndirection. Nevertheless, this correction was never applied to VIRGO\/SPM because the correction would have been negligible. \nAlso, in December 2001, NASA  discontinued the CDHF (Central Data Handling Facility), which was the facility in charge of processing, \nproducing and distributing the SoHO telemetry and the ancillary data products. The production of all these data were continued in\n others ways but the production of attitude data was stopped in September 2002. Indeed, when SoHO is in normal mode, the attitude\n  follows nominal attitude well enough for most purposes, and because the roll determination had large errors (because of certain procedural problems).\n\nEven knowing that, it would be very unlikely that the pointing maneuvres could  modulate any signal in the SoHO instruments; the three \nangles have been analysed and their time-evolution diagrams computed (see Figure~\\ref{pointing}).\n \nThe available attitude data concerning pitch, yaw and roll angles were obtained from the NASA archive from 1996 April 11 to 2002 September\n 22 and we built the time series of the three angles. These data have a cadence of 10 minutes and have been used with this sampling rate \n because it is good enough for our purposes. The length of these time series enables us to get 32 time series of 800 days (each shifted 50 days \n with respect to the previous one). This is approximately half of the time-evolution power diagrams used in the VIRGO\/SPM. This length is \n sufficient to see if any correlation exists between pointing and SPM signals during the common period (around 6 years).\n\nThe pitch angle has a very constant value of around -3.3 arcmin during the time span, with spikes of 5 arcmin and only a few of them \nwith higher values, between 6 and 13 arcmin. These latter  are probably due to spacecraft maneuvres. The associated time-evolution \ndiagram is shown in Figure~\\ref{pointing} ({\\it top}). Some power density has been found at a level of $10^{-5}  (arcmin)^2$\/$\\mu$Hz with\n no visible correlation with the VIRGO\/SPM signal.\n\nThe yaw angle is zero during practically the whole  time-span considered with some spikes around 1.7 arcmin and only a few between 6.8 to \n12 arcmin. This yields  a pure noise time-evolution diagram (see Figure~\\ref{pointing} ({\\it medium})) with a power density of around $10^{-6}  \n(arcmin)^2$\/$\\mu$Hz with also no visible correlation with the VIRGO\/SPM signal.\n\nFinally, as we have already said, the roll angle does not affect the data achieved by instruments that observe the Sun as a star (integrated \nlight) but, in any case, it has also been analysed. The roll angle changes following the Earth orbit between 7.16 and -7.16 degrees with some \nlarge spikes that have been removed (the roll angle sometimes has large errors) and the time-evolution diagram is shown in \nFigure~\\ref{pointing} ({\\it bottom}). The power density is around $1 (arcmin)^2$\/$\\mu$Hz and, once again, no correlation with the measurements \nof VIRGO\/SPM has been found.\n\n\\begin{figure*}[!htb]\n\\centering\n\\begin{tabular}{cc}\n\t\\includegraphics[width=12pc,angle =90]{f1a.eps} \n\t\\includegraphics[width=12pc,angle =90]{f9.eps} \\\\\n  \t\\includegraphics[width=12pc,angle =90]{f10.eps} \n\t\\includegraphics[width=12pc,angle =90]{f11.eps}\\\\ \n\t\\includegraphics[width=12pc,angle =90]{f12.eps} \n  \t\\includegraphics[width=12pc,angle =90]{f13.eps}\\\\ \n\n\n  \n\n\n  \n\\end{tabular}\n\\caption{\\label{Temps}Time-evolution power diagram of the HK temperatures of the VIRGO\/SPM package. To simplify the \ncomparison we have repeated, on the same scale, the time-evolution diagram of the VIRGO\/SPM Blue channel.}\n\\end{figure*} \n\n\\subsection{VIRGO\/SPM temperatures}\n\n\\begin {itemize}\n\\item{VIRGO\/SPM sensor temperatures}\n\nThe most important VIRGO\/SPM temperature is the temperature sensor. Each of the three VIRGO\/SPM channels (blue, green and red) are \ncorrected by a quantity proportional to each of the temperature sensors (sensor blue, green and red). This correction is applied in \nthe level 1 software, so the data we are handling are already multiplied by this quantity.  This correction is:\n \\begin{equation}\nSPM_{channel}=(1+C_{channel}(TS_{channel}-293.15))\n\\end{equation}\n where ``channel'' means blue, green or red; $C_{channel}$ is a  constant for each channel and $TS_{channel}$ is the temperature of each of the three sensors.\n\nIn Figure~\\ref{Temps} the time-evolution power diagram  of the temperature sensor of the blue channel is shown (top of the right column) . \nThe fluctuation of this temperature is two orders of magnitudes smaller that the VIRGO\/SPM signals at frequencies around 220$\\mu$Hz and no \nclear correlation  with VIRGO\/SPM signals is visible.  \n\n\n\n\\item{VIRGO\/SPM electronic temperature}\n\nThe temperature of the SPM electronics has been also analysed to see if there exists some modulation that could produce a periodic variation \nin the output voltage of the low-noise electrometer amplifiers (or in the input current). If this exists, a modulation would go to the \nVoltage Frequency Converters (VFC) of the Data Acquisition System (DAS) and could produce a modulation in the output signal.  \n\nThe time-evolution power diagram for the VIRGO\/SPM electronic temperature is shown in Figure~\\ref{Temps} (middle of the left column). \nThe power density is of the same order as in the VIRGO\/SPM channels but no correlation with the signal at 220.7 $\\mu$Hz has been found.\n\n\\item{Data Acquisition System (DAS) temperature}\n\nThe Data Acquisition System (DAS) of VIRGO comprises the Onboard Data Handling System (interface for telemetry, telecommands and timing \nsignals), multiplexers, Voltage Frequency Converters (VFC) and counters. If the DAS temperature, the VFC or the counters have a periodic \nbehaviour, the output number of counts could contain that periodicity. The DAS temperature time-evolution diagram is shown in Figure~\\ref{Temps} \n(middle of right column). The power density is an order of magnitude higher than the SPM signal  but again there is no correlation with the signal at 220.7 $\\mu$Hz .\n\n\\item{VIRGO\/SPM Heatsink and DC\/DC temperatures}\n \nThe temperature variations of the VIRGO Heatsink and the VIRGO Power Supply (DC\/DC) have been also analysed for security. The Heatsink \ntime-evolution diagram (Figure~\\ref{Temps})(bottom of the left column) is ten times smaller than the VIRGO\/SPM and that corresponding \nto the DC\/DC is ten times larger (Figure~\\ref{Temps} ,bottom of the right column). In both cases no correlation is found with the 220.7 $\\mu$Hz signal.\n\n\n\\begin{figure*}[!hptb]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=35pc,angle =0]{f14.eps}\n\\end{tabular}}\n\\caption{\\label{zoom} Zoom of the time-evolution power diagrams of the VIRGO\/SPM Blue and the VIRGO housekeeping analysed in this research \nfor the time series 35 to 55 where the 220.7 $\\mu$Hz has higher amplitudes. This zoom helps to clarify the darker parts of some time-evolution \ndiagrams produced by the colour scales. None of these temperatures can explain the observed signal at ~220.7 $\\mu$Hz .}\n\\end{figure*} \n\nFinally, if a signal is the result of a certain temperature modulation, the temperature variation would be higher just where the power of \nthe signal is higher. In Figure~\\ref{zoom} the SPM\/Blue and VIRGO HK time-evolution power diagrams are plotted together but only between time \nseries 35 to 55, in which the power of the  220.7 $\\mu$Hz signal is stronger in the VIRGO\/SPM data. This zoom helps us to see the darker parts of \nsome HK time-evolution diagrams produced by the colour scales. None of the temperatures analysed in this section can explain the observed signal \nat 220.7 $\\mu$Hz.\n\n\n\\end{itemize}\n\n\\section{The 220.7 $\\mu$Hz signal in the others VIRGO instruments}\n\nAs  was mentioned in section 2.1 the VIRGO package comprises the SPM Sunphotometers and also two types of absolute radiometers (VIRGO\/DIARAD \nand VIRGO\/PMO6-V) and one Luminosity Oscillation Imager (VIRGO\/LOI). In this section we study the 220.7 $\\mu$Hz signal in these instruments.\n\n\\begin {itemize}\n\\item{Luminosity Oscillation Imager (VIRGO\/LOI)}\n\nVIRGO\/LOI measures  the radiance in 12 pixels over the solar disc. We convert these 12 pixels into one by simply adding all of them. From \nthe raw time series the same analysis as in VIRGO\/SPM has been carried out.  Figure~\\ref{loi} ({\\it Top}) shows its time-evolution diagram. \nIt looks similar to the VIRGO\/SPM and with the same visible signal at 220.7 $\\mu$Hz. In the VIRGO\/LOI observations, the signal is weaker \nthan in VIRGO\/SPM but with the same characteristics, for example, at time series 60 the signal slightly changes its frequency. However, \nwith this instrument, the peak seems to be like a doublet instead of only one signal concentrated in a couple of bins.\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f15.eps} \\\\\n  \t\\includegraphics[width=12pc,angle =90]{f16.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f17.eps} \\\\\n\\end{tabular}\n\\caption{\\label{loi}Time-evolution power diagram of the Luminosity Oscillation Imager (VIRGO\/LOI),  the VIRGO\/DIARAD absolute radiometer \nand the VIRGO\/PMO6-V absolute radiometer, respectively from top to bottom. All these instruments are part of the VIRGO package.}\n\\end{figure} \n\n\\item{Absolute radiometers (VIRGO\/DIARAD and VIRGO\/PMO6-V)}\n\nAbsolute radiometers use a quite different technique from that of VIRGO\/SPM and VIRGO\/LOI, which are silicon detectors measuring the \nspectral irradiance and the radiance respectively. Absolute radiometers are based on the measurements of the heat flux by using an \nelectrically calibrated heat flux transducer to measure the total solar irradiance (solar constant). Once again, from the raw time \nseries we have performed the same analysis. Figure~\\ref{loi} ({\\it medium and bottom}) shows the results for VIRGO\/DIARAD and VIRGO\/PMO6-V. \nTime-evolution power diagrams are similar to the previous ones, although the 220.7 $\\mu$Hz is weaker in both radiometers but the 220.7 $\\mu$Hz signal is still present.\n\n\\end{itemize}\n\n\\section{Analysis using Doppler velocity instrumentation}\n\nUp to now we have not found any instrumental origin for the 220.7 $\\mu$Hz signal observed in all the VIRGO package. We can now study this \nregion in other helioseismic instruments. We will start by analysing the signal of the other two instruments on board SoHO and we finish by\n using the GONG ground-based network.\n\n\\begin{itemize}\n\\item{GOLF} is the other Sun-as-a-star instrument on board SoHO. We have analysed the velocity time series following the same procedure \nemployed in the VIRGO analysis and we have computed the time-evolution power diagram shown in Figure \\ref{Doppler} {\\it (top)}. As  \n mentioned in the introduction, the 220.7 $\\mu$Hz signal was first observed by GOLF during the first years of the mission and it was flagged \nas a ``g-mode'' candidate by \\cite{STC04} and, after 4182 days, it is still visible as part of a quadruplet above a 90\\% confidence level \n\\citep{2008AN....329..476G}. Figure~\\ref{Doppler} {\\it (Top)} shows that the evolution with time of the signal, although weaker than in VIRGO, \nis still there. The signal in GOLF has an interval between time series 12 and 19, where it disappears, and it corresponds to the place  where \nthe signal in VIRGO\/SPM is the weakest (see figure~\\ref{spms}). Therefore, we can conclude that the 220.7 $\\mu$Hz is also observed in velocity \nmeasurements using GOLF data but with a smaller signal-to-noise ratio (SNR).\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f18.eps} \\\\\n  \t\\includegraphics[width=12pc,angle =90]{f19.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f20.eps} \\\\\n\\end{tabular}\n\\caption{\\label{Doppler}Time-evolution power diagram of the GOLF, MDI and GONG instruments respectively from top to bottom.}\n\\end{figure} \n\n\\item{}Disc-averaged MDI velocity signals from the calibrated level-1.4 MDI LOI-proxy Doppler images were obtained using integrated spatially \nweighted masks following \\cite{Henney99}. These time series from 1996 May 25 till  2007 October 28 have been analysed and the time-evolution \ndiagram plotted in Figure~\\ref{Doppler} {\\it (medium)}. There are no fingerprints of the presence of the 220.7 $\\mu$Hz signal in this data set. \nThis could be due to a lower SNR in the MDI LOI-proxy as compared to GOLF. Indeed, \\cite{HenneyUlrich99} showed that for the lowest measurable \np-modes, the GOLF instrument has a higher SNR than this particular MDI mask. To go further, we need to apply our methodology to specifically \ndesigned g-mode masks such as those derived by \\cite{Watcher}.\n\n\\item{}The radial velocity of GONG used in this work started on 1995 May 7 and finished on 2006 March 9. These series are shifted by a year compared to the \nSoHO data but they were the longest we could use. The disc-integrated data provided by the GONG Team are very noisy at low frequency and were not \nsuited for our studies. Therefore, we have decided to use the $\\ell$=2 spherical-harmonic series. These are optimized for acoustic modes of this \ndegree and they have the advantage of having a much stabler behaviour at low frequency. We preferred these series to the $\\ell$=1 mode because the \nclosest theoretical frequency to the target frequency of 220.7 $\\mu$Hz corresponds to an $\\ell$=2 g mode. In Figure~\\ref{Doppler} {\\it(bottom)} \nthe time-evolution diagram of GONG is shown. Although the SNR at this frequencies in the GONG data is very low, it seems to be a trace of the \n220.7 $\\mu$Hz signal in the GONG data, especially between series 36 and 52, which corresponds to a maximum in the GOLF between series 43 and 60 \n(7 subseries shift, i.e.\\ $\\sim$ 350 days). However, looking only at this time-evolution diagram, it is impossible to disentangle the feature at \n220.7 $\\mu$Hz from others visible in the analysed region. Nevertheless, It would be extremely important to be able to confirm the visibility of \nsuch a peak using ground-based data because that would directly mean that the nature of this 220.7 $\\mu$Hz signal has a solar origin.\n\\end{itemize}\n\nTo compare the averaged behavior of the 220.7 $\\mu$Hz signal in the GOLF, GONG and VIRGO\/SPM data sets we  computed the collapsograms of the \ntime-evolution power diagram, i.e.\\ to average the 66 power spectra used to produce the time-evolution power diagrams. The resultant graphs \nare plotted in Figure~\\ref{Colapso}. A similar structure appears around  the target frequency of  220.7 $\\mu$Hz,   this peak being the highest in \nthe three instruments. However, in the case of GONG, it is at noise level. \n\n\\begin{figure}[!hbt]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=18pc,angle =0]{f21.eps}\n\\end{tabular}}\n\\caption{\\label{Colapso} Collapsograms of the time-evolution power diagrams of VIRGO\/blue, GOLF and GONG. The 220.7  $\\mu$Hz structure is\n present in the three different instruments although in GONG it is at noise level.}\n\\end{figure} \n\n\n\n\n\\section{Conclusions} \nIn the present paper we have studied a peak that appears around the frequency of 220.7 $\\mu$Hz in the VIRGO\/SPM data. This peak has a more \nthan 90$\\%$ confidence level  of not being due to noise in the full spectrum of 4098 days. A detailed study of its nature revealed that this \npeak existed since the very begining of the mission in a continuous way for the last 11 years and only at the very end of the  \ntime series considered does it seem to  change slightly in frequency. By Monte Carlo simulations we have computed the confidence level of such kinds of behaviour\n and we found that it is really unlikely (more than 99$\\%$) that it is due to a noise with the same statistical characteristics as the convective\n  noise. Therefore, we  checked all the available housekeeping data from the VIRGO package as well as a detailed analysis of the SoHO \n  spacecraft attitude control, looking for an instrumental origin. None of these studies was able to explain the presence of a peak in the  \n  region studied. Indeed, this study seems to rule out this possibility. The  origin should therefore be solar. We  then studied Doppler velocity data \n  from another instrument on board SoHO, GOLF, and we  found that the peak is also present (with lower SNR). Even though  analysis of data from the GONG \n  ground-based network revealed a very noisy spectrum, the highest peak in a 10 $\n\\mu$Hz region around the 220.7 $\\mu$Hz signal is precisely that peak. However, it is not significant enough for us to claim that we have a positive \ndetection using this instrument.\n\nThe present study has proved the solar origin of the peak at 220.7 $\\mu$Hz. Two solar phenomena could be responsible of such a peak. The first could be \n convection, in particular  granulation motions. However, it is very unlikely that a turbulent displacement of plasma on the solar surface \nwith a typical time scale of 10 minutes give a stable frequency during more than 10 years in the power spectrum of the disc-integrated data. On \nthe other hand, gravity modes propagate inside the radiative region of the Sun and  are expected to have long lifetimes (at least longer \nthan the period of measurements). Thus, the properties of the peak that we found are similar to those expected for a g mode. Using the principle of Ockham's \nrazor  (or the {\\it Lex Parsimoniae} principle) in which the explanation of any phenomenon should make as few assumptions as possible, we \ncan conclude that if this peak is not noise it should be a component of a g mode.  Analysing in detail the structure of this possible g-mode \ncomponent, Figure~2 reveals a peak structure containing several bins. Indeed, Figure~1 might also show the presence of a parallel component at \naround 220.64 $\\mu$Hz with high amplitudes in several of the  series considered. Thus, a possible explanation of such behaviour might be the presence \nof an inner magnetic field that could slightly split the component of the g-mode multiplet in some peaks. Another possibility might be that the \ng-mode power could be spread into several bins as a consequence of a smaller  than expected lifetime or due to a change in the size of the resonant \ncavity (for example due to a displacement of the position of the tachocline during the activity cycle). This latter effect is particularly interesting \nbecause it seems that the 220.7 signal follows a small change in frequency over the entire time-span with the lowest frequency (220.68 $\\mu$Hz) \nreached around time series 35---corresponding to the maximum of the activity cycle--- and then increasing the frequency again towards the two \nperiods with minimum activity (at the beginning and the end of the series). In any case,  assuming a faster rotation in the core than in the \nrest of the radiative envelope (as suggested by Garc\\'\\i a et al. 2007), the 220.7 $\\mu$Hz peak could be or a component of the $\\ell$=2, n=-3 g\n mode, or a component of the $\\ell$=3, n=-5 or a bitting between this latter and the $\\ell$=5, n=-8. Whatever the true answer is, there is still \n an important question to be answered: why is this particular peak so excited when there are no other visible g-mode components? More work will be \n necessary before solving the solar g-mode puzzle. \n\n\n\\begin{acknowledgements}\nThe authors want to thank the members of the PHOEBUS group present at the first ISSI (International Space Science Institute) meetings for \ntheir useful comments and discussions. This work has been partially supported by the Spanish grant PNAyA2007-62651 and the CNES\/GOLF grant at the SAp\/CEA-Saclay.\nThe authors also thank all their colleagues (scientists, engineers and technicians) involved with the GOLF, VIRGO and MDI instruments\n aboard SoHO which is a space mission of international cooperation between ESA and NASA. This work utilizes data obtained by the Global \n Oscillation Network Group (GONG) program, managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative \n agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude \n Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrof\\'\\i sica de Canarias, and Cerro Tololo Interamerican Observatory. One of the authors (AJ) would like to thank M.Ortiz for invaluable help with the last part of this article. \n \n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.37]{pl-overview.pdf}\n    \\caption{\n        An example of how parallel news documents can be used to train a model that is capable of making educated guesses on what the question is asking, and how it may help to derive a better answer.\n       \n    }\n    \\label{fig:overview}\n\\end{figure}\n\nAnswering questions often involves making educated guesses: we do not necessarily have accurate facts but can use common sense to understand what most questions are asking and what kinds of knowledge are needed. For example (see Fig.~\\ref{fig:overview}), we can understand the question ``\\emph{Is Albany, GA more crowded than Albany, NY?}'' involves comparing the size and population of two cities without knowing the specific numbers to compare.\nIt is often desirable to make such a decomposition because a city's population is usually much easier to acquire than a direct answer to the original question. \n\nExisting approaches to end-to-end question-answering (QA) assume that pre-trained language models (LMs) are capable of both robust question understanding of this type and acquiring the relevant facts. Much recent evidence, however, has revealed limitations in the commonsense and compositional reasoning abilities of current transformers \\cite{zhou2019going,liu2021challenges}, in part due to \\emph{reporting biases} (e.g., relating the semantics of ``more crowded'' and ``overpopulation'' can be difficult given that such contexts rarely co-occur in single document on which models are pre-trained) and other \\emph{dataset artifacts} \\cite{gururangan2018annotation}. This is even more evident in recent datasets with complex questions that are designed to require decomposition. For example, GPT-3 \\cite{Brown2020LanguageMA}, a language model with 175 billion parameters, only achieves mid-60s accuracy on StrategyQA \\cite{geva2021did}, a binary QA benchmark with a random baseline at around 50. Moreover, such datasets are often small in size and scope, which makes it difficult to overcome knowledge gaps in LMs through fine-tuning and developing general-purpose decomposition models. \n\n\n\nIn this paper,\\footnote{\\url{http:\/\/cogcomp.org\/page\/publication_view\/992}} we attempt to bridge the gap of reporting biases, which hinders LMs from learning implicit connections between questions and decompositions (e.g., ``crowded'' and ``population''). We do this through intermediate pre-training on distant supervision, following recent attempts to distill common sense into transformers via distant supervision \\cite{zhou2021temporal}.\nSpecifically, we use collections of article pairs with parallel descriptions of similar news events from different angles as our distant supervision. As illustrated in Fig.~\\ref{fig:overview}, large collections of comparable texts (\\S\\ref{sec:distant-supervision-intuitions}) contain a wide variety of commonsense implications needed for decomposition. We extract 2.6 million sentence pairs (\\S\\ref{sec:pl-extraction}) for this purpose, and then train \\mbox{\\textsc{DecompT5}}{} (\\S\\ref{sec:pretrain}), a T5 \\cite{Raffel2020ExploringTL} model that is further-pre-trained on our distant supervision instances. In \\S\\ref{sec:intrinsic-experiments}, we show that \\mbox{\\textsc{DecompT5}}{}, while simple, serves as a more effective model than the base language model on general question understanding through experiments on Overnight \\cite{Wang2015BuildingAS} and TORQUE \\cite{Ning2020TORQUEAR} semantic parsing tasks, achieving 22-32\\% absolute improvements. \n\nSince smaller language models cannot sufficiently memorize facts (e.g., the exact population of Albany), they are often used in conjunction with external knowledge retrieval for more complicated tasks such as QA. To bridge this gap, we design a novel QA pipeline using \\mbox{\\textsc{DecompT5}}{} at its core (\\S\\ref{sec:e2e-qa-system}). The full model and pipeline, called \\mbox{\\textsc{DecompEntail}}{}, first generates explicit question decompositions, then makes factual corrections on the decomposed statements with GPT-3. As a final step, \\mbox{\\textsc{DecompEntail}}{} employs an entailment model that derives the final answer with the generated decomposition as the premise and the question and candidate answer as the hypothesis. \n\nIn \\S\\ref{sec:e2e-experiments}, we show that \\mbox{\\textsc{DecompEntail}}{}, despite its relatively small size, can generate good decomposition chains and outperforms GPT-3 on both StrategyQA and a binary portion of HotpotQA by 4\\% and 8\\%, respectively.\nThis shows that we can improve baseline language models or even much larger reasoners with explicit decomposition, which has the advantage of enhanced interpretability and transferability. On the other hand, \\mbox{\\textsc{DecompT5}}{} only relies on supporting fact annotations instead of explicit reasoning steps, which is more common in datasets and can be better applied for joint learning.\n\n\\noindent \\textbf{Contributions.} In summary, our contributions are three-fold: 1) we collect distant supervision from parallel news to encourage robust semantic understanding for question decomposition, 2) we train a general decomposition model called \\mbox{\\textsc{DecompT5}}{} with our collected distant supervision that significantly improves over the baseline language models on intrinsic evaluations, and 3) we propose a decomposition-based QA pipeline called \\mbox{\\textsc{DecompEntail}}{} that relies on \\mbox{\\textsc{DecompT5}}{} at its core. We show that \\mbox{\\textsc{DecompEntail}}{} has improved performance over several baselines on decomposition-based QA.\n\n\\section{Related Work}\n\nOur work relates to the literature on multi-hop reasoning \\cite{yang2018hotpotqa}, which has recently produced new annotation schemes (e.g., \\emph{QDMR} from \\citet{wolfson2020break} and \\emph{strategy question decomposition} annotations from \\citet{geva2021did}) and datasets for complex reasoning that target explicit model decomposition \\cite{talmor2018web,wolfson2020break,geva2021did,khot2022hey}. We take inspiration from systems that build explicit reasoning paths, such as semantic parsers \\cite{Liang2011LearningDC, Berant2013SemanticPO}, and their modern variations \\cite{Andreas2016NeuralMN,Gupta2020NeuralMN,khot2020text}. \n\\citet{Min2019MultihopRC,Perez2020UnsupervisedQD} aim to build general question decomposition models, however, focusing on simpler tasks than our study.  \n\n\nOur work is also related to sentence-pair datasets collected from comparable texts \\cite{Fader2013ParaphraseDrivenLF, Zhang2019PAWSPA, Reimers2019SentenceBERTSE}. Compared to most of these works, our extraction does not use human annotation, and produces clean and diverse signals for question understanding.\n\nPrevious work has also discussed using large-scale further pre-training to improve language models \\cite{Zhou2020TemporalCS, zhou2021temporal, zhao2021effective}. We follow a similar general scheme with novel extraction sources and focus on a general representation for questions, which resembles some idea in existing work \\cite{Khashabi2020UnifiedQACF}.\n\n\n\n\n\n\\section{Distant Supervision for Decomposition}\nIn \\S\\ref{sec:distant-supervision-intuitions}, we describe our intuitions on why question decomposition is important and what is missing from existing pre-trained language models for them to do well. Following that, we describe how we collect distant supervision signals to improve the process of learning to decompose in \\S\\ref{sec:pl-extraction}. In \\S\\ref{sec:pretrain}, we propose \\mbox{\\textsc{DecompT5}}{}, a T5-based model that is further pre-trained on the collected distant supervision using standard seq-to-seq training objectives.\n\n\\subsection{Intuitions}\n\\label{sec:distant-supervision-intuitions}\n\\noindent \\textbf{Educated Guesses in QA.} We, as humans, need to answer questions all the time, but we may not possess all the facts. For example, an ordinary person may not know the exact populations of Albany to answer ``Is Albany, GA more crowded than Albany, NY'', or the density of corgis to answer `Will a corgi float on water''. However, that person may search for ``population'' or ``density'' instead of the original question to find the answer because we know that it is much easier to find the ``population of a city'' than to find an answer to the original question. The human capacity for guessing what the question is asking and how that question can be decomposed to simpler concepts by associating \\textit{crowded} with \\textit{population}, and \\textit{float} with \\textit{density} is crucial for solving day-to-day tasks. However, making such connections can be very challenging for pre-trained language models because of reporting biases. Written texts rarely make such connections explicit in single documents, as most authors expect readers to make many trivial inferences.\n\n\\noindent \\textbf{Parallel News.} In this work, we aim to bridge this decomposition gap in pre-trained language models through incidental supervision \\cite{Roth2017IncidentalSM} from comparable corpora \\cite{Klementiev2006WeaklySN}. We find news articles reporting the same news event but from different authors and angles. Related sentences in such parallel news often complement each other and provide new information. This complementary information is often more sophisticated and diverse than paraphrasing, because it contains implications and causal relations. Fig.~\\ref{fig:overview} shows an example of how a pair of articles describing Tokyo from slightly different angles may help decompose the running example question. One article mentions that Tokyo is crowded, while the other expresses similar points but focuses on area size and population descriptions. Intuitively, a model may benefit from such connections to learn that ``crowded'' is related to ``size'' and ``count''. It is rare, however, for a single document to contain both aspects, causing difficulties for LMs that primarily learn from single documents.\n\n\\subsection{Parallel News Extraction}\n\\label{sec:pl-extraction}\nWe use the RealNews corpus \\cite{Zellers2019DefendingAN} as the source corpus because it contains cleaned, date-marked new articles from diverse domains. We aim to select news article pairs that describe the same main event and find sentence pairs within these document pairs that are likely to contain complementary information to each other.\n\n\\noindent \\textbf{Filter Article Pairs.} We select article pairs within a 2-day window of publication because the same news events are typically covered within a relatively short period. We then employ a pre-trained entailment model from SentenceBert \\cite{Reimers2019SentenceBERTSE} to check the titles of each article pair and retain those pairs whose titles have a cosine similarity greater than $0.8$.\n\n\\noindent \\textbf{Find Sentence Pairs.} We then find sentence pairs across each selected article pairs that are related and complementary to each other. To do this, we run the same sentence similarity model and retain all sentence pairs with a similarity score between $0.6$ and $0.9$. The lower bound is to make sure the sentences are approximately related. Even though $0.6$ is considerably a loose bound for many tasks (e.g., paraphrasing), it is suitable in our case because we have a strong assumption that the articles are closely related because of date and title similarities. This lower bound is sufficient to guarantee that the vast majority of sentence pairs above this threshold contain complementary information to each other. For example, the similarity score between ``The US Military has already started withdrawal from Syria'' and ``The US is only moving non-essential equipment out of Syria, because precipitous withdrawal would shatter US policy in Syria and allow IS to rebuild'' is only $0.6$. However, the second sentence provides non-paraphrasing but complementary information to the first sentence. A model may learn that troops in other countries are linked with foreign policy, which is the type of information that is often implicit in single documents. The upper-bound $0.9$ is to filter out sentence pairs that are too similar or simply paraphrasing each other, as these pairs do not provide much additional information to facilitate question understanding. \n\n\\noindent \\textbf{Filtering with tf-idf.} We employ an additional filtering process based on sentence topics to keep the final dataset's diversity. To do this, we calculate the inverse document frequency (idf) of each word in the vocabulary based on Wikipedia and multiply that with the term frequency (tf) of each word within the sentence pairs. Next, we use the top three words ranked by td-idf scores of each sentence pair as the ``signature'' and randomly keep ten sentence pairs with identical signatures at most. 2.6 million sentence pairs remain after this step. Finally, we format the data as a standard seq-to-seq training task, where the input sentence is one of the sentences in the pair, while the model is trained to generate the other sentence in the pair. The order is randomly decided. \n\n\\noindent \\textbf{Data for Language Modeling Objective.} Beyond the sentence pairs, we also inject some data from Project Gutenberg\\footnote{\\url{https:\/\/www.gutenberg.org\/}} and format it to the language model pre-training format (e.g., the denoising objective for T5 \\cite{Raffel2020ExploringTL}). We sample around 900K sentences for this purpose.\n\n\\subsection{Comparisons with Similar Data Sources}\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lccccc}\n\\toprule\nMetric \/ Data & Ours & P-auto & P-inc. & NLI & QA \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\nLength $\\uparrow$ & 52 & 42 & 20 & 31 & 40\\\\ \nLength-diff $\\uparrow$ & 9 & 1 & 2 & 10 & 20\\\\\nEmbed-sim $\\downarrow$ & 0.7 & 1.0 & 0.9 & 0.6 & 0.6\\\\\nSem-sim $\\downarrow$ & 0.7 & 1.0 & 0.9 & 0.7 & 0.7\\\\\nCost $\\downarrow$ & low & low & mid & high & high \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Comparisons between our data and other sources for reasoning tasks. \\textit{P-auto} is paraphrasing data from automatic (distant) collection, \\textit{P-inc.} is paraphrasing data from incidental supervision. \\textit{Sem-sim} is semantic similarity. $\\uparrow$\/$\\downarrow$ marks the direction for each metric to present a more diverse data source.}\n\\label{tab:naive-comparison}\n\\end{table}\nWe compare our data collected in \\S\\ref{sec:pl-extraction} with other sources that may similarly be used, including paraphrasing, textual entailment (NLI), and question-answering (QA). Paraphrasing data can be collected either automatically (e.g., PAWS \\cite{Zhang2019PAWSPA}), or from incidental but human-involved processes (e.g., Quora duplicated questions\\footnote{\\url{https:\/\/quoradata.quora.com\/}}). We use these two datasets to represent each category respectively. In addition, we use the MNLI dataset \\cite{Williams2018ABC} for NLI, and StrategyQA (question+answer\/supporting-facts) for QA. We randomly sample 10k sentence pairs from each source. We compare basic statistics, including sentence pair length and the length difference between the two sentences. We also compare sentence similarity via averaged word embeddings \\cite{Pennington2014GloVeGV} and sentence-level semantic embeddings \\cite{Reimers2019SentenceBERTSE}.\\footnote{We use the ``average\\_word\\_embeddings\\_glove.840B.300d'' and ``all-MiniLM-L6-v2'' models, respectively.}\nTable~\\ref{tab:naive-comparison} shows that our data source provides richer and more diverse information while not requiring any human annotation. This observation aligns with our intuitions in \\S\\ref{sec:distant-supervision-intuitions}.\n\n\\subsection{Pre-training with Distant Supervision}\n\\label{sec:pretrain}\n\nWe use T5-large \\cite{Raffel2020ExploringTL} as our base language model due to its sequence-to-sequence architecture and relatively small parameter size (containing 770m parameters) for easier pre-training. We train the base language model on our distant supervision dataset for one epoch and call the resulting model \\mbox{\\textsc{DecompT5}}{}. We expect, however, that this pre-training technique with our collected dataset is beneficial to most existing pre-trained language models, as it bridges the reporting bias gap in general language modeling objectives.\n\n\\section{Decomposition-based QA Pipeline}\n\\label{sec:e2e-qa-system}\n\n\nOur proposed model \\mbox{\\textsc{DecompT5}}{} has two uses: it can be \\textbf{directly fine-tuned} on tasks that require query understanding and decomposition, as we later show in \\S\\ref{sec:intrinsic-experiments}. It can also be applied in a pipeline fashion to \\textbf{produce meaningful decompositions} that help with more complicated tasks that require external knowledge, such as general question answering. This section focuses on the design challenges and choices for such a QA pipeline. We first explain the intuitions in \\S\\ref{sec:pipeline-intuition}, then describe and propose \\mbox{\\textsc{DecompEntail}}{} in \\S\\ref{sec:pipeline}. We evaluate our proposed QA pipeline in \\S\\ref{sec:e2e-experiments}.\n\n\\subsection{Intuitions and Design Choices}\n\\label{sec:pipeline-intuition}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.37]{stqa-example.pdf}\n    \\caption{\n        An example StrategyQA \\citep{geva2021did} instance that includes a question annotated with decomposed questions and their corresponding facts.\n    }\n    \\label{fig:stqa-example}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcc}\n\\toprule\nAdditional Information & \\#Train & Accuracy \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\nNone & 2061 & 53.3 \\\\\nAspects & 2061 & 59.7 \\\\\nFacts (Impossible) & n\/a & n\/a \\\\\nAspects, Facts & 2061 & 84.5 \\\\\nAspects, Facts, Indirect & 15296 & \\textbf{90.2} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{T5-3B accuracy on StrategyQA dev set when different information is provided for both training and evaluation. \\textit{Aspects} refers to the knowledge dimensions (without values) that are involved with each question. \\textit{Facts} refers to the actual knowledge involved, which is not possible to acquire without knowing the corresponding aspects. \\textit{Indirect} contains additional supervision of paraphrasing and entailment. Details are in \\S\\ref{sec:sanity_check}.}\n\\label{tab:sanity_check}\n\\end{table}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.5]{pipeline-overview.pdf}\n    \\caption{\n        An overview of our proposed \\mbox{\\textsc{DecompEntail}}{} pipeline. The final decomposition is an actual output from the pipeline. See more examples in Fig.~\\ref{fig:manual-analysis-examples}. \n       \n    }\n    \\label{fig:pipeline-overview}\n\\end{figure*}\n\nAs we argue in \\S\\ref{sec:distant-supervision-intuitions}, an agent can decompose complex questions into simpler and more controlled forms by linking a question to all relevant \\textit{aspects} of that question (e.g., the relevant sub-queries related to the input question). With such aspects or components, the agent can make easier knowledge retrieval to acquire the specific values of the aspects, which we call relevant \\textit{facts}. As shown in Fig.~\\ref{fig:stqa-example}, StrategyQA provides two kinds of supporting annotations for each question. The decomposed questions do not contain the answers and thus approximate the \\textit{aspects} of each question. The annotated facts answers the sub-questions with accurate values, so they approximate \\textit{aspects+facts}. \n\nIn \\S\\ref{sec:sanity_check}, we conduct an experiment for sanity checking purposes, with results shown in Table~\\ref{tab:sanity_check}. We see that T5 does not improve much when given only the \\textit{aspects} (+6\\%) but gains much more (31\\%-37\\%) when provided with \\textit{aspects+facts} and additional indirect supervision. When given \\textit{aspects+facts}, the model is in effect doing textual entailment. The 90.2\\% accuracy shows that this entailment part of deciding how to use the facts is a much smaller bottleneck than finding the proper aspects and their values. At the same time, relatively small LMs such as T5 do not gain much from only seeing the \\textit{aspects} because of their poor memorization (e.g., even if the model knows that the population of a city is needed, it cannot produce the correct number without external resources). This observation serves as the motivation for building a binary QA pipeline that first generates accurate \\textit{aspects+facts} (\\textbf{decompose}) and then decides the final answer with an entailment model  (\\textbf{entail}).\n\nThe \\textbf{decompose} step can be approached in two ways: i) generating the \\textit{aspects} first, then perform information retrieval (IR) and compose a new statement for \\textit{aspects+facts}; ii) generating \\textit{aspects+facts} directly, then perform some factual correction because small LMs cannot memorize well. We choose the second approach for the following three reasons. 1) Our basis \\mbox{\\textsc{DecompT5}}{} is trained on parallel news, which are natural language statements that approximate the \\textit{aspects+facts} together (see Fig.~\\ref{fig:overview}). 2) Generating \\textit{aspects+facts} together allows the model to adhere to its beliefs and generate self-consistent logic chains because decomposition may be inter-dependent (e.g., in Fig.~\\ref{fig:pipeline-overview}, the country that Cyril represents plays an important role in the next generation step). 3) Supporting facts are a much more common type of annotation (e.g., in HotpotQA) than \\textit{aspects}-only annotations, which allows us to explore transfer and joint learning with other existing datasets.\n\n\\subsection{Factual Correction for Generated Facts}\n\\label{sec:factual-correction}\n\nIn order for generating \\textit{aspects+facts} to work, we need to correct any factual errors in the generated facts. This is crucial because relatively small LMs such as T5 cannot generate accurate facts, and wrong information will hinder the performance of the entailment model when deciding the final answer. Standard information retrieval (IR) approaches aim to find a specific piece of text from a knowledge base \\cite{Karpukhin2020DensePR} and tailor the correct information in the retrieved text to specific needs. However, this will not work well in our scenario because doing IR on \\textit{aspects} and incorrect \\textit{facts} will lead to much noise. Moreover, certain commonsense information, such as the weight of a six-year-old, are often missing from standard IR resources such as Wikipedia. \n\nTo this end, we propose to use large-scale language models such as GPT-3 \\cite{Brown2020LanguageMA} directly as a fact-checker, as we have found that GPT-3 does reasonably well on memorizing and retrieving the majority of well-known facts. Furthermore, when given appropriate prompts, GPT-3 simultaneously performs retrieval and new statement synthesis, allowing us to inspect the reasoning capability of our decomposition model directly and more efficiently. Therefore, we design a prompt that starts with \\textit{``Fix the input sentence with correct facts if there are factual errors''} followed by six examples listed in Appendix~\\ref{sec:appendix-prompts}. \n\nWe emphasize that GPT-3 is only used as a fact-checker in our pipeline. It does not add any information on how to find the \\textit{aspects} because it does not see the original question, rather the output of single-step generated facts. As a result, we view our ``reasoning'' component much smaller than GPT-3 as we disentangle these two parts. We discuss this more in \\S\\ref{sec:manual-analysis} and Appendix~\\ref{sec:appendix-examples}.\n\n\\subsection{\\mbox{\\textsc{DecompEntail}}{} QA Pipeline}\n\\label{sec:pipeline}\n\n\\noindent \\textbf{Decompose.} Since \\mbox{\\textsc{DecompT5}}{} hasn't been pre-trained on questions, we fine-tune it on [question, supporting-fact] annotations from relevant datasets to generate \\textit{aspects+facts} for each question. Because supporting facts are usually composed of multiple sentences, we formulate a step-by-step generation. \nFor $n$ training facts, we formulate $n$ training instances from time $1$ to time $n$. At time $t$, a model sees an input sequence that is the question and all supporting facts with indices smaller than $t$ concatenated. The output sequence (learning target) is the supporting fact at index $t$. During evaluation time, the model generates one fact at a time, which then goes through the factual correction process in \\S\\ref{sec:factual-correction}. At time $t$, the model receives an input sequence including the original question and all current generated facts (after correction) before time $t$, and generates the $t^{th}$ supporting fact.\n\nWe design the specific input sequence as \\keywordCode{[Q]Decompositions:[G(current)]}, and output sequences as \\keywordCode{[G(next)]}. \\keywordCode{[G(current)]} is the concatenation of all current generations, which is empty before generating the first fact. \\keywordCode{[G(next)]} is the immediate next fact to be generated. \n\n\\noindent \\textbf{Entail.} With the generated facts from \\textbf{decompose}, we derive binary answers for questions with the \\textit{aspects+facts+indirect} model as seen in Table~\\ref{tab:sanity_check}.\n\n\n\n\\subsection{Inference}\n\\label{sec:voting}\nWe sample the top five generation candidates at each generation step via diverse beam search \\cite{Vijayakumar2016DiverseBS}. We select one randomly based on their $\\mathrm{softmax}$ probabilities. We generate at most three facts (i.e., $t=3$ as specified in \\S\\ref{sec:pipeline}) or early stops for each chain if all candidates at a generation step are very similar to the current generations, determined by the SentenceBert paraphrasing model with 0.95 as the threshold. We run the three-fact generation five times for each question due to randomness in the underlying generation selection process. As a result, we will have five chains of at most three generated facts for each question. We run the entailment model individually on each chain and derive a final answer based on majority voting from each chain. The majority voting is weighted with the confidence score of the entailment model's decisions on each chains.\n\n\\section{Intrinsic Experiments}\n\\label{sec:intrinsic-experiments}\nIn this section, we conduct two intrinsic experiments with \\mbox{\\textsc{DecompT5}}{} that directly evaluate its general decomposition capability through fine-tuning task-specific input\/output sequences. We compare with T5-large as it is the base LM, and such a comparison reveals how much we improve through pre-training with parallel news distant signals. We do not compare our model with GPT-3 because few-shot learning might not be enough for it to learn the complete grammar of different tasks' decomposition. This is an advantage of fine-tuning relatively small but capable models over directly using much bigger ones in few-shot settings. All experiments use a 5e-5 learning rate, and they are repeated and averaged with three seeds.\\footnote{We use $10,20,30$ as the seeds for all experiments.}\n\n\\subsection{Overnight}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\nSystem & Hit@1 & Hit@5 & Hit@10 \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\nT5-large & 21.8 & 51.6 & 63.1 \\\\\n\\mbox{\\textsc{DecompT5}}{} & 48.6 & 78.9 & 85.4 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Hit@K performances on Overnight decomposition generation. Hit@K is the percentage of instances where the top K generations contains at least one exact match. \\mbox{\\textsc{DecompT5}}{} is from this work.}\n\\label{tab:overnight}\n\\end{table}\n\n\\noindent \\textbf{Dataset and Metrics.} We evaluate and compare our model's capability to produce intermediate decomposition on the Overnight dataset \\cite{Wang2015BuildingAS}. It is a semantic parsing dataset that aims to parse natural language queries into a formal parsing that can be programmatically executed to denotations. In between the natural language query and the formal parsing, it annotates an intermediate ``canonical'' form with semi-formal language, which has recently been used for work on text-based semantic parsing with transformers \\cite{shin2021constrained} that we take inspiration from. For example, the annotated intermediate form of ``biggest housing unit that allows dogs'' is ``housing unit that has the largest size and that allows dogs''.\nWe evaluate the performance of mapping natural language queries to such intermediate forms with three domains that contain 3.8K training instances and 972 test cases. Both models are trained with three epochs. We use the same inference for both T5-large and \\mbox{\\textsc{DecompT5}}{}, which generates ten candidates using beam search. Following previous work, the generation is also constrained by possible ``next words'', that is, we assume that we know controlled output space beforehand.\n\n\\noindent \\textbf{Results and Analysis.} Table~\\ref{tab:overnight} details the performance of our \\mbox{\\textsc{DecompT5}}{} compared to its base model, T5-large. Our model doubles the performance on the exact match of the top prediction, which translates to a much higher denotation accuracy because multiple decompositions can be executed to the same denotation. Our model can find the exact match decomposition 78.9\\% of the time with only five candidates to consider, showing much higher potential for end-to-end tasks that may improve through iterative learning. On the other hand, T5-large can barely cover more than half of the queries with top-five candidates and only improves to 63.1\\% with more candidates (top-ten). This shows that \\mbox{\\textsc{DecompT5}}{} is much better at making commonsense connections (e.g., ``biggest'' to ``largest size'') after fine-tuning, thanks to the pre-training process on our parallel news corpus. \n\n\\subsection{TORQUE}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lc}\n\\toprule\nSystem & Exact Match \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\nT5-large & 50.3 \\\\\nT5-large-paraphrase & 72.2 \\\\\n\\mbox{\\textsc{DecompT5}}{} & 82.8  \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Exact match accuracy of different models on custom-annotated TORQUE. T5-large-paraphrase is first fine-tuned on paraphrasing supervision.}\n\\label{tab:torque}\n\\end{table}\n\n\\noindent \\textbf{Dataset.} TORQUE \\cite{Ning2020TORQUEAR} is a temporal question-answering dataset. For example, ``what happened before...'' asks the model to find all events with a start time before that of the given event, and ``what ended before...'' should be answered with events with end times before the start time of the given event. Compared to traditional temporal relation extraction tasks, this format is more challenging to existing temporal reasoning models, as they now have to parse the question and understand what aspects (e.g., start or end times) the question is asking first. To this end, we evaluate if our proposed model can better parse the question into correct temporal phenomena.\n\n\\noindent \\textbf{Annotate Decomposition.} Because TORQUE does not come with an intermediate annotation specifying the temporal properties required for each question, we need to annotate TORQUE questions with a form of intermediate decomposition to evaluate if a model understands the questions correctly. We adopt Overnight grammar for this purpose. For example, ``what started before [X]'' can be written as ``find all events whose start time is smaller than the start time of [X]''. Luckily, TORQUE uses several question templates during its annotation process. As a result, the intermediate decomposition of many questions can be automatically labeled. We create a training set of 15K question-decomposition pairs from 10 templates that are \\textbf{only} about events' start time comparisons. On the other hand, we create an evaluation set of 624 questions from 11 templates, and 9 of them compare events' end times, which a model will not see during training. We do this to evaluate models' capability of ``universal'' decomposition by generalizing to unseen relations in a ``zero-shot'' fashion. For a model to do well, it must have a pre-existing representation of what the question is asking.\n\n\\noindent \\textbf{Results and Analysis.} Table~\\ref{tab:torque} reports the exact match accuracy on our custom TORQUE evaluation. In addition to the T5 baseline, we use the same hyper-parameters as \\mbox{\\textsc{DecompT5}}{} to fine-tune a T5-large on the distant supervision portion from PAWS \\cite{Zhang2019PAWSPA}, containing 320K sentence pairs. We do this to compare the data quality of our distant supervision and that from paraphrasing since TORQUE requires a higher level of question understanding than Overnight. All models are trained for one epoch because the training data is highly repetitive, and generate one sequence via greedy decoding. We see that our model improves more than 30\\% over the baseline model, and 10\\% over the paraphrasing-supervised model. More importantly, this shows that \\mbox{\\textsc{DecompT5}}{} develops a solid implicit representation for query understanding from the pre-training, which allows it to generalize to end-time queries from supervisions that are only about start times. On the other hand, T5-large cannot correctly produce anything about end-time queries as expected.\n\n\\section{Sanity Check Experiments}\n\\label{sec:sanity_check}\nIn this section, we describe the details of the sanity check experiment mentioned and analyzed in \\S\\ref{sec:pipeline-intuition}.\n\n\\subsection{Dataset and Settings}\n\n\\noindent\\textbf{Dataset.}\nWe use StrategyQA, a QA dataset with high requirements for question understanding and knowledge acquisition. It contains questions that can be answered with either ``yes'' or ``no'', and is divided into 2061\/229 train\/dev, and an additional 490 test questions. Each question in the training and development sets is annotated with two types of supporting evidence as shown in Fig.~\\ref{fig:stqa-example}: decomposed questions and annotated facts. We use the decomposed questions as the \\textit{aspects} of a question \n, and the annotated facts as \\textit{aspects+facts}, as they provide specific values for the aspects. \n\n\\noindent \\textbf{Indirect Supervision.} Under the \\textit{aspects+facts} setting, the model is performing general textual entailment (TE) with the given facts as the premise and the question as the hypothesis, which allows us to use indirect supervision inspired by TE. We first augment each training instance in StrategyQA with five additional instances where all supporting facts are replaced with one of their paraphrases obtained with an off-the-shelf paraphrasing model.\\footnote{\\url{https:\/\/huggingface.co\/tuner007\/pegasus_paraphrase}} We then add additional supervision from e-SNLI's development set \\cite{Camburu2018eSNLINL}. We also add supervision from HotpotQA \\cite{Yang2018HotpotQAAD} with its annotated supporting facts.\n\n\\subsection{Training and Results} \nWe formulate a sequence-to-sequence task with input sequences as \\keywordCode{[Q]Decompositions:[D]} and output sequences of either \\keywordCode{yes\/no}. \\keywordCode{[Q]} is the question, and \\keywordCode{[D]} is the additional information such as supporting facts. We fine-tune T5-3B models for three epochs under each supervision setting and evaluate with the same gold information provided during test time. Each experiment is averaged over three random seeds. Table~\\ref{tab:sanity_check} details the performances on StrategyQA's development set. We have analyzed this result in \\S\\ref{sec:pipeline-intuition}.\n\n\\section{Decomposition QA Experiments}\n\\label{sec:e2e-experiments}\n\nWe detail two experiments that evaluates the QA pipeline \\mbox{\\textsc{DecompEntail}}{} proposed in \\S\\ref{sec:pipeline}. \n\\subsection{Datasets} \n\nAs argued in \\S\\ref{sec:pipeline-intuition}, our proposed pipeline benefits from any question-answering dataset that annotates supporting facts. To demonstrate this property, we use StrategyQA and HotpotQA jointly as supervision, and evaluate on both datasets. Because our pipeline setting is mostly designed for binary questions, we select questions that can be answered with either ``yes'' or ``no'' from HotpotQA, which accounts for 5430 questions from the training set. We use 300 binary questions from the development set of HotpotQA as evaluation. Because the supporting fact annotation in StrategyQA is human-written instead of Wikipedia sentences, it is shorter and more precise. To this end, we want the decomposition model to primarily rely on such annotations, and we duplicate each set of supporting facts in StrategyQA five times with shuffled order. These together produce around 35K decomposition instances for training.\n\n\\subsection{Settings and Baselines} \n\nWe compare with T5-large under the same joint supervision setting (denoted as ``S+H''). We also compare with RoBERTa*-IR as described in \\citet{geva2021did} on StrategyQA. It uses BoolQ \\cite{clark2019boolq} as additional supervision, which is denoted as ``S+B''. We also include GPT-3 baselines, one in a regular few-shot setting and another with a few-shot chain-of-thought \\cite{Wei2022ChainOT} supplement (denoted as GPT-3 CoT). Both prompts are available in Appendix~\\ref{sec:appendix-prompts}. \nWe report an aggregated performance (i.e., voting with all seeds as described in \\S\\ref{sec:voting}) on StrategyQA's development set. However, we report a single best seed's\\footnote{We determine the best seed based on the StrategyQA's development set.} performance on the test set as well as HotpotQA because of both StrategyQA leaderboard's limitation and cost considerations of using GPT-3. Experiments are repeated with three random seeds, trained for three epochs with 5e-5 learning rates.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcccc}\n\\toprule\nSystem & Source & Dev & Test & Hotpot \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\nT5-Large & S+H & 55.9 & - & 56.0 \\\\\nRoBERTa*-IR & S+B & 65.8 & 64.9 & -\\\\\nGPT-3 & Few & 62.5 & 64.1 & 70.0 \\\\\nGPT-3 CoT & Few & 65.9 & 63.7 & 73.0  \\\\\nOurs & S+H & \\textbf{70.3} & \\textbf{67.4} & 81.0 \\\\\n\\cmidrule(lr){1-5}\nOurs -pretrain & S+H & 67.2 & - & 80.7 \\\\\nOurs -correction & S+H & 62.9 & - & 69.0 \\\\\nOurs -joint & S or H & 65.5 & - & \\textbf{81.3} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Accuracy on StrategyQA and HotpotQA. Ours refers to the \\mbox{\\textsc{DecompEntail}}{} pipeline.}\n\\vspace{-0.2cm}\n\\label{tab:strategyqa_test}\n\\end{table}\n\n\\subsection{Results}\nTable~\\ref{tab:strategyqa_test} shows the performances with different baselines on StrategyQA and HotpotQA. On StrategyQA, \\mbox{\\textsc{DecompEntail}}{} outperforms all baseline models by 4\\%, proving that our model benefits the most, and more efficiently, from existing human-annotated resources on complicated questions. On HotpotQA's binary questions, our proposed pipeline outperforms the chain-of-thought variant of GPT-3 by over 8\\%, and the T5 baseline by 25\\%. This shows that explicit decomposition is better than reasoning in a black box, as we achieve better performances with a decomposition model that is over 200 times smaller.\\footnote{There are 770M parameters in T5-large and 175B parameters in GPT-3.}\n\n\\subsection{Ablation Studies}\nWe conduct ablation studies on three variants of the proposed pipeline: without the further pretraining described in \\S\\ref{sec:pretrain} (-pretrain), without the factual correction in \\S\\ref{sec:factual-correction} (-correction), and without the joint learning with both datasets (-joint). Table~\\ref{tab:strategyqa_test} details the performances of ablation models. Similarly, we evaluate the ablation models on StrategyQA's development set with three random seeds and vote with all seeds, but HotpotQA only once due to cost limitations. We see that pretraining with our parallel news corpus accounts for over 3\\% gain on StrategyQA. This aligns with our intuition and intrinsic experiments in \\S\\ref{sec:intrinsic-experiments} because StrategyQA requires advanced question understanding. Factual correction is also significant in our pipeline, which makes a 7\\% difference on StrategyQA and 12\\% on HotpotQA. On the other hand, joint learning contributes to the performances on StrategyQA but not on HotpotQA, which might be because HotpotQA experiments are run with single seeds.\n\n\\subsection{Manual Analysis}\n\\label{sec:manual-analysis}\nWe argue that the core of our improvement is producing proper decompositions instead of the use of GPT-3. We conduct a manual analysis on 20 questions\\footnote{We use the first 20 questions in the dev set that have agreeable annotated facts, without looking at the predictions.} from StrategyQA's dev set and inspect the raw decomposition before factual correction. We find that \\mbox{\\textsc{DecompT5}}{} fails to produce at least one decomposition with all necessary aspects on only two. This suggests that \\mbox{\\textsc{DecompT5}}{} does well in understanding ~90\\% of the questions without GPT-3, even though we need factual correction for the entailment model to produce the correct answer. Moreover, the analysis shows that GPT-3 does not provide anything beyond correcting any factual errors in the statement generated by \\mbox{\\textsc{DecompT5}}{}, as it only sees one decomposition at a time without seeing the actual question. We provide some actual output examples in Fig.~\\ref{fig:manual-analysis-examples} for more insights.\n\n\\section{Conclusion}\nThis work proposes a novel method that extracts distant and incidental signals from parallel news to facilitate general question representation. Such parallel news signals intuitively bridge the reasoning gap in pre-trained language models due to reporting biases. To support this intuition, we train a model named \\mbox{\\textsc{DecompT5}}{} on such distant supervision and show that it improves 20\\%-30\\% on two semantic parsing benchmarks, namely Overnight and TORQUE, that directly evaluate query understanding. With \\mbox{\\textsc{DecompT5}}{} as the basis, we design a well-motivated question-answering pipeline \\mbox{\\textsc{DecompEntail}}{} that follows a decomposition, correction, and entailment scheme. We show that \\mbox{\\textsc{DecompEntail}}{} improves on StrategyQA and HotpotQA by 3.7\\% and 8\\%, respectively.\n\n\\section*{Acknowledgments}\nThis research is based upon work supported in part by the office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA Contract No. 2019-19051600006 under the BETTER Program, and by Contract FA8750-19-2-1004 with the US Defense Advanced Research Projects Agency (DARPA). The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government. We also thank the Aristo team at the Allen Institute for AI for valuable support and feedback throughout the entire research process.\n\n\\section{Limitations}\nIn this section, we discuss some of the limitations of our work, and motivate future works.\n\n\\noindent \\textbf{Limited Question Formats.} Our proposed QA pipeline operates on binary \\textit{yes\/no} questions. While binary questions are very general, as most other questions can be re-written into similar forms, such transformations have not been designed or evaluated, which motivates future works.\n\n\\noindent \\textbf{Limited Factual Correction Coverage.} We use GPT-3 as the backbone for our factual correction step. Although it is shown to be effective, it is not as deterministic as Wikipedia-based IR approaches, and we cannot easily interpret why it makes mistakes and understand how to improve.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe eigenvalues of fourth-order differential operators are central in\ndetermining mechanical properties of rigid bodies. This paper considers\nthe small amplitude out-of-plane vibrations of a thin elastic\nplate~\\cite{RE}. The vibrational frequencies $\\lambda>0$ and modes\n$u(\\mathbf{x})$ satisfy the bi-Laplacian eigenvalue problem\n\\begin{subequations}\\label{eqn:intro}\n  \\begin{equation}\n  \\label{eqn:introA} \\Delta^2 u = \\lambda u, \\qquad \\mathbf{x} \\in\\Omega; \\qquad  \\int_{\\Omega} u^2\\, d\\mathbf{x} = 1,\n  \\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a closed planar region\nrepresenting the extent of the plate, $\\mathbf{x} = (x,y)$, and $\\Delta^2 u:=\nu_{xxxx} + 2 u_{xxyy} + u_{yyyy}$. Conditions on the boundary\n$\\partial\\Omega$ are application specific, with a common condition\nbeing that the plate is \\emph{clamped} on its periphery which stipulates that \n  \\begin{equation}\\label{eqn:introB} \n  u = \\partial_\\mathbf{n} u = 0, \\qquad \\mathbf{x} \\in\\partial\\Omega,\n  \\end{equation} \nwhere $\\partial_{\\mathbf{n}}$ is the outward facing normal derivative. A wide\nvariety of engineering systems utilize thin perforated plates in their\nconstruction. Examples include heat exchangers~\\cite{Nilles95,\nVenkatarathnam1996, Krishnakumar2003}, porous elastic materials, and\nacoustic tilings~\\cite{Atalla2007, wang2010, Jaouen2011}. The specific\nplacement of these perforations permits the manipulation of acoustic\nand vibrational properties of the plate while economizing on weight and\nmaterial cost. Homogenization theories have been proposed to replace\nthe natural elastic modulus of the plate with an effective\nmodulus~\\cite{BH,ADK}, however, an averaging approach omits the\npronounced localizing effects that clamping has on vibrational\nmodes~\\cite{FM}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.9\\textwidth]{HoleDiagram.pdf}\n\\parbox{0.75\\textwidth}{\\caption{In the limit of vanishing hole radius ${\\displaystyle \\varepsilon}\\to0$, a point constraint $u(\\mathbf{x}_j)=0$ must be enforced at each of the hole centers for $j = 1,\\ldots, M$.\\label{fig:IntroShrink}}}\n\\end{figure}\n\nIn the present work, we consider a finite collection of $M$ defects or punctures on~\\eqref{eqn:introA} with the conditions\n\\begin{equation}\\label{eqn:introC} \nu(\\mathbf{x}_j) = 0, \\qquad j = 1,\\ldots, M.\n\\end{equation}\n\\end{subequations}$\\!$\nThese \\emph{point constraints} arise in singular perturbation studies\nof~\\eqref{eqn:introA} in the presence of $M$ small circular\nperforations of radius ${\\displaystyle \\varepsilon}$ (cf.~Fig.~\\ref{fig:IntroShrink}). As the\nradius ${\\displaystyle \\varepsilon}$ of the perforations shrink to zero, the behavior of the\nlimiting eigenvalue $\\lambda_{{\\displaystyle \\varepsilon}}$ as\n${\\displaystyle \\varepsilon}\\to0$ satisfies~\\cite{KLW,LWK,LHS,CN01}\n\\begin{equation}\\label{HoleBehavior}\n  \\lambda_{{\\displaystyle \\varepsilon}} = \\lambda + 4\\pi \\nu \\sum_{j=1}^M \n  |\\nabla u(\\mathbf{x}_j)|^2 + \\mathcal{O}(\\nu^2), \\qquad \\nu = -\\frac{1}{\\log{\\displaystyle \\varepsilon}},\n\\end{equation}\nwhere $(\\lambda,u)$ satisfies (\\ref{eqn:introA}-\\ref{eqn:introB}) plus the point\nconstraints~\\eqref{eqn:introC}. In the degenerate case $\\sum_{j=1}^M\n|\\nabla u(\\mathbf{x}_j)|^2 = 0$, equation~\\eqref{HoleBehavior} is not valid\nand a separate limiting form can be derived~\\cite{CN01,KLW}.  The fact\nthat the clamping condition on each perforation leaves an imprint as\nthe radius shrinks to zero (Fig.~\\ref{fig:IntroShrink}) implies that no\nmatter how small a perforation is, the vibrational characteristics are\ndistinct from the no hole problem\n\\begin{equation}\\label{HoleFree}\n  \\Delta^2 u^{\\star} = \\lambda^{\\star} u^{\\star}, \n  \\quad \\mathbf{x} \\in\\Omega; \n  \\qquad u^{\\ast} = \\partial_\\mathbf{n} u^{\\star} = 0, \n  \\quad \\mathbf{x} \\in\\partial\\Omega; \n  \\qquad \\int_{\\Omega} {u^{\\star}}^2 d\\mathbf{x} = 1.\n\\end{equation}\n\n\nThe discontinuous limiting behavior of~\\eqref{HoleBehavior} is\nqualitatively different from the spectral problem for the Laplacian in\nthe presence of small perturbing holes~\\cite{F,KTW,O,WHK,WK}. A\nconsequence of the point constraints~\\eqref{eqn:introC} is that the\neigenfunctions $u(\\mathbf{x})$ are not necessarily smooth but satisfy local\nconditions\n\\begin{equation}\\label{behaviorLocal}\n  u(\\mathbf{x}) = \\alpha_j |\\mathbf{x} - \\mathbf{x}_j|^2 \\log|\\mathbf{x} - \\mathbf{x}_j| + \n    \\mathcal{O}(1), \\qquad \\mathbf{x}\\to\\mathbf{x}_j; \\qquad j = 1,\\ldots,M,\n\\end{equation}\nwhere the constants $\\{\\alpha_j\\}_{j=1}^M$ reflect the strength of each\npuncture and depend on the domain $\\Omega$ and the clamping locations\n$\\{\\mathbf{x}_j\\}_{j=1}^M$. The difference between the punctured eigenvalues\n$\\lambda$ of~\\eqref{eqn:intro} and the puncture free eigenvalues\n$\\lambda^{\\star}$ of~\\eqref{HoleFree} satisfies (cf.~\\cite{LHS}) \n\\begin{align}\n  \\label{EigDifference}\n  (\\lambda - \\lambda^{\\star}) \\braket{u,u^{\\star}} = \n  -8\\pi\\sum_{j=1}^M \\alpha_j u^{\\star}(\\mathbf{x}_j), \\qquad \n  \\braket{u,u^{\\star}} = \\int_{\\Omega} u(\\mathbf{x}) u^{\\star}(\\mathbf{x})\\, d\\mathbf{x}.\n\\end{align}\nThe presence of clamped locations also has a profound localizing effect\non the eigenfunctions. In a rectangular domain with a single clamped\npoint located along the long axis, the effect of clamping\non~\\eqref{eqn:intro} has been observed (cf.~\\cite{FM}) to partition\n$\\Omega$ into two distinct domains on the left and right of the clamping\nlocation, as shown in Fig~\\ref{fig:introClamp}. One aim of this work is\nto numerically investigate the global effects that point constraints\nhave on the eigenfunctions of~\\eqref{eqn:intro} in a variety of\ndifferent planar geometries.\n \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.45\\textwidth]{introClamped.png}\n\\parbox{0.75\\textwidth}{ \\caption{The localization of two eigenfunctions\nby a single clamped point, located at the black point, in a rectangular\ndomain. In each case, the eigenfunction is essentially zero on one side\nof the clamping point. See~\\cite{FM} and Sec.~\\ref{sec:rectangle} for\nmore details.  \\label{fig:introClamp} }}\n\\end{figure}\n \nFourth-order eigenvalue problems~(Equations~\\eqref{eqn:intro} and\n\\eqref{HoleFree}) exhibit other qualitatively different properties\ncompared to the well-understood Laplacian counterpart. For example, the\nfundamental eigenfunction of~\\eqref{eqn:intro}, ie.~the mode associated\nwith the lowest eigenvalue, is not necessarily single signed~\\cite{CD1,\nCD2, S, Coffman82, Gazzola2010, Grunau2014}. In contrast, the\nfundamental eigenfunction of the Laplacian is always single signed and\nthe corresponding eigenvalue is simple~\\cite{Evans2010,Gilbarg1998}.\nAn elementary example of this phenomenon is the annular domain\n${\\displaystyle \\varepsilon}<r<1$ in which the radially symmetric and mode $1$ eigenvalues of\nthe bi-Laplacian cross at ${\\displaystyle \\varepsilon}^{-1}\\approx762.36$~\\cite{CD2}.\nCorrespondingly, for ${\\displaystyle \\varepsilon}^{-1}>762.36$, the fundamental eigenfunction\nhas multiplicity two and one nodal line. Also, in domains with a\ncorner, the first eigenfunction may possess an infinite number of nodal\nlines~\\cite{Coffman82}. Many numerical methods have been developed to\ntreat fourth-order eigenvalue problems in view of these\ncharacteristics~\\cite{brown2000, AA, CD, LC2010,\njia-kro-qua2013,Zhao2002}.\n\nThe main goal of this paper is to introduce a novel high-order boundary\nintegral equation method for the numerical solution of~\\eqref{eqn:intro}\nin the presence of a finite collection of punctures \\eqref{eqn:introC}.\nHigh-order methods for computing eigenvalues of the Laplacian and\nHelmholtz equations in two and three dimensions have been developed with\ndomain decomposition methods~\\cite{bet2007, des-tol1983, dri1997},\nradial basis functions~\\cite{pla-dri2004}, boundary integral\nequations~\\cite{bac2003, ste-ung2009, dur-ned-oss2009}, the method of\nparticular solution~\\cite{bar2009, fox-hen-mol1967, kar2001}, the\nDirichlet to Neumann map~\\cite{bar-has2014}, and chebfun~\\cite{drum}.\nThe method of fundamental solutions has also been used to compute\neigenvalues of the biharmonic equation~\\cite{mar-les2005, AA}.  However,\nnone of these works consider the eigenvalue problem with clamped points.\nWe extend the work of one of the previous authors~\\cite{LHS} where a\nfinite difference method coupled with an inexact Newton method is used\nto solve~\\eqref{eqn:intro} in the unit circle with symmetrically chosen\nclamped points.  Owing to the accuracy and robustness of the boundary\nintegral equation methods, our new method forms third-order solutions\nof~\\eqref{eqn:intro} in smooth two-dimensional geometries, including\nmultiply-connected geometries (Figure~\\ref{fig:couetteModes}), and with\na large assortment of clamping locations.  \n\nUsing our new method, we demonstrate the dependence of $\\lambda$ on the\nnumber $M$ and locations $\\{\\mathbf{x}_1, \\ldots,\\mathbf{x}_M \\}$ of the puncture\nsites for a variety of planar regions $\\Omega \\subset \\mathbb{R}^2$.\nIn particular, we investigate two effects that clamped points have on\nthe vibrational properties of plates with various regular and irregular\ngeometries. Our first observation is that by specific location of\npunctures, the vibrational properties can be dramatically altered---in\nparticular, undesirable frequencies of vibration can be tuned out by\ndeliberate location of clamped points at nodal lines of the unclamped\neigenfunction $u^{\\star}$ of \\eqref{HoleFree}. Our second observation,\nextending previous results in~\\cite{FM} for rectangular domains, is\nthat mode confinement occurs in a variety of two dimensional geometries.\n\nThe outline of the paper is as follows. In Section~\\ref{sec:methods} we\ndescribe the details of a boundary integral method for\nsolving~\\eqref{eqn:intro}. In Section~\\ref{sec:algorithms}, the\nimplementation details are discussed and third-order convergence of the\nmethod is verified for a closed-form solution of~\\eqref{eqn:intro}.  In\nSection~\\ref{sec:numerics}, we apply our method to a disk, rectangles,\nan ellipse, a non-symmetric shape, and a multiply-connected region.\nFinally, in Section~\\ref{sec:conclusions} we discuss the results and\nareas of future investigations.\n\n\n\\section{Integral equation formulation of the clamped eigenvalue\nproblem}\n\\label{sec:methods}\nIn this section, we first compute and analyze the fundamental solution\nof the modified biharmonic operator $\\Delta^2 - \\lambda$.  We then use\nthe fundamental solution to reformulate equation~\\eqref{eqn:intro} as a\nsystem of second-kind boundary integral equations with compact integral\noperators.\n\n\\subsection{Fundamental solution}\n\\label{sec:fundSoln}\nWe require the fundamental solution $G(\\mathbf{x},\\mathbf{y})$ of the modified biharmonic operator satisfying\n\\begin{align*}\n  \\Delta^2 G - \\mu^4 G = \\delta(\\mathbf{x}-\\mathbf{y}), \\qquad \\mathbf{x} \\in \\mathbb{R}^2,\n\\end{align*}\nwhere $\\lambda = \\mu^4$.  The factorization $\\Delta^2 - \\mu^4 =\n(\\Delta - \\mu^2)(\\Delta + \\mu^2)$, and the fact the fundamental\nsolution is radially symmetric, imposes that $G(\\mathbf{x},\\mathbf{y})$ is a linear\ncombination of the Bessel functions $J_0(\\mu \\rho)$, $Y_0(\\mu \\rho)$,\n$I_0(\\mu \\rho)$, and $K_0(\\mu \\rho)$, where $\\rho = |\\mathbf{x}-\\mathbf{y}|$.  Using\na linear combination of the two singular Bessel functions that decay as\n$r \\rightarrow \\infty$, the fundamental solution centered at $\\mathbf{y}$ is\nof the form\n\\begin{align*}\n  G(\\mathbf{x},\\mathbf{y}) = c_{1}Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) + c_{2}K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|).\n\\end{align*} \nTo find the appropriate constants $c_1, c_2$, we use the identities\n$(\\Delta + \\mu^{2})Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) = -4\\delta(\\mathbf{x}-\\mathbf{y})$ and $(\\Delta\n- \\mu^{2})K_{0}(\\mu|\\mathbf{x} - \\mathbf{y}|)  = -2\\pi\\delta(\\mathbf{x}-\\mathbf{y})$, and compute\nthe fundamental solution by solving\n\\begin{align*}\n  (\\Delta - \\mu^{2})(\\Delta + \\mu^{2})c_{1}Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) + \n  (\\Delta + \\mu^{2})(\\Delta - \\mu^{2})c_{2}K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|)=\n  \\delta(\\mathbf{x}-\\mathbf{y}).\n\\end{align*}\nThis calculation reveals that the fundamental solution of $\\Delta^{2} -\n\\mu^{4}$ centered at $\\mathbf{y}$ is \n\\begin{align}\n  \\label{FundamentalGreens}\n  G(\\mathbf{x},\\mathbf{y}) = -\\frac{1}{8\\mu^{2}} Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) - \n            \\frac{1}{4\\pi\\mu^{2}} K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|).\n\\end{align}\nWe will be using $G$ in an indirect integral equation formulation, and\nthis will require the behavior of the fundamental solution when $\\mathbf{x}\n\\rightarrow \\mathbf{y}$.  Without loss of generality, we take $\\mathbf{y}=\\textbf{0}$\nand expand the fundamental solution for small $|\\mathbf{x}|$.  Using small\nargument approximations of the Bessel functions\n(cf.~\\cite{abr-ste1964}), we have\n\\begin{align*}\n  G(\\mathbf{x},\\textbf{0}) =  \n      \\frac{|\\mathbf{x}|^2}{8\\pi}\\log |\\mathbf{x}| \\left(1 +\n      \\mathcal{O}(|\\mathbf{x}|^{4})\\right)\n      +\\frac{|\\mathbf{x}|^{2}}{8\\pi}\\left(-1 + \\gamma +\n      \\log\\left(\\frac{\\mu}{2}\\right) + \\mathcal{O}(|\\mathbf{x}|^{4})\\right),\n      \\quad \\mbox{as} \\quad |\\mathbf{x}|\\to0,\n\\end{align*}\nwhere $\\gamma \\approx 0.5772156649$ is Euler's constant. As mentioned\nin the introduction, a key behavior of the solution\nof~\\eqref{eqn:intro} is the local behavior~\\eqref{behaviorLocal} near\neach of the defects.  Since the fundamental solution satisfies this\nrequired behavior, the solution of~\\eqref{eqn:intro} can be written as\n\\begin{align}\\label{eqn:LinSep}\n  u(\\mathbf{x}) = u_S(\\mathbf{x})+ u_R(\\mathbf{x}), \\qquad u_S(\\mathbf{x}) = 8\\pi\\sum_{j=1}^M \\alpha_j G(\\mathbf{x},\\mathbf{x}_j),\n\\end{align}\nwhere $G(\\mathbf{x},\\mathbf{y})$ is given in~\\eqref{FundamentalGreens}. In Section\n\\ref{sec:Newton}, we describe an inexact Newton method to find the\nstrength of the defects $\\{\\alpha_{j}\\}_{j=1}^M$ and the eigenvalues\n$\\lambda$.  The decomposition~\\eqref{eqn:LinSep} of the solution as the\nsum of a singular and regular part allows for the local\nbehavior~\\eqref{behaviorLocal} to be precisely enforced while the\nregular part $u_R$ satisfies the homogeneous fourth-order PDE\n\\begin{subequations}\\label{eqn:regularPDE}\n\\begin{gather}\n  \\label{eqn:regularPDE_A}\n  \\Delta^{2}u_{R} - \\lambda u_{R} = 0, \\quad \\mathbf{x} \\in \\Omega; \\\\[5pt]\n  \\label{eqn:regularPDE_B}  u_{R} = -u_{S}, \\qquad \\partial_{n} u_{R} = -\\partial_{n} u_{S}, \n  \\quad \\mathbf{x} \\in \\partial \\Omega,\n\\end{gather}\n\\end{subequations}$\\!$\nwhere $u_S$ is specified in \\eqref{eqn:LinSep}. We note that in~\\cite{LHS}, the singular part was chosen to be\n\\begin{align*}\n  u_{S}(\\mathbf{x}) = \\sum_{j=1}^{M} \\alpha_{j} |\\mathbf{x} - \\mathbf{x}_{j}\n    |^2\\log |\\mathbf{x} - \\mathbf{x}_{j}|.\n\\end{align*}\nWhile this choice has the correct local behavior~\\eqref{behaviorLocal},\nit leads to a forcing term in the PDE for $u_R$ that, for a boundary\nintegral equation method, is prohibitive. However, the boundary\nconditions~\\eqref{eqn:regularPDE_B} in our new formulation depends\nnonlinearly on the unknown eigenvalue $\\lambda$.\n\nOnce the functions $u_S$ and $u_R$ are computed, they can be easily\nevaluated at the locations of the clamped points.  This is used to\niteratively solve the non-linear equation (Section~\\ref{sec:Newton})\n\\begin{align}\n\\label{eqn:mismatch}\n  F(\\mathbf{z}) = \\left[\n  \\begin{array}{c}\n    u_{S}(\\mathbf{x}_{1}) + u_{R}(\\mathbf{x}_{1}) \\\\\n    \\vdots \\\\\n    u_{S}(\\mathbf{x}_{M}) + u_{R}(\\mathbf{x}_{M}) \\\\\n    \\alpha_{1}^{2} + \\cdots + \\alpha_{M}^{2} - 1\n  \\end{array}\n  \\right] = \\left[\n  \\begin{array}{c}\n    0 \\\\ \\vdots \\\\ 0 \\\\ 0\n  \\end{array}\n  \\right],\n\\end{align}\nwhere $\\mathbf{z} = (\\alpha_1,\\ldots,\\alpha_M,\\lambda)$.  The particular\nnormalization condition $\\sum_{j=1}^M \\alpha_j^2 = 1$ is chosen purely\nfor ease of implementation. Once a solution is obtained, the\neigenfunction can be normalized according to~\\eqref{eqn:introA} or any\nother condition.\n\n\\subsection{Computing the regular solution $u_R$}\n\\label{sec:layer_pots}\nEquation~\\eqref{eqn:regularPDE} is linear and homogeneous, so it can be\nrecast in terms of a boundary integral equation.  In this section, we\ndescribe appropriate layer potentials. Since the PDE is fourth-order, a\nsum of two linearly independent layer potentials must be used.  The\nregular part $u_R$ is written as\n\\begin{align}\n  u_{R}(\\mathbf{x}) = \\int_{\\partial\\Omega} G_{1}(\\mathbf{x},\\mathbf{y})\\sigma_{1}(\\mathbf{y}) ds_{\\mathbf{y}} +\n               \\int_{\\partial\\Omega} G_{2}(\\mathbf{x},\\mathbf{y})\\sigma_{2}(\\mathbf{y}) ds_{\\mathbf{y}},\n  \\label{eqn:layerPot}\n\\end{align}\nwhere $G_{1}$ and $G_{2}$ are linear combinations of $G$ and its partial\nderivatives.  The choice of $G_{1}$ and $G_{2}$ determines the nature of\nthe boundary integral equation which plays a crucial role on the\nconditioning of the linear system that arises after discretization.  In\nparticular, $G_{1}$ and $G_{2}$ should be chosen so that the resulting\nboundary integral equation is of the second-kind with compact integral\noperators.  This means that the limiting values of the layer potential\nansatz~\\eqref{eqn:layerPot} must have jumps that are proportional to\n$\\sigma_1$ and $\\sigma_2$ as $\\mathbf{x} \\rightarrow \\partial\\Omega$, and the\nkernels must be integrable.\n\nTo find kernels $G_1$ and $G_2$ with these desired results, we use the\nwork of Farkas~\\cite{far1989} who formulated the desired second-kind\nintegral equations for the fourth-order biharmonic equation.  For the\nbiharmonic equation with Dirichlet and Neumann boundary conditions,\nFarkas proposed the kernels\n\\begin{align*}\n  G_{1}(\\mathbf{x},\\mathbf{y}) &= G_{\\mathbf{n}\\nn\\mathbf{n}} + 3G_{\\mathbf{n}\\boldsymbol{\\tau}\\ttau}, \\\\\n  G_{2}(\\mathbf{x},\\mathbf{y}) &= \\Delta G - 2G_{\\mathbf{n}\\nn},\n\\end{align*}\nwhere the normal vector $\\mathbf{n}$ and tangent vector $\\boldsymbol{\\tau}$ are taken with\nrespect to the source point $\\mathbf{y}$.  Since the leading order singularity\nof $G$, $\\frac{1}{8\\pi}|\\mathbf{x}|^{2}\\log|\\mathbf{x}|$, is equal to the fundamental\nsolution of the two-dimensional biharmonic equation, the jumps in the\nlayer potential~\\eqref{eqn:layerPot} agree, to a first approximation,\nwith the jumps found by Farkas.  In particular, any additional jumps in\n$G_1$ and $G_2$ will result from the higher-order terms in the expansion\nof $G$.  Since the higher-order terms contain singularities of strength\nno less than $|\\mathbf{x}|^{6}\\log|\\mathbf{x}|$, no additional jumps will be present\nas long as $G_1$ and $G_2$ do not involve derivatives of order six or\nhigher.  Since the derivatives $G_1$ and $G_2$ are no more than\nthird-order, the jumps of $G_1$ and $G_2$ will agree with those computed\nby Farkas.\n\n\\subsection{Explicit expressions of the kernels}\n\\label{sec:kernels}\nFor $\\mathbf{x},\\mathbf{y} \\in \\partial\\Omega$, we require the four kernels\n\\begin{align*}\n  G_{11}(\\mathbf{x},\\mathbf{y}) &= G_{1}(\\mathbf{x},\\mathbf{y}), \\\\\n  G_{12}(\\mathbf{x},\\mathbf{y}) &= G_{2}(\\mathbf{x},\\mathbf{y}), \\\\\n  G_{21}(\\mathbf{x},\\mathbf{y}) &= \\pderiv{}{\\mathbf{n}_{\\mathbf{x}}}G_{1}(\\mathbf{x},\\mathbf{y}), \\\\\n  G_{22}(\\mathbf{x},\\mathbf{y}) &= \\pderiv{}{\\mathbf{n}_{\\mathbf{x}}}G_{2}(\\mathbf{x},\\mathbf{y}).\n\\end{align*}\nSubstituting the fundamental solution~\\eqref{FundamentalGreens} into\nthese expressions, and using the identities\n\\begin{align*}\n  \\pderiv{}{\\mathbf{n}}(\\mathbf{r} \\cdot \\nn) = -1, \\qquad\n  \\pderiv{}{\\mathbf{n}}\\rho = -2 \\frac{\\mathbf{r} \\cdot \\nn}{\\rho^{2}}, \\qquad\n  \\pderiv{}{\\mathbf{n}_\\mathbf{x}}(\\mathbf{r} \\cdot \\nn) = 1, \\qquad\n  \\pderiv{}{\\mathbf{n}_\\mathbf{x}}\\rho = -2 \\frac{\\mathbf{r} \\cdot \\nn_{\\mathbf{x}}}{\\rho^{2}},\n\\end{align*}\nwhere $\\mathbf{r} = \\mathbf{x} - \\mathbf{y}$, $\\rho = |\\mathbf{r}|$, and similar identities for\nthe tangential derivatives, the kernels $G_{11}$ and $G_{12}$ are\n\\begin{align*}\n  G_{11} = -\\frac{1}{4\\pi\\mu^2} &\\left(\n    3\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho} - \n    2\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^3} +\n    6\\mu^{2} K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{2}} \\right. \\\\\n    &\\left.\n    -8\\mu^{2} K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{4}} - \n    16\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{5}} + \n    12\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{3}}\n    \\right) \\\\\n    -\\frac{1}{8\\mu^{2}}&\\left(\n    -3\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho} + \n    2\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^3} -\n    6\\mu^{2} Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{2}} \\right. \\\\\n    &\\left.\n    +8\\mu^{2} Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{4}} - \n    16\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{5}} + \n    12\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{3}}\n    \\right), \\\\\n  G_{12} = -\\frac{1}{4\\pi}&\\left(1 - 2\\frac{(\\mathbf{r} \\cdot \\nn)^2}{\\rho^{2}}\\right)\n    \\left(K_{0}(\\mu\\rho) + \\frac{2}{\\mu\\rho}K_{1}(\\mu\\rho)\\right) \n    +\\frac{1}{8}\\left(1 - 2\\frac{(\\mathbf{r} \\cdot \\nn)^{2}}{\\rho^{2}}\\right)\n    \\left(Y_{0}(\\mu\\rho) - \\frac{2}{\\mu\\rho}Y_{1}(\\mu\\rho)\\right).\n\\end{align*}\nThe expressions for $G_{21}$ and $G_{22}$ require one additional\nderivative of $G_{11}$ and $G_{12}$.  For completeness, these lengthy expressions are given in Appendix~\\ref{app:kernels}.\n\n\\subsection{The boundary integral equation}\n\\label{sec:BIE}\n\nAs discussed in Section~\\ref{sec:fundSoln}, all four kernels $G_{ij}$\nhave the same asymptotic behavior as the fundamental solution of the\nbiharmonic equation.  Therefore, the boundary integral equation for\n$\\boldsymbol{\\sigma}$ is identical to the boundary integral equation for the\nbiharmonic equation~\\cite{far1989},\n\\begin{align}\n  \\label{eqn:bie}\n  D(\\mathbf{x})\\boldsymbol{\\sigma}(\\mathbf{x}) + \\int_{\\partial\\Omega} A(\\mathbf{x},\\mathbf{y})\\boldsymbol{\\sigma}(\\mathbf{y}) ds_{\\mathbf{y}}\n    =\\gg(\\mathbf{x}),\n\\end{align}\nwhere\n\\begin{align*}\n  D(\\mathbf{x}) = \\left(\n  \\begin{array}{cc}\n    \\displaystyle\\frac{1}{2} & 0 \\\\ -\\kappa(\\mathbf{x}) & \\displaystyle\\frac{1}{2}\n  \\end{array}\n  \\right),\n\\end{align*}\n$\\kappa(\\mathbf{x})$ is the curvature of $\\partial\\Omega$ at $\\mathbf{x}$, and \n\\begin{align*}\n  \\gg = -\\left(\n  \\begin{array}{c}\n    u_S \\\\ \\partial_{\\mathbf{n}}u_S\n  \\end{array}\n  \\right),\n  \\quad \n  \\boldsymbol{\\sigma} = \\left(\n  \\begin{array}{c}\n    \\sigma_{1} \\\\ \\sigma_{2}\n  \\end{array}\n  \\right),\n  \\quad\n  A = \\left(\n  \\begin{array}{cc}\n    G_{11} & G_{12} \\\\ \n    G_{21} & G_{22}\n  \\end{array}\n  \\right).\n\\end{align*}\nTo apply quadrature formulae, the limiting values of $G_{ij}$ as $\\mathbf{x}\\to\\mathbf{y}$ are\nrequired.  These can be found by applying L'H\\^{o}pital's rule to each\nof the four kernels.  For $\\mathbf{x}$, $\\mathbf{y}$ on $\\partial\\Omega$ we have\n\\begin{equation}\n  \\label{eqn:limits}\n  \\begin{aligned}\n    \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{11}(\\mathbf{x},\\mathbf{y}) &= 0, \\\\\n    \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{12}(\\mathbf{x},\\mathbf{y}) &=\n    \\frac{1}{4\\pi}\\kappa(\\mathbf{x}), \\\\\n    \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{21}(\\mathbf{x},\\mathbf{y}) &=\n    -\\frac{3}{4\\pi}\\kappa(\\mathbf{x})^2, \\\\\n    \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{22}(\\mathbf{x},\\mathbf{y}) &=\n    \\frac{1}{2\\pi}\\kappa(\\mathbf{x}).\n  \\end{aligned}\n\\end{equation}\n\n\n\\section{Numerical Methods}\n\\label{sec:algorithms}\nHere we describe a numerical method for solving the boundary integral\nequation~\\eqref{eqn:bie} (Section~\\ref{sec:bie_dis}), applying an\ninexact Newton method for~\\eqref{eqn:mismatch}\n(Section~\\ref{sec:Newton}), and an algorithm for tracing the first\neigenvalue, $\\lambda$, as clamped points are smoothly moved through the\ngeometry $\\Omega$ (Section~\\ref{sec:initial_guess}). \n\n\\subsection{Discretization of the integral equation}\n\\label{sec:bie_dis}\nWe apply a standard collocation method to solve the second-kind boundary\nintegral equation~\\eqref{eqn:bie}.  The boundary, $\\partial\\Omega$, is first\ndiscretized at collocation points $\\mathbf{x}_{i}$,\n$i=1,\\ldots,N$.  To satisfy the boundary integral equation at these\ncollocation points, we require\n\\begin{align}\\label{eq:integralMain}\n  D(\\mathbf{x}_i)\\boldsymbol{\\sigma}(\\mathbf{x}_i) + \\int_{\\partial\\Omega} A(\\mathbf{x}_i,\\mathbf{y})\\boldsymbol{\\sigma}(\\mathbf{y}) ds_{\\mathbf{y}}\n    =\\gg(\\mathbf{x}_i).\n\\end{align}\nThe integral in~\\eqref{eq:integralMain} is approximated with the\ntrapezoid rule where the abscissae are the collocation points which\nyields the dense linear system\n\\begin{align*}\n  D(\\mathbf{x}_{i}) \\boldsymbol{\\sigma}_i + \\sum_{j=1}^{N} A(\\mathbf{x}_{i},\\mathbf{x}_{j})\\Delta s_{j}\n    \\boldsymbol{\\sigma}_{j} = \\gg_{i},\n\\end{align*}\nwhere $\\boldsymbol{\\sigma}_i = \\boldsymbol{\\sigma}(\\mathbf{x}_{i})$, $\\gg_{i} = \\gg(\\mathbf{x}_{i})$, and\n$\\Delta s_{j}$ is the Jacobian of the curve at point $\\mathbf{x}_{j}$.  The\nlimiting values from~\\eqref{eqn:limits} are used for the diagonal terms\n$A(\\mathbf{x}_i,\\mathbf{x}_i)$ of the linear system.\n\nThe convergence order of the method depends on the regularity of the\nkernels $G_{ij}$.  The regularity of the kernels can be computed by\ntaking a simple geometry, such as the unit circle, fixing $\\mathbf{x}$, and\ncomputing the limit as $\\mathbf{y} \\rightarrow \\mathbf{x}$ of $G_{ij}(\\mathbf{x},\\mathbf{y})$ and\nits derivatives.  These calculations reveal that\n\\begin{align*}\n  G_{11} \\in C^3, \\qquad G_{12} \\in C^3, \\qquad G_{21} \\in C^1, \\qquad\n  G_{22} \\in C^3.\n\\end{align*}\nThe accuracy of the trapezoid rule for a periodic $C^k$ function is\n$k+2$, so we expect third-order convergence because of the $C^1$\nregularity of $G_{21}$.  Higher-order accuracy can be achieved by using\nspecialized quadrature~\\cite{alp1999, kap-rok1997} designed for\nfunctions with weak logarithmic singularities. Once values for the\ndensity function $\\boldsymbol{\\sigma}_j$ are computed, we can compute $u_R(\\mathbf{x})$ for\nany $\\mathbf{x} \\in \\Omega$ with spectral accuracy.  In particular, we compute\nthe value at the clamped locations with the trapezoid rule to yield that\n\\begin{align*}\n  u_{R}(\\mathbf{x}) &= \\int_{\\partial\\Omega} G_{1}(\\mathbf{x},\\mathbf{y})\\sigma_{1}(\\mathbf{y}) ds_{\\mathbf{y}} +\n             \\int_{\\partial\\Omega} G_{2}(\\mathbf{x},\\mathbf{y})\\sigma_{2}(\\mathbf{y}) ds_{\\mathbf{y}}\\\\[5pt]\n             &\\approx \\frac{2\\pi}{N} \\sum_{j} \\left( G_{1}(\\mathbf{x},\\mathbf{y}_j) \\sigma_{1_j} +\n                G_{2}(\\mathbf{x},\\mathbf{y}_j) \\sigma_{2_j}\n                \\right)\\Delta s_j.\n\\end{align*}\nIf a target point $\\mathbf{x}$ is sufficiently close to $\\partial\\Omega$, then the\naccuracy of the trapezoid rule will be diminished due to large derivatives\nin $G_{i}(\\mathbf{x},\\mathbf{y})$.  In this case, we simply upsample the geometry and\ndensity functions so that sufficient accuracy can be achieved at the\nclamped locations.\n\n\\subsection{Nonlinear solvers}\\label{sec:Newton}\nTo solve the nonlinear equation~\\eqref{eqn:mismatch} for\n$\\{\\alpha_j\\}_{j=1}^{M}$ and $\\lambda$, we apply one of two strategies.\nFirst, in symmetric cases such as the disk geometry, if the clamped\npoints are equidistributed in the azimuthal direction at a fixed radius,\nthen $\\alpha_1 = \\cdots = \\alpha_M$.  Therefore, $\\alpha_j =\nM^{-\\frac{1}{2}}$ for $j = 1,\\ldots,M$, and the only parameter remaining\nis $\\lambda$. For such a case, and any scenario in which symmetry\nconsiderations reduce the unknown to just $\\lambda$, a bisection method\ncan be applied to reliably solve~\\eqref{eqn:mismatch} since convergence\nto the desired root is guaranteed for an appropriately chosen initial\ninterval. This method is generally preferred in cases where all the\n$\\alpha_j$ are equal and the single unknown is the eigenvalue itself.\n\nSecond, when symmetry can not be assumed, we apply an inexact Newton's\nmethod to~\\eqref{eqn:mismatch}.  In our calculations, the Jacobian\nmatrix $J$ of $F$ is formed by finite difference approximations which we\nhave found to be accurate and efficient.\n\nWe validate the method with the unit disk geometry. A closed-form\nsolution of~\\eqref{eqn:intro} can be developed in the special case $M=1$\nand $\\mathbf{x}_1 =(0,0)$. In a similar manner to the construction of the\nfundamental solution~\\eqref{FundamentalGreens}, a linear combination of\n$K_0$ and $Y_0$ can be chosen to eliminate the logarithmic singularity\nat the origin.  Therefore radially symmetric eigenfunctions\nof~\\eqref{eqn:intro} are a combination of $Y_0(\\mu \\rho),$ $K_0(\\mu \\rho)$, \n$J_0(\\mu \\rho)$ and $I_0(\\mu \\rho)$ with $\\rho = |\\mathbf{x}|$. The eigenfunctions that are finite at the\norigin and satisfy $u(0)=0$ and $u(1) = \\partial_\\rho u(1)=0$ are\n\\begin{equation*}\nu(\\rho) = A \\left[J_0(\\mu_{0,n} \\rho) - I_0(\\mu_{0,n} \\rho) -\n\\left(\\frac{J_0(\\mu_{0,n}) - I_0(\\mu_{0,n})}{\\frac{2}{\\pi} K_0(\\mu_{0,n})\n+ Y_0(\\mu_{0,n})} \\right) \\left(\\frac{2}{\\pi} K_0(\\mu_{0,n} \\rho) +\nY_0(\\mu_{0,n} \\rho)\\right) \\right],\n\\end{equation*}\nwhere $A$ is a normalization constant and the eigenvalues $\\lambda_{0,n}\n= \\mu_{0,n}^4$ satisfy the relationship\n\\begin{equation}\\label{disk_exact}\n\\big(J_0(\\mu_{0,n}) - I_0(\\mu_{0,n})\\big) \\left( \\frac{2}{\\pi}\nK_1(\\mu_{0,n}) + Y_1(\\mu_{0,n}) \\right) = \\big(J_1(\\mu_{0,n}) +\nI_1(\\mu_{0,n})\\big) \\left( \\frac{2}{\\pi} K_0(\\mu_{0,n}) + Y_0(\\mu_{0,n})\n\\right).\n\\end{equation}\nThe smallest positive solution of~\\eqref{disk_exact} gives rise to the\neigenvalue $\\lambda_{\\textrm{true}} \\approx 516.9609.$  This solution\nprovides a benchmark against which the efficacy of our numerical method\ncan be verified.  We compute the relative error\n$\\mathcal{E}_{\\textrm{rel}}$ between the numerically determined value of\n$\\lambda_{\\textrm{num}}$ and the exact value $\\lambda_{\\textrm{true}}$.\nIn Fig.~\\ref{fig:ErrorDisk}, the numerical error scales $\\mathcal{O}(N^{-3})$\nas the number of boundary points $N$ increases which agrees with our\nexpected third-order convergence.  In this example, the bisection method\nwas used, and the strength of the singularity is $\\alpha = 1$.\n\n\\begin{figure}[htbp]\n\\centering\n\\input{ErrorDisk.tikz}\n\\parbox{0.75\\textwidth}{\\caption{The relative error (black) of our\nnumerical method when using the bisection method to find the first\neigenvalue of \\eqref{eqn:intro} with a single clamped point at the\ncenter of the unit disk.  A line of slope $-3$ (red) indicates the\nexpected third-order convergence.\\label{fig:ErrorDisk}}}\n\\end{figure}\n\n\\subsection{Initialization, parameterization of puncture patterns, and\narclength continuation}\n\\label{sec:initial_guess}\n\nThe solution of the nonlinear system~\\eqref{eqn:mismatch} by Newton's\nMethod relies on good initial iterates. In addition, a careful selection\nof the initial guess is necessary to reliably locate the lowest mode of\nthe punctured problem~\\eqref{eqn:intro}. For the unit circle, we start\nwith the clamped points at the center of the circle and initialize\nNewton iterations for~\\eqref{eqn:mismatch} with the known eigenvalue\n$\\lambda \\approx 516.9609$ for a single clamped point at the origin.\nFor other geometries, we start the clamped point near $\\partial\\Omega$.  In\nthis scenario, equation~\\eqref{eqn:mismatch} is initialized with a mode\nof the unclamped problem~\\eqref{HoleFree} calculated from a low-accuracy\nfinite element approximation~\\cite{KI78}.  Once a solution\nof~\\eqref{eqn:intro} has been generated, the punctures are gradually\nmoved, and~\\eqref{eqn:mismatch} is repeatedly solved until the punctures\noccupy a specified target set.\n\nIn the examples that follow, we compute eigenvalues $\\lambda =\n\\lambda(r)$ of~\\eqref{eqn:intro} for families of puncture patterns\ndescribed by a single parameter $r\\geq0$. For reasons of efficiency\nand to provide robustness to the Newton iterations, we use arc-length\nadaptively to focus resolution at  sharp peaks of the curve as compared\nto the surrounding areas. The algorithm to find points on the curve\n$\\lambda = \\lambda(r)$ is initialized with a relatively large step size\n$\\mathrm{d}r$ with the concavity monitored until proximity to an extrema\nis detected. Once an extrema of the curve is detected, $\\mathrm{d}r$ is\nreduced based on the current slope up to a minimum allowable step size. \n\n\n\n\\section{Numerical Examples}\n\\label{sec:numerics}\n\nIn this section we demonstrate the effectiveness of the method on\nregular and irregular domains. To understand the role of clamping in the\neigenvalue problem, and interpret the results obtained with our\nnumerical method, we recall from~\\eqref{EigDifference} that\n\\begin{align*}\n (\\lambda - \\lambda^{\\star}) \\braket{u,u^{\\star}} = \n  -8\\pi\\sum_{j=1}^M \\alpha_j u^{\\star}(\\mathbf{x}_j), \\qquad \n  \\braket{u,u^{\\star}} = \\int_{\\Omega} u(\\mathbf{x}) u^{\\star}(\\mathbf{x})\\, d\\mathbf{x},\n\\end{align*}\nwhich relates the modes $(\\lambda,u)$ of~\\eqref{eqn:intro} to the\nunclamped modes $(\\lambda^{\\star},u^{\\star})$ of~\\eqref{HoleFree}.  In\neach of the examples that follow, we will use a low accuracy finite\nelement method~\\cite{KI78} to obtain the required solutions\nof~\\eqref{HoleFree}. \n\n\n\n\\subsection{Unit circle}\nThe relationship~\\eqref{EigDifference} shows how the distinct\neigenvalues and eigenfunctions of the clamped and unclamped\nproblems,~\\eqref{eqn:intro} and~\\eqref{HoleFree}, respectively, are\nrelated. For each domain it is therefore important to consider the\nsolutions $(\\lambda^{\\star},u^{\\star})$ to understand the effect of\npuncture configurations.\n\nFor the unit disk case, the solutions of problem \\eqref{HoleFree} are\nfound by first factorizing $\\Delta^2 - \\mu^4 = (\\Delta -\\mu^2)(\\Delta +\n\\mu^2)=0$ which indicates that the basis for the space of eigenfunctions\nis\n\\begin{equation*}\n e^{im \\theta} \\{  J_m(\\mu_{m,n} \\rho), Y_m(\\mu_{m,n} \\rho), K_m(\\mu_{m,n} \\rho), I_m(\\mu_{m,n} \\rho) \\}, \\qquad \\mu_{m,n} = \\lambda_{m,n}^{1\/4},\n\\end{equation*}\nwhere $\\rho = |\\mathbf{x}|$. The indices $m = 0, \\pm 1, \\pm2, \\ldots$ indicate\nthe angular wavenumber (and number of angular nodal lines) where as\n$n=0,1,2,\\ldots$ counts the number of radial nodal lines for each\nwavenumber. In the unclamped problem~\\eqref{HoleFree}, the smooth\neigenfunctions satisfying $u^{\\star} = \\partial_{\\rho} u^{\\star} = 0$\non $\\rho=1$ are\n\\begin{align*}\n  u^{\\star}_{m,n}(\\rho,\\theta) = e^{im\\theta} \\left[ J_m(\\mu^{\\star}_{m,n}\\rho) - \n    \\frac{J_m(\\mu^{\\star}_{m,n})}{I_m(\\mu^{\\star}_{m,n})} \n    I_m(\\mu^{\\star}_{m,n}\\rho) \\right],\n\\end{align*}\nwith the eigenvalues $\\mu^{\\star}_{m,n}$ determined by the relationship\n\\begin{align}\\label{NoHolesExact_b}\nJ'_m(\\mu^{\\star}_{m,n})I_m(\\mu^{\\star}_{m,n}) = I'_m(\\mu^{\\star}_{m,n}) J_m(\\mu^{\\star}_{m,n}).\n\\end{align}\nThe first four eigenvalues $\\lambda^{\\star}_{m,n} =\n(\\mu^{\\star}_{m,n})^4$, found from the numerical solution\nof~\\eqref{NoHolesExact_b}, are\n\\begin{equation}\\label{eigsDiskPFree}\n\\lambda^{\\star}_{0,0} = 104.4, \\qquad \\lambda^{\\star}_{1,0} = 452.0, \\qquad \\lambda^{\\star}_{2,0} = 1216.4, \\qquad \\lambda^{\\star}_{0,1} = 1581.7.\n\\end{equation}\nIn Fig.~\\ref{fig:NoHoles}, the first few eigenfunctions are plotted with the nodal lines along which $u^{\\star}=0$ highlighted.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.7\\textwidth]{EigsDisc.pdf}\n\\parbox{0.75\\textwidth}{\\caption{The contour lines of the first $15$\nmodes of the unclamped problem~\\eqref{HoleFree} on the unit disk.  The\nnodal lines ($u^{\\star}=0$) are plotted in black.  Eigenfunctions are\nrepeated according to their multiplicity.  \\label{fig:NoHoles}}}\n\\end{figure}\nFor punctures away from the origin, we seek solutions\nof~\\eqref{eqn:intro} with parameterized puncture sets to minimize the\nnumber of unknowns over which nonlinear iterations are processed. In the\ndisk case, our first example is a single ring of punctures given\nexplicitly as\n\\begin{align}\n  \\label{singleRing}\n    \\mathbf{x}_j =  r \\left( \\cos\\frac{2\\pi j}{M}, \n    \\sin \\frac{2\\pi j}{M} \\right), \\qquad j = 1,\\ldots, M.\n\\end{align}\n\nThere is now a single parameter $r$ over which various configurations\ncan be investigated from $0<r<1$. From the eigenvalue $\\lambda\\approx\n516.96$, which is the lowest positive solution of \\eqref{disk_exact}, we\nform an initial guess for the Newton iterations. Since our search\npattern~\\eqref{singleRing} is radially symmetric, the puncture strengths\n$\\{\\alpha_j\\}_{j=1}^M$ can be assumed to be identical in this case. \n\\begin{figure}[htbp]\n\\centering\n{\\subfigure[Single puncture ring pattern]{\\includegraphics[width =\n0.32\\textwidth]{SchematicDisk.pdf}}\\label{fig:SingleRing_A}}\n\\hspace{0.25in} {\\subfigure[The lowest eigenvalue against ring\nradius]{\\includegraphics[width =\n0.45\\textwidth]{SingleRingDisk.pdf}}\\label{fig:SingleRing_B}}\n\\parbox{0.75\\textwidth}{ \\caption{Left: A disk with a single ring\npuncture configuration (red dots). Right: The lowest radially symmetric\neigenvalue as a function of the puncture ring radius $r$.  The integral\nequation~\\eqref{eqn:bie} is discretized with $N=128$ boundary points.\n\\label{fig:SingleRing}} }\n\\end{figure}\n\nIn Fig.~\\ref{fig:SingleRing} we see that as the eigenvalue is varied,\n$\\lambda$ attains a maximum value depending on the number of punctures\nand the radius of the ring. In Table~\\ref{tab:SingleRing}, we display\nthe maximum value of the lowest eigenvalue $\\lambda_c$ and the critical\nradius $r_c$ of puncture ring where it is attained. The results in\nTable~\\ref{tab:SingleRing} are in good agreement with~\\cite{LHS} and\ncorroborate the observation that the maximum eigenvalue saturates as the\nnumber of puncture points $M$ increases.\n\nTo explain the saturation effect, we recall\nequation~\\eqref{EigDifference} that relates the puncture and puncture\nfree modes. Consider a mode $(\\lambda_k^{\\star},u_k^{\\star})$ of the\npuncture free eigenfunction problem \\eqref{HoleFree} such that\n$u_k^{\\star}(\\mathbf{x}_j) = 0$, for $j = 1,\\ldots,M$. Then from\n\\eqref{EigDifference} we have that $(\\lambda - \\lambda_k^{\\star})\n\\braket{u,u_k^{\\star}} = 0$.\nThis implies that either $\\lambda = \\lambda_k^{\\star}$ or\n$\\braket{u,u_k^{\\star}} = 0$ so that if $u\\to u^{\\star}_k$, then\n$\\lambda \\to \\lambda_k^{\\star}$. From this we conclude that by centering\npunctures on the nodal set of a puncture free eigenfunction\n$u_k^{\\star}$, then that eigenfunction becomes a mode of\n\\eqref{eqn:intro} thereby eliminating other modes form the spectrum.\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{|c|ccccccc|}\n\\hline\n$M$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$\\\\\n\\hline\n$r_c$ & $0.222$ & $0.348$ &  $0.379$&  $0.379$&  $0.379$&  $0.379$&  $0.379$\\\\\n\\hline\n$\\lambda_c$ & $734.96$ & $1264.2$ & $1581.5$ & $1581.5$ & $1581.5$ & $1581.5$ & $1581.5$\\\\\n\\hline\n\\end{tabular}\n\\parbox{0.75\\textwidth}{ \\caption{The maximum value $\\lambda_c$ of the\nlowest eigenvalue of the unit disk with a single ring of punctures.  The\nvalues are obtained by numerical simulations with $N=128$ boundary\npoints.  \\label{tab:SingleRing}} }\n\\end{table}\n\nIn this disk case with a single ring of punctures, saturation occurs\nwhen the punctures are placed on the nodal set of\n$u^{\\star}_{0,1}(\\mu^{\\star}_{0,1} \\rho)$---the mode with zero angular\nnodal lines and one radial nodal line. From~\\eqref{eigsDiskPFree}, we\nsee that the corresponding eigenvalue is $\\lambda_{0,1}^{\\star} =1581.7$\nwhich agrees closely with the saturating value in Table~\\ref{tab:SingleRing}.\nThe critical radius $r_c$ is found by solving\n\\begin{align*}\n  u^{\\star}_{0,1}(\\mu^{\\star}_{0,1} r_c) = 0 \\quad \n    \\Longrightarrow \\quad r_c \\approx 0.379,\n\\end{align*}\nwhich agrees closely with the numerical results in\nTable~\\ref{tab:SingleRing}. \n\nThe practical importance of this phenomenon is that undesirable\nfrequencies of the plate can be removed by specific placement of\nclamping locations. For example, the placement of five clamped points\nequally spaced on a circle of radius $r_c \\approx 0.379$ yields that\nthe lowest mode of~\\eqref{eqn:intro} becomes $u^{\\star}_{0,1} \\approx\n1585.3$, and the first four (including multiplicities) vibrational\nfrequencies in Fig.~\\ref{fig:NoHoles} have been tuned out. For four\nequally spaced clamping points, the lowest mode would have been\n$\\lambda^{\\star}_{2,0} \\approx1218.9$ resulting in the lowest three\nmodes being tuned out.\n\n\n\\subsection{Rectangular Domain}\\label{sec:rectangle}\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[The first two unclamped modes.]{\\includegraphics[height =\n2in]{RectangleFig2.pdf} \\label{fig:rectangles_a}}\\qquad\n\\subfigure[The first clamped mode along the\ncenterline]{\\includegraphics[height = 2in]{RectangleFig1.pdf} \\label{fig:rectangles_b} }\n\\parbox{0.75\\textwidth}{\\caption{Results for the rectangular\ndomain~\\eqref{eqn:rectangle} with $a=\\sqrt{2}$, $b=1\/\\sqrt{2}$ and a\nsingle clamped point on the horizontal axis. The eigenvalue attains its\nmaximum when clamping is at the origin.\n\\label{fig:rectangles}}}\n\\end{figure}\nIn this section, we consider rectangular domains and demonstrate a\nqualitative agreement with a previous study~\\cite{FM} on the\nlocalization of eigenfunctions of~\\eqref{eqn:intro}.  We parameterize\nthe boundary as\n\\begin{equation}\\label{eqn:rectangle}\n  \\partial\\Omega = (x,y) = (a\\,r(\\theta) \\cos\\theta, b\\,r(\\theta) \\sin\\theta), \\qquad\n  \\theta \\in [0,2\\pi), \\qquad\n  r(\\theta) = \\left[ \\cos^{p}\\theta  + \\sin^{p} \\theta \\right]^{-\\frac{1}{p}},\n\\end{equation}\nwhere $p$ is a parameter which regularizes the corners of the rectangle\nwhile $a$ and $b$ give the aspect ratio. We use $p=16$ in our\ncalculations. Our first simulation investigates the effect of a single\nclamping location on the eigenvalue $\\lambda$ of~\\eqref{eqn:intro}.  We\ntake a single clamped point and vary its location on the horizontal axis\nwhile calculating $\\lambda(r)$ for $r\\in(0,1)$ where $r=0$ and $r=1$\ncorrespond to the left and right hand boundaries. In the curve\nFig.~\\ref{fig:rectangles_b} we see that the lowest eigenvalue attains a\nmaximum at $r=1\/2$ when the clamping occurs at the origin. The value of\n$\\lambda$ at that peak corresponds to the second unclamped mode seen in\neigenfunction of Fig.~\\ref{fig:rectangles_a}. Therefore a single\nclamping point placed at the center of the rectangle completely removes\nthe first mode from the spectrum of~\\eqref{HoleFree}.\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[One clamped point.]{\\includegraphics[width= 0.45\\textwidth]{OneClampedRectangle.png} \\label{fig:ClampedRectangles_a}}\\qquad\n\\subfigure[Two clamped points.]{\\includegraphics[width= 0.45\\textwidth]{TwoClampedRectangle.png} \\label{fig:ClampedRectangles_b} }\n\\parbox{0.75\\textwidth}{\\caption{Mode confinement effect with one (left\npanel) and two (right panel) point constraints on higher (bottom) and\nlower (top) modes of~\\eqref{eqn:intro} in the rectangular\ndomain~\\eqref{eqn:rectangle} with $a=2$, $b= 1\/2$. Clamping occurs on\nthe horizontal axis at locations $\\pm(1.2,0)$.  The simulations use\n$N=512$ discretization points of the domain's boundary.\n\\label{fig:ClampedRectangles} }}\n\\end{figure}\n\nIn our second simulation, we investigate the effect clamping points has\non the eigenfunctions oscillations (cf.~\\cite{FM}). In\nFig.~\\ref{fig:ClampedRectangles} we display two modes\nfor~\\eqref{eqn:intro} for the rectangular domain with $a=2$, $b = 1\/2$\nwith one and two clamped points. The main observation is that a result\nof clamping is a confinement region, i.e.~the eigenfunction is\neffectively zero on either the left or right of the clamping point.\n\n\\subsection{Elliptical Domain}\nIn this section, we consider equation~\\eqref{eqn:intro} on the\nelliptical domain defined parametrically as\n\\begin{equation}\\label{def:ellipse}\n\\partial\\Omega = (x,y) = (a \\cos\\theta,   b \\sin\\theta), \\qquad\n  \\theta \\in [0,2\\pi).\n\\end{equation}\nAs with the disk problem, the defect free eigenfunctions $u^{\\star}$\nsatisfying equation~\\eqref{HoleFree} gives insight into the effect that\npuncturing will have on the modes. In Fig.~\\ref{fig:NoHolesEllipse}, we\ndisplay the first $15$ modes of~\\eqref{HoleFree} for $a = \\frac{3}{2}$,\n$b = \\frac{2}{3}$ with nodal lines.  The choice $a = b^{-1}$ is made so\nthat the ellipse has area $\\pi$.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.95\\textwidth]{EigsEllipse.pdf}\n\\parbox{0.75\\textwidth}{\\caption{The first $15$ clamped modes\nof~\\eqref{HoleFree} of an ellipse~\\eqref{def:ellipse} with $a= 3\/2$, $b\n= 2\/3$. The nodal lines ($u^{\\star}=0$) are in black.\n\\label{fig:NoHolesEllipse}}}\n\\end{figure}\n\nAs with the disk domain, we determine solutions of~\\eqref{eqn:intro}\nover one parameter families of puncture patterns. In this example we\nchoose inscribed ellipses, circular patterns, and rectangular patterns.\nFor the elliptical and rectangular cases, the aspect ratio of the\npuncture pattern is chosen to be the same as the outer ellipse and the\npattern is stretched in a uniform way by a single parameter $r$\n(cf.~Fig.~\\ref{fig:EllipsePatterns}).\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[Ellipse puncture patterns.]{\\includegraphics[width =\n0.475\\textwidth]{ellipsePattern1.pdf}}\\qquad\n\\subfigure[Rectangular puncture patterns.]{\\includegraphics[width =\n0.475\\textwidth]{ellipsePattern2.pdf}}\n\\parbox{0.75\\textwidth}{\\caption{Elliptical and rectangular puncture\npatterns for $M=4,\\ldots,12,$ inside an ellipse \\eqref{def:ellipse} with\n$a = 3\/2$, $b = 2\/3$. The aspect ratio of the patterns matches\nthat of the outer ellipse. \\label{fig:EllipsePatterns} }}\n\\end{figure}\n\nThe results for the puncture patterns corresponding to\nFig.~\\ref{fig:EllipsePatterns} and the circular pattern\nFig.~\\ref{fig:SingleRing} are given in Fig.~\\ref{fig:EllipseResults}.\nThe curves show that the pattern that generates the maximum eigenvalue\nvaries for different $M$. The horizontal axis is a positive parameter\n$r$ which is a scale factor controlling the size of the pattern. The\nelliptical puncture pattern generates the largest eigenvalue for all $M$\ntested except for $M=5$ where the rectangular patter generates a higher\nvalue. The circular pattern generally results in a lower maximum\neigenvalue than the other patterns tested, expect in the $M=4$ case\nwhere the circular example generates a slightly larger value than the\nrectangle.\n\nThe discussion from the previous circular and rectangular examples,\ntogether with formula~\\eqref{EigDifference}, suggest that clamping\nalong the nodal lines of the unperturbed mode results in a maximum\ndeviation of the eigenvalue $\\lambda$ of~\\eqref{eqn:intro}. This\nsuggests that the better performance of the elliptical pattern in\nmaximizing the eigenvalue is that it more closely places punctures on\nthe nodal lines (cf.~Fig.~\\ref{fig:EllipsePatterns}) of\n$u^{\\star}(\\mathbf{x})$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = \\textwidth]{EllipseResults.pdf}\\\n\\parbox{0.75\\textwidth}{\\caption{The first eigenvalue of \\eqref{eqn:intro} for the ellipse \\eqref{def:ellipse} with $a= 3\/2$, $b = 2\/3$ and $M=4,\\ldots,12,$ punctures. Curves for rectangular (blue), elliptical (black) and circular (red) patterns. \\label{fig:EllipseResults} }}\n\\end{figure}\n\n\\subsection{Non-symmetric geometry}\\label{sec:nonsymmetric}\n\nHere we consider the asymmetric domain whose boundary $\\partial\\Omega$\nis specified parametrically as\n\\begin{equation*}\n  \\partial\\Omega = (r(\\theta)\\cos\\theta,r(\\theta)\\sin\\theta), \\qquad \n  \\theta\\in[0,2\\pi); \\qquad \n  r(\\theta) = 1 + 0.25\\sin \\theta + 0.15\\cos 3\\theta,\n\\end{equation*}\nand investigate the solution of~\\eqref{eqn:intro} with a single clamped\npoint. As in the previous examples, the first step is to consider the\neigenvalues of the hole free problem~\\eqref{HoleFree}. The first two\nmodes are shown in Fig.~\\ref{fig:asym_a}.\n\nTo initiate Newton iterations for this example, we begin with a single\nclamped point near the boundary so that the unperturbed mode\n$\\lambda^{\\star}\\approx 118.3$ in Fig.~\\ref{fig:asym_a} provides a good\ninitial guess for the system. From this start point, we vary the\npuncture location along a straight line (blue line in\nFig.~\\ref{fig:asym_a}) through the center of the domain to the opposing\nboundary. The maximum of this curve is attained when the clamping\nlocation coincides with the maximum of the first eigenfunction,\noccurring at the solid dot shown in Fig.~\\ref{fig:asym_a}.\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[The first and second unpunctured\nmodes.]{\\includegraphics[width=0.5\\textwidth]{AsymTogether.pdf}\\label{fig:asym_a}}\\qquad\n\\subfigure[The first punctured\neigenvalue.]{\\includegraphics[width=0.32\\textwidth]{asym3.pdf}\\label{fig:asym_c}}\n\\parbox{0.75\\textwidth}{\\caption{Panel (a): The first two modes of the hole free problem~\\eqref{HoleFree}. Panel (b): The first eigenvalue of~\\eqref{eqn:intro} for a clamped point located a distance $r$ along the blue line in panel (a). \\label{fig:asym} }}\n\\end{figure}\nIn this example we again see that the first frequency can be tuned out of the vibrational characteristics of the plate by careful placement of just one clamping point.\n\\subsection{Non-simply connected geometry}\nIn this section, we consider a non-simply connected domain comprised of\nthe unit disk with a circle of radius $0.2$ and center $(-0.3,0)$\nremoved.  This multiply-connected domain has a confinement zone in the\neigenfunction (see Fig.~\\ref{fig:couette1}). Throughout this confinement\nzone (shaded region of Fig.~\\ref{fig:couette1a}), the eigenfunction is\nvery close to zero indicating again that small defects (holes, clamping\npoints) cause a global perturbation to the modal characteristics in the\nfourth-order problem \\eqref{eqn:intro}.\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[Domain and first unclamped mode.]{\\includegraphics[width = 0.3\\textwidth,clip]{CouetteDiagram.png} \\label{fig:couette1a}}\\hspace{1in}\n\\subfigure[Effect of clamping along an ellipse.]{\\includegraphics[width\n= 0.4\\textwidth,clip]{Couette_Ellipses.pdf} \\label{fig:couette1b}}\n\\parbox{0.75\\textwidth}{\\caption{Non simply-connected domain example: Left: The\nfirst mode corresponding to $\\lambda^{\\star} = 454.1$ with the shaded\nconfinement zone in which\n$u^{\\star} \\approx 0$. Right: The clamped eigenvalue for a single point\non inner ellipses with $a = 0.95$ and $b = 0.5$ (red curve), $0.6$ (blue curve), $0.7$ (green curve). The clamped\nmodes corresponding to the black squares are shown in Fig.~\\ref{fig:couetteModes}.\n\\label{fig:couette1}}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$b = 0.5$.]{\\includegraphics[width = 0.4\\textwidth,clip]{Couette_b1-2th145} \\label{fig:couetteModesa}}\\hspace{0.5in}\n\\subfigure[$b = 0.7$.]{\\includegraphics[width = 0.4\\textwidth,clip]{Couette_b7e-1th} \\label{fig:couetteModesb}}\n\\parbox{0.75\\textwidth}{\\caption{The clamped modes corresponding to\nsolid squares in Fig.~\\ref{fig:couette1b}. The mode changes sign in the\nvicinity of the clamping point. Left: The single clamped point is $\\mathbf{x}_1\n= (0.114, 0.496)$ and $\\lambda = 494.5$. Right: The single clamped point\nis $\\mathbf{x}_1 = (0.648, -0.512)$ and $\\lambda = 503.4$.}\n\\label{fig:couetteModes} }\n\\end{figure}\n\nTo investigate the effect of clamping, we vary the location of a single\nclamped point along an inner ellipse with parameterization\n\\eqref{def:ellipse} for $a=0.95$ and the three values $b= 0.5$, $0.6$,\n$0.7$. The results, displayed in Fig.~\\ref{fig:couette1b}, show the\neigenvalue as a function of angular position on the ellipse. The flat\nregion in the center of each curve corresponds to the transit of the\nclamped point through the confinement zone. In this region the\neigenfunction is very close to zero ($u^{\\star}\\approx 0$), therefore\nthe formula~\\eqref{EigDifference} shows that the eigenvalue should\ncorrespond to the unperturbed mode of Fig.~\\ref{fig:couette1a}.\n\nIn Fig.~\\ref{fig:couetteModes} we show two typical clamped modes which correspond to the particular clamping locations marked in Fig.~\\ref{fig:couette1b}. In each case a large confinement zone is seen to the left of the hole while the mode is also suppressed in the vicinity of the clamped point.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nThis paper has analyzed the modes of vibrations of thin elastic plates\nwith multiple point constraints. We have developed and validated a novel\nboundary integral method for determining the eigenvalues of the\nfourth-order bi-Laplacian problem~\\eqref{eqn:intro} with multiple\nclamped point constraints. This method is third-order accurate and can\nbe easily applied to a variety of symmetric, asymmetric and\nmultiply-connected geometries in two dimensions.\n\nOur results indicate that the number and location of clamping points has\ntwo profound effects on the modes of vibrations of thin plates. First,\nby placing the clamping locations at the nodal lines of the unclamped\neigenfunctions, certain eigenvalues can be removed from the spectrum of\nthe problem.  This implies that the vibrational characteristics of the\nplate can be manipulated or tuned by judicious placement of a small\nnumber clamping sites. A particularly important consequence of this\neffect in engineering applications is that undesirable frequencies of\nvibration can be completely removed.  Second, clamping can have the\neffect of partitioning the plate into multiple subdomains in which\nvibrational modes are largely confined to a subset of those smaller\nspatial regions.  This localization effect, previously seen only in\nrectangular plates with a single clamping location~\\cite{FM}, has been\nobserved here in other geometries and for multiple clamping locations. \n\nThere are many possibilities for future work that arise from this study.\nThe order of the numerical method could be improved by adopting\nquadrature methods (cf.~\\cite{alp1999, kap-rok1997}) suited to integrals\nwith logarithmic singularities.  In addition, the boundary integral\nequation could be more efficiently solved with a fast summation method\nsuch as the kernel-independent fast multipole\nmethod~\\cite{yin-bir-zor2004}.  A more significant challenge is to\ndevelop a numerical method for the point constraint eigenvalue\nproblem~\\eqref{eqn:intro} which does not require the solution of a\nnonlinear system, such as~\\eqref{eqn:mismatch}, but still enforces the\nweak logarithmic singularity on the eigenfunction. This would eliminate\nthe use of Newton's method and the need to carefully control the initial\nconditions of the system. Such a method would easily accommodate a much\nlarger number of clamping locations and allow for more reliable\nevaluation of the high frequency modes of~\\eqref{eqn:intro}.\n\n\n\n\\section*{Acknowledgements} A.E.L.~was supported by the National Science\nFoundation under grant DMS 1516753.  B.Q.~was supported by startup funds\nat Florida State University.\n\n\\begin{appendices}\n\\section{Kernels}\n\\label{app:kernels}\n\nThe four kernels that appear in the integral equation~\\eqref{eqn:bie}\nare $G_{11} = G_{1}$, $G_{12} = G_{2}$, $G_{21} =\n\\partial{\\mathbf{n}_{\\mathbf{x}}}G_{1}$, and $G_{22} = \\partial{\\mathbf{n}_{\\mathbf{x}}}G_{2}$.\nExpressions for $G_{11}$ and $G_{12}$ are in Section~\\ref{sec:kernels}.\nHere we compute their normal derivatives with respect to the target\npoint.\n\n\\begin{align*}\n  G_{21} = -\\frac{1}{4\\pi\\mu^{2}}&\\left(\n    -3\\mu^{4}K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{2}}\n    -12\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n    +3\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho}\n  \\right. \\\\\n  &\\left.\n    +2\\mu^{4}K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{4}}\n    +16\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n    -6\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n  \\right. \\\\\n  &\\left.\n    +6\\mu^{2}K_{0}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{2}} \n    -24\\mu^{2}K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^4}\n    -24\\mu^{2}K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{4}}\n  \\right. \\\\\n  &\\left.\n    +48\\mu^{2}K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{6}}\n    +96\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{7}}\n    -48\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{5}}\n  \\right. \\\\\n  &\\left.\n     -48\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n     +12\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n  \\right) \\\\\n  -\\frac{1}{8\\mu^{2}}&\\left(\n    -3\\mu^{4}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{2}}\n    +12\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n    -3\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho}\n  \\right. \\\\\n  &\\left.\n    +2\\mu^{4}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{4}}\n    -16\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n    +6\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n  \\right. \\\\\n  &\\left.\n    -6\\mu^{2}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{2}} \n    +24\\mu^{2}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^4}\n    +24\\mu^{2}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{4}}\n  \\right. \\\\\n  &\\left.\n    -48\\mu^{2}Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{6}}\n    +96\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{7}}\n    -48\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{5}}\n  \\right. \\\\\n  &\\left.\n    -48\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n    +12\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n  \\right).\n\\end{align*}\n\\begin{align*}\n  G_{22} = -\\frac{1}{4\\pi}&\\left( \n    8 K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^2(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{4}}\n    +\\frac{16}{\\mu}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n    -4K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{2}}\n    \\right. \\\\\n    &\\left.\n    -\\frac{8}{\\mu}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n    -\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho} \n    -\\frac{4}{\\mu}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n    \\right. \\\\\n    &\\left.\n    -2K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{2}} \n    +2\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n  \\right) \\\\\n  +\\frac{1}{8}&\\left( \n    8 Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^2(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{4}}\n    -\\frac{16}{\\mu}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{5}}\n    -4Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{2}}\n    \\right. \\\\\n    &\\left.\n    +\\frac{8}{\\mu}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)(\\mathbf{n} \\cdot \\mathbf{n}_{\\mathbf{x}})}{\\rho^{3}}\n    -\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho} \n    +\\frac{4}{\\mu}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n    \\right. \\\\\n    &\\left.\n    -2Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{2}} \n    +2\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{2}(\\mathbf{r} \\cdot \\nn_{\\mathbf{x}})}{\\rho^{3}}\n  \\right).\n\\end{align*}\n\\end{appendices}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe prehistoric stars whose formation epochs lie beyond the redshift\naccessible by the Hubble Ultra Deep Field, have been found en masse in\nlittle satellites around the Milky Way. Other less fortunate dwarf\ngalaxies have been pulled apart by gravity to furnish the diffuse\nGalactic halo. These recently uncovered relics of the ancient dwarf\ngalaxy population may play a vital role in the pursuit of\nreconstructing the formation of the Galaxy. Its path from the distant\npre-reionisation era, through the most active growth periods to the\npresent day, can be gleaned by studying the chemical composition and\nthe phase-space density distribution of these halo denizens. How this\nGalactic {\\it archaeological} record is collected and analyzed, and\nwhat aspects of galaxy (dwarf and otherwise) formation and evolution\nit illuminates is the topic of this review.\n\nThe field of Galactic and Local Group studies has enjoyed a decade of\nunprecedented busyness thanks to the abundance of data supplied by\nseveral generations of wide-angle sky surveys and the coming of age of\nN-body and hydro-dynamic computer simulations of galaxy\nformation. Accordingly, there have been several fresh in-depth reviews\nof the matters relevant to the topic of this article. In particular,\nthe star-formation histories and abundances of the Milky Way dwarf\ngalaxies have been scrutinized by \\citet{Tolstoy2009}.\n\\citet{Mcconnachie2012} has painstakingly assembled a homogenized\ndatabase of the properties of all known dwarfs within 3\nMpc. \\citet{Walker2012} has written down a scrupulous account of the\ncurrent evidence for the presence of Dark Matter in dwarfs, while\n\\citet{kravtsov2010} has reviewed the progress in reconciling the\nmismatch in the appearance of the real dwarf satellites and the toy\nones built into simulated Dark Matter sub-structure. The quest for the\nleast luminous galaxies, so-called {\\it ultra-faint dwarfs}, has been\ndocumented by \\citet{Willman2010}. To complement these, a review of\nthe structure, chemistry and dynamics of the Galactic stellar halo can\nbe found in \\citet{Helmi2008}.  Finally, \\citet{Ivezic2012} explain\nexactly how the large surveys like the SDSS have revolutionized the\nway research into the Galactic stellar populations is conducted.\n\nThis review will attempt to avoid boring the reader with re-stating\nthe facts already discussed thoroughly in the works above. Instead,\nits purpose is to report on the most recent progress in the area of\nthe Galactic Archaeology and to list some of the burning questions\nthat are destined to be answered with the upcoming sky surveys.\n\n\\section{Little galaxies. Big questions.}\n\nThe main premise of the current galaxy evolution theory, which itself\nexists within the broader theory of the Universal structure formation\na.k.a. $\\Lambda$CDM, postulates that all galaxies are born, live and\ndie inside dark matter (DM) halos. $\\Lambda$CDM uses Cold Dark Matter\nto provide nucleation sites for the subsequent budding of galaxies of\nall sizes. Small lumps in the primordial DM gravy are the most\nnumerous and develop the quickest, therefore the first baryonic\nsystems to appear are the dwarf galaxies. What happens later is the\ncompetition between the expansion of the Universe and the\ngravitational pull of the emerging DM halos. In this game, the\nmajority of the dwarfs eventually lose: they are dragged into deeper\npotential wells, where they get undone and their matter, dark and\notherwise, is subsumed to form bigger galaxies.\n\nA small fraction of the aboriginal dwarf satellite population that\nsurvives the tidal disruption during the accretion should, in\nprinciple, be detectable in and around the Galaxy today. $\\Lambda$CDM\nis a young theory and it is perfectly reasonable that its fundamental\nproperties are still being actively questioned. One key prediction of\nthe theory is the existence of the large number of small DM halos\n(called {\\it sub-halos} to indicate the hierarchical nature of all\nstructures in this paradigm) in a galaxy like our own. The theory then\npostulates that dwarf satellites are the sub-halos that have accreted\nenough gas and have held on to it long enough to cool it down and form\nstars. Those dwarfs that have survived until $z=0$ must have either\nstayed well away from the strong tides of the central Milky Way or are\njust arriving to their final destination \\citep[see\n  e.g.][]{Bullock2005}.\n\n\\subsection{Galactic dwarf census}\n\nGalactic Archaeology provides the observational evidence of the\naccretion onto the Milky Way and the statistics of the survived and\nthe destroyed dwarfs. This data can be interpreted within the current\nhierarchical structure formation model provided the processes to do\nwith star formation and interaction between the baryons and the DM are\nwell understood. For example, \\citet{Klypin1999} demonstrate how\nmerely the count of the Galactic dwarf galaxies can be used to\nchallenge the very foundations of the accepted galaxy formation\nparadigm. As they show, it is possible that to reconcile the\nmeasurement and the prediction of the satellite mass function, a\ncut-off in the power spectrum of the primordial matter fluctuations on\nsmall scales is required. Alternatively, they point out, it is\nperfectly feasible that the suppression of star formation in low-mass\nsystems is stronger than expected, thus leaving the vast majority of\nsub-halos dark forever.\n\nThe number of the low-mass galaxies around the Milky Way has more than\ndoubled in the last ten years solely due to the supply of high quality\nall-sky data from the SDSS. Yet, any attempt to corroborate the theory\nof dwarf formation based on these recent discoveries, would be\nhopeless without first quantifying how much the rapidly growing\nsatellite sample is influenced by the selection effects.  All Galactic\ndwarfs (excluding the Sagittarius and the Sextans dSphs) discovered\nbefore the first SDSS data releases and now conventionally known as\n{\\it Classical} were found by simply eye-balling the photographic\nplates. The SDSS has changed the game completely: today the\nscience-ready catalogs containing hundreds of millions of stars and\ngalaxies can be trawled through quickly, making the search efficient\nand quantifiable. Using the early released data from only two SDSS\nruns, \\citet{Willman2002} estimates the sensitivity of the full survey\ndataset to resolved Galactic companions and predicts that the\nsatellite Luminosity Function can be straightforwardly constructed\nfacilitating the first unbiased comparison between the theory and\nobservations. Accordingly, \\citet{Koposov2008} present the\ncompleteness calculation for the SDSS DR5 based on the fully-automated\nsatellite detection algorithm and a large suite of realistic mock\nobservations of dwarf galaxies of various sizes and luminosities. The\nresults presented by \\citet{Koposov2008} show clearly that there are\nlarge swathes of the parameter space where the Milky Way satellites\ncan be detected with nearly $100\\%$ efficiency.  Outside these\nregions, the detectability drops sharply to 0. The transition boundary\nis controlled by the surface brightness limit of the SDSS $\\mu_{\\rm\n  lim}$, the luminosity of the satellite and its\ndistance. Alternatively, for all objects with $\\mu_{\\rm lim} < 30$ mag\narcsec$^{-2}$, the SDSS completeness can be expressed in terms of the\nvolume within which it discovered {\\it all} satellites of a particular\nluminosity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{lf2.pdf}\n\\caption{Dependence of the Galactic dwarf luminosity function on the\n  assumed radial density profile of the satellites. This shows the\n  cumulative luminosity functions (the total number of objects as a\n  function of their absolute magnitude) for the Milky Way satellites\n  in the spherical volume with radius $R=280$ kpc (filled circles),\n  which is similar to the volume used in \\citet{Koposov2008}; and with\n  $R=400$ kpc (squares), similar to the maximal distance used in\n  \\citet{Tollerud2008}. Red curves give the LFs obtained with the\n  satellite radial profile proportional to $r^{-3}$, while the LFs\n  given in black assume that the number of satellites decays as\n  $r^{-2}$. Blue curve is the LF based on the radial distribution of\n  the Via Lactea sub-halos. All density laws are truncated at $R=10$\n  kpc, the satellite detection efficiency applied is from\n  \\citet{Koposov2008}. Only $\\sim 100$ dwarfs are predicted to inhabit\n  the Galaxy if their radial number density follows the $r^{-3}$ law\n  (red curves), i.e. a NFW distribution with small scale radius, see\n  also \\citet{Koposov2008}. Substantially more ultra-faint objects are\n  anticipated if the inverse square law is adopted, particularly if\n  the dwarf population extends to distances as large as $R=400$ kpc\n  (black squares). In Via Lactea simulation, in the inner 60-80 kpc,\n  the number density of sub-halos is even flatter as shown in the\n  Supplementary Figure 1 of \\citet{Diemand2008}. Accordingly, most of\n  the faintest satellites are supposed to be undetected by the SDSS,\n  as reflected in the quick rise of the LF (in blue) at $M_V \\sim -3$,\n  see also \\citet{Tollerud2008}. Figure courtesy of Sergey Koposov,\n  IoA.}\n\\label{fig:lf}\n\\end{figure}\n\nAccording to \\citet{Koposov2008}, the sample of dwarf satellites with\nluminosities brighter than $M_{V}\\sim -5$ is essentially complete out\nto the Galactic virial radius $r_{\\rm vir}=280$ kpc within the SDSS\nDR5 field of view. However, the accessible volume plummets fast with\ndecreasing dwarf luminosity, which means that for satellites as faint\nas Segue 1 with $M_V\\sim-3$, only few percent of the virial volume\nhave been probed. Importantly, using thus-calculated fraction of the\ntotal Galactic volume sampled, it is now possible to predict the\ncomplete number of the satellites if their distribution with radius is\nassumed. Naturally, the flatter the radial distribution of the\nsatellites the bigger is faction of objects remaining to be\ndetected. \\citet{Koposov2008} surmise that if the number of dwarfs\ndecays in a NFW-like fashion ($\\propto r^{-3}$ at large distances)\nthen the Luminosity Function (LF) of the Milky Way satellites goes as\n$\\sim 10^{0.1(M_V+5)}$. According to this LF, the objects with\nluminosities between $M_V=-2$ and $M_V=-5$ contribute just under a\nhalf of the total $\\sim 100$ dwarfs. Some 4 times more satellites is\npredicted by \\citet{Tollerud2008} who take advantage of the\ncompleteness calculation published by \\citet{Koposov2008}, but choose\nto adopt the radial distribution of sub-halos from the Via Lactea\nN-body simulation \\citep{Diemand2008}. In the inner 60-80 kpc, the\ndistribution of sub-halos in Via Lactea drops very slowly with radius.\nIn fact, for several tens of kpc it is almost flat according to the\nSupplementary Figure 1 of \\citet{Diemand2008}. The effect of this\nlarge ``core'' in the satellite number density profile on the dwarf LF\nis particularly striking for the faintest of the satellites, those\ndetectable by the SDSS only out to 30-40 kpc. Out of the fiducial\n$\\sim$ 400 satellites predicted by \\citet{Tollerud2008}, almost $90\n\\%$ are those with $-2 < M_V < -5$. The drastic dependence of the\nGalactic dwarf LF on the assumed radial distribution of satellites is\nillustrated in Figure~\\ref{fig:lf}. From these first attempts to gauge\nthe size of the Galactic satellite body, the important role played by\nthe faintest of the dwarfs emerges.\n\n\\subsection{Unexplored variety of hierarchical galaxy formation}\n\nIn the decade following the publication of the two whistle-blowing\npapers by \\citet{Klypin1999} and \\citet{Moore1999}, the {\\it sub-halo\n  abundance matching} (SHAM) technique that relies on assigning higher\nstellar luminosities to the DM sub-halos with higher maximal masses,\nhas been optimized near to perfection. The success of the abundance\nmatching has created a brief period of the comforting lull with the\ngaping void between dwarf galaxies and sub-halos (known as the {\\it\n  missing satellites problem}) seemingly breached\n\\citep[e.g.][]{Koposov2009,Maccio2010}. However, pitted against the\nexistent Galactic satellites, these mock dwarfs do not stand a close\nscrutiny: the density profiles in the most massive systems are not\ncorroborated by the available kinematics \\citep[see\n  e.g.][]{Boylan-Kolchin2012}. It turns out, the overall\nstar-formation efficiency is now such a strong function of the\nsub-halo mass that the dwarfs with modest stellar masses and velocity\ndispersions are forced to inhabit disproportionally massive and dense\nhalos. What is missing from this picture drawn with the help of SHAM?\nIt appears quite a few of the vital ingredients might be lacking.\n\nTo begin with, the influence of the rather significant baryonic disk\nin the Galaxy has been surprisingly overlooked in many of the SHAM\nstudies. Yet, as demonstrated in a recent succession of papers\n\\citep[e.g.][]{Taylor2001,Read2006,Penarrubia2010,D'onghia2010}, disks\naide the tidal destruction of satellites thus seriously depleting the\nnumber of the dwarf survivors. One crucial detail is noted by\n\\citet{Penarrubia2010}: dwarf galaxies with cored density profiles are\nless likely to survive the devastating action of the\ndisk. $\\Lambda$CDM on its own does not permit cored dark matter\nprofiles, but \\citet{Read2006} and \\citet{Pontzen2012} show that the\ngas flow due to the persistent supernova feedback can pull dark matter\nalong to the outskirts of dwarf galaxies. Accordingly, a plausible\nsetup in which the satellite survival rate at $z=0$ is regulated by\nthe interplay between the strong stellar feedback evacuating the\ncenters of sub-halos and the enhanced tidal disruption due to the disk\nis described in the work by \\citet{Brooks2013}. Here, by means of\napplying a simple correction to the central masses of semi-analytical\ndwarfs, as originally proposed by \\citet{Zolotov2012}, many of the\nmassive satellite galaxies are wrecked, and the tension between the\ndata and the theory seems to be alleviated once again. Note, however,\nthat the total amounts of supernova energy required to cause\nappreciable damage to the DM central density cusps have been deemed\nexcessive by many authors \\citep[e.g.][]{penarrubia2012,gk2013}.\n\nAs the physics of star formation is just starting to be explored,\nthere does not exist a single hydro-dynamical simulation of the Milky\nWay run at the resolution appropriate to resolve the gas infall and\ncooling at all epochs from high redshifts to the present day. Instead,\non a star by star basis, the processes that play the most important\nrole (like cooling and feedback) are gleaned, synopsized and\nsubsequently incorporated into the simulations as {\\it sub-grid}\nrecipes to be followed together with the laws of the Newtonian gravity\n(and sometimes hydrodynamics). In this approach, it is believed that,\nthe evolution of the dark matter density on the relevant scales has\nbeen fully captured with the latest pure N-body simulations. However,\nas the DM particles are followed from the distant past to the current\nday, the actual sequence of accretion events the Milky Way prototype\ngoes through varies considerably from host to host. This has profound\nimplications on the final shape of the DM sub-halo mass function (MF),\nand ultimately on the properties of the Galaxy's dwarf satellite\npopulation.\n\n\\subsubsection{Host mass and concentration}\n\nWhen faced with the myriad of DM halos to furnish a Milky Way together\nwith its satellites, the conventional choice is to select the host by\nmatching its virial mass to that of the Galaxy. Note that according to\nFigure 8 in \\citet{Springel2008}, in the 6 host halos with different\nvirial masses, the sub-halo number counts in the bins of mass scaled\nto the the mass of the host lie exactly on top of each\nother. Accordingly, it is established that the relative MF scales as\n$(M_{sub}\/M_{50})^{-1.9}$ and the absolute sub-halo abundance\nnormalization includes another factor of $M_{50}$ for the host\nmass. Thus, for the hosts whose masses are different by $100\\%$ the\ntotal sub-halo counts would also disagree by a factor of 2. Alas, the\nmass of the real Milky Way is not known with the accuracy as high as\n$100\\%$. Even though, depending on the method used, the formal\nuncertainties can be as low as $30\\%$, there are serious disagreements\nbetween the measurements making the systematic error much higher. For\nexample, according to \\citet{Watkins2010}, the plausible range for the\nMilky Way mass within 300 kpc is approximately from $1 \\times 10^{12}\nM_{\\odot}$ to $3 \\times 10^{12} M_{\\odot}$, which would imply a factor\nof $\\sim$3 difference in the total number of sub-halos.\n\nThe present uncertainty in the measurements of the Milky Way's\nconcentration is $\\sim 100\\%$, which is perhaps even more appalling\nsince the allowed range of concentrations is much smaller. The\nconcentration $c=r_{vir}\/r_s$ of a DM halo describes how dense its\ninner parts (within the scale radius $r_s$) are compared to the halo\noverall (out to the virial radius $r_{vir})$. For the halos of similar\nvirial mass, their concentrations are ultimately linked to the shape\nof the host's mass assembly history, with those peaking at very early\ntimes producing higher concentrations \\citep[see\n  e.g.][]{Wechsler2002}. For the population of dwarf satellites at\nredshift zero, having a host halo with a high concentration is a\ndouble whammy. The first implication is obvious: if the accretion\nhistory settled into the quiescent phase at high redshifts, at later\ntimes, fewer dwarfs will be accreted. Second, having a dense pile up\nof dark matter (and baryons) means more efficient tidal disruption and\nthe lower satellite survival rate. According to the N-body\nsimulations, for the halos with Milky Way-like masses, the\nconcentration is predicted to be of order of $c=12 \\pm 3$ \\citep[see\n  e.g.][]{Boylan-Kolchin2010}.\n\nThere are four techniques available today to measure the Galactic\nmass, each with its own assumptions and inherent limitations. The\nfirst three rely on the kinematics of a sample of the gravitational\npotential tracers, and therefore can only be applied straightforwardly\nwithin few tenths of the Galactic virial radius. The first method uses\nstars or gas to determine the run of the Galactic rotation velocity\n$v_{\\rm rot}=\\sqrt{r\\frac{{\\rm d}\\Phi}{{\\rm d}r}}$ (where $\\Phi$ is\nthe underlying potential) with Galacto-centric radius $r$. Naturally,\nthe rotation curve can only be sampled within $r < 20$ kpc, as there\nis no indication that the disk continues much beyond that. Having the\nfull 6D information is rare, the latest such attempt presents the\nmeasurements of trigonometric parallaxes for a dozen or so masers,\nfrom which the circular rotation speed at the Sun's location is\ndeduced \\citep{Reid2009}.  Alternatively, \\citet{Bovy2012} shows that\nthe circular velocity can be inferred by marginalizing over poorly\nconstrained distances and unknown proper motions for a large\n($>3,000$) set of (mostly) disk giant stars with accurate\nline-of-sight velocities.\n\nAn equally rewarding, but perhaps yet a more challenging approach is\nto gauge the Galactic escape speed $v_{\\rm esc}=\\sqrt{2|\\Phi|}$ by\nanalyzing the tail of the stellar velocity distribution. The results\nare sensitive to the quality of the distance and the proper motion\ndata, in particular, imperfect proper motion measurements are so\ndetrimental that, normally, they are avoided altogether. Instead, a\nvelocity distribution function is chosen, whose exact shape is\ncontrolled by a small number of parameters that get simultaneously\nconstrained in the process of the likelihood maximization. For\nexample, using a relatively small sample (16) of high-velocity stars\nprovided by the earlier releases of the RAVE survey, \\citet{Smith2007}\nmeasure the local escape speed. Conveniently, given the Galactic\nescape speed and assuming the contributions of the bulge and the disk\nto the total potential, the mass and the concentration of the Milky\nWay's halo also can be extracted. The analysis by \\citet{Smith2007}\nseems to prefer the Galaxy with the mass as low as $0.9 \\times 10^{12}\nM_{\\odot}$ and the concentration as high as 24. While the\napplicability of both the circular speed and the escape speed\ntechniques is restricted to the inner Galaxy, the latter has the\nadvantage of probing the Galactic mass out of the disk plane.\n\nMost of the Milky Way's mass lies beyond the extent of the disk, hence\nat large Galacto-centric distances, a different approach is required.\nGiven enough mass tracers (stars or satellites) in a wide range of\nlocations throughout the Galaxy, the total mass profile can be\nobtained by means of Jeans modelling of the tracer kinematics\n\\citep[see e.g.][]{Battaglia2005}. The terms that enter the spherical\nJeans equation are: the tracer density, the tracer velocity dispersion\nand the tracer velocity anisotropy. At large distances, only one of\nthese might be available, namely the line-of-sight velocity\ndispersion. Making the Jeans analysis of the far reaches of the\nGalactic halo possible clearly falls within the realm of Galactic\nArchaeology which can both deliver the most distant tracers as well as\nconstrain the overall tracer density distribution.  The stumbling\nblock, however, is the scarcity of tracers with the tangential\ncomponents of the velocity measured. As a consequence, the anistropy\nis normally treated as a nuisance parameter since the most datasets\navailable lack in accuracy and breadth to constrain it. Even with the\narrival of Gaia, the situation will only improve for the nearby\nobjects, leaving the distant ones wanting in more precise proper\nmotions. While assigning anistropy to a tracer population is a\nsolution far from ideal, presently, it is the Jeans modelling together\nwith its variants that provides the most stringent constraints on the\ntotal mass of the Milky Way \\citep[e.g.][]{Xue2008}.\n\nFinally, a new, conceptually different method to probe the matter\ndistribution in the Galaxy is now coming of age. Compared to the three\napproaches discussed above, it does not rely at all on the\ninstantaneous kinematic properties of large samples of tracers, and\nthus, for example, needs no assumption of their velocity\nanisotropy. Stellar {\\it tidal streams} are shown to align closely\nwith the obit of their disrupting (or disrupted) progenitor and\ntherefore give an almost direct way of measuring the underlying\npotential. Recently, the power of the method has been demonstrated\nbeautifully by \\citet{Koposov2010} who, using the 6D data of the GD-1\nstream, measured the Galactic rotation curve locally. This type of\nanalysis can, in principle, be extended to distances beyond the\npredicted Galaxy's scale radius $r_s$. The prime source of degeneracy\nin recovering the Galactic potential using tidal tails, is the length\nof the stream available. However, to date, for several distant streams\nthere exists sufficient data covering tens \\citep[Orphan Stream with\n  the maximal distance of $\\sim 50$ kpc][]{Belokurov2007a,\n  Newberg2010} or even hundreds of degrees \\citep[Sagittarius Stream\n  with the maximal distance of $\\sim 100$ kpc, e.g.][]{ Majewski2004,\n  Newberg2003,Belokurov2006b, Yanny2009, Belokurov2013}. Given the\nmagnitude limit of the on-going imaging surveys like SDSS or\nPan-STARRS, for stellar streams to be detected so far out in the halo,\nthe progenitor's luminosity, and therefore mass, ought to be\nsubstantial. This bias implies that the currently known distant\nstreams can not be appropriately modeled using simple orbit\napproximation, the circumstance that now can be mitigated with the\narrival of more sophisticated modeling techniques\n\\citep[e.g.][]{Eyre2011,Sanders2013}\n\n\\subsubsection{Mass assembly history and environment}\n\nThe computational expense of running numerical simulations of Galactic\nhalos at the resolution adequate to capture the properties of the halo\nsub-structure is prohibitively high. Hence, the comparison between DM\nsub-halos and the observed dwarfs has been based on the analysis of\nonly 8 N-body simulations: a sample of 6 Aquarius halos\n\\citep{Springel2008}, complemented by halos of Via Lactea II\n\\citep{Diemand2008} and GHalo \\citep{Stadel2009}. For this reason, the\nhost-to-host variation of the dark and the luminous sub-structure\nremains largely un-studied. As well as improving the resolution and\nthe speed of the simulations, there is an ongoing effort to quantify\nthe complex diversity of structures forming within $\\Lambda$CDM with a\nhandful of key parameters, e.g. host halo mass, shape of the accretion\nhistory and significance of the overdensity of the local\nvolume. These, of course, are inter-related: the mass of the DM halo\nhosting a Milky Way galaxy at redshift $z=0$ is the sum total over its\naccretion history, which in turn is dictated by the whereabouts of the\nhalo within the cosmic Large Scale Structure. While the importance of\nnot knowing such an elementary property of the Galaxy like its mass is\nnow accepted, the impact of the location of the Milky Way within the\nlarger cosmic structure and the details of its accretion history are\njust beginning to be investigated.\n\nToday, there exist two intriguing constraints on the Milky Way's\naccretion history.  First is the observation that the Galactic disk\nprobably has to survive intact for some 7-10 Gyr \\citep[e.g. Figure 18\n  of][]{Burnett2011}. This, therefore, potentially excludes any\nsignificant mergers between $z \\sim 1$ and now. Second is the new\nobservational and numerical evidence for the late infall of the\nMagellanic Clouds \\citep[e.g.][]{Besla2010}. This signifies the end of\nthe quiescent phase in the Galactic accretion history and can be\nexploited to place useful constraints on the mass assembly of the\nGalaxy \\citep[e.g.][]{Busha2011}. What happened before the quiescent\nphase, why did it begin and why did it end?  How common is this\nparticular shape of the {\\it mass assembly history} (MAH) amongst\nother disk galaxies of similar mass?  Was the early accretion\ndominated by small satellite infall and was it synchronized?  Or\nperhaps, was the bulk of the Galactic matter instead acquired in one\nor two mergers with massive nearby fragments?  Unfortunately, these\nquestions remain largely unanswered and therefore, a variety of loose\nends continues to confuse the current picture of the Galaxy formation\nand muddle the modelling of the nearby dwarfs. For example, if many\nsmall satellites are accreted early on, enough should survive and be\ndetectable today. On the contrary, massive mergers usually lead to an\nentirely different outcome: in this case, the dynamical friction is\nstrong enough to slow the dwarf down thus boosting its plunge into the\ninner Galaxy where it is quickly disrupted. These two scenarios can be\nidentical in terms of the epoch of accretion and the total mass\naccreted, yet they can produce dramatically different dwarf satellite\npopulations at $z=0$.\n\nAn attempt to quantify the amplitude of the host-to-host scatter in\nthe properties of artificial Galactic dwarfs using analytic models is\npresented in \\citet{Purcell2012}. The p\\`iece-de-r\\'esistance of the\u2260\u2260\u2260\u2260\nmethod is the Monte-Carlo sampling of an arbitrary large number of\ndifferent accretion histories \\citep[as described\n  in][]{Zentner2005b}. Using this technique, it can be demonstrated\nthat the scatter in the possible MAHs is naturally large enough for\nthe Milky Way-like halo to host a satellite population consistent with\nthe observed one in 10\\%-20\\% of cases. These results, within the\nlimitations of the method, shed light onto the statistical\nsignificance of the ``too-big-to-fail'' problem\n\\citep{Boylan-Kolchin2012}: there does not have to be a serious excess\nof massive invisible sub-halos in the Galaxy. Interestingly, together\nwith the recently invoked lower Galaxy mass\n\\citep[e.g.][]{Vera-Ciro2013} and the strong stellar feedback\n\\citep[e.g.][]{Brooks2013}, this is now the third solution for the\npotential problem identified by \\citet{Boylan-Kolchin2012}. It would\nseem that if all three methods are as efficient as described, there\ncould be very few satellites left around the Galaxy! It is, therefore,\nthe most urgent task for the Galactic Archaeology to provide new\nobservational constraints of the Milky Way's accretion history through\nstudies of the spatial and the chemo-dynamical distributions of the\nancient stellar halo populations.\n\nThe Milky Way is not a solitary field spiral: together with its\nneighbor of approximately the same mass, Andromeda and its satellites,\nit makes up the small slightly over-dense region of the Universe known\nas the Local Group of galaxies. The so-called {\\it assembly bias}\nstipulates an excess of probability of finding a massive satellite\nsub-halo around hosts situated in higher density regions as compared\nto those in under-dense environments\n\\citep[e.g.][]{Wechsler2006}. Possibly, this effect could go some way\nto explaining the presence around the Milky Way satellites as massive\nas the Magellanic Clouds. According to \\citet{Busha2011b}, while for\nthe field halo of Milky Way-like mass, the probability to host LMC\/SMC\npair is of order of $5\\%$-11$\\%$, having another host halo of similar\nmass in the vicinity boosts it up to 25$\\%$. This is good news, but\nare these sub-halos on their first (or perhaps second) passage around\nthe simulated Galaxies as the Milky Way observations seem to indicate?\nA unique investigation is described in \\citet{Forero-Romero2011} who\nuse a suite of so-called constrained simulations of the Local Group\n(CLUES, see http:\/\/www.clues-project.org\/) in which the broad-brush\nfeatures of the Milky Way-Andromeda pair are reproduced, to study the\nassembly history of either host halo. They find that i) both galaxies\nhad their last significant accretion event some 10-12 Gyr ago, and\nthat ii) this particular common accretion history is quite rare (from\n1$\\%$ to 3$\\%$) amongst the pairs of host halos in Bolshoi\nsimulation. This conclusion appears to be in contradiction with the\nstudies in which the Clouds are just being accreted.\n\n\\subsection{Tidal origin of the local dwarf galaxies}\n\nIt is inspiriting that there exists at least one alternative, and,\nimportantly, testable scenario of the formation of dwarf satellites in\nand around the Milky Way. \\cite{Lynden-Bell1976} first pointed out the\nproximity of the several of the Galactic dwarfs to the LMC's orbital\nplane as defined by the gaseous stream leading the Cloud. The\nhypothesis then put forward is of a Greater Magellanic Galaxy that had\nbeen torn apart as it interacted with the Milky Way, giving birth to\nthe Large and Small Clouds, as well as to a litter of smaller\ndwarfs. A quarter of a century later, with the measurement of the\nspace velocities of the satellites in hand, the surprising\njuxtaposition of the orbital planes of the LMC, SMC, UMi and Dra is\nconfirmed \\citep[e.g.][]{Palma2002}. This motivates \\citet{Kroupa2005}\nto claim that the observed distribution of the Galactic satellites is\ntoo anisotropic to fit seamlessly within the CDM paradigm. In the\nauthors' opinion, such alignment (dubbed later as the ``disk of\nsatellites'', DoS) is prohibitively rare in computer simulations of\ngalaxy formation in the Universe full of Dark Matter: the accreted\nsub-halos should have had enough time to relax in the Milky Way's\npotential, thus erasing any signs of coherence.\n\nIt is, however, certainly too naive to believe that in $\\Lambda$CDM\nUniverse, the distribution of dwarf satellites around a Milky Way-like\nhost is always isotropic. \\citet{Zentner2005} show that through the\ncombined effect of i) filamentary accretion and ii) the alignment of\nsub-halo orbits with the major axis of the triaxial host halo, the\nprobability of choosing the simulated sub-halo populations from an\nisotropic distribution is as low as $10^{-4}$. The success of these\nsimulations in assembling anisotropic satellite distributions is\ncurious since these particular host galaxies do not posses disks. The\npresence of a baryonic disk should help to get rid of the satellites\norbiting near it, thus making the satellite distribution more\nanisotropic. \\cite{Libeskind2005} use a slightly different numerical\nsetup to generate their host halos as well as their satellite\ngalaxies but come to the same conclusion: a good fraction of the\nbrightest satellites is bound to end up in a plane-like arrangement\nhaving arrived to the host through 1 or 2 primary filaments.\n\nWhile \\cite{Lynden-Bell1976} only briefly mentions a possible scenario\nin which the parent galaxy dissolves to leave several smaller\nfragments behind to be observed today as dwarf satellites,\n\\citet{Kroupa2005} go further to suggest the exact mechanism\nresponsible for their production. They speculate that the creation and\nthe subsequent compression of the gaseous tidal tails is followed by\ntail fragmentation and active star-formation. It is claimed that the\nstellar systems born in this violent process, also known as {\\it tidal\n  dwarf galaxies} can survive long enough. If they do, their\nanisotropic distribution on the sky is merely the consequence of the\nproximity of their birthplaces in the tidal tail that is now\nvanished. This dSph formation mechanism advocated by\n\\citet{Kroupa2005} harks back to their earlier dynamical work\n\\citep{Kroupa1997}, in which a quasi-stable solution for a dSph-like\nDM-free stellar system is discovered. With the help of a suite of\nsimple N-body simulations, it is argued that a tidal dwarf galaxy in\nthe last throws of disruption can posses apparent surface brightness\nand velocity dispersion not unlike those observed in dSphs around the\nMilky Way. As \\citet{Kroupa1997} argues such high velocity dispersions\nwould lead to over-estimated masses and therefore to highly inflated\nmass-to-light ratios, while the actual $M\/L$ remains quite\nlow. \\citet{Metz2007} re-run the experiment and show that their\nsimulated tidal dwarf remnants and the Galactic dwarfs can look alike,\nespecially within the region of the structural parameter space\noccupied by the ultra-faint satellites. Even though the fact of the\nexistence of such out-of-equilibrium satellite configurations in\nnumerical simulations is established, as of today, no evidence has\nbeen found that they can persevere for longer than a 1-2 Gyrs\n\\citep[see e.g.][]{Casas2012}.\n\nAs the census of the sub-structure in the halos of the Milky Way and\nthe Andromeda galaxies is being filled in fast, the growing sample of\nsatellites and streams allows for more rigorous tests of possible\nanistropies in their spatial and kinematic distributions. For example,\n\\citet{Pawlowski2012} extend the study of the Galactic ``disk of\nsatellites'' to include the known stellar and gaseous streams. Their\nargument in support of the previously found DoS orientation is that 7\nout the 14 streams they analyse align well with the disk. With this\nobservation in hand, they claim that it is not merely the ``disk of\nsatellites'' that surrounds the Milky Way, but rather a ``vast polar\nstructure'' (VPOS) appears to dominate the Galactic sub-system\ndistribution at all radii. Once again the conclusion is reached that\nthe presence of such structures is in contradiction with the $\\Lambda\nCDM$ theory. Before the probability of encountering this so-called\nVPOS is worked out for the current galaxy formation paradigm, it is\nworth noting that while the number of the streams contributing to it\nseems large (a half of the total considered), their combined mass is\nminuscule. Therefore, these (in particular stellar) streams contribute\nclose to nothing to the significance of the supposed anisotropy in the\nGalactic halo.\n\nCuriously, in the case of the M31, \\citet{Ibata2013} exhibit plausible\nevidence for the planar alignment of nearly half of the dwarf\nsatellites. Moreover, these appear to be co-rotating around Andromeda\nin a semblance of a disk, which contains the line connecting the host\ngalaxy and the Milky Way. This discovery is responsible for another\nattempt to debunk $\\Lambda CDM$ this time by \\citet{Hammer2013} who\ndevelop their earlier idea of a major merger at the M31 location\n\\citep[see e.g.][]{Hammer2007} and suggest that most of the dwarf\ngalaxies, including the Magellanic Clouds have formed as a result of\nthis upheaval. \n\nOverall, it seems that the hypothesis in which dwarf satellites are\nborn in major merger events can give a convincing account of the\nobserved distribution of satellites on the sky. However, currently the\ntheory does not stack up against the entirety of the observational\nevidence, both locally (e.g. the extended star-formation histories and\nthe extremely old stellar populations of the Milky Way dwarfs) as well\nas outside the Galaxy (e.g. low major merger rates for L$_*$ hosts).\n\n\\section{Archaeologist's toolbox}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.93\\linewidth]{tracers.pdf}\n\\caption{Stellar tracer selection in the SDSS database. {\\it Left:}\n  Density of stars in the plane of surface gravity $\\log g$ and\n  effective temperature $T_{\\rm eff}$ for $\\sim 180,000$ DR8 spectra\n  with $15 < g < 17.5$. {\\it Right:} Stars with spectroscopy from the\n  left column are plotted on the plane of $u-g$ and $g-r$ color. {\\bf\n    Top:} overview of the sample, darker shades of grey indicate\n  higher density. {\\bf Middle:} Selecting the tracers. BHB (blue),\n  Blue Straggler (violet), MSTO (green) and M-giant (red) stars are\n  chosen in the left column based on their temperature and surface\n  gravity. Density of selected stars is then over-plotted in $u-g$,\n  $g-r$ space using the same color scheme. {\\bf Bottom:} Metallicity\n  distribution in the sample. This shows false RGB images (left and\n  right) constructed with 3 grey-scale density distributions of stars\n  picked based on their $[Fe\/H]$. Red component is for metal-rich stars\n  with $-0.75 <[Fe\/H]< 0$, green (intermediate) $-1.5 <[Fe\/H] <\n  -0.75$ and blue (metal-poor) $-3 <[Fe\/H] <\n  -1.5.$} \\label{fig:tracers}\n\\end{figure}\n\nLow-mass stars (around $\\sim 1 M_{\\odot}$) shine for billions of\nyears, and therefore keep the record of historical events in the Milky\nWay. To be able to read into the Galactic diary, collections of stars\nwith comparable chemistry, age or, at least, similar luminosity class\nmust be identified. The distributions of such {\\it stellar tracers} in\ntwo (positions on the sky), three (place on the sky and along the line\nof sight), four (location in space and in radial velocity) or even\nseven (configuration space and velocity space coordinates together\nwith chemistry) dimensions are then measured to benchmark, with some\nhelp from Galactic Dynamics, the theories of structure formation.\n\nThe Galaxy endlessly churns the pieces of smaller satellites it\nacquires, continuously smoothing the spatial densities of the debris.\nThe rate at which the Galactic blender operates decreases from the\ncentre outwards. Far out in the halo, where the orbital periods reach\ngiga-years, unbound stellar sub-structures can maintain superficial\nspatial coherence for eons. However, closer to the Solar radius, extra\n(dynamical or chemical) information is required to filter out\nparticular debris from the smooth mess. Therefore, the interplay\nbetween the number of useful stellar tracers, the information content\nper star, and the overall volume probed is what determines the\nrelevance of a Galactic halo survey.\n\nIn the not-so-distant future, with the data from the Gaia astrometric\nspace mission and a host of planned large-area spectroscopic surveys,\nit should be possible to paint the unambiguous picture of the events\nthat took place in the Galaxy between redshift $z=20$ and redshift\n$z=0$. At the moment, we will have to make do with what we have\ngot. The observational advances in Galactic Archaeology made in the\nlast few years happened thanks to a handful of wide area imaging\nsurveys, namely 2MASS and SDSS, and massive spectroscopic efforts such\nas Segue and RAVE.\n\nOf the several sky surveys of past decade, the SDSS appears to have\nbeen operating in a sweet spot: it turns out a 54 second exposure is\nlong enough to reach Main Sequence stars at distances of several tens\nof kpc from the Sun, and thus yield an unprecedented 100 million\nobject database; yet short enough to see plenty of the sky in limited\namount of time. The now classic $ugriz$ filter set encodes the stellar\nspectral energy distribution (SED) into a compact form, but preserves\nenough frequency diversity to study in detail a variety of stellar\npopulations. This section therefore mostly concentrates on the\nobserved properties of the Galactic stellar halo as seen by the SDSS\n(and its extensions) outside the Solar radius.\n\n\\subsection{Stellar tracers of the Galactic halo in the SDSS}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{bhb_msto.pdf}\n\\caption{Absolute magnitude of stellar tracers. {\\it Left:} Blue\n  Horizontal Branch star candidates in 11 Galactic star clusters. Each\n  dot represents one BHB, stars from different clusters are marked\n  with different color. Cluster name and the color convention are\n  shown in the inset. Once a model for the slight variation of the\n  luminosity with color has been applied, the absolute magnitude of a\n  BHB star can be estimated with accuracy $\\lesssim 0.1$ mag. {\\it\n    Right} Stars with $g-r < 0.4$ in 11 Galactic star clusters. Apart\n  from the variation by $\\pm 0.5$ mag around the mean magnitude of the\n  turn-off $M_g\\sim 4$ due to age and metallicity differences between\n  clusters, stars on the MS with lower luminosity as well as Sub-giant\n  stars bright with higher luminosity are picked up by this $g-r$\n  cut. This results in the overall asymmetric spread of $\\sim 3$ mag\n  in $M_g$.}\n\\label{fig:bhb_msto}\n\\end{figure}\n\nThere are at least three species of stellar tracers available in the\nSDSS photometric data that a Galactic archaeologist can put to\nwork. In order of decreasing population size, increasing luminosity\nand decreasing contamination, these are: Main Sequence Turn Off (MSTO)\nstars, Blue Horizontal Branch (BHB) stars and M giant\nstars. Figure~\\ref{fig:tracers} gives the whereabouts of each of these\nthree in the space of stellar atmosphere parameters and the space of\nbroad-band colors.\n\nThe left column of the Figure shows the logarithm of density of a\nsample of bright ($15<g<17.5$) stars in the spectroscopic Data Release\n8 of the SDSS \\citep{Aihara2011} on the plane of surface gravity $\\log\ng$ and effective temperature $T_{\\rm eff}$.  The density distribution\nof stars in this analog of the familiar Hertzsprung-Russell diagram is\ndominated by the pitch-black ribbon of the Main Sequence (MS),\ncovering the range of $6500 {\\rm K} < T_{\\rm eff} < 4500 {\\rm K}$. The\nsharp edge to the MS feature at high temperatures corresponds to the\nMS turn-off - these are the brightest of the MS stars and so, ideal\nfor the halo exploration. The two faint and fuzzy clouds at\ntemperatures above 7000 K are the helium burning Blue Horizontal\nBranch stars, with $3.7 < \\log g < 3$, and the ``reinvigorated''\nhydrogen burning pseudo-MS stars also known as Blue Stragglers, with\n$4 < \\log g < 5$. Finally, the cool and inflated stars populate the\nRed Giant Branch, attached to the MS at around $T_{\\rm eff} \\sim 6,500\n{\\rm K}$ and reaching as high up as $\\log g \\sim 1.5$. There, right at\nthe tip sits the small group of M giants, with $T_{\\rm eff} < 5000$\nK. To guide the eye, the stellar populations mentioned above are\nmarked in color in the middle panel of the Figure.\n\nAs of DR8, only $\\sim$0.2\\% of all detected stars have been targeted\nwith SDSS spectroscopy. Therefore, to make surveying the Galaxy's halo\npractical, stellar tracers need to be identified by means of\nbroad-band photometry only. To illustrate the photometric selection,\nthe right column of Figure~\\ref{fig:tracers} gives the logarithm of\nthe stellar density in the color-color space of $u-g$, $g-r$. Exactly\nthe same bright stars, those with SDSS spectra, as used for the\ncreation of the left column are plotted here. As expected, the\nbehavior of the broad-band color distribution is to do with the\nlocally measured slope of the SED, and hence is driven by the stars'\ntemperature. Dwarf and giant stars are not easily separable anymore as\nthey collapse to form one stellar locus, running from $g-r \\sim 0.3$\nto $g-r \\sim 1.3$. However, it transpires that around the corners of\nthe locus, the familiar populations can be picked up with ease.\n\n\nThe MSTO stars, being the hottest denizens of the MS, are clumped\nright at the blue edge of the $g-r$ distribution as evidenced by the\ntight green cluster in the right middle panel of the Figure. It is\nobvious that a simple $g-r< 0.4$ cut would produce a relatively clean\nsample of the MSTO tracers.  Still bluer in $g-r$, deviating downwards\nfrom the MSTO corner of the stellar locus, lies a ``claw'' of\nA-colored but old stars, BHBs and BSs. On further look, following\ntheir loci (marked in blue and violet) as they curve in $u-g$, $g-r$\nspace, some overlap between the two populations is visible, but more\nimportantly, in $u-g$ color primarily, the BHB and the BS ridge-lines\nstand separated by some 0.15 mag. \\citet{yanny2000} provide an\nefficient $u-g$,$g-r$ cut which yields a sample of BHB tracers with\nminimal contamination from BS or MSTO stars. Unlike the ubiquitous\nMSTO stars, the BHBs are manifestations of an old and metal-poor\nstellar population, as represented by their abundance in the Galactic\nglobular clusters.\n\nEven though telling a dwarf star from a giant star photometrically is\npretty much impossible across a wide range of SDSS color, luminous and\nmetal rich M giants stars peel away and redward from the stellar locus\nat around $u-g \\sim 2.5$ as the red streak in the right middle panel\nof Figure~\\ref{fig:tracers} indicates. Equation 1 in \\citet{Yanny2009}\nstipulates the M giant selection boundaries in the SDSS\ncolors. Interestingly, age-wise M giants provide a probe of the halo\ncomplementary to that offered by old MSTO and BHB stars. As shown by\n\\citet{Bellazzini2006} the stars ages range between 5 and 9.5 Gyr,\nwith an average age of 8 Gyr.\n\n\\subsection{Chemical abundances of the SDSS stellar halo tracers}\n\nWhat are the metallicity biases induced by the particular choice of\nthe stellar tracers described above? The bottom row of the\nFigure~\\ref{fig:tracers} shows the metallicity $[Fe\/H]$ distribution\nof the bright SDSS DR8 stars with spectra. Stars are split in three\ngroups according to their $[Fe\/H]$ and greyscale density distributions\nin $\\log g, T_{\\rm eff}$ and $u-g, g-r$ are produced. Then the three\ngreyscale pictures are combined to produce two false-color images, one\nfor the left column and one for the right. The red components in the\nfalse RGB images are based on metal-rich stars with $-0.75 < [Fe\/H] <\n0$, for the green components intermediate metallicity stars are\nselected with $-1.5 < [Fe\/H] < 0.75$, finally the most metal-poor\nstars with $-3 < [Fe\/H] -1.5$ contribute to the blue components of the\nimages. Therefore, clumps of stars that are mostly blue in the bottom\nrow of the Figure are mostly metal-poor, the red features are made up\nof mostly metal-rich stars, with other colors corresponding to\n$[Fe\/H]$ mixtures in between. In particular, stars in all three\nmetallicity bins contributed roughly equal amounts to pixels with dark\ngrey or almost black color.\n\nThe lower left panel of Figure~\\ref{fig:tracers} confirms that, as\nexpected, the BHBs are predominantly metal-poor, the M giants are\nmetal-rich and the MSTO have no particular metallicity bias. The right\npanel in this row showcases beautifully the discriminating power of\nthe SDSS broad-band filters: the pixels on the stellar locus can be\nseen to change their color dramatically depending on their $u-g, g-r$\nvalues. This means that a unique $[Fe\/H]$ value can be assigned to a\nMS star given its $ugr$ measurements. The idea of photometrically\nderived metallicities is the same idea that is behind the UV-excess\nmethod first applied to interpret the chemo-dynamical properties of\nthe Galactic halo stars by \\citet{Eggen1962}. \\citet{Ivezic2008}\ndevelop polynomial models (updated recently by \\citet{Bond2010}) to\ncalculate photometric metallicities from SDSS $ugr$ measurements for F\nand G Main Sequence stars in the range of $0.2 < g-r < 0.6$.\n\n\\subsection{Absolute magnitudes of stellar tracers}\n\\label{sec:abs_mag}\n\nOne of the primary advantages of mapping the galaxy in which we\nactually reside is the access to the third spatial dimension. While\nmost other galaxies appear to us in a cartoonish 2D, distances to the\nMilky Way stars can be measured using the annual parallax or with much\ncheaper (but less accurate) photometric parallax \\citep[see\n  e.g.][]{Juric2008}.\n\nFigure~\\ref{fig:bhb_msto} reveals exactly how much uncertainty there\nexists in determining the luminosities of BHB and MSTO stars using\ntheir broad-band colors only. The left panel of the Figure shows\nvariation in $g$-band absolute magnitude $M_g$ as a function of $g-r$\ncolor for BHB candidate stars in 11 Galactic star clusters imaged in\nthe SDSS $ugriz$ filters \\citep{An2008}. Regardless of the (small)\nmetallicity differences and irrespective of the (modest) age spread,\nthe BHBs form a tight sequence with a gentle slope in $g-r$. The\nchanges in $M_g$ from slightly above $M_g=0.5$ at red colors to\nslightly below $M_g$ at blue colors can be approximated with a simple\npolynomial model to give the BHB absolute magnitude within 0.1 mag\n\\cite[see e.g.][]{Deason2011a}.\n\nThe simple $g-r < 0.4$ cut picks up a whole slew of stars of various\nluminosities as evidenced by the right panel of\nFigure~\\ref{fig:bhb_msto}. These range from bright sub-giants at\n$M_g\\sim 3.5$, through the actual MSTO stars with $3.5 < M_g < 4.5$,\nto dwarfs on the Main Sequence that are as faint as $M_g\\sim\n6$. Additionally, even though the star clusters in the sample\nconsidered do not cover the whole range of metallicity or age,\nmatching quite well the old and metal-poor Galactic halo, the $[Fe\/H]$\nand age differences result in significant shifts in both $g-r$ and\n$M_g$ around the MS turn-off. As a result, the overall absolute\nmagnitude spread for the tracers selected is of the order of 3\nmagnitudes.\n\nIt is obvious from the right panel of Figure~\\ref{fig:bhb_msto} that\nfor a star on the Main Sequence, an estimate of the absolute magnitude\ncan be obtained from the value of its $g-r$ color. The dimming of\ndwarf stars with lowering temperature is the basis for the so-called\nphotometric parallax method, which actually does not have anything in\ncommon with the annual parallactic motion, apart from the fact that it\nalso delivers the stellar distance. \\citet{Juric2008} takes advantage\nof the superb quality of the SDSS photometry and calibrates the\nabsolute magnitude of MS stars using the color-magnitude behavior of\nstars in the Galactic globular clusters (GC) with well-determined\ndistances. Such distance estimate can be further improved, as shown by\n\\citep{Ivezic2008}, if the photometrically-derived metallicity is\nincluded. However, as shown by \\citet{Smith2009, Smith2012}, around\nthe MSTO the absolute magnitude calibration provided by\n\\citet{Ivezic2008} suffers from considerable bias. To remedy this,\n\\citet{Smith2009, Smith2012} offer an appropriately flexible method to\ntune the absolute magnitudes of stars around the turn-off according to\ntheir metallicity.\n\nFinally, as regards to M giants, these stars are too luminous, too\nmetal-rich and too young to be found in the Milky Way's globular\nclusters, and hence, the methods of absolute magnitude calibration\ndiscussed above do not apply. However, \\citet{Yanny2009} show that\nusing the pieces of the Sagittarius stream based on the distances\nmeasured with RR Lyrae one can calibrate the M gaint absolute\nmagnitude to obtain $M_g \\sim -1$. Note, however, that this is only\nvalid for a particular mix of metallicity and age similar to that of\nthe Sagittarius debris.\n\n\n\\subsection{Matched Filter approach}\n\nRather than selecting stars in a particular luminosity class and\/or\nmetallicity range to trace the stellar halo sub-structure, an\nalternative popular approach is to use the entirety of the stellar\npopulations belonging to the satellite that is assumed to be\ndisrupting or disrupted. The probability of any star in the halo to\ncome from the desired population is obtained by simply taking the\nratio of stellar density in bins of color and magnitude of the\nsatellite and of the background. These probabilities (or weights) are\nthen binned on the sky and the smooth slowly-varying component of the\ndensity contributed by the background is subtracted. This so-called\nMatched Filter technique as pioneered by \\citet{Grillmair1995} has\nbeen employed with great success to isolate extra-tidal congregations\nof stars around many Milky Way companions\n\\citep[e.g.][]{Odenkirchen2001,Rockosi2002,no2010,Sollima2011}. The\nmethod can deliver superb results, but has two inherent breaking\npoints: i) for satellites that are completely dissolved in the\nGalactic gravitational potential, no template color-magnitude density\nis available, and ii) as the stars from a disrupting object normally\ncover a large area on the sky their heliocentric distances change and\ntherefore the probabilities assigned by the method will not match\nthose in the debris everywhere. While the first problem can be easily\novercome by searching for the best-matching CMD template by trial and\nerror as demonstrated beautifully by \\citet{Grillmair2006a}, there is\nno simple (and elegant) remedy to the issue of evolving distance.\n\n\\section{Stellar halo of the Galaxy}\n\n\\subsection{Evidence of sub-structure in the stellar halo}\n\n\\subsubsection{The Field of Streams}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{fos_dr9.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_components_0.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_components_1.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_components_2.jpg}\n\\caption{Map of the stellar halo of the Milky Way in Equatorial\n  coordinates (centered on RA$=180^{\\circ}$) using the data from SDSS\n  DR9. {\\it Top:} False-color composite image uses $\\sim$ 16,000,000\n  predominantly MS and MSTO stars selected with a simple color cut\n  $g-i<0.6$ and split into three equal magnitude bins between $i=19$\n  and $i=22.5$. Given the $g-i$ cut, the color of the pixel can be\n  interpreted as a rough estimate of the average heliocentric distance\n  of the stars contributing to it. White corresponds to high stellar\n  densities in all three magnitude bins. Black corresponds to areas\n  with missing data.{\\it Bottom:} Grey-scale images each showing the\n  stellar tracer density in a particular magnitude bin are then\n  combined in the order that maps the densities of bright,\n  intermediate and faint stars onto blue, green and red channels as\n  shown in the Top panel. }\n\\label{fig:fos_equ}\n\\end{figure}\n\nThe broad-brush spatial properties of the sub-structure in the Milky\nWay's halo as seen by the SDSS in its DR9 incarnation \\citep{Ahn2012}\nare displayed in Figures~\\ref{fig:fos_equ} and ~\\ref{fig:fos_gal} in\nEquatorial and Galactic coordinates respectively. The SDSS DR5 version\nof this false-color halo map dubbed the Field of Streams is published\nby \\citet{Belokurov2006a}. Included in this map are predominantly MS\nand MSTO stars, which are selected with a simple color cut $g-i<0.6$\n(a selection almost identical to the described above $g-r<0.4$). These\nstars are then split into three equal magnitude bins between $i=19$\nand $i=22.5$. Three grey-scale images each showing stellar tracer\ndensity in a particular magnitude bin (see bottom row of the Figures)\nare then combined in the order that maps bright-intermediate-faint\nstars onto blue-green-red channels. Given the fixed $g-i$ cut, the\nfalse RGB color of each pixel can be interpreted as a rough estimate\nof the average heliocentric distance of the stars contributing to it.\nAssuming, very approximately, that the typical $M_i \\sim 4.$ for the\nselected tracers, given the magnitude range of $19 < i < 22.5$, the\nrange of the heliocentric distances probed is $10< D {\\rm (kpc)} <\n50$.\n\nMost of the contiguous sky coverage in the SDSS footprint falls around\nthe North Galactic Cap. The arc dominating the density map at the high\nGalactic latitudes in the North is the leading tidal tail emanating\nfrom the Sagittarius dwarf galaxy. The Sgr debris is also the only\nprominent structure in the stellar halo in the Galactic South, where\nthe trailing tail of the dwarf can be seen. In\nFigure~\\ref{fig:fos_equ}, the leading stream changes color from red to\ngreen-blue as its heliocentric distance drops from $\\sim 50$ kpc at\nDec$=220^{\\circ}$ (just behind the Galactic centre) to $\\sim 15$ kpc\nat Dec$=130^{\\circ}$ \\citep{Belokurov2006a}. There, at the Galactic\nanti-centre, the Sgr stream crosses the prominent Galactic Anti-Center\nStellar Structure seen in the Figure as a violet-blue tilted band with\nstriation. This complex configuration is due to the multiple\ncomponents of the GASS: the so-called Monoceros Ring\n\\citep[e.g.][]{Newberg2002}, the Anti-Center Stream\n\\citep[e.g.][]{Grillmair2006d, Grillmair2008} and the Eastern Banded\nStructure \\citep{Grillmair2006d}. The fuzzy green haze directly\nunderneath the Sgr stream at around RA$\\sim 180^{\\circ}$ is the large\ncloud of stars dubbed Virgo overdensity \\citep[e.g.][]{Juric2008}. The\ncentral, dense regions of the stellar halo can be seen as bright\nwhite-blue glow on either size of the Galactic disk at $320^{\\circ} <$\nRA $< 220^{\\circ}$. As \\citet{Belokurov2007a} point out, the\ndistribution of the MS and the MSTO stars does not peak in the\ndirection of the Galactic centre as it seems offset towards the\npositive Galactic $l$. Moreover, on closer examination there appears\nto be a substantial asymmetry in the counts of MS\/MSTO stars with\nheliocentric distances $10 < D {\\rm (kpc)}< 20$ between the Galactic\nNorth and the South. These observations lead the authors to the\nconclusion that a sizable portion of the central Milky Way halo could\nbe due to yet another massive stellar sub-structure, so-called\nHercules-Aquila Cloud.\n\nA Galactic projection of the same map (see Figure~\\ref{fig:fos_gal})\nhelps to make sense of some of the stellar halo density patterns. For\nexample, it shows that the Sagittarius leading tail nearly misses the\nGalactic North, the GASS is mostly confined to Galactic latitude $b <\n30^{\\circ}$. Additionally, even though the SDSS coverage at positive\nand negative Galactic longitude is not equal, it is clear that at\n$l>0^{\\circ}$ at high $b>30^{\\circ}$ there does not exist a\ncounter-part to the Virgo overdensity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{fos_dr9_galactic.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_0.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_1.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_2.jpg}\n\\caption{Same as Figure~\\ref{fig:fos_gal} but in Galactic\n  coordinates. Galactic $l=0^{\\circ}, b=0^{\\circ}$ is at the centre of\n  the Figure.}\n\\label{fig:fos_gal}\n\\end{figure}\n\n\\subsubsection{The big 4}\n\\label{sec_big4}\n\nThe Sagittarius Stream, the Galactic Anti-center Stellar Structure,\nthe Virgo and the Hercules-Aquila Clouds are the four largest stellar\nstructures in the halo of the Milky Way. Out of these four, only the\nSgr Stream lies predominantly outside the Galactic disk making it\npossible to estimate its total extent and the overall stellar\nmass. The Stream consists of two tails, the leading and the trailing,\nflowing from the Sgr dwarf galaxy, which currently lies on the\nopposite side of the Galaxy, behind the bulge, several degrees under\nthe disk. The dwarf is falling onto the disk and has just passed its\npoint of the nearest approach at $\\sim$15 kpc from the Galactic\ncenter. The two tails appear bifurcated \\citep[see\n  e.g.][]{Belokurov2006a,Koposov2012} and extend each at least as far\nas $\\sim$ 180$^{\\circ}$ away from the progenitor (see\nFigures~\\ref{fig:fos_equ} and \\ref{fig:fos_gal}). The leading tail is\ntraced as far as 50 kpc from the Galactic center, while the\napo-galacticon of the trailing debris is probably as far as 60-100\nkpc. Both the Sgr remnant and the stream host a range of stellar\npopulations with different ages and metallicities. In particular,\nalong the stream, a substantial population gradient is observed\n\\citep[e.g.][]{Chou2007,Yanny2009,Bell2010,Chou2010,Keller2010,Carlin2012},\nwhich, within any sensible model of the dwarf disruption, would mean a\nsimilarly pronounced abundance and age gradient in the\nprogenitor. Using a variety of stellar tracers across the sky,\n\\citet{No2010b} map the Sgr debris and, correcting for the distance\nand the abundance gradients estimate the total stellar luminosity of\nthe progenitor prior to disruption. They find that, before merging\nwith the Galaxy, the dwarf was as bright as $1.5\\times 10^8 M_{\\odot}$\nor just under $M_V \\sim -16$, but today it has lost as much as $70\\%$\nof its stars to the Galactic tides.\n\nThe Virgo Cloud can be seen as green haze directly underneath the Sgr\nStream at around $RA\\sim 12^h$. While early glimpses of this structure\nare reported in several studies, based on the SDSS DR4 imaging data,\n\\citet{Juric2008} provide the first large scale map of the Cloud and\nemphasize its gigantic extent on the sky of least $\\sim\n1000$ deg$^2$. From the inspection of Figure~\\ref{fig:fos_equ}, it is\nobvious that the portion of the Virgo Cloud analyzed by\n\\citet{Juric2008} is only the tip of the structure that appears to\ncontinue to lower Declinations as far as the SDSS\/Segue imaging\nstripes can go. Accordingly, \\citet{Bonaca2012b} take advantage of the\nextra imaging in the SDSS DR8 and claim that the true extent of the\nCloud is somewhere between 2000 deg$^2$ and 3000 deg$^2$. The\ndebris cover an enormous portion of the sky, but given the typical\ndistance and the low surface brightness, the total luminosity of the\nVirgo Cloud is estimated to be modest $< 10^6 M_{\\odot}$\n\\citep{Bonaca2012b}.\n\nThe Galactic Anti-Center Stellar Structure and the Hercules-Aquila\nCloud have most of their stars at low Galactic latitudes: within $|b|\n< 40^{\\circ}$, GASS can be found at roughly $120^{\\circ} < l <\n240^{\\circ}$ and HAC at $20^{\\circ} < l < 70^{\\circ}$ (see\nFigure~\\ref{fig:fos_gal}. In fact, both of these structures appear to\nbe stuck right in the plane of the disk as their candidate member\nstars are detected in both Northern and Southern hemispheres. Given\nsuch an awkward location in the Galaxy, it is still questioned whether\nall, or at least some of the signal attributed to these two can be\nexplained away invoking variants of the known components of Milky\nWay. For example, it is claimed that parts of the GASS can well be\nascribed to the Galactic flare and\/or the warp\n\\citep[e.g.][]{Ibata2003}, and the HAC is really nothing but the\nasymmetric thick disk \\citep[e.g.][]{Larsen2008,Larsen2011}. However,\nthere exists additional observational data within which stellar\nover-densities are clearly seen in the directions of both the GASS and\nthe HAC in tracers unlikely to populate either of the disks. For\nexample, the distant portion of the GASS, the And-Tri stream is traced\nwith M giants at distances of the order of 30 kpc. HAC can be picked\nup with RR Lyrae in the SDSS Stripe 82 dataset\n\\citep[e.g.][]{Watkins2009, Sesar2010a} at $10 < D < 20$ kpc.  As most\nof the light in both GASS and HAC is hidden in the Galactic plane,\nonly very approximate estimates of their total stellar masses exist in\nthe literature. \\citet{Belokurov2007b} give a conservative estimate of\n$\\sim 10^7 L_{\\odot}$ for the Hercules-Aquila Cloud. For the closer\nportion of the GASS, \\citet{Yanny2003} get the total stellar mass in\nthe range of $0.2 - 5 \\times 10^8 M_{\\odot}$, with the larger value\nobtained assuming that i) the GASS follows an exponential profile as a\nfunction of z and ii) encompasses the entire Milky Way. Several\nfollow-up studies present the updated measurements of the structure\nand the stellar populations of the pieces of GASS visible in the SDSS\n\\citep[e.g.][]{Dejong2010,Grillmair2011,Li2012} and in the deeper\nimaging \\citep[e.g.][]{Conn2012}. According to the body of work\npublished so far, the components of the GASS most consistent with the\naccretion scenario \\citep[see e.g.][]{Penarrubia2005} have, overall,\nmuch flatter density distribution as a function of Galactic $|b|$ or\n$|z|$. If true, this observation would lead to the substantial\nreduction of the overall luminosity of the GASS. Perhaps, the\nfollowing simple argument can be constructed to provide a\ncomplementary guess as to the total stellar mass in the GASS. Given\nthat the parts of the GASS detected within the SDSS field of view\ntypically have similar or lower surface brightness as compared to the\nSgr Stream, but are on average closer by a factor of 2-5, it is not\nunlikely that the structure, in fact, contains more than $10^8\nM_{\\odot}$.\n\n\n\\subsubsection{Ultra-faint satellites}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{dwarfs_lb_dr8.jpg}\n\\caption{Distribution of the classical dwarf galaxies (blue filled\n  circles) and the SDSS ultra-faint satellites (red filled circles),\n  including three ultra-faint star clusters, in Galactic\n  coordinates. The SDSS DR8 imaging footprint is shown in grey. Dashed\n  line marks the tentative orbit of the Sgr dwarf galaxy. Galactic\n  $l=0^{\\circ}, b=0^{\\circ}$ is at the centre of the Figure.}\n\\label{fig:dwarfs_lb}\n\\end{figure}\n\nVisible as bright dots of different colors in the maps in\nFigures~\\ref{fig:fos_equ} and~\\ref{fig:fos_gal} are the compact\nstellar over-densities corresponding to the Galactic satellites that\ngive the impression of being still intact. The brightest of these\n``hot pixels'' correspond to the well-known star clusters and\nclassical dwarf galaxies, while the very faint and barely visible\nsmall-scale over-densities mark the locations of the so-called\nultra-faint satellites of the Milky Way. Although several of these,\nincluding Boo I, Boo III, CVn I and UMa II, are seen in this picture\nwith a naked eye, the rest of the population of these objects is too\ninsignificant and can only be unearthed via an automated over-density\nsearch. The first example of such an automated stellar over-density\ndetection procedure is presented in \\citet{Irwin1994} who apply the\nmethod to the data from the photographic plates of the POSS I\/II and\nUKST surveys scanned at the APM facility in Cambridge. A vast area of\n20,000 square degree of the sky is searched but only one new nearby\ndwarf galaxy is detected, namely the Sextans dSph. A variant of the\nprocedure is used, albeit with a little less luck, by\n\\citet{Kleyna1997}, and subsequently by \\citet{Willman2005a,\n  Willman2005b} who actually find the two very first examples of\nultra-faint objects in the SDSS data. The ease with which these\nsystems reveal themselves in a stellar halo density map akin to the\n``Field of Streams'' \\citep[see][]{Zucker2006a, Belokurov2006c} helped\nto re-animate the search for new Milky Way satellites and more than a\ndozen of new discoveries have been reported in quick succession\n\\citep{Zucker2006b,Belokurov2007c,Irwin2007,Koposov2007,Walsh2007,\n  Belokurov2008,\n  Belokurov2009,Grillmair2009,Belokurov2010}. Figure~\\ref{fig:dwarfs_lb}\nmaps the distribution of all presently known SDSS ultra-faint\nsatellites on the Galactic sky.\n\nThe accuracy and the stability of the SDSS photometry makes it\npossible for the over-density detection algorithms to reach\nexceptionally faint levels of surface brightness across gigantic areas\nof the sky. However, even though genuine Galactic satellites can be\nidentified in the SDSS as groups of only few tens of stars, their\nstructural parameters can not be established with adequate accuracy\nusing the same data. Deep follow-up imaging on telescopes like INT,\nCFHT, LBT, Magellan, MMT, Subaru and most recently HST, has played a\nvital role in confirming the nature of the tiny stellar blobs in the\nSDSS, as well as in pinning down their precise sizes, ellipticities\nand their stellar content. The most recent, deep and wide photometric\nstudies of a significant fraction of the new SDSS satellites are\npublished by \\citet{Okamoto2012} and \\citet{Sand2012}. They point out\nthat even at distances $D>100$ kpc from the Galactic centre, the outer\ndensity contours of CVn II, Leo IV and Leo V display extensions and\nperturbations that are probably due to the influence of the Milky Way\ntides. Similarly, there is now little doubt that both UMa II and Her\nare excessively stretched, as their high ellipticities as first\nglimpsed at discovery \\citep{Zucker2006a, Belokurov2007c} are\nconfirmed with deeper data \\citep{Munoz2010, Sand2009}. Note, however\nthat apart from these two obvious outliers there does not seem to be\nany significant difference in the ellipticity distributions of the\nUFDs and the Classical dwarfs contrary to the early claims of\n\\citet{Martin2008}. This is convincingly demonstrated by\n\\citet{Sand2012} with the help of the imaging data at least 2\nmagnitudes deeper than the original SDSS.  They, however, detect a\nmore subtle sign of the tidal harassment: the preference of the\ndensity contours of the SDSS satellites to align with the direction to\nthe Galactic centre.\n\nAs far as the current data is concerned, the SDSS dwarfs do not appear\nto form a distinct class of their own, but rather are the extension of\nthe population of the Classical dwarfs to extremely faint absolute\nmagnitudes. However, as more and more meager luminosities are reached,\nit becomes clear how extreme the faintest of the UFDs are. The\nbrightest of the group, CVn I and Leo T show the usual for their\nClassical counter-parts signs of the prolonged star-formation. For\nexample, CVn I hosts both Blue Horizontal Branch and Red Horizontal\nBranch populations, while Leo T shows off a sprinkle of Blue Loop\nstars. However, the rest of the ensemble appears to have narrow CMD\nsequences with no measurable color spread around the conventional\ndiagnostic features, e.g. MSTO and\/or RGB, thus providing zero\nevidence for stellar populations born at different epochs\n\\citep[e.g.][]{Okamoto2012}. The CMDs of the UFDs have revealed no\nsecrets even under the piercing gaze of the HST: all three objects\nstudied by \\citet{Brown2012} appear to be as old as the ancient\nGalactic globular cluster M92. Yet the low\/medium and high-resolution\nfollow-up spectroscopy reveals a rich variety of chemical abundances\nsomewhat unexpected for such a no-frills CMD structure. The first\nlow-resolution studies of \\citet{Simon2007} and \\citet{Kirby2008}\nalready evince the existence of appreciable $[Fe\/H]$ spreads in the\nSDSS dwarfs with the metallicity distribution stretching to extremely\nlow values. Analyzing the medium and high resolution spectra of the\nBoo I system, \\citet{Norris2010} measure the spread in $[Fe\/H]$ of\n$\\sim$1.7 and the $[Fe\/H]$ dispersion of $\\sim$0.4 around the mean\nvalue of -2.55 at $M_V\\sim -6$. It seems that this behavior of\ndecreasing mean metallicity with luminosity while maintaining a\nsignificant enrichment spread is representative of the UFD sample as a\nwhole \\citep[see also][]{Lai2011,Koch2013,Vargas2013}. Crucially,\nthese spectroscopic observations require that, notwithstanding their\nlow stellar luminosities at the present day, these satellites had\nenough total mass in the past to hold on to some of the enriched gas\nafter the first supernovae explosions and subsequently produce more\nstars. Additionally, in the UFDs, the downwards shift of the mean\nmetallicity with decreasing stellar mass reveals that they can not\nsimply be direct analogs of the Classical dwarfs stripped off the bulk\nof their stellar content.\n\nOf the 16 ultra-faint satellites currently known, only 5 systems have\na handful of stars studied with high-resolution spectroscopy. More\nspecifically, one star in Leo IV \\citep{Simon2010}, two stars in Her\n\\citep{Koch2008}, 3 stars in each of UMa II and Com \\citep{Frebel2010}\nand 7 in Boo I \\citep{Gilmore2013} have been measured so far. It is\nperhaps too early to draw far-reaching conclusions from these highly\nincomplete data, nonetheless an interesting picture seems to be\nemerging from the detailed abundance work. Although wanting in\nquantity, these high-resolution high-quality spectroscopic data do\nrobustly confirm the key properties of the UFD chemical enrichment\nhistories hinted at by the analysis of the low-resolution (and at\ntimes, low-S\/N) samples. The SDSS dwarfs are indeed characterized by\nremarkably low levels of the overall iron enhancement as well as the\nheterogeneity of the individual stellar abundances (in each of the 4\nsatellites that have more than 1 star measured). Additionally, the\nvery first high-resolution study of a UFD by \\citet{Koch2008} reported\na depletion of heavy neutron capture elements. RGB stars with low\nabundance levels of barium are also found in Leo IV, Com, UMa II and\nBoo I \\citep{Simon2010, Frebel2010, Gilmore2013}. Moreover, in Boo I,\nseveral extremely metal-poor stars are demonstrated to have increased\nlevels of carbon \\citep[see e.g.][]{Norris2010}. Potentially, there\nare at least two notable implications of these enrichment\npatters. First, carbon-enhanced metal-poor stars are common denizens\nof the Galactic stellar halo, yet if there occur any in the classical\ndSphs, they have so far eluded the detection. The existence of such\nstars in both the UFDs and the MW stellar halo may signify the\ncommonality of the chemical evolution paths of the halo progenitor(s)\nand the ultra-faint satellites. Second, as several authors have\npointed out \\citep[e.g.][]{Koch2008,Simon2010, Frebel2010}, the\nenhancement in $\\alpha$-elements together with the depletion in\nneutron-capture elements at low metallicities can be linked to the\nproducts of the Population III SNe, therefore implying that a good\nfraction of the stellar content in the UFDs could be direct\ndescendants of the first stars \\citep[see also][]{Frebel2012}.\n\nIt is evidently not possible to come up with a sensible theory of the\nUFD formation without an idea of their total masses. Such a\nmeasurement, which necessarily involves accurate kinematics for a\nlarge enough sample of the satellite members, is, however, not\nstraightforward. This is simply due to the fact that, as illustrated\nby \\citet{Koposov2008}, the majority of these objects are discovered\nvery close to the detection boundary, implying that the over-density\nsignal is dominated by the stars close to the SDSS detection limit of\n$r\\sim 22$. At these magnitudes, only half a handful of facilities in\nthe world are capable of obtaining absorption spectra of\nsignal-to-noise sufficient to measure the line-of-sight velocities of\nindividual stars. Even if the kinematic signal is present in the data,\nwinnowing it out from the low-resolution spectra of low-metallicity\nstars is a challenge. An even harder challenge is figuring out the\nuncertainties of the velocity measurements. For most ultra-faints, the\ntypical member velocity uncertainty is of the order of, or larger\nthan, the intrinsic velocity dispersion of the system. Under or\nover-estimating the measurement error by a small fraction can lead to\na substantial systematic velocity dispersion bias, and as a\nconsequence, a wrong aperture mass. Despite the above mentioned\ndifficulties of the task at hand, several teams report the results of\ntheir heroic attempts to gauge the central masses of the UFDs\n\\citep[e.g.][]{Martin2007b, Simon2007, Walker2009, Belokurov2009,\n  Simon2011, Koposov2011}\n\nThe structural parameters of the faintest of the SDSS satellites,\ne.g. Willman 1, Segue 1 and 2, Boo II are dangerously similar to those\nof the most diffuse star clusters in the Milky Way and M31. It is not\nconceivable, purely on the basis of their size or luminosity, to come\nup with the most likely scenario of their formation. Therefore, their\nkinematic and chemical properties are the most important clue. Today,\nfor the faintest objects, it is just possible, after many hours spent\non Keck and VLT, to build datasets with radial velocities for a dozen\nor two of the MSTO members and a trickle of the Red\nGiants. Accordingly, the most recent and the most thorough kinematic\nanalysis of Willman 1, Segue 1 and Segue 2 can be found in\n\\citet{Willman2011, Simon2011} and \\citet{Kirby2013}\ncorrespondingly. Moreover, \\citet{Norris2010} independently carries\nout a thorough chemical study of Segue 1 using a different combination\nof the telescope, the instrument and the analysis techniques. For\nthese three best studies objects, the picture does not appear to be as\nclear-cut as for their more luminous peers. For example, the evolution\nof the line-of-sight velocity with radius in Willman 1, where the\ninner-most stars are offset by some 8 km\/s from the outer-most ones is\nunusual, and is, perhaps, a sign of the advanced stage of tidal\ndisruption. There is also an evidence of the spread in [Fe\/H], but\nunfortunately it is based on the measurements of only two Red Giant\nBranch stars.\n\nSegue 1, the best studied of the three, has a substantial velocity\ndispersion at 3.7$^{+1.4}_{-1.1}$ km\/s and an impressive metallicity\nspread. There are however some quirks with regards to both the\nvelocity and the metallicity dispersion measurements, such as the fact\nthat the velocity dispersion calculated using the brightest members\nonly (5 red giants stars) is essentially consistent with zero, or the\nfact that some of the most metal-poor stars also lie several\nhalf-light radii away from Segue 1's center\n\\citep[see][]{Norris2010}. Perhaps more significant is the observation\nby \\citet{Newberg2010} that the Orphan stellar stream runs at the\nidentical distance and velocity only $\\sim$2 degrees away from the\nposition of Segue 1. Given the width of the stream of 1 degree, a\nsignificant contamination of spectroscopic samples at Segue 1's\nlocation is not very likely. Yet, the dynamical association between\nthe two is, however, quite possible: both the progenitor of the Orphan\nStream and Segue 1 itself might have been parts of a bigger system\nwhich is now completely disrupted.\n\nThe evidence of such an accretion event is even more dramatic in the\ncase of Segue 2. Taking into account the observations reported in\n\\citet{Majewski2004,Rocha2004}, Segue 2 is immersed in the debris of\nthe Triangulum-Andromeda stream, which is interpreted as the distant\n(at $\\sim$ 30 kpc compared to $34$ kpc for Segue 2) counter-part of\nthe Monoceros stream and part of the larger Galactic Anti-Center\nStellar Structure. As published by \\citet{Rocha2004}, the velocities\nof M giant members of Tri-And structure are $0 < V_{GSR} < 60$ in the\nrange of longitudes $160^{\\circ} < l < 130^{\\circ}$ at the Galactic\nlatitudes slightly lower than that of Segue 2. This velocity\ndistribution can be modeled as a Gaussian that peaks around $V_{GSR}\n\\sim 30$ km s$^{-1}$ which is a good match to the measurement of the\nsatellites line-of-sight velocity $V_{GSR} \\sim 40$ km s$^{-1}$. The\ncoverage of the area with the spectroscopic M giants is sparse, and\nthe SDSS spectroscopic footprint is seriously incomplete\nhere. However, \\citet{Belokurov2009} present an unambiguous kinematic\nevidence for the stream's existence using the spectra obtained with 1\ndegree wide field Hectochelle instrument on MMT. They claim that the\nstream's stars are more metal-rich on average and their velocity\ndistribution can be described with a broader Gaussian, namely 15 km\/s\nvs $\\sim$3 km\/s for Segue 2. Most recently, \\citet{Kirby2013}\nre-evaluated the spectroscopic properties of Segue 2 albeit with a\ndifferent observational setup and a smaller field of view as compared\nto the original study of \\citet{Belokurov2010}. They claim no\ndetection of the stream signal, which is perhaps not surprising given\nthe targeting strategy and the area of the sky surveyed. Intriguingly,\nthey measure much lower velocity dispersion (essentially consistent\nwith zero), thus markedly reducing the central mass of the satellite.\n\n\n\\subsubsection{Star cluster streams}\n\nThe large undissolved stellar clouds (Virgo, Hercules-Aquila) and\nbroad long streams (Monoceros, Sagittarius) described earlier are the\nprimary contributors to the Galactic halo in terms of the stellar\nmass. In the past decade, an assortment of much narrower, often\nshorter and significantly less luminous streams has been\nidentified. It seems most likely that these would have originated in\nstar clusters. Some of these wispy tidal tails are discernible in\nFigures~\\ref{fig:fos_equ} and~\\ref{fig:fos_gal}, such as the tidal\ntails of the Palomar 5 globular cluster\n\\citep{Odenkirchen2001,Grillmair2006b}. However, in their majority\nthese feathery streams require a more subtle approach and are best\nseen with the help of the Matched Filter technique. Some of the star\ncluster debris have obvious progenitors like the short stubby tails\nvisible around e.g. NGC 5466 \\citep{Belokurov2006b}, NGC 5053\n\\citet{Lauchner2006}, Pal 14 \\citep{Sollima2011}, Pal 1\n\\citep{no2010}. For the others, typically extending many degrees on\nthe sky, no suitable progenitor has been discovered yet, e.g. the GD-1\nstream \\citep{Grillmair2006a}, a group of four streams Styx, Acheron,\nCocytos, Lethe \\citep{Grillmair2009} and the most recently identified\nPisces Stellar Stream \\citep{Bonaca2012, Martin2013}.\n\nIt is interesting to estimate the total number of star clusters that\nhave disrupted so far and whose stars are now part of the Galactic\nhalo. While such a count is valuable as it gives an idea of the\nfraction of the halo that is comprised of the GC debris, it is not\nstraightforward as it requires the knowledge of the Cluster Initial\nMass Function (CIMF) and a model of the cluster evolution in the Milky\nWay tidal field. An example of such calculation is presented in\n\\citet{Poul2013} who approximate the CIMF with a power-law\ndistribution and apply the semi-analytic model of \\citet{Gieles2011}\nfor the star cluster evolution in a logarithmic Galactic\npotential. They find that, of the several models they consider, the\nRoche volume under-filling model with a flat CIMF (power law index 0)\nreproduces the present day properties of the Milky Way's GCs the\nbest. While the authors do not give the exact number of dissolved\nclusters, it is clear that the flat mass function evolution can only\nproduce a moderate number of star cluster streams in the Galactic\nhalo, perhaps orders of magnitude less as compared to the rising power\nlaws (e.g. -2). Alternatively, the number of the GC streams detected\nso far with the SDSS could be translated into a Galaxy total if there\nexisted an estimate of the stream detection efficiency. However, it is\npossible that a significant fraction of the known long and narrow\nstellar streams may have been produced as a result of only a few\naccretion events. For example, given the noticeable alignment of their\norbital planes, it is feasible that the progenitors of the Styx,\nAcheron, Cocytos and Lethe streams arrived to the Galaxy together with\na much bigger satellite. The fact that the GC accretion is most likely\nlinked to the infall of more massive Galactic satellites is another\nreason to believe that the total number of GC streams is relatively\nlow given the evidence for the uneventful Milky Way's mass assembly\nhistory.\n\n\\subsubsection{Orphan and Styx. Streams from ultra-faint satellites?}\n\nThe tidal stream's cross-section on the sky is normally a giveaway of\nthe progenitor's mass. The low-density disrupting star clusters with\nsmall internal velocity dispersion $\\sigma \\lesssim 1$ km s$^{-1}$\ntypically produce tails that are only $\\sim 0.1^{\\circ}$ wide. On the\nother hand, a galaxy as massive as Sgr dwarf with its current $\\sigma\n\\lesssim 20$ km s$^{-1}$ \\citep[see e.g.][]{Ibata2009} gives rise to\nstreams that are at least 10$^{\\circ}$ across (see\nFigure~\\ref{fig:fos_equ} for example). This rule of thumb of course\nassumes comparable distances to the tidal tails and not hugely\ndifferent dynamical ages. Depending on how aspherical the\ngravitational potential of the Galaxy is and how long ago the debris\nwere stripped, even originally narrow tails can puff up with time.\n\nAmongst the panoply of stellar substructure recently discovered in the\nGalactic halo, there are at least two streams that seem to occupy the\nparameter space intermediate between the star clusters and dwarf\ngalaxies. These are the Orphan stream \\citep{Belokurov2006a,\n  Belokurov2007b, Grillmair2006c} visible in Figure~\\ref{fig:fos_equ}\nas almost vertical streak of orange color crossing the Sgr debris at\naround $140^{\\circ} <$ RA $< 160^{\\circ}$, and the Styx stream\n\\citep{Grillmair2009}, the faint blue nebulous smear running at almost\nconstant Dec$=30^{\\circ}$ from RA$=250^{\\circ}$ to RA$=220^{\\circ}$\nwhere it starts to drop in Dec towards the Sgr stream. Curiously, both\nOrphan and Styx run in a close vicinity of the several of the Galactic\nultra-faint satellites. The sky projection of the orbit of the Orphan\nstream takes it right through the position of the UMa II dwarf. The\nfeasibility of such association is explored in \\citet{Fellhauer2007}\nwho conclude that UMa II could well be the stream's\nprogenitor. However, as convincingly shown in \\citet{Newberg2010}, the\nearly tentative estimates of the stream's radial velocity were\nincorrect and that the actual orbit of the stream is much more\nconsistent with the 4D location of Segue 1. As regards to the Styx\nstream, when tracing its debris to the lower RA, \\citet{Grillmair2009}\ndiscovers a pronounced stellar clump within the stream's path. Dubbed\nBootes III and subsequently confirmed with spectroscopy\n\\citep{Carlin2009} this is the most diffuse of all ultra-faints found\nso far.\n\n\\subsubsection{Broad and Invisible}\n\nAs the proper motion, spectroscopy and the variability wide-area\nsurveys slowly catch up with the rapidly advancing sky imaging\ncampaigns, it is possible to gauge the presence of stealth stellar\nstructures, so diffuse that they elude detection in stellar halo maps\nakin to those described above. These detections are reminiscent of the\noriginal discovery of the Sgr dwarf \\citep{Ibata1994} that is too\nfaint and spread out to be seen on a photographic plate but produces a\nbooming signal in radial velocities.\n\nTrinagulum-Andromeda is an extended stellar structure located at\nseveral tens of kpc from the Galactic centre \\citep{Rocha2004}. It is\ninitially picked up as a faint excess of 2MASS M-giant stars, and\nlater confirmed with the help of proper motion data and follow-up\nspectroscopy. As judged by the radial velocities of its members, the\nTri-And cloud seems to be connected to the Southern Galactic\ncounterpart of the Monoceros stream, and thus forms the more distant\nwraps of the Galactic Anti-centre Stellar Structure\n\\citep{Newberg2002, Ibata2003, Rocha2003,\n  Yanny2003}. \\citet{Majewski2004} and \\citet{Martin2007} report the\ndetection of the Main Sequence stars in the Tri-And cloud, thus\nridding of the last shreds of doubts as to the reality of its\nexistence. Curiously, the recently discovered ultra-faint satellite\nSegue 2 \\citep{Belokurov2009} appears immersed in the debris of what\nvery well might be the Tri-And cloud.\n\nThe recently discovered Cetus Polar Stream \\citep{Newberg2009} has\navoided detection thanks to its low density and the overlap in\nprojection with much brighter Sagittarius trailing stream. However,\ntaking advantage of the SDSS spectroscopy available over a large\nportion of the Southern Galactic sky, \\citet{Newberg2009} present a\nconvincing argument in favor of a distinct stellar sub-structure,\ncolder and more metal-poor than the Sgr debris. \\citet{Koposov2012}\nprovide the first sky map of the Cetus Polar Stream debris, and having\nobtained accurate measurements of the stream's distance and velocity\ngradients they argue that the sense of direction of the orbital motion\nof the CPS is opposite to that of Sgr. In their maps, the structure\nappears to be at least 20$^{\\circ}$ wide and some 40$^{\\circ}$ long,\nyet with only 0.1 mag width along the line of sight.\n\nThe charting of the Galactic halo at distances beyond 50 kpc has been\nsomewhat sluggish due to the obvious lack of suitably bright tracers\ncovering a large enough area of the sky. A small fraction of the SDSS\nfootprint, so-called Stripe 82 has been imaged repeatedly during the\nSupernovae campaign. \\citet{Watkins2009} explores this multi-epoch\ndataset to identify RR Lyra stars. They find a significant\nover-density of RR Lyrae in the constellation of Pisces at\ngalacto-centric distances of $D\\sim 90$~kpc, thus discovering the most\ndistant sub-structure known in the Milky Way halo. \\citet{Sesar2010a}\nconfirm the discovery with a more sophisticated analysis of the same\nSDSS data, while \\citet{Kollmeier2009,Sesar2010b} present the\nspectroscopic confirmation of the structure by obtaining velocities\nfor several RR Lyra members. As of today the true extent of the Pisces\nOver-density is not known, but from the distribution of the RR Lyrae\nit subtends at least $10^{\\circ}$ on the sky making it some 15 kpc\nwide.\n\n\\subsection{Quantifying the amount of sub-structure}\n\nWithin the $\\Lambda$CDM paradigm, the global properties of the\nGalactic stellar halo, namely the total luminosity, the shape, the\nradial profile as well as the amount of sub-structure are simply the\nconsequences of the Milky Way's accretion history and as such all have\na straightforward interpretation. Observationally, however, these\nproperties are awkward to pinpoint. For example, to gauge the\nflattening and the shape of the radial density profile, data across\nlarge portions of the Northern and the Southern Galactic sky are\nrequired. With pencil-beam surveys, the halo flattening or, more\ngenerally any deviation from spherical symmetry (e.g. triaxliaity), is\nimpossible to determine and there is always a good chance of hitting\nunknowingly a stellar stream or a cloud, hence biasing the estimates\nof the density profile. Yet, in photometric studies, a robust global\ndensity model is vital when quantifying the amount of\nsub-structure. As the density distribution in the 6D phase-space,\nwhere the individual accreted fragments are readily identifiable, is\ncollapsed onto the 3 spatial dimensions (or sometimes 2.5 or 2), the\nsignal is diluted as a result of super-position of many\nstructures. Therefore, even a small bias in the background properties\ncan affect dramatically the amplitude of sub-structure. Of course, the\n``background'' itself, in this picture, is nothing else but the\nstellar debris jumbled up more efficiently. Accordingly, the global\nlaw parameterizing the behavior of the background provides crucial\ninformation in which the mass of the satellites contributing to it and\nthe time of their accretion is encoded.\n\n\\subsubsection{Spatial inhomogeneities}\n\nWith plenty of deep multi-band photometry in both Galactic\nhemispheres, the SDSS is an ideal resource to use to infer the global\nproperties of such an immense structure as the Milky Way's stellar\nhalo. A series of fits to the principal Galactic components as traced\nby the MS stars in the SDSS DR5 is presented in\n\\citet{Juric2008}. This sample is dominated by the faint MS dwarfs\nand, therefore, can not trace the volume density in the Milky Way much\nfurther than 20 kpc. Within this radius, the halo appears to be well\ndescribed by a single power law density model with the index $n\\sim\n2.8$. Importantly, this study confirms earlier indications of a\nsubstantial vertical flattening of the stellar halo $q\\sim0.6$. The\nresults of \\citet{Juric2008} are corroborated by the modelling of the\nSDSS DR8 data with increased Southern Galactic hemisphere coverage\npublished by \\citet{Bonaca2012b}. An attempt to delve deeper into the\nstellar halo can be found in \\citet{Bell2008}, where a simple\ncolor-cut (similar to that illustrated in the right panel of\nFigure~\\ref{fig:bhb_msto}) is used to isolate the brightest of the old\nMS stars in the halo. Using these blue, metal-poor turn-off stars,\nwith typical $M_g \\sim 4$, it is possible to discern halo structures\nas far as 30-40 kpc away from the Sun. However, as explained in\nSection~\\ref{sec:abs_mag}, the spread in the intrinsic luminosities of\nthe stars selected is as large as 3 magnitudes. There are two\nimportant consequences of such blurred vision. First, convolving the\nstellar halo distribution with large non-Gaussian errors in tracer\ndistances can have strong destructive effects on the accuracy of the\nvolume density inference. Second, when estimating the amplitude of\nsmall scale deviations from the background, a debris at one particular\ndistance appears in several apparent magnitude bins (and hence\ndistances), thus biasing high the total amount of sub-structure across\nthe range probed. This effect is exacerbated at magnitudes close to\nthe survey limit, as well as for stars with different age and\/or\nmetallicity.\n\nWhile troubled by a number of issues outlined above, the analysis of\n\\citet{Bell2008} is the first of its kind. Taking advantage of the\nimpressive sky coverage and depth of the SDSS imaging, they provide a\nquantitative interpretation of the inhomogeneous stellar halo glimpsed\nby the earlier works. The main conclusions of the study by\n\\citet{Bell2008} are as follows. First, a smooth density model for the\nMSTO tracers within 40 kpc is not appropriate for the Milky Way halo,\nwith most of the model parameters poorly constrained (see their\nFigures 4, 7 and 9). Second, even after excising the major known\ndebris pile-ups such as Sagittarius stream and Virgo overdensity, the\namount of sub-structure $\\sigma\/{\\rm total}$, parameterized in terms\nof the scaled rms deviation $\\sigma$ of the data around the smooth\nmodel, stays just under $40\\%$ from $r\\sim 19$ mag to $r \\sim 22$\nmag. In the presence of these large stellar halo structures, the\n$\\sigma\/{\\rm total}$ statistic grows with apparent magnitude (roughly\nproportional to distance) and reaches $>50\\%$ at $r\\sim\n21.5$. Finally, \\citet{Bell2008} compare the values of $\\sigma\/{\\rm\n  total}$ for the Milky Way halo traced by faint metal-poor MSTO stars\nin the SDSS to those obtained for the semi-analytic stellar halo\nsimulations of the Galaxy by \\citet{Bullock2005}. The 11 model halos\nare made entirely of accreted stars, and show a minimal level of\nsub-structure $\\sigma\/{\\rm total} > 20\\%$. Accordingly, the final\nverdict is: the amount of sub-structure in the Galactic halo matches\nthat in the hierarchical galaxy formation models, and, therefore,\nsatellite accretion is the primary mode of the Milky Way's halo\ncreation.\n\n\\citet{Helmi2011} aims to improve the analysis of \\citet{Bell2008} by\ni) coming up with a more robust sub-structure quantification, and ii)\ncomparing the SDSS data to the most recent stellar halo\nsimulations. Similarly to \\citet{Bell2008} they measure the stellar\ndensity scatter in bins of apparent magnitude and the two celestial\ncoordinates. However, instead of calculating the amount of residual\ndeviation between the data and the best-fit smooth parametric model,\n\\citet{Helmi2011} work out the RMS around the mean stellar density in\nthe bin. Predictably, the amount of sub-structure computed in this\nfashion is lower compared to that obtained by \\citet{Bell2008}, albeit\nonly slightly. According to \\citet{Helmi2011}, across the apparent\nmagnitude range of $18.5 < r < 22.5$, the normalized scatter ${\\rm\n  rms}(\\rho)\/<\\rho>$ in the SDSS DR7 MSTO star density is at the level\nof 30\\% to 40\\%. These rather serious levels of inhomogeneity found in\nthe SDSS data nonetheless appear low when contrasted with the degree\nof sub-structure in simulated stellar halos. For the comparison with\nthe data, \\citet{Helmi2011} examine the smoothness of the mock stellar\nhalos produced by \\citet{Cooper2010}. These are built into the\nAquarius DM-only halos \\citep{Springel2008} by tagging 1\\% of the\nmost-bound particles in selected sub-halos and following them to\nredshift 0. Compared to the mock ``Milky Ways'' of\n\\citet{Bullock2005}, these have the obvious advantages of being\nfabricated in the Cosmological setting, and with a superior\nresolution. However, there are disadvantages too. First, the Aquarius\nsuite explores only half as many accretion histories, in fact, in the\nend, there are only 4 stellar halos analyzed in \\citet{Helmi2011},\ncompared to 11 in \\citet{Bullock2005}. Second, these Galaxy analogs do\nnot posses disks. A quick glance at the Figure 5 of \\citet{Helmi2011}\nreveals: all stellar halos of \\citet{Cooper2010} are highly\nirregular, with $50\\% < \\frac{{\\rm rms}(\\rho)}{<\\rho>} < 150\\%$. The\nauthors raise concern that some of the data-model discrepancy could be\ndue to the combined effects of the MSTO sample contamination and the\npresence of a smooth, in-situ formed stellar halo\ncomponent. Nonetheless, they conclude that there exists considerable\ntension between the observations of the Galactic stellar halo\nsub-structure and the predictions of the simple but high-resolution\nmodel. Even though the halos of both the real and the mock Galaxy are\nvery inhomogeneous, the simulations easily reach 2-3 times the\nobserved scatter on scales as small as few degrees.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.02\\linewidth]{data_model_lb_deason.pdf}\n\\caption{Stellar halo of the Milky Way traced by the BHB stars. {\\it\n    Left} Distribution of the SDSS DR8 BHB candidates in the Galactic\n  $l$ and $b$. {\\it Right} Best-fit model of the stellar halo density\n  distribution shown in the Left panel, from \\citet{Deason2011a}. The\n  model halo is flattened with $q\\sim 0.6$ and has a break in the\n  radial density profile at $r\\sim 27$ kpc where the power-law index\n  changes from -2.3 to -4.6. Figure courtesy of Alis Deason,\n  IoA\/UCSC.} \\label{fig:halo_bhb}\n\\end{figure}\n\nThe picture of the utter chaos in the inner parts of the Galactic\nstellar halo is re-visited in \\citet{Deason2011a}. Instead of using\nthe more abundant MSTO stars, they choose to model the halo volume\ndensity with Blue Horizontal Branch stars. While these stars are\nrarer, their higher intrinsic luminosities, lower levels of\ncontamination and accurate absolute magnitude calibration independent\nof age and chemistry all make these a better fit for the task. There\nare, nonetheless, several limitations to the use of BHBs as\ntracers. For example, being some $\\sim 4$ magnitudes brighter and at\nleast two orders of magnitude less frequent as MSTO, these come in\nparticularly low numbers at bright apparent magnitudes due to the size\nof the volume probed. Additionally, while their blue color makes them\nstand out dramatically compared to most other stellar populations at\nhigh Galactic latitudes, there is one troublesome impostor. Blue\nStragglers (see Figure~\\ref{fig:tracers}) have close to identical\nbroad-band colors but are $\\sim 1.5$ mag fainter. Outnumbering the\nBHBs by a factor of 2 on average, these may pose a serious problem by\nscrambling the tracer counts as a function of apparent\nmagnitude. \\citet{Deason2011a} solve both the problem of the limited\ndynamic range and of the contamination by including the BS stars in\nthe model. For all ``blue'' stars in the SDSS DR8, i.e. $-0.25 < g-r <\n-0$, the probability of belonging to the BHB or the BS population is\nassigned based on their $u-g$ and $g-r$ colors. As a result, the\nnumber density of stars in volume elements of the space spanned by\nposition of the sky, color and apparent magnitude can be modeled\nsimply as the sum of the contributions from BHBs and BSs, weighted by\ntheir conditional probabilities.\n\nThe results of the maximum-likelihood analysis presented in\n\\citet{Deason2011a} are summarized for the impatient reader in the\narticle's title ``Squashed, broken but smooth''. In other words: out\nto 40 kpc, the Galactic stellar halo appears to be highly flattened,\nthe density profile follows closely the broken power law and, most\ninterestingly, the overall level of sub-structure detected using the\nBHB tracers is rather low. At small and intermediate distances,\n$\\sigma\/{\\rm total}$ rises from as low $10\\%$ to at most $20\\%$\nirrespective of the spatial scale of density perturbations. At large\ndistance, $\\sigma\/{\\rm total}$ is close to $20\\%$ on most scales, but\nrises to $40\\%$ for the angular sizes of several hundreds of\ndegrees. These numbers are obtained by excluding from the modeling the\nregions of the sky with known large-scale halo overdensities. Even\nwhen these are included, the small-size inhomogeneities are only $10\n\\% < \\sigma\/{\\rm total} < 30\\%$. While, superficially, these estimates\ndiffer significantly from those quoted in \\citet{Bell2008}, there are\nseveral possible solutions to this discrepancy. Both methods have\ntheir weak points. It is quite likely that some of the halo mess\nobserved by \\citet{Bell2008} is simply due to the limitations of the\nMSTO stars as tracers. On the other hand, the average number of BHBs\nin a $1^{\\circ} \\times 1^{\\circ}$ pixel is small, hence limiting the\nareas of the sky tested by \\citet{Deason2011a} to those towards the\ninner Galaxy where mixing is more efficient. On slightly larger\nangular scales (several degrees or so), it is, however, safe to\nconclude that the inner stellar halo is indeed smooth.\n\n\\subsubsection{Phase-space sub-structure. Spaghetti, ECHOS and SKOs}\n\nAs it is much easier to identify the accreted satellite debris in the\nphase-space compared to simple sky density maps or 3D spatial maps,\nseveral attempts have been made to search for the surviving Galactic\nsub-structure in the datasets of wide area spectroscopic surveys. The\nSpaghetti survey \\citep[e.g.][]{Morrison2000} is the first brave\nendeavor to collect substantial numbers of genuine halo tracers in a\nlarge distance range. It is set up to gather photometry and the\nfollow-up spectroscopy in several tens of ``pencil-beam'' fields over\nthe area covering many tens of degrees. The analysis dealing with the\nquantification of the presence of sub-structure in the final set of\n101 giants with spectra covering distances up to 100 kpc is presented\nin \\citet{Starkenburg2009}. They report the detection of 1 group and 6\npairs of clumped stars and conclude that their findings of 10\\% of\nsub-structure in the halo are consistent with the accretion scenarios\nin which early and\/or massive satellite infall leads to the creation\nof broad phase-space features.\n\nThe SEGUE survey that has taken $\\sim$240,000 spectra in $>$200\npointings spread over $\\sim$11,000 square degrees is the ideal source\nof data to carry out a systematic search for un-relaxed\nsub-structure. \\citet{Schlaufman2009} do exactly that, and detect in\n137 lines of sight studied 10 high-confidence ECHOS, elements of cold\nhalo substructure as traced by metal-poor MSTO stars with distances in\nthe range $10 <D~({\\rm kpc})< 20$. As the ECHOS identification\nalgorithm described is automated, the work also contains the results\nof the completeness calculation for the sub-structure search in the\nvelocity space. These are then used to turn the numbers of detections\ninto the predictions of sub-structure fractions existent in the halo:\nthey conclude that within 1\/3 of the volume of the Galactic halo,\nthere ought to be of the order of 10\\% of ECHOS.\n\n\\citet{Xue2011} use the spectroscopic sample of $\\sim 4,000$ SDSS BHB\nstars to gauge the power spectrum of the kinematic sub-structure in\nthe stellar halo by measuring the excess of close stellar pairs whose\nrelative distances are measured in the 4D comprised of 3 spatial\ncoordinates and the line-of-sight velocity. The presence of\nsub-structure is clearly detected when the observed close-pair\ndistribution is compared to that of the mock smooth halo created by\nscrambling the heliocentric distances of the SDSS BHBs. Moreover, as\npredicted, the outer halo ($20 < D < 40$ kpc) exhibits a stronger\nsub-structure signal as compared to the inner one ($5 < D < 20$\nkpc). However, when comparing to the simulations of\n\\citet{Bullock2005}, \\citet{Xue2011} find appreciably less signal\noverall, in particular, at the intermediate 4D distance scales.\n\nFinally, at moderate distances from the Sun, relatively accurate\nstellar proper motion measurements have been available for some\ntime. Combined with photometric parallax, this opens up the\npossibility to calculate all 6 of the phase-space\ncoordinates. Accordingly, \\citet{Smith2009} exploit the accurate\nproper motions calculated by \\citet{Bramich2008} using the multi-epoch\ndata in the SDSS Stripe 82 region, to compile a catalog of $\\sim$\n1,700 MS sub-dwarfs within 5 kpc from the Sun with full space\nvelocities and 3D coordinates measured. In estimating the tangential\ncomponents of the stars velocity, the main source of error is the\ndistance uncertainty, which is actually present at the rather modest\nlevel of 10\\%. After subtracting a smooth Galaxy model from the\ndistribution of the angular momenta $J_z, J_{\\perp}$ 3 significant\nclumps are detected, one of which has been previously discovered in\nthe pioneering work by \\citet{Helmi1999}.\n\n\n\\section{Putting the puzzle together}\n\nIt is obvious why in a framework which builds the Universe from bottom\nup, the smallest galaxies are under double scrutiny: such structure\nformation paradigm risks losing its credibility if its basic blocks\nlook unlifelike. For $\\Lambda$CDM, predicting the properties of the\nlocal dwarf galaxies, while seemingly completely straightforward, has\nturned out to be a trying quest. As this review argues, before a\nfaithful comparison between the dwarf galaxies formed according to the\nCold Dark Matter model and the observed satellites can be carried out, a\nnumber of loose ends need to be tied up. First, the host-to-host\nvariations in the properties of the DM sub-halo populations due to the\nparent halo accretion history and the environment need to be\nquantified. Also, it is not an overstatement to say that our\nunderstanding of the baryonic processes to do with cooling and\nfeedback, although developing swiftly, is not yet mature and, thus\ndeserves further improvement. Observationally, it seems now possible\nto provide better measurements of the basic properties of the host\nsuch as the Galactic total mass, the mass of the disk and, possibly\neven the concentration of the DM halo. Additionally, as the\nquantitative comparison is presently based around the satellite\nluminosity function, it is crucial to figure out the genesis of the\nsmallest objects dominating the census, the ultra-faints. Finally, it\nis clear that the same DM and baryonic physical processes that have\nsculpted the $z=0$ dwarf satellite population are also responsible for\nthe creation of the stellar halo. Therefore, a successful galaxy\nformation model is expected to get both the survived and the perished\nsatellites right.\n\n\\subsection{Light Galactic DM halo with high concentration} \n\n\\citet{Deason2012b} tackle the issue of the dearth of reliable tracers\nin the outer halo.  They take advantage of the availability of the\nmultiple epoch imaging data across the SDSS field of view. Due to the\nsurvey geometry, a considerable number of imaging ``stripes'' overlap\n(mostly around the survey poles) and the photometry from the multiple\nruns can be combined to yield higher signal-to-noise magnitude\nmeasurements.  This is especially valuable for the faint BHB candidate\nstars, whose single-epoch $u$-band magnitudes are simply too\nunreliable for the conventional broad-band BHB\/MS\/MSTO\nclassification. Using the overlaps as well as the multi-epoch data\nfrom the SDSS Stripe 82, \\citet{Deason2012b} find 43 distant BHB\ncandidates with $20 < g < 22$ and follow these up with deep\nspectroscopy on the ESO's VLT. Once a handful the QSO interlopers are\nremoved, of the remaining 38 A-colored stars, 7 are genuine BHBs in\nthe distance range 80-150 kpc. Note that, at these faint magnitudes,\neven the remaining 31 BS stars populate the poorly explored regime\nbetween 30 and 90 kpc. Their final sample also contains 8 cool carbon\nstars that span a distance range 80 to 160 kpc. This combined\nspectroscopic sample is the largest collection to date of the halo\ntracers with distances beyond 60 kpc. Curiously, outside 100 kpc from\nthe Galactic center, the line-of-sight velocity dispersion plummets to\na rather low 50-60 km s$^{-1}$. Unless the stellar tracers considered\nhave a significant tangential bias and\/or their density drops much\nfaster than the already rather steep power law with index -4.5, this\nmeasurement implies the mass of the Milky Way in the range of $5-10\n\\times 10^{11} M_{\\odot}$ within 150 kpc.\n\nTo figure out how concentrated the DM halo of the Galaxy is, the total\nmatter density has to be mapped out in a sufficiently wide range of\ndistances around or beyond the Solar neighborhood (inside $R_0$, the\ndisk dominates the mass budget). \\citet{Deason2012a} exploit the sheer\nvolume probed by the SDSS BHB stars with known radial velocities to\nsimultaneously constrain the halo velocity anistropy and the matter\ndistribution. By approximating the gravitational potential of the\nGalaxy as a power-law and adopting the tracer density distribution\npinned down in \\citet{Deason2011a}, they find that the rotation curve\nof the Galaxy has to start falling appreciably already at around 30\nkpc from the center. They ague that the preferred value of the\nnormalization and the power law index of the gravitational potential\nare inconsistent with massive, i.e. $2 \\times 10^{12} M_{\\odot}$ DM\nhalos with concentrations around $c\\sim 10$. Instead, a lighter and\nmore concentrated, i.e. $c\\sim 20$ dark halo is favored.\n\n\\subsection{Smooth stellar halo and signatures of early accretion}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{bullock_johnston.jpg}\n\\caption{Three of the 11 Galactic stellar halos simulated by\n  \\citet{Bullock2005}. Particles are color-coded according to the mass\n  of the progenitor, with most massive in red and least massive in\n  blue. The smooth, flattened Halo 7 has a density break at 24 kpc and\n  represents the simulation closest to the observed Galactic stellar\n  halo. Halo 8 gives a clue of how the stellar halo in M31 might look\n  like.}\n\n\\label{fig:bj2005}\n\\end{figure}\n\nThe stellar masses in the known large-scale over-densities in the\nGalactic halo (see Section~\\ref{sec_big4}) sum up to the approximate\ntotal of $\\sim 3 \\times 10^8 M_{\\odot}$. This is to be compared with\nthe estimates of the total stellar halo mass \\citep[e.g.][]{Bell2008,\n  Deason2011a} of the order of $\\sim 10 \\times 10^8\nM_{\\odot}$. Therefore, perhaps as much as $70 \\%$ of the stellar halo\nwithin 40 kpc is in a smooth or, more accurately, well-mixed,\ncomponent, which can be described by a broken power-law density\nprofile with a flattening of $q\\sim 0.6$. On contrary, the Andromeda's\nstellar halo harbors no such feature in its density profile over twice\nas large range of distances, albeit it does not look nearly as smooth\n\\citep{Gilbert2012}. \\citet{Deason2013} investigate whether stellar\nhalos with density breaks similar to that of the Milky Way can be\nassembled purely through satellite accretion, and if yes, what\ncontrols the break prominence and the radius. By studying a set of 11\nN-body simulations published by \\citet{Bullock2005} they come to the\nfollowing conclusion. In the simulations studied, the radial profiles\nof stars from individual accretion events can be described by single\npower law, double power law or can be so ill-defined that neither of\nthe simple models works. Ancient ($>10$ Gyr \\footnote{In this\n  definition, the age marks the time of when the stars became unbound,\n  which implies slightly earlier epochs for the arrival of the\n  progenitor.}) debris have had plenty of time to mix and therefore at\n$z=0$ the radial profile is comfortably fit with a single\npower-law. Old (7-10 Gyr) debris have spread out over a range of\nGalacto-centric distances but around the progenitor's apo-centre, the\ndrop in stellar density remains. Recent ($<6$ Gyr) mergers have not yet\nfilled the entire volume inward of the apo-centre and their radial\ndistribution still peaks at $R>0$.\n\nThe stellar halo (in this model) is just a superposition of the debris\nfrom the individual events across the entire accretion history. The\ncombined stellar profile can have a distinct break (at the average\napo-centre of the most massive accreted satellites) only if the most\nsignificant merger(s) happened at the right time, i.e. 8-10\nGyr. Additionally, it is required to dampen the accretion rate at the\nsubsequent epochs: as the Galaxy grows, the satellites that arrive\nwith increasingly larger apo-centers thus can flatten out the density\nprofile around and beyond the break radius, thus erasing this feature\naltogether. The hypothesis that the density break in the Galactic\nstellar halo reflects the apo-center(s) of the massive satellite(s)\naccreted at early epochs can be tested with 3D kinematics. Radial\nvelocities of stars tend to zero around the apo-center of the orbit,\ntherefore the radial velocity dispersion of the stellar halo should\nhave a dip around the break radius as well as an increase in the\ntangential anisotropy. Moreover, there exists a potentially powerful\ndiagnostic to decipher the properties of this old merger. Namely, if\nthe metallicities of the stellar halo tracers around the break radius\n(i.e. $20<R<30$ kpc) are available, then it is possible to distinguish\nbetween the accretion of one or two massive satellite(s) and the\naccretion of a group of dwarfs. If only one system contributed the\nbulk of the debris within the break then the radial velocity\ndispersion dip around the break radius should be most visible in stars\nwith the chemical abundance of that satellite. The density break\ncreated via superposition of the debris from many different satellites\nis not dominated by any particular stellar population, and hence, the\ndrop in radial velocity dispersion should occur, albeit weakly, across\nthe metallicity range.\n\nFigure~\\ref{fig:bj2005} shows the examples of three Galactic stellar\nhalos created in simulations by \\citet{Bullock2005}. Here, the X-Z\ndistributions of particles color-coded according to the mass of the\nprogenitor (red for most massive, blue for the least massive) are\nshown for the inner 100 kpc (left panel) and the inner 50 kpc (right\npanel). The differences in the structures of Halo 7 and Halo 8 provide\nvisual clues as to the findings of \\citet{Deason2013}. Halo 7 has a\npeaked accretion history, with the bulk of the stellar halo assembled\nquickly around 8 Gyr ago. Within 40 kpc from the center of the Galaxy,\nthe stellar halo seems smooth, flattened and strongly aligned with the\ndisk (compare to the view of the Galactic stellar halo in\nFigure~\\ref{fig:halo_bhb}). Note that Halo 7 has a prominent break in\nthe radial density profile at 24 kpc, closely matching the observed\nproperties of the Milky Way halo as traced by the BHBs. Halo 8, on\ncontrary, shows no evidence of the break - this turns out to be a\ngiveaway of its accretion history and the present day appearance. Halo\n8 can be categorized by a continuous infall of satellites with three\nparticularly massive fragments disrupting at 11, 7 and 1-2 Gyr. The\nstellar distribution is evidently more extended and substantially\nmessier (even in the inner parts) as compared to that of the ``quiet''\nHalo 7. Additionally, neither obvious vertical flattening nor\nalignment with the disk can be seen.\n\n\\subsection{Nature of the faintest of the ultra-faint satellites}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{dwarf_sequence.jpg}\n\\caption{Dwarfs accreting dwarfs. This shows the SDSS image mosaics of\n  the three Local Volume dwarf galaxies (with luminosities similar to\n  that of the SMC) disrupting their satellites with masses as low as a\n  hundredth of the host's. The images are processed to enhance the\n  low-surface brightness features. From Left to Right: the\n  dwarf-to-dwarf accretion sequence. The satellite of the NGC 4449 is\n  showing the first signs of tidal interaction, the satellite of LSBC\n  F567-01 is critically stretched, very close to being fully\n  destroyed, the satellite of MCG-01-08-001 is taken apart and\n  incorporated into the host, but still visible as faint stellar plume\n  around the Southern edge of host's halo.}\n\n\\label{fig:dwarf_accretion}\n\\end{figure}\n\nThe absolute majority of the stellar populations in the ultra-faint\ndwarf satellites are as old as the oldest Galactic globular clusters\nand are similarly metal-poor, as revealed, for example, by their deep\nColor-Magnitude Diagrams. Given their low total stellar masses, the\nvelocity dispersions in the range of $2 < \\sigma < 9$ km s$^{-1}$ and\nthe recently detected spreads in metallicity, there is now little\ndoubt that these are the remnants of the galaxies born at high\nredshifts in low-mass DM halos. What is not yet completely clear is\nthe exact mapping between the galaxies of a particular luminosity at\n$z=0$ and the original sub-halo mass at the epoch of formation. This\nis simply due to the fact that for such faint stellar systems, there\nis not enough stars currently available to trace the total matter\ndistribution out to large distances from the center. Hence, even\nthough their present-day central masses are being constrained, the\ntotal mass and the extent of their DM halos is still an enigma.\n\nAll semi-analytic models of the dwarf galaxy formation predict that\nmost satellites lose significant amounts of their dark and luminous\nmatter to the host's tides. The precise mass loss is the crucial\nmiddle part in the model that has at one end a variety of dark hosts\nwith the mass function steeply rising at low masses, and at the other,\nthe observed population of Galactic survivors. Relying on the fact\nthat the details of the tidal harassment are captured at the\nappropriate level in the high resolution N-body simulations, the\nluminosity function of the Milky Way satellites is inverted to reveal\nthe physics of star formation in the early Universe. However, it is\nnow apparent that the inclusion of the baryons (perhaps even as simple\nas adding a disk component to the host galaxy) can alter the satellite\nsurvival rates dramatically. Additionally, in the hierarchical\nUniverse, the smallest satellites have a non-zero chance of being\nfirst accreted onto the more massive dwarf galaxies before finally\nmerging with their ``Milky Way'' host. Such {\\it satellites of\n  satellites} with large host-to-satellite mass ratios have been known\nto exist in the galaxy formation simulations, but only now begin to be\ndiscovered in nature. For example, \\cite{Martinez-Delgado2012} report\nthe discovery of the early stages of the accretion of a low-luminosity\ndwarf by the galaxy as massive as the SMC, namely NGC\n4449. Figure~\\ref{fig:dwarf_accretion} shows this and two more\nexamples of different stages of the accretion of dwarfs onto dwarfs.\n\nTo increase the dynamic range, the images in the three panels of\nFigure~\\ref{fig:dwarf_accretion} are composed of two versions of the\nsame SDSS mosaic image: the original one in color and the greyscale\none obtained by applying the median filter to triples of pixels in the\nthree SDSS filters to minimize the ``patchwork'' effects of the sky\nslightly misaligned between different SDSS runs. The original SDSS\ncolor image and the stretched and inverted greyscale image are then\ncombined by choosing the pixel value to be the highest one between the\ntwo images. This simple image processing \\citep[also employed by\n  e.g.][]{Martinez-Delgado2010, Martinez-Delgado2012} makes it\npossible to show simultaneously the central object where most of the\nlight is as well as the further low surface brightness features that\nare to do with the ongoing tidal interactions. While the left panel\nshows the beginning of the process of accretion, the middle and the\nright panels of the Figure display more advanced stages of the dwarf\ninfall and disruption. LSBC F567-01 in the center is seen taking apart\na low-luminosity satellite which looks prominently stretched and\nperhaps close to its total disintegration. Finally, MCG-01-08-001 in\nthe right panel, is clearly about to finish eating up a smaller\nsatellite, which presently can only be seen as extended faint plume of\nstellar debris in the Southern parts of the host.\n\nTaking into account the existing evidence of a likely association\nbetween objects like Segue 1, Segue 2, Bootes II and III and the known\nlarge-scale sub-structure in the Galactic stellar halo, the following\nscenario is perhaps possible. The faintest of the currently known\nMilky Way satellites were born as sub-systems of larger dwarfs and\nwere subsequently pre-processed by their first hosts' tidal\nfields. They have probably lost much of their original mass but their\nremnant cores survived as part of the bigger dwarf galaxy until the\nwhole system was accreted by the Milky Way. Their parent galaxies were\nsufficiently massive to be dragged into the inner Milky Way where they\nwere destroyed and are visible today only as streams and clouds of\nstars in the halo. Some of the satellites of satellites persisted in\nthe Milky Way halo and can now be observed as the faintest of the\nultra-faint dwarfs. In this scheme, the current stellar and the DM\nmasses are severely truncated: thanks to the pre-processing in the\ngravitational field of the parent dwarf their current luminosities\ncould be much lower than what is attainable by the lowest mass dwarfs\nin the field. This in turn, would imply that the fast drop in the\nefficiency of the dwarf galaxy formation actually happens at the\nsub-halo masses that are higher than previously envisaged. Because the\nsatellites with absolute magnitudes around $-2 > M_V > -4$ can only\n(or mostly) exist as part of bigger dwarf systems, their distribution\nin the Galaxy is different from that of the accreted field dwarf\npopulation. Their radial density profile should be strongly radially\nconcentrated due to the combination of the two effects. First, their\nparent galaxies were massive enough to end up close to the center due\nto the dynamical friction. Second, in the Galaxy most of the large\nsystems (apart from the Sgr dwarf) were accreted as early as 8-10 Gyr,\nwhen the mass and the virial mass of the Milky Way were much\nsmaller. Taking these effects into account, much lower numbers of\nsatellites as faint as Segue I or II are predicted to be discovered by\nthe future deep all sky surveys.\n\n\n\n\\pagebreak\n\n\\section*{Acknowledgments} \nV. Belokurov thanks The Royal Society the support. The work on this\nreview has received funding from the European Research Council under\nthe European Union's Seventh Framework Programme (FP\/2007-2013) \/ ERC\nGrant Agreement n. 308024. The author has enjoyed conversations with\nA. Deason, W. Evans, A. Helmi, M. Irwin, S. Koposov, P. Kroupa,\nJ. Norris, M. Smith, E. Starkenburg and E. Tolstoy.\n\n\\pagebreak\n\n\\vspace*{2cm}\n\n\\noindent\n\n\\bibliographystyle{elsarticle-harv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{intr}\n\nHyperk\\\"{a}hler manifolds first appeared within the framework of differential geometry as Riemannian manifolds with holonomy group of special restricted group. Nowadays, hyperk\\\"{a}hler geometry forms a separate research subject fusing traditional areas of mathematics such as differential and algebraic geometry of complex manifolds, holomorphic symplectic geometry, Hodge theory and many others. \n\nOne of the latest links can be found in theoretical physics: In 2009, Gaiotto, Moore and Neitzke \\cite{gaiotto} proposed  a new construction of hyperk\\\"{a}hler metrics $g$ on target spaces $\\mathcal{M}$ of quantum field theories with $d = 4, \\mathcal{N} = 2$ superysmmetry. Such manifolds were already known to be hyperk\\\"{a}hler (see \\cite{seiberg}), but no known explicit hyperk\\\"{a}hler metrics have been constructed.\n\nThe manifold $\\mathcal{M}$ is a total space of a complex integrable system and it can be expressed as follows. There exists a complex manifold $\\mathcal{B}$, a divisor $D \\subset \\mathcal{B}$ and a subset $\\mathcal{M}' \\subset \\mathcal{M}$ such that $\\mathcal{M}'$ is a torus fibration over $\\mathcal{B}' := \\mathcal{B} \\backslash D$. On the divisor $D$, the torus fibers of $\\mathcal{M}$ degenerate, as Figure \\ref{nodtorus} shows.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{nodal_torus.eps}\n\t\\caption{Hyperk\\\"{a}hler manifolds realized as torus fibrations}\n\t\\label{nodtorus}\n\\end{figure}\n\nModuli spaces $\\mathcal{M}$ of Higgs bundles on Riemann surfaces with prescribed singularities at finitely many points are one of the prime examples of this construction. Hyperk\\\"{a}hler geometry is useful since we can use Hitchin's twistor space construction \\cite{hitchin} and consider all $\\mathbb{P}^1$-worth of complex structures at once. In the case of moduli spaces of Higgs bundles, this allows us to consider $\\mathcal{M}$ from three distinct viewpoints:\n\n\\begin{enumerate}\n\t\\item (Dolbeault) $\\mathcal{M}_{\\text{Dol}}$ is the moduli space of Higgs bundles, i.e. pairs $(E, \\Phi)$, $E \\to C$ a rank $n$ degree zero holomorphic vector bundle and $\\Phi \\in \\Gamma(\\text{End}(E) \\otimes \\Omega^1)$ a Higgs field.\n\t\n\t\\item (De Rham) $\\mathcal{M}_{\\text{DR}}$  is the moduli space of flat connections on rank $n$ holomorphic vector bundles, consisting of pairs $(E, \\nabla)$ with $\\nabla : E \\to \\Omega^1 \\otimes E$ a holomorphic connection and\n\t\n\t\\item (Betti) $\\mathcal{M}_{\\text{B}} = \\text{Hom}(\\pi_1(C) \\to \\text{GL}_n(\\mathbb{C}))\/\\text{GL}_n(\\mathbb{C})$ of conjugacy classes of representations of the fundamental group of $C$. \n\\end{enumerate}\n\n\\noindent All these algebraic structures form part of the family of complex structures making $\\mathcal{M}$ into a hyperk\\\"{a}hler manifold. \n\nTo prove that the manifolds $\\mathcal{M}$ from the integrable systems are indeed hyperk\\\"{a}hler, we start with the existence of a simple, explicit hyperk\\\"{a}hler metric $g^{\\text{sf}}$ on $\\mathcal{M}'$. Unfortunately, $g^{\\text{sf}}$ does not extend to $\\mathcal{M}$.  To construct a complete metric $g$, it is necessary to do ``quantum corrections'' to $g^{\\text{sf}}$. These are obtained by solving a certain explicit integral equation (see \\eqref{inteq} below). The novelty is that the solutions, acting as Darboux coordinates for the hyperk\\\"{a}hler metric $g$, have discontinuities at a specific locus in $\\mathcal{B}$. Such discontinuities cancel the global monodromy around $D$ and is thus feasible to expect that $g$ extends to the entire $\\mathcal{M}$.\n\nWe start by defining a Riemann-Hilbert problem on the $\\mathbb{P}^1$-slice of the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$. That is, we look for functions $\\mathcal{X}_\\gamma$ with prescribed discontinuities and asymptotics. In the language of Riemann-Hilbert theory, this is known as \\textit{monodromy data}. Rather than a single Riemann-Hilbert problem, we have a whole family of them parametrized by the $\\mathcal{M}'$ manifold. We show that this family constitutes an \\textit{isomonodromic deformation} since by the Kontsevich-Soibelman Wall-Crossing Formula, the monodromy data remains invariant. \n\nAlthough solving Riemann-Hilbert problems in general is not always possible, in this case it can be reduced to an integral equation solved by standard Banach contraction principles. We will focus on a particular case known as the ``Pentagon'' (a case of Hitchin systems with gauge group $\\text{SU}(2)$). The family of Riemann-Hilbert problems and their methods of solutions is a topic of independent study so we leave this construction to a second article that can be of interest in the study of boundary-value problems.\n\nThe extension of the manifold $\\mathcal{M}'$ is obtained by gluing a circle bundle with an appropriate gauge transformation eliminating any monodromy problems near the divisor $D$. The circle bundle constructs the degenerate tori at the discriminant locus $D$  (see Figure \\ref{pinch}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.15\\textwidth]{part_torus2.ps}\n\t\\caption{Construction of degenerate fibers}\n\t\\label{pinch}\n\\end{figure}\n\nOn the extended manifold $\\mathcal{M}$ we prove that the solutions $\\mathcal{X}_\\gamma$ of the Riemann-Hilbert problem on $\\mathcal{M}'$ extend and the resulting holomorphic symplectic form $\\varpi(\\zeta)$ gives the desired hyperk\\\"{a}hler metric $g$.\n\nAlthough for the most basic examples of this construction such as the moduli space of Higgs bundles it was already known that $\\mathcal{M}'$ extends to a hyperk\\\"{a}hler manifold $\\mathcal{M}$ with degenerate torus fibers, the construction here works for the general case of $\\dim_\\mathbb{C} \\mathcal{B} = 1$. Moreover, the functions $\\mathcal{X}_\\gamma$ here are special coordinates arising in moduli spaces of flat connections, Teichm\\\"{u}ller theory and Mirror Symmetry. In particular, these functions are used in \\cite{chan} for the construction of holomorphic discs with boundary on special Lagrangian torus fibers of mirror manifolds.\n\nThe organization of the paper is as follows. In Section \\ref{intsys} we introduce the complex integrable systems to be considered in this paper. These systems arose first in the study of moduli spaces of Higgs bundles and they can be written in terms of initial data and studied abstractly. This leads to a formulation of a family of Riemann-Hilbert problems, whose solutions provide Darboux coordinates for the moduli spaces $\\mathcal{M}$ considered and hence equip the latter with a hyperk\\\"{a}hler structure. In Section \\ref{ov} we fully work the simplest example of these integrable systems: the Ooguri-Vafa case. Although the existence of this hyperk\\\"{a}hler metric was already known, this is the first time it is obtained via Riemann-Hilbert methods. In Section \\ref{gmetric}, we explicitly show that this metric is a smooth deformation of the well-known Taub-NUT metric near the singular fiber of $\\mathcal{M}$ thus proving its extension to the entire manifold. In Section \\ref{pent} we introduce our main object of study, the Pentagon case. This is the first nontrivial example of the integrable systems considered and here the Wall Crossing phenomenon is present. We use the KS wall-crossing formula to apply an isomonodromic deformation of the Riemann-Hilbert problems leading to solutions continuous at the wall of marginal stability. Finally, Section \\ref{sfiber} deals with the extension of these solutions $\\mathcal{X}_\\gamma$ to singular fibers of $\\mathcal{M}$ thought as a torus fibration. What we do is to actually complete the manifold $\\mathcal{M}$ from a regular torus fibration $\\mathcal{M}'$ by gluing circle bundles near a discriminant locus $D$. This involves a change of the torus coordinates for the fibers of $\\mathcal{M}'$. In terms of the new coordinates, the $\\mathcal{X}_\\gamma$ functions extend to the new patch and parametrize the complete manifold $\\mathcal{M}$. We finish the paper by showing that, near the singular fibers of $\\mathcal{M}$, the hyperk\\\"{a}hler metric $g$ looks like the metric for the Ooguri-Vafa case plus some smooth corrections, thus proving that this metric is complete.\n\n\\textbf{Acknowledgment:} The author likes to thank Andrew Neitzke for his guidance, support and incredibly helpful conversations. \n\n\\section{Integrable Systems Data}\\label{intsys}\n\nWe start by presenting the complex integrable systems introduced in \\cite{gaiotto}. As motivation, consider the moduli space $\\mathcal{M}$ of Higgs bundles on a complex curve $C$ with Higgs field $\\Phi$ having prescribed singularities at finitely many points. In \\cite{wkb}, it is shown that the space of quadratic differentials $u$ on $C$ with fixed poles and residues is a complex affine space $\\mathcal{B}$ and the map $\\text{det} : \\mathcal{M} \\to \\mathcal{B}$ is proper with generic fiber $\\text{Jac}(\\Sigma_u)$, a compact torus obtained from the \\textit{spectral curve} $\\Sigma_u : = \\{(z, \\phi) \\in T^*C : \\phi^2 = u\\}$,  a double-branched cover of $C$ over the zeroes of the quadratic differential $u$. $\\Sigma_u$ has an involution that flips $\\phi \\mapsto -\\phi$. If we take $\\Gamma_u$ to be the subgroup of $H_1(\\Sigma_u, \\mathbb{Z})$ odd under this involution, $\\Gamma$ forms a lattice of rank 2 over $\\mathcal{B}'$, the space of quadratic differentials with only simple zeroes. This lattice comes with a non-degenerate anti-symmetric pairing $\\left\\langle , \\right\\rangle$ from the intersection pairing in $H_1$. It is also proved in \\cite{wkb} that the fiber $\\text{Jac}(\\Sigma_u)$ can be identified with the set of characters $\\text{Hom}(\\Gamma_u, \\mathbb{R}\/2\\pi \\mathbb{Z})$. If $\\lambda$ denotes the tautological 1-form in $T^* C$, then for any $\\gamma \\in \\Gamma$,\n\\[ Z_\\gamma = \\frac{1}{\\pi}  \\oint_\\gamma \\lambda \\]\ndefines a holomorphic function $Z_\\gamma$ in $\\mathcal{B}'$. Let $\\{\\gamma_1, \\gamma_2\\}$ be a local basis of $\\Gamma$ with $\\{\\gamma^1, \\gamma^2\\}$ the dual basis of $\\Gamma^*$. Without loss of generality, we also denote by $\\left\\langle ,  \\right\\rangle$ the pairing in $\\Gamma^*$. Let $\\left\\langle dZ \\wedge dZ \\right\\rangle$ be short notation for $\\left\\langle \\gamma^1, \\gamma^2 \\right\\rangle dZ_{\\gamma_1} \\wedge dZ_{\\gamma_2}$. Since $\\dim_\\mathbb{C} \\mathcal{B}' = 1$, $\\left\\langle dZ \\wedge dZ \\right\\rangle = 0$.\n\nThis type of data arises in the construction of hyperk\\\"{a}hler manifolds as in \\cite{gaiotto} and \\cite{notes}, so we summarize the conditions required:\n \n\nWe start with a complex manifold $\\mathcal{B}$ (later shown to be affine) of dimension $n$ and a divisor $D \\subset \\mathcal{B}$. Let $\\mathcal{B}' = \\mathcal{B} \\backslash D$. Over $\\mathcal{B}'$ there is a local system $\\Gamma$ with fiber a rank $2n$ lattice, equipped with a non-degenerate anti-symmetric integer valued pairing $\\left\\langle \\, , \\right\\rangle$. \n\nWe will denote by $\\Gamma^*$ the dual of $\\Gamma$ and, by abuse of notation, we'll also use $\\left\\langle \\, , \\right\\rangle$ for the dual pairing (not necessarily integer-valued) in $\\Gamma^*$. Let $u$ denote a general point of $\\mathcal{B}'$. We want to obtain a torus fibration over $\\mathcal{B}'$, so let $\\text{TChar}_u(\\Gamma)$ be the set of twisted unitary characters of $\\Gamma_u$\\footnote{Although we can also work with the set of unitary characters (no twisting involved) by shifting the $\\theta$-coordinates, we choose not to do so, as that results in more complex calculations}, i.e. maps $\\theta : \\Gamma_u \\to \\mathbb{R}\/2\\pi \\mathbb{Z}$ satisfying\n\\begin{equation*}\n\\theta_\\gamma + \\theta_{\\gamma'} = \\theta_{\\gamma + \\gamma'} + \\pi \\left\\langle \\gamma, \\gamma' \\right\\rangle.\n\\end{equation*}\nTopologically, $\\text{TChar}_u(\\Gamma)$ is a torus $(S^1)^{2n}$. Letting $u$ vary, the $\\text{TChar}_u(\\Gamma)$ form a torus bundle $\\mathcal{M}'$ over $\\mathcal{B}'$. Any local section $\\gamma$ gives a local angular coordinate of $\\mathcal{M}'$ by ``evaluation on $\\gamma$'', $\\theta_\\gamma : \\mathcal{M}' \\to \\mathbb{R}\/2\\pi \\mathbb{Z}$.\n\nWe also assume there exists a homomorphism $Z : \\Gamma \\to \\mathbb{C}$ such that the vector $Z(u) \\in \\Gamma^*_u \\otimes \\mathbb{C}$ varies holomorphically with $u$. If we pick a patch $U \\subset \\mathcal{B}'$ on which $\\Gamma$ admits a basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ of local sections in which $\\left\\langle , \\right\\rangle$ is the standard symplectic pairing, then (after possibly shrinking $U$) the functions\n\\[ f_i = \\text{Re}(Z_{\\gamma_i}) \\]\nare real local coordinates. The transition functions on overlaps $U \\cap U'$ are valued on $\\text{Sp}(2n, \\mathbb{Z})$, as different choices of basis in $\\Gamma$ must fix the symplectic pairing. This gives an affine structure on $\\mathcal{B}'$.\n\n By differentiating and evaluating in $\\gamma$, we get 1-forms $d\\theta_\\gamma, d Z_\\gamma$ on $\\mathcal{M}'$ which are linear on $\\Gamma$. For a local basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ as in the previous paragraph, let $\\{\\gamma^1, \\ldots, \\gamma^{2n}\\}$ denote its dual basis on $\\Gamma^*$. We write $\\left\\langle dZ \\wedge dZ \\right\\rangle$ as short notation for\n \\begin{equation}\\label{dzdz}\n  \\left\\langle \\gamma^i , \\gamma^j \\right\\rangle dZ_{\\gamma_i} \\wedge dZ_{\\gamma_j},\n  \\end{equation}\n where we sum over repeated indices. Observe that the anti-symmetric pairing $\\left\\langle \\, , \\right\\rangle$ and the anti-symmetric wedge product of 1-forms makes \\eqref{dzdz} symmetric.  We require that:\n \n\\begin{equation}\\label{dz0}\n\\left\\langle dZ \\wedge dZ \\right\\rangle = 0,\n\\end{equation}\n\nBy \\eqref{dz0}, near $u$, $\\mathcal{B}'$ can be locally identified with a complex Lagrangian submanifold of $\\Gamma^* \\otimes_\\mathbb{Z} \\mathbb{C}$.\n\nIn the example of moduli spaces of Higgs bundles, as $u$ approaches a quadratic differential with non-simple zeros, one homology cycles vanishes (see Figure \\ref{nodtorus}). This cycle $\\gamma_0$ is primitive in $H_1$ and its monodromy around the critical quadratic differential is governed by the Picard-Lefschetz formula. In the general case, let $D_0$ be a component of the divisor $D \\subset \\mathcal{B}$. We also assume the following:\n\n\\begin{itemize}\n\t\\item $Z_{\\gamma_0}(u) \\to 0$ as $u \\to u_0 \\in D_0$ for some $\\gamma_0 \\in \\Gamma$.\n\t\n\n\t\n\t\\item $\\gamma_0$ is primitive (i.e. there exists some $\\gamma'$ with $\\left\\langle \\gamma_0, \\gamma'\\right\\rangle = 1$).\n\t\n\t\\item The monodromy of $\\Gamma$ around $D_0$ is of ``Picard-Lefschetz type'', i.e.\n\t\\begin{equation}\\label{piclf}\n\t\\gamma \\mapsto \\gamma + \\left\\langle \\gamma, \\gamma_0 \\right\\rangle \\gamma_0\n\t\\end{equation}\n\\end{itemize}\n\n\nWe assign a complex structure and a holomorphic symplectic form on $\\mathcal{M}'$ as follows (see \\cite{notes} and the references therein for proofs). Take a local basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ of $\\Gamma$. If $\\epsilon^{ij} = \\left\\langle \\gamma_i , \\gamma_j\\right\\rangle$ and $\\epsilon_{ij}$ is its dual, let\n\\begin{equation}\\label{ome}\n\\omega_+ = \\left\\langle dZ \\wedge d\\theta \\right\\rangle = \\epsilon_{ij} \\, dZ_{\\gamma_i} \\wedge d\\theta_{\\gamma_j}.\n\\end{equation}\nBy linearity on $\\gamma$ of the 1-forms, $\\omega_+$ is independent of the choice of basis. There is a unique complex structure $J$ on $\\mathcal{M}'$ for which $\\omega_+$ is of type (2,0). The 2-form $\\omega_+$ gives a holomorphic symplectic structure on $(\\mathcal{M}', J)$. With respect to this structure, the projection $\\pi: \\mathcal{M}' \\to \\mathcal{B}'$ is holomorphic, and the torus fibers $\\mathcal{M}'_u = \\pi^{-1}(u)$ are compact complex Lagrangian submanifolds. \n\nRecall that a positive 2-form $\\omega$ on a complex manifold is a real 2-form for which $\\omega(v,Jv) >0$ for all real tangent vectors $v$. From now on, we assume that $\\left\\langle dZ \\wedge d\\overline{Z} \\right\\rangle$ is a positive 2-form on $\\mathcal{B}'$. Now fix $R > 0$. Then we can define a 2-form on $\\mathcal{M}'$ by\n\\begin{equation*}\n\\omega_3^{\\text{sf}} = \\frac{R}{4} \\left\\langle dZ \\wedge d\\overline{Z} \\right\\rangle - \\frac{1}{8\\pi^2 R}\\left\\langle d\\theta \\wedge d\\theta \\right\\rangle.\n\\end{equation*}\nThis is a positive form of type (1,1) in the $J$ complex structure. Thus, the triple $(\\mathcal{M}', J, \\omega_3^{\\text{sf}})$ determines a K\\\"{a}hler metric $g^{\\text{sf}}$ on $\\mathcal{M}'$. This metric is in fact hyperk\\\"{a}hler (see \\cite{freed}), so we have a whole $\\mathbb{P}^1$-worth of complex structures for $\\mathcal{M}'$, parametrized by $\\zeta \\in \\mathbb{P}^1$. The above complex structure $J$ represents $J(\\zeta = 0)$, the complex structure at $\\zeta = 0$ in $\\mathbb{P}^1$. The superscript ${}^\\text{sf}$ stands for ``semiflat''. This is because $g^{\\text{sf}}$ is flat on the torus fibers $\\mathcal{M}'_u$.\n\nAlternatively, it is shown in \\cite{gaiotto} that if\n\\begin{equation}\\label{xsfr}\n\\mathcal{X}_\\gamma^{\\text{sf}}(\\zeta) = \\exp\\left( \\frac{\\pi R Z_\\gamma}{\\zeta} + i\\theta_\\gamma + \\pi R \\zeta \\overline{Z_\\gamma} \\right)\n\\end{equation}\nThen the 2-form\n\\[ \\varpi(\\zeta) = \\frac{1}{8\\pi^2 R} \\left\\langle d\\log \\mathcal{X}^{\\text{sf}}(\\zeta) \\wedge d\\log \\mathcal{X}^{\\text{sf}}(\\zeta) \\right\\rangle \\]\n(where the DeRham operator $d$ is applied to the $\\mathcal{M}'$ part only) can be expressed as\n\\[ -\\frac{i}{2\\zeta}\\omega_+ + \\omega^{\\text{sf}}_3 -\\frac{i \\zeta}{2} \\omega_-, \\]\nfor $\\omega_- = \\overline{\\omega_+} = \\left\\langle d\\overline{Z} \\wedge d\\overline{\\theta} \\right\\rangle$, that is, in the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$ of \\cite{hitchin}, $\\varpi(\\zeta)$ is a holomorphic section of $\\Omega_{\\mathcal{Z}\/\\mathbb{P}^1} \\otimes \\mathcal{O}(2)$ (the twisting by $\\mathcal{O}(2)$ is due to the poles at $\\zeta = 0$ and $\\zeta = \\infty$ in $\\mathbb{P}^1$). This is the key step in Hitchin's twistor space construction. By \\cite[\\S 3]{gaiotto}, $\\mathcal{M}'$ is hyperk\\\"{a}hler.\n\nWe want to reproduce the same construction of a hyperk\\\"{a}hler metric now with corrected Darboux coordinates $\\mathcal{X}_\\gamma(\\zeta)$. For that, we need another piece of data. Namely, a function $\\Omega : \\Gamma \\to \\mathbb{Z}$ such that $\\Omega(\\gamma;u) = \\Omega(-\\gamma;u)$. Furthermore, we impose a condition on the nonzero $\\Omega(\\gamma;u)$. Introduce a positive definite norm on $\\Gamma$. Then we require the existence of $K > 0$ such that\n\\begin{equation}\\label{support}\n\\frac{|Z_\\gamma|}{\\left\\| \\gamma \\right\\|} > K\n\\end{equation}\nfor those $\\gamma$ such that $\\Omega(\\gamma; u) \\neq 0$. This is called the \\emph{Support Property}, as in \\cite{gaiotto}.\n\nFor a component of the singular locus $D_0$ and for $\\gamma_0$ the primitive element in $\\Gamma$ for which $Z_{\\gamma_0} \\to 0$ as $u \\to u_0 \\in D_0$, we also require\n\\[ \\Omega(\\gamma_0; u) = 1 \\text{ for all $u$ in a neighborhood of $D_0$} \\]\n\nTo see where these invariants arise from, consider the example of moduli spaces of Higgs bundles again. A quadratic differential $u \\in \\mathcal{B}'$ determines a metric $h$ on $C$. Namely, if $u = P(z)dz^2$, $h = |P(z)| dz d\\overline{z}$. Let $C'$ be the curve obtained after removing the poles and zeroes of $u$. Consider the finite length inextensible geodesics on $C'$ in the metric $h$. These come in two types:\n\n\\begin{enumerate}\n\t\\item \\textit{Saddle connections}: geodesics running between two zeroes of $u$. See Figure \\ref{saddlec}.\n\t\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{saddle_connection.ps}\n\t\\caption{Saddle connections on $C'$}\n\t\\label{saddlec}\n\\end{figure}\n\n\\item \\textit{Closed geodesics}: When they exist, they come in 1-parameter families sweeping out annuli in $C'$. See Figure \\ref{clgeod}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{closed_loop.ps}\n\t\\caption{Closed geodesics on $C'$ sweeping annuli}\n\t\\label{clgeod}\n\\end{figure}\n\n\\end{enumerate}\n\nOn the branched cover $\\Sigma_u \\to C$, each geodesic can be lifted to a union of closed curves in $\\Sigma_u$, representing some homology class $\\gamma \\in H_1(\\Sigma_u, \\mathbb{Z})$. See Figure \\ref{lift}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{double_cover.ps}\n\t\\caption{Lift of geodesics to $\\Sigma_u$}\n\t\\label{lift}\n\\end{figure}\n\n\n In this case, $\\Omega(\\gamma,u)$ counts these finite length geodesics: every saddle connection with lift $\\gamma$ contributes $+1$ and every closed geodesic with lift $\\gamma$ contributes $-2$.\n\n\nBack to the general case, we're ready to formulate a Riemann-Hilbert problem on the $\\mathbb{P}^1$-slice of the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$. Recall that in a RH problem we have a contour $\\Sigma$ dividing a complex plane (or its compactification) and one tries to obtain functions which are analytic in the regions defined by the contour, with continuous extensions along the boundary and with prescribed discontinuities along $\\Sigma$ and fixed asymptotics at the points where $\\Sigma$ is non-smooth. In our case, the contour is a collection of rays at the origin and the discontinuities can be expressed as symplectomorphisms of a complex torus:\n\nDefine a ray associated to each $\\gamma \\in \\Gamma_u$ as:\n\\[ \\ell_\\gamma(u) = Z_\\gamma \\mathbb{R}_-. \\]\nWe also define a transformation of the functions $\\mathcal{X}_{\\gamma'}$ given by each $\\gamma \\in \\Gamma_u$:\n\\begin{equation}\\label{kjump}\n\\mathcal{K}_\\gamma \\mathcal{X}_{\\gamma'} = \\mathcal{X}_{\\gamma'} (1- \\mathcal{X}_{\\gamma})^{\\left\\langle \\gamma', \\gamma \\right\\rangle}\n\\end{equation}\nLet $T_u$ denote the space of twisted complex characters of $\\Gamma_u$, i.e. maps $\\mathcal{X} : \\Gamma_u \\to \\mathbb{C}^{\\times}$ satisfying\n\\begin{equation}\\label{xprop}\n \\mathcal{X}_\\gamma \\mathcal{X}_{\\gamma'} = (-1)^{\\left\\langle \\gamma, \\gamma'\\right\\rangle} \\mathcal{X}_{\\gamma + \\gamma'}\n\\end{equation}\n$T_u$ has a canonical Poisson structure given by\n\\[ \\{ \\mathcal{X}_\\gamma,  \\mathcal{X}_{\\gamma'} \\} = \\left\\langle \\gamma, \\gamma' \\right\\rangle \\mathcal{X}_{\\gamma + \\gamma'}\\]\nThe $T_u$ glue together into a bundle over $\\mathcal{B}'$ with fiber a complex Poisson torus. Let $T$ be the pullback of this system to $\\mathcal{M}'$. We can interpret the transformations $\\mathcal{K}_\\gamma$ as birational automorphisms of $T$.\nTo each ray $\\ell$ going from 0 to $\\infty$ in the $\\zeta$-plane, we can define a transformation\n\\begin{equation}\\label{stkfac}\nS_\\ell = \\prod_{\\gamma : \\ell_\\gamma(u) = \\ell} \\mathcal{K}_\\gamma^{\\Omega(\\gamma;u)}\n\\end{equation}\nNote that all the $\\gamma$'s involved in this product are multiples of each other, so the $\\mathcal{K}_\\gamma$ commute and it is not necessary to specify an order for the product.\n\nTo obtain the corrected $\\mathcal{X}_\\gamma$, we can formulate a Riemann-Hilbert problem for which the former functions are solutions to it.  We seek a map $\\mathcal{X} : \\mathcal{M}'_u \\times \\mathbb{C}^{\\times} \\to T_u$ with the following properties:\n\\begin{enumerate}[label=\\textnormal{(\\arabic*)}]\n\t\\item $\\mathcal{X}$ depends piecewise holomorphically on $\\zeta$, with discontinuities only at the rays $\\ell_\\gamma(u)$ for which $\\Omega(\\gamma;u) \\neq 0$.\n\t\\item The limits $\\mathcal{X}^{\\pm}$ as $\\zeta$ approaches any ray $\\ell$ from both sides exist and are related by\n\t\\begin{equation}\\label{invjmp}\n\t\\mathcal{X}^+ = S_\\ell^{-1} \\circ \\mathcal{X}^-\n\t\\end{equation}\n\t\\item $\\mathcal{X}$ obeys the reality condition\n\t\\begin{equation}\\label{realcond}\n\t\\overline{\\mathcal{X}_{-\\gamma}(-1\/\\overline{\\zeta})} = \\mathcal{X}_\\gamma(\\zeta)\n\t\\end{equation}\n\t\\item For any $\\gamma \\in \\Gamma_u$, $\\lim_{\\zeta \\to 0} \\mathcal{X}_\\gamma(\\zeta) \/ \\mathcal{X}^{\\text{sf}}_\\gamma(\\zeta)$ exists and is real. \\label{asymptotic}\n\\end{enumerate}\n\nIn \\cite{gaiotto}, this RH problem is formulated as an integral equation:\n\\begin{equation}\\label{inteq}\n\\mathcal{X}_\\gamma(u,\\zeta) = \\mathcal{X}^{\\text{sf}}_\\gamma(u,\\zeta)\\exp\\left[ -\\frac{1}{4\\pi i} \\sum_{\\gamma'} \\Omega(\\gamma';u) \\left\\langle \\gamma, \\gamma' \\right\\rangle \\int_{\\ell_{\\gamma'(u)}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta}\\log\\left( 1 - \\mathcal{X}_{\\gamma'}(u,\\zeta')\\right)\\right],\n\\end{equation}\nOne can define recursively, setting $\\mathcal{X}^{(0)} = \\mathcal{X}^{\\text{sf}}$:\n\\begin{equation}\\label{recurs}\n\\mathcal{X}^{(\\nu+1)}_\\gamma(u,\\zeta) = \\mathcal{X}^{\\text{sf}}_\\gamma(u,\\zeta)\\exp\\left[ -\\frac{1}{4\\pi i} \\sum_{\\gamma'} \\Omega(\\gamma';u) \\left\\langle \\gamma, \\gamma' \\right\\rangle \\int_{\\ell_{\\gamma'(u)}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta}\\log\\left( 1 - \\mathcal{X}^{(\\nu)}_{\\gamma'}(u,\\zeta')\\right)\\right],\n\\end{equation}\n\nMore precisely, we have a family of RH problems, parametrized by $u \\in \\mathcal{B}'$, as this defines the rays $\\ell_\\gamma(u)$, the complex torus $T_u$ where the symplectomorphisms are defined and the invariants $\\Omega(\\gamma;u)$ involved in the definition of the problem.\n\nWe still need one more piece of the puzzle, since the latter function $\\Omega$ may not be continuous. In fact, $\\Omega$ jumps along a real codimension-1 loci in $\\mathcal{B}'$ called the ``wall of marginal stability''. This is the locus where 2 or more functions $Z_\\gamma$ coincide in phase, so two or more rays $\\ell_{\\gamma}(u)$ become one. More precisely:\n\\[ W = \\{u \\in \\mathcal{B}': \\exists \\gamma_1, \\gamma_2 \\text{ with } \\Omega(\\gamma_1;u) \\neq 0, \\Omega(\\gamma_2;u) \\neq 0, \\left\\langle \\gamma_1, \\gamma_2\\right\\rangle \\neq 0, Z_{\\gamma_1}\/Z_{\\gamma_2} \\in \\mathbb{R}_+\\}\\]\n The jumps of $\\Omega$ are not arbitrary; they are governed by the Kontsevich-Soibelman wall-crossing formula.\n\nTo describe this, let $V$ be a strictly convex cone in the $\\zeta$-plane with apex at the origin. Then for any $u \\notin W$ define\n\\begin{equation}\nA_V(u) = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma : Z_\\gamma(u) \\in V} \\mathcal{K}_\\gamma^{\\Omega(\\gamma;u)} = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\ell \\subset V} S_\\ell\\footnote{This product may be infinite. One should more precisely think of $A_V(u)$ as living in a certain prounipotent completion of the group generated by $\\{\\mathcal{K}_\\gamma\\}_{\\gamma : Z_\\gamma(u) \\in V}$ as explained in \\cite{kont}}\n\\end{equation}\n\nThe arrow indicates the order of the rational maps $\\mathcal{K}_\\gamma$. $A_V(u)$ is a birational Poisson automorphism of $T_u$. Define a $V$-\\textit{good path} to be a path $p$ in $\\mathcal{B}'$ along which there is no point $u$ with $Z_\\gamma(u) \\in \\partial V$ and $\\Omega(\\gamma;u) \\neq 0$. (So as we travel along a $V$-good path, no $\\ell_\\gamma$ rays enter or exit V.) If $u, u'$ are the endpoints of a $V$-good path $p$, the wall-crossing formula is the condition that $A_V(u), A_V(u')$ are related by parallel transport in $T$ along $p$. See Figure \\ref{partrnpt}.\n\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.80\\textwidth]{wall_crossing.ps}\n\t\\caption{For a good path $p$, the two automorphisms $A_V(u), A_V(u')$ are related by parallel transport}\n\t\\label{partrnpt}\n\\end{figure}\n\n\n\n\\subsection{Statement of Results}\\label{results}\n\nWe will restrict in this paper to the case $\\dim_\\mathbb{C} \\mathcal{B} = 1$, so $n = \\dim \\Gamma = 2$. We want to extend the torus fibration $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with degenerate torus fibers.  To give an example, in the case of Hitchin systems, the torus bundle $\\mathcal{M}'$ is not the moduli space of Higgs bundles yet, as we have to consider quadratic differentials with non-simple zeroes too. The main results of this paper center on the extension of the manifold $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with an extended fibration $\\mathcal{M} \\to \\mathcal{B}$ such that the torus fibers $\\mathcal{M}'_u$ degenerate to nodal torus (i.e. ``singular'' or ``bad'' fibers) for $u \\in D$.\n\nWe start by fully working out the simplest example known as Ooguri-Vafa \\cite{cecotti}. Here we have a fibration over the open unit disk $\\mathcal{B} := \\{u \\in \\mathbb{C} : |u| < 1 \\}$. At the discriminant locus $D : = \\{ u = 0 \\}$, the fibers degenerate into a nodal torus. The local rank-2 lattice $\\Gamma$ has a basis $(\\gamma_m, \\gamma_e)$ and the skew-symmetric pairing is defined by $\\left\\langle \\gamma_m, \\gamma_e \\right\\rangle = 1$. The monodromy of $\\Gamma$ around $u = 0$ is $\\gamma_e \\mapsto \\gamma_e, \\gamma_m \\mapsto \\gamma_m + \\gamma_e$. We also have functions $Z_{\\gamma_e}(u) = u, Z_{\\gamma_m}(u) = \\frac{u}{2\\pi i }( \\log u - 1) + f(u)$, for $f$ holomorphic and admitting an extension to $\\mathcal{B}$. Finally, the integer-valued function $\\Omega$ in $\\Gamma$ is here: $\\Omega(\\pm \\gamma_e; u) = 1$ and $\\Omega(\\gamma; u) = 0$ for any other $\\gamma \\in \\Gamma_u$. There is no wall of marginal stability in this case. The integral equation \\eqref{inteq} can be solved after just 1 iteration.\n\nFor all other nontrivial cases, in order to give a satisfactory extension of the $\\mathcal{X}_\\gamma$ coordinates, it was necessary to develop the theory of Riemann-Hilbert-Birkhoff problems to suit these infinite-dimensional systems (as the transformations $S_\\ell$ defining the problem can be thought as operators on $C^\\infty(T_u)$, rather than matrices). It is not clear that such coordinates can be extended, since we may approach the bad fiber from two different sides of the wall of marginal stability and obtain two different extensions. To overcome this first obstacle, we have to use the theory of isomonodromic deformations as in \\cite{boalch} to reformulate the Riemann-Hilbert problem in \\cite{gaiotto} independent of the regions determined by the wall.\n\nHaving redefined the problem, we want our $\\mathcal{X}_\\gamma$ to be smooth on the parameters $\\theta_{\\gamma_1},  \\theta_{\\gamma_{2}}$ and $u$,\naway from where the prescribed jumps are. Even at $\\mathcal{M}'$, there was no mathematical proof that such condition must be true. In the companion paper \\cite{rhprob}, we combine classical Banach contraction methods and Arzela-Ascoli results on uniform convergence in compact sets to obtain:\n\\begin{theorem}\\label{smooth}\nIf the collection $J$ of nonzero $\\Omega(u; \\gamma)$ satisfies the support property \\eqref{support} and if the parameter $R$ of \\eqref{xsfr} is large enough (determined by the values $|Z_\\gamma(u)|, \\gamma \\in J$), there exists a unique collection of functions $\\mathcal{X}_\\gamma$ with the prescribed asymptotics and jumps as in \\cite{gaiotto}. These functions are smooth on $u$ and the torus coordinates $\\theta_1, \\theta_{2}$ (even for $u$ at the wall of marginal stability), and piecewise holomorphic on $\\zeta$\n\\end{theorem}\n\nSince we're considering only the case $n=1$, $\\Gamma$ is a rank-1 lattice over the Riemann surface $\\mathcal{B}'$ and the discriminant locus $D$ where the torus fibers degenerate is a discrete subset of $\\mathcal{B}'$. \n\nFrom this point on, we restrict our attention to the next nontrivial system, known as the Pentagon case \\cite{notes}. Here $\\mathcal{B} = \\mathbb{C}$ with 2 bad fibers which we can assume are at $u = -2, u = 2$ and $\\mathcal{B}'$ is the twice-punctured plane. There is a wall of marginal stability where all $Z_\\gamma$ are contained in the same line. This separates $\\mathcal{B}$ in two domains $\\mathcal{B}_\\text{out}$ and a simply-connected $\\mathcal{B}_\\text{in}$. See Figure \\ref{aplane}.\n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{walls.ps}\n\t\\caption{The wall $W$ in $\\mathcal{B}$ for the Pentagon case}\n\t\\label{aplane}\n\\end{figure}\n\n\nOn $\\mathcal{B}_\\text{in}$ we can trivialize $\\Gamma$ and choose a basis $\\{\\gamma_1, \\gamma_2\\}$ with pairing $\\left\\langle \\gamma_1, \\gamma_2\\right\\rangle = 1$. This basis does not extend to a global basis for $\\Gamma$ since it is not invariant under monodromy. However, the set $\\{\\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}$ is indeed invariant so the following definition of $\\Omega$ makes global sense:\n  \\begin{align*}\n  \\text{For $u \\in \\mathcal{B}_\\text{in}$}, \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n    1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2\\}\\\\\n    0 & \\text{otherwise}\n  \\end{array} \\right. \\\\\n \\text{For $u \\in \\mathcal{B}_\\text{out}$} , \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n    1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}\\\\\n    0 & \\text{otherwise}\n  \\end{array}  \\right.\n  \\end{align*}\n  \n \n  \n \n  \n   The Pentagon case appears in the study of Hitchin systems with gauge group $\\text{SU}(2)$. The extension of $\\mathcal{M}'$ was previously obtained by hyperk\\\"{a}hler quotient methods in \\cite{biquard}, but no explicit hyperk\\\"{a}hler metric was constructed. \n\nOnce the $\\{ \\mathcal{X}_{\\gamma_i} \\}$ are obtained by Theorem \\ref{smooth},\nit is necessary to do an analytic continuation along $\\mathcal{B}'$ for the particular $\\mathcal{X}_{\\gamma_i}$ for which $Z_{\\gamma_i} \\to 0$ as $u \\to u_0 \\in D$. Without loss of generality, we can assume there\nis a local basis $\\{\\gamma_1, \\gamma_{2}\\}$ of $\\Gamma$ such that $Z_{\\gamma_2} \\to 0$ in $D$. After that, an analysis of the possible divergence of $\\mathcal{X}_\\gamma$ as $u \\to u_0$ shows the necessity of performing a gauge transformation on the torus coordinates of the fibers $\\mathcal{M}_u$ that allows us to define an integral equation even at $u_0 \\in D$. This series of transformations are defined in \\eqref{newmp}, \\eqref{outmp}, \\eqref{outmp2} and \\eqref{fingauge}, and constitute a new result that was not expected in \\cite{gaiotto}. We basically deal with a family of boundary value problems for which the jump function vanishes at certain points and\nsingularities of certain kind appear as $u \\to u_0$. As this is of independent interest, we leave the relevant results to \\cite{rhprob} and we show that our solutions contain at worst branch singularities at 0 or $\\infty$ in the $\\zeta$-plane. As in the case of normal fibers, we can run a contraction argument to obtain Darboux coordinates even at the singular fibers and conclude\n\n\\begin{theorem}\\label{extbf}\nLet $\\{\\gamma_{1}, \\gamma_2\\}$ be a local basis for $\\Gamma$ in a small sector centered at $u_0 \\in D$ such that $Z_{\\gamma_2} \\to 0$ as $u \\to u_0 \\in D$. For the Pentagon integrable system, the local function $\\mathcal{X}_{\\gamma_1}$ admits an analytic continuation $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ to a punctured disk centered at $u_0$ in $\\mathcal{B}$. There exists a gauge transformation $\\theta_1 \\mapsto \\widetilde{\\theta}_1$ that extends the torus fibration $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ that is locally, for each point in $D$, a (trivial) fibration over $\\mathcal{B} \\times S^{1}$ with fiber $S^1$ coordinatized by $\\theta_1$ and with one fiber collapsed into a point. For $R > 0$ big enough, it is possible to extend $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ and $\\mathcal{X}_{\\gamma_2}$ to $\\mathcal{M}$, still preserving the smooth properties as in Theorem \\ref{smooth}.\n\\end{theorem}\n\n\n\n\nAfter we have the smooth extension of the $\\{ \\mathcal{X}_{\\gamma_i} \\}$ by Theorem \\ref{extbf}, we can extend the holomorphic symplectic form $\\varpi(\\zeta)$ labeled by $\\zeta \\in \\mathbb{P}^1$ as in \\cite{hitchin} for all points except possibly one at the singular fiber. From $\\varpi(\\zeta)$ we can obtain the hyperk\\\"{a}hler metric $g$ and, in the case of the Pentagon, after a change of coordinates, we realize $g$ locally as the Taub-NUT metric plus smooth corrections, finishing the construction of $\\mathcal{M}$ and its hyperk\\\"{a}hler metric. The following is the main theorem of the paper.\n\n\\begin{theorem}\\label{smfrm}\nFor the Pentagon case, the extension $\\mathcal{M}$ of the manifold $\\mathcal{M}'$ constructed in Theorem \\ref{extbf} admits, for $R$ large enough, a hyperk\\\"{a}hler metric $g$ obtained by extending the hyperk\\\"{a}hler metric on $\\mathcal{M}'$ determined by the Darboux coordinates $\\{ \\mathcal{X}_{\\gamma_i} \\}$.\n\\end{theorem}\n\n\n\n   \n\\section{The Ooguri-Vafa Case}\\label{ov}\n\n\\subsection{Classical Case}\\label{clasov}\n\nWe start with one of the simplest cases, known as the Ooguri-Vafa case, first treated in \\cite{cecotti}. To see where this case comes from, recall that by the SYZ picture of K3 surfaces \\cite{gross}, any K3 surface $\\mathcal{M}$ is a hyperk\\\"{a}hler manifold. In one of its complex structures (say $J^{(\\zeta = 0)}$) is elliptically fibered, with base manifold $\\mathcal{B} = \\mathbb{P}^1$ and generic fiber a compact complex torus. There are a total of 24 singular fibers, although the total space is smooth. See Figure \\ref{k3}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{k3.ps}\n\t\\caption{A K3 surface $\\mathcal{M}$ as an elliptic fibration}\n\t\\label{k3}\n\\end{figure}\n\nGross and Wilson \\cite{gross2} constructed a hyperk\\\"{a}hler metric $g$ on a K3 surface by gluing in the Ooguri-Vafa metric constructed in \\cite{oovf} with a standard metric $g^{\\text{sf}}$ away from the degenerate fiber. Thus, this simple case can be regarded as a local model for K3 surfaces.\n \nWe have a fibration over the open unit disk $\\mathcal{B} := \\{a \\in \\mathbb{C} : |a| < 1 \\}$. At the locus $D : = \\{ a = 0 \\}$ (in the literature this is also called the \\textit{discriminant locus}), the fibers degenerate into a nodal torus. Define $\\mathcal{B}'$ as $\\mathcal{B} \\backslash D$, the punctured unit disk. On $\\mathcal{B}'$ there exists a local system $\\Gamma$ of rank-2 lattices with basis $(\\gamma_m, \\gamma_e)$ and skew-symmetric pairing defined by $\\left\\langle \\gamma_m, \\gamma_e \\right\\rangle = 1$. The monodromy of $\\Gamma$ around $a = 0$ is $\\gamma_e \\mapsto \\gamma_e, \\gamma_m \\mapsto \\gamma_m + \\gamma_e$. We also have functions $Z_{\\gamma_e}(a) = a, Z_{\\gamma_m}(a) = \\frac{a}{2\\pi i }( \\log a - 1)$. On $\\mathcal{B}'$ we have local coordinates $(\\theta_m, \\theta_e)$ for the torus fibers with monodromy $\\theta_e \\mapsto \\theta_e, \\theta_m \\mapsto \\theta_m + \\theta_e - \\pi$. Finally, the integer-valued function $\\Omega$ in $\\Gamma$ is here: $\\Omega(\\pm \\gamma_e, a) = 1$ and $\\Omega(\\gamma, a) = 0$ for any other $\\gamma \\in \\Gamma_a$. There is no wall of marginal stability in this case.\n\n We call this the ``classical Ooguri-Vafa'' case as it is the one appearing in \\cite{oovf} already mentioned at the beginning of this section. In the next section, we'll generalize this case by adding a function $f(a)$ to the definition of $Z_{\\gamma_m}$.\n\nLet\n\\begin{equation}\\label{xesf}\n \\mathcal{X}^{\\text{sf}}_\\gamma(\\zeta, a) := \\exp\\left( \\pi R \\zeta^{-1} Z_\\gamma(a) + i\\theta_\\gamma + \\pi R \\zeta \\overline{Z_\\gamma(a)}\\right)\n\\end{equation}\nThese functions receive corrections defined as in \\cite{gaiotto}. We are only interested in the pair $(\\mathcal{X}_m, \\mathcal{X}_e)$ which will constitute our desired Darboux coordinates for the holomorphic symplectic form $\\varpi$. The fact that $\\Omega(\\gamma_m, a) = 0$ gives that $\\mathcal{X}_e = \\mathcal{X}^{\\text{sf}}_e$. As $a \\to 0$, $Z_{\\gamma_e}$ and $Z_{\\gamma_m}$ approach 0. Thus $\\mathcal{X}_e|_{a = 0} = e^{i\\theta_e}$. Since $\\mathcal{X}_e = \\mathcal{X}^{\\text{sf}}_e$ the actual $\\mathcal{X}_m$ is obtained after only 1 iteration of \\eqref{recurs}. For each $a \\in \\mathcal{B}'$, let $\\ell_+$ be the ray in the $\\zeta$-plane defined by $\\{\\zeta : a\/\\zeta \\in \\mathbb{R}_- \\}$. Similarly, $\\ell_- : = \\{\\zeta : a\/\\zeta \\in \\mathbb{R}_+\\}$.\n\nLet\n\\begin{equation}\\label{defxm}\n\\mathcal{X}_m = \\mathcal{X}^{\\text{sf}}_m \\exp \\left[ \\frac{i}{4\\pi} \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' +\n \\zeta}{\\zeta' - \\zeta} \\log[1 - \\mathcal{X}_e(\\zeta')] -  \\frac{i}{4\\pi} \\int_{\\ell_-} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' +\n \\zeta}{\\zeta' - \\zeta} \\log[1 - \\mathcal{X}_e(\\zeta')^{-1}] \\right].\n\\end{equation}\nFor convenience, from this point on we assume $a$ is of the form  $sb$, where $s$ is a positive number, $b$ is fixed and $|b| = 1$. Moreover, in $\\ell_+$, $\\zeta' = -tb$, for $t \\in (0, \\infty)$, and a similar parametrization holds in $\\ell_-$.\n\\begin{lemma}\nFor fixed $b$, $\\mathcal{X}_m$ as in \\eqref{defxm} has a limit as $|a| \\to 0$.\n\\end{lemma}\n\\begin{proof}\nWriting $\\dfrac{\\zeta' + \\zeta}{\\zeta'(\\zeta' - \\zeta)} = \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta}$, we want to find the limit as $a \\to 0$ of\n\\begin{align}\n& \\int_{\\ell_+} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& - \\int_{\\ell_-} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(-\\pi Ra\/\\zeta' - i\\theta_e - \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{integs}.\n\\end{align}\nFor simplicity, we'll focus in the first integral only, the second one can be handled similarly. Rewrite:\n\\begin{align}\n& \\int_{\\ell_+} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta'  \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{-1}{\\zeta'} + \\frac{2}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\dfrac{-1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\dfrac{1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{0}^{-b} \\dfrac{2}{\\zeta' - \\zeta} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{4sums}\n\\end{align}\n\n\\noindent Observe that\n\\begin{align*}\n& \\int_{0}^{-b} \\dfrac{-1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta'\\\\\n& = -\\int_0^1 \\frac{1}{t} \\log[1 - \\exp(-\\pi Rs(t + 1\/t))] dt\\\\\n\\intertext{and after a change of variables $\\tilde{t} = 1\/t$, we get}\n& = -\\int_1^\\infty \\frac{1}{\\tilde{t}} \\log[1 - \\exp(-\\pi Rs(\\tilde{t} + 1\/\\tilde{t}))] d\\tilde{t}\\\\\n& = -\\int_{-b}^{-b\\infty} \\dfrac{1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta'.\n\\end{align*}\nThus, (\\ref{4sums}) reduces to\n\\begin{align}\n& \\int_{0}^{-b} \\dfrac{2}{\\zeta' - \\zeta} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{2sums}.\n\\end{align}\nIf $\\theta_e = 0$, (\\ref{integs}) diverges to $-\\infty$, in which case $\\mathcal{X}_m = 0$. Otherwise, $\\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})]$ is bounded away from 0. Consequently,\n$|\\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})]| < C < \\infty$ in $\\ell_+$.\nAs $a \\to 0$, the integrals are dominated by\n\\[ \\int_0^{-b} \\dfrac{2C}{|\\zeta' - \\zeta|} |d\\zeta'| + \\int_{-b}^{-b\\infty} \\frac{C|\\zeta\/b|}{|\\zeta'(\\zeta' - \\zeta)|} |d\\zeta'| < \\infty \\]\nif $\\theta_e \\neq 0$. Hence we can interchange the limit and the integral in (\\ref{2sums}) and obtain that, as $a \\to 0$, this reduces to\n\n\\begin{align}\n & 2\\log(1 - e^{i \\theta_e})\\left[\\int_{0}^{-b} \\frac{d\\zeta'}{\\zeta' - \\zeta} + \\int_{-b}^{-b\\infty} d\\zeta' \\left\\{ \\frac{1}{\\zeta' - \\zeta} - \\frac{1}{\\zeta'}\\right\\} \\right] \\notag\\\\\n & = 2\\log(1 - e^{i \\theta_e})[F(-b) + G(-b)], \\label{odes}\n \\end{align}\nwhere\n\\[ F(z) := \\log\\left( 1 - \\dfrac{z}{\\zeta}\\right), G(z) := \\log\\left( 1 - \\dfrac{\\zeta}{z}\\right) \\]\n are the (unique) holomorphic solutions in the simply connected domain $U := \\mathbb{C} - \\{z : z\/\\zeta \\in \\mathbb{R}_+\\}$ to the ODEs\n\\[ F'(z) = \\frac{1}{z - \\zeta}, F(0) = 0 \\hspace{10 mm} G'(z) = \\frac{1}{z - \\zeta} - \\frac{1}{z}, \\lim_{z \\to \\infty} G(z) = 0. \\]\nThis forces us to rewrite (\\ref{odes}) uniquely as\n\\begin{equation}\\label{prinbran}\n2\\log(1 - e^{i \\theta_e})\\left[\\log\\left(1 + \\frac{b}{\\zeta}\\right) - \\log\\left(1 + \\frac{\\zeta}{b}\\right)\\right]\n\\end{equation}\nHere $\\log$ denotes the principal branch of the log in both cases, and the equation makes sense for $\\{b \\in \\mathbb{C} : b \\notin \\ell_+ \\}$ (recall that by construction, we have the additional datum $|b| = 1$). We want to conclude that\n\\begin{equation}\\label{logfusion}\n\\log(1 + b\/\\zeta) - \\log(1 + \\zeta\/b) = \\log(b\/\\zeta),\n\\end{equation}\nstill using the principal branch of the log. To see this, define $H(z)$ as $F(z) - G(z) - \\log(-z\/\\zeta)$. This is an analytic function on $U$ and clearly $H'(z) \\equiv 0$. Thus $H$ is constant in $U$. It is easy to show that the identity holds for a suitable choice of $z$ (for example, if $\\zeta$ is not real, choose $z = 1$) and by the above, it holds on all of $U$; in particular, for $z = -b$.\n\nAll the arguments so far can be repeated to the ray $\\ell_-$ to get the final form of (\\ref{integs}):\n \\begin{equation}\\label{fextov}\n  2\\left\\{\\log\\left[\\frac{b}{\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n   -\\log\\left[\\frac{- b}{\\zeta}\\right]\\log(1 - e^{-i\\theta_e}) \\right\\}, \\hspace{3 mm} \\theta_e \\neq 0.\n  \\end{equation}\n   This yields that (\\ref{defxm}) simplifies to:\n\\begin{align}\n\\mathcal{X}_m & = \\mathcal{X}^{\\text{sf}}_m \\exp\\left( \\frac{i}{2\\pi} \\left\\{ \\log\\left[\\frac{b}{\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n   -\\log\\left[\\frac{- b}{\\zeta}\\right]\\log(1 - e^{-i\\theta_e})\\right\\} \\right) \\notag\\\\\n  & = \\mathcal{X}^{\\text{sf}}_m \\exp\\left( \\frac{i}{2\\pi} \\left\\{ \\log\\left[\\frac{a}{|a|\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n   -\\log\\left[\\frac{- a}{|a|\\zeta}\\right]\\log(1 - e^{-i\\theta_e})\\right\\} \\right)  \\label{xmnice}\n\\end{align}\nin the limiting case $a \\to 0$.\n\\end{proof}\n\nTo obtain a function that is continuous everywhere and independent of $\\arg a$, define regions I, II and III in the $a$-plane as follows: $\\mathcal{X}^{\\text{sf}}_m$ has a fixed cut in the negative real axis, both in the $\\zeta$-plane and the $a$-plane. Assuming for the moment that $\\arg \\zeta \\in (0,\\pi)$, define region I as the half plane $\\{a \\in \\mathbb{C} : \\text{Im}\\left( a\/\\zeta\\right) < 0 \\}$. Region II is that enclosed by the $\\ell_-$ ray and the cut in the negative real axis, and region III is the remaining domain so that as we travel counterclockwise we traverse regions I, II and III in this order (see Figure \\ref{3reg}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions.ps}\n\t\\caption{The three regions in the $a$-plane, as we traverse them counterclockwise}\n\t\\label{3reg}\n\\end{figure}\n\n\nFor $a \\neq 0$, Gaiotto, Moore and Neitzke \\cite{gaiotto} proved that $\\mathcal{X}_m$ has a continuous extension to the punctured disk of the form:\n\n\\begin{equation}\\label{regmod}\n \\widetilde{\\mathcal{X}}_m = \\left\\{ \\begin{array}{ll}\n\t   \\mathcal{X}_m & \\text{in region I}\\\\\n\t   (1 - \\mathcal{X}^{-1}_e) \\mathcal{X}_m & \\text{in region II}\\\\\n\t   - \\mathcal{X}_e (1 - \\mathcal{X}^{-1}_e) \\mathcal{X}_m = (1 - \\mathcal{X}_e)\\mathcal{X}_m  & \\text{in region III}\n\t   \\end{array} \\right. \n\\end{equation}\n\nIf we regard $\\mathcal{M}'$ as a $S^1$-bundle over $\\mathcal{B}' \\times S^1$, with the fiber parametrized by $\\theta_m$, then we seek to extend $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ by gluing to $\\mathcal{M}'$ another $S^1$-bundle over $D \\times (0,2\\pi)$, for $D$ a small open disk around $a = 0$, and $\\theta_e \\in (0,2\\pi)$. The $S^1$-fiber is parametrized by a different coordinate $\\theta'_m$ where the Darboux coordinate $\\widetilde{\\mathcal{X}}_m$ can be extended to $\\mathcal{M}$. This is the content of the next theorem. \n\n\\begin{theorem}\\label{mprtom}\n$\\mathcal{M}'$ can be extended to a manifold $\\mathcal{M}$ where the torus fibers over $\\mathcal{B}'$ degenerate at $D = \\{a = 0\\}$ and $\\widetilde{\\mathcal{X}}_m $ can be extended to $D$, independent of the value of $\\arg a$.\n\\end{theorem}\n\\begin{proof}\nWe'll use the following identities:\n\\begin{align}\n\\log(1 - e^{i\\theta_e}) & = \\log(1 - e^{-i\\theta_e}) +i(\\theta_e - \\pi), \\hspace{5mm} \\text{for } \\theta_e \\in (0, 2\\pi) \\label{logs}\\\\\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n     \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in region I}\\\\\n\t   \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in regions II and III}\n\t   \\end{array} \\right. \\label{regions}\\\\\n\t   \\log [a\/\\zeta] & = \\left\\{ \\begin{array}{ll}\n     \\log a - \\log \\zeta & \\text{in regions I and II}\\\\\n\t   \\log a - \\log \\zeta + 2\\pi i & \\text{in region III}\n\t   \\end{array} \\right. \\label{breaklog}\n\\end{align}\nto obtain a formula for $\\widetilde{\\mathcal{X}}_m$ at $a = 0$ independent of the region. Formula \\eqref{breaklog} can be proved with an argument analogous to that used for the proof of \\eqref{logfusion}. Starting with region I, by \\eqref{xmnice}, (\\ref{regmod}), (\\ref{logs}) and (\\ref{regions}):\n\\begin{align*}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right)  \\right] \\hspace{5 mm} \\text{in region I.}\\\\\n\\intertext{By \\eqref{breaklog},}\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right]\n\\end{align*}\nIn region II, by our formulas above, we get\n\\begin{align*}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] - \\frac{1}{2} \\log \\left(1 - e^{-i\\theta_e}\\right)  \\right]\\left(1 - e^{-i\\theta_e}\\right) \\\\\n& =  \\exp\\left[i\\theta_m -  \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] - \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) + \\log\\left(1 - e^{-i\\theta_e}\\right)  \\right]\\\\\n & = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right]  \\text{in region II.}\n\\end{align*}\nFinally, in region III, and making use of \\eqref{logs}, \\eqref{regions}, \\eqref{breaklog}:\n\\begin{align}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right]  - \\frac{1}{2} \\log \\left(1 - e^{-i\\theta_e}\\right)  \\right]\\left(1 - e^{i\\theta_e}\\right) \\notag\\\\\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta - i(\\theta_e - \\pi) \\notag \\right. \\\\\n&  \\left. \\hspace{14 mm} - \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) + \\log\\left(1 - e^{-i\\theta_e}\\right) + i(\\theta_e - \\pi)   \\right] \\notag \\\\\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi}  (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log\n \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right)  \\right] \\label{xmany}.\n\\end{align}\n\n Observe that, throughout all these calculations, we only had to use the natural branch of the complex logarithm. In summary, (\\ref{xmany}) works for any region in the $a$-plane, with a cut in the negative real axis.\n\nThis also suggest the following coordinate transformation\n\\begin{equation}\\label{thetapr}\n \\theta'_m = \\theta_m + \\frac{i(\\theta_e - \\pi)}{4\\pi} \\left( \\log\\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right) \n\\end{equation}\nHere $\\Lambda$ is the same cutoff constant as in \\cite{gaiotto}. Let $\\varphi$ parametrize the phase of $a\/|a|$. Then \\eqref{thetapr} simplifies to\n\\begin{equation}\\label{nicethm}\n\\theta'_m = \\theta_m - \\frac{(\\theta_e - \\pi)\\varphi}{2\\pi}\n\\end{equation}\n\nOn a coordinate patch around the singular fiber, $\\theta'_m$ is single-valued.\nThus, the above shows that we can glue to $\\mathcal{M}'$ another $S^1$-bundle over $D \\times (0,2\\pi)$, for $D$ a small open disk around $a = 0$, and $\\theta_e \\in (0,2\\pi)$. The $S^1$-fiber is parametrized by $\\theta'_m$ and the transition function is given by \\eqref{nicethm}, yielding a manifold $\\mathcal{M}$. In this patch, we can extend $\\widetilde{\\mathcal{X}}_m$ to $a = 0$ as:\n\n\\begin{equation}\\label{reblow}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = e^{i\\theta'_m} \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation}\nwhere the branch of $\\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}}$ is determined by the natural branch of the logarithm in the $\\zeta$ plane. Note that when $\\theta_e = 0$, $\\widetilde{\\mathcal{X}}_m \\equiv 0$ in \\eqref{reblow} and by definition, $\\mathcal{X}_e \\equiv 1$. Since these two functions are Darboux coordinates for $\\mathcal{M}$, the $S^1$ fibration over $D \\times (0, 2\\pi)$ we glued to $\\mathcal{M}'$ to get $\\mathcal{M}$ degenerates into a point when $\\theta_e = 0$. \n\nNow consider the case that $\\arg \\zeta \\in (-\\pi, 0)$. Label the regions as one travels counterclockwise, starting with the region bounded by the cut and the $\\ell_-$ (See Figure \\ref{negarg}). We can do an analytic continuation similar to \\eqref{regmod} starting in region I, but formulas \\eqref{regions}, \\eqref{breaklog} become now:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions_negative_argument.ps}\n\t\\caption{The three regions in the case $\\arg \\zeta < 0$.}\n\t\\label{negarg}\n\\end{figure}\n\n\\begin{align*}\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n     \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in region II}\\\\\n\t   \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in regions I and III}\n\t   \\end{array} \\right. \\\\\n\t   \\log [a\/\\zeta] & = \\left\\{ \\begin{array}{ll}\n     \\log a - \\log \\zeta & \\text{in regions I and II}\\\\\n\t   \\log a - \\log \\zeta - 2\\pi i & \\text{in region III}\n\t   \\end{array} \\right.\n\\end{align*}\n\nBy an argument entirely analogous to the case $\\arg \\zeta > 0$, we get again:\n\\begin{equation*}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = e^{i\\theta'_m} \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation*}\n\nThe case $\\zeta$ real and positive is even simpler, as Figure \\ref{zeroarg} shows. Here we have only two regions, and the jumps at the cut and the $\\ell_+$ ray are combined, since these two lines are the same. Label the lower half-plane as region I and the upper half-plane as region II. Start an analytic continuation of $\\mathcal{X}_m$ in region I as before, using the formulas:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions_zero_argument.ps}\n\t\\caption{Only two regions in the case $\\arg \\zeta = 0$.}\n\t\\label{zeroarg}\n\\end{figure}\n\n\\begin{align*}\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n     \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in region II}\\\\\n\t   \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in region I}\n\t   \\end{array} \\right. \\\\\n\t   \\log [a\/\\zeta] & = \\log a - \\log \\zeta \\hspace{4 mm} \\text{in both regions}\n\\end{align*}\n\nThe result is equation \\eqref{reblow} again. The case $\\arg \\zeta = \\pi$ is entirely analogous to this and it yields the same formula, thus proving that \\eqref{reblow} holds for all $\\zeta$ and is independent of $\\arg a$.\n\\end{proof}\n\n\\subsection{Alternative Riemann-Hilbert problem}\\label{altrh}\n\nWe may obtain the function $\\mathcal{X}_m$ (and consequently, the analytic extension $\\widetilde{\\mathcal{X}}_m$) at $a = 0$ through a slightly different formulation of the Riemann-Hilbert problem stated in \\eqref{defxm}. Namely, instead of defining a jump of $\\mathcal{X}_m$ at two opposite rays $\\ell_+, \\ell_-$, we combine these into a single jump at the line $\\ell$ defined by $\\ell_+$ and $\\ell_-$, as in Figure \\ref{onejump}. Note that because of the orientation of $\\ell$ one of the previous jumps has to be reversed.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{singlejump.ps}\n\t\\caption{The reversed orientation on $\\ell_+$ inverts the jump.}\n\t\\label{onejump}\n\\end{figure}\n\nFor all values $a \\neq 0$, $\\mathcal{X}_e = \\mathcal{X}_e^{\\text{sf}}$ approaches 0 as $\\zeta \\to 0$ or $\\zeta \\to \\infty$ along the $\\ell$ ray due to the exponential decay in formula \\eqref{xesf}. Thus, the jump function\n\\[ G(\\zeta) := \\left\\{ \\begin{array}{ll} 1-\\mathcal{X}^{-1}_e & \\text{ for $\\zeta = t a, 0 \\leq t \\leq \\infty$} \\\\ \n                                     1- \\mathcal{X}_e & \\text{ for $\\zeta = t a, -\\infty \\leq t \\leq 0$} \\end{array} \\right. \\]                                     \nis continuous on $\\ell$ regarded as a closed contour on $\\mathbb{P}^1$, and it approaches the identity transformation exponentially fast at the points $0$ and $\\infty$.\n\n\n\n\nThe advantage of this reformulation of the Riemann-Hilbert problem is that it can be extended to the case $a = 0$ and we can obtain estimates on the solutions $\\mathcal{X}_m$ even without an explicit formulation. If we fix $\\arg a$ and let $|a| \\to 0$ as before, the jump function $G(\\zeta)$ approaches the constant jumps\n\\begin{equation}\\label{jumpd}\n\\left. G(\\zeta) \\right|_{|a|=0} := \\left\\{ \\begin{array}{ll} 1-e^{-i\\theta_e} & \\text{ for $\\zeta = t a, 0 < t < \\infty$} \\\\ \n                                     1- e^{i\\theta_e} & \\text{ for $\\zeta = t a, -\\infty < t < 0$} \\end{array} \\right.\n\\end{equation}\nThus, $\\left. G(\\zeta) \\right|_{|a|=0}$ has two discontinuities at $0$ and $\\infty$. If we denote by \n\\[ \\Delta_0 = \\lim_{t \\to 0^+} G(\\zeta) - \\lim_{t \\to 0^-} G(\\zeta), \\qquad  \\Delta_\\infty = \\lim_{t \\to \\infty^+} G(\\zeta) - \\lim_{t \\to \\infty^-} G(\\zeta), \\]\nthen, by \\eqref{jumpd},\n\\begin{equation*}\n\\Delta_0 = -\\Delta_\\infty\n\\end{equation*}\n\n\n\n\nLet $D^+$ be the region in $\\mathbb{P}^1$ bounded by $\\ell$ with the positive, counterclockwise orientation. Denote by $D^-$ the region where $\\ell$ as a boundary has the negative orientation. We look for solutions of the homogeneous boundary problem\n\\begin{equation}\\label{bcond}\nX_m^+(\\zeta) = G(\\zeta) X_m^-(\\zeta)\n\\end{equation}\nwith $G(\\zeta)$ as in \\eqref{jumpd}. This is Lemma 4.1 in \\cite{rhprob}.\n\n \n \n The solutions $X_m^\\pm$ obtained therein are related to $\\mathcal{X}_m$ via $\\mathcal{X}_m (\\zeta) = \\mathcal{X}^{\\text{sf}}_m (\\zeta) X_m (\\zeta)$. Uniqueness of solutions of the homogeneous Riemann-Hilbert problem shows that these are the same functions (up to a constant factor) constructed in the previous section. Observe that the term $\\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}}$ appears naturally due to the nature of the discontinuity of the jump function at 0 and $\\infty$. The analytic continuation around the point $a = 0$ and the gauge transformation $\\theta_m \\mapsto \\theta'_m$ are still performed as before.\n                                      \n\n\n\n\\subsection{Generalized Ooguri-Vafa coordinates}\\label{genvafa}\n\n\nWe can generalize the previous extension to the case $Z_{\\gamma_m} := \\frac{1}{2\\pi i}a \\log a + f(a)$, where $f : \\mathcal{B}' \\to \\mathbb{C}$ is holomorphic and admits a holomorphic extension into $\\mathcal{B}$. In particular,\n\\begin{equation}\\label{newxmsf}\n\\mathcal{X}_m^\\text{sf} = \\exp \\left( \\frac{-iR}{2\\zeta}a \\log a + \\frac{\\pi R f(a)}{\\zeta} + i \\theta_m + \\frac{i \\zeta R}{2} \\overline{a} \\log \\overline{a} + \\pi R \\zeta \\overline{f(a)}\\right)\n\\end{equation}\nThe value at the singular locus $f(0)$ does not have to be 0. All the other data remains the same. \n\nThe first thing we observe is that $\\mathcal{X}_e$ remains the same. Consequently, the corrections for the generalized $\\mathcal{X}_m$ are as before. Using the change of coordinates as in \\eqref{nicethm}, we can thus write\n\\begin{equation}\\label{genov}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = \\exp\\left[ \\frac{\\pi R f(0)}{\\zeta} + i\\theta'_m + \\pi R \\zeta f(0) \\right] \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation}\n\n\\section{Extension of the Ooguri-Vafa metric}\\label{gmetric}\n\n\\subsection{Classical Case}\\label{clasmet}\n\n\\subsubsection{A $C^1$ extension of the coordinates}\\label{c1ext}\n\nIn section \\ref{clasov} we extended the fibered manifold $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with a degenerate fiber at $a = 0$ in $\\mathcal{B}$. We also extended $\\widetilde{\\mathcal{X}}_m$ continuously to this bad fiber. Now we extend the metric by enlarging the holomorphic symplectic form $\\varpi(\\zeta)$. Recall that this is of the form\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge  \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nClearly there are no problems extending $d \\log \\mathcal{X}_e$, so it remains only to extend $d \\log \\widetilde{\\mathcal{X}}_m$.\n\n\\begin{lemma}\nLet $\\widetilde{\\mathcal{X}}_m$ denote the analytic continuation around $a = 0$ of the magnetic function, as in the last section. The 1-form\n\\begin{equation}\\label{c1dlog}\nd \\log \\widetilde{\\mathcal{X}}_m = \\frac{d \\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m},\n\\end{equation}\n(where $d$ denotes the differential of a function on the torus fibration $\\mathcal{M}'$ only) has an extension to $\\mathcal{M}$\n\\end{lemma}\n\\begin{proof}\nWe proceed as in section \\ref{clasov} and work in different regions in the $a$-plane (see Figure \\ref{3reg}), starting with region I, where $\\widetilde{\\mathcal{X}}_m = \\mathcal{X}_m$. Then observe that we can write the corrections on $\\mathcal{X}_m$ as a complex number $\\Upsilon_m(\\zeta) \\in (\\mathcal{M}'_a)^{\\mathbb{C}}$ such that\n\\[ \\mathcal{X}_m = \\exp\\left( \\frac{-i R }{2\\zeta}(a\\log a - a) + i \\Upsilon_m + \\frac{i\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} )\\right). \\]\nThus, by \\eqref{c1dlog} and ignoring the $i$ factor, it suffices to obtain an extension of \n\\begin{align}\n& d\\left[ \\frac{- R }{2\\zeta}(a\\log a - a) +  \\Upsilon_m + \\frac{\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} ) \\right] \\notag \\\\\n& = \\frac{ -R}{2\\zeta} \\log a \\, da +  d \\Upsilon_m + \\frac{ \\zeta R}{2} \\log \\overline{a} \\, d\\overline{a}. \\label{3ext}\n\\end{align}\nUsing \\eqref{defxm},\n\\begin{align*}\nd \\Upsilon_m  = d\\theta_m & - \\frac{1}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e}\\left( \\frac{\\pi R}{\\zeta'} da +id\\theta_e+ \\pi R \\zeta' d\\overline{a}\\right) \\notag\\\\\n& +\\frac{1}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\\left( -\\frac{\\pi R}{\\zeta'} da -id\\theta_e - \\pi R \\zeta' d\\overline{a}\\right).\n\\end{align*}\nWe have to change our $\\theta_m$ coordinate into $\\theta'_m$ according to \\eqref{nicethm} and differentiate to obtain:\n\\begin{align}\nd \\Upsilon_m  & =  \nd\\theta'_m - \\frac{i(\\theta_e - \\pi)}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\overline{a}}{\\overline{a}}\\right)+\\frac{\\arg a}{2\\pi}d\\theta_e \\notag \\\\\n& - \\frac{1}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e}\\left( \\frac{\\pi R}{\\zeta'} da +id\\theta_e+ \\pi R \\zeta' d\\overline{a}\\right) \\notag\\\\\n& +\\frac{1}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\\left( -\\frac{\\pi R}{\\zeta'} da -id\\theta_e - \\pi R \\zeta' d\\overline{a}\\right) \\label{dups1}\n\\end{align}\nRecall that, since we have introduced the change of coordinates $\\theta_m \\mapsto \\theta'_m$, we are working on a patch on $\\mathcal{M}$ that contains $a = 0$ with a degenerate fiber here. It then makes sense to ask if \\eqref{3ext} extends to $a =0$. If this is true, then every independent 1-form extends individually. Let's consider the form involving $d\\theta_e$ first. By \\eqref{dups1}, this part consists of:\n\\begin{equation}\\label{dthpart}\n\\frac{\\arg a}{2\\pi}d\\theta_e  - \\frac{i}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} d\\theta_e - \\frac{i}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e} d\\theta_e.\n\\end{equation}\nWe can use the exact same technique in section \\ref{clasov} to find the limit of \\eqref{dthpart} as $a \\to 0$. Namely, split each integral into four parts, use the symmetry of $\\dfrac{\\mathcal{X}_e}{1- \\mathcal{X}_e}$ between $0$ and $\\infty$ to cancel two of these integrals and take the limit in the remaining ones. The result is:\n\\begin{align}\n& \\frac{\\arg a}{2\\pi} - \\frac{ie^{i\\theta_e}}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{e^{i \\arg a}}{\\zeta} \\right] - \n\\frac{ie^{-i\\theta_e}}{2\\pi(1-e^{-i\\theta_e})}\\log\\left[ \\frac{-e^{i \\arg a}}{\\zeta} \\right]  \\notag \\\\\n& = \\frac{\\arg a}{2\\pi} - \\frac{ie^{i\\theta_e}}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{e^{i \\arg a}}{\\zeta} \\right] + \n\\frac{i}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{-e^{i \\arg a}}{\\zeta} \\right] \\label{dthe}\n\\end{align}\nin region I (we omitted the $d\\theta_e$ factor for simplicity). Making use of formulas \\eqref{regions} and \\eqref{breaklog}, we can simplify the above expression and get rid of the apparent dependence on $\\arg a$ until finally getting:\n\\[ -\\frac{i\\log \\zeta}{2\\pi} - \\frac{1}{2(1-e^{i\\theta_e})}, \\hspace{7 mm} \\theta_e \\neq 0. \\]\nIn other regions of the $a$-plane we have to modify $\\widetilde{\\mathcal{X}}_m$ as in \\eqref{regmod}. Nonetheless, by \\eqref{regions} and \\eqref{breaklog}, the result is the same and we conclude that at least the terms involving $d\\theta_e$ have an extension to $a=0$ for $\\theta_e \\neq 0$.\n\nNext we extend the terms involving $da$. By \\eqref{3ext} and \\eqref{dups1}, these are:\n\\begin{equation*}\n\\frac{ -R}{2\\zeta} \\log a \\, da  - \\frac{i(\\theta_e - \\pi)}{4\\pi a} da - \\frac{R}{4}\\int_{\\ell_+} \\dfrac{d\\zeta'}{(\\zeta')^2} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} da - \\frac{R}{4}\\int_{\\ell_-} \\dfrac{d\\zeta'}{(\\zeta')^2} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e} da \n\\end{equation*}\nIn what follows, we ignore the $da$ part and focus on the coefficients for the extension. The partial fraction decomposition\n\\begin{equation}\\label{parfrac}\n\\frac{\\zeta'+\\zeta}{(\\zeta')^2(\\zeta'-\\zeta)} = \\frac{2}{\\zeta'(\\zeta'-\\zeta)} - \\frac{1}{(\\zeta')^2}\n\\end{equation}\nsplits each integral above into two parts. We will consider first the terms\n\\begin{equation}\\label{dap1}\n- \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{R}{4}\\int_{\\ell_+} \\dfrac{d\\zeta'}{(\\zeta')^2}\\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} + \\frac{R}{4}\\int_{\\ell_-} \\dfrac{d\\zeta'}{(\\zeta')^2}  \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}.\n\\end{equation}\nUse the fact that $\\mathcal{X}_e$ (resp. $\\mathcal{X}^{-1}_e$) has norm less than 1 on $\\ell_+$ (resp. $\\ell_-$) and the uniform convergence of the geometric series on $\\zeta'$ to write \\eqref{dap1} as:\n\\begin{align*}\n- \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{R}{4}\\sum_{n=1}^\\infty \\left\\{ \\vphantom{\\int_{\\ell_+}} \\right. & \\int_{\\ell_+}\\frac{d\\zeta'}{(\\zeta')^2}\\exp\\left(\n\\frac{\\pi R n a}{\\zeta'} +i n \\theta_e +\\pi R n \\zeta' \\overline{a}\\right) + \\\\\n& \\left.\n\\int_{\\ell_-}\\frac{d\\zeta'}{(\\zeta')^2}\\exp\\left(\n\\frac{-\\pi R n a}{\\zeta'} -i n \\theta_e -\\pi R n \\zeta' \\overline{a}\\right)\\right\\},\n\\end{align*}\n\\begin{align*}\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\left(\\frac{R}{4}\\right) \\left( \\frac{-2|a|}{a}\\right)\\sum_{n=1}^\\infty \\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)K_1(2\\pi R n |a|)\\\\\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} - \\frac{R|a|}{2a}\\sum_{n=1}^\\infty \\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)K_1(2\\pi R n |a|).\n\\end{align*}\nSince $K_1(x) \\thicksim 1\/x$, for $x$ real and $x \\to 0$, we obtain, letting $a \\to 0$:\n\\begin{align*}\n& - \\frac{i(\\theta_e - \\pi)}{4\\pi a} - \\frac{R|a|}{2a\\cdot 2\\pi R |a|} \\sum_{n=1}^\\infty \\frac{\\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)}{n}\\\\\n& =  - \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{1}{4\\pi a}[\\log(1-e^{i\\theta_e})-\\log(1-e^{-i\\theta_e})]\\\\\n\\intertext{and by \\eqref{logs},}\n& =  - \\frac{i(\\theta_e - \\pi)}{4\\pi a} +\\frac{i(\\theta_e -\\pi)}{4\\pi a} = 0.\n\\end{align*}\nTherefore this part of the $da$ terms extends trivially to 0 in the singular fiber.\n\nIt remains to extend the other terms involving $da$. Recall that by \\eqref{parfrac}, these terms are (after getting rid of a factor of $-R\/2$):\n\\begin{equation}\\label{last3}\n\\frac{ \\log a}{\\zeta} + \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} +  \\int_{\\ell_-} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}.\n\\end{equation}\n\nWe'll focus in the first integral in \\eqref{last3}. As a starting point, we'll prove that as $a \\to 0$, the limiting value of this integral is the same as the limit of\n\\begin{equation}\\label{simpler}\n \\int_{\\ell_+}  \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e +\\pi R \\zeta' \\overline{a}\\right)}.\n\\end{equation}\nIt suffices to show that\n\\begin{equation}\\label{lebes}\n \\int_{\\ell_+}  \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'}  \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e +\\pi R \\zeta' \\overline{a}\\right)} [1-\\exp(\\pi R \\zeta' \\overline{a})] \\to 0, \\hspace{5 mm} \\text{as $a \\to 0$, $\\theta_e \\neq 0$}\n\\end{equation}\nTo see this, we can assume $|a| < 1$. Let $b = a\/|a|$. Observe that in the $\\ell_+$ ray, $|\\exp(\\pi Ra\/\\zeta')| < 1$, and since $\\theta_e \\neq 0$, we can bound \\eqref{lebes} by\n\\[ \\text{const} \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} [1-\\exp(\\pi R \\zeta' \\overline{b})] < \\infty. \\]\nEquation \\eqref{lebes} now follows from Lebesgue Dominated Convergence and the fact that $1-\\exp(\\pi R \\zeta' \\overline{a}) \\to 0$ as $a \\to 0$. A similar application of Dominated Convergence allows us to reduce the problem to the extension of\n\\begin{equation}\\label{secsimp}\n\\int_{\\ell_+}  \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}.\n\\end{equation}\nIntroduce the real variable $s = -\\pi R a \/ \\zeta'$. We can write \\eqref{secsimp} as:\n\\begin{align}\n& e^{i\\theta_e}\\int_0^\\infty \\frac{ds}{s\\left[ \\frac{-\\pi R a}{s} - \\zeta \\right]} \\frac{e^{-s}}{1-e^{i\\theta_e-s}} \\notag\\\\\n& = -\\frac{1}{\\zeta}\\int_0^\\infty \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{e^{-s}}{e^{-i\\theta_e}-e^{-s}} \\notag\\\\\n& = \\frac{1}{\\zeta}\\int_0^\\infty \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{1}{1-e^{s-i\\theta_e}} \\label{doublez}\n\\end{align}\nThe integrand of \\eqref{doublez} has a double zero at $\\infty$, when $a \\to 0$, so the only possible non-convergent part in the limit $a=0$ is the integral\n\\[ \\frac{1}{\\zeta}\\int_0^1 \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{1}{1-e^{s-i\\theta_e}}. \\]\nSince\n\\[ \\int_0^1 \\frac{ds}{s} \\left[ \\frac{1}{1-e^{s-i\\theta_e}} - \\frac{1}{1-e^{-i\\theta_e}}\\right] < \\infty, \\]\nwe can simplify this analysis even further and focus only on\n\\begin{align}\n& \\frac{1}{\\zeta(1-e^{-i\\theta_e})} \\int_0^1 \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\\\\n& = -\\frac{\\log (\\pi R a \/\\zeta)}{\\zeta(1-e^{-i\\theta_e})}.\n\\end{align}\nWe can apply the same technique to obtain a limit for the second integral in \\eqref{last3}. The result is\n\\[ -\\frac{\\log (-\\pi R a \/\\zeta)}{\\zeta(1-e^{i\\theta_e})}, \\]\nwhich means that the possibly non-convergent terms in \\eqref{last3} are:\n\\begin{equation}\\label{cancel}\n\\frac{\\log a}{\\zeta} - \\frac{\\log a}{\\zeta(1-e^{-i\\theta_e})} - \\frac{\\log a}{\\zeta(1-e^{i\\theta_e})} = 0.\n\\end{equation}\nNote that the corrections of $\\mathcal{X}_m$ in other regions of the $a$-plane as in \\eqref{regmod} depend only on $\\mathcal{X}_e$, which clearly has a smooth extension to the singular fiber.\n\n\nThe extension of the $d\\overline{a}$ part is performed in exactly the same way as with the $da$ forms. We conclude that the 1-form\n\\[ \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nhas an extension to $\\mathcal{M}$; more explicitly, to the fiber at $a=0$ in the classical Ooguri-Vafa case. This holds true also in the generalized Ooguri-Vafa case since here we simply add factors of the form $f'(a)da$ and it is assumed that $f(a)$ has a smooth extension to the singular fiber.\n\\end{proof}\n\nIn section \\ref{sfiber}, we will reinterpret these extension of the derivatives of $\\mathcal{X}_m$ if we regard the gauge transformation \\eqref{nicethm} as a contour integral between symmetric contours. It will be then easier to see that the extension can be made smooth.\n\n\\subsubsection{Extension of the metric}\\label{extmetric}\n\n\nThe results of the previous section already show the continuous extension of the holomorphic symplectic form\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge  \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nto the limiting case $a = 0$, but we excluded the special case $\\theta_e = 0$. Here we obtain $\\varpi(\\zeta)$ at the singular fiber with a different approach that will allow us to see that such an extension is smooth without testing the extension for each derivative. Although it was already known that $\\mathcal{M}'$ extends to the hyperk\\\"{a}hler manifold $\\mathcal{M}$ constructed here, this approach is new, as it gives an explicit construction of the metric as we will see. Furthermore, the Ooguri-Vafa model can be thought as an elementary model for which more complex integrable systems are modeled locally (see \\S \\ref{sfiber}).\n\\begin{theorem}\nThe holomorphic symplectic form $\\varpi(\\zeta)$ extends smoothly to $\\mathcal{M}$. Near $a = 0$ and $\\theta_e = 0$, the hyperk\\\"{a}hler metric $g$ looks like a constant multiple of the Taub-NUT metric $g_{\\text{Taub-NUT}}$ plus some smooth corrections.\n\\end{theorem}\n\\begin{proof}\n By \\cite{gaiotto}, near $a = 0$,\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\left[ id\\theta_m + 2\\pi i A + \\pi i V \n\\left(\\frac{1}{\\zeta}da - \\zeta d\\bar{a}\\right)\\right], \\]\nwhere\n\\[ A = \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{R}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\sum_{n \\neq 0} (\\text{sgn} \\,n) e^{in\\theta_e} |a|\n   K_1(2\\pi R|na|) \\]\nshould be understood as a $U(1)$ connection over the open subset of $\\mathbb{C} \\times S^1$ parametrized by $(a,\\theta_e)$ and $V$ is given by Poisson re-summation as\n\\begin{equation}\\label{potent}\nV = \\frac {R}{4\\pi}\\left[ \\frac{1}{\\sqrt{R^2|a|^2 + \\frac{\\theta_e^2}{4\\pi^2}}} + \\sum_{\\substack{n = -\\infty \\\\ n \\neq 0}}^\\infty \\left( \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right) \\right].\n\\end{equation}\nHere $\\kappa_n$ is a regularization constant introduced to make the sum convergent, even at $a = 0, \\theta_e \\neq 0$. The curvature $F$ of the unitary connection satisfies\n\\begin{equation}\\label{curv}\ndA = *dV.\n\\end{equation}\nConsider now a gauge transformation $\\theta_m \\mapsto \\theta_m + \\alpha$ and its induced change in the connection $A \\mapsto A' = A - d\\alpha\/2\\pi$ (see \\cite{gaiotto}). We have $id\\theta'_m + 2\\pi i A' = id\\theta_m + id\\alpha + 2\\pi i A - id\\alpha = id\\theta_m + 2\\pi i A$. Furthermore, for the particular gauge transformation in (\\ref{thetapr}), at $a = 0$ and for $\\theta_e \\neq 0$:\n\\begin{align*}\nA' & = A - \\frac{d\\alpha}{2\\pi}\\\\\n& = \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{1}{8\\pi^2} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) \\left[ \\sum_{n = 1}^\\infty \\frac{e^{in\\theta_e}}{n} - \\sum_{n = 1}^\\infty \\frac{e^{-in\\theta_e}}{n} \\right]\\\\\n& - \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right),\\\\\n\\intertext{(here we're using the fact that $K_1(x) \\to 1\/x$ as $x \\to 0$)}\n& = \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) - \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) = 0.\\\\\n\\intertext{since the above sums converge to $-\\log(1 - e^{i\\theta_e}) + \\log(1 - e^{-i\\theta_e}) = -i(\\theta_e - \\pi)$ for $\\theta_e \\neq 0$.}\n\\end{align*}\n\nWriting $V_0$ (observe that this only depends on $\\theta_e$) for the limit of $V$ as $a \\to 0$, we get at $a = 0$\n\\begin{align*}\n\\varpi(\\zeta) & = -\\frac{1}{4\\pi^2 R} \\left( \\frac{\\pi R}{\\zeta}da + id\\theta_e + \\pi R \\zeta d\\bar{a} \\right) \\wedge \\left(\n  id\\theta'_m + \\pi i V_0 \\left( \\frac{da}{\\zeta} - \\zeta d\\bar{a}\\right) \\right)  \\\\\n  & = \\frac{1}{4\\pi^2 R} d\\theta_e \\wedge d\\theta'_m + \\frac{iV_0}{2}da \\wedge d\\bar{a} -\\frac{i}{4\\pi \\zeta}da \\wedge d\\theta'_m - \\frac{V_0}{4\\pi R\\zeta}da \\wedge d\\theta_e  \\\\\n  & - \\frac{i\\zeta}{4\\pi} d\\bar{a} \\wedge d\\theta'_m + \\frac{V_0 \\zeta}{4\\pi R} d\\bar{a} \\wedge d\\theta_e.\n\\end{align*}\n\nThis yields that, at the singular fiber,\n\\begin{align}\n\\omega_3 & = \\frac{1}{4\\pi^2 R} d\\theta_e \\wedge d\\theta'_m + \\frac{iV_0}{2}da \\wedge d\\bar{a} \\label{symp3}\\\\\n\\omega_+ & = \\frac{1}{2\\pi} da \\wedge \\left( d\\theta'_m - \\frac{iV_0}{R}d\\theta_e \\right)\\label{symp+}\\\\\n\\omega_- & = \\frac{1}{2\\pi} d\\bar{a} \\wedge \\left( d\\theta'_m + \\frac{iV_0}{R}d\\theta_e \\right)\\label{symp-}\n\\end{align}\n\nFrom the last two equations we obtain that $d\\theta'_m - iV_0\/R d\\theta_e$ and $d\\theta'_m + iV_0\/R d\\theta_e$ are respectively (1,0) and (0,1) forms under the complex structure $J_3$. A $(1,0)$ vector field dual to the $(1,0)$ form above is then $\\dfrac{1}{2}\\left(\\partial_{\\theta'_m} + iR\/V_0 \\partial_{\\theta_e}\\right)$. In particular,\n\\[ J_3(\\partial_{\\theta'_m}) = -\\frac{R}{V_0} \\partial_{\\theta_e}, \\hspace{5 mm} J_3 \\left(-\\frac{R}{V_0}\\partial_{\\theta_e} \\right) = -\\partial_{\\theta'_m}. \\]\nWith this and (\\ref{symp3}) we can reconstruct the metric at $a = 0$. Observe that\n\\begin{align*}\ng(\\partial_{\\theta_e}, \\partial_{\\theta_e}) & = \\omega_3(\\partial_{\\theta_e}, J_3(\\partial_{\\theta_e})) = \\omega_3\\left(\\partial_{\\theta_e}, \\frac{V_0}{R}\\partial_{\\theta'_m}\\right) = \\frac{V_0}{4\\pi^2 R^2} \\\\\ng(\\partial_{\\theta'_m}, \\partial_{\\theta'_m}) & = \\omega_3(\\partial_{\\theta'_m}, J_3(\\partial_{\\theta'_m})) = \\omega_3\\left(\\partial_{\\theta'_m}, -\\frac{R}{V_0}\\partial_{\\theta_e}\\right) = \\frac{1}{4\\pi^2 V_0}\n\\end{align*}\n\nConsequently,\n\\[ g = \\frac{1}{V_0} \\left( \\frac{d\\theta'_m}{2\\pi}\\right)^2 + V_0 d\\vec{x}^2, \\]\nwhere $a = x^1 + ix^2, \\theta_e = 2\\pi R x^3$. Since $V_0(\\theta_e)$ is undefined for $\\theta_e = 0$, we have to check that $g$ extends to this point. Let $(r,\\vartheta, \\phi)$ denote spherical coordinates for $\\vec{x}$. The formula above is the natural extension of the metric given in \\cite{gaiotto} for nonzero $a$:\n\\[ g = \\frac{1}{V(\\vec{x})} \\left( \\frac{d\\theta'_m}{2\\pi} + A'(\\vec{x})\\right)^2 + V(\\vec{x}) d\\vec{x}^2 \\]\nTo see that this extends to $r =0$, we rewrite\n\\begin{align}\nV & = \\frac{R}{4\\pi}\\left[ \\frac{1}{\\sqrt{R^2 |a|^2 + \\frac{\\theta_e^2}{4\\pi^2}}} + \\sum_{n \\neq 0} \\left(\n \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right)\\right] \\notag\\\\\n & = \\frac{1}{4\\pi}\\left[ \\frac{1}{\\sqrt{ |a|^2 + \\frac{\\theta_e^2}{4R^2 \\pi^2}}} + R\\sum_{n \\neq 0} \\left(\n \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right) \\right] \\notag\\\\\n & = \\frac{1}{4\\pi} \\left( \\frac{1}{r} + C(\\vec{x}) \\right), \\label{vtaub}\n\\end{align}\nwhere $C(\\vec{x})$ is smooth and bounded in a neighborhood of the origin.\n\nSimilarly, we do Poisson re-summation for the unitary connection\n\n\\[ A' = - \\frac{1}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) \\left[ \\frac{i(\\theta_e - \\pi)}{2\\pi} + R \\sum_{n \\neq 0} (\\text{sgn} \\,n) e^{in\\theta_e} |a| K_1(2\\pi R|na|) \\right]. \\]\n\nUsing the fact that the inverse Fourier transform of $(\\text{sgn }\\xi)e^{i\\theta_e \\xi}|a|K_1(2\\pi R|a\\xi|)$ is\n\\[ \\frac{i(\\frac{\\theta_e}{2\\pi} + t)}{2R\\sqrt{R^2|a|^2 + ( \\frac{\\theta_e}{2\\pi} + t)^2}}, \\]\nwe obtain\n\\begin{align}\nA' & = - \\frac{i}{8\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\sum_{n = -\\infty}^\\infty \\left( \\dfrac{ \\frac{\\theta_e}{2\\pi} + n}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n  \\right) \\notag\\\\\n& = \\frac{1}{4\\pi}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\left[-\\frac{i\\theta_e}{4\\pi \\sqrt{R^2 |a|^2 + \\left(  \n \\frac{\\theta_e}{2\\pi}\\right)^2}} - \\frac{i}{2}\\sum_{n \\neq 0} \\left( \\dfrac{ \\frac{\\theta_e}{2\\pi} + n}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n  \\right)\\right] \\notag\\\\\n \\intertext{since $d\\phi = d\\arg a = -id\\log \\dfrac{a}{|a|} = -\\dfrac{i}{2}\\left(\\dfrac{da}{a} - \\dfrac{d\\bar{a}}{\\bar{a}} \\right)$ and $\\cos \\vartheta = \\dfrac{x^3}{r}$, this simplifies to:}\n & = \\frac{1}{4\\pi}(\\cos \\vartheta + D(\\vec{x}))d\\phi. \\label{ataub}\n\\end{align}\nHere $\\kappa_n$ is a regularization constant that makes the sum converge, and $D(\\vec{x})$ is smooth and bounded in a neighborhood of $r = 0$. By (\\ref{vtaub}) and (\\ref{ataub}), it follows that near $r = 0$\n\\begin{align*}\ng & = V^{-1}\\left( \\frac{d\\theta'_m}{2\\pi} + A' \\right)^2 + Vd\\vec{x}^2\\\\\n& = 4\\pi \\left( \\frac{1}{r} + C \\right)^{-1} \\left( \\frac{d\\theta'_m}{2\\pi} + \\frac{1}{4\\pi}\\cos \\vartheta d\\phi + D d\\phi \\right)^2 + \\frac{1}{4\\pi} \\left( \\frac{1}{r} + C \\right) d\\vec{x}^2\\\\\n& = \\frac{1}{4\\pi}\\left[ \\left( \\frac{1}{r} + C \\right)^{-1} \\left( 2d\\theta'_m + \\cos \\vartheta d\\phi + \\tilde{D} d\\phi \\right)^2\n + \\left( \\frac{1}{r} + C \\right) d\\vec{x}^2 \\right]\\\\\n& = \\frac{1}{4\\pi} g_{\\text{Taub-NUT}} + \\text{smooth corrections}.\n\\end{align*}\nThis shows that our metric extends to $r = 0$ and finishes the construction of the singular fiber.\n\\end{proof}\n\n\\subsection{General case}\\label{genextmtr}\n\nHere we work with the assumption in subsection \\ref{genvafa}. To distinguish this case to the previous one, we will denote by $\\varpi_\\text{old}, g_\\text{old}$, etc. the forms obtained in the classical case.\n\nLet $C := -i\/2 + \\pi f'(0)$ and let\n\\[ B_0 = V_0 + \\frac{R \\, \\text{Im }C}{\\pi}. \\]\nWe will see that, to extend the holomorphic symplectic form $\\varpi(\\zeta)$  and consequently the hyperk\\\"{a}hler metric $g$ to $\\mathcal{M}$, it is necessary to impose a restriction on the class of functions $f(a)$ on $\\mathcal{B}$ for the generalized Ooguri-Vafa case.\n\n\\begin{theorem}\nIn the General Ooguri-Vafa case, the holomorphic symplectic form $\\varpi(\\zeta)$ and the hyperk\\\"{a}hler metric $g$ extend to $\\mathcal{M}$, at least for the set of functions $f(a)$ as in \\S \\ref{genvafa} with $f'(0) > B_0$.\n\\end{theorem}\n\\begin{proof}\n By formula \\eqref{newxmsf},\n\\begin{equation}\\label{dlogxm}\nd \\log \\mathcal{X}_m^\\text{sf} = d \\log \\mathcal{X}_{m, \\text{old}}^\\text{sf} + \\frac{R}{\\zeta}\\left( -\\frac{i}{2} + \\pi f'(a) \\right)da + R\\zeta \\left( \\frac{i}{2} + \\pi \\overline{f'(a)} \\right)d\\overline{a}\n\\end{equation}\n\nRecall that the corrections of $\\mathcal{X}_m$ are the same as the classical Ooguri-Vafa case. Thus, using \\eqref{dlogxm}, at $a = 0$\n\\begin{equation*}\n\\varpi(\\zeta) = \\varpi_\\text{old}(\\zeta) + \\frac{iR }{2\\pi} \\text{Im } C da \\wedge d\\overline{a} + \\frac{i C}{4\\pi^2 \\zeta}\n   da \\wedge d\\theta_e + \\frac{i \\zeta \\overline{C}}{4\\pi^2} d\\overline{a} \\wedge d\\theta_e.\n\\end{equation*}\n\nDecomposing $\\varpi(\\zeta) = -i\/2\\zeta \\omega_+ + \\omega_3 -i\\zeta \/2 \\omega_-$, we obtain:\n\\begin{align}\n\\omega_3 & = \\omega_{3, \\text{old}} + \\frac{i R}{2\\pi} \\text{Im } C da \\wedge d\\overline{a}, \\label{omeg3}\\\\\n\\omega_+ & = \\omega_{+, \\text{old}} - \\frac{C}{2\\pi^2} da \\wedge d\\theta_e \\label{newomp}\\\\\n\\omega_- & = \\omega_{-, \\text{old}} - \\frac{\\overline{C}}{2\\pi^2} d\\overline{a} \\wedge d\\theta_e \\label{newomm}\n\\end{align}\n\nBy \\eqref{newomp} and \\eqref{newomm},\n\\[ d\\theta'_m - \\frac{i}{R}\\left( V_0 - \\frac{iRC}{\\pi} \\right)d\\theta_e \\hspace{6 mm} \\text{and} \\hspace{6 mm} d\\theta'_m + \\frac{i}{R}\\left( V_0 + \\frac{iR\\overline{C}}{\\pi} \\right)d\\theta_e \\]\nare, respectively, (1,0) and (0,1) forms. It's not hard to see that\n\\begin{align*}\n\\frac{-V_0 \\pi -iR \\overline{C}}{R\\pi}\\partial_{\\theta'_m} &  - i\\partial_{\\theta_e} \\\\\n\\intertext{or, rearranging real parts,}\n\\left( -\\frac{V_0}{R} -\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m} & -i \\left( \\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e}\\right)\n\\end{align*}\nis a $(1,0)$ vector field. This allow us to obtain\n\\begin{align*}\nJ_3\\left[ \\left( -\\frac{V_0}{R} -\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m} \\right] & = \\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e}\\\\\nJ_3\\left[\\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e} \\right] & = \\left( \\frac{V_0}{R} +\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m}.\n\\end{align*}\nBy linearity,\n\\begin{align*}\nJ_3(\\partial_{\\theta'_m}) & = \\text{const} \\cdot \\partial_{\\theta'_m} - \\frac{R\\pi}{V_0 \\pi + R\\text{Im }C} \\partial_{\\theta_e}\\\\\nJ_3(\\partial_{\\theta_e}) & = \\left( \\frac{V_0 \\pi + R\\text{Im }C }{\\pi R} + \\frac{(\\text{Re }C)^2 R}{\\pi(V_0 \\pi \n  + R\\text{Im }C)} \\right)\\partial_{\\theta'_m} + \\text{const} \\cdot \\partial_{\\theta_e}.\n\\end{align*}\nWith this we can compute\n\\begin{align*}\ng(\\partial_{\\theta'_m}, \\partial_{\\theta'_m}) & = \\omega_3(\\partial_{\\theta'_m}, J_3(\\partial_{\\theta'_m}))\\\\\n& = \\frac{1}{4\\pi(V_0 \\pi + R\\text{Im }C)}\\\\\ng(\\partial_{\\theta_e}, \\partial_{\\theta_e}) & = \\omega_3(\\partial_{\\theta_e}, J_3(\\partial_{\\theta_e}))\\\\\n& = \\frac{V_0 \\pi + R\\text{Im }C}{4\\pi^3 R^2} + \\frac{(\\text{Re }C)^2}{4\\pi^3(V_0 \\pi + R\\text{Im }C)}\\\\\n& = \\frac{B_0}{4\\pi^3 R^2} + \\frac{(\\text{Re }C)^2}{4\\pi^3 B_0}\n\\end{align*}\n\nWe can see that, if $B_0 > 0$, the metric at $a = 0$ is\n\\begin{equation}\ng = \\frac{1}{B_0} \\left( \\frac{d\\theta'_m}{2\\pi}\\right)^2 + B_0 d\\vec{x}^2 + \\left(\\frac{R\\cdot\\text{Re }C}{\\pi}\\right)^2 \\frac{dx_3^2}{B_0}.\n\\end{equation}\nThis metric can be extended to the point $\\theta_e = 0$ ($r = 0$ in \\S \\ref{clasmet}) exactly as before, by writing $g$ as the Taub-NUT metric plus smooth corrections and observing that, since $\\lim_{\\theta_e \\to 0} B_0 = \\infty$,\n\\[ \\lim_{\\theta_e \\to 0} \\left(\\frac{R\\cdot\\text{Re }C}{\\pi}\\right)^2 \\frac{dx_3^2}{B_0} = 0. \\]\n\\end{proof}\n\n\n\\section{The Pentagon case}\\label{pent}\n\n\n\n\\subsection{Monodromy Data}\\label{solut}\n\nNow we will extend the results of the Ooguri-Vafa case to the general problem. We will start with the Pentagon example. This  example is presented in detail in \\cite{notes}. By \\cite{wkb}, this example represents the moduli space of Higgs bundles with gauge group $\\text{SU}(2)$ over $\\mathbb{P}^1$ with 1 irregular singularity at $z = \\infty$.\n\nHere $\\mathcal{B} = \\mathbb{C}$ with discriminant locus a 2-point set, which we can assume is $\\{-2,2\\}$  in the complex plane. Thus $\\mathcal{B}'$ is the twice-punctured plane. $\\mathcal{B}$ is divided into two domains $\\mathcal{B}_{\\text{in}}$ and $\\mathcal{B}_{\\text{out}}$ by the locus\n\\[ W = \\{u : Z(\\Gamma_u) \\text{ is contained in a line in } \\mathbb{C} \\} \\subset \\mathcal{B} \\]\nSee Figure \\ref{walls}. Since $\\mathcal{B}_{\\text{in}}$ is simply connected $\\Gamma$ can be trivialized over $\\mathcal{B}_{\\text{in}}$ by primitive cycles $\\gamma_1, \\gamma_2$, with $Z_{\\gamma_1} = 0$ at $u = -2$, $Z_{\\gamma_2} = 0$ at $u = 2$. We can choose them also so that $\\left\\langle \\gamma_1, \\gamma_2 \\right\\rangle = 1$. \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{walls.ps}\n\t\\caption{The wall $W$ in $\\mathcal{B}$ for the Pentagon case}\n\t\\label{walls}\n\\end{figure}\n\nTake the set $\\{\\gamma_1, \\gamma_2\\}$. To compute its monodromy around infinity, take cuts at each point of $D = \\{-2,2\\}$ (see Figure \\ref{moninf}) and move counterclockwise. By \\eqref{piclf}, the jump of $\\gamma_2$ when you cross the cut at $-2$ is of the form $\\gamma_2 \\mapsto \\gamma_1 + \\gamma_2$. As you return to the original place and cross the cut at $2$, the jump of $\\gamma_1$ is of the type $\\gamma_1 \\mapsto \\gamma_1 - \\gamma_2$.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{monodromy_infinity.ps}\n\t\\caption{The monodromy around infinity of $\\Gamma$}\n\t\\label{moninf}\n\\end{figure}\n\n\nThus, around infinity, $\\{\\gamma_1, \\gamma_2\\}$ transforms into $\\{-\\gamma_2, \\gamma_1 + \\gamma_2\\}$. The set $\\{\\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}$ is therefore invariant under monodromy at infinity and it makes global sense to define\n\n  \\begin{align}\n  \\text{For $u \\in \\mathcal{B}_\\text{in}$}, \\hspace{5 mm} \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n    1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2\\}\\\\\n    0 & \\text{otherwise}\n  \\end{array} \\right. \\notag\\\\\n \\text{For $u \\in \\mathcal{B}_\\text{out}$} , \\hspace{5 mm} \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n    1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}\\\\\n    0 & \\text{otherwise}\n  \\end{array}  \\right. \\label{omgpar}\n  \\end{align}\n\n\n\n\nLet $\\mathcal{M}'$ denote the torus fibration over $\\mathcal{B}'$ constructed in \\cite{notes}. Near $u=2$, we'll denote $\\gamma_1$ by $\\gamma_m$ and $\\gamma_2$ by $\\gamma_e$ (the labels will change for $u = - 2$). To shorten notation, we'll write $\\ell_{e}, Z_e$, etc. instead of $\\ell_{\\gamma_e}, Z_{\\gamma_e}$, etc. Let $\\theta$ denote the vector of torus coordinates $(\\theta_e, \\theta_m)$. With the change of variables $a := Z_e(u)$ we can assume, without loss of generality, that the bad fiber is at $a = 0$ and\n\\begin{equation}\\label{zmnotz}\n\\lim_{a \\to 0} Z_m(a) = c \\neq 0.\n\\end{equation}\nLet $T$ denote the complex torus fibration over $\\mathcal{M}'$ constructed in \\cite{gaiotto}. By the definition of $\\Omega(\\gamma; a)$, the functions $(\\mathcal{X}_e, \\mathcal{X}_m)$ both receive corrections. Recall that by \\eqref{recurs}, for each $\\nu \\in \\mathbb{N}$, we get a function $\\mathcal{X}_\\gamma^{(\\nu)}$, which is the $\\nu$-th iteration of the function $\\mathcal{X}_\\gamma$. We can write\n\\[ \\mathcal{X}_\\gamma^{(\\nu)}(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\theta)C_{\\gamma}^{(\\nu)}(a, \\zeta, \\theta). \\]\nIt will be convenient to rewrite the above equation as in \\cite[C.17]{gaiotto}. For that, let $\\Upsilon^{(\\nu)}$ be the map from $\\mathcal{M}_a$ to its complexification $\\mathcal{M}_a^{\\mathbb{C}}$ such that\n\\begin{equation}\\label{upsi}\n \\mathcal{X}_\\gamma^{(\\nu)}(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\Upsilon^{(\\nu)}).\n\\end{equation}\n\nWe'll do a modification in the construction of \\cite{gaiotto} as follows: We'll use the term ``BPS ray'' for each ray $\\{\\ell_\\gamma : \\Omega(\\gamma,a) \\neq 0 \\}$ as in \\cite{gaiotto}. This terminology comes from Physics. In the language of Riemann-Hilbert problems, these are known as ``anti-Stokes'' rays. That is, they represent the contour $\\Sigma$ where a function has prescribed discontinuities.\n\nThe problem is local on $\\mathcal{B}$, so instead of defining a Riemann-Hilbert problem using the BPS rays $\\ell_\\gamma$, we will cover $\\mathcal{B}'$ with open sets $\\{U_\\alpha : \\alpha \\in \\Delta \\}$ such that for each $\\alpha$, $\\overline{U_\\alpha}$ is compact, $\\overline{U_\\alpha} \\subset V_\\alpha$, with $V_\\alpha$ open and $\\left. \\mathcal{M}' \\right|_{V_\\alpha}$ a trivial fibration. For any ray $r$ in the $\\zeta$-plane, define $\\mathbb{H}_r$ as the half-plane of vectors making an acute angle with $r$. Assume that there is a pair of rays $r, -r$ such that for all $a \\in U_\\alpha$, half of the rays  lie inside $\\mathbb{H}_r$ and the other half lie in $\\mathbb{H}_{-r}$. We call such rays \\textit{admissible rays}. If $U_\\alpha$ is small enough, there exists admissible rays for such a neighborhood. We are allowing the case that $r$ is a BPS ray $\\ell_\\gamma$, as long as it satisfies the above condition. As $a$ varies in $U_\\alpha$, some BPS rays (or anti-Stokes rays, in RH terminology) converge into a single ray (wall-crossing phenomenon) (see Figures \\ref{3rays} and \\ref{2rays}). \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{3raysbeforewall.ps}\n\t\\caption{3 anti-Stokes rays before hitting the wall}\n\t\\label{3rays}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{2raysafterwall.ps}\n\t\\caption{At the other side of the wall there are only 2 anti-Stokes rays}\n\t\\label{2rays}\n\\end{figure}\n\n\n For $\\gamma \\in \\Gamma$, we define $\\gamma > 0$ (resp. $\\gamma < 0$) as $\\ell_\\gamma \\in \\mathbb{H}_r$ (resp. $\\ell_\\gamma \\in \\mathbb{H}_{-r}$). Our Riemann-Hilbert problem will have only two anti-Stokes rays, namely $r$ and $-r$. The specific discontinuities at the anti-Stokes rays for the function we're trying to obtain are called \\textit{Stokes factors} (see \\cite{boalch}). In \\eqref{invjmp}, the Stokes factor was given by $S^{-1}_\\ell$.\n \n In this case, the Stokes factors are the concatenation of all the Stokes factors $S^{-1}_\\ell$ in \\eqref{stkfac} in the counterclockwise direction:\n\\begin{align*}\nS_+ & = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma > 0}{\\mathcal{K}^{\\Omega(\\gamma; a)}_\\gamma}\\\\\nS_- & = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma < 0}{\\mathcal{K}^{\\Omega(\\gamma; a)}_\\gamma}\n\\end{align*}\n\nWe will denote the solutions of this Riemann-Hilbert problem by $\\mathcal{Y}$. As in \\eqref{upsi}, we can write $\\mathcal{Y}$ as\n\\begin{equation}\\label{thet}\n \\mathcal{Y}_\\gamma(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\Theta),\n\\end{equation}\nfor $\\Theta : \\mathcal{M}_a \\to \\mathcal{M}_a^\\mathbb{C}$.\n\nA different choice of admissible pairs $r', -r'$ gives an equivalent Riemann-Hilbert problem, where the two solutions $\\mathcal{Y}, \\mathcal{Y}'$ differ only for $\\zeta$ in the sector defined by the rays $r,r'$, and one can be obtained from the other by analytic continuation.\n\nIn the case of the Pentagon, we have two types of wall-crossing phenomenon. Namely, as $a$ varies, $\\ell_e$ moves in the $\\zeta$-plane until it coincides with the $\\ell_m$ ray for some value of $a$ in the wall of marginal stability (Fig. \\ref{3rays} and \\ref{2rays}). We'll call this type I of wall-crossing. In this case we have the Pentagon identity\n\\begin{equation}\\label{pentid}\n \\mathcal{K}_e \\mathcal{K}_m = \\mathcal{K}_m \\mathcal{K}_{e+m} \\mathcal{K}_e,\n\\end{equation}\nAs $a$ goes around 0, the $\\ell_e$ ray will then intersect with the $\\ell_{-m}$ ray now. Because of the monodromy $\\gamma_m \\mapsto \\gamma_{-e+m}$ around 0, $\\ell_m$ becomes $\\ell_{-e+m}$. This second type (type II) of wall-crossing is illustrated in Fig. \\ref{2rays2} and \\ref{3rays2}.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{2raysafterwalltoo.ps}\n\t\\caption{2 anti-Stokes rays before hitting the wall}\n\t\\label{2rays2}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{3raysbeforewalltoo.ps}\n\t\\caption{At the other side of the wall there are now 3 anti-Stokes rays}\n\t\\label{3rays2}\n\\end{figure}\n\nThis gives a second Pentagon identity\n\\[ \\mathcal{K}_{-e} \\mathcal{K}_m = \\mathcal{K}_m \\mathcal{K}_{-e+m} \\mathcal{K}_{-e} \\]\n\n\nIn any case, the Stokes factors above remain the same even if $a$ is in the wall of marginal stability. The way we defined $S_+, S_-$ makes this true for the general case also.\n\nSpecifically, in the Pentagon the two Stokes factors for the first type of wall-crossing are given by the maps:\n\\begin{align}\n\\left. \\begin{array}{ll}\n          \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}_e(1-\\mathcal{Y}_m))^{-1} \\\\\n          \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}_m)\n          \\end{array}  \\right\\} & S_+ \\label{newj1}\\\\\n\\intertext{and, similarly}\n\\left. \\begin{array}{ll}\n          \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e(1-\\mathcal{Y}^{-1}_m))\\\\\n          \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}^{-1}_m)^{-1}\n          \\end{array} \\right\\} & S_- \\label{newj2}\n\\end{align}\n\n\\noindent For the second type:\n\n\\begin{align}\n\\left. \\begin{array}{ll}\n          \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e) \\\\\n          \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e))\n          \\end{array}  \\right\\} & S_+ \\label{2newj1}\\\\\n\\left. \\begin{array}{ll}\n          \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}_e)^{-1}\\\\\n          \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}^{-1}_m(1-\\mathcal{Y}_e))^{-1}\n          \\end{array} \\right\\} & S_- \\label{2newj2}\n\\end{align}\n\n\\subsection{Solutions}\n\nIn \\cite{rhprob} we prove the following theorem (in fact, a more general version is proven).\n\n\\begin{theorem}\\label{yfunctions}\nThere exist functions $ \\mathcal{Y}_m(a, \\zeta, \\theta_e, \\theta_m),  \\mathcal{Y}_e(a, \\zeta, \\theta_e, \\theta_m)$ defined for $a \\neq 0$,  smooth on $a$, $\\theta_e$ and $\\theta_m$. The functions are sectionally analytic on $\\zeta$ and obey the jump condition\n\\[ \\begin{array}{rll}\n\t\\mathcal{Y}^+ & = S_+ \\mathcal{Y}^-, & \\qquad \\text{along $r$} \\\\\n\t\\mathcal{Y}^+ & = S_{-}  \\mathcal{Y}^-, & \\qquad \\text{along $-r$}\n\t\\end{array} \\]\nMoreover, $\\mathcal{Y}_m,  \\mathcal{Y}_e$ obey the reality condition \\eqref{realcond} and the asymptotic condition \\ref{asymptotic}.\n\\end{theorem}\n\n\\begin{remark}\nOur construction used integrals along a fixed admissible pair $r,-r$ and our Stokes factors are concatenation of the Stokes factors in \\cite{gaiotto}. Thus, the coefficients $f^{\\gamma'}$ are different here, but they are still obtained by power series expansion of the explicit Stokes factor. In particular, it may not be possible to express\n\\[ f^{\\gamma'} = c_{\\gamma'} \\gamma' \\]\nfor some constant $c_{\\gamma'}$. For instance, in the pentagon, wall-crossing type I, we have, for $0\\leq j\\leq i$ and $\\gamma' = \\gamma_{ie +jm}$:\n\\[ f^{\\gamma'} = \\frac{(-1)^{j}\\binom{i}{j}}{i^2} \\gamma_{ie}. \\]\nBecause of this, we didn't use the Cauchy-Schwarz property of the norm in $\\Gamma$ in the estimates above as in \\cite{gaiotto}. Nevertheless, the tameness condition on the $\\Omega(\\gamma',a)$ invariants still give us the desired contraction.\n\\end{remark}\n\nObserve that, since we used admissible rays, the Stokes matrices don't change at the walls of marginal stability and we were able to treat both sides of the wall indistinctly. Thus, the functions $\\mathcal{Y}$ in Theorem \\ref{yfunctions} are smooth across the wall.\n\n\n\nLet's reintroduce the solutions in \\cite{gaiotto}. Denote by $\\mathcal{X}_e, \\mathcal{X}_m$ the solutions to the Riemann-Hilbert problem with jumps of the form $S_\\ell^{-1}$ at each BPS ray $\\ell$ with the same asymptotics and reality condition as $\\mathcal{Y}_e, \\mathcal{Y}_m$. In fact, we can see that the functions $\\mathcal{Y}$ are the analytic continuation of $\\mathcal{X}$ up until the admissible rays $r, -r$. \n\nIn a patch $U_\\alpha \\subset \\mathcal{B}'$ containing the wall of marginal stability, define the admissible ray $r$ as the ray where $\\ell_e, \\ell_m$ (or $\\ell_e, \\ell_{-m}$) collide. Since one is the analytic continuation of the other, $\\mathcal{X}$ and $\\mathcal{Y}$ differ only in a small sector in the $\\zeta$-plane bounded by the $\\ell_e, \\ell_m$ ($\\ell_e, \\ell_{-m}$) rays, for $a$ not in the wall. As $a$ approaches the wall, such a sector converges to the single admissible ray $r$. Thus, away from the ray where the two BPS rays collide, the solutions $\\mathcal{X}$ in \\cite{gaiotto} are continuous in $a$.\n\n\n\\section{Extension to the singular fibers}\\label{sfiber}\n\nIn this paper we will only consider the Pentagon example and in this section we will extend the Darboux coordinates $\\mathcal{X}_e, \\mathcal{X}_m$ obtained above to the singular locus $D \\subset \\mathcal{B}$ where one of the charges $Z_\\gamma$ approaches zero.\n\nLet $u$ be a coordinate for $\\mathcal{B} = \\mathbb{C}$. We can assume that the two bad fibers of $\\mathcal{M}$ are at $-2,2$ in the complex $u$-plane. For almost all $\\zeta \\in \\mathbb{P}^1$, the BPS rays converge in a point of the wall of marginal stability away from any bad fiber:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{raysinB.ps}\n\t\\caption{For general $\\zeta$, there is only 1 pair of rays at each fiber}\n\t\\label{raysinb}\n\\end{figure}\n\n\n\n\nIt is assumed that $\\lim_{u \\to 2} Z_{\\gamma_1}$ exists and it is nonzero. If we denote this limit by $c = |c|e^{i\\phi}$, then for $\\zeta$ such that $\\arg \\zeta \\to \\phi + \\pi$, the ray $\\ell_{\\gamma_1}$ emerging from -2 approaches the other singular point $u = 2$ (see Figure \\ref{oneside}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatoneside.ps}\n\t\\caption{The BPS rays in $\\mathcal{B}$ nearly coalesce at the singular locus}\n\t\\label{oneside}\n\\end{figure}\n\n\nWhen $\\arg \\zeta = \\phi + \\pi$, the locus $\\{ u : Z_{\\gamma}(u)\/\\zeta \\in \\mathbb{R}_-\\}$, for some $\\gamma$ such that $\\Omega(\\gamma;u) \\neq 0$ crosses $u = 2$. See Figure \\ref{otherside}.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatotherside.ps}\n\t\\caption{For $\\zeta$ in a special ray, the rays intersect $u = 2$}\n\t\\label{otherside}\n\\end{figure}\n\nAs $\\zeta$ keeps changing, the rays leave the singular locus, but near $u = 2$, the tags change due to the monodromy of $\\gamma_1$ around $u=2$. Despite this change of labels, near $u = 2$ only the rays $\\ell_{\\gamma_2}, \\ell_{-\\gamma_2}$ pass through this singular point. See Figure \\ref{finalside}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatrightside.ps}\n\t\\caption{After the critical value of $\\zeta$, the rays leave $u = 2$ and their tags change}\n\t\\label{finalside}\n\\end{figure}\n\nIn the general case of Figures \\ref{raysinb}, \\ref{oneside} or \\ref{finalside}, the picture near $u = 2$ is like in the Ooguri-Vafa case, Figure \\ref{3reg}.\n\nIn any case, because of the specific values of the invariants $\\Omega$, it is possible to analytically extend the function $\\mathcal{X}_{\\gamma_1}$ around $u = 2$. The global jump coming from the rays $\\ell_{\\gamma_2}, \\ell_{-\\gamma_2}$ is the opposite of the global monodromy coming from the Picard-Lefschetz monodromy of $\\gamma_1 \\mapsto \\gamma_1 - \\gamma_2$ (see \\eqref{piclf}). Thus, it is possible to obtain a function $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ analytic on a punctured disk on $\\mathcal{B}'$ near $u = 2$ extending $\\mathcal{X}_{\\gamma_1}$.\n\nFrom this point on, we use the original formulation of the Riemann-Hilbert problem using BPS rays as in \\cite{gaiotto}. We also use $a = Z_{\\gamma_2}(u)$ to coordinatize a disk near $u = 2$, and we label $\\{\\gamma_1, \\gamma_2\\}$ as $\\{\\gamma_m, \\gamma_e\\}$ as in the Ooguri-Vafa case. Recall that, to shorten notation, we write $\\ell_e, \\mathcal{X}_e$, etc. instead of $\\ell_{\\gamma_e}, \\mathcal{X}_{\\gamma_e}$, etc.\n\nBy our work in the previous section, solutions $\\mathcal{X}_\\gamma$ (or, taking logs, $\\Upsilon_\\gamma$) to the Riemann-Hilbert problem are continuous at the wall of marginal stability for all $\\zeta$ except those in the ray $\\ell_m = Z_{m}\/\\zeta \\in \\mathbb{R}_- = \\ell_e$ (to be expected by the definition of the RH problem). We want to extend our solutions to the bad fiber located at $a=0$. We'll see that to achieve this, it is necessary to introduce new $\\theta$ coordinates.\n\n\n\nFor convenience, we rewrite the integral formulas for the Pentagon in terms of $\\Upsilon$ as in \\cite{notes}. We will only write the part in $\\mathcal{B}_\\text{in}$, the $\\mathcal{B}_{\\text{out}}$ part is similar.\n\\begin{align}\n\\Upsilon_e(a,\\zeta) & = \\theta_e -\n\\frac{1}{4\\pi}\\left\\{ \\int_{\\ell_m} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_m^{\\text{sf}}(a,\\zeta', \\Upsilon_m) \\right] - \\int_{\\ell_{-m}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_{-m}^{\\text{sf}}(a,\\zeta', \\Upsilon_{-m})\\right] \\right\\}, \\label{upsefor}\\\\\n\\Upsilon_m(a,\\zeta) & = \\theta_m +\n\\frac{1}{4\\pi}\\left\\{ \\int_{\\ell_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right]  - \\int_{\\ell_{-e}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_{-e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{-e}) \\right] \\right\\} \\label{upsmfor}\n\\end{align}\n\n\nWe can focus only on the integrals above, so write $\\Upsilon_\\gamma(a,\\zeta) = \\theta_\\gamma + \\dfrac{1}{4\\pi} \\Phi_\\gamma(a,\\zeta)$, for $\\gamma \\in \\{\\gamma_m, \\gamma_e\\}$. To obtain the right gauge transformation of the torus coordinates $\\theta$, we'll split the integrals above into four parts and then we'll show that two of them define the right change of coordinates (in $\\mathcal{B}_{\\text{in}}$, and a similar transformation for $\\mathcal{B}_{\\text{out}}$) that simplify the integrals and allow an extension to the singular fiber.\n\nBy Theorem \\ref{yfunctions}, both $\\Upsilon_m, \\Upsilon_e$ satisfy the ``reality condition'', which expresses a symmetry in the behavior of the complexified coordinates $\\Upsilon$:\n \\begin{equation}\\label{corresp}\n \\overline{\\Upsilon_\\gamma(a, \\zeta)} = \\Upsilon_\\gamma \\left(a, -1\/\\overline{\\zeta}\\right), \\qquad a \\neq 0\n \\end{equation} \n If we write as $\\Upsilon_0$ (resp. $\\Upsilon_\\infty$) the asymptotic of this function as $\\zeta \\to 0$ (resp. $\\zeta \\to \\infty$) so that\n \\[ \\Upsilon_0 = \\theta + \\frac{1}{4\\pi} \\Phi_0,  \\]\n for a suitable correction $\\Phi_0$. A similar equation holds for the asymptotic as $\\zeta \\to \\infty$. By the asymptotic condition \\ref{asymptotic}, $\\Phi_0$ is imaginary. \n\nCondition \\eqref{corresp} also shows that $\\Phi_0 = - \\Phi_\\infty$. This and the fact that $\\Phi_0$ is imaginary give the reality condition\n \\begin{equation}\\label{reali}\n \\Upsilon_0 = \\overline{\\Upsilon_\\infty}\n \\end{equation}\n \nSplit the integrals in \\eqref{upsmfor} into four parts as in \\eqref{integs}. For example, if we denote by $\\zeta_e := -a\/|a|$, the intersection of the unit circle with the $\\ell_e$ ray, then\n\\begin{align}\n& \\int_{\\ell_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) = \\notag\\\\\n& -\\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\notag\\\\\n& + \\int_0^{\\zeta_e} \\frac{2 d\\zeta'}{\\zeta'-\\zeta} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right)  + \\int_{\\zeta_e}^{\\zeta_e \\infty} 2d\\zeta' \\left\\{\\frac{1}{\\zeta'-\\zeta} -\\frac{1}{\\zeta'}\\right\\} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\label{4ints}\n\\end{align}\n\nWe consider the first two integrals apart from the rest. If we take the limit $a \\to 0$ the exponential decay in $\\mathcal{X}_e^{\\text{sf}}$:\n\\[ \\exp\\left( \\frac{\\pi R a}{\\zeta'} + \\pi R \\zeta' \\overline{a} \\right)\\]\nvanishes and the integrals are no longer convergent. \n \n By combining the two integrals with their analogues in the $\\ell_{-e}$ ray we obtain:\n\\begin{align}\n & -\\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\notag\\\\\n &  \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - {\\mathcal{X}_e^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_e) \\right) - \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - {\\mathcal{X}_e^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_e) \\right)  \\label{gen4int}\n \\end{align}\nThe parametrization in the first pair of integrals is of the form $\\zeta' = t\\zeta_e$, and in the second pair $\\zeta' = -t\\zeta_e$. Making the change of variables $\\zeta' \\mapsto 1\/\\zeta'$, we can pair up these integrals in a more explicit way as:\n\\begin{align}\n & -\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a})  \\right) \\right] \\right. \\notag\\\\\n & \\left. + \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) - i\\Upsilon_e(a,\\frac{1}{t} e^{i\\arg a})  \\right) \\right] \\right\\} \\notag \\\\\n &  + \\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-\\frac{1}{t} e^{i\\arg a}) \\right) \\right] \\right. \\notag\\\\\n & \\left. + \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right) \\right]  \\right\\} \\label{4intreal}\n \\end{align}\nBy \\eqref{corresp}, the integrands come in conjugate pairs. Therefore, we can rewrite \\eqref{4intreal} as:\n \\begin{align*}\n -2\\int_0^1 \\frac{dt}{t} \\text{Re } \\left\\{\\vphantom{\\int_0^1}\\right. & \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right) \\right] -  \\\\\n &  \\left. \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a})  \\right) \\right] \\right\\}\n \\end{align*}\n \\begin{equation}\\label{simpl4}\n = -2\\int_0^1 \\frac{dt}{t} \\log \\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a})  \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right)} \\right|\n \\end{equation}\n \n\nObserve that \\eqref{simpl4} itself suggest the correct transformation of the $\\theta$ coordinates that fixes this. Indeed, for a fixed $a \\neq 0$ and $\\theta_e$, let $Q$ be the map\n\\begin{equation*}\nQ(\\theta_m) =  \\theta_m + \\psi(a,\\theta),\n\\end{equation*}\nwhere\n \\begin{align}\n \\psi_{\\text{in}}(a,\\theta)& = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a})  \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a})  \\right)} \\right| \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| \\label{newmp}\n \\end{align}\n for $a \\in \\mathcal{B}_{\\text{in}}$. For $a \\in \\mathcal{B}_{\\text{out}}$ where the wall-crossing is of type I, let $\\varphi = \\arg (Z_{\\gamma_e + \\gamma_m}(a))$, with $\\zeta'= -t e^{i\\varphi}$ parametrizing the $\\ell_{e + m}$ ray:\n \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a})  \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a})  \\right)} \\right| \\right. \\notag\\\\\n & + \\left. \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |Z_{\\gamma_e + \\gamma_m}| \\left(t^{-1} + t \\right) +i\\Upsilon_{e +m}(a,-te^{i\\arg \\varphi})  \\right)}{1 - \\exp\\left( -\\pi R |Z_{\\gamma_e + \\gamma_m}| \\left(t^{-1} + t \\right) -i\\Upsilon_{e+m}(a,te^{i\\arg \\varphi})  \\right)} \\right|  \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{e+m}\\right](-te^{i\\varphi})}{1 - \\left[\\mathcal{X}_{-e-m}\\right](te^{i\\varphi})} \\right| \\right\\} \\label{outmp}\n \\end{align}\n \n Similarly, for wall-crossing of type II, $\\varphi = \\arg (Z_{\\gamma_{-e} + \\gamma_m}(a))$, with $\\zeta'= -t e^{i\\varphi}$ for the $\\ell_{-e + m}$ ray:\n  \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a})  \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a})  \\right)} \\right| \\right. \\notag\\\\\n & + \\left. \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |Z_{\\gamma_{-e} + \\gamma_m}| \\left(t^{-1} + t \\right) +i\\Upsilon_{-e +m}(a,-te^{i\\arg \\varphi})  \\right)}{1 - \\exp\\left( -\\pi R |Z_{\\gamma_{-e} + \\gamma_m}| \\left(t^{-1} + t \\right) -i\\Upsilon_{-e+m}(a,te^{i\\arg \\varphi})  \\right)} \\right|  \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{-e+m}\\right](-te^{i\\varphi})}{1 - \\left[\\mathcal{X}_{e-m}\\right](te^{i\\varphi})} \\right| \\right\\} \\label{outmp2}\n \\end{align}\n \n As $a$ approaches the wall of marginal stability $W$, $\\arg a \\to \\varphi$. We need to show the following\n\\begin{lemma}\nThe two definitions $\\psi_{\\text{in}}$ and $\\psi_{\\text{out}}$ coincide at the wall of marginal stability.\n\\end{lemma} \n\\begin{proof}\n  First let $a$ approach $W$ from the ``in'' region, so we're using definition \\eqref{newmp}. Start with the pair of functions $(\\mathcal{X}_e, \\mathcal{X}_m)$ in the $\\zeta$-plane and let $\\widetilde{\\mathcal{X}}_e$ denote the analytic continuation of $\\mathcal{X}_e$. See Figure \\ref{jumpxe}. When they reach the $\\ell_e$ ray, $\\mathcal{X}_e$ jumped to $\\mathcal{X}_e(1-\\mathcal{X}_m)$ by \\eqref{kjump} and \\eqref{invjmp}. Thus $\\mathcal{X}_e = \\widetilde{\\mathcal{X}}_e(1-\\mathcal{X}_m)$ along the $\\ell_e$ ray.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{2raysafterwalljumps2.eps}\n\t\\caption{Jump of $\\mathcal{X}_e$}\n\t\\label{jumpxe}\n\\end{figure}\n\nTherefore, \n\\begin{equation*}\n\\psi_{\\text{in}}(a,\\theta) = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e(1-\\mathcal{X}_m)\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}(1-\\mathcal{X}_m)^{-1}\\right](te^{i\\arg a})} \\right|\n\\end{equation*}\n\nNow starting from the ``out'' region, and focusing on the wall-crossing of type I for the moment, we start with the pair $(\\mathcal{X}_e, \\mathcal{X}_m)$ as before. This time, $\\mathcal{X}_e$ at the $\\ell_e$ ray has not gone to any jump yet. See Figure \\ref{jumpxem}. Only $\\mathcal{X}_{e+m}$ undergoes a jump at the $\\ell_{e+m}$ ray and it is of the form $\\mathcal{X}_{e+m} \\mapsto \\mathcal{X}_{e+m}(1-\\mathcal{X}_{e})^{-1}$.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{3raysbeforewalljumps2.eps}\n\t\\caption{Only $\\mathcal{X}_{e+m}$ has a jump}\n\t\\label{jumpxem}\n\\end{figure}\n\nWhen $a$ hits the wall $W$, $\\varphi = \\arg a$ and the integrals are taken over the same ray. Thus, we can combine the logs and obtain:\n\n \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) &  = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{e+m}(1-\\mathcal{X}_{e})^{-1}\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e-m}(1-\\mathcal{X}_{e})\\right](te^{i\\arg a})} \\right| \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e(1-\\mathcal{X}_m)\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}(1-\\mathcal{X}_m)^{-1}\\right](te^{i\\arg a})} \\right|\n \\end{align}\n \n\\noindent and the two definitions coincide. For the wall-crossing of type II the proof is entirely analogous.\n\n \n\n\\end{proof}\n \n \\begin{theorem}\\label{homeps}\n $Q$ is a reparametrization in $\\theta_m$; that is, a diffeomorphism of $\\mathbb{R}\/2\\pi \\mathbb{Z}$.\n \\end{theorem}\n \\begin{proof}\n To show that $Q$ is injective, it suffices to show that $\\left|\\frac{\\partial \\psi}{\\partial \\theta_m}\\right| < 1$. We will show this in the $\\mathcal{B}_{\\text{in}}$ region. The proof for the $\\mathcal{B}_{\\text{out}}$ region is similar.\n \n To simplify the calculations, write\n  \\begin{equation}\\label{psii}\n  \\psi(a,\\theta) = 2\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1-Cf(\\theta_m)}{1-Cg(\\theta_m)}\n     \\right|\n   \\end{equation}\n  for functions $f, g$ of the form $e^{i\\Upsilon_\\gamma}$ for different choices of $\\gamma$ (they both depend on other parameters, but they're fixed here) and a factor $C$ of the form\n  \\[ C = \\exp\\left( -\\pi R |a| (t^{-1} + t)\\right) \\]\n  Now take partials in both sides of \\eqref{psii} and bring the derivative inside the integral. After an application of the chain rule we get the estimate\n  \\[ \\left| \\frac{\\partial \\psi}{\\partial \\theta_m} \\right| \\leq 2\\int_0^1 \\frac{dt}{t} |C| \\left\\{ \\frac{|f||\\frac{\\partial \\Upsilon_e(t)}{\\partial \\theta_m}|}{|1-Cf|} + \\frac{|g||\\frac{\\partial \\Theta_e(-t)}{\\partial \\theta_m}|}{|1-Cg|} \\right\\} \\] \n By the estimates in \\cite[\\S 3.2]{rhprob}, $\\left|\\frac{\\partial \\Upsilon_e}{\\partial \\theta_m}\\right| < 1$. In \\cite[Lemma 3.2]{rhprob}, we show that $|f|, |g|$ can be bounded by 2. The part $C$ has exponential decay so if $R$ is big enough we can bound the above by 1 and injectivity is proved. For surjectivity, just observe that $\\psi(\\theta_m + 2\\pi) = \\psi(\\theta_m)$, so $Q(\\theta_m + 2\\pi) = \\theta_m + 2\\pi$. \n \\end{proof}\n \nWith respect to the new coordinate $\\theta'_m$, the functions $\\Upsilon_e, \\Upsilon_m$ satisfy the equation:\n\n\\begin{align}\n\\Upsilon_e(a,\\zeta)  = \\theta_e +\n\\frac{1}{4\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_e, \\gamma' \\right\\rangle   & \\int_{\\gamma'} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right]  \\label{inteq1} \\\\\n\\Upsilon_m(a,\\zeta)  = \\theta'_m +\n\\frac{1}{2\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_m, \\gamma' \\right\\rangle  \\left\\{ \\vphantom{\\int_0^b} \\right.\n& \\int_{0}^{b'} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] + \\notag \\\\\n& \\left.  \\int_{b'}^{b' \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right\\} \\label{inteq2} ,\n\\end{align}\nfor $b'$ the intersection of the unit circle with the $\\ell_{\\gamma'}$ ray. The $\\Omega(\\gamma';a)$ jump at the wall, but in the Pentagon case, the sum is finite.\n\nIn order to show that $\\Upsilon$ converges to some function, even at $a = 0$, observe that the integral equations in \\eqref{inteq1} and \\eqref{inteq2} still make sense at the singular fiber, since in the case of \\eqref{inteq1}, $\\lim_{a \\to 0} Z_m = c \\neq 0$ and the exponential decay is still present, making the integrals convergent. In the case of \\eqref{inteq2}, the exponential decay is gone, but the different kernel makes the integral convergent, at least for $\\zeta \\in \\mathbb{C}^\\times$. The limit function $\\lim_{a \\to 0} \\Upsilon$ should be then a solution to the integral equations obtained by recursive iteration, as in \\cite[\\S 3]{rhprob}.\n\nWe have to be specially careful with the Cauchy integral in \\eqref{inteq2}. It will be better to obtain each iteration $\\Upsilon^{(\\nu)}_m$ when $|a| \\to 0$ by combining the pair of rays $\\ell_{\\gamma'}, \\ell_{-\\gamma'}$ into a single line $L_{\\gamma'}$, where in the case of the Pentagon, $\\gamma'$ can be either $\\gamma_e$ or $\\gamma_{e+m}$, depending on the side of the wall we're at. We formulate a boundary problem over each infinite curve $L_{\\gamma'}$ as in \\S \\ref{altrh}. As in the Ooguri-Vafa case, the jump function\\footnote{Since we do iterations of boundary problems, we abuse notation and use simply $G(\\zeta)$ where it should be $G^{(\\nu)}(\\zeta)$. This shouldn't cause any confusion, as our main focus in this section is how to obtain \\emph{any} iteration of $\\mathcal{X}_m$} $G(\\zeta)$ has discontinuities of the first kind at 0 and $\\infty$, but we also have a new difficulty: For $\\theta_e$ close to 0, the jump function $G(\\zeta) = 1-e^{i\\Upsilon^{(\\nu - 1)}_{\\gamma'}}(\\zeta)$ may be 0 for some values of $\\zeta$.\n\nSince the asymptotics of $\\Upsilon^{(\\nu)}_{e}$ as $\\zeta \\to 0$ or $\\zeta \\to \\infty$ are $\\theta_e \\pm i\\phi_e \\neq 0$, the jump function $G(\\zeta)$ can only attain the 0 value inside a compact interval away from 0 or $\\infty$, hence these points are isolated in $L_{\\gamma'}$. By the symmetry relation expressed in Lemma \\ref{corresp}, the zeroes of $G(\\zeta)$ come in pairs in $L_{\\gamma'}$ and are of the form $\\zeta_k, -1\/\\overline{\\zeta_k}$. By our choice of orientation for $L_{\\gamma'}$, one of the jumps is inverted so that $G(\\zeta)$ has only zeroes along $L_{\\gamma'}$ and no poles.\n\nThus, as in \\S \\ref{altrh}, we have a Riemann-Hilbert problem of the form\\footnote{To simplify notation, we omit the iteration index $\\nu$ in the Riemann-Hilbert problem expressed. By definition, $\\mathcal{X}_m = \\mathcal{X}^{\\text{sf}}_m X_m$, for \\emph{any} iteration $\\nu$}\n\\begin{equation}\\label{rhpent}\nX_m^+(\\zeta) = G(\\zeta) X_m^-(\\zeta)\n\\end{equation}\n\nIn \\cite[Lemma 4.2]{rhprob}, we show that the solutions of \\eqref{rhpent} exist and are unique, given our choice of kernel in \\eqref{inteq2}. We thus obtain each iteration $\\Upsilon_m^{(\\nu)}$ of \\eqref{inteq2}. Moreover, since by \\cite{rhprob}, $\\mathcal{X}_m^+ = 0$ at points $\\zeta$ in the $L_e$ ray where $G(\\zeta) = 0$, $\\Upsilon_m^{(\\nu)+}$ has a logarithmic singularity at such points.\n\n\n\n\n\\subsection{Estimates and a new gauge transformation}\n\n\n\n\n\n\n\n\n\n\n\n\n\nAs we've seen in the Ooguri-Vafa case, we expect our solutions $\\lim_{a \\to 0} \\Upsilon$ to be unbounded in the $\\zeta$ variable.\nDefine a Banach space $\\mathrsfs{X}$ as the completion under the sup norm of the space of functions $\\Phi: \\mathbb{C}^\\times \\times \\mathbb{T} \\times U \\to \\mathbb{C}^{2n}$  that are piecewise holomorphic on $\\mathbb{C}^\\times$, smooth on $\\mathbb{T} \\times U$, for $U$ an open subset of $\\mathcal{B}$ containing $0$ and such that \\eqref{inteq1}, \\eqref{inteq2} hold.\n\nLike in the Ooguri-Vafa case, let $a \\to 0$ fixing $\\arg a$. We will later get rid of this dependence on $\\arg a$ with another gauge transformation of $\\theta_m$. The following estimates on $\\Upsilon^{(\\nu)}$ will clearly give us that the sequence converges to some limit $\\Upsilon^{(\\nu)}$.\n\n\\begin{lemma}\\label{estbadfib}\nIn the Pentagon case, at the bad fiber $a = 0$:\n\\begin{align}\n\\Upsilon_e^{(\\nu + 1)} & = \\Upsilon_e^{(\\nu)} + O\\left( e^{-2\\pi \\nu R |Z_m|} \\right), \\hspace{5 mm} \\nu \\geq 2 \\label{elstm}\\\\\n\\Upsilon_m^{(\\nu + 1)} & = \\Upsilon_m^{(\\nu)} + O\\left( e^{-2\\pi \\nu R |Z_m|} \\right), \\hspace{5 mm} \\nu \\geq 1 \\label{mgstm}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nAs before, we prove this by induction. Note that $\\Upsilon^{(1)}_m = \\Upsilon^{\\text{OV}}$, the extension of the Ooguri-Vafa case obtained in \\eqref{xmnice}, and $\\Upsilon^{(1)}_m$ differs considerably from $\\theta_m$ because of the $\\log \\zeta$ term. Hence the estimates cannot start at $\\nu = 0$. Because of this reason, $\\Upsilon^{(2)}_e$ differs considerably from $\\Upsilon^{(1)}_e$ since this is the first iteration where $\\Upsilon^{(1)}_m$ is considered.\n\nLet $\\nu = 1$. The integral equations for $\\Upsilon_e$ didn't change in this special case. By Lemma 3.3 in \\cite{rhprob}, we have for the general case:\n\n\\begin{equation}\\label{frappx}\n\\Upsilon^{(1)}_e = \\theta_e + \\sum_{\\gamma'} \\Omega(\\gamma',a) \\left\\langle \\gamma_e, \\gamma' \\right\\rangle \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{4\\pi i \\sqrt{R |Z_{\\gamma'}|}}\\frac{\\zeta_{\\gamma'} + \\zeta}{\\zeta_{\\gamma'} - \\zeta} e^{i\\theta_{\\gamma'}} + O\\left( \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{R}\\right)\n\\end{equation}\nwhere $\\zeta_{\\gamma'} = -\\frac{Z_{\\gamma'}}{|Z_{\\gamma'}|}$ is the saddle point for the integrals in \\eqref{inteq1}, and $\\zeta$ is not $\\zeta_{\\gamma'}$. Note that there is no divergence if $\\zeta \\to 0$ or $\\zeta \\to \\infty$. If $\\zeta = \\zeta_{\\gamma'}$, again by Lemma 3.3 in \\cite{rhprob}, we obtain estimates as in \\eqref{frappx} except for the $\\sqrt{R}$ terms in the denominator.\n\nIn any case, for the Pentagon, the $\\gamma'$ in \\eqref{frappx} are only $ \\gamma_{\\pm m},  \\gamma_{\\pm (e+m)}$, depending on the side of the wall of marginal stability. At $a = 0$, $Z_{e+m} = Z_m$, so \\eqref{frappx}  gives that $\\log[1 - e^{i \\Upsilon^{(1)}_e}] = \\log[1 - e^{i\\theta_e}] + O(e^{-2\\pi R |Z_m|})$ along the $\\ell_e$ ray, and a similar estimate holds for $\\log[1 - e^{-i \\Upsilon^{(1)}_e}]$ along the $\\ell_{-e}$ ray. Plugging in this in \\eqref{inteq2}, we get \\eqref{mgstm} for $\\nu = 1$.\n\nFor general $\\nu$, a saddle point analysis on $\\Upsilon^{(\\nu)}_e$ can still be performed and obtain as in \\eqref{frappx}:\n\\begin{equation}\\label{appgnu}\n\\Upsilon^{(\\nu+1)}_e = \\theta_e + \\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i\\Upsilon^{(\\nu)}_m(\\zeta_m)} -  \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i\\Upsilon^{(\\nu)}_m(-\\zeta_m)} \\right\\} + O\\left( \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{R}\\right),\n\\end{equation}\nfrom one side of the wall. On the other side (for type I) it will contain the extra terms\n\\begin{equation}\n\\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i(\\Upsilon^{(\\nu)}_m(\\zeta_m) + \\Upsilon^{(\\nu)}_e(\\zeta_m))} -  \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i(\\Upsilon^{(\\nu)}_m(-\\zeta_m) - \\Upsilon^{(\\nu)}_e(-\\zeta_m))} \\right\\}.\n\\end{equation}\n\\noindent Observe that for this approximation we only need $\\Upsilon^{(\\nu)}$ at the point $\\zeta_m$. By the previous part, for $\\nu = 2$,\n\\[  e^{i\\Upsilon^{(2)}_m(\\zeta_m)} = e^{i\\Upsilon^{(1)}_m(\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\]\nThus, for $\\nu = 2$,\n\\begin{align}\n\\Upsilon^{(3)}_e & = \\theta_e + \\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i\\Upsilon^{(1)}_m(\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\right. \\notag\\\\\n& -  \\left. \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i\\Upsilon^{(1)}_m(-\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\right\\} + O\\left( R^{1\/2}\\right)\\notag\\\\\n& = \\Upsilon^{(2)}_e + O\\left( e^{-4\\pi R|Z_m|}\\right) \\label{appgnu2}\n\\end{align}\nand similarly in the other side of the wall. For general $\\nu$, the same arguments show that \\eqref{elstm}, \\eqref{mgstm} hold after the appropriate $\\nu$.\n\\end{proof}\n\n\n\n\nThere is still one problem: the limit of $\\widetilde{\\mathcal{X}}_m$ we obtained as $a \\to 0$ for the analytic continuation of $\\mathcal{X}_m$ was only along a fixed ray $\\arg a = $ constant. To get rid of this dependence, it is necessary to perform another gauge transformation on the torus coordinates $\\theta$. Recall that we are restricted to the Pentagon case. Let $a \\to 0$ fixing $\\arg a$. Let $\\zeta_{\\gamma}$ denote $Z_{\\gamma}\/|Z_{\\gamma}|$. In particular, $\\zeta_e = a\/|a|$ and this remains constant since we're fixing $\\arg a$. Also, $\\zeta_m = Z_m\/|Z_m|$ and this is independent of $\\arg a$ since $Z_m$ has a limit as $a \\to 0$. The following lemma will allow us to obtain the correct gauge transformation.\n \n\\begin{lemma}\nFor the limit $\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a=0}$ obtained above, its imaginary part is independent of the chosen ray $\\arg a = c$ along which $a \\to 0$.\n\\end{lemma}\n\\begin{proof}\nLet $\\widetilde{\\Upsilon}_m$ denote the analytic continuation of $\\Upsilon_m$ yielding $\\widetilde{\\mathcal{X}}_m$. Start with a fixed value $\\arg a \\equiv \\rho_0$, for $\\rho_0$ different from $\\arg Z_m(0), \\arg (-Z_m(0))$. For another ray $\\arg a \\equiv \\rho$, we compute $\\left. \\Upsilon_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho}} - \\left. \\Upsilon_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho_0}}$ (without analytic continuation for the moment).\n\nThe integrals in \\eqref{inteq2} are of two types. One type is of the form\n\\begin{equation}\\label{1sttype}\n\\int_0^{\\zeta_{\\pm e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm e}(\\zeta')} \\right] +  \\int_{\\zeta_{\\pm e}}^{\\zeta_{\\pm e}\\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm e}(\\zeta')} \\right]\n\\end{equation}\nThe other type appears only in the outside part of the wall of marginal stability. Since $Z : \\Gamma \\to \\mathbb{C}$ is a homomorphism, $Z_{\\gamma_e + \\gamma_m} = Z_{\\gamma_e} + Z_{\\gamma_m}$. At $a = 0$, $Z_e = a = 0$, so $Z_{e+m} = Z_m$. Hence, $\\ell_m = \\ell_{e+m}$ at the singular fiber. This second type of integral is thus of the form\n\\begin{equation}\\label{2ndtype}\n\\int_0^{\\zeta_{\\pm m}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm (e+m)}(\\zeta')} \\right] +  \\int_{\\zeta_{\\pm m}}^{\\zeta_{\\pm m}\\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm (e+m)}(\\zeta')} \\right]\n\\end{equation}\nSince the $\\ell_m$ stays fixed at $a = 0$ independently of $\\arg a$, \\eqref{2ndtype} does not depend on $\\arg a$, so this has a well-defined limit as $a \\to 0$. We should focus then only on integrals of the type \\eqref{1sttype}. For a different $\\arg a$, $\\zeta_e$ changes to another point $\\widetilde{\\zeta}_e$ in the unit circle. See Figure \\ref{zetilde}. The paths of integration change accordingly. We have two possible outcomes: either $\\zeta$ lies outside the sector determined by the two paths, or $\\zeta$ lies inside the region.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{argadependence.ps}\n\t\\caption{As $\\arg a$ changes, the paths of integration change}\n\t\\label{zetilde}\n\\end{figure}\n\nIn the first case ($\\zeta_1$ on Figure \\ref{zetilde}), the integrands\n\\begin{equation}\\label{2kernl}\n \\frac{\\log[1-e^{i\\Upsilon_{\\pm e}(\\zeta')}]}{\\zeta'-\\zeta}, \\hspace{5 mm} \\frac{\\zeta\\log[1-e^{i\\Upsilon_{\\pm e}(\\zeta')}]}{\\zeta'(\\zeta'-\\zeta)}\n\\end{equation}\nare holomorphic on $\\zeta'$ in the sector between the two paths. By Cauchy's formula, the difference between the two integrals is just the integration along a path $C_{\\pm e}$ between the two endpoints $\\zeta_{\\pm e}, \\widetilde{\\zeta}_{\\pm e}$. If $f(s)$ parametrizes the path $C_e$, let $C_{-e} = -1\/\\overline{f(s)}$. The orientation of $C_{e}$ in the contour containing $\\infty$ is opposite to the contour containing 0. Similarly for $C_{-e}$. Thus, the difference of $\\Upsilon_m$ for these two values of $\\arg a$ is the integral along $C_{e}, C_{-e}$ of the difference of kernels \\eqref{2kernl}, namely:\n\\begin{equation}\\label{realdiff}\n\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i\\Upsilon_e(\\zeta')}] - \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i\\Upsilon_e(\\zeta')}]\n\\end{equation} \n\nEven if $e^{i\\Upsilon_e(\\zeta')} = 1$ for $\\zeta'$ in the contour, the integrals in \\eqref{realdiff} are convergent, so this is well-defined for any values of $\\theta_e \\neq 0$. By symmetry of $C_e, C_{-e}$ and the reality condition \\eqref{corresp}, the second integral is the conjugate of the first one. Thus \\eqref{realdiff} is only real.\n\nWhen $\\zeta$ hits one of the contours, $\\zeta$ coincides with one of the $\\ell_e$ or $\\ell_{-e}$ rays, for some value of $\\arg a$. The contour integrals jump since $\\zeta$ lies now inside the contour ($\\zeta_2$ in Figure \\ref{zetilde}). The jump is by the residue of the integrands \\eqref{2kernl}. This gives the jump of $\\mathcal{X}_m$ that the analytic continuation around $a = 0$ cancels. Therefore, only the real part of $\\Upsilon_m$ depends on $\\arg a$.\n\\end{proof}\n\nBy the previous lemma, $\\left. \\widetilde{\\Upsilon}_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho}} - \\left. \\widetilde{\\Upsilon}_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho_0}}$ is real and is given by \\eqref{realdiff}. Define then a new gauge transformation:\n\\begin{equation}\\label{fingauge}\n\\widetilde{\\theta}_m = \\theta'_m - \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i\\Upsilon_e(\\zeta')}] + \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i\\Upsilon_e(\\zeta')}] \\right\\}\n\\end{equation}\n\nThis eliminates the dependence on $\\arg a$ for the limit $\\left. \\widetilde{\\mathcal{X}}_m \\right|_{a=0}$. As we did in \\S \\ref{clasov} in Theorem \\ref{mprtom}, we can extend the torus fibration $\\mathcal{M}'$ by gluing a $S^1$-fiber bundle of the form $D \\times (0, 2\\pi) \\times S^1$ for $D$ a disk around $a = 0$, $\\theta_e \\in (0,2\\pi)$ and $\\widetilde{\\theta}_m$ the new coordinate of the $S^1$ fibers. Using Taub-NUT space as a local model for this patch, the trivial $S^1$ bundle can be extended to $\\theta_e = 0$ where the fiber degenerates into a point (nevertheless, in Taub-NUT coordinates the space is still locally isomorphic to $\\mathbb{C}^2$). Since $\\widetilde{\\mathcal{X}}_m \\equiv 0$ if $\\theta_e = 0$ as in \\S \\ref{clasov}, in this new manifold $\\mathcal{M}$ we  thus obtain a well defined function $\\widetilde{\\mathcal{X}}_m$.\n\n\n\n\\subsection{Extension of the derivatives}\\label{exderv}\n\\index{Extension of the derivatives@\\emph{Extension of the derivatives}}%\n\nSo far we were able to extend the functions $\\mathcal{X}_e, \\widetilde{\\mathcal{X}}_m$ to $\\mathcal{M}$. Unfortunately, we can no longer bound uniformly on $\\nu$ the derivatives of $\\widetilde{\\mathcal{X}}_m$ near $a = 0$, so the Arzela-Ascoli arguments no longer work here. Since there's no difference on the definition of $\\mathcal{X}_e$ at $a = 0$ from that of the regular fibers, this function extends smoothly to $a = 0$.\n\nWe have to obtain the extension of all derivatives of $\\widetilde{\\mathcal{X}}_m$ directly from its definition. It suffices to extend the derivatives of $\\mathcal{X}_m$ only, as the analytic continuation doesn't affect the symplectic form $\\varpi(\\zeta)$ (see below).\n\n\\begin{lemma}\n$\\log \\mathcal{X}_m$ extends smoothly to $\\mathcal{M}$, for $\\theta_e \\neq 0$.\n\\end{lemma}\n\\begin{proof}\nFor convenience, we rewrite $\\Upsilon_m$ with the final magnetic coordinate $\\widetilde{\\theta_m}$:\n\\begin{align*}\n\\Upsilon_m & = \\widetilde{\\theta_m} + \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i \\Upsilon_e(\\zeta')}] -\n\\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i \\Upsilon_e(\\zeta')}] \\right\\}\\\\\n& + \\frac{1}{2\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_m, \\gamma' \\right\\rangle  \\left\\{ \\vphantom{\\int_0^b}  \\int_{0}^{\\zeta_{\\gamma'}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right. +  \\\\\n& \\left.  \\int_{\\zeta_{\\gamma'}}^{\\zeta_{\\gamma'} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right\\}\n\\end{align*}\nwhere $e^{i \\Upsilon_e(\\zeta')}$ is evaluated only at $a = 0$. For $\\gamma'$ of the type $\\pm \\gamma_e \\pm \\gamma_m$, $\\mathcal{X}_{\\gamma'}$ and its derivatives still have exponential decay along the $\\ell_{\\gamma'}$ ray, so these parts in $\\Upsilon_m$ extend to $a =0$ smoothly. It thus suffices to extend only\n\n\\begin{align}\n\\Upsilon_m & = \\widetilde{\\theta_m} + \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i \\Upsilon_e(\\zeta')}] -\n\\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i \\Upsilon_e(\\zeta')}] \\right. \\notag\\\\\n& +  \\int_{0}^{\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{e}) \\right] + \\int_{\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{e}) \\right] \\notag\\\\\n& -\\left. \\int_{0}^{-\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - {\\mathcal{X}_{e}^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_{e}) \\right] -  \\int_{-\\zeta_{e}}^{-\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - {\\mathcal{X}_{e}^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_{e}) \\right]\\right\\}\\label{upsmcompl}\n\\end{align}\ntogether with the semiflat part $\\pi R \\frac{Z_m}{\\zeta} + \\pi R \\zeta \\overline{Z_m}$, which we assume is as in the Generalized Ooguri-Vafa case, namely:\n\\begin{equation}\\label{xmcomplet}\n \\mathcal{X}_m =  \\exp\\left( \\frac{-i R }{2\\zeta}(a\\log a - a + f(a)) + i \\Upsilon_m + \\frac{i\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} + \\overline{f(a)} )\\right)\n\\end{equation}\nfor a holomorphic function $f$ near $a = 0$ and such that $f(0) \\neq 0$. The derivatives of the terms involving $f(a)$ clearly extend to $a = 0$, so we focus on the rest, as in \\S \\ref{c1ext}.\n\nWe show first that $\\dfrac{\\partial \\log \\mathcal{X}_m}{\\partial_{\\theta_e}}, \\dfrac{\\partial \\log \\mathcal{X}_m}{\\partial_{\\theta_m}}$ extend to $a = 0$. Since there is no difference in the proof between electric or magnetic coordinates, we'll denote by $\\partial_\\theta$ a derivative with respect to any of these two variables.\n\nWe have:\n\\begin{align*}\n\\frac{\\partial}{\\partial \\theta}\\log \\mathcal{X}_m & = \\frac{-i}{2\\pi} \\left\\{ \\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\frac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}} \\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} - \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\frac{e^{-i \\Upsilon_e(\\zeta')}}{1-e^{-i \\Upsilon_e(\\zeta')}} \\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right.\\\\\n& +  \\int_{0}^{\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta}\\\\\n& \\left. +  \\int_{0}^{-\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{-\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right\\}\n\\end{align*}\n\n\nwhen $a \\to 0$, $ \\dfrac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} \\to \\dfrac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}}$. The integrals along $C_e$ and $C_{-e}$ represent a difference of integrals along the contour in the last integrals and a fixed contour, as in Figure \\ref{zetilde}. Thus, when $a = 0$,\n\n\\begin{align}\n\\left. 2\\pi i \\frac{\\partial}{\\partial \\theta}\\log \\Upsilon_m\\right|_{a =0} & =  \\int_{0}^{b} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{b}^{b \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\notag\\\\\n& \\left. +  \\int_{0}^{-b} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{-b}^{-b \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right\\} \\label{polec}\n\\end{align}\nfor a fixed point $b$ in the unit circle, independent of $a$. If $\\Upsilon_e(\\zeta') = 1$ for a point $c$ in the line $L$ passing through the origin and $b$, then as seen in \\cite[Lemma 4.2]{rhprob}, the function $\\mathcal{X}_m$ develops a zero on the right side of such line. Nevertheless, the analytic continuation $\\widetilde{\\mathcal{X}}_m$ around $a = 0$ introduces a factor of the form $(1 - \\mathcal{X}_e)^{-1}$ when $a$ changes from region III to region I in Figure \\ref{3reg}, so the pole at $c$ on the right side of $L$ for the derivative $\\dfrac{\\partial}{\\partial \\theta}\\log \\Upsilon_m$ coming from the integrand in \\eqref{polec} is canceled by analytic continuation. Hence, the integrals are well defined and thus the left side has an extension to $a = 0$.\n\nNow, for the partials with respect to $a, \\overline{a}$, there are two different types of dependence: one is the dependence of the contours, the other is the dependence of the integrands. The former dependence is only present in \\eqref{upsmcompl}, as the contours in Figure \\ref{zetilde} change with $\\arg a$. A simple application of the Fundamental Theorem of Calculus in each integral in \\eqref{upsmcompl} gives that this change is:\n\n\\begin{align*}\n\\left. -2\\pi i  \\frac{\\partial}{\\partial \\arg a}\\log \\Upsilon_m\\right|_{a =0}  & = \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}]  - \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}]\\\\\n& - \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}] + \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}] = 0,\n\\end{align*}\nwhere we again used the fact that the integrals along $C_e$ and $C_{-e}$ represent the difference between the integrals in the other pairs with respect to two different rays, one fixed. By continuity on parameters, the terms are still 0 if $\\Upsilon_e(\\zeta_e) = 0$. Compare this with \\eqref{dap1}, where we obtained this explicitly.\n\nThen there is the dependence on $a, \\overline{a}$ on the integrands and the semiflat part. Focusing on $a$ only, we take partials on $\\log \\mathcal{X}_m$ in \\eqref{xmcomplet} (ignoring constants and parts that clearly extend to $a = 0$). This is:\n\n\\begin{equation}\\label{logadiv}\n\\frac{\\log a}{\\zeta} + \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} + \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\n\\end{equation}\n\nThis is the equivalent of \\eqref{last3} in the general case. In the limit $a \\to 0$, we can do an asymptotic expansion of $\\dfrac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}} = \\dfrac{e^{i \\Upsilon_e(0)}}{1-e^{i \\Upsilon_e(0)}} + O(\\zeta')$. Clearly when we write this expansion in \\eqref{logadiv}, the only divergent term at $a = 0$ is the first degree approximation in the integral. Thus, we can focus on that and assume that the $\\dfrac{\\mathcal{X}_e}{1-\\mathcal{X}_e}$ (resp. $\\dfrac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}$) factor is constant. If we do the partial fraction decomposition, we can run the same argument as in Eqs. \\eqref{simpler} up to \\eqref{cancel} and obtain that \\eqref{logadiv} is actually 0 at $a = 0$. The only identity needed is\n\\[ \\frac{1}{1-e^{i\\Upsilon_e(0)}} +  \\frac{1}{1-e^{-i\\Upsilon_e(0)}}  =  1 \\]\n\nThe argument also works for the derivative with respect to $\\overline{a}$, now with an asymptotic expansion around $\\infty$ of $\\Upsilon_e$.\n\nThis shows that $\\widetilde{\\mathcal{X}_m}$ extends in a $C^1$ way to $a = 0$. For the $C^\\infty$ extension, derivatives with respect to any $\\theta$ coordinate work in the same way, all that was used was the specific form of the contours $C_e, C_{-e}$. The same thing applies to the dependence on the contours $C_e, C_{-e}$. For derivatives with respect to $a, \\overline{a}$ in the integrands, we can again do an asymptotic expansion of $\\Upsilon_e$ at 0 or $\\infty$ and compare it to the asymptotic of the corresponding derivative of $a \\log a - a$ as $a \\to 0$.\n\n\\end{proof}\n\nNothing we have done in this section is particular of the Pentagon example. We only needed the specific values of $\\Omega(\\gamma;u)$ given in \\eqref{omgpar} to obtain the Pentagon identities at the wall and to perform the analytic continuation of $\\mathcal{X}_m$ around $u = 2$.  For any integrable systems data as in section \\ref{intsys}  with suitable invariants $\\Omega(\\gamma;u)$ allowing the wall-crossing formulas and analytic continuation, we can do the same isomonodromic deformation of putting all the jumps at a single admissible ray, perform saddle-point analysis and obtain the same extensions of the Darboux coordinates $\\mathcal{X}_\\gamma$. This finishes the proof of Theorem \\ref{extbf}.\n\nWhat is exclusive of the Pentagon case is that we have a well-defined hyperk\\\"{a}hler metric $g_\\text{OV}$ that we can use as a local model of the metric to be constructed here.\n\nThe extension of the holomorphic symplectic form $\\varpi(\\zeta)$ is now straightforward. We proceed as in \\cite{gaiotto} by first writing:\n\\begin{equation*}\n\\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\mathcal{X}_m}{\\mathcal{X}_m}\n\\end{equation*} \n\nWhere we used the fact that the jumps of the functions $\\mathcal{X}_\\gamma$ are via the symplectomorphisms $\\mathcal{K}_{\\gamma'}$ of the complex torus $T_a$ (see \\eqref{kjump}) so $\\varpi(\\zeta)$ remains the same if we take $\\mathcal{X}_m$ or its analytic continuation $\\widetilde{\\mathcal{X}_m}$.\n\nWe need to show that $\\varpi(\\zeta)$ is of the form\n\\begin{equation}\n-\\frac{i}{2\\zeta}\\omega_+ + \\omega_3 -\\frac{i \\zeta}{2} \\varpi_-\n\\end{equation}\nthat is, $\\varpi(\\zeta)$ must have simple poles at $\\zeta = 0$ and $\\zeta = \\infty$, even at the singular fiber where $a = 0$.\n\nBy definition, $\\mathcal{X}_e = \\exp(\\frac{\\pi R a}{\\zeta} + i\\Upsilon_e + \\pi R \\zeta \\overline{a})$. Thus\n\n\\begin{equation*}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} = \\frac{\\pi R da}{\\zeta} + i d\\Upsilon_e(\\zeta) +\\pi R \\zeta d\\overline{a}\n\\end{equation*}\n\nBy \\eqref{inteq1}, and since $\\lim_{a \\to 0} Z_m \\neq 0$, $\\mathcal{X}_m$ (resp. $\\mathcal{X}_{-m}$) of the form $\\exp(\\frac{\\pi R Z_m(a)}{\\zeta} + i\\Upsilon_m + \\pi R \\zeta \\overline{Z_m(a)})$ still has exponential decay when $\\zeta$ lies in the $\\ell_m$ ray (resp. $\\ell_{-m}$), even if $a = 0$. The differential $d \\Upsilon_e(\\zeta)$ thus exists for any $\\zeta \\in \\mathbb{P}^1$ since the integrals defining it converge for any $\\zeta$.\n\nAs in \\cite{gaiotto}, we can write\n\\[ \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\mathcal{X}_m}{\\mathcal{X}_m} = \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\left( \\frac{d\\mathcal{X}^{\\text{sf}}_m}{\\mathcal{X}^{\\text{sf}}_m} + \\mathcal{I_{\\pm}} \\right), \\]\nfor $\\mathcal{I}_\\pm$ denoting the corrections to the semiflat function. By the form of $\\mathcal{X}^{\\text{sf}} =  \\exp(\\frac{\\pi R Z_m(a)}{\\zeta} + i\\theta_m + \\pi R \\zeta \\overline{Z_m(a)})$, the wedge involving only the semiflat part has only simple poles at $\\zeta = 0$ and $\\zeta = \\infty$, so we can focus on the corrections. These are of the form\n\\begin{align*}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\mathcal{I_{\\pm}} & = \\frac{-i}{2\\pi} \\left\\{ \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} \\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\right.\\\\\n& + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta'-\\zeta)}\\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\\\\n& + \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} \\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\\\\n& + \\left. \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta'-\\zeta)}\\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')} \\right\\}\n\\end{align*}\n\nIn the ``inside'' part of the wall of marginal stability. A similar equation holds in the other side. We can simplify the wedge products above by taking instead\n\\begin{equation}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\left(\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} - \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')} \\right) = \\pi R \\left[ \\left( \\frac{1}{\\zeta} - \\frac{1}{\\zeta'}\\right)da + (\\zeta - \\zeta')d\\overline{a}\\right] +i \\left( d\\Phi_e(\\zeta) - d\\Phi_e(\\zeta') \\right)\n\\end{equation}\n\nRecall that $\\Phi_e$ represents the corrections to $\\theta_e$, so $\\Upsilon_e = \\theta_e + \\Phi_e$. By \\S \\ref{solut}, $\\Phi_e$ and $d\\Phi_e$ are defined for $\\zeta = 0$ $\\zeta = \\infty$ even if $a = 0$, since $\\lim_{a \\to 0} Z_m(a) \\neq 0$ and the exponential decay in $\\mathcal{X}_m^\\text{sf}$ still present guarantees convergence of the integrals in \\ref{inteq1}. Hence, the terms involving $d\\Phi_e(\\zeta) - d\\Phi_e(\\zeta')$ are holomorphic for any $\\zeta \\in \\mathbb{P}^1$. It thus suffices to consider the other terms. After simplifying the integration kernels, we obtain\n\n\\begin{align*}\n\\frac{\\pi R da}{\\zeta} \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} & +\\pi R da \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{(\\zeta')^2} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\\\\n\\frac{\\pi R da}{\\zeta} \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} & +\\pi R da \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{d\\zeta'}{(\\zeta')^2} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\\\\n-\\pi R d\\overline{a} \\int_0^{\\zeta_e} d\\zeta' \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} & -\\pi R \\zeta d\\overline{a} \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\\\\n-\\pi R d\\overline{a} \\int_0^{\\zeta_e} d\\zeta' \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} & -\\pi R \\zeta d\\overline{a} \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\\\\n\\end{align*}\n\nThe only dependence on $\\zeta$ is in the factors $\\zeta, 1\/\\zeta$. Thus $\\varpi(\\zeta)$ has only simple poles at $\\zeta = 0$ and $\\zeta = \\infty$.\n\nFinally, the estimates in Lemma \\ref{estbadfib} show that if we recover the hyperk\\\"{a}hler metric $g$ from the holomorphic symplectic form $\\varpi(\\zeta)$ as in \\S \\ref{extmetric} and \\S \\ref{genextmtr}, we obtain that the hyperk\\\"{a}hler metric for the Pentagon case is the metric obtained in \\ref{extmetric} for the Ooguri-Vafa case plus smooth corrections near $a = 0, \\theta_e = 0$, so it extends to this locus.\n\n This gives Theorem \\ref{smfrm}.\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA  triple $(E,+,\\circ)$, where $(E,+)$ and $(E, \\circ)$ are groups is said to be a left skew brace if \n$$a \\circ (b+c)=a\\circ b-a+a \\circ c$$\nholds for all $a,b,c \\in E$, where $-a$ denotes the inverse of $a$ in $(E, +)$.  In 2007,  Rump \\cite{WR07} introduced classical braces to study involutive and non-degenerate solutions of the Yang-Baxter equation.  Later,  Guarnieri and Vendramin \\cite{GV17}  generalized this concept to skew brace  to study the non-degenerate solution of the Yang-Baxter equation, which is further generalized to semi-braces by Catino, Colazzo, and Stefanelli in \\cite{FMP21} to study non-bijective solutions of the Yang-Baxter equation.  In  \\cite{DG16},  Ben David and Ginosar investigated extensions of bijective $1$-cocycles.  Carter, Elhambadi and Satio in \\cite{CES} developed homology and cohomology theories for solution sets of the Yang-Baxter equations. Different homology theories for various structures related to solutions of the Yang-Baxter equations were investigated extensively by Lebed and Vendramin \\cite{LV17}. Cohomology and extensions of  linear cycle sets with trivial actions is studied by Lebed and Vendramin \\cite{LV16}.   Recently generalized  by Jorge A. Guccione,  Juan J. Guccione and Christian Valqui  \\cite{GG21} to non trivial actions. Various type of products like matched product,  semi-direct product, asymmetric product has been defined for the solutions of Yang-Baxter equation [see \\cite{DB18}, \\cite{BCJO19},  \\cite{CCS},  \\cite{CCS1},  ,\\cite{CCS2},   \\cite{WR08}].  In \\cite{NMY},  M. K. Yadav and author developed the theory of skew brace extensions for skew brace extensions by an abelian group  and developed the Well's type exact sequence for skew braces.  This work can be thought as a generalization of  \\cite{DB18},  \\cite{LV17} at the level of extensions.  The fundamental exact sequence of Wells for groups was introduced by C Wells in \\cite{W71}.  The fundamental exact sequence of Wells with various applications is carried out in all fine details in \\cite[Chapter 2]{PSY18}. A similar exact sequence for cohomology, extensions and automorphisms of quandles was constructed in \\cite{BS20}.\n In this paper, we define a new product for the skew braces and construct few examples.  We give constructions for skew braces  similar to that of group theory and generalize the Well's type exact sequence for the trivial skew brace.\n\n\n\n\n\\section{Preliminaries}\n\nAn algebraic structure $(E, + , \\circ)$ is said to be a \\emph{left skew brace} if $(E,  +)$ and $(E, \\circ)$ are a group and the following compatibility condition holds:\n\\begin{equation}\\label{bcomp}\na \\circ (b  + c ) = a \\circ b -a + a\\circ c\n\\end{equation}\nfor all $ a, b , c \\in E$,  where $-a$ denotes the inverse of $a$ with respect to `$+ $'. \nNotice that the identity element $0$ of $(E,  +)$ coincides with the identity element  of $(E, \\circ)$.\n\nFor a left skew brace $E$ and $a \\in E$, define a map $\\lambda_a : E \\to E$ by\n$$\\lambda_a(b) = -a + (a \\circ b)$$\nfor all $b \\in E$. The automorphism group  of a group $G$ is denoted by $\\operatorname{Aut} (G)$. \nWe have the following  result for skew braces .\n\\begin{lemma}\nLet $(E,+, \\circ)$ be a left skew brace, then for each $a \\in E$, the map $\\lambda_a$ is an automorphism of $(E, +)$ and the map $\\lambda : (E, \\circ) \\to \\operatorname{Aut} (E, +)$ given by $\\lambda(a) = \\lambda_a$ is a group homomorphism.\n\\end{lemma}\n\nA  sub skew brace $I$ of a left  skew brace $E$ is said to be a \\emph{left ideal} of $E$ if $\\lambda_a(y) \\in I$ for all $a \\in E$ and $y \\in I$.  A left ideal of $E$ is said to be an \\emph{ideal} if $(I, \\circ)$ is  a normal subgroup of $(E, \\circ)$. The Socle of a skew brace $E$ is defined as $\\operatorname{Soc} (E) = \\operatorname{Ker}  \\lambda$  $\\cap $ $\\operatorname{Z}  (E, +)$, where $\\operatorname{Z} (E, +)$ represents the centre of the group $(E, +)$ and the annihilator of $E$ is defined as $\\operatorname{Ann} (E)=\\operatorname{Soc} (E) \\cap \\operatorname{Z} (E, \\circ)$.\n\n\nThe following is an easy but important observation, which will be used several times in what follows.\n\\begin{lemma}\nLet $E$ be a left skew brace. Then for all $a, b \\in E$, the following hold:\n\n(i) $a + b = a \\circ \\lambda^{-1}_a(b)$.\n\n(ii) $a \\circ b = a + \\lambda_a(b)$.\n\\end{lemma}\n\n\nLet $E_1$ and $E_2$ be two  left skew braces. A map $f : E_1 \\to E_2$ is said to be a \\emph{skew brace homomorphism} if $f(a + b) = f(a) + f(b)$ and $f(a\\circ b) = f(a) \\circ f(b)$ for all $a, b \\in E_1$.  A one-to-one and onto skew brace homomorphism from $E_1$ to itself is called an \\emph{automorphism} of $E_1$. The \\emph{kernel} of a homomorphism $f : E_1 \\to E_2$ is defined to be the subset $\\{a \\in E_1 \\mid f(a) = 0\\}$ of $E_1$. It turns out that $\\operatorname{Ker} (f)$, the kernel of $f$, is an ideal of $E_1$. The set of all skew brace automorphisms of a left skew brace $E$, denoted by $\\operatorname{Autb} (E)$, is a group.\n\n\n\nLet $H$ and $I$ be two left skew  braces. By an \\emph{extension} of $H$ by $I$, we mean  a left skew brace $E$ with an exact sequence \n$$\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0,$$ \nwhere $i$ and $\\pi$ are injective and surjective brace homomorphisms, respectively.   Thereafter,  we denote the image of $y$ under $i$ by $y$ itself for all $y \\in I$.  A set map $s : H \\to E$ is called a \\emph{set-theoretic section} of $\\mathcal{E}$ if $\\pi(s(h)) = h$ for all $h \\in H$ and $s(0) = 0$. The abbreviation `st-section' will be used for `set-theoretic section' throughout. We call $\\mathcal{E}$ to be split exact sequence of skew braces if there exist a st-section of $\\mathcal{E}$ which is a skew brace homomorphism. \n\n\\section{split extensions of skew brace}\n\nLet $H$ and $I$ be two left skew braces. Let $ \\mu: (H, +) \\rightarrow Aut(I, +)$,  $\\sigma : (H, \\circ) \\rightarrow Aut(I, \\circ)$ be anti-homomorphisms, and $\\nu: (H, \\circ) \\rightarrow Aut (I, +)$ be a homomorphism.  Let $\\mu, \\sigma $ and $\\nu$ satisfy the following compatibility  condition\n\\noindent \\begin{align}\\label{SE}\n\\nu_{h_1\\circ(h_2 + h_3)}(\\sigma_{h_2 + h_3}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2 + h_3}(\\mu_{h_3}(y_2)+ y_3))& = \\mu_{-h_1+(h_2 \\circ h_3)}(\\nu_{h_1 \\circ h_2}(\\sigma_{h_2}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2}(y_2))-y_1) \\notag \\\\\n& +  \\nu_{h_1 \\circ h_3}(\\sigma_{h_3}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_3}(y_3)) \n \\end{align}\n \nfor all $y_1, y_2 , y_3 \\in I $ and $ h_1, h_2, h_3 \\in H$.\n\n\\begin{thm}\nLet  $H$ and $I$ be two skew braces with $(\\nu ,\\mu,\\sigma),$ as defined above and satisfying \\eqref{SE}, then the operations \n\\begin{align}\n (h_1, y_1) + (h_2,  y_2)&=(h_1 + h_2, \\mu_{h_1}(y_1) + y_2), \\label{sb+} \\\\\n (h_1, y_1) \\circ (h_2,  y_2)&=(h_1 \\circ h_2, \\nu_{h_1 \\circ h_2}(\\sigma_{h_2}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2}(y_2)) \\label{sbcirc}\n\\end{align}\ndefine a left skew brace structure on $ H \\times I $.\n\\end{thm}\n\\begin{proof}\nIt is easy to check that the given operations define group structure on $H \\times I$ and the condition \\eqref{bcomp}  follows from the compatibility condition of $(\\nu, \\mu, \\sigma)$.   \n\\hfill $\\Box$\n\n\\end{proof}\n\nWe call this structure a  \\emph{split semi-direct product} of $H$ by $I$ with respect to the triplet $( \\nu,\\mu, \\sigma)$ and denote it by $(H, I, \\nu,\\mu, \\sigma)$.\n\n\\begin{lemma}\nLet $(H, I, \\nu,\\mu, \\sigma)$ be  split semi-direct product of $H$ by $I$ with respect to some triplet $(\\nu,\\mu,\\sigma)$.  Then the following short exact sequence of skew braces\n$$ \\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  (H, I, \\nu, \\mu,\\sigma) \\stackrel{\\pi}{\\to} H \\to 0 $$\n splits, where $i$ and $\\pi$ are natural injection and projection respectively.\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to check that the map $ s: H \\rightarrow (H, I, \\mu, \\sigma,\\nu)$ given by $s(h)=(h,0)$  is both  a homomorphism of skew braces and  a st-section of $\\mathcal{E}$ simultaneously. \n\\hfill $\\Box$\n\n\\end{proof}\n\n\\begin{thm}\nLet $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ be a split short exact sequence of skew braces.  Then $E$ is a split semi-direct product  of $H$ by $I$.\n\\end{thm}\n\n\\begin{proof}\nLet the short exact sequence $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ split.  Then there exists a st-section $s :H \\rightarrow E$,  which is also a skew brace homomorphism.  Define $\\mu: H \\longrightarrow Aut(I, +)$,  $\\sigma : H \\rightarrow Aut(I, \\circ)$, and $\\nu : H \\rightarrow Aut(I, +) $ by\n\\begin{align}\\label{actions}\n\\nu_h(y) & =-s(h) + (s(h) \\circ y),\\notag\\\\\n\\mu_h(y) & =-s(h)+ y+ s(h),  \\\\\n\\sigma_h(y) & =s(h)^{-1} \\circ y \\circ s(h).\\notag\n\\end{align}\nSince $E$ is a skew brace,  we have \n\\begin{equation}\\label{sbc1}\n(s(h_1) + y_1) \\circ \\big{(}s(h_2) + y_2 + s(h_3) +y_3 \\big{)} = (s(h_1)+y_1) \\circ (s(h_2) + y_2) - (s(h_1) + y_1) + (s(h_1) + y_1) \\circ (s(h_3) + y_3). \n\\end{equation}\nUsing \\eqref{sbc1} and linearity of $s$ in `$+$' and `$\\circ$',  we can easily establish that $(\\nu, \\mu, \\sigma)$ satisfies \\eqref{SE}.  Hence we have semi-direct product  $(H,I,\\mu,\\sigma,\\nu)$.  We know that every element $ x \\in E$ can be uniquely written as $x=s(h)+ y$.  Define $\\phi : E \\rightarrow (H,I,\\mu,\\sigma,\\nu)$ by $\\phi(s(h)+ y)=(h, y)$.  Then $\\phi$ is an isomorphism of skew braces and the diagram\n$$\\begin{CD}\n 0 @>i>> I @>>> E @>{{\\pi} }>> H  @>>> 0\\\\\n &&  @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>i'>> I @>>> (H,I,\\nu,\\mu,\\sigma) @>{{\\pi^\\prime} }>> H @>>> 0\n\\end{CD}$$\ncommutes,  where $i^\\prime$ and $\\pi^\\prime$ are natural injection and projection,  respectively.  This completes the proof.\n\n\\hfill $\\Box$\n\n\\end{proof}\n\n\\section{Examples}\n\n\nIn this section we provide some examples of split semi direct product of skew braces.   We have used GAP to compute $\\nu, \\mu$ and $\\sigma$.\n\n\\noindent \\textbf{Example 1}  Let $\\mathbb{Z}$ and $\\mathbb{C}$ be trivial skew braces,  Define $\\nu, \\mu, \\sigma : \\mathbb{Z} \\rightarrow Aut(\\mathbb{C})$ by $\\nu_1(x)=\\mu_1(x)=\\sigma_1(x)=-x$. Using  \\eqref{sb+} and \\eqref{sbcirc},  we can define a skew brace structure on $\\mathbb{Z} \\times \\mathbb{C}$ by  \n\\begin{align*}\n(l, \\ y_1)+(m, \\ y_2) & = \\big(l+m,   \\ (-1)^{m+n}y_{1}+y_2\\big),\\\\\n(l, \\ y_1)\\circ(m, \\ y_2) & = \\big(l+m,  \\ y_1+(-1)^{m+n}y_2\\big).\n\\end{align*}\n\n\n\\noindent \\textbf{Example 2}  Let $H=D_{2n}= \\langle a,b\\hspace{.1cm}|\\hspace{.1cm} a^{2n}=b^2=e,  bab=a^{-1}\\rangle$ and $I=\\mathbb{Z}_p$ be trivial skew braces, where $D_{2n}$  and $\\mathbb{Z}_p$ denotes dihederal group of order $4n$ and cyclic group of order $p$ respectively.  Define $\\nu, \\mu, \\sigma : D_{2n} \\rightarrow Aut(\\mathbb{Z}_p)$ by $\\nu_a(x)=\\mu_a(x)=\\sigma_a(x)=-x$ and $\\nu_b(x)=\\mu_b(x)=\\sigma_b(x)=-x$. Using  \\eqref{sb+} and \\eqref{sbcirc},  we can define a skew brace structure on $D_{2n} \\times \\mathbb{Z}_p$ by  \n\\begin{align*}\n(a^{i}b^{j}, y_1)+(a^{m} b^{n},  y_2) & = \\big(a^{i}b^{j}a^{m} b^{n},   \\ y_2+(-1)^{m+n}y_{1}\\big),\\\\\n(a^{i}b^{j}, y_1)\\circ(a^{m} b^{n},  y_2) & = \\big(a^{i}b^{j}a^{m} b^{n},  \\ y_1+(-1)^{i+j}y_2\\big).\n\\end{align*}\nIf we have trivial skew brace $H=D_n$, where $n$ is odd and $I$ be the same as above,  then we can define $\\nu, \\mu, \\sigma : D_{n} \\rightarrow Aut(\\mathbb{Z}_p)$ by $\\nu_a(x)=\\mu_a(x)=\\sigma_a(x)=x$ and $\\nu_b(x)=\\mu_b(x)=\\sigma_b(x)=-x$.\n\n\\\n\n\\noindent \\textbf{Example 3} Let $H=\\mathbb{Z}_8$ be trivial skew brace and $I=\\mathbb{Z}_3 \\times \\mathbb{Z}_2$  be skew brace of order $6$ defined in \\cite{EM20} by the following operations\n\\begin{align*}\n (n,m)+(s,t) & =(n+2^{m}s, m+t),\\\\\n(n,m) \\circ (s,t) & =(2^{t}n+2^{m}s,m+t).\n\\end{align*}\nWe have $(I,+)=\\langle  (1,0), (0,1)  \\rangle    \\cong S_3$ and $(I, \\circ)=\\langle (1,1) \\rangle \\cong \\mathbb{Z}_6$. We take $\\mu_{a}(n,m) = (n,m)$,  $ \\sigma_{a}(n,m)=(n,m)^{-1}=(2n,m), $  $\\nu_{a}(n,m)=(2n,m) $,  for all $(n,m) \\in I$, where $a$  is a generator of $H$.  Hence the additive group of skew brace structure on $(H, I, \\nu, \\mu, \\sigma)$ is just direct product of their respective additive groups and multiplicative group is given as follows\n\\begin{eqnarray*}\n(a^k, (n,m)) \\circ (a^l,(s,t))=(a^{k+l},  ((2n,m) \\circ (2s,t)^{k}).\n\\end{eqnarray*} \n\\\n\n\n\\noindent \\textbf{Example 4} Let $H$ be brace of order $4$ defined in \\cite{DB15} by $(H,+)=\\mathbb{Z}_2 \\times \\mathbb{Z}_2$,  $(H, \\circ)=\\langle(0,1) \\rangle \\cong \\mathbb{Z}_4$ and $I$ be a brace such that $(I,+)=\\mathbb{Z}_4$,  $(I,\\circ)=\\langle 1,2 \\rangle \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2 $. Consider $\\mu_x=\\operatorname{Id}$ for all $x \\in H$,  where $\\operatorname{Id}$ denotes the identity mapping on $I$ and $\\nu_{(0,1)}(x)=\\sigma_{(0,1)}(x)=-x$. Then split semi- direct product of $I$ by $H$ is given by the skew brace with additive group as direct product of $H$ and $I$ and multiplicative group structure is given as follows\n\\begin{eqnarray*}\n\\big((0,1)^k, l) \\circ ((0,1)^n,  m\\big)=\\big((0,1)^{k+n},  l + (-1)^{k}m+(-1)^{n}lm\\big).\n\\end{eqnarray*}\n\\\n\n\\noindent \\textbf{Example 5} Let $H$ be the  brace of order $8$ with additive group $\\mathbb{Z}_8$ having Socle of order $2$ and $(H,\\circ)=\\langle 1,2 \\rangle \\cong \\mathbb{Z}_4 \\times \\mathbb{Z}_2$  defined in \\cite{DB15} and $I$ be brace of order $4$ as defined in Example 3. Then we have total $8$ different split semi-direct products of $I$ by $H$,  interestingly with $\\mu_x=Id$ for all $x \\in H$ in all cases. We list few cases\n\\begin{enumerate}\n\\item[(i)]\n$\\nu_1(x)=x^{-1}, \\hspace{.1cm} \\nu_2(x)=x \\hspace{.1cm} and \\hspace{.1cm} \\sigma_h(x)=x \\hspace{.1cm}for \\hspace{.1cm}all \\hspace{.1cm} x \\in I, \\hspace{.1cm} h \\in H. $\\\\\n\n\n\\item[(ii)]\n$\\nu_1(x)=x^{-1}, \\hspace{.1cm} \\nu_2(x)=x^{-1} \\hspace{.1cm} and \\hspace{.1cm} \\sigma_2(2)=3 ,\\hspace{.1cm} \\sigma_1(x)=x \\hspace{.1cm}  for \\hspace{.1cm}all \\hspace{.1cm} x \\in I.$ \n\n\\end{enumerate}\n\n\n\n\n\n\n\\section{general extensions of skew braces}\n\nLet $(H, +,  \\circ)$ and $(I, + , \\circ) $ be two skew braces,  $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension of $H$ by $I$. and let $s : H \\rightarrow E$ be  an st-section of $\\mathcal{E}$.  Corresonding to $s$, consider the pair $(\\beta, \\tau)$, where $\\beta$ and $\\tau$ are defined as  \n\\begin{align}\n\\beta(h_1, h_2) &:= - s(h_1 + h_2) + s(h_1) + s(h_2),\\label{cocycle1 sb}\\\\\n\\tau(h_1, h_2) &:= s(h_1 \\circ h_2)^{-1} \\circ s(h_1) \\circ s(h_2).\\label{cocycle2 sb}\n\\end{align} \nIt is easy to see that $ \\nu, \\mu $ and $ \\sigma$,  defined in \\eqref{actions},  need not be homomorphisms in general, but they satisfy the following identities\n\n\\begin{align}\n\\nu_{h_1 \\circ h_2}&=\\nu_{h_1}  \\nu_{h_2} \\lambda^{-1}_{\\tau(h_1, h_2)},\\label{action1 }\\\\\n \\mu_{h_1 + h_2} &= i^{+}_{-\\beta(h_1, h_2)} \\mu_{h_2} \\mu_{h_1}, \\label{action2 }\\\\\n\\sigma_{h_1 \\circ h_2}&=i^{\\circ}_{\\tau(h_1, h_2)^{-1}} \\sigma_{h_2} \\sigma_{h_1}, \\label{action3}\n \\end{align}\nwhere \n \n\\begin{align}\ni^{+}_y(z)&:=y+z-y, \\\\\ni^{\\circ}_y(z)&:=y \\circ z \\circ y^{-1},\n \\end{align}\nare inner automorphisms of $(H,+)$ and $(H, \\circ)$, respectively, and $\\beta$ and $\\tau$ are as defined above in \\eqref{cocycle1 sb} and \\eqref{cocycle2 sb}.\n \nLet $N$ be the smallest normal subgroup of $\\operatorname{Aut} (I, +)$ generated by the set $\\{ \\lambda_y \\hspace{.1cm}| \\hspace{.1cm} y \\in I\\}$. Let $ \\operatorname{Inn} (I , +)$ and $ \\operatorname{Inn} (I, \\circ)$ be the inner automorphism subgroups  of $\\operatorname{Aut} (I, +)$ and $\\operatorname{Aut} (I, \\circ)$ respectively. Then we have the maps  $\\bar{\\nu} : (H, \\circ ) \\rightarrow \\operatorname{Aut} (I, +)\/N$, $\\bar{\\mu} : (H, +) \\rightarrow \\operatorname{Aut} (I, )\/ \\operatorname{Inn} (I, +)$ and $\\bar{\\sigma} : (H, \\circ) \\rightarrow \\operatorname{Aut} (I, \\circ)\/ \\operatorname{Inn} (I, \\circ)$ defined by $\\nu , \\mu$ and $\\sigma$ composing with natural projections respectively.  We call the triplet  $\\chi:=(\\nu, \\mu, \\sigma)$ satisfying (\\ref{action1 }), (\\ref{action2 }), and (\\ref{action3}) an action of $H$ on $I$ and corresponding triplet $\\bar{\\chi}:=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ will be called a coupling from $H$ to $I$ corresponding to $(\\nu,\\mu, \\sigma)$.  Let $\\chi =(\\nu, \\mu, \\sigma)$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be two actions. Then we say that $\\bar{\\chi}^\\prime\\approx\\bar{\\chi}$ if there exists a map $\\theta: H \\rightarrow I$ such that $\\theta(0)=0$ and $\\nu^{\\prime}_{h}=\\nu_{h} \\lambda_{\\theta(h)}$,  $\\mu^\\prime_{h}=i^+_{\\nu_h(-\\theta(h))} \\mu_h$ and $\\sigma^\\prime=i^{\\circ}_{\\theta(h)^{-1}} \\sigma_{h}.$\n\n{\\bf{Remark:}} If $\\bar{\\chi}^\\prime\\approx\\bar{\\chi}$ then $\\bar{\\chi}^\\prime =\\bar{\\chi}$ but converse need not be true.  Note that the map $\\theta : H \\rightarrow I$ mentioned above need not be unique. \n\nWith this setting, we have\n\\begin{prop}\\label{well-def-act-coc}\nLet $0 \\to I \\stackrel{}{\\to} E \\stackrel{\\pi}{\\rightarrow}  H \\to 1$ be an extension of a left skew brace $I$ by $H$. Then the following hold:\n \n(1) The coupling $\\bar{\\chi}$ is independent of the choice of an st-section. \n \n(2) Equivalent extensions have the same coupling.\n\n\\end{prop}\n\\begin{proof}\n(1) Let $s_1$ and $s_2$ be two st-sections of $\\pi$. We know that two sections differ by an element of $I$, hence for an element $h \\in H$, there exist $y_h \\in I$ such that $s_2(h)=s_1(h)\\circ y_h$.  Let $\\chi=(\\nu, \\mu ,  \\sigma)$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be actions corresponding to $s_1$ and $s_2$ respectively. Define $\\theta: H \\rightarrow I$ be  $\\theta(h)=y_h$.  It can be easily seen that $\\bar{\\chi}^\\prime \\approx \\bar{\\chi}$  using $\\theta$ as a required map.\n \n(2) Let $E$ and $E^\\prime$ be two equivalent extensions. Then there exist a skew brace homomorphism $\\phi: E^\\prime \\rightarrow E$ such that the following diagram commutes\n$$\\begin{CD}\n 0 @>>> I @>>> E^\\prime @>{{\\pi^\\prime} }>> H  @>>> 0\\\\\n &&  @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> E @>{{\\pi} }>> H @>>> 0.\n\\end{CD}$$\nLet $s : H \\rightarrow E^\\prime$ be any st-section of  the extension $E^\\prime$.  Then $\\phi s : H \\rightarrow E$ is a st-section of extension the $E$. Let $\\chi=(\\nu, \\mu, \\sigma)$ be actions of $E$ corresponding to $\\phi s$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be actions of $E^\\prime$ corresponding to $ s$, respectively.  Then we have $\\nu=\\nu^\\prime$, $\\mu = \\mu^\\prime$ and $\\sigma=\\sigma^\\prime$.  Hence $\\bar{\\chi} \\approx \\bar{\\chi^\\prime}$ by taking $\\theta: H \\rightarrow I$ to be $\\theta(h)=0$ for all $h \\in H$.  As we have already proved that coupling is independent of an st-section,  so this holds for all st-sections of $E$ and $E^\\prime$.\n\\hfill $\\Box$\n\n\\end{proof}\nLet $\\operatorname{Ext} (H,I)$ denote the set of  equivalence classes of all  extensions of $H$ by $I$.  Equivalence class of an extension $\\mathcal{E} : 0 \\to I \\to E \\to H \\to 0$ is denoted by $[\\mathcal{E}]$. As a consequence of the preceding proposition, it follows that each equivalence class of extension of $H$ by $I$ admits a unique coupling $\\bar{\\chi}=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ corresponding to actions $\\chi=(\\nu, \\mu, \\sigma)$ of $H$ on $I$.  Let $\\operatorname{Ext} _{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})}(H, I)$ denote the equivalence class of those extensions of $H$ by $I$ whose corresponding coupling is $(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$.  We can easily establish\n \n\n\\begin{cor}\\label{cor 1}\n $\\operatorname{Ext} (H, I) = \\bigsqcup_{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})} \\operatorname{Ext} _{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})}(H, I)$.\n \\end{cor}\n\n\n\\begin{prop}\\label{prop 2}\nLet $\\mathcal{E}$ be a extension of $H$ by $I$. Then the following hold \\\\\n\n\\begin{itemize}\n\\item[1)]\nLet $s$ be an st-section of $\\mathcal{E}$. Then the pair $(\\beta, \\tau)$ corresponding to $s$ together with action defined in \\eqref{actions} satisfies\n\n\\begin{equation}\\label{cocycle 1}\n\\beta(h_1, h_2+h_3)+\\beta(h_2, h_3)-\\beta(h_1+h_2, h_3)-\\mu_{h_3}(\\beta(h_1, h_2))=0,\n\\end{equation}\nand \n\\begin{equation}\\label{cocycle 2}\n\\tau(h_1, h_2 \\circ h_3) \\circ \\tau(h_2, h_3) \\circ \\tau(h_1 \\circ h_2, h_3)^{-1} \\circ (\\sigma_{h_3}(\\tau(h_1, h_2)))^{-1}=0.\n\\end{equation}\n\n\\item[2)]\nLet $s_1$ and $s_2$ be two st-sections of $\\mathcal{E}$, and let $(\\beta_1, \\tau_1)$ and $(\\beta_2, \\tau_2)$ be the pairs corresponding to $s_1$ and $s_2$, respectively. Let $(\\prescript{}{1}{\\nu}, \\prescript{}{1}{\\mu}, \\prescript{}{1}{\\sigma})$  and $(\\prescript{}{2}{\\nu}, \\prescript{}{2}{\\mu}, \\prescript{}{2}{\\sigma})$ be actions corresponding to $s_1$ and $s_2$, respectively. Then there exists a map $\\theta : H \\rightarrow I$ such that \n$$\ns_2(h)=s_1(h) \\circ \\theta(h)=s_1(h) + \\prescript{}{1}{\\nu}_{h}(\\theta(h)),\n$$ \n\\begin{equation}\\label{equi 1}\n\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))=\\beta_2(h_1, h_2),\n\\end{equation}\nand\n\\begin{equation}\\label{equi 2}\n\\theta(h_1 \\circ h_2)^{-1} \\circ \\tau_1(h_1, h_2) \\circ \\prescript{}{1}{\\sigma}_{h_2}(\\theta(h_1)) \\circ \\theta(h_2)=\\tau_2(h_1, h_2),\n\\end{equation}\nfor all $h, h_1, h_2 \\text{ and } h_3 \\in H$.\\\\\n\n\\item[3)]\nLet $\\mathcal{E}_1$ and $\\mathcal{E}_2$ be two equivalent extensions of $H$ by $I$, and let $s_1$ and  $s_2$ be st-sections of $\\mathcal{E}_1$ and $\\mathcal{E}_2$, respectively. Let $(\\beta_1, \\tau_1)$ and $(\\beta_2, \\tau_2)$ be the pairs corresponding to $s_1$ and $s_2$, respectively. Then there exists a map $\\theta : H \\rightarrow I$ satisfying \\eqref{equi 1} and \\eqref{equi 2}.\n\n\\end{itemize}\n\n\n\n\\end{prop}\n\\begin{proof}\nIt is easy to see that (1) and (2) follows directly from definitions. Now we will prove (3). \nSince $\\mathcal{E}_1 := 0 \\to I \\stackrel{}{\\to}  E_1 \\stackrel{\\pi_1}{\\to} H \\to 0$ and $\\mathcal{E}_2:= 0 \\to I \\stackrel{i}{\\to}  E_2 \\stackrel{\\pi_2}{\\to} H \\to 0$ are two equivalent extensions, there exists an isomorphism $\\phi : E_1 \\rightarrow E_2$ such that the following diagram commutes\n$$\n\\begin{CD}\n 0 @>>> I @>>> E_1 @>{{\\pi_1} }>> H  @>>> 0\\\\\n &&  @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> E_2 @>{{\\pi_2} }>> H @>>> 0.\n\\end{CD}\n$$\n\nLet $s_1$ be an st-section of $\\mathcal{E}_1$. Then $\\phi s_1$ is an st-section of $\\mathcal{E}_2$.  Let $(\\beta_1, \\tau_1)$ and $(\\beta^\\prime, \\tau^\\prime)$ be the pairs corresponding to $s_1$ and $\\phi s_1$, respectively. By the commutativity of the above diagram, we have $\\phi(y)=y$, for all $y \\in I$, hence $\\beta_1=\\beta^\\prime$ and $\\tau_1=\\tau^\\prime$. Let $s_2$  be an st-section of $\\mathcal{E}_2$ and $(\\beta_2, \\tau_2)$ be the pair corresponding to $s_2$. Using \\eqref{equi 1}, \\eqref{equi 2} for $s_2$ and $\\phi s_1$, we get the desired result. \n\\hfill $\\Box$\n\n\\end{proof}\n\\begin{defn}\nLet $\\chi=(\\nu, \\mu, \\sigma)$ be an action of $H$ on $I$, and $\\beta, \\tau : H \\times H \\rightarrow I$ such that $\\beta$ and $\\tau$ together with $\\chi$ satisfies \\eqref{cocycle 1} and \\eqref{cocycle 2}, respectively. Then the ordered pair $(\\beta, \\tau)$ is a $2$-cocycle with action $\\chi$. \n\n{\\bf{Remark:}} Note that the pair $(\\beta, \\tau)$ defined by the \\eqref{cocycle1 sb} and \\eqref{cocycle2 sb} is a $2$-cocycle corresponding to st-section $s$ with action defined by \\eqref{actions}. \n\\end{defn}\n\nLet  $\\mathcal{E}:= 0 \\to I \\stackrel{}{\\to}  E\\stackrel{\\pi}{\\to} H \\to 0$ be an skew brace extension of $H$ by $I$. Let $s : H \\rightarrow E$ be an st-section of  $\\mathcal{E}$. Due to the compatiblity condition of a left skew brace $E$, we have\n$$\n(s(h_1) \\circ y_1) \\circ (s(h_2) \\circ y_2+s(h_3) \\circ y_3)=s(h_1) \\circ y_1) \\circ (s(h_2) \\circ y_2)-(s(h_1) \\circ y_1)+(s(h_1) \\circ y_1) \\circ (s(h_3) \\circ y_3).\n$$ \nFrom the above equality we see that the triple $(\\nu, \\mu, \\sigma)$ (defined  in \\eqref{actions}) together with $(\\beta, \\tau)$ satisfy\n\\begin{eqnarray}\\label{parent relation}\n \\nu_{h_1 \\circ (h_2+h_3)}(\\tau(h_1, h_2+h_3) \\circ \\sigma_{h_2+h_3}(y_1) \\circ \\nu^{-1}_{h_2+h_3}(\\beta(h_2, h_3) +\\mu_{h_3}(\\nu_{h_2}(y_2))+\\nu_{h_3}(y_3))) = A,\n\\end{eqnarray}\nwhere\n\\begin{align*}\nA = &\\beta (h_1 \\circ h_3-h_1, h_1 \\circ h_3)+\\mu_{h_1 \\circ h_3}(\\beta(h_1 \\circ h_2, -h_1)+ \\mu_{-h_1}(\\nu_{h_1 \\circ h_2}(\\tau(h_1, h_2) \\circ \\sigma_{h_2}(y_1) \\circ y_2))\\\\\n& -\\nu_{h_1}(y_1)- \\beta(h_1, -h_1)) +\\nu_{h_1 \\circ h_3}(\\tau(h_1, h_3) \\circ \\sigma_{h_3}(y_1) \\circ y_3),\n\\end{align*}\nfor all $h_1, h_2, h_3 \\in H$ and $y_1, y_2, y_3 \\in I.$\n\n\n{\\bf{Remark:}} Note that if $I$ is an abelian group equipped with trivial skew brace structure, then the above condition will simplify to the condition defined for good triplet of actions in \\cite[Pg.5]{NMY}.\n\n For $\\alpha=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ a coupling from $H$ to $I$.  Define\n\\begin{align*}\\label{stZ^2}\n\\mathcal{Z}^2_{\\alpha}(H, I):=\\Bigg\\{(\\chi,\\beta, \\tau)\\hspace{.1cm} \\Big| \\hspace{.1cm} \\hspace{.1cm}\\substack{\\chi \\mbox{ is an action of } H \\mbox{ on } I, \\hspace{.1cm}  \\bar{\\chi} \\approx \\alpha, \\hspace{.1cm}  \\mbox{and} \\hspace{.1cm}(\\beta,\\tau)\\hspace{.1cm} \\mbox{ia a 2-cocycle} \\hspace{.1cm}\\mbox{with action}  \\\\ \\hspace{.1cm} \\chi  \\hspace{.1cm} \\mbox{and satisfy}  \\hspace{.1cm} \\eqref{parent relation}\n  }\\Bigg\\}.\n\\end{align*}\nLet $(\\chi_1, \\beta_1, \\tau_1)$ and $(\\chi_2 \\beta_2, \\tau_2)$ be two elements of  $\\mathcal{Z}^2_{\\alpha}(H, I)$, where $\\chi_1=(\\prescript{}{1}{\\nu}, \\prescript{}{1}{\\mu}, \\prescript{}{1}{\\sigma})$ and $\\chi_2=(\\prescript{}{2}{\\nu}, \\prescript{}{2}{\\mu}, \\prescript{}{2}{\\sigma})$.  We say that  $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$ if there exits a map $\\theta: H \\rightarrow I$ such that $\\bar{\\chi_2} \\approx \\bar{\\chi_1} $ by $\\theta$ and $\\beta_1$, $\\beta_2$ satisfy (\\ref{equi 1}), $\\tau_1, \\tau_2$ satisfy (\\ref{equi 2}) with respect to $\\theta$. \\\\\n\n\n\\begin{prop}\nThe  relation  `$\\sim$' defined in above para is an equivalence relation.\n\\end{prop}\n\n\\begin{proof}\nReflexivity is easy to see by taking $\\theta:H\\rightarrow I$ given by $\\theta(h)=0$ for all $h \\in H$. Now we will show that the above relation is symmetric. Let $ (\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$, where $\\chi_i=(\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$ for $i=1,2$.  We know that there exist a map  $\\theta : H \\rightarrow I$ such that $\\bar{\\chi_1} \\approx\\bar{\\chi_2}$ by $\\theta$ and  \\eqref{equi 1}, \\eqref{equi 2}.  Define $\\psi:H \\rightarrow I$ by $\\psi(h)=\\theta(h)^{-1}$.  Then we have $\\prescript{}{1}{\\nu}_h=\\prescript{}{2}{\\nu}_h \\lambda_{\\psi(h)}$, and hence $\\prescript{}{2}{\\nu}_h(-\\psi(h))=\\prescript{}{1}{\\nu}_h(\\theta(h))$.  This proves that $\\bar{\\chi_1}=\\bar{\\chi_2} $ by $\\psi$. Similarly we can prove that $\\beta_2, \\beta_1$ satisfy (\\ref{equi 1}) and $\\tau_1, \\tau_2$ satisfy (\\ref{equi 2}) with respect to $\\psi$. Next we prove  transitivity.  Let  $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$  (by $\\theta_1$) and $ (\\chi_2, \\beta_2, \\tau_2)$ $\\sim$ $(\\chi_3, \\beta_3, \\tau_3)$ (by $\\theta_2$),  where\n$\\chi_i=(\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$ for $i=1,2,3$.  We claim that  $\\bar{\\chi_3} \\approx \\bar{\\chi_1}$ by $\\phi:H \\rightarrow I $ defined by $\\psi(h)=\\theta_1(h) \\circ\\theta_2(h)$. We have \n\\begin{eqnarray}\\label{first relation}\n\\prescript{}{2}{\\nu_h} =\n\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)},\n\\end{eqnarray}\nand \n\\begin{eqnarray*}\n\\prescript{}{3}{\\nu}_h =\n\\prescript{}{2}{\\nu_h} \\lambda_{\\theta_2(h)}.\n\\end{eqnarray*}\n\nCombining these two equations we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\nu}_h &=&\n\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)}\\lambda_{\\theta_2(h)}\\\\\n&=& \\prescript{}{1}{\\nu_h} \\lambda_{\\phi(h)}.\n\\end{eqnarray*} \nFor additive action we have\n\n\\begin{eqnarray*}\n\\prescript{}{2}{\\mu}_{h}=i^+_{\\prescript{}{1}{\\nu_h}(-\\theta_1(h))} \\prescript{}{1}{\\mu}_h\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h}=i^+_{\\prescript{}{2}{\\nu_h}(-\\theta_2(h))} \\prescript{}{2}{\\mu}_h.\n\\end{eqnarray*}\n\nCombining the above two equations we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{2}{\\nu_h}(-\\theta_2(h))}i^+_{\\prescript{}{1}{\\nu_h}(-\\theta_1(h))} \\prescript{}{1}{\\mu}_h.\\\\\n\\end{eqnarray*}\nUsing (\\ref{first relation}) we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)(-\\theta_2(h))}}\ni^+_{\\prescript{}{1}{\\nu_h}(-\\theta_2(h))} \\prescript{}{1}{\\mu}_h.\\\\\n\\end{eqnarray*}\n\nFinally, using the relation $a+\\lambda_a(b)=a \\circ b$, we get \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{1}{\\nu_h}(-\\phi(h))}\\prescript{}{1}{\\mu}_h.\n\\end{eqnarray*}\n\n\nSimilarly we can prove that $\\prescript{}{3}{\\sigma}_h=i^{\\circ}_{\\phi(h)^{-1}} \\prescript{}{1}{\\sigma_h}$,  which shows that $\\bar{\\chi_3} \\approx \\bar{\\chi_1}$ by $\\phi$.  Next we prove that $\\beta_1$ and $\\beta_3$ also satisfy (\\ref{cocycle 1}) with respect to $\\phi$.  We have\n\n\\begin{align*}\n\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta_1(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))=\\beta_2(h_1, h_2),\\\\\n\\prescript{}{2}{\\nu}_{h_1+h_2}(-\\theta_2(h_1+h_2))+\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))=\\beta_3(h_1, h_2).\n\\end{align*}\n\nCombining these two equations and using the fact that $\\prescript{}{1}{\\nu}_{h}(\\theta_1(h))+\\prescript{}{2}{\\nu}_{h}(\\theta_2(h))=\\prescript{}{1}{\\nu}_{h}(\\theta_1(h)\\circ \\theta_2(h)) $,  we have \n\n\\begin{align*}\n\\beta_3(h_1, h_2)& =  \\prescript{}{2}{\\nu}_{h_1+h_2}(-\\theta_2(h_1+h_2))+\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta_1(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))\n \\\\ \n &\\hspace*{4mm} +\\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))+ \\prescript{}{2}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-(\\theta_2(h_1+h_2)\\circ\\theta_1(h_1+h_2)))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))\\\\\n& \\hspace*{4mm} + \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+ \\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-(\\theta_2(h_1+h_2)\\circ\\theta_1(h_1+h_2)))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1) \\circ \\theta_2(h_1) ))\\\\\n& \\hspace*{4mm} + \\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2)\\circ \\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-\\phi(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\phi(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\phi(h_2)).\n\\end{align*}\n\n\n\n\nSimilar calculation shows that  $\\tau_1, \\tau_3$ satisfy (\\ref{cocycle 2}) with respect to  $\\phi$. Hence the relation `$\\sim$' is an equivalence relation.\n\\hfill $\\Box$\n\n\\end{proof}\n\nDefine  $$\\mathcal{H}^2_{\\alpha}(H, I):=\\mathcal{Z}^2_{\\alpha}(H, I)\/ \\sim$$ and denote $[(\\chi,\\beta, \\tau)] \\in \\mathcal{H}^2_{\\alpha}(H, I)$, the equivalence class of $(\\chi,\\beta, \\tau)$. This concept will be used in next section.\n\n\\section{action of cohomology group on extensions}\nIn this section, we define a faithful group action of $\\operatorname{H} _{N}^2(H, \\operatorname{Z} (I))$  \\cite[Pg.6]{NMY} on $\\operatorname{Ext} _{\\alpha}(H, I)$ and we will show that this action is transitive whenever $I$ is  trivial skew brace.\n\\begin{thm} \\label{main}\nLet $\\alpha$ be a coupling from $H$ to $I$. Then there exists a bijection between $\\operatorname{Ext} _{\\alpha}(H, I)$ and  $\\mathcal{H}^2_{\\alpha}(H, I)$.\n\\end{thm}\n\n\\begin{proof}\nDefine $\\phi : \\operatorname{Ext} _{\\alpha}(H, I) \\rightarrow \\mathcal{H}^2_{\\alpha}(H, I)$ as follows. Let $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension with coupling $\\alpha$.  Fix an st-section $s$, then there exists an action $\\chi=(\\nu, \\mu, \\sigma)$  as we defined in \\eqref{actions} such that $\\bar{\\chi}=\\alpha$; also we have $(\\beta, \\tau)$ as we defined in (\\ref{cocycle1 sb}) and (\\ref{cocycle2 sb}),  together they satisfy equation (\\ref{parent relation}).  Set \n$$ \n\\phi([\\mathcal{E}])=[(\\chi, \\beta, \\tau)].\n$$\nThen by Proposition \\ref{well-def-act-coc} and Proposition \\ref{prop 2} the map $\\phi$ is well defined.\nNext we define a map $\\psi : \\mathcal{H}^2_{\\alpha}(H, I) \\rightarrow Ext_{\\alpha}(H, I)$ as follows.  Given an element  $(\\chi, \\beta, \\tau)$ of $\\mathcal{Z}^2_{\\alpha}(H, I),$ we define  binary operations on the set $H \\times I$ by setting\n\n(1) $(h_1, y_1)+(h_2, y_2)=(h_1+h_2, \\nu^{-1}_{h_1+h_2}(\\beta(h_1, h_2)+\\mu_{h_2}(\\nu_{h_1}(y_1))+\\nu_{h_2}(y_2)))$,\n\n(2) $(h_1, y_1) \\circ (h_2, y_2)=(h_1 \\circ h_2,\\tau(h_1, h_2) \\circ \\sigma_{h_2}(y_1) \\circ y_2).$\n\nIt is easy to check that (\\ref{cocycle 1}) and (\\ref{cocycle 2}) gives  the associativity of `$+$' and `$\\circ$', respectively, which is enough to see that $(H \\times I, +)$ and $(H  \\times I , \\circ)$ are groups  and (\\ref{parent relation}) proves that `$+$' and `$\\circ$' defined here satisfy \\eqref{bcomp}.  We denote this left skew brace structure by $(H, I , \\chi, \\beta, \\tau)$.  Now Consider the extension \n$$\\mathcal{E}(\\chi, \\beta, \\tau) := 0 \\to I \\stackrel{i}{\\to}  (H, I , \\chi, \\beta, \\tau) \\stackrel{\\pi}{\\to} H \\to 0,$$ where $i(y)=(0, y)$ and $\\pi(h, y)=h$ for all $h \\in H$ and $y \\in I$.  Define $\\psi$ by setting $$\\psi([(\\chi, \\beta, \\tau)])= [\\mathcal{E}(\\chi, \\beta, \\tau)].$$ \nWe show  that the map $\\psi$ is well defined. Let $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$, then there exist a map $\\theta : H \\rightarrow I $ such that $\\bar{\\chi_1} \\approx \\bar{\\chi_2}$ by $\\theta$ and $\\beta_1, \\beta_2$ satisfy \\eqref{equi 1} and $\\tau_1, \\tau_2$ satisfy \\eqref{equi 2}, respectively.  Define $\\zeta: \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) \\rightarrow \\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)$ given by \n$$\n\\zeta(h, y)=(h, \\theta(h) \\circ y).\n$$\n\nWe have\n\\begin{align*}\n\\zeta((h_1, y_1)+(h_2, y_2))=& (h_1+h_2, \\ \\theta(h_1+h_2) \\circ \\prescript{}{2}{\\nu}^{-1}_{h_1+h_2}(\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=&(h_1+h_2, \\ \\theta(h_1+h_2)+ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))\\\\\n&+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+ \\prescript{}{1}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))\\\\\n&+ \\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1) \\circ y_1))\\\\\n&+ \\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2) \\circ y_2)))\\\\\n=&(h_1, \\ \\theta(h_1) \\circ y_1)+(h_2, \\ \\theta(h_2) \\circ y_2).\n\\end{align*}\nWhich shows that $\\zeta$ is linear in `$+$'. Similarly we can show that $\\zeta$ is linear in `$\\circ$' as well. It is easy to see that $\\zeta$ is an isomorphism and the following diagram commutes\n$$\n\\begin{CD}\n 0 @>>> I @>>> \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) @>{{\\pi_1} }>> H  @>>> 0\\\\\n &&  @V{\\text{Id}}VV @V{\\zeta}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) @>{{\\pi_2} }>> H @>>> 0,\n\\end{CD}\n$$\nwhere $\\pi_1$ and $\\pi_2$ are natural projections. Hence, $\\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)$ and $\\mathcal{E}(\\chi_2, \\beta_2, \\tau_2)$ are equivalent extensions. That shows that the map $\\psi$ is well defined. It is easy to check that $\\psi$ is well-defined and $\\psi$ and $\\phi$ are inverses of each other. The proof is now complete.\n\\hfill $\\Box$\n\n\\end{proof}\n\nThe elements of $\\mathcal{Z}^2_{\\alpha}(H, I)$ are called associated triplets as every element of $\\mathcal{Z}^2_{\\alpha}(H, I)$ is associated to some extension in view of Theorem \\ref{Main thm}.\n\n\\begin{thm}\\label{action change}\nLet $H$ and $I$ be two skew braces and let $(\\chi, \\beta, \\tau) \\in \\mathcal{Z}^2_{\\alpha}(H, I)$ be an associated triplet.  If $\\chi^\\prime$ is an action of $H$ on $I$ for which  $\\bar{\\chi} \\approx\\bar{\\chi}^\\prime$, then there exist maps $\\beta^\\prime, \\tau^\\prime : H \\rightarrow I$ such that $(\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)$ is an associated triplet and $[(\\chi, \\beta, \\tau)]= [ (\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)]$.\n\\end{thm}\n \\begin{proof}\n In the view of Theorem \\ref{main} there exists an extension $$\\mathcal{E}(\\chi, \\beta, \\tau) := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0,$$ corresponding to the associated triplet $(\\chi, \\beta, \\tau)$. Let $s : H \\rightarrow E$ be a st-section inducing $(\\chi, \\beta, \\tau)$. Since $\\bar{\\chi} \\approx\\bar{\\chi^\\prime}$, there exist a map $\\theta : H \\rightarrow I$ such that $\\theta(0)=0$, $\\nu^{\\prime}_{h}=\\nu_{h} \\lambda_{\\theta(h)}$,  $\\mu^\\prime_{h}=i^+_{\\nu_h(-\\theta(h))} \\mu_h$ and $\\sigma^\\prime=i^{\\circ}_{\\theta(h)^{-1}} \\sigma_{h}$. Define an st-section  $s^\\prime(h)=s(h) \\circ \\theta(h)$ for all $ x \\in H$. Consequently we have an associated triplet $(\\chi_1, \\beta^\\prime, \\tau^\\prime)$ corresponding to the st-section $s^\\prime$ and it is easy to see that $\\chi^\\prime=\\chi_1$. Hence $[(\\chi, \\beta, \\tau)]=[(\\chi_1, \\beta^\\prime, \\tau^\\prime)])=[(\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)]$ as $(\\chi, \\beta^\\prime, \\tau^\\prime)$ and $(\\chi_1,  \\beta^\\prime, \\tau^\\prime)$ are associated triplet of the  same extension by different st-sections. This completes the proof.\n \\hfill $\\Box$\n\n\\end{proof}\n We now state a theorem analogous  to  \\cite[Theorem 3.6]{NMY}.  Let $H$ be a skew brace and $I$ be an abelian group.  \n \n Define\n\\begin{align*}\n \\operatorname{Z} _N^2(H, I)=\\Bigg\\{(g ,f) \\hspace{.1cm} \\Big \\vert  \\hspace{.1cm}g,f:H \\times H \\rightarrow I, \\hspace{.1cm} \\substack{ g,f \\hspace{.1cm} \\mbox{sastisy}\\hspace{.1cm} (\\ref{cocycle 1}) \\hspace{.1cm}\\mbox{and}\\hspace{.1cm} (\\ref{cocycle 2}),\\hspace{.1cm} \\mbox{respectively, and} \\\\ \\hspace{.1cm}\\mbox{vanish on degenerate tupples}}  \\Bigg\\},\n\\end{align*}\nand $\\operatorname{B} _N^2(H, I)$ is the collection of the pairs $(g, f) \\in  \\operatorname{Z} _N^2(H, I)$ such that there exists a map $\\theta$ from $H$ to $I$ with $g= \\nu_{h_1+h_2}(-\\theta(h_1+h_2))+\\mu_{h_2}((\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2))$ and $f=-\\theta(h_1 \\circ h_2) + \\sigma_{h_2}\\theta(h_1)) + \\theta(h_2)$.\n\nPut\n\\begin{align*}\n\\operatorname{Z} _N^1(H, I)=\\Bigg\\{\\substack{ \\theta \\hspace{.1cm}\\mbox{is a map from} \\hspace{.1cm} H \\hspace{.1cm} \\mbox{to} \\hspace{.1cm} I \\hspace{.1cm} \\mbox{such that }\\hspace{.1cm} \\theta(h_1 \\circ h_2)= \\sigma_{h_2}(\\theta(h_1)+\\theta(h_2) \\hspace{.1cm} \\\\ \\mbox{and} \\hspace{.1cm} \\nu_{h_1+h_2}(\\theta(h_1 + h_2))= \\mu_{h_2}(\\nu_{h_1}(\\theta(h_1))+\\nu_{h_2}(\\theta(h_2))}   \\Bigg\\},\n\\end{align*}\nthe set $\\operatorname{Z} _N^1(H, I)$ is called as the set of derivations, and\n$$\n\\operatorname{H} ^2_N(H, I):=\\operatorname{Z} _N^2(H, I)\/\\operatorname{B} _N^2(H, I)\n$$\n is the second cohomology group of $H$ by $I$.\n\n\\begin{thm} \nLet $H$ be a skew brace and let $I$ be an abelian group equiped with trivial brace structure. Let $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension.  Then the coupling and action are  same, and  there is a bijection between $\\operatorname{Ext} _{(\\nu, \\mu, \\sigma)}(H, I)$ and $\\operatorname{H} ^2_N(H, I)$. \n\\end{thm}\n\n\n\n\\begin{thm}\\label{Main thm}\nLet $[\\mathcal{E}] \\in  \\operatorname{Ext} _{\\alpha}(H, I)$ and $(\\chi, \\beta, \\tau)$ be an associated triplet of $\\mathcal{E}$. Then  for $[(\\beta_1, \\tau_1)] \\in \\operatorname{H} ^2_{N}(H, \\operatorname{Ann} (I))$, the operation $$[(\\beta_1, \\tau_1)] [\\mathcal{E}(\\chi, \\beta, \\tau)]= [\\mathcal{E}(\\chi, \\beta_1+\\beta, \\tau_1+\\tau)] $$ defines a free  action of the group $\\operatorname{H} ^2_{N}(H, \\operatorname{Ann} (I))$ on the set $\\operatorname{Ext} _{\\alpha}(H, I)$. If $I$ is trivial skew brace, then this action becomes transtivite. \n\\end{thm}\n\n\\begin{proof}\nIt is easy to check that the  action under consideration is well defined.  Let $[\\mathcal{E}(\\chi, \\beta, \\tau)] \\in Ext_{\\alpha}(H, I)$ and $[(\\beta_1, \\tau_1)] \\in  \\operatorname{H} _N^2(H, \\operatorname{Ann} (I))$ be such that $[(\\beta_1, \\tau_1)] [\\mathcal{E}(\\chi, \\beta, \\tau)]=[\\mathcal{E}(\\chi, \\beta, \\tau)]$. Then $$[(\\chi, \\beta_1+\\beta, \\tau_1+\\tau)]=[(\\chi, \\beta, \\tau)],$$ and therefore there exist a map $\\theta: H \\rightarrow I$ such that $\\theta(0)=0$ and $\\bar{\\chi} \\approx\\bar{\\chi}$ by $\\theta$, which implies that $\\theta(h) \\in \\operatorname{Ann} (I)$ for all $h \\in H$, and\n\\begin{equation*}\n-\\nu_{h_1+h_2}(\\theta(h_1+h_2))+\\beta(h_1, h_2)+ \\mu_{h_2}(\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2))=\\beta(h_1, h_2)+\\beta_1(h_1, h_2)\n\\end{equation*}\nand\n\\begin{equation*}\n\\theta(h_1 \\circ h_2)^{-1} \\circ \\tau(h_1, h_2) \\circ \\sigma_{h_2}(\\theta(h_1)) \\circ \\theta(h_2)=\\tau(h_1, h_2)+\\tau_1(h_1, h_2)\n\\end{equation*}\nfor all $h_1, h_2 , h_3 \\in H$(using the fact that $\\theta(h) \\in \\operatorname{Ann} (I) $ for all $h \\in H$).\n\n We have \n \\begin{align*}\n \\beta_1(h_1, h_2)&= \\nu_{h_1+h_2}(-\\theta(h_1+h_2))+\\mu_{h_2}((\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2)),\\\\ \n\\tau_1(h_1, h_2)&=-\\theta(h_1 \\circ h_2) + \\sigma_{h_2}\\theta(h_1)) + \\theta(h_2).\n \\end{align*}\nThus $[(\\tau_1, \\beta_1)]=1$, and hence the action is free.\n\nLet $\\mathcal{E}_1$ and $\\mathcal{E}_2$ be two elements in $Ext_{\\alpha}(H,I)$. Then for $i=1,2$,  $[\\mathcal{E}_i]=[\\mathcal{E}_i(\\chi_i, \\beta_i,\\tau_i)]$ for some associated trilpet $(\\chi_i, \\beta_i,\\tau_i)$, where $\\chi_i= (\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$.  By Theorem \\ref{action change}, we can construct $(\\beta^\\prime,\\tau^\\prime)$ such that $(\\chi_2, \\beta^\\prime, \\tau^\\prime)$ is an associated triplet with\n$$ [\\mathcal{E}_1(\\chi_1, \\beta_1, \\tau_1)]=[\\mathcal{E}_1(\\chi_2, \\beta^\\prime, \\tau^\\prime)].$$\nWe set \n$$\\beta_3(h_1, h_2)=\\beta^\\prime(h_1,h_2)-\\beta_2(h_1, h_2)$$ \nand \n$$\\tau_3(h_1, h_2)=\\tau^\\prime(h_1, h_2)\\circ \\tau_2(h_1, h_2)^{-1}.$$\nNow $(\\beta^\\prime, \\tau^\\prime)$ and $(\\beta_2, \\tau_2)$ are $2$-cocycles with the same action $\\chi_2$.  Then from (\\ref{action1 }), (\\ref{action2 }) and (\\ref{action3}) it follows that $\\beta_3(h_1, h_2) \\in Z(I,+)$ and $\\tau_3(h_1, h_2) \\in Soc(I)$ for all $h_1 , h_2 \\in H$. If we take $I$ to be trivial skew brace,  then $\\operatorname{Z} (I,+)= \\operatorname{Soc} (I)=\\operatorname{Ann} (I)$. Finally we get $\\beta_3, \\tau_3 : H \\rightarrow Ann(I)$. It is easy to see that $(\\beta_3, \\tau_3)$ is $2-cocycle$ with respect to the action $\\chi_2$ and $(\\beta_3, \\tau_3)[\\mathcal{E}(\\chi_2, \\beta_2, \\tau_2)]=[\\mathcal{E}(\\chi_2, \\beta^\\prime, \\tau^\\prime)]=[\\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)]\n$. Hence the action is transtive, which completes the proof.\n\\hfill $\\Box$\n\n\\end{proof}\n\nAs a consequence, we get\n\\begin{thm} \\label{main 1}\nLet $H$ be a skew brace and  let $I$ be a trivial skew brace with a fixed coupling $\\alpha$.  Then there exists a bijection between $Ext_{\\alpha}(H,I)$ and $Ext_{\\alpha}(H, \\operatorname{Z} (I))$.\n\\end{thm}\n\n\\section{action of automorphism group on extensions}\n\nThroughout this section we consider  $I$ to be a trivial skew brace and $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to}  E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension of skew braces.  Then $ \\nu :H \\rightarrow Aut(I,+)$ as defined in (\\ref{actions}),  is independent of the choice of an st-section.  Also $\\bar{\\nu}=  \\nu$ and  $\\bar{\\mu}:(H,+) \\rightarrow \\operatorname{Out} (I)$,  $\\bar{\\sigma}: (H, \\circ) \\rightarrow \\operatorname{Out} (I)$,  where $Out(I)$ represents the group of outer-automorphisms of $I$. For a pair  $(\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$ of skew brace automorphisms and an  extension  \n$$\\mathcal{E} :  0 \\rightarrow I \\stackrel{i}{\\rightarrow} E \\stackrel{\\pi}{\\rightarrow} H \\rightarrow 0$$\nof $H$ by $I$,  we can define a new extension\n$$\\mathcal{E}^{(\\phi, \\theta)} : 0 \\rightarrow I \\stackrel{i\\theta}{\\longrightarrow} E \\stackrel{\\phi^{-1} \\pi}{\\longrightarrow} H \\rightarrow 0$$\nof $H$ by $I$. Thus, for a given $(\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$, we can define a map from $\\operatorname{Ext} (H, I)$ to itself given by \n\\begin{equation}\\label{act1 sb}\n[\\mathcal{E}] \\mapsto  [ \\mathcal{E}^{(\\phi, \\theta)}].\n\\end{equation}\n If $\\phi$ and $\\theta$ are identity automorphisms, than obviously $\\mathcal{E}^{(\\phi, \\theta)} = \\mathcal{E}$. It is also easy to see that  \n$$[\\mathcal{E}]  ^{(\\phi_1, \\theta_1) (\\phi_2, \\theta_2)}=  \\big([\\mathcal{E}]^{(\\phi_1, \\theta_1)}\\big)^{(\\phi_2, \\theta_2)}.$$\nWe conclude that the association \\eqref{act1 sb} gives an action of the group $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$  on the set $\\operatorname{Ext} (H, I)$. From Corollary \\ref{cor 1} ,  we know that  $\\operatorname{Ext} (H, I) = \\bigsqcup_{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$.  \\emph{Let $(\\nu, \\bar{\\mu}, \\bar{\\sigma})$ be an arbitrary but fixed coupling from $H$ to $I$.}  Let $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ denote the stabiliser of $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ in $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$; more explicitly\n$$\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} = \\{ (\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I) \\mid \\nu_h=\\theta^{-1}\\nu_{\\phi(h)}\\theta,  \\bar{\\mu}_h = \\theta^{-1}\\bar{\\mu}_{\\phi(h)}\\theta \\mbox{ and } \\bar{\\sigma}_h = \\theta^{-1}\\bar{\\sigma}_{\\phi(h)}\\theta \\}.$$ \nIt is easy to see that $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$  is a subgroup of $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$.  For details see \\cite[Pg.15]{NMY}.\n\nNext we consider an action of $ \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H, Z(I))$ (same as in \\cite{NMY}) by \n\\begin{equation} \\label{act3 sb}\n[(g, f)] \\mapsto [\\big(g^{(\\phi, \\theta)}, f^{(\\phi, \\theta)}\\big)],\n\\end{equation}\nwhere $g^{(\\phi, \\theta)}(h_1, h_2)=\\theta^{-1}(g(\\phi(h_1), \\phi(h_2))$.  This action of $  \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H, Z(I))$  is same as the action of $  \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$  transferred on $\\operatorname{H} ^2_N(H, Z(I))$ through bijection of Theorem \\ref{main 1} . Using this action we can define the semi-direct product $\\Gamma =   \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})} \\ltimes \\operatorname{H} ^2_N(H,Z(I))$.  We wish to define an action of $\\Gamma$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$. For $(c, h) \\in \\Gamma$ and $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$, define \n\\begin{equation}\\label{act4 sb}\n[\\mathcal{E}]^{(c, h)} = ([\\mathcal{E}]^c)^h.\n\\end{equation}\n\n\\begin{lemma}\\label{wells2 sb}\nThe rule  in \\eqref{act4 sb}   gives an action of  $\\Gamma$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows on the  lines of   \\cite[Lemma 5.2]{NMY}.\n\n\\end{proof}\n\n Let $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ be a fixed extension.  Since the action of $\\operatorname{H} ^2_N(H,Z(I))$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ is transitive and faithful, for  each $c \\in \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$, there exists a unique element (say) $h_c$  in  $\\operatorname{H} ^2_N(H,Z(I))$ such that  \n $$[\\mathcal{E}]^{c} = [\\mathcal{E}]^{h_c}.$$\n We thus have a well defined map $ \\omega(\\mathcal{E}): \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})} \\rightarrow \\operatorname{H} ^2_N(H,Z(I))$ given by\n \\begin{equation}\\label{wells-map sb}\n \\omega(\\mathcal{E})(c)=h_c\n \\end{equation}\n for $c \\in \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$. \n \n\\begin{lemma}\\label{wells3 sb}\nThe map $ \\omega(\\mathcal{E}): \\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} \\rightarrow \\operatorname{H} ^2_N(H,Z(I))$ defined in \\eqref{wells-map sb} is a derivation with respect to the action of $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H,Z(I))$ given in \\eqref{act3 sb}.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows on the  lines of  \\cite[Lemma  5.3]{NMY}.\n\\end{proof}\n\nLet \n$$\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H $$\nbe an extension of a left skew brace $H$ by a trivial skew brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\mu, \\sigma)}(H,I)$.\nLet  $\\operatorname{Autb} _I(E)$ denote the subgroup of $\\operatorname{Autb} (E)$ consisting of all automorphisms of $E$ which normalize $I$, that is,\n$$\\operatorname{Autb} _I(E) := \\{ \\gamma \\in \\operatorname{Autb} (E) \\mid \\gamma(y) \\in I \\mbox{ for all }  y \\in I\\}.$$ \nFor $\\gamma \\in \\operatorname{Autb} _I(E)$, set $\\gamma_I := \\gamma |_I$, the restriction of $\\gamma$ to $I$, and $\\gamma_H$ to be the automorphism of $H$ induced by $\\gamma$. More precisely, $\\gamma_H(h) = \\pi(\\gamma(s(h)))$ for all $h \\in H$, where $s$ is an st-section of $\\pi$. Notice that the definition of $\\gamma_H$ is independent of  the choice of an st-section. Define a map $\\rho(\\mathcal{E}) :  \\operatorname{Autb} _I(E) \\rightarrow  \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$ by\n$$\\rho(\\mathcal{E})(\\gamma)=(\\gamma_H, \\gamma_I).$$ \nAlthough $\\omega(\\mathcal{E})$ is not a homomorphism, but we can still talk about its set theoretic kernel, that is,\n$$\\operatorname{Ker} (\\omega(\\mathcal{E})) = \\{c \\in C_{(\\nu, \\bar{\\mu},\\bar{\\sigma})} \\mid [\\mathcal{E}]^c=[\\mathcal{E}]\\}.$$\n\n\\begin{prop}\\label{wells4 sb}\nFor the extension $\\mathcal{E}$,   $\\operatorname{Im} (\\rho(\\mathcal{E})) \\subseteq \\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ and  \n$\\operatorname{Im} (\\rho(\\mathcal{E})) = \\operatorname{Ker} (\\omega(\\mathcal{E}))$.\n\\end{prop}\n\n\\begin{proof}\nThe proof follows on the  lines of \\cite[Proposition 5.4]{NMY}.\n\\end{proof}\n\nContinuing with the above setting, set $\\operatorname{Autb} ^{H, I}(E) := \\{\\gamma \\in \\operatorname{Autb} _I(E) \\mid \\gamma_I = \\operatorname{Id},  \\gamma_H = \\operatorname{Id}\\}$. Notice that  $\\operatorname{Autb} ^{H,I}(E)$ is precisely the kernel of $\\rho(\\mathcal{E})$. Hence, using Proposition \\ref{wells4 sb}, we get\n\n\n\\begin{thm}\\label{wells5 sb}\nLet $\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be a extension of a left skew  brace  $H$ by a trivial skew brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H,I)$. Then we have the following exact sequence of groups \n$$0 \\rightarrow \\operatorname{Autb} ^{H,I}(E) \\rightarrow \\operatorname{Autb} _I(E) \\stackrel{\\rho(\\mathcal{E})}{\\longrightarrow} \\operatorname{C} _{(\\nu,\\bar{\\mu}, \\bar{\\sigma})} \\stackrel{\\omega(\\mathcal{E})}{\\longrightarrow} \\operatorname{H} ^2_N(H,Z(I)),$$\nwhere $\\omega(\\mathcal{E})$ is, in general,  only a derivation.\n\\end{thm}\n\nFurther we have\n\\begin{prop}\\label{wells6 sb}\nLet  $\\mathcal{E} :  0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be an extension of $H$ by $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$.  Then  $\\operatorname{Autb} ^{H,I}(E) \\cong \\operatorname{Z} ^1_N(H,Z(I))$.\n\\end{prop}\n\n\\begin{proof}\nThe map $\\psi : \\operatorname{Z} ^1_N(H,Z(I)) \\rightarrow \\operatorname{Autb} ^{H,I}(E)$ defined by $\\psi(\\lambda)(s(h) \\circ y)= s(h) \\circ \\lambda(h) \\circ y$ is the  required isomorphism. Rest proof follows on the lines of \\cite[Proposition 5.6]{NMY}\n\\end{proof}\n\nWe finally get the following Wells' like exact sequence for skew braces.\n\\begin{thm}\\label{wells7 sb}\nLet $\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be an extension of a left skew brace  $H$ by a trivial skew  brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H,I)$. Then we have the following exact sequence of groups \n$$0 \\rightarrow \\operatorname{Z} ^1_N(H,\\operatorname{Z} (I)) \\rightarrow \\operatorname{Autb} _I(E) \\stackrel{\\rho(\\mathcal{E})}{\\longrightarrow} \\operatorname{C} _{(\\nu,\\bar{\\mu}, \\bar{\\sigma})} \\stackrel{\\omega(\\mathcal{E})}{\\longrightarrow} \\operatorname{H} ^2_N(H,\\operatorname{Z} (I)),$$\nwhere $\\omega(\\mathcal{E})$ is, in general,  only a derivation.\n\\end{thm}\n\n\\section{Acknowledgements}\n\nI am grateful to my supervisor Prof.  M. K.  Yadav for his constant support, comments and suggestions while doing this project.  I would like to thank Prof.  L.  Vendramin for his kind help in writing GAP code.  The author acknowledge Harish-Chandra Research institute for fantastic facilities and for the serene ambience that it facilitates.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUnderstanding the universal behavior of classical many-body systems near their critical points is a central goal %\nof classical statistical mechanics. Although this is a difficult problem in general, in one and two spatial dimensions, significant insights have been provided by exactly solved models~\\cite{baxter2016exactly}. One important open problem is to generalize these solutions to three-dimensional~(3D) systems with realistic short-range interactions. Despite a long effort with some preliminary results~\\cite{suzuki1972solution,zamolodchikov1980tetrahedra,bazhanov1992new,huang1997exact,dhar2008exact,mangazeev2013integrable}, no physical 3D model has been exactly solved that displays a genuinely 3D phase transition~\\footnote{Among the models constructed in Refs.~\\cite{suzuki1972solution,zamolodchikov1980tetrahedra,bazhanov1992new,huang1997exact,dhar2008exact,mangazeev2013integrable}, only the models in  Refs.~\\cite{suzuki1972solution,huang1997exact} have phase transitions, and in these the 3D partition function factorizes into a product of partition functions of 2D systems, giving the phase transitions an essentially 2D character.}.  \n\n\n\nIn this paper, we make progress in this direction by exactly solving a classical Ising model on a special 3D lattice, as depicted in Fig.~\\ref{fig:hc0}, \nalthough with the caveat that the model has imaginary coupling constants. \nThe transfer matrix of this system has a  structure  similar to  a non-Hermitian version of the 2D Kitaev honeycomb model~\\cite{Kitaev2006}, and the partition function can be obtained using the representation theory of the so($2N$) Lie algebra and the corresponding Lie group.\nThe solution displays a third order phase transition between two distinct phases, and near the critical point we can exactly obtain a critical exponent of the model. \n\nThe phases are interesting in their own right, as they are distinguished by topological properties. Specifically, there is a family of loop observables whose \n expectation values distinguish the two phases and are equal to some rational numbers~($0$, $1$, or $1\/3$) depending on the topology of the loop. \n\nDespite its complex coupling constants~(also a complication of some previous approaches~\\cite{zamolodchikov1980tetrahedra,bazhanov1992new}), our findings have physical relevance. First, we show in Sec.~\\ref{sec:physical_model} that the topological features discovered in one of the phases of the model with complex couplings also exist in a similar exactly solvable model with real-valued couplings. More speculatively, it is possible more generally that the long-distance property of our model belongs to the same universality class of certain physical 3D classical systems. \nIt remains an open question whether the other phase of our model can also be  reproduced in a physical system, but if there indeed exists a physical classical system that has the two phases mentioned above and a phase transition between them, then the concept of universality suggests that the long-distance behaviors and the critical exponent we obtain here will apply to\nsuch physical systems.\n\nAs another point of physical relevance for the model with complex couplings, in Sec.~\\ref{sec:quantum_amplitude} we show two constructions that realize the partition function $Z$ of our model in certain dynamical processes of a 3D quantum system: one is to map $Z$ to the transition amplitude between a family of product states, the other is to realize $Z$ as the coherence of a probe spin coupled to the whole 3D system. Both constructions in principle allow the free energy to be experimentally measured, albeit with an exponentially small signal. Under these mappings, the phase transition of our model corresponds to a dynamical quantum phase transition~(DQPT)~\\cite{heyl2013dynamical,Heyl_2018}, a phenomenon that has gained much attention recently. Statistical mechanics with complex configuration energies also appears in  the study of Lee-Yang zeros~\\cite{yang1952statistical,lee1952statistical,wei2012lee,peng2015experimental}, non-Hermitian quantum systems~\\cite{moiseyev2011non,Gong2018Topological,Ashida2020Non}, and complex conformal field theories~\\cite{faedo2020holographic}.   \n\n\n\n\n\nOur paper is organized as follows. In Sec.~\\ref{sec:model} we define our model and a family of loop observables of interest. In Sec.~\\ref{sec:solution} we present the exact solution of  the model: in Sec.~\\ref{sec:TM} we derive the transfer matrix of the classical model, in Sec.~\\ref{sec:map_fermion} we use a spin-fermion mapping to reduce the problem to a free fermion problem, in Sec.~\\ref{sec:solve_fermion_TM} we solve the eigenvalues of the free fermion transfer matrix and calculate the thermodynamic free energy, in Sec.~\\ref{sec:phase_boundary} we obtain the phase diagram, in Sec.~\\ref{sec:critical_exp} we calculate a critical exponent, and in Sec.~\\ref{sec:TPloop} and Sec.~\\ref{sec:loop_observables} we calculate the expectation values of loop observables and demonstrate their topological properties. In Sec.~\\ref{sec:justification} we give two physical implications of our model: the existence of a physical classical phase with similar topological behaviors~(Sec.~\\ref{sec:physical_model}), and realizations of the partition function in quantum dynamical processes~(Sec.~\\ref{sec:quantum_amplitude}). %\nIn Sec.~\\ref{sec:summary} we summarize our results. \nThe Appendices contain technical results used throughout our arguments. \n\n\\section{The Model}\\label{sec:model}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.9\\linewidth]{3dbrickwall-yellowthin-loops-large-RGB.png}}%\n\t\\caption{\\label{fig:hc0} Definition of the model and loop observables. The classical system sits on a 3D stacking of the brick wall lattice, of arbitrarily large extent in each direction. Classical spins lie on vertices, and they only interact via the thicker links. The horizontal links~(red, blue, black) have real coupling constants $J_x,J_y,J_z$, for $x$-planes, $y$-planes, and $z$-planes, respectively. The coupling constant $J_\\perp$ for the vertical links~(pink) and the  external field $h$ are imaginary when the solvability condition Eq.~\\eqref{cond:exact} is met. The yellow shaded cuboid shows an example of the loop observable $\\sigma[\\mathfrak{L}_{(xy)}]$ for a contractible loop $\\mathfrak{L}$~(here being an elementary plaquette), which is equal to the product of Ising spins on the larger yellow vertices~[see Eq.~\\eqref{eq:def_loop_observable}]. Similarly, the green shaded rectangle shows an example of $\\sigma[\\mathfrak{L}_{(yz)}]$ for a noncontractible loop, extended infinitely to the right and to the left.\n\t}\n\\end{figure}\nIn this section we define our model and the class of physical observables we are interested in.\nThe model is defined on a 3D stacking of the 2D brick wall lattice, with classical Ising spins, $\\sigma_j \\in \\{-1, +1\\}$, lying on vertices $j$, as shown in Fig.~\\ref{fig:hc0}, and we use periodic boundary conditions~(PBC) for all the three directions for simplicity. Nearest neighbor Ising-type interactions exist only on a subset of links in this lattice, which are shown in Fig.~\\ref{fig:hc0} as thick red, blue, black, and pink links. The  energy of the system for a specific classical spin configuration is\n\\begin{eqnarray}\\label{eq:HIK}\n  H[\\{\\sigma\\}]&=&-J_x\\sum_{\\langle ij\\rangle\\in \\mathbf{X}}\\sigma_{i}\\sigma_{j}-J_y\\sum_{\\langle ij\\rangle\\in \\mathbf{Y}}\\sigma_{i}\\sigma_{j}\\\\\n  &&-J_z\\sum_{\\langle ij\\rangle\\in \\mathbf{Z}}\\sigma_{i}\\sigma_{j}-J_\\perp\\sum_{\\langle ij\\rangle\\in\\boldsymbol{\\perp}}\\sigma_{i}\\sigma_{j}+h\\sum_i\\sigma_i,\\nonumber\n \\end{eqnarray}\nwhere $\\mathbf{X}$ denotes the set of all thick links on $x$-planes, and similarly for $\\mathbf{Y},\\mathbf{Z}$, while $\\boldsymbol{\\perp}$ is the set of all the vertical links in Fig.~\\ref{fig:hc0}, and the external field $h$ acts on all spins. \nThe goal is to find the partition function\n\\begin{eqnarray}\\label{eq:Z}\nZ(K_x,K_y,K_z,K_\\perp,\\beta h)=\\sum_{\\{\\sigma\\}}e^{-\\beta H[\\{\\sigma\\}]},\n\\end{eqnarray}\nwhere $K_i=\\beta J_i,i=x,y,z,\\perp$. The free energy is related to the partition function by\n\\begin{equation}\\label{def:free_energy0}\n\tF=-k_B T \\ln Z.\n\\end{equation}\nThe model is exactly solvable when the following conditions hold:\n\\begin{equation}\\label{cond:exact}\n4J_\\perp\\beta\\equiv \\pi i~(\\mathrm{mod}~2\\pi i),~~~2h\\beta\\equiv \\frac{\\pi i}{2}~(\\mathrm{mod}~ 2\\pi i).\n\\end{equation}\nAfter imposing these solvability conditions, there remains a three-dimensionless-parameter space~$(K_x,K_y,K_z)$ of solutions.\n\n\nBeyond the free energy~(and its derivatives), we also consider the thermal expectation values of a family of loop observables that are products of $\\sigma_j$s on closed loops, defined by the following procedure:\\\\\n(1) Choose a loop $\\mathfrak{L}$ on the 2D brick wall lattice~($\\mathfrak{L}$ must consist of edges of the brick wall lattice);\\\\\n(2) Choose two nearest neighbor planes of type $\\alpha$ and $\\beta$ of the 3D lattice, denoted $(\\alpha \\beta)$, which can be $(xy),(yz)$ or $(zx)$;\\\\\n(3) Denote by $\\mathfrak{L}_{(\\alpha\\beta)}$ the graph consisting of all sites in the loop $\\mathfrak{L}$ of both $\\alpha$ and $\\beta$ planes and the edges of the lattice joining pairs of these sites;\\\\ \n(4) For a lattice site $i\\in \\mathfrak{L}_{(\\alpha\\beta)}$, denote by $\\bar{i}$ the same site of the other plane~(if $i\\in \\alpha$, then $\\bar{i}\\in \\beta$ and vice versa);\\\\\n(5) For  $i\\in \\mathfrak{L}_{(\\alpha\\beta)}$, define $n(i)$ to be the number of thick horizontal edges in $\\mathfrak{L}_{(\\alpha\\beta)}$ linked to $i$~[notice that $n(i)\\in\\{0,1\\}$];\\\\\n(6) The loop product is defined as \n\\begin{equation}\\label{eq:def_loop_observable}\n\t\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]=\\prod_{i\\in \\mathfrak{L}_{(\\alpha\\beta)}} \\sigma_i^{n(\\bar{i})}.\n\\end{equation}\nIn Fig.~\\ref{fig:hc0} we illustrate the definition of $\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]$ for a contractible and a noncontractible loop $\\mathfrak{L}$.\nIn Sec.~\\ref{sec:TPloop} we will compute their thermal expectation values\n\\begin{eqnarray}\\label{eq:thermal_loop}\n\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle=\\frac{1}{Z}\\sum_{\\{\\sigma\\}}\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}] e^{-\\beta H[\\{\\sigma\\}]}.\n\\end{eqnarray}\nWe will see  that the expectation values of these observables are sensitive to the topology of the loop $\\mathfrak{L}_{(\\alpha\\beta)}$. Namely, for a contractible loop $\\mathfrak{L}_{(\\alpha\\beta)}$ we have $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle=\\pm 1$~(and the same loop $\\mathfrak{L}_{(\\alpha\\beta)}$ takes the same value for different phases), while for a non-contractible loop $\\mathfrak{L}_{(\\alpha\\beta)}$, $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is equal to $0$ in one phase~(the $A$-phase) and $-1\/3$ in another phase~(the $B$-phase). Therefore, noncontractible loop observables can be used as order parameters of this model.\n\n\\section{The solution}\\label{sec:solution}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.75\\linewidth]{brickwall-unitcell-RGB.png}}\n\t\\caption{\\label{fig:brickwall-unitcell} The 2D brick wall lattice on which the transfer matrix Eq.~\\eqref{eq:def_TM} is defined. A unit cell is shown in the shaded square. The conserved loop operator $\\hat{W}_p$ acts on the six spins of the elementary plaquette $p$, and the conserved noncontractible loop $\\hat{\\Phi}_x$~($\\hat{\\Phi}_y$) acts on a row~(column) of spins. The $\\pm \\kappa$ shown next to each link is the real part of the small perturbation to the link coupling constant needed to gap the fermionic spectrum of the $B$-phase. }\n\\end{figure}  \n\\subsection{The Transfer Matrix}\\label{sec:TM}\nThe first step  to solve this model is to find the transfer matrix $\\hat{T}$ for each period of $x,y,z$ planes, as shown in Fig.~\\ref{fig:hc0}, defined so that $Z=\\mathrm{Tr}[\\hat{T}^M]$, where $M$ is the total number of periods. We will show that when the conditions~\\eqref{cond:exact} are satisfied, the transfer matrix is\n\\begin{eqnarray}\\label{eq:def_TM}\n  \\hat{T}&=&\\exp\\left(K_x\\sum_{\\langle ij\\rangle\\in X_{2D}}\\hat{\\sigma}^x_{i}\\hat{\\sigma}^x_{j}\\right)\\exp\\left(K_y\\sum_{\\langle ij\\rangle\\in Y_{2D}}\\hat{\\sigma}^y_{i}\\hat{\\sigma}^y_{j}\\right)\\nonumber\\\\\n  &&\\times\\exp\\left(K_z\\sum_{\\langle ij\\rangle\\in Z_{2D}}\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}\\right),\n\\end{eqnarray}\nwhere $\\hat{\\sigma}^{x,y,z}_i$ are Pauli operators acting on the spin located at site $j$ of the 2D brick wall lattice shown in Fig.~\\ref{fig:brickwall-unitcell}, $X_{2D}$ denotes the set of all the $x$-links shown in Fig.~\\ref{fig:brickwall-unitcell}, and similarly for $Y_{2D},Z_{2D}$. Henceforth, we use $\\sum_{x}$, $\\sum_{y}$, and $ \\sum_{z}$ as abbreviations for  $\\sum_{(i,j) \\in  X_{2D}}$, $\\sum_{(i,j) \\in  Y_{2D}}$, and $\\sum_{(i,j) \\in  Z_{2D}}$, respectively.\nWe prove Eq.~\\eqref{eq:def_TM} by inserting resolutions of identity on each plane in $Z=\\mathrm{Tr}[\\hat{T}^M]$ in Eq.~\\eqref{eq:def_TM} and showing that it reproduces\nEq.~\\eqref{eq:Z}. The trick here is that when inserting resolution of identity, we use the $\\hat{\\sigma}^x$ basis $|X\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_x$ on $x$-planes, $\\hat{\\sigma}^y$ basis $|Y\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_y$ on $y$-planes, and $\\hat{\\sigma}^z$ basis $|Z\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_z$ on $z$-planes, where $\\hat{\\sigma}^x_j|\\sigma_j\\rangle_x=\\sigma_j|\\sigma_j\\rangle_x$ and similarly for $|\\sigma_j\\rangle_y,|\\sigma_j\\rangle_z$. Therefore, we have\n\\begin{widetext}\n\\begin{eqnarray}\\label{eq:quantum_classical_map}\n  \\mathrm{Tr}[\\hat{T}^M]&=&\\sum_{\\substack{X_1,Y_1,Z_1,\\ldots,\\\\ X_M,Y_M,Z_M}}\\langle X_1|e^{K_x\\sum_{x}\\hat{\\sigma}^x_i\\hat{\\sigma}^x_j}|Y_1\\rangle \\langle Y_1|e^{K_y\\sum_{y}\\hat{\\sigma}^y_i\\hat{\\sigma}^y_j}|Z_1\\rangle \\langle Z_1|e^{K_z\\sum_{z}\\hat{\\sigma}^z_i\\hat{\\sigma}^z_j}|X_2\\rangle\\langle X_2|\\cdots\\nonumber\\\\\n                        &&~~~~~~~~{}\\times|X_M\\rangle\\langle X_M|e^{K_x\\sum_{x}\\hat{\\sigma}^x_i\\hat{\\sigma}^x_j}|Y_M\\rangle \\langle Y_M|e^{K_y\\sum_{y}\\hat{\\sigma}^y_i\\hat{\\sigma}^y_j}|Z_M\\rangle \\langle Z_M|e^{K_z\\sum_{z}\\hat{\\sigma}^z_i\\hat{\\sigma}^z_j}|X_1\\rangle\\nonumber\\\\\n                        &=&\\sum_{\\{\\sigma\\}}\\exp\\left(K_x\\sum_{\\langle ij\\rangle\\in \\mathbf{X}} \\sigma_i \\sigma_j+K_y\\sum_{\\langle ij\\rangle\\in \\mathbf{Y}} \\sigma_i \\sigma_j+K_z\\sum_{\\langle ij\\rangle\\in \\mathbf{Z}} \\sigma_i \\sigma_j\\right)\\nonumber\\\\\n                        &&~~~~{}\\times\\langle X_1|Y_1\\rangle \\langle Y_1|Z_1\\rangle \\langle Z_1|X_2\\rangle\\cdots\n  \\langle X_M|Y_M\\rangle \\langle Y_M|Z_M\\rangle \\langle Z_M|X_1\\rangle.\n\\end{eqnarray}\n\\end{widetext}\nThe first factor corresponds to the classical Boltzmann weight contributed by all the horizontal links. For the overlap matrices in the last line of Eq.~\\eqref{eq:quantum_classical_map}, using a suitable phase convention for basis states \n\\begin{eqnarray}\n\t|\\pm 1\\rangle_z&=&\\{|\\uparrow\\rangle,e^{\\pi i\/4}|\\downarrow\\rangle \\},\\nonumber\\\\\n\t|\\pm 1\\rangle_x&=&\\{\\frac{|\\uparrow\\rangle+|\\downarrow\\rangle}{\\sqrt{2}}e^{3\\pi i\/4},\\frac{|\\uparrow\\rangle-|\\downarrow\\rangle}{\\sqrt{2}}e^{\\pi i\/2} \\}\\nonumber\\\\\n\t|\\pm 1\\rangle_y&=&\\{\\frac{|\\uparrow\\rangle+i|\\downarrow\\rangle}{\\sqrt{2}}e^{-3\\pi i\/4},\\frac{|\\uparrow\\rangle-i|\\downarrow\\rangle}{\\sqrt{2}}e^{-\\pi i\/2} \\},\\nonumber\n\\end{eqnarray}\nwe have $${}_x\\langle \\sigma|\\sigma'\\rangle_y={}_y\\langle \\sigma|\\sigma'\\rangle_z ={}_z\\langle \\sigma|\\sigma'\\rangle_x  =\\frac{1}{\\sqrt{2}}e^{\\frac{\\pi i}{4}(\\sigma\\sigma'-\\frac{\\sigma+\\sigma'}{2}+3)}.$$ The overlaps give the Boltzmann weights contributed by the vertical links and external field terms with $\\beta J_\\perp=\\pi i\/4,\\beta h=\\pi i\/4$, up to an irrelevant constant shift of the energy. Also, one can show that adding $2\\pi i $ to $4J_\\perp\\beta$ or $2h\\beta$ will only multiply the partition function by an irrelevant overall constant phase factor, since whenever we flip a spin $\\sigma_j$, the imaginary part of $\\beta H[\\{\\sigma\\}]$ changes by $\\pm 2( 2s J_\\perp-  h )\\beta $, where $s=(\\sigma'_j+\\sigma''_j)\/2\\in\\{-1,0,+1\\}$, and $\\sigma'_j$~($\\sigma''_j$) is the neighbor of $\\sigma_j$ lying above~(below) it. Therefore the model Eq.~\\eqref{eq:HIK} has transfer matrix Eq.~\\eqref{eq:def_TM} when Eq.~\\eqref{cond:exact} is satisfied.  \n\n\n \nNow that we have obtained the transfer matrix $\\hat{T}$ of our model, the next step is to calculate the largest~(in magnitude) eigenvalue $\\Lambda_{\\max}$ of $\\hat{T}$, which governs the free energy in the thermodynamic limit \n\\begin{equation}\\label{def:free_energy}\n\tF =-M k_B T \\ln\\Lambda_{\\max}+O(\\Lambda_1^M\/\\Lambda_{\\max}^M),\n\\end{equation}\nwhere $\\Lambda_1$ is the next-to-largest~(in magnitude) eigenvalue of $\\hat{T}$. We will calculate the eigenvalues of $\\hat{T}$ in two steps: in Sec.~\\ref{sec:map_fermion} we map the transfer matrix $\\hat{T}$ to a free fermion transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Tbondfermion}, and then in Sec.~\\ref{sec:solve_fermion_TM} we solve the eigenvalues of this free fermion transfer matrix.\n\n\n\\subsection{Mapping to a free fermion problem}\\label{sec:map_fermion}\nOur goal in this section is to map the transfer matrix $\\hat{T}$ to a free fermion transfer matrix $\\hat{T}'$, written in terms of Majorana fermion bilinear operators. While this can be accomplished by Kitaev's original technique~\\cite{Kitaev2006}, or by using a Jordan-Wigner transformation~\\cite{Feng2007JordanWigner},\nhere we use the algebraic method developed in Refs.~\\cite{Nussinov2009bond,Cobanera2011bond,Chapman2020characterizationof,Ogura2020geometric}, which is far simpler. The key idea of this technique is that, instead of considering the mapping of each individual spin operators, we view the interaction term on each link $\\langle ij\\rangle$ as a whole, and consider the algebra generated by all these terms. We write the transfer matrix as \n\\begin{equation}\\label{eq:Tbond}\n\t\\hat{T}=e^{K_x\\sum_{x}\\hat{\\gamma}_{ij}}e^{K_y\\sum_{y}\\hat{\\gamma}_{ij}}e^{K_z\\sum_{z}\\hat{\\gamma}_{ij}},\n\\end{equation}\nwhere the bond operators are defined as $\\hat{\\gamma}_{ij}=\\hat{\\sigma}_i^\\alpha\\hat{\\sigma}_j^\\alpha$ if $\\langle ij\\rangle$ is an $\\alpha$-link in the 2D brick wall lattice. We now construct another transfer matrix\n\\begin{eqnarray}\\label{eq:Tbondfermion}\n\t\\hat{T}'&=&e^{K_x\\sum_{x}\\hat{\\gamma}'_{ij}}e^{K_y\\sum_{y}\\hat{\\gamma}'_{ij}}e^{K_z\\sum_{z}\\hat{\\gamma}'_{ij}}\\nonumber\\\\\n\t&\\equiv& e^{K_x\\sum_{x}u_{ij}i\\hat{c}_i\\hat{c}_j}e^{K_y\\sum_{y}u_{ij}i\\hat{c}_i\\hat{c}_j}e^{K_z\\sum_{z}u_{ij}i\\hat{c}_i\\hat{c}_j},\n\\end{eqnarray}\nwhich has exactly the same exponential structure  and the same set of parameters as $\\hat{T}$, but has the bond operators replaced by Majorana fermion bilinears $\\hat{\\gamma}'_{ij}\\equiv u_{ij}i\\hat{c}_i\\hat{c}_j$ on each link, where $\\hat{c}^\\dagger_i=\\hat{c}_i$, and $\\{\\hat{c}_i,\\hat{c}_j\\}=2\\delta_{ij}$.\nHere $u_{ij}$ is a real number defined independently on each link,  whose value is to be determined later. Notice that the ordering of Majorana operators $\\hat{c}_i\\hat{c}_j$ matters in the sum since they anti-commute; throughout this paper, we use the convention that whenever we sum~(or product) over links, each link $\\langle ij\\rangle$ appears only once in the sum, with $i$ representing an even site~(black dots in Fig.~\\ref{fig:brickwall-unitcell}) and $j$ representing an odd site ~(white open circles in Fig.~\\ref{fig:brickwall-unitcell}), and we always order $\\hat{c}_i$ to the left unless otherwise stated. \n\nThe goal now is to choose these real coefficients $\\{u_{ij}\\}$ \\textit{such that the algebra generated by $\\{\\hat{\\gamma}_{ij}\\}$ is isomorphic to the algebra generated by $\\{\\hat{\\gamma}'_{ij}\\}$.} Once this is done, Refs.~\\cite{Nussinov2009bond,Cobanera2011bond,Chapman2020characterizationof,Ogura2020geometric} claim that there exists a unitary mapping $\\hat{U}$ between the two systems such that $\\hat{\\gamma}'_{ij}=\\hat{U} \\hat{\\gamma}_{ij}\\hat{U}^\\dagger$ for all links $\\langle ij\\rangle$~(we will also need to check that the Hilbert space dimensions of the two systems are the same), leading to $\\hat{T}'=\\hat{U}\\hat{T}\\hat{U}^\\dagger$, i.e. $\\hat{T}$ and $\\hat{T}'$ have the same eigenvalues.  Requiring the two algebras to be isomorphic  means that any algebraic relation satisfied by the generators $\\{\\hat{\\gamma}_{ij}\\}$, say $f(\\{\\hat{\\gamma}_{ij}\\})=0$, must be satisfied by $\\{\\hat{\\gamma}'_{ij}\\}$ as well, $f(\\{\\hat{\\gamma}'_{ij}\\})=0$, and vice versa. In our case, this leads to the following four families of relations:\\\\\n\\textit{Relation 1.} We have $\\hat{\\gamma}^2_{ij}=1$ for each link $\\langle ij\\rangle$, and therefore we must require $\\hat{\\gamma}'^2_{ij}=u_{ij}^2=1$, which constrains $u_{ij}$ to be $\\pm 1$.\\\\\n\\textit{Relation 2.} Two bond operators anti-commute if and only if they share exactly one vertex, otherwise, they commute. It is straightforward to check that this is satisfied by both $\\{\\hat{\\gamma}_{ij}\\}$ and $\\{\\hat{\\gamma}'_{ij}\\}$, so this condition puts no  constraints on $\\{u_{ij}\\}$.\\\\\n\\textit{Relation 3.} The product of $\\hat{\\gamma}'_{ij}$ on any closed loop $\\mathfrak{L}$ is equal to a constant, so the  product of $\\hat{\\gamma}_{ij}$ on $\\mathfrak{L}$ must be equal to the same constant. It is enough to require this constraint only on all the elementary plaquettes $\\mathfrak{L}_p$ along with two large loops $\\mathfrak{L}_x$ and $\\mathfrak{L}_y$ winding around the torus~(as shown in Fig.~\\ref{fig:brickwall-unitcell}),\nsince the product on other loops decompose into products on these elementary loops. The product of $\\hat{\\gamma}'_{ij}$ on these loops are equal to \n\\begin{eqnarray}\\label{eq:Wpphixphiy}\n\tW_p&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} u_{ij},\\nonumber\\\\%\\hat{\\gamma}'_{01}\\hat{\\gamma}'_{21}\\hat{\\gamma}'_{23}\\hat{\\gamma}'_{43}\\hat{\\gamma}'_{45}\\hat{\\gamma}'_{05}\n\t\\Phi_x&\\equiv & \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} u_{ij},\\nonumber\\\\\n\t\\Phi_y&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} u_{ij},\n\\end{eqnarray}\nfor every plaquette $p$, and we order the product of operators according to their linear order in the loop~(the orientation of the loop and the initial point do not affect the result of the product).\n\nThe product of $\\hat{\\gamma}_{ij}$ on these loops are equal to \n\\begin{eqnarray}\\label{eq:phixphiy}\n\t\\hat{W}_p&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} \\hat{\\gamma}_{ij}=-\\hat{\\sigma}^z_0\\hat{\\sigma}^y_1\\hat{\\sigma}^y_2\\hat{\\sigma}^z_3\\hat{\\sigma}^x_4\\hat{\\sigma}^x_5,\\nonumber\\\\\n\t\\hat{\\Phi}_x&\\equiv & \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} \\hat{\\gamma}_{ij}=-\\prod_{i\\in \\mathfrak{L}_x} \\hat{\\sigma}^y_i,\\nonumber\\\\\n\t\\hat{\\Phi}_y&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} \\hat{\\gamma}_{ij}=-\\prod_{i\\in \\mathfrak{L}_y} \\hat{\\sigma}^z_i,\n\\end{eqnarray}\nwhere $0,1,2,3,4,5$  label the sites of the plaquette $p$, as shown in Fig.~\\ref{fig:brickwall-unitcell}~(and similarly for all other plaquettes).\nAlthough the RHS of Eq.~\\eqref{eq:phixphiy} are not constants, one can check that these operators mutually commute, and they commute with all the bond operators $\\hat{\\gamma}_{ij}$, and therefore they commute with the transfer matrix $\\hat{T}$. They play the role of conserved observables, and their common eigenspaces are invariant under the action of $\\hat{T}$. Further, since $\\hat{W}_p^2=\\hat{\\Phi}_x^2=\\hat{\\Phi}_y^2=1$, their eigenvalues can only be $\\pm 1$. \nTo guarantee the algebraic isomorphism between the algebras $\\{\\hat{\\gamma}_{ij}\\}$ and $\\{\\hat{\\gamma}'_{ij}\\}$, we need to \nmap the spin model transfer matrix $\\hat{T}$ in each common eigenspace of $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$ to a different fermionic transfer matrix  $\\hat{T}'$, with the $u_{ij}$ chosen in such a way that their loop products $\\{W_p,\\Phi_x,\\Phi_y\\}$ equal the eigenvalues of $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$.\\\\\n\\textit{Relation 4.}\nOn a closed manifold, the product of all $\\{\\hat{\\gamma}_{ij}\\}$ on the lattice equals a constant:\n\\begin{eqnarray}\\label{eq:prod_all_bond}\n\\prod_{\\text{all }\\langle ij\\rangle} \\hat{\\gamma}_{ij}=i^{4L_xL_y}=1,\n\\end{eqnarray}\nwhere $L_x$~($L_y$) is the system size in the $x$-~($y$-) direction.\nSimilarly, the product of all $\\{\\hat{\\gamma}'_{ij}\\}$ is\n\\begin{eqnarray}\\label{eq:prod_all_bond_f}\n\\prod_{\\text{all }\\langle ij\\rangle} \\hat{\\gamma}'_{ij}=\n\\hat{P}_f\\prod_{\\text{all }\\langle ij\\rangle}  u_{ij},\n\\end{eqnarray}\nwhere\n$\\hat{P}_f\\equiv \\prod_{z}\\left(-i\\hat{c}_i\\hat{c}_j\\right)$\nis the conserved fermion parity operator. Therefore the algebraic isomorphism restricts the fermion model to the eigen-subspace of $\\hat{P}_f$ with eigenvalue\n\\begin{equation}\\label{eq:parity_restriction}\n\tP_f= \\prod_{\\text{all }\\langle ij\\rangle} u_{ij}.\n\\end{equation}\n\\textit{Summary and consistency check.} In summary, the mutually commuting conserved operators $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$ split the full Hilbert space into a direct sum of their common eigen-subspaces, and the transfer matrix $\\hat{T}$ leaves each subspace invariant. In the subspace labeled by the conserved eigenvalues $\\{W_p,\\Phi_x,\\Phi_y\\}$, the transfer matrix $\\hat{T}$ is mapped to a fermionic transfer matrix $\\hat{T}'$ defined in Eq.~\\eqref{eq:Tbondfermion} where the parameters $u_{ij}=\\pm 1$ are chosen to satisfy Eq.~\\eqref{eq:Wpphixphiy}~\\footnote{While there are exponentially many solutions $\\{u_{ij}\\}$ to  Eq.~\\eqref{eq:Wpphixphiy} for a fixed configuration $\\{W_p,\\Phi_x,\\Phi_y\\}$, all of them are equivalent up to a gauge transformation, and the spectrum of $\\hat{T}'$ only depends on the values of $\\{W_p,\\Phi_x,\\Phi_y\\}$.}, and $\\hat{T}'$ is restricted to a fixed fermion parity sector satisfying Eq.~\\eqref{eq:parity_restriction}. \n\nAs a consistency check, let us verify that the subspace dimension of the spin and fermionic systems, mapped to each other by the above algebraic isomorphism, are the same. For the spin system, we have $4L_xL_y$ qubit degrees of freedom~(d.o.f.) in total; in each subspace, the constraint Eq.~\\eqref{eq:phixphiy} removes $2L_x L_y-1+2$ qubit d.o.f~($-1$ because the product of all $\\hat{W}_p$ is a constant, so only $2L_x L_y-1$ of them are independent), leaving us with $2L_x L_y-1$ qubit d.o.f. For the fermionic system, we have $4L_xL_y$ Majorana fermions in total, which amounts to $2L_xL_y$ Dirac fermion d.o.f.; the fermion parity restriction Eq.~\\eqref{eq:parity_restriction} further removes one of them, leaving us $2L_x L_y-1$ Dirac fermion d.o.f.. Therefore the Hilbert space dimension of the two systems are the same, both equal to $2^{2L_x L_y-1}$.\n\n\\subsection{Solving the free fermion transfer matrix}\\label{sec:solve_fermion_TM}\nIn the last section we mapped the transfer matrix $\\hat{T}$ in each sector labeled by $\\{W_p,\\Phi_x,\\Phi_y\\}$ to a free fermion transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Tbondfermion}, where $u_{ij}=\\pm 1$ are chosen to satisfy Eq.~\\eqref{eq:Wpphixphiy}, and the fermion parity satisfies Eq.~\\eqref{eq:parity_restriction}. Now we solve these free fermion problems in each sector to get the full spectrum of $\\hat{T}$. The difficulty here is that there are exponentially many such sectors~($2^{2L_x L_y+1}$ in total), most of which are not translationally invariant and can only be solved numerically. Fortunately we are most interested in the sector that contains the principal eigenvalue $\\Lambda_\\mathrm{max}$ of $\\hat{T}$, i.e. the sector $\\{W_p,\\Phi_x,\\Phi_y\\}$ where the principal eigenvalue of $\\hat{T}'$ is largest, since $\\Lambda_\\mathrm{max}$~(and the corresponding principal eigenstate $|\\Lambda_\\mathrm{max}\\rangle$) determines the thermodynamic properties of the original classical system. \nIn App.~\\ref{appen:Lieb} we prove a generalization of Lieb's optimal flux theorem~\\cite{lieb1994flux} for the transfer matrix $\\hat{T}'$, which shows that for real $K_x,K_y,K_z$, the principal eigenvalue of $\\hat{T}'$ is maximized by a configuration $\\{W_p,\\Phi_x,\\Phi_y\\}$ where all $W_p$ are equal to $+1$. From now on we will call such a configuration  vortex-free, and for a configuration with some $W_p=-1$ we say it has a vortex excitation at $p$. This leaves four sectors to consider, corresponding to $(\\Phi_x,\\Phi_y)=(++),(+-),(-+),(--)$~[we use $(++)$ as a shorthand for $(+1,+1)$, and similarly for the other three]. These four sectors can be treated in an identical way, which we do in the following.\n\nWe first need to find a solution $\\{u_{ij}\\}$ to Eq.~\\eqref{eq:Wpphixphiy}. For the $(++)$ sector, we can simply take  $u_{ij}=+1$ for all links $\\langle ij\\rangle$. To obtain solutions for the other three vortex-free sectors, %\nnotice that we can flip the sign of $\\Phi_x$ or $\\Phi_y$ by flipping the signs of $u_{ij}$ on a large~(i.e. non-contractible) loop of links, without changing the value of any $W_p$. For example, if we flip  all the $z$-links between $x=L_x-1\/2$ and $x=0$~(denote this set of links by $Z_{L_x-1\/2,0}$), then we can flip the sign of $\\Phi_x$ without flipping any of  the $W_p$. Similarly we can flip the sign of $\\Phi_y$ by flipping the signs of all the $y$-links between $y=L_y-1\/2$ and $y=0$~(denote this set of links by $Y_{L_y-1\/2,0}$). In this way, the solution for the sector $(\\Phi_x, \\Phi_y)$ can be taken as $u_{ij}=1$ for $\\langle ij\\rangle\\notin Z_{L_x-1\/2,0} \\cup Y_{L_y-1\/2,0} $, $u_{ij}=\\Phi_x$ for $\\langle ij\\rangle\\in Z_{L_x-1\/2,0} $, and $u_{ij}=\\Phi_y$ for $\\langle ij\\rangle\\in Y_{L_y-1\/2,0}$.  %\n\nThe transfer matrix defined in Eq.~\\eqref{eq:Tbondfermion} for all these four sectors can be written in a translationally invariant way provided that we use suitable boundary conditions for the Majorana operators.  To this end, we use $i=(\\vec{r},\\lambda)$ to label lattice sites, where $\\vec{r}$ labels the unit cells, and  $\\lambda=0,1,2,3$ label the sites in a unit cell, as shown in Fig.~\\ref{fig:brickwall-unitcell}. We define $\\hat{c}_{(L_x,y),\\lambda}=\\Phi_x\\hat{c}_{(0,y),\\lambda}$ and $\\hat{c}_{(x,L_y),\\lambda}=\\Phi_y\\hat{c}_{(x,0),\\lambda}$, corresponding to periodic or antiperiodic boundary conditions.\nThen the transfer matrices for all the four vortex-free sectors have the same expression\n\\begin{equation}\\label{eq:Ttilde}\n\t\\hat{T}'=e^{K_x\\sum_{x}i\\hat{c}_i\\hat{c}_j}e^{K_y\\sum_{y}i\\hat{c}_i\\hat{c}_j}e^{K_z\\sum_{z}i\\hat{c}_i\\hat{c}_j}.\n\\end{equation}\nwhere the above boundary condition on $\\hat{c}_i,\\hat{c}_j$ is used, and it is understood that the lattice coordinates of $i,j$ for each link $\\langle ij\\rangle$ should be consecutive numbers, e.g. the term on a flipped $z$-link is understood as $ \\hat{c}_{(L_x-1,y),\\lambda} \\hat{c}_{(L_x,y),\\lambda'}$ instead of $\\hat{c}_{(L_x-1,y),\\lambda} \\hat{c}_{(0,y),\\lambda'}$.\n\nThe rest of the task is to find the eigenvalues of the translationally invariant vortex-free transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Ttilde} under the four possible boundary conditions $(++),(+-),(-+),(--)$. To this end, we introduce the Fourier transform  of the Majorana operators\n\\begin{eqnarray}\\label{eq:Ftrans}\n  \\hat{a}_{\\vec{q},\\lambda}&=&\\frac{1}{\\sqrt{2N}}\\sum_{\\vec{r}} e^{-i\\vec{q}\\cdot\\vec{r}}\\hat{c}_{\\vec{r},\\lambda},\\nonumber\\\\\n  \\hat{c}_{\\vec{r},\\lambda}&=&\\sqrt{\\frac{2}{N}}\\sum_{\\vec{q}} e^{i\\vec{q}\\cdot\\vec{r}}\\hat{a}_{\\vec{q},\\lambda},\n\\end{eqnarray}\nwhere $N=L_x L_y$ is the total number of unit cells. \nThe quasi-momentum in the $\\alpha$-direction $q_\\alpha$ is quantized as  $2n\\pi\/L_\\alpha$ where $n\\in\\mathbb{Z}$ if $\\Phi_\\alpha=+1$ and $n\\in\\mathbb{Z}+1\/2$ if $\\Phi_\\alpha=-1$ . The operators $\\hat{a}_{\\vec{q},\\lambda}$ satisfy $\\hat{a}_{\\vec{q},\\lambda}^\\dagger=\\hat{a}^{\\phantom{\\dagger}}_{-\\vec{q},\\lambda}$ and $\\{\\hat{a}^{\\phantom{\\dagger}}_{\\vec{p},\\lambda},\\hat{a}^\\dagger_{\\vec{q},\\mu}\\}=\\delta_{\\vec{p},\\vec{q}}\\delta_{\\lambda,\\mu}$. We can now rewrite $\\hat{T}'$ as\n\\begin{eqnarray}\\label{eq:TtildeFF}\n  \\hat{T}'&=&\\exp\\left[2K_x\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},1}+i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},3})\\right]\\nonumber\\\\\n             &&\\times\\exp\\left[2K_y\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},1}e^{iq_y}+i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},3}e^{-iq_y})\\right]\\nonumber\\\\\n  &&\\times\\exp\\left[2K_z\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},1}+i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},3}e^{iq_x})\\right],\\nonumber\\\\\n  &\\equiv &\\tilde{T}_{0}\\prod_{\\vec{q}+}\\tilde{T}_{\\vec{q}},\n\\end{eqnarray}\nwhere $\\tilde{T}_{0}$ contains all the terms with $\\vec{q}\\equiv -\\vec{q}~(\\mathrm{mod}~ 2\\pi)$, and $\\prod_{\\vec{q}+}$ is the product over $\\vec{q}$ with $\\vec{q}\\not\\equiv -\\vec{q}~(\\mathrm{mod}~ 2\\pi)$ such that each pair $\\pm\\vec{q}$ appears exactly once, and in the last line we have rearranged terms of different $\\vec{q}$ modes using $[\\tilde{T}_{\\vec{q}},\\tilde{T}_{0}]=0$, and $[\\tilde{T}_{\\vec{q}},\\tilde{T}_{\\vec{p}}]=0$ for $\\vec{q}\\neq\\pm \\vec{p}$. Because of this commutativity, all the $\\tilde{T}_{\\vec{q}}$ and $\\tilde{T}_{0}$ can be simultaneously diagonalized. We treat $\\tilde{T}_{\\vec{q}}$ first, which can be written as \n\\begin{eqnarray}\\label{def:Tq}\n\\tilde{T}_{\\vec{q}}&=&e^{2K_x \\sum_{\\lambda,\\mu}P^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}}e^{2K_y \\sum_{\\lambda,\\mu} Q^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}}\\nonumber\\\\\n&&\\times e^{2K_z \\sum_{\\lambda,\\mu} R^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}},\n\\end{eqnarray}\nwhere the $4\\times 4$ matrices $P^{(\\vec{q})},Q^{(\\vec{q})},R^{(\\vec{q})}$ are~(we drop the superscript $\\vec{q}$ when there is no confusion)\n\\begin{eqnarray}\\label{def:PQR}\nP&=&\\begin{pmatrix}\n\t0 & i & 0 & 0 \\\\\n\t-i& 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & i \\\\\n\t0 & 0 & -i& 0\n\\end{pmatrix},\nR=\\begin{pmatrix}\n\t0 & 0 & 0 & ie^{-iq_x} \\\\\n\t0 & 0 & -i & 0 \\\\\n\t0 & i & 0 & 0 \\\\\n\t-ie^{iq_x} & 0 & 0 & 0\n\\end{pmatrix},\\nonumber\\\\\nQ&=&\\begin{pmatrix}\n\t0 & ie^{-iq_y} & 0 & 0 \\\\\n\t-ie^{iq_y}& 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & ie^{iq_y} \\\\\n\t0 & 0 & -ie^{-iq_y}& 0\n\\end{pmatrix}.\n\\end{eqnarray}\nNotice that the fermion bilinears $\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}$ in Eq.~\\eqref{def:Tq} form the basis of an $\\mathfrak{sl}(4)$ Lie algebra, so $\\tilde{T}_{\\vec{q}}$ is an element of the corresponding $\\mathrm{SL}(4)$ Lie group. Using the relation between the fundamental representation and the free fermion representation of this Lie algebra and group, (similar to the method in App.~\\ref{appen:numerical_method}), one can show that \n\\begin{equation}\\label{eq:Tq_diagonal}\n\t\\tilde{T}_{\\vec{q}}=e^{\\epsilon_{\\vec{q},1}(\\hat{n}_{\\vec{q},1}-\\hat{n}_{\\vec{q},\\bar{1}})+\\epsilon_{\\vec{q},2}(\\hat{n}_{\\vec{q},2}-\\hat{n}_{\\vec{q},\\bar{2}})},\n\\end{equation}\nwhere $ e^{\\pm\\epsilon_{\\vec{q},1}}, e^{\\pm\\epsilon_{\\vec{q},2}}$ are the eigenvalues of the matrix $T_{\\vec{q}}=e^{2K_x P}e^{2K_yQ}e^{2K_zR}$, which is the representation of $\\tilde{T}_{\\vec{q}}$ in the fundamental representation of the $\\mathrm{SL}(4)$ Lie group, and $\\hat{n}_{\\vec{q},j},\\hat{n}_{\\vec{q},\\bar{j}}$ ~(with $j\\in \\{1,2\\}$) are mutually commuting fermion number operators. The single mode energies $\\epsilon_{\\vec{q},j}$ can be analytically calculated by solving the quartic equation $P_{T_{\\vec{q}}}(x)=0$, where $P_{T_{\\vec{q}}}(x)$ is the degree four characteristic polynomial  of the $4\\times 4$ matrix  $T_{\\vec{q}}$. This quartic equation can be simplified to a quadratic one $z^2+Az+B=0$, where $z=(x+1\/x)\/2=\\cosh\\epsilon_{\\vec{q},j}$~(for $j=1,2$), and \n\\begin{eqnarray}\\label{eq:chepsilon}\nA&=&-2c_3(c_1c_2+s_1s_2 \\cos q_y),\\nonumber\\\\\nB&=&\\frac{1}{8}S_1S_2(3+C_3-2s_3^2\\cos q_x)\\cos q_y+\\frac{1}{2} s_1^2 s_2^2\\cos(2q_y)\\nonumber\\\\\n&&{}+\\frac{1}{4}s_3^2(1-C_1C_2)\\cos(q_x)+\\frac{C_1+C_2+3C_3}{8}\\nonumber\\\\\n&&{}+\\frac{C_1C_2}{4}+\\frac{C_1C_2C_3}{8},\n\\end{eqnarray}\nwhere $c_j=\\cosh 2K_j,s_j=\\sinh 2K_j,C_j=\\cosh4K_j$, and $S_j=\\sinh4K_j$.\nSince the eigenvalues of $T_{\\vec{q}}$ come in pairs $\\pm\\epsilon_{\\vec{q},1}, \\pm\\epsilon_{\\vec{q},2}$, we can assume without loss of generality that $0\\leq \\mathrm{Re}[\\epsilon_{\\vec{q},1}]\\leq \\mathrm{Re}[\\epsilon_{\\vec{q},2}]$. Then the maximal eigenvalue of $\\tilde{T}_{\\vec{q}}$ is $e^{\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}}$.\n\nThe term $\\tilde{T}_{0}$ in the last line of Eq.~\\eqref{eq:TtildeFF} is defined by %\n$\\tilde{T}_{0}=\\prod_{\\vec{q}\\equiv \\vec{0}~(\\mathrm{mod}~\\pi)}\\tilde{T}_{0,\\vec{q}}$ with\n\\begin{equation}\\label{eq:T_0qxqy}\n\t\\tilde{T}_{0,\\vec{q}}=e^{2(K_x+K_ye^{iq_y})i\\hat{a}_{\\vec{q}0}\\hat{a}_{\\vec{q}1}(1-\\hat{P}_{\\vec{q}})} e^{2K_zi\\hat{a}_{\\vec{q}1}\\hat{a}_{\\vec{q}2}(1+\\hat{P}_{\\vec{q}}e^{iq_x})},\n\\end{equation}\nwhere $\\hat{P}_{\\vec{q}}=4\\hat{a}_{\\vec{q},0}\\hat{a}_{\\vec{q},1}\\hat{a}_{\\vec{q},2}\\hat{a}_{\\vec{q},3}$. \nUsing $\\hat{a}^\\dagger_{\\vec{q},\\lambda}=\\hat{a}_{\\vec{q},\\lambda}, \\hat{a}_{\\vec{q},\\lambda}^2=1\/2$,\nthe eigenvalues of $\\tilde{T}_{0,\\vec{q}}$ can be straightforwardly obtained by diagonalizing  Eq.~\\eqref{eq:T_0qxqy}, and one can show that the largest one happens to be equal to $e^{(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})\/2}$.  %\n\nWe have not yet taken into account the fermion parity restriction in Eq.~\\eqref{eq:parity_restriction}. However, as we will see in Sec.~\\ref{sec:TPloop},\nthis constraint changes $\\ln\\Lambda_{\\mathrm{max}}$ by at most $O(\\epsilon_{\\vec{q},j})$, and therefore does not affect the free energy density in the thermodynamic limit. The largest eigenvalue $\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of $\\hat{T}'$ is\n\\begin{eqnarray}\\label{eq:lamdba_max}\n\t\\ln\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}&=&\\frac{1}{2}\\sum_{\\vec{q}}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}),%\n\\end{eqnarray}\nwhere $(\\Phi_x,\\Phi_y)\\in\\{(++),(+-),(-+),(--)\\}$, and the RHS implicitly depends on $(\\Phi_x,\\Phi_y)$  through the quantization of $\\vec{q}$. %\nThe largest eigenvalue $\\Lambda_{\\mathrm{max}}$ of $\\hat{T}$ is the largest of these four. Regardless of which one is the largest, the free energy density~(per site) in the thermodynamic limit is\n\\begin{eqnarray}\\label{eq:free_energy}\nf\\equiv\\frac{F}{12MN}&=&-\\frac{k_BT}{24N}\\sum_{\\vec{q}}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})\\\\\n&=&-\\frac{k_BT}{96\\pi^2}\\iint_{[-\\pi,\\pi]^2}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})~d^2q,\\nonumber\n\\end{eqnarray}\nwhere the free energy $F$ is defined in Eq.~\\eqref{def:free_energy}.\n\n\\subsection{Excitations and phase boundaries}\\label{sec:phase_boundary}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.7\\linewidth]{critical.png}}\n\t\\caption{\\label{fig:critical} A 2D section of the 3D phase diagram of our model,  with the intersecting plane $K_x+K_y+K_z=\\beta(J_x+J_y+J_z)=\\mathrm{const.}$ The parameters $(K_x,K_y,K_z)$ of an arbitrary point in the diagram is given by the distance from that point to the three sides of the triangle. \n\t%\n\tThe $B$-phase~(shaded) has a gapless transfer matrix, which acquires a gap after a small perturbation is introduced~(Sec.~\\ref{sec:phase_boundary}).\n\tThe $A$-region has a gapped transfer matrix and consists of three disjoint phases $A_x, A_y,A_z$. In Sec.~\\ref{sec:critical_exp} we study the critical behavior of the free energy  as we approach the phase boundary  from the $A_z$-phase $\\eta\\equiv K_z - K_x - K_y \\to 0^+$. }\n\\end{figure}\nIn this section we study other eigenvalues of the transfer matrix $\\hat{T}$ beyond the principal eigenvalue, and, using this, determine the phase diagram of our model. It is useful to define an effective non-Hermitian Hamiltonian \n$\t\\hat{H}=-\\ln \\hat{T}.$\nIn this way the principal eigenstates of $\\hat{T}$ are mapped to the ground states of $\\hat{H}$ and the eigenvalues $\\Lambda_j$ of $\\hat{T}$  are related to excitation energies $E_j-E_0$ of $\\hat{H}$ by  $E_j-E_0=\\ln \\Lambda_{\\text{max}}-\\ln\\Lambda_j$. For the rest of this paper, we use the term ``excitation spectrum of $\\hat{T}$'' to mean the excitation spectrum of $\\hat{H}$, and call the transfer matrix ``gapped''~(``gapless'') if $\\mathrm{Re}[E_j-E_0]$ is gapped~(gapless) in the thermodynamic limit. The spectral gap $\\Delta=\\min_{j\\neq 0}\\mathrm{Re}[E_j-E_0]$ plays an important role  in the physical properties of the original classical Ising model. First, as we will see in a moment, the phase boundary of our model is determined by regions where $\\Delta$ vanishes. Secondly, although we do not calculate in this paper, we claim that two point connected correlations $\\langle \\sigma_i\\sigma_j\\rangle_c$~(or more generally, $\\langle O_i O_j\\rangle_c$ where $O_i$ is a product of classical  spins in a local region) decay exponentially in distance when $\\Delta>0$, while there are algebraically decaying correlations when $\\Delta=0$. \n\n\n\nThere are two types of excitations: fermionic excitations, corresponding to the positive energy eigenmodes of the fermionic transfer matrix $\\hat{T}'$, %\nand vortex excitations, corresponding to eigenstates of $\\hat{T}'$ in a different sector $\\{W_p,\\Phi_x,\\Phi_y\\}$ where some of $W_p$s are equal to $-1$. Vortices can only be created in pairs. \nA pair of vortices can be created by first drawing a segment connecting the two vortices~(the segment should avoid passing through lattice sites) and then flipping $u_{ij}$ on all the lattice edges intersecting with this segment~(similar to Kitaev's honeycomb~\\cite{Kitaev2006} and toric code~\\cite{Kitaev2003Fault} models).  Our analysis in App.~\\ref{appen:Lieb} and the numerical results in App.~\\ref{appen:numerical_vgap} suggest that the vortices have  gapped and positive excitation energies. %\nOn the other hand, the fermionic excitations can become gapless for certain values of $(K_x,K_y,K_z)$, and this determines the phase boundary of our model. \n\nWe emphasize that it is the gap closing of the real part of $\\epsilon_{\\vec{q},1}$  that determines the phase boundary~\\footnote{In fact, for real $K_x,K_y,K_z$, the single fermion energies $\\epsilon_{\\vec{q},1},\\epsilon_{\\vec{q},2}$ are real; so the distinction between  $\\epsilon_{\\vec{q},1}$ and $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ is unimportant here. In particular, one obtains the same phase diagram even if $\\epsilon_{\\vec{q},1}=0$ is used as a criterion for phase transition. }. This claim is based on the analysis in App.~\\ref{appen:proof_analyticity}, where we rigorously prove that the free energy $f$ defined in Eq.~\\eqref{eq:free_energy} is complex analytic in all its parameters when  $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]>0~\\forall \\vec{q}\\in [-\\pi,\\pi]^2$.  The proof also suggests that when the gap closes $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]=0$, there are branch points in $\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}$ that leads to non-analytic behavior of $f$, which we calculate directly in Sec.~\\ref{sec:critical_exp}.%\n\nWe find two distinct phases corresponding to whether $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ is gapped or gapless. The phase boundary is determined as follows. One can show that for fixed $q_y$ the minimum of \n$\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ occurs at $q_x=0$~[since $\\partial_{q_x}\\epsilon_{\\vec{q}}=f(\\epsilon_{\\vec{q}},q_y) \\sin q_x $ for some positive function $f(\\epsilon_{\\vec{q}},q_y) $]. Furthermore, in the gapped phase the minimum of $\\mathrm{Re}[\\epsilon_{(0,q_y),1}]$ occurs either at $q_y=0$ or $q_y=\\pi$. %\nTherefore, the phase transition occurs when $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ vanishes at either $\\vec{q}=(0,0)$ or $\\vec{q}=(0,\\pi)$, \nwhich happens when one of $K_x,K_y,K_z$ equals the sum of the other two~[this can be seen by diagonalizing $\\tilde{T}_{\\vec{q}}$ in Eq.~\\eqref{def:Tq} at $\\vec{q}=(0,0)$ or $(0,\\pi)$]. When $K_x,K_y,K_z$ form three sides of a triangle~(we call this the $B$-region, shown as the shaded triangle in Fig.~\\ref{fig:critical}), the spectrum is gapless, and when one of $K_x,K_y,K_z$  is bigger than the sum of the other two, the spectrum is gapped~(we call this the $A$-region, consisting of three disjoint white triangles in Fig.~\\ref{fig:critical}). The phase diagram in terms of $K_x,K_y, K_z$ is shown in Fig.~\\ref{fig:critical}, which is identical to the phase diagram of Kitaev's honeycomb model~\\cite{Kitaev2006}. \n\nThe fermionic spectrum of the $B$-phase  can be gapped by adding suitable perturbations. For example, we can add small imaginary parts to $J_x, J_y$, so that $K_x\\to K_x+i\\kappa,K_y\\to K_y-i\\kappa$, and then add a small real part to the coupling constants of the $x,y$ links that break the lattice reflection symmetry, in the pattern shown in Fig.~\\ref{fig:brickwall-unitcell}. Here $\\kappa$ is a small real number $|\\kappa|\\ll |K_{i}|,i=x,y,z$. (Notice that this corresponds to modifying the link coupling constants of the original classical statistical model on all the $x$ and $y$ planes, which breaks the reflection symmetry of the 3D lattice.)  App.~\\ref{appen:gap_phases_B} proves that a subregion of the $B$-phase is gapped by this perturbation. More specifically, when $|K_z|\/2<|K_x|=|K_y|$, we have $\\Delta=\\min_{\\vec{q}}\\mathrm{Re}[\\epsilon_{\\vec{q},1}]\\propto\\kappa^2$. This fact will be useful for Sec.~\\ref{sec:TPloop} where we calculate the topological degeneracy of $\\Lambda_{\\max}$ and Sec.~\\ref{sec:loop_observables} where we find loop observables whose expectation values distinguish the two phases. Notice that while our proof of Lieb's theorem in App.~\\ref{appen:Lieb} assumes real $K_x,K_y,K_z$, as long as the vortices are gapped, the principal eigenstate is still in the vortex-free sector if $\\kappa$ is sufficiently small, which we  assume throughout this paper. \n\n\\subsection{Critical exponents}\\label{sec:critical_exp}\nIn this section we study the critical behavior of our model near the phase boundary between the $A$ and $B$ phases, and show that this is a third order phase transition. Specifically, we parameterize the distance to the phase boundary by $\\eta = K_z - K_x - K_y$ and show that as the phase boundary is approached from the $A$-phase side, $\\eta \\to 0^+$, the leading singular part of the free energy is $f \\sim \\eta^{5\/2}$~\\footnote{We are approaching the phase boundary strictly inside the big triangle, i.e. the parameters $K_x,K_y,K_z$ are all nonzero. If one instead approaches the point where two phase boundaries meet from along a side of the big triangle, then one can show that the transition is in 2D Ising universality class, where $f \\sim \\eta^{2}\\ln\\eta$.}.\n\nWe start from the expression in Eq.~\\eqref{eq:free_energy}. Near the phase boundary, the leading singular part of $f$ is contributed by the integration near $\\vec{q}=(0,\\pi)$ where $\\epsilon_{\\vec{q},1}$ approaches zero. Letting $\\vec{q}=(p_x,\\pi+p_y)$ where $p_x,p_y\\ll 1$, we expand $\\epsilon^2_{\\vec{q},1}$ in powers of the small parameters $p_x,p_y$, and $\\eta$. Using Eq.~\\eqref{eq:chepsilon} and $\\cosh \\epsilon\\approx 1+\\epsilon^2\/2$ for $\\epsilon\\ll 1$, we have\n\\begin{eqnarray}\n\t\\epsilon^2_{\\vec{q},1}%\n\t&=& \\frac{s^2_3}{4} p_x^2+2 \\frac{s_1s_2}{s_3} \\eta p_y^2+4\\eta^2+\\frac{s_1^2s_2^2}{4s_3^2}p_y^4\\\\\n\t&&{}+O(p_x^4)+O(p_x^2\\eta)+O(p_x^2p_y^2)+O(p_y^4\\eta), \\nonumber\n\\end{eqnarray}\nwhere the neglected terms will not affect the leading-order singularity. The leading singular part of $f$ is \n\\begin{eqnarray}\\label{eq:f_singular}\n\tf&\\sim&-\\frac{1}{48\\pi^2\\beta s_1s_2}\\iint \\sqrt{ p_x^2+2 \\eta p_y^2+4\\eta^2+\\frac{p_y^4}{4}} ~dp_x dp_y\\nonumber\\\\\n\t&\\sim &\\frac{1}{48\\pi^2\\beta s_1s_2}\\int 2(p_y^2+4\\eta)^2\\ln(p_y^2+4\\eta) ~ dp_y\\nonumber\\\\\n\t&\\sim &\\frac{64}{45\\beta\\pi s_1s_2}\\eta^{\\frac{5}{2}},\n\\end{eqnarray}\nwhere in the first line we rescale the integration variables $p_x,p_y$, the integration range is a fixed-length interval passing through the origin, say $[-\\epsilon,\\epsilon]^2$ with $0<\\epsilon\\ll 1$, and we use $\\sim$ to indicate that an unimportant analytic part has been ignored. %\nTherefore, the third derivative $\\partial^3_\\eta f$ diverges as $\\eta\\to 0^+$, i.e., the phase transition is third order. \n\\subsection{Topological degeneracy}\\label{sec:TPloop}\nIn this section we show that the largest eigenvalues of the transfer matrix $\\hat{T}$ of our original spin model are topologically degenerate, and the degeneracy depends on the phase. This topological degeneracy gives rise to the topological behaviors of the loop observables presented in the next section.\n \n %\nTo this end, we need to compare the values  $\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of  the four sectors $(\\Phi_x,\\Phi_y)\\in\\{(++),(+-),(-+),(--)\\}$, given in Eq.~\\eqref{eq:lamdba_max}. Let us focus on regions where $\\mathrm{Re}[\\epsilon_{\\vec{q}}]$ is gapped, i.e. the $A$-region and the $B$-region with the perturbation discussed in Sec.~\\ref{sec:phase_boundary}. \nIn App.~\\ref{appen:FSE} we show that the largest eigenvalues\n $\\ln\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of each of the four sectors are equal up to an exponentially small correction $O(e^{-L\/\\xi})$, where  $\\xi$ is a fixed correlation length. %\nThis suggests a 4-fold topological degeneracy since all the fermion and vortex excitations are gapped. However, we have not taken into account the fermion parity constraint yet. \nAs we discussed in Sec.~\\ref{sec:map_fermion}, only those eigenstates of $\\hat{T}'$ that satisfy the fermion parity constraint Eq.~\\eqref{eq:parity_restriction} correspond to eigenstates of $\\hat{T}$. So the actual degeneracy of $\\hat{T}$ is the number of ``parity-compatible'' sectors, i.e. sectors whose principal eigenstate $|\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}\\rangle$ satisfies the fermion parity constraint.\nThe fermion parity constraint Eq.~\\eqref{eq:parity_restriction}, written in terms of $\\hat{a}_{\\vec{q}\\lambda},\\Phi_x,\\Phi_y$, becomes\n\\begin{equation}\\label{eq:parity_restriction2}\n(-1)^{(L_x-1)L_y}\\prod_{\\substack{\\vec{q}+}}P_{\\vec{q},0} P_{\\vec{q},1}P_{\\vec{q},2}P_{\\vec{q},3}\\prod_{\\vec{q}\\equiv -\\vec{q}} P_{\\vec{q}} \n= \\Phi_x^{L_y},\n\\end{equation}\nwhere $P_{\\vec{q},\\lambda}=(1-2n_{\\vec{q},\\lambda})$, $P_{\\vec{q}}=4\\hat{a}_{\\vec{q},0}\\hat{a}_{\\vec{q},1}\\hat{a}_{\\vec{q},2}\\hat{a}_{\\vec{q},3}$ and $\\equiv$ is equality $\\mathrm{mod}~2\\pi$.  As we mentioned above Eq.~\\eqref{eq:free_energy}, the principal eigenstate of $\\tilde{T}_{\\vec{q}}$ in Eq.~\\eqref{def:Tq} always has $n_{\\vec{q},1}=n_{\\vec{q},2}=1$ and $n_{\\vec{q},{\\bar 1}}=n_{\\vec{q},{\\bar 2}}=0$, so we have  $P_{\\vec{q},0} P_{\\vec{q},1}P_{\\vec{q},2}P_{\\vec{q},3}=+1$ for $\\vec{q}\\not\\equiv -\\vec{q}$. Therefore, whether a sector $(\\Phi_x, \\Phi_y)$ is parity-compatible or not is determined by the values of $P_{\\vec{q}}$ where $\\vec{q}\\equiv -\\vec{q}$. \n\nThere are only four possible $\\vec{q}$ that can satisfy $\\vec{q}\\equiv -\\vec{q}$~: $(0,0),(0,\\pi),(\\pi,0),(\\pi,\\pi)$. For the rest of this section, we assume that  $L_x,L_y$ are both even numbers~[we treat the other cases in App.~\\ref{appen:TPD}; the conclusions are the same], in which case these four modes appear in the $(++)$ sector only. This means that Eq.~\\eqref{eq:parity_restriction2} is trivially satisfied for the sectors $(+-),(-+),(--)$, i.e. $\\hat{T}$ has at least a 3-fold degeneracy. For the $(++)$ sector, Eq.~\\eqref{eq:parity_restriction2} becomes $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=+1$.\nThe value of $P_{\\vec{q}}$ for these four Majorana modes in the principal eigenstate  $|\\Lambda_{\\mathrm{max}}^{(++)}\\rangle$ is determined by maximizing the $\\tilde{T}_{0,\\vec{q}}$ term in Eq.~\\eqref{eq:T_0qxqy}.\nIt is straightforward to see that $P_{\\pi0}=P_{\\pi\\pi}=-1$, $P_{00}=[K_z>K_x+K_y]$, and $P_{0\\pi}=[K_z>|K_x-K_y|]$, where $[S]=+1$ if the statement $S$ is true and $[S]=-1$ otherwise.  In the $A$-phases, $P_{00}, P_{0\\pi}$ are both $-1$~(for $A_x, A_y$) or both $+1$~(for $A_z$), so $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=+1$ and $\\hat{T}$ has a 4-fold degeneracy.  In the $B$-phase we have $P_{00}=-1, P_{0\\pi}=+1$, so $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=-1$, i.e. the sector $(++)$ is parity-incompatible, and $\\hat{T}$ has a 3-fold degeneracy.\n\n\n\n\n\\subsection{Loop Observables}\\label{sec:loop_observables}\nIn this section we compute the thermal expectation value of the family of loop observables $\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]$ defined in Eq.~\\eqref{eq:def_loop_observable}, and verify our earlier claim that it is equal to $\\pm 1$ for contractible loops,  $0$ for large loops in the $A$-phase, and $1\/3$ for large loops in the gapped $B$-phase. \n\nWe begin with a contractible loop $\\mathfrak{L}_p$ being an elementary plaquette of the brickwall lattice. Using the transfer matrix method, we find\n\\begin{eqnarray}\\label{eq:loop_transfer_matrix}\n\t\\langle\\sigma[\\mathfrak{L}_{p,(\\alpha\\beta)}]\\rangle&=&\\mathrm{Tr}[\\hat{W}_p \\hat{T}^M]\/\\mathrm{Tr}[\\hat{T}^M]\\nonumber\\\\%\\equiv \\langle \\hat{W}_p \\rangle,\n\t&\\underset{M\\to\\infty}{=}& \\frac{1}{D}\\sum^D_{j=1}\\langle \\Lambda^{(L)}_{\\max,j}|\\hat{W}_p|\\Lambda^{(R)}_{\\max,j}\\rangle\\nonumber\\\\\n\t&=&+1. \n\\end{eqnarray}\nwhere $(\\alpha\\beta)\\in\\{(xy),(yz),(zx)\\}$, the sum is over all the $D$-fold degenerate principal eigenstates, $\\langle \\Lambda^{(L)}_{\\max,j}|$ and $|\\Lambda^{(R)}_{\\max,j}\\rangle$ are the left and right principal eigenstates of $\\hat{T}$, respectively. The last line of Eq.~\\eqref{eq:loop_transfer_matrix} follows from the fact that the principal eigenstates of $\\hat{T}$ are eigenstates of the conserved operator $\\hat{W}_p$ with eigenvalue $+1$. The value of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ on larger contractible loops can be calculated in a similar way, and the result is~(up to a possible minus sign) the expectation value of the product of $\\hat{W}_p$ for all the plaquette $p$ enclosed by $\\mathfrak{L}$. Since the $\\hat{W}_p$ mutually commute and have eigenvalue $+1$ on the principal eigenstates, $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is $\\pm 1$ for contractible loops.  %\n\nThe behavior of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is more interesting on non-contractible loops. For a large loop $\\mathfrak{L}_y$ parallel to the $y$-direction, as shown in Fig.~\\ref{fig:hc0}, we have\n\\begin{eqnarray}\\label{eq:loop_Ly_transfer_matrix}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\mathrm{Tr}[\\hat{\\Phi}_y \\hat{T}^M]\/\\mathrm{Tr}[\\hat{T}^M]\\nonumber\\\\%\\equiv \\langle \\hat{W}_p \\rangle,\n\t&\\underset{M\\to\\infty}{=}& -\\frac{1}{D}\\sum^D_{j=1}\\langle \\Lambda^{(L)}_{\\max,j}|\\hat{\\Phi}_y|\\Lambda^{(R)}_{\\max,j}\\rangle.\t\n\\end{eqnarray} \nFor $A$-phases, this is\n\\begin{eqnarray}\\label{eq:loop_Ly_A}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\frac{\\langle \\hat{\\Phi}_y\\rangle_{++}+\\langle \\hat{\\Phi}_y\\rangle_{+-}+\\langle \\hat{\\Phi}_y\\rangle_{-+}+\\langle \\hat{\\Phi}_y\\rangle_{--}}{4}\\nonumber\\\\\n\t&=&0\n\\end{eqnarray} \nwhile for the gapped $B$-phase,\n\\begin{eqnarray}\\label{eq:loop_Ly_B}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\frac{\\langle \\hat{\\Phi}_y\\rangle_{+-}+\\langle \\hat{\\Phi}_y\\rangle_{-+}+\\langle \\hat{\\Phi}_y\\rangle_{--}}{3}\\nonumber\\\\\n\t&=&\\frac{1}{3}.\n\\end{eqnarray} \nThe value of $\t\\langle\\sigma[\\mathfrak{L}_{x,(\\alpha\\beta)}]\\rangle$ for a large loop $\\mathfrak{L}_x$ parallel to the $x$-direction is mapped to $-\\langle \\hat{\\Phi}_x\\rangle$ [Eq.~\\eqref{eq:phixphiy}] and can be calculated in an identical way, leading to the same result. \nWe see that the value of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ indeed distinguish between contractible and  non-contractible loops, are always quantized at rational values, and can be used as a (nonlocal) order parameter that distinguishes the phases. \n\nIn order for the topological features to be a universal characteristic of the phase, rather than an accidental property (arising, for example, due to the model's solvability), they must be in some way robust against small, local perturbations. We argue that this is likely the case.\nNotice that a local perturbation, e.g. a small real magnetic field term $B\\sum_j \\sigma_j$, in the original classical Ising model can be mapped to a local perturbation in the transfer matrix in  Eq.~\\eqref{eq:def_TM}. The classical loop observables defined in Eq.~\\eqref{eq:def_loop_observable} stills maps to the loop operators $\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y$, but they no longer commute with the perturbed $\\hat{T}$, and when they act on $|\\Lambda^{(R)}_{\\max,j}\\rangle$ they create excitations along the loop. Consequently we expect the expectation value $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ to decay exponentially in the length of $\\mathfrak{L}$. \n\nHowever, based on the robustness of the topological phases of the  2D quantum systems~(defined by the transfer matrix $\\hat{T}$), we expect that there exists a family of perturbed loop observables~(whose definition depends on the perturbation) that have exactly the same properties shown above. The argument is based on the idea of quasi-adiabatic continuation~\\cite{Hastings2005Quasiadiabatic}. For simplicity, let us assume $K_x,K_y,K_z\\ll 1$ so that $\\hat{T}$ can be approximated as a Hermitian operator. Then Ref.~\\cite{Hastings2005Quasiadiabatic} shows that there exists a quasi-local unitary transformation $\\hat{U}_\\lambda$ that evolves the unperturbed principal eigenstates  to the perturbed ones $|\\Lambda^{(R)}_{\\max,j}\\rangle_\\lambda=\\hat{U}_\\lambda|\\Lambda^{(R)}_{\\max,j}\\rangle_{\\lambda=0}$, where $\\lambda$ is the strength of the perturbation. [Roughly speaking, $\\hat{U}_\\lambda$ is a finite-time evolution by a locally-interacting Hamiltonian $\\sum_{i}\\hat{h}_i$ such that $t\\|\\hat{h}_i\\|=O(\\lambda)$, where $t$ is the total time duration.] Then the perturbed loop operators $\\hat{U}_\\lambda\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}\\hat{U}_\\lambda^\\dagger$ have exactly the same expectation values in the perturbed principal eigenstates as in the unperturbed solvable model shown above. And due to the quasi-locality of $\\hat{U}_\\lambda$, Lieb-Robinson bounds~\\cite{Lieb1972,hastings2010locality} show that these perturbed operators are finite-width~(of order $v_{\\text{LR}}t$, where $v$ is the Lieb-Robinson speed) extensions of the unperturbed ones. So we do expect robustness in this sense, essentially the same robustness of loop observables in quantum topological phases.\n\n\n \n\n\n\\iffalse\n\\subsection{Correlation functions}\nIn the following we discuss the distinction between the different phases  in terms of expectation value of observables and correlation functions. \n\\iffalse\nThe energy is\n\\begin{eqnarray}\\label{eq:energy}\n  E=-\\frac{\\partial}{\\partial \\beta}\\ln Z\n=\\frac{MN}{2}\\frac{1}{A_{1\\mathrm{BZ}}}\\frac{\\partial}{\\partial \\beta}\\int_{\\mathrm{1BZ}}\\mathrm{Re}[\\epsilon_{\\vec{q}}]~d^2q.\n\\end{eqnarray}\nNotice that the derivative has to be taken when $\\beta J_\\perp,\\beta h$ are fixed, so the energy does not include the $J_\\perp$ and $h$ terms, however, as we will show soon, the latter have zero expectation values and give no contribution to total energy. Entropy can be obtained by $S=(E-F)\/T$.\n\\fi\nThe general $n$-point correlation function is given by\n\\begin{eqnarray}\\label{eq:npoint}\n  \\langle\\sigma_1\\sigma_2\\ldots\\sigma_n\\rangle=\\frac{1}{Z}\\mathrm{Tr}\\{\\sigma^{\\alpha_1}_1T_{12}\\sigma^{\\alpha_2}_2T_{23}\\ldots\\sigma^{\\alpha_n}_nT_n T^{M-k}\\}\\nonumber\\\\\n  =\\frac{1}{\\Lambda^k_{\\mathrm{max}}}\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha_1}_1T_{12}\\sigma^{\\alpha_2}_2T_{23}\\ldots\\sigma^{\\alpha_n}_nT_n|\\Lambda_{\\mathrm{max}}\\rangle\n\\end{eqnarray}\nwhere $\\alpha_j=x,y,z$ if $\\sigma_j$ is on $x,y,z$-type plane, respectively, $T_{jj+1}$ is the transfer matrix between planes of $\\sigma_j$ and $\\sigma_{j+1}$~(suppose $\\sigma_j$ are arranged in increasing order in $t$-direction), $k=\\floor{m\/3}$ where $m$ is the total number of layers spanned by $\\sigma_1,\\sigma_2,\\ldots,\\sigma_n$.\n\nIt is easy to see that single spin expectation value vanishes $\\langle\\sigma\\rangle=\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha}|\\Lambda_{\\mathrm{max}}\\rangle=0$ since $\\sigma^\\alpha$ creates two vortices. Two point correlation $\\langle\\sigma_1\\sigma_2\\rangle$ also vanishes unless $\\sigma_1,\\sigma_2$ are on the same type of plane and are either on the same site or nearest neighbors when projected to the same plane, in this case the correlation is\n\\begin{eqnarray}\\label{eq:2point}\n  \\langle\\sigma_1\\sigma_2\\rangle=\\sum_s\\left(\\frac{\\Lambda_{s}}{\\Lambda_{\\mathrm{max}}}\\right)^d\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha}_1|\\Lambda_{s}\\rangle\\langle \\Lambda_{s}|\\sigma^{\\alpha}_2|\\Lambda_{\\mathrm{max}}\\rangle,\n\\end{eqnarray}\nwhere the summation is over all excited states with 2 vortices created by $\\sigma^{\\alpha}_1$, $d$ is the vertical distance between $\\sigma_1,\\sigma_2$.\nThe four-point correlation function between two bonds is given by\n\\begin{equation}\\label{eq:4point}\n  \\langle\\sigma_i\\sigma_j\\sigma_k\\sigma_l\\rangle_c=\\sum_s\\left(\\frac{\\Lambda_{s}}{\\Lambda_{\\mathrm{max}}}\\right)^d\\langle \\Lambda_{\\mathrm{max}}|i\\hat{c}_i\\hat{c}_j|\\Lambda_{s}\\rangle\\langle \\Lambda_{s}|ic_kc_l|\\Lambda_{\\mathrm{max}}\\rangle,\n\\end{equation}\nwhere the summation is over vortex-free excited states~(fermionic excitations).\n\nIn general, the non-vanishing condition for $n$-point function~\\eqref{eq:npoint} is that the operator product $\\prod_j \\sigma^{\\alpha_j}_j$ appears as a monomial in the expansion of the transfer matrix. \n\\fi\n\n\\section{Physical relevance of complex coupling constants }\\label{sec:justification}\nAlthough the complex coupling constants of Eq.~\\eqref{cond:exact} appear unphysical, this section argues that the model nevertheless gives insights into genuine physical systems.\n\nForemost, we expect the general strategy of this paper -- finding 3D classical models whose transfer matrices can be solved using techniques previously applied to solvable 2D  quantum models -- to be a fruitful idea that may lead to a wealth of new solvable models, some of which may have real-valued energy. For example, Refs.~\\cite{Chapman2020characterizationof,Ogura2020geometric,elman2020free} have classified families of quantum spin models that can be solved by mapping to free fermions, and these provide a fertile source for new 3D solvable models. \n\nAs an example of this strategy, Sec.~\\ref{sec:physical_model} shows that the $A$-phase of our model can be realized in a model with real coupling constants. This provides a physical model showing the topological properties.  As a speculative aside, we also note that this demonstrates that even models with complex-valued couplings may have the same universal physics as real-valued physical models, and thus the former may serve as windows into the latter. \n\nAdditionally, Sec.~\\ref{sec:quantum_amplitude} shows two different realizations of the partition function of our complex parameter Ising model in certain dynamical processes of a 3D quantum spin system. Both in principle allow the free energy of our model to be measured experimentally. They suggest that the statistical mechanics of Eqs.~(\\ref{eq:HIK},\\ref{eq:Z}) gives a solvable model of 3D DQPT~\\cite{Heyl_2018} that display topological features. %\n\n\\subsection{Realization of $A$-phase in a model with real energy}\\label{sec:physical_model}\n\nThe $A$-phase can be realized in a physical model with real energies, as we now show. Specifically, the model has a phase that reproduces the $A$-phase's topological properties, that contractible  loops have expectation value $\\pm 1$ while noncontractible loops have expectation value $0$. \n \n\\begin{figure}\n\t\\center{\\includegraphics[width=0.7\\linewidth]{KTC4-3.png}}\n\t\\caption{\\label{fig:KTC4} The model Eq.~\\eqref{eq:H_KTC4} lies on a 3D cubic lattice where the classical spins sit on links in the $x$ and $y$ directions. There are four spin interactions $\\sigma_{v,1}\\sigma_{v,2}\\sigma_{v,3}\\sigma_{v,4}$ between spins around every lattice vertex $v$~(shown as blue diamond) and eight spin interactions $\\epsilon[\\sigma_{u(c)}, \\sigma_{l(c)}]$ around every elementary cube $c$~(shown as orange cube). Example of a contractible loop $L_1$ is shown as black  square, and a noncontractible loop $L_2$ is shown as purple solid line.}\n\\end{figure}\nConsider a 3D square lattice where there is one classical Ising spin on each link in the $x$ and $y$ directions, but no spins live on the links in the $z$ direction, as shown in Fig.~\\ref{fig:KTC4}. The energy of a spin configuration $\\{\\sigma\\}$ is given by\n\\begin{equation}\\label{eq:H_KTC4}\nE[\\{\\sigma\\}]=-\\sum_v \\sigma_{v,1}\\sigma_{v,2}\\sigma_{v,3}\\sigma_{v,4}-\\sum_{c}\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}],\n\\end{equation}\nwhere the first sum is over all vertices $v$, $\\sigma_{v,1},\\sigma_{v,2},\\sigma_{v,3},\\sigma_{v,4}$ denote the four spins linked to the vertex $v$, the second sum is over all cubes $c$, and $\\{u(c)\\},\\{l(c)\\}$ denote the upper and lower plaquettes of $c$, respectively. We use $\\sigma_{\\{p\\}}=(\\sigma_{p,1},\\sigma_{p,2},\\sigma_{p,3},\\sigma_{p,4})$ to denote the configurations of the four spins of the plaquette $p$. The energy of the cube $c$ is defined as $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=\\ln \\cosh(1)$ if $\\sigma_{\\{u(c)\\}}=\\sigma_{\\{l(c)\\}}$, $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=\\ln\\sinh(1)$ if $\\sigma_{\\{u(c)\\}}=-\\sigma_{\\{l(c)\\}}$ while $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=-\\infty$ otherwise. \n\nThe partition function is \n\\begin{equation}\\label{eq:Z_KTC4}\nZ=\\sum_{\\{\\sigma\\}}e^{- E[\\{\\sigma\\}]}=\\mathrm{Tr}[\\hat{T}^M],\n\\end{equation}\nwhere the transfer matrix $\\hat{T}$ is an operator acting on quantum spins lying on a 2D slice of the lattice, defined by\n\\begin{equation}\\label{eq:T_KTC4}\n\\hat{T}=\\exp\\left(\\sum_v \\hat{\\sigma}^z_{v,1}\\hat{\\sigma}^z_{v,2}\\hat{\\sigma}^z_{v,3}\\hat{\\sigma}^z_{v,4}+\\sum_{p}\\hat{\\sigma}^x_{p,1}\\hat{\\sigma}^x_{p,2}\\hat{\\sigma}^x_{p,3}\\hat{\\sigma}^x_{p,4}\\right),\n\\end{equation}\nwhich is simply $e^{-\\hat{H}}$ where $\\hat{H}$ is the Hamiltonian of Kitaev's toric code model. The principal eigenstates of $\\hat{T}$ are the 4-fold degenerate ground states of $\\hat{H}$. \n\nFig.~\\ref{fig:KTC4} shows the family of loop observables we are interested in. Using the same method as in Sec.~\\ref{sec:loop_observables}, these classical loop observables can be mapped to the conserved loop operators of the quantum toric code, and\nthe thermal expectation values of the former are mapped to quantum expectation values of the latter. Averaging over the four topologically degenerate principal eigenstates, we find that the expectation value of contractible loops is $+1$ while non-contractible loops have expectation value $0$. This  reproduces the topological behavior of the $A$-phase of the Ising model presented in  Sec.~\\ref{sec:loop_observables}.\n\n\\subsection{Realizing the partition function in quantum dynamics}\\label{sec:quantum_amplitude}\n\nAnother way in which classical statistical models with complex energy can be physically relevant is that the partition function $Z$ can be mapped to measurable quantities of certain~(unitary) quantum dynamical processes in 3D (not 2D) quantum systems. In this section we show two such constructions: Sec.~\\ref{sec:transition_amplitude} shows how to realize $Z$ as a transition amplitude, while Sec.~\\ref{sec:probe_spin_coherence} shows that $Z$ gives the quantum coherence of a probe spin-$1\/2$ coupled to the whole system. The phase transition we studied in our model is then mapped to a DQPT in these quantities.\n\n\\subsubsection{Interpreting the partition function as a transition amplitude}\\label{sec:transition_amplitude}\nConsider a 3D quantum spin system on the same lattice as Fig.~\\ref{fig:hc0}, and with a Hamiltonian given by Eq.~\\eqref{eq:HIK} with all $\\sigma_i$ replaced by $\\hat{\\sigma}_i^z$, and we will take all the parameters $J_x,J_y,J_z,J_\\perp,h$ to be real to guarantee hermiticity. The quantum transition amplitude between two arbitrary states is\n\\begin{equation}\n\t\\langle A|e^{-it\\hat{H}}|B\\rangle=\\sum_{\\{\\sigma\\}} e^{-it H[\\{\\sigma\\}]}\\langle A|\\{\\sigma\\}\\rangle\\langle\\{\\sigma\\}|B\\rangle,\n\\end{equation}\nwhere on the RHS we inserted a complete set of $\\hat{\\sigma}^z$ basis states. If the states $|A\\rangle,|B\\rangle$ are of the following form\n\\begin{equation}\\label{eq:initial_final_states}\n\t|A\\rangle=\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{X}}|\\psi(A_x)\\rangle_{ij}\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Y}}|\\psi(A_y)\\rangle_{ij}\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Z}}|\\psi(A_z)\\rangle_{ij},\n\\end{equation}\nwhere $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{X}}$ is over all the red thick $x$-links in  Fig.~\\ref{fig:hc0}, and similarly for $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Y}}$ and $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Z}}$, and the local state on each link $\\langle ij\\rangle$ is defined as $|\\psi(A)\\rangle_{ij}=\\frac{1}{2\\sqrt{\\cosh 2\\mathrm{Re}(A)}}\\sum_{\\sigma_i,\\sigma_j}e^{A\\sigma_i\\sigma_j}|\\sigma_i,\\sigma_j\\rangle$. Note that Eq.~\\eqref{eq:initial_final_states} defines product states since the thick links $\\mathbf{X},\\mathbf{Y},\\mathbf{Z}$ are non-overlapping. Then we have\n\\begin{equation}\\label{eq:transition_amplitude}\n\t\\langle A|e^{-it\\hat{H}}|B\\rangle=\\mathrm{const.}\\times Z(K_x,K_y,K_z,it J_\\perp,it h),\n\\end{equation}\nwhere $K_j=A^*_j+B_j+itJ_j,j=x,y,z$. Therefore, when $tJ_\\perp\\equiv \\pi\/4~(\\mathrm{mod}~ \\pi\/2),t h\\equiv \\pi\/4~ (\\mathrm{mod} ~\\pi)$, the transition amplitude is given by the results we derived previously. %\n\nQuantum transition amplitudes, or closely related objects called dynamical partition functions $f(t)\\propto-\\ln \\langle A|e^{-it\\hat{H}}|B\\rangle$, are the central objects in the study of DQPTs~\\cite{heyl2013dynamical,Andraschko2014dynamical,Heyl2014dynamical,Vosk2014dynamical,Heyl2015dynamical,Schmitt2015dynamical,Heyl_2018}.  In this literature, a dynamical phase transition typically referes to a singularity of the dynamical evolution of a physical quantity~[e.g. $f(t)$] at a critical time. In our model, the time is fixed at special values e.g. $t_0=\\pi\/(4J_\\perp)=\\pi\/(4h)$ to guarantee solvability, and the singularity occurs in $f(t_0)$ %\nas we tune the parameters $K_x,K_y,K_z$ across the phase boundary shown in Fig.~\\ref{fig:critical}. Although the situation is slightly different, the analogy is clear, and we also expect that if we fix $K_x,K_y,K_z$ to be exactly at the phase boundary, say $K_z=K_x+K_y$, and let the system evolve in time, then there will likely be a singularity in $f(t)$ at $t_0$, i.e. a DQPT in the usual sense. \n\nAlthough quantum transition amplitudes are much harder to measure experimentally compared to local observables, there are promising experimental setups~\\cite{Jurcevic2017direct,Tian2020observation} that  measure this quantity in relatively small systems, and are capable of  observing signatures of dynamical phase transition. \n\n\\subsubsection{Mapping the partition function to a probe spin coherence}\\label{sec:probe_spin_coherence}\nWe can also realize the partition function as a probe spin coherence, based on the idea of measuring Yang-Lee zeros in the classical Ising model~\\cite{wei2012lee,peng2015experimental}. To this end we couple a probe spin-$1\/2$ to the whole  3D (quantum) spin system~(bath) shown in Fig.~\\ref{fig:hc0}, with probe-bath interaction %\n\\begin{eqnarray}\\label{eq:probe-bath_interaction}\n\tH_I%\n\t&=& \\hat{\\tau}^z\\otimes\\left(-J_\\perp\\sum_{\\langle ij\\rangle\\in\\boldsymbol{\\perp}}\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}+h\\sum_i\\hat{\\sigma}^z_i\\right)\\nonumber\\\\\n\t&=&\\frac{1}{2}\\hat{\\tau}^z \\hat{B}\n\\end{eqnarray}\nwhere $\\hat{\\tau}^z$ acts on the probe spin, and $J_\\perp$ and $h$ are real. The probe spin is initialized in a superposition state $(|\\uparrow\\rangle+|\\downarrow\\rangle)\/\\sqrt{2}$, and the system~(bath) is initially in equilibrium at temperature $T$ with only interactions in the horizontal $x,y,z$ links, described by the canonical ensemble in Eq.~\\eqref{eq:Z}  with $J_\\perp=h=0$. When we turn on the probe-bath interaction in Eq.~\\eqref{eq:probe-bath_interaction}, the thermal\nfluctuation of the field $\\hat{B}$ induces decoherence of the probe spin~(due to a random phase $Bt$). The probe spin coherence, defined as the ensemble average\nof $e^{i\\hat{B} t}$, is mapped to~\\cite{wei2012lee}\n\\begin{equation}\\label{eq:probe_spin_decoherence}\n\tL(t)\\equiv\\langle e^{i\\hat{B} t}\\rangle=\\frac{Z(\\beta J_x,\\beta J_y, \\beta J_z, i J_\\perp t,i h t)}{Z(\\beta J_x,\\beta J_y, \\beta J_z,0,0)}.\n\\end{equation}\nTherefore, when $tJ_\\perp\\equiv \\pi\/4~(\\mathrm{mod}~ \\pi\/2)$ and $t h\\equiv \\pi\/4~ (\\mathrm{mod} ~\\pi)$, $L(t)$ is given by our exact solution in Sec.~\\ref{sec:solution}~[notice that the denominator of Eq.~\\eqref{eq:probe_spin_decoherence} can be calculated easily and has no singularity], and has a topological phase transition when the parameters $J_x,J_y,J_z$ are tuned across the phase boundary in Fig.~\\ref{fig:critical}.  \nThis kind of probe spin coherence has been measured experimentally in an Ising model of 10 spins~\\cite{peng2015experimental}. \n\n\\section{Summary and Outlook}\\label{sec:summary}\nWe exactly solved a 3D classical Ising model on a special 3D lattice, which has some of its coupling constants fixed to imaginary values. The solution exploits the special structure of the transfer matrix, which can be mapped to free fermions using a method similar to the solution of Kitaev's honeycomb model. The analytic solution reveals two distinct phases, with a third order phase transition between them. The two phases can be distinguished by measuring the  product of spins on certain loops, the expectation value of which is quantized to certain rational values~($0$, $1$, or $1\/3$), depending only on the phase and the topology of the loop. \nWe therefore see that the model not only gives insight into interacting many-body systems in 3D,  but that the behavior it shows is particularly interesting: there are phases with topological properties, and a continuous phase transition between them. \n\nWe expect the topological character of the phases to be universal, as discussed in Sec.~\\ref{sec:loop_observables}. \nWe also expect universality in some other correlations we have not calculated in this paper. For example, the gapless $B$-phase has power-law decaying two-point correlations. For the gapped $B$-phase~(i.e. with the $\\kappa$-perturbation introduced in Sec.~\\ref{sec:phase_boundary}), if we put the system on a large cylinder~(with axis parallel to the $z$-direction), due to the existence of gapless chiral edge modes on the boundary of the 2D quantum system~(defined by the transfer matrix $\\hat{T}$), we expect that the Ising model has power-law decaying correlations on the cylinder boundary even though all two-point correlations in the bulk decay exponentially. We expect the universality in these power-law exponents~(i.e. remain the same when local perturbations are present).\n\nDespite the unphysical complex coupling constants, we described two connections to physical systems. First, the universal long-distance properties of the two phases and the phase transition may be reproduced in a physical 3D system. We demonstrated this by explicitly constructing another 3D classical statistical model with positive Boltzmann weights that has  topological properties identical to the $A$-phase of our 3D Ising model. More speculatively, this suggests that physical systems may have the same universal behavior as models with complex couplings independent of whether the corresponding real-coupling models can be explicitly found or solved. \nWe are unsure if the $B$-phase can be realized in a physical classical system, but we expect this to be challenging if at all possible, since Ref.~\\cite{Ringel2017Nogothm} suggests the prevalence of sign-problems in a family of closely related phases.    \nSecond, the partition function of our model can be realized in certain dynamical processes of a 3D quantum spin system, either as a transition amplitude or as a probe spin coherence, allowing the free energy to be experimentally measured in principle, and the phase transitions studied in our model are related to DQPTs in these 3D quantum systems. \n\nOur model may have other connections to real physical systems beyond the above two. First, when $K_x, K_y, K_z$ are purely imaginary, our transfer matrix $\\hat{T}$ in Eq.~\\eqref{eq:def_TM} becomes the unitary evolution operator of a periodically driven Kitaev model studied in Ref.~\\cite{FloquetKitaev2017}, so our technique of diagonalizing $\\hat{T}$ may be useful in studying certain properties of that system. Second, when $K_x, K_y,K_z\\to \\pm\\infty$, $\\hat{T}$ becomes a projection operator representing the sequential measurement of $\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}, \\hat{\\sigma}^y_{i}\\hat{\\sigma}^y_{j}$, and $\\hat{\\sigma}^x_{i}\\hat{\\sigma}^x_{j}$ on all the $z$-, $y$-, and $x$-links, respectively, which is reminiscent of the measurement process of the honeycomb quantum memory code proposed in Ref.~\\cite{hastings2021dynamically}.\n\nOur results may also provide hints for constructing a genuinely 3D--i.e. one which does not factorize into decoupled 2D models--classical statistical model with positive Boltzmann weights and a continuous phase transition, a problem that has been studied for more than 60 years but never solved. As one possible direction, we note that our model can be straightforwardly generalized to a large family of solvable 3D classical statistical models, whose transfer matrix is similar to one of the generalized Kitaev models~\\cite{Yao2007Exact,Yang2007Mosaic,SI2008Anyonic,Mandal2009Exactly,Yao2009Algebraic,Wu2009Gamma,Ryu2009Three,Tikhonov2010Quantum,Lai2011SU2,Yao2011Fermionic,Barkeshli2015Generalized} that can also be solved by mapping to free fermions. As free-fermion solvable spin models have been systematically classified recently~\\cite{Chapman2020characterizationof,Ogura2020geometric,elman2020free}, it is natural to ask if one of them can be promoted to a transfer matrix that corresponds to a physical 3D classical statistical model.\n\n\n\\acknowledgements\nZ.W. is especially grateful to Zongping Gong who suggested the idea in Sec.~\\ref{sec:probe_spin_coherence}. We also thank Sarang Gopalakrishnan, Bhuvanesh Sundar, and Maxim Olchanyi for helpful discussions. This work was supported in part by the Welch Foundation~(C-1872) and the National Science Foundation~(PHY-1848304).\nK.H.'s contribution benefited from discussions at the KITP, which was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. %\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe recent growth of sensitive optical time-domain surveys has revealed and expanded exciting new classes of stellar explosions. These include superluminous supernovae, which can be up to 10--100 times more luminous than ordinary massive star explosions (e.g.~\\citealt{Quimby2011,nicholl2013,inserra2013,howell2013,decia2018,lunnan2018,quimby2018}; see \\citealt{Gal-Yam2019} for a recent review). Conventionally, the optical emission from most core-collapse supernovae is powered by the radioactive decay of $^{56}$Ni (Type Ib\/c) and by thermal energy generated via shock heating of the stellar envelope (Type IIL, IIp). However, the peak luminosities of SLSNe greatly exceed the luminosity expected from those conventional mechanisms, and the origin of the energy is still debated. \n\nA popular model for powering the time-dependent emission of SLSNe, particularly the hydrogen-poor Type I class (SLSN-I), involves energy input from a young central engine, such as a black hole or neutron star, formed in the explosion. For example, the accretion onto the compact object from bound debris of the explosion could power an outflow which heats the supernova ejecta from within (\\citealt{Quataert2012,woosley2012,margalit2016,moriya2018}). Alternatively, the central engine could be a strongly magnetized neutron star with a millisecond rotation period, whose rotationally powered wind provides a source of energetic particles which heat the supernova ejecta \\citep{kasen2010,woosley2010,dessart2012a,metzger2015a,sukhbold2016}. The magnetar\\footnote{To remain consistent with the SLSNe literature, the term magnetar is used throughout this paper. Magnetars generally have large dipole magnetic fields $B\\simeq$ \\SIrange{e13}{e15}{G} with a rotation period of a few seconds. In the case of the SLSN magnetar model, the radiation is extracted from the rotational energy of the young millisecond pulsars, but with large magnetic fields characteristic of magnetars.} model provides a good fit to the optical light curves of most SLSNe-I \\citep{inserra2013,nicholl2017d}. Furthermore, analyses of the nebular spectra of hydrogen-poor SLSNe \\citep{nicholl2019,jerkstrand2017} and Type-Ib SNe \\citep{milisavljevic2018} support the presence of a persistent central energy source, consistent with an energetic neutron star.\n\nThe details of how the magnetar would couple its energy to the ejecta are uncertain. Several models consider that the rotationally powered wind from a young pulsar inflates a nebula of relativistic electron\/positron pairs and energetic radiation behind the expanding ejecta \\citep{kotera2013,metzger2014b,murase2015}. At the wind termination shock, the pairs are heated and radiate X-rays and gamma rays with high efficiency via synchrotron and inverse-Compton processes. Photons which evade absorption via $\\gamma\\textrm{-}\\gamma$ pair creation in the nebula can be ``absorbed\" by the ejecta further out, thermalizing their energy and directly powering the supernova's optical emission (e.g.~\\citealt{metzger2014b, vurm2021}). \n\nThermalization of the nebular radiation will be most efficient at early times, when the column through the ejecta shell and ``compactness'' of the nebula are at their highest. At these times one would expect the optical light curve to faithfully track the energy input of the central engine. However, as the ejecta expand, the radiation field dilutes and the shell becomes increasingly transparent to high-energy  and very-high-energy photons. The increasing transparency, and correspondingly decreasing thermalization efficiency, eventually causes the supernova's optical luminosity to drop below the rate of energy injection from the central engine \\citep{wang2015,chen2015}, with the remaining radiation escaping directly from the nebula as gamma rays or X-rays (the putative ``missing\" luminosity). \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2015bn_vs_time.pdf}\n    \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2017egm_vs_time_weightlate_v2.pdf}\n    \\caption{Optical depth at different photon energies as a function of time, calculated for ejecta properties (mass $M_{\\odot}$, mean velocity, etc) derived from observations of SN2015bn \\citep{nicholl2018a} and SN2017egm \\citep{nicholl2017c} shown in Table \\ref{tab:event_physical_params}. Top: SN2015bn. Bottom: SN2017egm. The horizontal dotted line represents $\\tau_{\\rm eff} = 1$. The cross-sections for photon-photon and photon-matter pair production opacities are taken from \\citet{zdziarski1989}. The solid lines correspond to target blackbody radiation temperature $T_{\\rm eff} = (L_{\\rm opt}\/4\\pi R^2)^{1\/4}$, where $L_{\\rm opt}$ and $R$ are the optical luminosity and ejecta radius, respectively. The dashed lines are computed with a temperature floor of $T = 4000$~K, to mimic the approximate spectrum in the nebular phase. Below ${\\sim}10$~GeV the opacity is dominated by photon-matter pair production at all times. Above $100$~GeV, pair production on the thermal target radiation field dominates up to a few years.}\n    \\label{fig:tau}\n\\end{figure}\n\nAs the ejecta expand and the spin down luminosity weakens, the conditions for various processes responsible for photon energy loss change and impact the effective optical depth. Within a few months, the effective optical depth to high-energy (HE; 100 MeV to 100 GeV) photons emitted from the central engine nears unity, and at several hundred days it reaches unity for very-high-energy (VHE; 100 GeV \u2013 100 TeV) photons. \n\nFigure~\\ref{fig:tau} shows examples of the effective optical depth through the ejecta for photons of various energies as a function of time. They have been calculated using time-dependent properties for the supernova ejecta and radiation field motivated by the observations of SN2015bn and SN2017egm, both particularly well-studied SLSNe-I explored in \\cite{vurm2021}.\n\nThe dominant processes involved in the calculation of the gamma-ray optical depth include photon-matter and photon-photon interactions, particularly pair production on the nuclei and soft radiation fields in the ejecta. An accurate treatment considering the radiation transport is discussed in depth in \\citet{vurm2021}. The standard version of the magnetar model \\citep{kasen2010, woosley2010, nicholl2017d} does not consider this time-dependent calculation and relies on constant effective  opacities to optical and high-energy photons. Figure~\\ref{fig:tau} provides a useful guiding timescale for when to consider gamma-ray emission at various energies, calculated with the model in \\citet{vurm2021} using the ejecta properties fit to the optical data in Table \\ref{tab:event_physical_params}.\n\nGiven its comparatively nearby distance at z=0.1136, SN2015bn is an excellent candidate event to test the magnetar hypothesis. The optical light curve shows a steepening from $\\propto t^{-2}$ decay to $\\propto t^{-4}$ around ${\\sim}200$ days \\citep{nicholl2018a}. This behavior is consistent with a leakage of high-energy radiation from a magnetar nebula \\citep{nicholl2018a}. A deep search in the ${\\sim}0.1-10$ keV X-ray band resulted in non-detections \\citep{bhirombhakdi2018}, eliminating the possibility that leakage from the nebula occurs in the softer X-ray bands.\n\n\\citet{margutti2018b} present a similar search for late-time X-ray emission from a larger sample of SLSNe-I, mostly resulting in upper limits; however, see \\citet{levan2013} for an X-ray detection of the SLSN-I SCP 06F6 that could still support the magnetar hypothesis. X-ray non-detections are not surprising, because the ejecta are likely to still be opaque in the $\\lesssim 10$ keV band due to photoelectric absorption in the hydrogen-poor ejecta \\citep{margalit2018a}. Intriguingly, \\citet{Eftekhari2019} detected radio emission from the location of the SLSN PTF10hgi at 7.5 years after the explosion and argued that the emission could be synchrotron emission from an engine-powered nebula.\n\nSome effort has been underway to search for nebular leakage in the gamma-ray band. \\citet{renault-tinacci2018} obtained upper limits on the $0.6-600$ GeV luminosities from SLSNe by a stacked analysis of 45 SLSNe with {\\it Fermi}-LAT. The majority of their sample were SLSNe-I, the most likely class to be powered by a central engine; however the results were dominated by a single, extremely close Type II event (SLSN-II), CSS140222. Hydrogen-rich SLSNe make up the Type II class (SLSN-II), which are suggested to be powered by the interaction of the circumstellar medium with the supernova ejecta.  Nevertheless, even with CSS140222 included, the upper limits are at best marginally constraining on the inferred missing luminosity. \n\nIn this paper, the search is expanded to gamma-ray emission from SLSNe-I in the HE to VHE bands using the {\\it Fermi} Gamma-Ray Space Telescope and the ground-based VERITAS observatory. In particular, observations of SN2015bn and SN2017egm are presented here. SN2017egm is the closest SLSN-I to date in the Northern Hemisphere at z=0.0310 \\citep{nicholl2017c,bose2017}. Observations of young supernovae with gamma-ray telescopes have been few, with no detections so far. Some tantalizing candidates like iPTF14hls and SN 2004dj have been explored with {\\it Fermi}-LAT but are unconfirmed due to large localization regions overlapping with other gamma-ray candidates \\citep{yuan2018,xi2020}. MAGIC carried out observations of a Type I SN \\citep{ahnen2017a}. HESS observed a sample of core-collapse SNe \\citep{Abdalla2019}, and later obtained upper limits on SN 1987A \\citep{theh.e.s.s.collaboration2015}. Our observations are the first of superluminous supernovae. \n\nThroughout this paper, a flat $\\Lambda$CDM cosmology is used, with $H_0 = \\SI{67.7}{km.s^{-1}.Mpc^{-1}}$, $\\Omega_{M}=0.307$, and $\\Omega_{\\Lambda} = 0.6911$ \\citep{planckcollaboration2016}. The corresponding luminosity distances to SN2015bn and SN2017egm are  \\SI{545.37}{\\mega\\pc} (z=0.1136) \\citep{nicholl2016} and \\SI{139.29}{\\mega\\pc} (z=0.0310) \\citep{bose2017}.\n\n\n\n\n\n\\section{Observations \\& Methods} \\label{sec:obs}\n\\label{sec:observations}\nThe superluminous supernovae SN2015bn and SN2017egm were observed with {\\it Fermi}-LAT and VERITAS during 2015--2016 and 2017--2020, respectively. SN2015bn is a SLSN-I explosion from 23 Dec 2014 (MJD 57014) and it peaked optically on 19 Mar 2015 (MJD 57100) \\cite{nicholl2016a}. SN2017egm is a SLSN-I explosion from 23 May 2017 (MJD 57896) and it peaked optically on 18 Jun 2017 (MJD 57922) \\cite{bose2017}. Some properties of the SLSNe are given in Table \\ref{tab:event_physical_params}. Details regarding the optical, {\\it Fermi}-LAT and VERITAS observations and the data-analysis methods are below.  \n\n\n\n\\begin{table}[ht]\n    \\begin{center}\n        \\caption{Properties of the SLSNe considered in this paper. The quantities $P_{0}$, $B$, $M_{\\rm ej}$, $\\kappa$, $E_{\\rm SN}$, $v_{\\rm ej}$, $\\kappa_{\\gamma}$ and $M_{\\rm NS}$ were obtained from a best-fit to the UVOIR supernova light curves, with errors found in \\cite{nicholl2017c, nicholl2017d}. }\n        \\begin{tabular}{cl|rr}\n        \\multicolumn{1}{l}{Parameter} & {[}unit{]}           & {SN2015bn} & {SN2017egm} \\\\ \\hline\n        RA                            & $^{\\circ}$           & 173.4232 & 154.7734  \\\\\n        Dec                           & $^{\\circ}$           & 0.725    & 46.454    \\\\\n        z                             & -                    & 0.1136   & 0.0310    \\\\\n        $t_{0}^{(a)}$                 & MJD                  & 57014    & 57896     \\\\\n        $t_{pk}^{(b)}$                & MJD                  & 57100    & 57922     \\\\ \\hline\n        $P_0^{(c)}$                   & ms                   & $2.50^{+0.29}_{-0.17}$     & $5.83^{+0.73}_{-0.70}$      \\\\\n        $B^{(d)}$                     & $10^{14}$ G          & $0.26^{+0.07}_{-0.05}$     & $0.94^{+0.13}_{-0.16}$      \\\\\n        $M_{ej}^{(e)}$                & $M_{\\odot}$          & $10.8^{+0.83}_{-1.34}$     & $2.99^{+0.30}_{-0.23}$      \\\\\n        $\\kappa^{(f)}$                & cm$^{2}$g$^{-1}$     & $0.18^{+0.01}_{-0.02}$     & $0.12^{+0.04}_{-0.06}$      \\\\\n       \n        $v_{ej}^{(h)}$                & $10^{8}$ cm s$^{-1}$ & $5.68^{+0.16}_{-0.14}$     & $10.3^{+0.35}_{-0.27}$      \\\\\n        $\\kappa_{\\gamma}^{(i)}$       & cm$^{2}$g$^{-1}$     & $0.008^{+0.01}_{-0.01}$    & $0.080^{+0.15}_{-0.06}$     \\\\\n        $M_{\\rm NS}^{(j)}$            & $M_{\\odot}$          & $1.84^{+0.28}_{-0.23}$     & $1.57^{+0.25}_{-0.29}$      \n        \\end{tabular}\n        \\label{tab:event_physical_params}        \n    \\end{center}\n    $^{(a)}$Epoch of explosion; $^{(b)}$Epoch of optical flux peak; $^{(c)}$Initial spin-period; $^{(d)}$magnetic field strength of magnetar; $^{(e)}$Total mass, $^{(f)}$effective opacity; $^{(g)}$kinetic energy; $^{(h)}$mean velocity of supernova ejecta; $^{(i)}$gamma-ray effective opacity; and $^{(j)}$neutron star mass.\n\\end{table}\n\n\\subsection{{\\it Fermi}-LAT} \\label{subsec:Fermi}\nThe Large Area Telescope (LAT) on board the {\\it Fermi} satellite has operated since 2008 \\citep{Atwood2009}. It is sensitive to photons between \\SI{{\\sim}20}{\\MeV} and \\SI{{\\sim}300}{\\GeV} and has ${\\sim}60\\degree$ field of view, enabling it to survey the entire sky in about three hours. \n\nThe data were analyzed using the publicly available {\\it Fermi}-LAT data with the \\texttt{Fermitools} suite of tools provided by the {\\it Fermi} Science Support Center (FSSC). Using the \\texttt{Fermipy} analysis package \\citep{wood2017}\n\\footnote{\\url{https:\/\/fermipy.readthedocs.io\/en\/latest\/} ; v0.19.0}, the data were prepared for a binned likelihood analysis in which a spatial spectral model is fit over the energy bins. The data were selected using the SOURCE class of events, which are optimized for point-source analysis, within a region of $15\\degree$ radius from the analysis target position. Due to the effect of the Earth, a $90\\degree$ zenith angle cut was applied to remove any external background events. The standard background models were applied to the test model, incorporating an isotropic background and a galactic diffuse emission model without any modifications. The standard 4FGL catalog was then queried for sources within the field of view and their default model parameters \\cite{abdollahi2020}. \n \nAdditional putative point sources were added to each field of view as needed to support convergence of the fit. These sources were added for all analysis time scales. This process continued until the distribution of test statistics for the field of view was Gaussian with standard deviation near 1 and mean centered at zero, and the residual maps were near uniformly zero without strong features. These conditions indicate the appropriate coverage of spectral sources within the analysis was reached and no putative sources are missing. The fitting process is performed in discrete energy bins while optimizing the spectral shape, but the distribution of test statistics is evaluated with the stacked data spanning the full energy range. With the improvements to {\\it Fermi}-LAT low-energy sensitivity in PASS8 reconstruction, the low energy bin covering $100-612$ MeV was also added. \n \nIn the case of both SN2015bn and SN2017egm, the data were fitted with a power-law spectral model, $N(E) = N_{0} E^{\\Gamma}$, with a free prefactor and a fixed photon index $\\Gamma$ of -2.0.  From the fit, the reported flux upper limit was found using a 95\\% confidence level with the bounded Rolke method \\citep{rolke2005}. In all cases reported here, the upper limit reported is the integral energy flux, integrated over the energy ranges described for each case, which has units of \\si{MeV cm^{-2}.s^{-1}}. This flux is converted to luminosity with the adopted distance for each event.\n \n\nSN2015bn was observed from 23 Dec 2014 to 23 Mar 2018. This observation period begins after the explosion, and is binned in a few windows to account for the absorption of low-energy gamma rays by the ejecta at early times (Figure~\\ref{fig:tau}). The first $\\sim90$ days is observed to make sure there are no early emission during the expected absorption period. The data were thereafter binned in time intervals of six months to maximize observation depth and sensitivity to time dependent variation. SN2017egm was observed 23 May 2017 to 21 Aug 2020. Again, this period covers the 3.5 years from the discovery date, starting with $\\sim90$ days after the explosion and split into six 6-month bins thereafter. The 3.5 year observation period is selected to cover approximately 1000 days after the explosion. After this period, it is expected that the predicted luminosity will have decreased below the {\\it Fermi}-LAT detectable limit. \n\nSN2015bn is within 5\\textdegree\\ of the Sun each year in August, so a one-month time cut is applied to each relevant time bin (to cover a $\\sim15$\\textdegree\\ radius field of view). SN2017egm is not near the the path of the Sun, so this cut was not applied.\n\n\\subsection{VERITAS} \\label{subsec:veritas}\nThe Very Energetic Radiation Imaging Telescope Array System (VERITAS) is an imaging atmospheric cherenkov telescope (IACT) array at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona, USA \\citep{weekes2002, Holder2006}. It consists of four 12-m telescopes separated by approximately $100$ m, and the observatory is sensitive to photons within the energy range $\\backsim$\\SI{100}{GeV} to $\\backsim$\\SI{30}{TeV}. The instrument has an angular resolution (68\\% containment) of $\\backsim$0.1\\degr\\ at \\SI{1}{TeV}, an energy resolution of $\\backsim$15\\% at \\SI{1}{TeV}, and 3.5\\degr\\ field of view. \n\n\nVERITAS serendipitously observed SN2015bn for a total of 1.01 hours between 7 May 2015 and 22 May, 2015, approximately 135 days from explosion (49 days from the date of peak magnitude), as a part of an unrelated campaign. Another 1.7 hours were taken between 25 May 2016 and 30 May 2016. Data were taken in good weather and dark sky conditions. Since SN2015bn was not the target source, its sky position averages 1.4\\textdegree~from the center of the camera.  \n\nVERITAS directly observed SN2017egm for 8.7 hours between 24 Mar 2019 and 5 Apr 2019, under dark sky conditions, as part of a Directors Discretionary Time (DDT) campaign, approximately 670 days from explosion. This target was triggered based on the predicted gamma-ray luminosity (see section \\ref{sec:MagnetarSpinDown} and appendix \\ref{sec:appendix} for a description) derived from the optical observation. Although it was almost two years after the explosion, the nearby distance yielded a gamma-ray luminosity prediction still within reach of VERITAS, making this an enticing target to follow up.\n\nThe SN2017egm data in this paper were taken using ``wobble\" pointing mode, where the source is offset from the center of the camera by $0.5\\degree$. This mode creates space for a radially symmetric off region to be used for background estimation in the same field of view, saving time from targeted background observations that contain the same data observing conditions. The data were processed with standard VERITAS calibration and reconstruction pipelines, and then cross-checked with a separate analysis chain \\citep{Maier2017,cogan2008}. \n \n \nUsing an Image Template Method (ITM) to improve event angular and energy reconstruction \\citep{christiansen2017}, analysis cuts are determined with a set of a priori data selection cuts optimized on sources with a moderate power-law index (from -2.5 to -3).\n\nUnfortunately, the large offset on SN2015bn due to the serendipitous observation precludes us from using ITM in the analysis, so in that case SN2015bn is analyzed without templates by calculating image moments directly from candidate images triggered by the camera \\citep{Maier2017,cogan2008}.   In both cases, the signal and background counts are determined using the reflected region method. \n\n\nThe upper limit is calculated for both SN2015bn and SN2017egm. The bounded Rolke method for upper limit calculation is used, assuming a power law spectrum with index of -2.0 and 95\\% confidence level \\citep{rolke2005}. Since the calculation of the upper limit depends on the underlying spectral model, a range of power-law spectral indices from -2 to -3 was computed to estimate impact of the model dependence. In all cases reported here, the upper limit reported is the integral photon flux, integrated over the energy ranges described for each case, which has units of \\si{cm^{-2}.s^{-1}}. This flux is converted to integral energy flux using the same spectral model so that the luminosity can be computed with the adopted distance.\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics{SN2015bn_ebl_lc.pdf}\n    \\caption{Light curves of SN2015bn spanning 30 to 1500 days after explosion. Curves shown include (1) the (thermal) supernova luminosity, $L_{\\rm opt}$, fit to UVOIR bolometric luminosity data (in red; \\citealt{nicholl2018a}) to obtain the magnetar parameters; (2) magnetar spin down luminosity, $L_{\\rm mag}$ (green dotted lined); and (3) predicted gamma-ray luminosity that escape the ejecta, $L_{\\gamma}$ (pink dot-dashed line; Equations \\ref{eq:Lmag}, \\ref{eq:trapped} and \\ref{eq:leaking}). Black bars show {\\it Fermi}-LAT upper limits reported for six 180 day bins starting ${\\sim}90$ days after explosion. The olive open box shows the VERITAS integral energy flux} upper limit taken ${\\sim}135$ days after the explosion, with EBL absorption correction applied. Upper limits on the 0.2-10 keV X-ray luminosity from {\\it Chandra} are from \\citet{bhirombhakdi2018} in green. Grey shaded regions labeled ``$\\tau_{\\gamma} <1$\" show the approximate time after which gamma rays of the indicated energy should escape ejecta, based on Figure \\ref{fig:tau}. A purple dot-dashed line shows the engine luminosity, $L_{\\mathrm{BH}}$ (Eq.~\\ref{eq:L_BH}), in an alternative model in which the supernova optical luminosity is powered by fall-back accretion onto a black hole. All upper limits denote the 95\\% confidence level.\n    \\label{fig:SN2015bn_lc}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics{SN2017egm_lc_ebl_corrected.pdf}\n    \\caption{The SN2017egm light curve spanning 10 to 1300 days after explosion, following the same format as Figure~\\ref{fig:SN2015bn_lc}. UVOIR data are shown in red \\citep{bose2017, nicholl2017c}. Integral energy flux upper limits from {\\it Fermi}-LAT are reported for six 180 day bins starting ${\\sim}90$ days after the explosion. Integral energy flux upper limits are shown for VERITAS data taken ${\\sim}670$ days after explosion, with EBL absorption correction applied. The maximum luminosity of the black hole accretion model $L_{\\mathrm{BH}}$ (Eq. \\ref{eq:L_BH}) is shown in purple. All upper limits denote the 95\\% confidence level.}\n    \\label{fig:SN2017egm_lc}\n\\end{figure*}\n\n\\begin{table*}[ht]\n    \\begin{center}\n        \\caption{Results from VERITAS observations for both epochs of SN2015bn, and SN2017egm. Shown are the quality selected livetime, number of gamma-ray-like events in the on and off-source regions, the normalization, the observed excess of the gamma-rays and the statistical significance. The integral flux upper limit is shown for the given energy threshold, without EBL absorption correction, integrated up to \\SI{30}{\\TeV}.}\n        \\label{tab:veritas_total_results}\n        \\begin{tabular}{cl|rrr}\n        \\multicolumn{1}{l}{Parameter} & {[}unit{]} & SN2015bn$_{1}$ & SN2015bn$_{2}$ & SN2017egm \\\\ \n        \\hline\n        \\hline\n        Start (MJD) & [day] & 57149 & 57533 & 58566 \\\\\n        End (MJD) & [day] & 57164 & 57538 & 58578 \\\\\n        Livetime & [hour] & 1.0 & 1.8 & 8.7 \\\\\n        On & [event] & 4 & 10 & 49\\\\\n        Off & [event] & 179 & 188 & 596 \\\\\n        $\\alpha^{(a)}$ & - & 0.0286 & 0.0299 & 0.0634 \\\\\n        Excess & [event] & -1.1 & 4.4 & 11.2 \\\\\n        Significance & [$\\sigma$] & -0.5 & 1.7 & 1.6 \\\\\n        Flux UL & [$\\SI{e-13}{cm^{-2}~s^{-1}}$] & 28.5 & 27.8 & 10.2 \\\\\n        $E_{\\rm  threshold}$ & [GeV] & $>320$ & $>420$ & $>350$ \\\\ \n        \\end{tabular}\n    \\end{center}\n    $^{(a)}$ Ratio of relative exposure for On and Off regions.\n\\end{table*}\n\n\\section{Results}\n\\label{sec:results}\n\nNo statistically significant detections were made of either SN2015bn or SN2017egm across the energy range 100 MeV to 30 TeV. Integral energy upper limits are reported for the energy ranges given for each instrument. Figure~\\ref{fig:SN2015bn_lc} and Figure~\\ref{fig:SN2017egm_lc} show the {\\it Fermi}-LAT and VERITAS upper limits in comparison to the supernova optical light curves and the theoretically-predicted escaping luminosity from the magnetar model. \n\\subsection{Optical}\\label{sec:optical_result}\nThe SN2015bn integrated ultraviolet-optical-infrared (UVOIR) light curve data are reproduced here from previous analyses \\citep{nicholl2016, nicholl2016a, nicholl2018a}. To produce these bolometric light curves, the multi-band optical data were interpolated and integrated at each epoch using the code \\texttt{superbol} \\citep{nicholl2018}. \n\nSimilarly, the SN2017egm UVOIR data are also reproduced here with \\texttt{superbol} \\citep{bose2017, nicholl2017c}.\n\n\\subsection{{\\it Fermi}-LAT}\nBoth SN2015bn and SN2017egm are not statistically significant sources in the first $\\sim90$ days or the subsequent 6-month bin starting 90 days after the explosion. These sources also remain undetected in any of the following 6-month bins, and in the multi-year data sets. \n\nThe evaluation of the integral energy flux upper limit for the {\\it Fermi}-LAT observations within each time bin was performed assuming a powerlaw spectral model with an index of -2. The model dependence of this calculation naturally impacts the interpretations in section \\ref{sec:discussion}, so the the fit was performed with indices 2, 2.5 and 3 to find the impact of the model on the final upper limit. An uncertainty of about $10\\%$ was found based on varying the index.\n\nSN2015bn is found to have test statistic (TS) of 0.06,\nwith 12 predicted events above the isotropic diffuse background $\\simeq \\num{4.8e4} $ events over the entire period. The flux upper limit is $\\SI{1.6e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.5e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{1.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS \\simeq 0$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<2$. \n\nSN2017egm is found to have $TS=4.4$,\nwith 43 predicted events above the isotropic diffuse background $\\simeq \\num{5.9e4} $ events. The flux upper limit is $\\SI{1.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{4.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS=\\num{10.1}$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<1$.\n\n\\subsection{VERITAS}\nTable \\ref{tab:veritas_total_results} reports the results from VERITAS observations of SN2015bn and SN2017egm. Each observation is consistent with a non-detection. The significance of each excess of observed events above background is below 2 standard deviations (sigma). The flux upper limits are also given, calculated by integrating above the threshold energy of the instrument.\n\nThe statistical significance of an excess is estimated using Equation 17 of Li \\& Ma \\citep{li1983}. SN2015bn has significance value of $-0.5 \\sigma$ in the first epoch observation. The integral flux upper limit from \\SIrange{0.32}{30}{\\TeV} for SN2015bn is \\SI{2.85e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.27e44}{\\erg.s^{-1}} at a redshift of 0.1136.  Due to the serendipitous nature of the observation, SN2015bn is significantly off-axis, which lowers the instrument sensitivity at the energy threshold of \\SI{320}{\\GeV}.  Additionally, a 10\\% systematic uncertainty is added to the flux normalization and reported energy threshold due to instrument degradation during the period of 2012-2015 \\cite{nievasrosillo2021}. This uncertainty is derived empirically from the observation of the Crab Nebula over the same period. During the second observation in 2016, SN2015bn was found to have a significance of 1.7. The integral flux upper limit from \\SIrange{0.42}{30}{\\TeV} for SN2015bn is \\SI{2.78e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.60e44}{\\erg.s^{-1}}. \n\nFor SN2017egm, the Li \\& Ma significance value is $0.2 \\sigma$ and an integral upper limit from \\SIrange{0.35}{30}{\\TeV} is \\SI{1.0238e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{3.54e42}{\\erg. s^{-1}} above the energy threshold of \\SI{350}{\\GeV} at redshift z=0.0310. The systematic correction due to instrument degradation during the period of 2012-2019 is applied automatically with the use of the throughput-calibrated analysis templates \\citep{nievasrosillo2021}. In the cases of both SN2015bn and SN2017egm, the impact of varying the power law model index parameter from -2 to -5 is about 10\\%, which is a negligible in the context of their respective light curves.\n\nVHE photons are absorbed by the extragalactic background light (EBL) throughout the universe, so the flux must be corrected to account for the missing photons. This absorption is energy and redshift dependent. Deabsorption is applied to the flux using the model of \\citet{Dominguez2011}.  \nThe EBL deabsorption factor was convolved with the upper limit calculation, assuming the same spectral shape (a power law with the photon index of -2.0). \nThe deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.32}{30}{\\TeV}, is \\SI{3.36e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.49e44}{\\erg.s^{-1}}. For the second observation, the deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.42}{30}{\\TeV}, is \\SI{3.30e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.91e44}{\\erg.s^{-1}}. For SN2017egm, with a slightly smaller energy range \\SIrange{0.350}{30}{\\TeV}, the deabsorbed integral photon flux is \\SI{1.07e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{3.70e42}{\\erg.s^{-1}}. These EBL corrected values are plotted in Figure \\ref{fig:SN2015bn_lc} and Figure \\ref{fig:SN2017egm_lc}.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe source of the extra luminosity powering SLSNe-I may be found in the signature of its late time gamma-ray emission. This section explores the HE to VHE emission hundreds of days after the explosion. The following models with a gamma-ray emission component for the powering mechanism are discussed: 1) magnetar central engine (see section \\ref{sec:MagnetarSpinDown}), 2) black hole central engine (see section \\ref{sec:BlackHole}), and 3) circumstellar interaction (see section \\ref{sec:Circumstellar}).\n\n\\subsection{Magnetar Central Engine} \\label{sec:MagnetarSpinDown}\n\nThe most promising mechanism for powering SLSNe-I is the rotational energy input from a central magnetar. In this scenario, a young pulsar or magnetar inflates a nebula of relativistic particles, which radiate high-energy gamma rays and X-rays. This section initially explores a simple implementation of the magnetar model (see Appendix \\ref{sec:appendix} for full description), followed by a more complete model described in detail in \\citet{vurm2021} for both SN2015bn and SN2017egm. The application of this so-called self-consistent model is necessary to directly predict the energy-dependent luminosities within the energy ranges of the {\\it Fermi}-LAT and VERITAS observations, a major contribution that is not possible with simpler implementation described in the appendix.\n\nAt early times after the explosion (around and immediately after the maximum in the optical emission) the gamma rays are absorbed and thermalized by the expanding supernova ejecta. At these times, the luminosity and shape of the optical light curve can be used to constrain the parameters of the magnetar. In this model, the radiation of an input energy reservoir (the spin down luminosity of a rotating magnetar) diffuses through the ejecta following the analytical solution by \\citet{arnett1982} (equation \\ref{eq:trapped}). \n\nThe time evolution of the magnetar's spin-down luminosity can be modeled by assuming a rotating dipole magnetic field whose energy loss is dominated by emission of radiation in the gamma-ray and X-ray bands (see Appendix \\ref{sec:appendix} for details). \n\nThis luminosity depends on the magnetar initial spin period, surface dipole magnetic field strength, and neutron star mass, $L_{\\rm mag}(t, P_0, B, M_{\\rm NS})$ (equation \\ref{eq:Lmag}). The emitted radiation thermalizes as it diffuses through the ejecta. The conditions of the ejecta determine the optical and gamma-ray outputs, dominated by the values of the ejecta mass, ejecta velocity, and optical and gamma-ray opacities to form $L_{\\rm opt}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa,\\kappa_{\\gamma})$ (equation \\ref{eq:leaking}) and $L_{\\gamma}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa, \\kappa_{\\gamma})$ (equation \\ref{eq:escape}). \n\nFor SN2015bn and SN2017egm, the parameters for the magnetar and the supernova ejecta properties were found by fitting their integrated ultraviolet-optical-infrared (UVOIR) light curves, shown with red points in Figure~\\ref{fig:SN2015bn_lc} and Figure~\\ref{fig:SN2017egm_lc}. All fits were conducted using non-linear least squares minimization\\footnote{\\texttt{scipy.optimize.curve\\_fit}}.\nThe best-fit parameters with errors for the magnetar model are given in Table \\ref{tab:event_physical_params}. The redshifts and time of peak optical magnitude are shown in the table as listed in The Open Supernova Catalog \\citep{Guillochon2016}\\footnote{\\url{https:\/\/sne.space}}.\n\nThese parameters are consistent with the results of previous fits \\citep{nicholl2018a,nicholl2017c} that took into account both the optical spectral energy distribution and light curve using the open source code \\texttt{MOSFiT} \\footnote{\\url{https:\/\/mosfit.readthedocs.io\/en\/latest\/}}. \nThe relative statistical errors on these fit parameters may be optimistic at $\\sim10\\%$, and the systematic errors will still need to be incorporated for a better understanding the magnetar parameter space. The largest contributor to the magnetar power are the period and magnetic field values, which determine the overall magnitude of the luminosity. The ejecta mass and velocity determine the time to optical peak by the diffusion of the emission through the ejecta.\n\nA particularly important shortfall of this model is the constant effective opacity to both optical and gamma-ray photons, rather than a time-dependent treatment of the opacity. TeV gamma rays interact preferentially with optical photons, so at the time of the peak optical emission, $\\gamma\\gamma$ absorption by optical photons will be high, reducing any predicted gamma-ray emission by this model. Equation \\ref{eq:leaking} is a bolometric luminosity, so it does not take into account the energy and time dependent opacity, instead fitting a constant effective $\\kappa$ and $\\kappa_{\\gamma}$ to generate the time dependent optical depth. \n\nTherefore, Figure~\\ref{fig:tau} is used as a guide for when to expect $L_{\\gamma}$ to provide an appropriate estimate for the gamma-ray emission. The shaded regions in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} estimate the time periods when photons of the given energies can escape. It is important to reiterate that this model is energy independent, representing the bolometric luminosity not thermalized by the ejecta. This model cannot distinguish the emission between LAT and VERITAS energy bands since it does not consider the physical model of the nebula; the self-consistent model described by \\cite{vurm2021} and discussed below will be an attempt to do so explicitly.\n\nFollowing the methodology in Appendix \\ref{sec:appendix} with the magnetar parameters for each SLSN, $L_{\\rm mag}(t)$, $L_{\\rm opt}(t)$, and $L_{\\gamma}(t)$ were calculated and are shown in comparison to the gamma-ray limits in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}.\n\nFor SN2015bn (Figure \\ref{fig:SN2015bn_lc}), neither the {\\it Fermi}-LAT upper limits nor the VERITAS upper limit constrain the predicted escaping luminosity. \nSimilarly, for SN2017egm (Figure \\ref{fig:SN2017egm_lc}), both the VERITAS and {\\it Fermi}-LAT upper limits are not deep enough to constrain the predicted escaping luminosity. An important caveat to these upper limits is that the escaping luminosity may also be emitted at energies not explored here, such as hard X-rays or gamma-rays greater than \\SI{ 30}{\\TeV}.\n\nThe optimal time to observe with a pointed instrument sensitive at a particular photon energy results from a trade-off between the dropping ($\\propto t^{-2}$) magnetar luminosity and the rising transparency of the ejecta; predicting the optimal time post-peak to observe requires knowledge of the evolution of the optical spectrum. It is possible to accumulate enough optical data within a few weeks after the optical peak to fit the magnetar model for a reliable prediction of the gamma-ray luminosity. In the case of SN2017egm, the gamma-ray luminosity prediction was anchored by the late optical data points about 1 year after the explosion. This means that had the VERITAS observations been taken at that point (more than a year earlier than the original observation), they would have been deeply constraining to the magnetar model.\n\n\nGoing beyond these relatively model independent statements to compare to a more specific spectral energy distribution for the escaping magnetar nebula requires a detailed model for the nebula emission and its transport through the expanding supernova ejecta. Such a model offers preliminary support that a significant fraction of $L_{\\gamma}$ may come out in the VHE band \\citep{vurm2021}. In this case, the VHE limits on SN2015bn and SN2017egm do not strongly constrain the parameters of the magnetar model, such as the nebular magnetization.\n\n\nThe model of \\citet{vurm2021} self-consistently follows the evolution of high-energy electron\/positron pairs injected into the nebula by the magnetar wind and their interaction with the broadband radiation and magnetic fields. They found that the thermalization efficiency and the amount of gamma-ray leakage depends strongly on the nebular magnetization, $\\varepsilon_B$, i.e. the fraction of residual magnetic energy in the nebula relative to that injected by the magnetar.\n\nThe model is simulated for dimensionless $\\varepsilon_B$ values set between  $10^{-6}$ and $10^{-2}$; the higher magnetizations lead to greater synchrotron efficiencies, which dominate within a few hundred days, and lead to the optical emission tracking the spin-down luminosity. Lowering the magnetization to $10^{-7}-10^{-6}$ for SLSN-I events like those in this work delays the transition to synchrotron-dominated thermalization, so that the predicted optical emission actually tracks the observed data.\n\nThe theoretical light curves and gamma-ray upper limits are shown in Figure ~\\ref{fig:Indrek}. \\citet{vurm2021} concluded that the predicted low magnetizations constrained by the optical data alone presents new challenges to the theoretical framework regarding the dissipation of the nebular magnetic field. This may invoke magnetic reconnection ahead of the wind termination shock or near the termination shock through forced reconnection of alternating field stripes described in \\citet{komissarov2013}, \\citet{lyubarsky2003}, \\citet{margalit2018b}. It is also possible that the true luminosity of the central engine decreases faster in time than the simpler $\\propto t^{-2}$ magnetic spin down, such that escaping VHE emission is not necessary to explain the model. These VHE upper limits do not rule out this model, and do not settle the challenges inferred by the low magnetization required to fit the optical data. Further observations are needed to probe the nebular magnetization and synchrotron efficiency, and deep VHE observations will contribute to these constraints.\n\nThe non-detection of x-rays for both events is consistent with the predictions of \\cite{margalit2018a} of a fully ionized ejecta. Even under the most optimistic conditions - an engine that puts 100\\% of its spin-down luminosity into ionizing photons of ideal energies - cannot reduce the opacity enough to allow x-rays to escape under the usual assumptions (e.g. spherically symmetric ejecta shell). \n\n\n\n\n\\begin{figure}[ht]\n    \\includegraphics[width=1.1\\columnwidth]{SN2015bn_lc_ebl_model.pdf}\n    \\includegraphics[width=1.1\\columnwidth]{SN2017egm_lc_ebl_model.pdf}\n    \\caption{Model light curve for nebular magnetization (from  \\cite{vurm2021}) for SN2015bn with $\\varepsilon_B = 10^{-7}$ (top panel) and SN2017egm with $\\varepsilon_B=10^{-6}$ (bottom panel). }\n    \\label{fig:Indrek}\n\\end{figure}\n\n\\subsection{Black Hole Central Engine}  \\label{sec:BlackHole}\n\nInstead of forming a neutron star like a magnetar, a SLSN-I might form a black hole, in which case the optical peak of the light curve could be powered by energy released from the fallback accretion of ejecta from the explosion (e.g.~\\citealt{dexter2013}). Even if a black hole does not form immediately, it could form at late times once the magnetar accretes enough fallback material \\citep{moriya2016a}. The main practical difference as compared to a magnetar in section \\ref{sec:MagnetarSpinDown} is that the black hole central engine power would be predicted to decay with the fall-back accretion rate $\\dot{M}_{\\rm fb} \\propto t^{-5\/3}$ instead of $\\propto t^{-2}$. Thus, in principle, for the same luminosity at the time of the optical maximum $t_{\\rm pk}$, the central engine output at times $t \\gg t_{\\rm pk}$ could be enhanced by a factor $\\propto (t\/t_{\\rm pk})^{1\/3} \\sim 2$ for $t \\sim 1$ year and $t_{\\rm pk} \\sim 1$ month, thus tightening our constraints.\n\nIn Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}, a rough estimate of the maximal engine luminosity in the BH accretion scenario is shown, which is calculated as \n\n\\begin{align}\nL_{\\mathrm{BH}}=\\frac{2^{5\/3} L^{\\rm pk}_{\\rm opt}}{\\left(1+\\frac{t}{t_{\\rm pk}}\\right)^{5\/3}} \\label{eq:L_BH},\n\\end{align}\n\nwhere $L^{\\rm pk}_{\\rm opt}$ is the peak optical luminosity, scaled so that $L_{\\rm BH} = L_{\\rm opt}$ around the optical peak.\n\nOn the other hand, while gamma rays are naturally expected from the ultra-relativistic spin-down powered nebula of a magnetar, it is less clear this would be the case for a black hole engine. For instance, the majority of the power from a black hole engine could emerge in a mildly relativistic wind from the black hole accretion disk instead of an ultra-relativistic spin-down powered pulsar wind.\n\nAs seen in both Figure \\ref{fig:SN2015bn_lc} (SN2015bn) and Figure \\ref{fig:SN2017egm_lc} (SN2017egm), the gamma-ray emission in the black hole scenario is not constrained in the {\\it Fermi}-LAT and VERITAS energy bands.\n\n\n\\subsection{Circumstellar Interaction} \\label{sec:Circumstellar}\n\nAn alternative model for powering the light curve of SLSNe is to invoke the collision of the supernova ejecta with a slower expanding circumstellar shell or disk surrounding the progenitor at the time of the explosion (e.g.~\\citealt{smith2006,chevalier2011,Moriya2013a}). Features of this circumstellar model (CSM), such as the narrow hydrogen emission lines that indicate the interaction of a slow-moving gas, provide compelling evidence for this being a powering mechanism for many but not all of the hydrogen-rich class of SLSNe (SLSNe-II; e.g.~\\citealt{smith2007,nicholl2020}). \n\nShock interaction could in principle also power some hydrogen-poor SLSNe (SLSNe-I), particularly in cases where the circumstellar interaction is more deeply embedded and less directly visible (e.g.~\\citealt{sorokina2016,kozyreva2017}). \nThere is growing evidence for hydrogen-poor supernovae showing hydrogen features from the interaction in their late-time spectra \\citep{Milisavljevic2015,Yan2015,Yan2017,Chen2018,Kuncarayakti2018,Mauerhan2018}. The light echo from iPTF16eh \\citep{lunnan2018} implies a significant amount of hydrogen-poor circumstellar medium in a SLSN-I at ${\\sim}10^{17}$ cm. However, this material is too distant for the ejecta to reach by the time of maximum optical light and hence cannot be responsible for boosting the peak luminosity.\n\nIn principle, the gamma-ray observations of SLSNe can constrain shock models. In many cases, this may not work out since most of the emission from shock-heated plasma is either expected to: (1) come out in the X-ray band, as is well studied in other CSM-powered supernovae such as SNe IIn like SN 1998S \\citep{Pooley2002}, SN 2006jd \\citep{Chandra2012}, and SN 2010jl \\citep{Chandra2015}, and SNe Ib\/c \\citep{Chevalier2006}; or (2) be absorbed by the surrounding ejecta and reprocessed into the optical band. Thus, these VHE limits on SLSNe do not constrain the bulk of the shock power. \n\nHigher-energy radiation can be produced if the shocks accelerate a population of non-thermal relativistic particles which interact with ambient ions or the supernova optical emission to generate gamma rays (e.g. via the decay of $\\pi^{0}$ generated via hadronic interactions with matter and radiation; e.g., \\citealt{murase2011}). However, because shocks typically place a fraction $\\epsilon_{\\rm rel} \\lesssim 0.1$ of their total power into relativistic particles (or even less; \\citealt{steinberg2018,fang2019}), the predicted gamma-ray luminosities (matching the same level of optical emission as magnetar models) would be at least 10 times lower than $L_{\\gamma}$ predicted by the magnetar nebula scenario, thus rendering our VHE upper limits unconstraining on non-thermal emission from shocks on SN2015bn and SN2017egm. This is consistent with upper limits from the Type IIn SN 2010j from {\\it Fermi}-LAT, which \\citet{Murase2019} used to constrain $\\epsilon_{\\rm rel} \\lesssim 0.05-0.1$. \n\n\\section{Future Prospects}\n\\label{sec:future}\n\nThese results demonstrate that high-energy gamma-ray observations of SLSN-I are on the brink of enabling constraints on the light curves and even spectral energy distribution of magnetar models. Given the rarity of bright, nearby SLSN-I, and the need to take observations in the optimal window (when $L_{\\gamma}$ is near maximum), careful planning will be required to make progress going ahead \\citep{prajs2017,Quimby2011,mccrum2015}. The strategy outlined below will focus only on SLSN-I, as type II SLSN are likely to be powered by a mechanism that requires a different consideration of the temporal and spectral evolution of the gamma-ray emission.\n\nStandard arrays of IACTs provide an improved instantaneous sensitivity to gamma-ray emission over {\\it Fermi}-LAT due to $10^4$ to $10^5$ larger effective area, counterbalanced in part by the pointed nature of their observations. To propose a strategy, we firstly re-visited the characteristics of a large sample of observed SLSNe and performed a systematic study.\n\n\\citet{nicholl2017d} fit a sample of 38 SLSNe light curves using MOSFiT to obtain a distribution of magnetar model parameters. This sample is a selection of SLSNe with well observed events classified as Type-I with published data near the optical peak, forming a representative sample of good SLSNe-I for a population study. For each event in this sample, the following was calculated: the escaping gamma-ray luminosity $L_{\\gamma}$ following the procedure outlined in Appendix \\ref{sec:appendix} and the flux $F_{\\gamma} = L_{\\gamma}\/4\\pi D_{\\rm L}^{2}$ based on the source luminosity distance $D_{L}$. In performing this analysis, rather than fitting the value of $\\kappa_{\\gamma}$ individually to each optical light curve (as done in \\citealt{nicholl2017d}), the value $\\kappa_{\\gamma} = \\SI{0.01}{cm^{2} g^{-1}}$ is fixed in all events, based on the best-fit to SN2015bn (given its particularly high-quality late-time data, which provides the most leverage on $\\kappa_{\\gamma}$). \n\nThe results for $F_{\\gamma}(t)$ are shown in the top panel of Figure~\\ref{fig:distributions}. In the magnetar model, the predicted gamma-ray flux could emerge anywhere across the HE to VHE bands and hence it represents an upper limit on flux in the bands accessible to {\\it Fermi}-LAT and IACTs. The bottom two panels of Figure~\\ref{fig:distributions} show the distribution of the peak escaping flux $F_{\\rm \\gamma, max}$ and time of the peak flux relative to the explosion. For most SLSNe-I presented here, $F_{\\rm \\gamma, max}$ is well below the sensitivity of VERITAS and even the future Cherenkov Telescope Array (CTA) \\citep{thectaconsortium2019}. Also note that the characteristic timescale to achieve the peak gamma-ray flux is $\\approx 2-3$ months from the explosion. This timescale occurs approximately at the same time as when the optical depth of the ejecta to VHE emission falls below unity, when the VHE photons can escape (Figure~\\ref{fig:tau}). \n\n\\begin{figure}[ht!]\n    \\centering\n   \n   \n   \n    \\includegraphics[width=\\columnwidth]{Combined_LCs_flux_dist.pdf}\n    \\caption{Top: Escaping gamma-ray luminosity $L_{\\gamma}(t)$ for the sample of SLSNe fit by \\citet{nicholl2017d}. Five well studied SN are highlighted in blue, including SN2015bn. Overplotted are the VERITAS and CTA sensitivity curves for various exposures.  Middle: Distribution of peak escaping gamma-ray flux $F_{\\rm \\gamma,max} = {\\rm max}[L_{\\gamma}]\/4\\pi D^{2}$, for the light curves from the top panel where $D$ is the distance to each source. Again, VERITAS and CTA sensitivities for different exposures are shown as vertical dashed lines. Bottom: Distributions of times since explosion to reach the maximum gamma-ray flux $F_{\\gamma,max}$ from $F_{\\gamma}$ above.} \n   \n    \\label{fig:distributions}\n\\end{figure}\n\nFigure~\\ref{fig:Optical_to_Escaping_Fluxes} shows $F_{\\rm \\gamma,600 d}$ as a function of the peak optical magnitude of the SLSNe-I from the same sample as in Figure~\\ref{fig:distributions}. The selection of fluxes at \\SI{600}{\\day} approximates the time when the effective opacity to 1 TeV photons reaches 1, based on Figure~\\ref{fig:tau}. The top axis also gives the all-sky rate of SLSNe-I above a given peak optical magnitude, which is estimated using the magnitude distribution of SLSNe-I and assuming they occur at a comoving volumetric rate of $ R(z)=19(1+z)^{3.28}\\SI{}{\\, Gpc^{-3}\\, yr^{-1}}$ following \\citet{nicholl2017b,lunnan2018,decia2018}. This estimation captures the general volumetric rate of events, but is unreliable for exceptionally bright events such as SN2017egm due to the small population for estimating the magnitude normalization. A bright event like SN2017egm may actually happen more often than once a century.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=.95\\textwidth]{Optical_to_Escaping_Fluxes_kap_1TeV.pdf}\n    \\caption{\n    Blue dots show the peak optical apparent magnitudes of a sample of SLSNe-I \\citep{nicholl2017d} as a function of their predicted maximum gamma-ray luminosity at 600 days after explosion ($F_{\\gamma, 600d}$). The top axis shows the approximate rate of events above the given peak optical magnitude, calculated using the method described in the main text. Peak maximum gamma-ray luminosities are calculated from fits of optical data with fixed $\\kappa_{\\gamma} = 0.01$ cm$^{2}$ g$^{-1}$. Integral sensitivities of various instruments are overplotted for different exposures. Solid lines: VERITAS 10 and 50 hour integral sensitivities above 220 GeV. Dotted lines: CTA (in development) 10 and 50 hour integral sensitivities above 125 GeV as estimated from 50 hour Monte Carlo simulations of the southern array \\citep{thectaconsortium2019} and extrapolated to 10 hours. Similar extrapolation is done for {\\it Fermi}-LAT from 10 years to 6 months \\citep{nolan2012} (dashed line). Proposed project AMEGO integral sensitivity above 100 MeV for 6 month observation window is also plotted (dash-dot line) \\citep{kierans2020}. \n    }\n    \\label{fig:Optical_to_Escaping_Fluxes}\n\\end{figure*}\n\nShown for comparison in Figure~\\ref{fig:Optical_to_Escaping_Fluxes} are the integral sensitivities of various gamma-ray instruments for different exposures. For IACT instruments such as VERITAS and the future CTA, sensitivity is defined as the minimum flux necessary to reach $5 \\sigma$ detection of a point-like source, requiring at least 10 excess gamma rays and the number of signal counts at least $5\\%$ of the number of background counts. For VERITAS, the sensitivity was calculated using observed Crab Nebula data to estimate the rates of signal and background photons with cuts optimized for a $\\Gamma = -2.5$ power-law spectrum, and then re-scaled for the appropriate observation time \\citep{Park2015}. For CTA, Monte Carlo simulations were used to derive angular resolution, background rates and energy dispersion features -- the instrument response functions (IRF) -- based on the Prod3b-v2 telescope configuration for the Southern site and its atmosphere \\citep{cherenkovtelescopearrayobservatory2016}. These IRFs are publicly available and were analyzed using the open-source CTOOLS\\footnote{\\url{http:\/\/cta.irap.omp.eu\/ctools\/}} \\citep{Knodlseder2016}. A power law spectral model was used to estimate the integral sensitivity above \\SIlist{0.125;1}{\\TeV} each for observations of \\SIlist{10;50}{\\hour} (see \\citealt{Fioretti2016} for further discussion on CTA integral sensitivity).\n\nBased on this systematic study, we propose the following observation strategy: 1) Receive automated public alert and Type I classification of SLSN from a survey instrument such as the Zwicky Transient Facility (ZTF). Classification is generally determined by identification of early spectral components such as OII absorption features. 2) During the multi-day rise and fall of bolometric optical light curve, fit the magnetar model ($L_{\\rm opt}$, yielding parameters for $L_{\\rm mag}$ and $L_{\\gamma}$) 3) Compare $L_{\\gamma}$ to the telescope sensitivity at the appropriate day when the effective $\\gamma$-$\\gamma$ opacity falls below ${\\sim1}$ for the telescope's sensitive energy range (see Figure~\\ref{fig:tau}). In the case of IACTs sensitive to energies above $\\SI{100}{\\GeV}$, the gamma rays will escape the magnetar a few hundred days after explosion, requiring a bright SLSN-I that will power gamma rays for as much as two years. \n\nEstimating ${\\sim}35\\%$ of all-sky visibility at VERITAS due to Sun, Moon, and seasonal weather cut, and above 60\\textdegree~ elevation, VERITAS is capable of detecting up to ${\\sim}0.4$ and ${\\sim}4$ SLSNe-I per year for \\SI{10}{h} and \\SI{50}{h} exposures, respectively. The next-generation CTA observatory will be able to detect as many as ${\\sim}8$ and ${\\sim}80$ events for \\SI{10}{hr} and \\SI{50}{h}, respectively, assuming a larger sky visibility fraction of ${\\sim}80\\%$ when both North and South arrays are included. On the other hand, SLSNe at greater distances also imply a stronger role of $\\gamma-\\gamma$ interactions on the EBL in suppressing the $\\gtrsim$ TeV emission, decreasing the observed integral flux by as much as 60 times at redshifts near 0.5 in the VERITAS energy range. \n\nFigure~\\ref{fig:TeV_gamma_flux_sens} shows the distribution of fluxes at \\SI{200}{\\day} and \\SI{600}{\\day} which are approximate average dates when the opacity to \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons falls below 1, respectively, and they are able to escape the ejecta. Accounting for this time delay for the opacity to drop, the expected rate of bright events drops by another 3 to 15 times. While past observations have not been followed up until this publication, the distribution of predicted gamma-ray fluxes hints that, particularly for \\SI{100}{\\GeV} photons, future SLSN-I will be observable with current and planned observatories. \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Escaping_Fluxes_kappa_gamma_dist.pdf}\n    \\caption{ Distribution of gamma-ray luminosities $L_{\\gamma}$ at $t=\\SI{200}{\\day}$ (top) and $t=\\SI{600}{\\day}$ (bottom), when the optical depth for \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons drops below 1, calculated for a sample of 38 SLSNe \\citep{nicholl2017d}}\n    \\label{fig:TeV_gamma_flux_sens}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nSLSN-I are potential gamma-ray emitters, and this paper provides the first upper limits at different times after the optical outburst for two good candidates. The reported upper limits approach the magnetar spin-down luminosity limit of SN2015bn and SN2017egm. While the expected gamma-ray luminosity in either the magnetar central-engine scenario or the shock-acceleration scenario is not constrained by these limits, a relativistic jet powered by fall-back accretion onto a black hole is disfavored in both cases. We explore prospects for obtaining improved VHE gamma-ray constraints in the future by current and planned IACTs. We estimate the Type-I SLSNe rate for VERITAS and CTA, considering observation constraints and the time delay due to the optical depth. For sufficiently nearby and bright SLSN-I, 0.4 and 4 events per year can be observed by VERITAS from 10-hr and 50-hr observation, respectively, and similarly rates of 8 and 80 events per year can be expected by CTA. \n\n\n\n\n\\acknowledgments \n{This research is supported by grants from the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, and by the Helmholtz Association in Germany. MN is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.~948381) and by a Fellowship from the Alan Turing Institute. IV acknowledges support by the ETAg grant PRG1006 and by EU through the ERDF CoE grant TK133. VVD's work is supported by NSF grant 1911061 awarded to the University of Chicago (PI: Vikram Dwarkadas). We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. \nThis research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see \\url{http:\/\/www.cta-observatory.org\/science\/cta-performance\/} (version prod3b-v2) for more details.\n}\n\n\\software{fermipy (v0.19), \\citep{wood2017},\nastropy \\citep{Robitaille2013,Price-Whelan2018},\nCTOOLs, \\citep{Knodlseder2016a}, superbol \\citep{nicholl2018} , EventDisplay \\citep{Maier2017}}, VEGAS \\citep{cogan2008}\n\n\\facilities{VERITAS, {\\it Fermi}-LAT}\n\n\\clearpage\n\n\n\\section{Introduction}\\label{sec:intro}\n\nThe recent growth of sensitive optical time-domain surveys has revealed and expanded exciting new classes of stellar explosions. These include superluminous supernovae, which can be up to 10--100 times more luminous than ordinary massive star explosions (e.g.~\\citealt{Quimby2011,howell2013,inserra2013,nicholl2013,decia2018,lunnan2018,quimby2018}; see \\citealt{Gal-Yam2019} for a recent review). Conventionally, the optical emission from most core-collapse supernovae is powered by the radioactive decay of $^{56}$Ni (Type Ib\/c) and by thermal energy generated via shock heating of the stellar envelope (Type IIL, IIp). However, the peak luminosities of SLSNe greatly exceed the luminosity expected from those conventional mechanisms, and the origin of the energy is still debated. \n\nA popular model for powering the time-dependent emission of SLSNe, particularly the hydrogen-poor Type I class (SLSN-I), involves energy input from a young central engine, such as a black hole or neutron star, formed in the explosion. For example, the accretion onto the compact object from bound debris of the explosion could power an outflow which heats the supernova ejecta from within (\\citealt{woosley2012,Quataert2012,margalit2016,moriya2018}). Alternatively, the central engine could be a strongly magnetized neutron star with a millisecond rotation period, whose rotationally powered wind provides a source of energetic particles that heat the supernova ejecta \\citep{kasen2010,woosley2010,dessart2012a,metzger2015a,sukhbold2016}. The magnetar\\footnote{To remain consistent with the SLSNe literature, the term magnetar is used throughout this paper. Magnetars generally have large dipole magnetic fields $B\\simeq$ \\SIrange{e13}{e15}{G} with a rotation period of a few seconds. In the case of the SLSN magnetar model, the radiation is extracted from the rotational energy of the young millisecond pulsars, but with large magnetic fields characteristic of magnetars.} model provides a good fit to the optical light curves of most SLSNe-I \\citep{inserra2013,nicholl2017d}. Furthermore, analyses of the nebular spectra of hydrogen-poor SLSNe \\citep{jerkstrand2017,nicholl2019} and Type Ib SNe \\citep{milisavljevic2018} support the presence of a persistent central energy source, consistent with an energetic neutron star.\n\nThe details of how the magnetar would couple its energy to the ejecta are uncertain. Several models consider that the rotationally powered wind from a young pulsar inflates a nebula of relativistic electron\/positron pairs and energetic radiation behind the expanding ejecta \\citep{kotera2013,metzger2014b,murase2015}. At the wind-termination shock, the pairs are heated and radiate X-rays and gamma rays with high efficiency via synchrotron and inverse-Compton processes. Photons that evade absorption via $\\gamma\\textrm{-}\\gamma$ pair creation in the nebula can be ``absorbed\" by the ejecta further out, thermalizing their energy and directly powering the supernova's optical emission (e.g.~\\citealt{metzger2014b, vurm2021}). \n\nThermalization of the nebular radiation will be most efficient at early times, when the column through the ejecta shell and ``compactness'' of the nebula are at their highest. At these times one would expect the optical light curve to faithfully track the energy input of the central engine. However, as the ejecta expand, the radiation field dilutes and the shell becomes increasingly transparent to high-energy  and very-high-energy photons. The increasing transparency, and correspondingly decreasing thermalization efficiency, eventually causes the supernova's optical luminosity to drop below the rate of energy injection from the central engine \\citep{chen2015,wang2015}, with the remaining radiation escaping directly from the nebula as gamma rays or X-rays (the putative ``missing\" luminosity). \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2015bn_vs_time.pdf}\n    \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2017egm_vs_time_weightlate_v2.pdf}\n    \\caption{Optical depth at different photon energies as a function of time, calculated for ejecta properties (mass $M_{\\odot}$, mean velocity, etc.) derived from observations of SN2015bn \\citep{nicholl2018a} and SN2017egm \\citep{nicholl2017c} shown in Table \\ref{tab:event_physical_params}. Top: SN2015bn. Bottom: SN2017egm. The horizontal dotted line represents $\\tau_{\\rm eff} = 1$. The cross-sections for photon-photon and photon-matter pair production opacities are taken from \\citet{zdziarski1989}. The solid lines correspond to target blackbody radiation temperature $T_{\\rm eff} = (L_{\\rm opt}\/4\\pi R^2)^{1\/4}$, where $L_{\\rm opt}$ and $R$ are the optical luminosity and ejecta radius, respectively. The dashed lines are computed with a temperature floor of $T = 4000$~K, to mimic the approximate spectrum in the nebular phase. Below ${\\sim}10$~GeV the opacity is dominated by photon-matter pair production at all times. Above $100$~GeV, pair production on the thermal target radiation field dominates up to a few years.}\n    \\label{fig:tau}\n\\end{figure}\n\nAs the ejecta expand and the spin-down luminosity weakens, the conditions for various processes responsible for photon energy loss change and impact the effective optical depth. Within a few months, the effective optical depth to high-energy (HE; 100 MeV to 100 GeV) photons emitted from the central engine nears unity and, at several hundred days it reaches unity for very-high-energy (VHE; 100 GeV \u2013 100 TeV) photons. \n\nFigure~\\ref{fig:tau} shows examples of the effective optical depth through the ejecta for photons of various energies as a function of time. They have been calculated using time-dependent properties for the supernova ejecta and radiation field motivated by the observations of SN2015bn and SN2017egm, both particularly well-studied SLSNe-I explored in \\cite{vurm2021}.\n\nThe dominant processes involved in the calculation of the gamma-ray optical depth include photon-matter and photon-photon interactions, particularly pair production on the nuclei and soft radiation fields in the ejecta. An accurate treatment considering the radiation transport is discussed in depth in \\citet{vurm2021}. The standard version of the magnetar model \\citep{kasen2010, woosley2010, nicholl2017d} does not consider this time-dependent calculation and relies on constant effective  opacities to optical and high-energy photons. Figure~\\ref{fig:tau} provides a useful guiding timescale for when to consider gamma-ray emission at various energies, calculated with the model in \\citet{vurm2021} using the ejecta properties fit to the optical data in Table \\ref{tab:event_physical_params}.\n\nGiven its comparatively nearby distance at z=0.1136, SN2015bn is an excellent candidate event to test the magnetar hypothesis. The optical light curve shows a steepening from $\\propto t^{-2}$ decay to $\\propto t^{-4}$ around ${\\sim}200$ days \\citep{nicholl2018a}. This behavior is consistent with a leakage of high-energy radiation from a magnetar nebula \\citep{nicholl2018a}. A deep search in the ${\\sim}0.1-10$ keV X-ray band resulted in nondetections \\citep{bhirombhakdi2018}, eliminating the possibility that leakage from the nebula occurs in the softer X-ray bands.\n\n\\citet{margutti2018b} present a similar search for late-time X-ray emission from a larger sample of SLSNe-I, mostly resulting in upper limits; however, see \\citet{levan2013} for an X-ray detection of the SLSN-I SCP 06F6 that could still support the magnetar hypothesis. X-ray nondetections are not surprising, because the ejecta are likely to still be opaque in the $\\lesssim 10$ keV band due to photoelectric absorption in the hydrogen-poor ejecta \\citep{margalit2018a}. Intriguingly, \\citet{Eftekhari2019} detected radio emission from the location of the SLSN PTF10hgi at 7.5 yr after the explosion and argued that the emission could be synchrotron emission from an engine-powered nebula.\n\nSome effort has been underway to search for nebular leakage in the gamma-ray band. \\citet{renault-tinacci2018} obtained upper limits on the $0.6-600$ GeV luminosities from SLSNe by a stacked analysis of 45 SLSNe with {\\it Fermi}-LAT. The majority of their sample were SLSNe-I, the most likely class to be powered by a central engine; however the results were dominated by a single, extremely close Type II event (SLSN-II), CSS140222. Hydrogen-rich SLSNe make up the Type II class (SLSN-II), which are suggested to be powered by the interaction of the circumstellar medium with the supernova ejecta.  Nevertheless, even with CSS140222 included, the upper limits are at best marginally constraining on the inferred missing luminosity. \n\nIn this paper, the search is expanded to gamma-ray emission from SLSNe-I in the HE to VHE bands using the {\\it Fermi} Gamma-Ray Space Telescope and the ground-based VERITAS observatory. In particular, observations of SN2015bn and SN2017egm are presented here. SN2017egm is the closest SLSN-I to date in the Northern Hemisphere at z=0.0310 \\citep{bose2017,nicholl2017c}. Observations of young supernovae with gamma-ray telescopes have been few, with no detections so far. Some tantalizing candidates like iPTF14hls and SN 2004dj have been explored with {\\it Fermi}-LAT but are unconfirmed due to large localization regions overlapping with other gamma-ray candidates \\citep{yuan2018,xi2020}. MAGIC carried out observations of a Type I SN \\citep{ahnen2017a}. HESS observed a sample of core-collapse SNe \\citep{Abdalla2019}, and later obtained upper limits on SN 1987A \\citep{theh.e.s.s.collaboration2015}. Our observations are the first of superluminous supernovae. \n\nThroughout this paper, a flat $\\Lambda$CDM cosmology is used, with $H_0 = \\SI{67.7}{km.s^{-1}.Mpc^{-1}}$, $\\Omega_{M}=0.307$, and $\\Omega_{\\Lambda} = 0.6911$ \\citep{planckcollaboration2016}. The corresponding luminosity distances to SN2015bn and SN2017egm are  \\SI{545.37}{\\mega\\pc} (z=0.1136) \\citep{nicholl2016} and \\SI{139.29}{\\mega\\pc} (z=0.0310) \\citep{bose2017}.\n\n\n\\section{Observations and Methods} \\label{sec:obs}\n\\label{sec:observations}\nThe superluminous supernovae SN2015bn and SN2017egm were observed with {\\it Fermi}-LAT and VERITAS during 2015--2016 and 2017--2020, respectively. SN2015bn is a SLSN-I explosion from 23 Dec 2014 (MJD 57014) and it peaked optically on 19 Mar 2015 (MJD 57100) \\cite{nicholl2016a}. SN2017egm is a SLSN-I explosion from 23 May 2017 (MJD 57896) and it peaked optically on 18 Jun 2017 (MJD 57922) \\cite{bose2017}. Some properties of the SLSNe are given in Table \\ref{tab:event_physical_params}. Details regarding the optical, {\\it Fermi}-LAT and VERITAS observations and the data-analysis methods are below.  \n\n\n\\begin{table}[ht]\n    \\begin{center}\n        \\caption{Properties of the SLSNe considered in this paper. }\n        \\begin{tabular}{cl|rr}\n        \\multicolumn{1}{l}{Parameter} & {(}Unit{)}           & {SN2015bn} & {SN2017egm} \\\\ \\hline\n        R.A.                            & $^{\\circ}$           & 173.4232 & 154.7734  \\\\\n        decl                           & $^{\\circ}$           & 0.725    & 46.454    \\\\\n        z                             & -                    & 0.1136   & 0.0310    \\\\\n        $t_{0}^{(a)}$                 & MJD                  & 57014    & 57896     \\\\\n        $t_{pk}^{(b)}$                & MJD                  & 57100    & 57922     \\\\ \\hline\n        $P_0^{(c)}$                   & ms                   & $2.50^{+0.29}_{-0.17}$     & $5.83^{+0.73}_{-0.70}$      \\\\\n        $B^{(d)}$                     & $10^{14}$ G          & $0.26^{+0.07}_{-0.05}$     & $0.94^{+0.13}_{-0.16}$      \\\\\n        $M_{ej}^{(e)}$                & $M_{\\odot}$          & $10.8^{+0.83}_{-1.34}$     & $2.99^{+0.30}_{-0.23}$      \\\\\n        $\\kappa^{(f)}$                & cm$^{2}$ g$^{-1}$     & $0.18^{+0.01}_{-0.02}$     & $0.12^{+0.04}_{-0.06}$      \\\\\n       \n        $v_{ej}^{(h)}$                & $10^{8}$ cm s$^{-1}$ & $5.68^{+0.16}_{-0.14}$     & $10.3^{+0.35}_{-0.27}$      \\\\\n        $\\kappa_{\\gamma}^{(i)}$       & cm$^{2}$ g$^{-1}$     & $0.008^{+0.01}_{-0.01}$    & $0.080^{+0.15}_{-0.06}$     \\\\\n        $M_{\\rm NS}^{(j)}$            & $M_{\\odot}$          & $1.84^{+0.28}_{-0.23}$     & $1.57^{+0.25}_{-0.29}$      \n        \\end{tabular}\n        \\label{tab:event_physical_params}        \n    \\end{center}\n    Notes. The quantities $P_{0}$, $B$, $M_{\\rm ej}$, $\\kappa$, $E_{\\rm SN}$, $v_{\\rm ej}$, $\\kappa_{\\gamma}$ and $M_{\\rm NS}$ were obtained from a best-fit to the UVOIR supernova light curves, with errors found in \\cite{nicholl2017c, nicholl2017d}. \n    $^{(a)}$Epoch of explosion; $^{(b)}$Epoch of optical flux peak; $^{(c)}$Initial spin-period; $^{(d)}$Magnetic field strength of magnetar; $^{(e)}$Total mass, $^{(f)}$Effective opacity; $^{(g)}$Kinetic energy; $^{(h)}$Mean velocity of supernova ejecta; $^{(i)}$Gamma-ray effective opacity and $^{(j)}$Neutron star mass.\n\\end{table}\n\n\\subsection{{\\it Fermi}-LAT} \\label{subsec:Fermi}\nThe Large Area Telescope (LAT) on board the {\\it Fermi} satellite has operated since 2008 \\citep{Atwood2009}. It is sensitive to photons between \\SI{{\\sim}20}{\\MeV} and \\SI{{\\sim}300}{\\GeV} and has an ${\\sim}60\\degree$ field of view, enabling it to survey the entire sky in about 3 hours. \n\nThe data were analyzed using the publicly available {\\it Fermi}-LAT data with the \\texttt{Fermitools} suite of tools provided by the {\\it Fermi} Science Support Center (FSSC). Using the \\texttt{Fermipy} analysis package \\citep{wood2017}\n\\footnote{\\url{https:\/\/fermipy.readthedocs.io\/en\/latest\/} ; v0.19.0}, the data were prepared for a binned likelihood analysis in which a spatial spectral model is fit over the energy bins. The data were selected using the SOURCE class of events, which are optimized for point-source analysis, within a region of $15\\degree$ radius from the analysis target position. Due to the effect of the Earth, a $90\\degree$ zenith angle cut was applied to remove any external background events. The standard background models were applied to the test model, incorporating an isotropic background and a galactic diffuse emission model without any modifications. The standard 4FGL catalog was then queried for sources within the field of view and their default model parameters \\cite{abdollahi2020}. \n \nAdditional putative point sources were added to each field of view as needed to support convergence of the fit. These sources were added for all analysis time scales. This process continued until the distribution of test statistics for the field of view was Gaussian with standard deviation near 1 and mean centered at 0, and the residual maps were near uniformly 0 without strong features. These conditions indicate the appropriate coverage of spectral sources within the analysis was reached and no putative sources are missing. The fitting process is performed in discrete energy bins while optimizing the spectral shape, but the distribution of test statistics is evaluated with the stacked data spanning the full energy range. With the improvements to {\\it Fermi}-LAT low-energy sensitivity in PASS8 reconstruction, the low-energy bin covering $100-612$ MeV was also added. \n \nIn the case of both SN2015bn and SN2017egm, the data were fitted with a power-law spectral model, $N(E) = N_{0} E^{\\Gamma}$, with a free prefactor and a fixed photon index $\\Gamma$ of -2.0.  From the fit, the reported flux upper limit was found using a 95\\% confidence level with the bounded Rolke method \\citep{rolke2005}. In all cases reported here, the upper limit reported is the integral energy flux, integrated over the energy ranges described for each case, which has units of \\si{MeV cm^{-2}.s^{-1}}. This flux is converted to luminosity with the adopted distance for each event.\n \n\nSN2015bn was observed from 2014 December 23 to 2018 Mar 23. This observation period begins after the explosion, and is binned in a few windows to account for the absorption of low-energy gamma rays by the ejecta at early times (Figure~\\ref{fig:tau}). The first $\\sim90$ days is observed to make sure there are no early emission during the expected absorption period. The data were thereafter binned in time intervals of six months to maximize observation depth and sensitivity to time-dependent variation. SN2017egm was observed 2017 May 23 to 2020 Aug 21. Again, this period covers the 3.5 yr from the discovery date, starting with $\\sim90$ days after the explosion and split into six 6 month bins thereafter. The 3.5 yr observation period is selected to cover approximately 1000 days after the explosion. After this period, it is expected that the predicted luminosity will have decreased below the {\\it Fermi}-LAT detectable limit. \n\nSN2015bn is within 5\\textdegree\\ of the Sun each year in August, so a one-month time cut is applied to each relevant time bin (to cover an $\\sim15$\\textdegree\\ radius field of view). SN2017egm is not near the the path of the Sun, so this cut was not applied.\n\n\\subsection{VERITAS} \\label{subsec:veritas}\nThe Very Energetic Radiation Imaging Telescope Array System (VERITAS) is an imaging atmospheric cherenkov telescope (IACT) array at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona, USA \\citep{weekes2002, Holder2006}. It consists of four 12 m telescopes separated by approximately $100$ m, and the observatory is sensitive to photons within the energy range $\\backsim$\\SI{100}{GeV} to $\\backsim$\\SI{30}{TeV}. The instrument has an angular resolution (68\\% containment) of $\\backsim$0.1\\degr\\ at \\SI{1}{TeV}, an energy resolution of $\\backsim$15\\% at \\SI{1}{TeV}, and 3.5\\degr\\ field of view. \n\n\nVERITAS serendipitously observed SN2015bn for a total of 1.01 hr between 2015 May 7 and 2015 May 22, approximately 135 days from explosion (49 days from the date of peak magnitude), as a part of an unrelated campaign. Another 1.7 hr were taken between 2016 May 25 and 30. Data were taken in good weather and dark sky conditions. Since SN2015bn was not the target source, its sky position averages 1.4\\textdegree~from the center of the camera.  \n\nVERITAS directly observed SN2017egm for 8.7 hr between 2019 Mar 24 and 2019 Apr 5, under dark sky conditions, as part of a Directors Discretionary Time (DDT) campaign, approximately 670 days from explosion. This target was triggered based on the predicted gamma-ray luminosity (see section \\ref{sec:MagnetarSpinDown} and appendix \\ref{sec:appendix} for a description) derived from the optical observation. Although it was almost two years after the explosion, the nearby distance yielded a gamma-ray luminosity prediction still within reach of VERITAS, making this an enticing target to follow up.\n\nThe SN2017egm data in this paper were taken using ``wobble\" pointing mode, where the source is offset from the center of the camera by $0.5\\degree$. This mode creates space for a radially symmetric off region to be used for background estimation in the same field of view, saving time from targeted background observations that contain the same data observing conditions. The data were processed with standard VERITAS calibration and reconstruction pipelines, and then cross-checked with a separate analysis chain \\citep{cogan2008,Maier2017}. \n \nUsing an Image Template Method (ITM) to improve event angular and energy reconstruction \\citep{christiansen2017}, analysis cuts are determined with a set of a priori data-selection cuts optimized on sources with a moderate power-law index (from -2.5 to -3).\n\nUnfortunately, the large offset on SN2015bn due to the serendipitous observation precludes us from using ITM in the analysis, so in that case SN2015bn is analyzed without templates by calculating image moments directly from candidate images triggered by the camera \\citep{cogan2008,Maier2017}.   In both cases, the signal and background counts are determined using the reflected region method. \n\n\nThe upper limit is calculated for both SN2015bn and SN2017egm. The bounded Rolke method for upper limit calculation is used, assuming a power law spectrum with index of -2.0 and 95\\% confidence level \\citep{rolke2005}. Since the calculation of the upper limit depends on the underlying spectral model, a range of power-law spectral indices from -2 to -3 was computed to estimate impact of the model dependence. In all cases reported here, the upper limit reported is the integral photon flux, integrated over the energy ranges described for each case, which has units of \\si{cm^{-2}.s^{-1}}. This flux is converted to integral energy flux using the same spectral model so that the luminosity can be computed with the adopted distance.\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics{SN2015bn_ebl_lc.pdf}\n    \\caption{Light curves of SN2015bn spanning 30-1500 days after explosion. Curves shown include (1) the (thermal) supernova luminosity, $L_{\\rm opt}$, fit to UVOIR bolometric luminosity data (in red; \\citealt{nicholl2018a}) to obtain the magnetar parameters; (2) magnetar spin-down luminosity, $L_{\\rm mag}$ (green dotted lined); and (3) predicted gamma-ray luminosity that escape the ejecta, $L_{\\gamma}$ (pink dotted-dashed line; Equations \\ref{eq:Lmag}, \\ref{eq:trapped} and \\ref{eq:leaking}). Black bars show {\\it Fermi}-LAT upper limits reported for six 180 day bins starting ${\\sim}90$ days after explosion. The olive open box shows the VERITAS integral energy flux} upper limit taken ${\\sim}135$ days after the explosion, with EBL absorption correction applied. Upper limits on the 0.2-10 keV X-ray luminosity from {\\it Chandra} are from \\citet{bhirombhakdi2018} in green. Gray shaded regions labeled ``$\\tau_{\\gamma} <1$\" show the approximate time after which gamma rays of the indicated energy should escape ejecta, based on Figure \\ref{fig:tau}. A purple dotted-dashed line shows the engine luminosity, $L_{\\mathrm{BH}}$ (Eq.~\\ref{eq:L_BH}), in an alternative model in which the supernova optical luminosity is powered by fallback accretion onto a black hole. All upper limits denote the 95\\% confidence level.\n    \\label{fig:SN2015bn_lc}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics{SN2017egm_lc_ebl_corrected.pdf}\n    \\caption{The SN2017egm light curve spanning 10-1300 days after explosion, following the same format as Figure~\\ref{fig:SN2015bn_lc}. UVOIR data are shown in red \\citep{bose2017, nicholl2017c}. Integral energy flux upper limits from {\\it Fermi}-LAT are reported for six 180 day bins starting ${\\sim}90$ days after the explosion. Integral energy flux upper limits are shown for VERITAS data taken ${\\sim}670$ days after explosion, with EBL absorption correction applied. The maximum luminosity of the black hole accretion model $L_{\\mathrm{BH}}$ (Eq. \\ref{eq:L_BH}) is shown in purple. All upper limits denote the 95\\% confidence level.}\n    \\label{fig:SN2017egm_lc}\n\\end{figure*}\n\n\\begin{table*}[ht]\n    \\begin{center}\n        \\caption{Results from VERITAS observations for both epochs of SN2015bn, and SN2017egm. }\n        \\label{tab:veritas_total_results}\n        \\begin{tabular}{cl|rrr}\n        \\multicolumn{1}{l}{Parameter} & {(}Unit{)} & SN2015bn$_{1}$ & SN2015bn$_{2}$ & SN2017egm \\\\ \n        \\hline\n        \\hline\n        Start (MJD) & [day] & 57149 & 57533 & 58566 \\\\\n        End (MJD) & [day] & 57164 & 57538 & 58578 \\\\\n        Live time & [hr] & 1.0 & 1.8 & 8.7 \\\\\n        On & [event] & 4 & 10 & 49\\\\\n        Off & [event] & 179 & 188 & 596 \\\\\n        $\\alpha^{(a)}$ & - & 0.0286 & 0.0299 & 0.0634 \\\\\n        Excess & [event] & -1.1 & 4.4 & 11.2 \\\\\n        Significance & [$\\sigma$] & -0.5 & 1.7 & 1.6 \\\\\n        Flux UL & [$\\SI{e-13}{cm^{-2}~s^{-1}}$] & 28.5 & 27.8 & 10.2 \\\\\n        $E_{\\rm  threshold}$ & [GeV] & $>320$ & $>420$ & $>350$ \\\\ \n        \\end{tabular}\n    \\end{center}\n    Notes. Shown are the quality selected livetime, number of gamma-ray-like events in the on and off-source regions, the normalization, the observed excess of the gamma-rays and the statistical significance. The integral flux upper limit is shown for the given energy threshold, without EBL absorption correction, integrated up to \\SI{30}{\\TeV}.\n    $^{(a)}$ Ratio of relative exposure for on and off regions.\n\\end{table*}\n\n\\section{Results}\n\\label{sec:results}\n\nNo statistically significant detections were made of either SN2015bn or SN2017egm across the energy range 100 MeV to 30 TeV. Integral energy upper limits are reported for the energy ranges given for each instrument. Figure~\\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} show the {\\it Fermi}-LAT and VERITAS upper limits in comparison to the supernova optical light curves and the theoretically predicted escaping luminosity from the magnetar model. \n\n\\subsection{Optical}\\label{sec:optical_result}\nThe SN2015bn integrated ultraviolet-optical-infrared (UVOIR) light-curve data are reproduced here from previous analyses \\citep{nicholl2016, nicholl2016a, nicholl2018a}. To produce these bolometric light curves, the multiband optical data were interpolated and integrated at each epoch using the code \\texttt{superbol} \\citep{nicholl2018}. \n\nSimilarly, the SN2017egm UVOIR data are also reproduced here with \\texttt{superbol} \\citep{bose2017, nicholl2017c}.\n\n\\subsection{{\\it Fermi}-LAT}\nBoth SN2015bn and SN2017egm are not statistically significant sources in the first $\\sim90$ days or the subsequent 6 month bin starting 90 days after the explosion. These sources also remain undetected in any of the following 6 month bins, and in the multiyear data sets. \n\nThe evaluation of the integral energy flux upper limit for the {\\it Fermi}-LAT observations within each time bin was performed assuming a power-law spectral model with an index of -2. The model dependence of this calculation naturally impacts the interpretations in section \\ref{sec:discussion}, so the the fit was performed with indices 2, 2.5 and 3 to find the impact of the model on the final upper limit. An uncertainty of about $10\\%$ was found based on varying the index.\n\nSN2015bn is found to have test statistic (TS) of 0.06,\nwith 12 predicted events above the isotropic diffuse background $\\simeq \\num{4.8e4} $ events over the entire period. The flux upper limit is $\\SI{1.6e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.5e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{1.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS \\simeq 0$, consistent with a nondetection. All of the following 6 month bins reported nondetections with $TS<2$. \n\nSN2017egm is found to have $TS=4.4$,\nwith 43 predicted events above the isotropic diffuse background $\\simeq \\num{5.9e4} $ events. The flux upper limit is $\\SI{1.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{4.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS=\\num{10.1}$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<1$.\n\n\\subsection{VERITAS}\nTable \\ref{tab:veritas_total_results} reports the results from VERITAS observations of SN2015bn and SN2017egm. Each observation is consistent with a nondetection. The significance of each excess of observed events above background is below 2 standard deviations ($\\sigma$). The flux upper limits are also given, calculated by integrating above the threshold energy of the instrument.\n\nThe statistical significance of an excess is estimated using Equation 17 of Li \\& Ma \\citep{li1983}. SN2015bn has significance value of $-0.5 \\sigma$ in the first epoch observation. The integral flux upper limit from \\SIrange{0.32}{30}{\\TeV} for SN2015bn is \\SI{2.85e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.27e44}{\\erg.s^{-1}} at a redshift of 0.1136.  Due to the serendipitous nature of the observation, SN2015bn is significantly off axis, which lowers the instrument sensitivity at the energy threshold of \\SI{320}{\\GeV}.  Additionally, a 10\\% systematic uncertainty is added to the flux normalization and reported energy threshold due to instrument degradation during the period of 2012-2015 \\cite{nievasrosillo2021}. This uncertainty is derived empirically from the observation of the Crab Nebula over the same period. During the second observation in 2016, SN2015bn was found to have a significance of 1.7. The integral flux upper limit from \\SIrange{0.42}{30}{\\TeV} for SN2015bn is \\SI{2.78e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.60e44}{\\erg.s^{-1}}. \n\nFor SN2017egm, the Li \\& Ma significance value is $0.2 \\sigma$ and an integral upper limit from \\SIrange{0.35}{30}{\\TeV} is \\SI{1.0238e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{3.54e42}{\\erg. s^{-1}} above the energy threshold of \\SI{350}{\\GeV} at redshift z=0.0310. The systematic correction due to instrument degradation during the period of 2012-2019 is applied automatically with the use of the throughput-calibrated analysis templates \\citep{nievasrosillo2021}. In the cases of both SN2015bn and SN2017egm, the impact of varying the power-law model index parameter from -2 to -5 is about 10\\%, which is a negligible in the context of their respective light curves.\n\nVHE photons are absorbed by the extragalactic background light (EBL) throughout the universe, so the flux must be corrected to account for the missing photons. This absorption is energy and redshift dependent. Deabsorption is applied to the flux using the model of \\citet{Dominguez2011}.  \nThe EBL deabsorption factor was convolved with the upper limit calculation, assuming the same spectral shape (a power law with the photon index of -2.0). \nThe deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.32}{30}{\\TeV}, is \\SI{3.36e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.49e44}{\\erg.s^{-1}}. For the second observation, the deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.42}{30}{\\TeV}, is \\SI{3.30e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.91e44}{\\erg.s^{-1}}. For SN2017egm, with a slightly smaller energy range \\SIrange{0.350}{30}{\\TeV}, the deabsorbed integral photon flux is \\SI{1.07e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{3.70e42}{\\erg.s^{-1}}. These EBL-corrected values are plotted in Figure \\ref{fig:SN2015bn_lc} and Figure \\ref{fig:SN2017egm_lc}.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe source of the extra luminosity powering SLSNe-I may be found in the signature of its late-time gamma-ray emission. This section explores the HE to VHE emission hundreds of days after the explosion. The following models with a gamma-ray emission component for the powering mechanism are discussed: (1) magnetar central engine (see section \\ref{sec:MagnetarSpinDown}), (2) black hole central engine (see section \\ref{sec:BlackHole}), and (3) circumstellar interaction (see section \\ref{sec:Circumstellar}).\n\n\\subsection{Magnetar Central Engine} \\label{sec:MagnetarSpinDown}\n\nThe most promising mechanism for powering SLSNe-I is the rotational energy input from a central magnetar. In this scenario, a young pulsar or magnetar inflates a nebula of relativistic particles, which radiate high-energy gamma rays and X-rays. This section initially explores a simple implementation of the magnetar model (see Appendix \\ref{sec:appendix} for full description), followed by a more complete model described in detail in \\citet{vurm2021} for both SN2015bn and SN2017egm. The application of this so-called self-consistent model is necessary to directly predict the energy-dependent luminosities within the energy ranges of the {\\it Fermi}-LAT and VERITAS observations, a major contribution that is not possible with simpler implementation described in the appendix.\n\nAt early times after the explosion (around and immediately after the maximum in the optical emission) the gamma rays are absorbed and thermalized by the expanding supernova ejecta. At these times, the luminosity and shape of the optical light curve can be used to constrain the parameters of the magnetar. In this model, the radiation of an input energy reservoir (the spin-down luminosity of a rotating magnetar) diffuses through the ejecta following the analytical solution by \\citet{arnett1982} (equation \\ref{eq:trapped}). \n\nThe time evolution of the magnetar's spin-down luminosity can be modeled by assuming a rotating dipole magnetic field whose energy loss is dominated by emission of radiation in the gamma-ray and X-ray bands (see Appendix \\ref{sec:appendix} for details). \n\nThis luminosity depends on the magnetar initial spin period, surface dipole magnetic field strength, and neutron star mass, $L_{\\rm mag}(t, P_0, B, M_{\\rm NS})$ (equation \\ref{eq:Lmag}). The emitted radiation thermalizes as it diffuses through the ejecta. The conditions of the ejecta determine the optical and gamma-ray outputs, dominated by the values of the ejecta mass, ejecta velocity, and optical and gamma-ray opacities to form $L_{\\rm opt}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa,\\kappa_{\\gamma})$ (equation \\ref{eq:leaking}) and $L_{\\gamma}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa, \\kappa_{\\gamma})$ (equation \\ref{eq:escape}). \n\nFor SN2015bn and SN2017egm, the parameters for the magnetar and the supernova ejecta properties were found by fitting their integrated ultraviolet-optical-infrared (UVOIR) light curves, shown with red points in Figures~\\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}. All fits were conducted using nonlinear least squares minimization\\footnote{\\texttt{scipy.optimize.curve\\_fit}}.\nThe best-fit parameters with errors for the magnetar model are given in Table \\ref{tab:event_physical_params}. The redshifts and time of peak optical magnitude are shown in the table as listed in The Open Supernova Catalog \\citep{Guillochon2016}\\footnote{\\url{https:\/\/sne.space}}.\n\nThese parameters are consistent with the results of previous fits \\citep{nicholl2018a,nicholl2017c} that took into account both the optical spectral energy distribution and light curve using the open-source code \\texttt{MOSFiT} \\footnote{\\url{https:\/\/mosfit.readthedocs.io\/en\/latest\/}}. \nThe relative statistical errors on these fit parameters may be optimistic at $\\sim10\\%$, and the systematic errors will still need to be incorporated for a better understanding the magnetar parameter space. The largest contributor to the magnetar power are the period and magnetic field values, which determine the overall magnitude of the luminosity. The ejecta mass and velocity determine the time to optical peak by the diffusion of the emission through the ejecta.\n\nA particularly important shortfall of this model is the constant effective opacity to both optical and gamma-ray photons, rather than a time-dependent treatment of the opacity. TeV gamma rays interact preferentially with optical photons, so at the time of the peak optical emission, $\\gamma\\gamma$ absorption by optical photons will be high, reducing any predicted gamma-ray emission by this model. Equation \\ref{eq:leaking} is a bolometric luminosity, so it does not take into account the energy and time-dependent opacity, instead fitting a constant effective $\\kappa$ and $\\kappa_{\\gamma}$ to generate the time-dependent optical depth. \n\nTherefore, Figure~\\ref{fig:tau} is used as a guide for when to expect $L_{\\gamma}$ to provide an appropriate estimate for the gamma-ray emission. The shaded regions in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} estimate the time periods when photons of the given energies can escape. It is important to reiterate that this model is energy independent, representing the bolometric luminosity not thermalized by the ejecta. This model cannot distinguish the emission between LAT and VERITAS energy bands since it does not consider the physical model of the nebula; the self-consistent model described by \\cite{vurm2021} and discussed below will be an attempt to do so explicitly.\n\nFollowing the methodology in Appendix \\ref{sec:appendix} with the magnetar parameters for each SLSN, $L_{\\rm mag}(t)$, $L_{\\rm opt}(t)$, and $L_{\\gamma}(t)$ were calculated and are shown in comparison to the gamma-ray limits in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}.\n\nFor SN2015bn (Figure \\ref{fig:SN2015bn_lc}), neither the {\\it Fermi}-LAT upper limits nor the VERITAS upper limit constrain the predicted escaping luminosity. \nSimilarly, for SN2017egm (Figure \\ref{fig:SN2017egm_lc}), both the VERITAS and {\\it Fermi}-LAT upper limits are not deep enough to constrain the predicted escaping luminosity. An important caveat to these upper limits is that the escaping luminosity may also be emitted at energies not explored here, such as hard X-rays or gamma-rays greater than \\SI{ 30}{\\TeV}.\n\nThe optimal time to observe with a pointed instrument sensitive at a particular photon energy results from a trade-off between the dropping ($\\propto t^{-2}$) magnetar luminosity and the rising transparency of the ejecta; predicting the optimal time post-peak to observe requires knowledge of the evolution of the optical spectrum. It is possible to accumulate enough optical data within a few weeks after the optical peak to fit the magnetar model for a reliable prediction of the gamma-ray luminosity. In the case of SN2017egm, the gamma-ray luminosity prediction was anchored by the late optical data points about 1 yr after the explosion. This means that had the VERITAS observations been taken at that point (more than a year earlier than the original observation), they would have been deeply constraining to the magnetar model.\n\n\nGoing beyond these relatively model-independent statements to compare to a more specific spectral energy distribution for the escaping magnetar nebula requires a detailed model for the nebula emission and its transport through the expanding supernova ejecta. Such a model offers preliminary support that a significant fraction of $L_{\\gamma}$ may come out in the VHE band \\citep{vurm2021}. In this case, the VHE limits on SN2015bn and SN2017egm do not strongly constrain the parameters of the magnetar model, such as the nebular magnetization.\n\n\nThe model of \\citet{vurm2021} self-consistently follows the evolution of high-energy electron\/positron pairs injected into the nebula by the magnetar wind and their interaction with the broadband radiation and magnetic fields. They found that the thermalization efficiency and the amount of gamma-ray leakage depends strongly on the nebular magnetization, $\\varepsilon_B$, i.e. the fraction of residual magnetic energy in the nebula relative to that injected by the magnetar.\n\nThe model is simulated for dimensionless $\\varepsilon_B$ values set between  $10^{-6}$ and $10^{-2}$; the higher magnetizations lead to greater synchrotron efficiencies, which dominate within a few hundred days, and lead to the optical emission tracking the spin-down luminosity. Lowering the magnetization to $10^{-7}-10^{-6}$ for SLSN-I events like those in this work delays the transition to synchrotron-dominated thermalization, so that the predicted optical emission actually tracks the observed data. \n\nThe theoretical light curves and gamma-ray upper limits are shown in Figure ~\\ref{fig:Indrek}. \\citet{vurm2021} concluded that the predicted low magnetizations constrained by the optical data alone presents new challenges to the theoretical framework regarding the dissipation of the nebular magnetic field. This may invoke magnetic reconnection ahead of the wind-termination shock or near the termination shock through forced reconnection of alternating field stripes described in \\citet{komissarov2013}, \\citet{lyubarsky2003}, \\citet{margalit2018b}. It is also possible that the true luminosity of the central engine decreases faster in time than the simpler $\\propto t^{-2}$ magnetic spin down, such that escaping VHE emission is not necessary to explain the model. These VHE upper limits do not rule out this model, and do not settle the challenges inferred by the low magnetization required to fit the optical data. Further observations are needed to probe the nebular magnetization and synchrotron efficiency, and deep VHE observations will contribute to these constraints.\n\nThe nondetection of X-rays for both events is consistent with the predictions of \\cite{margalit2018a} of a fully ionized ejecta. Even under the most optimistic conditions - an engine that puts 100\\% of its spin-down luminosity into ionizing photons of ideal energies - cannot reduce the opacity enough to allow X-rays to escape under the usual assumptions (e.g. spherically symmetric ejecta shell). \n\n\\begin{figure}[ht]\n    \\includegraphics[width=1.1\\columnwidth]{SN2015bn_lc_ebl_model.pdf}\n    \\includegraphics[width=1.1\\columnwidth]{SN2017egm_lc_ebl_model.pdf}\n    \\caption{Model light curve for nebular magnetization (from  \\cite{vurm2021}) for SN2015bn with $\\varepsilon_B = 10^{-7}$ (top panel) and SN2017egm with $\\varepsilon_B=10^{-6}$ (bottom panel). }\n    \\label{fig:Indrek}\n\\end{figure}\n\n\\subsection{Black Hole Central Engine}  \\label{sec:BlackHole}\n\nInstead of forming a neutron star like a magnetar, a SLSN-I might form a black hole, in which case the optical peak of the light curve could be powered by energy released from the fallback accretion of ejecta from the explosion (e.g.~\\citealt{dexter2013}). Even if a black hole does not form immediately, it could form at late times once the magnetar accretes enough fallback material \\citep{moriya2016a}. The main practical difference as compared to a magnetar in section \\ref{sec:MagnetarSpinDown} is that the black hole central-engine power would be predicted to decay with the fallback accretion rate $\\dot{M}_{\\rm fb} \\propto t^{-5\/3}$ instead of $\\propto t^{-2}$. Thus, in principle, for the same luminosity at the time of the optical maximum $t_{\\rm pk}$, the central-engine output at times $t \\gg t_{\\rm pk}$ could be enhanced by a factor $\\propto (t\/t_{\\rm pk})^{1\/3} \\sim 2$ for $t \\sim 1$ yr and $t_{\\rm pk} \\sim 1$ month, thus tightening our constraints.\n\nIn Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}, a rough estimate of the maximal engine luminosity in the BH accretion scenario is shown, which is calculated as \n\n\\begin{align}\nL_{\\mathrm{BH}}=\\frac{2^{5\/3} L^{\\rm pk}_{\\rm opt}}{\\left(1+\\frac{t}{t_{\\rm pk}}\\right)^{5\/3}} \\label{eq:L_BH},\n\\end{align}\n\nwhere $L^{\\rm pk}_{\\rm opt}$ is the peak optical luminosity, scaled so that $L_{\\rm BH} = L_{\\rm opt}$ around the optical peak.\n\nOn the other hand, while gamma rays are naturally expected from the ultra-relativistic spin-down-powered nebula of a magnetar, it is less clear that this would be the case for a black hole engine. For instance, the majority of the power from a black hole engine could emerge in a mildly relativistic wind from the black hole accretion disk instead of an ultra-relativistic spin-down-powered pulsar wind.\n\nAs seen in both Figure \\ref{fig:SN2015bn_lc} (SN2015bn) and Figure \\ref{fig:SN2017egm_lc} (SN2017egm), the gamma-ray emission in the black hole scenario is not constrained in the {\\it Fermi}-LAT and VERITAS energy bands.\n\n\\subsection{Circumstellar Interaction} \\label{sec:Circumstellar}\n\nAn alternative model for powering the light curve of SLSNe is to invoke the collision of the supernova ejecta with a slower expanding circumstellar shell or disk surrounding the progenitor at the time of the explosion (e.g.~\\citealt{smith2006,chevalier2011,Moriya2013a}). Features of this circumstellar model (CSM), such as the narrow hydrogen emission lines that indicate the interaction of a slow-moving gas, provide compelling evidence for this being a powering mechanism for many but not all of the hydrogen-rich class of SLSNe (SLSNe-II; e.g.~\\citealt{smith2007,nicholl2020}). \n\nShock interaction could in principle also power some hydrogen-poor SLSNe (SLSNe-I), particularly in cases where the circumstellar interaction is more deeply embedded and less directly visible (e.g.~\\citealt{sorokina2016,kozyreva2017}). \nThere is growing evidence for hydrogen-poor supernovae showing hydrogen features from the interaction in their late-time spectra \\citep{Milisavljevic2015,Yan2015,Yan2017,Chen2018,Kuncarayakti2018,Mauerhan2018}. The light echo from iPTF16eh \\citep{lunnan2018} implies a significant amount of hydrogen-poor circumstellar medium in a SLSN-I at ${\\sim}10^{17}$ cm. However, this material is too distant for the ejecta to reach by the time of maximum optical light and hence cannot be responsible for boosting the peak luminosity.\n\nIn principle, the gamma-ray observations of SLSNe can constrain shock models. In many cases, this may not work out since most of the emission from shock-heated plasma is expected to either (1) come out in the X-ray band, as is well studied in other CSM-powered supernovae such as SNe IIn like SN 1998S \\citep{Pooley2002}, SN 2006jd \\citep{Chandra2012}, SN 2010jl \\citep{Chandra2015}, and SNe Ib\/c \\citep{Chevalier2006} or (2) be absorbed by the surrounding ejecta and reprocessed into the optical band. Thus, these VHE limits on SLSNe do not constrain the bulk of the shock power. \n\nHigher-energy radiation can be produced if the shocks accelerate a population of nonthermal relativistic particles that interact with ambient ions or the supernova optical emission to generate gamma rays (e.g. via the decay of $\\pi^{0}$ generated via hadronic interactions with matter and radiation; e.g., \\citealt{murase2011}). However, because shocks typically place a fraction $\\epsilon_{\\rm rel} \\lesssim 0.1$ of their total power into relativistic particles (or even less; \\citealt{steinberg2018,fang2019}), the predicted gamma-ray luminosities (matching the same level of optical emission as magnetar models) would be at least 10 times lower than $L_{\\gamma}$ predicted by the magnetar nebula scenario, thus rendering our VHE upper limits unconstraining on non-thermal emission from shocks on SN2015bn and SN2017egm. This is consistent with upper limits from the Type IIn SN 2010j from {\\it Fermi}-LAT, which \\citet{Murase2019} used to constrain $\\epsilon_{\\rm rel} \\lesssim 0.05-0.1$. \n\n\\section{Future Prospects}\n\\label{sec:future}\n\nThese results demonstrate that high-energy gamma-ray observations of SLSN-I are on the brink of enabling constraints on the light curves and even spectral energy distribution of magnetar models. Given the rarity of bright, nearby SLSN-I, and the need to take observations in the optimal window (when $L_{\\gamma}$ is near maximum), careful planning will be required to make progress going ahead \\citep{Quimby2011,mccrum2015,prajs2017}. The strategy outlined below will focus only on SLSN-I, as type II SLSN are likely to be powered by a mechanism that requires a different consideration of the temporal and spectral evolution of the gamma-ray emission.\n\nStandard arrays of IACTs provide an improved instantaneous sensitivity to gamma-ray emission over {\\it Fermi}-LAT due to $10^4$ to $10^5$ larger effective area, counterbalanced in part by the pointed nature of their observations. To propose a strategy, we firstly revisited the characteristics of a large sample of observed SLSNe and performed a systematic study.\n\n\\citet{nicholl2017d} fit a sample of 38 SLSNe light curves using MOSFiT to obtain a distribution of magnetar model parameters. This sample is a selection of SLSNe with well-observed events classified as Type I with published data near the optical peak, forming a representative sample of good SLSNe-I for a population study. For each event in this sample, the following was calculated: the escaping gamma-ray luminosity $L_{\\gamma}$ following the procedure outlined in Appendix \\ref{sec:appendix} and the flux $F_{\\gamma} = L_{\\gamma}\/4\\pi D_{\\rm L}^{2}$ based on the source luminosity distance $D_{L}$. In performing this analysis, rather than fitting the value of $\\kappa_{\\gamma}$ individually to each optical light curve (as done in \\citealt{nicholl2017d}), the value $\\kappa_{\\gamma} = \\SI{0.01}{cm^{2} g^{-1}}$ is fixed in all events, based on the best-fit to SN2015bn (given its particularly high-quality late-time data, which provides the most leverage on $\\kappa_{\\gamma}$). \n\nThe results for $F_{\\gamma}(t)$ are shown in the top panel of Figure~\\ref{fig:distributions}. In the magnetar model, the predicted gamma-ray flux could emerge anywhere across the HE to VHE bands; hence, it represents an upper limit on flux in the bands accessible to {\\it Fermi}-LAT and IACTs. The bottom two panels of Figure~\\ref{fig:distributions} show the distribution of the peak escaping flux $F_{\\rm \\gamma, max}$ and time of the peak flux relative to the explosion. For most SLSNe-I presented here, $F_{\\rm \\gamma, max}$ is well below the sensitivity of VERITAS and even the future Cherenkov Telescope Array (CTA) \\citep{thectaconsortium2019}. Also note that the characteristic timescale to achieve the peak gamma-ray flux is $\\approx 2-3$ months from the explosion. This timescale occurs approximately at the same time as when the optical depth of the ejecta to VHE emission falls below unity, when the VHE photons can escape (Figure~\\ref{fig:tau}).  \n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Combined_LCs_flux_dist.pdf}\n    \\caption{Top: escaping gamma-ray luminosity $L_{\\gamma}(t)$ for the sample of SLSNe fit by \\citet{nicholl2017d}. Five well-studied SN are highlighted in blue, including SN2015bn. Overplotted are the VERITAS and CTA sensitivity curves for various exposures.  Middle: distribution of peak escaping gamma-ray flux $F_{\\rm \\gamma,max} = {\\rm max}[L_{\\gamma}]\/4\\pi D^{2}$, for the light curves from the top panel where $D$ is the distance to each source. Again, VERITAS and CTA sensitivities for different exposures are shown as vertical dashed lines. Bottom: distributions of times since explosion to reach the maximum gamma-ray flux $F_{\\gamma,max}$ from $F_{\\gamma}$ above.} \n    \\label{fig:distributions}\n\\end{figure}\n\nFigure~\\ref{fig:Optical_to_Escaping_Fluxes} shows $F_{\\rm \\gamma,600 d}$ as a function of the peak optical magnitude of the SLSNe-I from the same sample as in Figure~\\ref{fig:distributions}. The selection of fluxes at \\SI{600}{\\day} approximates the time when the effective opacity to 1 TeV photons reaches 1, based on Figure~\\ref{fig:tau}. The top axis also gives the all-sky rate of SLSNe-I above a given peak optical magnitude, which is estimated using the magnitude distribution of SLSNe-I and assuming they occur at a comoving volumetric rate of $ R(z)=19(1+z)^{3.28}\\SI{}{\\, Gpc^{-3}\\, yr^{-1}}$ following \\citet{nicholl2017b,lunnan2018,decia2018}. This estimation captures the general volumetric rate of events, but is unreliable for exceptionally bright events such as SN2017egm due to the small population for estimating the magnitude normalization. A bright event like SN2017egm may actually happen more often than once a century.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=.95\\textwidth]{Optical_to_Escaping_Fluxes_kap_1TeV.pdf}\n    \\caption{\n    Blue dots show the peak optical apparent magnitudes of a sample of SLSNe-I \\citep{nicholl2017d} as a function of their predicted maximum gamma-ray luminosity at 600 days after explosion ($F_{\\gamma, 600d}$). The top axis shows the approximate rate of events above the given peak optical magnitude, calculated using the method described in the main text. Peak maximum gamma-ray luminosities are calculated from fits of optical data with fixed $\\kappa_{\\gamma} = 0.01$ cm$^{2}$ g$^{-1}$. Integral sensitivities of various instruments are overplotted for different exposures. Solid lines: VERITAS 10 and 50 hr integral sensitivities above 220 GeV. Dotted lines: CTA (in development) 10 and 50 hr integral sensitivities above 125 GeV as estimated from 50 hr Monte Carlo simulations of the southern array \\citep{thectaconsortium2019} and extrapolated to 10 hours. Similar extrapolation is done for {\\it Fermi}-LAT from 10 yr to 6 months \\citep{nolan2012} (dashed line). Proposed project AMEGO integral sensitivity above 100 MeV for 6 month observation window is also plotted (dashed-dotted line) \\citep{kierans2020}. \n    }\n    \\label{fig:Optical_to_Escaping_Fluxes}\n\\end{figure*}\n\nShown for comparison in Figure~\\ref{fig:Optical_to_Escaping_Fluxes} are the integral sensitivities of various gamma-ray instruments for different exposures. For IACT instruments such as VERITAS and the future CTA, sensitivity is defined as the minimum flux necessary to reach $5 \\sigma$ detection of a point-like source, requiring at least 10 excess gamma rays and the number of signal counts at least $5\\%$ of the number of background counts. For VERITAS, the sensitivity was calculated using observed Crab Nebula data to estimate the rates of signal and background photons with cuts optimized for a $\\Gamma = -2.5$ power-law spectrum, and then rescaled for the appropriate observation time \\citep{Park2015}. For CTA, Monte Carlo simulations were used to derive angular resolution, background rates and energy dispersion features -- the instrument response functions (IRF) -- based on the Prod3b-v2 telescope configuration for the Southern site and its atmosphere \\citep{cherenkovtelescopearrayobservatory2016}. These IRFs are publicly available and were analyzed using the open-source CTOOLS\\footnote{\\url{http:\/\/cta.irap.omp.eu\/ctools\/}} \\citep{Knodlseder2016}. A power-law spectral model was used to estimate the integral sensitivity above \\SIlist{0.125;1}{\\TeV} each for observations of \\SIlist{10;50}{\\hour} (see \\citealt{Fioretti2016} for further discussion on CTA integral sensitivity).\n\nBased on this systematic study, we propose the following observation strategy: (1) Receive automated public alert and Type I classification of SLSN from a survey instrument such as the Zwicky Transient Facility (ZTF). Classification is generally determined by identification of early spectral components such as $O_{II}$ absorption features. (2) During the multiday rise and fall of bolometric optical light curve, fit the magnetar model ($L_{\\rm opt}$, yielding parameters for $L_{\\rm mag}$ and $L_{\\gamma}$) (3) Compare $L_{\\gamma}$ to the telescope sensitivity at the appropriate day when the effective $\\gamma$-$\\gamma$ opacity falls below ${\\sim1}$ for the telescope's sensitive energy range (see Figure~\\ref{fig:tau}). In the case of IACTs sensitive to energies above $\\SI{100}{\\GeV}$, the gamma rays will escape the magnetar a few hundred days after explosion, requiring a bright SLSN-I that will power gamma rays for as much as two years. \n\nEstimating ${\\sim}35\\%$ of all-sky visibility at VERITAS due to Sun, Moon, and seasonal weather cut, and above 60\\textdegree~ elevation, VERITAS is capable of detecting up to ${\\sim}0.4$ and ${\\sim}4$ SLSNe-I per year for \\SI{10}{h} and \\SI{50}{h} exposures, respectively. The next-generation CTA observatory will be able to detect as many as ${\\sim}8$ and ${\\sim}80$ events for \\SI{10}{hr} and \\SI{50}{h}, respectively, assuming a larger sky visibility fraction of ${\\sim}80\\%$ when both North and South arrays are included. On the other hand, SLSNe at greater distances also imply a stronger role of $\\gamma-\\gamma$ interactions on the EBL in suppressing the $\\gtrsim$ TeV emission, decreasing the observed integral flux by as much as 60 times at redshifts near 0.5 in the VERITAS energy range. \n\nFigure~\\ref{fig:TeV_gamma_flux_sens} shows the distribution of fluxes at \\SI{200}{\\day} and \\SI{600}{\\day} which are approximate average dates when the opacity to \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons falls below 1, respectively, and they are able to escape the ejecta. Accounting for this time delay for the opacity to drop, the expected rate of bright events drops by another 3 to 15 times. While past observations have not been followed up until this publication, the distribution of predicted gamma-ray fluxes hints that, particularly for \\SI{100}{\\GeV} photons, future SLSN-I will be observable with current and planned observatories. \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Escaping_Fluxes_kappa_gamma_dist.pdf}\n    \\caption{ Distribution of gamma-ray luminosities $L_{\\gamma}$ at $t=\\SI{200}{\\day}$ (top) and $t=\\SI{600}{\\day}$ (bottom), when the optical depth for \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons drops below 1, calculated for a sample of 38 SLSNe \\citep{nicholl2017d}}\n    \\label{fig:TeV_gamma_flux_sens}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nSLSN-I are potential gamma-ray emitters, and this paper provides the first upper limits at different times after the optical outburst for two good candidates. The reported upper limits approach the magnetar spin-down luminosity limit of SN2015bn and SN2017egm. While the expected gamma-ray luminosity in either the magnetar central-engine scenario or the shock-acceleration scenario is not constrained by these limits, a relativistic jet powered by fallback accretion onto a black hole is disfavored in both cases. We explore prospects for obtaining improved VHE gamma-ray constraints in the future by current and planned IACTs. We estimate the Type I SLSNe rate for VERITAS and CTA, considering observation constraints and the time delay due to the optical depth. For sufficiently nearby and bright SLSN-I, 0.4 and 4 events per year can be observed by VERITAS from 10 hr and 50 hr observation, respectively, and similarly rates of 8 and 80 events per year can be expected by CTA. \n\n\n\\acknowledgments \n{This research is supported by grants from the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, and by the Helmholtz Association in Germany. M.N. is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No.~948381) and by a Fellowship from the Alan Turing Institute. I.V. acknowledges support by the ETAg grant PRG1006 and by EU through the ERDF CoE grant TK133. V.V.D.'s work is supported by NSF grant 1911061 awarded to the University of Chicago (PI: Vikram Dwarkadas). We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. \nThis research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see \\url{http:\/\/www.cta-observatory.org\/science\/cta-performance\/} (version prod3b-v2) for more details.\n}\n\n\\software{fermipy (v0.19), \\citep{wood2017},\nastropy \\citep{Robitaille2013,Price-Whelan2018},\nCTOOLs, \\citep{Knodlseder2016a}, superbol \\citep{nicholl2018} , EventDisplay \\citep{Maier2017}}, VEGAS \\citep{cogan2008}\n\n\\facilities{VERITAS, {\\it Fermi}-LAT}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe spread of misinformation is increasingly being recognized as an enormous problem. In recent times, misinformation has been reported to have grave consequences such as causing accidents \\cite{ma2016detecting}, while fake news around election times have reportedly reached millions of people \\cite{allcott2017social} causing concerns as to whether they might have influenced the electoral outcome. {\\it Post-Truth} was recognized as the Oxford Dictionary Word of the Year in 2016\\footnote{https:\/\/en.oxforddictionaries.com\/word-of-the-year\/word-of-the-year-2016}. These have spawned an extensive interest in the data analytics community in devising techniques to detect fake news in social and online media leveraging content, temporal and structural features (e.g., \\cite{kwon2013prominent}). A large majority of research efforts on misinformation detection has focused on the political domain within microblogging environments (e.g., \\cite{zhao2015enquiring,ma2016detecting,ma2017detect,qazvinian2011rumor,zubiaga2016learning,castillo2013predicting,zhang2017detecting}) where structural (e.g., the user network) and temporal propagation information (e.g., re-tweets in Twitter) are available in plenty. \n\nFake news and misinformation within the health domain have been increasingly recognized as a problem of immense significance. As a New York Times article suggests, {\\it `Fake news threatens our democracy. Fake medical news threatens our lives'} \\footnote{https:\/\/www.nytimes.com\/2018\/12\/16\/opinion\/statin-side-effects-cancer.html}. Fake health news is markedly different from fake news in politics or event-based contexts on at least two major counts; first, they originate in online websites with limited potential for dense and vivid digital footprints unlike social media channels, and secondly, the core point is conveyed through long and nuanced textual narratives. Perhaps in order to aid their spread, the core misinformation is often intertwined with trustworthy information. They may also be observed to make use of an abundance of anecdotes, conceivably to appeal to the readers' own experiences or self-conscious emotions (defined in \\cite{tracy2004putting}). This makes health misinformation detection a challenge more relevant to NLP than other fields of data analytics. \n\nWe target detection of health fake news within quasi-conventional online media sources which contain information in the form of articles, with content generation performed by a limited set of people responsible for it. We observe that the misinformation in these sources is typically of the kind where scientific claims or content from social media are exaggerated or distilled either knowingly or maliciously (to attract eyeballs). Some example headlines and excerpts from health fake news articles we crawled are shown in Table~\\ref{tab:examples}; these illustrate, besides other factors, the profusion of trustworthy information within them and the abundantly emotion-oriented narrative they employ. Such sources resemble newspaper websites in that consumers are passive readers whose consumption of the content happens outside social media platforms. This makes fake news detection a challenging problem in this realm since techniques are primarily left to work with just the article content - as against within social media where structural and temporal data offer ample clues - in order to determine their veracity.\n\n\\begin{table*}[!htb]\n\\caption{Examples of health fake news headlines and excerpts from them}\n\\label{tab:examples}      \n\\centering\n\\resizebox{0.9\\linewidth}{!}{%\n\\begin{tabular}{p{15cm}}\n\\hline\\noalign{\\smallskip}\n\\textbf{Wi-Fi: A Silent Killer That Kills Us Slowly!} \\\\\nWiFi is the name of a popular wireless networking technology that uses radio waves to provide wireless high-speed Internet and network connections. People can browse the vast area of internet through this wireless device.  A common misconception is that the term Wi-Fi is short for ``wireless fidelity'', however this is not the case. WiFi is simply a trademarked phrase that means IEEE 802.11x. The first thing people should examine is the way a device is connected to the router without cables. Well, wireless devices like cell phones, tablets, and laptops, emit WLAN signals (electromagnetic waves) in order to connect to the router. However, the loop of these signals harms our health in a number of ways. The British Health Agency conducted a study which showed that routers endanger our health and the growth of both, people and plants.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textbf{Russian Scientist Captures Soul Leaving Body; Quantifies Chakras} \\\\\nIt uses a small electrical current that is connected to the fingertips and takes less than a millisecond to send signals from. When these electric charges are pulsed through the body, our bodies naturally respond with a kind of `electron cloud' made up of light photons.  Korotkov also used a type of Kirlian photography to show the exact moment someone's soul left their body at the time of death! He says there is a blue life force you can see leaving the body. He says the navel and the head are the first parts of us to lose their life force and the heart and groin are the last. In other cases, he's noted that the soul of people who have had violent or unexpected deaths can manifest in a state of confusion and their consciousness doesn't actually know that they have died.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textbf{Revolutionary juice that can burn stomach fat while sleeping} \\\\\nHaving excess belly fat poses a serious threat to your health. Fat around the midsection is a strong risk factor for heart disease, type 2 diabetes, and even some types of cancers. Pineapple-celery duo is an ideal choice for those wanting to shed the fat deposits around the stomach area due to the presence of enzymes that stimulate the fat burning hormones. All you need to do is drink this incredible burn-fat sleeping drink and refrain from eating too much sugar and starch foods during the day.\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\subsection{Our Contribution}\n\\label{sec:1.1}\n\nIn this paper, we consider the utility of the affective character of article content for the task of health fake news identification, a novel direction of inquiry though related to the backdrop of fake news identification approaches that target exploiting satire and stance \\cite{rubin2016fake,chopra2017towards}. We posit that fake and legitimate health news articles espouse different emotional characters that may be effectively utilized to improve fake news identification. We develop a simple method to amplify emotion information within documents by leveraging emotion lexicons, and empirically illustrate that such amplification helps significantly in improving the accuracy of health fake news identification within both supervised and unsupervised settings. Our emotion-enrichment method is intentionally of simple design in order to illustrate the generality of the point that emotion cognizance improves health fake news detection. While the influence of emotions on persuasion has been discussed in recent studies \\cite{vosoughi2018spread,majeed2017want}, our work provides the first focused data-driven analysis and quantification of the relationship between emotions and health fake news. Through illustrating that there are significant differences in the emotional character of fake and legitimate news in the health domain in that exaggerating the emotional content aids techniques that would differentiate them, our work sets the stage for further inquiry into identifying the nature of the differences in the emotional content. \n\nThe objective of our study is motivated by the need to illustrate the generality of the point that emotion cognizance improves fake news detection (as indicated or informally observed in various studies e.g., \\cite{vosoughi2018spread,majeed2017want}). Accordingly, we devise a methodology to leverage external emotion lexicons to derive emotion-enriched textual documents. Our empirical study in using these emotion-enriched documents for supervised and unsupervised fake news identification tasks establish that emotion cognizance improves the accuracy of fake news identification. This study is orthogonal but complementary to efforts that rely heavily on non-content features (e.g., \\cite{wu2018tracing}). \n\n\\section{Related Work}\n\\label{sec:rel}\n\nOur particular task, that of understanding the prevalence of emotions and its utility in detecting fake news in the health domain, has not been subject to much attention from the scholarly community. Herein, we survey two streams of related work very pertinent to our task, that of general fake news detection, and secondly, those relating to the analysis of emotions in fake news. \n\n\\subsection{Fake News Detection}\n\\label{sec:2.1}\n\nOwing to the emergence of much recent interest in the task of fake news detection, there have been many publications on this topic in the last few years. A representative and a non-comprehensive snapshot of work in the area appears in Table~\\ref{tab:lit}. As may be seen therein, most efforts have focused on detecting misinformation within microblogging platforms using content, network (e.g., user network) and temporal (e.g., re-tweets in Twitter) features within the platform itself \\cite{anoop2019leveraging}; some of them, notably \\cite{wu2018tracing}, target scenarios where the candidate article itself resides outside the microblogging platform, but the classification task is largely dependent on information within. An emerging trend, as exemplified by \\cite{ma2017detect,wu2018tracing}, has been to focus on how information propagates within the microblogging platform, to distinguish between misinformation and legitimate ones. Unsupervised misinformation detection techniques \\cite{zhang2017detecting,zhang2016distance} start with the premise that misinformation is rare and of differing character from the large majority, and use techniques that resemble outlier detection methods in flavor. Of particular interest is a recent work \\cite{guo2019exploiting} that targets to exploit emotions for fake news detection within microblogging platforms. This makes extensive usage of the {\\it publisher emotions}, the emotions expressed in the content, and {\\it social emotions}, the emotions expressed in responses, in order to improve upon the state-of-the-art in fake news detection accuracies. To contrast with this stream of work on fake news detection, it may be noted that our focus is on the health domain where information is usually in the form of long textual narratives, with limited information on the responses, temporal propagation and author\/spreader\/reader network structure available for the technique to make a veracity decision. \n\n\\begin{table}[!htb]\n\\caption{Overview of Related Literature}\n\\label{tab:lit}      \n\\centering\n\\begin{tabular}{llccc}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{Work} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Target\\\\ Domain\\end{tabular}} & \\multicolumn{3}{c}{Features used} \\\\ \\cline{3-5} \\noalign{\\smallskip} \n & & Content   & Network   & Temporal \\\\ \n\\hline \\noalign{\\smallskip}\n\\multicolumn{5}{c}{Task Setting: Supervised} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\t\\cite{kwon2013prominent} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n    \\cite{zubiaga2017exploiting} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n    \\cite{qazvinian2011rumor} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n    \\cite{wu2018tracing} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n    \\cite{ma2016detecting} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n    \\cite{zhao2015enquiring} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n    \\cite{ma2017detect} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n    \\cite{guo2019exploiting} & Weibo & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    \\multicolumn{5}{c}{Task Setting: Unsupervised} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}    \n    \\cite{zhang2017detecting}& Weibo & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n    \\cite{zhang2016distance} & Weibo & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Emotions and Fake News}\n\\label{sec:2.2}\n\nFake news is generally crafted with the intent to mislead, and thus narratives powered with strong emotion content may be naturally expected within them. \\cite{bakir2018fake} analyze fake news vis-a-vis emotions and argue that what is most significant about the contemporary fake news furore is what it portends: the use of personally and emotionally targeted news produced by journalism referring to what they call as ``empathic media''. They further go on to suggest that the commercial and political phenomenon of empathically optimised automated fake news is on the near-horizon, and is a challenge needing significant attention from the scholarly community. A recent study, \\cite{paschen2019investigating}, conducts an empirical analysis on 150 real and 150 fake news articles from the political domain, and report finding significantly more negative emotions in the titles of the latter. Apart from being distinctly different in terms of domain, our focus being health (vs. politics for them), we also significantly differ from them in the intent of the research; our work is focused not on identifying the tell-tale emotional signatures of real vis-a-vis fake news, but on providing empirical evidence that there are differences in emotional content which may be exploited through simple mechanisms such as word-addition-based text transformations. In particular, our focus is on establishing that there are differences, and we keep identification of the nature of differences outside the scope of our present investigation. A recent tutorial survey on fake news in social media, \\cite{shu2019detecting}, also places significant emphasis on the importance of emotional information within the context of fake news detection. \n\n\\subsection{Our Work in Context} \n\\label{sec:2.3}\n\nTo put our work in context, we note that the affective character of the content has not been a focus of health fake news detection so far, to our best knowledge. Our effort is orthogonal but complementary to most work described above in that we provide evidence that emotion cognizance in general, and our emotion-enriched data representations in particular, are likely to be of much use in supervised and unsupervised fake news identification for the health domain. As observed earlier, identifying the nature of emotional differences between fake and real news in the health domain is outside the scope of our work, but would evidently lead to interesting follow-on work. \n\n\\section{Emotionizing Text}\n\\label{sec:3}\nThe intent in this paper is to provide evidence that the affective character of fake news and legitimate articles differ in a way that such differences can be leveraged to improve the task of fake news identification. First, we outline our methodology to leverage an external emotion lexicon to build emotion amplified (i.e., {\\it emotionized}) text representations. The methodology is designed to be very simple to describe and implement, so any gains out of emotionized text derived from the method can be attributed to emotion-enrichment in general and not to some nuances of the details, as could be the case if the transformation method were to involve sophisticated steps. The empirical analysis of our emotionized representations {\\it vis-a-vis} raw text for fake news identification will be detailed in the next section. \n\n\\subsection{The Task}\n\\label{sec:3.1}\nThe task of emotionizing is to leverage an emotion lexicon $\\mathcal{L}$ to transform a text document $D$ to an emotionized document $D'$. We would like $D'$ also to be similar in format to $D$ in being a sequence of words so that it can be fed into any standard text processing pipeline; retaining the document format in the output, it may be noted, is critical for the uptake of the method. In short:\n\\begin{center}\n$\nD, \\mathcal{L} \\xrightarrow[]{Emotionization} D'\n$\n\\end{center}\nWithout loss of generality, we expect that the emotion lexicon $\\mathcal{L}$ would comprise of many 3-tuples, e.g., $[w, e, s]$, each of which indicate the affinity of a word $w$ to an emotion $e$, along with the intensity quantified as a score $s \\in [0,1]$. An example entry could be $[unlucky, sadness, 0.7]$ indicating that the word {\\it unlucky} is associated with the {\\it sadness} emotion with an intensity of 0.7. \n\n\\begin{table*}[!htb]\n\\caption{Emotionized Health Fake News Excerpts (added emotion labels in bold)}\n\\label{tab:emotionize_example}      \n\\centering\n\\resizebox{0.9\\linewidth}{!}{%\n\\begin{tabular}{p{15cm}}\n\\hline\\noalign{\\smallskip}\nWi-Fi: A Silent Killer {\\bf fear} That Kills {\\bf fear} Us Slowly! \\\\\nWiFi is the name of a popular wireless networking technology that uses radio waves to provide wireless high-speed Internet and network connections. People can browse the vast area of internet through this wireless device.  A common misconception {\\bf fear} is that the term Wi-Fi is short for ``wireless fidelity'', however this is not the case. WiFi is simply a trademarked phrase that means IEEE 802.11x. The first thing people should examine is the way a device is connected to the router without cables. Well, wireless devices like cell phones, tablets, and laptops, emit WLAN signals (electromagnetic waves) in order to connect to the router. However, the loop of these signals harms {\\bf fear} our health in a number of ways. The British Health Agency conducted a study which showed that routers endanger {\\bf fear} our health and the growth {\\bf joy} of both, people and plants.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nRussian Scientist Captures Soul Leaving {\\bf sadness} Body; Quantifies Chakras \\\\\nIt uses a small electrical current that is connected to the fingertips and takes less than a millisecond to send signals from. When these electric charges are pulsed through the body, our bodies naturally respond with a kind of `electron cloud' made up of light {\\bf joy} photons.  Korotkov also used a type of Kirlian photography to show the exact moment someone's soul left their body at the time of death {\\bf sadness}! He says there is a blue life force you can see leaving {\\bf sadness} the body. He says the navel and the head are the first parts of us to lose {\\bf sadness} their life force and the heart and groin are the last. In other cases, he's noted that the soul of people who have had violent {\\bf anger} or unexpected deaths {\\bf sandess} can manifest in a state of confusion and their consciousness doesn't actually know that they have died {\\bf sadness}.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nRevolutionary juice that can burn stomach fat while sleeping \\\\\nHaving excess belly fat poses a serious threat {\\bf anger} to your health. Fat around the midsection is a strong risk {\\bf fear} factor for heart disease {\\bf fear}, type 2 diabetes, and even some types of cancers {\\bf sadness}. Pineapple-celery duo is an ideal choice for those wanting to shed the fat deposits around the stomach area due to the presence of enzymes that stimulate the fat burning hormones. All you need to do is drink this incredible burn-fat sleeping drink and refrain from eating too much sugar and starch foods {\\bf joy} during the day.\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\subsection{Methodology}\n\\label{sec:3.2}\nInspired by recent methods leveraging lexical neighborhoods to derive word \\cite{mikolov2013distributed} and document \\cite{le2014distributed} embeddings, we design our emotionization methodology as one that alters the neighborhood of highly emotional words in $D$ by adding emotion labels. As illustrated in Algorithm~\\ref{algorithm}, we sift through each word in $D$ in order, outputting that word followed by its associated emotion from the lexicon $\\mathcal{L}$ into $D'$, as long as the word emotion association in the lexicon is stronger than a pre-defined threshold $\\tau$. In cases where the word is not associated with any emotion with a score greater than $\\tau$, no emotion label is output into $D'$. In summary, $D'$ is an `enlarged' version of $D$ where every word in $D$ that is strongly associated with an emotion additionally being followed by the emotion label. This ingestion of `artificial' words is similar in spirit to {\\it sprinkling} topic labels to enhance text classification \\cite{DBLP:conf\/acl\/HingmireC14}, where appending topic labels to document is the focus.\n\n\\begin{algorithm}[!htb]\n\\small\n\\SetKwInOut{Input}{input}\n\\SetKwInOut{Output}{output}\n\\SetKwInOut{Initialization}{Initialize}\n\\Input{Document $D$, Emotion-Lexicon $\\mathcal{L}$}\n\\Output{Emotionized Document $D'$}\n\\Parameter{Threshold $\\tau$}\n\\BlankLine\n\\caption{Emotionization \\label{algorithm}}\n\\renewcommand{\\nl}{\\let\\nl\\oldnl} Let $D = [w_1, w_2, \\ldots, w_n]$ \\;\ninitialize $D'$ to be empty \\;\n\\For{$(i = 1;\\ i \\leq n;\\ i++)$}{\n    write $w_i$ as the next word in $D'$ \\;\n    \\uIf{$(\\exists [w_i,e,s] \\in \\mathcal{L} \\wedge s \\geq \\tau)$}{\n    write $e$ as the next word in $D'$ \\;\n}\n}\noutput $D'$\n\\end{algorithm}\n\\noindent\nA sample of article excerpts and their emotionized versions appear in Table~\\ref{tab:emotionize_example}. \n\n\\section{Empirical Study}\n\\label{sec:4}\nGiven our focus on evaluating the effectiveness of emotionized text representations over raw representations, we consider a variety of unsupervised and supervised methods (in lieu of evaluating on a particular state-of-the-art method) in the interest of generality. Data-driven fake news identification, much like any analytics task, uses a corpus of documents to learn a statistical model that is intended to be able to tell apart fake news from legitimate articles. Our empirical evaluation is centered on the following observation: {\\it for the same analytics model learned over different data representations, differences in effectiveness (e.g., classification or clustering accuracy) over the target task can intuitively be attributed to the data representation}. In short, if our emotionized text consistently yields better classification\/clustering models over those learned over raw text, emotion cognizance and amplification may be judged to influence fake news identification positively. We first describe our dataset, followed by the empirical study settings and their corresponding results. \n\n\\subsection{Dataset and Emotion Lexicon}\n\\label{sec:4.1}\n\nWith most fake news datasets being focused on microblogging websites in the political domain making them less suitable for content-focused misinformation identification tasks as warranted by the domain of health, we curated a new dataset of fake and legitimate news articles within the topic of {\\it health and well being} which will be publicly released upon publication to aid future research. For legitimate news, we crawled $500$ health and well-being articles from reputable sources such as CNN, NYTimes, New Indian Express and many others. For fake news, we crawled $500$ articles on similar topics from well-reported misinformation websites such as BeforeItsNews, Nephef, MadWorldNews, and many others. These were manually verified for category suitability. The detailed dataset statistics is shown in Table \\ref{tab:dataset}. \n\nFor the lexicon, we use the NRC Intensity Emotion Lexicon \\cite{mohammad2017word} which has data in the 3-tuple form outlined earlier. For simplicity, we filter the lexicon to retain only one entry per word, choosing the emotion entry with which the word has the highest intensity; this filtered out around 22\\% of entries in our lexicon. This filtering entails that each word in $D$ can only introduce up to one extra token in $D'$; the emotionization using the filtered corpus was seen to lengthen documents by an average of 2\\%, a very modest increase in document size. To put it in perspective, only around one in fifty words triggered the lexicon label attachment step. Interestingly, there was only a slight difference in the lengthening of document across the classes; while fake news documents were seen to be enlarged by $2.2\\%$ on average, legitimate news articles recorded an average lengthening by $1.8\\%$. For e.g., out of 1923 word sense entries that satisfy the threshold $\\tau = 0.6$, our filter-out-non-best heuristic filtered out 424 entries (i.e., 22\\%); thus, only slightly more than one-fifth of entries were affected. This heuristic to filter out all-but-one entry per word was motivated by the need to ensure that document structures be not altered much (by the introduction of too many lexicon words), so assumptions made by the downstream data representation learning procedure such as document well-formedness are not particularly disadvantaged. \n\n\\begin{table*}[!htb]\n\\caption{Dataset Details}\n\\label{tab:dataset}      \n\\centering\n\\begin{tabular}{lccccc}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{1}{c}{Dataset}& Class & Documents & Average Words & Average Sentences & Total Words \\\\ \\hline\n\\noalign{\\smallskip}\n\\multirow{2}{*}{Health and Well Being} & Real  & 500  & 724  & 31   & 362117      \\\\ \n\\noalign{\\smallskip}\n& Fake  & 500   & 578 & 28  & 289477      \\\\ \n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Supervised Setting - Conventional Classifiers}\n\\label{sec:4.2}\nLet $\\mathcal{D} = \\{ \\ldots, D, \\ldots \\}$ be the corpus of all news articles, and $\\mathcal{D}' = \\{ \\ldots, D', \\ldots \\}$ be the corresponding emotionized corpus. Each document is labeled as either fake or not (0\/1). With word\/document embeddings gaining increasing popularity, we use the DBOW doc2vec model\\footnote{https:\/\/radimrehurek.com\/gensim\/models\/doc2vec.html} to build vectors over each of the above corpora separately, yielding two datasets of vectors, correspondingly called $\\mathcal{V}$ and $\\mathcal{V}'$. While the document embeddings are learnt over the corpora ($\\mathcal{D}$ or $\\mathcal{D}'$), the output comprises one vector for each document in the corpus that the learning is performed over. The doc2vec model uses an internal parameter $d$, the dimensionality of the embedding space, i.e., the length of the vectors in $\\mathcal{V}$ or $\\mathcal{V}'$. \n\nEach of these vector datasets are separately used to train a conventional classifier using train and test splits within them. By conventional classifier, we mean a model such as random forests, kNN, SVM, Naive Bayes, Decision Tree or AdaBoost. The classification model learns to predict a class label (one of {\\it fake} or {\\it real}) given a d-dimensional embedding vector. We use multiple train\/test splits for generalizability of results where the chosen dataset (either $\\mathcal{V}$ or $\\mathcal{V}'$) is partitioned into $k$ random splits (we use $k=10$); these lead to $k$ separate experiments with $k$ models learnt, each model learnt by excluding one of $k$ splits, and evaluated over their corresponding held-out split. The accuracies obtained by $k$ separate experiments are then simply averaged to obtain a single accuracy score for the chosen dataset ($Acc(\\mathcal{D})$ and $Acc(\\mathcal{D}')$ respectively). The quantum of improvement achieved, i.e., $Acc(\\mathcal{D}') - Acc(\\mathcal{D})$ is illustrative of the improvement brought in by emotion cognizance. \n\n\\subsection{Supervised Setting - Neural Networks}\n\nNeural network models such as LSTMs and CNNs are designed to work with vector sequences, one for each word in the document, rather than a single embedding for the document. This allows them to identify and leverage any existence of sequential patterns or localized patterns respectively, in order to utilize for the classification task. These models, especially LSTMs, have become very popular for building text processing pipelines, making them pertinent for a text data oriented study such as ours. \n\nAdapting from the experimental settings for the conventional classifiers in Section~\\ref{sec:4.2}, we learn LSTM and CNN classifiers with learnable word embeddings where each word would have a length of either $100$ or $300$. Unlike in Section~\\ref{sec:4.2} where the document embeddings are learnt separately and then used in a classifier, this model interleaves training of the classifier and learning of the embeddings, so the word embeddings are also trained, in the process, to benefit the task. The overall evaluation framework remains the same as before, with the classifier-embedding combo being learnt separately for $\\mathcal{D}$ and $\\mathcal{D}'$, and the quantum by which $Acc(\\mathcal{D}')$ surpasses $Acc(\\mathcal{D})$ used as an indication of the improvement brought about by the emotionization. \n\n\n\\subsubsection{Results and Discussion}\nTable~\\ref{tab:classification} lists the classification results of a variety of standard classifiers as well as those based on CNN and LSTM, across two values of $d$ and various values of $\\tau$.  $d$ is overloaded for convenience in representing results; while it indicates the dimensionality of the document vector for the conventional classifiers, it indicates the dimensionality of the word vectors for the CNN and LSTM classifiers. Classification {\\it models learned over the emotionized text are seen to be consistently more effective for the classification task}, as exemplified by the higher values achieved by $Acc(\\mathcal{D}')$ over $Acc(\\mathcal{D})$ (highest values in each row are indicated in bold). While gains are observed across a wide spectrum of values of $\\tau$, the gains are seen to peak around $\\tau \\approx 0.6$. Lower values of $\\tau$ allow words of low emotion intensity to influence $D'$ while setting it to a very high value would add very few labels to $D'$ (at the extreme, using $\\tau=1.0$ would mean $D=D'$). Thus the observed peakiness is along expected lines, with $\\tau \\approx 0.6$ achieving a middle ground between the extremes. The quantum of gains achieved, i.e., $|Acc(\\mathcal{D}')-Acc(\\mathcal{D})|$, is seen to be significant, sometimes even bringing $Acc(\\mathcal{D}')$ very close to the upper bound of $1.0$; this establishes that emotionized text is much more suitable for supervised misinformation identification. It is further notable that the highest accuracy is achieved by AdaBoost as against the CNN and LSTM models; this may be due to the lexical distortions brought about addition of emotion labels limiting the emotionization gains in the LSTM and CNN classifiers that attempt to make use of the word sequences explicitly.\n\n\\begin{table}[!htb]\n\\caption{Classification Results (NB = Naive Bayes, RF = Random Forest, DT = Decision Tree, AB = AdaBoost)}\n\\label{tab:classification}      \n\\centering\n\\resizebox{1\\linewidth}{!}{%\n\\begin{tabular}{lcccccc}\n\\hline\\noalign{\\smallskip}\n    \\multirow{2}{*}{{\\small Method}} & \\multirow{2}{*}{$Acc(\\mathcal{D})$} &  \\multicolumn{5}{c}{$Acc(\\mathcal{D}')$} \\\\ \\noalign{\\smallskip} \\cline{3-7} \\noalign{\\smallskip}\n    & & $\\tau=0.0$ & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    \\multicolumn{7}{c}{Classification Results for $d = 100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    NB & 0.770 & 0.780 & 0.780 & 0.785 & {\\bf 0.790} & 0.775 \\\\\n    KNN & 0.750 & 0.750 & 0.755 & 0.760 & {\\bf 0.925} & 0.750 \\\\\n    SVM & 0.500 & 0.650 & 0.750 & 0.750 & {\\bf 0.900} & 0.700 \\\\\n    RF & 0.630 & 0.710 & 0.700 & 0.720 & {\\bf 0.840} & 0.805 \\\\\n    DT & 0.680 & 0.690 & 0.700 & 0.780 & {\\bf 0.940} & 0.785 \\\\\n    AB & 0.550 & 0.570 & 0.700 & 0.710 & {\\bf 0.965} & 0.825 \\\\\n    CNN & 0.870 & 0.880 & 0.900 & 0.880 & {\\bf 0.910} & 0.880 \\\\\n    LSTM & 0.905 & 0.900 & 0.910 & 0.910 & {\\bf 0.920} & {\\bf 0.920} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    \\multicolumn{7}{c}{Classification Results for $d = 300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    NB & 0.770 & 0.800 & 0.810 & 0.790 & {\\bf 0.830} & 0.780 \\\\\n    KNN & 0.720 & 0.740 & 0.750 & 0.760 & {\\bf 0.910} & 0.745 \\\\\n    SVM & 0.600 & 0.670 & 0.720 & 0.740 & {\\bf 0.890} & 0.720 \\\\\n    RF & 0.650 & 0.700 & 0.730 & 0.715 & {\\bf 0.820} & 0.750 \\\\\n    DT & 0.600 & 0.650 & 0.730 & 0.780 & {\\bf 0.905} & 0.750 \\\\\n    AB & 0.550 & 0.550 & 0.720 & 0.810 & {\\bf 0.945} & 0.750 \\\\\n    CNN & 0.912 & 0.910 & {\\bf 0.927} & 0.920 & 0.920 & 0.910 \\\\\n    LSTM & 0.900 & 0.902 & 0.900 & 0.902 & {\\bf 0.907} & 0.900 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\subsection{Unsupervised Setting}\n\\label{sec:4.3}\nThe corresponding evaluation for the unsupervised setting involves clustering both $\\mathcal{V}$ and $\\mathcal{V}'$ (Ref. Sec.~\\ref{sec:4.2}) using the same method and profiling the clustering against the labels on the clustering purity measure\\footnote{https:\/\/nlp.stanford.edu\/IR-book\/html\/htmledition\/evaluation-of-clustering-1.html}; as may be obvious, the labels are used only for evaluating the clustering, clustering being an unsupervised learning method. We used K-Means \\cite{macqueen1967some} and DBSCAN \\cite{ester1996density} clustering methods, two very popular clustering methods that come from distinct families. K-Means uses a top-down approach to discover clusters, estimating cluster centroids and memberships at the dataset level, followed by iteratively refining them. DBSCAN, on the other hand, uses a more bottom-up approach, forming clusters and enlarging them by adding proximal data points progressively. Another aspect of difference is that K-Means allows the user to specify the number of clusters desired in the output, whereas DBSCAN has a substantively different mechanism, based on neighborhood density. For K-Means we measured purities, averaged over 1000 random initializations, across varying values of $k$ (desired number of output clusters); it may be noted that purity is expected to increase with $k$ with finer clustering granularities leading to better purities (at the extreme, each document in its own cluster would yield a purity of $1.0$). For DBSCAN we measured purities across varying values of $ms$ (minimum samples to form a cluster); the {\\it ms} parameter is the handle available to the user within the DBSCAN framework to indirectly control the granularity of the clustering (i.e., the number of clusters in the output). Analogous to the $Acc(.)$ measurements in classification, the quantum of purity improvements achieved by the emotionized text, i.e., $Pur(\\mathcal{D}')-Pur(\\mathcal{D})$, indicate any improved effectiveness of emotionized representations.\n\n\\subsubsection{Results and Discussion} Table~\\ref{tab:clustering} lists the clustering results in a format similar to that of the classification study. With the unsupervised setting posing a harder task, the quantum of improvements ($Pur(\\mathcal{D}')-Pur(\\mathcal{D})$) achieved by emotionization is correspondingly lower. We believe the cause of low accuracy is because most conventional combinations of document representation and clustering algorithm are suited to generate topically coherent clusters, and thus fare poorly on a substantially different task of fakeness identification. However, the trends are consistent with the earlier observations in that emotionization has a positive effect, with gains peaking around $\\tau \\approx 0.6$. \n\n\\begin{table}[!htb]\n\\caption{Clustering Results}\n\\label{tab:clustering}      \n\\centering\n\\resizebox{1\\linewidth}{!}{%\n\\begin{tabular}{lcccccc}\n\\hline\\noalign{\\smallskip}\n    \\multirow{2}{*}{} & \\multirow{2}{*}{$Pur(\\mathcal{D})$} &  \\multicolumn{5}{c}{$Pur(\\mathcal{D}')$} \\\\ \\noalign{\\smallskip} \\cline{3-7} \\noalign{\\smallskip}\n    & & $\\tau=0.0$ & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    k & \\multicolumn{6}{c}{K-Means Clustering Results for $d=100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    2 & 0.523 & 0.524 & 0.523 & 0.523 & {\\bf 0.561} &  0.529 \\\\\n    4 & 0.781 & 0.780 & 0.786 & 0.793 & {\\bf 0.816} &  0.793 \\\\\n    7 & 0.850 & 0.857 & 0.852 & 0.851 & {\\bf 0.869} &  0.856 \\\\\n    10 & 0.853 & 0.851 & 0.851 & 0.851 & {\\bf 0.877} & 0.857 \\\\\n    15 & 0.852 & 0.853 & 0.851 & 0.851 & {\\bf 0.878} & 0.858 \\\\\n    20 & 0.852 & 0.852 & 0.850 & 0.851 & {\\bf 0.887} & 0.857 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    k & \\multicolumn{6}{c}{K-Means Clustering Results for $d=300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    2 & 0.513 & 0.520 & 0.520 & 0.520 & {\\bf 0.555} &  0.520 \\\\\n    4 & 0.771 & 0.778 & 0.781 & 0.789 & {\\bf 0.815} &  0.785 \\\\\n    7 & 0.840 & 0.840 & 0.850 & 0.849 & {\\bf 0.869} &  0.846 \\\\\n    10 & 0.850 & 0.850 & 0.850 & 0.850 & {\\bf 0.871} & 0.851 \\\\\n    15 & 0.851 & 0.853 & 0.851 & 0.851 & {\\bf 0.875} & 0.852 \\\\\n    20 & 0.850 & 0.852 & 0.850 & 0.850 & {\\bf 0.880} & 0.850 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    ms & \\multicolumn{6}{c}{DBSCAN Clustering Results for $d=100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    20 & 0.610 & 0.620 & 0.620 & 0.620 & {\\bf 0.650} &  0.619 \\\\\n    40 & 0.627 & 0.655 & 0.645 & 0.581 & {\\bf 0.665} &  0.650 \\\\\n    60 & 0.716 & 0.721 & 0.720 & 0.725 & {\\bf 0.725} &  0.725 \\\\\n    80 & 0.851 & 0.850 & 0.851 & 0.856 & {\\bf 0.860} &  0.856 \\\\\n    100 & 0.845 & 0.841 & 0.848 & 0.847 & {\\bf 0.860} & 0.840 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    ms & \\multicolumn{6}{c}{DBSCAN Clustering Results for $d=300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    20 & 0.610 & 0.615 & 0.610 & 0.610 & {\\bf 0.635} &  0.620 \\\\\n    40 & 0.635 & 0.663 & 0.665 & 0.669 & {\\bf 0.670} &  0.655 \\\\\n    60 & 0.675 & 0.701 & 0.705 & 0.710 & {\\bf 0.715} &  0.700 \\\\\n    80 & 0.780 & 0.810 & 0.819 & 0.820 & {\\bf 0.825} &  0.808 \\\\\n    100 & 0.755 & 0.800 & 0.800 & 0.800 & {\\bf 0.805} & 0.800 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\section{Conclusions}\n\\label{sec:con}\nIn this paper, we considered the utility of the affective character of news articles for the task of misinformation detection in the health domain. We illustrated that amplifying the emotions within a news story (and in a sense, uplift their importance) helps downstream algorithms, supervised and unsupervised, to identify health fake news better. In a way, our results indicate that fake and real news differ in the nature of emotional information within them, so exaggerating the emotional information within both stretch them further apart in any representation, helping to distinguish them from each other. In particular, our simple method to emotionize text using external emotion intensity lexicons were seen to yield text representations that were empirically seen to be much more suited for the task of identifying health fake news. In the interest of making a broader point establishing the utility of affective information for the task, we empirically evaluated the representations over a wide variety of supervised and unsupervised techniques and methods over varying parameter settings, across which consistent and significant gains were observed. This firmly establishes the utility of emotion information in improving health fake news identification. \n\n\\subsection{Future Work} \n\\label{sec:5.1}\nAs a next step, we are considering developing emotion-aware end-to-end methods for supervised and unsupervised health fake news identification. Secondly, we are considering the use of lexicons learned from data \\cite{bandhakavi2014generating} which may be better suited for fake news identification in niche domains. Third, we are exploring the usage of the affective content of responses to social media posts. \n\n\n\n\n\n\n\n\\bibliographystyle{spmpsci}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{section.1_intro}\n\nThe study of the physical properties and evolution of massive stars (M\\,>\\,8-9~M$_{\\odot}$) is crucial for many aspects of our understanding of the Universe. They play an important role in the chemodynamical evolution of the galaxies \\citepads{2012ceg..book.....M} and were key players in the epoch of reionization of the Universe (\\citeads{1999ApJ...527L...5B}; \\citeads{2000ApJ...540...39A}). They are the precursors of hyperenergetic supernovae, long-duration $\\gamma$-ray burst (see \\citeads{2012ARA&A..50..107L}, and references therein), and the recently detected gravitational wave events (e.g., \\citeads{2016PhRvL.116f1102A}; \\citeads{2017PhRvL.119p1101A}; \\citeads{2020arXiv200201950A}). Their high luminosities make them observable individually at large distances, and they are thus optimal tools for access to invaluable information about abundances and distances in galaxies at up to several megaparsec (e.g., \\citeads{2003ApJ...584L..73U}; \\citeads{2008A&A...485...41C}; \\citeads{2013ApJ...779L..20K}). Moreover, through their feedback into the interstellar medium in the form of ultraviolet radiation and stellar winds, massive stars critically affect the star formation process by both triggering the formation of new generations of stars and stopping mass accretion in the surrounding forming stars.\n\nMost massive stars are found within or are linked to young open clusters and the so-called OB associations (\\citeads{2003ARA&A..41...57L}; \\citeads{2010ARA&A..48..431P}). These stellar groupings are therefore perfect laboratories to study them. \n\n\\citetads{1978ApJS...38..309H} compiled a catalog of all known supergiants (Sgs) and O stars in associations and clusters of the Milky Way, including over 1000 objects of this type. Among the list of associations quoted in that paper, Per~OB1, which also includes the famous $h$~and~$\\chi$~Persei double cluster, clearly stands out as one of the richest. In particular, it is one of the few Galactic OB associations in which, given its age ($\\sim$\\,13\\,--\\,14\\,Myr, \\citeads{2002ApJ...576..880S}; \\citeads{2019ApJ...876...65L}), a massive star population covering a wide range of evolutionary stages can be found (e.g., it harbors 23 red Sgs and several dozen blue Sgs). In addition, it is also relatively close to us (d\\,$\\sim$\\,2.2\\,--\\,2.4\\,kpc, \\citeads{2018A&A...616A..10G}; \\citeads{2019MNRAS.486L..10D}) and is characterized by a moderate extinction (E(B-V)$\\sim$0.6, \\citeads{2002ApJ...576..880S}). This unique combination of characteristics makes Per~OB1 a very interesting testbed for the study of a large interrelated population of evolved massive stars from an evolutionary point of view.\n\nPer~OB1 has attracted the attention of the astrophysical community for many years and has been the subject of studies from many different fronts. We highlight the investigation of how the association could have been formed (\\citeads{2008ApJ...679.1352L}); the membership of stars to the association (\\citeads{1970ApJ...160.1149H}; \\citeads{1978ApJS...38..309H}; \\citeads{1992A&AS...94..211G}; \\citeads{2008ApJ...679.1352L}; \\citeads{2009MNRAS.400..518M}) and, in particular, to $h$~and~$\\chi$~Persei (\\citeads{2002PASP..114..233U}; \\citeads{2010ApJS..186..191C}); the characterization of the kinematics of the region (\\citeads{2017MNRAS.472.3887M}; \\citeads{2019A&A...624A..34Z}); the identification of blue Sg binaries (\\citeads{1973ApJ...184..167A}); or the spectroscopic characterization of different samples of blue stars in the region (including the determination of rotational velocities, stellar parameters and surface abundances \\citeads{1968ApJ...154..933S};  \\citeads{1988A&A...195..208L}; \\citeads{1995A&A...298..489K}; \\citeads{1996A&A...310..564K}; \\citeads{2005AJ....129..809S}; \\citeads{2019ApJ...876...65L}), also reaching the red Sg domain (\\citeads{2014ApJ...788...58G}).\n\nDespite all the information compiled about the Per~OB1 association, and particularly, $h$~and~$\\chi$~Persei, we still lack a complete homogeneous empirical characterization (that also takes environmental and kinematical information into account) of the physical and evolutionary properties of its massive star population. This is the main objective of this series of papers, which is based on a set of high-quality observations including high-resolution, multi-epoch spectroscopy (mostly gathered in the framework of the IACOB project, see \\citeads{2015hsa8.conf..576S} and references therein), and astrometric information delivered by the {\\em Gaia} mission (\\citeads{2018A&A...616A...1G}; \\citeads{2018A&A...616A...2L}). The compiled empirical information resulting from the analysis of this observational dataset will allow us to proceed in our understanding of massive star evolution, and also investigate some long-standing and new open questions in this important field of stellar astrophysics. These questions include the evolutionary status of the blue supergiants, or the effect that binarity and rotation have on the evolution of massive stars.\n\nIn this first paper, we carry out a membership analysis of a sample of 88 blue and red Sgs located within 4.5\\,deg from the center of the Per~OB1 association, and we also investigate some of its kinematical properties. In Sect.~\\ref{section.2_samobs} we present the sample of stars and the main characteristics of the compiled observations. \nIn Sect.~\\ref{section.3_rv} we describe the strategy we followed to derive reliable radial velocities (RVs). Sect.~\\ref{section.4_results} presents the results extracted from the analysis of the observations, mainly referring to parallaxes, proper motions, RV measurements, and the identification of spectroscopic signatures of binarity and other types of spectroscopic variability phenomena.\nIn Sect.~\\ref{section.5_discus} we use all these results to establish and apply our membership criteria to all stars in the sample, and we also identify outliers for each of the considered quantities, in particular, binary and runaway stars. We also analyze some global features of Per~OB1, and discuss some individual cases of interest. The main conclusions of this work and some future prospects are provided in Sect.~\\ref{section.6_summary}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig1.png}\n\\caption{Sky map with the complete sample of stars. Purple, blue, cyan, golden, and red symbols represent the O-, B-, A-, F-, and K- and M-type stars, respectively. This color code is the same in all the plots unless otherwise specified. The central green cross denotes the center of the Per~OB1 association taken from \\citetads{2017MNRAS.472.3887M}. The large green circle indicates a 4.5-degree circle around the center. The small green circles show the positions of $h$~and~$\\chi$~Persei. The background image, used for reference, was taken from DSS-red.}\n\\label{figure.fig1}\n\\end{figure}\n\n\n\n\n\\section{Sample definition and observations}\n\\label{section.2_samobs}\n\nIn this section, we describe the process we have followed to build the sample under study, and to compile the associated observations. The latter mainly refers to high-quality spectroscopy obtained with the FIbre-fed Echelle Spectrograph (FIES), \\citepads{2014AN....335...41T} and the High Efficiency and Resolution Mercator Echelle Spectrograph (HERMES) \\citepads{2011A&A...526A..69R} high-resolution\nspectrographs attached to the 2.56~m Nordic Optical Telescope (NOT) and the 1.2~m Mercator telescope, respectively, and astrometric and photometric data delivered by the {\\em Gaia} mission in the second data release \\citepads[DR2, ][]{2018A&A...616A...1G, 2018A&A...616A...2L, 2018A&A...616A...4E}.\n\n\n\\subsection{Sample definition}\n\\label{subsection.21_sample}\nThe final sample of targets considered for this work comprises 88 blue and red massive stars located within 4.5\\,deg from the center of the Per~OB1 association (as defined in \\citeads{2017MNRAS.472.3887M}). \nTo restrict the sample to the most massive stars, the luminosity classes (LCs) were limited to bright giants (Gs) and Sgs (LC II and I, respectively) in the case of the O- and B-type stars, and to Sgs when we refer to A- and later-type stars. In addition, the sample includes a few O and early-B Gs for which we already had available observations in the IACOB spectroscopic database (see Sect.~\\ref{subsection.22_specobs}).\n\nTable~\\ref{table.A1} summarizes the list of targets, separated and ordered by spectral type (SpT). We note that the quoted spectral classifications were carefully revised using the spectra with the best signal-to-noise ratio (S\/N) of our own spectroscopic observations (see Sect.~\\ref{subsection.22_specobs}) following the criteria explained in \\cite{Negueruela2020}, in prep.) and \\citeads{2018A&A...618A.137D}, for the case of the blue and red Sg samples, respectively.\nIn addition, Fig.~\\ref{figure.fig1} shows their location on the sky. We also indicate as a large green circle the search area of 4.5\\,deg around the center of Per~OB1, marked as a green cross. Most stars, including those from the $h$~and~$\\chi$~Persei double cluster (indicated as two small green circles), are concentrated along the diagonal of the image. In addition, our sample includes four stars lying within one degree from the center of IC\\,1805 (the Heart nebula, located in the top left corner of the figure). The top panel in Fig.~\\ref{figure.fig2} depicts the histogram of SpT of the sample, which shows that the majority of stars are B~Sgs.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig2a.png}\n\\includegraphics[width=.47\\textwidth]{figures\/Fig2b.png}\n\\caption{Histograms by SpT separated with colors (top) and number of spectra separated by SpT and stacked (bottom).} \n\\label{figure.fig2}\n\\end{figure}\n\nTo assemble this sample of stars, we considered several bibliographic sources, including the works by \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, \\citetads{2010ApJS..186..191C}, and \\citetads{2014ApJ...788...58G}. In a first step, we used the Topcat\\footnote{\\href{Topcat}{http:\/\/www.star.bris.ac.uk\/~mbt\/topcat\/}} Virtual Observatory tool to cross-match all the stars that are quoted in these four papers and fulfilled the criteria indicated above and the list of targets with spectra available in the IACOB spectroscopic database. In a second step, we tried to obtain new spectra of as many of the missing stars as possible using the NOT or Mercator telescopes (see Sect.~\\ref{subsection.22_specobs}).\n\nFrom the original lists of luminous stars in Galactic OB associations quoted in \\citetads{1978ApJS...38..309H} and \\citetads{1992A&AS...94..211G}, we found 207 targets that are located within 4.5\\,deg of the center of Per~OB1. Only 109 of these fulfill our luminosity class criteria; the rest are either dwarfs, (sub)giants, or do not have a defined luminosity class. Our sample includes 82 of these, but we miss spectra for another 12 (7 B and 5 M Sgs).\n\nWe also used the list of targets quoted in the extensive study of the stellar population of $h$~and~$\\chi$~Persei by \\citetads{2010ApJS..186..191C} to find suitable candidates. From the complete list of several ten thousand stars, only 23 were found to have luminosity classes I or II. We currently have spectra for 17 of them. Of the remaining 6 (all of them B~Sgs), one was identified previously when we cross-matched our observations with the list of targets in \\citetads{1978ApJS...38..309H} and \\citetads{1992A&AS...94..211G}. This means that we lack spectra for another 5 blue Sgs at the time of writing.\n\nLast, our sample includes all the red Sgs of those listed in \\citetads{2014ApJ...788...58G}. In summary, the sample of stars we discuss here comprises all the blue and red Sgs (except for 12 B and five M Sgs, listed at the end of Table~\\ref{table.A1} for future reference) that are quoted in the abovementioned papers and are located within 4.5\\,deg around the center of Per~OB1. Further notes on the actual completeness of our sample can be found in Sect.~\\ref{subsection.43_compl}.\n\n\n\\subsection{Spectroscopic observations}\n\\label{subsection.22_specobs}\n\nThe spectroscopic observations of the stars in the sample come from different observing runs performed between November 2010 and December 2019 using either the FIES (NOT) or the HERMES (Mercator) instruments.\n\nThe first observations, comprising an initial sample of B, A, and M Sgs in Per~OB1 selected from \\citetads{1978ApJS...38..309H}, were obtained in 2010 during an observing run of three nights with Mercator (PI. M.A. Urbaneja). The O stars in the sample were targeted by the IACOB project (P.I. S. Sim\\'on-D\\'iaz) as part of a more general objective of observing all O stars in the Northern Hemisphere up to $V_{mag}$ = 9. These observations, obtained with both HERMES and FIES, include a minimum of three epochs per target (see more details in \\citeads{2018A&A...613A..65H}; \\citeads{Holgado2019}; \\citeads{2020arXiv200505446H}). We also benefit from the multi-epoch observations available for a subsample of O and B~Sgs as gathered by the IACOB project as part of a subproject aimed at investigating line-profile variability phenomena in the OBA Sg domain and its relation with pulsational-type phenomena (see, e.g., \\citeads{2010ApJ...720L.174S}; \\citeads{2017A&A...597A..22S}; \\citeads{2018A&A...612A..40S}; \\citeads{2017A&A...602A..32A}; \\citeads{2018MNRAS.476.1234A}). The time span of these observations covers several years. We also count on multi-epoch observations of red Sgs obtained during several of our observing runs with HERMES. Last, all these observations have more recently been complemented by FIES spectroscopy obtained as part of the time granted to A. de Burgos in 2018 by the Spanish time-allocation committee, and through internal service observations performed in 2019 and 2020 by A. de Burgos. In addition, we were able to obtain a new epoch for a large fraction of stars in the sample during an observing run with Mercator in December 2019. \n\nFIES is a cross-dispersed high-resolution \\'echelle spectrograph mounted at the 2.56~m Nordic Optical Telescope (NOT), located at the Observatorio del Roque de los Muchachos on La Palma, Canary Islands, Spain. The observations made with FIES were taken with different fibers\/resolutions from R\\,$\\sim$\\,25000 to R\\,$\\sim$\\,67000, and with a wavelength coverage of 370-830\\,nm. \n\nHERMES is a fibre-fed prism cross-dispersed \\'echelle spectrograph mounted at the 1.2~m Mercator Telescope, also located at the Observatorio del Roque de los Muchachos. It provides a spectral resolution of R\\,$\\sim$\\,85000 and wavelength coverage of 377-900 nm, similar to FIES. \n\nThe FIES and HERMES spectrographs provide good mechanical and thermal stability that allows for a good precision in RV measurements. For FIES, the RV y accuracy\\footnote{http:\/\/www.not.iac.es\/instruments\/fies\/fies-commD.html} for the high-resolution fiber has been proved to be 5-10\\,m\/s, regardless of the atmospheric conditions. For the medium-resolution fiber under poor conditions, the precision reaches 150\\,m\/s. In the case of HERMES, the precision obtained for the low- and high-resolution fibers is 2.5 and 2\\,m\/s, respectively (\\citeads{2011A&A...526A..69R}). In both cases this precision is well above the precision required for this work, as we expect variations of several \\kms\\ for the blue Gs\/Sgs, and a few \\kms\\ for the red Sgs. \n\nAll the spectra were reduced using the FIESTool (\\citeads{2017ascl.soft08009S}) and HermesDRS\\footnote{http:\/\/www.mercator.iac.es\/instruments\/hermes\/drs\/} dedicated pipelines. Both pipelines provide merged wavelength-calibrated spectra. In addition, we used our own programs, implemented in IDL, to normalize the spectra and compute the heliocentric velocity to be applied to each spectrum before the associated RV was measured (see Sect.~\\ref{section.3_rv}).\n\nAs indicated above, we have multi-epoch spectroscopy for a large fraction of the stars in our sample. The bottom panel in Fig.~\\ref{figure.fig2} summarizes this characteristic of our observations, showing the histogram of the collected number of spectra per star. In addition, Table~\\ref{table.A2} quotes all those stars for which we have five or more spectra. This table includes the time span covered by the spectra, together with the total number of spectra for each of these stars, separated by SpT. It is important to remark that the cadence of the spectra taken for each star is very inhomogeneous, as they were gathered during different observing runs, as described at the beginning of this section. \n\n\n\\subsection{Photometric and astrometric data}\n\\label{subsection.23_gaiaobs}\n\nFor all the stars in the sample, Table~\\ref{table.A1} quotes the {\\em Gaia} $G_{mag}$ and $BP_{mag}$\\,-\\,$RP_{mag}$, parallaxes ($\\varpi$) and proper motions ($\\mu_{\\alpha}$,$\\mu_{\\delta}$), as well as associated errors, retrieved from {\\em Gaia} DR2. Sources in the {\\em Gaia} catalog were identified using Topcat, defining a radius threshold of 2~arcsec. \n\nWe adopted a parallax zero-point offset of $-0.03$\\,mas (see \\citeads{2018A&A...616A...2L}), which is already applied to all values quoted in Table~\\ref{table.A1} and used to generate the various figures in the paper. We note, however, that some other authors push this value up to $-0.08$\\,mas (see \\citeads{2018ApJ...862...61S}; \\citeads{2019MNRAS.486L..10D}). \n\nThe {\\em Gaia} DR2 renormalized unit weight error (RUWE) is also included in the last column of Table~\\ref{table.A1}. The value of this quantity is used to estimate the goodness of the {\\em Gaia} astrometric solution for each individual target. Following recommendations by the {\\em Gaia} team for the known issues\\footnote{https:\/\/www.cosmos.esa.int\/web\/gaia\/dr2-known-issues}, we decided to adopt a RUWE\\,=\\,1.4 to distinguish between good and bad solutions.\n\nSeven stars (or 8$\\%$ of the sample) have an associated RUWE higher than this value. Their parallaxes and proper motions are indicated in parentheses in Table~\\ref{table.A1}. Hereafter, we call them stars with \"unreliable astrometry\" or \"unreliable astrometric solution\". For all the stars with a RUWE $<$ 1.4, the top panel of Fig.~\\ref{figure.fig3} shows the $G_{mag}$ against the {\\em Gaia} error in parallax, and the bottom panel shows the {\\em Gaia} error in total proper motion against the {\\em Gaia} error in parallax. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig3.png}\n\\caption{(Top) $G_{mag}$ against the {\\em Gaia} error in parallax. (Bottom) {\\em Gaia} error in total proper motion against the error in parallax. Both panels include all stars in our sample except for the seven targets with {\\em Gaia} RUWE > 1.4 (see Sect.~\\ref{subsection.23_gaiaobs}).}\n\\label{figure.fig3}\n\\end{figure}\n\nThe $G_{mag}$ of the stars in our working sample ranges between 5.1 and 9.7\\,mag. It has been shown that bright sources ($G_{mag}$ < 6) also result in unreliable astrometric solutions because of uncalibrated CCD saturation (\\citeads{2018A&A...616A...2L}). In the sample, four stars have magnitudes lower than 6, and they are discussed in detail in Sect.~\\ref{subsection.51_memb}. In order to verify the {\\em Gaia} DR2 parallaxes and proper motions for the brightest stars in the sample, we also retrieved the values provided in the {Hipparcos} (\\citeads{2007A&A...474..653V}), and TGAS (\\citeads{2015A&A...574A.115M}) catalogs; however, the results were not better. \n\nThe {\\em Gaia} errors in parallax range between 0.032, and 0.121\\,mas, while the errors in total proper motion range between 0.046, and 0.308\\,mas\/yr. Six stars have uncertainties in parallax $\\sim$0.08\\,mas or larger. The same have uncertainties in total proper motion larger than 0.18\\,mas\/yr. They are all red Sgs except for HD\\,14489 (the A~Sg in the upper right corner). The explanation for their large errors lies in the combined effect of large size and variability for the red Sgs, and the high brightness for HD\\,14489 ($G_{mag}$ = 5.1). In both cases, the {\\em Gaia} astrometric solution is affected (see Sect.~\\ref{subsection.51_memb}). \nOf the red Sgs, HD\\,14528 (in the upper right corner) has the largest errors and also a relatively high RUWE value (1.25), followed by HD\\,14489, which in comparison has a RUWE = 0.81. In particular, for HD\\,14528, we adopt the results from \\citetads{2010ApJ...721..267A} from this point on, who used the very long baseline interferometry (VLBI) technique to derive the astrometric parameters.\n\nFor the stars with RUWE $>$ 1.4, errors in parallax range between 0.086, and 0.384\\,mas with a mean of 0.170\\,mas, and errors in total proper motion range between 0.171, and 0.501\\,mas\/yr, with a mean of 0.347\\,mas\/yr. We note that as expected, all these stars have larger errors than those associated with the main concentration of stars in the bottom panel of Fig.~\\ref{figure.fig3}. \n\n\n\n\\section{Radial velocity measurements}\n\\label{section.3_rv}\n\nWe first generated various suitable lists of spectral lines, optimized for the different SpT, using information available in the Atomic Line List interface\\footnote{https:\/\/www.pa.uky.edu\/~peter\/newpage\/} \\citep{2018Galax...6...63V}, and the SpectroWeb\\footnote{\\href{SpectroWeb}{http:\/\/spectra.freeshell.org\/whyspectroweb.html}} database. Each line list comprises a few to several dozen strong (log(gf) > --0.5), unblended lines covering the full 390\\,--\\,650~nm spectral window (or 510\\,--\\,870~nm in the case of the red Sgs). \n\nFor early and mid O-type stars, a few lines of N~{\\sc iii-v} and O~{\\sc iii} were used. In addition, we also included some He~{\\sc i} lines to compensate for the lower number of available metal lines. For the late O-type stars, we added some lines of Si~{\\sc iv} and O~{\\sc ii}. The situation improves for the B and A Sgs, were a much larger sample of lines is available, including lines from Si~{\\sc ii-iv}, N~{\\sc ii-iii}, O~{\\sc ii-iii}, S~{\\sc ii-iii}, C~{\\sc ii}, Mg~{\\sc ii}, and Fe~{\\sc ii}. Last, in the case of red Sgs, we mostly used lines from Mg~{\\sc i}, Ti~{\\sc i}, Fe~{\\sc i}, Ca~{\\sc i}, Cr~{\\sc i}, Ni~{\\sc i}, and V~{\\sc i}.\n\nWe then used our own tool (developed in Python\\,3.6) to perform a RV analysis. For each star, the corresponding list of lines was selected based on its SpT. For each line, an iterative normalization of the surrounding local continuum was made. Then, each line was fit to either a Gaussian or a Gaussian plus a rotational profile, depending on the first estimate of the full width at half maximum (FWHM) of the line. The measured central wavelength was then used to calculate the RV of each individual line in the initial line list (see above). From all the identified lines we removed those with equivalent widths lower than 25\\,{m\\AA} directly before we carried out an iterative sigma clipping (using a threshold of 2\\,$\\sigma$) to remove potential poorly fit lines or incorrect identifications. The RVs of the surviving lines were then averaged, and we calculated the standard deviation of the final RV. This process was repeated for each spectrum and for each star in the sample. \n\nThe measurements of the individual RVs, together with the number of lines used for each spectrum, are listed in Table~\\ref{table.rvstable}. For O-type stars, the average number of lines is 12, the final average number of lines after sigma clipping is 6, and the typical uncertainties associated with the dispersion of RV measurements obtained after sigma clipping is $\\sim$3.9\\,\\kms. For the B-type stars, these values are 37 and 22 lines and $\\sim$0.9\\,\\kms , respectively. For A\/F-type stars, they are 42 and 32 lines, and $\\sim$0.26\\,\\kms. Finally, for the K\/M-type stars, they are 31 and 24 lines, and $\\sim$0.17\\,\\kms. This error is larger for the O-type stars for two main reasons: the first is that fewer lines are available, and the second reason is related to the broadening of the diagnostic lines, which is much larger for the O-type stars than in the cooler B, A, and red Sgs.\n\nThe RV results for the spectra with the best S\/N are shown in the last column of Table~\\ref{table.A1}. For each star, we also searched for double-lined spectroscopic binaries (SB2) by looking at different key diagnostic lines (e.g., \\ioni{He}{i}~$\\lambda$5875, \\ioni{Si}{iii}~$\\lambda$4552, \\ioni{O}{iii}~$\\lambda$5592, \\ioni{C}{ii}~$\\lambda$4267, \\ioni{and Mg}{ii}~$\\lambda$4481). \n\nWe were able to measure individual RVs for the two components in three of the five SB2. We used the spectrum of maximum separation between them. Their values are listed in Table~\\ref{table.A1}.\n\nFor each star with four or more spectra, an average RV was calculated as the mean of the RVs obtained for each individual spectrum. In addition to the associated standard deviation, the peak-to-peak amplitude of variability in RV (RV$_{\\rm PP}$) was calculated as the difference between the highest and lowest individual RVs, and its error was calculated as the square root of the sum of the squares of the their individual uncertainties. The results for the stars for which multi-epoch spectroscopy is available are listed in Table~\\ref{table.A2}. \n\nLast, we also visually inspected the line-profile variability in each star with available multi-epoch spectroscopy. By doing this we were able to identify those cases in which any detected variability is more likely due to stellar oscillations than to (single-line) spectroscopic binarity (see Sect.~\\ref{subsubsection.442_multi}) \n\n\n\n\n\\section{Results}\n\\label{section.4_results}\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=.72\\textwidth]{figures\/Fig4.png}\n\\caption{{\\em Top pannel}: Sky map of all the stars in the sample overplotted over a DSS-red image of the region. Dashed yellow line indicates the galactic plane, and the central green cross marks the center of Per~OB1 (as defined in \\citetads{2017MNRAS.472.3887M}). Green circles indicated the location of the $h$~and~$\\chi$ Persei double cluster. Colored vectors indicate the individual proper motion of each star. {\\em Middle and bottom panels}: Parallax and RV of the spectrum with the highest S\/N, respectively, for each star in the sample against their position in right ascension. Open circles and square symbols indicate stars that deviate more than 2$\\sigma$ from the mean of the distribution of parallaxes and RVs, respectively (see Sects.~\\ref{subsection.41_plx&pm} and \\ref{subsubsection.441_bestsnr}). Stars with bad astrometry (see Sect.~\\ref{subsection.23_gaiaobs}) are indicated with a plus, and no proper motion vectors are overplotted.}\n\\label{figure.fig4}\n\\end{figure*}\n\nFig.~\\ref{figure.fig4} summarizes all the compiled information on astrometry and RVs (except for the information we extracted from the multi-epoch spectroscopy, which is presented in Fig.~\\ref{figure.fig9}). The top panel of the figure shows the position of the stars in the sky, and the corresponding proper motions are indicated with arrows. For reference, we also indicate the location of the $h$~and~$\\chi$~Persei double cluster (green circles at the center of the image) and the Galactic plane (dashed yellow line).\n\nThis image is complemented with another two panels, in which the distribution of parallaxes and RVs (as derived from the best S\/N spectrum of each star) is plotted against the right ascension (middle and bottom panels, respectively). These two panels allow us to better identify the location in the sky of the outliers of both distributions, and to easily connect the information of the three investigated quantities.\n\nFrom a first visual inspection of this summary figure, it becomes clear that generally speaking, the stars in our sample (including those located in the $h$~and~$\\chi$~double cluster) belong to a connected population in terms of proper motions, parallaxes, and RVs. In addition, there is a non-negligible number of outliers that we discuss in detail in the next sections. They are potential nonmembers of the Per~OB1 association, and\/or runaway stars and binary systems.\n\n\n\\subsection{Parallaxes and proper motions}\n\\label{subsection.41_plx&pm}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig5.png}\n\\caption{Total proper motions against parallax for the sample of stars except for those labeled \"unreliable astrometry\" (see Sect.~\\ref{subsection.23_gaiaobs}). The 2$\\sigma$ boundaries of the distribution are shown as a rectangle. Empty colored circles show outliers of the distribution of any of the two quantities, and the associated uncertainties are overplotted. The mean and standard deviation obtained from the stars within the 2$\\sigma$ box are shown in the top right corner.}\n\\label{figure.fig5}\n\\end{figure}\n\nFigure~\\ref{figure.fig5} shows again the results for proper motions and parallaxes ($\\varpi$) from a different perspective. The central panel of the figure depicts the combined distribution of these two quantities, this time using the modulus of the proper motion ($\\mu$), defined as the square root of the sum of the squares of the proper motion in right ascension and declination. Stars labeled \"unreliable astrometric solution\" (see Sect.~\\ref{subsection.23_gaiaobs}) are excluded from this figure. \n\nMost of the stars are grouped together around $\\varpi$\\,$\\approx$\\,0.4~mas and $\\mu$\\,$\\approx$\\,1.2~mas\\,yr$^{-1}$. This is also shown in the left and bottom panels of Fig.~\\ref{figure.fig5}, where histograms of both parallax and total proper motion are shown. \n\nAn iterative 2$\\sigma$ clipping of these distributions results in $\\varpi$ = 0.40 $\\pm$ 0.07~mas, and $\\mu$ = 1.22 $\\pm$ 0.26~mas\\,yr$^{-1}$, and the identification of a total of 18 outliers (i.e., deviating more than 2$\\sigma$ from the mean of the distribution). The 2$\\sigma$ boundaries of the distribution (0.265 < $\\varpi$ < 0.540~mas, and 0.706 < $\\mu$ < 1.740~mas\\,yr$^{-1}$, respectively) and the outliers are highlighted in Fig.~\\ref{figure.fig5}. The latter are also indicated in the second and third columns of Table~\\ref{table.A4} and are discussed in Sect.~\\ref{subsection.51_memb}.\n\nThese results assume that no different local substructures exist in the region, especially in terms of parallax. To investigate this statement further, we show again in Fig.~\\ref{figure.fig6} an image of the region with the proper motions overplotted, but this time using the mean proper motion obtained by considering the 16 stars located within 15\\,arcmin from the center of $h$~and~$\\chi$~Persei, respectively, and having reliable astrometry (see the black arrow in the bottom right corner of the figure, corresponding to $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.47$ and $\\mu_{\\delta}$ = $-0.99$\\,mas\\,yr$^{-1}$). \n\nThis figure is complemented with the information provided in Table~\\ref{table.radiplxpm}, where we summarize the resulting means and standard deviations of parallaxes and proper motions when the sample is divided into circular regions around the center of $h$~and~$\\chi$~Persei. The first region only includes the double cluster. The other regions extend outward by one degree each, starting at a distance of 30 arcmin from the center of the double cluster.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig6.png}\n\\caption{Same as the top panel of Fig.~\\ref{figure.fig4}, but this time, the individual proper motion of each star is referred to the mean proper motion of 16 stars in $h$~and~$\\chi$~Persei with good astrometric solution (black arrow in the bottom right corner of the top panel).}\n\\label{figure.fig6}\n\\end{figure}\n\n\n\\begin{table}\n  \\centering\n  \\caption[]{Mean and standard deviation of parallaxes and proper motions for different groups of stars located at increased distance from the center of $h$~and~$\\chi$~Persei. Proper motions are referred to the mean of the proper motions of stars within 15 arcmin of each of the clusters and with good astrometric solution.} \n    \\label{table.radiplxpm}\n      \\begin{tabular}{cccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nRadius [deg]  & N$_{\\rm stars}$  & $\\varpi$ [mas] &$\\mu$ [mas\/yr] \\\\  %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$h$~and~$\\chi$ Persei & 16 & 0.43 $\\pm$ 0.06 & 0.31 $\\pm$ 0.13 \\\\\n0.5 < R < 1.5  &19  &0.39 $\\pm$ 0.07  & 0.43 $\\pm$ 0.22 \\\\ %\n1.5 < R < 2.5  & 9  &0.34 $\\pm$ 0.04  & 0.54 $\\pm$ 0.33 \\\\ %\n2.5 < R < 3.5  & 8  &0.40 $\\pm$ 0.07  & 0.60 $\\pm$ 0.23 \\\\ %\n3.5 < R < 4.5  &14  &0.40 $\\pm$ 0.07  & 0.74 $\\pm$ 0.29 \\\\\n\\hline\n      \\end{tabular}\n\\end{table}\n\nBased on the results presented in this section, we conclude the following: There is some empirical evidence of the existence local substructures in the spacial distribution of proper motions (see further discussion in Sect.~\\ref{subsubsection.521_pm}). These subgroups of stars have a compatible distribution of parallaxes and proper motions. As a result, this justifies the decision to use the whole sample of stars to obtain the mean values and standard deviations of these two quantities to characterize this population of stars, as well as to identify potential outliers in parallax (i.e., nonmembers) and proper motion (i.e., runaway stars).\n\n\n\\subsection{Comparison with previous works}\n\\label{subsection.compa}\n\n\n\\subsubsection{Distance}\n\\label{subsection.compa_plx}\n\nWe have obtained an average value for the parallax of $\\varpi$ = 0.398 $\\pm$ 0.066\\,mas (adopting a zero offset of $-0.03$~mas). This value represents the mean of all stars in the sample with good astrometric solution that are not outliers in parallax and total proper motion.\n\nBased on the corrected computed distances to these stars from \\citetads{2018AJ....156...58B}, we obtain an average of $d$ = 2566 $\\pm$ 432\\,pc. This is compatible with the distance obtained using the inverse of our derived parallax: $d$ = 2510 $\\pm$ 415\\,pc. When we assume this distance, the projected distance extends up to $\\sim$180\\,pc for the furthest stars in the association. In particular for the stars in the double cluster used in Table~\\ref{table.radiplxpm}, we obtain a distance of $d$ = 2340 $\\pm$ 328\\,pc using the inverse of the derived parallax. \n\nThis result agrees well with previous estimates for $h$~and~$\\chi$~Persei using different approaches. In addition, it also indicates that the parallax zero-point offset correction proposed by \\citetads{2018A&A...616A...2L} is adequate.\n\nTo give some examples, \\citetads{2002PASP..114..233U} obtained an average double cluster distance of $d$ = 2014 $\\pm$ 46\\,pc using the ZAMS fitting approach. \\citetads{2010ApJS..186..191C} used main-sequence stars with a very large sample and obtained a distance to each cluster of $d_{h}$ = 2290$^{+87}_{-82}$\\,pc and $d_{\\chi}$ = 2344$^{+88}_{-85}$\\,pc. The previously mentioned work by \\citetads{2010ApJ...721..267A} estimated a distance to HD\\,14528 of $d_{h}$ = 2420$^{+110}_{-90}$\\,pc using high-precision interferometric observations. More recently, \\citetads{2018A&A...616A..10G} published mean parallaxes for a broad selection of open clusters using {\\em Gaia} DR2 including $h$~and~$\\chi$~Persei. By applying a --0.03\\,mas zero-point offset, they obtained $d_{h}$ = 2239\\,pc and $d_{\\chi}$ = 2357$^{+88}_{-85}$\\,pc. Finally, \\citetads{2019MNRAS.486L..10D} estimated the distance to $h$ Persei in $d_{h}$ = 2250$^{+160}_{-140}$\\,pc, adopting an offset of $-0.05$\\,mas for the {\\em Gaia} parallaxes. \n\nThe aim of this work is not to provide a better estimate, but to ensure that the stars selected here based on their parallax belong to the association. Only a few works provide distances to the Per~OB1 association. For instance, \\citetads{2019ApJ...882..180S} used the photometric distance and {\\em Gaia} parallaxes for a selection of O-type stars to derive a distance to the association of $d$ = 2.99\\,$\\pm$\\,0.14\\,kpc and $d$ = 2.47\\,$\\pm$\\,0.57\\,kpc, respectively.\n\n\n\\subsubsection{Proper motions}\n\\label{subsection.compa_proper}\n\nFor the stars that are not outliers in proper motion and parallax, we obtain mean values and standard deviation for the individual components of the proper motion of $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.51$ $\\pm$ 0.48\\,mas\/yr, $\\mu_{\\delta}$ = $-1.00$ $\\pm$ 0.31\\,mas\/yr. This result agrees quite well with previous results obtained in the literature by other authors and different samples of stars. For example, \\citetads{2019A&A...624A..34Z} investigated a sample of more than 2100 stars (covering a much wider range in mass than our study) located within 7.5 degrees around the $h$~and~$\\chi$~Persei double cluster. They found for each cluster $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.71$ $\\pm$ 0.18\\,mas\/yr and $\\mu_{\\delta}$ = $-1.12$ $\\pm$ 0.17\\,mas\/yr, respectively. Similar results were also obtained by \\citetads{2017MNRAS.472.3887M} and \\citetads{2019ApJ...876...65L}.\n\n\n\\subsection{Completeness of the sample}\n\\label{subsection.43_compl}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig7.png}\n\\caption{Color-magnitude diagram (using {\\em Gaia} photometry) of stars located within 4.5~degrees from the center of the Per~OB1 association. Colored stars shows the stars in our sample, gray circles represent the remaining stars from {\\em Gaia}, green diamonds show 17 blue and red supergiants quoted in the literature for which we lack spectra (see the last part of Table~\\ref{table.A1}). Two isochrones and a reddening vector are also included for reference purposes. See Sect.~\\ref{subsection.43_compl} for explanation.}\n\\label{figure.fig7}\n\\end{figure}\n\nAs indicated in Sect.~\\ref{subsection.21_sample}, our sample of 88 stars includes almost all blue and red Sgs (LC I and II) quoted in \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, \\citetads{2010ApJS..186..191C}, and \\citetads{2014ApJ...788...58G}, plus a few LC III objects (Gs) with late-O and early-B spectral types. In particular, from a total of 107 targets quoted in these four papers that meet our selection criteria, we only lack spectra for 12 B and 5 M Sgs.\n\nTo further evaluate the completeness of our sample, we benefit from photometry provided by {\\em Gaia} and the results about parallaxes and proper motions described in Sect.~\\ref{subsection.41_plx&pm}. To this aim, we retrieved all the stars in the {\\em Gaia} DR2 catalog with $G_{mag}$ brighter than 10.5 whose parallaxes and total proper motions lie within 2$\\sigma$ of the distributions depicted in Fig.~\\ref{figure.fig5}. We then removed all stars with RUWE larger than 1.4, and those classified by the SIMBAD Astronomical Database\\footnote{\\href{SIMBAD}{http:\/\/simbad.u-strasbg.fr\/simbad\/}} as dwarfs or subgiants (luminosity classes V and IV). \n\nThe results are presented in a color-magnitude diagram (CMD) in Fig.~\\ref{figure.fig7}, where we use the same color-code as in previous figures for the stars in our sample, but this time, we also highlight the 17 stars that are classified as LC III or II-III stars in light green. \nFor reference purposes, we also include a $A_{\\rm v}$ = 1.7\\,mag reddening vector and two reddened 14\\,Myr isochrones\\footnote{Downloaded from the {\\em Mesa Iscochrones and Stellar Tracks} interface, MIST (\\citeads{2016ApJS..222....8D,2016ApJ...823..102C}).} (solid lines) shifted to a distance of 2.5\\,kpc (or, equivalently, a distance modulus of 12\\,mag.). The values of reddening for the isochrones (A$_{\\rm v}$ = 1.0 and 2.7, respectively) were selected to embrace the main-sequence band, corresponding to the region of the CMD with higher density of gray points in the bottom left corner. \n\nFrom inspection of this figure we can conclude that the level of completeness in our sample is very high, specially when we concentrate on the region of the CMD where the blue and red Sgs are located (purple, dark blue, cyan, and red stars). Interestingly, we also find that a high percentage of the 12 B~Sgs quoted in \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, and \\citetads{2010ApJS..186..191C} are likely B~Gs, instead of B~Sgs. These refer to all green diamonds with $G_{mag}$ < 9, most of them classified as B Sgs in \\citetads{2010ApJS..186..191C} (see the last rows of Table~\\ref{table.A1}).\n\nIn addition, Fig.~\\ref{figure.fig7} allows us to conclude that the blue and red Sg population of Per~OB1 is affected by a variable reddening that ranges from $A_{\\rm v}$~$\\sim$~1.0 to 2.7\\,mag (in agreement with previous findings by \\citeads{2019ApJ...876...65L}), and that the age associated with the blue and red Sg population is not compatible because the higher mass present in the 14~Myr isochrone is $\\sim$14\\,$M_{\\odot}$, while all O, B, and A Gs\/Sgs included in our sample are expected to have masses higher than 20\\,$M_{\\odot}$. This latter result will be further investigated in the next paper of this series, after information about the stellar parameters of the full working sample is included.\n\n\n\\subsection{Radial velocities}\n\\label{subsection.44_rv}\n\nBy following the strategy described in Sect.~\\ref{section.3_rv}, we obtained RV estimates for all the available spectra in our sample of stars. These measurements are used (1) to investigate the RV distributions resulting from the best S\/N spectra, (2) to provide empirical constraints on intrinsic spectroscopic variability typically associated with the various types of stars, and (3) to identify spectroscopic binaries and runaway candidates.\n\n\n\\subsubsection{Best S\/N spectra}\n\\label{subsubsection.441_bestsnr}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.4\\textwidth]{figures\/Fig8.png}\n\\caption{RV distributions associated with the different SpT groups resulting from the analysis of the best S\/N spectra. The orange bin in the second panel from the bottom is HD\\,12842, the F Sg.}\n\\label{figure.fig8}\n\\end{figure}\n\nThe bottom panel of Fig.~\\ref{figure.fig4} shows the RVs of all stars in the sample, obtained from the best S\/N spectra, as a function of the position of the stars in right ascension. The associated distributions, this time separated by SpTs, are depicted in the form of histograms in Fig.~\\ref{figure.fig8}, with the mean and standard deviation associated with each RV distribution (after performing an iterative 2$\\sigma$ clipping) indicated at the top of the various panels. The corresponding outliers in each distribution are indicated as open squares in the bottom panel of Fig.~\\ref{figure.fig4} and listed in the fourth column of Table~\\ref{table.A4}.\n\nFrom a visual inspection of Fig.~\\ref{figure.fig8} we can conclude that except for the case of O-type stars, which has a flatter and more scattered distribution, the other three distributions are quite similar (when the outliers are eliminated), following a more or less clear Gaussian shape. (For the A\/F-type stars, only the two situated on the right-most side of Fig.~\\ref{figure.fig8} are outliers. The consequence of having fewer stars than for the B- and K\/M-type stars results in a poorer Gaussian shape.) The mean values of these three distributions are compatible within the uncertainties, with a difference smaller than 2~\\kms. Interestingly, the standard deviation of the distributions significantly drops from O- to B- and A-type stars, and continues to decrease to the K\/M-type stars (see further notes in Sects.~\\ref{subsubsection.442_multi} and \\ref{subsubsection.443_binaries}).\n\n\n\\subsubsection{Multi-epoch spectra: intrinsic variability}\n\\label{subsubsection.442_multi}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig9.png}\n\\caption{Measured RVs (subtracted from their mean) for a sample of 15 stars (ordered by SpT) for which we have five or more spectra, and whose detected variability in RV is more likely produced by intrinsic variability than by the orbital motion in a binary system.}\n\\label{figure.fig9}\n\\end{figure}\n\nAs indicated in the bottom panel of Fig.~\\ref{figure.fig2}, we have more than one spectrum for 73 of the stars in the sample. These observations can be used to identify binaries; however, as extensively discussed in \\cite{SimonDiaz2020}, in prep.), the effect of intrinsic variability also needs to be taken into account to minimize the spurious detection of single-line spectroscopic binaries (SB1) in the blue supergiant domain (see also further notes regarding the red supergiant domain in \\citeads{2019A&A...624A.129P,2020A&A...635A..29P}).\n\nSome examples of the type of spectroscopic variability phenomena produced by stellar oscillations or the effect of a variable stellar wind in the OBA Sg domain can be found in \\citetads{1996ApJS..103..475F, 2004A&A...418..727P, 2006A&A...457..987P, 2006A&A...447..325K, 2015A&A...581A..75K, 2017A&A...597A..22S,2018A&A...612A..40S, 2017A&A...602A..32A, 2018MNRAS.476.1234A}, for example. This effect is also illustrated in Fig.~\\ref{figure.fig9} using a subsample of 15 stars in PerOB1 for which five or more spectra are available, and whose detected variability in RV is more likely produced by intrinsic variability than by the orbital motion in a binary system (see Table~\\ref{table.A2} and further notes in Sect.~\\ref{subsubsection.443_binaries}). \n\nThese results warn us about the dangers of using a single snapshot observation to associate the outliers detected in the RV distributions shown in Fig.~\\ref{figure.fig8} with potential runaway stars or spectroscopic binaries. Some of these cases might even correspond to a single measurement in a specific phase of the intrinsic variability of the star instead of being associated with the orbital motion in a binary system or with a single star with an anomalous RV due to an ejection event. They also partially explain why the standard deviation of the RV distributions presented in Fig.~\\ref{figure.fig8} becomes smaller when moving from the blue to the red Sgs. This is just a consequence of the behavior of the characteristic amplitude of spectroscopic variability with SpT (see Table~\\ref{table.variab} and \\cite{SimonDiaz2020}, in prep.). Last, it also affects the fraction of detected SB1 stars using multi-epoch observations, or the final sample of outliers in RV (see further notes in Sects.~\\ref{subsubsection.443_binaries} and \\ref{subsubsection.523_rw&bin}, respectively).\n\nTo evaluate the effect that including information about multi-epoch spectroscopy has on the identification of outliers in the RV distribution, we have repeated the same exercise as in the case of the single-snapshot observations (Sect.~\\ref{subsubsection.441_bestsnr}), but modifying the individual measurements (obtained from the analysis of the best S\/N spectra) of stars for which four or more spectra are available by the mean of the multi-epoch RV measurements. Results of this exercise are presented in the \"RV multi\" column of Table~\\ref{table.A4}. Although the number of stars with a modified outlier status in RV is small in this specific example (only HD\\,13402 and HD\\,12953), the results presented in Fig.~\\ref{figure.fig9} indicate that it could have been larger if other epochs of the time series had been selected as single-snapshot observations.\n\n\\begin{table}\n  \\centering\n  \\caption[]{Summary of detected variability (mean and maximum of peak-to-peak amplitude of RV in each SpT group) for the sample of 15 stars depicted in Fig.~\\ref{figure.fig9}. RVs in \\kms.} \n    \\label{table.variab}\n      \\begin{tabular}{lcccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT group & N$_{\\rm stars}$ & $\\overline{\\rm N}_{\\rm spectra}$ & $\\overline{RV}_{\\rm PP}$ & RV$_{\\rm PP, max}$ \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO-type   &  2 &  6 & 21.3 $\\pm$ 0.4  & 21.7 \\\\ \nB-type   &  9 & 13 &  8.8 $\\pm$ 3.5  & 16.1 \\\\  %\nA\/F-type &  2 & 22 &  7.9 $\\pm$ 2.5  & 10.4 \\\\  %\nK\/M-type &  2 & 16 &  1.6 $\\pm$ 0.4  &  2.0 \\\\  %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table}\n\n\n\\subsubsection{Multi-epoch spectra: spectroscopic binaries}\n\\label{subsubsection.443_binaries}\n\nGiven the stability of the FIES and HERMES instruments, and the accuracy reached in the RV measurements for most of the stars in the sample with multi-epoch spectroscopy, it might be tempted to assign the SB1 status to all stars showing a RV$_{\\rm PP}$ above a few \\kms. However, as indicated in Sect.~\\ref{subsubsection.442_multi}, the intrinsic variability in single blue supergiants can reach amplitudes of a few dozen \\kms; hence, many of these identification may lead to spurious results.   \n\nTo avoid this situation as much as possible, and in order to identify the most secure candidates to be SB1, we performed a careful inspection of the type of line-profile variability detected in each of the stars with more than one spectrum. To this aim, we mainly considered the following diagnostic lines, whenever available: \\ioni{He}{i}~$\\lambda$5875, \\ioni{Si}{iii}~$\\lambda$4552, \\ioni{O}{iii}~$\\lambda$5592, \\ioni{C}{ii}~$\\lambda$4267, \\ioni{and Mg}{ii}~$\\lambda$4481. For the case of the two red supergiants with multi-epoch observations, we found that the measured RV$_{\\rm PP}$ is lower than 2\\,\\kms, which we directly attribute to intrinsic variability.\n\nThe list of clearly identified SB1 is presented in Tables~\\ref{table.A3}. In addition to the four SB1 stars quoted there, we found five SB2 systems (some of them directly detected from a single-snapshot observation) and labeled \"LPV\/SB1?\" another five cases in which we are not entirely sure if the detected variability is due to binarity or intrinsic variability. All this information is also added to \\ref{table.A4} (column \"Spec. variability\"). \n\nWe also performed a bibliographic search for previously identified binaries in our sample of blue and red supergiants. We mainly concentrated in the works by \\citetads{2018A&A...613A..65H, 2020arXiv200505446H} and \\citetads{2019A&A...626A..20M} for the case of O-type stars, and \\citetads{1973ApJ...184..167A,1985AbaOB..58..313Z}, and \\citetads{2017A&A...598A.108L} for the B supergiant sample. In addition, we made use of \\textit{The International Variable Star Index (VSI)}\\footnote{\\href{VSI}{https:\/\/www.aavso.org\/vsx\/index.php}}. \n\nIn total, we found that six out of our sample of ten detected SB1 or SB2 systems from this work were previously identified in any of these references as such (HD\\,16429 is actually a triple system \\citepads{2003ApJ...595.1124M}). This implies four newly detected binaries: HD\\,13969, HD\\,14476, HD\\,17378 (all SB1), and HD\\,13402 (SB2).\nWe also found three binaries in the literature that were not detected from our available spectra because of short time-coverage: BD\\,+56578, an eclipsing binary \\citepads{2016AstL...42..674T, 2017A&A...598A.108L}, plus HD\\,17603 and HD\\,14956, identified as SB1 by \\citetads{2018A&A...613A..65H} and \\citetads{1973ApJ...184..167A}, respectively. All of them are labeled \"(lit.)\" in the corresponding column of Table~\\ref{table.A4}.\n\nThe stars classified as \"LPV\/SB1?\" are HD\\,13036, HD\\,13854, HD\\,13267, HD\\,14542, HD\\,12953, and HD\\,17378.\nHD\\,13036 (B0.2~III), HD\\,13267 (B6~Iab) and HD\\,14542 (B8~Iab) have RV$_{\\rm PP}$ = 10 -- 14\\,\\kms; although this value is at the boundary of the expected variability due to pulsations, which may indicate an SB1 classification, we cannot conclude after visual inspection of their line-profile variability. HD\\,13854 (B1~Ia-Iab), mostly looks like a pulsational variable, but we do not discard the possibility entirely that  this star might be a SB1. We note that \\citetads{1973ApJ...184..167A} provide RV$_{\\rm PP}$ = 24.8\\,\\kms; however, they did not consider it as a binary. For HD\\,12953 (A1~Iae), we measured RV$_{\\rm PP}$ = 10.4\\,\\kms, the largest variability in the A supergiant sample; however, after visual inspection of its line profile variability, we cannot conclude whether this is a SB1 system. \\citetads{1973ApJ...184..167A} found RV$_{\\rm PP}$ = 15.8\\,\\kms for this star, which would favor that it is an SB1. HD\\,17378 (A6~Ia) has RV$_{\\rm PP}$ = 8\\,\\kms, which is large enough to consider it as potential binary. However, we only have three spectra. \n\nLast, we found that although HD\\,14956 (B2~Ia) was classified as an SB1 with a period of $P$ = 175\\,days, and ${RV}_{\\rm PP}$ = 27.0\\,\\kms \\citepads{1973ApJ...184..167A} , and \\citetads{2017A&A...598A.108L} classified this star as $\\alpha$\\,Cygni variable, we do not see such signs of SB1 variations, as we measure RV$_{\\rm PP}$ = 5.5\\,\\kms. However, we do not have enough spectra (three) to discard this possibility.\n\nThese results about detected spectroscopic binaries, along with the RV distributions obtained from the analysis of the best S\/N spectra (i.e., obtained from a single-snapshot observation), allow us to evaluate the extent to which these distributions can be used to identify spectroscopic binaries among the outliers. We find that only four out of all the SB1\/SB2 systems detected by means of multi-epoch spectroscopy are outliers in the abovementioned distributions. In addition, some outliers have not been detected as spectroscopic binaries although more than four spectra are available for them (e.g., HD\\,13268, O8.5~IIIn, RV = $-106.2$\\,\\kms). These results can be explained when we take into account (1) that the best S\/N of some of the spectroscopic binaries correspond to an orbital phase in which the RV is close to the systemic velocity, and (2) that some outliers in RV might be runaways and not necessarily binaries. The latter situation is the case of HD\\,13268, a well-known runaway star (see also Sect.~\\ref{subsection.51_memb}). This means that if a given star is an outlier in RV, it is useful to first investigate its runaway nature (by means of its proper motion) before marking it as a potential spectroscopic binary, and vice versa; for example, although the measured RV of the B1~Ib-II star HD\\,14052 ($-90.8$\\,\\kms) deviates more than 3$\\sigma$ from the mean, this star is not an outlier in proper motion, and so we may conclude that it is more likely a spectroscopic binary than a runaway. This is confirmed through access to multi-epoch spectroscopy. We further discuss the percentage of spectroscopic binaries in our sample of stars in Sect.~\\ref{subsubsection.523_rw&bin}.\n\n\n\n\n\\section{Discussion}\n\\label{section.5_discus}\n\nTable~\\ref{table.A4} compiles and summarizes some information of interest for the discussion about membership and final identification of spectroscopic binaries and runaway stars. Columns \"$\\varpi$\" and \"$\\mu$\" indicate if a given star is part of the bulk distribution of parallaxes and proper motions, respectively, or if it is detected as an outlier of these distributions (Sect.~\\ref{subsection.41_plx&pm}). Columns \"RV best\" to \"RV final\" provide similar information for the case of RV estimates obtained from the best S\/N spectra (Sect.~\\ref{subsubsection.441_bestsnr}) for stars with four or more spectra (Sect.~\\ref{subsubsection.442_multi}), or the final distribution of RVs (Sect.~\\ref{subsubsection.521_pm}), respectively. In all these cases, different symbols are used to identify secure or doubtful cases.\n\nFor completeness, we also add to Table~\\ref{table.A4} information about confirmed spectroscopic binaries, our final decision on cluster membership status (columns \"Spec. variability\" and \"Member\"), as well as some other comments of interest for the final interpretation of results (column \"Comments\").\n\n\n\\subsection{Cluster membership}\n\\label{subsection.51_memb}\n\nAs discussed in Sect.~\\ref{subsection.41_plx&pm}, most of the stars in the sample with good astrometry (81 stars) are grouped together in the proper motion versus parallax diagram. The mean and standard deviation of the distribution of these two quantities are $\\varpi$ = 0.40 $\\pm$ 0.07\\,mas, and $\\mu$ = 1.22 $\\pm$ 0.26\\,mas\\,yr$^{-1}$ , respectively. All the stars that are located within the 2$\\sigma$ boundaries of the distribution (64 in total) are directly considered as members and labeled with filled circles in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}. The remaining 17 stars are marked with an open circle or a cross in Table~\\ref{table.A4} depending on whether they deviate by 2\\,--\\,3$\\sigma$ or more than 3$\\sigma$, respectively. We note that in this case, columns \"$\\varpi$\" and \"$\\mu$\" include information about the remaining seven stars that were not included in Fig.~\\ref{figure.fig5}: those labeled \"unreliable astrometry\" (or RUWE > 1.4). Because the information about parallaxes and proper motions is uncertain for them, we exclude these stars for the moment and mark them using brackets surrounding the corresponding symbols in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}.\n\nThese are not the only stars with unreliable parallaxes. Figs.~\\ref{figure.fig3} and \\ref{figure.fig5} include a small sample of 6 K\/M-type supergiants that despite a RUWE value well below 1.4 have larger errors than the rest of stars in the sample, and interestingly, all of them are systematically shifted to larger parallaxes (although except for one, all have total proper motions within the 2$\\sigma$ boundaries and proper motion vectors compatible with the bulk of member stars, Fig.~\\ref{figure.fig4}). They are also all marked with brackets in Table~\\ref{table.A4}.\n\nThis is likely connected to an already known problem that affects the reliability of the {\\em Gaia} DR2 astrometric solution. In brief, as pointed out by \\citetads{2011A&A...532A..13P} and \\citetads{2018A&A...617L...1C}, the position of the centroid changes on timescales of several months or a few years because of the large size and strong intrinsic photocentric variability of red supergiants. This effect leads to unreliable parallaxes and errors. \n\nA particular example of interest regarding this issue with the astrometric solution of {\\em Gaia} for the case of red supergiants is the the highly variable star HD~14528 (S~Per, $\\varpi_{Gaia}$\\,=\\,0.25\\,$\\pm$ 0.12\\,mas, $\\mu_{total\\,Gaia}$ = 2.57 $\\pm$ 0.31\\,mas\/yr). This star has an average angular size of 6.6\\,mas (\\citeads{2012A&A...546A..16R}). It was monitored for six years by \\citetads{2010ApJ...721..267A} with VLBI. The authors obtained an independent parallax of 0.413$\\pm$0.017~mas, which is just at the center of the distribution. We therefore cannot discard completely  that these six K\/M-type supergiants, which are outliers in parallax using data from {\\em Gaia} DR2, are members\nof Per~OB1.\n\nThe last star that we place in brackets is the A-type supergiant HD\\,14489. This is the brightest stars in our sample, with $G_{mag}$ = 5.1. As shown in Fig.~\\ref{figure.fig3}, this star also has much larger errors in parallax and proper motions than the bulk of stars in the sample. This may be related to the current limitation of {\\em Gaia} DR2 regarding the reliability of the astrometric solutions for stars brighter than $G_{mag} \\lesssim$ 6 \\citepads{2018A&A...616A...2L}. Another three stars share this issue, but their associated astrometric errors are much smaller and their magnitudes are close to $G_{mag}$ = 6; therefore we decided to consider their astrometric solutions reliable.\n\nTaking all this information into account, we decided to following strategy below to evaluate the membership to Per~OB1 of each star in our sample. Stars with reliable values on parallax and proper motion (i.e., not marked with parentheses in columns 2 and 3 of Table~\\ref{table.A4}) are considered as {\\em \\textup{confirmed members}} if they do not deviate more than 2$\\sigma$ from the mean of the distribution of parallaxes. For stars with unreliable values of parallax and proper motion (i.e., highlighted with brackets in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}), we adopted the following: if they are not outliers in parallax, they are considered {\\em \\textup{likely members}}; if they are outliers in parallax, we consider them {\\em \\textup{candidate members}}, except for the K\/M-type stars, which remain likely members because of the arguments provided above. Last, stars with reliable astrometry that are outliers in parallax (as well as those stars in IC\\,1805, see below) are considered {\\em \\textup{nonmembers}}.\n\nMost of the stars are properly classified using these criteria. However, a few cases deserve further attention.\n\\paragraph{HD~13022 (O9.7~III) and HD~12842 (F3~Ib):} These two stars are classified as members following the guidelines above, but they are outliers in proper motion (Fig~\\ref{figure.fig5}). Interestingly, they have a very small proper motion compared to the rest of the stars in the sample (see in Fig.~\\ref{figure.fig4} the two stars with very small vectors located at (RA, DEC)\\,$\\sim$\\,(32, 58.5)\\,deg). Awaiting a more detailed study of these two stars, we continue considering them members for the moment.\n\\paragraph{HD~16691 (O4~If), HD~15642 (O9.5~II-IIIn), HD~13745 (O9.7~II(n)), and HD\\,13268 (ON8.5~IIIn):} These four O-type stars are clear outliers in proper motion (see Figs.~\\ref{figure.fig5} and \\ref{figure.fig4}). We consider the first three runaway members  because their parallaxes lie within the 2$\\sigma$ boundaries. The fourth (HD~13268) is an interesting case; although this star has a somewhat larger parallax, it has a RV of $\\sim$ 105\\,\\kms. Therefore, given its spectral classification and this high RV pointing to us, it can still be considered a runaway member of Per~OB1. This star is a well-known fast-rotating nitrogen-rich O-type runaway star (e.g., \\citeads{1972AJ.....77..138A}; \\citeads{1989A&AS...81..237M}; \\citeads{2014A&A...562A.135S}; \\citeads{2015A&A...578A.109M}; \\citeads{2017A&A...603A..56C,2017A&A...604A.123C})\n\\paragraph{HD~14322 (B8~Iab):} This star is an outlier in parallax with a value of $\\varpi$ = 0.21$\\pm$0.04\\,mas. Although the TGAS catalog provides a value for it of $\\varpi$ = 0.44$\\pm$0.38\\,mas (within the boundaries of $\\varpi$), the error is much larger. This inconsistency caused us to modify its status from nonmember to member candidate while awaiting {\\em Gaia} DR3.\n\n\\paragraph{HD~14489 (A1~Ia):} This is a bright A-type star ($G_{mag}$ = 5.1), outlier in parallax, and with the largest parallax error. Although it has a RUWE = 0.81, we do not trust its {\\em Gaia} astrometry, as explained before, because of its brightness. The result from TGAS provides a parallax of $\\varpi$ = 0.45 $\\pm $0.94\\,mas, and although it is within the adopted boundaries of Per~OB1, the error is very large. This star is also an outlier in RV and close to the 2$\\sigma$ boundary in proper motion. Therefore we decide to label it a runaway member candidate.\n\n\\paragraph{BD+56724 (M4-M5~Ia-Iab):} This star has the largest parallax in Fig.~\\ref{figure.fig5}, and a RUWE = 0.93. Although the reliability of {\\em Gaia} DR2 parallaxes for the K\/M-type stars may be low, its large deviation from the mean of the distribution could mean that this star is not a member. It is also an outlier in proper motion, but its magnitude and RV are similar to other red supergiants in the sample. We therefore retain this star as member candidate for the moment.\n\n\\paragraph{HD\\,15570 (O4~If), HD15558 (O4.5~III(f)), HD16429 (O9~II(n)), and BD~+60493 (B0.5~Ia):} All these stars are located within or in the surroundings of IC\\,1805. Interestingly, all of them but one are O-type stars. Although they are located within the 2$\\sigma$ boundaries of the parallax and proper motion distribution (except for HD~16429, but this is a triple system with a RUWE = 8.8), we decided to mark them nonmembers based on their separated location in the sky and their direct connection with the surrounding H~{\\sc ii} region. They seem to be linked to a younger star-forming region located at higher galactic latitudes (but at the same distance). Most of them are also outliers in RV (see Table~\\ref{table.A4}), but this is likely due to their binary nature. \n\nThe final result of this classification, also taking into account the comments on some individual stars presented above, is summarized in column \"Member\" of Table~\\ref{table.A4}. In total, we have 70 confirmed members, 9 likely members, 5 member candidates, and 4 nonmembers. Interestingly, only stars in IC\\,1805 are finally classified nonmembers. The remaining 84 stars likely belong to the Per~OB1 association (although some of them are identified as runways, see Sect.~\\ref{subsubsection.523_rw&bin}).\n\n\n\\subsection{Kinematics.}\n\\label{subsection.52_kinem}\n\nIn Sect.~\\ref{subsection.41_plx&pm} and \\ref{subsection.44_rv} we provided a global overview of the results about proper motions and RVs for the complete sample of stars, also including some information about identified spectroscopic binaries. In this section we discuss these results more in detail. We also refer to  \\citetads{2017MNRAS.472.3887M, 2019A&A...624A..34Z, 2020MNRAS.493.2339M} for complementary (and in some cases more detailed) information about the global and internal kinematical properties of stars in the Per~OB1 association.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=.95\\textwidth]{figures\/Fig10.png}\n\\caption{Radial distance from the center of Per~OB1 against the mean RV for the stars with more than one spectrum, or RV for the stars with only one spectrum, excluding the SB2 binaries. The stars within 1\\,degree from IC\\,1805 were also excluded. The filled gray rectangle shows 2$\\sigma$ of all the RVs, excluding those from stars identified as binaries (SB1 or SB2). The filled colored circles denote stars identified as SB1. The filled colored crosses denote starts outside 2$\\sigma$ from the mean that are not identified as binaries. The colored stars show the remaining stars. The colored error bars present RV$_{\\rm PP}$ for stars with multi-epoch data, except for SB1. The open triangles show stars that are outliers in proper motion and were therefore identified as runaways.} \n\\label{figure.fig10}\n\\end{figure*}\n\n\n\\subsubsection{Proper motions}\n\\label{subsubsection.521_pm}\n\nFigure~\\ref{figure.fig5} and the top panel of Fig.~\\ref{figure.fig4} provide a global overview of the distribution of proper motions in the whole star sample. These figures show that (except for a few outliers) most of the stars in our sample that are located below the Galactic plane (among them, those in $h$~and~$\\chi$ Persei) can be considered a dynamically connected population of stars. This result perfectly agrees with previous findings by \\citetads{2008ApJ...679.1352L}. Using proper motions from the {\\em Hipparcos} mission, these authors showed that the luminous members of the Per~OB1 association exhibit a bulk motion away from the Galactic plane, such that their average velocity increases with height above the Galactic plane.\n\nFurthermore, inspection of the results for proper motions (relative to h\\,and\\,$\\chi$~Persei) and parallaxes presented in Fig.~\\ref{figure.fig6} and Table~\\ref{table.radiplxpm} allows us to conclude that the distributions of parallaxes associated with stars located at increasing distances from the center of the double cluster are all compatible (at least we find no clear subgrouping in terms of parallax at least given the accuracy of {\\em Gaia} DR2 astrometry -- except maybe the stars in $h$~and~$\\chi$~Persei because their parallax is  somewhat larger or their distance is somewhat closer). We also conclude that the mean and standard deviation of the distribution of proper motions in $h$~and~$\\chi$~Persei is much smaller than in the more extended population. \n\nMoreover, the spatial distribution of proper motions in the extended population of blue and red supergiants in Per~OB1 does not follow an expanding structure centered in the $h$~and~$\\chi$~Persei double cluster. Instead, the local proper motions of most of the stars located north of these clusters seem to point outwards from an imagined center located at about 1 degree north of the double cluster (see also \\citeads{2018ARep...62..998M}, \\citeads{2020MNRAS.493.2339M}). These results from the proper motion, linked with the results by \\citetads{2008ApJ...679.1352L} mentioned above, are compatible with a scenario in which the halo population of blue and red supergiants around the double cluster has been formed from a more diffuse region of interstellar material compared to the denser region associated with the clusters themselves. \n\nIn addition, four O-type stars south of the region can be clearly considered runaways based on the size and direction of their proper motions (see also Fig.~\\ref{figure.fig5}). Interestingly, their proper motion vectors do not point outward from $h$~and~$\\chi$~Persei, but to a far more extended region of the Galactic plane (see also the discussion in Sect.~\\ref{subsubsection.523_rw&bin}). Finally, as also indicated in Sect.~\\ref{subsection.51_memb}, the stars located within or near IC~1805 likely belong to a younger population of stars that is not necessarily connected with the remaining stars in Per~OB1.\n\n\n\\subsubsection{Radial velocities}\n\\label{subsubsection.522_rv}\n\nFigure~\\ref{figure.fig8} and the bottom panel of Fig.~\\ref{figure.fig4} summarize the RV results obtained with the best S\/N spectra. The analysis of these spectra has allowed us to characterize the RV distributions for the different SpT groups and to identify potential spectroscopic binary systems and runaway stars among the outliers of the distributions (see columns \"RV best\" and \"RV multi\" in Table~\\ref{table.A4}). We then illustrated in Sect.~\\ref{subsubsection.442_multi} (see also Fig.~\\ref{figure.fig9}) the importance of incorporating information from the analysis of multi-epoch observations for the correct interpretation of the RV distribution, and in particular, to avoid the spurious identification of spectroscopic binaries (either from a single epoch or from multi-epoch observations) due to the effect on the measured RVs of the intrinsic variability caused by stellar oscillation and\/or wind variability in the blue supergiant domain.\n\nLast, we learned that after eliminating outliers associated with confirmed spectroscopic binaries (via multi-epoch spectroscopy) and runways (via proper motions), the RV distributions for the B, A\/F, and K\/M~Sgs are fairly compatible in terms of mean values and standard deviations. In addition, we found that most of the O-type stars in the sample are either (1) runaways, as detected from the proper motions, (2) spectroscopic binaries, or (3) are considered nonmembers because they are located nearby IC\\,1805, far away from the main distribution of stars in Per~OB1. As a result, the RV distribution of the O-type sample is remarkably broader than for those associated with the other SpT.\n\nWe now take all these results into account to provide final information about RVs in Fig.~\\ref{figure.fig10} and in the column \"RV final\" of Table~\\ref{table.A4}. To do this, we first replaced the list of measurements obtained from the best S\/N spectra by the mean value resulting from the analysis of the multi-epoch observations for those cases for which we have more than one spectrum. Then we used this list of values, except for all the SB2 binaries, and the stars identified as nonmembers (see Sect.~\\ref{subsection.51_memb}), to obtain the mean and standard deviation by performing an iterative 2$\\sigma$ clipping. \n\n\\begin{table*}\n  \\centering\n  \\caption[]{Summary of the number of outliers in proper motion and RV that are used for the final identification of runaway stars. In the case of the proper motion, we indicate cases that deviate by more than 2$\\sigma$ from the mean of the distribution for each individual component and the total proper motion. In the case of RV, we separate cases that deviate by more than 2$\\sigma$ and 4$\\sigma$, respectively. In parentheses, we indicate targets whose outlier characteristic is not entirely clear from the available data. The last column indicates the final number and percentage of clearly detected runaways for each SpT group.}\n    \\label{table.rw}\n      \\begin{tabular}{lccccccccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT & \\multicolumn{3}{c}{PM} & & \\multicolumn{2}{c}{RV} & & \\multicolumn{2}{c}{Runaways} \\\\ %\n\\cline{2-4} \\cline{6-7} \\cline{9-10}\n\\noalign{\\smallskip}\n      &  $\\mu_{\\alpha}\\cos{\\delta}$      & $\\mu_{\\delta}$   & $\\mu_{\\rm Total}$ &  &   $>$2$\\sigma$ &   $>$4$\\sigma$ &  & \\# & \\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO-type   &  3  &  5    & 5    & & 7~(+1) & 3 & & 5 & 45 \\\\\nB-type   &  1  &  2    & 2    & & 9~(+6) & 1 & & 2 &  5 \\\\ %\nA\/F-type &  1  &  1(+1)& 1(+1)& & 3 & 1 & & 1(+1) &  1(+1) \\\\ %\nK\/M-type & (1) & (1)   & (1)  & & 2 & 0 & & (1) &  (5) \\\\ %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table*}\n\nThe results of this process are presented in Fig.~\\ref{figure.fig10}, where the RVs of all stars that were not excluded from the list are presented against the radial distance from the center of Per~OB1. The obtained mean and standard deviation are shown in the top right corner ($-42.9$ $\\pm$ 3.5\\,\\kms) of the figure, and the horizontal gray band indicates the 2$\\sigma$ boundaries. \n\nAlthough most of the stars in the sample are concentrated within the central 100\\,pc, we observe that except for a few cases, the remaining stars also lie within the 2$\\sigma$ boundaries. Therefore, once more, and as was suggested by \\citetads{2010ApJS..186..191C} and \\citetads{2019A&A...624A..34Z}, the extended population of blue and red Sgs in Per~OB1 (up to 200~pc, i.e., relatively far away from $h$~and~$\\chi$~Persei) seems to have a common origin in terms of kinematics. No global gradient (as a function of distance to the center of the association) or local substructures are observed in the distribution of RVs. \n\nAs indicated above, we obtain $\\overline{RV}$ = $-42.9$ $\\pm$ 3.5\\,\\kms\\ using the whole sample of stars that are not excluded from the list. Regarding $h$~and~$\\chi$~Persei, we obtained average values of RV$_{\\rm \\chi\\,Per}$ = $-44.4$ $\\pm$ 1.4\\,\\kms and RV$_{\\rm h\\,Per}$ = $-41.1$ $\\pm$ 2.6\\,\\kms, respectively. These results agree well with those previously obtained by other authors. For the association as a whole, \\citetads{2017MNRAS.472.3887M} provided a mean value of $\\overline{RV}$ = $-43.2$ $\\pm$ 7.0\\,\\kms\\ using available information of member stars from the TGAS catalog. For the individual clusters, \\citetads{1991AJ....102.1103L} provided RV$_{\\rm \\chi\\,Per}$ = $-44.4$ $\\pm$ 0.7\\,\\kms and RV$_{\\rm h\\,Per}$ = $-46.8$ $\\pm$ 1.7\\,\\kms, respectively, using a sample of cluster stars (mainly early type, more specifically, B- and A-type stars). In the particular case of $h$\\,Per, the fact that we have only three suitable stars to compute the mean may explain the poorer agreement. \n\nAs in the case of the analysis of the best S\/N spectra and the multi-epoch observations, we provide in column \"RV final\" of Table~\\ref{table.A4} a list of identifiers to separate the outliers of the distribution of final values of RV from stars within the 2$\\sigma$ boundaries. This information is used in the next section to determine additional SB1 stars that have not previously been identified based on the available multi-epoch spectra.    \n\n\n\\subsubsection{Runaway and binary stars.}\n\\label{subsubsection.523_rw&bin}\n\n\n\\begin{table*}\n  \\centering\n  \\caption[]{Summary of the number of binary stars in the sample (see Table~\\ref{table.A4}). For columns \"SB1\" and \"SB2\", the percentage shows the fraction with respect to the total number for each SpT. We split the B-type stars into two groups to separate giants from supergiants. Column \"Lit.\" counts the number of binary stars found in the literature. Column \"SB1?\" counts the sum of the stars labeled \"LPV\/SB1?\" in column \"Spec. variability\" and \"SB1?\" in column \"Comments\" (we note that if a star is labeled both as \"LPV\/SB1?\" and \"SB1?\", we only count the first). The total number of stars are in column N$_{\\rm All}$. Column \"\\% bin\" gives the percentage of total and potential binary stars with respect to the total number of stars.} \n    \\label{table.bin}\n      \\begin{tabular}{cccccccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT & SB1 & SB2 & Lit. & SB1? & N$_{\\rm All}$ & \\% bin. \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO             & 0        & 2~(15\\%)   & 1 & 1 & 13 & 15 -- 30\\% \\\\\nB~I \\& II     & 3~(8\\%)  & 1~(3\\%)    & 1 & 5 & 37 & 10 -- 27\\% \\\\ %\nA\/F          & 0         & 0          & 0 & 2 & 11 & 0 -- 18\\% \\\\ %\nK\/M          & 0         & 0          & 0 & 2 & 18 &  0 -- 10\\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nB~III         & 2~(22\\%) & 2~(22\\%)   & 0 & 1 &  9 & 45 -- 55\\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table*}\n\n\nThe last two columns of Table~\\ref{table.rw} summarize the final number and percentage of identified runaways in each of the four SpT groups. We proceed as follows to obtain this. First, we assigned the runaway status to all outliers in proper motion (meaning that the magnitude of any of their individual components, or the total proper motion, deviates by more than 2$\\sigma$ from the mean of the corresponding distribution). Then we considered the possibility of identifying additional runaway stars through their RVs. In this case, we decided to only label them (if not detected as SB2) clear runaways if their RV deviates by more than 15~\\kms from the mean of the final RV distribution presented in Fig.~\\ref{figure.fig10}. We made this decision based on two arguments: the first refers to the result presented in the fifth column of Table~\\ref{table.rw}; namely, the number of identified runaways when all outliers in RV are considered that deviate by more than 2$\\sigma$ is too large when compared with those detected through PMs. The second argument is based on the results of RV$_{\\rm PP}$ that is expected to be produced by intrinsic variability, which can be as high as 10\\,--\\,20~\\kms\\ in some cases (see Table~\\ref{table.variab}).\n\nThese arguments are supported by the fact that it is very unlikely to find a runaway star that is an outlier in RV, but not in at least one of the components of the proper motion. In contrast, as described above, intrinsic variability can lead to single-snapshot RV measurement that can easily deviate by up to 10\\,--\\,20~\\kms (or, equivalently, about 4\\,--\\,5$\\sigma$ in this specific sample of stars). Furthermore, this situation can be even more dramatic for large-amplitude SB1 systems for which only a low number of spectra is available. It is therefore more likely that a star that is not an outlier in proper motion but is an outlier in RV is an SB1 than a runaway. Alternatively, if the deviation in RV is smaller than the typical intrinsic variability corresponding to the associated SpT, it might not even be a binary star.   \n\nA practical example of the latter situation is the BN2~II-III star BD\\,+56578, for which we only have three spectra that cover a very short time-span (one day). This star is not an outlier in proper motion, but is an outlier in RV (it deviates by 13$\\sigma$ from the mean of the RV distribution). Based on what we have described above, this star should be labeled as a potential binary, a suspicion that is confirmed from the literature (\\citeads{2017A&A...598A.108L}).\n\nFollowing these arguments, all stars in Fig.~\\ref{figure.fig10} whose RV measurements deviate by more than 2$\\sigma$ (i.e., which lie outside the gray band) and up to 10\\,--\\,20~\\kms and that have not previously been detected as runaways through proper motions (open triangles) or as spectroscopic binaries through multi-epoch spectroscopy (filled circles) are quite likely single pulsating stars.\n\nOverall, we identify a total of 11 runaway stars. The group of stars with a larger number of runaways (45\\%) are the O-type stars. This is followed by the B and K\/M Sgs, with 5\\% each. (We note, however, that the runaway status of the M Sg BD\\,+56724 can be a spurious result because it is based on the $Gaia$-DR2 proper motion, which may not be as reliable as for the other stars because of the problems regarding size and variability of the red Sgs.) Last, the lower percentage of runaways is found for the A Sgs, with only 1 or 2\\%, depending on whether we trust the $Gaia$-DR2 proper motion of the bright star HD14489, which also has a much larger parallax than the remaining stars in Per\\,OB1.\n\nIt thus becomes clear again that a high percentage of the O-type stars in the Per~OB1 region can be considered a dynamically distinct group. However, in contrast to previous assumptions (see, e.g., \\citeads{2002AJ....124..507W}), the fact that all of them are found within the 2$\\sigma$ boundaries of the parallax distribution indicates that they belong to the same grouping as the remaining blue and red supergiants in Per~OB1, and not to a more distant, dispersed association. Although further confirmation is needed, the most likely origin of the O-star runaways is a dynamical kick by a supernova explosion in a previously bounded binary system. This hypothesis is reinforced by the fact that none of the detected runaways are identified as binary systems (and the other way round).  \n\nTable~\\ref{table.bin} summarizes the results for the detected binaries, again separated by SpT group, and this time differentiating the B Sgs from the B Gs because they represent the evolutionary descendants of main-sequence stars in two different mass domains. We refer to Sect.~\\ref{subsubsection.442_multi} for a description of how the SB1 and SB2 stars where identified. The targets labeled \"SB1?\" include targets fulfilling any of the two following criteria. On the one hand, stars with five or more spectra for which we cannot clearly decide whether the detected line-profile variability is due to intrinsic variability or orbital motion. On the other hand, following the arguments above, we identified stars as \"SB1?\" whose RV$_{\\rm PP}$ is larger than the typical intrinsic variability expected for their SpT (see Fig.~\\ref{figure.fig9}).\n\nThe main conclusions from inspection of the results presented in Table~\\ref{table.bin} (and Table~\\ref{table.rw}) are summarized as follows. First, the percentage of detected spectroscopic binaries decreases toward later SpT, or equivalently as the massive star evolution proceeds. This result agrees with recent findings by \\citetads{2017IAUS..329...89B}; \\citetads{2019A&A...624A.129P,2020A&A...635A..29P}; Sim\\'on-D\\'iaz et al. (2020, subm.). When we assume that the detected runaways indicate a past binary evolution, the total percentage of clear binaries (excluding those labeled \"SB1?\") would decrease from $\\sim$60\\% to $\\sim$15\\% when the O star and B~Sg samples are compared, and further below $\\sim$5\\% when the cooler Sgs are considered. Second, while the decreasing tendency remains in both cases, the exact behavior of the percentage of detected spectroscopic binaries is different depending on whether we also include the stars labeled \"SB1?\" stars. Therefore it is critical to confirm or dismiss our suspicion that most of the stars with RV$_{\\rm PP}$ below 10-15~\\kms\\ are actually single pulsating stars and not spectroscopic binaries. Access to multi-epoch data for the whole sample of star is therefore crucial to obtain reliable empirical information about the relative percentage of binaries throughout the massive star evolution. Finally, as an aside, the percentage of spectroscopic binaries is much higher among the B Gs than in the B Sgs.\n\nThis clearly shows that any further attempt to interpret the empirical properties of this sample of massive stars in an evolutionary context must take into account that a large fraction of the O stars is or likely has been part of a binary or multiple system. In addition, some of the other more evolved targets may also have been affected by binary evolution. \n\n\n\n\n\\section{Summary and future prospects.}\n\\label{section.6_summary}\n\nOur study has provided all the necessary environmental information that will be used in a forthcoming paper, in which we will also incorporate results obtained from a quantitative spectroscopic analysis of the whole sample (including stellar parameters and surface abundances) to perform a complete homogeneous characterization of the physical and evolutionary properties of the massive star population of the Per~OB1 association.\n\nIn this paper, we have studied a sample of 88 massive stars located within 4.5\\,deg from the center of the Per~OB1 association using high-resolution multi-epoch spectroscopy, and astrometric information from the {\\em Gaia} second data release (DR2). \n\nWe have investigated membership of all star in the sample to the Per\\,OB1 association, resulting in 70 members, 9 likely members, and another 5 candidates that require further investigation, while the other 4 were considered nonmembers as they belong to IC\\,1805.\n\nWe have found eight clear and two likely runaway stars, most of them O-type stars. We also identified 5 SB1 and five SB2 stars (these include three and one new binary systems, respectively), plus another 11 potential SB1 stars that we propose are single pulsating stars. \n\nTo obtain these results, we took their parallaxes and proper motions (as compiled from {\\em Gaia} DR2) into account, and the RV estimates obtained from the available multi-epoch and\/or single snapshot spectra. In addition, we also considered the reliability of the astrometry provided by {\\em Gaia} through the RUWE value, the potential decrease in reliability of {\\em Gaia} astrometry in the case of the red Sgs because of their large size and photocentric variability, and the expected amplitude of spectroscopic variability produced by stellar pulsations and\/or wind variability when spectroscopic binaries are identified based on their RV measurements. \n\nWe have also analyzed some global properties of the sample and obtained averages in parallax, total proper motion, and RV of $\\varpi$ = 0.40 $\\pm$ 0.07\\,mas, $\\mu$ = 1.22 $\\pm$ 0.26\\,mas\\,yr$^{-1}$ ($\\mu_{\\alpha}\\cos{\\delta}$ = -0.50 $\\pm$ 0.48, $\\mu_{\\delta}$ = -0.99 $\\pm$ 0.31), and $-42.9$ $\\pm$ 3.5\\,\\kms. All these results agree relatively well with previous studies based on different stellar samples comprising the Per~OB1 association (some of them focused on the h~and~$\\chi$ Persei clusters). \n\nGenerally speaking, no important differences are detected in the distribution of parallaxes, proper motions, and RVs when stars in h~and~$\\chi$ Persei or the full sample are considered, which suggests a very extended dynamically interrelated population. However, a few clear outliers in the proper motion and RV distributions are also found. A large fraction of these are O-type stars (almost 50\\%). The further analysis of their proper motions and RVs indicates that they are runaway stars, probably resulting from the kick of a supernova explosion in a previously bounded binary system.\n\nFinally, we have found that the percentage of secure binaries decreases from the hotter to the cooler Sgs. In particular, this percentage decreases from 15\\% to 10\\% when the O star and B~Sg samples are compared (or alternatively, from 60\\% to 15\\% when we consider the runaway stars as previous binaries), and it practically vanishes in the A\/F and K\/M Sgs. Further investigation of the potential connection between this result and merging processes that occur during the evolution of massive stars is an interesting direction of future work.\n\n\n\n\\begin{acknowledgements}\n\nBased on observations made with the Nordic Optical Telescope, operated by NOTSA, and the Mercator Telescope, operated by the Flemish Community, both at the Observatorio de El Roque de los Muchachos (La Palma, Spain) of the Instituto de Astrof\\'isica de Canarias. We acknowledge funding from the Spanish Government Ministerio de Ciencia e Innovaci\u00f3n through grants PGC-2018-091\\,3741-B-C211\/C22, SEV 2015-0548, and CEX2019-000920-S and from the Canarian Agency for Research, Innovation and Information Society (ACIISI), of the Canary Islands Government, and the European  Regional Development Fund (ERDF), under grant with reference ProID2017010115. This research made use of the SIMBAD, operated at Centre de Donn\\'ees astronomiques de Strasbourg, France, and NASA's Astrophysics Data System. The background images were taken from The STScI Digitized Sky Survey (\\href{http:\/\/archive.stsci.edu\/dss\/copyright.html}{Copyright link)}.\n\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nInternet of Things (IoT) is an omnipresent technological field with a focus on ease of use as well as a high degree of device autonomy. However, it presents a host of new issues regarding data security as well as substantively increasing the attack surface of the average end user's device ecosystem. The purpose of these IoT devices is to incorporate `smart' automation into the everyday life of the end user, but just as the function of these devices is automatic, the access control mechanisms are as well. As IoT devices become more prevalent in normal user environments, as well as integrated into our societal infrastructure, this automatic data sharing becomes a potential weak-point in the chain of data security. \n\nSecuring the data handled by smart autonomous devices becomes more important as the ubiquity of connected devices grows. If one home in a neighborhood has an IoT connected thermostat, there is not much incentive for adversaries to develop technologies to exploit potential underlying weaknesses. However, if every other house contains a plethora of smart devices, each of which is continuously gathering and transmitting data, there is a much higher potential gain from compromising these devices\\cite{205156}. Reality is becoming more reflective of this hypothetical every day. The amount of smart devices worldwide increased by one billion from 2019 to 2020. The global IoT market is projected to nearly triple between 2020 to 2030, from 8.74 to 25.44 billion devices\\cite{holst_2021}. \n\nWith this substantial increase in amount of connected devices, the infection rate of these devices is similarly growing. In 2019, compromised IoT devices made up 16.17\\% of all infected devices connected to mobile networks, that number more than doubled to 32.72\\% in 2020\\cite{onestore}. \nThe type of data being secured is also changing as these technologies are adopted across different fields, and therefore so are the consequences of data being compromised. Manufacturing environments become `smarter' every day as they integrate IoT devices to increase efficiency and decrease production time. In the US alone, manufacturing is a 2.3 trillion dollar industry which accounts for 11.39\\% of GDP\\cite{nam}. Interruption to these processes on a macro scale could very well lead to fiscal losses in the billions. Further, smart internet connected cars are becoming more widespread every year, with 51.1 million being sold in 2019 and a projected 76.3 million to be sold in 2023\\cite{wagner_2020}. This far-reaching growth leads to higher quality and convenient cars for consumers, however internet connected cars must have sufficient security protocols in place. The reliability and integrity of smart and autonomous car data is critical when user's lives depend on the vehicle functioning as designed. \n\nThe gravity of these security issues highlight the need for more secure frameworks and practices regarding the handling of data generated by IoT devices and connected ecosystem. One way to accomplish this is the integration of \\textbf{digital twins} \\cite{digitaltwins} into device control and data acquisition. Digital twins, or device shadows\\footnote{While digital twin refers to the whole encapsulation of a physical device in software and device shadow refers to the JSON data structure holding a representation of device state, these terms are very similar and are used interchangeably in this paper.}, are the virtual counterpart to physical objects which introduce a layer of abstraction between higher level control of devices and device specific actuation and sensing methods. These shadows can be used to facilitate separation between the object and cloud services layer (detailed in the background section), as well as enabling separation of IoT data into subsets. Digital twins also lead to more consistent interaction between higher level layers and physical devices. Device state as well as current connection status can always be accessed by higher level layers due to the persistent nature of the twin. It should be noted that digital twins comprise only a portion of the overarching architecture. The separation of device communication into layers within an access control oriented (ACO) architecture (discussed in the related work section) is necessary and present. Authentication and subsequently authorization must happen between discrete layers to ensure the architecture as a whole is secure. \n\nIoT devices often have multiple state data such as user settings, manufacturer configuration, and operational status. Each of these state data needs to be accessed by different users or at differing frequencies. Usually, there is one to one mapping between physical devices and virtual objects, meaning, one can only associate a single virtual object to a single device and are required to store all sets of device state data in one shadow. As a limiting consequence, all users will have access to the entire shadow and can consequently read and update state data they should not have. Further, data tagging plays a large role in our access control mechanisms and serves as the basis for the separation of data in the architecture. Tags are attached to the data generated in the system and are the attribute on which data is separated. This allows for easy and computationally inexpensive grouping of like data and adds further classification for data type. In this way, tags are the metric on which data is separated, and the digital twins are the receptive containers for that data. Once data has been separated and distributed based on the tags it carries, access control is centered around granting access to individual shadows. The data present in a given shadow is directly related to the tags applied to that data, therefore granting access to individual subsets of data within shadows is a form of Tag Based Access Control (TBAC). \n\nIn this paper, we propose a novel approach of data security by using multiple shadows (digital twins) for one physical object, with the intent of separating data among different virtual objects based on tags assigned on the fly, which are then used to limit access to different data points by any authorized users\/applications. The proposed solution is deployed at the edge, supporting low latency and real time security mechanisms with minimal overhead, and is light-weight as discussed in the implementation section. The implementation described in this paper is built on version 5.0 of the MQTT\\footnote{https:\/\/mqtt.org\/} protocol, and therefore communication occurs within topics in a publisher-subscriber model. While the referenced implementation applies access control to these topics, any model must include similar access control techniques with regard to flow of information between layers. For proof-of-concept, we focus on the integration of digital twins and TBAC in two industry applications: smart vehicles and smart manufacturing. We will examine the mechanisms for secure data sharing between digital twins, the advantages of tagging data in a digital twin system, and the performance impacts of the proposed data separation scheme. \n The key contributions of this paper are as follows:\n\\begin{itemize}\n\\item Attachment of tags directly to device state information in order to reduce `distance' between access control mechanisms and device data itself.\n\\item Dynamic and on-the-fly subdivision of device state at the local edge according to attached tags.\n\\item Limiting data exposure to authorized entities via subdivision of data in a many-to-one relationship between digital twins and physical devices.\n\\item Implementation of the proposed architecture to reflect the plausibility and efficiency, together with\nbrief comparative discussion on performance metrics.\n\\end{itemize}\n\n\n\n\n\nThe remainder of the paper is structured as follows. Section \\ref{sec:background} discusses relevant background such as established access control oriented (ACO) architectures and IoT literature. Section \\ref{sec:need-for-edge} demonstrates the necessity of security at the local edge in abstract principle as well as in applications such as intelligent vehicles and smart manufacturing. Existing industry solutions and their limitations are also examined in this section. Section \\ref{sec:tbac-proposal} defines the proposed architecture and the mechanisms for the attachment of tags within the context of TBAC. Section \\ref{sec:evaluation} presents implementation and associated performance metrics. Finally, section \\ref{sec:conclusion} summarizes our work and looks ahead to future work in this field. \n\n\n\\section{Relevant Background}\n\\label{sec:background}\nThis section reviews primitive building blocks of cloud and edge assisted smart connected systems. In addition, we will also reflect on relevant literature which has offered some security solutions and approaches for IoT and CPS ecosystems. \n\n\\begin{figure*}[th!]\n  \\centering\n  \\includegraphics[width=.7\\textwidth]{Diagrams\/Architectures.png}\n  \\caption{Multi-layered Access Control Focused Architectures}\n  \\label{Architecture_Layers}\n\\end{figure*}\n\\subsection{Access Control Oriented Architectures}\nSeveral access control oriented (ACO) architectures have been proposed in the literature for IoT \\cite{7809752,gupta2018authorization,Alshehri2018AccessCM,8673782,weijia2018,weijia2021,userauthIoT,celik_tan_mcdaniel_2019,franch2020,yahyazadeh2019,yahyazadeh2020,gupta2020attribute,dgupta2020access} and cyber physical systems (CPS) such as smart cars \\cite{gupta2018authorization}\\cite{guptaABAC2019}, intelligent transportation \\cite{gupta2020secure} and smart manufacturing \\cite{9502070}, which focus on the separation of systems into layers as illustrated in Figure \\ref{Architecture_Layers} (a). As shown in\nFigure \\ref{Architecture_Layers} (b), ACO architecture (proposed by Alsehri and Sandhu\\cite{7809752}) has four layers - object, virtual object, cloud services and application \u2013 with users and administrators\ninteracting at object and application layers. In addition, communication can happen\nwithin a layer (shown as self loop in Figure \\ref{Architecture_Layers} (b) and the adjacent\nlayers above and below. It should be noted that the extended access control oriented (E-ACO) \\cite{gupta2018authorization} architecture shown in Figure \\ref{Architecture_Layers} (b) is an extension to the generic ACO architecture with some additional components as discussed in the following section. \n\nThe \\textit{object layer} is comprised of the physical devices which either sense or actuate the environment within which they reside. These devices can be individual or clustered into larger objects (shown in Figure \\ref{Architecture_Layers} (b)) which contain many sensors, actuators, etc. There are several examples of clustered objects such as smart cars, mobile phones, or production lines; all of which contain many smart devices connected to a network. The physical objects in the object layer communicate with their digital twins (aka virtual objects) in the virtual object layer. These devices can communicate with other devices using different communication technologies\nincluding Bluetooth, WiFi, Zigbee, LAN, LTE or 5G. Physical devices communicate with their cyber counterparts (virtual objects) using protocols like HTTP, MQTT, DDS or CoAP. Users can also\ndirectly access physical objects at this layer. In an extended access control oriented architecture (E-ACO) as shown in Figure \\ref{Architecture_Layers} (b), clustered objects (COs) are introduced, which are objects with multiple sensors, and allow for possible interaction between sensors in same CO or between different object's sensors. These COs, such as smart cars, also have applications associated with them which offer services to users, in this case drivers. For example, a rear vision system is an application in cars to get rear-view, which gets data from the rear camera (an object) to provide dashboard view to the driver. These applications\nin the object layer of E-ACO are add-on's to the object layer in ACO architectures.\n\nThe \\textit{virtual object} layer holds the digital twins for all of the physical objects in the system. Digital twins in this layer communicate directly with their associated physical objects, the other virtual objects (VOs) present, and the cloud layer. The VOs in this layer hold the last received state of the physical object they represent, as well as processing desired states for those objects. These desired states can be received from other VOs, or the cloud layer. There may also be many virtual objects associated with one physical object. Virtual objects can hold the entire data set generated by their physical object, or subsets of that data. The virtual object layer in E-ACO architecture can have one or many cyber entities (virtual object or digital twins) for both clustered and individual objects. These twins can be created in the cloud layer, or local edge layer to support real time communication. For example, when sensors s$_{1}$ and s$_{2}$  across different clustered objects\ncommunicate with each other, the sequence of communication via\nvirtual object layer should follow starting s1 to vs$_{1}$  (digital twin of\ns$_{1}$), vs$_{1}$  to vs$_{2}$  and vs$_{2}$  to physical sensor s$_{2}$ . \n\n The \\textit{cloud layer} is the location of long term storage of device state, as well as more complex processing of received device data. Computationally intensive operations can be performed at this layer, thereby easing the burden of devices themselves as well as the hardware at the edge. These operations could include, but are not limited to: image processing with the intent of facial or object recognition, machine learning in order to fine tune a system's efficiency, or data visualizations. This layer manages communication with the virtual object and application layers, and is responsible for propagating control signals entered by the user as well as generating control signals based on the aforementioned data processing. Communication between clouds can also take place within this layer to enable big-data analytics or the union of discrete but related implementations. Single or multiple cloud scenarios can exist to support\nfederation or trusted collaboration between them. Some important IoT cloud platforms include Amazon AWS\\footnote{https:\/\/aws.amazon.com\/}, Microsoft\nAzure\\footnote{https:\/\/azure.microsoft.com\/en-us\/} IoT Hub, and Google Cloud IoT Core\\footnote{https:\/\/cloud.google.com\/iot-core}. An important use for cloud layer in IoT\/CPS involves defining security policies for authorized\ncommunication among different objects. \n\nThe \\textit{application layer}, is responsible for both displaying system information to the user and for user input. This layer needs to communicate with the layer directly below it to pass on control signals and receive visualizations and system state information. Users and administrators can remotely send commands and instructions to smart devices residing within the bottom layer using these applications, but such\ninteraction must propagate through the other two ACO middleware layers (cloud services and virtual object)\n\nThe layered access control oriented (ACO) structure discussed was proposed by Alsehri and Sandhu\\cite{7809752} with a focus on clarifying the middleware layers' function and form in IoT architectures. The distinction between the virtual object layer and the cloud layer lends itself to integration of heterogeneous objects into the system, as well as giving a well defined framework for access control techniques. This work also supports computation at the edge, as opposed to the cloud layer, by delineating the differences therein. Edge computation is necessary in industry with a focus on low latency that necessitate fast response times i.e. autonomous cars, or dynamic agricultural monitoring systems such as drones.\n\n\\subsection{Related Work}\n\nRecent extensive analysis of IoT technologies into the field of agriculture has been published by Gupta et al \\cite{gupta2020security}. The authors found that computation at the edge is a requirement for many systems with a focus on real time analysis and dynamic behavior. However, the assignment of responsibility at the edge comes with an increased attack surface due to the array of heterogeneous physical devices deployed\\cite{sina2020farming}. These devices are usually not designed with security as a chief concern\\cite{o2016insecurity}, and are a major security liability if configured incorrectly. The deployment of cryptographic security measures are difficult at the device level due to the computational constraints of most IoT enabled devices. While solutions do exist \\cite{dhanda_singh_jindal_2020}, they are relatively novel and have not yet found widespread implementation. They propose a lightweight multi-factor authentication protocol in the form of an independent Certificate Authority (CA). This allows for dynamic authentication and meets the complexity needs at the device level. It is worth noting however, that this solution does not detail practices to limit what data is being shared, only how to grant authorization. \n\nAnother area smart connectivity can greatly improve performance and efficiency is manufacturing. Kusiak\\cite{kusiak2018smart} makes the case that due to the trend of ever-increasing integration of smart sensors into manufacturing environments, the utilization of that data will drive further integration of smart actuators and data analysis into manufacturing processes. The employment of this novel data will lead to more accurate and complex modelling, optimization, and simulation. These models will give insight into potential fine-tuning practices to increase manufacturing efficiency, and the analysis of equipment monitoring will lead to predictive maintenance and prevention of equipment failure\\cite{s20195480}. This comes at the cost of increased cyber-security and safety concerns. As companies become reliant on modelling and IoT device infrastructure the value of these technologies goes up, therefore their security becomes paramount to continued profit and growth. In regard to safety, as automation and autonomous smart decision-making becomes integral to manufacturing centers, the responsibility of equipment to function correctly continuously shifts to lie upon the cyber-physical implementation.   \n\n\\section{Need for Edge Centric Secure Data Sharing}\n\\label{sec:need-for-edge}\n\nImplementations of IoT technologies at scale involve the generation of large quantities of data, which are used to affect system state by adjusting IoT actuators present in the system. The metrics for this state change are system specific but all systems require the sharing of data generated by local physical devices. This sharing can take place directly from virtual device to virtual device, virtual device to the local edge, or local edge to cloud. Which type of sharing takes place is determined by the level of computation necessary before the system state is affected. \n\nIn all aspects of device data sharing in a smart IoT connected system the local edge is critical and extensively utilized. These edge systems ensure\nlow latency and real time communication much needed in\nmost smart applications without bandwidth issues. In such scenarios, the edge plays a role in virtual device to virtual device sharing because all shadow clients in these systems reside on these local edge. Therefore even if the hardware of the edge is unneeded for computations more complex than device hardware can handle, the mechanisms of data sharing between virtual device clients still reside on, and are controlled by the edge which works as a middle man and relay the data. Virtual device to local edge sharing is required to facilitate computations exceeding physical device hardware, aggregation of device data in order to manage the system as a whole, or simply for comprehensive logging of system state. In the case of local edge to cloud data sharing, the local edge acts as a data pass-through in order to supply system information to cloud resources for computations that exceed local edge hardware capacity. These computations may include, but are not limited to, facial or object recognition, complex image processing, or machine learning algorithms. \n\nDue to the local edge's involvement in all data sharing which take place within an IoT system, the security of edge and the data it holds is of the utmost importance. The architecture proposed in this paper focuses on securing data in the system by managing the allocation of individual pieces of data into dynamic subsets based on tags. This is a form of TBAC with a focus on reducing the `distance' between tags assigned to data and the data itself. The implementations of TBAC currently present at the industry level utilize rules to tag data and independently apply tags to resources. This creates separation between the data and the tags applied to that data, as well as the containers that data will be placed within. We aim to improve this by directly applying tags to data and distributing data into digital twins based on those tags. Therefore each digital twin will have a set of tags defining what subset of data it will hold, and data will be distributed into each twin based on tags attached directly to that data. \n \n \\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=8cm, height=2.8in]{Diagrams\/External_Car_Communication.png}\n  \\caption{External Smart Car Communication}\n  \\label{Car_comm}\n\\end{figure}\n\\subsection{Motivating Use cases}\n\\subsubsection{Smart Cars and Intelligent Transportation}\n Smart vehicles require low-latency with high-volume data sharing. The internal network-connected sensors and actuators present in the car must be continuously sharing their data with the edge. This data is processed to allow functionality such as lane assistance systems, emergency collision avoidance, or full autonomous navigation. Externally, the car may be communicating with roadway infrastructure such as traffic lights, speed limit transmitters, or construction zone signalling shown in Figure \\ref{Car_comm}. Sharing data with other smart vehicles offers many benefits as well, in the form of automated lane merging protocols, increased speed limits due to increased reliability of surrounding vehicles, and shared awareness of roadway hazards. These factors culminate in smart cars prioritizing internal sensor-to-edge and external edge-to-edge sharing.\n\n While local and edge-to-edge sharing is prioritized, there is also utilization of the cloud layer in both logging data and implementation of more complex algorithms. User usage data such as location, driving habits, and maintenance history can be stored in the cloud for later retrieval. Performance data generated by the vehicle can also be sent to the cloud for processing by machine learning algorithms in order to monitor system health and send preemptive maintenance alerts. \n\n\\subsubsection{Smart Manufacturing}\n Smart manufacturing environments can take advantage of IoT technologies by distributing large quantities of internet connected smart sensors throughout the production pipeline. The local edge can be used to monitor system health by ensuring that sensor values fall within acceptable operating ranges. The cloud layer ensures system health by employing machine learning algorithms which monitor system efficiencies as reported by sensors in the system and give predictive points of failure. This architecture considers the necessity for low latency response times in the event of critical failure via utilization of the edge as a monitoring system, while also encouraging long-term health of the system via utilization of machine learning resources in the cloud. \n\n\\subsection{Threat Model}\n\\par The adversary threat model considered in this paper is heavily influenced by the security research put forward by the USDOT Intelligent Transportation Systems Office \\footnote{https:\/\/www.its.dot.gov\/factsheets\/cybersecurity.htm}. We have chosen to consider this research in developing our threat model because the environment it studies, smart transportation, is one of the most dynamic and difficult to secure. It is also the most industry applicable environments for IoT requiring edge based solution, as described earlier. The threats and vulnerabilities we address in the proposed solution include:\n\\noindent\n\\begin{itemize}[leftmargin=*]\n    \\item Entities authorized to read or affect system state of objects may get access to extraneous data which they should not have. As an example, roadway infrastructure such as speed limit transmitters should be allowed to affect maximum speed of a smart vehicle, but should not be able to read or write data such as location, personal user data, vehicle specifications, or maintenance information.  In traditional IoT digital twin architectures access is granted as a binary, where users are authorized to view and affect contents of a digital twin as a whole or not at all. This exposes even authorized entities to an excess of data, and is less secure than giving access to individually tagged pieces of data. \n    \\item Due to the large number of IoT devices in ecosystem such as smart factories, ITS, or smart homes, it is a near certainty that some of these devices will malfunction. In all of these objects failure may have severe consequences, therefore quick and efficient realization of device malfunction is a necessity. The attachment of tags directly to pieces of data allows for consistent processing and verification regarding the value of that data by the associated digital twin. For example, all values tagged 'temperature' within a system could have bounds implemented as rules such as: temperature should be a positive integer, and temperature should never exceed 100 units. If a piece of data exceeds or falls below these bounds then it is safe to assume that the physical device is malfunctioning and system state is compromised.\n\\end{itemize}\nThis paper proposes an edge based solution addressing these security concerns via data distribution into multiple digital twins foundationally built on TBAC. We also support and build upon security properties addressed by USDOT ITS research. We focus on \\textbf{Authenticity \\& Trust} by implementing open source software such as Mosquitto\\footnote{https:\/\/mosquitto.org\/} which maintains support for multiple forms of authorization including username\/password, PSK (Pre-Shared Key), and external plugin support. This allows for system specific authorization schemes to be implemented, while also providing built in authorization methods. \\textbf{Confidentiality \\& Privacy} is supported in this architecture by the subdivision of data into multiple digital twins. Data exposure is limited by allowing authorized entities to view only the subset of data they require to function, thereby keeping the information in the system confidential and private.\n\n\\subsection{Some Industry Solutions and Limitations}\n\\subsubsection{Microsoft Azure}\n Microsoft Azure IoT Hub allows attachment of tags to digital twins and physical devices but they are static informational metadata such as device specific location\/properties and do not serve a security function nor do they delineate pieces of data. Queries can be used to route data into digital twins based on tags, but are not dynamic and queries must be added to process additional tags. Digital twins in this architecture may not receive subsets of generated data as tags are applied to physical devices, not individual pieces of data. Therefore digital twins may be tagged in order to authorize reception of device data, however this authorization is purely a binary: either they will receive the full device message if tags are matching, or they will receive nothing. This limitation is not present in our implementation because the tags are attached to each key-value pair in every message and therefore messages may be subdivided based on tags.\n\\subsubsection{Google IoT Core}\nGoogle IoT Core offers a highly scalable industry IoT solution, however does not implement distribution of data based on tags. Tags in their architecture can be applied to physical devices and serve as device identifiers specifying metadata information such as: serial number, location, or manufacturer information. Tags may also be applied at the digital twin level in order to grant access to users authorized to view individual tags. However due to the lack of data distribution based on data present, all data is collected in one digital twin. Therefore subsets of data cannot be accessed and in order to view device data a user must be authorized to view all tags present. This is subversive to the limitation of data sharing in the system and is less dynamic than access control granted to individual shadows and therefore tags. \n\\subsubsection{AWS IoT Core}\nAmazon Web Services IoT Core supports a many-to-many digital twin-to-physical device relationship in the form of named digital twins accompanying a base unnamed twin. Physical devices may publish data directly to their named shadow counterparts, or publish all data to the base unnamed shadow which can then manage publications to named shadows. The purpose of named shadows is to hold subsets of physical device data in order to minimize data exposure and system malleability upon authorization of a resource regarding access to the shadow. This division of data comes closest to our proposed architecture, however there is no support for tagging discrete pieces of data. Rules can be implemented to distribute data to named shadows, however due to the lack of tag attachment to data these rules must work on data value, associated key, or other system information. This means data can be subdivided in the system, but like data can not be effectively grouped dynamically. Rules must be defined to sort individual data keys into named shadows resulting in a less scalable and more implementation specific system. \n\\subsubsection{Oracle IoT Asset Monitoring Cloud Service}\n Oracle's cloud IoT service allows the creation of digital twins to hold device information, as well as predictive twins to hold the results of complex analysis of device performance such as machine learning and neural networks. They also allow simple creation of rules regarding alerts and system functioning such as location-based rules which activate when a device enters or exits defined locations, threshold-based rules which trigger when a devices reported data either exceeds or falls below set values, and alert-based reactions which trigger physical device actions given alerts present in the system. However the tags which can be attached to devices are purely descriptive and serve no security or access control centric function. Therefore the division of data in this architecture is difficult, as individual pieces of data are not delineated in any way other than their associated keys. Highly dynamic environments may suffer security consequences as authentication in this architecture is a binary of full access or no access. \n\n\\section{Proposed Multiple Digital Twins with \\\\ Tags Based Access Control}\n\\label{sec:tbac-proposal}\n It is clear at this point that IoT environments generate and subsequently share large amounts of data. Mechanisms for sharing relevant and required information facilitate correct data apportionment between resources, as well as limiting the amount of data shared as much as possible. Minimizing data sharing within the architecture both increases security and decreases the burden on networking hardware. \n Our approach to controlling data sharing implements subdivision of data generated by physical sensors, and grants individual access to those subsets. This employs the security principle of least privilege by giving access to only the information required by the authorized resource, and allowing system malleability on the smallest surface possible. This increases system security as well as efficiency by minimizing the size of data flowing in the system from producers to consumers. \n\n\\begin{figure*}[t!]\n\\centering\n    \\includegraphics[scale=0.51]{Diagrams\/Sub-JSON-Generation-new-edited.png}\n    \\caption{Propagation of Reported States to Sub-JSONs}\n    \\label{sub_json_generation}\n\\end{figure*}\nDigital twins are the source of this subdivision, as they can exist in a many-to-one relationship with their physical counterparts. Each shadow instance holds a subset of the data present and can independently grant access to resources. These resources may query the shadow for the current system state, or publish desired states to the system. The resulting architecture leads to a distribution of data, and prevents a single MQTT client assuming all interaction with resources wishing to read or affect system state. The modularity of the separation of data into many separate digital twins also affords flexibility because not all clients must be active at any given point in time. Twins have the potential to be spun up or spun down as necessitated by resources in a form of load balancing. If a digital twin registers long periods of disconnection or inactivity from its associated device, the client could be halted until the device either has a state to report or the subset of information the client holds is requested by an external resource. This reactivity could be converted to a highly dynamic and scalable system which manages the number of active twins in real time based on demand. \n\nThe implementation of physical devices is as straightforward in this architecture as it is in a one-to-one device-to-twin structure. Due to the centralized nature of MQTT, physical devices need only subscribe to topics following a pre-defined API (Application Programming Interface) structure to receive state change control signals. Authorization to publish to those topics may be handled by the broker, giving a central point at which access control can be done regarding all digital twins. This ensures security of the channels in which interaction takes while requiring few subscriptions from the physical device.\n\n\\subsection{Proposed Architecture}\nThe distribution and subdivision of data in our architecture is facilitated by the application of tags. Each key-value pair in the system holds a key string describing the meaning of the data held in the object and a value array containing the sensor value and tags attached to the object. These tags identify the function of that data within the implementation, provide structure for groupings of related data, and are the central mechanism for access control.\nIn this architecture tags support grouping of data by allowing similar data to be quickly associated and divided into subsets. Figure \\ref{sub_json_generation} shows the processing of reported states (from the physical device to base shadow) with attached tags, and the division of data (from base shadow to multiple sub JSONs) based on those tags. For example in smart cars there are many different sets of data that could be produced such as speed, location, pressure, temperature, etc. All sensors in the car would then attach `pressure' to data measuring a pressure in the car. Additionally, more specific subsets can be made in order to grant external resources access to only the information they require. Therefore pressure data being monitored associated with the tires of the car may be tagged `tire' as well as `pressure' in order to differentiate it and allow more specific data sharing. Tags serve to group the data into most specific subset possible, after which the key associated denotes exactly what that data represents in the system.\n \nTags and key-value pairs hold a many-to-many relationship where one tag may be applied to many key-value pairs and conversely one key-value pair may hold many tags. This relationship allows data values to be distributed and held by many shadows, and also one shadow may receive many data values at once if a single tag is distributed to multiple key-value pairs in a message. As discussed earlier, where many key-value pairs are tagged `pressure' in a single message and subsequently distributed to the 'pressure' digital twin.\n\\begin{figure}[t!]\n \n  \\includegraphics[width=\\linewidth]{Diagrams\/System_Architecture-edited.png}\n  \\caption{Implemented System Architecture}\n  \\label{des_state_resolution}\n\\end{figure}\n\\subsection{Assignment of Tags}\n The assignment of tags within a TBAC architecture must follow proper security practices, as assigned tags are the basis of access control. If tags are improperly assigned and therefore data is distributed to digital twins in which it should not reside, then resources that are given access to those twins will be served data they are not authorized to view. The attachment of low-security classification tags to high-security pieces of data is a simple way to gain access to critical data within the system. For example, if there exists a `timing' tag that functions as a benchmark to synchronize elements of the system then all resources would be able to access the digital twin containing `timing' information. If administrator is able to attach the `timing' tag to a piece of sensitive information such as location, or user data, it will enable unauthorized data read via the `timing' digital twin. \n\n Tags should only be malleable to a few key authenticated resources in the system. They may be applied by the physical device itself based on characteristics of the data being generated. This is the foundation of on-the-fly dynamic tag attachment within the architecture. For example, a smart temperature sensor may have a ceiling at which the recorded temperature is no longer safe. When the recorded temperature exceeds that ceiling a `warning' tag could be applied to the data in order to trigger a safety system or inform the user. Additional levels may be present as well, so if the temperature exceeds another threshold a `critical' tag may be applied. Therefore response behavior can vary dependent upon the level of device failure. These tags can be attached when a value exceeds or falls below a predefined set-point and are device specific. The attachment of these tags means the associated key-value pair will be placed within the `warning' or `critical' digital twins, which allows all system health to be monitored via a small number of digital twins. This functionality is shown in Figure \\ref{sub_json_generation}, where both tire pressure sensors are reporting values which have system health tags attached. The driver-side sensor has applied a `warning' tag which may be applied when the tire falls below the recommended specification by a relatively small margin. The passenger-side sensor has applied a `critical' tag which may be applied when the pressure falls too far e.g. below 30. \n\n System administrators should also be able to add tags to device data as necessary. Therefore tags published to the device by the base unnamed shadow are applied to all further data generated by the device. Only the base shadow should have this functionality as it is the most controlled due to the centralized nature of the data it holds, and therefore administrators should be the only users with access. \n\n\n\\begin{table}[t!]\n\\begin{center}\n\\caption{Raspberry Pi 4 Model B Specifications}\n\\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{ | p{0.60in} | p{2.25in} | } \n \\hline\n Operating System & Raspberry Pi OS, May 7, 2021 \\\\ \n \\hline\n CPU & Broadcom BCM2711, Quad core Cortex-A72 (ARM v8) 64-bit SoC @ 1.5GHz \\\\\n \\hline\n RAM & 4 GB LPDDR4-3200 SDRAM \\\\\n \\hline \n Network Interface & Gigabit Ethernet, 2.4GHz and 5GHz 802.11b\/g\/n\/ac Wi-Fi \\\\\n \\hline\n\\end{tabular}\n\\label{tab:Raspberry-pi-specs}\n\\end{center}\n\\end{table}\n\n\\section{Implementation and Evaluation}\n\\label{sec:evaluation}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=7cm, height=3.4in]{Diagrams\/Experimentation_Struture-edited-v2.png}\n  \\caption{Transformation from Five to Six Key-Value Pairs}\n  \\label{experiment_state_transform}\n\\end{figure}\nThe proposed architecture has been evaluated using a lightweight digital twin implementation written in python and emulating AWS MQTT topic structure. This allows for large scalability and integration into industrial environments with minimal modification of the current code base. In this section, mechanism of interaction with devices in this architecture will be explained, then the performance of our implementation will be evaluated, finally application to industry will be discussed.\n\n Device state is divided into three subgroups: \\textit{reported}, \\textit{desired}, and \\textit{delta}. The physical device modifies the reported subgroup when it is connected and reports its current state. Desired states may be pushed to the digital twin created in the local edge by authorized external clients in order to request a change in physical device state. If a difference is determined between the reported and desired states of the device, then the differing keys are added to the delta subgroup. When the device is connected, the calculated delta state is published to the physical device. Once the device receives these keys and transitions state, it reports the new state. Upon reception of a reported state matching a given desired state, the digital twin acknowledges that state as resolved and removes the associated key from both the desired and delta subgroups. The key-value pairs present in the reported state are then divided based on attached tags and distributed to their associated base shadows in the local edge, as shown in Fig. \\ref{des_state_resolution}. The shadow and device python clients were run on a Raspberry Pi 4 for data collection, the specifications of which can be seen in Table \\ref{tab:Raspberry-pi-specs}. The open source broker Mosquitto\\footnote{https:\/\/mosquitto.org\/} was used for authentication as it includes username\/password, PSK (Pre-Shared Key), and external plugin support. The system was not stressed with the entirety of the architecture running on one device, and this should not affect the timing data collected as only the processing time of the digital twin is being evaluated.\n\n\n\n\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\linewidth, height=2.15in]{Diagrams\/Dynamic_graph_pi_99ci-edited.png}\n  \\caption{Average Dynamic Tag Assignment Processing Time}\n  \\label{dynamic_processing_time_graph}\n\\end{figure}\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\linewidth, height=2.5in]{Diagrams\/static_graph_pi-edited.png}\n  \\caption{Average Static Tag Assignment Processing Time}\n  \\label{static_processing_time_graph}\n\\end{figure}\n\nIn order to evaluate the system's performance, the shadow linearly scales the relationship between the number of key-value pairs and tags associated with each pair in the system at a given point, as shown in Fig. \\ref{experiment_state_transform}. For example, when there are five key-value pairs in the system there will be five tags attached to each of those pairs. Upon reception of a reported state from the simulated device, the shadow introduces a new key-value pair as well as increments the number of tags attached to all other pairs previously present. Therefore once the five pair state is reported by the device, a sixth pair is added and a new tag is appended to each of the five pairs already present. The new desired state of the system is then added to delta subgroup and subsequently published to simulated device. Once the simulated device conforms to the received desired state and reports its current state, this cycle continues. \nTo effectively measure the efficiency of our architecture, we calculate time only while the shadow is resolving messages. This avoids introduction of variance from network delay as well as abstracts our evaluation away from device specific implementations. Different IoT devices will have widely varying computational hardware as well as polling rates depending on implementation specific variables. The exclusion of device interaction from collected timings leads to more consistent results and a stronger examination of our implementation of the proposed architecture.\n\n\n\n\nThere are three distinct function calls included in the timings collected: `update', `delta', and `parse\\_tags'. Update function processes the incoming message and places relevant key-value pairs into their designated position within the JSON structure. The delta function balances the JSON structure to ensure it retains continuity and consistency regarding the AWS-style. This encompasses functionality discussed earlier, such as removing keys from desired subgroup once a matching reported state has been received. Once the JSON has been cleaned the remaining keys in delta subgroup are published to device. The final call is to `parse\\_tags', which compiles a list of all tags attached to key-value pairs in the message and generates and publishes sub-JSONs to associated shadows.\n\n \n\n\n\nFigure \\ref{dynamic_processing_time_graph} shows the processing time of the system averaged over 500 trials where each trial incremented the number of key-value pairs in the system from zero to forty and then emptied the system. Each point is the average performance time of the system associated with that many key-value pairs present. The attached error bars indicate a 99\\% confidence interval and show that with 1600 total tags present in the system (40 attached to 40 key-value pairs), the average processing time will rarely exceed 36 ms.\n Figure \\ref{static_processing_time_graph} displays the variation in system function with a static number of tags attached to each key-value pair present. Each data series represents an incrementation of number of key-value pairs from zero to one hundred with each additional pair containing specified number of tags, e.g. at data point 80 in the `5 Tags' series there are 80 key-value pairs each of which has 5 attached tags. Each data series is the average of 500 trials where each trial represents the filling of the system from zero to one hundred pairs. \n\n\n\nThese results are promising for real time and edge centric industry applications as they show minimal increase in processing time at the digital twin level, while largely scaling the number of tags and data pairs present in the system. Due to the exponential nature of the `parse\\_tags' function, it is the largest bottleneck of the system. However, if the number of tags applied to each key-value pair is kept low the system remains scalable and suffers minor performative degradation. \n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis research demonstrates edge centric access control structure in IoT environments by proposing a novel TBAC architecture focused on the division of data into multiple digital twins. This architecture fills a gap in environments where on the fly and real time limited data exposure is highly critical, and allows for complete subdivision of data based on tags attached directly to present data. Complex device relationships are supported via the many-to-many relationship between tags and data, allowing implementations to model peculiar environments with little additional complexity. We discuss the usefulness of this architecture in smart environments such as manufacturing and internet-connected vehicles, and give an example of the flow of tagged data in these environment. Industry solutions currently offered have been examined regarding their integration of TBAC as well as their capacity to divide data into subsets. The weaknesses and strengths of offered services are discussed in relation to the proposed architecture. We deployed a local implementation of our architecture and examined the effect of number of attached tags on performance. We envision further exploration regarding access control on the tagged shadows, and the application of this data distribution to other smart environments. \n\n\n\\section*{Acknowledgement}\nThis research is supported by NSF CREST Center Grant\nHRD-1736209 at UTSA, and by the Grant 2025682 at TTU.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNon--linear integrable differential systems have been subject to intensive\ninvestigations since last century. In the last few years,\none of the most exciting\ndevelopments in this field is the revelation of its close relationship\nwith 2--dimensional exactly sovable field theories, in particular\n2--dimensional quantum gravity  and string theory. This relation enables us to extract\nnon--perturbative properties of non--critical string. More precisely,\nthe discretization of 2--dimensional quantum gravity  can be reformulated as matrix models. Through\nthe {\\it double scaling limit}, one can prove that 1--matrix model  with even\npotential is described by {KdV hierarchy } and string equation.\nRecently it has been shown that the double scaling limit is not an inevitable\nstep. In other words, the {KdV hierarchy } is not merely an occasional effect of double\nscaling limit but intrinsic in matrix model formulation.\nThe idea is as follows: we represent matrix models as certain\ndiscrete linear\nsystem(s), from which we can extract lattice integrable hierarchies, finally\nwe can directly extract differential hierarchies from these lattice\nhierarchies. Using this approach, one can prove that\nthe 1--matrix model  with general potential is characterized by { non--linear Schr\\\"odinger hierarchy }(NLS), which is\nthe two bosonic field representation of { KP hierarchy }.  In other words, the 1--matrix model  gives\na new solution for  the $\\tau$--function  of { KP hierarchy }\\cite{BX1}.\nIn the same way, we can apply this procedure to multi--matrix models  to obtain their\nfull differential integrable hierarchies. In this process, we are led to\nconsider a possible generalization of { KP hierarchy }. The study of this generalization\nis the main purpose of this letter. Actually another enlargement of { KP hierarchy } was\nconsidered some years ago by adding additional flows. However, although\neach of the additional flows commutes with all the old KP flows, they don't\ncommute among themselves\\cite{Dickey}. Thus this generalization is not\nconsistent. Fortunately, we will see that properly introducing new flows\nwe can obtain an integrable hierarchy, in which all the flows are\ncommutative. We will explain the realization of such generalized { KP hierarchy }\nin multi--matrix models  elsewhere\\cite{BX2}.\n\n\n\n\\section{KP hierarchy}\n\nLet us begin with a pseudo--differential operator(PDO) of arbitrary order\n\\begin{eqnarray}\nA=\\sum_{-\\infty}^n a_i(x)\\partial_i.\\label{pseudodp}\n\\end{eqnarray}\nwhere $x$ is {\\it the space coordinate}, while\n${\\partial}^{-1}$ is formal integral operation over $x$.\nAll the pseudo--differential operators form an algebra $\\wp$ under the\ngeneralized Leibnitz rules\n\\begin{eqnarray}\n&&{\\partial} a(x)=a(x){\\partial}+a'(x),\\qquad [{\\partial}, x]=1,\\nonumber\\\\\n&&\\partial^{-1}\\partial=\\partial\\partial^{-1}=1,\\label{leibnitz}\\\\\n&&\\partial^{-j-1}a(x)=\\sum_{l=0}^{\\infty}(-1)^l{{j+l}\\choose l}\na^{(l)}(x)\\partial^{-j-l-1},\\nonumber\n\\end{eqnarray}\nwhere $a^{(l)}(x)$ denotes ${{{\\partial}^l a(x)}\\over{{\\partial} x^l}}$.\nThe algebra $\\wp$ has two sub--algebras:\n\\begin{eqnarray}\n\\wp=\\wp_+\\oplus\\wp_-.\\nonumber\n\\end{eqnarray}\nwhere $\\wp_+$ denotes the algebra of pure differential operators,\nwhile $\\wp_-$ means the algebra of pure integration operations.\n\nFor any given pseudo--differential operator $A$ of type (\\ref{pseudodp}), we\ncall $a_{-1}(x)$ its residue, denoted by\n\\begin{eqnarray}\n\\rm res_{{\\partial}}A=a_{-1}(x)\\qquad \\hbox{ or}\\qquad A_{(-1)}\\nonumber\n\\end{eqnarray}\nand we define the following functional\n\\begin{eqnarray}\n<A>=\\int a_{-1}(x)dx.\\label{innerproduct}\n\\end{eqnarray}\nwhich naturally gives an inner scalar product on the algebra\n$\\wp$\\cite{Babelon}.\n\n\\subsection{The integrable structure}\n\nNow let $L$ be a pseudo--differential operator of the first order\n\\begin{eqnarray}\nL={\\partial}+\\sum_{i=0}^{\\infty}u_i(x){\\partial}^{-i}\\label{PDO}\n\\end{eqnarray}\nwhich we will call KP(or Lax) {\\it operator}.\nWe call $u_i$'s KP {\\it coordinates}.\n$(L-{\\partial})\\in \\wp_-$, so we can represent\n a functional of KP coordinates as\n\\begin{eqnarray}\nf_X(L)=<LX>,\\qquad\\qquad X\\in\\wp_+\\nonumber\n\\end{eqnarray}\n\\def${\\cal F}(\\wp_-)$ {${\\cal F}(\\wp_-)$ }\nwhich  span a functional space ${\\cal F}(\\wp_-)$ . The remarkable fact is that ${\\cal F}(\\wp_-)$  is\ninvariant under the co--adjoint action of $\\wp_+$, consequently the algebraic\nstructure on $\\wp_+$ determines the Poisson structure on ${\\cal F}(\\wp_-)$ \n\\begin{eqnarray}\n\\{f_X, f_Y\\}_1(L)=L([X,Y])\\label{poisson1}\n\\end{eqnarray}\nThe infinite many conserved quantities (or Hamiltonians) are\n\\begin{eqnarray}\nH_r={1\\over r}<L^r>\\qquad \\forall r\\geq1\\label{hamiltonian}\n\\end{eqnarray}\nThey generate infinite many flows,\n\\begin{eqnarray}\n\\ddt r L=[L^r_+, L]\\label{KPequation}\n\\end{eqnarray}\nwhere the subindex ``+\" indicates choosing the non--negative powers of ${\\partial}$.\nSince\n\\begin{eqnarray}\n[L^r_+, L]=[L, L^r_-]\\in{\\cal P}_-,\\nonumber\n\\end{eqnarray}\nwe see that all the flows preserve the form of KP operator ``$L$\", and\nthey all commute with each other. This commutativity implies\nthe ``{\\it zero curvature representation}\"\n\\begin{eqnarray}\n\\ddt m L^n_+-\\ddt n L^m_+=[L^m_+, L^n_+],\\qquad \\forall n,m\n\\label{zerocurvature}\n\\end{eqnarray}\nBy { KP hierarchy } we mean the set of differential equations (\\ref{KPequation}) or\n(\\ref{zerocurvature}).\nIn fact the { KP hierarchy } possesses another Poisson structure\\cite{Watanabe}\n\\begin{eqnarray}\n\\{f_X, f_Y\\}_2(L)&=&<(XL)_+YL>-<(LX)_+LY>\\nonumber\\\\\n&+&\\int [L,Y]_{(-1)}\\Bigl({\\partial}^{-1}[L,X]_{(-1)}\\Bigl)\n\\label{poisson2}\n\\end{eqnarray}\nWith respect to these Poisson brackets,\nthe KP coordinates $u_i$ form $W$--infinity algebras.\nThe important point is that these two Poisson structures are compatible\nin the sense\n\\begin{eqnarray}\n\\{f, H_{r+1}\\}_1=\\{f, H_r\\}_2\\qquad \\forall\\qquad\\hbox{function f}\n\\end{eqnarray}\nThis compatibility ensures the integrability of { KP hierarchy }. Generally speaking,\nfor a system of infinite many degrees of freedom, for example, the { KP hierarchy },\nwe may  find various definitions of integrability\\cite{Babelon}.\nThe essential point is that there must exist infinite many conserved\nquantities in involution. Therefore we can list  some of the definitions\nbelow\n\\begin{enumerate}\n\\item\n{\\it There exist two compatible Poisson brackets}\n({\\it or bi-hamiltonian structure}).\n\\item\n{\\it The flows are all commutative}.\n\\item\n{\\it There  exists the zero curvature representation}(\\ref{zerocurvature}).\n\\end{enumerate}\nFor different purposes we may use different definitions. For example,\nbi-Hamiltonian structure can exhibit the  Poisson algebraic structure of the\nsystem. But in the next two sections we will mainly use the second definition\nto prove the integrability of the generalized { KP hierarchy } due to its simplicity.\n\n\n\\subsection{The associated linear system}\n\nThe KP operator $L$ can be expressed in terms of the ``dressing\" operator\n\\begin{eqnarray}\nL=K{\\partial} K^{-1}\\qquad K=1+\\sum_{i=1}^{\\infty}w_i{\\partial}^{-i}\\nonumber\n\\end{eqnarray}\nAfter defining\n\\begin{eqnarray}\n\\xi(t,\\lambda)=\\sum_{r=1}^{\\infty}t_r\\lambda^r\n\\end{eqnarray}\nand introducing the Baker--Akhiezer function\n\\begin{eqnarray}\n\\Psi(t,\\lambda)=Ke^{\\xi(t,\\lambda)}\\label{baker}\n\\end{eqnarray}\nwe can associate to the KP hierarchy (\\ref{KPequation}) a linear system\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL\\Psi=\\lambda\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt r \\Psi=L^r_+\\Psi.\n\\end{array}\\right.\n\\label{flow}\n\\end{eqnarray}\n\n\\subsection{ The $\\tau$--function}\n\nOne of the important ingrediants of the KP system is its $\\tau$--function,\nwhich can be introduced through Baker--Akhiezer function\n\\begin{eqnarray}\n\\Psi(t,\\lambda)={{\\tau(t_1-{1\\over {\\lambda}},t_2-{1\\over\n{2\\lambda^2}},\\ldots)}\n\\over{\\tau(t)}}e^{\\xi(t,\\lambda)}\n\\end{eqnarray}\nOne can prove that\n\\begin{eqnarray}\n{{\\partial}^2\\over{{\\partial} t_1{\\partial} t_r}}\\ln\\tau=\\rm res_{{\\partial}}L^r,\\qquad \\forall r\\geq1\n\\end{eqnarray}\nIf we define a set of new functionals\n\\begin{eqnarray}\nJ_r=\\int \\rm res_{{\\partial}}L^r dx,\\qquad \\forall r\\geq1\n\\end{eqnarray}\nthen\n\\begin{eqnarray}\n \\ddt s J_r=0,\\qquad\\quad \\forall r,s\\geq1\n\\end{eqnarray}\nSo $J_r$'s are the conservation laws of the KP hierarchy.\n\n\\section{Generalization of KP hierarchy }\n\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{footnote}{0}\n\nNow we come to discuss the generalization of the { KP hierarchy }, which we promised\nin the introduction.\n\n\\subsection{The additional flows}\n\nOur purpose is to show\nthat we may introduce other series of flows. In order to do so, we define\na new operator\\cite{Chenlee},\n\\begin{eqnarray}\nM\\equiv K(\\sum_{i=1}^{\\infty}rt_r{\\partial}^{r-1})K^{-1}=\\sum_{i=-\\infty}^{\\infty}\nv_i{\\partial}^i.\\label{mpdo}\n\\end{eqnarray}\nwhich is conjugate to the KP operator $L$ in the sense that\n\\begin{eqnarray}\n[L, M]=K [{\\partial}, \\sum_{i=1}^{\\infty}rt_r{\\partial}^{r-1}]K^{-1}=1,\\label{conjugate}\n\\end{eqnarray}\nWe can derive the equations of motion for $M$,\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddt r M=[L^r_+, M],\\\\\\noalign{\\vskip10pt}\n{\\partial \\over {\\partial \\lm}}\\Psi=M\\Psi.\n\\end{array}\\right.\n\\end{eqnarray}\nSo we see that $L$ and $M$ are nothing but the operatorial expressions of\n${\\lambda},{\\partial \\over {\\partial \\lm}}$ (acting on  $\\Psi(t,{\\lambda})$).\n\nAs we know, the basic requirement for new flows is that they should preserve\nthe form of KP operator $L$. So we can define new flows like\n\\begin{eqnarray}\n\\ddt {mn} L=[L, (M^mL^n)_-],\\qquad\\forall m,n.\\label{additional}\n\\end{eqnarray}\nOne can show that each flow commutes with KP flows (\\ref{KPequation}), but\nthese additional flows do not commute among themselves\\cite{Dickey}. Our aim\nis to show that properly choosing combinations of these additional flows, we\ncan define new flows which commute with the old KP flows and among themselves.\n\n\\subsection{Another series of flows}\n\nOur  starting  remark is that the $t$--series of perturbations is due to\nthe fact that $[L, L^r_-]\\in{\\cal P}_-,\\forall r\\geq1$. Now\nwe also have $[L, M^r_-]\\in{\\cal P}_-,\\forall r\\geq1$,\nso we could introduce a new series of deformation\nparameters\\footnote{That is to say, the KP coordinates $u_i$'s and $v_i$'s\ndepend on both $t$ and $y$.} $y_1,y_2,y_3,\\ldots$, such that\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r \\Psi=-M^r_-\\Psi.\n\\end{array}\\right.\n\\end{eqnarray}\nAll of these equations together result in the following enlarged KP system\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddt r L=[L^r_+, L],\\\\\\noalign{\\vskip10pt}\n\\ddt r M=[L^r_+, M],\\\\\\noalign{\\vskip10pt}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r M=[M^r_+, M].\n\\end{array}\\right.\\label{gen}\n\\end{eqnarray}\n\nNow we  should prove  that these new series of perturbations do not distroy\nconsistency, that is to say,\nwe should check the commutativity of all these flows. In the following we only\nconsider an example, i.e.\n\\begin{eqnarray}\n\\ddt r\\bigl(\\ddy s L\\bigl)=\\ddy s\\bigl(\\ddt r L\\bigl).\\label{comflows}\n\\end{eqnarray}\nUsing eqs.(\\ref{gen}), we see that the left hand side is\n\\begin{eqnarray}\n&&\\quad \\ddt r\\bigl(\\ddy s L\\bigl)=\\ddt r[L, M^s_-]\\nonumber\\\\\n&&=[[L^r_+, L], M^s_-]+[L,[L^r_+, M^s]_-]\\nonumber\\\\\n&&=[[L^r_+, L], M^s_-]+[L,[L^r_+, M^s_-]]-[L,[L^r_+, M^s_-]_+]\\nonumber\\\\\n&&=[L^r_+, [L, M^s_-]]+[[L^r,M^s_-]_+, L]\\nonumber\\\\\n&&={\\rm r.h.s.}\\nonumber\n\\end{eqnarray}\nThe other cases can be checked in the similar way.\nSo the perturbations we introduced before indeed give an enlarged { KP hierarchy }.\nIts associated linear system is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL\\Psi={\\lambda}\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt r \\Psi=L^r_+\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddy r \\Psi=-M^r_-\\Psi,\\\\\\noalign{\\vskip10pt}\nM\\Psi={\\partial \\over {\\partial \\lm}}\\Psi.\n\\end{array}\\right.\\label{lmpsi}\n\\end{eqnarray}\n\nThe usual KP hierarchy (\\ref{KPequation}) is a particular case\nof eqs.(\\ref{gen}) by fixing the $y$--series of the perturbations.\n\n\\subsection{The new basic derivative and the new bi--hamiltonian structure}\n\nAs we remarked a moment ago, when we disgard the $y$--series of flows, we\nrecover the usual { KP hierarchy }, whose hamiltonians are\n\\begin{eqnarray}\nH_{r(L)}={1\\over r}<L^r>,\\nonumber\n\\end{eqnarray}\nhere we use the subindex $(L)$ to indicate that the Hamiltonians are\nconstructed\nfrom the KP operator $L$. We may  also use the same symbol to denote the\nPoisson\nbrackets, $\\{, \\}_{(L)}$.\n\nNow if we fix all the $t$--series of parameters,  then we get another subset of\nthe enlarged hierarchy (\\ref{gen}), that is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r M=[M^r_+, M].\n\\end{array}\\right.\\label{geny}\n\\end{eqnarray}\nThe second equation is in fact a { KP hierarchy } with KP operator $M$ of the form\n(\\ref{mpdo}). Since all these flows commute, this is an integrable\nsystem, and there should  exist two compatible Poisson brackets written\nin terms of coordinates $v_i$'s. However, this bi--hamiltonian structure\nis unknown due to the  fact that the positive powers of ${\\partial}$ in $M$ go to\ninfinity.\n\n\nFortunately, we may overcome the difficulty by introducing a new basic\nderivative. To this end, we recall that in our\nprevious analysis we treated $t_1$ as the space coordinate.\nFor later convenience, we denote $\\ddy 1$ by ${\\tilde\\d}$.\n{}From the $y_1$--flows of $\\Psi$, we may extract an operator identity\n\\begin{eqnarray}\n{\\tilde\\d}=-M_-=\\sum_{i=1}^{\\infty}\\Gamma_i{\\partial}^{-i}.\\label{dtilde}\n\\end{eqnarray}\nSince any positive powers of ${\\tilde\\d}$ belongs\nto ${\\cal P}_-({\\partial})$, so $\\{{\\tilde\\d}^i;i\\geq1\\}$\nforms a basis of ${\\cal P}_-({\\partial})$.\nWe may invert the relation (\\ref{dtilde}) to express ${\\partial}$ in\nterms of the new derivative\\footnote{\nRigorously speaking, this is only true when it acts on the function $\\Psi$.\nBut we  may think of it in the following way, starting from\n\\begin{eqnarray}\n{\\tilde\\d}\\Psi=\\sum_{i=1}^{\\infty}\\Gamma_i{\\partial}^{-i}\\Psi.\\nonumber\n\\end{eqnarray}\nproperly choosing the combinations of ${\\tilde\\d}$ such that we can\nreexpress the ${\\partial}^{-1}\\Psi$ in terms of new derivatives\n${\\tilde\\d}$, we replace  all the derivatives ${\\partial}$ in the linear system\n(\\ref{lmpsi}) by ${\\tilde\\d}$. So we may interpret $y_1$ as another space\ncoordinate.} ${\\tilde\\d}$\n\\begin{eqnarray}\n{\\partial}=\\sum_{i=1}^{\\infty}{\\tilde \\Gamma}_i{\\tilde\\d}^{-i}.\\label{d}\n\\end{eqnarray}\nUsing this fact, we get\n\\begin{eqnarray}\nM=-({\\tilde\\d}+\\sum_{i=1}^{\\infty}{\\tilde v}_i{\\tilde\\d}^{-i})=-{\\tilde K}{\\tilde\\d}{\\tilde K}^{-1}.\n\\end{eqnarray}\nwith new dressing operator ${\\tilde K}$ and new KP coordinates ${\\tilde v}_i$'s.\nObviously\n\\begin{eqnarray}\nM^r_-({\\partial})=M^r_+({\\tilde\\d}),\\qquad \\forall r\\geq1,\n\\end{eqnarray}\nwhere LHS is expanded in powers of ${\\partial}$, while the RHS is expanded in\npowers of ${\\tilde\\d}$.  Using eq.(\\ref{d}),\nwe can reexpress all the formulas (\\ref{gen}) in terms of this\nnew derivative ${\\tilde\\d}$, i.e.\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r (-M({\\tilde\\d}))=(-1)^{r+1}[(-M)^r_+({\\tilde\\d}),\n(-M)({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddy r L({\\tilde\\d})=(-1)^{r+1}[(-M)^r_+({\\tilde\\d}), L({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddt r (-M)({\\tilde\\d})=-[(-M)({\\tilde\\d}), L^r_-({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddt r L({\\tilde\\d})=[L^r_+({\\tilde\\d}), L({\\tilde\\d})].\n\\end{array}\\right.\n\\end{eqnarray}\nApart from some additional signs, these equations are isomorphic to\neqs.(\\ref{gen}).\nThis reminds us that we can even consider $(-M)$ as a KP operator, and\nalternatively interpret $y_1$ as space coordinate, all the other parameters\nas time parameters.\nTherefore we can define two compatible Poisson brackets\nfor KP operator $(-M)$ by simply replacing $L$ in (\\ref{poisson1}) and\n(\\ref{poisson2}) by $(-M)$, which shows that\non the space $y_1$, the fields ${\\tilde v}_i$'s form\n$W_{\\infty}$ algebras too.\n\n\\section{Further perturbations and the full generalized { KP hierarchy }}\n\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{footnote}{0}\n\nIn the previous section we have shown that the { KP hierarchy } can be perturbed by\nthe conjugate operator $M$ of the KP operator $L$.\nIn fact, the KP system allows further deformations.\n\n\\subsection{The new series of the flows}\n\nIn order to explain the further perturbations just mentioned,\nwe change a little bit our notation. Denote $t_r$'s and $y_r$'s  by\n$t_{1r}$ and $t_{2r}$ respectively. Furthermore define\n\\begin{eqnarray}\n&&L(1)\\equiv L,\\qquad\\qquad V(1)\\equiv\\sum_{r=1}^{\\infty}rt_{1r}L^{r-1}(1)\\nonumber\\\\\n&&L(2)\\equiv -{1\\over {c_{12}}}M\\qquad V(2)\\equiv\\sum_{r=1}^{\\infty}\nrt_{2r}L^{r-1}(2)\\nonumber\n\\end{eqnarray}\nNow let us introduce new operators in the following way\n\\ai\n&&L(\\alpha)\\equiv-{1\\over c_{\\alpha-1,\\alpha}}\\Bigl(c_{\\alpha-2,\\alpha-1}\nL(\\alpha-2)+V(\\alpha-1)\\Bigl)\\label{loperator}\\\\\n&&V(\\alpha)=\\sum_{r=1}^{\\infty}rt_{\\alpha,r}L^{r-1}(\\alpha),\\qquad \\alpha=3,4,\n\\ldots,n\n\\bj\nwhere $c_{{\\alpha},{\\alpha}+1}$'s are arbitrary constants, which amount to rescaling\n the space coordinates, and $n$ is an arbitrary positive integer.\nThen, in the same way, we can perturb the system further as follows\n\\ai\n&&\\ddt {\\beta r} L(\\alpha)=[L^r_+(\\beta), L(\\alpha)],\\qquad 1\\leq\\beta<\\alpha\n\\label{flowa}\\\\\n&&\\ddt {\\beta r} L(\\alpha)=[L(\\alpha), L^r_-(\\beta)],\\qquad\n\\alpha\\leq\\beta\\leq n\\label{flowb}\n\\bj\nNow in order to justify the consistency of these perturbations, we once again\nshould prove that all the flows commute among themselves. Let us check one\nexample,\n\\begin{eqnarray}\n\\ddt{{\\alpha} l}\\bigl(\\ddt{\\beta m}L(\\gamma)\\bigl)\n=\\ddt{\\beta m}\\bigl(\\ddt{{\\alpha} l}L(\\gamma)\\bigl),\\qquad{\\alpha}<\\beta<\\gamma.\\nonumber\n\\end{eqnarray}\nUsing the above hierarchy and Jacobi identities, we see that\n\\begin{eqnarray}\n{\\rm l.h.s.}=\\ddt{{\\alpha} l}[L^m_+(\\beta), L(\\gamma)]\n=[[L^l_+({\\alpha}), L^m(\\beta)], L(\\gamma)]+[L^m_+(\\beta), [L^l_+({\\alpha}),\nL(\\gamma)]].\\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\rm r.h.s.}=\\ddt{\\beta m}[L^l_+({\\alpha}), L(\\gamma)]\n=[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]+[L^l_+({\\alpha}),\n[L^m_+(\\beta), L(\\gamma)]].\\nonumber\n\\end{eqnarray}\nThe first term of ``l.h.s\" can be written as\n\\begin{eqnarray}\n{\\rm the~~1st~~term}&=&[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]\n+[[L^l_+({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\\nonumber\\\\\n&=&[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]+\n[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)],\\nonumber\n\\end{eqnarray}\ntherefore\n\\begin{eqnarray}\n{\\rm l.h.s.}\n&=&[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\n+[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]+\n[L^m_+(\\beta), [L^l_+({\\alpha}), L(\\gamma)]]\\nonumber\\\\\n&=&[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\n+[L^l_+({\\alpha}), [L^m_+(\\beta), L(\\gamma)]]\\nonumber\\\\\n&=&{\\rm r.h.s.}\\nonumber\n\\end{eqnarray}\nAll the other cases can be done in the same way. Therefore eqs.(4.2)\nreally define an integrable system.\nThe associated linear system is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL(1)\\Psi={\\lambda}\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt {1,r}\\Psi=L^r_+(1)\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt {{\\alpha},r}\\Psi=-L^r_-({\\alpha})\\Psi,\\qquad {\\alpha}=2,3,\\ldots,n,\\\\\\noalign{\\vskip10pt}\nM\\Psi={\\partial \\over {\\partial \\lm}}\\Psi.\n\\end{array}\\right.\\label{lmpsigen}\n\\end{eqnarray}\nIn fact we can rewrite this\nlinear system in a better way by choosing a new function\n\\begin{eqnarray}\n\\Psi({\\lambda},t)\\Longrightarrow\\xi({\\lambda},t)=\n\\exp(-\\sum_{r=1}^{\\infty}t_{1,r}{\\lambda}^r_1)\\Psi({\\lambda},t),\\nonumber\n\\end{eqnarray}\nthen all the flows can be\nsummarized by a single equation\n\\begin{eqnarray}\n\\ddt {{\\alpha}, r}\\xi=-L^r_-({\\alpha})\\xi.\n\\end{eqnarray}\nThe consistency conditions of the above linear system exactly give the\nhierarchy (4.2) We would like to remark\nthat the hierarchy (4.2) have several important\nsub--hierarchies.\n\n$(i)$. ${\\alpha}=\\beta=1$, the eqs.(\\ref{flowb}) are nothing but the\nusual { KP hierarchy } (\\ref{KPequation}).\n\n$(ii)$. $2\\leq{\\alpha}=\\beta\\leq  n$, the eqs.(\\ref{flowb}) give $(n-1)$\nKP hierarchies whose KP operator possess the form (\\ref{loperator}).\n\n$(iii)$. All the  flows commute.\n\n$(iv)$. When $n\\longrightarrow\\infty$, the full hiearchy (4.2)\ncontains all possible combinations of additional flows\n(\\ref{additional}).\n\nWe may  conclude that  the hierarchy (4.2)\npossess $n$ bi--hamiltonian structures, each  of  them generates a { KP hierarchy },\nall of these hierarchies couple together. The integrability of the system\nis guaranteed by the commutativity of the flows. However, we are not sure\nwhether the  hamiltonians in\ndifferent series are commutative.\n\n\\subsection{New bi--hamiltonian structures}\n\nIn the above analysis, all the operators are expanded in terms of ${\\partial}$.\nHowever,\nif we  use the same trick as the one in previous section,\nit is not difficult to reexpress them in terms of any one of $\\ddt {{\\alpha},1}$'s.\nLet us define\n\\begin{eqnarray}\n{\\partial}_{{\\alpha}}\\equiv {{\\partial}\\over{{\\partial} t_{{\\alpha},1}}}\n\\end{eqnarray}\nand expand $L_-({\\alpha})$ in powers of ${\\partial}$\n\\begin{eqnarray}\nL_-({\\alpha})=-\\sum_{i=1}^{\\infty}\\Gamma^{({\\alpha})}_i{\\partial}^{-i}\n\\end{eqnarray}\nthen the first flows of the linear system (\\ref{lmpsigen}) suggest\n\\begin{eqnarray}\n{\\partial}_{{\\alpha}}=\\sum_{i=1}^{\\infty}\\Gamma^{({\\alpha})}_i{\\partial}^{-i}\n\\end{eqnarray}\nsimilar to the argument in the previous section, we can invert these\nrelations, such that\n\\begin{eqnarray}\n{\\partial}=\\sum_{i=1}^{\\infty}{\\tilde\\Gamma}^{({\\alpha})}_i{\\partial}^{-i}_{{\\alpha}}\n\\end{eqnarray}\nSubstituting them into the formulas (\\ref{loperator}), we get the expansions\nof $L({\\alpha})$ in $\\ddt {\\beta,1}$ ({\\it for any ${\\alpha},\\beta$}). In particular\n$L({\\alpha})$ expanded in $\\ddt {{\\alpha},1}$ is also a KP operator,\n\\begin{eqnarray}\nL({\\alpha})=-({\\partial}_{{\\alpha}}+\\sum_{i=1}^{\\infty}v^{({\\alpha})}_i{\\partial}^{-i}_{{\\alpha}})\n\\end{eqnarray}\nand its ${\\alpha}-th$ series of flows is nothing but the ordinary KP hierarchy\n\\begin{eqnarray}\n\\ddt {{\\alpha}, r} L(\\alpha)=(-1)^{r+1}[L^r_+(\\alpha), L({\\alpha})]\n\\end{eqnarray}\nwhere the operators are expanded in powers of ${\\partial}_{{\\alpha}}$, and the additional\nsign indicates rescaling of the parameters. Of course,\nfor this subsystem, we can\nconstruct its integrable structure, by replacing $L$ in (\\ref{poisson1})\nand (\\ref{poisson2}) by $L({\\alpha})$.\nTherefore, we may say\nthat KP system (4.2) possesses multi bi--hamiltonian structures,\nand it contains ``$n$\" coupled ordinary KP hierarchies.\nThe coupling comes from the dynamical equations (4.2)\nwith ${\\alpha}\\neq\\beta$.\n\n\\subsection{The $\\tau$--function of the generalized { KP hierarchy }}\n\nUsing eqs.(4.2), we get\n\\begin{eqnarray}\n\\ddt {\\beta,s} \\rm res_{{\\partial}}L^r({\\alpha})\n=\\ddt {{\\alpha},r} \\rm res_{{\\partial}}L^s(\\beta),\\qquad \\forall {\\alpha},\\beta;\\quad r,s.\n\\end{eqnarray}\nThese equalities imply the existence of $\\tau$--function\n\\begin{eqnarray}\n{{\\partial}^2\\over{{\\partial} t_{1,1}{\\partial} t_{{\\alpha},r}}}\\ln\\tau=\\rm res_{{\\partial}}L^r({\\alpha}),\\qquad\n\\forall {\\alpha},r.\\label{taumulti}\n\\end{eqnarray}\nUsing this $\\tau$--function, we can introduce a series of the Baker--Akhiezer\nfunctions,\n\\begin{eqnarray}\n\\Psi_{{\\alpha}}(t,{\\lambda}_{{\\alpha}})={{\\tau(t_{{\\alpha},1}-{1\\over {{\\lambda}_{{\\alpha}}}},t_{{\\alpha},2}-\n{1\\over {2{\\lambda}^2_{{\\alpha}}}},\n\\ldots)}\\over{\\tau(t)}}e^{\\xi(t,{\\lambda}_{{\\alpha}})}\n\\end{eqnarray}\nwhere ${\\alpha}=1,2,\\ldots, n$. To each $\\Psi_{{\\alpha}}$ we can associate a linear\nsystem. Among them, the ${\\alpha}=1$ case was discribed above. The other\ncases can be analysed in the similar way\\cite{Thesis}.\n\n\\section{Discussion}\n\nWe have shown that the KP hierarchy can be extended to a much larger hierarchy\nby introducing additional KP operators. This generalized hierarchy can be\nconsidered as several coupled KP hierarchies. For each\nof the KP operators, we have constructed its  bi--hamiltonian structure\nby introducing new basic derivatives. Although we do not\nknow if all these hamiltonians are in involution,\nThe commutativity of the flows guarantees the integrability of the system.\n\nAs we know in the ordinary KP hierarchy case, the series of flows reflects the\nlarge symmetry of the system generated  by its Hamiltonians. In our case,\nthe multi--series of flows imply that this new hierarchy (4.2)\nshould possess a much larger symmetry.\nHowever we are not sure what this large symmetry is.\n\nIt is not clear if this new hierarchy relates to the multi--component\nKP hierarchy. Another interesting problem is to reduce this hierarchy to the\nknown hierarchies like generalized KdV hierarchies. This is under\ninvestigation.\n\n\\vskip0.8cm\n\\noindent\n{\\bf Achnowledgement}\n\n\\vskip 0.2cm\n\nI would like to thank Prof. L. Bonora for his constant encouragement,\nvaluable suggestions and fruitful discussions.\n\n\n\\renewcommand{\\Large}{\\normalsize} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nLet ${\\mathcal O}$ be a discrete valuation ring with field of fractions $K$.\nTropicalization is a procedure which takes as input a $d$-dimensional subvariety of an algebraic torus over $K$,\n$X\\subset (K^*)^n$, and associates to it a balanced weighted rational \n$d$-dimensional polyhedral complex, $\\operatorname{Trop}(X)\\subseteq{\\mathbb R}^n$.  \nSeveral questions naturally arise in this framework; for instance, one may ask what combinatorial properties\nof $\\operatorname{Trop}(X)$ correspond to geometric properties of $X$.\nOne may also ask what constraints being a tropicalization places on the topology of a polyhedral complex.  In \n\\cite{Hacking}, Hacking proved that if $X$ is a subvariety of $({\\mathbb C}^*)^n$ satisfying a certain \ngenericity condition, then the link of the fan $\\operatorname{Trop}(X)$ only has reduced rational homology in \nthe top dimension.  Hacking's result holds for a number of examples, including generic \nintersections of ample hypersurfaces in projective toric varieties.  In \\cite[Sec. 10]{Sp3}, \nSpeyer showed that if $C$ is a genus $g$ curve in $(K^*)^n$ satisfying a genericity condition \nthen there exists a balanced metric graph $\\Gamma$ with $b_1(\\Gamma)\\leq g$ and a parameterization \n$i:\\Gamma\\rightarrow \\operatorname{Trop}(C)$ that is affine-linear on edges.  Our results include an \nanalogue of Hacking's result for varieties defined over $K$ or as a higher-dimensional \ngeneralization of Speyer's result.  One may consider the monodromy action of $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$\non the \\'{e}tale cohomology $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ and ask what properties of the monodromy action are encoded in $\\operatorname{Trop}(X)$.  This is the algebraic analogue of the monodromy action of a family of varieties defined over a punctured disk.  The work of the second-named author with Hannah Markwig and Thomas Markwig \\cite{KMM} relating the valuation of the $j$-invariant of a plane elliptic curve to the length of the cycle in its tropicalization can be seen in this light.  We give a generalization of that result. \n\n\n\nAll of our results require that the variety $X$ be {\\em sch\\\"{o}n}, a natural condition introduced \nin \\cite{Tevelev} and generalized by \\cite{LQ} to the nonconstant coefficient case.\nThis condition means that the ambient torus $(K^*)^n$ may be compactified to a \ntoric scheme ${\\mathbb P}$ over an extension of $\\operatorname{Spec} {\\mathcal O}$ such that the intersection of ${\\mathcal X}=\\overline{X}\\subset {\\mathbb P}$ \nwith each open torus orbit $U_P$ is smooth of the expected dimension.  For appropriate ${\\mathbb P}$, ${\\mathcal X}$ is then a simple normal\ncrossings degenerations of $X$ (c.f.~\\ref{prop:degeneration}).   The construction of the desired toric scheme ${\\mathbb P}$ follows from Proposition \\ref{prop:nc}, a technical result in polyhedral geometry which has not appeared, to our knowledge, in the literature before.\n\nThe existence of such simple normal crossings degenerations for a sch\\\"on $X$ allows us to construct \na natural ``parameterizing space'' $\\Gamma_X$.  This generalizes a construction introduced by\nSpeyer \\cite{Sp3} when $X$ has dimension $1$.  The space $\\Gamma_X$ is closely related to the\ndual complex of an appropriate degeneration ${\\mathcal X}$ of $X$; in this guise it already appears implicitly\nin \\cite{Hacking}, as well as in \\cite{HKT}.  Kontsevich-Soibelman use rigid analytic techniques\nto construct a similar polyhedral complex, with an integral affine structure,\nfrom a suitable degeneration of $X$ in~\\cite{KS}.\n\nThe space $\\Gamma_X$ we construct is {\\em independent} of a choice of model ${\\mathcal X}$ for $X$;\nit depends only on $X$ and its embedding in the torus.  Moreover, $\\Gamma_X$ comes equipped with a canonical\nmap to $\\operatorname{Trop}(X)$.  A choice of sufficiently fine triangulation of $\\operatorname{Trop}(X)$ gives \n$\\Gamma_X$ the structure of a polyhedral complex.  When $\\Gamma_X$ is viewed in such a way,\nthe natural parameterization $\\Gamma_X\\rightarrow \\operatorname{Trop}(X)$ is affine-linear on polyhedra.  \nThis parameterization has several nice properties.\nFor instance, it is natural under monomial morphisms: if $X$ and $Y$ are sch\\\"{o}n \nsubvarieties of tori $T$ and $T'$ and $\\phi:T\\rightarrow T'$ is a homomorphism taking $X$ to $Y$, then \nthere is an induced map of complexes $\\Gamma_X\\rightarrow\\Gamma_Y$ that commutes with parameterizations.  \nMoreover, $\\Gamma_X$ satisfies a balancing condition analogous to the one satisfied by all \ntropical varieties.  Finally, it is ``not far'' from $\\operatorname{Trop}(X)$: if the intersections of ${\\mathcal X}$\nwith open torus orbits $U_P$ in ${\\mathbb P}$ satisfy certain connectedness hypotheses,\nwe may equate the cohomology of $\\Gamma_X$ and $\\operatorname{Trop}(X)$ in certain \ndegrees.  We hope that parameterizing complexes will be seen as a \nfundamental object in tropical geometry.\n\nOur main results (principally Theorem~\\ref{thm:main}, Corollary~\\ref{cor:CI}, and Proposition\\ref{prop:npower}) relate\nthe cohomology of $\\Gamma_X$ to geometric invariants of $X$.  In particular we consider the\n{\\'e}tale cohomology $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$; this cohomology comes equipped with a\nnatural filtration, called the weight filtration.  We construct a natural isomorphism between the\n``weight $0$'' subquotient $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ arising from this filtration\nand the cohomology $H^r(\\Gamma_X, {\\mathbb Q}_l)$ of $\\Gamma_X$.  We then use this to show\nthat if $X$ is the generic intersection of ample hypersurfaces in a toric scheme ${\\mathbb P}$, \nthen $H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l)$ vanishes for $1\\leq r<\\dim X$, a non-constant coefficient analogue of \nHacking's result.  We also prove results about the monodromy action on the middle cohomology $H^n_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\nThe main tool we use is the Rapoport-Zink weight spectral sequence \\cite{RZ}.  Under the sch\\\"{o}n \ncondition, after a finite base-extension ${\\mathcal O}^{\\prime}$ of ${\\mathcal O}$, we may compactify the ambient torus to \na toric scheme ${\\mathbb P}$ defined over ${\\mathcal O}^{\\prime}$ so that the central fiber of the closure\n${\\mathcal X}$ of $X$ in ${\\mathbb P}$ is a divisor with simple normal crossings.  The divisor, a degeneration of $X$, \nhas a stratification coming from intersections of its irreducible components.  The Rapoport-Zink\nspectral sequence then gives a very explicit formula for the cohomology on $X$, together with its\nweight filtration, in terms of these strata.  The $E_1$-term \nof the weight spectral sequence is formed from the cohomology groups of the strata with \nboundary maps built from the data of restriction maps and Gysin maps.  The spectral sequence \nconverges to the cohomology of the general fiber, and the induced filtration is the weight filtration.  \nMoreover, the weight spectral sequence \ndegenerates at $E_2$.  We thus obtain an explicit description of the smallest nontrivial piece of \nthe filtration which is isomorphic to the cohomology groups of $\\Gamma_X$.\n\nIt is interesting to compare this result to results of Berkovich \\cite{B} on rigid analytic spaces.\nIn particular, Berkovich shows that the cohomology group \n$H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}$ arising in our result is isomorphic to\nthe cohomology of the Berkovich space $X^{\\mbox{\\rm \\tiny an}}$ attached to $X$.  Our result thus suggests \na strong link between $\\Gamma_X$ and $X^{\\mbox{\\rm \\tiny an}}$.  In fact, Speyer \\cite{Speyer} constructs a\nnatural map from $X^{\\mbox{\\rm \\tiny an}}$ to $\\operatorname{Trop}(X)$.  This map factors through the map $\\Gamma_X \\rightarrow \\operatorname{Trop}(X)$,\nand it is natural to ask if the resulting map $X^{\\mbox{\\rm \\tiny an}} \\rightarrow \\Gamma_X$ map is a homotopy \nequivalence.  Links between tropical geometry and rigid\ngeometry have also appeared in works of Einsiedler-Kapranov-Lind~\\cite{EKL}, and Payne~\\cite{P}.\n\nUnder additional hypotheses, one can relate the results above to questions involving monodromy.\nA variety defined over $\\operatorname{Spec} K$ is analogous to a family of varieties defined \nover a punctured disk.  \nThe fundamental group of the punctured disk acts on the cohomology of a general fiber of such a\nfamily by monodromy.  The analogue of this monodromy action for varieties over $\\operatorname{Spec} K$ is the\naction of the inertia group $I_K$ of $K$ on the \\'{e}tale cohomology \n$H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  After a \npossible finite base-extension of ${\\mathcal O}$, this action is unipotent, and is \ngiven by the {\\em monodromy operator}\n$$N: H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l(-1))$$\nan endomorphism of the \\'etale cohomology that is essentially the (matrix) logarithm of\nthe action.  (We refer the reader to section~\\ref{sec:monodromy} for precise\ndefinitions).  The action of $N$ induces an increasing filtration on the cohomology.\nThe weight-monodromy conjecture asserts that this filtration coincides (up to a shift in degree) \nwith the weight filtration\ndescribed above.  Although it is not completely settled, this conjecture is known to be\ntrue in many cases of interest; for instance, it is known $X$ is a curve, surface, or\nan abelian variety.  Ito \\cite{Ito} has proven the weight-mondromy conjecture when ${\\mathcal O}$ has\nequal characteristic.  Thus, in these situations, one can interpret Theorem~\\ref{thm:main}\nas an isomorphism between the cohomology of $\\Gamma_X$ and the smallest\nnontrivial piece of the monodromy filtration of the cohomology of $\\overline{X}_K$, the closure \nof $X_K$ in the generic fiber of ${\\mathbb P}$:\n$$H^r(\\Gamma_X, {\\mathbb Q}_l) \\cong H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}.$$\nAs a consequence, Corollary \\ref{cor:bound} bounds \nthe Betti numbers of $\\Gamma_X$ in terms of those of $X$:\n$$b_r(\\Gamma_X) \\leq \\frac{1}{r+1} b_r(X)$$\ngeneralization Speyer's result on curves.\n  \nWe apply our techniques to get a description of some of the monodromy action and not just of the monodromy filtration.  In Proposition \\ref{prop:npower}, we give an interpretation of the top power of monodromy operator acting on the middle-dimensional cohomology\n\\[N^d:H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}\\rightarrow H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}(-d)\\]\nusing the isomorphisms \n\\begin{eqnarray*}\nH^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}&\\cong&H_d(\\Gamma,{\\mathbb Q}_l)(-d)\\\\ \nH^d_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}&\\cong&H^d(\\Gamma_X, {\\mathbb Q}_l).\n\\end{eqnarray*}\nThe operator can be viewed as a bilinear pairing on $H_d(\\Gamma_X, {\\mathbb Q}_l)$ in which case it is the volume pairing that takes a pair of top-dimensional cycles to the oriented volume of their intersection.  This specializes to the length pairing in the case of curves.  In the case of genus $1$ curves, by a straightforward application of the conductor-discriminant formula, we are able to recover the spatial sch\\\"on analogue of a result proved by the second-named author with H. Markwig andT. Markwig \\cite{KMM}: the valuation of the $j$-invariant of an elliptic curve $X$ with potentially multiplicative reduction is equal to $-a$ where $a$ is the length of the unique cycle in $\\Gamma_X$.  \n\nOur techniques are very similar to those of Hacking and Speyer.  Hacking uses a spectral sequence coming \nfrom a weight filtration on a complex of differential forms while we use the Rapoport-Zink spectral \nsequence.  Speyer's results use a resolution of the structure sheaf of a degeneration of a curve $C$ coming \nfrom a stratification induced by a toric scheme while we use a locally constant sheaf.\n\nWe should mention some related results.  In \\cite{GS}, Gross and Siebert construct \na scheme $X_0$ from an integer affine manifold and a toric polyhedral decomposition.  If $X_0$ is \nembedded in a family ${\\mathcal X}$ over ${\\mathbb C}[[t]]$, they are able to determine the limit mixed Hodge structure \nin terms of the combinatorial data.  In \\cite{TMNF}, the second-named author and Stapledon give a description of the Hodge-Deligne polynomial of the limit mixed Hodge structure of a sch\\\"{o}n family of varieties over a punctured disk in terms of $\\operatorname{Trop}(X)$ and initial degenerations of $X$.\n\nWe would like to thank Brian Conrad, Richard Hain, Kalle Karu, Sean Keel, Sam Payne, Zhenhua Qu, Bernd Siebert, Martin Sombra, \nDavid Speyer, Alan Stapledon, and Bernd Sturmfels for valuable discussions.\n\n\\section{Toric Schemes}\n\nWe begin by reviewing a construction that attaches a degenerating family of\ntoric varieties over a discrete valuation ring to a rational polyhedral complex in ${\\mathbb R}^n$.\nThis has appeared several times in the literature~\\cite{Speyer}~\\cite{NS}\n~\\cite{Smirnov}.  We follow the approach of~\\cite{NS} here.  Fix a discrete\nvaluation ring ${\\mathcal O}$, with field of fractions $K$ and residue field $k$, and \na uniformizer $\\pi$ of ${\\mathcal O}$.\n\n\\begin{defn} A rational polyhedral complex in ${\\mathbb R}^n$ is a collection \n$\\Sigma$ of finitely many convex rational polyhedra\n$P \\subset {\\mathbb R}^n$ with the following properties:\n\\begin{itemize}\n\\item If $P \\in \\Sigma$ and $P^{\\prime}$ is a face of $P$, then $P^{\\prime}$\nis in $\\Sigma$.\n\\item If $P,P^{\\prime} \\in \\Sigma$ then $P \\cap P^{\\prime}$ is a face of both\n$P$ and $P^{\\prime}$.\n\\end{itemize}\nIf, in addition, the union $\\bigcup_{P\\in\\Xi} P$  is equal to ${\\mathbb R}^n$, then $\\Sigma$ is said to be {\\em complete}.\n\\end{defn}\n\n\nGiven a $\\Sigma$ as above, we can construct a fan ${\\tilde \\Sigma}$ in \n${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ as follows: for each $P \\in \\Sigma$ let\n${\\tilde P}$ be the closure in ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ of the set\n$$\\{(x,a) \\subset {\\mathbb R}^n \\times {\\mathbb R}_{>0} : \\frac{x}{a} \\in P\\}.$$\nThen ${\\tilde P}$ is a rational polyhedral cone in ${\\mathbb R}^n \\times {\\mathbb R}_{>0}$.  Its \nfacets come in two types:\n\\begin{itemize}\n\\item cones of the form ${\\tilde P}^{\\prime}$, where $P^{\\prime}$ is a facet of $P$, and\n\\item the cone $P_0 = {\\tilde P} \\cap ({\\mathbb R}^n \\times \\{0\\})$, which is the limit as $a$ goes to zero\nof the polyhedron $aP$ in ${\\mathbb R}^n$.\n\\end{itemize}\nWe let ${\\tilde \\Sigma}$ be the collection of cones of the form ${\\tilde P}$ and $P_0$ for $P$ in $\\Sigma$.\nIf $\\Sigma$ is a subcomplex of a complete rational polyhedral complex, we say that $\\Sigma$ is {\\em completable} and, by Corollary 3.12 of \\cite{BGS}, ${\\tilde \\Sigma}$ is a rational polyhedral fan in ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$.   Note that\n$\\Sigma = {\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{1\\})$.  On the other hand\nthe fan $\\Sigma_0$ given by ${\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{0\\})$ is the limit as\n$a$ approaches zero of the polyhedral complexes $a\\Sigma$.\n\n\\begin{rem} \\rm In fact, by results of \\cite{BGS}, the association $\\Sigma \\mapsto {\\tilde \\Sigma}$ defines a bijection\nbetween the set of complete polyhedral complexes in ${\\mathbb R}^n$ and the set of complete fans in \n${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ for which every cone contained in ${\\mathbb R}^n \\times \\{0\\}$\nis the boundary of a cone that meets ${\\mathbb R}^N \\times {\\mathbb R}_{>0}$.  Its inverse is\n${\\tilde \\Sigma} \\mapsto {\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{1\\}).$\n\\end{rem}\n\nLet $X({\\tilde \\Sigma})_{{\\mathbb Z}}$ be the toric scheme over ${\\mathbb Z}$ associated to the fan ${\\tilde \\Sigma}$.  (The\nconstruction associating a toric variety to a fan is usually given over a field, but works\njust as well with coefficients in ${\\mathbb Z}$.)\nProjection from ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ to ${\\mathbb R}_{\\geq 0}$ induces a map of \nfans from ${\\tilde \\Sigma}$ to the fan\n$\\{0,{\\mathbb R}_{\\geq 0}\\}$ associated to ${\\mathbb A}^1_{{\\mathbb Z}}$.  This gives rise to a map of\ntoric varieties $\\pi_{{\\mathbb Z}}: X({\\tilde \\Sigma})_{{\\mathbb Z}} \\rightarrow {\\mathbb A}^1_{{\\mathbb Z}}$.  As remarked in~\\cite{NS},\nthis map is flat and torus equivariant.  Let $\\iota: \\operatorname{Spec} {\\mathcal O} \\rightarrow {\\mathbb A}^1_{{\\mathbb Z}}$\nbe the map corresponding to the map ${\\mathbb Z}[t] \\rightarrow {\\mathcal O}$ that takes $t$ to $\\pi$. \nWe let $X({\\tilde \\Sigma})$ be the scheme over ${\\mathcal O}$ obtained by base change from $X({\\tilde \\Sigma})_{{\\mathbb Z}}$\nvia $\\iota$, and let $\\pi: X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ be the projection.\n\nWe summarize results of~\\cite{NS} concerning this construction:\n\\begin{itemize}\n\\item The general fiber $X({\\tilde \\Sigma}) \\times_{\\operatorname{Spec} {\\mathcal O}} \\operatorname{Spec} K$ is isomorphic\nto the toric variety over $K$ associated to $\\Sigma_0$.\n\\item If $\\Sigma$ is {\\em integral}, i.e. the vertices of every polyhedron in $\\Sigma$\nlie in ${\\mathbb Z}^n$, then the special fiber $X({\\tilde \\Sigma})_k = X({\\tilde \\Sigma}) \\times_{\\operatorname{Spec} {\\mathcal O}} \\operatorname{Spec} k$\nis reduced.   \n\\item There is an inclusion-reversing bijection\nbetween closed torus orbits in $X({\\tilde \\Sigma})_k$ and polyhedra $P$ in $\\Sigma$; the irreducible \ncomponents of $X({\\tilde \\Sigma})_k$ correspond to vertices in $\\Sigma$; the intersection of a collection\nof irreducible components corresponds to the smallest polyhedron in $\\Sigma$ containing all\nof their vertices.\n\\end{itemize}\n\nNote that adjoining a $d$th root of $\\pi$ to ${\\mathcal O}$ has the effect\nof rescaling $\\Sigma$ by $d$; that is, if ${\\mathcal O}^{\\prime} = {\\mathcal O}[\\pi^{\\frac{1}{d}}]$, then\nthe base change of the family $X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ is the family \n$X(\\widetilde{(d\\Sigma)}) \\rightarrow {\\mathcal O}^{\\prime}$.  \nIn particular,\ngiven any toric scheme coming from a polyhedral complex $\\Sigma$, we can choose $d$\nsuch that $d\\Sigma$ is integral; after taking a suitable ramified base change of ${\\mathcal O}$\nthe special fiber of the family $X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ will be reduced.\n\nWe will be particularly interested in degenerations of toric varieties in which the special\nfiber is a divisor with simple normal crossings.  These are easy to construct, because the\nboundary of a smooth toric variety is always a divisor with simple normal crossings.  We make use of the following result in polyhedral combinatorics:\n\n\\begin{prop} \\label{prop:nc}\nLet $\\Sigma$ be a complete rational polyhedral complex in ${\\mathbb R}^n$. \nThere exists an integer $d$, and a subdivision $\\Sigma^{\\prime}$ of $d\\Sigma$ such that the general\nfiber of the scheme $X({\\tilde \\Sigma}^{\\prime})$ is a smooth toric variety\nand the special fiber of $X({\\tilde \\Sigma}^{\\prime})$ is a divisor with simple normal crossings.  Moreover, if the recession fan $\\Sigma_0$ is already simplicial and unimodular, $\\Sigma^{\\prime}$ can be chosen to have $\\Sigma_0'=\\Sigma_0$.\n\\end{prop}\n\\begin{proof} Choose an integer $l_1$ such that $l_1\\Sigma$ is integral.\nFulton~\\cite[Sec. 2.6]{Fulton} gives an algorithm for constructing a subdivision\n${\\tilde \\Sigma}^{\\prime}$ of the fan $\\widetilde{l_1\\Sigma}$ such that all the cones of ${\\tilde \\Sigma}_1$\nare simplicial and unimodular.  Pick an integer $l_2$ sufficiently divisible so $\\Sigma_1={\\tilde \\Sigma}^{\\prime} \\cap ({\\mathbb R}^n \\times \\{l_2\\})$ is integral.  $\\Sigma_1$ is a subdivision of ${\\mathbb R}^n$ with the property that each of its recession cones is simplicial.  Because ${\\tilde \\Sigma}_1$ is simplicial each of its cones is of the form $\\tilde{P}+Q_0$ where $P$ is a \nsimplex in $\\Sigma_1$, $Q_0$ is a cone in the recession fan $(\\Sigma_1)_0$, and $\\tilde{P}\\cap Q_0=\\{0\\}$.    Consequently, the corresponding polyhedron of $\\Sigma_1$ is $(\\tilde{P}+Q_0)\\cap ({\\mathbb R}^n\\times\\{l_2\\})=P+Q_0$.\nFor a polyhedron $F$ in ${\\mathbb R}^k$, let $N_F={\\mathbb Z}^k\\cap {\\operatorname{Span}}_{\\mathbb R}(F-w)$ \nwhere $w$ is a point of $F$.  If $F$ is a rational polytope, then $N_F$ has the property a basis of it can be extended to a basis of ${\\mathbb Z}^k$.\n\nWe claim $N_P+N_{Q_0}=N_{P+Q_0}$ for every cone $\\tilde{P}+Q_0$ of ${\\tilde \\Sigma}_1$.  It is clear that $N_P+N_{Q_0}\\subseteq N_{P+Q_0}$.  The inclusion ${\\mathbb Z}^n\\hookrightarrow{\\mathbb Z}^{n+1}={\\mathbb Z}^n\\times{\\mathbb Z}$ given by $x\\mapsto (x,l_2)$ identifies $N_{P+Q_0}$ with the intersection of $N_{\\tilde{P}+Q_0}$ with ${\\mathbb Z}^n\\times \\{l_2\\}$.   Since $\\tilde{P}+Q_0$ is unimodular, \n$N_{\\tilde{P}}+N_{Q_0}$ is equal to $N_{\\tilde{P}+Q_0}$.    Therefore, if $x\\in N_{P+Q_0}$, $(x,l_2)=x_{\\tilde{P}}+x_{Q_0}$ where $x_{\\tilde{P}}\\in \\tilde{P}$ and $x_{Q_0}\\in Q_0$.  Since the last coordinate of $x_{Q_0}$ is $0$, $x_{\\tilde{P}}=(x_P,l_2)$ for $x_P\\in N_P$.  It follows that $x=x_P+x_{Q_0}$.\n\nLet $\\Sigma_1^b$ be the union of the bounded polyhedra of $\\Sigma_1$. \nBy an important step of the proof of semi-stable reduction \\cite[Ch. 3, Thm 4.1]{KKMS}, there is an integer $l_2$ and a unimodular triangulation ${\\Sigma_1^b}'$ of $l_2\\Sigma_1^b$.  This induces a subdivision of $l_2\\Sigma_1$ where the polyhedra whose relative interior is contained in the relative interior of $l_2(P+Q_0)$  are of the form $P'+Q_0$ where $P'$ is a simplex in ${\\Sigma_1^b}'$ whose relative interior is contained in the the relative interior of $l_2P$.  We call this subdivision $\\Sigma'$.  It is simplicial by construction.  We claim that it is also unimodular.   It suffices to show that maximal cones in ${\\tilde \\Sigma}'$ are unimodular.  Let $\\tilde{P}'+Q_0$ be a  maximal cone in ${\\tilde \\Sigma}'$.  Then the relative interior of $P'$ is contained in the relative interior of $l_2P$ with $\\dim P=\\dim P'$.   Since $P'$ is unimodular, its ${\\mathbb Z}$-affine span is $N_{P'}$.    Consequently, since $N_{P'}+N_{Q_0}=N_P+N_{Q_0}=N_{P+Q_0}$, we see that any element of $N_{P+Q_0}$ can be written as integer combination $\\sum m_i v_i+\\sum n_iw_i$ where $v_i$ are vertices of $P'$, $w_i$ are the primitive vectors along the rays of $Q_0$, and $\\sum m_i=1$.  Consequently, any element of $N_{P+Q_0}\\times \\{1\\}$ can be written as  an integer combination of the primitive vectors along the rays of $\\tilde{P'}+Q_0$.  Consequently these vectors generate $N_{\\tilde{P}+Q_0}\\subset{\\mathbb Z}^{n+1}$.  Therefore $\\tilde{P}'+Q_0$ is smooth.\n\nIf $\\Sigma_0$ was simplicial and unimodular to begin with, none of these steps would have affected the cones $Q_0$ of $\\Sigma_0$\n\nThe upshot is that $X({\\tilde \\Sigma}^{\\prime})$ is a smooth toric variety\nwith a birational morphism $X({\\tilde \\Sigma}^{\\prime}) \\rightarrow X({\\tilde \\Sigma})$.  The induced map\n$X({\\tilde \\Sigma}^{\\prime}) \\rightarrow {\\mathcal O}$ is the toric scheme associated to\nthe integral polyhedral complex $\\Sigma^{\\prime}$.\nThe general fiber of $X({\\tilde \\Sigma}^{\\prime})$ over ${\\mathcal O}$ corresponds to the fan \n$\\Sigma^{\\prime}_0$, and is therefore smooth.  The special fiber is a union of\nirreducible components of the boundary of $X({\\tilde \\Sigma}^{\\prime})$ and is therefore\na divisor of $X({\\tilde \\Sigma}^{\\prime})$ with simple normal crossings.\n\\end{proof}\n\n\\section{Tropical Degenerations}\n\nWe now describe the applications of tropical geometry to the study of degenerations of varieties over $K$.\nThese techniques have their origins in the Speyer's thesis \\cite{Speyer}.  \nThe approach we take here is due to Tevelev \\cite{Tevelev} in the ``constant coefficient case''; \nthe extension of Tevelev's work to the case of an arbitrary DVR done by Luxton and Qu \\cite{LQ}.\n\nLet $\\overline{K}$ be an algebraic closure of $K$.  There is a unique valuation \n$$\\operatorname{ord}: \\overline{K} \\rightarrow {\\mathbb Q}$$\nsuch that $\\operatorname{ord}(\\pi) = 1$.  \n\nLet ${\\mathcal T} \\cong {\\mathbb G}_m^n$ be a split $n$-dimensional torus over ${\\mathcal O}$, and let\n$T = {\\mathcal T} \\times_{{\\mathcal O}} K$ be the corresponding torus over $K$.  \nThe valuation $\\operatorname{ord}$\ngives rise to a map\n$$\\operatorname{val}: {\\mathcal T}(\\overline{K}) \\rightarrow {\\mathbb Q}^n,$$\nby fixing an isomorphism of ${\\mathcal T}$ with ${\\mathbb G}_m^n$ (and hence an isomorphism of ${\\mathcal T}(\\overline{K})$ \nwith $(\\overline{K}^*)^n$.)\nLet $X$ be a closed subvariety of $T$, defined over $K$.\n\n\\begin{defn}[\\cite{EKL}, 1.2.1]: The tropical variety $\\operatorname{Trop}(X)$ associated to $X$ is\nthe closure of $\\operatorname{val}(X(\\overline{K}))$ in ${\\mathbb R}^n$.\n\\end{defn}\n\nGiven such an $X$, one can ask for a well-behaved compactification $\\overline{X}$ of $X$,\nand a well-behaved degeneration of $\\overline{X}$ over ${\\mathcal O}$.  The problem of\nfinding such a degeneration is intimately connected to the set $\\operatorname{Trop}(X)$.\n\nLet $\\Sigma$ be a completable rational polyhedral complex in ${\\mathbb R}^n$, and\nlet ${\\mathbb P}$ be the corresponding toric scheme over ${\\mathcal O}$.  Identify\nthe group of cocharacters of ${\\mathcal T}$ with ${\\mathbb Z}^n$ in ${\\mathbb R}^n$; this identifies ${\\mathbb T}$ with \nthe open torus orbit on ${\\mathbb P}$.\n\nWe can thus take the closure of ${\\mathcal X}$ of $X$ in ${\\mathbb P}$.  By \\cite{Speyer}, 2.4.1, \nthe scheme ${\\mathcal X}$ is proper over ${\\mathcal O}$ if, and only if, $\\operatorname{Supp} \\Sigma$ contains\n$\\operatorname{Trop}(X)$.  We assume henceforth that $\\operatorname{Supp} \\Sigma$ contains $\\operatorname{Trop}(X)$.  Let $\\overline{X}$ be the fiber\nof ${\\mathcal X}$ over $K$, and ${\\mathcal X}_k$ be the special fiber of ${\\mathcal X}$.  The natural multiplication map\n$${\\mathcal T} \\times_{{\\mathcal O}} {\\mathbb P} \\rightarrow {\\mathbb P}$$ \nrestricts to a multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}.$$ \n\n\\begin{defn} The pair $(X,{\\mathbb P})$ is {\\em tropical} if the map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$\nis faithfully flat, and ${\\mathcal X} \\rightarrow {\\mathcal O}$ is proper.\n\\end{defn}\n\nWe then have the following, due to Tevelev \\cite{Tevelev} in the constant coefficient case.\nIn the non-constant coefficient case they can be found in \\cite{LQ}.\n\n\\begin{prop} A subvariety $X\\subset(\\overline{K}^*)^n$ admits a tropical pair $(X,{\\mathbb P})$.  \n\\end{prop}\n\n\\begin{prop}\nSuppose $(X,{\\mathbb P})$ is tropical and let\n${\\mathbb P}^{\\prime} \\rightarrow {\\mathbb P}$ be a morphism of toric schemes corresponding\nto a refinement $\\Sigma^{\\prime}$ of $\\Sigma$.  Then $(X,{\\mathbb P}^{\\prime})$ is also tropical. \n\\end{prop}\n\n\\begin{prop} If $(X,{\\mathbb P})$ is a tropical pair then $\\operatorname{Supp} \\Sigma = \\operatorname{Trop}(X)$.\n\\end{prop}\n\nFollowing Speyer (\\cite{Speyer}, 2.4) If $(X,{\\mathbb P})$ is a tropical pair, we call $\\overline{X}$ a \n{\\em tropical compactification} of $X$, and ${\\mathcal X}_k$ a {\\em tropical degeneration} of $X$.\n\nThe combinatorics of the special fiber of a tropical degeneration of $X$ is closely\nrelated to the combinatorics of $\\operatorname{Trop}(X).$  In particular if $(X,{\\mathbb P})$ is a tropical\npair, and ${\\mathcal X}$ is the corresponding tropical degeneration, then a polyhedron $P$\nof $\\Sigma$ corresponds to the closure of a torus orbit in the special fiber of ${\\mathbb P}$.  \nCall this torus orbit closure ${\\mathbb P}_P$.  Then the intersection ${\\mathcal X}_P$ of ${\\mathcal X}$ with ${\\mathbb P}_P$ \nis a closed subscheme of ${\\mathcal X}_k$.  Moreover, if $P$ and $P^{\\prime}$ are\npolyhedra of $\\Sigma$, and $Q$ is the smallest polyhedron in $\\Sigma$\ncontaining both $P$ and $P^{\\prime}$, then the intersection of ${\\mathcal X}_P$ and ${\\mathcal X}_{P^{\\prime}}$\nis ${\\mathcal X}_Q$.\n\nLet $U_P$ be the open torus orbit corresponding to $P$.\nFix a point $p$ in $U_P$, and consider the fiber over $p$\nof the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}.$$\nOn the one hand, $m^{-1}(p)$ is nonempty of dimension equal to the dimension of $X$,\nsince $m$ is flat and surjective.  On the other hand, by projection onto ${\\mathcal X}$, $m^{-1}(p)$ \nis isomorphic to the product ${\\mathcal T}_P\\times ({\\mathcal X} \\cap U_P)$, where ${\\mathcal T}_P$ is the subgroup \nof ${\\mathcal T}_0$ that acts trivially on $U_P$.  Since $({\\mathcal X} \\cap U_P)$ is dense in ${\\mathcal X}_P$ we find that\n${\\mathcal X}_P$ is nonempty of dimension equal to the dimension of $X$ minus the dimension of $P$.\n\nOn the other hand, let $w$ be a point in the relative interior of $P$.  Then $w$ corresponds\nto a cocharacter of $T$, and $w(\\pi)$ specializes to a point $p$ in $U_P$.   Projection\nonto ${\\mathcal T}$ identifies $m^{-1}(p)$ with the mod $\\pi$ reduction $\\operatorname{in}_w X$ of $w(\\pi)X$.\n(More formally, $\\operatorname{in}_w X$ can be defined as the special fiber of the closure in ${\\mathcal T}$\nof the subscheme $w(\\pi)X$ of $T$.  Note that this depends only on $X$ and $T$, not on our choice of\n$\\Sigma$.) In particular we have \n$$\\operatorname{in}_w X \\cong {\\mathcal T}_P \\times ({\\mathcal X} \\cap U_P)$$ \nfor any $w$ in the relative interior of $P$.  We have thus shown:\n\n\\begin{lemma} \\label{lemma:comp}\nThe space $\\operatorname{in}_w X$ is a torus bundle over ${\\mathcal X} \\cap U_P$.  In particular, if\n$C(\\operatorname{in}_w X)$ is the set of connected components of $(\\operatorname{in}_w X)_{\\overline{k}}$,\nand $C({\\mathcal X} \\cap U_P)$ is the set of connected components of $({\\mathcal X} \\cap U_P)_{\\overline{k}}$,\nthen the maps\n$$\\operatorname{in}_w X \\cong m^{-1}(p) \\rightarrow {\\mathcal X} \\cap U_P$$  \ngive a natural bijection of $C(\\operatorname{in}_w X)$ with $C({\\mathcal X} \\cap U_P)$.\n\\end{lemma}\n\nWe will be particularly interested in tropical pairs $(X,{\\mathbb P})$ where the multiplication\nmap $m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$ is {\\em smooth}.  This condition is\ndue to Tevelev.\n\n\\begin{defn} A subvariety $X$ of ${\\mathcal T}$ is {\\em sch\\\"on}\nif there exists a tropical pair $(X,{\\mathbb P})$ such that the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ is smooth.\n\\end{defn}\n\nOne then has (\\cite{LQ}, 6.7):\n\\begin{prop} If $X$ is sch\\\"on, then for {\\em any} tropical pair\n$(X,{\\mathbb P})$, the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ \nis smooth.\n\\end{prop}\n\nNote that if $X$ is sch\\\"on then it is smooth (consider the preimage of the identity in T\nunder the multiplication map.)  In fact, we have:\n\n\\begin{prop} \\label{prop:schon} The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is sch\\\"on.\n\\item $\\operatorname{in}_w X$ is smooth for all $w \\in \\operatorname{Trop}(X)$.\n\\item For any tropical pair $(X,{\\mathbb P})$, and any polyhedron $P$ in $\\Sigma$,\n${\\mathcal X} \\cap U_P$ is smooth.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nStatements 2) and 3) are clearly equivalent since we have seen that $\\operatorname{in}_w X$ is the product\nof a torus with ${\\mathcal X} \\cap U_P$, where $P$ is the polyhedron in $\\Sigma$ that contains $w$ in its\nrelative interior.\n\nAs for the equivalence of 1) and 2), fix a tropical pair $(X,{\\mathbb P})$.  We have seen that the \nfibers of the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ \nare isomorphic to $\\operatorname{in}_w X$ as $w$ ranges over $\\operatorname{Trop}(X)$.  So 1) implies 2) is clear.\nFor the converse, note that since $m$ is faithfully\nflat, to show it is smooth it suffices to show that it has smooth fibers.  \n\\end{proof}\n\nIt is easy to construct tropical degenerations of sch\\\"on varieties $X$ in which the special\nfiber is a divisor with simple normal crossings.  In particular we have:\n\n\\begin{prop}[c.f.~\\cite{Hacking}, proof of 2.4] \\label{prop:degeneration}\nLet $X$ be sch\\\"on.  There exists an integer $d$ and a tropical pair $(X,{\\mathbb P})$ over \n${\\mathcal O}[\\pi^{\\frac{1}{d}}]$ such that\n$\\overline{X}$ is smooth over $K[\\pi^{\\frac{1}{d}}]$, and ${\\mathcal X}_k$ is a divisor in ${\\mathcal X}$ with \nsimple normal crossings.  \n\\end{prop}\n\\begin{proof}\nLet $(X,{\\mathbb P})$ be any tropical pair over ${\\mathcal O}$, and let $\\Sigma$ be the rational polyhedral\ncomplex corresponding to ${\\mathbb P}$.  By Proposition~\\ref{prop:nc}, we can\nfind a refinement $\\Sigma^{\\prime}$ of $\\Sigma$ and an integer $d$ such that the corresponding toric \nscheme ${\\mathbb P}^{\\prime}$ (viewed over ${\\mathcal O}[\\pi^{\\frac{1}{d}}]$)\nhas smooth general fiber, and special fiber a divisor with simple normal crossings.\n\nThen $(X,{\\mathbb P}^{\\prime})$ is also tropical, and the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X}^{\\prime} \\rightarrow {\\mathbb P}^{\\prime}$$\nis smooth by the previous proposition.  Since the special fiber of ${\\mathbb P}^{\\prime}$ is a divisor\nwith simple normal crossings, so is the special fiber of ${\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X}^{\\prime}$.\nHence the special fiber of ${\\mathcal X}^{\\prime}$ is a divisor with simple normal crossings as well.\nSimilarly, the general fiber of ${\\mathcal X}^{\\prime}$ is smooth because the general fiber of\n${\\mathbb P}^{\\prime}$ is smooth.\n\\end{proof}\n\n\\begin{defn} We call a pair $(X,{\\mathbb P})$ of the sort produced by Proposition~\\ref{prop:degeneration}\na {\\em normal crossings pair}.  If $(X,{\\mathbb P})$ is a normal crossings pair, and $\\Sigma$\nis the polyhedral decomposition of $\\operatorname{Trop}(X)$ corresponding to ${\\mathbb P}$, we say that\n$\\Sigma$ is a {\\em normal crossings decomposition} of $\\operatorname{Trop}(X)$.\n\\end{defn}\n\n\\begin{rem} \\rm In much of what follows, we will often need to attach a normal\ncrossings pair to a sch\\\"on variety $X$ over ${\\mathcal O}$.  To do this we may need to replace\n${\\mathcal O}$ with a ramified extension ${\\mathcal O}[\\pi^{\\frac{1}{d}}]$; this is harmless and we often\ndo so without comment.\n\\end{rem}\n\n\\section{Parameterized Tropical Varieties} \\label{sec:param}\n\nIn this section, given a sch\\\"on subvariety $X$ of ${\\mathcal T}$, we construct a\nnatural parameterization of $\\operatorname{Trop}(X)$ by a topological space $\\Gamma_X$.  This\nparameterization is functorial in a sense we make precise below.  Moreover, we\nwill see in the next section that the space $\\Gamma_X$ encodes more precise information\nabout the cohomology of $X$ than $\\operatorname{Trop}(X)$ does.  Our approach generalizes a construction\nof Speyer (\\cite{Sp3}, proof of Theorem 10.8) when $X$ has dimension $1$.\n\nSuppose we have a normal crossings pair $(X,{\\mathbb P})$ so $\\operatorname{Supp}(\\Sigma)=\\operatorname{Trop}(X)$.  We associate to $(X,{\\mathbb P})$\na polyhedral complex $\\Gamma_{(X,{\\mathbb P})}$ as follows: its $k$-cells are pairs\n$(P,Y)$, where $P$ is a $k$-dimensional polyhedron in $\\Sigma$ and $Y$ is an irreducible component of\n${\\mathcal X}_P$.  The cells on the boundary of $(P,Y)$ are the cells of the form $(P_i,Y_i)$,\nwhere $P_i$ is a facet of $P$ and $Y_i$ is the irreducible\ncomponent of ${\\mathcal X}_{P_i}$ containing $Y$ (there is exactly one such irreducible component\nbecause ${\\mathcal X}_{P_i}$ is smooth, so its irreducible components do not meet).  The complex \n$\\Gamma_{(X,{\\mathbb P})}$ maps naturally to $\\Sigma$ by sending $(P,Y)$ to $P$.\n\n\\begin{prop} \\label{prop:independence}\nThe underlying topological space of $\\Gamma_{(X,{\\mathbb P})}$ depends only on $X$.\n\\end{prop}\n\\begin{proof}\nAny two polyhedral decompositions of $\\operatorname{Trop}(X)$ have a common refinement; we can further\nrefine this to be a normal crossings decomposition of $\\operatorname{Trop}(X)$.  It thus suffices to show that if \n$\\Sigma$ and $\\Sigma^{\\prime}$ are normal crossings decompositions of $\\operatorname{Trop}(X)$, with associated\nnormal crossings pairs $(X,{\\mathbb P})$ and $(X,{\\mathbb P}^{\\prime})$, and\n$\\Sigma^{\\prime}$ refines $\\Sigma$, then the underlying topological spaces of\n$\\Gamma_{(X,{\\mathbb P})}$ and $\\Gamma_{(X,{\\mathbb P}^{\\prime})}$ are homeomorphic.\n\nSince $\\Sigma^{\\prime}$ is a refinement of $\\Sigma$, we have a map ${\\mathbb P}^{\\prime} \\rightarrow {\\mathbb P}$.\nLet ${\\mathcal X}^{\\prime}$ be the degeneration corresponding to the pair $(X,{\\mathbb P}^{\\prime})$.\nIf $P$ is a polyhedron of $\\Sigma$, and $P^{\\prime}$ is a polyhedron of $\\Sigma^{\\prime}$\ncontained in $P$, then this map induces a map of ${\\mathcal X}^{\\prime}_{P^{\\prime}}$ to ${\\mathcal X}_P$.\nIn particular, for every pair $(P^{\\prime},Y^{\\prime})$ of $\\Gamma_{X,{\\mathbb P}^{\\prime}}$,\nthe image of $Y^{\\prime}$ in ${\\mathcal X}$ is contained in an unique irreducible component $Y$ of\n${\\mathcal X}_P$.  The map taking $(P^{\\prime},Y^{\\prime})$ to $(P,Y)$ is then a map of polyhedral\ncomplexes $\\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\Gamma_{(X,{\\mathbb P})}.$\n\nWe have a commutative diagram:\n\\begin{diagram}\n\\Gamma_{(X,{\\mathbb P}^{\\prime})} & \\rTo & \\operatorname{Trop}(X) \\\\\n\\dTo & \\ruTo & \\\\\n\\Gamma_{(X,{\\mathbb P})} & & \n\\end{diagram}\nThe fiber of $\\pi: \\Gamma_{(X,{\\mathbb P})}\\rightarrow \\operatorname{Trop}(X)$ over a point $w$ is in canonical\nbijection with the set $C({\\mathcal X}_P)$ of (geometric) connected components of ${\\mathcal X}_P$.  Similarly, the fiber\nof $\\pi^{\\prime}: \\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\operatorname{Trop}(X)$ over $w$ is in canonical bijection\nwith $C({\\mathcal X}^{\\prime}_{P^{\\prime}})$.  By Lemma~\\ref{lemma:comp}, both of these sets of connected components\nare in bijection with $C(\\operatorname{in}_w X)$; in fact, we have a commutative diagram:\n\\begin{diagram}\nC(\\operatorname{in}_w X) & \\rTo & (\\pi^{\\prime})^{-1}(w)\\\\ \n\\dEq & & \\dTo\\\\\nC(\\operatorname{in}_w X) & \\rTo & \\pi^{-1}(w)\n\\end{diagram}\nThus the map $\\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\Gamma_{(X,{\\mathbb P})}$\nis bijective, and is therefore a homeomorphism on the underlying topological\nspaces of $\\Gamma_{(X,{\\mathbb P}^{\\prime})}$ and $\\Gamma_{(X,{\\mathbb P})}$.\n\\end{proof}\n\nIn light of this proposition, we denote by $\\Gamma_X$ the underlying topological\nspace of $\\Gamma_{(X,{\\mathbb P})}$ for {\\em any} normal crossings pair $(X,{\\mathbb P})$.  \nWe think of $\\Gamma_X$, together\nwith its natural map to $\\operatorname{Trop}(X)$, as a ``parameterized tropical variety.''\nNote that $\\Gamma_X$ inherits an integral affine structure by pullback from\n$\\operatorname{Trop}(X)$; more precisely, for any normal crossings pair $(X,{\\mathbb P})$,\nthe map $\\Gamma_X \\rightarrow \\operatorname{Trop}(X)$ is\nlinear on any polyhedron in $\\Gamma_{(X,{\\mathbb P})}$.\n\nNote that for any $w \\in \\operatorname{Trop}(X)$, the number of preimages\nof $w$ in $\\Gamma_X$ is equal to the number of connected components of\n$\\operatorname{in}_w X$.  Therefore, if $\\Sigma$ is a normal crossings decomposition \nof $\\operatorname{Trop}(X)$, and $w$ is in the relative interior\nof a top dimensional cell $P$ of $\\Sigma$, then the number of preimages of\n$w$ is equal to the {\\em multiplicity} of $P$ in $\\operatorname{Trop}(X)$.  This suggests\nthat we should give $\\Gamma_X$ the structure of a weighted polyhedral complex\nby giving every polyhedron on $\\Gamma_X$ weight one.  \n\nIf we do this, then $\\Gamma_X$ satisfies a ``balancing condition'' analogous\nto the well-known balancing condition on $\\operatorname{Trop}(X)$.  Fix a normal\ncrossings decomposition $\\Sigma$ of $\\operatorname{Trop}(X)$, with corresponding normal\ncrossings pair $(X,{\\mathbb P})$.  Consider a polyhedron $(P,Y)$ of $\\Gamma_{(X,{\\mathbb P})}$\nof dimension $\\dim X - 1$, and let $\\{(P_i,Y_i)\\}$ be the top dimensional\npolyhedra of $\\Gamma_{(X,{\\mathbb P})}$ containing $(P,Y)$.\n\nFix a point $w$ with rational coordinates in the relative interior of $P$, and let $V_P$\nbe the linear span, ${\\operatorname{Span}}(P - w)$.  Similarly, for each $P_i$,\nlet $V_i$ be the positive span of ${\\operatorname{Span}}^+(P_i - w)$.\nThen $V_i\/V_P$ is a ray in ${\\mathbb R}^n\/V_P$; this collection of rays is the fan attached to\nthe toric variety ${\\mathbb P}_P$.  Let $v_i$ be the smallest integer vector along the ray $V_i\/V_P$.\n\n\\begin{prop} \\label{prop:balancing}\nThe $v_i$'s satisfy the ``balancing property'':\n$$\\sum_{(P_i,Y_i)} v_i = 0.$$\n\\end{prop}\n\\begin{proof}\nTorus-equivariant rational functions on ${\\mathbb P}_P$ correspond to lattice vectors $u$ in\nthe space $({\\mathbb R}^n\/V_P)^*$ dual to ${\\mathbb R}^n\/V_P$.  The valuation of $u$ along the divisor\nof ${\\mathbb P}_P$ corresponding to $v_i$ is simply $u(v_i)$.\n\nNow restrict $u$ to the curve ${\\mathcal X}_P$.  For any polyhedron $P^{\\prime}$ of $\\Sigma$ containing\n$P$, ${\\mathcal X}_P$ intersects the boundary divisior ${\\mathbb P}_{P^{\\prime}}$ in one point for each\ncell $(P_i,Y_i)$ of $\\Gamma_{(X,{\\mathbb P})}$ with $P_i = P^{\\prime}$.\nThe divisor of $u$ is therefore equal to\n$$\\sum_{(P_i,Y_i)} u(v_i) Y_i,$$\nas ${\\mathcal X}_P$ intersects each boundary divisor ${\\mathbb P}_{P_i}$ transversely.\nThis divisor is a principal divisor and thus has degree zero.\n\\end{proof}\n\nWe have thus attached to any sch\\\"on subvariety $X$ of ${\\mathcal T}$, a canonical, multiplicity\nfree parameterization by the topological space $\\Gamma_X$.  Moreover, this construction is \nfunctorial:  let ${\\mathcal T}$ and ${\\mathcal T}^{\\prime}$ be tori over ${\\mathcal O}$, and let\n$T$ and $T^{\\prime}$ be their general fibers.  Suppose we have sch\\\"on subvarieties\n$X$ and $Y$ of $T$ and $T^{\\prime}$, respectively, and a homomorphism of tori\n$T \\rightarrow T^{\\prime}$ that takes $X$ to $Y$.  We then have a natural map\n$f:\\operatorname{Trop}(X) \\rightarrow \\operatorname{Trop}(Y)$.\n\n\\begin{prop} \\label{prop:functoriality}\nThere is a natural map\n$\\Gamma_X \\rightarrow \\Gamma_Y$ that makes the diagram\n\\begin{diagram}\n\\Gamma_X & \\rTo & \\Gamma_Y\\\\\n\\dTo & & \\dTo\\\\\n\\operatorname{Trop}(X) & \\rTo & \\operatorname{Trop}(Y)\n\\end{diagram}\ncommute.\n\\end{prop}\n\\begin{proof}\nLet $\\Sigma^{\\prime}$ be a normal crossings decomposisition of $\\operatorname{Trop}(Y)$. \nBy proposition~\\ref{prop:degeneration}\nwe can find a normal crossings decomposition $\\Sigma$ of $\\operatorname{Trop}(X)$ \nsuch that the image of any cell of $\\Sigma$ under the map $f$ \nis contained in a cell of $\\Sigma^{\\prime}$.\nLet $(X,{\\mathbb P})$ and $(Y,{\\mathbb P}^{\\prime})$ be the tropical pairs corresponding to\n$\\Sigma$ and $\\Sigma^{\\prime}$, and let ${\\mathcal X}$ and ${\\mathcal Y}$ denote the associated\ntropical degenerations.  Since each cell of $\\Sigma$ maps into a cell of\n$\\Sigma^{\\prime}$, we obtain a map from ${\\mathcal X}$ to ${\\mathcal Y}$ extending the map\n$X \\rightarrow Y$.\n\nNow let $P$ be a polyhedron in $\\Sigma$, and $P^{\\prime}$ be the polyhedron\nof $\\Sigma^{\\prime}$ containing the image of $P$.  Then our map\n${\\mathcal X} \\rightarrow {\\mathcal Y}$ induces a map ${\\mathcal X}_P \\rightarrow {\\mathcal Y}_{P^{\\prime}}$.\n\nIf $(P,X_i)$ is a polyhedron of $\\Gamma_{(X,{\\mathbb P})}$, then by definition\n$X_i$ is a connected component of ${\\mathcal X}_P$.  The image of $X_i$\nin ${\\mathcal Y}_{P^{\\prime}}$ is contained in a connected component $Y_i$ of\n${\\mathcal Y}_{P^{\\prime}}$.  We can thus construct a map of polyhedral complexes\n$$\\Gamma_{(X,{\\mathbb P})} \\rightarrow \\Gamma_{(Y,{\\mathbb P}^{\\prime})}$$\nthat maps $(P,X_i)$ to $(P^{\\prime},Y_i)$ by the map $P \\rightarrow P^{\\prime}$.\nThe induced map $\\Gamma_X \\rightarrow \\Gamma_Y$ on underlying topological\nspaces is clearly continuous and makes the diagram commute.  \n\nTo see that it is independent of choices, let $\\pi_X$ and $\\pi_Y$\nbe the projections of $\\Gamma_X$ and $\\Gamma_Y$ to $\\operatorname{Trop}(X)$ and $\\operatorname{Trop}(Y)$\nrespectively.  We then have canonical bijections between\n$\\pi_X^{-1}(w)$ and $C(\\operatorname{in}_w X)$, and between $\\pi_Y^{-1}(f(w))$ \nand $C(\\operatorname{in}_{f(w)} Y)$.  The map $X \\rightarrow Y$ induces a natural map\n$\\operatorname{in}_w X \\rightarrow \\operatorname{in}_{f(w)} Y$, and the diagram\n\\begin{diagram}\nC(\\operatorname{in}_w X) & \\rTo & \\pi_X^{-1}(w)\\\\\n\\dTo & & \\dTo\\\\\nC(\\operatorname{in}_{f(w)} Y) & \\rTo & \\pi_Y^{-1}(f(w))\n\\end{diagram}\ncommutes.  As the left hand side is independent of the choices of $\\Sigma$\nand $\\Sigma^{\\prime}$, the result follows.\n\\end{proof}\n\n\\begin{rem} \\rm Although Proposition~\\ref{prop:functoriality} is stated for\nmaps $X \\rightarrow Y$ that are {\\em monomial morphisms} (i.e., that arise from\nmorphisms of the ambient tori), we can avoid this issue if $X$ and $Y$\nare intrinsically embedded.  Recall that $X$ is {\\em very affine} if it can\nbe embedded as a closed subscheme of a torus $T$.  In this case (c.f.\n~\\cite{Tevelev}, section 3) there is an intrinsic torus $T_X$ associated to $X$\na {\\em canonical} embedding of $X$ in $T_X$.  Moreover, if $X$ and $Y$ are\nvery affine and $f: X \\rightarrow Y$ is a morphism, there is a morphism\nof tori $T_X \\rightarrow T_Y$ that induces $f$.\n\\end{rem}\n\nWe also record, for later use, the following result relating the cohomology of\n$\\Gamma_X$ to that of $\\operatorname{Trop}(X)$:\n\n\\begin{lemma} \\label{lemma:cohomology}\nLet $X$ be sch\\\"on, and let $\\Sigma$ be a normal crossings decomposition of $\\operatorname{Trop}(X)$.\nSuppose that for each polyhedron $P$ in $\\Sigma$, ${\\mathcal X}_P$ is either connected or\nhas dimension zero.  Then the natural\nmap \n$$H^r(\\operatorname{Trop}(X),{\\mathbb Z}) \\rightarrow H^r(\\Gamma_X,{\\mathbb Z})$$\nis an isomorphism for $0 \\leq r < \\dim X$, and an injection for $r = \\dim X$.\n\\end{lemma}\n\\begin{proof}\nLet $(X,{\\mathbb P})$ be the normal crossings pair attached to $\\Sigma$, so that\n$\\Gamma_{(X,{\\mathbb P})}$ is a triangulation of $\\Gamma$.  The polyhedra $P$ in $\\Gamma_{(X,{\\mathbb P})}$\nwith $\\dim P < \\dim X$ are, by construction, in bijection with the polyhedra in $\\Sigma$  \nwith $\\dim P < \\dim X$.  Thus $\\Gamma_{(X,{\\mathbb P})}$ is obtained from $\\Sigma$ by adding\nadditional top-dimensional cells; the result follows immediately.\n\\end{proof}\n\n\\section{Weight filtrations and the weight spectral sequence} \\label{sec:weight}\n\nOur goal will be to relate the combinatorial structure of $\\Gamma_X$ to geometric\ninvariants of $X$.  The invariants that appear arise from Deligne's theory of weights, which\nwe now summarize.  Recall (c.f.~\\cite{WeilII}, 1.2) that if $F$ is a finite field of order $q$, a continuous $l$-adic \nrepresentation $\\rho$ of $\\operatorname{Gal}(F^{\\operatorname{sep}}\/F)$ has weight $r$ if all the eigenvalues of the geometric \nFrobenius of $F$ are algebraic integers $\\alpha$, all of whose Galois conjugates have complex absolute \nvalue equal to $q^{r\/2}$.  If $A$ is a finitely generated ${\\mathbb Z}$-algebra, an {\\'e}tale sheaf \n${\\mathcal F}$ on $\\operatorname{Spec} A$ \nhas weight $r$ if for each closed point $s$ of $\\operatorname{Spec} A$, the stalk ${\\mathcal F}_s$ has weight $r$ when \nconsidered as a $\\operatorname{Gal}(k(s)^{\\operatorname{sep}}\/k(s))$-module.\n\nFollowing Ito (\\cite{Ito}, 2.2), we extend this definition to the case where $F$\nis a purely inseparable extension of a finitely generated extension of ${\\mathbb F}_p$ or ${\\mathbb Q}$.  For\nsuch $F$, one can find a finitely generated ${\\mathbb Z}$-subalgebra $A$ of $F$ such that $F$ is a purely\ninseparable extension of the field of fractions of $A$.  \n\nIn this setting, a representation $\\rho$ of $\\operatorname{Gal}(F^{\\operatorname{sep}}\/F)$ has weight $r$ if there is an \nopen subset $U$ of $\\operatorname{Spec} A$, and a smooth ${\\mathcal F}$ on $U$ of weight $r$, such that $\\rho$ arises from ${\\mathcal F}$\nby pullback to $\\operatorname{Spec} F$.  The Weil conjectures imply that for any proper smooth\nvariety $X$ over $F$, and any $l$ prime to the characteristic of $F$,\n$H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{F^{\\operatorname{sep}}},{\\mathbb Q}_l)$ has weight $r$. \n\nWe henceforth assume that the residue field $k$ of ${\\mathcal O}$ is a purely inseparable\nextension of a finitely generated extension of ${\\mathbb F}_p$ or ${\\mathbb Q}$.  \nWe also fix an $l$ prime to the characteristic of $k$.\n\nLet $G$ be the absolute Galois group of the field $K$.  Then $G$ admits a\nsurjection $G \\rightarrow \\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$, whose kernel is the inertia group $I_K$\nof $K$.  If $M$ is a $G$-module on which $I$ acts through a finite quotient,\nthere is a finite index subgroup $H$ in $G$ such that $H \\cap I$ acts trivially\non $M$.  Thus $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k^{\\prime})$ acts on $M$ for some finite extension\n$k^{\\prime}$ of $k$.  We say $M$ is pure of weight $r$ if it has weight $r$ as a\n$\\operatorname{Gal}(k^{\\operatorname{sep}}\/k^{\\prime})$-module.  Note that this is independent of $k^{\\prime}$.\n\nThe {\\'e}tale cohomology of a variety over $K$ with semistable reduction has a filtration by subquotients\nwhich are pure in the above sense.  More precisely, let ${\\mathcal X}$ be a proper scheme over ${\\mathcal O}$, of relative \ndimension $n$,\nwhose fiber $X_K$ over $\\operatorname{Spec} K$ is smooth and whose fiber ${\\mathcal X}_k$ over $\\operatorname{Spec} k$ is a divisor with\nsimple normal crossings.  Then the Rapoport-Zink weight spectral sequence relates the {\\'e}tale cohomology \nof $X_{K^{\\operatorname{sep}}}$ to the geometry of the special fiber ${\\mathcal X}_k$.  More\nprecisely, let ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(r)}$ denote the disjoint union of $(r+1)$-fold intersections\nof irreducible components of ${\\mathcal X}_{k^{\\operatorname{sep}}}$; it is smooth of dimension $n-r$ over $k^{\\operatorname{sep}}$.\nWe then have:\n\n\\begin{thm}[\\cite{RZ}, Satz 2.10; see also~\\cite{Ito}]\nThere is a spectral sequence:\n$$E_1^{-r,w+r} = \\bigoplus_{s \\geq \\max(0,-r)} H^{w-r-2s}_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(2s + r)}, {\\mathbb Q}_l(-r-s))\n\\Rightarrow H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l).$$\n\\end{thm}\n\nHere ${\\mathbb Q}_l(n)$ is the $n$th ``Tate twist'' of the constant sheaf ${\\mathbb Q}_l$; that is, it is the\ntensor product of ${\\mathbb Q}_l$ with the $n$th tensor power of the sheaf ${\\mathbb Z}_l(1)$, where\n${\\mathbb Z}_l(1)$ is the inverse limit of the sheaves $\\mu_{l^n}$ of $l$-power roots of unity.  Note that ${\\mathbb Z}_l(1)$\nis pure of weight $-2$.\n\nThe boundary maps of this spectral sequence are completely explicit, and can be described as follows: \nup to sign, they are direct sums of restriction maps\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$$\nwhere $Y$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j)}$ and\n$Y^{\\prime}$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j+1)}$ contained in $Y$,\nor Gysin maps\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)) \\rightarrow H^{i+2}_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1))$$\nwhere $Y$ and $Y^{\\prime}$ are as above.\n\nMore precisely, each term $E_1^{p,q}$ is a direct sum of terms of the form\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m))$ for some irreducible component $Y$ of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j)}$.\nIf $Y^{\\prime}$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j+1)}$, then we have:\n\\begin{itemize}\n\\item Whenever $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p+1,q}$, then\nthe corresponding direct summand of the boundary map $E_1^{p,q} \\rightarrow E_1^{p+1,q}$\nis (up to sign) the restriction\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)).$$\n\\item Whenever $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1))$ is a direct summand of $E_1^{p+1,q}$, then\nthe corresponding direct summand of the boundary map $E_1^{p,q} \\rightarrow E_1^{p+1,q}$\nis (up to sign) the Gysin map\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)) \\rightarrow H^{i+2}_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1)).$$\n\\end{itemize}\n\nWe refer the reader to example \\ref{ex:curves} for a description of the weight spectral\nsequence in the case when $X$ is a smooth curve.\n\nNote that the term $E_1^{-r,w+r}$ of the weight spectral sequence is pure of weight $w+r$.\nAs the only map between ${\\mathbb Q}_l$-sheaves that are pure of different weights is the zero map,\nthis implies that the weight spectral sequence degenerates at $E_2$.  Moreover, the\nsuccessive quotients of the filtration on $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ induced\nby the weight spectral sequence are pure.  The filtration arising in this way is called\nthe {\\em weight filtration} on $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\nWe say a $G$-module $M$ is {\\em mixed} if $M$ admits an increasing $G$-stable filtration\n$$\\dots \\subset W_rM \\subset W_{r+1}M \\subset \\dots$$\nsuch that $W_rM\/W_{r-1}M$ has weight $r$ for all $r$.  (Such a filtration, if it exists, will\nbe unique.)  We say $M$ is mixed of weights between $r$ and $r^{\\prime}$ if $M$ is mixed\nand the quotients $W_iM\/W_{i-1}M$ are nonzero only when $r \\leq i \\leq r^{\\prime}$.\nThe above result shows that the cohomology of any scheme over $K$ with semistable reduction\nis mixed.  More generally, one has:\n\n\\begin{thm} (c.f.~\\cite{Ito}, 2.3) Let $X$ be a smooth, proper $n$-dimensional variety over \n$K$.  Then $H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ is mixed of weights between\n$\\max(0,2r-2n)$ and $\\min(2n,2r)$.\n\\end{thm}\n\\begin{proof}\nIf $X$ has strictly semistable reduction, i.e., $X$ is isomorphic to the general fiber of a \nscheme ${\\mathcal X}$ that is proper over ${\\mathcal O}$, and whose special fiber is a divisor with \nsimple normal crossings, then this follows from the weight spectral sequence.  \nThe general case follows by de Jong's theory of alterations~\\cite{dJ}.\n\\end{proof}\n\n\\begin{prop} \\label{prop:cohomology}\nLet $X$ be a smooth $n$-dimensional variety over $K$, and $\\overline{X}$ a compactification\nof $X$ such that $\\overline{X} - X$ is a divisor with simple normal crossings.  Then for $r \\leq n$,\n$H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ is mixed of weights between $0$ and $2r$, and the natural map\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l) \\rightarrow W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$$\nis an isomorphism.\n\\end{prop}\n\\begin{proof}\nLet $D$ be the divisor $\\overline{X} \\setminus X$, and let $\\overline{D}_1, \\dots, \\overline{D}_r$ be \nits irreducible components.  Let $X_i$ be the open subset $X \\setminus \n\\{\\overline{D}_1 \\cup \\dots \\cup \\overline{D}_i\\}$.  We proceed\nby induction on $i$; the case $i = 0$ is clear. \n\nSuppose the proposition is true for $i$.  Define \n$$D_i = \\overline{D}_{i+1} \\setminus \\{\\overline{D}_1 \\cup \\dots \\cup \\overline{D}_i\\},$$\nso that $X_i \\setminus X_{i+1} = D_i$.\nBy the inductive hypothesis the spaces $H^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l)$\nand $H^r_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l)$\nare mixed of weights between $0$ and $2r$ for $r \\leq n$.  We have a Gysin sequence:\n$$\n\\begin{array}{cccccc}\nH^{r-2}_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l(-1)) & \\rightarrow &\nH^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l) & \\rightarrow & \nH^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l) & \\rightarrow \\\\ \nH^{r-1}_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l(-1)) & \\rightarrow & \\dots,\n\\end{array}\n$$\nand the first and last terms are mixed of weights between $2$ and $2r$.  It follows that\n$H^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l)$ is mixed of weights between $0$ and $2r$ as\nrequired.  We also obtain an isomorphism\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l) \\cong W_0H^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l)$$ \nand hence by induction the desired isomorphism\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_L, {\\mathbb Q}_l) \\cong W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(X_L, {\\mathbb Q}_l).$$\n\\end{proof}\n\n\\section{Cohomology of sch\\\"on varieties} \\label{sec:schon}\n\nThe control that tropical geometry gives over the degenerations of sch\\\"on subvarieties\n$X$ of $T$ has significant consequences on the level of cohomology.  In particular the theory\nof vanishing cycles allows one to relate the {\\'e}tale cohomology of a nice tropical compactification \nof $X$ to that of its tropical degeneration.  When the degeneration is a divisor with\nsimple normal crossings, this relationship is given by the Rapoport-Zink weight spectral\nsequence. \n\nWe apply this sequence in the setting of tropical geometry.  Let $X$ be sch\\\"on.  By\nProposition~\\ref{prop:degeneration} there is a polyhedral complex $\\Sigma$, with\nsupport equal to $\\operatorname{Trop}(X)$ and corresponding toric scheme ${\\mathbb P}$, such that the pair \n$(X,{\\mathbb P})$ is tropical, the corresponding compactification $\\overline{X}$ of $X$\nis smooth with simple normal crossings boundary, and the special fiber of the corresponding\ntropical degeneration ${\\mathcal X}$ is a divisor with simple normal crossings.\n\nThe polyhedral complex $\\Gamma_{(X,{\\mathbb P})}$ encodes the combinatorics of the special fiber\n${\\mathcal X}_{k^{\\operatorname{sep}}}$.  In particular ${\\mathcal X}_{k^{\\operatorname{sep}}}$ is a union of smooth \nconnected varieties ${\\mathcal X}_v$, where $v$ runs over the vertices of $\\Gamma_{(X,{\\mathbb P})}$.  The varieties\n${\\mathcal X}_{v_1}, \\dots, {\\mathcal X}_{v_r}$ meet if and only if $v_1, \\dots, v_r$ are the vertices of\na polyhedron in $\\Gamma_{(X,{\\mathbb P})}$.  \n[Note that since ${\\mathcal X}_k$ is a simple normal crossings divisor, if $Y_0, \\dots, Y_r$ intersect\nin codimension $r$ then they are the only irreducible components of ${\\mathcal X}_{k^{\\operatorname{sep}}}$ containing \ntheir intersection.]  \n\nWe have a natural map $\\Gamma_{(X,{\\mathbb P})} \\rightarrow \\Sigma$. \nSince $\\Sigma$ is a triangulation of $\\operatorname{Trop}(X)$, and $\\Gamma_{(X,{\\mathbb P})}$ is a triangulation\nof $\\Gamma_X$, this induces a natural map\n$$H^r(\\operatorname{Trop} X, {\\mathbb Q}_l) \\rightarrow H^r(\\Gamma_X, {\\mathbb Q}_l).$$\nBy the proof of Lemma \\ref{lemma:cohomology}, this map is an isomorphism if ${\\mathcal X}_P$ is connected for\nevery polyhedron $P$ in $\\Sigma$, or (equivalently) if $\\operatorname{in}_w X$ is connected\nfor every $w$ in $\\operatorname{Trop}(X)$. \n\n\\begin{thm} \\label{thm:main} There is a natural isomorphism \n$$H^r(\\Gamma_X, {\\mathbb Q}_l) \\cong W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l),$$\nand hence a natural map\n$$H^r(\\operatorname{Trop}(X), {\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l).$$\nThis map is an isomorphism if ${\\mathcal X}_P$ is connected for\nevery polyhedron $P$ in $\\Sigma$.\n\\end{thm}\n\\begin{proof}\nThe bottom nonzero row of the $E_1$ term of the Rapoport-Zink spectral sequence (i.e., the $w=-r$ row)\nis the complex:\n$$\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)},{\\mathbb Q}_l) \\rightarrow\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)},{\\mathbb Q}_l) \\rightarrow\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(2)},{\\mathbb Q}_l) \\rightarrow \\dots\n$$\nin which the horizontal maps are restriction maps.\nThis is simply the coboundary complex of the polyhedral complex formed by the bounded \ncells of $\\Gamma_{(X,{\\mathbb P})}$.  This polyhedral complex is homotopy equivalent to \n$\\Gamma_{(X,{\\mathbb P})}$.  We thus have a natural isomorphism\n$$E_2^{r,0} \\cong H^r(\\Gamma_X, {\\mathbb Q}_l).$$\n\\end{proof}\n\n\\begin{rem} \\rm Theorem~\\ref{thm:main} shows in particular that the space\n$W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$, which {\\em a priori} depends on\n$\\overline{X}$ and thus a choice of $\\Sigma$, in fact depends only on $X$ and is\n{\\em independent} of $\\Sigma$.  Proposition~\\ref{prop:cohomology} establishes this \ndirectly on the level of cohomology.\n\\end{rem}\n\nThe above results allow us to translate results about the cohomology of complete intersections \nin toric varieties into results about their tropicalizations.  For instance:\n\n\\begin{cor} \\label{cor:CI}\nLet $X$ be a sch\\\"on subvariety of $T$, and ${\\mathbb P}_K$ a smooth projective toric variety of $T$ such that:\n\\begin{enumerate}\n\\item the closure $Z$ of $X$ in ${\\mathbb P}_K$ is a smooth complete intersection of ample divisors, and\n\\item the boundary $Z \\setminus X$ is a divisor with simple normal crossings.\n\\end{enumerate}\nThen $H^r(\\Gamma_X, {\\mathbb Q}_l) = 0$ for $1 \\leq r < \\dim X$.\n\\end{cor}\n\\begin{proof}\nBy Proposition~\\ref{prop:degeneration} and Theorem~\\ref{thm:main} there is a tropical\npair $(X,{\\mathbb P}^{\\prime})$, with corresponding compactification $\\overline{X}$ of $X$, such that\n$H^r(\\Gamma_X, {\\mathbb Q}_l)$ is isomorphic to $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.\nBy Proposition~\\ref{prop:cohomology} the latter is isomorphic to\n$W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(Z_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  \n\nSince $Z$ is a complete intersection in ${\\mathbb P}_K$, the Lefschetz hyperplane theorem shows that\nfor $r < \\dim X$, the restriction map\n$$H^r_{\\mbox{\\rm \\tiny \\'et}}({\\mathbb P}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^r_{\\mbox{\\rm \\tiny \\'et}}(Z_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$ \nis an isomorphism.  But ${\\mathbb P}_K$ is a smooth toric variety, and hence has good reduction.\nThe weight spectral sequence thus shows that $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}({\\mathbb P}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) = 0$\nfor $r > 0$.\n\\end{proof}\n\nUnder more restrictive hypotheses on $X$, we can turn the above result into a result about the\ncohomology of $\\operatorname{Trop}(X)$.  This will be the main goal of section~\\ref{sec:CI}.\n\n\\section{Monodromy} \\label{sec:monodromy}\n\nIn many situations, the weight filtration has an alternative interpretation in terms of monodromy.\nLet $X$ be a variety over $K$, and consider the base change $X_{K^{\\operatorname{sep}}}$ of $X$ to $K^{\\operatorname{sep}}$.\nThe group $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$ admits a map to $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$; the kernel is the\ninertia group $I_K$.  The group $I_K$ is profinite; if $l$ is prime to the characteristic of $k$\nthen the pro-$l$ part $I_K^{(l)}$ of $I_K$ is isomorphic to ${\\mathbb Z}_l(1)$.  (The Tate twist here refers\nto the fact that the quotient $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$ acts on $I_K^{(l)}$ by conjugation in the\nsame way that it acts on the inverse limit of the roots of unity $\\mu_{l^n}$.\n\nThe group $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$ acts on the {\\'e}tale cohomology\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ for any prime $l$.  This action is quasi-unipotent, i.e.\na subgroup of $H$ of $I_K$ of finite index acts unipotently on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.\n(And thus the action of $H$ factors through $I_K^{(l)}$.)\nIn particular there is a nilpotent map \n$$N: H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l(-1))$$\ncalled the {\\em monodromy operator} such that for all $\\sigma \\in H$, $\\sigma$ acts on\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ by $\\exp(t_l(\\sigma)N)$, where $t_l$ is the map\n$I_K \\rightarrow I_K^{(l)} \\cong {\\mathbb Z}_l(1).$\n\nNow, if $V$ is any finite dimensional vector space, with a nilpotent endomorphism $N$ such that $N^r = 0$, \nthen there is a unique increasing filtration $\\{V_i\\}$ on $V$ such that:\n\\begin{itemize}\n\\item $V_r = V$,\n\\item $V_{-r} = 0$,\n\\item $N$ maps $V_i$ to $V_{i-2}$, and\n\\item $N^i$ induces an isomorphism $V_i\/V_{i-1} \\rightarrow V_{-i}\/V_{-i-1}$.\n\\end{itemize}\n(see~\\cite{WeilII} I, 1.7.2 for details.)  We thus obtain a natural filtration, called the\nmonodromy filtration, on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\n\\begin{rem} \\rm If $V$ consists of a single Jordan block of dimension $r$, one sees easily that\n$V_i\/V_{i-1}$ is one-dimensional for $i \\in \\{r-1, r-3, \\dots, -r+1\\}$, and zero otherwise.  \nMoreover, $V_{r-1-2k}$ is the image of $N^k$ for $0 \\leq k \\leq r-1$.\nIt is thus straightforward to read off the filtration coming from an arbitrary $V$ and $N$ from\na Jordan normal form for $N$.  The filtration is independent of choices, even though the Jordan\nnormal form of $N$ is not.\n\\end{rem}\n\nWhen $X$ has a semistable model, one can read the monodromy action on $X$ off\nof the weight spectral sequence $E^{p,q}$.  More precisely, there is a\nmonodromy operator\n$N: E_1^{p,q} \\rightarrow E_1^{p+2,q-2}(-1)$, which converges to the the monodromy operator $N$\non $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.  It is easily described:\nif $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m))$ occurs as a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m+1))$ occurs as a direct summand of $E_1^{p+2,q-2}$, then\nthe corresponding direct summand of $N$ is the identity \n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m+1))(-1).$$\nAll other direct summands of $N$ are the zero map.\n\nThe following conjecture (the ``weight-monodromy conjecture'') relates the weight filtration to the \nmonodromy filtration in this situation:\n\n\\begin{conj} \\label{conj:w-m}\nThe top nonzero power of the monodromy operator:\n$$N^r: E_2^{-r,w+r} \\rightarrow E_2^{r,w-r}$$\nis an isomorphism for all $r,w$.  In particular the weight filtration on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$\n$E$ coincides (up to a shift in degree) with the monodromy filtration; that is,\n$$H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}\/H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r-1} \\cong \nW_{w-r} H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l).$$\n\\end{conj}\n\nThe weight-monodromy conjecture is well-known to hold for curves and surfaces.  If\n${\\mathcal O}$ is an equal characteristic discrete valuation ring, it is a difficult theorem of\nIto (\\cite{Ito}, Theorem 1.1).  It is open in general when ${\\mathcal O}$ has mixed characteristic.\n\nFor the remainder of this section we assume we are in a situation where Conjecture~\\ref{conj:w-m}\nholds.  The following result, due to Speyer (\\cite{Sp3}, Theorem 10.8) for curves, follows \nimmediately:\n\n\\begin{cor} \\label{cor:bound} \nLet $b_r(\\Gamma_X)$ and $b_r(X)$ denote the $r$th Betti numbers of $\\Gamma_X$ and $X$,\nrespectively.  Then we have:\n$$b_r(\\Gamma_X) \\leq \\frac{1}{r+1} b_r(X).$$\n\\end{cor}\n\\begin{proof}\nTheorem~\\ref{thm:main}, together with the weight-monodromy conjecture, shows that $b_r(\\Gamma_X)$ \nis the dimension of $H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}$.  The dimension of this\npiece of the monodromy filtration counts the number of Jordan blocks of size $r+1$ \nin a Jordan normal form for $N$ acting on $H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  \nIn particular the dimension of the latter is at least $r+1$ times the dimension of the \nformer.\n\\end{proof}\n\nSuppose $X_K$ is an $n$-dimensional variety.  There is a geometric interpretation of the action of the $n$th power of the monodromy operator on the middle-dimensional cohomology.\n\\begin{prop} \\label{prop:npower} The $d$th power of the monodromy map acting on middle cohomology,\n\\[N^d:H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}\\rightarrow H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}(-d).\\]\nis the map\n\\[H_d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\rightarrow H^d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\]\ninduced from the ``volume pairing'' on the parameterizing complex $\\Gamma_{(X,{\\mathbb P})}$ which takes a pair of (integral) $d$-dimensional cycles to the (oriented) lattice volume of their intersection.  \n\\end{prop}\n\n\\begin{proof}\nThe term $H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}$ is computed by the $(-d,d)$-entry in the Rapoport-Zink spectral sequence.  The $d$th row is:\n$$\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d)},{\\mathbb Q}_l)(-d) \\rightarrow\nH^2_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d-1)},{\\mathbb Q}_l)(-d+1) \\rightarrow\\dots\\rightarrow\nH^{2d}_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)},{\\mathbb Q}_l)$$\nwhere the horizontal map is the Gysin map of ${\\mathcal X}^{(k)}\\rightarrow {\\mathcal X}^{(k-1)}$.  Since each component of ${\\mathcal X}^{(k)}$ is an $(d-k)$-dimensional smooth variety, this is the chain complex formed by the bounded cells of $\\Gamma_{(X,{\\mathbb P})}$.  Consequently, $E_2^{-k,d}\\cong H_k(\\Gamma,{\\mathbb Q}_l)(-k)$.  Now, \n\\[N^d:E_2^{-d,d}\\cong H_d(\\Gamma,{\\mathbb Q}_l)(-d)\\rightarrow E_2^{0,0}(-d)\\cong H^d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\] is induced from the identity map on $H^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d)},{\\mathbb Q}_l)$.  In the language of homology and cohomology of $\\Gamma_{(X,{\\mathbb P})}$, it comes from the \nmap $C_d(\\Gamma_{(X,{\\mathbb P})})\\rightarrow C^d(\\Gamma_{(X,{\\mathbb P})})$ taking a simplex $F$ to the cocycle $\\delta_F:C^d(\\Gamma_{(X,{\\mathbb P})})\\rightarrow{\\mathbb Z}$ that is the indicator function of $F$.  Consequently, if we view $N^d$ as  a bilinear pairing on $H_d(\\Gamma,{\\mathbb Q}_l)$, it is the volume pairing as every bounded top-dimensional cell of $\\Gamma_{(X,{\\mathbb P})}$ has volume $1$.\n\\end{proof}\n\n\\begin{ex} \\label{ex:curves}\n\\rm Suppose that $X_K$ is a curve of genus $g$.  Then ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)}$ is the \nnormalization of ${\\mathcal X}_{k^{\\operatorname{sep}}}$; it is a disjoint union of smooth curves $C_i$ of genus \n$g_i$.  On the other hand, ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}$ is the set of singular points of \n${\\mathcal X}_{k^{\\operatorname{sep}}}$; each such point lies on exactly two of the $C_i$.  The corresponding\nweight spectral sequence is nonzero only for $-1 \\leq r \\leq 1$ and $0 \\leq w+r \\leq 2$; it \nlooks like:\n$$\n\\begin{array}{ccccc}\n\\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l(-1) & \\rightarrow & \\bigoplus_i H^2_{\\mbox{\\rm \\tiny \\'et}}(C_i, {\\mathbb Q}_l) & & 0\\\\\n0 & & \\bigoplus_i H^1_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l) & & 0\\\\\n0 & & \\bigoplus_i H^0_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l) & \\rightarrow & \\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l\n\\end{array}\n$$\nThe sequence clearly degenerates at $E_2$.  \nThe monodromy operator $N$ is nonzero only from $E_1^{-1,2}$ to $E_1^{1,0}(-1)$; \nit is simply the identity map on $$\\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l(-1).$$\nWe thus find that the middle quotient of\nthe monodromy filtration on $H^1_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ is isomorphic to\nthe direct sum of $H^1_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l)$, whereas the top and bottom quotients are isomorphic\nto $H_1(\\Gamma,{\\mathbb Q}_l)$, (resp. $H^1(\\Gamma,{\\mathbb Q}_l)$) where $\\Gamma$ is the dual graph of \n${\\mathcal X}_{k^{\\operatorname{sep}}}$.  As above, the map $N: H_1(\\Gamma,{\\mathbb Q}_l) \\rightarrow H^1(\\Gamma,{\\mathbb Q}_l)$\ncan be interpreted as the length pairing on $H_1(\\Gamma,{\\mathbb Q}_l)$. \\end{ex}\n\nThis example has a more classical interpretation.  If we let $J$ be the Jacobian of\n$\\overline{X}$, then the connected component of the identity in the\nspecial fiber of the N{\\'e}ron model of $J$ over ${\\mathcal O}$ is an extension of an abelian variety\nby a torus; let $\\chi$ be the character group of this torus.  Then $\\chi$\nis naturally isomorphic to $H_1(\\Gamma,{\\mathbb Z})$.  Moreover, one has a monodromy pairing\n$\\chi \\times \\chi \\rightarrow {\\mathbb Z}$ (see \\cite{SGA} for details.)  If one identifies\n$\\chi$ with $H_1(\\Gamma,{\\mathbb Z})$, the resulting pairing on $H_1(\\Gamma,{\\mathbb Z})$ is precisely\nthe length pairing.\n\nTo summarize:\n\\begin{prop} If $X$ is a sch\\\"on open subset of a smooth proper curve $\\overline{X}$ over $K$, then\nthe ``length pairing'' on $\\Gamma_X$ coincides with the monodromy pairing on\nthe character group $\\chi$ associated to the Jacobian of $\\overline{X}$.\n\\end{prop}\n\nThis has connections to Mikhalkin's construction of tropical Jacobians.  Given a\ntropical curve $\\Gamma$, which Mikhalkin interprets as a metric graph, the length\npairing on $\\Gamma$ induces a map $H_1(\\Gamma) \\rightarrow \\operatorname{Hom}(H_1(\\Gamma),{\\mathbb Z})$; Mikhalkin\ndefines the tropical Jacobian of $\\Gamma$ to be the torus $\\operatorname{Hom}(H_1(\\Gamma),{\\mathbb R})\/H_1(\\Gamma)$.\nThis torus has a natural integral affine structure induced from that on $\\operatorname{Hom}(H_1(\\Gamma),{\\mathbb R})$.\nSee~\\cite{MZ} for details.\n\nMikhalkin's definition is purely combinatorial but has a nice interpretation in\nterms of the uniformization of abelian varieties: if $J$ is the Jacobian of\n$\\overline{X}$ then there is a pairing $\\chi \\times \\chi \\rightarrow \\overline{K}^*$\nwhose valuation is the monodromy pairing.  This pairing gives an embedding of\n$\\chi$ as a lattice in the torus $\\operatorname{Hom}(\\chi, \\overline{K}^*)$; the quotient\n$\\operatorname{Hom}(\\chi,\\overline{K}^*)\/\\chi$ is a rigid space isomorphic to $J$.\nIf we ``tropicalize'' this space by taking valuations, we obtain the\nspace $\\operatorname{Hom}(\\chi,{\\mathbb R})\/\\chi$, where $\\chi$ embeds into $\\operatorname{Hom}(\\chi,{\\mathbb R})$ by\nthe monodromy pairing.  In particular the ``tropicalization'' of $J$ is\nthe tropical Jacobian of $\\Gamma_X$.\n\nThe upshot is that- provided we are careful about what we mean by tropicalization-\n``tropicalization'' commutes with taking Jacobians.\n\nThe following result of \\cite{KMM} is another easy consequence of this point of view:\n\\begin{prop}[\\cite{KMM}, Theorem 6.4] Let $X$ be a sch\\\"on open subset of an\nelliptic curve $\\overline{X}$ over $K$ with potentially multiplicative reduction.\nThen $H_1(\\Gamma_X,{\\mathbb Z})$ is isomorphic to ${\\mathbb Z}$, and valuation of the $j$-invariant \n$j(\\overline{X})$ is equal to $-a$, where $a$ is the length of the unique cycle in $\\Gamma_X$.\n\\end{prop}\n\\begin{proof}\nReplacing ${\\mathcal O}$ with a suitable ramified extension we may assume that $\\Gamma_X$ is integral.\nThen $\\overline{X}$ has split multiplicative reduction.  This base change scales both $\\Gamma_X$ and\nthe valuation of $j(\\overline{X})$ by the degree $d$ of the extension.  Now the tropical\ndegeneration ${\\mathcal X}$ of $X$ associated to $\\Gamma_X$ gives a model of $\\overline{X}$ whose special\nfiber contains a cycle of rational curves, of length equal to the lattice length $a$ of the unique\ncycle in $\\Gamma_X$.  The conductor-discriminant formula (\\cite{Si}, Theorem 11.1) then shows that\nthe valuation of $j(X)$ is equal to $-a$.\n\\end{proof}\n\nIn fact, it is easy to see that any smooth curve $\\overline{X}$ contains a sch\\\"on open subset:\ntake a semistable model of $\\overline{X}$, embed it in ${\\mathbb P}^n_{{\\mathcal O}}$, let ${\\mathcal T}$ be the complement of\n$n+1$ hyperplanes in general positionin ${\\mathbb P}^n_{{\\mathcal O}}$, and take $X = {\\mathcal T} \\cap \\overline{X}$.  \nThen the compactification $\\overline{X}$ of $X$ in ${\\mathbb P}^n_{{\\mathcal O}}$ is tropical, and one verifies\neasily that the multiplication map is smooth.  Thus the above result applies to all elliptic curves\nwith potentially multiplicative reduction.\nLuxton and Qu~\\cite{LQ} have shown that any variety over a field of characteristic $0$ contains a\nsch\\\"on open subset.\n\n\\section{Complete Intersections} \\label{sec:CI}\n\nIn the constant coefficient case, a (Zariski) general hyperplane section of a sch\\\"on variety is\nsch\\\"on.  Unfortunately this is no longer true in the nonconstant coefficient case.  For instance,\nlet $X_k$ be a singular hypersurface in $T_k$.  Then any hypersurface $X$ in $T_K$ that reduces\nmodulo $\\pi$ to $X_k$ has $\\operatorname{in}_{(0,\\dots,0)} X = X_k$, and hence cannot be sch\\\"on.  The set\nof such $X$ is a rigid analytic open subset of the projective space of hypersurfaces of fixed degree.\n\nAs this example suggests, to study loci of sch\\\"on varieties in a nonconstant coefficient setting,\none needs to work with the rigid analytic topology\nrather than the Zariski topology.  (For the basics of the theory of rigid analytic spaces we refer the\nreader to~\\cite{EKL} or~\\cite{Schneider}; we use very little here.)  \n\nTo make precise the connection to rigid geometry, we first observe:\n\n\\begin{lemma} \\label{lemma:transversality}\nLet ${\\mathbb P}$ be a toric scheme, proper over ${\\mathcal O}$, and let $X$ be a subvariety of \nthe open torus $T$ in ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} K$.\nSuppose that for all polyhedra $P$ in the polyhedral complex $\\Sigma$ corresponding to $P$, the \nclosure ${\\mathcal X}$ of $X$ in ${\\mathbb P}$ intersects ${\\mathbb P}_P$ transversely.  Then $X$ is sch\\\"on, and\n$(X, {\\mathbb P}^{\\prime})$ is a normal crossings pair, where ${\\mathbb P}^{\\prime}$ is the open subset\nof ${\\mathbb P}$ obtained by deleting all torus orbits that do not meet ${\\mathcal X}$.\n\nConversely, if $X$ is sch\\\"on and there exists a toric open subset ${\\mathbb P}^{\\prime}$ of ${\\mathbb P}$\nsuch that $(X,{\\mathbb P}^{\\prime})$ is a normal crossings pair, then the closure of $X$ intersects ${\\mathbb P}_P$ \ntransversely for all polyhedra $P$ in $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the multiplication map\n$$m:{\\mathcal T} \\times {\\mathcal X} \\rightarrow {\\mathbb P}^{\\prime}.$$\nIf $y$ is a point in ${\\mathbb P}^{\\prime}$ in the torus orbit corresponding to a polyhedron $P$ in\nthe subcomplex $\\Sigma^{\\prime}$ of $\\Sigma$ corresponding to ${\\mathbb P}^{\\prime}$, then the fiber over \n$y$ is isomorphic to the product\n${\\mathcal X} \\cap {\\mathbb P}^{\\prime}_P$ with a torus.  By assumption,\nthis is smooth, so $m$ has smooth fibers.  The argument of~\\cite{Hacking}, Lemma 2.6\nthen shows that $m$ is smooth.  It follows that $X$ is sch\\\"on and $(X,{\\mathbb P}^{\\prime})$\nis a normal crossings pair.  The converse is clear.\n\\end{proof}\n\nNote that the lemma implies that $\\operatorname{Trop}(X)$ will be equal to the support of $\\Sigma^{\\prime}$ \nfor all such $X$.  One can therefore use this result to study the space of sch\\\"on subvarieties of \na toric variety over $K$ with a given tropicalization.  We will not pursue this here, beyond a few\nstraightforward observations.\n\nSuppose ${\\mathbb P}$ is projective.  Fix an $X$ as in the lemma, and let $\\operatorname{Hilb}({\\mathbb P})$ be the Hilbert\nscheme over ${\\mathcal O}$ parameterizing subschemes of ${\\mathbb P}$ with the same Hilbert polynomial as the closure\nof $X$.  Complex points of $\\operatorname{Hilb}({\\mathbb P})$ correspond to subschemes of the special fiber of ${\\mathbb P}$; those \nthat meet each ${\\mathbb P}_P$ transversely form an open subset $U_0$ of \n$\\operatorname{Hilb}({\\mathbb P}) \\times_{{\\mathcal O}} \\operatorname{Spec} k$.\n\nNow if $y$ is a point of $\\operatorname{Hilb}({\\mathbb P})(\\overline{K})$, then $y$ corresponds to a subscheme $X_y$ of the \ngeneral fiber of ${\\mathbb P}$ over a finite extension of $K$.  Then $X_y \\cap T$ will satisfy the hypotheses\nof Lemma~\\ref{lemma:transversality} if, and only if, $y$ specializes to a point $y_0$\non the special fiber of $\\operatorname{Hilb}({\\mathbb P})$ that lies in $U_0$.  The set of points\n$y$ that specialize to $U_0$ forms a ``neighborhood of $U_0$'' in the rigid analytic topology\non $\\operatorname{Hilb}({\\mathbb P}).$  More precisely, let $\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$ denote the rigid analytic space\nassociated to the general fiber of $\\operatorname{Hilb}({\\mathbb P})$; then $\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$ is equipped\nwith a ``reduction mod $\\pi$'' map\n$$\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}} \\rightarrow \\operatorname{Hilb}({\\mathbb P}) \\times_{{\\mathcal O}} \\operatorname{Spec} k.$$\nThe preimage of $U_0$ under this map is an admissible open subset $U^{\\mbox{\\rm \\tiny rig}}$ of\n$\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$, and those $y \\in \\operatorname{Hilb}({\\mathbb P})(\\overline{K})$ such that $X_y \\cap T$ satisfies the \nhypotheses of Lemma~\\ref{lemma:transversality} are precisely the $\\overline{K}$-points of $U^{\\mbox{\\rm \\tiny rig}}$.\n\nIf we restrict our attention to complete intersections, we can say more than this.\nIn particular fix a projective toric scheme ${\\mathbb P}$ over ${\\mathcal O}$, and ample line\nbundles $L_1, \\dots, L_s$ on ${\\mathbb P}$.  The space ${\\mathcal H}$ parameterizing tuples\n$(D_1, \\dots, D_s)$ such that for each $i$, $D_i$ is an effective divisor in the linear system\ncorresponding to $L_i$, and all the $D_i$'s intersect transversely, is an open subset\nof a product of projective spaces over ${\\mathcal O}$.  \n\nBy Bertini's theorem, the set of points in ${\\mathcal H}(k)$ that correspond to divisors\n$(D_1, \\dots, D_s)$ in ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} k$ such that $D_1 \\cap \\dots \\cap D_s$ \nintersects each stratum ${\\mathbb P}_P$ of ${\\mathbb P}$ transversely is an open dense subset $U_0$ of\nthe special fiber of ${\\mathcal H}$.  The preimage of $U_0$ under the reduction map\n$${\\mathcal H}^{\\mbox{\\rm \\tiny rig}} \\rightarrow {\\mathcal H} \\times_{{\\mathcal O}} \\operatorname{Spec} k$$\nis a (necessarily nonempty) admissible open subset $U^{\\mbox{\\rm \\tiny rig}}$ of ${\\mathcal H}^{\\mbox{\\rm \\tiny rig}}$; the\npoints of $U^{\\mbox{\\rm \\tiny rig}}$ correspond precisely to those complete intersections\n$D_1 \\cap \\dots \\cap D_s$ whose intersection with $T$ satisfies the conditions of \nLemma~\\ref{lemma:transversality}.\n\nMoreover, if $(D_1, \\dots, D_s)$ is a $K$-point of $U^{\\mbox{\\rm \\tiny rig}}$, and $X$ is the corresponding\ncomplete intersection $D_1 \\cap \\dots \\cap D_s \\cap T$ in $T$, then for each polyhedron $P$ in\n$\\Sigma$, ${\\mathcal X}_P = D_1 \\cap \\dots \\cap D_s \\cap {\\mathbb P}_P$ is the intersection of ample divisors\nin the smooth toric variety ${\\mathbb P}_P$, and is therefore either zero-dimensional or connected.\n\nLemma~\\ref{lemma:cohomology} and Theorem~\\ref{thm:main} now have immediate implications\nfor the cohomology of $\\operatorname{Trop}(X)$:\n\n\\begin{thm} \\label{thm:CI}\nLet $(D_1, \\dots, D_s)$ be a $K$-point of $U^{\\mbox{\\rm \\tiny rig}}$, and set\n$$X = D_1 \\cap \\dots \\cap D_s \\cap T.$$\nThen $H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l)$ vanishes for $1 \\leq r < \\dim X$, and the natural map:\n$$H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$\nis injective for $r = \\dim X$.\n\\end{thm}\n\\begin{proof}\nThe above discussion shows that $X$ is sch\\\"on and $(X,{\\mathbb P})$ is a normal crossings pair.\nWe thus apply Theorem~\\ref{thm:main} and Lemma~\\ref{lemma:cohomology} to see that the\nmap\n$$H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$\nis an isomorphism for $0 \\leq r < \\dim X$ and injective for $r = \\dim X$.  On the other\nhand, $\\overline{X}$ is a complete intersection in the general fiber of the smooth toric\nvariety ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} K$.  The result thus follows from Corollary~\\ref{cor:CI}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\label{sec:intro}\nOver the past several years many dedicated experiments\nhave been used to detect\nthe Sunyaev--Zel'dovich (SZ) effect \\citep{Sunyaev1970}\nfrom galaxy clusters at radio wavelengths [e.g.,\nBerkeley-Illinois-Maryland Association (BIMA) \\citep{Dawson2006};\nCombined Array for Research in Millimeter-wave Astronomy (CARMA) \\citep{Muchovej2012, Mantz2014};\nthe South Pole Telescope (SPT) \\citep{Reichardt2013};\nthe N\\'eel IRAM KIDs Array (NIKA) \\citep{Adam2014};\nthe Atacama Pathfinder EXperiment Sunyaev--Zel'dovich Instrument (APEX-SZ) \\citep{Dobbs2006, Bender2016};\nthe Arcminute Microkelvin Imager (AMI) \\citep{Zwart2008, Rumsey2016};\n{\\em Planck} Surveyor  \\citep{PlanckCollaboration2011};\nthe Atacama Cosmology Telescope (ACT) \\citep{Hasselfield2013};\nArray for Microwave Background Anisotropy (AMiBA) \\citep{Lin2016}].\nWithin the next few years, new observational facilities\nwill become operational and will search for galaxy clusters,\ncomplementing the galaxy cluster census across\nthe Universe\n[e.g.,\nNew IRAM KID Array 2 (NIKA2) on\nthe Institut de Radio Astronomie Millimetrique 30~m telescope \\citep{Calvo2016}].\n\n\n\nThe One Centimeter Receiver Array (OCRA)\n\\citep{Browne2000,Peel2011} is one of the\nexperiments capable of detecting the SZ effect at 30~GHz using\nbeam-switching radiometers installed on a 32-meter radio\ntelescope \\citep{Lancaster2007,Lancaster2011}.\nOCRA will be mostly sensitive to SZ clusters with virial size $> 3'$\nand hence to clusters at redshifts in the range $0.1 < z < 0.5$ and with\nmasses $M_{\\rm{vir}} > 3\\times 10^{14} M_{\\odot}\/h$ \\citep{Lew2015}.\nHowever, a single\nfrequency, beam-switching system may suffer from confusion with\nthe primordial cosmic microwave background (CMB) or suffer from\nsystematic error when observing extended sources.\n\nConfusion effects due to the CMB were investigated in detail by\n\\cite{Melin2006} for AMI, SPT and {\\em Planck} Surveyor.  It was\nfound that for single frequency instruments, such as AMI (a\n15~GHz interferometer), the photometric accuracy that contributes\nto the accuracy of the reconstructed comptonization parameter is\nstrongly limited due to primary CMB confusion.\n\n\nIn \\cite{Lew2015} the impact of CMB flux density confusion at 30\nGHz was investigated, in particular for the OCRA\/RT32 (32~m Radio\nTelescope in Toru\\'n, Poland) experiment.  It was found that the\n$1\\sigma$ thermal SZ (tSZ) flux density uncertainty due to CMB\nconfusion should be of the order of $10\\%$ for the range of\nclusters detectable with OCRA.  However, in that work, the impact\non the reconstructed comptonization parameter in the presence of\nthe CMB and radio sources was not calculated directly for the\ncase of dual-beam differential observations.\n\n\nThe $\\approx 3'$ separation of OCRA beams is very effective in\nCMB removal, but large correcting factors are required to\ncompensate for the missing SZ signal (after accounting for point\nsources) \\citep{Lancaster2007}.  Thus, there is a trade-off\nbetween compromising photometry by the primary CMB signal versus\nlosing flux due to the differential beam pattern.  In between\nthese extremes, there should exist an optimal separation of\ndifferential beams that would need to be defined by criteria that\naim to maximize CMB removal and minimize SZ flux density removal.\n\nIn this paper, we reconsider the issue of systematic effects on\nthe reconstructed comptonization parameter from single frequency,\nbeam-switched observations performed with a cm~wavelength\nradiometer.  We consider a particular instrumental setting for\nthe OCRA\/RT32 experiment and an extension to the standard\nobservation scheme that previously involved only the angular\nscales defined by the receiver feeds. The extension adds\nadditional beam pointings that map cluster peripheries, further\nfrom the central core than the initial pointings.\n\nThe {\\em kinetic} SZ (kSZ) may significantly modify the\nbrightness of the cluster peripheries that are integrated with\nthe reference beam. The significance of this effect depends on a\ncombination of the peculiar velocities of the intra-cluster\nmedium (ICM) and internal gas clumps, but at cm wavelengths, the\nkSZ only weakly modifies the central brightness.\n\nWith dual-beam observations, the reference beam background\ncoverage improves while integrating along arcs around the cluster\ncenter as the field of view (FOV) rotates.  However, due to the\nsmall angular size of the arcs, the chance of zeroing the average\nbackground may be low, depending on the alignment with the CMB\npattern. We investigate the significance of this effect depending\non observational strategy.\n\nFor experiments limited by the size of the focal plane array the\nintegration time required to generate a radio map and to probe\nthe outer regions of a galaxy cluster is significant and can make\nthe observation prohibitive.  Therefore, previously, the method\nof reconstructing comptonization parameters from OCRA\nobservations of cluster central regions required inclusion of\nX-ray luminosity data in order to find the best fitting\n$\\beta$-model for each cluster, and correction for the SZ power\nlost due to the close beam separation.  However, this approach\nrelies on the cluster model assumptions and makes the radio SZ\nmeasurements dependent on X-ray measurements of the cluster.\nAnother possible approach is to observe SZ clusters out to larger\nangular distances but retain averaging over a range of\nparallactic angles.  This is done at the cost of incurring extra\nnoise due to weak tSZ in cluster peripheries and stronger\nsystematic effects due to CMB.\n\n\nAn OCRA-SZ observational program is presently underway. In\nsupport of this and similar efforts, we also investigate the\npossibility of mitigating CMB confusion by using the available\n{\\em Planck} data.  Finally, we calculate the astrometric\npointing and tracking accuracy requirements needed to attain a\ngiven accuracy in flux density reconstruction.\n\n\nIn Section~\\ref{sec:strategy} we review the current observing\nstrategy and discuss its possible extensions.  In\nSection~\\ref{sec:sims} we briefly outline our numerical\nsimulation setting.  Section ~\\ref{sec:CMBsims} describes the\nconstruction of CMB templates from the currently available {\\em\nPlanck} data.  Section~\\ref{sec:sample} describes simulated\nsamples of galaxy clusters used for the flux-density\nanalyses. The main results are in Sec.~\\ref{sec:results}.  Final\nremarks and conclusions are in Sections~\\ref{sec:discussion} and\n~\\ref{sec:conclusions} respectively.\n\n\\section{Observational strategy}\n\\label{sec:strategy}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_1.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_2.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_3.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_4.eps}\n\\caption{ ({\\em Top-left}) Projected map of the the line-of-sight\n(LOS) integrated comptonization parameter for a selected halo,\nand ({\\em top-right}) simulated CMB temperature fluctuations\n$\\Delta T$ including both primary CMB as well as tSZ and kSZ\neffects induced by the halo.  ({\\em Bottom-left}) 217~GHz {\\em\nPlanck} resolution simulation -- the Gaussian CMB $\\Delta T$\nsignal smoothed with a half-power beam width (HPBW) of $\\approx\n5'$ and including a realistic {\\em Planck} noise realization\n(see Sec.~\\ref{sec:CMBsims}). This map is used to make a high\nresolution CMB template by means of a smooth-particle\ninterpolation.  ({\\em Bottom-right}) Residual map: primary CMB\nincluding tSZ minus interpolated {\\em Planck} simulation\n(Sec.~\\ref{sec:CMBsims}).  The variance of the residual map\n(without SZ effects) is about one order of magnitude smaller\nthan that measured using maps that include CMB.  In the {\\em\ntop-right} panel the circles denote OCRA beamwidths\ntraversing over the background CMB for a typical observational\nscheme (see Sec.~\\ref{sec:strategy} for details).  In the {\\em\ntop-left} panel black circles represent the full focal plane\nof OCRA receivers.  }\n\\label{fig:templates}\n\\end{figure*}\n\n\nThe common position-switching mode of observing\n\\citep{Lancaster2011} is that in which the reference beam\nnon-uniformly (due to varying FOV rotation speed) integrates the\nbackground along arcs $\\approx 3'$ from the source.  In\nFig.~\\ref{fig:templates} ({\\em top-right} panel), circles denote\nOCRA beamwidths for a typical observational scheme\n\\citep{Birkinshaw2005}.  First, the beam pair ``A--B'' measures\nthe difference signal between the cluster center and periphery,\nrespectively.  The beam pair is then ``switched'',\ni.e. translated to configuration ``C--A'' with a swap of the\nroles of the primary and reference beams, so that beams ``C'' and\n``A'' now trace the cluster periphery and center, respectively.\nThe position switching cycle is closed by returning to the\ninitial configuration ``A--B'' and the cycle is repeated.\n\nAs the Earth rotates, the reference beams sweep arcs around the\ncluster center and probe different off-center background regions\n(beam B becomes B' and C becomes C').  The beam\nposition-switching reduces fluctuations due to atmospheric\nturbulence on time scales of a few tens of seconds.  At shorter\ntime scales fluctuations due to receiver-gain instability and\natmospheric absorption are reduced by switching and\ndifferentiating signals in receiver arms by means of\nelectronically-controlled phase switches\n\\citep{Peel2010,Lancaster2011}.\\footnote{Beam switching is\nrealized at the rate of 277~Hz which improves the $1\/f$ knee of\nthe resulting difference signal power spectrum roughly by an\norder of magnitude; typically down to frequencies $0.1{\\rm\nHz}<f_{\\rm knee}<1{\\rm Hz}$.}\n\nAn extension to this pattern can be realized by adding an extra\nbeam pointing ``D''.  In this case an observation cycle would be\nextended so that a beam pair would observe differences between\n``A--B'', then ``B--D'' and followed by ``A--B'', again probing\nthe cluster peripheries for varying parallactic\nangles.  This scheme can be extended to both sides of the cluster\nand to larger distances from the cluster center.  In\nFig.~\\ref{fig:templates} ({\\em top-left} panel) the projected\nbeamwidths of the OCRA-f receiver focal plane are shown (black\ncircles) overlaid on a nearby galaxy cluster seen through the SZ\neffect. The extra beam pointing ``D'' for the whole array is\nshown with blue circles.  In the following sections we\ninvestigate the implications of such an extended observation\nscheme using numerical simulations.\n\n\n\n\n\n\\section{Simulations}\n\\label{sec:sims}\n\n\n\\subsection{LSS and SZ effect}\n\\label{sec:LSSsims}\n\nFor the main results in this work we use the simulation approach\ndescribed in \\cite{Lew2015}, with a few modifications.  In\nparticular, for the same field of view ($\\approx 5.2^\\circ$) we use\nan increased map resolution of $\\approx 0.9''$.  In observational\npractise, often only the central comptonization parameter value\nis quoted, so we include the kSZ signal calculated as a\ncontribution to the measured Compton $y$-parameter for the\nappropriate frequency.\n\nIn this analysis we neglect the large-scale foreground\ngalactic synchrotron, free--free and dust emissions.  We \nassume these to be smooth enough to be \nremoved in differential observations.\n\nWe assume that atmospheric effects and receiver noise are\nmitigated by sufficiently long integrations (see\nSect.~\\ref{sec:strategy} and \\cite{Birkinshaw2005}).\nSystematic errors in\nflux density estimation that we neglect include feed and elevation dependent beam\nresponse, feed and elevation dependent sidelobes, and elevation\ndependent antenna gain.  We defer treatment of these effects to a\nseparate analysis of end-to-end full OCRA focal plane\nsimulations.  Although the simulated maps include some of the\neffects of halo--halo LOS projections, the significance of the\nprojection effects on the $y$-parameter photometry are not the\nmain focus of the present study.  These\neffects should be small in this study, since\nwe only consider the most massive systems at the\nFOV generation stage.\n\n\nA cluster is not necessarily observed at the position that\nmaximises the SZ signal.  For an X-ray selected galaxy cluster,\nSZ observations can be centered at the maximum of the X-ray\nsignal, but this may have a small offset with respect to the\nmaximum of the SZ signal, although the difference should be\nrather small.  This will effectively translate into pointing\nerrors that we model in this work.\n\nThe OCRA beam separation is $s_{\\rm OCRA} = 0.0526\\pm\n0.0010^\\circ$ (Fig.~\\ref{fig:templates}), which we adopt in this\nwork.  Within the measurement accuracy this does not depend on\nsky direction.\n\n\\subsection{CMB templates}\n\\label{sec:CMBsims}\n\nThe {\\em Planck} mission has provided full sky maps of the CMB\ntemperature fluctuations at several frequencies, including\n217~GHz, where the tSZ signal is minimal.  This opens up the\npossibility to correct the differential SZ observations at other\nfrequencies by removing a customized CMB template and thus reduce\nthe CMB confusion in single frequency observations, provided\nthat: (i) the small scale noise level of the template is low,\n(ii) the angular resolution of the data is sufficient and (iii)\ndiffuse Galactic foregrounds can be neglected.  In practise, \nrequirement (iii) may exclude all or most of the sky regions\ncovered by {\\em Planck} frequency maps on either side of the\nGalactic Plane.\n\nIn order to test the usefulness of {\\em Planck} data in\nminimizing the CMB-induced variance in the measured flux\ndensities with a single-frequency instrument, we generate CMB\nsimulations for {\\em Planck}'s 217~GHz frequency band using\nsimulated CMB power spectrum \\citep{Lewis2000} and assuming\ncosmological parameters as in \\cite{Lew2015}.  We assume a\nGaussian beam transfer function defined by a full-width half\nmaximum FWHM~=~$5.02'$ \\citep{PlanckCollaboration2016b} and use\nrealistic and publicly available 217~GHz {\\em Planck} receiver\nnoise\nsimulations\\footnote{\\protect{\\url{http:\/\/pla.esac.esa.int\/pla\/\\#maps}}}\nat Healpix \\citep{Gorski2005} resolution $n_s=2048$.\n\nWe simulate the primary CMB up to $\\ell_{\\rm\nmax}=3500$,\\footnote{The variance lost due to neglecting even\nhigher multipoles is negligibly small ($\\lesssim 0.004\\%$) and\nwell below the cosmic variance uncertainty.}  i.e. the OCRA\nbeam separation.  In order to create the final CMB template with\n$\\approx 0.9''$ resolution in small FOVs, we use a\nsmooth-particle interpolation of the projected field (as\ndiscussed in \\citealt{Lew2015}).\n\nReal observations at 217~GHz will contain kSZ contributions,\nwhich we ignore for generation of simulated templates.  For each\nrealization of the simulated template, we store maps with CMB\nsignal, and maps with the CMB smoothed with the instrumental beam\nand contributed by a {\\em Planck} noise realization\n(Fig.~\\ref{fig:templates}).  The former is used for simulating\nastrophysical signals (SZ) in FOVs, while the latter is used for\nremoving a {\\em Planck}-compatible version of CMB contamination,\nfor the same FOVs.  Each FOV simulation is made at fixed galactic\nlatitude $b=40^\\circ$, but at a different galactic longitudes to\naccount for variations due to direction dependent properties of\n{\\em Planck} noise.\n\nThe single frequency maps from {\\em Planck} mission are\ncontaminated by foregrounds other than the cluster tSZ and kSZ\nsignals, especially at low galactic latitudes.  Since in the\ncurrent work we do not investigate galactic foregrounds we\nadditionally analyze another set of simulations based on the\nforeground-reduced map generated with the needlets-based internal\nlinear combination (NILC) algorithm \\citep{PlanckCollaboration2016a}.  The\nalgorithm \\citep{Delabrouille2009} provides a very clean primary\nCMB map outside of masked regions which mask Galactic Plane and\nbright point sources (together $\\approx 3.6\\%$ of the full sky).\nWe simulate the NILC map using the published beam transfer\nfunction, and use the full mission NILC rendition of ringhalf-1\nand ringhalf-2 half difference maps to generate a {\\em Planck}\nNILC-compatible noise realization.  We verified that the\nresulting NILC simulations are compatible with the {\\em Planck}\nNILC map in terms of their angular pseudo-power spectra.  Slight\ndifferences in the high-$\\ell$ regime of up to a few percent are\npresent due to our choice of cosmological parameter values that\nis consistent with the WMAP9 results.\n\nThe NILC map resolution simulations are similar to those of the\n217~GHz map, so in Fig.~\\ref{fig:templates} we only show the map\nfor the 217~GHz case.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_5.eps}\n\\caption{ Distribution of the solid-angle--integrated\ncomptonization parameter $Y$ integrated within the angular\nradius $\\theta<0.75'$ from the cluster center ({\\em left});\ndistribution of cluster total mass measured within a volume\nwith mean mass density $500$ times higher than the critical\ndensity of the Universe at the cluster's redshift ({\\em\nmiddle}); and distribution of redshifts ({\\em right}) for the\nsimulated cluster samples (solid lines) and for the SPT cluster\nsample \\citep{Bleem2015} (dashed lines).  For each simulated\nsample only halos with $M_{\\rm vir}>M_{\\rm vir,\\min}$ and\n$z>z_{\\min}$ are chosen (see Table~\\ref{tab:samples}).  The\nvertical dashed line shows the division into high-$z$ and\nlow-$z$ sub-samples that is used later in the analysis.  }\n\\label{fig:sample}\n\\end{figure}\n\n\n\n\\subsection{Simulated galaxy cluster samples}\n\\label{sec:sample}\n\n\n\n\\begin{table}[t]\n\\caption{Selection criteria used for constructing galaxy cluster samples.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\n\nParameter & \\multicolumn{2}{c}{Sample\/Value}\\\\\n& \\sampleHS & \\sampleDF \\\\\n$z_{\\min}$ & 0.05 & 0.0 \\\\\n$M_{\\rm vir,\\min}\\,[10^{14} M_\\odot\/h]$ & 4.0 & 2.0 \\\\\nhalo selection & full simulation volume& $5.2^\\circ\\times 5.2^\\circ$ FOV \\\\\nhalo count\\footnotemark[1] & 475 & 361 \\\\\nSub-samples &&&& \\\\\n\\multicolumn{1}{r}{low-$z\\leq0.4$} & 426 & 214  \\\\\n\\multicolumn{1}{r}{high-$z>0.4$}   & 49  & 147\n\n\\footnotetext[1]{~The actual number of halos\nused in statistical analyses are slightly different\nas they are further screened for halos that lie well within the\nprojected FOV, which is required for simulating dual-beam\nobservations at all possible parallactic angles and beam separations in a consistent way.}\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:samples}\n\\end{table}\n\n\nFor the analyses presented in Sect.~\\ref{sec:results}, we\nconstruct two galaxy cluster samples. The first one, hereafter\nreferred as \\sampleHS (Fig.~\\ref{fig:sample} thick solid lines)\nis constructed by selecting the heaviest halos [$M_{\\rm vir}>\n4\\times 10^{14} M_\\odot\/h$ (see Table~\\ref{tab:samples})], from\neach independent simulation volume and using each recorded\nsimulation snapshot.  We impose a low redshift cut-off\n$z>z_{\\min}$ to remove very extended clusters.  The choice of\nredshifts for which simulation snapshots are taken is made such\nthat the simulation volume continuously fills comoving space out\nto the maximal redshift (see Fig.1 of~\\cite{Lew2015}).  For each\nsimulation volume we apply random periodic coordinate shifts of\nthe particles within, and we apply random coordinate switches.\nThis (i) improves redshift space coverage and (ii) yields cluster\nSZ surface brightness profiles in different projections, at the\ncost of generating a partially correlated sample.\n\nThe second sample, hereafter referred to as \\sampleDF\n(Fig.~\\ref{fig:sample} thin solid lines), is generated using a\nblind survey approach (as in ~\\cite{Lew2015}). We generate 37 FOV\nrealizations each $\\approx 27\\,{\\rm deg}^2$ together covering a sky\narea of $\\approx 1000\\, {\\rm deg}^2$.  From each realization we\nselect halos with virial masses $M_{\\rm vir,c} > 2\\times 10^{14}\nM_\\odot\/h$.\n\nThe solid angle integrated comptonization for any given halo\n(${\\protect Y=\\int y(\\hat {\\mathbf n}) d\\Omega}$) depends on a\ncombination of halo redshift and mass.  The \\sampleDF sample is\ndominated by lighter halos than those found in the SPT sample\n(Fig.~\\ref{fig:sample}), although redshift space distributions of\nthe two are similar. Hence, the bulk of the \\sampleDF sample\nhalos yields lower $Y(\\theta<0.75')$ values than those in the SPT\nsample (Fig.~\\ref{fig:sample}).  Although increasing the lowest\nmass limit for the halos of the \\sampleDF sample tends to make\nits mass and redshift distributions more consistent with those of\nthe SPT sample, it reduces the numbers of halos, thus increasing\nPoisson noise.  For the statistical analysis in\nthis work, larger simulations and more FOV realizations than are\ncurrently available would be required to reach\nconsistency. Therefore, we use this sample for tSZ analyses of\nsimulated dual beam observations, bearing in mind that in this\nlimit of weak SZ effects, CMB confusion is expected to be the\nmost significant. On the other hand, the \\sampleHS sample is\nexpected to be less affected by CMB confusion.\n\n\n\nIn order to investigate the differences between compact and\nextended SZ clusters we further split our cluster samples by\nredshift at $z=0.4$ (Table~\\ref{tab:samples}).  This split roughly\ncorresponds to half of the radial comoving distance to $z=1.0$,\nbeyond which we do not observe any heavy (Fig.~\\ref{fig:sample})\nhalos in our simulations.\nWe find that the low-$z$ and high-$z$\nsamples mainly differ due to the strength of the SZ effects,\nand due to the presence of sub-structures,\nbeing respectively stronger and more abundant in the low-$z$ subset.\nExamples for halos from low-$z$ and high-$z$ samples are shown\nbelow in Fig.~\\ref{fig:halos}.\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure_6.eps}\n\\caption{Simulated fractions ($F$) of SZ effect flux density at\n30~GHz recovered from difference, dual beam observations of\nclusters from \\sampleDF sample as a function of beam angular\nseparation $s$ and redshift range.  The shaded\/hatched regions\nmap the 68\\% confidence regions (CR) in the distribution of\n$F$.  The scatter in $F$ calculated from maps containing only\ntSZ signal is shown in gray.  The backslash-hatched region\nshows the effects of primary CMB on biasing the tSZ flux\ndensity measurements.  The forward-slash--hatched region shows\nthe intrinsic scatter due to kSZ when converted and embedded\ninto the 30~GHz thermal SZ effect maps.  The green region shows\nthe improvements in decreasing the intrinsic scatter in $F$ as\na result of subtracting the Planck CMB template from CMB+tSZ\nsimulated maps prior to flux density calculations.  The median\n$F$ values are shown as lines.  The 68\\% confidence regions\nabout the medians become asymmetric as the beam separation\nincreases (simulation sample error also becomes obvious in the\nTSZ+CMB case by comparing upper to lower plots; the TSZ,\nTSZ+KSZ, and TSZ+CMB cases are statistically equivalent between\nthe upper and lower panels).  The vertical dashed line marks\nthe actual separation of OCRA beams fixed by the telescope\noptics.  It is assumed that the reference beam covers an\nannulus around a galaxy cluster within parallactic angle range\n$[0^\\circ,180^\\circ]$ on either side of the central direction,\nand that pointing error $\\epsilon_p=0$ (see\nSec.~\\ref{sec:pointing}).  }\n\\label{fig:medianfDF}\n\\end{figure*}\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure_7.eps}\n\\caption{As in Fig.~\\ref{fig:medianfDF} but for the \\sampleHS sample.}\n\\label{fig:medianfHS}\n\\end{figure*}\n\n\n\\section{Analysis and results}\n\\label{sec:results}\n\n\n\n\\subsection{Systematic effects from beam separation}\n\\label{sec:beam_separation}\n\n\nDual beam difference observations capture only a fraction of the\nintrinsic flux density, depending on the physical extent of the\nsource, its redshift and the angular separation of the beams.  We\ndefine this fraction as\n\\begin{equation}\n\\label{eq:F}\nF(\\theta_b, s,q_{\\rm max}) = \\bigg\\langle \\frac{S_0^{\\rm x}(\\theta_b,s) - S_r^{\\rm x}(\\theta_b,s,\\mathbf{\\hat n_i})}{S_0^{\\rm tSZ}(\\theta_b)} \\bigg\\rangle_i,\n\\end{equation}\nwhere $S_0^{\\rm x}$ is the measured central flux density per beam\ninduced by effect ``x'', e.g. x~=~tSZ, $S_0^{\\rm tSZ}$ is the\ntrue central flux density per beam due to tSZ (neglecting CMB,\npoint sources, and other effects), $S_{\\mathrm r}^{\\rm x}$ is the\nflux density per beam due to effect ``x'' in the reference beam\ndirection ($\\mathbf{\\hat n}_i$), $\\theta_b$ is the instrumental\nhalf-power beam width (HPBW), and $s$ is the angular separation\nof the beams.  $N_{\\mathrm r}=500$ reference beam directions\n($\\mathbf{\\hat n}_i$) are chosen randomly from a uniform\ndistribution of parallactic angles ($q \\in [0,q_{\\max}]$), where\nthe upper limit $q_{\\max}$ is a free parameter.\n\nFor each halo, we measure these fractions $F$ by integrating\nspecific intensity directly from high resolution maps, and using\nthe mean over the $N_{\\mathrm r}$ values of $q$.  If there is no\nCMB contamination, i.e. setting x~=~tSZ, so that\n$S_0^{\\rm x} = S_0^{\\rm tSZ}$\nand\n$S_{\\mathrm r}^{\\rm x} = S_{\\mathrm r}^{\\rm tSZ}$,\nthen $F\\leq 1$ and $1-F$ represents the fraction of the signal\nlost only due to the closeness of the beam angular separation.\n\nThe impact of beam separation on the dual-beam observations is\nshown in Figs.~\\ref{fig:medianfDF} and~\\ref{fig:medianfHS} for\nthe \\sampleDF and \\sampleHS samples respectively, for an\nidealistic case of exact pointing---i.e., no pointing\ninaccuracies are allowed ($\\epsilon_p=0$).  In these figures, the\nmedian $F$ (from all halos matching the selection criteria) is\nplotted along with a 68\\% confidence region.  Clearly, dual-beam\nobservations at larger beam separations are less biased than\nobservations at smaller beam separations, and the 68\\% confidence\nrange generally shrinks as $s$ increases.\n\nThe significance of the primary CMB fluctuations for the\ndual-beam observations is estimated by setting x~= tSZ$+$CMB,\ni.e.,\n$S_0^{\\mathrm x}=S_0^{\\mathrm{tSZ+CMB}}$, \n$S_{\\mathrm r}^{\\mathrm x}=S_{\\mathrm r}^{\\mathrm{tSZ+CMB}}$.\nWhile the median $F$ does not differ significantly from the pure\ntSZ case, the 68\\% confidence region significantly increases with\nbeam separations due to primary CMB confusion.  For example,\nsince a primordial CMB fluctuation has a good chance of being of\nthe same sign as the SZ signal at the cluster center but of the\nopposite sign in a distant reference beam, $F$ can easily be\ngreater than unity, as is clear in Fig.~\\ref{fig:medianfDF}.  As\nexpected, the increase is stronger in the \\sampleDF\nsample\/high-$z$ sub-sample than in the \\sampleHS sample\/low-$z$\nsub-sample, due to differences in amplitudes of SZ effects\ncompared to the level of CMB fluctuations.\n\n\n\nComparing Figs.~\\ref{fig:medianfDF} and ~\\ref{fig:medianfHS} it\nis clear that the main difference is the relative significance of\nthe CMB as a source of confusion and the amount of residual\nbiasing.  However, for any individual high-$z$ and\/or low-mass\ncluster observation, the measured flux density can be biased\nsubstantially. \nThis can be inferred from the size of the\n$1\\sigma$ tSZ+CMB confidence region. Even observations of the\nmost massive clusters, which are the least affected by the\npresence of the CMB, can be biased substantially depending on the\nangular scales being measured ($s$) (Fig.~\\ref{fig:medianfHS}\nleft panels).  In the figure, the trade-off\nbetween CMB confusion due to observations at larger angular\nscales and the level of biasing ($F$) in the limit of small $s$\nis clearly seen.\n\n\nFor clusters that are small relative to the beam size,\nmeasurements far away from the cluster center are not really\nneeded as the $F$ values approach unity relatively fast\n(e.g. \\sampleDF\/ high-$z$ sample in Fig.~\\ref{fig:medianfDF}).\nAt the OCRA beam separation (the vertical line in the figures)\nthe primary CMB does not strongly contribute to the scatter in\nflux density measurements.  This is even more so in the case of\nthe \\sampleHS sample of heavy and low-$z$ clusters.  On the other\nhand, the most massive halos (Fig.~\\ref{fig:medianfHS}) require\nsignificant ($> 10\\%$) flux density corrections even at\nlarge beam separations (although these may partially be generated\nby projection effects discussed in Sec.~\\ref{sec:discussion}).\n\nIt is clear that in the two cluster samples, kSZ \nonly slightly increases the scatter in $F$ at the OCRA beam\nseparation, as expected at 30~GHz.\n\n\nThe impact of {\\em Planck} based CMB template removal is shown in green.\nThe calculation is done by setting x~= tSZ$+$CMB$-$template \nin Eq.~\\ref{eq:F},\ni.e.,\n$S_0^{\\mathrm x}=S_0^{\\mathrm{tSZ+CMB-template}}$, \n$S_{\\mathrm r}^{\\mathrm x}=S_{\\mathrm r}^{\\mathrm{tSZ+CMB-template}}$.\nFrom Figs.~\\ref{fig:medianfDF} and~\\ref{fig:medianfHS} it is\nclear that at the OCRA beam separation, and for the full range of\nparallactic angles, the {\\em Planck} template does not\nsignificantly help, or does not help at all, in reducing the\nconfusion due to primary CMB. However, in observations that probe\nlarger angular separations, the CMB template removal can\nsubstantially reduce the $1\\sigma$ contours. The template removal\nmay also be useful for observations of high-$z$ massive clusters\nfor which mapping larger angular distances away from the central\ndirections still appears to be well motivated. Both in the\nhigh-$z$ and low-$z$ sub-samples of the \\sampleHS sample the\ntemplate reduces the tSZ+CMB scatter nearly down to the level\nlimited by the intrinsic tSZ scatter for the full range of $s$\nstudied here (Fig.~\\ref{fig:medianfHS}).\n\n\n\n\\subsection{Parallactic angle dependence}\n\\label{sec:PA}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_8.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_9.eps}\n\\caption{As in Figs~\\protect\\ref{fig:medianfDF} and\n\\protect\\ref{fig:medianfHS}, but for observations where the\nreference beam covers an annulus around a galaxy cluster within\nparallactic angle range $[0^\\circ,22.5^\\circ]$ on either side\nof the cluster direction ({\\rm left}), or only on one side of\nthe cluster direction ({\\rm right}) for the \\sampleDF (top) and\n\\sampleHS (bottom) samples.  }\n\\label{fig:F_vs_PA_DF}\n\\end{figure}\nIn practise, the OCRA observations exploit beam and position\nswitching (Sect.~\\ref{sec:strategy}) but it is unrealistic to\ncover the full parallactic angle range: i.e. $q\\in\n[0^\\circ,q_{\\max}]$ where $q_{\\max}=360^\\circ$.\n\nIt was already known that position switching\n\\citep{Birkinshaw2005} significantly mitigates atmospheric\ninstabilities over the time scale of tens of seconds by (i)\nsubtracting linear drifts caused by large-scale precipitable\nwater vapor (PWV) fluctuations \\citep{Lew2016}, (ii) accounting\nfor beam response asymmetries, and (iii) maximizing the\nprobability of avoiding (masking out) intervening radio sources\nthat can significantly bias the SZ measurement.  In this section,\nwe show that position switching is also efficient in mitigating\nthe confusion due to primordial CMB, even with a very modest\ncoverage of parallactic angles.\n\nThere should not be any statistical correlation between\nprimordial CMB fluctuations and the locations of heavy\nhalos. Moreover, galaxy clusters have small angular sizes\ncompared those representing most of the CMB power.  Thus,\nclusters should mostly lie on slopes rather than peaks or troughs\nin the CMB map. Hence, sampling SZ flux density differences at\nopposite sides of a galaxy cluster core should help average out\nthe primordial CMB in comparison to one-sided observations.  We\nconfirm that this is indeed the case and find that this\nimprovement is reached at even moderate values of $q_{\\max}$.\n\nWe calculate $F(s)$ [Eq.~(\\ref{eq:F})] for maps containing tSZ\nand CMB using mean flux density estimates either according to the\nposition switching observation scheme or without it.  As before,\neach measurement is an average of $500$ dual-beam pointings at\ndifferent $q$ but drawn randomly from within the range\n$[0^\\circ,q_{\\max}]$ where\n$q_{\\max}\\in\\{180^\\circ,90^\\circ,45^\\circ,22.5^\\circ\\}$.\n\nThe result is shown in Fig.~\\ref{fig:F_vs_PA_DF} for\n$q_{\\max}=22.5^\\circ$.  By comparing the left panel of this\nfigure with the top-left panel of Fig.~\\ref{fig:medianfDF} it is\nclear that even strongly incomplete coverage of the parallactic\nangles does not cause significant broadening of the 68\\%\nconfidence level (CL) contours.  However, when position switching\nis not used (right panels in Fig.~\\ref{fig:F_vs_PA_DF}), the\nconfusion due to primordial CMB is stronger. As before, the\n\\sampleHS sample of the heaviest halos is less affected by the\npresence of the CMB, but the effect of not using position\nswitching is still visible, even at $s=s_{\\rm OCRA}$ (vertical\nline in Fig.~\\ref{fig:F_vs_PA_DF}, bottom panels).\n\n\n\\subsection{Systematic effects in redshift space}\n\\label{sec:Fz}\n\nThe $F$ factor depends on a cluster's angular size, which in turn\ndepends on the cluster's redshift.  We model the dependence of\n$F(\\theta_b, s,p)$ (Eq.~\\ref{eq:F}) on redshift by defining:\n\\begin{equation}\n\\label{eq:Fz}\nF_{\\mathrm m}(\\theta_b, s,\\beta,\\theta_c) =\n1 - \\frac{\\int b(\\mathbf{\\hat n},\\theta_b,s) \\, I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)\\d\\Omega}{\\int b(\\mathbf{\\hat n},\\theta_b,s=0) \\, I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)\\d\\Omega},\n\\end{equation}\nwhere $b(\\mathbf{\\hat n},\\theta_b,s)$ is a Gaussian beam profile\nwith beam width $\\theta_b$ , offset by angular distance $s$ from\nthe cluster center direction $\\mathbf{\\hat n_0}$.  We choose\n$s=s_{\\rm OCRA}$ to simulate the position of the OCRA reference\nbeam when the primary beam points at the cluster center.\n$I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)$ is a normalized,\nLOS integrated $\\beta$ profile that represents the SZ effect\nsurface brightness:\n\\begin{equation}\n\\label{eq:beta}\nI_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta, \\theta_c) \\propto\n\\left(1+ \\frac{\\theta^2}{\\theta_c^2} \\right)^{\\frac{1}{2}-\\frac{3}{2}\\beta},\n\\end{equation}\nwhere $\\theta_c=2 r_c\/d_A(z)$ is the angular diameter of the\nobserved galaxy cluster defined in terms of its core size $r_c$,\n$\\theta$ is the angle from $\\mathbf{\\hat n}$ to $\\mathbf{\\hat\nn_0}$, and $d_A(z)$ is the angular diameter distance.\n$F_{\\mathrm m}$ depends on the choice of cosmological parameters\nand on the chosen cluster density profile. We calculate\n$F_{\\mathrm m}$ for $\\Lambda$CDM cosmological parameters:\n$h=0.7$, $\\Omega_m=0.3$, $\\Omega_\\Lambda=0.7$, and for an\nEinstein--de~Sitter cosmological model. For our redshift range we\nfind that the dependence on cosmological parameters is weak\ncompared to the dependence on the halo density profile\n(Fig.~\\ref{fig:f_z}).  We also calculate $F_{\\mathrm m}$ for the\ncase of a Gaussian halo but find that such profile is strongly\ndisfavored by simulations as $F_{\\mathrm m}$ approaches unity at\nfairly low redshifts.\n\n\nThe simplest $\\beta$-model does not allow for the steepening of\ndensity profiles with increasing $\\theta$.  However, X-ray\nobservations suggest that such steepening is real\n(e.g. \\citealt{Vikhlinin2006}), and it is expected that at large\ndistances from cluster cores (or higher redshifts) the\n$\\beta$-model yields lower $F_{\\mathrm m}(z)$ values than those\npredicted by simulations, as seen in Fig.~\\ref{fig:f_z}.\n\nClearly, the \\sampleHS sample has a large scatter in $F$ values\nat high redshifts. Some of that scatter is due to projection\neffects, which we discuss latter.  Heavy clusters of the\n\\sampleHS sample appear more compatible with the $\\beta$-model at\nlower $\\beta$ values than the lower-mass clusters of the\n\\sampleDF sample.  At the OCRA beam separation, the low-mass\nclusters in both samples show very weak effects of biasing\n($F\\approx 1$) at the highest redshifts.  On the other hand heavy\nhalos require large corrections, some of which do not result from\nsimple projection effects. In the next section, a selection of\nhalos are investigated individually.\n\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_10.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_11.eps}\n\\caption{Simulated fractions ($F$) as a function of redshift and\nvirial mass for clusters from \\sampleDF (left) and \\sampleHS\n(right) samples and for observations at effective beams angular\nseparation ${\\protect s=0.0526^\\circ}$ (the separation of OCRA\nbeams).  The fractions were measured from maps containing tSZ\nsignal only.  The lines trace the dependence for a halo\ndescribed by a $\\beta$-model according to Eq.~\\ref{eq:Fz} with\nparameters given in the plot legend.  The rectangles ``1'' and\n``2'' mark strong outliers and some halos from the main group\nthat are inspected individually (see text for discussion).  }\n\\label{fig:f_z}\n\\end{figure*}\n\n\n\\subsection{Analysis of individual clusters}\n\n\\begin{table}[t]\n\\caption{Parameters of halos selected from Fig.~\\ref{fig:f_z}.\nThe parameter $y_0$ is the maximal value of the LOS integrated\ncomptonization parameter.  }\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\n\nID & $z$ & $F$ & $M_{\\rm vir}$ & $y_0\\times 10^5$ & comment\\footnotemark[1]\\\\\n& & & $[10^{14} M_\\odot\/h]$  &&\\\\\n\\hline\n\\multicolumn{6}{c}{\\sampleHS sample (rect. ``1'' selection)}\\\\\n1  & 0.552 & 0.39 & 4.6  & 1.38 & P \\\\\n2  & 0.520 & 0.43 & 4.7  & 3.40 & P \\\\\n3  & 0.885 & 0.49 & 4.4  & 0.71 & P, D \\\\\n4  & 0.429 & 0.47 & 4.6  & 2.10 & P, E \\\\\n5  & 0.366 & 0.44 & 5.1  & 1.75 & P \\\\\n6  & 0.370 & 0.53 & 4.9  & 2.10 & P \\\\\n7  & 0.349 & 0.50 & 5.1  & 3.10 & P \\\\\n8  & 0.337 & 0.34 & 5.4  & 3.70 & P \\\\\n9  & 0.388 & 0.50 & 4.4  & 2.2  & P \\\\\n\\multicolumn{6}{c}{\\sampleHS sample ($M_{\\rm vir}>12.7\\times 10^{14} M_\\odot\/h$)}\\\\\n10 & 0.103 & 0.41 & 13.8 & 11.2 & D \\\\\n11 & 0.109 & 0.56 & 14.6 & 25.6 & R, P \\\\\n12 & 0.193 & 0.49 & 12.8 & 5.6  & E, D, S, P\\\\\n\\multicolumn{6}{c}{\\sampleDF sample (rect. ``1'' selection)}\\\\\n13 & 0.367 & 0.44 & 2.1  & 0.93  & P \\\\\n\\multicolumn{6}{c}{\\sampleHS sample (rect. ``2'' selection)}\\\\\n14 & 0.182 & 0.64 & 11.3 & 6.68  & D \\\\\n15 & 0.268 & 0.77 & 12.6 & 9.50  & D \\\\\n16 & 0.169 & 0.75 & 10.1 & 10.3  & R, S\\\\\n\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{P - reference beam flux density contamination\nfrom another halo due to LOS projection; D - disturbed\nmorphology; E - elongated shape; R - regular morphology\n(virialized halo); S - sub-halo(s) present;}\n\\label{tab:halos}\n\\end{table}\n\nIn Fig.~\\ref{fig:f_z} some of the halos are selected by\nrectangles in the $z-F$ diagram.  The properties of some of these\nhalos are given in Table.~\\ref{tab:halos}.  Fig.~\\ref{fig:f_z}\nshows that only the lightest halos in our samples are found to be\nstrong outliers, which is unsurprising.\n\nWe visually inspected all the clusters listed in\nTable~\\ref{tab:halos} and verified that each of the clusters from\nrectangle ``1'' (halo IDs from 1 to 9, and 13) lie at sky\npositions that are partially within another cluster's atmosphere\nand also within the angular distance of the reference beam. An\nexample of such overlap is shown in Fig.~\\ref{fig:halos} (top\npanels).\n\nInspection of the three highest mass clusters in the \\sampleHS\nsample (halo IDs 10, 11, and 12; black dots in right panel of\nFig.~\\ref{fig:f_z}) show that two of them (IDs 11 and 12) are\nalso affected to some degree by a LOS projection, but the\nmorphology of halo 10 shows no signs of another halo in the\ncomposite high-resolution map.  Instead, the SZ signature has a\ndisturbed morphology with angular extents larger than a single\nOCRA beam separation even though all three are at redshift\n$z>0.1$. This results in small $F$ values, and motivates\nmeasurements at larger angular separations.\n\nIn order to test whether high redshift clusters that\nsignificantly contribute to the scatter in the $F$--$z$ plane\n(Fig.~\\ref{fig:f_z}) could also benefit from observations out to\nangular distances beyond $s_{\\rm OCRA}$, we investigate the three\nmost massive clusters from rectangle ``2'' (Fig.~\\ref{fig:f_z},\nIDs: 14,15 and 16).  Their corresponding $F$ values\n(Tab.~\\ref{tab:halos}) do not seem to result from projection\neffects. Instead, these clusters have extended atmospheres and\/or\nstrongly disturbed and asymmetric SZ profiles (e.g. cluster 14,\nFig.~\\ref{fig:halos2}).\n\n\nSome of the heavy clusters have surface brightness profiles\n(Fig.~\\ref{fig:halos2}) that are strongly inconsistent with an\naxially-symmetric $\\beta$-profile. This necessitates using more\nsophisticated two-dimensional profiles at the data analysis stage\n\\citep{Lancaster2011,Mirakhor2016}.  Clearly, heavy halos\ngenerate low $F$ values and require large flux density\ncorrections with an OCRA type standard observational strategy\n(Sec.~\\ref{sec:strategy}).  These low $F$ values may partially\nstem from spurious projection effects (e.g. halos 11 and 12)\nwhich arise at the FOV generation stage for halos from the\n\\sampleHS sample (see Sec.~\\ref{sec:discussion}).\n\nThe outlying halos (rectangle ``1'') are either mergers (close\npairs of SZ-strong halos), or have elongated of disturbed\nmorphology (e.g. halos 3 and 4), or have small scale\nsub-structures.  However, in many cases these properties occur\nat spatial scales that will not be resolved in OCRA~SZ\nobservations and\/or may be relevant only as galaxy scale SZ\neffects that are too faint to be detected.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_12.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_13.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_14.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_15.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_16.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_17.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_18.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_19.eps}\n\\caption{Selection of simulated Compton $y$-parameter profiles\n(in arbitrary units) for halos from Table.~\\ref{tab:halos}. The\npanels show profiles for individual halos in physical\ncoordinate space ({\\em left}), and their coarse-grained version\nobtained from high resolution maps in angular space with\ncontributions from other halos along the LOS ({\\em right}).\nThe position of halos in the left-hand side panels is defined\nby a box size that contains all FOF particles of the halo\nassociated with a given cluster. In the right-hand side panels\nthe SZ peak for the cluster is located in the plot center. For\nany given cluster the flux density calculation is done at the\nsky position of the peak. For each cluster the black circles\ndenote OCRA FWHMs and their relative separation ($s_{\\rm\nOCRA}$).  }\n\\label{fig:halos}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_20.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_21.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_22.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_23.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_24.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_25.eps}\n\\caption{As in Fig.~\\ref{fig:halos} but for the selection of halos from rectangle ``2''. See Table.~\\ref{tab:halos} for details.}\n\\label{fig:halos2}\n\\end{figure}\n\n\n\n\n\\subsection{Practical aspects of using CMB templates}\n\\label{sec:template}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{figure_26.eps}\n\\caption{Residual histogram (i.e. observational pixel\nfrequencies minus median pixel frequencies estimated from an\nensemble of Gaussian NILC map simulations) of the CMB\ntemperature fluctuations at and around the {\\em Planck} SZ\ngalaxy clusters, measured in the {\\em Planck} NILC inside\ncircular apertures of radius $r$ centered at the clusters'\npositions (solid); and $1\\sigma$, $2\\sigma$ and $3\\sigma$\nconfidence contours of the pixel frequencies in these\nsimulations (dashed).}\n\\label{fig:NILC-test}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{figure_27.eps}\n\\caption{As in Fig.~\\ref{fig:NILC-test} but for the {\\em Planck}\n217~GHz frequency map and only for clusters at galactic\nlatitude $b>60^\\circ$.}\n\\label{fig:217GHz-test}\n\\end{figure}\n\n\nBy subtracting the templated version of the CMB map\n(Sec.~\\ref{sec:CMBsims}) from the pure CMB simulation, it is easy\nto estimate the upper limit of the residual CMB signal captured\nin the OCRA difference beam observations.  The {\\em Planck}\n217~GHz and NILC maps have enough pixels to create a template of\nresolution of the order of an arcminute, and at least the former\nshould contain only a negligible tSZ signal.\n\nAlthough subtracting CMB templates from the pure CMB maps\ndecreases the large-scale variance by an order of magnitude, the\nresidual variance in the map is carried by high frequency noise\nthat will generate a small amount of dispersion in difference\nobservations (Fig.~\\ref{fig:templates} bottom-right panel).\nHowever, the residual small-scale noise should approximately\naverage out under rotation of the beams in the sky, and since the\nlarge-scale power is effectively removed, increasing the\neffective separation should not suffer from exponential variance\ngrowth due to primordial CMB at arcminute angular scales.\n\nHow reliable are the 217~GHz or NILC {\\em Planck} templates in\ncorrecting single frequency SZ observations for confusion with\nthe primordial CMB?  The 217~GHz map is foreground contaminated\nand the NILC map, although foreground cleaned, still may contain\nresidual tSZ signals at scales least optimized in the needlet\nspace.\n\nIn order to quantify the foregrounds and residual tSZ\ncontaminaiton in each map, we calculate histograms of the\ntemperature fluctuation distribution outside of a mask that\nremoves the full sky except for the directions towards {\\em\nPlanck}-detected galaxy clusters from the PCSS SZ union R.2.08\ncatalog \\citep{PlanckCollaboration2015f}.  Each non-masked region\nis a circular patch of radius $a=\\{2.5',5',10',15'\\}$.\nForegrounds will generate strong positive skewness in the\ntemperature distribution, while the presence of residual tSZ in\nthe NILC map should manifest itself by either a positive or\nnegative skew depending on the frequency weights in the internal\nlinear combination.\n\nWhile the results of the test for the {\\em Planck} NILC map\n(Fig.~\\ref{fig:NILC-test}) do not give strong deviations from\nGaussian simulations, the 217~GHz map generally does.  The data\nare inconsistent with Gaussian simulations even at high galactic\nlatitudes (Fig.~\\ref{fig:217GHz-test}), although the significance\nof the foregrounds seems to depend on the size of the circular\npatch. This implies that the 217~GHz frequency map cannot readily\nbe used to mitigate the confusion due to CMB in OCRA observations\nwithout further assumptions on the foregrounds' frequency\ndependence.  However, it should be interesting to quantify the\nsignificance of the arcminute scale Galactic foregrounds at\n30~GHz at high and intermediate latitudes for OCRA difference\nobservations with small beam separations.  {\\em Planck}-LFI data\nmight also help reduce these foregrounds, though we do not study\nthis here.\n\nSince the foreground cleaned NILC map is statistically consistent\nwith Gaussian simulations\n\\citep{PlanckCollaboration2015a,PlanckCollaboration2015d} towards\nthe {\\em Planck}-detected galaxy clusters\n(Fig.~\\ref{fig:NILC-test}), it should also be suitable for\nmitigating CMB confusion in OCRA observations in directions\noutside of the mask where clusters undetected by {\\em Planck}\nlie.\n\n\\subsection{Pointing requirements}\n\\label{sec:pointing}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figure_28.eps}\n\\includegraphics[width=0.45\\textwidth]{figure_29.eps}\n\\caption{Systematic effects in tSZ flux density reconstruction\nfrom dual-beam observations as a function of telescope pointing\nerrors ($\\epsilon_p$) and beam separation $s$ for the\n\\sampleDF~sample (left) and for the \\sampleHS~sample (right).\nThe reference beam is assumed to cover all possible parallactic\nangles for any given galaxy cluster.  }\n\\label{fig:F_vs_pointing}\n\\end{figure*}\n\nIn order to quantify the implications of telescope pointing\nerrors on the reconstruction of Compton $y$-parameters, and to\ndefine pointing requirements, we introduce a pointing precision\nparameter $\\epsilon_p$ that defines the maximal angular distance\nthat a primary beam can have from the intended position, and then\nwe repeat the analysis of Sec.~\\ref{sec:beam_separation}.  The\npointing error $p$ is drawn from a uniform distribution on\n$[0,\\epsilon_p]$, since the RT32 pointing and tracking are\ndominated by systematic errors, and we investigate different\nvalues of $\\epsilon_p$.  Since galaxy cluster SZ profiles are\ntypically steep functions of angular separation, any pointing\ninaccuracy will lead to biasing measurements of the central\ncomptonization parameter when taking averages from multiple\nobservational sequences.\n\nFigure~\\ref{fig:F_vs_pointing} shows that for the \\sampleHS\nsample, i.e. typically heavy clusters, pointing error up to\n$\\epsilon_p\\approx {\\rm HPBW}\/4$ should not lead to strong\n($>10\\%$) extra biases relative to the $\\epsilon_p=0$ case.\nMeasurements of the \\sampleDF sample, i.e. typically less massive\nclusters, are more sensitive to pointing errors, but if the\npointing accuracy is better than $\\epsilon_p=0.005^\\circ$ (${\\rm\n\\theta_b^{\\rm OCRA}}\\approx 1.2'$) the additional systematic\neffects will be smaller than $10\\%$.  However, larger pointing\nerrors should be taken into account at the data analysis stage. An\nobservational campaign is currently under way to improve RT32 pointing\naccuracy.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe map-making procedure that has been tested for recovering the\nsource intensity distribution from OCRA difference measurements\n($s=s_{\\rm OCRA}$) assumes a flat background. This is not a\nproblem for reconstructions of comptonization parameters from SZ\nobservations of heavy clusters, as with the standard OCRA\nbeam-pair separation the corrections due to CMB background are\nsmall.  However, reconstructing cluster SZ profiles out to larger\nangular distances could benefit from correcting the difference\nmeasurements according to the {\\em Planck} CMB template.  For\nexample, Fig.~\\ref{fig:medianfHS} (left panel) shows that an\nobservation at $s=2 s_{\\rm OCRA}$ decreases bias by $\\Delta\nF\\approx 0.2$. At the same time CMB confusion broadens the 68\\%\nCR by $\\Delta F \\approx 0.1$, but applying a {\\em Planck} CMB\ntemplate reverses this effect almost down to the intrinsic\ntSZ+kSZ scatter.\n\nAs discussed in Sec.~\\ref{sec:sample} the \\sampleDF and \\sampleHS\nsamples represent quite opposite observational\napproaches. However, since halos of the \\sampleHS sample were\nselected from full simulation volumes (rather than from\nlight-cone sections), mock maps for this sample contain clusters\nwith angular sizes calculated according to their redshifts and\nphysical extents, as in the case of FOV simulations, but are\nplaced in the map at rectilinearly projected locations.  This\ncontaminates the resulting maps with halos that would not fall\ninto the assumed FOV in the standard light-cone approach.  These\nspurious halo--halo overlaps may somewhat enlarge the 68\\% CR\ncontours of various $F$ distributions (e.g. x=tSZ or x=tSZ+CMB).\nA possible modification of the calculation scheme for the\n\\sampleHS sample would be to consider each halo independently,\nthus completely ignoring the intrinsic projection effects that\nexist in the light-cone approach, or by extending the FOV to a\nhemisphere (which would probably require implementing adaptive\nresolution maps to maintain the angular resolution of the present\ncalculations).\n\nThe cluster samples that we analyze were not screened to select\nvirialized clusters.  Although we analyzed sub-samples selected\nusing a virialization criterion (based on ratios of potential to\nkinetic energy of FOF halo particles) the results presented here\nare based on the full sample in order to retain a morphological\nvariety of SZ galaxy cluster profiles (Figs.~\\ref{fig:halos} and\n~\\ref{fig:halos2}), and to expose the complexity of SZ flux\ndensity reconstructions from observations that do not intend to\ncreate multi-pixel intensity maps.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe quantify the significance of systematic effects arising in\ndual-beam, differential observations of Sunyaev-Zel'dovich (SZ)\neffect in galaxy clusters.  We primarily focus on effects\nrelevant to the reconstruction of comptonization parameters from\nsingle frequency flux-density observations performed with the One\nCentimeter Receiver Array (OCRA) -- a focal plane receiver with\narcminute scale beamwidths and arcminute scale beam separations\n-- installed on the 32~m radio telescope in Toru\\'n.\n\nUsing numerical simulations of large scale structure formation we\ngenerate mock cluster samples (i) from blind surveys in small\nfields of view and (ii) from volume limited targeted observations\nof the most massive clusters (Sec.~\\ref{sec:sample}).  Using mock\nintensity maps of SZ effects we compare the true and recovered SZ\nflux densities and quantify systematic effects caused by the\nsmall beam separation, by primary CMB confusion and by\ntelescope pointing accuracy.\n\nWe find that for massive clusters the primary CMB confusion does\nnot significantly affect the recovered SZ effect flux density\nwith OCRA beam angular separation of $\\approx 3'$. However, these\nobservations require large corrections due to the differential\nobserving strategy. On the other hand, measurements of SZ-faint\n(or high redshift $0.4<z<1.0$) clusters may have their SZ\nphotometry erroneously estimated by $10\\%$ or more due to CMB\nconfusion, which becomes stronger in observations that map larger\nangular distances from cluster centers.\n\nWe investigate the possibility of mitigating the CMB confusion in\nSZ observations that map scales beyond $3'$ from cluster centers\nby using {\\em Planck} CMB 217~GHz and foreground reduced NILC\nmaps as primary CMB templates. Using simulations we find that\nthese templates have sufficiently high angular resolution\/low\nnoise to significantly mitigate CMB confusion in 30~GHz\nobservations of high-$z$ clusters, given that the templates do\nnot contain residual foregrounds. Using a simple one-point\nstatistic, targeted towards directions of known clusters, we\nverify that at least the {\\em Planck} NILC map should also be\nsufficiently free from foregrounds to serve as a primary CMB\ntemplate (Sec.~\\ref{sec:template}) that could improve OCRA-SZ or\nsimilar observations extended to larger angular scales\n(Sec.~\\ref{sec:strategy}).\n\nFinally, we find that RT32 telescope pointing and tracking\naccuracy $\\epsilon_p\\ < 0.005^\\circ$ should keep systematic\nerrors in recovered SZ flux densities (comptonization parameters)\nbelow $\\approx 5\\%$ (after correcting for other systematic\neffects) even in observations of SZ-weak clusters\n(Sec.~\\ref{sec:pointing}).\n\n\\section*{Acknowledgments}\nThank you to Mark Birkinshaw for discussion on OCRA observational\nstrategies, and to an anonymous referee for useful comments.\nThis research has made use of a modified version of the\nGPL-licensed ccSHT library.  We also acknowledge use of the\nmatplotlib plotting library \\citep{Hunter2007}.\nThis work was financially supported by the Polish National\nScience Centre through grant DEC-2011\/03\/D\/ST9\/03373.  A part of\nthis project has made use of computations made under grant 197 of\nthe Pozna{\\'n} Supercomputing and Networking Center (PSNC).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nRecently, deep convolutional neural network (DCNN)-based approaches have shown remarkable performance on image classification tasks \\cite{He2015ResNet, Szegedy2016InceptionNet, Huang2017DenseNets}.  State-of-the-art image classification networks often contain thousands or even millions of parameters which need to be learned during training.  While it has been argued in the literature that high-dimensionality (data, parameterization, model complexity) is more amenable for classification \\cite{Anderson2014BlessingDimensionalityGaussianMixtures,Breiman2001StatisitcalModeling,Gorban2019BlessingofDimensionality,Huang2019UnderstandingGeneralization}, methods taking advantage of high-dimensionality are often inscrutable.  One approach for handling the nuances associated with high-dimensionality is simply to reduce the dimensionality of the data\/model.   \n\n\\textit{Manifold Learning} or \\textit{Dimensionality Reduction} (DR) has proven to be a critical tool in many machine learning applications, such as: visualizing data, removing redundancy, minimizing memory usage, and reducing the effects of the Curse of Dimensionality. \\cite{VanDerMaaten2009DRReview, Oza2001DecorrelationDimReduction,Becht2019Dimensionality}.  The goal of manifold learning can be posed as discovering intrinsic (often lower-dimensional) features from the data which meet an overarching objective, such as: preserving variance, finding compressed representations of data, maintaining global or local structure or promoting class discriminability in the embedding space \\cite{VanDerMaaten2009DRReview,Bengio2014RepLearningReview, Geng2005SupNonlinearDimRed, Thorstensen2009ManifoldThesis}.  Most studies perform classification or regression after applying unsupervised dimensionality reduction.  However, it has been shown that there are advantages to learning the low-dimensional representations and classification\/regression models simultaneously \\cite{Chao2019RecentAdvancesSupervisedDimRed,Rish2008SupDimRedGLM}.  Specific to classification, the goal of DR is to discover embedding functions that take data from the input feature space and transform it into a lower-dimensional coordinate system or \\textit{latent space}. Ideally, the low-dimensional features capture ``useful\" properties of the data while enforcing constraints such as topological ordering and class separability \\cite{Vural2018StudySupervisedManifoldLearning}.\n\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=.67\\textwidth]{Figures\/DREN_v2.PNG}\n  \\caption{Architecture for the proposed divergence regulated encoder networks (DRENs).}\n  \\label{fig:model}\n\\end{figure*}\n\nThe representation of data is a critical factor in determining machine learning classification performance \\cite{Bengio2014RepLearningReview}.  For images, one can generally represent information through color, shape, and texture features \\cite{hiremath2007content}. As shown in the literature \\cite{liu2019bow}, texture features often serve as the most powerful descriptor of the three. As currently constructed, DCNNs cannot directly model texture information without the use of extra layers and parameters. To capture texture features in DCNNs, \\textit{histogram layer} networks \\cite{peeples2020histogram} were introduced to characterize the distribution of features in the model. We hypothesize that incorporating texture features into our proposed approach will assist in providing meaningful representations of the data for classification. \n\n\n\nBased on these motivations, we introduce a neural classification approach which inherently learns compressed feature representations optimized for class discriminability.  \nThe contributions of this work are summarized as the following:\n\\begin{itemize}\n\n  \\item  We propose a neural classification scheme which finds discriminative, low-dimensional representations of the input data.  The network learns from a t-SNE based objective, yet outperforms the data representations found by classical t-SNE in terms of classification accuracy.\n  \\item The proposed method is constructed from a histogram layer network backbone which extracts highly descriptive texture features. \n  \\item The proposed approach can readily embed out-of-sample points, allowing it to be applied universally to other image classification datasets. In our experiments, we verified the effectiveness of the proposed method on two datasets: KTH-TIPS-2b \\cite{KTHTIPS_Databases} and Describable Texture Dataset (DTD) \\cite{cimpoi14describing}.\n\\end{itemize}\n\n\n\n\n\n\n\n    \n\n\n\n\n\\section{Methods}\n\\label{sec:methods}\n\\subsection{Proposed Approach}\nIn order to jointly perform classification and DR, our objective function is comprised of two terms: classification loss, $L_{class}$ , and a divergence measure, $L_{div}$:\n\\begin{equation}\n    L_{total} = (1-\\lambda)L_{class}(\\hat{\\mathbf{Y}},\\mathbf{Y}) +\\lambda L_{div}(\\mathbf{P},\\mathbf{Q})\n\\end{equation}\nwhere $\\mathbf{P} \\in \\mathbb{R}^{N \\times N}$ is the joint probability distribution on the high dimensional input features $\\mathbf{X} \\in \\mathbb{R}^{N \\times D}$ for $N$ samples in a mini-batch, $\\mathbf{Q} \\in \\mathbb{R}^{N \\times N}$ is the joint probability distribution on the lower dimensional embeddings $\\mathbf{Z} \\in \\mathbb{R}^{N \\times d}$, $\\hat{\\mathbf{Y}} \\in \\mathbb{R}^{N \\times C}$ are the sample predictions for $C$ classes, and $\\mathbf{Y} \\in \\mathbb{R}^{N \\times C}$ are the true class labels. We add a weight, $\\lambda$, to leverage the contribution of each term to the overall objective. We define $L_{class}$ as the popular cross-entropy loss:\n\\begin{equation}\n   L_{class}(\\hat{\\mathbf{Y}},\\mathbf{Y}) = -\\frac{1}{N}\\sum_{i=1}^{N}\\sum_{c=1}^{C}y_{ic}log\\hat{y}_{ic}\n\\end{equation}\n\nFor t-SNE, Kullback-Leibler (KL) divergence is typically used to minimize the divergence between the higher and lower dimensional joint probability distributions:\n\\begin{equation}\n    L_{div}(\\mathbf{P},\\mathbf{Q}) = \\sum_{i=1}^{N}\\sum_{j=1}^{N} p_{ij}log\\frac{p_{ij}}{q_{ij}}\n\\end{equation}\nFollowing the standard t-SNE \\cite{VanDerMaaten2008Visualizing} implementation, we used a normalized radial basis kernel and Student t-distribution with a single degree of freedom to estimate the neighbor probability matrices $\\mathbf{P}$ and $\\mathbf{Q}$ in the high and low dimensional feature spaces, respectively. Our approach is agnostic to the selection of a divergence measure. Hence, we used other divergence measures such as Wassertein-$1$ (\\textit{i.e.}, earth mover's distance) \\cite{levina2001earth} and Renyi's divergence \\cite{van2014renyi} with $\\alpha = 0.5$. There was not a statistically significant difference in the classification performance of our model. Therefore, results presented in this work are representative of using KL divergence.\n\\begin{figure*}[t]\n    \\centering\n\t\\begin{subfigure}{.40\\textwidth}{\n\t\t\t\\includegraphics[width=\\textwidth]{Figures\/KTH_Base_DREN.png}\n\t\t\t\\caption{$D=512$, 78.77\\% $\\pm$ 2.86\\%}\n\t\t\t\\label{fig:KTH_Base}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\\begin{subfigure}{.40\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/KTH_Hist_DREN.png}\n\t\t\t\\caption{$D=1024$, 77.97\\% $\\pm$ 2.12\\%}\n\t\t\t\\label{fig:KTH_Hist}\n\t\t}\n\t\\end{subfigure}\n\t\n\t\\centering\n\t\\begin{subfigure}{.40\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/DTD_Base_DREN_V2.png}\n\t\t\t\\caption{$D=2048$, 62.79\\% $\\pm$ 0.94\\%}\n\t\t\t\\label{fig:DTD_Base}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\\begin{subfigure}{.40\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/DTD_Hist_DREN_V2.png}\n\t\t\t\\caption{$D=4096$, 62.78\\% $\\pm$ 1.12\\%}\n\t\t\t\\label{fig:DTD_Hist}\n\t\t}\n\t\\end{subfigure}\n\t\\caption{Encoder experiments performance of the different variations of DREN for KTH-TIPS-2b (\\ref{fig:KTH_Base}-\\ref{fig:KTH_Hist}) and DTD (\\ref{fig:DTD_Base}-\\ref{fig:DTD_Hist}). In each sub caption, the high dimensional size and performance without the encoder is shown. For clarity, the error bars are not shown.}\n\t\\label{fig:PKNN_examples} \n\\end{figure*}\n\n\\subsection{Implementation}\nFor our divergence regulated encoder network (DREN), we used two pretrained models as the backbones, ResNet18 and ResNet50 \\cite{He2015ResNet} for KTH-TIPS-2b \\cite{KTHTIPS_Databases} and DTD \\cite{cimpoi14describing}, respectively.  We added a histogram layer after the last convolutional layer for each of the pretrained models. From these models, we extracted the high dimensional feature vectors, $\\mathbf{X}$, for each sample. For the baseline and histogram models used for KTH-TIPS-2b and DTD, $D$ was $512$, $1024$, $2048$, and $4096$ respectively. Our encoder was comprised of four fully connected layers of size $128$, $64$, $32$, and $d$. We used ReLU activation functions for the first three layers and feed the $d$-dimensional embedding to a softmax output layer for classification. \n\nThe embeddings, $\\mathbf{Z}$, were used to compute $L_{div}$ and to update the weights of the model via backprogation of errors. To create a fair comparison to t-SNE applied to the high dimensional features, we only updated the lower-dimensional embeddings.  Therefore, $\\mathbf{P}$ was removed from the computational graph such that only the gradient information from $\\mathbf{Q}$ was used. The overall structure of DREN is shown in Figure  \\ref{fig:model} and the code for our work is publicly available [code provided in supplemental material].\n\n\\section{Experimental Setup}\n\t\n\nOur experiments consisted of two main components: encoder performance and comparisons to t-SNE. For our encoder experiments, we investigated the selection of hyperparameters $\\lambda$ and $d$ for the proposed DREN models. The typical train\/test splits for DTD (train on 80 images per class and test on 40 images per class) and KTH-TIPS-2b (train on three samples, test on one sample) were used for our experiments \\cite{liu2019bow}. We held 10\\% of the training data for validation and applied each model to the holdout test set. We performed four and ten runs of data splits for KTH-TIPS-2b and DTD respectively. The average classification accuracy is reported across the different folds. We followed a similar training procedure and data augmentation from previous works \\cite{xue2018deep,peeples2020histogram}, except Adam optimization was used. We also only updated newly added layers (\\textit{i.e.}, histogram layers, encoder, output layer) and kept the pretrained weights fixed. The number of bins used in all histogram models was 16.\n\nThe weight in the objective term, $\\lambda$, was varied from $0$ to $1$ in steps of $0.1$. The embedding dimension, $d$, was set to $2$, $3$, $4$, $8$, and $16$. After our encoder experiments, we compared the training and testing embeddings learned by DREN to t-SNE. In order to embed test samples for t-SNE, we used an approach similar to Locally Linear Embedding \\cite{Roweis2000LLE}, which assumes that a sample can be represented as a linear combination of its nearest neighbors. Thus, low-dimensional coordinates of unseen test points were found as constrained combinations of their nearest neighbors in the training set. We evaluated each embedding a) qualitatively and b) quantitatively by reporting the classification accuracy of a $K$-NN classifier on the test data ($K=3$). \n\n\\section{Results and Discussion}\n\\label{sec:Results}\n\\begin{figure*}[t]\n    \\centering\n\t\\begin{subfigure}{.156\\textwidth}{\n\t\t\t\\includegraphics[width=\\textwidth]{Figures\/KTH_TSNE_embedding.png}\n\t\t\t\\caption{t-SNE}\n\t\t\t\\label{fig:KTH_TSNE_embed}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\\begin{subfigure}{.162\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/KTH_Base_embedding.png}\n\t\t\t\\caption{ResNet18}\n\t\t\t\\label{fig:KTH_Res_embed}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\\begin{subfigure}{.162\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/KTH_Hist_embedding.png}\n\t\t\t\\caption{HistRes18}\n\t\t\t\\label{fig:KTH_Hist_embed}\n\t\t}\n\t\\end{subfigure} \n\t\\centering\n\t\\begin{subfigure}{.163\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/DTD_TSNE_embedding_V2.png}\n\t\t\t\\caption{t-SNE}\n\t\t\t\\label{fig:DTD_TSNE_embed}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\t\\begin{subfigure}{.164\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/DTD_Base_embedding_V2.png}\n\t\t\t\\caption{ResNet50}\n\t\t\t\\label{fig:DTD_Res_embed}\n\t\t}\n\t\\end{subfigure}\n\t\\centering\n\t\\begin{subfigure}{.153\\textwidth}{\n\t\\includegraphics[width=\\textwidth]{Figures\/DTD_Hist_embedding_V2.png}\n\t\t\t\\caption{HistRes50}\n\t\t\t\\label{fig:DTD_Hist_embed}\n\t\t}\n\t\\end{subfigure}\n\t\\caption{2D embeddings of test images from t-SNE out-of-sample approach and different variations of DREN for KTH-TIPS-2b (\\ref{fig:KTH_TSNE_embed}-\\ref{fig:KTH_Hist_embed}) and DTD (\\ref{fig:DTD_TSNE_embed}-\\ref{fig:DTD_Hist_embed}). The embeddings are shown from models trained and validated on the same data. The colors represent different classes from each dataset. In the 2D space, the HistRes features are more separable according to an angular measure, and are also much more compact, \\textit{i.e.}, each class mostly varies across a single principal vector, and the principal vectors are well-separated by an angular margin.}\n\t\\label{fig:Embeddings} \n\\end{figure*}\n\n\\begin{table*}[htb]\n\\centering\n\\caption{Test classification performance using $K$-NN ``trained\" on embeddings from t-SNE and DREN models. For the DREN models, the weight corresponding to the best average performance is reported. The best average result is bolded.}\n\\label{tab:KNN}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nDataset                      & Embedding Dimension ($d$) & t-SNE              & DREN ResNet               & DREN HistRes             \\\\ \\hline\n\\multirow{2}{*}{KTH-TIPS-2b} & 2                         & 76.70\\%$\\pm$2.88\\% & \\textbf{76.79\\%$\\pm$ 3.18\\%} ($\\lambda=.2$) & 74.60\\%$\\pm$3.47\\% ($\\lambda=.1$) \\\\ \\cline{2-5} \n                             & 3                         & 76.91\\%$\\pm$3.40\\% & 78.18\\%$\\pm$3.40\\% ($\\lambda=0$)  & \\textbf{79.23\\%$\\pm$3.63\\%} ($\\lambda=.5$) \\\\ \\hline\n\\multirow{2}{*}{DTD}         & 2                         & \\textbf{43.37\\%$\\pm$1.01}\\% & 29.35\\%$\\pm$1.22\\% ($\\lambda=.2$)   & 40.44\\%$\\pm$1.64\\% ($\\lambda=0$) \\\\ \\cline{2-5} \n                             & 3                         & 43.55\\%$\\pm$1.22\\% & 44.28\\%$\\pm$1.57\\% ($\\lambda=0$)   & \\textbf{53.02\\%$\\pm$1.24}\\% ($\\lambda=.1$) \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\\subsection{Encoder Experiments}\nFor the two hyperparameters, $\\lambda$ and $d$, the performance of the DREN models depended more on the selection of the embedding dimension. Ideally, as the dimensionality of the feature space becomes larger, inter-class separability should increase and intra-class variations should decrease leading to improved performance (as shown in our experiments for the DREN models). However, our DREN models achieved statistically comparable performance to the baseline models without an encoder at significantly lower dimensions (except for DREN with ResNet50 for DTD).  Our method is also stable to the selection of $\\lambda$ except for when only the $L_{div}$ is considered ($\\lambda$=1). In this case, the model is an unsupervised approach and will only learn embeddings that minimize the divergence between higher and lower dimension. In our implementation, the error was not backpropagated through the output layer once $\\lambda=1$. In this instance, the output layer will not be updated, leading to poor classification performance. \n\nIn Figures \\ref{fig:KTH_Base} and \\ref{fig:KTH_Hist}, the performance of the DREN models with and without the histogram layers for KTH-TIPS-2b were comparable. KTH-TIPS-2b primarily consisted of images collected in a controlled environment \\cite{liu2019bow}. As a result, the local, texture features extracted by the histogram layers may not add as much information for the classification. However, the DREN models with the histogram layers performed better on DTD across the different combinations of hyperparameters as shown in Figures \\ref{fig:DTD_Base} and \\ref{fig:DTD_Hist}. DTD contained images collected ``in-the-wild\" and also has more classes (47) than KTH-TIPS-2b (11). The embeddings learned by the DREN histogram models retained useful information from the texture features.nal joint probability distributions without considering class information. \n\n\n\\vspace{-2mm}\n\\subsection{t-SNE Comparisons}\nIn Table \\ref{tab:KNN}, the test performance for the $K$-NN classifiers showed the utility of the DREN embeddings for not only visualization but also classification. Our DREN models also have an advantage over t-SNE in that new images can be embedded without a separate out-of-sample approach. In Figure \\ref{fig:Embeddings}, the t-SNE embedding of the test points are not as visually compact and separable as the DREN models. The test samples embeddings produced from the DREN models also seem to have ``angular\" feature distributions. As noted in other works \\cite{choi2020amc}, models trained using cross-entropy appear to learn these unique, intrinsic feature coordinates in comparison to the t-SNE embeddings that are learned by only using KL divergence. For the DREN histogram models, the ``angular\" embeddings may also be produced as a result of normalization as the features are also centered around $0$. The features captured by the histogram layer binning function are naturally between $0$ and $1$ while the features from the global average pooling layer are followed by batch normalization \\cite{peeples2020histogram}.\n\n\\section{Conclusion}\nIn this work, we presented DREN models for joint dimensionality reduction and classification. The proposed approach learns discriminative features at lower dimensions that can be used for different tasks. The approach has several advantages and is a general framework that can use various classification\/divergence measures and deep learning models.  \n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\nThis text presents research results of the IAP programme P7\/12 initiated by the Belgian-state \nFederal Services for Scientific, Technical and Cultural Affairs. \nP. D. acknowledges the support of F.R.S.-FNRS (Belgium) and of FAPESP (Brazil). M. S. H. acknowledges partial support of the FAPESP, of the CNPq and of the INCT-IQ (Brazilian agencies).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\\bf\nWe consider the dynamics of the non-Hermitian Su-Schrieffer-Heeger model arising as the no-click limit of a continuously monitored free fermion chain where particles and holes are measured on two sublattices. \nThe model has $\\mathcal{PT}$-symmetry, which we show to spontaneously break as a function of the strength of measurement  backaction, resulting in a spectral transition where quasiparticles acquire a finite lifetime in patches of the Brillouin zone. We compute the entanglement entropy's dynamics in the thermodynamic limit and demonstrate an entanglement transition between volume-law and area-law scaling, which we characterize analytically. Interestingly we show that the entanglement transition and the $\\mathcal{PT}$-symmetry breaking do not coincide, the former occurring when the entire decay spectrum of the quasiparticle becomes gapped. }\n\n\n\n\\vspace{10pt}\n\\noindent\\rule{\\textwidth}{1pt}\n\\tableofcontents\n\\thispagestyle{fancy}\n\\noindent\\rule{\\textwidth}{1pt}\n\\vspace{10pt}\n\n\n\n\\section{Introduction\\label{sec:introduction}}\n\nUnitarity reflects core aspects of quantum mechanics, such as probability conservation and the linearity of the Schr\\\"dinger equation~\\cite{Shankar:102017}, and constrains how quantum information and correlations spread in a many-body system, as demonstrated by the extensive study of non-equilibrium dynamics of isolated quantum many-body systems~\\cite{lieb72thefinite,calabrese2005evolution,polkolnikov2011colloquium,cheneau2012lightcone,Gogolin_2016}.\n\nHowever in several instances of theoretical and experimental relevance one is interested in non-unitary quantum dynamics. A notable example is the recent interest on monitored quantum many-body systems, where unitary evolution competes with quantum measurements, resulting in a rich pattern of entanglement and correlation dynamics. In this context several works have highlighted a  measurement-induced entanglement phase transition, separating an error-correcting phase, in which the quantum information encoded in the system is resilient to the external probes, and a quantum Zeno phase, where frequent measurements continuously collapse the dynamics to a restricted manifold~\\cite{skinner2019measurementinduced,li2018quantum,li2019measurementdriven,chan2019unitaryprojective,boorman2022diagnostics,Biella2021manybodyquantumzeno,szyniszewski2020universality,barratt2022transitions,zabalo2022infinite,barratt2022field,lunt2020measurement,turkeshi2021measurementinduced,zabalo2022operator,iaconis2021multifractality,Sierant2022dissipativefloquet,bao2020theory,choi2020quantum,szyniszewski2019entanglement,block2022measurementinduced,jian2020measurementinduced,agrawal2021entanglement,gullans2020scalable,sharma2022measurementinduced,zabalo2020critical,vasseur2019entanglement,li2021conformal,turkeshi2020measurementinduced,lunt2021measurementinduced,sierant2022universal,gullans2020dynamical}. \nThe scaling of entanglement at the stationary state characterizes the dynamical phases and their transitions. For example, random quantum circuits follow a volume-law scaling in the error-correcting phase, while it has an area-law behavior in the quantum Zeno phase~\\cite{fisher2022quantum,lunt2021quantum,potter2021entanglement}. Conversely, monitored free fermionic systems typically follow a logarithmic entanglement scaling in the error-correcting phase~\\cite{alberton2021entanglement,buchhold2021effective,muller2022measurementinduced,turkeshi2021entanglement,turkeshi2021measurementinduced2,Kalsi_2022,kells2021topological,fleckenstein2022nonhermitian,Zhang2022universal,turk}. This phenomenon is considered an effect of their Gaussian nature, leading to more drastic measurement effects on the system~\\cite{Fidkowski2021howdynamicalquantum}. \n\nAnother example of non-unitarity is provided by non-Hermitian quantum mechanics, which has a long tradition dating back to works on noninteracting electronic disordered systems~\\cite{hatano1996localization,efetov1997directed}, to studies of non-unitary conformal field theory~\\cite{cardy1985conformal,fisher1978yanglee}, of few body-systems with parity-time (PT) reversal symmetry~\\cite{Bender1998} or exceptional points~\\cite{Heiss_2012}. In quantum optics non-Hermitian evolution naturally arises when a measurement apparatus monitors the system, which stems from the measurements and their backaction, and the subsequent post-selection of quantum trajectories corresponding to no-measurement events (no-click limit)~\\cite{Wiseman2009,carmichael2009open, gardiner2004quantum,daley2014quantum,Jacobs2014}. \n\nRecently non-Hermitian quantum many-body systems are attracting widespread interest for their exotic properties, including quantum criticality and topology (see Refs.~\\cite{ashida2020non,bergholtz2021exceptional} for a comprehensive review).  Their dynamical behavior has also received attention recently, in particular for  $\\mathcal{PT}$-symmetric systems such as Luttinger Liquids~\\cite{dora2020quantum,moca2021universal} or non-interacting lattice models tuned at an exceptional point. In this latter case the entanglement entropy has been shown to grow linearly in time and to saturate to a volume-law ~\\cite{ashida2018full,bacsi2021dynamics,dora2021correlations}, similarly to their Hermitian counterpart. In presence of interactions it has been proposed that an entanglement transition from an extensive to a sub-extensive scaling could take place in correspondence with $\\mathcal{PT}$-symmetry breaking~\\cite{gopalakrishnan2021entanglement}. Other mechanisms for entanglement transitions have been discussed, such as the non-Hermitian skin effect~\\cite{Kawabata22} or a spectral transition between a gapless and a gapped phase in the non-Hermitian XY chain, leading to an entanglement transition between a logarithmic and area-law scaling as the strength of the measurement backaction is tuned~\\cite{turkeshi2022nonhermitian}. Overall, a full understanding of the entanglement patterns of non-Hermitian systems is still missing and exact calculations on an analytically treatable quantum system can provide important insights.\n\n\\begin{figure}[t!]  \n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{phase_diag_1.png}\n     \\caption{ a) Cartoon of the setting under study, giving rise to a non-Hermitian SSH model. A one dimensional chain with two sublattices (A and B, respectively blue and orange circles) and staggered hopping $-J\\pm\\frac{h}{2}$ is coupled to a monitoring environment measuring particle and holes in the two sublattices. The measurement backaction induces a staggered imaginary chemical potential $\\pm i\\gamma$ leading to the non-Hermitian SSH in Eq.~(\\ref{eq:Heff}) (See Appendix \\ref{app:deriv}). b) Phase diagram in the ($\\gamma$, $h$) plane summarizing spectral and entanglement transitions in the system. The light blue area corresponds to a purely real spectrum, as indicated by the percentage $\\phi$ of real eigenvalues in the spectrum,  which is the $\\mathcal{PT}$-symmetric phase, and the dark blue one to a purely imaginary spectrum. The gradient area presents a mixed spectrum with both purely real and purely imaginary eigenvalues. Two critical lines emerge,  $\\gamma_{PT}$ (orange line) and $\\gamma_c$ (black line), corresponding respectively to the $\\mathcal{PT}$-symmetry breaking and to the opening of a gap in the imaginary part of the spectrum. The entanglement transition from volume to area law scaling occurs in correspondence with the second transition at $\\gamma_c$.  The three markers will be used in the following figures to characterize the properties of the system.\n     }\n     \\label{fig:phase_diag:1b}\n    \\end{figure} \n\nIn this work we reconsider the role of $\\mathcal{PT}$-symmetry and its breaking for entanglement transitions in a non-interacting free fermionic systems. Taking a non-Hermitian Su-Schrieffer\u2013Heeger (SSH) model~\\cite{ashida2018full,chang2020entanglement,bacsi2021dynamics} as working example we compute exactly its spectral properties and entanglement dynamics in the thermodynamic limit. The resulting dynamical phase diagram, plotted in Fig.~\\ref{fig:phase_diag:1b}, shows the existence of a volume-to-area-law entanglement transition as a function of the strength of the measurement backaction. Quite interestingly this turns to be distinct in general from the $\\mathcal{PT}$-symmetry breaking transition.\n\nThe remaining is structured as follows. In Sec.~\\ref{sec:model} we introduce the model and the dynamical protocol we consider, particularly discussing how the non-Hermitian dynamics can arise from full stochastic evolution. In Sec.~\\ref{sec:non_hermitian}, we diagonalize the model with periodic boundary conditions and discuss its spectrum, showing that it displays a transition associated with a breaking of $\\mathcal{PT}$ symmetry. In Sec.~\\ref{sec:entdyn} we present our results for the entanglement dynamics and the associated entanglement transition from volume to area law. We present a discussion of our results in Sec.~\\ref{sec:discussion} and our conclusions and future directions in Sec.~\\ref{sec:conclusion}. \n \n\n\\section{Non-Hermitian Su-Schrieffer\u2013Heeger Model\\label{sec:model}} \n \nThis section introduces and discusses the model of interest for this paper: the non-Hermitian SSH chain~\\cite{ashida2018full,chang2020entanglement,bacsi2021dynamics}. We consider a periodic chain made of $L$ units, each one containing two inequivalent sites $A,B$. We assume periodic boundary conditions, and label the fermionic degrees of freedom of the chain with two indices $A$ and $B$ corresponding to the two sublattices (Fig.~\\ref{fig:phase_diag:1b}). \nWe are interested in the quantum dynamics generated by the non-Hermitian Hamiltonian\n\\begin{equation}\\label{eq:Heff}\n H_\\mathrm{eff}   =   H - i \\gamma \\sum_{i=1}^L( c^\\dagger_{A,i} c_{A,i} - c^\\dagger_{B,i} c_{B,i} ).\n\\end{equation}\nwhere $H$ describes a conventional SSH chain\n\\begin{equation} \n    H  =    \\sum_{j=1}^{L} \\left[ -\\left(J-\\frac{h}{2}\\right)c_{A,j}^{\\dagger}c_{B,j-1} - \\left(J + \\frac{h}{2}\\right) c_{A,j}^{\\dagger}c_{B,j} +\\mathrm{h.c.} \\right] \\label{eq:model}\n\\end{equation} \nwith periodic boundary conditions and a staggered hopping for $h\\neq0$, while the last term in Eq.~(\\ref{eq:Heff}) is a purely imaginary staggered chemical potential which accounts for the backaction of a measurement protocol trying to monitor particle and holes in the two sublattices (See Appendix~\\ref{app:deriv} for a derivation of $ H_\\mathrm{eff} $ from quantum jumps dynamics.) The system evolution under $H_\\mathrm{eff} $ reads\n\\begin{equation}\n    |\\Psi(t)\\rangle = \\frac{e^{-i H_{\\mathrm{eff}} t}|\\Psi(0)\\rangle}{\\| e^{-i H_{\\mathrm{eff}} t}|\\Psi(0)\\rangle\\| }.\\label{eq:nonHermSSH}\n\\end{equation}\nwhere the normalization takes into account the fact that for non-Hermitian systems the norm is not conserved due to the lack of unitarity. Equivalently, one can show that this amounts to solve a non-linear Schrodinger equation\n\\begin{equation}\n    {d| \\Psi(t) \\rangle  }  =   -i H_\\mathrm{eff}dt |\\Psi(t)\\rangle - i \\frac{dt}2 \\langle H_\\mathrm{eff} - H_\\mathrm{eff}^\\dagger\\rangle_t |\\Psi(t)\\rangle \n\\end{equation} \nEq.~\\eqref{eq:nonHermSSH} is the central object we will investigate in this manuscript. \nThroughout this paper, we fix the energy scale $J=1$ without loss of generality. For convenience, we fix the initial state $|\\Psi(0)\\rangle$ to be the ground state of $H$ (cf. Eq.~\\eqref{eq:model}) with $h=0$ (the XX spin chain).\nNevertheless, the discussion presented in this paper is extendable to other initial states, provided they have sub-extensive entanglement content.\n\n\n\n\n\\section{Spectral transition and $\\mathcal{PT}$-Symmetry Breaking\\label{sec:non_hermitian}}\n\nIn this section, we study the spectral properties of the non-Hermitian Hamiltonian in Eq.~\\eqref{eq:Heff}. \n\nA fundamental property of systems described by a non-Hermitian Hamiltonian is that their spectrum is generally complex, with the imaginary part corresponding to excitations with a finite lifetime.  This fact is not surprising since we emphasize that our system is open and in contact with an environment even within the no-click limit. A relevant exception to this paradigm is provided by $\\mathcal{PT}$-symmetric non-Hermitian Hamiltonians~\\cite{bender1998real}, where a combination of parity ($\\mathcal{P}$), i.e. space-reflection, and time-reversal ($\\mathcal{T}$) symmetry provides a sufficient condition for the spectrum to be purely real.  In the context of our non-Hermitian SSH the action of parity and time-reversal reads respectively\n\n\\begin{equation}\n    \\mathcal{{P}}c_{A,j}\\mathcal{{P}}    =  c_{B,L-j+1},\\qquad \\mathcal{{P}}c_{B,j}\\mathcal{{P}}   = c_{A,L-j+1} ,\\qquad \\mathcal{{T}}i\\mathcal{{T}}   =  -i.\\label{eq:ptsym}\n\\end{equation} \n With these actions $\\mathcal{{P}} ^2 = 1$ and $\\mathcal{{P}}$ and $\\mathcal{{T}}$ commutes. The $\\mathcal{{P}{T}}$ transformation applied to the non-Hermitian SSH Hamiltonian in Eq.~(\\ref{eq:nonHermSSH}) then fulfills  $(\\mathcal{{P}}\\mathcal{{T}})H_{eff}(\\mathcal{{T}}\\mathcal{{P}}) = H_{eff}$ which means that $H_{eff}$ presents a $\\mathcal{PT}$-symmetry and can have purely real eigenvalues. \n\nThe spectrum is purely real when the eigenvectors of $H_\\mathrm{eff}$ are eigenvectors of the $\\mathcal{PT}$-symmetry operation, then the $\\mathcal{PT}$-symmetry is unbroken~\\cite{Bender1998} and the system is said to be in a $\\mathcal{PT}$-symmetric phase. \nConversely, when this condition does not hold, the spectrum presents complex conjugate eigenvalues (which in our case are purely imaginary Fig.~\\ref{fig:spectrums}), and the system is in a $\\mathcal{PT}$-symmetry broken phase.\n\n\\begin{figure}[t!]  \n\\centering\n\\includegraphics[width=\\columnwidth]{spectrum_comp.png}    \n \\caption{Evolution of the spectrum of the Non-Hermitian SSH model along the line $h=1$ (see Fig.~\\ref{fig:phase_diag:1b}). For \n $\\gamma = 0.5$ (panel a) the spectrum is purely real and gapped at $k=\\pm \\pi$. Above $\\gamma_{PT}=h=1$ the $\\mathcal{PT}$-symmetry breaks and a certain fraction of eigenvalues become purely imaginary, corresponding to $k-$modes near the zone boundaries acquiring a finite lifetime. The system has two exceptional points at $k=\\pm k_{EP}(\\gamma)$. As $\\gamma$ increases further above $\\gamma_c=2$ (panel c) the full spectrum becomes purely imaginary and gapped. }\n \\label{fig:spectrums}\n\\end{figure} \n\nOur non-Hermitian SSH Hamiltonian in Eq.~\\eqref{eq:Heff}  is quadratic which allows us to compute exactly its spectrum.\nWe perform the Fourier transform $c_{X,j}  = \\sum_{k} e^{ikj} c_{X,k}\/\\sqrt{L}$ where the sum of $k$ runs over the Brillouin zone $[-\\pi,\\pi]$ and $X=A,B$. Neglecting irrelevant constant contributions, we have\n\\begin{equation}\n\\begin{split}\n      H_\\mathrm{eff}  &=   \\sum_{k} \n\\begin{pmatrix}\nc_{A,k}^{\\dagger} && c_{B,k}^{\\dagger} \\\\\n\\end{pmatrix}\n\\mathcal{H}_k\n\\begin{pmatrix}\nc_{A,k}  \\\\\nc_{B,k}  \\\\\n\\end{pmatrix} \\\\ \\mathcal{H}_k &= \\begin{pmatrix}\n    i \\gamma & \\left(-1+\\frac{h}{2}\\right)e^{-ik} - \\left(1+\\frac{h}{2}\\right) \\\\ \n    \\left(-1+\\frac{h}{2}\\right)e^{+ik} - \\left(1+\\frac{h}{2}\\right) & -i\\gamma\n\\end{pmatrix}.\n\\end{split}\\label{eq:diagonal}\n\\end{equation}\n\nThe eigenvalues of the single-particle Hamiltonian $\\mathcal{H}_k$ fixes the spectral properties of the system, \nDiagonalizing $\\mathcal{H}_k$ we get\n\n\\begin{equation}\n    E_k  =   \\pm \\sqrt{h^2 - \\gamma^2 + (4 -h^2)\\cos^2{\\left(\\frac{k}{2}\\right)}}.\\label{eq:eigvals}\n\\end{equation}\\label{eq:E_k}\nAs clear in Eq.~\\eqref{eq:eigvals}, depending on the values of $\\gamma$ and $h$, $E_k$ can be either real or imaginary and the system may be in a $\\mathcal{PT}$-symmetric or broken phase.\nWe illustrate the phase diagram in Fig.~\\ref{fig:phase_diag:1b} employing the density of real eigenvalues, defined as\n\\begin{align}\n    \\Phi& = \\int_{-\\pi}^{\\pi}\\frac{dk}{2\\pi} \\Theta\\left(h^2 - \\gamma^2 + (4 -h^2)\\cos^2{\\left(\\frac{k}{2}\\right)}\\right),\n\\end{align}\nwith $\\Theta(x)=1$ for $x>0$ and $\\Theta(x)=0$ for $x<0$ the Heaviside function. \n\nWe can fix $h=1$ and discuss the qualitative features of the spectrum varying $\\gamma$ for reference, and discuss how the real and imaginary part of $E_k = E_k^\\mathrm{Re}+i E_k^\\mathrm{Im}$ are affected. As we show in Fig.~\\ref{fig:spectrums} (panel a) for small values of $\\gamma$ the spectrum is purely real and has a gap at $k=\\pm \\pi$ as in the conventional SSH Hermitian case. This regime corresponds to the $\\mathcal{PT}$-symmetric phase. As one increases $\\gamma$, the gap at the edge of the Brillouin zone decreases and ultimately closes at $\\gamma_{PT}=h=1$, where two exceptional points (EPs) emerge at $k=\\pm \\pi$~\\cite{chang2020entanglement,bacsi2021dynamics}. The $\\mathcal{PT}$-symmetry breaking occurs when $\\gamma$ is increased further above $\\gamma_{PT}$ (panel b) and part of the $k$-modes acquire a finite lifetime given by the imaginary part of the energy $E_k^\\mathrm{Im}$. In this regime the spectrum is gapless both in its real and imaginary part and has two EPs at $k=\\pm k_{EP}(\\gamma)$ whose value can be read from the spectrum in Eq.~(\\ref{eq:E_k}) and it is given (for $h<2$) by\n\\begin{align}\\label{eq:kEP}\nk_{EP}(\\gamma)=\\pm 2\\arccos{\\sqrt{\\frac{\\gamma^2-h^2}{4-h^2}}}\n\\end{align}\nAs the strength of $\\gamma$ is increased further the EPs in Eq.~(\\ref{eq:kEP}) move towards $k=0$ and finally merge for $\\gamma_c=2$. This signals a second spectral transition above which all the spectrum becomes purely imaginary and gapped (panel c).  To summarize, the analysis of the spectrum reveals two separate transitions, where first $\\mathcal{PT}$-symmetry breaks and some eigenvalues acquire an imaginary part, and then the full spectrum becomes imaginary and gapped. A qualitatively similar behavior occurs for other values of $h$ and leads to the spectral phase diagram reported in Figure~\\ref{fig:phase_diag:1b}. We note that for $h>2$ the $\\mathcal{PT}$-symmetry breaking transition occurs at $\\gamma_{PT}=2$, independently of $h$, while the gapless-to-gapped transition occurs at $\\gamma_c=h$ (See Fig.~\\ref{fig:phase_diag:1b}). Remarkably,  as we are going to show in the next section, the spectral properties of the system reflect the non-trivial entanglement dynamics and stationary states of our model. In particular, we will see how the entanglement transition occurs in correspondence of the critical point $\\gamma_c$ where the spectrum becomes fully imaginary and gapped.\n\n\n\n\n\n\\section{Entanglement Transition\\label{sec:entdyn}}\n\n\nThis section discusses the dynamics and stationary-state behavior of the entanglement entropy of the non-Hermitian SSH as a function of system parameters and in particular the emergence of an entanglement transition from volume to area law scaling. \n\nTo this extent we first discuss how entanglement entropy can be efficiently computed in free fermionic systems through the correlation matrix.\nAfterward, we present the analytic computation of the entanglement dynamics and the stationary state entanglement entropy. In particular, we obtain an explicit expression for the leading order entanglement scaling with system size. \n\n\n\n\\subsection{Dynamics of the correlation matrix and entanglement entropy\\label{sec:thermo_limit}}\n\n\nOur non-Hermitian Hamiltonian is quadratic therefore starting from a Gaussian fermionic state, by Gaussianity, the dynamics is entirely encoded in the correlation matrix \n\\begin{equation}\n    G_{m,n} = \\begin{pmatrix}\n    \\langle c_{A,m}^{\\dagger}c_{A,n} \\rangle_t & \\langle c_{A,m}^{\\dagger}c_{B,n} \\rangle_t \\\\\n    \\langle c_{B,m}^{\\dagger}c_{A,n} \\rangle_t & \\langle c_{B,m}^{\\dagger}c_{B,n} \\rangle_t\n\\end{pmatrix}\\label{eq:corrmat}.\n\\end{equation}\nIn particular, Eq.~\\eqref{eq:nonHermSSH} translates to an equation of motion for $G$. \n\nRelying on translational invariance, we can focus on the Fourier transform \n\\begin{equation}\n{G}_{m,n}(t) = \\frac{1}{L} \\sum_{k} e^{ik(m-n)} {G}_k(t), \\label{eq:defGk}\\qquad G_k(t)=\\begin{pmatrix}\n    \\langle c_{A,k}^{\\dagger}c_{A,k} \\rangle_t & \\langle c_{A,k}^{\\dagger}c_{B,k} \\rangle_t \\\\\n    \\langle c_{B,k}^{\\dagger}c_{A,k} \\rangle_t & \\langle c_{B,k}^{\\dagger}c_{B,k} \\rangle_t\n\\end{pmatrix}.\n\\end{equation}\nEq.~\\eqref{eq:defGk} states that $G_k(t)$ is the generating symbol for $G_{m,n}(t)$, which is a block Toeplitz matrix. This fact will lead to finding a closed expression for the stationary state entanglement entropy using the Szeg\\\"o theorem. \\\\\nThe time evolution Eq.~\\eqref{eq:nonHermSSH} induces the equation of motion for $G_k^{X,Y}$ ($X,Y\\in\\{A,B\\}$)\n\\begin{equation}\n    \\partial_t G_k^{X,Y} = i\\langle H^\\dagger_\\mathrm{eff} c_{X,k}^{\\dagger}c_{Y,k} - c_{X,k}^{\\dagger}c_{Y,k} H_\\mathrm{eff} \\rangle_t - i \\langle H_\\mathrm{eff}- H_\\mathrm{eff}^\\dagger \\rangle_t G_k^{X,Y}.\n\\end{equation}\nA straightforward computation  lead to the exact correlation matrix~\\cite{bacsi2021dynamics}\n\\begin{equation}\n    G_k(t)= G_k(0) + \\frac{1}{ \\mathcal{N}_k(t)}\n\\begin{pmatrix}   B_k(t)    & e^{ik\/2}A_k(t)(  - C_k   +2JiD_k)   \\\\\n-e^{-ik\/2}A_k(t)( C_k   +2JiD_k)   &   - B_k(t) \n\\end{pmatrix}   \\label{eq:solGk}\n\\end{equation}\nwhere we have defined \n\\begin{align}\n { G}_k(0) &= \\frac{1}{2} \\begin{pmatrix}   1    & e^{ik\/2}   \\\\\ne^{-ik\/2}   &   1 \n\\end{pmatrix}, \\quad   \\mathcal{N}_k(t) = 1 + (1+ C_k )A_k(t) \\nonumber\\\\ A_k(t)&=\\frac{ \\gamma^2 -  h^2\\sin(k\/2)^2   }{2|E_k|^2}(1- \\cos(2 E_k t)),\\qquad  B_k(t) = \\frac{ \\gamma -  h \\sin(k\/2)  }{2|E_k|}\\sin(2E_k t),\\\\  C_k  &= \\frac{\\gamma -  h\\sin(k\/2)}{\\gamma +  h\\sin(k\/2)},\\quad D_k = \\frac{\\cos(k\/2)}{ \\gamma +  h \\sin(k\/2)}.\\nonumber\n\\end{align}\nThe dynamics of correlations in the non-Hermitian SSH was studied in Ref.~\\cite{ashida2018full,bacsi2021dynamics} revealing the emergence of supersonic modes propagating with integer multiples of the Fermi velocity. Here we focus on the dynamics of entanglement entropy that can be obtained directly from the correlation matrix.\n\\begin{figure}[t!]  \n\\centering\n\\includegraphics[width=\\columnwidth]{time_evol_com.png}\n \\caption{Time evolution of entanglement entropy in the thermodynamic limit, for different subsystem size $\\ell$ and for increasing values of $\\gamma$ along the $h=1$ line (corresponding to the points in Fig.~\\ref{fig:phase_diag:1b}. The dynamic is obtained by diagonalizing numerically the exact correlation matrix of the subsystem of size at each time step. (a-b) The growth is linear in time and saturates to a value $S_{\\infty}$ which scale linearly with the subsystem size (inset), (c) The entanglement quickly settles to a small constant value $S_{\\infty}$ which do no depend of system size (inset).\n     }\n     \\label{fig:entanglement_scalling_finite_size_non_hermi}\n\\end{figure} \nGiven a bipartition $A\\cup B$ the entanglement entropy is defined as $S = - \\text{Tr}(\\rho_A \\log{\\rho_A})$ with $\\rho_A = \\text{Tr}_B(\\rho)$. With a pure state $|\\Psi(t)\\rangle$, the entanglement entropy is computable from the spectrum of $G^A_{m,n}(t) = G_{m,n}(t)$ for $m,n\\in A$. \nWe consider $B$ an infinite system and $A$ a finite interval of length $\\ell$. From the reduced correlation matrix, the entanglement is given by ~\\cite{korepin, peschel,Peschel_2009}\n\\begin{equation}\n    S_A = \\sum_{n} s(\\nu_n)\\qquad s(x) \\equiv -x\\ln x - (1-x) \\ln (1-x),\\label{eq:entdefinit}\n\\end{equation}\nwhere $\\nu_n$ are the eigenvalues of $G^A$. Eq.~\\eqref{eq:entdefinit} is efficiently implementable and requires polynomial computational resources, whereas the Hilbert space would scale exponentially in system size. It requires computing the matrix elements of $G^A$ through Eqs.~\\eqref{eq:defGk} and~\\eqref{eq:solGk}, and its diagonalization.\n\nIn Fig.~\\ref{fig:entanglement_scalling_finite_size_non_hermi}, we present the time evolution of the entanglement entropy in the thermodynamic limit, for different subsystem sizes $\\ell$ and three different values of $\\gamma$, along the line $h=1$. For small values of the monitoring strength $\\gamma$ (panel a, $\\gamma=0.5$) the entanglement grows linearly in time and saturates to a long-time value which scales with the size of the subsystem (see inset), corresponding to a volume-law scaling~\\cite{bacsi2021dynamics}. This behavior, similar to the unitary case~\\cite{calabrese2005evolution}, essentially persists upon increasing the value of $\\gamma$ even beyond the $\\mathcal{PT}$-symmetry breaking transition at $\\gamma_{PT}=1$. In fact, as we see in panel b for $\\gamma=1.5$, the entanglement entropy still grows in time, although slower than in the $\\mathcal{PT}$-symmetric phase and possibly with a different dynamical behavior, and saturates to a steady-state value showing a clear volume-law scaling (see inset). A qualitative change in the behavior of the entanglement entropy occurs instead when $\\gamma$ is increased further above $\\gamma_c=2$ (panel c). Here we see that the dynamics rapidly reaches a stationary state which is independent on subsystem size, compatibly with an area law scaling as expected on the Zeno side of an entanglement transition. Our numerical results on the entanglement dynamics shows therefore that a volume to area law entanglement transition emerges in the non-Hermitian SSH as a function of the measurement strength $\\gamma$. In the next subsection we will confirm this result by computing analytically the leading entanglement entropy contribution in the large subsystem size limit $\\ell\\rightarrow\\infty$.\n\n\n\\subsection{Entanglement entropy of the stationary state}\n\nThe results of previous section for the entanglement entropy were obtained numerically for a subsystem of size $\\ell$ in a thermodynamically large chain. Here we show that the leading large-$\\ell$ entanglement contribution can be obtained in closed form using manipulations similar to ones done for the unitary case~\\cite{calabrese2005evolution}.\n\nTo begin, we recast Eq.~\\eqref{eq:entdefinit} for the entanglement entropy using the Cauchy theorem\n\\begin{equation}\n    S_A = \\frac{1}{4\\pi i}\\oint_\\mathcal{C}d\\lambda e(0^+,\\lambda) \\frac{d}{d\\lambda} \\ln \\det \\tilde{G}^A(\\lambda),\\label{eq:cauchy}\n\\end{equation}\nwhere, using the $\\ell$-dimensional identity matrix $\\mathbb{1}_{\\ell}$, we defined\n\\begin{align}\n    \\tilde{G}^A(\\lambda) &= \\lambda\\mathbb{1}_{2\\ell}  - G^A(\\infty) ,\\qquad \n    G^A(\\infty) \\equiv \\lim_{\\tau\\to\\infty}\\lim_{t\\to\\infty} \\frac{1}{\\tau}\\int_t^{t+\\tau}d\\tau G^A(t),\n\\nonumber \\\\\n    \\tilde G_k(\\lambda) &= \\lambda\\mathbb{1}_2 - G_k(\\infty), \\qquad G_k(\\infty) \\equiv \\lim_{\\tau\\to\\infty}\\lim_{t\\to\\infty} \\frac{1}{\\tau}\\int_t^{t+\\tau}d\\tau G_k(t),\\\\ \n    e(y,x) & = -(y+x)\\ln \\left( y+x \\right) - (1+y-x)\\ln \\left(1+y-x\\right).\\nonumber\n\\end{align}\nThe key observation is that $\\tilde{G}^A$ is a block Toeplitz matrix with generator $\\tilde{G}_k$. The asymptotic form of $\\mathcal{D}^A(\\lambda)\\equiv \\ln \\det \\tilde{G}^A(\\lambda)$ then is given by Szeg\\\"o theorem\n\\begin{equation}\n   \\mathcal{D}^A(\\lambda) \\simeq_{\\ell\\gg 1}  \\ell \\int_{-\\pi}^{\\pi}\\frac{dk}{2\\pi} \\ln\\det\\tilde G_k . \\label{eq:szego}\n\\end{equation}\nAfter simple manipulations, we have\n\\begin{equation}\n    \\frac{d}{d\\lambda}\\mathcal{D}^A(\\lambda)= \\ell \\int_{-\\pi}^{\\pi}\\frac{dk}{2\\pi} \\frac{2\\lambda-1}{(\\lambda-\\nu_+(k))(\\lambda-\\nu_-(k))},\\label{eq:szego2}\n\\end{equation}\nwith $\\nu_\\pm(k)$ the eigenvalues of $G_k$. In the stationary limit these eigenvalues can be exactly computed and depend on the energy $E_k$ being real or imaginary.\nWhen $E_k$ is real\n\\begin{equation}\n\\nu_{\\pm}(k) = \\frac{1 \\pm \\nu_{k}}{2} ,\\qquad \n\\nu_{k} = \\sqrt{4 \\chi_k^2 \\left(C_k^2+4 D_k^2\\right)-4 \\chi_k  C_k+ 1},\n\\end{equation}\nwhere we have introduced the auxiliary functions\n\\begin{equation}\n\\begin{split}\n A_k^{\\infty} &\\equiv \\frac{\\gamma^2-h^2 \\sin^2(k\/2)}{2 |E_k|^2}\\\\\n\t\\chi_k&\\equiv \\lim_{\\tau\\to\\infty}\\lim_{t\\to\\infty} \\frac{1}{\\tau}\\int_t^{t+\\tau}d\\tau \\frac{A_k(t)}{N_k(t)} = \\frac{1}{1+C_k}   \\left(1- \\frac{1}{  \\sqrt{2(1+C_k)A_k^{\\infty}+1}} \\right) \n\\end{split}\n\\end{equation}\nSince $\\nu_{k,\\pm}\\neq 0 $, these eigenvalues contribute in Eq.~\\eqref{eq:szego2} and hence in Eq.~\\eqref{eq:cauchy}.\nConversely, for imaginary $E_k$ we have $\\nu_+=1$ and $\\nu_-=0$, and as a result imaginary modes do not contribute to the stationary state volume-law entanglement. \n\n\n\\begin{figure}[t!]  \n\\centering\n \\includegraphics[width=\\columnwidth]{vh_comp2.png}  \n    \\caption{ Left-Panel : prefactor of the volume law entanglement entropy $v(h,\\gamma)=S_A\/\\ell$ as a function of $\\gamma$ along the line $h=1$ as obtained from the numerical evaluation and the analytical formula in Eq.~\\eqref{eq:vh_coeff}.\n     We notice a perfect agreement between the two methods, and a clear volume-to-area law transition at $\\gamma_c = 2$. Right-Panel :   Derivative $\\partial_{\\gamma} v(h=1,\\gamma)$ presenting interestingly two divergences, first at $\\gamma_{PT} = 1$ when the $\\mathcal{PT}$-symmetry breaking occurs and then at $\\gamma_c =2$ when the volume-to-area law transition happens.}\n     \\label{fig:final}\n\\end{figure} \n\nEvaluating the integral we obtain a closed form expression for the leading behavior of the entanglement entropy which reads\nThe final expression for the entanglement entropy reads therefore $ S_A = v(h,\\gamma) \\ell+O(1)$ where the slope $v(h,\\gamma)$ of the volume-law entanglement contribution reads\n\\begin{align}\n   v(h,\\gamma) = \\int_{-\\pi}^\\pi \\frac{dk}{2\\pi}   \\Theta\\left(h^2 - \\gamma^2 + (4 -h^2)\\cos^2{\\left(\\frac{k}{2}\\right)}\\right) s(\\nu_{k,+}).\n    \\label{eq:vh_coeff}\n\\end{align}\nFrom this result we see that  $v(h,\\gamma)$ depends on system parameters and it is written in closed form in term of an integral of an entropy function $s(\\nu_{k,+})$, with the integration domain restricted to those momenta with purely real-eigenvalues,\ni.e. those enclosed within the pair of EPs in the spectrum (see Fig.~\\ref{fig:spectrums}). \n\nWe plot this quantity and compare it with the numerical late time one in Fig.~\\ref{fig:final} (left panel) for $h=1$, finding perfect agreement. This result confirms the entanglement transition from volume to area law, already evoked from the dynamics, which is sharply characterized here by the vanishing of the slope coefficient $v(h,\\gamma)$ at $\\gamma_c=2$ for $h=1$. From our exact result we can see that the vanishing of the slope coefficient $v(h,\\gamma)$ is in part driven by the merging of the exceptional points $k_{EP}(h,\\gamma)$ at the gapless to gapped transition $\\gamma_c$, when the support of the integral in Eq.~(\\ref{eq:vh_coeff}) shrinks to zero as $k_{EP}(h,\\gamma)\\sim \\sqrt{\\gamma_c-\\gamma}$, and in part by the vanishing of the entropy function as $\\nu_{k,+}\\rightarrow 1$, resulting in a linear behavior near $\\gamma_c$ as seen in Fig.~\\ref{fig:final}. \n\nThis volume-to-area entanglement transition is clearly separated from the $\\mathcal{PT}$-symmetry breaking point, which occurs for $h=1$ at $\\gamma_{PT}=h=1$. Interestingly, we see that nevertheless a non-analytic behavior emerges at $\\gamma_{PT}$, which can be interpreted as a weaker volume-to-volume \"transition\". This is identified by looking at the derivative $\\partial_{\\gamma}v(h,\\gamma)$ which diverges at $\\gamma_{PT}$ as we show in the right panel of Fig.~\\ref{fig:final}. \n\n\n\n\n\\section{Discussion\\label{sec:discussion}}\n\nIn this section we discuss our results, comment on their broader implications for the entanglement properties of non-unitary quantum many-body systems and on related results in the literature. \n\nOur main result is the existence of a volume-to-area law entanglement transition in the non-Hermitian SSH model and its analytical characterization in terms of the spectral properties of the system.  As our exact results show clearly, the origin of this transition can be qualitatively understood in terms of quasiparticles acquiring a finite life-time due to the measurement backaction and thus not contributing to the volume law scaling (See Eq.~\\ref{eq:vh_coeff}). While this process starts at the $\\mathcal{PT}$-symmetry breaking, when first imaginary parts appear in the spectrum,  it is only when all the quasiparticle modes become short-lived and a gap opens up in the spectrum of decay modes (imaginary part of the complex energy) that a true entanglement transition to an area law arises. One could therefore wonder whether a similar mechanism could hold more generally beyond the SSH case, for non-Hermitian quantum many-body systems with $\\mathcal{PT}$-symmetry breaking.  In this respect we could speculate that as long as the $\\mathcal{PT}$-symmetry breaking occurs \\emph{gradually}, with imaginary parts remaining zero for a finite density of modes, then a volume law phase could be sustained since these mode would dephase and heat up.\nOn the other hand, if the spectrum at the $\\mathcal{PT}$-symmetry breaking acquires a finite imaginary part leading to a gap in the spectrum of decay modes, then we could expect the entanglement transition into an area law to coincide with the $\\mathcal{PT}$-symmetry breaking. \n\nOur results for the dynamics of non-Hermitian SSH model complement the results of Ref.~\\cite{bacsi2021dynamics}, which had focused only on a quench at the $\\mathcal{PT}$-symmetry breaking point finding volume law scaling. Furthermore tt is worth emphasizing that that our results concern entanglement entropy \\emph{dynamics} after a quench of the dissipation and thus are different than those obtained in Ref.~\\cite{chang2020entanglement}. This work in fact takes a different approach to non-Hermitian systems based on bi-orthogonal quantum mechanics and therefore\nfocuses on the entanglement entropy of low-energy eigenstates of the model in the $\\mathcal{PT}$-symmetric phase or at the critical points. This results in a logarithmic scaling of the entanglement and negative central charges as opposed to our volume vs area law scaling.\n\nFinally, it is worth mentioning that the non-Hermitian version of the SSH model discussed here, including the $\\mathcal{PT}$-symmetry breaking, has been experimentally implemented in the field of topological photonics, in particular using photonic waveguide arrays~\\cite{PhysRevLett.115.040402,weimann2017topologically,ozawa2019topological}. There the non-Hermitian quantum dynamics is simulated as propagation along the axial direction of the waveguide, using the analogy between paraxial Maxwell equations and Schrodinger equation. A staggered imaginary chemical potential in the two-sublattices can be easily implemented using alternating gain and losses. It could be interesting to discuss whether an analog dynamics for the correlation matrix could be achieved with these platforms, which could give access to the dynamics of the entanglement entropy.\n\n\n\\section{Conclusion\\label{sec:conclusion}}\n\nIn this work we have studied the entanglement dynamics in a non-Hermitian SSH free fermionic model, arising as the no-click limit of a quantum jump master equation. We analytically find two types of critical behavior: a spectral $\\mathcal{PT}$ symmetry breaking transition and a volume-to-area law entanglement transition for the stationary state. Importantly, the two do not coincide, despite at the $\\mathcal{PT}$-phase transition the volume law entanglement prefactor exhibits singular behavior.\n\nSeveral open questions remain to be addressed and will be the subject of future investigations. For what concerns the non-Hermitian SSH the exact time evolution of the entanglement entropy can possibly be computed in the scaling limit, following the arguments in Ref.~\\cite{fagotti2008evolution}. Furthermore it would be interesting to discuss the entanglement dynamics of this model for open-boundary conditions. From one side it is known that the $\\mathcal{PT}$-symmetric phase is a topological non-Hermitian phase~\\cite{chang2020entanglement,bergholtz2021exceptional} hosting boundary modes and characterized by a non-zero complex Berry phase, therefore it would be interesting to see whether signatures of this non-trivial topology emerge in the dynamics of the entanglement. Similar questions have been raised for monitored quantum systems~\\cite{fleckenstein2022nonhermitian,kells2021topological}. In addition, it is known that for non-Hermitian systems changing the boundary conditions can have important effects on the physics of the problem, including entanglement properties~\\cite{Kawabata22}.\n\nThe existence of a volume to area law entanglement transition in a free non-Hermitian model is an interesting result by itself, which suggests that the patterns of entanglement in these systems are richer than in the unitary case and calls for further studies. The results of this work, together with those obtained for the non-Hermitian Ising chain~\\cite{turkeshi2021entanglement}, point towards a close connection between spectral properties of the system and entanglement dynamics which would be interesting to put on a firmer grounds through a phenomenological quasiparticle picture for entanglement in non-Hermitian systems.\n\nFinally, the scaling of the entanglement in the complete quantum jumps protocol beyond the no-click limit needs to be studied. In particular we would like to understand whether an entanglement transition will survive in that case and if the error-correcting phase will preserve a volume-law entanglement entropy. This would be particularly interesting in light of the recently proposed quasiparticle picture for monitored quantum many-body systems~\\cite{turkeshi2021entanglement}.\n\n\n\n\n\\paragraph{Funding information}\nWe acknowledge support from the ANR grant ``NonEQuMat''\n(ANR-19-CE47-0001) and computational resources on the Coll\\`ege de France IPH cluster.\n\n\n\\begin{appendix}\n\n\n\\section{Non-Hermitian Quantum Quenches from Quantum Jumps\\label{app:deriv}}\n\n \n\n\nIn this appendix we discuss how the non-Hermitian SSH model that we considered in the main text arises as the no-click limit of a monitored SSH model. Specifically we consider a conventional SSH chain with Hamiltonian $H$ defined in the main text, coupled to local measurement apparatus.\nThese continuously monitor, stochastically and independently, the local density of particles on sublattice $A$, $n_{A,i}=c^\\dagger_{A,i} c_{A,i}$, and the local density of holes, $1-n_{B,i}= c_{B,i} c^\\dagger_{B,i}$, on sublattice $B$. We consider a quantum jump protocol, where the evolution of the system is described by  the stochastic Schr\\\"odinger equation (see e.g.~\\cite{turkeshi2022nonhermitian} and references therein for a detailed derivation)\n \\begin{equation}\n \\begin{split}\n     {d| \\Psi(t) \\rangle  }  &=   -i H dt |\\Psi(t)\\rangle - i \\frac{dt}2 \\langle H_\\mathrm{eff} - H_\\mathrm{eff}^\\dagger\\rangle_t |\\Psi(t)\\rangle  \\\\&\\qquad + \\sum_{i=1}^L \\left[dN_{A,i}^t\\left(\\frac{n_{i,A}}{\\sqrt{\\langle n_{i,A}\\rangle_t}}-1\\right) + dN_{B,i}^t\\left(\\frac{1-n_{i,B}}{\\sqrt{\\langle 1-n_{i,B}\\rangle_t}}-1\\right)\\right] |\\Psi(t)\\rangle\n \\end{split} \\label{eq:stocha_equa}\n \\end{equation} \n where $\\langle \\circ\\rangle_t\\equiv \\langle\\Psi(t)|\\circ|\\Psi(t)\\rangle$ is the expectation value, $dN_{A,i}^t,dN_{B,i}^t \\in \\{0,1\\}$ are independent Poisson processes with $\\overline{dN_{X,i}} = \\delta p_{X,i}$, $dN_{X,i} dN_{Y,i} = \\delta_{i,j}\\delta_{X,Y} dN_{X,i}$ with $X,Y \\in \\{A,B\\}$. \n The probability of a quantum jump are uncorrelated for any site and fermionic type and given by\n \\begin{equation}\n     \\delta p_{A,i} = 2\\gamma \\delta t\\langle \\Psi(t)| c^\\dagger_{A,i} c_{A,i} |\\Psi(t) \\rangle \\quad\\text{and}\\quad\\delta p_{B,i} = 2\\gamma \\delta t\\langle \\Psi(t)| c_{B,i} c^\\dagger_{B,i} |\\Psi(t) \\rangle \\,.\n \\end{equation}\n In Eq.~\\eqref{eq:stocha_equa} we have introduced the non-Hermitian Hamiltonian\n \\begin{equation}\n     H_\\mathrm{eff}   =   H - i h \\sum_{i=1}^L( c^\\dagger_{A,i} c_{A,i} - c^\\dagger_{B,i} c_{B,i} ).\n \\end{equation} \n This is the evolution that follows the post-selected trajectory $dN^t_{X,i} =0$ for all $t,i$ and $X=A,B$. In this case, the evolution is deterministic and given by\n \\begin{equation}\n     |\\Psi(t)\\rangle = \\frac{e^{-i H_{\\mathrm{eff}} t}|\\Psi(0)\\rangle}{\\| e^{-i H_{\\mathrm{eff}} t}|\\Psi(0)\\rangle\\| }.\n \\end{equation}\n\n\\end{appendix}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter*{Publications}\\label{chapter:publications}\n\\addcontentsline{toc}{chapter}{Publications}\n\nThis thesis is partially based on work presented in the following publications.\n\n\\begin{enumerate}[leftmargin=*]\n\t\\item Quantum parameter estimation with imperfect reference frames~\\cite{safranek2015quantum}\n\t\\item Quantum parameter estimation using multi-mode Gaussian states~\\cite{Safranek2015b}\n\t\\item Ultimate precision: Gaussian parameter estimation in flat and curved spacetime~\\cite{Safranek2015a}\n\t\\item Optimal probe states for the estimation of Gaussian unitary channels~\\cite{safranek2016optimal}\n\\end{enumerate}\n\n\n\\chapter*{Introduction}\\addcontentsline{toc}{chapter}{Introduction}\n\nThe aim of this thesis is to provide an elegant and useful basis for future quantum technology and to take small but significant steps towards experimental testing of theories in the overlap of quantum mechanics and general relativity. The importance and future impact of quantum technologies has been recognized not only by governments but also by several large financially-savvy corporations. The world is heading towards the second quantum evolution. Quantum technologies will serve as platform for secure communication, quantum computers will offer highly parallel computations which for certain tasks outperform classical computers,\nquantum simulation will allow for safe and inexpensive modeling of chemistry experiments in the comfort of one's own living room, quantum clocks will provide precision in global positioning system which will lead to precise construction and possibly even space-controlled transportation, hand-held devices will be able to measure small distortions in gravity, lasers will be able to measure slight changes in the atmosphere allowing us to predict\nextreme weather conditions and ultimately save lives.\n\nWe are still in the beginning though. Currently only a few genuinely practical applications have been developed. On the other hand, the recent amazing discovery of gravitational waves has demonstrated that such technology is feasible~\\cite{abbott2016observation}. The new generation of quantum enhanced gravitational wave detectors have already delivered improvement by a factor of $2$~\\cite{Demkowicz2013a} and soon we will hear about more such astonishing achievements.\n\n\nIn this thesis we develop practical tools for the optimal estimation of special class of quantum states -- called Gaussian states -- which are relatively easy to prepare and manipulate in experiments, and thus can serve as an effective building block for this new generation of quantum sensors. We use these tools to show which states to use for which tasks, we show the optimal states. In essence, we give a prescription of how to build the core of such quantum sensors. Moreover, we study how such sensors perform in the estimation of space-time parameters such as proper time, Schwarzschild radius, amplitude of a gravitational wave, or proper acceleration. Finally, we use the powerful tools of quantum metrology to show how to overcome certain issues in distance-communication or distance-sensoring when the reference frame of a sensor and the observer can easily become misaligned. This could prove particularly useful for a new generation of space-based experiments such as eLISA~\\cite{amaro2012low} where the detector is in deep space while the operator who reads the data stays on Earth.\n\nThis thesis is structured as follows: in Part I of the thesis we introduce and overview tools which have been developed in previous literature, although we believe that some results included have not been published before. For example, we describe the discontinuity of the quantum Fisher information and show that the Bures metric in general does not coincide with the quantum Fisher information matrix, we derive the form of a general Gaussian unitary in the phase-space formalism, the full parametrization of two- and three-mode Gaussian states, and formulae for arbitrary order of the continuous Bogoliubov transformations. Part II of the thesis consists entirely of original work.\n\nIn the first chapter~\\ref{chap:QM} we overview the necessary tools of quantum metrology that we will use in the main part. The second~\\ref{chap:GS} chapter focuses on Gaussian states. In particular we introduce the phase-space formalism of Gaussian states, Gaussian transformation, parametrization of Gaussian states, and state-of-the-art quantum metrology in the phase-space formalism. In the third chapter~\\ref{chap:operations_in_QFT} we quickly overview how space-time distortions affect quantum states and show that such transformation are indeed Gaussian transformations. In the fourth chapter~\\ref{chap:QM_GS} we develop new expressions for the optimal estimation of Gaussian states. Namely we focus on the quantum Fisher information as the figure of merit. We also use the derived expressions to devise a method for finding optimal probe states for Gaussian channels and we unravel how different characteristics of a probe state affect the estimation precision. The fifth chapter~\\ref{chap:QFT_metrology} shows an application of quantum metrology of Gaussian states in quantum field theory in curved space-time. We provide general formulae which show how squeezed thermal states perform as probes for channels that encode space-time parameters.\nThe last chapter~\\ref{chap:reference_frames} is focused on how the estimation precision changes when two parties, one which encodes a parameter into a quantum state and one which decodes, do not share a common reference frame. We show that sending a quantum reference frame in the communication channel could significantly improve the precision with which the parameter is decoded. Finally, we discuss open problems and suggest possible future directions, and we conclude with some short remarks. The table of frequently used notation can be found in appendix~\\ref{app:notations}.\n\n\n\\chapter*{Part I}\n\\addcontentsline{toc}{chapter}{Part I}\n\n\\chapter{Introduction to quantum metrology}\\label{chap:QM}\n\nMetrology is the science of measurement. Metrology aims to determine the highest possible precision in measuring parameters of a physical system. It also provides tools to reach that limit of precision. Since measurement plays a central role in quantum physics and cannot be taken out of consideration, measurement theory in quantum theory becomes especially significant.\n\nThe groundwork on quantum metrology has been set up by Holevo and Helstrom in their seminal papers\\cite{Holevo2011a,meystre1977quantum}. They developed and used many tools from probability theory and statistics and applied it to quantum systems. Their aim was to improve communication protocols with a particular interest in the problem of aligning measurement basis. This problem arises when two parties who wish to communicate cannot do so, because the information the first party sends is encoded with respect to a certain reference frame and the second party does not know what this reference frame is. Put simply, the direction that the first party calls the right direction can be called the left by the other party. If two parties cannot agree on what is left or right, it is impossible for the second party to decode received information. To align a basis between two parties, A and B, some information must be exchanged between the two. This can be achieved in different ways. It has been noted that exchanging information encoded in a quantum state with properties such as superposition, entanglement, and squeezing can be much more effective than exchanging classical information. In fact, the number of qubits needed to reach a given amount of alignment scales as the square root of the number classical bits needed for the same task.\nSuch a significant advantage is the manifestation of advantages of quantum physics.\n\nThe problem of aligning two measurement basis can be viewed as a problem of estimating an angle in which one basis is rotated with respect to the other. As mentioned before, information about this angle is encoded in a quantum state sent from party A to party B. For the first party the task then is to encode the angle efficiently into the quantum state, while for the second party it is to estimate this parameter by measuring the received quantum state. This is a typical metrological setting.\n\nBecause aligning measurement basis and estimating angles is essentially the same task, results in the theory of aligning measurement basis can be translated to the theory of estimating angles or phases. But this means that using quantum states to estimate angles, or in fact any parameter~\\cite{Giovannetti2004a,Giovannetti2006a}, can yield far better results than any classical method. In other words, the ability to construct devices using quantum systems could lead to a significant improvement in sensing. With this in mind it is no surprise that the field of quantum metrology has begun to grow rapidly. This is illustrated in various review articles and books~\\cite{Giovannetti2011a,Toth2014a,Paris2009a,kok2010introduction}.\n\nThis chapter is structured as follows: first we give a brief overview on different areas of theoretical quantum metrology. We then quickly delve into the local estimation theory, which is the most developed part of quantum metrology and the main focus of this thesis. We introduce some mathematical results of the current state-of-the-art quantum metrology, its interpretation and use, and its connection to statistics.\n\n\\section{Overview}\n\nIn this section we sketch the structure quantum metrology. Quantum metrology can be divided into following subfields:\n\n\\begin{itemize}[leftmargin=*]\n\\item \\emph{Discrete problems}: With a given input from a discrete set of elements (usually a finite set of quantum states), and usually some a priori probability distribution over this set, the task is to determine which element of the set has been sent. Examples are:\n    \\begin{itemize}\n    \\item \\emph{Quantum state discrimination}: The task to discriminate between quantum states using an appropriate measurement basis. The method aims to minimize the probability that our guess about the input state is wrong, or more generally, to minimize the cost function. However, using this method that we are never entirely certain that our guess is correct.\n    \\item \\emph{Unambigious discrimination}: The task is, again, to discriminate between quantum states. It differs from the previous method in a way that we choose our measurement basis that allows some measurement results to give an indefinite answer about the state. Put simply, in numerous cases the measurement results do not give us any information about the state we received (so we cannot discriminate), however, after receiving some result we can be certain that our guess is correct.\n    \\end{itemize}\n\\item \\emph{Continuous problems}: The task is to estimate a parameter or parameters encoded in a quantum state, while the type of dependence of the state on the parameter is usually known. This is exactly the same as to discriminate between a continuum of quantum states. Examples of two different approaches such estimation are:\n    \\begin{itemize}\n    \\item \\emph{Maximum likelihood estimation}: Given a set of measurement results, MLE assumes that the best guess for the parameter is such that maximizes the probability of receiving those measurement results.\n    \\item \\emph{Bayesian estimation}: Given a set of measurement results, Bayesian estimation assumes that the best guess for the parameter is such that minimizes a cost function.\n    \\end{itemize}\n    Also, it is important to point out that continuous estimation problems divide into two subfields:\n    \\begin{itemize}\n    \\item \\emph{Global (Bayesian) estimation theory}: Global estimation theory provides general methods how to deal with the estimation of a parameter about what we have not any a priori knowledge about the distribution over the values the parameter can take. The measurement basis do not depend on the parameter we are trying to estimate. The figure of merit in this is the cost function we are trying to minimize. However, this is usually not an easy task. Quantum state tomography can be also viewed as an example of the global estimation theory, where the number of parameters to be fitted is equal to the dimension of the Hilbert space of a quantum system.\n    \\item \\emph{Local (frequentist) estimation theory}: If we know that the parameter is localized around certain approximate value, we may move to the local estimation theory. This theory gives us an optimal measurement which depends on that approximate value, and helps us to estimate the parameter in the shortest amount of time\/with minimal resources. The optimal measurement are usually written to be dependent on the unknown parameter we want to estimate -- but in that case we take the previously mentioned approximate value around which the real value is localized.\n    \\end{itemize}\n\\end{itemize}\n\nIn the following we will focus only on the continuous problems and the local estimation theory.\n\n\n\\section{Continuous problems in quantum metrology}\n\nLet us consider $N$ copies of the same density matrix $\\hat{\\rho}_\\epsilon$ dependent on the same parameter. The task is to estimate the parameter $\\epsilon$ which comes from an uncountable set, usually an interval.\nFirst we measure each matrix separately in an appropriate measurement basis. That way we obtain $N$ measurement outcomes $(x_1,\\dots,x_N)$. Then we choose an estimator which will give us an estimate of the parameter considering the measurement outcomes. Mathematically, an estimator $\\hat{\\epsilon}$ is a function which maps the set of possible outcomes into the interval where the parameter lies,\n\\[\n\\hat{\\epsilon}:(x_1,\\dots,x_N)\\longrightarrow\\hat{\\epsilon}(x_1,\\dots,x_N).\n\\]\n\nProperly chosen estimator will approximate the real value of the parameter after several measurements, $\\hat{\\epsilon}(x_1,\\dots,x_N)\\approx\\epsilon$. Such an appropriate estimator is called consistent. By definition, when number of measurements $N$ goes to infinity, the value of the consistent estimator converges to the real value $\\epsilon$. Another class of important estimators are locally unbiased estimators for which the overestimated value and underestimated value balance each other, i.e., such an estimator gives a correct value on average. Defining $p(x|\\epsilon)$ as the probability distribution of obtaining outcome $x$ given the value $\\epsilon$, the locally unbiased estimator is defined by\n\\[\n\\mean{\\hat{\\epsilon}}:=\\int \\dif{x}\\ p(x|\\epsilon)\\hat{\\epsilon}(x)=\\epsilon.\n\\]\nAs we will show later, mean squared error of such an estimator is bounded below via the Cram\\'er-Rao bound~\\eqref{eq:Cramer_Rao}. The last type of estimators are called efficient estimators. Such estimators saturate the minimal value given by the Cram\\'er-Rao bound. Asymptotically efficient estimators saturate the Cram\\'er-Rao bound in the limit of large $N$. Although efficient estimators might not always exist, asymptotically efficient always do. Examples of asymptotically efficient estimators are the Bayes estimator and the maximum likelihood estimator\\cite{meystre1977quantum,BraunsteinCaves1994a,barndorff2000fisher}. For more information about the non-quantum estimation theory see for example \\cite{lehmann2006theory,amari2007methods}.\n\n\\begin{example}\\label{ex1}\n\\emph{\nWe assume $\\epsilon\\in[0,\\frac{\\pi}{2}]$. Let us have $N$ copies of the state $\\ket{\\psi}=\\cos\\epsilon\\ket{0}+\\sin\\epsilon\\ket{1}$. To estimate the parameter, we may decide to measure in the computational basis $\\{\\ket{0},\\ket{1}\\}$. Since the probability of outcome $0$ is $p_0=\\cos^2\\epsilon$, a good choice of an estimator is $\\hat{\\epsilon}(x_1,\\dots,x_N)=\\arccos{\\sqrt{\\frac{N_0}{N}}}$, where $N_0$ is a number of times we receive the measurement result $0$. This estimator is clearly consistent. Note that we could have chosen a completely different estimator, for example $\\hat{\\epsilon}(x_1,\\dots,x_N)=N_0+N+4$, but such an estimator is not consistent and does not give an appropriate estimate.\n}\n\\end{example}\n\n\\section{Local estimation theory}\n\nLocal estimation theory enters the parameter estimation in its latest stage, i.e., when the parameter is localized around certain known approximate value. Then methods which are the most effective for that approximate value are used. For example, optimal measurements will be different for different values as well as optimal probe states for channels encoding the unknown parameter. Since local estimation theory enters in the latest stage of estimation, it also provides the ultimate limit of precision with what we can estimate the parameter. This is given by the Cram\\'er-Rao bound\\cite{BraunsteinCaves1994a,Paris2009a,jarzyna2015true}, also elegantly proven in~\\cite{kok2010introduction}. This is the lower bound on the mean squared error of any locally unbiased estimator $\\hat{\\epsilon}$ with certain regularity conditions and reads\n\\[\\label{eq:Cramer_Rao}\n\\mean{\\Delta\\hat{\\epsilon}^2}\\geq\\frac{1}{N F(\\epsilon)}.\n\\]\nFor the full statement of the theorem see appendix~\\ref{app:CR}. $F(\\epsilon)$ is a quantity called the Fisher information which we will define in Eq.~\\eqref{eq:the_Fisher_information}, and $N$ is the number of measurements performed on $N$ identical copies of the same quantum state. $\\mean{\\Delta\\hat{\\epsilon}^2}$ is the mean squared error of the estimator defined as\n\\[\n\\mean{\\Delta\\hat{\\epsilon}^2}=\\int \\dif{x_1}\\dots\\dif{x_N}(\\hat{\\epsilon}(x_1,\\dots,x_N)-\\epsilon)^2p(x_1|\\epsilon)\\dots p(x_N|\\epsilon).\n\\]\nThe Cram\\'er-Rao bound says that in average our guess $\\hat{\\epsilon}(x_1,\\dots,x_N)$ cannot be closer to the real value $\\epsilon$ than the value given by inverse of the Fisher information and the number of measurements performed. Although Cram\\'er-Rao bound is entirely general and holds for any parameter-dependent probability distribution $p(x|\\epsilon)$, in quantum physics $p(x|\\epsilon)$ is the probability of obtaining the measurement result $x$ given a density matrix $\\hat{\\rho}_\\epsilon$. Mathematically, assuming we are going to perform a measurement (positive-operator valued measure -- POVM) in basis $M=\\{M_x\\}_x$, $\\sum_x M_x=I$, $M_x\\geq0$, where $M_x$ is an element of the POVM (while in the special case of projective measurement the operator $M_x$ is a projector onto the Hilbert space given by the eigenvalue $x$), this probability is defined as\n\\[\\label{def:prob_distribution}\np(x|\\epsilon)=\\mathrm{tr}[M_x\\hat{\\rho}_\\epsilon].\n\\]\nThe Fisher information is then defined as\n\\[\\label{eq:the_Fisher_information}\nF(\\epsilon):=\\int\\!\\!\\dif{x}\\ \\frac{\\big(\\partial_\\epsilon p(x|\\epsilon)\\big)^2}{p(x|\\epsilon)}.\n\\]\n$\\partial_\\epsilon$ denotes the partial derivative with respect to $\\epsilon$, $\\partial_\\epsilon p(x|\\epsilon)=\\mathrm{tr}[M_x\\partial_\\epsilon\\hat{\\rho}_\\epsilon]$, and integral goes over all values of $x$ such that $p(x|\\epsilon)>0$. The Fisher information measures how much information a random variable with by the probability distribution $p(x|\\epsilon)$ carries about the parameter $\\epsilon$.\nThe above definition of the Fisher information comes from the proof of the Cram\\'er-Rao bound. In the countable space of possible outcomes (the sample space) the integral is exchanged for the sum over all possible measurement outcomes.\n\nStrictly speaking, the Fisher information should have been written as $F(\\hat{\\rho}_\\epsilon,M)$, to reflect the fact that the Fisher information depends on a particular structure of density matrix and the choice of measurement to be performed. Here, however, we use the simple notation $F(\\epsilon)$ as is common in the literature.\n\n\\begin{example}\n\\emph{Using definition~\\eqref{eq:the_Fisher_information}, we calculate the Fisher information for example~\\ref{ex1} to be equal to 1. The Cram\\'er-Rao bound for the estimation of $\\epsilon$ reads $\\mean{(\\Delta\\hat{\\epsilon})^2}\\geq\\frac{1}{N}$.\n}\n\\end{example}\n\n\nThe Fisher information is a measure how much information about the unknown parameter $\\epsilon$ can be extracted from the quantum state given a choice of the measurement basis. However, one can rather ask how much information the quantum state itself yields about the parameter. In other words, how much information is in principle extractable from the quantum state. For that reason one can define the quantum Fisher information, which is obtained by maximizing the Fisher information over all possible measurements $M$,\n\\[\\label{def:QFI}\nH(\\epsilon):=\\underset{M}{\\mathrm{sup}}\\ F(\\hat{\\rho}_\\epsilon,M),\n\\]\nwhere $\\mathrm{sup}_{M}$ denotes supremum. This definition naturally implies $F(\\epsilon)\\leq H(\\epsilon)$, and from the Cram\\'er-Rao bound we immediately obtain the quantum Cram\\'er-Rao bound,\n\\[\\label{def:quantum_Cramer_Rao}\n\\mean{\\Delta\\hat{\\epsilon}^2}\\geq\\frac{1}{N H(\\epsilon)}.\n\\]\n\nDefinition~\\eqref{def:QFI} is one of many possible definitions of the quantum Fisher information and it does not provide an effective formula to calculate such quantity. It is also not clear whether the maximum can be saturated with some optimal measurement under which the Fisher information will be equal to the quantum Fisher information. For a single parameter estimation it is, in fact, possible by doing a projective measurement~\\cite{Paris2009a}, a feature which does not hold in general when estimating multiple param-\neters. This optimal measurement might not be unique, however, one of the possible optimal measurements is given by projectors $P_x$ constructed from eigenvectors of the symmetric logarithmic derivative $L_\\epsilon$. The symmetric logarithmic derivative is an operator defined as a solution to operator equation\n\\[\\label{def:SLDsolution}\n\\frac{L_\\epsilon\\hat{\\rho}_\\epsilon+\\hat{\\rho}_\\epsilon L_\\epsilon}{2}=\\partial_\\epsilon\\hat{\\rho}_\\epsilon.\n\\]\nAn alternative and completely equivalent definition~\\cite{Paris2009a} of the quantum Fisher information is then obtained by inserting projectors $P_x\\equiv M_x$ into Eq.~\\eqref{def:prob_distribution} and evaluating Eq.~\\eqref{eq:the_Fisher_information}\\footnote{Here it is important to point out that although the symmetric logarithmic derivative $L_\\epsilon$ in essence depends on the unknown parameter $\\epsilon$, so does its spectral decomposition, the derivative of~\\eqref{def:prob_distribution} is still given by $\\partial_\\epsilon p(x|\\epsilon)=\\mathrm{tr}[P_x\\partial_\\epsilon\\hat{\\rho}_\\epsilon]$, and not by $\\partial_\\epsilon p(x|\\epsilon)=\\mathrm{tr}[\\partial(P_x\\hat{\\rho}_\\epsilon)]$. This is because $\\partial_\\epsilon p(x|\\epsilon)$ measures the change of the probability distribution $p(x|\\epsilon)$ when $\\epsilon$ is slightly varied while the measurement basis $P_x$ is fixed. But how to reconcile with the fact that $P_x$ seems to be $\\epsilon$-dependent? In practice, in the local estimation theory the symmetric logarithmic derivative is evaluated at the approximate value $\\epsilon_{\\mathrm{app}}$ of the parameter, and the optimal measurement basis $P_x$ is fixed at this approximate value. Therefore, $P_x$ does not depend on the real value $\\epsilon$.}  (see appendix~\\ref{app:CR}), which yields\n\\[\\label{def:H_using_L}\nH(\\epsilon)=\\mathrm{tr}[\\partial_\\epsilon\\hat{\\rho}_\\epsilon L_\\epsilon]=\\mathrm{tr}[\\hat{\\rho}_\\epsilon L_\\epsilon^2].\n\\]\nThe above definition must be the same as the definition given by Eq.~\\eqref{def:QFI} because all Fisher informations are upper bounded by $\\mathrm{tr}[\\hat{\\rho}_\\epsilon L_\\epsilon^2]$ (see~\\cite{Paris2009a}), and as mentioned before a special pick of the measurement will results in the Fisher information to be equal to $\\mathrm{tr}[\\hat{\\rho}_\\epsilon L_\\epsilon^2]$.\n\n\\begin{example}\n\\emph{Why the name symmetric logarithmic derivative? From the definition~\\eqref{def:SLDsolution} it is clear why it is called symmetric. For the other part let us for simplicity assume that $L_\\epsilon$ and $\\hat{\\rho}_\\epsilon$ commute (so we can sum the symmetric part of the definition) and that $\\hat{\\rho}_\\epsilon$ is full rank operator (so $\\hat{\\rho}_\\epsilon^{-1}$ exists). Under those conditions it is easy to find a solution to Eq.~\\eqref{def:SLDsolution}, $L_\\epsilon=\\partial_\\epsilon\\log{\\hat{\\rho}_\\epsilon}$.\n}\n\\end{example}\n\nSolving equation~\\eqref{def:SLDsolution} is not easy in general. However, the solution has been found~\\cite{hubner1993computation} for the case when the spectral decomposition of the density matrix is known,\n\\begin{equation}\\label{SLD}\nL_\\epsilon=2\\!\\!\\!\\!\\sum_{\\substack{k,l \\\\p_k+p_l> 0}}\\!\\!\\!\\!\\frac{\\langle \\psi_k|\\partial_{\\epsilon}\\hat{\\rho}_{\\epsilon}|\\psi_{l}\\rangle}{p_k+p_l} |\\psi_{k}\\rangle\\langle\\psi_{l}|,\n\\end{equation}\nwhere the vectors  $\\{|\\psi_k\\rangle\\}$ are the eigenvectors of $\\hat{\\rho}_{\\epsilon}$, i.e. $\\hat{\\rho}_{\\epsilon}=\\sum_k p_k|\\psi_k\\rangle\\langle\\psi_k|$. The summation involves only elements for which $p_i+p_j>0$. The quantum Fisher information is then\n\\begin{equation}\\label{QFI}\nH(\\epsilon)=2\\!\\!\\!\\!\\sum_{\\substack{k,l \\\\p_k+p_l> 0}}\\!\\!\\!\\!\\frac{\\left|\\langle \\psi_k|\\partial_{\\epsilon}\\hat{\\rho}_{\\epsilon}|\\psi_{l}\\rangle\\right|^2}{p_k+p_l}.\n\\end{equation}\n\n\\begin{example}\\label{ex:pure}\n\\emph{Now we can derive an elegant formula for the quantum Fisher information of pure states. Assuming $\\hat{\\rho}_\\epsilon=\\pro{\\psi}{\\psi}$ we sum over all elements in Eq.~\\eqref{QFI}. It is important not to forget eigenvectors with zero eigenvalue (there will be $\\mathrm{dim}\\mathcal{H}-1$ many of them). Then we use the Parseval identity and the normalization condition $\\braket{\\psi}{\\psi}=1$. By differentiating the normalization condition twice we find that $\\braket{\\psi}{\\partial_{\\epsilon}\\psi}$ is purely imaginary. Using this property we finally derive \\[\\label{eq:QFI_pure}\nH(\\epsilon)=4(\\braket{\\partial_{\\epsilon}\\psi}{\\partial_{\\epsilon}\\psi}-\\norm{\\braket{\\psi}{\\partial_{\\epsilon}\\psi}}^2).\\]\nA relatively easier alternative route is to prove that $L_\\epsilon=2(\\pro{\\psi}{\\partial_{\\epsilon}\\psi}+\\pro{\\partial_{\\epsilon}\\psi}{\\psi})$ solves Eq.~\\eqref{def:SLDsolution} and then to use Eq.~\\eqref{def:H_using_L}. The third option for deriving this formula is through the Uhlmann fidelity, see example~\\ref{ex:pure2}.\n}\n\\end{example}\n\n\\section{Estimating channels: encoding operations, probes, and uncertainty relations}\n\nIn the previous section we introduced a formalism which can be used to find the ultimate limit of precision with what a parameter encoded in a quantum state can be estimated. In here, we will introduce a common metrological scenario aimed to estimate quantum channels.\n\nEstimating channels in quantum metrology has numerous stages. These stages can be depicted as follows,\n\\[\\label{scheme_for_estimating_channels}\n\\xrightarrow{preparation}\\hat{\\rho}_0\\xrightarrow{channel(\\epsilon)}\\hat{\\rho}_\\epsilon\\xrightarrow{measurement}(x_1,\\dots,x_N)\\xrightarrow{estimation}{\\hat{\\epsilon}(x_1,\\dots,x_N)\\approx\\epsilon}\n\\]\nFirst, the probe state $\\hat{\\rho}_0$ is prepared. This probe state is fed into a channel which encodes the unknown parameter $\\epsilon$. The structure of the channel is usually known. For example, it is known that it is a phase-changing channel. However, it is not known how much phase-change is introduced by that channel. It is the phase we are trying to estimate.\n\nAfter the parameter is encoded, an appropriate measurement basis is chosen. Repeating this procedure on $N$ identical states $\\hat{\\rho}_\\epsilon$ we obtain a statistics of measurement results. Those results are then turned into an estimate of the parameter through an estimator $\\hat{\\epsilon}$.\n\nThe task of quantum metrology is then three-folded: First, it is finding the optimal state $\\hat{\\rho}_0$ for probing the channel, i.e., the state which is the most sensitive to the channel. Within the local estimation theory, this is usually done by maximizing the quantum Fisher information under some fixed condition on the probe state. For example, finding the best probe state given a fixed amount of energy. The second task of quantum metrology is to find the optimal measurement. The optimal measurement is such that it produces a statistics of the measurement results which is the most informative about the parameter, i.e., statistics which leads to the lowest mean squared error on the parameter we want to estimate. An optimal measurement is given by the condition that the Fisher information for that particular measurement is equal to the quantum Fisher information. One of the optimal measurements can be always found by diagonalizing the symmetric logarithmic derivative. Third task of quantum metrology is to choose an appropriate estimator, which gives an appropriate meaning to the estimate with respect to the real value $\\epsilon$. Two obvious choices are previously mentioned MLE estimator, which gives the value has the highest probability to produce the measurement results, or the Bayes estimator, which minimizes the risk that the estimated value is far away from the real value.\n\nThis thesis will focus on the mathematical formalism of the first stage, as well as finding optimal Gaussian probe states, both being discussed in the next chapters. The other two stages will not be discussed in detail, but we will point out certain directions when appropriate. Now we will present some basic results about the channel estimation.\n\nAssuming the encoding operation is a unitary in an exponential form, or more precisely one-parameter unitary group, $\\hat{\\rho}_\\epsilon=e^{-i\\hat{K}\\epsilon}\\hat{\\rho}_0e^{+i\\hat{K}\\epsilon}$, where $\\hat{K}$ is a Hermitian operator, the quantum Fisher information is a constant independent on the parameter we want to estimate. This serves as a good check when calculating the quantum Fisher information for such channels. Note, however, that the symmetric logarithmic derivative still depends on the parameter, and so does the optimal POVM. Channels represented by one-parameter unitary group are very common. Examples include phase-changing channels, mode-mixing channels, squeezing channels, or a displacing channel.\n\nBecause of the unitarity of the channel, pure states remain pure after the parameter is encoded. We can use the result of example~\\ref{ex:pure} and derive the quantum Fisher information for pure states undergoing such unitary channels,\n\\[\\label{pureK}\nH(\\epsilon)=4\\bra{\\psi_0}\\Delta\\hat{K}^2\\ket{\\psi_0}=:4\\mean{\\Delta\\hat{K}^2},\n\\]\nwhere $\\Delta\\hat{K}:=\\hat{K}-\\bra{\\psi_0}\\hat{K}\\ket{\\psi_0}$. This interesting result shows that the quantum Fisher information scales quadratically with the encoding operator $\\hat{K}$. This means, if the encoding is twice as fast, the quantum Fisher information is four times bigger, and the mean squared error with what we can estimate the parameter is $\\frac{1}{4}$ of the previous mean squared error. Also, the right hand side is essentially the variance of the observable $\\hat{K}$. But this immediately says that the best probe states are such which maximize the variance in the observable $\\hat{K}$, which is the generator of translations in the parameter we want to estimate. For example, the encoding operator can be the Hamiltonian and the encoded parameter time. Because of identity~\\eqref{pureK}, the best time-probes are such which have the widest distribution in energy. These are called the GHZ states and play an important role in the estimation theory.\n\nFor one measurement ($N=1$) and SI units ($\\hbar\\neq 1$), the quantum Cram\\'er-Rao bound~\\eqref{eq:Cramer_Rao} gives an interesting relation for the pure states,\n\\[\n\\mean{\\Delta\\hat{K}^2}\\mean{\\Delta\\hat{\\epsilon}^2}\\geq\\frac{\\hbar^2}{4}.\n\\]\nThe quantum Cram\\'er-Rao bound thus represents a type of the Heisenberg uncertainty relations. However, this is not between two observables as it is usually considered, but between one observable and one parameter. If for example the encoding operator is the Hamiltonian and the parameter time, this inequality states that the mean squared error on the time estimate multiplied by the square root of the variance in energy of the quantum state cannot go below $\\frac{\\hbar}{2}$ in a single-shot experiment.\n\nThe equivalent formulae for the mixed states exist and can be found for example in~\\cite{Paris2009a}. However, the quantum Fisher information for mixed states is not equal to four times the variance as it is for pure states. In fact, it is always lower. For more details on the Heisenberg uncertainty relations, as well as on its connection to the speed of evolution of quantum states, see for example~\\cite{kok2010introduction,taddei2013quantum}.\n\n\\section{Classical and Heisenberg scaling}\n\nAs briefly mentioned in the previous section, finding optimal probe states usually encompasses fixing a certain parameter, usually the mean energy of the probe state, or equivalently, the mean number of particles in the probe state. In the beginning of this chapter we assumed we have $N$ identical copies of the same state dependent on the parameter we want to estimate, i.e., in total we have $\\hat{\\rho}_\\epsilon^{\\otimes N}$. A single measurement of such large state can extract as much information as $N$ measurements on the $N$ identical subsystems. For a single-parameter estimation, if each subsystem represents one particle, there is no difference between measuring all of these particles at the same time, or each particle separately.\\footnote{Nevertheless it is important to note that entangled measurements can improve the estimation of multiple parameters~\\cite{fujiwara1995quantum}.} The improvement on the precision of the estimated parameter then scales as central limit theorem dictates for any identical and identically distributed variables. That is why the quantum Fisher information scales as $H(\\epsilon)\\sim N$ for such states, called the shot-noise limit. However, one can consider an alternative input state with the same `cost' which performs much better. If the probe state is an entangled state such as GHZ state, the task is no longer equivalent to measuring each particle separately and central limit theorem does not apply. The quantum enhancement is possible. Such states can then scale as $H(\\epsilon)\\sim N^2$, called the Heisenberg limit.\n\n\\begin{example}\\label{ex:Heisenberg_scaling}\n\\emph{Consider the phase-changing encoding operator $e^{-i\\hat{N}\\epsilon}$, where $\\hat{N}$ is the total number operator. We can use results of example~\\ref{ex:pure} or equation~$\\eqref{pureK}$ to calculate the quantum Fisher information for the probe states. First, $\\hat{\\rho}_0=\\hat{\\rho}_{0s}^{\\otimes N}$, where $\\hat{\\rho}_{0s}=\\pro{\\psi_{0s}}{\\psi_{0s}}$ is the pure state defined as $\\ket{\\psi_{0s}}=\\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{1})$. Second, the GHZ probe state $\\hat{\\rho}_0=\\pro{\\psi_0}{\\psi_0}$, where $\\ket{\\psi_0}=\\frac{1}{\\sqrt{2}}(\\ket{0,\\dots,0}+\\ket{1,\\dots,1})$. Although both states have the same mean energy $\\frac{N}{2}$, the quantum Fisher information of the separable state achieves the shot-noise limit, $H(\\epsilon)=N$, in contrast to the entangled state which achieves the Heisenberg limit, $H(\\epsilon)=N^2$.\n}\n\\end{example}\n\nNote however, despite its name, whether the Heisenberg limit is or is not a fundamental limit depends on the particular definition one chooses to use. For the definitions which define the scaling of the quantum Fisher information with respect to the mean energy of the state, or with respect to the maximum energy of the state, a sequences of states can be found to achieve super-Heisenberg scaling~\\cite{luis2004nonlinear,boixo2008quantum}. Constructing such states is particularly easy in the Fock space, since its infinite-dimensional structure allows for states to have an arbitrarily large variance in energy while having an arbitrarily low mean value of energy. We will discuss this issue in more detail in section~\\ref{sec:Heisenberg_limit_Gaussian}. On the other hand, when one considers the scaling with respect to the amount of resources needed to prepare such a probe state, the Heisenberg limit is indeed the fundamental limit~\\cite{zwierz2010general}.\n\n\n\n\n\\section{Geometry of estimation and multi-parameter estimation}\\label{QMsec:multi}\n\nWe have considered only one parameter to be estimated so far. But there are tasks where it is important to estimate multiple parameters. These scenarios include for example simultaneous estimation of a phase and the decoherence or estimation of multiple spins pointing in different directions. The theory outlined in previous sections can be naturally generalized to multi-parameter estimation. However, there are some problems which are connected to the impossibility of estimating the parameters simultaneously. This is tied to the fact that optimal measurements for estimation of different parameters do not necessarily commute. Higher precision in one parameter induces a trade-off on the precision in others. For that reason, it is not even clear what figure of merit to maximize. Whether to maximize the total variance on the parameters, which gives each parameter the same importance, a weighted sum of variances, or a covariance between the two parameters. For an introduction to the multi-parameter estimation see for example~\\cite{Paris2009a,szczykulska2016multi}.\n\nAssuming the density matrix depends on a vector of parameters $\\boldsymbol{\\epsilon}=(\\epsilon_1,...,\\epsilon_n)$, in the analogy of Eq.~\\eqref{def:SLDsolution} we define symmetric logarithmic derivatives,\n\\[\n\\frac{L_{i}\\hat{\\rho}_{\\boldsymbol{\\epsilon}}+\\hat{\\rho}_{\\boldsymbol{\\epsilon}} L_{i}}{2}=\\partial_{i}\\hat{\\rho}_{\\boldsymbol{\\epsilon}}.\n\\]\nWe use a simplified notation $\\partial_{i}\\equiv\\partial_{\\epsilon_i}$.\nThe quantum Fisher information matrix is then a symmetric positive or symmetric positive semi-definite matrix given by\n\\[\\label{def:Information_matrix}\nH^{ij}(\\boldsymbol{\\epsilon})=\\mathrm{tr}\\left[\\frac{L_{i}L_{j}+L_{j}L_{i}}{2}\\hat{\\rho}_{\\boldsymbol{\\epsilon}}\\right],\n\\]\nfrom which it is possible to derive a multi-parameter equivalent of Eq.~\\eqref{QFI},\n\\begin{equation}\\label{QFI_multi}\nH^{ij}(\\boldsymbol{\\epsilon})=2\\!\\!\\!\\!\\sum_{\\substack{k,l \\\\p_k+p_l> 0}}\\!\\!\\!\\!\\frac{\\Re(\\langle \\psi_k|\\partial_i\\hat{\\rho}_{\\boldsymbol{\\epsilon}}|\\psi_{l}\\rangle\\langle \\psi_l|\\partial_j\\hat{\\rho}_{\\boldsymbol{\\epsilon}}|\\psi_{k}\\rangle)}{p_k+p_l},\n\\end{equation}\nwhere $\\Re$ denotes the real part. Performing $N$ identical measurements on $N$ identical copies of a quantum state the multi-parameter quantum Cram\\'er-Rao bound reads\n\\[\n\\mathrm{Cov}[\\hat{\\boldsymbol{\\epsilon}}]\\geq\\frac{1}{N}{H}^{-1}(\\boldsymbol{\\epsilon}),\n\\]\nwhere $\\mathrm{Cov}[\\hat{\\boldsymbol{\\epsilon}}]=\\mean{\\hat{\\epsilon}_i\\hat{\\epsilon}_j}-\\mean{\\hat{\\epsilon}_i}\\mean{\\hat{\\epsilon}_j}$ is the covariance matrix of the locally unbiased estimator vector $\\hat{\\boldsymbol{\\epsilon}}$ (i.e., matrix with variances of single parameters on the diagonal and correlation coefficients being the non-diagonal elements), and $H^{-1}(\\boldsymbol{\\epsilon})$ the inverse of the matrix defined in Eq.~\\eqref{def:Information_matrix}. The above equation should be understood as an operator inequality. It states that $\\mathrm{Cov}[\\hat{\\boldsymbol{\\epsilon}}]-\\frac{1}{N}{H}^{-1}$ is a positive semi-definite or a positive definite matrix.\n\nIn contrast to the one-parameter scenario for which an optimal measurement can be always found, it is not always possible to find the optimal measurement the multi-parameter quantum Cram\\'er-Rao bound, i.e., it is not always possible to find a measurement for which the Fisher information matrix $\\Big($defined as $F^{ij}(\\boldsymbol{\\epsilon})=\\int\\!\\!\\dif{x}\\ \\frac{\\partial_{i} p(x|\\boldsymbol{\\epsilon}) \\partial_{j} p(x|\\boldsymbol{\\epsilon})}{p(x|\\boldsymbol{\\epsilon})}$$\\Big)$ equals the quantum Fisher information matrix. This is because projectors $P_x^{(i)}$ from spectral decompositions of different symmetric logarithmic derivatives do not necessarily commute which is a general problem in the multi-parameter estimation.\nSeveral advancements in the attainability of the multi-parameter bound are reviewed in~\\cite{szczykulska2016multi}.\n\nThe quantum Fisher information matrix is connected to an important statistical measure called the Bures metric~\\cite{Bures1969a}. To define this metric we first introduce the Bures distance. The Bures distance is a measure of distinguishability between two quantum states $\\hat{\\rho}_{1,2}$ and is defined through the Uhlmann fidelity~\\cite{Uhlmann1976a}\n\\[\\label{def:Uhlmann_Fidelity}\n\\mathcal{F}({\\hat{\\rho}}_{1},{\\hat{\\rho}}_{2})\\,:=\\,\\big(\\mathrm{tr}\\sqrt{\\sqrt{{\\hat{\\rho}}_{1}}\\,{\\hat{\\rho}}_{2}\\,\\sqrt{{\\hat{\\rho}}_{1}}}\\big)^{2}\n\\]\nas\n\\[\\label{def:bures_distance}\nd_B^2(\\hat{\\rho}_1,\\hat{\\rho}_2)=2\\big(1-\\sqrt{\\mathcal{F}(\\hat{\\rho}_1,\\hat{\\rho}_2)}\\big).\n\\]\nThe Bures distance gives rise to the Bures metric $g^{ij}$ which measures the amount of distinguishability of two close density matrices in the coordinate system $\\boldsymbol{\\epsilon}$ through the definition for the line element,\n\\[\\label{eqn:bures}\nd_B^2(\\hat{\\rho}_{\\boldsymbol{\\epsilon}},\\hat{\\rho}_{\\boldsymbol{\\epsilon}+\\boldsymbol{\\mathrm{d}\\epsilon}})=\\sum_{i,j}g^{ij}(\\boldsymbol{\\epsilon})\\mathrm{d}\\epsilon_i\\mathrm{d}\\epsilon_j.\n\\]\n\nIt is usually thought and it is mentioned in~\\cite{Paris2009a} that the quantum Fisher information matrix~\\eqref{def:Information_matrix} is a multiple of the Bures metric. Although usually true, this is not always the case. This belief is based on derivations of explicit formulae for the Bures metric in~\\cite{hubner1992explicit} and \\cite{sommers2003bures}. The first derivation assumes that the density matrix is invertible, while the second lacks particular details concerning the treatment of problematic points. Moreover, both derivations are finding expressions for\ninfinitesimal distance $d_B^2(\\hat{\\rho},\\hat{\\rho}+\\mathrm{d}\\hat{\\rho})$ which is ill-defined at the boundary of the convex space of density matrices (when $\\hat{\\rho}^{-1}$ does not exist), because certain choices of $\\mathrm{d}\\hat{\\rho}$ can cause $\\hat{\\rho}+\\mathrm{d}\\hat{\\rho}$ not to be a density matrix anymore. In general, however, for parameterized quantum states $\\hat{\\rho}_{\\boldsymbol{\\epsilon}}$ in which a slight change in the parameter $\\boldsymbol{\\epsilon}$ results in an eigenvalue of the density matrix to vanish (or, equivalently, results in an eigenvalue to `pop out'), there is an extra term in the Bures metric\nwhich needs to be accounted for. As we show in appendix~\\ref{app:discontinuity_of_QFI}, the quantum Fisher information matrix is connected to the Bures metric through relation\n\\[\\label{eq:bures_metric}\ng^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{4}\\Big(H^{ij}(\\boldsymbol{\\epsilon})+2\\!\\!\\!\\!\\!\\!\\sum_{k:\\,p_k(\\boldsymbol{\\epsilon})=0}\\!\\!\\!\\!\\!\\!\\partial_{i} \\partial_{j}p_k(\\boldsymbol{\\epsilon})\\Big).\n\\]\nBy $k:\\,p_k(\\boldsymbol{\\epsilon})=0$ we mean that the sum goes over all eigenvalues $p_k$ such that their value vanishes at point $\\boldsymbol{\\epsilon}$. From the above relation we can see that the (four times) Bures metric and the quantum Fisher information matrix do not coincide only at certain points $\\boldsymbol{\\epsilon}$, at which an eigenvalue vanishes. When change of the parameter does not result in the change of purity, for example when the operation encoding $\\boldsymbol{\\epsilon}$ is a unitary operation, the (four times) Bures metric and the quantum Fisher information matrix are identical. It is worth noting that the Hessian $\\mathcal{H}_k^{ij}(\\boldsymbol{\\epsilon}):=\\partial_{i} \\partial_{j}p_k(\\boldsymbol{\\epsilon})$ is a positive or a positive semi-definite matrix, because $p_k$ reaches the local minimum at point $\\epsilon$ such that $p_k(\\epsilon)=0$. Therefore the following matrix inequality holds,\n\\[\n4g\\geq H\n\\]\nand $4g= H$ if and only if for all $\\boldsymbol{\\epsilon}$ and $k$ such that $p_k(\\boldsymbol{\\epsilon})=0$, $\\mathcal{H}_k(\\boldsymbol{\\epsilon})=0$.\n\nExpression~\\eqref{eq:bures_metric} has a surprising interpretation. It can be shown from Eq.~\\eqref{QFI} that even for analytical functions $\\hat{\\rho}_\\epsilon$ the quantum Fisher information can be discontinuous at points $\\boldsymbol{\\epsilon}$ for which $p_k(\\boldsymbol{\\epsilon})=0$. This discontinuity is however removable. It is possible to redefine these points in a way which makes the quantum Fisher information matrix continuous in the following sense: for $\\hat{\\rho}_{\\boldsymbol{\\epsilon}}\\in C^{(2)}$ every element of the redefined matrix $H_c^{ij}$ is a continuous function in parameter $\\epsilon_i$ while all other parameters $\\epsilon_k, k\\neq i$ are kept fixed. The same holds for the parameter $\\epsilon_j$. Such a redefinition leads exactly to the expression defined by the Bures metric,\n\\[\\label{eq:connection_between_Hc_and_H}\nH_c^{ij}(\\boldsymbol{\\epsilon})=4g^{ij}(\\boldsymbol{\\epsilon})=H^{ij}(\\boldsymbol{\\epsilon})+2\\!\\!\\!\\!\\!\\!\\sum_{k:\\,p_k(\\boldsymbol{\\epsilon})=0}\\!\\!\\!\\!\\!\\!\\partial_{i} \\partial_{j}p_k(\\boldsymbol{\\epsilon}).\n\\]\nIn the end, it is important to point out that although the continuous version of the quantum Fisher information seems to have nicer properties, the quantum Cram\\'er-Rao bound holds only for the (possibly discontinuous) quantum Fisher information matrix $H$. For more details see appendices~\\ref{app:CR} and~\\ref{app:discontinuity_of_QFI}.\n\nCombining the above equation with Eq.~\\eqref{eqn:bures} we obtain the expression for the continuous quantum Fisher information matrix in terms of fidelity,\n\\[\\label{eq:QFI_matrix_using_fidelity}\n\\sum_{i,j}H_c^{ij}\\mathrm{d}\\epsilon_i\\mathrm{d}\\epsilon_j\n=8\\big(1-\\sqrt{\\mathcal{F}(\\hat{\\rho}_{\\boldsymbol{\\epsilon}},\\hat{\\rho}_{\\boldsymbol{\\epsilon}+\\boldsymbol{\\mathrm{d}\\epsilon}})}\\big).\n\\]\nThis means that we can calculate the continuous quantum Fisher information matrix by expanding the Uhlmann fidelity to the second order in infinitesimal parameters. For a single parameter we can simply write\n\\[\\label{QFI_using_fidelity}\nH_c(\\epsilon)=8\\lim_{\\mathrm{d}\\epsilon\\rightarrow0}\\frac{1-\\sqrt{\\mathcal{F}(\\hat{\\rho}_\\epsilon,\\hat{\\rho}_{\\epsilon+\\mathrm{d}\\epsilon})}}{\\mathrm{d}\\epsilon^{2}}.\n\\]\n\n\\begin{example}\\label{ex:pure2}\n\\emph{\nIn the case of pure states $H$ equals $H_c$ because the purity does not change. Therefore, using Eq.~\\eqref{QFI_using_fidelity} we can derive the quantum Fisher information for pure states one more time. The result should coincide with the results of example~\\ref{ex:pure}.\n}\n\\end{example}\n\nThe next chapter will introduce the current state-of-the art quantum metrology on Gaussian states. We build on these results and take them even further in chapter~\\ref{chap:QM_GS} in which we derive new formulae for the parameter estimation and we develop a new and effective formalism for finding optimal Gaussian probe states. We will also discuss the issue of discontinuity of the quantum Fisher information matrix in the context of Gaussian states. Chapter~\\ref{chap:reference_frames} then sheds light on the quantum metrology in the context of quantum reference frames. There we will show how having misaligned or imperfect reference frames affects the estimation precision.\n\n\\chapter{Gaussian states}\\label{chap:GS}\n\nGaussian states are of great use in experimental quantum physics, mainly because they combine several useful properties. They are relatively straightforward to prepare and handle, especially in optical systems~\\cite{andersen201530}, and they are resistant to decoherence~\\cite{demkowicz2013fundamental}. Although they resemble some properties of classical fields, they also exhibit quantum phenomena such as entanglement and thus can be used for quantum information protocols, for instance quantum teleportation~\\cite{furusawa2007quantum} and quantum cryptography~\\cite{jouguet2013experimental,huang2016long}. Moreover, Gaussian states are also simple to handle mathematically via elegant phase-space formalism.\n\nThere has been extensive literature published on Gaussian states. Let us mention for example lecture notes on continuous variable quantum information~\\cite{ferraro2005gaussian}, PhD thesis on entanglement of Gaussian states~\\cite{adesso2007entanglement} or Gaussian channels~\\cite{schafer2013information}, and review articles on Gaussian states~\\cite{wang2007quantum,Weedbrook2012a,Adesso2014a}. In this introductory chapter we focus on aspects of continuous variable quantum information directly related to finding optimal probe states for the estimation of Gaussian channels.\n\nThis chapter is organized as follows: we first introduce the Fock space of a bosonic field which is necessary to define Gaussian states. We introduce the phase-space formalism of Gaussian states and Gaussian channels. In particular, we provide symplectic matrices in the real and the complex phase-space formalism for the most common Gaussian unitary channels. We introduce basic Gaussian states and fully parametrize one-, two-, and three-mode Gaussian states. This parametrization will be used in section~\\ref{sec:estimation_of_channels} to find the optimal Gaussian probe states for the estimation of Gaussian channels. Finally, we give an overview on the current state-of-the-art in the estimation of Gaussian states in the phase-space formalism. Later in chapter~\\ref{chap:QM_GS} we build on these results and derive new easy-to-use formulae.\n\n\\section{Fock space of a bosonic field}\\label{sec:Fock_space}\nBosons are particles which follow Bose-Einstein statistics and are characterized by an integer spin. Examples include photons -- particles of light, W and Z bosons -- particles mediating the weak interaction, phonons -- excitations of a vibrational field, or Cooper pairs -- bound states of electrons responsible for super-conductivity. An important property of bosons is that their statistics gives no restriction on the number of indistiguishable particle occupying the same quantum state. That is why a quantum description of many such particles offer a rich structure, described by a bosonic Fock space.\n\nLet $\\mathcal{H}$ be a single particle Hilbert space. The (bosonic) Fock space is the direct sum of the symmetric tensor powers of the Hilbert space $\\mathcal{H}$,\n\\[\nF(\\mathcal{H})=\\bigoplus_{n=0}^\\infty S\\left(\\mathcal{H}^{\\otimes n}\\right).\n\\]\n$\\mathcal{H}^{\\otimes 0}=\\mathbb{C}$ and $S$ is the operator which symmetrizes the Hilbert space, i.e., $S\\left(\\mathcal{H}^{\\otimes n}\\right)$ consists of such states $\\ket{\\psi}\\in\\mathcal{H}^{\\otimes n}$ which are completely symmetric with respect to the exchange of particles.\n\nAssuming $\\mathcal{H}=\\mathrm{span}\\{\\ket{\\psi_1},\\ket{\\psi_2},\\dots\\}$, we can construct an elegant way of how to write a basis of the Fock space. $F(\\mathcal{H})=\\mathrm{span}\\{\\ket{n_1^{(1)},n_2^{(2)},\\dots}\\}_{n_1,n_2,\\dots}$ where $n_1^{(1)}$ denotes that there are $n_1$ particles in the state $\\ket{\\psi_1}$, $n_2^{(2)}$ denotes that there are $n_2$ particles in the state $\\ket{\\psi_2}$ and so forth. Because the particles are indistinguishable, exchanging any two particles in the same state of the single-particle Hilbert space $\\mathcal{H}$ should not change the full state in the Fock space. Hence a pure state in the Fock space can be fully described only by the number of particles in each single-particle state and this notation is consistent. Any other state in the Fock space can be described as a linear combination of these number vectors. Construction of the number basis will be clarified in the following example.\n\n\\begin{example}\\label{ex:twomode_Fock_space}\n\\emph{\nLet $\\mathcal{H}=\\mathrm{span}\\{\\ket{\\uparrow},\\ket{\\rightarrow}\\}$ be a Hilbert space a polarized photon with $\\ket{\\uparrow}$ representing a vertically polarized photon and $\\ket{\\rightarrow}$ representing a horizontally polarized photon. The Fock space is\n\\[\nF(\\mathcal{H})=\\mathbb{C}\\oplus S\\left(\\mathcal{H}\\right) \\oplus S\\left(\\mathcal{H}\\otimes\\mathcal{H}\\right)\\oplus S\\left(\\mathcal{H}\\otimes\\mathcal{H}\\otimes\\mathcal{H} \\right)\\oplus\\cdots,\n\\]\nwhere $S(\\mathcal{H})=\\mathcal{H}$ denotes the single particle Hilbert space, $S(\\mathcal{H}\\otimes\\mathcal{H})$ the symmetrized two particles Hilbert space, $S(\\mathcal{H}\\otimes\\mathcal{H}\\otimes\\mathcal{H})$ the symmetrized three particles Hilbert space. Number states are then constructed as\n\\begin{align}\n\\ket{00}&=1\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\cdots,\\nonumber\\\\\n\\ket{10}&=0\\oplus\\ket{\\uparrow}\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\cdots,\\nonumber\\\\\n\\ket{01}&=0\\oplus\\ket{\\rightarrow}\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\cdots,\\nonumber\\\\\n\\ket{20}&=0\\oplus\\boldsymbol{0}\\oplus\\ket{\\uparrow}\\otimes\\ket{\\uparrow}\\oplus\\boldsymbol{0}\\oplus\\cdots,\\\\\n\\ket{11}&=0\\oplus\\boldsymbol{0}\\oplus\\frac{1}{\\sqrt{2}}(\\ket{\\uparrow}\\otimes\\ket{\\rightarrow}+\\ket{\\rightarrow}\\otimes\\ket{\\uparrow})\\oplus\\boldsymbol{0}\\oplus\\cdots,\\nonumber\\\\\n\\ket{02}&=0\\oplus\\boldsymbol{0}\\oplus\\ket{\\rightarrow}\\otimes\\ket{\\rightarrow}\\oplus\\boldsymbol{0}\\oplus\\cdots,\\nonumber\\\\\n\\ket{30}&=0\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\ket{\\uparrow}\\otimes\\ket{\\uparrow}\\otimes\\ket{\\uparrow}\\oplus\\cdots,\\nonumber\\\\\n\\ket{21}&=0\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\n\\frac{1}{\\sqrt{3}}(\\ket{\\uparrow}\\otimes\\ket{\\uparrow}\\otimes\\ket{\\rightarrow}+\\ket{\\uparrow}\\otimes\\ket{\\rightarrow}\\otimes\\ket{\\uparrow}+\\ket{\\rightarrow}\\otimes\\ket{\\uparrow}\\otimes\\ket{\\uparrow})\\oplus\\cdots\\nonumber\\\\\n&\\cdots\\nonumber\n\\end{align}\nClearly, these vectors are linearly independent and any vector of the Fock space can be written as their linear combination. They form a basis of the Fock space. The zero vector in the Fock space commonly denoted as $0$ is defined as\n\\[\n0=0\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\boldsymbol{0}\\oplus\\cdots.\\\\\n\\]\n}\n\\end{example}\n\nEach Fock space can be equipped with a set of annihilation and creation operators which are necessary to define Gaussian states. We assign one annihilation $\\hat{a}_i$ and one creation operator $\\hat{a}_i^\\dag$ to each single-particle basis vector $\\ket{\\psi_i}$. The action of these field operators is to either annihilate or to create a particle in the state $\\ket{\\psi_i}$,\n\\begin{subequations}\n\\begin{align}\n\\hat{a}_i\\ket{\\dots,n_{i-1},n_i,n_{i+1},\\dots}&=\\sqrt{n_i}\\ket{\\dots,n_{i-1},n_i-1,n_{i+1},\\dots},\\\\\n\\hat{a}_i\\ket{\\dots,n_{i-1},0,n_{i+1},\\dots}&=0,\\\\\n\\hat{a}_i^\\dag\\ket{\\dots,n_{i-1},n_i,n_{i+1},\\dots}&=\\sqrt{n_i+1}\\ket{\\dots,n_{i-1},n_i+1,n_{i+1},\\dots}.\n\\end{align}\n\\end{subequations}\nThe field operators satisfy the commutation relations $[\\hat{a}_i,\\hat{a}_j^\\dag]=\\delta_{ij}\\mathrm{id}$, where $\\delta_{ij}$ denotes Kronecker delta and $\\mathrm{id}$ denotes the identity element of the algebra.\n\n\\begin{example}\\label{ex:building_a_Fock_basis}\n\\emph{\nHaving defined the action of creation operators, it is clear that all basis vectors introduced in example~\\ref{ex:twomode_Fock_space} can be written as\n\\[\n\\ket{n_1^{(1)}n_2^{(2)}}=\\frac{(\\hat{a}_1^\\dag)^{n_1}}{\\sqrt{n_1!}} \\frac{(\\hat{a}_2^\\dag)^{n_2}}{\\sqrt{n_2!}} \\ket{00}.\n\\]\nLater we will focus on quantum field theory where even more compact notation is used. This is due to the fact that there are often infinitely many modes, making it impossible to write write them all into one long vector. The vacuum state will be denoted as $\\ket{0}:=\\ket{00}$ and the state of $n_1$ particles in the first mode as $\\ket{n_1^{(1)}}:=\\ket{n_1^{(1)} 0}$.\n}\n\\end{example}\n\nNow we will switch to the more elegant notation which will be appropriate for an effective description of Gaussian states. Assuming there is a finite number of basis vectors $\\ket{\\psi_i}$ -- from now on called modes -- of the single particle Hilbert space $\\mathcal{H}=\\mathrm{span}\\{\\ket{\\psi_1},\\dots,\\ket{\\psi_N}\\}$, we collect their associated annihilation and creation operators into a vector $\\boldsymbol{\\hat{A}}:=(\\hat{a}_1,\\dots,\\hat{a}_N,\\hat{a}_1^\\dag,\\dots,\\hat{a}_N^\\dag)^T$. The commutation relations between the operators can be written in compact form,\n\\[\\label{def:commutation_relation}\n[\\boldsymbol{\\hat{A}}^{i},\\boldsymbol{\\hat{A}}^{j\\dag}]=K^{ij}\\mathrm{id}\\quad\\!\\Rightarrow\\quad\\! K=\n\\begin{bmatrix}\nI & 0 \\\\\n0 & -I\n\\end{bmatrix},\n\\]\nwhere $I$ denotes the identity matrix. This equation also defines matrix $K$ to which we will later refer to as to the symplectic form. This matrix has numerous useful properties, namely $K^{-1}=K^\\dag=K$ and $K^2=I$.\n\n\n\\section{Gaussian states in the phase-space formalism}\n\nQuantum states are usually described by a positive semi-definite operator called the density matrix $\\hat{\\rho}$, however, for bosonic systems an alternative and completely equivalent description exists which is particularly useful for a description of Gaussian states. Given a state $\\hat{\\rho}$ we define \\emph{the symmetric characteristic function} as\n\\[\n\\chi(\\boldsymbol{\\xi})\\,=\\,\\mathrm{tr}[\\hat{\\rho}\\,\\hat{D}(\\boldsymbol{\\xi})],\n\\]\nwhere $\\hat{D}(\\boldsymbol{\\xi})\\,=\\,e^{\\boldsymbol{\\hat{A}}^{\\dag}K\\boldsymbol{\\xi}}$ is the \\emph{Weyl displacement operator} with the variable of the form $\\boldsymbol{\\xi}\\,=\\,\\boldsymbol{\\gamma}\\oplus\\overline{\\boldsymbol{\\gamma}}$. Gaussian states are those whose characteristic function is, by definition, of Gaussian form, i.e.,\n\\[\n\\chi(\\boldsymbol{\\xi})\\,=\\,e^{-\\frac{1}{4}\\boldsymbol{\\xi}^{\\dag}\\sigma\\boldsymbol{\\xi}-i\\,{\\boldsymbol{d}}^{\\dag}K\\boldsymbol{\\xi}}.\n\\]\n\nIn the analogy of classical probability theory, Gaussian states are completely described by the first and the second statistical moments $\\boldsymbol{d}$ and $\\sigma$ of the field. The \\emph{displacement vector} $\\boldsymbol{d}$ and the \\emph{covariance matrix} $\\sigma$ are defined as~\\cite{Weedbrook2012a},\n\\begin{subequations}\\label{def:covariance_matrix}\n\\begin{align}\n\\boldsymbol{d}^i&=\\mathrm{tr}\\big[\\hat{\\rho}\\boldsymbol{\\hat{A}}^i\\big],\\\\\n\\sigma^{ij}&=\\mathrm{tr}\\big[\\hat{\\rho}\\,\\{\\Delta\\boldsymbol{\\hat{A}}^i,\\Delta\\boldsymbol{\\hat{A}}^{j{\\dag}}\\}\\big].\n\\end{align}\n\\end{subequations}\nThe density operator $\\hat\\rho$ specifies the state of the field and $\\{\\!\\cdot,\\cdot\\!\\}$ denotes the anti-commutator, and $\\Delta\\boldsymbol{\\hat{A}}:=\\boldsymbol{\\hat{A}}-\\boldsymbol{d}$.\n\nFrom the definition~\\eqref{def:covariance_matrix} we can observe the following structure of the first and second moments:\n\\[\\label{def:first_and_second_moments}\n\\boldsymbol{d}=\n\\begin{bmatrix}\n\\boldsymbol{\\tilde{d}} \\\\ \\overline{\\boldsymbol{\\tilde{d}}}\n\\end{bmatrix},\\quad\n\\sigma\\,=\\,\\begin{bmatrix}\nX & Y \\\\\n\\overline{Y} & \\overline{X}\n\\end{bmatrix},\n\\]\nwhere \\emph{bar} denotes the complex conjugation. The covariance matrix is a positive-definite Hermitian matrix, $\\sigma^\\dag=\\sigma$, i.e., $X^\\dag=X$ and $Y^T=Y$, and further satisfying~\\cite{simon1994quantum}\n\\[\\label{eq:sigma_K_positivity}\n\\sigma+K\\geq0\n\\]\nwhich is a consequence of the commutation relations~\\eqref{def:commutation_relation}.\n\nSometimes we can be interested in a subsystem of the Gaussian states. In the density matrix formalism, the density matrix of a subsystem is obtained by the partial tracing, i.e., tracing over all states we are not interested in. Partial tracing in the covariance matrix formalism is very simple. Tracing over modes we are not interested in is done simply by taking away the rows and columns of the covariance matrix and elements of the displacement vector associated with those modes.\n\nWe emphasise that Eq.~\\eqref{def:covariance_matrix} defines the complex form of the covariance matrix, which is defined by the anti-commutator of annihilation and creation operators. Most authors use the real form, which is defined in terms of position and momenta operators. Defining vector of position and momenta operators\n$\\boldsymbol{\\hat{Q}}:=(\\hat{x}_1,\\dots,\\hat{x}_N,\\hat{p}_1,\\dots,\\hat{p}_N)^T$, where $\\hat{x}_i:=\\frac{1}{\\sqrt{2}}(\\hat{a}^\\dag_i+\\hat{a}_i)$, $\\hat{p}_i:=\\frac{i}{\\sqrt{2}}(\\hat{a}^\\dag_i-\\hat{a}_i)$, the real form displacement and the real form covariance matrix are defined as\n\\begin{subequations}\\label{def:covariance_matrix_real}\n\\begin{align}\n\\boldsymbol{d}_{\\Re}^i&=\\mathrm{tr}\\big[\\hat{\\rho}\\boldsymbol{\\hat{Q}}^i\\big],\\\\\n\\sigma_{\\Re}^{ij}&=\\mathrm{tr}\\big[\\hat{\\rho}\\,\\{\\Delta\\boldsymbol{\\hat{Q}}^i,\\Delta\\boldsymbol{\\hat{Q}}^j\\}\\big],\n\\end{align}\n\\end{subequations}\nwhere $\\Delta\\boldsymbol{\\hat{Q}}:=\\boldsymbol{\\hat{Q}}-\\boldsymbol{d}_{\\Re}$.\n\nOther notations exist which adds to the confusion in literature. One other common example includes different ordering of the quadrature operators, $\\boldsymbol{\\hat{Q}}:=(\\hat{x}_1,\\hat{p}_1,$$\\hat{x}_2,\\hat{p}_2,$$\\dots,$ $\\hat{x}_N,\\hat{p}_N)^T$. In this thesis we will consistently use the complex form unless specified differently in concrete examples. This is because the complex form expose the inner symmetries in more detail than the real form and because some formulae and matrices are much more elegantly expressed in the complex form. Also, the complex form is generally easier to work with at a small cost of admitting complex numbers. For more information about the real form and its connection to the complex form see appendix~\\ref{app:real_covariance}, or~\\cite{Arvind1995a,Adesso2014a}.\n\n\\section{Gaussian unitaries and symplectic geometry}\n\nGaussian transformation is an transformation which maps Gaussian states into Gaussian states. Gaussian unitary is a Gaussian transformation which is represented by a unitary operator, i.e., it transforms Gaussian state $\\hat{\\rho}$ into Gaussian state $\\hat{\\rho}'=\\hat{U}\\hat{\\rho}\\hat{U}^\\dag$. All such operators can be generated via an exponential map with the exponent at most quadratic in the field operators~\\cite{Weedbrook2012a},\n\\[\\label{def:Gaussian_unitary}\n\\hat{U}=\\exp\\big(\\tfrac{i}{2}\\boldsymbol{\\hat{A}}^\\dag W \\boldsymbol{\\hat{A}}+\\boldsymbol{\\hat{A}}^\\dag K \\boldsymbol{\\gamma}\\big),\n\\]\nwhere $W$ is a Hermitian matrix of the form following the same structure as the covariance matrix~\\eqref{def:first_and_second_moments},\n\\[\\label{def:W_for_Gaussian_unitary}\nW=\\begin{bmatrix}\nX & Y \\\\\n\\ov{Y} & \\ov{X}\n\\end{bmatrix},\n\\]\n$\\boldsymbol{\\gamma}$ a complex vector of the form $\\boldsymbol{\\gamma}=(\\tilde{\\boldsymbol{\\gamma}},\\ov{\\tilde{\\boldsymbol{\\gamma}}})^T$, and $K$ is the matrix defined in Eq.~\\eqref{def:commutation_relation}. In the case that $W=0$, the Gaussian operator~\\eqref{def:Gaussian_unitary} corresponds to the~\\emph{Weyl displacement operator} $\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})$, while for $\\boldsymbol{\\gamma}=0$ we obtain other Gaussian transformations such as the phase-changing operator, one- and two-mode squeezing operators, or mode-mixing operators depending on the particular structure of $W$. For more details see section~\\ref{sec:list_of_Gaussian_unitaries}.\n\n\\subsection{Transformation of the first and the second moments}\nUnder the unitary channel~\\eqref{def:Gaussian_unitary} the first and the second moments transform according to rule\n\\[\\label{def:transformation}\n\\boldsymbol{d}'=S\\boldsymbol{d}+\\boldsymbol{b},\\ \\ \\sigma'=S\\sigma S^\\dag,\n\\]\nwhere, as we prove in appendix~\\ref{app:derivation_of_S_and_b},\n\\[\\label{eq:S_and_b}\nS=e^{iKW},\\ \\\n\\boldsymbol{b}=\\Big(\\!\\int_0^1e^{iKWt}\\mathrm{d}t\\!\\Big)\\ \\!\\boldsymbol{\\gamma}.\n\\]\nThe above identities together with transformation relations~\\eqref{def:transformation} are central for the effective description of Gaussian states. They allow us to transform the density matrix description of Gaussian states to the phase-space formalism, which immensely simplify every calculation. In the density matrix formalism, Gaussian states can be usually written only in terms of Taylor series in operators, while in the phase-space formalism they are represented by one vector and one matrix.\n\n\\subsection{Symplectic group and the Lie algebra}\nThe matrix $S$ from Eq.~\\eqref{eq:S_and_b}, called the \\emph{symplectic matrix}, has the same structure as $W$ and satisfies the relation\n\\[\\label{def:structure_of_S}\nS=\n\\begin{bmatrix}\n\\alpha & \\beta \\\\\n\\ov{\\beta} & \\ov{\\alpha}\n\\end{bmatrix},\\ \\ SKS^\\dag=K.\n\\]\nThese two properties define the complex representation of the real symplectic group $Sp(2N,\\mathbb{R})$. Note that transformations for which $\\beta=0$ are called passive, while transformation with $\\beta\\neq0$ are called active. This is because symplective matrix representing a passive transformation commutes with the total number operator, $\\hat{N}=\\sum_{i+1}^N\\hat{a}_i^\\dag\\hat{a}_i$, i.e., states before and after passive transformation contain the same mean number of bosons. In contrast, active transformation either create or annihilate particles.\n\nWhen describing quantum metrology on Gaussian states, the Lie algebra associated the symplectic group will prove to be very useful. The complex form of the Lie algebra associated with the real symplectic group $Sp(2N,\\mathbb{R})$ is defined by properties\n\\[\\label{def:P_1}\nP=\n\\begin{bmatrix}\nR & Q \\\\\n\\ov{Q} & \\ov{R}\n\\end{bmatrix},\\ \\ PK+KP^\\dag=0.\n\\]\nThe second property implies that $R$ is skew-Hermitian, $R^\\dag=-R$, and $Q$ is symmetric (and complex in general), $Q^T=Q$. Note that we used the definition of the Lie algebra more common to mathematical literature~\\cite{helgason1979differential}: for any $t\\in\\mathbb{R}$, $S=e^{Pt}$ is symplectic. Some authors~\\cite{Arvind1995a,deGosson2006a} define the Lie algebra by the property for any $t\\in\\mathbb{R}$, $S=e^{iPt}$ is symplectic matrix leading to $PK-KP^\\dag=0$. This is rather a cosmetic difference and does not affect any of the results of this thesis.\n\n\\begin{example}\n\\emph{\\emph{Properties of the symplectic group $Sp(2N,\\mathbb{R})$.}\n\\begin{itemize}[leftmargin=*]\n\\item The symplectic group $Sp(2N,\\mathbb{R})$ is connected, non-compact, simple Lie group.\n\\item Both defining properties~$\\eqref{def:structure_of_S}$ are necessary to define the real symplectic group. This group is a subgroup of the the more general pseudo-unitary group~\\cite{Mostafazadeh2004a} which is defined only by the second property, $U(N,N)=\\{S\\in GL(2N,\\mathbb{C})|SKS^\\dag=K\\}$.\n\\item Eqs.~\\eqref{def:structure_of_S} can be rewritten in two useful ways. The first one is actually identical to the definition of the Bogoliubov transformations used in the quantum field theory in curved space-time~\\cite{Birrell1984a},\n\\begin{subequations}\\label{def:Bogo_id}\n\\begin{align}\n\\alpha\\A^\\dag-\\beta\\B^\\dag&=I,\\label{def:first_Bogo_id}\\\\\n\\alpha\\beta^T&=\\beta\\alpha^T.\\label{def:second_Bogo_id}\n\\end{align}\n\\end{subequations}\nEq.~\\eqref{def:first_Bogo_id} implies that $\\alpha^{-1}$ always exists. We can therefore define $\\C:=\\alpha^{-1}\\beta$ and obtain even simpler form,\n\\begin{subequations}\n\\begin{align}\n\\alpha^{-1}(\\alpha^{-1})^\\dag+\\C\\C^\\dag&=I,\\\\\n\\C&=\\C^T.\n\\end{align}\n\\end{subequations}\n\\item Let $S$ be a symplectic matrix. $\\alpha$ is unitary $\\Leftrightarrow$ $\\beta=0$ $\\Leftrightarrow$ $S$ is unitary. \\emph{Proof:} $\\beta=0\\Rightarrow\\alpha$: follows trivially from Eq.~\\eqref{def:first_Bogo_id}. $\\beta=0\\Leftarrow\\alpha$: For unitary $\\alpha$ Eq.~\\eqref{def:first_Bogo_id} implies $\\beta^\\dag\\beta=0$. Sequence of implications follows:\n    $$\n    \\beta^\\dag\\beta=0\\ \\Rightarrow\\ \\forall\\ket{\\psi},\\ |\\!|\\beta\\ket{\\psi}|\\!|^2=\\bra{\\psi}\\beta^\\dag\\beta\\ket{\\psi}=0\\ \\Rightarrow\\ \\forall\\ket{\\psi},\\ \\beta\\ket{\\psi}=0\\ \\Rightarrow\\ \\beta=0.\n    $$\n    $\\beta=0$ $\\Leftrightarrow$ $S$ follows the same logic. \\qed\n\\item $\\mathrm{dim}(Sp(2N,\\mathbb{R}))=2N^2+N$. \\emph{Proof:} The dimension of a matrix group is the same as the dimension of the associated Lie algebra defined in Eq.~\\eqref{def:P_1}. Because number of real parameters needed to fully characterize the element of the Lie algebra $P$ is $N^2$ for the skew-hermitian matrix $R$ and $N^2+N$ for the symmetric matrix $Q$ respectively, the dimension of the Lie algebra is $2N^2+N$.\\qed\n\\item $\\mathrm{det}(S)=1$. \\emph{Proof:} A simple proof\\footnote{This simple proof is attributed to Jan Kohlrus.} involves transforming the symplectic matrix into the real form using appendix~\\ref{app:real_covariance}, and applying a property of the Pfaffian, $\\mathrm{pf}(S_\\Re\\Omega S_\\Re^T)=\\mathrm{pf}(\\Omega)\\mathrm{det}(S_\\Re)$. A complicated proof can be found in~\\cite{Arvind1995a}. \\qed\n\\item $S^T\\in Sp(2N,\\mathbb{R})$, $S^{-1}=KS^\\dag K\\in Sp(2N,\\mathbb{R})$, $K\\in Sp(2N,\\mathbb{R})$.\n\\item Any symplectic matrix can be decomposed using Euler's decomposition. For more details see section~\\ref{sec:parametrization_of_Gaussian_states}.\n\\end{itemize}}\n\\end{example}\n\nAll definitions and properties in this section can be of course rewritten in terms of the real representation of the real symplectic group. For details see appendix~\\ref{app:real_covariance} or~\\cite{Arvind1995a}.\n\n\\subsection{A list of Gaussian unitaries}\\label{sec:list_of_Gaussian_unitaries}\nIn this section we provide a list of basic Gaussian unitaries. We parametrize one-mode and two-mode Gaussian unitaries and provide their symplectic matrices.\n\nThe simplest Gaussian unitary which acts only on the displacement vector and leave the covariance matrix invariant is previously mentioned displacement operator \\[\\label{eq:Weyl_displacement_operator}\n\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})=\\exp\\big(\\boldsymbol{\\hat{A}}^\\dag K \\boldsymbol{\\gamma}\\big),\n\\]\nwhere $\\boldsymbol{\\gamma}=(\\tilde{\\boldsymbol{\\gamma}},\\ov{\\tilde{\\boldsymbol{\\gamma}}})^T$. According to Eq.~\\eqref{def:transformation}, this operator acts as $\\boldsymbol{d}'=\\boldsymbol{\\gamma}$, $\\sigma'=\\sigma$. Other transformations described in this section will characterized by $\\boldsymbol{\\gamma}=\\boldsymbol{0}$ and will act as $\\boldsymbol{d}'=S\\boldsymbol{d}$, $\\sigma'=S\\sigma S^\\dag$.\n\nFor one-mode states ($N=1$), the Hermitian matrix $W$ in Eq.~\\eqref{def:W_for_Gaussian_unitary} describing the Gaussian unitary can be fully parametrized as\n\\[\\label{eq:W1}\nW=\\begin{bmatrix}\n-\\theta & i r e^{i \\chi} \\\\\n-i r e^{-i \\chi} & -\\theta\n\\end{bmatrix}.\n\\]\nFor $r=0$ and $\\boldsymbol{\\gamma}=\\boldsymbol{0}$ the Gaussian unitary~\\eqref{def:Gaussian_unitary} represents a one-mode phase-shift $\\hat{R}(\\theta)=\\exp(-i\\theta\\hat{a}^\\dag \\hat{a})$. We will denote its correspondent symplectic matrix derived using Eq.~\\eqref{eq:S_and_b} as $S=R(\\theta)$. Choosing $\\theta=0$ instead, we obtain one-mode squeezing at angle $\\chi$, $\\hat{S}(r,\\chi)=\\exp(-\\frac{r}{2}(e^{i\\chi}\\hat{a}^{\\dag2}-e^{-i\\chi}\\hat{a}^{2}))$. Squeezing at angle zero will be denoted as $\\hat{S}(r)$ and its symplectic matrix equivalent will be denoted as $S(r)$.\n\nIn the analogy with one-mode Gaussian channels, for two-mode states ($N=2$) we parametrize the Hermitian matrix $W$ as\n\\[\\label{eq:W2}\nW=\\begin{bmatrix}\n-\\theta_1 & -i\\theta_{B}e^{i \\chi_{B}}  & i r_1 e^{i \\chi_1} & i r_{T} e^{i \\chi_{T}} \\\\\ni\\theta_{B}e^{-i \\chi_{B}} & -\\theta_2 & i r_{T} e^{i \\chi_{T}} & i r_2 e^{i \\chi_2} \\\\\n-i r_1 e^{-i \\chi_1} & -i r_{T} e^{-i \\chi_{T}} & -\\theta_1 & i\\theta_{B}e^{-i \\chi_{B}}\\\\\n-i r_{T} e^{-i \\chi_{T}} & -i r_2 e^{-i \\chi_2} & -i\\theta_{B}e^{i \\chi_{B}} & -\\theta_2\n\\end{bmatrix}.\n\\]\nSetting all parameters apart from $\\theta_1$ to zero, the Gaussian unitary~\\eqref{def:Gaussian_unitary} represents the one-mode phase-shift operator $\\hat{R}_1(\\theta_1)=\\exp(-i\\theta_1\\hat{a}_1^\\dag \\hat{a}_1)$, and we write $S=R_1(\\theta_1)$. Similarly, for $\\theta_2$ we have $S=R_2(\\theta_2)$. Setting all parameters apart from $\\theta_{B}$ and $\\chi_{B}$ to zero, we obtain the general mode-mixing channel $\\hat{B}(\\theta_{B},\\chi_{B})=\\exp(\\theta_{B}(e^{i\\chi_{B}}\\hat{a}_1^\\dag\\hat{a}_2-e^{-i\\chi_{B}}\\hat{a}_2^\\dag\\hat{a}_1))$, where $\\chi_{B}$ represents the angle of mode-mixing. For $\\chi_{B}=0$ we obtain the usual beam-splitter with transmissivity $\\tau=\\cos^2\\theta_{B}$, denoted $\\hat{B}(\\theta_{B})$. Following the same logic, parameters $r_1$ and $r_2$ represent the one-mode squeezing of the first and the second mode as defined in the previous section, denoted $\\hat{S}_1(r_1,\\chi_1)$, $\\hat{S}_2(r_2,\\chi_2)$, and parameter $r_{T}$ represents the two-mode squeezing at angle $\\chi_{T}$, $\\hat{S}_{T}(r_{T},\\chi_{T})=\\exp(-r_{T}(e^{i\\chi_{T}}\\hat{a}_1^\\dag\\hat{a}_2^\\dag-e^{-i\\chi_{T}}\\hat{a}_1\\hat{a}_2))$.\n\nMulti-mode channels ($N\\geq3$) can be obtained generalizing the same parametrization which has been used in~\\eqref{eq:W2}. This essentially means there are not any other Gaussian unitary operators other than phase-changing, mode-mixing and single- and two-mode squeezing channels and their combinations. Number of parameters needed for fully parametrize a Gaussian unitary is $2N^2+N$ for the symplectic matrix and $2N$ for the displacement vector, thus $2N^2+3N$ in total.\n\n\\subsubsection{The phase-space representation of Gaussian unitaries}\\label{sec:The_phase_space_Gaussian_unitaries}\n\nNow we provide a list of the introduced Gaussian unitaries in both the complex or the real form matrices defined by Eq.~\\eqref{def:covariance_matrix}, Eq.~\\eqref{def:covariance_matrix_real} respectively. Symplectic matrices in most other commonly used notations are obtained by rearranging some rows and columns of either complex or the real form of the symplectic matrix. For example, the symplectic matrix in the real form~\\eqref{def:covariance_matrix_real} given by `$xxpp$' vector transforms into `$xpxp$' form given by $\\boldsymbol{\\hat{Q}}:=(\\hat{x}_1,\\hat{p}_1,\\hat{x}_2,\\hat{p}_2)^T$ as\n\\[\nS_\\Re=\\begin{bmatrix}\nS_{x_1x_1} & S_{x_1x_2}  & S_{x_1p_1} & S_{x_1p_2} \\\\\nS_{x_2x_1} & S_{x_2x_2} & S_{x_2p_1} & S_{x_2p_2} \\\\\nS_{p_1x_1} & S_{p_1x_2} & S_{p_1p_1} & S_{p_1p_2} \\\\\nS_{p_2x_1} & S_{p_2x_2} & S_{p_2p_1} & S_{p_2p_2}\n\\end{bmatrix}\\ \\longrightarrow\\\nS_{\\Re,xpxp}=\\begin{bmatrix}\nS_{x_1x_1} & S_{x_1p_1}  & S_{x_1x_2} & S_{x_1p_2} \\\\\nS_{p_1x_1} & S_{p_1p_1} & S_{p_1x_2} & S_{p_1p_2} \\\\\nS_{x_2x_1} & S_{x_2p_1} & S_{x_2x_2} & S_{x_2p_2} \\\\\nS_{p_2x_1} & S_{p_2p_1} & S_{p_2x_2} & S_{p_2p_2}\n\\end{bmatrix}\n\\]\nIn addition, it is often convenient to consider one-mode operations acting on a multi-mode state. One-mode operations which leave the other modes invariant are easily lifted into multi-mode operations by adding identities onto suitable places as illustrated on Eq.~\\eqref{eq:phase_operator}.\\\\\n\n\\noindent\n\\emph{Rotation\/phase-change} $\\hat{R}(\\theta)=\\exp(-i\\theta\\hat{a}^\\dag \\hat{a})$, $\\hat{R}_1(\\theta)=\\exp(-i\\theta\\hat{a}_1^\\dag \\hat{a}_1)$,\n\\begin{flalign}\\label{eq:phase_operator}\nR(\\theta)&=\\begin{bmatrix}\ne^{-i\\theta} & 0 \\\\\n0 & e^{i\\theta}\n\\end{bmatrix},\n\\quad R_\\Re(\\theta)=\\begin{bmatrix}\n\\cos\\theta & \\sin\\theta \\\\\n-\\sin\\theta & \\cos\\theta\n\\end{bmatrix},&&\\\\\n\\quad R_1(\\theta)&=\\begin{bmatrix}\ne^{-i\\theta} & 0  & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & e^{i\\theta} & 0 \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix},\n\\quad\nR_{1\\Re}(\\theta)=\\begin{bmatrix}\n\\cos\\theta & 0  & \\sin\\theta & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n-\\sin\\theta & 0 & \\cos\\theta & 0 \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}.&&\\nonumber\n\\end{flalign}\n\\emph{One-mode squeezing} $\\hat{S}(r,\\chi)=\\exp(-\\frac{r}{2}(e^{i\\chi}\\hat{a}^{\\dag2}-e^{-i\\chi}\\hat{a}^{2}))$,\n\\begin{flalign}\\label{eq:squeezing_operator}\nS(r,\\chi)&=\\begin{bmatrix}\n\\cosh r & -e^{i\\chi}\\sinh r \\\\\n-e^{-i\\chi}\\sinh r & \\cosh r\n\\end{bmatrix},\\\\\n\\quad S_\\Re(r,\\chi)&=\\begin{bmatrix}\n\\cosh r-\\cos\\chi\\sinh r & -\\sin\\chi\\sinh r \\\\\n-\\sin\\chi\\sinh r & \\cosh r+\\cos\\chi\\sinh r\n\\end{bmatrix}.&&\n\\end{flalign}\n\\emph{Mode-mixing} $\\hat{B}(\\theta,\\chi)=\\exp(\\theta(e^{i\\chi}\\hat{a}_1^\\dag\\hat{a}_2-e^{-i\\chi}\\hat{a}_2^\\dag\\hat{a}_1))$,\n\\begin{flalign}\\label{eq:mode_mixing_operator}\nB(\\theta,\\chi)&=\\begin{bmatrix}\n\\cos\\theta & e^{i\\chi}\\sin\\theta  & 0 & 0 \\\\\n-e^{-i \\chi}\\sin\\theta & \\cos\\theta & 0 & 0 \\\\\n0 & 0 & \\cos\\theta & e^{-i \\chi}\\sin\\theta \\\\\n0 & 0 & -e^{i \\chi}\\sin\\theta & \\cos\\theta\n\\end{bmatrix},&&\\\\\n\\quad B_\\Re(\\theta,\\chi)&=\\begin{bmatrix}\n\\cos\\theta & \\cos\\chi\\sin\\theta & 0 & -\\sin\\chi\\sin\\theta \\\\\n-\\cos\\chi\\sin\\theta & \\cos\\theta & -\\sin\\chi\\sin\\theta & 0 \\\\\n0 & \\sin\\chi\\sin\\theta & \\cos\\theta & \\cos\\chi\\sin\\theta\\\\\n\\sin\\chi\\sin\\theta & 0 & -\\cos\\chi\\sin\\theta & \\cos\\theta\n\\end{bmatrix}.&&\\nonumber\n\\end{flalign}\n\\emph{Two-mode squeezing} $\\hat{S}_{T}(r,\\chi)=\\exp(-r(e^{i\\chi}\\hat{a}_1^\\dag\\hat{a}_2^\\dag-e^{-i\\chi}\\hat{a}_1\\hat{a}_2))$,\n\\begin{flalign}\\label{eq:twomode_squeezing_operator}\nS_T(r,\\chi)&=\\begin{bmatrix}\n\\cosh r & 0  & 0 & -e^{i\\chi}\\sinh r \\\\\n0 & \\cosh r & -e^{i\\chi}\\sinh r & 0 \\\\\n0 & -e^{-i\\chi}\\sinh r & \\cosh r & 0 \\\\\n-e^{-i\\chi}\\sinh r & 0 & 0 & \\cosh r\n\\end{bmatrix},&&\\\\\nS_{T\\Re}(r,\\chi)&=\\begin{bmatrix}\n\\cosh r & -\\cos \\chi\\sinh r  & 0 & -\\sin \\chi\\sinh r \\\\\n-\\cos \\chi\\sinh r & \\cosh r & -\\sin \\chi\\sinh r & 0 \\\\\n0 & -\\sin \\chi\\sinh r & \\cosh r & \\cos \\chi\\sinh r \\\\\n-\\sin \\chi\\sinh r & 0 & \\cos \\chi\\sinh r & \\cosh r\n\\end{bmatrix}.&&\\nonumber\n\\end{flalign}\n\n\\section{Common Gaussian states}\\label{sec:common_Gaussian_states}\nIn this section we introduce the most common Gaussian states. As we will see in section~\\eqref{sec:parametrization_of_Gaussian_states}, characteristics of all other Gaussian states are mixtures of characteristics of these basic ones. In that sense the following list is complete.\n\n\\subsection{Thermal state}\\label{sec:thermal_state}\nThe simplest Gaussian state is the thermal state. Assuming the single particle Hilbert space is spanned by $N$ states -- modes, each mode is characterized by the energy $E_i$ of the state $\\ket{\\psi_i}$. We assume that each mode is thermally populated, i.e., number of particles in each mode is given by the thermal distribution, $\\hat{\\rho}_{{\\mathrm{th}}i}=\\frac{1}{Z}\\mathrm{exp}(-\\frac{E_i}{kT}\\hat{n}_i)$, where $\\hat{n}_i=\\hat{a}_i^\\dag\\hat{a}_i$ denotes the number operator associated with mode $i$, $k$ is the Boltzmann constant, and $Z=\\mathrm{tr}[e^{-\\frac{E_i}{kT}\\hat{n}_i}]$ defines the partition function\nThe full thermal state is then a tensor product of the thermal states of each mode, $\\hat{\\rho}_{\\mathrm{th}}=\\hat{\\rho}_{\\mathrm{th}1}\\otimes\\cdots\\otimes\\hat{\\rho}_{\\mathrm{th}N}$. The displacement vector of the thermal state is equal to zero and the covariance matrix in both complex and the real form is a diagonal matrix,\n\\[\n\\boldsymbol{d}=\\boldsymbol{0}, \\quad \\sigma_{\\mathrm{th}}=\\mathrm{diag}(\\lambda_1,\\dots,\\lambda_N,\\lambda_1,\\dots,\\lambda_N).\n\\]\n$\\lambda_i=\\coth(\\frac{E_i}{2kT})$ are called symplectic eigenvalues for reasons described in the next section. They can be also expressed in terms of the mean number of thermal bosons, $\\lambda_i=1+2n_{{\\mathrm{th}}i}$, where $n_{{\\mathrm{th}}i}:=\\mathrm{tr}[\\hat{n}_i\\hat{\\rho}_{\\mathrm{th}}]$.\nLarger temperatures and smaller energies correspond to larger symplectic eigenvalues. For each $i$, $\\lambda_i\\geq1$ and $\\lambda_i=1$ for $T=0$. Thermal state corresponding to $T=0$ is the lowest-energy state called vacuum and is described by the identity matrix $\\sigma=I$.\n\n\\subsection{Coherent state}\nA Gaussian state characterized only by its displacement vector is the coherent state,\n\\[\n\\ket{\\alpha}=e^{-\\frac{\\abs{\\alpha}^2}{2}}\\sum_{n=0}^\\infty \\frac{\\alpha^n}{\\sqrt{n!}}\\ket{n}.\n\\]\nCoherent state is an eigenvector of the annihilation operator, $a\\ket{\\alpha}=\\alpha\\ket{\\alpha}$. Coherent states typically describe beams of light emitted by a laser~\\cite{zhang1990coherent}. Mathematically, coherent state can be created by the action of the Weyl displacement operator~\\eqref{eq:Weyl_displacement_operator} on the vacuum (thus an equivalent name would be a single-mode displaced vacuum), $\\ket{\\alpha}=\\hat{D}(\\alpha)\\ket{0}$. The first and the second moments can be easily derived using Eq.~\\eqref{def:transformation},\n\\[\n\\boldsymbol{d}=(\\alpha,\\ov{\\alpha})^T, \\quad \\sigma=I.\n\\]\n\n\\subsection{Single-mode squeezed state}\nSqueezed state is created by an action of the squeezing operator~\\eqref{eq:squeezing_operator} on the vacuum, $\\ket{S(r,\\chi)}=S(r,\\chi)\\ket{0}$. For $\\chi=0$ this state takes the form~\\cite{kok2010introduction}\n\\[\n\\ket{S(r)}=\\frac{1}{\\sqrt{\\cosh\\abs{r}}}\\sum_{n=0}^\\infty\\frac{\\sqrt{(2n)!}}{n!}\\left(\\frac{-r}{2\\abs{r}}\\right)^n\\tanh^n\\abs{r}\\ket{2n}\n\\]\nSuch states for example from a laser light by going through an optical parametric oscillator~\\cite{breitenbach1997measurement,Lvovsky2014squeezed}.\nThe first and the second moments are\n\\[\n\\boldsymbol{d}=\\boldsymbol{0}, \\quad \\sigma=S(r,\\chi)S^\\dag(r,\\chi)=S(2r,\\chi).\n\\]\n\n\\subsection{Two-mode squeezed state}\nTwo-mode squeezed states are entangled two-mode states created by an action of the two mode squeezing operator~\\eqref{eq:twomode_squeezing_operator} on the vacuum, $\\ket{S_T(r,\\chi)}=S_T(r,\\chi)\\ket{0}$. For $\\chi=0$ this state takes the form~\\cite{kok2010introduction}\n\\[\n\\ket{S_T(r)}=\\frac{1}{\\cosh\\abs{r}}\\sum_{n=0}^\\infty\\left(\\frac{-r}{\\abs{r}}\\right)^n\\tanh^n\\abs{r}\\ket{n,n}\n\\]\nPhysically, two-mode squeezed states are prepared by sending squeezed and anti-squeezed state (squeezed with the negative squeezing) through a beam-splitter. The first and the second moments are\n\\[\n\\boldsymbol{d}=\\boldsymbol{0}, \\quad \\sigma=S_T(r,\\chi)S_T^\\dag(r,\\chi)=S_T(2r,\\chi).\n\\]\n\n\\begin{example}\n\\emph{\nIt is easy to show that tracing over one-mode of a two-mode squeezed state $\\ket{S_T(r)}$ leaves us with a thermal state.\n}\n\\end{example}\n\n\\section{Number of particles in a Gaussian state}\nFor some applications it is useful to know the mean number of particles in a Gaussian state or the mean energy of a Gaussian state. For example, in quantum metrology we are usually interested how well the sensitivity of a Gaussian probe state scales with its energy. Calculating this quantity is very simple when using the complex form of the covariance matrix. Defining the mean number of particles in mode $i$, $0\\leq i\\leq N$, as $n_i:=\\mathrm{tr}[\\hat{a}_i^\\dag \\hat{a}_i\\hat{\\rho}]$ we can use the definition of the covariance matrix~\\eqref{def:covariance_matrix} and the commutation relations~\\eqref{def:commutation_relation} to derive\n\\[\nn_i=\\frac{1}{2}\\big(\\sigma^{ii}+2\\ov{d}^id^i-1\\big).\n\\]\nThe mean energy of the probe state is then $\\mean{E}=\\sum_{i=1}^Nn_iE_i$ where $E_i$ is the energy of a particle in mode $i$. The mean number of particles in a Gaussian state can be calculated as\n\\[\\label{eq:mean_number_of_particles}\nn:=\\sum_{i=1}^Nn_i=\\frac{1}{2}\\left(\\frac{1}{2}\\mathrm{tr}[\\sigma]+\\boldsymbol{d}^\\dag \\boldsymbol{d}-N\\right).\n\\]\n\n\\section{Williamson's decomposition of the covariance matrix}\\label{sec:Williamson's_decomposition}\n\nIn the section~\\ref{sec:common_Gaussian_states} we have illustrated that covariance matrices can be constructed by applying symplectic matrices on diagonal matrix. In this section will show that the opposite is also true. We introduce a theorem which is crucial for understanding structure of Gaussian states, and which will be later used for a parametrization of Gaussian states. This is the Williamson's decomposition of the positive definite matrices. According to the Williamson's theorem~\\cite{Williamson1936a,deGosson2006a,Simon1998a}, any positive-definite matrix can be diagonalized by symplectic matrices of the form introduced in Eq.~\\eqref{def:structure_of_S},\n\\[\\label{def:Williamson's_decomposition}\n\\sigma=SDS^\\dag.\n\\]\n$D$ is the diagonal matrix consisting of \\emph{symplectic eigenvalues},\\\\\n$D=\\mathrm{diag}(\\lambda_1,\\dots,\\lambda_N,\\lambda_1,\\dots,\\lambda_N)$.\n\nSymplectic eigenvalues can be found by solving the usual eigenvalue problem for the matrix\n\\[\\label{def:A}\nA:=K\\sigma,\n\\]\nwhere $K$ is the symplectic form defined by commutation relations~\\eqref{def:commutation_relation}. Eigenvalues of $A$ always appear in pairs. If $\\lambda_i$ is an eigenvalue of $A$, then also $-\\lambda_i$ is an eigenvalue of the same operator. The symplectic spectrum is then defined as a collection of the positive eigenvalues of $A$. In other words, $\\lambda_i$ is a symplectic eigenvalue of $\\sigma$ if and only if it is positive and $\\pm\\lambda_i$ are the eigenvalues of the operator $A$. Combining Eqs.~\\eqref{def:structure_of_S} and~\\eqref{def:Williamson's_decomposition} we find $\\mathrm{tr}[A]=\\mathrm{tr}[A^3]=0$. This together with the expansion of determinant gives analytical formulae for the symplectic eigenvalues of a single mode Gaussian state,\n\\[\\label{eq:singlemode_symplectic_eigenvalue}\n\\lambda=\\sqrt{\\mathrm{det}(A)}=\\frac{1}{\\sqrt{2}}\\sqrt{\\mathrm{tr}[A^2]},\n\\]\nand of a two-mode Gaussian state,\n\\[\\label{eq:twomode_symplectic_eigenvalues}\n\\begin{split}\n\\lambda_{1,2}&=\\frac{1}{2}\\sqrt{\\mathrm{tr}[A^2]\\pm\\sqrt{(\\mathrm{tr}[A^2])^2-16\\mathrm{det}(A)}}\\\\\n&=\\frac{1}{2}\\sqrt{\\mathrm{tr}[A^2]\\pm\\sqrt{4\\mathrm{tr}[A^4]-(\\mathrm{tr}[A^2])^2}}.\n\\end{split}\n\\]\nDiagonalizing symplectic matrices $S$ can be found for example by a method described in~\\cite{Simon1998a}.\n\nWe have seen in the previous section that the thermal state was represtented only by symplectic eigenvalues. These eigenvalues were connected with purity of the state. Temperature equal to zero -- symplectic eigenvalues equal to one -- results in vacuum, which is a pure state. Higher temperature lead to a mixed state. But that is true not only for a thermal state, but any Gaussian state. Every symplectic eigenvalue of a Gaussian state is larger than one, $\\lambda_i\\geq1$, which is a consequence of Eq.~\\eqref{eq:sigma_K_positivity}. Purity of a Gaussian state can be calculated as\n\\[\n\\mu(\\hat{\\rho})=\\prod_{i=1}^{N}\\lambda_{i}^{-1}=\\frac{1}{\\sqrt{\\mathrm{det}(A)}}.\n\\]\nA Gaussian state is pure if all symplectic eigenvalues are equal to one. We say that mode $i$ is pure if $\\lambda_i=1$.\n\n\\section{Parametrization of Gaussian states}\\label{sec:parametrization_of_Gaussian_states}\n\nIn this section we will use the Williamson's decomposition to fully parametrize Gaussian states of a given number of modes. But to do that, we need to fully parametetrize symplectic matrices first. Any symplectic matrix~\\eqref{def:structure_of_S} can be decomposed using Euler's decomposition~\\cite{Arvind1995a,Weedbrook2012a} as\n\\[\\label{def:S_decomposition}\nS=\n\\begin{bmatrix}\nU_1 & 0 \\\\\n0 & \\ov{U}_1\n\\end{bmatrix}\n\\begin{bmatrix}\n\\cosh{M_{\\boldsymbol{r}}} & -\\sinh{M_{\\boldsymbol{r}}} \\\\\n-\\sinh{M_{\\boldsymbol{r}}} & \\cosh{M_{\\boldsymbol{r}}}\n\\end{bmatrix}\n\\begin{bmatrix}\nU_2 & 0 \\\\\n0 & \\ov{U}_2\n\\end{bmatrix},\n\\]\nwhere $U_1$ and $U_2$ denote unitary matrices, and $M_{\\boldsymbol{r}}=\\mathrm{diag}(r_1,\\dots,r_N)$ is the diagonal matrix of the squeezing parameters. This shows that any symplectic matrix can be decomposed into two passive operations and one active, which is consisted of single mode squeezers. This is important from an experimental point of view because that means there does not need to be any direct two mode squeezing operation as long as there are single mode squeezers and beam splitters.\n\nWith a full parametrization of unitary matrices $U_1$ and $U_2$, one can use this decomposition to fully parametrize the covariance matrix via Eq.~\\eqref{def:Williamson's_decomposition}. Moreover, since the displacement vector is fully parametrized by its elements, we have a full parametrization of Gaussian states. Note, however, that some parameters may not add any additional complexity and can be removed. This is a consequence of the fact that in Eq.~\\eqref{def:Williamson's_decomposition} some parts of (the decomposition of) $U_2$ vanish, because they commute with the diagonal matrix $\\mathrm{diag}(\\lambda_1,\\dots,\\lambda_N)$. Since the parametrizations of unitary matrices up to $N=3$ are known, we can explicitly write the most general single-, two-, and three-mode Gaussian states.\n\nThe most general one-mode Gaussian state is the one-mode squeezed rotated displaced thermal state~\\cite{Weedbrook2012a},\n\\[\\label{eq:general_1mode_state}\n\\hat{\\rho}_0=\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})\\hat{R}(\\theta)\\hat{S}(r)\\rho_{\\mathrm{th}}(\\lambda)\\hat{S}^\\dag(r)\\hat{R}^\\dag(\\theta)\\hat{D}^\\dag(\\tilde{\\boldsymbol{\\gamma}}),\n\\]\nwhere the variable in the Weyl displacement operator $\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})$ is of the form $\\tilde{\\boldsymbol{\\gamma}}=\\norm{{d}}e^{i\\phi_d}$. The first and the second moments of this state are\n\\[\n\\boldsymbol{d}=(\\tilde{\\boldsymbol{\\gamma}},\\ov{\\tilde{\\boldsymbol{\\gamma}}})^T,\\quad \\sigma=R(\\theta)S(r)D(\\lambda)S(r)^\\dag R^\\dag(\\theta),\n\\]\nwhere $D(\\lambda)=\\mathrm{diag}(\\lambda,\\lambda)$.\n\nApplying the parametrization of the general $2\\times2$ unitary matrix to Eq.~\\eqref{def:S_decomposition} we find the most general two-mode Gaussian state,\n\\[\\label{eq:general_2mode_state}\n\\begin{split}\n\\hat{\\rho}_0=&\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})\\hat{R}_1(\\phi_1)\\hat{R}_2(\\phi_2)\\hat{B}(\\theta_2)\\hat{R}_{\\mathrm{as}}(\\psi_2)\\hat{S}_1(r_1)\\hat{S}_2(r_2)\\\\\n&\\hat{R}_{\\mathrm{as}}(\\psi_1)\\hat{B}(\\theta_1)\\hat{\\rho}_{\\mathrm{th}}(\\lambda_1,\\lambda_2)(\\ \\cdot\\ )^\\dag,\n\\end{split}\n\\]\nwhere we define $\\hat{R}_{\\mathrm{as}}(\\psi):=\\hat{R}_1(\\psi)\\hat{R}_2(-\\psi)$ and $\\tilde{\\boldsymbol{\\gamma}}=(\\norm{{d}_1}e^{i\\phi_{d1}},\\norm{{d}_2}e^{i\\phi_{d2}})$. The displacement vector and the covariance matrix are obtained in analogy to the single mode state by removing `hats' while the displacement operator affects only the displacement vector, $\\boldsymbol{d}_0=(\\tilde{\\boldsymbol{\\gamma}},\\ov{\\tilde{\\boldsymbol{\\gamma}}})^T$.\n\nA general $3\\times3$ unitary matrix can be fully parametrized using the $C\\!K\\!M$ matrix (Cabibbo--Kobayashi--Maskawa~\\cite{chau1984comments}). Assuming $\\hat{B}_{ij}(\\theta,\\chi)$ is the mode-mixing operation between modes $i$ and $j$ (3-mode generalizations of Eq.~\\eqref{eq:mode_mixing_operator}), $\\hat{B}_{ij}(\\theta):=\\hat{B}_{ij}(\\theta,0)$ the beam-splitter operation respectively, we can define $C\\!K\\!M$ operator as\n\\[\n\\hat{C\\!K\\!M}(\\theta_1,\\theta_2,\\theta_3,\\chi_1):=\\hat{B}_{23}(\\theta_1)\\hat{B}_{13}(\\theta_2,\\chi_1)\\hat{B}_{12}(\\theta_3).\n\\]\nWe also denote a collection of single mode rotations and a collection single mode squeezers as\n\\begin{subequations}\n\\begin{align}\n\\hat{R}(\\phi_1,\\phi_2,\\phi_3)&:=\\hat{R}_1(\\phi_1)\\hat{R}_2(\\phi_2)\\hat{R}_3(\\phi_3),\\\\ \\hat{S}(r_1,r_2,r_3)&:=\\hat{S}_1(r_1)\\hat{S}_2(r_2)\\hat{S}_3(r_3).\n\\end{align}\n\\end{subequations}\nThe most general three-mode Gaussian state is\n\\[\n\\begin{split}\n\\hat{\\rho}_0=&\\hat{D}(\\tilde{\\boldsymbol{\\gamma}})\\hat{R}(1,\\phi_1,\\phi_2)\\hat{C\\!K\\!M}(\\theta_1,\\theta_2,\\theta_3,\\chi_1)\\hat{R}(\\phi_3,\\phi_4,\\phi_5)\\hat{S}(r_1,r_2,r_3)\\\\\n&\\hat{R}(1,\\phi_6,\\phi_7)\\hat{C\\!K\\!M}(\\theta_4,\\theta_5,\\theta_6,\\chi_2)\\hat{\\rho}_{\\mathrm{th}}(\\lambda_1,\\lambda_2,\\lambda_3)(\\ \\cdot\\ )^\\dag,\n\\end{split}\n\\]\nwhere $\\tilde{\\boldsymbol{\\gamma}}=(\\norm{{d}_1}e^{i\\phi_{d1}},\\norm{{d}_2}e^{i\\phi_{d2}},\\norm{{d}_3}e^{i\\phi_{d3}})$.\n\nThe number of parameters $\\#(N)$ needed to fully parametrize $N$-mode Gaussian states is $5$ for a one-mode state, $14$ for a two-mode state, and $27$ for a three-mode state. In general the following formula holds,\n\\[\\label{eq:number_of_parameters}\n\\#(N)=2N^2+3N.\n\\]\nInterestingly, this means that number of parameters needed to fully parametrize a Gaussian state is the same as the number of parameters needed to fully parametrize a Gaussian unitary~\\eqref{def:Gaussian_unitary}. We can prove this expression by studying properties of the displacement and the covariance matrix~\\eqref{def:first_and_second_moments}. Because the covariance matrix is a Hermitian matrix its sub-block $X$ is also a Hermitian matrix and its sub-block $Y$ is ad (complex) symmetric matrix. But Hermitian matrices of size $N\\times N$ are fully parametrized by $N^2$ parameters and symmetric matrices are fully parametrized by $N^2+N$ parameters, i.e., the covariance matrix is fully parametrized by $2N^2+N$ parameters. The displacement vector is parametetrized by $N$ absolute values of the displacement and $N$ phases. Summed up, this gives Eq.~\\eqref{eq:number_of_parameters}.\n\nPure Gaussian states are characterized by a significantly smaller number of parameters,\n\\[\\label{eq:number_of_parameters_pure}\n\\#_{\\mathrm{pure}}(N)=N^2+3N.\n\\]\nThis comes from the the Euler's decomposition~\\eqref{def:S_decomposition} of the symplectic matrix and the Williamson's decomposition of the covariance matrix~\\eqref{def:Williamson's_decomposition}. In case of pure states all symplectic eigenvalues are equal to one, and the unitary matrix $U_2$ commutes with the diagonal matrix representing the vacuum state. Therefore, we have $N^2$ parameters needed to parametrize the unitary matrix $U_1$, $N$ squeezing parameters, and $2N$ parameters of the displacement vector. Summed up, this gives Eq.~\\eqref{eq:number_of_parameters_pure}.\n\n\\section{State-of-the-art quantum metrology in the phase-space formalism}\\label{sec:state_quantum_metrology}\n\nIn this section we review state-of-the-art methods of quantum metrology in the phase-space formalism.\n\nIn the first chapter we introduced several formulae for the quantum Fisher information. However, expressions introduced there were only for states represented by a density matrix. On the other hand, as we illustrated in section~\\ref{sec:common_Gaussian_states}, density matrices of Gaussian states can be usually expressed only in terms of relatively complicated infinite series. This is why calculating the quantum Fisher information -- the figure of merit of the local quantum estimation -- has been quite a difficult task for Gaussian states until recently. This has changed when new expressions using the phase-space formalism have been derived.\n\nThe first leap in deriving general formulae has been taken by Pinel et al.~\\cite{Pinel2012a}, who found an expression for the quantum Fisher information for pure states, i.e., for the states which are pure at point $\\epsilon$ and remain pure even if the $\\epsilon$ slightly changes. The same year Marian and Marian found the formula for the fidelity between one-mode and two-mode Gaussian states~\\cite{Marian2012a}, which allowed for the derivation of the general formula for the one-mode state~\\cite{Pinel2013b}. Also, Spedalieri et al.~found a formula for the fidelity between one pure and one mixed Gaussian state~\\cite{Spedalieri2013a}, from which one can derive a slightly more general formula for pure states, i.e., for the states which are pure at the point $\\epsilon$ but the small change in $\\epsilon$ introduces impurity. A different path has been followed by Monras~\\cite{Monras2013a}, who connected the quantum Fisher information to the solution of the so-called Stein equation. Using this approach, he derived the quantum Fisher information for a generalization of the pure states called iso-thermal states, and a general formula for any multi-mode Gaussian state in terms of an infinite series. Using the previous result, Jiang derived a formula~\\cite{Jiang2014a} for the Gaussian states in exponential form and simplified a known formula for pure states. Quite recently, Gao and Lee derived an exact formula~\\cite{Gao2014a} for the quantum Fisher information for the multi-mode Gaussian states in terms of the inverse of certain tensor products, elegantly generalizing the previous results, however with some possible drawbacks, especially in the necessity of inverting relatively large matrices. The last result from Banchi et al.~\\cite{Banchi2015a} provides a very elegant expression for the quantum Fisher information for multi-mode Gaussian states written in terms of inverses of certain super-operators.\n\nThe original results has been been published in many different notations. We translate all of them into the complex form, although we mention the real form version in some examples. In the following, we will give details on these results which relate to our work introduced later in the thesis.\n\nThe simplest case of a Gaussian state is a single mode Gaussian state. Making the identification $\\hat{\\rho}_{1}\\rightarrow(\\boldsymbol{d}_{1},\\sigma_{1})$ the Uhlmann fidelity between two one-mode states~\\cite{Marian2012a} is given by\n\\[\n\\mathcal{F}_{1}(\\hat{\\rho}_{1},\\hat{\\rho}_{2})\\,=\\,2\\,\\frac{e^{-(\\boldsymbol{d}_1-\\boldsymbol{d}_2)^{\\dag}(\\sigma_{1}+\\sigma_{2})^{-1}(\\boldsymbol{d}_1-\\boldsymbol{d}_2)}}{\\sqrt{\\Delta+\\Lambda}-\\sqrt{\\Lambda}},\n\\]\nwhere $\\Delta=\\det{{\\sigma}_{1}+{\\sigma}_{2}}$, $\\Lambda=\\det{{\\sigma}_{1}+{K}}\\det{{\\sigma}_{2}+{K}}$, and $\\det{\\cdot}:=\\mathrm{det}[\\cdot]$ denotes determinant. One can use this formula and the connection between the Uhlmann fidelity and the Quantum Fisher information~\\eqref{QFI_using_fidelity}, expand the determinants in the small parameter $\\mathrm{d}\\epsilon$, and derive the quantum Fisher information for a single mode state~\\cite{Pinel2013b}\n\\[\\label{eq:one_mode_quantum_fisher_information}\nH(\\epsilon)=\\frac{1}{2}\\frac{\\mathrm{tr}\\Big[\\big(A^{-1}\\dot{A}\\big)^{2}\\Big]}{1+\\det{A}^{-1}}\n+\\frac{1}{2}\\,\\frac{\\det{A}^{-1}\\mathrm{tr}[A^{-1}{\\dot{A}}]^{2}}{1-\\det{A}^{-2}}+2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}},\n\\]\nwhere $A:=K\\sigma$ for the complex form.\n\nA very elegant expression for the quantum Fisher information can be derived for pure states. Taking a different approach for finding this quantity -- solving equations for the symmetric logarithmic derivative~\\eqref{def:SLDsolution} -- has been taken in~\\cite{Monras2013a}. This equation translate into the Stein equation\\footnote{Stein equation for $X$, $X-F X F^\\dag=W$, is a discrete-time Lyapunov equation~\\cite{Bhatia2006a}.} which can be solved in terms of infinite series~\\cite{Bhatia2006a}. This series has been evaluated for \\emph{isothermal} (also called \\emph{isotropic}) which are defined as states with all eigenvalues being equal, $\\lambda_1=\\cdots=\\lambda_N=\\lambda$. The quantum Fisher information reads\n\\[\\label{eq:nu_pure}\nH(\\epsilon)=\\frac{\\lambda^{2}}{2(1+\\lambda^{2})}\\mathrm{tr}\\big[(A^{-1}\\dot{A})^2\\big]+2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}}.\n\\]\nAs noted in~\\cite{Jiang2014a}, using $\\sigma^{-1}=\\frac{1}{\\lambda^2}K\\sigma K$ this the expression can be further simplified,\n\\[\\label{eq:nu_pure_simplified}\nH(\\epsilon)=-\\frac{1}{2(1+\\lambda^{2})}\\mathrm{tr}\\big[\\dot{A}^2\\big]+\\frac{2}{\\lambda^2}\\dot{\\boldsymbol{d}}^\\dag A K\\dot{\\boldsymbol{d}}.\n\\]\nFor pure states we take $\\lambda=1$.\n\n\nFor some applications, an exact expression for the quantum Fisher information is not necessary. It can be easier to numerically obtain an approximate value of this quantity. The same method used to find the expression for pure states can be also used for general mixed states, however, in terms of an infinite sum. The quantum Fisher information for any number of modes reads\n\\[\\label{eq:Monras_QFI}\nH(\\epsilon)=\\frac{1}{2}\\mathrm{tr}\\big[\\dot{\\sigma}Y\\big]+2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}},\n\\]\nwhere $Y=-\\sum_{n=0}^\\infty (K\\sigma)^{-n}\\dot{(\\sigma^{-1})}(\\sigma K)^{-n}$. The limit converges if and only if all symplectic eigenvalues are larger than one, i.e., when all modes are mixed.\n\nA similar method of solving the equation for the symmetric logarithmic derivative has been used~\\cite{Gao2014a} to derive an exact formula for estimation multiple parameters for multi-mode mixed Gaussian states. This also generalizes the single parameter results of~\\cite{Monras2013a}. The original formula is written in terms of tensor elements. However, we notice the result can be expressed in an elegant matrix form. The quantum Fisher information matrix for Gaussian state $(\\boldsymbol{d},\\sigma)$ can be calculated as\n\\begin{subequations}\\label{eq:mixed_QFI}\n\\begin{align}\nH^{ij}(\\boldsymbol{\\epsilon})&=\\frac{1}{2}\\vectorization{\\partial_i\\sigma}^\\dag\\mathfrak{M}^{-1}\\vectorization{\\partial_j\\sigma}+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d},\\\\\n\\mathfrak{M}&=\\ov{\\sigma}\\otimes\\sigma-K\\otimes K,\n\\end{align}\n\\end{subequations}\nwhere $\\otimes$ denotes the Kronecker product, $\\vectorization{\\cdot}$ is a vectorization of a matrix, and $\\partial_i\\equiv\\partial_{\\epsilon_i}$. Again, this formula holds only for states for which all symplectic eigenvalues are larger than one. The symmetric logarithmic derivative reads\n\\[\n\\mathcal{L}_i=\\Delta \\boldsymbol{\\hat{A}}^\\dag\\mathcal{A}_i\\Delta \\boldsymbol{\\hat{A}}-\\frac{1}{2}\\mathrm{tr}[\\sigma\\mathcal{A}_i]+2\\Delta\\boldsymbol{A}^\\dag\\sigma^{-1}\\partial_i{\\boldsymbol{d}},\n\\]\nwhere $\\Delta \\boldsymbol{\\hat{A}}:=\\boldsymbol{\\hat{A}}-\\boldsymbol{d}$, $\\vectorization{\\mathcal{A}_i}:=\\mathfrak{M}^{-1}\\vectorization{\\partial_i\\sigma}$. The quantum Fisher information matrix is then defined as $H^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{2}\\mathrm{tr}[\\hat{\\rho}\\{\\mathcal{L}_i,\\mathcal{L}_j\\}]$. For the full derivation of the above matrix formulae and the real form version see appendix~\\ref{app:mixed_state}. Note that although the above multi-mode formula encompasses all previous formulae, it may be harder to use. For example, calculating the quantum Fisher information of a single-mode state with Eq.~\\eqref{eq:mixed_QFI} requires inverting $4\\times4$ matrix $\\mathfrak{M}$, while Eq.~\\eqref{eq:one_mode_quantum_fisher_information} only requires inverting $2\\times2$ matrix $A$.\n\nThe last formula we will present here is again the expression for multi-mode mixed Gaussian state~\\cite{Banchi2015a}. Defining a super-operator $S_Y(X):=YXY$, the quantum Fisher information matrix reads,\n\\[\nH^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{2}\\mathrm{tr}\\big[\\partial_i\\sigma(S_\\sigma-S_K)^{-1}\\partial_j\\sigma\\big]+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d}.\n\\]\nDespite a very elegant form this expression seems slightly impractical for actual mathematical calculations. This is because the task of inverting the super-operator $S_\\sigma-S_K$ leads to the same task as before -- solving the Stein equation.\n\nAll formulae for mixed states introduced here suffer of the same problem - they cannot be applied to states which have at least one symplectic eigenvalue equal to one.\\footnote{For example, this procedure sets the term $\\frac{1}{2}\\,\\frac{\\det{A}^{-1}\\mathrm{tr}[A^{-1}{\\dot{A}}]^{2}}{1-\\det{A}^{-2}} $ in Eq.~\\eqref{eq:one_mode_quantum_fisher_information} that is undefined for $\\det{A}=1$ to zero.} These are exactly cases where the continuous quantum Fisher information and the quantum Fisher information might not coincide, as shown by Eq.~\\eqref{eq:connection_between_Hc_and_H}. It turns out that in cases where this happens the solution is to use the regularization procedure~\\eqref{eq:regularization_procedure}. We discuss this problem in sections~\\ref{sec:when_Williamson's_decomposition} and~\\ref{sec:problems_at_pops}. Then, in analogy of the expression for a single mode Gaussian state, we derive the quantum Fisher information two-mode Gaussian states. We also simplify the limit formula~\\eqref{eq:Monras_QFI} and provide an estimate of the remainder of the series. Finally, we derive an elegant and useful expression for the quantum Fisher information for the case when the symplectic decomposition of the covariance matrix is known.\n\n\\chapter{Operations in quantum field theory and state-of-the-art in estimating space-time parameters}\\label{chap:operations_in_QFT}\n\nWith the enormous success of quantum theory the question arose how to combine this theory with special and general relativity and whether such theory is even possible. The first attempts were performed by Klein~\\cite{klein1926quantentheorie} and Gordon~\\cite{gordon1926comptoneffekt} who came up with an idea of deriving an equation of motion in a similar way to the Schr\\\"odinger equation -- simply by exchanging energy and momenta for its respective operators in the energy-momentum relation. This led to the Klein-Gordon equation which we now use to describe scalar fields of spinless particles. After numerous interpretational problems -- especially with the notion of particle -- quantum field theory was born. One of the most precise theories we have today successfully predicted and confirmed anomalous magnetic dipole moments, hyperfine splitting of energy levels of a hydrogen atom, and the quantum Hall effect. Assuming that the space is not necessarily flat has led to further generalization of the theory called quantum field theory in curved space-time. This theory attempts to describe quantum fields in large velocities and accelerations and on scales where gravity plays a role. The famous predictions of this theory are: Hawking radiation~\\cite{Hawking1974a} which says that particles can escape an enormous black hole potential behind the Schwarzchild horizon, the Unruh effect~\\cite{Unruh1979a} which illustrates that an accelerating observer sees more particles than an inertial observer, and the dynamical Casimir effect~\\cite{moore1970quantum} which shows particles can be created between two moving mirrors. Predictions of this theory, so far, have only been confirmed in analogue systems~\\cite{Wilson2011a}. Despite the practical success of this theory, it is not believed to be the final theory. This is simply because the theory attempts to describe quantum fields propagating on a fixed space-time. But gravity, which gives rise to the space-time, is itself provided by other quantized fields. These quantum properties of gravity are expected to have observable effects on either very small scales or in high energies and will be described by a future theory of quantum gravity.\n\nAn excellent although quite concise text on quantum field theory in curved space-time has been written by Birrel and Davies~\\cite{Birrell1984a}. A more mathematical approach can be found in~\\cite{ashtekar1975quantum} and a more pedagogical approach in~\\cite{wald1994quantum}.\n\nThis chapter is organized as follows: we first summarize the quantization of the Klein-Gordon field while omitting mathematical technicalities that can be found in the references above. Then we introduce Bogoliubov transformations which can describe how different observers perceive the field and how the field evolves. However, these transformations are not suitable for the description of continuous evolution. For that reason we follow on~\\cite{Bruschi2013b,Bruschi2013a} and show how the equations of motion for continuous transformations are constructed. Such equations are usually difficult to solve exactly and perturbation methods need to be used. Previous works considered only the first order correction to the solution of continuous Bogoliubov transformations in the small parameter of interest. We derive a general prescription on how to calculate these coefficients to any order which can later be used for more precise approximation of the quantum Fisher information. Finally, we overview the current state-of-the-art of quantum metrology applied in the estimation of space-time parameters.\n\n\\section{Quantization of the Klein-Gordon field}\n\nThe space-time in general relativity is described by a smooth manifold equipped with patches of \\emph{local coordinates}. Put simply, manifolds are objects which when viewed from a sufficiently small region resemble the flat space. Local coordinates are then a mathematical description of how an observer measures space and time in this sufficiently small region. In these local coordinates $x^\\mu=(t,\\boldsymbol{x})$, where $t$ represents time and $\\boldsymbol{x}$ position, we define a line element\n\\[\\label{def:metric}\n\\mathrm{d}s^2=g^{\\mu\\nu}\\mathrm{d}x_\\mu \\mathrm{d}x_\\nu,\n\\]\nwhere $g^{\\mu\\nu}$ are the elements of the metric tensor and $\\mu,\\nu=0,1,2,3$.\n\nThe simplest example of particles living on the manifold are spin-0 particles which can be described by either real or complex scalar field $\\phi$. The massless scalar field $\\phi$ obeys the Klein-Gordon equation~\\cite{Birrell1984a},\n\\[\\label{eq:klein_gordon_equation}\n\\nabla^{\\mu}\\nabla_{\\mu}\\,\\phi(t,\\boldsymbol{x})=0.\n\\]\nThe operator $\\nabla_{\\mu}$ is the covariant derivative defined with respect to the metric tensor $\\boldsymbol{g}$ in local coordinates $(t,\\boldsymbol{x})$.\n\nWhen space-time $\\boldsymbol{g}$ admits global or asymptotic time-like killing vector field, it is possible to quantize the field. This Killing vector field then splits the set of linearly independent solutions -- modes -- of Klein-Gordon Eq.~\\eqref{eq:klein_gordon_equation} to either positive frequency modes $u_{\\boldsymbol{k}}$ or negative frequency modes $\\ov{u}_{\\boldsymbol{k}}$. Because Eq.~\\eqref{eq:klein_gordon_equation} is a linear equation, the full solution is a linear combination of these positive and negative frequency modes,\n\\[\\label{eq:general_solution_to_KG}\n\\phi(t,\\boldsymbol{x})=\\int\\!\\!\\!\\mathrm{d}\\boldsymbol{k}\\ \\ a_{\\boldsymbol{k}}u_{\\boldsymbol{k}}(t,\\boldsymbol{x})+\\ov{a}_{\\boldsymbol{k}}\\ov{u}_{\\boldsymbol{k}}(t,\\boldsymbol{x}).\n\\]\nFollowing the standard quantization procedure~\\cite{Birrell1984a}, the coefficients $a_{\\boldsymbol{k}}$, $\\ov{a}_{\\boldsymbol{k}}$ are lifted into the annihilation and creation operators $a_{\\boldsymbol{k}}\\rightarrow\\hat{a}_{\\boldsymbol{k}}$, $\\ov{a}_{\\boldsymbol{k}}\\rightarrow\\hat{a}_{\\boldsymbol{k}}^\\dag$. In simplified terms, mode $u_{\\boldsymbol{k}}$ is usually associated with a particle, the creation operator $\\hat{a}_{\\boldsymbol{k}}^\\dag$ creates this particle, and the annihilation operator $\\hat{a}_{\\boldsymbol{k}}$ annihilates this particle. However, it is important to point out that this view is very tricky and in general the notion of particle in quantum field theory is still unsettled~\\cite{Birrell1984a}.\n\nWe will restrict ourselves to the real Klein-Gordon field quantized in a finite space (for example a box or a cavity) in $1+1$ dimensions. In that scenario the spectrum of modes is discrete and the field operator constructed from Eq.~\\eqref{eq:general_solution_to_KG} takes the form of an infinite sum\n\\[\\label{eq:field_expansion}\n\\hat{\\phi}(t,\\boldsymbol{x})=\\sum_n \\hat{a}_n u_n(t,\\boldsymbol{x})+\\hat{a}_n^\\dag \\ov{u}_n(t,\\boldsymbol{x}).\n\\]\n$n$ labels the mode and annihilation and creation operators satisfy the same commutation relations as introduced in Eq.~\\eqref{def:commutation_relation}. The Fock space describing the system of relativistic bosons is constructed identically to the non-relativistic case from section~\\ref{sec:Fock_space}.\n\n\\section{Bogoliubov transformation}\n\nThe field expansion~\\eqref{eq:field_expansion} is not unique. It can be written in different basis of solutions to the Klein-Gordon equation denoted $\\{v_n\\}_n$. We collect both sets of modes into a (possibly infinite) vectors $\\boldsymbol{u}=(u_1,u_2,\\dots)$, $\\boldsymbol{v}=(v_1,v_2,\\dots)$. Solutions to the Klein-Gordon equation form a linear vector space. Therefore every solution $u_n$ can be expressed as a linear combination of modes $v_n$, $\\ov{v}_n$,\n\\[\\label{eq:bogoliubov_transformation_definition}\n\\begin{bmatrix}\n\\boldsymbol{u} \\\\ \\overline{\\boldsymbol{u}}\n\\end{bmatrix}=\nS\n\\begin{bmatrix}\n\\boldsymbol{v} \\\\ \\overline{\\boldsymbol{v}}\n\\end{bmatrix},\n\\]\nwhere matrix\n\\[\\label{def:Bogoliubov_S}\nS=\n\\begin{bmatrix}\n\\alpha & \\beta \\\\\n\\overline{\\beta} & \\overline{\\alpha}\n\\end{bmatrix}\n\\]\nis called the Bogoliubov transformation and $\\alpha$ and $\\beta$ are the Bogoliubov coefficients.\n\nBogoliubov transformations are important in the quantum field theory in curved space-time, because they relate solutions to the Klein-Gordon equation written in one local coordinates to a different local coordinates. Some problems can be very hard to solve in certain coordinates, while they can be relatively easy in specially picked ones. Bogoliubov transformation can be used, for example, to derive many results of quantum field theory in curved space-time such as black-hole evaporation~\\cite{Hawking1974a,hawking1975particle,fabbri2005modeling}, the Unruh effect~\\cite{takagi1986vacuum,Unruh1979a}, and the dynamical Casimir effect~\\cite{moore1970quantum}. They also used to model the time evolution of quantum states, and quantum states from the point of view of different observers.\n\nThe field operator~\\eqref{eq:field_expansion} can be written in two different vector forms,\n\\[\\label{eq:two_field_expansion}\n\\hat{\\phi}(t,x)=\n\\begin{bmatrix}\n\\ov{\\boldsymbol{u}} \\\\ \\boldsymbol{u}\n\\end{bmatrix}^\\dag\n\\cdot\\begin{bmatrix}\n\\hat{\\boldsymbol{a}} \\\\ \\hat{\\boldsymbol{a}}^\\dag\n\\end{bmatrix}=\n\\begin{bmatrix}\n\\ov{\\boldsymbol{v}} \\\\ \\boldsymbol{v}\n\\end{bmatrix}^\\dag\n\\cdot\\begin{bmatrix}\n\\hat{\\boldsymbol{b}} \\\\ \\hat{\\boldsymbol{b}}^\\dag\n\\end{bmatrix}\n\\]\nCombining Eqs.~\\eqref{eq:two_field_expansion} and~\\eqref{eq:bogoliubov_transformation_definition} we derive transformation relations between two sets of annihilation and creation operators,\n\\[\\label{eq:transformation_of_field}\n\\begin{bmatrix}\n\\hat{\\boldsymbol{b}} \\\\ \\hat{\\boldsymbol{b}}^\\dag\n\\end{bmatrix}=\n\\ov{S}^\\dag\\begin{bmatrix}\n\\hat{\\boldsymbol{a}} \\\\ \\hat{\\boldsymbol{a}}^\\dag\n\\end{bmatrix}.\n\\]\nBoth sets of annihilation and creation operators must obey the commutation relations~\\eqref{def:commutation_relation} which gives a condition on the matrix $S$,\n\\[\nSKS^\\dag=K.\n\\]\nThis is equivalent to\n\\begin{subequations}\n\\begin{align}\n\\alpha\\A^\\dag-\\beta\\B^\\dag&=I,\\\\\n\\alpha\\beta^T&=\\beta\\alpha^T,\n\\end{align}\n\\end{subequations}\nknown as Bogoliubov identities. But these conditions are exactly the defining conditions of the symplectic group, Eqs.~\\eqref{def:structure_of_S} and~\\eqref{def:Bogo_id} respectively. In other words, Bogoliubov transformations are identical to Gaussian unitary transformations and they transform Gaussian states into Gaussian states. This is why the phase-space description of Gaussian states and can be easily applied in the quantum field theory in curved space-time which will be used in following chapters.\n\nThe Bogoliubov coefficients can be calculated using the (pseudo-)inner product of the Klein-Gordon equation~\\cite{Crispino2008} which depends on the metric~\\eqref{def:metric}\nas\n\\begin{subequations}\\label{eq:basic_bogos}\n\\begin{align}\n\\alpha_{mn}&\\,=\\, \\big(u_{m},v_{n}\\big)\\big|_{\\Sigma},\\\\\n\\beta_{mn}&\\,=\\, \\big(u_{m},\\overline{v}_{n}\\big)\\big|_{\\Sigma}.\n\\end{align}\n\\end{subequations}\nThese coefficients encode the information of a transformation between two sets of solutions of the Klein-Gordon equation on a given time-like hypersurface $\\Sigma$.\n\n\\section{Continuous Bogoliubov coefficients}\n\nThe Bogoliubov transformations in Eq.~\\eqref{eq:basic_bogos} are defined only for a fixed time, they will not in general be suitable to describe the continuous evolution of a quantum state. In particular, they are not suitable for describing the evolution of a quantum state which transform an initial state at time $\\tau_{0}$ to a final state at time $\\tau$. In this section we utilize perturbative methods and derive expressions for continuous Bogoliubov transformations up to any order in the expansion in a small parameter $\\epsilon$. The method was formerly described in~\\cite{Bruschi2013b,Bruschi2013a}, however, previous work considered only the first order correction. Here we derive expressions for perturbative coefficients up to any order.\n\nTo construct the continuous Bogoliubov transformations consider a function $h(\\tau)$ which parametrizes the physical scenario, for example a cavity moving through a curved space-time, a passing gravitational wave, or an accelerating cavity. For example, $h(\\tau)$ could describe the proper acceleration of the cavity. The Bogoliubov co-efficients, which describe the evolution of a quantum state inside of the cavity, would be then dependent on this function as $\\alpha[h(\\tau)]$ and $\\beta[h(\\tau)]$. The matrices $\\alpha[h(\\tau)]$ and $\\beta[h(\\tau)]$ represent the Bogoliubov transformation from one set of modes at time $\\tau_{0}$ to a new set of modes at time $\\tau$. After the Bogoliubov transformation has been applied, the evolution of the field is governed by the free Hamiltonian $H_f[h(\\tau)]$. By ``free'' here we mean that it governs the evolution the state would undergo if it is left alone, without adding any extra energy into the system, for example without adding any further acceleration.  For example, for a non-interacting field the free Hamiltonian this is given by the mode frequencies defined as $\\Omega_{j}[h(\\tau)]$. In other words, at time $\\tau$, each mode is characterized by a frequency which depends, for example, on the acceleration $h(\\tau)$. To construct the continuous transformations, consider a successive combination of transformations composed of: i) An initial transformation at time $\\tau=\\tau_{0}$, $\\mathbin{_{o}\\alpha}:=\\alpha[h(\\tau_{0})]$ and $\\mathbin{_{o}\\beta}:=\\beta[h(\\tau_{0})]$, ii) Evolution under the ``free\" Hamiltonian in the new modes for some time $\\Delta\\tau$, iii) Applying the inverse of the initial transformation for the coefficients $\\mathbin{_{o}\\boldsymbol{\\alpha}}$ and $\\mathbin{_{o}\\boldsymbol{\\beta}}$. Combining these successive transformations and letting each interval $\\Delta\\tau\\rightarrow 0$, while keeping the \\emph{total} proper time fixed, we can construct a continuous Bogoliubov transformation. Following this procedure, we can derive the following initial value problem for the \\emph{total} continuous Bogoliubov transformation~\\cite{Bruschi2013b},\n\\[\\label{eqn:exact_continuous_bogo_ODE}\n\\mathrm{d}_{\\tau}S(\\tau,\\tau_{0})=iKH[h(\\tau)] S(\\tau,\\tau_{0}),\\quad S(\\tau_{0},\\tau_{0})=I.\n\\]\nHere, we have defined the effective Hamiltonian of the system as\n\\[\\label{def:Hamiltonian_continuous_bogos}\nH=\\begin{bmatrix}\nA & B \\\\\n\\overline{B} & \\overline{A}\n\\end{bmatrix}.\n\\]\nThe form of Eq.~\\eqref{eqn:exact_continuous_bogo_ODE} is to be expected from the Lie group structure of symplectic transformations, i.e., matrix $iKH$ is an element of the Lie algebra associated with the symplectic group and satisfy Eq.~\\eqref{def:P_1}. Similar results have been pointed out before in the quantum field theory literature~\\cite{Brown2013a,Bruschi2013a}.\n\nTaking advantage of the block structure of the symplectic matrices~\\eqref{def:Bogoliubov_S}, Eq.~\\eqref{eqn:exact_continuous_bogo_ODE} can be rewritten as two coupled equations,\n\\begin{subequations}\n\\label{eqn:exact_continuous_bogo_ODE_alpha_beta}\n\\begin{align}\n\\mathrm{d}_{\\tau}\\alpha&= +i\\,\\Big(A\\alpha+B\\overline{\\beta}\\Big),\\quad\\alpha(\\tau_{0},\\tau_{0})=I,\\\\\n\\mathrm{d}_{\\tau}\\overline{\\beta}&= -i\\,\\Big(\\overline{A}\\,\\overline{\\beta}+\\overline{B}\\alpha\\Big),\\quad\\beta(\\tau_{0},\\tau_{0})=0.\n\\end{align}\n\\end{subequations}\nEq.~\\eqref{eqn:exact_continuous_bogo_ODE} or Eq.~\\eqref{eqn:exact_continuous_bogo_ODE_alpha_beta} give a concrete recipe to determine the symplectic transformation induced by the Bogoliubov transformations of our quantum field theory. If those Bogoliubov transformations describe the evolution of the state, once the matrices $\\alpha$ and $\\beta$ are determined one can use the full symplectic transformation $S$ to evolve the moments of the Gaussian state via Eq.~\\eqref{def:transformation}.\n\n\\begin{example}\n\\emph{\nThe Hamiltonian~\\eqref{def:Hamiltonian_continuous_bogos} depends on the particular method of construction of `gluing' together infinitesimal fixed-time Bogoliubov transformation and free-time evolution. Our construction can for example describe the previously mentioned case when the function $h(\\tau)$ describes the proper acceleration of an observer. Then the effective Hamiltonian reads,\n\\begin{subequations}\\label{eq:accelerating_cavity_Hamiltonian}\n\\begin{align}\nA&= \\mathbin{_{o}\\alpha^{\\dag}}\\Omega\\mathbin{_{o}\\alpha}+\\mathbin{_{o}\\beta^{T}}\\Omega\\mathbin{_{o}\\overline{\\beta}}, \\\\\nB&= \\mathbin{_{o}\\alpha^{\\dag}}\\Omega\\mathbin{_{o}\\beta}+\\mathbin{_{o}\\beta^{T}}\\Omega\\mathbin{_{o}\\overline{\\alpha}}.\n\\end{align}\n\\end{subequations}\nAbove, we have also introduced the free Hamiltonian given by the frequency matrix $\\Omega:=\\mathrm{diag}(\\Omega_{1}[h(\\tau)],\\Omega_{2}[h(\\tau)],\\ldots)$. In different scenarios, such as slowly moving cavities in Schwarzchild space-time, different methods of construction are more viable for describing the evolution of the physical system. This then leads to a different effective Hamiltonian.\n}\n\\end{example}\n\nThe general solution to Eq.~\\eqref{eqn:exact_continuous_bogo_ODE}, Eq.~\\eqref{eqn:exact_continuous_bogo_ODE_alpha_beta} respectively, can be formally written as the time-ordered exponential,\n\\[\nS=\\mathrm{Texp}\\left(i\\int_{\\tau_0}^\\tau KH\\ \\mathrm{d}\\tau\\right).\n\\]\nHowever, this solution is usually difficult to compute exactly and ODE methods, such as solving order by order in a parameter, have to be used. In later chapters we will use this perturbative method to estimate the ultimate limits of precision in estimating space-time parameters.\n\n\\subsection{Perturbative method of finding the continuous Bogoliubov coefficients}\\label{sec:perturbative_method}\nLet us assume the function $h(\\tau)$ introduced in the previous subsection can be factorized as $h(\\tau)=\\epsilon f(\\tau)$, where $\\epsilon$ is a fixed constant -- a small parameter, and $f(\\tau)$ is another function independent of $\\epsilon$. Because the effective Hamiltonian $H[h(\\tau)]$ depends on time only through the function $h(\\tau)$, setting $\\epsilon=0$ will make this Hamiltonian time-independent. We define this time-independent part of the effective Hamiltonian as $H_0:=H[0]$.\nWe look for the solution of Eq.~\\eqref{eqn:exact_continuous_bogo_ODE} in the form\n\\[\\label{def:tildeS}\nS=e^{iKH_0(\\tau-\\tau_0)}\\tilde{S},\\quad\n\\tilde{S}=\\begin{bmatrix}\n\\tilde{\\alpha} & \\tilde{\\beta} \\\\\n\\overline{\\tilde{\\beta}} & \\overline{\\tilde{\\alpha}}\n\\end{bmatrix}.\n\\]\nThis is of course only a mathematical substitution which we could have chosen differently. However, this choice of substitution ensures that for $\\epsilon=0$ we retrieve the exact solution of the free-time evolution instead of just perturbative expansion of such evolution. The Eq.~\\eqref{eqn:exact_continuous_bogo_ODE} transforms into\n\\[\\label{eqn:exact_continuous_bogo_ODE2}\n\\mathrm{d}_{\\tau}\\tilde{S}(\\tau,\\tau_{0})=iV(\\tau,\\tau_{0}) \\tilde{S}(\\tau,\\tau_{0}),\\quad \\tilde{S}(\\tau_{0},\\tau_{0})=I,\n\\]\nwhere we have defined the new effective Hamiltonian,\n\\[\nV(\\tau,\\tau_{0}):=e^{-iH_0(\\tau-\\tau_0)}KH[h(\\tau)]e^{iH_0(\\tau-\\tau_0)}-KH_0.\n\\]\nMatrix $iV$ is again the element of the Lie algebra associated with the symplectic group and satisfy Eq.~\\eqref{def:P_1}. This provides an internal structure of the matrix $V$,\n\\[\nV=\\begin{bmatrix}\nV_{11} & V_{12} \\\\\n\\overline{V}_{12} & \\overline{V}_{11}\n\\end{bmatrix},\\ \\ V_{11}=V_{11}^\\dag,\\ \\ V_{12}=-V_{12}^T.\n\\]\nFrom now on we assume every matrix $M$ has a perturbative expansion in $\\epsilon$,\n\\[\nM=\\sum_{k=0}^\\infty M^{(k)}\\epsilon^k.\n\\]\nClearly from the definition $H_0=H^{(0)}$ and $V^{(0)}=0$. By inserting expansions of matrix $\\tilde{S}$ and matrix $V$ to Eq.~\\eqref{eqn:exact_continuous_bogo_ODE2} we obtain a set of differential equations,\n\\begin{subequations}\n\\begin{align}\n\\sum_{k=0}^\\infty \\mathrm{d}_\\tau \\tilde{\\alpha}^{(k)}\\epsilon^k=i\\sum_{k=1}^\\infty V_{11}^{(k)}\\epsilon^k+i\\sum_{k,l=1}^\\infty\\left(V_{11}^{(l)}\\tilde{\\alpha}^{(k)}+V_{12}^{(l)}\\ov{\\tilde{\\beta}}^{(k)}\\right)\\epsilon^{k+l},\\\\\n\\sum_{k=0}^\\infty \\mathrm{d}_\\tau \\tilde{\\beta}^{(k)}\\epsilon^k=i\\sum_{k=1}^\\infty V_{12}^{(k)}\\epsilon^k+\ni\\sum_{k,l=1}^\\infty\\left(V_{11}^{(l)}\\tilde{\\beta}^{(k)}+V_{12}^{(l)}\\ov{\\tilde{\\alpha}}^{(k)}\\right)\\epsilon^{k+l}.\n\\end{align}\n\\end{subequations}\nRearranging terms in infinite summations gives\n\\begin{subequations}\n\\begin{align}\n\\sum_{k=0}^\\infty \\mathrm{d}_\\tau \\tilde{\\alpha}^{(k)}\\epsilon^k&= iV_{11}^{(1)}\\epsilon+i\\sum_{k=2}^\\infty \\left(V_{11}^{(k)}+\\sum_{l=1}^{k-1}\\left(V_{11}^{(l)}\\tilde{\\alpha}^{(k-l)}+V_{12}^{(l)}\\ov{\\tilde{\\beta}}^{(k-l)}\\right)\\right)\\epsilon^{k},\\\\\n\\sum_{k=0}^\\infty \\mathrm{d}_\\tau \\tilde{\\beta}^{(k)}\\epsilon^k&= iV_{12}^{(1)}\\epsilon+\ni\\sum_{k=2}^\\infty\\left(V_{12}^{(k)}+\\sum_{l=1}^{k-1}\\left(V_{11}^{(l)}\\tilde{\\beta}^{(k-l)}+V_{12}^{(l)}\\ov{\\tilde{\\alpha}}^{(k-l)}\\right)\\right)\\epsilon^{k}.\n\\end{align}\n\\end{subequations}\nFinally, by comparing coefficients of different powers of $\\epsilon$ we obtain recursive formulae for an arbitrarily high coefficient of a continuous Bogoliubov transformation,\n\\[\n\\begin{split}\\label{eqns:general_solution_arbitrarily_high_order}\n\\tilde{\\alpha}^{(0)}(\\tau,\\tau_0)&= I,\\\\\n\\tilde{\\beta}^{(0)}(\\tau,\\tau_0)&= 0,\\\\\n\\tilde{\\alpha}^{(1)}(\\tau,\\tau_0)&= i\\int_{\\tau_0}^{\\tau}\\!\\!\\!\\mathrm{d}t\\ V_{11}^{(1)}(t,\\tau_0),\\\\\n\\tilde{\\beta}^{(1)}(\\tau,\\tau_0)&= i\\int_{\\tau_0}^{\\tau}\\!\\!\\!\\mathrm{d}t\\ V_{12}^{(1)}(t,\\tau_0),\\\\\n\\tilde{\\alpha}^{(k)}(\\tau,\\tau_0)&= i\\int_{\\tau_0}^{\\tau}\\!\\!\\!\\mathrm{d}t\\ \\left(V_{11}^{(k)}+\\sum_{l=1}^{k-1}\\left(V_{11}^{(l)}\\tilde{\\alpha}^{(k-l)}+V_{12}^{(l)}\\ov{\\tilde{\\beta}}^{(k-l)}\\right)\\right)(t,\\tau_0),\\\\\n\\tilde{\\beta}^{(k)}(\\tau,\\tau_0)&= i\\int_{\\tau_0}^{\\tau}\\!\\!\\!\\mathrm{d}t\\ \\left(V_{12}^{(k)}+\\sum_{l=1}^{k-1}\\left(V_{11}^{(l)}\\tilde{\\beta}^{(k-l)}+V_{12}^{(l)}\\ov{\\tilde{\\alpha}}^{(k-l)}\\right)\\right)(t,\\tau_0).\n\\end{split}\n\\]\nExpressions for $\\tilde{\\alpha}^{(0)}$ and $\\tilde{\\beta}^{(0)}$ have been obtained from the initial condition in Eq.~\\eqref{eqn:exact_continuous_bogo_ODE2}. The full solution to the continuous Bogoliubov transformation is obtain by combining the above equations and Eq.~\\eqref{def:tildeS}.\n\n\\begin{example}\\label{ex:accelerated_cavity}\nAccelerating cavity first introduced in~\\cite{Bruschi2013b}.\n\\emph{The scenario is the following. Assume a quantum state inside of a non-moving cavity. Starting at proper time $\\tau_0=0$, the cavity goes through a period $\\tau$ of the proper acceleration $a$ (as measured in the centre of the cavity) and period $\\tau$ of retardation $-a$, stopping again at time $2\\tau$. The proper length of the cavity $L=1$ is considered constant during the whole procedure. We are going to expand in the small parameter $a$, i.e. $\\epsilon\\equiv a$.\nThe effective Hamiltonian is given by Eqs.~\\eqref{def:Hamiltonian_continuous_bogos} and~\\eqref{eq:accelerating_cavity_Hamiltonian}. The function is given by $h(t)=af(t)$, where $f(t)=1$ for $\\tau_0\\leq t\\leq\\tau$ and $f(t)=-1$ for $\\tau\\leq t\\leq2\\tau$. The frequencies of the free Hamiltonian $H_f=\\mathrm{diag}(\\Omega_1[a f(\\tau)],\\Omega_2[a f(\\tau)],\\dots,\\Omega_1[a f(\\tau)],\\Omega_2[a f(\\tau)])$ and the Bogoliubov coefficients which transforms the cavity from the still state to the accelerating state with proper acceleration $a$ are given by\n\\begin{subequations}\n\\begin{align}\n\\Omega_n[a f(t)]&\\,=\\,\\omega_n+\\mathcal{O}(a^2),\\quad \\omega_n=\\frac{n\\pi}{L},\\\\\n\\mathbin{_{o}\\alpha}_{mn}&\\,=\\, 1+\\mathcal{O}(a^2),\\ &m=n\\nonumber\\\\\n&\\,=\\, \\frac{(-1+(-1)^{m+n})\\sqrt{mn}}{(m-n)^3\\pi^2}a+\\mathcal{O}(a^2),\\ &m\\neq n\\\\\n\\mathbin{_{o}\\beta}_{mn}&\\,=\\, \\frac{(1-(-1)^{m+n})\\sqrt{mn}}{(m+n)^3\\pi^2}a+\\mathcal{O}(a^2),&\n\\end{align}\n\\end{subequations}\nThe continuous coefficients up to the first order are obtained combining Eqs.~\\eqref{eqns:general_solution_arbitrarily_high_order} and Eq.~\\eqref{def:tildeS},\n\\begin{subequations}\\label{eq:bogos_acceleration}\n\\begin{align}\n\\alpha_{mn}(a)&\\,=\\, e^{i\\omega_n2\\tau}+\\mathcal{O}(a^2),\\ &m=n\\nonumber\\\\\n&\\,=\\, -\\frac{8i\\sqrt{mn}}{(m-n)^3\\pi^2}e^{\\frac{1}{2}i\\pi(m+n-2m\\tau+6n\\tau)}\\sin{\\tfrac{(m+n)\\pi}{2}}\\sin^2{\\tfrac{(m-n)\\pi\\tau}{2}}\\,a+\\mathcal{O}(a^2),\\ &m\\neq n\\\\\n\\beta_{mn}(a)&\\,=\\, -\\frac{8i\\sqrt{mn}}{(m+n)^3\\pi^2}e^{\\frac{1}{2}i\\pi(m+n-2m\\tau+2n\\tau)}\\sin{\\tfrac{(m+n)\\pi}{2}}\\sin^2{\\tfrac{(m+n)\\pi\\tau}{2}}\\,a+\\mathcal{O}(a^2).&\n\\end{align}\n\\end{subequations}\n}\n\\end{example}\n\nThe method introduced here can be used to derive the Bogoliubov transformations depending on the arbitrarily space-time parameter, not only the proper acceleration. In chapter~\\ref{chap:QFT_metrology} we show how to derive the quantum Fisher information for estimating such parameters encoded by a general Bogoliubov transformations into squeezed and two-mode squeezed thermal states.\n\n\\section{State-of-the-art quantum metrology for estimating space-time parameters}\\label{sec:state_QM_in_qft}\n\nAfter successful efforts in combining quantum physics and general relativity on macroscopic scales leading to infamous black-hole evaporation~\\cite{Hawking1974a,hawking1975particle,fabbri2005modeling}, the Unruh effect~\\cite{takagi1986vacuum,Unruh1979a}, and the dynamical Casimir effect~\\cite{moore1970quantum}, the focus shifted to studying relativistic quantum effects from a more analytical perspective of quantum information. Pioneering articles in this newly established field focused on entanglement and its generation due to either gravity or non-inertial motion~\\cite{alsing2002lorentz,pachos2002generation,Peres2004quantum_information,fuentes2005alice,alsing2006entanglement} and on quantum information protocols such as quantum teleportation~\\cite{alsing2003teleportation,alsing2004teleportation} and communication channels~\\cite{Peres2004quantum_information}. Relativistic quantum information was booming. It has been discovered that relativistic motion could serve as a source for generating quantum gates~\\cite{friis2012quantum,Bruschi2013relativistic_motion,martin2014quantum}. To complement studies on entanglement, it was investigated how motion affects other quantum information figures of merit such as quantum discord~\\cite{datta2009quantum}. Moreover, it has been shown that continuous variable methods could prove fruitful in relativistic quantum information~\\cite{adesso2012continuous}.\nOne of the latest achievements was the introduction of quantum metrology into quantum field theory. As quantum metrology provides the limits of precision in measuring parameters, its methods can be used to validate the predictions in quantum field theory in curved space-time,  and also to demonstrate whether measuring space-time parameters is achievable with current and future technology.\n\nQuantum metrology was first applied to the scenario in which a quantum state in a cavity is used estimate the temperature of the Unruh-Hawking effect~\\cite{aspachs2010optimal}. This work showed that Fock states have an advantage as probes over Gaussian states. It was followed in~\\cite{wang2014quantum} where a pair of Unruh-DeWitt detectors were used instead of cavity modes, and it was found that the limit of precision depends on an effective coupling strength and a longer interaction time. Authors of~\\cite{Tian2015a} have evaluated the Fisher information and so, unlike the previous work, were taking realistic measurements into account. They showed that the optimal bound given by the quantum Fisher information could be achieved by the population measurement.\n\nOther applications of quantum metrology include relativistic quantum accelerometers. The first article~\\cite{dragan2011quantum} shows how a field in a cavity can be used to distinguish two scenarios that are kinematically indistinguishable: an accelerated cavity as seen by a stationary observer and an accelerated observer looking at a stationary cavity. Authors of article~\\cite{Yao2014quantum} have shown how the precision in estimating parameters encoded in a general two-qubit state changes under accelerated motion. Interestingly, they found that the quantum Fisher information converges to a constant as the acceleration goes to infinity. A similar analysis has been made for fermionic cavities~\\cite{shamsi2014quantum}. It has been illustrated in~\\cite{Ahmadzadegan2014a} that the internal atomic degrees of freedom are sensitive to the atom's spatial trajectory and could serve as a good accelerometer.\n\nQuantum metrology has been also used to study the precision of quantum clocks. It has been shown that the accelerated motion affects the precision in measuring time~\\cite{Lindkvist2015a}, and that the quantum Fisher information degrades with higher acceleration. When acceleration is applied coherent states are more robust against the loss of precision, despite the fact that squeezed states perform better than coherent states in the absence of motion. As proposed in~\\cite{lindkvist2014twin}, superconducting circuits could perform well in simulating the time dilation.\n\nQuantum metrology has been applied to determine relativistic parameters such as the Schwarzchild radius using squeezed light~\\cite{Bruschi2014a,kohlrus2015quantum,kish2016estimating} and the mass of a black hole~\\cite{Doukas2014a} using scattering experiments. There are proposals for implementation in analogue gravity, using optical waveguides. Finally, a gravitational wave detector using Bose-Einstein condensates has been proposed~\\cite{Sabin2014a}. It has been shown that phonon states which exist on a BEC can be altered by a passing gravitational wave. This is because the gravitational wave effectively changes boundary conditions of the BEC trap, leading to a similar effect as the dynamical Casimir effect. The changed phonon state could be measured and compared to its unchanged counterpart, effectively serving as a sensor for the gravitational wave. The initial proposal could achieve few orders of magnitudes higher precision than the current large interferometers~\\cite{Caron1995a,Abbott2004a,Grote2008a} in the ideal case. The follow up article~\\cite{Sabin2015a} analyzed the effects of finite temperature and concluded that there is no significant effect on the precision of such a proposed detector. However, what is still not taken into account is decoherence and practical aspects of the preparation of the initial state and the prospect of performing an optimal measurement.\n\nFinally, work has been done on estimating space-time parameters in general, providing useful formulae for the quantum Fisher information for any encoding Bogoliubov transformation. Authors of~\\cite{Ahmadi2014a} considered a general Bogoliubov transformation acting on two one-mode squeezed states. They derived the quantum Fisher information for estimating a parameter of such a transformation and applied the derived expression to estimating the proper acceleration of a cavity. Later, the same authors generalized this result to also incorporate the displaced squeezed states and two-mode squeezed states as probes~\\cite{Ahmadi2014a}. In article~\\cite{Friis2015a} a general procedure for finding the quantum Fisher information for initially pure probe states has been proposed. As this work also applies to non-Gaussian states it can be considered to be more general than previous contributions. However, as this work is done in the density matrix formalism it is impractical for the Gaussian states due to the complicated form of their density matrices.\n\nThe work presented in this section has been almost exclusively limited to pure initial states. But in real scenarios one cannot achieve perfect vacuum, nor is it possible to have perfect squeezed and coherent states. Achieving exactly pure states is, in fact, forbidden by the third law of thermodynamics. Probe states are always exposed to thermal fluctuations. In chapter~\\ref{chap:QFT_metrology} we provide expressions for the quantum Fisher information for an arbitrary channel acting on one- and two-mode squeezed thermal states, allowing us to study the effects of temperature on estimating space-time parameters.\n\n\\chapter*{Part II}\n\\addcontentsline{toc}{chapter}{Part II}\n\n\\chapter{Quantum metrology on Gaussian states}\\label{chap:QM_GS}\n\nIn this chapter we focus on quantum metrology of Gaussian states in the phase-space formalism. This chapter is divided into two sections. In the first section we derive numerous new formulae for the multi-parameter estimation of multi-mode Gaussian states and discuss problems of discontinuity of the figure of merit -- the quantum Fisher information. The first section can be also viewed as a continuation of section~\\ref{sec:state_quantum_metrology} in which we overview current state-of-the-art quantum metrology of Gaussian states. In the second section we apply the derived formulae to devise a practical method of finding optimal probe states for Gaussian unitary channels. We use this method to find optimal Gaussian probe states for common Gaussian channels and for some channels which have not been optimized before. We also discuss related issues such as how different parameters of the probe state affect the estimation precision and whether entanglement plays a significant role in quantum metrology.\n\n\\section{The quantum Fisher information in the phase-space formalism}\\label{sec:new_formulae\n\nIn this section we derive new formulae for the quantum Fisher information in the phase-space formalism following known formulae overviewed in section~\\ref{sec:state_quantum_metrology}. This section partially consists of results we published in~\\cite{Safranek2015b}. Some results are extended, for example, we provide expressions for the multi-parameter estimation. Also, many proofs and derivations are performed in a different and simpler way, mostly based on the matrix form of the quantum Fisher information matrix~\\eqref{eq:mixed_QFI}. First we derive the quantum Fisher information matrix for two-mode states. Then we use the quantum Fisher information matrix in terms of the Williamson's decomposition of the covariance matrix, which will beautifully expose the inner structure of this figure of merit. We also simplify the limit formula~\\eqref{eq:Monras_QFI} and provide an estimate for the remainder of the series, which is very useful for numerical calculations. Finally we show different versions of the quantum Fisher information for pure states and address the problems of the quantum Fisher information at the points of purity.\n\n\\subsection{Two-mode Gaussian states}\nIn the analogy of the derivation of the quantum Fisher information for single mode Gaussian states~\\eqref{eq:one_mode_quantum_fisher_information} we can derive the expression for two-mode Gaussian states. The calculations are more involved than in deriving the single mode case and the following derivation is shortened compared to the original version published in~\\cite{Safranek2015b}.\n\nThe derivation goes as follows. First, we use the expression for the Uhlmann fidelity between two two-mode Gaussian states derived in~\\cite{Marian2012a},\n\\[\\label{eq:fidelity_basic_formula}\n\\mathcal{F}(\\rho_1,\\rho_2)=\\frac{4e^{-\\delta\\boldsymbol{d}^{\\dagger}\\left(\\sigma_{1}\n+\\sigma_2\\right)^{-1}\\delta\\boldsymbol{d}}}{\\left(\\sqrt{\\Gamma}+\\sqrt{\\Lambda}\\right)-\\sqrt{\\left(\\sqrt{\\Gamma}+\\sqrt{\\Lambda}\\right)^{2}-\\Delta}},\n\\]\nwhere $\\delta\\boldsymbol{d}=\\boldsymbol{d}_1-\\boldsymbol{d}_2$ is a relative displacement and $\\Delta,\\Gamma,\\Lambda$ denotes three determinants defined as\n\\begin{subequations}\\label{eqs:GDL_unpolished}\n\\begin{align}\n\\Delta &=\\det{\\sigma_{1}+\\sigma_{2}},\\\\\n\\Gamma &=\\det{I+K\\sigma_{1}K\\sigma_{2}},\\\\\n\\Lambda &=\\det{\\sigma_{1}+K}\\det{\\sigma_{2}+K}.\n\\end{align}\n\\end{subequations}\nAccording to Eq.~\\eqref{QFI_using_fidelity} expanding fidelity between two close states $\\rho_\\epsilon$ and $\\rho_{\\epsilon+\\mathrm{d}\\epsilon}$ in the small parameter $\\mathrm{d}\\epsilon$ will give us the quantum Fisher information. First we need to expand determimants~\\eqref{eqs:GDL_unpolished} in the small parameter $\\mathrm{d}\\epsilon$. However doing that directly leads to numerous problems. That is why we rewrite these determimants in terms of the Williamson's decomposition~\\eqref{def:Williamson's_decomposition} of the covariance matrix, $\\sigma=SDS^\\dag$, and expand this decomposition in the small parameter $\\mathrm{d}\\epsilon$ instead. An element of the Lie algebra associated with the symplectic group will naturally appear,\n\\[\nP:=S^{-1}\\dot{S},\n\\]\nwhere \\emph{dot} denotes the derivative with respect to $\\epsilon$. Because $P$ satisfies identities of the Lie algebra~\\eqref{def:structure_of_S}, we use these identities to simplify the expression for the expansion of the fidelity and derive\n\\[\\label{GeneralQFISD}\n\\begin{split}\nH&(\\epsilon)=\\frac{1}{\\det{D}-1}\\bigg(\\det{D}\\Big(\\mathrm{tr}[P^2]-\\mathrm{tr}[D^{-1}KPDKP]\\Big)\\\\\n&\\!+\\!\\sqrt{\\det{C}}\\Big(\\mathrm{tr}[(C^{-1}P)^2]\\!+\\!\\mathrm{tr}[(C^{-1}DKP)^2]\\!-\\!\\mathrm{tr}[C^{-1}P^2]\\Big)\\!\\bigg)\\\\\n&+\\frac{1}{2}\\mathrm{tr}\\big[(D+K)^{-1}D^{-1}\\dot{D}^2\\big]+2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}},\n\\end{split}\n\\]\nwhere $C=I+D^2$. At this stage, we would like to obtain the quantum Fisher information in terms of the covariance matrix again. Defining $A:=K\\sigma$ introduced in Eq.~\\eqref{def:A} and using identities\n\\begin{subequations}\n\\begin{align}\n\\mathrm{tr}[(A^{-1}\\dot{A})^2]&=2\\mathrm{tr}[P^2]-2\\mathrm{tr}[D^{-1}KPDKP]+\\mathrm{tr}[D^{-1}\\dot{D}D^{-1}\\dot{D}]\\\\\n\\mathrm{tr}[((1+A^2)^{-1}\\dot{A})^2]&=2\\mathrm{tr}[(C^{-1}DKP)^2]+2\\mathrm{tr}[(C^{-1}P)^2]-2\\mathrm{tr}[C^{-1}(P)^2]\\nonumber\\\\\n&+\\mathrm{tr}[(C^{-1}\\dot{D})^2].\n\\end{align}\n\\end{subequations}\nwe derive the quantum Fisher information for two-mode Gaussian states,\n\\[\\label{GeneralQFI}\n\\begin{split}\nH(\\epsilon)&=\\frac{1}{2(\\det{A}-1)}\\Bigg(\\det{A}\\mathrm{tr}\\Big[\\big(A^{-1}\\dot{A}\\big)^2\\Big]+\\sqrt{\\det{I+A^2}}\\mathrm{tr}\\Big[\\big((I+A^2)^{-1}\\dot{A}\\big)^2\\Big]\\\\\n&+4\\big(\\lambda_1^2-\\lambda_2^2\\big\n)\\bigg(-\\frac{\\dot{\\lambda_1}^2}{\\lambda_1^4-1}\n+\\frac{\\dot{\\lambda_2}^2}{\\lambda_2^4-1}\\bigg)\\Bigg)+2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}}.\n\\end{split}\n\\]\nThe symplectic eigenvalues $\\lambda_{1,2}$ can be calculated via Eq.~\\eqref{eq:twomode_symplectic_eigenvalues}. The above formula is not directly applicable when one of the modes is pure, i.e., when at least one of the symplectic eigenvalues is equal to $1$. We will address this issue in following sections.\n\nSimilarly to the quantum Fisher information for single mode states~\\eqref{eq:one_mode_quantum_fisher_information}, formula for two-mode states~\\eqref{GeneralQFI} provides the advantage over the general quantum Fisher information for multi-mode states~\\eqref{eq:mixed_QFI} because it requires inverting much smaller matrices, i.e., $4\\times 4$ matrix $A$ in Eq.~\\eqref{eq:one_mode_quantum_fisher_information} matrix as compared to $16\\times 16$ matrix $\\mathfrak{M}$ in Eq.~\\eqref{eq:mixed_QFI}. We will use this computational advantage in chapter~\\ref{chap:QFT_metrology} to derive how two-mode Gaussian states perform as probes in estimating space-time parameters.\n\n\n\\subsection{When the Williamson's decomposition is known}\\label{sec:when_Williamson's_decomposition}\nIn section~\\ref{sec:Williamson's_decomposition} we have shown that every covariance matrix can be diagonalized using symplectic matrices. In this section we are going to derive an expression for the quantum Fisher information matrix for the case when the Williamson's decomposition of the covariance matrix is known. As we will see in section~\\eqref{sec:estimation_of_channels}, such expression can be very useful for example when we are trying to find the optimal probe state for the estimation of Gaussian unitary channels.\n\nWe define matrices $P_i:=S^{-1}\\partial_i{S}$ which are elements of the Lie algebra associated with the symplectic group~\\eqref{def:structure_of_S},\n\\[\\label{def:P_i}\nP_i=\n\\begin{bmatrix}\nR_i & Q_i \\\\\n\\ov{Q}_i & \\ov{R}_i\n\\end{bmatrix},\\ \\ P_iK+KP_i^\\dag=0.\n\\]\nRewriting Eq.~\\eqref{eq:mixed_QFI} in terms of the Williamson's decomposition of the covariance matrix, switching to element-wise notation, and simplifying using identities~\\eqref{def:structure_of_S} and~\\eqref{def:P_i}, we derive an exact expression for the quantum Fisher information matrix of Gaussian states in terms of the Williamson's decomposition of the covariance matrix,\n\\[\\label{eq:multimode_QFI}\n\\begin{split}\nH^{ij}(\\boldsymbol{\\epsilon})&=\\!\\!\\!\\sum_{k,l=1}^{N}\\!\\frac{(\\lambda_k\\!-\\!\\lambda_l)^2}{\\lambda_k\\lambda_l\\!-\\!1}\\Re[\\ov{R_{i}}^{kl}R_{j}^{kl}]+\\frac{(\\lambda_k\\!+\\!\\lambda_l)^2}{\\lambda_k\\lambda_l\\!+\\!1}\\Re[\\ov{Q_{i}}^{kl}Q_{j}^{kl}]\\\\\n&+\\sum_{k=1}^{N}\\frac{\\partial_i\\lambda_k\\partial_j\\lambda_k}{\\lambda_k^2-1}+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d},\n\\end{split}\n\\]\nwhere $\\Re$ denotes the real part, $R_i=\\alpha^\\dag\\partial_i{\\alpha}-\\overline{\\beta^\\dag\\partial_i{\\beta}}$ is the skew-Hermitian and $Q_i=\\alpha^\\dag\\partial_i{\\beta}-\\overline{\\beta^\\dag\\partial_i{\\alpha}}$ the (complex) symmetric matrix. This formula represents a multi-parameter generalization of the one-parameter result of~\\cite{Safranek2015b}. The full derivation can be found in appendix~\\ref{app:Williamson's_QFI}.\n\nBeauty of the above formula lies in the fact that we can see each contribution of the different parts of the Gaussian state to the quantum Fisher information. The first part consists of matrices $R_i$ and $Q_i$. These matrices are proportional to the derivative of the diagonalizing symplectic matrix $S$ which encode squeezing and the orientation of squeezing. If the symplectic matrix $S$ is very sensitive to the changes in $\\epsilon$, the derivatives of $S$ are large which leads to the higher quantum Fisher information. In other words, the first part of this equation shows the contribution to the ultimate precision from the changes of squeezing.  Moreover, these terms have constant factors given by symplectic eigenvalues. As we explained in section~\\ref{sec:Williamson's_decomposition}, the symplectic eigenvalues $\\lambda_i$ encode purity or equivalently temperature of the state. The second contribution is proportional to the changes in the symplectic eigenvalues and thus to the changes in purity or the temperature of the state. The third contribution is proportional to the changes in displacement. Summed up, how well we can estimate the vector of parameters $\\boldsymbol{\\epsilon}$ is given by how sensitive is the squeezing, purity, and the displacement of a Gaussian to the changes in the vector of parameters $\\boldsymbol{\\epsilon}$. We will talk more about the effects of temperature in section~\\ref{sec:effects_of_temperature}.\n\nSymplectic eigenvalues are larger than one, $\\lambda_i\\geq1$, and $\\lambda_i=1$ if and only if the mode is in the pure state. The multi-mode Gaussian state is pure when all symplectic eigenvalues are equal to one. To be able to apply Eq.~\\eqref{eq:multimode_QFI} to states which have some eigenvalues equal to one, we have to define these problematic points. There are two possible ways of how to define them. The first choice will give us the quantum Fisher information matrix~\\eqref{def:Information_matrix} for which the Cram\\'er-Rao bound holds, while the second choice will give us the continuous quantum Fisher information matrix~\\eqref{eq:connection_between_Hc_and_H}, which corresponds to the quantity defined by the Bures distance. Recalling the discussion in section~\\ref{QMsec:multi} of the first chapter, the quantum Fisher information matrix for states where some of the modes are pure is obtained simply by not summing over terms which are undefined.  Equivalently, to obtain the quantum Fisher information $H$, for $\\boldsymbol{\\epsilon}$ such that $\\lambda_k(\\boldsymbol{\\epsilon})=\\lambda_l(\\boldsymbol{\\epsilon})=1$ we define\n\\begin{subequations}\\label{def:problematic_points0}\n\\begin{align}\n\\frac{(\\lambda_k\\!-\\!\\lambda_l)^2}{\\lambda_k\\lambda_l\\!-\\!1}(\\boldsymbol{\\epsilon})\\!&:=0,\\\\\n\\frac{\\partial_i\\lambda_k\\partial_j\\lambda_k}{\\lambda_k^2-1}(\\boldsymbol{\\epsilon})\\!&:=0.\n\\end{align}\n\\end{subequations}\nNow we look at the second choice of defining these points which will give the continuous quantum Fisher information matrix $H_c$ (see Eq.~\\eqref{eq:connection_between_Hc_and_H}). Assuming $\\boldsymbol{\\epsilon}$ is such that $\\lambda_k(\\boldsymbol{\\epsilon})=1$ and $\\sigma(\\boldsymbol{\\epsilon})\\in C^{(2)}$, the function $\\lambda_k$ must achieve the local minimum at point $\\boldsymbol{\\epsilon}$ and the Taylor expansion must be of the form $\\lambda_k(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})=1+\\frac{1}{2}\\mathrm{d}\\boldsymbol{\\epsilon}^T\\mathcal{H}_k\\mathrm{d}\\boldsymbol{\\epsilon}+\\cdots$, where $\\mathcal{H}_k^{ij}=\\partial_i\\partial_j\\lambda_k$ is the positive semi-definite matrix called Hessian. To obtain the continuous quantum Fisher information $H_c$, for $\\boldsymbol{\\epsilon}$ such that $\\lambda_k(\\boldsymbol{\\epsilon})=\\lambda_l(\\boldsymbol{\\epsilon})=1$ we define the problematic terms as the limit of the values given by its neighborhood,\n\\begin{subequations}\\label{def:problematic_points}\n\\begin{align}\n\\frac{(\\lambda_k\\!-\\!\\lambda_l)^2}{\\lambda_k\\lambda_l\\!-\\!1}(\\boldsymbol{\\epsilon})\\!&:=\\!\\lim_{\\mathrm{d}\\boldsymbol{\\epsilon}\\rightarrow 0}\\frac{(\\lambda_k(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})\\!-\\!\\lambda_l(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon}))^2}{\\lambda_k(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})\\lambda_l(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})\\!-\\!1}=0,\\\\\n\\frac{\\partial_i\\lambda_k\\partial_j\\lambda_k}{\\lambda_k^2-1}(\\boldsymbol{\\epsilon})\\!&:=\\!\\lim_{\\mathrm{d}\\epsilon_i\\rightarrow 0}\\lim_{\\mathrm{d}\\epsilon_j\\rightarrow 0}\\lim_{\\overset{\\mathrm{d}\\epsilon_r\\rightarrow 0}{r\\neq i,j}}\\!\\frac{\\partial_i\\lambda_k(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})\\partial_j\\lambda_k(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})}{\\lambda_k^2(\\boldsymbol{\\epsilon}+\\mathrm{d}\\boldsymbol{\\epsilon})-1}=\\mathcal{H}_k^{ij}(\\boldsymbol{\\epsilon}).\n\\end{align}\n\\end{subequations}\nThis construction is identical to the way of how the proof of the continuity of the continuous Fisher information was built (see appendix~\\ref{app:discontinuity_of_QFI}, Eq.~\\eqref{eq:construction_of_continuous_QFI}). The above definitions~\\eqref{def:problematic_points0} and~\\eqref{def:problematic_points} allows us to use Eq.~\\eqref{eq:multimode_QFI} for any Gaussian state.\n\nWhen all symplectic eigenvalues are larger than one, we can define Hermitian matrix $\\widetilde{R}_i^{kl}:=\\frac{\\lambda_k-\\lambda_l}{\\sqrt{\\lambda_k\\lambda_l-1}}R_i^{kl}$, symmetric matrix $\\widetilde{Q}_i^{kl}:=\\frac{\\lambda_k+\\lambda_l}{\\sqrt{\\lambda_k\\lambda_l+1}}Q_i^{kl}$, and diagonal matrix $L:=\\mathrm{diag}(\\lambda_1,\\dots,\\lambda_N)$. We rewrite Eq.~\\eqref{eq:multimode_QFI} in an elegant way,\n\\[\\label{eq:exact_multimode_compact}\nH^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{2}\\mathrm{tr}\\big[\\widetilde{R}_i\\widetilde{R}_j^\\dag+\\widetilde{R}_j\\widetilde{R}_i^\\dag+\\widetilde{Q}_i\\widetilde{Q}_j^\\dag+\\widetilde{Q}_j\\widetilde{Q}_i^\\dag\\big]\n+\\mathrm{tr}\\big[(L^2-I)^{-1}\\partial_iL\\partial_jL\\big]+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d}.\n\\]\n\n\\subsection{Regularization procedure}\\label{sec:regularization_procedure}\nWe have seen so far that many expressions for the quantum Fisher information have problems when some of the symplectic eigenvalues is equal to one. Using the knowledge discovered in the previous section we can devise a method how to use the formulae for mixed states~\\eqref{eq:one_mode_quantum_fisher_information},~\\eqref{eq:mixed_QFI},~\\eqref{GeneralQFI},~\\eqref{eq:exact_multimode_compact} to calculate the quantum Fisher information matrix for any state. This regularization procedure is the Gaussian version of the regularization procedure described in appendix~\\ref{app:discontinuity_of_QFI}, Eq.~\\eqref{eq:regularization_procedure_density_matrix}. It goes as follows. First we multiply the covariance matrix by a regularization parameter $\\nu>1$, use any expression for the quantum Fisher information matrix of mixed Gaussian states and calculate $H(\\boldsymbol{\\epsilon})\\equiv H(\\hat{\\rho}_{\\boldsymbol{\\epsilon},\\nu})$, $\\hat{\\rho}_{\\boldsymbol{\\epsilon},\\nu}\\equiv(\\boldsymbol{d}(\\boldsymbol{\\epsilon}),\\nu\\sigma(\\boldsymbol{\\epsilon}))$, and in the end we perform the limit $\\nu\\rightarrow 1$. We need to make sure, however, that such method leads to the proper definition of the problematic points given either by Eq.~\\eqref{def:problematic_points0} or Eq.~\\eqref{def:problematic_points}. Assuming $\\lambda_k(\\boldsymbol{\\epsilon})=\\lambda_l(\\boldsymbol{\\epsilon})=1$ and performing the limit, both problematic terms are set to zero, $\\lim_{\\nu\\rightarrow1}\\frac{(\\nu\\lambda_k-\\nu\\lambda_l)^2}{\\nu\\lambda_k\\nu\\lambda_l-1}=\\lim_{\\nu\\rightarrow1}\\frac{0}{\\nu^2-1}=0$, $\\lim_{\\nu\\rightarrow1}\\frac{(\\nu\\partial_i{\\lambda}_k)^2}{(\\nu\\lambda_k)^2-1}=\\lim_{\\nu\\rightarrow1}\\frac{0}{\\nu^2-1}=0$. This corresponds to the definition~\\eqref{def:problematic_points0} for the quantum Fisher information matrix. To obtain the continuous quantum Fisher information matrix given by definition~\\eqref{def:problematic_points} we need to add a Hessian matrix for every $k$ for which $\\lambda_k(\\boldsymbol{\\epsilon})=1$. Together we write\n\\[\\label{eq:regularization_procedure}\nH, H_c(\\boldsymbol{\\epsilon})=\\lim_{\\nu\\rightarrow1}H\\big(\\boldsymbol{d}(\\boldsymbol{\\epsilon}),\\nu\\sigma(\\boldsymbol{\\epsilon})\\big)+c\\!\\!\\!\\!\\!\\!\\sum_{k:\\lambda_k(\\epsilon)=1}\\!\\!\\!\\!\\!\\!\\mathcal{H}_k(\\boldsymbol{\\epsilon}),\n\\]\nwhere $c=0$ corresponds to the quantum Fisher information matrix $H$, and $c=1$ corresponds to the continuous quantum Fisher information matrix $H_c$.\n\n\\subsection{Limit formula}\nIn previous sections we presented exact formulae, however, in some cases an approximate value is enough. Here we will simplify and generalize the limit expression for the quantum Fisher information given by Eq.~\\eqref{eq:Monras_QFI} to multi-parameter estimation. Defining matrix $A:=K\\sigma$ the limit expression for the quantum Fisher information matrix reads,\n\\[\\label{eq:limit_formula}\nH^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{2}\\sum_{n=1}^\\infty\\mathrm{tr}\\big[A^{-n}\\partial_i{A}A^{-n}\\partial_j{A}]+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d}.\n\\]\nWhen using approximate methods, it is convenient to estimate the error connected to the approximation. In complete analogy of the proof presented for the estimation of a single parameter~\\cite{Safranek2015b}, in appendix~\\ref{app:Remainder} we find a bound on the remainder of the series. Defining $R_M^{ij}:=\\frac{1}{2}\\sum_{n=M+1}^\\infty\\mathrm{tr}\\big[A^{-n}\\!\\partial_i{A}A^{-n}\\!\\partial_j{A}]$, we have\n\\[\n|R_M^{ij}|\\leq\\frac{\\sqrt{\\mathrm{tr}[(A\\partial_iA)^2]}\\sqrt{\\mathrm{tr}[(A\\partial_jA)^2]}}{2\\lambda_{\\mathrm{min}}^{2(M+1)}(\\lambda_{\\mathrm{min}}^2-1)},\n\\]\nwhere $\\lambda_{\\mathrm{min}}:=\\min_{k}\\{\\lambda_k\\}$ is the smallest symplectic eigenvalue of the covariance matrix $\\sigma$. The bound shows that the series converges as a geometric series. This is a very fast convergence -- it is enough to take few terms of the infinite summation and the quantum Fisher information will be very well approximated. On the other hand, the bound for the remainder is of no use when some symplectic eigenvalue is equal to one, i.e., when some of the modes is pure. In the case when some symplectic eigenvalue is equal to one, the infinite series~\\eqref{eq:limit_formula} converges, but it is not absolutely convergent. Moreover, this converging expression does not give the correct quantum Fisher information. Both of these statements can be checked by a careful analysis of the elements in the series given by Eq.~\\eqref{eq:expression_in_terms_of_P} which is explained in more detail in~\\cite{Safranek2015b}. Put simply, some terms which should contribute are identically zero as a consequence of the fact that the limit for the smallest eigenvalue $\\lambda_{\\mathrm{min}}\\rightarrow0$ and the upper limit in the infinite summation in Eq.~\\eqref{eq:limit_formula} do not commute. The correct expression for the states with at least one symplectic eigenvalue equal to one must be obtained using the regularization procedure~\\eqref{eq:regularization_procedure}.\n\nTo prove formula~\\eqref{eq:limit_formula} we show how it connects to Eqs.~\\eqref{eq:multimode_QFI} and~\\eqref{eq:mixed_QFI}. Using\n\\[\\label{eq:expression_in_terms_of_P}\n\\begin{split}\n&\\mathrm{tr}[A^{-n}\\partial_i{A}A^{-n}\\partial_j{A}]=2\\mathrm{tr}[D^{-n+1}\\!K^{-n+1}\\!P_{i}D^{-n+1}\\!K^{-n+1}\\!P_{j}]\\\\\n&-\\mathrm{tr}[D^{-n+2}\\!K^n\\!P_{i}D^{-n}\\!K^n\\!P_{j})]\n-\\mathrm{tr}[D^{-n+2}\\!K^n\\!P_{j}D^{-n}\\!K^n\\!P_{i})]\\\\\n&+\\mathrm{tr}[D^{-n}\\partial_i{D}D^{-n}\\partial_j{D}]\n\\end{split}\n\\]\nand changing to element-wise notation, the infinite sum~\\eqref{eq:limit_formula} turns out to be geometric series in powers of $\\lambda_k$'s which can be evaluated. Then, using $R_i^{kl}=-\\ov{R}_i^{lk}$, $Q_i^{kl}=Q_i^{lk}$ which follows from identity~\\eqref{def:P_i}, we prove that Eq.~\\eqref{eq:limit_formula} simplifies to Eq.~\\eqref{eq:multimode_QFI}. To obtain Eq.~\\eqref{eq:mixed_QFI}, we use $\\sigma^\\dag=\\sigma$, properties of vectorization $\\mathrm{tr}[A^\\dag B]=\\vectorization{A}^\\dag\\vectorization{B}$ and properties of Kronecker product~\\eqref{id:Kronecker_product_ids} to transform the infinite sum in Eq.~\\eqref{eq:limit_formula} into a Neumann series which can be evaluated,\n\\[\n\\begin{split}\n\\sum_{n=1}^\\infty\\mathrm{tr}\\big[A^{-n}\\partial_i{A}A^{-n}\\partial_j{A}]&=\n\\vectorization{\\partial_i\\sigma}^\\dag\\bigg(\\sum_{n=0}^\\infty(\\ov{A}\\otimes A)^{-n}\\bigg)\\big(\\ov{\\sigma}\\otimes\\sigma\\big)^{-1}\\vectorization{\\partial_j\\sigma}\\\\\n&=\\vectorization{\\partial_i\\sigma}^\\dag(I-\\ov{A^{-1}}\\otimes A^{-1})^{-1}\\big(\\ov{\\sigma}\\otimes\\sigma\\big)^{-1}\\vectorization{\\partial_j\\sigma}\\\\\n&=\\vectorization{\\partial_i\\sigma}^\\dag(\\ov{\\sigma}\\otimes\\sigma-K\\otimes K)^{-1}\\vectorization{\\partial_j\\sigma}\n\\end{split}\n\\]\nCombining the above expression with Eq.~\\eqref{eq:limit_formula} gives Eq.~\\eqref{eq:mixed_QFI}.\n\n\\subsection{Pure states}\\label{sec:pure_states_QM}\n\nCombining Eq.~\\eqref{eq:limit_formula}, the regularization procedure~\\eqref{eq:regularization_procedure}, and $A^2(\\boldsymbol{\\epsilon})=I$, which holds for pure states, we derive the quantum Fisher information matrix for states which are pure at point $\\boldsymbol{\\epsilon}$,\n\\[\\label{eq:pure_non-elegant}\nH^{ij},H_c^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{4}\\mathrm{tr}[\\sigma^{-1}\\partial_i\\sigma\\sigma^{-1}\\partial_j\\sigma]+c\\sum_{k}\\mathcal{H}_k^{ij}+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d}.\n\\]\nIf either $c=0$ (the definition corresponding to $H$), or if the state remains pure when the vector of parameters $\\boldsymbol{\\epsilon}$ is slightly varied, i.e., $\\mathcal{H}_k(\\boldsymbol{\\epsilon})=0$, the above expression reduces to the known formula for pure states given by Eq.~\\eqref{eq:nu_pure} which has been derived in~\\cite{Pinel2012a}.\n\nCalculating the continuous quantum Fisher information matrix $H_c$ from Eq.~\\eqref{eq:pure_non-elegant} requires to calculate derivatives of symplectic eigenvalues. However, it is possible to find an alternative form which avoids that need,\n\\[\\label{eq:pure_elegant}\nH_c^{ij}(\\boldsymbol{\\epsilon})=\\frac{1}{4}\\big(2\\mathrm{tr}[\\sigma^{-1}\\partial_i\\partial_j\\sigma]-\\mathrm{tr}[\\sigma^{-1}\\partial_i\\sigma\\sigma^{-1}\\partial_j\\sigma]\\big)+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d},\n\\]\nwhich generalizes the result of~\\cite{Safranek2015b} to the multi-parameter estimation. To show that Eq.~\\eqref{eq:pure_non-elegant} for $c=1$ and Eq.~\\eqref{eq:pure_elegant} are identical, one needs to use the Williamson's decomposition of the covariance matrix $\\sigma=SDS^\\dag$, find~$\\mathrm{tr}[\\sigma^{-1}\\partial_i\\partial_j\\sigma]$ and $\\mathrm{tr}[\\sigma^{-1}\\partial_i\\sigma\\sigma^{-1}\\partial_j\\sigma]$ in terms of $K,P_i,P_{ij}:=S^{-1}\\partial_i\\partial_jS$ (which gives similar expressions to Eq.~\\eqref{eq:expression_in_terms_of_P}), and use identities~\\eqref{def:structure_of_S},~\\eqref{def:P_i}, $P_{ij}K+P_iKP_j^\\dag+P_jKP_i^\\dag+KP_{ji}^\\dag=0$. Both~\\eqref{eq:pure_non-elegant} and~\\eqref{eq:pure_elegant} then reduce to the same expression,\n\\[\\label{eq:QFI_almost_pure}\nH_c(\\epsilon)=\\frac{1}{2}\\Big(\\mathrm{tr}\\big[P_iP_j\\big]-\\mathrm{tr}\\big[KP_iKP_j\\big]\\Big)+\\frac{1}{2}\\mathrm{tr}[\\partial_i\\partial_jD].\n\\]\nIn analogy of Eq.~\\eqref{eq:nu_pure_simplified}, Eqs.~\\eqref{eq:pure_non-elegant},~\\eqref{eq:pure_elegant} can be further simplified by using $\\sigma^{-1}=K\\sigma K=AK$ to avoid the necessity of inverting the covariance matrix.\n\nEq.~\\eqref{eq:QFI_almost_pure} is an expression for the quantum Fisher information in terms of the decomposition of the covariance matrix. This expression can be further simplified by expressing $P_i$ in terms of its submatrices. In analogy of Eq.~\\eqref{eq:exact_multimode_compact} we write\n\\[\\label{eq:exact_multimode_compact_pure}\nH^{ij},H_c^{ij}(\\boldsymbol{\\epsilon})=\\mathrm{tr}\\big[{Q}_i{Q}_j^\\dag+{Q}_j{Q}_i^\\dag\\big]\n+c\\,\\mathrm{tr}\\big[\\partial_i\\partial_jL\\big]+2\\partial_i\\boldsymbol{d}^\\dag\\sigma^{-1}\\partial_j\\boldsymbol{d}.\n\\]\n\n\\subsection{Problems at points of purity}\\label{sec:problems_at_pops}\nIn section~\\ref{sec:when_Williamson's_decomposition}  we showed two possible definitions of the problematic points of the quantum Fisher information and devised the regularization procedure in a way that takes this definition into account. Here we illustrate this difference more concretely on an example and show how different definitions relate to mixed states.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{problematicexample3.pdf}\n\\caption{The quantum Fisher information for the estimation of the parameter of the single mode state~\\eqref{eq:problematic_example}.}\\label{fig:problematic_example}\n\\end{figure}\n\n\\begin{example}\\label{ex:problematic_example}\n\\emph{Let $\\hat{\\rho}$ be a displaced two-mode squeezed thermal state with the covariance matrix $\\sigma=S_T(\\epsilon)\\lambda S_T^\\dag(\\epsilon)$ and the displacement vector $\\boldsymbol{d}=(\\epsilon,\\epsilon,\\epsilon,\\epsilon)^T$. We assume that experimenter has access only to the first mode, i.e., we trace over the second mode. The resulting state is a single mode state with moments\\footnote{We choose $\\boldsymbol{d}$ non-zero to avoid the problem with identifiability of $\\epsilon$. For $\\boldsymbol{d}=\\boldsymbol{0}$ values $\\epsilon$ and $-\\epsilon$ would produce exactly the same statistics because $\\cosh$ appearing in Eq.~\\eqref{eq:problematic_example} is an even function. Consequently an experimentalist is not be able to distinguish between the two values, which would suggest the quantum Fisher information should be zero. Such an example favors the first definition of problematic points~\\eqref{def:problematic_points0}. However, we wanted to illustrate differences between the two definitions on a less trivial example where the choice of an appropriate definition is not so clear.}\n\\[\\label{eq:problematic_example}\n\\boldsymbol{d}=\\begin{bmatrix}\n\\epsilon \\\\\n\\epsilon\n\\end{bmatrix}, \\quad \\sigma=\\begin{bmatrix}\n\\lambda\\cosh (2\\epsilon) & 0 \\\\\n0 & \\lambda\\cosh (2\\epsilon)\n\\end{bmatrix}.\n\\]\nThe quantum Fisher information for the estimation of the parameter $\\epsilon$ is obtained either from Eq.~\\eqref{eq:one_mode_quantum_fisher_information} or Eq.~\\eqref{eq:multimode_QFI},\n\\[\nH(\\lambda,\\epsilon)=\\frac{4\\lambda^2\\sinh^2(2\\epsilon)}{\\lambda^2\\cosh^2(2\\epsilon)-1}\n+\\frac{4}{\\lambda \\cosh(2\\epsilon)}.\n\\]\nThe first term corresponds to the Fisher information gained from the change of purity, while the second term corresponds to the Fisher information gained from the change of the displacement. The graph of this function is shown on Fig.~\\ref{fig:problematic_example}. This function of two variables is continuous everywhere apart from the point $(\\lambda,\\epsilon)=(1,0)$ where it is not defined. There are two reasonable ways of how to define this point.\nThe first way will ensure that for any $\\epsilon$ the quantum Fisher information is continuous in $\\lambda$, but it will not be continuous in $\\epsilon$ for $\\lambda=1$. We define,\n\\[\\label{def:problematic_points0_example}\nH(1,0):=\\lim_{\\lambda\\rightarrow1}\\lim_{\\epsilon\\rightarrow0}H(\\lambda,\\epsilon)=4.\n\\]\nSuch a definition corresponds to the choice of the definition of problematic points for the quantum Fisher information, Eq.~\\eqref{def:problematic_points0}. The quantum Fisher information is given as the limit of the quantum Fisher information of close mixed states $\\hat{\\rho}_{\\lambda+\\mathrm{d}\\lambda,\\epsilon}$.\nThe second way will ensure that for any $\\lambda$ the Quantum Fisher information is continuous in $\\epsilon$, but it will not be continuous in $\\lambda$ for $\\epsilon=0$. We define\n\\[\nH(1,0):=\\lim_{\\epsilon\\rightarrow0}\\lim_{\\lambda\\rightarrow1}H(\\lambda,\\epsilon)=8.\n\\]\nSuch a definition corresponds to the choice of the definition of problematic points for the continuous quantum Fisher information, Eq.~\\eqref{def:problematic_points}. The quantum Fisher information is given as the limit of the quantum Fisher information of close pure states $\\hat{\\rho}_{\\lambda,\\epsilon+\\mathrm{d}\\epsilon}$.}\n\\end{example}\n\nThe two choices of the definition of problematic points correspond to the difference on a more fundamental level. From the physical point of view, the pro for the first definition is that pure states, in fact, do not exist because their existence is forbidden by the third thermodynamical law. There are always some thermal fluctuations which will result for the state not to be pure. According to this argument, the quantum Fisher information for pure state should be calculated as the limit of mixed states. The pro for the second definition is that $\\epsilon=0$ is the point of measure zero and thus cannot be ever measured. The real -- measured -- value of $\\epsilon$ will be close to zero but never equal. It is important to remind, however, that the right figure of merit for the Cram\\'er-Rao bound is the quantum Fisher information given by the first definition of problematic points, Eqs.~\\eqref{def:problematic_points0} and \\eqref{def:problematic_points0_example} respectively\n\nSuch a problematic behavior of the quantum Fisher information can never be avoided when dealing with pure states which change its purity. The issue becomes especially problematic when we want to analyze the quantum Fisher information depending on two parameters as illustrated on example~\\ref{ex:problematic_example}. We formalize this problem in the following theorem,\n\\begin{theorem}\nLet quantum state $\\hat{\\rho}\\equiv(\\boldsymbol{d}(\\epsilon),\\sigma(p,\\epsilon))$ be a smooth function of two parameters, where $\\epsilon$ is the parameter we are going to estimate, and where parameter $p$ is encoded only in the symplectic eigenvalues of the covariance matrix. If there exists a symplectic eigenvalue $\\lambda_k(p,\\epsilon)$ of the covariance matrix such that $\\lambda_k(p,\\epsilon)=1$, but $\\lambda_k(p+\\mathrm{d} p,\\epsilon)>1$ and $\\partial_{\\epsilon\\epsilon}\\lambda_k(p,\\epsilon)\\neq 0$, then the multi-parameter Taylor expansion of the form\n\\[\\label{not_existing_form}\n\\begin{split}\nH(\\sigma(p+\\mathrm{d} p,\\epsilon+\\mathrm{d} \\epsilon))&=H(\\sigma(p,\\epsilon))+\\partial_pH(\\sigma(p,\\epsilon))\\mathrm{d} p+\\partial_\\epsilon H(\\sigma(p,\\epsilon))\\mathrm{d} \\epsilon\\\\\n&+\\mathcal{O}(\\mathrm{d} p^2,\\mathrm{d} p\\mathrm{d} \\epsilon, \\mathrm{d} \\epsilon^2)\n\\end{split}\n\\]\ndoes not exist for either choice of definition of problematic points.\n\\end{theorem}\nThe proof is based on the fact that either $\\partial_pH(\\sigma(p,\\epsilon))=-\\infty$ or $\\partial_\\epsilon H(\\sigma(p,\\epsilon))=+\\infty$ from which the theorem immediately follows. The full proof can be found in appendix~\\eqref{app:not_existing_form}. Intuition why this theorem should hold can be also obtained by studying the second term in Eq.~\\eqref{eq:H_expanded}, which shows how the quantum Fisher information behave around problematic points.\n\nThe problem of points of purity is not limited to Gaussian states. As we can see on Fig.~\\ref{BobsQFIexample3} or Fig.~\\ref{fig:Continuous_QFI}, a non-Gaussian state can suffer of the same discontinuity.\nWe will encounter this problem again in the next chapter where we are trying to expand the quantum Fisher information at the same time in the small space-time parameter $\\epsilon$ and in a small parameter connected to temperature $Z\\,=\\,e^{-E_{m}\/2T}$.\n\n\\subsection{Unitary encoding operations}\n\nLet us assume that a single parameter was encoded into an initial Gaussian state $\\hat{\\rho}_0$ via a Gaussian unitary transformation $\\hat{U}_\\epsilon$ which forms a one parameter group. We construct this unitary from the general Gaussian unitary~\\eqref{def:Gaussian_unitary},\n\\[\\label{def:Gaussian_unitary_epsilon}\n\\hat{U}_\\epsilon=\\exp\\big(\\big(\\tfrac{i}{2}\\boldsymbol{\\hat{A}}^\\dag W \\boldsymbol{\\hat{A}}+\\boldsymbol{\\hat{A}}^\\dag K \\boldsymbol{\\gamma}\\big)\\epsilon\\big).\n\\]\nThe final state $\\hat{\\rho}_\\epsilon=\\hat{U}_\\epsilon\\hat{\\rho}_0\\hat{U}_\\epsilon^\\dag$ is given by the first and the second moments\n\\begin{subequations}\\label{eq:moments_epsilon}\n\\begin{align}\n\\boldsymbol{d}_\\epsilon&=S_\\epsilon\\boldsymbol{d}_0+\\boldsymbol{b}_\\epsilon,\\\\\n\\sigma_\\epsilon&=S_\\epsilon \\sigma_0 S_\\epsilon^\\dag,\n\\end{align}\n\\end{subequations}\nwhere $S_\\epsilon=e^{iKW\\epsilon}$ and $\\boldsymbol{b}_\\epsilon=\\big(\\int_0^1e^{iKWt\\epsilon}\\mathrm{d}t\\big)\\boldsymbol{\\gamma}\\epsilon$. We have $\\dot{\\sigma}_\\epsilon=S_\\epsilon[iKW,\\sigma_0]S_\\epsilon^\\dag$, $\\dot{\\boldsymbol{d}}_\\epsilon=S_\\epsilon(iKW\\boldsymbol{d}_0+\\boldsymbol{\\gamma})$, and the quantum Fisher information is independent of $\\epsilon$.\n\nWe can use Eq.~\\eqref{eq:mixed_QFI} and properties of the Kronecker product~\\eqref{id:Kronecker_product_ids} to derive the quantum Fisher information for mixed Gaussian states,\n\\begin{subequations}\\label{eq:mixed_QFI_unitary}\n\\begin{align}\nH(\\epsilon)&=\\frac{1}{2}\\mathrm{vec}\\big[[iKW,\\sigma_0]\\big]^\\dag\\mathfrak{M}^{-1}\\mathrm{vec}\\big[[iKW,\\sigma_0]\\big]\n+2(iKW\\boldsymbol{d}_0+\\gamma)^\\dag\\sigma_0^{-1}(iKW\\boldsymbol{d}_0+\\gamma),\\\\\n\\mathfrak{M}&=\\ov{\\sigma}_0\\otimes\\sigma_0-K\\otimes K.\n\\end{align}\n\\end{subequations}\nSimilarly, the quantum Fisher information for pure Gaussian states is obtained from Eq.~\\eqref{eq:pure_non-elegant},\n\\[\nH(\\epsilon)=\\frac{1}{2}\\mathrm{tr}\\big[[iKW,\\sigma_0]\\ \\!\\sigma_0^{-1}iKW\\big]+2(iKW\\boldsymbol{d}_0+\\gamma)^\\dag\\sigma_0^{-1}(iKW\\boldsymbol{d}_0+\\gamma).\\\\\n\\]\nThe scenario in which the Willimson's decomposition of the covariance matrix is known is discussed in section~\\ref{sec:general_method}.\n\nThe above formulae can be generalized to multiparameter estimation where parameters are encoded via a Gaussian unitary $\\hat{U}_{\\boldsymbol{\\epsilon}}=\\mathrm{exp}\\big(\\sum_i\\hat{G}_i\\epsilon_i\\big)$. However, in the case when $\\hat{G}_i$ do not commute the quantum Fisher information matrix cannot be simplified and in general depends on the vector of parameters $\\boldsymbol{\\epsilon}$.\n\n\\section{Estimation of Gaussian unitary channels}\\label{sec:estimation_of_channels}\n\nWe have shown in the first chapter that finding optimal probe states for the estimation of quantum channels is one of the main tasks of quantum metrology. In this section we develop a practical method for finding optimal Gaussian probe states and illustrate this method on the estimation of the most common Gaussian unitary channels. This section is partially based on results we published in~\\cite{safranek2016optimal}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1\\linewidth]{scheme1.pdf}\n\\caption{Scheme of the usual metrology setting illustrated on a one-mode Gaussian probe state. First, we prepare the state by using various Gaussian operations, then we feed the state into the channel we want to estimate, perform an appropriate measurement, and an estimator $\\hat{\\epsilon}$ gives us an estimate of the true value of the parameter. In this section we are interested in optimizing over the preparation stage for a given encoding Gaussian unitary channel $\\hat{U}(\\epsilon)$.\n}\\label{fig:scheme}\n\\end{figure}\n\nThe typical setup is given by scheme~\\eqref{scheme_for_estimating_channels} which is illustrated on Fig.~\\ref{fig:scheme} where a single mode Gaussian states is being used to estimate a one-mode Gaussian channel. The probe state is fed into the channel, the channel encodes the parameter on the state of the system and, finally, measurements are performed with the aim of gaining maximal information about the parameter.\n\nGaussian states are usually not optimal probe states for estimating Gaussian unitary channels. Non-gaussian states such as GHZ states usually perform as better probes. However, previous theoretical studies show that Gaussian states can be still effectively used for the estimation of Gaussian channels such as phase-changing~\\cite{Monras2006a,Aspachs2008a,Sparaciari2015a,Sparaciari2015b}, squeezing~\\cite{Milburn1994a,Chiribella2006a,Gaiba2009a}, two-mode squeezing and mode-mixing channels~\\cite{Gaiba2009a}. Previous studies analyzed specific channels and for each channel \\emph{only one} probe state achieving the Heisenberg limit was found. Moreover, Gaussian state metrology is often restricted to pure states. Less attention has been given to thermal states, which are of great relevance in practice. In the laboratory, quantum states can never be isolated from the environment which thermalizes them. In this section we develop a formalism that can effectively be used to study \\emph{any} Gaussian probe state for \\emph{any} one- and two-mode Gaussian unitary channel. We also develop methods to find \\emph{all} optimal Gaussian probe states for these channels. We take advantage of recent progress in the phase-space formalism of Gaussian states. We make use of Euler's decomposition of symplectic matrices~\\eqref{def:S_decomposition}, the Williamson's decomposition of the covariance matrix in the complex form~\\eqref{def:Williamson's_decomposition}, and the expression for the quantum Fisher information in terms of the Williamson's decomposition of the covariance matrix~\\eqref{eq:multimode_QFI}. These techniques enable us to simplify expressions so that formulae can be easily used in practical applications. As an example, we derive optimal states for channels that, to our knowledge, have not been studied before. These are the channels such as generalized mode-mixing, two-mode squeezing, or combining the phase-change and squeezing. Interestingly, we find that in the estimation of two-mode channels, separable states consisting of two one-mode squeezed states perform as well as their entangled counterpart: two-mode squeezed states. This shows that entanglement between the modes does not enhance precision in this case.\n\nOur formalism also enables us to further our understanding of the effects of temperature in probe states. It has been reported in~\\cite{Aspachs2008a} that higher temperature in squeezed thermal states can enhance phase estimation, while higher temperature of displaced thermal states is detrimental. We show that the effects of thermalised probe states on the estimation of Gaussian channels are generic, i.e., for all Gaussian unitary channels, temperature effects are always manifested in multiplicative factors of four types. Two of the factors correspond to the ones previously found in~\\cite{Aspachs2008a}. The other two factors, that we discovered, show that not only temperature of the probe state, but also temperature difference between different modes of the probe state helps the estimation.\n\nThis section is organized as follows: first we present a general framework for finding optimal Gaussian probe states for Gaussian unitary channels. Then we study the effects of temperature on the estimation strategy and show that effects of temperature are generic. We apply our formalism to present concrete examples for one- and two-mode Gaussian unitary channels and we generalize bounds on the ultimate limit of precision of estimation of Gaussian channels found in~\\cite{Milburn1994a,Chiribella2006a,Monras2006a,Aspachs2008a,Gaiba2009a}. The previous examples included mixed states, however we design a simplified way of how to find optimal pure probe states for the estimation of an arbitrary Gaussian unitary channel. Finally we discuss the connection between entanglement of Gaussian probe states with the Heisenberg scaling and summarize our findings.\n\n\\subsection{General method}\\label{sec:general_method}\n\nIn this subsection we are going to describe the general method of finding the optimal probe states. Put simply, first we take parametrization of general one- or two- mode Gaussian states and use the formula for the quantum Fisher information in terms of the Williamson's decomposition derived in the previous section. Finally we maximize the quantum Fisher information by choosing appropriate parameters in the parametrization of the probe state.\n\nLet us assume we have full control over the preparation of the initial probe state $\\hat{\\rho}_0\\equiv(\\boldsymbol{d}_0,\\sigma_0)$, with the Williamson decomposition $\\sigma_0=S_0D_0S_0^\\dag$ of the covariance matrix. The diagonal matrix, $D_0$, represents a thermal state and the symplectic matrices, $S_0$, represent the operations we are going to perform on this thermal state. After the probe state is created, we feed it into a Gaussian channel encoding the parameter we want to estimate.\n\n\nUsing Eqs.~\\eqref{def:transformation} we find the final state is given by the first and the second moments\n\\begin{subequations}\\label{eq:moments_epsilon}\n\\begin{align}\n\\boldsymbol{d}_\\epsilon&=S_\\epsilon\\boldsymbol{d}_0+\\boldsymbol{b}_\\epsilon,\\\\\n\\sigma_\\epsilon&=S_\\epsilon S_0D_0S_0^\\dag S_\\epsilon^\\dag.\n\\end{align}\n\\end{subequations}\nAs the covariance matrix appears precisely in the form of the Williamson decomposition, we can use formula~\\eqref{eq:multimode_QFI} directly. Applying Eqs.~\\eqref{def:structure_of_S}, \\eqref{def:P_i}, and \\eqref{eq:moments_epsilon}, we derive\n\\begin{subequations}\\label{eq:eq_for_general_method}\n\\begin{align}\n&P=S_0^{-1}P_\\epsilon S_0,\\label{eq:first_part}\\\\\n&\\sum_{k=1}^N\\frac{\\dot{\\lambda_k}^2}{\\lambda_k^2-1}=0,\\label{eq:second_part}\\\\\n&2\\dot{\\boldsymbol{d}}^\\dag\\sigma^{-1}\\dot{\\boldsymbol{d}}=2(P_\\epsilon\\boldsymbol{d}_0+S_\\epsilon^{-1}\\dot{\\boldsymbol{b}}_\\epsilon)^\\dag\\sigma_0^{-1}(P_\\epsilon\\boldsymbol{d}_0+S_\\epsilon^{-1}\\dot{\\boldsymbol{b}}_\\epsilon),\\label{eq:displacement_part}\n\\end{align}\n\\end{subequations}\nwhere we have denoted $P_\\epsilon:=S_\\epsilon^{-1}\\dot{S}_\\epsilon$. Due to the unitarity of the channel the symplectic eigenvalues do not change, and the expression~\\eqref{eq:second_part} vanishes. This scheme can be used for any Gaussian unitary channel. However, in next sections we are going to study Gaussian unitary channels which form a one-parameter unitary group given by Eq.~\\eqref{def:Gaussian_unitary_epsilon}. In that case $P_\\epsilon=iKW$, $\\dot{\\boldsymbol{b}}_\\epsilon=S_\\epsilon\\boldsymbol{\\gamma}$, and the quantum Fisher information is independent of $\\epsilon$. The problem of finding the optimal states is thus reduced to finding the optimal parameters from the parametrization of the initial state for a given Gaussian channel represented by a constant matrix $W$ and a constant vector $\\boldsymbol{\\gamma}$.\n\n\\subsection{Effects of temperature}\\label{sec:effects_of_temperature}\n\nIt is interesting to note that the symplectic eigenvalues in Eq.~\\eqref{eq:multimode_QFI} appear only in a form of multiplicative factors, independent of other parameters and channels we estimate.\n\nThis is particularly interesting from a physical point of view because, as we explained in sections~\\ref{sec:thermal_state} and~\\ref{sec:Williamson's_decomposition}, the symplectic eigenvalues encode temperature. The symplectic eigenvalue describing a thermal state of the harmonic oscillator with frequency $\\omega_k$ is given by $\\lambda_k=\\coth(\\frac{\\omega_k\\hbar}{2kT})$, or alternatively, $\\lambda_k=1+2n_{{\\mathrm{th}}k}$ where $n_{{\\mathrm{th}}k}$ denotes the mean number of thermal bosons in each mode.\n\nIn Eq.~\\eqref{eq:multimode_QFI} we can identify four types of multiplicative factors given by symplectic eigenvalues, $\\frac{\\lambda_k^2}{1+\\lambda_k^2}$, $\\frac{(\\lambda_k+\\lambda_l)^2}{\\lambda_k\\lambda_l+1}$, $\\frac{(\\lambda_k-\\lambda_l)^2}{\\lambda_k\\lambda_l-1}$, and $\\frac{1}{\\lambda_k}$. First, let us focus on effects of temperature given by the first three types of factors which multiply matrices $R$ and $Q$. These represents sensitivity of squeezing and orientation of squeezing of the probe state with respect to the channel we estimate.\nThe first type of factor, $\\frac{\\lambda_k^2}{1+\\lambda_k^2}$, is one of the two to appear for the \\emph{isothermal} (sometimes called \\emph{isotropic}) states for which all symplectic eigenvalues are equal. This class also encompasses all pure states. Because $1\\leq\\lambda_k\\leq+\\infty$, we have $\\frac{1}{2}\\leq\\frac{\\lambda_k^2}{1+\\lambda_k^2}\\leq1$, where the lower bound is attained by pure states and the upper bound by thermal states with infinite temperature. This means that for isothermal states temperature helps the estimation with maximal enhancement of a factor of two, a fact already noted in~\\cite{Aspachs2008a}. Next, for mixed multi-mode states we have the second and third type of factors, $\\frac{(\\lambda_k-\\lambda_l)^2}{\\lambda_k\\lambda_l-1}$ and $\\frac{(\\lambda_k+\\lambda_l)^2}{\\lambda_k\\lambda_l+1}$. These terms become especially important when there is a large difference between the symplectic eigenvalues. Considering $\\lambda_l \\rightarrow 1$ we have\n\\begin{subequations}\n\\begin{align}\n\\frac{(\\lambda_k-\\lambda_l)^2}{\\lambda_k\\lambda_l-1}&\\longrightarrow\\lambda_k-1=2n_{{\\mathrm{th}}k},\\\\\n\\frac{(\\lambda_k+\\lambda_l)^2}{\\lambda_k\\lambda_l+1}&\\longrightarrow\\lambda_k+1=2(n_{{\\mathrm{th}}k}+1).\n\\end{align}\n\\end{subequations}\nGenerally, assuming $\\lambda_k\\gg \\lambda_l$ yields\n\\[\n\\frac{(\\lambda_k-\\lambda_l)^2}{\\lambda_k\\lambda_l-1}\\approx\\frac{(\\lambda_k+\\lambda_l)^2}{\\lambda_k\\lambda_l+1}\\approx\\frac{2n_{{\\mathrm{th}}k}}{2n_{{\\mathrm{th}}l}+1}.\n\\]\nThis shows that the enhancement by temperature difference is no longer bounded by some fixed value as in the previous case.\n\nIf we keep one mode sufficiently cool and the other hot, or if one mode has a high frequency and the other a low frequency, we can, in principle, achieve an infinite enhancement in the estimation of the unknown parameter. In general, states with a large variance in energy, which in this case is in the form of thermal fluctuations, have a higher ability to carry information, and thus can carry more information about the parameter we want to estimate. We will refer to this phenomenon as temperature-enhanced estimation.\n\nWe have shown that temperature and temperature difference enhances the first two terms in Eq.~\\eqref{eq:multimode_QFI} due to first three types of factors. However, the opposite behaviour is observed in the last term. This last term shows how sensitive the displacement is to the small changes in the parameter of the channel. Factors of the fourth type, $\\frac{1}{\\lambda_k}$, are hidden in the inverse of the initial covariance matrix in this last term as shown in Eq.~\\eqref{eq:displacement_part}. As temperature rises and the symplectic eigenvalues grow to infinity, this factor goes to zero and the precision in estimation diminishes.\n\nLet us look at what these factors mean physically for different probe states. Channels quadratic in the field operators do not affect the displacement of non-displaced states such as squeezed thermal states. This means that the precision in estimation of such channels when using non-displaced states will be affected only by factors of the first three types. When using a squeezed thermal state as a probe, temperature and temperature difference in different modes of this probe will always help the estimation. In contrast, when a displaced thermal state is used as a probe, the effect of quadratic channels on the squeezing of such probes is very minor. In other words, covariance matrix of displaced thermal states is almost unchanged by a quadratic channel and completely unchanged in the case of passive channels. Therefore the first three types of factor play a minor role. Quadratic channels will greatly change the displacement of a displaced thermal state therefore the factor of the last type $\\frac{1}{\\lambda_k}$ is of great relevance. Higher temperature in displaced thermal states decreases the precision of estimation of quadratic channels. Put simply, it is good to have either a hot squeezed state or a cold displaced state as a probe.\n\n\\subsection{Optimal probe states for the estimation of Gaussian unitary channels}\nIn this subsection we illustrate the general methods of finding optimal probe state for the estimation of parameters of special class of Gaussian unitary channels. These channels will be one- and two- mode  unitary channels generated by a purely quadratic Hamiltonian.\n\n\\subsubsection{Estimation of one-mode channels: combining squeezing and phase-change}\n\nFirst we are going to look at the estimation of one-mode Gaussian unitary channels with purely quadratic generators. These channels are fully parametrized by Eq.~\\eqref{eq:W1}. We will look on a channel which combines the squeezing and a phase-change which is constructed by substituting $\\theta\\rightarrow\\omega_p\\epsilon$, $r\\rightarrow \\omega_s\\epsilon$ in Eq.~\\eqref{eq:W1}. The resulting symplectic matrix $S_\\epsilon:=e^{iKW}$ then represents an encoding operator $\\hat{S}_\\epsilon=\\exp((-i\\omega_p\\hat{a}^\\dag \\hat{a}-\\frac{\\omega_s}{2}(e^{i\\chi}\\hat{a}^{\\dag2}-e^{-i\\chi}\\hat{a}^{2}))\\epsilon)$. $\\omega_p$ and $\\omega_s$ are the frequencies with which the state is rotated and squeezed respectively. We assume these frequencies and the squeezing angle $\\chi$ are known, so $\\epsilon$ is the only unknown parameter we are trying to estimate.\n\nUsing the general probe state~\\eqref{eq:general_1mode_state} we derive the quantum Fisher information,\n\\[\\label{eq:one_mode_channel_QFI}\n\\begin{split}\nH(\\epsilon)&=\\frac{4\\lambda_1^2}{1+\\lambda_1^2}\\Big(\\omega_s^2\\big(\\cos^2(2\\theta+\\chi)+\\cosh^2(2r)\\sin^2(2\\theta+\\chi)\\big)\\\\\n&+\\omega_p^2\\sinh^2(2r)-\\omega_s\\omega_p\\sin(2\\theta+\\chi)\\sinh(4r)\\Big)\\\\\n&+\\frac{4\\norm{{d}}^2}{\\lambda_1}\\Big(e^{2r}\\big(\\omega_s\\cos(\\theta-\\phi_d+\\chi)-\\omega_p\\sin(\\theta+\\phi_d)\\big)^2\\\\\n&+e^{-2r}\\big(\\omega_s\\sin(\\theta-\\phi_d+\\chi)+\\omega_p\\cos(\\theta+\\phi_d)\\big)^2\\Big).\n\\end{split}\n\\]\nAssuming all $\\omega_s,\\omega_p,r$ are positive, this function clearly achieves its maximum when $\\sin(2\\theta+\\chi)=-1$, $\\sin(\\theta-\\phi_d+\\chi)=1$, and $\\sin(\\theta+\\phi_d)=-1$. For example, these conditions are fulfilled when $\\theta=-\\frac{\\chi}{2}-\\frac{\\pi}{4}$, $\\phi_d=\\frac{\\chi}{2}-\\frac{\\pi}{4}$, which leads to\n\\[\\label{eq:one_mode_channel_QFI_max}\nH_{\\mathrm{max}}(\\epsilon)=\\frac{4\\lambda_1^2}{1+\\lambda_1^2}\\big(\\omega_s\\cosh(2r)+\\omega_p\\sinh(2r)\\big)^2+\\frac{4\\norm{{d}}^2}{\\lambda_1}e^{2r}\\big(\\omega_s+\\omega_p\\big)^2.\n\\]\nThis shows that both displacement and squeezing, if properly oriented, enhance the estimation precision. However, to study what strategy is the best when only a fixed amount of energy of the probe state is available we use the relation for the mean total number of Bosons given by Eq.~\\eqref{eq:mean_number_of_particles} and derive\n\\[\\label{eq:mean_number_of_bosons_one}\nn=n_{{d}}+n_{\\mathrm{th}}+(1+2n_{\\mathrm{th}})\\sinh^2r,\n\\]\nwhere $n_{{d}}:=\\norm{{d}}^2$ denotes the mean number of Bosons coming from the displacement. Together with the relation $\\lambda_1=1+2n_{\\mathrm{th}}$ Eq.~\\eqref{eq:one_mode_channel_QFI_max} transforms into\n\\[\n\\begin{split}\nH_{\\!\\mathrm{max}}(\\epsilon)&=\\frac{2\\Big(\\omega_s(2n\\!-\\!2n_{{d}}\\!+\\!1)+2\\omega_p\\sqrt{\\!n\\!-\\!n_{{d}}\\!-\\!n_{\\mathrm{th}}\\!}\\sqrt{\\!n\\!+\\!1\\!-\\!n_{{d}}\\!+\\!n_{\\mathrm{th}}\\!}\\Big)^{\\!2}\\!\\!\\!\\!}{1\\!+\\!2n_{\\mathrm{th}}(1\\!+\\!n_{\\mathrm{th}})}\\\\ &+\\frac{4n_{{d}}\\big(2n\\!-\\!2n_{{d}}\\!+\\!1\\!+\\!2\\sqrt{\\!n\\!-\\!n_{{d}}\\!-\\!n_{\\mathrm{th}}\\!}\\sqrt{\\!n\\!+\\!1\\!-\\!n_{{d}}\\!+\\!n_{\\mathrm{th}}\\!}\\big)^2\\!\\!\\!}{(1\\!+\\!2n_{\\mathrm{th}})^2}(\\omega_s\\!+\\!\\omega_p)^2\n\\end{split}\n\\]\nKeeping $n$ fixed, the maximum is achieved when $n_{\\mathrm{th}}=n_{{d}}=0$, i.e., when all available energy is invested into squeezing, which coincides with some special cases~\\cite{Aspachs2008a,Gaiba2009a}. The quantum Fisher information then reaches the Heisenberg limit,\n\\[\\label{eq:optimized_QFI_squeezing_phase}\nH_{\\mathrm{max}}(\\epsilon)=2\\big(\\omega_s(2n+1)+\\omega_p2\\sqrt{n}\\sqrt{1+n}\\big)^2.\n\\]\nOn the other hand, if we decide to invest only into the displacement (which corresponds to the coherent probe state), i.e., $n=n_{{d}}$, we obtain the shot-noise limit $H_{\\mathrm{max}}(\\epsilon)=2\\omega_s^2+4n(\\omega_s+\\omega_p)^2$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fiveCov2.pdf}\n\\caption{Estimation of the one-mode squeezing channel $\\hat{S}(\\epsilon)$ around point $\\epsilon=0$ using various squeezed states. Each squeezed state is represented by a covariance matrix depicted as an ellipse. Each state has the same energy, but the initial rotation varies. The initial squeezing was set to $r=0.8$, the initial displacement $\\tilde{d_0}=0$, and the final squeezing $\\epsilon=0$~(blue with full line) or $\\epsilon=0.1$~(orange with dashed line). The initial rotation from left to right $\\theta=0,\\frac{\\pi}{8},\\frac{\\pi}{4},\\frac{3\\pi}{8},\\frac{\\pi}{2}$. Covariance matrices with $\\theta=\\frac{\\pi}{4}$ can be easily distinguished allowing for the optimal estimation of the parameter $\\epsilon$.}\\label{fig:three_cov}\n\\end{figure}\n\nThe phase changing and squeezing channel are the special cases of the channel introduced above. The phase changing channel is given by $(\\omega_p,\\omega_s)=(1,0)$ and squeezing channel by $(\\omega_p,\\omega_s)=(0,1)$. Corresponding quantum Fisher information and optimized quantum Fisher information is given by Eq.~\\eqref{eq:one_mode_channel_QFI} and~\\eqref{eq:optimized_QFI_squeezing_phase}. It is interesting to note that the optimal probe state for a phase-changing channel is an arbitrary squeezed state achieving the quantum Fisher information of $H(\\epsilon)=8n(n+1)$, while for a squeezing channel it is the vacuum squeezed in the $45^\\circ$ from the angle from which the channel squeezes. Such state then achieves the Heisenberg scaling $H(\\epsilon)=2(2n+1)^2$. For illustration how different squeezed states perform in the estimation of the squeezing channel, see Fig.~\\ref{fig:three_cov}.\n\n\\subsubsection{Estimation of two-mode channels: mode-mixing and two-mode squeezing channels}\n\nIn this section we are going to study the estimation of two-mode Gaussian unitary channels with purely quadratic generators, namely two-mode squeezing and mode-mixing channels, using a wide class of two-mode mixed probe states and the general two-mode pure state. In the analogy with one-mode Gaussian channels, these channels are fully parametrized by Hermitian matrix $W$ given by Eq.~\\eqref{eq:W2}.\n\nAlthough analysis with the general probe state~\\eqref{eq:general_2mode_state} can be made, the results seem to be too complicated to be used effectively. Also, as the first three operations applied on the thermal state only swap and entangle the symplectic eigenvalues, we do not expect much generality will be lost when not considering them. Moreover, in the case of the isothermal states (which also covers all pure states), such operations do not have any effect. This is why we restrict ourselves to probe states which we write in the covariance matrix formalism as\n\\begin{subequations}\\label{eq:simplified_probe_state}\n\\begin{gather}\n\\boldsymbol{d}_0=(\\tilde{\\boldsymbol{\\gamma}},\\ov{\\tilde{\\boldsymbol{\\gamma}}})^T,\\\\\n\\sigma_0=R_1(\\phi_1)R_2(\\phi_2)B(\\theta)R_{\\mathrm{as}}(\\psi)S_1(r_1)S_2(r_2)D_0(\\cdot)^\\dag,\n\\end{gather}\n\\end{subequations}\nwhere $\\tilde{\\boldsymbol{\\gamma}}=(\\norm{{d}_1}e^{i\\phi_{d1}},\\norm{{d}_2}e^{i\\phi_{d2}})$ and $D_0=\\mathrm{diag}(\\lambda_1,\\lambda_2,\\lambda_1,\\lambda_2)$. Also, it is believed that mixed states cannot improve the\nquality of estimation when fixing the energy of the probe state, the optimal states are always pure.\\footnote{This belief comes fact that the quantum Fisher information is a convex function on the space of density matrices. However, there are some problems for more detailed discussion see section~\\ref{sec:are_pure_optimal}.\nAs Eq.~\\eqref{eq:simplified_probe_state} encompasses all pure states, it is enough to use this restricted class of states to find the optimal.\n\n\\textbf{Two-mode squeezing channel.}\nFirst we are going to study the optimal states for the estimation of the two-mode squeezing channel $\\hat{S}_{T}(\\epsilon,\\chi)$, assuming the direction of squeezing $\\chi$ is known. This channel was introduced in Eq.~\\eqref{eq:twomode_squeezing_operator}. Using the state from Eq.~\\eqref{eq:simplified_probe_state} we find only two cases which lead to significantly different results. In the first case a beam-splitter is not used ($\\theta=0$) in the preparation process, which corresponds to using two simultaneously sent, but non-entangled single-mode squeezed probe states. In the second case the balanced beam-splitter is used ($\\theta=\\pi\/4$), which corresponds to using two-mode squeezed-type probe states. The full expression for the quantum Fisher information is a mixture of these two qualitatively different cases. Defining $\\phi_\\chi:=\\phi_1+\\phi_2+\\chi$, $\\phi_{1\\chi}:=\\phi_1-\\phi_{d2}+\\chi$, $\\phi_{2\\chi}:=\\phi_2-\\phi_{d1}+\\chi$, the quantum Fisher information for the estimation of a two-mode squeezing channel $\\hat{S}_{T}(\\epsilon,\\chi)$ reads\n\\[\n\\begin{split}\nH(\\epsilon)&=2\\cos^2(2\\theta)\\bigg(\\!\\frac{(\\lambda_1\\!+\\!\\lambda_2)^2}{\\lambda_1\\lambda_2\\!+\\!1}\\big(\\!\\cos^2\\!\\!\\phi_\\chi\\cosh^2(\\!r_1\\!-\\!r_2\\!)\\!+\\!\\sin^2\\!\\!\\phi_\\chi\\cosh^2(\\!r_1\\!+\\!r_2\\!)\\big)\\\\\n&+\\frac{(\\lambda_1\\!-\\!\\lambda_2)^2}{\\lambda_1\\lambda_2\\!-\\!1}\\big(\\!\\cos^2\\!\\!\\phi_\\chi\\sinh^2(\\!r_1\\!-\\!r_2\\!)\\!+\\!\\sin^2\\!\\!\\phi_\\chi\\sinh^2(\\!r_1\\!+\\!r_2\\!)\\!\\big)\\!\\bigg)\\\\\n&+4\\sin^2(2\\theta)\\bigg(\\frac{\\lambda_1^2}{\\lambda_1^2+1}\\big(\\cos^2(\\phi_\\chi+2\\psi)+\\sin^2(\\phi_\\chi+2\\psi)\\cosh(2r_1)\\big)\\\\\n&+\\frac{\\lambda_2^2}{\\lambda_2^2+1}\\big(\\cos^2(\\phi_\\chi-2\\psi)+\\sin^2(\\phi_\\chi-2\\psi)\\cosh(2r_2)\\big)\\bigg)\\\\\n&+\\frac{4}{\\lambda_1}\\Big(e^{2r_1}\\big(\\norm{{d}_1}\\sin\\theta\\cos(\\phi_{2\\chi}+\\psi)-\\norm{{d}_2}\\cos\\theta\\cos(\\phi_{1\\chi}+\\psi)\\big)^2\\\\\n&+e^{-2r_1}\\big(\\norm{{d}_1}\\sin\\theta\\sin(\\phi_{2\\chi}+\\psi)-\\norm{{d}_2}\\cos\\theta\\sin(\\phi_{1\\chi}+\\psi)\\big)^2\\Big)\\\\\n&+\\frac{4}{\\lambda_2}\\Big(e^{2r_2}\\big(\\norm{{d}_1}\\sin\\theta\\cos(\\phi_{2\\chi}-\\psi)+\\norm{{d}_2}\\cos\\theta\\cos(\\phi_{1\\chi}-\\psi)\\big)^2\\\\\n&+e^{-2r_2}\\big(\\norm{{d}_1}\\sin\\theta\\sin(\\phi_{2\\chi}-\\psi)+\\norm{{d}_2}\\cos\\theta\\sin(\\phi_{1\\chi}-\\psi)\\big)^2\\Big).\\\\\n\\end{split}\n\\]\n\nAssuming both $r_1$ and $r_2$ are positive, one of the optimal states is given by setting $\\theta=0$, $\\phi_1=\\phi_2=\\frac{\\pi}{4}-\\frac{\\chi}{2}$, $\\phi_{d1}=\\phi_{d2}=\\frac{\\pi}{4}+\\frac{\\chi}{2}$ which corresponds to two one-mode squeezed displaced thermal states, squeezed in the angle of $45^\\circ$ from the squeezing angle of the channel. This probe states then leads to\n\\[\\label{eq:max_QFI_nonentangled_twomode}\n\\begin{split}\nH_{\\!\\mathrm{max}}\\!(\\epsilon)\\!\\!&=\\!\\!\\frac{2(\\lambda_1\\!\\!+\\!\\!\\lambda_2)^2\\!\\!}{\\lambda_1\\lambda_2\\!+\\!1}\\cosh^2\\!(\\!r_1\\!+\\!r_2\\!)\n+\\frac{2(\\lambda_1\\!\\!-\\!\\!\\lambda_2)^2\\!\\!}{\\lambda_1\\lambda_2\\!-\\!1}\\sinh^2\\!(\\!r_1\\!+\\!r_2\\!)\\\\\n&+\\frac{4\\norm{{d}_2}^2}{\\lambda_1}e^{2r_1}\n+\\frac{4\\norm{{d}_1}^2}{\\lambda_2}e^{2r_2}.\n\\end{split}\n\\]\n\nIn contrast, for $r_1\\leq0$, $r_2\\geq0$, one of the optimal states are given by setting $\\theta=\\frac{\\pi}{4}$, $\\phi_\\chi=0$, $\\psi=\\phi_{1\\chi}=\\phi_{2\\chi}=\\frac{\\pi}{4}$ and leads to\n\\[\\label{eq:max_QFI_entangled_twomodeb}\n\\begin{split}\nH_{\\mathrm{max}}(\\epsilon)&=\\frac{4\\lambda_1^2}{\\lambda_1^2+1}\\cosh^2(2r_1)\n+\\frac{4\\lambda_2^2}{\\lambda_2^2+1}\\cosh^2(2r_2)\\\\\n&+\\frac{2}{\\lambda_1}(\\norm{{d}_1}-\\norm{{d}_2})^2e^{-2r_1}\n+\\frac{2}{\\lambda_2}(\\norm{{d}_1}+\\norm{{d}_2})^2e^{2r_2}.\n\\end{split}\n\\]\nThis probe state corresponds to the two-mode squeezed displaced thermal state.\n\nHowever, optimizing over the energy of the probe using the relation for the mean number of particles,\n\\[\\label{eq:mean_number_of_bosons_two}\nn=n_{{d}_1}+n_{\\mathrm{th}1}+\\lambda_1\\sinh^2r_1\n+n_{{d}_2}+n_{\\mathrm{th}2}+\\lambda_2\\sinh^2r_2,\n\\]\nwhere $n_{{d}_i}:=\\norm{{d}_i}^2$, and $\\lambda_i=1+2n_{\\mathrm{th}i}$, $i=1,2$, we find that all states perform the best when all energy is invested into squeezing and the squeezing parameters are equal. Both Eq.~\\eqref{eq:max_QFI_nonentangled_twomode} and Eq.~\\eqref{eq:max_QFI_entangled_twomodeb} then lead to the same Heisenberg limit $H_{\\mathrm{max}}(\\epsilon)=4(n+1)^2$. To conclude, the optimal probe states for estimating two-mode squeezing channel are either two one-mode squeezed states or a two-mode squeezed state while squeezing is in the angle of $45^\\circ$ from the squeezing angle of the channel. This adds to the current knowledge of optimal states for squeezing channels, since until now the research has been focused only on one type of a two-mode squeezing channel and non-entangled probe states~\\cite{Gaiba2009a}. Investing all energy into the displacement of the state we obtain the shot-noise limit $H_{\\mathrm{max}}(\\epsilon)=4(n+1)$. The same shot noise limit is also achieved by any single-mode state used as a probe for the two-mode squeezing channel.\n\n\\textbf{Mode-mixing channel.}\nUsing the probe state from Eq.~\\eqref{eq:simplified_probe_state}, and defining $\\phi_\\chi:=\\phi_1-\\phi_2+\\chi$, $\\phi_{1\\chi}:=\\phi_1+\\phi_{d2}+\\chi$, $\\phi_{2\\chi}:=\\phi_2+\\phi_{d1}-\\chi$, the quantum Fisher information for the estimation of a mode-mixing channel $\\hat{B}(\\epsilon,\\chi)$ introduced in Eq.~\\eqref{eq:mode_mixing_operator} reads\n\\[\\label{eq:mode_mixing_full}\n\\begin{split}\nH(\\epsilon)&=4\\sin^2(2\\theta)\\sin^2\\!\\!\\phi_\\chi\\bigg(\\frac{\\lambda_1^2}{\\lambda_1^2+1}\\sinh^2(2r_1)+\\frac{\\lambda_2^2}{\\lambda_2^2+1}\\sinh^2(2r_2)\\bigg)\\\\\n&+\\frac{2(\\lambda_1\\!+\\!\\lambda_2)^2}{\\lambda_1\\lambda_2\\!+\\!1}\\!\\Big(\\!\\big(\\!\\cos(\\!2\\theta\\!)\\sin\\!\\phi_\\chi\\sin(\\!2\\psi\\!)\\!-\\!\\cos\\!\\phi_\\chi\\cos(\\!2\\psi\\!)\\!\\big)^2\\!\\!\\sinh^2(\\!r_1\\!-\\!r_2\\!)\\\\\n&+\\big(\\!\\cos(\\!2\\theta\\!)\\sin\\!\\phi_\\chi\\cos(\\!2\\psi\\!)\\!+\\!\\cos\\!\\phi_\\chi\\sin(\\!2\\psi\\!)\\!\\big)^2\\!\\!\\sinh^2(\\!r_1\\!+\\!r_2\\!)\\!\\Big)\\\\\n&+\\frac{2(\\lambda_1\\!-\\!\\lambda_2)^2}{\\lambda_1\\lambda_2\\!-\\!1}\\!\\Big(\\!\\big(\\!\\cos(\\!2\\theta\\!)\\sin\\!\\phi_\\chi\\sin(\\!2\\psi\\!)\\!+\\!\\cos\\!\\phi_\\chi\\cos(\\!2\\psi\\!)\\!\\big)^2\\!\\!\\cosh^2(\\!r_1\\!-\\!r_2\\!)\\\\\n&+\\big(\\!\\cos(\\!2\\theta\\!)\\sin\\!\\phi_\\chi\\cos(\\!2\\psi\\!)\\!+\\!\\cos\\!\\phi_\\chi\\sin(\\!2\\psi\\!)\\!\\big)^2\\!\\!\\sinh^2(\\!r_1\\!+\\!r_2\\!)\\\\\n&+\\frac{1}{2}\\cos(\\!2\\theta\\!)\\sin(\\!2\\phi_\\chi\\!)\\sin(\\!4\\psi\\!)\\sinh(2r_1)\\sinh(2r_2)\\!\\Big)\\\\\n&+\\frac{4}{\\lambda_1}\\Big(e^{2r_1}\\big(\\norm{{d}_1}\\sin\\theta\\cos(\\phi_{2\\chi}+\\psi)+\\norm{{d}_2}\\cos\\theta\\cos(\\phi_{1\\chi}+\\psi)\\big)^2\\\\\n&+e^{-2r_1}\\big(\\norm{{d}_1}\\sin\\theta\\sin(\\phi_{2\\chi}+\\psi)+\\norm{{d}_2}\\cos\\theta\\sin(\\phi_{1\\chi}+\\psi)\\big)^2\\Big)\\\\\n&+\\frac{4}{\\lambda_2}\\Big(e^{2r_2}\\big(\\norm{{d}_1}\\cos\\theta\\cos(\\phi_{2\\chi}-\\psi)-\\norm{{d}_2}\\sin\\theta\\cos(\\phi_{1\\chi}-\\psi)\\big)^2\\\\\n&+e^{-2r_2}\\big(\\norm{{d}_1}\\cos\\theta\\sin(\\phi_{2\\chi}-\\psi)-\\norm{{d}_2}\\sin\\theta\\sin(\\phi_{1\\chi}-\\psi)\\big)^2\\Big).\\\\\n\\end{split}\n\\]\n\nFor positive $r_1$ and $r_2$ one of the optimal states is given by $\\theta=0$, $\\phi_\\chi=\\frac{\\pi}{2}$, and $\\phi_{1\\chi}=\\phi_{2\\chi}=0$,\n\\[\\label{eq:max_QFI_nonentangled_bs}\n\\begin{split}\nH_{\\mathrm{max}}(\\epsilon)&=\\frac{2(\\lambda_1\\!\\!+\\!\\!\\lambda_2)^2\\!\\!}{\\lambda_1\\lambda_2\\!+\\!1}\\sinh^2(\\!r_1\\!+\\!r_2\\!)\n+\\frac{2(\\lambda_1\\!\\!-\\!\\!\\lambda_2)^2\\!\\!}{\\lambda_1\\lambda_2\\!-\\!1}\\cosh^2(\\!r_1\\!+\\!r_2\\!)\\\\\n&+\\frac{4\\norm{{d}_2}^2}{\\lambda_1}e^{2r_1}\n+\\frac{4\\norm{{d}_1}^2}{\\lambda_2}e^{2r_2}.\n\\end{split}\n\\]\nThese conditions are fulfilled for example for $\\phi_1=\\frac{\\pi}{4}-\\frac{\\chi}{2}$, $\\phi_{d1}=\\frac{\\pi}{4}+\\frac{\\chi}{2}$, $\\phi_2=-\\frac{\\pi}{4}+\\frac{\\chi}{2}$, $\\phi_{d2}=-\\frac{\\pi}{4}-\\frac{\\chi}{2}$. This corresponds to two-single mode squeezed displaced thermal states of which the angle of between squeezing of each mode of the probe is given by the angle of the mode-mixing channel $\\chi$.\n\nSetting $\\theta=\\psi=\\frac{\\pi}{4}$, $\\phi_\\chi=\\frac{\\pi}{2}$, and $\\phi_{1\\chi}=\\phi_{2\\chi}=-\\frac{\\pi}{4}$ in Eq.~\\eqref{eq:mode_mixing_full} we obtain\n\\[\\label{eq:QFI_entangled_BS}\n\\begin{split}\nH_{\\mathrm{max}}(\\epsilon)&=\\frac{4\\lambda_1^2}{\\lambda_1^2+1}\\sinh^2(2r_1)+\\frac{4\\lambda_2^2}{\\lambda_2^2+1}\\sinh^2(2r_2)\\\\\n&+\\frac{2}{\\lambda_1}\\big(\\norm{{d}_1}+\\norm{{d}_2}\\big)^2e^{2r_1}+\\frac{2}{\\lambda_2}\\big(\\norm{{d}_1}-\\norm{{d}_2}\\big)^2e^{2r_2},\n\\end{split}\n\\]\nThis state corresponds to the two-mode squeezed displaced thermal state.\n\nFor mode-mixing channels we find a unique phenomenon which does not occur with the squeezing channels, and which can be exploited only when using a beam-splitter in the preparation process. Setting $\\lambda_1=\\lambda_2=1$, $r_1=r_2=r$, $\\theta=\\psi=\\frac{\\pi}{4}$, and $\\phi_1+\\phi_2+\\phi_{d1}+\\phi_{d2}=-\\frac{\\pi}{2}$ in Eq.~\\eqref{eq:QFI_entangled_BS}, we derive\n\\[\\label{eq:QFI_universal_BS}\nH(\\epsilon)=4\\sinh^2(2r)+4\\Big(\\big(\\norm{{d}_1}^2+\\norm{{d}_2}^2\\big)\\cosh(2r)+2\\norm{{d}_1}\\norm{{d}_2}\\sinh(2r)\\Big).\n\\]\nAny free parameter has not been at this point set to be dependent on the angle of the mode-mixing $\\chi$. Also, the leading order here is identical to the energy-optimal probe states. In other words, we have found the universal optimal probe state for the mode-mixing channels $\\hat{B}(\\epsilon,\\chi)$. If we set the initial displacement $\\boldsymbol{d}_0$ to zero, according to Eq.~\\eqref{eq:simplified_probe_state} this probe state becomes the two-mode squeezed vacuum in the direction of $\\chi_T=\\frac{\\pi}{2}$, $\\hat{\\rho}_{0\n=\\hat{S}_{T}(r,\\tfrac{\\pi}{2})\\hat{S}_{T}^\\dag(r,\\tfrac{\\pi}{2})$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{one_and_optimal.pdf}\n\\caption{Estimation of the beam-splitter $\\hat{B}(\\epsilon)$ around point $\\epsilon=0$ using one of the optimal states $\\hat{\\rho}=\\hat{S}_1(r)\\hat{S}_2(-r)\\hat{S}_2^\\dag(-r)\\hat{S}_1^\\dag(r)$, and the single-mode state $\\hat{\\rho}=\\hat{S}(r_1)\\hat{S}^\\dag(r_1)\\otimes\\ket{0}\\bra{0}$, both with the same mean energy $n=2$. We plot the real form marginal covariance matrices showing correlations between positions in the first and the second mode $x_1$ and $x_2$, and momenta $p_1$ and $p_2$ in the real form phase-space, before(blue with full line) and after(orange with dashed line) beam-splitter $\\hat{B}(0.1)$ has been applied. There are no correlations between position and momentum. Clearly, the optimal state is more sensitive to the channel allowing the optimal estimation of the parameter $\\epsilon$.}\\label{fig:one_and_optimal}\n\\end{figure}\n\nOptimizing over the energy of the probe using the relation for the mean number of particles~\\eqref{eq:mean_number_of_bosons_two} we find that all probe states leading to Eqs.~\\eqref{eq:max_QFI_nonentangled_bs}, \\eqref{eq:QFI_entangled_BS}, and~\\eqref{eq:QFI_universal_BS} perform the best when all energy is, as in the case of estimating two-mode squeezing channel, invested into squeezing and the squeezing parameters are equal. All probe states then lead to the same Heisenberg limit $H_{\\mathrm{max}}(\\epsilon)=4n(n+2)$. In contrast, investing all energy into the displacement of the state we obtain the shot-noise limit $H_{\\mathrm{max}}(\\epsilon)=4n$. The same shot noise limit is also achieved by any one-mode state used as a probe for the mode-mixing channel. For illustration of how a one-mode state compares to the optimal state see figure~\\ref{fig:one_and_optimal}.\n\n\n\\subsection{Discussion on optimality of pure states and squeezed state\n}\\label{sec:are_pure_optimal}\n\nIn the previous sections we have studied how mixed states perform as probe states. We have shown that investing some energy into the temperature of the probe state can help the estimation. However, after optimizing over the mean energy we have shown the pure probe states are always optimal. Why this happens can be viewed in the following way: let us assume that we have the set of all states with a fixed mean energy. Mixed states of that set are states on which we (as observers) lack some information. Pure states, where we do not lack any information, should serve as better probes as we know how to retrieve the information about the parameter more accurately. However, the way in which to prove this statement mathematically seems unclear. What we would like to prove is that for any mixed state $\\hat{\\rho}_0$ there exists a pure state $\\ket{\\psi_0}$ with the same energy (i.e., the same mean value given by a positive semi-definite operator $\\hat{E}$) such that for any encoding channel $\\mathrm{C}_\\epsilon$ (CTPT map) the quantum Fisher information satisfies $H(\\mathrm{C}_\\epsilon(\\ket{\\psi_0}\\bra{\\psi_0}))\\geq H(\\mathrm{C}_\\epsilon(\\rho_0))$. One of states $\\ket{\\psi_0}$ could be the purification of the state $\\hat{\\rho}_0$. However the purification of a quantum state lies in a larger Hilbert space than the mixed state itself and the definition of the energy operator may not necessarily extend to this larger Hilbert space. We could extend the operator $\\hat{E}$ to the larger Hilbert space in any possible way as long as it reduces to the original $\\hat{E}$ on the former Hilbert space. In this way the energy of the purification should be larger unless we define the energy of all states in the extended Hilbert space as zero -- but this seems very arbitrary and unphysical as we cannot access those states in this extended Hilbert space anyway. Therefore the question is: ``For any mixed state $\\hat{\\rho}_0$ of energy $E$ from Hilbert space $\\mathcal{H}$, does there exist a pure state $\\ket{\\psi_0}$ from the same Hilbert space and with the same energy as the mixed state such that $H(\\mathrm{C}_\\epsilon(\\ket{\\psi_0}\\bra{\\psi_0}))\\geq H(\\mathrm{C}_\\epsilon(\\hat{\\rho}_0))$?''\n\nWe will not answer this question precisely but we can provide some insight. The quantum Fisher information is a convex function on the set of density matrices~\\cite{Toth2014a}, i.e., for any mixed state $\\hat{\\rho}_0=\\sum_{i}p_i\\ket{\\psi}\\bra{\\psi}$ and for any encoding unitary $\\hat{U}_\\epsilon=e^{-i\\hat{G}\\epsilon}$ the following inequalites are satisfied,\n\\[\\label{eq:convex_QFI}\nH\\Big(\\hat{U}_\\epsilon\\hat{\\rho}_0\\hat{U}_\\epsilon^\\dag\\Big)\\ \\!\\leq\\ \\!\\sum_{i}p_iH\\Big(\\hat{U}_\\epsilon\\ket{\\psi_i}\\bra{\\psi_i}\\hat{U}_\\epsilon^\\dag\\Big)\\ \\!\\leq\\ \\!\\max_{i}\\ \\! H\\Big(\\hat{U}_\\epsilon\\ket{\\psi_i}\\bra{\\psi_i}\\hat{U}_\\epsilon^\\dag\\Big).\n\\]\nThis shows that for any mixed state there exists a pure state in the same Hilbert space which provides better or equal precision in the estimation of the parameter $\\epsilon$. However, this state may not have the same energy as the original mixed state $\\hat{\\rho}_0$. Our original question thus remains unanswered. On the other hand, for Gaussian states we can use an elegant formula for the quantum Fisher information in terms of the Williamson's decomposition of the covariance matrix~\\eqref{eq:multimode_QFI}. We will illustrate the optimality of high energy single mode pure state in the estimation of a unitary channel $\\hat{U}_\\epsilon$. We will schematically show the maximal scaling with particle number for each element in Eq.~\\eqref{eq:multimode_QFI} and then combine these scalings into a single formula. We will assume all particle numbers are large, $n,n_{d},n_{\\mathrm{th}}\\gg 1$. Eq.~\\eqref{eq:mean_number_of_bosons_one} then becomes\n\\[\\label{eq:mean_number_of_bosons_one_large}\nn=n_{d}+n_{\\mathrm{th}}+(1+2n_{\\mathrm{th}})e^{2r}.\n\\]\nGiven the Euler's decomposition~\\eqref{def:S_decomposition}, elements of the symplectic matrices scale as $S_0\\sim e^r$, $S_0^{-1}\\sim e^r$. In combination with Eq.~\\eqref{eq:first_part} this implies $P\\sim e^{2r}$, $Q\\sim e^{2r}$, $R\\sim e^{2r}$, and $\\sigma_0^{-1}\\sim \\frac{e^{2r}}{2n_{\\mathrm{th}}+1}$. Temperature factor scales as $\\frac{\\lambda^2}{1+\\lambda^2}\\sim \\frac{n_{\\mathrm{th}}^2}{n_{\\mathrm{th}}^2}$ and the displacement scales as $\\boldsymbol{d}_0\\sim \\sqrt{n_{d}}$. Scaling of these quantities and Eq.~\\eqref{eq:mean_number_of_bosons_one_large} give us the scaling of the quantum Fisher information,\n\\[\\label{eq:scaling_QFI}\nH(\\epsilon)\\sim C_1\\frac{n_{\\mathrm{th}}^2}{n_{\\mathrm{th}}^2}\\left(\\frac{n-n_{\\mathrm{th}}-n_{d}}{2n_{\\mathrm{th}}+1}\\right)^2+C_2\\frac{n-n_{\\mathrm{th}}-n_{d}}{2n_{\\mathrm{th}}+1}(\\sqrt{n_{d}}+C_3)^2.\n\\]\nFactors $C_1,C_2,C_3$ are real constants which depend on the probe state we use and the channel we probe. Fixing the total mean number of particles $n$ of the probe state we can see that the maximum of the quantum Fisher information is achieved when the number of thermal bosons is the lowest, i.e., when the initial state is pure. This reasoning partially and schematically shows that pure states are indeed optimal states for the estimation of Gaussian unitary channels.\n\nEq.~\\eqref{eq:scaling_QFI} also shows that the optimal probe is either the squeezed state or a mixture of squeezed and displaced state. We assume the encoding channel is generated by a purely quadratic Hamiltonian for which $C_3=0$. Fixing the $n_{\\mathrm{th}}=0$ the Eq.~\\eqref{eq:scaling_QFI} becomes\n\\[\\label{eq:scaling_QFI}\nH(\\epsilon)\\sim\n\\left(n-n_{d}\\right)\\left(C_1n+(C_2-C_1)n_{d}\\right).\n\\]\nFor $C_2\\leq 2C_1$ the maximum is achieved when $n_{d}=0$, i.e., when the probe state is a squeezed state. For $C_2>2C_1$ the maximum is achieved for $n_{d}=\\frac{C_2-2C_1}{2(C_2-C_1)}$, i.e., when the probe state is a squeezed displaced state. In all channels studied we had $C_2=2C_1$ when assuming $r_1=r_2$, $\\norm{{d}_1}=\\norm{{d}_2}$ and $\\lambda_1=\\lambda_2=1$. As an empirical rule the optimal state is always a squeezed state. However a rigorous mathematical proof is still necessary.\n\n\n\\subsection{Simplified way of finding optimal probes}\\label{sec:optimizing_for_pure_state}\n\nWe illustrated in the previous section that it is likely that optimal probe states are always pure. If this is the case, we can restrict our search for optimal probe states to pure states and significantly simplify the algorithm.\n\nCovariance matrix for pure states can be always written as $\\sigma_0=S_0S_0^\\dag$. Considering the Euler's decomposition of the symplectic matrix $S_0$~\\eqref{def:S_decomposition}, the covariance matrix of a pure state is fully parametrized by\n\\[\\label{eq:S_0_pure}\nS_0=\n\\begin{bmatrix}\nU_1 & 0 \\\\\n0 & \\ov{U}_1\n\\end{bmatrix}\n\\begin{bmatrix}\n\\cosh{M_{\\boldsymbol{r}}} & -\\sinh{M_{\\boldsymbol{r}}} \\\\\n-\\sinh{M_{\\boldsymbol{r}}} & \\cosh{M_{\\boldsymbol{r}}}\n\\end{bmatrix}.\n\\]\nThe quantum Fisher information can be obtained by combining Eqs.~\\eqref{eq:S_0_pure},~\\eqref{eq:exact_multimode_compact_pure}, and~\\eqref{eq:eq_for_general_method},\n\\[\nH(\\epsilon)=2\\mathrm{tr}\\left[QQ^\\dag\\right]+2\\boldsymbol{v}^\\dag\\begin{bmatrix}\n\\cosh{M_{\\boldsymbol{2r}}} & \\sinh{M_{\\boldsymbol{2r}}} \\\\\n\\sinh{M_{\\boldsymbol{2r}}} & \\cosh{M_{\\boldsymbol{2r}}}\n\\end{bmatrix}\\boldsymbol{v},\n\\]\nwhere\n\\begin{subequations}\n\\begin{align}\nQ&=\\cosh{M_{\\boldsymbol{r}}}U_1^\\dag Q_\\epsilon \\ov{U}_1\\cosh{M_{\\boldsymbol{r}}}-\\sinh{M_{\\boldsymbol{r}}}\\ov{U}_1^\\dag \\ov{Q}_\\epsilon U_1\\sinh{M_{\\boldsymbol{r}}}\\nonumber\\\\\n&+\\sinh{M_{\\boldsymbol{r}}}\\ov{U}_1^\\dag\\ov{R}_\\epsilon \\ov{U}_1\\cosh{M_{\\boldsymbol{r}}}-\\cosh{M_{\\boldsymbol{r}}}U_1^\\dag R_\\epsilon U_1\\sinh{M_{\\boldsymbol{r}}},\\\\\n\\boldsymbol{v}&=\\begin{bmatrix}\nU_1^\\dag & 0 \\\\\n0 & \\ov{U}_1^\\dag\n\\end{bmatrix}\n\\left(P_\\epsilon\\boldsymbol{d}_0+S_\\epsilon^{-1}\\dot{\\boldsymbol{b}}_\\epsilon\\right),\\ \\\nP_\\epsilon:=S_\\epsilon^{-1}\\dot{S_\\epsilon}=\\begin{bmatrix}\nR_\\epsilon & Q_\\epsilon \\\\\n\\ov{Q}_\\epsilon & \\ov{R}_\\epsilon\n\\end{bmatrix}.\n\\end{align}\n\\end{subequations}\nFinding the optimal probe states then reduces to maximizing the quantum Fisher information over the unitary matrix $U_1$, the matrix of squeezing $M_{\\boldsymbol{r}}$, and the vector of displacement $\\boldsymbol{d}_0$. For the estimation of one- two- and three- mode Gaussian channels the unitary $U_1$ can be fully parametrized as discussed in section~\\eqref{sec:parametrization_of_Gaussian_states}.\n\nIn the case when the encoding operation is a one-parameter unitary group~\\eqref{def:Gaussian_unitary_epsilon} with the hermitian matrix $W$ of structure given by Eq.~\\eqref{def:W_for_Gaussian_unitary} we have\n$P_\\epsilon=iKW$, $R_\\epsilon=iX$, $Q_\\epsilon=iY$, and $\\boldsymbol{v}=\\big(\\tilde{\\boldsymbol{v}},\\ov{\\tilde{\\boldsymbol{v}}}\\big)^T$, $\\tilde{\\boldsymbol{v}}=U_1^\\dag\\big(iX\\tilde{\\boldsymbol{d}}_0+iY\\ov{\\tilde{\\boldsymbol{d}}}_0+\\tilde{\\boldsymbol{\\gamma}}\\big)$.\n\n\n\n\\subsection{Role of entanglement and the Heisenberg limit}\\label{sec:Heisenberg_limit_Gaussian}\n\nAs illustrated in previous sections entangled states such as two mode squeezed states do not achieve any advantage over the two single-mode squeezed states in the estimation of two-mode channels. In this section we first show why it is usually thought that entanglement in the probe state is necessary to achieve the Heisenberg limit, and why this reasoning is not applicable in the continuous variable states known as Gaussian states.\n\nAlthough there are many possible definitions of Heisenberg limit in quantum metrology~\\cite{Giovannetti2011a}, in this thesis we adopt the definition where the precision of estimation is compared to the mean energy of the probe state, mean number of Bosons respectively. Sequence of states $\\rho_m$ with ever-increasing mean value of energy $\\lim_{m\\rightarrow\\infty}\\mean{\\hat{E}}_{\\rho_m}=\\infty$ is said to reach the Heisenberg limit iff there exists a number $c>0$ such that\n\\[\\label{def:limit_Heisenberg}\n\\lim_{m\\rightarrow\\infty}\\frac{H(\\rho_m)}{(\\mean{\\hat{E}}_{\\rho_m})^2}=c.\n\\]\nIn the case of a Bosonic system the operator $\\hat{E}$ which measures the energy of the probe state is up to a scaling constant identical to the total number operator, $\\hat{E}\\equiv\\hat{N}$.\n\nWe first consider a Hilbert space $\\mathcal{H}$ such that for every state $\\rho\\in\\mathcal{H}$ the quantum Fisher information is bounded by the same value $B_H$, i.e.,\n\\[\\label{eq:bound_H}\n\\exists B_H>0,\\ \\ \\forall \\rho\\in \\mathcal{H},\\ \\ H(\\rho)\\leq B_H.\n\\]\nIt is not possible to create a sequence of states from such Hilbert space to achieve the Heisenberg limit, because by definition $\\lim_{m\\rightarrow\\infty}\\frac{H(\\rho_m)}{(\\mean{\\hat{E}}_{\\rho_m})^2}\\leq\\lim_{m\\rightarrow\\infty}\\frac{B_H}{\\mean{\\hat{E}}_{\\rho_m}}=0$.\nHowever, we can increase the quantum Fisher information by adding more particles, which corresponds to expanding the Hilbert space. We consider a (fully) separable state\n\\[\\label{eq:construction_separable_states}\n\\rho_m=\\sum_ip_i\\rho_i^{(1)}\\otimes\\rho_i^{(2)}\\otimes\\cdots\\otimes\\rho_i^{(m)}\\in\\mathcal{H}^{\\otimes m},\n\\]\nwhere $\\sum_ip_i=1$. Assuming that energy of each added state does not go below certain value, i.e.,\n\\[\\label{eq:bound_E}\n\\exists B_E>0,\\ \\ \\forall i,\\ \\forall k,\\ \\ \\mean{\\hat{E}}_{\\rho_i^{(k)}}\\geq B_E,\n\\]\nand using convexity of the quantum Fisher information and additivity under tensoring~\\cite{Toth2014a}, we derive\n\\[\\label{eq:proof_separable_states_QFI}\n\\lim_{m\\rightarrow\\infty}\\!\\frac{H(\\rho_m)}{(\\mean{\\hat{E}}_{\\rho_m})^2}\\leq\\!\\lim_{m\\rightarrow\\infty}\\!\\frac{\\sum_{i,k}p_iH(\\rho_i^{(k)})}{(\\sum_{i,k}p_i\\mean{\\hat{E}}_{\\rho_i^{(k)}}\\!)^2}\n\\leq\\!\\lim_{m\\rightarrow\\infty}\\!\\frac{mB_H}{m^2B_E}=0.\n\\]\nThis illustrates that under conditions~\\eqref{eq:bound_H} and~\\eqref{eq:bound_E}, the construction~\\eqref{eq:construction_separable_states} using separable states cannot lead to the Heisenberg limit and entangled states are necessary. This follows the proofs from~\\cite{Pezze2009a,Demkowicz2012a,Toth2014a} showing that existence of entanglement in an $m$-qubit state is necessary condition for the scaling of the quantum Fisher information larger than the shot-noise limit.\n\nAlthough $1$-qubit Hilbert space, from which the $m$-qubit Hilbert space is created, satisfies Eq.~\\eqref{eq:bound_H}, such condition is no longer satisfied by the Fock space representing a Bosonic system. There are states in the Fock space, such as squeezed states and coherent states, which can lead to an arbitrarily large precision in the estimation. Therefore proof~\\eqref{eq:proof_separable_states_QFI} does not apply anymore and entanglement is not necessary. As shown in previous sections, separable states such as squeezed states can also achieve the Heisenberg limit.\n\nMoreover, in the Fock space it is possible to construct states which achieve arbitrary scaling of the quantum Fisher information with the energy of the probe state. We will illustrate this on the the example of pure states and unitary (not necessarily Gaussian) encoding operation $e^{-i\\hat{K}\\epsilon}$ where $\\hat{K}$ is a Hermitian operator. For pure states the quantum Fisher information is simply the variance in the operator~\\eqref{pureK} and the ratio~\\eqref{def:limit_Heisenberg} reads\n\\[\n\\frac{H(\\rho_m)}{(\\mean{\\hat{E}}_{\\rho_m})^2}=\\frac{4\\mean{\\Delta\\hat{K}^2}_{\\rho_m}}{(\\mean{\\hat{E}}_{\\rho_m})^2}.\n\\]\nBut in the infinite dimensional Hilbert space the variance of a Hermitian operator $\\hat{K}$ is not bounded by the mean value of energy. Even for the phase estimation for which $\\hat{K}=\\hat{N}^2$ and $\\hat{E}=\\hat{N}$ the variance in particle numbers is not related to the mean value of the particle number. We can construct states with arbitrarily large variance and arbitrarily low mean. From such states we can construct a series $\\rho_m$ which achieve any scaling $f$,\n\\[\nH(\\rho_m)\\sim f(\\mean{\\hat{E}}_{\\rho_m}).\n\\]\nHowever, the states which achieve such extraordinary scaling cannot be Gaussian states. These states could nevertheless achieve extremely high precisions while having a very low energy, being extremely interesting from the experimental point of view.\n\nIn comparison to $m$-qubit systems, which use entanglement as a resource, the resources in Bosonic systems are rather highly superposed states spanning across all infinite-dimensional Hilbert space, while entanglement does not play a significant role anymore.\n\n\\section{Summary}\n\nIn the first section we derived new formulae for the quantum Fisher information matrix of Gaussian states for the estimation of the vector of parameters $\\boldsymbol{\\epsilon}$. This included expressions in terms of the Williamson's decomposition of the covariance matrix and the limit formula which can be used for effective numerical calculations. We also noted problems when a pure mode changes its purity due to small variations in the vector of parameters $\\boldsymbol{\\epsilon}$, and devised a regularization procedure to fix them. This also led to a new expression for pure states that takes into account possible changes in purity.\n\nIn the second section we took advantage of the derived formulae and we devised a method of finding optimal probe states for the estimation of Gaussian channels based on the phase-space formalism. We also simplified this method by restricting ourselves to pure probe states. We applied this method to the estimation of one- and two-mode Gaussian channels. We found that for every channel we studied the optimal states are either squeezed or two-mode squeezed states. Further, the entanglement of the probe state does not play any significant role, which corresponds to the findings of~\\cite{Gaiba2009a,Friis2015a}. This is not in contradiction with some previous studies that show that entanglement is necessary to achieve the Heisenberg limit~\\cite{Pezze2009a,Demkowicz2012a}, as assumptions taken there do not apply anymore to the Fock space describing these bosonic systems.\n\nIn estimating parameters of phase-changing, one-mode squeezing, mode-mixing, and two-mode squeezing channels ($\\hat{R},\\hat{S},\\hat{B},\\hat{S}_T$ respectively), the quantum Fisher information reaches the Heisenberg limits\n\\begin{subequations}\n\\begin{align}\nH_{R}(\\epsilon)&=2\\sinh^2(2r)=8n(n+1),\\\\\nH_{S}(\\epsilon)&=2\\cosh^2(2r)=2(2n+1)^2,\\\\\nH_{B}(\\epsilon)&=4\\sinh^2(2r)=4n(n+2),\\\\\nH_{S_T}(\\epsilon)&=4\\cosh^2(2r)=4(n+1)^2,\n\\end{align}\n\\end{subequations}\nwhere $r$ denotes the squeezing of one of the modes in the probe state, and $n$ is the mean total number of particles of the probe state.\nThese results generalize the precision bounds found in~\\cite{Milburn1994a,Chiribella2006a,Monras2006a,Aspachs2008a,Gaiba2009a}. Alternatively, if we choose coherent states as probe states, we obtain the shot-noise limits\n\\begin{subequations}\n\\begin{align}\nH_{R}(\\epsilon)&=4n,\\\\\nH_{S}(\\epsilon)&=2(2n+1),\\\\\nH_{B}(\\epsilon)&=4n,\\\\\nH_{S_T}(\\epsilon)&=4(n+1).\n\\end{align}\n\\end{subequations}\nThese are the same limits we find when using any one-mode state to probe two-mode Gaussian channels. In addition, we have shown that non-Gaussian probe states spanning over the full infinite-dimensional Fock space can achieve arbitrarily high scaling of the quantum Fisher information which can be of interest for experimental application.\n\n\n\nAuthors of~\\cite{Aspachs2008a} showed that the temperature of the probe state may enhance the estimation precision by a factor of two, and authors of~\\cite{Gaiba2009a} explored of how temperature acts in the estimation of mode-mixing channels. We demonstrated that effects of temperature are generic. Independent of which Gaussian unitary channel is probed, the effects of temperature always come in multiplicative factors of four types. The first three appear when the channel changes the squeezing or the orientation of squeezing of the probe state. The first one accounts for the absolute number of thermal bosons in each mode and corresponds to the one found in~\\cite{Aspachs2008a}. The next two take into account differences between thermal bosons in each mode. Larger differences then lead to higher precision in the estimation, while the enhancement factor scales with the ratio of the number of thermal bosons $\\frac{n_{{\\mathrm{th}}i}}{n_{{\\mathrm{th}}j}}$, for $n_{{\\mathrm{th}}i}\\gg n_{{\\mathrm{th}}j}\\gg 0$. The last type of factor is of the form $(2n_{{\\mathrm{th}}i}+1)^{-1}$ and appears when the Gaussian channel changes the displacement of the probe state.\n\nWe have shown how different aspects of a probe state affect the estimation precision, and have provided a framework that can be effectively used to study optimal probe states for the construction of new-era quantum detectors. In addition to applications for existing gravitational wave detectors~\\cite{Abbott2004a,Caron1995a}, our results may be useful for designing new gravimeters~\\cite{Snadden1998a,Altin2013a,Sabin2014a}, climate probes~\\cite{Tapley2004a}, or for the estimation of space-time parameters~\\cite{Danzmann1996a,Everitt2011a,Bruschi2014a}.\n\n\\chapter{Applications in quantum field theory: Estimating effects of space-time}\\label{chap:QFT_metrology}\n\nIn this chapter we derive the ultimate precision limits with which we can estimate parameters encoded by a general Bogoliubov transformation into one- and two- mode Gaussian probe states. As detailed in chapter~\\ref{chap:operations_in_QFT} such transformations can represent a quantum state from the point of view of an accelerated observer, cavities moving in curved space-time, action of an expanding universe on a quantum state, or a gravitational wave passing through a Bose-Einstein condensate. Each such transformation contains parameters\nwhich, upon correct estimation, could tell us more about heavenly bodies and gravity, how non-inertial observers see quantum particles, or the universe itself. The precise estimation of these parameters can lead to novel applications in gravimeters, space-time probes, and gravitational wave detectors. Moreover, since the predictive power of quantum field theory in curved space-time lies in the overlap of quantum physics and general relativity, measuring such parameters could either validate the theory or it could lead to a new theory of quantum gravity.\n\nPrevious work in this direction, as overviewed in section~\\ref{sec:state_QM_in_qft}, considered almost exclusively pure probe states. However, in realistic situations probe states are mixed. In this chapter we provide a framework for the computation of optimal precision bounds for mixed single- and two-mode Gaussian states within quantum field theory. This enables the estimation of space-time parameters in the case when the states are initially at some finite temperature.\n\nThis chapter is structured as follows: first we introduce a general method in which we will approach the problem. Then we compute the perturbative expression of the quantum Fisher information associated to one- and two-mode mixed Gaussian states as a function of the Bogoliubov coefficients. We compute explicitly the quantum Fisher information for the case where we wish to estimate a state parameter around the value $\\epsilon_{0}\\,=\\,0$. Finally, we apply these results to calculate the quantum Fisher information for the estimation of the proper acceleration using one- and two-mode squeezed thermal states.\n\n\\section{General method}\n\nThe general method is very similar to the one used for finding the optimal probe states for Gaussian channels: here, however, because of computational complexity we will not optimize over probe states.\n\nLet us consider a quantum state represented by a covariance matrix $\\tilde{\\sigma}_0$ of the field.This covariance matrix is infinite dimensional which reflects the infinite-dimensional nature of states in quantum field theory in curved space-time. It consists of the initial covariance matrix of a mode or modes of interest, $\\sigma_0:=\\mathrm{tr}_E[\\tilde{\\sigma}_0]$, and the covariance matrix of the remaining modes of the field (or ``environment'' $E$), $\\sigma_{E}:=\\mathrm{tr}_{\\neg E}[\\tilde{\\sigma}_{0}]$, which contains no initial correlation with the system modes $\\sigma_{0}$. Therefore, the initial state $\\tilde{\\sigma}_{0}$ is separable in the subsystem-environment bipartition. Its block structure is given as for any Gaussian state by Eq.~\\eqref{def:first_and_second_moments},\n\\[\n\\tilde{\\sigma}_0\\,=\\,\\begin{bmatrix}\nX_0 & Y_0 \\\\\n\\overline{Y}_0 & \\overline{X}_0\n\\end{bmatrix},\n\\]\nNote this infinite-dimensional matrix takes into account any modes of interest and also the remaining ``environment\" modes of the system. We will also assume the initial state has zero initial displacement, $\\boldsymbol{d}_0=0$.\n\nThe transformation between the initial and final state of the field, when restricted to the Gaussian case, is given by the (infinite-dimensional) Bogoliubov matrix $\\tilde{S}$. The covariance of the transformed state is calculated via Eq.~\\eqref{def:transformation} and the Bogoliubov matrix $\\tilde{S}$ is the same as transformations of the field operators~\\eqref{eq:transformation_of_field}, i.e.,\n\\[\n\\tilde{S}:=\\ov{S}^\\dag=\\begin{bmatrix}\n\\ov{\\alpha} & -\\ov{\\beta} \\\\\n-\\beta & \\alpha\n\\end{bmatrix}.\n\\]\nMatrix $S$ from which the matrix $\\tilde{S}$ is computed can be found by solving Eq.~\\eqref{eqn:exact_continuous_bogo_ODE} for example by using the perturbative method described in section~\\ref{sec:perturbative_method}.\\footnote{Note however that the definition and meaning of the matrix $\\tilde{S}$ in this section differs from the definition and meaning of the matrix $\\tilde{S}$ in section~\\ref{sec:perturbative_method}. While in here the matrix $\\tilde{S}$ represents the transformation of the field operators given by Eq.~\\eqref{eq:transformation_of_field}, $\\tilde{S}$ in section~\\ref{sec:perturbative_method} denotes a convenient substitution which has been used for the purposes of the perturbation method and is given by Eq.~\\eqref{def:tildeS}.} This perturbation method uses the expansion in the small parameter $\\epsilon$. Both $\\alpha=\\alpha(\\epsilon)$ and $\\beta=\\beta(\\epsilon)$ depend on this parameter, but to keep expressions sufficiently short we do not explicitly indicate this dependence. Finally, we describe the transformation from the initial subsystem state $\\sigma_{0}$ to a final subsystem state $\\sigma(\\epsilon)$ via a map $\\mathcal{E}$ defined as\n\\[\n\\label{eqn:QFI_quantum_channel}\n\\sigma(\\epsilon)=\\mathcal{E}[\\tilde{\\sigma}(\\epsilon)]=\\mathrm{tr}_{E}\\big[{\\tilde{S}}(\\epsilon)\\tilde{{\\sigma}}_{0}{\\tilde{S}}(\\epsilon)^{\\dag}\\big]\n=\\begin{bmatrix}\nX & Y \\\\\n\\ov{Y} & \\ov{X}\n\\end{bmatrix}.\n\\]\nThe Bogoliubov transformation between the initial and final states can be viewed as a quantum channel on the space of quantum states.\n\nThe elements of the final matrix $\\sigma(\\epsilon)$ for an arbitrary Gaussian state are given by\n\\begin{subequations}\n\\begin{align}\nX^{ij}&\\,=\\,\\sum_{a,b}\\Big(\\ov{\\alpha}^{ia}\\,X_0^{ab}\\,\\alpha^{jb}-\\ov{\\beta}^{ia}\\,\\overline{Y}_{0}^{ab}\\alpha^{jb}-\\ov{\\alpha}^{ia}Y_{0}^{ab}\\beta_{jb}+\\ov{\\beta}^{ia}\\overline{X}_{0}^{ab}\\beta^{jb}\\Big),\\\\\nY^{ij}&\\,=\\,\\sum_{a,b}\\Big(-\\ov{\\beta}^{ia}\\,\\overline{X}_{0}^{ab}\\,\\ov{\\alpha}^{jb}+\\ov{\\alpha}^{ia}\\,Y_{0}^{ab}\\,\\ov{\\alpha}^{jb}+\\ov{\\beta}^{ia}\\,\\overline{Y}_{0}^{ab}\\,\\ov{\\beta}^{jb}-\\ov{\\alpha}^{ia}\\,X_{0}^{ab}\\,\\ov{\\beta}^{jb}\\Big).\n\\end{align}\n\\end{subequations}\nThese expressions, coupled with the exact definitions of the quantum Fisher information and Bogoliubov co-efficients, can be used to compute the quantum Fisher information for any state within our quantum field theory framework. The resulting expressions are rather unwieldy and hence we have not written them out explicitly. However, in the following we will write the quantum Fisher information explicitly for the cases where the initial state is one- and two-mode squeezed thermal state and the environment is assumed to be a thermal state. The expressions provided will be perturbative expressions in the small parameter $\\epsilon$ we estimate, while the zeroth order will be exact for $\\epsilon=0$, i.e., $H^{(0)}=H(0)$. Note that when using the perturbative solution of the continuous Bogoliubov transformations~\\eqref{eqns:general_solution_arbitrarily_high_order}, having the transformation matrices expanded up to order $(n)$ in the unknown parameter $\\epsilon$ the quantum Fisher information can be expressed up to order $(n-1)$. This is because the quantum Fisher information is a function of at most first derivatives of the transformation matrix $S$, $H(\\sigma(\\epsilon))=H(\\tilde{\\sigma}_0,\\alpha,\\beta,\\tfrac{\\mathrm{d}}{\\mathrm{d}\\epsilon}{\\alpha},\\tfrac{\\mathrm{d}}{\\mathrm{d}\\epsilon}{\\beta})$.\n\nIn the following we also implicitly assume that the Bogoliubov coefficients have the following property\n\\begin{equation}\n\\label{eqn:simplifying_condition}\n\\frac{\\mathrm{d}}{\\mathrm{d}\\epsilon}\\alpha_{jj}\\big|_{\\epsilon\\,=\\,0}\\,=\\,\\frac{\\mathrm{d}}{\\mathrm{d}\\epsilon}\\beta_{jj}\\big|_{\\epsilon\\,=\\,0}\\,=\\,0.\n\\end{equation}\nThis is equivalent to the statement that the first order coefficients of the diagonal $\\boldsymbol{\\alpha}$ and $\\boldsymbol{\\beta}$ are zero i.e. $\\alpha^{(1)}_{jj}\\,=\\,\\beta^{(1)}_{jj}\\,=\\,0$. As an example, these assumptions hold when the symplectic operation is symmetric around zero, i.e., $\\alpha(\\epsilon)=\\alpha(-\\epsilon)$, $\\beta(\\epsilon)=\\beta(-\\epsilon)$, and also for the special case of $\\alpha_{mn},\\beta_{mn}\\in\\mathbb{R}$. Physically, this condition means that the channel does not affect the same mode up to the first order in $\\epsilon$, i.e., the channel is mostly mode-entangling channel. It is possible to generalise this work to cases where diagonal first order Bogoliubov coefficients are non-zero, however, since Bogoliubov coefficients considered in previous literature~\\cite{Bruschi2012a,Bruschi2013b,Friis2013a,Sabin2014a} all satisfy Eq.~\\eqref{eqn:simplifying_condition}, we will restrict to such case. In the following sections, we consider that all quantities (matrix and scalar) can be expanded in the form,\n\\begin{equation}\nf(\\epsilon)\\,=\\,f^{(0)}+f^{(1)}\\epsilon+f^{(2)}\\epsilon^{2}+\\mathcal{O}(\\epsilon^{3}).\n\\end{equation}\nThroughout this chapter we will be using Planck units $\\hbar\\,=\\,c\\,=\\,k_{B}\\,=\\,1$.\n\n\\section{Estimating space-time parameters with single-mode Gaussian states}\nWe first compute the quantum Fisher information of a single mode undergoing a Bogoliubov transformation that depends on the physical parameter to be estimated. We consider the following initial state, \\begin{equation}\n\\label{eqn:Single_mode_state}\n\\sigma_{0}\\,=\\,\\lambda_{m}\\,\n\\begin{bmatrix}\n\\cosh(2r) & \\sinh(2r) \\\\\n\\sinh(2r) & \\cosh(2r)\n\\end{bmatrix},\\quad\\sigma_{E}\\,=\\,\\bigoplus_{j\\ne m}\\lambda_{j}\\,I,\n\\end{equation}\nwhich corresponds to a single mode squeezed thermal state with squeezing parameter $r$, thermal parameter $\\lambda_{m}\\ge 1$ and all other modes in a separable thermal state. The temperature of the state, denoted by $T$, is related to the thermal parameter through $\\lambda_{m}\\,=\\,\\coth(E_{m}\/2T)$ where $E_{m}=\\omega_{m}$ is the energy of each mode. Note that for zero temperature, the thermal parameter reduces to $\\lambda_{m}\\,=\\,1$.\n\nExact elements of the the final state $\\sigma(\\epsilon)$ can be computed as\n\\begin{subequations}\n\\label{eqn:one_mode_XYsummed}\n\\begin{align}\nX^{mm}&=\\lambda_{m}\\,\\Big(\\cosh(2r)\\,\\big(|\\alpha^{mm}|^{2}+|\\beta^{mm}|^{2}\\big)-2\\,\\mathrm{Re}\\big[\\alpha^{mm}\\overline{\\beta}^{mm}\\big]\\sinh(2r)\\Big)\\nonumber\\\\\n&+\\sum_{a\\ne m}\\lambda_{a}\\,\\big(|\\alpha^{ma}|^{2}+|\\beta^{ma}|^{2}\\big),\\\\\nY^{mm}&=\\lambda_{m}\\Big(-2\\,\\cosh(2r)\\,\\overline{\\alpha}^{mm}\\overline{\\beta}^{mm}+\\big(\\overline{\\alpha}^{mm\\ \\!2}+\\overline{\\beta}^{mm\\ \\!2}\\big)\\sinh(2r)\\Big)-2\\,\\sum_{a\\ne m}\\lambda_{a}\\overline{\\alpha}^{ma}\\overline{\\beta}^{ma}.\n\\end{align}\n\\end{subequations}\nWe can write this expression as a series expansion in $\\epsilon$ around the point $\\epsilon_{0}\\,=\\,0$,\n\\begin{equation}\n\\sigma(\\epsilon)\\,=\\,\\sigma^{(0)}+\\sigma^{(1)}\\,\\epsilon+\\mathcal{O}(\\epsilon^{2}).\n\\end{equation}\nIt should also be noted that, in general, the covariance matrix elements $X^{(j)mn}$ and $X^{(j)mn}$ will depend on both squeezing, $r$, and the thermal parameters $\\lambda_{m}$. We will also denote phases acquired due to free time evolution as $G^{m}\\,=\\,e^{+i\\,\\omega_{m}\\tau}$ with $\\omega_{m}$ the zeroth order contribution to frequency of the mode $m$.\n\nWe now proceed to choose specific values for the temperature and squeezing to find analytically the quantum Fisher information in regimes of interest.\n\n\\subsection{Zero initial temperature}\nWe start by considering an initial state with zero temperature. The perturbative expansion of the quantum Fisher information in Eq.~\\eqref{eq:one_mode_quantum_fisher_information} needs particular attention. If we consider a state which is initially pure one finds that the denominators in Eqs.~\\eqref{eq:one_mode_quantum_fisher_information} vanish. This potentially problematic point can be handled in multiple ways which is discussed in more detail in section~\\ref{sec:problems_at_pops}. However, one can make a series expansion of each term and by applying L'H\\^{o}pital's rule one obtains a finite result,\n\\begin{equation}\n\\label{eqn:one_mode_qfi_zero_temp}\n\\begin{split}\nH_{1}(\\epsilon)\\,&=\\,X^{(2)mm}\\,\\cosh(2\\,r)-\\mathrm{Re}[(G^{m})^2\\,Y^{(2)mm}]\\,\\sinh(2r)\\\\\n&+\\frac{2}{3}\\Big(X^{(3)mm}\\,\\cosh(2\\,r)-\\mathrm{Re}[(G^{m})^2\\,Y^{(3)mm}]\\,\\sinh(2\\,r)\\Big)\\,\\epsilon+\\mathcal{O}(\\epsilon^{3})\n\\end{split}\n\\end{equation}\nFor convenience and clarity, we have left the second order covariance matrix elements written in the general form $X^{(2)mm}$ and $Y^{(2)mm}$. This elegant expression builds upon the zeroth order result of~\\cite{Ahmadi2014a} and extends it to the linear regime in $\\epsilon$. It should be noted that, for the expansion~\\eqref{eqn:one_mode_qfi_zero_temp} to be valid, the initial squeezing $r$ and parameter $\\epsilon$ must satisfy $e^{2r}\\,\\epsilon\\ll1$.\n\n\n\n\\subsection{Large initial temperature}\n\nIn this case we find that the zeroth order quantum Fisher information for a single mode is identically zero, i.e. $H^{(0)}_{1}\\,=\\,0$ at any non-zero temperature. This implies that the estimation of the parameter $\\epsilon$ around zero is impossible for a one-mode squeezed state with a non-zero temperature. The first non-trivial contribution comes at $\\mathcal{O}(\\epsilon^{2})$. The result is,\n\\begin{equation}\\label{one:mode:large:temperature:result}\n\\begin{split}\nH^{(2)}_{1}&=\\frac{|Y_{mm}^{(2)}|^{2}}{\\lambda_{m}^{2}+1}+\\frac{(X_{mm}^{(2)})^{2}}{\\lambda_{m}^{2}-1}\\,-\\, \\frac{2\\,\\lambda_{m}^{2}\\,X_{mm}^{(2)}\\mathrm{Re}[G_{m}^{2}\\overline{Y}_{mm}^{(2)}]}{\\lambda_{m}^{4}-1}\\sinh(4r)\\,\\\\\n&+\\frac{2\\,\\lambda_{m}^{2}\\Big((X_{mm}^{(2)})^{2}+\\mathrm{Re}[G_{m}^{4}(\\overline{Y}_{mm}^{(2)})^{2}]\\Big)}{\\lambda_{m}^{4}-1}\\sinh^{2}(2r)\n\\end{split}\n\\end{equation}\nClearly the condition $\\lambda_{m}>1$ is key and the equation holds in the regime $\\epsilon^{2}\\ll\\lambda_{m}-1$ or, in terms of the previous subsections notation, $\\epsilon^{2}\\ll Z^{2}$ by which the ``large\" temperature regime is defined.\n\nNote that the significant non-smooth difference between the zeroth order of the large temperature case and the small temperature case is given by a contribution from the change of the purity of the state as illustrated on example~\\ref{ex:problematic_example}.\n\n\\section{Estimating space-time parameters with two-mode Gaussian states}\n\nHere we compute the quantum Fisher information for thermal two-mode states with non-degenerate thermal parameters (i.e., the frequencies of the two modes are different). The initial state is the two-mode squeezed thermal state and has the form,\n\\begin{equation}\n\\label{eqn:Two_mode_state}\n\\sigma_{0}\\,=\\,\n\\begin{bmatrix}\nD^{mn} & 0 & 0 & C^{mn} \\\\\n0 & D^{nm} & C^{mn} & 0 \\\\\n0 & C^{mn} & D^{mn} & 0 \\\\\nC^{mn} & 0 & 0 & D^{nm}`\n\\end{bmatrix},\\,\\,\\sigma_{E}\\,=\\,\\bigoplus_{j\\ne m,n}\\lambda_{j}\\,I,\n\\end{equation}\nwhere we have introduced\n\\begin{subequations}\n\\begin{align}\nD^{mn}&:=\\lambda_{m}\\cosh^{2}(r)+\\lambda_{n}\\sinh^{2}(r), \\\\\nC^{mn}&:=(\\lambda_{m}+\\lambda_{n})\\cosh(r)\\sinh(r).\n\\end{align}\n\\end{subequations}\n\nExact elements of the the final state $\\sigma(\\epsilon)$ can be computed as\n\\begin{subequations}\n\\label{eqn:two_mode_XYsummed}\n\\begin{align}\n\\begin{split}X^{ij}&=D^{mn}(\\ov{\\alpha}^{im}\\alpha^{jm}+\\ov{\\beta}^{im}\\beta^{jm})-C^{mn}(\\ov{\\beta}^{im}\\alpha^{jn}+\\ov{\\alpha}^{im}\\beta^{jn})\n+D^{nm}(\\ov{\\alpha}^{in}\\alpha^{jn}+\\ov{\\beta}^{in}\\beta^{jn})\\\\\n&-C^{nm}(\\ov{\\beta}^{in}\\alpha^{jm}+\\ov{\\alpha}^{in}\\beta^{jm})\n+\\sum_{a\\neq m,n}\\lambda_{a}(\\ov{\\alpha}^{ia}\\alpha^{ja}+\\ov{\\beta}^{ia}\\beta^{ja}),\n\\end{split}\\\\\n\\begin{split}Y^{ij}&=-D^{mn}(\\ov{\\beta}^{im}{\\ov{\\alpha}}^{jm}+\\ov{\\alpha}^{im}{\\ov{\\beta}}^{jm})+C^{mn}(\\ov{\\alpha}^{im}{\\ov{\\alpha}}^{jn}+\\ov{\\beta}^{im}{\\ov{\\beta}}^{jn})\n-D^{nm}(\\ov{\\beta}^{in}{\\ov{\\alpha}}^{jn}+\\ov{\\alpha}^{in}{\\ov{\\beta}}^{jn})\\\\\n&+C^{nm}(\\ov{\\alpha}^{in}{\\ov{\\alpha}}^{jm}+\\ov{\\beta}^{in}{\\ov{\\beta}}^{jm})-\\sum_{a\\neq m,n}\\lambda_{a}(\\ov{\\beta}^{ia}{\\ov{\\alpha}}^{ja}+\\ov{\\alpha}^{ia}{\\ov{\\beta}}^{ja}),\n\\end{split}\n\\end{align}\n\\end{subequations}\nwhere $i,j=m,n$.\n\n\\subsection{Zero initial temperature}\n\nIn the two-mode case, the quantum Fisher information for zero temperature was computed using Eq.~\\eqref{GeneralQFI} and is given as a series expansion in $\\epsilon$. The resulting expressions are computable but considerably more involved. Here we present the results for the zero and first order contributions in $\\epsilon$. In the linear contribution, we present only the case of zero squeezing. The formula for non-zero squeezing is too long and therefore, we have chosen to focus on the quantitative behaviour.\n\\begin{subequations}\n\\begin{align}\n\\label{eqn:two_mode_qfi_zero_temperature_zeroth_order}\nH_{2}^{(0)}\\,=&\\,(X^{(2)mm}+X^{(2)nn})\\cosh(2r)\\nonumber\\\\\n&\\,-4|\\beta^{(1)mn}|^{2}-2\\,\\mathrm{Re}[G^{m}G^{n}Y^{(2)mn}]\\sinh(2r)\\nonumber\\\\\n&\\,-4\\,(|\\alpha^{(1)mn}|^{2}+\\mathrm{Im}[G^{m}\\,\\overline{\\beta}^{(1)mn}]^{2})\\sinh^{2}(2r), \\\\\n\\label{eqn:two_mode_qfi_zero_temperature_first_order}\nH_{2}^{(1)}\\big|_{r\\,=\\,0}\\,=\\,&\\frac{2}{3}\\,\\Big(6\\,\\mathrm{Re}[G^{n}\\beta^{(1)mn}Y^{(2)mn}]+X^{(3)mm}+X^{(3)nn}]\\Big).\n\\end{align}\n\\end{subequations}\nAt zeroth order, the quantum Fisher information depends on the squeezing parameter in the same way as in the single mode channels studied in the previous section. However, at first order, particle creation terms  $\\beta^{(1)mn}$ appear generating entanglement in the system. Therefore, we conclude that in this case entanglement does not provide an important improvement in precision. A highly squeezed single mode probe state could be enough to enable a good measurement strategy for the estimation of parameters uncoded in Bogoliubov transformations. Single mode states are usually more accessible in realistic experiments and this could provide an important advantage in quantum metrology for quantum fields.\n\n\n\n\n\\subsection{Large initial temperature}\n\nAs in the single mode case, the thermal parameters take values strictly greater than unity, i.e. $\\lambda_{j}>1$. The components of the state can be exactly computed and are given in Eq.~\\eqref{eqn:two_mode_XYsummed}. We find that the quantum Fisher information, including linear contributions, is given by $H_2(\\epsilon)\\,=\\,H^{(0)}_{2}+H^{(1)}_{2}\\epsilon+\\mathcal{O}(\\epsilon^{2})$, with coefficients,\n\\begin{equation}\n\\label{eqn:two_mode_QFI_zeroth_order}\n\\begin{split}\nH_{2}^{(0)}&\\,=\\,h_{00}+h_{02}\\,\\sinh^{2}(2r), \\\\\nh_{00}&\\,=\\,\\frac{2\\,(\\lambda_{m}-\\lambda_{n})^{2}|\\alpha^{(1)mn}|^{2}}{\\lambda_{m}\\lambda_{n}-1}+\\frac{2\\,(\\lambda_{m}+\\lambda_{n})^{2}|\\beta^{(1)mn}|^{2}}{\\lambda_{m}\\lambda_{n}+1}, \\\\\nh_{02}&\\,=\\,\\frac{2\\,(\\lambda_{m}+\\lambda_{n})^{2}((\\lambda_{m}\\lambda_{n}-1)^{2}+\\lambda_{m}^{2}+\\lambda_{n}^{2}-2)|\\alpha^{(1)mn}|^{2}}{(\\lambda_{m}^{2}+1)(\\lambda_{n}^{2}+1)(\\lambda_{m}\\lambda_{n}-1)}\n+\\frac{2\\,(\\lambda_{m}+\\lambda_{n})^{2}\\mathrm{Im}[G^{m}\\overline{\\beta}^{(1)mn}]^{2}}{\\lambda_{m}\\lambda_{n}+1}.\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\nH_{2}^{(1)}&\\,=\\,h_{10}+h_{11}\\,\\sinh(2r)+h_{12}\\,\\sinh^{2}(2r), \\\\\nh_{10}&\\,=\\,4\\,\\Bigg(\\frac{\\lambda_{n}-\\lambda_{m}}{\\lambda_{m}\\lambda_{n}-1}\\,\\mathrm{Re}[G_{n}\\overline{\\alpha}^{(1)mn} \\overline{X}^{(2)mn}]-\\frac{\\lambda_{m}+\\lambda_{n}}{\\lambda_{m}\\lambda_{n}+1}\\,\\mathrm{Re}[G_{n}\\beta^{(1)mn}Y^{(2)mn}]\\,\\cosh(2r)\\Bigg), \\\\\nh_{11}&\\,=\\, \\frac{2\\,(\\lambda_{m}+\\lambda_{n})\\mathrm{Re}[G^{m}\\overline{\\beta}^{(1)mn}]}{\\lambda_{m}\\lambda_{n}+1}(X^{(2)mm}+X^{(2)nn})\\\\\n&-\\frac{16\\,\\lambda_{m}\\lambda_{n}(\\lambda_{m}^{2}-\\lambda_{n}^{2})^{2}|\\alpha^{(1)mn}|^{2}\\mathrm{Re}[G^{m}\\overline{\\beta}^{(1)mn}]}{(\\lambda_{m}^{2}+1)(\\lambda_{n}^{2}+1)(\\lambda_{m}^{2}\\lambda_{n}^{2}-1)}\\cosh(2r) \\\\\n&-\\frac{2(\\lambda_{m}+\\lambda_{n})(\\lambda_{m}\\lambda_{n}+1)}{(\\lambda_{m}^{2}+1)(\\lambda_{m}^{2}+1)}\\mathrm{Re}[G^{n}\\overline{\\alpha}^{(1)mn}(\\overline{G}^{m}\\overline{G}^{n} \\overline{Y}^{(2)mm}-G^{m}G^{n}Y^{(2)nn})]\\\\\n&+\\frac{2(\\lambda_{m}-\\lambda_{n})(\\lambda_{m}+\\lambda_{n})^{2}}{(\\lambda_{m}^{2}+1)(\\lambda_{n}^{2}+1)(\\lambda_{m}\\lambda_{n}-1)}\\mathrm{Re}[G^{n}\\overline{\\alpha}^{(1)mn}(\\overline{G}^{m}\\overline{G}^{n} \\overline{Y}^{(2)mm}\\!\\!+\\!G^{m}G^{n}Y^{(2)nn})]\\cosh(2r),\\\\\nh_{12}&\\,=\\,\\frac{4\\,(\\lambda_{n}-\\lambda_{m})(\\lambda_{n}+\\lambda_{m})^{2}\\mathrm{Re}[G^{n}\\overline{\\alpha}^{(1)mn}\\,\\overline{X}^{(2)mn}]}{(1+\\lambda_{m}^{2})(1+\\lambda_{n}^{2})(\\lambda_{m}\\lambda_{n}-1)}.\n\\end{split}\n\\end{equation}\nIn general, the coefficients $\\alpha^{(1)mn}$,  $\\beta^{(1)mn}$ and the covariance matrix are time dependent. In the large temperature regime, to zeroth order, the quantum Fisher information for two-mode probe states is non-zero. This result is in contrast with the single-mode case. Even when the probe state  has zero squeezing, the quantum Fisher information is non-zero and it is proportional to the number of particles created after the Bogoliubov transformation~\\cite{Bruschi2013c}. We can also analyse the effect of temperature on the quantum Fisher information by varying the parameters $\\lambda_{m}$ and $\\lambda_{n}$.\nWe note that at when estimating around the point $\\epsilon\\,=\\,0$, the zero order expressions for the quantum Fisher information are exact, $H(0)=H^{(0)}$.\n\n\n\\section{Example: Estimation of the proper acceleration}\n\nTo illustrate the power of the derived formulae, we calculate the bound on the estimation of the proper acceleration using cavities. This section is a continuation of example~\\ref{ex:accelerated_cavity}. Assume a quantum state inside of a non-moving cavity. Starting at proper time $\\tau_0=0$, the cavity goes through a period $\\tau$ of the proper acceleration $a$ (as measured in the centre of the cavity) and period $\\tau$ of retardation $-a$, stopping again at time $2\\tau$. We wish to estimate the proper acceleration $a$, we thus identify $a\\equiv \\epsilon$. The proper length of the cavity $L=1$ is considered constant during the whole procedure. Bogoliubov transformation of the state in this scenario has been calculated using a continuous perturbative expansion in the small parameter $a$ and is given by Eq.~\\eqref{eq:bogos_acceleration}. Using these transformations, we calculate the zeroth order quantum Fisher information for a one-mode squeezed vacuum and a two-mode squeezed thermal state as shown on figures~\\ref{fig:one-modeQFI} and~\\ref{fig:two-modeQFI}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{onemode.png}\n\\caption{The zeroth order of the quantum Fisher information for the estimation of the acceleration parameter $a$ using a one-mode squeezed state with initial zero temperature. Calculated using Eq.~\\eqref{eqn:one_mode_qfi_zero_temp} and $m=1$ (using a Fock space corresponding to the first excited state within a cavity). The graph shows that to achieve the highest possible precision in estimation it is appropriate to measure at certain times ($\\tau=1,3,5,\\dots$). This periodic behavior is due to the fact that the information about the parameter moves into the modes we cannot access -- the environment -- and back. For times when the quantum Fisher information is the highest the estimation precision grows exponentially with the squeezing parameter $r$.}\\label{fig:one-modeQFI}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1\\linewidth]{mixL.png}\n\\caption{The zeroth order of the quantum Fisher information for the estimation of the acceleration parameter $a$ using a two-mode squeezed thermal state with initial ``large'' temperature. Calculated using Eq.~\\eqref{eqn:two_mode_QFI_zeroth_order} and $m=1$, $n=2$ (Fock spaces corresponding to the first and the second excited state within the cavity). Different combinations of initial temperatures are used, $\\nu_{m,n}=2,6,10$. Similarly to the one-mode scenario, it is appropriate to measure at certain times ($\\tau=1,3,5,\\dots$) when the estimation precision grows exponentially with the squeezing parameter $r$. Moreover, the graph shows that the highest precision in estimation is achieved with large temperature difference between the modes, i.e., when $\\nu_1=2$ and $\\nu_2=10$, or $\\nu_1=10$ and $\\nu_2=2$. The diagonal $\\nu_1=\\nu_2\\rightarrow\\infty$ quickly converges to the double of the two-mode squeezed vacuum value given by $\\nu_1=\\nu_2=1$. An opportunity of using temperature difference between the modes is not the only advantage of using the two-mode states. In contrast to the one-mode states, two-mode states also achieve one order higher precision with the same amount of squeezing.}\\label{fig:two-modeQFI}\n\\end{figure}\n\n\\section{Conclusion and Discussion}\\label{sec:QFT_conclusion}\n\nIn order to provide a general framework for estimating space-time parameters using quantum metrology, we have extended previous pure state analysis to the mixed case. The main motivation is that, for any practical and experimental purposes, quantum systems are always mixed. We have restricted our analysis to Gaussian probe states as the covariance matrix formalism provides a simple mathematical description. In particular, Gaussian states are also straightforward to prepare in quantum optical laboratories. We have computed general and exact expressions for the quantum Fisher information for one- and two- mode mixed Gaussian probe states undergoing arbitrary Bogoliubov transformations, and illustrated their use for the estimation of proper acceleration.\n\nBy expanding the Bogoliubov coefficients around the point $\\epsilon_{0}=0$, we were able to evaluate the quantum Fisher information for the case of one- and two-mode Gaussian probe states. We obtained exact expressions for the quantum Fisher information at point $\\epsilon\\,=\\,0$, and perturbative expressions for $\\epsilon\\neq 0$. In the single mode case, for a finite temperature, the quantum Fisher information is identically zero at $\\epsilon=0$. This implies that for states which are at some temperature other than absolute zero, one cannot distinguish between two states in the neighbourhood of $\\epsilon=0$. This can be explained in the following way: the assumption~\\eqref{eqn:simplifying_condition} says that there is no change in the same mode up to the first order in $\\epsilon$, i.e., the Bogoliubov transformation is a purely mode-entangling channel up to the first order in $\\epsilon$. In general, the quantum Fisher information depends only on the first derivatives of the parameter. Since these derivatives are zero for a single mode state, the zeroth order of the quantum Fisher information is also zero. The only exception is for pure states because, as explained in section~\\ref{sec:pure_states_QM}, there is a contribution from the change of purity given by the second derivative of the symplectic eigenvalues. This is why the zeroth order quantum Fisher information is non-zero in the zero and small temperature regime. For larger values of $\\epsilon$, the quantum Fisher information is non-zero for all cases and the quantum Cram\\'{e}r-Rao bound is finite. In the case of a thermal two-mode squeezed state there is, however, always the possibility of distinguishing between infinitesimally close states in the neighbourhood of $\\epsilon=0$.\n\nHigher squeezing and a high temperature difference of modes significantly improves the precision in estimation. We observed a similar behaviour in the estimation of unitary channels as studied in the previous chapter. Squeezed states are generally more sensitive to rotations and mode-mixing as well as particle creation when the squeezed state is appropriately oriented. The difference in temperature also helps because any mode-mixing channel given by a non-trivial passive coefficient $\\alpha$ will, in general, mix temperatures of different modes. This effect vanishes when the modes have the same temperature. Ultimately, this is due to the fact that squeezed thermal states have high variance in energy which, according to the general equation for the quantum Fisher information~\\eqref{pureK}, leads to a greater precision in estimation.\n\n\n\n\nOur results will enable researchers to evaluate how well space-time parameters, such as the amplitude of gravitational waves, accelerations and local gravitational fields, can be estimated in the presence of background temperature~\\cite{Sabin2014a,Sabin2015a}. We observe that strategies involving one- and two-mode probe states exhibit the same exponential gain for large squeezing. However, single mode thermal states do not perform well in the scenario when the channel is mostly mode-entangling, which is a common case in the literature~\\cite{Bruschi2012a,Bruschi2013b,Friis2013a,Sabin2014a}.\n\nOur results lead naturally to other important questions. The quantum Fisher information is the optimisation of the classical Fisher information over all possible measurements. One can therefore ask: ``What is the optimal measurement for our scheme?'' An analysis of the symmetric logarithmic derivative would certainly shed light on this and general knowledge in this direction has already been developed~\\cite{Monras2013a,Gao2014a}, Eq.~\\eqref{eq:SLD_Gaussian}. Furthermore, if the optimal measurement is found to be impractical then an analysis of more realistic measurements, such as homodyne and heterodyne measurements for Gaussian states, could prove fruitful. These questions are left for future work.\n\n\\begin{comment}\nQUANTUM REFERENCE FRAMES\n-quantum reference frames\n    -the integral and review\n    -reference frames in metrology\n    -derived formulas in a simple language and examples, adding what hasn't been added\n    -future guesses and possible directions (theorem about boob-shaped things, why everything goes to the same shape?), possible generalizations (remember what sam braunstein said??, look at the notes we did together)\n\\end{comment}\n\n\\chapter{Quantum metrology with imperfect reference frames}\\label{chap:reference_frames}\n\nThe last chapter of this thesis will be slightly different from the previous in scope. This chapter is not primarily focused on Gaussian states and the operations studied here are not Gaussian operations. However, we will use many tools of quantum metrology as introduced in chapter~\\ref{chap:QM}. In this chapter we show how not sharing a common reference frame between two parties affects the estimation precision of a parameter encoded in a quantum state which was sent from party A to party B. Moreover, we will show how this estimation precision can be improved using quantum reference frames. This chapter is mostly based on a paper we published in~\\cite{safranek2015quantum} which also contains full proofs.\n\nConsider the following scenario: Alice encodes the parameter of interest into a known quantum state. She sends this state to Bob and it is Bob's task to decode the parameter. This known quantum state has been, however, defined with respect to Alice's reference frame, i.e., with respect to the measurement basis Alice chose to use. Bob's reference frame may be different but if he knows how to relate his reference frame to Alice's, he can simply rearrange his measurement basis to match hers and measure the parameter with the same precision. The problem begins when Bob does not know the relative orientation of his reference frame with respect to Alice's reference frame. Then his knowledge of the quantum state of interest is only partial and some or all information about the quantum state is lost. Therefore, Bob's precision in estimation of the parameter will be lower than Alice's. Nevertheless, there is a way to counter that. Because Bob does not share a reference frame with Alice, Alice can send her original state of interest along with another quantum state which represents a piece of her reference frame. This additional quantum state is called the quantum reference frame~(QRF). Bob will then perform the measurement on this composite system, in other words, he will perform a composite measurement which can be interpreted as the measurement on the original quantum state relative to the attached quantum reference frame. Because the parameter of interest is now encoded in the internal degrees of freedom of the composite system some precision in estimation of this parameter is retrieved.\n\nQuantum reference frames had been first introduced in~\\cite{Peres2001Transmission,Peres2001entangled,bartlett2007reference,angelo2011physics}. A QRF is different from its classical counterpart in two ways: first, due to its quantum nature, it has an inherent uncertainty and the measurement results are only an approximation of what would be obtained using a classical reference frame.  For instance, if the reference frame describes a continuum of orientations in space, then states with different orientations are not perfectly distinguishable. Second, each time the QRF is used, it suffers a back-action, which causes future measurements to be less accurate. There has been extensive literature published on QRFs. We mention phase measurements using a QRF~\\cite{bartlett2006degradation}, degradation of a directional QRF~\\cite{bartlett2006degradation,poulin2007dynamics,ahmadi2010dynamics}, or a QRF considered as the resource state~\\cite{gour2008the,gour2009measuring,skotiniotis2012alignment,narasimhachar2014phase}. QRFs enable us to achieve quantum information processing tasks without first establishing a shared reference frame. A QRF allows us to perform tasks in the absence of a common classical reference frame, in the same way that entangled states allow for the possibility of performing non-local quantum operations.\n\nIn this chapter, we utilize the powerful machinery of quantum metrology to study the ultimate precision bounds in measurement of physical parameters with respect to QRFs. First we investigate how the ultimate precision in measurement of a parameter decreases due to inaccessibility of a perfect classical reference frame. We analyze the decrease in quantum Fisher information as a result. In particular, we provide necessary and sufficient conditions for two extreme cases that can occur in quantum parameter estimation. The first case is when the absence of a perfect reference frame does not affect the precision and the second case is when measurement of the parameter is no longer possible due to lack of access to a classical reference frame. We split the problem into two subproblems: the first case is when the encoding operator commutes with the operator representing the noise and the second is when it does not. Counter-intuitively, we show that the non-commuting case has some advantages over the commuting one. While the existence of ``decoherence-free subspaces'' is essential for encoding information in the commuting case~\\cite{lidar1998decoherence,lidar2001decoherence,bacon2001coherence,bartlett2007reference}, for non-commuting operators the estimation is possible even in the absence of such subspaces. The trade-off is, however, that the precision will in general depend on the parameter to be estimated.\nFinally, we present three examples to further clarify different aspects of quantum metrology with imperfect frames of reference.\n\n\\section{Mathematical framework of quantum reference frames}\n\nConsider $g\\in G$ to be the group element that describes the passive transformation from Alice's to Bob's reference frame. Alice prepares a state ${\\hat{\\rho}}_{A}$ relative to her local reference frame. In the Bob's reference frame this state looks as $\\hat{U}(g){\\hat{\\rho}}_A \\hat{U}(g)^\\dag$, where $\\hat{U}$ is a unitary operator parametrized by a group element $g$. However, if Bob is completely unaware of the relation between his local reference and Alice's local reference frame, we can assume that the group element $g$ is completely unknown. It follows that relative to Bob's reference frame this state is seen as\\footnote{We will restrict our attention to Lie-groups that are compact, so that they possess a group-invariant (Haar) measure ${\\mathrm{d}} g$. We refer the readers for more details to \\cite{bartlett2007reference}.} \\cite{bartlett2007reference}\n\\[\n\\label{gtwirl}\n{\\hat{\\rho}}_{B}={\\cal{G}}[{\\hat{\\rho}}_{A}]=\\int {\\mathrm{d}} g \\hat{U}(g) {\\hat{\\rho}} _{A}\\hat{U}(g)^{\\dag}.\n\\]\nTherefore, lacking such a shared reference frame is equivalent to having a noisy completely positive trace-preserving map which is known as the ``\\textsl{g-twirling map}'', i.e. $\\G({\\hat{\\rho}}_{A})$. The resulting density matrix $\\rho_B$ then represents the state of knowledge Bob has about the physical system. It follows from the construction that such state must be less or equally pure to the state of Alice, and no information is lost only if the $\\hat{\\rho}_A$ is invariant under the g-twirling map. The integral above is over the Haar measure ${\\mathrm{d}} g$ which in case of compact Lie groups is group-invariant.\n\nNow we formalize the metrological scenario discussed in the introduction: We assume Alice sends a state ${\\hat{\\rho}}_\\epsilon$ which depends on the parameter of interest $\\epsilon$. This state can represent either solely the original state with the encoded parameter ${\\hat{\\rho}}_\\epsilon={\\hat{\\rho}}_{A\\epsilon}$ or the original state plus the attached quantum reference frame, ${\\hat{\\rho}}_\\epsilon={\\hat{\\rho}}_{A\\epsilon}\\otimes{\\hat{\\rho}}_{QRF}$. We moreover assume that the unitary operator which connects states from the Alice's reference frame and the Bob's reference frame is a one-parameter unitary group $\\hat{U}(t)=e^{-i\\hat{G}t}$, where $\\hat{G}$ is a Hermitian operator. Eq.~\\eqref{gtwirl} becomes\n\\begin{equation}\\label{Htwirl}\n{\\hat{\\rho}}_{B}=\\G[{\\hat{\\rho}}_{\\epsilon}]=\\lim_{T\\rightarrow\\infty} \\frac{1}{T}\\int_{0}^{T} dt\\ \\hat{U}(t){\\hat{\\rho}}_{\\epsilon}\\hat{U}(t)^{\\dag}\n\\end{equation}\nParameter $t$ is the element of the group and $\\hat{G}$ is the generator of translations in this parameter. For example, if two parties do not have synchronized clocks, i.e. they do not share a time reference frame, the parameter $t$ represents time and generator of translation in time $\\hat{G}$ is the Hamiltonian of the system. $\\hat{G}$ and $t$ are thus determined by the type of reference frame that is lacking.\n\nAssuming the spectral decomposition $\\hat{G}=\\sum_{i}G_{i}\\hat{P}_{i}$, where the $\\hat{P}_{i}$s are the projectors into subspaces with eigenvalues $G_{i}$ and $\\sum_{i}\\hat{P}_{i}=I$, one can easily check that the state $\\G[{\\hat{\\rho}}_{\\epsilon}]$ in~\\eqref{Htwirl} can be written as\n\\[\\label{rhoB}\n\\G[{\\hat{\\rho}}_{\\epsilon}]=\\sum_{i}\\hat{P}_i{\\hat{\\rho}}_{\\epsilon}\\hat{P}_i.\n\\]\nThe full aim of this chapter motivated by quantum reference frames is quite simple: It is to understand how the quantum Fisher information behaves under the transformation~\\eqref{rhoB}. Because of the computational complexity we consider only states which are pure in Alice's reference frames, ${\\hat{\\rho}}=\\ket{\\psi_\\epsilon}\\bra{\\psi_\\epsilon}$. From now on we also drop the lower index $\\epsilon$.\n\n\\section{Loss of the estimation precision}\n\nAssuming ${\\hat{\\rho}}=\\ket{\\psi}\\bra{\\psi}$ is a pure state the quantum Fisher information is given by Eq.~\\eqref{eq:QFI_pure},\n\\[\\label{eq:QFI_pure2}\nH({\\hat{\\rho}})=4(\\braket{\\partial_{\\epsilon}\\psi}{\\partial_{\\epsilon}\\psi}-\\norm{\\braket{\\psi}{\\partial_{\\epsilon}\\psi}}^2).\n\\]\nUsing Eq.~\\eqref{QFI} for the quantum Fisher information, the properties of the quantum channel $\\G$~\\eqref{rhoB} and the Parseval identity (see appendix~\\ref{app:FirstFormula}) we derive Bob's quantum Fisher information as\n\\[\\label{HrhoB}\nH(\\G[{\\hat{\\rho}}])=4\\braket{\\partial_\\epsilon\\psi}{\\partial_\\epsilon\\psi}-\n4\\sum_i\\frac{(\\Im\\bra{\\psi}\\hat{P}_i\\ket{\\partial_\\epsilon\\psi})^2}{\\bra{\\psi}\\hat{P}_i\\ket{\\psi}},\n\\]\nwhere the summation is over the indices $i$ for which $p_i=\\bra{\\psi_{\\epsilon}}\\hat{P}_i\\ket{\\psi_{\\epsilon}}\\neq0$. Note that we will use this convention throughout the rest of this chapter.\n\nAs mentioned before, because Bob does not share a reference frame with Alice, his precision in estimation in estimation should be lower than Alice's. We formalise this statement in the theorem below together with the necessary and sufficient conditions for two extreme cases. The first case is where the precision in measurement of $\\epsilon$ remains the same both in the absence or the presence of a perfect reference frame and the second case is where the measurement of $\\epsilon$ is not possible anymore due to inaccessibility of such reference frames. The full proof can be found in appendix~\\ref{app:nonnegQFIl} or~\\cite{safranek2015quantum}.\n\\begin{theorem}\\label{theorem}\nLet ${\\hat{\\rho}}=\\pro{\\psi}{\\psi}$ be a pure initial state. Then the quantum Fisher information ${\\hat{\\rho}}_B=\\G[{\\hat{\\rho}}]$ is bounded,\n$0\\leq H(\\G[{\\hat{\\rho}}])\\leq H({\\hat{\\rho}})$.\n\nNo precision in the estimation is lost (no loss), i.e. $H(\\G[{\\hat{\\rho}}])= H({\\hat{\\rho}})$, if and only if\n\\[\\label{cnoloss}\n\\forall i,\\ \\Im\\bra{\\psi}\\hat{P}_i\\ket{\\partial_\\epsilon\\psi}=-i\\bra{\\psi}\\hat{P}_i\\ket{\\psi}\\braket{\\psi}{\\partial_\\epsilon\\psi}.\n\\]\n\n$\\epsilon$ cannot be estimated anymore (maximum loss), i.e. $H(\\G[{\\hat{\\rho}}])=0$, if and only if\n\\[\\label{maxlosstheorem}\n\\forall i,\\ \\Re\\bra{\\psi}\\hat{P}_i\\ket{\\partial_\\epsilon\\psi}=0\\ \\wedge\\ \\forall \\ket{\\phi_j},\\ \\braket{\\phi_j}{\\partial_\\epsilon\\psi}=0,\n\\]\nwhere $\\{\\frac{\\hat{P}_i\\ket{\\psi}}{\\sqrt{p_i}},\\ket{\\phi_j}\\}_{i,j}$ forms an orthonormal basis of the Hilbert space.\n\\end{theorem}\nWithout loss of generality in Eq.~\\eqref{cnoloss}, we can restrict our analysis to the terms for which  $p_i=\\bra{\\psi}\\hat{P}_i\\ket{\\psi}\\neq0$, since using the Cauchy-Schwarz inequality it can be checked that the condition~\\eqref{cnoloss} holds trivially if $p_i=0$. Also, the set of states $\\{\\ket{\\phi_j}\\}_j$ are orthonormal states which together with the set of normalised states $\\{\\frac{\\hat{P}_i\\ket{\\psi}}{\\sqrt{p_i}}\\}_i$ form a complete basis. We can always find the set of states $\\{\\ket{\\phi_j}\\}_j$ via the Gram-Schmidt process for orthonormalisation of a set of vectors. Assuming the eigenvectors of $\\hat{G}$ span the whole Hilbert space, $\\{\\ket{\\phi_j}\\}_j$ is exactly the set of the eigenvectors of $\\G({\\hat{\\rho}})$ with the respective eigenvalue $0$.\\\\\n\n\nUsing similar analysis we can find the symmetric logarithmic derivative as\n\\[\\label{NSLD}\nL(\\G({\\hat{\\rho}}))=\\sum_i\\pro{\\varphi_i}{\\psi_i}+\\pro{\\psi_i}{\\varphi_i},\n\\]\nwhere $|\\psi_{i}\\rangle$ and $|\\varphi_{i}\\rangle$ are defined as\n\\begin{equation}\n\\ket{\\psi_i}=\\frac{\\hat{P}_i\\ket{\\psi}}{\\sqrt{p_i}},\\quad\n\\ket{\\varphi_i}=\\frac{1}{\\sqrt{p_i}}\\left(2\\hat{P}_i\\ket{\\partial_\\epsilon\\psi}-\n\\langle{\\psi_i}\\ket{\\partial_\\epsilon\\psi}\\ket{\\psi_i}\\right).\n\\end{equation}\nAs discussed in the first chapter, eigenvectors of the symmetric logarithmic derivative give the optimal POVM for the estimation of the parameter.\n\n\\section{Unitary encoding operations}\n\nIn this section we will assume the parameter of interest has been encoded via one-parameter unitary group, $\\ket{\\psi}=e^{-i\\hat{K}\\epsilon}\\ket{\\psi_0}$.\nWe will analyse the quantum Fisher information, the no-loss and the and maximum-loss conditions in terms of the hermitian operator $\\hat{K}$. We also remind that the Hermitian operator $\\hat{G}$ is the generator of the the noisy channel $\\G$. This way we split the problem into two different cases. The first case is where the encoding process in general does not commute with the noisy channel, i.e. $[\\hat{K},\\hat{G}]\\neq0$. We call such noise non-commutative. The second is when when the noise is commutative. If the noise is commutative, it simply means that the noisy channel \\eqref{Htwirl} commutes with the encoding process, i.e. $[\\hat{K},\\hat{G}]=0$. In that case our results can be also applied on systems where the noise \\eqref{rhoB} precedes the encoding operation.\n\nSuprisingly, for the parameter encoded via one-parameter unitary group the quantum Fisher information takes a very elegant form. This form follows directly from Eq.~\\eqref{HrhoB} and is formalized in the following theorem.\n\\begin{theorem}\\label{nicetheorem}\nFor an initial pure state $\\ket{\\psi_0}$, a generator $\\hat{K}$ of a unitary encoding operator $\\hat{U}(\\epsilon)=\\text{exp}(-i\\hat{K}\\epsilon)$ and projectors $\\hat{P}_i$ of the g-twirling map in \\eqref{rhoB}, the quantum Fisher information of the state $\\G[{\\hat{\\rho}}_{\\epsilon}]$ is\n\\begin{equation}\\label{BobQFI}\n\\begin{split}\n H(\\G[{\\hat{\\rho}}_{\\epsilon}])\n=4\\mathrm{Var}_{{\\hat{\\rho}}_{\\epsilon}}(\\hat{K})-4\\sum_i\np_i\\left[\\mathrm{Cov}_{{\\hat{\\rho}}_{\\epsilon}}\\left(\\frac{\\hat{P}_i}{p_i},\\hat{K}\\right)\\right]^2,\n\\end{split}\n\\end{equation}\nwhere ${\\hat{\\rho}}_{\\epsilon}=|\\psi_{\\epsilon}\\rangle\\langle\\psi_{\\epsilon}|$, $\\ket{\\psi_{\\epsilon}}=\\hat{U}(\\epsilon)\\ket{\\psi_0}$ and $p_i=\\mean{\\hat{P_i}}_{{\\hat{\\rho}}_\\epsilon}$. The covariance between two operators $\\hat{A}$ and $\\hat{B}$ is defined as $\\mathrm{Cov}_{\\hat{\\rho}}(\\hat{A},\\hat{B})=\\frac{1}{2}\\mean{\\{\\hat{A},\\hat{B}\\}}_{\\hat{\\rho}}-\\mean{\\hat{A}}_{\\hat{\\rho}}\\mean{\\hat{B}}_{\\hat{\\rho}}$ and the variance can be written as $\\mathrm{Var}_{\\hat{\\rho}}(\\hat{A})=\\mathrm{Cov}_{\\hat{\\rho}}(\\hat{A},\\hat{A})$. Note that the first part of the expression is equal to the quantum Fisher information in Alice's reference frame, $H({\\hat{\\rho}})=4\\mathrm{Var}_{{\\hat{\\rho}}_{\\epsilon}}(\\hat{K})$.\n\nNo precision in the estimation is lost (no loss), i.e. $H(\\G[{\\hat{\\rho}}])= H({\\hat{\\rho}})$, if and only if\n\\[\\label{cnolossKG}\n\\forall i,\\ \\mathrm{Cov}_{\\hat{\\rho}}(\\hat{P}_i,\\hat{K})=0.\n\\]\n\n$\\epsilon$ cannot be estimated anymore (maximum loss), i.e. $H(\\G[{\\hat{\\rho}}])=0$, if and only if\n\\[\\label{maxlosstheoremKG}\n\\forall i,\\ \\bra{\\psi}[\\hat{P}_i,\\hat{K}]\\ket{\\psi}=0\\ \\wedge\\ \\forall \\ket{\\phi_j},\\ \\bra{\\phi_j}\\hat{K}\\ket{\\psi}=0,\n\\]\nwhere $\\{\\frac{\\hat{P}_i\\ket{\\psi}}{\\sqrt{p_i}},\\ket{\\phi_j}\\}_{i,j}$ forms an orthonormal basis of the Hilbert space.\n\\end{theorem}\n\nFrom Eq.~\\eqref{BobQFI} we deduce that the decrease in the quantum Fisher information is proportional to the mean of squared covariances between the normalized\\footnote{$\\mean{\\hat{P}_i\/{p_i}}_{{\\hat{\\rho}}}=1$} projectors ${\\hat{P}_i}\/{p_i}$ and the encoding operator $\\hat{K}$. This means that  the more projectors ${\\hat{P}_i}\/{p_i}$ are correlated with the encoding operator $\\hat{K}$, the more precision is lost. Roughly speaking, in order to lose the minimum amount of precision one should choose an encoding operator $\\hat{K}$ which is less correlated with the decoherence caused by the noisy channel $\\G$. Explicit examples will be presented in the next section.\n\nAnother useful theorem will show that in certain common scenarios the estimation of the parameter is impossible.\n\\begin{theorem}\\label{thirdtheorem}\nLet $\\hat{G}$ have a non-degenerate spectrum, i.e., each projector $\\hat{P}_i$ has rank $1$. If $[\\hat{K},\\hat{G}]=0$, then $H(\\G[{\\hat{\\rho}}])=0$, i.e., parameter $\\epsilon$ cannot be estimated anymore.\n\\end{theorem}\nBecause $\\hat{K}$ and $\\hat{G}$ commute if and only if $\\hat{K}$ commutes with every projector $\\hat{P}_i$ in the spectral decomposition of $\\hat{G}$, the proof immediately follows from Eq.~\\eqref{maxlosstheoremKG}. Physically, this theorem shows that non-degenerate simply wipe out all information about the parameter in the system. The above theorem can be also explained as follows. To be able to estimate the parameter in the presence of a commutative noise it is necessary to have decoherence-free subspaces in which the information about the parameter is stored. Decoherence-free subspaces are given by projectors associated with degenerate eigenvalues of $\\hat{G}$. These subspaces are spanned by the eigenvectors of these projectors associated with the eigenvalue 1, i.e., by the states which are invariant under the projection. However, decoherence-free subspaces are not needed for the parameter estimation in the presence of the non-commutative noise.\n\nFrom Eq.~\\eqref{cnolossKG} we can also derive a necessary condition for the scenario when no precision in estimation is lost. Summing over all projectors $\\hat{P}_i$ yields\n\\[\\label{noloss}\nH(\\G[{\\hat{\\rho}}])=H({\\hat{\\rho}})\\ \\Rightarrow\\ \\mathrm{Cov}_{\\hat{\\rho}}(\\hat{G},\\hat{K})=0.\n\\]\nThis means that if operators $\\hat{K}$ and $\\hat{G}$ are correlated with respect to the pure initial state ${\\hat{\\rho}}$, i.e. $\\mathrm{Cov}_{\\hat{\\rho}}(\\hat{G},\\hat{K})\\neq0$, then some precision is lost due to the misalignment of reference frames. It is worth emphasising that this condition is not sufficient. As an example consider the operators $\\hat{K}=\\pro{2}{2}$, $\\hat{G}=6\\pro{0}{0}+3\\pro{1}{1}+4\\pro{2}{2}$, and the initial state $\\ket{\\psi_0}=\\frac{1}{\\sqrt{6}}\\ket{0}+\\frac{1}{\\sqrt{3}}\\ket{1}+\\frac{1}{\\sqrt{2}}\\ket{2}$. In this example the covariance between $\\hat{G}$ and $\\hat{K}$  is zero, nevertheless, since $\\hat{K}$ and $\\hat{G}$ commute and the fact that no non-degenerate subspace exists, according to theorem~\\ref{thirdtheorem} we will not be able to extract any information about $\\epsilon$.\n\n\\section{Examples}~\\label{sec:Ex}\n\nIn the previous sections we analysed how the quantum Fisher information changes when of the party which decodes the parameter of interest does not share a local reference frame with the party which encoded the parameter. Now we will present some explicit examples. In the first example we show the scenario where the encoding operator $\\hat{K}$ commutes with the generator connecting the two misaligned reference frames $\\hat{G}$. In such scenario the decoherence-free subspaces are necessary. In the second example, however, due to the interaction between the system and the QRF $\\hat{K}$ and $\\hat{G}$ no longer commute. The third example is somewhat different. There we present the case where Alice and Bob share only one axis of their reference frames but not the others. We also find the optimal encoding operator $\\hat{K}$ which maximizes the amount of information extracted by Bob.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{SqDisvac.png}\n\\caption{Bob's quantum Fisher information in terms of mean photon number ${\\mean{\\hat{N}}}$ and $x$ for a squeezed, displaced vacuum state, i.e. $|\\psi_{QRF}\\rangle=|\\alpha,r\\rangle$, as the initial state of the QRF. Parameter $x$\ndenotes the fraction of mean energy due to displacing the vacuum, i.e. $x=\\frac{\\alpha^2}{\\mean{\\hat{N}}}$.}\\label{SQD}\n\\end{figure}\n\n\\subsection{Two non-interacting quantum harmonic oscillators}\\label{sec:two_nonint_osc}\n\nThe scenario that we consider in this example is as follows. Alice and Bob do not have access to synchronised clocks, i.e. they do not share a time reference frame. Alice prepares a state $|\\psi_{\\epsilon}\\rangle= \\hat{U}_{\\epsilon}|\\psi_{0}\\rangle$, where $\\hat{U}_{\\epsilon}=e^{-i \\hat{K}\\epsilon}$. Since the local clocks of the parties are not synchronised, in Bob's frame the state of the system is given by Eq.~\\eqref{gtwirl}, where $\\hat{U}(t)=e^{-i\\hat{H}t}$ and $\\hat{G}\\equiv\\hat{H}$ is the Hamiltonian of the qubit and the $QRF$. The operators $\\hat{P}_{i}$ are the projectors into subspaces with total energy $E_{i}$. We analyse the quantum Fisher information of the state ${\\hat{\\rho}}_{B}=\\G[{\\hat{\\rho}}]$ which shows how precise Bob will be able to measure $\\epsilon$.\\\\\n\nLet us consider the example of two non-interacting quantum harmonic oscillators with the Hamiltonian $H=\\hbar \\omega (\\hat{a}^{\\dag}\\hat{a}+\\hat{b}^{\\dag}\\hat{b})$,  where $\\hat{a}$ and $\\hat{a}^{\\dag}$ are the creation and annihilation operators corresponding to the first quantum harmonic oscillator and $\\hat{b}$ and $\\hat{b}^{\\dag}$ to the second respectively. One harmonic oscillator will serve as a carrier of the information about the parameter while the other will represent a quantum reference frame. The initial state is therefore of the product form $|\\psi_0\\rangle=|\\psi_q\\rangle \\otimes |\\psi_{QRF}\\rangle$, where $|\\psi_q\\rangle=\\frac{1}{\\sqrt{2}}(|0\\rangle+|1\\rangle)$ and $|0\\rangle$ and $|1\\rangle$ are the eigenstates of number operator $\\hat{N_q}=\\hat{a}^{\\dag}\\hat{a}$ with eigenvalues $0$ and $1$ respectively. We choose the generator of the unitary channel $\\hat{U}_{\\epsilon}$ to be number operator associated with the first harmonic oscillator, $\\hat{K}=\\hat{a}^{\\dag}\\hat{a}$. It is worth emphasising at this point that in this example $[\\hat{K},\\hat{H}]=0$. Note that this example is similar to the quantum communication scheme between two parties when they do not have a common phase reference frame as was considered in~\\cite{bartlett2009quantum_communication}.\n\nUsing Eq.~\\eqref{eq:QFI_pure2}, it is straightforward to find the quantum Fisher information in Alice's frame as $H({\\hat{\\rho}})=1$. Note that Alice's quantum Fisher information is independent of the state of the QRF. On the other hand, if we consider the state $|\\psi_{QRF}\\rangle=\\sum_{n=0}^{N-1}c_{n}|n\\rangle$, then using either Eq.~\\eqref{BobQFI} or Eq.~\\eqref{HrhoB}, we find the quantum Fisher information in Bob's frame as\n\\[\\label{foranystate}\nH({\\hat{\\rho}}_B)=2\\sum_{n=0}^{N-2}\\frac{\\abs{c_n}^2\\abs{c_{n+1}}^2}{\\abs{c_n}^2+\\abs{c_{n+1}}^2}.\n\\]\nIf Alice chooses a uniform superposition of Fock states, i.e. the state $|\\psi_{US}\\rangle=\\frac{1}{\\sqrt{N}}\\sum_{n=0}^{N-1}|n\\rangle$, then using~\\eqref{foranystate} we can easily compute Bob's quantum Fisher information as $H({\\hat{\\rho}}_B)=1-\\frac{1}{N}$. Using Eq.~\\eqref{SLD}, we find the elusive symmetric logarithmic derivative for this case as\n\\[\nL({\\hat{\\rho}}_{B,US})=\\sum_{n=1}^{N-1}ie^{i\\epsilon}\\ket{0}\\pro{n}{n-1}\\bra{1}-ie^{-i\\epsilon}\\ket{1}\\pro{n-1}{n}\\bra{0}.\n\\]\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{Coh-opt-US.pdf}\n\\caption{Bob's quantum Fisher information in terms of mean photon number in the initial state of the QRF for three different states. The solid-black, dashed-brown and dotted-green curves correspond to coherent state, uniform superposition state $|\\psi_{\\tiny{US}}\\rangle$, and the optimal state with finite cut-off $N=2\\mean{\\hat{N}}+1$.}\\label{comp}\n\\end{figure}\n\nLet us next consider a squeezed, displaced vacuum state~\\cite{gerry2005introductory} $|\\alpha,r\\rangle=D(\\alpha)S(r)\\ket{0}$ as the state of the QRF.\nThe mean energy of this state is equal to $\\mean{\\hat{N}}=\\alpha^2+\\sinh^{2}r$. We define parameter $x$ as the fraction of initial mean energy due to displacing the vacuum, i.e. $x=\\frac{\\alpha^2}{\\mean{\\hat{N}}}$. Note that with this definition, $x=0$ and $x=1$ represent a squeezed state and a coherent state respectively. In particular, noticing that in the Fock basis a squeezed state is of the form $|r\\rangle=\\sum_{n}c_{n}|2n\\rangle$ together with Eq.~\\eqref{gtwirl} we find that $H(\\G[\\ket{r}\\bra{r}])=0$, i.e. Bob will not be able to decode $\\epsilon$ if Alice prepares the QRF in a squeezed state.\\\\\n\nIn figure \\ref{SQD} we have plotted Bob's quantum Fisher information for the state $|\\alpha,\\xi\\rangle$ in terms of $x$ and the mean energy $\\mean{\\hat{N}}$. As can be seen in this figure, if we fix the mean energy of the QRF, then it is optimal to have zero squeezing in the initial state of the QRF, i.e. $x=1$. This corresponds to preparing the QRF in a coherent state. Using Eq.~\\eqref{foranystate} we find Bob's quantum Fisher information for a coherent state as\n\\begin{equation}\\label{CohQFI}\nH({\\hat{\\rho}}_B)=2\\frac{\\abs{\\alpha}^2}{1+\\abs{\\alpha}^2}M\\left(1,2+\\abs{\\alpha}^2,-\\abs{\\alpha}^2\\right),\n\\end{equation}\nwhere $M(a,b,z)$ is a confluent hypergeometric function. We derive the asymptotic expression for the limit of large\nmean energy, i.e. $\\abs{\\alpha}^2\\rightarrow\\infty$, as\n\\[\\label{CoherentH}\nH({\\hat{\\rho}}_B) \\approx 1-\\frac{1}{4(\\abs{\\alpha}^2+1)}.\n\\]\n\nIn figure \\ref{comp}, we compare Bob's quantum Fisher information for different QRFs. This figure shows that a coherent state outperforms the uniform superposition of Fock states. This is in complete agreement with the results of~\\cite{ahmadi2013the_Wigner} where it is shown that if Bob chooses the Maximum-likelihood estimation process to decode $\\epsilon$, then choosing a coherent state as the initial state of the QRF instead of the state $|\\psi_{US}\\rangle$ improves the efficiency of the communication protocol. We also optimize over the quantum reference frames numerically by maximizing Eq.~\\eqref{foranystate} over amplitudes $c_1,\\dots,c_{N-1}$.\\footnote{We found that the resulting optimized states, $\\ket{\\psi_{OPT}}=\\sum_{n=0}^{N-1}c_{n}|n\\rangle$, representing the quantum reference frames have always symmetric amplitudes around the value $\\mean{\\hat{N}}=\\frac{N-1}{2}$.} The green square-shaped dots in figure \\ref{comp} represent the quantum Fisher information for this optimal state. As can be seen from the figure the coherent state is nearly optimal in this case.\n\n\\subsection{Two interacting quantum harmonic oscillators}\\label{sec:two_int_osc}\n\nIn this example we again consider Alice and Bob not having an access to synchronised clocks. However, in this example we assume that the QRF interacts with the original state.\n\nLet us consider the example of two interacting quantum harmonic oscillators with the total Hamiltonian\n\\[\n\\hat{H}=\\hbar\\omega(\\hat{a}^{\\dagger}\\hat{a}+\\hat{b}^{\\dagger}\\hat{b})+\\hbar \\kappa(\\hat{a}^{\\dagger}\\hat{b}+\\hat{b}^{\\dagger}\\hat{a}),\n\\]\nwhere $\\kappa$ is the interaction strength. Similar to the example of two non-interacting quantum harmonic oscillators, we consider the generator of the unitary channel to be the number operator, i.e. $\\hat{K}=\\hat{a}^{\\dag}\\hat{a}$. Note that the two operators $\\hat{K}$ and $\\hat{H}$ do not commute in this case, $[\\hat{K},\\hat{H}]=\\kappa (\\hat{a}^{\\dag}\\hat{b}-\\hat{a}\\hat{b}^{\\dag})$. As mentioned earlier whenever these two operators do not commute, even in the absence of degenerate subspaces of total energy, we may still be able to estimate the parameter. For simplicity we also assume that frequency $\\omega$ is not a fraction of the interaction strength $\\kappa$, i.e.\n\\[\n\\forall P,R\\in \\mathbb{Z},\\ P\\omega\\neq R\\kappa.\n\\]\nThis assumption ensures that the hamiltonian $\\hat{H}$ does not possess any degenerate eigenvalues.  In order to make the computations easier, we change the basis by defining a new set of annihilation operators as \\cite{estes1968quantum}\n\\[\n\\hat{A}=\\frac{1}{\\sqrt{2}}(\\hat{a}+\\hat{b}),\\ \\ \\hat{B}=\\frac{1}{\\sqrt{2}}(\\hat{a}-\\hat{b}).\n\\]\nThis change of basis allows us to write the Hamiltonian as\n$\\hat{H}=\\hbar(\\omega+\\kappa)\\hat{A}^{\\dagger}\\hat{A}+\\hbar(\\omega-\\kappa)\\hat{B}^{\\dagger}\\hat{B}$\nwith the eigenvectors\n\\[\n\\ket{\\widetilde{m,n}}\\!=\\!\\frac{(\\hat{A}^{\\dagger})^m}{\\sqrt{m!}}\\frac{(\\hat{B}^{\\dagger})^n}{\\sqrt{n!}}\\ket{\\widetilde{0,0}}\\!,\n\\]\nwhich using $\\ket{\\widetilde{0,0}}=\\ket{0,0}$ and applying the creation operators can be written in terms of the Fock basis as\n\\[\n\\ket{\\widetilde{m,n}}=\\sum_{k=0}^m\\sum_{l=0}^n{{m}\\choose{k}}{{n}\\choose{l}}\\sqrt{\\frac{(k+l)!(m+n-k-l)!}{2^{m+n}m!n!}}\\ket{k+l,m+n-k-l}.\n\\]\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{QFIint.pdf}\n\\caption{Bob's quantum Fisher information vs. $\\epsilon$ for two interacting quantum harmonic oscillators. The initial state is considered as $|\\psi_{0}\\rangle= \\frac{1}{\\sqrt{2}}(|0\\rangle+|1\\rangle)\\otimes|\\psi_{\\tiny{US}}\\rangle$. The dashed(green), dotted(brown) and solid(black) curves correspond to $N=4,10$ and $300$ respectively.}\\label{nonvsint}\n\\end{figure}\n\nNow let us consider that the QRF is initially prepared in the uniform superposition of Fock states. Using Eq.~\\eqref{HrhoB}, we derive the quantum Fisher information of the averaged state as\n\\[\nH({\\hat{\\rho}}_B)=2-\\frac{8}{N}\\Bigg(\\sum_{m=1}^{\\lfloor\\frac{N}{2}\\rfloor}\\sum_{n=0}^{m-1}c_{m,n}(1-d_{m,n}(\\epsilon))\n-\\!\\!\\!\\!\\sum_{m=\\lfloor\\frac{N}{2}\\rfloor+1}^{N}\\!\\sum_{n=0}^{N-m}c_{m,n}-\\!\\!\\!\\!\\sum_{m=\\lfloor\\frac{N}{2}\\rfloor+1}^{N-1}\\!\\!\\!\\!\\sum_{n=0}^{N-m-1}c_{m,n}d_{m,n}(\\epsilon)\\Bigg),\n\\]\nwhere $d_{m,n}(\\epsilon)=\\frac{(m+n)((m-n)^2+m+n)\\sin^2\\epsilon}{((m-n)^2-m-n)^2+4(m+n)(m-n)^2\\sin^2\\epsilon}$, $c_{m,n}=\\frac{(m+n-1)!(m-n)^2}{2^{m+n+1}m!n!}$, and $\\lfloor\\cdot\\rfloor$ is the floor function.\n\nBecause $\\hat{K}$ and $\\hat{G}$ do not commute in this example, $H(\\G[{\\hat{\\rho}}])$ is $\\epsilon$-dependent as opposed to the first example where Bob's quantum Fisher information was independent of the encoded parameter $\\epsilon$. In figure \\ref{nonvsint} we have plotted the quantum Fisher information $H(\\G[{\\hat{\\rho}}])$ in terms of $\\epsilon$ for increasing values of the mean energy in the state of the QRF. The maximum and minimum of the quantum Fisher information occurs at $\\epsilon=\\pm \\frac{\\pi}{2}$ and $\\epsilon=0,\\pm\\pi$ respectively. Note that, even for very large $N$, the quantum Fisher information converges but it does not approach the ideal case. In other words, even in the limit of very large mean energy in the initial state of the quantum clock, we can not estimate the phase parameter $\\epsilon$ as precise as we could if we had access to a classical clock. This can be proved using the necessary conditions~\\eqref{noloss}. One can easily check that $\\mathrm{Cov}_{\\hat{\\rho}}(\\hat{G},\\hat{K})=\\frac{\\hbar\\omega}{4}$, which means that independent of $N$ and $\\epsilon$, the quantum Fisher information is always smaller than one, i.e. $H(\\G[{\\hat{\\rho}}])<1$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.30\\textwidth]{BobRotated.pdf}\n\\caption{Bob only shares his z-axis with Alice, i.e., he lacks the knowledge about the angle $t$ that relates his other two axes to Alice's axes.}\\label{BobRotated}\n\\end{figure}\n\n\\subsection{Direction indicator}\\label{RotatingObserver}\n\nIn the first two examples we observed how a QRF can help the estimation in the scenario when Alice and Bob do not have synchronized clocks. Here, we investigate the precision of estimation of an angle encoded in a qubit when Bob's measurement apparatus is rotated by an unknown angle with respect to Alice's measurement apparatus.\n\nLet us start with the case where Alice wishes to both encode and decode a parameter herself. She chooses a spin-$\\frac{1}{2}$ particle as the physical system to encode a parameter $\\epsilon$ and then she encodes this parameter using a unitary channel with the generator\n\\[\n\\hat{K}=\\frac{1}{2}\\vec{n}\\cdot\\vec{\\sigma}=\\frac{1}{2}(x\\sigma_x+y\\sigma_y+z\\sigma_z).\n\\]\nThis is the generator of a general rotation in the Bloch sphere around the axis $\\vec{n}=(x,y,z)$, where $x^2+y^2+z^2=1$ and $x,y,z$ are real parameters.\nFor simplicity we choose the initial state to be the eigenstate of $\\sigma_z$ with eigenvalue $1$, i.e. $\\ket{\\psi_0}=\\ket{0}$.\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{BlochSpheres.pdf}\n\\caption{Encoding $\\epsilon$ via rotating the fiducial state $\\ket{\\psi_0}=\\ket{0}$ around the unit vector $\\vec{n}$. For $\\vec{n}=(1,0,0)$ the state of the qubit in Bob's frame is ${\\hat{\\rho}}_B=\\cos^2(\\frac{\\epsilon}{2})\\pro{0}{0}+\\sin^2(\\frac{\\epsilon}{2})\\pro{1}{1}$ and Bob's quantum Fisher information is the same as Alice's. For $\\vec{n}=(0,\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}})$, ${\\hat{\\rho}}_B$ is ${\\hat{\\rho}}_B=(1-\\frac{1}{2}\\sin^2(\\frac{\\epsilon}{2}))\\pro{0}{0}+\\frac{1}{2}\\sin^2(\\frac{\\epsilon}{2})\\pro{1}{1}$. Note that ${\\hat{\\rho}}_{B}$ is the projection of $|\\psi\\rangle$ onto the z-axis. Also note that in the latter case Bob's quantum Fisher information is $\\epsilon$-dependent(See figure \\ref{BobsQFIexample3}).}\\label{BlochSpheres}\n\\end{figure}\n\nUsing Euler's formula for Pauli matrices $e^{-i\\hat{K}\\epsilon}=\\cos(\\frac{\\epsilon}{2})\\mathbb{I}-i\\sin(\\frac{\\epsilon}{2})(\\vec{n}\\cdot\\vec{\\sigma})$ we can write the state Alice prepared as\n\\begin{equation}\\label{EX3state}\n\\ket{\\psi_{\\epsilon}}=\\left(\\cos\\left(\\tfrac{\\epsilon}{2}\\right)-iz\\sin\\left(\\tfrac{\\epsilon}{2}\\right)\\right)\\ket{0}+\n(y-ix)\\sin\\left(\\tfrac{\\epsilon}{2}\\right)\\ket{1}.\n\\end{equation}\n\nThen using Eq.~\\eqref{eq:QFI_pure2}, the quantum Fisher information in Alice's frame reads as\n\\[\\label{example3AlicesH}\nH({\\hat{\\rho}})=1-z^2 .\n\\]\nNote that for $z=1$, the corresponding generator is $\\hat{K}=\\frac{1}{2}\\sigma_z$ which leaves the initial state invariant, i.e. $\\text{exp}(-i\\frac{\\sigma_{z}}{2})|0\\rangle=|0\\rangle$. Since the encoding process is not successful, the quantum Fisher information $H({\\hat{\\rho}})$ vanishes which simply means that a different generator needs to be used at the preparation stage. The quantum Fisher information achieves its maximum when when the parameter $\\epsilon$ is encoded via a rotation around any vector in the $xy$-plane, i.e. when $z=0$.\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{BobsQFIexample3.png}\n\\caption{Bob's quantum Fisher information in terms of $\\epsilon$ and $z$ for general $\\vec{n}=(x,y,z)$.}\\label{BobsQFIexample3}\n\\end{figure}\n\nNow suppose that Alice and Bob only share their $z$-axis, i.e. Bob is completely unaware of the relative angle $t$ between his other two axes and Alice's, as depicted in figure \\ref{BobRotated}. In this case, $\\hat{G}$ is the generator of rotations around $z$-axis, i.e. $\\hat{G}=\\frac{1}{2}\\sigma_z$. Using Eq.~\\eqref{BobQFI}, the quantum Fisher information in Bob's frame can be written as\n\\[\\label{example3BobsH}\nH({\\hat{\\rho}}_B)=\\frac{1-z^2}{1+z^2\\tan^2\\left(\\tfrac{\\epsilon}{2}\\right)}=\\frac{H({\\hat{\\rho}})}{1+z^2\\tan^2\\left(\\tfrac{\\epsilon}{2}\\right)}.\n\\]\n\nAgain note that for $z=1$, the quantum Fisher information is zero in Bob's frame. This is expected, since Bob lacks some information with respect to Alice, therefore Alice's inability in extracting information about $\\epsilon$ means that Bob will not be able to decode the message either, i.e. $H({\\hat{\\rho}}_{B})=0$. On the other hand, as can be seen from \\eqref{example3BobsH}, when $z=0$ the quantum Fisher information is the same in Alice's frame and Bob's frame. Figure \\ref{BlochSpheres} depicts the two cases of $\\vec{n}=(1,0,0)$ and $\\vec{n}=(0,\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}})$. For the former case, the efficiency of communication is $\\epsilon$-independent, whereas for the latter case it is $\\epsilon$-dependent, as can be seen in figure \\ref{BobsQFIexample3}. In this figure, we have plotted Bob's quantum Fisher information in terms of $\\epsilon$ and $z$ for general direction $\\vec{n}=(x,y,z)$. We observe that as $ \\epsilon$ approaches the value $\\pi$, the quantum Fisher information approaches its minimum value, i.e.\n$H({\\hat{\\rho}}_B)\\rightarrow 0$. In other words, for the chosen encoding operator $\\hat{K}$ and the fiducial state $|0\\rangle$, Bob will not be able to distinguish ${\\hat{\\rho}}_{\\pi}$ form its neighbouring states ${\\hat{\\rho}}_{\\pi\\pm\\mathrm{d}\\epsilon}$, where $\\mathrm{d}\\epsilon$ is a very small change in $\\epsilon$.\n\nAlso after some algebra and with the aid of Eq.~\\eqref{NSLD}, we find the symmetric logarithmic derivative that achieves the quantum Fisher information in~\\eqref{example3BobsH} as\n\\[\nL({\\hat{\\rho}}_B)=\\frac{(z^2-1)\\tan\\left(\\tfrac{\\epsilon}{2}\\right)}{1+z^2\\tan^2\\left(\\tfrac{\\epsilon}{2}\\right)}\\pro{0}{0}+\\cot\\left(\\tfrac{\\epsilon}{2}\\right)\\pro{1}{1}.\n\\]\nAgain the optimal POVM can be constructed from the eigenvectors of this operator, i.e. $\\{\\ket{0}\\langle 0|,\\ket{1}\\langle 1|\\}$. This simply means that the most informative measurement for Bob is the number-counting measurement.\n\n\\section{Conclusion}\n\nIn quantum metrological schemes the existence of a perfect classical reference frame is often assumed. In this chapter we analyzed how the ultimate limits of precision change due to the absence of such frames of reference, and how attaching a quantum reference frame can improve the estimation precision.\n\nWe considered effects of noise due to lack of a certain reference frame.\nWe have shown that more precision is lost when the encoding process resembles the nature of the noise. We observed two qualitatively different scenarios. The first scenario is when the the encoding operator commutes with the noise, and the second is when it does not. Interestingly, we demonstrated that choosing an encoding operator which does not commute with the generator of the noise may be advantageous in certain situations. For example when the noise is non-degenerate using a commutative encoding operation would lead to the complete inability of extracting the parameter. Moreover, we derived necessary and sufficient conditions for two extreme cases. One in which the parameter can no longer be estimated due to the lack of a reference frame, and the second in which the parameter can be extracted with the maximal possible precision.\n\nWe proved that the use of quantum reference frames is a feasible strategy for the estimation of quantum parameters in the scenario when a common reference frame is lacking. In certain cases, not using quantum reference frames would lead to the complete loss of precision in estimation. However, when using a QRF of large energy, it is often possible to extract the parameter with the highest possible precision.\n\n\\chapter{Final remarks}\\label{chap:final_remarks}\n\nIn this last chapter we discuss possible future directions and open questions. Then we conclude this thesis.\n\n\\section{Future research directions and open questions}\\label{sec:open_questions}\n\n\n\nThe following is a list of questions which arose during our work in quantum metrology. It is by no means a complete list but rather, in our view, a collection of the most interesting or most important questions to be answered.\n\n\\begin{itemize}[leftmargin=*]\n\\item \\emph{Full parametrization of Gaussian states.} In chapter~\\ref{chap:GS} we managed to fully parameterized one-, two-, and three-mode Gaussian states. To do this we have used the parametrization of unitary matrices up to size $3\\times 3$. However, a general parametrization of unitary matrices is not known. The natural question arises: how to parametrize Gaussian states consisting of $N\\geq4$ number of modes? Of course, it is possible to parametrize the covariance matrix directly~\\eqref{def:covariance_matrix} as its submatrices are Hermitian and symmetric matrices. However, there would be many conditions on these parameters~\\eqref{eq:sigma_K_positivity} in order for the matrix $\\sigma$ to be a valid covariance matrix of a Gaussian state. Answering this question would be especially useful in finding optimal probe states for the multi-mode channels, or for the multi-parameter estimation.\n\\item \\emph{Are Gaussian states extremal on the set of all quantum probe states?} Let us consider an arbitrary encoding channel $C_\\epsilon$ on a bosonic Fock space $\\mathcal{H}$. For any quantum state in the Fock space it is possible to calculate the first and second moments of the field operators~\\eqref{def:covariance_matrix}. The proposed statement is the following: from all quantum states $\\hat{\\rho}$ with a given displacement vector $\\boldsymbol{d}$ and covariance matrix $\\sigma$, it is the Gaussian state $\\hat{\\rho}_G$ with those moments which performs the worst as a probe, i.e., the inequality for the quantum Fisher information holds,\n    \\[\n    H(C_\\epsilon(\\hat{\\rho}))\\geq H(C_\\epsilon(\\hat{\\rho}_G)).\n    \\]\n    We strongly believe that such a statement holds for Gaussian encoding operations $C_\\epsilon$. This is because a Gaussian probe state is fully described by the first and the second moments even after the parameter $\\epsilon$ is encoded, therefore the higher moments of this state cannot provide any additional information about the parameter. In contrast, the first and the second moments of a non-Gaussian states change the same way as the moments of the Gaussian state but the higher moments could provide some additional information about the parameter. A similar statement about extremality of Gaussian states has been proven for other quantum information measures such as entropy~\\cite{wolf2006extremality}. The proof there is based on the super-additivity of such measures, however, the quantum Fisher information is not super-additive~\\cite{hansen2007wigner}.\n\\item \\emph{Are squeezed states the optimal probe states?} In all examples of probing Gaussian unitary channels we found that the optimal scaling with the number of particles is always achieved by some multi-mode squeezed state when restricting ourselves to Gaussian probe states. Is this always the case for any Gaussian channel? This topic was partially discussed in section~\\ref{sec:are_pure_optimal}.\n\\item \\emph{The quantum Fisher information of non-Gaussian states in terms of their moments.} We have derived various formulae for the quantum Fisher information of Gaussian states in terms of the first and second moments in the field operators. Is it possible to write the quantum Fisher information of non-Gaussian states in terms of their (possibly higher) moments? If possible, would the contribution to the quantum Fisher information from the first and second moments correspond to the terms in the known formulae for Gaussian states? Answering this question could also resolve the open problem introduced earlier -- the extremality of Gaussian probe states.\n\\item \\emph{Optimal measurements.} In this thesis we focused on finding optimal Gaussian probe states. It is known that the optimal measurement is given by the symmetric logarithmic derivative~\\eqref{eq:SLD_Gaussian}. Diagonalizing this quantity usually shows that the optimal measurement is a projective measurement on infinitely squeezed states. However, such states contain an infinite amount of energy and are thus unattainable in a lab. Is it possible to find some realistic measurements which achieve the ultimate limit of precision given by the quantum Fisher information? The first step towards this direction has been taken in~\\cite{Monras2013a} where it was proven that homodyne detection is the optimal measurement in case of measuring isothermal Gaussian states. However, the general result is still missing.\n\\item \\emph{Universality of quantum reference frames.} In section~\\ref{sec:Ex}, we showed examples in which a quantum reference frame that does not interact with the original state~\\ref{sec:two_nonint_osc}, and a quantum reference frame interacting with the original state~\\ref{sec:two_int_osc}. We observed a certain universality of QRFs. In both examples, the uniform superposition of Fock states and the coherent state used as a QRF ultimately led to the same quantum Fisher information as energy grows to infinity. In the non-interacting scenario the quantum Fisher information converges to a constant function, while in the interacting scenario the limit is a function depicted on Fig.~\\ref{nonvsint}. We also observed that not every quantum reference frame is useful. For example, the squeezed state did not lead to any improvement in precision in the non-interacting scenario. We therefore pose the following questions: is it possible to classify quantum reference frames in some practical way? Are there types of quantum reference frames that are in some sense universal? In other words, is it possible to always use only one type of a QRF and achieve the same precision as with any other ``equally good'' QRF? We formalize these questions in appendix~\\ref{app:QRF_questions}.\n\\item \\emph{Generalizations of not sharing a reference frame.} There are several possible way of how to generalize what it means not to share a reference frame. The first generalization is quite straigtforward: in Eq.~\\eqref{Htwirl} we considered only the complete lack of knowledge about the parameter of the reference frame. However, we could relatively easily incorporate some weight $p(t)$ representing the knowledge about this parameter,\n    \\[\n    \\G[{\\hat{\\rho}}_{\\epsilon}]=\\int_{-\\infty}^{+\\infty}dt\\ p(t)\\hat{U}(t){\\hat{\\rho}}_{\\epsilon}\\hat{U}(t)^{\\dag}=\\sum_{i,j}\\hat{p}\\Big(\\tfrac{G_i-G_j}{2\\pi}\\Big)\\hat{P}_i{\\hat{\\rho}}_{\\epsilon}\\hat{P}_j.\n    \\]\n    $\\hat{p}$ denotes the Fourier transform of the function $p$. When $p$ is a Gaussian function, transformation $\\G$ is a Gaussian transformation, and it should be possible to derive how such a $\\G$-twirling map translates into the phase-space formalism of Gaussian states, which gives the second possible generalization. The third generalization could be possible in the direction of quantum field theory in curved space-time. However, it is not entirely clear what this generalization should look like. This is because there is no general consensus of how to define a reference frame in general relativity, especially when combined with quantum physics. The simplest scenario involves constructing the $\\G$-twirling map by averaging over the group elements of the Lorenz or the Poincar\\'e group as suggested in~\\cite{bartlett2007reference}. The first steps towards this direction has been taken in~\\cite{ahmadi2015communication}. Other approaches could involve averaging over histories in the path-integral formulation of quantum field theory, averaging over certain local coordinates associated with an observer in general relativity, or averaging over tetrads in the tetrad formalism of general relativity.\n\\end{itemize}\n\n\\section{Conclusion}\n\nIn this thesis we combined tools of many different fields: quantum metrology, Gaussian states, quantum field theory, and quantum reference frames. We followed on from the current state-of-the-art quantum metrology and derived new formulae for the optimal estimation of parameters encoded in Gaussian states. These new formulae are such powerful tools for the treatment of Gaussian states that by using them we have managed to single-handedly generalize all previous bounds on the precision with which Gaussian unitary channels can be estimated. Moreover, these formulae provide a deep insight into the structure of estimation: they show which combinations of initial squeezing, displacement, and temperature of the probe state leads to the highest possible sensitivity. In other words, these formulae can be used to design the core of future quantum detectors.\n\nThe application of such mathematical framework can also provide strategies for the estimation of space-time parameters. Previous studies in this direction considered almost exclusively pure initial probe states. However, in real scenarios probe states are mixed. We derived the limits of precision in the estimation of a general space-time parameter using these realistic probes. With such expressions it is possible to determine which space-time parameters are within experimental reach of current and future technology, and which experimental paths are worth pursuing. The ultimate aim is to measure the elusive predictions of quantum field theory in curved space-time such as the Unruh effect, Hawking radiation, or the dynamical Casimir effect. These phenomena have been confirmed so far only in the analogue systems~\\cite{Wilson2011a,weinfurtner2011measurement}. Measuring these effects could either validate the theory or possibly lead to a new theory of quantum gravity.\n\n\nTo achieve the precision required to measure these space-time effects it may be necessary to construct such detectors in space. By doing so we can greatly increase the scale of such experiments, as well as avoiding various sources of noise. Such design could involve transmission of quantum parameters between the orbiting detector and the control centre on Earth. However, due to the motion of such space-based detectors, the reference frames of the detector and the operator on Earth can become easily misaligned. Such misalignment leads to the loss of precision in the estimation of the parameters. We discovered that attaching a quantum reference frame provides a feasible strategy for preventing this loss. Moreover, these results could help in designing novel ways of storage of quantum parameters for quantum information protocols, or develop satellite-based quantum key distribution.\n\n\n\n\nWe consider the following results as the main results of this thesis: connection between the quantum Fisher information matrix and the Bures metric at the boundary of the space of density matrices~\\eqref{eq:connection_between_Hc_and_H}; the quantum Fisher information matrix in the scenario when the Williamson's decomposition of the covariance matrix is known~\\eqref{eq:multimode_QFI}; the limit formula for the quantum Fisher information matrix of Gaussian states~\\eqref{eq:limit_formula}; a general method for finding optimal Gaussian probe states, section~\\ref{sec:general_method}; finding that effects of the temperature of the probe state on the estimation are generic, section~\\ref{sec:effects_of_temperature};\nexpression for the quantum Fisher information for the estimation of space-time parameters using two-mode squeezed thermal states~\\eqref{eqn:two_mode_QFI_zeroth_order};\nand finally Eq.~\\eqref{BobQFI} which shows how to encode the parameter in a quantum state such that the loss of precision in the estimation due to the lack of a shared reference frame is minimized.\n\nOur results can lead to applications in existing gravitational wave detectors~\\cite{Abbott2004a,Caron1995a}, they may be useful for designing new gravimeters~\\cite{Snadden1998a,Altin2013a,Sabin2014a}, climate probes~\\cite{Tapley2004a}, or sensors for the estimation of space-time parameters~\\cite{Danzmann1996a,Everitt2011a,Bruschi2014a}. Our results will enable researchers to evaluate how well space-time parameters, such as the amplitude of a gravitational wave, accelerations, and local gravitational fields, can be estimated in the presence of background temperature~\\cite{Sabin2014a,Sabin2015a}. Our studies on metrology with misaligned reference frames could prove useful for future space-based experiments such as the gravitational wave detector eLISA~\\cite{amaro2012low} or miniaturized satellites~\\cite{bedington2015deploying}. Finally, our results could help the efforts in bringing the new era of quantum technologies to the general public.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe  molecular ion T$_2^+$ composed of two tritium nuclei (tritons) and one electron is the heaviest isotopomer of H$_2^+$.  This three body  system  is the less studied molecular ion in the family of three-body molecular systems with Coulomb interaction composed of protons, deuterons, tritons  and one electron.  A complete quantitative description has been done is some papers  mainly for the ground state  ($v=0$ and $L=0$).  For this state, a complete list of various geometrical and energetically properties were obtained by Frolov~\\cite{Fr99,Fr02}. Polarizabilities and energies were  calculated with high precision  by Yan et al.~\\cite{YZL03} not only for the ground state but also for the lowest P state. In 2004 Yan et al.~\\cite{YZ04} reported non relativistic energies for those states with vibrational quantum number $v=0$ and  $L$ ranging  from 2 to 12.  However, at present time, the number of  rotational-vibrational bound states supported by the ground state is not known.  \n\nThe goal of this paper is to present a systematic description of the spectra for the lowest four vibrational states $v=0,1,2,3$ and all the bound rotational states without considering the Born-Oppenheimer approximation. A few rotational states beyond the dissociation limit are also presented. In order to do that the Lagrange mesh method is applied in the same way that was done for the homonuclear systems H$_2^+$~\\cite{OPB12} and D$_2^+$~\\cite{OP13}. This method is an approximate variational calculation using a basis of Lagrange functions and the associated Gauss quadrature. The main advantage of this method is its simplicity and high accuracy in the obtained energies and wave functions. With the analytical approximations for the wave functions provided by the Lagrange-mesh method allowed quadrupole transition probabilities are calculated. The CODATA 1986  fundamental constant $m_t= 5496. 921\\,58\\,m_e$ is used.\n\nBecause this method was applied successfully for the three body systems with Coulomb interaction H$_2^+$~\\cite{OPB12}, D$_2^+$~\\cite{OP13} and HD$^+$~\\cite{OPB13} the reader is  referred to  these references for details. \nIn Sec.~\\ref{s2} only the expression for the quadruple transition is presented.  In Sec. \\ref{s3}, energies are given for the lowest four vibrational levels over the full rotational bands and E2 transition probabilities are tabulated. Concluding remarks are presented in Sec.~\\ref{s4}. Throughout atomic units are used. \n \n\\section{Quadrupole transition probabilities}\n\\label{s2}\n\nThe electric quadrupole transition probability for spontaneous emission $(E_f < E_i)$ per time unit  (the atomic unit of time is $a_0\/ \\alpha c \\approx 2.418\\,8843\\times 10^{-17} s$) between an initial state $i$ and a final state $f$ is given by\n\\begin{equation}\nW_{i\\rightarrow f}^{(2)}=\\frac{1}{15}\\alpha^{5}(E_i-E_f)^5 \\frac{S_{if}^{(2)}}{2J_i+1},\n\\end{equation}\nwhere $\\alpha$ is the fine-structure constant, $S_{if}^{(2)}$ is the reduced matrix element and $J_i$ the total angular momentum of the initial state. The quadrupole oscillator strength is given by\n\\begin{equation}\nf_{i\\rightarrow f}^{(2)}=\\frac{1}{30}\\alpha^{2}(E_f-E_i)^3 \\frac{S_{if}^{(2)}}{2J_i+1}.\n\\end{equation}\nThe evaluation of the reduced matrix element $S_{if}^{(2)}$ makes necessary to have the wave functions of the final and initial states. In our approximation, the wave functions are obtained using the Lagrange-mesh method.\nIn order to apply this method, we use the center of mass coordinates. The six-dimensional wave function is presented as a product of two three-dimensional functions, ($i$) a function carrying all the angular dependence, and ($ii$) a function  describing the form of the triangle formed by the three particles. For this internal degrees of freedom, the function is expand in a  three-dimensional Lagrange functions which are noting but a product of three one-dimension Lagrange functions. The size of each one-dimensional bases are indicated by $N_x$, $N_y$ and $N_z$. Because of the symmetry between the two centers $N_x = N_y \\equiv N$. Three additional  parameters $h_x$, $h_y$ and $h_z$ are introduced in order to adjust the base to the physical problem. Due to the symmetry between the two centers $h_x=h_y\\equiv h$. Any state is labeled by the total angular momentum $L$, the vibrational quantum number $v$ and the parity $\\pi$: $(L^{\\pi},v)$.\nFor a detailed discussion see~\\cite{OPB12,OP13,OPB13}.\n\n\\section{T$_2^+$ bound and quasibound energies}\n\\label{s3}\n\nThe rotational-vibrational  spectra of the molecular ion T$_2^+$ is obtained using the Lagrange mesh method. Energies for the four lowest vibrational states ($v=0,1,2,3$) of the ground electronic state are presented in Table~\\ref{tab:1}.  An important part of this work is to obtain the quadrupole transition probability per time unit, then,  it is convenient to use a single three-dimensional mesh for all states. An excellent accuracy is obtained when the size of the bases are chosen as $N= 54$ and $N_z =18$ and  the scaling parameters as $h=0.08$ and $h_z =0.6$.  In this paper the triton mass value  $m_t = 5496.921\\,58\\,m_e$ is used. The  dissociation energy  is then at $E_d = -0.499\\,909\\,056\\,5$ a.u.   The first line for  each $L$-value in Table~\\ref{tab:1} presents the obtained energies. Comparison with previous results are possible only for the lowest twelve rotational states with vibrational quantum number $v=0$. The energy for the ground state  $(0^+,0)$ is well known and our result is in agreement in 13 significant digits with the most accurate results presented by Frolov~\\cite{Fr02} and Yan {\\it et al}~\\cite{YZL03}. The first rotational state $(1^-,0)$ was studied also by Yan {\\it et al}~\\cite{YZL03} and the agreement is in the 13 significant digit. Energies of the excited states $(L^{\\pi},0)$ for $L\\in(2,12)$ are reported by Yan {\\it et al}~\\cite{YZ04} and all of them have the same correspondence of 13 significant digits with our results. \n\nConvergence for the energies was tested by changing the values of $N$ and $N_z$. The accuracies for the first, second and third rotational bands are $10^{-11}$, $10^{-10}$ and $10^{-9}$, respectively.\nThe results  demonstrate that  the vibrational band with  quantum number $v=0$ supports 62 rotational states below the dissociation energy.  The number of rotational states below the dissociation energy supported  by excited vibrational bands decrease with the vibrational quantum number: 61, 60 and 58 for  $v=1,2,3$, respectively. The obtained spectrum is depicted in Figure~\\ref{fig:1}. \n\n\n\\begin{center}\n\\begin{longtable}{rllll}\n\\caption{Energies of the four lowest vibrational bound or quasibound states in the\n$\\Sigma_g$ rotational band of the T$_2^+$ molecular ion. Quasibound states are\nseparated from bound states by a horizontal bar.  For each $L$ value the first line\npresents the Lagrange-mesh results obtained with $N_x = N_y = 54$, $N_z = 18$\nand $h_x = h_y = 0.08$, $h_z = 0.6$. More\naccurate energies are given for some levels (a: \\cite{Fr02},\nb:\\cite{YZL03}, c: \\cite{YZ04}). The triton mass is taken as $m_t=5496.92158\\,m_e$.}\n\\label{tab:1}\\\\\n\\hline\n$L$& $v=0$          & $v=1$           &  $v=2$          &$v=3$\\\\\n\\hline\n\\endfirsthead\n\\multicolumn{5}{c}{{\\tablename} \\thetable{} -- Continuation}\\\\\n\\hline\n$L$& $v=0$          & $v=1$           &  $v=2$          &$v=3$\\\\\n\\hline\n\\endhead\n\\hline\n\\multicolumn{5}{l}{{Continued on Next Page\\ldots}}\\\\%Footer for all pages except the last page\n\\endfoot\n\\hline                                      \n\\endlastfoot\n 0 &-0.599\\,506\\,910\\,111\\,5 &-0.593\\,589\\,927\\,812 &-0.587\\,871\\,233\\,66 &-0.582\\,346\\,606\\,1\\\\\n    &-0.599\\,506\\,910\\,111\\,54$^{a}$&&&\\\\\n    &-0.599\\,506\\,910\\,111\\,541$^{b}$&&&\\\\\n 1 &-0.599\\,417\\,152\\,359\\,8 &-0.593\\,502\\,913\\,068 &-0.587\\,786\\,903\\,36 &-0.582\\,264\\,906\\,3\\\\\n    &-0.599\\,417\\,152\\,359\\,852$^{b}$&&&\\\\\n 2 &-0.599\\,237\\,876\\,293\\,2 &-0.593\\,329\\,117\\,147 &-0.587\\,618\\,470\\,66 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&-0.579\\,451\\,235\\,0\\\\\n    &-0.596\\,324\\,968\\,711\\,257$^{c}$&&&\\\\\n 9 &-0.595\\,544\\,949\\,299\\,1 &-0.589\\,749\\,502\\,104 &-0.584\\,149\\,773\\,62 &-0.578\\,741\\,750\\,4\\\\\n    &-0.595\\,544\\,949\\,299\\,086$^{c}$&&&\\\\\n10 &-0.594\\,685\\,295\\,136\\,5 &-0.588\\,916\\,340\\,727 &-0.583\\,342\\,552\\,27 &-0.577\\,959\\,962\\,1\\\\\n     &-0.594\\,685\\,295\\,136\\,511$^{c}$&&&\\\\\n11 &-0.593\\,748\\,087\\,259\\,4 &-0.588\\,008\\,065\\,087 &-0.582\\,462\\,608\\,44 &-0.577\\,107\\,799\\,7\\\\\n     &-0.593\\,748\\,087\\,259\\,376$^{c}$&&&\\\\\n12 &-0.592\\,735\\,549\\,928\\,4 &-0.587\\,026\\,843\\,285 &-0.581\\,512\\,056\\,21 &-0.576\\,187\\,325\\,4\\\\\n     &-0.592\\,735\\,549\\,928\\,400$^{c}$&&&\\\\\n13 &-0.591\\,650\\,035\\,282\\,8 &-0.585\\,974\\,967\\,849 &-0.580\\,493\\,130\\,81 &-0.575\\,200\\,719\\,5\\\\\n14 &-0.590\\,494\\,007\\,823\\,0 &-0.584\\,854\\,840\\,613 &-0.579\\,408\\,173\\,90 &-0.574\\,150\\,265\\,8\\\\\n15 &-0.589\\,270\\,028\\,927\\,2 &-0.583\\,668\\,957\\,635 &-0.578\\,259\\,618\\,84 &-0.573\\,038\\,337\\,6\\\\\n16 &-0.587\\,980\\,741\\,586\\,5 &-0.582\\,419\\,894\\,344 &-0.577\\,049\\,976\\,31 &-0.571\\,867\\,383\\,5\\\\\n17 &-0.586\\,628\\,855\\,526\\,3 &-0.581\\,110\\,291\\,083 &-0.575\\,781\\,820\\,19 &-0.570\\,639\\,913\\,7\\\\\n18 &-0.585\\,217\\,132\\,856\\,0 &-0.579\\,742\\,839\\,179 &-0.574\\,457\\,774\\,08 &-0.569\\,358\\,487\\,3\\\\\n19 &-0.583\\,748\\,374\\,370\\,8 &-0.578\\,320\\,267\\,657 &-0.573\\,080\\,498\\,44 &-0.568\\,025\\,699\\,1\\\\\n20 &-0.582\\,225\\,406\\,601\\,3 &-0.576\\,845\\,330\\,704 &-0.571\\,652\\,678\\,40 &-0.566\\,644\\,168\\,6\\\\\n21 &-0.580\\,651\\,069\\,688\\,7 &-0.575\\,320\\,795\\,939 &-0.570\\,177\\,012\\,45 &-0.565\\,216\\,528\\,3\\\\\n22 &-0.579\\,028\\,206\\,140\\,4 &-0.573\\,749\\,433\\,558 &-0.568\\,656\\,201\\,95 &-0.563\\,745\\,414\\,4\\\\\n23 &-0.577\\,359\\,650\\,502\\,1 &-0.572\\,134\\,006\\,365 &-0.567\\,092\\,941\\,54 &-0.562\\,233\\,456\\,7\\\\\n24 &-0.575\\,648\\,219\\,963\\,7 &-0.570\\,477\\,260\\,739 &-0.565\\,489\\,910\\,42 &-0.560\\,683\\,270\\,9\\\\\n25 &-0.573\\,896\\,705\\,902\\,6 &-0.568\\,781\\,918\\,497 &-0.563\\,849\\,764\\,56 &-0.559\\,097\\,450\\,9\\\\\n26 &-0.572\\,107\\,866\\,354\\,7 &-0.567\\,050\\,669\\,681 &-0.562\\,175\\,129\\,76 &-0.557\\,478\\,562\\,0\\\\\n27 &-0.570\\,284\\,419\\,389\\,4 &-0.565\\,286\\,166\\,215 &-0.560\\,468\\,595\\,63 &-0.555\\,829\\,135\\,2\\\\\n28 &-0.568\\,429\\,037\\,360\\,5 &-0.563\\,491\\,016\\,423 &-0.558\\,732\\,710\\,35 &-0.554\\,151\\,662\\,7\\\\\n29 &-0.566\\,544\\,341\\,992\\,9 &-0.561\\,667\\,780\\,362 &-0.556\\,969\\,976\\,24 &-0.552\\,448\\,592\\,9\\\\\n30 &-0.564\\,632\\,900\\,264\\,8 &-0.559\\,818\\,965\\,926 &-0.555\\,182\\,846\\,09 &-0.550\\,722\\,327\\,7\\\\\n31 &-0.562\\,697\\,221\\,036\\,4 &-0.557\\,947\\,025\\,683 &-0.553\\,373\\,720\\,25 &-0.548\\,975\\,219\\,3\\\\\n32 &-0.560\\,739\\,752\\,377\\,7 &-0.556\\,054\\,354\\,391 &-0.551\\,544\\,944\\,28 &-0.547\\,209\\,568\\,5\\\\\n33 &-0.558\\,762\\,879\\,545\\,2 &-0.554\\,143\\,287\\,149 &-0.549\\,698\\,807\\,35 &-0.545\\,427\\,622\\,9\\\\\n34 &-0.556\\,768\\,923\\,558\\,7 &-0.552\\,216\\,098\\,133 &-0.547\\,837\\,541\\,08 &-0.543\\,631\\,576\\,1\\\\\n35 &-0.554\\,760\\,140\\,328\\,2 &-0.550\\,274\\,999\\,878 &-0.545\\,963\\,319\\,07 &-0.541\\,823\\,567\\,5\\\\\n36 &-0.552\\,738\\,720\\,285\\,9 &-0.548\\,322\\,143\\,055 &-0.544\\,078\\,256\\,80 &-0.540\\,005\\,682\\,4\\\\\n37 &-0.550\\,706\\,788\\,477\\,6 &-0.546\\,359\\,616\\,697 &-0.542\\,184\\,412\\,02 &-0.538\\,179\\,952\\,4\\\\\n38 &-0.548\\,666\\,405\\,072\\,7 &-0.544\\,389\\,448\\,859 &-0.540\\,283\\,785\\,61 &-0.536\\,348\\,356\\,7\\\\\n39 &-0.546\\,619\\,566\\,254\\,9 &-0.542\\,413\\,607\\,645 &-0.538\\,378\\,322\\,77 &-0.534\\,512\\,823\\,4\\\\\n40 &-0.544\\,568\\,205\\,458\\,6 &-0.540\\,434\\,002\\,600 &-0.536\\,469\\,914\\,65 &-0.532\\,675\\,231\\,4\\\\\n41 &-0.542\\,514\\,194\\,920\\,6 &-0.538\\,452\\,486\\,430 &-0.534\\,560\\,400\\,24 &-0.530\\,837\\,412\\,8\\\\\n42 &-0.540\\,459\\,347\\,520\\,8 &-0.536\\,470\\,857\\,022 &-0.532\\,651\\,568\\,69 &-0.529\\,001\\,155\\,2\\\\\n43 &-0.538\\,405\\,418\\,889\\,7 &-0.534\\,490\\,859\\,764 &-0.530\\,745\\,161\\,89 &-0.527\\,168\\,205\\,1\\\\\n44 &-0.536\\,354\\,109\\,765\\,3 &-0.532\\,514\\,190\\,143 &-0.528\\,842\\,877\\,46 &-0.525\\,340\\,271\\,0\\\\\n45 &-0.534\\,307\\,068\\,587\\,2 &-0.530\\,542\\,496\\,620 &-0.526\\,946\\,371\\,95 &-0.523\\,519\\,027\\,4\\\\\n46 &-0.532\\,265\\,894\\,319\\,7 &-0.528\\,577\\,383\\,791 &-0.525\\,057\\,264\\,58 &-0.521\\,706\\,119\\,2\\\\\n47 &-0.530\\,232\\,139\\,504\\,5 &-0.526\\,620\\,415\\,839 &-0.523\\,177\\,141\\,20 &-0.519\\,903\\,166\\,6\\\\\n48 &-0.528\\,207\\,313\\,545\\,9 &-0.524\\,673\\,120\\,301 &-0.521\\,307\\,558\\,80 &-0.518\\,111\\,770\\,4\\\\\n49 &-0.526\\,192\\,886\\,243\\,2 &-0.522\\,736\\,992\\,180 &-0.519\\,450\\,050\\,49 &-0.516\\,333\\,518\\,6\\\\\n50 &-0.524\\,190\\,291\\,589\\,6 &-0.520\\,813\\,498\\,456 &-0.517\\,606\\,131\\,04 &-0.514\\,569\\,993\\,2\\\\\n51 &-0.522\\,200\\,931\\,870\\,4 &-0.518\\,904\\,083\\,055 &-0.515\\,777\\,303\\,23 &-0.512\\,822\\,778\\,9\\\\\n52 &-0.520\\,226\\,182\\,104\\,3 &-0.517\\,010\\,172\\,357 &-0.513\\,965\\,064\\,96 &-0.511\\,093\\,472\\,8\\\\\n53 &-0.518\\,267\\,394\\,886\\,3 &-0.515\\,133\\,181\\,376 &-0.512\\,170\\,917\\,53 &-0.509\\,383\\,695\\,9\\\\\n54 &-0.516\\,325\\,905\\,713\\,6 &-0.513\\,274\\,520\\,748 &-0.510\\,396\\,375\\,32 &-0.507\\,695\\,107\\,0\\\\\n55 &-0.514\\,403\\,038\\,899\\,0 &-0.511\\,435\\,604\\,758 &-0.508\\,642\\,977\\,24 &-0.506\\,029\\,420\\,6\\\\\n56 &-0.512\\,500\\,114\\,211\\,8 &-0.509\\,617\\,860\\,665 &-0.506\\,912\\,300\\,64 &-0.504\\,388\\,428\\,3\\\\\n57 &-0.510\\,618\\,454\\,433\\,7 &-0.507\\,822\\,739\\,742 &-0.505\\,205\\,978\\,29 &-0.502\\,774\\,027\\,1\\\\\n58 &-0.508\\,759\\,394\\,080\\,4 &-0.506\\,051\\,730\\,550 &-0.503\\,525\\,719\\,95 &-0.501\\,188\\,256\\,2\\\\\\cline{5-5}\n59 &-0.506\\,924\\,289\\,631\\,3 &-0.504\\,306\\,375\\,245 &-0.501\\,873\\,339\\,99 &-0.499\\,633\\,348\\,3\\\\\n60 &-0.505\\,114\\,531\\,741\\,2 &-0.502\\,588\\,290\\,012 &-0.500\\,250\\,794\\,05 &-0.498\\,111\\,800\\,5\\\\\\cline{4-4}\n61 &-0.503\\,331\\,560\\,100\\,5 &-0.500\\,899\\,191\\,343 &-0.498\\,660\\,229\\,06 &-0.496\\,626\\,48\\\\\\cline{3-3}\n62 &-0.501\\,576\\,881\\,907\\,7 &-0.499\\,240\\,930\\,722 &-0.497\\,104\\,054\\,16 &-0.495\\,180\\,78\\\\\\cline{2-2}\n63 &-0.499\\,852\\,095\\,376\\,7 &-0.497\\,615\\,541\\,903 &-0.495\\,585\\,045\\,64 &-0.493\\,778\\,9\\\\\n64 &-0.498\\,158\\,920\\,453\\,2 &-0.496\\,025\\,307\\,747 &-0.494\\,106\\,511 &-0.492\\,426\\\\\n65 &-0.496\\,499\\,240\\,184\\,4 &-0.494\\,472\\,859\\,0     &-0.492\\,672\\,5&-0.491\\,13\\\\\n66 &-0.494\\,875\\,158\\,447\\,9 &-0.492\\,961\\,329          &-0.491\\,288  &\\\\\n67 &-0.493\\,289\\,084\\,03        &-0.491\\,494\\,6              & & \\\\\n68 &-0.491\\,743\\,86                 &&&\\\\\n\\end{longtable}\n\\end{center}       \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{spectra.eps}\n\\caption{Four lowest $\\Sigma_g$ rotational bands of the T$_2^+$ molecular ion \nand dissociation energy $E_d$.  Arrows show how the direction of $L\n\\rightarrow L+2$  transitions between the two lowest bands changes along the band.}\n\\label{fig:1}\n\\end{center}\n\\end{figure}\n\n\nQuadrupole transitions per time unit are easily obtained using the wave functions obtained with the Lagrange-mesh method. Table~\\ref{tab:2} presents all the probabilities per second for transitions within the same rotational band, $L_f = L_i - 2$ and $v_f = v_i \\le 3$. Some transition probabilities involving quasibound states are presented and these are separated by a horizontal bar. The probabilities increase slowly with $L$  reaching a maximum at $L= 55, 53, 52$ and $50$ for $v=0,1,2$ and $3$, respectively.\n\n\\begin{center}\n\\begin{longtable}{rllll}\n\\caption{Quadrupole transition probabilities per second $W$ for transitions \nbetween states of a same rotational band ($v_f = v_i$, $L_f = L_i - 2$). \nResults are given with five digits followed by the power of 10. } \n\\label{tab:2} \\\\\n\\\\[-4.9ex]\n\\hline\n$L_i$&$v_i=0$&$v_i=1$&$v_i=2$&$v_i=3$\\\\\n\\hline\n\\endfirsthead\n\\multicolumn{4}{c}{{\\tablename} \\thetable{} -- Continuation}\\\\\n\\hline\n$L_i$&$v_i=0$&$v_i=1$&$v_i=2$&$v_i=3$\\\\\n\\hline\n\\endhead\n\\hline\n\\multicolumn{4}{l}{{Continued on Next Page\\ldots}}\\\\\n\\endfoot\n\\hline\n\\endlastfoot\n 2& 4.077\\,29-14& 4.087\\,46-14& 4.069\\,14-14& 4.024\\,12-14\\\\\n 3& 6.703\\,22-13& 6.717\\,68-13& 6.685\\,44-13& 6.609\\,46-13\\\\\n 4& 3.971\\,67-12& 3.978\\,26-12& 3.957\\,28-12& 3.910\\,52-12\\\\\n 5& 1.463\\,26-11& 1.464\\,72-11& 1.456\\,08-11& 1.438\\,00-11\\\\\n 6& 4.086\\,27-11& 4.086\\,99-11& 4.059\\,72-11& 4.006\\,31-11\\\\\n 7& 9.510\\,01-11& 9.502\\,44-11& 9.430\\,29-11& 9.297\\,96-11\\\\\n 8& 1.944\\,69-10& 1.940\\,97-10& 1.924\\,18-10& 1.895\\,23-10\\\\\n 9& 3.609\\,63-10& 3.598\\,19-10& 3.562\\,77-10& 3.505\\,11-10\\\\\n10& 6.213\\,50-10& 6.185\\,18-10& 6.116\\,15-10& 6.009\\,44-10\\\\\n11& 1.006\\,81-09& 1.000\\,69-09& 9.880\\,86-10& 9.694\\,84-10\\\\\n12& 1.552\\,29-09& 1.540\\,34-09& 1.518\\,55-09& 1.487\\,70-09\\\\\n13& 2.295\\,73-09& 2.274\\,07-09& 2.238\\,13-09& 2.189\\,10-09\\\\\n14& 3.277\\,02-09& 3.240\\,08-09& 3.183\\,20-09& 3.108\\,09-09\\\\\n15& 4.536\\,88-09& 4.477\\,00-09& 4.390\\,15-09& 4.278\\,75-09\\\\\n16& 6.115\\,72-09& 6.022\\,70-09& 5.894\\,26-09& 5.733\\,72-09\\\\\n17& 8.052\\,40-09& 7.913\\,12-09& 7.728\\,51-09& 7.503\\,01-09\\\\\n18& 1.038\\,31-08& 1.018\\,11-08& 9.922\\,49-09& 9.612\\,99-09\\\\\n19& 1.314\\,02-08& 1.285\\,55-08& 1.250\\,14-08& 1.208\\,55-08\\\\\n20& 1.635\\,14-08& 1.595\\,99-08& 1.548\\,53-08& 1.493\\,69-08\\\\\n21& 2.003\\,87-08& 1.951\\,24-08& 1.888\\,83-08& 1.817\\,78-08\\\\\n22& 2.421\\,80-08& 2.352\\,47-08& 2.271\\,81-08& 2.181\\,23-08\\\\\n23& 2.889\\,84-08& 2.800\\,16-08& 2.697\\,59-08& 2.583\\,81-08\\\\\n24& 3.408\\,19-08& 3.294\\,11-08& 3.165\\,59-08& 3.024\\,62-08\\\\\n25& 3.976\\,35-08& 3.833\\,42-08& 3.674\\,57-08& 3.502\\,11-08\\\\\n26& 4.593\\,08-08& 4.416\\,49-08& 4.222\\,61-08& 4.014\\,11-08\\\\\n27& 5.256\\,45-08& 5.041\\,08-08& 4.807\\,19-08& 4.557\\,85-08\\\\\n28& 5.963\\,87-08& 5.704\\,28-08& 5.425\\,20-08& 5.130\\,06-08\\\\\n29& 6.712\\,11-08& 6.402\\,66-08& 6.072\\,98-08& 5.726\\,95-08\\\\\n30& 7.497\\,37-08& 7.132\\,21-08& 6.746\\,44-08& 6.344\\,33-08\\\\\n31& 8.315\\,34-08& 7.888\\,51-08& 7.441\\,05-08& 6.977\\,67-08\\\\\n32& 9.161\\,25-08& 8.666\\,71-08& 8.151\\,96-08& 7.622\\,13-08\\\\\n33& 1.002\\,99-07& 9.461\\,67-08& 8.874\\,06-08& 8.272\\,68-08\\\\\n34& 1.091\\,60-07& 1.026\\,80-07& 9.602\\,01-08& 8.924\\,12-08\\\\\n35& 1.181\\,37-07& 1.108\\,00-07& 1.033\\,04-07& 9.571\\,18-08\\\\\n36& 1.271\\,71-07& 1.189\\,20-07& 1.105\\,36-07& 1.020\\,85-07\\\\\n37& 1.362\\,03-07& 1.269\\,83-07& 1.176\\,62-07& 1.083\\,10-07\\\\\n38& 1.451\\,72-07& 1.349\\,31-07& 1.246\\,26-07& 1.143\\,32-07\\\\\n39& 1.540\\,19-07& 1.427\\,07-07& 1.313\\,75-07& 1.201\\,02-07\\\\\n40& 1.626\\,84-07& 1.502\\,54-07& 1.378\\,55-07& 1.255\\,71-07\\\\\n41& 1.711\\,09-07& 1.575\\,18-07& 1.440\\,17-07& 1.306\\,91-07\\\\\n42& 1.792\\,38-07& 1.644\\,46-07& 1.498\\,11-07& 1.354\\,19-07\\\\\n43& 1.870\\,16-07& 1.709\\,89-07& 1.551\\,90-07& 1.397\\,11-07\\\\\n44& 1.943\\,90-07& 1.770\\,97-07& 1.601\\,12-07& 1.435\\,28-07\\\\\n45& 2.013\\,13-07& 1.827\\,27-07& 1.645\\,36-07& 1.468\\,33-07\\\\\n46& 2.077\\,36-07& 1.878\\,36-07& 1.684\\,22-07& 1.495\\,92-07\\\\\n47& 2.136\\,18-07& 1.923\\,84-07& 1.717\\,37-07& 1.517\\,76-07\\\\\n48& 2.189\\,17-07& 1.963\\,36-07& 1.744\\,48-07& 1.533\\,55-07\\\\\n49& 2.235\\,97-07& 1.996\\,58-07& 1.765\\,27-07& 1.543\\,04-07\\\\\n50& 2.276\\,23-07& 2.023\\,22-07& 1.779\\,47-07& 1.546\\,02-07\\\\\n51& 2.309\\,67-07& 2.042\\,99-07& 1.786\\,85-07& 1.542\\,28-07\\\\\n52& 2.336\\,01-07& 2.055\\,68-07& 1.787\\,20-07& 1.531\\,66-07\\\\\n53& 2.355\\,00-07& 2.061\\,05-07& 1.780\\,35-07& 1.513\\,99-07\\\\\n54& 2.366\\,45-07& 2.058\\,95-07& 1.766\\,13-07& 1.489\\,14-07\\\\\n55& 2.370\\,16-07& 2.049\\,19-07& 1.744\\,41-07& 1.456\\,98-07\\\\\n56& 2.366\\,00-07& 2.031\\,65-07& 1.715\\,05-07& 1.417\\,41-07\\\\\n57& 2.353\\,84-07& 2.006\\,20-07& 1.677\\,93-07& 1.370\\,29-07\\\\\n58& 2.333\\,55-07& 1.972\\,73-07& 1.632\\,94-07& 1.315\\,50-07\\\\\n59& 2.305\\,07-07& 1.931\\,13-07& 1.579\\,94-07& 1.252\\,87-07\\\\\n60& 2.268\\,30-07& 1.881\\,30-07& 1.518\\,77-07& 1.182\\,18-07\\\\\n61& 2.223\\,19-07& 1.823\\,10-07& 1.449\\,24-07& 1.103\\,12-07\\\\\n62& 2.169\\,64-07& 1.756\\,37-07& 1.371\\,06-07& 1.015\\,22-07\\\\\n63& 2.107\\,59-07& 1.680\\,90-07& 1.283\\,81-07& 9.177\\,09-08\\\\\n64& 2.036\\,91-07& 1.596\\,37-07& 1.186\\,85-07& 8.093\\,00-08\\\\\n65& 1.957\\,44-07& 1.502\\,31-07& 1.079\\,16-07& 6.874\\,67-08\\\\\n66& 1.868\\,92-07& 1.397\\,98-07& 9.589\\,20-08& \\\\\n67& 1.770\\,96-07& 1.282\\,13-07& &\\\\\n68& 1.662\\,90-07& &&\\\\\n\\end{longtable}\n\\end{center}\n\n\nQuadrupole transitions per time unit for other transitions are presented in Table~\\ref{tab:3}. The columns correspond to transitions between different vibrational states. For each $L_i$ value, the successive lines correspond to increasing values of $L_f$, i.e. to $L_f =L_i - 2$ for $L_i >1$, $L_f =L_i$ for $L_i > 0$, and $L_f =L_i+2$, respectively. The strongest transition from each state occurs in general towards the nearest vibrational state $(v_f = v_i -1)$ for $L_f = L_i - 2$. For $v_f = v_i -1$, in the vicinity of $L_i = 41$, and beyond, the $(L_i, v_i )\\rightarrow (L_i + 2, vi-1)$ transitions are replaced by $(L_i + 2, vi -1) \\rightarrow (L_i, v_i )$ transitions because the sign of the energy difference changes (see the arrows in Figure~\\ref{fig:1} for the $1 0 $ transitions). These numbers are indicated in italics in Table~\\ref{tab:3}. For example, the first number in the last line for $L_i = 41$ corresponds to the $(43, 0) \\rightarrow (41, 1)$ transition. Hence, the transition probabilities strongly vary in this region.\n\n\\begin{center}\n\\begin{longtable}{lllllll}\n\\caption{Quadrupole transition probabilities per second $W$ for transitions \nbetween different vibrational quantum numbers $(v_i \\ne v_f)$. \nThe three successive lines correspond to increasing $L_f$ values, i.e.\\ \n$L_f = L_i - 2$, $L_f = L_i$ and $L_f = L_i + 2$, respectively, for $L_i \\ge 2$. \nItalicized numbers for $(1 \\rightarrow 0)$, $(2 \\rightarrow 1)$ and $(3 \\rightarrow 2)$ \nmean that the initial and final states are exchanged \n(the first one is preceded in each case by a horizontal bar).} \n\\label{tab:3}\\\\\n\\\\[-4.9ex]\n\\hline\n$L_i$&$(1\\rightarrow\\,0)$&$(2\\rightarrow\\,0)$&$(2\\rightarrow\\,1)$\n     &$(3\\rightarrow\\,0)$&$(3\\rightarrow\\,1)$&$(3\\rightarrow\\,2)$\\\\\n\\hline\n\\endfirsthead\n\\multicolumn{7}{c}{{\\tablename} \\thetable{} -- Continuation}\\\\\n\\hline\n$L_i$&$(1\\rightarrow\\,0)$&$(2\\rightarrow\\,0)$&$(2\\rightarrow\\,1)$\n     &$(3\\rightarrow\\,0)$&$(3\\rightarrow\\,1)$&$(3\\rightarrow\\,2)$\\\\\n\\hline\n\\endhead\n\\hline\n\\multicolumn{7}{l}{{Continued on Next Page\\ldots}}\\\\\n\\endfoot\n\\hline\n\\endlastfoot\n 0& 2.352\\,92$-$08& 1.966\\,46$-$09& 4.209\\,56$-$08& 1.218\\,32$-$10& 5.449\\,55$-$09& 5.631\\,51$-$08\\\\\n1 & 1.076\\,05$-$08& 1.025\\,38$-$09& 1.920\\,69$-$08& 7.676\\,43$-$11& 2.817\\,57$-$09& 2.563\\,57$-$08\\\\\n  & 1.278\\,01$-$08& 9.691\\,38$-$10& 2.289\\,40$-$08& 5.114\\,59$-$11& 2.702\\,85$-$09& 3.066\\,63$-$08\\\\\n 2& 6.053\\,60$-$09& 6.477\\,38$-$10& 1.077\\,63$-$08& 5.616\\,33$-$11& 1.766\\,56$-$09& 1.434\\,44$-$08\\\\\n  & 7.668\\,00$-$09& 7.318\\,48$-$10& 1.368\\,58$-$08& 5.487\\,71$-$11& 2.010\\,80$-$09& 1.826\\,48$-$08\\\\\n  & 9.839\\,31$-$09& 6.702\\,09$-$10& 1.764\\,53$-$08& 2.889\\,74$-$11& 1.882\\,55$-$09& 2.366\\,10$-$08\\\\\n 3& 8.334\\,47$-$09& 9.577\\,09$-$10& 1.480\\,60$-$08& 9.016\\,18$-$11& 2.599\\,81$-$09& 1.966\\,74$-$08\\\\\n  & 7.131\\,55$-$09& 6.822\\,58$-$10& 1.272\\,66$-$08& 5.128\\,11$-$11& 1.874\\,29$-$09& 1.698\\,23$-$08\\\\\n  & 8.118\\,99$-$09& 4.906\\,61$-$10& 1.457\\,34$-$08& 1.621\\,87$-$11& 1.389\\,52$-$09& 1.955\\,89$-$08\\\\\n 4& 9.839\\,96$-$09& 1.209\\,31$-$09& 1.744\\,09$-$08& 1.224\\,32$-$10& 3.268\\,27$-$09& 2.311\\,43$-$08\\\\\n  & 6.913\\,64$-$09& 6.634\\,88$-$10& 1.233\\,56$-$08& 5.002\\,82$-$11& 1.822\\,39$-$09& 1.645\\,73$-$08\\\\\n  & 6.852\\,13$-$09& 3.618\\,13$-$10& 1.230\\,83$-$08& 8.249\\,22$-$12& 1.034\\,47$-$09& 1.653\\,02$-$08\\\\\n 5& 1.100\\,38$-$08& 1.441\\,38$-$09& 1.945\\,56$-$08& 1.557\\,27$-$10& 3.878\\,88$-$09& 2.571\\,96$-$08\\\\\n  & 6.784\\,85$-$09& 6.536\\,72$-$10& 1.210\\,31$-$08& 4.948\\,10$-$11& 1.795\\,02$-$09& 1.614\\,33$-$08\\\\\n  & 5.825\\,36$-$09& 2.636\\,09$-$10& 1.046\\,95$-$08& 3.389\\,76$-$12& 7.623\\,79$-$10& 1.406\\,77$-$08\\\\\n 6& 1.195\\,55$-$08& 1.664\\,33$-$09& 2.108\\,16$-$08& 1.906\\,95$-$10& 4.460\\,38$-$09& 2.779\\,29$-$08\\\\\n  & 6.687\\,94$-$09& 6.473\\,36$-$10& 1.192\\,71$-$08& 4.922\\,88$-$11& 1.777\\,14$-$09& 1.590\\,39$-$08\\\\\n  & 4.956\\,19$-$09& 1.874\\,69$-$10& 8.910\\,54$-$09& 8.223\\,10$-$13& 5.499\\,08$-$10& 1.197\\,65$-$08\\\\\n 7& 1.274\\,29$-$08& 1.880\\,78$-$09& 2.240\\,48$-$08& 2.273\\,62$-$10& 5.020\\,13$-$09& 2.944\\,95$-$08\\\\\n  & 6.602\\,78$-$09& 6.425\\,42$-$10& 1.177\\,15$-$08& 4.912\\,48$-$11& 1.763\\,42$-$09& 1.569\\,13$-$08\\\\\n  & 4.205\\,83$-$09& 1.286\\,50$-$10& 7.562\\,74$-$09& 3.163$-$15      & 3.842\\,83$-$10& 1.016\\,60$-$08\\\\\n 8& 1.338\\,44$-$08& 2.090\\,35$-$09& 2.345\\,90$-$08& 2.654\\,77$-$10& 5.557\\,41$-$09& 3.073\\,59$-$08\\\\\n  & 6.520\\,84$-$09& 6.384\\,48$-$10& 1.162\\,13$-$08& 4.910\\,45$-$11& 1.751\\,56$-$09& 1.548\\,51$-$08\\\\\n  & 3.553\\,06$-$09& 8.398\\,43$-$11& 6.388\\,81$-$09& 5.151\\,4 $-$13& 2.570\\,35$-$10& 8.587\\,19$-$09\\\\\n 9& 1.388\\,65$-$08& 2.291\\,35$-$09& 2.425\\,70$-$08& 3.046\\,33$-$10& 6.067\\,94$-$09& 3.167\\,10$-$08\\\\\n  & 6.438\\,02$-$09& 6.346\\,37$-$10& 1.146\\,91$-$08& 4.913\\,45$-$11& 1.740\\,42$-$09& 1.527\\,57$-$08\\\\\n  & 2.984\\,49$-$09& 5.105\\,00$-$11& 5.365\\,33$-$09& 2.015\\,57$-$12& 1.617\\,32$-$10& 7.209\\,46$-$09\\\\\n10& 1.425\\,08$-$08& 2.481\\,48$-$09& 2.480\\,33$-$08& 3.443\\,26$-$10& 6.545\\,88$-$09& 3.226\\,30$-$08\\\\\n  & 6.352\\,26$-$09& 6.308\\,78$-$10& 1.131\\,13$-$08& 4.919\\,57$-$11& 1.729\\,36$-$09& 1.505\\,82$-$08\\\\\n  & 2.490\\,52$-$09& 2.784\\,36$-$11& 4.475\\,49$-$09& 4.215\\,75$-$12& 9.305\\,25$-$11& 6.010\\,82$-$09\\\\\n11& 1.447\\,76$-$08& 2.658\\,16$-$09& 2.510\\,01$-$08& 3.839\\,88$-$10& 6.984\\,74$-$09& 3.251\\,72$-$08\\\\\n  & 6.262\\,43$-$09& 6.270\\,34$-$10& 1.114\\,59$-$08& 4.927\\,55$-$11& 1.718\\,00$-$09& 1.483\\,01$-$08\\\\\n  & 2.063\\,39$-$09& 1.263\\,30$-$11& 3.705\\,70$-$09& 6.871\\,68$-$12& 4.639\\,12$-$11& 4.973\\,47$-$09\\\\\n12& 1.456\\,72$-$08& 2.818\\,84$-$09& 2.515\\,01$-$08& 4.230\\,17$-$10& 7.378\\,14$-$09& 3.244\\,01$-$08\\\\\n  & 6.167\\,98$-$09& 6.230\\,15$-$10& 1.097\\,19$-$08& 4.936\\,50$-$11& 1.706\\,11$-$09& 1.459\\,01$-$08\\\\\n  & 1.696\\,33$-$09& 3.893\\,88$-$12& 3.044\\,04$-$09& 9.779\\,27$-$12& 1.768\\,15$-$11& 4.081\\,72$-$09\\\\\n13& 1.452\\,14$-$08& 2.961\\,08$-$09& 2.495\\,86$-$08& 4.608\\,01$-$10& 7.720\\,13$-$09& 3.204\\,18$-$08\\\\\n  & 6.068\\,67$-$09& 6.187\\,63$-$10& 1.078\\,89$-$08& 4.945\\,75$-$11& 1.693\\,54$-$09& 1.433\\,77$-$08\\\\\n  & 1.383\\,15$-$09& 2.803\\,80$-$13& 2.479\\,52$-$09& 1.277\\,08$-$11& 3.317\\,84$-$12& 3.320\\,98$-$09\\\\\n14& 1.434\\,34$-$08& 3.082\\,73$-$09& 2.453\\,38$-$08& 4.967\\,37$-$10& 8.005\\,53$-$09& 3.133\\,67$-$08\\\\\n  & 5.964\\,42$-$09& 6.142\\,34$-$10& 1.059\\,69$-$08& 4.954\\,74$-$11& 1.680\\,16$-$09& 1.407\\,28$-$08\\\\\n  & 1.118\\,03$-$09& 6.106\\,10$-$13& 2.001\\,77$-$09& 1.571\\,20$-$11& 1.196\\,23$-$13& 2.677\\,42$-$09\\\\\n15& 1.403\\,87$-$08& 3.181\\,97$-$09& 2.388\\,79$-$08& 5.302\\,5$-$10& 8.230\\,1$-$09& 3.034\\,38$-$08\\\\\n  & 5.855\\,30$-$09& 6.093\\,99$-$10& 1.039\\,60$-$08& 4.963\\,01$-$11& 1.665\\,90$-$09& 1.379\\,58$-$08\\\\\n  & 8.954\\,75$-$10& 3.857\\,94$-$12& 1.600\\,94$-$09& 1.849\\,87$-$11& 5.310\\,1$-$12& 2.137\\,81$-$09\\\\\n16& 1.361\\,45$-$08& 3.257\\,39$-$09& 2.303\\,62$-$08& 5.608\\,17$-$10& 8.390\\,8$-$09& 2.908\\,72$-$08\\\\\n  & 5.741\\,47$-$09& 6.042\\,34$-$10& 1.018\\,65$-$08& 4.970\\,17$-$11& 1.650\\,69$-$09& 1.350\\,70$-$08\\\\\n  & 7.103\\,10$-$10& 9.143\\,31$-$12& 1.267\\,71$-$09& 2.105\\,36$-$11& 1.649\\,90$-$11& 1.689\\,62$-$09\\\\\n17& 1.308\\,04$-$08& 3.308\\,04$-$09& 2.199\\,77$-$08& 5.879\\,7$-$10& 8.485\\,83$-$09& 2.759\\,50$-$08\\\\\n  & 5.623\\,14$-$09& 5.987\\,25$-$10& 9.968\\,78$-$09& 4.975\\,87$-$11& 1.634\\,51$-$09& 1.320\\,71$-$08\\\\\n  & 5.576\\,85$-$10& 1.572\\,69$-$11& 9.933\\,25$-$10& 2.332\\,30$-$11& 3.166\\,31$-$11& 1.321\\,02$-$09\\\\\n18& 1.244\\,73$-$08& 3.333\\,43$-$09& 2.079\\,44$-$08& 6.113\\,1$-$10& 8.514\\,4$-$09& 2.589\\,92$-$08\\\\\n  & 5.500\\,56$-$09& 5.928\\,59$-$10& 9.743\\,38$-$09& 4.979\\,82$-$11& 1.617\\,32$-$09& 1.289\\,68$-$08\\\\\n  & 4.331\\,11$-$10& 2.299\\,86$-$11& 7.696\\,66$-$10& 2.527\\,31$-$11& 4.912\\,29$-$11& 1.021\\,00$-$09\\\\\n19& 1.172\\,80$-$08& 3.333\\,5$-$09& 1.945\\,07$-$08& 6.305\\,3$-$10& 8.477\\,1$-$09& 2.403\\,50$-$08\\\\\n  & 5.374\\,02$-$09& 5.866\\,29$-$10& 9.510\\,84$-$09& 4.981\\,75$-$11& 1.599\\,11$-$09& 1.257\\,68$-$08\\\\\n  & 3.324\\,75$-$10& 3.046\\,69$-$11& 5.892\\,76$-$10& 2.688\\,68$-$11& 6.751\\,52$-$11& 7.794\\,50$-$10\\\\\n20& 1.093\\,63$-$08& 3.308\\,75$-$09& 1.799\\,29$-$08& 6.453\\,9$-$10& 8.375\\,8$-$09& 2.203\\,97$-$08\\\\\n  & 5.243\\,83$-$09& 5.800\\,32$-$10& 9.271\\,75$-$09& 4.981\\,44$-$11& 1.579\\,86$-$09& 1.224\\,81$-$08\\\\\n  & 2.520\\,58$-$10& 3.774\\,70$-$11& 4.454\\,05$-$10& 2.816\\,09$-$11& 8.576\\,37$-$11& 5.872\\,12$-$10\\\\\n21& 1.008\\,72$-$08& 3.259\\,9$-$09& 1.644\\,9$-$08& 6.557\\,60$-$10& 8.213\\,3$-$09& 1.995\\,20$-$08\\\\\n  & 5.110\\,33$-$09& 5.730\\,67$-$10& 9.026\\,73$-$09& 4.978\\,72$-$11& 1.559\\,59$-$09& 1.191\\,14$-$08\\\\\n  & 1.885\\,36$-$10& 4.454\\,87$-$11& 3.320\\,16$-$10& 2.910\\,27$-$11& 1.030\\,46$-$10& 4.360\\,81$-$10\\\\\n22& 9.196\\,0$-$09& 3.188\\,4$-$09& 1.484\\,74$-$08& 6.615\\,5$-$10& 7.993\\,4$-$09& 1.781\\,11$-$08\\\\\n  & 4.973\\,86$-$09& 5.657\\,36$-$10& 8.776\\,43$-$09& 4.973\\,42$-$11& 1.538\\,29$-$09& 1.156\\,77$-$08\\\\\n  & 1.389\\,74$-$10& 5.066\\,35$-$11& 2.437\\,80$-$10& 2.972\\,75$-$11& 1.187\\,62$-$10& 3.188\\,17$-$10\\\\\n23& 8.278\\,55$-$09& 3.095\\,7$-$09& 1.321\\,70$-$08& 6.628\\,1$-$10& 7.720\\,7$-$09& 1.565\\,62$-$08\\\\\n  & 4.834\\,76$-$09& 5.580\\,44$-$10& 8.521\\,50$-$09& 4.965\\,44$-$11& 1.515\\,99$-$09& 1.121\\,79$-$08\\\\\n  & 1.008\\,13$-$10& 5.595\\,26$-$11& 1.760\\,51$-$10& 3.005\\,69$-$11& 1.325\\,01$-$10& 2.291\\,12$-$10\\\\\n24& 7.350\\,4$-$09& 2.983\\,6$-$09& 1.158\\,6$-$08& 6.596\\,1$-$10& 7.400\\,6$-$09& 1.352\\,5$-$08\\\\\n  & 4.693\\,41$-$09& 5.499\\,98$-$10& 8.262\\,61$-$09& 4.954\\,69$-$11& 1.492\\,71$-$09& 1.086\\,30$-$08\\\\\n  & 7.185\\,10$-$11& 6.033\\,52$-$11& 1.248\\,31$-$10& 3.011\\,60$-$11& 1.440\\,11$-$10& 1.615\\,38$-$10\\\\\n25& 6.427\\,0$-$09& 2.854\\,39$-$09& 9.981\\,4$-$09& 6.521\\,4$-$10& 7.038\\,9$-$09& 1.145\\,4$-$08\\\\\n  & 4.550\\,15$-$09& 5.416\\,06$-$10& 8.000\\,44$-$09& 4.941\\,11$-$11& 1.468\\,46$-$09& 1.050\\,38$-$08\\\\\n  & 5.021\\,28$-$11& 6.377\\,79$-$11& 8.672\\,25$-$11& 2.993\\,26$-$11& 1.531\\,71$-$10& 1.114\\,92$-$10\\\\\n26& 5.523\\,16$-$09& 2.710\\,2$-$09& 8.429\\,80$-$09& 6.406\\,0$-$10& 6.641\\,95$-$09& 9.477\\,97$-$09\\\\\n  & 4.405\\,34$-$09& 5.328\\,79$-$10& 7.735\\,64$-$09& 4.924\\,68$-$11& 1.443\\,29$-$09& 1.014\\,14$-$08\\\\\n  & 3.432\\,55$-$11& 6.628\\,48$-$11& 5.887\\,73$-$11& 2.953\\,53$-$11& 1.599\\,65$-$10& 7.512\\,05$-$11\\\\\n27& 4.652\\,76$-$09& 2.553\\,44$-$09& 6.955\\,2$-$09& 6.252\\,7$-$10& 6.216\\,2$-$09& 7.627\\,4$-$09\\\\\n  & 4.259\\,35$-$09& 5.238\\,29$-$10& 7.468\\,89$-$09& 4.905\\,39$-$11& 1.417\\,23$-$09& 9.776\\,61$-$09\\\\\n  & 2.288\\,51$-$11& 6.788\\,90$-$11& 3.893\\,97$-$11& 2.895\\,29$-$11& 1.644\\,55$-$10& 4.924\\,11$-$11\\\\\n28& 3.828\\,79$-$09& 2.386\\,6$-$09& 5.579\\,9$-$09& 6.064\\,9$-$10& 5.768\\,3$-$09& 5.930\\,9$-$09\\\\\n  & 4.112\\,52$-$09& 5.144\\,70$-$10& 7.200\\,83$-$09& 4.883\\,29$-$11& 1.390\\,30$-$09& 9.410\\,36$-$09\\\\\n  & 1.482\\,59$-$11& 6.864\\,49$-$11& 2.498\\,85$-$11& 2.821\\,34$-$11& 1.667\\,68$-$10& 3.126\\,69$-$11\\\\\n29& 3.063\\,0$-$09& 2.212\\,26$-$09& 4.323\\,80$-$09& 5.845\\,8$-$10& 5.304\\,95$-$09& 4.413\\,36$-$09\\\\\n  & 3.965\\,19$-$09& 5.048\\,16$-$10& 6.932\\,10$-$09& 4.858\\,40$-$11& 1.362\\,57$-$09& 9.043\\,54$-$09\\\\\n  & 9.289\\,68$-$12& 6.862\\,23$-$11& 1.548\\,11$-$11& 2.734\\,34$-$11& 1.670\\,72$-$10& 1.912\\,68$-$11\\\\\n30& 2.365\\,97$-$09& 2.032\\,79$-$09& 3.204\\,24$-$09& 5.599\\,29$-$10& 4.832\\,60$-$09& 3.096\\,05$-$09\\\\\n  & 3.817\\,71$-$09& 4.948\\,82$-$10& 6.663\\,32$-$09& 4.830\\,79$-$11& 1.334\\,06$-$09& 8.677\\,02$-$09\\\\\n  & 5.596\\,29$-$12& 6.790\\,03$-$11& 9.199\\,19$-$12& 2.636\\,76$-$11& 1.655\\,62$-$10& 1.119\\,16$-$11\\\\\n31& 1.746\\,77$-$09& 1.850\\,63$-$09& 2.236\\,07$-$09& 5.329\\,18$-$10& 4.357\\,58$-$09& 1.996\\,57$-$09\\\\\n  & 3.670\\,39$-$09& 4.846\\,86$-$10& 6.395\\,09$-$09& 4.800\\,55$-$11& 1.304\\,83$-$09& 8.311\\,61$-$09\\\\\n  & 3.216\\,00$-$12& 6.656\\,34$-$11& 5.198\\,05$-$12& 2.530\\,91$-$11& 1.624\\,53$-$10& 6.204\\,20$-$12\\\\\n32& 1.213\\,15$-$09& 1.668\\,10$-$09& 1.431\\,46$-$09& 5.039\\,43$-$10& 3.885\\,89$-$09& 1.128\\,80$-$09\\\\\n  & 3.523\\,55$-$09& 4.742\\,43$-$10& 6.127\\,98$-$09& 4.767\\,77$-$11& 1.274\\,92$-$09& 7.948\\,14$-$09\\\\\n  & 1.744\\,50$-$12& 6.469\\,81$-$11& 2.760\\,53$-$12& 2.418\\,85$-$11& 1.579\\,65$-$10& 3.216\\,05$-$12\\\\\n33& 7.714\\,1$-$10& 1.487\\,39$-$09& 7.999\\,1$-$10& 4.733\\,98$-$10& 3.423\\,18$-$09& 5.029\\,3$-$10\\\\\n  & 3.377\\,49$-$09& 4.635\\,71$-$10& 5.862\\,57$-$09& 4.732\\,57$-$11& 1.244\\,37$-$09& 7.587\\,36$-$09\\\\\n  & 8.802\\,65$-$13& 6.238\\,95$-$11& 1.355\\,38$-$12& 2.302\\,43$-$11& 1.523\\,18$-$10& 1.529\\,98$-$12\\\\\n34& 4.264\\,6$-$10& 1.310\\,55$-$09& 3.483\\,6$-$10& 4.416\\,7$-$10& 2.974\\,70$-$09& 1.255\\,9$-$10\\\\\n  & 3.232\\,50$-$09& 4.526\\,88$-$10& 5.599\\,37$-$09& 4.695\\,05$-$11& 1.213\\,25$-$09& 7.230\\,01$-$09\\\\\n  & 4.045\\,46$-$13& 5.971\\,99$-$11& 6.005\\,87$-$13& 2.183\\,29$-$11& 1.457\\,23$-$10& 6.496\\,11$-$13\\\\\n35& 1.817\\,9$-$10& 1.139\\,50$-$09& 8.119$-$11& 4.091\\,4$-$10& 2.545\\,26$-$09& 5.494$-$16\\\\\n  & 3.088\\,87$-$09& 4.416\\,10$-$10& 5.338\\,90$-$09& 4.655\\,34$-$11& 1.181\\,59$-$09& 6.876\\,79$-$09\\\\\n  & 1.639\\,80$-$13& 5.676\\,68$-$11& 2.313\\,56$-$13& 2.062\\,86$-$11& 1.383\\,84$-$10& 2.354\\,82$-$13\\\\\n36& 3.960\\,1$-$11& 9.759\\,5$-$10& 4.209$-$13& 3.761\\,56$-$10& 2.139\\,20$-$09& 1.261\\,19$-$10\\\\\n  & 2.946\\,83$-$09& 4.303\\,56$-$10& 5.081\\,64$-$09& 4.613\\,58$-$11& 1.149\\,44$-$09& 6.528\\,36$-$09\\\\\n  & 5.563\\,95$-$14& 5.360\\,21$-$11& 7.277\\,03$-$14& 1.942\\,37$-$11& 1.304\\,86$-$10& 6.748\\,60$-$14\\\\\n37& 8.078$-$13& 8.214\\,7$-$10& 1.058\\,0$-$10& 3.430\\,7$-$10& 1.760\\,4$-$09& 5.009\\,0$-$10\\\\\n  & 2.806\\,65$-$09& 4.189\\,43$-$10& 4.828\\,03$-$09& 4.569\\,88$-$11& 1.116\\,85$-$09& 6.185\\,35$-$09\\\\\n  & 1.438\\,79$-$14& 5.029\\,16$-$11& 1.661\\,93$-$14& 1.822\\,85$-$11& 1.222\\,03$-$10& 1.313\\,86$-$14\\\\\n38& 6.516$-$11& 6.774\\,36$-$10& 3.950\\,1$-$10& 3.102\\,08$-$10& 1.412\\,30$-$09& 1.118\\,53$-$09\\\\\n  & 2.668\\,56$-$09& 4.073\\,87$-$10& 4.578\\,50$-$09& 4.524\\,39$-$11& 1.083\\,87$-$09& 5.848\\,33$-$09\\\\\n  & 2.331\\,32$-$15& 4.689\\,41$-$11& 2.118\\,82$-$15& 1.705\\,19$-$11& 1.136\\,88$-$10& 1.197\\,40$-$15\\\\\n39& 2.313\\,39$-$10& 5.450\\,6$-$10& 8.637\\,9$-$10& 2.778\\,6$-$10& 1.097\\,81$-$09& 1.970\\,66$-$09\\\\\n  & 2.532\\,76$-$09& 3.957\\,06$-$10& 4.333\\,45$-$09& 4.477\\,23$-$11& 1.050\\,54$-$09& 5.517\\,86$-$09\\\\\n  & 1.386\\,40$-$16& 4.346\\,17$-$11& 6.554\\,54$-$17& 1.590\\,12$-$11& 1.050\\,80$-$10& 1.160\\,54$-$17\\\\\n40& 4.970\\,1$-$10& 4.253\\,9$-$10& 1.506\\,18$-$09& 2.463\\,07$-$10& 8.194\\,6$-$10& 3.046\\,6$-$09\\\\\n  & 2.399\\,47$-$09& 3.839\\,15$-$10& 4.093\\,24$-$09& 4.428\\,52$-$11& 1.016\\,91$-$09& 5.194\\,44$-$09\\\\\\cline{7-7}\n  & 1.484\\,82$-$19& 4.004\\,02$-$11& 2.290\\,73$-$26& 1.478\\,21$-$11& 9.649\\,65$-$11& {\\it 3.553\\,47$-$19}\\\\\n41& 8.589\\,7$-$10& 3.193\\,17$-$10& 2.314\\,67$-$09& 2.158\\,00$-$10& 5.793\\,48$-$10& 4.333\\,86$-$09\\\\\n  & 2.268\\,86$-$09& 3.720\\,31$-$10& 3.858\\,22$-$09& 4.378\\,38$-$11& 9.830\\,22$-$10& 4.878\\,53$-$09\\\\\\cline{2-2}\\cline{4-4}\n  & {\\it 3.300\\,74$-$18}& 3.666\\,84$-$11& {\\it 5.047\\,82$-$17}& 1.369\\,94$-$11& 8.804\\,22$-$11& {\\it 3.384\\,62$-$16}\\\\\n42& 1.313\\,23$-$09& 2.275\\,68$-$10& 3.280\\,37$-$09& 1.865\\,7$-$10& 3.791\\,8$-$10& 5.817\\,89$-$09\\\\\n  & 2.141\\,12$-$09& 3.600\\,68$-$10& 3.628\\,71$-$09& 4.326\\,93$-$11& 9.489\\,15$-$10& 4.570\\,56$-$09\\\\\n  & {\\it 3.274\\,43$-$16}& 3.337\\,97$-$11& {\\it 1.605\\,87$-$15}& 1.265\\,68$-$11& 7.980\\,38$-$11& {\\it 5.325\\,31$-$15}\\\\\n43& 1.855\\,11$-$09& 1.507\\,49$-$10& 4.393\\,24$-$09& 1.588\\,36$-$10& 2.203\\,26$-$10& 7.482\\,77$-$09\\\\\n  & 2.016\\,39$-$09& 3.480\\,41$-$10& 3.404\\,97$-$09& 4.274\\,27$-$11& 9.146\\,32$-$10& 4.270\\,93$-$09\\\\\n  & {\\it 3.346\\,80$-$15}& 3.020\\,12$-$11& {\\it 1.187\\,03$-$14}& 1.165\\,67$-$11& 7.185\\,34$-$11& {\\it 3.049\\,73$-$14}\\\\\n44& 2.479\\,36$-$09& 8.933\\,87$-$11& 5.642\\,2$-$09& 1.327\\,8$-$10& 1.038\\,2$-$10& 9.311\\,25$-$09\\\\\n  & 1.894\\,81$-$09& 3.359\\,63$-$10& 3.187\\,29$-$09& 4.220\\,48$-$11& 8.802\\,10$-$10& 3.980\\,00$-$09\\\\\n  & {\\it 1.592\\,91$-$14}& 2.715\\,53$-$11& {\\it 4.847\\,99$-$14}& 1.070\\,13$-$11& 6.424\\,96$-$11& {\\it 1.092\\,26$-$13}\\\\\n45& 3.180\\,20$-$09& 4.370\\,6$-$11& 7.015\\,36$-$09& 1.085\\,86$-$10& 3.043$-$11& 1.128\\,50$-$08\\\\\n  & 1.776\\,53$-$09& 3.238\\,48$-$10& 2.975\\,88$-$09& 4.165\\,65$-$11& 8.456\\,87$-$10& 3.698\\,07$-$09\\\\\n  & {\\it 5.143\\,45$-$14}& 2.425\\,93$-$11& {\\it 1.432\\,61$-$13}& 9.791\\,66$-$12& 5.703\\,82$-$11& {\\it 2.980\\,88$-$13}\\\\\n46& 3.951\\,50$-$09& 1.412\\,8$-$11& 8.500\\,1$-$09& 8.641\\,2$-$11& 6.898$-$13& 1.338\\,4$-$08\\\\\n  & 1.661\\,64$-$09& 3.117\\,09$-$10& 2.770\\,96$-$09& 4.109\\,84$-$11& 8.110\\,97$-$10& 3.425\\,44$-$09\\\\\n  & {\\it 1.313\\,00$-$13}& 2.152\\,65$-$11& {\\it 3.453\\,00$-$13}& 8.928\\,43$-$12& 5.025\\,38$-$11& {\\it 6.814\\,32$-$13}\\\\\n47& 4.786\\,76$-$09& 8.061$-$13& 1.008\\,33$-$08& 6.641\\,4$-$11& 1.493\\,2$-$11& 1.559\\,01$-$08\\\\\n  & 1.550\\,27$-$09& 2.995\\,57$-$10& 2.572\\,72$-$09& 4.053\\,08$-$11& 7.764\\,74$-$10& 3.162\\,36$-$09\\\\\n  & {\\it 2.863\\,53$-$13}& 1.896\\,63$-$11& {\\it 7.236\\,25$-$13}& 8.111\\,81$-$12& 4.392\\,10$-$11& {\\it 1.375\\,59$-$12}\\\\\n48& 5.679\\,28$-$09& 3.879\\,6$-$12& 1.175\\,13$-$08& 4.873\\,7$-$11& 7.334\\,1$-$11& 1.788\\,08$-$08\\\\\n  & 1.442\\,48$-$09& 2.874\\,03$-$10& 2.381\\,32$-$09& 3.995\\,41$-$11& 7.418\\,49$-$10& 2.909\\,04$-$09\\\\\n  & {\\it 5.580\\,06$-$13}& 1.658\\,46$-$11& {\\it 1.369\\,78$-$12}& 7.341\\,58$-$12& 3.805\\,53$-$11& {\\it 2.533\\,21$-$12}\\\\\n49& 6.622\\,16$-$09& 2.344\\,7$-$11& 1.349\\,02$-$08& 3.352\\,95$-$11& 1.759\\,97$-$10& 2.023\\,55$-$08\\\\\n  & 1.338\\,38$-$09& 2.752\\,57$-$10& 2.196\\,90$-$09& 3.936\\,83$-$11& 7.072\\,52$-$10& 2.665\\,69$-$09\\\\\n  & {\\it 9.994\\,50$-$13}& 1.438\\,48$-$11& {\\it 2.400\\,60$-$12}& 6.617\\,19$-$12& 3.266\\,46$-$11& {\\it 4.348\\,19$-$12}\\\\\n50& 7.608\\,4$-$09& 5.958$-$11& 1.528\\,6$-$08& 2.094$-$11& 3.229\\,1$-$10& 2.263\\,2$-$08\\\\\n  & 1.238\\,01$-$09& 2.631\\,31$-$10& 2.019\\,58$-$09& 3.877\\,32$-$11& 6.727\\,13$-$10& 2.432\\,45$-$09\\\\\n  & {\\it 1.677\\,03$-$12}& 1.236\\,73$-$11& {\\it 3.961\\,54$-$12}& 5.937\\,86$-$12& 2.775\\,01$-$11& {\\it 7.061\\,89$-$12}\\\\\n51& 8.630\\,9$-$09& 1.123\\,5$-$10& 1.712\\,3$-$08& 1.113$-$11& 5.140\\,9$-$10& 2.504\\,97$-$08\\\\\n  & 1.141\\,43$-$09& 2.510\\,32$-$10& 1.849\\,46$-$09& 3.816\\,82$-$11& 6.382\\,56$-$10& 2.209\\,47$-$09\\\\\n  & {\\it 2.671\\,95$-$12}& 1.053\\,07$-$11& {\\it 6.230\\,94$-$12}& 5.302\\,60$-$12& 2.330\\,70$-$11& {\\it 1.097\\,12$-$11}\\\\\n52& 9.682\\,7$-$09& 1.818\\,6$-$10& 1.898\\,9$-$08& 4.278$-$12& 7.495\\,6$-$10& 2.746\\,45$-$08\\\\\n  & 1.048\\,70$-$09& 2.389\\,69$-$10& 1.686\\,62$-$09& 3.755\\,27$-$11& 6.039\\,07$-$10& 1.996\\,84$-$09\\\\\n  & {\\it 4.082\\,49$-$12}& 8.871\\,63$-$12& {\\it 9.425\\,75$-$12}& 4.710\\,29$-$12& 1.932\\,59$-$11& {\\it 1.643\\,99$-$11}\\\\\n53& 1.075\\,67$-$08& 2.682\\,1$-$10& 2.086\\,92$-$08& 5.923$-$13& 1.029\\,43$-$09& 2.985\\,41$-$08\\\\\n  & 9.598\\,52$-$10& 2.269\\,50$-$10& 1.531\\,12$-$09& 3.692\\,52$-$11& 5.696\\,88$-$10& 1.794\\,65$-$09\\\\\n  & {\\it 6.027\\,05$-$12}& 7.385\\,21$-$12& {\\it 1.380\\,93$-$11}& 4.159\\,68$-$12& 1.579\\,30$-$11& {\\it 2.391\\,37$-$11}\\\\\n54& 1.184\\,60$-$08& 3.716\\,0$-$10& 2.274\\,72$-$08& 3.184$-$13& 1.353\\,9$-$09& 3.219\\,47$-$08\\\\\n  & 8.749\\,05$-$10& 2.149\\,82$-$10& 1.383\\,02$-$09& 3.628\\,41$-$11& 5.356\\,22$-$10& 1.602\\,97$-$09\\\\\n  & {\\it 8.648\\,33$-$12}& 6.065\\,34$-$12& {\\it 1.970\\,25$-$11}& 3.649\\,46$-$12& 1.269\\,14$-$11& {\\it 3.394\\,27$-$11}\\\\\n55& 1.294\\,36$-$08& 4.923\\,0$-$10& 2.460\\,91$-$08& 3.757\\,7$-$12& 1.723\\,58$-$09& 3.446\\,19$-$08\\\\\n  & 7.938\\,82$-$10& 2.030\\,72$-$10& 1.242\\,34$-$09& 3.562\\,72$-$11& 5.017\\,25$-$10& 1.421\\,82$-$09\\\\\n  & {\\it 1.211\\,91$-$11}& 4.904\\,85$-$12& {\\it 2.749\\,85$-$11}& 3.178\\,28$-$12& 1.000\\,12$-$11& {\\it 4.721\\,21$-$11}\\\\\n56& 1.404\\,29$-$08& 6.307\\,32$-$10& 2.643\\,99$-$08& 1.128\\,6$-$11& 2.139\\,01$-$09& 3.663\\,00$-$08\\\\\n  & 7.167\\,94$-$10& 1.912\\,25$-$10& 1.109\\,10$-$09& 3.495\\,14$-$11& 4.680\\,16$-$10& 1.251\\,22$-$09\\\\\n  & {\\it 1.665\\,01$-$1}1& 3.895\\,79$-$12& {\\it 3.768\\,54$-$11}& 2.744\\,75$-$12& 7.700\\,17$-$12& {\\it 6.458\\,86$-$11}\\\\\n57& 1.513\\,72$-$08& 7.874\\,6$-$10& 2.822\\,44$-$08& 2.337\\,9$-$11& 2.601\\,26$-$09& 3.867\\,15$-$08\\\\\n  & 6.436\\,44$-$10& 1.794\\,46$-$10& 9.832\\,96$-$10& 3.425\\,27$-$11& 4.345\\,05$-$10& 1.091\\,18$-$09\\\\\n  & {\\it 2.250\\,18$-$11}& 3.029\\,52$-$12& {\\it 5.087\\,71$-$11}& 2.347\\,51$-$12& 5.764\\,32$-$12& {\\it 8.718\\,78$-$11}\\\\\n58& 1.622\\,00$-$08& 9.632\\,6$-$10& 2.994\\,65$-$08& 4.064\\,8$-$11& 3.111\\,74$-$09& 4.055\\,65$-$08\\\\\n  & 5.744\\,28$-$10& 1.677\\,39$-$10& 8.649\\,25$-$10& 3.352\\,61$-$11& 4.012\\,03$-$10& 9.416\\,67$-$10\\\\\n  & {\\it 3.000\\,06$-$11}& 2.296\\,89$-$12& {\\it 6.786\\,01$-$11}& 1.985\\,18$-$12& 4.167\\,99$-$12& {\\it 1.164\\,78$-$10}\\\\\n59& 1.728\\,49$-$08& 1.159\\,21$-$09& 3.158\\,95$-$08& 6.389\\,1$-$11& 3.672\\,30$-$09& 4.225\\,13$-$08\\\\\n  & 5.091\\,36$-$10& 1.561\\,06$-$10& 7.539\\,64$-$10& 3.276\\,45$-$11& 3.681\\,10$-$10& 8.026\\,58$-$10\\\\\n  & {\\it 3.956\\,26$-$11}& 1.688\\,35$-$12& {\\it 8.966\\,33$-$11}& 1.656\\,45$-$12& 2.884\\,14$-$12& {\\it 1.544\\,45$-$10}\\\\\n60& 1.832\\,55$-$08& 1.376\\,72$-$09& 3.313\\,53$-$08& 9.415\\,7$-$11& 4.285\\,26$-$09& 4.371\\,62$-$08\\\\\n  & 4.477\\,53$-$10& 1.445\\,49$-$10& 6.503\\,81$-$10& 3.195\\,86$-$11& 3.352\\,19$-$10& 6.741\\,05$-$10\\\\\n  & {\\it 5.172\\,91$-$11}& 1.193\\,95$-$12& {\\it 1.176\\,66$-$10}& 1.360\\,00$-$12& 1.884\\,37$-$12& {\\it 2.038\\,55$-$10}\\\\\n61& 1.933\\,53$-$08& 1.617\\,65$-$09& 3.456\\,35$-$08& 1.328\\,44$-$10& 4.953\\,30$-$09& 4.490\\,23$-$08\\\\\n  & 3.902\\,59$-$10& 1.330\\,65$-$10& 5.541\\,36$-$10& 3.109\\,51$-$11& 3.025\\,06$-$10& 5.559\\,48$-$10\\\\\n  & {\\it 6.722\\,11$-$11}& 8.034\\,38$-$13& {\\it 1.537\\,73$-$10}& 1.094\\,59$-$12& 1.138\\,78$-$12& {\\it 2.687\\,22$-$10}\\\\\n62& 2.030\\,78$-$08& 1.884\\,46$-$09& 3.585\\,00$-$08& 1.818\\,38$-$10& 5.679\\,09$-$09& 4.574\\,56$-$08\\\\\n  & 3.366\\,27$-$10& 1.216\\,48$-$10& 4.651\\,79$-$10& 3.015\\,43$-$11& 2.699\\,22$-$10& 4.481\\,18$-$10\\\\\n  & {\\it 8.702\\,42$-$11}& 5.061\\,86$-$13& {\\it 2.007\\,02$-$10}& 8.590\\,24$-$13& 6.155\\,96$-$13& {\\it 3.551\\,46$-$10}\\\\\n63& 2.123\\,60$-$08& 2.180\\,30$-$09& 3.696\\,49$-$08& 2.436\\,91$-$10& 6.464\\,00$-$09& 4.615\\,48$-$08\\\\\n  & 2.868\\,29$-$10& 1.102\\,88$-$10& 3.834\\,54$-$10& 2.910\\,50$-$11& 2.373\\,69$-$10& 3.505\\,35$-$10\\\\\n  & {\\it 1.125\\,29$-$10}& 2.911\\,00$-$13& {\\it 2.625\\,03$-$10}& 6.521\\,44$-$13& 2.804\\,86$-$13& {\\it 4.730\\,03$-$10}\\\\\n64& 2.211\\,19$-$08& 2.509\\,15$-$09& 3.786\\,75$-$08& 3.218\\,0$-$10& 7.304\\,37$-$09& 4.598\\,57$-$08\\\\\n  & 2.408\\,31$-$10& 9.895\\,93$-$11& 3.088\\,98$-$10& 2.789\\,31$-$11& 2.046\\,49$-$10& 2.630\\,97$-$10\\\\\n  & {\\it 1.457\\,69$-$10}& 1.464\\,45$-$13& {\\it 3.455\\,25$-$10}& 4.728\\,57$-$13& 9.553\\,73$-$14& {\\it 6.397\\,43$-$10}\\\\\n65& 2.292\\,60$-$08& 2.875\\,75$-$09& 3.849\\,73$-$08& 4.203\\,02$-$10& 8.178\\,2$-$09& 4.497\\,26$-$08\\\\\n  & 1.985\\,97$-$10& 8.762\\,06$-$11& 2.414\\,37$-$10& 2.640\\,78$-$11& 1.713\\,26$-$10& 1.856\\,32$-$10\\\\\n  & {\\it 1.898\\,57$-$10}& 5.956\\,39$-$14& {\\it 4.604\\,33$-$10}& 3.200\\,84$-$13& 1.803\\,21$-$14& \\\\\n66& 2.366\\,50$-$08& 3.284\\,65$-$09& 3.875\\,22$-$08&  & & \\\\\n  & 1.600\\,87$-$10& 7.618\\,61$-$11& 1.809\\,79$-$10& & & \\\\\n  & {\\it 2.498\\,29$-$10}& 1.652\\,49$-$14& {\\it 6.269\\,60$-$10}& & & \\\\\n67& 2.430\\,79$-$08& & & & &\\\\\n  & 1.252\\,58$-$10&  &  &  &  &  \\\\\n\\end{longtable}\n\\end{center}\n\nQuadrupole oscillator strength  are presented in Figure~\\ref{fig:2}, \\ref{fig:3} and \\ref{fig:4}.  All the points connected by a curve  corresponds to transitions with the same value of the initial $v_i$ and final $v_f$ vibrational quantum number. For all the cases there is a systematic grouping of the curves depending of the difference between of the vibrational quantum numbers. Roughly the oscillator strength with $v_i-v_f=1$ are larger by an order of magnitude than those with  $v_i-v_f=2$ and $v_i-v_f=2$ are larger by an order of magnitude than those with  $v_i-v_f=3$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{fdLp2.eps}\n\\caption{Oscillator strengths for $L_f = L_i+2$ transitions.}\n\\label{fig:2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{fdL00.eps}\n\\caption{Oscillator strengths for $L_f = L_i$ transitions.}\n\\label{fig:3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{fdLm2.eps}\n\\caption{Oscillator strengths for $L_f = L_i-2$ transitions.}\n\\label{fig:4}\n\\end{center}\n\\end{figure}\n\nLifetimes for all states considered are calculated with \n\\begin{equation}\n\\tau = \\left(\\sum_{E_f<E_i} W_{i  \\rightarrow f}^{(2)}\\right)^{-1},\n\\end{equation} \nand are displayed in Figure~\\ref{fig:4}. Using the allowed electric quadrupole transitions presented in tables \\ref{tab:2}  and \\ref{tab:3} we obtain, for the vibrational states $(L=0)$  $v=1,2$ and $3$: $ \\tau= 4.250\\,039\\times 10^7$\\,s, $2.269\\,525\\times 10^7$\\,s and $1.615\\,861\\times 10^7$\\,s,   respectively.  The zero-vibrational states $(L^{\\pi},0)$ have  the longer lifetime. Decreasing monotonically from $(2^{+},0)$  ($\\tau= 777\\,201$ years)  up to $L\\sim 25$. In this domain $L\\in[0-25]$, the lifetime for the excited  vibrational states  $v=1,2,3$  is around 9 months.  For $L>25$, the lifetime for all states decrease reaching a minimum around $L\\sim 54$ ($\\tau \\sim 55$ days) and then starting to increase again slowly. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{tlifev.eps}\n\\caption{Lifetimes  in seconds for the first four rotational bands $(v = 0-3)$.}\n\\label{fig:5}\n\\end{center}\n\\end{figure}\n\n\\section{Mass Dependence}\n\nThe mass dependence in the binding energy has been widely studied for three body Coulomb systems $m_1$, $m_2$ and $m_3$  and in particular for the symmetric configuration where $m_1=m_2$ (see for example \\cite{GR1982} and \\cite{RF2002} and references therein). The Lagrange-mesh method, briefly presented in section II, can easily be adapted to study the ground state energy of the symmetric three body systems with   $m_1=m_2=m$ and $m_3$. We consider the symmetric systems with one\/two electrons: $^{\\infty}$H$_2^+$ (infinite protons mass), Ps$^{-}$($e^+e^-e^-$), H$^-$, D$^-$, T$^-$, $^{\\infty}$H$^-$ (infinite proton mass) and the muonic systems $\\mu^+\\mu^+e$ and $\\mu^+e\\,e$. In the Lagrange-mesh approach, energies and several properties of the systems Ps$^{-}$ and $^{\\infty}$H$^-$  were  calculated in~\\cite{HB1999}.  Results are presented in Table~\\ref{tab:fit} together with those of the molecular ion H$_2^+$ and its isotopomers  D$_2^+$ and T$_2^+$. The values, in atomic units, of the particle masses  for the proton $m_p $, deuteron $m_d$, triton $m_t$ and muon $m_{\\mu}$ are\n\\begin{eqnarray*}\nm_p = 1836.152\\,701,&\\hspace{0.5cm} &m_d=3670.483\\,014,\\\\\nm_t = 5496.921\\,58\\,\\,\\,,& &m_{\\mu}=206.768\\,262\\,.\n\\end{eqnarray*}\n\n\\begin{center}\n\\begin{table}[!thb]\n\\caption{Ground state energies for symmetric three body systems $m_1=m_2=m$ and $m_3$ with Coulomb interaction. $\\beta=m_3\/(2m+m_3)$. Results for H$_2^+$ and  D$_2^+$ from $^{a}$\\cite{OPB12} and $^{b}$\\cite{OP13}. The symbol \"$\\infty$\" stands for an infinitely heavy particle. Comparison is made with $^{c}$\\cite{YZL03}, $^{d}$\\cite{HBGD2000}, $^{e}$\\cite{HNN2009}, $^{f}$\\cite{SBGS1998},  $^{g}$\\cite{Fro2005}, $^{h}$\\cite{FR2004}, $^{i}$\\cite{FS2003} ,  $^{j}$\\cite{Nak2007} (rounded).  }\n\\label{tab:fit}\n\\begin{tabular}{cllll}\n\\hline\\hline\nSystem   & $m$&  $\\beta$             & $E_t$\\,(present)  &  Reference\\\\\n\\hline\n$^{\\infty}$H$_2^+$ & $\\infty$&0.0                &-0.602\\,634\\,619\\,105 & \\\\  \n\\,\\,\\,\\,\\,\\,T$_2^+$& 5496.921\\,58&9.0951729$\\times10^{-5}$&-0.599\\,506\\,910\\,111\\,5 &-0.599\\,506\\,910\\,111\\,541$^{c}$\\\\\n\\,\\,\\,\\,\\,\\,D$_2^+$& 3670.483\\,014&1.3620330$\\times10^{-4}$&-0.598\\,788\\,784\\,330\\,7$^{a}$ &-0.598\\,788\\,784\\,330\\,68$^{d}$ \\\\\n\\,\\,\\,\\,\\,\\,H$_2^+$& 1836.152\\,701&2.7223437$\\times10^{-4}$&-0.597\\,139\\,063\\,123\\,41$^{b}$  &-0.597\\,139\\,063\\,123\\,405$^{e}$\\\\\n\\,\\,\\,\\,$\\mu^+\\mu^+e$& 206.768\\,262&2.4123327$\\times10^{-3}$&-0.585\\,126\\,097\\,219\\,20&-0.585\\,126\\,097\\,219\\,193$^{f}$\\\\\n\\,\\,\\,\\,Ps$^{-}$         & 1.0 &0.33333333&-0.262\\,005\\,070\\,232\\,97 & -0.262\\,005\\,070\\,232\\,980\\,$^{g}$\\\\\n$\\mu^+e\\,e$           & 1.0&0.99042000&-0.525\\,054\\,806\\,243\\,53&-0.525\\,054\\,806\\,243\\,526\\,$^{h}$\\\\\n\\,\\,\\,\\,\\,\\,H$^-$         & 1.0&0.99891195 &-0.527\\,445\\,881\\,114\\,18 &-0.527\\,445\\,881\\,114\\,179\\,$^{i}$\\\\\n\\,\\,\\,\\,\\,\\,D$^-$         & 1.0&0.99945541&-0.527\\,598\\,324\\,686\\,48&  -0.527\\,598\\,324\\,686\\,478\\,$^{i}$\\\\ \n\\,\\,\\,\\,\\,\\,T$^-$         & 1.0&0.99963629&-0.527\\,649\\,048\\,203\\,01&  -0.527\\,649\\,048\\,202\\,999\\,95\\,$^{i}$\\\\ \n\\,\\,$^{\\infty}$H$^-$ & 1.0&1.0 &-0.527\\,751\\,016\\,544\\,38 &             -0.527\\,751\\,016\\,544\\,377\\,$^{i,j}$\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\nFollowing Gur'yanov and Rebane~\\cite{GR1982} the ground state energy is expanded as follows\n\\begin{equation}\n\\label{expn}\nE(\\beta)  \\approx m\\beta f(\\beta)\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{expnf}\nf(\\beta)= \\sum_{j=0}^{n} C_j \\beta^{j\/2}\\,,\n\\end{equation}\nand  $\\beta=m_3\/(2m+m_3)$ varies between $[0,1]$. The parameter $C_0$ is fixed in terms of  the value of the ground state energy of the static hydrogen molecular ion $^{\\infty}$H$_2^+$. First  $5$ terms in (\\ref{expn}), $n=5$ reproduce no less than 7 s.d. in energies with parameters \n\\begin{center}\n\\begin{tabular}{lll}\n$C_1 =  0.64179582$,&$C_2 =  0.28372246$,&$C_3 = -0.15304332$,\\\\\n$C_4 = -0.21968058$,&$C_5 =  0.12472387$,&\n\\end{tabular}\n\\end{center}\nwhich are in agreement to the parameters found in~\\cite{GR1982}. Increasing number of terms up to $7$, we reproduce no less than $10$ s.d. in energy (see Table~\\ref{tab:fit}) with parameters \n\\begin{center}\n\\begin{tabular}{lll}\n$C_1 =  0.64177988217$,& $C_2 =  0.28534422820$,& $C_3 = -0.18731634808$,\\\\\n$C_4 = -0.04273755179$,& $C_5 = -0.24388850969$,& $C_6=0.33561102864$,\\\\\n$C_7=-0.11127450779$. &&\\\\\n\\end{tabular}\n\\end{center}\n\nThe expansion $f(\\beta)$ (\\ref{expnf})  is an increasing function of $\\beta \\in [0,1]$ and is presented in Figure~\\ref{fig:5}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{fitE.eps}\n\\caption{Expansion $f(\\beta)$ (\\ref{expnf}) for $n=7$ (line). Symbols indicate  all  $11$ systems present in Table~\\ref{tab:fit}.}\n\\label{fig:5}\n\\end{center}\n\\end{figure}\n\n\\section*{Conclusions}\n\\label{s4}\nSummarizing, in order to solve the non-relativistic Shr$\\ddot{\\rm o}$dinger equation for the three body system with Coulomb interaction composed of two tritons and one electron, the  Lagrange-mesh method  is applied.  The spectra for the four lowest vibrational sates $v=0,1,2,3$  is presented with $13$, $12$, $11$ and $10$ significant digits using $54$ mesh points in the $x-y$ perimetric coordinates and $18$ mesh points in the $z$ perimetric coordinate.  It is found that the vibrational states $v=0,1,2,3$ support $62$, $61$, $60$ and $58$ rotational states, respectively. Some quasi bound states are also given.  The vibrationless band $(L^{\\pi},0)$ supports $27$ ($11$) rotational bound states more than the  isotopomer H$_2^+$~\\cite{OPB12}  (D$_2^+$ ~\\cite{OP13}). Table~\\ref{tab:4} presents a complete comparison between these systems.  \n\nUsing the wave functions provided by the Lagrange mesh method it is easy to calculate the electric quadrupole transitions probabilities per second $W^{(2)}_{i\\rightarrow f}$. All possible transitions probabilities are presented with six significant digits. Quadrupole oscillator strength $f^{(2)}_{i\\rightarrow f}$ for $\\Delta L = 2, 0,-2$ are depicted in Figures \\ref{fig:2}, \\ref{fig:3} and \\ref{fig:4}, respectively. For $L_i = 41$  and beyond the $(L_i, 1 )\\rightarrow (L_i + 2, 0)$ transitions are replaced by $(L_i + 2, 0) \\rightarrow (L_i, 1 )$ transitions. The rotational state $L_i$ after which the initial and final states are exchanged for the three cases $v_{1\\rightarrow 0}$, $v_{2\\rightarrow 1}$ and $v_{3\\rightarrow 2}$ and comparison with the molecular ions H$_2^+$ and D$_2^+$ are shown  in Table~\\ref{tab:4}.\n \n\\begin{center}\n\\begin{table}[!thb]\n\\caption{Comparison between the isotopomers H$_2^+$, D$_2^+$ and T$_2^+$. $a)$ Number of rotational bound states for the  lowest three vibrational states and  $b)$ rotational state after which the initial and final states are exchanged.}\n\\label{tab:4}\n\\begin{tabular}{lllcccccccc}\n\\hline\n  &         & &\\multicolumn{4}{c}{Rotational bound states}&&\\multicolumn{3}{c}{Change direction}\\\\  \\cline{4-7} \\cline{9-11}  \n  &Mass &$(0^+,0)$&$v=0$&$v=1$&$v=2$&$v=3$&&$v_{1\\rightarrow 0}$&$v_{2\\rightarrow 1}$&$v_{3\\rightarrow 2}$\\\\\n\\hline\nH$_2^+$&$1836.152\\,701$&$-0.597\\,139\\,063\\,123\\,3$&35&34&33&31&& 23 & 23 & 22\\\\\nD$_2^+$&$3670.483\\,014$&$-0.598\\,788\\,784\\,330\\,7$&51&49&48&47&& 33 & 33 & 32\\\\\nT$_2^+$&$5496.921\\,58$  &$-0.599\\,506\\,910\\,111\\,5$&62&61&60&58 && 41 & 41 & 40\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n \nQualitatively the behaviour of the electric quadrupole transitions $W^{(2)}$ resembles to that of the electric quadrupole oscillator strength (see Figures \\ref{fig:2}, \\ref{fig:3} and \\ref{fig:4}).  Comparing with the isotopomers H$_2^+$~\\cite{OPB12} and D$_2^+$ ~\\cite{OP13}  the hierarchy  \n\\begin{equation}\nW_{{\\rm H}_2^+}^{(2)} > W_{{\\rm D}_2^+}^{(2)} > W_{{\\rm T}_2^+}^{(2)},\n\\end{equation}\nis found, except at some points  near to the minima  and at the ends of the curves. As a consequence, the lifetimes of T$_2^+$ are larger than those of D$_2^+$ and H$_2^+$,  around three times and 21 times, respectively.\n\nIn order to study the mass dependence for  the ground state of three body systems, we consider the systems $^{\\infty}$H$_2^+$, $^{\\infty}$H$^-$,  Ps$^{-}$, H$^-$, D$^-$, T$^-$, $\\mu^+\\mu^+e$, $\\mu^+e\\,e$  and those previously investigated H$_2^+$ and  D$_2^+$ (see Table \\ref{expn}).  Expanding the energy as (\\ref{expn}) and keeping $7$ terms, $n=7$, we reproduce no less than $10$ s.d.\n\n\\begin{acknowledgements}\nI would like to thank Professor A Turbiner for his valuable comments and suggestions, to  CONACyT (Mexico) for a postdoctoral grant and  the Instituto de Ciencias Nucleares (UNAM, Mexico City) for their kind hospitality.  This work was partially supported by the University Program FENOMEC (UNAM, Mexico).\n\\end{acknowledgements}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe paper studies  discrete time processes and their predictability and randomness\nin deterministic pathwise setting, without using  probabilistic assumptions on the ensemble.\n\\par\nUnderstanding of the pathwise randomness leads to many applications in Monte-Carlo methods, cryptography, and control systems.\nThere are many classical works devoted to the concept of pathwise randomness and the problem of distinguishability\n of random sequences; see the references in \\cite{LV,Dow}. \\index{Li and Vitanyi\n(1993) and  Downey (2004).} In particular, the approach from \\\n Borel (1909) \\cite{Bor}  , Mises (1919) \\cite{Mis}  , Church (1940) \\cite{Ch}   was   based  on\n limits of the sampling proportions of zeros in the binary sequences and subsequences; Kolmogorov\n(1965) \\cite{Ko65} and Loveland (1966) \\cite{Lo} developed a different  concept of the algorithmic randomness and compressibility;\nSchnorr (1971) \\cite{Sch} suggested approach based on predicability and  martingale properties. So far, the exiting theory is devoted to the problem of distinguishability\n of random sequences  and does not consider the problem  of quantification of the degree of randomness.\nThis paper studies randomness in the sense of\nthe pathwise  predicability and attempts to develop an approach for  quantification and separation of the ``noise\" for the\nsequences that are deemed to be random. The estimation of the degree on randomness is  a difficult  problem, since the task of\ndetecting the randomness  is  nontrivial itself.\n\\par\nThe paper  investigates randomness  and noise for the\nsequences in a more special setting originated from the linear filtering and prediction of stochastic processes\nrather than algorithmic randomness in the sprit of   Downey (2004).  We suggest exploring  the following straightforward pathwise\ncriterion:  a class of sequences that is\npredictable or such that its missing value can be recovered without error from observations of remaining values\n is assumed to consist of  non-random\nsequences.\n\n\n\nFor stationary discrete time processes,  there\nis a criterion of predictability and recoverability in the frequency domain setting\ngiven by the classical minimality criterion  \\cite{Ko41},\nTheorem 24, and the Szeg\\\"o-Kolmogorov theorem; see \\cite{Sz,V}\\index{\nSzeg\\\"o (1920, 1921), Verblunsky  (1936), Kolmogorov (1941),\\index{\nTheorem 2}} and recent literature reviews in \\cite{Bin,S}.\\index{ Simon (2011) and Bingham\n(2012).} By this theorem,  a stationary process  is\npredictable if its spectral density is vanishing with a certain rate at a point of the\nunit circle $\\{z\\in{\\bf C}:\\ |z|=1\\}$. In particular, it holds if the spectral\ndensity vanishes on an arc of\nthe unit circle, i.e., the  process is bandlimited.  There are many works devoted to smoothing in frequency domain and sampling; see, e.g.,\n\\cite{Alem,rema, F94, FKR, Leef, TH,W} and the bibliography here.\n  \\index{\\cite{jerry}, \\cite{pollock}, \\cite{PFG}, \\cite{PFG1}, \\cite{AU}.}  In \\cite{D10,D12a,D12b,D12c,D12d,D16}, predictability  was\nreaddressed in the deterministic setting for two-sided sequences for\nwith Z-transform vanishing in a point on ${\\mathbb{T}}$, and some predictors\nwere suggested.  These results were based on frequency characteristics of the entire\ntwo-sided sequences, since the properties of the Z-transforms were used.   Application of the two-sided Z-transform requires to select some\npast time at the middle of the time interval of the observations as the zero point for a model of  the two-sided sequence; this could be inconvenient.\n In many applications, it is more convenient\n to represent data flow\nas one-sided sequences such that $x(t)$  represents outdated observations with diminishing significance as\n$t\\to -\\infty$.\nThis leads to the analysis of the  one-sided sequences directed backward to the past. However,\nthe straightforward application of the one-sided\nZ-transform to the one-sided sequences  does not generate Z-transform vanishing on a part of the unit circle even for a band-limited underlying  sequence.\n\nThe paper suggests\nsome approaches to quantification of randomness based on frequency analysis of two-sided and one-sided sequences.\nIn addition, the paper suggests an extension of the notion\nof bandlimitiness on one-sided sequences and a  procedure allowing to represent an one-sided sequence as a sum of left-bandlimited and predictable sequences and a non-reducible noise.\n\n\n\n\n\n\\section{Definitions and background}\nWe use notation ${\\rm sinc\\,}(x)=\\sin(x)\/x$ and  ${\\mathbb{T}}=\\{z\\in{\\bf C}:\\  |z|=1\\}$, and we denote by ${\\mathbb{Z}}$  the set of all integers.\n\\par\n For a Hilbert space $H$, we denote by $(\\cdot,\\cdot)_{H}$ the\ncorresponding inner product. We denote by $L_2(D)$ the usual Hilbert space of complex valued\nsquare integrable functions $x:D\\to{\\bf C}$, where $D$ is an interval  in ${\\bf R}$.\n\nLet $\\tau\\in{\\mathbb{Z}}\\cup\\{+\\infty\\}$ and $\\theta<\\tau$; the case where\n$\\theta=-\\infty$  is not excluded. We denote by $\\ell_r(\\theta,\\tau)$ the\nBanach space of complex valued sequences $\\{x(t)\\}_{t=\\theta}^\\tau$\nsuch that\n$\\|x\\|_{\\ell_r(\\theta,\\tau)}=\\left(\\sum_{t=\\theta}^\\tau|x(t)|^r\\right)^{1\/r}<+\\infty$\nfor $r\\in[1,\\infty)$ or  $\\|x\\|_{\\ell_\\infty(\\theta,\\tau)}=\\sup_{t: \\theta-1<t<\\tau+1}|x(t)|<+\\infty$\nfor $r=+\\infty$.\n\\par\nLet $\\ell_r=\\ell_r(-\\infty,+\\infty)$ and\n$\\ell_r^-=\\ell_r(-\\infty,0)$.\n\\par\nFor  $x\\in \\ell_1$ or $x\\in \\ell_2$, we denote by $X={\\cal Z} x$ the\nZ-transform  \\begin{eqnarray*} X(z)=\\sum_{t=-\\infty}^{\\infty}x(t)z^{-t},\\quad\nz\\in{\\mathbb{T}}. \\end{eqnarray*} Respectively, the inverse Z-transform  $x={\\cal Z}^{-1}X$ is\ndefined as \\begin{eqnarray*} x(t)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi\nX\\left(e^{i\\omega}\\right) e^{i\\omega t}d\\omega, \\quad t=0,\\pm 1,\\pm 2,....\\end{eqnarray*}\nIf $x\\in \\ell_2$, then $X|_{\\mathbb{T}}$ is defined as an element of\n$L_2({\\mathbb{T}})$.\n\n\n\n\\par\n\\index{Let ${\\cal Y}_N$ be the Hilbert space of sequences $\\{y_k\\}_{k=\n-N}^{N}\\subset{\\bf C}$ provided with the $\\ell_2$-norm, i.e.,\n$\\|y\\|_{{\\cal Y}_N}=\\left(\\sum_{k\\in {\\mathbb{Z}}_N }|y_k|^2\\right)^{1\/2}<+\\infty$.}\n\n\\par\n\nFor a set $I\\subset (-\\pi,\\pi]$, we denote $I^c=(-\\pi,\\pi]\\backslash I$.\n\n\nLet ${\\cal J}$ be the set of all $I\\subset (-\\pi,\\pi]$ such  that the set $\\{e^{i\\omega}\\}_{\\omega\\in I}$ is a connected arc and $I^c\\neq \\emptyset$.\n\nFor any $I\\in{\\cal J}$,  we  denote by $\\omega_I$ the middle point $e^{i\\omega_I}$ of the arc $\\{e^{i\\omega}\\}_{\\omega\\in I}$.\n\n\nWe denote by ${\\mathbb{B}}_\\infty$  the set of all mappings $X:{\\mathbb{T}}\\to{\\bf C}$ such\nthat $X\\left(e^{i\\o}\\right) \\in L_2(-\\pi,\\pi)$ and $X\\left(e^{i\\o}\\right) =0$ for $\\omega\\notin I$. We will call  the corresponding processes $x={\\cal Z}^{-1}X$\n{\\em band-limited}. Let $\\ell_2^{\\scriptscriptstyle BL}(I)=\\{x\\in\\ell_2:\\ X={\\cal Z} x\\in {\\mathbb{B}}_\\infty\\}$.\n\\begin{definition}  Assume that there exists $I\\in{\\cal J}$ such that\n   $x\\in\\ell_2^-$ represents the  trace of a band-limited process $x_{{\\scriptscriptstyle BL}}\\in\\ell_2^{\\scriptscriptstyle BL}(I)$ with the spectrum on $I$, i.e., $x(t)=x_{{\\scriptscriptstyle BL}}(t)$ for $t\\le 0$, and $X_{{\\scriptscriptstyle BL}}={\\cal Z} x_{{\\scriptscriptstyle BL}}\\in {\\mathbb{B}}_\\infty$. We call the process  $x$\n\\underline{ left band-limited} with the spectrum on $I$.\n\\end{definition}\n\n\n Let $\\ell_2^{-,\\LBL}(I)$ be the subset of $\\ell_2^-$ consisting of semi-infinite sequences\n$\\{x(t)\\}_{t\\le 0}$ such that $x(t)=({\\cal Z}^{-1}\nX)(t)$ for $t\\le 0$ for some $X \\in {\\mathbb{B}}_\\infty $.\n\n\n\\section{Quantification of randomness for two-sided sequences}  \\label{Sec2Sided}\nLet us discuss first a straightforward approach where noise is associated with the high-frequency component.\nConsider a\nsequence $x\\in\\ell_{2}$ that does not feature predicability described in Lemma \\ref{lemmaPred}.\nLet\n\\begin{eqnarray*}\nX={\\cal Z} X,\\qquad Y_{\\scriptscriptstyle BL}|_{{\\mathbb{T}}}=({\\mathbb{I}}_I X)|_{{\\mathbb{T}}},\\qquad y_{\\scriptscriptstyle BL}={\\cal Z}^{-1} Y_{\\scriptscriptstyle BL}\n\\end{eqnarray*} for some given $I\\in{\\cal J}$. Here ${\\mathbb{I}}$ is the indicator function, i.e.,\n${\\mathbb{I}}_I\\left(e^{i\\o}\\right)=1$ if $\\omega\\in I$ and ${\\mathbb{I}}_I\\left(e^{i\\o}\\right)=0$ if $\\omega\\in I^c=(-\\pi,\\pi]\\backslash I$.\nIn many applications, it is acceptable to deem the process $n_{\\scriptscriptstyle BL}(t)=x(t)-y_{\\scriptscriptstyle BL}(t)$  with $I=I_0=(-\\Omega,\\Omega)$, where $\\Omega\\in(0,\\pi)$, to be a noise\naccompanying the systematic movement $y_{\\scriptscriptstyle BL}(t)$.\nHowever, estimation\nof $n_{\\scriptscriptstyle BL}(t)$ will not help to quantify the randomness of $x$, since\n\\begin{eqnarray}\n\\|n_{\\scriptscriptstyle BL}\\|_{\\ell_2}=\\|x-y_{\\scriptscriptstyle BL}\\|_{\\ell_2}\\to 0\\quad \\hbox{as}\\quad {\\rm mes\\,}(-\\pi,\\pi]\\setminus I)\\to 0\\quad \\hbox{for any}\\quad x\\in\\ell_2.\n\\label{2s}\\end{eqnarray}\nIn  addition, $n_{\\scriptscriptstyle BL}$  is also a predictable  band-limited process,\n\\begin{eqnarray*}\nn_{\\scriptscriptstyle BL}=x-y_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I^c} {\\cal Z} x)\\in \\ell_2^{\\scriptscriptstyle BL}(I^c),\n\\label{2}\n\\end{eqnarray*}\n In particular, this means that\nany two-sided sequence  $x\\in \\ell_2$ can be represented as\n as a sum \\begin{eqnarray}\nx=y_{\\scriptscriptstyle BL}+ n_{\\scriptscriptstyle BL}, \\quad y_{\\scriptscriptstyle BL}\\in \\ell_2^{\\scriptscriptstyle BL} (I),\\quad n_{\\scriptscriptstyle BL}\\in\\ell_2^{\\scriptscriptstyle BL} (I^c),\n\\label{xxy}\\end{eqnarray}\n  i.e., as   a sum of two two-sided  band-limited predictable  in the sense of Lemma \\ref{lemmaPred} sequences, and that can be done with any choice of $I$.\n\\index{In addition,   $(\\widehat x_{\\scriptscriptstyle BL},y_{\\scriptscriptstyle BL})_{\\ell_2}=0$ since $( x_{\\scriptscriptstyle BL},y_{\\scriptscriptstyle BL})_{\\ell_2}=0$ for any\n$x_{\\scriptscriptstyle BL}\\in \\ell_2^{\\scriptscriptstyle BL}(I)$ and $y_{\\scriptscriptstyle BL}\\in \\ell_2^{\\scriptscriptstyle BL}(I^c)$.}\nThis also does not lead to possibility to detect   and quantify randomness.\n\nWe suggest a different approach.\nWe will show below a meaningful  quantification of the randomness of $x\\in \\ell_2$ can be achieved with the value\n\\begin{eqnarray}\n\\sigma(x)=\\mathop{\\rm ess\\, inf}}%\\def\\col{{\\rm col \\,}_{\\omega\\in(-\\pi,\\pi]}|X\\left(e^{i\\o}\\right)|,\\quad X= {\\cal Z} x.\n\\label{sig}\n\\end{eqnarray}\n\n\\subsection{Randomness as a measure of non-predictability}\n\\subsubsection*{Some predictability results for two-sided sequences}\nTwo-sided band-limited sequences are predictable in the following sense.\n\\begin{theorem}\\label{lemmaPred} \\begin{enumerate}\n\\item\nLet ${\\cal X}\\subset \\ell_2^{\\scriptscriptstyle BL}(I)$ be a bounded set. Then, for any $\\varepsilon>0$, there exists a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that  $\\sup_{t\\in {\\mathbb{Z}}}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s\\le t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\item Let ${\\cal J}_1\\subset{\\cal J}$ be a set of $I$ such that $\\sup_{I\\in J_1}{\\rm mes\\,}(I)<2\\pi$.  Let ${\\cal X}\\subset \\cup_{I\\in {\\cal J}_1} \\ell_2^{\\scriptscriptstyle BL}(I)$ be a bounded set in $\\ell_2$. Then, for any $\\varepsilon>0$, there exists a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that  $\\sup_{t\\in {\\mathbb{Z}}}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s\\le t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\item\nLet ${\\cal J}_1$ be the set of $I\\in{\\cal J}$ such that $\\sup_{I\\in J}{\\rm mes\\,}(I)<2\\pi$, and  ${\\cal X}\\subset \\cup_{I\\in {\\cal J}_1} \\ell_2^{\\scriptscriptstyle BL}(I)$ be a bounded set in $\\ell_2$ such\nthat $\\sum_{t\\le \\tau}|x(t)|^2\\to 0$ as $\\tau\\to +\\infty$ uniformly over $x\\in {\\cal X}$. Then, for any $\\varepsilon>0$, there exists $\\tau<0$ and a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that  $\\sup_{t\\ge 1}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s=\\tau}^{t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{lemmaPred}(iii) states  that some predicability  based on finite sets of  observations also can be achieved if\nwe relax  predicability requirement to cover times $t\\ge 1$ only;  this would be a weaker version of predicability comparing with the one described in Theorem \\ref{lemmaPred} (ii).\n\nSome versions of this Theorem and some examples of predictable classes can be found in \\cite{D12a,D12b}.\n\nIn addition, it appears that the spectrum supporting sets $I$ can be estimated from the\nset of observations $\\{x(s)\\}_{s\\le \\tau}$ for any $\\tau<0$. More precisely, the following theorem holds.\n\\begin{theorem}\\label{ThO}\nLet ${\\cal X}\\subset  \\ell_2$ be a  set such that if $x\\in{\\cal X}$ then $x\\in \\ell_2^{\\scriptscriptstyle BL}(I)$ for some $I=I(x)\\in{\\cal J}$,\nand that $\\nu\\stackrel{{\\scriptscriptstyle \\Delta}}{=} 2\\pi-\\sup_{x\\in {\\cal X}}{\\rm mes\\,}(I(x))>0$. Let $\\widehat\\nu=\\nu\/3$. Then, for  any $\\tau<0$,\n there exists a mapping $F:\\ell_2(-\\infty,\\tau)\\to (-\\pi,\\pi]$ such that,\n for $\\widehat\\omega_c=F(x(t)|_{t\\le\\tau})$, ${\\mathbb{T}}_c\n \\subset \\{e^{i\\omega},\\ \\omega\\in I^c\\}$, where\n \\begin{eqnarray*}\n{\\mathbb{T}}_c=\\left\\{e^{i(\\omega+\\pi)}:\\ \\omega\\in (-\\pi,\\pi],\\ \\min_{k=0,\\pm 1}|\\widehat\\omega_c-\\omega+2k\\pi|\\le\\widehat\\nu\\right\\}.\\end{eqnarray*}\nIn other words, if $x\\in {\\cal X}$, then $x\\in\\ell_2^{\\scriptscriptstyle BL}(\\widehat I)$  and $I\\subset \\widehat I$, where \\begin{eqnarray*}\n\\widehat I=\\left\\{\\omega\\in (-\\pi,\\pi]: e^{i\\omega}\\notin{\\mathbb{T}}_c\\right\\}.\n\\end{eqnarray*}\n\\end{theorem}\nThe set $\\widehat I$ in Theorem \\ref{ThO} can be regarded as an estimate of $I$ based on observations of $\\{x(t)\\}_{t\\le \\tau}$.\n\n\nLet ${\\cal X}\\subset \\ell_2\\cap \\ell_1$ be a class of processes such that $\\sigma(x)>0$ for $x\\in{\\cal X}$ and that, for $x\\in{\\cal X}$ and $X=Z x$, for any $m>0$, the\nfunctions $X\\left(e^{i\\o}\\right)$ and $|X\\left(e^{i\\o}\\right)|^{-1}$ are differentiable in $\\omega\\in{\\bf R}$ and that\n$\\sup_{x\\in{\\cal X}}\\sup_{\\omega\\in[-\\pi,\\pi]}|dX\\left(e^{i\\o}\\right)\/d\\omega|<+\\infty$. For the purpose of the\ninvestigation of the predictability for $x$, this smoothness and assumed   without a loss generality: it is\nsufficient to replace $x$ by a faster vanishing processes with the\nsame predictability properties such that\n$x(t)\/(1+|t|^m)$, $m\\ge 2$. $\\sigma=\\min_{\\omega\\in[-\\pi,\\pi]}|X\\left(e^{i\\o}\\right)|>0$\n\n  We want to\n represent each $x\\in {\\cal X}$ as\n\\begin{eqnarray*}\nx=y_{\\scriptscriptstyle BL}+n,\n\\end{eqnarray*}\nwhere $y_{\\scriptscriptstyle BL}$ is a band-limited predictable process such that the class ${\\cal Y}=\\{y_{\\scriptscriptstyle BL}\\}_{x\\in{\\cal X}}$,\nis predictable in the sense of Lemma \\ref{lemmaPred}. In this case, each\n  $n= x-y_{\\scriptscriptstyle BL}$ is a non-predictable (random) noise.\n\n  We suggest the following  restrictions on the choice of $y_{\\scriptscriptstyle BL}$:\n  \\begin{enumerate}\n  \\item  \\begin{eqnarray}\n\\|X\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}=\\|Y_{\\scriptscriptstyle BL}\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}+\\|N\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)},\\quad d=1,+\\infty, \\label{N}\\end{eqnarray}\n where $Y_{\\scriptscriptstyle BL}={\\cal Z} y_{\\scriptscriptstyle BL}$ and  $N={\\cal Z} n'$.\n  \\item\n  $n$ does not allow a similar representation $n=y'_{\\scriptscriptstyle BL}+n'$, with a\nnon-random (predictable)   non-zero $y'_{\\scriptscriptstyle BL}$ such that \\begin{eqnarray*}\n\\|N\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}=\\|Y'\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}+\\|N'\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)},\\quad d=1,+\\infty, \\label{N'}\\end{eqnarray*}\nwhere $Y'_{\\scriptscriptstyle BL}={\\cal Z} y'_{\\scriptscriptstyle BL}$ and  $N'={\\cal Z} n'$.\n  \\end{enumerate}\nIt appears that  $n$ featuring these properties exists in some case\n and can be derived\nexplicitly from $X$. Let us show this.\n\n\n\nLet $\\o_0\\in(-\\pi,\\pi]$ be such\nthat $|X\\left(e^{i\\omega_I }\\right)| =\\sigma$, and\nlet  \\begin{eqnarray} \\gamma\\left(e^{i\\o}\\right)=\\frac{\\sigma(x)}{|X\\left(e^{i\\o}\\right)|},\\quad Y\\left(e^{i\\o}\\right)=[1-\\gamma\\left(e^{i\\o}\\right)]X\\left(e^{i\\o}\\right),\\quad\nN\\left(e^{i\\o}\\right)=\\gamma\\left(e^{i\\o}\\right) X\\left(e^{i\\o}\\right). \\label{YN}\\end{eqnarray}  Clearly,\n\\begin{eqnarray*}\nX=Y+N, \\quad Y\\left(e^{i\\omega_I }\\right)=0,\\quad |N\\left(e^{i\\o}\\right)|\\equiv\\sigma(x),\n\\end{eqnarray*}\nand\n  (\\ref{N}) holds with $d=1$ and $d=\\infty$.   By continuity  of  $X\\left(e^{i\\o}\\right)$ and\n$|X\\left(e^{i\\o}\\right)|^{-1}$, the function $Y\\left(e^{i\\o}\\right)$ is also continuous $\\omega$.\n\nIf $Y\\left(e^{i\\o}\\right)$ vanishes  fast enough when $\\omega\\to\\o_0$ (see \\cite{D12a}), then $y={\\cal Z}^{-1}Y$ is predictable; in this case, the set\n$\\{n\\}_{x\\in{\\cal X}}$ can be considered as the set of pathwise noises; therefore, this gives  a quantification  $n$ as  a norm of $n$ or $N$, such as\n\\begin{eqnarray}\n\\|N\\left(e^{i\\o}\\right)\\|_{L_1(-\\pi,\\pi)}=\\sigma(x).\n\\label{noise}\n\\end{eqnarray}\n\nHowever, it would be too restrictive to require  that the set ${\\cal X}$ is such that (\\ref{YN}) leads to $Y\\left(e^{i\\o}\\right)$ that vanishes so fast as $\\omega\\to\\o_0$\nthat $y_{\\scriptscriptstyle BL}$ is predictable. To overcome this, we suggest to replace (\\ref{YN}) by\n\\begin{eqnarray} &&\\gamma_\\varepsilon\\left(e^{i\\o}\\right)=1\\quad \\hbox{if}\\quad |e^{i\\omega}-e^{i\\o_0}|\\le \\varepsilon,\\nonumber\\\\\n&&\\gamma_\\varepsilon\\left(e^{i\\o}\\right)=\\frac{\\sigma(x)}{|X\\left(e^{i\\o}\\right)|}\\quad\\hbox{if}\\quad |e^{i\\omega}-e^{i\\o_0}|> \\varepsilon,\\quad\n\\nonumber \\\\&& Y_\\varepsilon\\left(e^{i\\o}\\right)=[1-\\gamma\\left(e^{i\\o}\\right)]X\\left(e^{i\\o}\\right),\\qquad\nN_\\varepsilon\\left(e^{i\\o}\\right)=\\gamma_\\varepsilon\\left(e^{i\\o}\\right) X\\left(e^{i\\o}\\right),\n\\label{g}\\end{eqnarray}\nwhere $\\varepsilon\\to 0$.  In this case,\n\\begin{eqnarray}\nx=y_\\varepsilon+n_\\varepsilon, \\quad y_\\varepsilon={\\cal Z}^{-1}Y_\\varepsilon\\in \\ell_2^{\\scriptscriptstyle BL}(I_\\varepsilon),\\quad n_\\varepsilon={\\cal Z}^{-1}N_\\varepsilon,  \\label{Yg}\\end{eqnarray}\nwhere $I_\\varepsilon=\\{\\omega:\\ |e^{i\\omega}-e^{i\\o_0}|\\le \\varepsilon\\}$,\n\\begin{eqnarray}  &&|N_\\varepsilon\\left(e^{i\\o}\\right)|= X\\left(e^{i\\omega}\\right),\\quad \\hbox{if}\\quad \\omega\\in I_\\varepsilon,\\nonumber\\\\\n&&|N_\\varepsilon\\left(e^{i\\o}\\right)|=|X\\left(e^{i\\o_0}\\right)|={\\rm const\\,}\\quad\\hbox{if}\\quad  \\omega\\notin I_\\varepsilon,\\quad\n\\end{eqnarray}\nWe regard $n_\\varepsilon$ as approximation of the noise as $\\varepsilon\\to 0+$.\n\nTo justify this description of   the noise,  we have to show that the set of band-limited processes  $\\{y_\\varepsilon\\}$ in (\\ref{g})-(\\ref{Yg}) is predictable in some sense. Theorem \\ref{lemmaPred}(i)-(ii) does not ensure predicability of this set, since it requires to know  the values $\\o_0$.\nThis would require to know $\\omega_{I_\\varepsilon}$, which is inconsistent   with the notion of predictability. However, Theorem \\ref{ThO}\nensures sufficient estimation of $I_\\varepsilon$  and $\\omega_{I_\\varepsilon}$ based on observations of $\\{x(t)\\}_{t\\le \\tau}$; we can take  select  $\\omega_{I_\\varepsilon}=\\widehat\\omega_{c}-\\pi$ if $\\widehat\\omega_c\\in (0,\\pi]$, and\n $\\omega_{I_\\varepsilon}=\\widehat\\omega_{c}+\\pi$ if $\\widehat\\omega_c\\in (-\\pi,0]$,  in the notations of Theorem \\ref{ThO}.\n This leads to the following two step procedure:\nthe  set $\\{x(s)\\}_{\\tau<s< t}$ is used for prediction of $x(t)$, and the set $\\{x(s)\\}_{s\\le \\tau}$ is used for the\nestimation of $\\widehat\\omega_{I_\\varepsilon}\\approx \\omega_{I_\\varepsilon}$. This allows to satisfy conditions\nof  Theorem \\ref{lemmaPred}(iii).\n\n\nTherefore, the set of band-limited processes  $\\{y_\\varepsilon\\}$ in (\\ref{g})-(\\ref{Yg}) is predictable in the sense of Theorem \\ref{lemmaPred}(iii). This  predictability  covers times $t\\ge 1$ only; it is a weaker version of predictability comparing with the one described in Lemma \\ref{lemmaPred}(i)-(ii).\n\nSince a norm for $N_\\varepsilon$ is approaching the norm for $N$, the norm of $N$ can be used for quantification\n of the\nrandomness of the two-sided sequences.\n\n\\index{It can be noted that the ``noise\" $n={\\cal Z}^{-1}N$ does not have to be a\nparticularly irregular process. For instance, if $X\\left(e^{i\\o}\\right)\\equiv 1$ then\n$Y\\left(e^{i\\o}\\right)\\equiv 0$ and $N\\left(e^{i\\o}\\right)\\equiv X\\left(e^{i\\o}\\right)\\equiv 1$, and the process\n$n={\\cal Z}^{-1}N$ does not look particularly irregular. However, as was\nmentioned above, this process in not allowed in typical classes  of\npredictable processes covered by Definition 1.}\n\nThe process  $y_\\varepsilon={\\cal Z}^{-1}Y_\\varepsilon$ can also be interpreted as an output of a\nsmoothing filter.\n\nThis support the choice of the value $\\sigma(x)$ for the quantification of $x$.\n\\subsection{Randomness as a measure of recoverability}\n\nBy recoverability,  we mean a possibility of constructing a  linear  recovering\noperator as described in the definition below.\n\nNote that $X\\left(e^{i\\o}\\right)$ is continuous in $\\omega$ for $x\\in\\ell_1$, $X={\\cal Z} x$.\n\nLet   $\\o_0\\in (0,\\pi]$ be given. For $\\sigma\\ge 0$,  let\n${\\cal X}_\\sigma=\\{x\\in\\ell_1:\\ \\min_{\\omega\\in(-\\pi,\\pi]} |X\\left(e^{i\\o}\\right)|=|X\\left(e^{i\\o_0 }\\right)|=\\sigma\\}$.\n\n\n\nFor $m\\in{\\mathbb{Z}}$, assume that the value $x(m)$ is not observable for $x\\in\\ell_1$ and that all other values of $x$\nare observable. We consider recovering problem for $x(m)$ as finding an estimate\n$\\widetilde x(m)=F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, where $F:\\ell_1(-\\infty,m-1)\\times \\ell_1(m+1,+\\infty)\\to{\\bf R}$ is some mapping.\n\n\\begin{theorem} \\label{Threc} For any  estimator $\\widetilde x(m)=F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, where $F:\\ell_1(-\\infty,m-1)\\times \\ell_1(m+1,+\\infty)\\to{\\bf R}$ is some mapping,\nwe have that\n\\begin{eqnarray}\n\\sup_{x\\in{\\cal X}_\\sigma}|\\widetilde x(m)-x(m)|\\ge \\sigma.\n\\label{optrec}\\end{eqnarray}\nIn addition, there exists an optimal estimator $\\widehat x(m)=\\widehat F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, where $\\widehat F:\\ell_1(-\\infty,m-1)\\times \\ell_1(m+1,+\\infty)\\to{\\bf R}$ is some mapping, such that\n\\begin{eqnarray}\n\\sup_{x\\in{\\cal X}_\\sigma}|\\widehat x(m)-x(m)|=\\sigma.\n\\label{opt}\n\\end{eqnarray}\n\\end{theorem}\n\n\\par \\def\\sigma{\\sigma}\n\nThis supports again  the choice of the value $\\sigma(x)$ for the quantification of the randomness for $x$.\n\n\n\\index{In  some numerical experiments, we calculated  the the normalized size of the noise, i.e. the value \\begin{eqnarray}\n\\frac{2\\pi\\sigma(x)}{\\|X\\left(e^{i\\o}\\right)\\|_{L_1(-\\pi,\\pi)}}.\\label{noise2}\\end{eqnarray}\nfor some historical financial time series (returns of stock prices). The value of\n(\\ref{noise2}) was found  to be in the range of 0.05-0.6 for\ndifferent stocks.\n}\n\n\\index{; this  was consistent with the volatility range for the prices.}\n\\index{In another experiment,  we matched value (\\ref{noise2}) with\nthe parameter $\\sigma$ used for simulated autoregressions\n$x(t+1)=x(t)+{\\rm const\\,}\\cdot  \\Delta +\\sigma \\Delta^{1\/2} \\xi(t),$ where\n$\\xi(t)$ was modelled as a discrete time white noise with ${\\bf E}\n\\xi(t)=0$, ${\\rm Var\\,} \\xi(t)^2=1$. These processes are used for financial\nmodelling; for instance,  the process $x(t)$ with $\\Delta=1\/250$ can\nrepresent daily returns for a stock with the price volatility $\\sigma$.\nThe results were quite interesting; more experiments are planned.}\n\n\\section{Separating the noise for one-sided sequences}\\label{Sec1Sided}\nUnfortunately, representation (\\ref{xxy}) does not lead toward a solution of  the predictability problem, since it would require to know the entire sequence $\\{x(t)\\}_{t=-\\infty}^{+\\infty}$ to calculate $X={\\cal Z} x$.\n\nOn the other hand, it is natural to  use one-sided sequences interpreted as available past observations  for predictability problems.\nFor this, we have to use the notion of left  bandlimitness for\none-sided sequences. We will use a modification of representation (\\ref{xxy}) that was stated for two-sided sequences.\n\nFor this, we have to of representation   to two-sided sequences. We suggest to replace   the \"ideal\" projections  $\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I} {\\cal Z} x)\\in \\ell_2$ for $x\\in\\ell_2$ and\n$y_{\\scriptscriptstyle BL}=x-\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I^c} {\\cal Z} x)$  by their \"optimal\" one-sided substitutes.\n\\subsection*{Uniqueness of the extrapolation for left band-limited processes}\n\\begin{lemma}\\label{propU} For any $I\\in{\\cal J}$ and\n  any $x\\in\\ell_2^{-,\\LBL}(I) $, there exists an\nunique  $x_{\\scriptscriptstyle BL}\\in\\ell_2^{\\scriptscriptstyle BL}(I)$ such that $x(t)=x_{{\\scriptscriptstyle BL}}(t)$ for $t\\le 0$.\n\\end{lemma}\n\\par\nBy Lemma \\ref{propU}, the future values $x_{\\scriptscriptstyle BL}(t)|_{t>0}$ of a\nband-limited process $x_{\\scriptscriptstyle BL}$,\n are uniquely defined by the trace\n$x_{\\scriptscriptstyle BL}(t)|_{t\\le 0}$.\n  This statement represent a reformulation in the deterministic setting\nof  the classical Szeg\\\"o-Kolmogorov Theorem for stationary Gaussian processes\n\\cite{Ko65,Sz,Sz1,V}.\n\n\\subsection*{Existence  of optimal\nband-limited approximation} Let $x\\in\\ell_2^-$ be a semi-infinite one-sided sequence representing available\nhistorical data, and let $I\\in{\\cal J}$.\n\\begin{theorem}\\label{Th1} There exists an unique optimal solution  $\\widehat x$\nof the minimization problem \\begin{eqnarray} &&\\hbox{Minimize}\\quad  \\sum_{t=-\\infty}^0|\\widehat\nx(t)-x(t)|^2  \\quad\\hbox{over}\\quad \\widehat x\\in \\ell_2^{-,\\LBL}(I) .\\label{min} \\end{eqnarray}\n\\end{theorem}\n\\par\nBy  Lemma \\ref{lemmaPred},  there exists a unique band-limited process  $x_{{\\scriptscriptstyle BL}}\\in\\ell_2^{\\scriptscriptstyle BL} (I)$\nsuch that $\\widehat x(t)|_{t\\le 0}= x_{{\\scriptscriptstyle BL}}(t)|_{t\\le 0}$. This offers a natural way  to extrapolate a left\nband-limited solution $\\widehat x\\in \\ell_2^-$ of problem (\\ref{min}) on the future\ntimes  $t>0$.\n\n\\index{It can\nbe interpreted  as the optimal forecast (optimal given $\\Omega$ and\n$N$).}\n\\subsubsection*{The optimal solution}\nLet $I\\in{\\cal J}$ be given, and let ${\\rm mes\\,}(I)=2\\Omega$ for some $\\Omega\\in (0,\\pi)$.\n\n\nLet $I_0=(-\\Omega,\\Omega)$, i.e.,  $\\omega_{I_0}=0$.\n\nFor $\\omega\\in[-\\pi,\\pi)$, let the operator $p_{\\omega}: \\ell_2^-\\to\\ell_2^-$ be defined as $\\bar x(t)=e^{i\\omega  t}x(t)$ for $\\bar x=p_\\omega x$.\n\n\nLet the operator ${\\cal Q}: {\\ell_2}\\to \\ell_2^{-,\\LBL}(I_0) $ be defined as $\\widehat x={\\cal Q} y={\\cal Z}^{-1}\\widehat X$, where\n\\begin{eqnarray}\n\\widehat X\\left(e^{i\\o}\\right) =\\sum_{k\\in\n{\\mathbb{Z}}}y_ke^{ik\\omega\\pi\/\\Omega}{\\mathbb{I}}_{\\{|\\omega|\\le\\Omega\\}},\n\\label{wX}\\end{eqnarray}  for the corresponding  $y=\\{y_k\\}\\in {\\ell_2}$.\nSimilarly to the classical sinc representation, we obtain that \\begin{eqnarray} \\widehat x(t)=\\frac{1}{2\\pi}\n\\int_{-\\Omega}^{\\Omega}\\left(\\sum_{k\\in {\\mathbb{Z}}}y_k e^{ik\\omega\\pi\/\\Omega}\\right)e^{i\\omega\nt}d\\omega\\nonumber\\\\ }\\def{\\nonumber\\\\&&} % Must be in the body after \\begin{documnet=\\frac{1}{2\\pi}\n\\sum_{k\\in {\\mathbb{Z}}}y_k\\int_{-\\Omega}^{\\Omega}e^{ik\\omega\\pi\/\\Omega+i\\omega t}d\\omega\\nonumber\\\\\n=\\frac{1}{2\\pi}\\sum_{k\\in {\\mathbb{Z}}}y_k \\frac{e^{ik\\pi+i\\Omega t}-\ne^{-ik\\pi-i\\Omega t}}{ik\\pi\/\\Omega+it}\\nonumber\\\\ }\\def{\\nonumber\\\\&&} % Must be in the body after \\begin{documnet=\\frac{\\Omega}{\\pi}\\sum_{k\\in\n{\\mathbb{Z}}_N }y_k {\\rm sinc\\,}(k\\pi+\\Omega t)=({\\cal Q} y)(t).\\label{sinc}\\end{eqnarray}\nIt follows that the ${\\cal Q}: {\\ell_2}\\to \\ell_2^{-,\\LBL}(I_0) $ is actually defined as\n\\begin{eqnarray*} \\widehat x(t)=({\\cal Q} y)(t)=\\frac{\\Omega}{\\pi}\\sum_{k\\in\n{\\mathbb{Z}}}y_k {\\rm sinc\\,}(k\\pi+\\Omega t).\\label{Qs}\\end{eqnarray*}  Consider the operator  ${\\cal Q}^*:\\ell_2^{-,\\LBL}(I_0) \\to {\\ell_2}$  being adjoint to the operator\n${\\cal Q}:{\\ell_2}\\to\\ell_2^{-,\\LBL}(I_0)$, i.e., such that\n\\begin{eqnarray}\n({\\cal Q}^*x)_k=\\frac{\\Omega}{\\pi}\\sum_{t\\in{\\cal T}}{\\rm sinc\\,}(k\\pi+\\Omega t)x(t).\n\\label{Q*}\\end{eqnarray}\n\nConsider a\nlinear bounded non-negatively defined  Hermitian operator $R:{\\ell_2}\\to {\\ell_2}$  defined as\n\\begin{eqnarray*}\nR={\\cal Q}^*{\\cal Q}.\n\\end{eqnarray*}\nConsider operator $P_I=p_{\\omega_I}{\\cal Q} R^{-1} {\\cal Q}^* p_{-\\omega_I}:\\ell_2\\to\\ell_2^{-,{\\scriptscriptstyle LBL}}(I)$.\n\\begin{theorem}\n\\label{ThP}\n\\begin{itemize}\\item[(i)] The operator $R:{\\ell_2}\\to{\\ell_2}$ has a bounded inverse\n operator $R^{-1}:{\\ell_2}\\to{\\ell_2}$.\n\\item[(ii)] Problem (\\ref{min}) has a unique solution\n \\begin{eqnarray}\n\\widehat x=P_Ix.\\label{wx}\n\\end{eqnarray}\n\\end{itemize}\n\\end{theorem}\n\\begin{theorem}\\label{Th1n} For any $I\\in{\\cal J}$, there exists $n_I\\in \\ell_2^-$ such that $P_In_I=0$ and $n_I\\neq 0$.\n\\end{theorem}\nThe processes $n_I$ can be considered as the noise component with respect to smooth processes with the spectrum on $I$, for a  given $I\\in{\\cal J}$.\n\\begin{corollary}\n\\label{corrxx}\nA process $x\\in\\ell_2^-$ is left-bandlimited with the spectrum $I$ if and only if $x=p_{\\omega_I}{\\cal Q} R^{-1} {\\cal Q}^* p_{-\\omega_I}x$.\n\\end{corollary}\n\\begin{remark}\nIt can be noted that $\\widehat x=p_{\\omega_I}{\\cal Q} {\\cal Q}^+ p_{-\\omega_I}x$, where ${\\cal Q}^+=R^{-1} {\\cal Q}^*:\\ell_2^-\\to{\\ell_2}$ is a Moore--Penrose pseudoinverse of the operator ${\\cal Q}:{\\ell_2}\\to\\ell_2^-$.\n\\end{remark}\nLet us elaborate equation (\\ref{wx}).  The optimal  process $\\widehat x$ can be expressed as  \\begin{eqnarray*} \\widehat x(t)=e^{i\\omega_I t}\\frac{\\Omega}{\\pi}\\sum_{k\\in {\\mathbb{Z}} }\\widehat y_k {\\rm sinc\\,}(k\\pi+\\Omega t). \\label{wxx}\\end{eqnarray*}\nHere $\\widehat y=\\{\\widehat\ny_k\\}_{k\\in{\\mathbb{Z}}}$ is defined as    \\begin{eqnarray} \\widehat y=R^{-1}{\\cal Q} p_{-\\omega_I} x.\\label{wy}\\end{eqnarray} The operator $R$ can be represented via a matrix\n$R=\\{R_{km}\\}$, where $k,m\\in{\\mathbb{Z}}$.  In this\nsetting, $(Ry)_k=\\sum_{k=-\\infty}^\\infty R_{km}y_m$, and  the components of the matrix $R$  are defined as  \\begin{eqnarray*} R_{km}=\n\\frac{\\Omega^2}{\\pi^2}\\sum_{j=-\\infty}^0{\\rm sinc\\,}(m\\pi+\\Omega j)\\,{\\rm sinc\\,}(k\\pi+\\Omega j)\n.\\label{R}\\end{eqnarray*}\nRespectively, the components of the vector  ${\\cal Q}^*x=\\{({\\cal Q}^*x)_k\\}_{k\\in{\\mathbb{Z}}}$   are defined as\n\\begin{eqnarray} ({\\cal Q}^*x)_{k}= \\frac{\\Omega}{\\pi} \\sum_{j=-\\infty}^0{\\rm sinc\\,}(k\\pi+\\Omega j)x(j) .\n\\label{r}\\end{eqnarray}\n\n\n\\subsection{A multi-step procedure  for one-sided sequences}\\label{subsecMS}\nUnfortunately, the approach described in Section \\ref{Sec2Sided} does not lead toward a solution of  the predictability problem, since it would require to know the entire sequence $\\{x(t)\\}_{t=-\\infty}^{+\\infty}$ to calculate\n$X={\\cal Z} x$ and quantitative characteristics suggested in Section \\ref{Sec2Sided}.\n\nOn the other hand, it is natural to  use one-sided sequences interpreted as available past observations  for predictability problems.\nIn this case, we have to use the notion of left  bandlimitness for\none-sided sequences. We will use a modification of representation (\\ref{xxy}) that was stated for two-sided sequences.\n\nFor this, we suggest to replace   the \"ideal\" projections  $\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I} {\\cal Z} x)\\in \\ell_2$ for $x\\in\\ell_2$\nby their \"optimal\" one-sided substitutes\n$\\widehat x=P_I x\\in\\ell_2^-$;   this substitution is  optimal on $\\{t\\le 0\\}$ in the sense of optimization problem (\\ref{min}).\nUnfortunately,  it may happen that\n\\begin{eqnarray*}\nx-\\widehat x\\notin \\ell_2^{-,{\\scriptscriptstyle LBL}}(I^c).\n\\end{eqnarray*}\n\n\nFor this, we suggest to replace   the \"ideal\" projections  $\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I} {\\cal Z} x)\\in \\ell_2$ for $x\\in\\ell_2$ and\n$y_{\\scriptscriptstyle BL}=x-\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I^c} {\\cal Z} x)$  by their \"optimal\" one-sided substitutes\n$\\widehat x=P_I x\\in\\ell_2^-$ and $\\widehat y=P_{I^c} (x-\\widehat x)\\in\\ell_2^-$;   this substitution is  optimal on $\\{t\\le 0\\}$ in the sense of optimization problem (\\ref{min}).\nUnfortunately, \\index{an analog of the connection between  (\\ref{1}) and (\\ref{2}) is not valid for onesided sequences,\ni.e.,} it may happen that\n\\begin{eqnarray*}\n\\widehat y=P_{I^c} (x-\\widehat x)\\notin \\ell_2^{-,{\\scriptscriptstyle LBL}}(I^c).\n\\end{eqnarray*}\n\nWe suggest a  multi-step procedure that to deal with this complication.\n\n\n\nAssume that we observe a semi-infinite one-sided sequence $\\{x(t)\\}_{t\\le 0}\\in \\ell_2^-$.\n\nConsider a sequence of sets $\\{I_k\\}_{k=0,1,2,..}\\subset{\\cal J}$, with the corresponding middle\npoints $\\omega_k\\in I_k$.\nFurther, let us consider the following  sequences of elements of  $\\ell_2^-$:\n\\begin{itemize}\n\\item Set \\begin{eqnarray*}\nx_0=x,\\qquad \\widehat x_0=P_{I_0}x_0,\\qquad y_0=x_0-\\widehat x_0,\\qquad \\widehat y_0=P_{I_0^c}y_0,\\quad x_1=y_0-\\widehat y_0.\n\\end{eqnarray*}\n\\item\nFor $k\\ge 1$, set\n \\begin{eqnarray*}\n\\widehat x_k=P_{I_k}x_k,\\qquad y_k=x_k-\\widehat x_k,\\qquad \\widehat y_{k}=P_{I_k^c}y_k,\\quad x_{k+1}=y_k-\\widehat y_k.\n\\end{eqnarray*}\n\\end{itemize}\nThe following lemma will be useful.\n\\begin{lemma}\\label{lemmaN}  For any $I\\in{\\cal J}$ and $x\\in\\ell_2^-$, the following holds:\n\\begin{enumerate}\n\\item $\\|x\\|\\ge \\|x-P_{I}x\\|$, and  \\item The equality in (i) holds if and only if $P_Ix=0$.\n\\end{enumerate}\n\\end{lemma}\n\\subsubsection*{Stopping upon arriving at a predictable process}\nIf there exists $k\\ge 0$ such that $y_k=0$ then\n\\begin{eqnarray}\nx=\\widehat x_0+y_0=\\widehat x_0+\\widehat y_0+x_1=\\widehat x_0+\\widehat y_0+\\widehat x_1 +y_1=...=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+...  +\\widehat x_k.\n\\label{xx}\\end{eqnarray}\nThis means that $x$ is a finite sum of left band-limited processes. These processes were calculated by the observer,\nand, in this sense, each of them can be deemed to be observed, with known (pres-selected) $I_k$;\nin particular, $x$ can be predicted without error. Similarly, if there exists $k\\ge 0 $ that  $x_{k+1}=0$, then\n\\begin{eqnarray}\nx=\\widehat x_0+y_0=\\widehat x_0+\\widehat y_0+x_1=\\widehat x_0+\\widehat y_0+\\widehat x_1 +y_1=...=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+...  +\\widehat y_k.\n\\label{xy}\\end{eqnarray}\n This means that $x$ again\n is a finite sum of observed  left band-limited processes. Again,  $x$ can be predicted without error. \\par\nThe norms $\\|\\eta_k\\|_{\\ell_2^-}$ and $\\|\\bar \\eta_k\\|_{\\ell_2^-}$ can be used for quantification of the\nrandomness of one-sided semi-infinite sequences.\n \\subsubsection*{The case of never stopping procedure}\n  It may happen that, for any $N>0$, there exists $k\\ge N$ such that either\n$\\|y_k\\|_{\\ell_2^-}+\\|x_k\\|_{\\ell_2^-}>0$. In this, the randomness can be quantified\nas\n\\begin{eqnarray*}\n\\max\\left(\\limsup_{k\\to +\\infty}\\|x_k\\|_{\\ell_2^-},\\limsup_{k\\to +\\infty}\\|y_k\\|_{\\ell_2^-} \\right).\n\\end{eqnarray*}\n \\subsubsection*{Arrival at a non-reducible noise}\n A process $x\\in\\ell_2^-$ is either left band-limited or not band-limited. Therefore, some processes cannot\n be represented as a finite sum of left bandlimited processes such as (\\ref{xx}) or (\\ref{xy}) with a finite $k$.\n In this case, the procedure will not be stopped according to the rule described above. It could be  beneficial\n to stop procedure using the following  rule.\n\\par\n Let \\begin{eqnarray*}\n\\delta_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=} \\|x_k\\|_{\\ell_2^-}-\\|x_k-\\widehat x_k\\|_{\\ell_2^-},\\qquad \\bar\\delta_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=} \\|y_k\\|_{\\ell_2^-}-\\|y_{k}-\\widehat y_k\\|_{\\ell_2^-},\n\\end{eqnarray*}\ni.e.,\n$\\delta_k=\\|x_k\\|_{\\ell_2^-}-\\|y_k\\|_{\\ell_2^-}$,  $\\bar\\delta_k=\\|y_k\\|_{\\ell_2^-}-\\|x_{k+1}\\|_{\\ell_2^-}$,\n \\begin{eqnarray*}\n\\|x_{k}\\|_{\\ell_2^-}=\\|y_k\\|_{\\ell_2^-}+\\delta_k=\\|x_{k+1}\\|_{\\ell_2^-}+\\delta_k+\\bar\\delta_k, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\|y_{k}\\|_{\\ell_2^-}=\\|x_{k+1}\\|_{\\ell_2^-}+\\bar\\delta_k=\\|y_{k+1}\\|_{\\ell_2^-}+\\delta_k+\\bar\\delta_k, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\n By Lemma \\ref{lemmaN},\n it follows that $\\delta_k\\ge 0$ and  $\\bar\\delta_k\\ge 0$ for all $k$, i.e., \\begin{eqnarray*}\n\\|x_{k}\\|_{\\ell_2^-}\\ge \\|y_k\\|_{\\ell_2^-}\\ge \\|x_{k+1}\\|_{\\ell_2^-}, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\nBy Theorem \\ref{Th1n}, it may happen that $\\delta_k=0$, i.e., $\\|x_{k}\\|_{\\ell_2^-}=\\|y_k\\|_{\\ell_2^-}$.\nTo save the resources, the procedure should be stopped when this occurs, since further steps will not improve the result.\nOn this step, $x$ is presented as\n\\begin{eqnarray*}\nx=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat x_k+ y_k=x_{p}^{(k)} + \\eta_k.\n\\end{eqnarray*}\nwhere $x_{p}^{(k)}=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+...+\\widehat x_k$ is a predictable  process since it is a finite sum of observed\nleft band-limited processes, and $\\eta_k=y_k$ is a noise. Given the selected set $\\{I_k\\}$, further reduction\nof the norm of this noise is impossible. Hence  we can call $y_k$\na non-reducible noise.\n\nSimilarly, it may happen that  $\\bar\\delta_k=0$ and $\\delta_k>0$, i.e. $\\|y_{k}\\|_{\\ell_2^-}=\\|x_{k+1}\\|_{\\ell_2^-}$.\nAgain, the procedure should be stopped when this occurs, since further steps will not improve the result.  This means that the procedure have\nto stop on the step where $x$ is presented as\n\\begin{eqnarray*}\nx=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat y_k+ x_{k+1}=y_{{\\scriptscriptstyle BL}}^{(k)} +\\bar\\eta_k.\n\\end{eqnarray*}\nHere $y_{p}^{(k)}=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat y_k$ is a predictable process again, and $\\bar\\eta_k=x_{k+1}$\nis a   non-reducible noise again.\n\n\\section{Proofs}\\label{SceP}\n For the case where $I_0=(-\\Omega,\\Omega)$, i.e.\n $\\omega_I=0$, the proofs of Theorem \\ref{lemmaPred}, Lemma \\ref{propU} and  Theorems \\ref{Th1}-\\ref{ThP},  can be found in \\cite{D12a}. Let us extend these proofs on case where $\\omega_I\\neq 0$.\n\nLet us observe that $x\\in \\ell_2^{-,\\LBL}(I)$ and $X={\\cal Z} x\\in {\\mathbb{B}}_\\infty$ if and only if $x_0\\stackrel{{\\scriptscriptstyle \\Delta}}{=} p_{-\\omega_I}x\\in \\ell_2^{-,\\LBL}(I_0)$ and\n$X_0\\stackrel{{\\scriptscriptstyle \\Delta}}{=} {\\cal Z} x_0\\in {\\X}(I_0)$. In this case, $x=p_{\\omega_I}x_0$, and\n \\begin{eqnarray*} X\\left(e^{i\\o}\\right)=\\sum_{t=-\\infty}^{\\infty}x(t)e^{-i\\omega t}=\\sum_{t=-\\infty}^{\\infty}x_0(t)e^{i\\omega_I t}e^{-i\\omega t}= X_0\\left(e^{i(\\omega_I -\\omega )t} \\right), \\quad\n\\omega\\in[0,2\\pi). \\end{eqnarray*}\nThen the proof of Theorem \\ref{lemmaPred}(i)-(ii) and Lemma \\ref{propU} follows.\n\n\\par\n {\\em Proof of Theorem \\ref{lemmaPred}} (iii) follows from the robustness of the predictor used in \\cite{D12a} with respect to truncation\nof inputs from $\\ell_2$. $\\Box$\n\nFurther, we have that\n \\begin{eqnarray*}  \\|\\widehat\nx-x\\|_{\\ell_2^-}=\\|p_{-\\omega_I}\\widehat\nx-p_{-\\omega_I} x\\|_{\\ell_2^-}  \\quad\\hbox{for any}\\quad \\widehat x, x\\in \\ell_2^-.\\label{minn} \\end{eqnarray*}\nHence the problem\n\\begin{eqnarray} &&\\hbox{Minimize}\\quad \\|p_{-\\omega_I}\\widehat\nx-p_{-\\omega_I} x\\|_{\\ell_2^-} \\quad\\hbox{over}\\quad \\widehat x\\in \\ell_2^{-,\\LBL}(I) \\label{minnn} \\end{eqnarray}\nhas the  same sets of solution as problem (\\ref{min}). Therefore, there is a bijection\nbetween the sets  of optimal solutions for\nproblem (\\ref{min}) and for the problem\n\\begin{eqnarray} &&\\hbox{Minimize}\\quad \\|\\widehat\ny-y\\|_{\\ell_2^-} \\quad\\hbox{over}\\quad \\widehat y\\in \\ell_2^{-,\\LBL}(I_0),\\label{minnnn} \\end{eqnarray}\nwhere $y= p_{-\\omega_I} x $. This bijection has the form  $\\widehat y= p_{-\\omega_I}\\widehat x $.\nTherefore, the proof for $\\omega_I\\neq 0$ follows from the proof for $\\omega_I=0$ from \\cite{D12a}.\nThen the proof of Theorem \\ref{Th1} and Theorem \\ref{ThP} follows. $\\Box$\n\n{\\em Proof of\nTheorem \\ref{ThO}}. \\index{Let $m$ the entire part of the number $2\\pi\/\\widehat\\nu +1$, and let $\\omega_k=-\\pi+2k\\pi\/m $, $k=1,...,m$.}\nIt is easy to see that there exists a finite set $\\{I_k\\}_{k=1}^M\\subset {\\cal J}$, $M<+\\infty$, such that  ${\\rm mes\\,}(I_k)\\le \\nu\/3$,\n$\\cup_{k=1}^M  I_k=(0,2\\pi]$, and that the intersections of two different $ I_k$ cannot contain two or more elements.\nLet $\\widehat I_k=(-\\pi,\\pi]\\setminus I_k$.\n\nLet $P_I$ be operators such as defined in Section \\ref{Sec1Sided}, with rather technical  adjustment: we assume that the\nset of times\n $\\{t\\le 0\\}$ in Theorem \\ref{Th1} is replaced by  the $\\{t\\le \\tau\\}$,  and that $\\ell_2^-$ replaced by $\\ell_2(-\\infty,\\tau)$.\nAs is shown in Theorem \\ref{ThP}, the values $d_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\\|{\\bf P}_{\\widehat I_k}x-x\\|_{\\ell_2(-\\infty,\\tau)}$ for $k=1,...,M$ can be found\nbased on observations of $\\{x(t)\\}_{t\\le\\tau}$. By the assumptions on $x$, there\nexists $m$ such that\n$d_m=0$. The  set $\\widehat I=\\widehat I_m$ is such as described in the Theorem; the point $\\widehat\\omega_c$ can be defined as\nselect  $\\widehat\\omega_{c}=\\widehat\\omega_{\\widehat I}-\\pi$ if $\\widehat\\omega_{\\widehat I}\\in (0,\\pi]$, and $\\widehat\\omega_{c}=\\widehat\\omega_{\\widehat I}+\\pi$ if $\\widehat\\omega_{\\widehat I}\\in (-\\pi,0]$.\nThen the proof of Theorem \\ref{ThO} follows. $\\Box$\n $\\Box$\n\n\n\n{\\em Proof of Theorem \\ref{Threc}}.\nLet $Y\\left(e^{i\\o}\\right) = \\sum_{k\\in{\\mathbb{Z}}\\setminus\\{m\\} } e^{-i\\omega k}x(k)$, $\\omega\\in(-\\pi,\\pi]$; this function to be observable. By the definitions, it follows that\n \\begin{eqnarray*}\n X\\left(e^{i\\o}\\right) -Y\\left(e^{i\\o}\\right) -e^{-im}x(m)\\equiv 0,\\quad \\omega\\in(-\\pi,\\pi].\n \\end{eqnarray*}\nHence\n\\begin{eqnarray*}\nx(m)=-e^{ im}Y\\left(e^{\\o_0}\\right)+ e^{ im}X\\left(e^{\\o_0}\\right)=-e^{ im}Y\\left(e^{\\o_0}\\right)+\\xi,\n \\end{eqnarray*}\nwhere $\\xi=e^{ im}X\\left(e^{\\o_0}\\right)$. Hence\n\\begin{eqnarray*}\n|x(m)+e^{ im}Y\\left(e^{\\o_0}\\right)|=  |\\xi|=\\sigma.\n\\end{eqnarray*}\nLet us accept the value $\\widehat x(m)=-e^{ im}Y\\left(e^{\\o_0}\\right)$ as the estimate of the\nmissing value $x(m)$. For this estimator, the size of the recovery error is $\\sigma$ for any $x\\in{\\cal X}_\\sigma$.\nIf $\\sigma=0$ then the estimator is error-free.  In a general case where $\\sigma\\ge 0$, we have that (\\ref{opt}) holds.\n\nLet us show that  this estimator\nis optimal in the following sense:\n\\begin{eqnarray*}\n\\sigma=\\sup_{x\\in{\\cal X}_\\sigma}|\\widehat x(m)-x(m)|\\le \\sup_{x\\in{\\cal X}_\\sigma}|\\widetilde x(m)-x(m)|\n\\label{optrec2}\\end{eqnarray*}\nfor any other  estimator $\\widetilde x(m)=F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, where $F:\\ell_2(-\\infty,m-1)\\times \\ell_2(m+1,+\\infty)\\to{\\bf R}$ is some mapping.\n\n\\index{Assume that $m=0$, $X_\\pm\\left(e^{i\\o}\\right) =\\pm \\sigma$, $x_\\pm={\\cal Z}^{-1}X_\\pm$, i.e. $x_\\pm(t)=\\pm \\sigma{\\mathbb{I}}_{\\{t\\neq 0\\}}$.\nClearly, $x_\\pm\\in{\\cal X}_\\sigma$ and $\\widetilde x_-=\\widetilde x_+$ for $\\widetilde x_\\pm= F\\left(x|_{t\\in{\\cal Z},\\ t\\neq 0}\\right)$, for any mapping $F$ such as described above.\nHence\n\\begin{eqnarray*}\n \\max(|\\widetilde x_-(0)-x_-(0)|,|\\widetilde x_+(0)-x_+(0)|)\\ge \\sigma.\n\\end{eqnarray*}}\n\nLet  $m\\in {\\mathbb{Z}}$ be fixed, and let  $X_\\pm\\left(e^{i\\o}\\right) =\\pm \\sigma e^{-im \\omega}$, $x_\\pm={\\cal Z}^{-1}X_\\pm$, i.e. $x_\\pm(t)=\\pm \\sigma{\\mathbb{I}}_{\\{t=m\\}}$.\nClearly, $x_\\pm\\in{\\cal X}_\\sigma$. Moreover, we have that $\\widetilde x_-=\\widetilde x_+$ for $\\widetilde x_\\pm= F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, for any mapping $F$ such as described above.\nHence\n\\begin{eqnarray*}\n \\max(|\\widetilde x_-(m)-x_-(m)|,|\\widetilde x_+(m)-x_+(m)|)\\ge \\sigma.\n\\end{eqnarray*}\n Then (\\ref{optrec}) follows. This completes the proof of Theorem \\ref{Threc}. $\\Box$\n\n\\par\n{\\em Proof of Theorem \\ref{Th1n}}. It suffices to observe that $\\ell_2^-\\setminus {\\cal Q}(\\ell_2)\\neq \\emptyset$, for the operator ${\\cal Q}:\\ell_2\\to \\ell_2^-$, since\n${\\cal Q}(\\ell_2)=\\ell_2^{-,{\\scriptscriptstyle LBL}}$.\n Hence the kernel of the adjoint operator   ${\\cal Q}^*:\\ell_2^-\\to\\ell_2$ contains non-zero elements.\n \\index{Since $R$ is invertible, we have that\n$R^{-1}{\\cal Q}^*x^\\bot\\neq 0_{\\ell_2}$. Clearly, ${\\cal Q} y\\neq 0_{\\ell_2^-}$ if $y\\neq 0_{\\ell_2}$. Hence  ${\\cal Q} R^{-1}{\\cal Q}^*x^\\bot\\neq 0_{\\ell_2^-}}\n$\\Box$\n\n\\par\n{\\em Proof of Lemma \\ref{lemmaN}}. Statement (i) follows from the choice of $P_{I}x$ as a solution of\noptimization problem (\\ref{min}). To prove statement (ii), it suffices to show that if $\\|x\\|=\\|x-P_{I}x\\|$ then $P_Ix=0$.\nIf $\\|x\\|=\\|x-P_{I}x\\|$ then  $\\|x-0_{\\ell_2^-}\\|=\\|x-P_{I}x\\|$. Hence both sequences $0_{\\ell_2^-}$ and $\\|P_{I}x\\|$\nare solutions of problem (\\ref{min}). We proved that the solution is unique, hence $\\|P_{I}x\\|=0_{\\ell_2^-}$.\nThis completes the proof. $\\Box$\n\\section{Possible applications and future development}\nThe  approach suggested in this paper  allows many modifications. We\noutline below some possible straightforward modifications as well as\n  more challenging problems and possible applications  that we leave for the future research.\n\\begin{enumerate}\n\\item It would be interesting to investigate sensitivity of the prediction results with respect to the choice\nof\n$\\{I_k\\}$. It would be interesting to find  an optimal choice of the set $\\{I_k\\}$ such as\n    \\begin{eqnarray*}\n    \\hbox{Maximize}\\quad  \\delta_k+\\bar \\delta_k\\quad \\hbox{over}\\quad  I\\in {\\cal J}\n    \\end{eqnarray*} for $k=1,2,..,$, with some constraints on the choice of $I_k$, for example, such that ${\\rm mes\\,}(I_k)$ is given.\n\\item It could be interesting  to try another basis in $L_2(I_0)$ for expansion in (\\ref{wX}).\n\\item Optimization problem in (\\ref{min}) is based on optimal approximation in\n$L_2(I)$ for Z-transforms. This approximation in  can be replaced by approximation in a  weighted\n$L_2$-space on $I$. This leads to modification of the optimization\nproblem; the weight will represent the relative importance of the\napproximation on different frequencies.\n\\item It is unclear if an analog of property (\\ref{N}) can be obtained with $d=2$ instead of $d=1,+\\infty$.\n\\end{enumerate}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe discovery of superconformal theories (SCFTs) in six and five dimensions has been one of the most surprising results emerging from string theory in the past few decades.  There are two types of 6d SCFTs, both of which are classified in terms of singular geometries: $\\mathcal N = (2,0)$ theories \\cite{Witten:1995zh} and $\\mathcal N =(1,0)$ theories \\cite{Heckman:2013pva,Heckman:2015bfa,Bhardwaj:2015xxa}.  Given the surprising effectiveness of geometry in describing 6d SCFTs, a natural next step is to attempt to classify 5d SCFTs in terms of singular geometries.\nIn some ways, 5d SCFTs are more rigid as there is only a single type of 5d SCFT corresponding to the 5d ${\\cal N}=1$ (i.e. eight supercharges) superconformal algebra. Many examples of 5d SCFTs have been realized in string theory using brane probes \\cite{Seiberg:1996bd}, M-theory on local Calabi-Yau 3-folds \\cite{Morrison:1996xf,Douglas:1996xp,Intriligator:1997pq}, and type IIB $(p,q)$ 5-brane webs \\cite{Aharony:1997bh,Aharony:1997ju,Leung:1997tw,Bergman:2014kza}.   \n\nThe classification of 6d $\\mathcal N = (1,0)$ theories led to a picture involving generalized `quiver-like' theories whose structures could by and large be anticipated from field theoretic reasoning.\nThere are of course exceptions to this idea and explicit geometric constructions in F-theory clarified which possible exceptions arise that evade field theoretic analysis \\cite{Heckman:2015bfa,Heckman:2013pva}.  Similarly, in the 5d case, one might expect field theoretic reasoning to be a powerful, albeit incomplete guide.  Indeed, as spearheaded in \\cite{Intriligator:1997pq} it has been clear for a long time that field theoretic tools combined with the constraints of supersymmetry provide an unexpectedly powerful method for deducing the existence of interacting UV fixed points.  More recently it was found in \\cite{Jefferson:2017ahm} that relaxing some of the assumptions in \\cite{Intriligator:1997pq} can resolve the conflict between the gauge theoretic classification  described in \\cite{Intriligator:1997pq} with low energy descriptions of some known stringy constructions, leading to a set of necessary (as opposed to sufficient) conditions for a 5d gauge theory to have a UV fixed point.  However, it is unclear whether or not there are additional conditions needed to guarantee the existence of gauge theories\nas consistent 5d SCFTs.  Moreover, there are known cases in which a 5d SCFT is not a gauge theory (for example, M-theory on a local $\\mathbb P^2$ embedded in a Calabi-Yau 3-fold)\\footnote{Despite the fact that these cases do not admit a Lagrangian description, they can nevertheless be obtained from a gauge theory by passing through phases where some non-perturbative degrees of freedom become massless.}.\nA reasonable follow-up to the field theoretic approach, then, is to try to check if the necessary gauge theoretic consistency conditions described in \\cite{Jefferson:2017ahm} are in fact also sufficient, by using other string constructions to engineer the same theories.  The main aim of this paper is to use geometric constructions of 5d SCFTs, realized as M-theory compactified on local Calabi-Yau (CY) 3-fold (and cross checked with dual constructions involving ($p,q$) 5-brane webs), to devise a classification scheme for 5d SCFTs. As a byproduct of our efforts, we are led to either validate or exclude various candidate 5d SCFTs predicted by the perturbative gauge theoretic analysis.\n\nThe basic mathematical setup leading to 5d SCFTs from M-theory on CY 3-folds involves studying how all compact 4-cycles (compact complex surfaces) inside a non-compact 3-fold can be shrunk to a point at a finite distance in moduli space; we call CY 3-folds engineering 5d SCFTs in this manner `shrinkable' 3-folds. This geometric picture can be schematically represented by a graph whose nodes are 4-cycles (surfaces) and whose edges denote the resulting intersecting 2-cycles (curves).\nWe note that a systematic study of the consistency conditions needed to construct such geometries has not been undertaken in the mathematics literature.  Starting from a collapsed set of 4-cycles, the condition that one can resolve the singularities and thereby bring the 4-cycles to finite volume restricts the admissible types of K\\\"ahler surfaces (i.e.\\ the nodes of the graph). We call the number of nodes of such a graph the \\emph{rank} of the 5d SCFT.  In particular, we show that the nodes of the graph must be rational or ruled surfaces (possibly blown up at a positive number of points)\\footnote{Rational and ruled surfaces are equivalent to (respectively) $\\mathbb P^2$ and ruled surfaces over genus $g$ curves (which we argue can be restricted to $g=0$)---see Section~\\ref{sec:gtrans} for additional details.} in the rank 2 case, and further conjecture this to be true for arbitrary rank.\nThe Calabi-Yau condition and the requirement of positive volumes place further restrictions on the allowed intersections of the surfaces (i.e.\\ the edges of the graph; see Figure \\ref{fig:graph}).  We thus devise a set of necessary critieria which must be satisfied for a 3-fold to engineer a 5d SCFT and conjecture that these criteria are sufficient to guarantee the existence of a 5d SCFT; this conjecture is supported by various cross checks using ($p,q$) 5-brane webs.  Furthermore, we conjecture that all 5d SCFTs can be realized in M-theory on CY 3-folds satisfying these criteria.  Similar to the 6d case, where F-theory compactified on elliptic 3-folds was used to classify $\\mathcal N = (1,0)$ theories and it was subsequently found that for a few exotic cases frozen singularities are necessary to realize $\\text{O7}^+$ planes in F-theory \\cite{Tachikawa:2015wka,Bhardwaj-progress}, we find that in the M-theory case it is also necessary to include frozen singularities to obtain a complete classification of 5d SCFTs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[draw,circle] (a) at (0,0) {$S_5$};\n\t\t\t\\node[draw,circle] (b) at (2,0) {$S_3$};\n\t\t\t\\node[draw,circle] (e) at (2,2) {$S_4$};\n\t\t\t\\node[draw,circle] (f) at (2,-2) {$S_2$};\n\t\t\t\\node[] (c) at (4,0) {$\\cdots$};\n\t\t\t\\node[draw,circle] (g) at (4,-2) {$S_1$};\n\t\t\t\\node[draw,circle] (d) at (6,0) {$S_r$};\n\t\t\t\\draw (b) --(c) -- (d);\n\t\t\t\\draw (a) -- (b);\n\t\t\t\\draw (b) -- (e);\n\t\t\t\\draw (e) -- (a);\n\t\t\t\\draw (b)--(f);\n\t\t\t\\draw (f) -- (g) -- (d);\n\t\t\\end{tikzpicture}\n\t\\end{center}\t\n\t\\caption{Graphical representation of a rank $r$ K\\\"ahler surface $S = \\cup S_i \\subset X$ embedded in local Calabi-Yau 3-fold $X$. The nodes of the graph correspond to 4-cycles $S_i$, while the edges $C_{i,i+1} = S_i \\cap S_{i+1}$ correspond to 2-cycles along which the nodes intersect.}\n\t\\label{fig:graph}\n\\end{figure}\t\t\n\nA complete classification of such CY 3-folds appears to be a rather daunting task.   For example, it is unknown whether or not the list of possible 5d SCFTs is finite for a given rank.  Luckily, it turns out that the rank 2 case is finite, permitting an exhaustive classification of physically distinct SCFTs.\n\nBy classifying rank 2 SCFTs in terms of Calabi-Yau geometry, we learn that all rank 2 gauge theories predicted in \\cite{Jefferson:2017ahm}, except for one family, are realized.\\footnote{We conjecture that all SCFTs admit at least one Coulomb branch parameter at the CFT point.  The missing family which is represented by $SU(3)$ at Chern-Simons level $k=8$ has no Coulomb branch parameter at the would-be CFT point\nand that is why we rule it out.  This family would have led to a putative CFT which allows a Coulomb branch deformation only after a mass deformation (i.e. turning on $1\/g^2$).}  Additionally, we are also able to pinpoint the non-perturbative physics missing in the gauge theoretic approach of \\cite{Jefferson:2017ahm} responsible for excluding this family of SCFTs. Furthermore, the geometric approach allows us to identify additional non-Lagrangian SCFTs whose existence motivates the existence of dual ($p,q$) 5-brane web configurations.\n\nGiven the significant practical challenges presented by this classification program, it is natural to ask if the insight we have gained from the rank 2 case can be used to streamline the classification of higher rank cases.  Indeed, a careful examination of the list of rank 2 theories reveals a beautifully simple picture: rank 2 SCFTs in 5d can be organized into four distinct families, related and interconnected by RG flows triggered by mass deformations---see Figure \\ref{tree}. Each family of 5d SCFTs has a parent 6d SCFT, where the parent 6d SCFT is related to a 5d descendant by circle compactification, up to a choice of automorphism twist (see \\cite{Apruzzi:2017iqe} for work on classifying such automorphism twists, and see \\cite{DelZotto:2015isa} for a discussion of additional discrete data characterizing circle compactifications of 6d SCFTs.)  Thus the rank 2 classification could have been anticipated entirely from the 6d perspective!  This result echoes a well-known property of rank 1 SCFTs: rank 1 5d SCFTs belong to a single family which descends from the 6d E-string theory via circle compactification.\n\nWe thus conjecture that {\\it all 5d SCFTs arise from 6d SCFTs compactified on a circle, possibly up to an automorphism twist}.\nMore precisely, we anticipate that all 5d SCFTs can be organized into distinct families, each of which arises from a 6d theory. For a fixed rank in 5d, the possible 6d SCFT parents are rather limited.  For example (ignoring the possible automorphism twist), the 6d SCFTs leading to rank $r$ 5d SCFTs will have $r-k$ dimensional tensor branches with rank $k$ gauge algebra.  This suggests a practical method to classify 5d SCFT families starting with the 6d classification: compactifying a 6d SCFT on a circle produces a 5d theory with a Kaluza Klein (KK) tower of states.  We call such theories `5d KK theories'; these theories are in some sense analogous to 6d little string theories.   To obtain non-trivial 5d SCFTs from 5d KK theories we need to turn on holonomies suitably tuned to trigger an RG flow to a nontrivial 5d SCFT in the infrared. Aspects of the phase structure of 5d theories arising from circle compactifications of 6d SCFTs were analyzed in \\cite{DelZotto:2017pti}.\n\nThe organization of this paper is as follows.  \nIn Section \\ref{sec:review} we discuss the preliminaries of 5d SCFTs, their effective gauge theory descriptions on the Coulomb branch, and their realizations in M-theory.\nIn Section \\ref{sec:algorithm} we discuss the mathematics of shrinkable 3-folds and explain the basic approach of our geometric classification program.\nIn Section \\ref{sec:classification} we repeat the classification of rank 1 5d SCFTs and extend the same methods to the rank 2 case.  We also discuss the connection to 6d $\\mathcal N = (1,0)$ SCFTs.  Some mathematical results essential for the rank 2 classification are collected in the appendices: Appendix \\ref{app:AG} contains an explicit description of the Mori cones of blowups of Hirzebruch surfaces; Appendix \\ref{app:bound} contains some numerical bounds constraining rank 2 shrinkable 3-folds; finally, Appendix \\ref{app:smooth} contains a detailed discussion of some smoothness assumptions which simplify the classification program.\n\n\\section{Effective Description of 5d SCFTs}\n\\label{sec:review}\n\nIn this section we discuss some of the preliminaries that set the stage for the classification of 5d SCFTs later in this paper.  The following discussion involves two perspectives on 5d $\\mathcal N =1$ theories:\nthe gauge theoretic perspective, and the geometric perspective of M-theory compactified on a Calabi-Yau 3-fold.  \n\n5d superconformal field theories (SCFTs) are strongly interacting systems with no marginal deformations \\cite{Cordova:2016emh} and no known Lagrangian description at the CFT fixed point. In order to study the physics of these conformal theories, one needs to use rather indirect approaches. 5d SCFTs admit supersymmetric relevant deformations which lead to several weakly interacting effective descriptions while preserving some amount of supersymmetry. Surprisingly, these effective descriptions can be powerful tools for studying the dynamics of the conformal point. There exist some CFT observables which are rigidly protected under the renormalization group (RG) flow triggered by these deformations.  Many BPS quantities are such observables: for example, the spectrum of BPS operators, supersymmetric partition functions, effective Lagrangians on the Coulomb branch, the Coulomb branch of moduli space, etc. In particular, BPS observables are protected by supersymmetry and thus we expect BPS quantities appearing in the effective theories to be a reliable description of the corresponding observables at the CFT fixed point.\n\nString theory provides many effective descriptions of 5d SCFTs. Multiple D4-brane systems in Type IIA string theory and ($p,q$) 5-brane webs in Type IIB string theory can engineer various 5d SCFTs as singularities. Away from the singularity, when mass parameters and gauge couplings are turned on, these brane systems often permit a gauge theory description of the corresponding 5d theories.\n\n5d SCFTs can also be engineered in M-theory: M-theory on a singular non-compact Calabi-Yau 3-fold is described at long distances by an SCFT living on the five-dimensional spacetime transverse to the 3-fold. In familiar cases, the Calabi-Yau singularity can be resolved by means of various K\\\"ahler deformations, which correspond to mass and Coulomb branch deformations in the corresponding gauge theory. \n\n\n\n \n\\subsection{Gauge theory description}\n\nGauge theories in five dimensions are non-renormalizable and flow to free fixed points at low energy. As a result, these theories are typically believed to be `trivial' theories. However, a large class of 5d gauge theories, mostly engineered in string theory, turn out to have interacting CFT fixed points in the UV \\cite{Seiberg:1996bd}. In such cases, 5d gauge theories are rather interesting since they can provide low energy effective descriptions of the CFT.\n\nIn this paper, we focus primarily on gauge theories which have 5d SCFTs as their UV completions. These theories preserve $\\mathcal{N}=1$ supersymmetry, and their massless field content consists of vector multiplets with gauge algebra $G$ and hypermultiplets in a representation $\\textbf{R} = \\oplus \\textbf{R}_j$ of $G$. These gauge theories might be further specified by topological data $k$ corresponding to classical Chern-Simons level, as in the case of $G = SU(N \\geq 3)$, or discrete $\\theta$-angle as in the cases $G= Sp(N)$. We can also consider the cases with product gauge algebra $G=\\prod_i G_i$. Once the data $G,\\textbf{R},k$ is fixed, the low energy gauge theory Lagrangian is uniquely determined by supersymmetry. Our notation for describing 5d gauge theories is\n\\begin{align}\nG_k + \\sum_j N_{\\textbf{R}_j} \\textbf{R}_j,\n\\end{align} \nwhere $\\textbf{R}_j$ is the representation under which the $j$-th matter hypermultiplet is charged, $N_{\\textbf{R}_j}$ is the number of hypermultiplets in the representation $\\textbf{R}_j$.\n\n5d $\\mathcal{N}=1$ gauge theories possesses a rich vacuum structure. The moduli space of vacua is parametrized by expectation values of various local operators. In particular, we are interested in the Coulomb branch of vacua parametrized by vacuum expectation values of scalar fields $\\phi$ in the vector multiplets. Here the scalar field $\\phi$ takes values in the Cartan subalgebra of the gauge group $G$. So the dimension of the moduli space of the Coulomb branch is given by the rank of group $G$, $r={\\rm rank}(G)$. \nBy abuse of notation, we will denote both a scalar field in the vector multiplet and its expectation value by $\\phi$ from now on. \n\nThere are global symmetries acting on the hypermultiplets. The classical Lagrangian has global symmetry algebra $F$ rotating the perturbative hypermultiplets and also a topological $U(1)_I$ symmetry for each gauge group. The objects charged under the $U(1)_I$ are non-perturbative particles called `instantons'. Surprisingly, this classical global symmetry is often enhanced in the CFT fixed point by non-perturbative instanton dynamics \\cite{Seiberg:1996bd,Douglas:1996xp}. The flavor symmetry of the perturbative hypermultiplets can combine with the topological $U(1)_I$ instanton symmetry and enhance to an even larger symmetry algebra in the UV CFT. One can turn on mass parameters $m_i$ associated to the global symmetry. Doing so breaks some of the global symmetry. In particular, the mass deformation with parameter $g^{-2}$ along the $U(1)_I$ instanton symmetry leads to a gauge theory description with gauge coupling $g$ at low energy.\n\nAt a generic point in the Coulomb branch, the gauge symmetry $G$ is broken to the maximal torus $U(1)^{r}$. Thus the low energy dynamics on the Coulomb branch can be effectively described by abelian gauge theories.\nThe low energy abelian action is determined by a prepotential $\\mathcal{F}$. The prepotential is 1-loop exact and the full quantum result is a cubic polynomial of the vector multiplet scalar $\\phi$ and mass parameters $m_j$, given by \\cite{Witten:1996qb,Intriligator:1997pq}:\n\\begin{equation}\n\\label{eqn:pre}\n\t\\mathcal{F} = \\frac{1}{2g^2}h_{ij}\\phi_i \\phi_j + \\frac{k}{6} d_{ijk}\\phi_i\\phi_j\\phi_k + \\frac{1}{12}\\left(\\sum_{e\\in {\\rm root}} |e\\cdot \\phi|^2 -\\sum_j \\sum_{w\\in {\\bf R}_j}|w\\cdot\\phi+m_j|^3\\right) \\ ,\n\\end{equation}\nwhere by abuse of notation ${\\bf R}_j$ denotes the set of weights of the $j$-th hypermultiplet representation of $G$, $h_{ij}={\\rm Tr}(T_iT_j)$, and $d_{ijk}=\\frac{1}{2}{\\rm Tr}_{\\bf F}(T_i\\{T_j,T_k\\})$ with ${\\bf F}$ in the fundamental representation. The first two terms in the prepotential are from the classical Lagrangian and the last two terms are 1-loop corrections coming from integrating out charged fermions in the Coulomb branch. We remark that the prepotential may have different values in the different sub-chambers (or phases) of the Coulomb branch due to the absolute values in the 1-loop contributions.\n\nThe 1-loop correction to the prepotential renormalizes the gauge coupling. The effective coupling in the Coulomb branch is simply given by a second derivative of the quantum prepotential which also fixes the exact metric on the Coulomb branch:\n\\begin{equation}\n\\label{eqn:der}\n\t(\\tau_{\\rm eff})_{ij} = (g^{-2}_{\\rm eff})_{ij} = \\partial_i\\partial_j\\mathcal{F} \\ , \\qquad ds^2 = (\\tau_{\\rm eff})_{ij}d\\phi_id\\phi_j \\ .\n\\end{equation}\nInterestingly, the exact spectrum of magnetic monopoles on the Coulomb branch can be easily obtained from the quantum prepotential. Since monopoles are magnetically dual to electric gauge bosons, tensions of magnetic monopole strings can be computed as\n\\begin{equation}\n\\label{eqn:mono}\n\t\\phi_{Di} = \\partial_i\\mathcal{F} \\ , \\quad i=1,\\cdots r \\ .\n\\end{equation}\nOne can also compute Chern-Simons couplings:\n\t\\begin{align}\n\t\tk_{ijk} = \\partial_ i \\partial_j \\partial_k \\mathcal F. \n\t\\end{align}\nTherefore, we can use $\\mathcal F $ to exactly compute some quantum observables such as the Coulomb branch metric and monopole spectrum.\n\nIn \\cite{Intriligator:1997pq,Jefferson:2017ahm}, the above supersymmetry protected data is used to attempt a classification of possible 5d SCFTs admitting low energy gauge theory descriptions. The main idea in these classification programs is that the quantum metric on the Coulomb branch should be positive semi-definite in the CFT limit, as required by unitarity. In \\cite{Intriligator:1997pq}, the positivity condition of the metric was imposed throughout the `perturbative' Coulomb branch and all sensible gauge theories were subsequently identified using this constraint. In this classification, the `perturbative' Coulomb branch is determined by forcing only \\emph{perturbative} particles to have positive masses. Under this condition, the number and type of hypermultiplets are strictly constrained and  quiver type gauge theories are ruled out; see \\cite{Intriligator:1997pq} for details. We refer to this classification as the `IMS classification'.\n\nHowever, it was pointed out later works \\cite{Aharony:1997ju,Bergman:2014kza,Hayashi:2015fsa,Gaiotto:2015una,Yonekura:2015ksa} that string theory can engineer many 5d gauge theories with non-trivial CFT fixed points not included among the theories in the IMS classification. It turns out that the condition of metric positivity throughout the entire perturbative Coulomb branch is too strong \\cite{Jefferson:2017ahm} and unnecessarily excludes many non-trivial 5d gauge theories. This suggests that the IMS classification is incomplete, and the gauge theories exceeding the IMS bounds lead us to revisit the problem of classifying 5d SCFTs. \n\nLet us briefly review the classification of \\cite{Jefferson:2017ahm}. One of the main results of this analysis is the observation that the `perturbative' Coulomb branch receives quantum corrections by light non-perturbative states \\cite{Aharony:1997ju}. It is possible that some of non-perturbative states can become massless somewhere in the perturbative Coulomb branch. These hyperplanes in the Coulomb branch where these light states become massless can be thought of as `non-perturbative' walls. Beyond such walls, the perturbative Coulomb branch breaks down. One way to see this is to note that the signature of the quantum metric on the Coulomb branch changes beyond these non-perturbative walls, which implies the metric cannot be trusted in these regions. However, the classification in \\cite{Intriligator:1997pq} imposes metric positivity on the whole perturbative Coulomb branch, even beyond non-perturbative walls. The result is that some theories are excluded because of the unreliability of the metric in these regions, and this leads to an incomplete classification. In order to obtain a complete classification, metric positivity should be applied only on the `physical' Coulomb branch, which can be computed by accounting for restrictions introduced by non-perturbative states.\n\nIn general, it is difficult to identify the correct physical Coulomb branch after taking into account non-perturbative effects since this necessarily involves studying the full non-perturbative spectrum. In particular, it is not easy to analyze the spectrum of gauge theory instantons. Only when we know a precise UV completion of the instanton moduli space, such as the ADHM construction, can we compute the exact spectrum using localization. For most gauge theories, such a convenient construction of the instanton moduli space is lacking.\n\nFortunately, the perturbative prepotential contains part of the exact spectrum of non-perturbative states. As noted in (\\ref{eqn:mono}), the full monopole spectrum can be obtained from the prepotential. We can use this information to identify some of the non-perturbative walls in the perturbative Coulomb branch. By relaxing the metric positivity constraint to apply only to the region interior to such non-perturbative walls, it was conjectured in \\cite{Jefferson:2017ahm} that all gauge theories having interacting CFT fixed points satisfy the metric positivity condition in the sub-locus of Coulomb branch where perturbative particles and monopole strings have positive masses. In \\cite{Jefferson:2017ahm}, it was also shown that a large class of known 5d gauge theories satisfy this criterion. It may be true that all the known 5d gauge theories having 5d SCFT fixed points satisfy this refined condition.\n\nIn addition, there are two more conjectures in \\cite{Jefferson:2017ahm} used to carry out the classification of 5d gauge theories with simple gauge algebras. The first conjecture is that if all perturbative particles and monopoles have positive masses \\emph{somewhere} in the Coulomb branch, the gauge theory has a UV CFT fixed point. The second conjecture is that perturbative prepotentials of all gauge theories with UV CFT fixed points are positive \\emph{everywhere} in the perturbative Coulomb branch.\nNote that the first conjecture is not sufficient to guarantee that all instanton particles have positive mass and also that the metric is positive in the same region. So this is simply a necessary condition. We will see later that certain theories predicted by this approach must be excluded because some non-perturbative particles acquire negative masses in the CFT limit.\nThe second conjecture is based on the convergence of the 1-loop sphere partition function of 5d CFTs, but there is neither physical nor mathematical motivation for this conjecture beyond its practical implications.\nUsing these two conjectures, non-trivial gauge theories with single gauge node were fully classified in \\cite{Jefferson:2017ahm}. This classification includes all known single gauge node theories and additionally predicts a large number of new gauge theories.\n\nIn this paper, we construct rank 1 and rank 2 CFTs using Calabi-Yau geometry. Rank 1 gauge theories arising from SCFTs were classified in \\cite{Seiberg:1996bd,Morrison:1996xf,Intriligator:1997pq,Katz:1996fh}; these theories have gauge algebra $SU(2)$ with $N_\\textbf{F}\\leq  7$. Geometrically, the rank 1 SCFTs can be engineered by del Pezzo surfaces embedded in a non-compact $3$-fold.\nThe families of rank 2 gauge theories predicted by the classification of \\cite{Jefferson:2017ahm} are displayed in Table \\ref{tb:rank2-gauge-theory-clssification}. The UV completions of the theories shown in Table \\ref{tb:rank2-gauge-theory-clssification} are all expected to be 6d theories, rather than 5d SCFTs; on the other hand, their descendants obtained by mass deformations are expected to have 5d CFT fixed points. \nMany of these theories in Table \\ref{tb:rank2-gauge-theory-clssification} are new theories, for example $SU(3)$ with $(N_{\\bf F},|k|)=(6,4),(3,\\frac{13}{2}),(0,9)$ in $(a)$.\n\n\nOne of the purposes of this paper is to check if the new rank 2 CFTs predicted in \\cite{Jefferson:2017ahm} (or descendants of theories in Table \\ref{tb:rank2-gauge-theory-clssification}) can be constructed geometrically. We will see that, surprisingly, almost all new theories in Table \\ref{tb:rank2-gauge-theory-clssification} admit geometric constructions, therefore their descendants indeed have interacting CFT fixed points. However, some theories do not correspond to geometries in their conformal limits due to subtle non-perturbative effects. Therefore, the geometric constructions of this paper indicate that the criteria described in \\cite{Jefferson:2017ahm} require additional non-perturbative corrections in order to be complete.\nWe hope to revisit the field theoretic approach of \\cite{Jefferson:2017ahm} in the near future with the benefit of our improved understanding.\n\n\\begin{table}\n\\centering\n\\begin{subtable}[t]{0.45\\linewidth}\n\\centering\n\\vspace{0pt}\n\\begin{tabular}{|c|c|c|}\n\t\\hline\n\t $N_{\\textbf{Sym}}$ & $N_{\\textbf F}$ & $|k|$  \\\\\n\t\\hline\n\t$1$ & $0$ & $\\frac{3}{2}$\\\\\n\t\\hline\n\t$1$ & $1$ & $0$ \\\\\n\t\\hline\n\t$0$ & $10$ & $0$\\\\\n\t\\hline\n\t$0$ & $9$ & $\\frac{3}{2}$\\\\\n\t\\hline\n\t$0$ & $6$ & $4$\\\\\n\t\\hline\n\t$0$ & $3$ & $\\frac{13}{2}$\\\\\n\t\\hline\n\t$0$ & $0$ & $9$\\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Marginal $SU(3)$ theories with CS level $k$, $N_{\\textbf{Sym}}$ symmetric and $N_{\\textbf F}$ fundamental hypermultiplets.}\n\t\\label{tb:SU3-classification}\n\n\\end{subtable}\\hfill\n\\begin{subtable}[t]{0.45\\linewidth}\n\\centering\n\\vspace{0pt}\n\\begin{tabular}{|c|c|}\n  \\hline\n  $N_{\\textbf{AS}}$ & $N_{\\textbf F}$ \\\\\n  \\hline\n  $3$ & $0$\\\\\n  \\hline\n  $2$ & $4$\\\\\n  \\hline\n  $1$ & $8$\\\\\n  \\hline\n  $0$ & $10$\\\\\n  \\hline\n\\end{tabular}\n\\caption{Marginal $Sp(2)$ gauge theories with $N_{\\textbf{AS}}$ anti-symmetric, $N_{\\textbf F}$ fundamental hypermultiplets. The theory with $N_{\\textbf{AS}}=3$ can have $\\theta=0,\\pi$.}\n\\label{tb:Sp2-classification}\n\n\\vspace{0.5cm}\n\n\\begin{tabular}{|c|c|c|}\n  \\hline\n  $N_{\\textbf F}$ \\\\\n  \\hline\n  $6$ \\\\\n  \\hline\n\\end{tabular}\n\\caption{A marginal $G_2$ gauge theory with $N_{\\textbf F}$ fundamental matters.}\n\\label{tb:G2-classification}\n\n\\end{subtable}\n\\caption{Rank 2 gauge theories.}\\label{tb:rank2-gauge-theory-clssification}\n\\end{table}\n\n\n\\subsection{M-theory compactifications}\n\\label{sec:Mth}\n\nString compactifications are an extraordinarily useful tool for realizing local, non-perturbative models of gauge sector physics in terms of brane dynamics. Consider in particular M-theory on a non-compact singular Calabi Yau variety $Y$, which is conjectured to be described at low energies by a 5d $\\mathcal N = 1$ SCFT. We are specifically interested in studying the Coulomb branch deformations of these 5d SCFTs. The heart of this analysis is the correspondence between the Coulomb branch $\\mathcal C$ and the extended K\\\"ahler cone $\\mathcal K(Y)$ of the singular threefold $Y$ \\cite{Witten:1996qb}:\n\t\\begin{align}\n\t\t\\mathcal C =\\mathcal K(Y).\n\t\\end{align}\t\n\nThe above correspondence is made more precise by establishing a dictionary between the geometry of the threefold and the BPS spectrum of the associated 5d theory, which we now describe in detail. Consider a smooth non-compact 3-fold $X$. The K\\\"ahler metric of $X$ depends on $h^{1,1}(X)$ moduli controlling the sizes of complex $p$ cycles in $X$.  In order to decouple gravitational interactions, it is necessary to scale the volume of $X$ to be infinitely large while keeping the volumes of all 2- and 4-cycles at finite size; this has the effect of sending the 5d Planck mass to infinity. Given a basis $D_i \\in H^{1,1}(X)$, one may therefore express the K\\\"ahler form $J$ as the linear combination \n\t\\begin{align}\n\t\tJ = \\phi_i D_i,~~ i = 1, \\dots, h^{1,1}(X),\n\t\\end{align}\nwhere the K\\\"ahler moduli $\\phi_{i=1,\\dots, r}$ associated to (cohomology classes dual to) compact 4-cycles $D_i = S_i$ are identified with Coulomb branch moduli, while the K\\\"ahler moduli $\\phi_{r+j,\\dots, r+M}=m_{j=1,\\dots, M}$ associated to non-compact 4-cycles $D_{r+j} = N_j$ are interpreted as mass parameters of the 5d theory. To align the discussion with the 5d field theoretic interpretation, we find it useful to partition the K\\\"ahler moduli into $r$ Coulomb branch parameters and $M$ mass parameters:\n\t\\begin{align}\n\t\th^{1,1}(X) = r + M. \n\t\\end{align}\t\nNote that when the associated 5d field theory admits a description as a gauge theory, $r$ coincides with the rank of the gauge group. \n\nThe BPS states of the 5d theory include electric particles and (dual) magnetic strings. Geometrically these states correspond to M2 branes wrapping holomorphic 2-cycles and magnetic dual M5 branes wrapping holomorphic 4-cycles, and the masses and tensions of these BPS degrees of freedom are proportional to the volumes of the corresponding holomorphic cycles. At a generic point $\\phi \\in \\mathcal C$ the spectrum of BPS states is massive, and this is reflected by the fact that the 2- and 4-cycles of $Y$ have finite volume. Since the conformal point $\\phi = 0$ is characterized by the appearance of interacting massless and tensionless degrees of freedom, we interpret the threefold $Y$ as a singular limit of the smooth threefold $X$ in which some collection of compact 4-cycles have collapsed to a point. Said differently, $X$ is a desingularization of $Y$.\n\n\nThe above discussion suggests that the data of the massive BPS spectrum is encoded in the geometry of $X$. Indeed this is the case, the main connection to geometry being the interpretation of the 5d prepotential (\\ref{eqn:pre}) as the cubic polynomial of triple intersection numbers of 4-cycles in $X$:\n\t\\begin{align}\n\t\\mathcal F = \\text{vol}(X) =\\frac{1}{3!} \\int_X J^3 =\\frac{1}{3!} \\phi_i \\phi_j \\phi_k\\int_X D_i \\wedge D_j \\wedge D_k.\n\t\\end{align}\nIn the previous section, we saw that various data characterizing the massive BPS spectrum can be expressed as derivatives of $\\mathcal F$. This data equivalently characterizes the geometry of $X$. In particular, the tensions (\\ref{eqn:der}) of elementary monopole strings are the volumes of the compact 4-cycles $S_i$:\n\t\\begin{align}\n\t\t\\phi_{Di} = \\partial_i \\mathcal F =\\text{vol}(S_i)=  \\frac{1}{2!} \\int_X J^2 \\wedge S_i,~~ 1 \\leq i \\leq r,\n\t\\end{align} \t\nthe matrix of effective couplings has as its components the volumes of various 2-cycles:\n\t\\begin{align}\n\t\t\\tau_{ij} = \\partial_i \\partial_j \\mathcal F= \\text{vol}(S_i \\cap S_j) = \\int_X J \\wedge S_i \\wedge S_j,~~ 1 \\leq i,j \\leq r,\n\t\\end{align}\nand the effective Chern-Simons couplings $k_{ijk}$ are triple intersection numbers:\n\t\\begin{align}\n\t\tk_{ijk} = \\partial_i \\partial_j \\partial_k \\mathcal F =  \\int_X D_i \\wedge D_j \\wedge D_k. \n\t\\end{align}\nThe K\\\"ahler cone $\\mathcal K$ of the singularity $Y$ can also be specified quite easily; $\\mathcal K$ is simply the set of all positive K\\\"ahler forms (parametrized by the moduli $\\phi$):\n\t\\begin{align}\n\t\\mathcal K(X \\backslash Y) = \\{ J = \\phi_i D_i ~|~\\int_{C} J > 0 ~~\\text{for all holomorphic curves $C \\subset X$} \\}. \n\t\\end{align}\nThus, it is possible to study Coulomb branch deformations of 5d SCFTs purely in terms of the geometry of a smooth 3-fold $X$. Generically there are multiple smooth 3-folds $X_i$ which share a common singular limit $Y$, so the extended K\\\"ahler cone is simply the closure of the union of K\\\"ahler cones,\n\t\\begin{align}\n\t\t\\mathcal K(Y) = \\overline{\\cup \\mathcal K(X_i \\backslash Y)}.\n\t\\end{align}\nThe extended K\\\"ahler cone has the structure of a fan, with pairs of cones separated by hypersurfaces in the interior of $\\mathcal K(Y)$. The boundaries of $\\mathcal K(X_i \\backslash Y)$ correspond to loci where the 3-fold $X_i$ develops a singularity. The interior boundaries are regions where a holomorphic curve collapses to zero volume and formally develops negative volume in the adjacent K\\\"ahler cone, signaling a flop transition (see Section (\\ref{sec:gtrans}) for further discussion.) By contrast, the boundaries of $\\mathcal K(Y)$ are loci where one of the 4-cycles can collapse to a 2-cycle or a point. The SCFT point is the origin of $\\mathcal K(Y)$, and corresponds to the singularity $Y$ which is characterized by a connected union of 4-cycles shrinking to a point. \n\nIn some cases the 5d theory associated to a 3-fold $X$ admits a description as a gauge theory. In such cases, the abelian gauge algebra is $H^2(X,\\mathbb R) \/ H^2(X ,\\mathbb Z)$ and enhances to a non-abelian gauge algebra in the singularity $Y$. The simple coroots of the gauge algebra correspond to the classes $S_i \\in H^2(X,\\mathbb Z)$, whereas the simple roots are generic fibers $f_j$ contained in $H_2(X,\\mathbb Z)$. More precisely, the W-bosons of the 5d theory correspond to M2-branes wrapping holomorphic curves $f_j$, and so the Cartan matrix $A_{ij}$ is the matrix of charges\n\t\\begin{align}\n\t\\label{eqn:Cartan}\n\t\tA_{ij} = - \\int_{f_j} S_i.\n\t\\end{align}\n\t\nIn practice, we work in an algebro-geometric setting in which volumes of holomorphic cycles can be computed as intersection products. Thus the volumes of 2-cycles $C_i \\subset H_2(X,\\mathbb Z)$ and 4-cycles $S_i \\subset H_4(X,\\mathbb Z)$ are expressed in terms of the intersection products of numerical classes of (resp.) complex curves $[C]$ and surfaces $[D]$. That is, $\\text{vol}(C) = (J \\cdot [C])_X$ and $\\text{vol}(S) = (J \\cdot J \\cdot S_i)_{X}$. We abuse notation and use the same symbols to denote $p$-cycles, their homology classes, and their numerical equivalence classes whenever the context is clear.\n\n\\section{Classification Program}\n\\label{sec:algorithm}\n\n\\subsection{Physical equivalence classes of 3-folds}\n\\label{subsec:constructshrink}\n\nIn this section we propose a classification of CY 3-folds defining 5d SCFTs via M-theory compactification. One way to approach this problem is to study singular 3-folds for which there exist desingularizations that preserve the Calabi-Yau condition (i.e. \\emph{crepant resolutions}.) However, the problem of classifying singular 3-folds admitting crepant resolutions is notoriously difficult. Rather than attempting to classify singularities, we instead classify \\emph{physical equivalence classes} of singularities. We define a pair of 3-folds to be physically equivalent (i.e. leading to the same SCFT, up to decoupled sectors) if they are related by a finite change in K\\\"ahler and complex parameters.\nThere is a conjectural aspect to this definition which we now clarify.\n\nIt is immediate from the above definition that normalizable K\\\"ahler and complex deformations do not change the physical equivalence class of a 3-fold, since these deformations do not change the singular limit (and hence do not change the SCFT). However, we also find it useful to identify 3-folds that differ by non-dynamical large complex deformations. While the singular limits of such 3-folds are not identical, we claim they are nevertheless closely related in that their SCFTs differ at most by decoupled free states\n\nAs we will see, the notion of physical equivalence dramatically simplifies the problem of classification. \n\n\\subsection{Shrinkable 3-folds}\n\\label{sec:shrinkable}\n\nIn this section we specify the necessary criteria a smooth 3-fold must satisfy in order to define a 5d SCFT.  Note that we assume all 5d SCFTs have a \\emph{maximal} Coulomb branch, meaning that there exists a phase in which the 5d theory has no dynamical massless hypermultiplets, possibly after turning on some mass parameters.   Geometrically this means that we assume there exists a smooth 3-fold which has no normalizable (dynamical) complex structure deformations. The geometry of such a 3-fold is thus controlled by three types of parameters:  normalizable K\\\"ahler (i.e. Coulomb branch) parameters, non-normalizable K\\\"ahler (i.e. mass) parameters, and non-dynamical non-normalizable complex structure deformation parameters (see Section \\ref{sec:transitions} for an example).\n\nBefore spelling out the necessary criteria, we recall the key features of the geometries which are the subject of our analysis. We are interested in smooth, non-compact CY 3-folds $X$ containing a finite number of compact 4-cycles $S_i$ and non-compact 4-cycles $N_j$.  As discussed in the previous section the number of independent compact 4-cycles is equal to the number of Coulomb branch parameters, while the number of mass parameters is identified with the number of non-normalizable K\\\"ahler deformations. The 4-cycles $S_i \\subset X$ are irreducible projective algebraic surfaces, hence K\\\"ahler. Moreover, $X$ also contains compact 2-cycles which can either be isolated or part of a family of compact 2-cycles belonging to one of the 4-cycles. \n\nFrom the physics perspective the natural condition for CY 3-folds to lead to SCFTs is that we can tune non-normalizable K\\\"ahler parameters (mass parameters) so that at a finite distance in normalizable K\\\"ahler moduli space we can reach a singular CY 3-fold which has no finite volume cycles or surfaces.  However, formulating this in algebro-geometric terms is not simple.  Instead we formulate it in a somewhat different way which we believe is equivalent to this.  Namely,\nin order for a 3-fold $X$ to define a 5d SCFT, $X$ must satisfy the property of being \\emph{shrinkable}, which we define below:\n\n\\medskip\\noindent\n{\\bf Definition.} \n\\label{def:shrinkability}\nLet $X$ be a smooth CY 3-fold modeled locally as the neighborhood of  a connected union of compact K\\\"ahler surfaces $S = \\cup S_i$. We say $X$ is \\emph{shrinkable} if there exists an intersecting (possibly empty) union of non-compact surfaces $N=\\cup N_j$ and a limit $Y$ of K\\\"ahler metrics such that:\n\t\\begin{enumerate}\n\t\t\\item $S$ (and all curves $C \\subset S$) have zero volume in $Y$;\n\t\t\\item $Y$ is at finite distance from a metric $X_0$ for which $N$ has zero volume while $S$ has positive volume.\n\t\\end{enumerate} \nBy abuse of terminology, we say the surface $S$ is shrinkable if $S$ is contained in a shrinkable 3-fold $X$ as a maximal compact algebraic surface.\n\\medskip\n\n\n\n\n\n\nLet us now translate the above definition of shrinkability into a set of necessary geometric conditions. We consider first the limit where all non-normalizable K\\\"ahler moduli have been set to zero.  In this limit we may have a singular 3-fold which is described by the K\\\"ahler class $J=\\phi_iS_i$. Our convention is to assume $\\phi_i\\ge0$ and compute volumes with respect to $-J$;  thus, the volume of a curve $C$ is given by $\\mathrm{vol}(C)= -J\\cdot C$ and the volume of a divisor $D$ is $\\text{vol}(D) = J^2 \\cdot D$.\\footnote{This choice of sign is consistent with \nthe description of K\\\"ahler classes $J$ on compact CY 3-folds, as the expansion of $J$ (or any other ample divisor class) in terms of $S_i$ will have non-positive coefficients.  A simple example illustrating this point is the rank~1 case, for which $S$ is a del Pezzo surface. Since $J\\cdot C = \\phi K_S \\cdot C$, it follows that $J$ has non-positive intersection with all curves $C \\in S$. We therefore have to change the sign in order for $J$ to be a limit of K\\\"ahler classes on $X$.}  Since we require $-J$ to define a K\\\"ahler metric which assigns postive volumes to complex $p$-cycles in $X$, a necessary condition for shrinkablity is \n\\begin{equation}\n  \\label{eq:shrinkability}\n \\mathrm{vol}(C)= -J\\cdot C\\ge0,~~\\forall C\\subset S.\n\\end{equation}\n\n\nWhat happens when the inequality (\\ref{eq:shrinkability}) is saturated? Suppose there exists a curve $C$, with $\\text{vol}(C)=0$. So far, we have only considered the case in which all non-normalizable K\\\"ahler moduli are set to zero. To give finite volume to $C$ requires a non-normalizable K\\\"ahler deformation, which in turn implies the existence of a non-compact 4-cycle $N$ attached to $S$ along $C$. Notice that since $C$ belongs to $N$, there may also be other compact curves $C'$ which are homologous to $C$ in $N$; in particular, the full set of curves homologous to $C$ can fiber over $N$. For each of these curves $C'$ it must be that $\\text{vol}(C')=0$, and thus $N$ can be said to have degenerated to a non-compact 2-cycle along its fibers.\\footnote{It would interesting to compare this defintion of shrinkability with the conjecture of \\cite{Xie:2017pfl} that canonical 3-fold singularities give 5d SCFTs, since it is known that the only noncompact 4-cycles in a Calabi-Yau (crepant) resolution of a canonical 3-fold singularity are ADE fibrations.  However, we do not need this for the description in our classification.}  By making a non-normalizable K\\\"ahler deformation, we can bring the curve $C = S \\cap N$ to finite volume, and we expect that we are again in a situation where the surface $S$ is contractible.\n\nWe believe that the above necessary criteria are in fact sufficient to define a shrinkable 3-fold:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Let $X$ be a smooth CY 3-fold modeled locally as the neighborhood of a connected union of compact K\\\"ahler surfaces $S= \\cup S_i$. Then $S$ is shrinkable provided that $- J \\cdot C \\geq 0$ for all curves $C \\subset S$ and that there is one $S_i$ with positive volume and the rest should have non-negative (possibly zero) volume.\n\\medskip\n\nElliptic Calabi-Yau 3-folds are immediately ruled out by these criteria. F-theory on an elliptic 3-fold engineers a 6d theory. In a 6d theory, cubic terms in the prepotential $\\mathcal{F}$ are trivial; they are non-trivial only when we compactify the 6d theory on a circle and turn on holonomies for gauge symmetries where the circle size is inversely proportional to a mass parameter (or a non-compact K\\\"ahler parameter). This means that the volumes of all 4-cycles in the associated 3-fold are zero when we turn off mass parameters (or equivalently, in the 6d limit). Therefore elliptic 3-folds are not shrinkable.\n\n\\subsection{Building blocks for shrinkable 3-folds}\n\\label{sec:buildingblocks}\nWe now argue in favor of a series of simplifying assumptions we make concerning the surfaces $S$ which are instrumental for our proposed classification of shrinkable rank 2 surfaces modulo physical equivalence. Observe that when the inequalities of (\\ref{eq:shrinkability}) are all strict, then $S$ is \\emph{contractible} \\cite{grauert}, so that $S$ can be contracted to an isolated singular point $p$ of a singular 3-fold $Y$.  In more precise mathematical terms, this means there exists a holomorphic map $f:X \\to Y$ with $f(S)=p$ such that $f$ restricts to an isomorphism away from $S$, i.e. $f|_{X-S}:X-S \\cong Y-p$.  Since $X$ is at finite distance from $Y$ in moduli space, it is evident that contractibility of $S \\subset X$ implies shrinkability of $X$. When a curve has zero volume, we expect that we can obtain a contractible surface by means of a non-normalizable K\\\"ahler deformation which involves bringing non-compact 4-cycles to finite volume. Hence, we conjecture that a holomorphic map $f$ exists when $S$ is shrinkable, as well:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Let $X$ be a shrinkable CY 3-fold modeled locally as a neighborhood  of a connected union of compact K\\\"ahler surfaces $S= \\cup S_i$ meeting a (possibly empty) collection of non-compact surfaces $N = \\cup N_j$. Then there exists a holomorphic map $f:X \\to Y$ sending $S$ to a point $p$ and $N$ to a collection of curves $C$ such that $\\left. f\\right|_{X - S - N} : X - S - N \\to Y - C$ is an isomorphism.\n\\medskip\n\nThe existence of a holomorphic map $f$ as described above permits a number of simplifying assumptions for the following reasons. Replacing the singular 3-fold $Y$ by its normalization if necessary, we can assume that the singularities of $Y$ are normal.  It follows that $Y$ has ``canonical singularities'', and moreover that $X$ is a crepant resolution of $Y$.  But it is known the components of the resolutions of canonical threefold singularities $Y$ are rational or ruled \\cite{can3f}.\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe next argue that we can further restrict the types of possible building blocks by exploiting physical equivalence:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Shrinkable surfaces are physically equivalent to a shrinkable surface $S=\\cup S_i$, where the irreducible components $S_i$ are either equal to $\\mathbb P^2$ or a blowup $\\text{Bl}_{p} \\mathbb F_n$ of a Hirzebruch surface at $p$ points intersecting one another (or self-intersecting) transversally.  Moreover, there exist non-negative integers $p_{\\text{max}}(n)$ such that $p \\leq p_{\\text{max}}(n)$.\n\\medskip\n\n\nWe briefly discuss the content of the above conjecture, deferring a more detailed discussion of the first two points to Section \\ref{sec:transitions}.  \nIn that section, we describe the rank~2 case only.  For higher rank, we have to also consider the situation where three surfaces can intersect transversally.\\footnote{Since four or more surfaces in a threefold cannot intersect nontrivially and transversally, we only need to consider intersections of three surfaces at a time.}  At such a point of intersection, called a triple point, the three intersecting surfaces have local equation $xyz=0$.  As part of the argument in Section \\ref{sec:transitions}, we blow up a point where two surfaces intersect, at which the intersecting surfaces have local equation $xy=0$, so our construction will not apply at a triple point.  To handle triple points, we simply supplement the argument in Section~\\ref{sec:gtrans} by noting that a complex structure deformation will keep a point to be blown up distinct from any of the triple points.  \n\n\\smallskip\\noindent\n\\begin{enumerate}\n \n\n\n\\item Using a combination of complex structure and K\\\"ahler deformations, it is possible\nto map a 3-fold containing a ruled surface over a genus $g$ to a 3-fold containing a Hirzebruch surface. We defer a detailed discussion to Section~\\ref{sec:transitions}.\n\n\\item In all examples that we have investigated, we have been able to bypass non-transverse intersections in one of two ways: either by a complex structure deformation, or by a K\\\"ahler deformation in the form of a flop.  The idea is that when we flop a curve (in $S_1$, say) which passes through a point of non-transversal intersection, the result is to blow up $S_2$ at that point, simplifying the singularity of the intersection curve and rendering it more transverse.\nWe therefore assume that a combination of complex and K\\\"ahler deformations will always suffice to produce a 3-fold containing transversally intersecting surfaces $S_i$.\n\n\\item We prove in Appendix \\ref{app:Mori}  that if $p>p_{\\text{max}}(n)$ there are infinitely many generators for rational curves. The presence of infinitely many generators is expected to indicate the presence of an infinite dimensional global symmetry group. An example of this is $\\text{dP}_9$ (note $p_{\\text{max}}(1)=7$), in which case the symmetry group permuting these generators is the affine $E_8$ Weyl group. In such a case, the Weyl group is infinite dimensional, and can be interpreted as a finite symmetry group of a 6d theory viewed from the 5d perspective. As we discussed above, geometries associated to 6d theories are not shrinkable. Since a CFT should not have an infinite dimensional global symmetry group, we claim that surfaces $S_i$ with an infinite number of Mori cone generators cannot be building blocks for 5d SCFTs and are thus excluded.\n\\end{enumerate}\n\n\n\n\n\\subsection{Consistency conditions for shrinkable 3-folds}\n\\label{sec:consistency}\n\nThe condition that $S$ is contained in a CY 3-fold imposes constraints on the curves of intersection of the components of $S$, which will be exploited in a crucial way in our classification program. \n\nLet $S_1$ and $S_2$ be two smooth surfaces glued along a curve $C = S_1 \\cap S_2$.   Now suppose that $S_1\\cup S_2$ is contained in a 3-fold $X$, and that the intersection of $S_1$  and $S_2$ is transverse in $X$.\nThen the normal bundle of $C$ in $X$ is given by $N_{C,X}=N_{C,S_1}\\oplus N_{C,S_2}$.  The Calabi-Yau condition then implies\n\\begin{equation}\n  \\label{eq:gluingcond}\n  C^2_{S_1}\\oplus C^2_{S_2}=2g-2,\n\\end{equation}\nwhere $g$ is the genus of $C$ and the subscripts on the right-hand side denote the irreducible surface in which the self-intersection takes place. The gluing curves must satisfy the adjunction formula for each surface $S_i$:\n\t\\begin{align}\n\t\\label{eq:adjunction}\n\t\t(K \\cdot C)_{S_i}   + C_{S_i}^2 = 2g - 2,\t\n\t\\end{align}\nwhere $K_{S_i}$ is the canonical class of the surface $S_i$. For the rank 2 case, which is the primary focus of this paper, we argue in Section \\ref{sec:rank2} that it suffices for our classification to assume that $g=0$.\n\nSuppose a compact connected holomorphic surface $S$ satisfies the above constraints on its curves of intersection. These constraints immediately imply that a CY 3-fold can be found containing a neighborhood in $S$ of the curves of intersection (for example, the total space of the normal bundle of $S_1 \\cap S_2$ in $X$ works, as the complement of $S_1 \\cap S_2 \\subset S$ is smooth).  Moreover, we can also find local CY 3-folds containing the complement of the intersection curves $S_1 \\cap S_2$ in $S$ (for example, just take the total space of the canonical bundle as before).  Therefore, it seems reasonable to expect that above two types of local models can be glued to form a local model of a CY 3-fold.  In other words,\ngiven smooth holomorphic surfaces $S_1$ and $S_2$ glued along a smooth curve $C$ and satisfying (\\ref{eq:gluingcond}), \na smooth CY 3-fold $X$ can be found containing $S=S_1\\cup S_2$.\nWhile we have not proven that such an $X$ can always be found if (\\ref{eq:gluingcond}) and (\\ref{eq:adjunction}) are satisfied, these conditions are consistent with all known examples and it is presumably not too difficult to rigorously prove this.\n\nWe emphasize here that the above gluing condition is a local condition that has no bearing on the overall topology of the surface $S$, and therefore permits a variety of interesting configurations. In principle there is nothing preventing, for example, gluing two surfaces together along multiple irreducible curves. Another interesting configuration involves two curves belonging to a single surface $S_i$ being glued together. However, we will see that the only gluing configurations which play a role in the rank 2 classification are pairwise transverse intersections between the irreducible components $S_1$ and $S_2$.\n\nThe above discussion plays an essential role in our classification because we do not need to actually construct $X$ to proceed; rather, we only require the existence of $X$ and the existence of a surface $S$ can be used as a proxy for the existence of a local 3-fold. Thus the problem of classifying shrinkable 3-folds can be reduced to the problem of classifying embeddable, shrinkable surfaces $S$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n   \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n   \n\n\n\n\n\\subsubsection*{A simple example: $S = \\mathbb F_0 \\cup \\mathbb F_2$}\n\nAn illustrative example of this construction is a simple complex surface $S=S_1 \\cup S_2$ with $S_1= \\mathbb F_0, S_2 = \\mathbb F_2$ as depicted in Figure \\ref{fig:F0F2}. Our rank 2 ansatz gives us \n\t\\begin{align}\n\t\\begin{split}\n\t\\label{eqn:geotrip}\n\t\tJ^3 &= S_1^3 \\phi_1^3 + S_2^3  \\phi_2^3 + 3 \\phi_1 \\phi_2 (J \\cdot S_1 \\cdot S_2)  =K_{S_1}^2 \\phi_1^3 + K_{S_2}^2 \\phi_2^3 - 3 \\phi_1 \\phi_2 \\text{vol}(S_1 \\cap S_2). \n\t\\end{split}\n\t\\end{align}\n\nThe first order of business is to determine an appropriate gluing. Gluing these two surfaces together requires us to identify an irreducible, smooth curve $C = S_1 \\cap S_2$ belonging to the Mori cone of both surfaces, satisfying (\\ref{eq:gluingcond}).  In the case of Hirzebruch surfaces $\\mathbb F_{n_i}$, the Mori cones are the positive linear spans $\\langle E_{i}, F_{i} \\rangle$, where the curve classes satisfy the intersections $F_i^2= 0, E_i \\cdot F_i =1, E_i^2 = -n_i$, so the range of possibilities is severely restricted. The gluing condition (\\ref{eq:gluingcond}) implies that the self intersection of one of the two gluing curves must be negative. Since the curve $E$ is the unique rational curve with negative self intersection \\cite{GH}, it therefore follows that we must select $C_{S_i}  = E_{i}$ for one of the two surfaces, say $C_{S_2} =E_{2}$. The other curve must then satisfy\n\t\\begin{align}\n\t\tC_{S_1}^2 = 0. \n\t\\end{align}\nAs a trial solution let us take $C_{S_1} = a F_{1} + b E_{1}$, so that $C_{S_1}^2 = 2 ab = 0$. Therefore, either $a = 0$ or $b = 0$. From the adjunction formula (\\ref{eq:adjunction}), we know that $(C \\cdot E_1 + C \\cdot F_1)_{S_1}= a+b= 1$, and therefore the remaining nonzero coefficient must be set equal to unity. To be concrete, we choose\n\t\\begin{align}\n\t\tC_{S_1} = F_{1},~~~ C_{S_2} = E_{2}. \n\t\\end{align}\n\nNow that we have constructed the surface $S$, we must check that the local 3-fold $X$ associated to this surface is shrinkable. We parametrize a K\\\"ahler class $J$ as follows:\n\t\\begin{align}\n\t\tJ = \\phi_1 [\\mathbb F_0] + \\phi_2 [ \\mathbb F_2],\n\t\\end{align}\nwhere $[\\mathbb F]$ is the class associated to the 4-cycle $\\mathbb F \\subset X$. The Mori cone of $X$ is the union of the Mori cones of the component surfaces $S_i$, namely the positive span $\\langle E_{1}, E_{2}, F_{2} \n\\rangle$ (we omit $F_{1}$ because the gluing identifies $F_1$ and $E_2$.) Therefore, the shrinkability condition (\\ref{eq:shrinkability}) implies \n\t\\begin{align}\n\t\t(\\text{vol}(E_1), \\text{vol}(E_2), \\text{vol}(F_2) ) = (2 \\phi_1 -\\phi_2 , 2 \\phi_1, -\\phi_1 + 2\\phi_2 ) \\geq 0. \n\t\\end{align}\nSince that the above conditions can be satisfied for a nontrivial set of Coulomb branch parameters $\\phi_i$, we conclude that the geometry $X$ corresponds to a 5d SCFT on the Coulomb branch. \n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[scale=.5]{F0-F2.pdf}\n\\end{center}\n\\caption{Example of a gluing construction of the K\\\"ahler surface  $S = \\mathbb F_0 \\cup \\mathbb F_2$. The gluing curves in both surfaces, $C_1, C_2$, are encircled by dashed lines in the left figure. The final geometry (on the right) is the result of identifying these two curves subject to the conditions described in Section \\ref{sec:algorithm}.}\n\\label{fig:F0F2}\n\\end{figure}\n\n \n\n\n\n\n\n\\subsection{Geometry of physical equivalences}\n\\label{sec:transitions}\n\nIn this section we discuss some important types of physical equivalences upon which our classification relies. Many of these equivalences identify 3-folds related by geometric transitions, i.e.\\ maps between smooth geometries which involve passing through an intermediate singularity.  Another type of physical equivalence identifies 3-folds related by a ``large\" change in the complex structure of non-dynamical modes, which interpolates between two singular geometries---this is a Hanany-Witten transition \\cite{Hanany:1996ie}.  We illustrate these two types of maps in turn.\n\n\\subsubsection{Geometric transitions}\n\\subsubsection*{Flop transitions}\n\\label{sec:gtrans}\nOne of the simplest and most thoroughly studied types of geometric transitions is a \\emph{flop transition}, which is a topology-changing transition $X \\rightarrow X'$ between two 3-folds $X, X'$  that is in practice typically realized by blowing down a $-1$ curve $C \\subset X$ and blowing up a different $-1$ curve $C' \\subset X'$ (see Figure \\ref{fig:flop}).   A flop is a birational map $X\\dashrightarrow X'$ which is an isomorphism away from curves $C,C'$, with $K_X\\cdot C=K_{X'}\\cdot C'=0$.  If $C$ and $C'$ are both isomorphic to ${\\mathbb P}^1$, the flop is called a simple flop.  Simple flops were classified in \\cite{km}.  \n\n\nIn field theoretic terms, a flop transition corresponds to a continuous change of the mass of a particular state in the matter hypermultiplet from positive to negative values; this change corresponds to a singular phase transition on the Coulomb branch.\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.7]{flop-trans.pdf}\n\t\\end{center}\n\t\\caption{A local illustration of a flop transition $X \\rightarrow X'$ between two CY 3-folds. The red lines in both diagrams correspond to the $-1$ curves in (respectively) $X$ and $X'$.}\n\t\\label{fig:flop}\n\\end{figure}\t\n\n\n\\subsubsection*{Genus reduction}\nWe saw in Section~\\ref{sec:buildingblocks} that the $S_i$ can be ruled surfaces over higher genus curves as well as genus 0.  Here we argue that by our notion of physical equivalences we can restrict to $g=0$ using geometric transitions.  This can be obtained by composing a complex structure deformation of a surface $S_i$ with a flop transition. This provides a map from a ruled surface over a curve of \ngenus $g$ to a self-glued Hirzebruch surface.\n\nThis type of geometric transition is particularly important because it exhibits the non-normalizable K\\\"ahler moduli of the local 3-fold defined by a ruled surface over a curve of genus $g$ as blowup parameters of the 3-fold defined by a self-glued surface $\\text{Bl}_{2g} \\mathbb F_n$. While we have not proven that the transition can always be achieved in the higher rank case due to the requirement that additional compact surfaces remain glued throughout the transition, we nevertheless believe this construction can be extended to higher rank surfaces with at most minor modifications.\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\\node(a) at (0,0) {$\n\\begin{tikzpicture}[yscale=.8,xscale=1.1]\n\\draw [thick] (0,0) to [out=90,in=180] (1,1) to [out=0,in=180] (2,.85) to [out=0,in=180] (3,1) to [out=0,in=90] (4,0) to [out=270,in=0] (3,-1) to [out=180,in=0] (2,-.85) to [out=180,in=0] (1,-1) to [out=180,in=270] (0,0);\n\\draw[thick] (.7,0) to [out=-20,in=180] (1.1,-.13) to [out=1,in=200] (1.5,0);\n\\draw[thick] (2.5,0) to [out=-20,in=180] (2.9,-.13) to [out=1,in=200] (3.3,0);\n\\draw[thick] (2.6,-.05) -- (3.2,-.05);\n\\draw[thick] (.8,-.05) -- (1.4,-.05);\n\\end{tikzpicture}\n$};\n\\node(b) at (6,0) {$\n\\begin{tikzpicture}[yscale=.8,xscale=1.1]\n\\draw [thick] (0,0) to [out=90,in=180] (1,1) to [out=0,in=180] (2,.85) to [out=0,in=180] (3,1) to [out=0,in=90] (4,0) to [out=270,in=0] (3,-1);\n\\draw[thick] (.8,-1) to [out=180,in=270] (0,0);\n\\draw[thick] (2.5,0) to [out=-20,in=180] (2.9,-.13) to [out=1,in=200] (3.3,0);\n\\draw[thick] (2.6,-.05) -- (3.2,-.05);\n\\draw[thick] (.9,-.09) -- (1.45,-.09);\n\\draw[thick](.7,-.05)  to [out=-10,in=90] (1.2,-.45) to [out=270,in=0] (.8,-1);\n\\draw[thick] (1.6,-.03) to (1.4,-.13) to  [out=200,in=90]  (1.2,-.5) to [out=270,in=180] (1.6,-1) to (3,-1);\n\\end{tikzpicture}\n$};\n\\draw[big arrow] (a) -- (b);\n\\end{tikzpicture}\n\\end{center}\n\\caption{A genus $g= 2$ Riemann surface degenerating into a $g= 1$ Riemann surface with a nodal singularity as the result of identifying two points. By identifying $g$ pairs of points in this manner, it is possible for a smooth curve of genus $g$ to degenerate into a rational curve with $g$ nodal singularities.}\n\\label{fig:degen}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.6]{self-gluing.pdf}\n\t\\end{center}\n\t\\caption{A transition from a ruled surface over a $g=1$ curve to a Hirzebruch surface. The red point in the second figure is a blowup point on a nodal curve and the red lines in the third figure are the exceptional curves. Two proper transforms of the fiber $F$ in a blown up Hirzebruch surface are glued together along the nodal curve.}\n\t\\label{fig:selfgluing}\n\\end{figure}\n\nBefore giving a detailed description of this geometric transition, we recall that by the irreducibility of the moduli space $\\overline{M}_g$\nof stable curves of genus $g$ the complex structure of a smooth curve $C$ of genus $g$ can be\ndegenerated to a rational curve $C_0$ with $g$ nodes (see Figure \\ref{fig:degen}.) The curve $C_0$ can be \nconstructed directly by identifying $g$ pairs of points of ${\\mathbb P}^1$. Note that this construction immediately extends to give a degeneration of a ruled surface\n$S$ over $C$ to a ruled surface $S_0$ over the singular curve $C_0$.  Conversely,\nthe degeneration of the ruled surface can be described by starting with ${\\mathbb P}^1$-bundle over\n${\\mathbb P}^1$ (i.e.\\ a Hirzebruch surface ${\\mathbb F}_n$) and identifying $g$ pairs of fibers\n$F \\subset \\mathbb F_n$. \n\nHowever, this description of $S_0$ is not completely satisfactory, as $S_0$ cannot be embedded into a CY 3-fold for the following reason. Let $F\\subset S_0$ be one of the singular fibers obtained by identifying $g$ pairs of fibers.  Locally, $S_0$ has two branches near $F$ with equation $xy=0$ (pulled back\nfrom the local equation $xy=0$ of a node of $C_0$). Being a fiber, $F$ has self-intersection 0 in each branch, So if $S_0$ were contained in a smooth\nthreefold, the normal bundle of $F$ would be ${\\cal O}_F\\oplus{\\cal O}_F$. Fortunately, the geometric transition naturally rectifies this problem by introducing blowups, in a manner which we describe below.\n\nConsider again the degeneration point of view, which can be described by a holomorphic map $\\pi:{\\cal S}\\to \\Delta$.  Here ${\\cal S}$ is a smooth\\footnote{Requiring ${\\cal S}$ to be smooth is not a problem; its local equation near a point of $F$ can be taken as $xy=t$, which is smooth.  This is the same local calculation which shows that  $\\overline{M}_g$ is smooth at the nodal curves (in the orbifold sense).} threefold, $\\Delta$ is a disk, $\\pi^{-1}(0)\\simeq S_0$, and $\\pi^{-1}(t)$ is diffeomorphic to\n$S$ for $t\\ne0$. We now pick a point $p\\in F\\subset S_0\\subset {\\cal S}$ and blow up $p$ to get $\\phi:\\widetilde{{\\cal S}}\\to {\\cal S}$. Via $\\pi\\circ\\phi$ we can view $\\widetilde{\\cal S}$ as a family over $\\Delta$.  However, $\\widetilde{\\cal S}$ and ${\\cal S}$ are isomorphic over $\\Delta-0$, so this gives another degeneration of $S$.  The singular limit is $(\\pi\\circ\\phi)^{-1}(0)$, which we now describe.\n\nBlowing up a point $p$ in a smooth threefold creates an exceptional divisor $E$ isomorphic to ${\\mathbb P}^2$, and blows up $S_0$ to a surface $\\widetilde{S_0}$.  We have $(\\pi\\circ\\phi)^{-1}(0)=\\widetilde{S}_0\\cup {\\mathbb P}^2$.  It remains to describe $\\widetilde{S}_0$ and how ${\\mathbb P}^2$ is attached to it.\n\nSince $S_0$ has local equation $xy=0$ at $p$, the exceptional curve of $\\widetilde{S}_0\\to S_0$ has $xy=0$ as its equation.  In this latter instance, the equation $xy=0$ is understood as a homogeneous equation in the exceptional\n${\\mathbb P}^2$ of the blown-up threefold.  In other words, ${\\mathbb P}^2$ meets $\\widetilde{S_0}$ in two intersecting projective lines $L,L'$;  each of these ${\\mathbb P}^1$'s can be thought of as arising from the blowup of $p$ in a corresponding branch of $S_0$ near $p$.\n\nThe point of intersection $q =  L \\cap L'$ also intersects the proper transform $\\widetilde{F}$ of the original singular fiber $F$.  The curve $\\widetilde{F}$ is still singular in $\\widetilde{S_0}$ and still has two branches in a local\ndescription, but now the blowup has reduced the self-intersection from $0$ to $\\widetilde F^2 = -1$ in each branch.  So if $\\widetilde{S_0}$ is contained in a smooth threefold, then the normal bundle of $\\widetilde{F}$ is ${\\cal O}_F(-1)\\oplus {\\cal O}_F(-1)$ and the threefold can be Calabi-Yau!\n\nWe can apply this construction to all of the $g$ singular fibers. Since $\\widetilde{F}$ has self-intersection $-1$ in each branch, we can view it as the gluing of a pair of exceptional ${\\mathbb P}^1$'s.  \nTherefore the\nresulting $\\widetilde{S}_0$ is a blown up Hirzebruch surface with $g$\npairs of exceptional curves identified.  Each singular fiber consists of a double curve with self-intersection $-1$ in each branch, glued at a common point $q$ to curves $L,L'$ of self-intersection $-1$ in each of the respective local branches\n(the surface $\\widetilde{S}_0$ is smooth along $L\\cup L'-\\{q\\}$).\n\nIn the degeneration described above, we also need to attach $g$ copies of ${\\mathbb P}^2$.  However, we are only concerned with the rank~2 case, so in our examples\nthese ${\\mathbb P}^2$'s can replaced by noncompact cycles containing $L\\cup L'$ and safely ignored.\n\nThe final step is to flop the $g$ curves $\\widetilde{F}_1,\\ldots\\widetilde{F}_g$, where we have added a subscript to $\\widetilde{F}$ to distinguish\nthese curves.  Let us investigate the birational transform of $\\widetilde{S_0}$ after the flops.\nWhen the curves $\\widetilde{F}_i$ are contracted, the points of intersection $q_i = L_i \\cap L_i'$ become conifolds.  When we complete the flops, new ${\\mathbb P}^1$'s appear in place of the $q_i$ and the curves $L_i,L'_i$ get separated. These curves\nbecome identified with fibers of a ruled surface over the desingularization $\\widetilde{C}_0$ of $C_0$, the fibers over the pairs of  points of $\\widetilde{C}_0$ which get identified to form a node of $C_0$. Since $\\widetilde{C}_0$ is isomorphic to ${\\mathbb P}^1$, the result\nis  a Hirzebruch surface in general with blowups.\n\n\\subsubsection*{An example of genus reduction: $G_2 + N_\\textbf{F} \\textbf{F}$ }\n\nAn illustrative example of complex deformations that exchange ruled surfaces over a curve of genus $g >0$ for self-glued Hirzebruch surfaces blown up at $2g$ points is the family of shrinkable 3-folds engineering $G_2 + N_{\\textbf{F}} \\textbf{F}$, as described in \\cite{Diaconescu:1998cn}. \n\nWe begin by recalling the form of the gauge theoretic 1-loop prepotential for $G_2 + N_{\\textbf{F}} \\textbf{F} + N_{\\textbf{adj}} \\textbf{adj}$:\n\t\\begin{align}\n\t\\begin{split}\n\t\\label{eqn:G2nomass}\n\t\t6 \\mathcal F_{\\text{1-loop}} &= ( 8 - 8 N_\\textbf{F} -8 N_\\textbf{adj}) \\phi_1^3 + ( 8 - 8 N_{\\textbf{adj}})  \\phi_2^3\\\\\n\t\t&~+ 3 \\phi_1 \\phi_2 [ (6 +3 N_\\textbf{F} - 6 N_\\textbf{adj} )\\phi_1 + (8 N_{\\textbf{adj}} - N_\\textbf{F} - 8 ) \\phi_2 ].\n\t\\end{split}\n\t\\end{align}\nWe set $N_{\\textbf{adj}} =0$ to be consistent with $\\mathcal N = 1$ supersymmetry. By giving a nonzero value to mass parameters in the hypermultiplet contributions to the prepotential, one can study the RG flow from $N_{\\textbf{F}}$ to $N_{\\textbf{F}}-1$ flavors. In order to decouple a massive hypermultiplet, the theory must pass through three phase transitions. These four phases have the following prepotentials (we omit mass parameter terms for brevity): \n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\\label{eqn:G2RG}\n\t\t\t6 \\mathcal F_{}^{(1)} &=(8- 8N_\\textbf{F}) \\phi_1^3 + 8 \\phi_2^3 + 3 \\phi_1 \\phi_2 [\\phi _1 \\left(3 N_{\\textbf F}+6\\right)-\\phi _2 \\left(N_{\\textbf F}+8\\right)]\\\\\n\t\t\t6 \\mathcal F_{}^{(2)} &=(16-8N_\\textbf{F}) \\phi_1^3 + 7 \\phi_2^3 + 3 \\phi_1 \\phi_2 [ \\phi _1 \\left(3 N_{\\textbf F}+2\\right)-\\phi _2 \\left(N_{\\textbf F}+6\\right)]\\\\\n\t\t\t6 \\mathcal F_{}^{(3)} &=(15 - 8 N_\\textbf{F}) \\phi_1^3  + 8 \\phi_2^3 + 3 \\phi_1 \\phi_2 [ \\phi _1 \\left(3 N_{\\textbf F}+3\\right)-\\phi _2 \\left(N_{\\textbf F}+7\\right) ]\\\\\n\t\t\t6 \\mathcal F_{}^{(4)} &=6 \\mathcal F_{N_\\textbf{F}-1}^{(1)}.\n\t\t\\end{split}\n\t\\end{align}\n\t\nWe determine a shrinkable K\\\"ahler surface $S$ that engineers this theory by setting the triple intersection polynomial (\\ref{eqn:geotrip}) equal to prepotential (\\ref{eqn:G2nomass}) and demanding that there exist an intersection matrix $f_i \\cdot S_j = (A_{G_2})_{ij}$ for some choice of fiber classes $f_i \\subset S_i$. Restricting the possible building blocks to be blowups of rational and ruled surfaces \\emph{without self-gluing}, the only solutions to these conditions are the geometries shown in Table \\ref{tab:G2geo}. For all of these surfaces we have $9n_2+6a=2g-2+n_1$, as required by (\\ref{eq:gluingcond}).  A key point here is that the surface $S_1$ must be a ruled surface of a curve of genus $g = N_{\\textbf{F}}$. This is precisely the geometric setup described in \\cite{Diaconescu:1998cn}. \n\n\\begin{table}\n\t\\begin{center}\n$\n\t\t\t\\begin{array}{|c|c|c|c|}\n\t\t\\hline\n\t\t\tg & a & (n_1,n_2)  \\\\\\hline\n\t\t\t0  & 1 & (8,0) \\\\\\hline\n\t\t\t1  & 0 & (9,1)  \\\\\\hline\n\t\t\t2  & 2 & (10,0)  \\\\\\hline\n\t\t\t3  & 1 & (11,1)  \\\\\\hline\n\t\t\t4  & 0 & (12,2)  \\\\\\hline\n\t\t\t4  & 3 & (12,0)  \\\\\\hline\n\t\t\t5  & 2 & (13,1)  \\\\\\hline\n\t\t\t6  & 4 & (14,0) \\\\\\hline\n\t\t\\end{array}\n\t\t$\n\t\\end{center}\n\t\\caption{Shrinkable surfaces $S = \\mathbb F^{g}_{n_1} \\cup \\mathbb F_{n_2}$ engineering $G_2 + N_\\textbf{F} \\textbf{F}$ gauge theories. The surface $\\mathbb F^g_{n_1}$ is a ruled surface over a curve $E$ with $g(E) = N_{\\textbf{F}}$ and satisfying $E^2 = -n_1$. The gluing curve $C = S_1 \\cap S_2$ is given by $C_{S_1} = E$ and $C_{S_2} = a F + 3 H$. The fiber classes are given by are $f_i = F_i$.}\n\t\\label{tab:G2geo}\n\t\\end{table}\n\n\n\nWe now demonstrate that we can engineer the same family of theories described above by replacing $S_1$ with the surface $S_1' =  \\text{Bl}_{ 2g} \\mathbb F_{n_1}^{(g)} $, where again $g = N_{\\textbf{F}}$ and the superscript notation indicates $S_1'$ is obtained by identifying $g$ pairs of exceptional curves in $\\text{Bl}_{2g} \\mathbb F_{n_1}$ (i.e. self-gluing; see Appendix \\ref{app:math} for some mathematical background.) This shrinkable surface not only reproduces the prepotential (\\ref{eqn:G2nomass}) and $G_2$ Cartan matrix, but also has the merit of exhibiting the RG flow (\\ref{eqn:G2RG}) in a very natural manner. The four phases, related by flops, have the following geometries: \n \\begin{enumerate}\n\t\t\\item $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g} \\cup  \\mathbb F_{n_2}$, where the blowups are all at special points\\footnote{Note that while we consider blowups at special points $F \\cap E \\subset \\mathbb F_n$ here for convenience, since we do not introduce any additional irreducible curves with self intersection less than $-1$, we can without loss of generality view a blowup of $\\mathbb F_n$ at $p$ special points as a blowup of $\\mathbb F_{n+p}$ at $p$ general points. We explore the distinction between special and general points in more depth in Section \\ref{sec:rank2}.} $F \\cap E$.\n\t\t\\item  $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{8- g} \\cup \\text{Bl}_1 \\mathbb F_{n_2}$.\n\t\t\\item  $\\text{Bl}_{2g-1} \\mathbb F^{(g-1)}_{8- g} \\cup  \\mathbb F_{n_2\\pm{} 1}$.\n\t\t\\item $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{9- g} \\cup \\mathbb F_{n_2 \\pm{} 1}$. \n\t\\end{enumerate}\n\n\tThe first phase is $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g} \\cup  \\mathbb F_{n_2}$, where we introduce $g$ self-gluings of $\\text{Bl}_{2g} \\mathbb F_p$ along the pairs of exceptional divisors $X_{2i}, X_{2i-1}, i = 1, \\dots, g$,\\footnote{Here and in the sequel, we use the notation $X_i$ to denote the exceptional divisor of the $i$-th blowup, since we reserve the more standard notation $E_i$ for sections of Hirzebruch surfaces.}the  where the gluing curve is defined by $C_{S_1} = E - \\sum_{i=1}^{2 g} X_i$ and $C_{S_2} =  F + 3 H$, so that $a=1$ in the notation adopted in the caption of Table~\\ref{tab:G2geo}. Since the canonical class\\footnote{More precisely, the dualizing sheaf of the singular surface $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g}$, pulled back to its natural desingularization $ \\text{Bl}_{2g} \\mathbb F_{8-g}$.} is given by $K_{\\mathbb F_{8 - g}} + 2\\sum_{i=1}^{N_\\textbf{F}} (X_{2i-1}+X_{2i})$, we find a perfect match with the first line of (\\ref{eqn:G2RG}), using the adjunction relation $9n_2+6-(8+g)=2g-2$.\n\t\n\tWe now describe the flop to the second phase. The matter curve with volume $2\\phi_1 - \\phi_2$ which shrinks is one of the self-gluing exceptional divisors, say $X_1$. Blowing down $X_1$ forces us to also blow down $X_2$.\nWe can blow up $\\mathbb F_{n_2}$ at a generic point $F_2 \\cap H_2$ if we eventually want to decrease $n_2$ to $n_2 -1$, or at a special point $F_2 \\cap E_2$ if we want to increase $n_2$ to $n_2 +1$ in the third phase.\n\t\nThe geometry of the second phase is $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{8- g} \\cup \\text{Bl}_1 \\mathbb F_{n_2}$, where $C_{S_1} = E -\\sum_{i=1}^{2g -2} X_i$ and $C_{S_2}=a F + 3 H - 2 Y_1$. Since the blowup of ${\\mathbb F}_{n_2}$ is at the double point of $E$ introduced by gluing $X_{2g-1}$ to $X_{2g}$, the coefficient of $Y$ in $C_{S_2}$ is $-2$. \n\nThe matter curve with volume $\\phi_2 - \\phi_1$ which we blow down is $F_2 - Y_1 \\subset \\text{Bl}_1 \\mathbb F_{n_2}$. Because $F - Y_1$ meets $C$ in one point, we must introduce an exceptional divisor $Y_2$ in the surface $S_1$, leading us to the third phase.\n\t\n\tThe geometry of the third phase is $\\text{Bl}_{2g-1} \\mathbb F^{(g-1)}_{8- g} \\cup  \\mathbb F_{n_2\\pm{} 1}$, where $C_{S_1} = E - \\sum_{i=1}^{2g-2} X_i - Y_2$. Concerning the gluing curve class $C \\subset \\mathbb F_{n_2 \\pm{} 1}$, there are two possible cases. In the case of a generic blowup, the proper transforms of $H, F \\subset S_2$ are  $H- Y_1, Y_1$, so we set $C_{S_2} = (a +1)  F + 3  H$, where now $ H^2_{S_2} = n_2 -1$. It follows that $C^2_{S_2} =((a +1)  F + 3  H)_{S_2}^2 = 6 (a + 1) + 9 (n_2-1) = 3 g + 3$, which is a nontrivial check that this geometry is consistent with the phase structure of the $G_2$ theory. On the other hand, in the case of a special blowup, the difference is that the proper transform of $H \\subset S_2$ is $H$, so that $C_{S_2} = H + (a - 2)  F$, where now $H_{S_2}^2 = n_2 +1$. We again confirm that $C^2_{S_2} = ((a -2)  F + 3  H)_{S_2}^2  = 6 (a - 2) + 9 (n_2 +1) = 3 g+ 3$.\n\t\t\n\t\tIn order to reach the fourth and final phase, the matter curve with volume $\\phi_1$ which we blow down is $F - Y_2 \\subset S_1$. The geometry of the fourth phase is $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{9- g} \\cup \\mathbb F_{n_2 \\pm{} 1}$. Keeping in mind the previous identity $n_1 = 8- g$ along with the fact that we blow down the curve $F - Y_2 \\subset S_1$, we compute the canonical class:\n\t\t\\begin{align}\n\t\t\\begin{split}\n\t\t\tK_{S_1} &= -2 H + (n_1 -2) F + 2\\sum_{i=1}^{g-1} (X_{2i-1} + X_{2i} ) + Y_2\\\\\n\t\t\t&= -2 H + ((n_1+1) -2) F + 2 \\sum_{i=1}^{g-1} (X_{2i-1} + X_{2i} ) .\n\t\t\\end{split}\n\t\t\\end{align}\n\tNote also that the self-intersection of $H \\subset S_1$ shifts from $8-g$ to $9-g$.\n\n\n\n\\subsubsection{Hanany-Witten transitions and complex deformations}\nThe next type of transition we will discuss is a \\emph{complex structure deformation}.\nIn particular, we concern ourselves with two types of complex structure deformations that preserve the rank of the 3-fold. The first type of complex structure deformation is a Hanany-Witten (HW) transition \\cite{Hanany:1996ie}. This type of transition is most easily understood in the setting of $(p,q)$ 5-brane webs, and involves interchanging the relative position of a $(p,q)$ 7-brane and a $(p,q)$ 5-brane. After the transition, despite the fact that the brane webs look different, in the low-energy decoupling limit the corresponding SCFTs describe the same physics up to decoupled free sectors. The example displayed in Figure \\ref{fig:HW} describes a geometric (or HW) transition from a local 3-fold $X$ with $S= \\mathbb F_2$ to another 3-fold $X'$ with $S' = \\mathbb F_0$. Therefore, $X$ and $X'$ are physically equivalent.\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[scale=.5]{F0-F2-HW.pdf}\n\\end{center}\n\\caption{Hanany-Witten transition from $\\mathbb F_2$ to $\\mathbb F_0$. The $\\otimes$ symbol denotes the location of a transverse $(0,1)$ 7-brane, and the dashed line denotes the location of the 7-brane monodromy cut.}\n\\label{fig:HW}\n\\end{figure}\n\nThis example can be geometrically described as follows:    ${\\mathbb F}_2$ is physically equivalent to ${\\mathbb F}_0$ by a (non-normalizable) complex structure deformation.  One way to see this is to first contract the curve $E$ in ${\\mathbb F}_2$ (with $E^2=-2$) to an $A_1$ singularity, which can be identified with the quadric cone $x^2+y^2+z^2=0$ in ${\\mathbb P}^3$.  A complex structure deformation takes this to a smooth quadric surface (e.g.\\ $w^2+x^2+y^2+z^2 =0$), which is isomorphic to ${\\mathbb P}^1\\times{\\mathbb P}^1={\\mathbb F}_0$.\n\nAnother type of complex structure deformation involves changing special type blow ups (i.e. blow ups on top of blow ups) to generic blow ups, where the blow up points are not on top of one another, unless the blow up curve is part of the identification between $S_i$'s.  We will show that in the rank 2 case this can be avoided and we can always assume general point blow ups.\n \n \n\n\n\n\n\n\n\\section{Classifications}\n\\label{sec:classification}\n\n\nLet $S=\\cup S_i$ be a connected union of surfaces contained in a CY 3-fold $X$. We classify all shrinkable $S$ for rank 1 and rank 2 according to the conjectures and algorithm described in Section \\ref{sec:algorithm}.\nWe first summarize the rank 1 and rank 2 classification results and in the next two subsections we present details of the classification. \n\nAll rank 1 and rank 2 shrinkable geometries (or SCFTs) belong to one or more families of geometric RG-flows, and the geometries in each RG-flow family are related by rank-preserving mass deformations (or blowdowns of -1 curves in geometric terminology), up to physical equivalence. The ideas of geometric RG-flow and rank-preserving mass deformations will be discussed later.\nBased on these ideas, we can start from a ``top'' geometry, which corresponds to a 5d CFT or a 6d CFT on a circle (equivalently, a 5d Kaluza-Klein (KK) theory), and obtain all other geometries in the same family by a finite sequence of geometric transitions or mass deformations. This UV geometry is at the top of the RG-flow in a given family and can therefore be a representative of the entire RG-flow family. We conjecture that all descendants of the top UV geometry engineer 5d SCFTs. When shrinkable, the top UV geometry itself also engineers a 5d SCFT.\n\nFor rank 1 geometries, we have only one RG-flow family corresponding to a local elliptic 3-fold defined by the del Pezzo surface $\\text{dP}_9$. All other rank 1 geometries are obtained by blowing down exceptional curves. The RG-flow family of $\\text{\\text{dP}}_9$ involves other del Pezzo surfaces $\\text{dP}_n$ with $n\\le 8$ and a Hirzebruch surface $\\mathbb{F}_0$; it is believed that these are the complete set of geometries leading to rank 1 5d SCFTs. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|}\n\t\\hline\n\t $S=S_1\\cup S_2$ & $G$  \\\\\n\t\\hline\n\n\n\t$(\\mathbb{F}_6\\cup \\text{dP}_4)^*$ & $Sp(2)_{\\theta=0} + 3\\textbf{AS}$ \\\\\n\t\\hline\n\t$(\\mathbb{F}_2\\cup \\text{dP}_7)^*$ & $SU(3)_4 + 6\\textbf{F}$ \\\\\n\t & $Sp(2)+ 4 \\textbf{F} + 2\\textbf{AS}$ \\\\\n\t & $G_2 + 6 \\textbf{F}$ \\\\\n\t\\hline\n\t$(\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0)^*$ & $SU(3)_{\\frac{3}{2}} + 9 \\textbf{F}$ \\\\\n\t & $Sp(2)+8\\textbf{F} + \\textbf{AS}$ \\\\\n\t\\hline\n\t$(\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0)^*$ & $SU(3)_0+10 \\textbf{F}$ \\\\\n\t & $Sp(2)+ 10\\textbf{F}$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Rank 2 geometries with maximal $M$. In the above table, $S$ is the rank 2 K\\\"ahler surface, while $G$ is the corresponding gauge theory description. These geometries denoted as $(\\cdot)^*$ are not shrinkable and correspond to 5d KK theories.}\n\t\\label{tb:rank2-classification}\n\\end{table}\nSimilarly, the top rank 2 geometries are summarized in Table \\ref{tb:rank2-classification}. We have identified four geometric RG-flow families represented by these top geometries. These geometries are not shrinkable; rather, we expect that these geometries have 6d UV completions and thus they engineer 5d KK theories. However, their descendants, obtained by blowing down $-1$ curves, are shrinkable and therefore give rise to 5d SCFTs.\nFor example, the geometry $\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0$ is ruled out from our CFT classification because its building block $\\text{Bl}_9\\mathbb{F}_4$ has an infinite number of Mori cone generators as explained in Appendix~\\ref{sec:mori}, violating our criterion in Section \\ref{sec:buildingblocks}. However, a geometric RG-flow from this geometry by blowing down an exceptional curve as well as a number of flop transitions leads to the geometry $\\text{Bl}_8\\mathbb{F}_3\\cup \\text{dP}_1$ which is now shrinkable and engineers a 5d SCFT. Similarly, other geometries in Table \\ref{tb:rank2-classification} are associated to KK theories, but their descendants are shrinkable. Therefore, we find that all rank 1 and 2 smooth 3-fold geometries engineering 5d SCFTs are mass deformations of 5d KK theories. See Section \\ref{sec:rank2} for further discussion.\n\nThis result confirms the existence of many new rank 2 SCFTs predicted in \\cite{Jefferson:2017ahm} which are listed in Table \\ref{tb:rank2-gauge-theory-clssification}.\nFor example, the $SU(3)_7$ gauge theory is predicted to exist in Table \\ref{tb:SU3-classification}. This theory turns out to have a geometric realization as $\\mathbb{F}_0\\cup \\mathbb{F}_8$ which is a descendant of $\\mathbb{F}_2\\cup \\text{dP}_7$. This implies that the gauge theory approach in \\cite{Jefferson:2017ahm}, which analyzes the magnetic monopole and  perturbative BPS spectrum, is quite powerful and capable of predicting new interacting 5d SCFTs.\n\nOur study also reveals that there are no smooth 3-fold geometries associated to the following gauge theories:\n\\begin{align}\n\\begin{split}\n\\label{eqn:ruleout}\n\t&SU(3)_{\\frac{1}{2}} + 1\\bf{Sym} \\ , \\\\\n\t& SU(3)_{7} +2 {\\bf F} \\ \\rightarrow \\ SU(3)_{\\frac{15}{2}}+1{\\bf F} \\ \\rightarrow \\ SU(3)_8 \\ .\n\\end{split}\n\\end{align}\nThese theories are expected to have interacting CFT fixed points by the perturbative gauge theory analysis in \\cite{Jefferson:2017ahm}. See Table \\ref{tb:SU3-classification}.\nThe SCFT of the first gauge theory indeed exists---this theory is a mass deformation of the $SU(3)_0$ theory with $N_{\\bf Sym}=1,N_{\\bf F}=1$ whose brane construction is given in \\cite{Bergman:2015dpa,Hayashi:2015vhy}. Our study of smooth 3-folds fails to capture this theory. The reason for this failure is because the corresponding geometry involves a `frozen' singularity. For example, the brane construction in \\cite{Bergman:2015dpa,Hayashi:2015vhy} contains O7$^+$-planes; indeed, constructions involving O7$^+$ planes are dual to frozen singularities involving non-geometric monodromies and a fractional M-theory 3-form background as discussed in \\cite{Tachikawa:2015wka}. Therefore, we do not expect that our analysis can capture this type of singularity, and hence the geometric classification in this paper is incomplete in this sense.\nWe nevertheless conjecture that our classification includes all 5d SCFTs coming from {\\it smooth} Calabi-Yau threefolds which do not involve frozen singularities dual to brane constructions involving O7$^+$ planes.\nIn the following sections, we classify smooth rank 1 and rank 2 3-fold geometries engineering 5d SCFTs in their singular limits.\n\nOn the other hand, we predict that there are no SCFTs corresponding to three gauge theories belonging to the RG flow in the second line of (\\ref{eqn:ruleout}). As we discuss in Section \\ref{sec:rank2}, despite the fact that these gauge theories can be realized geometrically using our algorithm, they are shrinkable only when we attach a number of non-degenerate non-compact 4-cycles to the compact surface $S$. Introducing these non-compact 4-cycles entails non-normalizable K\\\"ahler deformations which in the field theory setting corresponds to introducing nonzero mass parameters. We find that these mass parameters cannot be set to zero in the CFT limit---at small nonzero values, the corresponding geometries develop at least one 2-cycle with negative volume and therefore their singular limits do not engineer well-defined CFT fixed points. \nThis computation excludes the three gauge theories in the second line of (\\ref{eqn:ruleout}) as possible candidates for interacting 5d SCFTs. This is also an indication that the classification criteria described in \\cite{Jefferson:2017ahm} are necessary, but not sufficient to identify 5d SCFT fixed points. The criteria of \\cite{Jefferson:2017ahm} must be modified to account for non-perturbative BPS states (such as instantons in gauge theories) in order to be both necessary and sufficient. \n\nWe also remark that a single 3-fold $X$ can admit multiple gauge theory descriptions.\nThis is possible because some geometries admit more than one distinct choice of fiber class associated to charged gauge bosons. The existence of multiple gauge theoretic descriptions corresponding to a single geometry suggests that the gauge descriptions are dual to one another. Starting with the the ``top'' UV geometries in Table \\ref{tb:rank2-classification}, we predict the following dualities:\n\\begin{align}\n\\begin{split}\n\\label{eqn:dual}\n\t SU(3)_{5-\\frac{N_{\\bf F}}{2}} + N_{\\bf F} {\\bf F} ~&\\cong  ~Sp(2)+N_{\\bf F}{\\bf F} \\ , \\quad N_{\\bf F} \\le 10 \\\\\n\tSU(3)_{6-\\frac{N_{\\bf F}}{2}}+ N_{\\bf F}{\\bf F} ~&\\cong ~Sp(2)+1{\\bf AS}+(N_{\\bf F}-1){\\bf F} \\ , \\quad 1 \\le N_{\\bf F} \\le 9 \\\\\n\t SU(3)_{7-\\frac{N_{\\bf F}}{2}}+N_{\\bf F}{\\bf F} ~&\\cong ~ G_2 + N_{\\bf F}{\\bf F}   ~ \\overset{2\\le N_{\\bf F}}{\\cong}~ Sp(2)+ 2{\\bf AS}+(N_{\\bf F}-2){\\bf F} \\ , \\quad N_{\\bf F} \\le 6\n\\end{split}\n\\end{align}\nThe first and the second dualities in (\\ref{eqn:dual}) were conjectured already in \\cite{Gaiotto:2015una} and in \\cite{Jefferson:2017ahm}, respectively. So our construction provides concrete geometric evidence for these duality conjectures. On the other hand, the third duality is a new duality discovered by an explicit geometric construction in this section.\n\n\\subsection{Rank 1 classification}\n\\label{sec:rank1}\n\nWe warm up by starting with rank 1, recovering the result that all rank 1 5d SCFTs are geometrically engineered by local 3-folds containing a del Pezzo surface.  More precisely, our algorithm identifies del Pezzo surfaces as shrinkable, but also identifies additional shrinkable surfaces; however, each of these turns out to be physically equivalent to a del Pezzo surface.\n\n\nRecall that a del Pezzo surface $S$ is defined to be a smooth algebraic surface whose anticanonical bundle $-K_S$ is ample---this means that $-K_S \\cdot C > 0$ for all effective curves $ C \\subset S$.  The classification of del Pezzo surfaces is well known: $S$ is either $\\text{dP}_n$ for $0\\le n\\le 8$ or ${\\mathbb P}^1\\times{\\mathbb P}^1={\\mathbb F}_0$.  Such a surface satisfies (\\ref{eq:shrinkability}) as well as $K_S^2>0$, so is shrinkable.  We now set out to systematically classify rank 1 shrinkable surfaces up to physical equivalence.\n\n\n\n\nTo apply (\\ref{eq:shrinkability}), we need to know $K_S$, the generators of the Mori cone of curves on $S$, and the intersection numbers of the curves in $S$.  Our algorithm leads us to consider ${\\mathbb P}^2$, ${\\mathbb F}_n$, and their generic blowups. \n\n${\\mathbb P}^2$ is del Pezzo, but it is instructive to check shrinkability anyway. For ${\\mathbb P}^2$, the Mori cone is generated by the class $\\ell$ of a line, $\\ell^2=1$, and \n$K_{{\\mathbb P}^2}=-3\\ell$. So $K_{{\\mathbb P}^2}^2=9>0$ and $K_{{\\mathbb P}^2}\\ell=-3<0$, so ${\\mathbb P}^2$ is shrinkable.\n\nNext, we consider ${\\mathbb F}_0$,\\ ${\\mathbb F}_1$ and ${\\mathbb F}_{n \\geq 2}$ separately.  Since ${\\mathbb F}_1$ is the blowup of ${\\mathbb P}^2$ at a point, ${\\mathbb F}_1$ and its generic blowups are just the generic blowups of ${\\mathbb P}^2$.  Similarly, ${\\mathbb F}_0$ is del Pezzo, and the blowup of ${\\mathbb F}_0$ at a point is isomorphic to the blowup of ${\\mathbb P}^2$ at two points \\cite{GH}.  So the possibilities for $S$ can be reduced to either generic blowups of ${\\mathbb P}^2$, or ${\\mathbb F}_{n \\geq 2}$.\n\nAs usual, we denote by  $\\text{dP}_n$ the blowup of ${\\mathbb P}^2$ at general points\n$p_1,\\ldots,p_n$.  Let $X_1,\\ldots, X_n$ denote the corresponding\nexceptional ${\\mathbb P}^1$'s,\\footnote{As noted earlier, we reserve the more customary notation\n  $E$ for the curves on Hirzebruch surfaces described in Appendix~\\ref{app:Mori}.} and we let $\\ell$ denote\nthe class of the total transform in $\\text{dP}_n$ of a line in ${\\mathbb P}^2$.  The intersection numbers are\n\\begin{equation}\n  \\label{eq:intp2}\n  \\ell^2=1,~~\\ X_i\\cdot X_j = -\\delta_{ij},~~\\ \\ell\\cdot X_i=0 \n\\end{equation}\nand $K_{\\text{dP}_n}=-3\\ell+\\sum_{i=1}^nX_i$.  Then $K_{\\text{dP}_n}^2=9-n>0$ for $n\\le 8$.\n\nWe first observe that $\\text{dP}_n$ is not shrinkable for $n\\ge9$.  To see this, we simply observe that $K_{\\text{dP}_n}^2\\le0$ for $n\\ge9$ which implies that the string tensions are not positive.\n\nAgain, we can cite known results simply say that $\\text{dP}_n$ is shrinkable for $n\\le8$, but it is  instructive to work out details without assuming this fact.\nWe adopt a convenient shorthand to describe the generators of the Mori cone: Any curve\n$C\\subset \\text{dP}_n$ other than the $X_i$ will project to a curve $D\\subset {\\mathbb P}^2$ of some degree $d>0$.\nLet $m_i$ be the multiplicity of $D$ at $p_i$, so that $m_i=0$ if $p_i\\not\\in D$, $m_i=1$\nif $p$ is a nonsingular point of $D$, $m_i=2$ if $p$ is a node or cusp of $D$, etc.  Then \nthe class of $C$ is $d\\ell-\\sum_{i=1}^n a_i X_i$.  It is customary to abbreviate this class as\n$(d;m_1,\\ldots,m_n)$, as well as to omit any $m_i$ which are zero.  Then the\nMori cone of $\\text{dP}_n$ is generated by the classes\\footnote{Strictly speaking, we have only written the Mori generators for $n=8$.\nFor $n<8$, we modify (\\ref{eq:moridp}) by removing those generators which need more than $n$ exceptional divisors to define them. In addition, for\n$n=1$, we include $(1;1)$ as a generator.}\n\\begin{equation}\nX_i,\\ (1;1^2),\\ (2,1^5),\\ (3,2,1^6),\\ (4,2^3,1^5),\\ (5,2^6,1^2),\\ (6;3,2^7)\n  \\label{eq:moridp}\n\\end{equation}\nup to permuting the order of the $p_i$.  It follows from the adjunction formula (\\ref{eq:adjunction}) that each of the curve\nclasses $C$ in (\\ref{eq:moridp}) satisfies $K_{\\text{dP}_n}\\cdot C=-1$,\\footnote{ For $n=1$, we also check that $K_{\\text{dP}_1}\\cdot(\\ell-X_1)=-2$.} so $\\text{dP}_n$ is shrinkable. \n\n\nNext, consider the Hirzebruch surfaces $S={\\mathbb F}_n$.  \nUsing the notation in Appendix~\\ref{app:Mori}, there are two disjoint toric sections $E,H$\nand the fiber class $F$.  These classes satisfy\n\\begin{equation}\n  \\label{eq:intfn}\n  H^2=n,\\ E^2=-n,\\ H \\cdot E=0,\\ H \\cdot F=E\\cdot F=1,\\ F^2=0,\\ H=E+nF.\n\\end{equation}\nThe canonical bundle of ${\\mathbb F}_n$ is $K_{{\\mathbb F}_n}=-2H+(n-2)F$ and so\n   $K_{{\\mathbb F}_n}^2=8>0$.  Furthermore,  \n  the Mori cone of effective curves is generated by $E$ and $F$.  While $K_{{\\mathbb F}_n} \\cdot F=-2<0$,\n  we also have $K_{{{\\mathbb F}}_n}\\cdot E=n-2$, which is strictly negative for $n<2$,  zero for $n=2$, but\n  strictly positive for $n>2$.  Thus ${\\mathbb F}_2$ is shrinkable.  However, as discussed in section 3, this is physically equivalent to ${\\mathbb F}_0$.\n  The same reasoning combined with the earlier observation that $\\text{Bl}_1{\\mathbb F}_0\\simeq \\text{dP}_2$ shows that $\\text{Bl}_p \\mathbb F_2$ is physically equivalent to $\\text{dP}_{p+1}$.\n\n  In conclusion, all rank 1 shrinkable surfaces are physically equivalent to $\\text{dP}_n$ for some $n$ or ${\\mathbb F}_0$.\n\n\\subsection{Rank 2 classification}\n\\label{sec:rank2}\n\nThe main result of this paper is a full classification of shrinkable rank 2 geometries up to physical equivalence. We preface our result by arguing some further simplifying assumptions we make about the surface $S$ in order to make the classification into a manageable problem.\n\n\\subsubsection*{Three simplifications}\nIn this section we show that we can utilize the following three simplifying assumptions for classifying shrinkable rank 2 surfaces:\n\n\\begin{itemize}\n\\item $S_1 \\cap S_2$ is an irreducible curve.\n\\item $S_1 \\cap S_2$ is a rational curve. \n\\item The surfaces $S_i$ are equal to ${\\mathbb P}^2$ or Hirzebruch surfaces and their blowups at general points.\n\\end{itemize}\nWe now discuss these three simplifications in order.\n\nFirst, we argue that in the case of a rank 2 surface $S = S_1 \\cup S_2$, we can assume that $S_1$ is not glued to $S_2$ along multiple curves. Namely, there exists a single edge between two nodes.\nSuppose we glue two surfaces along $C_1,C_2$ with appropriate identifications. Since $S_1$ and $S_2$ should intersect transversally, we have $(C_1 \\cdot C_2)_{S_1} = (C_1 \\cdot C_2)_{S_2}= 0$. This means that $C_1, C_2$ do not intersect. \nWe claim there always exists an effective curve $D=d_1+d_2$ such that ${\\rm vol}(D) \\le 0$. If ${\\rm vol}(D)<0$, then $S$ is not shrinkable, so it suffices to consider the situation where ${\\rm vol}(D)=0$.  But in that case, we will further show below that we can arrange for the curve $D$ to be elliptic (i.e. $g(D) = 1$), which would contradict our conjectures. Therefore, the full surface is not shrinkable implying that we cannot glue two surfaces along two or more curves.\n\nIn order to show this, we first prove that there always exist curves $d_i \\subset S_i$ with  $K_{S_i}\\cdot d_i\\ge-2$ that intersect both $C_1$ and $C_2$. These classes $d_1$ and $d_2$ are identified as follows. \nFirst, if both $C_1$ and $C_2$ are not fiber classes, we can always find a curve $d_1$ satisfying these conditions among $\\{F, \\, F-X_i,\\, H-X_i-X_j \\}$\\footnote{For general $n$ we choose $d_1=F-X_i$ if $C_1=X_i$ or $C_2=X_i$, otherwise $d_1=F$. When $n=2$ and $C_1=X_1,C_2=X_2$, we choose $d_1=H-X_1-X_2$.}  in $\\text{Bl}_p\\mathbb{F}_n$, where $X_i$ are exceptional curves associated to the blowups of $\\mathbb F_n$ at $p$ general points.  When $n>2$, $C_1 =E$, otherwise the volume of the curve $E$ will be negative.\nNext, suppose $C_1$ or $C_2$ is a fiber class. This is possible only when $S_1=\\text{Bl}_p\\mathbb{F}_1$ or $\\text{dP}_n$, otherwise the class $E$, which has $E\\cdot C_1\\neq0$ or $E\\cdot C_2 \\neq 0$, will have negative volume thus preventing the surface $S$ from being shrinkable. In the case that $S_1=\\text{Bl}_p\\mathbb{F}_1$, when $C_1$ is a fiber class $F_1$, $C_2$ must be one of $X_i$'s, due to the assumption of transversal intersection. Then we can take $d_1=H-X_i$ with $H^2=1$.\nWith any choice of $d_1$ given here, we find that ${\\rm vol}(d_1)=m\\phi_1 - n\\phi_2$ with $m=1,2$ and $n\\ge2$ where $\\phi_i \\geq 0$.\nWe can choose $d_2 \\subset S_2$ in the same manner and then show that  ${\\rm vol}(d_2)=m'\\phi_1 - n'\\phi_2$ with $m'=1,2$ and $n'\\ge2$.\n\nThis proves ${\\rm vol}(D) \\le 0$ for an effective curve $D=d_1+d_2$. Now we will assume ${\\rm vol}(C_i)\\ge 0$ for all other curves $C_i$ because otherise the surface is not shrinkable and already ruled out. As already noted above, it is clear that the total surface is not shrinkable when ${\\rm vol}(D)<0$. Moreover, when ${\\rm vol}(D) = 0$, i.e. when $m=m'=n=n'=0$, the curves $d_1$ and $d_2$ are both fiber classes $F_i\\subset S_i$. In this case, the curve $F_1$ and $F_2$ can be deformed so that $F_1\\cap C_i=F_2\\cap C_i$ for $i=1,2$.  Then the curve $D=F_1+F_2$ is the union of two rational curves intersecting in two points, hence elliptic.  By further complex structure deformation if necessary, we can arrange that all fibers $F_1$ of $S_1$ meet all fibers $F_2$ of $S_2$ in two points, or in other words, that $S=S_1\\cup S_2$ is elliptically fibered.\n\nWe argue that we can deform the complex structure of $X$ if necessary so that $X$ is also elliptically fibered.  To see this, let $E$ be an elliptic fiber of $S$.  Since $E$ is part of an elliptic fibration of $S$, we have that $N_{E\/S}\\simeq\\mathcal{O}_E$.  Furthermore, $\\det(N_{E\/X})$ is trivial by the Calabi-Yau condition and the ellipticity of $E$.  Then the normal bundle sequence\n\\begin{equation}\n  \\label{eq:nbs}\n0 \\to N_{E\/S} \\to N_{E\/X} \\to N_{S\/X}|_E \\to 0  \n\\end{equation}\nis identified with\n\\begin{equation}\n  \\label{eq:Atiyah}\n  0 \\to \\mathcal{O}_E \\to N_{E\/X} \\to \\mathcal{O}_E \\to 0.\n\\end{equation}\nHowever, since $H^1(\\mathcal{O}_E) \\ne 0$, (\\ref{eq:Atiyah}) generically does not split\\footnote{The non-splitting of (\\ref{eq:Atiyah}) identifies $N_{E\/X}$ as the Atiyah bundle on $E$.} and dim $H^0(N_{E\/X})=1$.  The uniqueness of a normal direction says that $E$ moves in a 1-parameter family, enough deformations to fiber $S$ but not enough to fiber $X$.\n\nHowever, we can choose a complex structure deformation of $X$ so that (\\ref{eq:Atiyah}) splits, and then $N_{E\/X}\\simeq\\mathcal{O}_E^2$.  In this situation, $E$ moves in two independent directions and fibers $X$.\n\n   This justifies our claim, hence $S$ is not shrinkable. The same argument holds for cases with more than two edges (i.e. gluing curves) between $S_1$ and $S_2$. Therefore rank 2 geometries formed by two surfaces glued along two or more different curves are not shrinkable.\n\nSecond, we claim that the gluing curves must be rational. Suppose $C = S_1 \\cap S_2$ has $g>0$. In Appendix~\\ref{app:Mori} we explain that we must have finitely many Mori cone generators in each $S_i$ (which implies a bound on the number of blowups), hence we have finitely many Mori cone generators in $X \\supset S = S_1 \\cup S_2$.  We argue that this implies $C_{S_i}^2 \\ge 0$ as follows. \nWe assume $C_{S_i}^2<0$ and derive a contradiction.  Since $C_{S_i}^2+C \\cdot K_{S_i}=2g-2\\ge0$, we have $C\\cdot K_{S_i}>0$.  \nAnticipating the next bulleted claim that the building blocks are generic blowups of Hirzebruch surfaces at a bounded number of points, we show in Appendix~\\ref{app:Mori} that $C_{S_i} \\cdot K_{S_i}>0$ implies $C_{S_i}=E$.  This is a contradiction, since $g>0$.\nAlthough this argument is slightly circular in its current form depending as it does on the next bulleted claim, we believe that with further care we can independently justify $C_{S_i}^2\\ge0$.  Furthermore, an extensive computer search has revealed no counterexamples.\n\n\nLet us now return to the claim that the gluing curves are rational. Recalling equations (\\ref{eq:gluingcond}) and (\\ref{eq:adjunction}), we have\n\\begin{equation}\n\t C^2_{S_1} + C^2_{S_2}  = C_{S_i}^2 + K_{S_i}\\cdot C = 2g-2 \\ .\n\\end{equation}\nThese conditions tell us that $K_{S_i} \\cdot C\\ge0$. This implies that the volume of the intersection curve, ${\\rm vol}(C)=-\\phi_1 K_{S_1}\\cdot C -\\phi_2 K_{S_2}\\cdot C$, is negative unless $C^2_{S_1}=C_{S_2}^2=0$ and $g=1$, i.e. unless $C$ is an elliptic curve. This proves that rank 2 geometries containing two surfaces meeting in a curve with genus $g>0$ are not shrinkable.\n\n\nThird, we observe that many of the building blocks in our classification program are related to one another by maps (for instance, isomorphisms and complex deformations) which at the level of 5d SCFT physics constitute physical equivalences. Therefore, we observe that the full number of rank 2 surfaces that can be constructed from our list of building blocks dramatically overcounts the number of unique CFT fixed points, and hence we can reduce the complexity of the problem at the outset by restricting our attention to a minimal representative set of configurations capturing the full list of physical equivalence classes. We will argue in particular that we need only consider configurations $S = S_1 \\cup S_2$ for which $S_1$ is a blowup of $\\mathbb F_{n> 0}$ at $p$ generic points\\footnote{By ``generic point'', we mean a point not contained in any exceptional divisors, i.e.\\ rational curves with self intersection $-1$.} and $S_2$ is $\\text{dP}_m$ or $\\mathbb F_0$.  We summarize our simplifications by stating that {\\it every rank 2 shrinkable CY 3-fold can be realized locally as a neighborhood of} $S = S_1 \\cup S_2${\\it , for which \t}$S_1 = \\text{Bl}_{p} \\mathbb F_{n_1 > 0} $\\, {\\it  and} $S_2 = \\text{dP}_{n_2}$ {\\it  or }$\\mathbb F_0$. {\\it Moreover, the surfaces }$S_1, S_2$ {\\it are glued along a single smooth rational curve} $C =S_1 \\cap S_2$.\n\nWe argue the third simplification as follows. First, observe that all of the curves $C'$ with self intersection $C'{}^2 < -2$ which do not intersect the gluing curve $C$ have negative volume. Therefore, the only curves $C' \\neq C$ with negative self-intersection should have $C'{}^2 \\geq -2$. Suppose $C'{}^2 = -2$ and the surface $S$ is shrinkable. Then, it should follow that such a geometry is related via complex deformation to a physically-equivalent surface for which the only curves $C'$ of negative self-intersection have $C'{}^2 = -1$.  The idea is essentially identical to the description of a transitions already described in Section~\\ref{sec:transitions}: we perform a conifold transition.  Strictly speaking, this is only true up to physical equivalence, but that is good enough for us.  Hence, we may assume that the only component surfaces $S_i$ appearing in our representative classes are those for which all curves $C' \\ne C$ satisfy $C'{}^2 \\geq -1$. This already places a significant constraint on the possible configurations $S_1 \\cup S_2$. \n\nNext, recall that our list of possible building blocks includes $\\mathbb P^2$ and $\\text{Bl}_p \\mathbb F_n$, where the configuration of $p$ points can be special or generic. The gluing condition (\\ref{eq:gluingcond}) implies that one of the two gluing curves $C_{S_1}$ or $C_{S_2}$ must have negative self-intersection. Therefore, we are forced to fix one of the two surfaces, say $S_1 = \\text{Bl}_p \\mathbb F_{n_1}$. Observe that any blowup of $\\mathbb F_n$ at $p$ points $F \\cap E$ is always isomorphic to the blowup of $\\mathbb F_{n+p}$ at $p$ generic points, so (redefining $n$) we can always assume that $S_1$ is a blowup of $\\mathbb F_{n_1}$ at $p$ points away from the curve $E$ with self intersection $E^2 = -n_1$. \n\nAssume that $n \\geq 2$ and suppose we take such a surface $S_1$ and glue it to $S_2$ along some curve $C_{S_1} \\ne E$. Then this violates the condition that all curves $C' \\ne C_1$ satisfy $C'{}^2 \\geq -1$, in particular for $C' = E$. Hence, we are forced to set $C_{S_1} = E$, and moreover we are confined to surfaces $S_1 = \\text{Bl}_p \\mathbb F_{n_1}$ for which the configuration of points $p$ is a generic configuration (a special configuration of points would produce curves with self-intersection less than $-1$). \n\nLet us focus on $S_2$. If $n_1 \\geq 2$, then $S_2$ must be glued to $S_1$ along a curve $C_{S_2}$ with non-negative self intersection, $C_{S_2}^2 \\geq 0$. Since we may again assume that all $C' \\ne C_{S_2}$ satisfy $C'{}^2 \\geq -1$, it follows that $S_2 = \\text{dP}_{n_2}$ or $S_2 = \\mathbb F_0$. Returning to the remaining cases $n _1< 2$, we find these cases consist of gluing configurations for which $S_i = \\text{dP}_{n_i}$ glued along curves $C_{S_i}$ with $C_{S_i}^2 = -1$. However, $\\text{dP}_n \\cong \\text{Bl}_{n-1} \\mathbb F_1$, and therefore in order to avoid overcounting we assume that our configuration is again of the form conjectured above.\n\nFinally, we turn our attention to the case where one of the component surfaces $S_i$ is a ruled surface over a curve of genus $g >0$. As explained in Section \\ref{sec:transitions}, a ruled surface over a curve with genus $g >0$ is physically equivalent to a blowup of $\\mathbb F_n$ at $2g$ generic points with $g$ self-gluings. Notice that when $S_1$ is the $\\text{Bl}_{2g}\\mathbb{F}_n$ with $g$ self-gluings, the gluing curve $C_{S_1}$ should be the section $E$ (with $E^2=-n$) since otherwise $E$ has negative volume or leads to an elliptic fiber class. This implies due to the shrinkability condition that the second surface $S_2$ is again ${\\rm dP}_m$ or $\\mathbb{F}_0$. The self-gluing curves must always be exceptional curves, and hence we perform a flop transition in which we blow these curves down at the expense of blowing up another curve inside the surface $S_2$. Provided we always perform enough blow downs to completely eliminate the self-glued curves, we can always exchange a configuration involving a self-glued blowup of $\\mathbb F_n$ with one of the configurations described in the above conjecture. This completes our argument concerning the representative configurations for rank 2 surfaces $S=S_1 \\cup S_2$.\n\n\\subsubsection*{Endpoint classification: 0 and 1 mass parameters}\n\\label{subsec:endpoint}\n\nIn this section we show that we can first classify geometries which are blown down `as much as possible'; we refer to these as `endpoint geometries'.  The general\nclassification then follows by classifying endpoints and subsequently classifying their possible blowups.\n \nSuppose a SCFT admits mass deformations for its global symmetry. Then we can take a large mass limit and integrate out all the heavy degrees of freedom. This triggers an RG flow and it is expected that the SCFT below energy scales set by the masses flows to another SCFT with a lower rank global symmetry group commuting with the mass deformations of the UV SCFT. In general, such mass deformations can reduce the rank of the resulting theory. Another possibility is for the IR theory to be a trivial free theory. \n\nWe pay attention to a particular class of mass deformations which leads to interacting SCFTs while preserving the rank of the UV SCFT. Equivalently, we restrict our attention to mass deformations which do not change the dimension of the Coulomb branch. One can typically obtain a new interacting SCFT with the same rank by means of such `rank-preserving mass deformations'. We expect that RG flows of the UV SCFT triggered by such mass deformations can generate a family of SCFTs with the same rank but different global symmetries. SCFTs in the family are distinguished by their global symmetries (i.e. the number of mass parameters), as well as topological data such as the classical Chern-Simons level $k$ or ${\\mathbb Z}_2$-valued $\\theta$ angle.\n\nThese types of RG flows terminate in a class of interacting SCFTs which we will call `endpoint SCFTs'. An endpoint SCFT is defined to be a theory which does not admit any rank-preserving mass deformations. Thus these theories are `endpoints' of RG flows and they cannot flow to other SCFTs via rank-preserving deformations. Endpoint geometries engineer endpoint SCFTs.\n\nRank-preserving mass deformations and endpoint geometries are mathematically well-defined notions. We define distinct endpoint geometries to be surfaces which cannot be related to another smooth surface of the same rank via a large mass deformation. Rank-preserving mass deformations are defined as follows: suppose $S$ is shrinkable and $C\\subset S_j$ is a $-1$ curve which does not intersect any $S_k$ for $k\\ne j$. Then $S$ can be blown down to a surface $S'=\\cup S'_i$ with $S'_j$ the blowdown of the $-1$ curve of $S_j$ and $S'_k\\simeq S_k$ for $k\\ne j$. This type of blowdown is the geometric realization of a rank-preserving mass deformation.\n\n\nWe will now show that {\\it if }$S$ {\\it is shrinkable, then its endpoint geometry }$S'$ {\\it is also shrinkable.}\n  If $C'\\subset S'_i$, let $C\\subset S_i$ be its proper transform.  We have $K_{S'_i}^2=K_{S_i}^2+1$.  If $i\\ne j$\nwe have $K_{S_j}\\cdot C=K_{S'_j} \\cdot C'$, so we need only consider the case $i=j$.  \nLet $p\\in S'_j$ be the point that the $-1$ curve in $S_j$ blows down to, and suppose that $C'$ has multiplicity $m$ at $p$.  Then\n$K_{S'_i}\\cdot C'=K_{S_i}\\cdot C-m$.  The desired conclusion follows immediately.\n\nEndpoint SCFTs are interesting due to the following reasons. First, these theories are the simplest theories in their family of RG flows. Their parameter spaces are smaller, so they are comparatively easier to understand than other theories belonging to the same family.\nThe classification of endpoint SCFTs is therefore a much easier problem than the full classification, as we will see below.\nWe can thus regard the endpoint classification as a tutorial on our classification algorithm.\nSecond, all other SCFTs in the family of RG flows in principle can be obtained from endpoint theories by increasing the number of mass parameters. Namely, we can undo mass deformations, and retrace the RG flow to obtain an entire family of UV SCFTs. This could sound puzzling: we know that RG flow is irreversible. So it may be hard to accept the idea that we can restore UV theories starting from an IR theory. However, this turns out to be the case among 5d supersymmetric theories.\nSince 5d $\\mathcal{N}=1$ SCFTs are so strongly constrained by supersymmetry, one can control their RG flows by tuning discrete data such as (for theories with gauge theory descriptions) gauge algebra, matter representations, classical CS level, and discrete $\\theta$ angle.  We expect that this allows us to build a family of SCFTs starting from an endpoint theory.\n\n\nFrom the geometric standpoint, these constraints can be understood as arising from the Calabi-Yau condition. Mass deformations of a 3-fold correspond to blowups or blowdowns of exceptional curves. As discussed above, a large mass deformation corresponds to blowing down a $-1$ curve which is isolated from gluing curves and is in fact a reversible geometric transition---one can just as easily blow up the same curve to recover the original 3-fold. This means that by starting from an endpoint geometry, it is possible to obtain a family of local (smooth) 3-folds by blowing up all possible exceptional curves. In this sense, the study of endpoint geometries is a good starting point for the classification of 5d SCFTs.\n\nLet us now classify all rank 2 endpoint geometries by employing our classification algorithm. We learned above that rank 2 geometries are constructed by gluing $S_1=\\text{Bl}_p\\mathbb{F}_{m_1}$ and $S_2={\\rm dP}_{m_2}$ or $\\mathbb{F}_0$. This implies that endpoint geometries with $M=0,1$ will take the form $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$ or $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$. For being an endpoint geometry with $M>1$, there must be no irreducible exceptional curve which does not intersect with the gluing curves and no flop transitions  introducing such exceptional curve away from the gluing curves. This is possible only for ${\\rm dP}_2 \\cup {\\rm dP}_2$ with $C_1 = \\ell \\!-\\!X_1 \\!-\\!X_2$ and $C_2=\\ell \\!-\\!X_1\\!-\\!X_2$ which is shrinkable. We thus find that  ${\\rm dP}_2 \\cup {\\rm dP}_2$ is the only  endpoint geometry with $M>1$ \\footnote{We thank Sung-Soo Kim for pointing out that this geometry has no rank-preserving mass deformation}. Therefore the endpoint classification reduces to a simple classification of two types of geometries, $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$ for $M=0$ and $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ for $M=1$, other than ${\\rm dP}_2 \\cup {\\rm dP}_2$ with $M=3$.\n\nWe first classify geometries of the type $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$. We can choose a curve class $C_{S_1}=C_1=a \\ell$ in $\\mathbb{P}^2$ with a positive integer $a$ and $C_{S_2}=C_2=E$ in $\\mathbb{F}_n$ satisfying the gluing condition (\\ref{eq:gluingcond}). Since $C$ should be rational, the integer in $C_1$ is fixed to be either $a=1$ or $a=2$. Accordingly, the second surface is fixed to be $\\mathbb{F}_3$ or $\\mathbb{F}_6$ respectively. Hence we find only two geometries of this type:\n\\begin{align}\n\\begin{split}\n\t& \\mathbb{P}^2 \\cup \\mathbb{F}_3 \\quad {\\rm with} \\quad C_1 = \\ell \\ , \\ C_2 = E_3 \\ ,  \\\\\n\t &\\mathbb{P}^2 \\cup \\mathbb{F}_6 \\quad {\\rm with} \\quad C_1 = 2\\ell \\ , \\ C_2 = E_6 \\ .\n\\end{split}\n\\end{align}\nThese two geometries have brane constructions as depicted in Fig \\ref{fig:rank2-branes1}. These geometries have no mass parameter. Therefore we do not expect any gauge theory descriptions associated to these CFTs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.4]{rank2-branes-m0.pdf}\n\t\\end{center}\n\t\\caption{Brane configurations of rank 2 SCFTs with zero mass.}\n\t\\label{fig:rank2-branes1}\n\\end{figure}\n\nThe second type of endpoint geometry can be classified in the same manner. Due to the gluing condition (\\ref{eq:gluingcond}), a gluing curve in one of two Hirzebruch surfaces should have negative self-intersection. We choose $C_2=E_2$ in the second surface $\\mathbb{F}_{n_2}$. Then the gluing curve $C_1$ in the first surface $\\mathbb{F}_{n_1}$ needs to be a rational irreducible curve with self-intersection $n_2-2$. The curve $C_1$ takes the form of $C_1 = aF_1+bH_1$ with $a,b\\ge0$ or $C_1=E_1$, and must satisfy\n\\begin{equation}\n\tC_1^2 = n_2-2 \\ , \\quad C_1 \\cdot S_1 = -n_2 \\ .\n\\end{equation}\nWe now need to check shrinkability conditions. In both irreducible components $S_i = \\mathbb F_{n_i}$, the curve classes generating Mori cone are $E_i, F_i$.  When these curve classes have non-negative volumes with respect to the K\\\"ahler class $-J=-\\phi_1 S_1-\\phi_2 S_2$, the local 3-fold defined by $S$ is shrinkable and thus engineers a 5d SCFT. In this case, the criteria for shrinkability are\n\\begin{eqnarray}\n\t&&{\\rm vol}(E_1) =  (2-n_1)\\phi_1-a \\phi_2 \\ge 0 \\ , \\quad {\\rm vol}(F_1) = 2\\phi_1-b\\phi_2 \\ge 0 \\ , \\nonumber \\\\\n\t&&{\\rm vol}(E_2) =  (2a+2b-bn)\\phi_1+(2-n)\\phi_2 \\ge 0 \\ , \\quad {\\rm vol}(F_2) = -\\phi_1+2\\phi_2 \\ge 0 \\ ,\n\\end{eqnarray}\nwith $\\phi_1,\\phi_2>0$.\nWe can easily solve these conditions and the gluing condition (\\ref{eq:gluingcond}). Each solution will give a shrinkable geometry and thus a SCFT. The full list of shrinkable surfaces $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ (denoted by $(n_1,n_2)$) is given in Tables \\ref{tb:endpoint-F-F} and \\ref{tb:shirinkable-F-F}. Some of these geometries have brane constructions given in Figure \\ref{fig:rank2-branes2}. We find that only the six geometries in Table \\ref{tb:endpoint-F-F} are independent endpoint geometries.\n\\begin{table}\n\\centering\n\\begin{subtable}{.8\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\t\\hline\n\t $S_1\\cup S_2$ & $C_{S_1}$ & $C_{S_2}$\\\\\n\t\\hline\n\t$\\mathbb{P}^2\\cup \\mathbb{F}_3$ & $\\ell$ & $E$ \\\\\n\t\\hline\n\t$\\mathbb{P}^2 \\cup \\mathbb{F}_6$ & $2\\ell$ & $E$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Endpoint geometries with $M=0$.}\n\t\\label{tb:endpoint-P-F}\n\\end{subtable}%\n\\vspace{.5cm}\n\\begin{subtable}{.9\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c||c|c|c|}\n\t\\hline\n\t $(n_1,n_2)$ & $C_{S_1}$ & $G$ & $(n_1,n_2)$ & $C_{S_1}$ & $G$\\\\\n\t\\hline\n\t$(0,2)$ & $F$ & $SU(3)_1$ & $(0,8)$ & $F+3H$ & $SU(3)_7,G_2$  \\\\\n\t\\hline\n\t$(0,4)$ & $F+H$ & $SU(3)_3$ & $(1,1)$& $E$ & $SU(3)_0$ \\\\\n\t\\hline\n\t$(0,6)$ & $F+2H$ & $SU(3)_5,Sp(2)_{\\pi}$ & $(1,7)$& $2F+H$ & $SU(3)_6$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Endpoint geometries with $M=1$. Here $C_{S_2}=E$. These geometries have gauge theory descriptions with gauge group $G=SU(3)_k,Sp(2)_\\theta,G_2$ where $k$ is the classical CS level and $\\theta$ is the $\\mathbb Z_2$-valued $\\theta$ angle.}\n\t\\label{tb:endpoint-F-F}\n\\end{subtable}%\n\n\\vspace{.5cm}\n\\begin{subtable}{.8\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\t$(n_1,n_2)$ & $C_{S_1}$ & $G$ & Endpoint \\\\\n\t\\hline\n\t$(1,2)$ & $F$ & $SU(2)\\hat{\\times}SU(2)$ & $\\mathbb{P}^2\\cup \\mathbb{F}_3$ \\\\\n\t\\hline\n\t$(1,3)$& $H$ & $SU(3)_2$ & $\\mathbb{P}^2\\cup \\mathbb{F}_3$\\\\\n\t\\hline\n\t$(1,5)$& $F+H$ & $SU(3)_4$ & $\\mathbb{P}^2\\cup \\mathbb{F}_6$\\\\\n\t\\hline\n\t $(1,6)$ & $2H$  & $Sp(2)_{0}$ & $\\mathbb{P}^2\\cup \\mathbb{F}_6$\\\\\n\t \\hline\n\t \\hline\n\t$(2,4)$ & $H$ & $SU(3)_1$ & $\\cdot$\\\\\n\t\\hline\n\t$(0,10)$ & $F+4H$ & $SU(3)_9$ & $\\cdot$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Other geometries of $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$. The first four are not endpoints and flow to geometries in (a) by mass deformations. $(2,4)$ is an endpoint, but is also equivalent to $(0,4)$ by a HW transition. $(0,10)$ is an endpoint, but not shrinkable.}\n\t\\label{tb:shirinkable-F-F}\n\\end{subtable}\n\n\\caption{Classification of all rank 2 geometries with $M=0,1$.}\\label{tb:rank2-F-F-clssification}\n\\end{table}\n\n\n\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.35]{rank2-branes.pdf}\n\t\\end{center}\n\t\\caption{Brane configurations of rank 2 SCFTs with $M=1$.}\n\t\\label{fig:rank2-branes2}\n\\end{figure}\n\nIn fact, all the endpoint geometries in Table \\ref{tb:endpoint-F-F} have gauge theory descriptions with simple gauge group $G$. As explained in Section \\ref{sec:Mth}, a distinguished property of geometries corresponding to gauge theories is that the matrix of intersection numbers (\\ref{eqn:Cartan}) of holomorphic fiber classes $f_i$ with the surfaces $S_i$ is equal to (minus) the Cartan matrix of the gauge algebra. We remark here that the Hirzebruch surface $\\mathbb{F}_0$ has a base-fiber duality exchanging the base curve class $H$ and the fiber curve class $F$. Geometrically, this is an isomorphism between two geometries related by the exchange of $H$ and $F$. It is possible that the dual geometry often has different gauge theory realization from the gauge theory of the original geometry. In this case, the geometric duality leads to a duality between two different gauge theories. \n\nAside from studying the Cartan matrices, we can also compare the triple intersection polynomial $J^3$ to the perturbative expression for the prepotential given in (\\ref{eqn:pre}). For the geometries in Table \\ref{tb:endpoint-F-F} and \\ref{tb:shirinkable-F-F}, the prepotentials are \n\\begin{equation}\n\t6\\mathcal{F} = J^3 = 8\\phi_1^3 + 3\\phi_1\\phi_2(-n_2\\phi_1+(n_2-1)\\phi_2) + 8 \\phi_2^3\\ .\n\\end{equation}\nWe can compare these prepotentials against known gauge theory prepotentials as a means to identify the corresponding gauge theories.\n\nLet us first select the respective fibers $H,F$ for $\\mathbb{F}_{0}\\cup \\mathbb{F}_{n_2}$, and  $F,F$ for $\\mathbb{F}_{1}\\cup \\mathbb{F}_{n_2}$. The Cartan matrix $A_{ij}$ of the following geometries computed using these fiber classes is that of the gauge algebra $SU(3)$ as\n\\begin{equation}\n\t(A_{SU(3)})_{ij} ~:~(n_1,n_2) ~=~ (0,2) \\, , \\ (0,4) \\,, \\ (0,6) \\,, \\ (0,8) \\,, \\ (1,1) \\,, \\  (1,7) \\ ,\n\\end{equation}\nfor the choices of degrees $(n_1,n_2)$ of $\\mathbb F_{n_1} \\cup \\mathbb F_{n_2}$.\nMoreover, their triple intersections agree with gauge theory prepotentials of $SU(3)_k$ listed in Table \\ref{tb:endpoint-F-F}. Therefore, we expect that these endpoint geometries have $SU(3)_k$ gauge theory realizations.\n\n\nThe geometries $(0,6)$ and $(0,8)$ are particularly interesting, as they have two different gauge theory descriptions related by the base-fiber exchange of $\\mathbb{F}_0$. \nWhen we consider the fibers classes to be $F,F$, the two geometries $(0,6),(0,8)$ exhibit (respectively) $Sp(2),G_2$ Cartan matrices. On the other hand, if we choose fiber classes $H,F$, the geometries exhibit the $SU(3)$ Cartan matrix in both cases.\n\nStudying triple intersection numbers gives us a means to narrow down the precise gauge theory that corresponds to these geometries. The triple intersection polynomial $J^3$ of the geometry $(0,6)$ is identical to the prepotentials of both pure $SU(3)_5$ gauge theory and also pure $Sp(2)_\\theta$ theory, which can have either $\\theta=0$ or $\\theta=\\pi$. However, the prepotential cannot distinguish two $Sp(2)$ cases. We can instead determine the $\\theta$ angle using the known duality between $SU(3)$ and $Sp(2)$. In \\cite{Gaiotto:2015una}, it was conjectured that $SU(3)_5$ is dual to $Sp(2)_\\pi$. This suggests that the geometry $(0,6)$ corresponds to $Sp(2)_\\pi$ while $(1,6)$ corresponds to $Sp(2)_0$. Thus, the geometric construction provides yet additional evidence supporting the duality between the $SU(3)_5$ and $Sp(2)_{\\pi}$ gauge theories.\n\n\nAs another example of a duality between gauge theories, the triple intersections of $(0,8)$ agree with the prepotentials of $SU(3)_7$ and $G_2$ gauge theories. We thus conjecture that $SU(3)_7$ and $G_2$ theories are dual and describe the low energy physics of the SCFT corresponding to $\\mathbb{F}_0\\cup \\mathbb{F}_8$.\n\n\n\nAdditional (not necessarily endpoint) geometries of type $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ are displayed in Table \\ref{tb:shirinkable-F-F}. The first five geometries in Table \\ref{tb:shirinkable-F-F} are shrinkable. However, the first four geometries of these are not endpoints. They all can be obtained from other endpoint geometries, $\\mathbb{P}^2\\cup \\mathbb{F}_3$ or $\\mathbb{P}^2\\cup \\mathbb{F}_6$, by blowing up a point and performing flop transitions; see Figure \\ref{fig:P2-F-transition} for more details. We find that these geometries but $(1,2)$ have gauge theory descriptions as listed in Table \\ref{tb:shirinkable-F-F}. The geometry $(1,2)$ has gauge algebra $SU(2)\\hat{\\times}SU(2)$ where $\\hat{\\times}$ denotes that we gauge the $SU(2)$ global symmetry of another $SU(2)$ gauge theory which arises from the $U(1)_I$ instanton symmetry in the IR gauge theory.\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[scale=.45]{P2-F-transitions.pdf}\n\t\t\n\t\\caption{Geometric transitions from $\\mathbb{P}^2\\cup \\mathbb{F}_3$ and $\\mathbb{P}^2\\cup \\mathbb{F}_6$ to $\\mathbb{F}_1\\cup \\mathbb{F}_n$'s with $n=2,3,5,6$.}\n\t\\label{fig:P2-F-transition}\n\\end{figure}\n\nThe geometry $(2,4)$ in Table \\ref{tb:shirinkable-F-F} is an endpoint geometry admitting no additional rank preserving mass deformations. However, this geometry is equivalent to another endpoint geometry $(0,4)$ by a complex structure deformation, or a Hanany-Witten transition. Thus these two geometries belong to the same physical equivalence class.\n\nLastly, the geometry $(0,10)$ is not shrinkable. This geometry satisfies all other shrinkablity conditions, but we find that no 4-cycles have nonzero volume at any point in the K\\\"ahler cone. Thus $(0,10)$ is not shrinkable unless we make a non-normalizable K\\\"ahler deformation. This means the corresponding field theory possesses an intrinsic energy scale set by the K\\\"ahler parameter of the non-compact 4-cycle. Therefore, we do not expect that this geometry corresponds to a 5d SCFT. Indeed, in Section \\ref{sec:rank2}, we will argue that this geometry gives a 5d KK theory.\n\nWe have finished the full classification of rank 2 endpoint geometries (thus rank 2 endpoint SCFTs), which have $M=0,1$. The result is rather surprising---we observe that all rank 2 SCFTs are actually realized by gauge theories and their mass deformations. Note that geometries $\\mathbb{P}^2\\cup\\mathbb{F}_3$ and $\\mathbb{P}^2\\cup\\mathbb{F}_6$ corresponding to non-Lagrangian theories can also viewed as deformations of geometries which admit gauge theory descriptions, for example (respectively) $\\mathbb{F}_1\\cup\\mathbb{F}_2$ and $\\mathbb{F}_1\\cup\\mathbb{F}_5$. This seems to suggest that gauge theory descriptions are generally quite useful, even for 5d SCFTs of higher rank.\n\nFurthermore, all geometries in Table \\ref{tb:rank2-F-F-clssification} except for $(1,2)$ were already predicted in \\cite{Jefferson:2017ahm} using perturbative gauge theory analysis. In fact these geometric constructions confirm all predictions with $r=2$ and $M=1$ in \\cite{Jefferson:2017ahm} except for $SU(3)_8$. \nIt was conjectured in \\cite{Jefferson:2017ahm} that the $SU(3)_8$ theory exists and has an interacting UV fixed point. However, the existence of this theory appears to be ruled out by our geometric classification.\n\nLet us briefly discuss the geometry of the $SU(3)_8$ gauge theory.\nThis theory in fact has a geometric realization as the local 3-fold with K\\\"ahler surface $\\mathbb{F}_1\\cup \\mathbb{F}_9$, where we identify the 2-cycles $C_{S_1}=3F_1+H_1$ and $C_{S_2} = E_2$. However, this geometry is not shrinkable because at least one 2-cycle contained in $S$ has negative volume. For example, the volumes\n\\begin{equation}\n\t\\text{vol}(E_1) = \\phi_1 - 3\\phi_2 \\ , \\quad \\text{vol}(F_2) = 2\\phi_2 - \\phi_1 \\ \n\\end{equation}\nwith $\\phi_1,\\phi_2>0$ cannot be both non-negative. Therefore the Coulomb branch of this geometry is trivial and this geometry is not shrinkable.\nIn order to make the geometry shrinkable we need to attach a non-compact 4-cycle with non-zero K\\\"ahler parameter corresponding to bare gauge coupling constant $1\/g^2$.  This K\\\"ahler parameter cannot be tuned to zero while maintaining positivity of the K\\\"ahler metric.  So even though the IR gauge description with $1\/g^2\\not=0$ makes sense geometrically, we cannot take the $1\/g^2=0$ limit without taking the Coulomb branch parameter to $0$.  This means that if the point $1\/g^2=0$ is a CFT point, then it has no Coulomb branch deformation, and thus in conflict with a SCFT from this gauge theory based on our assumptions.  Thus we do not expect that this geometry has a CFT limit. The gauge theory analysis in \\cite{Jefferson:2017ahm} uses only the perturbative spectrum and monopole tensions and thus cannot capture the spectrum of M2-branes wrapping the curve $E_1 \\subset \\mathbb F_1$ (which correspond to instantons in the gauge theory). Missing non-perturbative states such as these are crucial for assessing whether or not a geometry is shrinkable. This again shows that the perturbative constraints used in \\cite{Jefferson:2017ahm} are necessary but not sufficient to guarantee the existence of CFT fixed points.\n\n\\subsubsection*{Full rank 2 classification}\n\nWe showed in the previous section that our classification program can be reduced to a classification of the following types of geometric configurations: $\\text{Bl}_{p_1} \\mathbb F_n \\cup \\text{dP}_{p_2}$ and $ \\text{Bl}_{p_1} \\mathbb F_n \\cup \\mathbb F_0$.  As already discussed $p_2$ and $p_1$ are bounded above by $p_{\\text{max}}(n)$, which we note depends upon both the degree $n$ and the type of gluing configuration.  However, we are still faced with the problem of restricting the range of (non-negative) integer $n$ for which there exist shrinkable configurations. It turns out that some necessary conditions of shrinkability allows us to derive a crude bound on $n$. From a physical perspective, the existence of such a bound is not surprising as it is closely tied to the existence of only a finite number of 5d interacting fixed CFT points for a fixed rank.  \n\nAppropriate bounds on $n$ can be determined in the two separate cases of $S_2 = \\text{dP}_{p_2}$ or $S_2= \\mathbb F_0$. For both cases, we need only consider $n \\geq 2$, since setting $n=0,1$ produces a geometric configuration isomorphic to $\\text{dP}_{p_1+1} \\cup \\text{dP}_{p_2}$. In the case of $S_2 = \\text{dP}_2$, we  find that $n \\leq 7$, while in the case of $S_2 = \\mathbb F_0$, we find that $n \\leq 8$. See Appendix \\ref{app:bound} for proofs of these bounds.\n\nWe present our classification of rank 2 K\\\"ahler surfaces associated to 5d UV interacting fixed points in Figures \\ref{fig:11}-\\ref{fig:0}. These results are organized by the number of mass parameters $M$, with $0 \\leq M \\leq 11$. Given $M >0$ mass parameters, a shrinkable geometry with $M-1$ mass parameters may be obtained by performing a blowdown of an exceptional divisor (possibly after a sequence of flops) in the surface $S$; in the associated field theory, blowing down an exceptional curve corresponds to integrating out a massive matter hypermultiplet. \n\n\nIn each figure, we list the K\\\"ahler surface $S= S_1 \\overset{C_{S_2}}{\\cup} S_2$, where $C_{S_2}=( S_1 \\cap S_2)_{S_2}$ is the curve along which the two surfaces are glued, restricted to the \\emph{second} surface $S_2$. Geometries marked with $( \\cdot )^*$ correspond to 5d KK theories. Beneath each geometry, we also list the associated gauge theory; geometries with no associated gauge system indicated do not admit a known description as a gauge theory. \n\nOur method for identifying gauge theoretic descriptions involves comparing the triple intersection $J^3$ with the gauge-theoretic prepotential $6\\mathcal F$ in (\\ref{eqn:pre}) for given gauge group and matter content in the K\\\"ahler cone, as well as identifying a geometric realization of the Cartan matrix of associated to the gauge algebra.\n\n\nThe Cartan matrices are determined up to sign by a choice of fibers\\footnote{In the present discussion, a \\emph{fiber} is a rational curve $f$ with self intersection $f^2= 0$.} $f_1\\subset S_1, f_2 \\subset S_2$ satisfying \n\t\t\\begin{align}\n\t\t\t(f_i \\cdot S_j)_{S_i} = - (A_G)_{ij}.\n\t\t\\end{align}\t\nGeometrically, these fibers are rational curves over which M2-branes may be wrapped to give rise to charged BPS vectors in the 5d spectrum.  In Figures \\ref{fig:11}-\\ref{fig:0}, we indicate to the left of each gauge description a possible choice of fibers giving rise to stated gauge algebra. We merely list all possible gauge theory descriptions and do not attempt to list all possible configurations of fibers. When there is more than one choice of fiber leading to different Cartan matrices (and hence different gauge symmetries), there are dualities between the associated gauge theory descriptions. For $\\text{dP}_{p_2 <8}$, the possible fibers are (using the same notation as in \\ref{eq:moridp})\n\t\\begin{align}\n\t\t(1;1)~,~(2;1^4)~,~(3;2,1^6)~,~(4;2^3,1^4)~,~(5;2^6,1).\n\t\\end{align}\nThe list of possible fibers in $\\text{Bl}_{p_1} \\mathbb F_n$ is significantly more complicated; see Appendix \\ref{app:fiber}.\n\n\n\tWe also note that the double arrows connecting pairs of different geometries $S$ indicate flop transitions mapping the geometries into one another. Each figure contains several clusters of geometries connected by arrows, with each cluster belonging to the same birational, and thus physical, equivalence class. Arrows decorated with the symbol $\\phi_1 \\leftrightarrow \\phi_2$ indicate that the flop transition requires us to reverse our identifications $S_1 \\leftrightarrow S_2$, and flip the sign of the Chern-Simons level, $k \\to - k$. \n\n\n\tFinally, we remark that the gluing curves $C_{S_2} \\in \\text{dP}_{p_2 \\geq 3}$ are only listed up to the action of the Weyl group $W(E_{p_2})$. Said differently, each choice of gluing curve displayed in the figures is a single element in the Weyl orbit. We now briefly describe the Weyl group action in $\\text{dP}_{p_2}$ and explain why in most cases we only need to distinguish geometric configurations whose gluing curves belong to the same Weyl orbit in a given surface. Given a simple root $\\alpha_i = X_i - X_{i+1}, i = 1, \\dots, p_2-1$, and an effective curve \n\t\t\\begin{align}\n\t\t\tC= d \\ell -  m_i X_i,\t\t\n\t\t\\end{align}\n\tthe Weyl reflections $w_{\\alpha_i}$ act by transposing exceptional divisors, $X_i \\leftrightarrow X_{i+1}$, while the reflection $w_{\\alpha_{p_2}}$ associated to the root $\\alpha_{p_2} = \\ell - \\sum_{i=1}^3 X_i$ acts on $C$ as follows:\n\t\t\\begin{align}\n\t\t\\begin{split}\n\t\t\tw_{\\alpha_{p_2}}(C) &= (2 d - m_1 - m_2 -m_3) \\ell - ( d - m_2 - m_3 ) X_1 - ( d - m_1 - m_3) X_2 \\\\\n\t\t\t&- (d - m_1 - m_2) X_3 - \\sum_{i > 3} m_i X_i.\n\t\t\\end{split}\n\t\t\\end{align}\n\tAs was shown in \\cite{Iqbal:2001ye}, the action of $W(E_{p_2})$ on a rational curve $C \\in \\text{dP}_{p_2}$ for $p_2 \\geq 4$ and degree $d_C \\equiv - K \\cdot C = C^2 + 2 = n$ in all cases studied in this paper is transitive. Therefore, since the Weyl action $w_{\\alpha}: C \\mapsto C + (C \\cdot \\alpha) C$ preserves intersection products, \n\t\t\\begin{align}\n\t\t\tC \\cdot C' = ( C + (C \\cdot \\alpha) \\alpha) \\cdot ( C' + (C'\\cdot \\alpha) \\alpha),\n\t\t\\end{align}\n\tit is sufficient to set the gluing curve $C_{S_2}$ equal to a single element of the Weyl orbit in order to understand the full intersection structure, as the intersection numbers are identical up to permutation for any two elements belonging to the same Weyl orbit. For $p_2 <3$, the Weyl group either has multiple orbits (as in the case of $p_2 =3$) or is otherwise undefined (as in the case of $p_2 <3$), and so for $p_2<4$ we only list gluing curves $C_{S_2}$ up to cyclic permutations of the exceptional divisors $X_i$.\n\nUpon mass deforming these SCFTs and flowing to the IR we get a tree of relations between these conformal theories which is summarized in the RG flow tree diagram in Figure \\ref{tree}.  The top theories of the RG families are related to 5d KK theories which are discussed in the next section.\n\n\\begin{figure}\n\\begin{center}\n\\noindent\\makebox[\\textwidth]{\n$\n\\begin{array}{c}\n\\begin{tikzpicture}[]\n\t\\node[yscale=1.2,xscale=1.1] at (0,0) {\\includegraphics[scale=.5]{rank2.pdf}};\n\\end{tikzpicture}\n\\end{array}\n$\n}\n\\end{center}\n\\caption[]{The diagram above shows the RG flow among rank 1 and rank 2 SCFTs obtained by mass deformations. The first and the second rows in each box correspond to the geometric and the gauge theoretic descriptions respectively of a 5d theory \\footnotemark. The parent theory in each branch is a 5d KK theory related to a 6d theory on $S^1$.}\n\\label{tree}\n\\end{figure}\n\\footnotetext{\\label{foot:GZ}\nWe note that while $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ has no gauge theory description, it is nonetheless related to $[SU(2)+5{\\bf F}]\\times SU(2)_0$ by a flop transition: a flop of $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ leads to the geometry $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$, which has gauge theory description $[SU(2)+5{\\bf F}]\\times SU(2)_0$. However, $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$ is not shrinkable, which implies that the BPS spectrum of the gauge theory will develop a negative mass before reaching a CFT fixed point. Nevertheless, this gauge theory theory makes sense as an effective description of the CFT from $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ through a flop transition to $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$ when mass parameters are turned on. We are greatful to Gabi Zafrir for pointing out that the CFT related to the $[SU(2)+5{\\bf F}]\\times SU(2)_0$ gauge theory should exist since an associated $(p,q)$ 5-brane system exists.}\n\n\n \\begin{figure}\n  \\begin{center}\n $\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.4]\n\t\t\\node[](a) at (-4,2) {$ \\begin{array}{c} (\\text{Bl}_{10} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1)^* \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 10 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i  , \\ell - X_1 &\\hat A_1\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,0) {$\\begin{array}{c} (\\text{Bl}_{9} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2)^*  \\\\\\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_1 & SU(3)_0 + 10 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 10\\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_2& \\hat A_1 \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](b2) at (4,2) {$\\begin{array}{c} (\\text{Bl}_{10} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0)^* \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 10\\textbf{F} \\\\\\hline F, E & SU(3)_0 + 10 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i ,F& \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (0,-2.5) {$\\begin{array}{c} (\\text{Bl}_{8} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3)^* \\\\ \\scalebox{.7}{$  \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_0 + 10 \\textbf{F} \\\\\\hline F,\\ell - X_3 & Sp(2) + 10\\textbf{F}\\\\\\hline H + 2 F - \\sum X_i, \\ell - X_3 & \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (0,-5) {$\\begin{array}{c} (\\text{Bl}_{7} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4)^*\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_0 + 10 \\textbf{F}  \\\\\\hline F,\\ell - X_4 & Sp(2) + 10\\textbf{F}\\\\\\hline H + 2 F - \\sum X_i, \\ell -X_4 & \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (0,-7.5) {$\\begin{array}{c} (\\text{Bl}_{6} \\mathbb F_2 \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5)^* \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &  SU(3)_0 + 10 \\textbf{F}  \\\\\\hline F,\\ell - X_5 & Sp(2) + 10\\textbf{F}\\\\\\hline H- X_1 - X_2, 2 \\ell-\\sum_{i=1}^4 X_i & [SU(2) + 4 \\textbf{F}] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline  H + 2 F - \\sum X_i,\\ell-X_5 & \\hat A_1 \\\\\\hline \\end{array} $} \\end{array}$};\n\t\t\\node[](f) at (0,-10) {$\\begin{array}{c} (\\text{Bl}_{5} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6)^*\\\\ \\scalebox{.7}{$  \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_0 + 10 \\textbf{F}  \\\\\\hline F, \\ell - X_6 & Sp(2) + 10\\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=2}^5 X_i & [SU(2) + 4 \\textbf{F}] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline f_1 \\cdot E= 2, \\ell- X_6 & \\hat A_1 \\\\\\hline \\end{array}  $}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n$\n\\end{center}\n\\caption{$M=11$ geometries.}\n\\label{fig:11}\n \\end{figure}\n  \n \n \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (7.5,10) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (5.5,8) {$\\begin{array}{c} \\text{Bl}_{8} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_1 & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 9\\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](b2) at (3.5,10) {$\\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 9\\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,6) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$  \\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_{1} & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}  \\\\\\hline F, \\ell - X_3 & Sp(2) + 9\\textbf{F}\\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](d) at (5.5,4) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}   \\\\\\hline F, \\ell - X_4 & Sp(2) + 9\\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (5.5,1.5) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_{1} &  SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}   \\\\\\hline F,\\ell - X_5 & Sp(2) + 9\\textbf{F}\\\\\\hline H - X_1-X_2, 2 \\ell- \\sum_{i=1}^4 X_i & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (-2,1.5) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF, \\ell- X_5  &SU(3)_{\\frac{1}{2}} + 9 \\textbf{F}  \\\\\\hline F, \\ell - X_6 & Sp(2) + 9\\textbf{F}\\\\\\hline f_1 \\cdot E = 0, \\ell-X_6 & [SU(2) + 3 \\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node[](a3) at (-2,11.5) {$ \\begin{array}{c} \\text{Bl}_{10} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\t\\node[] at (5.5,12) {$ \\begin{array}{c} (\\text{Bl}_{9} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0)^* \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline F,E& SU(3)_{-\\frac{3}{2}} + 9 \\textbf{F} \\\\\\hline H + 2 F - \\sum_{i=1}^8 X_i ,E & Sp(2) +  8 \\textbf{F}+ 1\\textbf{AS}   \\\\\\hline \\end{array}$} \\end{array} $};\n\t\t\\node[](b3) at (-2,10) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_1 & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F} \\\\\\hline H + 2 F- \\sum X_i , \\ell- X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c3) at (-2,8) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F}  \\\\\\hline H + 2 F- \\sum X_i , \\ell- X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d3) at (-2,6) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F}  \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_{1} & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e3) at (-2,4) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F}  \\\\\\hline H + 2 F - \\sum X_i, \\ell - X_{1} & Sp(2) + 9 \\textbf{F} \\\\\\hline H - X_1 -X_2 , 2 \\ell - \\sum X_i & [SU(2) + 4 \\textbf{F}] \\times [ SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\draw[big arrow] (a3) -- (b3);\n\t\t\\draw[big arrow] (e3) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (e3);\n\t\t\\draw[big arrow] (b3) -- (c3);\n\t\t\\draw[big arrow] (c3) -- (d3);\n\t\t\\draw[big arrow] (d3) -- (e3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (a3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (b3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (c3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e3) -- (d3);\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=10$ geometries.}\n \\label{fig:10}\n \\end{figure}\n \n\n \n  \\begin{figure}\n \t  \\begin{center}\n\\noindent\\makebox[\\textwidth]{ $\n \\begin{array}{c}\n \\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (9,-4) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (5,-4) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{0}+ 8 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (6,-6) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{0}+ 8 \\textbf{F}  \\\\\\hline H- X_1 - X_2 , 2\\ell - \\sum X_i & [SU(2) + 3 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (6,-8.5) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t F,\\ell-X_1 & SU(3)_0 + 8 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2l- \\sum_{i=1}^4 X_i & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t \\node[] at (4,-1.5) {$  \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-2} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum_{i=1}^7 X_i , \\ell - X_1 & Sp(2) + 7 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\\\\ \\\\\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (5,4.5) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (5,2) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_1 & SU(3)_{1} + 8 \\textbf{F}  \\\\\\hline F,\\ell - X_2 & Sp(2) + 8 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (5,-.5) {$\\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 8 \\textbf{F} \\\\\\hline F,E & SU(3)_{1} + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (.5,2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{1} + 8 \\textbf{F}   \\\\\\hline F, \\ell - X_3 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (.5,-.5) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{1} + 8 \\textbf{F}   \\\\\\hline F,\\ell - X_4 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (-6.5,-.5) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{1} + 8 \\textbf{F}  \\\\\\hline F,\\ell - X_5 & Sp(2) + 8 \\textbf{F} \\\\\\hline H- X_1 - X_2 , 2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 2 \\textbf{F} ] \\times [ SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](f) at (-6.5,2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{1} + 8 \\textbf{F}  \\\\\\hline F, \\ell - X_6 & Sp(2) + 8 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & [SU(2)+ 2 \\textbf{F} ] \\times [SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node(z) at (.5,6.5) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , F & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(y) at (.5,4.5) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell- X_1 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(x) at (-6.5,4.5) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_3  \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_{2} & Sp(2) + 8 \\textbf{F} \\\\\\hline H-X_1 -X_2 ,\t\\ell-X_1 & [SU(2) +4 \\textbf{F} ] \\times [ SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.7em}] (f) --node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$}  (x);\n\t\t\t\\draw[big arrow] (x)-- (f);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (x) -- (y);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (x) -- (y);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (y) -- (z);\n\t\t\t\\draw[big arrow] (z) -- (y);\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$}\n\\end{center}\n \\caption{$M=9$ geometries.}\n \\label{fig:9}\n \\end{figure}\n \n\n \n   \\begin{figure}\n \t  \\begin{center}\n\\noindent\\makebox[\\textwidth]{ $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.4]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\scalebox{.65}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 7 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\\\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 7 \\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F}   \\\\\\hline F,\\ell - X_3 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}} + 7 \\textbf{F}   \\\\\\hline F,\\ell - X_4 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F}  \\\\\\hline F, \\ell - X_5 & Sp(2) + 7 \\textbf{F} \\\\\\hline H - X_1 - X_2 ,2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 1\\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](f) at (0,4) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}} + 7 \\textbf{F}  \\\\\\hline F, \\ell - X_6 & Sp(2) + 7 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2l - \\sum_{i=1}^4 X_i & [SU(2) + 1 \\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](a4) at (-5,0) {$  \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_1 & Sp(2) + 7\\textbf{F}\\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\node[](a1) at (-5,2) {$  \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_2 & SU(3)_{-\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline H+ 2F - \\sum X_i , \\ell-X_2 & Sp(2) + 7 \\textbf{F} \\\\\\hline H -X_1 - X_2 , \\ell- X_1 & [SU(2) + 4\\textbf{F}]\\times [SU(2) + 1 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow] (a1)-- node[left,midway]{$\\phi_1 \\leftrightarrow \\phi_2$}(f);\n\t\t\\draw[big arrow] (a1)-- (a4);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a4) -- (a1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (a1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-2,0) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\n\t\t\\node[](b) at (-2,-2) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\\node[](c) at (-2,-4) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\n\t\t\\node[](d) at (-2,-6) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\\\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\\node[](e) at (-2,-8) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} &SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F}  \\\\\\hline H-X_1 -X_2 , 2\\ell - \\sum X_i & [SU(2) + 2 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (7,-8) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\\ell- X_1  & SU(3)_{\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 2 \\textbf{F}] \\times [ SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\t\\node[](a2) at (2.5,0) {$  \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_2 \\overset{ E}{\\cup} \\mathbb F_0 \\\\\\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-\\frac{5}{2}} + 7 \\textbf{F} \\\\\\hline H-X_1 -X_2 , E & [SU(2) + 5 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline H + 2F - \\sum_{i=1}^6 X_i , F & Sp(2) + 6 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node at (2.5,-3) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\t\\node (b3) at (7,-1) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c3) at (7,-3) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d3) at (7,-5) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_3  \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline H-X_1 - X_2 ,\\ell- X_1 & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (c3);\n\t\t\t\\draw[big arrow] (c3) -- (b3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (d3);\n\t\t\t\\draw[big arrow] (d3) -- (c3);\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (f) -- node[left,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (d3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$}\n\\end{center}\n \\caption{$M=8$ geometries. (See Footnote \\ref{foot:GZ} for a comment about $\\text{Bl}_{8} \\mathbb F_3 \\cup \\mathbb P^2$.)}\n \\label{fig:8}\n \\end{figure}\n \n\n    \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{2} + 6 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 6 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 6 \\textbf{F} \\\\\\hline F, E & SU(3)_{2} + 6 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{2} + 6 \\textbf{F}   \\\\\\hline F,\\ell - X_3 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{2} + 6 \\textbf{F}   \\\\\\hline F, \\ell - X_4 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{2} + 6 \\textbf{F}  \\\\\\hline F,\\ell - X_5 & Sp(2) + 6 \\textbf{F} \\\\\\hline H- X_1 - X_2 , 2 \\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (5.5,4.5) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} &SU(3)_{2} + 6 \\textbf{F}  \\\\\\hline F, \\ell - X_6 & Sp(2) + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0 , 2\\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [SU(2) + 4 \\textbf{F} ] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\n\t\t\t\\node(g) at (-2,4.5) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-2} + 6 \\textbf{F} \\\\\\hline  H+2F - \\sum X_i ,F & Sp(2) + 6 \\textbf{F} \\\\\\hline H-X_1-X_2, E & [SU(2) + 4 \\textbf{F}] \\times SU(2)_{\\pi} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(A) at (7.5,6) {$\\begin{array}{c} \\text{Bl}_7 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(B) at (3,6) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline H-X_1-X_2 , \\ell - X_1 & [SU(2) + 4 \\textbf{F} ] \\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(C) at (-3,6) {$ \\begin{array}{c} \\text{Bl}_5 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell-X_{1} &   [SU(2) + 4 \\textbf{F}]\\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (A) -- (B);\n\t\t\\draw[big arrow] (B) -- (A);\n\t\t\\draw[big arrow](C) -- (B);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}](B) --  (C);\n\n\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow] (f) -- node[above,pos=.5]{$\\phi_1 \\leftrightarrow \\phi_2$}  (g);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (g) -- (f);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=7$ geometries.}\n \\label{fig:7a}\n \\end{figure}\n \n\n \\begin{figure}\n \\begin{center}\n \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node (e) at (2.5,-1) {$\\begin{array}{c} (\\mathbb F_2 \\overset{\\ell - X_1}{\\cup} \\text{dP}_7 )^*\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline F, \\ell- X_2  & SU(3)_4 + 6 \\textbf{F}\\\\\\hline F, 2\\ell- \\sum_{i=2}^5 X_i & Sp(2) + 4 \\textbf{F} + 2 \\textbf{AS} \\\\\\hline  F,4\\ell - \\sum_{i=1}^4 X_i - 2 \\sum_{j=5}^7 X_j& G_2 + 6 \\textbf{F} \\\\\\hline  F,5\\ell - X_1 - 2 \\sum_{i=2}^7 X_i &A^{(2)}_2  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\\\\\\n\t\\begin{array}{c}\n\t\\begin{tikzpicture}[yscale=1,xscale=1.2]\n\t\t\\node[](a) at (3.2,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{1}+ 6 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{1}+ 6 \\textbf{F}  \\\\\\hline H-X_1-X_2 , 2\\ell - \\sum X_i & [SU(2)+ 1 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-8.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\tF,\\ell-X_1 & SU(3)_{1} + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2 \\ell - \\sum_{i=1}^4 X_i & [SU(2) + 1 \\textbf{F} ] \\times [ SU(2)\\times 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t  \\node[](a3) at (0,-11) {$  \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_2 & SU(3)_{-1} + 6 \\textbf{F} \\\\\\hline H-X_1-X_2, \\ell-X_1 & [SU(2) + 3 \\textbf{F}] \\times [ SU(2)+ 1 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[](a4) at (0,-13) {$  \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-1} + 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[] at (7.5,0) {$  \\begin{array}{c}  \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_7 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1   & SU(3)_{3} + 6 \\textbf{F} \\\\\\hline F,\\ell-X_2 & Sp(2) +  5 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node (b3) at (7.5,-3) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,E & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c3) at (7.5,-5) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_{1} & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d3) at (7.5,-7) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3  \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_{2} & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline H-X_1-X_2 , \\ell -X_1 & [SU(2) + 2 \\textbf{F} ] \\times [SU(2) \\times 2\\textbf{F}] \\\\\\hline\\end{array} $}\\end{array}$};\n\t\t\t\\node(e3) at (7.5,-9) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_{3} & SU(3)_0 + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, \\ell- X_{1} &  [SU(2) + 2 \\textbf{F} ] \\times [SU(2) \\times 2\\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (c3);\n\t\t\t\\draw[big arrow] (c3) -- (b3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (d3);\n\t\t\t\\draw[big arrow] (d3) -- (c3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (e3);\n\t\t\t\\draw[big arrow] (e3) -- (d3);\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (a3) -- (f);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (a3) -- (a4);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (a3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a4) -- (a3);\n\t\t\\end{tikzpicture}\n\t\t\\end{array}\n\t\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=7$ geometries, cont.}\n\\label{fig:7b}\n \\end{figure}\n \n \\clearpage\n \n \n\n     \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 5 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 5 \\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}    \\\\\\hline F, \\ell - X_3 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}    \\\\\\hline F,\\ell - X_4 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}   \\\\\\hline F,\\ell - X_5 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (-1,2) {$\\begin{array}{c}  \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$  \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F}   \\\\\\hline F, \\ell - X_6 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (4,-2) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F}  \\\\\\hline H - X_1 - X_2 , 2\\ell - \\sum X_i & SU(2)_\\pi \\times [SU(2) + 3 \\textbf{F} ] \\\\\\hline   \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-10.8) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t  F,\\ell - X_1 &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [ SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\n\t  \\node(g) at (0,-13) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-\\frac{3}{2}} + 5 \\textbf{F} \\\\\\hline H-X_1 - X_2 ,E& [SU(2) + 3\\textbf{F}] \\times SU(2)_\\pi \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (f) -- (g);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (g) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (f);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t \t \\node(A) at (8,-8) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(B) at (8,-10) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1  \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline H-X_1-X_2 , \\ell - X_1 & [SU(2) + 3 \\textbf{F} ] \\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(C) at (8,-12) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline  F,\\ell-X_{1} &   [SU(2) + 3 \\textbf{F}] \\times SU(2)_{0}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (A) -- (B);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (C) -- (B);\n\t\t\t\\draw[big arrow] (B) -- (A);\n\t\t\t\\draw[big arrow] (B) -- (C);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=6$ geometries.}\n \\label{fig:6a}\n \\end{figure}\n \n\n   \\begin{figure}\n \\begin{center}\n \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\n\t\t\t\\node (e) at (1.5,-2.5) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_6 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell - X_6  & SU(3)_{\\frac{9}{2}} + 5 \\textbf{F}\\\\\\hline F, 2\\ell - \\sum_{i=3}^6 X_{i} & Sp(2) + 3 \\textbf{F} + 2 \\textbf{AS}  \\\\\\hline F, 3 \\ell - \\sum_{i=1}^5 X_i - 2 X_6 & G_2 + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (f) at (9,-2.5) {$\\begin{array}{c} \\mathbb F_2 \\overset{\\ell - X_1}{\\cup} \\text{dP}_6 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline F,\\ell-X_2  & SU(3)_{\\frac{7}{2}} + 5 \\textbf{F}\\\\\\hline F,2 \\ell- \\sum_{i=3}^6 X_i & Sp(2) + 4 \\textbf{F} + 1 \\textbf{AS}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\\\\\n\t\\begin{array}{c}\n\t\\begin{tikzpicture}\n\t\t\\node (bn) at (8.1,-1) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (cn) at (8.1,-3) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (dn) at (8.1,-5.5) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3  \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline H- X_1 - X_2 , \\ell-X_1 &  [SU(2) + 1 \\textbf{F} ] \\times [SU(2) +2 \\textbf{F}] \\\\\\hline  \\end{array}$} \\end{array}$};\n\t\t\t\\node(en) at (8.1,-8) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,\\ell -X_4 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline f_1 \\cdot E =0 , \\ell-X_1 & [SU(2) + 1 \\textbf{F} ] \\times [SU(2) +2 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\node[](a1n) at (8.1,-13) {$  \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[](a2n) at (8.1,-10.5) {$  \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline H- X_1 -X_2 ,\\ell-X_1 & [SU(2) + 2 \\textbf{F}] \\times [ SU(2) + 1 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (bn) -- (cn);\n\t\t\t\\draw[big arrow] (cn) -- (bn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (cn) -- (dn);\n\t\t\t\\draw[big arrow] (dn) -- (cn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (dn) -- (en);\n\t\t\t\\draw[big arrow] (en) -- (dn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a1n) -- (a2n);\n\t\t\t\\draw[big arrow] (a2n) -- (a1n);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a2n) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (en);\n\t\t\t\\draw[big arrow] (en) -- (a2n);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=6$ geometries, cont.}\n\\label{fig:6b}\n \\end{figure}\n \n \n \\clearpage\n \n\n      \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_1 & SU(3)_{3} + 4 \\textbf{F}  \\\\\\hline F,\\ell - X_2 & Sp(2) + 4 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 4 \\textbf{F} \\\\\\hline F, E & SU(3)_{3} + 4 \\textbf{F}  \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F}      \\\\\\hline F,\\ell - X_3 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F}      \\\\\\hline \\ell - X_4 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c}  \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F}     \\\\\\hline F, \\ell - X_5 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{2}+ 4 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,\\dots,3} &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,\\dots,4} &SU(3)_{2}+ 4 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-10) {$ \\begin{array}{c}  \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t  F,\\ell - X_1 &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=5$ geometries.}\n \\label{fig:5a}\n \\end{figure}\n \n \\clearpage \n\n   \\begin{figure}\n \\begin{center}\n \\noindent\\makebox[\\textwidth]{ \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\node (b) at (0,-3) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\node (c) at (0,-5) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{1} + 4 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d) at (0,-7) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3  \\\\\\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_2 & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline H-X_1-X_2 , \\ell - X_1 & SU(2)_\\pi \\times [SU(2) + 2 \\textbf{F}]\\\\\\hline\\end{array}$} \\end{array}$};\n\t\t\t\\node(e) at (0,-9.5) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell- X_3 & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline f_1 \\cdot E =0, \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$}; \n\t\t\t\\node(f) at (0,-12.5) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-1} + 4 \\textbf{F} \\\\\\hline H-X_1-X_2 ,E&[SU(2) + 2 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline \\end{array}$} \\end{array}$};\n\n\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (e);\n\t\t\t\\draw[big arrow] (e) -- (d);\n\t\t\t\\draw[big arrow] (e) --(f);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) --  node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$}(e);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\end{array} ~~~~~~~ \\begin{array}{c}\\begin{tikzpicture} \\node(A) at (6,-7) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\\node(B) at (6,-9) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1  \\\\\\begin{array}{|c|c|} \\hline H-X_1-X_2,\\ell-X_1 &[SU(2) + 2 \\textbf{F}] \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\\node(C) at (6,-11) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell-X_{1} &  [SU(2) + 2 \\textbf{F}] \\times SU(2)_{0}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (A) -- (B);\n\t\t\\draw[big arrow] (B) -- (A);\n\t\t\\draw[big arrow] (B) -- (C);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (C) -- (B);\n\t\t\\end{tikzpicture}\\end{array}\\\\ \\\\\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (e) at (-2,-7.5) {$\\begin{array}{c} \\mathbb F_4 \\overset{2\\ell-X_1-X_2}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell- X_1  & SU(3)_{5} + 4 \\textbf{F}\\\\\\hline F, \\ell - X_5 & Sp(2) + 2 \\textbf{F} + 2 \\textbf{AS}  \\\\\\hline F, 2\\ell - \\sum_{i=2}^5 X_i & G_2 + 4 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (f) at (5,-7.5) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F, \\ell - X_1 & SU(3)_{4} + 4 \\textbf{F}\\\\\\hline F, 2\\ell - \\sum X_i & Sp(2) + 3 \\textbf{F} + 1 \\textbf{AS}  \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (-4,-4.5) {$  \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{0} + 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node (l) at (3.5,-4.5) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline F,\\ell-X_2 & SU(3)_{0} + 4 \\textbf{F}   \\\\\\hline H-X_1-X_2 , \\ell-X_1 & [SU(2) +1 \\textbf{F}] \\times[ SU(2) +1 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\node (m) at (3.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell - X_3 & SU(3)_{0} + 4 \\textbf{F}   \\\\\\hline f_1 \\cdot E = 0,\\ell- X_1 & [SU(2) +1 \\textbf{F}] \\times[ SU(2) +1 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n\t$}\n\\end{center}\n\\caption{$M=5$ geometries, cont.}\n\\label{fig:5b}\n \\end{figure}\n \n \\clearpage \n \n\n       \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{7}{2}} + 3 \\textbf{F}  \\\\\\hline F, \\ell - X_2 & Sp(2) + 3 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\\\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 3 \\textbf{F} \\\\\\hline F,E &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F}  \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F}    \\\\\\hline F, \\ell - X_3 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c}  \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F}     \\\\\\hline F,\\ell - X_4 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F}\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8) {$ \\begin{array}{c}  \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F}  \\\\\\hline \\end{array}$}\\end{array}$};\n\t\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t~~~~~~~~~~~~~~~~~~~\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (b) at (1,-3) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{3}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c) at (1,-5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 &SU(3)_{\\frac{3}{2}} + 3 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d) at (1,-7) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3  \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{3}{2}} + 3 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(e) at (1,-9) {$\\begin{array}{c}  \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_3 &SU(3)_{\\frac{3}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (e);\n\t\t\t\\draw[big arrow] (e) -- (d);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=4$ geometries.}\n \\label{fig:4a}\n \\end{figure}\n \n \\clearpage \n \n\n    \\begin{figure}\n \\begin{center}\n \t \\noindent\\makebox[\\textwidth]{ $\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node(a) at (-2,2) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (-2,0) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline H-X_1 -X_2 , \\ell - X_1 &[SU(2) + 1\\textbf{F}] \\times SU(2)_0\\end{array} \\\\\\hline \\end{array}$};\n\t\t\t\\node(C) at (7,0) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|}\\hline F,\\ell-X_{j=1,2} &   [SU(2) + \\textbf{F}] \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\draw[big arrow] (b) -- (C);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (C) -- (b);\n\t\t\n\t\t\t\\node (e) at (6,-7.5) {$\\begin{array}{c} \\mathbb F_4 \\overset{2\\ell-X_1-X_2}{\\cup} \\text{dP}_4 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell -X_1  & SU(3)_{\\frac{9}{2}} + 3 \\textbf{F}\\\\\\hline F,\\ell - X_3 & Sp(2) + 2 \\textbf{F} +  1\\textbf{AS}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (-2,-7.5) {$\\begin{array}{c} \\mathbb F_5 \\overset{2\\ell - X_1}{\\cup} \\text{dP}_4 \\\\ \\begin{array}{|c|c|} \\hline F , \\ell - X_1  & SU(3)_{\\frac{11}{2}} + 3 \\textbf{F}\\\\\\hline  F, \\ell - X_1 & Sp(2) +  \\textbf{F} + 2 \\textbf{AS}  \\\\\\hline F , 2\\ell - \\sum X_i & G_2 + 3 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (3,2.5) {$\\begin{array}{c} (\\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_4)^* \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 & Sp(2)_{ 0} + 3 \\textbf{AS}   \\\\\\hline F, 2\\ell - \\sum X_i & A^{(2)}_2 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (-2.5,-4.5) {$  \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (-2.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell -X_2 &SU(3)_{\\frac{1}{2}} + 3 \\textbf{F}  \\\\\\hline  H-X_1 - X_2 , \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 1 \\textbf{F}] \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node (m) at (6.5,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_3 &SU(3)_{\\frac{1}{2}} + 3 \\textbf{F}  \\\\\\hline f_1 \\cdot E =0, \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 1 \\textbf{F}] \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\t\\node(n) at (6.5,-4.5) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F, F & SU(3)_{-\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline H - X_1 - X_2, E  & [SU(2) + 1 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline  \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\t\\draw[big arrow] (m) -- (n);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (n) -- node[right,midway]{$\\phi_1\\leftrightarrow \\phi_2$} (m);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t$}\n\\end{center}\n\\caption{$M=4$ geometries, cont.}\n\\label{fig:4b}\n \\end{figure}\n \n \\clearpage \n \n\n        \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-4.5,-2) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & SU(3)_{3}+ 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{3}+ 2 \\textbf{F}   \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](d) at (5.5,-4.5) {$ \\begin{array}{c}  \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_{1} &SU(3)_{3}+ 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\n\t\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\n\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\\\\ \n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{4} + 2 \\textbf{F}  \\\\\\hline F,\\ell - X_2 & Sp(2) + 2 \\textbf{F}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|} \\hline \n\tF,\tF & Sp(2) + 2 \\textbf{F} \\\\\\hline F, E &SU(3)_{4} + 2 \\textbf{F}\\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c}  \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\  \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{4} + 2 \\textbf{F}    \\\\\\hline F, \\ell - X_3 & Sp(2) + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node (b) at (-5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (c) at (-.5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 &SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (d) at (4.5,-2.5) {$ \\begin{array}{c}  \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3  \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=3$ geometries.}\n \\label{fig:3b}\n \\end{figure}\n \n \\clearpage \n\n     \\begin{figure}\n \\begin{center}\n \t$\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node(C2) at (4,5) {$\\begin{array}{c} \\text{dP}_2 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|}\\hline \\ell- X_1, \\ell-X_{1} &  SU(2)_{0} \\times SU(2)_0  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(a) at (-2,5) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (-2,3) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline H-X_1-X_2, \\ell - X_1 & SU(2)_\\pi \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(C1) at (6,3) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline f_1 \\cdot E = 0, \\ell-X_{1} &  SU(2)_{\\pi} \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow] (b) -- (C1);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (C1) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\node(d1) at (-2.5,1) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{0} + 2 \\textbf{F} \\\\\\hline  H - X_1 - X_2 , E & SU(2)_\\pi \\times SU(2)_\\pi \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d2) at (5.5,1) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{0} + 2 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, \\ell - X_2 & SU(2)_\\pi \\times SU(2)_\\pi \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (-3.5,-3) {$\\begin{array}{c} \\mathbb F_5 \\overset{2\\ell - X_1}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|}\\hline  F,\\ell - X_1  & SU(3)_{5} + 2 \\textbf{F}\\\\\\hline F,\\ell - X_{2}  & Sp(2) +  1\\textbf{F} +  1\\textbf{AS}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (2,-3) {$\\begin{array}{c} \\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_3\\\\ \\begin{array}{|c|c|} \\hline F,\\ell -X_{1} & Sp(2)_{0}+ 2 \\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node[](k) at (-3,-1) {$  \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{1} + 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (2,-1) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell-X_2 &SU(3)_{1} + 2 \\textbf{F}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (m) at (7,-1) {$\\begin{array}{c} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_3 &SU(3)_{1} + 2 \\textbf{F}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (g) at (7,-3.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{3 \\ell - 2 X_1 - X_2 }{\\cup} \\text{dP}_3\\\\ \\begin{array}{|c|c|} \\hline F, \\ell-X_2 &Sp(2)_{\\pi} + 2 \\textbf{AS} \\\\\\hline F,\\ell - X_1 & SU(3)_6 + 2 \\textbf{F} \\\\\\hline F, \\ell- X_3 &G_2 + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (d1) -- (d2);\n\t\t\t\\draw[big arrow] (d2) -- (d1);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=3$ geometries, cont. Note that for the geometry $\\text{dP}_2 \\cup \\text{dP}_2$ at the top, the gluing curves in \\emph{both} surfaces are $C = \\ell - X_1 - X_2$, in contrast to the other geometries.}\n\\label{fig:3b}\n \\end{figure}\n \n \\clearpage \n \n\n        \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c} \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-5,-2) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{7}{2}}+1  \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](c) at (5,-2) {$ \\begin{array}{c} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{\\frac{7}{2}}+ 1 \\textbf{F}   \\\\\\hline \\end{array}\\end{array}$};\n\n\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\n\n\t\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\n\n\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (5,0) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) +  1\\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](b1) at (0,0) {$\\begin{array}{c} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{9}{2}} + 1 \\textbf{F}  \\\\\\hline F,\\ell - X_2 & Sp(2) + 1 \\textbf{F}  \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](b2) at (-5,0) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|} \\hline \n\tF,\tF & Sp(2) +  1\\textbf{F} \\\\\\hline F,E &SU(3)_{\\frac{9}{2}} +  1\\textbf{F}\\\\\\hline \\end{array} \\end{array}$};\n\t\n\n\t\n\n\t\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\n\t\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a);\n\n\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (b) at (-2.5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0  \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{5}{2}} +  1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (c) at (2.5,-2.5) {$ \\begin{array}{c}  \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2  \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 &SU(3)_{\\frac{5}{2}} +  1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\n\t\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\\\\\n\\begin{tikzpicture}\n\t\t\t\\node(a) at (-1,3) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (3,3) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\node(d1) at (.5,1.5) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F, F & SU(3)_{\\frac{1}{2}} +  1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d2) at (5.5,1.5) {$\\begin{array}{c}  \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{1}{2}} +  1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d3) at (7,3) {$\\begin{array}{c}  \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\end{array}$};\n\t\t\t\\draw[big arrow] (d3) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (d3);\n\t\t\n\t\t\n\t\t\t\\node (f) at (-2,-2.5) {$\\begin{array}{c} \\mathbb F_7 \\overset{3\\ell -2 X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline  F,\\ell - X_1  & SU(3)_{\\frac{13}{2}} +  1\\textbf{F}\\\\\\hline F, \\ell - X_2  & G_2 +  1\\textbf{F}   \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (3,-2.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 &Sp(2)_0 + 1\\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f1) at (8,-2.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{3\\ell - 2 X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1& SU(3)_{\\frac{11}{2}} + 1\\textbf{F}\\\\\\hline F,\\ell - X_2 & Sp(2)_\\pi +1 \\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (0,-.5) {$  \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{3}{2}} + 1\\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (5,-.5) {$\\begin{array}{c}  \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell -X_2 &SU(3)_{\\frac{3}{2}} +  1\\textbf{F}  \\\\\\hline \\end{array} \\end{array}$};\n\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\n\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (d1) -- (d2);\n\t\t\t\\draw[big arrow] (d2) -- (d1);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=2$ geometries.}\n \\label{fig:2a}\n \\end{figure}\n \n \n \\clearpage\n \n\n         \\begin{figure}\n \t  \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a1) at (-4,-3) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b1) at (-.5,-3) {$ \\begin{array}{c}  \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{4} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\draw[big arrow] (a1) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a1);\n\t\t\\node[](a) at (3,-3) {$ \\begin{array}{c}  \\text{Bl}_1 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2 \\end{array}$};\n\t\t\\node(ee) at (6.5,-3) {$\\begin{array}{c} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\end{array}$};\n\t\t\\node[](b2) at (1.5,.5) {$\\begin{array}{c} \\mathbb F_{2b} \\overset{F + (b-1) E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|c|} \\hline \n\t\tb=1 & F,E & SU(3)_1 \\\\\\hline b=2 &F,F  & SU(3)_3 \\\\\\hline b=3 &F, E & SU(3)_5 \\\\\\hline b = 3 &F,F & Sp(2)_{\\pi} \\\\\\hline b = 4 & F,E & SU(3)_7 \\\\\\hline b = 4 &F,F &G_2 \\\\\\hline b = 5  & F,E & SU(3)_9\\\\\\hline b = 5 & F, F & A^{(2)}_2 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (ee);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (ee) -- (a);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node at (-2.5,0) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell - X_1 & SU(3)_2 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node[](a) at (1,0) {$ \\begin{array}{c}  \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 &Sp(2)_0\\\\\\hline \\end{array}\\end{array}$};\n\t\t\t\\node at (4.5,0) {$\\begin{array}{c} \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell - X_1 & SU(3)_0 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node at (8,0) {$\\begin{array}{c} \\mathbb F_7 \\overset{3\\ell - 2X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 & SU(3)_6 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\n\t\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=1$ geometries.}\n \\label{fig:1}\n \\end{figure}\n \n\n  \\begin{figure}\n  \\centering\n  $\n  \t\\begin{array}{ccc}\n\t\t\\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2 \n\t& ~~~~&\n\t\\mathbb F_6 \\overset{2\\ell}{\\cup} \\mathbb P^2\n\t\\\\ \\\\\n\t\\end{array}\n$\n    \\caption{$M=0$ geometries.}\n\\label{fig:0}\n    \\centering\n  \\end{figure}\n  \n\n\n\n\n\n\\subsubsection*{6d Theories on a Circle}\nIn this section we show that the complicated web of theories we have uncovered are actually unified from the perspective of 5d Kaluza-Klein (KK) theories arising from 6d SCFTs compactified on a circle (up to possible automorphism twists and holonomies).\n\nAs discussed in Section \\ref{sec:rank1}, shrinkable rank 1 geometries are classified by del Pezzo surfaces $\\text{dP}_{n\\leq 8}$ and $\\mathbb{F}_0$ up to physical equivalence. Interestingly, all of them can be obtained via geometric RG flows from $\\text{dP}_9$ (equivalently, $\\frac{1}{2}$K3). The local $\\text{dP}_9$ model is an elliptic 3-fold engineering the 6d SCFT called the `E-string theory'. Therefore all rank 1 5d SCFTs are descendants (i.e. related by rank preserving mass deformations) of the 6d E-string theory compactified on a circle.\n  \nWe also find that all rank 2 5d SCFTs have 6d origin, but the rank 2 case is significantly more elaborate than the rank 1 case. Geometric constructions produce 5d SCFTs belonging to the four distinct families displayed in Table \\ref{tb:rank2-classification}. The geometries of type $(\\cdot)^*$ are not shrinkable but rather 5d KK theories~\\footnote{These theories are also called \\emph{marginal} theories \\cite{Jefferson:2017ahm}.}. We expect that these geometries correspond to 6d SCFTs compactified on a circle, possibly with automorphism twists.\n\n\nOne distinguished property of geometries corresponding to 5d KK theories  is that there must exist an elliptic curve class whose volume is not controlled by normalizable K\\\"ahler moduli. The M2-branes wrapping this elliptic class correspond to KK momentum states. For example, the canonical class $-K_{\\text{dP}_9} \\subset\\text{dP}_9$ is an elliptic class with zero volume associated to the KK momenta of the E-string theory compactified on a circle. Another important property is that some KK geometries contain fiber classes forming an affine gauge algebra. Namely, we can find fiber classes $f_i$ such that\n\\begin{equation}\n\t-f_i \\cdot S_j = (A_{\\hat{G}})_{ij},\n\\end{equation}\nwhere $\\hat{G}$ denotes an affine gauge algebra. This signals that the corresponding geometry is an elliptic geometry realizing a 5d KK theory.  We will now identify 6d origins of the geometries in Table \\ref{tb:rank2-classification} using these properties.\n\nWe begin with $\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0$. This geometry has two gauge theory descriptions, namely $SU(3)_0 + 10 \\textbf{F}$ and $Sp(2)+ 10 \\textbf{F}$. The 6d origin of these gauge theories is discussed in \\cite{Yonekura:2015ksa,Hayashi:2015fsa,Gaiotto:2015una,Hayashi:2016abm}. These theories are a circle reduction of the 6d $(D_5,D_5)$ conformal matter theory  introduced in \\cite{Heckman:2013pva,DelZotto:2014hpa}. The geometry $\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0$ realizes the circle compactification of this 6d theory. This theory has another duality frame in which an affine gauge algebra is manifest. To see this, choose the fiber classes $f_1=H+2F-\\sum_{i=1}^{10}X_i $ and $f_2= F$. These fiber classes indeed form the affine $\\hat{A}_1$ Cartan matrix:\n\\begin{equation}\n\t-(f_i \\cdot S_j) = \\begin{pmatrix} 2 & -2 \\\\ -2 & 2 \\end{pmatrix} .\n\\end{equation}\n\nAnother geometry $\\mathbb{F}_2\\cup \\text{dP}_7$ is interesting for similar reasons. This geometry admits three different gauge theory descriptions corresponding to the following choices of fiber classes:\n\\begin{align}\n\\begin{split}\n f_1&= F,~~  f_2= \\ell - X_2 \\  \\ \\rightarrow \\ \\ SU(3)_4+6{\\bf F}  \\ , \\\\\n f_1&= F,~~ f_2= 2\\ell - \\sum_{i=2}^5X_i \\ \\ \\rightarrow \\ \\ Sp(2)+2{\\bf AS}+4{\\bf F}  \\ ,  \\\\  \n  f_1&= F,~~ f_2= 3\\ell - \\sum_{i=2}^6X_i -2X_7 \\ \\ \\rightarrow \\ \\ G_2+6{\\bf F}  \\ .\n \\end{split} \n\\end{align}\nHere, the two surfaces are glued along the curves $C_{S_1}=E$ and $C_{S_2}=\\ell-X_1$. This implies new dualities between these three gauge theories and their descendants obtained by RG-flows induced by relevant mass deformations. In addition, we find another distinct duality frame:\n\\begin{equation}\n\tf_1 = F \\,, \\ f_2 = 5\\ell -X_1-2\\sum_{i=2}^7X_i \\ .\n\\end{equation}\nThe fiber classes in this last frame form the affine Cartan matrix $ A^{(2)}_2$:\n\\begin{equation}\n\t-(f_i\\cdot S_j) = \\left(\\begin{array}{cc} 2 & -1 \\\\ -4 & 2 \\end{array}\\right) \\ .\n\\end{equation}\nThis algebra $ A^{(2)}_2$ is obtained by an outer automorphism twist of the affine $A^{(1)}_2=\\hat{A}_2$ algebra which identifies ${\\bf 3}$ and $\\bar{\\bf 3}$ representations in $A_2\\subset \\hat{A}_2$. Therefore, one can expect that this geometry is also a KK geometry corresponding to a 6d $SU(3)$ gauge theory compactified on a circle with an outer automorphism twist. The unique 6d theory satisfying these properties is the 6d $\\mathcal{N}=(1,0)$ SCFT with $SU(3)$ gauge group and $N_\\textbf{F}=12$ fundamental hypermultiplets. Circle compactification of this 6d theory with an outer automorphism twist of the $SU(3)$ gauge algebra leads to a 5d theory with affine $A^{(2)}_2$ gauge algebra and 6 flavors. This interpretation agrees with the geometric model $\\mathbb{F}_2\\cup \\text{dP}_7$. Therefore, we conclude that $\\mathbb{F}_2\\cup \\text{dP}_7$ is a `KK geometry' engineering the circle compactification of the 6d $SU(3)$ theory with $N_\\textbf{F} = 12$.\n\n$\\mathbb{F}_6\\cup \\text{dP}_4$ is also a KK geometry. When one chooses the fiber classes $f_1=F_1,f_2=\\ell-X_1$ (with the gluing curve $C_{S_2}=2\\ell$), this geometry has a gauge theory description as $Sp(2)_{0}+3{\\bf AS}$. However, if we choose the fiber classes $f_1=F,f_2=2\\ell-\\sum_{i=1}^4X_i$, their intersections with the irreducible components $S_i$ form the affine $A^{(2)}_2$ Cartan matrix, up to sign. This suggests that $\\mathbb{F}_6\\cup \\text{dP}_4$ is a KK geometry. Indeed we find that the 6d $SU(3)$ gauge theory with $N_\\textbf{F}=6$ can give rise to the 5d KK theory associated to this geometry upon circle reduction with an outer automorphism twist.\n\n$\\mathbb{F}_{10}\\cup \\mathbb{F}_0$ is yet another KK geometry constructed by our building blocks. This geometry admits two dual descriptions related to the base-fiber exchange symmetry of $\\mathbb{F}_0$. One description is $SU(3)_9$, while the other is the $ A^{(2)}_2$ gauge theory description without matter hypermultiplets. We anticipate that this affine $ A^{(2)}_2$ gauge theory is the 5d KK theory coming from the 6d theory $\\mathcal{O}(-3)$ minimal SCFT with $SU(3)$ gauge group compactified on a circle with an outer automorphism twist of the $SU(3)$ gauge algebra.\n\nLastly, $\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0$ is a KK geometry. This geometry is formed by gluing two surfaces along $C_{S_1}=E$ in $\\text{Bl}_9\\mathbb{F}_4$ and $C_{S_2}=F+H$ in $\\mathbb{F}_0$. We find that this geometry involves an elliptic fiber class given by $E+2X$ (with $E^2=-4,X^2=-1,E\\cdot X=2$) in $\\text{Bl}_9\\mathbb{F}_4$ which signals that this geometry is an elliptic CY 3-fold. \nIn the 5d reduction, this geometry has two gauge theory descriptions as predicted in \\cite{Jefferson:2017ahm}: $SU(3)_{\\frac{3}{2}}$ with $N_{\\bf F}=9$ and $Sp(2)$ with $N_{\\bf AS}=1,N_{\\bf F}=8$. This geometry is associated to the 6d rank 2 E-string theory on a circle. This becomes clearer after a flop transition with respect to the exceptional curve $X$. The flop transition described in Section \\ref{sec:transitions} leads to $\\text{dP}_9\\cup \\mathbb{F}_0^{g=1}$ geometry where we glue the anticanonical class in $\\text{dP}_9$ to the elliptic class $E$ (with $E^2=0$) in $\\mathbb{F}_0^{g=1}$. This is the rank 2 generalization of $\\text{dP}_9$ (or the 6d rank 2 E-string theory).\n\n\nAll top geometries in Table \\ref{tb:rank2-classification} come from 6d SCFTs. \nWe also claim that all smooth rank 2 3-folds engineering 5d SCFTs belong to one of the RG-flow families exhibited in Table \\ref{tb:rank2-classification}.\nTherefore, we deduce the following conclusion: \\emph{All rank 2 5d SCFTs realized by smooth non-compact 3-folds have 6d SCFT origins.}\n\nThis is one of the most important lessons from our classification of rank 2 5d SCFTs.\nThe same conclusion may hold also for singular geometries involving $\\text{O7}^+$-planes. As mentioned earlier, the classification of smooth 3-folds misses a single geometry corresponding to the theory $SU(3)_{\\frac{1}{2}} + 1\\textbf{Sym}$, despite the fact that this theory is known to have a brane construction involving $\\text{O7}^+$-planes \\cite{Hayashi:2015vhy}. This theory may be the only rank 2 SCFT which cannot be engineered by a smooth 3-fold. But, we also know that this theory can be obtained from a KK theory with 6d origin, so we have found no counterexamples to the notion that all rank 2 5d SCFTs come from 6d SCFTs.\n\nThe above discussion motivates classifying automorphisms of 6d SCFTs which lead to 5d KK theories, as in \\cite{Apruzzi:2017iqe}.  Given the fact that 6d SCFTs are already classified (not counting frozen singularities involving $\\text{O7}^+$ planes), the possible automorphisms can be deduced from symmetries of the tensor branch diagrams of 6d SCFTs dressed by gauge symmetries which respect the automorphisms.\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank  Ron Donagi, Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Dave Morrison, Kantaro Ohmori and Gabi Zafrir for useful comments and discussions. We also like to thank SCGP summer workshop 2017 for hospitality during part of this work. The research of P.J. and H.K. and C.V. is supported in part by NSF grant PHY-1067976. S.K. is supported by NSF grant DMS-1502170.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTranslocation features of polymers through natural and artificial\npores is a current active research topic in biophysics and\nnanotechnology~\\cite{KasPNAS96,RMP,NL}. Motivated by many broad\ninterest experimental results, different models have been introduced\nto describe and study in a simple way this and related problems. For\ninstance, single barrier potentials \\cite{Pizz}, as well as flashing\nratchet models \\cite{linke}, have been studied to describe the polymer\ntranslocation and polymer transport dynamics. The passage of small\nmolecules through passive cell channels can be also modeled by\nstochastic and rachetlike forces \\cite{shulten}. In some cases the\ntransport phenomena involves not translocation through pores, but also\nmolecular motors, whose complex action has been recently addressed at\nhigh attention \\cite{Bust,Bust09}. In addition, nanotechnological\napplications try to emulate the complex biological process related to\nthe translocation dynamics \\cite{mickler, starikov}.\n\nRecently, we have studied different models for the 1d translocation of\na spring-bead polymer helped by a motor using a sinusoidal\nforce~\\cite{fjf-sine}. The introduction of a time dependent driving\nforce imposes a new time scale on the system, and provides new and\nricher phenomenology: for sinusoidal driving, the translocation time\nshows an oscillatory behavior as a function of the frequency.\n\nIn order to introduce stochasticity in the motor action and motivated\nby the relevant role played by dichotomous noise in biological\nproblems, in this manuscript we consider the case of a polymer driven\nby a two-state force: constant force which pushes the polymer chain in\none direction during the activity of the motor, and zero force which\nleaves the polymer to diffuse freely otherwise. This pure dichotomous\nmechanism constitutes a first approach in describing a machine working\ndichotomously between two on-off states~\\cite{gomez,fjf-damn}.\n\nThe motor modeled in \\cite{gomez,fjf-damn} acts during a fixed time,\nwhile the waiting times are exponentially distributed with a mean time\ndepending on the ATP concentration. In the present work a simpler\ndichotomous mechanism which can well point out, by contrast, the\nspecific behavior of the ATP based machines is studied.\n\nOn the other hand, pure dichotomous driving makes sense in the\nnanotechnological context as well as in the biophysical one.  In the\nfirst case the passage of a polymer can be induced through a graphene\npore or solid state channeling \\cite{han99,luan2010} by applying a\ndichotomous force between the two sides of the layer.  In the second\ncase, the model can describe the translocation of a linear molecule\nthrough a cell membrane gate having a chemical potential difference\nbetween their two sides. The driving is in this case induced by the\ntypical open\/close mechanism which follow the purely dichotomous\nswitching largely used in literature \\cite{shulten,Millonas,kargol}.\n\nThe purpose of our work is to model phenomenologically the possible\nphysical systems described above. We want to stress here the\nqualitative specific results connected to the purely dichotomous\ndriving.\n\nThus, differently from the sinusoidal case, no special behavior is\nobserved in the mean translocation time of the polymer for the case\nhere studied. However, for this problem, another observable parameter\ncan be studied. In fact, single molecule experiments are able to\ndetect and use the instantaneous velocity in order to quantify the\ntranslocation process in forced systems \\cite{Bust}.  Remarkably, we\nfind a non trivial behavior of the polymer translocation velocity as a\nfunction of the mean frequency $\\nu$ of the driving with the presence\nof a maximum, even if the translocation time shows only a monotonic\nbehavior. This difference reveals the importance of dealing with\nseveral measures to explore the complex behavior of the polymer\ntranslocation.\n\nThe dependence of the stall force $F_{stall}$ of the machine is also\ncalculated. We find, again, a strong nonmonotonic behavior of\n$F_{stall}$ with the frequency, similar to the one found in\n\\cite{fjf-sine}.\n\nThe paper is organized as follows: first we present the model for\npolymer and the properties of the stochastic driving force. The main\nproperties of the translocation process are then calculated:\ntranslocation time, mean velocity and stall force. Finally, we analyze\nthe dependence of the above properties with the chain stiffness.\n\n\\section{The model}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.2cm]{SchemaRTN.eps}\n\\includegraphics[angle=-90,width=9.5cm]{RTN.eps}\n\\caption{Scheme of a linear chain driven by a dichotomous force\n  restricted to a small space region (width $L_M$). $T$ is the mean\n  time during which the force maintains a same value\n  $\\{0,F_M\\}$.}\n \\label{schema}\n\\end{figure}\n\nThe polymer is modeled as a unidimensional chain of $N$\ndimensionless monomers connected by harmonic springs \\cite{Rouse}.\n\n1d models are suitable in order to describe the dynamics of polymers\nconstrained to move in confined channels \\cite{luan2010}. Also, in\nmany experimental situations [7] the polymer is stretched, thus\nremoving the dimensionality dependence of the measured\nquantities. Moreover, in this work, we want to fix our attention to\nthe motor activity in the translocation more than the delay given by\nother effects, such as entropic contributions.\n\nThe total potential energy is\n \\begin{equation}\n V_{\\rm har}=\\frac{k}{2}\\sum_{i=1}^{N} (x_{i+1}-x_i-d_0)^2,\n \\label{v-har}\n \\end{equation}\n\\noindent where $k$ is the elastic constant, $x_i$ the position of the\n$i$-th particle, and $d_0$ the equilibrium distance between adjacent\nmonomers.\n\nThe translocation is helped by the presence of a motor which is\nactivated dichotomously. The machine has a spatial working width $L_M$\nand the position $x=0$ represents the right edge of its action (see\nFig.~\\ref{schema}). Thus the monomers $i$ such that $x_i \\in [-L_M,0]$\nexperience a force made by the motor. We define $\\eta(x,t)$ to\nrepresent the dichotomous force, which fluctuates between two values\n$0$ (no force) and $F_M$. Thus \\begin{equation} {\\langle \\eta(t)\n  \\rangle}=\\frac{F_M}{2} \\; \\; \\, \\; {\\rm and} \\; \\; \\; {\\langle\n  \\eta(t) \\eta(t') \\rangle}=\\frac{F_M^2}{4}(1+ e^{-2\\frac{(t'-t)}{T}})\n   \\label{eta-corr}\n \\end{equation}\nHere $T$ gives the mean residence time in each state. With\n respect to the spatial dependence\n\\begin{equation} \\eta(x) = \\left\\{   \\begin{array} {lr}\n     F_M     &    x \\in [-L_M,0] \\vspace{0.2cm}\\\\\n     0       &    otherwise\n   \\end{array} \\right.\n   \\label{Potential-s}\n \\end{equation}\n\n\n\nThe dynamics of the $i^{\\rm th}$ monomer of the chain is then\ndescribed by the following overdamped Langevin equations:\n \\begin{equation}\n  \\dot{x}_i = -\\frac{\\partial{V_{har}}}{\\partial{x_i}} + \\eta(t,x_i) + \\xi_{i}(t)\n  \\label{lang}\n \\end{equation} where the viscosity parameter for each monomer is included in the\n normalized time units. $\\xi_{i}(t)$ stands for Gaussian uncorrelated\n thermal fluctuation and follows the usual statistical properties\n $\\langle\\xi_i(t)\\rangle=0$ and $\\langle\\xi_i(t)\\xi_j(t')\\rangle = 2 D\n \\delta_{i j}\\delta(t'-t)$.\n\n\n\\section{Results}\n\nWe performed a set of $N_{exp}=20,000$ numerical experiment with a\nstochastic Runge-Kutta algorithm, using a time step of $dt=0.01$.  The\npolymer is compound by $N$ monomers and starts with all the spring at\nthe rest length ($d_0=1$), and the last monomer of the chain lies at\n($x_N=0$), just in the final action range of the dichotomous\nforce. The noise intensity is held fixed at the value $D=0.001$,\n$L_M=5.5$, and $N=12$. The choice of the number of monomers $N$, or\nequivalently the length $L$, is arbitrary and this small number has\nbeen used for computational convenience. We note that in a previous\nwork \\cite{fjf-sine}, also with 1d chain, it was found that $\\tau$\nscales with $L^2=(N-1)^2$. Similarly we find that $v$ scales with\n$1\/N$.\n\nIn this first part, the elastic constant $k$ is held equal to 1, a\nmeaningful choice that corresponds to a not too rigid approximation\nfor the polymer. We will study the main observables of the system as a\nfunction of the mean frequency transition $\\nu=1\/T$.\n\n\\subsection{Translocation times}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{tauRTN-OD.eps}\n\\caption{Polymer translocation time $\\tau$ as a function of the\n  frequency $\\nu$ of the fluctuating force. The solid line shows the\n  theoretical prediction of Eq.~\\ref{theor}. In the inset is shown the\n  standard deviation of the exit times distribution.}\n\\label{RA}\n\\end{figure}\n\nThe translocation time $\\tau$ is computed as a mean first passage time\nof the center of mass of the polymer: the average over the $N_{exp}$\nrealizations of the time spent by the center of mass of the chain to\nreach the position $x=0$.  In Fig.~\\ref{RA} we see the value of $\\tau$\nplotted as function of the mean frequency of the driving, and, in the\ninset, the standard deviation $\\sigma$ whose values are of the same\norder of magnitude than the mean time values, as expected. We find\nthat $\\tau$ is a monotonic function of $\\nu$. This result is different\nfrom that for a periodic force (sinusoidal or square wave ones) where\nit is observed a minimum in the translocation for $\\nu \\sim 10^{-2}$\nand an oscillating behavior for higher frequencies~\\cite{fjf-sine}.\n\nIn contrast with the behavior of the translocation time, as we will\nsee, the velocity is not a monotonic function of $\\nu$ and a maximum\nis found in this function for $\\nu \\sim 10^{-2}$ (see\nFig.~\\ref{vRTN}). Both effects (minimum translocation\ntime~\\cite{fjf-sine} or maximum mean velocity) reveal some interesting similarities with the resonant activation phenomenon~\\cite{Pizz2,RA}.\n\nWe can make a simple analytical prediction for $\\tau(\\nu)$ in the low\nfrequency region which however is found to be valid in a broad\nfrequency range (see solid line in Fig.~\\ref{RA}). Let $\\tau_{on}$ be\nthe value of the exit time when a constant force $F_M$ is applied\nduring all the dynamics. In the $\\nu \\to 0$ limit we have to\ndistinguish between two cases depending on the initial value of the\nforce, $F_M$ or $0$. In the first case the translocation time is\n$\\tau_{on}$ corrected in a first approximation by a long waiting time\n$T$ if the system switches to the {\\em off} state before $\\tau_{on}$,\nwhich occurs with a probability $p_s=1-e^{\\tau_{on}\/T}$. This\ncorrection gives a contribution of $\\tau_{on} (1-p_s)+ (T+\\tau_{on})\np_s$ to the total time. In the second case there is an additional time $T$ in the off state for escaping. Thus the total translocation time\nis\n \\begin{equation}\n  \\tau \\simeq \\frac{1}{2}(\\tau_{on} + T (1-e^{\\tau_{on}\/T})) +\n \\frac{1}{2} (\\tau_{on} + T + T (1-e^{\\tau_{on}\/T}))\n \\label{theor1}\n \\end{equation}\nSince this equation is derived in the low frequency limit where\n$1-e^{\\tau_{on}\/T} \\simeq \\tau_{on}\/T$ we have\n \\begin{equation}\n  \\tau \\simeq 2 \\tau_{on} + T\/2\n  \\label{theor}\n \\end{equation}\nThe intermediate frequency region is characterized by the presence of\nthe constant force alternated by the absence of the force (diffusive\ndynamics) with an average time ratio between them different for\ndifferent values of the mean frequency. Surprisingly,\nEq.~(\\ref{theor}) also describes in a good way that frequency region.\nThe third region is instead characterized by a high frequency\nswitching rate between the two force states. There the translocation\ntime is much smaller than $T$, the polymer experiences a mean force\n$F_M\/2$, and $\\tau \\simeq 2 \\tau_{on}$. A careful observation of our\nnumerical results show that in this high frequency regime\n \\begin{equation}\n  \\tau \\simeq 2 \\tau_{on} + T ,\n  \\label{theorH}\n \\end{equation} not observable in Fig.\\ref{RA}.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{vRTN-OD.eps}\n \\caption{Mean velocity and number of monomers inside the motor while\n   in its active state (inset) as a function of the mean frequency\n   $\\nu$ of the fluctuating force.}\n \\label{vRTN}\n\\end{figure}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{Pt2.eps}\n\\caption{First passage time (left) and velocity (right) probability\n  distributions at four frequency values.}\n\\label{P}\n\\end{figure}\n\n\\subsection{Mean velocity}\n\nFig.~\\ref{vRTN} shows the mean velocity $v$ of the polymer as a\nfunction of $\\nu$. The inset of the figure shows the average number of\nmonomers inside the motor during the active states, $n_{mot}$. We can\nsee that this number is not constant for different values of the mean\nfrequency $\\nu$, at least for the value of the elastic constant $k=1$\nused in those calculations.\n\nThe main result in the velocity curves is the presence of a well\npronounced maximum, which put in evidence the qualitative difference\nbetween the calculation of the mean first passage time and the mean\nvelocity. In fact, the velocity is computed as $v_{cm}=1\/N_{exp}\n\\sum_i^{N_{exp}} L\/t_i$, where $t_i$ is the escape time in the $i$-th\nrealization.\n\nAs visible in Fig.~\\ref{P}, the exit times distribution changes in\nshape by changing the mean driving frequency $\\nu$. For high values of\n$\\nu$, the time distribution is very narrow around its mean value\n$\\tau$. The corresponding probability distribution function for the\nvelocity is also a narrow function. Decreasing the value of $\\nu$, the\ndistributions are more asymmetric and the width increases. The maximum\nof the time distribution moves toward lower values of time, but the\nasymmetry changes and higher and higher values of translocation times\nare involved. That's why in that region the mean first passage time\n$\\tau$ increases, although the time of the $P(t)$'s maximum\ndecreases. The distribution of the velocity change as well; but\nbecause of the increased width of the time distribution, the mean\nvalue of the velocity in that region does not follow the relation\n$v_{med}=1\/\\tau$ and increases with respect to the high frequency\nlimit, in opposite direction as the one expected from the time\nbehavior. The reason of this effect is that in the average, the\nsmaller times have a higher weight in the inverse $1\/t_i$ than bigger\nones. Thus the mean velocity increases up to a maximum. Decreasing\n$\\nu$ in the low frequency region, the average of the times continues\nrising up, because the distribution involves higher and higher\ntimes. The velocity, however, now decreases since the very high times\nescapes do not contribute importantly to the mean velocity.\n\nIn a first approach the translocation velocity in the high frequency\nlimit is given by \\begin{equation} v_l=\\frac{F_M}{2} \\frac{n_{mot}}{N}, \\end{equation} a\nfraction of monomers given by $n_{mot}\/N$ experience a force\n$F_M\/2$. Then, the corresponding translocation time is $2 \\tau_{on}$\n(remind that $\\tau_{on}$ is the escape time if the motor is always\nworking). On the contrary, in the low frequency limit one half of the\nrealizations give a very long escape time (and velocity goes to zero)\nand another half give $\\tau_{on}$. Thus for low frequencies we obtain\nthe same value of the velocity that for high frequencies.\n\nHowever, we can see in Fig.~\\ref{vRTN} that the low frequency limit of\nthe mean velocity does not satisfy the relationship just derived,\nbeing lower than the high frequency value $v_l$. This happens because\nthe force exerted on the polymer is affected by the number of monomers\ninside the motor which, as shown in the inset of the figure, also\ndepends on $\\nu$. We will see below a confirmation of the given\nrelation by using a strong elastic constant between the monomers,\nwhich guarantees a constant number of monomers inside the motor\nnevertheless the dynamical conditions are (see Fig.~\\ref{ka}).\n\nFrom Eq.~(\\ref{lang}) it is easy to derive the following equation for\nthe mean velocity \\begin{equation} v=\\frac{1}{N} \\frac{1}{N_{exp}}\n\\sum_{i=1}^{N_{exp}} \\langle \\eta_i(t) \\rangle_T = \\frac{F_M}{N}\n\\frac{1}{N_{exp}} \\sum_{i=1}^{N_{exp}} \\frac{n_{\\rm mot, i}^{\\rm\n    on}(\\nu) t^{on}_i} {t_i},\n \\label{v_t}\n \\end{equation}\nwhere, for each experiment $i$, $n_{\\rm mot, i}^{\\rm on} (\\nu)$ is the\naverage number of particles inside the motor when the motor is $on$,\n$t^{on}_i$ is the total motor working time, and $t_i$ is the\ntranslocation time of each realization.\n\nAt high frequency, $t_i^{on}=\\frac{1}{2}t_i$. However, decreasing the\nfrequency for most of the cases, $t_i^{on}>\\frac{1}{2}t_i$, during the\ntranslocation the motor spends more time activated that deactivated\nsince most translocations happen during the activation stage of the\nmotor. Thus both, the translocation time and the mean velocity\nincrease\\footnote{This is not the case at low values of $k$, where\n  the strong change in $n_{mot}$ with the frequency dominates the\n  overall behavior and suppress the velocity maximum as shown in\n  the inset of Fig.~\\ref{ka} for $k=0.1$.}. This behavior changes when $1\/\\nu \\sim\n\\tau_{on}$. Then $t_i^{on}$ remains constant in Eq.~(\\ref{v_t}), $t_i$\nincreases when $\\nu$ decreases and the velocity also decreases towards\nthe expected $v_l$ value moderate by the mean number of monomers in\nthe motor in the low frequency limit. This explains the presence of\nthe maximum in the velocity.\n\nA rough estimation of $n_{\\rm mot}$ is given by the fixed value\n$n_{\\rm mot}=5.5$, corresponding to the distribution of monomer inside\nthe motor in the case that they maintain the same relative distance,\nequal to the rest, over all the dynamics.  This condition will be\ncompletely satisfied for high values of the elastic constant $k$\n(rigid chain limit), when $n_{mot}$ becomes independent on $\\nu$. As\nwe will see below, both the high and low frequency limits for the mean\nvelocity take in that case the same value (see the inset of\nFig.~\\ref{ka})\n\n \\[v_{l, theor}= \\frac{0.2 \\cdot 5.5}{2 \\cdot 12}= 0.04583,\\]\nwhich is slightly higher than the limit value $v_l$ shown in the inset\nof Fig.~\\ref{RA} because $n_{mot}(k=1)<5.5$.\n\n\n\\subsection{Stall Force}\n\nThe stall force $F_{stall}$ is the force that we need to apply against\nthe motor in order to stop the polymer translocation. It is a measure\nof the strength of the motor and, in this model, it depends on the\nfrequency of the driving.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=9.5cm]{SchemaMotorePullCM.eps}\n\\caption{A force pull is applied at the first monomer to measure the\n  motor stall force.} \\label{stall2}\n\\end{figure}\nA set of simulations have been performed by applying a pull force\n$F_p$ (see Fig.~\\ref{stall2}) on the left extremum of the chain, in\nopposite direction to the motor driving force. \u00a0The initial condition\nfor the chain has been fixed with the polymer center of mass in the\ncenter of the motor. Then, the velocity of the center of mass is\nmeasured waiting for the exit on the left or on the right of the motor\nregion. That way, the force for which the mean velocity is zero gives\n$F_{stall}$.\n\nFig.~\\ref{StallForce} shows the stall force as a function of the\nfrequency. As shown in the lower inset, for a given frequency the mean\nvelocity decreases linearly with $F_p$. The upper inset, shows that\nfor pull forces of the order of the stall force the velocity presents\na minimum, contrary to the behavior at $F_p=0$ (Fig.~\\ref{vRTN}). Then\nthe stall force, which presents a similar trend, shows a clear minimum\nin the same frequency region.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{f_v_OD.eps}\n\\caption{Stall force as a function of the frequency $\\nu$ of the\n  driving.  The upper inset shows the mean velocity as a function of\n  $\\nu$ for the three pull force values $F_p=0.447, 0.454, 0.465$. The\n  bottom \u00a0inset shows the linear behavior of the mean velocity as a\n  function of the pull force for the three frequency values\n  $\\nu=0.032, 0.02, 0.1$. The other parameters are the same of\n  Fig.~\\ref{RA}.}\n\\label{StallForce}\n\\end{figure}\n\n\nAs in the oscillating case \\cite{fjf-sine}, the scale variation of the\nstall force is small (around $7.5\\%$), and an experimental\nverification of its behavior with the mean frequency the minimum could\nbe not immediately simple to perform.\n\n\\subsection{Elastic constant dependence}\n\nFinally, we investigate the dependence of translocation time and\nvelocity on the elastic constant $k$ of the polymer. A magnitude that\nstrongly depends on $k$ is the mean number of monomer inside the motor\nduring the pushing cycle, $n_{mot}$.  This number modules the velocity\nas it is show in Eq~(\\ref{v_t}). Results are plotted in Fig.~\\ref{ka},\nwhere the translocation time and velocity (in the inset) are presented\nfor different value of $k$.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{ka.eps}\n\\caption{Translocation time and mean velocity (in the inset) of the\n  polymer chain for different values of the elastic constant $k$.}\n\\label{ka}\n\\end{figure}\nWe can see that notable differences (especially visible in the mean\nvelocity plot) are evident by changing the value of $k$. For the case\n$k=0.1$ the velocity looses the maximum, which is always present for\nthe higher values of $k$. As expected, a clear saturating behavior of\nthe whole curve is evident by increasing $k$ when the chain behaves\nlike a rigid bar.  As announced before in the text, in this limit the\nmean velocity in the cases of both high and low switching frequency\ngives the expected value $v_{l,theor}$ given above. This limit is\nalready fulfilled for $k=5$.\n\n\n\\section{Conclusions}\nThe interest in the introduction of simple models is that they can\ncapture the more relevant features of different processes. In that\nway, they can result to be very useful for a coarse-grain description\nof different systems.\n\nThe model described here studies the translocation process of a\npolymer driven by a simple motor which exerts a dichotomous force.  We\nanalyze the dependence of the translocation time with the mean\nfrequency of the driving field, and find an analytical expression for\nthe low frequency regime.  In spite of the monotonic behavior of the\ntranslocation time, the velocity presents a clear maximum at a\nresonant value of the mean frequency. We argue that this maximum comes\nfrom the optimization of the \"on states\" duration of the driving\nforces with the corresponding translocation time. The detection of\nthis maximum, (also seen in the periodic case) could be tackled with\nthe recent single molecule experimental techniques.\n\nThe stall force able to stop the polymer translocation against the\nmotor has been also evaluated, finding in our calculations results\nvery close to the oscillating driving, previously studied. The stall\nforce show a very clear minimum at a resonant mean frequency of the\ndriving.\n\nThe model can have application in artificial nanotechnological devices\ndriven by dichotomously fluctuating fields, as well as biological pore\nmembrane with intrinsic noise.\n\n\n\n\\vspace{0.5cm}\nThis work has been supported by the project\nFIS2008-01240 of the Spain MICINN.\n\n\\vspace{0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgment} %\nThe research presented in this paper was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2075 - 390740016.\nThis work was possible due to kind funding by the Ministerium f\u00fcr Wissenschaft, Forschung und Kunst Baden-W\u00fcrttemberg via the project SiVeGCS-MWK.\nThe authors gratefully acknowledge the support and the computing time on ``Hawk'' provided by the HLRS through the project ``hpcdg'' and the support by the Stuttgart Center for Simulation Science (SimTech).\nMoreover, the authors gratefully acknowledge the SmartSim developers for their support.\n\n\n\n\n\\bibliographystyle{elsarticle-num}\n\n\\section{Reinforcement Learning - A Brief Outline}\\label{sec:rl}\n\nThe following section gives a brief outline of the general reinforcement learning (RL) paradigm.\nHowever, this summary is by no means exhaustive and provides only the bare fundamentals required to motivate our software implementation, which is introduced in \\secref{sec:software}.\nFor a more thorough discussion of RL, the reader is referred to \\cite{sutton2018reinforcement}.\nIn the RL paradigm, an agent trains by interacting with an environment, as illustrated in \\figref{fig:MDP}.\nIn each point in time $t$, the environment is in some state $s_t$, based on which the agent's (possibly parametrized) policy $\\pi_{\\theta}\\left(a\\:|\\:s_t\\right)$ prescribes which action $a_t$ the agent should perform.%\n\\footnote{In principal, the policy $\\pi_{\\theta}\\left(a\\:|\\:s=s_t\\right)$ is a random variable, which describes the conditional probability distribution of performing action $a$, given the state $s_t$. To keep the notation short, we will use $\\pi_{\\theta}\\left(a\\:|\\:s_t\\right) \\equiv \\pi_{\\theta}\\left(a\\:|\\:s=s_t\\right)$ and refer the reader again to \\cite{sutton2018reinforcement} for more details. The same holds for the transition function $\\mathcal{T}\\left(s_{t+1}\\:|\\:a_t,s_t\\right)\\equiv\\mathcal{T}\\left(s_{t+1}\\:|\\:a=a_t,s=s_t\\right)$.}\nThis action causes the environment to change its state to a new state $s_{t+1}$, which is determined by the environment's transition function $\\mathcal{T}(s_{t+1}\\:|\\:a_t,s_t)$.\nThe transition function thus encodes the dynamics of the environment, which could be for instance the spatial and temporal integration of the discretized Navier-Stokes equations.\nAlongside the new state $s_{t+1}$, the agent receives a reward $r_{t+1}=\\mathcal{R}(s_{t+1})$, which quantifies how favorable the transition is with respect to some performance metric.\nThe reward function $\\mathcal{R}(s)$ is highly problem-specific and has to be designed by a domain expert.\nBased on the new state $s_{t+1}$, the agent performs another action, until the environment reaches a final state $s_n$.\nEventually, such an episode results in a trajectory $\\tau$ of states, actions and rewards:\n\\begin{equation}\n  \\tau = \\left\\{ \\left(s_0,a_0\\right),\\left(s_1,a_1,r_1\\right),\\:......\\;,\\left(s_{n},a_{n},r_{n}\\right)\\right\\}.\n  \\label{eq:trajectory}\n\\end{equation}\nThis problem formulation is typically framed as a Markov decision process (MDP), which again can be interpreted as a discrete-time control task.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{fig\/RL_MDP.pdf}\n  \\caption{General outline of the Markov decision process (MDP). At step $t$, the environment is in state $s_t$. Following its policy $\\pi_{\\theta}(a\\:|\\:s_t)$, the agent performs an action $a_t$. In deep RL, the policy is a deep artificial neural network (ANN) with parameters $\\theta$. The action causes the environment to transition from state $s_t$ to a new state $s_{t+1}$ that is prescribed by the environment's transition function $\\mathcal{T}(s\\:|\\:a_t,s_t)$. Based on how desirable the new state is, the agent receives a reward, which is determined by the reward function $r_{t+1}=\\mathcal{R}(s_{t+1})$.}\n  \\label{fig:MDP}\n\\end{figure}\n\nThe behavior of the agent is described by its policy $\\pi_{\\theta}$, which determines the action the agent performs for each of the possible states of the environment.\nIn deep RL, this policy is represented by an ANN with the weights $\\theta$.\nThe key quantity to distinguish between favorable and unfavorable actions is the expected future return along a trajectory $\\tau$ with $n$ steps\n\\begin{equation}\n  R(\\tau) = \\sum_{t=1}^{n} \\gamma^t r_{t} .\n\\end{equation}\nHere, $\\gamma \\leq 1$ is the discount factor, which balances the importance of short-term and long-term rewards.\nThe overall goal of the RL algorithm is then to find the optimal policy, i.e. the set of optimal model parameters, which maximizes the expected return on all possible initial states.\n\nThe key purpose of an RL algorithm is to state an optimization task that allows optimization of the policy based on sampled interactions of the agent with the environment.\nRL algorithms thus differ primarily in the way those interactions are sampled and how these interactions are used to update the policy.\nThroughout this work, we employ the clipping version of proximal policy optmization (PPO) \\cite{schulman2017proximal} as our RL algorithm of choice.\nWe highly recommend \\cite{notter2021hierarchical} for a clear and concise summary of the PPO method.\n\n\n\\section{Software Architecture}\\label{sec:software}\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{fig\/Framework_paper.pdf}\n  \\caption{General architecture of Relexi. Before entering the training loop, the SmartSim IL is used to launch the SmartSim Orchestrator, which provides a database to exchange data between Relexi and the HPC workload. At the beginning of each training loop, the SmartSim IL starts a batch of FLEXI simulations and distributes them to the \\textit{worker} nodes. The simulations are initialized on a randomly drawn initial state and are then evolved in time by the flow solver. For our application in turbulence modeling, FLEXI sends each time interval $\\Delta t_{RL}$ its current state to the agent and receives a new action, i.e. a new set of parameters for the turbulence model. Since a syncronous RL algorithm is applied here, Relexi waits until all simulations have terminated. The collected episodes $\\tau^{(i)}$ are then used by the gradient ascent algorithm to improve the model weights $\\theta$. Here, $\\nabla_{\\theta} J(\\tau^{(i)})$ is the gradient of the loss function, evaluated on the sampled experience. The training loop is then repeated with the new set of weights, i.e. the new policy, until the policy converges.}\n  \\label{fig:relexi}\n\\end{figure*}\nIn this work, we present a novel and modular RL framework named Relexi\\footnote{Code available at: \\href{https:\/\/github.com\/flexi-framework\/relexi}{https:\/\/github.com\/flexi-framework\/relexi}} that implements an RL training loop for applications in the field of scientific computing, as illustrated in \\figref{fig:relexi}.\nAs discussed beforehand, coupling HPC codes with modern ML libraries is oftentimes tedious.\nThis holds especially for RL, where the interaction between the HPC application and the RL algorithm becomes much more intricate.\nA considerable amount of these difficulties stem from the differences in the required hardware and the different preferences in terms of programming languages.\nMany approaches in coupling HPC and ML tried to either rewrite basic ML capabilities in languages used for HPC or writing an HPC solver in Python from scratch.\nConsidering the enormous codebases on both sides and the enormous pace of progress in the ML community, this approach is, from our perspective, doomed to fail.\nIn addition, both families of methods require significant computational effort:\nWhile the parameter update of the policy requires access to the collected trajectories to compute the gradient vectors through backpropagation, running CFD simulations mandates high parallel efficiency, in particular for problems of practical interest.\nThus, compromising the performance of any one component, e.g. through re-writing, is not advisable.\nRelexi tries to bridge this gap differently by employing the SmartSim library to couple a state-of-the-art RL library with modern HPC solvers.\nSpecial focus is put on the modularity of the framework, which allows the exchange of RL algorithms and simulation environments with only minimal changes of the underlying code. \nRelexi is designed with application on HPC system in mind, scaling applications up to thousands of cores on modern HPC systems.\nRelexi comprises the following building blocks, which will be discussed in more detail in the following sections.\n\\begin{itemize}\n  \\item \\textbf{SmartSim}: The SmartSim library \\cite{partee2021using} fulfills two major tasks. Firstly, the library is used to start and manage the MPI-parallelized HPC workloads on the allocated hardware resources. Moreover, it implements the communication between the HPC workload and Relexi by deploying an in-memory database and providing communication clients for several programming languages.\n  \\item \\textbf{FLEXI}: In this work, the flow solver FLEXI \\cite{krais2021flexi} provides simulations of turbulent flow as a training environment for the agent. FLEXI is used here for illustrative purposes only and can be easily replaced by other simulation codes.\n  \\item \\textbf{TensorFlow\/TF-Agents}:  Relexi is build upon TensorFlow \\cite{tensorflow2015-whitepaper}, which provides a framework for efficient ML workflows and allows for distributed training on multiple GPUs in parallel. In addition, its RL extension TF-Agents \\cite{TFAgents} provides implementations of several RL algorithms and allows for custom environments that can be integrated in a straight-forward fashion.\n\\end{itemize}\n\n\n\\subsection{SmartSim}\\label{subsec:smartsim}\nThe SmartSim library \\cite{partee2021using} is a workflow library that simplifies the convergence of traditional HPC workloads and ML.\nThe library comprises two components: the SmartSim Infrastructure Library (IL) and SmartRedis.\nThe IL provides extensive functionalities to start, manage and distribute workloads in HPC environments as well as submitting workloads automatically as batch jobs to the job scheduler. \nIn Relexi, we employ the IL to repeatedly start the MPI-parallelized simulation instances on demand for each training iteration inside of a single allocated batch job.\nThe simulations can be either started individually or multiple simulations can be lauched within a single call by using the multiple-program-multiple-data (MPMD) paradigm.\nWe also use another key aspect of the SmartSim IL, which is its ability to configure and launch an in-memory, Redis-based datastore referred to as the Orchestrator. \nThis database serves as a data intermediary between the simulations and the main program.\nFor Relexi, we used a non-clustered Orchestrator, which is launched on the \\textit{head} node.\nSmartSim also supports the distribution of the Orchestrator across multiple nodes, but a single instance on the \\textit{head} node was sufficient for our application.\nThe simulation communicates with the Orchestrator using the  SmartRedis clients (available in Python, Fortran, C, and C++) which can send and receive data from the orchestrator or trigger actions within the Orchestrator, i.e. running scripts or ML model inference previously loaded into the database.\nIn our case, during training, FLEXI sends its current flow state via the Fortran SmartRedis client to the Orchestrator.\nIn addition, a scalar flag is sent which indicates whether FLEXI has reached its final state and will terminate.\nRelexi then uses the Python SmartRedis client to read this data from the Orchestrator.\nSubsequently, the agent's actions are sent by Relexi to the database via its Python SmartRedis Client and are read by FLEXI via its Fortran client.\nWhile SmartSim itself employs a Redis database by default, we used the multi-threaded fork of Redis called KeyDB, which provided significantly more performance for our application.\n\n\\subsection{FLEXI}\\label{subsec:flexi}\nThe open-source flow solver FLEXI \\cite{krais2021flexi} is based on the discontinuous Galerkin (DG) method, which can be seen as a hybrid of the finite element and the finite volume methods.\nFor the DG method, the computational domain is partitioned into individual elements.\nIn each element, the solution is represented by a polynomial basis of degree $N$ and the individual elements are then coupled by consistent surface fluxes at the element faces.\nThis results in a small communication stencil, which allows FLEXI to scale perfectly on hundreds of thousands of cores using a pure distributed-memory parallelization with MPI.\nA detailed description of FLEXI can be found in \\cite{krais2021flexi}.\nThe communication between FLEXI and the database is implemented by means of the SmartRedis clients provided by the SmartSim library.\nTo this end, SmartRedis is linked to FLEXI during compile time.\nImplementing the data transfer in FLEXI is straight-forward and requires only a few lines of additional code.\nSince FLEXI is parallelized with a distributed-memory approach, every rank contains only a chunk of the global flow state, which represents the environment's state $s_t$ in the RL formulation.\nTo communicate with the database, the flow state is thus first gathered across all ranks of the respective FLEXI instance before it is written to the database by the root rank.\nAnalogously, the predictions sent by Relexi are retrieved from the database by the root rank and are then scattered across the other ranks of the respective FLEXI instance.\nIt seems important to stress once again that FLEXI is used primarily for demonstration purposes and can be easily replaced by other simulation codes by using the SmartRedis Clients.\n\n\n\\subsection{Relexi}\\label{subsec:relexi}\nRelexi implements the main RL training loop by means of the TF-Agents \\cite{TFAgents} library, as illustrated in \\figref{fig:relexi}.\nIn TF-Agents, custom environments can be implemented by subclassing the provided environment class and implementing a few mandatory methods like initialization and time-stepping.\nSince the TF-Agents library interacts with the environment via these pre-defined function interfaces, all intricacies of the coupling with the HPC workload are ``hidden'' inside of the environment class.\nThis has the advantage that all RL algorithms and tools provided by TF-Agents can interact natively with the custom environment and thus, with the HPC workload.\nThe hardware resources are distributed as follows.\nRelexi itself is started as a single thread on a node termed \\textit{head}.\nThe \\textit{head} node executes all ML-specific work and thus should be equipped with a GPU.\nRelexi also supports setups with multiple GPUs on the \\textit{head} node by using the distribution strategies provided by TensorFlow.\nThe SmartSim Orchestrator is also started on the \\textit{head} node at the beginning of Relexi.\nDuring training, the HPC workload is repeatedly started with MPI on the available \\textit{worker} nodes.\nTo ensure that each MPI rank is placed correctly on the available hardware and to avoid double occupancy, Relexi generates rankfiles on-the-fly based on the available hardware resources.\nThese rankfiles are then passed to MPI to ensure the correct placement of the MPI ranks.\n \nAfter the simulations are launched, each simulation instance writes its initial states to the database.\nRelexi reads these states, provides the respective actions based on the current policy and writes them back to the database.\nEach FLEXI instance reads its prescribed actions from the database and proceeds with its simulation to obtain the next state. %\nIn the meantime, Relexi polls until the new state becomes available and reads it from the database.\nWith the new state, the agent can compute the reward and get the new actions from the current policy.\nThese steps are repeated until the required amount of experience is sampled, on which the policy then can be trained.\nThis algorithmic outline of Relexi is also summarized in \\algoref{alg:relexi}.%\n\\footnote{For the sake of clarity, \\algoref{alg:relexi} only gives the algorithm for a single environment. To generalize it to multiple environments in parallel, each of the lines 7 to 11 is simply executed for each individual environment.}\n\n\\begin{algorithm}\n  \\caption{Relexi}\n  \\begin{algorithmic}[1]\n    \\State Initialization\n    \\State Launch SmartSim Orchestrator\n    \\For{$i = 1,i_{max}$}  \\Comment{Run $i_{max}$ iterations}\n       \\State Start FLEXI instances\n       \\State Read $s_{0}$\n       \\For{$t=1,n$}  \\Comment{Run simulation for $n$ steps}\n         \\State $a_t \\leftarrow \\pi_{\\theta}(a\\:|\\:s_t)$\n         \\State Write $a_t$\n         \\State Polling for $s_{t+1} $\\Comment{Here, the HPC solver runs}\n         \\State Read $s_{t+1}$\n         \\State $r_{t+1}\\leftarrow \\mathcal{R}(s_{t+1})$\n       \\EndFor\n       \\State $\\tau \\leftarrow \\{ (s_0,a_0), (s_1,a_1,r_1), ..)\\}$\n       \\For{$j=1,n_{epochs}$}  \\Comment{Train ANN for $n_{epochs}$}\n         \\State $\\theta \\leftarrow \\theta + \\alpha \\, \\nabla_{\\theta} J(\\tau)$\n       \\EndFor\n    \\EndFor\n    \\State Shutdown SmartSim Orchestrator\n  \\end{algorithmic}\n  \\label{alg:relexi}\n\\end{algorithm}\n\n\nA potential bottleneck we identified is the overhead introduced by repeatedly starting hundreds of parallel environments with thousands of MPI ranks.\nFor some configurations, the time required for starting the simulations exceeded the actual simulation time.\nTo tackle this issue, we implemented two major improvements.\nFirst, we employed the multiple-program-multiple-data (MPMD) functionality provided by OpenMPI's implementation of MPI, which is also supported by SmartSim.\nWith MPMD, all simulations can be started with individual commandline arguments within a single call of MPI.\nSecondly, we implemented a functionality to copy all files required by the simulation, e.g. parameter files and restart files, to local drives located in the random access memory (RAM) of each node.\nThis reduced the access times compared to using a parallel file system like Lustre significantly.\nWith these improvments in place, the performance penality of launching large amounts of environments became negligible.\n\n\n\n\\section{Hardware Configuration}\\label{sec:hardware}\n\n    All benchmarks and experiments are performed on the HPE Apollo 9000 supercomputer (Hawk) and the Hawk-AI extension, a HPE Apollo 6500 Gen10 Plus at the High-Performance Computing Center Stuttgart (HLRS).\n    In the following, the hardware of both systems is given in detail.\n\n    \\subsection{Hawk -- HPE Apollo 9000}\n\n        Hawk consists of 5,632 dual socket nodes with 256 GiB of main memory each.\n        Each node is equipped with two 64-core AMD EPYC 7742 (Rome) processors with a base frequency of 2.25 GHz.\n        The nodes are connected via an enhanced 9D-hypercube.\n        For the node to node interconnect, the high-performance interconnect InfiniBand HDR200 is used.\n        This leads to a homogeneous massively parallel system with 720,896 compute cores and approximately 1.37 TiB of main memory.\n        Hawk has a theoretical peak-performance of 25.1 Pflop\/s.\n        The system reached 19.334 Pflop\/s in the LINPACK-benchmark and 334.65 TFlop\/s in the HPCG benchmark.\n        Hawk is connected to a Lustre file system with a capacity of about 25 PiB.\n\n    \\subsection{Hawk-AI -- HPE Apollo 6500 Gen10 Plus}\n        The Hawk-AI extension consists of 24 dual socket nodes, which are each equipped with two 64-core AMD EPYC 7702 processors, 1 TiB of main memory, a local hard disk of 15 TiB and eight Nvidia A100 GPUs.\n        20 Nodes are equipped with Nvidia A100 with 40 GiB of memory and four nodes are equipped with Nvidia A100 with 80 GiB of memory.\n        The nodes are also connected via the InfiniBand HDR200 interconnect.\n        The Hawk-AI extension is connected directly to the enhanced 9D-hypercube of Hawk and to the same Lustre filesystem.%\n\n\\section{Conclusions}\\label{sec:conclusion}\nSupervised learning is generally suitable for learning in situations when input--output pairs can be defined a priori.\nIn contrast, in reinforcement learning, training data is gathered in an online process, in which the policy is sampled for the current state of the dynamical system.\nThis makes RL more suitable for learning optimal behaviors in dynamical systems, e.g. those described by the equations of fluid dynamics, where an a priori definition of an admissible and complete training data space is illusive and would, at best, be cumbersome.\nFlow control problems thus lend themselves naturally to an RL approach, however, other modeling tasks can be expressed in this context as well.\nHere, we have chosen to interpret the task of finding an optimal eddy viscosity in space and time as a control problem, and have the RL-trained agent predict a strategy.\nOur results show that this approach outperforms existing, established models and indeed returns a near optimal behavior in the chosen reward norm.\nThis highlights the potential of combining CFD and RL into an inclusive optimization framework.\n\nHowever, before we can explore or leverage this potential, we need to enable training and deploying such algorithms at scale, recognizing that they pose different challenges for software development and HPC.\nIn RL, trajectory data consisting of both actions and states needs to be gathered along the solution evolution during training and made available to the gradient update.\nThe RL agent and the CFD scheme thus need to be coupled during training, and they need to exchange large datasets continuously. In addition, the current policy must be explored efficiently, meaning that many parallel runs of the environments are required.\nThus, in this work, we propose Relexi as a novel framework for coupling RL algorithms with essentially generic solvers for partial differential equations on heterogeneous hardware.\nThe framework is made up of three components:\nThe PDE solver of choice, in our case, the LES solver FLEXI, TensorFlow for the ML definition and training, and the SmartSim library, which handles job allocation and management and provides an in-memory database for communication and intermediate storage of solution trajectories and policy commands.\nThe PDE solver and TensorFlow can run independently on their chosen hardware, thus exploiting their full parallel potential.\nWe show two methods for improving the overall time-to-solution of the RL problem with Relexi:\nFirst of all, the weak scaling across the gradient estimator, that is increasing the amount of runs of the environment for a given policy.\nMore parallel runs explore the current policy more efficiently and thus allow an overall better gradient update and thus quicker convergence.\nOur scaling results here indicate that the framework scales up to hundreds of parallel environments and thousands of compute cores with very good performance.\nFor the second approach, we scaled the individual simulation runs across more MPI ranks, i.e. we exploit the strong scaling characteristics of the standalone PDE solver.\nHere, we recover the expected behavior up until the individual core load becomes too small.\nIn combination, these results demonstrate that the Relexi framework is capable of efficient training on HPC systems at scale and can enable RL-methods for CFD for complex flow cases.\nFurther work will focused on applying Relexi to more complex cases and to other combinations of RL and PDE solvers.\n\n\\section{Introduction}\nIn recent years, there have been increasing efforts to transfer advances in machine learning (ML) to the field of computational fluid dynamics (CFD) in order to enhance simulations for a myriad of different applications \\cite{brunton2020machine}, which cover the fields of turbulence modeling \\cite{beck2021perspective,kurz2022machine,beck2019deep,maulik2019subgrid}, shock detection \\cite{ray2019detecting,beck2020neural}, the formulation of turbulent inflow conditions \\cite{fukami2019synthetic,kim2020deep} or applications in flow control \\cite{paris2021robust}.\nAs of today, most of these advances are based on the supervised learning (SL) paradigm.\nIn SL, the ML model is trained based on a dataset that is obtained a priori from CFD simulations or experiments.\nDuring training, the parameters of the ML model are optimized to approximate the functional relationship between the input and output quantities of the dataset.\nSince the training dataset is fixed and does not interact with the predictions of the ML model, the training itself is thus, in a sense, static and offline.\nThis oftentimes leads to inconsistencies, if, during inference, the SL-trained model is confronted with a varying and dynamic environment.\nIn this case, the model's prior predictions affect how the system evolves and thus which input states the model will see in the future.\nEnsuring that all potential states of a non-linear dynamical system are sufficiently represented within the training dataset is by no means trivial and for many applications elusive.\nThe reinforcement learning (RL) paradigm addresses this issue by training ML models not on a static dataset, but instead trains models by letting the model interact with the actual environment it will later be deployed in.\nThus, the training process not only aims at predicting outputs, but it does so by taking dynamically generated inputs into account.\nThe goal of RL is to find an optimal strategy for moving forward from the current, observed state.\nIn the course of the training process, predictions of the model change the state of the environment and the model will be rewarded based on how beneficial this transition is.\nTraining based on such sequences of actions and transitions allows incorporation of the long-term implications of the model's predictions into the training process.\nThus, the training itself becomes dynamic and requires ``online'' joint runs of the environment and the model.\n\nCFD simulations are typically computationally expensive and thus rely heavily on high-performance computing (HPC) systems.\nCoupling such HPC flow solvers with the more recently emerged ML libraries is tough, since both rely on different hardware, algorithms and overall programming paradigms.\nFor instance, most HPC codes are still written and optimized for CPU architectures and are parallelized with the Message Passing Interface (MPI), while ML libraries run most efficiently on GPUs.\nThe HPC environment thus has to provide sufficient hardware resources for both components of the application and efficient communication between them.\nIn addition, most HPC codes are written in compiled languages like C\/C++ or Fortran.\nOn the other hand, ML libraries are oftentimes also written in compiled languages but are exposed to the user via a Python interface.\nAs these languages typically lack a well-defined interface, bridging the language gap in an efficient manner for HPC is tough.\nThese problems are less pronounced for SL tasks, since the generation of the training dataset with HPC codes and the training itself are mostly decoupled and can be performed separately on appropriate hardware.\nThe HPC solver and the ML model only have to be coupled for inference, i.e. when the trained ML model is evaluated in actual simulations.\nFor this, Maulik et al. \\cite{maulik2021deploying} proposed to link the TensorFlow library directly to the flow solver and execute the model via TensorFlow's C-API.\nOtt et al. \\cite{ott2020fortran} proposed the Fortran-Keras-Bridge, which allows convenient execution of trained Keras models from Fortran code, and is based on the Neural-Fortran library by Curcic \\cite{curcic2019parallel}.\nFor RL however, the integration of ML and HPC workloads is much more involved, since the training process itself requires running simulations and thus needs to run HPC simulations and the optimization of the ML model in parallel.\nAn efficient RL framework for HPC workloads needs to manage the simulations in the HPC environment and has to implement efficient communication between the simulation codes and the employed ML library.\nIn \\cite{novati2021automating}, Novati et al. used the smarties library to couple a flow solver and TensorFlow in order to train a data-driven turbulence model for large eddy simulation (LES) on an HPC system.\nBae and Koumoutsakos applied a similar framework to the wall-modeling of wall-bounded flows in \\cite{bae2021scientific}.\nPawar and Maulik \\cite{pawar2021distributed} developed a distributed RL framework called PAR-RL, which they applied to optimize the time-stepping of a CFD simulation.\nRabault and colleagues developed an RL framework for flow control \\cite{rabault2019accelerating,rabault2019artificial,tang2020robust}, which was also coupled with a spectral flow solver by Li and Zhang in \\cite{li2022reinforcement}.\nSimilarly, Fan et al. \\cite{fan2020reinforcement} coupled a spectral element LES solver with TensorFlow to control the flow around a cylinder.\n\nHowever, most of those applications are limited in the problem sizes that can be investigated, since the simulations cannot take advantage of the parallel computing resources provided by modern HPC systems.\nIn this paper, we present a scalable RL framework that overcomes the gap between numerical simulation and ML workflows on HPC systems providing both components with its required specialized hardware.\nMoreover, we demonstrate the prospects of the reinforcement learning paradigm in scientific computing by applying our framework to derive data-driven turbulence models for large eddy simulation. \n\nThe paper is structured as follows.\n\\secref{sec:rl} gives a brief outline of the reinforcement learning paradigm.\nIn \\secref{sec:software}, we present the different software components of our framework and how they are integrated.\nThe hardware environment for our benchmarks is presented in \\secref{sec:hardware}.\nIn \\secref{sec:numerics}, we outline how our framework can be applied to derive turbulence models for large eddy simulations.\nThe results of our scaling studies and the performance of the derived data-driven turbulence models are presented in \\secref{sec:results}.\n\\secref{sec:conclusion} summarizes and concludes the paper.\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Scaling}\\label{sec:scaling}\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure0.pdf}\n  \\caption{Scaling behavior of the Relexi framework on up to 16 Hawk compute nodes (2048 MPI ranks) and one Hawk-AI node for the HIT test case with 24 DOF and 32 DOF for 2, 4, 8 and 16 MPI ranks per FLEXI instance. The black line indicates perfect scaling.}\n  \\label{fig:scaling_hit}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure1.pdf}\n  \\caption{Strong scaling of FLEXI within the Relexi framework for the HIT test case with 24 DOF and 32 DOF for 2, 4, 8 and 16 MPI ranks per FLEXI instance. For clarity, only the runs for 2, 8, 32 and 128 parallel FLEXI environments are shown. The black line indicates perfect scaling.}\n  \\label{fig:strong_scaling_hit}\n\\end{figure*}\n\nTo analyze the scaling behaviour of our framework, we use the HIT test case for the two different configurations listed in \\tabref{tab:testcases}.\nFor the benchmarks, we used a single Hawk-AI node of the HPE Apollo 6500 Gen10 Plus as \\textit{head} node and up to 16 Hawk compute nodes of the HPE Apollo 9000 as \\textit{worker} nodes in a single batch job.\nThis results in up to 2048 compute cores used for generating training data.\nIn the scaling analysis, we investigate two types of scaling.\nFirst, we examine how the framework scales if the number of parallel environments is increased, while keeping the amount of MPI ranks per environment constant.\nThis corresponds to a weak scaling approach, since the parallel load and the available resources are increased simultaneously, while keeping the load per rank constant.\nSince the used resources, and thus also the simulation time, for a FLEXI simulation should theoretically stay unchanged, eventually emerging losses in parallel speedup can be attributed solely to the communication overhead, the limited throughput of the database, the cost of the data managment in Relexi, the increasing amount of policy evaluations and the start as well as the termination of the simulations.\nThe measured executation time includes launching the simulations and running the simulation with the policy until all simulations terminated.\nAs a performance metric, we use the ``Speedup'', which is computed as the quotient of the time needed for sampling the $n_{envs}$ parallel environments and the time that would be needed to run $n_{envs}$ environments sequentially, i.e. the speedup of running the environments in parallel instead of sequentially.\nFor each configuration, we ran Relexi in two separate jobs for 6 iterations each to account for fluctuations in hardware and communication performance and computed the mean of the 12 measurements.\nThe scaling is performed with either 2, 4, 8 or 16 MPI ranks for each FLEXI run.\nFurther, we start with two parallel FLEXI instances and double them until the 16 Hawk compute nodes are fully occupied.\nIt seems important to note that the results are not compared directly to the performance of standalone FLEXI simulations, since the performance differences in running a single FLEXI instance within Relexi and running it standalone were negligible.\n\nThe results in \\figref{fig:scaling_hit} demonstrate that the framework can scale efficiently up to a thousand parallel environments on thousands of cores.\nTwo major trends can be identified in the scaling results.\nAs mentioned before, the observed decrease in parallel efficiency when running more environments in parallel can be attributed mainly to the sequential work done by Relexi.\nThe framework should scale better if the FLEXI instance gets more time-consuming, since then, this sequential work becomes less relevant and the perfect scaling abilities of FLEXI can be recovered.\nIn contrast, if the necessary time to compute the FLEXI simulation decreases, i.e. FLEXI gets more ranks, the sequential work of Relexi becomes more dominant, which decreases the scaling efficiency.\nThis causes the runs with fewer ranks per FLEXI instance to scale better than the runs using more ranks.\nThe second interesting behavior is that the decrease in performance when switching from a single to two parallel environments is most pronounced for the FLEXI environment with only two cores, which is counterintuitive. \nWe attribute this to the limited memory bandwith and the hierarchical architecture of the processors used.%\n\\footnote{The used EPYC CPUs comprise several dies, which contain 8 cores each. All cores on a single die share the available memory bandwidth.}\nIf a single FLEXI instance with two ranks is spawned on a compute node, this instance gets all available memory bandwidth.\nIf however, a second FLEXI is spawned, both instances have to share the available bandwidth, which slows down the simulation as well as the interaction with the database.\nThis effect vanishes with an increasing amount of used cores.\nThe observed loss of parallel efficiency especially for the last data points, which correspond to using all 2,048 available cores, can in parts be attributed to single simulations running significant slower than the average.\nThese outliers can probably be attributed to fluctuations in the load of the interconnect. %\nThis issue is subject of current investigations.\n\n\nFor the strong scaling, we examine the configurations with 2, 8, 32 and 128 parallel FLEXI instances.\nThe results of the strong scaling given in \\figref{fig:strong_scaling_hit} match the general behavior observed in the weak scaling benchmark. %\nIn cases where the computational cost of the HPC workload is dominant, i.e. if only few MPI ranks are used per environment, the optimal scaling of FLEXI can be recovered.\nIf the amount of MPI ranks is increased, the time spent for the simulation decreases and the work done by Relexi becomes the limiting factor.\nThis causes the parallel efficiency to decrease for low simulation loads per core.\nIt seems important to note that for both cases, using 16 MPI runs per simulation falls quite below the optimal load per core for FLEXI.\nFor the more realistic cases of up to 8 ranks per FLEXI, most of the FLEXI performance can be recovered.\n\n\n\n\n\n\n\n\n\\subsection{Turbulence Modeling}\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure2.pdf}\n  \\caption{Training results of the 24 DOF configuration with 16, 32 and 64 sampled episodes per training update. Top left: Collected return averaged over all training runs and normalized with the maximum achievable return. The minimum and maximum return in each iteration is indicated by the shaded area. Top right: Normalized return on the unseen testing state, which was evaluated every 10 iterations. Bottom left: Spectra at the end of the simulation $t_{end}$ on the unseen test state after 4,000 training iterations for the configuration running 64 parallel environments. The resulting spectra of Smagorisky's model and the implicit model are shown for comparison. The shaded area around the time-averaged DNS spectrum indicates the maximum and minimum observed amplitudes during the DNS. Bottom left: Distribution of the model's predictions for $C_s$ during the simulation on the test state.}\n  \\label{fig:results_rl_24DOF}\n\\end{figure*}\n\nTo investigate the performance of the RL-based turbulence model against established analytical closure models, we trained the agent on the 24 DOF configuration for 4,000 Iterations, which corresponds to 20,000 gradient ascent steps for the policy overall.\nTo quantify the benefits of gathering more episodes per policy update, the training was carried out using 8 MPI ranks per environment and 16, 32 and 64 parallel FLEXI instances, respectively.\nFor the 16 and 64 runs configurations, the overall training required 20 and 30 hours, respectively.\nSampling the trajectories took 15 and 18 seconds per iteration, respectively, while updating the policy on a single GPU took 0.5 and 2 seconds, respectively.\nThe results in \\figref{fig:results_rl_24DOF} indicate that increasing the number of parallel simulations does indeed improve the training performance in several ways.\nFirstly, the training runs with more episodes converge faster, i.e. need less training iterations to achieve a given episode return. \nMoreover, the collected return of the 16 episode configuration increases less consistently after around 2,000 iterations than the training configurations with more parallel episodes.\nThis confirms our prior assumption that the gradient estimator becomes more reliable if more episodes are sampled, which again leads to more efficient training updates and thus to faster convergence.\nIn the same vein, the training runs using more episodes tend to converge to a higher total return.\nThis notion holds especially for the return collected during training, where the return is computed on a variety of different initial states.\nWhile the return on the unseen initial state for testing gives similar results, the reward of the 16 episodes configuration fluctuates between 2,000 and 3,000 iterations and temporarily exceeds the return of the 32 episodes configuration before converging to a stable return.\nThis behavior probably stems from the high variance of the 16 episodes training which causes the less consistent improvement on the unseen test state.\nThe results in \\figref{fig:results_rl_24DOF} also clearly demonstrate that for our application, the RL-based model outperforms both Smagorinsky's model and the implicit modeling strategy with regard to the energy spectrum.\nEspecially surprising is that the RL-enhanced model agrees with the high-fidelity data almost up to $k_{max}$ with deviations only at $k=6$.\nThis indicates that our novel data-driven model does not only replicate the vital flow physics related to the energy cascade, but also pushes the resolution limits of the underlying discretization.\nThe predictions of the initial untrained model appear to be almost normally distributed, as is expected from the initialization of the policy ANN.\nInterestingly, the trained model modifies this distribution heavily by predicting tiny $C_s$ parameters for the vast majority of the flow field while increasing the parameter only in selective elements. \nThe model also takes advantage of the entire admissable range of $C_s\\in[0,0.5]$.\nThe reported results demonstrate that the application of the RL paradigm to turbulence modeling (and control tasks in CFD in general) can contribute to major advancements of the state-of-the-art given that the necessary resources can be used efficiently on massively parallel systems.\n\n\n\\section{Application to Turbulence Modeling}\\label{sec:numerics}\n\n\\subsection{Turbulence Modeling}\nTurbulent flows are notoriously hard to resolve accurately, since turbulence is a multiscale phenomenon.\nThe resolution requirements for numerical simulations to resolve the wide range of active length scales are usually intractable.\nInstead, reduced order descriptions of turbulence can be derived, which only resolve the large energy-containing flow scales, while employing a turbulence model to account for the influence of the non-resolved fine scales.\nFrom a mathematical standpoint, this is equivalent to applying a low-pass filter to the flow field.\nThis approach is commonly referred to as large eddy simulation (LES).\nA myriad of different LES turbulence models have been proposed in literature.\nHowever, such models typically contain empirical parameters, which have to be tuned to the employed numerical discretization and the specific test case.\nAs a consequence, no universal or overall \\emph{best} model has been found to this date.\n\nThe most important property of the LES model is to mimic the energy drain from the large to the small flow scales.\nBased on this reasoning, a popular modeling strategy is to approximate the unresolved scales by introducing an additional turbulent viscosity $\\nu_t$ that is added to the physical viscosity.\nThe common model by Smagorinsky \\cite{smagorinsky1963general} computes this turbulent viscosity as\n\\begin{equation}\n  \\nu_t= \\left(C_s \\Delta\\right)^2 \\sqrt{2\\:\\overline{S}_{ij}\\overline{S}_{ij}} \\qquad \\text{with} \\quad \\overline{S}_{ij} = \\frac{\\partial \\overline{u}_i}{\\partial x_j}.\n  \\label{eq:smagorinsky}\n\\end{equation}\nHere, $\\overline{S}_{ij}$ is the rate-of-strain tensor of the coarse-scale velocity field $\\overline{u}_i$ with respect to the coordinates $x_j$ with $i,j=1,2,3$.\nThe filter width $\\Delta$ is a measure of the employed spatial resolution and $C_s$ is a model coefficient, which has to be tuned for each specific test case.\nAnother common modeling approach is the implicit modeling paradigm, which assumes that the numerical dissipation error introduced by the numerical scheme acts as an implicit LES model. The implicit LES model can obviously be seen as a special case of Smagorinsky's model with $C_s=0$.\n\nWith our RL framework, we strive to improve the performance of Smagorinsky's model given in \\eqref{eq:smagorinsky} by employing an RL algorithm to tune the model coefficient $C_s$ dynamically in space and time during the simulation.\nIn this regard, we show that RL represents a promising approach to enhance current simulation workloads and that our novel RL framework is capable of handling computationally intensive simulation environments on modern HPC architectures.\n\n\\subsection{Stating the Reinforcement Learning Task}\nAs test case, i.e. as training environment, we perform LES of homogeneous isotropic turbulence (HIT) at a Reynolds number of $Re_{\\lambda}\\approx 200$ with respect to the Taylor microscale.\nThe cubic domain of side length $2\\pi$ is equipped with periodic boundary conditions and is discretized by a Cartesian mesh with the resolutions given in \\tabref{tab:testcases}.\nThis test case of \\emph{turbulence in a box} can be seen as the building block of turbulence and describes freely decaying turbulence in the absence of boundaries.\nIn order to obtain a quasi-static solution, an isotropic linear forcing is employed as proposed by \\cite{lundgren2003linearly,de2015anisotropic} to balance the dissipation of the turbulence model.\nThis results in a quasi-stationary distribution of the turbulent kinetic energy in the system, which is mainly characterized by the energy drain from the large to the small scales.\nSince this energy cascade is a fundamental property of turbulence that the LES should reproduce, we define our optimization target for the RL task in terms of this energy spectrum.\nThe reward for the RL algorithm is thus computed based on the error of the instantaneous energy spectrum of the LES $E_{LES}(k)$ compared to the mean energy distribution of the underlying ground truth solution $E_{DNS}(k)$, which was obtained beforehand from a high-fidelity simulation.\nFor this, we use the mean relative error across the wavenumbers $k$ up to $k_{max}$, which can be computed as\n\\begin{equation}\n  \\ell = \\underset{k}{\\mathrm{mean}}\\left[\\left(\\frac{E_{DNS}(k)-E_{LES}(k)}{E_{DNS}\\left(k\\right)}\\right)^2\\right], \\quad k\\in\\left[1,k_{max}\\right].\n  \\label{eq:reward_error}\n\\end{equation}\nTo ensure that the reward is normalized to $r_t\\in[-1,1]$, the reward is eventually computed as\n\\begin{equation}\n  r_t = 2 e^{\\left(\\ell \/ \\alpha \\right)}-1,\n  \\label{eq:reward}\n\\end{equation}\nwith $\\alpha$ as a scaling parameter.\n\nWith the reward function in place, the optimization task for the RL algorithm is framed as follows.\nThe state of the environment observed by the agent is the current coarse-scale velocity field $\\overline{u}_i$.\nThe agent predicts as action a single $C_s$ coefficient for each element in the computational mesh solely based on the local flow state in the respective element.\nThe environment's state is then evolved with the flow solver for some time $\\Delta t_{RL}$ before requesting new predictions for the elementwise $C_s$.\nThe time inteval $\\Delta t_{RL}$ is generally chosen much larger than the computational timestep of the simulation.\nThe reward for the agent is computed based on the differences in energy distribution compared to the ground truth, as given in \\eqref{eq:reward}.\nThis loop is repeated until some final time $t_{end}$ is reached.\n\n\n\n\\subsection{Computational Setup}\nIn this work, we perform LES at different resolutions, which are listed in \\tabref{tab:testcases}.\nAll simulations are run up to $t_{end}=5$ and actions are performed with in a time interval of $\\Delta t_{RL}=0.1$, which corresponds to 50 predictions per simulation.\nThe initial state for each simulation run is drawn randomly from a set of flow states that are computed by filtering different realizations of the high-fidelity solution.\nA single initial state is kept hidden to evaluate the model performance on unseen test data.\nBased on these setups, two different approaches can be followed to reduce the required computational time for the training by using parallel HPC resources.\nFirst, the number of MPI ranks per environment can be increased to reduce the simulation time of the environments by exploiting the strong scaling capabilities of FLEXI.\nSecond, the number of simulated environments per training update can be increased, which can be seen as a weak scaling approach for the full framework.\nSince an increased number of sampled episodes might result in a better estimate of the gradient for the optimizer, this approach can reduce the number of iterations required for convergence and thus the necessary training time.\nSince both perspectives highlight crucial scaling properties of the underlying framework, both will be investigated in \\secref{sec:scaling}.\nHere, we employ up to 16 cores per FLEXI environment and up to 1024 parallel environments per training iteration using a maximum of 2,048 cores.\n\\begin{table}\n  \\centering\n  \\begin{tabular}{lccccc}\n    \\hline    \n          & N & \\#Elems & \\#DOF  & $k_{max}$ & $\\alpha$ \\\\\n    \\hline    \n    24 DOF & 5 &   $4^3$ & 13,824 &         9 &   0.4  \\\\\n    32 DOF & 7 &   $4^3$ & 32,768 &        12 &   0.2  \\\\\n    \\hline    \n  \\end{tabular}\n  \\caption{Investigated configurations for the LES of the HIT test case. The runs are named by the number of degrees of freedom (DOF) per spatial direction, which result from the given polynomial degree $N$ and the number of elements \\#Elems. The total number of DOF can be computed by $\\text{\\#DOF} = \\text{\\#Elems}\\:(N+1)^3$. The hyperparameters $k_{max}$ and $\\alpha$ refer to the maximum wavenumber and the scaling factor of the reward function, respectively.}%\n  \\label{tab:testcases}\n\\end{table}\n\nAs RL algorithm, we use the clipping variant of the proximal policy optimization (PPO) \\cite{schulman2017proximal} algorithm as already discussed in \\secref{sec:rl}.\nThis algorithm is synchronous, which means that the algorithm performs the steps of sampling experience and updating the policy in a sequential fashion.\nThis means that first, FLEXI simulations are performed based on the current policy and thereafter, the policy is updated based on the collected experience to maximize the reward on future runs.\nFor all experiments we used a discount factor of $\\gamma=0.995$ and a learning rate of $10^{-4}$ with the Adam optimizer \\cite{kingma2014adam} to train the policy for 5 epochs per iteration.\nFor the PPO algorithm, we used a clipping parameter of 0.2 and set the entropy coefficient to zero.\nThe employed policy ANN comprises around 3,300 parameters with its architecture given in \\tabref{tab:ann}.\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{lrrrr}\n    \\hline    \n    Layer   & Kernel & Filters & Padding  & Dimension \\\\\n    \\hline     \n    Input   &        &         &          &  $6\\times 6\\times 6\\times 3$ \\\\\n    Conv3D  &      3 &       8 &     zero &  $6\\times 6\\times 6\\times 8$ \\\\\n    Conv3D  &      3 &       8 &     none &  $4\\times 4\\times 4\\times 8$ \\\\\n    Conv3D  &      3 &       4 &     none &  $2\\times 2\\times 2\\times 4$ \\\\\n    Conv3D  &      2 &       1 &     none &  $1\\times 1\\times 1\\times 1$ \\\\\n    Scale   &        &         &          &  $                        1$ \\\\\n    \\hline    \n  \\end{tabular}\n  \\caption{Architecture of the policy ANN for $N=5$ with the dimensions of each layer's output. The model's input dimensions follow from the $(N+1)^3$ solution points in each element times the three velocity components $\\overline{u}_i$, which are used as input features. The ANN is build from three-dimensional convolutional layers with a specific kernel size and number of filters, each. The first layer uses zero-padding, while the rear layers employ no padding to convolve the high-dimensional input to a single scalar. All layers except the last convolutional layer employ the rectified linear unit (ReLU) as activation function. The final scaling layer uses the operation $y=\\frac{1}{2}\\sigma_s(x)$ to scale the input to the interval $[0,0.5]$ with $\\sigma_s(x)$ denoting the sigmoid activation function.}\n  \\label{tab:ann}\n\\end{table}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGiven a passage (context) and a question about it, a reading comprehension system should be able to read the passage and answer the question. While not a hard task for a human, it requires that the system both understand natural language and have knowledge about the world. \nBecause of the renaissance of neural networks and accessibility of large-scale datasets, great progress has recently been made in reading comprehension. For example, according to the leaderboard of SQuAD 1.0 \\cite{DBLP:conf\/emnlp\/RajpurkarZLL16}, over $80$ systems have been submitted, and human performance has been left behind.\nIn experiments, the reimplementation and comparison of these solutions are necessary but not easy tasks, because researchers usually build their blocks from scratch and in different environments. Meanwhile, the efficient construction of original prototypes is not possible, although reading comprehension models often share similar components and architectures.\n\nIn this paper, we present the Sogou Machine Reading Comprehension toolkit\\footnote{\\url{https:\/\/github.com\/sogou\/SMRCToolkit}}, which has the goal of allowing the rapid and efficient development of modern machine comprehension models, including both published models and original prototypes.\nFirst, the toolkit simplifies the dataset reading process by providing dataset reader modules that support popular datasets. Second, the flexible preprocessing pipeline allows vocabulary building, linguistic feature extraction, and operations to work in a seamless way.\nThird, the toolkit offers frequently used neural network components, a trainer module, and a save\/load function, which accelerates the construction of custom models. Last, but not the least, some published models are implemented in the toolkit, making model comparison and modification convenient. The toolkit is built based on the Tensorflow\\footnote{\\url{https:\/\/github.com\/tensorflow\/tensorflow}} library \\cite{DBLP:conf\/osdi\/AbadiBCCDDDGIIK16}.\n\n\\begin{figure*}[htbp] \n\\centering\n\\includegraphics[width=\\textwidth]{architecture.pdf}\n\\caption{Toolkit Architecture}\n\\label{architecture} \n\\end{figure*}\n\n\\section{Toolkit Framework}\nAs shown in Figure 1, the architecture of our toolkit mainly contains four modules: the\nDataset Reader, Data Preprocessing, Model Construction, and Model Training modules.  These four modules\nare designed as a pipeline flow and can be used for most machine reading comprehension tasks. In the following, we\nwill introduce each part in detail.\n\\subsection{Dataset Reader}\nOne reason that machine reading comprehension has made rapid progress, that cannot be ignored, is the release of a variety of large-scale and high-quality question answering datasets. In addition, preprocessing and evaluating are essential steps when doing research on these datasets.\n\n\\noindent \\textbf{Reader} \\indent To avoid repeating the development of dataset reading codes, the toolkit provides reader modules for some typical datasets SQuAD 1.0 \\cite{DBLP:conf\/emnlp\/RajpurkarZLL16}, SQuAD 2.0 \\cite{DBLP:conf\/acl\/RajpurkarJL18} and CoQA \\cite{DBLP:journals\/corr\/abs-1808-07042}. To enhance the language diversity, we also support a Chinese dataset, CMRC2018 \\cite{DBLP:journals\/corr\/abs-1810-07366}.\nThe reader modules first tokenize texts and generate labels (e.g., start\/end positions), and then transform data instances into nested structure objects, the fields of which are uniformly named. This makes data serialization\/deserialization convenient and helps in error analysis. By inheriting the base reader, users can develop custom readers for other datasets.\n\n\\noindent \\textbf{Evaluator} \\indent Most datasets offer official evaluation scripts. To ease the model validation and early stopping, we integrate these evaluation scripts into the toolkit and simplify the evaluation in the model training process.\n\n\\subsection{Data Preprocessing}\nTo prepare data for the training model, we need to build a vocabulary, extract linguistic features, and map discrete features into indices. The toolkit provides modules for fulfilling these requirements.\n\n\\noindent \\textbf{Vocabulary Builder} \\indent By scanning the training data from the dataset reader, Vocabulary Builder maintains a corresponding vocabulary of words (and characters if needed). Adding any special tokens or setting the whole vocabulary is allowed as well. Another important function of Vocabulary Builder is creating an embedding matrix from pretrained word embeddings. If you feed a pretrained embedding file, for example Glove\\footnote{\\url{https:\/\/nlp.stanford.edu\/projects\/glove\/}} \\cite{DBLP:conf\/emnlp\/PenningtonSM14}, to the Vocabulary Builder, it will produce a word embedding matrix for its inner vocabulary.\n\n\\noindent \\textbf{Feature Extractor} \\indent Linguistic features are used in many machine reading comprehension models such as DrQA \\cite{DBLP:conf\/acl\/ChenFWB17}, FusionNet \\cite{DBLP:journals\/corr\/abs-1711-07341} and have been proven to be effective. The Feature Extractor supports commonly used features, e.g., part-of-speech (POS) tags, called entity recognition (NER) tags, along with normalized term frequency (TF) and word-level exact matching. By simply adding new feature fields, Feature Extractor does not break the serializability and readability of data instance objects. Meanwhile, Feature Extractor also builds vocabularies for discrete features like POS and NER, which will be used in the next steps for index mapping.\n\n\\noindent \\textbf{Batch Generator} \\indent The last step of the preprocessing is to pack all of the features up and modify them to fit the form of the model input. In Batch Generator, we first map words and tags to indices, pad length-variable features, transform all of the features into tensors, and then batch them. To make these steps efficient, we implement Batch Generator based on the Tensorflow Dataset API\\footnote{\\url{https:\/\/www.tensorflow.org\/api_docs\/python\/tf\/data\/Dataset}}, which parallelizes data transformation and provides fundamental functions like dynamic padding and data shuffling, which make it behave consistently with the \"generator\" in Python. Batch Generator is designed to be flexible and compatible with the feature types frequently used in machine reading comprehension tasks.\n\n\\subsection{Model Construction}\nThe core part of the machine reading comprehension task is constructing an effective and efficient model for generating answers from given passages. The toolkit provides two methods: build your own model or use a built-in model. For the first one, we implement frequently used neural network components in the machine reading comprehension task. We follow the idea of functional API and wrap them as MRC specific supplements of Tensorflow layers.\n\n\\noindent \\textbf{Embedding} \\indent Besides a vanilla embedding layer, the toolkit also provides \\textit{PartiallyTrainableEmbedding}, as used in \\cite{DBLP:conf\/acl\/ChenFWB17} \\cite{DBLP:journals\/corr\/abs-1711-07341}, and pretrained contextualized representation layers, including \\textit{CoVeEmbedding}, \\textit{ElmoEmbedding}, and \\textit{BertEmbedding}.\n\n\\noindent \\textbf{Recurrent} \\indent \\textit{BiLSTM} and \\textit{BiGRU} are basic recurrent layers, and their CuDNN version \\textit{CudnnBiLSTM} and \\textit{CudnnBiGRU} are also available.\n\n\\noindent \\textbf{Similarity Function} \\indent Functions are available for calculating the word-level similarities between texts, e.g., \\textit{DotProduct}, \\textit{TriLinear}, and \\textit{MLP}.\n\n\n\\noindent \\textbf{Attention} \\indent Attention layers are usually used together with the Similarity Function, e.g., \\textit{BiAttention}, \\textit{UniAttention}, and \\textit{SelfAttention}.\n\n\\noindent \\textbf{Basic Layer} \\indent Some of the basic layers are used in machine reading comprehension models, e.g., \\textit{VariationalDropout}, and \\textit{Highway}, \\textit{ReduceSequence}.\n\n\n\\noindent \\textbf{Basic Operation} \\indent These are mainly masking operations, e.g., \\textit{masked\\_softmax}, \\textit{mask\\_logits}. By inheriting the base model class and combining the components above, developers should be able to construct most mainstream machine reading comprehension models. To build a custom model, a developer should define the following three member methods,\n\n\\begin{enumerate}\n  \\item \\textit{\\_build\\_graph}: Define the forward process of the model\n  \\item \\textit{compile}: Schedule the optimization of the model like the learning rate decay and gradient clipping\n  \\item \\textit{get\\_best\\_answer}: Transform the model output (probability) to answer text\n\\end{enumerate}\n\n\n\\noindent Training functions (\\textit{train\\_and\\_evaluate}, \\textit{evaluate}, and \\textit{inference}) should also be inherited if needed.\n\nThe toolkit also provides simple interfaces for using the built-in models. We will introduce the details in Section \\ref{built-in}.\n\\subsection{Model Training}\nWhen training a model, we usually care about how the metrics change on the train\/dev set, when to perform early stopping, how many epochs the model needs to converge, and so on. Because most models share a similar training strategy, the toolkit provides a Trainer module, with main functions that include baby-sitting the training, evaluation and inference processing, saving the best weights, cooperating with the exponential moving average, and recording the training summary. Each model also provides interfaces for saving and loading the model weights.\n\\section{Using Built-In Models}\\label{built-in}\n\n\\subsection{Have a Try}\n\nWe will show an example of running the BiDAF model on the SQuAD 1.0 dataset in this section.\n\nFirst, the data file of SQuAD 1.0 is loaded using SquadReader. Meanwhile, we also create an evaluator for validation.\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{reader.pdf}\n\\end{figure}\n\n\nSecond, we build a vocabulary and corresponding word embedding matrix.\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{vocab.pdf}\n\\end{figure}\n\nThird, data instances are fed to Batch Generator for the necessary preprocessing and batching.\n\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{batch.pdf}\n\\end{figure}\n\n\nLast, we use the built-in BiDAF model and compile it with default hyperparameters. \\textit{train\\_and\\_evaluate} will handle the training process and save the best model weights for inference.\n\\begin{figure*}[!htbp] \n\\centering\n\\includegraphics[width=\\textwidth]{model.pdf}\n\\end{figure*}\n\n\nWith our toolkit, users can try different machine reading comprehension models in a neat and fast way.\n\n\\subsection{Model Zoo}\nIn the section, we will briefly introduce the machine reading comprehension models implemented in this toolkit. \n\n\\noindent \\textbf{BiDAF} was introduced by \\cite{DBLP:journals\/corr\/SeoKFH16}. Unlike the attention mechanisms in previous work, the core idea of BiDAF is bidirectional attention, which models both the query-to-context and context-to-query attention.\n\n\\noindent \\textbf{DrQA} was proposed by \\cite{DBLP:conf\/acl\/ChenFWB17} and aims at tackling open-domain question answering. DrQA use word embedding, basic linguistic features, and a simple attention mechanism, and proves that simple models without sophisticated architectural designs can also achieve strong results in machine reading comprehension.\n\n\\noindent \\textbf{FusionNet} Based on an analysis of the attention approaches in previous work, \\cite{DBLP:journals\/corr\/abs-1711-07341} proposed FusionNet, which extends the attention from three perspectives. They proposed the use of the \"history of word\" and fully aware attention, which let the model combine the information flows from different semantic levels. In addition, the idea was also applied to natural language inference.\n\n\\noindent \\textbf{R-Net} The main contribution of R-Net was the self-matching attention mechanism. After the gating matching of the context and question, passage self-matching was introduced to aggregate evidence from the whole passage and refine the passage representation.\n\n\\noindent \\textbf{QANet} The architecture of QANet \\cite{DBLP:journals\/corr\/abs-1804-09541} was adapted from the Transformer \\cite{DBLP:conf\/nips\/VaswaniSPUJGKP17} and only contains the convolution and self-attention. By not using the recurrent Layers, QANet gains a 3\u201313-fold speed increase in the training time and 4\u20139-fold increase for the inference time.\n\n\\noindent \\textbf{IARNN} In our toolkit, two types of Inner Attention-based RNNs (IARNNs) \\cite{DBLP:conf\/acl\/WangL016} are implemented, which are advantageous for sentence representation and efficient in the answer selection task. IARNN-word weights the word representation of the context for the question before inputting into the RNN models. Unlike IARNN-word, which only achieves input word embedding, IARNN-hidden can capture the relationships between multiple words by adding additional context information to the calculation of the attention weights.\n\n\\noindent \\textbf{BiDAF++} \\cite{DBLP:conf\/acl\/GardnerC18} originally introduced a model for multi-paragraph machine reading comprehension. Based on BiDAF, BiDAF++ adds a self attention layer to increase the model capacity. We also apply the model to CoQA \\cite{DBLP:journals\/corr\/abs-1809-10735} for conversational question answering.\n\n\\noindent \\textbf{BERT} Pretrained models like BERT \\cite{DBLP:journals\/corr\/abs-1810-04805} and ELMo\\cite{DBLP:conf\/naacl\/PetersNIGCLZ18} have shown great efficacy in many natural language processing tasks. In our toolkit, we use BERT, ELMo, and Cove\\cite{DBLP:conf\/nips\/McCannBXS17} as embedding layers to provide a strong contextualized representation. Meanwhile, we also include the BERT model for machine reading comprehension, as well as our modified version. The results of the models in our toolkit are listed in Section \\ref{experiments}.\n\\section{Experiments}\\label{experiments}\n\nWe conducted experiments on a supported dataset with the models in the toolkit. By following the experimental settings in the original papers, we attempted to reproduce the results of the models on a different dataset. It is worth mentioning that slight modifications were applied when necessary, and the scripts and hyperparameters for producing the results shown below are included in the toolkit.\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score on SQuAD 1.0 dev set}\\label{tab:squad1}\n\\begin{tabular}{ | l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF & 77.3\/67.7  & 77.3\/67.7 \\\\ \\hline \n\tBiDAF+ELMo & 81.0\/72.1 & - \\\\ \\hline\n\tIARNN-Word & 73.9\/65.2 & - \\\\ \\hline\n\tIARNN-hidden &  72.2\/64.3& - \\\\ \\hline \n\tDrQA & 78.9\/69.4 & 78.8\/69.5  \\\\ \\hline \n  DrQA+ELMO&83.1\/74.4 & - \\\\ \\hline\n\tR-Net & 80.0\/71.6 & 79.5\/71.1  \\\\ \\hline \n\tBiDAF++ & 78.6\/69.2 & -  \\\\ \\hline \n\tFusionNet & 81.0\/72.0 & 82.5\/74.1  \\\\ \\hline \n\tQANet & 80.8\/71.8 & 82.7\/73.6  \\\\ \\hline \n\tBERT-Base & 88.3\/80.6 & 88.5\/80.8 \\\\ \\hline\n\n\\end{tabular}\n\\end{table}\n\nIn Table \\ref{tab:squad1}, we report the results of the implemented models on the development set of SQuAD 1.0. A sophisticated and effective attention mechanism is necessary for building a high-performance model, according to the table. In addition,  pretrained models like ELMo and BERT give reading comprehension a big boost and have become a new trend in natural language processing. Our toolkit also wraps commonly used attention and pretrained models in a high-level layer and allows flexible combinations.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score on SQuAD 2.0 dev set}\\label{tab:squad2}\n\\begin{tabular}{ | l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF & 62.7\/59.7 & 62.6\/59.8 \\\\ \\hline \n\tBiDAF++ & 64.3\/61.8 & 64.8\/61.9  \\\\ \\hline \n\tBiDAF++ + ELMo  & 67.6\/64.8& 67.6\/65.1 \\\\ \\hline\n\tBERT-Base & 75.9\/73.0 & 75.1\/72.0 \\\\ \\hline\n\\end{tabular}\n\n\\end{table}\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1 score on CoQA dev set}\\label{tab:coqa}\n\\begin{tabular}{| l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF++ & 71.7 & 69.2 \\\\ \\hline \n\tBiDAF++ + ELMo & 74.5 & 69.2 \\\\ \\hline \n\tBERT-Base & 78.6 & - \\\\ \\hline\n\tBERT-Base+Answer Verification & 79.5 & - \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nBecause SQuAD 2.0 and CoQA are different from SQuAD 1.0 in a variety of respects, the models are not directly transferrable between these datasets. Following \\cite{DBLP:conf\/conll\/LevySCZ17} and \\cite{DBLP:journals\/corr\/abs-1809-10735}, we implement several effective models.\nMoreover, our implemented BiDAF achieves Exact Match $35.0$ and F1 $57.01$ on the CMRC dataset, providing a strong baseline.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score of different embedding}\\label{tab:embedding}\n\\begin{tabular}{ | c | c | c |}\n\t\\hline\n\tEmbedding & BiDAF & DrQA \\\\ \\hline\n\trandom  & 70.3\/59.3 & 72.4\/62.5 \\\\ \\hline\n\tword2vec & 77.1\/67.7 & 78.2\/68.8 \\\\ \\hline\n\tglove   & 77.3\/67.7 & 78.9\/69.4 \\\\ \\hline\n\tfast-text-wiki & 77.1\/67.7 & 75.4\/66.0 \\\\ \\hline\n\tfast-text-crawl & 76.9\/67.4 & 77.0\/67.2 \\\\ \\hline\n\tELMo    & 79.9\/71.1 & 82.7\/74.3 \\\\ \\hline \n\\end{tabular}\n\\end{table}\n\nTo investigate the effect of the word representation, we selected two popular models and tested their performances with different embeddings. Table \\ref{tab:embedding} suggests that DrQA is more sensitive to the word embedding and ELMo helps improve the score consistently (here, when ELMo was used, no word embedding was concatenated).\n\n\\section{Conclusion and Future Work}\nIn the paper, we present the Sogou Machine Reading Comprehension toolkit, which has the goal of allowing the rapid and efficient development of modern machine comprehension models, including both published models and original prototypes.\n\nIn the future, we plan to extend the toolkit,  and make it applicable to more tasks, e.g. multi-paragraph and multi-document question answering, and provide more available models.\n\\bibliographystyle{acl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by D Alexander, Juliet Sutherland and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n Stories by American Authors\n\n VOLUME VIII\n\n _THE BRIGADE COMMANDER_\n BY J. W. DE FOREST\n\n _SPLIT ZEPHYR_\n BY HENRY A. BEERS\n\n _ZERVIAH HOPE_\n BY ELIZABETH STUART PHELPS\n\n _THE LIFE-MAGNET_\n BY ALVEY A. ADEE\n\n _OSGOOD'S PREDICAMENT_\n BY ELIZABETH D. B. STODDARD\n\n NEW YORK\n CHARLES SCRIBNER'S SONS\n 1896\n\n\n\n\nCOPYRIGHT, 1884, BY\n\nCHARLES SCRIBNER'S SONS\n\n\n\n\n_The Stories in this Volume are protected by copyright, and are\nprinted here by authority of the authors or their representatives._\n\n\n\n\n[Illustration: Elizabeth Stuart Phelps]\n\n\n\n\nTHE BRIGADE COMMANDER.\n\nBY J. W. DE FOREST.\n\n_New York Times._\n\n\nThe Colonel was the idol of his bragging old regiment and of the\nbragging brigade which for the last six months he had commanded.\n\nHe was the idol, not because he was good and gracious, not because he\nspared his soldiers or treated them as fellow-citizens, but because he\nhad led them to victory and made them famous. If a man will win\nbattles and give his brigade a right to brag loudly of its doings, he\nmay have its admiration and even its enthusiastic devotion, though he\nbe as pitiless and as wicked as Lucifer.\n\n\"It's nothin' to me what the Currnell is in prrivit, so long as he\nshows us how to whack the rrebs,\" said Major Gahogan, commandant of\nthe \"Old Tenth.\" \"Moses saw God in the burrnin' bussh, an' bowed down\nto it, an' worrshipt it. It wasn't the bussh he worrshipt; it was his\nGod that was in it. An' I worrship this villin of a Currnell (if he\nis a villin) because he's almighty and gives us the vict'ry. He's\nnothin' but a human burrnin' bussh, perhaps, but he's got the god of\nwar in um. Adjetant Wallis, it's a ---- long time between dhrinks, as\nI think ye was sayin', an' with rayson. See if ye can't confiscate a\ncanteen of whiskee somewhere in the camp. Bedad, if I can't buy it\nI'll stale it. We're goin' to fight to-morry, an' it may be it's the\nlast chance we'll have for a dhrink, unless there's more lik'r now in\nthe other worrld than Dives got.\"\n\nThe brigade was bivouacked in some invisible region, amid the damp,\nmisty darkness of a September night. The men lay in their ranks, each\nwith his feet to the front and his head rearward, each covered by his\novercoat and pillowed upon his haversack, each with his loaded rifle\nnestled close beside him. Asleep as they were, or dropping placidly\ninto slumber, they were ready to start in order to their feet and pour\nout the red light and harsh roar of combat. There were two lines of\nbattle, each of three regiments of infantry, the first some two\nhundred yards in advance of the second. In the space between them lay\ntwo four-gun batteries, one of them brass twelve-pounder \"Napoleons,\"\nand the other rifled Parrotts. To the rear of the infantry were the\nrecumbent troopers and picketed horses of a regiment of cavalry. All\naround, in the far, black distance, invisible and inaudible, paced or\nwatched stealthily the sentinels of the grand guards.\n\nThere was not a fire, nor a torch, nor a star-beam in the whole\nbivouac to guide the feet of Adjutant Wallis in his pilgrimage after\nwhisky. The orders from brigade headquarters had been strict against\nilluminations, for the Confederates were near at hand in force, and a\nsurprise was purposed as well as feared. A tired and sleepy youngster,\nalmost dropping with the heavy somnolence of wearied adolescence, he\nstumbled on through the trials of an undiscernible and unfamiliar\nfooting, lifting his heavy riding-boots sluggishly over imaginary\nobstacles, and fearing the while lest his toil were labor misspent. It\nwas a dry camp, he felt dolefully certain, or there would have been\nmore noise in it. He fell over a sleeping Sergeant, and said to him\nhastily, \"Steady, man--a friend!\" as the half-roused soldier clutched\nhis rifle. Then he found a Lieutenant, and shook him in vain; further\non a Captain, and exchanged saddening murmurs with him; further still\na camp-follower of African extraction, and blasphemed him.\n\n\"It's a God-forsaken camp, and there isn't a horn in it,\" said\nAdjutant Wallis to himself as he pursued his groping journey. \"Bet you\nI don't find the first drop,\" he continued, for he was a betting boy,\nand frequently argued by wagers, even with himself. \"Bet you two to\none I don't. Bet you three to one--ten to one.\"\n\nThen he saw, an indefinite distance beyond him, burning like red-hot\niron through the darkness, a little scarlet or crimson gleam, as of a\nlighted cigar.\n\n\"That's Old Grumps, of the Bloody Fourteenth,\" he thought. \"I've\nraided into his happy sleeping-grounds. I'll draw on him.\"\n\nBut Old Grumps, otherwise Colonel Lafayette Gildersleeve, had no\nrations--that is, no whisky.\n\n\"How do you suppose an officer is to have a drink, Lieutenant?\" he\ngrumbled.\n\n\"Don't you know that our would-be Brigadier sent all the commissary to\nthe rear day before yesterday? A canteenful can't last two days. Mine\nwent empty about five minutes ago.\"\n\n\"Oh, thunder!\" groaned Wallis, saddened by that saddest of all\nthoughts, \"Too late!\" \"Well, least said soonest mended. I must wobble\nback to my Major.\"\n\n\"He'll send you off to some other camp as dry as this one. Wait ten\nminutes, and he'll be asleep. Lie down on my blanket and light your\npipe. I want to talk to you about official business--about our\nwould-be Brigadier.\"\n\n\"Oh, _your_ turn will come some day,\" mumbled Wallis, remembering\nGildersleeve's jealousy of the brigade commander--a jealousy which\nonly gave tongue when aroused by \"commissary.\" \"If you do as well as\nusual to-morrow you can have your own brigade.\"\n\n\"I suppose you think we are all going to do well to-morrow,\" scoffed\nold Grumps, whose utterance by this time stumbled. \"I suppose you\nexpect to whip and to have a good time. I suppose you brag on fighting\nand enjoy it.\"\n\n\"I like it well enough when it goes right; and it generally does go\nright with this brigade. I should like it better if the rebs would\nfire higher and break quicker.\"\n\n\"That depends on the way those are commanded whose business it is to\nbreak them,\" growled Old Grumps. \"I don't say but what we are rightly\ncommanded,\" he added, remembering his duty to superiors. \"I concede\nand acknowledge that our would-be Brigadier knows his military\nbusiness. But the blessing of God, Wallis! I believe in Waldron as a\nsoldier. But as a man and a Christian, faugh!\"\n\nGildersleeve had clearly emptied his canteen unassisted; he never\ntalked about Christianity when perfectly sober.\n\n\"What was your last remark?\" inquired Wallis, taking his pipe from his\nmouth to grin. Even a superior officer might be chaffed a little in\nthe darkness.\n\n\"I made no last remark,\" asserted the Colonel with dignity. \"I'm not\na-dying yet. If I said anything last it was a mere exclamation of\ndisgust--the disgust of an officer and gentleman. I suppose you know\nsomething about our would-be Brigadier. I suppose you think you know\nsomething about him.\"\n\n\"Bet you I know _all_ about him,\" affirmed Wallis. \"He enlisted in the\nold Tenth as a common soldier. Before he had been a week in camp they\nfound that he knew his biz, and they made him a Sergeant. Before we\nstarted for the field the Governor got his eye on him and shoved him\ninto a Lieutenancy. The first battle h'isted him to a Captain. And the\nsecond--bang! whiz! he shot up to Colonel, right over the heads of\neverybody, line and field. Nobody in the old Tenth grumbled. They saw\nthat he knew his biz. I know _all_ about him. What'll you bet?\"\n\n\"I'm not a betting man, Lieutenant, except in a friendly game of\npoker,\" sighed Old Grumps. \"You don't know anything about your\nBrigadier,\" he added in a sepulchral murmur, the echo of an empty\ncanteen. \"I have only been in this brigade a month, and I know more\nthan you do, far, very far more, sorry to say it. He's a reformed\nclergyman. He's an apostatized minister.\" The Colonel's voice as he\nsaid this was solemn and sad enough to do credit to an undertaker.\n\"It's a bad sort, Wallis,\" he continued, after another deep sigh, a\nvery highly perfumed one, the sigh of a bar-keeper. \"When a clergyman\nfalls, he falls for life and eternity, like a woman or an angel. I\nnever knew a backslidden shepherd to come to good. Sooner or later he\nalways goes to the devil, and takes down whomsoever hangs to him.\"\n\n\"He'll take down the old Tenth, then,\" asserted Wallis. \"It hangs to\nhim. Bet you two to one he takes it along.\"\n\n\"You're right, Adjutant; spoken like a soldier,\" swore Gildersleeve.\n\"And the Bloody Fourteenth, too! It will march into the burning pit\nas far as any regiment; and the whole brigade, yes sir! But a\nbackslidden shepherd, my God! Have we come to that? I often say\nto myself, in the solemn hours of the night, as I remember my\nSabbath-school days, 'Great Scott, have we come to that?' A reformed\nclergyman! An apostatized minister! Think of it, Wallis, think of it!\nWhy, sir, his very wife ran away from him. They had but just buried\ntheir first boy,\" pursued Old Grumps, his hoarse voice sinking to a\nwhimper. \"They drove home from the burial-place, where lay the\nnew-made grave. Arrived at their door, _he_ got out and extended his\nhand to help _her_ out. Instead of accepting, instead of throwing\nherself into his arms and weeping there, she turned to the coachman\nand said, 'Driver, drive me to my father's house.' That was the end of\ntheir wedded life, Wallis.\"\n\nThe Colonel actually wept at this point, and the maudlin tears were\nnot altogether insincere. His own wife and children he heartily loved,\nand remembered them now with honest tenderness. At home he was not a\ndrinker and a rough; only amid the hardships and perils of the field.\n\n\"That was the end of it, Wallis,\" he repeated. \"And what was it while\nit lasted? What does a woman leave her husband for? Why does she\nseparate from him over the grave of her innocent first-born? There are\ntwenty reasons, but they must all of them be good ones. I am sorry to\ngive it as my decided opinion, Wallis, in perfect confidence, that\nthey must all be whopping good ones. Well, that was the beginning;\nonly the beginning. After that he held on for a while, breaking the\nbread of life to a skedaddling flock, and then he bolted. The next\nknown of him, three years later, he enlisted in your regiment, a smart\nbut seedy recruit, smelling strongly of whisky.\"\n\n\"I wish I smelt half as strong of it myself,\" grumbled Wallis. \"It\nmight keep out the swamp fever.\"\n\n\"That's the true story of Col. John James Waldron,\" continued Old\nGrumps, with a groan which was very somnolent, as if it were a twin to\na snore. \"That's the true story.\"\n\n\"I don't believe the first word of it--that is to say, Colonel, I\nthink you have been misinformed--and I'll bet you two to one on it.\nIf he was nothing more than a minister, how did he know drill and\ntactics?\"\n\n\"Oh, I forgot to say, he went through West Point--that is, nearly\nthrough. They graduated him in his third year by the back door,\nWallis.\"\n\n\"Oh, that was it, was it? He was a West Pointer, was he? Well, then,\nthe backsliding was natural, and oughtn't to count against him. A\nmember of Benny Havens' church has a right to backslide anywhere,\nespecially as the Colonel doesn't seem to be any worse than some of\nthe rest of us, who haven't fallen from grace the least particle, but\ntook our stand at the start just where we are now. A fellow that\nbegins with a handful of trumps has a right to play a risky game.\"\n\n\"I know what euchered him, Wallis. It was the old Little Joker; and\nthere's another of the same on hand now.\"\n\n\"On hand where? What are you driving at, Colonel?\"\n\n\"He looks like a boy. I mean she looks like a boy. You know what I\nmean, Wallis; I mean the boy that makes believe wait on him. And her\nbrother is in camp, got here to-night. There'll be an explanation\nto-morrow, and there'll be bloodshed.\"\n\n\"Good-night, Colonel, and sleep it off,\" said Wallis, rising from the\nside of a man whom he believed to be sillily drunk and altogether\nuntrustworthy. \"You know we get after the rebs at dawn.\"\n\n\"I know it--goo-night, Adjutant--gawblessyou,\" mumbled Old Grumps.\n\"We'll lick those rebs, won't we?\" he chuckled. \"Goo-night, ole\nfellow, an' gawblessyou.\"\n\nWhereupon Old Grumps fell asleep, very absurdly overcome by liquor, we\nextremely regret to concede, but nobly sure to do his soldierly duty\nas soon as he should awake.\n\nStumbling wearily blanketward, Wallis found his Major and regimental\ncommander, the genial and gallant Gahogan, slumbering in a peace like\nthat of the just. He stretched himself a-near, put out his hand to\ntouch his sabre and revolver, drew his caped great-coat over him,\nmoved once to free his back of a root or pebble, glanced languidly\nat a single struggling star, thought for an instant of his far-away\nmother, turned his head with a sigh, and slept. In the morning he was\nto fight, and perhaps to die; but the boyish veteran was too seasoned,\nand also too tired, to mind that; he could mind but one\nthing--nature's pleading for rest.\n\nIn the iron-gray dawn, while the troops were falling dimly and\nspectrally into line, and he was mounting his horse to be ready for\norders, he remembered Gildersleeve's drunken tale concerning the\ncommandant, and laughed aloud. But turning his face toward brigade\nheadquarters (a sylvan region marked out by the branches of a great\noak), he was surprised to see a strange officer, a fair young man in\nCaptain's uniform, riding slowly toward it.\n\n\"Is that the Boy's brother?\" he said to himself; and in the next\ninstant he had forgotten the whole subject; it was time to form and\npresent the regiment.\n\nQuietly and without tap of drum the small, battle-worn battalions\nfiled out of their bivouacs into the highway, ordered arms and waited\nfor the word to march. With a dull rumble the field-pieces trundled\nslowly after, and halted in rear of the infantry. The cavalry trotted\noff circuitously through the fields, emerged upon the road in advance\nand likewise halted, all but a single company, which pushed on for\nhalf a mile, spreading out as it went into a thin line of skirmishers.\n\nMeanwhile a strange interview took place near the great oak which had\nsheltered brigade headquarters. As the unknown officer, whom Wallis\nhad noted, approached it, Col. Waldron was standing by his horse ready\nto mount. The commandant was a man of medium size, fairly handsome in\nperson and features, and apparently about twenty-eight years of age.\nPerhaps it was the singular breadth of his forehead which made the\nlower part of his face look so unusually slight and feminine. His\neyes were dark hazel, as clear, brilliant, and tender as a girl's,\nand brimming full of a pensiveness which seemed both loving and\nmelancholy. Few persons, at all events few women, who looked upon him\never looked beyond his eyes. They were very fascinating, and in a\nman's countenance very strange. They were the kind of eyes which\nreveal passionate romances, and which make them.\n\nBy his side stood a boy, a singularly interesting and beautiful boy,\nfair-haired and blue-eyed, and delicate in color. When this boy saw\nthe stranger approach he turned as pale as marble, slid away from the\nbrigade commander's side, and disappeared behind a group of staff\nofficers and orderlies. The new-comer also became deathly white as he\nglanced after the retreating youth. Then he dismounted, touched his\ncap slightly and, as if mechanically, advanced a few steps, and said\nhoarsely, \"I believe this is Colonel Waldron. I am Captain Fitz Hugh,\nof the --th Delaware.\"\n\nWaldron put his hand to his revolver, withdrew it instantaneously, and\nstood motionless.\n\n\"I am on leave of absence from my regiment, Colonel,\" continued Fitz\nHugh, speaking now with an elaborate ceremoniousness of utterance\nsignificant of a struggle to suppress violent emotion. \"I suppose you\ncan understand why I made use of it in seeking you.\"\n\nWaldron hesitated; he stood gazing at the earth with the air of one\nwho represses deep pain; at last, after a profound sigh, he raised his\neyes and answered.\n\n\"Captain, we are on the eve of a battle. I must attend to my public\nduties first. After the battle we will settle our private affair.\"\n\n\"There is but one way to settle it, Colonel.\"\n\n\"You shall have your way if you will. You shall do what you will. I\nonly ask what good will it do to _her_?\"\n\n\"It will do good to _me_, Colonel,\" whispered Fitz Hugh, suddenly\nturning crimson. \"You forget _me_.\"\n\nWaldron's face also flushed, and an angry sparkle shot from under his\nlashes in reply to this utterance of hate, but it died out in an\ninstant.\n\n\"I have done a wrong, and I will accept the consequences,\" he said.\n\"I pledge you my word that I will be at your disposal if I survive\nthe battle. Where do you propose to remain meanwhile?\"\n\n\"I will take the same chance, Sir. I propose to do my share in the\nfighting if you will use me.\"\n\n\"I am short of staff officers. Will you act as my aid?\"\n\n\"I will, Colonel,\" bowed Fitz Hugh, with a glance which expressed\nsurprise, and perhaps admiration, at this confidence.\n\nWaldron turned, beckoned his staff officers to approach, and said,\n\"Gentlemen, this is Captain Fitz Hugh of the --th Delaware. He has\nvolunteered to join us for the day, and will act as my aid. And now,\nCaptain, will you ride to the head of the column and order it forward?\nThere will be no drum-beat and no noise. When you have given your\norder and seen it executed, you will wait for me.\"\n\nFitz Hugh saluted, sprang into his saddle and galloped away. A few\nminutes later the whole column was plodding on silently toward its\nbloody goal. To a civilian, unaccustomed to scenes of war, the\ntranquillity of these men would have seemed very wonderful. Many of\nthe soldiers were still munching the hard bread and raw pork of their\nmeagre breakfasts, or drinking the cold coffee with which they had\nfilled their canteens the day previous. Many more were chatting in an\nundertone, grumbling over their sore feet and other discomfits,\nchaffing each other, and laughing. The general bearing, however, was\ngrave, patient, quietly enduring, and one might almost say stolid. You\nwould have said, to judge by their expressions, that these sunburnt\nfellows were merely doing hard work, and thoroughly commonplace work,\nwithout a prospect of adventure, and much less of danger. The\nexplanation of this calmness, so brutal perhaps to the eye of a\nsensitive soul, lies mainly in the fact that they were all veterans,\nthe survivors of marches, privations, maladies, sieges, and battles.\nNot a regiment present numbered four hundred men, and the average was\nnot above three hundred. The whole force, including artillery and\ncavalry, might have been about twenty-five hundred sabres and\nbayonets.\n\nAt the beginning of the march Waldron fell into the rear of his staff\nand mounted orderlies. Then the Boy who had fled from Fitz Hugh\ndropped out of the tramping escort, and rode up to his side.\n\n\"Well, Charlie,\" said Waldron, casting a pitying glance at the yet\npallid face and anxious eyes of the youth, \"you have had a sad fright.\nI make you very miserable.\"\n\n\"He has found us at last,\" murmured Charlie in a tremulous soprano\nvoice. \"What did he say?\"\n\n\"We are to talk to-morrow. He acts as my aide-de-camp to-day. I ought\nto tell you frankly that he is not friendly.\"\n\n\"Of course, I knew it,\" sighed Charlie, while the tears fell.\n\n\"It is only one more trouble--one more danger, and perhaps it may\npass. So many _have_ passed.\"\n\n\"Did you tell him anything to quiet him? Did you tell him that we were\nmarried?\"\n\n\"But we are not married yet, Charlie. We shall be, I hope.\"\n\n\"But you ought to have told him that we were. It might stop him from\ndoing something--mad. Why didn't you tell him so? Why didn't you think\nof it?\"\n\n\"My dear little child, we are about to have a battle. I should like to\ncarry some honor and truth into it.\"\n\n\"Where is he?\" continued Charlie, unconvinced and unappeased. \"I want\nto see him. Is he at the head of the column? I want to speak to him,\njust one word. He won't hurt me.\"\n\nShe suddenly spurred her horse, wheeled into the fields, and dashed\nonward. Fitz Hugh was lounging in his saddle, and sombrely surveying\nthe passing column, when she galloped up to him.\n\n\"Carrol!\" she said, in a choked voice, reining in by his side, and\nleaning forward to touch his sleeve.\n\nHe threw one glance at her--a glance of aversion, if not of downright\nhatred, and turned his back in silence.\n\n\"He is my husband, Carrol,\" she went on rapidly. \"I knew you didn't\nunderstand it. I ought to have written you about it. I thought I would\ncome and tell you before you did anything absurd. We were married as\nsoon as he heard that his wife was dead.\"\n\n\"What is the use of this?\" he muttered hoarsely. \"She is not dead. I\nheard from her a week ago. She was living a week ago.\"\n\n\"Oh, Carrol!\" stammered Charlie. \"It was some mistake then. Is it\npossible! And he was so sure! But he can get a divorce, you know. She\nabandoned him. Or _she_ can get one. No, _he_ can get it--of course,\nwhen she abandoned him. But, Carrol, she _must_ be dead--he was _so_\nsure.\"\n\n\"She is _not_ dead, I tell you. And there can be no divorce. Insanity\nbars all claim to a divorce. She is in an asylum. She had to leave\nhim, and then she went mad.\"\n\n\"Oh, no, Carrol, it is all a mistake; it is not so, Carrol,\" she\nmurmured in a voice so faint that he could not help glancing at her,\nhalf in fury and half in pity. She was slowly falling from her horse.\nHe sprang from his saddle, caught her in his arms, and laid her on the\nturf, wishing the while that it covered her grave. Just then one of\nWaldron's orderlies rode up and exclaimed: \"What is the matter with\nthe--the Boy? Hullo, Charlie.\"\n\nFitz Hugh stared at the man in silence, tempted to tear him from\nhis horse. \"The boy is ill,\" he answered when he recovered his\nself-command. \"Take charge of him yourself.\" He remounted, rode onward\nout of sight beyond a thicket, and there waited for the brigade\ncommander, now and then fingering his revolver. As Charlie was being\nplaced in an ambulance by the orderly and a sergeant's wife, Waldron\ncame up, reined in his horse violently, and asked in a furious voice,\n\"Is that boy hurt?\"\n\n\"Ah--fainted,\" he added immediately. \"Thank you, Mrs. Gunner. Take\ngood care of him--the best of care, my dear woman, and don't let him\nleave you all day.\"\n\nFurther on, when Fitz Hugh silently fell into his escort, he merely\nglanced at him in a furtive way, and then cantered on rapidly to the\nhead of the cavalry. There he beckoned to the tall, grave, iron-gray\nChaplain of the Tenth, and rode with him for nearly an hour, apart,\nengaged in low and seemingly impassioned discourse. From this\ninterview Mr. Colquhoun returned to the escort with a strangely\nsolemnized, tender countenance, while the commandant, with a more\ncheerful air than he had yet worn that day, gave himself to his\nmartial duties, inspecting the landscape incessantly with his glass,\nand sending frequently for news to the advance scouts. It may properly\nbe stated here that the Chaplain never divulged to any one the nature\nof the conversation which he had held with his Colonel.\n\nNothing further of note occurred until the little army, after two\nhours of plodding march, wound through a sinuous, wooded ravine,\nentered a broad, bare, slightly undulating valley, and for the second\ntime halted. Waldron galloped to the summit of a knoll, pointed to a\nlong eminence which faced him some two miles distant, and said\ntranquilly, \"There is our battle-ground.\"\n\n\"Is that the enemy's position?\" returned Captain Ives, his\nAdjutant-General. \"We shall have a tough job if we go at it from\nhere.\"\n\nWaldron remained in deep thought for some minutes, meanwhile scanning\nthe ridge and all its surroundings.\n\n\"What I want to know,\" he observed, at last, \"is whether they have\noccupied the wooded knolls in front of their right and around their\nright flank.\"\n\nShortly afterward the commander of the scouting squadron came riding\nback at a furious pace.\n\n\"They are on the hill, Colonel,\" he shouted.\n\n\"Yes, of course,\" nodded Waldron; \"but have they occupied the woods\nwhich veil their right front and flank?\"\n\n\"Not a bit of it; my fellows have cantered all through, and up to the\nbase of the hill.\"\n\n\"Ah!\" exclaimed the brigade commander, with a rush of elation. \"Then\nit will be easy work. Go back, Captain, and scatter your men through\nthe wood, and hold it, if possible. Adjutant, call up the regimental\ncommanders at once. I want them to understand my plan fully.\"\n\nIn a few minutes Gahogan, of the Tenth; Gildersleeve, of the\nFourteenth; Peck, of the First; Thomas, of the Seventh; Taylor, of\nthe Eighth, and Colburn, of the Fifth, were gathered around their\ncommander. There, too, was Bradley, the boyish, red-cheeked chief of\nthe artillery; and Stilton, the rough, old, bearded regular, who\nheaded the cavalry. The staff was at hand, also, including Fitz Hugh,\nwho sat his horse a little apart, downcast and sombre and silent, but\nnevertheless keenly interested. It is worthy of remark, by the way,\nthat Waldron took no special note of him, and did not seem conscious\nof any disturbing presence. Evil as the man may have been, he was a\nthoroughly good soldier, and just now he thought but of his duties.\n\n\"Gentlemen,\" he said, \"I want you to see your field of battle. The\nenemy occupy that long ridge. How shall we reach it?\"\n\n\"I think, if we go at it straight from here, we shan't miss it,\"\npromptly judged Old Grumps, his red-oak countenance admirably cheerful\nand hopeful, and his jealousy all dissolved in the interest of\napproaching combat.\n\n\"Nor they won't miss us nuther,\" laughed Major Gahogan. \"Betther slide\nour infantree into thim wuds, push up our skirmishers, play away wid\nour guns for an hour, an' thin rowl in a couple o' col'ms.\"\n\nThere was a general murmur of approval. The limits of volunteer\ninvention in tactics had been reached by Gahogan. The other regimental\ncommanders looked upon him as their superior in the art of war.\n\n\"That would be well, Major, if we could do nothing better,\" said\nWaldron. \"But I do not feel obliged to attack the front seriously\nat all. The rebels have been thoughtless enough to leave that long\nsemicircle of wooded knolls unoccupied, even by scouts. It stretches\nfrom the front of their centre clear around their right flank. I shall\nuse it as a veil to cover us while we get into position. I shall throw\nout a regiment, a battery, and five companies of cavalry, to make a\nfeint against their centre and left. With the remainder of the brigade\nI shall skirt the woods, double around the right of the position, and\nclose in upon it front and rear.\"\n\n\"Loike scissors blades upon a snip o' paper,\" shouted Gahogan, in\ndelight. Then he turned to Fitz Hugh, who happened to be nearest him,\nand added, \"I tell ye he's got the God o' War in um. He's the burrnin'\nbussh of humanity, wid a God o' Battles inside on't.\"\n\n\"But how if they come down on our thin right wing?\" asked a cautious\nofficer, Taylor, of the Eighth. \"They might smash it and seize our\nline of retreat.\"\n\n\"Men who have taken up a strong position, a position obviously chosen\nfor defense, rarely quit it promptly for an attack,\" replied Waldron.\n\"There is not one chance in ten that these gentlemen will make a\nconsiderable forward movement early in the fight. Only the greatest\ngeniuses jump from the defensive to the offensive. Besides, we must\nhold the wood. So long as we hold the wood in front of their centre we\nsave the road.\"\n\nThen came personal and detailed instructions. Each regimental\ncommander was told whither he should march, the point where he should\nhalt to form line, and the direction by which he should attack. The\nmass of the command was to advance in marching column toward a knoll\nwhere the highway entered and traversed the wood. Some time before\nreaching it Taylor was to deploy the Eighth to the right, throw out a\nstrong skirmish line and open fire on the enemy's centre and left,\nsupported by the battery of Parrotts, and, if pushed, by five\ncompanies of cavalry. The remaining troops would reach the knoll, file\nto the left under cover of the forest, skirt it for a mile as rapidly\nas possible, enfold the right of the Confederate position, and then\nmove upon it concentrically. Counting from the left, the Tenth, the\nSeventh, and the Fourteenth were to constitute the first line of\nbattle, while five companies of cavalry, then the First, and then the\nFifth formed the second line. Not until Gahogan might have time to\nwind into the enemy's right rear should Gildersleeve move out of the\nwood and commence the real attack.\n\n\"You will go straight at the front of their right,\" said Waldron,\nwith a gay smile, to this latter Colonel. \"Send up two companies as\nskirmishers. The moment they are clearly checked, lead up the other\neight in line. It will be rough work. But keep pushing. You won't have\nfifteen minutes of it before Thomas, on your left, will be climbing\nthe end of the ridge to take the rebels in flank. In fifteen minutes\nmore Gahogan will be running in on their backs. Of course they will\ntry to change front and meet us. But they have extended their line a\nlong way in order to cover the whole ridge. They will not be quick\nenough. We shall get hold of their right, and we shall roll them up.\nThen, Colonel Stilton, I shall expect to see the troopers jumping into\nthe gaps and making prisoners.\"\n\n\"All right, Colonel,\" answered Stilton in that hoarse growl which is\napt to mark the old cavalry officer. \"Where shall we find you if we\nwant a fresh order?\"\n\n\"I shall be with Colburn, in rear of Gildersleeve. That is our centre.\nBut never mind me; you know what the battle is to be, and you know how\nto fight it. The whole point with the infantry is to fold around the\nenemy's right, go in upon it concentrically, smash it, and roll up\ntheir line. The cavalry will watch against the infantry being flanked,\nand when the latter have seized the hill, will charge for prisoners.\nThe artillery will reply to the enemy's guns with shell, and fire\ngrape at any offensive demonstration. You all know your duties, now,\ngentlemen. Go to your commands, and march!\"\n\nThe Colonels saluted and started off at a gallop. In a few minutes\ntwenty-five hundred men were in simultaneous movement. Five companies\nof cavalry wheeled into column of companies, and advanced at a trot\nthrough the fields, seeking to gain the shelter of the forest. The six\ninfantry regiments slid up alongside of each other, and pushed on in\nsix parallel columns of march, two on the right of the road and four\non the left. The artillery, which alone left the highway, followed at\na distance of two or three hundred yards. The remaining cavalry made a\nwide detour to the right, as if to flank the enemy's left.\n\nIt was a mile and a quarter--it was a march of fully twenty\nminutes--to the edge of the woodland, the proposed cover of the\ncolumn. Ten minutes before this point was reached a tiny puff of smoke\nshowed on the brow of the hostile ridge; then, at an interval of\nseveral seconds, followed the sound of a distant explosion; then,\nalmost immediately, came the screech of a rifled shell. Every man who\nheard it swiftly asked himself, \"Will it strike _me_?\" But even as the\nwords were thought out it had passed, high in air, clean to the rear,\nand burst harmlessly. A few faces turned upward and a few eyes glanced\nbackward, as if to see the invisible enemy. But there was no pause in\nthe column; it flowed onward quietly, eagerly, and with business-like\nprecision; it gave forth no sound but the trampling of feet and the\nmuttering of the officers, \"Steady, men! Forward, men.\"\n\nThe Confederates, however, had got their range. A half minute later\nfour puffs of smoke dotted the ridge, and a flight of hoarse humming\nshrieks tore the air. A little aureole cracked and splintered over\nthe First, followed by loud cries of anguish and a brief, slight\nconfusion. The voice of an officer rose sharply out of the flurry,\n\"Close up, Company A! Forward, men!\" The battalion column resumed its\neven formation in an instant, and tramped unitedly onward, leaving\nbehind it two quivering corpses and a wounded man who tottered\nrearward.\n\nThen came more screeches, and a shell exploded over the high road,\nknocking a gunner lifeless from his carriage. The brigade commander\nglanced anxiously along his batteries, and addressed a few words to\nhis chief of artillery. Presently the four Napoleons set forward at a\ngallop for the wood, while the four Parrotts wheeled to the right,\ndeployed, and advanced across the fields, inclining toward the left of\nthe enemy. Next, Taylor's regiment (the Eighth) halted, fronted, faced\nto the right, and filed off in column of march at a double-quick until\nit had gained the rear of the Parrotts, when it fronted again, and\npushed on in support. A quarter of a mile further on these guns went\ninto battery behind the brow of a little knoll, and opened fire. Four\ncompanies of the Eighth spread out to the right as skirmishers, and\ncommenced stealing toward the ridge, from time to time measuring the\ndistance with rifle-balls. The remainder of the regiment lay down in\nline between the Parrotts and the forest. Far away to the right,\nfive companies of cavalry showed themselves, maneuvering as if they\nproposed to turn the left flank of the Southerners. The attack on this\nside was in form and in operation.\n\nMeantime the Confederate fire had divided. Two guns pounded away at\nTaylor's feint, while two shelled the main column. The latter was\nstruck repeatedly; more than twenty men dropped silent or groaning out\nof the hurrying files; but the survivors pushed on without faltering,\nand without even caring for the wounded. At last a broad belt of green\nbranches rose between the regiments and the ridge; and the rebel\ngunners, unable to see their foe, dropped suddenly into silence.\n\nHere it appeared that the road divided. The highway traversed the\nforest, mounted the <DW72> beyond and dissected the enemy's position,\nwhile a branch road turned to the left and skirted the exterior of the\nlong curve of wooded hillocks. At the fork the battery of Napoleons\nhad halted, and there it was ordered to remain for the present in\nquiet. There, too, the Fourteenth filed in among the dense greenery,\nthrew out two companies of skirmishers toward the ridge, and pushed\nslowly after them into the shadows.\n\n\"Get sight of the enemy at once!\" was Waldron's last word to\nGildersleeve. \"If they move down the <DW72>, drive them back. But don't\ncommence your attack under half an hour.\"\n\nNext he filed the Fifth into the thickets, saying to Colburn, \"I want\nyou to halt a hundred yards to the left and rear of Gildersleeve.\nCover his flank if he is attacked; but otherwise lie quiet. As soon as\nhe charges, move forward to the edge of the wood, and be ready to\nsupport him. But make no assault yourself until further orders.\"\n\nThe two next regiments--the Seventh and First--he placed in _echelon_,\nin like manner, a quarter of a mile further along. Then he galloped\nforward to the cavalry, and had a last word with Stilton. \"You and\nGahogan must take care of yourselves. Push on four or five hundred\nyards, and then face to the right. Whatever Gahogan finds let him go\nat it. If he can't shake it, help him. You two _must_ reach the top of\nthe ridge. Only, look out for your left flank. Keep a squadron or two\nin reserve on that side.\"\n\n\"Currnell, if we don't raich the top of the hill, it'll be because it\nhasn't got wan,\" answered Gahogan. Stilton only laughed and rode\nforward.\n\nWaldron now returned toward the fork of the road. On the way he sent a\nstaff officer to the Seventh with renewed orders to attack as soon as\npossible after Gildersleeve. Then another staff officer was hurried\nforward to Taylor with directions to push his feint strongly, and\ndrive his skirmishers as far up the <DW72> as they could get. A third\nstaff officer set the Parrotts in rear of Taylor to firing with all\ntheir might. By the time that the commandant had returned to Colburn's\nambushed ranks, no one was with him but his enemy, Fitz Hugh.\n\n\"You don't seem to trust me with duty, Colonel,\" said the young man.\n\n\"I shall use you only in case of extremity, Captain,\" replied Waldron.\n\"We have business to settle to-morrow.\"\n\n\"I ask no favors on that account. I hope you will offer me none.\"\n\n\"In case of need I shall spare no one,\" declared Waldron.\n\nThen he took out his watch, looked at it impatiently, put it to his\near, restored it to his pocket, and felt into an attitude of deep\nattention. Evidently his whole mind was on his battle, and he was\nwaiting, watching, yearning for its outburst.\n\n\"If he wins this fight,\" thought Fitz Hugh, \"how can I do him a harm?\nAnd yet,\" he added, \"how can I help it?\"\n\nMinutes passed. Fitz Hugh tried to think of his injury, and to steel\nhimself against his chief. But the roar of battle on the right, and\nthe suspense and imminence of battle on the left, absorbed the\nattention of even this wounded and angry spirit, as, indeed, they\nmight have absorbed that of any being not more or less than human. A\nprivate wrong, insupportable though it might be, seemed so small amid\nthat deadly clamor and awful expectation! Moreover, the intellect\nwhich worked so calmly and vigorously by his side, and which alone of\nall things near appeared able to rule the coming crisis, began to\ndominate him, in spite of his sense of injury. A thought crossed him\nto the effect that the great among men are too valuable to be punished\nfor their evil deeds. He turned to the absorbed brigade commander, now\nnot only his ruler but even his protector, with a feeling that he must\naccord him a word of peace, a proffer in some form of possible\nforgiveness and friendship. But the man's face was clouded and stern\nwith responsibility and authority. He seemed at that moment too lofty\nto be approached with a message of pardon. Fitz Hugh gazed at him with\na mixture of profound respect and smothered hate. He gazed, turned\naway, and remained silent.\n\nMinutes more passed. Then a mounted orderly dashed up at full speed,\nwith the words, \"Colonel Major Gahogan has fronted.\"\n\n\"Has he?\" answered Waldron, with a smile which thanked the trooper and\nmade him happy. \"Ride on through the thicket here, my man, and tell\nColonel Gildersleeve to push up his skirmishers.\"\n\nWith a thud of hoofs and a rustling of parting foliage the cavalryman\ndisappeared amid the underwood. A minute or two later a thin, dropping\nrattle of musketry, five hundred yards or so to the front, announced\nthat the sharpshooters of the Fourteenth were at work. Almost\nimmediately there was an angry response, full of the threatenings and\nexecution of death. Through the lofty leafage tore the screech of a\nshell, bursting with a sharp crash as it passed overhead, and\nscattering in humming slivers. Then came another, and another, and\nmany more, chasing each other with hoarse hissings through the\ntrembling air, a succession of flying serpents. The enemy doubtless\nbelieved that nearly the whole attacking force was massed in the wood\naround the road, and they had brought at least four guns to bear upon\nthat point, and were working them with the utmost possible rapidity.\nPresently a large chestnut, not fifty yards from Fitz Hugh, was struck\nby a shot. The solid trunk, nearly three feet in diameter, parted\nasunder as if it were the brittlest of vegetable matter. The upper\nportion started aside with a monstrous groan, dropped in a standing\nposture to the earth, and then toppled slowly, sublimely prostrate,\nits branches crashing and all its leaves wailing. Ere long, a little\nfurther to the front, another Anak of the forest went down; and,\nmingled with the noise of its sylvan agony, there arose sharp cries of\nhuman suffering. Then Colonel Colburn, a broad-chested and ruddy man\nof thirty-five, with a look of indignant anxiety in his iron-gray\neyes, rode up to the brigade commander.\n\n\"This is very annoying, Colonel,\" he said. \"I am losing my men without\nusing them. That last tree fell into my command.\"\n\n\"Are they firing toward our left?\" asked Waldron.\n\n\"Not a shot.\"\n\n\"Very good,\" said the chief, with a sigh of contentment. \"If we can\nonly keep them occupied in this direction! By the way, let your men\nlie down under the fallen tree, as far as it will go. It will protect\nthem from others.\"\n\nColburn rode back to his regiment. Waldron looked impatiently at his\nwatch. At that moment a fierce burst of line firing arose in front,\nfollowed and almost overborne by a long-drawn yell, the scream of\ncharging men. Waldron put up his watch, glanced excitedly at Fitz\nHugh, and smiled.\n\n\"I must forgive or forget,\" the latter could not help saying to\nhimself. \"All the rest of life is nothing compared with this.\"\n\n\"Captain,\" said Waldron, \"ride off to the left at full speed. As soon\nas you hear firing at the shoulder of the ridge, return instantly and\nlet me know.\"\n\nFitz Hugh dashed away. Three minutes carried him into perfect peace,\nbeyond the whistling of ball or the screeching of shell. On the right\nwas a tranquil, wide waving of foliage, and on the left a serene\nlandscape of cultivated fields, with here and there an embowered\nfarm-house. Only for the clamor of artillery and musketry far behind\nhim, he could not have believed in the near presence of battle, of\nblood and suffering and triumphant death. But suddenly he heard to his\nright, assaulting and slaughtering the tranquillity of nature, a\ntumultuous outbreak of file-firing, mingled with savage yells. He\nwheeled, drove spurs into his horse, and flew back to Waldron. As he\nre-entered the wood he met wounded men streaming through it, a few\nmarching alertly upright, many more crouching and groaning, some\nclinging to their less injured comrades, but all haggard in face and\nghastly.\n\n\"Are we winning?\" he hastily asked of one man who held up a hand with\nthree fingers gone and the bones projecting in sharp spikes through\nmangled flesh.\n\n\"All right, Sir; sailing in,\" was the answer.\n\n\"Is the brigade commander all right?\" he inquired of another who was\nwinding a bloody handkerchief around his arm.\n\n\"Straight ahead, Sir; hurrah for Waldron!\" responded the soldier, and\nalmost in the same instant fell lifeless with a fresh ball through his\nhead.\n\n\"Hurrah for him!\" Fitz Hugh answered frantically, plunging on through\nthe underwood. He found Waldron with Colburn, the two conversing\ntranquilly in their saddles amid hissing bullets and dropping\nbranches.\n\n\"Move your regiment forward now,\" the brigade commander was saying;\n\"but halt it in the edge of the wood.\"\n\n\"Shan't I relieve Gildersleeve if he gets beaten?\" asked the\nsubordinate officer eagerly.\n\n\"No. The regiments on the left will help him out. I want your men and\nPeck's for the fight on top of the hill. Of course the rebels will try\nto retake it; then I shall call for you.\"\n\nFitz Hugh now approached and said, \"Colonel, the Seventh has attacked\nin force.\"\n\n\"Good!\" answered Waldron, with that sweet smile of his which thanked\npeople who brought him pleasant news. \"I thought I heard his fire.\nGahogan will be on their right rear in ten minutes. Then we shall get\nthe ridge. Ride back now to Major Bradley, and tell him to bring his\nNapoleons through the wood, and set two of them to shelling the\nenemy's centre. Tell him my idea is to amuse them, and keep them from\nchanging front.\"\n\nAgain Fitz Hugh galloped off as before on a comfortably safe errand,\nsafer at all events than many errands of that day. \"This man is\nsparing my life,\" he said to himself. \"Would to God I knew how to\nspare his!\"\n\nHe found Bradley lunching on a gun caisson, and delivered his orders.\n\"Something to do at last, eh?\" laughed the rosy-cheeked youngster.\n\"The smallest favors thankfully received. Won't you take a bite of\nrebel chicken, Captain? This rebellion must be put down. No? Well,\ntell the Colonel I am moving on, and John Brown's soul not far ahead.\"\n\nWhen Fitz Hugh returned to Waldron he found him outside of the wood,\nat the base of the long incline which rose into the rebel position.\nAbout the <DW72> were scattered prostrate forms, most numerous near the\nbottom, some crawling slowly rearward, some quiescent. Under the brow\nof the ridge, decimated and broken into a mere skirmish line sheltered\nin knots and, singly, behind rocks and knolls and bushes, lay the\nFourteenth Regiment, keeping up a steady, slow fire. From the edge\nabove, smokily dim against a pure, blue heaven, answered another\nrattle of musketry, incessant, obstinate, and spiteful. The combatants\non both sides were lying down; otherwise neither party could have\nlasted ten minutes. From Fitz Hugh's point of view not a Confederate\nuniform could be seen. But the smoke of their rifles made a long gray\nline, which was disagreeably visible and permanent; and the sharp\n_whit! whit!_ of their bullets continually passed him, and cheeped\naway in the leafage behind.\n\n\"Our men can't get on another inch,\" he ventured to say to his\ncommander. \"Wouldn't it be well for me to ride up and say a cheering\nword?\"\n\n\"Every battle consists largely in waiting,\" replied Waldron\nthoughtfully. \"They have undoubtedly brought up a reserve to face\nThomas. But when Gahogan strikes the flank of the reserve, we shall\nwin.\"\n\n\"I wish you would take shelter,\" begged Fitz Hugh. \"Everything depends\non your life.\"\n\n\"My life has been both a help and a hurt to my fellow-creatures,\"\nsighed the brigade commander. \"Let come what will to it.\"\n\nHe glanced upward with an expression of profound emotion; he was\nevidently fighting two battles, an outward and an inward one.\n\nPresently he added, \"I think the musketry is increasing on the left.\nDoes it strike you so?\"\n\nHe was all eagerness again, leaning forward with an air of earnest\nlistening, his face deeply flushed and his eye brilliant. Of a sudden\nthe combat above rose and swelled into higher violence. There was a\nclamor far away--it seemed nearly a mile away--over the hill. Then the\nnearer musketry, first Thomas' on the shoulder of the ridge, next\nGildersleeve's in front, caught fire and raged with new fury.\n\nWaldron laughed outright. \"Gahogan has reached them,\" he said to one\nof his staff who had just rejoined him. \"We shall all be up there in\nfive minutes. Tell Colburn to bring on his regiment slowly.\"\n\nThen, turning to Fitz Hugh, he added, \"Captain, we will ride forward.\"\n\nThey set off at a walk, now watching the smoking brow of the eminence,\nnow picking their way among dead and wounded. Suddenly there was a\nshout above them and a sudden diminution of the firing; and looking\nupward, they saw the men of the Fourteenth running confusedly toward\nthe summit. Without a word the brigade commander struck spurs into his\nhorse and dashed up the long <DW72> at a run, closely followed by his\nenemy and aid. What they saw when they overtook the straggling,\nrunning, panting, screaming pell-mell of the Fourteenth was victory!\n\nThe entire right wing of the Confederates, attacked on three sides at\nonce, placed at enormous disadvantage, completely outgeneraled, had\ngiven way in confusion, was retreating, breaking, and flying. There\nwere lines yet of dirty gray or butternut; but they were few, meagre,\nfluctuating, and recoiling, and there were scattered and scurrying\nmen in hundreds. Three veteran and gallant regiments had gone\nall to wreck under the shock of three similar regiments far more\nintelligently directed. A strong position had been lost because the\nheroes who held it could not perform the impossible feat of forming\nsuccessively two fresh fronts under a concentric fire of musketry. The\ninferior brain power had confessed the superiority of the stronger\none.\n\nOn the victorious side there was wild, clamorous, fierce exultation.\nThe hurrying, shouting, firing soldiers, who noted their commander\nriding among them, swung their rifles or their tattered hats at him,\nand screamed \"Hurrah!\" No one thought of the Confederate dead under\nfoot, nor of the Union dead who dotted the <DW72> behind. \"What are you\nhere for, Colonel?\" shouted rough old Gildersleeve, one leg of his\ntrousers dripping blood. \"We can do it alone.\"\n\n\"It is a battle won,\" laughed Fitz Hugh, almost worshipping the man\nwhom he had come to slay.\n\n\"It is a battle won, but not used,\" answered Waldron. \"We haven't a\ngun yet, nor a flag. Where is the cavalry? Why isn't Stilton here? He\nmust have got afoul of the enemy's horse, and been obliged to beat it\noff. Can anybody hear anything of Stilton?\"\n\n\"Let him go,\" roared old Grumps. \"The infantry don't want any help.\"\n\n\"Your regiment has suffered, Colonel,\" answered Waldron, glancing at\nthe scattered files of the Fourteenth. \"Halt it and reorganize it, and\nlet it fall in with the right of the First when Peck comes up. I shall\nreplace you with the Fifth. Send your Adjutant back to Colburn and\ntell him to hurry along. Those fellows are making a new front over\nthere,\" he added, pointing to the centre of the hill. \"I want the\nFifth, Seventh, and Tenth in _echelon_ as quickly as possible. And I\nwant that cavalry. Lieutenant,\" turning to one of his staff, \"ride off\nto the left and find Colonel Stilton. Tell him that I need a charge in\nten minutes.\"\n\nPresently cannon opened from that part of the ridge still held by the\nConfederates, the shells tearing through or over the dissolving groups\nof their right wing, and cracking viciously above the heads of the\nvictorious Unionists. The explosions followed each other with stunning\nrapidity, and the shrill whirring of the splinters was ominous. Men\nbegan to fall again in the ranks or to drop out of them wounded. Of\nall this Waldron took no further note than to ride hastily to the brow\nof the ridge and look for his own artillery.\n\n\"See how he attinds to iverything himself,\" said Major Gahogan, who\nhad cantered up to the side of Fitz Hugh. \"It's just a matther of\nplain business, an' he looks after it loike a business man. Did ye see\nus, though, Captin, whin we come in on their right flank? By George,\nwe murthered um. There's more'n a hundred lyin' in hapes back there.\nAs for old Stilton, I just caught sight of um behind that wood to our\nleft, an' he's makin' for the enemy's right rair. He'll have lots o'\nprisoners in half an hour.\"\n\nWhen Waldron returned to the group he was told of his cavalry's\nwhereabouts, and responded to the information with a smile of\nsatisfaction.\n\n\"Bradley is hurrying up,\" he said, \"and Taylor is pushing their left\nsmartly. They will make one more tussle to recover their line of\nretreat; but we shall smash them from end to end and take every gun.\"\n\nHe galloped now to his infantry, and gave the word \"Forward!\" The\nthree regiments which composed the _echelon_ were the Fifth on the\nright, the Seventh fifty yards to the rear and left of the Fifth, the\nTenth to the rear and left of the Seventh. It was behind the Fifth,\nthat is the foremost battalion, that the brigade commander posted\nhimself.\n\n\"Do _you_ mean to stay here, Colonel?\" asked Fitz Hugh, in surprise\nand anxiety.\n\n\"It is a certain victory now,\" answered Waldron, with a singular\nglance upward. \"My life is no longer important. I prefer to do my duty\nto the utmost in the sight of all men.\"\n\n\"I shall follow you and do mine, Sir,\" said the Captain, much moved,\nhe could scarcely say by what emotions, they were so many and\nconflicting.\n\n\"I want you other wheres. Ride to Colonel Taylor at once, and hurry\nhim up the hill. Tell him the enemy have greatly weakened their left.\nTell him to push up everything, infantry, and cavalry, and artillery,\nand to do it in haste.\"\n\n\"Colonel, this is saving my life against my will,\" remonstrated Fitz\nHugh.\n\n\"Go!\" ordered Waldron, imperiously. \"Time is precious.\"\n\nFitz Hugh dashed down the <DW72> to the right at a gallop. The brigade\ncommander turned tranquilly, and followed the march of his _echelon_.\nThe second and decisive crisis of the little battle was approaching,\nand to understand it we must glance at the ground on which it was to\nbe fought. Two hostile lines were marching toward each other along the\nbroad, gently rounded crest of the hill and at right angles to its\ngeneral course. Between these lines, but much the nearest to the Union\ntroops, a spacious road came up out of the forest in front, crossed\nthe ridge, swept down the smooth decline in rear, and led to a single\nwooden bridge over a narrow but deep rivulet. On either hand the road\nwas hedged in by a close board fence, four feet or so in height. It\nwas for the possession of this highway that the approaching lines were\nabout to shed their blood. If the Confederates failed to win it, all\ntheir artillery would be lost, and their army captured or dispersed.\n\nThe two parties came on without firing. The soldiers on both sides\nwere veterans, cool, obedient to orders, intelligent through long\nservice, and able to reserve all their resources for a short-range\nand final struggle. Moreover, the fences as yet partially hid them\nfrom each other, and would have rendered all aim for the present vague\nand uncertain.\n\n\"Forward, Fifth!\" shouted Waldron. \"Steady. Reserve your fire.\" Then,\nas the regiment came up to the fence, he added, \"Halt; right dress.\nSteady, men.\"\n\nMeantime he watched the advancing array with an eager gaze. It was a\nnoble sight, full of moral sublimity, and worthy of all admiration.\nThe long, lean, sunburned, weather-beaten soldiers in ragged gray\nstepped forward, superbly, their ranks loose, but swift and firm, the\nmen leaning forward in their haste, their tattered slouch hats pushed\nbackward, their whole aspect business-like and virile. Their line was\nthree battalions strong, far outflanking the Fifth, and at least equal\nto the entire _echelon_. When within thirty or forty yards of the\nfurther fence they increased their pace to nearly a double-quick, many\nof them stooping low in hunter fashion, and a few firing. Then Waldron\nrose in his stirrups and yelled, \"Battalion! ready--aim--aim low.\nFire!\"\n\nThere was a stunning roar of three hundred and fifty rifles, and a\ndeadly screech of bullets. But the smoke rolled out, the haste to\nreload was intense, and none could mark what execution was done.\nWhatever the Confederates may have suffered, they bore up under the\nvolley, and they came on. In another minute each of those fences, not\nmore than twenty-five yards apart, was lined by the shattered fragment\nof a regiment, each firing as fast as possible into the face of the\nother. The Fifth bled fearfully: it had five of its ten company\ncommanders shot dead in three minutes; and its loss in other officers\nand in men fell scarcely short of this terrible ratio. On its left the\nSeventh and the Tenth were up, pouring in musketry, and receiving it\nin a fashion hardly less sanguinary. No one present had ever seen, or\never afterward saw, such another close and deadly contest.\n\nBut the strangest thing in this whole wonderful fight was the conduct\nof the brigade commander. Up and down the rear of the lacerated Fifth\nWaldron rode thrice, spurring his plunging and wounded horse close to\nthe yelling and fighting file-closers, and shouting in a piercing\nvoice encouragement to his men. Stranger still, considering the\ncharacter which he had borne in the army, and considering the evil\ndeed for which he was to account on the morrow, were the words which\nhe was distinctly and repeatedly heard to utter. \"Stand steady,\nmen--God is with us!\" was the extraordinary battle-cry of this\nbackslidden clergyman, this sinner above many.\n\nAnd it was a prophecy of victory. Bradley ran up his Napoleons on the\nright in the nick of time, and, although only one of them could be\nbrought to bear, it was enough; the grape raked the Confederate left,\nbroke it, and the battle was over. In five minutes more their whole\narray was scattered, and the entire position open to galloping\ncavalry, seizing guns, standards, and prisoners.\n\nIt was in the very moment of triumph, just as the stubborn Southern\nline reeled back from the fence in isolated clusters, that the\nmiraculous impunity of Waldron terminated, and he received his death\nwound. A quarter of an hour later Fitz Hugh found a sorrowful group of\nofficers gazing from a little distance upon their dying commander.\n\n\"Is the Colonel hit?\" he asked, shocked and grieved, incredible as the\nemotion may seem.\n\n\"Don't go near him,\" called Gildersleeve, who, it will be remembered,\nknew or guessed his errand in camp. \"The Chaplain and surgeon are\nthere. Let him alone.\"\n\n\"He's going to render his account,\" added Gahogan. \"An' whativer he's\ndone wrong, he's made it square to-day. Let um lave it to his\nbrigade.\"\n\nAdjutant Wallis, who had been blubbering aloud, who had cursed the\nrebels and the luck energetically, and who had also been trying to\npray inwardly, groaned out, \"This is our last victory. You see if it\nain't. Bet you two to one.\"\n\n\"Hush, man!\" replied Gahogan. \"We'll win our share of um, though we'll\nhave to work harder for it. We'll have to do more ourselves, an' get\nless done for us in the way of tactics.\"\n\n\"That so, Major,\" whimpered a drummer, looking up from his duty of\nattending to a wounded comrade. \"He knowed how to put his men in the\nright place, and his men knowed when they was in the right place. But\nit's goin' to be uphill through the steepest part of hell the rest of\nthe way.\"\n\nSoldiers, some of them weeping, some of them bleeding, arrived\nconstantly to inquire after their commander, only to be sent quietly\nback to their ranks or to the rear. Around lay other men--dead\nmen, and senseless, groaning men--all for the present unnoticed.\nEverything, except the distant pursuit of the cavalry, waited for\nWaldron to die. Fitz Hugh looked on silently, with the tears of\nmingled emotions in his eyes, and with hopes and hatreds expiring in\nhis heart. The surgeon supported the expiring victor's head, while\nChaplain Colquhoun knelt beside him, holding his hand and praying\naudibly. Of a sudden the petition ceased, both bent hastily toward the\nwounded man, and after what seemed a long time exchanged whispers.\nThen the Chaplain rose, came slowly toward the now advancing group\nof officers, his hands outspread toward heaven in an attitude of\nbenediction, and tears running down his haggard white face.\n\n\"I trust, dear friends,\" he said, in a tremulous voice, \"that all is\nwell with our brother and commander. His last words were, 'God is with\nus.'\"\n\n\"Oh! but, man, _that_ isn't well,\" broke out Gahogan, in a groan.\n\"What did ye pray for his sowl for? Why didn't ye pray for his loife?\"\n\nFitz Hugh turned his horse and rode silently away. The next day he was\nseen journeying rearward by the side of an ambulance, within which lay\nwhat seemed a strangely delicate boy, insensible, and, one would say,\nmortally ill.\n\n\n\n\nSPLIT ZEPHYR.\n\nAN ATTENUATED YARN SPUN BY THE FATES.\n\nBY HENRY A. BEERS\n\n_Century Magazine, June, 1883._\n\n\nIt was the evening of Commencement Day. The old church on the green,\nwhich had rung for many consecutive hours with the eloquence of slim\nyoung gentlemen in evening dress, exhorting the Scholar in Politics or\ndenouncing the Gross Materialism of the Age, was at last empty and\nstill. As it drew the dewy shadows softly about its eaves and filled\nits rasped interior with soothing darkness, it bore a whimsical\nlikeness to some aged horse which, having been pestered all day with\nflies, was now feeding in peace along the dim pasture.\n\nIt was Clay who suggested this resemblance, and we all laughed\nappreciatively, as we used to do in those days at Clay's clever\nsayings. There were five of us strolling down the diagonal walk to\nour farewell supper at \"Ambrose's.\" Arrived at that refectory, we\nfound it bare of guests and had things quite to ourselves. After\nsupper, we took our coffee out in the little court-yard, where a\nfountain dribbled, and the flutter of the grape-leaves on the\ntrellises in the night wind invited to confidences.\n\n\"Well, Armstrong,\" began Doddridge, \"where are you going to spend the\nvacation?\"\n\n\"Vacation!\" answered Armstrong; \"vacations are over for me.\"\n\n\"You're not going to work for your living at once?\" inquired Berkeley.\n\n\"I'm going to work to-morrow,\" replied Armstrong, emphatically: \"I'm\ngoing down to New York to enter a law office.\"\n\n\"I thought you had some notion of staying here and taking a course of\ngraduate study.\"\n\n\"No, sir! The sooner a man gets into harness, the better. I've wasted\nenough time in the last four years. The longer a man loafs around in\nthis old place, under pretense of reading and that kind of thing, the\nharder it is for him to take hold.\"\n\nArmstrong was a rosy little man, with yellow hair and light eyes. His\nexpression was one of irresolute good nature. His temper was sanguine\nand expansive, and he had been noted in college for anything but\nconcentration of pursuit. He was gregarious in his habits, susceptible\nand subject to sudden enthusiasms. His good nature made him a victim\nto all the bores and idlers in the class, and his room became a\nfavorite resort for men on their way to recitation, being on the\nground floor and near the lecture-rooms. They would drop in about half\nan hour before the bell rang, and make up a little game of \"penny\nante\" around Armstrong's center-table. In these diversions he seldom\ntook part, as he had given it out publicly that he was \"studying for a\nstand\"; but his abstinence from the game in no wise damped the spirits\nof his guests. Occasionally his presence would receive the notice of\nthe company somewhat as follows:\n\n     No. 1. \"Make less noise, fellows: Charley is digging out\n     that Puckle lesson.\"\n\n     No. 2. \"You go into the bedroom, Charley, and shut the door,\n     and then you won't be bothered by the racket.\"\n\n     No. 3. \"Oh, hang the Puckle! Come and take a hand, Charley.\n     We'll let you in this pool without an ante.\"\n\n     No. 4. \"Why don't you get a new pack of cards, Charley? It's\n     a disgrace to you to keep such a dirty lot of old\n     pasteboards for your friends.\"\n\nIn face of which abuse, Armstrong was as helpless as Telemachus under\nthe visitation of the suitors. The resolute air with which he now\ndeclared his intention of grappling with life had therefore something\ncomic about it, and Berkeley said, rather incredulously:\n\n\"I suppose you'll keep up your reading along with your law?\"\n\n\"No,\" replied the other; \"Themis is a jealous mistress. No; I'm going\nto bone right down to it.\"\n\n\"Haven't you changed your ideal of life lately?\" asked Clay, a little\nscornfully.\n\n\"Perhaps I have,\" said Armstrong, \"perhaps I've had to.\"\n\n\"What _is_ your ideal of life?\" I inquired.\n\n\"Well, I'll tell you,\" he answered, draining his coffee-cup solemnly,\nand putting it down with the manner of a man who has made up his mind.\nThe rest of us arranged ourselves in attitudes of attention. \"My ideal\nis independence,\" began Armstrong. \"I want to live my own life; and as\nthe first condition of independence is money, I'm going for money.\nCulture and taste, and all that, are well enough when a man can afford\nit, but for a poor man it means just so many additional wants which he\ncan't gratify. My father is an educated man; a country minister with a\nsmall salary and a large family; and his education, instead of being a\nblessing, has been an actual curse to him. He has pined for all sorts\nof things which he couldn't have--books, engravings, foreign travel,\nleisure for study, nice people and nice things about him. I've made\nup my mind that, whatever else I may be, I won't be poor, and I\nwon't be a minister, and I won't have a wife and brats hanging\nto me. I tell you that, next to ill health, poverty is the worst\nthing that can happen to a man. All the sentimental grievances\nthat are represented in novels and poetry as the deepest of human\nafflictions,--disappointed ambitions, death of friends, loss of faith,\nestrangements, having your girl go back on you,--they don't signify\nvery long if a man has sound health and a full purse. The ministers\nand novel writers and fellows that preach the sentimental view of life\ndon't believe it themselves. It's a kind of professional or literary\nquackery with them. Just let them feel the pinch of poverty, and then\noffer them a higher salary or a chance to make a little 'sordid gain'\nin some way, and see how quick they'll accept the call to 'a higher\nsphere of usefulness.' Berk, hand over a match, will you; this cigar\nhas gone out.\"\n\n\"Loud cries of 'We will--we will'!\" said Berkeley. \"But can it be? Has\nthe poick turned cynic, and the sickly sentimentalist become a\nmaterialist and a misogynist?\"\n\n(Armstrong was our class poet, and had worried the official muse on\nPresentation Day to the utterance of some four hundred lines filled\nwith allusions to Alma Mater, Friendship's Altar, the Elms of Yale,\netc. His piece on that occasion had been \"pronounced, by a well-known\nliterary gentleman who was present, equal to the finest productions of\nour own Willis.\")\n\n\"I'll bet the cigars,\" said Doddridge, \"that Armstrong marries the\nfirst girl he sees in New York.\"\n\n\"Yes,\" said Clay, \"his boarding-house keeper's daughter.\"\n\n\"And has a dozen children before he is forty,\" added Berkeley; \"a\ndozen kids, and all of them girls. Charley is sure to be a begetter of\nwenches.\"\n\n\"And writes birthday odes 'To My Infant Daughter' for the 'Home\nJournal,'\" continued Clay.\n\n\"No, no,\" said the victim of this banter, shaking his head solemnly.\n\"I shall give no hostages to Fortune. I mean to live snug and carry as\nlittle sail as possible: to leave only the narrowest margin out for\nFate to tread on. The man who has the fewest exposed points leads, on\nthe whole, the happiest life. How can a man enjoy himself freely when\na piece of defective plumbing, the bursting of a toy pistol, the\ncarelessness of a nurse, may plunge him into a life-long sorrow? I\ndon't say it's a very noble life that I propose to myself, but it's a\nsafe one. I'm too nervous and anxious to stand the responsibilities of\nmatrimony.\"\n\n\"If you can't stand responsibility,\" said Doddridge, \"I don't see why\nyou choose the law for a profession. You don't seem to me cut out for\na lawyer anyway. I always thought you meant to be some kind of a\nliterary chap.\"\n\n\"Yes,\" said Berkeley, \"why don't you go for a snug berth under the\ngovernment, or study for a tutorship here? That's the life that would\nsuit you, old man.\"\n\n\"Not at all,\" answered Armstrong; \"I have a horror of any salaried\nposition, or of any position where a man is obliged to conform his\nhabits and opinions to other people's. It is the worst sort of\ndependence. Now a lawyer in successful practice, and especially if he\nis a bachelor, is about as independent as a man can be. His relations\nwith his clients are merely professional, and what he does or thinks\nprivately is nobody's business.\"\n\n\"If you are going to be a mere lawyer,\" asked Clay, \"what becomes of\nyour education and your intellectual satisfactions, etc.?\"\n\n\"A man can get his best intellectual satisfactions out of the work of\nhis profession,\" answered Armstrong. \"Besides, as to that, there's\ntime enough. Fifteen years of solid work will enable one to put by a\nfair competence, if he lives carefully and has no one but himself to\nsupport; and then he will be free to take up a hobby. Oh, I shall\ncultivate a hobby or two after awhile. It keeps the mind healthy to\nhave some interest of the kind outside of one's business. I may take\nto book-collecting or numismatics or raising orchids. Perhaps I may\nbecome an authority on ancient armor; time enough for that by and by.\nAnd then I can cut over to Europe every summer if I like, and no one\nto interfere with my down-sittings or my up-risings, my goings-out or\nmy comings-in. Do you know,\" he went on, after a pause, \"how I always\nlook to myself in the glass of the future? I figure myself like old\nTulkinghorn, in 'Bleak House,'--going down into his reverberating\nvaults for a bottle of choice vintage, after the work of the day, and\nthen sitting quietly in the twilight in his dusky, old-fashioned law\nchambers, sipping his wine while the room fills with the fragrance of\nsouthern grapes. The gay old silver-top!\"\n\nThere was silence for a few minutes after Armstrong had finished his\ndeclaration. It was broken by Berkeley, who had risen, and was walking\nup and down in front of the fountain with his hands thrust into his\npockets.\n\n\"You couldn't lead that sort of life if you tried,\" he said; \"you\naren't built for it.\"\n\n\"Don't you make any mistake,\" rejoined the other; \"it's the sort of\nlife I'm going to live.\"\n\n\"It's a cowardly life,\" retorted Berkeley.\n\n\"Did I say it wasn't? I said it was safe. You can call it what you\nlike.\"\n\n\"Well,\" replied Berkeley, seating himself again, \"my ideal career is\njust the opposite of that.\"\n\n\"Suppose you explain yours, then,\" said Armstrong.\n\nBerkeley hesitated a few moments before beginning. He was a lean,\ntallish fellow, with a Scotch cast of countenance, a small blue eye,\nhigh cheek bones, a freckled skin, and whity-brown hair. He had a dry,\ncautious humor, fed by much out-of-the-way reading. He had been\ndistinguished in college by methodical habits, a want of ambition, a\ndisposition to keep to himself, and a mixture of selfishness and\n_bonhomie_ which made him a cold friend but an agreeable companion.\nIt was therefore with some surprise that we heard him deliver himself\nas follows:\n\n\"I believe that the greatest mistake a man can make is in not getting\nenough out of life. I want to lead a full life, to have a wide\nexperience, to develop my whole nature to the utmost, to touch mankind\nat the largest possible number of points. I want adventure, change,\nexcitement, emotion, suffering even,--I don't care what, so long as it\nis not stagnation. Just consider what there is on this planet to be\nseen, learned, enjoyed, and what a miserably small share of it most\npeople appropriate. Why, there are men in my village who have never\nbeen outside the county and seldom out of the township; who have never\nheard a word of any language but English; never seen a city or a\nmountain or the ocean--or, indeed, any body of water bigger than Fresh\nPond or the Hogganum River; never been in a theatre, steamboat,\nlibrary, or cathedral. Cathedral! Their conception of a church is\nlimited to the white wooden meeting-house at 'the center.' Their\nart-gallery is the wagon of a travelling photographer. Their\nmetropolitan hotel is the stoop and bar-room of the 'Uncas House.'\nTheir university is the unpainted school-house on the hill. Their\nliterature is the weekly newspaper from the county town. But take the\nmajority of educated men even. What a rusty, small kind of existence\nthey lead! They are in a rut, just the same as the others, only the\nrut is a trifle wider. If I had my way I would never do the same work\nor talk with the same people--hardly live in the same place for two\ndays running. Life is too short to do a thing twice. When I come to\nthe end of mine I don't want to say _J'ai manque la vie_; but make my\nbrag, with the Wife of Bath,\n\n     'Unto this day it doth myn herte bote\n     That I have had my world as in my time.'\"\n\n\"Well, how are you going to do all those fine things?\" inquired\nArmstrong. \"For instance, that about not living in one place two days\nrunning. I'm afraid you'll find that inconvenient, not to say\nexpensive.\"\n\n\"Oh, you mustn't take me too literally. I may have to travel on foot\nor take a steerage passage, but I shall keep going all the same. I\nhaven't made any definite plans yet. I shall probably strike for\nsomething in the diplomatic line,--secretary of legation, or some\nsmall consulship perhaps. But the principle is the main thing, and the\nprinciple is: Don't do anything because it's the nearest and easiest\nand most obvious thing to do, but make up your mind to get the best.\nLook at the lazy way in which men accept their circumstances. There is\nthe matter of acquaintance, for instance--we let chance determine it.\nWe know the men that we can't help knowing,--the ones in the next\nhouse, cousins and second cousins, business connections, etc. Here at\ncollege, now, we get acquainted with the fellows at the eating club or\nin the same society, or those who happen to sit next us in the\nclass-room, because their names begin with the same letter. That's it;\nit's just a sample of our whole life. Our friendships, like everything\nelse about us, are determined by the alphabet. We go with the Z's\nbecause some arbitrary system of classification has put us among them,\ninstead of fighting our way up to the A's, where we naturally belong.\nThe consequence is that one's friends are mostly dreadful bores.\"\n\n\"I'm sure we are all much obliged to you,\" murmured Clay,\nparenthetically.\n\n\"There are about two or three thousand people in the world,\" continued\nBerkeley, \"supremely worth knowing. Why shouldn't _I_ know them?---- I\nwill! Everybody knows two or three thousand people,--mostly very\nstupid people,--or, rather, he lets them know him. Why shouldn't he\nuse some choice in the matter? Why not know Thackeray and Carlyle,\nLord Palmerston and the Pope, and the Emperor of China and all the\ngreat statesmen, authors, African explorers, military commanders,\nartists, hereditary nobles, actresses, wits and belles of the best\nsociety, instead of putting up with Tom, Dick, and Harry?\"\n\n\"Berkeley, 'with whom the bell-mouthed flask had wrought!'\" exclaimed\nClay. \"Decidedly, Berk, you should take your coffee without cognac.\"\n\n\"Let me suggest,\" put in Doddridge, \"that some of those parties you\nmentioned are not so easy to get introductions to.\"\n\n\"Oh, I say again, you mustn't take me too literally. But even the top\nswells are easier to know than you think. All that is wanted is a\nlittle cheek. But take it in a smaller way; say that we resolve to\ncultivate the best society within our reach. Doubtless there are\nnumbers of interesting and distinguished people right here in New\nHaven whose acquaintance it would be worth while to have. But how long\nwould you beggars live here without making the least effort to look\nthem out, and meanwhile put up with the same old every-day bores--like\nme, or Polisson here? And it's the same way with marriage. A fellow\nblunders into matrimony with the first attractive girl that gives him\nthe opportunity. He knows, if he takes the time to think about it,\nthat there are a thousand others better than she, if he will wait and\nlook through the world a little. 'Juxtaposition in fine,' as Clough\nsays.\"\n\n\"Of course, with such a brilliant destiny before you, _you'll_ never\nmarry,\" said I.\n\n\"Yes, I think I shall. I fancy that the noblest possibilities of life\nare never realized without marriage. Yes, I can think of nothing finer\nthan to have a lot of manly boys and sweet girls growing up around\none. But when I marry it shall be so as to give completeness and\nexpansion to life, not narrowness and dullness. I shall never marry\nand settle down. Settle down! What a damnable expression that is! A\nman ought to settle _up_. I mean to have my fling first, too. I should\nlike to gamble a bit at Baden-Baden. I should like to go out to\nColorado and have a lick at mining speculations. I want to rough it\nsome too, and see how life is lived close to the bone: ship for a\nvoyage before the mast; enlist for a campaign or two somewhere and\nhave joy of battle; join the gypsies or the Mormons or the Shakers for\nawhile, and taste all the queerness of things. And then I want to\nfloat for another while on the very top-most crest of society. I want\nto fight a duel or two, elope with a marquise, do a little of\neverything for the experience's sake, as a man ought to take opium\nonce in his life just to know how it feels.\"\n\nWhether it was indeed the cognac, or only the unusual excitement\nattending this outburst of pent-up fire, Berkeley's cheek had got\na flush upon it. Perhaps, too, it was owing to the influences of\nthe day and the hour, the splash of the fountain, the rustle of\nthe vine-leaves, and the wavering shadows which played about the\ncourt-yard as the gas-jets flickered in the breeze of night, that made\nhis boastful words seem less extravagantly out of character than they\notherwise would. The silence which followed his speech was broken by\nClay, who sat with his foot on the rim of the fountain, balancing on\nthe hind legs of his chair, and looking thoughtfully at the slender\njet as it rose and fell. He still wore the dress suit in which he had\nfigured on the Commencement platform in the afternoon, and which set\noff the aristocratic grace of his slight figure. There was a pale\nintellectual light in his face, and his black eyes had the glow of\ngenius.\n\n\"I think,\" he began, \"that Berkeley makes a mistake in confounding a\nfull life with a restless one. I believe in a full experience too, but\nthe satisfactions should be inward ones. Take the matter of foreign\ntravel, for one thing, on which you lay so much stress. It is a great\nstimulus to the imagination, no doubt; but then foreign countries are\naccessible to the imagination by other means--through books and art,\nfor example. I think it likely that the reality is, quite as often as\nnot, disappointing. Place, after all, is indifferent. 'The soul is its\nown place': you can't get rid of yourself by going abroad, and it's\nhimself that a man gets sooner tired of than of anything else. Then as\nto acquaintances, I don't know that I should care to know personally\nsuch men as Thackeray and Carlyle, and the big composers and\nartists and other people that you mentioned. It might be equally\ndisenchanting. They put the best of themselves into their books, or\npictures, or music. I certainly would not seek their society through\na formal introduction, at all events. It is hard for a small man to\nkeep his self-respect in face of a great man when he obtains his\nacquaintance as a special favor. If I could meet some of those\nfellows, quite naturally and accidentally, on equal terms, I might\nlike it, but not otherwise. But, leaving that point out of account, I\nthink that the career which Berkeley proposes to himself would turn\nout very hollow. It would result in the superficial gratification of\nthe curiosity and the senses; and, as soon as the novelty got rubbed\noff, what is there left?\"\n\n\"So then,\" said Berkeley, \"you've swung into line with Armstrong, have\nyou? You mean to plod along in some professional rut too. What has got\ninto all our idealists?\"\n\n\"Not by any means,\" answered Clay. \"Armstrong talks about\nindependence, and yet destines himself to the worst kind of\ndependence--slavery to money-getting. Most people, it seems to me,\nspend the best part of their lives not in living, but in getting the\nmeans to live. We'll give Armstrong, say twenty years, to lay up\nenough money to retire on and begin to live. What sort of a position\nwill he be in then to enjoy his independence? His nature will have got\nso subdued to what it works in that the only safety for him will be to\nkeep on at the law.\"\n\n\"All right! Then I'll keep on,\" interjected Armstrong.\n\n\"What the devil do _you_ mean to do then?\" asked Berkeley of Clay.\n\n\"I don't quite know yet,\" replied the latter. \"I shall 'loaf and\ninvite my soul' whenever I feel like it. I shall live as I go along,\nand not postpone it till I am forty. I sha'n't put myself into any\nmill that will grind me just so much a day. I need my leisure too\nbadly for that. I presume I shall spend most of my time at first in\nreading and walking. Then, whenever I think of anything to write I\nshall write it, and if I can sell what I write to some publisher or\nother, so much the better. If not, go on as before.\"\n\n\"Meanwhile, where will your bread and butter come from?\" asked\nArmstrong.\n\n\"Oh, I sha'n't starve. I can get some sort of hack work--something\nthat won't take much of my time, and which I can do with my left hand.\nBut the great point, after all, is to make your wants simple; to live\nlike an Arab, content with a few dates and a swallow from the gourd.\n'Lessen your denominator.' It's easier than raising your numerator,\nand the quotient is the same.\"\n\n\"No, it's not the same,\" Berkeley retorted. \"Renunciation and\nenjoyment are not the same. It makes a heap of difference whether you\nhave a thing or simply do without it. The plain living and high\nthinking philosophy may do for Clay, whose mind to him a kingdom is;\nbut a fellow like me, whose mind is only a small Central American\nrepublic, can't live on the revenues of the spirit. The fact is, Clay,\nyou've read too much Emerson. I went into that myself once, but I soon\nfound out that it wouldn't wear. I want mine thicker. The worst thing\nabout the career of a literary man or an artist is that if he fails\nthere are no compensations; and success is mighty uncertain. Nobody\ndoubts that you are smart enough, Clay, and I am sure we expect great\nthings of you, whatever line you take up. But, for the sake of the\nargument, suppose you have grubbed along in a small way, living on\ncrusts and water, till you are fifty, without doing any really good\nwork. Then where are you? You haven't had any fun. You've no other\nstring to your bow. You haven't that practical experience of the world\nwhich would enable you to turn your hand to something else. You have\nno influence or reputation; for, of all poor things, poor art of any\nkind is the worst--hateful to gods and men and columns. In short,\nwhere are you? You're out of the dance; you don't count.\"\n\n\"Yes,\" added Armstrong, \"and you've no professional success or solid\nstanding in the community; and, what's worse, you've no money, which\nmight make up for the want of all the rest.\"\n\n\"I don't think you get my meaning. I may fail,\" said Clay, proudly; \"I\nmay never even try to succeed, in your sense of the word. I decline\nall mean competitions and all low views of success. The noblest ideal\nof life--at least, the noblest to me--is self-culture in the high\nmeaning of the word; the harmonious development of one's whole nature.\nArmstrong has drawn a picture of his future in the likeness of old\nTulkinghorn. I suppose we are all accustomed to put our anticipations\ninto some such concrete shape before our mind's eye. The typical\nsituation which I am fond of imagining is something like this: I like\nto fancy myself sitting in a dark old upper room in some remote\nfarm-house, at the close of a winter day, after three or four hours of\nsteady reading or writing. The room is full of books--the _best_\nbooks. There is a little fire on the hearth, there is a dingy curtain\nat the window. It is solitary and still, and when the light gets too\nscant to let me read any more, I fill my pipe, and go and stand in the\nwindow. Outside, there is a row of leafless elms, and beyond that a\ndim, wide landscape of lakes and hills, and beyond that a red, windy\nsunset. I can sit in that window and smoke my pipe and have my own\nthoughts till the hills grow black. There is no one to say to me 'Go'\nor 'Come'; no patient to visit; no confounded case on the docket next\nmorning at nine; no distasteful, mean, slavish job of any kind. How\ncan I fail to have thoughts worth the thinking, and to live a rich and\nfree life when I breathe every day the bracing air of nature and the\ngreat poets? Isn't such a life in itself the best kind of success,\neven if a man accomplishes nothing in particular that you can put your\nhand on?\"\n\n\"Yes, I know,\" said Armstrong, taking a long breath. \"I have felt that\nway too. But a man has got to put all that sternly behind him and do\nthe world's work for the world's wages, if he means to amount to\nanything. It's only a finer kind of self-indulgence, after\nall--egoistic Hedonism and that sort of thing.\"\n\n\"It won't be all standing at windows and looking at sunsets,\" added\nDoddridge. \"Has it ever occurred to you that, before entering on a\nlife of self-denial and devotion to rather vague ideals, a man ought\nto be mighty sure of himself? Can you keep up the culture business\nwithout growing in on yourself unhealthily, and then getting sick of\ninaction? Don't you think there will be times of disappointment and\ndoubt when you look around and see fellows without half your talents\ngetting ahead of you in the world?\"\n\n\"Of course,\" answered Clay, \"I shall have to make sacrifices, and\nI shall have to stick to them when made. But there have always\nbeen plenty of people willing to make similar sacrifices for\nsimilar compensations. Men have gone out into the wilderness or\nshut themselves up in the cloister for opportunities of study or\nself-communion, or for other objects which were perhaps at bottom no\nmore truly devotional than mine. Nowadays such opportunities may be\nhad by any man who will keep himself free from the servitude of a\nbread-winning profession. It is not necessary now to cry _Ecce in\ndeserto_ or _Ecce in penetralibus_. Oh, I shall have my dark days; but\nwhenever the blue devils get thick I shall take to the woods and\nreturn to sanity.\"\n\n\"You mean to live in the country, then?\" I inquired.\n\n\"Yes; most of the time, at any rate. Nature is fully half of life to\nme.\"\n\nAgain there was a pause.\n\n\"Well, you next, Polisson,\" said Armstrong, finally. \"Let's hear what\nyour programme is.\"\n\n\"Oh, nothing in the least interesting,\" I replied. \"My future is all\ncut and dried. I shall spend the next two years in the south of\nFrance--mainly at Lyons--to learn the details of the silk manufacture.\nThen I shall come home to go into my father's store for a year as a\nclerk in the importing department. At the close of that year the\ngovernor will take me in as junior partner, and I shall marry my\nsecond cousin. We shall live with my parents, and I am going to be\nvery domestic, though, as a matter of form, I shall join one or two\nclubs. I shall go down town every morning at nine, and come up at\nfive.\"\n\n\"Quite a neat little destiny,\" said Armstrong. \"I wish I had your\nbacking. Come, Dodd, what's yours? You're the only man left.\"\n\n\"I haven't made up my mind yet,\" said Doddridge, slowly.\n\nHe was a large, spare man, with a swarthy skin, a wide mouth, a dark,\nsteady eye, and a long jaw. There was an appearance of power and will\nabout him which was well borne out by his character. He had been a\nsystematic though not a laborious student, and while maintaining a\nstand comfortably near the head of the class, had taken a course in\nthe Law School during Senior year, doing his double duties with\napparent ease. He was a constant speaker in the debates of the\nLinonian Society, and the few who attended the meetings of that\nmoribund school of eloquence spoke of Doddridge's speeches as oases\nin the waste of forensic dispute, being always distinguished by vigor\nand soundness, though without any literary quality, such as Clay's\noccasional performances had. Berkeley, who covered his own lazy and\nmiscellaneous reading with the mask of eclecticism, and proclaimed\nhis disbelief in a prescribed course of study, was wont to say\nthat Doddridge was the only man that he knew who was using the\nopportunities given by the college for all they were worth, and really\ngetting out of \"the old curric\" that mental discipline which it\nprofessed to impart. Though rather taciturn, he was not unsocial, and\nwas fond of his pipe in the evening. He liked a joke, especially if\nit was of a definite kind, and at some one's expense touching a\ncharacteristic weakness of the man. There was at bottom something a\nlittle hard about him, though every one agreed that he was a good\nfellow. We all felt sure that he would make a distinguished success in\npractical life; and we doubtless thought--if we thought about it at\nall--that with his clear foresight and habits of steady work, he had\nalready decided upon his career. His words were therefore a surprise.\n\n\"What! you don't mean to say that you are going to drift, Dodd?\"\ninquired Armstrong.\n\n\"Drift? Well, no; not exactly. I shall keep my steering apparatus well\nin hand, but I haven't decided yet what port to run for. There's no\nhurry. I have an uncle in the Northwest in the lumber business, who\nwould give me a chance. I may go out there and look about awhile at\nfirst. If it doesn't promise much, there is the law to fall back upon.\nMy father has a fruit farm at Byzantium in western New York,--where I\ncome from, you know,--and he is part owner of the Byzantium weekly\n'Bugle.' I've no doubt I could get on as editor, and go to the\nLegislature. Or I might do worse than begin on the farm; farming is\nlooking up in that section. I may try several things till I find the\nright one.\"\n\n\"That's queer,\" said Armstrong. \"I thought you had made up your mind\nto enter the Columbia Law School.\"\n\n\"Hardly,\" answered Doddridge, \"though I may, after all. The main point\nis to keep yourself in readiness for any work, and take the best thing\nthat turns up--like Berkeley here,\" he added, drily.\n\nArmstrong looked at his watch and remarked that it was nearly\nmidnight.\n\n\"Boys,\" said I, \"in fifteen years from to-night let's have a supper\nhere and see how each man of us has worked out his theory of life, and\nhow he likes it as far as he has got.\"\n\n\"Oh, give us twenty,\" said Doddridge, laughing, as we all arose and\nprepared to break up. \"No one accomplishes anything in this latitude\nbefore he is forty.\"\n\n       *       *       *       *       *\n\nIt was in effect just fifteen years from the summer of our graduation\nthat I started out to look up systematically my quondam classmates and\ncompare notes with them. The course of my own life had been quite\nother than I had planned. For one thing, I had lived in New Orleans\nand not in New York, and my occasions had led me seldom to the North.\nThe first visit I paid was to Berkeley. I had heard that he was still\nunmarried, and that he had been for years settled, as minister, over a\nsmall Episcopal parish on the Hudson. The steamer landed me one summer\nafternoon at a little dock on the west bank; and after obtaining from\nthe dock-keeper precise directions for finding the parsonage, I set\nout on foot. After a walk of a mile along a road skirted by handsome\ncountry seats, but contrasting strangely in its loneliness with the\nbroad thoroughfare of the river constantly occupied by long tows of\nbarges and rafts, I came to the rectory gate. The house was a stone\ncottage, covered with trailers, and standing well back from the road.\nIn the same inclosure, surrounded by a grove of firs, was a little\nstone chapel with high pitched roof and rustic belfry. In front of the\nhouse I spied a figure which I recognized as Berkeley. He was in his\nshirt-sleeves, and was pecking away with a hoe at the gravel walk,\nwhistling meanwhile his old favorite \"Bonny Doon.\" He turned as I came\nup the driveway, and regarded me at first without recognition. He, for\nhis part, was little changed by time. There was the same tall,\nnarrow-shouldered, slightly stooping figure; the face, smooth-shaved,\nwith a spot of wintry red in the cheek, and the old humorous cast in\nthe small blue eyes.\n\n\"You don't know me from Adam,\" I said, pausing in front of him.\n\n\"Ah!\" he exclaimed, directly. \"Polisson, old man, upon my conscience\nI'm glad to see you, but I didn't know you till you spoke. You've been\nhaving the yellow fever, haven't you? Come in--come into the house.\"\n\nWe passed in through the porch, which was covered with sweet-pea vines\ntrained on strings, and entered the library, where Berkeley resumed\nhis coat. The room was lined with book-shelves loaded to the ceiling,\nwhile piles of literature had overflowed the cases and stood about on\nthe floor in bachelor freedom. After the first greetings and\ninquiries, Berkeley carried my valise upstairs, and then returning,\nsaid:\n\n\"I'm a methodical though not methodistical person, or rather parson\n(excuse the Fullerism); and as you have got to stay with me till I let\nyou go, that is, several days at the least (don't interrupt), I'll\nkeep a little appointment for the next hour, if you will excuse me. A\nboy comes three times a week to blow the bellows for my organ\npractice. Perhaps you would like to step into the church and hear me.\"\n\nI assented, and we went out into the yard and found the boy already\nwaiting in the church porch. Berkeley and his assistant climbed into\nthe organ loft, while I seated myself in the chancel to listen. The\ninstrument was small but sweet, and Berkeley really played very well.\nThe interior of the little church was plain to bareness; but the sun,\nwhich had fallen low, threw red lights on the upper part of the\nundecorated walls, and rich shadows darkened the lower half. Through\nthe white, pointed windows I saw the trembling branches of the firs. I\nhad been hurrying for a fortnight past over heated railways, treading\nfiery pavements, and lodging in red-hot city hotels. But now the music\nand the day's decline filled me with a sense of religious calm, and\nfor a moment I envied Berkeley. After his practicing was over the\norganist locked the chapel door, and we paced up and down in the\nfir-grove on the matting of dark red needles, and watched the river,\nwhose eastern half still shone in the evening light. After supper we\nsat out on the piazza, which commanded a view of the Hudson. Berkeley\nopened a bottle of Chablis and produced some very old and dry Manilla\ncheroots, and, leaning back in our wicker chairs, we proceeded to\n\"talk Cosmos.\"\n\n\"You are very comfortably fixed here,\" I began; \"but this is not\nprecisely what I expected to find you doing, after your declaration of\nprinciples, fifteen years ago, you may remember, on our Commencement\nnight.\"\n\n\"Fifteen years! So it is--so it is,\" he answered, with a sigh. \"Well,\n_l'homme propose_, you know. I don't quite remember what it was that\nI said on that occasion: dreadful nonsense, no doubt. As Thackeray\nsays, a boy _is_ an ass. Whatever it was, it proceeded, I suppose,\nfrom some temporary mood rather than from any permanent conviction;\nthough, to be sure, I slipped into this way of life almost by accident\nat first. But, being in, I have found it easy to continue. I am rather\ntoo apt, perhaps, to stay where I am put. I am a quietist by\nconstitution.\" He paused, and I waited for him to enter upon a fuller\nand more formal apology. Finally, he went on much as follows:\n\n\"Just after I left college I made application through some parties at\nWashington for a foreign consulate. While I was waiting for the\napplication to be passed on (it was finally unsuccessful), I came up\nhere to visit my uncle, who was the rector of this parish. He was a\nwidower, without any children, and the church was his hobby. It is a\nqueer little affair, something like the old field-kirks or chapels of\nease in some parts of England. It was built partly by my uncle and\npartly by a few New York families who have country places here,\nand who use it in the summer. This is all glebe land,\" he said,\nindicating, with a sweep of his hand, the twilight fields below the\nhouse sloping down toward the faintly glimmering river. \"My uncle had\na sort of prescription or lien by courtesy on the place. There's not\nmuch salary to speak of, but he had a nice plum of his own, and lived\ninexpensively. Well, that first summer I moped about here, got\nacquainted with the summer residents, read a good deal of the time,\ntook long walks into the interior,--a rough, aboriginal country, where\nthey still talk Dutch,--and waited for an answer to my application.\nWhen it came at last, I fretted about it considerably, and was for\nstarting off in search of something else. I had an idea of getting a\nplace as botanist on Coprolite's survey of the Nth parallel, and I\nwrote to New Haven for letters. I thought it would be a good outdoor,\nhorseback sort of life, and might lead to something better. But that\nfell through, and meanwhile the dominie kept saying: 'My dear fellow,\ndon't be in too much of a hurry to begin. Young America goes so fast\nnowadays that it is like the dog in the hunting story,--a _leetle_ bit\nahead of the hare. Why not stay here for awhile and ripen--ripen?' The\ndominie had a good library,--all my old college favorites, old Burton,\nold Fuller, and Browne, etc., and it seemed the wisest course to\nfollow his advice for the present. But in the fall my uncle had a\nslight stroke of paralysis, and really needed my help for awhile; so\nthat what had been a somewhat aimless life, considered as loafing,\nbecame all at once a duty. At first he had a theological student, from\nsomewhere across the river, come to stay in the house and read service\nfor him on Sundays. But he was a ridiculous animal, whose main idea of\na minister's duties was to intone the responses in a sonorous manner.\nHe used to practice this on week days in his surplice, and I remember\nespecially the cadence with which he delivered the sentence: 'Yea,\nlike a broken _wall_ shall ye be and as a ruined _hedge_.'\n\n\"He got the huckleberry, as we used to say in college, on that\nparticular text, and it has stuck by me ever since. The dominie fired\nhim out after a fortnight, and one day said to me: 'Jack, why don't\n_you_ study for orders and take up the succession here? You are a\nbookworm, and the life seems to be to your liking.' Of course, I\ndeclined very vigorously in the beginning, though offering to stay on\nso long as the dominie needed my help. I used to do lay reading on\nSundays when he was too feeble. Gradually, 'the idea of the life did\nsweetly creep into my study of imagination.' The quaintness of the\nplace appealed to me. And here was a future all cut out for me: no\npreliminary struggle, no contact with vulgar people, no cut-throat\ncompetition, but everything gentlemanly and independent about it. I\nhad strong doubts touching my theology, and used to discuss them with\nmy uncle; but he said,--and said rightly, I now think,--'You young\nfellows in college fancy that it's a mighty fine, bold thing to effect\nradicalism and atheism, and the Lord knows what all; but it won't\nstick to you when you get older. Experience will soften your heart,\nand you'll find after awhile that belief and doubt are not matters of\nthe pure reason, but of the will. It is a question of _attitude_.\nBesides, the church is broad enough to cover a good many private\ndifferences in opinion. It isn't as if you were going to be a\nblue-nosed Presbyterian. You can stay here and make your studies with\nme, instead of going into a seminary, and when you are ready to go\nbefore the bishop I'll see that you get the right send-off.' In short,\nhere I am! My uncle died two years after, when I was already in\norders, and I've been here ever since.\"\n\n\"I should think you would get lonely sometimes, and make a strike for\na city parish,\" I suggested.\n\n\"Why--no, I don't think I should care for ordinary parish work. The\nbeauty of my position here is its uniqueness. In winter I keep the\nchurch open for the Aborigines till they get snowed up and stop\ncoming, and then I put down to New York for a month or two of work at\nthe Astor Library. Last winter I held service for two Sundays running\nwith one boy for congregation. Finally I announced to him that the\nchurch would be closed until spring.\"\n\n\"What in the----: well, what do you find to do all alone up here?\"\n\n\"Oh, there's always plenty to do, if you'll only do it. I've been\ncultivating some virtuosities, among other things. Remind me to show\nyou my etchings when we go in. Did you notice, perhaps, that little\nhead over the table, on the north wall? No? Then I smatter botany\nsome. I'll let you look over my _hortus siccus_ before you go. It has\nsome very rare ferns; one of them is a new species, and Fungus--who\nexchanges with me--swore that he was going to have it named after me.\nI sent the first specimen to have it described in his forthcoming\nreport. But doubtless all this sort of thing is a bore to you. Well,\nlately I have been going into genealogy, and I find it more and more\nabsorbing. Those piles of blank-books and manuscripts on the floor at\nthe south end are all crammed with genealogical notes and material.\"\n\n\"I should think you would find it pretty dry fodder,\" I said.\n\n\"That is because you take an outside, unsympathetic view of it. Now,\nto an amateur it's anything but dry. There is as much excitement in\nhunting down a missing link in a pedigree that you have been on the\ntrail of for a long time, as there is in the chase of any other kind\nof game.\"\n\n\"Do you ever get across the water? Travel, if I remember right, played\na large part in your scheme of life once.\"\n\n\"Yes; I've been over once, for a few months. But my income, though\nvery comfortable for the statics of existence, is rather short for the\ndynamics, and so I mostly stay at home.\"\n\n\"Did you meet any interesting people over there? Any of the crowned\nheads, famous wits, etc., whom you once proposed to cultivate?\"\n\n\"No; nobody in particular. I went in a very quiet way. I had some good\nletters to people in England, but I didn't present them. The idea of\nintroductions became a bore as I got nearer to it.\"\n\n\"And, of course, you didn't elope with the marquise?\"\n\n\"Was that in my scheme? Well--no, I did not.\"\n\n\"You might have done worse, old man. You ought to have a wife, to keep\nyou from getting rusty up here. And, besides, a fellow that goes so\nmuch into genealogy should take some interest in posterity. You ought\nto cultivate the science practically.\"\n\n\"Oh, I'm past all danger of matrimony now,\" said Berkeley, with a\nlaugh. \"There was a girl that I was rather sweet on a few years ago. I\nwas looking up a pedigree for her papa, and I found that I was related\nto her myself, in eight different ways, though none of them very near.\nI explained it to her one evening. It took me an hour to do it, and I\nfancy she thought it a little slow. At all events, when I afterward\nhinted that we might make the eight ways nine, she answered that our\nrelationship was so intricate already that she couldn't think of\ncomplicating it any further. No, you may put me down as safe.\"\n\nAfter this, we sat listening in silence to the distant beat of\npaddle-wheels where a steamer was moving up river.\n\n\"The river is a deal of company,\" resumed my host. \"Thirty-six\nsteamers pass here every twenty-four hours. That now is the _Mary\nPowell_.\"\n\n\"Well,\" I said, answering not so much to his last remark as to the\nwhole trend of his autobiography, \"I suppose you are happy in this\nway of life, since you seem to prefer it. But it would be terribly\nmonotonous to me.\"\n\n\"Happy?\" replied Berkeley, doubtfully. \"I don't know. Happiness is a\nsubjective matter. You _are_ happy if you think yourself so. As for\nme, I cultivate an obsolete mood--the old-fashioned humor of\nmelancholy. I don't suppose now that a light-hearted, French kind of\nchap like you can understand, in the least, what those fine, crusty\nold Elizabethans meant when they wrote,\n\n     'There's naught in this life sweet,\n     If man were wise to see't,\n     But only melancholy.'\n\nThis noisy generation has lost their secret. As for me, I am content\nwith the grays and drabs. I think the brighter colors would disturb my\nmood. I know it's not a large life, but it is a safe one.\"\n\nI did not at the moment remember that this had been Armstrong's very\nsaying fifteen years ago, but some unconscious association led me to\nmention him.\n\n\"Armstrong and you have changed places in one respect, I should\nthink,\" said I. \"He is keeping a boarding-school somewhere in\nConnecticut. And instead of leading a Tulkinghorny existence in the\nNew York University building, as he firmly intended, he has married\nand produced a numerous offspring, I hear.\"\n\n\"Yes, poor fellow!\" said Berkeley; \"I fancy that he is dreadfully\noverrun and hard up. There always was something absurdly domestic\nabout Armstrong. They say he has grown red, fat, and bald. Think of a\nman with Armstrong's education--and he had some talent, too--keeping a\nsort of Dotheboys Hall! I haven't seen him for eight or nine years.\nThe last time was at Jersey City, and I had just time to shake hands\nwith him. He was with a lot of other pedagogues, all going up to a\nteachers' convention, or some such dreary thing, at Albany.\"\n\nI had an opportunity for verifying Berkeley's account of Armstrong a\nfew days after my conversation with the former. The Pestalozzian\nInstitute, in the pleasant little village of Thimbleville, was\nsituated, as its prospectus informed the public, on \"one of the most\nelegant residence streets, in one of the healthiest and most beautiful\nrural towns of Eastern Connecticut.\" Over the entrance gate was a\nRoman arch bearing the inscription \"Pestalozzian Institute\" in large\ngilt letters. The temple of learning itself was a big, bare, white\nhouse at some distance from the street, with an orchard and kitchen\ngarden on one side, and a roomy play-ground on the other. The latter\nwas in possession of some small boys, who were kicking a broken-winded\nfoot-ball about the field with an amount of noise greatly in excess of\nits occasion. To my question where I could find Mr. Armstrong, they\nanswered eagerly: \"Mr. Armstrong? Yes, sir. You go right into the\nhall, and knock on the first door to the right, and he'll come--or\nsome one.\"\n\nThe door to the large square entry stood wide open, and through\nanother door opposite, which was ajar, I saw long tables, and heard\nthe clatter of dishes being removed, while a strong smell of dinner\nfilled the air. I knocked at the door on the right, but no one\nappeared. Finally, a chubby girl of about ten summers came running\nround the corner of the house and into the front door. She was eating\nan apple, and gazed at me wonderingly.\n\n\"Is Mr. Armstrong in?\" I asked.\n\n\"Yes, sir; he's about somewhere. Walk into the parlor, please, and sit\ndown, and I'll find him.\"\n\nI entered the room on the right, which was a bleak and\nofficial-looking apartment,--apparently the reception-room where\nparents held interviews with the instructor of youth, or tore\nthemselves from the parting embraces of homesick sons at the beginning\nof a new term. There is always something depressing about the parlor\nof an \"institution\" of any kind, and I could not help feeling sorry\nfor Armstrong, as I waited for him, seated on a sofa covered with\nfaded rep. At length the door of an inner room opened, and the\nprincipal of the Pestalozzian Institute waddled across the floor\nwith his hand held out, crying:\n\n\"Franky Polisson, how are you?\"\n\nHe certainly had grown stout, and his light hair had retreated from\nthe forehead. He wore glasses and was dressed in a suit of rusty\nblack, with a high vest which gave him a ministerial look--a much more\nministerial look than Berkeley had. His pantaloons presented that\nappearance which tailors describe as \"kneeing out.\" He sat down and we\nchatted for half an hour. The little girl had followed him into the\nroom, and behind her came another three or four years her junior. The\nolder one stood by his side, and he kept his arm around her, while he\nheld the younger on his knee. They were both pretty, healthy-looking\nchildren, and kept their eyes fixed on \"the man.\"\n\n\"Are those your own kids?\" I inquired presently.\n\n\"Yes, two of them. I have six, you know,\" he answered, with a fond\nsigh: \"five girls and one boy. The lasses are rather in the majority.\"\n\n\"I heard you were quite a _paterfamilias_,\" I said. \"Won't you come\nand kiss me, little girl?\"\n\nTo this proposal the elder answered by burying her head bashfully in\nher father's shoulder, while the smaller one simply opened her eyes\nwider and stared with more fixed intensity.\n\n\"Oh, by the way,\" exclaimed Armstrong, \"of course you'll take tea with\nus and spend the evening. I wish I could offer to sleep you here; but\nthe fact is, Mrs. Armstrong's sister is with us for a few days, and\nthe parents of one of my boys, who is sick, are also staying here; so\nthat my guest chambers are full.\"\n\n\"Don't mention it,\" I said. \"I couldn't stay over night. I've got to\nbe in New York in the morning, and must take the nine-o'clock train.\nBut I'll stay to supper and much obliged, if you are sure I sha'n't\ntake up too much of your time.\"\n\n\"Not the least--not the least. This is a half holiday, and nothing in\nparticular to do.\" He bustled to the door and called out loudly,\n\"Mother! Mother!\"\n\nThere was no response.\n\n\"Nelly,\" he commanded, \"run and find your mamma, and tell her that Mr.\nPolisson--from New Orleans--an old classmate of papa's, will be here\nto tea. That's a good girl. Polisson, put on your hat and let's go\nround the place. I want to show you what an establishment I've got\nhere.\"\n\nWe accordingly made the tour of the premises, Armstrong doing the\ncicerone impressively, and every now and then urging me with emphatic\nhospitality to come and spend a week--a fortnight--longer, if I chose,\nduring the summer vacation.\n\n\"Bring Mrs. Polisson and the kids. Bring 'em all,\" he said. \"It will\ndo them good; the air here is fine; eleven hundred feet above the sea.\nNo malaria--no typhoid. I laid out four hundred dollars last year on\nsewerage.\"\n\nIt being a half holiday, most of the big boys had gone to a pond in\nthe neighborhood for a swim, under the conduct of the classical\nmaster,--a Yale graduate, Armstrong explained, who had stood fourth\nin his class, \"and a very able fellow,--very able.\"\n\nBut while we sat at tea in Armstrong's family dining-room, which\nadjoined the school commons, we were made aware of the return of the\nswimming party by the constant shuffle and tramp of feet through the\nhall and the noise of feeding in the next room. At our table were\npresent Mrs. Armstrong, her sister (who had a frightened air when\naddressed and conversed in monosyllables), the parents of the sick\npupil, and Armstrong's two eldest children. I surmised that the\nyounger children had been in the habit of sharing in the social meal,\nand had been crowded out on this occasion by the number of guests; for\nI heard them _fremunt_ing _in carcere_ behind a door through which\nthe waitress passed out and in, bringing plates of waffles. The\nremonstrances of the waitress were also audible, and, when the wailing\nrose high, my hostess's face had a distrait expression, as of one\nprepared at any moment for an irruption of infant Goths.\n\nMrs. Armstrong was a vivacious little woman, who, I conjectured, had\nonce been a village belle, with some pretensions to _espieglerie_ and\nthe fragile prettiness common among New England country girls. But the\nbearing and rearing of a family of children, and the matronizing of a\nhouseful of hungry school-boys in such a way as to make ends meet, had\nsubstituted a faded and worried look for her natural liveliness of\nexpression. She bore up bravely, however, against the embarrassments\nof the occasion. In particular, it pleased her to take a facetious\nview of college life.\n\n\"Oh, Mr. Polisson,\" she cried, \"I am afraid that you and my husband\nwere very gay young men when you were at college together. Oh, don't\ntell me; I know--I know. I've heard of some of your scrapes.\"\n\nI protested feebly against this impeachment, but Armstrong winked at\nme with the air of a sly dog, and said:\n\n\"It's no use, Polisson. You can't fool Mrs. A. Buckingham and one or\ntwo of the fellows have been here to dinner occasionally, and I'm\nafraid they've given us away.\"\n\n\"Yes,\" she affirmed, \"Mr. Buckingham was one of you too, I guess,\nthough he _is_ the Rev. Mr. Buckingham now. Oh, he has told me.\"\n\n\"You remember old Buck?\" put in Armstrong. \"He is preaching near\nhere--settled over a church at Bobtown.\"\n\n\"Yes,\" I answered, \"I remember there was such a man in the class, but\nreally I didn't know that he was--ah--such a character as you seem to\ninfer, Mrs. Armstrong.\"\n\n\"Oh, he has quieted down now, I assure you,\" said the lady. \"He is as\nprim and proper as a Methodist meeting-house. Why, he _has_ to be, you\nknow.\"\n\nThis amusing fiction of the wildness of Armstrong's youth had\nevidently become a family tradition, and even, by a familiar process,\nan article of belief in his own mind. It reminded me grotesquely of\n_Justice Shallow's_ reminiscences with _Sir John Falstaff_: \"Ha,\nCousin Silence, that thou hadst seen that, that this knight and I have\nseen.... Jesu, Jesu, the mad days that I have spent!\"\n\nThe resemblance became still stronger when, as we rose from the table,\nthe good fellow beckoned me into a closet which opened off the\ndining-room, saying, in a hoarse whisper:\n\n\"Here, Polisson, come in here.\"\n\nHe was uncorking a large bottle half-filled with some red liquid, and\nas he poured a portion of this into two glasses he explained:\n\n\"I don't have this sort of thing on the table, you understand, on\naccount of the children and my--ah--position. It would make talk.\nBut I tell you this is some of the real old stuff. How!\" And he\nheld his glass up to the light, regarding it with the one eye of a\nconnoisseur, and then drank down its contents with a smack. I was\nconsiderably astonished, on doing the same, to discover that this dark\nbeverage--which, from Armstrong's manner, I had been prepared to find\nsomething at least as wicked as absinthe--was simply and solely\nBordeaux of a mild quality. After this Bacchanalian proceeding we went\nout into the orchard, which was reserved for family use, and sat on a\nbench under an apple-tree. Armstrong called his little boy who had\nbeen at supper with us and gave him a whispered message, together\nwith some small change. The messenger disappeared, and after a short\nabsence returned with two very domestic cigars, transparently bought\nfor the nonce from some neighboring grocer. \"Have a smoke,\" commanded\nmy host, and we solemnly kindled the rolls of yellow leaf, Armstrong\npuffing away at his with the air of a man who, though intrusted by\ndestiny with the responsibility of molding the characters of youth,\nhas not forgotten how to be a man of the world on occasion.\n\n\"Well, Charley,\" I began, after a few preliminary draughts, \"you seem\nto have a good thing of it. Your school is prosperous, I understand;\nthe work suits you; you have a mighty pretty family of children\ngrowing up, and your health appears to be perfect.\"\n\n\"Yes,\" he admitted; \"I suppose I ought to be thankful. I certainly\nenjoy great mercies. It's a warm, crowded kind of life; plenty of\naffection,--plenty of anxiety too, to be sure. I like to have the boys\naround me; it keeps one's heart fresh, though in a way it's sometimes\nwearing to the nerves. Yes, I like the young rascals--I like them.\nBut, of course, it has its drawbacks. Most careers have,\" he added, in\na burst of commonplace.\n\n\"It is not exactly the career that you had cut out for yourself,\" I\nsuggested, \"when we talked our plans over, you remember, that last\nevening at New Haven.\"\n\n\"No, it's not,\" he acknowledged; \"but perhaps it is a better one. What\nwas it I said then? I really don't recall it. Something very silly,\nno doubt.\"\n\n\"Oh, you said, in a general way, that you were going in for money and\ncelibacy and selfishness,--just as you have _not_ done.\"\n\n\"Yes, yes; I know, I remember now,\" he said, laughing. \"Boys are great\nfools with their brag of what they are going to do and be. Life knocks\nit out of them fast enough; they learn to do what they must.\"\n\n\"Do you ever write any poetry nowadays?\"\n\n\"No, no; not I. The muse has given me the go-by completely. Except for\nsome occasional verses for a school festival or something of the kind,\nwhich I grind out now and then, I've sunk my rhyming dictionary deeper\nthan ever plummet sounded. The chief disadvantage of running a big\nschool like this,\" he continued, with a sigh, \"is the want of leisure\nand retirement to enable a man to keep up his studies. Sometimes I\nactually ache for solitude--for a few weeks or months of absolute\nloneliness and silence. Mrs. Armstrong has fixed me up a nice little\nprivate study,--remind me to take you in there before you go,--where I\nkeep my books, etc. But the children will find their way in, and then\nI'm seldom undisturbed anywhere for more than an hour at a time;\nthere's always some call on me,--something wanted that no one else can\nsee to.\"\n\n\"You ought to swap places with Berkeley for awhile. He's got more\nleisure than he knows what to do with.\"\n\n\"Berkeley! Well, what's he up to now? Philately? Arboriculture? What's\nhis last fad? You've seen him lately, you said. I met him for a minute\nin New York, a few years ago, and he told me he was going to an old\nbook auction.\"\n\n\"He's got genealogy at present,\" I explained.\n\n\"Genealogy! What hay! What sawdust! Aren't there enough live people to\ntake an interest in, without grubbing up dead ones from tombstones\nand town clerks' records? Berkeley must be a regular old bachelor\nantiquary by this time, with all human sympathy dried out of him. No,\nI wouldn't change with _him_. Would we, fatty?\" he said, appealing to\na small offspring of uncertain sex which had just toddled out the door\nand across the gangway to kiss its papa good-night.\n\nI took leave of Armstrong and his interesting family with a sense of\nincreased liking. His worldliness, good nature, and simple little\nenthusiasms and self-satisfactions had somehow kept him young, and he\nseemed quite the old Armstrong of college days. I afterward learned\nthat the excellent fellow had just finished his law studies, and was\npreparing to enter upon practice, when his father's health failed,\nforcing him to give up his parish, and leaving a number of younger\nbrothers and sisters partly dependent on Armstrong. He had accordingly\ntaken the first situation that promised a fair salary, and, having got\nstarted upon the work of teaching, had been unable to let go until it\nwas too late; had, indeed, got deeper and deeper in, by falling in\nlove and impulsively marrying at the first opportunity, and finally\nsetting up for himself at the Pestalozzian Institute. Poor fellow!\nGood fellow! _Amico mio, non della fortuna._\n\nMy next call was upon Clay, who had rooms in the Babel building in New\nYork, and was reported to be something of a Bohemian. He received me\nin a smoking jacket and slippers. He had grown a full beard which hid\nhis finely cut features. His black eyes had the old fire, but his\nskin was sallower, and I thought that his manner had a touch of\nlistlessness mingled with irritability and defiance. He was glad to\nsee me; but inclined to be at first, not precisely distant, yet by\nno means confidential. After awhile, however, he thawed out and\nbecame more like the Clay whom I remembered--our college genius, the\nbrilliant, the admired, in those days of eager hero-worship. I told\nhim of my visits to Berkeley and Armstrong.\n\n\"Berkeley I see now and then in town,\" said Clay. \"It was rather queer\nof him to turn parson, but I guess he doesn't let his theology bother\nhim much. He has a really superior collection of etchings, I am told.\nArmstrong I haven't seen for years. I knew he was a pedagogue\nsomewhere in Connecticut.\"\n\n\"Don't you ever go to the class reunions?\" I asked.\n\n\"Class reunions? Well, hardly.\"\n\n\"I should think you would; you are so near New Haven.\"\n\n\"How charmingly provincial you are--you Southern chaps! Don't you know\nthat, to a man who lives in New York, nothing is near? Besides, as to\nmy classmates at old Yale and all that, I would go round a corner to\navoid meeting most of them.\"\n\nI expressed myself as duly shocked by this sentiment, and presently I\ninquired:\n\n\"Well, Clay, how are you getting on, anyway?\"\n\n\"That's a d---- general question. How do you want me to answer it?\"\n\n\"Oh, not at all, if you don't like.\"\n\n\"Well, don't get miffed. Suppose I answer, 'Pretty well, I thank you,\nsir.' How will that do?\"\n\n\"Are you writing anything now?\"\n\n\"I'm always scribbling something or other. At present, I've got the\nposition of dramatic critic on the 'Daily Boreas,' which is not a very\nbad bore, and keeps the pot boiling. And I do more or less work of a\nhack kind for the magazines and cyclopedias, etc.\"\n\n\"I thought you were on the 'Weekly Prig.' Berkeley or somebody told me\nso.\"\n\n\"So I was at one time, but I got out of it. The work was drying me\nup too fast. The concern is run by a lot of cusses who have failed\nin various branches of literature themselves, and undertake, in\nconsequence, to make it unpleasant for every one else who tries to\nwrite anything. I got so that I could sling as cynical a quill as the\nrest of them. But the trick is an easy one and hardly worth learning.\nIt's a great fraud, this business of reviewing. Here's a man of\nlearning, for instance, who has spent years of research on a\nparticular work. He has collected a large library, perhaps, on his\nsubject; knows more about it than any one else living. Then along\ncomes some insolent little whipper-snapper,--like me,--whose sole\nknowledge of the matter in hand is drawn from the very book that he\npretends to criticise, and patronizes the learned author in a book\nnotice. No, I got out of it; I hadn't the cheek.\"\n\n\"I bought your book,\"[A] said I, \"as soon as it came out.\"\n\n[Footnote A: Dialogues and Romances. By E. Clay. New York: Pater &\nSons, 1874.]\n\n\"That's more than the public did.\"\n\n\"Yes, and I read it, too.\"\n\n\"No! Did you, now? That's true friendship. Well, how did you like it?\nDid you get your money's worth?\"\n\nI hesitated a moment and then answered:\n\n\"It was clever, of course. Anything that you write would be sure to be\nthat. But it didn't appear to get down to hard-pan or to take a firm\ngrip on life--did it?\"\n\n\"Ah, that's what the critics said,--only they've got a set of phrases\nfor expressing it. They said it was amateurish, that it was in a\nfalsetto key, etc.\"\n\n\"Well, how does it strike you, yourself? You know that it didn't come\nout of the deep places of your nature, don't you? You feel that you've\ngot better behind?\"\n\n\"Oh, I don't know. A man does what he can. I rather think it's the\nbest I can do at present.\"\n\n\"Why don't you go at some more serious work; some _magnum opus_ that\nwould bring your whole strength into play?\"\n\n\"A _magnum opus_, my dear fellow!\" replied Clay, with a shade of\nirritation in his voice. \"You talk as if a _magnum opus_ could be done\nfor the wishing. Why don't _you_ do a _magnum opus_, then?\"\n\n\"Why don't _I_? Oh, I'm not a literary fellow--never professed to be.\nWhat a question!\"\n\n\"Well, no more am I, perhaps. I don't think any better of the stuff\nthat I scribble than you do. It's all an experiment with me. I'm\ntrying my brushes--trying my brushes. Perhaps I may be able to do\nsomething stronger some day, and perhaps not. But at all events I\nsha'n't force my mood. I shall wait for my inspiration. One thing I've\nnoticed, that as a man grows older he loses his spontaneity and gets\nmore critical with himself. I could do more, no doubt, if I would only\nlet myself go. But I'm like this meerschaum here,--a hard piece and\nslow in coloring.\"\n\n\"Well, meanwhile you might do something in the line of scholarship, a\nhistory or a volume of critical essays--'Hours with the Poets,' or\nsomething of that kind, that would bring in the results of your\nreading. Have you seen Brainard's book? It seemed to me work that was\nworth doing. But you could do something of the same kind, only much\nbetter, without taking your hands out of your pockets.\"\n\nBrainard was a painstaking classmate of ours, who had been for some\nyears Professor of Mental and Moral Philosophy, English Literature,\nand European History, in a Western university, and had recently\npublished a volume entitled \"Theism and Pantheism in the Literature of\nthe English Renaissance,\" which was well spoken of, and was already in\nits third edition.\n\n\"Yes, I've seen the stuff,\" said Clay. \"My unhappy country swarms with\nthat sort of thing: books about books, and books about other books\nabout books--like the big fleas and little fleas. It's not literature;\nit's a parasitic growth that infests literature. I always say to\nmyself, with the melancholy Jaques, whenever I have to look over a\nbook by Brainard or any such fellow, 'I think of as many matters as\nhe; but I give Heaven thanks and make no boast of them.' No, I don't\ncare to add anything to that particular rubbish heap. You know Emerson\nsaid that the worst poem is better than the best criticism of it. The\ntrouble with me is that what I want to do I can't do--at present; what\nI can do I don't think it worth while to do--worth my while, at\nleast. Some one else may do it and get the credit and welcome.\"\n\n\"But you do a good deal of work that you don't care about, as it is,\"\nI objected.\n\n\"Of course. A man must live, and so I do the nearest thing and the one\nthat pays quickest. I got eighty dollars, now, for that last screed in\n'The Reservoir.'\"\n\n\"But,\" I persisted, \"I thought that money-making had no part in your\nscheme. You could make more money in a dozen other businesses.\"\n\n\"So I could,\" he answered; \"but they all involve some form of slavery.\nNow, I am my own master. After all, every profession has its drudgery,\nand literary drudgery is not the worst.\"\n\n\"Well,\" I conceded, \"independent of what you accomplish, I suppose\nyour way of life furnishes as many daily satisfactions as any. I\nsometimes envy you and Berkeley your freedom from business cares and\nyour opportunities for study. What becomes of most men's college\ntraining, for example? By Jove! I picked up a Greek book the other\nday, and I couldn't read three words running. Now, I take it, you\nmanage to keep up your classics, among other things.\"\n\n\"Oh, my way of life has its compensations,\" he answered. \"But Sydney\nSmith--wasn't it?--said that life was a middling affair, anyway. As\nfor the classics, etc., I find that reading and study lose much of\ntheir stimulus unless they get an issue in action,--unless one can\napply them directly toward his own work. I often think that, if I\nwere fifteen or even ten years younger, I would go into some branch of\nnatural science. A scientific man always seems to me peculiarly happy\nin the healthy character of his work. He can keep himself apart from\nit. It is objective, impersonal, makes no demand on his emotions. Now\na writing man has to put himself into his work. He has to keep looking\nout all the time for impressions, material; to keep trying to enlarge\nand deepen his own experience, and he gets self-conscious and loses\nhis freshness in the process.\"\n\n\"I am surprised to find you in New York,\" said I, by way of changing\nthe subject. \"I thought you had laid out to live in the country. Do\nyou remember that pretty little word-picture of a winter afternoon\nthat you drew us--something in the style of an _Il Penseroso_\nlandscape? I expected to find you domesticated in a Berkshire\nfarm-house.\"\n\n\"Yes, I remember. I tried it. But I find it necessary, for my work, to\nbe in New York. The newspapers--confound 'em!--won't move into the\nwoods. But, after all, place is indifferent. See here; this isn't\nbad.\"\n\nHe drew aside the window curtain, and I looked out over a wilderness\nof roofs to the North River and the Palisades tinged with a purple\nlight. The ferry boats and tugs plying over the water in every\ndirection, the noise of the steam whistles, and the clouds of white\nvapor floating on the clear air, made an inspiriting scene.\n\n\"I'm up among the architects here,\" continued Clay; \"nothing but the\njanitor's family between me and the roof.\"\n\nWe talked awhile longer, and on taking leave, I said:\n\n\"I shall be on the lookout for something big from you one of these\ndays. You know what we always expected of you. So don't lose your\ngrip, old man.\"\n\n\"Who knows?\" he replied. \"It doesn't rest with me, but with the\n_daimon_.\"\n\nI was unable to visit Doddridge, the remaining member of our group. He\nlived in the thriving town of Wahee, Minnesota, and I had heard of\nhim, in a general way, as highly prosperous. He was a prominent lawyer\nand successful politician, and had lately been appointed United States\ndistrict judge, after representing his section in the State Senate for\na term or two. I wrote to him, congratulating him on his success and\nasking for details. I mentioned also my visits to Berkeley, Armstrong,\nand Clay. I got a prompt reply from Doddridge, from which I extract\nsuch portions as are material to this narrative:\n\n     \"The first few months after I left college I traveled pretty\n     extensively through the West, making contracts with the\n     farmers as agent for a nursery and seed-farm in my part of\n     the country, but really with the object of spying out the\n     land and choosing a place to settle in. Finally I lit on\n     Wahee, and made up my mind that it was a town with a future.\n     It was bound to be a railroad center. It had a first-rate\n     agricultural country around it, and a rich timber region a\n     little further back; and it already had an enterprising\n     little pop. growing rapidly. To-day Wahee is as smart a city\n     of its inches as there is in the Northwest. I squatted right\n     down here, got a little raise from the old man, and put it\n     all into building lots. I made a good thing of it, and paid\n     it all back in six years with eight per cent. interest.\n     Meanwhile, I went into Judge Pratt's law office and made my\n     salt by fitting his boy for college--till I learned enough\n     law to earn a salary. The judge was an old Waheer--belonged\n     to the time-honored aristocracy of the place, having been\n     here at least fifteen years before I came. He got into\n     railroads after awhile (is president now of the Wahee and\n     Heliopolis Bee-line), and left his law practice to me. I\n     married his daughter Alice in 1875. She is a Western girl,\n     but she was educated at Vassar. We have two boys. If you\n     ever come out our way, Polisson, you must put up with us for\n     as long as you can stay. I would like to show you the\n     country about here and have you ride after my team. I've got\n     a pair that can do it inside three minutes. Do you remember\n     Liddell of our class? He is an architect, you know. I got\n     him to come to Wahee, and he has all he can do putting up\n     business blocks. We have got some here equal to anything in\n     Chicago....\n\n     \"Yes, I am United States judge for this district. There is\n     not much money in it, but it will help me professionally by\n     and by. I shall not keep it long. Do I go into politics\n     much, you ask. I used to, but I've got through for the\n     present. The folks about here wanted to run me for Congress\n     last term, but I hadn't any use for it. As to what you are\n     kind enough to say about my 'success,' etc., whatever\n     success I have had is owing to nothing but a capacity for\n     hard work, which is the only talent that I lay claim to.\n     They want a man out here who will do the work that comes to\n     hand, and keep on doing it till something better turns\n     up....\n\n     \"So Berkeley has turned out a dilettante instead of an\n     African explorer. I heard he was a minister. He does not\n     seem to have much ambition even in that line of life. I\n     should think Armstrong had got the right kind of place for\n     him. He was a good fellow, but never had much practical\n     ability. You say very little about Clay. How is old\n     'Sweetness and Light,' any way? I saw some fluff of his in\n     one of the magazines,--a 'romance' I think he called it.\n     This is not an age for scribbling romances. The country\n     wants something solider. I never took much stock in\n     philosophers like Berkeley and Clay. There is the same thing\n     the trouble with them both: they don't want to do any hard\n     work, and they conceal their laziness under fine\n     names,--culture, transcendentalism, and what not? 'Feeble\n     and restless youths, born to inglorious days.'\"\n\nThis letter may be supplemented by another,--say Exhibit B,--which I\nreceived from Clay not long after:\n\n     \"MY DEAR POLISSON: It occurs to me that your question the\n     other day, as to how I was 'getting on,' did not receive as\n     candid an answer as it deserved. I am afraid that you\n     carried away an impression of me as of a man who suspected\n     himself to be a failure, but had not the manliness to\n     acknowledge it. You will say, perhaps, that there are all\n     degrees of half success short of absolute failure. But I say\n     no. In the career which I have chosen, to miss of\n     success--pronounced, unquestionable success--is to fail; and\n     I am not weak enough to hide from myself on which side of\n     the line I fall. The line is a very distinct one, after all.\n     The fact is, I took the wrong turning, and it is too late to\n     go back. I am a case of arrested development--a common\n     enough case. I might give plenty of excellent excuses to my\n     friends for not having accomplished what they expected me\n     to. But the world doesn't want apologies; it wants\n     performance.\n\n     \"You will think this letter a most extraordinary outburst of\n     morbid vanity. But while I can afford to have you think me a\n     failure, I couldn't let you go on thinking me a fraud. That\n     must be my excuse for writing.\n\n                         \"Yours, as ever,    E. CLAY.\"\n\nThis letter moved me deeply by its characteristic mingling of egotism\nwith elevation of feeling. As I held it open in my hand, and thought\nover my classmates' fortunes, I was led to make a few reflections.\nFrom the fact that Armstrong and Berkeley were leading lives that\nsquarely contradicted their announced ideas and intentions, it was an\nobvious but not therefore a true inference that circumstance is\nusually stronger than will. Say, rather, that the species of necessity\nwhich consists in character and inborn tendency is stronger than any\nresolution to run counter to it.\n\nBoth Armstrong and Berkeley, on our Commencement night, had spoken\nfrom a sense of their own limitations, and in violent momentary\nrebellion against them. But, in talking with them fifteen years later,\nI could not discover that the lack of correspondence between their\nideal future and their actual present troubled them much. It is matter\nof common note that it is impossible to make one man realize another's\nexperience; but it is often quite as hard to make him recover a past\nstage of his own consciousness.\n\nThese, then, had bent to the force of chance or temperament. But\nClay had shaped his life according to his programme, and had the\nresult been happier? He who gets his wish often suffers a sharper\ndisappointment than he who loses it. \"_So taeuscht uns also bald die\nHoffnung, bald das Gehoffte_,\" says the great pessimist, and Fate is\nnever more ironical than when she humors our whim. Doddridge alone,\nwho had thrown himself confidingly into the arms of the Destinies, had\nobtained their capricious favors.\n\nI cannot say that I drew any counsel, civil or moral, from these\ncomparisons. Life is deeper and wider than any particular lesson to be\nlearned from it; and just when we think that we have at last guessed\nits best meanings, it laughs in our face with some paradox which turns\nour solution into a new riddle.\n\n\n\n\nZERVIAH HOPE.\n\nBY ELIZABETH STUART PHELPS.\n\n_Scribner's Monthly, November, 1880._\n\n\nPRELUDE.\n\nIn the month of August, in the year 1878, the steamer _Mercy_, of the\nNew York and Savannah line, cast anchor down the channel, off a little\ntown in South Carolina which bore the name of Calhoun. It was not a\nregular part of her \"run\" for the _Mercy_ to make a landing at this\nplace. She had departed from her course by special permit to leave\nthree passengers, two men and one woman, who had business of a grave\nnature in Calhoun.\n\nA man, himself a passenger for Savannah, came upon deck as the\nsteamship hove to, to inquire the reason of the delay. He was a short\nman, thin, with a nervous hand and neck. His eyes were black, his hair\nwas black, and closely cut. He had an inscrutable mouth, and a\nforehead well-plowed rather by experience than years. He was not an\nold man. He was cleanly dressed in new, cheap clothes. He had been\ncommented upon as a reticent passenger. He had no friends on board the\n_Mercy_. This was the first time upon the voyage that he had been\nobserved to speak. He came forward and stood among the others, and\nabruptly said:\n\n\"What's this for?\"\n\nHe addressed the mate, who answered with a sidelong look, and none too\ncordially:\n\n\"We land passengers by the Company's order.\"\n\n\"Those three?\"\n\n\"Yes, the men and the lady.\"\n\n\"Who are they?\"\n\n\"Physicians from New York.\"\n\n\"Ah-h!\" said the man, slowly, making a sighing noise between his\nteeth. \"That means--that means--\"\n\n\"Volunteers to the fever district,\" said the mate, shortly, \"as you\nmight have known before now. You're not of a sociable cast, I see.\"\n\n\"I have made no acquaintances,\" said the short passenger. \"I know\nnothing of the news of the ship. Is the lady a nurse?\"\n\n\"She's a she-doctor. Doctors, the whole of 'em. There ain't a nurse\naboard.\"\n\n\"Plenty to be found, I suppose, in this place you speak of?\"\n\n\"How should I know?\" replied the mate, with another sidelong look.\n\nOne of the physicians, it seemed, overheard this last question and\nreply. It was the woman. She stepped forward without hesitation, and,\nregarding the short passenger closely, said:\n\n\"There are not nurses. This place is perishing. Savannah and the\nlarger towns have been looked after first--as is natural and right,\"\nadded the physician, in a business-like tone. She had a quick and\nclear-cut, but not ungentle voice.\n\nThe man nodded at her curtly, as he would to another man; he made no\nanswer; then with a slight flush his eye returned to her dress and\nfigure; he lifted his hat and stood uncovered till she had passed and\nturned from him. His face, under the influence of this fluctuation of\ncolor, changed exceedingly, and improved in proportion as it changed.\n\n\"Who is that glum fellow, Doctor?\"\n\nOne of the men physicians followed and asked the lady; he spoke to her\nwith an air of _camaraderie_, at once frank and deferential; they had\nbeen classmates at college for a course of lectures; he had theories\naverse to the medical education of women in general, but this woman in\nparticular, having outranked him at graduation, he had made up his\nmind to her as a marked exception to a wise rule, entitled to a candid\nfellow's respect. Besides, despite her diploma, Marian Dare was a\nlady--he knew the family.\n\n\"_Is_ he glum, Dr. Frank?\" replied Dr. Dare.\n\nBut the other young man stood silent. He never consulted with\ndoctresses.\n\nDr. Dare went below for her luggage. A lonely dory, black of\ncomplexion and skittish of gait, had wandered out and hung in the\nshadow of the steamer, awaiting the passengers. The dory was manned by\none <DW64>, who sat with his oars crossed, perfectly silent.\n\nThere is a kind of terror for which we find that animals, as well as\nmen, instinctively refrain from seeking expression. The face and\nfigure of the <DW64> boatman presented a dull form of this species of\nfear. Dr. Dare wondered if all the people in Calhoun would have that\nlook. The <DW64> regarded the _Mercy_ and her passengers apathetically.\n\nIt was a hot day, and the water seemed to be blistering about the\ndory. So, too, the stretching sand of the shore, as one raised the\neyes painfully against the direct noon-light, was as if it smoked. The\nlow, gray palmetto leaves were curled and faint. Scanty spots of shade\nbeneath sickly trees seemed to gasp upon the hot ground, like\ncreatures that had thrown themselves down to get cool. The outlines of\nthe town beyond had a certain horrible distinctness, as if of a sight\nthat should but could not be veiled. Overhead, and clean to the flat\nhorizon, flashed a sky of blue and blazing fire.\n\n\"Passengers for Calhoun!\"\n\nThe three physicians descended into the dory. The other\npassengers--what there were of them--gathered to see the little group\ndepart. Dr. Frank offered Dr. Dare a hand, which she accepted, like a\nlady, not needing it in the least. She was a climber, with firm, lithe\nankles. No one spoke, as these people got in with the <DW64>, and\nprepared to drift down with the scorching tide. The woman looked from\nthe steamer to the shore, once, and back again, northwards. The men\ndid not look at all. There was an oppression in the scene which no one\nwas ready to run the risk of increasing by the wrong word.\n\n\"Land me here, too,\" said a low voice, suddenly appearing. It was the\nglum passenger. No one noticed him, except, perhaps, the mate (looking\non with the air of a man who would feel an individual grievance in\nanything this person would be likely to do) and the lady.\n\n\"There is room for you,\" said Dr. Dare. The man let himself into the\nboat at a light bound, and the <DW64> rowed them away. The _Mercy_,\nheading outwards, seemed to shrug her shoulders, as if she had thrown\nthem off. The strip of burning water between them and the town\nnarrowed rapidly, and the group set their faces firmly landwards.\nOnce, upon the little voyage, Dr. Frank took up an idle pair of oars,\nwith some vaguely humane intent of helping the <DW64>--he looked so.\n\n\"I wouldn't, Frank,\" said the other gentleman.\n\n\"Now, Remane--why, for instance?\"\n\n\"I wouldn't begin by getting overheated.\"\n\nNo other word was spoken. They landed in silence. In silence, and\nsomewhat weakly, the <DW64> pulled the dory high upon the beach. The\nfour passengers stood for a moment upon the hot, white sands, moved\ntoward one another, before they separated, by a blind sense of human\nfellowship. Even Remane found himself touching his hat. Dr. Frank\nasked Dr. Dare if he could serve her in any way; but she thanked him,\nand, holding out her firm, white hand, said, \"Good-bye.\"\n\nThis was, perhaps, the first moment when the consciousness of her sex\nhad made itself oppressive to her since she ventured upon this\nundertaking. She would have minded presenting herself to the Relief\nCommittee of Calhoun, accompanied by gentlemen upon whom she had no\nclaim. She walked on alone, in her gray dress and white straw hat,\nwith her luggage in her own sufficient hand.\n\nThe reticent passenger had fallen behind with the <DW64> boatman, with\nwhom he walked slowly, closing the line.\n\nAfter a few moments, he advanced and hesitatingly joined the lady,\nbeginning to say:\n\n\"May I ask you--\"\n\n\"Ah,\" interrupted Dr. Dare, cordially, \"it is you.\"\n\n\"Will you tell me, madam, the best way of going to work to offer\nmyself as a fever nurse in this place? I want the _best_ way. I want\nreal work.\"\n\n\"Yes, yes,\" she said, nodding; \"I knew you would do it.\"\n\n\"I came from the North for this purpose, but I meant to go on to\nSavannah.\"\n\n\"Yes, I know. This is better; they need _everything_ in this place.\"\n\nShe looked toward the gasping little town through the relentless noon.\nHer merciful blue eyes filled, but the man's look followed with a dry,\nexultant light.\n\n\"There is no porter,\" he said, abruptly, glancing at her heavy bag and\nshawl-strap. \"Would you permit me to help you?\"\n\n\"Oh, thank you!\" replied Dr. Dare, heartily, relinquishing her burden.\n\nPlainly, this poor fellow was not a gentleman. The lady could afford\nto be kind to him.\n\n\"I know nothing how we shall find it,\" she chatted, affably, \"but I go\nto work to-night. I presume I shall need nurses before morning. I'll\nhave your address.\"\n\nShe took from her gray sacque pocket a physician's note-book, and\nstood, pencil in hand.\n\n\"My name,\" he said, \"is Hope--Zerviah Hope.\"\n\nShe wrote without comment, walking as she wrote; he made no other\nattempt to converse with her. The two physicians followed, exchanging\nnow and then a subdued word. The <DW64> dragged himself wearily over\nthe scorching sand, and thus the little procession of pity entered the\ntown of Calhoun.\n\nMy story does not deal with love or ladies. I have to relate no tender\npassages between the fever-physicians, volunteers from New York, for\nthe afflicted region of Calhoun. Dr. Marian Dare came South to do a\nbrave work, and I have no doubt she did it bravely, as a woman should.\nShe came in pursuit of science, and I have no doubt she found it, as a\nwoman will. Our chief interest in her at this time lies in the fact\nthat certain missing fragments in the history of the person known as\nZerviah Hope we owe to her. She hovers over the tale with a distant\nand beautiful influence, pervading as womanly compassion and alert as\na woman's eye.\n\nI have nothing further to say about the story before I tell it, except\nthat it is true.\n\n       *       *       *       *       *\n\nThat night, after the physicians had gone about their business,\nZerviah Hope wandered, a little forlornly, through the wretched town.\nScip, the <DW64> boatman, found him a corner to spend the night. It was\na passable place, but Hope could not sleep; he had already seen too\nmuch. His soul was parched with the thirst of sympathy. He walked his\nhot attic till the dawn came. As it grew brighter he grew calmer; and,\nwhen the unkindly sun burst burning upon the land, he knelt by his\nwindow and looked over the doomed town, and watched the dead-carts\nslinking away toward the everglades in the splendid color of the sky\nand air, and thought his own thoughts in his own way about this which\nhe had come to do. We should not suppose that they were remarkable\nthoughts; he had not the look of a remarkable man. Yet, as he knelt\nthere,--a sleepless, haggard figure blotted against the sunrise,\nwith folded hands and moving lips,--an artist, with a high type of\nimagination and capable of spiritual discernment, would have found in\nhim a design for a lofty subject, to which perhaps he would have given\nthe name of \"Consecration\" rather than of \"Renunciation,\" or of\n\"Exultance\" rather than of \"Dread.\"\n\nA common observer would have simply said: \"I should not have taken him\nfor a praying man.\"\n\nHe was still upon his knees when Dr. Dare's order came, \"Nurse wanted\nfor a bad case!\" and he went from his prayer to his first patient. The\nday was already deep, and a reflection, not of the sunrise, moved with\nhim as light moves.\n\nDoctor Dare, in her gray dress, herself a little pale, met him with\nkeen eyes. She said:\n\n\"It is a _very_ bad case. An old man--much neglected. No one will go.\nAre you willing?\"\n\nThe nurse answered:\n\n\"I am glad.\"\n\nShe watched him as he walked away--a plain, clean, common man, with\nunheroic carriage. The physician's fine eyes fired.\n\nTo Doctor Frank, who had happened in, she said:\n\n\"He will do the work of ten.\"\n\n     \"His strength was as the strength of ten,\n     Because his heart was pure,\"\n\nquoted the young man, laughing lightly. \"I don't know that I should\nhave thought it, in this case. You've taken a fancy to the fellow.\"\n\n\"I always respect an unmixed motive when I see it,\" she replied,\nshortly. \"But I've been in practice too long to take sudden fancies.\nThere is no profession like ours, Doctor, for putting the sympathies\nunder double picket guard.\"\n\nShe stiffened a little in her manner. She did not like to be thought\nan over-enthusiastic woman--womanish, unused to the world.\n\nThe weather, soon after the arrival of the _Mercy_, took a terrible\nmood, and a prolonged drought settled upon Calhoun. The days dawned\nlurid and long. The nights fell dewless and deadly. Fatal and\nbeautiful colors lurked in the swamps, and in the sifting dust, fine\nand hard, blown by siroccos across the glare of noon, like sands on\nthe shores of the Lake of Fire. The pestilence walked in darkness, and\nthe destruction wasted at midday. Men died, in that little town of a\nfew thousand souls, at the rate of a score a day--black and white,\npoor and rich, clean and foul, saint and sinner. The quarantine laws\ntightened. Vessels fled by the harbor mouth under full sail, and\nmelted like helpless compassion upon the fiery horizon. Trains upon\nthe Shore Line shot through and thundered past the station; they\ncrowded on steam; the fireman and his stoker averted their faces as\nthey whirled by. The world turned her back upon Calhoun, and the dying\ntown was shut in with her dead. Only, at long intervals, the _Mercy_,\ncasting anchor far down the channel, sent up by Scip, the weak, black\nboatman, the signs of human fellowship--food, physician, purse,\nmedicine--that spoke from the heart of the North to the heart of the\nSouth, and upheld her in those well-remembered days.\n\nZerviah Hope, volunteer nurse, became quickly enough a marked man in\nCalhoun. He more than verified Doctor Dare's prognosis. Where the\ndeadliest work was to be done, this man, it was observed, asked to be\nsent. Where no one else would go, he went. What no one else would do,\nhe did. He sought the neglected, and the <DW64>s. He braved the\nunclean, and the unburied. With the readiness of all incisive\ncharacter acting on emergencies, he stamped himself upon the place and\ntime. He went to his task as the soldier goes to the front under\nraking fire, with gleaming eyes and iron muscles. The fever of the\nfight was on him. He seemed to wrestle with disease for his patients,\nand to trample death beneath his feet. He glowed over his cures with a\npositive physical dilation, and writhed over his dead as if he had\nkilled them. He seemed built of endurance more than mortal. It was not\nknown when he slept, scarcely if he ate. His weariness sat upon him\nlike a halo. He grew thin, refined, radiant. In short, he presented\nan example of that rare spectacle which never fails to command\nspectators--a common man possessed by an uncommon enthusiasm.\n\nWhat passed with him at this time in that undiscovered sea which we\ncall a man's inner life, it would not be easy to assert. So far as we\ncan judge, all the currents of his nature had swelled into the great,\npulsing tide of self-surrender, which swept him along. Weakness,\nwrong, memory, regret, fear, grief, pleasure, hope,--all the little\nchannels of personal life,--ran dry. He was that most blessed of human\ncreatures, a man without a past and without a future, and living in a\npresent nobler than the one could have been or the other could become.\nHe continued to be a silent man, speaking little, excepting to his\npatients, and now and then, very gently, to the lady, Dr. Dare. He was\nalways pliable to the influence of a woman's voice or to womanly\nmanner. He had, in the presence of women, the quick responsiveness and\nsudden change of color and sensitiveness of intonation which bespeak\nthe man whose highest graces and lowest faults are likely to be owing\nto feminine power.\n\nThis was a quality which gave him remarkable success as a nurse. He\nwas found to be infinitely tender, and of fine, brave patience. It was\nfound that he shrank from no task because it was too small, as he had\nshrunk from no danger because it was too great. He became a favorite\nwith the sick and with physicians. The convalescent clung to him, the\ndying heard of him and sent for him, the Relief Committee leaned upon\nhim, as in such crises the leader leans upon the led. By degrees, he\nbecame greatly trusted in Calhoun; this is to say, that he became\ngreatly loved.\n\nI have been told that, to this day, many people personally unknown to\nhim, whose fate it was to be imprisoned in that beleaguered town at\nthat time, and who were familiar with the nervous figure and plain,\nintense countenance of the Northern nurse, as he passed, terrible day\nafter terrible day, to his post, cannot hear of him, even now, without\nthat suffusion of look by which we hold back tears; and that, when\nhis name took on, as it did, a more than local reputation, they were\nunable to speak it because of choking voices. I have often wished that\nhe knew this.\n\nIt was the custom in Calhoun to pay the nurses at short, stated\nintervals,--I think once a week, on Saturday nights. The first time\nthat Hope was summoned to receive his wages, he evinced marked\nemotion, too genuine not to be one of surprise and repugnance.\n\n\"I had not thought,--\" he began, and stood, coloring violently.\n\n\"You earn your five dollars a day, if anybody in Calhoun does,\" urged\nthe official, with kindly brusqueness.\n\n\"It is not right; I do not wish to take the money,\" said the nurse,\nwith agitation. \"I do not wish to be paid for--saving--human life.\nI did not come to the fever district to make money; I came to save\nlife--_to save life_!\" he added, in a quick whisper. He had not slept\nfor four nights, and seemed, they noticed, more than usually nervous\nin his manner.\n\n\"The money is yours,\" insisted the treasurer.\n\n\"Very well,\" said Hope, after a long silence; and no more was said\nabout it. He took his wages and walked away up the street, absorbed\nin thought.\n\nOne morning, he went to his lodgings to seek a little rest. It was\nabout six o'clock, and people were already moving in the hot, thirsty\nstreets. The apothecaries' doors were open, and their clerks were\nastir. The physicians drove or walked hastily, with the haggard look\nof men whose days and nights are too short for their work, and whose\nhope, and heart as well, have grown almost too small. Zerviah noticed\nthose young Northern fellows among them, Frank and Remane, and saw how\nthey had aged since they came South,--brave boys, both of them, and\nhad done a man's brave deed. Through her office window, as he walked\npast, he caught a glimpse of Dr. Dare's gray dress and blonde, womanly\nhead of abundant hair. She was mixing medicines, and patients stood\nwaiting. She looked up and nodded as he went by; she was too busy\nto smile. At the door of the Relief Committee, gaunt groups hung,\nclamoring. At the telegraph office, knots of men and women gathered,\nduly inspiring the heroic young operator,--a slight girl,--who had not\nleft her post for now many days and nights. Her chief had the fever\nlast week,--was taken at the wires,--lived to get home. She was the\nonly person alive in the town who knew how to communicate with the\nouter world. She had begun to teach a little brother of hers the Morse\nalphabet,--\"That somebody may know, Bobby, if I--can't come some day.\"\nShe, too, knew Zerviah Hope, and looked up; but her pretty face was\nclouded with the awful shadow of her own responsibility.\n\n\"We all have about as much as we can bear,\" thought Zerviah, as he\nwent by. His own burden was lightened a little that morning, and he\nwas going home to get a real rest. He had just saved his last\npatient--the doctor gave him up. It was a young man, the father of\nfive very little children, and their mother had died the week before.\nThe nurse had looked at the orphans, and said: \"_He's got to live._\"\nThis man had blessed him this morning, and called the love of heaven\non his head and its tender mercy on his whole long life. Zerviah\nwalked with quick step. He lifted his head, with its short, black,\ncoarse hair. His eyes, staring for sleep, flashed, fed with a food the\nbody knows not of. He felt almost happy, as he turned to climb the\nstairs that led to the attic shelter where he had knelt and watched\nthe dawn come on that first day, and given himself to God and to the\ndying of Calhoun. He had always kept that attic, partly because he had\nmade that prayer there. He thought it helped him to make others since.\nHe had not always been a man who prayed. The habit was new, and\nrequired culture. He had guarded it rigidly since he came South, as\nhe had his diet and regimen of bathing, air, and other physical needs.\n\nOn this morning that I speak of, standing with his almost happy face\nand lifted head, with his foot upon the stairs, he turned, for no\nreason that he could have given, and looked over his shoulder. A man\nbehind him, stepping softly, stopped, changed color, and crossed the\nstreet.\n\n\"I am followed,\" said the nurse.\n\nHe spoke aloud, but there was no one to hear him. A visible change\ncame over his face. He stood uncertain for a moment; then shut the\ndoor and crawled upstairs. At intervals he stopped on the stairs to\nrest, and sat with his head in his hands, thinking. By and by he\nreached his room, and threw himself heavily upon his bed. All the\nradiance had departed from his tired face, as if a fog had crept over\nit. He hid it in his long, thin, humane hands, and lay there\nfor a little while. He was perplexed--not surprised. He was not\nshocked--only disappointed. Dully he wished that he could get five\nminutes' nap; but he could not sleep. Not knowing what else to do, he\ngot upon his knees presently, in that place by the window he liked to\npray in, and said aloud:\n\n\"Lord, I didn't expect it; I wasn't ready. I should like to sleep long\nenough to decide what to do.\"\n\nAfter this, he went back to bed and lay still again, and in a little\nwhile he truly slept. Not long; but to those who perish for rest, a\nmoment of unconsciousness may do the work of a cup of water to one\nperishing of thirst. He started, strengthened, with lines of decision\nforming about his mouth and chin; and, having bathed and cleanly\ndressed, went out.\n\nHe went out beyond the town to the hut where Scip the boatman lived.\nScip was at home. He lived quite alone. His father, his mother and\nfour brothers had died of the plague since June. He started when he\nsaw Hope, and his habitual look of fear deepened to a craven terror;\nhe would rather have had the yellow fever than to have seen the\nNorthern nurse just then. But Zerviah sat down by him on the hot sand,\nbeside a rather ghastly palmetto that grew there, and spoke to him\nvery gently. He said:\n\n\"The _Mercy_ came in last night, Scip.--I know. And you rowed down for\nthe supplies. You heard something about me on board the _Mercy_. Tell\nme, Scip.\"\n\n\"He's a durn fool,\" said Scip, with a dull show of passion.\n\n\"Who is a durn fool?\"\n\n\"That dem mate.\"\n\n\"So it was the mate? Yes. What did he say, Scip?\"\n\n\"I never done believe it,\" urged Scip, with an air of suddenly\nrecollected virtue.\n\n\"But you told of it, Scip.\"\n\n\"I never told nobody but Jupiter, the durn fool!\" persisted Scip.\n\n\"Who is Jupiter?\"\n\n\"Doctor Remane's Jupiter, him that holds his hoss, that he brung up\nfrom the fever. He said he wouldn't tell. I never done believe it,\n_never_!\"\n\n\"It seems to me, Scip,\" said Zerviah, in a low, kind voice, \"that I\nwouldn't have told if I'd been you. But never mind.\"\n\n\"I never done mean to hurt you!\" cried Scip, following him into the\nroad. \"Jupiter the durn, he said he'd never tell. I never told nobody\nelse.\"\n\n\"You have told the whole town,\" said Zerviah Hope, patiently. \"I'm\nsorry, but never mind.\"\n\nHe stood for a moment looking across the stark palmetto, over the\ndusty stretch of road, across the glare, to the town. His eyes blinded\nand filled.\n\n\"It wouldn't have been a great while,\" he said. \"I wish you hadn't,\nScip, but never mind!\"\n\nHe shook the <DW64> gently off, as if he had been a child. There was\nnothing more to say. He would go back to his work. As he walked along,\nhe suddenly said to himself:\n\n\"She did not smile this morning! Nor the lady at the telegraph office,\neither. Nor--a good many other folks. I remember now.... Lord!\" he\nadded aloud, thought breaking into one of his half-unconscious\nprayers, which had the more pathos because it began with the rude\nabruptness of an apparent oath,--\"Lord! what in the name of heaven am\nI going to do about it?\"\n\nNow, as he was coming into the little city, with bowed head and broken\nface, he met Doctor Dare. She was riding on her rounds upon a patient,\nSouthern tackey, and she was riding fast. But she reined up and\nconfronted him.\n\n\"Mr. Hope! There is a hateful rumor brought from New York about you. I\nam going to tell you immediately. It is said--\"\n\n\"Wait a minute!\" he pleaded, holding out both hands. \"Now. Go on.\"\n\n\"It is said that you are an escaped convict,\" continued the lady,\ndistinctly.\n\n\"It is false!\" cried the nurse, in a ringing voice.\n\nThe doctor regarded him for a moment.\n\n\"I may be wrong. Perhaps it was not so bad. I was in a cruel hurry,\nand so was Doctor Frank. Perhaps they said a discharged convict.\"\n\n\"What else?\" asked Zerviah, lifting his eyes to hers.\n\n\"They said you were just out of a seven years' imprisonment for\nmanslaughter. They said you killed a man--for jealousy, I believe;\nsomething about a woman.\"\n\n\"What else?\" repeated the nurse, steadily.\n\n\"I told them I _did not believe one word of it_!\" cried Marian Dare.\n\n\"Thank you, madam,\" said Zerviah Hope, after a scarcely perceptible\npause; \"but it is true.\"\n\nHe drew one fierce breath.\n\n\"She was beautiful,\" he said. \"I loved her; he ruined her; I stabbed\nhim!\"\n\nHe had grown painfully pale. He wanted to go on speaking to this\nwoman, not to defend or excuse himself, not to say anything weak or\nwrong, only to make her understand that he did not want to excuse\nhimself; in some way, just because she _was_ a woman, to make her feel\nthat he was man enough to bear the burden of his deed. He wanted to\ncry out to her, \"You are a woman! Oh, be gentle, and understand how\nsorry a man can be for a deadly sin!\" but his lips were parched. He\nmoved them dryly; he could not talk.\n\nShe was silent at first. She was a prudent woman; she thought before\nshe spoke.\n\n\"Poor fellow!\" she said, suddenly. Her clear blue eyes overflowed. She\nheld out her hand, lifted his, wrung it, dropped it, and softly added,\n\"Well, never mind!\" much as if he had been a child or a patient,--much\nas he himself had said, \"Never mind!\" to Scip.\n\nThen Zerviah Hope broke down.\n\n\"I haven't got a murderer's heart!\" he cried. \"It has been taken away\nfrom me. I ain't so bad--_now_. I meant to be--I wanted to do--\"\n\n\"Hush!\" she said. \"You have, and you shall. God is fair.\"\n\n\"Yes,\" said the penitent convict, in a dull voice, \"God is fair, and\nso he let 'em tell of me. I've got no fault to find with _Him_. So\nnigh as I can understand Almighty God, He means well.... I guess\nHe'll pull me through some way.... But I wish Scip hadn't told just\nnow. I can't _help_ being sorry. It wasn't that I wanted to cheat,\nbut\"--he choked--\"_the sick folks used to like me_. Now, do you think\nI'd ought to go on nursing, Doctor? Do you think I'd ought to stop?\"\n\n\"You are already an hour late,\" replied the woman of science, in her\nusual business-like voice. \"Your substitute will be sleepy and\nrestless; that affects the patient. Go back to your work as fast as\nyou can. Ask me no more foolish questions.\"\n\nShe drew her veil; there was unprofessional moisture on her long,\nfeminine lashes. She held out her hearty hand-grasp to him, touched\nthe tackey, and galloped away.\n\n\"She is a good woman,\" said Zerviah, half aloud, looking down at his\ncold fingers. \"She touched me, and she knew! Lord, I should like to\nhave you bless her!\"\n\nHe looked after her. She sat her horse finely; her gray veil drifted\nin the hot wind. His sensitive color came. He watched her as if he had\nknown that he should never see her again on earth.\n\nA ruined character may be as callous as a paralyzed limb. A ruined and\nrepentant one is in itself an independent system of sensitive and\ntortured nerves.\n\nZerviah Hope returned to his work, shrinking under the foreknowledge\nof his fate. He felt as if he knew what kind of people would remind\nhim that they had become acquainted with his history, and what ways\nthey would select to do it.\n\nHe was not taken by surprise when men who had lifted their hats to the\npopular nurse last week, passed him on the street to-day with a cold\nnod or curious stare. When women who had reverenced the self-sacrifice\nand gentleness of his life as only women do or can reverence the\nquality of tenderness in a man, shrank from him as if he were\nsomething infectious, like the plague,--he knew it was just, though he\nfelt it hard.\n\nHis patients heard of what had happened, sometimes, and indicated a\nfeeling of recoil. That was the worst. One said:\n\n\"I am sorry to hear you are not the man we thought you,\" and died in\nhis arms that night.\n\nZerviah remembered that these things must be. He reasoned with\nhimself. He went into his attic, and prayed it all over. He said:\n\n\"Lord, I can't expect to be treated as if I'd never been in prison.\nI'm sorry I mind it so. Perhaps I'd ought not to. I'll try not to care\ntoo much.\"\n\nMore than once he was sure of being followed again, suspiciously or\ncuriously. It occurred to him at last that this was most likely to\nhappen on pay-days. That puzzled him. But when he turned, it was\nusually some idler, and the fellow shrank and took to his heels, as if\nthe nurse had the fever.\n\nIn point of fact, even in that death-stricken town, to be alive was\nto be as able to gossip as well people, and rumor, wearied of the\nmonotonous fever symptom, found a diverting zest in this shattered\nreputation.\n\nZerviah Hope was very much talked about in Calhoun; so much, that the\nRelief Committee heard, questioned, and experienced official anxiety.\nIt seemed a mistake to lose so valuable a man. It seemed a severity to\ndisturb so noble a career. Yet who knew what sinister countenance the\nmurderer might be capable of shielding beneath his mask of pity? The\nofficial mind was perplexed. Was it humane to trust the lives of our\nperishing citizens to the ministrations of a felon who had so\nskillfully deceived the most intelligent guardians of the public weal?\nThere was, in particular, a chairman of a sub-committee (on the water\nsupply) who was burdened with uneasiness.\n\n\"It's clear enough what brought _him_ to Calhoun,\" said this man.\n\"What do you suppose the fellow does with his five dollars a day?\"\n\nThe Committee on the Water Supply promptly divided into a\nSub-Vigilance, and to the Sub-Vigilance Committee Zerviah Hope's case\nwas referred. The result was, that he was followed on pay-day.\n\nOne Saturday night, just as the red-hot sun was going down, the\nsub-committee returned to the Relief Office in a state of high\nofficial excitement, and reported to the chief as follows:\n\n\"We've done it. We've got him. We've found out what the fellow does\nwith his money. He puts it--\"\n\n\"Well?\" for the sub-committee hesitated.\n\n\"Into the relief contribution-boxes on the corners of the street.\"\n\n\"_What!_\"\n\n\"Every dollar. We stood and watched him count it out--his week's\nwages. Every mortal cent that Yankee's turned over to the fund for the\nsufferers. He never kept back a red. He poured it all in.\"\n\n\"Follow him next week. Report again.\"\n\nThey followed, and reported still again. They consulted, and accepted\nthe astounding truth. The murderer, the convict, the miserable, the\nmystery, Zerviah Hope,--volunteer nurse, poor, friendless, discharged\nfrom Sing Sing, was proved to have surrendered to the public charities\nof Calhoun, every dollar which he had earned in the service of her\nsick and dying.\n\nThe Committee on the Water Supply, and the Sub-Vigilance Committee\nstood, much depressed, before their superior officer. He, being a just\nman, flushed red with a noble rage.\n\n\"Where is he? Where is Zerviah Hope? The man should be sent for. He\nshould receive the thanks of the committee. He should receive the\nacknowledgments of the city. And we've set on him like detectives!\nhunted him down! Zounds! The city is disgraced. Find him for me!\"\n\n\"We have already done our best,\" replied the sub-committee, sadly.\n\"We have searched for the man. He cannot be found.\"\n\n\"Where is the woman-doctor?\" persisted the excited chief. \"She\nrecommended the fellow. She'd be apt to know. Can't some of you find\nher?\"\n\nAt this moment, young Dr. Frank looked haggardly into the Relief\nOffice.\n\n\"I am taking her cases,\" he said. \"She is down with the fever.\"\n\n       *       *       *       *       *\n\nIt was the morning after his last pay-day--Sunday morning, the first\nin October; a dry, deadly, glittering day. Zerviah had been to his\nattic to rest and bathe; he had been there some hours since sunrise,\nin the old place by the window, and watched the red sun kindle, and\nwatched the dead-carts slink away into the color, and kneeled and\nprayed for frost. Now, being strengthened in mind and spirit, he was\ndescending to his Sabbath's work, when a message met him at the door.\nThe messenger was a <DW64> boy, who thrust a slip of paper into his\nhand, and, seeming to be seized with superstitious fright, ran rapidly\nup the street and disappeared.\n\nThe message was a triumphal result of the education of the freedmen's\nevening school, and succinctly said:\n\n     \"ive Gut IT. Nobuddy Wunt Nuss me. Norr no Docter nEther.\n\n     \"P. S. Joopiter the Durn hee sed he'd kerry This i dont\n     Serpose youd kum.   SCIP.\"\n\nThe sun went down that night as red as it had risen. There were no\nclouds. There was no wind. There was no frost. The hot dust curdled in\nthe shadow that coiled beneath the stark palmetto. That palmetto\nalways looked like a corpse, though there was life in it yet. Zerviah\ncame to the door of Scip's hovel for air, and looked at the thing. It\nseemed like something that ought to be buried. There were no other\ntrees. The everglades were miles away. The sand and the scant, starved\ngrass stretched all around. Scip's hut stood quite by itself. No one\npassed by. Often no one passed for a week, or even more. Zerviah\nlooked from the door of the hut to the little city. The red light lay\nbetween him and it, like a great pool. He felt less lonely to see the\ntown, and the smoke now and then from chimneys. He thought how many\npeople loved and cared for one another in the suffering place. He\nthought how much love and care suffering gave birth to, in human\nhearts. He began to think a little of his own suffering; then Scip\ncalled him, sobbing wretchedly. Scip was very sick. Hope had sent for\nDr. Dare. She had not come. Scip was too sick to be left. The nurse\nfound his duty with the <DW64>. Scip was growing worse.\n\nBy and by, when the patient could be left for a moment again, Zerviah\ncame to the air once more. He drew in great breaths of the now cooler\nnight. The red pool was gone. All the world was filled with the fatal\nbeauty of the purple colors that he had learned to know so well. The\nswamps seemed to be asleep, and to exhale in the slow, regular\npulsations of sleep. In the town, lamps were lighted. From a hundred\nwindows, fair, fine sparks shone like stars as the nurse looked over.\nThere, a hundred watchers tended their sick or dead, or their healing,\ndearly loved, and guarded ones. Dying eyes looked their last at eyes\nthat would have died to save them; strengthening hands clasped hands\nthat had girded them with the iron of love's tenderness, through the\nvalley of the shadow, and up the heights of life and light. Over\nthere, in some happy home, tremulous lips that the plague had parted\nmet to-night in their first kiss of thanksgiving.\n\nZerviah thought of these things. He had never felt so lonely before.\nIt seemed a hard place for a man to die in. Poor Scip!\n\nZerviah clasped his thin hands and looked up at the purple sky.\n\n\"Lord,\" he said, \"it is my duty. I came South to do my duty. Because\nhe told of me, had I ought to turn against him? It is a lonesome\nplace; he's got it hard, but I'll stand by him.... Lord!\"--his worn\nface became suddenly suffused, and flashed, transfigured, as he lifted\nit--\"Lord God Almighty! You stood by me! _I_ couldn't have been a\npleasant fellow to look after. You stood by _me_ in my scrape! I\nhadn't treated _You_ any too well.... You needn't be afraid I'll leave\nthe creetur.\"\n\nHe went back into the hut. Scip called, and he hurried in. The nurse\nand the plague, like two living combatants, met in the miserable place\nand battled for the <DW64>.\n\nThe white Southern stars blazed out. How clean they looked! Zerviah\ncould see them through the window, where the wooden shutter had\nflapped back. They looked well and wholesome--holy, he thought. He\nremembered to have heard some one say, at a Sunday meeting he happened\ninto once, years ago, that the word holiness meant health. He wondered\nwhat it would be like, to be holy. He wondered what kinds of\npeople would be holy people, say, after a man was dead. Women, he\nthought,--good women, and honest men who had never done a deadly deed.\n\nHe occupied his thoughts in this way. He looked often from the cold\nstars to the warm lights throbbing in the town. They were both company\nto him. He began to feel less alone. There was a special service\ncalled somewhere in the city that night, to read the prayers for the\nsick and dying. The wind rose feebly, and bore the sound of the\nchurch-bells to the hut. There was a great deal of company, too, in\nthe bells. He remembered that it was Sunday night.\n\n       *       *       *       *       *\n\nIt was Monday, but no one came. It was Tuesday, but the nurse and the\nplague still battled alone together over the <DW64>. Zerviah's stock\nof remedies was as ample as his skill. He had thought he should save\nScip. He worked without sleep, and the food was not clean. He lavished\nhimself like a lover over this black boatman; he leaned like a mother\nover this man who had betrayed him.\n\nBut on Tuesday night, a little before midnight, Scip rose, struggling\non his wretched bed, and held up his hands and cried out:\n\n\"Mr. Hope! Mr. Hope! I never done mean to harm ye!\"\n\n\"You have not harmed me,\" said Zerviah, solemnly. \"Nobody ever harmed\nme but myself. Don't mind me, Scip.\"\n\nScip put up his feeble hand; Zerviah took it; Scip spoke no more. The\nnurse held the <DW64>'s hand a long time; the lamp went out; they sat\non in the dark. Through the flapping wooden shutter the stars looked\nin.\n\nSuddenly, Zerviah perceived that Scip's hand was quite cold.\n\n       *       *       *       *       *\n\nHe carried him out by starlight, and buried him under the palmetto. It\nwas hard work digging alone. He could not make a very deep grave, and\nhe had no coffin. When the earth was stamped down, he felt extremely\nweary and weak. He fell down beside his shovel and pick to rest, and\nlay there in the night till he felt stronger. It was damp and dark.\nShadows like clouds hung over the distant outline of the swamp.\n\nThe Sunday bells in the town had ceased. There were no sounds but the\ncries of a few lonely birds and wild creatures of the night, whose\nnames he did not know. This little fact added to his sense of\nsolitude.\n\nHe thought at first he would get up and walk back to the city in the\ndark. An intense and passionate longing seized him to be among living\nmen. He took a few steps down the road. The unwholesome dust blew up\nthrough the dark against his face. He found himself so tired that he\nconcluded to go back to the hut. He would sleep, and start in the\nmorning with the break of the dawn. He should be glad to see the faces\nof his kind again, even though the stir of welcome and the light of\ntrust were gone out of them for him. They lived, they breathed, they\nspoke. He was tired of death and solitude.\n\nHe groped back into the hut. The oil was low, and he could not relight\nthe lamp. He threw himself in the dark upon his bed.\n\nHe slept until late in the morning, heavily. When he waked, the birds\nwere shrill in the hot air, and the sun glared in.\n\n\"I will go now,\" he said, aloud. \"I am glad I can go,\" and crept to\nhis feet.\n\nHe took two steps--staggered--and fell back. He lay for some moments,\nstricken more with astonishment than alarm. His first words were:\n\n\"Lord God! After all--after all I've gone through--Lord God Almighty,\nif You'll believe it--I've _got it_!\"\n\nThis was on Wednesday morning. Night fell, but no one came.\nThursday--but outside the hut no step stirred the parched, white dust.\nFriday--Saturday--no voice but his own moaning broke upon the sick\nman's straining ear.\n\nHis professional experience gave him an excruciating foresight of his\nsymptoms, and their result presented itself to him with horrible\ndistinctness. As one by one he passed through the familiar conditions\nwhose phases he had watched in other men a hundred times, he would\nhave given his life for a temporary ignorance. His trained imagination\nhad little mercy on him. He weighed his chances, and watched his fate\nwith the sad exactness of knowledge.\n\nAs the days passed, and no one came to him, he was aware of not being\nable to reason with himself clearly about his solitude. Growing weak,\nhe remembered the averted faces of the people for whom he had labored,\nand whom he had loved. In the stress of his pain their estranged eyes\ngazed at him. He felt that he was deserted because he was distrusted.\nPatient as he was, this seemed hard.\n\n\"They did not care enough for me to miss me,\" he said, aloud, gently.\n\"I suppose I was not worth it. I had been in prison. I was a wicked\nman. I must not blame them.\"\n\nAnd again:\n\n\"They would have come if they had known. They would not have let me\n_die_ alone. I don't think _she_ would have done that. I wonder where\nshe is? Nobody has missed me--that is all. I must not mind.\"\n\nGrowing weaker, he thought less and prayed more. He prayed, at last,\nalmost all his time. When he did not pray, he slept. When he could not\nsleep, he prayed. He addressed God with that sublime familiarity of\nhis, which fell from his lips with no more irreverence than the kiss\nof a child falling upon its mother's hand or neck.\n\nThe murderer, the felon, the outcast, talked with the Almighty\nHoliness, as a man talketh with his friends. The deserted, distrusted,\ndying creature believed himself to be trusted by the Being who had\nbestowed on him the awful gift of life.\n\n\"Lord,\" he said, softly, \"I guess I can bear it. I'd like to see\nsomebody--but I'll make out to get along.... Lord! I'm pretty weak. I\nknow all about these spasms. You get delirious next thing, you know.\nThen you either get better or you never do. It'll be decided by Sunday\nnight. Lord! Dear Lord!\" he added, with a tender pause, \"don't _You_\nforget me! I hope _You'll_ miss me enough to hunt me up.\"\n\nIt grew dark early on Saturday night. The sun sank under a thin,\ndeceptive web of cloud. The shadow beneath the palmetto grew long over\nScip's fresh grave. The stars were dim and few. The wind rose, and the\nlights in the city, where watchers wept over their sick, trembled on\nthe frail breeze, and seemed to be multiplied, like objects seen\nthrough tears.\n\nThrough the wooden shutter, Zerviah could see the lights, and the\nlonely palmetto, and the grave. He could see those few cold stars.\n\nHe thought, while his thoughts remained his own, most tenderly and\nlongingly of those for whom he had given his life. He remembered how\nmany keen cares of their own they had to carry, how many ghastly deeds\nand sights to do and bear. It was not strange that he should not be\nmissed. Who was he?--a disgraced, unfamiliar man, among their kin and\nneighborhood. Why should they think of him? he said.\n\nYet he was glad that he could remember them. He wished his living or\nhis dying could help them any. Things that his patients had said to\nhim, looks that healing eyes had turned on him, little signs of human\nlove and leaning, came back to him as he lay there, and stood around\nhis bed, like people, in the dark hut.\n\n\"_They loved me_,\" he said: \"Lord, as true as I'm alive, they did!\nI'm glad I lived long enough to save life, _to save life_! I'm much\nobliged to You for that! I wish there was something else I could do\nfor them.... Lord! I'd be willing to die if it would help them any. If\nI thought I could do anything that way, toward sending them a frost--\n\n\"No,\" he added, \"that ain't reasonable. A frost and a human life ain't\nconvertible coin. He don't do unreasonable things. May be I've lost\nmy head already. But I'd be glad to. That's all. I suppose I can\n_ask_ You for a frost. _That's_ reason.\n\n\"Lord God of earth and heaven! that made the South and North, the\npestilence and destruction, the sick and well, the living and the\ndead, have mercy on us miserable sinners! Have mercy on the folks that\npray to You, and on the folks that don't! Remember the old graves, and\nthe new ones, and the graves that are to be opened if this hellish\nheat goes on, and send us a blessed frost, O Lord, _as an act of\nhumanity_! And if that ain't the way to speak to You, remember I\nhaven't been a praying man long enough to learn the language very\nwell,--and that I'm pretty sick,--but that I would be glad to die--to\ngive them--a great, white, holy frost. Lord, a frost! Lord, a cool,\nwhite, clean frost, for these poor devils that have borne so much!\"\n\nAt midnight of that Saturday he dozed and dreamed. He dreamed of what\nhe had thought while Scip was sick: of what it was like, to be holy;\nand, sadly waking, thought of holy people--good women and honest men,\nwho had never done a deadly deed.\n\n\"I cannot be holy,\" thought Zerviah Hope; \"but I can pray for frost.\"\nSo he tried to pray for frost. But by that time he had grown confused,\nand his will wandered pitifully, and he saw strange sights in the\nlittle hut. It was as if he were not alone. Yet no one had come in.\n_She_ could not come at midnight. Strange--how strange! Who was that\nwho walked about the hut? Who stood and looked at him? Who leaned to\nhim? Who brooded over him? Who put arms beneath him? Who looked at\nhim, as those look who love the sick too much to shrink from them?\n\n\"I don't know You,\" said Zerviah, in a distinct voice. Presently he\nsmiled. \"Yes, I guess I do. I see now. I'm not used to You. I never\nsaw You before. You are Him I've heered about--God's Son! God's Son,\nYou've taken a great deal of trouble to come here after me. Nobody\nelse came. You're the only one that has remembered me. You're very\ngood to me.\n\n\"... Yes, I remember. They made a prisoner of _You_. Why, yes! They\ndeserted _You_. They let _You_ die by Yourself. What did You do it\nfor? I don't know much about theology. I am not an educated man. I\nnever prayed till I come South.... I forget--_What did You do it\nfor?_\"\n\nA profound and solemn silence replied.\n\n\"Well,\" said the sick man, breaking it in a satisfied tone, as if he\nhad been answered, \"I wasn't worth it ... but I'm glad You came. I\nwish they had a frost, poor things! _You_ won't go away? Well, I'm\nglad. Poor things! Poor things! I'll take Your hand, if You've no\nobjections.\"\n\nAfter a little time, he added, in a tone of unutterable tenderness and\ncontent:\n\n\"_Dear_ Lord!\" and said no more.\n\nIt was a quiet night. The stars rode on as if there were no task but\nthe tasks of stars in all the universe, and no sorrow keener than\ntheir sorrow, and no care other than their motion and their shining.\nThe web of cloud floated like exhaling breath between them and the\nearth. It grew cooler before the dawn. The leaves of the palmetto over\nScip's grave seemed to uncurl, and grow lax, and soften. The dust\nstill flew heavily, but the wind rose.\n\nThe Sunday-bells rang peacefully. The sick heard them, and the\nconvalescent and the well. The dying listened to them before they\nleft. On the faces of the dead, too, there came the look of those who\nhear.\n\nThe bells tolled, too, that Sunday. They tolled almost all the\nafternoon. The young Northerner, Dr. Remane, was gone,--a reticent,\nbrave young man,--and the heroic telegraph operator. Saturday night\nthey buried her. Sunday, \"Bobby\" took her place at the wires, and\nspelled out, with shaking fingers, the cries of Calhoun to the wide,\nwell world.\n\nBy sunset, all the bells had done ringing and done tolling. There was\na clear sky, with cool colors. It seemed almost cold about Scip's hut.\nThe palmetto lifted its faint head. The dust slept. It was not yet\ndark when a little party from the city rode up, searching for the\ndreary place. They had ridden fast. Dr. Frank was with them, and the\nlady, Marian Dare. She rode at their head. She hurried nervously on.\nShe was pale, and still weak. The chairman of the Relief Committee was\nwith her, and the sub-committee and others.\n\nDr. Dare pushed on through the swinging door of the hut. She entered\nalone. They saw the backward motion of her gray-sleeved wrist, and\ncame no farther, but removed their hats and stood. She knelt beside\nthe bed, and put her hand upon his eyes. God is good, after all. Let\nus hope that they knew her before they closed.\n\nShe came out, and tried to tell about it, but broke down, and sobbed\nbefore them all.\n\n\"It's a martyr's death,\" said the chief, and added solemnly, \"Let us\npray.\"\n\nHe knelt, and the others with him, between the buried <DW64> and the\nunburied nurse, and thanked God for the knowledge and the recollection\nof the holy life which this man had lived among them in their hour of\nneed.\n\n       *       *       *       *       *\n\nThey buried him, as they must, and hurried homeward to their living,\ncomforting one another for his memory as they could.\n\nAs for him, he rested, after her hand had fallen on his eyes. He who\nhad known so deeply the starvation of sleeplessness, slept well that\nnight.\n\nIn the morning, when they all awoke, these of the sorrowing city here,\nand those of the happy city yonder; when they took up life again with\nits returning sunrise,--the sick and the well, the free and the\nfettered, the living and the dead,--the frost lay, cool, white,\nblessed, on his grave.\n\n\n\n\nTHE LIFE-MAGNET.\n\nBY ALVEY A. ADEE.\n\n_Putnam's Magazine, August, 1870._\n\n\nThere was something about the wholesome sleepiness of Freiberg, in\nSaxony, that fitted well with the lazy nature of Ronald Wyde. So,\nhaving run down there to spend a day or two among the students and the\nmines, and taking a liking to the quaint, unmodernized town, he bodily\nchanged his plans of autumn-travel, gave up a cherished scheme of\nRussian vagabondage, had his baggage sent from Dresden, and made ready\nto settle down and drowse away three or four months in idleness and\nnot over-arduous study. And this move of his led to the happening of a\nvery strange and seemingly unreal event in his life.\n\nRonald Wyde was then about twenty-five or six years old, rather above\nthe medium height, with thick blue-black hair that he had an\nartist-trick of allowing to ripple down to his neck, dark hazel eyes\nthat were almost too deeply recessed in their bony orbits, and a\ntroublesome growth of beard that, close-shaven as he always was,\nshowed in strong blue outline through the thin and rather sallow skin.\nHis address was singularly pleasing, and his wide experience of life,\ntaught him by years of varied travel, made him a good deal of a\ncosmopolitan in his views and ways, which caused him to be looked upon\nas a not over-safe companion for young men of his own age or under.\n\nHaving made up his mind to winter in Freiberg, his first step was to\nquit the little hotel, with its mouldy stone-vaulted entrance and its\ncolumned dining-room, under whose full-centered arches close beery and\nsmoky fumes lingered persistently, and seek quieter student-lodgings\nin the heart of the town. His choice was mainly influenced by a\nthin-railed balcony, twined through and through by the shoots of a\nvigorous Virginia creeper, that flamed and flickered in the breezy\nOctober sunsets in strong relief against the curtains that drifted\nwhitely out and in through the open window. So, with the steady-going\nand hale old Frau Spritzkrapfen he took up his quarters, fully\npersuading himself that he did so for the sake of the stray\nhome-breaths that seemed to stir the scarlet vine-leaves more gently\nfor him, and ignoring pretty Lottchen's great, earnest Saxon eyes as\nbest he could.\n\nA sunny morning followed his removal to Frau Spritzkrapfen's tidy\nhome. There had been a slight rain in the early night, and the\nfootways were yet bright and moist in patches that the slanting\nmorning rays were slowly coaxing away. Ronald Wyde, having set his\nfavorite books handily on the dimity-draped table, which comprised for\nhim the process of getting to rights, and having given more than one\nglance of amused wonderment at the naive blue-and-white scriptural\ntiles that cased his cumbrous four-story earthenware stove, and smiled\nlazily at poor Adam's obvious and sudden indigestion, even while the\nuneaten half-apple remained in his guilty hand, he stepped out on his\nbalcony, leaned his elbows among the crimson leaves, and took in the\nhealthful morning air in great draughts. It was a Sunday; the bells of\nthe gray minster hard by were iterating their clanging calls to the\nsimple townsfolk to come and be droned to in sleepy German gutturals\nfrom the carved, pillar-hung pulpit inside. Looking down, he saw\nthick-ankled women cluttering past in loose wooden-soled shoes, and\ndumpy girls with tow-braids primly dangling to their hips, convoying\nsturdy Dutch-built luggers of younger brothers up the easy <DW72> that\nled to the church and the bells. Presently Frau Spritzkrapfen and\ndainty Lottchen, rosy with soap and health, slipped through the\ndoorway beneath him out into the little church-bound throng, and, as\nthey disappeared, left the house and street somehow unaccountably\nalone. Feeling this, Ronald Wyde determined on a stroll.\n\nSomething in the Sabbath stillness around him led Ronald away from the\nswift clang and throbbing hum of the bells and in the direction of the\nold cemetery. Passing through the clumsy tower-gate that lifts its\ngrimy bulk sullenly, like a huge head-stone over the grave of a dead\ntime of feudalism, he reached the burial-ground and entered the quiet\nenclosure. The usual touching reverence of the Germans for their dead\nwas strikingly manifest around him. The humbler mounds, walled up with\nrough stones a foot or two above the pathway level, carried on their\ncrests little gardens of gay and inexpensive plants; while on the tall\nwooden crosses at their head hung yellow wreaths, half hiding the\nhopeful legend, \"Wiedersehen.\" The more pretentious slabs bore vases\nfilled with fresh flowers; while in the grate-barred vaults, that\nskirted the ground like the arches of a cloister, lay rusty heaps of\nlong-since mouldered bloom, topped by newer wreaths tossed lovingly in\nto wilt and turn to dust in their turn, like those cast in before them\nin memory of that other dust asleep below.\n\nTurning aside from the central walk that halved the cemetery, Ronald\nstrolled along, his hands in his pockets, his eyes listlessly fixed on\nthe orange- fumes and rolling smoke that welled out of tall\nchimneys in the hollow beyond, an idle student-tune humming on his\nlips, and his thoughts nowhere, and everywhere, at once. Happening to\nlook away from the dun smoke-trail for an instant, he found something\nof greater interest close at hand. An old man stooped stiffly over a\nsimple mound, busied among the flowers that hid it, and by his side\ncrouched a young girl, perhaps fourteen years old, who peered up at\nRonald with questioning, velvet-brown eyes. The old man heard the\nintruder's steps crunching in the damp gravel, and slowly looked up\ntoo.\n\n\"Good morning, mein Herr,\" said Ronald, pleasantly.\n\nThe old man remained for an instant blinking nervously, and shading\nhis eyes from the full sunlight that fell on his face. A quiet face it\nwas, and very old, seamed and creased by mazy wrinkles that played at\naimless cross-purposes with each other, beginning and ending nowhere.\nHis thick beard and thin, curved nose looked just a little Jewish, and\nseemed at variance with his pale blue eyes that were still bright in\nspite of age. And yet, bearded as he was, there was a lurking\nexpression about his features that bordered upon effeminacy, and made\nthe treble of his voice sound even more thin and womanish as he\nanswered Wyde's greeting.\n\n\"Good morning, too, mein Herr. A stranger to our town, I see.\"\n\n\"Yes; but soon not to be called one, I hope. I am here for the\nwinter.\"\n\n\"A cold season--a cold season; our northern winters are very chilling\nto an old man's blood.\" And slouching together into a tired stoop, he\nresumed his simple task of knotting a few flowers into a clumsy\nnosegay. Ronald stood and watched him with a vague interest.\nPresently, the flowers being clumped to his liking, the old man pried\nhimself upright by getting a good purchase with his left hand in the\nsmall of his back, and so deliberately that Ronald almost fancied he\nheard him creak. The girl rose too, and drew her thin shawl over her\nshoulders.\n\n\"You Germans love longer than we,\" said Ronald, glancing at the\nflowers that trembled in the old man's bony fingers, and then\ndownwards to the quiet grave; \"a lifetime of easy-going love and a\nyear or two of easier-forgetting are enough for us.\"\n\n\"Should I forget my own flesh and blood?\" asked the old man, simply.\n\nRonald paused a moment, and, pointing downwards, said:\n\n\"Your daughter, then, I fancy?\"\n\n\"Yes.\"\n\n\"Long dead?\"\n\n\"Very long; more than fifty years.\"\n\nRonald stared, but said nothing audibly. Inwardly he whispered\nsomething about being devilish glad to make the wandering Jew's\nacquaintance, rattled the loose groeschen in his pocket, and turned to\nfollow the tottering old man and firm-footed child down the walk.\nAfter a dozen paces they halted before a more ambitious tombstone, on\nwhich Ronald could make out the well-remembered name of Plattner. The\nchild took the flowers and laid them reverently on the stone.\n\n\"It seems to me almost like arriving at the end of a pilgrimage,\" said\nRonald, \"when I stand by the grave of a man of science. Perhaps you\nknew him, mein Herr?\"\n\n\"He was my pupil.\"\n\n\"Whew!\" thought Ronald, \"that makes my friend here a centenarian at\nleast.\"\n\n\"My pupil and friend,\" the feeble voice went on; \"and, more than that,\nmy daughter's first lover, and only one.\"\n\n\"Ach so!\" drawled Ronald.\n\n\"And now, on her death-day, I take these poor flowers from her to him,\nas I have done all these years.\"\n\nSomething in the pathetic earnestness of his companion touched Ronald\nWyde, and he forthwith took his hands out of his pockets, and didn't\ntry to whistle inaudibly--which was a great deal for him to do.\n\n\"I know Plattner well by his works,\" he said; \"I once studied\nmineralogy for nearly a month.\"\n\n\"You love science, then?\"\n\n\"Yes; like every thing else, for diversion.\"\n\n\"It was different with him,\" quavered the old man, pointing unsteadily\nto the head-stone. \"Science grew to be his one passion, and many\ndiscoveries rewarded him for his devotion. He was groping on the track\nof a far greater achievement when he died.\"\n\n\"May I ask what it was?\" said Ronald, now fairly interested.\n\n\"The creation and isolation of the principle of Life!\"\n\nThis was too much for Ronald Wyde; down dived his restless hands into\nhis trowsers' pockets again, and the groeschen rattled as merrily as\nbefore.\n\n\"I have made quite a study of biology, and all that sort of thing,\"\nsaid he; \"and, although a good deal of a skeptic, and inclined to\nfollow Huxley, I can't bring myself to conceive of life without\norganism. Such theorizing is, to my mind, on a par with the illogical\nsearch for the philosopher's stone and a perpetual motor.\"\n\nThe old man's eyes sparkled as he turned full upon Ronald.\n\n\"You dismiss the subject very airily, my young friend,\" he cried; \"but\nlet me tell you that I--I, whom you see here--have grappled with such\nproblems through a weary century, and have conquered one of them.\"\n\n\"And that one is--\"\n\n\"The one that conquered Plattner!\"\n\n\"Do I understand you to claim that you have discovered the\nlife-principle?\"\n\n\"Yes.\"\n\n\"Will you permit an utter stranger to inquire what is its nature?\"\n\n\"Certainly. It is twofold. The ultimate principle of life is carbon;\nthe cause of its combination with water, or rather with the two\ngaseous elements of water, and the development of organized existence\ntherefrom, is electricity.\"\n\nRonald Wyde shrugged his broad shoulders a little, and absently\nreplied,\n\n\"All I can say, mein Herr, is, that you've got the bulge on me.\"\n\n\"I beg your pardon--\"\n\n\"Excuse me; I unconsciously translated an Americanism. I mean that I\ndon't quite understand you.\"\n\n\"Which means that you do not believe me. It is but natural at your\nage, when one doubts as if by instinct. Would you be convinced?\"\n\n\"Nothing would please me better.\"\n\nWith the same painful effort as before, the old man straightened\nhimself and made a piercing clairvoyant examination into and through\nRonald Wyde's eyes, as if reading the brain beyond them.\n\n\"I think I can trust you,\" he mumbled at last. \"Come with me.\"\n\nLeaning on the young girl's arm, the old philosopher faltered through\nthe cemetery and into the town, followed by Wyde, his hands again\npocketed for safety. Groups of released church-goers, sermon-fed, met\nthem, and once in a while some stout burgher would nod patronizingly\nto Ronald's guides, and get in response a shaky, sidelong roll of the\nold man's head, as if it were mounted on a weak spiral spring. Further\non they intersected a knot of students, who eyed them askance and\nexchanged remarks in an undertone. Keeping on deeper into the foul\nheart of the town, they passed through swarms of idle children playing\nsportlessly, as poverty is apt to play, in the dank shadows of the\nnarrow street. They seemed incited to mirth and ribaldry by the sight\nof Ronald's new friend, and one even ventured to hurl a clod at him;\nbut this striking Ronald instead, and he facing promptly to the\nhostile quarter from whence it came, caused a sudden slinking of the\ncrowd into unknown holes, like a horde of rats, and the street was for\na time empty save for the little party that threaded it. Ronald began\nto think that the old man's sanity was gravely called in doubt by the\ntownsfolk, and would readily have backed out of his adventure but for\nthe curiosity that had now got the upper hand of him.\n\nPresently the old man sidled into a dingy doorway, like a tired beast\nrun to earth, and Ronald followed him, not without a wish that the\narchitect had provided for a more efficient lighting of the sombre\npassage-way in which he found himself. A sharp turn to the right after\na dozen groping-paces, a narrow stairway, a bump or two against\nunexpected saliences of rough mortared wall, two steps upward and one\nvery surprising step downward through a cavernous doorway that took\naway Ronald's breath for a moment, and sent it back again with a hot,\ncreeping wave of sudden perspiration all over him, as is the way with\nmissteps, and two more sharp turns, brought the three into a black\nno-thoroughfare of a hall, whose further end was closed by a locked\ndoor. The girl here rubbed a brimstone abomination of a match into a\nmal-odorous green glow, and by its help the old man got a tortuous key\ninto the snaky opening in the great lock, creakily shot back its bolt,\nswung open the door, and motioned Ronald to enter.\n\nHe found himself in a long and rather narrow room, with a high\nceiling, duskily lighted by three wide windows that were thickly\nwebbed and dusted, like ancestral bottles of fine crusty Port.\nA veritable den it was, filled with what seemed to be the\nwrecks of philosophical apparatus dating back two or three\ngenerations--ill-fated ventures on the treacherous main of science.\nHere a fat-bellied alembic lolled lazily over in a gleamy sand-bath,\nlike a beach-lost galleon at ebb-tide; and there a heap of broken\nporcelain-tubing and shreds of crucibles lay like bleaching ship-ribs\non a sullen shore. Beyond, by the middle window, stood a furnace,\nfireless, and clogged with gray ashes. Two or three solid old-time\ntables, built when joiners were more lavish of oaken timber than\nnowadays, stood hopelessly littered with retorts, filtering funnels,\nlamps, ringstands, and squat-beakers of delicate glass, caked with\nlong-dried sediment, all alike dust-smirched. Ronald involuntarily\nsought for some huge Chaldaic tome, conveniently open at a favorite\nspell, or a handy crocodile or two dangling from the square beams\noverhead, but saw nothing more formidable than a stray volume of\n\"Kant's Critique of Pure Reason.\" Taking this up and glancing at its\nfly-leaf, he saw a name written in spidery German script, almost\nillegible from its shakiness--\"Max Lebensfunke.\"\n\n\"Your name?\" he asked.\n\n\"Yes, mein Herr,\" answered the old man, taking the volume and\ncaressing it like a live thing in his fumbling hands. \"This book was\ngiven to me by the great Kant himself,\" he added.\n\nReverently replacing it, he advanced a few steps toward the middle of\nthe room. Ronald followed, and, turning away from the windows, looked\nfurther around him. In striking contrast to the undisturbed disorder,\nso redolent of middle-age alchemy, was the big table that flanked the\nlaboratory through its whole length. It began with a powerful galvanic\nbattery, succeeded by a wiry labyrinth of coils and helices, with\nlittle keys in front of them like a telegraph-office retired from\nbusiness; these gave place to many-necked jars wired together by twos\nand threes, like oath-bound patriots plotting treason; beyond them\nstood a great glass globe, connected with a sizable air-pump, and\nfilled with a complexity of shiny wires and glassware; next loomed up\na huge induction-magnet, carefully insulated on solid glass supports;\nand at the further extremity of the table lay--a corpse.\n\nRonald Wyde, in spite of his many-sided experience of\ndissection-rooms, and morgues, and other ghastlinesses to which he had\nlong since accustomed himself from principle, drew back at the\nsight--perhaps because he had come to this strange place to clutch the\nworld-old mystery of the life-essence, and found himself, instead,\nconfronted on its threshold by the equal mystery of death.\n\nHerr Lebensfunke smiled feebly at this movement.\n\n\"A subject received this morning from Berlin,\" he said, in answer to\nWyde's look of inquiry. \"A sad piece of extravagance, mein Herr--a\nluxury to which I can rarely afford to treat myself.\"\n\nRonald Wyde bent over the body and looked into its face. A rough, red\nface, that had seemingly seen forty years of low-lived dissipation.\nThe blotched skin and bleary eyes told of debauchery and drunkenness,\nand a slight alcoholic foetidness was unpleasantly perceptible, as\nfrom the breath of one who sleeps away the effects of a carouse.\n\n\"I hope you don't think of restoring this soaked specimen to life?\"\nsaid Ronald.\n\n\"That is still beyond me,\" answered the old man, mournfully. \"As yet I\nhave not created life of a higher grade than that of the lowest\nzoophytes.\"\n\n\"Do you claim to have done as much as that?\"\n\n\"It is not an idle claim,\" said Herr Lebensfunke, solemnly. \"Look at\nthis, if you doubt.\"\n\n\"This\" was the great crystal globe that rose from the middle of the\nlong table, and dominated its lesser accessories, as some great dome\nswells above the clustered houses of a town. Tubes passing through its\nwalls met in a smaller central globe half filled with a colorless\nliquid. Beneath this, and half encircling it, was an intricate maze of\nbright wire; and two other wires dipped into it, touching the surface\nof the liquid with their platinum tips. Within the liquid pulsed a\nshapeless mass of almost transparent spongy tissue.\n\n\"You see an aggregation of cells possessed of life--of a low order, it\nis true, but none the less life,\" said the philosopher, proudly.\n\"These were created from water chemically pure, with the exception of\na trace of ammonia, and impregnated with liquid carbon, by the\ncombined action of heat and induced electricity, in vacuo. Look!\"\n\nHe pressed one of the keys before him. Presently the wire began to\nglow with a faint light, which increased in intensity till the coil\nflamed into pure whiteness. Removing his finger, the current ceased to\nflow, and the wire grew rapidly cool.\n\n\"I passed the whole strength of sixty cups through it to show you its\naction. Ordinarily, with one or two carbon cells, and refining the\ncurrent by triple induction, the temperature is barely blood-warm.\"\n\n\"Pardon an interruption,\" said Ronald. \"You spoke of liquid carbon;\ndoes it exist?\"\n\n\"Yes; here is some in this phial. See it--how pure, how transparent!\nhow it loves and hoards the light!\" The old man held the phial up as\nhe spoke, and turned it round and round. \"See how it flashes! No\nwonder, for it is the diamond, liquid and uncrystallized. Think how\nthese fools of men have called diamonds precious above all gems\nthrough these many weary years, and showered them on their kings, or\ntossed them to their mistresses' feet, never dreaming that the silly\nstone they lauded was inert, crystallized life!\"\n\n\"Can't you crystallize diamonds yourself?\" asked Wyde, \"and make\nFreiberg a Golconda and yourself a Croesus?\"\n\n\"It could be done, after the lapse of thousands of years,\" replied\nHerr Lebensfunke. \"Place undiluted liquid carbon in that inner globe,\nkeep the coil at a white heat, and if Adam had started the process,\nhis heir-at-law would have a koh-i-noor to-day, and a nice lawsuit for\nits possession.\"\n\nRonald Wyde bent toward the globe once more and examined the throbbing\nmass closely, whistling softly meanwhile.\n\n\"If you can create this cellular life, why not develop it still higher\ninto an organism?\"\n\n\"Because I can only create life--not soul. Years ago I was a\nfreethinker, now my discoveries have made me a deist; for I found that\nmy cells, living as they were, and possessing undoubted parietal\ncirculation, were not germs; and though they might cluster into a bulk\nlike this, as bubbles do to form froth, to evolve an animal or plant\nfrom them was far beyond me; that needs what we call soul. But, in\nsearching blindly for this higher power, I grasped a greater discovery\nthan any I had hoped for--the power to isolate life from its bodily\norganism.\"\n\n\"You have to keep the bottle carefully corked, I should imagine,\"\nlaughed Ronald.\n\n\"Not quite,\" said Herr Lebensfunke, joining in the laugh. \"Life is not\nglue. My grand discovery is the life-magnet.\"\n\n\"Which has the post of honor on your table here, has it not?\" inquired\nRonald, drawing his hand from his pocket and pointing to the insulated\ncoil.\n\nThe old man glanced keenly at his hand as he did so; at which Ronald\nseemed confused, and pocketed it again abruptly.\n\n\"Yes, that is the life-magnet. You see this bent glass tube surrounded\nby the helix? That tube contains liquid carbon. I pass through the\nhelix a current of induced electricity, generated by the action of\nthese sixty Bunsen cups upon a succession of coils with carbon cores,\nand the magnet becomes charged with soulless life. I reverse the\nstream--what was positive now is negative, and the same magnet will\nabsorb life from a living being to an extent only to be measured by\nthousands of millions.\"\n\n\"Then, what effect is produced on the body you pump the life from?\"\n\n\"Death.\"\n\n\"And what becomes of the soul?\"\n\n\"I don't quite know. I fancy, however, that the magnet absorbs that\ntoo.\"\n\n\"Can it give it back?\"\n\n\"Certainly; otherwise my life-magnet would belie its name, and be\nsimply an ingenious and expensive instrument of death. By reversing\nthe conditions, I can restore both soul and life to the body from\nwhich I drew them, or to another body, even after the lapse of several\ndays.\"\n\n\"Have you ever done so?\"\n\n\"I have.\"\n\nRonald looked reflectively downward to his boot-toe, but seemed to\nfind nothing there--except a boot-toe.\n\n\"I say, my friend,\" he spoke at last, \"haven't you got a pin you can\nstick in me? I'd like to know if I'm dreaming.\"\n\n\"I can convince you better than by pins,\" replied Herr Lebensfunke.\n\"Let me see that hand you hide so carefully.\"\n\nRonald Wyde slowly drew it from his pocket, as reluctantly as though\nit were a grudged charity dole, and extended it to the old man. Its\nlittle finger was gone.\n\n\"A defect that I am foolishly sensitive about,\" said he. \"A childish\nfreak--playing with edged tools, you know. A boy-playmate chopped it\noff by accident: I cut his head open with his own hatchet, and made an\nidiot of him for life--that's all.\"\n\n\"I could do this,\" said Herr Lebensfunke, pausing on each word as if\nit were somewhat heavy, and had to be lifted out of his cramped chest\nby force; \"I could draw your entity into that magnet, leaving you\nside by side with this corpse. I could dissect a finger from that\nsame corpse, attach it to your own dead hand by a little of that\npalpitating life-mass you have seen, pass an electric stream through\nit, and a junction would be effected in three or four days. I could\nthen restore you to existence, whole, and not maimed as now.\"\n\n\"I don't quite like the idea of dying, even for a day,\" answered Wyde.\n\"Couldn't you contrive to lend me a body while you are mending my\nown?\"\n\n\"You can take that one, if you like.\"\n\nRonald Wyde looked once more at the sodden features of the corpse, and\nsmiled lugubriously.\n\n\"A mighty shabby old customer,\" he said, \"and I doubt if I could feel\nat home in his skin; but I'm willing to risk it for the sake of the\nnovelty of the thing.\"\n\nThe old philosopher's thin face lit up with pleasure.\n\n\"You consent, then?\" he chuckled in his womanish treble.\n\n\"Of course I do. Begin at once, and have done with it.\"\n\n\"Not now, mein Herr; some modifications must be made in the\nconnections--mere matters of detail. Come again to-night.\"\n\n\"At what hour?\"\n\n\"At ten. Mein Voegelein, show the Herr the way out.\"\n\nThe girl, who had been moving restlessly about the room all this time,\nwith her wild brown eyes fixed now on Ronald, now on the old man, and\noftener in a shy, inquisitive stare on the corpse, lit a dusty\nchemical lamp and led the way down the awkward passages and stairs.\nRonald tried to start a conversation with her as he followed.\n\n\"You are too young, my birdling, to be accustomed to such sights as\nthis upstairs.\"\n\n\"Birdling is not too young, she's almost fourteen,\" said the girl,\nproudly. \"And she likes it, too; it makes her think of mother. Mother\nwent to sleep on that table, mein Herr.\"\n\n\"Poor thing! she's half-witted,\" thought Wyde as he passed into the\nstreet. \"By-by, birdie.\"\n\nHome he walked briskly, to be met under his flaming balcony by\nLottchen's kindly afternoon greeting. How had mein Herr passed his\nSabbath? she asked.\n\n\"Quietly enough, Lottchen. I met an old philosopher in the God's-Acre,\nand went home with him to his shop. Have you ever heard of Herr Doctor\nLebensfunke?\"\n\n\"Yes, mein Herr. Wrong here, they say;\" and she tapped her wide, round\nGerman forehead, and lifted her eyes expressively heavenward.\n\n\"Sold himself to the devil, eh?\" asked Wyde.\n\nLottchen was not quite sure on that point. Some said one thing, and\nsome another. There was undoubtedly a devil, else how could good\nDoctor Luther have thrown his inkstand at him? But he had never been\nseen in Doctor Lebensfunke's neighborhood; and, on the whole, Lottchen\nwas inclined to attribute the Herr Doctor's trouble to an indefinable\nsomething whose nature was broadly hinted at by more tapping of the\nforehead.\n\nRonald Wyde mounted the stairs, locked himself in his room, and wished\nhimself out of the scrape he was getting into. But, being in for it\nnow, he lit a cigar, and tried to fancy the processes he would have to\ngo through, and how he, a natty and respectable young fellow, would\nlook and feel in a drunkard's skin. His conjectures being too foggily\noutlined to please him, he put them aside, and waited impatiently\nenough for ten o'clock.\n\nA moonlight walk through the low streets, transfigured by the silver\ngleam into fairy vistas--all but the odor--brought him to Herr\nLebensfunke's house. Simple birdling, on the lookout for him, piloted\nhim through the unsafe channel, and brought him to anchor in the\ndimly-lit room.\n\n\"All is ready,\" said the philosopher, as he trembled forward and shook\nRonald's hand. \"See here.\" Zig-zags of silk-bound wire squirmed hither\nand thither from the life-magnet. Two of them ended in carbon points.\n\"And here, too, my young friend, is your new finger.\"\n\nIt lay, detached, in the central globe, and on its severed end atoms\nof protoplasm were already clustered. \"Literally a second-hand\narticle,\" thought Ronald; but, not venturing to translate the idiom,\nhe only bowed and said, \"Ach so!\" which means any thing and every\nthing in German.\n\nIt was not without a very natural sinking of the heart that Ronald\nWyde divested himself of his clothing, and took his position, by the\nold man's direction, on the stout table, side by side with the dead. A\nflat brass plate pressed between his shoulders, and one of the carbon\npoints, clamped in a little insulated stand, rested on his bosom and\nquivered with the quickened motion of the heart beneath it. The other\npoint touched the dead man's breast.\n\n\"Are you ready?\"\n\n\"Yes.\"\n\nThe old man pressed a key, and as he did so a sharp sting, hardly\nworse than a leech's bite, pricked Ronald Wyde's breast. A sense of\nlanguor crept slowly upon him, his feet tingled, his breath came\nslowly, and waves of light and shade pulsed in indistinct alternation\nbefore his sight; but through them the old man's eyes peered into his,\nlike a dream. Presently Ronald would have started if he could, for\ntwo old philosophers were craning over him instead of one. But as he\nlooked more steadily, one face softly dimmed into nothing, and the\nother grew brighter and stronger in its lines, while the room flushed\nwith an unaccountable light. The little key clicked once more;\na vague sensation that the current had somehow ceased to flow,\nroused him, and he raised himself on his elbow and looked in blank\nbewilderment at his own dead self lying by his side in the daylight,\nwhile the sunrise tried to peer through the webbed panes.\n\n\"Is it over?\" he asked, with a puzzled glance around him; and added,\n\"Which am I?\"\n\n\"Either, or both,\" answered Herr Lebensfunke. \"Your identity will be\nsomething of a problem to you for a day or two.\"\n\nAided by the old man, Ronald awkwardly got into the sleazy clothes\nthat went with the exchange--growing less and less at home each\nminute. He felt weak and sore; his head ached, and the wound left by\nthe fresh amputation of his little finger throbbed angrily.\n\n\"I suppose I may as well go now,\" he said. \"When can I get my own self\nthere back again?\"\n\n\"On Thursday night, if all works well,\" said the old man. \"Till then,\ngood-day.\"\n\nRonald Wyde's first impulse, as he shambled into the open air, was to\ngo home; but he thought of the confusion his sadly-mixed identity\nwould cause in Frau Spritzkrapfen's quiet household, and came to a\ndead stop to consider the matter. Then he decided to quit the town for\nthe interminable four days--to go to Dresden, or anywhere. His next\nstep was to slouch into the nearest beer-cellar and call for beer,\npen, and paper. While waiting for these, he surveyed his own\nreflection in the dingy glass that hung above the table he sat by--a\nglass that gave his face a wavy look, as if seen through heated air.\nHe felt an amused pride in his altered appearance, much as a\nmasquerader might be pleased with a clever disguise, and caught\nhimself wondering whether he were likely to be recognized in it.\nApparently satisfied of his safety from detection, he turned to the\ntable and wrote a beer-scented note to Frau Spritzkrapfen, explaining\nhis sudden absence by some discreet fiction. He got along well enough\ntill he reached the end, when, instead of his own flowing sign-manual,\nhe tipsily scrawled the unfamiliar name of Hans <DW62>. Tearing the\nsheet angrily across, he wrote another, and signed his name with an\neffort. He was about to seek a messenger to carry his note, when it\noccurred to him to leave it himself, which he did; and had thereby the\nkeen satisfaction of hearing pretty Lottchen confess, with a blush on\nher fair German cheek, that they would all miss Herr Wyde very much,\nbecause they all loved him. Turning away with a sigh that was very\nlike a hiccough, he trudged to the railway-station and took a ticket\nto Dresden, going third-class as best befitting his clothes and\nappearance.\n\nHe felt ashamed enough of himself as the train rumbled over the\nrolling land between Freiberg and the capital, and gave him time to\nthink connectedly over what had happened, and what he now was. His\nfellow-passengers cast him sidelong looks, and gave him a wide berth.\nEven the quaint, flat-arched windows of one pane each, that winked out\nof the red-tiled roofs like sleepy eyes, seemed to leer drunkenly at\nhim as they scudded by.\n\nRonald Wyde's account of those days in Dresden was vague and misty. He\ncrept along the bustling streets of that sombre, gray city, that\nseemed to look more natural by cloud-light than in the full sunshine,\nfeeling continually within him a struggle between the two incompatible\nnatures now so strangely blended. Each day he kept up the contest\nmanfully, passing by the countless beer-cellars and drinking-booths\nwith an assumption of firmness and resolution that oozed slowly away\ntoward nightfall, when the animal body of the late Hans <DW62> would\ncontrive to get the better of the animating principle of Ronald Wyde;\nthe refined nature would yield to the toper's brute-craving, with an\nawful sense of its deep degradation in so succumbing, and, before\nmidnight, Hans was gloriously drunk, to Ronald's intense grief.\n\nTime passed somehow. He had memories of sunny lounges on the\nBruhl'sche Terrace, looking on the turbid flow of the eddied Elbe, and\nwatching the little steamboats that buzzed up and down the city's\nflanks, settling now and then, like gad-flies, to drain it of a few\ndrops of its human life. Well-known friends, whose hands he had\ngrasped not a week before, passed him unheedingly; all save one,\nwho eyed him for a moment, said \"Poor devil!\" in an undertone, and\ndropped a silber-gro' into his maimed hand. He felt glad of even this\nlame sympathy in his lowness; but most of all he prized the moistened\nglance of pity that flashed upon him from the great dark eyes of a\nlovely girl who passed him now and then as he slouched along. Once, a\nbeing as degraded and scurvy as his own outward self, turned to him,\ncalled him \"Dutzbruder,\" asked him how he left them all in Berlin,\nstared at Ronald's blank look of non-recognition, and passed on with a\nmuttered curse on his own stupidity in mistaking a stranger, in broad\ndaylight, for his crony <DW62>.\n\nAnother memory was of the strange lassitude that seemed to almost\nparalyze him after even moderate exertion, and caused him to drop\nexhausted on a bench on the terrace when he had shuffled over less\nthan half its length. More than once the suspicion crept upon him\nthat only a portion of his vitality now remained to him, and that\nits greater part lay mysteriously coiled in Herr Lebensfunke's\nlife-magnet. And this, in turn, broadened into a doubting distrust\nof the Herr himself--a dread lest the old man might in some way\nappropriate this stock of life to his own use, and so renew his\nfast-expiring lease for a score or two of years to come. At last this\ndread grew so painfully definite, that he hurried back to Freiberg a\nday before his appointed time, and once more found his twofold self\nwandering through its devious streets.\n\nIt was long after dark, and a thin rain slanted on the slippery\nstones, as he again made his way through the deserted and sleepy paths\nof the town to the old philosopher's house. He was wet, chilled,\nweary, and sick enough at heart as he leaned against the cold stone\ndoorway and waited for an answer to his knock. The plash of the\nheavier rain-drops from the tiled leaves was the only sound he heard\nfor many minutes, until, at last, pattering feet neared him on the\ninside, and a child's voice asked who was there. To his friendly\nresponse the door was opened half-wide, and Voegelein's blank, pretty\nface peeped through.\n\nWas Herr Lebensfunke at home? No; he had said that he wasn't at home;\nbut then, she thought he was in the long room where mamma went to\nsleep. Could he be seen? No, she thought not; he was very tired, and,\nin her own--Voegelein's--opinion, he was going to sleep too, just as\nmamma did. And the wizened little face, with its eldritch eyes and\ntangled hair, was withdrawn, and the door began to close. Ronald\nforced himself inside, and grasped the child's arm.\n\n\"Voegelein, don't you know me?\"\n\nThe girl, in nowise startled, gravely set her flickering candle on the\ndoor-step, looked up at him wonderingly, as if he were an exhibition,\nand said she thought not, unless he had been asleep on the table.\n\n\"Good heavens!\" cried Ronald, \"can this child talk of nothing but\npeople asleep on a table?\"\n\nBut, as he spoke, a thought whirred through his brain. He drew the\npoor half-witted thing close to him and asked:\n\n\"Can Voegelein tell me something about mamma, and how she went to\nsleep?\"\n\nThe child rambled on, pleased to find a listener to her foolish\nprattle. All he could connect into a narrative was, that the girl's\nmother, some seven or eight years before, had been drained of her life\nby the awful magnet, and that, as the child said, \"the Herr Doctor\never since had talked just like mamma.\"\n\nHis dread was well founded, then. The old man's one dream and aim was\nto prolong his wretched life; could he doubt that he would not now\nmake use of the means he had so unwisely thrown in his way? He turned\nabout, half maddened.\n\n\"Girl!\" he cried, \"I must see the old man! Where is he?\"\n\nHe couldn't see him, she whined. He was asleep up there, on the table.\nAt one o'clock he had said he would wake up.\n\nHe pushed past her, mounted to the long room, pressed open the\nunfastened door, and entered.\n\nThe old man and the corpse of his former self lay together under the\nlight of a lamp that swung from the beam overhead. An insulated carbon\npoint was directed to each white, still breast. From the old man's\nhand a cord ran to a key beyond, arranged to make or break connection\nat a touch. By it stood a clock, with a simple mechanism attached that\nbore upon a second key like the first, evidently planned to press upon\nit when the hands should mark a given hour. The child had said that he\nwould wake at one, and it was now past midnight.\n\nRonald Wyde comprehended it all now. The wily old man's feeble life\nhad been withdrawn into the great magnet, and mixed therein with what\nremained of his own. In less than an hour the key would fall, and the\ndouble stream would flow into and animate his young body, which would\nthen wake to renewed life; while the cast-off shell beside it, worn to\nutter uselessness by a toilsome century, would be left to moulder as a\nmothed garment.\n\nSurely no time was to be lost; his life depended upon instant action.\nAnd yet, comprehending this, he went to work slowly, and as a\nsomnambulist might, acting almost by instinct, and well knowing that a\nblunder now meant irrevocable death.\n\nCarefully disengaging the cord from the old man's yet warm grasp, and\nsetting the carbon point aside, he lifted the shrivelled corpse and\nbore it away, to cast it on the white rubbish-heap in one corner.\nReturning to his work, he stripped himself, and laid down in the old\nman's place. As he did so, the distant Minster bells rang the\nthree-quarters.\n\nWas there yet time?\n\nHe braced his shoulders firmly against the brass plate under them,\nand moved the carbon point steadily back to its place, with its tip\nresting on his breast; the silk-wrapped wire that dangled between\nit and the magnet quivering, as he did so, as with conscious life.\nDrawing a long breath, he tightened the cord, and heard a faint click\nas the key snapped down.\n\nThe same sharp sting as before instantly pricked his breast, tingling\nthrills pulsed over him, beats of light and shadow swept before his\neyes, and he lost all consciousness. For how long he knew not. At last\nhe felt, rather than saw, the lamp-rays flickering above him, and\nopened his eyes as though waking from a tired sleep. Sitting up, he\ngave a fearful look around him, as if dreading what he might see. The\ndrunkard's body lay stretched and motionless beside him, and the clock\nmarked three. He was saved!\n\nSlipping down from his perilous bed, he resumed the old familiar\ngarments that belonged to him as Ronald Wyde, shuddering with emotion\nas he did so. Only pausing to give one look at the pale heap in the\nshadowy corner, and at the other sleeper under the now dying lamp, he\nquitted the room and locked its heavy door upon the two silent\nguardians of its life-secrets. When he reached the street, he found\nthe rain had ceased to drop, and that the cold stars blinked over the\nslumbrous town.\n\nBefore noon he had taken leave of Frau Spritzkrapfen, turned buxom\nLottchen scarlet all over by a hearty, sudden, farewell-kiss, and was\nfar on his way from Freiberg, with its red-vined balcony and its dark\nlaboratory, never again to visit it or them. And as the busy engine\ntoiled and shrieked, and with each beat of its mighty steam-heart\ncarried him further away, his thoughts flew back and clustered around\nwitless, brown-eyed birdling. Poor child, he never learned her fate.\n\n       *       *       *       *       *\n\nI heard this strange story from its hero, one sunny summer morning as\nwe swept over the meadowy reaches of the Erie Railway, or hung along\nthe cliffside by the wooded windings of the Susquehanna. When he had\nended it, he smiled languidly, and, showing me his still-mutilated\nhand, said that the old doctor's job had been a sad bungle, after all.\nIn fact, the only physical proof that remained to verify his story,\nwas a curved blue spot where the ingoing current from the magnet had\ncarried particles from the carbon point and lodged them beneath the\nskin. Psychologically, he was sadly mixed up, he said; for, since that\ntime, he had felt that four lives were joined in him--his own, the\nremnant of Herr Lebensfunke's miserable hoard merged in that of poor\nbirdling's mother, and, last of all, Hans <DW62>'s.\n\nHe left the cars soon afterward at Binghamton, watchfully followed by\na stout, shabby man with a three days' beard stubbling his chin, who\nhad occupied the seat in front of us, and had turned now and then to\nlisten for a moment to Ronald's rapid narration.\n\nA week later, and I heard that he was dead--having committed suicide\nin a fit of delirium soon after his admission to the Binghamton\nInebriate Asylum. The attendant who made him ready for burial noticed\na singular blue mark on his left breast, that looked, he said, a\nlittle like a horseshoe magnet.\n\n\n\n\nOSGOOD'S PREDICAMENT.\n\nBY ELIZABETH D. B. STODDARD.\n\n_Harper's Magazine, June, 1863._\n\n\nOsgood took a cane-bottomed chair whose edges had given way from the\napplication of boot-soles, cane and umbrella ferules, and studied his\npredicament. He commenced this necessary study early in the morning in\nhis room, which was in a boarding-house situated in this metropolis.\nThe early carts were taking their way down town through a blue haze,\nwhich in the country prefigured a golden day. The milkman, the\nwalk-sweeper, and the rag-picker, were the only creatures moving in\nOsgood's neighborhood. The time was propitious for meditation and\nresolve, but Osgood's head was not ready. The still Champagne that he\nhad drank the night before buzzed in his brain. With a glass of it in\nhis hand, under a side gas-light, in the drawing-room of his Aunt\nFormica, he had proposed marriage to a handsome dashing girl, and\nthe handsome dashing girl had accepted him. They swallowed the\nbubbles on the \"beaker's brim,\" thinking it was the Cup of Life\nthey were drinking from. Neither supposed that the moment was one\nof exhilaration or enthusiasm. Osgood never felt so serious, or so\ndetermined to face the music, as he called it, which was the short for\na philosophical design to march boldly through life, and shoulder its\nnecessities with a brave spirit and a martial air.\n\nOsgood was intelligent, agreeable, and handsome. He had advanced no\nfurther into life than to give this impression. He knew nothing more\nof himself than that he was intelligent, handsome, and \"plucky.\" He\nhad no father or mother, but he had an aunt who had married Mr.\nFormica; this pair, effete in themselves, belonged to that mysterious\nclass who are always able to get their relatives places under\nGovernment. When Osgood was eighteen they obtained a place in the\nSub-Treasury, which yielded him the income of fifteen hundred dollars.\nAunt Formica expected a great deal from him in the way of deportment\nand dress. The exigencies of his position, she observed, compelled him\nto do as those around him did. Of course he never laid up any of his\nsalary, but he kept out of debt, and in doing this he fulfilled the\nhighest duty that came within his province. His associates were young\nmen who had more money than he, and who expected him to spend as much\nas they spent. The houses he visited were inhabited by people who\ntook it for granted that all who came in contact with them were as\nrich as themselves. The Formica interest was large. When he went to\nWashington with his aunt, he went the rounds of the senators' houses\nand hotels in the way of calls, dinners, and parties. When he went\nto Boston with her he began his visits at the right hand of Beacon\nStreet, and branched into the streets behind it, where as good blood\nabides, though it has not the same advantage of the air of the Common.\nWherever he went expense was involved, in the way of gloves, bouquets,\ncards, fees to errand boys, exchange of civilities in lunches, cigars,\nale, brandy, sherry, stage, hack, and car fare, which he bore like a\nhero.\n\nLily Tree, the girl whom he proposed to marry, belonged to a family\nof the Formica species. It sailed through society all a-taut with\nconvention, and was _comme il faut_ from stem to stern. Lily and\nOsgood had always known each other. They passed through the season\nof hoop and ball, dancing-school, tableaux, and charades together;\nsympathized in each other's embryonic flirtations; and were such fast\nfriends that no one ever dreamed of any danger to them from love. But\nas the wagon that goes from the powder-mill in safety innumerable\ntimes at last carries the keg which explodes it, so Osgood and Lily at\nlast touched the divine spark which threw them out of their old world\ninto one they had not anticipated.\n\nThis was part of Osgood's predicament.\n\nWhat made him do as he had done?\n\nWhy had Lily accepted him?\n\nShe would never, he argued, consent to go out of the area which\nbounded her ideas, and which comprised a small portion of New York,\nBoston, Washington, and the tour of Europe, which meant a week in\nLondon, six months in Paris, and ten days in Rome. Unless he descended\nfrom the Sub-Treasury, and sought some business, such as making\nvarnish, glue, buttons, soap, sarsaparilla, or sewing machines, could\nhe marry? What shrewdness had he in the place of capital to bring to\nbear on the requirements of these Yankee callings? How he worried over\nthe prospect which looked so pleasant the night before! Champagne,\nflowers, light, and perfume were gone from it. He pitied himself in\nhis helplessness. The thought of Lily deprived of her delicate evening\ndresses, her diurnal bouquets, caramels, and her pecunious caprices,\nwas not pleasant. He could not see her in any light that made her so\nagreeable as in the light that he must certainly cause her to lose.\n\nSomething practical must be done.\n\nNaturally he looked into his pocket-book. There was eighteen dollars\nin it--all the money he had. It was the last day in the month,\nhowever, and he was entitled to draw one hundred and twenty-five\ndollars. He shut his pocket-book and looked into his closet. He\nfound there several pairs of patent-leather boots and a brilliant\ndressing-gown. \"Pooh!\" he said, peevishly, and shut the door. He then\nexamined his bureau: in its drawers were many socks, shirts, cravats,\nfour sets of studs and sleeve-buttons, and five scarf-pins. He rattled\nthe studs and buttons thoughtfully; but nothing came of it, and he\nclosed the drawers. His eye then fell on a dress-coat which he had\nworn for the first time the evening before. He resolved to take the\ncoat back to Wiedenfeldt, his tailor. This resolve was the nucleus\nprobably of his future undertakings. He finished dressing and left the\nhouse. Before reaching Wiedenfeldt he purchased and drank a bottle of\nCongress Water. He also stopped at a favorite restaurant and made an\nexcellent breakfast, and came away with a \"Relampagos\"--a small\ncigar of superior flavor--and three daily papers. His interview with\nWiedenfeldt was satisfactory; the coat was taken back, and when he had\nsettled the matter he felt as if a beginning had been made in a new\nand right direction.\n\nThat afternoon he drew his pay, and walked up town. The moment he\nentered his room his predicament fell upon him again, and his spirits\nsunk. He sat on the edge of his bed, so quiet in his misery that he\nbegan to hear the ticking of the watch in his pocket; it associated\nitself in his mind with the sound and motion of railroad-cars. He felt\nhimself traveling hundreds of miles away, listening all the while to a\nrhythmic sound, which said, \"Many a mile, many a mile.\" Why should he\nnot go \"many a mile, many a mile,\" in reality? He went out immediately\nand bought a valise. After that his demeanor was settled and\ntranquil. He then wrote three notes--to his chief, his Aunt Formica,\nand Lily. The first was a note of resignation; the second conveyed the\ninformation to his aunt that he was sick of his place, had thrown it\nup, and was going out of town for a change of air. He regretted, when\nhe began his note to Lily, that he had not sent her some flowers. A\nmomentary impulse to go and see her stayed his hand; but he remembered\nthat she must be at Mrs. Perche's \"sit-down supper\" that evening, and\nresumed writing. He begged her to enjoy herself, and not miss him\nwhile he was away. He did not know what to write besides, but put in\na few chaotic expressions which might or might not mean a great deal.\n\nWhile he put a few necessary articles in the valise he wondered where\nhe should go, never dropping the thought that he must go somewhere.\nThe remainder of his wardrobe, including the brilliant dressing-gown,\nhe packed in a trunk and locked it.\n\nHe rang the bell, and when the waiter came up asked for the landlady,\nMrs. Semmes. The waiter thought that it was not too late to see her in\nher own parlor, and lingered, with his hand on his chin and his eyes\non the valise.\n\n\"Jem,\" said Osgood, \"I have left some boots in the closet, and some\nshirts in the drawers, which are at your service.\"\n\nThe alacrity with which Jem changed his attitude and expression\nstruck Osgood with a sense of pain. \"How horribly selfish servants\nare!\" he thought, taking his way down stairs. Mrs. Semmes hoped there\nwas no trouble, and asked him to be seated. He looked at her\nearnestly; she was the only one to say farewell to. Never had he\nlooked Mrs. Semmes in the face before; he had only seen the hand into\nwhich he had placed the price of his board.\n\n\"I came to tell you, Mrs. Semmes, that I am about to leave town for\nthe present. Will you allow my trunk to remain here? If I do not\nreturn in a year and a day, break it open.\"\n\nMrs. Semmes promised to keep the trunk; took some money due her;\nwondered at his going away at that time of year, and asked him his\ndestination.\n\n\"I think I shall go to Canada,\" he answered, vaguely.\n\n\"There must be snow there, by the accounts.\"\n\n\"Where shall I go?\" he was about to say, but checked himself.\n\n\"If you were going East,\" she continued, \"you would find the ground\nbare enough, especially in the neighborhood of the sea: the sea-winds\nmelt the snow almost as soon as it falls.\"\n\n\"I think I will go East,\" he said, musingly. He sat so long without\nsaying any thing, staring straight before him, that Mrs. Semmes began\nto feel fidgety. She recalled him to the present by walking to the\nwindow. He started, bade her good-by, and retired.\n\nHe tossed about all night in a feverish sleep, tormented with dreams\nwhich transformed Lily into a small child which he was compelled to\ncarry in his arms, or furnished his Aunt Formica with a long spear,\nwith which she pursued him, and was forever on the point of overtaking\nhim.\n\nAt 8 o'clock A.M. he might have been seen by a detective at the\nTwenty-seventh Street depot. A few minutes after he was going through\nthe tunnel; and, emerging from that, he considered himself fairly\ndivided from New York. At the first station beyond the State-line\nof Massachusetts he consulted a map, and concluded to stop at the\njunction of the Old Colony Railroad. There he changed the route, and\nin the evening reached a town which seemed waiting to go somewhere\nelse, where he passed the night.\n\nThe next morning he started on his travels again toward Cape Cod. Five\nmiles beyond a large village, in a flat, sterile, gloomy region, he\nalighted with his baggage, and said, \"This is the place for me.\" The\ntrain went on, and the depot-master went into his little den without\nnoticing Osgood. Several tall school-girls, who had come to watch for\nthe train, strolled down a cross-road, and he was alone. He went to\nthe end of the platform and surveyed the country. He stood on the edge\nof a wide plateau along which ran the railroad-track. Beyond that a\nroad deviated through dismal fields, by unpainted houses, large barns,\nand straggling orchards. Below the plateau a wide marsh extended,\nintersected by crooked creeks, which gnawed into the black earth like\nworms. A rim of sea bordered the tongue of the marsh, but it was too\nfar off to add life to the scene. The sedge, giving up all hope of\nbeing moistened by the salt waves, had died in great circles, which\nlooked like mats of gray hair on some pre-Adamite monster's buried\nhead.\n\nOsgood determined to pursue the windings of the road. He plowed the\nsand for two miles, and at a sudden turn of the road came upon a\nhouse, with a number of barns and sheds attached to it. A dog with a\nstiff tail ran out from a shed and barked at him, and a pale-faced\nwoman in a muslin cap appeared at a window of the house. He knocked at\nthe door: she opened it.\n\n\"Will thee come in?\" she asked.\n\nHe entered, following her as he would have followed a ghost. She moved\na chair from the wall without the least noise, and he dropped upon it.\nAs he looked at her his identity seemed slipping away--seemed to be\nslipping into an atmosphere connected with her and her surroundings.\nShe brought him some water which she dipped from a pail near by, and\nheld the cocoa-nut dipper which contained it to his lips.\n\n\"Thee has come to us from strange parts, I reckon, from thy looks.\"\n\n\"Yes,\" he answered, absently; \"I needed change.\"\n\n\"There has been no change here since the Indians went away. If thee\nwill look across the road thee can see the ground is strewed with the\nbits of shells from their feasts.\"\n\nHe went to the window, and again remarked to himself, \"This is the\nplace for me.\"\n\n\"Could you,\" he asked, going toward her, \"let me stay with you a\nwhile?\"\n\n\"Did thee come to the Marsh End station this morning?\"\n\n\"Yes; my valise is there.\"\n\n\"Thy parents are rich?\"\n\n\"I have none.\"\n\n\"Thee has been well cared for, though.\"\n\n\"I have not left home because of any--\" Misfortune, he was about to\nsay, but that did not seem to be the right word; so he tried to think\nof something else to say. She saw his embarrassment, and said,\nquickly,\n\n\"I never have harbored a stranger; but if Peter likes, he may take\nthee.\"\n\nOsgood thanked her so pleasantly that she determined he should stay.\nShe asked him his name, his age, his place of residence, his business,\nand his intentions. Except in regard to the latter, his answer proved\nsatisfactory; and when Peter returned at noon from the distant shore\nwith a load of sea-weed, she introduced Osgood as if he were an old\nacquaintance of whom Peter was in a state of lamentable ignorance. He\npushed his hat on the back of his head, shook hands with Osgood, and\nsaid, \"Maria, will thee give me my dinner?\" taking no further notice\nof Osgood till she had placed it on the table. It consisted of stewed\nbeans, boiled beef, apple-pie, and cheese. Osgood ate half a pie, and\nestablished himself in Peter's good graces.\n\n\"Thee will learn that Maria's pie-crust beats all,\" he said.\n\n\"Thee is ready to consent,\" said his wife, \"to keep young Osgood a\nwhile?\"\n\n\"I don't know yet,\" answered Peter.\n\nBut after dinner he harnessed his horse and went to the depot for\nOsgood's valise, which he carried upstairs and deposited in the spare\nroom. He then invited Osgood to take a look at the premises. He wished\nto make his own investigations in regard to Osgood without Maria's\nintervention. They lingered by the pig-sty, and while Peter scratched\nthe pigs with a cord-wood stick, exchanged views of men and things.\nPeter saw the capabilities of Osgood's character, and easily divined\nthe manner of life he had led. He knew him to be selfish from\nignorance, and because he had early formed the habits which impose\nself-indulgence. Something in the young man's bearing won his heart--a\ncertain impetuous simplicity and frankness which made him long to be\nof service to a nature unlike his own. Osgood found Peter genial,\nshrewd, and sad. Such a man he had never met. It seemed to him that\nPeter could set him straight in his own estimation; there was no\nnonsense about the old man, and yet he could see deep feeling in his\ndark, cavernous eyes. The feeling which had oppressed him passed\naway, and another took its place which contained restoration, and\nfaith in the future. He got into Peter's way by attempting to help\nfodder the cattle and \"slick up\" the barn. When the work was done, and\nwhile Peter fastened the barn-doors with an ox-bow, Osgood looked\nabout him. It was a March afternoon; no wind blew, and no sun shone;\nbut the gray round of the sky, which neither woods nor hills hid from\nhis sight, rolled over him in soft commotion. The reddish, barren\nfields stretched in their flatness beyond his vision, and the narrow\nroads of yellow sand ran to nowhere. The world of God, he thought, he\nsaw for the first time; and, away from the world of men, felt himself\na _man_.\n\nHe looked so kindly upon Maria when he entered the house that she\ndelayed the stream of the tea-kettle which she held over the teapot to\nadmire him. The supper was the dinner--cold, with an addition of warm\nbiscuits; and again Osgood ate himself into Peter's good graces.\n\nThe evening was passed in silence. Peter smoked, Maria mended, and\nOsgood reflected. A violent storm arose in the night, which lasted\nthree days. They were improved by Maria and Peter in overhauling\ngarden-seeds in the garret, and in setting up a leach-tub in the\nwood-house. Osgood assisted. When he was alone with Maria she talked\nto him of the boy who was lost at sea, and of the girl who died in\nchildhood; with the hungry eyes of a bereaved mother she looked upon\nhim, and his heart was touched with a new tenderness. When he was\nalone with Peter the old man sounded the depths of the young man's\nsoul with wise, pathetic, quaint speech; he went over the ground of\nhis own life, which had been passed on the spot where he now was, with\nthe exception of several mackerel voyages, and one in a merchant\nvessel to some of the southern ports of Europe. But when together\nPeter and Maria never talked with Osgood on personal matters. Between\nthem a marital silence was kept, which was more expressive than the\nconjugal volubility which ordinarily exists; it proved that they had\npassed through profounder experiences.\n\nWhen the storm ceased Peter went to the station for his Boston\nnewspaper, which he read to Maria, who took it afterward and read it\nover to herself. Brother Quakers, Peter's neighbors, who lived out of\nsight, dropped in from time to time to exchange a word with Maria, or\nhold talks outside with Peter, with one foot in the rut and the other\non the wagon-step. The present subject of interest, Osgood discovered,\nwas the approaching Quarterly Meeting, and the mackerel fishery. Peter\nasked him to accompany himself and Maria to the town where the meeting\nwas to be. They breakfasted at sunrise, when the day arrived, in full\ndress--Peter in a snuff- suit, and Maria in a series of brown\narticles--dress, shawl, and bonnet. They started in good spirits in an\nopen wagon, with an improvised seat for Peter in front. Beyond a belt\nof pine woods stood the meeting-house, and a mile beyond the\nmeeting-house lay the town, before a vast bay. Osgood drove alone into\nthe town, and spent several hours there. He visited the shops to find\nsome trifle for Maria, and then went through the town down to the\nshore. How happy he grew in the pure wind and the gay morning light!\nThe gulls rode over the foaming wave-crests and dipped into their\ngreen walls, and hawks swooped between the steadfast sky and heaving\ndeep. The sea traveled round and round before his eyes with a mad joy,\nand tempted him to plunge into it. He wrote his name in the heavy sand\nwith a broken shell, and the water filtered out the letters; then he\npaved it in pebbles with the word _Strength_.\n\nPeter and Maria were waiting for him when he returned to the\nmeeting-house with the wagon.\n\n\"Thee has been skylarking,\" she said.\n\n\"After something for you,\" he answered, putting in her hand a handsome\nwork-basket.\n\n\"Has thee so much money that thee must waste it on me, Osgood?\"\n\nBut she was pleased with the gift. They rode home amicably. Peter, as\na favor, allowed Osgood to drive, while he imparted to Maria sundry\nbits of information gained at the meeting.\n\n\"Mackerel\" went in and out at Osgood's ears without gaining his\nattention, till he caught at something Peter said about the _Bonita_.\nHe listened. Three vessels were about to sail from the town on a\nmackerel voyage, and the _Bonita_ was one of them. He comprehended\nthat Peter owned half the _Bonita_, and a plan struck him. He inquired\ninto the subject, and obtained its history. That evening he proposed\ngoing on a mackerel voyage, which proposal so fired Peter that he\ndeclared he had a mind to go too; but Maria quenched his enthusiasm by\ngoing over the programme of work that must be done at home. She made\nno opposition to Osgood's going, but set before him in plain terms the\nhardships of such a voyage. He was not to be deterred, and Peter gave\nhis consent, promising him a small share of the profits.\n\nOsgood wrote to his Aunt Formica that night, assuring her that he\nalready felt much better, and that he was about to enter into a new\nbusiness, of which she should hear more. He also wrote Lily Tree a\nminute, lengthy epistle. He described his situation with Peter and\nMaria; told her how much board he paid--two dollars and fifty cents a\nweek--and how well he had learned to do chores. He fed the pigs every\nday; he wished that she could see how well they thrived on the diet\nlately introduced by Peter and himself--a dry mash of boiled potatoes\nand meal, with an occasional horseshoe thrown in as a relish. Would\nshe, he wondered, have enjoyed the day that he, Maria, and Peter made\nsoft soap? He mentioned his intended voyage, and asked her if she\nliked sailors. Could he have the hope, he continued, of her sympathy\nin his future enterprises, which perhaps would differ from those she\nhad thought of for him? He avowed a change in himself. Would it affect\nher?\n\nHe sealed his letters, and began pacing his little room. Writing home\nhad brought his old life near him again; the distance it had come to\nreach him seemed enormous.\n\n\"It was only a few days ago,\" he thought, \"and yet I am so different!\"\n\nHe rolled up his paper window-curtain and softly raised the window.\nThe moon made the landscape look more vast and desolate than it was in\nthe light of day. Under the horizon it revealed a strip of sea which\nshone as if it were the portal of another world whose light was\nreflected thereon. Osgood felt that he was an imprisoned soul this\nside of it. The light gave him an intimation of immortality. \"Where is\nLily's soul?\" he asked. \"Has she any dream beyond the life she is in?\"\n\nWhen Lily received Osgood's note she was angry; so was Mrs. Formica\nwhen she received hers. An intuition that Osgood repented his rashness\ntouched Lily's pride, and preserved her silence. When the second\nletter came, she thought he had the intention of experimenting with\nher; a test, she concluded, was unendurable, not to be submitted to.\nShould she test him, and proclaim the engagement she meditated? it\nwould be a relief to do something. She could not reach him with a\nletter, for he had gone on a mackerel voyage beyond the limits of the\npost-office. She decided differently according to the light she had.\nUnlike Osgood, she was chained to the place she was in. She was alone,\ntoo; her mother was occupied with neuralgia, and her father was out of\ntown half his time, on mysterious agencies which referred to canals.\nThe newspaper reporters at Albany were well acquainted with Mr. Tree's\nname while they were putting into short-hand the doings of the\nLegislature. Mrs. Formica had no suspicion that Lily was the cause of\nOsgood's disappearance; she would not have regretted his absence so\nmuch on these grounds, for a match with Lily was not desirable.\n\nWithin a month Lily's engagement to Mr. Barclay Dodge was announced.\nHe was a young man of fortune, whose father owed his rise in the world\nto corn starch, and who had made himself known by spending large sums\nof money on pictures, landscapes mostly, which had been indorsed by\nthe public in exhibitions.\n\nMr. Barclay Dodge was happy; he had for more than two years followed\nLily through all vicissitudes attendant upon the career of a young\ngirl in society. From an exhilaration the pursuit had become a\ndesperation. He had never suspected any man of being his rival, and\naccounted for the acquaintance between Lily and Osgood by believing\nthat Lily was related to the Formica family. How she managed so\nsuddenly to convince Barclay Dodge that it was safe for him to propose\nis a mystery which none but a disappointed, contrary woman may\nreveal. He had the usual penetration of his sex in regard to such\nmysteries; he was a man of sense and experience, but he was in love,\nand when a man is in love he only analyzes himself, and all that he\nlearns is, that his love must be gratified.\n\nIn the whirl of his attentions, and the congratulations of her\nfriends, the time passed quickly; not so quickly, however, as to avert\nthe plan by which the Fates were to bring her to a knowledge of\nherself.\n\nBarclay proposed an immediate marriage. Lily declined the proposal\nwith so much vehemence that he dared not insist. He pulled his\nmustache in rage after he left her, and wondered why he did not\ninsist. By what means, he cogitated, could he make her yield her will\nto his? Her resistance he set down to coyness; all women had freaks;\nthey were alike in such matters. He divined after a while that she\nwould let go the lasso at any moment if he proved restive; so he\nplayed the submissive to perfection. If she ever saw his eyes flame,\nor any gesture which contained a threat, he never knew it; but every\nrevelation from him was a revelation to her of herself, and this was\nto be her education and her punishment.\n\n\"Where is your friend Osgood?\" he asked once.\n\n\"He has been away a long time,\" she answered, looking him full in the\nface, but with rather a stony expression in her eyes.\n\n\"He is your relative?\"\n\n\"Oh no.\"\n\n\"No? I thought so, always seeing you in the same places.\"\n\n\"Our families have been acquainted always.\"\n\n\"Do you think he is handsome?\"\n\n\"Yes.\"\n\n\"He is too short\" (Barclay was tall), \"and his eyes have a wandering,\nunsettled look.\"\n\n\"He is following his destiny by them,\" she answered, bitterly. \"I wish\nthat I could follow mine as a man can.\"\n\n\"Do you mean that you would like to follow Osgood's eyes?\"\n\n\"By no means; I must see destiny by your eyes.\"\n\nThe words were pleasant, but the tone was malicious. It made his heart\nbound as if an invisible foe had come into his atmosphere to do battle\nwith him, and he could do nothing.\n\n       *       *       *       *       *\n\n     \"'With the vapors all around, and the breakers on our lee,\n     Not a light is in the sky, not a light is on the sea.'--\n\nbarring the lantern abaft,\" roared Osgood, from the deck of the\nschooner _Bonita_, which was tossing outside Cape Malabar.\n\n\"You may sing t'other side of your mouth afore long,\" bawled back the\nskipper. \"We ain't fur from the Cormorant Rocks; the wind p'r'aps will\nshove us on the ledge.\"\n\n\"What, when we are just going home with full barrels?\"\n\n\"The mackerel may be briled in Tophet for all we know.\"\n\nThe skipper was at the helm; Osgood and he were in the radius of a\nlantern which revealed their faces to each other. Outside of that was\npitch darkness; the rain drove in fierce slants against them, and the\nwind howled all round the sea.\n\nThe skipper did not look concerned, neither did Osgood; but they were\nboth wondering which would first break over the _Bonita_, the light of\nmorning or the sea.\n\n\"Them boys are asleep, I s'pose, wet to the bone?\" the skipper yelled.\n\n\"Yes.\"\n\n\"Let 'em sleep; there ain't a lanyard loose.\"\n\n\"What time must it be?\"\n\n\"Hard onto 'leven. My old woman's turned in long afore this, _she_\nhas; allus goes to bed on the stroke o' nine.\"\n\n\"She has thought of you to-night?\"\n\n\"She has give me a prayer or so; she's the strictest kind. Now I'll\nluff, there is a lull comin'; peskiest storms that have lulls in 'em.\nYou don't hear a swashing to a distance now?\"\n\n\"No.\"\n\n\"Hark!\"\n\nA sound, not of wind nor sea, approached them--a rapid, rushing,\ncutting sound.\n\n\"Up with the helm!\" shrieked the skipper to himself. \"God Almighty,\nshe is down on us!\"\n\nOsgood leaped up. The bowsprit of a large ship was over him; he threw\nup his arms instinctively and caught at something; he felt his feet\ndrawing over the skipper's head, and that he thumped it with his\nboots. He knew no more. The great ship crushed and plowed the _Bonita_\ninto the waves as easily as a plow buries in the sod the stubble\nof the corn-field. Nothing signaled her destruction except the\nexclamation of the skipper; nothing remained in the wide sea to show\nit. Her timbers and the sleeping crew went to the bottom together.\nMorning dawned on the wild scene, revealing no floating spar, no rib\nof boat, no stave of tub or barrel, no sailor's hat, no remnant of\nsail, no shred of clothing; the jaws of the sea had closed over all.\nThe ship, a Liverpool liner, driven out of her course by the storm,\ncruised round the spot for a few hours, and then went on her way,\ntaking Osgood with her. He had clung to the folds of the forward sail;\nand there he was found with his left wrist dislocated, his body\nstrained and sore, and his mind wandering. He was no romantic sight\nwith his red flannel shirt, fishy trowsers, cowhide boots, and hands\npickled in brine. Still the ship's surgeon took to him, and found,\nwhen Osgood came to himself, that he had taken to a gentleman. He\nlent him a suit of customary black, and introduced him to his\nacquaintances. Osgood would have enjoyed the voyage across the\nAtlantic but for the horror which had fallen on his mind from the\ncatastrophe of the _Bonita_.\n\n\"How old are you?\" the surgeon asked him.\n\n\"About the first of March I was twenty-three; since then I have grown\nso old I have lost the reckoning.\"\n\n\"I'll have to give you quinine, my boy.\"\n\n\"Give me some of the tincture of Lethe.\"\n\n\"It is of no use to one to forget; don't be soft.\"\n\n\"Let us reason together, Sawbones.\"\n\nThe Doctor agreed, and Osgood began his story with, \"Poor Peter,\" and\nfinished it with asking, \"Do you think I love her?\"\n\n\"I'll bet a guinea,\" said the Doctor, \"that she is married.\"\n\n\"She isn't,\" replied Osgood, indignantly.\n\n\"I am sure that she is engaged, as you call it, to somebody besides\nyourself.\"\n\n\"I know better.\"\n\n\"What do you propose doing when you get home?\"\n\n\"What can I do with thirty dollars, which I left with Peter\nby-the-way?\"\n\n\"We shall see what we shall see when we come face to face with Aunt\nFormica. I intend going the rounds with you in New York. I am a\nstudent.\"\n\nHe carried Osgood to his country-home beyond Liverpool, where they\nstaid till the ship was ready to sail again. He amused his mother and\nsisters with stories of Osgood's adventures on sea and land, and\nrepresented him in the light of a \"Jarley's wax-works\" hero, till he\nwas fairly cured of his melancholy.\n\nFive months from the day on which he left New York Osgood returned,\nand stood on his Aunt Formica's door-steps with Dr. Black. They looked\nlike a pair of Englishmen. Both had shiny, red noses, shiny, hard,\nnarrow-brimmed hats, and shiny, narrow-toed boots, and the nap had\nbrushed off their coats.\n\nOsgood looked into the familiar area with emotion, and the Doctor\nlooked at the windows with curiosity.\n\n\"They must be out of town,\" he said; \"the house has been put in brown\nhollands.\"\n\nBut Osgood knew the habits of his aunt--knew that from the first of\nJuly till the first of October the house was put on an out-of-town\nfooting; and that she skirmished between city and country, or\nwatering-place. The bell was answered by a servant he did not know.\n\n\"I wish to see Mrs. Formica,\" he said, brushing past her, and entering\nthe dark parlor. \"Dr. Black and friend say.\"\n\nMrs. Formica came in a moment after with a slight air of amazement,\nwhich increased to astonishment when she saw her nephew. She gave a\nlittle yelp as he embraced her, and said, \"Where _have_ you been?\"\n\n\"To Cape Cod, and to Europe. I have been shipwrecked, aunt--that is, I\nlost my mackerel venture, and have been taken care of by my noble\nfriend, Dr. Black.\"\n\nAunt Formica grew pale at the word \"shipwrecked,\" and turned to Dr.\nBlack. Something in his face made her extend her hand and give him a\nwarm welcome.\n\n\"Black may stay here while he is in port, mayn't he? He will amuse you\nwith yarns about me.\"\n\n\"Of course,\" she replied. \"Now tell me the whole story.\"\n\nBetween Osgood and the Doctor it was related.\n\n\"Why did you ever go from me?\" she asked, wiping away a real tear.\n\n\"I believe, aunt, I shall keep up the business of going--it suits me.\nI can never live through your conventional cramps.\"\n\nShe did not think it prudent to combat him just then; but made a\nmental memorandum that something must be done that would change his\nfoolish resolution. A plan developed at dinner that evening.\n\n\"I had a note yesterday from Mrs. Senator Conch,\" said Mrs. Formica.\n\"She will be in Saratoga this week, and begs me to meet her there.\nFormica and I have been talking it over, Osgood, and we think that it\nwill be pleasant for Dr. Black and you to go up for a week. You will\ngo, Doctor?\"\n\n\"Thank you, Madam, provided Osgood is not averse.\"\n\n\"Any of our set there?\" Osgood asked.\n\n\"The Trees went up last Saturday with Barclay Dodge. They are making\nan extensive tour this year.\"\n\n\"What's Barclay Dodge along for?\"\n\n\"He is engaged to Lily Tree!\"\n\n\"Ah!\" said Osgood, looking at the Doctor, who could not help giving\nhim a malicious grimace. \"How long since? It's a capital match, ain't\nit?\"\n\n\"The engagement must have been announced soon after you left.\"\n\nThis reply put Osgood in a brown study. What impulse, he mused, had\nprompted Lily to give herself to Barclay Dodge? Would _he_ have done\nso?\n\nDr. Black commented on Osgood's face, and considered himself in a fair\nway to make studies.\n\n\"As far as money goes,\" continued Mrs. Formica, \"it may be called a\ngood match; but certainly not as far as family goes.\"\n\n\"Family!\" echoed Dr. Black, softly.\n\n\"His father was a tradesman,\" explained Mr. Formica, \"while Lily can\ngo back to her great-grandfather before trade need be mentioned.\"\n\n\"Old Mr. Tree's father,\" remarked his wife, \"was a brigadier-general\nin the Revolution.\"\n\n\"He was a drover, for all that,\" said Osgood.\n\nMrs. Formica changed the theme, and talked of Saratoga.\n\n\"We'll go,\" Osgood said, crossly; \"but I must first go to my tailor.\"\n\nMrs. Formica held a private conversation with him after dinner, gave\nhim a check, and told him not to worry about the future: she had a\nplan in view.\n\n\"Plans go by contraries with me, aunt.\"\n\n\"You owe it to me not to be perverse.\"\n\n\"I can't pay any debt.\"\n\nPrevious to going to bed Dr. Black and Osgood smoked several cigars.\n\n\"You strike me,\" said the Doctor, \"as growing to the dramatic just\nnow. One event runs into another with monstrous rapidity among you\nAmericans. How you differ from the English! How is it that you catch\nfortune by the hair so?\"\n\n\"We are passionate and quick-witted.\"\n\n\"And then you repudiate with ease.\"\n\n\"Bah! you imitate Sydney Smith.\"\n\n\"I did not mean in the sense of State bonds precisely.\"\n\n\"I think,\" Osgood groaned, \"that I begin to feel like a snob again.\nWhat shall I do to be saved?\"\n\n\"Go on in the groove that is making for you. I'll stand by and be the\nchorus. When I hear thy plaints of misery I will let fall the tear;\nbut remember that 'laws determine even the fates.'\"\n\n\"Bosh!\"\n\nExcept a dispute between the Doctor and Osgood concerning a slouched\nhat, which the Doctor would not wear, the party succeeded in starting\nand arriving amicably at the Union in Saratoga. In a few hours Mrs.\nFormica knew who was there. The Trees were at the Union. Mrs. Senator\nConch had taken a cottage; but the Senator himself had stopped at\nAlbany for a day to confer with the Governor. Old Madam Funchal of\nPhiladelphia was at Congress Hall, with her train, and Mrs. Romeo\nPipps Bovis and husband, from Boston. All her friends were round her;\nthat is, the traveling set she was in the habit of meeting; and her\nspirits rose to the occasion. These particulars she detailed, in a\nwhite muslin morning-dress, to Osgood, who, dressed in a new\ncream- suit, lounged in the doorway of a small parlor off the\nhall. He shouldered round just in time to come face to face with Lily\nTree, who was passing on the arm of Barclay Dodge. She stopped, of\ncourse, to shake hands with Mrs. Formica, whose apparently warm kiss\nfell on the edge of a braid of her chestnut hair with the weight and\ncoldness of a snow-flake. Her face settled into rigidity when she\nturned to speak to Osgood, and, like a transparent boy, he looked,\nwith all the earnestness his gray eyes were capable of, straight into\nhers. Aunt Formica and Barclay read a story at once upon the text his\ncountenance furnished; but they both made the mistake of believing\nthat Lily had rejected him. Lily was too much occupied in managing her\nown feelings to divine Osgood's. The imperative necessity of\nconcealment, which all tutored women feel, governed her. She laughed a\ngreat deal, though nobody said a witty thing, and kept her eyes going\nbetween Mrs. Formica and Barclay with a steadiness which equaled the\nmovements of the wax women in the Broadway shop windows. Mr. Formica\nand Dr. Black added themselves to the party, and the relief of an\nintroduction to the Doctor came to Lily. She approached him, and his\nhonest face induced her to skirmish lightly with him; but not a word\ndid he utter of the whys and wherefores of his being with Osgood. He\nwould not, at any rate, extend his self-elected office of chorus so\nfar as to include her. He felt a dislike toward her. She was too thin,\nhe thought; there was an air of wear and tear about her which was not\npleasant. He felt, too, that she knew more than Osgood; and a woman,\nin his estimation, should never be the intellectual superior of a man\nshe might make choice of. But the Doctor was an Englishman; his ideas\nof women had been developed by the cynical Thackeray and the material\nDickens. There was a line between the two classes of women he only\nbelieved to exist--the bad capable woman and the good foolish\nwoman--which could never be crossed by one or the other. The elements\nwhich go to make up a man, of good and evil mixed, never enter into\nthe composition of the women of Englishmen of the present time. It is\npossible that Lily discovered Dr. Black's impression: she discovered\nit so nearly that she was certain Osgood had talked of her with him.\nWhy had he? she wondered.\n\nIn a few minutes the party fell apart as naturally as it had come\ntogether. Lily went on her walk with Barclay; after which she retired\nto dress for luncheon, but instead of appearing thereat kept her room\ntill evening.\n\nOsgood avoided every body; he was tormented with an idea that Lily had\nsuffered. There was no reason for his thinking so; he derived the idea\nfrom reasoning with himself--reasoning which meeting with her had put\nin play. In the evening he went to the drawing-room, and waited till\nhe saw her come in. Barclay, who was waiting too, darted toward her,\nbut Osgood reached her first. When Barclay saw Lily take the arm which\nOsgood offered her, he turned away; but changing his mind again went\nup to them.\n\n\"Osgood,\" he said, in a frank voice, \"you have not congratulated me on\nmy engagement to your friend Lily.\"\n\nTalk of heroes and martyrs; was not Lily both, at that moment,\nstanding between these two men, with her hair dressed by a barber, and\nwearing a pale blue silk?\n\nShe eyed with a dainty air a little bouquet she held in her hand, of\ntea-roses and geraniums, and applied it to her nose with great\ndeliberation. She felt an impetus from Osgood's arm. He had not\nanswered Barclay, but was dragging her decorously out of the\ndrawing-room. When they were alone he spoke to her.\n\n\"I have faced death since I saw you. I have grown a man; but until\nnow, I did not know that I loved you. Which man do you belong to?\"\n\n\"I have faced life since I saw you,\" she answered, in a silvery\nvoice, \"and I belong to Barclay Dodge.\"\n\n\"Let us go back.\"\n\nShe tossed her bouquet over the railing of the veranda with a\nvindictive smile which would have astonished Osgood had he seen it.\n\nBarclay was on the threshold; he looked at Lily and missed the\nbouquet; it was not in Osgood's button-hole--what could she have done\nwith it? He looked at Osgood, and saw that his teeth were set with a\npassion which he could understand. Lily sat down in the nearest chair,\nand the young men moved away together.\n\n\"There is no need of any nonsense between us,\" said Osgood; \"I was\nunder a wrong impression regarding your engagement. I do offer my\ncongratulations.\"\n\n\"Thank you,\" said Barclay, dubiously. And then they looked at each\nother with mad eyes. What a relief it would have been if they could\nhave fought to the death!\n\nOsgood left Barclay abruptly, and sought his Aunt Formica.\n\n\"Aunt!\" he said, in a mild voice, \"you need not ask Conch to blow any\nhorn for me. I am going to sea.\"\n\n\"You will be better when she is married,\" she answered, significantly.\n\n\"I intend to before that. Your surmise is incorrect. You do not know\nthat I ran away from Lily, as well as from you and the Sub-Treasury.\"\n\n\"What do you mean?\"\n\n\"I offered myself to her; she accepted me, and on the strength of it I\nleft her immediately. What do you think of me?\"\n\n\"_She_ is a little wretch. Did you care for her _very_ much?\"\n\n\"I thought she couldn't make a poor man a good wife, _after_ I had\nasked her to be such. And I thought a poor man wouldn't be a good\nhusband.\"\n\n\"It was the height of foolishness in both of you. It is most unwise\nfor two people who have had luxuries separately to join and give them\nup.\"\n\n\"Luxuries! I wish you knew Peter and Maria.\"\n\n\"Osgood, you are morbid.\"\n\n\"Now, aunt, hear me. I am resolved to choose my own life; you must let\nme go. Whatever way I go, I shall not disgrace you. Formica may give\nme a sailor's outfit, if he chooses. Meantime let us enjoy ourselves\nfor the remainder of the week.\" Notwithstanding she saw that he was\ndetermined, she applied to Senator Conch for a place, and he promised\nher one for Osgood in a department at Washington. When she told Osgood\nof it, he deigned no reply; but shook his head so fiercely that she\nforebore to trouble him.\n\nEvery day that he saw Lily she learned his nature by the contrast\nBarclay offered; she also learned to doubt herself. She never had been\nworthy of Osgood; it was fit that she should marry Barclay. She\ndoubted whether she could keep up the strain, which she knew Osgood's\nlove would impose upon her, of self-abnegation, self-denial,\nisolation, and independence. She was not sure that she did not prefer\nenervation with Barclay to action with Osgood. Barclay watched them\nboth. Jealousy gnawed his soul, not because he doubted Osgood, but\nbecause he had a suspicion that Lily once felt an interest in Osgood,\nwhich might be on the point of awakening. He tried experiments upon\nher feelings, pinched them, tore them up by the roots, extracted them\nwith wrenches of his will, applied slow fire; but he learned nothing.\nHis motive was so palpable to Osgood that he more than once felt on\nthe point of knocking him down, and had he seen any encouraging sign\nfrom Lily he would have done it. He sometimes sighed over Barclay's\nfailure, hateful as his conduct was.\n\nThrough the torture which Barclay applied to her she saw the passion\nwhich tortured him. Could a woman have been quailed into love she\nwould have been at his feet; for he broke loose from his feigned\nsubmission and savagely demanded an equal return of his love. Then\ncame the full measure of her punishment. She was incapable of rising\nto the strength, height, and abandon of Barclay's love. She was just\nas unworthy of him as she was of Osgood.\n\nHow she hated herself!\n\nSomehow she heard that Osgood was going to sea. It is probable that\nAunt Formica's feminine malice directed the disclosure to her ears.\nShe staggered Dr. Black a moment after she heard the report by asking\nif it was true.\n\n\"It is,\" he answered, with dignity, though inwardly scared.\n\nShe asked no other question of him, but snapped her fan together and\nwalked away.\n\n\"Lily does not want you to go to sea,\" he said, when next he saw\nOsgood.\n\nOsgood blew a ring of cigar smoke into the air and watched its\ndisappearance.\n\n\"If wedding rings would only disappear that way!\" said the Doctor.\n\nOsgood blew another. \"Include engagement rings,\" he said.\n\n\"One did vanish,\" replied the Doctor, slyly.\n\n\"I do not believe so. I swear she wears two this moment.\"\n\nHe left the Doctor, shut himself in his room, and wrote a long letter\nto Peter about himself, Lily, and Barclay, and posted it.\n\n\"Peter will understand me,\" he thought; \"and more than that, he will\nunderstand Lily.\"\n\nThe last day of the Formicas' stay in Saratoga came. Osgood and Dr.\nBlack appeared in traveling costume. Lily saw them enter the\nbreakfast-room, and followed them with her father. As she passed their\nchairs, she asked, \"Do you go to-day?\" Osgood bowed. Dr. Black engaged\nMr. Tree in making a remark.\n\n\"Why do you go?\" she asked.\n\n\"Because Barclay stays,\" he whispered.\n\nShe turned a fiery red and passed on. He looked across the table once\nand met her eyes. She thought they said \"_Farewell._\" A wild wish rose\nin her heart which compelled all her nature to give way to it, to\nspeak to him once more; to see him alone, and force him to tell her if\nhe loved her. She resolved to find him somewhere, at all hazards.\n\nDr. Black watched her also. His comment was, that she was \"coming to a\ncrisis,\" and was beautifully following out the laws which governed her\nsex. \"Why can't they be something without hysterics?\" he lamented.\n\"Osgood will break down if he is not got away.\" He mechanically turned\nback his wristbands.\n\nLily waited in an ante-room, whose door Osgood must pass on his way\nout, and when he came, beckoned to him.\n\n\"Say your farewell to me as you feel it,\" she said, her eyes in a\nblaze.\n\n\"I can not.\"\n\n\"You shall.\"\n\nHer eyes and her voice threw him into a tumult; had he followed the\ndesire which assailed him, he would have taken her in his arms and\ncarried her off. As it was, he looked at her, with a far-off look, as\nif he were calling some one to his aid.\n\n\"Osgood, Osgood!\" she cried.\n\n\"Lily!\"\n\nShe wrung her hands.\n\n\"Lily!\" he said again.\n\n\"No, no, you need not speak; you may go.\"\n\nBoth of them gained a victory.\n\n\"After I have gone,\" he said, \"if you think it proper, will you visit\nPeter and Maria?\"\n\n\"Peter and Maria?\"\n\n\"The friends I found when I left you, who helped me to find a better\nself--a self that at last finds _you_.\"\n\n\"I will go.\"\n\n\"To-morrow, then, I will write you of them.\"\n\nHe was gone.\n\nIn a few days she received a letter which contained the narrative of\nhis sojourn with Peter and Maria, and a letter of introduction to\nthem. She showed the letter to Barclay.\n\n\"Shall you meet him there?\"\n\nShe gave him no answer.\n\n\"On what terms are you with yourself?\" he continued.\n\n\"To answer candidly, bad terms.\"\n\n\"Could you marry that beggar on better?\"\n\n\"Alas! no.\"\n\n\"Tell me, are you satisfied with your choice?\"\n\nShe looked so irresolute that he trembled and was sorry that he had\nasked the question. Her better angel took wings, however, and she laid\nher hand on his shoulder, saying, \"I make no other.\"\n\nSo she went on her travels with Barclay in her train, and Osgood went\non a voyage in the _Stormy Petrel_ as third mate. When autumn came,\nand the travelers had returned to town, Lily grew miserable. One day\nshe told Barclay that she wanted to read him a poem. He composed\nhimself to listen, and she read \"The Palace of Art.\"\n\n     \"'What is it that will take away my sin,\n     And save me lest I die?'\"--\n\nshe repeated.\n\n\"Barclay,\" she entreated, \"let me throw _your_ royal robes away, and\ngo to those friends of Osgood's, where I may learn that I am either\nworthy of you or of him.\"\n\nA stormy scene ensued. He would neither allow her to go, he said, nor\nwould he give her back her promise to him. But she was firm, and said\nthat she must go. His imprecations and his tears agitated her, but did\nnot shake her resolution. She had a battle with her father also when\nshe mentioned the subject, but she triumphed over him so far as to\nmake him promise to accompany her. She sent the letter of introduction\nto Peter, and received a pithy reply from him. He advised her to come.\nWith Peter and Maria she learned why Osgood wished her to visit them.\nShe left them with a request that they should allow her to return\nwhenever she should wish.\n\nShe found Barclay sullen and unhappy; but in spite of himself she\nconvinced him that they were not intended for each other. It was a\nwork to persuade him to the contrary; but at last they parted not as\nfoes but friends.\n\nWhen the engagement was annulled she took pains to ascertain from the\nowners of the _Stormy Petrel_ what time she was expected home, and\nbefore the date of her arrival she went on a visit to Peter and Maria.\n\nThere she studied the Marine List till she saw that the _Stormy\nPetrel_ was in port. She said nothing of the fact to Peter, but as he\nread the Marine List too, he found it out for himself. He went away in\nhis wagon a few mornings afterward, and when he returned Osgood was\nbeside him.\n\n\"Thee is as white as a ghost, Lily,\" said Maria, after a few minutes.\n\nOsgood put his arm round her, and they kissed each other. Peter pushed\nhis hat on the back of his head, and kissed Maria, and said, \"Give me\nmy dinner.\"\n\n\n\n\nTRANSCRIBER'S NOTE:\n\nMinor changes have been made to correct typesetters' errors; otherwise\nevery effort has been made to remain true to the authors' words and\nintent.\n\n\n\n\n\nEnd of Project Gutenberg's Stories by American Authors, Volume 8, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nI\n\nII\n\nIII\n\nIV\n\nV\n\nVI\n\nVII\n\nVIII\n\nIX\n\nX\n\nXI\n\nXII\n\nXIII\n\nXIV\n\nAfterword\n**PRAISE FOR THE** _**NEW YORK TIMES**_ **BESTSELLING** _**MEN AT WAR** _**SERIES**\n\n\"WRITTEN WITH A SPECIAL FLAIR for the military heart and mind.\" _\u2014The Kansas Daily Courier_\n\n\"SHREWD, SHARP, ROUSING ENTERTAINMENT.\"\n\n_\u2014Kirkus Reviews_\n\n\"CAMEOS BY SUCH HISTORICAL FIGURES as William 'Wild Bill' Donovan, Joseph P. Kennedy, Jr., David Niven, and Peter Ustinov lend color . . . suspenseful.\" _\u2014Publishers Weekly_\n\n**W.E.B. GRIFFIN'S ACCLAIMED BESTSELLERS**\n\n_**BLOOD AND HONOR**_\n\n\"ROUSING . . . AN IMMENSELY ENTERTAINING ADVENTURE. \" _\u2014Kirkus Reviews_\n\n\"INTRICATELY PLOTTED and packed with those accurate details that fans of Griffin have come to expect.\" _\u2014Booklist_\n\n_**HONOR BOUND**_\n\n\"A TAUTLY WRITTEN STORY whose twists and turns will keep readers guessing until the last page.\" _\u2014Publishers Weekly_\n\n\"A SUPERIOR WAR STORY.\" _\u2014Library Journal_\n\n**W.E.B. GRIFFIN'S CLASSIC SERIES**\n\n_**THE CORPS**_ W.E.B. Griffin's bestselling saga of the heroes we call Marines . . .\n\n\"THE BEST CHRONICLER OF THE U.S. MILITARY EVER TO PUT PEN TO PAPER.\" _\u2014Phoenix Gazette_\n\n\"A BRILLIANT STORY . . . NOT ONLY WORTHWHILE, IT'S A PUBLIC SERVICE.\" _\u2014The Washington Times_\n\n\"GREAT READING. A superb job of mingling fact and fiction . . . [Griffin's] characters come to life.\" _\u2014The Sunday Oklahoman_\n\n\"THIS MAN HAS REALLY DONE HIS HOMEWORK . . . I confess to impatiently awaiting the appearance of succeeding books in the series.\" _\u2014The Washington Post_\n\n\"ACTION-PACKED . . . DIFFICULT TO PUT DOWN.\"\n\n_\u2014Marine Corps Gazette_\n\n_**BROTHERHOOD OF WAR**_\n\nA sweeping military epic of the United States Army that became a  \n _New York Times_ bestselling phenomenon . . .\n\n\"A MAJOR WORK . . . MAGNIFICENT . . . POWERFUL . . . If books about warriors and the women who love them were given medals for authenticity, insight and honesty, _Brotherhood of War_ would be covered with them.\" _\u2014_ William Bradford Huie, author of _The Klansman_ and _The Execution of Private Slovik_\n\n\" _Brotherhood of War_ gets into the hearts and minds of those who by choice or circumstances are called upon to fight our nation's wars.\" \u2014William R. Corson, Lt. Col. (Ret.) U.S.M.C., author of _The Betrayal_ and _The Armies of Ignorance_\n\n\"Captures the rhythms of army life and speech, its rewards and deprivations . . . A WELL-WRITTEN, ABSORBING ACCOUNT.\"\n\n_\u2014Publishers Weekly_\n\n\"REFLECTS THE FLAVOR OF WHAT IT'S LIKE TO BE A PROFESSIONAL SOLDIER.\" \u2014Frederick Downs, author of _The Killing Zone_\n\n\"LARGE, EXCITING, FAST-MOVING.\" \u2014Shirley Ann Grau, author of _The Keeper of the House_\n\n\"A MASTER STORYTELLER who makes sure each book stands on its own.\" \u2014 _Newport News Press_\n\n\"GRIFFIN HAS BEEN CALLED THE LOUIS L'AMOUR OF MILITARY FICTION, AND WITH GOOD REASON.\"\n\n_\u2014Chattanooga News-Free Press_\n\n_**BADGE OF HONOR**_\n\nW.E.B. Griffin's electrifying epic series of a big-city police force . . .\n\n\"DAMN EFFECTIVE . . . He captivates you with characters the way few authors can.\" \u2014Tom Clancy\n\n\"TOUGH, AUTHENTIC . . . POLICE DRAMA AT ITS BEST . . . Readers will feel as if they're part of the investigation, and the true-to-life characters will soon feel like old friends. Excellent reading.\" \u2014Dale Brown, bestselling author of _Storming Heaven_ and _Fatal Terrain_\n\n\"COLORFUL . . . GRITTY . . . TENSE.\"\n\n_\u2014The Philadelphia Inquirer_\n\n\"A REAL WINNER.\" _\u2014New York Daily News_\n\n\"NOT SINCE JOSEPH WAMBAUGH have we been treated to a police story of the caliber that Griffin gives us. He creates a story about real people in a real world doing things that are AS REAL AS TODAY'S HEADLINES.\" \u2014Harold Coyle, bestselling author of\n\n_Team Yankee_ and _Sword Point_\n\n\"FANS OF ED McBAIN'S 87TH PRECINCT NOVELS BETTER MAKE ROOM ON THEIR SHELVES . . . _Badge of Honor_ is first and foremost the story of the people who solve the crimes. The characters come alive.\" \u2014 _Gainsville (GA) Times_\n\n\"GRITTY, FAST-PACED . . . AUTHENTIC.\"\n\n\u2014Richard Herman, Jr., author of _The Warbirds_\n_**Titles by W.E.B. Griffin**_\n\n**HONOR BOUND** HONOR BOUND BLOOD AND HONOR SECRET HONOR\n\n**BROTHERHOOD OF WAR** BOOK I: THE LIEUTENANTS BOOK II: THE CAPTAINS BOOK III: THE MAJORS BOOK IV: THE COLONELS BOOK V: THE BERETS BOOK VI: THE GENERALS BOOK VII: THE NEW BREED BOOK VIII: THE AVIATORS BOOK IX: SPECIAL OPS\n\n**THE CORPS** BOOK I: SEMPER FI BOOK II: CALL TO ARMS BOOK III: COUNTERATTACK BOOK IV: BATTLEGROUND BOOK V: LINE OF FIRE BOOK VI: CLOSE COMBAT BOOK VII: BEHIND THE LINES BOOK VIII: IN DANGER'S PATH BOOK IX: UNDER FIRE BOOK X: RETREAT, HELL!\n\n**BADGE OF HONOR** BOOK I: MEN IN BLUE BOOK II: SPECIAL OPERATIONS BOOK III: THE VICTIM BOOK IV: THE WITNESS BOOK V: THE ASSASSIN BOOK VI: THE MURDERERS BOOK VII: THE INVESTIGATORS BOOK VIII: FINAL JUSTICE\n\n**MEN AT WAR** BOOK I: THE LAST HEROES BOOK II: THE SECRET WARRIORS BOOK III: THE SOLDIER SPIES BOOK IV: THE FIGHTING AGENTS BOOK V: THE SABOTEURS\n\n**PRESIDENTIAL AGENT** BOOK I: BY ORDER OF THE PRESIDENT BOOK II: THE HOSTAGE\n\n**THE BERKLEY PUBLISHING GROUP**   \n**Published by the Penguin Group**   \n**Penguin Group (USA) Inc.**   \n**375 Hudson Street, New York, New York 10014, USA**   \nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada  \n(a division of Pearson Penguin Canada Inc.)  \nPenguin Books Ltd., 80 Strand, London WC2R 0RL, England  \nPenguin Group Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.)  \nPenguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia  \n(a division of Pearson Australia Group Pty. Ltd.)  \nPenguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi\u2014110 017, India  \nPenguin Group (NZ), Cnr. Airborne and Rosedale Roads, Albany, Auckland 1310, New Zealand  \n(a division of Pearson New Zealand Ltd.)  \nPenguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196,  \nSouth Africa\n\nPenguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental. The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.\n\nTHE FIGHTING AGENTS\n\nA Jove Book \/ published by arrangement with the author\n\nCopyright \u00a9 1987 by W.E.B. Griffin.\n\nOriginally published under the pseudonym Alex Baldwin.\n\nAll rights reserved.  \nNo part of this book may be reproduced, scanned, or distributed in any printed or electronic form  \nwithout permission. Please do not participate in or encourage piracy of copyrighted materials in  \nviolation of the author's rights. Purchase only authorized editions.  \nFor information, address: G. P. Putnam's Sons,  \na division of Penguin Group (USA) Inc.,  \n375 Hudson Street, New York, New York 10014.\n\neISBN : 978-0-515-13052-2\n\nJOVE\u00ae  \nJove Books are published by The Berkley Publishing Group,  \na division of Penguin Group (USA) Inc.,  \n375 Hudson Street, New York, New York 10014.  \nJOVE is a registered trademark of Penguin Group (USA) Inc.  \nThe \"J\" design is a trademark belonging to Penguin Group (USA) Inc.\n\n<http:\/\/us.penguingroup.com>\n_For Lieutenant Aaron Bank, Infantry, AUS, detailed OSS (later, Colonel, Special Forces) and Lieutenant William E. Colby, Infantry, AUS, detailed OSS (later, Ambassador and Director, CIA)_\n\n_As Jedburgh Team Leaders operating in German-occupied France and Norway, they set the standards for valor, wisdom, patriotism, and personal integrity that thousands who followed in their steps in the OSS and CIA tried to emulate._\n**Prologue**\n\nSince General Douglas MacArthur's departure for Australia from the Fortress of Corregidor in Manila Bay was in compliance with a direct order from President and Commander -in-Chief Franklin Delano Roosevelt, it was the General's belief that the move was nothing more than a transfer of his headquarters. He believed, in other words, that the battered, outnumbered, starving U.S. and Philippine troops in the Philippine Islands would remain under his command.\n\nHe believed specifically that Lieutenant General Jonathan M. Wainwright, a tall, skinny cavalryman who had been his deputy, would, as regulations and custom prescribed, remain under his orders.\n\nGeneral MacArthur's last order to Wainwright\u2014on the small wooden wharf at Corregidor just before MacArthur, his wife, his son, and a small staff boarded the boats that would take them away\u2014was verbal: He told Wainwright to \"hold on.\" Wainwright understood this to mean that he was forbidden to surrender.\n\nSince he had been promised reinforcement and resupply of his beleaguered forces by Roosevelt himself, MacArthur believed that as long as the Fortress of Corregidor held out, Roosevelt would be forced to make good on his promise of reinforcement. The island of Luzon, including the capital city of Manila, had fallen to the Japanese. But there were upward of twenty thousand reasonably healthy, reasonably well-supplied troops under Major General William Sharp on the island of Mindanao. That force, MacArthur believed, could serve as the nucleus for the recapture of Luzon, once reinforcements came.\n\nMacArthur accepted the possibility that Corregidor might fall. But if that should happen, he believed that Wainwright should move his three-starred, red general's flag and the other colors to Mindanao, assume command of General Sharp's troops, and continue the fight.\n\nBefore MacArthur reached Brisbane, however, traveling first by PT boat and then by B-17 aircraft, General Wainwright began to receive orders directly from Washington, from General George Catlett Marshall, the Chief of Staff.\n\nGeneral MacArthur and General Marshall were not friends. For instance, some time before the war when Marshall was a colonel at Fort Benning, MacArthur, then Chief of Staff of the Army, had officially described Marshall as unfit for command of a unit larger than a regiment. Several such incidents did not bring the two closer.\n\nIt was made clear to General Wainwright by the War Department that he was no longer subject to General Mac-Arthur's orders, and that the conduct of resistance in the Philippines was entirely his own responsibility.\n\nWithout MacArthur's knowledge or consent, the decision had already been made by President Roosevelt, acting with the advice of General Marshall and Brigadier General Dwight D. Eisenhower (who had once served as Mac-Arthur's deputy in the Philippines), that not only was reinforcement of the Philippines impossible\u2014given the relative capabilities of the United States and Imperial Japanese navies\u2014but that the first priority in the war was the conflict against the Germans in North Africa and Europe.\n\nOn May 1, 1942, there were thirteen thousand American and Philippine troops (on a three-eighths ration) in the granite tunnels of Corregidor Island. These included a large number of wounded and all the nurses evacuated from Luzon in order to spare them rape at the hands of the Japanese. That day, Japanese artillery fired sixteen thousand rounds at Corregidor, one heavy shell landing every five seconds. And that many shells were fired the next day. And the next day. And the next.\n\nOn the night of May 5, 1942, when it became evident to General Wainwright that the Japanese were about to make an assault on the fortress, he radioed General Sharp and other commanders elsewhere in the Philippines, releasing them from his command.\n\nAlthough most of the heavy coast artillery cannon on the island had already been destroyed by Japanese artillery, there were enough smaller cannon and automatic weapons still available to Wainwright's forces so that Japanese losses in the assault were severe. But the Japanese were both determined and courageous, and a foothold was gained.\n\nThe fall of Corregidor was no longer in doubt.\n\nThere was nothing to be gained by further resistance. In fact, further resistance would have meant that the Japanese would have trained cannon at the mouth of Malinta Tunnel. These would have swept the tunnel clean of nurses and wounded and the rest of the garrison as effectively as a hose washing down a drainage pipe.\n\nWainwright sent his aide, carrying a white flag, and a staff officer to treat with the enemy.\n\nSoon after that, General Wainwright met with his counterpart, Lieutenant General Masaharu Homma, on the porch of a small, bullet-pocked frame house on Luzon. The shaven-headed Homma, although he spoke fluent English, addressed Wainwright through an interpreter.\n\nHomma was not interested in the surrender of Corregidor. He demanded the absolute, unconditional surrender of all American troops in the Philippine Islands. If General Wainwright were not prepared to offer absolute surrender of all U.S. forces, he would resume tactical operations. By this, he clearly meant wiping out the Corregidor garrison.\n\nAccompanied by a Japanese lieutenant named Kano, who had been educated in New Jersey, General Wainwright was taken in a captured Cadillac to the studios of radio station KZRH in Manila. There he broadcast a message to all commanders of all U.S. military and naval forces in the Philippines. As senior U.S. officer in the Philippines, he ordered all American forces to immediately suspend hostile action and to make all preparations to surrender to the Imperial Japanese Army.\n\nNot all Americans chose to obey General Wainwright's final order.\n**I**\n\n#  **1**\n\n##  **HEADQUARTERS, MINDANAO-VISAYAN FORCE UNITED STATES FORCES IN THE PHILIPPINES 28 DECEMBER 1942**\n\nBrigadier General Wendell W. Fertig, Commanding, Mindanao-Visayan Force, wore two items not commonly seen on general officers of the U.S. Army: a goatee with mustache and a cone-shaped, woven-reed hat perched at a cocky angle on his head. From this dangled what looked like a native bracelet.\n\nGeneral Fertig, a trim, red-haired man of forty-one, was not a professional soldier. He had not gone to West Point; rather, he had entered the military service of the United States just over a year before, directly commissioned as Captain, Corps of Engineers, U.S. Army Reserve. The U.S. Army in the Philippines had been delighted to have the services of an experienced civil engineer, in particular one who was familiar with the Philippines. When he had entered the Army, Fertig had sent his wife and family to safety in Colorado.\n\nFrom the time of the Japanese invasion until the surrender ordered by General Wainwright on May 5, 1942, Fertig had been primarily involved in the demolition\u2014usually by explosive\u2014of roads, bridges and tunnels, supply and petrol dumps, and other facilities to deny their use to the enemy. Many of the facilities he destroyed he had built before the war.\n\nOn May 5, 1942\u2014by then twice promoted\u2014Lt. Colonel Fertig willfully and with full knowledge of the consequences elected to disobey the lawful order of his military superior, Lt. General Jonathan Wainwright, to immediately cease hostile action against the Imperial Japanese Army and to make all preparations to surrender.\n\nHe went instead into the mountains of Mindanao, with every intention of waging what hostile action he could against the Japanese. With him at the beginning were Captain Charles Hedges, another newly commissioned reserve officer of the Army, Chief Petty Officer Ellwood Orfett, USN, and Private Robert Ball, USA.\n\nThings did not go well at first for the little group. To avoid Japanese capture, they had to live in the jungle, eating what they could find there. Or else they ate the native food Moro tribesmen furnished them every now and again\u2014at the risk of their lives.\n\nOnce, they watched from the jungle as a long line of American prisoners\u2014their officers bareheaded and with their arms tied behind them\u2014were moved to a prison camp.\n\nAlthough they encountered some yet-to-surrender Philippine troops, there was no rush to Fertig's colors. Most of the Filipinos, in and out of uniform, sadly suggested to them that the war was over and that the only logical course for the ragtag quartet to follow was to surrender.\n\nBut Fertig, if personally modest, had a somewhat grand notion of the role he could play in the war. He kept a diary, which has survived, and in it, in a rice paddy near Moray, he wrote:\n\n\"I am called on to lead a resistance movement against an implacable enemy under conditions that make victory barely possible. . . . But I feel . . . my course is charted and that only success lies at the end of the trail. . . . If we are to win only part of the time and gain a little each time, in the end we will be successful.\"\n\nLt. Colonel Fertig gave a good deal of thought to the reluctance of the Filipinos and other Americans who had not surrendered to join him. He finally concluded that this was because they quite naturally thought he was simply one more middle-level brass hat, one more American civilian temporarily commissioned into the Army.\n\nThey would, on the other hand, follow a real soldier, he realized. He improved on this: If there were a _general officer_ who announced himself as the official representative of the United States and Philippine governments, that individual would command the respect of everybody.\n\nOn October 1, 1942, on the back of a Delinquent Tax Notice, Fertig wrote a proclamation in pencil and nailed it to a tree:\n\nA Moro silversmith hammered out two five-pointed stars\u2014the rank insignia of a brigadier general\u2014from silver dollars, and Fertig pinned them to his collar points.\n\nIt was likely, Fertig knew, that his proclamation would be blown by the wind from the tree before anyone saw it. Or if it stayed on the tree (the distribution list, for instance, was a bluff; the delinquent tax form was the only sheet of paper he had), that whoever read it would either laugh or conclude there was a crazy American running loose.\n\nBut two days later, as the quartet was walking along the beach beside a Mindanao jungle, ready to rush in and hide if Japanese soldiers appeared, a wiry little Moro wearing vestiges of a uniform and carrying a Model 1917 Enfield U.S. Army rifle stepped into view. And then others appeared, until there were almost two hundred of them.\n\nThe wiry little Moro saluted crisply and in the best English he could manage informed General Fertig that he and his men were at the General's orders, and with respect, could he suggest they go into the jungle, for there were Japanese just a short distance down the beach.\n\nSoon other Filipinos appeared, as well as other Americans who had decided to take their chances in the mountains and the jungles rather than enter Japanese captivity. No one seemed to question the stars on Fertig's collar points; they all seemed happy to be able to place themselves under the orders of someone who knew what he was doing.\n\nA reasonably safe headquarters was established. Though it was not defensible, it was in a location that would be invisible from the air and difficult to locate on the ground. And even if located, it would be very difficult to surround. If Japanese appeared, Fertig and his forces would be able to vanish into the mountains before the Japanese got close.\n\nRemaining free was the first priority.\n\nThe second priority, as Fertig saw it, was to make his presence known to others who had not surrendered and who could join his forces; to the Japanese, who would be obliged to tie down forces on a ratio of at least seven to one in order to look for and contain him; and to the U.S. Army.\n\nThere were risks involved in making the U.S. Army aware of what he was doing. For one thing, he simply might be ordered to surrender. He thus decided that if such an order came, he would not acknowledge it. For another, the U.S. Army was likely to frown both on his self-promotion to brigadier general and on the authority he had vested in himself to take command of Mindanao and proclaim martial law.\n\nFertig decided that these risks had to be taken. There was simply no way he could arm a guerrilla force as large as he envisioned by stealing arms from the Japanese. And the only possible source of arms was the U.S. Army, which could either make airdrops or possibly send a submarine. And then on top of that, just about as important as arms was medicine, especially quinine. And the only possible source of medicine was the Army.\n\nWhat he really needed most of all was money. Not green-backs. Gold. Preferably twenty-dollar gold coins. Lots of twenty-dollar gold coins. With them he could pay his troops, which would lend sorely needed credence to Brigadier General Fertig and his authority. And he could buy food and possibly medicine, and make gifts to Moro chieftains and others who could thereby be persuaded to help him.\n\nThere was one major problem with informing the U.S. Army of the existence of the Mindanao-Visayan force of United States forces in the Philippines: Headquarters, USFIP, had no radio. And if it could somehow get hold of a radio, it had no generator to power it. And if USFIP came into possession of a radio and a generator, and could somehow begin to transmit, there was a very good possibility that the U.S. Army Signal Corps radio operators in the States would not reply. They would presume that the Japanese were playing games with them, because any message from legitimate American forces would be encrypted, that is, sent in code.\n\nActing on the authority he had vested in himself, Fertig commissioned Chief Petty Officer Orfett and Private Ball as second lieutenants. Lieutenant Orfett was put in charge of a deserted coconut-oil mill. Coconut oil could be sold or bartered. Lieutenant Ball was appointed signal officer, USFIP, and ordered to establish communications with the U.S. Army in Australia. He was to use his own judgment in determining how this could be best accomplished.\n\nLieutenant Ball appointed as his chief radio operator a Filipino high school boy by the name of Gerardo Almendres. Almendres, before war came, had completed slightly more than half of a correspondence course in radiotelephony. Using the correspondence course schematic diagrams as a guide, Almendres set about building a shortwave transmitter. Most of his parts came from the sound system of a motion picture projector that had been buried to keep it out of Japanese hands.\n\nA boatload of recruits from Luzon arrived. It comprised the remnants of a Philippine Scout Explosive Ordnance Disposal Detachment: six master sergeants, one of them an American. With them they had an American captain who had deserted USAFFE, U.S. Army Forces, Far East, and taken to the jungles, rather than face certain capture on Corregidor.\n\nThe captain, Horace B. Buchanan, USMA '34, a slight, balding man showing signs of malnutrition, provided the second item necessary to establish communication with the U.S. Army in Australia. It was a small metal box bearing a brass identification tag on which was stamped:\n\nSECRET Device, cryptographic, m94 serial number 145. It is absolutely forbidden to remove this device from its assigned secure cryptographic facility SECRET\n\nGeneral Fertig had never seen one before. He found it fascinating.\n\nIt consisted of twenty-five aluminum disks. Each disk was about the size of a silver dollar and just a little thicker. The disks were stacked together and laid on their edges, so they could rotate independently on an axle. The stack of disks was about five inches long. On the outside of each disk there was printed an alphabet, sometimes A, B, C in proper sequence and sometimes with the characters in a random order.\n\n\"How does it work?\" Fertig asked.\n\nCaptain Buchanan showed him.\n\nEach of the disks was rotated until they all spelled out, horizontally on the \"encrypt-decrypt line,\" the first twenty-five characters of the message they were to transmit. That left the other lines spelling out gibberish.\n\nCryptographic facilities were furnished a Top Secret document, known as the SOI (Signal Operating Instructions) . Among other things, the SOI prescribed the use of another horizontal line, called the \"genatrix,\" for use on a particular day. The gibberish on the genatrix line was what was sent over the air.\n\nActually, Buchanan explained, the SOI provided for a number of genatrix lines, for messages usually were far longer than twenty-five characters. The genatrix lines were selected at random. One day, for example, Lines 02, 13, 18, 21, 07, and so on were selected, and Lines 24, 04, 16, 09, 09, and so on, the next.\n\nWhen the message was received, all the decrypt operator had to do was consult his SOI for that day's genatrix lines. He would then set the first twenty-five characters of the gibberish received on that genatrix line on his Device, Cryptographic, M94, and the decrypted message would appear on the encrypt-decrypt line. He would then move to the next prescribed genatrix line and repeat the process until the entire message had been decrypted.\n\nThe forehead of the red-goateed brigadier general creased thoughtfully.\n\nBuchanan read his mind.\n\n\"In an emergency, Sir,\" Buchanan said, \"in the absence of an SOI, there is an emergency procedure. A code block . . .\"\n\n\"A what?\" Fertig asked.\n\n\"A five-character group of letters, Sir,\" Buchanan explained, \"is included as the third block of the five five-character blocks in the first twenty-five characters. That alerts the decrypt operator to the absence of an SOI.\"\n\n\"And then what?\"\n\n\"First, there is a standard emergency genatrix line sequence. The message will then be decrypted. The receiving station will then attempt to determine the legitimacy of the sender by other means.\"\n\n\"Such as?\"\n\n\"His name, for one thing. Then the maiden name of his wife's mother, the name of his high school principal, or his children. Personal data that would not be available to the enemy.\"\n\nGeneral Fertig nodded.\n\n\"You are a very clever fellow, Buchanan,\" Fertig said. \"You are herewith appointed cryptographic officer for United States forces in the Philippines.\"\n\nThat left two connected problems. The first was to get Gerardo Almendres's International Correspondence School transmitter-receiver up and running. That would require electrical power, and that translated to mean a generator would be required.\n\nBuchanan had no idea how that could be handled, but both he and Lt. Ball suggested that perhaps Master Sergeant George Withers might be of help. Withers was the NCOIC (noncommissioned officer in charge) of the Explosive Ordnance Disposal Detachment on whose boat Buchanan had escaped from Luzon. He was a competent fellow; master sergeants of the Regular U.S. Army are almost by definition highly knowledgeable and resourceful. He had, after all, managed to acquire and hide the boat and bring his detachment safely to Mindanao on it.\n\nMaster Sergeant Withers was summoned.\n\nHe was obviously uncomfortable, and after some gentle prodding, General Fertig got him to blurt out:\n\n\"The truth of the matter is, General, I'm not sure I'm a master sergeant.\"\n\n\"Would you care to explain that, Sergeant?\"\n\nWithers explained that he had been a staff sergeant assigned to an Army ammunition depot on Luzon when he had been suddenly transferred to a Philippine Scout Explosive Ordnance Disposal Detachment.\n\n\"There was fifteen Scouts, General . . . we lost ten before we finally got out. Anyway, Sir, two of them was technical sergeants. They didn't know nothing about explosives, they'd come out of the Twenty-sixth Cavalry with Lieutenant Whittaker when it got all shot up and was disbanded.\"\n\n\"Lieutenant Whittaker? A cavalry officer? Was he killed, too?\" General Fertig asked.\n\n\"No, Sir, and he wasn't a cavalry officer, either. He was a fighter pilot. They put him in the cavalry after they ran out of airplanes, and then they put him to work blowing things up when the Twenty-sixth Cavalry got all shot up and they butchered their horses for rations. He was a fucking artist with TNT . . .\"\n\n\"What happened to him?\"\n\n\"I don't know,\" Withers said. \"The brass on Corregidor sent for him. That's where we got Captain Buchanan. He was sent to fetch Lieutenant Whittaker, and he talked Lieutenant Whittaker into letting him come with us.\"\n\n_It made sense,_ Fertig thought, _that a demolitions expert . . . \"a fucking artist with TNT\" . . . would be summoned to Corregidor to practice his art just before the fortress fell. Poor bastard, if he wasn't dead, he was now in a prison camp. With a little bit of luck, he could be here, and free. USFIP could use a fucking artist with TNT._\n\n\"You were telling me, Sergeant,\" General Fertig said, \"about your rank.\"\n\n\"Yes, Sir. Well, Lieutenant Whittaker thought that since I knew about explosives, and the Scouts didn't, it would be awkward with two of the Scouts outranking me, so he said, right when I first reported to him, that I had been promoted to master sergeant. I'm not sure he had the authority to do that, Sir. I wasn't even on the technical sergeant promotion list.\"\n\nSgt. Withers looked at General Fertig for the general's reaction. His face bore the look of a man who has made a complete confession of his sins and has prepared himself for whatever fate is about to send his way.\n\n\"Sergeant Withers,\" General Fertig said. \"You may consider that your promotion in the field, by my authority, has been confirmed and is now a matter of record.\"\n\n\"Yes, Sir,\" Sgt. Withers said. \"Thank you, General.\"\n\n\"The reason I asked you in here, Sergeant,\" General Fertig said, \"is to ask for your thoughts on a problem we have. We have need of a source of electrical power.\"\n\n\"What for, Sir?\"\n\n\"To power our radio transmitter.\"\n\nWithers hardly hesitated.\n\n\"There's a diesel on the boat\u2014\"\n\n\"We sank the boat.\"\n\n\"We sunk it before on Luzon,\" Withers said, undaunted. \"The engine's sealed. I'll take my Scouts down there and get it.\"\n\n\"And how will you get it up here?\"\n\n\"We'll steal a water buffalo and make a travois . . . like the Indians had? . . . No problem, General.\"\n\n\"The sooner the better, Sergeant,\" General Fertig said.\n\n#  **2**\n\n##  **NAVAL COMMUNICATIONS FACILITY MARE ISLAND NAVY YARD SAN FRANCISCO, CALIFORNIA 5 JANUARY 1943**\n\nThe radioman second looked to be about seventeen years old. He was small and slight, and his light brown hair was cropped close to his skull. He wore government-issue metal-framed glasses, and his earphones made his head look very small.\n\nBut he was good at his trade, capable of transcribing the International Morse Code coming over his Hallicrafters receiver far faster than it was being sent. He had time, in other words, to read what he was typing instead of just serving as a human link in the transmission process.\n\nHe raised one hand over his head to signal his superior while with the other, with practiced skill, he took the sheet of paper in his typewriter out and fed a fresh sheet.\n\nThe lieutenant junior grade who came to his station looked very much like the radioman second, except that he was perhaps four years older and just a little heavier. But he was slight, too, and wore glasses and looked very young.\n\nHe took the sheet of yellow paper from the radioman second and read it:\n\nMFS FOR US FORCES AUSTRALIA  \nMFS FOR US FORCES AUSTRALIA  \nACNOW BRTSS DXSYT QRSHJ ERASH  \nPOFTP QOPOQ CHTFS SDHST ALITS  \nCGHRZ QMSGL QROTX VABCG LSTYE  \nACNOW BRTSS DXSYT QRSHJ ERASH  \nPOFTP QOPOQ CHTFS SDHST ALITS  \nCGHRZ QMSGL QROTX VABCG LSTYE  \nMFS STANDING BY FOR US FORCES AUSTRALIA  \nMFS STANDING BY FOR US FORCES AUSTRALIA\n\n\"What the hell is this?\"\n\n\"Look at the third block, Sir, \" the radioman second said.\n\n\"What about it?\"\n\n\"It was the emergency code, no SOI, when the Army was still using the old M94,\" the radioman second said.\n\n\"Who's MFS?\" the j.g. asked.\n\n\"There's no such station, Sir,\" the radioman second said.\n\n\"What do you think?\"\n\n\"I think it's the Japs playing games,\" the radioman said.\n\n\"Well, what the hell, I'll send it over to the Presidio,\" the j.g. said. \"Maybe they've still got an M94 around someplace. \"\n\n\"You don't think I should give them a call back?\"\n\n\"They weren't trying to reach us, they were calling Australia. Let Australia call them back.\"\n\n#  **3**\n\n##  **THE NATIONAL INSTITUTES OF HEALTH BUILDING WASHINGTON, D.C. 10 JANUARY 1943**\n\nMotor Machinist's Mate First Class Charles D. Staley, USN, in compliance with his orders, presented himself at the National Institutes of Health building.\n\nFive weeks before, Staley had been running the tune-up shop at the Great Lakes Naval Training Center motor pool, outside Chicago. It was a hell of a thing for a first class petty officer with eighteen years' service to be doing with a war on; but Staley was a Yangtze River Patrol sailor, and he had learned that Yangtze River Patrol sailors who had managed to make it back to the States\u2014instead of either getting killed or captured in the Philippines\u2014seemed to get dumb billets like that. The Navy didn't seem to know what to do with them, so it gave them billets like running a motor pool, shit that had to be done but had little to do with ships or fighting a war.\n\nAnd then the personnel chief had called him in and said there was a levy down from BuPers\u2014the Navy Bureau of Personnel\u2014for someone with his rate, who had been a China Sailor, and who was unmarried. The personnel chief said he had to volunteer, for the billet was \"classified and hazardous.\" Reasoning that anything had to be better than cleaning carburetors, Staley volunteered.\n\nFive days later, his orders came through. For the first time in his service, Staley was flown somewhere in a Navy airplane. He was flown to Anacostia Naval Air Station in Washington, where a civilian driving a Plymouth station wagon met him and took him to a large country estate in Virginia about forty miles from Washington. Some very rich guy's house\u2014there was a mansion, and a stable, and a swimming pool, set on 240 acres in the middle of nowhere\u2014had been taken over by the government for the duration.\n\nA real hard-nosed civilian sonofabitch named Eldon C. Baker had given him and ten other guys a short speech, saying the purpose of the training they were about to undergo was to determine if they met the standards of the OSS. Staley didn't know what the hell the OSS was, but he'd been in the service long enough to know when to ask questions and when not to ask questions, and this was one of the times not to ask questions.\n\nBaker, as if he had been reading his mind, almost immediately made that official.\n\n\"This is not a summer camp,\" Baker said, \"where you will make friends for life. You are not to ask questions about the backgrounds, including girlfriends and families, of other trainees, and if a trainee asks you questions that do not directly concern what is going on at the school, you will report that immediately to one of the cadre.\"\n\nBaker had made it clear that if you reported it, the trainee who had asked the questions would be immediately \"relieved\" (which Staley understood to mean thrown out on his ass), and if you didn't report it, _you_ would be relieved.\n\nThey would be restricted to the camp, Mr. Baker told them, for the length of the course, or unless \"sooner relieved for cause.\"\n\nThe training itself had been part boot camp\u2014running around and learning about small arms; part how to fight like a Shanghai pimp\u2014in other words, with a knife, or by sticking your thumbs into a guy's eyes, or kicking him in the balls; part how to blow things up; and part how to be a radio operator. Staley hadn't had any trouble with any of it, but some of the other guys had had a hell of a time, and although they had said as little as possible about themselves, Staley had been able to figure out that most of the other guys were college guys, and he would have laid three to one that at least three of them were officers. Of the twelve guys who started, six made it through. Three got thrown out, one broke his leg climbing up the side of a barn, and two just quit.\n\nSome Army full-bull colonel, a silver-haired Irishman wearing the blue-starred ribbon of the Medal of Honor (the first one Staley had ever seen actually being worn), came to the estate just before they were through with the course and shook their hands; Staley was able to figure out from that that whatever was going on involved more than one service.\n\nTwo days before, the cadre had loaded them all in station wagons, taken them to Washington, and handed them $300 and a list of \"recommended civilian clothing.\" Staley had bought two suits, six shirts, a pair of shoes, and some neckties.\n\nThe night before, one at a time, Baker had called everybody in and given them their orders, which they were not to discuss with anyone else. Staley didn't know what to make of his. He was ordered to report in civilian clothing to the National Institutes of Health, in Washington, D.C.\n\nThey had brought him there in one of the station wagons.\n\nThere was a receptionist in the lobby, and a couple of cops.\n\nHe went to the receptionist, not sure what to do about his orders. They were stamped SECRET, and you don't go around showing SECRET orders to every dame behind a plate-glass window with a hole in it.\n\n\"I was told to report here,\" Staley said, when she finally looked at him.\n\n\"May I have your name, Sir?\" she asked.\n\nWhen he gave it to her, she looked at a typewritten list, then handed him a cardboard badge with VISITOR printed on it and an alligator clip on the back of it so that he could pin it to the lapel of his new suit. Then she called one of the cops over.\n\n\"Would you take Mr. Staley to Chief Ellis, please?\" she said.\n\nThe cop smiled and made a _come with me_ gesture with his hand. Staley followed him to an elevator, and they rode up in it and then went down a corridor until they came to a door with a little sign reading \"Director.\" Inside that door was an office with a couple of women clerks pushing typewriters, an older woman who was obviously in charge, and a door with another sign reading \"Director\" on it.\n\n\"This is Mr. Staley,\" the cop said.\n\n\"The Chief expects him,\" the gray-haired woman said with a smile. Then she looked at Staley. \"Go on in,\" she said.\n\nStaley stopped at the door and, conditioned by long habit of the proper way to report to a commanding officer, knocked and waited to be told to enter.\n\n\"Come in,\" a male voice called.\n\nThere was another office beyond that door, furnished with a large, glistening desk, a red leather couch, and two red leather chairs. Sitting at the desk, sidewards, so he could rest his feet on the open lower drawer of the desk, was a chief boatswain's mate, USN, smoking a cigar and reading a newspaper.\n\n\"Whaddayasay, Staley?\" the Chief said. \"Getting any lately?\"\n\nIt took a moment before Staley was sure who the Chief was, then he said, \"Jesus H. Christ! Ellis!\"\n\nEllis swung around in his high-backed chair and pushed a lever on an intercom box.\n\n\"Could somebody bring us some coffee?\" he asked. Then he turned to Staley and gestured toward the red leather couch. \"Sit down,\" he said. \"Take a load off.\"\n\nChief Boatswain's Mate J. R. Ellis, USN, was wearing a brand-new uniform. There were twenty-four years' worth of hash marks on the sleeve. The uniform was his Christmas present to himself. It was custom-made. He had had custom-made uniforms before, but in China, when he'd been with the Yangtze River Patrol. But he hadn't been a chief then, and custom-made uniforms cost a hell of a lot less in China than they did in the States. Chief Ellis had figured, what the hell, he had never even expected that he would make chief, why the hell not get a stateside custom-made uniform. He could afford it.\n\nThe last time Staley had seen Ellis had been in Shanghai, and Ellis had been right on the edge of getting busted from bosun's mate first and maybe even getting his ass kicked out of the Navy. Ellis had been on the _Panay_ when the Japs sank it in December 1937. After he'd swum away from the burning _Panay,_ Ellis just hadn't given much of a damn for anything. Staley understood that: How the hell could you take pride in being a sailor if your government didn't do a goddamn thing to the goddamn Japs after they sank a U.S. man-of-war and killed a lot of sailors while they were at it?\n\nBut he had never expected to see Ellis as a chief, and certainly not in a billet where he was obviously some kind of a big wheel.\n\nOne of the typists came in with two cups of coffee, in nice cups and saucers, not mugs.\n\n\"There's cream and sugar,\" she said, smiling at Staley, \"but Chief Ellis never uses what he calls 'canned cow.' \"\n\n\"Black's just fine, Ma'am,\" Staley said.\n\nWhen she left, curiosity got the better of him.\n\n\"What the hell is going on around here, Ellis?\" he asked.\n\n\"I've been trying to figure out how to tell you that,\" Ellis said. \"I guess the quickest way is the chain of command.\"\n\n\"Huh?\"\n\n\"Tell me about the chain of command.\"\n\nStaley looked at him in confusion. Ellis was obviously dead serious.\n\n\"Tell me,\" Ellis repeated.\n\n\"Well,\" Staley said, \"I'm first class, and you're a chief, so I report to you, and you report to some officer, and he reports to some senior officer, and it works its way to the top, all the way, I suppose, to the Chief of Naval Operations. \"\n\n\"All the way to the President,\" Ellis corrected him. \"The Chief of Naval Operations reports to the Chairman of the Joint Chiefs of Staff, and _he_ reports to the President, who is Commander-in-Chief.\"\n\n\"So?\" Staley said.\n\n\"The way it is here,\" Ellis said, \"is that you report to me, and I report to the Colonel . . . you met him, he was out to look things over in Virginia. . . .\"\n\n\"The guy with the Medal of Honor?\"\n\n\"Colonel William J. Donovan,\" Ellis said. \"I work for him, and he works for the President. I mean, directly. He gets his orders from the President. Nobody else can tell him what to do.\"\n\nStaley said, \"No shit?\"\n\n\"You're going to have to learn to watch your language around here, Charley,\" Ellis said, almost primly.\n\n\"Sorry,\" Staley said. \"Where do I fit in around here?\"\n\n\"You're going to be the Colonel's driver,\" Ellis said. \"And don't look down your nose at it. There's more to it than driving a car.\"\n\n\"Such as?\"\n\n\"There's a lot of people would like to see him dead, for one thing. Your first job is to see that don't happen.\"\n\n\"Like a bodyguard, you mean? Is that what all that crap in Virginia was for?\"\n\nEllis nodded, but then explained. \"Baker got to the Colonel,\" he said. \"Everybody who comes into the OSS gets run through that school. For a while, I thought they were going to make me go.\"\n\n\"What exactly is this 'OSS'?\"\n\n\"It stands for 'Office of Strategic Services,' \" Ellis said. \"It's sort of like the FBI and Office of Naval Intelligence put together, plus Errol Flynn in one of them war movies where he parachutes behind enemy lines and takes on the whole Jap army by himself.\"\n\n\"Give me a for example,\" Staley said.\n\n\"The school was supposed to teach you Rule One around here,\" Ellis said. \"You don't ask questions. If they figure you should know something, they'll tell you. You ask the wrong questions around here, and you'll wind up counting snowballs on Attu.\"\n\n\"Can I ask what you do around here?\" Staley asked.\n\n\"I'm on the books as 'Special Assistant to the Director, ' \" Ellis said. \"What that means is that I do everything and anything that makes life easier for him, and keeps him from wasting his time. And what you're going to do is help me do that.\"\n\n\"Plus being a bodyguard, you said,\" Staley said.\n\n\"We don't talk about that,\" Ellis said. \"He's got body-guards, mostly ex-FBI guys and ex-Secret Service guys. And he ducks away from them whenever he can. _That's_ when you cover him. Get the picture?\"\n\nStaley nodded. \"I get the feeling you get along pretty good with him.\"\n\n\"I never met anybody smarter or nicer,\" Ellis said flatly. \"Or who works harder.\"\n\n\"How come I got this job?\"\n\n\"The Colonel came in here about two weeks ago,\" Ellis said, \"and found me working about midnight. And he said, 'I thought I told you to get some help.' And he sounded like he meant it. So I asked myself, Do I want some FBI guy who looks down his nose at a sailor and is going to be pissed when he has to take orders from me? And unless I could think of something else, that's what was going to happen. So I called the Navy, BuPers, and told them to find me ex-China Sailors in the States.\"\n\n\"You _told_ the Navy?\" Staley asked.\n\nEllis, grunting, took a small leather wallet from his hip pocket and handed it to Staley.\n\n\"It means what it says on there,\" he said. \"You carry one of those things, everybody in the government, civilian agencies, as well as any military, has got to give you what you ask for. If they don't like it, they can bitch, later, after they give you what you ask for.\"\n\n\"Jesus Christ!\" Staley said, and handed the OSS credentials back.\n\n\"You're going to get one of those,\" Ellis said. \"You fuck up with it, Charley, we'll send you someplace that'll make Portsmouth navy prison look like heaven. And no second chances. You read me?\"\n\n\"Loud and clear, Chief,\" Staley said.\n\n\"You're also going to get a badge and credentials saying you're a deputy U.S. marshal. That's in case anybody asks why you're carrying a gun. You try to get by with that. I mean, you don't show the OSS credentials until you don't have any other choice. You understand?\"\n\nStaley nodded.\n\n\"Same thing applies to the marshal's credentials. Fuck up with them once, and you're finished.\"\n\n\"Okay, okay,\" Staley said.\n\n\"So like I was saying, the Navy found you in Great Lakes, and I remembered that we always got along pretty good, and that you weren't as dumb as you look, so I told them to see if you would volunteer. And you did. And you got through the school all right, and here you are.\"\n\n\"Yeah,\" Staley said. \"Here I am.\"\n\n\"You can walk out of here right now, Charley,\" Ellis said. \"I'll get you any billet you want in the Navy. But if you stay, you're here for good. And there's liable to be more to it before we're done than driving the Colonel's Buick.\"\n\nHe looked at Staley and waited for a response.\n\n\"I'm in, Chief,\" Staley said.\n\nEllis nodded and then dialed one of the three telephones on his desk.\n\n\"I'm sending a guy named Staley down there,\" he said. \"Get him credentials, and take him by the arms room and get him a .45 and a shoulder holster, and then take him over to the house.\"\n\nHe hung the phone up.\n\n\"You'll get a rations and quarters allowance from the Navy,\" Ellis said, \"and a rations and quarters allowance from us. Otherwise you would wind up sleeping on a park bench and starving. Until you can find someplace to live, we'll put you up in the garage at the house.\"\n\n\"The house?\"\n\n\"It's a mansion over in Rock Creek we have,\" Ellis explained. \"There's a couple of apartments over the garage. Nice. Get yourself settled, and then come back here in the morning. I probably shouldn't have to tell you this, but I will. There's two women at the house. They're absolutely off-limits.\"\n\n\"Got it,\" Staley said.\n\n\"You fixed all right for money?\" Ellis asked.\n\n\"Fine.\"\n\nEllis pushed a lever on the intercom.\n\n\"Will you have somebody take Staley to the photolab, please?\" he said, then gestured for Staley to leave.\n\nEllis was pleased with the way things had turned out with Staley. It had been a risk, recruiting him. But he'd done well in the school (that sonofabitch Baker had even been impressed; he'd called and said he had a job for Staley if what he was going to do in Washington was \"relatively unimportant\"), and now that Ellis had talked to him, he thought he could handle what was expected of him here, and, very important, that he would get along with the Colonel. He hadn't been worried about how Staley would get along with Captain Peter Douglass, Sr., USN, Donovan's deputy (a Navy petty officer and a Navy officer would understand each other), but the Colonel might have been a problem.\n\nColonel \"Wild Bill\" Donovan had been one hell of a soldier in his day. He'd won the Medal of Honor in France with the \"Fighting 69th,\" the National Guard regiment from New York City. Between wars, he'd been a rich and powerful lawyer in New York City and Washington. He had little patience for people he decided were fools. But Staley was no fool. The way he'd handled himself at the school and the way he acted now had proved that. He would fit in.\n\nEllis thought of his responsibilities\u2014now to be shared with Staley and maybe even a couple of others, if he could find the right men\u2014rather simply: It was his job to make things easier for the Colonel. Sometimes that meant he would fry up ham and eggs in the kitchen of the Colonel's Georgetown town house. And sometimes it meant that he went around the world with the Colonel, serving as bodyguard and confidant and sort of private secretary and transportation officer. You name it, he did it.\n\nAnd he got to learn a lot. He was _supposed_ to read everything the Colonel read, so that if he had to do something for the Colonel, the Colonel wouldn't have to waste his time explaining things. Some of the stuff he had to read was really pretty dull, but sometimes it was interesting. As far as he had been able to figure out, there was only one secret the Colonel knew that he didn't. Ellis had concluded that Captain Douglass knew that secret, because when Ellis had started getting nosy, Douglass got his back up.\n\nThat secret had something to do with what an Army brigadier general named Leslie Groves was doing at a secret base in the Tennessee mountains with something called uranium. That's what he'd asked Captain Douglass, \"What's uranium?\"\n\nThat's what had gotten Douglass's back up.\n\n\"Now hear this clearly, Chief. You don't ask that question. You don't mention the word 'uranium' to me, or to Colonel Donovan, and certainly not to anyone else. You understand that?\"\n\n\"Aye, aye, Sir.\"\n\nEllis was confident that when the time came, he would find out what uranium was, and what General Groves was doing with it.\n\nSome of the interesting things that came with the job had nothing to do with secrets.\n\nWhat he had been doing when Staley had reported in, for example. He had been reading the _Mainichi._ He didn't think there were very many other people who got to do that. The _Mainichi_ was the English-language newspaper published in Tokyo. The edition he had in his hands was only ten days old. Ellis wondered how the hell they managed to get one in ten days halfway around the world from the Jap capital. But they did. And they did it regularly.\n\nIt was full of bullshit, of course.\n\nFor example, there was a story in the _Mainichi_ today that troops under some Jap general with an unpronounceable name had destroyed the headquarters of Major General Fertig on Mindanao, killed General Fertig, and sent the rest of his troops running off to the mountains to starve.\n\nThe reason Ellis knew the story was pure bullshit was that he had been at a briefing in the situation room when guerrilla activity in the Philippines had been discussed. A full-bull colonel\u2014a guy who had gotten out of the Philippines with MacArthur and then had been sent to Washington as a liaison officer and who should know what he was talking about\u2014had said that while there was a chance that small units of a dozen or so men could evade Japanese capture for as long as several months, there was no possibility of organized \"militarily significant\" guerrilla activity in the Philippines.\n\nAnd there was no General Fertig. Ellis had checked that out himself. The only guy named Fertig in the Philippines was a light colonel, a reserve officer reliably reported to have blown himself up taking down a bridge.\n\nAccording to the _Mainichi,_ this nonexistent general had at least a regiment, which the Japs wiped out to the last man at least once a week.\n\nThe messenger appeared in Chief Ellis's office with the distribution. The messenger was an Army warrant officer in civilian clothes. There was no love lost between them. The warrant officer naturally wondered how come he was wandering around the halls of the National Institutes of Health, delivering the mail like a PFC clerk, while this swab-jockey got to sit around with his feet on a desk reading a newspaper.\n\nEllis signed his name twenty-seven times, acknowledging receipt of twenty-seven Top Secret documents, each of which had to be accounted for separately, and then signed twice more for a batch of Secret, and Confidential, Files.\n\nWhen the messenger had gone, he scanned the titles of the Top Secret documents. He recognized every one of them. They had been here before. Then he read the titles of the Secret documents and scanned through the half dozen he had not seen before. Finally, he turned to the Confidential titles and saw nothing of interest except the regular of-possible -interest memorandum, which Ellis thought of as the \"What-the-Hell-Is-This? List.\"\n\nThis was a compilation of intelligence data that didn't fit into any of the established categories. A report that the Germans had bought a ferryboat in Spain, for example. Or that the Italian Gendarmerie had lost another battle against the Mafia in Sicily. It had come to someone's attention in one of the intelligence agencies. He hadn't known what to do with it, but maybe somebody else could make something of it. When that happened, it was circulated on the of-possible-interest memorandum.\n\nEllis read it faithfully. And his eyebrows went up when he came to item six:\n\n1:6. The Presidio of San Francisco has received from Mare Island Communications Facility an encrypted message transmitted by an unknown station operating in the 20-meter band. The message was encrypted using an apparently captured M94 encryption device.\n\nThe message was addressed to \"U.S. Forces in Australia.\"\n\nThe decrypted message follows: We Have the Hot Poop from the Hot Yanks in the Phils. Fertig Brig Gen\n\nThe station identified itself with the call letters MFS and reported itself standing by.\n\nComment: There is no station with call letters MFS. There is no General officer in the USA or USMC by the name of Fertig. This is therefore presumed to be a Japanese subterfuge. No attempt to contact the calling station has been made.\n\nChief Ellis called the office of the adjutant general in the Pentagon, where he ascertained that there was no confirmed report of the death of Lt. Colonel Wendell W. Fertig, or that he had been captured. His status was missing and presumed dead. He got the name and telephone number of Colonel Fertig's next of kin, Mrs. Mary Fertig, his wife, in Golden, Colorado.\n\nAnd then he took a red grease pencil and drew a box around Item 6 on the What-the-Hell-Is-This? List, tore that sheet from the file, and moved it to the top of the stack of Top Secret documents. Then he carefully scissored the clipping about the glorious victory of Japanese forces over Major General Fertig from the _Mainichi_ and stapled that to the What-the-Hell-Is-This? List.\n\nTwenty minutes later, Colonel William Donovan marched into the office, his face betraying that the morning session at the White House had been difficult.\n\n\"I would kill for a cup of coffee,\" he greeted Ellis as he walked past his desk.\n\nWhen Ellis carried the coffee into the office, Donovan was dangling the page torn from the What-the-Hell-Is-This? List between his thumb and forefinger.\n\n\"What the hell is this?\" he asked.\n\n\"I think it's interesting,\" Ellis said.\n\n\"You want to try to call that station back?\" Donovan asked.\n\nEllis nodded.\n\n\"Have it done,\" Colonel Donovan ordered.\n\n\"Colonel, things get lost in proper channels,\" Ellis said.\n\nDonovan considered that a moment.\n\n\"Meaning you want to go out to California?\"\n\n\"I could be back in three days,\" Ellis said. \"Before it got there through channels.\"\n\n\"You have a gut feeling, Chief?\" Donovan asked.\n\n\"Yes, Sir, you could put it that way.\"\n\n\"Okay,\" Donovan said.\n\nChief Ellis called the chief at Flight Operations at Anacostia Naval Air Station, on the other side of the District of Columbia.\n\n\"Hey, Chief, how they hanging? This is Chief Ellis.\"\n\n\"How's my favorite China Sailor? What are you trying to beat me out of today?\"\n\n\"I need a seat for somebody very important on the next plane to Mare Island.\"\n\n\"Is he self-important, or just very important?\"\n\n\"Actually, he's a pretty good guy.\"\n\n\"Reason I ask is I got a half-dozen torpedo bombers being ferried from Baltimore to load on a carrier at Mare Island. If this guy's not too ritzy to ride in a torpedo bomber . . .\"\n\n\"From Anacostia or Baltimore?\"\n\n\"Here. They're picking up people here. That's how I know about it.\"\n\n\"When?\"\n\n\"How soon can he get here?\"\n\n\"He's on his way.\"\n\n#  **4**\n\n##  **UNITED STATES NAVY BASE, MARE ISLAND SAN FRANCISCO, CALIFORNIA 12 JANUARY 1943**\n\nThe radioman second had seen the base commander only once before, and then he had been riding by in his Navy gray Packard Clipper with its three-starred vice admiral's plate.\n\nAnd now here he was, in the radio room, looking right at him.\n\n\"Stand at ease, son,\" the Admiral said, almost kindly. \"This is Chief Ellis, and he wants to ask you some questions. \"\n\n\"You picked up a message from somebody calling themselves MFS, right?\"\n\n\"That's right, Chief.\"\n\n\"You heard them again?\"\n\n\"They're on every day, for ten, sometimes twenty minutes, \" the radioman second said. \"They were on, oh, hell, twenty minutes ago.\"\n\n\"See if you can raise them,\" Ellis said.\n\nThe Vice Admiral's eyes went up, but he said nothing. He had seen the card signed by the Chairman of the Joint Chiefs of Staff.\n\n\"Go on,\" Ellis repeated. \"See if you can raise them.\"\n\nThe radioman second turned to his key and moved it quickly.\n\n\"I sent 'KSF calling MFS,' \" he replied.\n\n\"I read code,\" Ellis said, not arrogantly.\n\nThere was no immediate reply.\n\nThe radioman second tapped his key again. When the transmitter was activated, the receiver was automatically shut down. When he turned the transmitter to standby, the receiver was issuing a series of dots and dashes.\n\nThe radioman second, without thinking, tapped it out on his typewriter. The Vice Admiral leaned over to read:\n\nMFS STANDING BY FOR KSF\n\n\"Send this,\" Ellis said, and handed the radioman a sheet of paper, on which was typed\n\nKSF TO MFS SEND ENCRYPTED FOLLOWING FIRST NAME OF FERTIG SECOND NEXT OF KIN NAME AND DATE OF BIRTH KSF BY\n\n\"Send it twice, and then wait,\" Ellis ordered. \"If he's using one of these things, it'll take him a minute.\"\n\nHe held up a Device, Cryptographic, M94. He'd had a hell of a time finding one and had annoyed the Presidio of San Francisco no end by requisitioning theirs.\n\nFive minutes later, MFS came back on the air, and the radioman second quickly typed it.\n\nMFS TO KSF QEWRG SJTRE SDIQN SPIID CVKQJ MFS BY\n\nIt didn't take Ellis long to work the Device, Cryptographic, M94; there had been one on the _Panay._\n\n\"Hot damn!\" he said, after a minute. Then he ordered: \"Send 'We are ready for your traffic,' \" and then he corrected himself. \"No, send 'Welcome to the net, we are ready for your traffic.' \"\n\nThen, without asking permission, Chief Ellis picked up the telephone and told the Navy operator to get him Mrs. Mary Fertig in Golden, Colorado.\n\nThe telephone operator said that no long-distance calls could be placed without the authority of the communications officer and an authorization number.\n\n\"I'm going to need an authorization number,\" Ellis said to the communications officer.\n\nThe Admiral motioned for Ellis to hand him the telephone.\n\n\"This is Admiral Sendy,\" he said to the telephone. \"Put the call through.\"\n\nIn Golden, Colorado, Mrs. Mary Fertig answered her telephone.\n\n\"Ma'am,\" Ellis said. \"This is Chief Ellis. You remember me?\"\n\nOf course she remembered him. He had telephoned late the night before and said he couldn't tell her why he wanted to know, but could she give him the full name and date of birth of her oldest child? He had woken her up, and she hadn't been thinking too clearly, so she had given it to him. Later, she had worried about it. There were all kinds of nuts and sick people running loose.\n\n\"Yes, I remember you, Chief,\" Mrs. Fertig said somewhat warily. \"What do you want now?\"\n\n\"Ma'am,\" the salty old chief bosun's mate said, \"we're in contact with your husband. I thought maybe you'd want to say something to him.\"\n\n\"Where is he?\" she asked, very softly.\n\n\"Somewhere in the Philippines, that's all we know,\" Ellis said. Then he said, \"Wait a minute.\"\n\nThe radioman second had handed him a brief decrypted message.\n\nFOR MRS FERTIG QUOTE PINEAPPLES FOR  \nBREAKFAST LOVE END QUOTE\n\nEllis read it over the telephone.\n\nIt took Mrs. Fertig a moment to reply, and then, when she spoke, it was with an audible effort to control her voice.\n\n\"My husband, Chief Ellis,\" she said, \"is on the island of Mindanao. We used to go there to play golf at the course on the Dole Plantation. And we ate pineapples for breakfast. \"\n**II**\n\n#  **1**\n\n##  **SHEPHEARD'S HOTEL CAIRO, EGYPT 23 JANUARY 1943**\n\nCaptain James M. B. Whittaker, U.S. Army Air Corps, was twenty-five years old. He was tall, pale blond, and slender, with leopard-like moves. He was wearing a superbly tailored pink-and-green uniform and half Wellington boots. The uniform and the boots had both come from Savile Row in London. The boots had cost just about as much money as the Air Corps paid Captain Whittaker each month, and the uniform had cost a little more than the boots.\n\nWhittaker had never considered what the uniform and boots had cost, mostly because he really had no idea how much money he had. Whatever his civilian income was, it was more than he could spend. There was a lawyer in New York who looked after his affairs and saw to it that there was always a comfortable balance in his Hanover Trust checking account.\n\nThis is not to suggest that Whittaker was simply a rich young man who happened to be in uniform. There were silver pilot's wings on the breast of his green blouse. He was checked out (qualified to fly) in fighter, bomber, and transport aircraft. Beneath the wings were ribbons representing the award of the Silver Star, the Distinguished Flying Cross, several lesser awards for valor, and brightly colored ribbons indicating that he had had overseas service in both the European and Pacific Theaters of Operation.\n\nAt the moment, Captain James M. B. Whittaker, Harvard University '39, was solemnly considering what he believed to be irrefutable evidence that he was a miserable, amoral, good-for-nothing sonofabitch.\n\nThis solemn consideration sometimes came upon him when he'd taken a drink or two more than he should have. When he had a load on (and he had been drinking, more or less steadily, for the last three days), truth raised its ugly head, and he could see things with a painful clarity.\n\nHe had started drinking before he'd boarded the MATS (Military Air Transport Service) C-54 at London's Croydon Airfield.\n\nTaking leave of Liz Stanfield had been very painful. He loved Liz and she loved him, and there were certain problems with that. For one thing, Captain Elizabeth Alexandra Mary, the Duchess of Stanfield, WRAC (Women's Royal Army Corps), a pale-skinned, splendidly bosomed, lithe woman in her middle thirties, was not really free to love him. There was a husband, Wing Commander the Duke Stanfield, RAF. He was down somewhere, \"missing in action, \" the poor sonofabitch.\n\nOnly a miserable, amoral, good-for-nothing sonofabitch, such as himself, Capt. Whittaker reasoned, would carry on the way he had with a married woman whose husband was missing in action, and a fellow airman to boot. That was really low and rotten.\n\nAnd it wasn't as if he was free, either. He was in love himself. Her name was Cynthia Chenowith, and he had loved her from the time he was thirteen and she was eighteen, and he had gotten a look at her naked breast as she hauled herself out of his uncle Chesty's swimming pool at the winter place in Palm Beach.\n\nIt didn't matter that Cynthia professed not to love him (that was the age difference, he had concluded): _He_ loved her. And a man who loves a woman with his entire soul, who wants to spend the rest of his life with her, caring for her, making babies, is not supposed to go around fucking married women. Unless, of course, he is a miserable, good-for -nothing sonofabitch.\n\nCapt. Whittaker had had the foresight to bring with him on the MATS C-54 three quart bottles of single-malt Scotch whiskey. Half of one had gotten him to Casablanca, and the other half had sustained him from Casablanca to Cairo.\n\nSince he had been in Cairo, he'd worked his way through all of the second bottle and one quarter of the third. The airplane was broken. The pilot had told Capt. Whittaker, as a courtesy to a fellow flyer, that he'd lost oil pressure on Number Three and had no intention of taking off again until they had replaced\u2014rather than repaired\u2014 the faulty pump. One was being flown in from England. When it had been installed, they would continue on their flight, which would ultimately terminate in Brisbane, Australia.\n\nUntil the airplane was repaired, there was a good deal to see and do in Cairo.\n\nMadame Jeanine d'Autrey-Lascal\u2014who was thirty, tall for a French woman, blond, blue-eyed, and who saw no need to wear a brassiere\u2014leaned close to Capt. Whittaker and laid her hand on his.\n\nMadame d'Autrey-Lascal had been left behind in Cairo when her husband, who had been managing director of the Banc d'Egypte et Nord Afrique, had gone off to fight with the Free French under General Charles de Gaulle. She had been in the bank lobby when Capt. Whittaker had appeared to change money and to see if the bank, with which his family's firm had had a long relationship, could do something about getting him into a decent hotel. He had spent the previous night in the transient officers' quarters at the airfield and really didn't want to do that again.\n\nThey had been introduced quite properly, after which it had seemed to Madame d'Autrey-Lascal simply the courteous thing to do to offer to drive him to Shepheard's Hotel. The bank would call in as many favors as it could to get him accommodation in Shepheard's. No promises. The place was always jammed.\n\nThe assistant manager who greeted them said that he would try to find something. No promises. But perhaps if the Captain would not mind waiting for a bit in the bar . . .\n\nIt had seemed to Madame d'Autrey-Lascal that simple courtesy dictated that she not just leave him stranded high and dry in the bar at Shepheard's. If the bank's influence could not get him into Shepheard's, then something else would have to be arranged.\n\nCapt. Whittaker spoke French, which was unexpected of an American, and they chatted pleasantly. She told him that her husband was off with General de Gaulle, and he told her a story about de Gaulle that took her a moment to understand. It seemed that General de Gaulle had declined an invitation to visit with President Roosevelt, on the grounds that it was too long a walk.\n\nBut finally she understood and laughed, and then he told her about London. She hadn't been in London since 1939, and she found what he told her very interesting.\n\nBy the time they had had three drinks from his bottle of single-malt Scotch whiskey, it occurred to Madame d'Autrey-Lascal that it didn't look as though the assistant manager was going to be able to find a room for him in Shepheard's (and if he did, it would be little more than a closet), and that there was absolutely no reason she couldn't put him up overnight, or for a day or two, at her house.\n\nThe first time she suggested this, Capt. Whittaker smiled at her (and she noticed his fine, even teeth) and told her that she was very kind, but he wouldn't think of imposing.\n\nShe told him it would be no imposition at all; the house was large, and at the moment empty, for her children were spending the night with friends.\n\nHe repeated that he wouldn't think of imposing. And then he lapsed into silence, broken only when she laid her hand on his.\n\n\"Sorry,\" Whittaker said. \"I was thousands of miles away.\"\n\n\"Thousands of miles away, you would probably have a hotel room,\" Madame Jeanine d'Autrey-Lascal said. \"Here, you don't. I think you are very sweet for not wanting to impose on me, and very foolish for not believing me when I say it will not be an imposition.\"\n\nHe turned his hand over and caught hers in it.\n\n\"And you are very kind to a lonely traveler,\" he said.\n\n_And I knew the moment I saw you in the bank manager's office that you had an itch in your britches, and miserable, amoral, no-good sonofabitch that I am, given half a chance, that I would wind up scratching it._\n\n\"You have such sad eyes,\" Madame d'Autrey-Lascal said, very softly, as she looked into them.\n\nAnd then, finally, she reclaimed her hand and stood up.\n\n\"Shall we go?\" she asked.\n\nWhittaker followed her out of the crowded bar. As they walked across the lobby, she took his arm.\n\n#  **2**\n\n##  **OSS STATION CAIRO SAVOY HOTEL, OPERA SQUARE 24 JANUARY 1943**\n\nThe Chief, Cairo Station, was Ernest J. Wilkins, thirty-six, a roly-poly man whose face darkened considerably whenever he was upset. He was upset now, and smart enough to know that he was. Before speaking, he went to his window and looked out at the statue of Ibrahim, sitting on his horse in the middle of Opera Square. And then he looked at the Opera building itself, until he was sure he had his temper under control.\n\nThen he turned and faced the three men standing in front of his desk. They were his deputy, his administrative officer, and his liaison officer to the British.\n\n\"Well, where the hell could he be?\" he asked.\n\n\"I think,\" his administrative officer said, \"that we can no longer overlook the possibility of foul play.\"\n\n\"Horseshit,\" Wilkins snapped. \"If anything had happened to him, we would have heard it by now. And since nobody knew he was coming, how the hell could they get anything like that going so quick?\"\n\nHis administrative officer had no response to that and said nothing. Wilkins had hoped that he would say something, so that he could jump his ass.\n\nWilkins lost his temper again.\n\n\"Jesus Christ,\" he flared. \"Do you realize how goddamned inept this makes us look?\" He saw the message on his desk and picked it up and read it aloud:\n\nURGENT  \nFROM OFFICE OF THE DIRECTOR WASHINGTON  \nTO CAIRO FOR WILKINS  \nINTERCEPT CAPTAIN JAMES M. B. WHITTAKER USAAC EN  \nROUTE LONDON TO BRISBANE VIA MATS FLIGHT 216  \nSTOP REDIRECT WASHINGTON FIRST AVAILABLE AIR  \nTRANSPORT STOP ADVISE COMPLIANCE AND ETA  \nWASHINGTON STOP DONOVAN\n\n\"You'll notice,\" Wilkins said, \"that it's signed 'Donovan. ' Not 'Douglass for Donovan,' or 'Chenowith for Donovan,' or even 'Ellis for Donovan.' 'Donovan' himself, goddamnit. And what he's asked us to do isn't going to be written up in a history of intelligence triumphs of the Second World War. All Colonel Donovan asks is that we find some Air Corps captain that he knows is on a MATS flight and send the sonofabitch to Washington.\"\n\n\"Skipper,\" his deputy said to Wilkins (in deference to Wilkins's pre-OSS service as a Naval officer), \"I'll lay even money he's off somewhere getting his ashes hauled.\"\n\n\"Where, for Christ's sake? In the bushes in Al Ezbekia Park, no doubt? For three goddamn days? He's not in a hotel, we know that. And he's not with any high-class whore, or we'd know that, too . . . and goddamn, I found it embarrassing to have to call the Egyptian cops and ask them to check their whores for him. . . .\"\n\nHe stopped and looked out the window at Opera Square again.\n\n\"The Chrysler here?\" he asked, reasonably calmly, when he turned around a moment later.\n\n\"Yes, Sir,\" his deputy said.\n\n\"Nobody stole the wheels? The driver is present and sober?\"\n\n\"Yes, Sir.\"\n\n\"I'll be back,\" Wilkins said, and headed for the door.\n\n\"Going to the airport, Sir?\"\n\nWilkins glared at what he considered to be a stupid question.\n\n\"I'll lay even money he'll show up for the flight, Skipper, \" his deputy said reassuringly.\n\n\"And if he doesn't? What if he got tired of waiting for them to fix the engine and hitchhiked a ride to Brisbane? That MATS flight isn't the only plane headed in that direction. How the hell am I going to say anything to Donovan without looking like a horse's ass?\"\n\nWith an effort, Wilkins kept from slamming the door after him.\n\nThe 1941 Chrysler Imperial was equipped with the very latest in automotive transmission technology. This was called \"fluid drive.\" In theory, it eliminated the need to shift gears. In practice, it didn't work, the result being that it crawled away from a stop. The Chrysler was, Wilkins decided on the way from Opera Square to the airfield, north-east of Cairo, probably the worst possible automobile in the world for Cairo traffic, less practical than a water buffalo pulling a wooden-wheeled cart.\n\nAt the MATS terminal, he sought out the military police captain in charge of security, showed him his OSS identification, and said that it was absolutely essential that he locate one Captain Whittaker, James M. B., USAAC.\n\nTen minutes later, three military police brought Captain Whittaker and a strikingly beautiful woman to the MP captain's office. A flyboy, Wilkins decided somewhat sourly. A good one, to judge by the DFC. He wondered what the OSS wanted from a flyboy.\n\n\"This gentleman wishes to see you, Captain,\" the MP captain said.\n\nWhittaker smiled.\n\n\"As long as it won't take long,\" Whittaker said with a smile. \"They're loading my plane.\"\n\n\"You won't be making that flight, Captain,\" Wilkins said.\n\n\"Says who?\"\n\n\"Says me.\"\n\n\"And who are you?\"\n\n\"That's not really important,\" Wilkins said. \"You'll have to take my word for it. You're coming with me.\"\n\nWhittaker looked at him with amusement in his eyes, his left eyebrow cocked quizzically.\n\n\"That just won't wash,\" Whittaker said.\n\nWilkins took his OSS identity card and held it out.\n\nCaptain Whittaker fumbled in his pockets and came out with a nearly identical card and held it out. Wilkins saw that there were two differences in the cards. His own card bore the serial number 1109 and was signed \"for the Chairman, The Joint Chiefs of Staff\" by Captain Peter Douglass, Sr., USN. Whittaker's card bore the serial number 29 and was signed by Colonel W. J. Donovan, GSC, USA. Obviously, this handsome flyboy had been in the OSS almost from the beginning.\n\n\"What is all this, mon cher?\" the Frenchwoman asked, softly, in French.\n\n\"Nothing at all,\" Whittaker replied, in French, and then looked at Wilkins, waiting for an explanation.\n\nWilkins handed him the radiogram from Donovan.\n\n\"I'll be damned,\" Whittaker said. \"When's my plane?\"\n\n\"Tomorrow,\" Wilkins said. \"At 0915. You had a seat on this morning's flight, but you missed it.\"\n\n\"It appears,\" Whittaker said to the Frenchwoman in French, \"that we're going to have to climb the Great Pyramid again.\"\n\nShe blushed attractively.\n\n\"There are quarters available, if you've checked out of your hotel,\" Wilkins said.\n\n\"That's very kind of you, Sir,\" Whittaker said. \"But that won't be necessary. I'll be staying with a friend.\"\n\nThe Frenchwoman blushed attractively again.\n\n\"War is hell, isn't it?\" Whittaker, smiling broadly, asked Mr. Wilkins.\n\n#  **3**\n\n##  **VIRGINIA HIGHWAY 234 NEAR WASHINGTON, D.C. 25 JANUARY 1943**\n\nThere were four men in the 1942 black Buick Roadmaster, riding in silence.\n\nThere had been a little snow, but the road was clear, and the illuminated needle of the speedometer pointed just past seventy miles per hour. There was virtually no traffic on the road, not even the glow of distant headlights over the gentle hills before them.\n\nWhen the flashing red signal lantern suddenly appeared in the road before them, Chief Ellis was startled. But, even as the driver started stabbing at the brakes, Ellis reached under the seat and came out with a Thompson machine-pistol.\n\nIn the backseat, Colonel William J. Donovan looked up from the document he was reading. Ellis had rigged a really nice reading light on a flexible shaft. The light turned automobile rides into work sessions rather than wastes of time.\n\n\"What is it?\" Captain Peter Douglass asked.\n\n\"Dunno,\" Ellis replied, and then, almost immediately, \"It's the fucking cops!\"\n\n\"How fast were we going?\" Donovan asked calmly.\n\n\"About seventy, Sir,\" Staley, the driver, said.\n\nStaley was in civilian clothing. Ellis was in uniform, except for his brimmed chief's cap, which was on the seat beside him. But in his blue, insignia-less overcoat, he appeared at casual glance to be a portly, ruddy-faced civilian.\n\nEllis shoved the Thompson back under the front seat as the driver pulled onto the shoulder.\n\nThe Virginia state trooper, in a stiff-brimmed hat, swaggered up to the car.\n\n\"May I see your license and registration, please, Sir?\" he asked, with ritual courtesy.\n\nThey were handed over.\n\n\"Sir, are you Charles D. Staley, of this Q Street, Northwest, address, in the District?\"\n\n\"Yes, Sir,\" the driver said.\n\n\"And this vehicle is the property of . . .\" He paused to examine the registration with his flashlight. \". . . W. J. Donovan?\"\n\n\"Yes, it is.\"\n\n\"Does Mr. Donovan know you are driving his vehicle?\"\n\n\"I'm Donovan,\" Donovan said. The trooper flashed his light in Donovan's face.\n\n\"Yes, Sir,\" he said. He returned his attention to the driver. \"Sir, you went through a speed-detection area. You were clocked, over a measured quarter mile, at seventy-three point six miles per hour.\"\n\n\"I didn't realize I was going that fast,\" the driver said.\n\n\"Two state troopers will testify that you were, Sir,\" the trooper said. \"I'm going to have to issue you a citation. You will be charged with reckless driving. The law is that any speed twenty miles in excess of the posted speed limit is considered reckless driving. Are you aware, Sir, that in order to conserve gasoline and rubber for the war effort, the speed limit across the nation is now thirty-five miles per hour?\"\n\n\"I heard about that,\" the driver said dryly.\n\n\"If you are found guilty in a court of law\u2014the place and time of your required appearance will be on the citation I am about to give you\u2014your local ration board will be notified of this violation. You have a C sticker, which means that you agreed in writing to make a genuine effort to conserve the gasoline authorized for you. I think you will agree that driving seventy-three point six miles per hour does not conserve fuel.\"\n\n\"I was in sort of a hurry,\" the driver said.\n\n\"So are our boys in uniform,\" the state trooper said. \"In a hurry to get the war over. And personally, I think we should do all we can to help them.\"\n\n\"Ellis!\" Donovan warned softly.\n\n\"Can I go now?\" Staley asked, taking the citation.\n\n\"Yes, Sir,\" the state trooper said, and marched off.\n\nThe driver cranked up the window.\n\n\"Sorry about that, Colonel,\" he said.\n\n\"Hell, I told you to step on it,\" Donovan said. \"Ellis, give Staley money to pay the fine. If there are any other complications, let Captain Douglass know.\"\n\n\"Yes, Sir,\" Ellis said.\n\n\"And as soon as we're over the next hill,\" Donovan said, \"step on it.\"\n\nTwenty minutes later, the Buick was in the Rock Creek section of the District of Columbia, moving down Q Street, Northwest. They came to an estate surrounded by an eight-foot -high brick wall. The driver switched from low beam to high beam and back again, and a moment later turned off Q Street, stopping the Buick with its nose against a heavy, solid gate in the wall.\n\nA muscular man in civilian clothing stepped out of the shadows and walked to the car. The driver turned the interior lights on for a moment, and then off again.\n\nThe muscular man touched the brim of his snap-brim hat. A moment later, the double gate swung inward. As soon as the car was inside, the gates closed after it.\n\n\"Ellis,\" Donovan said, \"I hate to make you an orderly, but it would save us a lot of time if you went by my house and packed a bag. And get your own while you're at it. Then we can go from here to Union Station.\"\n\n\"Yes, Sir.\"\n\n\"The Secret Service sent over the passes?\" Donovan asked.\n\n\"I'll check on that, too, Sir,\" Ellis said.\n\n\"I don't want to find myself waving bye-bye on the platform as the President goes off to Georgia by himself,\" Donovan said.\n\n\"No, Sir, I'll see we're aboard the train,\" Ellis said.\n\nDonovan and Douglass got out of the car and entered the turn-of-the-century mansion through the kitchen door. The kitchen was enormous and furnished with restaurant-size stoves and refrigerators.\n\nA tall young woman with blond hair hanging to her shoulders came into the room. She wore a simple black dress, a single string of pearls, and just above her right breast a miniature pair of pilot's wings. Captain Douglass's eyes betrayed a moment's surprise and special interest in the wings. He was sure he knew their source: His wife had an identical pair, sent from London by their son. What seemed like last week, their son had seemed an eager-eyed West Point cadet; and now, at twenty-five, he was a lieutenant colonel. His son also liked this girl very much.\n\n\"Good evening,\" Charity Hoche said with a radiant smile. Her accent betrayed her origins: Charity Hoche had been raised on a twenty-acre estate in Wallingford, which was one of the plusher suburbs of Philadelphia, and educated at Bryn Mawr.\n\n\"Hello, Charity,\" Donovan said. \"Mr. Hoover here?\"\n\n\"No, Sir,\" she said. \"And no calls, either. From him.\"\n\n\"Time and J. Edgar Hoover wait for no man,\" Donovan said. \"What are we going to feed him?\"\n\n\"Capon,\" she said. \"And wild rice.\"\n\n\"Good.\" Donovan chuckled. \"Eating chicken with a knife and fork is not one of J. Edgar's strong points. He always makes me feel he'd rather eat one with his hands. After biting off the head, of course.\"\n\n\"And,\" Charity said, \"a very nice Chateau de Long Chablis, '35.\"\n\n\"Where the hell did we get that?\" Donovan asked.\n\n\"Actually, I brought it from home,\" Charity said. \"I knew this was important.\"\n\n\"And you wanted to butter up the boss, too,\" Donovan said.\n\n\"Guilty,\" Charity said with a smile.\n\n\"I might decide to keep you here for your father's cellar, \" Donovan said.\n\n\"As opposed to what?\" Douglass asked.\n\n\"Charity wants to go to England,\" Donovan said. \"I can't imagine why.\"\n\nCharity chuckled deep in her throat.\n\nA very sexual young woman, Captain Douglass thought. Not quite what he had hoped for Peter Douglass, Jr. He wanted for Doug a girl just like the girl who had married dear old dad when he'd been an ensign fresh from Annapolis. Not this Main Line socialite who was used to spending more money on her clothing than Doug (even as an Air Corps lieutenant colonel drawing flight pay) made in a year. And who, according to the FBI's CBI (Complete Background Investigation) on her, was a long way from having any claim to a virginal white bridal dress.\n\nHe was really worried, he thought, that Charity looked on Doug as this year's chic catch, a dashing hero, rather than as someone whose life she would share.\n\n\"There have been some cables from London,\" Charity said. \"Nothing important, except that Fulmar and Fine have left for Lisbon. And there's one from Cairo, with Jimmy Whittaker's ETA.\"\n\n\"Good,\" Donovan said. \"I wasn't sure we could catch him.\"\n\n\"Apparently, they had some trouble finding him,\" Charity said. \"The cable said that he had not checked in with them, which is why he wasn't on an earlier plane.\"\n\n\"I wonder what her name was?\" Donovan chuckled.\n\n\"Jeanine d'Autrey-Lascal,\" Charity furnished. \"Her husband ran a bank there before the war and is now with General de Gaulle.\"\n\n\"Wilkins sent that, too?\" Donovan chuckled. \"Thorough, isn't he?\"\n\n\"Wilkins described her as Jimmy's 'good friend,' \" Charity said.\n\n\"Pilots do get around, don't they, Charity?\" Donovan teased.\n\n\"Until they're finally forced to land,\" Charity said. \"What goes up, they say, has to come down. Eventually, if they're lucky, a Delilah comes into their lives.\"\n\n\"As in Samson-and?\" Donovan chuckled. \"You're planning on giving young Douglass a haircut?\"\n\n\"I don't really think that's what Delilah did to Samson,\" Charity said. \"But if that's what it takes . . .\"\n\nBoth Donovan and Douglass laughed, but Douglass's laughter seemed a little strained. If he had correctly understood Charity, and he was afraid he had, she had as much as said that she was going to drain Doug sexually to the point where straying would be physically impossible.\n\nA buzzer buzzed four times.\n\n\"The Director has arrived,\" Charity said. \"Are you going to meet him outside, or would it be better if we all prostrated ourselves in the entrance foyer?\"\n\nDonovan laughed heartily. He genuinely enjoyed Charity Hoche.\n\n\"Let's meet him outside and bring him in through the kitchen,\" Donovan said.\n\nThey went back to the cobblestone driveway that separated the mansion from the stable\u2014still so called, although it had been converted to a five-car garage\u2014as a Cadillac limousine, bristling with shortwave radio antennae, rolled majestically in.\n\nThere were two neatly dressed young men in the front seat, one of whom jumped out to open the door the instant the car stopped.\n\nJ. Edgar Hoover, the Director of the Federal Bureau of Investigation, got out.\n\n\"Hello, Edgar,\" Donovan said. \"I'm glad you could find the time.\"\n\n\"It's always a pleasure, Bill,\" Hoover said, firmly shaking his hand. He nodded curtly to Captain Douglass. \"Douglass, \" he said.\n\n\"Mr. Director,\" Douglass said.\n\n\"And you know Miss Hoche, I believe, Edgar?\"\n\nHoover beamed.\n\n\"How nice to see you, my dear,\" he said. \"And how is your father?\" Before Charity could open her mouth, he went on, \"You be sure to give both your mother and father my kindest regards.\"\n\n\"Of course,\" Charity said.\n\n\"Would you like a little belt, Edgar?\" Donovan asked. \"Or would you rather go right in to dinner?\"\n\n\"This is one of those days when I would dearly like a little taste,\" Hoover said, \"and just don't have the time.\"\n\n\"Well, we'll give you a rain check,\" Donovan said. \"I'm trying to be very nice to you, Edgar.\"\n\n\"That sounds as if you want something,\" Hoover said, jovially, as they entered the house through the kitchen.\n\n\"Actually,\" Donovan said, \"I was hoping you might have a contact with the state police in Virginia.\"\n\n\"I can probably help,\" Hoover said. \"What is it you need?\"\n\n\"You know somebody that can fix a speeding ticket?\" Donovan asked.\n\nHoover looked at him in genuine surprise.\n\n\"Seventy-three-point-six in a thirty-five-mile zone,\" Donovan said, straight-faced. \"The cop said that we'd probably lose our C-ration sticker, too.\"\n\nHoover smiled.\n\n\"Darn you, Bill,\" he said. \"You really had me going there for a minute.\"\n\n\"Oh, Edgar, you know better than that. I'd never ask you to fix a speeding ticket.\"\n\n\"You didn't really get one, did you?\" Hoover asked.\n\n\"Less than an hour ago,\" Donovan said. \"On the way here. But don't worry about it, Edgar. I'm going to ask the boss for a presidential pardon.\"\n\nHoover's smile was now strained.\n\n\"As soon as we get our business out of the way, Edgar, we're headed for Warm Springs,\" Donovan said. \"On his way down there, Franklin's always in a very good mood. _He'll_ take care of the speeding ticket, I'm sure.\"\n\nHoover marched ahead of him toward the dining room. He knew the way.\n\nDonovan glanced at Charity Hoche. She smiled and gave him a nod of approval. He had put Hoover off balance, and with consummate skill that Charity appreciated. First, by the suggestion of an insult: that the nation's ranking law-enforcement officer, Mr. G-Man himself, would fix a speeding ticket, and then with the announcement that he was going to Warm Springs with President Roosevelt (whom he was privileged to call by his first name) on a trip on which Hoover had obviously not been invited.\n\nThere were very few people who could discomfit J. Edgar Hoover. Donovan, Charity thought, could play him like a violin.\n\nThe table was set for three.\n\nCharity waited until they were seated, then started to leave.\n\n\"I'll serve now, if that would be all right,\" she said.\n\n\"Fine,\" Donovan said, and then, as if he had just thought of it, \"Oh, Charity, there was one more cable from London, a personal to me from Stevens.\"\n\n\"Something I should know about?\"\n\n\"I want you to get it decoded,\" Donovan said. \"The message is 'Katharine Hepburn's Fine by Me.' \"\n\nShe smiled at him. It needed no decoding. Donovan had apparently cabled Lt. Colonel Ed Stevens, Deputy Chief of London Station, asking how he felt about Charity's being transferred there. Making light of her Main Line Philadelphia accent, Charity was known as \"Katharine Hepburn.\"\n\n\"Oh, Uncle Bill,\" Charity blurted, and ran to him and kissed him wetly on the cheek. \" _Thank_ you!\"\n\n\"Serve dinner, Miss Hoche,\" Donovan said. \"The Director looks hungry.\"\n\nHoover did not turn over his glass when a middle-aged maid produced the bottle of Chateau de Long '35.\n\nDonovan interpreted this as a good sign: that Hoover had not come to this meeting with a litany of OSS offenses against the FBI.\n\nThe relationship between the Director of the FBI and the Director of the OSS was complex. When a new broom had been needed to sweep out the scandal-ridden Federal Office of Investigation, the post had been offered to Donovan, both because of his public image as a war hero of untainted honesty, and because of his political influence. He had declined, and taken some effort to see that the job went to J. Edgar Hoover, then a young Justice Department lawyer. When the FBI was established in 1935, Hoover\u2014again with Donovan's support\u2014was named its first director.\n\nBy the time Donovan returned to public service, shortly before the war, as the $1.00-per-annum Coordinator of Information, the predecessor organization to the OSS, Hoover had become a highly respected fixture in Washington, very nearly above criticism.\n\nThe FBI was without question the most efficient law-enforcement agency the nation had ever known, and the credit was clearly Hoover's.\n\nAnd when the idea of a superagency to sit atop all the other governmental intelligence agencies came up, Hoover perhaps naturally presumed that it would fall under the FBI. He was bitterly disappointed when that role was given to the Office of the Coordinator of Information, and his old friend and mentor Bill Donovan was named as its head.\n\nHoover was a skilled political infighter with many friends on Capitol Hill and within Roosevelt's inner circle. He did not simply roll over and play dead. He got President Roosevelt to agree that the FBI should retain its intelligence and counterintelligence roles, not only within the United States but in Latin and South America as well. And he got Roosevelt to keep Bill Donovan's agents in South America under his own control by claiming the right to \"coordinate\" all their activities. Clearly, he could not coordinate their activities unless they made frequent and detailed reports of their activities to the FBI.\n\nDonovan, because he acknowledged the battle as lost, or perhaps because Latin and South America were low in his priorities, gave Hoover his way. Not completely, of course, but he paid lip service to the notion that Hoover had been given North and South America as his area of operations.\n\nHoover saw Donovan for what he was: a highly competent man with a sense of morality and patriotism that was close to his own\u2014and a good friend. But he also saw Donovan as someone who was challenging his (the FBI's) authority in all things concerned with espionage. And this was especially galling because Donovan had the same access to the President's ear that Hoover did. Despite their sharp political and ideological differences, Donovan and Roosevelt had been friends since they had been students at the Columbia School of Law.\n\nAnd, with consummate skill, Roosevelt played games with them\u2014Hoover and Donovan\u2014sometimes pitting one against the other, and other times assuring one that the other regarded him as the greatest patriot and most efficient employee on the government payroll.\n\nAnd both Hoover and Donovan understood that the most dangerous thing that could happen to either was to force Roosevelt to choose between them. As confident of their own ability and their own influence with Roosevelt as they each were, neither was assured that the other would ever be asked for his resignation.\n\nTonight, with nothing specific on the agenda, they exchanged tidbits. Hoover told Donovan and Douglass what his agents had uncovered in Latin and South America. Donovan heard nothing he thought was very important. Much of what Hoover told him, he had heard before.\n\nHoover, only half joking, said that he was on the edge of doubling his security force at Oak Ridge, Tennessee, where the refining of uranium was getting under way in a top-secret plant. He would use half the force, he said, to keep the Germans from finding out what was going on, and the other half to keep the scientists\u2014fifty percent of whom, he said, were \"pinkos\"\u2014from passing what they knew and were learning to the Soviets. There was no question in his mind, Hoover said, that the scientist in charge, J. Robert Oppenheimer, was as left of center as Vladimir Lenin.\n\n\"And it's delicate, you know, Bill, with the Boss,\" Hoover said. \"If he has one flaw in his political judgment, it has to do with the Russians. He thinks Joe Stalin is sort of the Russian senator from Georgia. And that he can buy him off with a dam or a highway.\"\n\nDonovan laughed.\n\n\"You think there's a genuine danger of somebody actually spilling the beans to the Russians?\" he asked.\n\n\"Not so long as I'm in charge of security,\" Hoover said. \"Instead of, for example, Henry Wallace.\"\n\nHe said it with a smile, but Donovan understood that Hoover regarded the Vice President and several of the people around him as bona fide threats to the one great secret of the war: that the United States was engaged in building a bomb that would use as its explosive force nuclear energy, a force\u2014presuming theory could be turned into practice\u2014that would give one five-thousand-pound bomb the destructive force of twenty thousand tons of TNT.\n\n\"Henry doesn't know about Oak Ridge,\" Donovan said. \"And the President tells me he has no intention of telling him.\"\n\n\"Franklin Roosevelt has been known to change his mind to fit the circumstances of the moment,\" Hoover said, adding dryly, \"I'm surprised you haven't noticed.\"\n\nDonovan chuckled appreciatively.\n\n\"On the subject of Oak Ridge, Edgar,\" Donovan said, \"there's something coming up\u2014\"\n\n\"Oh?\" Hoover interrupted.\n\n\"We are going to try to bring some German scientists here,\" Donovan said.\n\n\"You mean, get them out of Germany?\" Hoover asked, surprised. \"Can you do that?\"\n\n\"In the next couple of days,\" Donovan said, \"we're going to make sort of a trial run.\" He waited for Hoover to interrupt him again, and when he didn't, went on. \"The first man we're going to bring out is a metallurgist\u2014\"\n\nNow Hoover obliged him. \"Why a metallurgist?\"\n\n\"I've told you about the German flying bombs and jet-propulsion engines,\" Donovan said. \"I finally managed to convince the President that they pose a real threat, no matter what the Air Corps says, to our plans for the massive bombing of Germany. I have permission to do what I can to at least slow down the production of jet-propulsion and rocket engines. Both require special metal alloys and special techniques to machine the special alloys. The idea is that when we find out what kind of special metal and what kind of special techniques are required for the necessary machining, we will just put those locations on the top of the bombing priority list.\"\n\nHoover grunted, then asked, \"What's this got to do with Groves's bomb? With Oak Ridge?\"\n\n\"If we succeed in getting the metallurgist safely out, and see how much attention the Germans pay to his disappearance, we'll start bringing out the mathematicians and physicists we need . . . or whose services we don't want the Germans to have.\"\n\n\"And if they catch you bringing out the metallurgist, the Germans won't connect it with the Manhattan Project?\" Hoover asked.\n\n\"Precisely,\" Donovan said. \"If we get to the point where we do bring nuclear people out, once they get to this country, they'll be your responsibility, protecting them at either Oak Ridge or White Sands. I thought perhaps, presuming we get the metallurgist out, you might want to use him as sort of a dry run yourself.\"\n\n\"You keep saying 'if' and 'presuming' you can get him out,\" Hoover said. \"There's some question in your mind that you will? Or do you believe the operation won't work?\"\n\n\"We have high hopes, of course,\" Donovan said, and went on to explain that the OSS had set up a new escape route \"pipeline,\" which ran through Hungary and Yugoslavia, for the sole purpose of getting the \"special category\" people out of Germany. The normal, in-place pipelines took people off the European continent through Holland and France to England.\n\nHoover displayed a deep curiosity in the details of the new pipeline, and Donovan explained the operation to him, wondering if the FBI Director's curiosity was professional or personal. Hoover, he knew, liked to think of himself as an agent rather than an administrator. Donovan suspected that Hoover was vicariously crossing the border from Germany into Hungary, and then walking out of Yugoslavia in the company of Yugoslavian partisans.\n\nWhen the explanation was finally over, Hoover grunted, then looked at Captain Douglass.\n\n\"You don't seem to have much to say, Douglass,\" he said.\n\n\"I ask Pete to sit in on the more important meetings, Edgar, so I don't have to spend time repeating to him what was said.\"\n\n\"I was thinking along those lines myself,\" Hoover said. \"That it's going to take me some time to repeat all this to Tolson.\" Clyde Tolson was Deputy Director of the FBI and Hoover's closest friend. They shared a house.\n\n\"If Clyde was cleared for the Manhattan Project,\" Donovan said, \"I'd be the first to say bring him along.\"\n\n\"Clyde knows about the Manhattan Project,\" Hoover said. \"He's my Deputy.\"\n\nDonovan was not surprised that Hoover had made Tolson privy to the secrets of the Manhattan Project, but he was surprised that Hoover had admitted it so openly to him. Tolson, like Vice President Wallace, was not on the shortlist of people authorized access to information concerning the atomic bomb.\n\n\"Then you should have brought him with you, Edgar,\" Donovan said. \"Clyde's an old pal. He doesn't need a formal invitation to break bread with us.\"\n\nHoover, Donovan realized, had just put him on a spot. Should he run, as he was supposed to, to Roosevelt and tattle that the head of the FBI had taken it upon himself to breach security? If he did, would it turn out that Hoover had gotten permission from Roosevelt to tell Tolson? Which would make him look the fool. And if he didn't go, would Roosevelt find out, and be justifiably angry that he had known and said nothing?\n\nHe decided that this was one of those rare instances where it was necessary to be very open with Hoover.\n\n\"Edgar, does Roosevelt know you've decided it was necessary to brief Clyde?\"\n\n\"No,\" Hoover said, and met his eyes. \"Are you going to tell him?\"\n\n\"Certainly,\" Donovan said. \"I've been hearing rumors about Clyde. He's supposed to be about as pinko as Henry Wallace.\"\n\nHoover laughed, but his smile was strained.\n\n_I'll let you worry about whether or not I'm going to tell Roosevelt,_ Donovan thought. _That hand went to me. Another proof of the theory that when you really don't know what to do, try telling the truth._\n\nHoover looked at his wristwatch and stood up.\n\n\"I had no idea how late it is,\" he said.\n\n\"I'll walk you to your car, Edgar,\" Donovan said.\n\n#  **4**\n\n##  **UNION STATION WASHINGTON, D.C. 30 JANUARY 1943**\n\nStaley had no trouble picking Capt. James M. B. Whittaker out of the crowd of people walking away from the train, although many of them were in uniform, and almost a dozen of those in uniform were captains of the U.S. Army Air Corps.\n\n\"Look for a guy who looks like an Air Corps recruiting poster,\" Chief Ellis had told him. \"Tall, good-looking, and either the sloppiest officer you ever saw, or the sharpest. Depends on how he feels right then.\"\n\nCapt. Whittaker, Staley concluded, had decided to be sharp. He was wearing a perfectly tailored pink-and-green uniform, and he was in the process of putting on a camel's-hair short coat when Staley spotted him. His brimmed cap had the fifty-mission crush, an affectation of a fighter pilot, but except for that, he looked as if he had just walked out of a clothing store window.\n\nStaley intercepted him, catching himself just before he started to salute. He was not quite used to wearing civilian clothes and acting like a civilian.\n\n\"Captain Whittaker?\"\n\n\"Guilty,\" Whittaker said, smiling at him.\n\n\"I'm standing in for Chief Ellis, Sir,\" Staley said. \"Let me give you a hand with your gear.\"\n\n\"Since you're foolish enough to volunteer,\" Whittaker said, \"you can have the heavy one. Where's Ellis?\"\n\n\"He's in Georgia, Sir,\" Staley said.\n\n\"With the Colonel? And our commander-in-chief?\"\n\n\"Yes, Sir,\" Staley said, wondering how Whittaker could have known that.\n\nWhen they were in the Buick, Whittaker said, \"Well, I appreciate you meeting me, but I could have taken a cab.\"\n\n\"To Virginia?\" Staley blurted. Ellis had told him that Whittaker was rich, that, in fact, he owned the house on Q Street, but the notion of taking a forty-mile taxi ride startled him.\n\n\"Virginia? I'm talking about Q Street.\"\n\n\"Sir, I'm supposed to take you to Virginia,\" Staley said.\n\n\"I'm going to the house on Q Street,\" Whittaker said firmly. \"If that makes it awkward for you, just drop me at the next corner. I'll catch a cab, and we'll say you couldn't find me at Union Station.\"\n\n\"They expect you in Virginia,\" Staley protested.\n\n\"In a word, fuck 'em,\" Whittaker said, then, quickly, \"Right over there, there's a cab.\"\n\n\"I'll take you to the house,\" Staley said. \"Nobody said anything about me making you go to Virginia. But if you tell them I told you . . .\"\n\n\"I'll cover you,\" Whittaker said. \"You know what goes on in Virginia, I suppose? They do all sorts of obscene things out there, like push-ups and running for miles before they have breakfast.\"\n\nStaley laughed. \"I went through it.\"\n\n\"Then you must know a prick by the name of Eldon C. Baker,\" Whittaker said, \"which is another reason I'm not going to Virginia.\"\n\n\"I know him,\" Staley said.\n\nWhen they got to the house on Q Street, Northwest, the guard would not pass the Buick through the gate until Whittaker showed him his credentials.\n\nAnd when they walked into the kitchen, Charity Hoche, in her bathrobe, was waiting for them.\n\n\"You're not supposed to be here, Jimmy,\" she said.\n\n\"Jesus, and I was hoping for something along the lines of 'Welcome home, Jimmy.' \"\n\n\"They expect you in Virginia,\" Charity said.\n\n\"I hope they're not holding their breath,\" Whittaker said. \"Aren't you going to ask me about Doug?\"\n\n\"How's Doug?\"\n\n\"Bearing up rather well, considering,\" he said.\n\n\"Bearing up rather well considering what?\"\n\n\"That he's the official stud for the London-area Red Cross girls,\" Whittaker said. \"Some of them are real man-eaters. \"\n\n\"Damn you,\" she said.\n\n\"Actually, the last time I saw him, he was staring moodily off into space, muttering Browning sonnets,\" Whittaker said. \" 'How do I love Charity? Let me count the ways. . . . I love her . . .' \"\n\n\"That's better,\" Charity said. \"I'm going over there. I found out a couple of days ago.\"\n\n\"Well, that should certainly change his social life,\" Whittaker said, and then he asked the question that had been on his mind since he first saw Charity.\n\n\"Where's the regular house mother?\"\n\n\"Cynthia's at the place in Virginia,\" Charity said.\n\n\"What's she doing there?\"\n\n\"Going through the course,\" Charity said.\n\n\"What course?\"\n\n\"The regular course,\" Charity said.\n\n\"What the hell is that all about?\" he asked.\n\n\"What do you think?\" Charity asked.\n\nThe notion that Cynthia was undergoing training to become an agent was so preposterous that he didn't pursue it.\n\n\"I'll go out there in the morning,\" he said. \"Is my car here?\"\n\n\"It is, but I'm not sure they allow you to have a car out there,\" Charity said.\n\n\"I'll take my chances,\" he said. \"Now, if you will give me some whiskey to drink, I'll brief you on the competition you're going to face when you get to England. And just for the record, Charity, I came here over the very strenuous objections of this gentleman.\"\n\n\"Staley's my name, Captain,\" Staley said, and offered his hand. Staley liked Whittaker. Ellis had said he would. He himself hadn't been so sure. Officers are officers. But there was something about this guy that made him special.\n\n\"Over the strenuous objections of Mr. Staley,\" Whittaker said. \"And now can I have some booze?\"\n\nHe woke early, his body clock confused by the distances he'd covered, and aware that sometime around two in the afternoon, he would get very sleepy. Worse, he thought, his mind would be dulled. And he wanted to be sharp when he saw Cynthia.\n\nHe took a shower in the large, tiled, two-headed shower where legend had it that Chesley Haywood Whittaker, his uncle \"Chesty,\" had died of a stroke. The truth was that Chesty Whittaker had died in the saddle, on Pearl Harbor Day, and that Chief Ellis had manhandled the body over here so that it could be \"found\" in his own shower rather than in the bed of a young woman, the daughter of a college classmate, with whom he had had a two-year affair. The young woman's name was Cynthia Chenowith.\n\nOnly a few people knew what had really happened: Wild Bill Donovan\u2014who had been Chesty's lifelong crony and with whom he had flown to Washington when Donovan had been summoned to the White House\u2014knew. And Captain Douglass knew. And Chief Ellis. And Dick Canidy, Whittaker's school chum and now number-three man in London for the OSS. And, of course, Jimmy Whittaker knew. He didn't think Cynthia knew he knew, and that was the way he wanted to keep it. It didn't matter to him, he told himself\u2014and most of the time, he believed, it didn't.\n\nBut he thought about it in the shower, and he thought about it when he backed the Packard out of the garage. The 1941 Packard 280 convertible coupe had been Chesty's. Presumably, Chesty and Cynthia had been in it together on many happy occasions. He didn't think they had made the beast with two backs in the backseat, but it was reasonable to presume that they had held hands, and kissed, and that sort of thing.\n\nDespite the cold, when he was out of the District, he pulled to the side of the road and put the roof down. He had the heater going full blast, and he left the windows up, and it was really rather pleasant.\n\nA quarter of a mile off the state highway into the Virginia property, well out of sight of the highway, a guard post had been erected, and Whittaker learned that Charity had been right about the car. They expected him, but not at the wheel of a car.\n\n\"I really don't know what the hell to say,\" the guard, wearing the uniform of a member of the National Park Service police, said. \"I got your name on the list, Captain, but as a trainee, and trainees can't have private cars.\"\n\n\"But as I've shown you, I'm not a trainee,\" Whittaker said. \"Look, call Baker and tell him I'm here, and driving a car.\"\n\nThe guard went into his little shack and a moment later came out again, and said, \"Mr. Baker says come right to his office, Captain. It's in the main house. You can't miss it.\"\n\nThe road wound through a stand of pine trees, and as he was coming out of it, he passed a group of twelve or fifteen trainees taking a run. They were carrying, in front of them, at \"Port Arms,\" Springfield Model 1903 caliber .30-06 rifles, not that it was expected they would ever use one, but to make the physical conditioning a little tougher.\n\nHe slowed down and glanced out the side window at them as he passed them. And saw Cynthia Chenowith. She had her hair hidden under a GI fatigue habit, and the truth was that he saw her breasts flopping around under her fatigue jacket and marveled at that for a moment before he recognized her.\n\n\"Oh, shit!\" he said with great disgust, then stepped on the accelerator.\n\nEldon Baker's office was in what had been the breakfast room of the mansion, a rather small room whose floor-length doors opened onto a flagstone patio, and beyond that to a flat grassy area that Whittaker remembered as having been a putting green.\n\nBaker was sitting behind a government-issue gray metal desk when Whittaker walked in. He was a pudgy-faced man in his thirties. He was wearing fatigues, but where an officer would have worn the insignia of his rank and branch of service, there was a square insignia embroidered in blue: a triangle within the square, and the letters \"U.S.\" It was the insignia worn by civilian experts attached to the U.S. Army in the field. Baker had been a State Department intelligence officer before joining the OSS, where he was listed on the OSS Table of Organization as \"Chief, Recruitment and Training.\" So far as Whittaker knew, he had never been in the service.\n\n\"Well, hello, Jim,\" Baker said. \"We rather expected you last night.\"\n\n\"You look very military, Eldon,\" Whittaker said. \"Am I expected to salute?\"\n\n\"We don't salute around here,\" Baker said. \"Neither do we wear insignia of rank or branch of service.\"\n\n\"Can I ask you a question?\" Whittaker asked.\n\n\"What are you doing here? Well, that's very simple. You haven't gone through the course and . . .\"\n\n\"What is Cynthia Chenowith doing running around in fatigues and carrying a Springfield?\"\n\n\"Isn't that self-evident? She's going through the course. And doing rather well. Frankly, much better than I expected she would.\"\n\n\"To what end?\" Whittaker asked.\n\n\"Again, isn't that self-evident?\"\n\n\"You're out of your fucking mind, Eldon,\" Whittaker said matter-of-factly. \"What the hell is the matter with you?\"\n\n\"I had hoped that our relationship would be amicable,\" Baker said. \"You're making that difficult.\"\n\n\"Are you telling me you seriously propose to send that girl out operationally?\" Whittaker asked.\n\n\"Nothing specific at the moment, but when the opportunity presents itself . . .\"\n\n\"And Bill Donovan's going along with that insane notion? \"\n\n\"Obviously, it has Colonel Donovan's approval,\" Baker said. \"And, as obviously, it's really none of your business, is it?\"\n\n\"I'm making it my business,\" Whittaker said.\n\n\"Have you some explanation for not coming here as you were directed to do?\" Baker said. \"You will notice I have changed the subject.\"\n\n\"I don't have to explain anything to you, Eldon,\" Whittaker said. \"I don't work for you. I don't even know what I'm doing in the States.\"\n\nThere was more, Whittaker thought, than simple chemistry to explain why he had disliked Eldon Baker from the moment he had met him. He could prepare a long list of Things-Wrong-with-Eldon-Baker, headed by Baker's ruthlessness, and running down to such items as pompous, overbearing, and the compleat bureaucrat, but it was the chemistry primarily responsible for the inevitable verbal flare-ups whenever they were together.\n\nBaker now chose to tolerate Whittaker.\n\n\"There's a mission envisioned for you,\" he said.\n\n\"What kind of a mission?\"\n\nBaker ignored the question.\n\n\"Prior to which it has been decided that you will go through the course.\"\n\n\"Decided by whom?\"\n\n\"It's OSS policy,\" Baker said, \"that everybody will go through the school.\"\n\n\"You're weaseling,\" Whittaker said. \"Donovan doesn't know you expect me to go through this school of yours for spies, does he? You were just going to tell me that's the 'way it is.' Screw you, Eldon. That won't work. Canidy told me that Donovan told him that neither one of us had to do this. For Christ's sake, I was _running_ the school in England. \"\n\n\"You have no training in infiltration by rubber boat from a submarine,\" Baker said. \"Obviously, it was not my intention to send you through the whole course . . .\"\n\n\"Oh?\"\n\n\"And actually, I had planned to ask you to teach a few hours. I thought it would really get and keep the men's attention if they understood they were being taught by someone who had been operational.\"\n\n\"If that's a bone you're throwing, gnaw on it yourself,\" Whittaker said. He started out of the room, then turned and stopped at the door. \"I'm going back to Washington,\" he said. \"And it's going to take Wild Bill himself to order me back here. And then I may not come.\"\n\n\"Obviously, there's no purpose in debating this with you,\" Baker said.\n\nWhen he went outside the building, determined to find Cynthia, Whittaker saw her immediately. In the time it had taken him to go through the confrontation with Baker, her group of trainees had run from where he had seen them on the road to the mansion.\n\nPresumably, he decided, they had run all the way. Cynthia and another woman, both of them red-faced and heaving from the exertion, were sitting on the ground, their backs against a wall.\n\nHe walked over to her. She looked up at him but said nothing.\n\nOne of the senior trainees walked quickly up to him. He was tall and muscular and very handsome, and looked somehow familiar to Whittaker.\n\n\"May I help you, Sir?\" he asked.\n\n\"Take a walk,\" Whittaker said.\n\nHe met Cynthia's eyes. \"What the hell do you think you're doing?\"\n\n\"What does it look like?\" she replied.\n\n\"Jesus Christ, if it wasn't so stupid, it would be funny,\" he said.\n\n\"Jimmy, why don't you just turn around and walk away from here?\" Cynthia asked.\n\nInstead, he reached down and grabbed her wrist and jerked her to her feet.\n\n\"What do you think you're doing?\" she snapped.\n\nHe kissed her, moving so quickly there was no time for her to avert her face, and so surprising her that it was a moment before she twisted free.\n\nOne of the trainees laughed and applauded.\n\n\"What was that all about?\" Cynthia said, seeming torn between outrage and tears. \"Why did you do that?\"\n\n\"Two reasons,\" he said. \"To remind you that you're a woman. And because I love you.\"\n\n\"Damn you!\" Cynthia said, fighting an infuriating urge to cry.\n\n\"Now, just a minute here!\" the senior trainee said.\n\n\"Greg, don't!\" Cynthia called quickly. \"He's crazy. He'll kill you!\"\n\nThe trainee looked at him warily and with great interest.\n\n\"Relax,\" Whittaker said. \"I'm a lover, not a fighter.\" Then, feeling very pleased with himself, he walked over to the Packard, got in, and started it up.\n**III**\n\n#  **1**\n\n##  **HEADQUARTERS, 344TH FIGHTER GROUP ATCHAM ARMY AIR CORPS STATION, ENGLAND 31 JANUARY 1943**\n\nRank hath its privileges. In this case that meant that the commanding officer of the 344th Fighter Group was driven in a jeep from the final briefing to the revetment where his aircraft was parked. The other pilots rode jammed together in the backs of trucks.\n\nThe commanding officer of the 344th Fighter Group, Eighth United States Air Force, was Lieutenant Colonel Peter (\"Doug\") Douglass, Jr., USMA '39, a slight, pleasant -appearing officer who looked, until you saw his eyes, much too young to be either a fighter group commander or a lieutenant colonel. He was, in fact, twenty-five years old.\n\nHe was wearing a horsehide A-2 jacket, which had a zipper front and knit cuffs. On its back was painted the flag of the Republic of China and a legend in Chinese stating that the wearer had come to China to fight the Japanese invader, and that a reward in gold would be paid for his safe return in case he fell from the sky.\n\nDoug Douglass had been a member of the American Volunteer Group in China and Burma, a \"Flying Tiger,\" one of a small group of pilots who, before the United States had entered the war, were recruited from the Army Air Corps, the Marines, and the Navy to fly Curtiss P-40 fighters against the Japanese. On the nose of his P-38F there were painted ten small Japanese flags, called \"meatballs, \" each signifying a Japanese kill. There were also painted six swastikas, representing the kills of six German aircraft, and the representation of a submarine.\n\nWhile attacking the German submarine pens at Saint-Lazare, then-Major Douglass had attempted to skip-bomb a five-hundred-pound aerial bomb into the mouth of the pens. He hadn't made it. But his bomb had struck, literally by accident, a U-boat tied to a wharf just outside the mouth of the pen. It had penetrated the hull in the forward torpedo room, and what was known as a \"sympathetic explosion\" had occurred. The explosives in the bomb and in God-alone -knew-how-many torpedoes had combined, and the submarine had simply disappeared, leaving few recognizable pieces.\n\nDouglass and his group had been accompanied on the mission by photo reconnaissance aircraft, and there was a motion picture record of the five-hundred-pound bombs dropping from Douglass's wings, and of one of them striking the submarine, and of large chunks of the submarine hull floating lazily through the air. There was no question about it, mistakes counted, it was a confirmed kill.\n\nNewly promoted Lieutenant Colonel Douglass had given in to the \"suggestion\" by his division commander that he paint a submarine on the nose of his P-38F not because he considered it a victory but because it signified that he had been on the Saint-Lazare raid. He had lost forty percent of his aircraft\u2014and his pilots\u2014on that raid.\n\nA story made the rounds that after the raid Douglass had walked into Eighth Air Force Headquarters and decked the Plans & Training officer who had ordered the mission. And that the bloody nose he'd given the chair-warmer had given the brass a choice between court-martialing a West Pointer who was a triple ace or promoting him, and they'd opted in favor of the promotion.\n\nToday, there was with him in the jeep as it made its way down the parking ramp at Atcham another pilot wearing an identical A-2 jacket with the Chinese flag and calligraphy painted on its back. He was taller and heavier than Douglass, and, at twenty-six, a year older. His name was Richard Canidy, and he had been Lt. Col. Douglass's squadron leader in the Flying Tigers.\n\nHe was not a member of the 344th Fighter Group, nor, despite the gold leaves of a major pinned to his A-2 jacket epaulets, even an officer of the Army Air Corps. Canidy (BS, Aeronautical Engineering, MIT '38) had first been recruited from his duty as a lieutenant junior grade, USNR, instructor pilot to be a Flying Tiger, and from the Flying Tigers to be a \"technical consultant\" to the Office of the Coordinator of Information.\n\nThe Office of the Coordinator of Information had been redesignated the Office of Strategic Services, and Canidy was now officer in charge, Whitbey House Station, OSS-England, which made him the third-ranking OSS officer in England. Civilians, in a military environment, attract attention. But little attention is paid, particularly at the upper levels of the military hierarchy, to majors. It had been arranged with the Army Air Corps to issue \"Technical Consultant Canidy\" an AGO card from the Adjutant General's Office, identifying him as a major, and to ensure that if inquiries were made at Eighth Air Force or SHAEF (Supreme Headquarters, Allied Expeditionary Force) there would be a record of a Canidy, Major Richard M., USAAC.\n\nCanidy was not supposed to be flying with the 344th Fighter Group on this mission. Indeed, if either he or Lt. Col. Douglass had asked their superiors for permission for him to come along, the request would have been denied.\n\nDouglass wasn't sure why Canidy wanted to go. He guessed that it had something to do with Jimmy Whittaker getting his ass shipped to Australia, and with Eric Fulmar and Stanley Fine having disappeared suddenly from Whitbey House, destination and purpose unspecified. Canidy's old gang, with the exception of Lt. Commander Eddie Bitter, USN (another ex-Flying Tiger), and of course Douglass himself, had been broken up. A deal like that, being with your buddies, was of course too good to last.\n\nOnce, at Whitbey House, Douglass with most of a quart of Scotch in him, had looked at the others with a sudden wave of warmth: _They were good guys, the best, and they were his buddies; he would never, as long as he lived, have better friends._ And then he had made what had seemed in his condition to be a profound philosophical observation: _\"War, like politics, makes strange bedfellows.\"_\n\nThe undisputed leader of the gang, the best natural commander Douglass had ever seen\u2014and the test had been combat\u2014was Canidy. And Canidy was not, like Douglass (West Point) and Bitter (Annapolis), a professional warrior, but almost the antithesis, an MIT-trained aeronautical engineer who made no secret that he found most of the traditions sacred to the professional military hilarious.\n\nThe wise man, the philosopher so to speak, of the gang was Captain Stanley S. Fine, a tall ascetic Jew who had been a Hollywood lawyer before he had been recruited for the OSS from command of a B-17 Squadron. If closing with the enemy and killing him with bare hands was the ultimate description of a warrior, then the gang's most ferocious members were unlikely warriors. Eric Fulmar was the son of a movie star and a German industrialist, and Jimmy Whittaker was a wealthy socialite who addressed the President of the United States as \"Uncle Franklin.\"\n\nDouglass knew that if coincidence had thrown these men together in any normal military organization, and if, improbably, they had become buddies there, any commanding officer with enough sense to find his ass with both hands would have broken up the gang and transferred them as far from each other as possible\u2014as awesome threats to \"good military order and discipline.\"\n\nBut they weren't in any normal military organization. They were in the Office of Strategic Services.\n\nLt. Col. Douglass knew more about the OSS than he had any right to know. He wasn't even supposed to know about Whitbey House, much less spend most of his free time in the requisitioned mansion, the ancestral home of the Dukes of Stanfield. But he was a special case. Not only had he been Dick Canidy's wingman in the Flying Tigers, but his father was Captain Peter Douglass, Sr., deputy director of the OSS, Colonel Wild Bill Donovan's number two.\n\nDavid Bruce, Chief of London Station, and his deputy, Lt. Col. Ed Stevens, simply ignored Douglass's illegal presence at Whitbey House when they saw him there. Canidy and the others didn't talk about what they were doing in Douglass's presence, or tried hard not to, but it was difficult to remember all the time that Douglass didn't have the Need-to-Know, and things slipped out.\n\nWhen Canidy had hinted that he wouldn't mind getting checked out in the P-38F, Douglass had known that the next inevitable step would be for him to go along on a mission. But it would have been difficult to tell his old squadron commander, on whose wing he had first experienced aerial combat, that that was against regulations and therefore impossible. It would have been difficult if he had wanted to say \"no,\" and he didn't want to say no.\n\nHe was the group commander, and no one asked questions when they saw him personally showing an Air Corps major around a P-38F, or when he scheduled a couple of P- 38Fs for training flights and went along with the major.\n\nIf Dick dumped a P-38F while he was learning, Douglass decided, he would just say that he was flying it. That would work unless Canidy killed himself, in which case it wouldn't matter. That fear turned out to have been academic. Canidy hadn't had any trouble with the P-38F. He was a good pilot, and an experienced one. He had several thousand hours in the air. Many of Douglass's pilots had less than two hundred fifty.\n\nWhen the jeep stopped in front of the revetment in which waited the P-38F that Canidy would fly today, and Canidy started to get out, Douglass touched his arm.\n\n\"I'll fly your wing, if you like, Skipper,\" he said.\n\nCanidy smiled at him, touched by the gesture.\n\n\"I'm just going along for the ride, thank you, _Colonel,_ \" he said.\n\nDouglass nodded and motioned for the driver to continue.\n\nCanidy walked into the revetment. The crew chief, a young technical sergeant, threw him a casual salute.\n\n\"Good morning, Major,\" he said.\n\n\"Morning,\" Canidy said. \"You've wound both rubber bands, I presume?\"\n\n\"Yes, Sir,\" the crew chief said.\n\nCanidy, with the crew chief trailing him, walked around the airplane, making the preflight check. He found nothing wrong and nodded his approval of the aircraft's condition.\n\nThey walked back to the nose of the aircraft, where the crew chief held out a sheepskin flying jacket to Canidy, and then when Canidy had put his arms into it, steadied him as he pulled sheepskin trousers over his uniform trousers.\n\nCanidy started to climb the ladder to the cockpit, which sat between the twin engines. And for the first time he saw what was painted on the nose. The Flying Tiger's shark's jaw, and \"Dick Canidy,\" in flowing script, and beneath it five meatballs.\n\n\"That was very nice of you, Sergeant,\" Canidy said. \"Thank you very much.\"\n\n\"The Colonel thought you'd like it, Major,\" the crew chief said. \"He was your squadron CO in the Flying Tigers, wasn't he?\"\n\n\"Right,\" Canidy said. It was not the time for historical accuracy.\n\nHe climbed into the cockpit. The crew chief climbed the ladder after him, carrying sheepskin boots. Canidy, not without difficulty, put them on, and then the crew chief helped him with the parachute straps, and finally handed him the leather helmet and oxygen mask, with its built-in microphone.\n\n\"Go get a couple, Major,\" the crew chief said. \"God go with you.\"\n\nCanidy smiled and nodded.\n\nThe crew chief climbed down the ladder, then removed it from where it hooked on the cockpit. Another crew member, as Canidy ran the controls through their limits, rolled up a fire extinguisher. Then he and the crew chief looked up at the cockpit, waiting for Canidy's next order.\n\nCanidy looked down and saw they were ready for him.\n\n_This is not the smartest thing I have ever done,_ Canidy thought. _I know better. Only a goddamn fool goes off voluntarily into the wild blue yonder, from which he stands a good chance indeed of dying in flames._\n\nThe alternative was sitting around Whitbey House going nuts. Christ only knew what Donovan had in mind for Jimmy Whittaker. And at this moment, Eric Fulmar was somewhere in Germany wearing the uniform of an SS-Obersturmf\u00fchrer (first lieutenant). If the SS caught him in that, they would be inspired to see that his death was preceded by their most imaginative interrogation techniques.\n\nIt was either this\u2014which by stretching a point could be considered flying a reconnaissance mission himself that otherwise the Air Corps would have to make\u2014or drink. Or go nuts.\n\nHe flipped the Main Power Buss on, then adjusted the richness control of the left engine. He looked down from the cockpit.\n\n\"Clear!\" he called.\n\n\"Clear, Sir,\" the crew chief called back.\n\nCanidy leaned forward and held the ENGINE START LEFT toggle switch against the pressure of its spring.\n\nThe left engine began to grind, and the prop began to turn, very slowly. Then the engine caught for a moment, bucked, and spit. The prop became a silver blur.\n\nThere had been time to think. He was just along for the ride. He was riding Douglass's wing, throttled back at 25,000 feet so as not to outrun the bomber stream of B- 17Es at 23,000 feet. Douglass had the responsibility for the flock of sheep. All Canidy had to do was maintain his position relative to Doug.\n\nThe first thing he thought was that this was where he really belonged. He was a pilot, and a good one, a combat-experienced pilot. And also an aeronautical engineer. He knew what he was doing here. He should have fought this war as a pilot.\n\nBut other thoughts intruded. Experience was relative to somebody else's experience. Relatively speaking, he was an old-timer in the intelligence business, not because he'd done so much but because hardly anybody else had done anything at all. The Americans, as the British were so fond of pointing out whenever they found the opportunity, were virgins in the intelligence business.\n\nThere had been a cartoon one time on the bulletin board at MIT in Cambridge: \"Last Weak I Cudn't Even Spell 'Enginnear' And Now I Are One.\"\n\nThere should be one on his corkboard in his office, he thought: \"Last Year, I Didn't Even Know What An Action Officer Was, But Look At Me Now!\"\n\n_And I am now possessed of knowledge,_ he thought, _that would scare the shit out of those guys in the bombers. They have been told so often\u2014by people who believe what they are saying\u2014that the \"box\" tactic\u2014_ which provided a theoretically impenetrable fire zone of .50-caliber machine-gun fire _\u2014is going to keep them safe from harm that they tend to believe it._\n\n_They question what they are told, of course. They're smart enough to figure out\u2014or have learned from experience\u2014that German fighters will get past the fighter escort and then penetrate the box. But they hope that the fighter escorts will grow more skilled and the .50-caliber fire zones will be refined so that things will get better, not worse, and that all they will really have to worry about is flak._\n\n_I know that the Germans have flight-tested fighter aircraft propelled not by airscrews but by jets of hot air. I know that these aircraft will fly two or three hundred miles_ _per hour faster than our fighters, which means the Germans will be able to just about ignore our fighter escorts. And I know that the best aerial gunner in the world isn't going to be able to hit a small fighter approaching at closing speeds over eight hundred miles per hour._\n\n_And I know that unless we can stop the Germans from getting their jet fighters operational, there is going to be an unbelievable bloodbath up here._\n\n_It is for that reason that I can intellectually, if not emotionally, justify sending Eric Fulmar into Germany. If we can find out from the guy he's bringing out what the Germans need to build their jet engines, maybe we can bomb their factories out of existence before they can start turning out engines. In the cold, emotionless logic of my profession, that justifies dispatching an agent, even running the risk that if he is caught, the Sicherheitsdienst will begin his interrogation by peeling the skin from his wang, before they get down to serious business._\n\n\"Dawn Patrol Leader,\" Douglass's voice came over the air-to-air. \"Dawn Patrol Two. We just crossed the German border.\"\n\nUnder the black rubber oxygen mask that covered the lower half of his face, Canidy smiled. What seemed like a very long time ago, when he and Doug had been assigned to fly patrols at first light looking for Japanese bombers on their way to attack Chungking, they had, feeling very clever about it, chosen \"Dawn Patrol\" as their air-to-air identity. Errol Flynn had recently played a heroic fighter pilot in a movie with that name.\n\n\"If you see Eric, wave,\" Canidy said to his microphone. He immediately thought, _Now, that wasn't too smart, was it?_\n\n\"No shit?\" Douglass replied. This time Canidy didn't reply.\n\nFive minutes later, Douglass came on the air again.\n\n\"Blue Group Leader. We have what looks like two squadrons of ME-109s at ten o'clock. Baker and Charley flights, hold your positions. Able will engage. Able, follow me.\"\n\nCanidy looked for the German fighters and found them, maybe twenty-five black specks in a nose-down attitude, obviously intending to strike the bomber stream from behind and above.\n\nThe Germans preferred to attack from above, preferably from above and to the rear, but from above. Diving at the P-38Fs on their way to the bomber stream beneath would give the Messerschmitts a considerable advantage. With the American fighters between the B-17s and the Germans, the B-17 gunners would have their fields of fire restricted unless they wanted to run the risk of hitting the P-38Fs. And with just a little bit of luck, machine-gun and cannon fire directed at the P-38Fs might strike one of the bombers beyond them.\n\nCanidy waited until Douglass was out of the way, then tested his guns (he had tested them over the English Channel, but it was better to test them again than to find himself nose up against a Messerschmitt with a bad solenoid and no guns) and pushed the nose up and to the left and stayed on Douglass's wing.\n\nHe felt his hands sweating inside his gloves, and knew that it was a manifestation of fear.\n\nThe attacking Messerschmitts split into two groups, one to continue the attack on the bomber stream, the other to engage the American fighters. The tactic had obviously been preplanned.\n\nThe P-38Fs had not been able to gain much speed from the time they left their original position to rise to the attack, but the Germans were running with their needles on the DO NOT EXCEED red line, and the closing speed was greater than Canidy expected. He was sure that his three-second burst had missed the Messerschmitt he had aimed at.\n\nTurning outside of Douglass, he felt the world grow red, then almost black, as the centrifugal forces of the turn drained the blood from his head.\n\nThe twin 1,325-horsepower Allison engines, with their throttles shoved forward to FULL EMERGENCY MILITARY POWER indent, were screaming. Full Emergency Military Power was hell on fuel consumption, and cut deeply into the operational life of engines, but the extra power, when needed, was worth the cost. When they came out of the 360-degree turn, they were running a little faster than the Messerschmitts. They gained on them slowly and followed them through the bomber stream.\n\nThe tracers from the bombers' guns seemed to fill the sky; there was a real possibility that he would be hit, and that prospect was frightening. But the fear was overcome by what Canidy, very privately, thought of as the animal urge to kill. Man\u2014because he fancied himself civilized\u2014 liked to pretend he entered combat reluctantly. And he prepared for combat reluctantly. But once he was in it, he was far less removed from the savage than he liked to believe. He wallowed in the prospect of killing the enemy.\n\nThe pair of Messerschmitts he and Douglass were chasing pulled out of their dives. To be sure of a killing burst from his battery of eight .50-caliber Brownings (the mark, Canidy thought approvingly, of the experienced fighter pilot; \"don't shoot until you can see the whites of their eyes\"), Douglass, who had crept ever closer to the German before him, was taken by surprise. His P-38F could not respond in time, and he had lost his opportunity to fire.\n\nCanidy was two hundred yards behind him. Without thinking of what he was doing, he moved the nose of his P38-F from the Messerschmitt he had been following to the one that had gotten away from Doug. The plane vibrated for a moment from the recoil of eight heavy machine guns, then he aimed at the first plane, this time firing a three-second burst.\n\nHe saw his tracer stream move from just in front of the Messerschmitt to the engine cowling, then to the left wing. There was a hint of orange, and then the wing tank exploded.\n\nCanidy pulled up abruptly and looked around for the other fighter. He couldn't find it for a moment, and then he saw it, smoke pouring from its engine nacelle as it spun toward the cloud cover below. He looked for a parachute but didn't see one.\n\nAnd then Douglass was on his wing.\n\n\"Two more,\" Douglass's voice came over the air-to-air. \"How the hell are we going to explain that?\"\n\n\"That'll make seventeen for you, won't it, Colonel?\" Canidy replied.\n\n\"Bullshit!\" Douglass said, then switched frequencies. \"Blue Group Able, this is Blue Group Leader. Form on me in Position A.\"\n\nThe P-38Fs scattered all over the sky began to turn and to resume their original protective positions over the B-17 stream.\n\nCanidy reached inside his sheepskin jacket, then inside his uniform jacket and came out with a Map, US Army Corps of Engineers: Germany.\n\nIt wasn't an aerial navigation chart, but rather one intended for use by ground troops. It could also be used by a pilot who intended to navigate by flying close enough to the ground and following roads and rivers. Canidy had taken it with him to the final briefing, and copied onto it the course the bomber stream would fly. Once they had joined the bomber stream, over a known location, it was not difficult to plot from that position and time where the head of the bomber stream would be at a given time.\n\nIt wasn't precise, but Canidy had had experience in China navigating with a lot less. He looked at his watch, then scrawled some arithmetic computations on the map. He put a check mark on the map. The way he had it figured, the lead aircraft of the bomber stream was now passing over a relatively unpopulated area of Germany, southeast of Dortmund. He made some more marks on the map, then touched his air-to-air microphone switch.\n\n\"Dawn Patrol Two,\" he called.\n\n\"Go ahead,\" Douglass replied a moment later.\n\n\"There's something I want to see,\" Canidy said.\n\n\"Say again?\"\n\n\"I say again, I'm going to have a look at something I want to see,\" Canidy said. \"I'll be back in about two zero minutes.\"\n\n\"Dick, are you all right?\" Douglass asked, the concern in his voice clear even over the clipped tones of the radio.\n\n\"Affirmative,\" Canidy said.\n\n\"Permission to leave the formation is denied,\" Douglass said.\n\nCanidy ignored him. He dropped the nose of the P-38F and headed east. He knew that Douglass could not simply ignore his responsibilities as fighter escort commander; so Douglass would not follow him.\n\nCanidy dropped through the bomber stream, more than a little surprised that at least one gunner didn't get excited and take a shot at him. In a P-51 or a P-47, that probably would have happened. But the twin-engine, twin tail boom shape of the P-38F was distinctive. There was no German plane that looked even remotely like it.\n\nWhen he passed through 11,000 feet, he took the oxygen mask from his face and rubbed the marks it had made on his cheeks and nose, and under his chin. He loosened the snaps of the sheepskin jacket. It was cold, but not nearly as bitter cold as it had been at 25,000 feet, nearly five miles up.\n\nHe dropped to 2,000 feet and trimmed it up to cruise at 300 miles per hour. If the air were still, that would have moved him across the ground at five miles per minute. The air wasn't still, of course, but it still helped to have that stored in the back of his mind. He was making, roughly, a mile every twelve seconds.\n\nHis chronometer showed that he had left the formation thirteen minutes before when he found what he was looking for. The River Eder had been dammed near Bad Wildungen, making a lake with a distinctive shape. He passed east of it, far enough so that if there was antiaircraft protecting the dam, he would not be in its range.\n\nHe reset the second counter on the chronometer.\n\nAlmost exactly six minutes later, which would put him thirty miles from the dam, he spotted what had to be the River Lahn. Right, he thought, where it should be.\n\nHe banked sharply to follow the river south, and dropped even lower toward the ground. He would be very vulnerable if he was attacked from behind and above. He was counting on not being detected until he had seen what he had come looking for. He was also counting on the probability that whatever Germans were airborne would be directing their efforts toward the bomber stream and its escorts, rather than trying to look for one lone fighter on the deck.\n\nHe saw first the medieval castle on the hill in Marburg, then he dropped his eyes just ahead of him and to the right.\n\nAnd there it was, the Marburg Werke of Fulmar Elektrische GmbH.\n\nHe retarded his throttle and extended the flaps, and when it was safe, lowered his wheels. Technically, lowering the gear was a sign of surrender. But in order to surrender, there had to be someone to surrender to, and there was no German in sight. He lowered his wheels to slow the P- 38F down. He wanted as good a look as he could get.\n\nHe passed so low and so slow that he thought he could see surprise on the faces of the workmen who were erecting fences and framework for camouflage netting around and over the new, square, windowless concrete block building.\n\nAnd then he was past it. He shoved the throttle ahead and retracted his gear and flaps and pulled back on the stick.\n\nHe wondered if Eric was down there and could hear, or perhaps even see, the American fighter as it climbed steeply into the sky.\n\nHe hadn't seen anything of great significance. And he wasn't even sure that the Germans really intended to use the Marburg facility's electric furnaces to make the special alloy steel parts for their jet-propulsion engines. But it was important that he have a look for himself. Now that he'd done so, he was glad that he had\u2014even if his fund of knowledge was not appreciably greater than it had been.\n\nThe odds were that he would be responsible for mounting a mission against that particular factory. He wanted to know what something that would certainly cost American lives looked like. And he would now be in a position to recommend the path attackers would take. Having been there, he was now an expert.\n\nAs he came out of a cloud layer at 15,000 feet, he saw the bomber stream above him. When he reached 20,000 feet, .50-caliber tracers from several of the bombers began to arc in his direction.\n\nThat was bad, but worse would follow. There was a mob instinct. _If the guy in the next plane is firing at that airplane, maybe he can see something I can't, like Maltese crosses on the wings. Why take a chance?_\n\nCanidy put the P-38F into a steep dive away from the bomber stream, to get out of range, and when he felt safe, he went to 23,000 feet and caught up with the fighter escort. He got there just as a swarm of Messerschmitts based near Frankfurt began their attack, and the fighter formations broke up to repel it. He didn't find Douglass until long after the bombing run, when they were headed home.\n\nWhen he pulled beside him, Douglass took off his glove so there would be no question but that he was giving Canidy the international aviation hand signal known as \"The Finger.\"\n\nThe commanding officer of the 344th Fighter Group went on the air-to-air.\n\n\"You goddamn sonofabitch,\" he said. \"I was worried about you.\"\n\n#  **2**\n\n##  **EAST RAILWAY STATION BUDAPEST, HUNGARY 1145 HOURS 31 JANUARY 1943**\n\nWhen the Opel Admiral was found parked in the reserved area of the East Railway Station, it quite naturally caused a certain curiosity among the Gestapo agents assigned to the station.\n\nFor one thing, there were few Admirals\u2014which was to the line of Adam Opel GmbH automobiles as Cadillac was to General Motors\u2014around anywhere, and possession of one was a symbol of power and authority. This one, moreover, bore Berlin license plates, a CD (Corps Diplomatique) plate, and affixed to the Berlin license tag where the tax sticker was supposed to go, a sticker signifying that taxes had been waived because the automobile was in the service of the German Reich, and specifically in the service of the SS-SD.\n\nObviously, whoever had parked the car was someone of high importance. The question was just who he was.\n\nFirst things first. Josef Hamm, the ranking Gestapo agent, ordered that the Hungarian railway police be \"requested\" to station a railway policeman to watch the car. If there was one thing known for sure, it was that, whoever the high official was, he would not be at all pleased to return to his car and find that someone had taken a key or a coin and run it along the fenders and doors. There had been a good deal of that, lately. A number of Hungarians took offense at the Hungarian-German alliance generally, and at the large\u2014and growing\u2014presence of German troops and SS in Budapest specifically, and expressed their displeasure in small, nasty ways.\n\nThen Hamm called the security officer at the German embassy and asked whom the car belonged to.\n\n\"It probably belongs to von Heurten-Mitnitz,\" the security officer said. \"That would explain the SD sticker, and he's the type to have an Admiral.\"\n\n\"Who's von Heurten-Mitnitz?\"\n\n\"Helmut von Heurten-Mitnitz,\" the security officer said. \"He's the new first secretary.\"\n\n\"How does he rate an SD sticker?\"\n\n\"Because when he's bored with wearing striped pants, he can wear the uniform of a Brigadef\u00fchrer SS-SD,\" the security officer said. \"You could say that von Heurten-Mitnitz is a very influential man. His brother is a great friend of the F\u00fchrer. If you'd like, I can check the license plate number by teletype with Berlin.\"\n\n\"How long would that take?\"\n\n\"Thirty, forty minutes,\" the security officer said.\n\n\"I'll call you back in an hour,\" Josef Hamm said. \"Thank you, Karl.\"\n\nWhen he called back, Hamm was told that von Heurten-Mitnitz did not own the Admiral. It was owned by Standartenf\u00fchrer (Colonel) Johann M\u00fcller, of the SS-SD.\n\n\"Do you think he knows von Heurten-Mitnitz is driving it?\"\n\n\"I think if it was stolen, Josef,\" the security officer said sarcastically, \"they probably would have said something. M\u00fcller is with the F\u00fchrer at Wolf's Lair. Nobody takes a personal car there. So maybe he loaned it to von Heurten-Mitnitz.\"\n\n\"Have you seen this von Heurten-Mitnitz? What's he look like?\"\n\n\"Tall, thin, sharp-featured. Classy dresser. If you're thinking, Josef, of asking von Heurten-Mitnitz what he's doing with M\u00fcller's car, I wouldn't.\"\n\n\"I'm thinking of finding the new First Secretary when he comes back and telling him that if he will be so good, when he leaves his car at the station, as to tell us, we will do our very best to make sure some Hungarian doesn't piss on his engine or write a dirty word on the hood with a pocketknife.\"\n\nThe security officer chuckled. \"You're learning, Josef,\" he said, and then hung up.\n\nJosef Hamm and two of his men were waiting at the end of the platform when the 1705 from Vienna pulled in. The two men positioned themselves at opposite ends of the three first-class cars, and, when one of them spotted a \"tall, sharp-featured, classy dresser\" getting off, he signaled to Josef Hamm by taking off his hat and waving it over his head, as if waving at someone who had come to meet him at the train.\n\nHamm saw that Helmut von Heurten-Mitnitz was indeed a classy dresser. He wore a gray Homburg and an overcoat with a fur collar. With him were three people, an Obersturmf\u00fchrer-SS and a man and woman who looked like father and daughter.\n\nWhen they had almost reached the police checkpoint at the end of the platform, Hamm walked around it and up to von Heurten-Mitnitz.\n\n\"Heil Hitler!\" Hamm said, giving a quick, straight-armed salute. Von Heurten-Mitnitz made a casual wave in return.\n\n\"Herr Brigadef\u00fchrer von Heurten-Mitnitz?\" Hamm asked.\n\n\"Yes,\" von Heurten-Mitnitz said, but did not smile.\n\n\"Josef Hamm at your service, Herr Brigadef\u00fchrer,\" he said. \"I have the honor to command the Railway Detachment, Gestapo District Budapest.\"\n\n\"What can I do for you, Herr Hamm?\" von Heurten-Mitnitz asked, obviously annoyed to be detained.\n\n\"First, let me get you past the checkpoint,\" Hamm said.\n\n\"This officer and these people are with me,\" von Heurten-Mitnitz said.\n\nThe young SS officer raised his hand in a sloppy salute.\n\n\"Make way for the Brigadef\u00fchrer and his party!\" Hamm called out as he led them to and past the checkpoint.\n\n\"Very kind of you,\" von Heurten-Mitnitz mumbled. \"Now, what's on your mind?\"\n\n\"Herr Brigadef\u00fchrer,\" Hamm began, \"if you would be so kind as to notify one of my men whenever you park your car here at the station\u2014\"\n\n\"Why would I want to do that?\" von Heurten-Mitnitz interrupted.\n\n\"\u2014then I can make sure that no one bothers it while you are gone.\"\n\nHelmut von Heurten-Mitnitz looked at Hamm without speaking, but a raised eyebrow asked, What the hell are you talking about?\n\n\"There have been unfortunate incidents, Herr Brigadef\u00fchrer, \" Hamm explained, \"where cars have been . . . _defiled_ . . . by unsavory elements among the Budapest population. Paint scratched. Worse.\"\n\nVon Heurten-Mitnitz seemed to consider this a moment, and then he smiled.\n\n\"I believe I am beginning to understand,\" he said. \"You saw my car parked, and took the trouble to find out whose it was, and then to meet me. How very obliging of you, Herr Hamm! I am most grateful.\"\n\n\"It was my pleasure, Herr Brigadef\u00fchrer,\" Hamm said.\n\n\"You can do me one other courtesy,\" von Heurten-Mitnitz said. \"Please do not use my SS rank when addressing me. The less well known it is in Budapest, the better, if you take my meaning. I also hold the rank of minister.\"\n\n\"That was thoughtless of me, Herr Minister,\" Hamm said. \"I beg the Herr Minister's pardon.\"\n\n\"Don't be silly, my dear Hamm,\" von Heurten-Mitnitz said. \"How could you have known?\"\n\n\"Is there any other way in which I can help the Herr Minister?\" Hamm said.\n\n\"I can't think of one,\" von Heurten-Mitnitz said after a moment's hesitation. He offered his hand. \"I am touched by your courtesy, Herr Hamm, and impressed with your thoroughness. I shall tell the Ambassador what you've done for me.\"\n\nThey were by then standing beside the Admiral. Hamm opened both doors and, after the father and daughter had gotten into the backseat, closed them. The young SS officer walked around the rear of the car and slipped in beside von Heurten-Mitnitz. Hamm gave another salute, which von Heurten-Mitnitz returned casually, and with a smile, and then Hamm stood back as von Heurten-Mitnitz backed the Admiral out of its parking space.\n\n_All things considered,_ Hamm thought, _I handled that rather well._\n\nWhen they were a few yards from the station, the tall, gray-haired man in the backseat spoke. \"My God, when he stopped you, I thought I was going to faint.\"\n\n\"You really don't faint when you're frightened, Professor, \" the young SS officer said. \"Fear causes adrenaline to flow, and that increases, not decreases, the flow of blood to the brain. Shutting off blood to the brain is what makes you faint.\"\n\n\"Oh, my God!\" the young woman in the backseat said with infinite disgust.\n\nHelmut von Heurten-Mitnitz chuckled.\n\n\"How very American,\" he said.\n\nThe young SS officer carried an identification card that identified him as Obersturmf\u00fchrer-SS Baron von Fulmar, of the personal staff of the Reichsf\u00fchrer-SS. It was a forgery, a very good one. In a safe at Whitbey House, Kent, there was a bona fide identity card issued by the Adjutant General's Office, U.S. Army, identifying him as FULMAR, Eric, 1st Lt., Infantry, Army of the United States.\n\n\"Where are we going?\" the gray-haired man asked. He was Professor Doktor Friedrich Dyer, until two days before of the Metallurgy Department, College of Physics, the University of Marburg. His name was now being circulated over SS-SD and police teletypes. He was being sought for questioning regarding the murder of SS Hauptsturmf\u00fchrer (Captain) Wilhelm Peis. The teletype message said that he was probably accompanied by his daughter Gisella, that it was possible that they would try to flee the country, and that authorities in ports along the English Channel should consequently be on special alert.\n\n\"To Batthyany Palace,\" von Heurten-Mitnitz said. \"It's on Holy Trinity Square. Not far from here.\"\n\n\"And what happens there?\" Professor Dyer asked.\n\n\"I don't know about anybody else,\" Fulmar said. \"But I intend to go to work on a bottle of brandy.\"\n\n\"That's not what I meant,\" Professor Dyer snapped.\n\n\"You'll be told what you have to know, Professor,\" Fulmar said, \"when you have to know it. The less you know, the better. I thought I'd made that plain.\"\n\nProfessor Dyer exhaled audibly and slumped against his seat. His daughter flashed a look of contempt at the back of Fulmar's head, and shook her own head in resignation.\n\nBatthyany Palace, directly across Holy Trinity Square from St. Matthias's Church, had been built at approximately the same time (1775-77) as the royal castle (1715-70) atop Castle Hill. Twelve-foot statues of barechested men on the facade appeared to be carrying the upper stories on their shoulders, earning the admiration of ten-foot, large-bosomed granite women twined around pillars at each of three identical double doors.\n\nThe door at the left was a fake. The center door opened into the entrance foyer of the palace, and the door at the right was the carriage entrance. Von Heurten-Mitnitz turned off the square and stopped the Admiral with its nose against the right door and blew the horn. A moment later, one by one, the double doors opened. He drove through, and the doors closed after him.\n\nBeatrice, Countess Batthyany and Baroness von Steighofen, was standing in a vestibule waiting for them. She was a tall, generously built woman in her early thirties. She was wearing a sable coat that reached nearly to her ankles and a matching sable hat under which a good deal of dark red hair was visible. Von Heurten-Mitnitz drove past her into a courtyard, turned around, and returned to the vestibule, where he stopped.\n\nThe Countess went to the rear door and pulled it open.\n\n\"I'm the Countess Batthyany,\" she said. \"Won't you please come in?\"\n\nProfessor Dyer and his daughter got out of the car and, following the direction indicated by the Countess's outstretched hand, walked into the building. The Countess turned to smile at Fulmar. \"And you must be dear cousin Eric,\" she said, dryly. \"How nice to finally meet you.\"\n\nFulmar laughed. \"Hello,\" he said.\n\nShe turned to von Heurten-Mitnitz, who had walked around the front of the car.\n\n\"I see everything turned out all right,\" she said.\n\n\"The Gestapo man at the station personally led us past the checkpoint,\" he said.\n\n\"Oooh,\" she said. \"I suppose you could use a drink.\"\n\n\"I could,\" Fulmar said.\n\nShe turned to look at him again.\n\n\"You look like Manny,\" she said. \"You even sound like him. That terrible Hessian dialect.\"\n\nHe chuckled.\n\n\"Let's hope you are luckier,\" the Countess said as she started into the house.\n\n\"Let's hope there's some of his clothing here, and that it fits,\" Fulmar said. \"Particularly shoes.\"\n\nShe turned and looked at him again, this time appraisingly.\n\n\"You're a little larger than Manny was,\" she said. \"But there should be something. I gather you want to get out of that uniform?\"\n\n\"They're looking for an Obersturmf\u00fchrer who looks like me,\" Fulmar said. \"There was a Gestapo agent at the border who thought he had found him.\"\n\n\"That close?\" she asked.\n\n\"I think it's been smoothed over,\" von Heurten-Mitnitz said. \"It was close, but I think it . . . is smoothed over.\"\n\nThe Countess considered what he had said and nodded her head.\n\nHeating the enormous old palace had under the best of circumstances always been difficult. Now, without adequate supplies of coal, it had proved impossible. It wasn't as if she didn't have coal. There were half a dozen coal mines running around the clock on Batthyany property, and she could have all the coal she wanted. The problem was getting the coal from the mines to Batthyany Palace. That required trucks, and she had been allocated one truckload per month. She didn't always get that, and even when she did, one truckload was nowhere near enough to heat the palace.\n\nShe didn't even bother to try to heat the entire lower floor of the palace, nor the two upper floors. They had been shut off with rather ugly and really not very effective wooden barriers over the stairwells. Only the first floor was occupied (in America, the second floor). The Countess was living in a five-room apartment overlooking Holy Trinity Square, but she often thought she might as well be living in the basement for all she got to look at the square. Most of the floor-to-ceiling windows had been timbered over to preserve the heat from the tall, porcelain-covered stoves in the corners of the rooms. The two windows (leading to the balcony over the square and the garden in the rear) that were not covered over with timber were covered with seldom -opened drapes.\n\nThe Dyers, not knowing where to go and looking uncomfortable, waited for the others to catch up with them at the foot of what had been the servants' stairway to the first floor. The Countess went up ahead of them. They came out in the large, elegantly furnished sitting room overlooking the square.\n\nFulmar immediately sat down on a fragile-looking gilded wood Louis XIV sofa and began to pull his black leather boots off.\n\nThe Countess looked askance at him, but von Heurten-Mitnitz sensed there was something wrong.\n\n\"Something wrong with your feet?\" he asked.\n\n\"These goddamned boots are four sizes too small,\" Fulmar said. \"I soaked them with water, but it didn't help a whole hell of a lot.\"\n\nWhen he had the boots off, he pulled a stocking off and, holding his foot in his lap, examined it carefully.\n\n\"Goddamn, look at that!\" he said. The skin was rubbed raw, and was bleeding in several places.\n\nThe Countess walked to the sofa, dropped to her knees, and took the foot in her hand.\n\n\"How did you manage to walk?\" she asked.\n\n\"Why, Cousin,\" Fulmar said, \"I simply considered the alternative.\"\n\n\"You'll have to soak that in brine,\" she said. \"It's the only thing that will help.\"\n\n\"By brine, you mean salt in water?\" he asked, and she nodded.\n\n\"Before we do that, I would like a very large cognac,\" he said, and pulled off the other sock. The other foot was worse. The blood from the sore spots had flowed more copiously, and when it had dried, it had glued the sock to the wounds. He swore as he pulled the stocking off.\n\nThe Countess walked to a cabinet and returned with a large crystal brandy snifter.\n\n\"I'll heat some water,\" she said. \"And make a brine.\"\n\n\"And pickle my feet,\" Fulmar said dryly. \"Thank you, Cousin, ever so much.\"\n\n\"Why do you call her 'cousin'?\" Professor Dyer asked.\n\n\"We are, by marriage,\" the Countess said. \"My late husband and Eric are, or were, cousins.\"\n\n\"Your late husband?\" the professor asked.\n\n\"The professor tends to ask a lot of questions,\" Fulmar said mockingly.\n\n\"My husband, the late Oberstleutnant [Lieutenant Colonel] Baron Manfried von Steighofen, fell for his fatherland on the eastern front,\" the Countess said dryly.\n\n\"And you're doing this?\" the professor asked.\n\n\"It's one of the reasons I'm doing 'this,' my dear Herr Professor,\" the Countess said.\n\n\"And the other?\" Fulmar asked.\n\n\"Is it important?\"\n\n\"I'm curious,\" Fulmar said. \"If I were in your shoes, I would be rooting for the Germans.\"\n\n\"If I thought they had a chance to win, I probably would be,\" she said matter-of-factly. \"But they won't win. Which means that the Communists will come to Budapest. If they don't shoot me, I'll find myself walking the square outside asking strangers if they're looking for a good time.\"\n\n\"Beatrice!\" von Heurten-Mitnitz exclaimed.\n\n\"Face facts, my dear Helmut,\" the Countess said.\n\n\"The flaw in your logic,\" Fulmar said, \"is that you are helping the Russians to come here.\"\n\n\"In which case, I can only hope that you and Helmut will still be alive and in a position to tell the Commissar what a fearless anti-fascist I was,\" she said. \"There's a small chance that would keep them from shooting me out of hand.\" There was a moment's silence, and then she went on. \"What I'm really hoping for is that there will be a coup d'\u00e9tat by people like Helmut against the Bavarian corporal, and in time for whoever takes over to sue for an armistice. If there's an armistice, perhaps I won't lose everything.\"\n\n\"Huh,\" Fulmar grunted.\n\n\"And what has motivated you, my dear Eric,\" the Countess said, \"to do what you're doing?\"\n\nIt was a moment before he replied. \"Sometimes I really wonder,\" he said.\n\nThe Countess nodded, then turned to Gisella Dyer.\n\n\"Would you help me, please?\" she said. \"I made a guly\u00e1s, and if you would help serve it, I'll heat some water to 'pickle' Eric's feet.\"\n\nThe sting of the warm salt water on his feet was not as painful as Eric Fulmar had expected, and he wondered if this was because he was partially anesthetized by the Countess's brandy, or whether his feet were beyond hurting. The guly\u00e1s was delicious, and he decided that was because it was delicious and not because of the cognac\u2014or because they'd had little to eat save lard and dark bread sandwiches since leaving Marburg an der Lahn.\n\nVon Heurten-Mitnitz waited until they were finished and Fulmar was pouring a little brandy to improve his small, strong cup of coffee, and then he said:\n\n\"I think it would be best if I knew precisely what has happened since you entered Germany, Eric.\"\n\n\"A synopsis would be that everything that could go wrong, did,\" Fulmar said.\n\n\"What about the Gestapo agent? Did you have to kill him?\"\n\n\"I killed him when he opened the luggage that had been left on the train for me,\" Fulmar said matter-of-factly, \"and found the Obersturmf\u00fchrer's uniform. And then the boots didn't fit.\"\n\nVon Heurten-Mitnitz nodded. \"And in Marburg, was what happened there necessary?\"\n\n\"Yes, of course it was,\" Fulmar said impatiently. \"I don't like scrambling people's brains.\"\n\n\"You could learn some delicacy,\" the Countess said.\n\n\"We are not in a delicate business, Cousin,\" Fulmar said.\n\n\"But that's it? There's nothing else I don't know about?\" von Heurten-Mitnitz asked.\n\nFulmar's hesitation was obvious.\n\n\"What else?\" von Heurten-Mitnitz persisted.\n\n\"I was recognized on the train,\" he said. \"Before I got to Frankfurt. On the way to Marburg.\"\n\n\"By whom?\"\n\nThere was another perceptible hesitation.\n\n\"Christ, I really hate to tell you,\" he said. \"I don't want you playing games with her.\"\n\n\"I think I have to know,\" von Heurten-Mitnitz said.\n\n\"Fuck you,\" Fulmar said. \"You have to know what I goddamn well decide to tell you.\"\n\nVon Heurten-Mitnitz stiffened. He was not used to being talked to like that. But he kept control of himself.\n\n\"Someone you knew when you were at Marburg?\" he asked reasonably. And then, when Fulmar remained silent, he added, \"I don't want to sound melodramatic, but I will be here when you are safe in England.\"\n\n\"Tell him, Eric,\" the Countess said. \"As you pointed out, we are not in a delicate business.\"\n\n\"I don't want you trying to use her, you understand me? Her, or her father.\"\n\n\"Who recognized you?\" von Heurten-Mitnitz persisted gently.\n\n\"Elizabeth von Handleman-Bitburg,\" Fulmar said.\n\nVon Heurten-Mitnitz's eyebrows went up. The Countess looked at him with a question in her eyes.\n\n\"Generaloberst von Handleman-Bitburg's daughter?\" von Heurten-Mitnitz asked.\n\nFulmar nodded.\n\n\"Possibly it's meaningless,\" von Heurten-Mitnitz said. \"She met a young Obersturmf\u00fchrer whom she had once known. Was there any reason you think she was suspicious? \"\n\n\"Her father had told her that I was seen in Morocco in an American uniform,\" Fulmar said. \"She knew.\"\n\n\"And what do you think she will tell her father?\" von Heurten-Mitnitz asked.\n\n\"Nothing,\" Fulmar said. \"She won't tell him a thing.\"\n\n\"I wish I shared your confidence,\" von Heurten-Mitnitz said.\n\n\"The only reason I'm telling you this,\" Fulmar said, \"is because I don't want you to protect your ass by taking her out.\"\n\n\"Telling me what?\"\n\n\"We spent the night together,\" Fulmar said. \"Okay? Get the picture?\"\n\n\"Yes, I think I do,\" von Heurten-Mitnitz said.\n\n\"If anything happens to her,\" Fulmar said. \"I will . . .\"\n\n\"Don't be childish,\" von Heurten-Mitnitz said.\n\n\"I was about to say something childish,\" Fulmar said. \"Like I will come back here and kill you myself. But I won't have to do that. All I'll have to do is make sure the Sicherheitsdienst finds out about you.\"\n\n\"My God!\" von Heurten-Mitnitz said.\n\n\"I made a mistake in telling you,\" Fulmar said.\n\n\"No, you didn't, Eric,\" the Countess said. She walked to von Heurten-Mitnitz and put her arm in his, then stood on her tiptoes and kissed his cheek. \"Helmut understands that even in the midst of this insanity, people fall in love.\"\n\nFulmar looked through them, then chuckled.\n\n\"Well, I'll be goddamned,\" he said. \"The Merry Widow in the flesh. \"\n**IV**\n\n#  **1**\n\n##  **THE MAYLAYBALAY-KIBAWE HIGHWAY ISLAND OF MINDANAO COMMONWEALTH OF THE PHILIPPINES 4 FEBRUARY 1943**\n\nThe mountainous center of the island of Mindanao is virtually inaccessible by motor vehicle, and accessible by foot only with great difficulty. It was for that reason that Brigadier General Wendell Fertig Commanding, U.S. Forces in the Philippines, had elected to place his headquarters and the bulk of his force in the mountains: the Japs had a hell of a hard time getting in there, and when they tried it, he was always notified in plenty of time to plan his defensive strategy.\n\nAlmost without exception, that strategy was to evacuate his headquarters and, from positions in the mountainous jungle nearby, observe how close the Japanese had come to finding it.\n\nSo far they had failed, although on occasion they had come across outposts or villages where he had stationed small detachments of his guerrilla force. That was, he knew, a somewhat grandiose manner of describing the six, or eight, or a dozen armed men living in those villages and earning their support from the villagers by working in the fields.\n\nWhen the Japanese had proof (or strongly suspected) that a village was harboring guerrillas, they burned it to the ground. They would have shot the village leaders, had they caught them, but the villagers\u2014men, women, and children, as well as the guerrillas\u2014invariably found safety in the surrounding jungle when Japanese appeared.\n\n_Pour l'encouragement de les autres,_ the elders of several villages that had not been housing guerrillas had been shot, and their villages burned down by the Japanese. The result of this had been to increase the number of natives willing to support U.S. forces in the Philippines. The remaining men would have been happy to enlist in USFIP, but Fertig had neither food to feed them nor arms with which to equip them.\n\nThe Japanese had quickly learned, too, that their expeditions into the mountains were very expensive\u2014and did little good. They were almost always engaged by Fertig's guerrillas. Not in pitched battles, not even in situations that could be considered an armed engagement. Although Fertig liked to think that he was doing to the Japanese what the Minutemen had done to the English on their way back from Concord\u2014causing them serious harm by attacking their formations with accurate rifle fire from the surrounding forests\u2014all he was really able to do was harass the Japanese patrols.\n\nWhen it was absolutely safe to do so\u2014in the sense that there was a sure escape route into the impenetrable jungle\u2014and when there was an absolutely sure target, two or three or half a dozen shots would ring out from the jungle, and one or two or three sweating Japanese soldiers marching along a trail would be killed or wounded.\n\nWith some exceptions\u2014there were some guerrillas who had as much as one hundred rounds of ammunition, which they were unwilling to share), most of Fertig's troops had no more than twenty-five rounds of ammunition for their Model 1917 Enfield .30-06-caliber rifles, or their Arisaka 7.7mm-caliber captured Japanese rifles, or their Winchester or Savage hunting rifles, or their Browning and Remington shotguns.\n\nFertig's guerrillas were not equipped to engage Japanese forces in battle.\n\nBefore long, the Japanese, who were not fools, had for all practical purposes abandoned their expeditions into the mountains. Fertig wasn't posing any bona fide military threat to their occupation. He was contained. And they could live with him until such time as the Filipinos came to understand that it was in their best interest to cooperate with the Japanese, to enter willingly into the Japanese Greater Co-Prosperity Sphere. At that point, they would stop feeding and supporting Fertig's guerrillas, and the threat would be over.\n\nThe Japanese had turned to winning the hearts and minds of the people. Propaganda detachments, protected by company-size detachments of riflemen, began to visit villages on the periphery of Fertig's mountainous jungle area of operations. The propaganda detachments carried with them 16mm motion picture projectors and generators and gifts of food and candy. They would set up a screen and show Charlie Chaplin and Bugs Bunny motion pictures, along with newsreels of the fall of Singapore, and of Lieutenant General Jonathan Wainwright surrendering to General Homma, and of long lines of American soldiers\u2014 hands in the air in surrender\u2014entering Japanese captivity.\n\nAnd then there would be a speech, or speeches, most often by Filipinos already convinced that the future of the Philippine people lay with their Japanese brothers. The speeches would invariably contain sarcastic references to General Fertig and his so-called U.S. forces in the Philippines.\n\n_Where were they? If they hadn't already died of starvation, hiding out like rats in the jungle, why weren't they attacking the Japanese?_\n\nGeneral Fertig was aware of the problem, and aware that it had to be dealt with. With some reluctance, he had concluded that the only way to deal with it was by doing exactly what he believed he was probably incapable of doing: engaging a Japanese company-strength unit in a battle. A battle in which there would be a winner and a loser, not just a dozen shots fired from concealment in the jungle.\n\nThe Japanese cooperated in two ways that helped Fertig's plans. First, they were methodical. Their propaganda detachments had a schedule. And Fertig obtained a copy of it from a Filipino woman who had been employed by the Japanese as a typist. Secondly, when it had become apparent to the Japanese that Fertig was unwilling to attack the propaganda detachment convoys, they had grown a little careless.\n\nWhen the first convoys had gone out, fully expecting to be attacked, they had moved slowly and with great caution. They had sent a point ahead and they were prepared to fight at any moment. Now, as a general rule of thumb, the troops in the trucks did their best to sleep when they were on the road. Their officers indulged them, for they believed that if Fertig were going to attack, he would do so at night. The way to preclude that was to establish a strong perimeter guard. That required the use of wide-awake soldiers. It was better that the troops get what sleep they could when they could, so they would be wide-awake guards at night.\n\nTwo highways crossed the main portion of the island of Mindanao, both running north-south, one to the west of the mountains, the other to the east. There was no highway running east-west through the mountains. The terrain was difficult, construction would be practically impossible, and there was no economic justification to build such highways.\n\nThe place Fertig picked for the attack on the convoy was almost exactly equidistant between Maylaybalay and Kibawe on the highway that crossed Mindanao to the west of the mountains. The nearest Japanese reinforcements would be twenty-three miles north in Maylaybalay, or twenty-one miles south in Kibawe. In one possible scenario\u2014where one of the trucks would escape the ambush and run for help\u2014it would be anywhere from an hour and a half to two hours before Japanese reinforcements could reach the ambush location.\n\nIn another\u2014and much worse\u2014possible scenario, they would not be able to totally overwhelm the Japanese in twenty minutes. In that scenario, the Japanese troops would be equipped with both machine guns and mortars. If they were not able to knock out the mortars and machine guns in the first minute or two of the ambush, overwhelming the Japanese would be difficult and time-consuming.\n\nAnd overwhelming the Japanese quickly was absolutely necessary. The initial attack would consume a great percentage of available ammunition, including their entire stock of fourteen fragmentation hand grenades. Fertig's only possible source of resupply was from the bodies of vanquished Japanese. There would be no question of breaking off the attack and making for the mountains. And the longer it took to overwhelm the Japanese, the more time they would have to defend themselves, which meant the more ammunition they would expend, and the less there would be for the guerrillas to capture.\n\nThere were other problems, of course. For one thing, statistically\u2014and this was not a reflection on the Filipinos' loyalty generally\u2014he had to assume that several of his troops were in Japanese service. A father, or a wife, or a child was in Japanese \"protection,\" with the understanding that as soon as proof came of the \"loyalty\" of the guerrilla the father or wife or child would be released. Loyalty could be proved by getting word to the Japanese of where and when there would be an ambush of Japanese forces, or where and when Fertig or one of his senior officers could be found.\n\nIt was not black and white. The same guerrilla who would decide that his greater loyalty lay to his family, and that therefore he should let the Japanese know where they could find Fertig, could more often than not be counted upon to be willing to lay his life on the line sniping at a Japanese patrol.\n\nWhat this situation required was keeping secret the actual place and time of the planned attack until virtually the last minute, so that the guerrilla with a member of his family in Japanese \"protection\" would not have the opportunity to communicate with the Japanese.\n\nTo assemble the 120-150-man force he considered the optimum for the ambush of the propaganda detachment, therefore, Fertig had to pick several sites within two hours' march of the ambush site. In the event, he picked five different sites, then sent word by runner to various guerrilla cells\u2014numbering in the aggregate just over two hundred men\u2014to assemble into five larger groups at the designated sites.\n\nHis experience had taught him that about sixty percent of the guerrillas summoned would appear at the designated site at the proper time.\n\nFive hours before the propaganda detachment and its company of guards was scheduled to reach the ambush site, a second group of runners was sent to the five assembly areas, bearing orders for the men to come to the final assembly point. From the moment the runners reached the five sites, it was presumed that anyone leaving intended to betray the troops to the Japanese. If someone ran and it was impossible to capture him, the operation would be called off, and the guerrillas would disperse. If someone ran and was caught, he would be beheaded. Beheading with a heavy, razor-sharp machete was supposed to be more or less painless, and it did not expend ammunition.\n\nOne hour before the Japanese were to pass the ambush site, the last group of guerrillas arrived. No one had disappeared, or tried to. The force now totaled 136 men; and two of the guerrillas, formerly Philippine Scouts, had brought with them BARs\u2014Browning Automatic Rifles\u2014and seven loaded magazines.\n\nFertig was of two minds about using the BARs. They were splendid weapons, and God knew his troops needed something to counter the Japanese Namimba machine guns the guards would certainly have. But he had only seventy rounds per gun\u2014three and a half magazines. And every round that ripped through the BARs with such speed could be fired one at a time from an Enfield in sniping fire, where the kill-per-cartridge rate was so much more effective.\n\nIn the end, he decided that the more fire expended at the beginning of the assault, the sooner the Japanese would be overwhelmed, and thus the more ammunition could be taken from their bodies.\n\nFertig then explained the tactics of the attack, which were very simple.\n\nThe force would be divided into two elements, with two-thirds of the force close to one side of the road. From there a devastating fire could be delivered at close range. The second element, commanded by Fertig and consisting of the remaining third of the force, with both BARs and ten of the fourteen fragmentation grenades, would be on the opposite side of the road.\n\nOn signal, which would be when Fertig and a former Philippine Scout opened fire with their Enfield rifles on the driver of the first vehicle in the convoy, the smaller force would bring BAR fire to bear on the trucks carrying the troops. Other riflemen would disable the last truck in the convoy, preferably by killing its driver.\n\nAt this point, Fertig authorized the throwing of one\u2014 only\u2014fragmentation grenade at each troop-carrying truck.\n\nThe Japanese convoy would thus be immobilized, and it was to be hoped that many, if not most, of the truck-borne troops would be killed before they exited the trucks.\n\nSome, of course, would survive. Most, Fertig believed, would exit toward the ditch and forest opposite the direction from which they had been attacked.\n\nThey would then present themselves as targets to the bulk of the ambush force. Meanwhile, the third of the force that had opened the attack would rapidly divide itself in half, half going to the head of the convoy, and half to the tail. This would get them out of the line of fire of the larger ambush force and leave them in a position to fire upon any Japanese from the sides.\n\nFertig did his best to impress upon his men the absolute necessity of aimed fire. They were dangerously short of ammunition, and there was absolutely no excuse for a guerrilla to fall from a bullet fired by another guerrilla.\n\nEveryone seemed to accept his reasoning. But Fertig knew that even the most phlegmatic of people got excited once the crack of small-arms fire filled the air. And by no stretch of the imagination could his force be called at all phlegmatic.\n\nIn the engagement that followed, the ambush force of United States forces in the Philippines, Brigadier General W. W. Fertig commanding, triumphed over the 1104th Army Information Detachment and Company 3, 505th Infantry Regiment of the Imperial Japanese Army. There were no Japanese survivors.\n\nUSFIP suffered eleven dead (including the Philippine Scout who had opened the engagement at General Fertig's side, and of whom he had been extraordinarily fond) and thirty-six wounded. Of the thirty-six wounded, twenty would subsequently die. USFIP had virtually no medical supplies.\n\nThe Japanese, once they overcame their initial surprise, had fought gallantly and well. It was more than half an hour before the last of them had died for his Emperor. By the time the engagement was over, the Japanese had expended a large part of their ammunition.\n\nOn balance, USFIP had more weapons after the ambush than before, including two 60mm mortars and sixty rounds for them, several Nambu pistols, nearly two hundred Arisaka rifles, and one Namimba machine gun. Countering this increase was the expenditure of .30-06 ammunition and hand grenades. An Enfield or a BAR without .30-06 ammunition is simply a finely machined piece of steel, not a weapon. And the Japanese had expended all of their hand grenades before they were overwhelmed, the last dozen of them as instruments of suicide.\n\nJust before he disappeared back into the jungle, General Fertig took a last look at the carnage on the highway.\n\nThe Japanese, if for no other reason than to save face, would rush reinforcements up here. Patrols would be sent into the jungle.\n\nThere would be an opportunity for other ambushes, perhaps not as overwhelmingly successful as this one, but successful enough to kill many Japanese, to force the Japanese to expend fuel and manpower on one patrol after another\u2014 and to lose face.\n\nThere was a caveat. To conduct other ambushes, he would need ammunition. He had come out of the ambush with only marginally greater stocks of ammunition than he had going in, and that was for the Japanese Arisaka rifles, not the Enfields and the BARs.\n\nHe turned and entered the jungle. He would now go back into hiding.\n\n_How the hell can I wage a war if they won't supply me with what I need? Supply me with what I need? The sons-ofbitches won't even talk to me!_\n\n#  **2**\n\n##  **THE HOUSE ON Q STREET, NORTHWEST WASHINGTON, D.C. 4 FEBRUARY 1943**\n\nChief Ellis found Captain James M. B. Whittaker in the billiards room in the basement. There were two tables in the darkly paneled room: a standard English billiards table, and a somewhat smaller pocket billiards table. Whittaker was alone at the smaller table.\n\n\"Anchors aweigh, Chief,\" Whittaker said, looking up from the table when he saw Ellis. He had carefully arranged balls at the lip of each of the pockets on the table. What he was trying to do was sink as many of them as he could with one shot.\n\nEllis waited until he had made the shot\u2014sinking four of the six balls\u2014before replying.\n\n\"I hear you've been a bad boy again, Captain Whittaker, \" Ellis said.\n\n\"Was Baker waiting for you when you got back?\" Whittaker asked, and then, before Ellis could reply, he asked, \"Who's your friend?\"\n\nEllis had with him a Navy white hat, a small man made to look even smaller by his waist-length Navy blue peacoat. He wore round-framed GI glasses. He looked, Whittaker thought, like a Sea Scout.\n\n\"Radioman Second Joe Garvey, say hello to Captain Jim Whittaker,\" Ellis said.\n\nThe sailor snatched off his white hat and came to attention.\n\n\"How do you do, Sir?\" he asked.\n\n\"Poorly, now that you ask,\" Whittaker said, smiling at him. \"Didn't your mother warn you to avoid evil companions when you joined the Navy?\"\n\nThen he saw that his joke had fallen flat and that the young sailor was uncomfortable, not amused. Whittaker came quickly around the pool table and, smiling, offered his hand.\n\n\"Hello, Garvey,\" he said. \"If you're with Chief Ellis, you must be somebody special. I'm happy to meet you.\"\n\nGarvey shook his hand and smiled uneasily.\n\n\"You ever know somebody named Fertig?\" Ellis asked.\n\nWhittaker thought it over. \"There is a faint tinkle of the bell of memory,\" he said.\n\n\"In the Philippines?\"\n\n\"I put that together,\" Whittaker said, \"but that's as far as it goes. Is there some reason I should know him?\"\n\n\"He's still in the Philippines,\" Ellis said.\n\n\"Poor sonofabitch,\" Whittaker said.\n\n\"Garvey's been talking to him on the radio,\" Ellis said.\n\nWhittaker's face fit up with curiosity.\n\n\"He's in the mountains of Mindanao,\" Ellis said. \"He says there's an army sergeant named Withers with him.\"\n\n\"I knew a guy named Withers over there,\" Whittaker said.\n\n\"You want to find out if it's the same one?\" Ellis said.\n\n\"I don't think this is just idle curiosity on your part,\" Whittaker said.\n\nEllis shrugged.\n\n\"How could we do that?\" Whittaker asked.\n\n\"You got time to take a ride over to the Navy commo facility in Virginia?\" Ellis asked.\n\n\"You're starting to act like Captain Douglass,\" Whittaker said. \"You answer questions with another question.\"\n\n\"Well, I don't 'manifest a belligerent and uncooperative attitude,' \" Ellis said.\n\n\"Is that what that sonofabitch said?\" Whittaker asked.\n\n\"There was more,\" Ellis said. \"There was something about 'subjecting a trainee to a humiliating public display of affection.' Two pages, single spaced.\"\n\n\"Has the Colonel seen it?\" Whittaker asked.\n\n\"Not yet,\" Ellis said. \"I intercepted it. I can lose it, but Baker's going to expect some kind of a reply, so you better start thinking about that. And about the fact that the Colonel thinks you're in Virginia running around in the woods.\"\n\n\"Hmmm,\" Whittaker said, considering that.\n\n\"You want to take a run over to Virginia?\" Ellis asked.\n\n\"Nothing would give me greater pleasure,\" Whittaker said. He turned to put the pool cue in its rack. \"We'll have lunch on the way,\" he said. \"I want to go to that three-for-a -quarter hamburger place.\"\n\n\"White Castle?\" Ellis asked incredulously.\n\n\"White Castle,\" Whittaker confirmed happily. \"And eat a dollar's worth, with a large fries and a Dr Pepper.\"\n\n\"Maybe Baker's right,\" Ellis said. \"He says he thinks you may be crazy.\"\n\n\"In that case, you can buy your own hamburgers,\" Whittaker said as he took his tunic from a bentwood coatrack.\n\nAn hour and a half later, a lieutenant commander signed them into his log, then took them past a Marine MP guarding access to a gray painted steel door with RADIO ROOM\u2014 POSITIVELY NO UNAUTHORIZED VISITORS painted on it.\n\nThe officer on watch, a young lieutenant j.g. with a blond crew cut, got up from his desk and walked to meet them.\n\n\"These people wish to use one of your transmitters,\" the lieutenant commander said. \"They have their own operator. \"\n\n\"Sir?\" the j.g. asked, not sure he had heard correctly.\n\n\"We'd like to use that Collins, Lieutenant,\" Chief Ellis said, nodding his head toward one of a row of transmitters lining the wall.\n\nThe j.g. looked at the lieutenant commander for instructions. Strange people coming into the transmitter room was unusual; it was absolutely out of the lieutenant's experience that they should be given access to the equipment.\n\n\"Do it, Mr. Fenway,\" the lieutenant commander said.\n\n\"Aye, aye, Sir,\" the j.g. said, and motioned Garvey to follow him. He led him to a small cubicle holding a telegrapher's key, a typewriter, and a control panel. Garvey, still wearing his peacoat, pulled up a chair and reached for a set of earphones.\n\nHe tapped the key tentatively, then adjusted set screws on its base and tried it again. He rolled paper into the typewriter, then tuned both the receiver and the transmitter.\n\nThen he started to tap the key.\n\nEllis and Whittaker walked and stood behind him, and looked over his shoulder.\n\n\"All they've got is an old M94,\" Ellis said. \"There's no sense even trying to encrypt. We're talking in the clear.\"\n\n\"I have no idea what you're talking about,\" Whittaker said.\n\n\"It's a coding device,\" Ellis explained. \"But we have to presume the Japs got at least one of them.\"\n\n\"Oh,\" Whittaker said.\n\n\"When we raise them, you're going to have to think of some way to find out if this Withers guy is the one you were with, and do it so the Japs will be as confused as possible. \"\n\n\"Ask him if he still has the watch,\" Whittaker said. \"Call him Sergeant Boomboom. Sign it, Polo.\"\n\nGarvey's fingers flew over the typewriter keys. It was an automatic reaction to what he had heard in his earphones. Ellis and Whittaker looked at what he had typed:\n\nMFS FOR KGS BY\n\n\"Send 'For Sergeant BoomBoom,' \" Ellis ordered, \" 'Have you got the watch. Signed Polo.' \"\n\nGarvey tapped the message out with his key.\n\n\"What's with the watch?\" Ellis asked.\n\n\"I gave him my watch, just before I left,\" Whittaker said.\n\nThere was a long wait before Garvey started typing again.\n\nMFS FOR KGS AFFIRMATIVE WHERE POLO MFS BY\n\n\"Send 'Polo Washington,' \" Whittaker ordered. \" 'Where Scarface.' \"\n\nMFS FOR KGS SCARFACE EVERYBODY HERE MFS BY\n\n\"Send 'Send Third Letter Scarface Last Name,' \" Whittaker ordered.\n\nMFS FOR KGS VVVVVVVVVVVVVVV MSF BY\n\n\"Send 'Glad You All Made It,' \" Whittaker said.\n\nMFS FOR KGS FOR POLO FROM SCARFACE VAYA CON DIOS MFS BY\n\n\"Send,\" Whittaker began, and then his voice broke, and when Ellis turned to look at him, he saw tears running down his cheeks.\n\n\"Send,\" Whittaker went on, \" 'Hold On. The Twenty-sixth Will Ride Again. God Bless You All. Polo.' \"\n\nMFS FOR KGS MFS OUT\n\nCaptain James M. B. Whittaker, rather loudly, blew his nose. When he spoke, he had his voice under control.\n\n\" 'Scarface' is Master Sergeant Victor Alvarez, late of the Twenty-sixth Cavalry, Philippine Scouts. He was in the habit of calling Sergeant Withers 'Sergeant BoomBoom' because Withers blew things up.\"\n\n\"Clandestine station in the Philippines?\" the lieutenant commander asked. Whittaker nodded. \"Poor bastards!\"\n\n\"Thank you for your assistance, Commander,\" Whittaker said formally. \"Let's get out of here, Ellis.\"\n\nWhen they got in the Buick Roadmaster, Ellis reached into the glove compartment and came out with a pint bottle of Old Overholt. He handed it to Whittaker.\n\n\"Good for the sinuses,\" he said.\n\n\"I wish I had gone with you to Warm Springs, Ellis,\" Whittaker said tensely. \"It would have given me a chance to ask Uncle Franklin why the hell we have abandoned those guys.\"\n\n\"I suppose that's why the Colonel wanted you to run around in the woods in Virginia,\" Ellis said. \"Every time you tell off your uncle Franklin, he has to pick up the pieces.\"\n\n\"And what, exactly, he plans to do about it,\" Whittaker said.\n\n\"You might as well hear this now,\" Ellis said. \"They asked for money. There is Army brass, both here and in Australia, who are against it, because they think the Japs are using those people . . . what the Colonel calls 'turned agents.' \"\n\n\"How much did they ask for?\" Whittaker asked.\n\nEllis thought it was a strange question, but told him.\n\n\"A million, in gold, gold coins, for openers.\"\n\n\"They say what for?\"\n\n\"We're talking in the clear, Captain,\" Ellis said. \"You can't expect them to offer details.\"\n\n\"When can I get to see the Colonel?\" Whittaker asked.\n\n\"He said that I should go to Virginia and pick you up and see if we could raise MFS,\" Ellis said. \"I think he wanted to see if you thought they were being controlled by the Japs. To answer your question, Captain, that's where we're going now.\"\n\n#  **3**\n\n##  **OFFICE OF STRATEGIC SERVICES THE NATIONAL INSTITUTES OF HEALTH BUILDING WASHINGTON, D.C. 4 FEBRUARY 1943**\n\nColonel William J. Donovan was in civilian clothing: a well-cut, double-breasted Glen plaid suit, a crisp white shirt, and a red-and-blue finely patterned necktie. He looked, Whittaker thought, like a successful lawyer about to sue Chrysler or DuPont for a lot of money.\n\nWhen Whittaker entered the office, Donovan walked around his desk with his hand extended, and then the handshake gave way to a quick embrace.\n\n\"Good to see you, Jimmy,\" he said. \"How did you find the place in Virginia?\"\n\n\"I'd been there before,\" Jimmy said. \"And Staley drew a map. No problem.\"\n\n\"Why do I suspect you purposely misunderstood me?\" Donovan asked.\n\n\"You mean 'what did I think of the place'?\"\n\nDonovan nodded.\n\n\"Baker and I crossed swords again,\" Whittaker said. \"He seems to feel I 'manifested a belligerent and uncooperative attitude.' I also 'subjected a trainee to public humiliation.' \"\n\n\"Oh, Jimmy,\" Donovan said, both angry and resigned. \"What the hell was that all about?\"\n\n\"Well, the belligerent and uncooperative attitude is something that seems to happen when I get in the same room with Baker,\" Whittaker said. \"It seems to be contagious. Canidy has the same thing happen to him.\"\n\n\"We're talking about you, not Dick Canidy,\" Donovan said. \"What happened with the trainee? What was he doing so wrong you felt you had to humiliate him?\"\n\n_\"Her,\"_ Whittaker corrected him. \"I kissed _her._ \"\n\n\"Cynthia?\" Donovan asked. Whittaker nodded. \"I don't know why I'm smiling,\" Donovan added. \"I'm sure she didn't think it was funny. You'll notice that I am assuming she didn't want to be kissed.\"\n\n\"That girl doesn't know what she wants,\" Whittaker said. \"For example, she has some absurd notion that she wants to go operational. When I saw her, she was all dressed up in fatigues and carrying a Springfield at port arms. I found her irresistible. I wonder what a psychiatrist would make of that?\"\n\n\"You made your peace with Baker?\" Donovan asked.\n\n\"I left,\" Whittaker said. \"He's probably still mad.\"\n\n\"You left?\" Donovan asked, confused. \"You mean, when Ellis came for you?\"\n\n\"I left about thirty minutes after I got there,\" Whittaker said. \"I've been at the house.\"\n\n\"I left orders that you were to be taken out there,\" Donovan said coldly.\n\n\"Staley told me,\" Whittaker said. \"He was pretty insistent. \"\n\nDonovan looked at him coldly, waiting for a further explanation.\n\n\"I could offer some excuse, like I would probably have broken Baker's arms if I stayed, but the real reason I left was that Baker was acting as if he was controlling me.\"\n\n\"That's what he's paid to do,\" Donovan said sharply.\n\n\"I don't know what you've got planned for me, why I'm here and not in Australia, but if it means that Baker is my control, you're going to have to get yourself another boy.\"\n\n\"You can be a real pain in the ass sometimes, Jim,\" Donovan said. \"And this is one of them. Just who the hell do you think you're talking to?\"\n\nWhittaker's reply came a long moment later.\n\n\"I know I'm talking to the head of the OSS,\" he said. \"Not Uncle Bill, who used to bounce me on his knee. I'm not asking for any special treatment. I don't know what my alternatives are, but whatever they are, I'll take them, rather than go anywhere with him as my control.\"\n\nDonovan glared at him.\n\n\"You have a reason for feeling that way, I presume?\"\n\n\"There are two kinds of controls,\" Whittaker said. \"Both profess great sadness when somebody gets bagged. One kind means it. Baker is the other kind. Baker is too willing to accept risks with somebody else's life. He sees 'the big picture' much too clearly.\"\n\nThey locked eyes for a moment, and then Donovan asked, \"Did Ellis mention anything about dinner tonight?\"\n\nThe question surprised Whittaker.\n\n\"No,\" he said. \"He didn't.\" Then he thought a moment. \"Don't tell me I'm to have dinner with Baker?\"\n\n\"Not with Baker,\" Donovan said. And then, when he was sure in his own mind that Ellis hadn't said anything about the dinner and that Whittaker in fact did not know, he added, \"With the President.\"\n\n\"Oh?\" Whittaker said.\n\n\"There will be no repetition, nothing remotely resembling a repetition of what happened the last time you had dinner with him,\" Donovan said.\n\n\"I was a little crazy the last time,\" Whittaker said. \"And I don't want to find myself locked up in a loony bin again.\"\n\n\"You take my point,\" Donovan said evenly.\n\nWhittaker nodded. \"Is dinner his idea, or yours?\" he asked.\n\n\"His idea,\" Donovan said. \"But when I told him you were in Washington, I was pretty sure he'd want to see you.\"\n\n\"You're being devious again,\" Whittaker said.\n\n\"Trust me, Jimmy,\" Donovan said, smiling.\n\n\" _You,_ I trust,\" Whittaker said.\n\n\"Ellis has some dossiers, and some other material, I want you to look at,\" Donovan said. \"By the time you're finished, I should be finished here; and we can go over to the house.\"\n\nThe President of the United States traveled from 1600 Pennsylvania to Embassy Row in a four-car convoy: There was a District of Columbia police car with flashing red lights; then a black Chevrolet full of Secret Service agents; a 1939 Packard limousine (not _the_ presidential limousine); and finally another Chevrolet packed with Secret Service agents.\n\nThe gate in the wall was already open when the convoy arrived. The police car and the tailing Secret Service car pulled to the curb and stopped. The lead Secret Service car and the Packard drove through the gate, which closed immediately after them.\n\nWhen the two cars stopped, two burly Secret Service agents half trotted to the limousine. One of them reached in and swung the President's feet outward. Then he hauled him from the car and erect. Then he and the other agent, with an ease born of practice, made a cradle of their locked arms and carried him to and up the kitchen stairs. By the time they got there, a third Secret Service agent had taken a collapsible wheelchair from the trunk of the Chevrolet, trotted with it to the kitchen, and had it unfolded and waiting when the President was carried to it.\n\n\"One of you,\" the President of the United States said, \"smells of something that didn't come out of an after-shave bottle. 'My Sin'?\"\n\nThe burly Secret Service agent now pushing the wheelchair chuckled.\n\n\"No comment, Mr. President,\" he said.\n\nThe other agent trotted ahead and pushed open doors until he reached the double sliding doors to the library, both of which he slid open.\n\n\"Is this the place with the booze?\" the President asked as he was rolled in.\n\nDonovan and Whittaker, who had been sitting on identical couches at right angles to a carved sandstone fireplace, stood up.\n\n\"Good evening, Mr. President,\" Donovan said.\n\n\"That'll be all, Casey,\" the President said. \"If I need it, the Colonel can push me around.\"\n\nThe Secret Service agent left the room, closing the double doors carefully behind him.\n\n\"Well, Jimmy,\" the President said. \"You look a hell of a lot better than the last time I saw you.\"\n\n\"Good whiskey and fast women, Uncle Frank,\" Whittaker said.\n\nHe went to Roosevelt and offered his hand. Roosevelt ignored it. He gripped his arms with both hands, and with strength that always surprised Whittaker, forced his body down so that his face was level with Roosevelt's. Roosevelt studied him intently for a moment, and then, nodding his head in approval, let him go.\n\n\"Chesty would be very proud of you,\" the President said. \"I am.\"\n\nHe let that sink in a moment, then changed the tone. \"I had a letter from Jimmy,\" he said. \"You know about Jimmy?\" James Roosevelt, the President's eldest son, was commissioned in the USMC. He was second in command of the Marine Raiders in the Pacific.\n\n\"Somebody talked him into joining the Marines,\" Whittaker said. \"I thought he was smarter than that.\"\n\nRoosevelt laughed heartily.\n\n\"I think he was taken with the uniform,\" he said. \"Anyway, he asked about you.\"\n\n\"Give him my regards,\" Whittaker said.\n\nDonovan handed the President a martini glass.\n\n\"I think you'll like that, Franklin,\" he said. \"Basically, it's frozen gin.\"\n\nRoosevelt sipped the martini and nodded his approval.\n\nRoosevelt asked about England, first generally, and then specifically about David Bruce, the OSS Chief of Station in London, and finally about Canidy.\n\n\"Your friend Canidy's all right?\"\n\n\"Just fine,\" Whittaker said.\n\n\"I'm sorry that Bill and I can't tell you why, Jimmy,\" Roosevelt said, \"but that Congo mission the two of you flew was of great importance.\"\n\n\"I thought it probably was of enormous importance,\" Whittaker said.\n\n\"Why did you think that?\" Roosevelt asked. His famous smile was just perceptibly strained.\n\n\"The airplane Canidy and I flew was a brand-new C-46, fitted out like the Taj Mahal, and intended to fly Navy brass around the Pacific.\"\n\n\"Nothing is too good for our boys in the OSS,\" Roosevelt joked, exchanging a quick glance with Donovan.\n\nThe mission, ordered by Roosevelt himself, had been to bring ten tons of bagged ore from Kolwezi in the Katanga Province of the Belgian Congo. Only four people\u2014the President; Donovan; Capt. Peter Douglass, Donovan's deputy; and Brig. General Leslie R. Groves, director of something called \"The Manhattan Project\"\u2014knew that the ore was uraninite. The Manhattan Project was intended, in the great secret of the Second World War, to refine the uraninite into uranium 235, and from the uranium 235 to construct a bomb, an \"atomic bomb\" that would have the explosive equivalent of twenty thousand tons of TNT.\n\nRoosevelt's, and Donovan's, great fear was that the Germans, among whose scientists were some of the greatest physicists in the world, and who were known to be conducting their own nuclear research, would learn of the American effort and increase their own research effort. Whoever could produce the first nuclear weapons would win the war.\n\n\"Canidy,\" Donovan said very quickly, to shut off any possibility that Whittaker\u2014now that he'd made his little joke\u2014might ask why it was of great importance and that the President just might tell him, \"shot down two German fighters, Messerschmitts, near Dortmund three days ago.\"\n\n\"Good for him!\" the President said, pleased to change the subject.\n\n\"Bad for him,\" Donovan said. \"He's not supposed to be flying missions as a fighter pilot.\"\n\n\"He must have had his reasons,\" Whittaker said loyally.\n\n\"You and Dick always have your reasons,\" Donovan said dryly.\n\n\"Come on, Bill,\" the President said. \"You're just jealous. I'm sure that you would rather be in the field with a regiment than doing what you're doing.\"\n\n\"I do what I'm told,\" Donovan said. \"And I naively expect people who work for me to do what they're told.\"\n\n\"Did I hear a subtle reprimand?\" Whittaker asked. \"Or is that just my guilty conscience?\"\n\n\"Well, Jimmy, what have you been doing that you shouldn't?\" Roosevelt asked.\n\nDonovan walked to Roosevelt and topped off the President's martini from a heavy crystal mixer.\n\n\"Not doing what he should have been doing, Franklin,\" Donovan said.\n\n\"What was that?\" Whittaker asked.\n\n\"Learning how to get into a rubber boat from a submarine, \" Donovan said.\n\n\"Why would I want to do that?\" Whittaker asked.\n\n\"Scheduled Pan American service to the Philippines has been temporarily suspended,\" Donovan said. \"A submarine's the only way we know to get you into the Philippines.\"\n\n\"Is that where I'm going?\" Whittaker asked.\n\n\"That hasn't been decided yet,\" the President said coldly. \"Whether you or anybody else is going into the Philippines.\"\n\n\"Now that you mention it, Uncle Frank . . .\" Whittaker said.\n\n\"I don't think I'm going to like it, Jim,\" Roosevelt said. \"But finish that.\"\n\n\"Why have we abandoned the people in the Philippines? \" Whittaker asked.\n\n\"What makes you think we have?\" Roosevelt replied, just a little indignantly. He was not used to having his decisions questioned by anyone. \"There was simply no way to reinforce MacArthur before the Japanese overwhelmed him, and there is simply no way, at this time, that we can consider an invasion. It's just too far away, and we just don't have the logistical capability.\"\n\n\"I'm talking about the guerrillas,\" Whittaker said. \"The people who haven't quit. The ones in the mountains.\"\n\nIt was a moment before Roosevelt replied.\n\n\"I was about to say, Jim, that you are emotionally involved, and that unfortunately I can't always do what my emotions tell me I should. But then it occurred to me that you have a greater right to be emotionally involved than most people. So I will not change the subject. The answer to your question is that the best advice I can get is that there are no guerrillas. I tend to place faith in that advice, because it comes to me from Douglas MacArthur and George Marshall, and it is the first thing I can think of that they have agreed upon since 1935.\"\n\n\"There are at least ten guerrillas, Uncle Frank,\" Whittaker said.\n\n\"How can you possibly know that?\"\n\n\"I talked to them on the radio this afternoon,\" Whittaker said.\n\n\"You did?\"\n\n\"I did,\" Whittaker said.\n\nRoosevelt looked at Donovan.\n\n\"You arranged that, Bill? He's talking about this self-appointed general . . . what's the name?\"\n\n\"Fertig,\" Donovan said.\n\n\"Fertig,\" Roosevelt repeated. \"Jim,\" he said kindly, \"it is the opinion of everybody but Bill Donovan that the Japanese, for whatever reason, are using prisoners, attempting something. Most likely, that they hope to get us to send them a million dollars in gold by submarine. Whereupon, they will take the million dollars and sink the submarine.\"\n\n\"Uncle Frank, I talked on the radio this afternoon with two of _my_ men.\"\n\n\"What do you mean, 'your' men?\"\n\n\"When MacArthur ordered me from Luzon to Corregidor, I gave my wristwatch to my sergeant, a guy named George Withers. And I told him when Luzon fell, he should make his way, him and the Philippine Scouts I had, to Mindanao. I talked to him and to one of the Philippine Scouts this afternoon. They're on Mindanao and waiting for help.\"\n\n\"And they said what their Japanese captors told them to say.\"\n\n\"There is no way the Japanese could know what they used to call me and I used to call them,\" Whittaker said. \"They're on Mindanao, and they're free, and God damn it, we have a duty to help them.\"\n\n\"You mean, send them the million dollars?\"\n\n\"And a radio, and quinine, and ammunition,\" Whittaker said.\n\n\"They _have_ a radio,\" Roosevelt said. \"You talked to them.\"\n\n\"They need an encryption device,\" Whittaker said. \"So the Japs won't be able to listen in.\"\n\n\"Bill?\" the President asked.\n\n\"We need to send somebody in there who can separate fact from fantasy, and then come out armed with facts on which further decisions can be made,\" Donovan said. \"The basic fact of guerrilla warfare is that one guerrilla can tie down at least seven troops. . . .\"\n\n\"So you keep telling me,\" Roosevelt said. \"And you think Jimmy is the man to go to the Philippines, have a look around, and then come out?\"\n\n\"Yes,\" Donovan said.\n\n\"And since the Japanese are listening to the guerrilla radio, and since there is no way we can code what we are sending, how do you propose to let the people in the Philippines know where and when he's coming? With the Japanese listening in, I mean?\"\n\n\"We're working on that, Franklin,\" Donovan said.\n\n\"The translation of which is, 'we hope to think of something'? \"\n\nDonovan didn't reply.\n\n\"And you're willing to put your neck in the noose again, Jimmy?\" Roosevelt asked.\n\n\"Being very cold-blooded about it,\" Whittaker said, \"I seem to be the round peg for that round hole.\"\n\n\"You already escaped once from the Philippines,\" Roosevelt said. \"How often do you think you can do that?\"\n\n\"I hear that Jimmy nearly got himself blown away during the Makin Island Raid,\" Whittaker replied.\n\n\"'Blown away'?\" Roosevelt said. \"Interesting euphemism. \" It was obvious that he was making his decision.\n\n\"All right,\" he said finally. \"Do it. I'll avoid telling George Marshall as long as I can. And I don't think we should tell Douglas MacArthur until you come out.\"\n\n\"Thank you, Mr. President,\" Donovan said.\n\nRoosevelt was not through. \"And come out you will, Jimmy. You understand that? You will go in there, and have a look around, and come out. You may consider that a direct order.\"\n\n\"I suppose that means I'll have to go freeze my ass learning how to get into a rubber boat from a submarine?\" Whittaker asked.\n\nRoosevelt and Donovan chuckled.\n\n\"Now we get down to price,\" Whittaker said. \"I have a price.\"\n\n\"Everybody else seems to,\" Roosevelt said dryly. \"What's yours?\"\n\n\"Cynthia Chenowith is my control,\" Whittaker said. \"Reporting directly to Colonel Donovan.\"\n\n\"I think I see a hook in there,\" Roosevelt said. \"What's all that about?\"\n\n\"Cynthia was the control for the Kolwezi operation,\" Whittaker said.\n\n\"Cynthia is going through the agents' course,\" Donovan said. \"That runs against Jimmy's notions of the proper role of women.\"\n\nRoosevelt chuckled. \"Mine, too,\" he said. \"Eleanor, maybe. But Cynthia?\"\n\nWhittaker laughed.\n\n\"That would be your decision, of course, Bill,\" the President said.\n\n\"Okay,\" Donovan said. \"You win, Jimmy. I think she'll be furious, Jimmy, but that's your worry.\"\n\n\"She'll be alive,\" Whittaker said simply. \"I would much rather have her pissed and alive than happy, heroic, and dead.\"\n\n\"Are you getting hungry, Franklin?\" Donovan asked. \"Or would you rather have some more frozen gin?\"\n\n\"Why does it have to be either\/or?\" Roosevelt asked, holding his glass up to be refilled.\n\nDonovan pushed the servant call button twice, then went to refill Roosevelt's glass.\n\n#  **4**\n\n##  **OSS LONDON STATION BERKELEY SQUARE LONDON, ENGLAND 5 FEBRUARY 1943**\n\nHelene B. Dancy, Captain, WAC (Women's Army Corps), U.S. Army, administrative assistant to London Chief of Station David Bruce, was of two minds about Richard Canidy. When she didn't see him for a while, she began to mirror her boss's opinion of him: that Canidy wasn't a team player, that he was often doing things\u2014going off with Capt. Douglass's son as a fighter pilot was the most recent example\u2014that brought into question the wisdom of his having as much authority and autonomy as he did.\n\nBut when she was with him, most of her disapproval seemed to vanish. It was absurd to think that anything could happen between them\u2014Helene B. Dancy had been commissioned in the WAC from her job as executive secretary to the senior vice president for real estate, the Prudential Insurance Company, thirty-six hours before she turned thirty and became ineligible because of her age\u2014but she privately admitted that Richard Canidy was the most desirable male she had ever seen. And when she'd been with The Rock, she'd seen a large number of desirable men.\n\nShe thought you could tell a lot about a man by his eyes, and when she looked into Canidy's eyes, she saw gentleness and strength and compassion. And when she did that, she felt about nineteen years old.\n\n\"Good morning, Dancy,\" Canidy greeted her. \"What's the latest fire from the dragon's mouth all about?\"\n\n\"Good morning, Major Canidy,\" Capt. Dancy said.\n\n\"Well, have I done something new, or is he still mad from the last time?\"\n\n\"You really did put him on a spot with the Air Corps, Major,\" Capt. Dancy said.\n\n\"I know,\" Canidy said, smiling at her.\n\nHelene thought he had very nice teeth, which gave him a very nice smile.\n\n\"White is black, up is down,\" Canidy went on, \"and I am supposed to apologize for taking a shot at the bad guys.\"\n\n_There is,_ Helene thought, _a certain undeniable logic to what he says. You'd think they'd want to give him a medal for shooting down enemy planes, not be furious with him._\n\n\"Major Canidy,\" she said chastisingly.\n\n\"I let all the pretty girls call me 'Dick,' \" he said.\n\n\"You are impossible,\" she said. \"This is supposed to be a military organization.\"\n\nCanidy's face registered great surprise.\n\n\"You're kidding!\" he said.\n\n\"Mr. Bruce is down in crypto,\" she said. \"You are to wait.\"\n\n\"And you're not going to tell me what I've done wrong, are you, Dancy?\"\n\n\"No,\" she said, unable to resist smiling back at him. \"But it may have something to do with this.\"\n\nShe opened her drawer and took from it a TOP SECRET cover sheet.\n\nAs he took it from her, she said softly, \"If it doesn't come up, it would probably be better if you didn't mention I'd shown you that.\"\n\nCanidy raised the cover sheet and read the partially decrypted message. Even if the Germans intercepted the message and succeeded in decrypting the text, they would not know the meaning of the code words.\n\nEXLAX FOUR PROCEEDING ALL WELL YACHTSMAN\n\n\"Speaking to you both as your military superior, Captain, \" Canidy said, \"and as someone you _know_ has the Need-to-Know, have there been any developments in the Balkans I should know about?\"\n\nShaking her head and smiling, Capt. Dancy said, \"You have it in your hand.\"\n\n\"Well, now you're off the hook with the dragon,\" Canidy said. \"I _asked_ you for this. You had no choice but to give it to me.\"\n\nShe smiled at him. She thought that was nice of him.\n\n\"Have you got a copy of the OPPLAN [Operations Plan] here, or am I going to have to root around in the basement? \"\n\nCapt. Dancy walked to a sturdy safe from which, quite unnecessarily, for the door was ajar, hung a sign reading \"Open\" and took from it a manila folder with TOP SECRET stamped on it.\n\nCanidy unfolded a map. On it was drawn in grease pencil Eric Fulmar's route into Germany, and his escape route. Along it were marked, in Roman numerals, the stages of the route. There was a I at Marburg an der Lahn, in Germany. There was a II beside Vienna on the map, and a III beside Budapest. The fourth leg of the route ended at P\u00e9cs, in southwest Hungary.\n\nP\u00e9cs was the site of the Batthyany family coal mines. Most of the coal in Hungary is low-grade \"brown\" coal. The mines at P\u00e9cs produced a high-grade anthracite that for hundreds of years had contributed to the Batthyany wealth. Now it was of value because one of the heavy, multiwheeled Tatra trucks that had carried bagged anthracite to Budapest (including, through the influence of Helmut von Heurten-Mitnitz, some to Batthyany Palace) had returned to P\u00e9cs with Eric Fulmar and Professor Dyer and his daughter concealed in a box under a stack of coal bags.\n\nProfessor Dyer was a physicist. There was a tenuous connection between physics the science and physics as in laxative. Hence, \"Ex-Lax.\" In the planning stages of the operation, when they were picking code names, David Bruce had reluctantly admitted that the Germans would probably be baffled by references to a laxative, although he privately thought Canidy's suggestion was one more indication that Canidy was not as serious as he should be.\n\n\"Yachtsman\" was an OSS agent in Hungary. He was a first-generation American from Hamtramck, Michigan, who had learned Hungarian from his mother. Equipped with the appropriate forged identity documents, he was employed with relatives as a deckhand on a Danube River barge. It permitted him to move around the country and, when necessary, to disappear from the barge for a couple of hours, or days.\n\nCompletely decoded, Yachtsman's message meant that Fulmar and the Dyers had made it from Budapest safely to P\u00e9cs, and were proceeding to V. This leg of the route was by barge. \"Ex-Lax\" would travel down the barge canal built under the auspices of Emperor Franz Josef of Austro-Hungary to transport coal from P\u00e9cs to the Danube.\n\nThe barge canal crossed the border between Hungary and Croatia (Yugoslavia) in a sparsely populated region near Ben Manastir, and joined the Danube at Batina. Shortly before reaching Ba\u010dka Palanka, where the Danube turned east toward Belgrade in another desolate, unpopulated area, there would be a signal\u2014in response to lights arranged in a special way on the barge\u2014from the western shore of the Danube.\n\nThe barge would then move close enough to the bank for Fulmar and the Dyers to jump off and pass into the hands of \"Postman,\" the senior of four OSS agents with the guerrilla forces of Colonel Dra\u017ea Mihajlovi\u0107 late of the Royal Yugoslav Army.\n\nCanidy had a little trouble with the bland assurances by radio of Postman\u2014an American of Yugoslavian parentage who had literally been a mail carrier in the States\u2014that this leg of the trip could be safely and conveniently accomplished by truck. According to Postman, the trucks (and the diesel fuel to run them) had been captured by Mihajlovi\u0107 from the Germans, and the Colonel's warning system was so effective that he ran them up and down forest and mountain roads of Croatia and Bosnia and Hercegovina on regular supply and transport missions as if the Germans weren't there and actively looking for him.\n\nVI was the town of Metkovi\u0107 on the Neretva River, fifteen miles from Neretljanski Kanal, a sheltered, natural body of water that opened onto the Adriatic Sea. At Metkovi\u0107, Ex-Lax would be turned over to an agent of the British Special Operations Executive who would arrange for their transport by fishing boat to the island of Vis, VII. The SOE agent's code name, \"Saint Peter,\" was another Canidy suggestion to which David Bruce had somewhat uneasily agreed.\n\nVis was entirely in British hands, though the Germans, who made periodic sweeps of the island, did not suspect it. There was a hidden wharf, onto which supplies could be off-loaded from submarines for transshipment to the mainland. And, between two hills, there was a 4,900-foot runway. A stream flowing across the field seemed to entirely discount the notion that the long valley could be used as a landing strip. But the stream had been altered. There was a twenty-yard-wide stretch where the water was only a foot deep. To observers both on the ground and in the air, it looked for all intents and purposes to be just an area of turbulent water.\n\nExlax will be transported from VII to Cairo, Malta, or such other final destination as the circumstances at the time dictate by U.S. aircraft. In the event this is impossible, Exlax will be evacuated from VII by Royal Navy submarine on a space-available basis.\n\n\"You look deep in thought, Richard,\" David Bruce said as he came into the office, trailed by Lt. Col. Edmund T. Stevens, his deputy. Bruce and Stevens were tall and erect and well-tailored. There was a West Point ring on Stevens's hand. He had resigned from the Army before the war and had been in England when the war broke out, running his wife's food and wine import-export business.\n\n\"Either of you ever collect stamps when you were kids?\" Canidy asked. \"Ever have any from Bosnia-Hercegovina? \"\n\n\"I don't really recall,\" Bruce said impatiently.\n\n\"They had some that were triangular,\" Canidy said, \"that intrigued me.\"\n\n\"I remember those,\" Col. Stevens said.\n\n\"Come on in, Richard,\" Bruce said. \"I fear we are about to have another of our arguments.\"\n\n\"What have I done now?\" Canidy asked, folding the map and handing it to Capt. Dancy.\n\n\"I presume you have the Yachtsman message?\" Bruce asked, after he'd taken a look at the folder.\n\n\"Captain Dancy gave it to me with great reluctance,\" Canidy said, \"only after I threatened to write her name and phone number in phone booths in pubs all over town.\"\n\n\"Major Canidy,\" Capt. Dancy said, \"you're impossible.\" But she was smiling.\n\nBruce closed his office door after they were inside.\n\n\"It isn't what you've done . . . unless, of course, there's something I don't know about yet . . . it's what you are planning to do.\"\n\n\"What would that be?\"\n\n\"Go to Vis to pick up Ex-Lax yourself,\" Bruce said.\n\n\"Have you made up your mind about that, or are you open to my reasoning?\"\n\n\"I'm always willing to listen,\" Bruce said with a smile, \"even when you make it difficult. But this, you should be forewarned, is coloring my thinking.\"\n\nHe took a sheet of yellow foolscap from his desk drawer and handed it to Canidy.\n\nROUTINE FROM OSS WASH DC FOR OSS LONDON PERSONAL BRUCE PLEASE RELAY CANIDY QUOTE CONGRATULATIONS ON DOUBLE KILL UNQUOTE STOP PRESUME HE HAD REASONS FOR BEING WHERE HE WAS STOP REGARDS STOP DONOVAN\n\n\"Looks like he's giving me the benefit of the doubt,\" Canidy said. \"In my experience, the Colonel is not at all subtle. That message could just as easily have read, 'Ground the sonofabitch.' \"\n\nStevens chuckled, earning himself a dirty look from Bruce.\n\n\"Grounding you might make sense, Richard,\" Stevens said. \"From this side of the desk, perceptions are a little different.\"\n\n\"The arguments I made are still valid,\" Canidy argued. \"And to refresh your memory they were (a) that the Air Corps is already bitching about our photorecon missions; and (b) that laying on a mission we would have had to fight over would have called unwanted attention to the Fulmar Werke.\"\n\n\"So are my counterarguments that you're pretty far up in the scheme of things for us to lose you if you get shot down,\" Bruce said. \"But that's over. What you have to do now is convince me there are reasons why we should not just tell the Eighth Air Force what we need, and have them do it. Or even why it is necessary to bring Ex-Lax out by air at all. Why shouldn't they come out on a British submarine? \"\n\n\"Arrogance,\" Canidy said.\n\n\"I beg your pardon? My arrogance, or yours?\" Bruce asked.\n\n\"Mine.\" Canidy chuckled. \"I want to take a good look at the field on Vis myself,\" Canidy said. \"I arrogantly don't trust anybody else's enthusiastic opinion of how good it is. I don't want to lose Ex-Lax, or whoever we bring out later, at stop VII because of pilot error. I want to make that landing and takeoff by myself, so I can tell somebody else how to do it.\"\n\nThe look on Bruce's face, Canidy thought, was not one of acceptance, but he thought Stevens understood.\n\n\"I can also argue,\" Canidy continued, \"that we don't want to involve the English in this operation any more than we have to. If we start demanding space on their submarines, they are going to want justification.\"\n\nHe stopped again and looked at Bruce. After a moment, Bruce made a \"give me more\" gesture with his hand.\n\n\"We have the B-25,\" Canidy said, \"already rigged for this sort of passenger-haul mission, with auxiliary fuel tanks and even seats. If we ask the Air Corps, they're going to have to modify one of their aircraft, and they will naturally ask questions.\"\n\n\"Unless we let them use our B-25,\" Bruce said.\n\n\"I was afraid you'd think of that,\" Canidy said. \"And I'm prepared. I think we would have trouble getting it back from them. If they get their hands on it, David, they're liable to remember it's on loan. Think 'lawn mower,' as in borrowed from next-door neighbor.\"\n\nBruce shook his head.\n\n\"And for a crew?\"\n\n\"I thought about asking for an Eighth Air Force volunteer, \" Canidy said. \"If he turns out okay, we can draft him, permanently. If he doesn't, we send him back.\"\n\n\"Just a copilot?\" Stevens asked.\n\n\"No,\" Canidy said. \"Before we sent him to Switzerland, I was planning to take Stanley Fine. And then, before we sent him to Australia, I was going to take Jimmy Whittaker. Now, I think Dolan.\"\n\nBruce's eyebrows rose again.\n\n\"Why Dolan?\" he asked.\n\n\"He's an old pilot\u2014\" Canidy began.\n\n\"That's what I mean,\" Bruce interrupted reasonably.\n\nChief Aviation Motor Machinist's Mate\u2014formerly, until physically disqualified, Chief Aviation Pilot\u2014John B. Dolan, USN, had, after twenty-six years of service, retired from the Navy to go to Burma and China with the Flying Tigers as a maintenance officer. Afterward, he had managed to acquire a reserve commission in the Navy as lieutenant commander and had been sent by the Navy to England as the aviation maintenance officer for Operation Aphrodite. That was the code name for an attempt to convert worn-out B-17 aircraft into radio-controlled flying bombs, to be used against the German submarine pens at Saint-Lazare, which had proven immune to attack by conventional aerial bombardment.\n\nEisenhower, his patience with Air Corps-Navy squabbling exhausted, had turned Project Aphrodite over to the OSS. Dolan had been delighted. Canidy had been put in charge of the project, and he had known Canidy at the Pensacola, Florida, Naval Air Station when they had both been in the American Volunteer Group. Dolan had correctly guessed that Canidy would not watch his every move the way the Air and Navy brass had been doing.\n\n\"We intrepid birdmen have a saying,\" Canidy said. \" 'There are old pilots, and there are bold pilots, but there are no old, bold pilots.' \"\n\n\"Very interesting,\" David Bruce said.\n\nColonel Stevens gave in to the temptation. \"And where, Richard, would you say that profound observation leaves you?\" he asked innocently.\n\n\"Why, I thought you knew, Colonel,\" Canidy said, smiling broadly, \"that I intend to be a very old pilot.\"\n\n\"Not the way you're going, you're not,\" Stevens said. \"But, okay, Richard, you have . . . just barely . . . made your point.\"\n\n\"I presume Commander Dolan is physically up to it?\" Bruce asked. \"Specifically, that he's had a recent flight physical?\"\n\n\"It's in his records,\" Canidy said. \"Look for yourself.\"\n\n\"I just might,\" Bruce said.\n\nThere was a Report of Physical Examination (Flight) in Lt. Commander Dolan's records. Canidy did not think that David Bruce would notice the astonishing similarity between the handwriting of Commander A. J. Franklin, Medical Corps, USNR, who had signed the examination, and that of Lt. Commander John B. Dolan, USNR.\n\nCanidy intended to see that the old sailor didn't overexert himself on the flight. But he really wanted the old \"Flying Chief\" with his eight-thousand-plus hours in the air with him, heart condition or not. Experience was far more valuable than youth and health on a flight like this.\n\n\"It just makes sense for me to go,\" Canidy argued. \"It accomplishes what has to be done with the least fuss.\"\n\nBruce studied him thoughtfully for a moment, then asked, \"Ed?\"\n\n\"You will take good care of Commander Dolan, won't you, Dick?\" Stevens asked, and when Canidy looked at him, Canidy knew that he knew who had signed Dolan's flight physical.\n\n\"It'll be the other way around, Colonel,\" Canidy said.\n\n\"I think we should defer to Dick's judgment,\" Stevens said.\n\n\"So be it,\" Bruce said resignedly.\n\nCanidy thanked Stevens with a slight nod of his head. Stevens responded with a slight shrug of his shoulders. The message was clear. He had meant what he had said about deferring to Canidy's judgment.\n\nCanidy stopped by Capt. Dancy's desk on his way out.\n\n\"Would you ask the Air Corps to furnish us with short-and long-term weather forecasts for from here to Casablanca, and from Casa to Malta, and from Malta to the Adriatic, starting right now?\" he asked.\n\n\"I was afraid you'd talk him into it,\" she said. \"You want them here, or do you want me to send them out to Whitbey House with the courier?\"\n\n\"Send them to Dolan,\" Canidy said.\n\n\"Will he know what they're for?\"\n\n\"He will after I tell him,\" Canidy said. \"I'm going out there now.\"\n\n\"I thought you would be staying in London,\" she said.\n\n\"No reason for me to do that,\" Canidy said.\n\n\"Yes, there is,\" Capt. Dancy said. \"She's back. She called earlier.\"\n\n\"You didn't tell me,\" Canidy said. It was more of a question than a reprimand.\n\n\"She said that she would be at Broadcast House until half past five, and after that at her apartment, if I happened to see you,\" Capt. Dancy said.\n\nSometimes, Capt. Dancy realized, she was just a little jealous of Ann Chambers, for being young and pretty, and for being able to light up Dick Canidy's eyes at the mere mention of her. And sometimes, like now, she felt like Canidy's sister, or for that matter like his mother, happy that he had a nice, decent girl.\n\n\"You will call in when you decide where you're going to spend the night?\" Capt. Dancy asked.\n\n\"Yeah, sure,\" Canidy said. Then he suddenly leaned across Capt. Dancy's desk and kissed her on the forehead.\n\n\"Major Canidy,\" Capt. Dancy said. \"You're impossible.\"\n\n#  **5**\n\n##  **WOBURN MANSIONS, WOBURN SQUARE LONDON, ENGLAND 5 FEBRUARY 1943**\n\nBefore the war, the private park in the center of Woburn Square had been an area of manicured lawns and flower beds and curving walks beneath ancient trees, all surrounded by a neat fence. Now, only the fence and the trees were left. A bomb shelter had been excavated, and several corrugated sheds had been erected by the Fire Protection Service to store firefighting equipment.\n\nIt had been needed. There were ugly gaps in the rows of limestone-faced houses where German bombs had landed. There had been twenty-four entrances on all four sides of Woburn Square in 1940. Now there were fourteen.\n\n16, Woburn Mansions had not been hit, although the limestone facade had been darkened by the furious fires that had raged down the street on both sides; and there was plywood nailed over what once had been beveled glass windows in the entrance door.\n\nBut inside, it was much as it had always been, a quietly elegant building holding five large, floor-size apartments. The basement apartment and the one on the top floor were smaller than the three main apartments, but they all had large, high-ceilinged rooms and central heating, which was an uncommon luxury.\n\nThe first-floor flat, which would have been the second-floor flat in America, was occupied by Miss Ann Chambers. Technically, it was assigned to the Chambers News Service and intended to house all Chambers News Service female employees in London. The SHAEF billeting officer had been informed that the Chambers News Service ultimately planned to have six to eight female employees with correspondent status stationed in London. That would effectively fill the three bedrooms with the regulation two officer -equivalent persons per room.\n\nThe SHAEF billeting officer had not been told the truth, the whole truth, and nothing but the truth, which was that the Chambers News Service had no plans at all to station any additional female correspondents in London. Brandon Chambers, Chairman of the Board of the Chambers Publishing Company, did not believe that women should go to war as correspondents or anything else. The rule was bent only in the case of his daughter, and that was not really nepotism. Rather, Brandon Chambers had believed his daughter when she told him that either he send her to London as a war correspondent, or she would go to work for Gardiner Cowles\u2014the publisher of, among other things, _Look_ magazine\u2014with whom he had carried on a running feud for twenty years, and who was just the kind of a sonofabitch to give Ann a job just because he knew it would annoy her father.\n\nAnn Chambers had had the London bureau chief tell the billeting officer the story of the five to seven soon-to-arrive female accredited correspondents not because she was the spoiled daughter of a very rich man who considered herself entitled to private quarters (in fact, the other two bedrooms were more often than not occupied by roomless journalists of both sexes), but because Ann intended to share her own bed, whenever possible, with Richard Canidy, and she didn't want anybody around when that might happen.\n\nIf she had a permanent roommate, or roommates, it would not have been possible, for example, to do what she and Richard Canidy were doing now, which was recovering from an enthusiastic, wholly satisfying roll in the hay (actually a roll on a dozen large pillows covered with Chinese silk) at quarter to six in the evening before the fire-place in the sitting room.\n\n\"I don't suppose,\" Ann said, her face against his chest, \"that I will have to ask if you have been a good boy while I was gone, will I?\"\n\n\"If you don't ask, I won't have to lie about it,\" Canidy said.\n\n\"You bastard!\" she said, and jerked a hair from his chest.\n\n\"Two can play at that game,\" he warned.\n\n\"And you would, too,\" she said, shifting her midsection to avoid his searching hand. She failed.\n\n\"You've heard the expression 'by the short hairs'?\" he asked.\n\n\"Let go,\" she said. \"I'll be good.\"\n\n\"Who wants good?\" he asked.\n\n\"Wicked?\" she asked.\n\n\"You got it,\" he said, and let her go.\n\nShe got to her feet and walked out of the room, with an exaggerated shake of her tail. In a moment she was back. She tossed him a dressing gown and shrugged into a sheepskin high-altitude flyer's jacket. It was far too large for her, but it was warm.\n\n\"You look like you should be painted on the fuselage of a B-17,\" Canidy said. \" 'Dick's Delight' or something like that.\"\n\n\"Is that a compliment or a complaint?\" she asked.\n\n\"Compliment,\" he said.\n\n\"You like me to wear it because when I bend over you can see my fanny,\" she said.\n\n\"And everything else,\" he said. \"That's why you wear it, to excite me.\"\n\n\"So what else is new?\" Ann said.\n\n\"You're about to get a roommate,\" he said.\n\n\"You'll be spending some time in London?\"\n\n\"No,\" he said. \"As a matter of fact, I've got a little trip to make. I'll be gone a week or ten days.\"\n\n\"Where are you going?\" she asked quickly, softly.\n\n\"You're not curious about your roommate?\" he asked, ignoring the question.\n\n\"Where are you going, Dick?\" she insisted.\n\n\"Come on, Annie,\" he said. \"You know the rules.\"\n\n\"To hell with the rules, and don't call me 'Annie,' \" she said.\n\n\"Yes, Ma'am.\"\n\n\"After Fulmar?\" Ann asked.\n\n\"Who?\"\n\nShe dropped to her knees on the pillows beside him.\n\n\"He's all right, isn't he?\" she challenged. \"I know you\u2014\"\n\n\"And I know you, as Moses said to the slave girl.\"\n\n\"And if he wasn't, you'd be miserable. And if you didn't know, you'd be all tense. You're relaxed and making jokes, and that means that you've heard something good.\"\n\n\"That's not why I'm relaxed, as Samson said to Delilah,\" Canidy said. \"But, yeah, honey, he's all right. I was a little worried, but the rough part of what he was doing is over.\"\n\n\"Oh, baby, I'm happy for you,\" she said.\n\n\"And you're not curious about your roommate?\"\n\n\"I don't know what you're talking about,\" she said. \"I don't have roommates. If I had a roommate, I couldn't greet you at the door wearing nothing but a sheepskin jacket and a smile. So I don't want a roommate. Get the idea?\"\n\n\"What about good ol' Chastity?\"\n\n\"Charity,\" she corrected him automatically. Then, \"Charity? She's coming here?\"\n\n\"In the next couple of days,\" Canidy said. \"What I was thinking was that maybe you could take a couple of days off.\"\n\n\"For what purpose?\" she asked suspiciously.\n\n\"So she could stay here with Doug Douglass,\" Canidy said.\n\n\"If she moved in here, I'd never get rid of her,\" Ann said. \"How long is she going to be in London, anyway?\"\n\n\"Permanently,\" he said.\n\n\"Then no, period,\" Ann said. \"Charity cannot stay here. She would move in, and I wouldn't have the heart to throw her out, and that would be the end of us making love on the pillows.\"\n\n\"In that case, screw her,\" Canidy said. \"Your logic is irrefutable. \"\n\nShe threw herself at him and nibbled his ear.\n\n\"You keep that up, you know what's going to happen,\" he said.\n\n\"I hope, I hope, I hope,\" Ann said. Then she said, \"Damn, I'm glad Eric's all right. I love you when you're like this.\"\n\n\"Like what?\"\n\n\"Happy and horny,\" Ann said. \"Where is he?\"\n\n\"Ah, come on, Mata Hari,\" he said.\n\n\"I was just trying to find out how long you'd be gone, and where you'll be going.\"\n\n\"Eric at this very moment is somewhere on the European landmass, riding down a forest road between towering pines,\" he said. \"That tell you anything?\"\n\n\"No,\" she said. \"And I don't really mean to pry.\"\n\n\"I know,\" he said.\n\nEric Fulmar, at that very moment, was walking down a basement corridor in the municipal jail in P\u00e9cs, Hungary. He was handcuffed to Professor Friedrich Dyer, and both of them wore chain hobbles.\n\nA member of the Black Guard, an SS-like organization owing its allegiance to Admiral Horthy, the Regent of Hungary, stopped them by a cell, unlocked the handcuffs, and pushed Professor Dyer inside. Then he pushed Fulmar into motion again, until he came to the next cell door. He retrieved his handcuffs, then pushed Fulmar into the cell.\n**V**\n\n#  **1**\n\n##  **OSS VIRGINIA STATION 5 FEBRUARY 1943**\n\nCynthia Chenowith had elected to skip the evening meal. When she had finished her bath, she would dine on Ritz crackers and canned Vienna sausages and Nescaf\u00e9 from the PX store. The Vienna sausages tasted like soap and would more than likely give her indigestion, and boiling water for the Nescaf\u00e9 (indeed, possessing an electric hot plate) was a specific violation of station regulations for trainees, but she desperately needed a bath, and she didn't want to go to supper, or for that matter to leave the privacy of her room.\n\nHis name was Horace G. Hammersmith. It had been impossible in the case of Lt. Horace G. Hammersmith, Signal Corps, U.S. Army, to obey either the spirit or the letter of the regulation that forbade any interest in, or discussion of, the private life of fellow trainees. Horace Hammersmith was also known as Greg Hammer, and Greg Hammer was a movie star in private life. He wasn't up there with Clark Gable or Tyrone Power, but his rough-hewn face, his astonishingly golden wavy hair, and his football player's build had left no question in any of the trainees' minds from the moment they first saw him that Lt. Horace G. Hammersmith was _really him!_\n\nAnd from the moment Lt. Hammersmith had seen Miss Chenowith, he had made it plain that he found her fascinating. At first, Cynthia had thought it was simply a case of movie-staritis. Without arrogance, as a simple statement of fact, she realized that she was the best looking of the half-dozen women at Virginia Station. As a movie star accustomed to the adoration of his female fans, Cynthia reasoned, Hammersmith had come to believe that the pick of the herd, or the pride, or the flock, or whatever word fitted the half-dozen women at Virginia Station, was his.\n\nHis Training Group had begun training six weeks before Cynthia's. The way the school was set up (before she had come to Virginia Station as a trainee, Cynthia had read Eldon Baker's training syllabus), incoming trainees were placed under the supervision of trainees who had finished their training and were awaiting assignment. The announced purpose was to spare the training staff the mundane work of seeing to the issue of equipment, the first painful hours of calisthenics, the explanation of the rules, and so on. The real reason was so that the training staff could judge how well the \"senior\" trainees dealt with subordinates\u2014to see if they could inspire cooperation. There was no place on an operational OSS team for someone who antagonized, intentionally or otherwise, the others on the team.\n\nLt. Horace G. Hammersmith had been as good and as natural a leader of his peers at Virginia Station as Greg Hammer had been a leader in the movies. Despite herself, Cynthia had come to like him. And she found that her first snap judgment of him had been almost entirely wrong. She had found Hammersmith to be really shy, rather than being arrogant. And she learned that, rather than being awed with himself as a movie star, he thought the whole movie business was rather funny.\n\nOver the weeks, she had learned that he was an electrical engineer who had been sent to Los Angeles by the Murray Hill division of the Bell Telephone Laboratories to supervise the installation of a recording studio at Continental Studios.\n\n\"Lana Turner,\" he told her one afternoon while they were taking a five-minute break on a ten-mile run, \"was discovered in Schwab's Drug Store. I was discovered having dinner with a vice president of Continental Studios, Stan Fine, at the Villa Friscati.\"\n\n\"Stanley Fine?\" she asked, genuinely surprised.\n\n\"Uh-huh,\" he said.\n\n\"We're not supposed to be talking about our private lives, you know,\" she said.\n\n\"I know,\" he said, \"and I also know you know Stan.\"\n\nThen he'd looked at his watch, and the five-minute break was over, and he'd jumped to his feet and blown his whistle, and they'd resumed the ten-mile run. That night, at supper, he had sat down beside her and resumed the conversation where he'd broken it off.\n\n\"Over a steak, which Bell Labs was paying for, I was explaining to Stanley why it was going to cost Continental Studios a bunch of money more than they expected to get what they wanted, when this fat little bald-headed man walked up to the table and said, in an accent you could cut with a knife, 'So tell me, Stanley, who's your friend? And vy I haven't zeen any film?' \"\n\n\"Max Liebermann,\" Cynthia said, laughing at Hammersmith's apt mimicry of the founder and chairman of the board of Continental Studios.\n\n\"Right,\" Hammersmith said. \"But I didn't know who he was. So Stanley said, 'Uncle Max, he's the engineer from Bell Telephone.' \"\n\n\" 'What I vant to know is can he ride a horz?' Max said,\" Hammersmith went on. \" 'If he can ride a horz, I tink he's Major Porter. We god a hell uf a problem wit dat, Stanley, if I god to tell you.' \"\n\nBy then, Cynthia was giggling at the mimicry.\n\n\"It didn't take much to corrupt me,\" Hammersmith had gone on. \"All it took to get me before the cameras was as much by the week, on a year's contract, as Bell Labs was paying me by the month. And luckily, I could ride a 'horz.' \"\n\n\"I saw _Calvary Raid,_ \" Cynthia said. \"You were very good.\"\n\n\"That's because my only lines were 'Yes, Sir,' and 'Sound the Charge!' \" Hammersmith said. \"Anyway, Stan and I became pals. And he got me into this, and he wrote me a letter saying if I got to Washington and desperately needed a place to stay, I should call a Miss Cynthia Chenowith and say I was a friend of his. Unless there is another Cynthia Chenowith?\"\n\nHorace G. Hammersmith had not so much as touched her hand, except in the line of duty. But neither had he for long taken his eyes off her whenever they were around each other.\n\nAnd now he was going. He was going operational. She wondered where, and doing what. And she just wasn't up to spending his last night here with him. In the morning, she would have breakfast with him, and maybe even go to the station wagon with him, and kiss his cheek.\n\nBut she didn't want to see him tonight. Tonight, there would be just too much of a temptation to give him what he wanted, even if he didn't ask for it. She didn't want him to go operational with her on his mind. She didn't love him, but she really liked him, and she was almost sure he thought he was falling in love with her. Whatever they were going to have him doing, the one thing he didn't need was her on his mind any more than she already was.\n\nThe bathtub was full. So when Cynthia sensed the water was cooling, she had to let water out before filling it again with hot water. She bent her left leg, in order to get a good look at her foot, then vigorously rubbed away a layer or two on the calluses. Then she repeated the operation on the right foot.\n\nAnd finally she stepped out of the tub and toweled herself dry. Then she took the towel and wiped the condensation from the full-length mirror on the door and examined herself in it.\n\nShe \"made muscles,\" as she had seen men do, and was surprised\u2014and not sure whether she was pleased or disappointed\u2014that she could see no development in her biceps. With all the push-ups and pull-ups she'd done, she had expected some.\n\nShe had bruised, ugly blue areas in several places. The largest area was in her right shoulder, from the recoil of the Springfield rifle, and the Garand rifle, and the Winchester shotgun, and the Thompson submachine gun she had fired on the range. She had fallen twice on the obstacle course. There was a bruised area on her lower stomach, a souvenir of an encounter with a peeled log when she had tripped running up an obstacle, and another on her right leg, just above her knee. She had earned that battle stripe just by stumbling, exhausted, and landing on the goddamned Springfield.\n\nFinally, there was a raw spot on the web of her right hand, where the Colt .45 automatic pistol had \"bitten\" her.\n\nShe dried that spot very carefully with a wad of toilet tissue and then applied Merthiolate and a Band-Aid. And then she took a large economy-size tube of Ben-Gay and applied it liberally to all the bruised areas.\n\n_If Greg should come up here,_ she thought, _I will smell like the men's locker room, and maybe that will dampen his ardor._\n\nStill naked, she washed and dried her hair, wrapped her head in a towel, and then finally put on what she considered a grossly unfeminine set of pajamas. They were from the PX, too. Flannel, with a particularly ugly red-and-brown pattern. She put a bathrobe over the pajamas, examined herself a final time in the mirror, stuck her tongue out at herself, and then went into her bedroom.\n\nShe sat down at a government-issue gray metal desk, which was conspicuously ugly in comparison to the rest of the furniture, turned on the desk lamp, and took a brown-paper -bound book from a rack. The book was titled _U.S. Field Manual, FM 21-10: The Law of Land Warfare._\n\nThere would be a written examination to make sure the trainees knew what the Hague and Geneva Conventions had had to say about where the line was between a soldier, who was entitled to treatment as a prisoner of war, a partisan, and a spy. Under the law of land warfare, partisans and spies could be shot.\n\nCynthia had serious doubts that either the Germans or the Japanese were going to pay much attention to the fine print, but the course was a part of the curriculum, and she had to pass it to graduate. And she was determined to graduate.\n\nThirty minutes later, just after she had opened a can of Vienna sausages and was trying without much success to get one of the tightly packed little obscenities out of the can, there was a knock at her door.\n\nShe didn't respond. If it was Horace G. Hammersmith and she didn't respond to his knock, he might take the hint and go away.\n\nBut after a moment, there was another knock, this time far more demanding.\n\n\"Who is it?\"\n\n\"Eldon Baker.\"\n\n\"Come on in,\" Cynthia called.\n\nBaker entered the room.\n\n\"Studying,\" Cynthia said unnecessarily.\n\nShe saw that Baker had seen the hot plate and the jar of Nescaf\u00e9 and wondered if he would turn her in. He knew that she had a close relationship with Colonel Donovan and Captain Douglass; the other training personnel did not.\n\n\"Have you got a minute so that we can talk?\" Baker asked.\n\n\"I should study, Eldon,\" she said, \"but sure.\"\n\n\"Don't worry about the examination,\" he said as he closed the door. \"You won't be taking it.\"\n\n\"Oh?\"\n\n\"I have just had a telephone call from Chief Ellis,\" Baker said. \"You are to go to Washington to the house on Q Street with the station wagon in the morning.\"\n\n''Oh?\" she repeated.\n\n\"You will take your things with you,\" Baker said. \"According to the Chief, you will not be coming back. At least as a trainee.\"\n\n\"What's this all about?\" Cynthia asked.\n\nShe was sure she knew.\n\n_Oh, goddamn you, Jimmy!_\n\n\"Chief Ellis did not elect to tell me,\" Baker said. \"But I think we can both make an educated guess, can't we?\"\n\n\"Whittaker?\" Cynthia asked.\n\n\"Doesn't it seem that way to you?\" Baker said. \"I can't tell you how annoyed this makes me.\"\n\n\"Why should it bother you? He's not offering you his unasked-for male protection.\"\n\n\"After some thought,\" Baker said, \"after Captain Whittaker's visit, I decided I could not overlook it. That, in other words, I had to make an official issue of it.\"\n\n\"I don't think I quite follow you,\" Cynthia said.\n\n\"In addition to what he did to you,\" Baker said, \"he had a run-in with me. He was insubordinate. Technically, I suppose, he's AWOL. He was ordered to report here for training. He decided, on his own, that he'd really rather not do that. I wrote a letter reporting what had transpired to Colonel Donovan.\"\n\nCynthia wondered why that bothered her, why she felt a surprising flash of anger. Baker was right. Jimmy Whittaker was an Air Corps officer. Officers do what they are told to do. And there was absolutely no excuse for his having kissed her the way he had, making a fool of her in front of the others.\n\n\"It would appear that the rules which apply to everyone else in the OSS, myself included, do not apply to Captain Whittaker.\"\n\n\"We don't know that's what's happened,\" she said.\n\n\"I felt sure that Colonel Donovan would understand my motives in making an official report of what happened,\" Baker said. \"That, rather than trying to get Whittaker in trouble, my concern was for the overall discipline of the organization. I felt confident he would understand that it was not a personality clash between Captain Whittaker and myself, but rather an impersonal incident in which an agent willfully disobeyed his superior, with the result that the authority of the Director of Training was seriously undermined. \"\n\nHe waited for her to respond to that, and then, when she did not, went on, \"Obviously, I was wrong. The only response to my letter was the telephone call just now. When Whittaker left here, after telling me that I was 'out of my mind' for having you in the school, he said that he was going to see the colonel. I had the impression he meant both about his coming here and about you.\"\n\n\"He's known Colonel Donovan all his life,\" Cynthia said softly.\n\n\"And so have you,\" Baker said.\n\nCynthia looked at him.\n\n\"You want me to go to Colonel Donovan?\" she asked.\n\n\"I thought you might consider it,\" Baker said. \"For what a fraternal organization would call 'the good of the order.' \"\n\n\"I'm going to see Colonel Donovan,\" Cynthia said. \"I intend to graduate from this school.\"\n\n\"I thought perhaps you could make it clear to him why this whole sequence of events is so distressing to me,\" Baker said.\n\nCynthia's mind was rushing ahead.\n\n\"If I'm to go to Washington in the morning,\" she said, \"what do I do about turning in my equipment, settling things?\"\n\n\"I'll take care of that for you myself,\" Baker said.\n\n#  **2**\n\nIt had taken a long time for Cynthia to go to sleep, and she had gone to sleep angry.\n\nAnd she had awakened still angry, and had grown angrier with the realization that there was not going to be time to pack and dress and eat breakfast, too, and that she was just going to have to miss breakfast.\n\nThere was a small silver lining to the black cloud, she thought. It would be the first time that Greg had seen her dressed up in anything fancier than a skirt and a sweater, or wearing any makeup except a faint touch of lipstick. She had a moment to enjoy that before thinking that it probably would be better if he didn't get to see her that way. It would fuel what she suspected he felt for her.\n\nWhen she carried her luggage downstairs, he was in the entrance foyer. It was the first time she had seen him dressed up, too. He was in his pink-and-green lieutenant's uniform, wearing his new silver parachutist's wings.\n\nHe smiled when he saw her.\n\n\"Baker said you would be going to Washington,\" he said. \"He didn't say why, and he didn't tell me how pretty you are in your civilian clothing.\"\n\n\"Good morning, Greg,\" she said.\n\nShe wondered what his destination was, and when they had passed the checkpoint, she asked him.\n\n\"I don't know,\" he said.\n\nCynthia leaned forward and asked the driver, \"Where are you taking Lieutenant Hammersmith?\"\n\n\"The house on Q Street,\" the driver replied. \"He's to see Chief Ellis.\"\n\n\"What's the 'house on Q Street'?\" Greg Hammersmith asked.\n\n\"It's a mansion near Rock Creek Park,\" she said. \"We use it as both a safe house and sort of a hotel for transients.\"\n\n\"You've been there before, I gather.\"\n\n\"I used to run it,\" she said.\n\n\"And am I permitted to ask where you're going?\" he asked.\n\n\"I'm going there too,\" she said.\n\n\"And am I permitted to ask why?\"\n\n\"No,\" she said. \"I'm sorry.\"\n\n\"Then, in the short time remaining to us, Miss Chenowith\u2014\" he began.\n\n\"Don't, Greg,\" she said. \"Please don't\u2014\"\n\n\"What I was going to say, you have apparently figured out all by yourself,\" he said.\n\nShe looked at him and met his eyes, then averted her eyes and avoided looking at him on the rest of the way to Washington.\n\nWhen she walked into the kitchen, she asked the cook if Chief Ellis was around.\n\n\"In the dining room with Captain Whittaker,\" the cook replied.\n\n\"Come on, Greg,\" Cynthia said, aware that her temper was up and not caring.\n\nCaptain Whittaker and Chief Ellis were eating either a late breakfast or an early lunch. They were having eggs with their steaks, she saw, so it had to be breakfast.\n\n\"I think you know Miss Chenowith, Chief,\" Whittaker said when he saw her. \"Otherwise known as 'Super-woman. 'And I don't know the name of the gentleman with her, but he is the one who almost came to her aid when I publicly humiliated her.\"\n\n\"Damn you!\" Cynthia flared.\n\n\"My name is Hammersmith,\" Greg said coldly.\n\n\" 'My name is Hammersmith, _Sir,_ ' \" Whittaker said. \"We try very hard to observe the military amenities around here, don't we, Chief?\"\n\n\"Yes, Sir,\" Ellis said. \"That we do, Sir.\"\n\n\"Sit down, Cynthia,\" Whittaker said. \"Take a load off. Have a bite to eat. We have several hours to kill.\"\n\nGlowering at him, she walked to the head of the table and stabbed the call button on the floor with her toe.\n\n\"For a moment, there, I thought she was going to slug me with her purse,\" Whittaker said. \"Didn't it look that way to you?\"\n\n\"You sonofabitch,\" Cynthia said.\n\n\"Nice to see you, too, Miss Chenowith,\" Whittaker said.\n\nThe cook appeared.\n\n\"Yes, Ma'am?\"\n\n\"I'd like some breakfast,\" Cynthia said. \"Greg, are you hungry?\"\n\n\"I missed breakfast,\" he said.\n\n\"Bring us, please, the same thing they had,\" Cynthia said.\n\n\"You may sit down, Lieutenant,\" Whittaker said.\n\nLieutenant Hammersmith didn't move.\n\n\"I'll rephrase,\" Whittaker said. \"Sit down, Lieutenant.\"\n\n\"Damn you, play your games with me, but leave Greg alone.\"\n\n\"'Greg'?\" Whittaker parroted mockingly. \"Wonder-woman to the rescue of 'Greg'?\"\n\n\"You really are a bastard, Jimmy,\" she said.\n\n\"You miss the point, Cynthia,\" Whittaker said. \"The one thing I demand of my subordinates when I'm off saving the world for democracy is what they call instant, cheerful obedience.\"\n\n\"What is that supposed to mean?\" Cynthia flared.\n\n\"I'm about to go into the Philippines,\" Whittaker said. \"If the lieutenant here is half the radio wizard Douglass tells me he is, and if I'm convinced he'll take orders, he's going with me.\"\n\n\"That's operational information,\" Cynthia flared. \"That's Top Secret. I'm going to tell Colonel Donovan you've been running off at the mouth again, and Ellis, damn you, too, you're my witness.\"\n\n\"Oh, you've got the Need-to-Know, Cynthia,\" Whittaker said. \"You're the control.\"\n\nShe looked at him and saw in his eyes that he was telling the truth.\n\n\"I'm not thrilled about you being my control, frankly,\" Whittaker said. \"But it was the only way I could think of to get you out of that school.\"\n\n\"Why did you do that?\" Cynthia snapped. \"What gave you the right?\"\n\n\"I already told you,\" he said. \"I love you, and all's fair in love and war. This seems to be both, so anything goes.\"\n\n\"Damn you, Jimmy!\" she said, furious that she felt like crying.\n\n\"That may pose certain problems between us, Captain,\" Hammersmith said.\n\n\"How is that?\" Whittaker asked.\n\n\"I'm in love with her, too,\" Greg Hammersmith said.\n\n\"Oh, Greg!\" Cynthia said.\n\n\"From this point, then, Lieutenant, you are advised not to turn your back on me,\" Whittaker said.\n\n\"Fair enough,\" Hammersmith said.\n\n\"You look vaguely familiar to me, Lieutenant,\" Whittaker said. \"Do we know each other?\"\n\n\"No, Sir,\" Hammersmith said.\n\n\"He's the actor, Captain,\" Chief Ellis said. \"Greg Hammer? \"\n\n\"Oh, yeah,\" Whittaker said. \"I'll be damned. How'd a movie star get in the OSS?\"\n\n\"I'm a friend of Stan Fine's,\" Hammersmith said. \"When the Army announced that I would be stationed as an instructor at Fort Monmouth for the indefinite future, I asked him to get me out of it.\"\n\n\"I'm really sorry you told me that,\" Whittaker said. \"I always find it difficult to cut the throats of friends of friends of mine.\"\n\n\"Catch me asleep,\" Hammersmith said. \"I'm very vulnerable when I'm asleep.\"\n\n\"You just volunteered to run around in the Philippines, Lieutenant,\" Whittaker said. \"How do you feel about that?\"\n\n\"I thought I had to prove I was a radio wizard first,\" Hammersmith said.\n\n\"That was before you told me you have the hots for our girl . . . ,\" Whittaker said.\n\n\"Damn you!\" Cynthia said.\n\n\"Obviously,\" Whittaker went on, \"I could not go off to run around in the jungle and eat monkeys and leave you here to pursue yon fair maiden by yourself.\"\n\n\"Obviously not,\" Hammersmith said, and chuckled.\n\n_Damn it,_ Cynthia thought, _they like each other!_\n\n#  **3**\n\n##  **FERSFIELD ARMY AIR CORPS STATION BEDFORDSHIRE, ENGLAND 7 FEBRUARY 1943**\n\nFirst Lieutenant Henry \"Hank\" Darmstadter, U.S. Army Air Corps, a stocky, round-faced young officer of twenty-three, was not sure why he had volunteered for a \"classified assignment involving great personal risk\" or why he had been accepted.\n\nAs a simple statement of fact, rather than from modesty, he understood that he was not the world's greatest airplane driver. There was proof of this. He had twice\u2014once in basic and again in advanced\u2014been sent before the elimination board. The first time, the reason had been simple. He had suffered airsickness.\n\nThe only reason he had not been eliminated in basic and sent to navigator's or bombardier's school, or for that matter to aerial gunner's school, was that his class had an extraordinary number of cadets who also suffered from airsickness, plus half a dozen guys who had just quit. The elimination board had considered all those cadets who had an airsickness problem and decided that Darmstadter, H., was the least inept of the inept.\n\nThey really couldn't eliminate all of those who under other circumstances should have been eliminated. Pilots were in short supply, and the demand was growing. When he had been given another \"probationary period\" by the elimination board, it had two conditions. The first was official: that he \"demonstrate his ability to perform aerobatic maneuvers without manifesting signs of illness or disorientation. \" Translated, that meant that he do a loop without getting airsick. The second, unofficial, unspoken condition was that he understand he would not get to be a fighter pilot or a bomber pilot, and that there was a good likelihood, presuming he got his wings, that he would be assigned to a liaison squadron, flying single-engine two-seaters. Or even be assigned to the Artillery to fly Piper Cubs directing artillery fire.\n\nHank Darmstadter had conquered his airsickness. He wasn't sure whether this was because he had grown accustomed to the world turning at crazy angles or to being upside down, or because he had simply stopped eating when he knew that he was going to be flying.\n\nHe had been given his wings and his second lieutenant's gold bar and sent to advanced training. Not in P-51s or P- 38s or B-17s or B-24s, but in C-45s. The C-45 was a small, twin-engine aircraft built by Beech. It had several missions in the Army Air Corps, none of them connected directly with bringing aerial warfare to the enemy. It was used as a small passenger transport, and it was used as a flying classroom to train navigators and bombardiers.\n\nTwo weeks before Hank Darmstadter was to graduate from advanced training in the C-45 aircraft, he had, flying solo, dumped one. He had lost the right engine on takeoff, and if he had had one hundred feet less altitude, he would have gone into the ground. But the hundred feet made the difference, and he had been able to stand it on its wing and make a 360-degree turn and get it back onto the runway, downwind and with the wheels up, just as the second engine cut out.\n\nThirty seconds after he had scrambled out the small door in the fuselage, there had been a dull rumble, and then a larger explosion as the fuel tanks ignited and then exploded.\n\nWhen he appeared before that elimination board, they had discussed the accident and Darmstadter, 2nd Lt. H., as if he were not there. In the opinion of one of his examiners, if he was that far along in the course, he should have known and demonstrated the proper procedure to follow in the case of engine failure on takeoff. And the proper procedure was not to make a dangerous 360 and land the wrong way on the runway as Darmstadter had done, but to make the proper adjustments for flight on one engine, then to circle the field and gain sufficient altitude to make a proper approach (that is, from the other direction, into the wind).\n\nAnother of his examiners, to Darmstadter's considerable surprise, had taken the position that since no one was with him in the cockpit, they didn't know what had happened, and that it wasn't really fair to assume that he had done what he had done from panic; that he was entitled to the benefit of the doubt; and that Darmstadter's best judgment had been to do what he had done.\n\nThere had been seven officer pilots on the elimination board. The vote\u2014it was supposed to be secret, but the president of the board told him anyway\u2014was four to three not to eliminate him. He would be permitted to graduate and to transition to Douglas C-47 aircraft.\n\nThe C-47\u2014the Army Air Corps version of the Douglas DC-3 airliner\u2014was supposed to be the most forgiving aircraft, save the Piper Cub, in the Army Air Corps. Douglas was building them by the thousands, and each of them needed two pilots. They were used as personnel transports and cargo aircraft. Most of the C-47s being built would be used in support of airborne operations, both to carry paratroopers and to tow gliders.\n\nHank Darmstadter had understood that his glamorous service as an Air Corps pilot would be in the right\u2014copilot's\u2014seat of a C-47 Gooney Bird. He would work the radios and the landing gear and the flaps, while a more skillful pilot would do the flying.\n\nAnd that's what he had done at first when he'd come to England. But then the system had caught up with him. He had received an automatic promotion to first lieutenant, based solely on the length of his service. It was the policy of the troop carrier wing commander that the pilot-in-command, whenever possible, be senior to the copilot. And Darmstadter had picked up enough hours, and enough landings and takeoffs as a copilot, to be qualified as an aircraft commander.\n\nTen days before, when his squadron had returned from a practice mission\u2014in empty aircraft practicing low-level formation flight as required for the dropping of parachutists\u2014the troop carrier wing commander had gathered the pilots in a maintenance hangar and told them Eighth Air Force was looking for twin-engine qualified pilots for a \"classified mission involving great personal risk\" and that those inclined to volunteer should see the adjutant.\n\nOnly three Gooney Bird pilots had volunteered. The other two were pilots who desperately wanted to be fighter pilots, and believed that unless they did something, anything, to get out of Gooney Birds, they would spend the war in a Gooney Bird cockpit.\n\nHank Darmstadter, who himself would have loved to be a fighter pilot, didn't think there was any chance at all of getting to be one by volunteering for this \"classified mission. \" He could think of no good, logical reason for his having volunteered. Without false heroics, he understood that there was hazard enough in either towing gliders or dropping parachutists when there were a hundred Gooney Birds all doing the same thing at the same time in a very small chunk of airspace.\n\nThe one reason he had volunteered was that he had wanted to, and he was perfectly willing to admit that it was probably a goddamned dumb thing to do.\n\nWhen he saw the adjutant, there was a short questionnaire to fill out. It asked the routine questions, and a couple of strange ones. One question was to rate his own ability as a pilot, with five choices from \"completely competent\" down through \"marginally competent.\" Darmstadter had judged himself in the middle: \"reasonably competent, considering experience and training.\" Another question wanted to know if he spoke a foreign language, and if so, which one and how well. And the last question was whether or not he had any relatives, however remote the connection, living on the European continent, and if so, their names and addresses.\n\nHe was tempted to answer \"no\" to both questions, but in the end, he put down that he understood German, and that he had a great-uncle, Karl-Heinz Darmstadter, and presumably some other relatives, in Germany but that he didn't know where.\n\nHe hadn't quite forgotten about having volunteered, but he had put it out of his mind. For one thing, he felt pretty sure if they were making a selection of volunteers, they would probably have a dozen better qualified people than a Gooney Bird driver to pick, and for another, considering the Army Air Corps bureaucracy, it would take three weeks or a month before they told him \"thanks, but no thanks.\"\n\nAt four o'clock this morning, the charge of quarters had come to his Quonset hut, and told him the adjutant wanted to see him. The adjutant had handed him a teletype message:\n\nPRIORITY  \nHQ EIGHTH US AIR FORCE  \nCOMMANDING OFFICER 312TH TROOP CARRIER WING  \n1ST LT HENRY G. DARMSTADTER 03434090 2101 TROOP  \nCARRIER SQUADRON TRANSFERRED AND WILL  \nIMMEDIATELY PROCEED FERSFIELD ARMY AIR CORPS  \nSTATION REPORTING UPON ARRIVAL THEREAT TO  \nCOMMANDING OFFICER 402ND COMPOSITE SQUADRON  \nFOR DUTY. OFFICER WILL CARRY ALL SERVICE RECORDS  \nAND ALL PERSONAL PROPERTY. CO 312TH TCW DIRECTED  \nTO PROVIDE MOST EXPEDITIOUS AIR OR GROUND  \nTRANSPORTATION.  \nBY COMMAND OF LT GENERAL EAKER  \nA.J. MACNAMEE COLONEL USAAC ADJUTANT GENERAL\n\nAt 0400 there was soup thick enough to cut with a knife, and the weather forecast said \"snow and\/or freezing rain,\" so the most expeditious air or ground transportation had been a jeep. It had been a five-hour drive, and Darmstadter had been stiff with cold when they were passed inside the Fersfield gate by an MP wearing his scarf wrapped around his head against the cold.\n\n\"The 402nd's way the hell and gone the other end of the field, Lieutenant. When you see a B-17 graveyard, you found it,\" the MP said.\n\nAs they drove down a road paralleling the north-south runway, past lines of B-17s in revetments, Darmstadter was surprised to hear an aircraft approaching, engines throttled back for landing. He stuck his head out the side of the jeep and looked at the sky. It was neither raining nor snowing, but conditions were far below what he thought of as minimums of visibility.\n\nAnd then he saw the airplane. It was a B-25, and for a moment he thought the pilot had overshot the runway and would have to go around. But the pilot set it down anyway.\n\n_Damned fool!_ Darmstadter thought, professionally.\n\nThey reached the end of the runway. There was, as the MP had said, a B-17 graveyard: fifteen, maybe twenty, battered and wrecked and skeletal B-17s, some missing engines, some with no landing gear, their fuselages sitting on the ground. Three battered B-17s, Darmstadter saw with confused interest, were still flyable, to judge by their positions near the taxi ramp and by the fire extinguishers and other ground equipment near them. But the tops of their fuselages, except for portions of the pilots' windshields, were gone, as if someone had simply taken a cutting torch and cut them away. Someone, for reasons Darmstadter could not imagine, had turned three B-17s into open-cockpit aircraft.\n\nThere were half a dozen Quonset huts and a homemade arrangement of tent canvas and wooden supports that obviously served as some sort of hangar, or at least a means to work on engines out of the snow and rain.\n\nAs the jeep approached the area, the B-25 he had seen land taxied down a dirt taxiway, turned around with a roar of its engines, and stopped. Three sailors\u2014it took Darmstadter a moment to be sure that's what they really were\u2014 trotted up to the B-25 and started to tie it down and put chocks in place. The crew door dropped open and an Air Corps officer jumped to the ground. Darmstadter waited for the rest of the crew to come out, and then, when the pilot turned and pushed the door closed, he was forced to conclude that, in violation of regulations\u2014and, as far as he was concerned, common sense\u2014the B-25 had been flown without either a copilot or a flight engineer.\n\nThe jeep, all this time, had been moving.\n\n\"This must be it, Lieutenant,\" the jeep driver said, and pointed to a small sign reading simply ORDERLY ROOM nailed to the door of one of the Quonsets.\n\n\"I'll see,\" Darmstadter said, and got out of the jeep and walked to the Quonset.\n\nHe knocked and was told to come in. Inside were two Navy enlisted men, three Air Corps enlisted men, and three naval officers, all three of them wearing gold naval aviator's wings. Two of them were wearing USN fur-collared leather, zipper jackets. The third wore a navy blouse, with pilot's wings, the gold sleeve stripes of a lieutenant commander, and an impressive row of ribbons. Some of them Darmstadter had never seen before, but he recognized both the Distinguished Flying Cross and the Purple Heart.\n\nDarmstadter saluted.\n\n\"Sir, I'm looking for the 402nd Composite Wing.\"\n\n\"You've found it, Lieutenant,\" the Navy flyer with the DFC said. He offered his hand. \"I'm Commander Bitter.\"\n\n\"How do you do, Sir,\" Darmstadter said.\n\n\"You must be Darmstadter,\" the lieutenant commander said.\n\n\"Yes, Sir,\" Darmstadter said. He handed over a Certified True Copy of the teletype message from Eighth Air Force.\n\nThe door opened and a tall Air Corps officer, a major, the one Darmstadter had seen climb out of the B-25, entered the Quonset hut. For the first time, Darmstadter got a good look at his leather A-2 jacket. There was a Chinese flag and what was apparently some kind of a message in Chinese characters painted on the back.\n\n\"What the hell are you doing flying in that shit?\" one of the other Navy flyers said. He was the oldest of the three, a ruddy-faced middle-aged man.\n\n\"Oh, ye of little faith!\" the Air Corps major said, then turned to Darmstadter. \"You must be Darmstadter.\"\n\n\"Yes, Sir,\" Darmstadter said.\n\n\"I could tell because you looked confused,\" the major said. \"And like the kind of guy who would dump a C-45.\" He paused a moment. \"You're in good company, Lieutenant. Commander Bitter also dumped one, didn't you, Commander?\"\n\nThe middle-aged Navy flyer laughed.\n\n\"Goddamn, I'd forgotten about that,\" he said. \"He did, didn't he?\"\n\n\"Presumably,\" Commander Bitter said, his voice revealing that he was a little annoyed at the reference to a dumped C-45, \"you're going to explain what this is all about?\"\n\n\"I'm going to borrow Dolan for a couple of days,\" the major said, and then, as if he had just remembered his manners, offered his hand to Darmstadter. \"I'm Dick Canidy, Darmstadter. Welcome aboard.\"\n\n\"Sir,\" Darmstadter said, \"I'm a little confused.\"\n\n\"So am I,\" Commander Bitter said. \"Where are you and Dolan going?\"\n\n\"An island called Vis in the Adriatic Sea,\" Canidy said, then turned to Darmstadter. \"You checked out in the B-25, Darmstadter?\"\n\n\"No, Sir,\" Darmstadter said. \"I've never even been in one.\"\n\n\"Fine,\" the major said. \"I was afraid you might have picked up some bootleg time.\"\n\nDarmstadter was now wholly confused.\n\n\"No, Sir,\" he said.\n\n\"Eric needs a ride home,\" Canidy said. \"We're going to take Lieutenant Darmstadter along with us.\"\n\n\"He just said he's never even been in a B-25,\" Commander Bitter said.\n\n\"That's the whole idea,\" Canidy replied. He turned to face Darmstadter. \"What I want to find out is whether a pilot with about your level of skill can be taught to land and take off from a dirt runway with a stream running through the middle of it.\"\n\n\"Sir?\"\n\n\"It'll be two or three days before we go,\" Canidy said, \"time enough for Commander Dolan to check you out in the B-25. That is, presuming you're still an eager volunteer? \"\n\n\"Sir, I'm still confused,\" Darmstadter said.\n\n\"But maybe you've heard enough to rethink a little? Reconsider volunteering? If you want to walk, you can walk right now. No hard feelings, and no black marks on your record.\"\n\n\"You aren't pulling my leg, are you, Major?\" Darmstadter said. \"You're making a joke of it, but you really meant everything you said, didn't you?\"\n\nCanidy nodded.\n\n\"And that's all I'm going to be told, isn't it?\"\n\nThe major nodded again.\n\n\"In or out, Darmstadter?\" Canidy asked. \"It's up to you.\"\n\n\"In, Sir,\" Darmstadter said.\n\n\"Commander Dolan,\" the major said, \"may I suggest we follow that delightful naval custom of splicing the main brace to welcome a new officer to the wardroom?\"\n\n\"Aye, aye, Sir,\" Commander Dolan said, and took a bottle of bourbon from a file cabinet.\n\n\"For Christ's sake,\" Commander Bitter said, \"it's half past ten in the morning!\"\n\n\"I'm Joe Kennedy,\" the third naval aviator said to Darmstadter, offering his hand. The gold letters below the aviator's wings on the leather patch sewn to his flight jacket identified him as LT. J. P. KENNEDY, JR., USNR. \"It's a little crazy around here, but you get used to it.\"\n\nDolan passed around glasses that had once contained Kraft cheese spread. They now held a good two inches of the bourbon. Commander Bitter shook his head but took one.\n\nCanidy took a small swallow of the whiskey.\n\n\"Rule One around here, Darmstadter,\" he said, \"is that you don't write home to Mommy about what you're doing or what you've seen. And you don't tell your pals, either. The Second Great Commandment is like unto the first. You don't ask questions. But before we put that into effect, you can have one question.\"\n\nThere were at least a dozen questions spinning around in Darmstadter's mind. He was surprised at the one he blurted:\n\n\"Why are the tops cut off those B-17s?\"\n\n\"That's not the question I expected,\" Canidy said. \"I thought you'd ask what's going on around here. Then I would have told you that you have just joined the OSS on a probationary status. If you turn out, you'll join the OSS's private air corps. If you don't . . . you won't like what will happen if you don't. Not a threat, a statement of fact.\"\n\nDarmstadter had heard about the OSS. Very hush-hush, involved in all sorts of things involving espionage and sabotage and dropping agents behind enemy lines.\n\nCanidy saw the shock on Darmstadter's face and smiled.\n\n\"As far as the B-17s are concerned,\" Canidy went on, \"what we're trying to do with them is turn them into radio-controlled flying bombs. We fill them with an English explosive called Torpex. Then Joe gets in, fires it up, and takes off. We cut the roof off so he can bail out. The plane is then flown to the target by radio control. If we can get the sonofabitch to work twice in a row, we're going to fly one into the German submarine pens at Saint-Lazare. So far we haven't been able to get it to work twice in a row.\"\n\nDarmstadter looked into Canidy's face and saw that he had been told the truth. \"You've had your question,\" Canidy said. \"I answered it. That's all you get.\"\n\n\"I understand, Sir,\" Darmstadter said seriously.\n\nThe door of the Quonset creaked again as it opened. Darmstadter saw an enormous Packard limousine sitting outside. It had been adopted for military service by having a serial number stenciled onto the hood and the words U.S. ARMY on the doors. But it still looked, Darmstadter thought, as if it should be rolling up to Buckingham Palace and not a Quonset hut in a B-17 graveyard.\n\nA tall, attractive woman wearing the uniform of a sergeant of the Women's Royal Army Corps came in. The uniform was of rough woolen material and ill-fitting, but it did not hide the fact that beneath it was a very well set-up female, indeed.\n\nShe looked curiously, hesitantly, at Darmstadter.\n\nIn the prescribed British manner, the WRAC sergeant came to stiff attention and stamped her foot.\n\n\"Sir,\" she said to Canidy. \"Sorry to be late, Sir. There was a dreadful smash-up on the way.\"\n\n\"It's all right, Agnes, he's now one of us. Lieutenant Darmstadter, Sergeant Agnes Draper.\"\n\n\"Hello,\" Sgt. Draper said. Her smile was dazzling.\n\n\"To answer your unspoken question, Commander Bitter, \" Canidy said dryly. \"Yes, Sergeant Draper and I can find time in our busy schedule to take lunch with you. And how lucky for you both that I have just given Darmstadter the 'no questions allowed' speech.\"\n\nCommander Bitter's face tightened in anger. Commander Dolan and Lt. Kennedy laughed. Sgt. Draper blushed.\n\n\"Damn you, Dick,\" Sgt. Draper said.\n\n\"Military courtesy around here, you may have noticed, Lieutenant Darmstadter, is sometimes a bit lax. In the future, Sergeant Draper, you will make that 'damn you, _Sir._ ' \"\n\n\"Oh, go to hell,\" she said, but she smiled at him.\n\n#  **4**\n\n##  **PETTY OFFICERS' CLUB NAVY YARD, WASHINGTON, D.C. 2130 HOURS 7 FEBRUARY 1943**\n\nRadioman Second Class Joe Garvey, USN, moved his beer glass in little circles on the bar, spreading the little puddle of condensation in ever-larger circles. Joe Garvey was more than a little drunk. He had been drinking in the petty officers' club since half past five, when he'd come to the club from the petty officers' mess. And he was not used to drinking. Sometimes, out at Mare Island, after he'd made radioman third, he had a beer. It was bad enough in boot camp being a skinny little guy with glasses who had never been afloat on anything bigger than a whaleboat, without getting the reputation for being a teetotaler too. Real sailors drank. It was as simple as that.\n\nJoe Garvey hadn't wanted to be a radioman when he joined the Navy. He had wanted to go to sea as maybe a gunner on a twin-Bofors 20mm, something like that, maybe on a destroyer. Maybe even in a submarine. If he had known more about the Navy, he would have kept his mouth shut about having a ham license. But he'd been a boot, and when they'd asked him, he'd told about being a ham. So they gave him a code test, at twenty words per minute, and he'd flown through that; he'd been copying forty words a minute since he was fifteen.\n\nSo he'd gone right from Great Lakes Naval Training Station to Mare Island as a radioman striker\u2014a USN enlisted man working to qualify for a rating\u2014instead of going to sea. And they'd made him seaman first and given him the exam for radioman third, and he'd passed that with a 98.5. And then he'd been on the next promotion list. And six months after that, he'd made 97.4 on the exam for radioman second.\n\nAnd when he'd asked his chief about maybe getting sea duty, his chief told him the Navy needed him right where he was; there weren't all that many guys around who could handle a key the way he could; and it made more sense to have the best operators in an important commo center, rather than afloat, where they might average maybe fifteen minutes a day on the air.\n\nThe first interesting thing that had happened to him since he'd been in the Navy was the Chief coming to him and telling him to pack his gear, that he'd been placed on TDY to Washington, and that they were holding the courier plane for him.\n\nA couple of times at Mare Island, when he couldn't think of a way to get out of it, he'd sometimes had two beers, or even three, but he was not used to just sitting at a bar and drinking one beer after another.\n\nThey had been treating him real well at the Navy Yard. Instead of what he expected\u2014a bunk and a wall locker in one of the big bays reserved for in-transit white hats\u2014he had a private room, with a desk and even a telephone.\n\n\"These are chief's quarters,\" the master-at-arms had told him. \"If anybody asks what you're doing in them, you tell them to see me.\"\n\n\"What am I doing in them?\" Garvey had asked.\n\n\"Let's just say that's where Chief Ellis said to put you,\" the master-at-arms said.\n\n\"What about formations?\"\n\n\"You don't have to stand no formations,\" the master-at-arms said. \"All you got to do is be available, in case they need you. You can go anywhere you want to go, so long as there's a telephone where you're going and I know where you are and what the number is\u2014and you can get back here in thirty minutes. You want to go get your ashes hauled, Garvey, just make sure she's got a telephone and that you'll be able to pull your pants on and get back here in thirty minutes.\"\n\nJoe Garvey had not been summoned, and neither had he gotten his ashes hauled. The truth of the matter was that they had shown him a Technicolor movie in boot camp that had scared the hell out of him. Guys with balls as big as basketballs, and guys with their dicks rotting off. And the chief who had given that lecture had said that if you didn't want to get promoted and wanted to spend the rest of your time in the Navy cleaning grease traps or chipping paint, catching a dose of clap was a good way to do that.\n\nThe smart thing to do, the chief had said, was to keep your pecker in your pocket and wait until you got home and could stick it in some nice, clean, respectable girl you knew wasn't going to give you nothing that would fuck up your life permanently.\n\nThere were a couple of nice girls Joe Garvey knew back in Louisville, but none who had given him any hint that they would go to the movies with him, much less let him do _that_ to them, but he had decided to keep his pecker in his pocket anyway. He didn't want his dick rotting off before he had a chance to use it.\n\nAnd he wanted to get promoted. He was already a petty officer second, and if you were a skinny little shit who wore glasses, he knew that was a good thing to be. What he had wanted most out of life, at least until they'd put him on a plane at Mare Island and flown him here, was to make chief radioman. That wasn't such an impossible dream. Not only was he one hell of a radio operator\u2014he could knock out fifty words a minute and read sixty\u2014but he _knew_ about radios.\n\nThere were a lot of radiomen who were good operators, and there were a lot of radiomen who were good technicians, but there weren't all that many who were both. Since the Navy wasn't going to send him to sea, the next best thing was to make chief radioman. Nobody would believe that a chief radioman had never been to sea. Or if that came out, people would understand that the Navy had its reasons for keeping him ashore. If he was a chief, it wouldn't matter that he was a skinny little shit who wore glasses. A chief was a chief, period.\n\nAnd making radioman first was going to be easier than he had thought it would be. He was going to go back to Mare Island when they were through with him with a letter of commendation from a goddamned Navy captain.\n\n\"Makes you sound like John Paul Jones, Garvey,\" Chief Ellis had told him. \"I know, 'cause I wrote it.\"\n\nThe next time the promotion board sat, he was probably going to be the only radioman second going for first with a letter of commendation like that. He had already taken the radioman first examination, and he'd made 91.5. If he just kept his mouth shut, he was going to make radioman first, and a little later, he would make chief radioman.\n\nBut that was no longer good enough. He didn't want to sit out the war in the commo section at Mare Island. He wanted to get into the war. When somebody asked him, later, what he'd done in the war, he didn't want to have to tell them he'd been at Mare Island, period.\n\nAnd he thought he had figured out what to do about it.\n\n\"Fuck it!\" Radioman Second Joe Garvey said aloud, which made the bartender look at him strangely.\n\nThen he got off the bar stool, shrugged his arms into his peacoat, put his hat at a jaunty angle on his head, and walked, somewhat unsteadily, out of the bar of the petty officers' club.\n\nHe didn't stop to pick up his Liberty Card. He was afraid the master-at-arms would smell the beer on him and not give it to him. He had been given an \"any hour in and out\" duty card, which would get him past the Marine MP at the gate.\n\nAs he went through the gate, a taxicab rolled up and an officer got out. Joe Garvey saluted and got in.\n\n''Q Street, Northwest,\" he ordered. \"I'll show you where.\"\n\nOn the way, he fell asleep, and the cabdriver had to stop the cab and reach in the back and shake him awake when they were on Q Street.\n\n\"Further down,\" Joe told him, and the cab drove slowly down the street until Joe recognized the brick wall.\n\n\"Right there,\" he said, and handed the cabdriver a five-dollar bill. \"Keep the change.\"\n\nHe had almost made it to the door in the gate when a large man in a heavy overcoat appeared out of nowhere.\n\n\"Hold it right there, sailor!\"\n\n\"It's all right,\" Garvey said. \"I'm to report to Chief Ellis.\"\n\n\"You missed him, then,\" the man said. \"He left an hour ago.\"\n\nAnother, equally burly man appeared.\n\n\"What have you got, Harry?\" he asked.\n\n\"I got me a drunken sailor,\" the first man said. \"The sonofabitch can barely stand up.\"\n\n\"Fuck you,\" Joe Garvey said.\n\n\"I got me a belligerent drunken sailor,\" the man said, laughing. He put his hand on Garvey's arm.\n\n\"What the hell do we do with him?\"\n\n\"I'll take him inside and ask the duty officer,\" the first man said. \"He says he's supposed to report to Ellis.\"\n\n\"Kid,\" the second man said. \"I think you just fucked up by the numbers.\"\n\nThe first man, firmly gripping Garvey's arm, propelled him a hundred yards farther down the street, then through the automobile gate to the property, then up the drive, and finally into the kitchen.\n\nJoe Garvey recognized the two men in shirtsleeves sitting at the kitchen table drinking coffee. As well as he could, he came to attention and saluted. \"Sir,\" he said (it came out \"Shir\"), \"Radioman Second Class Garvey, J., requests permission to speak to the captain, Sir.\"\n\n\"What have we here?\" 1st Lt. Horace G. Hammersmith, Signal Corps, U.S. Army, asked, smiling.\n\n\"He just got out\u2014fell out\u2014of a cab,\" the burly man said.\n\n\"Garvey, my boy,\" Capt. James M. B. Whittaker said, \"if one didn't know better, one would suspect that you have been communing with John Barleycorn.\"\n\n\"You know him?\" the burly man asked.\n\nWhittaker nodded.\n\n\"Sir, I wish to volunteer,\" Garvey said, very thickly.\n\n\"Volunteer? For what?\"\n\n\"You're going into the Philippines,\" Garvey said. \"I want to go with you.\"\n\n\"So much for the big secret,\" Lt. Hammersmith said, chuckling.\n\n\"You're drunk, Garvey,\" Whittaker said.\n\n\"No, I'm not,\" Garvey said righteously.\n\n\"I'll take care of Garvey,\" Whittaker said. \"Thank you.\"\n\n\"I don't know, Captain,\" the burly man said. \"I think I better see what the duty officer has to say.\"\n\n\"Hey,\" Whittaker said, smiling, but with a layer of steel just beneath the surface. \"I said, I'll take care of Garvey.\"\n\n\"Not only am I a much faster operator than the lieutenant, \" Garvey said, \"but you'll be working a Navy net\u2014\"\n\n\"Garvey!\" Whittaker said, sharply.\n\n\"Shir?\"\n\n\"Sssshhhh,\" Whittaker said.\n\n\"Yes, shir,\" Garvey said obediently. Hammersmith laughed. Garvey looked at him with hurt eyes.\n\n\"That will be all, thank you,\" Whittaker said to the burly man.\n\n\"You understand, Captain, that I'll have to make a report of this,\" the burly man said.\n\n\"You just report that you turned him over to me,\" Whittaker said evenly. \"Okay?\"\n\n\"Yes, Sir,\" the burly man said after a moment's hesitation. Then he left the kitchen.\n\nGarvey was making a valiant and unsuccessful effort to stand at attention. He swayed.\n\n\"If I may make a suggestion?\" Lt. Hammersmith said.\n\n\"By all means,\" Capt. Whittaker said.\n\n\"Why don't we each take one arm and guide him to a place of rest? Before he falls down, I mean?\"\n\n\"Splendid suggestion, Lieutenant,\" Whittaker said, as he made for Garvey.\n\nThey had just about made it to the kitchen door when it swung inward and Cynthia Chenowith came in.\n\n\"What in the world?\" she demanded.\n\n\"You remember Garvey, of course, Cynthia?\" Whittaker said.\n\n\"He's drunk!\" Cynthia said.\n\n\"Didn't I tell you Cynthia was perceptive?\" Whittaker said.\n\n\"What's he doing here?\" Cynthia said. \"Where are you taking him?\"\n\n\"We're putting him to bed,\" Hammersmith said.\n\n\"Not here, you're not,\" Cynthia said. \"I'm going to get Chief Ellis back here and let him handle this.\"\n\n\"Don't be a bitch, Cynthia,\" Whittaker said. \"Make a real effort.\"\n\n\"Now, just a minute, Jimmy!\" Cynthia said.\n\n\"Cynthia?\" Whittaker said.\n\n\"What?\"\n\n\"Sssshhhh,\" Whittaker said, and by that time, Whittaker and Hammersmith were through the kitchen door, with Garvey more or less suspended between them.\n**VI**\n\n#  **1**\n\n##  **FERSFIELD ARMY AIR CORPS STATION BEDFORDSHIRE, ENGLAND 0615 HOURS 12 FEBRUARY 1943**\n\nCanidy was late. He had been expected at 0600. And Lt. Hank Darmstadter had been waiting to go since he had awakened, after a restless night, at quarter to four. When he looked out the window, there was thick fog, so thick that flight in his Troop Carrier Squadron would not even have been considered. It was likely that the fog would keep them from flying, but there was no one at 0345 whom he could ask.\n\nDolan knocked on Darmstadter's door at 0500 and seemed surprised to find him wearing the high-altitude flight gear over his uniform.\n\n\"Why don't you leave that sheepskin gear here?\" Dolan suggested. \"I thought we'd ride over and get breakfast in the Air Corps mess.\"\n\nDolan ate a hearty, air-crew-about-to-go-on-a-combat-mission breakfast, complete with real eggs and a slice of ham. Darmstadter's Troop Carrier Squadron had not gone on combat missions and consequently had been issued no fresh eggs, so they should have been a real treat. But he was so nervous he had no appetite, and he ate them only because he told himself he needed the nutrition.\n\nA jeep took them to the revetment where the B-25G had been readied for flight. Dolan made a careful, if leisurely, preflight examination of the aircraft, then hoisted himself onto the hood of the jeep and waited for Canidy to show up.\n\n\"You think we're going to go, Commander?\" Darmstadter asked. When Dolan's eyes rose in question, Darmstadter added, \"The fog?\"\n\n\"What I'm wondering about is where's Canidy?\" Dolan said.\n\nFor lack of some better way to kill time, Darmstadter walked around the airplane again. Knowing that he was not only to be checked out in the B-25G but that they were about to make a long-distance flight in it, Darmstadter had studied at length and with great interest _TM 1-B-25-G, Flight Operations Manual, B-25G (Series) Aircraft._\n\nHe had realized the moment Commander Dolan had taken him out to the airplane for his first ride that most of his dedicated study had been a waste of time.\n\n\"You'll notice,\" Dolan had told him, \"that we've modified this one a little.\"\n\nIt was a massive understatement.\n\nThe B-25G had been delivered to the Eighth Air Force with a twin .50-caliber machine-gun position in the tail; with another pair of .50s in a rotating turret on top of the fuselage at the leading edge of the wing; with two single .50-caliber machine-gun positions\u2014\"waist guns\"\u2014in the sides of the fuselage; and with two fixed .50s and a 75mm M4 cannon in the nose.\n\nAll of the guns had been removed and their positions faired over. The bomb-dropping racks and mechanism were gone, and the bomb-bay doors were riveted permanently closed. Auxiliary fuel tanks had been installed in what had been the bomb bay, where the bombs were supposed to be.\n\nIn the fuselage aft of the trailing edge of the wing, where the radio operator's and waist gunner's positions had been, there were now five\u2014as many as would fit\u2014light brown leather civilian airliner passenger seats.\n\nThe seats had been \"salvaged,\" Dolan told Darmstadter, from a U.S. Navy Boeing \"Strato-Cruiser\" transport, that Canidy had \"dumped in Africa.\"\n\nDarmstadter was very curious to learn more about that, but he had come to understand that while Major Canidy and the others seemed to make jokes about everything else, Canidy had been dead serious about the \"Ask No Questions\" rule.\n\nDolan had given Darmstadter seven hours of in-flight instruction in the B-25G, which was really more than it sounded like, because with the exception of the first takeoff and landing, Dolan had never touched the controls again.\n\nSomewhat to Darmstadter's surprise, he had been an apt pupil. Dolan's only criticism had come right at the start, \"Don't try so hard. It's not that hard to fly, and you're a better pilot than you think you are.\"\n\nHe had made mistakes, of course, but after Dolan had shown him what he was doing wrong, he had not made that particular mistake again. He had had the most trouble, not surprisingly, in landing. The B-25G came in a lot hotter than the C-47, and if the power settings were not right on the mark, it dropped like a stone. The Gooney Bird was a very forgiving aircraft; the B-25 was not.\n\nBut he'd shot hour after hour of touch-and-go landings until his technique satisfied Dolan. Then he'd spent another two hours trying to touch down right at the end of the runway and to bring it to a complete stop as quickly as possible. He was aware that he had not been able to accomplish that to Dolan's satisfaction. And he was embarrassed about that, even after he told himself that he should not be. What Dolan was asking would have been difficult for a good, experienced pilot, and he knew he was neither.\n\nThey heard the crunch of automobile tires a minute before they could see the glow of headlights in the fog. But then the distinctive grille of the Packard limousine appeared.\n\n\"I stopped to get the latest forecast,\" Canidy said by way of greeting. \"I presume that the rubber bands are all wound up and we can go?\"\n\n\"It'll take five minutes to light the runway,\" Dolan said.\n\n\"It'll take that long to warm it up,\" Canidy said. \"Tell them to light it.\"\n\nDarmstadter was confused by that. There were no landing field lights at Fersfield. If there were, he thought, he would have seen them.\n\nCommander Bitter and Lt. Kennedy drove up in a jeep.\n\n\"I would suggest that you wait until you've got at least a thousand feet,\" Bitter said. \"But Weather says it's going to be this way until noon, maybe later.\"\n\n\"I think we can get off,\" Canidy said. He turned to Darmstadter. \"Get aboard, Darmstadter,\" he said. \"Strap yourself in the seat that faces backward.\"\n\nThen he gestured for Dolan to precede him aboard. It was more than a gesture of courtesy, Darmstadter saw. He was telling Dolan that Dolan would function as aircraft commander.\n\nAs Darmstadter was strapping himself in, Canidy appeared momentarily in the cabin to wedge a canvas Valv-Pak between one of the seats and the fuselage ribs. Then he disappeared. The plane shook as the left engine started to turn, then caught.\n\nFrom where he was sitting, Darmstadter could look out the small window where the waist gunner position had been faired over with Perspex. Though he couldn't see much, he did see Sgt. Draper standing beside Commander Bitter, both of them with their hands raised in farewell. And then there was nothing to see but the edge of the taxiway as the B-25G trundled to the threshold of the runway. Then he saw a fire at the end of the runway. He unstrapped himself for a better look, and saw that it was a GI can\u2014a No. 10 tin can\u2014and that the fire burning in it was gasoline. Pressing his head against the Perspex, he looked as far as he could down the runway. It was lined at fifty-foot intervals with flaming GI cans.\n\nHe realized that the burning sand-and-gasoline-filled cans were the runway lights Canidy and Dolan had been talking about. They would not \"light\" the runway, in the sense of illuminating it, but they would provide an indication where the runway was. He quickly counted cans. He got to fourteen. That meant seven hundred feet. Not nearly enough to take off.\n\nAnd at that moment, having completed the run-up of engines, the B-25 started to move.\n\nAs Darmstadter watched with something approaching terror, the dull glow of another burning can appeared through the fog, and then another. Despite the thick fog, he realized, it would be possible to take off by staying on the runway between two lines of burning GI cans.\n\nAnd then the rumbling of the undercarriage suddenly stopped. A moment later the nose of the B-25 lifted, so steeply that he fell against the seat that he was supposed to be strapped into, and he heard the whine of the hydraulics as the gear was retracted.\n\nThe reddish glow of the burning cans disappeared; there was nothing whatever to be seen through the Perspex window now but gray.\n\nDarmstadter found the heavy sheepskin flying gear, put it on, and plugged it in. Then he put earphones over his ears and adjusted the oxygen mask, with its built-in microphone, over his lower face.\n\n\"Do you read?\" he asked.\n\n\"We have been calling you, Lieutenant,\" Canidy's dry voice came through the earphones, \"with no response. We thought maybe you'd had a last-minute change of heart.\"\n\n\"Sorry, Sir,\" Darmstadter said. \"I was putting on the sheepskins.\"\n\n\"We're passing through eight thousand,\" Canidy said. \"I'll let you know when we pass through ten. Make sure the oxygen is working.\"\n\nDarmstadter opened the valve and felt the cold oxygen in his nostrils and throat.\n\n\"Oxygen okay,\" he said.\n\n\"Couple of things,\" Canidy said. \"Make sure you've got a walk-around bottle and a spare. We're going way up, so stay on oxygen.\"\n\n\"Yes, Sir.\"\n\n\"If you feel like it,\" Canidy went on, \"and it might be a good idea, move around a little. Wave your arms, bend your legs. But don't work up a sweat. If you do that, the sweat will freeze and weld your skin to the oxygen mask. Then it will smart when you try to take it off.\"\n\n\"Yes, Sir,\" Darmstadter said, chuckling.\n\n\"And stop calling me 'Sir,' \" Canidy said.\n\nIt grew colder very quickly as the B-25 maintained its climb.\n\nAnd by the time the B-25 leveled off, and the sound of the engines changed as they throttled back and leaned off for cruising, it was bitter cold in the fuselage, and the bulky, sheepskin, electrically heated flying suits and boots did not provide comfort, only protection from frostbite and freezing.\n\nEvery fifteen minutes or so, Darmstadter got out of the leather-upholstered, civilian airline seat and, within the limits of movement the flexible oxygen hose gave him, stamped his feet and flailed his arms around. Carefully, for he believed what Canidy had said about working up a sweat and freezing the mask to his face.\n\nThey had been airborne an hour when Canidy came over the intercom and asked him to bring up some coffee. Darmstadter hooked up a portable oxygen bottle and found the wooden crate that held two narrow-mouthed stainless-steel thermos bottles of coffee and one much larger, wide-mouthed thermos holding sandwiches in waxed paper. He took one of the thermos bottles and two china mess-hall cups forward.\n\nHe poured coffee and handed a mug to Canidy, who indicated with a jerk of his thumb that it should go to Dolan. Dolan took it, moved his mask away for a moment, sipped the coffee, and then put the mask back on.\n\n\"Shit,\" his voice came over the earphones. \"Burned my fucking lip!\"\n\nDarmstadter glanced at the altimeter, then looked at it again, more closely, to be sure he had read it right. It indicated 27,500 feet, which was three thousand five hundred feet higher than the \"maximum service altitude\" for a fully loaded B-25G, according to _TM 1-B-25-G_.\n\nHad Canidy rigged the engines so they would function at that altitude? he wondered. Or was the greater altitude possible because the weight of the guns and the parasitic drag of their turrets and mounts was gone?\n\nThen he thought that the only thing he knew for sure to explain what he was doing at 27,500 feet over the Atlantic Ocean was that they were headed for an island called Vis. He had a hundred questions in his mind about that, including how come there was a landing field in an area shaded in red\u2014indicating \"enemy occupied\"\u2014on every map he had ever seen of the Adriatic area.\n\nAnd, of course, there was the big question: Why had they picked a C-47 pilot with a mediocre record like his to go along? It was almost impossible to accept the reason Canidy had offered, that they wanted to see if a pilot of his skill level could manage a takeoff and a landing on a strip that had a stream running through the middle of it.\n\nCanidy surprised him by getting out of the copilot's seat and motioning him into it, then pointing to the altimeter, then handing him the chart.\n\nThat was the first time he'd seen the chart. They had politely but carefully kept him from seeing it before they'd left. Dolan had even kept him from attending the final weather briefing at Fersfield by going there before he came to Darmstadter's room to wake him up.\n\nThe chart for the first leg of the flight showed a course leading out to sea in a general south-southwest direction so they would pass no closer than two hundred miles to the coast of France. Then it turned southeast, with Casablanca, Morocco, as their destination.\n\nThere were cone-shaped areas drawn on the chart, the small end in France, the wide end over the Atlantic. Canidy explained that they indicated the normal patrol areas for German Messerschmitt ME109F fighters, based in France. There were larger cones, which Canidy identified as the patrol areas for German Heinkel bombers used as long-range reconnaissance aircraft. The larger cones covered much of the B-25's projected route.\n\n\"The theory,\" Canidy said dryly, \"is that the Heinkels fly at about ten thousand feet, which gives them their best look for convoys and the best fuel consumption. And we hope that if one of their pilots happens to look up here and see us, he will decide that prudence dictates he keep looking for ships.\"\n\n\"But what if one of them sees us?\"\n\n\"We have two defenses,\" Canidy said. \"We're a little faster. If that doesn't work, Brother Dolan will lead us in prayer.\"\n\n\"We're faster because you removed the guns? That weight is gone?\" Darmstadter asked.\n\n\"The weight, sure, but primarily because of the parasitic drag,\" Canidy said. \"By taking the two turrets out of the slipstream, we picked up twenty knots at twenty thousand feet. We got another five or six knots when we faired over the waist-gun position. We can go either faster or farther at the same fuel-consumption rate.\"\n\n\"Clever,\" Darmstadter said. \"The engineers obviously knew their stuff.\"\n\n\"Thank you,\" Canidy said, smiling.\n\n\"You did it? You're an engineer?\" Darmstadter blurted, remembering as he spoke that it was a question and questions were against the rule. But Canidy didn't jump on him.\n\n\"You will doubtless be awed to hear that you are dealing with R. Canidy, BS, Aeronautical Engineering, the Massachusetts Institute of Technology, '39.\"\n\nDarmstadter bit off just in time the question that popped to his lips: \"How'd you get involved in something like this?\"\n\nHe was beginning to understand that there were questions he could ask, but that asking personal questions was taboo.\n\nThe answer, anyway, seemed self-evident. Whatever the OSS really did\u2014some of the stories he'd heard about the OSS simply couldn't be true\u2014it obviously had a high priority for personnel and equipment. The big brass had apparently decided that an MIT-trained aeronautical engineer could do more good working and flying for the OSS than he could, say, as a maintenance officer in a troop carrier or heavy bombardment wing.\n\nCanidy connected a portable oxygen bottle to his face mask, then went into the cabin. Ten minutes later, he returned.\n\n\"I'll sit there awhile, John,\" he said to Dolan, motioning him out of the pilot's seat. \"Take a nap.\"\n\nWhen Dolan had hooked up a portable oxygen mask and gone back into the fuselage, Canidy's voice came metallically over the intercom.\n\n\"Dolan's a hell of a fine pilot,\" he said. \"He was a gold-stripe chief aviation pilot before the war.\"\n\nDarmstadter had heard that both the Navy and the Marines had enlisted pilots in peacetime, and the legend was that they were better pilots than most of the officers because all they did was fly.\n\n\"And then he got a commission?\" Darmstadter asked.\n\n\"No,\" Canidy said. \"First, they took him off flight status. Bad heart. Then he got out of the Navy and went to China with the American Volunteer Group as a maintenance officer. _Then_ he got a commission.\"\n\n\"But he's flying!\"\n\n\"How Commander Dolan passed a flight physical, Darmstadter, is one of those questions you're not supposed to ask,\" Canidy said. \"When you were in preflight, and they were giving you those fascinating lectures on military tactics, did they touch on 'conservation of assets'?\"\n\nDarmstadter thought about it, then shook his head.\n\n\"I don't remember,\" he said.\n\n\"What you're supposed to do, if you're a general or an admiral and about to enter battle, is decide what 'asset' you absolutely have to have if things get tough. Then you squirrel that asset away so it's ready when you need it. I just sent my asset back for a nap. If anybody can sit this thing down safely on a mountain strip with a stream running across the runway, Dolan can. You follow?\"\n\n\"Yes, Sir,\" Darmstadter said. He was more than a little uncomfortable. Canidy was obviously a highly skilled B- 25 pilot and comfortable doing things with it that most people would not try (his solo flight of the B-25 through the soup the day Darmstadter had first met him was proof of that). And he had just admitted that he didn't think he could make the landing on the island of Vis.\n\n\"There is an additional problem,\" Canidy said. \"Commander Dolan thinks he is still twenty-two years old and that the doctors are dead wrong about the condition of his heart. He will take affront unless handled properly. Kid gloves are required.\"\n\n\"I understand, Sir,\" Darmstadter said.\n\n\"And I told you before, stop calling me 'Sir,' \" Canidy said.\n\nSix hours and fifteen minutes after taking off from Fersfield, the B-25G landed at Casablanca. Darmstadter made the landing. He had to tell himself there was no reason to be nervous. Landing on the wide, concrete runway of a commercial airport on a bright, sunny afternoon should be a snap, compared to landing on the rough, narrow gravel runways at Fersfield. But he was aware that it was sort of a test. Major Canidy was in effect giving him a check ride to see how well Dolan had done as an instructor pilot.\n\nDarmstadter was enormously pleased and relieved that the landing was a greaser.\n\nA Follow Me jeep, painted in checkerboard black and white and flying an enormous checkerboard flag, met them at the end of the runway and led them away from the terminal to a remote corner of the field. There was an old hangar there with the legend AIR FRANCE barely legible through a layer of rust.\n\nAs they approached, the doors opened and a ground crewman gave Darmstadter hand signals, directing him to taxi to the doorway and then shut it down. The moment the engines died, a dozen Air Corps ground crewmen manhandled the B-25 inside the hangar and closed the doors.\n\n#  **2**\n\n##  **THE MARK HOPKINS HOTEL SAN FRANCISCO, CALIFORNIA 12 FEBRUARY 1943**\n\nIt had been decided in Washington that Whittaker, Hammersmith, and Garvey would spend the night at Mare Island. Cynthia, to avoid the curiosity and comment that a civilian woman in the Mare Island Female Officers' Quarters would cause, would stay in a San Francisco hotel.\n\n\"I know someone who can get you into the Mark Hopkins, \" Jimmy Whittaker had said, innocently, when the issue of where she would stay in San Francisco came up in Captain Douglass's office. \"What the hell, you might as well go first class.\"\n\n\"Go ahead and do it, Jim,\" Captain Douglass had answered for her. \"Hotel rooms are in damned short supply in San Francisco.\"\n\nWhen they arrived in San Francisco, by commercial air, they went first to the hotel. Cynthia's reserved \"room\" turned out to be the Theodore Roosevelt Suite, four elegantly furnished rooms on an upper floor.\n\n\"It was all they had available,\" Jimmy said innocently.\n\nCynthia knew that simply wasn't true. What had happened was that Jimmy had told the hotel something like \"I'd like something very nice for a very good friend of mine,\" and the hotel had come up with the Theodore Roosevelt Suite. The hotel had been very obliging to Jim Whittaker because Jimmy was a very rich man, and the hotel knew it.\n\nJimmy's father and his two uncles had inherited the Whittaker Construction Company from their father. There was more to it than the construction company, though God knew that was enough. The Whittaker fortune was based in railroads. They had built them before the Civil War, and grown very rich during the war building and operating railroads for the Union Army.\n\nAfter the Civil War, there had been more railroads. And harbors, and heavy construction. Whenever they could, which was most often, they took part of their pay in stock of whatever they were building. The company had large real estate holdings in New York City and elsewhere. It was even possible, Cynthia thought, looking around the Theodore Roosevelt Suite, that Jimmy had an interest in the hotel.\n\nJimmy's father had been killed in World War I. And his third of Whittaker Construction had gone to his only son. Both Jimmy's uncle Jack and his uncle Chesty had died childless. Jack Whittaker's third would pass to Jimmy on the death of his widow. Jimmy had already inherited the house on Q Street, Northwest, from Chesty, as well as some other property.\n\nChesty Whittaker, Jimmy's uncle and Cynthia's lover, had told her all about the financial position of James M. B. Whittaker. Not subtly. Chesty had thought she should marry Jimmy.\n\n\"You've got to think of the future, my darling,\" Chesty had said. \"We can't go on.\"\n\n\"Why can't we?\"\n\n\"Well, for one thing, I'm a little long in the tooth. You'll still be a young woman when I am long gone.\"\n\n\"Goddamn you!\" she had screamed. \"This is obscene. You're not going to die, and I'm not going to marry Jimmy. Jimmy's a kid.\"\n\n\"There is only three years' difference\u2014\"\n\n\"Four,\" she had snapped.\n\n\"Four years,\" he'd said. They had looked at each other for a moment, before he went on, \"Presumably, you meant it when you said you didn't want my wife to ever find out about us.\"\n\n\"The way I put it was 'I'd rather die than have her find out,' \" Cynthia had said. \"Yes, of course I meant it.\"\n\n\"The reality of our situation is that you are as poor as a church mouse,\" Chesty had said. \"And what do you think she would think if I made provision for you in my will? In addition to her many other virtues, she is intelligent and perceptive.\"\n\n\"Then don't 'make provision' for me,\" Cynthia had said.\n\n\"I love you,\" he'd said. \"I could not not do that.\"\n\n\"And the convenient way to do it is to marry me off to Jimmy? Damn you, Chesty.\"\n\n\"Jimmy stopped off here on his way to Randolph Field,\" Chesty Whittaker had said. \"He said that it was his intention, when he graduated, to ask you to marry him, and what did I think of that?\"\n\n\"What did you say?\" she'd asked.\n\n\"I told him I thought it was a splendid idea,\" Chesty'd said. \"Actually, what I said, making my little joke, was 'name the first son after me.' \"\n\n\"Oh, damn you!\" she'd said, and she'd started to cry, and he'd held her.\n\nThree months after that happened, Chesty Haywood Whittaker had dropped dead. And he had not made provision for her in his will, and she was as poor as a church mouse.\n\nCynthia decided not to make an issue of the Theodore Roosevelt Suite. It would be pointless to protest, for one thing, and for another, it wasn't as if there was a suggestion he would share it with her. He had just made a generous gesture. In the family tradition, she thought. In many ways, Jimmy reminded her of Chesty.\n\nThe Navy sent a Plymouth staff car to carry them from the Mark Hopkins to Mare Island. Waiting for them in a hangar there, guarded by a platoon of Marines under a gunnery sergeant, was a five-foot-high stack of wooden crates that would at 0500 the next morning be loaded aboard the Naval Air Transport Service Douglas C-54 that would carry them to Pearl Harbor in the Hawaiian Islands.\n\nJimmy, very seriously, ordered Radioman Second Class Joe Garvey to take charge of the guard detail. Cynthia had to restrain a smile at the slight sailor's obvious feeling of importance at being given the responsibility.\n\nGarvey's status was still undecided. Since he had correctly deduced that Whittaker and Hammersmith were going into the Philippines, he could not be simply returned to duty. But on the other hand, it had not been decided that he would go with Whittaker and Hammersmith. For the meantime, taking him with them to San Francisco and Hawaii would serve two purposes. An extra hand was going to be helpful, and he already knew what was going on. And if he was with them, he was considered to be secure. He could, at any point, be put on ice if it was ultimately decided not to take him to Mindanao.\n\nThey then went to the Mare Island Officers' Club for dinner. Whittaker ordered a steak dinner with all the trimmings to go, and sent their Navy driver to the hangar to deliver it to Garvey.\n\nThere was an orchestra in the club. After dinner, after first, with great mock courtesy, asking Whittaker's permission, Greg asked Cynthia to dance. Whittaker graciously gave his permission, then rose and gave a little bow as Greg led her off to the dance floor.\n\nThen it was Jimmy's turn to dance with her. Thirty seconds after he had put his arms around her, she had felt his erection stabbing at her stomach. He didn't grab her and press her close or try to move his hands so they would come against her breasts, but he had an erection, and it was obvious that he was not only not embarrassed by it, but seemed pleased that she had no choice but to be aware of it.\n\nAnd since she had learned in a class euphemistically called \"Human Hygiene\" in college that the male erection was an \"involuntary vascular reaction,\" she had not been able to tell him to \"stop that.\"\n\nHe held her hand as they returned to the table.\n\nJimmy picked up his glass and, smiling, looked over the rim of it at Greg.\n\n\"I have been thinking, Ronald Reagan\u2014\" he began.\n\n\"I saw the furrowed brow,\" Hammersmith interrupted, \"and it's 'Greg Hammer.' Ronny Reagan is the one they call the 'Errol Flynn of the B movies.' \"\n\n\"Right,\" Whittaker said. \"Hammer, as in the baking soda.\"\n\n\"Now you've got it,\" Hammersmith said. \"What have you been thinking, O worthy leader?\"\n\n\"That despite my initial unflattering impression of you, you may be reasonably trustworthy after all.\"\n\n\"Oh, _thank_ you, Sir.\"\n\n\"To the point where I would feel comfortable in leaving you in sole charge of Radioman Garvey while I escort the lady to her hotel.\"\n\n\"I can get to the hotel by myself,\" Cynthia protested.\n\nThey ignored her.\n\n\"So that you can protect our girl from the unwanted attentions of sailors in the Mark Hopkins?\"\n\n\"Correct,\" Whittaker said. \"I have heard all sorts of tales about sex-starved naval officers making indecent proposals to unaccompanied young ladies such as Miss Chenowith, right in the lobby of the Mark Hopkins.\"\n\n\"We couldn't have that, could we?\" Greg replied. \"You sneaky sonofabitch.\"\n\n\" 'You sneaky sonofabitch, _Sir,_ ' \" Whittaker corrected him.\n\nThe two men, pleased with their own wit, smiled at each other, which infuriated Cynthia.\n\n\"I don't need an escort,\" Cynthia said.\n\n\"The way she says that,\" Whittaker replied, \"you'd think she thinks I have designs on her body, wouldn't you?\"\n\n\"I don't think you're funny, Jimmy,\" she said.\n\n\"Let's go,\" he said. \"We have an early day tomorrow.\"\n\nHe took her arm when he put her into the Navy car, but as soon as Greg had gotten out at the hangar, he slid away from her on the seat, so that their hips were no longer pressing together. And he did not try to hold her hand, put his arm around her, or kiss her on the way to the hotel.\n\nHe did speak to the driver:\n\n\"How are we going to get Miss Chenowith back out to Mare Island in the morning?\"\n\n\"My orders are to stick with you, Sir, until you get on the plane.\"\n\n\"All night?\"\n\n\"Yes, Sir.\"\n\n\"Well, you go get yourself some sleep,\" Whittaker ordered. \"Be at the hotel at 0400. I'll catch a cab back out there tonight.\"\n\n\"Why don't you go back out with him?\" Cynthia asked.\n\nWhittaker ignored her for a moment, then somewhat lamely said, \"I want to check in with Ellis. I'd rather do that from your room than try to get a long-distance authorization at Mare Island or feed quarters to a pay phone.\"\n\nHe might, indeed, actually call Ellis from the Theodore Roosevelt Suite, once he gets there, Cynthia thought, but he obviously just thought up that excuse to get into the room.\n\nThere was also a good chance that the moment he got her behind the closed doors of the Theodore Roosevelt Suite, he would make a play for her, she thought. She really didn't want that. But she didn't want to make an issue of it now. If it happened, she could handle him.\n\nWhen they got to the suite, he went directly to the telephone on the table in front of the couch and put in a call to Chief Ellis in Washington.\n\nHe seemed genuinely disappointed that Ellis was not immediately available.\n\n\"I'm in Miss Chenowith's room in the Mark Hopkins,\" he said to the telephone. \"I'll wait here for his call.\"\n\nHe put the telephone in its cradle.\n\n\"Not there? That's surprising,\" Cynthia said.\n\n\"He would have been there if I had called when I was supposed to,\" he said. \"I didn't even think of calling him until I needed an excuse to be alone with you.\"\n\nShe smiled at him.\n\n_That should have been my cue, she thought, to say something cutting\u2014\"Don't get any ideas, Jimmy,\" something like that. I wonder why I didn't?_\n\nIt was, she decided, because his honesty disarmed her. And then she realized there was more than that. She had tried to force the thought from her mind whenever it had appeared. But that had been hard, and it kept reappearing, as it was doing now.\n\nThe thought was that the clock was running down, like the clock at a basketball game. Very soon, Jimmy and Greg\u2014and maybe even Garvey, whom she thought of as \"the boy in the sailor suit\"\u2014would get on the submarine and try to establish contact with this man Fertig and his guerrillas in the Philippines. There was a very good chance that they would be caught, and if they were caught, they could count on being executed. Cynthia had seen photographs of Japanese executions of Americans. It was done ritually, according to the Japanese warriors' code of Bushido, which prescribed execution by beheading.\n\nAnd this was followed by another thought, alarming in its implications: There seemed to be little morally wrong with going to bed with a man who stood a very good chance of being executed by beheading in the very near future. It seemed little enough to do for him.\n\nBut that presumed he would be executed. Jimmy, God bless him, seemed to have an incredible ability to stay alive. And if he stayed alive, he would be back. And he would interpret her taking him into her bed as a reciprocation of love. And he would want to marry her.\n\nThere were a number of reasons she couldn't marry Jimmy. For openers, she was convinced that the love she felt for him was not the sort of love a woman should have for the man she would marry, whose children she would bear. He was younger than she was. And she had been his uncle's mistress. She sometimes thought that she owed the love she felt for Jimmy simply to his likeness, in so many subtle ways, to Chesty Whittaker. Sometimes, when he looked at her, it was as if Chesty was behind the eyes.\n\nAnd she didn't reciprocate Greg's affection, either. Greg said it jokingly, but she believed that he thought he loved her. And she didn't want to sleep with him, either.\n\nIt would be better all around if she were a slut, she thought every so often. Not an absolute, four-star slut, but just a little bit of a slut, like Charity Hoche. The situation Cynthia found herself in would pose no great problems for Charity. If Charity believed that two men like these, both of them handsome and rich, and head and shoulders above most other men, thought they were in love with her, and if she was as fond of both of them as Cynthia was, Charity would sleep with both of them. One at a time, of course, but with both of them.\n\n\"I think we should talk about Joe Garvey,\" Cynthia said. \"Ellis will want to know when he calls back.\"\n\nWhittaker nodded.\n\n\"On the one hand, you need a backup for Greg,\" Cynthia said, all business.\n\n\"And on the other, Joe Garvey looks and acts as if he should be working the lights for the senior play,\" Whittaker said.\n\nHe walked to the bar and made himself a drink, then returned to the couch and sat down, slumped against the rear cushion, his legs stretched out straight in front of him, holding his glass on his stomach.\n\n\"He's not trained for anything like this,\" Cynthia said.\n\n\"Neither am I, according to good old Eldon Baker,\" Whittaker said.\n\n\"You're going out of your way to be difficult, aren't you?\"\n\n\"I'm about to start,\" he said.\n\n\"Excuse me?\"\n\n\"While I was off in Merry Old England,\" Whittaker said, \"I was fucking a duchess.\"\n\n\"For God's sake, Jimmy!\"\n\n\"Elizabeth Alexander Mary Alexandra, Her Grace the Duchess of Stanfield,\" he said. \"Her family owns Whitbey House. He's in the RAF. Missing in action. I'm sure there is a word for what I was doing. And it was my fault, not hers.\"\n\nHe met her eyes until she averted them.\n\n\"And then, when I was in Cairo, I was fucking another married woman. Her husband was off with Charles de Gaulle and the Free French.\"\n\n\"Why are you telling me this?\" Cynthia asked. \"You think it's funny?\"\n\n\"There's a punch line,\" he said.\n\n\"I don't think I want to hear it,\" she said.\n\n\"I used to ask myself, Cynthia,\" Whittaker said, looking at her, \"sometimes at _very_ inappropriate moments, 'Why are you doing this? If you love Cynthia, why the hell are you screwing somebody else?' \"\n\nHe looked at her as if he expected a response.\n\n\"No answer came, Cynthia,\" he said. \"The conclusion to be drawn, therefore, is that I am an unprincipled sonofabitch. \"\n\n\"Another possibility is that you don't really love me,\" she said. \"Not that way. For God's sake, Jimmy, we have known each other since we were kids. I used to take care of you when you were a little boy.\"\n\n\"I have loved you since you were about fourteen,\" he said, matter-of-factly. \"You were climbing out of Chesty's pool in Palm Beach, and I got a look down your bathing suit. My heart stopped, and then jumped. My heart still stops and then jumps sometimes when I look at you. What this equation means, I'm afraid, is that I do in fact love you. _That way._ \"\n\n\"What about Garvey?\" she said.\n\nWhittaker nodded his head as if he expected not only her change of subject but even that particular question.\n\n\"She said,\" he said, \"changing the subject.\"\n\nHe drained his drink, then stretched across the couch to put the empty glass on a table.\n\n\"I'm not going to let you off the hook there, Cynthia,\" he said, and started to cross the room to the bar.\n\n\"What the hell is that supposed to mean?\"\n\n\"There's more to playing Mata Hari, my dear Cynthia, than running around the woods in Virginia with a rifle, or flashing your OSS credentials to impress people.\"\n\n\"Now, that's a cheap shot!\"\n\n\"It involves things like making decisions,\" he said. \"For example, 'Do I send a nice little boy in a sailor suit off someplace where he is liable to drown, or have his head sliced off with a sword?' \"\n\n_My God, he's seen those pictures! He knows what he's getting himself into. He's frightened!_\n\nHe looked at her out of Chesty's eyes.\n\n\"Goddamn you!\" she said.\n\nHe didn't reply. He walked back to the couch and sat down.\n\nShe felt a sudden infuriating urge to cry. She fought it down, then went to the bar and poured an inch of brandy into a snifter.\n\nShe wondered why Whittaker was being such a sonofabitch about Garvey. Why he didn't just say, \"We'll take him,\" or \"We better not take him.\" He damned well was equipped to decide whether the contribution Garvey could make to the mission overrode his youth, and inexperience, and lack of training, and, for that matter, physical stamina.\n\nThat's what had to be judged. Whether Garvey was drowned or beheaded was important only insofar as it would affect the mission.\n\nClearly, Garvey should go. Why had Jimmy been unwilling to come out and say that?\n\nBecause, she suddenly understood, he was being a sonofabitch again\u2014a _male_ sonofabitch. He was simply unable to understand that she thought as he did. He still thought she was playing at being a spy; the bastard had even called her \"Mata Hari\" and accused her of flashing her OSS credentials to impress people.\n\n_Goddamn him!_\n\n\"Garvey will go,\" she announced.\n\nHe nodded.\n\nTheir eyes met.\n\n\"If I asked you a straight question, could I have a straight answer?\" Cynthia heard herself ask.\n\n\"That would depend on the question,\" he said.\n\nThe telephone rang. It was Ellis.\n\n\"Sorry I didn't call earlier, Ellis,\" he said. \"I just forgot.\"\n\nHe reported that the material was on hand, that the weather was good, and unless Ellis heard to the contrary, they would depart Mare Island for Hawaii on schedule.\n\n\"And we're taking Garvey,\" he concluded. \"Get him transferred officially as soon as you can. Get him overseas pay, and hazardous-duty pay . . . whatever you can.\"\n\nEllis said something else, to which Whittaker replied:\n\n\"Thanks, Chief, I'll damned well try.\"\n\nCynthia knew that Ellis had told him to take care of himself.\n\nWhittaker hung the phone up again.\n\n\"You were asking?\" he said, meeting her eyes.\n\n\"Are you afraid?\"\n\n\"I'll tell you what I'm afraid of,\" he said seriously, after a pause. \"I'm afraid I'll answer that dumb question the wrong way, and that'll give you your excuse to throw me out of here.\"\n\n\"Are you afraid, Jimmy?\" Cynthia asked.\n\n\"This is probably the wrong answer, but fuck it. Truth time. No, I'm not. I'm good at this sort of thing. There's a thrill, Cynthia. It's even better than flying.\"\n\nShe looked at him first in disbelief, then in astonishment when she realized he was telling the truth.\n\n\"The wrong answer, I gather?\" he asked dryly.\n\n\"It wasn't the answer I expected,\" she said.\n\n\"Do I get to stay?\"\n\nShe felt her face flush. She felt faint. There was a contraction at the base of her stomach.\n\nShe forced herself to look at him.\n\n\"If you like,\" she said very softly.\n\nAnd then, more quickly than she would have thought possible, he erupted from the couch and came to her.\n\nEmbarrassed, she averted her face.\n\nHis hand came up, and the balls of his fingers touched her cheek and gently turned her face to his. She met his eyes.\n\nHis fingers moved down her cheek, and down her neck, and onto her shoulders. He buried his face in her hair. She felt his arms around her, pressing her to him, and then felt his body shudder.\n\nAnd then he picked her up and carried her into the bedroom.\n\n#  **3**\n\n##  **ST. GERTRUD'S MUNICIPAL PRISON P\u00c9CS, HUNGARY 12 FEBRUARY 1943**\n\nThere was just barely room enough for the Tatra diesel dump truck to pass through the tunnel to the courtyard of St. Gertrud's. Scrape marks on the granite walls of the tunnel and on the fenders of the truck testified that the drivers didn't always make it through on the first try.\n\nThe Tatra pulled into the courtyard and, with a great clashing of gears and bursts of sooty diesel exhaust, backed up to within ten feet of an interior door.\n\nThe heavy wooden door opened inward and three guards came out. They were middle-aged men in gray uniforms and black boots. Carrying billy clubs and small .32-caliber automatic pistols in closed-top holsters, two of them took up positions facing each other between the truck and the door. The third, holding a clipboard in gray woolen gloved hands, stood to one side by the door. As the prisoners came out of the door and started to climb onto the truck, he checked their names off on a roster.\n\nThe prisoners, of various ages and sizes, wore loose-fitting black duck jackets and trousers over whatever clothing they had been wearing when they were arrested. On their heads were black cotton caps with brims, universally too large. These covered their ears as well as the tops of their heads. There were more than thirty of them, more than the Tatra's dump body could comfortably accommodate sitting down. It was necessary for them to line the three walls of the truck bed (the rear wall of the dump truck was slanted) standing up and hanging on to the wall and each other.\n\nIt was just after six in the morning, and they had just been fed. Breakfast had been a hunk of dark bread and a veal, potato, and cabbage soup. It was hearty fare and tasty. The intention of the prison authorities was obviously to provide adequate nutrition for the prisoners. There would be a second meal, bread and lard, and a third at night, always a guly\u00e1s (stew). This sometimes had paprika, making the traditional Hungarian stew, and sometimes just chunks of meat floating in a rich broth with potatoes and cabbage.\n\nWhen all the prisoners had climbed onto the Tatra truck, the guard with the clipboard went back inside the prison. The other two guards went to a small BMW motorcycle, kicked it into life, and waited for the truck to leave the courtyard. Then they followed it, ten or fifteen yards behind, making a series of slow turns on the cobblestones so they would not catch up with it and have to stop.\n\nSt. Gertrud's prison was on the edge of P\u00e9cs. Three minutes after leaving the prison, the truck was groaning in low gear as it climbed a narrow and winding cobblestone street. The motorcycle had to come to a stop three times to wait for the truck to get ahead.\n\nThe truck climbed to the top of a hill, then started down the other side, equally steep and winding. The truck moved very slowly, in low gear, for it had snowed the night before, and there was a layer of slush over the cobblestones. When the road was clear, the truck went down the hill at a terrifying rate.\n\nWhen it had almost reached the bottom of the hill, the truck turned off onto a road that appeared to be paved with coal. There was a dirt road under the coal, but coal falling from trucks had then been crushed under other trucks, so that there was in fact a three-inch-deep layer of coal paving the road.\n\nWhen the truck reached the mine head and stopped, the prisoners, without being told what to do, got off the truck and walked to the shaft head. There, suspended from an enormous wheel, like a monstrous water bucket over a well, was a steel-framed elevator. The prisoners filed onto it until they closely packed it.\n\nThen the basket descended into the mine.\n\nFifty feet from the surface, it began to get dark. At one hundred feet, they could see nothing at all; it was like being blind. By three hundred feet, however, their pupils had reacted to the absence of light and dilated to the point where some sight returned.\n\nAt five hundred feet, when the basket stopped with a groan and then bounced up and down until the elasticity of the cables had expended itself, there were faintly glowing electric lights.\n\nThe prisoners were issued carbide headlamps by a foreman. They gathered around a table to clean them. Then they filled the brass fuel tanks with fingernail-size pellets of carbide, added water, and quickly screwed the covers in place. The headlamps began to hiss as the water reacted chemically on the carbide and produced gas. The prisoners ignited the escaping gas from a lamp burning on the table, then adjusted the lamps to their heads.\n\nThe foreman looked over the prisoners and gestured at two of them. They went to him as the others walked into a tunnel.\n\n_I have been selected to shovel donkey shit,_ First Lieutenant Eric Fulmar, Infantry, Army of the United States, thought. _I wonder why. That job usually goes to the old men; shoveling donkey shit and spreading straw doesn't require as much strength as wielding picks or sledgehammers or coal shovels._\n\nThe basic motive power in the mines was donkeys. They were hitched to a coal car and dragged the full car to the elevator. They were then unhitched, the coal car manhandled onto the basket, and the basket hauled to the surface.\n\nThe donkeys were then hitched to an empty coal car, which they dragged back along the rails to be filled again with coal.\n\nEric at first had been horrified at what appeared to be cruel and inhumane treatment of the animals, even though he was aware that, in the circumstances, there was little room for him to pity anything, human or animal. He had then expected any minute that the Gestapo or the SS\u2014or the Hungarian version thereof, the Black Guard\u2014would show up and introduce themselves by knocking him down and kicking his teeth out to put him in the right frame of mind for the interrogation to follow.\n\nBut that had not happened. Except for one man, the last Black Guards he had seen were the ones who had carried him and Professor Dyer to St. Gertrud's prison. That man had been a corporal or a sergeant (Fulmar was not sure about their rank insignia) he had seen the next morning. That morning, the one Black Guard had been sitting backward on a chair watching, as prison guards went through the paperwork.\n\nA prison guard had dumped on the table the contents of a gray paper envelope containing all the personal property taken from him when they had arrested him on the barge. Except for his wristwatch and his money. The prison guard, in soft German, had told him to identify the property taken from him, and to sign a form he handed him. It had not seemed to be a propitious time to bring up the missing money or the wristwatch.\n\n\"Your property will be returned to you at the completion of your sentence,\" the guard had said.\n\nFulmar had said nothing, praying that his relief would not be evident on his face. He had quickly come up with a scenario that seemed to make sense, but was frightening because it seemed to be too good to be true: He and Dyer had been arrested not because the Gestapo and the SS-SD were looking for them all over German-occupied Europe, but because they seemed to be black marketeers who had come to Hungary with a good deal of money in search of foodstuffs.\n\nPainfully aware that it was wishful thinking, he began to realize that the Black Guards who had stopped and searched the barge and found them had been looking for black marketeers\u2014not to bring them before the bar of justice, but to find them with large amounts of cash that could \"disappear\" between the time they were arrested and the time they got to the police station.\n\nIf the Black Guards charged them with black marketing, which was a serious crime, requiring a formal trial, the state would take the money Fulmar had with him. If, on the other hand, they were charged with \"unauthorized travel,\" the euphemism for Austrians and Germans who came privately to Hungary to buy sausage and smoked ham and salami for their own use, there was no need for the subject of the money to come up at all.\n\n\"May I ask, Sir, what my sentence is?\" Eric had asked very carefully.\n\n\"You have been sentenced by the Municipal Magistrate to three months' confinement at hard labor for unauthorized travel to P\u00e9cs,\" the prison guard had said.\n\n\"Yes, Sir,\" Fulmar said. \"Thank you, Sir.\"\n\n\"Three months in the mines,\" the Black Guard had said, in barely understandable German, \"will be good for you. And maybe it will even teach you that you can't slip things past the river patrol.\"\n\nThere was a suggestion there that if he had offered the Black Guard on the boat a little money, he would not have been arrested at all.\n\nThere was a terrible temptation to press his luck, to offer them more money to let them go. But he realized in time that he was so overexcited by fear that he couldn't trust his own judgment. He was deeply aware that a vein on his temple was pulsing in time with his heart. And his ears rang.\n\n\"I will remember that, Sir,\" Fulmar said, managing a weak smile.\n\nSmiling, the prison guard waved him out of the little office.\n\nAs quickly as the first scenario had come to him, others followed, and they were not nearly as pleasant. A hundred things could go wrong: Professor Dyer might panic. He might decide to try to save his own skin by turning on Fulmar. And Gisella had not been arrested. So he might decide that turning himself and\/or Fulmar in would somehow help her.\n\nBut above all, there was the alarm sounded for all of them by the Gestapo and the SS-SD. It was wishful thinking gone mad to hope that no connection would be made between the two men the entire German security services were looking for and the two \"persons traveling to P\u00e9cs without authorization.\"\n\nBut there had been nothing to do about that possibility but pray.\n\nOn his second day in the mines, Professor Dyer had crushed his fingers under the wheels of one of the coal cars. He had been taken from the mine, howling in pain. It had been easy then to imagine that the accident would attract the authorities to him, but that hadn't happened, either.\n\nDyer's hand had been treated and bandaged. And he now spent his days one-handedly sweeping out the cells in St. Gertrud's and replacing the straw in the mattresses.\n\nEvery night, when he got back, Fulmar had to display a confidence that he did not feel at all. He had to reassure Dyer they had nothing to worry about, that all they had to do was avoid attracting attention to themselves, and they would be turned free.\n\nAnd every morning, he gave the professor what he hoped was an encouraging wink as he filed out of the cell block to get on the truck.\n\nThe donkeys in their stalls stood waiting stoically to be led out and hitched to the coal cars. They didn't seem to mind, obviously, doing what was expected of them. Being in the mines, for them, was the way things were.\n\nThe mine corridor where the donkeys had their stalls was several hundred feet long; the donkey stalls occupied the center portion. It smelled, not unpleasantly, of donkey manure. There was a sharp odor on top of that, ammonia-like, from donkey urine.\n\nThree-quarters of the way down the line of stalls the donkey -shit car sat waiting for attention. As they approached it, Fulmar understood why he and another muscular young prisoner had been selected from the line of incoming miners. There was more than donkey shit to be loaded aboard the donkey-shit car today. There was a dead donkey.\n\n\" _Tot_ [dead],\" the foreman said, quite unnecessarily.\n\nThen he showed them how one of the sides of the donkey -shit car could be removed, and how, with the aid of a block and tackle, they were to load the carcass onto the car. The donkey's eyes were open, a curious white. And he was already starting to decompose, and to smell. When they got the block and tackle in place and hauled him out of the stall onto the tracks, the movement caused the contents of his lower bowel, not ordinary donkey shit, but a foul-smelling, bluish semiliquid, to pass from his anus.\n\nMore of it came out after they had rearranged the block and tackle and dragged him onto the car. Fulmar felt nauseated, tried to fight it down, and failed.\n\nThe foreman laughed at him and said he could tell that he was a city boy who had never lived on a farm.\n\nAfter they got the donkey carcass into the car and closed the side, they went down the line of donkeys and shoveled the donkey shit into the car. By the time they were finished, you couldn't see the donkey carcass.\n\nAnd then they hooked a donkey to the car to drag the car to the elevator.\n\nFulmar had another unpleasant thought. He didn't know how long he had been in jail and working in the mine, and therefore did not know how much longer he would be in the mines. He thought he was a damned fool for not having made a scratch on his cell wall once a day. Then he would have known.\n\nThen he thought it really didn't matter. Long before his ninety-day sentence was up, they would find out that he wasn't a black marketeer.\n\nAnd soon after that, some other prisoner would roll his dead body off somewhere in a cart, just as he was doing with the donkey. The donkey, Fulmar thought, was actually better off than he was. The donkey had not had the ability to stand around imagining what was going to happen to him.\n**VII**\n\n#  **1**\n\n##  **HEADQUARTERS, COMMANDER-IN-CHIEF, PACIFIC PEARL HARBOR NAVAL BASE OAHU ISLAND, TERRITORY OF HAWAII 0915 HOURS 13 FEBRUARY 1943**\n\nLieutenant Commander Stuart J. Collins, United States Navy, Cryptographic Officer, Headquarters, CINCPAC, was aware that the lieutenant commander in the crisp white uniform in the outer office of CINCPAC was looking askance at his uniform. Commander Collins's khaki uniform was mussed and wilted, and there were sweat stains under the armpits.\n\nThe cryptographic section, in the basement of the neatly white-painted, red-tile-roofed headquarters office building, was of course air-conditioned. But it had been air-conditioned in 1937, when no one could have guessed how many people and how much equipment it would be necessary to stuff into the three small rooms. It was hot down there, and people sweated.\n\nIf the commander in the crisp white uniform in the admiral's cool and spacious office didn't like his sweaty, shapeless uniform, fuck her. Goddamn women in the Navy, anyway.\n\n\"The Admiral will see you, Commander,\" the WAVE Lieutenant Commander said, quite unnecessarily. Commander Collins was not deaf; he had heard the Admiral tell her, over the intercom, to send him in.\n\nCommander Collins walked into the CINCPAC's office.\n\n\"Good afternoon, Sir,\" he said, and extended a clipboard to the Admiral, who scrawled his name on the form, acknowledging receipt of Top Secret Incoming Message 43- 2-1009. Commander Collins then handed him the message, hidden beneath a TOP SECRET cover sheet.\n\nCINCPAC read it:\n\nURGENT  \nTOP SECRET  \nFROM CHIEF OF NAVAL OPERATIONS WASHINGTON DC  \nTO [EYES ONLY] COMMANDER IN CHIEF PACIFIC, PEARL  \nHARBOR TERR HAWAII\n\nDP YOU WILL MAKE AVAILABLE GATO CLASS SUBMARINE FOR SUCH TIME AND FOR SUCH MISSION AS SPECIFIED BY C. J. CHENOWITH OF THE OFFICE OF STRATEGIC SERVICES. CHENOWITH AND PARTY OF THREE [3] EN ROUTE BARBERS POINT NAS ABOARD NATS FLIGHT 232 ETA 1530 HOURS 14 FEBRUARY. CARGO ACCOMPANYING CHENOWITH PARTY OF APPROXIMATELY TWO [2] TONS GROSS WEIGHT IN THIRTY TWO [32] WOODEN CRATES WILL REQUIRE TREATMENT AS TOP SECRET MATERIAL. OCNO DOES NOT DESIRE TO DISCUSS THIS ORDER. OCNO WILL BE ADVISED IN DETAIL BY MOST EXPEDITIOUS MEANS OF REASONS FOR INABILITY TO COMPLY WITH THIS ORDER. BY DIRECTION: SOLOMON VICE ADMIRAL.\n\nCINCPAC looked up at Lt. Commander Collins.\n\n\"No reply, Commander,\" he said.\n\n\"Yes, Sir,\" Collins said, and started to do an about-face.\n\n\"Collins?\" CINCPAC said.\n\nCollins faced CINCPAC again.\n\n\"Hot in the basement?\"\n\n\"Yes, Sir.\"\n\n\"You talk to the engineer about it?\"\n\n\"Yes, Sir.\"\n\n\"And what did he say?\"\n\n\"He said that the ambient temperature is within the operating range of the equipment, Admiral, and there's no way he can authorize more air-conditioning.\"\n\n\"Collins,\" CINCPAC said. \"There's a Chief Kellerman over in Civil Engineering. We were aboard the old _Des Moines_ together. You go see him, tell him I sent you, and ask him to cool your shop down.\"\n\n\"Yes, Sir,\" Commander Collins said. \"Thank you, Admiral. \"\n\n\"And on your way out, ask Commander Oster to get COMSUBFORPAC in here just as soon as possible.\"\n\n\"Aye, aye, Sir.\"\n\nCOMSUBFORPAC, Rear Admiral (Upper Half) Geoffrey H. Keene, USN, a ruddy-faced, freckled man of forty-three, who looked much younger, was a professional officer, and thus accustomed to carrying out any order given with cheerful, willing obedience.\n\n\"Gerry, what boat, or boats, Gato class, have you got here ready for sea?\"\n\n\"None this minute, Sir,\" Admiral Keene said. \"But the _Drum_ 's just about through with her sea trials. She's off Kahoolawe Island right now, and she's scheduled to go on patrol in three or four days, as soon as they correct what needs fixing.\"\n\n\"There will be a mission for her,\" CINCPAC said. \"Apparently, a people-carrying mission.\"\n\n\"Yes, Sir?\" Admiral Keene said. His tone made it clear he wanted more information.\n\n\"If the _Drum_ is all that's available, it'll have to be the _Drum,_ \" CINCPAC said.\n\n\"Admiral, may I suggest that the _Narwhal_ will shortly be available? She's about to leave Diego.\"\n\n\"It'll have to be the _Drum,_ Admiral,\" CINCPAC said. \"And if you had anything special planned for her, it will have to be put on the back burner.\"\n\nCOMSUBFORPAC could not help but question the wisdom of using a multimillion-dollar naval vessel and its highly trained crew as a kind of seagoing taxicab. Transporting people somewhere was something that submariners did from time to time\u2014but at the pleasure of the submariners, if and when that could be reasonably fitted into the normal duty of submariners: That, first, last, and always, was the destruction of enemy men-of-war and the interdiction and destruction of enemy shipping.\n\nBut CINCPAC had addressed Keene as \"Admiral,\" rather than by his Christian name, a subtle reminder that he was giving an order.\n\n\"Aye, aye, Sir,\" COMSUBFORPAC said.\n\nCINCPAC handed him the Top Secret folder.\n\n\"If you can find the time, Gerry,\" CINCPAC said, \"it might be a good idea if you met this Mr. Chenowith at the airfield. Present my compliments, and as tactfully as possible, let him know that I would be grateful to learn what the hell this is all about.\"\n\n\"Aye, aye, Sir,\" Admiral Keene said.\n\n#  **2**\n\n##  **WAIKAHALULU BAY, KAHOOLAWE ISLAND TERRITORY OF THE HAWAIIAN ISLANDS 0945 HOURS 13 FEBRUARY 1943**\n\nThe Alenuihaha Channel (depths of at least 1,000 fathoms) runs between the Hawaiian Islands of Hawaii, Maui, and Kahoolawe.\n\nThere is a shelf approximately forty miles off the southern coast of Kahoolawe Island, where the depth changes abruptly from about 1,400 fathoms to 650. Then, five miles off the Kahoolawe shore, the depth changes again abruptly to approximately forty fathoms.\n\nThe final sea trial after refitting of the USS _Drum\u2014SS- 228,_ a 311-foot-long submarine of the Gato class\u2014required her to approach the Alenuihaha Channel from the open Pacific, on the surface, in the hours of darkness, navigating by celestial navigation.\n\nShe would remain on the surface, crossing the channel until she reached the shelf, whereupon she would submerge to maximum operating depth on a course that would bring her off Waikahalulu Bay. She would then rise to near periscope depth and maintain that depth and course in the forty-odd-fathom water until visual contact with their assigned target was established, by periscope, in daylight.\n\nShe missed Waikahalulu Bay by five miles. Her skipper, Lieutenant Commander Edwin R. Lennox, USN, a stocky, round-faced, sandy-haired officer who had three days before celebrated his thirtieth birthday, was disappointed, but not surprised. There was really no good way to read the currents of the Alenuihaha Channel or the offshore waters of the island.\n\nWhen his periscope picked up the targets, without taking his eyes from the rubber eyepieces of the periscope, Commander Lennox softly ordered, \"Battle stations, Mr. Rutherford. Gun crews to stand by.\"\n\n\"Aye, aye, Sir,\" Lieutenant William G. Rutherford, USNR, the _Drum_ 's twenty-seven-year-old executive officer, a tall, black-haired, skinny man, said. He pushed the heel of his hand against a round brass knob. A bell clanged throughout the submarine, and there was frenzied activity everywhere but around the periscope itself.\n\n\"Steer zero eight five,\" Commander Lennox ordered.\n\n\"Coming to zero eight five, it is, Sir,\" the helmsman said. And a moment later, \"Sir, the course is zero eight five.\"\n\n\"Periscope down,\" Commander Lennox said. \"Take her to one hundred feet.\"\n\nCommander Lennox slapped the handles of the periscope in the up position.\n\n\"Down periscope,\" he ordered, and the periscope moved downward.\n\n\"One hundred feet, Sir,\" the chief of the boat reported.\n\n\"Hold her so,\" Commander Lennox ordered. He crossed the crowded area and pushed down on the lever that activated the public address system.\n\n\"This is the captain speaking,\" he said. \"If I have to say it again, and I think I do, the way to achieve speed is to be sure of what you're doing, and then to do it carefully. We will lose time if somebody falls down a ladder or over the side.\"\n\nThere was a murmur of chuckles throughout the boat.\n\n\"Gun crews standing by, Sir,\" the chief of the boat said.\n\n\"Very well,\" Commander Lennox said. \"Bring her around to two sixty-five.\"\n\n\"Coming to two six five it is, Sir,\" the helmsman replied. The _Drum_ banked like an airplane as she changed course. And then she straightened up.\n\n\"Up periscope,\" the captain ordered, and the periscope rose.\n\n\"Sir, the course is two six five,\" the helmsman reported.\n\n\"Keep her so,\" Commander Lennox said, and turned to the executive officer. \"Got your watch, Bill?\"\n\n\"Yes, Sir.\"\n\n\"Punch it,\" Commander Lennox said, then: \"Surface, surface!\"\n\nTwenty seconds later, in boiling water, the bow of the _Drum_ emerged from the sea.\n\nThere was a burst of black smoke as she went from battery to diesel power.\n\nCommander Lennox, Lt. Rutherford, and a talker came onto the conning tower.\n\n\"Make turns for ten knots,\" Commander Lennox ordered. \"Gun crews man your guns, report when ready.\"\n\nThe talker repeated his orders into his microphone.\n\nBluejackets in steel helmets and life vests poured from hatches in the conning tower. Some made their way to the five-inch cannon mounted forward of the conning tower, and began to prepare it for firing. Others went to a rapid-firing 40mm cannon mounted on a platform just below where the skipper, the exec, and the talker stood. A third group went to the 20mm rapid-firing cannon mounted on the rear of the conning tower.\n\nOther sailors formed a human chain, passing ammunition from the submarine to the guns.\n\nOne by one, the guns signaled (the gun chiefs raising a hand overhead) their readiness to open fire.\n\n\"The guns are ready to fire, Sir,\" the exec reported, and then added, \"one hundred eighteen seconds.\"\n\n\"Commence firing,\" Commander Lennox ordered.\n\n\"Commence firing,\" the talker repeated.\n\nCommander Lennox and the exec put binoculars to their eyes and trained them on the shore of Waikahalulu Bay. There were targets in place, wooden frameworks covered with canvas, fairly credible replicas of oil storage tanks.\n\nThe five-inch fired five rounds; one fell nowhere near the targets, but the other four went where they were supposed to go. Meanwhile, the 40mm and 20mm rapid-firing cannon fired continuously, the 20mm in a rapid staccato, the 40mm in a slower, more measured cadence. The targets were obscured by dust and smoke.\n\nCommander Lennox counted the five-inch rounds. The moment he saw the muzzle flash of the fifth round, without taking his eyes from his binoculars, he ordered, \"Cease fire, secure the guns, clear the decks.\"\n\nThe talker repeated the orders. The sailors at the guns now prepared them for submersion. The crews of the rapid-firing cannon began to pass unfired ammunition back into the hull, and then they all went below.\n\n\"Sir,\" the talker said, \"chief of the boat reports gun crews secure from firing.\"\n\n\"Dive!\" the captain ordered.\n\n\"Dive!\" the talker said. \"Dive!\"\n\nA Klaxon sounded. The exec, the talker, and finally the captain went through the hatch and secured it after them. By then, the decks were already awash.\n\n\"Take her to a hundred and fifty feet,\" Commander Lennox ordered.\n\n\"One fifty feet, aye,\" Lt. Rutherford repeated.\n\n\"What have we got, Helmsman?\" Commander Lennox asked a minute later.\n\n\"Sir, we are steering two six five degrees. . . .\" The helmsman paused and waited until the needle on the depth gauge was where it was supposed to be, and then went on, \"at one five zero feet, sir.\"\n\n\"Keep her so,\" Commander Lennox ordered, and then he stepped to the public address system again.\n\n\"This is the captain speaking,\" he said formally. \"For a bunch of Kansas hayseeds and Brooklyn thugs, that wasn't half bad. And the chief of the boat would have told me by now if somebody had gone over the side.\"\n\nChuckles and laughter ran through the boat.\n\nLeaving the microphone open, Commander Lennox said, \"Take her up, make turns for sixteen knots, and set us on a course for Pearl Harbor.\"\n\nHe let the spring-loaded microphone switch go and motioned for the chief of the boat to come to him.\n\n\"Chief,\" Commander Lennox said, \"I would not be too upset, when you check the guns, if you were to find something that would take, say, thirty-six hours to fix.\"\n\n\"Aye, aye, Sir,\" the chief of the boat said.\n\n\"And, of course, if the men aren't needed to help with the repair, there's no reason I can see why they shouldn't be given liberty.\"\n\n\"Aye, aye, Sir,\" the chief of the boat said.\n\n\"Surface, surface!\" Lt. Rutherford ordered.\n\n#  **3**\n\n##  **HEADQUARTERS, U.S. FORCES IN THE PHILIPPINES MISAMIS OCCIDENTAL PROVINCE, MINDANAO 14 FEBRUARY 1943**\n\nThey had worked out a cipher:\n\nOn the fifth of February KSF had sent a message, as opposed to responding to one of Fertig's messages. So far, all that establishing a radio link with the United States had done was to enable Fertig to get word to his wife that he was alive and not in a Japanese POW camp.\n\nKSF FOR MFS NAMES OF TOWN AND STATE WHERE PATRICIA LIVES WILL BE USED AS CODE PHRASES FOR DOUBLE TRANSPOSITION STOP SEND TEST MESSAGE IMMEDIATELY KSF BY\n\nPatricia, Fertig's daughter, was living with her mother in Golden, Colorado.\n\nUsing that as the basis for a rudimentary double transposition code, Fertig's homemade transmitter sent a meaningless phrase to KSF. Receipt of the message was acknowledged, but the reply, in the new code was only:\n\nKSF FOR MFS NO TRAFFIC FOR YOU AT THIS TIME KSF OUT\n\nTwo days later, on February 11, 1943, there had been another message for MFS:\n\nYOUR STATION DESIGNATED WYZB REPEAT WYZB STOP ALL REPEAT ALL FUTURE TRAFFIC WILL BE WITH KAZ REPEAT KAZ STOP KAZ HAS FILE OF ALL PAST TRAFFIC KSF OUT\n\nKAZ was the call sign of General Douglas MacArthur's General Headquarters, Southwest Pacific Command, in Australia. They heard KAZ on the air all the time, but had been unable to get KAZ to respond to their calls.\n\nNow things might be different. But several hours of calls to KAZ had produced no response whatever. There were several possible explanations for that, the most likely that radiations from Gerardo Almendres's homemade transmitter were for some reason unable to reach Australia. Fertig did not permit himself to dwell on the possibility that MacArthur did not want to talk to him.\n\nWhile Fertig did not personally know MacArthur, he had a number of friends who did. To a man, they reported that Douglas MacArthur, onetime Army Chief of Staff, later Marshal of the Philippine Army, and now once again in U.S. Army uniform, had an ego on a par with, say, Charlemagne's.\n\nWhile Fertig did not believe that the fall of the Philippines was MacArthur's fault\u2014indeed, he had acquired a deep respect for MacArthur's military ability; MacArthur's delaying actions with his limited resources had been undeniably brilliant\u2014he suspected that MacArthur was personally shamed by his defeat.\n\nIf that were the case, that shame might be deepened by proof that not all American officers and Philippine forces had hoisted the white flag and marched docilely into Japanese captivity.\n\nDuring his brief service as an officer, Fertig had quickly learned an old soldier's requisitioning trick. If you need something for one hundred men, and you want to be sure you get it, you requisition a quantity sufficient for two hundred. Or four hundred. Then, when the supply authorities cut your requisition by fifty percent, or seventy-five percent, you still wind up with what you really need.\n\nFertig had been \"generous\" in his communications with KSF with regard to his estimated strength report for the troop strength of the U.S. force in the Philippines. Not dishonest, just generous. He had elected to take the word of Philippine army officers who had not elected to surrender (putting his own serious doubts aside), when they told him how many men they had at their disposal, and how anxious\u2014providing he could supply and pay them\u2014they were to put themselves and their men under the command of Brigadier General Wendell W. Fertig and the U.S. forces in the Philippines.\n\nIf they told him, for example, that they had five hundred troops just waiting for the arms and food that would permit them to engage the Japanese, he took them at their word, even if it looked to him as if the five-hundred-man force consisted of a couple of officers and maybe sixty Philippine Scouts.\n\nHe had added up all the Philippine forces he was told were anxious to place themselves under his command and come up with a figure just in excess of six thousand officers and men.\n\nHis \"requisitions\" for arms and food and gold coins had been based on this strength figure.\n\nMacArthur, according to the radio message from San Francisco, had been made aware of this troop strength.\n\nFertig wondered how Douglas MacArthur was going to react to learning that, after he had reported his forces had fought to the last man and the last bullet, there were six thousand troops under a brigadier general still fighting on Mindanao.\n\nWhen Second Lieutenant (formerly Private) Robert Ball of USFIP came to report that MacArthur (or at least KAZ, his radio station) was finally being heard from, Brigadier General Fertig, a Thompson submachine gun beside him, was drinking a cup of tea on the shaded veranda of his combined headquarters and quarters. The tea was Lipton's. It had been grown in the Far East, sent to the United States, blended, put in tea bags, and then sent back to the Far East. How it had passed into the hands of the Moro tribal chief who had given it to Fertig, Fertig didn't know.\n\nAll he knew was that Lipton was putting out a better product than he had previously suspected. The tea bag that had produced the tea he was now drinking was on its fourth brewing cycle. (Brew, dry, brew again, dry, et cetera.) He knew this because he was a methodical man, and each time he drenched the tea bag in boiling water, he tore one of the corners of the tea-bag-tag off. The tea-bag-tag drying on the bamboo railing beside him was cornerless.\n\nHe felt that it behooved him to conceal from his subordinate staff the excitement he felt now that MacArthur was finally being heard from.\n\n\"Thank you, Ball,\" he said, with as much savoir-faire as he could muster. \"How long do you think it will take Captain Buchanan to decrypt the message?\"\n\n\"About thirty minutes, Sir,\" Ball said.\n\n\"Fine,\" Fertig said. \"I expect to be here in half an hour, when Captain Buchanan is finished.\"\n\nForty-five minutes later, Captain Horace Buchanan handed Brigadier General Fertig the two sheets of paper on which he had neatly lettered (Signal Section, HQ, USFIP, did not possess a typewriter) the decrypted message. From the look on Buchanan's face\u2014disappointment and embarrassment\u2014Fertig knew that there was little good news in the radio message.\n\n\"Thank you,\" Fertig said, and read the message:\n\nKAZ FOR MFS  \nONE LT COL WENDELL W. FERTIG CORPS OF  \nENGINEERS US ARMY RESERVE DETAILED INFANTRY  \nTWO COLONEL MARCARIO PERALTA PHILIPPINE  \nSCOUTS DESIGNATED MILITARY GUERRILLA CHIEF OF  \nTEMPORARILY OCCUPIED ENEMY TERRITORY  \nTHREE THE ISSUANCE OF MILITARY SCRIP IS  \nEXPRESSLY FORBIDDEN REPEAT EXPRESSLY  \nFORBIDDEN  \nFOUR COMMAND OF GUERRILLA FORCES WILL BE  \nEXECUTED ONLY BY OFFICERS PRESENTLY IN DIRECT  \nCOMMAND OF SAME  \nFIVE THIS HEADQUARTERS WILL ENTERTAIN  \nREQUISITIONS FOR SMALL IN SIZE URGENTLY  \nNEEDED EQUIPMENT ONLY  \nBY COMMAND OF GENERAL DOUGLAS MACARTHUR  \nCOMMANDER IN CHIEF SOUTHWEST PACIFIC  \nCOMMAND  \nWILLOUGHBY BRIGADIER GENERAL USA\n\nFertig looked up and met Buchanan's eyes.\n\n\"I took out the 'stops' and stuff, General,\" Buchanan said.\n\nThere had been a faint hesitation, Fertig noticed, before Buchanan had called him \"General.\"\n\n_It wasn't only a little bad news, it was all bad news._\n\nAs far as MacArthur was concerned, he was a reserve lieutenant colonel in the Corps of Engineers, not a brigadier general in command of U.S. forces in the Philippines.\n\nColonel Marcario Peralta was \"military guerrilla chief of temporarily occupied enemy territory.\" Fertig did know Peralta. Peralta had been a successful lawyer in Manila before the war. The last Fertig had heard, just before the surrender, Peralta had been a major. Now he was a colonel, which meant that Fertig was supposed to be subordinate to him.\n\nThat could explain why MacArthur had pointedly reminded him that he was a lowly lieutenant colonel.\n\nThere was another possibility: If he had not promoted himself, and thus offended MacArthur's sense of the military proprieties, it was possible (now that he thought of it, even likely) that he would have been promoted to colonel and named \"military guerrilla chief of temporarily occupied enemy territory.\"\n\nThe really worrisome paragraph was the one about forbidding him to issue scrip. He'd been issuing the scrip, signing each one-, five-, and ten-dollar bill himself; and the crude money had been accepted by the Filipinos; they had taken him at his word that, when the war was over and the Japanese had been driven from the Philippines, it would be redeemed at face value.\n\nAnd since MacArthur obviously was not about to send him gold, the scrip he was \"expressly forbidden repeat expressly forbidden\" to issue was the only way he had to pay the troops and to buy whatever the natives were willing to sell.\n\nThat was even more important than his rank, or Colonel Peralta's appointment as \"military guerrilla chief.\" Peralta was on the island of Panay. There was little or no chance that he would try to exercise command over Fertig. Peralta was no fool; he knew that Fertig would simply ignore him.\n\n\"Captain Buchanan,\" Fertig said, \"I presume that no one but you has seen the contents of this message?\"\n\n\"No, Sir.\"\n\n\"It is herewith classified Top Secret,\" Fertig said, and put a match to it. \"No one else is to be made privy to its contents.\"\n\n\"Yes, Sir.\"\n\n\"You may tell Lieutenant Ball and whomever else you wish,\" Fertig said, \"that the message dealt with our reinforcement in the future.\"\n\n\"Yes, Sir,\" Buchanan said. \"Sir, what do I call you?\"\n\n\"That would seem, Captain Buchanan,\" Fertig said, looking at him, \"to be entirely up to you.\"\n\nThere was a just-perceptible hesitation before Buchanan spoke. Then he said, \"Will there be a reply, General Fertig?\"\n\n\"No, no reply,\" Fertig said. \"That will be all, Captain, thank you.\"\n\n\"Permission to withdraw, General?\"\n\n\"Granted,\" Fertig said. Then, suddenly, \"Yes, there will be a reply, Captain.\" Fifteen minutes later, MFS went on the air:\n\nMFS FOR KAZ  \nPERSONAL FOR GENERAL MACARTHUR  \nREFERENCE PARA FIVE YOUR VALENTINES DAY  \nMESSAGE STOP URGENTLY REQUEST VIA FIRST  \nAVAILABLE TRANSPORTATION NECESSARY DRUGS  \nTREAT VENEREAL DISEASE CONTRACTED BY KEY  \nPERSONNEL STOP FERTIG\n\n#  **4**\n\n##  **CROYDON AIRFIELD LONDON, ENGLAND 14 FEBRUARY 1943\u2014ST. VALENTINE'S DAY**\n\n\"I think the thing to do with Charity Hoche, Helene,\" Lt. Colonel Stevens had said to Helene Dancy earlier that morning, \"is for you to meet her at the airport, run her past the officer's sales store, get her into uniform, and take her out to Whitbey House. She is a young lady who attracts a great deal of attention, and to the extent we can, I think we ought to keep her out of sight.\"\n\nColonel Stevens had then decided that it would be best to put Charity Hoche into the uniform of a WAC first lieutenant.\n\n\"We'll think about actually getting her a commission,\" Stevens had said. \"In the long run, that might be the thing to do. But for the short run, anyway, I think it makes more sense than putting her into a civilian specialist's uniform. That attracts attention.\"\n\nThe first impression Capt. Helene B. Dancy had of Miss Charity Hoche was not particularly favorable.\n\nMiss Hoche descended the stairway from the door of the ATC C-54, \"the Washington Courier,\" wearing the uniform of a War Department civilian, with the uniform cap perched perkily atop a mass of long golden hair. Neither Capt. Dancy nor Colonel Stevens had expected that Miss Charity Hoche would arrive in England in a civilian specialist's uniform.\n\nShe also managed to display a good deal of shapely thigh and lace-hemmed black petticoat as she came daintily down the stairs. She wore the gabardine uniform topcoat over her shoulders.\n\nTwo officers (one of them, in Capt. Dancy's opinion, old enough to know better) hovered solicitously around her. They were rewarded for their efforts with a radiant display of perfect white teeth between lips that Capt. Dancy thought had entirely too much lipstick of a too dazzling shade.\n\nA double-decker London bus had been driven onto the field to transport the arriving passengers to SHAEF Billeting. There they would be given a two-hour orientation lecture, known as the \"Be Kind to Our English Cousins speech.\" The trouble with Americans, in the opinion of many Englishmen, was that they were \"overpaid, oversexed, and over here.\"\n\nThe purpose of the orientation lecture was to remind the newly arrived Americans that England had been at war for more than three years; that there was a \"ration scheme\" for practically everything the English needed to live; and that the British quite naturally resented the relative luxury in which the American taxpayer was supporting its citizens in the United Kingdom.\n\nThe lecture, Capt. Dancy decided, seemed to have been prepared with Miss Charity Hoche in mind. But she would not hear it.\n\nCapt. Dancy showed her identification card to the guard and walked out of the terminal building and intercepted Charity Hoche as she was being escorted to the bus.\n\n\"Miss Hoche?\" she said. \"I'm Capt. Dancy. Will you come with me, please?\"\n\nThe pudgy lieutenant colonel who was carrying Charity's makeup kit looked crushed.\n\nCapt. Dancy happened to meet Charity Hoche's eyes and found herself being examined very carefully by very intelligent eyes.\n\n\"My luggage?\" Charity asked.\n\n\"It'll be taken care of,\" Capt. Dancy said.\n\nCharity said good-bye to the two officers and followed Capt. Dancy into the terminal, then to the Ford staff car.\n\n\"Where are we going?\" Charity asked when she was in the car, and then, without waiting for a reply, \"Is it hard to drive one of our cars on the wrong side of the road?\"\n\n\"The 'other' side of the road is the way I think of it,\" Capt. Dancy said. \"And the answer is 'no, you have to be careful, but you soon get used to it.' \"\n\n\"How did I get off on the wrong foot with you so soon, Captain?\" Charity challenged.\n\n_Because you're young and spectacularly beautiful and look and act as if a serious thought and a cold drink of water would kill you._\n\n\"If I gave that impression, Miss Hoche, I'm sorry,\" Capt. Dancy said. \"Where we're going is to my billet. There, we're going to put your hair up, take some of that makeup off, and do whatever else is necessary to make you credible as a WAC officer.\"\n\nCharity Hoche seemed oblivious to the reproof.\n\n\"Captain Douglass thought you might want to put me in a WAC uniform, but he wasn't sure. I've got the insignia and AGO card of a first lieutenant in my purse.\"\n\nDancy looked at her in surprise.\n\n\"So, all we'll have to do, then,\" Charity said sweetly, \"is pin on the insignia, put my hair up, and take some of the makeup off, right?\"\n\nShe gave Capt. Dancy a dazzling smile.\n\n\"But before we do that,\" Charity went on, just as sweetly, \"I think we should go by Berkeley Square. Not only do I have three 'Eyes Only' for Mr. Bruce, but I have crossed the Atlantic with a Colt 'Banker's Special' hanging from my bra strap. It hurts like hell, and I want to get rid of it.\"\n\n\"I'll be damned,\" Capt. Helene Dancy said.\n\n\"Won't we all be, sooner or later?\" Charity asked.\n\n\"Apparently, I was wrong about you,\" Capt. Dancy said.\n\n\"I don't know about that,\" Charity said, \"but you were wrong about Colonel Stevens. You should have known he wouldn't have let me come over here if I was a complete fool.\"\n\n#  **5**\n\n##  **OSS LONDON STATION BERKELEY SQUARE LONDON, ENGLAND 1610 HOURS 14 FEBRUARY 1943**\n\nDavid Bruce, Chief of London Station, was surprised to sense his office door being quietly opened, and when he looked up, to see the face of Capt. Helene Dancy waiting to catch his attention.\n\n\"Sorry to disturb you, Sir,\" Capt. Dancy said.\n\nBruce's eyebrows rose in question.\n\n\"Miss Hoche is here,\" Capt. Dancy said.\n\nBruce frowned. He didn't want to see Charity Hoche. He wanted, in fact, to nip in the bud any idea of hers that she would enjoy with him the same close personal relationship she was supposed to have with Bill Donovan.\n\nHe had directed that Helene Dancy pick the girl up at Croydon and take her directly to Whitbey House in one of the station's 1941 olive-drab Ford staff cars. En route, Helene was supposed to relay his orders to her to make herself useful wherever Lieutenant Robert Jamison felt she would fit in.\n\nJamison was Adjutant of Whitbey House Station. His job had been to relieve Canidy of as much of the administrative burden as he could. He had done a good job, but not only was he admittedly unhappy with what he called his chief clerk's role, but he was also qualified, in Bruce's opinion, to assume greater operational responsibility.\n\nJamison wanted to go operational, which was different from assuming greater operational responsibility.\n\nBruce had already decided that was out of the question, not because Jamison couldn't do it but because he knew too much for the OSS to risk having him captured. With Canidy the exception that proved the rule, OSS personnel privy to OSS plans and intentions in more than one\u2014their own\u2014case were not permitted to go operational.\n\nNo attempt had been made to brief Jamison on any particular operation, but he did the paperwork, and he was as bright as a new dime. There was no question in David Bruce's mind that Jamison knew far too much about too many things to send him off somewhere where he was likely to find himself being interrogated by the Sicherheitsdienst.\n\nBut Bruce had always thought there were areas where Jamison's intelligence and other talents could be put to better use than requisitioning sheets and towels and keeping abreast of the paperwork. Canidy had been giving him jobs of greater importance than these. And he had accomplished them admirably.\n\nJamison had handled, for example, and handled well, a project in connection with \"Operation Aphrodite\":\n\nThere was only one way to test the practicality of the drone bomber project, and that was by setting up a target and trying to blow it up with an explosives-laden, radio-controlled B-17. This, of course, had to be done with as much secrecy as possible, so when they finally flew the flying bombs against the German submarine pens, they would have the necessary element of surprise.\n\nJamison had scoured the maps of the United Kingdom until he found a lonely bay in Scotland that could be used as a target range. It had required coordination with the English, the local Scottish government, the U.S. Army (from whom he had borrowed a detachment of Engineers to build a target, a mock-up of the entrance to the Saint-Lazare submarine pens), and the U.S. Navy (who had provided ships to clear the area, and a yard boat to be available to pluck \"Operation Aphrodite\" aviators from the water, if that should prove necessary).\n\nAnd Jamison had carried this responsibility (which was, of course, in addition to his \"chief clerk\" duties) with a skill, imagination, and discretion that had pleased Bruce. Jamison had come up with a different cover story for each set of outsiders involved, with just enough truth in each to make it credible, and far enough from the real truth to keep the secret of what was really going on away from German agents.\n\nWhen the first Personal\u2014Eyes Only message from Colonel William J. Donovan regarding Miss Charity Hoche had come to Berkeley Square asking Ed Stevens if he could find useful work for her, Bruce had seen in it a solution to the problem of more efficient utilization of the talents of First Lieutenant Robert Jamison. She would be assigned first as Jamison's assistant. There she would do such things as learn how to requisition flour to bake bread\u2014or a similar-looking white powder that had extraordinary explosive power when detonated, say, against the supports of a bridge in France or Yugoslavia.\n\nThe sooner she could take the paperwork burden from Bob Jamison's shoulders, the sooner Jamison could be put to work doing other, more important things.\n\n\"Why is she here?\" Bruce asked. There was more than a hint of displeasure, even reproof in his voice.\n\n\"She has three Eyes Only for you,\" Captain Dancy said.\n\n\"Oh?\" Bruce was surprised that Charity Hoche had been put to work as a courier. Couriers were most often officers traveling to Europe for assignment, or sometimes warrant officers whose duty it was to travel around the world, providing armed, personal guard to documents that could not be trusted to the mail pouches.\n\n\"Send her in, please,\" Bruce said.\n\n\"She's in the ladies' room,\" Capt. Dancy said, then added, \"taking off her pistol.\"\n\nCharity Hoche appeared a minute later. She had three letter-size envelopes in her right hand and a Colt \"Banker's Special\" .38 Special revolver in her left.\n\nShe was stunning. She exuded, David Bruce thought personally, a subtle sexuality, even a sort of refined lewdness that would make an archbishop tend to forget his vows. Professionally, David Bruce had wondered if all of his happy plans to have this young woman relieve Jamison of his administrative chores might be shot out of the water by her blatant sex appeal.\n\nBruce had been amused to learn that the Army had _officially_ approved the policy of inserting slides of attractive and scantily attired or nude young women into slide trays containing other slides demonstrating the proper technique of waterproofing a truck or assembling a pontoon bridge. It caught the men's attention, woke them up, got the blood flowing.\n\nBruce was genuinely concerned about the degree to which Charity Hoche's simple presence among the men in training and awaiting assignment at Whitbey House would catch the men's attention. There were some women at Whitbey House, and some local women, but not nearly enough of the opposite sex to go around.\n\nMiss Charity Hoche, Bruce suspected, would wake them up and get their blood flowing to an undesirable degree.\n\n\"Mr. Bruce,\" Charity said in a low and sexy voice, \"I'm Charity Hoche. Daddy said when I saw you to give you his best regards.\"\n\nShe thrust the envelopes at him. They were of lightweight, airmail paper, double enveloped, the outer envelopes stamped TOP SECRET.\n\nThey were warm to the touch. After a moment, he figured that out. She had been carrying them on her person. In her girdle, specifically; there was no other place where they could have been carried unfolded. It made sense, of course, but there was still something unsettling about it.\n\nBruce forced his thoughts from Charity's girdle to the pistol. The way she was holding it\u2014upside down, her finger nowhere near the trigger, not waving it around, the muzzle pointed safely toward the floor\u2014showed that she was quite at home with firearms. But one did not expect to see a snub-nosed revolver in the soft white hands of a long-haired blonde with a face that brought to mind candlelight dinners.\n\nCharity Hoche saw the surprise in his eyes. She flashed Bruce a dazzling smile.\n\n\"I didn't mean to startle you, Mr. Bruce,\" she said. \"But I . . . I can't tell you where I've had the damned thing for the last thirty-six hours . . . just had to get it out of there. I'm scarred for life.\"\n\nDavid Bruce had been a little chagrined at how eagerly his mind considered in glorious Technicolor the various places Miss Hoche might have had the pistol concealed on her person for the past thirty-six hours.\n\n\"Not at all,\" David Bruce said, somewhat lamely.\n\nCharity handed him next three Receipt for Classified Top Secret Documents forms, and watched as he compared the numbers of the forms with the numbers on the outer envelopes, then signed them. When he gave them back to her, she folded them into a small wad and stuffed them inside her uniform blouse. He averted his eyes in a gentlemanly fashion as she did this.\n\n\"Let me take a quick look at these,\" David Bruce said, furious with himself for acting like a high-school boy before this stunning young woman. \"And then we'll have a little chat.\"\n\n\"Yes, Sir,\" said Charity Hoche.\n\n''Helene,\" Bruce heard himself say, \"why don't you get us some coffee?\"\n\nShe went to get the coffee, but he saw the look on her face and reminded himself again that although she was functioning as his secretary, she was a commissioned officer of the United States Army, and aware that captains are not sent to fetch coffee.\n\nThe first of the three Personal\u2014Eyes Only messages from the Director of the Office of Strategic Services dealt with logistic matters. He glanced at it, then opened the second. That dealt with the suspicions held by the FBI that a technical sergeant recruited for the OSS (and, he recalled from a remote portion of his memory, about to finish training at Whitbey House) had uncomfortably close connections with the Communist Party, USA. As he replaced that one in its envelope, he thought he would have to read that one very carefully indeed. Then he opened the third Eyes Only. It dealt with Miss Charity Hoche:\n\nDear David:\n\nWhile I would suggest that we leave intact the in-house gossip that Charity Hoche has been sent to you because she batted her eyes at Uncle Bill, and the old softie gave in, the truth of the matter is something else.\n\nBeneath the very attractive facade is an unusually bright (genius-level IQ) young woman with a master's degree in political science earned in four years, summa cum laude. As this came out, first as Charity proved far more useful working at the house on Q Street than frankly I thought she would be, and then officially, from a belatedly administered background investigation, Pete Douglass and I began to involve her in more and more higher-level operations.\n\nThe last time I was in England, I brought Ed Stevens into one such operation, together with a direct order that he was not to tell you I had done so. I should not have to tell you the decision to keep you out of this was not in any way a reflection on you. I will tell you that it is the only operation currently under way in Europe to which you are not fully privy, and that those, including Charity, who are privy to it are a very small number of people personally approved by the President.\n\nAnd neither Ed nor Charity is privy to all the details. I brought Ed into it, with the President's permission, because the operation is of such importance that nothing else being done can be permitted to interfere with it. He was told what he has been told solely so that he can make sure nothing that happens over there will get in the way. His orders are to reason with you, first, to see if he can talk you out of whatever it is you plan to do that might get in the way, and, failing that, to communicate directly with either myself or Pete Douglass. We would then, without explanation, cancel the planned operation. We have done that twice.\n\nCharity was brought into it, again with Franklin Roosevelt's specific permission, for the simple reason that this operation's in-house administration cannot be conducted through our normal channels, as secure as we believe them to be. Pete and I needed, in other words, a clerk-typist and file clerk with not only a Top Secret Presidential clearance, but one with the intellectual ability to comprehend the implications of the project, and to deal with the people involved.\n\nIt was only, frankly, after I pointed out to the President that none of the other people he proposed, in particular one Navy captain of our mutual acquaintance, to assume responsibility for in-house administration and liaison for this project could type or file, and that adding the Navy captain to the cleared list would leave us no better off than we then were, that he approved adding Charity to the list of those cleared for the project.\n\nThat situation has now changed, as a result of growth in the project. We now have the Navy captain, and he has an administrative staff of two. And as both the project, and your operations, have grown, so has the possibility that you will undertake something that could get in the way, and that it would somehow slip past Ed Stevens's attention.\n\nWe cannot take that risk. My recommended solution to the problem was what I thought to be the obvious one, to add your name to the list. Unfortunately, I made it hours after the President had become aware that, on his own, one individual on the list had made his deputy privy to some details of the project.\n\nRoosevelt was enraged . . . at the time I didn't know why . . . at my suggestion that we add \"every Tom, Dick, and Harry\" to the list, and, at my persuasive best, when I told him what I considered to be the risk of something slipping past Ed Stevens in London, all I could get from him was permission to send someone already on the list over there to keep that from happening.\n\nThat boiled down to one of the Navy captain's men, a commissioned warrant officer, absolutely trustworthy, but a sailor to the core, or Charity.\n\nMy decision is to send you Charity. On my authority, she has the presumed Need-to-Know anything concerned with any of your projects, to the same degree as Ed Stevens. I have instructed her, should something come to her attention that she feels has missed Ed's, to first bring it to his attention, and then to yours, and finally, if it comes to this, to communicate directly with Pete Douglass or me.\n\nHow you arrange for this is of course up to you, and I don't think I have to tell you this project review function of hers is to go no further than you or Ed.\n\nI am, of course, David, uncomfortable with keeping you in the dark, and can only hope that you will forgo judgment until the time when I can tell you what's been going on; when, I really believe, you will understand why all this has been necessary.\n\nYou may have noticed the strikeovers and other symptoms of amateur typing. This is because neither Miss Broyle, nor even the ever-faithful Chief Ellis, are in on this either, and this has been writ by hand by\n\nYour old friend,\n\n_Wild Bill_\n\nDavid Bruce recognized that, despite Wild Bill Donovan's liberally dispensed soft soap, his reaction to learning that the President of the United States, an old friend, had decided there were some secrets with which he could not be trusted was mixed hurt and anger.\n\nAnd he realized he was hurt and angered by learning that Ed Stevens, of whom he was very fond and whom he considered a true friend, had been involved in a months-long deception.\n\nAnd he realized that he was humiliated to learn that while he couldn't be trusted with this great goddamned secret, whatever it was, the long-haired blonde who had crossed the Atlantic with Top Secret\u2014Personal\u2014Eyes Only documents in her girdle enjoyed the confidence of the President. And Donovan.\n\nBruce was a man of great will. He forced the anger and humiliation down, succeeding after a long moment in convincing himself that the President must have his reasons, and that it was his duty not to question his judgment.\n\nCapt. Helene Dancy entered the office with three cups of coffee and coffee accoutrements on a tray.\n\n\"Miss Hoche,\" David Bruce said, \"I presume you are familiar with the Eyes Only that deals with you?\"\n\n\"In general terms, Sir,\" Charity Hoche said. \"I haven't read it. I've read the other two.\"\n\n\"I think you should read it,\" Bruce said, and handed it to her. He heard the sound of his voice, and told himself to be careful. He was still acting emotionally.\n\nHe looked at Helene Dancy and saw in her eyes that she sensed that something extraordinary was going on. He looked again at Charity Hoche as she read Donovan's letter. Twice, her eyebrows went up, apparently in surprise.\n\nThen she looked at him, and met his eyes.\n\n\"Captain Dancy,\" Bruce said, \"would you ask Colonel Stevens to come in, please?\"\n\n\"Yes, Sir,\" Helene Dancy said. \"Would you like me to log those Eyes Onlys in?\"\n\n_Meaning, of course,_ Bruce thought, _that your curiosity is aroused and that you'll get a quick look at them between here and the safe._\n\n\"You can take these two, Helene,\" Bruce said, looking at Charity Hoche. \"I'm not sure about the third.\"\n\n\"I don't mean to be forward, Sir,\" Charity Hoche said, \"but I think it would be better if Captain Dancy saw that letter.\"\n\nBruce handed it over. He saw that Charity Hoche was watching Helene Dancy's face as carefully as he was for her reaction. And they were both disappointed. Her face showed no reaction. She did look at Charity, however, as she folded the letter and stuffed it back into the envelope.\n\n\"May I make a suggestion?\" Capt. Dancy asked.\n\n\"Certainly,\" Bruce said.\n\n\"If you were to tell Lieutenant Jamison that Miss . . . or Lieutenant, which would probably be better . . . that _Lieutenant Hoche_ will be devoting half of her time to dealing with female personnel at Whitbey House for me, there would be no reason not to go ahead and send her out there as originally planned.\"\n\n\"Good idea,\" Bruce said after a moment. \"We'll just have to get Jamison some other help.\"\n\n\"I would say that it would take her two or three days to read the files here,\" Dancy said. \"In the meantime, she can stay with me.\"\n\n\"That's very kind,\" Charity said.\n\n\"Not at all,\" Captain Dancy said. \"I'm going to run you by the bar in the Dorchester. Maybe I can latch on to one of your rejects.\"\n\nCharity laughed with delight. They smiled at each other.\n\n_Womanly smiles,_ Bruce thought. _Even girlish._\n\nBut there was more to both of them than that. He reminded himself that another of his weaknesses was underestimating the female animal.\n\n\"I'll go fetch Colonel Stevens, Sir,\" Capt. Dancy said.\n\n#  **6**\n\n##  **PEARL HARBOR U.S. NAVAL BASE OAHU ISLAND, TERRITORY OF HAWAII 1615 HOURS 15 FEBRUARY 1943**\n\nCommander Edwin R. Lennox, wearing the trousers and shirt of a tropical worsted uniform\u2014the blouse hung from a protruding bolt on the _Drum_ 's conning tower\u2014 watched as the last of the fresh food was carried aboard. An hour before, an officer courier had delivered his sailing orders. They were in two sealed envelopes, numbered \"1\" and \"2.\"\n\nThe first order, by authority of COMSUBFORPAC, directed Lennox to take the _Drum_ to sea at 0600 16 February 1943. He was to sail to coordinates that would put him two hundred miles south-southwest of Pearl. Upon arrival there, he was directed to open envelope \"2.\" The second envelope would define the area the _Drum_ was to patrol, engaging enemy naval forces and shipping \"until such time as the expenditure of torpedoes, fuel and victuals, in your sole judgment, dictates your return to Pearl Harbor.\"\n\nAs soon as the last of the fresh food was stowed aboard, it was Lennox's intention to go ashore, mail his last letter to his wife, and then go to the officers' club for a steak and as many drinks of Kentucky sour mash bourbon as he could handle and still make it back to the _Drum_ under his own power by midnight.\n\nA Navy gray Plymouth sedan came onto the wharf and stopped beside the ton-and-a-half rations truck. A white hat jumped out from behind the wheel, opening the rear door and then standing to attention as a full commander in a crisp white uniform got out and walked to the center of three gangplanks laid from the wharf to the deck of the _Drum._ The thick golden rope of an aide to a flag officer hung from the shoulder of the crisp white uniform.\n\nThe admiral's aide walked down the gangplank, stopped, and crisply saluted the officer of the deck, who was wearing shorts, a T-shirt, an incredibly dirty brimmed cap he thought was a lucky piece, and a .45 in a holster slung low on his hip like a gunfighter's.\n\n\"Request permission to come aboard, Sir,\" the admiral's aide said in the prescribed nautical manner.\n\n\"Permission granted,\" the officer of the deck said, returning the salute far more casually than it had been rendered. There was in it faint overtones of the scorn felt by submarine officers about to go back on patrol for officers who walked around Pearl Harbor in crisp white uniforms dog-robbing for an admiral.\n\nThe admiral's aide saluted the colors and stepped onto the deck.\n\n\"I wish to see the captain, Sir,\" the aide said.\n\n\"Ask the commander to come up,\" Lennox called down. He didn't want to go into the hull. It was hot down there, and he was freshly showered and in a fresh uniform.\n\nVery carefully, so as not to soil his uniform, the admiral's aide climbed the ladder welded to the side of the conning tower.\n\n\"What can I do for you, Commander?\" Lennox asked.\n\n\"I have two documents for you, Captain,\" the admiral's aide said. \"Your operational order has been revised. May I suggest we go to your cabin?\"\n\n\"Yes, Sir,\" Lennox said. \"You want the original back?\"\n\n\"Please,\" the admiral's aide said.\n\n\"Watch yourself,\" Commander Lennox said as he entered the conning tower. \"It's pretty greasy in here.\"\n\nThey made their way to the captain's cabin, which was the size of a small closet. Lennox worked the combination of the safe and exchanged envelope \"2\" in it for an identical envelope handed him by the admiral's aide.\n\n\"Can I lock it?\" Lennox asked. \"You said 'two documents'? \"\n\n\"You can lock it,\" the admiral's aide said, and, when Lennox had closed the safe and twirled the dial, handed him a second envelope.\n\nLennox opened it and looked at it incredulously.\n\nMR. AND MRS. H. FREDERICK DENNISON REQUEST THE HONOR OF THE PRESENCE OF\n\nLt. Commander Edwin R. Lennox, USN\n\nAT COCKTAILS AND DINNER 5:30 p.m. February 15, 1943 411 OCEAN DRIVE, WAIKIKI\n\n\"What the hell is this?\" Lennox blurted.\n\n\"Beautiful place,\" the admiral's aide said. \"Mr. Dennison owns most of the movie theaters in Hawaii. And some other things, like maybe half of downtown Honolulu.\"\n\n\"Well, would you please express my regrets to Mr. Dennison? \" Lennox said. \"I have other plans.\"\n\n\"The Admiral thought you might,\" the admiral's aide said. \"That's why he sent me to deliver the invitation. It is the Admiral's desire, Commander, that you accept Mr. Dennison's invitation.\"\n\n\"I'm sailing at 0600,\" Lennox said.\n\n\"The Admiral is aware of that, Commander,\" the aide said.\n\n\"He's going to be there?\" Lennox asked.\n\n\"Oh, yes,\" the admiral's aide said. \"The Dennisons really know how to throw a party. Ever been to a luau, Commander? I mean a real one?\"\n\n\"Oh, what the hell!\" Lennox said. \"But why me?\"\n\n\"The Dennisons like to do what they can for the fleet,\" the admiral's aide said. \"I don't suppose you've got whites, do you?\"\n\n\"No, I don't,\" Lennox said.\n\n\"Pity,\" the admiral's aide said. \"You about ready to go?\"\n**VIII**\n\n#  **1**\n\n##  **\"ROLLING WAVES\" WAIKIKI BEACH, OAHU, TERRITORY OF HAWAII 15 FEBRUARY 1943**\n\nIt was a forty-five-minute drive from Pearl Harbor to the Dennison estate on the beach at Waikiki. The party was well under way by the time Lennox got there. The red-brick curved driveway before the long, low house was packed with cars, more than half of them military and naval staff cars. Lennox saw that many of the service cars had what looked like a second license plate covered with a canvas sleeve. He knew what they concealed: the starred plates identifying the passengers as admirals and generals.\n\nLennox realized that not only was he going to be out of place in his tropical worsted uniform but outranked by a platoon of brass hats and their entourages. This was no place for a simple submarine sailor to be.\n\nAnd when they were inside, and a houseboy had led them to a two-bartender bar set up by a large swimming pool, he saw two movie stars. Floating around in the pool with sort of inner tubes under their arms and drinks in their hands were Lana Turner and one of those too-handsome, too-perfect actors. It took him a minute to place the guy as Greg Hammer.\n\n_How does a large, splendid physical specimen like that avoid his draft board?_\n\nHe realized there must be two hundred people in the Dennison mansion. One in five was female. For woman-scarce Hawaii, that was an unusual percentage of females. Some of them were wives, but many were unattached.\n\n_Why am I surprised? Where did I expect the pretty girls to be, in downtown Honolulu trying to pick up sailors?_\n\nHe saw COMSUBFORPAC, which wasn't surprising, and CINCPAC, which was. He wondered why the hell COMSUBFORPAC had wanted him at the party. Probably, he thought somewhat bitterly, to give the condemned man a last hearty meal.\n\nCOMSUBFORPAC saw him, nodded, and gave him a quick smile, but made it clear by quickly looking away that Lennox was not expected to pay his respects to him in person at that time.\n\nAnd then the Admiral's aide disappeared, and Lennox was left alone. He finished his first drink, had the bartender make him another, and then wandered around until he came to the buffet.\n\nWhat he would do, he decided, was eat. They weren't serving the steak he had been looking forward to, but it was beyond reasonable argument a hearty, luxurious meal. There were roast pigs, \"steamboat\" restaurant rounds of roast beef, fish, and chicken. He tried to remember where he had seen a more luxuriant display of food, but nothing came to him.\n\nHe carried his tray outside the building and sat on a low brick wall beyond which was the white sand beach and the ocean. The food turned out to taste as good as it appeared, and he ate everything he had heaped on his plate.\n\nLennox had just lit a cigar when the Admiral's aide came for him.\n\n\"I wondered what had happened to you,\" the aide said.\n\n\"I was about to come looking for you, Commander,\" Lennox said. \"I've got to think about getting back to Pearl.\"\n\n\"We'll get you back to the _Drum,_ \" the aide said. \"But right now, will you come with me, please?\"\n\n\"Where are we going?\"\n\nThe aide did not reply. Lennox followed him around the pool, then through a long, high-ceilinged living room, and then down a corridor. The aide stopped before a door and knocked.\n\n\"Come!\" a male voice said.\n\nIt was a den, a private office.\n\nInside were CINCPAC, COMSUBFORPAC, CINC-PAC'S aide, a very good-looking young woman, an Air Corps captain, and movie star Greg Hammer in the uniform of a first lieutenant of the Army's Signal Corps.\n\nLennox was a little embarrassed about what he had imagined when he saw Hammer floating around in the pool. He was clearly not a draft dodger. But not too embarrassed. He'd heard about Hollywood movie stars going into the services. There was a Marine aviation squadron with Macdonald Carey and Tyrone Power in it, conveniently stationed in Diego, where they had rented a hotel so they wouldn't be forced to put up with the discomfits of a BOQ. Clark Gable had been commissioned a lieutenant in the Air Corps. Ronald Reagan was making training films in Hollywood as a first lieutenant. It was therefore not surprising to find Greg Hammer in an officer's uniform.\n\n\"Miss Chenowith,\" CINCPAC said, \"may I present Commander Lennox, captain of the _Drum?_ \"\n\nCynthia Chenowith gave him her hand and said she was glad to meet him. Her hand was the first female hand Lennox had touched in a year, and it was warm and soft, and he unkindly wondered who was privileged to jump Miss Chenowith.\n\n\"Miss Chenowith is connected with Continental Studios, \" CINCPAC said. \"And I'm sure you recognize Lieutenant Greg Hammer?\"\n\n\"Yes, of course,\" Lennox said, shaking the movie star's hand.\n\n\"And this is Captain Whittaker, of the Air Corps,\" CINCPAC said.\n\n\"How are you, Commander?\" Whittaker said, and gave Lennox his hand.\n\nLennox couldn't remember having seen Whittaker in a movie, but then he had never paid all that much attention to Hollywood pretty boys. At least Whittaker had gone to flight school; there were aviator's wings, if no ribbons, on his blouse.\n\n\"You may have wondered, Commander,\" CINCPAC said, making his little joke, \"why I have called this meeting. \"\n\nLennox laughed, dutifully.\n\n\"Yes, Sir,\" he said, \"I have.\"\n\n\"Continental Studios,\" CINCPAC said, \"has decided to make a motion picture documentary of a submarine patrol. The Navy has promised its full cooperation, and, after consulting with Admiral Keene, I have selected the _Drum_ to participate.\"\n\n\"I don't quite understand, Sir,\" Lennox said. He didn't quite believe what he was hearing.\n\n\"Captain Whittaker and Lieutenant Hammer will be sailing with you, Lennox. Plus a Navy enlisted photographer's assistant.\"\n\n\"On patrol, Sir?\" Lennox asked, incredulously.\n\n\"As I understand the way it will work,\" CINCPAC said, \"Greg Hammer will serve as narrator, Captain Whittaker will function as director\/producer, and the white hat will operate the camera.\"\n\n_If you open your mouth and say one word, Lennox, it will run away with you and you will tell CINCPAC, COMSUBFORPAC, and the pretty lady with the gorgeous breasts precisely what you think of the dumbest fucking idea you have ever heard of._\n\n\"Yes, Sir,\" Commander Lennox said.\n\nAnd then, in desperation, he thought of something that just might keep them from putting this idiotic idea into practice.\n\n\"I presume that you gentlemen and the sailor have gone through the school at New London?\" Lennox asked.\n\n\"No,\" Captain Whittaker said. \"We thought about it, but we couldn't find time in the schedule.\"\n\n\"Sir, may I respectfully suggest that poses a pretty severe problem?\" Lennox said. \"We have no way of knowing if these gentlemen can take the atmospheric pressures of the boat.\"\n\n\"We checked with the fleet surgeon about that, Lennox,\" COMSUBFORPAC said. \"He feels that, after examining their last physical examinations, there is no reason they will have trouble.\"\n\n\"Sir, may I suggest there are psychological considerations as well? There is the question of confinement, claustrophobia . . . \"\n\n\"Perhaps Admiral Keene didn't make himself clear,\" CINCPAC said, a little sharply. \"The potential medical problems have been considered, and judged to be manageable. \"\n\n\"Yes, Sir,\" Lennox said.\n\n\"Captain Whittaker and Lieutenant Hammer, and the white hat, will come aboard the _Drum_ at 0530,\" COMSUBFORPAC said. \"Their gear will be loaded aboard between now and then.\"\n\n\"Their gear, Sir?\" Lennox asked.\n\n\"Their cameras and recording equipment and film,\" COMSUBFORPAC said.\n\n\"And the rubber boats,\" Captain Whittaker said. \"And their outboard motors.\"\n\n\"We plan to inflate them when we're at sea,\" Greg Hammer offered, \"for what we call long shots, location shots.\"\n\n\"I don't know where we're going to find the room to store any rubber boats,\" Lennox said.\n\n\"Perhaps,\" CINCPAC said, \"it might be a good idea for you, Lennox, to go aboard now and supervise the loading yourself.\"\n\n\"Aye, aye, Sir,\" Lennox said. \"Your permission to withdraw, Sir?\"\n\n\"Granted,\" CINCPAC said. He offered Lennox his hand. \"Good hunting, Commander.\"\n\n\"Thank you, Sir,\" Lennox said. He nodded at the others and walked out of the room.\n\n_\"Good hunting\"? Jesus H. Christ! How the hell can I hunt for anything with a couple of second-rate movie stars and a photographer on board? What the fuck did I do to deserve this?_\n\nThe admiral's aide followed him back down the corridor and through the living room and to the bar by the swimming pool, where Lennox ordered a double bourbon and drank it neat.\n\nHe looked the admiral's aide in the eye.\n\n\"Have they lost their fucking minds, or what? If it's so important to make a fucking movie, why not send a couple of photographer's mates, submarine-qualified photographer's mates? Two fucking movie stars? It's absolutely insane! \"\n\n\"Yours not to reason why, Commander,\" the aide said. \"Yours but to do and die\u2014meanwhile being very courteous to your passengers. They have friends in high places.\"\n\nHe was never to know how close he came to being decked by the captain of the USS _Drum._\n\nWhen the Plymouth dropped him off at the wharf where the _Drum_ was tied up, there were half a dozen sailors staggering under the weight of small wooden boxes.\n\nLennox went aboard.\n\n\"What the hell is going on, Skipper?\" the officer of the deck asked.\n\n\"We are taking two movie stars, plus a movie cameraman, with us,\" Lennox said.\n\n\"What?\"\n\n\"There are supposed to be rubber boats and outboard motors,\" Lennox said, ignoring the question.\n\n\"I put two rubber boats with motors in the aft torpedo room,\" the officer of the deck said. \"I don't know how the hell anybody will be able to move in there. For sure, we won't be able to load the tubes with the boats in there.\"\n\n\"And the rest of their equipment?\"\n\n\"That wasn't so hard to store,\" the officer of the deck said. \"There were a couple of boxes maybe five feet long. Everything else is in those little boxes. They're heavy as hell. What's in them?\"\n\n\"What does it say on the boxes?\"\n\n\" 'Photographic Film. Do Not X-Ray.' \"\n\n\"Then, presumably, they contain motion picture film,\" Lennox said. \"See the chief of the boat, and tell him we'll have one more white hat with us. The movie stars will share bunks with the officers.\"\n\n\"Aye, aye, Sir,\" the officer of the deck said. \"May I ask which movie stars?\"\n\n\"Greg Hammer is one of them,\" Lennox said. \"The other is a guy named Whittaker. Never heard of him. An anonymous celebrity, so to speak.\"\n\n\"I know Hammer,\" the officer of the deck said.\n\n\"By the time this patrol is over, you will know him intimately, \" Lennox said. \"Good night, Mr. Downey.\"\n\n\"Good night, Skipper.\"\n\n#  **2**\n\n##  **FORD ISLAND, PEARL HARBOR NAVY YARD OAHU, TERRITORY OF HAWAII 16 FEBRUARY 1943**\n\nAt five minutes to six, twenty-five minutes late, CINC-PAC'S Cadillac limousine came onto the wharf. CINC-PAC'S aide, the two movie stars, and the woman from Continental Studios were in the back, CINCPAC's aide sitting on a jump seat. There was a very slight, bespectacled, very boyish-looking sailor in front with the driver.\n\nThe driver opened the door for them, and then, as they waved cheerfully at Lennox, the boyish-looking sailor took two small canvas bags from the trunk and carried them aboard.\n\nThe crew looked at the wharf in unabashed curiosity.\n\nCapt. Whittaker suddenly grabbed Miss Chenowith and kissed her on the mouth. The crew of the _Drum_ whistled and cheered.\n\nMiss Chenowith freed herself, turned to Lt. Hammer, and kissed him on the mouth.\n\nThe crew whistled and cheered again.\n\nWhittaker and Hammer walked down the gangplank and stepped onto the deck of the _Drum_. They did not salute the officer of the deck, nor ask permission to come aboard. They just walked on board and went into the conning tower as if they were boarding the Staten Island Ferry.\n\n\"Make all preparations to get under way,\" Commander Lennox ordered.\n\nThe Navy band on the wharf, following tradition, began to play \"Anchors Aweigh.\"\n\n\"Remove the gangplanks, loosen up all lines fore and aft,\" Lt. Rutherford ordered.\n\nCommander Lennox sensed movement behind him. He turned and saw Capt. Whittaker's head and shoulders coming through the hatch.\n\n\"Morning,\" Whittaker said cheerfully.\n\nA moment later, Lt. Hammer came through the hatch.\n\nWith a massive effort, Commander Lennox smiled.\n\n\"If you gentlemen will be good enough to stand back there,\" he said, pointing.\n\n\"Sure,\" Whittaker said. \"We don't want to be in the way.\"\n\nBoth of them waved at the girl on the wharf. Both of them, Lennox saw, wore evidence of her lipstick. She waved back.\n\n\"Cast off all lines,\" Lennox said. \"Secure all deck hatches. Half left rudder. Ahead dead slow.\"\n\nThe _Drum_ shuddered just perceptibly as the engines engaged. Very slowly, she moved away from the dock.\n\nWhen they were in the channel, moving past Battleship Row, Lennox turned to Rutherford.\n\n\"You have the conn, Mr. Rutherford,\" he said. \"Take us to sea.\"\n\n\"Aye, aye, Sir.\"\n\n\"And if you gentlemen don't mind, I would like a word with you in my cabin.\"\n\nCommander Lennox delivered a brief, precise, and pungent lecture on the customs of the Naval Service as they applied to submarine service, starting with the information that one was supposed to ask permission before boarding a Naval vessel and touching on such items as the prohibition from entering the bridge without the specific permission of the captain.\n\nAnd then he warmed to his subject.\n\nSo far as he was concerned, he told them, this movie documentary was the dumbest goddamned thing he had heard of in his eight years in the Navy.\n\nIn addition to that, he didn't like the attitude of either of them. He was the captain of a vessel at sea, and when they spoke with him, they would call him either \"Captain\" or \"Sir.\" But for the time being, he said, he would be pleased if they didn't speak to him unless spoken to, and he would consider it a personal favor if they would take their meals in the wardroom when he was not there. Movie actors in officers' uniforms ruined his appetite.\n\nAs far as he was concerned, his business was sinking Japanese ships, not making some kind of bullshit movie. They should conduct themselves accordingly.\n\nCapt. Whittaker and Lt. Hammer took the speech without comment, which Lennox found disturbing. He had hoped they would argue with him, which would have given him the chance to really eat ass, and possibly even an excuse to throw their goddamned rubber boats and movie cameras over the side.\n\n\"Sir,\" Whittaker said respectfully, \"we will do our best to keep out of your way.\"\n\n\"See that you do,\" Lennox said. \"You are dismissed.\"\n\nOnce he had finished blowing his top, Lennox was a little ashamed of himself. He told himself they had their orders, too, even if those orders were to make a fucking movie. And now that he had calmed down a little, he understood that he had been something of a prick to them.\n\nThey were still several hundred miles from the position in the Pacific where he was authorized to open envelope \"2,\" but he went to the safe and got it anyway. Maybe, once he knew where they were going, he would be able to suggest to the movie stars something they could take pictures of. Maybe that would make up for his having acted like a horse's ass.\n\nHe tore the envelope open.\n\n**TOP SECRET COMMANDER SUBMARINE FORCE PACIFIC PEARL HARBOR, TERRITORY OF HAWAII**\n\nTo: Commanding Officer USS Drum SS228\n\n1. By Direction of the President, you will proceed to the Island of Mindanao, Territory of the Philippines, and there put ashore, at such place and at such time as he may designate, Captain James M. B. Whittaker, USAAC, and such personnel and equipment as he may desire.\n\n2. While the nature of Captain Whittaker's duties while ashore in the Philippines are classified and are not to be inquired into, you are hereby informed that his duties have the highest priority, and that the entire efforts of the Drum and its crew are to be devoted to its accomplishment, to the exclusion of all else.\n\n3. After putting Captain Whittaker and his party ashore, you will put out to sea to a position determined by Captain Whittaker where you will maintain a radio communications schedule with Captain Whittaker, or his designate, at such times as he may require.\n\n4. On receipt of the appropriate orders from Captain Whittaker, you will take him, and whomsoever else he designates, together with whatever material and\/or equipment he may designate, from the shore of Mindanao at such time and place as he may designate. You will then transport him and boarded personnel and\/or equipment and material to such destination as he designates.\n\n5. You are specifically forbidden to engage in any action against the enemy unless specifically authorized to engage by Captain Whittaker.\n\n6. You are directed to ensure by whatever means necessary that your officers and crew understand both the priorities of this mission, its classification, and the absolute necessity that it remain TOP SECRET.\n\nBy direction:\n\n_G. H. Keene_\n\nGeoffrey H. Keene, Rear Admiral, USN\n\nCommander Lennox said, \"Oh, _shit!_ \" so loudly and with such fervor that his voice penetrated the baize curtain that served as the door to his cabin and could be heard above the rumble of the diesel.\n\nThe chief of the boat put his head past the curtain.\n\n\"You called, Captain?\"\n\n\"Moaned was more like it,\" Lennox said. \"Would you tell the exec to come here right away, Chief? And then ask the Army officers to join me at their convenience?\"\n\n\"Aye, aye, Sir,\" the chief of the boat said.\n\n\"And I'll want you in on this, too, Chief,\" Lennox said.\n\nEveryone was there in a matter of minutes.\n\n\"Chief, I don't want anybody using the passage while this is going on,\" Lennox said. \"Put some guards out, and then come back in here.\"\n\nWhen they were all crowded into the tiny cabin, waiting to hear what he had to say, Lennox said:\n\n\"Except to announce that I really showed my ass a while back, for which I apologize, I don't really know what to say. May I have your permission to show my orders to my exec and the chief of the boat, Captain Whittaker?\"\n\n\"I think that would be a good idea,\" Whittaker said.\n\nThe chief of the boat read the orders over the exec's shoulder. Both of them registered surprise on their faces but said nothing.\n\n\"No questions?\" Whittaker asked.\n\n\"What's in the boxes?\" the chief of the boat asked.\n\n\"The long ones are packed with carbines,\" Hammersmith answered.\n\n\"And half the others are filled with ammo,\" Whittaker added.\n\n\"And the other half?\"\n\n\"A million dollars' worth of gold coins,\" Whittaker said.\n\nThe chief of the boat accepted that stoically.\n\n\"Gonna be a bitch getting that stuff ashore in rubber boats,\" he said. \"I don't suppose the people who'll be meeting you would have boats, real boats, something big enough to handle that weight?\"\n\n\"That's one of our problems, Chief,\" Whittaker said. \"Nobody knows we're coming.\"\n\n\"Holy shit!\" the chief of the boat said, and then immediately got control of himself. \"Well, we'll figure something out, Captain.\"\n\n#  **3**\n\n##  **16 DEGREES 20 MINUTES NORTH LONGITUDE 43 DEGREES 5 MINUTES NORTH LATITUDE (OVER THE ADRIATIC SEA) 1520 HOURS 16 FEBRUARY 1943**\n\nThe B-25G \"Mitchell\" had been alone for hours high in the bright blue sky, its passage around the heel of the Italian boot and up the center of the Adriatic marked by twin trails of condensation behind it. Far beneath it was an unbroken bed of clouds, stretching as far as the eye could see, looking like a vast layer of cotton wool.\n\nDolan was at the controls, Canidy in the copilot's seat, and Darmstadter was sitting on a fold-down jump seat immediately behind the pilots' seats. It was uncomfortable on the jump seat, but the foam-rubber and leather seats in the fuselage had little appeal for Darmstadter. When he was alone in the fuselage, he had too great an opportunity to think of what could go wrong. He was finding what reassurance he could from being close to Canidy and Dolan.\n\nDarmstadter had been in the left seat when they left Malta and had made the takeoff. But Canidy had taken over the controls after they had left the ground, and he was the one who had set the course and rate of climb and fine-tuned the engines and the mixture.\n\nAnd then, matter-of-factly, he had told Darmstadter where they were going\u2014but not why\u2014and pointed out their course on a chart.\n\nAnd then he had told him, patiently, even kindly, as a flight instructor teaches a student pilot, how it was planned for them to find Vis and what would happen if things went wrong.\n\nCanidy explained that the OSS agent with the British SOE force on Vis had a radio transmitter-receiver capable of operating on the frequencies used for aviation. Using the radio direction-finding equipment on the B-25G, they would home in on Vis very much as they would home in on Newark Airport after a flight from Washington.\n\nWith several significant exceptions:\n\n\"The trouble with RDF transmitters,\" Canidy said, \"is that they can be picked up by anybody tuned to that frequency. For example, German or Italian aircraft. A curious Luftwaffe pilot looking for the way home from a patrol over the Adriatic might come across the signal from Vis and wonder what the hell it was.\"\n\n\"The worst possible scenario is that two pilots, or for that matter, two ground stations, might hear the Vis transmission at the same time and mark their position and the relative position of the Vis transmitter on a chart. If they did that, all that would have to be done would be to put the chart marks together. Triangulation. You with me?\"\n\nDarmstadter nodded. He knew that without actually following a signal to its source, the location of the transmitter could be easily determined. \"Triangulation\" simply meant the drawing of straight lines on a chart from two different points of reception toward the source of the signal. The intersection of the straight lines indicated the location of the transmitter.\n\n\"So what they're going to do to reduce the odds of getting caught,\" Canidy said, \"is to go on the air as little as possible. The first signal we'll listen for when we get close enough will be on the air for only five minutes. Then it will go off and come back on fifteen minutes later for sixty seconds on a different frequency and using different call letters.\"\n\nHe handed Darmstadter a typewritten list.\n\nThere were three columns. The first gave times, starting at 1500 and ending at 1745. Sometimes there was nineteen minutes between transmissions, and sometimes as little as eleven minutes. But there were no two intermissions alike. The second column listed the frequency of the transmissions. No two of these were alike. The third column listed the three-letter identification code that the transmitter would send, endlessly repeating them for the period of time it would be on the air.\n\n\"Clever,\" Darmstadter said.\n\n\"It presumes our guy on Vis has the transceiver, and that it's working, and that we'll be able to pick it up when we have to,\" Canidy said.\n\n\"And if we don't?\" Darmstadter asked.\n\n\"That could pose some problems,\" Canidy said. \"You'll notice that the Point of No Return on the chart is here, and the point where we hope we can pick up the Vis RDF transmitter is here.\"\n\nDarmstadter saw that the first place they could hope to pick up the direction-finding signal was at least two hundred miles from the Point of No Return.\n\n\"And if we don't get the RDF signal?\"\n\n\"Then we go down on the deck and try to find it by dead reckoning,\" Canidy said.\n\n\"That would be kind of hard, wouldn't it?\" Darmstadter asked.\n\n\"Think positively, Darmstadter,\" Canidy said dryly. \"But since you posed the question, I think it would be impossible. \"\n\n\"And then what?\" Darmstadter asked.\n\n\"Then you have a choice,\" Canidy said. \"You can take the airplane over the Yugoslav mainland, bail out, and take your chances that the partisans might get you before the Germans do. If the partisans get you, you're home free. If they don't, you'll have to take your chances with the Germans. \"\n\n\"What do you mean by that?\"\n\n\"You tell them you were on a bombing raid, got lost, and bailed out when you ran out of gas. If they believe you, you sit out the war in a Stalagluft [a prisoner-of-war camp for aviation personnel]. If they don't, you're in trouble.\"\n\n\"And where are you and Dolan going to be while I'm taking my chances with the partisans?\"\n\n\"Dolan and I will have drawn the 'Go Directly to Jail, Do Not Pass \"Go\" and Do Not Collect $200' card,\" Canidy said matter-of-factly. \"We can't get captured.\"\n\n\"Why not?\" Darmstadter blurted.\n\n\"Because the Germans can find out anything they want to know from anybody, if they put their mind to it,\" Canidy said. \"And there are certain things that Dolan and I know that you don't, and that the Germans shouldn't.\"\n\n\"What are you going to do,\" Darmstadter asked, horrified, \"to keep from getting captured?\"\n\nCanidy ignored the question. Instead, he handed Darmstadter another typewritten sheet of paper.\n\n\"There will be a bombing raid by B-25 aircraft on the boot of Italy,\" he said. \"Here're the details, what you would be expected to know if you had gone on the mission. Memorize as much as you can, especially your unit, your aircraft number, your departure field. Use your imagination for the names of the crew. I think you can probably get away with it.\"\n\n\"And what, exactly, are you and Dolan going to do?\" Darmstadter asked.\n\n\"To coin a phrase,\" Canidy said, \"we'll cross that bridge when we get to it.\"\n\n\"I'd really like to know,\" Darmstadter persisted.\n\nCanidy thought it over a moment before replying.\n\n\"They gave us a pill,\" he said. \"Actually, it's a small glass vial, filled with what looks like watery milk. When you bite it, it's supposed to work before you feel the little pieces of glass in your mouth. The idea is that we're supposed to bite it when it becomes clear we're not going to make it to Vis. But what I think we'll do is bail out over the mainland and take a chance the partisans will find us before the Germans do. If we land in the lap of the Wehrmacht, then we'll bite the pill.\"\n\n\"What the hell do you know that makes suicide necessary? \" Darmstadter blurted.\n\nCanidy had not responded.\n\nJust over an hour before, Canidy had turned on the radio direction finder. By then, the three of them had relieved for each other at the controls at roughly hourly intervals, and Dolan was then sitting in the pilot's seat. At first, the signal strength indicator needle on the instrument panel had made no response as Canidy turned the crank that rotated the loop antenna mounted atop the fuselage.\n\nThen the needle jumped, just perceptibly, and he reversed his cranking motion, aiming the antenna at the source of the radiation. The needle on the signal-strength indicator crept very slowly, barely perceptibly, upward as the signal strength increased.\n\nAnd then, very faintly, over the static in his earphones, Darmstadter began to be able to recognize one Morse code letter, Dah-Dah-Dah, D, and then another, and finally a third, until there was in his earphones, endlessly repeated Dah-Dah-Dah Dit-Dit-Dit-Dit Dah-Dah-Dah. He wondered if DHD meant something, or whether it had been selected because it was a long, readily recognizable string of letters.\n\n\"I don't think,\" Canidy's voice came dryly and metallically over the earphones, \"that's what they call 'right on the money.' \"\n\nDolan looked up at the roof of the cabin, at the needle on the antenna rotating mechanism. Then he put the B-25G into a very gentle turn, in a very slightly nose-down attitude, and made small adjustments to the throttle and richness controls.\n\nFinally, his voice came metallically over the earphones.\n\n\"Fuck you, Canidy.\"\n\nA moment later, he straightened the B-25 on a course corresponding with that indicated on the radio direction finder, made a minute adjustment of the trim wheel, and then touched his intercom mike button again.\n\n\"And if you can refrain from walking up and down, Darmstadter, like a passenger on a ferry boat, I would be obliged.\"\n\nThen he folded his arms on his chest.\n\nThe B-25 dropped very slowly toward the layer of cotton wool far below them.\n\nThe indicator needle on the signal-strength meter suddenly dropped back to the peg.\n\n\"You've lost the signal,\" Darmstadter said.\n\n\"That's probably because they've stopped transmitting,\" Dolan said dryly.\n\nThe B-25 flew on, in a very shallow descent.\n\nEleven minutes later, when they were still above the cloud cover, there was a one-minute transmission from Vis, and Dolan made a tiny course correction to line the plane up again on course before the signal-strength meter fell back to its peg again.\n\nThey were in the cloud bank when Vis came on the air again. Darmstadter could see about one inch past the windshield. There were a dozen or so drops of condensation on the window frame just past the Plexiglas, for some mysterious aerodynamic reason undisturbed by the air through which they were passing at an indicated 290 knots. But beyond the drops of condensation there was nothing but a gray mass.\n\n\"You don't want to go down to the deck and see if we can get out of this shit?\" Canidy asked. It was a question, Darmstadter understood, not a suggestion, certainly not an order.\n\nDolan shook his head, \"no,\" in reply, and then, a full minute later, spoke.\n\n\"If it looks like it's working, don't fuck with it,\" he said.\n\nIt sounded more as if Dolan was thinking aloud than replying to Canidy, or, Darmstadter thought a moment later, as if Dolan had called that old pilot's clich\u00e9 from the recesses of his memory to reassure himself.\n\nThe point of the needle on the vertical speed indicator was indicating a descent only on close examination; on casual glance, it seemed to indicate level flight. The needle on the altimeter moved counterclockwise very slowly. But it was moving, and they were going down.\n\nTwenty-odd minutes later, during another Vis transmission, Canidy said, \"I wish that transmitter wasn't working quite that well.\"\n\nIt took Darmstadter a moment to understand what he meant. Then he did. The needle on the signal-strength meter was now resting against the upper-limit peg; there was no way to judge if they were moving ever closer to the transmitter. The signal-strength meter was accepting all the signal strength it was capable of.\n\nWhen the altimeter indicated 12,000 feet, Canidy pulled his oxygen mask free from his face and rubbed his cheeks and under his chin with his fingers. When Darmstadter removed his own mask, the fresh air passing through his nostrils and mouth seemed warm and moist. Dolan did not take his mask off. Darmstadter wondered if this was a manifestation of the declaration he'd made earlier, \"if it looks like it's working, don't fuck with it,\" or if Dolan's concentration was on other things and he simply hadn't noticed they were at an altitude where it was safe to fly without bottled oxygen.\n\nAnd then, suddenly, startlingly, they dropped out of the cloud cover. There was an ocean down there, and land to the front and the sides.\n\nCanidy frantically searched through his aviator's briefcase and came up with a handful of eight-by-ten-inch glossy photographs. Dolan ripped his oxygen mask off.\n\n\"What was that you were saying, Dick, about 'right on the money'?\" he asked.\n\n\"Jesus,\" Canidy said. \"And I was right on the edge of agreeing with David Bruce that they shouldn't let old men like you fly.\"\n\nThe two looked at each other and beamed.\n\n\"Take her down to the deck, and make your approach around that hill on the left,\" Canidy said.\n\n\"Hey,\" Dolan said, annoyed, \"I'm driving.\"\n\nBut he lowered the nose of the B-25, until they were no more than a thousand feet off the choppy seas of the Adriatic, and made a wide, sweeping turn around the hill Canidy had indicated.\n\nWhen they crossed the rocky beach, they immediately encountered the steep hills of Vis; so an indicated altitude of one thousand feet, which was based on sea level, put them no more than two or three hundred feet over the side of the hill, and then the level valley on shore.\n\n\"Go strap yourself in,\" Dolan ordered. \"Quickly.\"\n\nReluctantly, Darmstadter made his way back to the leather-upholstered passenger chairs in the fuselage. He had just sat down, and was fumbling for the seat belt, when the nose of the B-25 lifted abruptly. Ignoring the seat belt, he pressed his nose against the Plexiglas.\n\nThere were fifteen or twenty people on a crude runway, their arms waving in a greeting.\n\nThen Dolan stood the B-25 on its wing and began a one-hundred -eighty-degree turn. As the plane leveled off, there came the sound of hydraulics as the flaps and gear came down, and the engines changed pitch.\n\nDarmstadter got his seat belt in place just as the plane touched down. There was a far louder than he expected rumble from the landing carriage, followed immediately by the change of pitch as the engine throttles were retarded. And then the plane lurched as if something had grabbed it.\n\nInstantly, Darmstadter's view through the Plexiglas disappeared in a gross distortion, and then almost as quickly the distortion seemed to be wiped away. He realized that what had happened was that water, a great deal of water, had splashed against the window.\n\nThe plane was now braking hard. Darmstadter felt himself being pressed against the upholstery of the rear-facing chair.\n\nAnd then it stopped for a moment and then turned around. As Darmstadter unfastened his lap belt, the engines died. The silence, broken only by the faint pings and moans of cooling metal, was surprising.\n\n\"Vis International Aerodrome,\" Canidy called cheerfully from the cockpit. \"Connections to Budapest, Voodapest, Zoodapest, and all points east. _Thank you_ for flying Balkan Airlines.\"\n\nChuckling, Darmstadter got to the access hatch in the floor behind the cockpit just after Canidy had dropped through it to the ground. Darmstadter jumped after him.\n\n#  **4**\n\n##  **HEADQUARTERS, 344TH FIGHTER GROUP ATCHAM ARMY AIR CORPS STATION, ENGLAND 1650 HOURS 16 FEBRUARY 1943**\n\nWhen Lt. Colonel Peter Douglass, Jr., returned to his quarters from the postmission debriefing, the Underwood typewriter and the service record were waiting for him on the old and battered desk in his room.\n\nIt was SOP, Standard Operating Procedure. There was a system. There had to be a system. The SOP Lt. Colonel Douglass had set up was that in the case of pilots within a section, their section leaders wrote the letters, subject to review by squadron commanders. In the case of section leaders, squadron commanders wrote the letters, subject to review by the group exec. In the case of squadron commanders, or squadron executive officers, the group commander wrote the letters himself.\n\nDouglass kicked off his sheepskin flying boots, sending them sailing across the small room in the curved-ceilinged Quonset hut. He took off his battered, leather-brimmed hat and skimmed it three feet toward a hook on the wall. It touched the hook, but bounced off and fell to the floor. He made no move to pick it up.\n\nHe reached into the pocket of the sheepskin flying jacket and came out with two miniature bottles of Old Overholt rye whiskey. Eighth Air Force SOP provided for the \"post-mission issue of no more than two bottles, 1.6 ounces, bourbon or rye whiskey 86 proof or 100 proof to flight crew personnel, when, in the opinion of the attending flight surgeon, such issue is medically indicated.\"\n\nThe Eighth Air Force SOP went on to stipulate that \"in no case is the issue of more than two bottles permitted\" and that \"wherever possible, the issue of medicinal whiskey will be made only after flight crews have undergone postmission debriefing.\"\n\nAnd finally, the Eighth Air Force SOP stated that \"medicinal whiskey so issued will be ingested in the presence of the prescribing flight surgeon.\"\n\nTranslated, that meant that unless you watched those crazy pilots, or, in the case of bombers, navigators, bombardiers, flight engineers, and aerial gunners, they were liable to hoard their \"bottles, 1.6 ounces\" of medicinal whiskey until they had enough to tie a load on, or worse, share it with people not entitled to medicinal whiskey.\n\nLt. Colonel Douglass walked to the battered desk, pulled the drawer open, and carefully laid his miniature bottles in it. There were already a dozen other bottles there. It was the 344th Fighter Group commander's unofficial SOP to pass out his ration of medicinal booze to his pilots when he thought such issue was indicated for morale purposes. Sometimes he passed it out to the enlisted men, too, in contravention of the spirit and letter of the Eighth Air Force SOP.\n\nIt bothered the hell out of the ground crews when their plane and pilot didn't come home. And some took it worse than others.\n\nSaving the miniatures to pass out as he saw fit did not represent any sacrifice, booze-wise, on the part of Lt. Colonel Douglass. He had his own out-of-supply-channels source of booze, and when he had a couple of medicinal postmission nips, he took them from a bottle of Scotch.\n\nHe shrugged out of the sheepskin, high-altitude flying jacket and threw it toward his bed. It, too, fell short of the target and slid to the floor. He left it there, then pushed the suspenders holding up the sheepskin trousers off his shoulders. He stood on one leg to pull the trousers off, then on the other leg to get them completely off. Then he threw them toward his bed. This time he made it.\n\nHe then picked up a telephone.\n\n\"Meteorology,\" he said when the operator came on the line. And then, a moment later, \"What have we got, Dick?\"\n\nHis weather officer predicted perfect\u2014that is absolutely unflyable\u2014weather in England and over the European landmass for not less than forty-eight hours, and probably for as much as seventy-two or ninety-six hours.\n\n\"There's a stationary front, Colonel, a massive chunk of arctic air, which, meeting with an equally massive chunk of warm air from the Mediterranean\u2014\"\n\n\"What your colonel had in mind, Captain,\" Lt. Colonel Douglass interrupted him, \"is whether or not it would be safe for him to get drunk for a day or two.\"\n\n\"In my professional meteorological opinion, Sir,\" the weather officer said, \"you have that option.\"\n\n\"Thank you,\" Douglass said.\n\n\"Colonel, I'm sorry about Major Till,\" the weather officer said.\n\n\"Yeah,\" Lt. Colonel Douglass said after a moment. \"Thank you.\"\n\nThen he hung up.\n\nHe went to a large, sagging-to-one-side wardrobe and worked the combination of the long-shafted bicycle padlock that, looped through two eye-rings, locked it. He opened the left door and looked inside, and then, frowning, the right door.\n\n_One lousy, half-empty imperial quart of Scotch! What the hell had happened to the rest of it?_\n\nHe didn't like his own answer. _I have drunk the rest of it, that's what has happened to it. A couple of little nips here,_ _and a couple more there, and the four imperial quarts of straight malt Scotch have evaporated._\n\nWell, what the hell, there was more where that came from. There was a sturdily locked room at Whitbey House stacked to its high ceiling with booze. Canidy ran the OSS Station at Whitbey House on the philosophy that unless his people were now given by a grateful nation the best available in the way of booze and food, there was a good chance that his people would not be around to get it later.\n\nHe would just have to run over to Whitbey House and replenish the larder, that was all there was to it. Canidy had declared him to be an Honorary Spook, with all the rights and privileges thereunto pertaining, such as access to the booze larder.\n\nAnd then he remembered that Canidy was gone. He was off on one of his nobody-knows-anything-about-it missions in his souped-up B-25G. Canidy had given Douglass no details, of course, other than that he \"would be away for a couple of days.\" But then Douglass had learned that Dolan was off somewhere, too. And he'd flown over Whitbey House, and the B-25G normally parked there was gone.\n\nErgo. Canidy and Dolan were off somewhere doing something secret and important in the souped-up B-25G.\n\nThere was a steady, sometimes nearly overwhelming, temptation for Douglass to ask Canidy\u2014or, probably smarter, to ask OSS London Station Chief David Bruce\u2014 to have him transferred to the OSS. And there was little question in his mind that it could be easily arranged: For one thing, if the OSS wanted somebody, they got him. No matter what assignment an officer\u2014or, for that matter, an enlisted man\u2014had, it was not considered as essential to the war effort as an assignment to the OSS.\n\nAnd he was sure that David Bruce had at least considered that Lt. Colonel Peter Douglass, Jr., knew far more about the OSS and its personnel and operations than he was supposed to.\n\nDouglass had flown with Canidy and Bitter with the Flying Tigers in China and Burma, where their airplanes had been maintained by \"Mr.\" John Dolan. It made no sense to indulge the notion that any of them would regard Doug Douglass as someone who couldn't be trusted with classified information, even if all of them, in fact, tried to keep him in the dark.\n\nHe had learned, for example, that Eric Fulmar was in Germany. He hadn't asked. Canidy had told him. He hadn't asked what Fulmar was doing in Germany. And he had tried, unsuccessfully, not to put two and two together. So he had come up with the answer that if Canidy and Dolan had gone off somewhere in the B-25G, it was very likely that they had gone to bring Fulmar home.\n\nFinally, the Deputy Director of the Office of Strategic Services was Captain Peter Douglass, Sr., USN, Doug's father. Considerations of nepotism aside, it made sense to have Peter Douglass, Jr., in the OSS, since he knew so much about it.\n\nThere were reasons Douglass had not asked to be taken in. He would have been embarrassed to speak them out loud, for they would, he thought, seem both egotistical and overly noble. But in his own mind, he was one hell of a fighter pilot and one hell of a commander. By staying where he was, he believed that he was probably saving lives.\n\nHe did not allow himself to dwell on the counterargument, that Canidy and Bitter and Jimmy Whittaker and the others were also saving lives. Not directly, by shooting down a Messerschmitt on the tail of one of his pilots, nor even less directly, by doing the things that a good commander does to keep his men alive, but in an almost abstract sense. If what the OSS was doing could shorten the war by a week, or a day, or even by six hours, that would mean that the guns would fall silent around the world, and more lives would be saved in six hours than he could hope to save by being a good fighter group commander for the rest of the war.\n\nThat argument seemed to be buttressed by the fact that Canidy and Bitter and Whittaker had proven themselves as fighter pilots.\n\nDouglass understood that he would not be asked to join the OSS. If they wanted him in the OSS, he would have been transferred into it long ago. He was going to have to submit an application, no matter how informal, and he didn't want to do that.\n\nLt. Colonel Doug Douglass carried what was left of the imperial quart of Scotch whiskey to the battered desk. He unscrewed the top, took a healthy swig from the neck, and then set the bottle on the desk.\n\nHe sat down and rolled a sheet of printed stationery into the typewriter. Then he typed the date.\n\nHe would, he thought wryly, have been one hell of a squadron clerk.\n\nHe opened the service record and found what he was looking for. His fingers began to fly over the keys.\n\nHeadquarters, 344th Fighter Group APO 86344, New York\n\n16 February 1943\n\nMr. and Mrs. J. Howard Till  \n711 Country Club Road  \nSpringfield, N.J.\n\nDear Mr. and Mrs. Till:\n\nBy now, you will have been notified by the Adjutant General that David has been killed in action.\n\nHe was my executive officer and my friend, and I share your grief.\n\nThe 344th Fighter Group was assigned the mission of protecting B-17 and B-24 bombers of the Eighth Air Force on a heavy bombardment mission to Frankfurt, Germany. The Group was divided into two echelons. David commanded one, and I the other.\n\nSome distance from the target, we were engaged by a large group of German Messerschmitt fighter aircraft. In the engagement that followed, David shot down two German fighters. He was going to the aid of another pilot when his aircraft came under fire from several Messerschmitts. David's aircraft was hit in the fuel tanks, which then exploded.\n\nDavid was instantly killed, probably without warning. He died, I think, as he would have wanted to, in aerial combat, leading his men as they protected other men.\n\n\"Greater Love Hath No Man Than He Lay Down His Life for Another.\"\n\nThe two German fighter aircraft he shot down brought his total kills to six. The posthumous award of the Air Medal (6th Award) has been approved. I have, in addition, just been informed by Eighth Air Force that David will also be awarded the Distinguished Flying Cross, and the citation will reflect his flying skill, devotion to duty and courage, not only on his last flight but during the entire period of his assignment to the 344th Fighter Group.\n\nI am aware that military decorations are small consolation to you at this time, and can only hope that you will accept them as a token of the respect and affection in which David was held, not only by the officers and men of the 344th Fighter Group, but by the highest echelons of the Eighth Air Force.\n\nDavid was a splendid officer and a fine human being. He will be missed.\n\nIf there is anything that I can do for you, please do not hesitate to let me know.\n\nSincerely,  \n _Peter Douglass, Jr._   \nPeter Douglass, Jr.  \nLt. Col., USAAC  \nCommanding\n\nWhen he had finished typing, he rolled the sheet of paper out of the typewriter and read it.\n\nThen he ran an envelope into the machine and typed the envelope. He folded the letter, put it into the envelope, and then wrote \"Free\" on the envelope where a stamp would normally go.\n\nHe picked up the telephone and, when the operator came on the line, said, \"Find Captain Delaney and get him over here, will you?\"\n\nHe walked to a small door beside the washbasin. Beyond was a small cubicle holding a shower and an ancient English water closet with a warped and cracked wooden seat. The shower consisted of a rusting showerhead pointing straight down from the slanted ceiling to the brick floor of the shower. A three-tier layer of bricks kept the shower water in place, and a shower curtain, cut from a condemned parachute, hung from a wooden rod.\n\nAn oil-temperature gauge, somehow modified by Douglass's crew chief, who had also laid the bricks and found the crapper somewhere, was mounted on the wall. The needle, pointing to a green \"OK\" strip, indicated 280 degrees Fahrenheit, but it had been explained to Douglas that he should ignore the indicated temperature; when the needle pointed to the \"OK\" strip, the water was at the proper temperature for a shower.\n\nDouglass went to the wardrobe and took out fresh underwear and a clean uniform. Then he stripped. As he pulled his T-shirt over his head, he winced at the sharp, acrid odor. He knew what it was. It was the enduring odor of sweat-while-terrified. Literally, the smell of fear.\n\nHe relived for a moment the absolute terror he had felt for about twenty seconds when it had looked like the pilot of the Messerschmitt on his tail was going to succeed in turning inside Douglass's turn. It had been as if time had somehow slowed down, like a movie newsreel in slow motion; and while things had been in slow motion, he had been able to see the stream of German tracers moving ever closer to him.\n\nAnd then the stream of tracers had stopped when the German pilot, who was good and knew his trade, realized that he wasn't going to make it. He had turned and dived sharply to the left.\n\nAs Douglass had turned to try to get on the German's tail, he had become aware that he was sweat-soaked.\n\n\"Jesus H. Christ!\" Douglass said disgustedly, throwing the T-shirt to the floor.\n\nHe went to his shower and turned it on full. It was hot, hotter than he liked, even too hot for comfort, but he stood under it, furiously rubbing red Lifebuoy soap over his skin, and then rinsing himself until the entire fifty-five gallons of the water supply in a former oil drum on the roof was exhausted.\n\nHe shut the head off and quickly opened a valve that would replenish the water in the drum. He heard a momentary hiss as the cold water struck whatever it was his crew chief had installed in the drum to heat the water, and he remembered that the crew chief had sternly warned him never to use all the water in the drum, otherwise the heating element would burn out.\n\n\"I've probably fucked that up, too,\" Douglass said aloud.\n\n\"Sir?\"\n\n\"Nothing.\"\n\nDouglass wondered how long he had kept Delaney waiting.\n\nHe wrapped a gray-white towel around his middle and went into his bedroom.\n\nDelaney was a serious-faced Irishman from someplace in Iowa, a devout Roman Catholic with a wife and several kids, although he was only twenty-two or twenty-three years old. He had been sitting in the chair by the desk and had gotten up when Douglass entered the room.\n\n\"Sit!\" Douglass said, and walked to his bed and pulled a clean T-shirt over his head.\n\n\"Who do you recommend to assume command of your squadron, Major Delaney?\" Douglass asked.\n\n\"Sir?\"\n\n\"By the authority vested in me by Eighth Air Force, you have been appointed executive officer of the 344th Fighter Group,\" Douglass said. \"The job carries with it a gold leaf.\"\n\n\"I'm sorry about Major Till, Sir,\" Delaney said.\n\n\"Yeah,\" Douglass said. \"I asked you a question, Major.\"\n\n\"I'm not sure I can handle it, Colonel,\" Delaney said.\n\n\"I made that decision,\" Douglass said. He had his undershorts on by then and was in the process of working his feet into half Wellington boots. When he had them on, he walked to the desk and unscrewed the cap on the imperial quart of Scotch.\n\n\"Till, you unlucky bastard,\" he said, holding the bottle up. \"I hope you went quick.\"\n\nHe handed the bottle to Delaney.\n\nDelaney wiped the neck on his blouse jacket and took a swig.\n\n\"Maybe he was dead before he went in,\" Delaney said. \"Needham followed him down, and he said he never got the canopy open.\"\n\n\"I just wrote his family that his ship blew up,\" Douglass said. \"One of the things a field grade officer must know, Major Delaney, is when to lie.\"\n\nDelaney looked at him and nodded, but said nothing.\n\n\"I will be gone for the next twenty-four to thirty-six hours,\" Douglass said. \"You will tell whatever lie you think you can get away with if there are inquiries as to my whereabouts. I'm going to leave a number where I can be reached. You will use it only if necessary, and you are to give it to no one.\"\n\n\"May I ask, Sir, where you will be?\"\n\n\"Repeating the caveat that you are to tell no one, I will be at Whitbey House in Kent. It's where the OSS hangs out.\"\n\n\"Yes, Sir.\"\n\n_There was relief in the way he said that, and on his face. The moral sonofabitch was afraid that I was going to tell_ _him that I was going to be shacked up somewhere_.\n\n\"It takes me about an hour and a half to get back here from there,\" Douglass said. \"In case I am needed. I will not be needed to fly. I have checked the weather, and nobody will be flying.\"\n\n\"Yes, Sir.\"\n\n\"If you keep up that 'yes, Sir' crap,\" Douglass said, \"you will almost, but not quite, succeed in making me feel guilty for leaving my new executive officer in charge.\"\n\nDelaney gave him a hesitant smile.\n\n\"Am I allowed to ask what you'll be doing with the OSS?\"\n\n\"I am going to get drunk, Major Delaney,\" Douglass said. \"I do that sometimes when something like Dave Till happens. It ill behooves a commanding officer to get shit-faced somewhere where his subordinates can see him in that condition.\"\n\n\"Yes, Sir,\" Delaney said.\n\n\"You will personally see to Till's personal effects,\" Douglass said. \"Collect them, go through them to make sure there are no dirty pictures, love letters, or anything else that might suggest he was a healthy young male. Make an inventory of what's left, and leave it on my desk.\"\n\n\"Yes, Sir,\" Delaney said.\n\n\"How are you fixed for money?\"\n\n\"Sir?\"\n\n\"You have been promoted, Major,\" Douglass said. \"It is a hoary tradition of the service that you have a promotion party.\"\n\n\"I have money, Sir,\" Delaney said. \"But thank you.\"\n\n\"In this case,\" Douglass explained, \"your party will also serve to keep our young warriors on the base tonight. You will lie again. You will tell them that just before the colonel left for High Wycombe and the headquarters of the Eighth Air Force, he left word that twenty-four-hour passes for pilots are authorized as of\u2014and not before\u20140400 tomorrow. It has been my experience that if I turn them loose after a mission like the one we flew today, they tend to behave in a manner unbefitting officers and gentlemen. And as you are about to find out, there is a good deal of paperwork involved when one of our young heroes punches out an English cop, or steals a taxicab.\"\n\n\"I understand, Sir,\" Delaney said.\n\n\"You do, Jack, you really do. That's why I gave you the job.\"\n\n\"I hope I can measure up to your expectations, Sir,\" Delaney said.\n\n\"You may leave, Major,\" Douglass said. \"And you may take the Scotch with you.\" Delaney looked surprised.\n\n\"If I took it with me,\" Douglass said, \"I would never make it to Whitbey House.\"\n\n\"Thank you,\" Delaney said.\n\n\"By the time I get back, Jack,\" Douglass said, \"I expect you to have made up your mind about who'll take over your squadron.\"\n**IX**\n\n#  **1**\n\n##  **THE ISLAND OF VIS 1615 HOURS 16 FEBRUARY 1943**\n\nFour men were on hand to greet Canidy, Dolan, and Darmstadter in the B-25.\n\nOne was a British officer wearing the red beret of a parachutist. The pips of a captain were on the shoulders of a sweater. Around his neck he wore a white silk scarf. There were two other Englishmen in British uniform. They were hatless and without insignia of rank. All three of the English had Sten submachine guns. The fourth man was in civilian clothing, a tieless white shirt, a double-breasted, heavy suit jacket, and baggy, unmatching trousers.\n\nThe British officer came to attention and saluted, an almost parade-ground salute, his hand, palm outward, quivering as he touched his temple with his fingertips.\n\n\"Afternoon, gentlemen,\" he said casually. \"My name is Hughson. Welcome to Vis.\"\n\nCanidy returned the salute.\n\n\"You're the aircraft commander, Major?\" Captain Hughson asked.\n\nCanidy jerked his thumb upward to the cockpit of the B- 25G.\n\n\"Commander Dolan's the aircraft commander,\" he said.\n\n\"With his permission, of course,\" Captain Hughson said, \"I would suggest the thing to do is get the aircraft under cover.\"\n\n\"How do we do that?\" Canidy asked.\n\nHughson gestured toward the hillside. Darmstadter saw there was a short, steep-sided indentation in the rocky hillside, a natural revetment, and that above it were rolls of camouflage netting.\n\nAs if reading his mind, the British officer said, \"Except as netting, the camouflage isn't worth a damn. Unless, of course, we wish to give the impression that a North Africa wadi has been miraculously transplanted to the island.\"\n\n\"What do you do?\" Canidy asked, chuckling.\n\n\"We artistically arrange local evergreens atop the netting, \" Hughson said. \"And devoutly pray that it works.\"\n\n\"Let's get at it, then,\" Canidy said.\n\nCapt. Hughson raised his hand above his head and snapped his fingers.\n\nEight Englishmen, in various combinations of uniform, trotted up. One of them, with sergeant's chevrons sewn to his rough woolen jacket, stamped his foot and gave the captain a quivering-hand salute.\n\n\"Sir!\" he barked.\n\nDarmstadter saw Canidy's eyebrows go up at the non-com's parade-ground behavior.\n\n\"Would you have the chaps roll the aircraft into the revetment?\" the British officer asked conversationally.\n\n\"Sir!\" the sergeant barked, and stamped his boot again.\n\nThe English soldiers, without further orders, went to the B-25G and started to push it. When they had trouble getting it moving, Canidy went to the left wheel, put his back against it, and tried to help. Darmstadter went to the other wheel and did the same thing. As he heaved, he saw that neither the British officer nor the civilian was helping. They even seemed surprised that Canidy and Darmstadter were lending a hand.\n\nOnce the initial inertia was overcome, their help was no longer needed, and they walked back to where the captain and the civilian stood.\n\nDarmstadter saw Dolan finally drop through the access hatch, and then, taking a quick look around to see what was going on, begin to give directions to the pushers.\n\n\"Commander _Dolan,_ you say?\" the British captain asked.\n\n\"Right,\" Canidy said, \"and this is Lieutenant Darmstadter. \"\n\nThe two shook Darmstadter's hand.\n\n\"I didn't catch your name,\" Canidy said to the civilian.\n\n\"Ferniany,\" the civilian said.\n\n\"Yachtsman,\" Canidy said, confirming his suspicion that the civilian was the OSS agent.\n\n\"We try not to use that identification unless we have to,\" Ferniany said.\n\n\"We're among friends, I think,\" Canidy said.\n\nThey all watched as the B-25G was turned and then rolled backward into the natural revetment. And they continued watching as the British soldiers, with a skill that could only have come from practice, unrolled the camouflage net and propped it up over the airplane with trunks of young pine trees, then covered the camouflage netting with branches.\n\nThen Dolan walked over to them, and there was an exchange of salutes between Dolan and the British officer. Darmstadter saw that Dolan was as surprised by the display of parade-ground military courtesy as Canidy had been.\n\n\"This is Yachtsman,\" Canidy said.\n\nDolan smiled and shook Ferniany's hand.\n\n\"Where's Fulmar?\" Dolan asked. He chuckled. \"Or what is it we're calling him, 'Ex-Lax'?\"\n\n\"I was about to ask,\" Canidy said.\n\n\"There is a minor problem with Fulmar,\" Ferniany said. \"Actually, it's almost funny.\"\n\n\"What's almost funny?\" Canidy snapped.\n\n\"He's doing ninety days in the coal mine at P\u00e9cs,\" Ferniany said. \"For black marketing. He and the professor. The girl is here.\"\n\n\"Go over that again,\" Canidy snapped. \"Spare me the humor.\"\n\n\"The barge we were to travel on was boarded, just before we were supposed to leave P\u00e9cs,\" Ferniany said. \"That happens sometimes. They found a lot of money on Fulmar. They naturally concluded that he was a black marketeer and hauled him and the professor off.\"\n\nDarmstadter saw that the B-25G was now well hidden from where they stood; from the air, it would be invisible. And the soldiers who had erected the netting over it were now walking down the \"runway\" where they had landed, sweeping the tire tracks with pine branches. Then he saw something that for a moment baffled him.\n\nTwo of the soldiers were rolling a boulder onto the center of the \"runway.\" The boulder was taller than they were. There was no way that a bulldozer, much less two men, could move a boulder that size with such ease. Unless, of course, it was phony, like the boulders that careen down a mountain in the movies. That's obviously what it had to be, Darmstadter realized, and then saw three more boulders farther down the field on the far side of the stream that cut the runway in half.\n\n\"May I suggest, gentlemen,\" Captain Hughson said, \"that we go to our digs? Every once in a while, Jerry flies a Storch over for a look. It would probably arouse his curiosity to see us all standing about in this deserted meadow.\"\n\nThey followed him toward the hillside, where, hidden behind a bush, was the start of a narrow, steep path that wound its way up through the boulders and stunted trees. After they had climbed for five minutes, they came to the first of what turned out to be a series of caves in the side of the hill.\n\nCaptain Hughson led them into one of them.\n\nA hissing Coleman lantern inside illuminated a small stone altar and crude paintings of people with halos on the cave walls.\n\n_They don't look like Jesus,_ Canidy thought. _They must be saints_.\n\nHe thought that his father would know whom the paintings depicted, what sort of Christian had painted them on the wall here, and when. The Reverend Dr. George Crater Canidy was an expert on early Christianity. It was the first time he had thought of his father recently. Whenever he did, he thought that his father would disapprove, if he knew what his son was doing.\n\nThe British SOE captain saw his interest.\n\n\"Orthodox,\" Hughson said. \"I don't know _what_ orthodox, but orthodox. They tell me that they came here after training in a monastery, and they carved out these caves, and then spent the rest of their lives in silence and prayer. Communal farm, that sort of thing, but all they did otherwise was think and pray. Rather unsettling, what, to think about it?\"\n\n\"Well, at least they left us their bomb shelters,\" Canidy said, and then looked for Ferniany. When he had his attention, he went on, \"Who carried Fulmar off where?\"\n\n\"The Black Guard and some local police,\" Ferniany said. \"To the municipal jail in P\u00e9cs. That happens all the time, with legitimate black marketeers, I mean . . . how about that? A 'legitimate' black marketeer . . . \"\n\n\"Hey!\" Canidy said sharply. \"I've had about all of your scintillating wit I can handle.\"\n\n\"Just who the hell do you think you are?\" Ferniany said.\n\n\"My name is Canidy. I'm both the action officer and your control, okay?\"\n\n\"I thought you said Commander Dolan was the aircraft commander,\" Ferniany said, half accusingly.\n\n\"I did,\" Canidy said. \"He _is_ the aircraft commander.\"\n\n\"Major,\" Ferniany said, \"I'm really sorry. It never entered my mind that you would show up here.\"\n\n\"A lot of things apparently 'never entered your mind,' \"Canidy said. \"Now, what the hell happened, one step at a time?\"\n\n\"The cops in Hungary are like the cops in Hamtramck, Michigan, Major,\" Ferniany said. \"They have their hands out. They want a slice of the pie, and then they look the other way. So far as they're concerned, if a Hungarian farmer sells a ham or a couple of salamis to a 'tourist,' instead of selling it to the state, that's his business, providing they get their cut. They make sure that everybody understands the rules by picking people up every once in a while and putting them in jail. Like the cops raid the whorehouses in Cicero on a scheduled basis. You understand? \"\n\n\"And Fulmar got picked up . . . on a schedule?\"\n\n\"The Black Guard had a good day with him,\" Ferniany said. \"I saw them counting the money they took away from him. How much did he have, anyway?\"\n\nCanidy ignored the question.\n\n\"How come they didn't pick you up?\" he asked. \"And you said the Dyer girl's here?\"\n\n\"I'm not making my point,\" Ferniany said. \"And it's important that I do.\"\n\n\"So make it,\" Canidy said.\n\n\"They didn't pick me up, or anybody else on the barge, because that would be killing the goose that lays the golden egg. They picked up Fulmar because he hadn't paid the toll.\"\n\n\"You mean beforehand?\" Canidy asked. Ferniany nodded. \"Well, if you knew about this system, why didn't you pay whatever had to be paid?\"\n\n\"I had a decision to make,\" Ferniany said. \"I decided it would be worth the risk . . . the word I got, presumably from you, Major . . . was to keep this operation as quiet as possible. I decided the best way to do that was to try to slip them through without paying off the cops.\"\n\n\"You should have paid the cops,\" Canidy said.\n\n\"When you pay the cops, it's for a round-trip,\" Ferniany said. \"They would have been curious when these people didn't head back to Vienna with suitcases full of salami and ham.\"\n\n\"Your orders, Captain,\" Canidy said icily, \"were to see that under no circumstances were Fulmar and Professor Dyer to fall into German hands.\"\n\n\"You mean, I was supposed to 'eliminate' them?\" Ferniany asked. \"The thing is, Major, I'm new at this. I'm not used to the euphemisms: 'eliminate' for 'kill,' specifically. So far, it hasn't been necessary for me to kill people on our side. I don't know, frankly, what I would have done if I had thought they were going to be turned over to the Sicherheitsdienst or the Gestapo.\"\n\nCanidy, his face rigid, looked at Ferniany a long time before he spoke.\n\n\"I don't know if I could have done it, either,\" he said finally, softly. \"It's easier to order people to do something like that than it is to do it yourself.\"\n\n\"Major, it's five-to-one that long before their ninety days is up, they'll be turned loose. They're not making any money for the cops in the coal mines. The coal mines are a lesson, you understand?\"\n\n\"I know what you're trying to tell me,\" Canidy said. \"But there's more to this than you understand.\"\n\n\"Like what?\"\n\n\"Like there was a very good reason for the elimination order,\" Canidy said.\n\n\"Are we back to that?\"\n\nCanidy didn't reply. He walked away from the others for a few moments, thinking. Then he came back to the group and turned to Captain Hughson.\n\n\"There is avgas here? Nothing 'almost funny' has happened to that?\"\n\n\"There are twenty-five fifty-five-gallon drums of aviation gasoline, Major,\" the British officer said. \"Twelve, thirteen hundred American gallons.\"\n\n\"And some kind of a pump?\" Canidy pursued.\n\n\"Hand pumps,\" Ferniany said. \"Three of them.\"\n\n\"Are we sure it's clean gas?\" Dolan said.\n\n\"The tanks are sealed,\" Ferniany said. \"And there's both metallic filters and chamois.\"\n\n\"You better get on the refueling right away, John,\" Canidy said. \"At first light tomorrow, after you put Darmstadter through a couple of touch-and-gos, I want you to make for Cairo.\"\n\nDolan accepted the order without question, with a nod of his head. But he was curious:\n\n\"Why Cairo? And aren't you coming with us?\"\n\n\"Cairo because we have a pretty good radio link with our station chief there, and no, I can't go back with you.\"\n\n\"We have radio contact with London, Major,\" the SOE captain said.\n\nCanidy ignored him.\n\n\"While you're fueling the plane,\" he went on, \"I'll start encrypting a message for London. You give it, personally, to the station chief. His name is Wilkins, Ernest J. Wilkins, and he's a lot more competent than he looks. Tell him to get it right out, and then you wait there for further orders.\"\n\nDolan nodded.\n\n\"You'll take the Dyer girl with you,\" Canidy said. \"If the decision is for you to go on to London, take her with you. If it isn't, turn her over to Wilkins, and have her put on ice. _His_ ice. Make damned sure he understands that. She is not a prisoner, but I don't want her talking to anybody but you and the station chief.\"\n\nDolan nodded again. \"Daylight will be at 0513,\" he said. \"Say twenty minutes to shoot two or three touch-and-gos, another half an hour to land, top off the tanks, and put the girl aboard. That'll get us out of here at no more than quarter past six.\"\n\n\"Fifteen minutes to shoot two touch-and-gos, and you'll be on your way at half past five. It's fifteen hundred miles, give or take a hundred, from here to Cairo. Presuming no bad head wind, that'd put you into Cairo in six hours, say noon Cairo time.\"\n\n\"Two other presumptions,\" Dolan said dryly. \"That you have your reasons for taking a passenger while Darmstadter's shooting touch-and-gos, and that you have your reasons for us not to make a refueling stop at Malta.\"\n\n\"There are reasons, John,\" Canidy said, \"but none you can't figure out yourself.\"\n\n\"Right,\" Dolan said.\n\nCanidy turned to Captain Hughson.\n\n\"How do you cook your meat here, Captain?\" he asked.\n\nThe British officer's eyebrows went up.\n\n\"Actually, there are two methods,\" he said. \"We usually heat the tins in boiling water. But sometimes, if the meat is your Spam, we take it from the tins and fry it for a treat.\"\n\n\"Could you rig up some sort of a spit over a fire?\" Canidy asked.\n\n\"I'm sure you have a reason for asking,\" Captain Hughson said.\n\n\"There's four hundred pounds of Four-in-One beef on the plane,\" Canidy said. \"I thought perhaps SOE might like to entertain its visitors with the roast beef of Merry Old England.\" Four-in-One was boned beef packed for the U.S. Army Quartermasters Corps, prepared so that it could be roasted whole, cut into steaks, chunked for stew, or ground.\n\nFor the first time, Captain Hughson smiled.\n\n\"Well, we'll give it a bloody good try, Major,\" he said.\n\n\"There's also some vegetables, but God only knows if they survived the cold,\" Canidy said. \"You stick around, Ferniany,\" he ordered, \"while I do the paperwork.\"\n\n\"Yes, Sir,\" Ferniany said.\n\nIt took Canidy longer than he thought it would to get what details he needed from Ferniany, then to write his report, then to edit it down to as short a version as possible for encryption, and then for the encryption itself.\n\nHe carried with him simple transposition codes on water -soluble tissue paper, one for each day, each five-letter code block representing a word or a phrase he and the OSS cryptographic officer had thought might be useful. But they had not considered the possibility that Fulmar and Professor Dyer would be locked up in a Hungarian municipal prison as petty criminals, so coming up with paraphrases for that situation from the available words and phrases was difficult. He had to laboriously build a second code from the code he had available, and by the time he had finally transferred the message Dolan would carry to Cairo for transmission, and had burned his notes and that day's code, a lot of time had passed. It was dark when they walked out of the cave.\n\nThey stood in the dark for a minute, until their eyes adjusted to the darkness, and then they followed their noses farther up the hill to the cave from which came the smell of roasting beef.\n\n#  **2**\n\n##  **OSS STATION WHITBEY HOUSE KENT, ENGLAND 1905 HOURS 16 FEBRUARY 1943**\n\nCaptain the Duchess Elizabeth Alexandra Mary Stanfield, WRAC, liaison officer of His Majesty's Imperial General Staff to OSS Station Whitbey House, liked First Lieutenant Charity Hoche, WAC, newly appointed assistant adjutant, from the moment she had first seen her getting out of the Ford staff car in front of Whitbey House.\n\nWhy she liked her, she could not explain. There were some women the Duchess liked at first sight, and some she didn't. But by and large, her snap-judgment first impressions were proven correct. Maybe in this case it was because Charity Hoche, although she looked up and somewhat shyly smiled at the Duchess and Lieutenant Bob Jamison as they started down the wide shallow stairs toward her, she did not ask for help, hauled her luggage from the backseat, and, staggering under the weight, started to carry it up the stairs herself. And then with a look of chagrin on her face\u2014and an \"Ooops!\"\u2014Charity Hoche put down the right suitcase and saluted.\n\nThe Duchess returned the salute.\n\n\"Welcome to Whitbey House,\" the Duchess said. \"And thank you for the salute, but we do rather little of that around here.\"\n\n\"I'm Bob Jamison,\" Jamison said. \"Let me give you a hand with your bags.\"\n\n\"What a marvelous house,\" Charity said, reaching to take the Duchess's extended hand.\n\n\"Small and unpretentious,\" Jamison said dryly, \"but comfy. Sometime, when you have a free week or ten days, I'll show you around.\"\n\nThe Duchess liked Charity's smile and peal of laughter.\n\n\"My name is Elizabeth Stanfield,\" the Duchess said.\n\n\"Charity Hoche,\" Charity said. \"How do you do?\"\n\n\"Have you eaten?\" the Duchess asked.\n\n\"Colonel Stevens took me by the Savoy Grill,\" Charity said, \"for a final lecture on the conduct expected of me as an officer and a gentlewoman.\"\n\n\"Well, I think, under the circumstances, you're doing quite well,\" the Duchess said as they entered the foyer.\n\nJamison had been informed, and he had informed the Duchess, of the decision to put Charity into an officer's uniform.\n\nThe Duchess found Charity's eyes on hers and saw in them both gratitude and appraisal. This was a highly intelligent woman, the Duchess decided. She wondered what her real role at Whitbey House was to be. There was a reason for the decision to put her into an officer's uniform, and it had nothing to do with the one offered: \"that it would make things a little easier when she's dealing with the female personnel.\"\n\nCharity laughed again, a pleasant peal of laughter, when she saw the signpost erected at the foot of the main staircase. It was ten feet tall and festooned with lettered arrows, and it gave the direction and miles to Washington, Berlin, Tokyo, Moscow, as well as to the mess, the club, and the officers' and billeting areas within the huge mansion.\n\n\"Don't laugh,\" Jamison said. \"You'll need it. We have three bloodhounds who do nothing but search for people who get lost on the premises.\"\n\nJamison set Charity's suitcases down in the corridor outside his office and motioned Charity inside.\n\n\"Before we go through the paperwork,\" Jamison said, \"let me make it official. On behalf of our beloved commanding officer, Major Richard Canidy, who is regrettably not available at the moment, let me welcome you to Whitbey House.\"\n\n\"Thank you very much.\" Charity smiled.\n\nThe Duchess saw on Charity's face that Charity had known that Canidy would not be here. And then she had the sure feeling that Charity knew why Canidy wasn't here, and very probably where he was and what he was doing.\n\nThere were documents for Charity to sign, and Jamison handed her an identity card overprinted with diagonal red stripes and sealed in plastic.\n\n\"The red stripes are what we call 'anyplace, anytime' stripes,\" Jamison explained, \"meaning you go anywhere on the station whenever you wish. You'll probably be asked for the card a lot, until the security people get to know you, and you _will_ be asked for it whenever you leave the inner and outer perimeters.\"\n\nCharity nodded her understanding, glanced at the card, and tucked it in the breast pocket of her uniform tunic.\n\n\"That, except for the question of your billet, is it,\" Jamison said. \"You have two choices. You can have a private room in the female officers' wing on the second floor, or you can move in with Captain Stanfield in the servants' quarters on the third floor.\"\n\n\"I'm in what used to be the apartment provided for . . .\" she hesitated just perceptibly, and then went on, \"the Duchess's personal maid. There are two bedrooms and a sitter, and a private bath with a bathtub. There are only showers in the female officers' quarters.\"\n\n\"That's very kind of you,\" Charity said, \"and I think I'd prefer that. But it raises a question.\"\n\n\"What's that?\" the Duchess asked.\n\n\"You're my very first duchess,\" Charity said. \"I knew a baroness one time, at school. But I don't know what to call you.\"\n\n\"Elizabeth, or Liz, will do just fine,\" the Duchess said.\n\nStevens had told her, the Duchess decided. Or David Bruce. Or possibly she had known even before she had arrived in England that the Imperial General Staff Liaison officer to OSS Whitbey House Station had before the war occupied the house as the Duchess Stanfield.\n\n\"I'm perfectly prepared,\" Charity said with a smile, \"to curtsy . . . for that matter to prostrate myself . . . if it means access to a hot bath. What I had in London was a trickle of rusty tepid water. More like a bad leak than a shower.\"\n\nThe Duchess laughed.\n\n\"Well, come on, then, we'll get you a hot bath. And you won't have to prostrate yourself, either.\"\n\nThe Duchess was surprised, almost astounded, to see what Charity Hoche's heavy suitcases contained. There was one spare uniform and several spare shirts, but the rest of the space was filled with cosmetics, soap, perfume, underwear, and silk stockings.\n\nCharity saw the surprise on the Duchess's face.\n\n\"We have a marvelous old sailor in Washington,\" she said. \"Chief Ellis. He told me what to bring. He said that I could get anything GI over here without any trouble, but that if I wanted 'lady-type things,' I should take them with me.\"\n\n\"You were given good advice,\" the Duchess said. \"That's the first time I've seen more than three pairs of silk stockings at once in years.\"\n\n\"Help yourself,\" Charity said.\n\n\"Oh, I couldn't,\" the Duchess said.\n\n\"Oh, I wish you would,\" Charity said. \"Sooner or later, there will be a chance for you to scratch my back. And there's three dozen pair, more than I can possibly use before Mommy sends me some more.\"\n\n\"Would you like me to prostrate myself now, or later?\" the Duchess asked.\n\nThey smiled at each other, and the Duchess understood that her snap judgment of Charity Hoche had been on the money. A good woman, and a nice one. Charity handed her a dozen pair of silk stockings.\n\n\"Wear them in good health,\" Charity said.\n\nCharity went to the tub, put in the stopper, and started to fill it. She then somewhat discomfited the Duchess by taking off all her clothes and walking around the bedroom starkers as she loaded her treasure of \"lady-type things\" into a chest of drawers.\n\nThen she got into the tub. The Duchess went to her room, threw away with great pleasure her remaining two pairs of silk stockings\u2014which had runs in them\u2014and put on a pair that Charity had given her. They made her feel good.\n\nThen she saw her own hoard of \"lady-type things.\" It primarily consisted of twenty-two jars of Elizabeth Arden bubble bath. Her eyes teared. Just before he'd gone off wherever the hell he was, Jimmy Whittaker had helped himself to her last half-tin of bubble bath, and she had been furious.\n\nNot too furious, she recalled, to accept his invitation to join in the bubbles. In fact, she'd probably really been more sad than angry. She had resigned herself to doing without bubble bath as she had resigned herself to doing without Jimmy Whittaker.\n\nAnd then Bob Jamison had called her into his office, handed her a U.S. Army package from the National Institutes of Health, Washington, D.C.\u2014which was how the OSS identified its packages\u2014stamped URGENT AIR PRIORITY SHIPMENT and a shipping label reading \"Crystals, Soluble, Non-Explosive,\" and addressed to the \"Officer-in-Charge, Agricultural Research Facility, Whitbey House, Kent.\"\n\n\"I think this is for you,\" Jamison had said.\n\nIt was a case of twenty-four bottles of Elizabeth Arden bubble-bath crystals.\n\n_God, how I miss Jimmy!_\n\n_And to hell with thinking about the illegal use of scarce air-freight facilities and interfering with the war effort._\n\nThe Duchess took one of the bottles and carried it into the bathroom. Charity was slumped down in the tub, so that only her chin and her nipples broke the surface of the water.\n\n\"How about a little bubble bath?\" the Duchess asked.\n\n\"Oh, I see it got here,\" Charity said. \"I was afraid to ask.\"\n\n\"You know where it came from?\"\n\n\"Yes,\" Charity said, \"I know.\"\n\n\"I won't ask where Jimmy is,\" the Duchess said.\n\n\"I'm glad, because I can't tell you,\" Charity said.\n\nThe Duchess filed that away, professionally. Charity Hoche was privy to upper-echelon secrets. And knew how to keep them. And then she was a little ashamed for being professional.\n\n\"Actually, I had something specific in mind before,\" Charity said, \"when I said there would come a time when you could scratch my back.\"\n\n\"Tell me,\" the Duchess said.\n\n\"How would I get Lieutenant Colonel Peter Douglass, Jr., on the telephone?\"\n\n\"Doug's a friend of yours?\"\n\n\"Understatement,\" Charity said. \"At least on my part.\"\n\n\"The way we do that,\" the Duchess said, \"is I get on the telephone, and when I have Colonel Douglass on the line, I bring the phone in here to you.\"\n\n\"Oh, nice!\"\n\nTwo minutes later, the Duchess went back into the bathroom.\n\n\"Colonel Douglass is not available,\" she said. \"He will not be available for the next thirty-six hours. I'm sorry.\"\n\n\"Damn,\" Charity said. She sat up abruptly, splashing water. \"That means he's out spreading pollen.\"\n\n\"I don't think so,\" the Duchess said.\n\n\"Oh, come on,\" Charity said. \"He doesn't know I'm here. And if you know him, you know he's just like the others. I'm not complaining. If I was in his shoes, I'd probably be doing the same thing. 'Live today . . .' \"\n\n\"I would guess that he's off somewhere getting drunk,\" the Duchess said.\n\n\"Oddly enough, that would make me happy. Compared to what I really think he's up to. Why do you say that?\"\n\nThe Duchess hesitated.\n\n\"Oddly enough, it's classified,\" she said.\n\n\"Oddly enough,\" Charity said, \"I'm cleared for anything going on around here. Didn't Jamison tell you?\"\n\n\"No,\" the Duchess said. \"Are you really?\"\n\n\"Yes, I am,\" Charity said. \"Does that mean you're not?\"\n\n\"I am accused,\" the Duchess said, \"of being the resident spy for the Imperial General Staff. There is a grain of truth in the accusation. But I know about this.\"\n\n\"I really am cleared,\" Charity said. \"Am I going to have to get Jamison up here to confirm that?\"\n\n\"He'd love that, dressed as you are.\" The Duchess chuckled. \"We'd better not.\"\n\nCharity Hoche was obviously telling the truth.\n\n\"Probably because of his father,\" the Duchess said. \"Or maybe just because he's Dick's good buddy, and Dick just uses that for an excuse, whenever Doug goes off on a mission, Eighth Air Force tells us. And they tell us when he comes back. TWX to Berkeley Square with info copy here. He flew a mission today. He made it back, but his executive officer was killed. I saw the TWX just before you got here. Under the circumstances, I don't think he's out . . . how did you put it? . . . 'spreading pollen.' \"\n\n\"Thank you,\" Charity said, almost solemnly.\n\n\"You want the bubble bath?\" the Duchess asked.\n\n\"What I would really like is a drink,\" Charity said, suddenly standing up and reaching for the flexible-pipe showerhead to rinse herself off. \"I'll save the bubble bath for sometime when it'll be useful.\"\n\n\"That I can offer,\" the Duchess said. \"We have a nice bar here, and sometimes even a piano player.\"\n\n#  **3**\n\nLieutenant Ferenc \"Freddy\" J\u00e1nos, the piano player, was a very large man. Which was, he thought, the reason he had broken his ankle. If one was six feet four inches tall and weighed two hundred and thirty pounds, one could not expect to be lowered to the ground by parachute as gently as could someone who weighed, say, one hundred sixty pounds.\n\nAnd it wasn't really that bad. The doctor had, perhaps predictably, told him that it \"could have been a lot worse.\" It had hurt like hell on the drop zone, and while the medics, heaving with the exertion, had carried him to the ambulance. But once they'd gotten a cast on it, there had been virtually no pain. A maddening itch under the cast, but no pain.\n\nAnd the X rays had shown a simple fracture of one of the major bones; he'd been told that \"knitting, for someone of your age and physical condition,\" would be rapid. It was an inconvenience, nothing more. It had, of course, kept him from going operational. The bad landing and the resulting broken ankle had taken him off the team. He had been replaced by a lieutenant flown hastily from the United States.\n\nGoing operational would have to wait until they took the cast off\u2014in three days; today was Tuesday, and the cast would come off on Friday\u2014and probably for a couple of weeks after that; a week to become intimate with a new team, and however long it took after that to schedule and arrange for a mission.\n\nThe major problem that faced Lt. Ferenc \"Freddy\" J\u00e1nos, as he saw it, was arranging to get laid between the time the cast came off and the time he went operational. That would require getting to London, and that was going to pose a problem, for the OSS did not like its people going into London once they had been made privy to a certain level of classified operational information.\n\nHe had been made privy to that level of classified information two days before the bad landing. It had then been intended that the men on his team parachute into Yugoslavia three days later. They had been taught\u2014and had committed to memory in case the drop had not gone as planned\u2014several alternate means to establish contact with the guerrilla forces of Colonel Dra\u017ea Mihajlovi\u0107.\n\nThis information was quite sensitive, and those in possession of it could not be trusted to go off and tie one on in London, or for that matter, anywhere off the Whitbey House estate. Freddy J\u00e1nos understood the reasoning, for lives were literally at stake, and he was perfectly willing to grant that liquor loosened tongues, especially his. But he thought it would be a truly unfortunate circumstance if he had to jump in Yugoslavia following a long period of enforced celibacy. God alone knew how he could get his ashes hauled in Yugoslavia.\n\nIt wasn't that there were not a number of females here at Whitbey House\u2014including two leaning on the piano at that moment as he played\u2014who could with relatively little effort be enticed into his room. But he had what he thought of as his standards. For one thing, he did not think officers should make the beast with two backs with enlisted women.\n\nThis belief had not come from The Officer's Guide, which had euphemistically dealt with the subject, but from Lt. J\u00e1nos's own experience as an enlisted man. He had been enraged when he had suspected that his officers were dazzling enlisted women into their beds with their exalted position, and he was unwilling to enrage the enlisted men here by doing the same thing himself. He had even gone further than that. He had had a word with several officers about the matter; he had let them know their behavior displeased him, and that when he was displeased, he sometimes had trouble keeping his displeasure nonviolent.\n\nThere were three American female officers and one British at Whitbey House, but the American WACs did not measure up to Freddy J\u00e1nos's standard for a bed partner, and the British officer, Captain the Duchess Elizabeth Alexandra Mary Stanfield, WRAC, whom Freddy J\u00e1nos would have loved to know much better, had proven to be the exception to the rule that upper-class women, when he looked at them with his large, sad, dark eyes, usually wished to comfort him with all the means at their disposal.\n\nFreddy J\u00e1nos had learned about the effect of his large, sad, dark eyes on women when he was fifteen. At fifteen, he was already nearly six feet tall and pushing one hundred eighty pounds. He had been accepted as a \"prot\u00e9g\u00e9\" piano student at the Juilliard School of Music in New York City. He had still spoken with something of an accent then, his father having brought them from Budapest to accept an appointment as concertmaster of the Cleveland Symphony only four years before.\n\nArrangements had been made for him to stay with friends of the family in a large and comfortable apartment on Riverside Drive overlooking the Hudson River. The friends had also been Hungarians and musicians, and it was their custom to hold Sunday-afternoon musicales, in the European manner, sometimes trios, sometimes quartets, sometimes quintets; and he was naturally asked to play when a piano was required.\n\nAfter one musicale, Mrs. Lizbeth Vernon, the lady in 6-B, one floor up, a tall, lithe woman of thirty-four, whom he had noticed smiling softly at him when he played, came to him and told him how much she had enjoyed his playing. And she went on to say that sometime when he had a few minutes, she hoped he would drop by her apartment and see if her piano was in tune. She had just had it tuned, but it didn't sound right, and she wanted a second opinion before she called Steinway & Sons and complained.\n\nWhen he went to her apartment the next day after school, Lizbeth Vernon answered the door in a thin silk robe and told him that she had been under the sunlamp and hoped he wasn't embarrassed. Lizbeth also told him that she thought he was lonely, that she had seen it in his eyes, and that she understood his loneliness, because her husband, a regional manager for Merrill Lynch, the stockbrokers, was on the road from Monday to Thursday, so she was lonely herself.\n\nThere were a couple of awkward moments that afternoon, after Lizbeth learned that not only was he only fifteen but that he had never been with a woman before.\n\n\"Jesus Christ,\" Lizbeth said, horrified, as they lay sated in the biggest bed he had ever seen.\n\nBut she quickly recovered.\n\n\"Well, I'll say this,\" Lizbeth said, laughing deep in her throat as she grabbed him, \"you _are_ big for your age. And you are a prot\u00e9g\u00e9, aren't you?\"\n\nAnd Lizbeth told him that what had \"driven me crazy\" from the first moment she'd seen him was his eyes.\n\nThat had been, from beginning to end, a fine relationship. And it had lasted long after his \"prot\u00e9g\u00e9\" status had ended. Two years at Juilliard had convinced everybody, his father included, that despite his \"early promise,\" he just didn't have what it would take to become a concert pianist.\n\nHe had often come down to Manhattan to visit Lizbeth\u2014when her husband was out of town over a weekend, when Freddy had been at Yale, working toward a degree in European history with a minor in Slavic languages\u2014and there'd been harsh words between them only twice: once when she had come to New Haven to surprise him and had found him in bed with a red-haired, white-skinned, Irish Sarah Lawrence student who had amazingly freckled breasts; and the other, in January 1942, when he had told her that he was going to enlist rather than wait until he graduated the following June.\n\nLizbeth had told him\u2014actually screamed at him\u2014that he was going to regret it when he woke up and found out what the Army was all about. If he had any sense at all, he would at least stay in school until he graduated and could get an officer's commission.\n\nThe Army had sent him to Fort Dix, New Jersey, for basic training, and then to Fort Knox, Kentucky, for tank training. He had loved all of it, even basic training. There was something about it that had made him feel for the first time in his life\u2014out of bed\u2014like a man. Piano prot\u00e9g\u00e9s play pianos, not baseball or football, and as lousy as the Yale football team was, there had been no place on it for someone even of his size who had never handled a football.\n\nHe had made expert with the Garand rifle in basic training, the first firearm he had ever touched, and to his great delight and satisfaction had proven to be just as skilled firing the 75mm tube on the M4A3 tank on the ranges at Knox. His record, education, and physical condition quickly got him into Officer Candidate School, and he was the Honor Graduate of his class of \"ninety-day wonders.\"\n\nBut instead of being ordered to a tank company, Second Lieutenant Ferenc J\u00e1nos was ordered to the 576th Military, Government Detachment at Fort Benjamin Harrison, at Indianapolis, Indiana. There, a very military lieutenant colonel who four months before had never worn a uniform crisply informed him he was an officer now, and the Army made the decision about officer assignments. He spoke Hungarian and Croatian and German, and his services would be required to govern a defeated Germany and\/or a defeated Hungary.\n\nHe had been compiling a list\u2014because of his musical background\u2014of German, Austrian, Hungarian, Bohemian, Moravian, and Yugoslavian church organs of historical and\/or cultural importance when he had seen a notice on the bulletin board at Fort Benjamin Harrison that applications from officers speaking any of a list of foreign languages would be accepted for an unspecified assignment involving \"great personal risk.\"\n\nThe lieutenant colonel who had told him that the Army made the decisions about officer assignment now told him that his application \"bordered on the disloyal\" and that he felt he should tell him that he would do everything in his power to have the application disapproved.\n\nTwo weeks later, Freddy J\u00e1nos had found himself reporting to a requisitioned estate in Virginia, known as OSS Virginia Station. As far as Freddy J\u00e1nos was concerned, it was even better than Forts Dix and Knox. Here he was taught really fascinating things, such as how to blow up bridges, and parachute from airplanes, and kill people with your bare hands.\n\nAnd then, just before he was to go back to Hungary, he broke his goddamned ankle.\n\n\"Hey, Freddy!\" an officer called in disgust from across the room. \"Jesus Christ!\"\n\nIt took Freddy J\u00e1nos just a moment to understand the nature of the complaint. Lost in thought, wallowing in self-pity over his enforced celibacy, he had without thinking gone from Gershwin to Prokofiev. He listened to what his subconscious had selected for him to play. He smiled. It was the Sonatina in G Minor, Opus 54, Number Two, from \"Visions Fugitives.\" Very appropriate.\n\n\"You just ain't got no couth, Sanderson,\" Freddy called back, and then segued into \"I'm Gonna Buy a Paper Doll.\"\n\nHe smiled at the two women leaning on the piano.\n\nAnd then he looked beyond them to the bar. Captain the Duchess Stanfield was walking up to it, and she was not alone.\n\n_Absolutely gorgeous! God was obviously feeling good when he made that one!_\n\n_And an officer! No restrictions!_\n\nWhat he would do, Freddy decided, was wait until they got their drinks and sat down someplace. Then he would just wander over and say, \"Hi!\"\n\nIt was too much to hope that they would come by the piano, where he would have a chance to dazzle the absolutely ravishing blonde with some piano pyrotechnics and then smile sadly at her.\n\nBut they did just that.\n\n_God is on my side! Virtue is its own reward!_\n\n\"Hello, Freddy,\" the Duchess said as she hopped onto the piano itself.\n\n\"Hello,\" he said.\n\n\"Charity, this is Freddy J\u00e1nos,\" the Duchess said.\n\n\"Hello,\" Charity said, smiling at him, giving him her hand, meeting his eyes.\n\n\"I'm overcome,\" Freddy said, taking her hand, marveling at the softness of it, the warmth, the _utter femininity_ of it.\n\n\"Freddy has a broken ankle,\" the Duchess said. \"I'd get my hand back if I were you, but after that you're fairly safe; he can't run at all well.\"\n\n\"How did you break your ankle?\" Charity asked, compassion and sympathy in her eyes and voice.\n\n_And where there is compassion and sympathy, can passion be far behind?_\n\n\"Small accident, landing by parachute,\" Freddy said, with a smile and what he thought was just the right touch of becoming modesty.\n\n\"Oh, Jesus!\" Charity breathed.\n\nFreddy hadn't expected quite that reaction and looked at her in surprise. She wasn't looking at him, but over his shoulder at the door.\n\nA flyboy had come into the bar. Freddy had seen him before. He was a buddy of Canidy, the headman. It was rumored that he was the son of some big shot in the higher echelons of the OSS. He was also supposed to be an ex- Flying Tiger. He was also pretty goddamned young to be a lieutenant colonel.\n\nHe headed straight for the bar, without looking toward the piano.\n\n\"Doug!\" the absolutely stunning blonde called. Or tried to call. She seemed to be having trouble with her voice.\n\nHe didn't hear her.\n\n\"Colonel Douglass!\" the Duchess called in her clear, crisp voice.\n\nThe flyboy looked for her, found her, and waved casually, dismissing her.\n\nAnd then did a double take.\n\nThen he walked to the piano, right to the blonde. He didn't look at anyone else, and he didn't speak.\n\nHe put his hand up, very slowly, very carefully, as if afraid when he made contact that the apparition would disappear, as does a soap bubble when touched, and touched the blonde's cheek.\n\n\"Doug,\" the blonde said again, as if she was about to cry.\n\nThe flyboy took his fingers from the blonde's cheek and reached down and caught her hand, and led her wordlessly out of the room.\n\n\"Sorry about that,\" the Duchess said. \"I saw your eyes light up.\"\n\n\"One gathers they have met before,\" Freddy said.\n\nThe Duchess chuckled.\n\n\"Did my eyes really light up?\" he asked.\n\n\"Yes, they did,\" she said.\n\n\"Why are you so sure they didn't light up for you?\"\n\nShe met his eyes.\n\n\"Sorry, Freddy,\" the Duchess said.\n\nShe had sad eyes, he saw. There was something in them that made him want to comfort her. Really comfort her, not screw her. Well, maybe both, but first comfort her. And then he saw in her eyes that she was neither going to let him comfort her nor screw her.\n\n\"Me, too,\" Freddy said.\n\n#  **4**\n\n##  **THE ISLAND OF VIS 0525 HOURS 17 FEBRUARY 1943**\n\nCanidy was sitting on a ten-foot-tall boulder, half buried in the side of the valley, his legs dangling over the side, sipping coffee from a gray pottery mug. Ferniany was sitting beside him, and Capt. Hughson was standing behind them.\n\nCanidy winced when the B-25 on its landing roll came to the shallow stream in the middle of the runway and set up an enormous cascade of water.\n\nBut the B-25 did not deviate from its path.\n\nIt rolled another thirty yards, braking hard, so that inertia depressed the piston on the nose gear almost completely. Then it stopped and turned, and began taxiing back down the runway.\n\nWhen it passed the boulder, Dolan, in the copilot's seat, made a \"what now?\" gesture with his hands, holding them out palms up, and shrugged.\n\nCanidy made a \"take it up\" gesture, followed by a \"bye-bye\" wave. Dolan nodded and smiled, then put his hands over his face in an _Oh my God! we're going to crash!_ gesture.\n\nThe B-25 reached the inland end of the runway sixty seconds later, turned, ran up its engines, and then started to move. As it passed the boulder, Canidy could see the expressionless face of Gisella Dyer through the Plexiglas window in the fuselage. He waved at her. There was no response.\n\nThere was another eruption of water when the B-25 passed through the stream again, and it visibly slowed. But then it picked up speed again quickly, the nosewheel left the ground, and a moment later it was airborne.\n\nThe wheels went up, and the flaps, and then it climbed steeply.\n\nCanidy watched for a minute until the plane was barely visible, and then he stood up, draining the coffee mug.\n\n\"Okay, Ferniany,\" he said, \"let's get our show on the road.\"\n\nThey walked off the top of the boulder where it joined the wall of the valley, then slid rather than walked to the valley floor. A three-wheel German Hanomag truck, sort of an oversize three-wheel motorcycle, was parked there. The Hanomag had a canvas-covered truck bed; Canidy and Ferniany got in the back and closed the canvas tail-curtain over them, then Hughson kicked the engine into life and got behind the steering wheel.\n\nThey made their way about four miles down a path that turned first into a narrow cobblestone road and then into a rough macadam street. In a little while, they turned off onto a steep, narrow dirt path that led them to the water's edge.\n\nWhen Canidy climbed out of the Hanomag, he saw a thirty-eight-foot, high-prowed fishing boat two hundred yards offshore dragging a net to the regular explosive snorting of a two-cylinder diesel engine. Just as he thought he saw the glint of binoculars in the small wheelhouse, the sound of the diesel engine changed pitch, the fishing boat slowed and then went dead in the water, and men started to retrieve the net.\n\nWhen it was aboard, the boat headed for the beach in a wide curve.\n\n\"I don't know how he's going to like this,\" Ferniany said.\n\n\"I hadn't planned to ask him,\" Canidy snapped. \"Maybe he'll be smart enough not to volunteer an opinion.\"\n\nThe moment he said it, he was a little sorry. There was something in the chemistry between him and Ferniany that produced dislike without real reason. But that wasn't why he had snapped at him. The reason for that was that Ferniany was close to the truth. \"Saint Peter,\" the OSS agent on the fishing boat, was probably not going to like what he was about to learn. Nor would Stevens and Bruce, and if it got that far, Capt. Douglass or Colonel Donovan.\n\nThe OSS agents on the scene would be annoyed both by having their thunder stolen by a visiting brass hat and by the extra risk his grandstanding would mean. And Stevens and Bruce would bitterly question his decision to go into Hungary himself. First and foremost was the question of his running the risk of falling into German hands. And right on the heels of that was the equally valid question of whether he could do what had to be done any better than Yachtsman and Saint Peter could do it.\n\nCaptain Hughson touched Canidy's arm.\n\n\"There's a rock over the water,\" he said. \"You can jump from it to the boat.\"\n\nHe nodded toward it.\n\n\"Would you like to take this with you?\" Hughson asked, unslinging his Sten submachine gun from his shoulder and offering it to Canidy.\n\n\"Have you got another one?\"\n\n\"Actually,\" Hughson said, \"there's a Schmeisser in my cell I've been looking for an excuse to carry.\"\n\n\"Then thank you, Hughson,\" Canidy said, and took the submachine gun from him.\n\n\"You will be a good chap, won't you, Major, and make an effort to return the Sten to me, in person?\" Hughson said.\n\n\"Despite what everybody apparently thinks,\" Canidy said, \"I am _not_ charging foolhardy into the valley of death.\"\n\n\"No, of course you aren't,\" Hughson said. He put out his hand, and Canidy took it.\n\nThe boat nosed in to the rock. First Ferniany and then Canidy jumped onto the deck. Immediately, the boat headed offshore.\n\nThere were two men in the wheelhouse, both dark-haired and dark-skinned, both needing a shave, and both dressed in dark blue fisherman's trousers and rough brown sweaters. It was only when one of them spoke in English to Ferniany that Canidy had any idea which was the genuine fisherman and which the SOE agent with the code name \"Saint Peter.\"\n\n\"And what, might one dare inquire, is one supposed to do with this downed, if intrepid, aviator?\" Saint Peter asked in an upper-class British accent.\n\nFerniany chuckled. \"Major Canidy, may I introduce Lieutenant J.V.M. Beane-Williams, late of the Household Cavalry?\"\n\n\"How'd'ja do?\" Lt. Beane-Williams said with a smile, offering his hand. \"I hate to put it to you so bluntly, Major, but you have, so to speak, just entered the 'Out' door. England . . . I presume you came from England . . . is in quite the opposite direction.\"\n\nCanidy chuckled. He liked this Englishman.\n\n\"Hughson tells me that you can put us ashore on the mainland,\" Canidy said.\n\n\"I presume there is a reason?\" Saint Peter said.\n\n\"Someplace where we can make contact with Mihajlovi\u0107's guerrillas,\" Canidy said. \"Our ultimate destination is Budapest, and the sooner we can get there, the better.\"\n\n\"Budapest is rather nasty this time of year,\" Saint Peter said. \"Snow and slush, and ever-increasing numbers of the Boches. But I daresay you've already considered that, haven't you?\"\n\nWithout waiting for a reply, he entered into a conversation with the Yugoslavian captain.\n\nFinally, he turned to Canidy.\n\n\"T\u00f3dor suggests we put you ashore at Plo\u010de,\" he said. \"He has a first cousin twice removed there. Or did he say a 'second cousin, once removed'? He also asked that I express his practically boundless admiration for your wristwatch. \"\n\nCanidy looked at the Yugoslavian captain, who was smiling warmly at him, exposing two gold and two missing teeth.\n\nThen he unstrapped his chronometer and handed it to him.\n\nThe Yugoslavian said something, and Saint Peter translated.\n\n\"He says, 'Oh, I couldn't.' \"\n\n\"Tell him I insist,\" Canidy said.\n\nThe Yugoslav unstrapped his cheap watch and handed it to Canidy.\n\n\"He says,\" Saint Peter said, \"that if you insist . . . \"\n\nCanidy chuckled.\n\n\"It's sixty miles, or thereabouts, to Plo\u010de,\" Saint Peter said. \"If we're not stopped, it should take us four, perhaps four and a half hours.\"\n\n\"And if we're stopped?\"\n\n\"Then none of us will get to visit Plo\u010de's many historical and cultural attractions,\" Saint Peter said.\n**X**\n\n#  **1**\n\n##  **CAIRO, EGYPT 1220 HOURS 17 FEBRUARY 1943**\n\nFirst Lieutenant Hank Darmstadter was riding in the copilot's seat working the radios when Commander John Dolan suddenly reached over and grasped his upper arm in a very tight grip.\n\nStartled, Darmstadter looked at him. Dolan's face was white and beaded with sweat. He seemed to be in pain.\n\n\"Indigestion,\" Dolan said with a terrible effort. \"There's a bottle of medicine in my briefcase. Get it, will you?\"\n\nThe first thing Darmstadter remembered, as he hastily unfastened his seat and shoulder harness, was that Dolan had been medically retired from the Navy before the war because of a heart condition.\n\n_Jesus, he's having a heart attack!_\n\nDolan's black leather Navy-issue briefcase was on a shelf in the passageway between the cockpit and the auxiliary fuel tanks that had been installed in the bomb bay. Its contents expanded the accordion folds, and Darmstadter grunted with the effort it took to open the catch and the straps that held it closed.\n\nAs he started rummaging through the briefcase, he glanced past the auxiliary fuel tanks into the fuselage. The German girl was looking at him. She had her hair done up in braids, which she had then coiled on the sides of her head. Darmstadter wondered who she was and why getting her and her father out of Germany had been worth all the effort it had cost.\n\nThey had been introduced, and she had politely shaken hands, but had remained silent. From the way her eyes had followed the conversation, however, Darmstadter had known that she at least understood English. And yet she had asked no questions, not even about where they were taking her. He wondered if she was in some kind of emotional shock, or simply acknowledging that for the moment she had no voice whatever in what happened to her.\n\nThen he had a strange thought. He wondered what she had done during the flight about taking a leak. There was a relief tube in the cockpit, but that wouldn't have done her any good, even if she had known about it and asked for it.\n\nHe returned his attention to Dolan's briefcase. There was everything in it, from a copy of _TM B-25-1 Flight Operation B-25 Series Aircraft_ to a change of socks and underwear and a toilet kit. And a pint bottle of a bright red liquid with a label reading \"Medical Corps, U.S. Army\" and the typewritten message: \"Lt. Commander J. B. Dolan, USNR, Take As Required for Indigestion.\"\n\nDarmstadter hurried back to the cockpit.\n\nDolan reached for the bottle. Darmstadter unscrewed the cap and handed it to him.\n\n\"Sit down and take the airplane,\" Dolan ordered. Then he waited until Darmstadter had gotten back into the copilot's seat, fastened his seat and shoulder belts again, and nodded to show his readiness to fly the airplane before he put the bottle of bright red liquid to his lips.\n\nHe took a large swallow, hesitated, and then took a second. In a moment, the look of pain on his face went away, and he managed a weak smile.\n\nDarmstadter looked at the instrument panel. They had been homing in on the Cairo RDF for the past thirty minutes. The needle on the signal-strength gauge was almost at the upper peg. They were flying ten degrees to the left of the direction indicated by the needle on the RDF antenna indicator.\n\nDarmstadter made the course correction and then looked at Dolan again. The startling paleness was gone from his face.\n\n\"You better start letting down,\" Dolan ordered. \"Thousand feet a minute.\"\n\nDarmstadter nodded, then reached over his head for the trim wheel and lowered the nose. After that, he retarded the throttle just a hair.\n\nThere was time to reconsider his first alarmed conclusion that Dolan was having a heart attack. That had been, he decided, a fear reaction. What was wrong with Dolan was what Dolan had told him: an attack of indigestion. He probably had them often, for he was carrying the bright red indigestion medicine with him.\n\nDolan said something, and Darmstadter missed it.\n\n\"Excuse me?\"\n\n\"I said it must have been Canidy's goddamned steaks,\" Dolan said, leaning over to make himself heard over the roar of the engines. \"Every time I eat charred meat, it does it to me.\"\n\nDarmstadter nodded.\n\nHe was right back to _Dolan was having, had had, a heart attack_. He'd smelled Dolan's breath when the older man had leaned over. Whatever was in that bottle, bright red or not, usually came in a narrow-necked bottle with a label reading \"Sour Mash Bourbon.\"\n\n\"You better sit it down,\" Dolan said, leaning over again and sending Darmstadter another cloud of bourbon fumes. Then he slumped back against the cushions of the pilot's seat and took another healthy swallow of \"indigestion medicine.\"\n\nDarmstadter reached for the microphone and put it before his lips.\n\n\"Cairo, Army Four Three Three.\"\n\nA voice with the unmistakable tones of Brooklyn came over the earphones.\n\n\"This is Cairo, go ahead, Army Four Three Three.\"\n\n\"Army Four Three Three, a B-25 aircraft, is passing through niner thousand about thirty miles north of your station. Request approach and landing.\"\n\n\"Four Three Three, Cairo. The winds are from the north at ten, gusting to twenty. Visibility is unlimited. The altimeter is Two Niner Niner Niner. Descend to three thousand feet and report when you have the airfield in sight.\"\n\n\"Cairo, Four Three Three. Understand three thousand,\" Darmstadter said, and hung his microphone up.\n\nThen Dolan's voice came over his earphones, and he turned and saw that he had his microphone in front of his lips.\n\n\"Cairo,\" Dolan said. \"Four Three Three. Four Three Three is Ninth Air Force flight Four Zero Five. Acknowledge. \"\n\nDarmstadter wondered what the hell that meant. It didn't surprise the Cairo tower.\n\n\"Four Three Three,\" the operator with the Brooklyn accent said, \"Cairo. Roger your Flight Four Zero Five.\"\n\nDarmstadter could see three large pyramidal structures to his left.\n\n_My God, those are the pyramids!! The real ones!_\n\nAnd then he looked to his left and picked up his microphone again.\n\n\"Cairo, Four Three Three, I am at four thousand five hundred. I have the field in sight.\"\n\n\"Four Three Three, Cairo. Maintain present course and rate of descent. You are cleared as number one to land on Runway Three Four. The altimeter is Two Niner Niner Niner. The winds are from the north at ten, gusting to fifteen. Report on final.\"\n\n\"Four Three Three, roger.\"\n\nDarmstadter looked at Dolan as he reached for the throttle quadrant. Now there was a sort of dazed look on his face. And he had not reached for the plastic sealed landing checkoff list hanging from the instrument panel.\n\nDarmstadter realized that he was going to have to land the airplane himself, without help. But he was more concerned about Dolan's condition than he was about getting the flaps and gear down without help.\n\nHe turned to the right, then the left.\n\n\"Cairo, Four Three Three on final.\"\n\n\"Roger, Four Three Three. You are number one to land. Look out for the C-47 on the threshold.\"\n\nDarmstadter put on twenty degrees of flaps, then lowered the gear. He came in low and slow and put it on the ground within a hundred yards of the threshold.\n\n\"Four Three Three on the ground.\"\n\n\"Four Three Three, take Taxiway Five, a Follow Me will meet you.\"\n\n\"Roger,\" Darmstadter said.\n\nTaxiway Five was the last turnoff. As he taxied down the runway to it, Darmstadter saw a jeep racing down a taxiway parallel to the runway. The jeep was painted in a black-and-white checkerboard pattern, with a huge checkerboard flag above it flapping in the wind.\n\nWhen he turned the B-25 off the runway, the jeep was there waiting for him. It led him to a remote corner of the field. There was a large hangar there whose doors were being opened as they arrived.\n\nThe Follow Me jeep stopped, and a ground handler hopped out and signaled for Darmstadter to move to the hangar doors. When the nose of the B-25 was ten feet from them, he gave the throat-cutting sign to stop engines.\n\nImmediately, a dozen GIs came out of the hangar and manhandled the B-25 inside the hangar. Darmstadter sensed, from the decreasing light inside the hangar, that the doors were being closed.\n\nHe looked at Dolan.\n\n\"You all right, Commander?\" he asked.\n\n\"The word you have to keep in mind, Darmstadter,\" Dolan said, \"is 'indigestion.' Am I going to have trouble with you about that?\"\n\n\"No, Sir,\" Darmstadter said after a moment.\n\n\"Thank you,\" Dolan said, simply and sincerely.\n\n\"What happens now?\" Darmstadter asked.\n\n\"I don't know,\" Dolan said. \"Canidy gave me that 'Flight Four Zero Five' message just before we took off. I expect somebody will show up shortly. In the meantime, you might have them start to refuel it.\"\n\nWhen Darmstadter dropped from the belly of the B-25, he saw that there were two military policemen, armed with Thompson submachine guns, guarding the airplane. And there was a captain, wearing an AOD (Aerodrome Officer of the Day) brassard.\n\nDarmstadter walked over to him and saluted.\n\n\"I'd like to get this fueled,\" he said.\n\n\"Someone will be along for you shortly, Lieutenant,\" the AOD said. \"In the meantime, nothing comes into, or goes out of, this hangar.\"\n\n\"We have a female passenger aboard,\" Darmstadter said. \"She has to use the can.\"\n\n\"I don't know if there's one available,\" the AOD said.\n\n\"There has to be something,\" Darmstadter said.\n\n\"Jesus Christ!\" the AOD said in annoyance.\n\n\"Sorry as hell to inconvenience you,\" Darmstadter flared.\n\nThe AOD glared at him.\n\n\"Who the hell do you think you are, Lieutenant?\"\n\n\"I'm only a lieutenant,\" Darmstadter said, \"but I can ask Commander Dolan to come down here if you have to have that as an order.\"\n\n\"Sergeant!\" the AOD said, and one of the submachine-gun -armed MPs came over.\n\n\"There is a female aboard this aircraft who needs the facilities, \" he said. \"Take her there and bring her back.\"\n\nDarmstadter climbed back into the aircraft.\n\n\"Would you like to . . . \"\n\n\"I must have the ladies' room,\" Gisella Dyer said in precise, if uneasy, English.\n\n\"Come with me,\" Darmstadter said.\n\nFive minutes later, before Gisella had come out of the men's room at the rear of the hangar, a side door opened and two men in U.S. Army civilian technician uniforms came in.\n\nThe AOD indicated Darmstadter with a nod of his head. One civilian walked up to him and held a leather folder in front of Darmstadter's eyes. They were OSS credentials, but Darmstadter had never seen any before, and it took him a moment to realize what they were.\n\nThe man's name was Ernest J. Wilkins.\n\n\"You're the flight Four Zero Five?\" Wilkins asked.\n\n\"That's right,\" Darmstadter said.\n\n\"You want to tell me what this is all about? Before that, you want to show me your identification?\"\n\n\"I think maybe you better go aboard and talk to Commander Dolan,\" Darmstadter said. \"I'm just an airplane driver.\"\n\n\"Why don't _you_ go aboard and ask Commander Dolan to join us?\" Wilkins said sarcastically.\n\n\"He's a little under the weather,\" Darmstadter said.\n\n\"What's wrong with him?\"\n\n\"Indigestion,\" Darmstadter said.\n\n\"Jesus H. Christ!\" Wilkins said, but he went to the access hatch and climbed aboard the B-25.\n\nThere was a marked change in Wilkins's attitude when he climbed back down from the airplane.\n\n\"Captain,\" he said to the AOD. \"Get on the horn and get an ambulance over here. There is no medical emergency, we will not need a physician. I will require one of the MPs to come with us. This airplane is to be refueled and kept under guard in this hangar. I presume you have cautioned your men to keep their mouths shut?\"\n\n\"Yes, Sir,\" the AOD said.\n\nGisella Dyer, trailed by the MP sergeant, walked up.\n\n\"Good afternoon, Miss Dyer,\" Wilkins said to her in fluent German. \"Welcome to Egypt. We're going to go from here to a place where you'll be staying for a while. I'm afraid, for reasons of security, that you'll have to travel by ambulance. It'll be a little warm in the back, but we don't have far to go.\"\n\nThirty minutes later, Dolan, Darmstadter, and Wilkins were in what had once been the pool house by the swimming pool of a wealthy Egyptian banker. The blue-tile-walled room now held an impressive array of communications equipment under the supervision of a gray-haired, distinguished-looking man who wore a ring, an amethyst surrounded by the legend \"20 Years Service AT&T.\"\n\nDolan seemed to be completely recovered from his \"indigestion. \" The color was back in his face, and he was no longer tensed with pain.\n\nDarmstadter was uncomfortable. There was no doubt in his mind that there was a hell of a lot more wrong with the old sailor than indigestion. What was his duty, to tell Wilkins\u2014who had identified himself as Station Chief, Cairo\u2014so that Wilkins could, by force if necessary, get him medical attention? Or to obey Dolan's admonition to \"keep in mind that the word was indigestion\"?\n\nDolan himself answered the question.\n\nWhen London acknowledged receipt of the encrypted message from Canidy and ordered Cairo to stand by while the message was decrypted, Dolan handed the man with the AT&T ring a sheet of paper.\n\n\"Encrypt that, and send it, urgent, before they get off the air,\" he ordered.\n\nWhen the communications officer had run the message through the encryption device and begun to transmit the encoded message, Dolan reclaimed the sheet of paper and handed it to Darmstadter.\n\nTO OSS LONDON STATION. EYES ONLY BRUCE AND STEVENS. SUFFERING SEVERE INTESTINAL DISTRESS AND FEVER. PROBABLY RECURRENCE OF MALARIA. HAVE MADE DARMSTADTER AWARE OF ALL REPEAT ALL OPERATIONAL DETAILS IN CASE HIS ASSUMPTION COMMAND NECESSARY. DOLAN, LIEUTENANT COMMANDER, USNR.\n\nWhen Darmstadter looked at him, Dolan shrugged.\n\n\"What the hell, kid,\" the old sailor said. \"You didn't really want to go back to flying Gooney Birds, did you?\"\n\n#  **2**\n\n##  **OSS LONDON STATION BERKELEY SQUARE LONDON, ENGLAND 1105 HOURS 17 FEBRUARY 1943**\n\nThere is a three-hour time difference between Cairo and London. The message transmitted from the pool house of the villa in Cairo at 1405 Cairo time was acknowledged by London at 1110 London time. The second acknowledgment (confirming satisfactory decryption in London) was sent to Cairo at 1124, London time, and the second acknowledgment of Dolan's message at 1141, London time.\n\nBoth encrypted messages had come out of the encryption \/decryption device in Berkeley Square in the form of punched tape. It was necessary to feed the punched tape into another machine (a converted teletype machine), which then typed out a copy on paper. The messages were next entered in the Classified Documents Log, and finally they were put, separately, inside two cover sheets. The outer was the standard Top Secret cover sheet, and the inner one was stamped with both TOP SECRET and EYES ONLY BRUCE AND STEVENS.\n\nIt was by then 1158.\n\nRank hath its privileges, and the privilege the senior cryptographic officer of OSS London Station, twenty-six-year -old Captain Paul J. Harrison, Signal Corps, had claimed for himself was the day shift, 0800 to 1600. And just as soon as he could get the personnel section at SHAEF (Supreme Headquarters, Allied Expeditionary Force) off their ass to pin second lieutenants' bars on two of his sergeants, he intended to take no shift at all. But now, with his perfectly qualified sergeants barred from acting as cryptographic duty officer by a bullshit directive from David Bruce, he had the duty.\n\nAs was his custom with Eyes Onlys\u2014the forty-page SOP for classified documents made no specific reference to who should physically carry messages\u2014Capt. Harrison personally took both messages up from the cryptographic room in the subbasement to the Station Chief's office.\n\nCapt. Helene Dancy told him that David Bruce had moments before left the building. He was to have lunch with Lieutenant General Walter Bedell Smith, General Dwight D. Eisenhower's deputy at SHAEF. \"Beetle\" Smith and David Bruce were friends as well as professional associates. Knowing this, Eisenhower had for all practical purposes given General Smith carte blanche in dealing with the OSS.\n\n\"Where's the Colonel?\" the cryptographic officer asked.\n\n\"Whitbey House,\" Helene Dancy replied. \"What have you got?\"\n\n\"An Eyes Only Operational Immediate . . . two of them . . . for Bruce and the Colonel. From Canidy and Dolan.\"\n\nThe SOP was very clear on the handling of Operational Immediate messages:\n\n16. [b]. Operational immediate messages will be immediately delivered to the addressee, or in his absence, to the senior officer present possessing the appropriate security clearance. In no circumstances will a delay of more than ten [10] minutes between decryption and delivery be tolerated.\n\n\"Can I see it?\" Helene Dancy asked.\n\n\"You're not next on the list,\" Captain Harrison said reluctantly, obviously uncomfortable.\n\n\"That's right,\" she said, just a little tartly. She picked up her telephone.\n\n\"Sergeant, do you know where Captain Fine is?\" she asked a moment later, and then, when there had been a reply: \"Send someone for him, please. Get him back here as soon as you can.\"\n\n\"Well, who's next on the list after Fine?\" Harrison asked.\n\n\"Oddly enough, I am,\" Capt. Dancy said, a little ice in her voice. She put out her hand for the documents.\n\n\"Hey, Dancy,\" Capt. Harrison said as he handed them to her. \"I don't make the rules. I just try to obey them.\"\n\n\"I know,\" Helene Dancy said. \"Damn, why does everybody have to be gone at once?\" And then she quickly glanced at the first message: Canidy's.\n\n\"Oh, Christ!\" she said.\n\n\"My thought exactly,\" Capt. Harrison said.\n\nShe flipped back the cover sheet on the second Eyes Only: Dolan's.\n\n\"I think you'd better get both of these off, Operational Immediate, to Washington, Eyes Only, Donovan and Douglass, \" Capt. Dancy said.\n\nShe saw the look on his face.\n\n\"Okay, I'll make it official. As the senior officer present, I order you to transmit these messages to Washington, Eyes Only, Donovan and Douglass.\"\n\n\"I'm not trying to be chickenshit about this,\" Harrison said. \"You heard Bruce eat my ass out the last time I 'acted without thought and authority . . .' \"\n\n\"Well, I just took you off the hook for this time,\" she said.\n\n\"Yeah,\" Harrison said. \"Helene, I'm not asking you to make it official, but should I try to run down Bruce at SHAEF?\"\n\n\"That would make a second copy necessary,\" she said. \"The sergeant major will get Fine in here in a couple of minutes.\"\n\nThe SOP was specific about that, too:\n\n16. [f]. In no case, except with the specific permission of the chief of station, or the deputy chief of station, will more than one [1] copy of an eyes only document be prepared. It is emphasized that addressees of eyes only documents, with the exception of the chief of station and deputy chief of station, are specifically forbidden to make copies of eyes only documents for their own files, or for any other purpose.\n\n\"What the hell,\" Capt. Harrison said. \"How mad can Bruce get?\"\n\n\"Pretty mad,\" she said. \"I don't know, Paul.\"\n\n\"Is Bruce eating at the SHAEF general's mess?\" Capt. Harrison asked, having made up his mind.\n\n\"He wasn't sure,\" Capt. Dancy said. \"When he can get Beetle Smith out of the building for an hour or so, he likes to butter him up at the Savoy Grill.\"\n\n\"And if I call either place to find out, no one will tell me,\" Harrison said. \"I think I'll take a chance on the Savoy.\"\n\nFive minutes later, after having made copies of the Eyes Only messages and ordered their transmission to Washington, Capt. Harrison went onto Berkeley Square to get into a Ford staff car. There he saw Capt. Stanley S. Fine getting out of a jeep driven by the sergeant major.\n\nHe waved at Fine, but said nothing to him about the Eyes Onlys, or about where he was going. If he told Fine, Fine might forbid him\u2014he had the authority\u2014to take copies of the Eyes Onlys to Bruce. More likely, once he'd explained the situation, Fine would also decide it was the thing to do, and to hell with Bruce's SOP. That would put him in the line of fire if Bruce didn't like the decision, and that wasn't necessary. Fine was a good guy.\n\nThe ma\u00eetre d'h\u00f4tel of the Savoy Grill blandly denied the presence of either Lt. General Walter Bedell Smith or Mr. David Bruce. He smilingly announced he hadn't seen either of them in days.\n\nCapt. Harrison looked around the large, elegantly appointed room and found what he was looking for. A major having a solitary lunch at the far end of the room. Behind the major was an ornately carved movable screen, so placed that it could conceal a table for two. And hanging from the epaulets of the major's green tunic was the golden rope of an aide-de-camp.\n\n\"Thank you very much,\" Harrison said to the ma\u00eetre d'h\u00f4tel. And then he ducked past the ma\u00eetre d' and headed for the screen. The ma\u00eetre d' scurried after him, but unless he broke into a run, Harrison knew he wouldn't catch up with him.\n\nBut Beetle Smith's aide-de-camp saw him and rose quickly to his feet, obviously intending to block his path. Harrison reached in his pocket and was enormously relieved to find his OSS credentials there. He was terrified of the consequences of losing them, and since he rarely had need of them, he usually kept them in the Top Secret safe.\n\nHe got them out of his pocket and held them up for General Smith's aide-de-camp to see.\n\n\"Just a moment,\" the aide-de-camp said. \"I'll tell Mr. Bruce you're here.\"\n\nHarrison smiled and kept going.\n\nDavid Bruce looked at him with surprise and annoyance.\n\nThe Chief of London Station and the Deputy Commander of SHAEF were lunching on small steaks, oven-browned potatoes, and asparagus. Harrison knew that the steaks and asparagus had come from OSS stocks. The usual fare at the Savoy Grill was broiled fish and Brussels sprouts. The Savoy was, however, happy to prepare whatever ingredients a guest might wish to send to its kitchen ahead of time. The price charged was the same as if they had furnished the ingredients.\n\nWhat that meant was that Bruce, as Helene Dancy had suggested, was buttering up Beetle Smith by providing an unusually nice luncheon at the Savoy. And that meant he was likely to be greatly annoyed to have the nice luncheon interrupted.\n\n\"I'm sorry to disturb you, sir,\" Paul Harrison said. \"But I could see no other choice.\"\n\nHe thrust a large manila envelope at him.\n\n\"Captain Fine is not available?\" Bruce asked, courteously enough.\n\n\"He was sent for, Sir,\" Harrison said. \"He was out of the office.\"\n\n\"Oh, excuse me,\" Bruce said. \"Beetle, this is Captain Harrison. And this is General Smith. Or do you know each other?\"\n\n_Hell, yes, we're old pals. How the hell are you, Beetle?_\n\n\"No, Sir,\" Capt. Harrison said. \"How do you do, Sir?\"\n\nGeneral Smith smiled and offered a firm handshake.\n\n\"Captain,\" Smith said. \" 'Harrison,' is it?\"\n\n\"Yes, Sir.\"\n\nBruce tore the envelope open carefully, glanced inside, then took the Eyes Only documents from it.\n\n\"I'm happy to meet you, Captain,\" General Smith said.\n\nHarrison could not think of a reply.\n\n_Dear Harriet, You'll never guess who I met at lunch at the Savoy Hotel._\n\nSmith, naturally curious, turned his attention to David Bruce.\n\n\"Important, David?\" General Smith asked.\n\n\"Not particularly,\" Bruce said. And then he corrected himself. \"I don't mean to suggest that you should not have brought this to my attention here, Harrison. That was the correct thing to do.\"\n\n\"Yes, Sir,\" Harrison said.\n\n\"You said that Captain Fine has been sent for?\"\n\n\"Yes, Sir.\"\n\n\"I don't think there's any point in involving Captain Fine in this, Captain,\" Bruce said. \"What I think you should do is see that Washington gets a copy of this as quickly as you can. And then get in touch with Colonel Stevens and ask him to be in my office at four. A little earlier, if he can make it. And I think it might be a good idea if you were to ask him to bring Lieutenant Hoche with him.\"\n\n\"Yes, Sir,\" Harrison said.\n\nLt. Hoche, Capt. Harrison recalled, was the newly arrived, absolutely splendiferous blonde who was supposed to be Helene Dancy's man . . . woman . . . at Whitbey House.\n\n_What the hell has she got to do with this?_\n\nBruce returned the documents to the envelope and handed it back to Harrison.\n\n\"Thank you, Captain,\" he said.\n\nHarrison was wondering whether or not the Customs of the Service required him to salute a three-star general in a hotel dining room, when General Smith solved the problem.\n\nHe gave Harrison his hand.\n\n\"Pleasure to have met you, Captain,\" he said. \"I look forward to seeing you again.\"\n\n\"Yes, Sir,\" Harrison said. \"Thank you, Sir.\"\n\n#  **3**\n\n##  **OFFICE OF STRATEGIC SERVICES NATIONAL INSTITUTES OF HEALTH BUILDING WASHINGTON, D.C. 0655 HOURS 17 FEBRUARY 1943**\n\nChief Boatswain's Mate J. R. Ellis, USN, pushed open the plate-glass door, marched into the lobby of the building, and crossed to the elevator, his metal-tapped heels making a ringing noise on the marble floor.\n\nHe was almost at the elevator when a guard, whose nose had been in the sports section of the _Washington Star,_ spotted him. The guard, in a blue, police-type uniform, erupted from his chair.\n\n\"Hey!\"\n\nEllis looked over his shoulder and saw the guard headed for him.\n\n\"Where do you think you're going?\" the guard demanded as he caught up with Ellis and put his hand on Ellis's arm.\n\nEllis fished in his trousers pocket with his free hand and came up with an identity badge sealed in plastic and fitted with an alligator clip. He held it out for the guard to see. The card bore his photograph, diagonal red \"anytime, anyplace\" stripes, his name, and in the Duty Assignment box, the words \"Office of the Director.\"\n\nThe guard was satisfied with Ellis's bona fides, but not mollified.\n\n\"You're supposed to wear that badge, you know,\" he said.\n\n\"Sorry,\" Ellis said. \"I forgot.\"\n\nEllis got on the elevator and rode up.\n\nWhen the second lobby guard returned from the men's room, the guard who had stopped Ellis was curious enough to ask him, \"Who the hell is the sailor with the anytime, anyplace badge?\"\n\n\"Navy chief? Big guy? Ruddy face?\"\n\n\"That's him. He walked in here like he owned the place.\"\n\n\"He almost does,\" the second guard told him. \"That's Chief Ellis. Donovan's shadow. Nice guy. Just don't fuck with him. The best way to handle him is to remember the only people around here who tell him what to do are Colonel Donovan and Captain Douglass.\"\n\nUpstairs, Ellis got off the elevator and walked down the marble-floored corridor to the Director's office.\n\n\"Good morning, Sir,\" he said to the slight, balding man in his late thirties sitting at Colonel Donovan's secretary's desk.\n\nWilliam R. Vole was in civilian clothes, but he was a chief warrant officer of the Army Security Agency, a cryptographer, on what had turned out to be permanent loan to the OSS. The Army Security Agency monitored Army radio and wire communications nets to ensure that no classified information was being transmitted in such a manner that it would become available to the enemy. It had also developed a capability, however, to intercept enemy radio transmissions and to break enemy codes.\n\nThere were eight such cryptographic experts assigned to the OSS in Washington, and one of them was always available to the office of the Director. They had become de facto duty officers in the Director's office, in addition to their cryptographic duties. It had been made official by Colonel Donovan, at Ellis's suggestion. Ellis had pointed out that their cryptographic duties had already made them privy to the contents of incoming and outgoing encrypted messages, so they would learn little they already didn't know by keeping the Director's office manned around the clock. And there were other ways they could make themselves useful in the Director's office.\n\n\"Chief,\" CWO Vole responded with a smile.\n\nVole liked Ellis, and felt a certain kinship with him as well. They both had long enlisted service before the war. And unlike many of his peers, he did not resent Ellis's authority to speak for Colonel Donovan, or Donovan's deputy, Captain Peter Douglass. He had been around the OSS long enough to see how Ellis used that authority, and he had never seen him abuse it.\n\nAnd there was enough vestigial enlisted man in Chief Warrant Officer Vole to take some pleasure in the annoyance and discomfiture of a long line of brass hats who had tried and failed to pull rank on the salty old chief. Vole could not remember an incident where Ellis had not been backed up by Captain Douglass when some brass hat had complained to him about a decision of Ellis's, and he had several fond memories of incidents where some brass hat, having gotten no satisfaction from Captain Douglass, had gone over Douglass's head to Colonel Donovan.\n\nThe response then had been a furious, if brief, ass-chewing of the brass hat, done with the skill and finesse only a former infantry regimental commander\u2014as Donovan had been in the First War\u2014could hand out.\n\nEllis took off his brimmed cap and hung it atop a bentwood clothing rack. Then he removed a white silk scarf and folded it very neatly and hung it on a wooden coat hanger. Finally, he took off his blue overcoat and hung that carefully on the hanger. Then he turned and looked at the ASA warrant officer.\n\n\"The Colonel's home,\" Chief Warrant Officer Vole reported. \"Staley's with him. The Captain's home. I sent Marmon with a car for him. He's going to the Pentagon and will be in about ten, maybe a little later.\"\n\nMarmon was a former District policeman who served as combination chauffeur and bodyguard to Captain Peter Douglass.\n\n\"That's it?\" Ellis asked.\n\n\"Mrs. Foster's going to be in late,\" Vole continued. \"She has a dental appointment, but says she can reschedule if you need her. Miss Haley, she says, can handle everything she knows about.\"\n\n\"Fine,\" Ellis said.\n\n\"And I just made a pot of coffee,\" the ASA warrant officer said.\n\n\"And can I use one!\" Ellis said. \"It's as cold as a witch's teat outside.\"\n\nHe went to the small closet where the coffeepot sat on an electric hot plate and poured a cup.\n\nWhen he came out, the ASA warrant officer had taken the overnight messages from the safe and laid them out, together with the forms for the receipt of classified documents, on an oak table. Ellis sat down at the table.\n\n\"Anything interesting in here?\" he asked as he began to sign the forms.\n\n\"Mostly routine,\" Vole said. \"The Philippines have been heard from again, but that's about all.\"\n\nEllis looked at him with a question on his face.\n\n\"Seventeen,\" the ASA warrant officer said.\n\nWhen Ellis had finished signing the receipts and pushed the receipt forms away from him, he picked up file number seventeen and opened it. The first thing he saw was that it was an intercept, rather than a message intended for the OSS.\n\nOn his own authority, as \"Special Assistant to the Director, \" he had sent a \"Request for Intercept\" to the ASA, asking that the OSS be furnished with whatever ASA intercept operators around the world heard on either American or enemy frequencies that had anything to do with American guerrilla activity in the Philippine Islands. Inasmuch as the ASA and every other military and naval organization knew that the alternative to not giving the OSS whatever it asked for was explaining to the Chairman of the Joint Chiefs of Staff why this could not be done, the \"request\" had been in fact an order.\n\nEllis had decided that if Douglass or Donovan asked him why he had done so, and he didn't think they would, he would tell them it was because of the Whittaker mission. That was logical, of course. But the truth was that Ellis had put in the Request for Intercept long before it had been decided to send Whittaker into the Philippines. He had suspected that the reason there had been no reply to Fertig's original transmissions to MacArthur's headquarters in Australia was that some brass hats of MacArthur's palace guard, or perhaps even MacArthur himself, considered the very existence of guerrillas embarrassing. MacArthur's liaison officer to Washington had flatly announced that \"effective guerrilla operations were impossible. \"\n\nThe ASA intercept operators were good. They had furnished Ellis with the radio message from MacArthur appointing Philippine Scout Major Marcario Peralta \"military guerrilla chief of temporarily occupied enemy territory,\" and with Fertig's response to that, a request for drugs to cure venereal disease\u2014as much as telling MacArthur he considered himself fucked.\n\nToday's message showed that Fertig had his temper under control and was thinking:\n\nURGENT FROM WYZB FOR KSF  \nPASS TO SECRETARY OF WAR WASHINGTON DC\n\nAS SENIOR AMERICAN OFFICER IN THE PHILIPPINE ISLANDS I HAVE ASSUMED COMMAND OF MINDANAO AND VISAYAS WITH RANK OF BRIGADIER GENERAL.\n\nI HAVE REACTIVATED UNITED STATES FORCES IN THE PHILIPPINES.\n\nUSFIP HAS REESTABLISHED PHILIPPINE CIVIL GOVERNMENT IN THE HANDS OF ELECTED COMMONWEALTH OF PHILIPPINES OFFICIALS.\n\nLAWFUL GOVERNMENT OF PHILIPPINES IN AREA OF RESPONSIBILITY OF USFIP IS PRINTING AND PLACING INTO CIRCULATION MONEY.\n\nUSFIP IS BORROWING NECESSARY OPERATIONAL FUNDS FROM COMMONWEALTH OF PHILIPPINES GOVERNMENT.\n\nUSFIP URGENTLY REQUIRES MINIMUM ONE MILLION DOLLARS IN GOLD.\n\nUSFIP URGENTLY REQUIRES FOR MORALE OF PHILIPPINE POPULATION ANY SORT OF AID. MEDICINE FIREARMS AND AMMUNITION PREFERABLE.\n\nFERTIG BRIG GENERAL USA COMMANDING USFIP\n\nEllis frowned.\n\n\"What the hell is that all about?\" Vole asked.\n\n\"Fertig is being fucked by the system,\" Ellis said. \"But he's too mean to lie down and take it.\"\n\nThe telephone rang. Vole answered it, and then held his hand over the microphone.\n\n\"There's an Eyes Only Operational Immediate for either Donovan or Douglass,\" he said. \"They want to know if anybody's here that can take it.\"\n\n\"Decrypted?\" Ellis asked.\n\n\"Yeah. Dispatched at 1207 London time.\"\n\n\"Would you run down there and get it?\" Ellis asked.\n\nVole nodded, and took his hand away from the telephone microphone.\n\n\"Put it in a cover,\" he said. \"I'll be right down.\"\n\nVole was gone no more than five minutes. By the time he returned, Ellis had gone through the overnight messages and arranged those he felt Colonel Donovan should personally see in the order of their importance.\n\nHe took the two Eyes Onlys from Vole.\n\n\"I thought you said one Eyes Only,\" he said.\n\n\"They're related,\" Vole said.\n\nHe opened Dolan's message first, read it, and grunted. Then he opened the message Canidy had laboriously encrypted in the monks' cave on the Island of Vis.\n\nTOP SECRET  \nOPERATIONAL IMMEDIATE\n\nOSS LONDON STATION OSS WASHINGTON  \nEYES ONLY COLONEL DONOVAN; CAPTAIN DOUGLASS\n\nFOLLOWING FROM CANIDY RECEIVED 1110 LONDON TIME FORWARDED AUTHORITY DANCY CAPT WAC.\n\nBRUCE AND\/OR STEVENS WILL HAVE MESSAGE IN HANDS NO LATER THAN 1230 LONDON TIME.\n\nQUOTE TOP SECRET OPERATIONAL IMMEDIATE EYES ONLY BRUCE AND STEVENS\n\n1. ON SAFE ARRIVAL STATION VII INFORMED BY YACHTSMAN EXLAX AND TINCAN ONE IN HANDS OF CIVIL AUTHORITIES STATION V. TINCAN TWO SAFE WELL STATION VII.\n\n2. SURPRISE BOARDING BY BLACK GUARD AND RIVER POLICE YACHT STATION V RESULTED DISCOVERY EXLAX OPERATIONAL FUNDS. HUNGARIANS PRESUMABLY BELIEVE FUNDS INTENDED FOR PURCHASE BLACK MARKET FOOD. EXLAX AND TINCAN ONE ARRESTED AS BLACK MARKETEERS. SENTENCED NINE ZERO DAYS HARD LABOR COAL MINES STATION V.\n\n3. YACHTSMAN REPORTS DOCUMENTS NOT REPEAT NOT QUESTIONED.\n\n4. YACHTSMAN STATES SITUATION FAIRLY COMMON. ABSENCE PREPAYMENT GRAFT BLACK GUARD AND RIVER POLICE REGULARLY ARREST BLACK MARKETEERS CONFISCATE GOODS OR MONEY CONFINE LOCAL JAIL AT MINE HARD LABOR AS LESSON. YACHTSMAN BELIEVES THEY WILL BE RELEASED WITHOUT FURTHER DIFFICULTY PRIOR COMPLETION SENTENCE.\n\n5. HAVE TAKEN FOLLOWING ACTION.\n\nA. WILL REMAIN HERE PENDING DECISIONS ACTIONS ENUMERATED LATER HEREIN.\n\nB. TINCAN TWO FLOWN CAIRO FOR ICING THERE. RECEIPT THIS MESSAGE WILL CONFIRM SAFE ARRIVAL.\n\nC. YACHTSMAN ORDERED TO STATION V TO PERSONALLY CONFIRM LOCATION OF EXLAX AND TINCAN ONE AND TO EXPLORE POSSIBILITY ESCAPE OR RELEASE BY FORCE. EXPECTED TRAVEL TIME FOUR REPEAT FOUR DAYS. STATION V TO STATION VII COMMUNICATIONS SLOW AND UNRELIABLE REPEAT UNRELIABLE.\n\n6. REQUEST PERMISSION EFFECT RELEASE EXLAX AND TINCAN ONE BEST MEANS AT MY DISCRETION. IF SO REQUIRE IMMEDIATE DISPATCH VIA STATION VIII NEXT AVAILABLE HUNGARIAN SPEAKING TEAM. STANDARD\n\nTEAM EQUIPMENT SHOULD BE AUGMENTED WITH THIRTY POUNDS COMPOSITION C2 AND EQUIVALENT TWENTY THOUSAND US DOLLARS IN HUNGARIAN, GERMAN AND YUGOSLAVIAN CURRENCY. TEAM SHOULD HAVE HUNGARIAN AND OR YUGOSLAVIAN IDENTIFICATION DOCUMENTS.\n\n7. IN VIEW NECESSARY ABSENCE EXLAX CONTROLLER SUGGEST FINE AS TEMPORARY REPLACEMENT.\n\nCANIDY  \nEND QUOTE  \nTOP SECRET\n\n\"Oh, _shit!_ \" Chief Ellis said.\n\nHe picked up the telephone and dialed a number from memory.\n\nStaley's familiar voice came on the line: \"Capitol 3-1991.\"\n\n\"Is he up yet?\" Ellis asked.\n\n\"I heard the crapper flush,\" Staley reported.\n\n\"Well, don't say nothing unless he tells you to go anywhere but here,\" Ellis said. \"If he does, say I called and said I think he should come here straight from there.\"\n\n\"What's up, Ellis?\" Colonel Wild Bill Donovan's voice asked.\n\n\"There's something I think you ought to see as soon as you can, Sir.\"\n\n\"Will it wait until after breakfast, would you say?\"\n\n\"Yes, Sir, it'll keep that long.\"\n\n\"We'll be there inside of forty-five minutes,\" Donovan said, and the line went dead.\n\nEllis tapped the cutoff button on the telephone with his finger and dialed another number from memory.\n\n\"Capitol 3-2772,\" a male voice answered.\n\n\"Captain Douglass?\" Ellis asked.\n\n\"Who's calling, please?\" the man asked.\n\n\"Marmon, goddamn you, is that you?\"\n\n\"You don't have to bite my ass off, Chief,\" Marmon said righteously. \"I thought I recognized your voice.\"\n\n\"Is the Captain there?\"\n\n\"You want me to get him?\"\n\n\"No. Shit! I'm taking a census.\"\n\nIn a moment, Captain Douglass came on the line.\n\n\"Good morning, Chief,\" he said. \"What's up?\"\n\n\"I don't know what's going on where you're going, but if you can put it off, I think it would be a good idea if you came in.\"\n\n\"He ask for me?\"\n\n\"No, Sir, but I think he probably will.\"\n\n\"I'll be there in half an hour,\" Captain Douglass said. \"Thank you, Chief.\"\n\nEllis hung the telephone up.\n\n\"That important, huh?\" Warrant Officer Vole asked.\n\nEllis looked at him.\n\n\"If you're fishing for an explanation,\" Ellis said, \"don't.\"\n\n\"I read the decrypt,\" Vole protested.\n\n\"That's only because we haven't figured out a way for you to decrypt stuff without reading it,\" Ellis said matter-of-factly.\n\nHe got up and walked to the safe and worked the combination. From a two-foot-high stack of folders piled precariously in the bottom, he pulled a thick one with a TOP SECRET cover sheet and EXLAX written on it with a thick pointed pen.\n\nHe carried it to the desk and started going through it. There was no more of a question in his mind that the Colonel would want the paperwork in front of him than there was that he would want to talk Canidy's Eyes Only Operational Immediate over with Captain Douglass. By the time either of them walked into the office, the paperwork would be ready for them.\n\nEllis's eye fell on the overnight traffic. He should get that out of the way before he laid this stuff out.\n\nThen he had another thought. He opened a drawer and took out a lined pad and a pencil and wrote quickly on it.\n\n\"You want to make yourself useful,\" he said to Vole. \"Get this encoded and out right away. And then stick around. I think there will be a reply to the Eyes Onlys.\"\n\nVole took the sheet of lined paper from Ellis and read it.\n\n_Urgent_   \n_via K S F for W Y Z B_   \n_For Hq US Forces in Philippines_   \n_Attention Brigadier General Fertig_\n\nKeep your shirt on stop   \n_J. R. Ellis_   \nChief USN Stop End\n\n\"You really want me to send this?\" Vole asked.\n\n\"Just that way,\" Chief Ellis said.\n\n#  **4**\n\n##  **OFFICE OF THE STATION CHIEF OSS LONDON STATION BERKELEY SQUARE, LONDON 1600 HOURS 17 FEBRUARY 1943**\n\n\"Is something wrong, David?\" Lt. Colonel Edmund T. Stevens asked.\n\nBruce looked at him with his eyebrows raised.\n\n\"I would say so, wouldn't you?\" he replied dryly.\n\n\"I mean, right now, here,\" Stevens said. \"You were frowning. \"\n\n\"Oh,\" Bruce said, and then managed a faint smile. He gestured vaguely around his office. \"Actually, I was thinking, paraphrasing Churchill, that 'never have so few been commanded by so many.' \"\n\nThe three visitors' chairs in the office were occupied by Colonel Stevens, Capt. Helene Dancy, and Lt. Charity Hoche. Capt. Stanley S. Fine was leaning against the wall.\n\n\"I don't see that it could be avoided,\" Stevens said.\n\n\"No,\" Bruce agreed, then: \"I presume this is one of those things in which Miss Hoche has a special interest?\"\n\nStevens nodded.\n\n\"Well, let's get on with it, then,\" Bruce said. \"You first, Charity, please.\"\n\nShe didn't seem surprised, but neither did she say anything.\n\n\"The way we do this, Charity,\" Bruce explained, \"is 'in the military manner.' That is to say, the junior member of this panel is asked for his . . . her . . . opinion first, so that it will not be influenced by that of more senior members. \"\n\nCharity nodded.\n\n\"I don't see that we have any choice but to give Dick Canidy what he's asked for,\" she said, and then quickly added, \"at least until we hear to the contrary from Washington. \"\n\n\"That doesn't address the question of authorizing him to try to get Fulmar and Professor Dyer out of the jail in P\u00e9cs,\" Bruce said.\n\n\"I think we'll be told what to do about that,\" Charity said. Bruce looked at Stevens, who just perceptibly nodded his head in agreement.\n\n\"What he's asked for, specifically, is the next available Hungarian-speaking team, thirty pounds of C-2, and twenty thousand dollars in mixed currency,\" Bruce said. \"That's what you mean?\"\n\nCharity nodded. \"That, and Captain Fine to step in as control.\"\n\n\"We'll start with that, then,\" Bruce said. \"Unless I hear an objection, I will ask Fine if there is some reason he cannot, or thinks he should not, take over as control.\"\n\nHe looked at Stevens, then at Helene Dancy, and finally at Fine.\n\n\"No, Sir,\" Fine said.\n\n\"So ordered,\" Bruce said.\n\n\"One thing, Stanley,\" Stevens said. \"Charity is cleared for this. All the way.\"\n\n\"Yes, Sir,\" Fine said.\n\n\"I want to clarify that, Stan,\" Bruce said. \"Charity is to be brought into anything connected with this that Colonel Stevens and myself are.\"\n\n\"Yes, Sir,\" Fine repeated.\n\n\"Well, why don't you sit here,\" Bruce said, \"and take over this meeting?\"\n\n\"I don't mind standing, Sir,\" Fine said.\n\n\"I'd rather walk around,\" Bruce said, and gestured for Fine to sit down.\n\nFine sat down at Bruce's desk, put a lined pad on the green blotter in front of him, and took a pencil from a dozen sitting, point up, in a gray pottery orange-marmalade jar.\n\n\"Helene,\" he said. \"You'll take care of the money? Is that going to pose any problem?\"\n\n\"We don't have that much,\" Capt. Dancy said. \"But I can have it by, say, nine in the morning.\"\n\n\"And the C-2?\"\n\n\"I'm sure there's at least that much at Whitbey House,\" Colonel Stevens said.\n\n\"There should be,\" Helene Dancy agreed. \"But I'll check.\"\n\n\"That, then, brings us to the team,\" Fine said.\n\n\"First,\" Charity Hoche said. \"To the question of their documents. Canidy said Hungarian and\/or Yugoslavian. If we can, I think we should give them both.\"\n\nFine's face was expressionless, but Colonel Stevens thought he saw in his eyes a hint of surprise, even annoyance.\n\n\"Helene?\" Fine asked.\n\n\"Documents Section can handle that,\" Capt. Dancy said. \"They'll need four hours.\"\n\n\"Why so long?\" Charity asked.\n\n\"They don't have very much of the proper paper for the photographs,\" Dancy explained. \"We have to use their paper; it produces a characteristic grain and image flatness. The Hungarian is different from the Yugoslavian. And the only place we can get it is on the local black markets. It is also lousy paper, and it takes that much time to be sure. In case they have to print the photographs twice, or even three times.\"\n\n\"But they will be able to come up with what we need?\" David Bruce asked.\n\n\"Probably in forty-five minutes,\" Helene Dancy said. \"I'm using the worst possible scenario.\"\n\n\"Have we got a team to photograph?\" Fine asked.\n\n\"They have all been photographed, Stanley,\" Helene Dancy said. \"Several times, in work clothes, suits, even in Black Guard uniforms. _Printing_ is the problem.\"\n\n\"That's not what I really meant,\" Fine said. \"I'll rephrase. Is a team available? If there is more than one available, which is the better of them?\"\n\n\"I was out there when this came up, Stanley,\" Colonel Stevens said. \"There are two teams finished with training, one in the last week.\"\n\n\"Did Jamison give you any indication which was better? \" Fine asked.\n\n\"There is a problem,\" Stevens said. \"The teams that have finished their training have been trained to go in to Tito, not Mihajlovi\u0107.\"\n\n\"God!\" Bruce said. There were two major guerrilla forces in Yugoslavia. Colonel Dra\u017ea Mihajlovi\u0107 led a force of Royalists, and Josip Broz, who called himself \"Tito,\" led a larger force of Communists.\n\n\"It was necessary, David,\" Stevens said. \"We had to appear even-handed, and that meant sending teams to Tito.\"\n\n\"What about the team in training?\" Bruce asked.\n\n\"Mihajlovi\u0107,\" Stevens said. \"But it's a communications team. No specific training for something like this.\"\n\n\"What about J\u00e1nos?\" Helene Dancy asked. \"When does he get out of his cast?\"\n\n\"Friday,\" Charity said.\n\n\"Who the hell is J\u00e1nos?\" Bruce asked.\n\n\"The first lieutenant who broke his ankle,\" Dancy furnished.\n\n\"Well, we can hardly take his cast off one day and jump him in the next,\" Bruce said.\n\n\"J\u00e1nos was trained to go in to Mihajlovi\u0107,\" Dancy said.\n\n\"And we're not going to jump them in anyway,\" Charity said. Bruce snapped his head around to look at her. Charity smiled, and added, \"Are we?\"\n\n\"We may have to,\" Stevens said. \"With Dolan under the weather, I think we have to leave him out of the equation. And that means there's no one but that young pilot . . . whatsisname?\"\n\n\"Darmstadter,\" Helene Dancy furnished.\n\n\"Darmstadter,\" Stevens said, \". . . to fly the B-25. Which means either parachuting them onto Vis or, for that matter, into Hungary, or sending them by submarine.\"\n\n\"I can fly the B-25,\" Fine said.\n\n\"No,\" Bruce said. \"You're the control.\"\n\n\"Doug Douglass can fly the B-25,\" Charity said.\n\nBruce looked at her.\n\n\"He's not . . . assigned to us,\" he said.\n\n\"Can't that be arranged?\" Charity asked. \"TDY or something?\"\n\n\"We're getting ahead of ourselves,\" Fine said. \"The first decision that has to be made is about the team. Do we send in a Tito team?\"\n\nCharity looked at Colonel Stevens.\n\n\"No,\" Stevens said flatly.\n\n\"What's the difference?\" Helene Dancy asked.\n\n\"For the men, none,\" Stevens said. \"But we will not send an officer on this who has been selected to go in to Tito.\"\n\n\"May I ask why, Sir?\" Fine asked.\n\n\"No, I'm afraid I can't tell you, Stan,\" Stevens said.\n\nFine looked curious, but shrugged.\n\n\"It looks as though we're back to J\u00e1nos,\" he said. \"And to flying him in rather than jumping him in, because of his ankle. And since I can't fly the B-25, we're also back to Douglass. May I ask that you reconsider, Sir, my flying the B-25?\"\n\n\"Out of the question,\" Bruce said.\n\n\"And we're not even sure of J\u00e1nos,\" Stevens said. \"Helene, get the medical officer at Whitbey House on the horn and get a report on J\u00e1nos's ankle. Specifically, what shape he will be in when they take the cast off, and for how long.\"\n\n\"While she's doing that,\" Fine said, \"what about transport of the team from here to Cairo? If they're going to Cairo?\"\n\n\"What do you mean by that?\" Stevens asked.\n\n\"I thought perhaps Malta,\" Fine said.\n\n\"Oh,\" Stevens said.\n\n\"We'll send them to Cairo,\" Bruce said. \"They'll attract less attention there. And we'll send them on the ATC courier. There's a daily flight. If we send one man a day, starting right now, they should attract no attention at all. Wilkins is good at distracting attention.\"\n\n\"Mr. Bruce,\" Fine said, almost hesitantly. \"I'm afraid that you will think I'm rephrasing a request that has already been denied\u2014\"\n\n\"What, Fine?\" Bruce asked impatiently.\n\n\"Project Aphrodite has two new B-17s,\" Fine said.\n\n\"One of which you would like to fly to Cairo?\" Bruce asked, icily sarcastic.\n\n\"May I explain my thinking, Sir?\"\n\n\"No,\" Bruce said.\n\n\"I would like to hear it,\" Charity said.\n\nBruce glared at her and opened his mouth to speak. Before he could, there was a knock at the door, and instead of saying what he'd intended, Bruce said, his voice angry and impatient:\n\n\"We are not to be disturbed!\"\n\n\"Operational Immediate Eyes Only for you, Sir,\" Capt. Harrison's voice came through the door.\n\n\"Oh, hell,\" Bruce said, \"now what?\" He raised his voice. \"Bring it in, Harrison!\"\n\nHarrison came into the room, extended a clipboard with a Receipt for Classified Document on it, and, when Bruce had signed it, handed him a document with a TOP SECRET cover sheet on it.\n\n\"Thank you,\" Bruce said. \"I didn't mean to snap at you, Paul.\"\n\n\"No problem, Sir,\" Harrison said. He made no move to leave.\n\n\"That'll be all, thank you,\" Bruce said.\n\n\"An action is required, Sir,\" Harrison said.\n\nBruce snorted, and lifted the cover sheet.\n\nTOP SECRET  \nOPERATIONAL IMMEDIATE\n\nOFFICE OF THE DIRECTOR OSS WASHINGTON  \nFOR OSS LONDON EYES ONLY BRUCE STEVENS  \nFOR OSS CAIRO EYES ONLY WILKINS\n\nRELAY FOLLOWING CANIDY MOST EXPEDITIOUS MEANS:\n\nQUOTE\n\n1. OSS LONDON AND CAIRO DIRECTED AS HIGHEST PRIORITY TO SUPPORT WITH ALL MEANS AVAILABLE ATTEMPT FREE EXLAX AND TINCAN ONE.\n\n2. RESCUE WILL BE ATTEMPTED AT EARLIEST POSSIBLE TIME AND IN ANY CASE NOT LATER THAN TEN REPEAT TEN DAYS FROM RECEIPT OF THIS MESSAGE.\n\n3. IN EVENT ATTEMPT IMPOSSIBLE OR ATTEMPT FAILS EXLAX AND TINCAN ONE WILL BE TERMINATED REPEAT WILL BE TERMINATED.\n\n4. NO DISCUSSION OF THIS ORDER WILL BE ENTERTAINED.\n\nUNQUOTE  \nEND QUOTE\n\nSTATION CHIEFS LONDON AND CAIRO WILL ACKNOWLEDGE RECEIPT AND COMPREHENSION OF CANIDY MESSAGE.\n\nSTATION CHIEF CAIRO ADDITIONALLY WILL REPORT TIME AND PLACE OF DELIVERY OF MESSAGE TO CANIDY WITH INFO COPY TO LONDON.\n\nDONOVAN\n\nBruce handed the message to Colonel Stevens, and then looked at Capt. Harrison.\n\n\"Would you please message Colonel Donovan that Colonel Stevens and myself acknowledge receipt and comprehension of this message?\" he said, almost formally.\n\n\"Yes, Sir,\" Harrison said.\n\n\"My God!\" Stevens said, when he had read the message. He extended it to Bruce. \"May I see that, please?\" Charity Hoche asked.\n\n\"Oh,\" Stevens said, as if he had just remembered she was present. \"Sure.\"\n\nWhen she had read it, she handed it to Fine. He frowned when he read it, but said nothing.\n\n\"You were saying, Stan,\" Charity said, \"something about new B-17s?\"\n**XI**\n\n#  **1**\n\n##  **OSS WHITBEY HOUSE STATION KENT, ENGLAND 1815 HOURS 17 FEBRUARY 1943**\n\nFirst Lieutenant Robert Jamison found First Lieutenant Ferenc J\u00e1nos where he thought he would be at this time of day, in the bar, at the piano, with a drink adding yet another scar to the varnished finish of the piano, and with two of the WRAC drivers listening to him play.\n\n\"Freddy, could I see you for a moment?\" Jamison asked.\n\n\"Ladies,\" Freddy J\u00e1nos said, \"duty calls.\"\n\n\"Not here, J\u00e1nos,\" Jamison said.\n\nJ\u00e1nos's eyebrows rose in curiosity, but he didn't say anything. He hoisted himself from the piano stool with a grunt, reached for his drink, finished it, and then squatted on his one good leg to pick up his crutches from the floor.\n\nHe followed Jamison down the center corridor of the first floor of the left wing of the mansion to the dispensary, which had been set up in what had been the ballroom. There were sixteen beds, eight on each side of the high-ceilinged room. Eleven of them were occupied.\n\nAt the far end of the ballroom was a small, flat-roofed \"building,\" roughly framed in with exposed two-by-fours and plywood. It held a simple, if surprisingly complete, operating room, an X-ray room, a pharmacy, two examining cubicles, a dentist's chair and equipment, and an office for the two physicians attached to Whitbey House station.\n\nBoth of them, and a nurse, were waiting for Jamison and J\u00e1nos.\n\n\"What's going on?\" J\u00e1nos asked when he saw where Jamison had led him.\n\n\"First things first,\" Jamison said. \"We want an X-ray of your ankle.\"\n\n\"I will repeat, Lieutenant Jamison,\" one of the doctors, a captain, said, \"that I don't like this.\"\n\n\"Whether or not you approved didn't come up, Doctor,\" Jamison said, \"when the Colonel said to do it.\"\n\nThe doctor gave him a cold look.\n\nThe nurse took J\u00e1nos into the X-ray room, motioned for him to hop on the table, and then took his crutches and leaned them on the wall. She wordlessly arranged his leg on the table under the X-ray apparatus, then stepped behind a crudely built, raw-lumber six-foot wall.\n\n\"Don't move,\" she ordered, and there was a whirring sound from the X-ray apparatus. She made six X-rays before she was finished, then issued another order: \"You'd better stay there until I get these out of the soup.\"\n\nThe two doctors and Jamison came into the room.\n\n\"What's going on?\" J\u00e1nos asked from the X-ray table.\n\n\"We want to see if we can take your cast off safely,\" Jamison said.\n\n\"Why?\"\n\n\"If we can, I'll tell you,\" Jamison said.\n\nJ\u00e1nos, who had been lying down, sat up on the table and let his legs hang over the side.\n\nThe nurse returned with still damp eleven-by-fourteen-inch X-ray photographs, put three (all it would handle) on a viewer, and turned it on.\n\nThe two doctors examined the X rays and then replaced them on the viewer with the other three.\n\nThe captain turned to Jamison.\n\n\"It appears to have healed and knitted satisfactorily,\" he said.\n\n\"The question, Doctor,\" Jamison said, \"is, in your professional medical opinion, can the cast be safely removed? \"\n\n\"There's a difference, Jamison, between taking it off and declaring this officer fit for duty.\"\n\n\"Can it be safely removed?\" Jamison replied. \"If so, please remove it.\"\n\n\"Jesus Christ,\" the other doctor, a lieutenant, said disgustedly.\n\n\"Would you get me the cutter, Nurse?\" the captain asked.\n\nJ\u00e1nos didn't like what he saw when the cast was removed. The skin beneath, where it was not marked with angry red marks, was unhealthily white, and although he couldn't be sure without actually comparing it side-by-side with his good ankle, it looked to him to be much thinner.\n\nBoth doctors manipulated the ankle and the foot. There was no pain, but it was uncomfortable.\n\n\"Well?\" Jamison asked.\n\n\"The fractures,\" the captain said, \"seem to have knitted satisfactorily. There is no pain or impediment of movement that I can detect.\"\n\n\"He can walk, in other words?\" Jamison asked.\n\n\"Before he can be determined to be physically fit for duty,\" the captain said, \"he will require therapy. Do you concur, Doctor?\"\n\n\"Jamison,\" the younger doctor said, \"there is muscle atrophy\u2014\"\n\n\"What kind of therapy?\" Jamison asked.\n\n\"Walking, actually,\" the captain said. \"Short walks, gradually extended. Manipulation of the foot and ankle to restore movement.\"\n\n\"That'll be all, J\u00e1nos,\" Jamison said. \"Thank you.\"\n\n\"You said you would tell me,\" J\u00e1nos protested.\n\n\"You're being considered for an operation,\" Jamison said.\n\n\"When and if it is decided you're going, you'll be told about it.\"\n\n\"When will that decision be made?\"\n\n\"Tomorrow morning, probably,\" Jamison said. \"Do you think you can manage without your crutches?\"\n\n\"I don't know,\" J\u00e1nos said.\n\n\"Give it a try,\" Jamison said. \"If you can, leave the crutches here. If you go back to the bar, go easy on the booze. I don't want you falling down and breaking it again.\"\n\nThe lieutenant, shaking his head, chuckled.\n\nThe captain said, \"Jamison, I might as well tell you, the moment Major Canidy returns, I'm going to protest this.\"\n\n\"Captain,\" Jamison said, \"all I'm doing is obeying my orders. That's what you do when you put a uniform on, obey orders.\"\n\nHe turned and walked out of the room.\n\nThe captain called after him. \"Jamison, in my capacity as the senior medical officer present, I absolutely forbid this officer to participate in a parachute jump.\"\n\n\"Your position has been noted, Doctor,\" Jamison called, over his shoulder.\n\nJ\u00e1nos got off the X-ray table and gingerly lowered his bare, sick, white foot to the floor.\n\n\"Any pain?\" the captain asked.\n\n\"No,\" J\u00e1nos said.\n\n\"Fuck him,\" the captain said, \"You use the crutches, J\u00e1nos. You start using that leg carefully. I'll deal with Jamison. \"\n\nJ\u00e1nos hoisted himself back onto the X-ray table and removed his other shoe and sock.\n\n\"With the shoe off,\" J\u00e1nos said, \"I think I can manage.\"\n\nHe lowered himself to the floor again, and then, awkwardly and carefully, walked very slowly out of the X-ray room.\n\n#  **2**\n\n##  **OSS WHITBEY HOUSE STATION KENT, ENGLAND 0600 HOURS 18 FEBRUARY 1943**\n\nFirst Lieutenant Ferenc J\u00e1nos marched into the office of the commanding officer, came to attention, and saluted. He was wearing a wool OD (olive drab) Ike jacket and trousers. Parachutist's wings were on the jacket, and his trousers were bloused over glistening Corcoran jump boots. His woolen \"overseas\" cap was tucked in an epaulet of his jacket.\n\n\"Sir, Lieutenant J\u00e1nos reporting as directed, Sir.\"\n\nLt. Colonel Edmund T. Stevens returned the salute.\n\n\"Stand at ease, Lieutenant,\" he said.\n\nJ\u00e1nos was surprised to see the good-looking blond WAC lieutenant in the room. He wondered why. The story about her (which had quickly circulated through Whitbey House) was that she would work for Jamison, taking care of the women.\n\n\"How's your ankle, J\u00e1nos? Straight answer, please,\" Stevens said.\n\n\"With the boot on, sir,\" J\u00e1nos said, \"no problem.\"\n\n\"How far do you think you could walk on it?\" Stevens asked.\n\n\"As far as I have to,\" J\u00e1nos said.\n\n\"An overestimate of capability is dangerous, J\u00e1nos,\" Colonel Stevens said.\n\n\"Yes, Sir,\" J\u00e1nos said.\n\n\"A mission of the very highest priority has come up,\" Stevens said. \"You have already expressed your willingness to participate in a mission involving great personal risk in enemy-occupied territory. You were also made aware that if you were captured, you would be treated not as a prisoner of war but as a spy. I ask you here and now if you still wish to volunteer for such a mission?\"\n\n\"Yes, Sir,\" J\u00e1nos said.\n\n\"From this point, Lieutenant,\" Stevens said, \"this conversation is classified Top Secret. Divulging what I am about to tell you to anyone, or discussing it with anyone not now present in this room, will constitute a general court-martial offense. Do you fully understand that?\"\n\n\"Yes, Sir.\"\n\n\"The mission is to free certain people from confinement in the hands of civil authorities in Hungary. I am now going to pose a question to you that I want you to think over very carefully before replying,\" Stevens went on. \"If the mission goes sour, or if the mission cannot be accomplished within a set time frame, you will be required to eliminate, by which I mean kill, or cause to have killed, the people presently imprisoned. Now, are you willing to accept the mission, knowing that may be necessary?\"\n\nJ\u00e1nos hesitated, but not for long.\n\n\"Yes, Sir,\" he said. He became aware that the good-looking blond WAC was looking at him. More than looking at him, he realized\u2014evaluating him and doing that very coldly.\n\n\"You believe you would be able to . . . and this is the only phrase that fits the situation . . . kill in cold blood the people presently imprisoned. And possibly a substantial number of others who can only be accurately described as 'innocent bystanders'?\"\n\n\"You're not going to tell me what this is all about?\"\n\n\"Just please answer my question,\" Stevens said.\n\n\"With your assurance that it's a military necessity, Sir,\" J\u00e1nos said.\n\nStevens nodded.\n\n\"Charity?\" he asked.\n\n\"Even, Freddy,\" Charity Hoche asked, \"if the people who had to be eliminated were known to you? Even if you had met them here?\"\n\n\"Holy Mother of God,\" J\u00e1nos blurted, and then found control again. \"With the same caveat as before, that Colonel Stevens assures me this is militarily necessary.\"\n\nThere was a knock at the door.\n\n\"Yes?\" Stevens called impatiently.\n\n\"Colonel Douglass is on the phone for Lieutenant Hoche, Sir,\" a male voice said.\n\n\"I guess I better take it,\" Charity said after a moment's thought. \"He probably just got his orders and wonders what they're all about.\"\n\nShe walked out of the office.\n\n\"That was _the_ important question,\" Colonel Stevens said. \"But there is another important question. For reasons I cannot go into, it is impossible for us to send Lieutenant Shawup on this mission. But the team that he commands will make it. There will be a certain resentment on their part toward you. Can you handle it?\"\n\n\"Yes, Sir,\" J\u00e1nos said without hesitation.\n\n\"They will resent\u2014after having received promises to the contrary\u2014not being under Shawup's leadership. And they will resent being told . . . they will not be asked, they will be told . . . that elimination of the people being held may be necessary. They will resent that, too.\"\n\n\"They'll do what I tell them to do,\" J\u00e1nos said confidently.\n\n\"You sound very sure of yourself,\" Stevens said.\n\n\"Look at me, Colonel,\" J\u00e1nos said. \"As big as I am, wouldn't you hate to make me mad?\"\n\nStevens's face went blank for a moment, and then he chuckled.\n\n\"Yes, I guess I would,\" he said.\n\nHe leaned over the desk and offered J\u00e1nos his hand.\n\n\"I have every confidence that you can handle this, Lieutenant J\u00e1nos,\" he said. \"Good luck!\"\n\n#  **3**\n\n##  **FERSFIELD ARMY AIR CORPS STATION BEDFORDSHIRE, ENGLAND 1200 HOURS 18 FEBRUARY 1943**\n\nWhen the P-38 flashed over them, Lieutenant Commander Edwin W. Bitter, USN, Captain Stanley S. Fine, USAAC, and Lieutenant j.g. Joseph P. Kennedy, Jr., USNR, were sitting on folding wooden chairs outside the Quonset hut that served officially as the orderly room of the 402nd Composite Squadron and secretly as the headquarters for Operation Aphrodite.\n\nThey were taking the sun. There was precious little sun in England in February, and when it did pop out, everyone who could take the time tried to get out in it.\n\n\"I have been told by everybody from Bill Donovan to that ferocious WAC captain in David Bruce's office that asking questions is like farting in the Sistine Chapel,\" Kennedy said, \"but I would still dearly like to know where the hell you are taking my brand-new airplane.\"\n\n\"Come on, Joe,\" Commander Bitter said, a mild reproof.\n\n\"Yours not to reason why, Lieutenant,\" Fine said, smiling at him, \"yours but to take yon fighter jockey aloft and see how much you can teach him in an hour or two about driving the B-17.\"\n\nHe gestured in the direction of the P-38, which the pilot had stood on its wing to line it up with the main Fersfield runway.\n\n\"I am also just a little curious why that is necessary,\" Kennedy said, \"since here sit Commander Bitter and myself, both fully qualified B-17 pilots, and in my case at least, an extraordinary 'Look Ma, No Copilot' 17 chauffeur. \"\n\nBitter and Fine laughed.\n\n\"Your country, Lieutenant,\" Fine said, \"is saving you for more important things.\"\n\n\"You aren't going to tell me, are you, you sonofabitch?\" Kennedy said.\n\n\"I can't, Joe,\" Fine said seriously.\n\nThey stood up to watch the P-38 land. It came in hot, in a crab, lining up with the runway at the last moment before touching down.\n\n\"If yon fighter jockey tries that in a 17,\" Kennedy said dryly, \"we will have one more to park over there.\"\n\nHe pointed to the \"graveyard\" where remnants of more than two dozen crashed and shot-up B-17s were scattered around.\n\n\"Without any whistling-in-the-dark self-confidence,\" Kennedy went on, \"what are our chances of getting that 17 back?\"\n\n\"That will depend on how much you can teach Doug,\" Fine said.\n\nA Follow Me jeep had driven out to the taxiway to meet the P-38. Fine started to walk toward the revetment in which it would be parked, and Bitter and Kennedy followed him.\n\n\"I think I'll go along in the 17,\" Bitter said. \"Maybe I could help Joe.\"\n\n\"No,\" Fine said, politely enough, but there was no mistaking it was an order. \"We want to keep you around to fly the other new one.\"\n\nThey reached the revetment as the P-38 taxied up to it.\n\nA ground crewman made a throat-cutting signal with his hand, and the engines died. A ground crewman laid a ladder against the cockpit, and Lt. Colonel Peter Douglass, Jr., climbed down it.\n\nHe was wearing a pink Ike jacket, matching trousers, a battered, oil-spotted, fur-felt brimmed cap with the crown stiffener removed on the back of his head, half Wellington boots, and a parachute-silk scarf in the open collar of a gabardine shirt.\n\n_He is absolutely, totally, out of uniform,_ Fine mused. And then he corrected himself. _No, that is the uniform prescribed by fighter pilots for themselves. And there is no question that Doug is one hell of a fighter pilot. There were Japanese meatballs and German swastikas painted in three neat rows on the cockpit nose, plus a submarine._\n\n_And something brand new. Douglass had named his airplane \"Charity.\"_\n\n\"Where the hell is my brass band?\" Douglass asked, wrapping his arm around Commander Bitter's shoulders and (because he knew it annoyed Bitter immensely) kissing him wetly on the temple.\n\nFine and Kennedy smiled.\n\n\"Who's Charity?\" Kennedy asked.\n\n\"As in 'Faith, Hope and,' \" Douglass said. \"if I don't get a band, how about lunch? I'm starved.\"\n\n\"You're going flying with Lieutenant Kennedy,\" Fine said. \"You can have lunch when you come back.\"\n\n\"Where am I going flying with you, Kennedy?\" Douglass asked.\n\n\"Up and down, up and down,\" Kennedy smiled. \"Fine wants me to teach you to line an airplane up with the runway while you're still in the air.\"\n\n\"Only bomber pilots have to do that,\" Douglass said. \"It's because their reflexes are so slow. You're serious about this, aren't you? _Before_ I have lunch?\"\n\n\"If you're a good boy, I'll have a surprise for you when you get back,\" Fine said.\n\n\"I already talked to her,\" Douglass said, \"which raises the question of Rank Hath Its Privileges.\"\n\n\"How?\" Fine asked.\n\n\"A senior officer such as myself,\" Douglass said, \"cannot be expected to share a room with low-grade underlings such as you guys. Do I make my point?\"\n\n\"Oh, I think Commander Bitter will be happy to accommodate you, Colonel, Sir,\" Kennedy said, chuckling. \"He already has had the troops spiffing up the transient female quarters. You'll notice the smile of anticipation on his face.\"\n\n\"Doug,\" Bitter said very seriously, changing the subject, \"if you really want something to eat, I'll have some sandwiches prepared and get them to the aircraft.\"\n\n\"Shame on you, Lieutenant Kennedy,\" Douglass said, \"you are embarrassing the commander.\"\n\nFor a moment, looking at Bitter, Fine was afraid the situation was going to get out of hand, but with a visible effort, Bitter finally managed a smile.\n\nDouglass looked at his watch.\n\n\"The girls are due here at two-fifteen,\" he said. \"That gives you just about two hours to teach me all you know, Kennedy. That shouldn't be a problem.\"\n\nDouglass and Kennedy flew for nearly two hours before landing a final time and taxiing the B-17F back to the 402nd Composite Squadron area. As they stood by the aircraft with the crew chief, giving him a list of things to check to prepare the plane for flight, a small convoy rolled past the B-17 graveyard and stopped before the Quonset hut.\n\nThe convoy consisted of an Austin Princess limousine, a Packard limousine, and a three-quarter-ton Dodge weapons-carrier. The Packard and the Austin Princess were driven by sergeants of the WRAC, and the canvas-bodied Dodge by a U.S. Army sergeant.\n\nLt. Colonel Ed Stevens and Lt. Charity Hoche got out of the Princess, and five men in olive-drab U.S. Army uniforms got out of the Packard.\n\n\"Let that be a lesson to you, Lieutenant Kennedy,\" Douglass said, \" 'Virtue is its own reward.' If you had allowed me to land this aerial barge when I wanted to, I wouldn't have had to stand around panting until just now.\"\n\n\"One gathers that the Colonel would be panting over the blond lieutenant?\" Kennedy asked. \"Who the hell is she, anyway?\"\n\n\"A senior officer such as myself,\" Douglass said, \"does not of course discuss either ladies or his personal affairs with a junior officer. But I will say this, Kennedy. If it were to come to my attention that anyone\u2014say, a lowly reserve swabby officer\u2014paid any but official attention to a certain WAC officer while I am off saving the world for democracy, I would feed him his balls.\"\n\n\"That's Charity,\" Kennedy said.\n\n\"That's Charity,\" Douglass confirmed possessively.\n\n\"I hate to tell you this, Colonel,\" Kennedy said. \"But the lady doesn't seem prone to throw herself in your arms.\"\n\n\"That's because she doesn't want to make you feel jealous, \" Douglass said.\n\nThey smiled at each other.\n\n\"Thanks for the lessons,\" Douglass said. \"How did a fair-to-middling airplane driver like you wind up flying aerial barges?\"\n\n\"Just lucky, I guess,\" Kennedy said. \"And just for the hell of it, Colonel, if that were a check ride, you would have passed it.\"\n\nThey smiled at each other again.\n\n\"Let's go see if we can make Bitter blush again,\" Douglass said.\n\n#  **4**\n\n##  **BUDAPEST, HUNGARY 0350 HOURS 19 FEBRUARY 1943**\n\nCanidy didn't see the policeman with his hand held up until he was almost on him.\n\nHe had been too busy watching the road in front of him. It had been a long time since he had ridden a bicycle, and while it was true, he had found out, that once you learned how, you never forgot, it was also true that pedaling a bicycle required muscles he hadn't used in a long time. Even moving as slowly as they had been riding, his calves and upper thighs were heavy with exhaustion.\n\nAnd the road was covered with frozen slush, which caught the wheel of the bicycle when it rode in one of the ruts. He had taken four spills, and one of them was a bad one, throwing him heavily on his right shoulder and bruising his right knee.\n\nThere was no chance to stop before he got to the policeman, although he made a valiant effort. And, he saw, there was no place to run either, no corner to duck around. The policeman had appeared from nowhere because he had been inside a small, wooden guard shack almost hidden by the buttresses of the \u00c1rp\u00e1d Bridge. There was nothing ahead but the bridge itself, and if the policeman couldn't run him down on foot, which seemed likely, then he would have no trouble shooting at him.\n\nThe policeman got out of his way, as Canidy locked the hand brakes and skidded to a stop on the icy slush, the bike slipping out from under him.\n\nHe heard Ferniany laugh behind him as Canidy fell to his knees.\n\nAnd then the policeman said something. Canidy had no idea what he said, but he thought there was a tone of laughter in it.\n\nCanidy got to his feet, picked up the bicycle, and walked to where the policeman was now examining Ferniany's identity documents. Canidy rested the bicycle against his leg, reached inside his ragged shepherd's coat for his papers, and held them ready in his hand until the policeman was ready to take them.\n\nHe looked toward the far end of the bridge. He could not tell if there was another policeman in another hidden shack at the far end. Probably not. The \u00c1rp\u00e1d Bridge crossed a branch of the Danube between Pest and Margit Island. The Margit Bridge crossed the other branch of the Danube to Buda. If there was another guard shack, it would be on the Margit Bridge, not at the end of this one.\n\nIf it became necessary to kill this policeman\u2014by breaking his neck or cutting his throat\u2014it would still be possible to continue across the Danube here.\n\nThe policeman handed Ferniany's papers back and turned to Canidy. He was shaking his head. He said something. Canidy had no idea what it was, but he shrugged.\n\nThe policeman took his papers. Canidy saw Ferniany take his garrote from his pocket.\n\nThe policeman returned Canidy's papers with what could have been a courteous bow. Then he turned Canidy around and unfastened the straps of the rucksack Canidy had on his back. He came out with a small cheese and a small sausage.\n\nCanidy gestured that he was welcome to it. The policeman smiled and then politely fastened the straps on the rucksack. Then he went to Ferniany's bicycle and began to unfasten the straps holding a limp rucksack over the fender. Canidy put his hands up his sleeves, hoping it looked as if he were trying to warm his hands. He jerked the strap around the hilt of his Baby Fairbairn free and tested to see if he could quickly get it out of its sheath. It was a dagger that had been developed by Captain Bruce Fairbairn of the Shanghai Municipal Police. The \"Baby\" was the smaller of two versions and was used when concealment was desirable.\n\nFulmar and Whittaker had given him a quick course in assassination. Neither of them liked the garrote. ( _\"What if the wire gets hung on a button or something?\"_ Fulmar had calmly argued. _\"Or if he gets his fingers under the wire before you can bury it in his neck? Put your hand over his mouth and stick him behind the ear. As soon as you scramble his brains, you can let him go. It takes a hell of a long time to strangle somebody.\"_ )\n\nWhittaker's preferred technique of assassination was throat-cutting. ( _\"Once you cut into the throat, all they can do is gargle,\"_ Whittaker had said. _\"I don't trust the ittybitty point on the Fairbairn, especially the little one. You hit a bone or something, and it breaks, and there you are with your hand over the mouth of some highly pissed-off character you can't put down.\"_ )\n\nCanidy had decided the Fairbairn was best, because it was far more concealable than a throat-cutting knife, and because Jimmy Whittaker had somewhat reluctantly conceded that there was a lot of blood when you cut some-one's throat and very little when you scrambled his brains.\n\nCanidy felt bile in his throat at the prospect that he might now have to put theory into practice, but it did not become necessary. The policeman helped himself to a tub of butter from Ferniany's rucksack and waved them on.\n\nThey rode to the end of the bridge and then crossed Margit Island. He could see what looked to him like an amusement park closed for the winter: small wooden shacks in a line; an oblong building that could have concealed a dodgem ride; a larger round building that almost certainly contained a merry-go-round.\n\nThere was no policeman at the Buda end of the Margit Bridge.\n\nTwo blocks into Buda, the cobblestone street became too steep and too slippery to pedal the bicycles, and they got off and pushed. And for some reason, here the slush had begun to melt (Canidy wondered about this and decided they were over a tunnel of some kind, maybe a sewer, that gave off enough heat to melt the frozen slush). So his feet, in rough leather work shoes and thick cotton socks, quickly became wet and then even colder than they had been.\n\nBetween the Margit Bridge and Batthyany Palace, they passed two more policemen, but neither of them showed any interest in the bicyclists.\n\nWhen Ferniany finally pushed his bicycle off the street and onto the sidewalk before the facade of what looked like a museum, Canidy was sweat-soaked from exertion and annoyed that Ferniany seemed immune to both fatigue and cold.\n\nThe doorbell was just that, a handle which when pulled caused a bell somewhere inside the building to just audibly tinkle.\n\nBy the time a small door built into the larger door opened a crack, Canidy had his breath back, but his sweat-soaked clothing had chilled, and he was shivering and his feet hurt.\n\nA small old man with white hair and very bright eyes exchanged a few words with Ferniany, then opened the door to let them pass.\n\nThere were more cobblestones inside the door, and at the end of a passageway a courtyard. The little old man led them into a huge kitchen and said something to Ferniany, apparently an order to wait. The kitchen, Canidy saw, was not in use. There was a huge icebox, and each of its half-dozen doors was wedged open. More important, none of the three wood-burning stoves held a fire.\n\nA door opened, and a rather startling redhead came into the kitchen. Her hair, a magnificent mop of dark red, hung below her shoulders. She was wrapped in an ankle-length, somewhat bedraggled, Persian lamb coat. The hem of a woolen nightgown was exposed at the bottom, and her feet were in what Canidy at first thought were half Wellington boots, but which he saw after a moment were really sheepskin -lined jodhpurs.\n\nShe shook Ferniany's hand, and they had a brief exchange. Then she turned to Canidy. She spoke British-accented English.\n\n\"I am the Countess Batthyany,\" she said. \"How may I be of service, Major?\"\n\n\"I'm Pharmacist,\" Canidy said.\n\nHer eyebrows rose in genuine surprise.\n\n\"You would be far more welcome,\" she said, \"if I didn't suspect that you wouldn't be here unless there is trouble.\"\n\n\"Have you got any brandy?\" Canidy said. \"I'm chilled to the bone.\"\n\n\"Yes, of course,\" she said. \"Forgive me.\"\n\nShe turned and motioned for them to follow her. There was a narrow, rather steep flight of stairs, and then a door. They stepped into a dimly lit room. The room was well furnished, and when Canidy glanced around, he saw that the door they had come through was cut through the paneling of the room so that it would fit in with the decor. A servants' passageway, he decided.\n\nWhen he turned around again, there was a man in the room. Tall, aristocratic, wearing a silk dressing gown. He held a Walther Ppk .32 ACP pistol in his hand. It was pointed at the floor.\n\n\"Was ist los?\" he asked.\n\n\"Liebchen, this is Major Canidy,\" the Countess said, adding, \" _Pharmacist_. Major, may I introduce His Excellency Brigadef\u00fchrer-SS von Heurten-Mitnitz?\"\n\nVon Heurten-Mitnitz's expression did not change, but he spent a long moment examining Canidy before he spoke.\n\n\"The major and his friend look frozen,\" he said. \"Could you ring for some brandy? Something for them to eat?\"\n\n\"Yes, of course,\" the Countess said.\n\nThen von Heurten-Mitnitz looked at Canidy again.\n\n\"You don't happen to know Putzi's son's name, do you?\"\n\n\"I was wondering if you were going to ask,\" Canidy said, then gave his part of the prearranged countersign. \"Ergon.\"\n\nVon Heurten-Mitnitz nodded coolly and managed a brief smile.\n\n\"My next question,\" he said, \"obviously, would be to ask what brings you here. But I'm a little afraid to ask.\"\n\n\"Eric Fulmar and Professor Dyer are in the municipal jail in P\u00e9cs,\" Canidy said. \"You didn't know?\"\n\n\"Jesus, Maria, und Josef!\" the Countess breathed.\n\n\"No,\" von Heurten-Mitnitz said, \"I didn't.\"\n\n\"We're done for,\" the Countess said matter-of-factly.\n\n\"Can you at least get Helmut and me out? Is that what you've come for?\"\n\n\"I came in to arrange for a site into which we can paradrop a team,\" Canidy said.\n\n\" 'Paradrop' ?\" von Heurten-Mitnitz asked. \"You mean parachute?\"\n\nCanidy nodded.\n\n\"You've got to get us out!\" the Countess said furiously.\n\n\"That may not be necessary,\" Canidy said. \"Fulmar and the professor have been arrested as black marketeers.\"\n\n\"How do you know that?\" von Heurten-Mitnitz asked calmly.\n\n\"I was there when they were arrested,\" Ferniany said.\n\n\"Then there is a _chance,_ \" von Heurten-Mitnitz said, searched for the words, and smiled wryly, \" 'that the jig is not up?' \"\n\n\"There's a chance,\" Canidy said. \"Ferniany is more confident about that than I am.\"\n\n\"The function of your team will be to get them out of prison?\" von Heurten-Mitnitz asked.\n\n\"The team leader will have my orders, I'm sure,\" Canidy said. \"I don't know what they will be.\"\n\nCanidy saw in von Heurten-Mitnitz's eyes that he would not have to explain that his orders might be to make absolutely sure that neither Fulmar nor Professor Dyer would be available for interrogation by the SS or the Gestapo. And when he looked at the Countess Batthyany, he saw in her face that she understood, too.\n\n\"I want to try to get them out,\" Canidy said.\n\n\"A question of priorities, then?\" von Heurten-Mitnitz said.\n\n\"Yes,\" Canidy said.\n\n\"And where on that list would be the priority to get out the Countess, or, for that matter, me?\"\n\n\"If it comes to that,\" Canidy said, \"we'll get you out.\"\n\n\"We will go out,\" the Countess said, \"or stay, together.\"\n\nVon Heurten-Mitnitz looked at her for a moment, then at his wristwatch.\n\n\"It's too early,\" he said. \"But later, I will call M\u00fcller and ask him to pick me up here.\" He saw the look on Canidy's face. \"It is necessary.\"\n\nAfter a moment, Canidy nodded.\n\n\"Just so long as he understands that I will make the decision about trying to get Fulmar and the professor out.\"\n\n\"I thought you implied that decision will be made by your superiors?\" von Heurten-Mitnitz asked.\n\n\"I'll decide,\" Canidy said flatly.\n\n#  **5**\n\n##  **FERSFIELD ARMY AIR CORPS STATION BEDFORDSHIRE, ENGLAND 0410 HOURS 19 FEBRUARY 1943**\n\n\"There's no reason for you to get up,\" Douglass said as he sat up in the narrow bed and swung his feet out onto the floor.\n\nCharity sat up in bed.\n\n\"I've been pretending that we're in Bala-Cynwyd . . . ,\" she said.\n\n\"Where?\" he asked, chuckling.\n\n\"It's a suburb of Philadelphia,\" she said. \" . . . and that the alarm clock has just gone off, and that you're going to get up and put on a suit, and that when you have had breakfast you'll kiss the children. And then I'll drive you to the station, and you'll get on the commuter train and go in to your office in Philadelphia . . . \"\n\n\"What kind of an office?\"\n\n\"You're a lawyer, like my father,\" she said.\n\n\"Why a lawyer?\"\n\n\"Because when lawyers leave their loving wives and adoring children to go to their offices, they know they'll be coming home that same night, not going off to some impossible island nobody ever heard about. . . .\"\n\n\"Stanley's a lawyer,\" Douglass said.\n\n\"Damn you, come back to me,\" Charity said.\n\n\"I'll have to, to make you an honest woman,\" he said.\n\n\"And to give the baby a name,\" Charity said.\n\n\"What baby?\"\n\n\"The one I think we made last night,\" Charity said.\n\n\"Last night, or ten minutes ago?\" he replied.\n\n\"I hope we did. Whenever,\" Charity said. \"How do you like them apples, Colonel?\"\n\n\"Hey, is this the right time to discuss something like that?\" Douglass asked.\n\n\"The best time,\" Charity said. \"If a man doesn't believe that a woman loves him after she says she wants his baby, he'll never believe it. I want you to _know_ it, Doug.\"\n\nHe stopped in the act of pulling his shorts on and went to the bed and sat on it.\n\n\"Me, too,\" he said.\n\n\"That's close,\" Charity said.\n\n\"I love you,\" he said.\n\n\"Correct,\" she said. \"That wins you your choice of a trip to the sunny and romantic Adriatic isle of Vis, a cement bicycle, or whatever else your little heart desires. Me, for example. \"\n\n\"Jesus, honey, they're waiting for me.\"\n\n\"I thought RHIP.\"\n\n\"It does,\" he said. \"Fuck 'em, let 'em wait.\"\n\n\" ''em? 'em?' \" Charity asked.\n\n#  **6**\n\n##  **HEADQUARTERS, U.S. FORCES IN THE PHILIPPINES MISAMIS OCCIDENTAL PROVINCE, MINDANAO 19 FEBRUARY 1943**\n\nThere was now some official stationery available to Headquarters, U.S. Forces in the Philippines. It was a good-quality, twenty-four-pound watermarked bond paper, with an engraved letterhead. The letterhead read,\n\nTHE DOLE CORPORATION Pineapple Plantation Three \"There Are None Finer\" Mindanao, Territory of the Philippine Islands\n\nHeadquarters, United States Forces in the Philippines used the blank side of the paper, but only for important official documents. After some thought, General Fertig decided that it was necessary to maintain certain files, and to use his available stock of stationery (one and one half boxes, totaling precisely 741 sheets of paper) to do so.\n\nUSFIP had acquired some other desperately needed supplies from the mountainside cottage of the manager of the Dole Corporation's Pineapple Plantation Number Three. The cottage, some miles from the plantation itself, had been the manager's private retreat. It had somehow escaped Japanese attention, and so it had held a dozen sets of bed linen\u2014which USFIP converted into bandages; a Winchester single-shot, bolt-action .22-caliber rifle and three and a half boxes of .22 shells; a motley collection of inexpensive tableware and pots and pans; a mixed assortment of condiments and canned delicacies (such as Planter's Peanuts, martini olives, and miniature onions); a Zenith portable radio; and a Smith-Corona \"Student's\" portable typewriter with a nearly new ribbon.\n\nGeneral Fertig had his staff prepare copies for the record of the several pronouncements he had made as Commanding General, USFIP; the commissions he had bestowed upon certain members of his staff; and memorandums of record of the money issued by the Provisional Government of Misamis Occidental Province and which he had borrowed for USFIP.\n\nAnd he instructed his cryptographic officer, Capt. Horace B. Buchanan, to assume personal responsibility for the Smith-Corona and the stock of stationery, and, aside from making copies of outgoing and incoming messages, to make sure that no one used either paper or typewriter in a manner that could by any stretch of the imagination be considered profligate.\n\nWhen Capt. Buchanan went to General Fertig's quarters with the two messages that had come in within five minutes of each other, the General was having his evening cocktail. Second Lieutenant (ex-chief petty officer, USN) Ellwood Orfett, whom Fertig had placed in charge of a deserted coconut oil mill, had revealed another talent. He could convert mashed pineapple meat into alcohol, producing a lethal-smelling transparent intoxicant with the kick of a mule, but which, when mixed with pineapple juice, didn't taste half bad.\n\n\"Would you like a little taste, Buchanan?\" Fertig asked as Buchanan came up the bamboo stairs of the General's quarters, shaking the whole building.\n\n\"Don't mind if I do, Sir,\" Buchanan said, and helped himself to a glass of the mixture. He poured it from a pottery mug in the shape of a cow's head. This was originally intended for milk, and was also salvaged from the pineapple plantation manager's cottage.\n\nFertig read the two messages, which were both on the same sheet of paper:\n\nPRIORITY FROM KAZ FOR WYZB  \nATTENTION LT COL FERTIG\n\nYOUR RADIO MESSAGE OF 15 FEBRUARY 1943 FOR SECWAR WASHINGTON HAS COME TO THE ATTENTION OF THIS HEADQUARTERS.\n\nALL REPEAT ALL COMMUNICATIONS FROM YOUR DETACHMENT OF WHATEVER NATURE WILL BE DIRECTED TO THIS HEADQUARTERS. NO DEVIATION FROM THIS POLICY WILL BE TOLERATED.\n\nBY COMMAND OF GENERAL MACARTHUR. WILLOUGHBY BRIG GEN\n\nURGENT\n\nFROM JOINT CHIEFS OF STAFF WASH DC  \nVIA KSF FOR WYZB HQ US FORCES IN PHILIPPINES\n\nATTENTION BRIGADIER GENERAL FERTIG\n\nKEEP YOUR SHIRT ON\n\nJ. R. ELLIS CHIEF USN\n\n\"I rather expected the first one,\" Fertig said. Then he read the second message.\n\n\"I rather like the sound of the second,\" Fertig said, \"even if I haven't the faintest idea what it means.\"\n\n\"I'd say it's the reason General Willoughby sounds just a little pissed,\" Capt. Buchanan said. \"The one from Washington\u2014from the Joint Chiefs\u2014is addressed to 'General Fertig,' you'll notice.\"\n\n\"You think Willoughby knows about it?\" Fertig asked.\n\n\"He knew about our message to the Secretary of War,\" Buchanan said. \"Sure, I think he heard about it. He's probably got the whole message.\"\n\n\"What do you mean by that?\" Fertig asked curiously.\n\n\"The signature on the message is incomplete,\" Buchanan said. \"There had to be more to it than 'Chief USN.' Chief of something. What?\"\n\n\"I thought it meant 'chief petty officer,' \" Fertig said.\n\n\"Chief petty officers don't sign messages from the Joint Chiefs of Staff,\" Buchanan said. \"Admirals and generals do that.\"\n\nHe remembered\u2014and then was a little ashamed of the memory\u2014that General Fertig, who had been a civilian eighteen months ago, knew damned little about the military services.\n\n\"Then what the hell does it mean?\" Fertig asked. \" 'Keep your shirt on' doesn't sound at all military, does it?\"\n\nBuchanan filled his glass again before replying.\n\n\"I thought about that, General,\" he said. \"It may be . . . maybe even probably is . . . a reply to your message to the Secretary of War. And it just might mean exactly what it says.\"\n\n\"That we should be patient, that they are sending help?\"\n\n\"I wonder at what point you want something so desperately that you lose sight of reality and imagine you see what you're looking for behind every bush,\" Buchanan said.\n\n\"But?\"\n\n\"The message is from the Joint Chiefs,\" Buchanan said. \"And we have a 'we are pissed' message from MacArthur. Which just might mean MacArthur has been asked to explain why no help has been sent to us. Or even that he has been ordered to get off his ass and send some.\"\n\n\"Yes,\" Fertig said softly, thoughtfully. \"Could be.\"\n\n\"And if I wanted to get a message to somebody who doesn't have any cryptographic equipment worth a damn,\" Buchanan went on, \"it would run through my mind to send a message in slang, in the clear, and hope that the Japs wouldn't understand the slang, and would try to decode the slang.\"\n\n\"We have heard from MacArthur about the Secretary of War message,\" Fertig said, \"and there was no reply to our message about VD medicine.\"\n\n\"That might be because it would be beneath the Generalissimo's dignity to acknowledge. Nobody talks to MacArthur that way.\"\n\n\"You really think there was more to that message than what we got\u2014specifically, a rank and a job title?\" Fertig asked.\n\n\"I think there just had to be.\"\n\n\"If there was a message, it seems common decency would have required MacArthur, or Willoughby, to relay it to us. To make sure we got it.\"\n\nBuchanan shrugged.\n\n\" 'Common decency,' \" he parroted bitterly.\n\n\"The last time I saw the Generalissimo,\" Fertig said, \"was in the Manila Club. There was a buffet. MacArthur, of course, and his queen and the crown prince didn't stand in the line. But I went through it with my wife. And as we walked to our table, we had to squeeze around their table. He was in a planter's white suit. I had a large bowl of shrimp bisque. I will regret for the rest of my life not having had an accident with it.\"\n\nBuchanan laughed.\n\n\"It may be, Buchanan,\" Fertig said, \"that help is on the way. But I think it more likely that you and I are sitting here with five ounces of Orfett's pineapple white lightning in us, seeing things we want to see behind bushes that just aren't there. I don't want any of this to go any further than you or me.\"\n\n\"No, Sir,\" Buchanan said. And then he blurted, \"But sooner or later, Christ, they're going to have to do something, aren't they?\"\n\n\"Sooner or later,\" Fertig said.\n\n#  **7**\n\n##  **BATTHYANY PALACE BUDAPEST, HUNGARY 0820 HOURS 20 FEBRUARY 1943**\n\nStandartenf\u00fchrer SS-SD Johann M\u00fcller came into the sitting room of the Countess's apartment in Batthyany Palace and quickly glanced around the room, taking in Canidy and Ferniany, who were sitting on a couch before a gilt coffee table.\n\nThere was no expression on his face.\n\n\"'Tag,\" he said, then started to unbutton his black leather overcoat. He hung it carefully on the back of a Louis XIV chair and then moved the chair to a position near one of the two white porcelain stoves. Then he moved the chair a foot farther away.\n\n\"If you get it too close, it cracks and dries the leather,\" he explained.\n\nAnd then he looked at Helmut von Heurten-Mitnitz, his eyebrows raised in question.\n\n\"Johann,\" von Heurten-Mitnitz said in German, \"this is Major Canidy of the United States Army. _'Pharmacist_. _'_ \"\n\nM\u00fcller examined Canidy carefully, then did the same thing to Ferniany. He was subjected to the same kind of an examination by the Americans.\n\n\"And him?\" M\u00fcller asked.\n\n\"Ferniany's my name,\" Ferniany said.\n\n\"Canidy's man on the scene, I gather,\" von Heurten-Mitnitz explained.\n\n\"Everybody speak slowly,\" Canidy said. \"My German is pretty weak.\"\n\n\"He was telling him who we are,\" Ferniany said.\n\n\"I got that,\" Canidy said. \"But go slow. I don't want to miss anything.\"\n\n\"Gott in Himmel!\" M\u00fcller said, exasperated. \"They send someone in who doesn't even speak the language!\"\n\n\"It was necessary,\" Canidy said.\n\n\"Why?\"\n\n\"Eric Fulmar and Professor Dyer are in the P\u00e9cs city jail,\" Canidy said.\n\nThis produced the first hint of excitement in M\u00fcller.\n\n\"And Gisella?\" he demanded. \"Fr\u00e4ulein Dyer?\"\n\n\"She's safe,\" Canidy said.\n\n\"Safe where?\"\n\n\"Cairo,\" Canidy said.\n\n\"So what happened?\" M\u00fcller demanded. He was back in control of his emotions, but there had been enough for Canidy to decide that there was more than a casual relationship between the SS-SD officer and Dyer's daughter.\n\n\"The barge was boarded by the River Police and the Black Guard,\" Ferniany said. \"They found a lot of money on Fulmar and decided he was a black marketeer. They helped themselves to the money and arranged for them to get ninety days in the coal mines.\"\n\n\"It's only a matter of time until somebody finds out who they are,\" M\u00fcller said. \"I had a teletype yesterday\u2014addressed to me personally, not to the senior SS-SD officer\u2014 from Von Hymme, Himmler's adjutant, telling me to personally make sure that 'the investigation was being pursued with all diligence.' \"\n\n\"They think they're here?\"\n\n\"From the time they found the Gestapo agent's body, they really closed down the borders of Germany to the occupied countries and to the neutrals. Himmler's mouth ran away with him again, and he said he could personally state that nobody got out that way. That leaves only here.\"\n\n\"Another question,\" Canidy said. \"Why all the interest?\"\n\n\"It would be enough,\" M\u00fcller replied dryly, \"that the Reichsf\u00fchrer-SS has showed his ass by not catching them long before this. And on top of that, our friend Eric used his knife on a Gestapo agent, which has the Gestapo in a rage. And then he used it on Peis, the SS-SD commander in Marburg an der Lahn, which has the SS-SD upset.\"\n\n\"Tell me about 'pursuing the investigation with all diligence, ' \" Canidy said.\n\n\"After you round up 'all the usual suspects,' which was done and which came up with nothing,\" M\u00fcller said, \"you start to recheck things like travel permits, hospital admittances, and jails. When I got the teletype, I ordered that done. I don't know how long it will take them to check the P\u00e9cs jail, but it won't be long. If I was doing it . . . instead, I mean, of having to ask the Black Guard for their cooperation . . . I would have them by now.\"\n\nM\u00fcller waited for that to sink in, and then went on, \"If you've got some idea of getting them out of that jail, you had better do it now.\"\n\n\"Major Canidy has asked for a team of specialists,\" von Heurten-Mitnitz said. \"The reason he's here is to arrange for a landing site.\"\n\n\"A landing site? You mean for parachutists?\" M\u00fcller asked.\n\nCanidy nodded.\n\n\"How long will that take? What's wrong with using the underground?\" M\u00fcller asked.\n\n\"The underground can't be involved in this,\" Canidy said.\n\n\"How long will it take to get your 'specialists' in here?\"\n\n\"Forty-eight hours, maybe twenty-four, after we find a place to drop them,\" Canidy said.\n\n\"The story I get,\" M\u00fcller said, \"is that there are parachutists dropping all over Yugoslavia and Hungary.\"\n\n\"This has to be kept separate from that,\" Canidy said.\n\n\"We may not have forty-eight hours,\" M\u00fcller said. \"We may not even have twenty-four.\" He looked at Canidy. \"If they catch Fulmar, he knows von Heurten-Mitnitz and me. And, sooner or later, he would tell them everything he knows.\"\n\n\"And me,\" the Countess said. \"He knows me.\"\n\n\"We'll arrange to get you out,\" Canidy said.\n\n\"Von Heurten-Mitnitz's family would probably be all right if he disappeared,\" M\u00fcller went on, \"and the Countess doesn't have anything to lose. But they would go after my mother and my brothers and sisters.\"\n\n\"Then the thing to do is get Eric and the professor out of the jail, isn't it?\" Canidy replied.\n\n\"Under the circumstances,\" M\u00fcller said, \"I would say the thing to do is arrange for them to be shot while being arrested,\" M\u00fcller said.\n\n\"If they are to be shot, I'll make that decision,\" Canidy said.\n\n\"I really don't need your permission, Herr Major,\" M\u00fcller said.\n\n\"How large an area do you need for your parachutists, Major?\" the Countess asked.\n\nM\u00fcller glared at her.\n\n\"For the time being, Johann,\" von Heurten-Mitnitz said, \"we will go along with Major Canidy.\"\n\n\"A minimum of eight hundred meters by three,\" Canidy said.\n\n\"So large?\" she asked, disappointed, and then went on: \"There is a field, a meadow, in the mountains above P\u00e9cs. We have a hunting lodge there. But it's not that big.\"\n\n\"What's around it?\" Canidy asked.\n\n\"A forest,\" she said.\n\n\"Would a low-flying aircraft attract attention?\"\n\n\"Of course,\" she said.\n\nCanidy exhaled.\n\n\"If that's all there is, we'll have to use it,\" Canidy said. \"Could you find it on a map?\"\n\n\"I don't know,\" she said. \"And I don't have a map.\"\n\nCanidy gestured impatiently at Ferniany, who went to his sheepskin coat, dug into a pocket, and came out with a map.\n\nWith some difficulty, the Countess found the meadow she was looking for on the map.\n\n\"It's damned small, and it's thirteen miles from P\u00e9cs,\" she said.\n\n\"But it hasn't been used, has it?\" Canidy said.\n\n\"No,\" Ferniany said. \"There's that.\"\n\n\"Take the coordinates,\" Canidy ordered, \"and then burn the map. And then you better get going.\"\n\n\"Where's he going?\" M\u00fcller asked.\n\n\"To radio the location of the drop zone,\" Canidy said. \"And to make arrangements to move the professor and Eric once we get them out.\"\n\n\"And what do you plan to do?\" von Heurten-Mitnitz asked.\n\n\"The next problem is to get me from here to the Countess's hunting lodge,\" Canidy said.\n\n\"How do you plan to do that?\" M\u00fcller asked.\n\n\"Gisella told me you have an Opel Admiral,\" Canidy said. \"How about that?\"\n\n\"I can't afford to be seen anywhere near P\u00e9cs,\" M\u00fcller said.\n\n\"No,\" Canidy said. \"You are going to be at the Austro-Hungarian border, noisily 'pursuing the investigation with all diligence.' \"\n\nM\u00fcller snorted.\n\n\"And the Countess and I will go to the hunting lodge?\" von Heurten-Mitnitz said thoughtfully, \"in M\u00fcller's car? With you in the luggage boot?\"\n\n\"Unless you've got a better idea,\" Canidy said.\n\n\"The plane will attract attention,\" von Heurten-Mitnitz said. \"And it will come out that we were there.\"\n\n\"The day before, maybe two days before,\" Canidy said.\n\n\"But it will come out,\" von Heurten-Mitnitz repeated.\n\n\"Unless you've got a better idea,\" Canidy repeated.\n\nM\u00fcller snorted again.\n\nCanidy looked at him coldly.\n\n\"And in case you think you have a better idea, Standartenf\u00fchrer M\u00fcller,\" he said, \"I think I had better tell you that if this operation goes sour, Reichsf\u00fchrer-SS Himmler will receive, mailed from Sweden, an hour-by-hour report of how you spent your last forty-eight hours in Morocco. With photographs, showing you with Eric in his U.S. Army uniform.\"\n\nM\u00fcller's eyes, very cold, met Canidy's, but he didn't say anything.\n\n\"At the risk of repeating myself,\" Canidy said, \"it may be necessary to do whatever is necessary to keep Fulmar and the professor from falling into the hands of the SS. But I will make that decision.\"\n\nM\u00fcller snorted again, and pursed his lips.\n\n\"When I first saw you, Major,\" M\u00fcller said finally, \"what I thought was they had sent an amateur. Obviously, I was wrong.\"\n\n#  **8**\n\n##  **CAIRO, EGYPT 1225 HOURS 20 FEBRUARY 1943**\n\nCaptain Stanley S. Fine resisted the temptation to let Lt. Colonel Peter Douglass, Jr., who was riding as copilot, land the B-17F. Doug Douglass, despite the expected fighter pilot's denigration of the \"flying barge,\" was obviously fascinated with the bomber. He would have liked to make the landing, and he probably would have handled it onto the wide and long runway without any trouble at all. He was an experienced pilot, and he had been an apt pupil.\n\nBut the moment they had taken off from Fersfield, Fine had been very much aware that they had crossed a line. From now on, everything was dead serious. There was no excuse whatever for taking any kind of a chance, no matter how slight.\n\nNothing had been said between them, but Douglass had seemed to understand and had conducted himself as a copilot should, making no control movements at all without first getting Fine's permission.\n\nFine set the B-17F down smoothly within two hundred feet of the threshold, then lowered it gently down onto the tail wheel.\n\n\"Call the tower?\" Douglass's voice came over the intercom.\n\n\"Please,\" Fine said.\n\n\"Cairo, Army Triple Zero Four on the ground at twenty-five past the hour,\" Douglass said. \"Request taxi instructions. \"\n\n\"Triple Zero Four, take Taxiway Two Right and find yourself a place to park with the other B-17s on the line.\"\n\nDouglass looked over at Fine. He was surprised. Ten minutes before, Cairo had acknowledged the \"This is Eighth Air Force Flight Five Six Six\" message that was supposed to alert OSS Cairo that they had arrived. Douglass did not expect the B-17 to be ordered to find itself a place to park with the other 17s on the line.\n\nFine looked surprised, too.\n\nDouglass pressed the mike button.\n\n\"Cairo, Triple Zero Four, say again your last transmission. You were garbled.\"\n\nCairo repeated the order.\n\nThere were a dozen B-17s and B-24s, and twice that many other transient aircraft on the parking line, but there was no sign of Canidy's B-25.\n\nFine taxied the B-17 to the end of the line, parked it in a line with another B-17, shut it down, and prepared the flight documents.\n\nA gas truck, a brand-new General Motors semitrailer, stopped just off the taxiway in front of them, and a crew got off and began to unroll fueling hoses.\n\n\"I'll go see what's going on,\" Fine said, unstrapping his harness. \"I think we had better keep our passengers aboard.\"\n\nFine opened the access hatch and lowered himself through it. Douglass went through the bomb bay into the rear of the fuselage. The team was peering out the gun ports.\n\n\"Colonel?\" J\u00e1nos asked. \"Can we get off?\"\n\n\"Not yet,\" Douglass said. \"Somebody fucked up. There's nobody here to meet us.\"\n\n\"That figures,\" J\u00e1nos said.\n\nIt was already getting hot in the fuselage; Douglass felt sweat under his arms and on his forehead as he saw it pop out on J\u00e1nos's face.\n\n\"Fuck it,\" he said. \"I don't see any point in melting. Get out, get in the shade of the wing, but don't stray off. And don't take anything with you.\"\n\nHe went to the side door in the fuselage and opened it, then waited until the last of the team had gotten out before getting out himself.\n\nThe team was gone when he got outside, and he saw that a Dodge ambulance had been backed up to the nose of the B-17. Normally, Dodge ambulance bodies had huge red crosses painted on their sides and roof; this one did not.\n\n\"You get to ride in front, Colonel,\" a voice called, and he saw a hand gesture toward the front of the vehicle.\n\nDouglass walked to the ambulance and got in.\n\nThe driver was a sergeant, and Douglass had his mouth open to ask him where they were being taken when a familiar voice spoke.\n\n\"The shit's hitting the fan.\"\n\nDouglass looked into the back of the ambulance. The narrow benches on each side were jammed with people, and one of them was Lt. Commander John Dolan.\n\n\"Canidy went into Hungary,\" Dolan went on.\n\n\"Jesus!\" Douglass said, then: \"How are you? There was word you had a terminal case of the GIs.\"\n\n\"I'm better,\" Dolan said.\n\n\"Where are we going?\"\n\n\"They got a villa,\" Dolan said. \"Very nice, swimming pool and everything.\"\n\n\"Does anybody know why Dick went into Hungary?\" Douglass asked.\n\n\"Does anybody know why he does anything?\" Dolan replied. \"They're trying to get a message to him to get his ass out of there. Everything's on hold until we see if that works.\"\n\n\"Who's 'they're'?\" Douglass asked.\n\n\"Donovan himself,\" Dolan said. \"They're apparently really pissed.\"\n\nWilkins, the Cairo Station Chief, was waiting for them at the villa. A lunch had been laid out for everyone at the side of the pool. There was no sense of urgency, and both Fine and Douglass were annoyed. But as they were eating, a distinguished-looking man in a stiffly starched but tieless shirt came to the table and handed Wilkins a sheet of paper.\n\nWilkins glanced at it, then handed it to Douglass.\n\n\"Sorry, Colonel,\" he said. \"But I didn't know where exactly you fitted into this.\"\n\nDouglass read it.\n\nTOP SECRET\n\nFROM OSS WASHINGTON TO OSS CAIRO\n\nLT COL PETER DOUGLASS JR USAAC IS AUTHORIZED ACCESS TO SUCH CLASSIFIED MATERIAL IN CONNECTION WITH CURRENT MISSION AS IS DEEMED ABSOLUTELY NECESSARY REPEAT ABSOLUTELY NECESSARY BY STATION CHIEF CAIRO AND PHARMACIST TWO DONOVAN\n\n\"Well,\" Douglass said, \"it's nice to know I'm to be trusted, if absolutely necessary.\"\n\nWilkins did not seem amused.\n\n\"I understand Donovan is pissed at Canidy,\" Douglass said.\n\n\"I don't think 'pissed' is the word,\" Wilkins said.\n\n\"If Canidy went into Hungary, he had his reasons,\" Douglass said loyally.\n\n\"I hope he finds his reasons worth it,\" Wilkins said.\n\n\"I don't think I follow you,\" Douglass said.\n\n\"Come with me,\" Wilkins said, \"and you, too, Fine.\"\n\nHe led them to the pool house, where two radio operators sat with earphones on their heads. He motioned Fine and Douglass into chairs, then dropped to his knees, worked the combination of a safe, and handed Fine a cover sheet stamped TOP SECRET.\n\nTOP SECRET\n\nOPERATIONAL IMMEDIATE\n\nFROM OSS WASHINGTON FOR OSS CAIRO EYES ONLY WILKINS\n\nPASS FOLLOWING TO PHARMACIST TWO ON ARRIVAL CAIRO STOP\n\nQUOTE YOU ARE HEREBY APPOINTED EXLAX CONTROL CANIDY RELIEVED STOP APPOINTMENT IS PERMANENT STOP CANIDY WILL NOT REPEAT NOT RESUME AUTHORITY OVER EXLAX UNDER ANY CONDITIONS STOP EVERYTHING POSSIBLE INCLUDING TERMINATION REPEAT INCLUDING TERMINATION WILL BE DONE TO PREVENT CANIDY FALLING INTO ENEMY HANDS STOP YOU WILL ACKNOWLEDGE TIME AND DATE OF RECEIPT DONOVAN END QUOTE\n\nFine read it and handed it to Douglass.\n\n\"Jesus, he is mad,\" Douglass said.\n\n\"You have a code word for a situation like this?\" Wilkins asked Fine. \"To acknowledge receipt?\"\n\n\"Yes, I do,\" Fine said.\n\n\"You want to give it to me?\" Wilkins asked, on the edge of sarcasm.\n\n\"I don't think I will,\" Fine said. \"I don't want to acknowledge that message.\"\n\n\"What?\" Wilkins asked incredulously.\n\n\"I'm not sure that was sent by Donovan,\" Fine said. \"Before I acknowledge it, I want confirmation.\"\n\n\"That will take hours,\" Wilkins said.\n\n\"It doesn't sound like Donovan to me, either,\" Douglass said loyally.\n\n\"What the hell are you trying to pull, Captain?\" Wilkins demanded.\n\n\"Canidy was there,\" Fine said. \"And he's not a fool. I certainly won't double-guess him, and I don't think Colonel Donovan would, either.\"\n\nWilkins opened his mouth to argue, but didn't get a chance to speak. One of the operators called out.\n\n\"Hey, I got something from Vis. . . . \"\n\n\"What does it say?\"\n\n\"It's not in the clear, for Christ's sake,\" the operator said, furiously pounding his typewriter.\n\nEight minutes later, the decryption process was completed:\n\nFROM POSTMAN FOR CAIRO VIA STATION VIII PHARMACIST REQUESTS EARLIEST POSSIBLE DROP RESCUE TEAM AT COORDINATES SEVEN FOUR NINE NINE THREE EIGHT ONE EIGHT STOP DROP MUST REPEAT MUST TAKE PLACE AT FIRST LIGHT STOP ADVISE\n\nIt took another five minutes to find the map of Hungary and then mark the location indicated by the coordinates.\n\n\"We're how far from Vis?\" Fine wondered aloud.\n\n\"Four hours thirty,\" Douglass said immediately. \"In the B-25.\"\n\nUsing his thumb and little finger as a compass, Fine measured the distance between Vis and P\u00e9cs.\n\n\"That's about an hour and a quarter,\" he said. \"Maybe a little less.\"\n\n\"What about that 'must take place at first light' business? \" Douglass asked.\n\n\"Jesus,\" Fine said. \"You're asking, how do we take off from P\u00e9cs in the dark?\"\n\n\"Yeah,\" Douglass said. \"But we don't have to take off from P\u00e9cs. We can take off from here.\"\n\n\"We don't have the range,\" Fine said.\n\n\"More than enough, if we sit down at P\u00e9cs on the way back,\" Douglass said.\n\nFine was silent for a moment.\n\nThen he said, \"Message Pharmacist as follows. Team will be available for drop first light tomorrow.\"\n**XII**\n\n#  **1**\n\n##  **P\u00c9CS, HUNGARY 1330 HOURS 20 FEBRUARY 1943**\n\nWhat Canidy had imagined was going to be adequate accommodation in the large trunk of Standartenf\u00fchrer-SS M\u00fcller's Opel Admiral quickly proved to be mildly, and then excruciatingly, uncomfortable.\n\nDespite the generous proportions of the Admiral's trunk, he could not stretch his legs without arcing his torso painfully, nor raise himself on his elbows without simultaneously lowering his head so that his chin rested on his upper chest.\n\nAnd the thick goose-down comforters and pillows that the Countess Batthyany had put into the trunk to keep him warm and serve as cushions had not been as helpful as everyone had cheerfully, almost gaily, believed. The comforter had quickly crushed down under him, so that he could feel every ridge and indentation in the trunk floor. And the comforter he had wrapped around himself for warmth, and the pillows on which he had planned to cushion his head, made things worse than nothing at all, for they retained enough bulk to get in the way when he shifted his body again and again to relieve the strain on his muscles.\n\nHe became uneasy, nervous, worried, and he began to wonder if he had some previously unsuspected problem with claustrophobia. He reasoned that through and decided his nervousness was perfectly reasonable: He was in the dark, and nobody liked that.\n\nMore important, it was fifty-fifty that von Heurten-Mitnitz was wrong when he said he \"rather doubted they would be stopped at all, or subjected to more than the most perfunctory examination if they were.\" There was a fifty-fifty chance that the trunk lid would suddenly open and he would find himself looking up at a Black Guard, a Hungarian cop, or even a Gestapo agent. If that happened, he was not going to be in a position to do much about it. The Sten submachine gun Captain Hughson had given him in Vis was now in the hands of an admiring Yugoslav partisan. Canidy was armed now only with the Fairbairn and a snub-nosed Smith & Wesson .38, neither of which would be of any real use if the car was stopped and checked. If that happened, in addition to being nearly paralyzed by the goddamned trunk, he would be blinded by the sudden light and helpless.\n\nThere had been time to remember where he had gotten the snub-nosed .38, and that hadn't helped his morale either. Jimmy Whittaker had given it to him just before they'd taken off on the mission to the Belgian Congo. Moments before that, Jimmy had taken it away from the flight engineer. The flight engineer had been given the pistol by the Chief, OSS London Station, together with an order that he use it on Canidy the moment it looked as if Canidy was going to fall into enemy hands.\n\nIt was not difficult to proceed from that to the logical conclusion that if an elimination order\u2014to keep him from failing into enemy hands\u2014had been issued then, a similar order had doubtless also been issued to cover this circumstance. He knew now more information that the Germans shouldn't know than he had known when he and Jimmy had flown off to the Belgian Congo.\n\nHe wondered where Whittaker was at that moment. In Australia, more than likely, dazzling the Australian women with his good looks and all-pink uniform. Whittaker, he thought, should have been a sailor; he already had a girl in every airport.\n\nAnn came to mind then, and he wallowed for a moment in the memory of the smell of her, and the feel, and the touch of her hand on him, and then he forced Ann from his mind.\n\nAnd then he got a headache. He was suddenly aware of it, a real bitch of a headache behind his eyes and across the base of his skull. He realized that he had been aware of _getting_ a headache for some time.\n\n\"Oh, shit!\" he said aloud.\n\nHe tried to look at his wristwatch to see how long he had been in the trunk. The Hamilton chronometer with the glowing hands was now adorning the wrist of the fishing boat captain. He couldn't even see the watch he had been given in return, much less tell what time it was.\n\nIn that ten seconds, the headache seemed to have grown even worse.\n\nAnd then he knew why he had a headache.\n\n\"Pull over!\" Canidy shouted. \"Let me out of here!\"\n\nThere was no reply. They apparently hadn't heard him. He could hear them talking. He couldn't make out what they were saying, but he could hear them.\n\nHe tried shouting again, and again there was no response. His voice was being muffled, he realized, by the thickly padded leather upholstery in the backseat of the Admiral; and what got through was not audible over the whistling of the wind on the convertible roof and the sound of the engine.\n\nThen there was a momentary wave of terror. He was going to die in this fucking trunk, be quietly asphyxiated by carbon monoxide from the exhaust. When they got to the Countess's hunting lodge and opened the trunk, they would find him dead.\n\nHe thought first of his pistol. If he fired that, they would hear it.\n\nBut where was he to fire it? Out the top of the trunk, so there would be a bullet hole for the cops to become fascinated with? Into the trunk floor, where it would pierce the fuel tank?\n\nAnd what would firing a pistol in the confined area of the trunk do to his ears?\n\nHe put both hands to his head and pressed inward as hard as he could against the pain of the carbon-monoxide-induced headache.\n\nAnd then he twisted around, shoving to the side the goose-down comforter under him. He felt the floor of the trunk. It was covered with some kind of padding. He found the edge, and with a great deal of effort managed to pry the edge loose. Finally, there was enough loose so that he could grip it. He gave a mighty heave and it came loose. Now there was nothing there but sheet metal.\n\nHe balled his fist and struck the floor of the trunk with all of his strength. And then did so again, and again, and again.\n\nAnd finally, he sensed that the Admiral was slowing, and then there was the sound of gravel under the tires. The car stopped, and Canidy heard a door open. And then the trunk opened, just a crack. But the light coming through the two-inch opening was so painful that Canidy closed his eyes against it.\n\n\"Are you all right in there?\" von Heurten-Mitnitz asked.\n\n\"I'm being asphyxiated,\" Canidy said. \"Is it clear? Can I get out?\"\n\n\"Asphyxiated?\" von Heurten-Mitnitz asked doubtfully.\n\n\"The goddamned muffler leaks,\" Canidy said.\n\n\"Just a moment,\" von Heurten-Mitnitz said. From his tone of voice, Canidy knew that he now believed him.\n\nAnd the trunk opened wide. Canidy heard the sound of the hinges and was aware of more light through his closed eyelids.\n\n\"Your lips are blue,\" von Heurten-Mitnitz said. \"Here, take my hand.\"\n\nCanidy opened his eyes just enough to see the hand, grabbed it, and closed his eyes again. Von Heurten-Mitnitz pulled him out of the trunk and led him to the curbside door.\n\n\"Lie on the seat,\" he ordered. \"Beatrice, there's a flask in the map box. Give it to him.\"\n\n\"He's sick?\" she asked.\n\n\"Exhaust poisoning,\" von Heurten-Mitnitz said. He closed the trunk, then got behind the wheel and started off.\n\nCanidy felt something cold and metallic at his lips. He took the flask from the Countess and took a deep pull.\n\nHe felt the warmth spread through his body, and then something else.\n\n\"I think I'm going to be sick,\" he said.\n\n\"Oh, please don't,\" the Countess Batthyany said practically. \"You never can get that smell out of a car!\"\n\nCanidy fought down the urge to vomit and took slow, deep breaths. The desire to vomit passed, and he was able after a while to keep his eyes open. He found himself looking into the Countess's face.\n\n\"You're getting color back,\" she said. \"You'll be all right now.\"\n\nThere was genuine relief on her face, Canidy saw, and then decided it almost certainly wasn't for him.\n\nThere was another queasy feeling in his stomach. He fought it by sitting up, and it passed, but there was a wave of sharp pain behind his eyes.\n\nHe took another pull at the silver brandy flask and looked out the windshield. They were all alone on a narrow, curving road cut through a dense forest of mature pines.\n\n\"Where are we?\" he asked. \"How long was I in the trunk?\"\n\n\"It's another couple of hours to P\u00e9cs,\" the Countess said. \"We left Budapest at half past nine. You were back there about two hours.\"\n\n\"What's next on the road?\" Canidy asked. \"Am I going to have to get back in the trunk?\"\n\n\"We just went through Dunaf\u00f6ldv\u00e1r,\" the Countess said. \"There's a couple of small towns between here and P\u00e9cs, Sioag\u00e1rd and P\u00e9csv\u00e1rad, hardly more than villages. You'll be all right in the back, I think.\"\n\n\"Do we go through P\u00e9cs itself?\"\n\n\"There's a way around,\" she said. \"But it's dirt roads, and there's no telling how muddy they would be this time of year. And we would attract attention.\"\n\n\"I was wondering whether we could run by the jail,\" Canidy said, \"and then trace the route the truck takes moving the prisoners to the mine.\"\n\n\"We'll take that road anyway,\" she said. \"But it would be a detour to go past Saint Gertrud's.\"\n\n\"A conspicuous detour?\" Canidy asked.\n\nShe thought that over before replying, \"No. It's on the edge of town. But we wouldn't be more conspicuous there than we're going to be anyway.\"\n\n\"Then please tell Herr von Heurten-Mitnitz how to get there,\" Canidy said. \"I want a look.\"\n\nAt quarter to two, the tires leaving a path across previously unbroken snow, the Opel Admiral pulled up before the hunting lodge. It was a long, low wooden building with elaborate scrollwork, now covered with dripping icicles, along the roofline. There was a chimney at each end and a much larger one in the middle. Smoke rose from one of the end chimneys, and as Canidy got out of the car, he could smell wood smoke.\n\n\"I think it would be better if you spoke German,\" the Countess said.\n\n\"Who's in the house?\" Canidy asked.\n\n\"The caretaker and his wife,\" she said. \"And there are foresters in small houses behind the lodge.\"\n\n\"And they can't be trusted?\" Canidy asked.\n\n\"Of course they can be trusted,\" she said. \"They have been with my family for hundreds of years. But if the Black Guard comes here, I don't want to ask them to lie any more than necessary. They don't speak German, but they recognize it. I want them to be able to report they saw me with two German-speaking men.\"\n\n\"They're going to know what's going on,\" Canidy said.\n\n\"They will do what I ask them to do,\" the Countess said, \"and then, because I ask them to, they will forget having done it.\"\n\nCanidy's disbelief showed on his face.\n\n\"My father was active in the Independent Hungary movement,\" the Countess said. \"Crown Prince Rudolf used to come here secretly. If my people could forget that he was here, they can forget you.\"\n\nThe look on his face confused her.\n\n\"Crown Prince Rudolf was the . . . ,\" she started to explain.\n\n\"Heir apparent to the Austro-Hungarian throne,\" Canidy filled in. \"The one who shot his girlfriend, and then himself. At Mayerling.\"\n\n\"Like Standartenf\u00fchrer M\u00fcller,\" von Heurten-Mitnitz said, \"the Countess seems to have underestimated you, Canidy.\"\n\n\"And not you?\"\n\n\"A good diplomat never underestimates anyone,\" von Heurten-Mitnitz said.\n\nAs they approached the hunting lodge, the door was opened by a hefty, large-bosomed woman with jet-black hair. The hair was parted in the middle and done up in elaborate braids.\n\nShe curtsied to the Countess, then to the men.\n\n\"She says,\" the Countess said, \"that if she had known we were coming, her husband would have of course been here, and there would be a meal prepared. As it is, all there is is simple boar guly\u00e1s. Paprika guly\u00e1s.\"\n\nAfter they had eaten, Canidy was outfitted, from a wide selection, with a green loden cloth coat and lace-up boots, which were, he suspected, older than he was. Laughing, the Countess added a black cap of heavy wool.\n\n\"A real Magyar!\" she said.\n\nThe caretaker showed up as the Countess was lacing up her boots. With him was a man Canidy's age, with a double -barreled shotgun hanging upside down on a woven leather strap from his shoulder.\n\n\"This is Alois, the chief hunter,\" the Countess explained. \"His great-grandfather was my great-grandfather's chief hunter. We will take him with us to the meadow. If there is anything that has to be done, he will see that it is done, and then he will forget that he ever saw you.\"\n\n\"How big a place do you have, Countess?\" Canidy asked. \"In other words, how about the neighbors?\"\n\n\"This estate is roughly an oblong,\" she replied matter-of -factly. \"It is twenty-three kilometers long and about fourteen wide. There are no neighbors, and the local authorities are my tenants. If I do not wish them to see me, or anything else, they will not see me, or anything else.\"\n\n\"You sound very confident of that,\" Canidy said.\n\n\"I am,\" she said.\n\nA ten-minute walk over light snow brought them to the meadow. It did not meet any of the criteria for a drop zone. It was far too small, and it was surrounded on three sides by a mature pine forest, into which anybody who missed the drop zone would land.\n\nBut trees had been harvested at one end of the meadow, where the land dropped precipitously off toward a stream.\n\nWhen Ferniany arrived with the radios and the panels, tomorrow or the next day, Canidy would arrange the panels either at the edge of the meadow by the forest or at the stream, depending on the wind. With a little bit of luck, they would be able to put three or four of the five parachutists down in the meadow. The others would have to take their chances on landing on his just-cut-over steep land at the end of the meadow.\n\nThere would be time to talk to the plane. Darmstadter had dropped parachutists before. He would know how to drop them here, once he had been told of the conditions by radio.\n\nCanidy thought of the emergency backup procedures. There was always that in the planning. Here, in the case of radio failure or if there were no opportunity to put the signal panels in place, it was a smoky fire at the point in the drop zone that would indicate where the first parachutist in the string was supposed to land.\n\nHowever, it didn't seem to make a hell of a lot of sense to bother about that particular backup. For one thing, there would be a chance to put the panels out and talk with the plane by radio. For another, unless the drop could be discussed with the plane, there would be no point in making the jump; it would be too risky.\n\nBut in the end, Canidy asked the Countess to have her chief hunter arrange for a five-foot-high stack of pine boughs at both ends of the drop zone. He showed, with his hands, how large the piles should be.\n\n\"And two cans of kerosene, preferably, or else gasoline, by each stack,\" he said.\n\nShe translated that for him.\n\nAnd then, as if they were two old friends out for a walk in the woods, she took his arm and they walked back to the hunting lodge.\n\n#  **2**\n\n##  **CAIRO, EGYPT 1715 HOURS 20 FEBRUARY 1943**\n\nThe first thing Freddy J\u00e1nos realized when he saw that the bomb-bay doors of the B-25 in the hangar were not functional was that he was going to have a hell of a hard time dropping out of the crew-access door when the time came.\n\nThen he measured the access door with his hands and realized that there was no way _any_ of the team could exit the aircraft wearing all their equipment.\n\n\"Something wrong, J\u00e1nos?\" Lt. Colonel Douglass asked him.\n\n\"That hatch isn't big enough,\" J\u00e1nos said. \"There's no way we can drop through that little hole.\"\n\n\"We've dropped people through that hole before,\" Douglass said.\n\n\"Only Fulmar,\" Capt. Stanley S. Fine said, entering the conversation. \"The others went out the bomb bay. Before Canidy removed the racks and the door-opening mechanism. And Fulmar jumped in with a British chute. No spare. And it took him a long time to get through the door. _If_ we could get them through the door, it would take so long they would land all over Hungary.\"\n\n\"Jesus Christ!\" Douglass said furiously. \"What the hell do we do now? How come this is the first time anybody thought about this?\"\n\n\"The B-17 can't land on Vis,\" Fine said, answering that question before it was asked.\n\n\"What's Vis?\" Freddy J\u00e1nos asked.\n\nFine and Douglass looked at each other before Fine answered, \"An island in the Adriatic. Where we will pick you up when this operation is over.\"\n\n\"Pick us up? We're not going to stay?\"\n\n\"No,\" Fine said. \"It has been decided to bring you out right away.\"\n\n\"Can I ask why?\"\n\n\"You can ask, but I can't tell you,\" Fine said.\n\n\"I must be out of my mind,\" J\u00e1nos said. \"But that sort of pisses me off.\"\n\n\"Jesus, that's all we need, a hero,\" Douglass said.\n\nJ\u00e1nos felt his face turn warm with anger. With an effort, he fought it down by telling himself that Douglass, by any criterion, was a hero, and thus had the right to mock the word.\n\n\"I guess that sounded pretty dumb,\" he said.\n\n\"Yes, it did,\" Douglass said, not backing off. \"I just hope you can restrain your heroic impulses when you do get in there, and that you do just what you're told, and nothing more.\"\n\nThey locked eyes for a moment. J\u00e1nos, for the first time, saw that Douglass could have very cold and calculating eyes. And he sensed suddenly that Douglass was judging him, and that if Douglass found him wanting\u2014if Douglass concluded that there was a risk he would foolishly take once he was in Hungary\u2014there was a good chance he would be left behind.\n\n\"Can a Gooney Bird land on this island?\" J\u00e1nos asked.\n\nThere was no response from Douglass. He continued to look at J\u00e1nos with cold, calculating eyes.\n\n\"What the hell,\" Douglass said finally. There was even the flicker of a smile. \"When all the clever ideas fail, be desperate. Go by the book. Use a parachutist-dropping airplane to drop parachutists.\"\n\n\"Can we get our hands on a C-47?\" J\u00e1nos asked.\n\n\"Yes,\" Fine said, almost impatiently. He had seen a dozen of the twin-engine transports sitting on the field. There would probably be one they could have simply by asking for it. And if there was a problem, one would have to be \"diverted from other missions.\" The OSS had the ultimate priority. \"But does a C-47 have the range?\"\n\n\"I don't think it does,\" Douglass said. \"I'm not even sure it will make it to Hungary. There's no way one of them could make it to P\u00e9cs and then to Vis.\"\n\n\"Where's Darmstadter?\" Fine asked. \"He ought to know.\"\n\n\"He and Dolan are checking the weather,\" Douglass said.\n\n\"What's the priority?\" Fine asked rhetorically.\n\n\"To get J\u00e1nos's team on the ground in one piece,\" Douglass said.\n\n\"We could . . . ,\" Fine began. \"I don't know what I'm talking about, and I won't until I know just what the Gooney Bird can do.\"\n\n\"Well,\" Douglass said, nodding toward a small door in one of the wide hangar doors where an MP, armed with a Thompson submachine gun, was checking the identification of Lt. Commander John Dolan, USNR, Lt. Henry Darmstadter, and Ernest J. Wilkins, \"here comes the expert. \"\n\n\"Well,\" Wilkins said, cheerfully confident, as he walked up to them. \"God loves us, apparently. The immediate and twenty-four-hour weather over the drop zone is going to be perfect.\"\n\nDouglass laughed nastily.\n\n\"Darmstadter,\" Fine asked. \"What's the range of a Gooney Bird? Would a Gooney Bird make it one way to P\u00e9cs?\"\n\n\"No,\" Darmstadter said immediately.\n\n\"What's wrong with the B-25?\" Dolan asked.\n\n\"Canidy has cleverly modified the B-25 so that you can't drop parachutists from it,\" Douglass said, \"or at least not a team of them, without scattering them all over Hungary. \"\n\n\"Good God!\" Wilkins said.\n\n\"And we can't put the 17 into Vis,\" Dolan said.\n\n\"Right,\" Fine said.\n\n\"Jesus, now what?\" Douglass asked. \"Canidy expects us at daybreak.\"\n\n\"So we use the B-17 for the drop,\" Dolan said. \"And it comes back here. And we send the B-25 to Vis. No problem. \"\n\n\"No,\" Wilkins said.\n\n\"What do you mean, 'no'?\" Fine asked.\n\n\"Maintenance found landing-gear problems,\" he said. \"They called me and told me it would take twenty-four hours, maybe a little more, to replace what was broken.\"\n\n\"Then you'll have to get us another 17,\" Fine said.\n\n\"There will be a lot of questions asked why someone wants to borrow a bomber,\" Wilkins said.\n\nDarmstadter's mind had been racing. He thought he saw a solution. But he was reluctant to offer it. _These people,_ he told himself, _know what they're doing. I'm just a mediocre Gooney Bird pilot._\n\nAnd then he thought, _Fuck it!_\n\n\"If there would be only the team, five men, on the Gooney Bird,\" he said, \"it would be very light. It would take another ton and a half, maybe two, before it got close to Max Over Gross.\"\n\n\"If you're talking about fuel,\" Dolan said, not unkindly, \"we just don't have time to rig auxiliary fuel tanks.\"\n\n\"I'm talking about fifty-five-gallon drums,\" Darmstadter plunged on, \"and hand pumps to replenish the fuel in the main tanks as it's burned off.\"\n\n\"Hey!\" Dolan said after a moment's thought.\n\n\"Would that work, John?\" Fine asked.\n\n\"Eight fifty-five-gallon drums would weigh thirty-two hundred pounds,\" Dolan said. \"A little over a ton and a half. And that would be another four hundred gallons. More than enough to get a Gooney Bird from here to P\u00e9cs, and then to Vis.\"\n\n\"And you can get a Gooney Bird into Vis?\" Douglass said.\n\nDolan thought that over a moment before replying.\n\n\"Yeah,\" he said after a moment, \"I think Brother Darmstadter and I could sit a Gooney Bird down on Vis in one piece.\" He caught Darmstadter's eye and went on. \"We'll have to get the tail wheel down before we hit the stream, going in. If we were still up on the main gear, we'd go over on our nose. Getting out will be easier; we'll just keep the tail wheel on the ground till we're through the water.\"\n\nDarmstadter nodded his understanding.\n\n\"Could Brother Darmstadter and me sit one down in one piece?\" Douglass asked.\n\nDolan looked at him.\n\n\"You don't have hardly any Gooney Bird time, Colonel,\" Dolan said, after a moment.\n\n\"But I don't have dysentery, either,\" Douglass said. \"Canidy told me about your 'dysentery,' John.\"\n\n\"Canidy has a big mouth,\" Dolan said. \"And I'm all right.\"\n\n\"I don't think we can take a chance on that, John,\" Douglass said.\n\n\"I'm missing something here,\" Wilkins said.\n\n\"I'm afraid Commander Dolan will not be able to go,\" Douglass said. \"Whatever plans we make will have to exclude him.\"\n\n\"First of all, that'd be Fine's decision,\" Dolan said. \"And you haven't heard me out.\"\n\n\"Go ahead, Commander,\" Fine said, and immediately wondered why he had called Dolan by his rank.\n\n\"Darmstadter knows more about dropping . . . what is it they say? 'sticks' . . . sticks of paratroopers than anybody else. And he's also the only one of us with any experience to speak of flying a Gooney Bird on the deck. And the only way we're going to be able to find P\u00e9cs and not get ourselves shot down is to go in on the deck.\"\n\n\"Okay, that takes care of Darmstadter,\" Douglass said. \"He flies the Gooney Bird. We're talking about who goes with him. We're talking about your 'dysentery,' Dolan.\"\n\n\"I was flying cross-country using a road map before anybody else here was out of diapers,\" Dolan said. \"I'm the only one here who can, for sure, find this meadow Canidy has picked out for us.\"\n\n\"That presumes you don't have another . . . attack of dysentery,\" Douglass said.\n\n\"If, for example, you were to go in the Gooney Bird,\" Dolan went on, ignoring him, \"that would leave me and Fine to fly the 25 to Vis. Captain Fine is not what you could call an experienced B-25 pilot. I hate to think what would happen if he had to try to land the B-25 on Vis.\"\n\n\"Dolan, do you think Colonel Douglass could land the 25 on Vis?\"\n\n\"He stands a much better chance than you do,\" Dolan said. \"And the kid doesn't need him in the C-47.\"\n\n\"And what if you're not 'available' in the C-47?\" Fine challenged.\n\n\"That's the chance we have to take, that by me just sitting there in the right seat and letting the kid fly, my dysentery won't come back.\"\n\nDouglass looked at Fine.\n\n\"I think we have to go with Dolan,\" Fine said. \"His main advantage, I think, is that he's the one with the best chance . . . maybe even the only one with a chance . . . of finding the drop zone.\"\n\n#  **3**\n\n##  **P\u00c9CS, HUNGARY 0515 HOURS 21 FEBRUARY 1943**\n\nLt. Hank Darmstadter thought that the most difficult part of the flight so far had been taxiing to the end of the runway in Cairo. They had taken off at 2100, which would put them over the meadow outside P\u00e9cs at just after daylight. The airfield at Cairo was blacked out, and while Wilkins had been able to arrange for the runway lights to be turned on long enough for them to take off, they had had to be led to the runway from the hangar by a man holding a flash-light in the back of a jeep.\n\nThe flashlight-in-the-jeep had been very hard to follow. It was almost impossible to see directly ahead out over the nose of a C-47 with its tail wheel on the ground. C-47 pilots learned to taxi by looking out the side and by swinging the nose from side to side to provide a look ahead through the side windows.\n\nIt was difficult following the jeep, but they'd made it to the end of the runway all right, sometimes flicking the landing lights on to make sure of their position. Darmstadter had been a little surprised and flattered that Dolan had not taken over the controls and done the taxiing, but Dolan had left that to Darmstadter.\n\nAnd from the moment they had lined up with the centerline of the runway, things had gone without a hitch.\n\nDolan had waited until he'd run the final mag check for the engines, and then he'd called the tower for the lights, and they had come on immediately.\n\nDespite what had turned out because of the air temperature to be four hundred pounds over Max Over Gross, the takeoff had been no problem at all. The only way Darmstadter could tell how heavy they were was a reluctance to pick up altitude. But they had never come close to a stall, and the climb was steady, if slow.\n\nThe first leg, the longest, was on a west-northwest course across the desert to the Mediterranean, and then across the Mediterranean far enough south of Crete to avoid a chance encounter with German aircraft based on the island. And then they turned north across the Ionian Sea.\n\nThere was almost a half moon, providing what Dolan described as the most they could ask for, enough light for them to make out landmasses and shorelines, but not enough to make it easy for anyone to spot them.\n\nThe Strait of Otranto, which separates the heel of the Italian boot from Albania and the Adriatic from the Ionian Sea, came into view just when they expected it to, and they could see both shorelines for a while.\n\nDolan had planned that that leg of the flight would take six hours and twenty-five minutes. It actually took six hours and two, meaning that they were making better time than anticipated, even with the engines thinned back as much as possible for fuel economy.\n\nOnce they had crossed the Strait, Darmstadter had raised the nose slightly, starting a slow climb to 9,000 feet, and Dolan had begun to peer intently out the window looking for the narrow strip of land that ran between the Adriatic and Lake Scutari on the Yugoslav-Albanian border.\n\nDolan had told him, jokingly, but meaning it, that the secret of \"road map\" navigation was to look for something on the ground that was large enough to be easily seen and that couldn't be confused with anything else.\n\nLake Scutari fit the bill. It was twenty-five miles long and was separated from the Adriatic by a strip of land as narrow as seven miles. It could be easily found, and it could not be mistaken for anything else.\n\n\"Steer straight north from the end of the lake,\" Dolan said when they had found Lake Scutari, and then he got out of his seat. \"I think it's time to get rid of another drum.\"\n\nLt. J\u00e1nos had been shown how to pump fuel from the fifty-five-gallon drums into the main tanks. One of the drums had been \"semipermanently\" installed, with a line running from its bottom to the main aircraft tank. Fuel from it had been pumped into the main tank, and then that fuel was replenished from other fifty-five-gallon barrels.\n\nThe empty tanks didn't weigh much, but they could not be completely drained, and Dolan was worried that the avgas sloshing around in them would create fumes that would be dangerous. He had gone back into the cabin several times to make sure that as soon as each drum had been emptied, J\u00e1nos had thrown it out.\n\nThe ground seemed to glow white about that time, and after a moment Darmstadter figured out what it was\u2014the moonlight reflecting back from snow on the ground. That meant they were approaching the mountains in Montenegro, the highest of which was about 7,500 feet. There would be at least 1,500 feet between them and the highest peak, but it was important that they know when they passed over it, so they could safely descend.\n\nDarmstadter had been worried that Dolan would want the controls after they started down and were flying on the deck. There was no question that Dolan was a better and more experienced pilot. But there was also no doubt that he had had a heart attack and might have another.\n\nBut Dolan lived up to what he had promised Douglass: that he would \"work the road map in the right seat and let the kid fly.\"\n\nThe only specific instructions Dolan gave him were course changes, and several times the \"suggestion\" that it would be \"okay to go down another couple hundred feet.\"\n\nAccording to the Corps of Engineers' map (which the Corps had apparently borrowed from _Le Guide Michelin_ ), this part of Hungary was sparsely populated. There were here and there a few lights to be seen, but there was no way of telling whether they were a few lights in violation of a village blackout, or lights in single farmhouses.\n\nAt 0500, as the sky to the east was starting to glow dull red, Dolan unstrapped himself again and got off the copilot's seat.\n\n\"In eight minutes, maybe ten,\" he said, \"we should see a few lights. That'll be P\u00e9cs. Or maybe Athens. If you see something round, that'll be Rome.\"\n\nDarmstadter knew he was expected to laugh, and did.\n\n\"This has gone so well, I'm afraid to believe it,\" Dolan said. \"I'll go back and tell our passengers. J\u00e1nos said he wanted fifteen minutes to suit up.\"\n\nDolan was back in his seat before they came onto P\u00e9cs, and he was the first to see it.\n\n\"Go down on the deck,\" Dolan now ordered. \"Put that line of hills between us and P\u00e9cs. It's damned near impossible to tell the direction of an airplane if you can't see it. And the more confused we can leave these people, the better. \"\n\nDarmstadter concentrated on flying as close to the ground as he dared between lines of hills. It was light enough now to make out individual trees, and here and there a road and fields.\n\nAnd then, surprising him, he flashed over a stream, then a cut-over section of hillside, then above that a meadow on a plateau.\n\n\"Christ, is that it?\"\n\n\"It should be,\" Dolan said, \"but I don't see any panels.\"\n\nDarmstadter glanced quickly at him. Dolan had a headset on and was working the controls of the radio.\n\n\"Not a goddamned thing,\" he said.\n\n\"What do I do?\"\n\n\"Stay on the deck under the hill lines,\" Dolan ordered. \"And make another pass over it. I'll go see what I can see from the door.\"\n\nFive minutes later, from the other direction, the C-47 approached the meadow.\n\nThere was no doubt now that they had found their destination. A pile of tree limbs was burning furiously at the near end of the meadow by the cut-over area, the wind blowing the smoke across the meadow and into the forest.\n\nDolan came into the cockpit.\n\n\"It's up to you now, kid,\" he said. \"The next pass is all we're going to get, or everybody will think we're having an air show up here.\"\n\nDarmstadter smiled uneasily.\n\nDolan went back into the fuselage. There he would strap himself into a harness and take up a position by the open door. When Darmstadter turned the red light on\u2014there were supposed to be red and green lights, but the green wasn't working\u2014and then off, he would push the first of the parachutists through the door. When they were all gone, he would throw the three equipment bags after them.\n\nDarmstadter made his approach very carefully, slowing the C-47 down as much as he dared, coming in very low and shallow over the tips of the trees in the forest, one hand on the Gooney Bird's wheel, the other on the toggle switch for the light for the door.\n\nAnd then he flicked the toggle switch.\n\nHe thought he could sense a slight change in the controls, which would mean that he had lost 1,000 pounds of weight\u2014five parachutists\u2014from his gross weight, and that the loss had changed the center of balance.\n\nHe had a strange, wild, arrogant thought.\n\n_I could have landed this sonofabitch in that meadow! The way the wind is blowing up from the stream, I was making maybe forty knots over the ground. I was going so slow I could see Canidy's face! And I could have stopped it in plenty of time._\n\nHe looked over his shoulder into the aisle for Dolan.\n\nHe couldn't see him at first, and then he did.\n\nDolan was on the cabin floor on his side, curled up. Darmstadter looked out the windshield, then back. Dolan straightened, grew almost stiff, and then went limp.\n\n#  **4**\n\n##  **150 DEGREES 20 MINUTES WEST LONGITUDE 08 DEGREES 35 MINUTES NORTH LATITUDE 1725 HOURS 20 FEBRUARY 1943**\n\nThere were four people on the bridge of the conning tower of the USS _Drum_ as she made fifteen knots on a course of 275 degrees through oil-smooth, gently rolling seas. They were almost exactly halfway around the world from the Adriatic Sea and Budapest, Hungary, where at that moment it was 5:25 A.M., February 21, \"the next day.\"\n\nThe _Drum_ 's captain, Lt. Commander Edwin R. Lennox, USN, and Capt. James M. B. Whittaker, USAAC, were in clean and pressed but unstarched khakis. Commander Lennox wore a battered brimmed cap whose cover was once white, but was now nearly brown with oil stains. Captain Whittaker was hatless.\n\nThe talker, with a headset and microphone device over his head, was also hatless. He wore a light blue denim shirt and a darker-shade pair of denim trousers, as did the lookout, who also wore a blue sailor's cap, the brim of which he had turned down all around.\n\nThe lookout, Commander Lennox, and Capt. Whittaker all had identical Navy-issue Bausch & Lomb ten-power binoculars on leather straps around their necks.\n\nCommander Lennox looked at his wristwatch, and then, with a sailor's eye, at the darkening sky.\n\n\"Anytime you're ready, Jim,\" Commander Lennox said, \"you can go below.\"\n\nWhittaker smiled.\n\n\"Aye, aye, Sir,\" he said. \"Permission to leave the bridge?\"\n\n\"Granted,\" the _Drum_ 's captain replied, smiling back.\n\nThey had grown to like each other on the voyage from Pearl Harbor. Lennox had thought about the growing friendship a good deal during that time\u2014remembering what he had been told by a full lieutenant when he'd been an ensign aboard the _Kingfisher:_ He'd been told that her skipper wasn't really such a hard-nosed sonofabitch as he seemed, but that a skipper couldn't afford to have friends, that command was indeed a lonely thing.\n\nHe had accepted that then because he was an ensign, and ensigns believe what they are told by full lieutenants. But it was only after they had given him the _Drum,_ his first command, that he'd really understood it. The master of a man-of -war could _not_ have friends. He could be civil and courteous, but there had to be a wall between the skipper and everybody else aboard. It had a little to do with \"familiarity breeds contempt,\" but there was more to it than that. The captain had to appear omniscient to his crew, and one of the best ways to do that, especially if you were convinced that at least two of your officers were far smarter than you were and better leaders of men, was to be aloof, to be somewhat mysterious, to share no opinion or confidences with anybody.\n\nLennox had seen in Whittaker somebody much like himself in character, and with similar command responsibilities, and with an _understanding_ of command. Very early on, Lennox had decided that having Whittaker aboard was very much what it must be like to be captain of a cruiser flying an admiral's flag. Where the cruiser and the accompanying task force went, and what it would do, was the admiral's responsibility. But the operation of the cruiser was the cruiser captain's responsibility.\n\nAnd Whittaker had acted as Lennox believed a good admiral would behave. Despite the authority the orders from COMSUBFORPAC had given Whittaker\u2014which had in effect made the _Drum_ his personal taxicab\u2014he had leaned over backward to avoid even the suggestion of giving Lennox orders.\n\nHe had asked questions, and \"wondered if it would be possible to\" do what he had the clear authority to order done. He had always scrupulously referred to Lennox as \"Captain\" or \"Skipper,\" even long after Lennox had started calling him \"Jim.\"\n\nAnd the night before, when they were alone with the talker on the bridge, Whittaker had asked \"if it would be possible to\" have a dry run of what would take place when they were off Mindanao.\n\n\"They assure me, Skipper,\" Whittaker said, \"that the outboards have been tuned by an expert. But cynical sonofabitch that I am, and with no reflection intended, Sir, on the U.S. Navy, I'd like to check that out.\"\n\n\"What you would really like, Jim, right, is a dry run?\"\n\n\"Yes, Sir,\" Whittaker asked. \"Is that going to be possible? \"\n\n\"Does the Army use the phrase 'SOP'?\" Lennox asked.\n\n\"Yes, Sir,\" Whittaker said.\n\n\"I violate mine,\" Lennox said. \"The SUBFORPAC SOP clearly states that when we are within the operating range of Japanese aircraft and proceeding on the surface, we will always be in a 'prepared to dive' condition. That means all hatches except the one here will be secured, and that we will be making sufficient headway so that the sub's diving planes will have effect in case we have to make an emergency dive.\"\n\nThey had, during the voyage, exchanged technical lore. Whittaker had been surprised to learn that the diving planes on the _Drum_ functioned like the ailerons of an airplane, controlling up and down movement of the submerged submarine. He knew that because of the dynamic forces acting upon the diving planes, the faster a submarine was moving across the surface of the ocean, the quicker it could be submerged.\n\n\"In other words, Skipper,\" Whittaker said, \"a dry run is a lousy idea?\"\n\n\"In these waters, if I follow the SOP,\" Lennox said, \"what I get is a boat ready to make a dive, and a crew of sweat-soaked, temperature-exhausted sailors not only getting on each other's nerves, but not able to function fast when they have to. So what I do is leave the hatches open when I can in waters like these, stationing men by the hatches to close them if they have to, and I make damned sure my lookout has the eyes of a hawk.\"\n\n\"And to conduct a dry run would mean stopping the boat,\" Whittaker said, \"increasing the time it would take you to submerge if a Jap plane spotted you.\"\n\nLennox nodded. \"Spotted _us_.\"\n\nWhittaker shrugged.\n\n\"Okay, if that's\u2014\"\n\nLennox interrupted him.\n\n\"Another unpleasant situation that comes to mind,\" he said, \"is us sitting on the surface a half mile or so offshore of Mindanao, and unable to submerge because there's a trio of Army guys in rubber boats with outboard motors they can't start.\"\n\nWhittaker looked at him but didn't say anything.\n\n\"And while I am being the high priest of doom and gloom,\" Lennox said, \"I have another scenario. There we are off Mindanao, and we get the boats out of the torpedo room, blow them up, and they leak. Since I can think of no other way to get those heavy little boxes ashore, that would mean we would have come all this way only to have to go all the way back for more rubber boats.\"\n\n\"I'd like to add to that gloom-and-doom scenario, if I might, Sir,\" Whittaker said.\n\n\"Go ahead, Jim,\" Lennox said.\n\n\"We are on the surface off Mindanao, the boats have inflated properly, and the outboards have even started. Then the Army guys\u2014whose total experience with rubber boats is limited to Lieutenant Hammersmith's time with an inner tube in a swimming pool\u2014start loading those heavy boxes into the rubber boats and drop the boxes over the side, fall overboard themselves, and I'll let you figure out the rest yourself.\"\n\n\"You've had no training?\" Lennox asked, surprised and concerned.\n\n\"No, Sir,\" Whittaker said. \"There wasn't time.\"\n\n\"Well, then,\" Lennox said, \"the question is not _if_ we do a dry run, but when.\"\n\n\"I think, if it's possible,\" Whittaker said, \"we should.\"\n\nLennox looked at Whittaker.\n\n_If I hadn't been so obliging,_ he wondered, _would you have pulled the rank the COMSUBFORPAC orders give you?_\n\n\"You told me, Jim,\" he said, \"that to a pilot, darkness rises from the ground.\"\n\n\"Yes, Sir, it does.\"\n\n\"Then I think we should do the dry run tomorrow, at dusk,\" Lennox said.\n\n\"Thank you, Skipper.\"\n\nThe day had been spent preparing for the dry run. This was mostly a good thing for the boat, Lennox realized, though it was risky. The morale of the crew was helped by the chance not only to do something constructive, but to get out on deck. The risk of being spotted by a Japanese patrol plane was no greater with them there, but submerging would take longer because of the people and the equipment on deck.\n\nLennox posted extra lookouts and ordered the manning of the machine gun and Bofors cannon. He didn't plan to use them, but it gave their crews a chance to get on deck and to feel useful, and he decided the price, the extra forty-five or sixty seconds it would take the gun crews to drop through the hatches and close them, was worth it.\n\nThe rubber boats themselves, as Lennox had supposed they would, posed the greatest problems. If the chief of the boat, who by default became the rubber boat expert, had any thoughts about the idiocy of sending people with no training or experience with rubber boats to make a landing through the surf on an enemy-held shore, he kept them to himself.\n\nThe first problem was to get the boats from the forward torpedo room through the hatch and onto the deck. The chief of the boat considered his options and decided that because of the weight and ungainly bulk it would make more sense to uncrate them below and pass them through the hatch, despite the risk that they would be impaled and torn on something sharp on the way.\n\nThe boats, which carried their own air bottles, were designed to be inflated with the bottles. Even if the boats were thrown over the side uninflated and sank, if the pull cord for the air bottles was pulled, the boats would inflate and pop to the surface.\n\nAlthough spare air bottles had been provided, the chief of the boat decided that the smart thing to do was not to use the bottles until it was necessary. He called for the air hose normally used to charge the air bottles in torpedoes, and when he had the first boat unrolled and lying limp on the deck, filled it with compressed air.\n\nWhen that boat was expanded, he ran soapy water over it to check for leaks. When he found none, he opened the exhaust valves, and as they hissed and the boat collapsed, he looked at it thoughtfully.\n\nThen he went aft and stood with his hands on his hips and spoke with Lennox and Whittaker, who were on the bridge.\n\n\"Two things, Skipper,\" he said.\n\n\"Go ahead, Chief,\" Lennox said.\n\n\"I think we could stow the boats aft of the conning tower,\" the chief of the boat said. \"Properly stowed, we could even submerge with them.\"\n\n\"Good idea,\" Lennox immediately agreed.\n\n\"Second, there's no way the boats will carry all that weight.\"\n\n\"Then we'll have to use the spares, too,\" Whittaker said.\n\n\"I meant using the spares,\" the chief of the boat said. \"The first time you flexed the boat in the surf, that weight'd rip the deck . . . or the bilge, whatever they call that sheet of rubberized canvas . . . free of the inflation chambers. If it didn't rip through before you got to the surf.\"\n\n\"What do you suggest, Chief?\" Lennox asked.\n\n\"We got a hundred and sixty percent of life jackets aboard,\" the chief said.\n\n\"I don't know what that means,\" Whittaker said.\n\n\"It means we got sixty percent more life jackets aboard than there is people,\" the chief said.\n\n\"And?\" Lennox asked.\n\n\"They're rated at two hundred pounds,\" the chief said. \"Which is just about what them 'film' boxes weigh.\"\n\n\"You mean put a life jacket around a film box,\" Whittaker asked, \"in case the bottom lets go?\"\n\n\"I mean wrap jackets around the boxes, tie lines to them, and tow them ashore,\" the chief said. \"And around them boxes with the weapons and the ammo, too.\"\n\n\"Could they be towed?\"\n\n\"There's only one way to find out, Skipper,\" the chief of the boat said.\n\n\"Put people on it, Chief,\" Lennox ordered.\n\n\"Carefully, Chief,\" Whittaker said. Both the chief and Lennox looked at him in surprise and annoyance, but then smiled when Whittaker went on. \"If we were to lose just one of those 'film' boxes out here, your beloved captain and myself would spend the rest of our days in Alcatraz.\"\n\n\"I take your meaning, Sir,\" the chief said with a smile.\n\nBy midafternoon, each of the boats had been brought on deck, inflated, checked for leaks, deflated, and then stowed, firmly tied to the mount of the twin Bofors aft of the conning tower.\n\nThe top was cut from an empty fifty-five-gallon oil drum, and then the drum three-quarters filled with seawater. Each outboard motor was test-run for five minutes, the noise incredible inside the hull.\n\nThe chief torpedoman was placed in charge of floating the \"film\" boxes. He cut the flotation packets from life preservers and tied them around the wooden boxes. The available light line was soon exhausted, and two sailors made what was needed by first sawing through a length of four-inch manila hawser and then untwisting the strands.\n\nAfter that, there was nothing to do but wait until dusk fell.\n\nCommander Lennox waited until he was sure that Whittaker was in the control room, and then he started the dry run.\n\n\"Close all hatches and watertight doors,\" he said, and the talker repeated the order.\n\nLennox could see the hatches on the deck closing, and he could hear a dull metallic clanging from all over the boat. With the exception of the hatch from the bridge, which would be his responsibility to close, the boat should now be watertight.\n\n\"All hatches and watertight doors secured, Sir,\" the talker confirmed.\n\n\"Prepare to dive,\" Lennox ordered. \"Clear the bridge!\"\n\n\"Prepare to dive,\" the talker repeated. \"Bridge being cleared.\"\n\n\"Dive!\" Lennox ordered\n\n\"Dive! Dive! Dive!\" the talker said, and dropped through the hatch. Lennox followed him, then closed the hatch after him.\n\nThe sound of the Klaxon hurt his ears.\n\n\"Take her to one hundred feet,\" the captain ordered, and put his hand out to steady himself as the bow of the _Drum_ nosed downward.\n\nTen minutes later, the bow of the _Drum_ broke the surface again.\n\nThe moment it did, Lennox started his stopwatch.\n\nAs soon as he was on the bridge, with water still spilling over the deck, he started issuing orders.\n\n\"Battle stations,\" he ordered.\n\nThe talker repeated the command, and the Klaxon went off.\n\n\"Man all cannon,\" Lennox ordered.\n\nSubmariners erupted from the hatches and went to the guns.\n\n\"All astern one-third,\" he ordered. \"Make her dead in the water.\"\n\nThe pitch of the just-started diesels changed.\n\nIt was time for another command, but there was nothing standard that Lennox could recall that fit the situation.\n\n\"Make all preparations to launch the rubber boats,\" he finally ordered.\n\nNow there was activity from every hatch on the deck.\n\nAs crewmen freed the rubber boats from the Bofors mount and handed them to crewmen on the deck, other crewmen emerged from other hatches. The weapons and ammunition boxes were first placed on the deck in a line, then tied together with ten-foot lengths of line.\n\nBy the time the crewmen carrying the limp boats had reached the forward deck, others had air hoses waiting. It took what seemed like a long time for the boats to be inflated, and by the time they were, Whittaker, Hammersmith, and Radioman Second Joe Garvey had come onto the deck, wearing their gear, and were waiting.\n\nThe chief of the boat and the chief torpedoman put the rubber boat over the side themselves, lowering it with ropes until it touched the nearly horizontal section of the hull, then they jumped down onto it with ropes around their waists.\n\nThen they pushed the boat off the hull into the water and raised their hands to help Whittaker from the deck to the sloping part of the hull and into the boat itself.\n\nWhittaker jerked the starting rope of the outboard motor. When he had it running, he checked to see that the line tied to a grommet in the heavy black rubber was in place. Then he put the motor in gear, and the boat started off. When the line tied to the grommet drew taut, crewmen slid the first of the two larger ammunition and weapons boxes (now wrapped with life preserver flotation packs) into the water, then skidded the line of small \"film\" boxes after it.\n\nThen the process was repeated for the second boat, except that both Hammersmith and Joe Garvey got into that one.\n\nThe atmosphere had been tense: to see if the boats could be launched and whether or not the flotation packets would keep the weapons and film boxes afloat.\n\nThen Lennox heard a guffaw, then a belly laugh, and then a high-pitched giggle. The first thing he thought, angrily, was that someone had fallen over the side. That, despite the genuine threat to life, was always good for a laugh from his men.\n\nAnd then he saw the object of the amusement.\n\nJim Whittaker was fifty yards off the bow, making a wide turn to return to the _Drum._ The strain on the line towing the boxes behind the rubber boat, plus the weight of the outboard motor and of Whittaker himself, had caused the bow to rise almost straight up out of the water. The outboard was open full bore, but it was just barely moving, and Whittaker himself looked as if he was about to sink into the water.\n\nSound carries well over water, and Whittaker heard the laughter of the crew.\n\nHe rose to the occasion. Balancing himself precariously, he saluted crisply.\n\n\"Man overboard!\" a shout went up, followed by a bellow of laughter.\n\nLennox looked quickly to see what had happened. The chief torpedoman had lost his footing and gone into the water. The chief of the boat was trying, with absolutely no success, to haul him back aboard by the rope around his waist.\n\nThe captain of the USS _Drum_ picked up his electric hailer and started to put it to his lips. Then he took it down and slammed it painfully against his leg until the pain was such that he was no longer overcome with hysterics.\n\n\"Attention on the deck,\" he finally announced. \"Prepare to recover rubber boats!\" And then the temptation was too much. \"And while you're at it, see if you can recover the chief torpedoman.\"\n**XIII**\n\n#  **1**\n\n##  **P\u00c9CS, HUNGARY 0500 HOURS 21 FEBRUARY 1943**\n\nCanidy woke in the dark in a large bedroom in the Countess Batthyany's hunting lodge. He was buried deep in goose down, his nostrils full of perfume.\n\nBut then he realized it wasn't perfume, it was something he had found in a bottle in his surprisingly ornate bathroom. The bottle bore a \"Lanvin Paris-London-New York\" label underneath the words \"Pour les Hommes.\" His French was good enough to understand what that meant, and the stuff hadn't smelled half bad when he sniffed at the bottle neck, and so he had liberally splashed it over himself after he'd wiped himself dry with a thick towel about the size of a pup tent.\n\nThe cologne would be a nice change from the way he had smelled after the fishing boat from Vis to the mainland, and after the farm truck\u2014redolent of horse manure\u2014 which had carried him across Yugoslavia to the neighborhood of the Hungarian border.\n\nIt was only when he had put on a pair of silk pajamas and the odor of the \"Pour les Hommes\" had not diminished\u2014had, in fact, seemed to intensify\u2014that he began to suspect the legend on the bottle was directed to the gentle sex. If they doused themselves in \"Pour les Hommes,\" men would be drawn to the smell like moths to a candle.\n\nIt had confirmed the somewhat cynical impression he had formed not long after they'd first shown him his room that the Batthyany family had apparently not only done their hunting in considerable comfort, but also that when they returned from the vigors of the field, the comfort they'd received then had been furnished by females. In his bathroom, he had found a bidet, and in a heavy bookcase by the bedside was a collection of leather-bound photo albums, the photographs portraying handsome men and women in their birthday suits performing what could only be described as sexual gymnastics.\n\nHe had at first wondered whether the albums had been purchased\u2014they looked professionally done\u2014or whether the Counts Batthyany had been unusually skilled amateur photographers. But when he got into the second volume, he recognized the huge fireplace in the main room of the lodge behind three dark-haired beauties and a hairy, skinny, mustachioed gentleman.\n\nThe thought passed through his mind that it might be fun to peel several of the neatly matted photographs free of the albums and take them home for Ann. It might brighten her day, he thought. But then he decided against that. Ann took sex very seriously. But then he was sure that as far as Ann was concerned, dirty pictures would be as high on her taboo list for him as carrying on with Her Gracefulness, the Duchess of Stanfield.\n\nThe next thought he had was that he would bring some of the dirty pictures back with him, to include them with his official report.\n\n_\"The photographs attached as Enclosures 16 through 26 are included in the belief that they might suggest exploitable character flaws in the Hungarian aristocracy possibly useful in future operations.\"_\n\nThat would shake up the system. Dave Bruce's near-glacial dignity would crack; he might even blush. He would certainly hem, haw, and stammer.\n\nAnd then he realized that he was already in enough trouble for having come to Hungary, without adding fuel to the fire. Did he need another demonstration that he didn't have the right attitude? Hardly.\n\nObviously, he thought, suddenly chagrined, he _did not_ have the right attitude. Instead of sitting here drooling over dirty pictures like some high-school junior, he should be wondering how to get Eric Fulmar and Professor Dyer out of St. Gertrud's prison without having to \"terminate\" them.\n\nHe put the leather-bound albums back in their case and went to sleep thinking over what he had just about decided to do\u2014the final decision to be made after talking it over with Ferniany and whoever London sent in to command the team.\n\nFerniany would be here tomorrow, probably around noon. He would have with him two of his people, Hungarians he had recruited, and the signal panels, and the radio, and the Sten gun Captain Hughson had loaned him just before he left Vis. Canidy would be glad to have that back. There was plenty of room in the Lodge to put Ferniany and his men up for however long it took London to get off its ass and send him the team, and the worst possible scenario for that was five days.\n\nVon Heurten-Mitnitz and the Countess would return to Budapest tomorrow. Canidy saw no problem with that. He didn't need the Countess now: She had told her servants they were to do what he asked. And he didn't think there would be any suspicion directed toward the Countess and von Heurten-Mitnitz for having been in P\u00e9cs several days before the prisoners had escaped from St. Gertrud's. Or several days before an unexplained explosion had destroyed a mine shaft in the Batthyany coal mine.\n\nIt would be a coincidence, nothing more, that His Excellency had been enjoying the overnight hospitality of the Countess at the Countess's rustic love nest ten or so miles away.\n\nThe most serious potential problem, Canidy had gone to sleep thinking, was not how to get Eric and the professor out of the hands of Hungarians, but how to do it without calling a hell of a lot of attention to the operation. He had been disturbed by Standartenf\u00fchrer M\u00fcller's report that the SS not only had not grown bored with looking for Fulmar and the professor, but quite the reverse, had intensified the examination.\n\nSt. Gertrud's prison would be swarming with SS and Gestapo just as soon as word got out that two prisoners had not only escaped but had been rescued by what it would take them about five minutes to figure out was a highly skilled team under the hands of either the SOE or the OSS.\n\nWhen he woke up smelling like a Hungarian courtesan, Canidy rested on his back in the dark for several minutes in the hope that, as sometimes happened, his subconscious had been working on the problem while he slept and that there would be new solutions, or new questions, or both.\n\nBut none came.\n\nHe fumbled for the bedside lamp, turned it on, then got out of bed and got dressed in the hunting clothes he had worn the day before. If nothing else, he decided, he would walk back through the woods to the drop zone and see for himself what it looked like at dawn.\n\nThen he would come back to the house and see about something to eat.\n\nHe sensed, when he entered the main room of the lodge, that there was someone there, someone watching him.\n\nThe room was lit now only by embers in the huge fire-place before which in happier times the aristocracy had staged their little _tableaux vivants_. He looked around, but he saw nothing.\n\nThen Alois, the chief hunter, rose out of a huge upholstered chair near the fireplace. Its bulk and high sides had hidden him. He was fully dressed and had apparently slept overnight in the chair as a sort of guard. He was wearing a heavy poncholike garment of gray wool, and he had his shotgun.\n\n\"Good morning,\" Canidy said, smiling.\n\nAlois grunted.\n\n\"I need a flashlight,\" Canidy said.\n\nThere was confusion on Alois's face.\n\nCanidy mimed a flashlight, and lighting a path with one.\n\nAlois grunted again and left the room. He returned with two flashlights, a square light with a handle, and a tiny two-cell that looked like a child's toy. He extended both to Canidy, offering him his choice.\n\nCanidy took the larger light and walked to the door. Alois didn't move, but by the time Canidy had unlatched the chains and dead bolts, he became aware that Alois had moved soundlessly across the room and was standing behind him.\n\nSomewhere, far off, there was the sound of aircraft engines.\n\nThe beam of his light picked out their footsteps in the snow from the day before, and Canidy, with Alois following him, walked away from the lodge toward the forest and the meadow beyond it.\n\nConcentrating on not losing the path or his footing in the dark, Canidy didn't pay much attention to the sound of the aircraft engines far away\u2014until they suddenly seemed much closer.\n\nHe looked up into the sky.\n\n_Jesus! Those sound like Twin Wasps!_\n\nHe broke into a trot, slipping and sliding on the frozen snow.\n\nWhen he reached the meadow, it was light enough to see the meadow and the area beyond. But there was no aircraft in sight, and it was only when he strained his ears that he could convince himself that he could just barely hear the sound of faraway engines.\n\n_Whatever it was, it was not for me. I should have known better. There's no way that could have been a Gooney Bird; no way they could have gotten a team here this quick. Now I look like a horse's ass in front of Alois_.\n\nHe met the large Hungarian's eyes and shrugged.\n\nAnd then he was sure the sound of the engines receding had changed, that it was growing louder. And it kept going in and out, growing louder then fainter, then louder again.\n\nAnd all of a sudden, it was very loud. A Gooney Bird appeared at the end of the meadow where the trees had been cut, its engine roar now deafening, and flashed overhead no more than two hundred feet off the ground. And there was no mistaking the star-in-a-bar U.S. identification painted on the wing.\n\n\"Jesus, Maria, und Josef!\" Alois said.\n\nThe Gooney Bird banked, then disappeared from sight.\n\nCanidy stuck two fingers in his mouth, then raised them over his head to confirm his suspicion that the wind was coming from the direction of the stream and the cut-over area.\n\nHe ran to the pile of pine boughs. He could just make out a shining glint underneath that had to be the kerosene.\n\nHe dug it out. It was a five-gallon tin can, bearing a Shell logotype. A sealed tin can, he saw when he unscrewed the cap. There was a seal over the hole he would have to pry out before he could pour the kerosene.\n\nHe changed his mind and threw the can atop the pile of boughs. And then he gestured to Alois.\n\n\"Shoot the sonofabitch, Alois!\" he said.\n\nAlois looked confused.\n\nCanidy gestured.\n\n\"Bang! Bang!\" Canidy shouted as he mimed the action.\n\nAlois looked confused, but he raised his shotgun and looked to Canidy for approval.\n\n\"Right! Yes! Ja! Schiessen!\"\n\nThe shotgun barked, and the can erupted. Canidy felt droplets of kerosene in the air.\n\nAlois looked at Canidy, as if he was afraid he had misunderstood him and done the wrong thing.\n\nCanidy smiled at him, then ran to him and reached for the shotgun. Alois debated for a moment parting with the shotgun, but finally handed it over. Canidy found a puddle of kerosene, put the barrel to it, and fired the other barrel.\n\nThere was a dull flicker of fire for a moment, and then the kerosene that had vaporized when the can had erupted ignited in a whoosh. A thick cloud of black smoke quickly formed.\n\n_Christ, I hope they just haven't given up! That somebody sees that!_\n\nThe pine boughs were burning now, and noisily.\n\nCanidy had just about decided that he could not hear the Twin Wasps at all anymore, when the Gooney Bird appeared, flaps and wheels down, right on the edge of a stall.\n\nAnd then very quickly, surprising him, something fell\u2014 five somethings fell\u2014from the door. And then the first canopy opened, and the second, and then one at a time all the rest, and five parachutes floated toward the ground.\n\nThe Gooney Bird pulled up its flaps and its gear and was gone.\n\n_A Gooney Bird! How the hell did they get a Gooney Bird this far?_\n\nCanidy ran toward the first parachutist, who was just about to touch down. He heard Alois plodding behind him.\n\nThe parachutist, a big guy, landed badly. He screamed.\n\nCanidy ran to him.\n\n\"I broke my fucking ankle again!\" J\u00e1nos said furiously. \"Jesus Christ!\"\n\n\"Was hat ihr gesacht?\" Alois asked in rough German.\n\n\"I said I broke my fucking ankle,\" J\u00e1nos said in Hungarian.\n\nAlois smiled sympathetically, then stooped over and scooped J\u00e1nos up in his arms like a baby. He looked at Canidy and nodded at the forest and then looked stone-faced at Canidy.\n\nWhen there was no immediate response, he spoke to J\u00e1nos, who translated:\n\n\"He wants to carry me into the woods, okay?\"\n\nCanidy nodded his head. \"Ja!\"\n\nThe other parachutists were on the ground now, and they ran over to Canidy. They were all armed, he saw, with .30-caliber carbines with folding stocks.\n\n\"Who are you?\" one of them demanded.\n\n\"That's Major Canidy,\" another said, recognizing him.\n\n\"Pick up your chutes and put them on the fire,\" Canidy said. \"And then\u2014\"\n\nHe interrupted himself. The sound of the Twin Wasps was back.\n\n_The equipment drop. Why the hell hadn't the jumpmaster kicked that crap out the door after he dropped the jumpers?_\n\nThe Gooney Bird appeared again over the cut-down area, its flaps and gear down again. He was now even lower than he had been before, when he'd buzzed the meadow.\n\n_If you stall it, friend, you're going to land here in this meadow!_\n\nThe Gooney Bird didn't stall. But the pilot chopped the engines, and the Gooney Bird touched down. He bounced once, then stayed down, and Canidy saw smoke from the gear as the pilot braked it.\n\n_Dolan, you sonofabitch! If I had wanted you to land here, I would have said so. You're too fucking old to be a hotshot pilot!_\n\nCanidy ran down the meadow and to the rear door of the Gooney Bird, and looked in.\n\nAnd Lt. Commander John Dolan, USNR, lying on the cabin floor, looked back at him out of sightless eyes.\n\n#  **2**\n\n##  **CROYDON AIRFIELD LONDON, ENGLAND 1130 HOURS 21 FEBRUARY 1943**\n\nIt was raining, and there had been fog, and there had been serious doubt that the Washington courier would be able to get in that day at all. Late the previous day, the ATC C-54 had managed to make it into Prestwick, Scotland, ahead of the front, but too late to try for London.\n\nThere had been a break in the weather, and an arctic blast of dry air moving down over Scotland had cleared the skies enough at 0930 for the C-54 to take off. But by then London had been socked in. The question had then been whether the break would close in again at Prestwick before the fog cleared at London.\n\nIt was decided in the end to take off and head for London in the hope that it would clear.\n\nAt Croydon, it had been necessary to \"light the burners.\" The theory was\u2014and damn the cost\u2014that if enough gasoline were burned in devices set up alongside a runway, the heat generated would cause the air mass and the fog it contained to rise, clearing the runway. In practice, as now, what the burners did for pilots was serve as sort of a super-beacon. If you could see the glow of the burners, you knew that the runway was somewhere down there, and with a little bit of luck, when you went down low enough, you could find the runway.\n\nThe C-54, flown by a commissioned TWA pilot who had lots of experience finding San Francisco in the fog, came in low and slow toward the glow on his horizon over London and found the Croydon runway on his second pass.\n\nAs he taxied toward the terminal, it was raining so hard that he had trouble seeing out the windshield. The ground crew who came out to meet them were wearing yellow rubber coats, hats, and trousers, and looked, the pilot thought, like so many misplaced sailboat sailors.\n\nThe first passenger to come down the ladder was a chief petty officer of the U.S. Navy. He had a Valv-Pak in each hand and smaller pieces of luggage under his arms.\n\nAs he came down the stairs, an Austin Princess limousine drove up close to him. The chief opened the front door and tossed the luggage inside, then backed out and held the rear door open.\n\n\"Get in, Ellis!\" Colonel William Donovan said as he came down the stairs from the C-54.\n\n\"In here, Ellis,\" Lt. Colonel Edmund T. Stevens said, motioning with his hand. \"You're getting soaked.\"\n\nEllis got in the backseat, and a moment later Donovan got in beside him and closed the door.\n\nDonovan gave Stevens his hand.\n\n\"Well, Ed,\" he said, \"how are you?\"\n\n\"Just fine, thank you, Bill,\" Stevens said. \"David said he hopes you will understand that he would have met you if he could.\"\n\nDonovan's reply surprised Stevens. Donovan was usually not only polite but manifested the lawyer's ability to say the unpleasant in the nicest possible way.\n\nDonovan said, \"I didn't want to see him anyway. Not just now.\"\n\nAnd then Donovan leaned forward and cranked down the divider separating the backseat from the chauffeur's compartment.\n\n\"Young lady, would you drive up to the terminal and get out, please? I'm sorry, but you're about to be put out in the rain.\"\n\n\"Yes, Sir,\" the driver, a WRAC sergeant, said.\n\n\"You call the office and have them send a car for you,\" Stevens said.\n\n\"There's a bus, Sir,\" the WRAC sergeant replied. \"I can take that.\"\n\n\"Do what Colonel Stevens said,\" Donovan said. \"The bus doesn't go near Berkeley Square.\"\n\nThe WRAC pulled the nose of the Princess close to a door of the terminal, pulled on the parking brake, jumped out, and ran into the building. Ellis climbed over Donovan and got in the front seat behind the wheel.\n\n\"She forgot her purse,\" Ellis announced.\n\n\"No problem,\" Donovan said. \"We'll probably be at Berkeley Square before she gets there. Get us off the field and drive in wide circles.\"\n\n\"Yes, Sir,\" Ellis said, and backed the Princess away from the terminal building. \"Colonel, you put the window down.\"\n\n\"It's all right, I want you to hear this anyway,\" Donovan said.\n\nBut then he didn't say anything else until they had left the field and were driving through Thorton Heath toward the Thames on Highway A235.\n\n\"Get off the highway, Ellis,\" he ordered.\n\nEllis made the next right turn.\n\n\"The ostensible purpose of my visit,\" Donovan said, \"is to smooth things over between you and SOE. 'Representations have been made at the highest levels' to the effect that you are not only being uncooperative but are interfering with their smooth operation. All of which proves that you are doing what I told you to do.\"\n\n\"Anything specific, Colonel?\" Stevens asked.\n\n\"No, just general allegations about your being uncooperative, which I interpret to mean you have both locked them out of our cupboard and have turned a deaf ear to the pronouncements of the professionals,\" Donovan said. \"But you'll have to arrange for me to see them, as soon as you can.\"\n\n\"This afternoon?\"\n\n\"Fine,\" Donovan said. \"And let's do it on our turf. Either at Berkeley Square or at Whitbey House. I don't want to give them the impression that I have been summoned for a dressing-down on their carpet.\"\n\n\"What about the apartment in the Dorchester?\"\n\n\"Fine,\" Donovan said. \"And let's do it over drinks and hors d'oeuvres. As fancy as we can manage.\"\n\n\"I'll get Helene Dancy to set it up,\" Stevens said. \"Better yet, Charity. She's at Berkeley Square.\"\n\nDonovan grunted approval.\n\n\"Ellis,\" Stevens said, \"there's a radio up there.\"\n\n\"I can hear it, Sir.\"\n\n\"We're Birddog,\" Stevens said. \"Call Foxhunt, Captain Dancy's monitoring it, and tell her to have Charity set up a fancy do for half past five at the Dorchester, details to follow. \"\n\n\"Aye, aye, Sir,\" Ellis said, and reached for the microphone.\n\n\"Napoleon said,\" Donovan said, \"that an army marches on its stomach. This one marches on hors d'oeuvres.\"\n\nStevens chuckled.\n\n\"My real purpose, of course,\" Donovan said, still conversationally, but very seriously, \"is to be near what's happening in Hungary. So you better start by telling me what _is_ happening, Ed.\"\n\n\"You got the message where Canidy asked for a team?\"\n\nDonovan nodded.\n\n\"It went in at 0500 this morning, or thereabouts,\" Stevens said. \"We've had no word how that went.\"\n\n\"This morning? God, that was fast! How did you arrange that?\"\n\n\"We flew the team\u2014specifically Stan Fine and young Douglass flew\u2014the team to Cairo in one of the new B-17s we got for Operation Aphrodite.\"\n\n\"And then used Canidy's B-25 to drop the team? That's why you involved young Douglass, to fly the B-25?\"\n\n\"That was the idea, but something went wrong. The last radio from Wilkins said that the team was being dropped by a C-47, flown by Dolan and a C-47 pilot we borrowed from the Air Corps, and that the B-25 with Douglass and Fine in it was going to Vis.\"\n\n\"Where'd you get the C-47?\" Donovan asked. And then went on without waiting for a reply, \"I didn't know a C-47 had that kind of range.\"\n\n\"It doesn't,\" Stevens said. \"I called Joe Kennedy and asked him about that, and he said that it's possible to refill the main tanks of a C-47 from barrels of fuel carried in the cabin. He also said that it's dangerous as hell, but apparently that's what they have done. Wilkins borrowed the C- 47 at Cairo.\"\n\nDonovan grunted.\n\n\"It's time we thought of the worst possible scenario,\" he said. \"That should be plural. The first thing that can go badly wrong\u2014and I am frankly surprised this hasn't already happened\u2014is that they will find out who Fulmar and the Professor really are. . . . \"\n\n\"Colonel,\" Stevens began.\n\n\"Let me finish, please, Ed,\" Donovan said. \"The best we could hope for in that situation would be that the Germans would decide we wanted Dyer for what he knows about jet- and rocket-engine metallurgy. That they would not suspect that what we're really after is getting nuclear-useful people out of Germany.\"\n\n\"Yes, Sir,\" Stevens said.\n\n\"The second thing that could go wrong would be for Canidy to be captured. Quite aside from what else he knows, I think we have to consider that the Germans know full well who he is\u2014that he's the number three here\u2014and would decide that we are either very interested in Professor Dyer, or, I'm afraid, that there is more to all this activity than is immediately apparent.\"\n\nStevens didn't reply.\n\n\"I think I have to say this, Ed,\" Donovan said. \"On reflection, I think I made an error in judgment. I think what I should have ordered\u2014to cut our losses to the minimum\u2014 was to give the Germans Fulmar and the professor.\"\n\nStevens didn't reply.\n\n\"Or alternatively, to arrange for them to be eliminated. On reflection, that's what should have been done. There are two ways to do that. The first would be to message Canidy to do it. I don't know if that would work. If he went in there without orders, in direct defiance of orders, I don't think we can expect him to obey any other order he doesn't like.\"\n\n\"Canidy is not a fool,\" Stevens said loyally.\n\n\"Sometimes I wonder about that,\" Donovan said. \"The second way to ensure that the Germans don't get to question Fulmar and the professor is to bomb St. Gertrud's prison.\"\n\n\"Canidy's thought of that. He asked for Composition C- 2.\"\n\n\"I meant by aircraft,\" Donovan said. \"A raid on Budapest. Failing to reach the target, a squadron of B-17s would bomb an alternative target. A target of opportunity. P\u00e9cs. That happens all the time.\"\n\n\"That's a little far-fetched, isn't it?\" Stevens said.\n\n\"It's laid on for tomorrow,\" Donovan said. \"Presuming the weather permits. If not tomorrow, the day after. I have been assured\u2014there is only minimal antiaircraft around P\u00e9cs, they can go in low\u2014that there is a seventy-five-percent chance that the prison can be taken out completely. Totally destroyed.\"\n\n\"My God!\"\n\n\"You know what's involved with this,\" Donovan said. \"I don't see I have any alternative. Do you?\"\n\n\"No, Sir,\" Stevens said after a moment.\n\n\"With that scenario,\" Donovan said, \"there is the possibility that the team, and Canidy, can get out.\"\n\n\"Yes, Sir.\"\n\n\"If he does,\" Donovan said, \"by the time I've finished with him, he may wish he was still in Hungary.\"\n\n\"Sir,\" Stevens said. \"From his perspective, I'm sure he thought he was doing the right thing.\"\n\nAfter a moment, Donovan said, \"I'm surprised to hear you say that, Ed. I thought by now you would have figured out that 'the right thing' has absolutely no meaning for the OSS. We do what has to be done, and 'right' has absolutely nothing to do with that.\"\n\nHe raised his voice.\n\n\"You can take us to Berkeley Square now, please, Ellis.\"\n\nWhen they got there, Captain Helene Dancy was waiting for them with a just-decrypted message:\n\nTOP SECRET  \nOPERATIONAL IMMEDIATE  \nFROM STATION VIII FOR OSS LONDON  \nC47 THREE HOURS OVERDUE HERE STOP TOTAL FUEL  \nEXPENDITURE OCCURRED NOT LATER THAN 0800  \nLONDON TIME STOP MUST PRESUME AIRCRAFT LOST  \nSTOP INASMUCH AS SUCCESSFUL DROP SIGNAL  \nUNRECEIVED MUST PRESUME FAILURE STOP UNABLE  \nESTABLISH CONTACT YACHTSMAN OR PHARMACIST STOP  \nADVISE STOP PHARMACIST II\n\nDonovan read it, then handed it to Stevens.\n\nThe C-47 with Dolan and Darmstadter was lost. And the worst possible scenario: before they had been able to drop the OSS team.\n\n\"I think you'd better radio him to come home,\" Donovan said. \"And message Wilkins to arrange for a ferry crew for the B-17. I don't want to lose that, too.\"\n\n#  **3**\n\n##  **127 DEGREES 20 MINUTES WEST LONGITUDE 07 DEGREES 35 MINUTES NORTH LATITUDE 0600 HOURS 21 FEBRUARY 1943**\n\nThe _Drum_ was on the surface. In these waters, off the eastern shore of Mindanao, the risk of a submarine on the surface being spotted by Japanese aircraft and patrol boats was almost unacceptable. But surfacing had been necessary. There was no way to attempt to contact the American guerrilla radio station from a submerged boat.\n\nIn these circumstances, when the life of his boat was literally at stake, Lt. Commander Edwin R. Lennox ordinarily would have exercised command from the bridge on the conning tower, where he could make the decisions (including the ultimate decision: to dive and run or stay and fight). But Lt. Bill Rutherford, the _Drum_ 's exec, was on the bridge and had the conn, and Lennox was below, leaning against the bulkhead. He, Captain Whittaker, and Lt. Hammersmith were watching as Radioman Second Joe Garvey tried to establish contact with U.S. forces in the Philippines.\n\nOnce he had learned that Joe Garvey was not really a motion-picture photographer, Lennox had wondered how good a radioman Garvey could be\u2014he looked to be about seventeen years old\u2014and how the boyish sailor was going to fare when they put him ashore on Mindanao.\n\nThe first question had been answered when they had been under way only a few days. The _Drum_ 's chief radioman, into whose care Garvey had been entrusted, a salty old submariner not given to complimenting his peers, had volunteered the information that \"Garvey really knows his stuff.\" From the chief radioman, that was tantamount to comparing Garvey to Marconi.\n\nLennox had noticed the two of them together frequently after that, with the innards of a radio spread out in front of them, and he had overheard several of their conversations, of which he had understood very little.\n\nBut he understood the problem Garvey and his chief radioman were trying to solve. The first part of it was that the American guerrillas were operating a homemade radio, and establishing contact with it using the radios available on the _Drum_ might prove difficult.\n\nAnd then once\u2014if\u2014they made it safely ashore, the next problem was the radio Garvey was carrying. They intended to replace the guerrillas' homemade radio with equipment capable of reliable communications to Australia, Hawaii, and the States. What they had was a new, apparently not fully tested \"transceiver,\" a device weighing only sixty pounds, including an electrical generation system that was pedaled like a stationary bicycle.\n\nBut that was several steps away. What had to be done now was to let the guerrillas know, and to keep the Japanese from learning, that Whittaker and his team were coming ashore\u2014and where, and when.\n\nSolving that problem had nothing to do with the esoterics of radio-wave propagation in the twenty-meter band.\n\nJoe Garvey had been sending a short message twice, and then listening for a response, and then sending twice again, and then listening again:\n\nKFH FOR WYZB  \nFOR GENERAL FERTIG\n\nRELAY WRISTWATCH  \nQUOTE POLO COMING FOR NORTH PUERTO RICAN  \nCOCKTAILS TODAY END QUOTE  \nACKNOWLEDGE KFH BY\n\nThe message, Captain Jim Whittaker had explained, would be delivered to Master Sergeant George Withers, whom he had left on Bataan, and who was now with Fertig on Mindanao. \"Wristwatch\" made reference to the watch Whittaker had taken from his wrist and given to Withers just before he had left him.\n\n\"Polo\" was simple. Jim Whittaker had been a polo player, and was known by that nickname.\n\nWhittaker was sure that Withers and Fertig would understand that \"cocktails\" meant \"at the cocktail hour.\" Whether they interpreted that to mean five P.M., or any hour up to eight or nine, didn't matter. If they were on the beach where Polo was coming at the cocktail hour, they would wait until the last hope he was coming was gone.\n\nThe tricky part of the message was \"Puerto Rican cocktails.\" Whittaker said he was banking on Whithers being initially baffled by that, saying aloud to find a meaning.\n\n_Puerto Rico? Puerto Rico? Puerto Rico?_\n\n\"Word association, Skipper,\" Whittaker had said. \"What's the first thing that pops into your mind when you think 'Puerto Rico'?\"\n\n\"Rum,\" Commander Lennox said immediately.\n\n\"Think geographically,\" Whittaker said.\n\n\"San Juan, I guess,\" Lennox had said. \"But I knew about San Juan.\"\n\nIt was Whittaker's intention to go ashore north of the small city of San Juan on the eastern shore of Mindanao at six, just before darkness fell.\n\n\"They will be thinking geographically,\" Whittaker said firmly. \"They'll get it, all right. The message isn't what's bothering me.\"\n\n\"Something is bothering you?\" Lennox asked sarcastically. \"I can't imagine what that would be.\"\n\n\"Well, for one thing, we don't seem to be getting any reply, \" Whittaker said dryly, \"which could mean that either Garvey's radio isn't working; or that Fertig's radio isn't working; or that Fertig's people just aren't listening; or if you insist on taking counsel of your fears, that they have been killed or captured by the Japanese.\"\n\n\"And what if they have been, Jim?\" Lennox asked, very seriously. \"What are you going to do if you can't raise them on the radio? Try again tomorrow?\"\n\n\"I've thought about that,\" Whittaker said, now as serious as Lennox. \"Garvey tells me that the signal he is sending is strong enough to be picked up all over the island. That means that other Americans, or at least Filipinos friendly to him, have heard the message and will get it to him. And so, of course, have the Japanese. I don't want to give the Japanese any more time to play word association than I already have. I want to go ashore at six tonight.\"\n\nLennox nodded.\n\nIt was, he realized, the first order Whittaker had given him that was not open to suggestion or argument.\n\n\"I think I'm going to go up to the bridge,\" he said, then added without thinking about it, \"if you don't need me?\"\n\n\"No, go ahead,\" Whittaker said absently.\n\nCommander Lennox had just reached the ladder to the conning tower when the Klaxon sounded and the speaker's voice came over the loudspeakers:\n\n\"Japanese aircraft ninety degrees three miles! Dive! Dive!\"\n\n#  **4**\n\n##  **DROP ZONE ASPIRIN NEAR P\u00c9CS, HUNGARY 0535 HOURS 21 FEBRUARY 1943**\n\nLt. Hank Darmstadter walked down the slanting floor of the C-47 to where Canidy knelt, with his ear to the chest of Lt. Commander John Dolan, USNR.\n\n\"Is he dead?\" he asked softly.\n\nCanidy straightened, still on his knees, and nodded.\n\n\"What the hell were you thinking of, sitting down?\" Canidy asked.\n\n\"He had an attack just before we landed at Cairo from Vis,\" Darmstadter said, and then answered Canidy's question: \"I couldn't kick the equipment bags out myself.\"\n\nTwo of the parachutists appeared at the door of the aircraft. They had stripped out of their black coveralls and except for the carbines they held in their hands looked like civilians.\n\n\"Jesus!\" one of them said when he saw Dolan.\n\nCanidy got off his knees and looked around the cabin for something to put over Dolan's body. He saw nothing.\n\n\"Give them the equipment bags,\" Canidy said to Darmstadter, then turned to the team. \"Take them into the woods. I don't suppose there's an ax in there?\"\n\n\"Whole fucking kit of engineer tools. Even a power saw,\" one of them replied as Darmstadter lowered one of the long, padded bags onto his shoulders.\n\n\"And C-2?\" Canidy asked.\n\n\"Hundred pounds of C-2, in two-pound blocks,\" the parachutist said as he headed for the cover of the pine forest, staggering under the weight.\n\nThe second parachutist took a bag as the other two members of the team trotted up.\n\n\"The lieutenant's in pain,\" he said. \"Pretty bad. Should we give him morphine?\"\n\n\"Not yet,\" Canidy said.\n\nThe parachutist gave Canidy a dirty look.\n\n\"Christ, he hurts! They never should have made him make this fucking jump!\"\n\n\"He's not dead,\" Canidy said. \"We'll be, if we don't get this airplane out of here before it's spotted.\"\n\nThen he looked at Darmstadter.\n\n\"You _can_ get it out of here?\"\n\n\"No problem,\" Darmstadter said immediately, confidently.\n\nA wild thought popped into Canidy's mind, and he asked the question:\n\n\"Loaded?\"\n\n\"With what?\"\n\n\"People. The team. Three others.\"\n\n\"Yeah,\" Darmstadter said, and then anticipated the next question: \"I've got about two hours' fuel aboard. If I can find Vis, that gives me a thirty-minute reserve.\"\n\n\"What do you mean, if you can find it?\"\n\nDarmstadter pointed out the door. Canidy looked. It had begun to snow: large, soft-looking flakes.\n\n\"Dolan was navigating by reference to the ground,\" Darmstadter said. \"Roads and railroads. I won't be able to see the ground. And I'm not sure I can find Vis just using a compass.\"\n\n\"That kind of snow won't last long,\" Canidy said reassuringly.\n\n_But,_ he thought angrily, _that fucking snow is just what we don't need!_\n\nAnd then he realized that exactly the opposite was true. The snow was just what he _did_ need. It would obscure the tracks the landing gear had made on the meadow. And, if he was right, and it left just a dusting of fresh snow atop the inch or two on the ground, it wouldn't interfere with a takeoff.\n\n\"Start it up,\" he ordered. \"I'm going to find a place to hide this big sonofabitch.\"\n\nAs he ran into the center of the meadow, looking for a break in the trees, someplace where the C-47 could be taxied to, he wondered whether his decision to use the Gooney Bird to get out of here was based on sound military reason (Darmstadter couldn't find Vis\u2014he could; it was an available asset and should be used) or whether he subconsciously saw it as a lifeboat with himself as a drowning sailor, and was irrationally refusing to let it go, as drowning sailors will fight to get into an already loaded lifeboat, not caring that their weight will swamp it.\n\nHe snapped out of that by telling himself the decision had been made and there was no going back on it now.\n\nHe found no place to hide the airplane, now sitting where it had stopped with engines idling and Darmstadter looking out the window, waiting for instructions.\n\nCanidy ran back to it and signaled Darmstadter to turn it around, then guided him to the edge of the forest, stopping him only when the nose was in the trees and the propeller on the right engine was spinning two feet from a thick pine trunk.\n\nThree of the team members were watching him. He wondered if they were simply curious or had already decided he was crazy.\n\n\"You said there was a power saw,\" he said. \"Get it. Cover as much of this thing as you can with the largest boughs you can.\"\n\n\"Why don't you just blow it?\" one of them, the one who was so concerned about J\u00e1nos being in pain, said. \"You already got one fire.\"\n\n\"Everybody gets one question,\" Canidy said. \"That was yours. I don't want to hear another. The answer to your question is we're going to get out of here on that Gooney Bird.\"\n\n\"You'll never get that off the ground in that short a distance, \" the parachutist said.\n\n\"That was an opinion,\" Canidy said icily. \"You get one, only, of those, too. The next time I want to see your mouth open is when I ask you a question.\"\n\nThe parachutist glared at him but said nothing.\n\n\"Get going!\" Canidy said. \"I want the snow to cover the boughs.\"\n\n\"There's an auxiliary fuel system,\" Darmstadter said. \"A fifty-five-gallon barrel connected to the main tanks. You want me to try to get it out?\"\n\n\"That and anything else heavy we don't absolutely need.\"\n\n\"You're not talking about Commander Dolan?\" Darmstadter flared.\n\n\"No,\" Canidy said. \"We'll take Dolan with us.\"\n\nThe Countess's housekeeper appeared in the main room of the lodge when Canidy, Alois, and Freddy J\u00e1nos, white-faced, his arms around their shoulders, walked into it.\n\nShe put a balled fist to her mouth. Canidy could not tell whether she was manifesting sympathy or fear.\n\n\"Major,\" J\u00e1nos said, embarrassed, \"I think I'm going to pass out.\"\n\n\"I'm going to give you something for pain just as soon as I get you in bed,\" Canidy said. \"Tell him to tell her to keep her mouth shut.\"\n\nThey half carried J\u00e1nos to the bed in which Canidy had slept and laid him flat on it. Canidy, as gently as he could, cut the boot from his leg, then pulled a coarsely woven cotton sock\u2014Hungarian, rather than GI wool-cushion-soled\u2014from it. Somewhere in J\u00e1nos's gear was a pair of Hungarian shoes that the plan called for him to put on once he was on the ground. The notion that jump boots might protect his ankle hadn't worked.\n\nThe ankle was blue and swollen, but there didn't seem to be any bones threatening to break through the skin.\n\nCanidy opened a flat metal can, sealed with tape, and took a morphine syringe from it. He pushed J\u00e1nos's trouser leg up as far as he could and shoved the needle into his calf. It would take a little longer for the morphine to take effect that way, but it would be less painful for J\u00e1nos than moving his body around to get at his upper arm or buttock.\n\n\"That'll take a minute or two,\" Canidy said. \"I'll be back.\"\n\n\"I'm getting sick to my stomach,\" J\u00e1nos said.\n\n\"Tell him,\" Canidy said, nodding at Alois. \"He'll get you something to throw up in.\"\n\nThen he went looking for the Countess and von Heurten-Mitnitz.\n\nIt was not necessary under the circumstances, he decided, to bother knocking on doors and politely waiting for permission to enter.\n\nHe found them behind the third door he opened, nearly hidden under a goose-down comforter.\n\n\"Good morning,\" he said.\n\nHelmut von Heurten-Mitnitz suddenly erupted from under the comforter, reaching for his Walther pistol as his eyes swept around the room.\n\nThe movement took the comforter off both of them. They were both naked.\n\nThe Countess, as Canidy had thought she might be, was a baroque work of art. His Excellency was a white-skinned, skinny man, from whose chest sprouted no more than a dozen long black hairs.\n\n\"What's all this?\" von Heurten-Mitnitz demanded in outrage as he put the pistol down and pulled the comforter over himself and the Countess.\n\n\"The team is here,\" Canidy said.\n\n\"I presume you mean Ferniany,\" von Heurten-Mitnitz said.\n\n\"No, I mean the team,\" Canidy said. \"They were dropped about thirty minutes ago. I think you ought to get dressed and get out of here right away.\"\n\n_I have just decided,_ Canidy realized, _that I am not going to tell them about the Gooney Bird_.\n\n\"Did everything go all right?\" the Countess Batthyany asked.\n\n\"One of them has a broken ankle,\" Canidy said. \"I brought him here.\"\n\n\"Where did you put him?\" she asked.\n\n\"In my bed,\" Canidy said.\n\nThe Countess slid out from under the comforter, modestly turned her back to Canidy, and wrapped herself in a dressing gown. She found shoes, worked her feet into them, and, brushing her magnificent mop of red hair off her face, walked out of the room.\n\nHelmut von Heurten-Mitnitz got out the other side of the bed and started to dress. Naked, Canidy thought, and in his underwear\u2014a sleeveless undershirt and baggy drawers, plus stockings held up by rubber suspenders on his skinny calves\u2014von Heurten-Mitnitz was not at all impressive.\n\n\"We have one dead man, too,\" Canidy said.\n\n\"What happened?\" von Heurten-Mitnitz asked.\n\n\"Natural causes,\" Canidy said. \"A heart attack.\"\n\nVon Heurten-Mitnitz didn't seem at all surprised by that announcement, which surprised Canidy.\n\n\"What are you going to do with the body?\" von Heurten-Mitnitz asked. \"Or the man with the injured . . . leg, you said?\"\n\n\"Ankle,\" Canidy said. \"I haven't made up my mind yet. The first priority, I think, is for you and the Countess to get back to Budapest.\"\n\n\"I think you're right,\" von Heurten-Mitnitz said.\n\nCanidy returned to his room.\n\n\"You landed the airplane,\" the Countess greeted him, looking up from the bed, where she was prodding and pulling on the ankle of the now unconscious J\u00e1nos.\n\nAlois had apparently told her, and she would now certainly tell von Heurten-Mitnitz.\n\n\"Yes,\" Canidy said.\n\n\"I will remain here while Herr von Heurten-Mitnitz returns to Budapest,\" she said. \"It would be better, if I were here when . . . if . . . the authorities come.\"\n\n\"I think it would be better if you went to Budapest,\" Canidy said. \"Just as soon as you can.\"\n\nShe ignored him.\n\n\"I have sent for rubber bandage,\" she said. \"I'm sure there's some here. I think about all we can do for this man is to wrap the ankle tightly, then stiffen the ankle. You take my meaning?\"\n\n\"Splint it,\" Canidy said, nodding. \"Thank you.\"\n\nAlois came into the room with von Heurten-Mitnitz on his heels.\n\n\"Their airplane landed,\" the Countess said.\n\nVon Heurten-Mitnitz looked at Canidy, surprised.\n\n\"Intact?\" he asked.\n\n\"Yes,\" Canidy said.\n\n\"And you plan to use it to leave?\" von Heurten-Mitnitz asked.\n\nCanidy nodded. \"If we can.\"\n\n\"I think it would be best if you took Beatrice with you,\" von Heurten-Mitnitz said.\n\n\"No,\" the Countess said. \"I am staying here to do what I can while you go to Budapest. But I am not leaving with them.\"\n\n\"I don't see any way that what has happened here can be hidden,\" von Heurten-Mitnitz said.\n\n\"Then you leave, too,\" the Countess said.\n\n\"There is a good chance that no one knows about either the drop or the plane landing,\" Canidy said.\n\n\"I think that is highly unlikely,\" von Heurten-Mitnitz said.\n\n\"You and the Countess slept through two passes and the landing itself,\" Canidy said.\n\nVon Heurten-Mitnitz grunted, reluctantly granting the point.\n\n\"I don't want to have to worry about you, Countess,\" Canidy said, \"while we're getting Eric and the professor out of St. Gertrud's. I want you to go to Budapest, and now.\"\n\nShe met his eyes for a moment.\n\n\"All right,\" she said finally. \"Just let me do what I can for him.\"\n\nTwenty minutes later, the Opel Admiral drove away from the lodge. By then, it had stopped snowing. Canidy wondered if enough snow had fallen to conceal the tracks the C-47 had made on the meadow, or to obscure the outline of the aircraft under the pine boughs.\n\nSince Ferniany hadn't shown up, there was nothing else to do, so he went to see.\n\n#  **5**\n\n##  **1715 HOURS 21 FEBRUARY 1943**\n\nFerniany drove up to the hunting lodge at the wheel of a small, canvas-bodied Tatra truck about the size of an American pickup. Canidy, summoned from the kitchen by Alois, went out to meet him. Ferniany had three men from the Hungarian underground with him, but that was about all.\n\nThere had been \"a little trouble,\" he told Canidy. The Germans, or maybe even the Hungarians, he didn't know which, had had radio direction-finding trucks in operation, and they had located the radio transmitter from which he had radioed the drop-zone coordinates.\n\nThere had been enough warning that the trucks were moving around, together with cars full of police, for him to get away before the police got to the hidden transmitter, but he had had to leave everything behind.\n\nThe police by now had found the signal panels, the radio, and the weapons, including the Sten submachine gun Captain Hughson had loaned Canidy on Vis.\n\n\"Where did the truck come from?\"\n\n\"We stole it,\" Ferniany replied, just a little smugly.\n\n\"How do you plan to get rid of it?\" Canidy asked.\n\nFerniany looked at him, making it clear he didn't think much of the question.\n\n\"Abandon it, when we're through with it.\"\n\n\"How many trucks do you think are stolen in Budapest and then abandoned in P\u00e9cs?\" Canidy asked. \"Did it occur to you that the police might find that curious? Or that the SS, now that they're aware there are people in here with transmitters and signal panels and English weapons, might be absolutely fascinated to learn that a truck had been stolen in Budapest and abandoned here?\"\n\n\"We'll hide it in the forest,\" Ferniany said lamely. \"Bury it, even.\"\n\n\"The damage is done,\" Canidy said. \"As soon as the team has gotten our people out of St. Gertrud's, you do whatever you can about the truck. Either, preferably, get it back to Budapest and abandon it there or take it someplace else. But get it away from here.\"\n\nFerniany did not seem to understand that stealing the truck had been a stupid thing to do. If they had been caught in the act of stealing it, or once they had it in their possession, even the dumbest Hungarian cop would have made the connection between someone barely escaping from the radio-detection operation and someone heading out of town in a stolen truck.\n\nAnd if he sensed that Canidy was furious, he showed no sign of it.\n\n\"You said, 'as soon as the team' gets our people out . . . ,\" Ferniany challenged.\n\n\"Yes, I did.\"\n\n\"Major,\" Ferniany explained patiently, almost tolerantly, \"without the signal panels and the radio, there's no way we can expect the team to get in here,\" Ferniany said. \"We're going to have to do this ourselves.\"\n\n\"You've got some kind of a plan?\" Canidy asked. It was all he trusted himself to say.\n\n\"Prisons are designed keep people _in,_ \" Ferniany said, solemnly announcing a great philosophical truth.\n\n\"And?\"\n\n\"From seven o'clock at night until five o'clock in the morning, there are on duty only six people: five guards and a sort of clerk. _And_ there is only one guard on the motor pool where they keep the trucks and motorcycles.\"\n\n\"You mean the mine trucks, the ones they carry the prisoners back and forth to the mine in?\"\n\n\"Right,\" Ferniany said.\n\n\"So what you're going to do is knock over the guard at the motor pool, steal a mine truck, and drive it to the prison. You'll be a little early, but they'll recognize the truck and pass you inside, whereupon you and your three men will take on the five guards and the clerk, grab Fulmar and Professor Dyer, and make your escape?\"\n\n\"I detect a little sarcasm,\" Ferniany said.\n\n\"Not a little,\" Canidy said.\n\nHe let that sink in, and waited for an angry response. He was surprised when none came.\n\n\"Right up there in importance with getting Fulmar and Dyer out,\" Canidy said, \"is getting them out without calling anybody's attention to the fact that they are anything but what they were\u2014thanks to your stupidity, we should keep that in mind\u2014arrested for: black marketeers. Don't you think the Germans would be goddamned curious to learn why two people\u2014who just happen to fit the descriptions of two men the whole goddamned SS is looking for\u2014were busted out of an obscure Hungarian prison with more shooting and dead bodies strewn all over than in a Jimmy Cagney gangster movie?\"\n\nFerniany's face colored with anger.\n\n\"I'm right on the edge of telling you to go fuck yourself, Canidy,\" Ferniany said.\n\n\"You really wouldn't want to do that, would you?\" Canidy asked primly.\n\n\"Why wouldn't I?\" Ferniany said. \"According to you, I don't do anything right.\" He paused, but then was carried along by his momentum: \"Fuck you, Canidy. Stick this whole operation up your ass. I'd like to hear how you plan to get them out, you wiseass sonofabitch.\"\n\n\"Now you've gone and done it,\" Canidy said, even more primly.\n\n\"Done what?\" Ferniany said, curiously, a smile forming on his face.\n\n\"Used naughty words in front of the enlisted men,\" Canidy said, gesturing to Alois and the men from the underground, who had been fascinated by the angry exchange, not a word of which they understood. \"Whatever will they think?\"\n\nFerniany looked at the four Hungarians. Then, although he tried not to, the innocent curiosity on their faces made him laugh.\n\nThat seemed to reassure the Hungarians. The looks of puzzlement were replaced by broad smiles.\n\n\"I would be fascinated, Major Canidy, Sir,\" Ferniany said, \"to learn precisely how the Major plans to carry out this mission.\"\n**XIV**\n\n#  **1**\n\n##  **NEAR SAN JUAN, ISLAND OF MINDANAO COMMONWEALTH OF THE PHILIPPINES 1815 HOURS 21 FEBRUARY 1943**\n\nThe commanding general of United States forces in the Philippines had climbed a tree. It wasn't a very tall tree, and he hadn't been able to climb very far up it, but it was on the highest point he could find on a bluff thirty feet above a narrow sandy beach, and he was sure that it was giving him the best possible view of the sea.\n\nIt was growing dark. In fifteen minutes, it would be completely dark. Moving through the jungle at any time was difficult, and when it was dark, damned near impossible.\n\nHe knew he had made a bad decision coming here at all. What he should have done was send Withers and one or two of his men down here to see what happened, not come himself.\n\nBut he had wanted so desperately to believe that something would happen. So he had come himself, and brought an unnecessarily large force with him. He knew it was because he wanted witnesses that his hopes had come true. But what else was there for him to do?\n\nHe put the one and only pair of binoculars in the hands of U.S. forces in the Philippines to his eyes.\n\nHe would search the open sea one more time, until his eyes started to tear from fatigue, he decided, and then he would order the withdrawal of this force by night to the mountains, and on the way maybe he'd think of one more credible excuse why \"the aid\" hadn't come this night either, one more reason to hope that maybe tomorrow\u2014\n\nThere wasn't one miserable fucking thing on the surface of the water.\n\nSomebody tugged on his shoe. He looked down in annoyance.\n\nIt was Master Sergeant Withers. He was pointing down at the beach, his hand shaking, and with tears running down his cheeks.\n\nThere was a submarine down there, in far closer to the beach than Fertig would have believed it possible for a submarine to maneuver. Torrents of water still gushed from ports in its side, but there were people on the conning tower, and then the colors went up on a mast over the conning tower.\n\nFertig's eyes filled with tears.\n\n\"I'll be a sonofabitch, there they are!\" Withers said.\n\nThere was all sorts of activity on the submarine now. Sailors ran purposefully about the narrow decks, objects were handed up through hatches.\n\nThe commanding general of USFIP slid down the tree trunk and slid down the bluff to the beach.\n\nThey had to wait for what seemed like an hour, but what was really not more than five minutes, before a rubber boat appeared close to the surf.\n\nHalf a dozen of his men ran out in water to their shoulders to reach it, to help it ashore.\n\nFertig thought, idly, that they seemed to be having a hell of a hard time pulling it.\n\nAnd then somebody jumped out of the rubber boat, and Fertig walked into the receding surf to meet him, although he had told himself he would not, the salt water would be hell on already deteriorating boots.\n\nHe was a tall and good-looking young man in khakis.\n\nHe splashed through the surf to Fertig.\n\nAnd then he stopped, still in water to his knees, and came to attention and saluted.\n\n\"Captain Whittaker, General,\" he said. \"United States Army Air Corps.\"\n\n\"Welcome to Mindanao, Captain,\" Fertig said, returning the salute crisply, controlling his voice with a massive effort, glad now that it was dark enough that Whittaker wouldn't be able to see the tears on his cheeks.\n\n\"Sir . . . Sergeant Withers?\"\n\nFertig pointed to the second rubber boat coming through the surf. With the same apparent difficulty that those helping the first boat had had, Withers was trying to hurry it ashore.\n\n\"Excuse me, sir,\" Whittaker said, and ran into the surf. He returned with a very small sailor riding on his shoulders.\n\n\"Send 'B,' \" Whittaker ordered as he set the small sailor onto the beach.\n\nJoe Garvey flashed the Morse code signal for \"B,\" a dash and three dots, from a flashlight with an angled head.\n\nThere was an immediate response from a signaling light on the conning tower of the submarine. Garvey hurriedly took a pad from his pocket and wrote it down.\n\n\"What was that?\" Fertig asked.\n\n\"Garvey sent them 'B,' \" Whittaker explained. \" 'B' is 'safely ashore, with equipment, in contact with U.S. forces in the Philippines.' \"\n\n\"Sir,\" Radioman Second Joe Garvey reported, \" _Drum_ messages, 'Aloha. God Bless.' \"\n\nFertig looked out at the submarine. It was under way. The colors had already been hauled down. Its deck was already awash. It was going back under.\n\nIt didn't matter. If one came, others could. Others _would_.\n\n\"My men seem to be having a time getting your boats ashore, Captain,\" General Fertig said, trying valiantly to sound nonchalant.\n\n\"We've got medicine for you, General,\" Whittaker said. \"And some small arms and ammunition. And a million dollars in gold coins. You wouldn't believe how much a million dollars weighs until you try to tow it around in a rubber boat with a five-horse outboard motor.\"\n\n#  **2**\n\n##  **ST. GERTRUD'S PRISON P\u00c9CS, HUNGARY 0630 HOURS 22 FEBRUARY 1943**\n\nThe Tatra dump truck scraped the stones in the tunnel between the courtyard and the street with the left edge of its bumper.\n\nA little harder than usual, Eric Fulmar, riding against the cab in the bed of the truck, thought idly. And then there was immediately another proof that it was going a little harder than usual. Instead of squeaking on through, the truck jerked to a stop and, with a clash of gears, backed up.\n\n_Oh, Christ, now what?_\n\nThen the gears clashed again, and the truck moved forward, and they were through the tunnel and onto the street.\n\nIt had snowed again overnight, not much, just a white dusting over the slush. Fulmar had hoped for freezing rain. That made the ride to the mine more interesting. He had concluded that all the truck drivers he had met since they had been locked up shared one quality: They had all learned how to drive last week and tried to hide this by driving as fast as the trucks would go.\n\nOn the slippery cobblestone streets on the way to the mines, they often skidded the truck into a ditch or into something hard enough to bend the fenders into the tires. This was routinely followed by marvelous displays of Hungarian temper and absolutely marvelous attempts to get the trucks out of the ditches by doing precisely the wrong thing.\n\nSometimes, as much as two hours would be lost. It was more pleasant than handling a donkey in the mines, and Fulmar looked forward to icy road conditions. He was disappointed this morning when the driver managed to negotiate a turn that had several times seen the truck skid into a ditch so steep that the rear wheels of the truck left the ground.\n\nThey were maybe a kilometer away from the mine when he felt the brakes lock, and the truck skid, and then jolt to a halt.\n\nHe could not see over the cab, so he had no idea what they'd hit.\n\nA moment later, there was a call in Hungarian for everybody to get out.\n\nGetting everybody out to push was routine, too. And while it wasn't as interesting as watching the Hungarians try to get the wheels of a dump truck back on the ground by swearing and throwing stones at it, it would still delay the journey into the mines.\n\nIt wasn't until he had slid from the truck bed and turned around that Eric saw that whatever was happening was not routine.\n\nThere were men behind the truck, Hungarian civilians with pistols; and the two Keystone Kops on the motorcycle who trailed the truck were on the ground, spread-eagled. As Fulmar watched, the driver and his assistant were brought to the rear of the truck and forced onto the ground beside the cops.\n\nOne of the men with pistols motioned the prisoners into a line, and then into two lines, then three, prodding the slow ones with the barrel of his pistol. And then another man came down the line and rudely jerked people out of line by grabbing their shoulders.\n\n_If I wasn't so afraid, this would be funny._\n\nThe man reached him, jerked Fulmar out of line, and marched him toward the front of the truck. Fulmar saw what had stopped the truck. A tree lay across the road. At first he thought it had been sawed, but then he saw that it had been taken down by somebody who knew how to use Primacord.\n\nStanding near the cab of the truck were more Hungarians. One of them, in a large soft black woolen hat, looked somehow familiar.\n\n\"You do not recognize me,\" Canidy ordered quietly when Fulmar was dragged before him.\n\nFulmar shook his head in wonderment and smiled, but said nothing.\n\n\"We don't have much time,\" Canidy said. \"Just tell me which of the others would escape if they had half a chance?\"\n\nFulmar looked confused.\n\n\"You heard me,\" Canidy said. \"I need to know who are the serious criminals.\"\n\nFulmar was as much confused by the question as he was surprised to see Canidy. But he finally understood that the question was important for reasons he could not imagine.\n\n\"These guys are petty criminals,\" Fulmar said. \"If they weren't in jail, they'd probably starve. No real criminals, if that's what you're asking.\"\n\n\"Damn,\" Canidy said. \"Now, is Professor Dyer one of the people we pulled out of there?\"\n\nFulmar looked.\n\n\"Second from the end,\" he said, \"with the glasses.\"\n\nCanidy waved another of the Hungarians over and spoke softly to him in English.\n\n\"No gangsters,\" he said. \"We'll just have to take half a dozen of them with us, that's all there is to it. You saw Dyer?\"\n\n\"Yeah, but I don't think he recognized me.\"\n\n\"Let's try to keep it that way for the time being,\" Canidy said. \"You go ahead and get them to uncover the plane.\"\n\n\"The plane?\" Fulmar blurted. \"You've got an _airplane?_ \"\n\n\"Take Loudmouth here with you,\" Canidy said. \"He insists on talking English.\"\n\nThere was a sharp cracking noise, followed a moment later by a creaking, tearing noise, and finally a great crashing sound.\n\nFulmar realized that another tree, its trunk severed by Primacord, had been dropped across the road.\n\n\"Let's go, Lieutenant,\" the man Canidy had spoken to said softly, and Fulmar followed him off the road and into the forest.\n\nIt was a long way across steep, heavily forested hills from where the prison truck had been stopped to the meadow; and when they got there, Fulmar was sweat-soaked and panting from the exertion.\n\nHe didn't see an airplane. All he saw was a Hungarian standing at the far end of the meadow beside two of the largest horses he had ever seen. The horses wore whatever horses used so they could pull a wagon or a plow, but there was nothing around for them to pull.\n\nAnd then, as they crossed the meadow, he saw a round red light sticking out of a snow-covered mound. And he understood that he was looking at the top of an aircraft vertical stabilizer.\n\nAn American pilot wearing a leather A-2 jacket and with a Thompson submachine gun in his hands came out of the woods.\n\n\"This is Fulmar,\" Ferniany told Darmstadter. \"Canidy's bringing the other one.\"\n\nDarmstadter looked with unabashed curiosity at Fulmar.\n\n_This young guy in blue work clothes was the purpose of this whole operation?_\n\n\"Hello,\" Fulmar said.\n\nThat shocked Darmstadter into action.\n\nHe looked around for someplace to put the Thompson down and finally hung it from a brass horn on the harness of one of the horses. Ferniany watched him, then shrugged and put his pistol in his pocket and went to the mound of snow-covered brush.\n\nWhen the branches were off the tail section, Alois hitched a stout rope to the tail wheel and the huge horses pulled the C-47 far enough out of the forest to turn the airplane around.\n\nIt took half an hour to remove all the branches from the C-47. Some of them had frozen to the wings and fuselage, and small branches had wedged into the openings of the movable control surfaces.\n\nDarmstadter started the engines, to make sure they would start. The engines started without difficulty, but when he tried to run the controls through their operating range, he found that snow had melted and then frozen the controls cables.\n\nHe let the engines run until they had reached operating temperature, then shut them down. Then he went after the ice in the ailerons and other movable control surfaces while Fulmar and Ferniany hammered at the ice on the wings. They quickly learned the best way to get it off was to stamp on it with their feet or slam it with their fists. The aluminum would then flex enough to free the ice, which could then be pushed or kicked out of the way.\n\nThey were still working on the airplane when the team, the Hungarian underground, Canidy, Dyer, and six wholly confused and terrified petty criminals from St. Gertrud's prison arrived.\n\n\"Wind it up,\" Canidy ordered. \"We're going. Get those people aboard.\"\n\n\"We're taking them?\" Fulmar asked incredulously.\n\n\"Instant immigration,\" Canidy said. \"Get them aboard.\"\n\nCanidy stood by the door of the airplane as the Hungarians and the team and Professor Dyer got aboard. He collected the weapons and passed them to the Hungarians. Darmstadter started one engine and then the other.\n\n\"Get on, Eric,\" Canidy ordered.\n\nFerniany and Canidy looked at each other a moment, wordlessly.\n\n\"You aren't really such a horse's ass after all,\" Canidy finally said. \"Take care.\"\n\n\"You are,\" Ferniany said with a smile. \"A horse's ass, I mean.\"\n\nThen he slapped Canidy on the back and ran to get out of the prop blast.\n\nCanidy climbed into the Gooney Bird. As he closed the door, Darmstadter started to taxi to the absolute end of the meadow.\n\nCanidy slid into the copilot's seat as Darmstadter turned the Gooney Bird around.\n\nDarmstadter locked the brakes, checked the mags, and then ran both engines up to takeoff power. The Gooney Bird trembled and bounced. He took the brakes off, and the airplane began to roll, first with maddening slowness, and then picking up speed. But not quite enough to get it off the ground.\n\nAs they reached the end of the meadow, Darmstadter pulled it into the air. There was not enough velocity to maintain flight, and it started to stall. Darmstadter pushed the nose down, getting it out of the incipient stall; and the Gooney Bird now followed the contour of the cut-over hillside down toward the stream. It was flying, but only barely.\n\nAnd then he pulled back on the wheel again, and this time, having picked up just enough speed, the Gooney Bird was willing to fly for real.\n\n\"Very impressive,\" Canidy's voice came over the earphones. Thinking it was sarcasm, Darmstadter snapped his head toward him.\n\nCanidy was beaming and making an \"OK\" sign of approval with his left hand.\n\nAnd then Canidy's face registered genuine surprise, and the \"OK\" sign changed into a finger pointing out the windshield. Darmstadter followed it.\n\nThere were sixteen B-17 aircraft flying in five staggered Vs at what was probably eight thousand feet. Their bomb bays were open, and as Darmstadter and Canidy watched, streams of 500-pound bombs began to drop.\n\n\"They're bombing P\u00e9cs,\" Darmstadter said. \"What the hell is there in P\u00e9cs worth bombing with a squadron of B- 17s?\"\n\nCanidy didn't respond to that.\n\n\"I think you had better get back on the deck,\" he said. \"Steer one nine zero.\"\n\n#  **3**\n\n##  **OSS LONDON STATION BERKELEY SQUARE LONDON, ENGLAND 1630 HOURS 22 FEBRUARY 1943**\n\nLieutenant Joseph P. Kennedy, Jr., came into David Bruce's office. Kennedy looked, Colonel Wild Bill Donovan thought, not unlike his father as a young man.\n\n\"Hello, Joe,\" Donovan said. \"How are you?\"\n\n\"Not very cheerful, Colonel,\" Kennedy said, raising a package in his hand. \"Dolan's personal items. I didn't know what to do with them.\"\n\n\"I'll take them, Mr. Kennedy,\" Chief Ellis said. \"I'll see that they get to his next of kin.\"\n\n\"Does he have any?\" Kennedy asked. \"I never heard him talk about a family.\"\n\n\"I'm sure there's a brother or a sister or somebody,\" Donovan said.\n\n\"And what do I do about Darmstadter?\" Kennedy said. \"Write the letter myself, or let his old outfit do it? He was on TDY to the composite squadron, officially.\"\n\nHe was, Donovan thought, approvingly, already assuming the responsibilities of command.\n\n\"You write it, Joe,\" Donovan ordered. \"Be vague. But let them know he went in as a volunteer doing something important. \" He thought about saying something else, realized that he shouldn't, but said it anyway: \"I wish we could report them KIA. Until we have positive word, of course, they'll have to be carried as MIA. But I don't think there's any real hope.\"\n\n\"Yes, Sir,\" Kennedy said.\n\nDonovan had been avoiding making the decision what to do about taking the necessary action about Dick Canidy and Ferniany. At the very least, they were missing in action. It might even be better to hope that they were dead. Just before it went off the air, interrupting a code block, the OSS radio station had sent the code for \"Station discovered, in immediate danger of being captured.\"\n\nIt was reasonable to presume that Ferniany had been captured in Budapest. If that was true, and he was lucky, he would be dead. If that was true, and he was unlucky, he was alive and in the hands of the SS; and it might be some time before they were through with him and shot him. Or hanged him with a length of piano wire.\n\nIf they had caught him alive, it had to be presumed that he had given them Canidy's location and told them what he knew. No matter how little that was, it was certain to be damaging to von Heurten-Mitnitz, the Countess Batthyany, and the whole Hungarian pipeline.\n\nThere seemed to be little doubt that Fulmar and Professor Dyer were dead. The last B-17 had carried photographers, and there was proof beyond question that St. Gertrud's prison and three square blocks around it had been bombed into rubble.\n\nCanidy, to be sure, might still be alive, on the run somewhere in the forests near P\u00e9cs. He had as many lives as a cat.\n\nIt was the particularly obscene nature of this business, Donovan thought, that I am forced to hope that he is dead. If he is dead, what he knows will not become known to the Germans.\n\nHe had decided that when he made up his mind to do it, he would personally write to the Reverend Doctor George Crater Canidy. He knew that it would be important, that Canidy would really want his father to believe he had died saving lives, not taking them. In a sense that was true, and maybe, Donovan decided, he would be able to make that point.\n\nA more immediate problem was telling Ann Chambers. She had no legal right to know, of course. But legality had nothing to do with it. Donovan wanted her to hear it from him, and that meant he would have to tell her in the next couple of hours, before he got on the Washington plane.\n\n\"Joe,\" he said, \"you understand, of course, that Operation Aphrodite is now your responsibility?\"\n\n\"Yes, Sir.\"\n\n\"When Stan Fine gets back, he will fill the role Canidy had. You will report to him.\"\n\n\"Yes, Sir.\"\n\n\"There's more to it than the sub pens at Saint-Lazare,\" Donovan said.\n\n\"I assumed there was,\" Kennedy said matter-of-factly.\n\nDonovan's eyebrows rose.\n\n\"I'll have Colonel Stevens fill you in,\" Donovan said.\n\n\"Yes, Sir.\"\n\n\"We have to expect setbacks, Joe,\" Donovan said, wondering if he was talking as much to himself as he was to Kennedy. \"And not everything has gone wrong. Just before you came, there was word that Jimmy Whittaker is safely ashore in the Philippines.\"\n\n\"Sir?\" Kennedy asked, confused.\n\n_I am more emotionally upset by all this than I like to think I am; there was no reason for me to tell Kennedy that, and I should have known that he didn't know what was planned for Whittaker._\n\n\"That's out of school, Joe,\" Donovan said. \"You don't have the Need-to-Know.\"\n\n\"You sent Jimmy _back_ to the Philippines?\" Kennedy asked incredulously.\n\n\"He volunteered to go,\" Donovan said.\n\n_That's pretty lame, Donovan, and you know it. You did indeed send Jimmy back, knowing full well the risks._\n\nThe door opened. Capt. Helene Dancy walked in.\n\n\"I asked not to be disturbed,\" Donovan said, coldly angry. \"Do I have to lock the door to keep from being interrupted? \"\n\n_Just because you don't like yourself right now is no reason to jump all over her_.\n\nCapt. Dancy did not reply. White-faced, obviously hurt and angry, she marched to his desk, laid a TOP SECRET cover sheet on it, and marched back out of the office.\n\nTOP SECRET  \nOPERATIONAL IMMEDIATE  \nFROM STATION VII  \nTO OSS LONDON EYES ONLY BRUCE AND STEVENS\n\nEXLAX AND TINCAN ONE ALIVE AND WELL STATION VII STOP GOONEYBIRDING STATION VIII STOP WILL REQUIRE IMMEDIATE AIR TRANSPORTATION STATION VIII DASH LONDON SIX HUNGARIAN CRIMINALS AND REMAINS LT CMDR JOHN DOLAN STOP\n\nCANIDY\n\nIt took Donovan a moment before he trusted his voice.\n\n\"I think, Joe,\" he said finally, handing him the message, \"that you had better hold off on writing Lieutenant Darmstadter's family until we can get this sorted out.\"\n\nAs Kennedy read the message, Donovan added, \"Let Chief Ellis see it when you're finished.\"\n\n\" 'Hungarian criminals'?\" Kennedy asked. \"Is that some kind of a code?\"\n\n\"Not as far as I know,\" Donovan said.\n\n\"I wonder what happened to Dolan,\" Ellis said.\n\n\"You were friends, Ellis?\" Donovan asked.\n\n\"Not really _friends,_ \" Ellis said. \"Yeah, well, maybe. A couple of old sailors. I liked him.\"\n\nThe door opened again.\n\n\"Yes, Sir?\" Capt. Helene Dancy asked.\n\n\"First, Helene, I'm sorry I jumped on you,\" Donovan said.\n\n\"That's perfectly all right, Sir,\" she said.\n\nShe's still mad.\n\n\"I think you had better message Wilkins, over my signature, and tell him to give Canidy whatever he wants when he gets there. You don't know what 'Hungarian criminals' means, do you?\"\n\n\"No, Sir. I presumably do not have the Need-to-Know.\"\n\n\"Neither do I, apparently, Helene,\" Donovan said. He smiled at her, and finally she cracked and smiled back.\n\n\"In that case, Sir,\" she said. \"I think we have to presume that Major Canidy, for reasons he will certainly explain to us, is going to have six Hungarian criminals with him.\"\n\nDonovan chuckled.\n\n\"Will that be all, Colonel?\"\n\n\"Lieutenant Kennedy has Commander Dolan's personal effects,\" Donovan said. \"Will you see if you can come up with a next of kin name and address?\"\n\n\"I've already inquired. Nothing yet. I'll keep trying. Anything else?\"\n\n\"You might tell Ann Chambers that Canidy is on his way home. If you think she'd be interested.\"\n\n#  **4**\n\n##  **THE NATIONAL INSTITUTES OF HEALTH BUILDING WASHINGTON, D.C. 1830 HOURS 24 FEBRUARY 1943**\n\nChief Ellis was tired, unshaven, and mussed. It had been almost forty hours before the ATC C-54 from London had touched down at Anacostia. But he had ignored Colonel Donovan's orders to \"go home and get some sleep, there's nothing that won't wait until tomorrow.\"\n\nThere was always something that wouldn't wait.\n\n\"You look like shit, Ellis,\" Staley greeted him.\n\n\"I feel like shit,\" Ellis said. \"How come you aren't all dressed up in new chief's blues?\"\n\n\"Captain Douglass said he thought it would be nice if the Colonel made it official,\" Staley said.\n\n\"Yeah, hell, why not?\" Ellis said.\n\n\"But you done it, Ellis,\" Staley said. \"Thank you.\"\n\n\"We old China Sailors got to stick together,\" Ellis said. \"And you're at the age where you look silly in bell-bottoms.\"\n\nHe tossed his overcoat on a chair, pushed his cap back on his head, sat down at the desk, and slid the stack of classified documents in front of him.\n\n\"Anything interesting in here?\"\n\n\"Yachtsman is alive and well,\" Staley said. \"That came operational immediately from London yesterday. What's it mean?\"\n\n\"It's damned good news,\" Ellis said. \"You don't have to know why. The Colonel will be happy as hell.\"\n\n\"Whittaker's ashore in the Philippines,\" Staley said.\n\n\"We heard that,\" Ellis said.\n\n\"And the radio works,\" Staley said. \"There's a whole bunch of messages from Fertig.\"\n\n\"And anything else?\"\n\n\"Two things for you,\" Staley said uncomfortably. \"I opened the telegram. I figured it might be important. It's on the bottom.\"\n\nEllis lifted the stack of cover sheets and found the Western Union telegram envelope.\n\nWESTERN UNION TELEGRAM\n\nUS GOVT WASHINGTON DC 4 PM FEB 23  \nCHIEF PETTY OFFICER JOHN R. ELLIS  \nC\/O THE NATIONAL INSTITUTES OF HEALTH  \nWASHINGTON DC\n\nTHE SECRETARY OF THE NAVY REGRETS TO INFORM YOU THAT YOUR FRIEND LIEUT COMMANDER JOHN DOLAN USNR DIED FEBRUARY 21 WHILE ON OVERSEAS SERVICE. FULL DETAILS WILL BE FURNISHED TO YOU WHEN AVAILABLE. YOU WILL BE SHORTLY CONTACTED BY NAVY OFFICIALS WITH REGARD TO YOUR SURVIVORS BENEFITS.\n\nFRANK KNOX, JR\n\nSECRETARY OF THE NAVY\n\n\"Jesus H. Christ!\" Ellis said. \"I guess he didn't have a family.\"\n\n\"It means you get the ten thousand insurance,\" Staley said.\n\nEllis gave him a look of disgust.\n\n\"There's a letter for you, too. Where the Western Union was.\"\n\nThe white envelope bore the neatly typewritten message, \"To Be Delivered to Chief Ellis in the Event of My Demise. Lt. Commander J. B. Dolan, USNR.\"\n\nEllis tore it open. It was undated and short.\n\nDear Chief Ellis:\n\nIt's my professional judgment that one of these Torpex-filled airplanes is going to sooner or later blow up with me in it.\n\nIf you get this, I was right.\n\nNo complaints. It's a lot better way to go than sitting around the Old Sailors' Home waiting for it.\n\nI have a cousin. I never could stand the sonofabitch.\n\nUnless I named you as my beneficiary, he would have gotten the insurance.\n\nHoist one for me, if you think about it some time.\n\nRegards,\n\n_John B. Dolan_\n\nJohn B. Dolan  \nChief Aviation Pilot, USN, Retired  \n(Temporary Lt. Commander, USNR)\n\nEllis folded the letter and put it back in the envelope.\n\n\"What's it say?\" Staley asked.\n\n\"I'm going to catch a shave and put on a fresh uniform,\" Ellis said. \"Then I'm going to go to the Chiefs' Club at the Navy Yard and tie one on. You want to come along?\"\n\n\u2022 \u2022 \u2022\n\n[For a complete list of this author's books click here or visit  \nwww.penguin.com\/griffinchecklist](http:\/\/www.penguin.com\/griffinchecklist?CMP=OTC-GRIFFINCHKLST)\n**Afterword**\n\nReaders who have read others of my books may find this of interest.\n\nI was privileged to know Wendell Fertig, and another reserve officer, commissioned in the Philippines as Fertig was, who also became a guerrilla, Major Ralph Fralick. Fralick and I were good friends. It was my sad honor to deliver his eulogy when he was buried in the National Cemetery in Pensacola.\n\nI have always thought that the Army's refusal to promote Wendell Fertig above the rank of colonel was outrageous. He had more than thirty thousand men under arms, under his command, when the Army returned to Mindanao. Thirty thousand men is just about the strength of two divisions. Major generals command divisions. Three-star lieutenant generals command Army Corps, which are defined as tactical units consisting of two or more divisions.\n\nIf\u2014as far as I'm concerned\u2014Brigadier General Fertig felt any resentment that he had to take the stars from his epaulets and exchange them for the silver eagles of a colonel, once the war was won, he never let it show.\n\nHe was frequently seen at Fort Bragg, North Carolina, in the company of Colonel Arthur \"Bull\" Simon, lending his expertise to the training and doctrine of the Green Berets, and elsewhere around the Army, including Fort Rucker, Alabama, where his friend Colonel Jay D. Vanderpool was in charge of Combat Developments for Army Aviation.\n\nI don't think the story of a bona fide and unquestioned hero like Wendell Fertig can be told too often, and I make no apologies for telling his story in both this book and in _Behind the Lines._ In this book, a fictitious Army officer went ashore on Mindanao from a submarine to establish contact with General Fertig. In _Behind the Lines,_ a fictitious Marine officer did the same thing.\n\nWhen that book was published, I got a somewhat angry telephone call from the editor of the coincidentally titled _Behind the Lines,_ which is sort of a professional magazine for Special Operations people. He asked what I had against Jay Vanderpool. I assured him I had nothing whatever against him; that I flattered myself to think we had been friends, and why the question?\n\n\"You should have said something about him in your book.\"\n\n\"Why?\"\n\n\"You mean you don't know?\"\n\n\"Know what?\"\n\n\"Jay Vanderpool was the guy they sent in by submarine to establish contract with Wendell Fertig.\"\n\nNo. I didn't know.\n\nSorry, Jay. If I had known, I damned well would have said something.\n\nW.E.B. Griffin  \nBuenos Aires, 18 July 1999\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Buenos Aires\n\n# Contents\n\n## Plan Your Trip\n\n### Welcome to Buenos Aires\n\n### Buenos Aires' Top\n\n### What's New\n\n### Need to Know\n\n### Top Itineraries\n\n### If You Like...\n\n### Month by Month\n\n### With Kids\n\n### Eating\n\n### Drinking & Nightlife\n\n### Entertainment\n\n### Tango\n\n### Shopping\n\n### Sports & Activities\n\n## Explore Buenos Aires\n\n### Neighborhoods at a Glance\n\n### The Center\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n### Puerto Madero\n\n####  Sights\n\n####  Eating\n\n####  Entertainment\n\n### Congreso & Tribunales\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n### San Telmo\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n### La Boca\n\n####  Sights\n\n####  Eating\n\n####  Entertainment\n\n####  Shopping\n\n### Retiro\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n### Recoleta & Barrio Norte\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n### Palermo\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Shopping\n\n####  Sports & Activities\n\n### South of Palermo\n\n####  Sights\n\n####  Eating\n\n####  Drinking & Nightlife\n\n####  Entertainment\n\n####  Sports & Activities\n\n### Day Trips from Buenos Aires\n\n### Sleeping\n\n## Understand Buenos Aires\n\n### Buenos Aires Today\n\n### History\n\n### Music\n\n### Literature & Cinema\n\n### Art & Architecture\n\n## Survival Guide\n\n### Transportation\n\n### Directory A\u2013Z\n\n## Special Features\n\n### Neighborhood Walk Through the Heart of BA\n\n### Neighborhood Walk Historical Saunter\n\n### Neighborhood Walk Death, Art & Shopping\n\n### Neighborhood Walk Walking the Green\n\n### Cafes of Buenos Aires\n\n### Fileteado Porte\u00f1o\n\n### The Immortal Evita\n\n### Street Art\nBuenos Aires Maps\n\n###\n\nMap Index\n\n The Center\n\n Puerto Madero\n\n Congreso & Tribunales\n\n San Telmo\n\n La Boca\n\n Recoleta & Barrio Norte\n\n Retiro\n\n Palermo\n\n South of Palermo\nWelcome to  \nBuenos Aires\n\nSexy, alive and supremely confident, this beautiful city gets under your skin. Like Europe with a melancholic twist, Buenos Aires is unforgettable.\n\n###  Tango\n\nArgentina's famous dance is possibly the country's greatest contribution to the outside world, a steamy strut that's been described as 'making love in the vertical position'. Folklore says it began in the bordellos of long-ago Buenos Aires, when men waiting for their 'ladies' passed time by dancing amongst themselves. Today, glamorized tango shows are supremely entertaining with their grand feats of athleticism. You'll also find endless venues for perfecting your moves, from salons to _milongas_ (dance events) to cafes. Just realize that some people become addicted \u2013 and can spend a lifetime perfecting this sensual dance.\n\n###  Food\n\nFancy some Indian, French, Vietnamese, Lebanese, Thai, Scandinavian, Mexican, Brazilian, Peruvian, Armenian or American cuisine during your stay in BA? No problem. But for many travelers, it's the city's carnivorous pleasures that shine. Satisfying a craving for juicy steak isn't hard to do in the land that has perfected grilling those wonderfully flavorful sides of beef. _Parrillas_ (steak\u00adhouses) sit on practically every corner and will offer up myriad cuts, from _bife de chorizo_ (sirloin) to _vacio_ (flank steak) to _ojo de bife_ (rib eye). Vegetarians, you've been warned!\n\n###  Nightlife\n\nTake a cat nap, down your coffee and be prepared to stay up all night \u2013 this is a city that never sleeps! Restaurants open at 9pm, bars at midnight and clubs at 2am \u2013 at the very earliest. If you're cool, of course, you'll show up after 4am and dance till dawn. International DJs are the rage, spinning electronica and house to legions of hip, trendy and well-dressed crowds. But you can also enjoy live music such as rock, blues, jazz and even folk \u2013 just remember that you'll be doing it all very late!\n\n###  Shopping\n\nIt's no joke: Buenos Aires is a shopper's paradise. The city is laced with shopping streets lined with heaps of clothing and shoe stores, leather shops and nearly every\u00adthing else you can think of. Large shopping malls are modern and family-friendly, offering designer goods, food courts and even children's play areas. But perhaps the city's best shopping is in Palermo Viejo, home to countless upscale boutiques; these offer avant-garde clothing, unique housewares and fun souvenirs. And let's not forget San Telmo, where antiques aficionados flock; the Sunday fair here is famous and entertaining, and will easily fill up a long afternoon.\n\nGaler\u00edas Pac\u00edfico (Click here), Microcentro  \nDAN HERRICK \/ GETTY IMAGES \u00a9\n\n###  Why I Love Buenos Aires\n\nBy Sandra Bao, Author\n\nBA is an amazing city, and I'm proud to call it my birthplace. It's an astonishing metropolis that looks a bit like Europe, but with an edgy Latin American twist. _Porte\u00f1os_ (the city's residents) are passionate, opinionated and \u2013 once you get past their hard-city facade \u2013 very friendly. I love walking among them along the busy avenues, taking in the street life and often the craziness that goes along with it. This is a very special place, beautiful in its own unique way, resilient and creative. It's the kind of city travelers fall in love with, dream about and then move to!\n\nSee authors for much more.\nBuenos Aires' Top 10\n\nCementerio de la Recoleta\n\nTRAVEL IMAGES\/UIG \/ GETTY IMAGES \u00a9\n\n1 Meander through the maze of narrow lanes lined with elaborate mausoleums in what must be the world's most ostentatious necro polis. This 'city of the dead' was BA's first pub lic cemetery, though it quickly became exclusive; Argentina's most illustrious historical figures are buried here, including Ev a Per\u00f3n ('Evita'). Myriad styles decorate the crumbling tombs, from art nouveau and art deco to neoclassical and neo-Gothic. There are also wonderfully flamboyant statues to discover, so pay your respects to Evita before getting lost among the marble angels.\n\n  Recoleta & Barrio Norte\n\n## Buenos Aires' Top 10\n\nTango Show\n\nJUPITERIMAGES \/ GETTY IMAGES \u00a9\n\n2 Nothing captures the essence of Buenos Aires like the sensual and melancholy tango, and no visit to the city is complete without experiencing tango in some form. Watch it in a San Telmo or La Boca street fair, see a slick show at a theater or join a _milonga_ (dance event) at dozens of venues. Tango classes are often held before _milongas,_ so take part \u2013 or just watch, but don't look too long at that handsome stranger across the room; a stare is an invitation to dance, and you could be breaking some hearts.\n\n  Tango\n\n## Buenos Aires' Top 10\n\nFootball Game\n\nCrowd at a match between Boca Juniors and River Plate at La Bombonera Stadium  \nLONELY PLANET \/ GETTY IMAGES \u00a9\n\n3 In Buenos Aires, _f\u00fatbol_ isn't just a game. The national pastime inspires near-religious passion in _porte\u00f1os_ , clearing the streets and sending spectators into fits of ecstasy and anguish as they huddle around TV screens or brave the explosive stadium crowds. The atmosphere is particularly boisterous (read: out of control) when River Plate and arch-rivals Boca Juniors face off during the much-anticipated _supercl\u00e1sico_ games. The tension is palpable, and for two hours on a Sunday afternoon here, nothing else really matters.\n\n  Sports & Activities\n\n## Buenos Aires' Top 10\n\nSteak Dinner\n\nJILL SCHNEIDER \/ GETTY IMAGES \u00a9\n\n4 Believe the hype: Argentine beef is some of the best in the world. Eat, drink and be merry at one of BA's hundreds of _parrillas_ (steakhouses), where a leisurely meal begins with waiters pouring Malbec and carving generous slabs of prime beef. _Parrillas_ run the gamut from neighborhood joints to classic establishments to upscale restaurants, so there's a price for every pocket. One thing is certain: expect some of the best meat you've ever eaten.\n\n  Eating\n\n## Buenos Aires' Top 10\n\nPlaza de Mayo\n\nCHAD EHLERS \/ GETTY IMAGES \u00a9\n\n5 Founded in 1580, Plaza de Mayo is the stage on which many of the dramatic events in Argentina's history were played out, from military bombings in 1955 to Evita's emotional speeches to massive union demonstrations (still going today). Most of the time, however, it's a peaceful place where families feed pigeons and the odd pickpocket makes off with a tourist's camera (stay sharp and you'll be OK!). If you're here on a Thursday afternoon, you might witness Las Madres de Plaza de Mayo: mothers peacefully marching for social-justice causes.\n\n  The Center\n\n## Buenos Aires' Top 10\n\nSan Telmo Stroll\n\nBRENT WINEBRENNER \/ GETTY IMAGES \u00a9\n\n6  The neighborhood of San Telmo is a beguiling mix of faded grandeur and bohemian spirit. The elegant belle-epoque architecture and crumbling villas are throwbacks to the district's 19th-century heyday, before yellow fever and cholera sent the aristocratic masses to higher ground. Today, you can wander along Defensa or Balcarce streets toward leafy Parque Lezama, taking in picturesque vistas of romantic facades and drooping balconies as you window-shop for antiques. There's definitely been gentrification (Starbucks has discovered San Telmo!), but much of the old-world atmosphere remains.\n\n  San Telmo\n\n## Buenos Aires' Top 10\n\nCycling in Palermo's Parks\n\nParque 3 de Febrero (Bosques de Palermo)  \nTRAVELSTOCK44 \/ ALAMY \u00a9\n\n7  Buenos Aires could hardly be called bike-friendly, but things are slowly changing. Bike lanes now exist on some avenues, there's a bike-sharing program and Critical Mass (where cyclists take over certain streets in a semi-organized ride) happens every month. For the traveler, a bike ride around Buenos Aires \u2013 especially in Palermo's green parks \u2013 is a great way to experience cycling in this big, vehicle-\u00addominated city. Here, miles of safe bike lanes exist, and you can enjoy the green grass instead of the gray concrete.\n\n  Palermo\n\n## Buenos Aires' Top 10\n\nEl Caminito\n\nHIROSHI HIGUCHI \/ GETTY IMAGES \u00a9\n\n8 Rough-and-tumble La Boca, with its brightly painted metal houses, was the city's principal port until Puerto Madero was created at the close of the 19th century. El Caminito ('the little walkway') is a curved pedestrian street lined with artists selling paintings and other creations. It has the air of a tourist trap, especially with the numerous souvenir shops nearby, but it's fun nevertheless. For something more upscale, head to Fundaci\u00f3n Proa (Click here), one of BA's fanciest art galleries, across from the (rather strongly scented) Riachuelo River.\n\n  La Boca\n\n## Buenos Aires' Top 10\n\nMuseo de Arte Latinoamericano de Buenos Aires (Malba)\n\nDAVID CHEREPUSCHAK \/ ALAMY \u00a9\n\n9 The vibrant Malba is a must, both for its brilliant modern Latin American and Argentine art, and for the stunning contemporary building that houses it. This significant collection of Argentine real- estate tycoon Eduardo Costantini spans the modernist, avant-garde, surrealist and abstract movements and includes work by Frida Kahlo and Diego Rivera. There are also temporary international art exhibits and a very fine restaurant-cafe with a lovely patio.\n\n  Palermo\n\n## Buenos Aires' Top 10\n\nFeria de Mataderos\n\nLONELY PLANET \/ GETTY IMAGES \u00a9\n\n10 Folk music emanates from the outdoor stage, local couples take to the streets to perform traditional _chacarera_ and _chamam\u00e9_ (folk dances and music), food stalls dish out hearty country dishes such as _locro_ (a meaty stew), deep-friend _empanadas_ and _humitas_ (a kind of tamale) \u2013 this is Feria de Mataderos, an authentic celebration of Argentine country traditions. You might also see gauchos demonstrating their horse-riding skills by playing _sortija,_ a game where they stand in their saddles and ride at full speed to spear a tiny dangling ring. Catch it on weekends during the summer.\n\n  South of Palermo\nWhat's New\n\n######  Pop-Up Restaurants\n\nThese one-time eating events are ideal for experimental chefs. Among others, look for GAJO (www.facebook.com\/gajococina).\n\n######  Metrobus & Bike Lanes\n\nA new bus rapid transit system has improved traffic flow and provided faster service. Meanwhile, expanded bike lanes and a free bike-share program have encouraged two-wheeled recreation and commuting.\n\n###### Faena Arts Center\n\nPuerto Madero's newest darling is this art space \u2013 housed in an old flour mill \u2013 that's meant to highlight gigantic, larger-than-life installations.\n\n######  Food Festivals\n\nPopular new food fairs include Feria Masticar (www.feriamasticar.com.ar), put on by some of BA's most famous chefs, and Feria Ra\u00edz (www.facebook.com\/Raiz FestivalGastronomico), an Argentine-food celebration.\n\n###### New Concert Space\n\nThe Centro Cultural del Bicentenario, located in the beaux arts post office building, wasn't open at research time, but when it is it'll seat 2000 spectators.\n\n###### Mercado Azul\n\nBuenos Aires' _mercado azul_ (literally 'blue market') for US dollars offers nearly twice the number of pesos over the official rate. The downsides are scams, fake bills and fly-by-night _cambios_ (unofficial exchange houses). Some hotels, restaurants and shops will give discounts for US-dollar transactions and sometimes change the dollars themselves.\n\n######  Molecular Gastronomy\n\nIn this interesting style of cuisine, multiple courses of just a few bites each are served to diners who want a unique food experience that encompasses taste combinations, textures and unforgettable visual appeal.\n\n###### Usina del Arte\n\nLa Boca isn't known for upscale buildings, but this gorgeous new music venue, located in a remodeled electricity factory, is meant to start a gentrifying process in one of BA's shadiest hoods.\n\n###### Villa Crespo\n\nThis neighborhood just south of Palermo has gained popularity as an affordable home for creative new restaurants, hotels and shops.\n\nFor more recommendations and reviews, see lonelyplanet.com\/buenos-aires\nNeed to Know\n\n###### For more information, see Survival Guide\n\n######  Currency\n\nPeso (AR$), though some upscale hotels quote in US dollars (US$).\n\n######  Language\n\nSpanish\n\n######  Visas\n\nGenerally not required for US, Canadian, Australian, New Zealand and most Western European citizens. However, Americans, Canadians and Australians must pay a significant 'reciprocity fee' to enter Argentina.\n\n######  Money\n\nATMs and exchange houses are common downtown. Major credit cards are generally accepted, though travelers checks are not.\n\n######  Mobile Phones\n\nNearly everyone in Buenos Aires uses a mobile phone. Local SIM chips can be bought for unlocked international phones, and credit added as needed.\n\n######  Time\n\nArgentina is three hours behind GMT and generally does not observe daylight-saving time, but this situation may change.\n\n######  Tourist Information\n\nTourist information offices and kiosks are dotted across the city, especially in neighborhoods popular with tourists.\n\n###  WHEN TO GO\n\nSpring (September to November) and fall (March to May) are the best temperature-wise; winter (June to August) is cold but not freezing.\n\n###  Daily Costs\n\n######  Budget: Less than US$60\n\n\u00bb Dorm bed: US$15\n\n\u00bb Double room in budget hotel: US$80\n\n\u00bb Cheap main dish: US$8\n\n######  Midrange: US$60-$120\n\n\u00bb Three-star hotel room: US$100-175\n\n\u00bb Average main dish: US$10-15\n\n\u00bb Museum admission: US$1-5\n\n######  Top End:\n\n######  US$120 and up\n\n\u00bb Five-star hotel room: US$200\n\n\u00bb Fine main dish: US$15-20\n\n\u00bb Taxi trip across town: US$15\n\n###  Advance Planning\n\nTwo months before Book accommodations if traveling during busy times and if your hotel is popular.\n\nOne month before Check reviews for tours or activities and decide on a schedule; make reservations if necessary.\n\nOne week before Pack smart, comfortable clothing. _Porte\u00f1os_ are a well-dressed lot, and you'll stick out as a tourist in loud shirt, shorts and flip-flops.\n\n###  Useful Websites\n\nArgentina Independent (www.argentinaindependent.com) Current affairs and culture, plus events listing and musings about expat life.\n\nArgentine Post (www.argentinepost.com) Useful wide-ranging articles on BA and Argentina.\n\nPick up the Fork (www.pickupthefork.com) Restaurant and bar reviews, plus where to shop for ingredients.\n\nLonely Planet (www.lonelyplanet.com) Forums, travel news, updates and more.\n\n###  Arriving in Buenos Aires\n\nEzeiza airport (EZE; officially Aeropuerto Internacional Ministro Pistarini) Shuttle services are a good way to get downtown (AR$80); the transport booth area is beyond customs and has a couple of companies with frequent shuttles. Penny pinchers can take a local bus (AR$6), which takes two hours to get downtown.\n\nFor taxis, go outside the transport booth area (taxis are overpriced here) to the reception area. Avoid taxi touts like the plague; a few steps beyond, find the city's official taxi stand (a blue sign says 'Taxi Ezeiza'). In late 2013 it charged AR$270 to the center, including tolls.\n\nAeroparque airport (officially Aeroparque Jorge Newbery) Close to downtown and easily accessible by local bus or taxi.\n\nSee arrival for much more.\n\n###  Getting Around\n\nDespite Buenos Aires' heavy traffic, the city's public-transportation system is efficient and usually a better way to get around than driving.\n\n\u00bb Bus The city has hundreds of bus lines that can take you within a few blocks of any destination. Sorting it all out is another matter; buy a pocket bus guide or use the 'city buses' link at www.omnilineas.com.\n\n\u00bb Subte BA's underground, or subway, is not difficult to figure out and a quick way to get around \u2013 though it gets hot and very crowded during rush hour.\n\n\u00bb Taxi Black and yellow street taxis are ubiquitous and generally fine, though some people prefer _remises_ (radio taxis) for more safety.\n\nSee getting around for much more.\n\n###  Sleeping\n\nBuenos Aires has a very wide range of accommodations, including hostels, guesthouses, B&Bs, homestays, apartments and hotels of all stripes and budgets. Services range widely; some hostels' private rooms are nicer than many budget hotels', and can cost more. Boutique hotels are a dime a dozen in Palermo, while five-star luxury is easily found in the Retiro and Recoleta neighborhoods.\n\nNovember through February are busy times, so book ahead if your hotel is a popular one.\n\n######  Useful Websites\n\n\u00bb Oasis Collections (www.oasisba.com) The cream of the crop for luxury stays.\n\n\u00bb Spare Rooms Buenos Aires (www.spareroomsba.com) When you need just a room.\n\n\u00bb San Telmo Loft (www.santelmoloft.com) A handful of artsy vacation rentals in San Telmo.\n\n\u00bb Craigslist (<http:\/\/buenosaires.en.craigslist.org>) Endless listings for rooms, sublets, apartments, houses.\n\nSee sleeping for much more.\nTop Itineraries\n\n##  Day One\n\n###### The Center\n\n  From leafy Plaza San Mart\u00edn, walk south on pedestrian Florida and experience masses of people shopping, busking, selling or just power-walking to their next destination. You'll eventually come within a block of Plaza de Mayo, the heart of Buenos Aires. This historic plaza is surrounded by Casa Rosada, Catedral Metropolitana and Museo del Cabildo.\n\n  Lunch Outside seating and fresh offerings at the Central Market (Click here).\n\n###### Puerto Madero\n\n  Lined with renovated old brick warehouses, Puerto Madero is replete with fancy lofts and apartment towers, plus some of the city's most expensive (and some say overpriced) restaurants. It's a very scenic and pleasantly vehicle-free place to stroll the cobbled paths along the dikes. Art lovers shouldn't miss Colecci\u00f3n de Arte Amalia Lacroze de Fortabat, highlighting the collection of Argentina's wealthiest woman. For a shot of nature, visit Reserva Ecol\u00f3gica Costanera Sur, a land-filled wetlands.\n\n  Dinner Casual and excellent Peruvian can be had at Chan Chan (Click here).\n\n###### Congreso & Tribunales\n\n  Take in an opera, ballet or classical-music show at Teatro Col\u00f3n, Buenos Aires' premier theater. The traditional entertainment district of Av Corrientes still hops these days, showcasing many films, art events and plays.\n\n## Itineraries\n\n##  Day Two\n\n###### San Telmo\n\n  Stroll through this colonial neighborhood's cobbled streets and window-shop for antiques. Come on a Sunday, when the famous Feria de San Telmo street market takes over the neighborhood. Book a tour to the stunning El Zanj\u00f3n de Granados for a peek into the city's origins.\n\n  Lunch A sidewalk table at Bar Plaza Dorrego (Click here) for great people-watching.\n\n###### La Boca\n\n  The colorful corrugated houses along El Caminito are indeed photo\u00adgenic, though this area is a bit of a tourist trap. It's still fun, however; check out the souvenir shops and artists' paintings and perhaps catch a street-tango show. Art lovers shouldn't miss Fundaci\u00f3n Proa, a cutting-edge gallery, while soccer fans can head to La Bombonera stadium and visit the Boca team's Museo de la Pasi\u00f3n Boquense. But whatever you do, don't stray too far from the tourist hordes; La Boca is at times a rough neighborhood.\n\n  Dinner Try exceptional international cuisine at Caf\u00e9 San Juan (Click here).\n\n###### San Telmo\n\n  Catch a tango show at El Viejo Almac\u00e9n, a long-running venue. Or go drinking at one of the many great watering holes in the area, from Gibraltar (an English pub) to Doppelg\u00e4nger (an upscale cocktail bar).\n\n## Itineraries\n\n##  Day Three\n\n###### Recoleta & Barrio Norte\n\n  Explore Recoleta's famous cemetery; you can wander for hours among the crumbling sacrophagi and marble angels. It's a veritable city of the dead, fascinating and mysterious; to seek out Evita's tomb, just follow everyone else.\n\n  Lunch _Empanadas_ at El Sanjuanino (Click here), one of the few cheap eateries.\n\n###### Recoleta & Barrio Norte\n\n  Check out the Museo Nacional de Bellas Artes, Argentina's top classic arts museum. Just north is the cool sculpture _Floralis Gen\u00e9rica_ , a giant metal flower whose petals open during the day and close at night (when the gears are working!). And if you've got the bucks, the city's most expensive boutiques are along Av Alvear, worth a stroll to also eyeball some huge old mansions. Visit Museo de Arte Hispanoamericano Isaac Fern\u00e1ndez Blanco and Palacio Paz, in nearby Retiro, if you like gorgeous palaces filled with antiques.\n\n  Dinner Looking for the locals? Then head to classic Rodi Bar (Click here).\n\n###### Recoleta & Barrio Norte\n\n  Time to drink up: Mili\u00f3n is a bar-restaurant in a beautiful old mansion, Casa Bar is a sports pub popular with expats, and La Biela is a traditional cafe-restaurant where the upper classes loiter over lattes.\n\n## Itineraries\n\n##  Day Four\n\n###### Palermo\n\n  Walk (or take a bike ride) along Palermo's Parque 3 de Febrero, where you can also visit a zoo, botanical garden and Japanese garden. Bike paths are laced throughout, and on Sunday the ring road around the Rose Garden is closed to vehicles.\n\n  Lunch Malba's (Click here) cafe-restaurant has a great patio for sunny days.\n\n###### Palermo\n\n  Visit Malba, a beautiful and contemporary art museum showcasing the collection of art patron Eduardo F Costantini. The Museo Nacional de Arte Decorativo is another must-see; it's a beaux arts mansion that once belonged to a Chilean aristocrat, and is full of his posh belongings. Finally, Evita fans can't miss Museo Evita, which chronicles the life of Argentina's most internationally famous woman.\n\n  Dinner For an excellent steakhouse, get a table at Don Julio (Click here).\n\n###### Palermo\n\n  Palermo is nightlife central. There are dozens of bars to check out, and people come from all over to dance at the famous clubs here. For outdoor action, head to Plaza Serrano, which is surrounded by restaurants and bars, along with dozens of their outdoor tables; it's _the_ place to see and be seen.\nIf You Like...\n\n####  Museums\n\nMuseo de Arte Latinoamericano de Buenos Aires Gorgeous, glassy art museum showcasing the private collection of art patron Eduardo F Costantini.\n\nMuseo Nacional de Bellas Artes From European impressionists to Latin American maestros, this national art museum covers them all.\n\n Museo Nacional de Arte Decorativo Beautiful beaux arts mansion strewn with the posh belongings of a Chilean aristocrat.\n\n Colecci\u00f3n de Arte Amalia Lacroze de Fortabat See what the collection of Argentina's wealthiest woman has to offer.\n\n Fundaci\u00f3n Proa Cutting-edge gallery-museum with contemporary art exhibits, plus a rooftop cafe with a view of La Boca.\n\n Palacio Paz Gorgeous European palace with ornate rooms, salons and guilded details.\n\nHistoric Places\n\n Plaza de Mayo Buenos Aires' original main square, dating to the 1580s and surrounded by significant buildings.\n\n El Zanj\u00f3n de Granados Beautifully renovated, underground architectural site of the city's first settlements.\n\n Plaza San Mart\u00edn Pleasant leafy park that was once home to Spanish governors, slave quarters, a bullring and a battlefield.\n\n Parque Lezama Considered to be the very spot where Buenos Aires was founded, back in 1536.\n\n Manzana de las Luces Taking up a whole city block, this was BA's most important center of culture and learning during colonial times.\n\nGreen Spaces\n\n Reserva Ecol\u00f3gica Costanera Sur Low-lying, 350-hectare landfill site that's become a haven for wildlife and nature-seekers.\n\n Parque 3 de Febrero Laced with miles of bike trails, this large green park also has a rose garden, planetarium and small lakes.\n\n Jard\u00edn Japon\u00e9s Tidy green oasis of tranquility in Palermo, complete with sushi restaurant and cultural offerings.\n\n Jard\u00edn Zool\u00f3gico A decent big-city zoo with lawns, lakes, trees and many natural enclosures for the critters.\n\n Jard\u00edn Bot\u00e1nico Carlos Thays Surrounded by busy avenues, this modest botanical garden offers a peaceful break from the city.\n\nFree Stuff\n\n Cementerio de la Recoleta Buenos Aires' most popular tourist attraction and a must-visit for its amazing tombs and statues.\n\n Museo Nacional de Bellas Artes Spend an afternoon at this large and excellent national art museum.\n\nStreet Fairs BA has several; be sure to hit San Telmo's Sunday fair Click here, and the one in Mataderos if you can get out there. Click here\n\nStreet Tango See these (donation) tango shows at the San Telmo Sunday fair Click here and on El Caminito in La Boca. Click here\n\n Reserva Ecol\u00f3gica Costanera Sur Marshy lands located in Puerto Madero near the city center, but miles away in atmosphere.\n\n Cementerio de la Chacarita A larger, less flashy, less accessible and less touristed version of Recoleta's cemetery.\n\nFree City Tour Locals who love their city offer free walking tours in English (tips appreciated but not mandatory; www.bafreetour.com).\n\nUnusual Tours\n\nBiking Buenos Aires Pedal around Palermo's parks and on bike lanes; a fun and easy way to tour the city (www.bikingbuenos\u00adaires.com).\n\nGraffitimundo See Buenos Aires through its colorful and dynamic street-art scene (www.graffitimundo.com).\n\nFoto Ruta Unique and self-guided tour via photographing clues around BA's neighborhoods (www.foto-ruta.com).\n\nThe Man Tour Smoke a cigar, get a straight razor shave and shop for a handmade hat (www.landingpadba.com\/the-man-tour-buenos-aires).\n\nParilla Tour Explore the city's off-the-beaten-track _parrillas_ (steakhouses) and learn about Argentina's food and culture (www.parrillatour.com).\n\nNarrative Tango Tour Get the scoop on tango via classes, _milongas_ and shows (www.narrativetangotours.com).\n\nUrban Running Tour Run around BA (literally) with a guide; they'll adapt to your pace (www.urbanrunningtours.com.ar).\n\nThe Offbeat\n\n Tierra Santa Visit this kitschy 'world's first religious theme park' and witness the resurrection every half-hour.\n\n Museo del Patrimonio Aguas Argentinas Pretty tiles, ceramic pipes and old bidets and toilets are highlighted at this small, quirky museum.\n\n Learn Polo Not many travelers can say they've hopped on a horse and learned to play polo while on vacation.\n\n Closed-Door Restaurants The menu is fixed, as is the dinner time, and you won't get the address until you book \u2013 so why are they so popular?\n\n Museo de la Polic\u00eda Federal Exhibits on cockfighting, drug paraphernalia and hacked-up murder victims \u2013 only at the Police Museum.\n\nPato Match Like weird sports? Then try watching this one (in December's Palermo championship), originally played on horseback by gauchos.\n\nFor more top Buenos Aires spots, see the following:\n\n\u00bb Eating\n\n\u00bb Drinking & Nightlife\n\n\u00bb Entertainment\n\n\u00bb Shopping\n\n\u00bb Sports & Activities\n\nElephants at the Jard\u00edn Zool\u00f3gico, Palermo (Click here)  \nKRZYSZTOF DYDYNSKI \/ GETTY IMAGES \u00a9\nMonth by Month\n\n###  TOP EVENTS\n\nFestival y Mundial de Tango August\n\nLa Rural July\n\nCreamfields November\n\nVinos y Bodegas September\n\nCampeonato Abierto Argentino de Polo December\n\n##  February\n\nIt's still summer, but vacationing _porte\u00f1os_ __start their return home. There are plenty of tourists in the city, some passing through on their way to or from Patagonia.\n\n######    Carnaval\n\nUsually occurring in late February, BA's Carnaval is a small affair compared to Rio's or Bahia's, but it's still lots of fun. Catch some Brazilian-flavored _murga_ groups (traditional Carnaval ensembles), with dancing and drumming in different neighborhoods around the city.\n\n######    Chinese New Year\n\nYes, Buenos Aires has a Chinatown, but it's only about four blocks long \u2013 on Arribe\u00f1os street in Belgrano. During Chinese New Year, expect plenty of food, firecrackers and festivities. Dates vary depending on the lunar calendar.\n\n######    Buenos Aires Fashion Week\n\nFour days of clothing stalls and catwalk action show off the city's latest threads and their makers. It takes place at Palermo's La Rural in February (fall\/winter collections) and August (spring\/summer collections). Expect plenty of beautiful people \u2013 includin g models, of course. (www.bafweek.com.ar)\n\n##  April\n\nIt's fall in BA, and one of the best times to visit \u2013 but always be prepared for a downpour. There are still plenty of activities as the city heads into low season.\n\n######    Feria Internacional del Libro\n\nBA's annual book fair attracts tens of thousands of book lovers for three weeks in April and May. Famous authors do readings and sign books, while publishers hawk their wares. Look for it at the La Rural building in Palermo. (www.el-libro.org.ar)\n\n######    Festival Internacional de Cine Independiente\n\nThis independent film festival highlights national and international films, with awards given out in separate categories. Over 100 films are screened, with a main venue being the Abasto shopping mall. (www.bafici.gov.ar)\n\n##  May\n\nLate autumn has hit and it's pleasantly cool as the rains die back a bit. Look for travel deals as low season starts in earnest.\n\n######    Arte Ba\n\nArte Ba features exhibitions from hundreds of art galleries, dealers and organizations, with both national and international contemporary art on display. Conferences, presentations and discussions make the rounds, while young new artists get exposure. It's at Palermo's La Rural building. (www.arteba.com)\n\n##  July\n\nIt's high winter, so be prepared with warm layers. Locals who can afford it head to the ski slopes down south.\n\n######    Exposici\u00f3n de Ganader\u00eda, Agricultura e Industria Internacional (La Rural)\n\nThe mother of all livestock fairs, where prize cows, sheep, goats, horses and \u2013 especially \u2013 bulls, all strut their stuff. Gaucho shows provide entertainment. It takes place for two weeks in late July at Palermo's La Rural building. (www.exposicionrural.com.ar)\n\n##  August\n\nIt's still cold, so keep those layers on, but it's also a great time to explore the city's theaters, museums and art galleries.\n\n######    Festival y Mundial de Tango\n\nTaking place in mid-August, this two-week-long tango festival offers a great way to see some of the country's best tango dancers and musicians do their thing. Plenty of competitions, classes and workshops take place. (www.tangobuenosaires.gob.ar)\n\n##  September\n\nSpring has sprung and it's a lovely time to be in BA. Polo season begins and the tourists start returning.\n\n######    Vinos y Bodegas\n\nA can't-miss event for wine aficionados, with vintages from dozens of Argentine _bodegas_ (wineries). Mix with thousands of sommeliers, restaurateurs, journalists and general wine-lovers at Palermo's La Rural building. Expect cooking demonstrations and live music too. (www.expovinosybodegas.com.ar)\n\n##  November\n\nIt's pretty darn near perfect weather in BA, and the jacaranda trees are showing off their gorgeous purple blooms. High season has arrived, so reserve your accommodation ahead.\n\n######    Marcha del Orgullo Gay\n\nIt's nothing like San Francisco's or Sydney's, but BA has its own gay pride march. Each year on the first Saturday in November, thousands of BA's gays, lesbians, transgenders and more strut their way through the city's center. (www.marchadelorgullo.org.ar)\n\n######    Creamfields\n\nBuenos Aires' premier electronic-music festival happens in early November. Tens of thousands of ravers party non-stop for hours on end, with famous local and international DJs spinning their best. (www.creamfieldsba.com)\n\n######    D\u00eda de la Tradici\u00f3n\n\nThe closest thing to authentic gaucho culture you'll probably ever witness, with traditional foods, feats of horsemanship, and folk music and dancing. It happens in San Antonio de Areco, a day trip from BA; call the Areco tourist office (www.arecoturismo.com.ar) for exact dates, which vary yearly. If you miss it, head to Feria de Mataderos, a weekly street fair outside BA's center.\n\n##  December\n\nSummer in BA means hot and humid temperatures, and many _porte\u00f1os_ head to the coast. There's still plenty going on in the city, however.\n\n######    Campeonato Abierto Argentino de Polo\n\nArgentina boasts the world's best polo, and the Abierto is the world's premier polo event. It takes place at Palermo's Campo Argentino de Polo. For exact dates and details, contact the Asociaci\u00f3n Argentina de Polo (www.aapolo.com).\n\n######    Buenos Aires Jazz Festival Internacional\n\nThis jazz festival takes place over five days in venues all over the city. Jazz musicians of all kinds are featured \u2013 emerging and established, avant-garde and traditional, national and international. Concerts and films also take place. (www.buenosairesjazz.gob.ar)\nWith Kids\n\nAlthough it's a megalopolis, BA is remarkably child-friendly. On sunny weekends Palermo's parks bustle with families taking walks and picnicking, while shopping malls fill with strollers. Zoos, museums and theme parks are also popular destinations \u2013 and don't forget those fun street fairs!\n\n####  Eating & Sleeping\n\nMany restaurants welcome kids, but if a place looks a bit too fancy, ask if they take children. And most offer a wide selection of food suitable for kids (like pizza, pasta, meats and vegetables); a few even have children's menus. Waiters are accustomed to providing extra plates and cutlery for little tykes, though you may not always find booster seats or high chairs.\n\nNote that Buenos Aires is a very late-night city; most restaurants don't open until 9pm, so you'll likely have to adjust your timetable during your travels here.\n\nDon't forget to take the kids out for ice cream \u2013 it's a real Argentine treat (Click here). Other local sweets to try include _alfajores_ (sandwich cookies usually covered in chocolate, available at corner stores) and _dulce de leche_ (a milk caramel often used in desserts).\n\nSmall boutique hotels, hostels or guesthouses are sometimes not the best places for rambunctious kids, but most hotels accept them. Some hotel rooms come with kitchenettes; apartment rentals are another good option (Click here).\n\n####  In Public\n\nOnce children are old enough to cross the street safely and find their way back home, _porte\u00f1o_ parents will often send unaccompanied pre-adolescents on errands or on visits to friends or neighbors. This is also a country where people frequently touch each other, so your children may be patted on the head by friendly strangers. In general, you can count on your children's safety in public places, though it's always a good idea to keep an eye on them.\n\n_Porte\u00f1os_ can be helpful on public transportation. Often someone will give up a seat for a parent and young child. Baby strollers on the crowded and uneven sidewalks of BA's downtown center are a liability, however; consider a baby carrier instead.\n\nPoorly maintained public bathrooms lacking baby-changing facilities or counter-\u00adtop space are common. Always carry toilet paper and wet wipes.\n\n####  Green Spaces\n\nBuenos Aires has numerous plazas and public parks, many with playgrounds, and these are always popular gathering spots for families. If you're downtown and need a nature break, try the Reserva Ecol\u00f3gica Costanera Sur (Click here), a large nature preserve with good birdwatching, pleasant dirt paths and no vehicles; bike rentals are sometimes available on summer weekends.\n\nUp north, the most attractive green spots are the wide open spaces of Palermo, especially Parque 3 de Febrero (Click here). This huge park has a planetarium, a Japanese Garden and a nearby zoo. And on Sunday vehicular traffic isn't allowed on the ring road around the rose garden; you can rent bikes, boats and inline skates and range freely without worrying about cars!\n\n####  Fun Museums\n\nMake sure to visit the Museo Participativo de Ciencias (Click here) in the Centro Cultural Recoleta. This science museum has interactive displays that focus on fun learning \u2013 signs say _'prohibido no tocar'_ (not touching is forbidden). And in San Telmo, the Museo Argentino del T\u00edtere (  4307-6917; www.museoargdeltitere.com.ar; Estados Unidos 802; admission free;   9:30am-12:30pm & 3-6pm Tue, Wed & Fri, 3-6pm Thu, Sat & Sun) is a small puppet museum with a fascinating collection of international and Argentine puppets, but it's the inexpensive shows that will amuse the little ones. Call beforehand to get hours and show times, as they vary widely.\n\nOutside the center in Caballito is the Museo Argentino de Ciencias Naturales (Click here), with myriad rooms containing giant dinosaur bones, dainty seashells, scary insects and amusing stuffed animals and birds.\n\n####  Animals\n\nAbout a 45-minute drive outside the city, in Escobar, is the exceptional zoo Parque Temaik\u00e9n (  034-8843-6900; www.temaiken.com.ar; RP 25, Km 1, Escobar; adult\/child 3-10 AR$82\/64;   10am-7pm Tue-Sun Dec-Feb, to 6pm Mar-Nov). Only the most charming animal species (like meerkats and tigers) are on display here, roaming freely around spacious natural enclosures. The beautiful grounds are tidy and park-like, and exceptional exhibits include a butterfly house, a fine aquarium and a large aviary (with parrots and toucans galore). Inter\u00adactive areas provide mental stimulation, and services include stroller rentals, gift stores and restaurants. Just outside Temaik\u00e9n is a large playground run by Helader\u00eda Munchi. Tuesday admission is half-off (unless it's a holiday).\n\nFor something much closer in, head to Palermo's Jard\u00edn Zool\u00f3gico (Click here). It's a fairly pleasant zoo for Latin America, and on sunny weekends it fills with families.\n\n####  Playgrounds\n\nSome of BA's outdoor parks have playgrounds, always popular with families. However, many large modern shopping malls have indoor playgrounds (often on the top floor), along with video arcades, multiplexes and toy shops. Paseo Alcorta (Click here) has plenty of mechanical rides next to the large food court, while Mercado de Abasto (Click here) boasts a full-blown 'Museo de los Ni\u00f1os' (more like a playground than a museum) where kids enter a miniature city complete with post office, hospital and even TV station. Abasto also has a mini-amusement park. On rainy days, these are great places to be with little ones.\n\n####  Amusement Parks\n\nHeading to Tigre (Click here), just north of the center, makes a great day excursion. Hop on the fun Tren de la Costa to get there; it ends at Parque de la Costa (Click here), a typical amusement park with rides and activities.\n\nChristian parents might want to take the kids to Tierra Santa (  4784-9551; www.tierrasanta-bsas.com.ar; Av Costanera R Obligado 5790; admission AR$40;   call for hours), a religious theme park unlike anywhere you've ever been. Not far away is Parque Norte (  4787-1382; www.parquenorte.com; Avs Cantilo & Guiraldes; admission AR$70 Mon-Fri, AR$90 Sat, AR$100 Sun;   9am-8pm Mon-Fri, 8am-10pm Sat & Sun), a large water park that's perfect on a hot day.\n\nEl Caminito, La Boca (Click here)  \nDEMETRIO CARRASCO \/ GETTY IMAGES \u00a9\n Eating\n\nArgentines take barbecuing to heights you cannot imagine. Their best pizzas and pastas vie with those of New York and Naples. They make fabulously tasty wines and impossibly delectable ice cream. And ethnic cuisine is rampant in Buenos Aires. In fact, you'll eat so well here that you'll need to power-walk between lunch and dinner to work off the excess calories.\n\nStreet asado (Argentine barbecue) seller  \nMICHAEL TAYLOR \/ GETTY IMAGES \u00a9\n\n######  Staples & Specialties\n\n######  BEEF\n\nArgentines have perfected the art of grilling beef on the _asado_ (barbecue). This involves cooking with coals and using only salt to prepare the meat. On the grill itself, slanted runners funnel the excess fat to the sides, and an adjustable height system directs the perfect amount of heat to the meat. The _asado_ is a family institution, often taking place on Sunday in the backyards of houses all over the country.\n\nA traditional _parrillada_ (mixed grill) is a common preparation at _parrillas_ (steakhouses) and offers a little bit of everything. Expect _chorip\u00e1n_ (a sausage appetizer), _pollo_ (chicken), _costillas_ (ribs) and _carne_ (beef). It can also come with more exotic items such as _chinchulines_ (small intestines), _mojellas_ (sweetbreads) and _morcilla_ (blood sausage).\n\nCommon steak cuts:\n\n\u00bb Bife de chorizo Sirloin; a popular thick and juicy cut.\n\n\u00bb Bife de costilla T-bone or Porterhouse steak.\n\n\u00bb Bife de lomo Tenderloin; a tender though less flavorful piece.\n\n\u00bb Cuadril Rump steak; often a thin cut.\n\n\u00bb Ojo de bife Ribeye; a choice smaller morsel.\n\n\u00bb Tira de asado Short ribs; thin, crispy strips of ribs.\n\n\u00bb Vac\u00edo Flank steak; textured, chewy and flavorful.\n\nIf you don't specify how you want your steak cooked, it will come _a punto_ (medium to well done). Getting a steak medium rare or rare is harder than you'd imagine. If you want some pink in the center, order it _jugoso;_ if you like it truly rare, try _vuelta y vuelta_.\n\nDon't miss _chimichurri,_ a tasty sauce made with olive oil, garlic and parsley \u2013 it adds a tantalizing spiciness. Occasionally you can also get _salsa criolla,_ a condiment made of diced tomatoes, onion and parsley.\n\n###  STEAK \u2013 OUTSIDE THE BOX\n\nGoing to a _parrilla_ is probably on every BA visitor's to-do list, but if you want to eat meat in a different way, try these options:\n\nAdentro (www.adentrodinnerclub.com) At this _puerta cerrada_ (closed-door restaurant) it's like being at a good friend's _asado._ You'll get stuffed on juicy _empanadas_ , delicious grilled shrimp and veggies, and then some amazing meat.\n\nArgentine Experience (www.theargentineexperience.com) Learn the meaning of local hand gestures, the story of Argentina's beef and how to make _empanadas_ and _alfajores_. Plus you'll eat a supremely tender steak.\n\nSteaks by Luis (www.steakbuenosaires.net) An upscale _asado_ experience where you'll nibble on cheese and sip boutique wine while watching large hunks of meat being grilled.\n\nParrilla Tour (www.parrillatour.com) Meet your knowledgeable guide at a restaurant for a _choripan_ (traditional sausage sandwich), then an _empanada_. You'll finish at a local _parrilla_.\n\n######  ITALIAN\n\nThanks to Argentina's Italian heritage, the national cuisine has been highly influenced by Italian immigrants who entered the country during the late 19th century. Along with an animated set of speaking gestures, they brought their love of pasta, pizza, g elato and more.\n\nMany restaurants make their own pasta \u2013 look for _pasta casera_ (handmade pasta). Some of the varieties of pasta you'll encounter are ravioles, _sorrentinos_ (large, round pasta parcels similar to ravioli), _\u00f1oquis_ (gnocchi) and _tallerines_ (fettuccine). Standard sauces include _tuco_ (tomato sauce, sometimes with meat), _estofado_ (beef stew, popular with ravioli) and _salsa blanca_ (b\u00e9chamel). Occasionally the sauce is not included in the price of the pasta \u2013 you choose and pay for it separately.\n\nPizza is sold at _pizzer\u00edas_ throughout the country, though many regular restaurants offer it as well. It's generally very cheesy and excellent, so go ahead and order a slice or three! Other common Italian-based treats include _fugazzeta_ (similar to focaccia) and _fain\u00e1_ (garbanzo flatbread).\n\nChimichurri sauce  \nMARYELLEN BAKER \/ GETTY IMAGES \u00a9\n\n######  OTHER ETHNIC\n\nSpanish cooking is less popular than Italian but is another cornerstone of Argentine food. In BA's Spanish restaurants, many of them found in the Congreso neighborhood, you'll find _paella,_ as well as other typically Spanish seafood dishes.\n\nThe Palermo Viejo neighborhood offers a wide range of Armenian, Brazilian, Mexican, French, Indian, Japanese, Southeast Asian and Middle Eastern cuisines (among many others). If you're craving spicy food (anathema to most Argentines), this is the place to come.\n\n\u00d1oquis with tomato sauce  \nSHYMAN \/ GETTY IMAGES \u00a9\n\n######  Vegetarians & Vegans\n\nMost restaurants, including _parrillas,_ serve a few items acceptable to most vegetarians, such as green salads, omelets, mashed potatoes, pizza and pasta. Words to look out for include _carne_ (beef), _pollo_ (chicken), _cerdo_ (pork) and _cordero_ (lamb), though all meat cuts are described in different words. S _in carne_ means 'without meat', and the phrase _soy vegetariano\/a_ ('I'm a vegetarian') comes in handy. _Pescado_ (fish) and _mariscos_ (seafood) are sometimes available for pescatarians.\n\nVegetarian restaurants have become trendy in recent years, along with health-food shops with bulk grains, wholewheat pasta, dried fruit, nuts and bakery goods.\n\nVegans will have a much harder time; there isn't a Spanish word for 'vegan'. Make sure homemade pasta doesn't include egg, and that fried vegetables aren't cooked in lard ( _grasa;_ _manteca_ means butter). You'll need to be creative to survive here. One tip: look for accommodations with a kitchen, so you can shop for and cook your own food.\n\nPizza Margherita  \nMONCHERIE \/ GETTY IMAGES \u00a9\n\n######  The Sweet Stuff\n\nOne of Argentina's most definitive treats is _dulce de leche,_ a milk-caramel sauce that is dripped on everything from flan to cake to ice cream. _Alfajores_ (round, cookie-type sandwiches) are also delicious \u2013 Argentina's version of the candy bar. The most upscale and popular brand is Havanna (also a coffee-shop chain), but kiosks carry many other kinds.\n\nBecause of Argentina's Italian heritage, Argentine _helado_ is comparable to the best ice cream anywhere in the world. Amble into a _helader\u00eda_ (ice-cream shop), order up a cone (usually you pay first) and the creamy concoction will be artistically swept up into a mountainous peak and handed over with a small plastic spoon tucked in the side. Important: _granizado_ means with chocolate flakes.\n\nSome of the best _helader\u00eda_ chains \u2013 with branches all over the city \u2013 are Persicco, Freddo and Una Altra Volta, but many smaller independent shops are excellent too.\n\nYoung man drinking mate  \nSANTIAGO SOTO MONLLOR \/ GETTY IMAGES \u00a9\n\n######  Drinks\n\n######  WINE\n\nBy now you've probably heard: Argentine wines are world-class. Most famous is malbec, that dark, robust plum-flavored wine that has solidly stomped the region of Mendoza on every oenophile's map (the Mendoza region produces 60% of the country's wine). But Argentina has other fine varietals that are very worthy of a sip or three \u2013 fresh _torront\u00e9s_ , fruity bonarda and earthy pinot noir.\n\nSo how to know which to try? They say there's a perfect Argentine wine for every occasion and a good _vinoteca_ (wine boutique) will help you find it. In Palermo, try Lo de Joaquin Alberdi (Click here), in San Telmo there's Vinotango (  4361-1101; Estados Unidos 488;   10:30am-9pm). Aldo's Vinoteca (Click here) is a restaurant that sells wines at retail prices \u2013 even when you eat there.\n\nSupermarket selections are usually adequate, though you miss out on the tailored advice. Among the mainstay brands are Norton, Trapiche, Zuccardi and Santa Julia, with different lines that cater to every price range. Spend a bit more to try the elegant Rutini (from Bodega La Rural) or Luigi Bosca.\n\nFor private wine tastings, your best bet is with Anuva Wines (  15-5768-8589; www.anuvawines.com). Try five boutique vintages with food pairings; they'll also ship your wine purchases to the USA. For informal tastings, inquire at Pain et Vin (  4832-5654; Gorriti 5132), a casual wine and bread shop. Bar du Marche (  4778-1050; www.bardumarchepalermo.com; Nicaragua 5946;   9:30am-midnight Mon-Sat) is a low-key bistro offering 50 wines by the glass, while Gran Bar Danz\u00f3n (Click here) is an upscale lounge-restaurant that also has a good selection of wines by the glass.\n\nMany _puertas cerradas_ (closed-door) restaurants offer fine wines with their meals; Casa Coupage (  4833-6354; www.casacoupage.com.ar), run by an Argentine sommelier couple, is especially wine-oriented. Finally, if you're really into wines consider staying at Miravida Soho (Click here) in Palermo; it has a wine bar, cellar and tastings for their guests.\n\n_Helader\u00eda_ (ice-cream shop), Microcentro  \nLONELY PLANET \/ GETTY IMAGES \u00a9\n\n######  MATE\n\n_Mate_ (mah-teh) is Argentina's unofficial national beverage. More than a simple drink like tea or coffee, _mate_ is more like an elaborate ritual shared among family and friends.\n\nThere's an informal etiquette to preparing and drinking _mate_. The _cebador_ (server) fills the gourd with _yerba,_ then pours in very hot water. Each participant drinks the gourd dry, then the _cebador_ refills it and hands it to the next person. Germaphobes beware: the _bombilla_ (a silvery straw with built-in filter), used to sip the _mate,_ is shared by everyone.\n\nAn invitation to drink _mate_ is a cultural treat you shouldn't turn down, though it's definitely an acquired taste. The tea is grassy, bitter and very hot; adding sugar can help. Saying ' _gracias_ ' is a sign you want to stop drinking. And remember not to hold the _mate_ too long before passing it on!\n\nBecause it is such a personal ritual, not many restaurants offer _mate_ on the menu \u2013 but a few do, so try it if you can.\n\nMalbec wine  \nBRIAN DOBEN \/ GETTY IMAGES \u00a9\n\n######  BEER, COFFEE & WATER\n\nIf Argentina has a national beer, it's Quilmes. Order a _porr\u00f3n_ and you'll get a half-liter bottle, or a _chopp_ and you'll get a frosty mug of draft.\n\nArgentines love their _caf\u00e9 con leche_ (coffee with milk). An espresso with a drop of milk is a _caf\u00e9 cortado_. Black and herbal teas are also commonly available.\n\nIn Buenos Aires, the _agua de canilla_ (tap water) is drinkable. In restaurants, however, most people order bottled mineral water \u2013 ask for _agua con gas_ (with bubbles) or _agua sin gas_ (without). In older, more traditional restaurants, carbonated water in a spritzer bottle ( _un sif\u00f3n de soda_ ) is great for drinking, though Argentines often mix it with cheap wine.\n\nMedialunas (croissants)  \nEDUARDO MORCILLO \/ GETTY IMAGES \u00a9\n\n######  Eat Like a Local\n\nArgentines eat little for breakfast \u2013 usually just coffee with _medialunas_ , either _de manteca_ (sweet) or _de grasa_ (plain). _Tostadas_ (toast) with _manteca_ (butter) or _mermelada_ (jam) is an alternative, as are _facturas_ (pastries). Most hotels offer this basic breakfast, but some higher-end hotels have breakfast buffets.\n\nArgentines make up for breakfast at lunch and dinner, and they love to dine out. Every neighborhood has basic restaurants serving the staples of pasta, pizza and steak (though for the best meats, head to a _parrilla_ ).\n\nCafes (which serve snacks, light meals and sometimes more) and _confiter\u00edas_ (restaurant-cafes) are open all day and into the night. Bars or pubs usually have a more limited range of snacks and meals, though some offer full meals. A _tenedor libre_ (literally, 'free fork') is an all-you-can-eat restaurant; quality is usually decent, but drinks are often mandatory and cost extra.\n\nLarge, modern, chain supermarkets are common, and they'll have whatever you need for self-catering, including (usually) a takeout counter with a decent range of offerings. Smaller, local grocery stores \u2013 usually family-run \u2013 are also ubiquitous, though they won't have takeout.\n\nThe most thorough online guide to BA restaurants is www.guiaoleo.com (in Spanish); for listings in English, try www.pickupthefork.com\n\n###  COOKING COURSES\n\nTaking a small-group cooking class or private class is probably the best option for short-term visitors who don't speak Spanish.\n\n\u00bb Norma Soued (www.argentinecookingclasses.com) Cook Argentine cuisine like _empanadas_ , traditional stews and _alfajores_.\n\n\u00bb Cooking with Teresita (www.try2cook.com) Partake of _asados_ (barbecues) and _empanadas_ ; she'll also take you to local markets.\n\nIf you have time, speak Spanish and are considering making cooking a profession, try the highly regarded Instituto Argentino de Gastronom\u00eda (IAG; www.iag.com.ar) or Mausi Sebess (www.mausi\u00adsebess.com), located in BA's suburb of Vicente L\u00f3pez.\n\n###  NEED TO KNOW\n\n####  Price Ranges\n\n$ mains under AR$80\n\n$$ mains AR$80-130\n\n$$$ mains over AR$130\n\nLook for the _menu ejecutivo_ (set lunch menu); this usually includes dessert and a drink and is a good deal.\n\n####  Opening Hours\n\n\u00bb Restaurants are generally open daily from noon to 3:30pm for lunch and 8pm to midnight or 1am for dinner.\n\n\u00bb A sure bet for that morning _medialuna_ (croissant) and _cortado_ (coffee with milk) are the city's many cafes, which often stay open from morning to late at night without a break.\n\n####  Tipping\n\nTip 10% for standard service; make it 15% for exceptional service. Tips usually cannot be added to credit-card purchases. The word for tip in Spanish is _propina._\n\n####  Reservations\n\nReserve at popular restaurants, especially on weekends. If you don't speak Spanish, ask a staff member at your hotel to make the call for you. Or check out www.restorando.com.ar.\n\n####  Etiquette\n\n\u00bb Most _porte\u00f1os_ eat no earlier than 9pm (later on weekends).\n\n\u00bb Ask for your bill by saying, ' _la cuenta, por favo_ r' ('the bill, please') or making the 'writing in air' gesture. Be aware that not all restaurants accept credit cards \u2013 always ask first.\n\n\u00bb At upscale restaurants, a per-person _cubierto_ (cover charge), usually AR$10 to AR$30, is tacked on to the bill. This covers the use of utensils and bread \u2013 it does not relate in any way to the tip.\n\n#####  Eating by Neighborhood\n\n\u00bb  The Center Many restaurants here cater to the business crowd, with quick takeout and power lunches.\n\n\u00bb  Puerto Madero Upscale \u2013 and some say overpriced \u2013 restaurants here offer great dockside atmosphere and traditional cuisine.\n\n\u00bb  Congreso & Tribunales BA's political center, come here for Spanish cuisine, along with the odd Chinese, Korean or Peruvian gem.\n\n\u00bb  San Telmo Traditionally has supported many _parrillas,_ though gentrification over the years translates to wider horizons.\n\n\u00bb  Recoleta & Barrio Norte BA's most exclusive neighborhood means expensive restaurants; those near the cemetery cater to tourists.\n\n\u00bb  Palermo Ground zero for the city's most creative and ethnically diverse dining scene.\n\n#####  Lonely Planet's Top Choices\n\n Hern\u00e1n Gipponi Restaurant Beautifully prepared cuisine and memorable brunch.\n\n Caf\u00e9 San Juan Tattooed celebrity chef serves up fabulously tasty dishes.\n\n Sarkis Everything is excellent at this famous Armenian restaurant.\n\n Le Sud Exquisite French cuisine in one of BA's best hotels.\n\n Aramburu Molecular gastronomy \u2013 not for everyone, but for some it's a peak experience.\n\n#####  Best by Budget\n\n######  $\n\n Chan Chan Colorful, casual Peruvian eatery with great _seviche_ (seafood cured in citrus).\n\n Rodi Bar Recoleta institution serving something for everyone.\n\n Cuman\u00e1 Upscale rustic joint specializing in northern Argentine cuisine.\n\n######  $$\n\n Las Pizarras Delicious, very creative dishes that can change daily.\n\n Astor Multicultural celebrity chef starts his own excellent restaurant.\n\n Malv\u00f3n Excellent breakfast and brunches, but you'll have to wait.\n\n######  $$$\n\n Elena Probably one of the best meals you'll have in BA.\n\n Unik Contemporary, highly sophisticated dishes in an upscale atmosphere.\n\n Tomo 1 Five-star restaurant, five-star food, five-star prices.\n\n#####  Best by Cuisine\n\n######  Steak\n\n Don Julio Great traditional steakhouse with classy service.\n\n La Cabrera Overly popular and touristy \u2013 but worth the wait.\n\n Miranda Modern atmosphere and pleasant sidewalk seating.\n\n Parrilla Pe\u00f1a No-nonsense, well priced and excellent meats.\n\n El Desnivel Long-running, touristy and still reasonably priced.\n\n######  Seafood\n\n Oviedo Upscale dining room, fine service and amazing cuisine.\n\n Crizia One of BA's best for fresh oysters and fish.\n\n Casal de Catalunya Spanish restaurant with old-time atmosphere.\n\n La Rosa N\u00e1utica Peruvian specialties like marinated octupus and _seviche_.\n\n######  Italian\n\n Siamo nel Forno Awesome Naples-style, thin-crust pizza.\n\n La Parolaccia Trattoria Tasty homemade pastas with a water view.\n\n Il Matterello Famous for its exceptional pastas and sauces.\n\n Filo Trendy restaurant with a variety of Italian specialities.\n\n######  Asian\n\n Sudestada Consistently good, Asian-fusion cuisine that can be spicy.\n\n Green Bamboo Vietnamese-style foods in BA-trendy dining room.\n\n Furai-Bo Japanese ramen and other delicacies, in authentic surroundings.\n\n Bi Won Good, long-running and no-nonsense Korean restaurant.\n\n Comedor Nikkai A good place for sushi, tempura and teriyaki.\n\n######  Vegetarian\n\n Bio Sophisticated and very tasty vegetarian dishes.\n\n Arevalito Tiny hippy-ish joint with great salads, sandwiches and tarts.\n\n Granix Modern cafeteria catering to meat-weary office workers.\n\n Vita Colorful, healthy and organic cafe, and they're friendly too.\n\n Abuela Pan Small, very casual eatery serving daily vegetarian specials.\n\nAlfajores de Maicena pastries filled with dulce de leche at Las Violetas Cafe (Click here)  \nVIVIANE PONTI \/ GETTY IMAGES \u00a9\n Drinking & Nightlife\n\nBuenos Aires' nightlife is legendary around the world. What else could you expect from a country where dinner rarely starts before 10pm? In some neighborhoods, finding a good sports bar, trendy cocktail lounge, atmospheric old cafe or upscale wine bar is as easy as walking down the street. And dancers will be in heaven, as BA boasts spectacular nightclubs showcasing top-drawer DJs.\n\n######  The Local Scene\n\n_Porte\u00f1os_ hardly ever imbibe to the point of drunkenness \u2013 it's just not cool \u2013 but they do like to go out drinking, especially in groups, and always stay up late. Walk into any corner bar or cafe in the city and you'll see groups of friends or family sitting around a table, sipping tiny white cups of espresso or splitting a bottle of Quilmes (a popular local beer). More fashionable bars, pubs and breweries draw more of a mixed crowd of party-going tourists, with style-conscious men trying to impress their dates or girlfriends celebrating a special occasion.\n\nHow to handle the late-night scene like a _porte\u00f1o_ ? If you're going out clubbing (some clubs open at 2am), take a nap after dinner and go easy on the booze \u2013 it will help you avoid conking out too early.\n\n######  Gay & Lesbian Buenos Aires\n\nIn July 2010 Argentina became the first Latin American country to legalize same-sex marriage. Since then, Buenos Aires has become a huge gay destination, lending momentum to local events such as the Marcha del Orgullo Gay (Gay Pride Parade; www.marchadelorgullo.org.ar;   Nov) and the Queer Tango Festival (www.festivaltangoqueer.com.ar).\n\nAn especially gay-friendly accommodation is Lugar Gay (  4300-4747; www.lugargay.com.ar; Defensa 1120; dm US$25, s US$50-70, d US$80-95), a casual guesthouse that also acts as an information center. Good general websites are www.thegayguide.com.ar and www.nighttours.com\/buenosaires.\n\nCurrent hot gay parties include Fiesta Plop (www.plop-web.com.ar;   Fri), the monthly Fiesta Dorothy (www.fiestadorothy.com) and Rheo (www.rheo.com.ar).\n\nCasa Brandon (www.brandongayday.com.ar; Luis Mar\u00eda Drago 236) is an art gallery-cultural center. And for a fun night of guided drinking and partying, there's Out & About Pub Crawl (www.outandaboutpubcrawl.com).\n\nFinally, gay classes and _milongas_ are given at La Marshall (  4300-3487; www.lamarshall.com.ar; Av Independencia 572) and Tango Queer (www.tangoqueer.com).\n\n######  Bars\n\nBars abound in every neighborhood of Buenos Aires, and they come in all shapes, sizes and styles. You can choose from sports bars, cocktail lounges, Irish pubs, microbreweries, local holes-in-the-wall and more. Many of the city's upscale restaurants and hotels also have lively bars worth a visit.\n\nMost bars serve beer, hard alcohol and wine, plus coffee and juice. Some make cocktails, and many offer a fair range of finger foods or even main dishes. Microbreweries and beer bars are catching on, offering decent selections of the hoppy stuff \u2013 but Argentina has a ways to go before competing globally, so don't expect anything mind-blowing.\n\nYounger travelers and backpackers looking to bar-hop in a group should check out Buenos Aires Pub Crawl (  15-5464-1886; www.pubcrawlba.com).\n\n######  Cafes\n\nCafes are an integral part of porte\u00f1o life, and you shouldn't miss popping into one of these beloved hangouts for an afternoon break. Many cafes are old classics that have been around for more than a hundred years, and undoubtedly will take you back in time. Others are contemporary or bohemian joints with sidewalk tables \u2013 perfect spots to take a load off while sightseeing or to delve into Borges' short stories at a corner table.\n\nMost cafes serve all meals and everything in between (including a late-night snack). For more on the classic places see Click here.\n\n######  Clubbing\n\nBuenos Aires is famous for its _boliches_ (nightclubs). Every weekend \u2013 and even on some weeknights \u2013 the city's clubs come alive with beautiful people moving to electronic and house music. Some of the most impressive nightlife hot spots are located in grandiose restored theaters, warehouses or factories \u2013 or perched on the banks of the Rio de la Plata where party-goers can watch the sun rise over the water as the festivities wind down. Clubs are spread out over the city, with main clusters in Palermo and on the Costanera Norte.\n\n######  Electronica in Buenos Aires\n\nBuenos Aires might be known for its tango, but there is something else to keep you dancing until dawn in this late-night city that generally looks to Europe for its trends. Since 1990 the electronic-music scene of BA has grown to become a major force in the music world. Touting some of the world's best venues and biggest crowds, Buenos Aires is listed by many DJs as a favorite place to play.\n\nOne of the most internationally acclaimed homegrown DJs is Hern\u00e1n Cattaneo, who began his professional career in the early '90s playing commercial clubs of the time, such as El Cielo and Cinema. Several years later he secured a residency for the Clubland night at Pach\u00e1, where legend has it he was discovered and whisked off to international stardom by UK legend Paul Oakenfold. The success of Cattaneo and Pach\u00e1 marked the beginning of a new era, when electronica emerged into mainstream pop culture.\n\nNowadays, when the weather warms up in spring, enormous events with up to 50,000 people take place, such as Creamfields (www.creamfieldsba.com) and the South American Music Conference, while newer, smaller-scale festivals like Festival Ciudad Emergente (www.ciudademergente.gob.ar) pack venues with thousands of young people and feature electronica performances. In addition to these annual events, the club and underground scene is alive and well, although somewhat less kicking (and with good reason) due to stricter rules since the 2004 Cromagnon club tragedy in which almost 200 people lost their lives in a fire.\n\nHouse music (referred to as ' _punchi, punchi'_ because of the relentless kick drum) is no longer the only option. You'll find a variety of sounds thanks to early diversification within Argentina's veteran underground DJ collective, DJ UNION, composed of Carla Tintore (www.carlatintore.com), Dr Trincado (www.drtrincado.com) and Diego Ro-k. Notoriously wild parties such as the Age of Communication and Ave Porco helped pave the way to a diverse underground tradition, which you can experience at clubs like Cocoliche (Click here).\n\nThe original DJ collectives and electronica parties have paved the way for another generation of musical stylings: whether it's progressive house, breakbeat, techno, IDM, deep house, drum and bass or even experimental _cumbia_ (Colombian music), Buenos Aires has it. Some of the DJs who experiment with styles are DJ Joven (<http:\/\/djoven.blogspot.com>) and Djs Pareja (www.djspareja.com.ar), the famous electronica twosome who spin a fusion of retro acid house techno and pop at clubs and parties around town; DJ Daleduro and his partner DJ Gone, who form the duo Groovedealers and do two-step garage and dubstep; Franco Cinelli, who plays minimal sounds and clicks; and Chancha Via Circuito, an experimental cumbia artist who got his start at Zizek (www.zzkrecords.com).\n\n###  NEED TO KNOW\n\n####  Opening Hours\n\n\u00bb Bars Vary widely depending on location and clientele, but most are usually open in the evening into the early-morning hours.\n\n\u00bb Cafes Usually from around 6am or 7am to 2am or 3am.\n\n\u00bb Clubs From 2am to dawn.\n\n####  Costs\n\n'After office' is often a term for happy hour _._\n\n\u00bb _Chopp_ (draft beer): AR$25\n\n\u00bb Pint of craft beer: AR$40\n\n\u00bb Cocktail: AR$60\n\n\u00bb Cup of coffee: AR$15\n\n\u00bb Club cover charge: varies widely depending on the club, your gender, the day of the week, the time of night and the DJ's fame. Bring cash, as credit cards aren't often accepted.\n\n####  Club Door Policies\n\nAll clubs have bouncers. Dress well \u2013 smart casual is good enough at most clubs. To get in more easily, try playing up your foreign accent \u2013 _porte\u00f1os_ love the exotic (though foreigners are now common in BA). You can also sign up in advance via online-reservation forms that some clubs keep; this sometimes gets you in more easily and\/or offers discounts.\n\n####  Resources\n\nMany newspapers have entertainment supplements published on Friday; the _Buenos Aires Herald_ (an English-language publication) is particularly handy. Also check www.vuenosairez.com (in Spanish) and www.argentinaindependent.com (in English) for current happenings.\n\n#####  Drinking & Nightlife by Neighborhood\n\n\u00bb  The Center Irish pub knock-offs cater to the business crowd, though there are several good cafes and clubs.\n\n\u00bb  Congreso & Tribunales This neighborhood, the political center of BA, has a few interesting bars and cafes.\n\n\u00bb  San Telmo Gentrified San Telmo is host to many fancy modern spots mixing it up with a few old classics.\n\n\u00bb  Retiro A wide choice of drinking holes attract business people during the day and into the evening, and the traveler-expat crowd at night.\n\n\u00bb  Recoleta Across from the cemetery is a two-block strip of restaurants, cafes and bars with great patios.\n\n\u00bb  Palermo BA's hippest nightlife lives here, especially near Plaza Serrano. Las Ca\u00f1itas, a sub-neighborhood, is also very lively.\n\n\u00bb  South of Palermo As Palermo becomes more expensive, these blue-collar neighborhoods are gentrifying and attracting their own attention.\n\n#####  Lonely Planet's Top Choices\n\n Florer\u00eda Atl\u00e1ntico Currently BA's hottest bar, and possibly its most oddly located.\n\n Las Violetas Gorgeous, traditional cafe full of _porte\u00f1os_ sipping tea and nibbling house-made pastries.\n\n Niceto Club Famous for Club 69, its Thursday-night show highlighting drag queens, strippers and breakdancers.\n\n Antares Atmospheric chain restaurant-bar with good food and craft beer.\n\n Museo Evita Restaurante Upscale restaurant-cafe with a patio that's perfect on a warm day.\n\n#####  Best Bars\n\n Florer\u00eda Atl\u00e1ntico 'Secret' basement bar located inside a flower shop!\n\n Verne Fancy drinking hole with awesome cocktails and smoking patio.\n\n Frank's Bar Elegant speakeasy with classic cocktails; enter via the telephone booth.\n\n Magdalena's Party Casual corner bar serving American comfort food.\n\n Mili\u00f3n Glamorous setting in a lovely mansion with an elegant garden.\n\n#####  Best Traditional Cafes\n\n Las Violetas BA's most beautiful cafe, with stained-glass awnings and afternoon tea.\n\n Caf\u00e9 de los Angelitos Ironically named for the mobsters that used to hang out here.\n\n Esquina Homero Manzi Traditional cafe off the tourist track, with a lovely atmosphere.\n\n La Biela Best for its excellent people-watching front patio on a warm sunny day.\n\n Caf\u00e9 Tortoni Very historic, very scenic and very touristy \u2013 but a classic that can't be ignored.\n\n#####  Best Clubs\n\n Niceto Club Best for its raucous, Thursday-night, over-the-top burlesque show.\n\n Pach\u00e1 The temple of electronica, with pretty people and international DJs.\n\n Boutique Famous 'after-office' party on Wednesday eves, in a building designed by Eiffel.\n\n Crobar Electronica, Latin beats and more at this perennially popular Palermo club.\n\n Kika Draws Tuesday-night crowds with its well-known 'Hype' party.\n\n#####  Best Beer Bars\n\n Antares Popular restaurant-bar serving tasty beers from Mar del Plata.\n\n Buller Brewing Company Recoleta microbrewery with six kinds of beer.\n\n Cervecer\u00eda Cossab Dedicated beer bar that boasts over 50 beers.\n\n Cruzat Beer House Has a nice patio to enjoy craft beer from all around Argentina.\n\nBroeders Not actually a beer bar, Broeders is an excellent craft beer only available at Fukuro Noodle Bar (www.fukuronoodlebar.com) or NOLA (www.nolabuenosaires.com).\n\n#####  Best Museum Cafes\n\n Museo Evita Restaurante Excellent patio cafe-restaurant with sophisticated cuisine.\n\n Museo Nacional de Arte Decorativo Upscale French bistro with pleasant outdoor tables.\n\n Fundaci\u00f3n Proa Fancy cafe with awesome rooftop terrace offering La Boca views.\n\n Malba Not cheap, but a nice atmosphere and good people-watching.\n\n Museo del Bicentenario Great for an indoor cafe break near Plaza de Mayo.\n\n#####  Best Gay Hot Spots\n\n Alsina Monthly Fiesta Dorothy packs in the cute guys.\n\n Glam Casual yet very sexy club in an old mansion.\n\n Crobar Saturday night's Rheo is the place to be.\n\n Amerika Rough and tumble, with all-you-can-drink nights.\n\n Pride Cafe Casual San Telmo coffee shop.\n Entertainment\n\nThe entertainment scene in Buenos Aires has always been lively, but there was an outburst of creative energy in the decade following the economic crisis of 2001. Filmmakers began producing quality works on shoestring budgets, troupes performed in new avant-garde theaters and live-music groups played in more mainstream venues. Today nearly every neighborhood offers great entertainment options.\n\n######  Live Music\n\nThere are some fine venues that only feature live music, but many theaters, cultural centers, bars and cafes also put on shows. Centro Cultural Torquato Tasso (  4307-6506; www.torquatotasso.com.ar; Defensa 1575) is an especially good choice for tango-music performances. For more on tango shows Click here.\n\n######  CLASSICAL\n\nSeveral venues offer classical-music concerts. Teatro Col\u00f3n (Click here) is the grandest and most famous; everyone who's anyone has played, acted, sung or danced here. It often features guest conductors from throughout Latin America. The classical-music scene takes a break from December to February, and is best from June to August.\n\n######  ROCK, BLUES & JAZZ\n\nBuenos Aires boasts a thriving rock-music scene. Smaller venues, like La Trastienda (Click here), showcase mostly local groups; when huge international stars come to town they tend to play soccer stadiums or Luna Park (Click here).\n\nBlues and jazz aren't as popular as rock but still have their own loyal following. Thelonious Bar (Click here) and Notorious (Click here) are top-notch venues for jazz concerts.\n\n######  FOLK\n\n_M\u00fasica folkl\u00f3rica_ definitely has its place in Buenos Aires. There are several _pe\u00f1as_ (traditional music clubs) in the city, including Los Cardones (Click here) and Pe\u00f1a del Colorado (Click here), but other venues \u2013 such as Cl\u00e1sica y Moderna (Click here) \u2013 occasionally host folk performances.\n\n######  Cinema\n\nBA's traditional cinema districts are along ped\u00adestrian Lavalle (west of Florida) and on Av Corrientes. Newer cinemas are in shopping malls throughout the city. Most cinemas offer big discounts for matinees, midweek shows or first screenings of the day. There is usually a _trasnoche_ (midnight or later showing) scheduled for Friday and Saturday night.\n\nCheck the English-language _Buenos Aires Herald_ for the original titles of English-language films. The entertainment sections of all the major newspapers will have movie listings as well, but be aware that Spanish translations of English-language film titles often don't translate directly. Except for children's films and cartoon features, which are dubbed, foreign films usually appear in their original language with Spanish subtitles.\n\nCosmos-UBA (  4953-5405; www.cosmosuba.wordpress.com; Av Corrientes 2046) and Sala Leopoldo Lugones (  0800-333-5254; www.teatrosanmartin.com.ar\/cine; Av Corrientes 1530) \u2013 in Teatro General San Mart\u00edn \u2013 often show retrospectives, documentaries, foreign film cycles and art-house movies. Espacio INCAA (  4371-3050; www.incaa.gov.ar; Av Rivadavia 1635) screens Ibero-American films only (essentially from Spanish- or Portuguese-speaking countries).\n\nSome cultural centers have their own small cinemas, while places such as Alianza Francesa and the British Arts Centre showcase movies in their respective languages.\n\n######  Theater\n\nTheater is big in Buenos Aires. There are more than 100 venues and annual attendance is in the hundreds of thousands. Productions range from classic plays to multimedia performances to lavish cabarets, and the acting is of a professional level across the board. Note that, unsurprisingly, performances tend to be in Spanish.\n\nTraditionally, the center for theater has been Av Corrientes between Av 9 de Julio and Callao, but there are now dozens of venues all over the city. The _Buenos Aires Herald_ and other local newspapers have good listings of major productions.\n\nMany alternative (or 'off-Corrientes') theater companies and independent troupes receive relatively little attention from the mainstream media, but they're worth seeking out if you're looking for something different. If you read Spanish, www.alternativateatral.com is a good source for current non-mainstream performances.\n\nTickets are generally affordable, but check _carteleras_ for bargain seats. The season is liveliest in winter (June through August), when upwards of 100 events may take place, but you can find a good variety of shows any time. Many of the most popular shows move to the provincial beach resort of Mar del Plata for the summer.\n\n######  Circo Moderno\n\nA popular movement in Argentina that found international fame through the Broadway performance of the De la Guarda troupe is _circo moderno_ (contemporary circus). This combination of traditional circus and contemporary dance and theater features a lot of aerial action, acrobatics and no words \u2013 great for those who don't speak Spanish. Cirque du Soleil is a well-known example of this modern gymnastic theater.\n\nIn 2005, Diqui James, one of the creators of De la Guarda, launched his solo act Fuerzabruta (www.fuerzabruta.net). It's a jaw-dropping, mind-blowing show of lights, electronic music, aerial dancing and water \u2013 and often the performance is above you. If you go to a show, you could get wet. The troupe is often on tour around the world, so check its website for listings.\n\n###  NEED TO KNOW\n\n####  Opening Hours\n\nShow times can vary widely, but this is a city that stays up all night, so expect to be out late. Restaurants usually open around 9pm \u2013 and 10pm is a more common dinner time \u2013 so many shows start around midnight.\n\n####  Resources\n\nMany newspapers publish entertainment supplements on Friday; the _Buenos Aires Herald_ has a particularly handy one. Also check www.vuenosairez.com (in Spanish) and www.argentinaindependent.com (in English).\n\n####  Discount Tickets & Booking\n\nMajor entertainment venues often require booking through Ticketek (  5237-7200; www.ticketek.com.ar). The service charge is about 10% of the ticket price.\n\n_Carteleras_ (discount-ticket offices) sell a limited number of discounted tickets for many events, such as movies, theater and tango shows, with savings of 20% to 50%. Try Cartelera Baires (www.cartelerabaires.com; Av Corrientes 1382, Inside Galer\u00eda Apolo), Cartelera Vea M\u00e1s (  6320-5319; www.veamasdigital.com.ar; Av Corrientes 1660, Local 2) or Cartelera Espect\u00e1culos (  4322-1559; www.123info.com.ar; Lavalle 742). Buy tickets as far in advance as possible, but if you want to see a show or movie at short notice \u2013 especially midweek \u2013 you can also drop by to check what's available.\n\n#####  Entertainment by Neighborhood\n\n\u00bb  The Center Has a little of everything \u2013 theater, live music, cinemas and tango shows.\n\n\u00bb  Congreso & Tribunales Home to Av Corrientes, BA's traditional theater district; also has several cinemas and flamenco venues.\n\n\u00bb  San Telmo Some live music and tango spots.\n\n\u00bb  Palermo A few tango _milongas_ and live-music venues.\n\n\u00bb  South of Palermo Ground zero for BA's avant-garde theater.\n\n#####  Lonely Planet's Top Choices\n\n Teatro Col\u00f3n Landmark, seven-story theater seating 2500 and boasting renowned acoustics.\n\n Usina del Arte Old electricity factory remodeled into a premier symphony hall.\n\n Centro Cultural Borges One of BA's top cultural centers, with countless offerings.\n\n#####  Best Theaters\n\n Teatro Col\u00f3n Buenos Aires' grandest entertainment concert hall and a gorgeous building.\n\n Teatro San Mart\u00edn Large venue that's great for classic theater and much more.\n\n Teatro Nacional Cervantes Traditional old venue showing contemporary productions.\n\n#####  Best Live Music\n\n Usina del Arte Amazing and beautiful new performance venue in La Boca seating 1200.\n\n Ciudad Cultural Konex Famous for its one-of-a-kind, Monday-night percussion parties.\n\n Centro Cultural Torquato Tasso Excellent venue for concerts, including tango music.\n\n#####  Best Cultural Centers\n\n Centro Cultural Borges Quality art galleries, cinema, workshops, music and shows.\n\n Centro Cultural Recoleta Many free or inexpensive events, plus a science museum for kids.\n\n Centro Cultural San Mart\u00edn Large cultural center with galleries, concerts, exhibitions and shows.\n\n#####  Best Flamenco\n\n \u00c1vila Bar Small stage inside a traditional Spanish restaurant.\n\n Cantares Basement venue that once hosted the poet Federico Garc\u00eda Lorca.\n\n Tiempo de Gitanos Palermo restaurant offering classic Spanish food and an intimate stage.\n Tango\n\nOnce a furtive dance relegated to the red-light brothels of early-1900s Buenos Aires, tango has experienced great highs and lows throughout its volatile lifespan. These days, however, the sensual dance is back with a vengeance. Everyone from Seattle to Shanghai is slinking their way down the parquet floor, trying to master those elusive dance steps and rhythm that make it so damn hard to perfect.\n\nTango dancers  \nALTRENDO IMAGES \/ GETTY IMAGES \u00a9\n\n######  Origins\n\nIn the words of its poet laureate Disc\u00e9polo, the 'tango is a sad thought you can dance to'. Though the exact origins can't be pinpointed, the dance is thought to have started in Buenos Aires in the 1880s. Legions of European immigrants, mostly lower-class men, arrived here to seek their fortune. They settled on the capital's fringes, such as La Boca and Barracas, but missing their motherlands and the women they left behind, sought out cafes and bordellos to ease the loneliness. Here (so the myth goes), these immigrant men danced with each other while they waited for their paramours to become available \u2013 women were scarce back then!\n\nThe perceived vulgarity of the dance that mainly belonged to the poor southern barrios was deeply frowned upon by the reigning porte\u00f1o elites of the plush northern suburbs, but it did manage to influence some brash young members of the upper classes. These rebel jet setters, known as _ni\u00f1os bien_ , took the novelty to Paris and created a craze \u2013 a dance that became an acceptable outlet for human desires, expressed on the dance floors of elegant cabarets. The trend spread around Europe and even to the USA, and 1913 was considered by some as 'the year of the tango'. When the evolved dance, now refined and famous, returned to Buenos Aires, it finally earned the respectability it deserved. And so the golden years of tango began.\n\nIn 1955, however, Argentina became a military state intolerant of artistic or ' nationalistic' activities \u2013 including the tango, which had been highly popular with the people. Some tango songs were banned, and the dance was forced underground due to curfews and a limit on group meetings. The dance didn't resurface until 1983, when the junta fell \u2013 and once it was back in the open again, it underwent a renaissance. After being constrained by the rigors of military rule, Argentines suddenly wanted to experience new life, be creative and move. The tango became popular once again \u2013 and remains so to this day.\n\nTango lesson  \nJAVIER PIERINI \/ GETTY IMAGES \u00a9\n\n######  Tango for Export \u2013 the Shows\n\nIf there's one thing Buenos Aires isn't short of it's tango shows. The best known are the expensive, tourist-oriented spectacles that are very entertaining and awe-inspiring, and showcase amazing feats of grace and athleticism. However, they are highly glamorized and not what purists consider 'authentic' tango.\n\nThe theatrical shows usually include various tango couples, an orchestra, a couple of singers and possibly some folkloric musicians. They last about 1\u00bd hours and come with a dinner option \u2013 the food is usually good. VIP options mean a much higher price tag for better views, meal choices and refreshments. Nearly all of them require reservations; some offer modest online discounts and pick-up from your hotel. (Many hotels will book shows for you \u2013 which is fine, since sometimes the price is similar to what you'd pay at the venue anyway.)\n\nMore modest shows cost far less; some are even free but require you to order a meal or drink at the restaurant. For free (or rather, donation) tango, head to San Telmo on a Sunday afternoon \u2013 or sometimes other days. Dancers do their thing in the middle of Plaza Dorrego, though you have to stake out a spot early to snag a good view. Another sure bet is weekends on El Caminito in La Boca; some restaurants have couple dancing for customers. Many _milongas_ also have good, affordable shows.\n\nOne thing to note: nearly all tango shows are touristy by nature. They've been sensationalized to make them more exciting for observers. 'Authentic' tango (which happens at _milongas_ ) is a very subtle art, primarily done for the pleasure of the dancers. It's not something to be observed so much as experienced, and not particularly interesting for casual spectators. Going to a _milonga_ just to watch isn't all that cool, either: folks are there to dance. So feel free to see a more flashy tango show and enjoy those spectacular high kicks \u2013 be wowed like the rest of the crowd.\n\nIf you like listening to live tango music, head to Centro Cultural Torquato Tasso (Click here). It's one of BA's best live-music venues, so don't expect any dancing.\n\nDancing the tango, San Telmo  \nROBERT FRERCK \/ GETTY IMAGES \u00a9\n\n###  CLASSES\n\nTango classes are available just about everywhere, from youth hostels to general dance academies to cultural centers to nearly all _milongas_. Even a few cafes and tango shows offer them.\n\nThere are also several tango schools in town, such as Escuela Argentina de Tango. It has two main locations: Talcahuano 1052 and San Mart\u00edn 768 (  4312-4990; www.eatango.org); the latter is in Galer\u00edas Pac\u00edfico.\n\nPrivate teachers are also ubiquitous; there are so many good ones that it's best to ask someone you trust for a recommendation. And with so many foreigners flooding into Buenos Aires, many teach in English or other languages.\n\n######  The Real Tango \u2013 Milongas\n\nTango's popularity is booming at both amateur and professional levels, and among all ages and classes. And _milongas_ are the dance events where people strut their stuff. The atmosphere at these venues can be modern or historical, casual or traditional. Most have tango DJs that determine musical selections, but a few utilize live orchestras. The dance floor is surrounded by many tables and chairs, and there's often a bar to the side.\n\nAt a proper, established _milonga,_ choosing an adequate partner involves many levels of hidden codes, rules and signals that dancers must follow. After all, no serious _bailarina_ (female dancer; the male equivalent is a _bailar\u00edn_ ) wants to be caught out dancing with someone stepping on her toes (and expensive tango heels). In fact, some men considering asking an unknown woman to dance will do so only after the second song, to avoid being stuck for the three to five songs that make a session. These sessions (known as _tandas_ ) alternate between tango, _vals_ (the Argentine version of the waltz) and _milonga_ ; they're followed by a _cortina_ (a short break when non-tango music is played). It's considered polite to dance an entire _tanda_ with any partner, so if you are given a curt _gracias_ after just one song, consider that partner unavailable for the rest of the night.\n\nNot easy to describe, tango needs to be seen and experienced for its full effect. The upper bodies are traditionally held upright and close, with faces almost touching. The man's hand is pressed against the woman's back, guiding her, with his other hand and one of hers held together and out. The lower body does most of the work. The woman swivels her hips, her legs alternating in short or wide sweeps and quick kicks, sometimes between the man's legs. The man guides, a complicated job since he must flow with the music, direct the woman, meld with her steps and avoid other dancers, all at once. He'll add his own fancy pivoting moves, and together the couple flows in communion with the music. Pauses and abrupt directional changes punctuate the dance. It's a serious business that takes a good amount of concentration, so while dancing the pair often wear hard expressions. Smiling and chatting are reserved for the breaks between songs.\n\nYour position in the area surrounding the dance floor can be critical. At some of the older _milongas_ , the more established dancers have reserved tables. Ideally, you want to sit where you have easy access to the floor and to other dancers' line of sight. You may notice couples sitting further back (they often dance just with each other), while singles sit right at the front. If a man comes into the room with a woman at his side, she is considered 'his' for the night. For couples to dance with others, they either enter the room separately, or the man signals his intent by asking another woman to the floor. Then 'his' woman becomes open for asking.\n\nThe signal to dance, known as _cabeceo,_ involves a quick tilt of the head, eye contact and uplifted eyebrows. This can happen from way across the room. The woman to whom the _cabeceo_ is directed either nods yes and smiles or pretends not to have noticed (a rejection). If she says yes the man gets up and escorts her to the floor. A hint: if you're at a _milonga_ and don't want to dance with anyone, don't look around too much \u2013 you could be breaking some hearts.\n\nSo why is it that tango becomes so addictive for some? Experienced dancers will tell you this: the adrenaline rush you get from an excellent performance is like a successful conquest. Some days it lifts you up to exhilarating heights and other days it can bring you crashing down. You fall for the passion and beauty of the tango's movements, trying to attain a physical perfection that can never be fully realized. The best you can do is to make the journey as graceful and passionate as possible.\n\nPlaying the bandone\u00f3n  \nMAGAL\u00cd IZAGUIRRE \/ GETTY IMAGES \u00a9\n\n###  EVOLVING TANGO MUSIC\n\nNuevo tango, born in the late 1990s, was seeded by \u00c1stor Piazzolla in the 1950s when he incorporated jazz and classical beats into traditional tango music. Dancers improvised new moves into their traditional base steps, utilizing a more open embrace and switching leads (among other things). Neo tango, the latest musical step in tango's changing landscape, fuses the dance with electronica for some decidedly nonstodgy beats that have done a superlative job of attracting the younger generation to this astounding dance. For much more on tango music, Click here.\n\n######  More Tango Info\n\nSome of the most complete tango listings are in free tango booklets around town, including El Tanguata (www.eltanguata.com) and La Milonga Argentina (www.lamilongaargentina.com.ar). All have basic information on the city's _milongas,_ classes, teachers and shows. They're often available at tango venues or tourist offices. You can also check the Caseron Porte\u00f1o (www.caseronporteno.com) tango map for _milongas_ locations.\n\nFor a very practical book on tango in BA, check out Sally Blake's _Happy Tango: Sallycat's Guide to Dancing in Buenos Aires_ (2nd edition). It has great information on _milongas_ \u2013 how to dress for them and act in them and who you can expect to see \u2013 plus much more.\n\nIf you don't mind hiring a dance partner for classes or _milongas,_ check out www.tango\u00adtaxidancers.com. There are, of course, many tango clothing and shoe stores in BA. Several accommodations cater to tango enthusiasts, including Caser\u00f3n Porte\u00f1o (Click here). All offer on-site classes. Finally, if you're in town in mid- to late August, don't miss the tango festival (www.tangobuenosaires.gov.ar).\n\n###  NEED TO KNOW\n\n\u00bb For discount tickets, show and venue descriptions and some reviews, check out www.tangotix.com.\n\n\u00bb _Milongas_ either start in the afternoon and run until 11pm or start at around midnight and run until the early-morning light (arrive late for the best action). They're affordable, and classes are often offered beforehand.\n\n\u00bb For a unique outdoor experience, head to the bandstand at the Barrancas de Belgrano, where the casual _milonga_ La Glorieta (www.glorietadebelgrano.com.ar) takes place on Saturday and Sunday evenings around 7pm (and possibly other evenings). Tango classes are also given.\n\n#####  Lonely Planet's Top Choices\n\n Caf\u00e9 de los Angelitos Well-put-together, imaginative show with great visual appeal.\n\n Feria de San Telmo Best for its casual ambience and price \u2013 a few coins!\n\n Salon Canning Traditional, very popular and well-located _milonga_.\n\n#####  Best Fancy Shows\n\n Caf\u00e9 de los Angelitos Well choreographed, with impressive costumes and props.\n\n Rojo Tango Very intimate, cabaret-style show that's supremely sexy.\n\n El Viejo Almac\u00e9n Great athleticism, small venue and great folkloric segment.\n\n La Ventana Good overall show with comedic gaucho swinging _boleadoras_ (hunting weights).\n\n#####  Best Less-Fancy Shows\n\n Feria de San Telmo It's street-donation tango at San Telmo's Sunday market \u2013 hustle for a good view!\n\n Caf\u00e9 Tortoni Decent basement show in BA's oldest, most traditional cafe.\n\nLos 36 Billares Another very historic cafe with tango show, but much less touristy than Tortoni.\n\n#####  Best Milongas\n\n Salon Canning Famous, popular and stylish _milonga_ in Palermo, with good music.\n\n Confiter\u00eda Ideal BA's most historic tango venue and the set for Sally Potter's _The Tango Lesson_.\n\n La Catedral Casual, bohemian warehouse space that attracts hip young dancers.\n\n La Marshall Gay-friendly _milonga_ where everyone is welcome \u2013 and where role reversals are OK.\n\nPainting of Carlos Gardel (Click here) near Museo Casa Carlos Gardel  \nDANITA DELIMONT \/ GETTY IMAGES \u00a9\n Shopping\n\nDespite a global recession and a drop in the purchasing power of the Argentine peso, porte\u00f1os continue to shop as if there's no tomorrow. A peek into the nearest mall on a weekend will make you wonder how people who seem to be making so little can spend so much. As the saying goes, 'An Argentine will make one peso and spend two'.\n\n######  Specialties & Souvenirs\n\nWine is one of the more obvious gifts, though it's hard to carry. Some stores will ship outside Argentina; expect to pay a premium for this service. Food items that make nice gifts are _dulce de leche_ (a delicious milk caramel that Argentines have perfected) and _alfajores,_ cookie sandwiches usually bathed in chocolate (Havanna is a popular brand and available at Ezeiza Airport). _Mate_ gourds are also good, and they're small and light.\n\nArgentina is known for its leather goods. There are leather stores all over the city, but for the best prices head to Calle Murillo. Silverwork is also high quality, and many items are gaucho-inspired. Looking for a gift for that aristocratic friend? There are a few polo stores with items that might fit the bill \u2013 whether or not he or she plays polo.\n\nFinally, soccer memorabilia always make popular souvenirs \u2013 especially from Boca, the most well-known team.\n\n######  Street Markets\n\nWandering through a weekend _feria_ (street market) is a quintessential BA experience. Artisans display their wares while buskers, mimes and tango dancers entertain. Often there are nearby restaurants with sidewalk tables for people-watching. At some of the more touristed markets, especially Feria de San Telmo, watch for pickpockets.\n\n\u00bbFeria Plaza Francia Click here\n\n\u00bbFeria de Mataderos Click here\n\n\u00bbFeria de San Telmo Click here\n\n\u00bbFeria Plaza Serrano Click here\n\n######  Antique Markets & Shopping Malls\n\nA couple of antique markets might be worth your time. Try Mercado de las Pulgas (Click here) or Mercado de San Telmo (Click here). Don't expect dirt-cheap bargains, though you might find a cool glass soda bottle or vintage lamp. Feria de San Telmo is a fun place to look for old coins and jewelry, though there's a lot of kitsch as well. The San Telmo neighborhood has some pricey antique stores too.\n\nMany of the bigger shopping malls in BA are slick and modern; some cater to families with children by offering special play areas and video arcades. Paseo Alcorta (Click here) has an especially large kids' playground on the 3rd floor, while Mercado de Abasto (Click here) sports an excellent children's museum and small amusement park complete with rides. Almost all of these malls also have multiplex cinemas and large food courts complete with fast-food outlets and ice-cream parlors. Expect all the popular chain stores; some even offer health clubs, beauty shops and internet cafes.\n\n######  High Fashion\n\nInterested in clothing design? Then make a beeline for Palermo Soho (Click here), where avant-garde fashion designers' boutiques grace the pretty tree-shaded streets. After the 2001 economic crash, dozens of young designers emerged from the woodwork to set up shop in this then-affordable neighborhood (rents have gone way up since then, driving some out). Some made it big, maturing into fully fledged designers with luxury sportswear lines and outposts in the US, Europe and Asia. Names you may come across include Maria Cher (known for deconstructed garments with an urban twist), Jazm\u00edn Chebar (with playful, feminine designs) and Mart\u00edn Churba (known for recycling fabrics). Cora Groppo and Jessica Trosman are other big names with chain stores in Buenos Aires malls and elsewhere.\n\nIf you're looking for leather bargains, avoid Calle Florida and head to the shops on Calle Murillo's 600 block, in the neighborhood of Villa Crespo. This is the best place in town to snag a relatively cheap but high-quality leather jacket and accessories. Bargain like mad, especially if you're paying in cash. One of the nicer (and pricier) shops is Murillo 666.\n\nFor outlet shopping there's the 800 block of Calle Aguirre, with deals on shoes and clothes. Ladies, check out the Pr\u00fcne outlet for stylish leather bags. There are also lots of other outlets on nearby Av C\u00f3rdoba.\n\nThe largest concentration of jewelry shops is on Libertad south of Av Corrientes.\n\n###  NEED TO KNOW\n\n####  Opening Hours\n\nStore hours generally run from 9am or 10am to 8pm or 9pm weekdays, with many open for a few hours on Saturday. Most stores close on Sunday.\n\n####  Taxes & Refunds\n\nTaxes are included in prices; what you see is what you pay.\n\nIf you buy more than AR$70 in merchandise from a store that displays a 'Tax Free Shopping' sticker, you're entitled to a tax refund. Just ask the merchant to make out an invoice for you (you'll need ID); upon leaving the country show the paperwork to a customs official, who'll stamp it and tell you where to obtain your refund. Give yourself some extra time at the airport for this transaction.\n\n####  Bargaining\n\nBargaining is not acceptable in stores, except possibly for high-price items like jewelry and leather jackets (in some places). Some shops will give a _descuento_ (discount) for cash payments. At street markets you can try negotiating, but keep in mind you may be talking to the artists themselves.\n\nBe clear about whether the vendor is quoting in pesos or dollars. Always check your change before walking away with your purchase, and keep a lookout for fake bills (Click here).\n\n#####  Shopping by Neighborhood\n\n\u00bb  The Center The area on and around Calle Florida offers modern shops selling pretty much everything.\n\n\u00bb  Congreso & Tribunales Not known for its shopping, though there are discount bookstores along Av Corrientes.\n\n\u00bb  San Telmo _The_ place for antique stores, with clothing and other boutiques here and there.\n\n\u00bb  Retiro Bustling Av Santa Fe starts here and heads through Palermo, lined the whole way with shops of every kind.\n\n\u00bb  Recoleta & Barrio Norte Upscale stores selling the city's most expensive threads and leather products live on Av Alvear.\n\n\u00bb  Palermo Best known for its locally designed clothing stores, with plenty of housewares shops and boutiques.\n\n\u00bb  South of Palermo Av Pueyrred\u00f3n near Once train station has cheap goods made in countries like China.\n\n#####  Lonely Planet's Top Choices\n\n Walrus Books A terrific range of new and used books, plus Argentine classics translated into English.\n\n Zival's Music store with a great selection of tango, jazz and classical.\n\n Wildlife Meets all your outdoor-gear needs.\n\n Lo de Joaquin Alberdi A wine-lover's paradise; offers tastings too.\n\n Gil Antiguedades Gorgeous vintage clothes are the star here, though there are _objets_ too.\n\n Autor\u00eda Ingenious, edgy, high-quality art and accessories, with an emphasis on Argentine designers' work.\n\n#####  Best for Clothing\n\n Rapsodia At times exotic clothing utilizing various genres and different textiles.\n\n Juana de Arco Frilly, silky, cute, sexy and very feminine items for the girl inside every woman.\n\n Hermanos Estebecorena Cutting-edge, creative and stylish clothes for men.\n\n Bolivia Metrosexual designs for men who aren't afraid of patterns and pastels.\n\n Punto Sur Dozens of designers stock the racks here with awesome, creative clothing.\n\n#####  Best for Wacky Gifts\n\n Materia Urbana Funky things like leather animal desk accessories and wood jewelry.\n\n Autor\u00eda Creative, high-quality and well-priced contemporary items made by local designers.\n\n Cualquier Verdura Expect the unexpected at this fun shop with eclectic and novelty gifts.\n\n L'Ago Kitschy home decor like colorful metal _mate_ sets, paper lamps and vintage-look pillows.\n\n Calma Chicha Cow- and sheepskin rugs, fun tablecloths, leather bags and more.\n\n#####  Best for Argentine Souvenirs\n\n Feria de San Telmo Everything-goes, multi-block street fair selling anything you can think of.\n\n Arte y Esperanza Fair-trade Argentine souvenirs handmade by Argentina's indigenous peoples.\n\n Feria Plaza Serrano Fun hippie products created by local craftspeople.\n\n Harapos Patagonia Woollen goods, alpaca jewelry, wood and ceramics from Patagonia.\n\n Nobrand T-shirts, mugs and notebooks stamped with iconic Argentine symbols.\n Sports & Activities\n\nWhen it comes to spectator sports, only one thing really matters to most porte\u00f1os \u2013 _f\u00fatbol_ (soccer). If you go to a game \u2013 or even watch one on TV \u2013 you'll witness human passion to the core. But other spectator sports also exist in Buenos Aires. And for those who'd rather play than watch, you'll have opportunities to run, bike, swim and even rock climb \u2013 though some activities will be harder to seek out than others.\n\n######  Spectator Sports\n\n######  F\u00daTBOL\n\n_F\u00fatbol_ is a national obsession, and witnessing a live game is an integral part of the BA experience. This is no amateur league \u2013 Argentina's national team won the World Cup in both 1978 and 1986 (one of only eight nations to have ever won the cup). The men's team also walked away with gold at the 2004 and 2008 summer Olympics. And Lionel Messi (Click here), currently Argentina's most famous player, has won FIFA's World Player of the Year (or Ballon d'Or) award _four_ times \u2013 from 2009 to 2012.\n\nArgentines are avid fans of the sport, and on game day (and there are many) you'll see TVs everywhere tuned to the soccer channels. Cheers erupt when goals are scored, and after a big win, cars sporting team flags go honking by \u2013 especially around the Obelisco.\n\nFor more information on Argentine _f\u00fatbol,_ see www.futbolargentino.com and www.afa.org.ar. Or check Daniel Schweimler's musings (via the team Argentina Juniors) at www.handofdan.com.\n\n###  DON'T JUST WATCH \u2013 PLAY F\u00daTBOL!\n\nInspired by watching professional _f\u00fatbol_ teams play the game? Well, you can partake yourself \u2013 just contact FC Buenos Aires F\u00fatbol Amigos (www.fcbafa.com) to join fellow travelers, expats and locals for fun on the pitch. There's a modest charge for the experience, but _asados_ (barbecues) often lie at the end of the _f\u00fatbol_ rainbow \u2013 and the sporty memories can be priceless.\n\n######  GOING TO A GAME\n\nIn a land where Maradona (Click here) is God, going to see a _f\u00fatbol_ game can be a religious experience. The _supercl\u00e1sico_ match between the Boca Juniors and River Plate has been called the number-one sporting event to see before you die, but even the less-celebrated games will give you insight into Argentina's national passion.\n\nAttending a regular match isn't too difficult. Keep an eye on the clubs' websites, which inform when and where tickets will be sold; often they're sold at the stadium before the game. You'll get a choice between _populares_ (bleachers) and _plateas_ (seats). Avoid the _populares_ , as these can get far too rowdy and sometimes dangerous.\n\nIf you want to see a _cl\u00e1sico_ \u2013 a match between two major teams \u2013 getting a ticket will be much harder. Plus Boca doesn't even put tickets for its key matches on sale; all tickets go to _socios_ (members). Instead, you're better off going with an agency such as Tangol (Click here) or via organizations like www.fcbafa.com or www.landingpad.com. It won't be cheap, but it's much easier (and safer) getting a ticket this way; fake tickets do exist.\n\nIf you want to chance getting your own _cl\u00e1sico_ or _supercl\u00e1sico_ ticket, however, you can always look online at www.buenosaires.craigslist.org or www.mercadolibre.com.ar. And if you're confident in your bargaining skills, scalpers will always exist.\n\nDress down, and try to look inconspicuous when you go. Take only minimum cash and keep your camera close. You probably won't get in with water bottles, and food and drink in the stadium is meager and expensive. Arrive early to get a good seat and enjoy the insane build-up to the game. And most importantly: don't wear the opposing team's colors.\n\nThe following are some of the clubs based in Buenos Aires:\n\n\u00bb Estadio Argentinos Juniors (  4551-6887; www.argentinosjuniors.com.ar; Gavil\u00e1n 2151)\n\n\u00bb Boca Juniors Map Click here;   4309-4700; www.bocajuniors.com.ar; Brandsen 805 A popular club in Buenos Aires.\n\n\u00bb Club Atl\u00e9tico V\u00e9lez S\u00e1rsfield (  4641-5663; www.velezsarsfield.com.ar; Juan B Justo 9200)\n\n\u00bb Club Deportivo Espa\u00f1o (  4619-1516; www.cde.com.ar; Santiago de Compostela 3801)\n\n\u00bb Club Ferro Carril Oeste (  4431-8282; www.ferrocarriloeste.org.ar; Federico G Lorca 350)\n\n\u00bb Club Hurac\u00e1n (  4911-0757; www.clubahuracan.com.ar; Av Caseros 3159)\n\n\u00bb River Plate (  4789-1200; www.cariverplate.com; Alcorta 7597)\n\n\u00bb San Lorenzo de Almagro (  4918-4237; www.sanlorenzo.com.ar; Varela 2680)\n\n######  BASKETBALL\n\nThe basketball scene in Buenos Aires has been picking up significantly since 2002, when Argentina's men's team played in the World Basketball Championship in Indianapolis. They only won silver but made history by beating the US 'Dream Team' in international competition. Then, with a similar roster, they defeated the US squad again (along with Italy in the finals) to win gold in the 2004 summer Olympics \u2013 their first Olympic medal in basketball ever. No team had beaten the Americans in the Olympics since 1992, when pro basketball players were allowed to play. They also won the FIBA Americas Championship in 2011.\n\nArgentina's best players include Emanuel 'Manu' Ginobili, Fabricio Oberto, Andr\u00e9s Nocioni, Luis Scola, Pablo Prigioni, Walter Herrmann and Carlos Delfino, all of whom have played for or currently play in the NBA.\n\nToday BA has several major squads, the most popular being Boca Juniors. You can watch them play in La Boca at Estadio Luis Conde (La Bombonerita; www.boca-basket.com.ar; Arzobispo Espinosa 600). Other popular basketball teams include Obras Sanitarias and Ferro Carril Oeste.\n\n######  RUGBY\n\nRugby is getting more popular by the year in Argentina, in part because the country's national team \u2013 Los Pumas \u2013 has done well in past years. After placing third at the Rugby World Cup in 2007 (no mean feat), Los Pumas was rated the best rugby team in the Americas. And at the 2011 Rugby World Cup it put in a pretty decent showing.\n\nIn Buenos Aires, the long-running Club Atl\u00e9tico de San Isidro (www.casi.org.ar) is the capital's best rugby team; in 1935 it gave birth to its own biggest rival, the San Isidro Club (www.sanisidroclub.com.ar).\n\nRugby season runs from April to October; contact the Uni\u00f3n de Rugby de Buenos Aires (www.urba.org.ar) for current happenings. Fanatics can visit the Museo de Rugby (  4732-2547; www.museodelrugby.com; Juan Bautista de Lasalle 653) in San Isidro.\n\n######  HORSE RACING\n\nRaces in BA are held at the Hip\u00f3dromo Argentino, a grand building designed by French architect Louis Faur\u00e9-Dujarric that dates from 1908 and holds up to 100,000 spectators. Race times vary, so check the schedule for details. The most important races take place in November, both here and at San Isidro's famous grass racetrack.\n\n######  POLO\n\nAdd Argentina's history of gauchos and horses to its past British influence, and you'll understand why the best polo in the world is played right here. The country has dominated the sport for over 70 years, boasting most of polo's top players. Forget those British princes: the world's best player is considered to be the handsome Adolfo Cambiaso.\n\nMatches take place in Buenos Aires from September to mid-November. They culminate in the annual Campeonato Argentino Abierto de Polo (Argentine Open Polo Championship) \u2013 the world's most prestigious polo tournament \u2013 in Palermo's Campo Argentina de Polo. For current information, contact the Asociaci\u00f3n Argentina de Polo (  4777-6444; www.aapolo.com), which keeps a schedule of polo-related activities throughout the country.\n\nFor polo camps (all outside BA) where you can learn to play yourself, check out www.argentinapoloday.com.ar, www.poloelite.com and www.lasofiapolo.info.\n\n######  PATO\n\nOf gaucho origins, the polo-like game of _pato_ (literally 'duck') takes its name from the original game ball \u2013 a live duck encased in a leather bag. The unfortunate fowl has since been replaced by a ball with leather handles, and players no longer face serious injury in what was once a very violent sport.\n\nFor information on _pato_ matches and tournaments (which usually take place 30km outside the city in the Campo Argentino de Pato), contact the Federaci\u00f3n Argentina de Pato (  4342-5271; www.pato.org.ar). The national championships occur in December, and are more centrally located in Palermo's polo grounds.\n\n###  PER\u00da BEACH\n\nAn interesting sports complex for those seeking outdoor activities is Per\u00fa Beach (  4793-5986; www.peru-beach.com.ar; Elcano 794;   8am-midnight). Short soccer fields, a covered roller rink, a freestanding climbing wall and water sports such as kayaking all bring in the crowds. In addition there's also a grassy lawn and outdoor tables for refreshments \u2013 great on a sunny day. It's more of a social scene than anything else, and families are welcome. Per\u00fa Beach is located in Acassuso, a suburb way north of Buenos Aires' center, just across from the Tren de la Costa's Barrancas station.\n\n######  Activities\n\nBuenos Aires is a big concrete city, so you'll have to seek out the outdoor spots in which to work out. Extensive greenery in Palermo provides good areas for recreation, especially on weekends when the ring road around the rose garden is closed to motor vehicles. Recoleta also has grassy parks, but not as extensive. Best of all is the Reserva Ecol\u00f3gica Costanera Sur (Click here), an ecological paradise just east of Puerto Madero that might just make you forget you're in a big city; it's excellent for walks, runs, leisurely bike rides and even a bit of wildlife viewing.\n\n######  CYCLING\n\nThe city's new bike lanes are making cycling in the center a safer proposition, but there are better places in which to spin your wheels. Bike paths run along many roads in Parque 3 de Febrero (Click here) \u2013 here, bicycle rentals are available in good weather on weekends, when the ring road is closed to motor vehicles. Look for rental companies along Av de la Infanta Isabel; four-wheeled pedal carts and inline skates can also be rented.\n\nFor safe family cycling, head to Nuevo Circuito KDT in Palermo's Parque General Belgrano. Here, Sprint Haupt (  4804-2870; www.sprinthaupt.com.ar; Salguero 3450;   9am-8:30pm Tue & Thu, to 7pm Wed, Fri, Sat & Sun) rents bicycles for use around a plain, 1250m-long concrete bike path (bring your passport). Helmets available. Look for the overpass parking lot, then go past the pedestrian bridge.\n\nThe Reserva Ecol\u00f3gica Costanera Sur on the eastern side of Puerto Madero along the coast, is green and tranquil and has some flat dirt roads that are great to bike on. Cheap bicycle rentals are available in good weather on weekends, just outside either entrance.\n\n######  GOLF\n\nBA's most convenient course is the 18-hole Campo Municipal de Golf (  4772-7261; Tornquist 6397;   7:30am-5pm Tue-Sun); be sure to reserve your spot in advance. Practice your long shots at the Costa Salguero Driving Range (  4805-4732; www.costasalguerogolf.com.ar; Avs Costanera R Obligado & Salguero), which also has a golf store, a cafe and a nine-hole, family-friendly course.\n\n###  PARQUE NORTE\n\nWhen the temperatures and humidity skyrocket, head north to this large water park. Parque Norte (Click here) is great for families, with huge shallow pools (perhaps 4ft at their deepest), plus a large water slide and lots of umbrellas and lounge chairs (both cost extra). There are plenty of grassy areas in which to enjoy a picnic or _mate_. Bring your own towels.\n\n######  SWIMMING\n\nSome upscale hotels have decent-size pools, but they charge hefty prices for nonguests (so hefty you might as well stay there). The fee generally includes gym use, at least. Try the Panamericano Hotel (www.panamericano.us), whose pool has the best view in BA.\n\nA more economical option is to find a health club with an indoor pool; Megatlon (www.megatlon.com) is a popular gym with many branches. You can also try the pool at Parque General Belgrano (  4807-7918; Salguero 3450; park AR$8, pool AR$20;   10am-7pm Sat & Sun Jan, daily Feb), in Palermo. For a more casual environment, especially with kids, head to Parque Norte.\n\n######  TENNIS\n\nA few places in BA offer courts, such as Parque General Belgrano (  4807-7879; Salguero 3450; park entry AR$8, court hire per hr AR$50-60;   8:30am-midnight Mon-Fri, to 8pm Sat & Sun), in Palermo. Bring your own racquet from home if you're serious about getting in touch with the Nalbandian or del Potro inside you.\n\n######  HORSEBACK RIDING\n\nIf you want to get out of town for a few hours and hop on a horse, forget those touristy estancias (ranches) and check out Caballos a la Par (  15-5248-3592, 4384-7013; www.caballos-alapar.com). Guided, private rides are given in a provincial park about an hour's drive from central Buenos Aires, and it's not just one of those 'follow-the-horse-in-front' deals. They'll take you around woodsy lanes and fields, and you'll have fun learning how to ride and even gallop on the fine horses.\n\n###  NEED TO KNOW\n\n\u00bb Bike rentals are available at Palermo's Parque 3 de Febrero (where you can also rent inline skates, quadricycles and pedal boats), Reserva Ecol\u00f3gica Costanera Sur and via some bike-tour companies (Click here).\n\n\u00bb Speaking of bikes, Masa Critica (Critical Mass; a semi-organized bike ride for thousands) is alive and well here \u2013 and a heap of fun to partake in (Click here).\n\n\u00bb In November, Buenos Aires' Marathon (www.maratondebuenosaires.org) is the southern hemisphere's biggest, attracting nearly 27,000 runners annually.\n\n\u00bb Feel the need for yoga ? There are many places that offer it, including some with instructors that speak English. Try Buena Onda Yoga (www.buenaondayoga.com), started by expats; it offers classes in Palermo and San Telmo.\n\n\u00bb Rock climbers should head to Punto Cumbre (www.puntocumbre.com), a small climbing wall located inside a Megatlon gym. You'll be provided with belays if you need them. There's a small bouldering cave, too, along with classes and excursions.\n\n#####  Sports & Activities by Neighborhood\n\n\u00bb Puerto Madero Reserva Ecol\u00f3gica Costanera Sur is great for running, bicycling or walking. These activities are also possible along the dikes' cobbled lanes.\n\n\u00bb Palermo Provides most of central BA's green spaces, along with tennis and golf courses, running paths and bicycling lanes.\n\n# Explore Buenos Aires\n\n### The Center\n\n#### Top Sight: Plaza de Mayo\n\n### Puerto Madero\n\n#### Top Sight: Reserva Ecol\u00f3gica Costanera Sur\n\n### Congreso & Tribunales\n\n#### Top Sight: Teatro Col\u00f3n\n\n### San Telmo\n\n#### Top Sight: Plaza Dorrego\n\n### La Boca\n\n#### Top Sight: El Caminito\n\n### Retiro\n\n#### Top Sight: Palacio Paz\n\n### Recoleta & Barrio Norte\n\n#### Top Sight: Cementerio de la Recoleta\n\n### Palermo\n\n#### Top Sight: Museo de Arte Latinoamericano de Buenos Aires (Malba)\n\n#### Top Sight: Parque de Febrero\n\n### South of Palermo\n\n### Day Trips\n\n### Sleeping\nNeighborhoods at a Glance\n\n The Center\n\nBuenos Aires' Center is where bustle meets hustle and endless lines of business suits move hastily along the narrow streets in the shadow of skyscrapers and old European buildings. Stretching from Retiro to San Telmo, this downtown area is the heart and brains of the city, and made up of the sub-neighborhoods of the Microcentro and Montserrat.\n\n Puerto Madero\n\nBA's youngest and least conventional barrio, Puerto Madero is home to old brick warehouses that have been converted into some of the city's trendiest lofts, offices, hotels and restaurants. Cobbled promenades make walking a pleasure for pedestrians, and there are plenty of upscale restaurants and cafes to check out.\n\n Congreso & Tribunales\n\nCongreso is an interesting neighborhood full of old-time cinemas, theaters and bustling commerce tinged with a hard-core political flavor. The buildings still hold a European aura, but there's more local feel, faded-glory atmosphere and grittiness than in the Center. It's a great place to wander around and explore.\n\n San Telmo\n\nSan Telmo is a lovely neighborhood full of cobbled streets, colonial mansions and rich history. Only a quick walk south of Plaza de Mayo, it's like stepping 100 years into the past. As a popular tourist destination it's been gentrifying fast, and some wonder if it might become the next Palermo.\n\n La Boca\n\nBlue-collar and raffish to the core, La Boca is very much a locals' neighborhood. Its colorful shanties are often portrayed as a symbol of Buenos Aires, while El Caminito is the barrio's most famous street, full of art vendors, buskers and tango dancers twirling for your spare change.\n\n Retiro\n\nWell located right between the Center and Recoleta, exclusive Retiro is one of the ritz\u00adiest neighborhoods in Buenos Aires. Giant mansions, art deco apartments and other landmark buildings characterize this area, along with Plaza San Mart\u00edn \u2013 a pleasant grassy park on a hill overlooking the Retiro train and bus stations.\n\n Recoleta & Barrio Norte\n\nRecoleta is where the rich live in luxury apartments and mansions while spending their free time sipping coffee at elegant cafes and shopping in expensive boutiques. Full of lush parks, grand monuments, French architecture and wide avenues, Recoleta is also famous for its cemetery.\n\n Palermo\n\nPalermo's large, grassy parks \u2013 regally punctuated with grand monuments \u2013 are popular destinations on weekends, when families fill the shady lanes, cycle the bike paths and paddle on the peaceful lakes. The sub-neighborhood of Palermo Viejo is home to dozens of ethnic restaurants, bars, nightclubs and shops, along with the city's largest selection of boutique hotels.\n\n South of Palermo\n\nThe neighborhoods south of Palermo are part of the 'real' Buenos Aires largely unaffected by the tourist trade. Villa Crespo is up and coming, benefiting from its proximity to Palermo; Abasto and Once are cultural melting pots and busy commercial districts; and Boedo has bohemian flavor and some very traditional cafes.\n\n# The Center\n\nMICROCENTRO | MONTSERRAT\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nThe Center\n\nMICROCENTRO | MONTSERRAT\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Hanging out at  Plaza de Mayo, steeped in history and surrounded by some of Buenos Aires' most important buildings, including the main cathedral, the Cabildo, the Museo del Bicentenario and \u2013 last but not least \u2013 Casa Rosada, where Argentina's president's office is located.\n\n Strolling down  Calle Florida to see BA's hustle and bustle at its most intense.\n\n Shopping at  Galer\u00edas Pac\u00edfico, a beautiful shopping mall with an amazing painted ceiling.\n\n Visiting historic  Manzana de las Luces, a symbol of the city's culture and higher learning.\n\n Taking a break at  Caf\u00e9 Tortoni, one of BA's most traditional \u2013 and touristy \u2013 cafes.\n\n###  Explore: the Center\n\nDuring the day, the Center is a heaving mass of humanity moving hastily along narrow streets in the shadow of skyscrapers and old European buildings \u2013 but in the evening, it's practically deserted. Stretching from Retiro to San Telmo (and flanked by Congreso and Puerto Madero), this area is the heart and brain of the city. It's made up of the Microcentro and Montserrat neighborhoods.\n\nPlaza de Mayo is a good place to start. Here you'll see the Casa Rosada presidential palace, with the Museo del Bicentenario right behind it; plan ahead if you want to visit, as it's closed Monday and Tuesday. The Cathedral Metropolitana is nearby \u2013 stop by for a Pope Francis souvenir \u2013 and the Cabildo has a nice back patio that's good for a break. If you want to see the Madres de la Plaza de Mayo, time your visit for Thursday afternoon (Click here).\n\nFrom here you can head south just one block, crossing over into Montserrat, and visit the Manzana de las Luces, a city block full of historic buildings. There are also a couple of small but interesting museums to visit around here. Further south a few more blocks is San Telmo (Click here).\n\nIf you're going north, walk on Calle Florida. This very busy pedestrian street is packed during the day with business men and women, street vendors, tourists, beggars, buskers and hustlers. Every few feet you'll hear _arbolitos_ (street money changers, called 'little trees' because they stand in place) chanting ' _cambio, cambio, cambio_ '. Use their services at your own risk! If you prefer more peace, Reconquista and Suipacha are also p edestrian.\n\n###  Local Life\n\n\u00bb Shopping Pedestrian streets Florida and Lavalle are lined with shops and services, and there's great people-watching too.\n\n\u00bb Hanging out Stop for a cup of coffee or \u2013 if it's later \u2013 a stiff drink at one of the many downtown cafes and bars, alongside locals who're taking a break.\n\n\u00bb Tango True aficionados head to historic Confiter\u00eda Ideal (Click here) for classes, shows and _milongas_ (dance events).\n\n###  Getting There & Away\n\n\u00bb Bus Take bus 29 from San Telmo; 29, 64 and 152 from La Boca; 59 from Recoleta; 29, 59, 64 and 152 from Palermo's Plaza Italia.\n\n\u00bb Subte Nearly all Subte lines radiate from the Center, going either north\u2013south from Retiro to San Telmo, or towards Palermo and other points west.\n\n####  Lonely Planet's Top Tip\n\nBecause there are so many business people in the Center, many restaurants offer _menu ejecutivos_ \u2013 or lunch specials \u2013 to attract this valuable clientele. These set lunches are offered weekdays and usually consist of a main course with dessert and drink, all for a reasonable fixed price. Sometimes an appetizer is included as well. It's a good way to try out otherwise pricey restaurants.\n\nParts of downtown can be a bit sketchy at night, so tread carefully.\n\n  Best Places to Eat\n\n\u00bbTomo 1\n\n\u00bbAldo's Vinoteca\n\n\u00bbGranix\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbCaf\u00e9 Tortoni\n\n\u00bbLa Cigale\n\n\u00bbLondon City\n\nFor reviews, Click here \u00bb\n\n  Best Museums\n\n\u00bbMuseo del Bicentenario\n\n\u00bbMuseo de la Ciudad\n\n\u00bbMuseo Etnogr\u00e1fico Juan B Ambrosetti\n\nFor reviews, Click here \u00bb\n\nWALTER BIBIKOW \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nPLAZA DE MAYO\n\nPlaza de Mayo is the political, social and symbolic center of Buenos Aires. Surrounded by the Casa Rosada, the Cabildo and the city's main cathedral, this plaza is ground zero for the city's political rallies and protests \u2013 both peaceful and vehement. When the plaza isn't full of activists, however, it attracts camera-toting tourists enjoying the sights \u2013 along with the occasional camera thief.\n\n###  DON'T MISS...\n\n\u00bb The bullet holes left on the side of the Ministerio de Econom\u00eda as a symbolic reminder of past intolerance.\n\n\u00bb The balconies where Juan and Evita Per\u00f3n preached to their loving masses.\n\n\u00bb The interior of the Banco de la Naci\u00f3n building \u2013 but only if you're an architecture fan!\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb cnr Av de Mayo & San Mart\u00edn\n\nWhen Juan de Garay refounded Buenos Aires in 1580, he laid out the large Plaza del Fuerte (Fortress Plaza) in accordance with Spanish law. Later called the Plaza del Mercado (Market Plaza), then the Plaza de la Victoria (after victories over British invaders in 1806 and 1807), the plaza acquired its present name of Plaza de Mayo after the date Buenos Aires declared independence from Spain: May 25, 1810.\n\nAt the center of the plaza is the Pir\u00e1mide de Mayo, a white obelisk built to mark the first anniversary of BA's independence from Spain. Looming on the plaza's northern side is the headquarters of Banco de la Naci\u00f3n (1939), the work of famed architect Alejandro Bustillo. Most other public buildings in this area belong to the late 19th century, when the Av de Mayo first connected the Casa Rosada with the Plaza del Congreso, obliterating most of the historic and dignified Cabildo in the process.\n\nPlaza de Mayo is famously known as being the preferred site of many civil protests; note the unsightly barricades separating the plaza in two, meant to discourage large numbers of _piqueteros_ (picketers) from congregating. But these barricades haven't prevented the Madres de la Plaza de Mayo \u2013 the mothers of the 'disappeared children' during the the Dirty War military dictatorship between 1976 and 1983 \u2013 from marching around the plaza every Thursday afternoon at 3:30pm since 1977. Originally they demanded a full account of the atrocities that occurred during this war, but in 2006 they declared a truce with N\u00e9stor Kirchner's presidency, as he was sympathetic to their requests. To this day they march on, but as a reminder of the past \u2013 and for other social-justice causes.\n\nCasa Rosada\n\nTaking up the whole eastern side of the Plaza de Mayo is the unmistakeable pink facade of the Casa Rosada MAP GOOGLE MAP (Pink House;), the presidential palace that was begun during the presidency of Domingo F Sarmiento. It occupies a site where colonial riverbank fortifications once stood; today, however, after repeated landfills, the palace stands more than 1km inland. The offices of 'La Presidenta' Cristina Kirchner are here (a small raised pennant, under Argentina's national flag, notes her presence in the building), but the presidential residence is in the calm suburb of Olivos, north of the center.\n\nThe side of the palace that faces Plaza de Mayo is actually the back of the building. It's from these balconies that Juan and Eva Per\u00f3n, General Leopoldo Galtieri, Ra\u00fal Alfons\u00edn and other politicians have preached to throngs of impassioned Argentines when they felt it necessary to demonstrate public support. Madonna also crooned from here for her movie _Evita_.\n\nThe salmon-pink color of the palace, which positively glows at sunset, could have come from President Sarmiento's attempt at making peace during his 1868\u201374 term (by blending the red of the Federalists with the white of the Unitarists). Another theory, however, is that the color comes from painting the palace with bovine blood, which was a common practice back in the late 19th century.\n\nOff-limits during the military dictatorship of 1976\u201383, the Casa Rosada is now reasonably accessible to the public. Free half-hour tours (  4344-3600;   10am-6pm Sat & Sun) are given.\n\nUnderneath the Casa Rosada, excavations have unearthed remains of the Fuerte Viejo, a ruin dating from the 18th century. These are accessible via entry to the Museo del Bicentenario.\n\nIn 1955 naval aircraft strafed the Casa Rosada and other nearby buildings during the Revoluci\u00f3n Libertadora, which toppled Juan Per\u00f3n's regime. On the northern side of the appropriately bureaucratic Ministerio de Econom\u00eda, an inconspicuous plaque commemorates the attacks (look for the bullet holes to the left of the doors). The inscription translates as, 'The scars on this marble were the harvest of confrontation and intolerance. Their imprint on our memory will help the nation achieve a future of greatness'.\n\nTowering above the Casa Rosada, just south of Parque Col\u00f3n on Av Col\u00f3n, is the army headquarters at the Edificio Libertador, the real locus of Argentine political power for many decades. It was built by military engineers inspired by the beaux arts Correo Central. A twin building planned for the navy never got off the ground.\n\nMuseo del Bicentenario\n\nBehind the Casa Rosada you'll notice a glassy wedge marking this airy and sparkling \u2013 and free! \u2013 underground museum (  4344-3802; www.museobicentenario.gob.ar; cnr Avs Paseo Col\u00f3n & Hip\u00f3lito Yrigoyen;   10am-6pm Wed-Sun Apr-Nov, 11am-7pm Wed-Sun Dec-Mar) housed within the brick vaults of the old _aduana_ (customs house). Head down into the open space, which has over a dozen side rooms \u2013 each dedicated to a different era of Argentina's tumultuous political history. There are mostly videos (in Spanish) and a few artifacts to see, along with temporary art exhibitions and an impressive restored mural by Mexican artist David Alfaro Siqueiros. A pleasant cafe-restaurant provides nourishment and rest.\n\nCatedral Metropolitana\n\nThis solemn cathedral (  7:30am-6:30pm Mon-Fri, 9am-7pm Sat & Sun) was built on the site of the original colonial church and not finished until 1827. It's a significant religious and architectural landmark, and carved above its triangular facade and neoclassical columns are bas-reliefs of Jacob and Joseph. The spacious interior is equally impressive, with baroque details and an elegant rococo altar.\n\nMore importantly, however, the cathedral is a national historical site that contains the tomb of General Jos\u00e9 de San Mart\u00edn, Argentina's most revered hero. In the chaos following independence, San Mart\u00edn chose exile in France, never returning alive to Argentina (although in 1829 a boat on which he traveled sighted Buenos Aires on its way to Montevideo). Outside the cathedral you'll see a flame keeping his spirit alive.\n\nTours of the church and crypt (in Spanish) were suspended at research time, but check during your tenure to see if they're being offered again. If you want a Pope Francis souvenir (he's Argentine, after all), explore the tiny gift shop inside. Occasional free choir concerts are also on the docket.\n\nCabildo\n\nThis mid-18th-century town hall building is now a museum (  4342-6729; www.cabildonacional.com.ar; Bol\u00edvar 65; admission AR$10;   10:30am-5pm Wed-Fri, to 6pm Sat & Sun). It used to have colonnades that spanned Plaza de Mayo, but, unfortunately, the building of surrounding avenues destroyed them. Inside you'll find a few mementos of early-19th-century British invasions, some paintings in colonial and early independence style, and the occasional temporary exhibit. On Thursday and Friday a lively crafts market sets up in the patio \u2013 and the cafe here is a great place to relax. Tours in Spanish offered.\n\nThe Old Mustard Trick\n\nIt's one of the oldest tricks in the book. You're in a tourist hot spot like Plaza de Mayo, minding your own business, and suddenly someone tells you that there are 'bird droppings' (or another substance) on your clothing. While this kind stranger takes out their surprisingly handy tissues to clean you up, their friend is cleaning out your pockets or stealing your bag.\n\nMadonna sang 'Don't Cry for Me Argentina' from the Casa Rosada's balconies for her movie _Evita_. It was a controversial film as many Argentines were not happy that the actress \u2013 associated with skimpy outfits and sex \u2013 was chosen to represent their beloved heroine. Also from the Casa Rosada's balconies, a triumphant Diego Maradona hoisted the 1986 World Cup, to the delight of thousands of soccer fans below. And on December 20, 2001, President Fernando de La Rua fled the Casa Rosada's roof by helicopter as the economic crash climaxed.\n\n The Center\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SIGHTS\n\n###    Microcentro\n\nPLAZA DE MAYO PLAZA\n\nSee Click here.\n\nGALER\u00cdAS PAC\u00cdFICO LANDMARK\n\nMAP GOOGLE MAP\n\n(  5555-5110; cnr Avs Florida & C\u00f3rdoba;   10am-9pm) Covering an entire city block, this beautiful French-style building has fulfilled the commercial purpose that its designers envisioned when they constructed it in 1889. Galer\u00edas Pac\u00edfico is now a shopping center \u2013 dotted with lovely fairy lights at night \u2013 and boasts upscale stores along with a large food court (which has longer hours than the stores). The excellent Centro Cultural Borges takes up the top floor. Tours are offered at 11:30am from Monday to Friday, in English and Spanish.\n\nWhen you step inside, check out the ceiling. In 1945 the completion of a central cupola made space for a dozen paintings by muralists Antonio Berni, Juan Carlos Castagnino, Manuel Colmeiro, Lino Spilimbergo and Demetrio Urruch\u00faa. All were adherents of the _nuevo realismo_ (new realism) school, heirs of an earlier social-activist tendency in Argentine art. For many years the building went semi-abandoned, but a joint Argentine-Mexican team repaired and restored the murals in 1992.\n\nCENTRO CULTURAL BORGES CULTURAL CENTER\n\nMAP GOOGLE MAP\n\n(  5555-5359; www.ccborges.org.ar; cnr Viamonte & San Mart\u00edn) One of the best cultural centers in BA, with inexpensive but high-quality art exhibitions and galleries, cinema, music, lectures, classes and workshops. Tango lessons are also available.\n\nIGLESIA SANTA CATALINA CHURCH\n\nMAP GOOGLE MAP\n\n(  5238-6040; www.santacatalina.org.ar; Plaza San Mart\u00edn 705;   L\u00ednea B Florida, L\u00ednea C Lavalle) Santa Catalina was founded in 1745, when it became Buenos Aires' first convent. In 1806 British troops invaded the city, and in July 1807 they took shelter in the convent. The soldiers holed up here for two days, and despite damaging the property did not hurt the nuns. Today Santa Catalina is a church, and a peek inside reveals beautiful gilded works and a baroque altarpiece created by Isidro Lorea, a Spanish carver.\n\nCENTRO CULTURAL DEL BICENTENARIO NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Ex-Correo Central; www.ccb.gov.ar; Sarmiento 151) It took 20 years to complete the massive Correo Central (main post office; 1928), which fills an entire city block. This beaux arts structure was originally modeled on New York City's main post office; the mansard roof was a later addition. The building is now being turned into a cultural center with concert space for the national philharmonic orchestra (and holding nearly 2000 spectators), but no one knows when it will open; check it out during your tenure and cross your fingers.\n\nMUSEO MUNDIAL DEL TANGO MUSEUM\n\nMAP GOOGLE MAP\n\n(  4345-6967; Av de Mayo 833, 1st fl; admission AR$20;   2:30-7:30pm Mon-Fri) Located below the Academia Nacional del Tango is this tango museum \u2013 for fans of the dance only. Just a couple of large rooms are filled with tango memorabilia, from old records and photos to historic literature and posters. Tango shoes are also featured, but the highlight has to be one of Carlos Gardel's famous fedora hats.\n\nMUSEO MITRE MUSEUM\n\nMAP GOOGLE MAP\n\n(  4394-8240; San Mart\u00edn 336; admission AR$15;   1-5pm Mon-Fri) This museum is located in the colonial house where Bartolom\u00e9 Mitre \u2013 Argentina's first legitimate president elected under the constitution of 1853 \u2013 resided with his family. Mitre's term ran from 1862 to 1868, and he spent much of it leading the country's armies against Paraguay. Two courtyards, salons, an office, a billiards room and Mitre's old bedroom are part of this complex. Since part of the museum is open air, you may find it closed during heavy rain.\n\nMUSEO DE LA POLIC\u00cdA FEDERAL MUSEUM\n\nMAP GOOGLE MAP\n\n(  4394-6857; San Mart\u00edn 353, 7th fl;   2-7pm Tue-Fri) F This quirky police museum displays a whole slew of uniforms and medals, along with 'illegal activities' exhibits (cockfighting and gambling), drug paraphernalia (including a fake arm stuck with a needle!) and even a stuffed police dog. The forensic room way in the back was being remodeled at research time \u2013 it may or may not keep its grisly photos and dummies of hacked-up murder victims; something to keep in mind if you bring the kids.\n\n###  POPE FRANCIS\n\nAfter Cardinal Jorge Mario Bergoglio, the archbishop of Buenos Aires, was named successor to Pope Benedict XVI in March 2013, he took the name Francis I. Not only was he the first pontiff to bear that moniker (adopted to honor St Francis of Assisi), he was also the first to hail from the Americas and the first to belong to the Jesuit order, which incidentally was expelled from most of South America for 47 years (1767\u20131814). It's a fair bet that he's also the first pope to have grown up drinking _mate_ , tangoing at _milongas_ and ardently supporting the San Lorenzo _f\u00fatbol_ club.\n\nFrancis has taken charge at a particularly low point in the church's modern history. It has been rocked by a seemingly endless series of sexual-abuse scandals, and subjected to investigations into charges of high-level corruption and financial malfeasance. These events and widening parishioner dissatisfaction with the Vatican's stance on homosexuality, divorce, abortion and the role of women in the church have caused congregations to shrink, a problem compounded in the Americas by the increasing popularity of various Pentecostal, evangelical and other denominations.\n\nWhile it remains to be seen what direction Francis will take on various aspects of Catholic doctrine, he has roundly criticized the structure and workings of the church at its highest levels, and vowed to make them more transparent and outward-looking, and less closed and hierarchical.\n\nFrancis appointed a special commission to delve into the workings of the Vatican bank, which has been under pressure from the Council of Europe's anti-money laundering committee to submit to independent supervision. In October 2013, the bank published an annual report for the first time in its history.\n\nIn a more headline-worthy move, Francis summoned Limburg's 'Bishop of Bling', free-spending prelate Franz-Peter Tebartz-van Elst, to Rome to explain how he managed to spend tens of millions of euros renovating his official residence. After eight days of cooling his heels, the bishop was granted a 20-minute audience with Francis, who ended up suspending him from his duties for an 'indefinite period'.\n\nThis pope doesn't just talk the talk, he walks the walk. After arriving in Rome for his 2001 anointment as cardinal by John Paul II, Archbishop Bergoglio left the modest priests' quarters on foot the morning of his ceremony, arriving at the Vatican accompanied only by his assistant and a couple of relatives. No surprise, as he had already eschewed the archbishop's palace in Olivos, remaining in his modest apartment and getting around Buenos Aires by bus and the Subte rather than with a car and driver.\n\nHe has continued these habits as Francis I, emulating his namesake and personal hero, the saint from Assisi who once renounced all worldly possessions including his clothing. These humble aspects, coupled with the very personable humanity Francis displays, have made him an extremely popular pontiff. Many Catholics speak of feeling 'understood' by him, and his popularity extends beyond the faithful. The church has a new face, just when it needed it the most.\n\n###    Montserrat\n\nMANZANA DE LAS LUCES NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Block of Enlightenment;   4342-3964; www.manazadelasluces.org; Per\u00fa 272; tours AR$15;   tours 3pm Mon-Fri, 3pm, 4:30pm & 6pm Sat & Sun) In colonial times, the Manzana de las Luces was Buenos Aires' most important center of culture and learning. Even today, this collection of buildings still symbolizes high culture in the capital. On the northern side of the block are two of the five original buildings; Jesuit defensive tunnels were discovered in 1912. Tours (in Spanish) are available, and a cultural center on the premises offers classes, workshops and theater.\n\nThe first people to occupy the Manzana de las Luces were the Jesuits, who built several structures including the Procuradur\u00eda (1730; administrative headquarters), part of which still survives today. (Unfortunately for the Jesuits, they were eventually expelled from the premises \u2013 and Argentina \u2013 in 1767 by the Spanish, who felt politically threatened by them.) Along with housing offices, these buildings hosted converted indigenous people from the provinces. Later, during the 19th century, they were home to various museums, legislative offices, schools and universities.\n\nThe city's oldest church, the Iglesia San Ignacio (1734), is also located here, originally built in adobe in 1661 and rebuilt or remodelled several times since. Today there remains only a single original cloister; it shares a wall with the Colegio Nacional de Buenos Aires (1863), a prep school where generations of the Argentine elite still send their children to receive secondary schooling. The city's oldest bookstore, La Librer\u00eda de Avila, is also nearby at Alsina and Bol\u00edvar.\n\nMUSEO DE LA CIUDAD MUSEUM\n\nMAP GOOGLE MAP\n\n(  4343-2123; Defensa 219; admission AR$1;   11am-7pm Mon-Fri, 10am-8pm Sat & Sun) This upstairs museum was closed at research time due to major restoration, but in the future you should expect both permanent and temporary exhibitions on porte\u00f1o life and history here. Downstairs is a large hall showcasing salvaged doors and ancient hardware. Nearby, at the corner of Alsina and Defensa, is the Farmacia de la Estrella (1835), a functioning homeopathic pharmacy with gorgeous woodwork and elaborate late-19th-century ceiling murals depicting health-oriented themes. Occasionally the museum opens via its Alsina 412 door.\n\nMUSEO ETNOGR\u00c1FICO JUAN B AMBROSETTI MUSEUM\n\nMAP GOOGLE MAP\n\n(  4331-7788; www.museoetnografico.filo.uba.ar; Moreno 350; admission AR$4;   1-7pm Tue-Fri, 3-7pm Sat & Sun) This small but attractive anthropological museum was created by Juan B Ambrosetti not only as an institute for research and university training but also as an educational center for the public. On display are archaeological and anthropological collections from the Andean Northwest and Patagonia. Beautiful indigenous artifacts are also featured, while an African and Asian room showcases some priceless pieces.\n\n###  LUNA PARK\n\nIf unique large-scale spectacles such as the Beijing Circus, the New York Ballet, the Philadelphia Philharmonic, Julio Iglesias or Tom Jones come to town, the dressing rooms of Luna Park MAP GOOGLE MAP (  5279-5279; www.lunapark.com.ar; cnr Bouchard & Av Corrientes) are probably their destination. Bordered by the thoroughfares of Lavalle, Bouchard, Av Corrientes and Madero, Luna Park was originally a boxing stadium built on the old grounds of the Pacific Railway. Finished in 1931, the venue gradually became a mecca for public events needing large spaces. When Carlos Gardel died in a plane crash in 1935, his wake was held here for the thousands of grieving fans. In 1944, a relatively unknown actress named Eva Duarte first met general Juan Per\u00f3n here during a benefit for victims of an earthquake in San Juan province. And on Nov\u00adember 7, 1989, Diego Maradona was married here before 11,000 fans.\n\nBut Luna Park never forgot its roots; throughout its history 25 boxing titles have been decided within its walls. Many other sports, including volleyball, basketball and tennis, are also occasionally highlighted at this stadium, and productions such as fashion shows, ice-skating spectacles and mass religious baptisms have found their way here as well. With a capacity of 15,000 (it's Argentina's largest enclosed stadium), Luna Park can easily handle these crowds, which also come to see recent big-time performers such as Liza Minnelli, Luciano Pavarotti, Norah Jones, Ricky Martin, David Byrne and Chrissie Hynde.\n\n The Center\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  EATING\n\nYou won't find Buenos Aires' best cuisine in the Center, as most restaurants here cater to business power-lunches or quick takeout. Some eateries don't even open for dinner since the working masses beeline home after the day is done. Even bars tend to open and close relatively early here. All this doesn't mean you won't find a decent bite to eat, however, and vegetarians especially might find some good choices. Also, five-star hotels often house top-notch restaurants that are worth a visit \u2013 and are definitely open for dinner. Nearby options include Le Sud (Click here) in the Sofitel, Elena (Click here) at the Four Seasons, La Bourgogne in the Alvear Palace (Click here) and El Mercado and El Bistr\u00f3 in the Faena (Click here).\n\nGRANIX VEGETARIAN $\n\nMAP GOOGLE MAP\n\n(  4343-4020; Florida 165, 1st fl; all-you-can-eat AR$85;   11am-3:30pm Mon-Fri;  ) Stepping into this large, modern lacto-ovo-vegetarian eatery will make you wonder if _porte\u00f1os_ have had enough steak already. Pick from the many hot appetizers and mains; there's also a great salad bar and plenty of desserts. It's only open for weekday lunches, and located in a shopping mall (look for the stairs on the right). Takeout is available.\n\nVITA HEALTH FOOD $\n\nMAP GOOGLE MAP\n\n(  4342-0788; www.vitamarket.com.ar; Hip\u00f3lito Yrigoyen 583; mains AR$34-36;   8am-8pm Mon-Fri, 10am-8pm Sat, noon-7pm Sun;  ) Here's a hippie-ish, casual and health-oriented eatery offering tasty vegetarian dishes like organic soy _milanesas_ , zucchini and eggplant lasagne and tofu in mustard sauce with mashed potatoes. Various freshly mixed juices and _licuados_ are available (with the option of adding a wheatgrass shot) and there are plenty of gourmet salads. Organic coffee is also served.\n\nA few shelves are lined with health-\u00adoriented products for sale.\n\nCALIFORNIA BURRITO COMPANY (CBC) MEXICAN $\n\nMAP GOOGLE MAP\n\n(  4328-3057; www.cbcburrito.com; Lavalle 441; mains AR$60-100;   11am-3pm Mon-Fri) This convenient burrito joint is popular with the business crowd at lunchtime and is passable if you don't expect authentic Mexican. Flour tortillas are loaded up with your choice of meat, rice, beans and salsa, and rolled into large burritos that will sustain you well into the evening. Tacos, fajitas and margaritas available.\n\nBranches in Palermo and Recoleta (see the website for addresses).\n\nALDO'S VINOTECA ARGENTINE $$\n\nMAP GOOGLE MAP\n\n(  4334-2380; Moreno 372; mains AR$70-100;   11am-midnight Sun-Thu, to 1am Fri & Sat) Located under the Moreno Hotel, this restaurant\u2013wine shop is an upscale eatery serving a small but tasty menu of meat, seafood and pasta dishes, all amidst walls lined with wine. What makes this place unique, however, is that the wine is sold at _retail_ prices \u2013 thus making it easier to sample (and buy) the nearly 500 labels available.\n\nFURAI-BO JAPANESE $$\n\nMAP GOOGLE MAP\n\n(  4334-3440; Adolfo Alsina 429; mains AR$80-120;   noon-11pm Sun-Thu, to midnight Fri & Sat) Walk up the staircase of this old building into a calm space meant to resemble a Buddhist temple; on weekend evenings, live instrumentalists set the mood with ambient ceremonial music. The house specialty is homemade ramen noodles with pork. The menu also includes excellent sushi and _katsu_ (a type of cutlet), plus unusual sweet treats like ginger ice cream.\n\nD'ORO ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4342-6959; Per\u00fa 159; mains AR$90-150;   noon-7pm Mon-Thu, to midnight Fri) Despite its downtown location, D'Oro is a serious Italian wine bar and restaurant to rival others in more gastronomically famous neighborhoods. Come for thin, crispy oven-baked pizzas, mushroom risotto, fettucine with shellfish, garlic-topped focaccia and capellini tossed with fresh basil and tomatoes. To go with it all, there's also a short but well-chosen selection of wines by the glass.\n\nBROCCOLINO ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4322-7754; www.broccolino.com; Esmeralda 776; mains AR$50-120;   noon-11:30pm) Pick from over 20 sauces (including squid ink!) for your pasta, with a choice of rigatoni, fusilli, pappardelle and all sorts of stuffed varieties. If you can't decide on your topper, try the delicious Sicilian sauce (spicy red peppers, tomato and garlic) or the pesto with mushrooms and garlic. Portions are large and the bread homemade.\n\n  TOMO 1 MODERN ARGENTINE $$$\n\nMAP GOOGLE MAP\n\n(  4326-6695; Carlos Pellegrini 521; mains AR$190-320;   noon-3pm & 5:30pm-midnight Mon-Fri, 5:30pm-midnight Sat) At renowned Tomo 1, European-influenced Chef Federico Fialayre promotes a blend of Italian and Spanish cooking methods in dishes featuring seasonal produce, homemade pasta and fresh fish. For a splash-out, sample his famed cuisine with a three-course _prix fixe_ menu (AR$370 lunch, AR$430 dinner); it comes with amuse-bouches, two glasses of wine, mineral water, coffee and petits fours.\n\n### 8 KEEPING YOUR PE\u00d1AS & MAYOS STRAIGHT\n\nSome first-time (or maybe second-time) visitors may get confused with certain similar-sounding street and attraction names. Keep them straight:\n\n25 de Mayo Street that goes north\u2013south from Retiro to Plaza de Mayo (Mayo is Spanish for the month of May).\n\nAv de Mayo Large avenue that goes east\u2013west from Plaza del Congreso to Plaza de Mayo.\n\nPlaza de Mayo BA's most important plaza.\n\nDiagonal Roque S\u00e1enz Pe\u00f1a Diagonal street that stretches from Plaza de Mayo to the Obelisco.\n\nLuis S\u00e1enz Pe\u00f1a Street that goes from Plaza del Congreso through Constituci\u00f3n.\n\nRodriguez Pe\u00f1a Street that goes from Recoleta to Plaza del Congreso.\n\n The Center\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  DRINKING & NIGHTLIFE\n\nMany watering holes in the Center are Irish-pub knock-offs that cater to the business crowd on weekdays. Because of this, some might close a bit earlier than in other neighborhoods, but the most popular ones stay packed all night long. The Center also has some of the oldest cafes in town, delightfully atmospheric venues that offer a welcome break while you're wandering around.\n\nCAF\u00c9 TORTONI CAFE\n\nMAP GOOGLE MAP\n\n(  4342-4328; www.cafetortoni.com.ar; Av de Mayo 829;   8am-2am Mon-Fri, 9am-2am Sat & Sun) BA's oldest and most famous cafe, the classic Tortoni has become so popular with foreigners that it's turned into a tourist trap. Still, it's practically an obligatory stop for any visitor to town: order a couple of _churros_ (fried pastry dough) with your hot chocolate and forget about the inflated prices. There are also tango shows nightly (AR$150) \u2013 reserve ahead.\n\nLA CIGALE BAR\n\nMAP GOOGLE MAP\n\n(  4893-2332; 25 de Mayo 597;   noon-4pm & 6pm-close) This upstairs bar-restaurant is very popular with both office workers (during the day) and music-industry folks (later in the evening). There's either live music or DJs most nights, but it's best known for its 'French Tuesday', when electronica and exotic cocktails draw heavy crowds. Fusion foods are served for both lunch and dinner.\n\nLONDON CITY CAFE\n\nMAP GOOGLE MAP\n\n(  4343-0328; Av de Mayo 599) This classy and historic cafe (being remodelled at research time) has been serving java enthusiasts for over 50 years, and claims to have been the spot where Julio Cort\u00e1zar wrote his first novel. Your hardest work here, however, will most likely be choosing which pastry to try with your fresh cup of coffee.\n\nNEW BRIGHTON BAR\n\nMAP GOOGLE MAP\n\n(  4322-1515; Sarmiento 645;   8am-close Mon-Sat) This beautifully restored landmark and historic gem feels like the well-kept secret of refined local gentlemen who gather here after work. A doorman welcomes guests in while bartenders stir and shake drinks behind a polished-wood bar; during mealtimes, a pianist entertains on the baby grand. Order a classic cocktail and enjoy the tray of elegant finger food that comes with it.\n\nLA PUERTO RICO CAFE\n\nMAP GOOGLE MAP\n\n(  4331-2215; www.lapuertoricocafe.com.ar; Adolfo Alsina 416;   7am-8pm Mon-Fri, 8am-7pm Sat, noon-7pm Sun) One of the city's most historic cafes, La Puerto Rico has been going strong since 1887 but remains miraculously un-touristy. Located a block south of Plaza de Mayo, the place serves great coffee and pastries, the latter baked on the premises. Old photos on the walls hint at a rich past and the Spanish movies that have been filmed here.\n\nALSINA CLUB\n\nMAP GOOGLE MAP\n\n(  4331-3231; www.palacioalsina.net; Adolfo Alsina 940;   Sun, plus 1 Fri per month) One of BA's biggest gay parties is Alsina's Fiesta Dorothy, on just one Friday a month. Expect the city's majority population of cute gay guys in attendance as DJs crank up the house \u2013 a beautiful building with three floors of open balconies, chandeliers and thick drapes \u2013 with dance, hip-hop and techno riffs. Sunday's Club One is more hetero-friendly.\n\nBAHREIN CLUB\n\nMAP GOOGLE MAP\n\n(  6225-2731; www.bahreinba.com; Lavalle 345;   Wed, Fri & Sat) Attracting a good share of BA's tattooed youth, Bahrein is a hugely popular downtown club housed in an old bank (check out the 'vault' in the basement). On the ground floor is the lounge-like Funky Room where resident DJs spin house music and electronica. Downstairs is the happening Xss discotheque, an impressive sound system and a dance floor for hundreds.\n\nCOCOLICHE CLUB\n\nMAP GOOGLE MAP\n\n(  4342-9485; Av Rivadavia 878;   Fri & Sat) An effortlessly cool DJ club in BA is this electronic-music paradise, based in a slightly run-down old mansion. It's the downstairs basement, gritty and nearly always packed, that holds the main stage, a fantastic sound system and a state-of-the-art light show. Breakbeat, drum and bass, reggaeton and electronic _cumbia_ (Colombian music) entertain; when you need a break, head to the 2nd-floor chill-out room.\n\n####   Neighborhood Walk  \n **Through the Heart of BA**\n\n**Start** Plaza San Mart\u00edn\n\n**End** Plaza de Mayo\n\n**Length** 3km; three hours\n\nStart at the leafy  Plaza San Mart\u00edn, the green heart of Retiro and a haven for loungers on a sunny day. At the bottom of the hill there's a monument to the fallen from the Falklands War.\n\nNow cross Av Santa Fe to the striking  Palacio Paz (Click here). Time it right so you can catch a tour and take in the grandeur of a long-ago era. On the same block is the  Museo de Armas (Click here), an astounding collection of guns, swords and cannon.\n\nFind your way to pedestrian Calle Florida and walk south to the elegant  Galer\u00edas Pac\u00edfico (Click here) shopping mall, one of the capital's most beautiful. Even if you don't like to shop, you should take a peek inside at the gorgeously painted ceiling murals.\n\nNow head west a few blocks on pedestrian Lavalle and cross Av 9 de Julio, considered by many the world's widest street. Your destination is the impressive  Teatro Col\u00f3n (Click here), BA's opera house and a major source of pride for _porte\u00f1os_.\n\nThe  Obelisco is the city's premier landmark. Not only is it used as ground zero for measuring distances from the city center, it's the place to honk your car's horn when your soccer team wins a major victory.\n\nBack in the day  Av Corrientes was BA's main theater district, and some of the city's largest theaters are still found here. It's also known for its many bookstores.\n\nHit Florida again and make your way south to Diagonal Roque S\u00e1enz Pe\u00f1a. You'll end up at historic  Plaza de Mayo (Click here), where you'll want to linger and take in the historic atmosphere. Behind the Cabildo is a cafe where you can sit and relax after your long walk.\n\n The Center\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nCAF\u00c9 TORTONI TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4342-4328; www.cafetortoni.com.ar; Av de Mayo 829; show AR$150-200) Nightly tango shows (reserve ahead) take place at this historic yet very touristy place, and they're fine if you don't expect too much. The overpriced food isn't included. If you come earlier for the cafe, you may have to line up outside beforehand. Despite these downfalls, the Tortoni is BA's most famous cafe and still offers a beautiful atmosphere.\n\nGet your ticket the day of or one day beforehand at the cafe between 11am and 5pm (cash only).\n\nCONFITER\u00cdA IDEAL TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4328-7750; www.confiteriaideal.com; Suipacha 384, 1st fl) This institution (since 1912) is the mother of all historic tango halls, with classes and _milongas_ offered daily. Live orchestras occasionally accompany dancers, and there are dinner-tango shows on Friday and Saturday. The actual cafe section could use a facelift, as it's a bit dim, stodgy and impersonal, but it remains a classic. Featured in the film _The Tango Lesson_.\n\nEL QUERAND\u00cd TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  5199-1770; www.querandi.com.ar; Per\u00fa 322; show from US$60, dinner & show from US$130) This large corner venue is also an elegant restaurant boasting an upscale atmosphere. This show follows tango's evolution from its bordello origins to cabaret influences to _milongas_ and modernism. There's more low-key dancing than at other shows \u2013 and also more singing and musical interludes \u2013 so don't expect overly athletic moves. One minus: columns can block some views.\n\nTEATRO OPERA THEATER\n\nMAP GOOGLE MAP\n\n(  4326-1335; www.operaciti-teatro.com.ar; Av Corrientes 860) This classic theater, which boasts an art-deco exterior, offers nearly 2000 seats and has performances that range from piano recitals to rock concerts to tango and ballet. It served many years as a cinema, later becoming a live-theater venue.\n\nTEATRO GRAN REX THEATER\n\nMAP GOOGLE MAP\n\n(  4322-8000; www.teatro-granrex.com.ar; Av Corrientes 857) A huge theater seating 3300, this place hosts myriad national and international musical productions, from Cyndi Lauper to Kenny G to Bj\u00f6rk.\n\n###  TAKING IT TO THE STREETS\n\nJust like the tango and _dulce de leche,_ street protests are a well-known pastime for _porte\u00f1os_. Whether the city is booming or in the midst of a depression, unless there's martial law, someone is out on the street demonstrating against something. Plaza de Mayo has long been the focal point of protests.\n\nThe best-known voices of dissent are the famed Madres de la Plaza de Mayo (the Mothers of Plaza de Mayo). On April 30, 1977, 14 mothers whose children had disappeared in the Dirty War marched on the Plaza de Mayo. They demanded to know what had happened to their missing children. The military government dismissed them, claiming that their children had simply moved abroad, but the women continued to march in their iconic white handkerchiefs every Thursday. They played an essential historical role as the first group to openly oppose the military junta and they opened the doors for later protests. In 1986 the Madres split into two groups. Asociaci\u00f3n Madres de Plaza de Mayo announced that the group would stop participating in _la marcha de la resistencia_ around Plaza de Mayo. The other group, Madres de Plaza de Mayo L\u00ednea Fundadora, still marches every Thursday.\n\nEven in 1996, when the economy was good and the country was under civilian control, a number of protests broke out against corruption and the reform of pensions. Senior citizens hurled eggs at government buildings and were chased by trucks mounted with water cannon. The protests after the economic collapse in 2001 were particularly large and vociferous. Thousands of people \u2013 in the poorer areas as well as middle-class neighborhoods \u2013 spontaneously gathered in public parks in Buenos Aires. To the shouts of ' _\u00a1Que se vayan todos\u00a1_ ' (get rid of them all), they banged pots and pans \u2013 an act known as a _cacerolazo_. Both the economic minister and the president eventually stepped down, and some of the politicians who hadn't fled the country were beaten in the streets.\n\nThere are still occasional grievances on Plaza de Mayo, whether it's a protest against the price of beef and tomatoes, or against the closure of a hospital. You can always count on protests being loud, but these days they're usually peaceful.\n\n The Center\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SHOPPING\n\nThe main shopping street in the Center is Florida. Most travelers to Buenos Aires take the obligatory stroll down this heaving pedestrian street, lined with shops and vendors selling clothes, shoes, jewelry, housewares and cheesy souvenirs. Touts zero in on tourists, offering currency exchange and leather jackets. We'll tell you now: you won't find the cheapest prices on leather jackets here (try Calle Murillo or Calle Aguirre instead) and you should definitely avoid changing money on the streets \u2013 fake bills and other scams are an occasional problem here.\n\nEL ATENEO BOOKS, MUSIC\n\nMAP GOOGLE MAP\n\n(  4325-6801; Florida 340 & 629;   9am-8pm Mon-Fri, to 5pm Sat) Buenos Aires' landmark bookseller stocks a few books in English (including some Lonely Planet guidebooks) and also has a decent selection of CDs. There are several branches within the city, including the gorgeous Grand Splendid (Av Santa Fe 1860;   9am-10pm Mon-Thu, to midnight Fri & Sat, noon-10pm Sun), a renovated cinema where Carlos Gardel got his career started; soak up the glamour at the cafe, located on the old stage.\n\nARTE Y ESPERANZA CRAFTS & TEXTILES\n\nMAP GOOGLE MAP\n\n(  4343-1455; www.arteyesperanza.com.ar; Balcarce 234;   9am-6pm Mon-Fri) This store sells fair-trade, handmade products that include many from Argentina's indigenous craftspeople. Shop for silver jewelry, pottery, textiles, _mate_ gourds, baskets, woven bags and animal masks. Also in Retiro (Suipacha 892).\n\nEL COLECCIONISTA MUSIC\n\nMAP GOOGLE MAP\n\n(  4322-0359; www.elcoleccionistacd.com.ar; Esmeralda 562;   noon-7:30pm Mon-Fri) This music store has an eclectic selection of jazz, blues, salsa, Celtic and symphonic rock CDs. It will buy used musical instruments, so trade in that guitar or drum you're tired of lugging around for a cool _bandone_ \u00f3 _n_ (the accordion-esque instrument you'll hear in every tango band). Staff members are knowledgeable.\n\nThe Center\n\n  Top Sights\n\n1Plaza de Mayo D5\n\n  Sights\n\n2Cabildo C5\n\n3Casa Rosada E5\n\n4Catedral Metropolitana D5\n\n5Centro Cultural Borges C1\n\n6Centro Cultural del Bicentenario E3\n\n7Galer\u00edas Pac\u00edfico C1\n\n8Iglesia Santa Catalina C1\n\n9Manzana de las Luces C6\n\n10Museo de la Ciudad D6\n\n11Museo de la Polic\u00eda Federal C3\n\n12Museo del Bicentenario E5\n\n13Museo Etnogr\u00e1fico Juan B Ambrosetti D6\n\n14Museo Mitre C3\n\n15Museo Mundial del Tango B5\n\n  Eating\n\n16Aldo's Vinoteca D6\n\n17Broccolino B1\n\n18California Burrito Company (CBC) D2\n\n19D'Oro C5\n\n20Furai-Bo D6\n\n21Granix C4\n\n22La Panader\u00eda de Pablo D6\n\n23Tomo 1 A2\n\nVita (see 43)\n\n  Drinking & Nightlife\n\n24Alsina A6\n\n25Bahrein D2\n\nCaf\u00e9 Tortoni (see 32)\n\n26Cocoliche B5\n\n27La Cigale D2\n\n28La Puerto Rico D6\n\n29London City C5\n\n30New Brighton C3\n\n  Entertainment\n\n31Asociaci\u00f3n Argentina de Polo C5\n\n32Caf\u00e9 Tortoni B5\n\n33Confiter\u00eda Ideal B3\n\n34El Querand\u00ed C6\n\n35Instituto Goethe D3\n\n36La Marshall B3\n\n37Luna Park E2\n\n38Teatro Gran Rex B3\n\n39Teatro Opera B3\n\n  Shopping\n\n40Arte y Esperanza E6\n\n41El Ateneo C3\n\n42El Coleccionista B2\n\n  Sports & Activities\n\n43Academia Buenos Aires C5\n\n44Alianza Francesa A1\n\nEscuela Argentina de Tango (see 5)\n\n45Expanish C4\n\n46Instituto Cultural Argentino-Norteamericano B2\n\n  Sleeping\n\n47Claridge Hotel C2\n\n48Gran Hotel Hispano B5\n\n49Hotel Avenida C5\n\n50Hotel Fac\u00f3n Grande D2\n\n51Hotel Lafayette D2\n\n52Hotel Maip\u00fa B1\n\n53Portal del Sur B5\n\n54V & S Hostel Club B2\n\n# Puerto Madero\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Entertainment\n\n###   Map\nPuerto Madero\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\nNeighborhood Top Five\n\n Escaping the city's hustle and bustle with a walk or bike ride through the unique  Reserva Ecol\u00f3gica Costanera Sur, wild wetlands where you can spot birds or perhaps a nutria or iguana.\n\n Strolling down the scenic and vehicle-free cobbled lanes next to Puerto Madero's dikes.\n\n Visiting the  Colecci\u00f3n de Arte Amalia Lacroze de Fortabat, the contemporary museum showing the collection of Argentina's richest woman.\n\n Eating lunch or dinner at an upscale restaurant with a water view \u2013 it'll be pricey but good.\n\n Seeing what's being exhibited at the slick new  Faena Arts Center.\n\n###  Explore: Puerto Madero\n\nPut on your most comfortable walking shoes, because you'll be on your feet all day here. You can start walking pretty much anywhere and make a big loop around the dikes, though if you start at the tourist office (under an old cargo crane at Dique 4), you can pick up some practical information and a map.\n\nThere are a few interesting museums around the dikes, including the Colecci\u00f3n de Arte Amalia Lacroze de Fortabat (also called Museo Fortabat). This shiny, glassy museum is in a cutting-edge building and art lovers shouldn't miss it. Another quirky place is the Museo Fragata Sarmiento. Walk the plank, pay your ticket and explore all the fascinating holds of this naval vessel. The Corbeta Uruguay, a couple blocks away, is another similar 'ship' museum. In between these two is the Puente de la Mujer \u2013 a pedestrian bridge that you'll be tempted to cross.\n\nWhen you get towards the south, cut east on R Vera Pe\u00f1aloza and look for the elegant fountain called La Fuente de las Nereidas. Just beyond is the southern entrance to the marshy Reserva Ecol\u00f3gica Costanera Sur, which offers the only real nature walk (or bike ride) in central BA. It's a peaceful place full of reedy lagoons, wildlife and dirt paths, and you can get a close-up look at the muddy waters of the R\u00edo de la Plata. The reserve is a sharp contrast to the upscale lofts, restaurants and hotels nearby, and thankfully it's available to everyone for no cost at all \u2013 just be sure you're not there on a Monday, when it's closed.\n\n###  Local Life\n\n\u00bb Food stands The road lining the Reserva Ecol\u00f3gica Costanera Sur is dotted with several barbecue stands selling cheap _chorip\u00e1n_ (spicy sausage sandwiches) or _bondiolas_ (pork sandwiches). You'll be mingling with the locals at sidewalk tables, and the view of the reserve is pleasant enough.\n\n###  Getting There & Away\n\n\u00bb Bus Buses 64, 126 and 152 run along LN Alem\/Paseo Col\u00f3n, which gets you within three blocks of Puerto Madero.\n\n\u00bb Subte The closest Subte stops are LN Alem (L\u00ednea B) and the end lines of L\u00edneas A, D and E, which terminate at Plaza de Mayo.\n\n\u00bb Tram BA's short light-rail system parallels Puerto Madero's docks, and may someday extend to La Boca.\n\n####  Lonely Planet's Top Tip\n\nIf you want to explore every corner of the Reserva Ecol\u00f3gica Costanera Sur, it's much easier on a bike. On weekends in good weather there are rentals outside the north and south entrances, but if you want to make sure you get a bike you might want to rent one beforehand at one of several bike tour companies in BA (or via the city's free bike share program, Click here).\n\n  Best Places to Eat\n\n\u00bbi Central Market\n\n\u00bbLa Rosa N\u00e1utica\n\nFor reviews, Click here \u00bb\n\n  Best Museums\n\n\u00bbColecci\u00f3n de Arte Amalia Lacroze de Fortabat\n\n\u00bbFragata Sarmiento\n\nFor reviews, Click here \u00bb\n\nJON ARNOLD IMAGES LTD \/ ALAMY \u00a9\n\n##### TOP SIGHT\n\nRESERVA ECOL\u00d3GICA COSTANERA SUR\n\nThe beautiful marshy land of this nature preserve makes it a popular site for sunny weekend outings, when hundreds of picnickers, cyclists and families come for fresh air and natural views. If you're lucky, you may spot a river turtle, iguana or nutria (semi-aquatic rodent); bird-watchers will adore the 250-plus bird species that pause to rest here.\n\n###  DON'T MISS...\n\n\u00bb Walking the trails.\n\n\u00bb Birdwatching and bike riding.\n\n\u00bb Views of the R\u00edo de la Plata.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb   4893-1588\n\n\u00bb Av Rodr\u00edguez 1550\n\n\u00bb   8am-7pm Tue-Sun Nov-Mar, to 6pm Apr-Oct\n\nDuring the military Proceso of 1976\u201383, access to the Buenos Aires waterfront was limited, as the area was diked and filled with sediments dredged from the R\u00edo de la Plata. While plans for a new satellite city across from the port stalled, trees, grasses, birds and rodents took advantage and colonized this low-lying, 350-hectare area that mimics the ecology of the Delta del Paran\u00e1.\n\nIn 1986 the area was declared an ecological reserve. Mysterious arson fires, thought to have been started by those with financial interests in the prime real estate, have occasionally been set. But permanent scars haven't remained \u2013 this beautifully lush marshy land survives hardily, and the reserve has become a popular site for outings and walks. Bring binoculars if you're a birder \u2013 ducks, swans, woodpeckers, parakeets, hawks, flycatchers and cardinals are just a few kinds of the feathered critters that can be spotted. Further in at the eastern shoreline of the reserve you can get a close-up view of the R\u00edo de la Plata's muddy waters \u2013 a rare sight in Buenos Aires.\n\nTours are available on weekends; monthly Friday night full-moon tours are also available (call for schedules). On warm weekends and holidays you can rent bikes just outside either the northern or southern entrances.\n\n Puerto Madero\n\nSights | Eating | Entertainment\n\n  Sights\n\nRESERVA ECOL\u00d3GICA COSTANERA SUR NATURE RESERVE\n\nSee Click here.\n\nCOLECCI\u00d3N DE ARTE AMALIA LACROZE DE FORTABAT MUSEUM\n\nMAP GOOGLE MAP\n\n(Museo Fortabat;   4310-6600; www.coleccionfortabat.org.ar; Olga Cossettini 141; admission AR$35;   noon-8pm Tue-Sun) Rivaling Palermo's Malba for cutting-edge looks is this stunning art museum, prominently located at the northern end of Puerto Madero. It shows off the collection of billionairess, philanthropist and socialite Amalia Lacroze de Fortabat, Argentina's wealth\u00adiest woman. There are galleries devoted to Antonio Berni and Ra\u00fal Soldi (both famous Argentine painters) and works by international stars like Dal\u00ed, Klimt, Rodin and Chagall; look for Warhol's colorful take on Fortabat herself in the family portrait gallery.\n\nThe building was designed by renowned Uruguayan architect Rafael Vi\u00f1oly, and is a creation of steel, glass and concrete \u2013 the last a most appropriate material considering its patroness (Fortabat is the major stockholder of Argentina's largest cement company). Finished in 2008, it encompasses over 6000 sq meters, with several airy floors showcasing works by famous Argentine and international artists. The most interesting thing about the museum itself, however, might be the movable aluminum panels above the glassy ceiling. They tilt open and shut, keeping sun off the delicate artworks. Lacroze requested this feature so that she could see her collection and the stars at the same time.\n\nSpanish tours are given Tuesday to Sunday at 3pm and 5pm; call ahead for group tours in English.\n\nFAENA ARTS CENTER ARTS CENTER\n\nMAP GOOGLE MAP\n\n(  4010-9233; www.faenaartscenter.org; Aime Paine 1169; admission AR$40, Mon free;   varies depending on exhibition) This very large, airy art space \u2013 in a beautifully renovated flour mill \u2013 highlights the contemporary dreams of local and international artists and designers. You should expect the most cutting-edge exhibits that utilize these spaces to the maximum \u2013 think rope nets hanging from the ceiling or light pyramids reaching for the sky. Check the website for upcoming shows.\n\nPUENTE DE LA MUJER BRIDGE\n\nMAP GOOGLE MAP\n\n(Dique 3) The striking Puente de la Mujer (Bridge of the Woman) is the barrio's signature monument. Unveiled in 2001, this gleaming, white structure spans Dique 3 and resembles a sharp fishhook or even a harp \u2013 but is supposed to represent a couple dancing the tango. Designed by acclaimed Spanish architect Santiago Calatrava and mostly built in Spain, this 160m-long pedestrian bridge cost AR$6 million and rotates 90 degrees to allow water traffic to pass.\n\nFRAGATA SARMIENTO MUSEUM\n\nMAP GOOGLE MAP\n\n(  4334-9386; Dique 3; admission AR$2;   10am-7pm) Over 23,000 Argentine naval cadets and officers have trained aboard this 85m sailing vessel, which traveled around the world 37 times between 1899 and 1938. On board are detailed records of its lengthy voyages, plenty of nautical items including old uniforms, and even the stuffed remains of Lampazo (the ship's pet dog). Peek into the ship's holds, galley and engine room and note the hooks where sleeping hammocks were strung up.\n\nBuilt in Birkenhead, England, in 1897 at a cost of \u00a3125,000, this impeccably maintained ship never participated in combat. US president Theodore Roosevelt (look for his photo) was a distinguished guest on board, but perhaps the greatest test of the ship's seaworthiness was the visit of Roosevelt's successor, William Howard Taft, who weighed more than 140kg.\n\nCORBETA URUGUAY MUSEUM\n\nMAP GOOGLE MAP\n\n(  4314-1090; Dique 4; admission AR$2;   10am-7pm) This 46-meter-long military ship did surveys along Argentina's coast and supplied bases in Antarctica until it was decommissioned in 1926, after 52 years of service. Displayed below the main deck are interesting relics from Antarctica expeditions, such as crampons and snowshoes, along with historical photos and nautical items. Check out the tiny kitchen, complete with _mate_ supplies (of course).\n\n###  A FAILED PORT\n\nBuenos Aires' waterfront was an object of controversy in the mid-19th century, when competing commercial interests began to fight over the location of a modernized port for Argentina's burgeoning international commerce. Two ideas came to light. One was to widen and deepen the channel of the Riachuelo to port facilities at La Boca, which indeed happened as planned. The other was proposed by Eduardo Madero, a wealthy exporter with strong political ties and solid financial backing. Madero proposed transforming the city's mudflats into a series of modern basins and harbors consistent with the aspirations and ambitions of a cosmopolitan elite. This also occurred, but not quite as he had planned.\n\nBy the time of its completion in 1898 (four years after Madero's death), Puerto Madero had exceeded its budget and Madero himself had come under scrutiny. Suspicions arose from Madero's attempts to buy up all the landfill in the area and from his links to politicians who had acquired nearby lands likely to increase in value. And the practical side of the scheme didn't go so well either. By 1910 the amount of cargo was already too great for the new port, and poor access to the rail terminus at Plaza Once made things even worse. New facilities in a rejuvenated La Boca partly assuaged these problems, but congressional actions failed to solve the major issues \u2013 until the 1926 completion of Retiro's Puerto Nuevo.\n\n Puerto Madero\n\nSights | Eating | Entertainment\n\n  Eating\n\nNearly all of Puerto Madero's restaurants are upscale and expensive, and many sport covered outdoor terraces with views of the nearby _diques_ (dikes). You won't get the best bang for your buck in this elegant strip and the cuisine is more traditional than inspired, but it's the location that counts. Caba\u00f1a Las Lilas is the most famous _parrilla_ restaurant here, but many consider it a tourist trap \u2013 a bit overrated and way overpriced. We don't review it here, but if you have money to burn, by all means try it.\n\nI CENTRAL MARKET MODERN ARGENTINE $$\n\nMAP GOOGLE MAP\n\n(  5775-0330; Macacha G\u00fcemes 302; mains AR$80-135;   8am-midnight) In the morning, this pleasant airy restaurant has a coffee counter for espresso and scones; in the afternoon there are paninis, a gourmet deli and a kitchenwares shop to poke around; and by night, the fancy dining room serves contemporary Argentinian dishes. Great waterfront seating for people-watching. Also in Dique 3 at Villaflor 300 (  5775-0330; www.icentralmarket.com.ar; Villaflor 300;   8am-midnight).\n\nLA PAROLACCIA TRATTORIA ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4343-1679; www.laparolaccia.com; Av Alicia Moreau de Justo 1052; mains AR$80-140;   noon-midnight Sun-Thu, to 2am Fri & Sat) This popular Italian eatery specializes in delicious homemade pastas. Reserve one of the few tables with a water view, then enjoy sweet potato gnocchi, gorgonzola ravioli or _cappelletti_ (a small stuffed pasta) in four cheeses. If you're here at midday, the lunch menu is a great deal (available Monday to Saturday). A nearby branch, La Parolaccia del Mare MAP GOOGLE MAP (Av Alicia Moreau de Justo 1170), specializes in seafood.\n\nLA ROSA N\u00c1UTICA PERUVIAN $$$\n\nMAP\n\n(  4311-5560; Av Alicia Moreau de Justo 246; mains AR$80-160;   noon-1am) This branch of the Lima (Peru) restaurant is worth a shot if you like seafood and fusion foods. Start with the house specialty \u2013 the marinated octopus _carpaccio_ (in looks only \u2013 it's not raw) with Parmesan and olive oil. Then move on to the _seviche_ , grilled fish or perhaps a Japanese sushi roll. It's not cheap, but it'll be tasty.\n\n Puerto Madero\n\nSights | Eating | Entertainment\n\n  Entertainment\n\nROJO TANGO TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4952-4111; www.rojotango.com; Faena Hotel & Universe, Martha Salotti 445; show AR$210, dinner & show AR$290) This sexy performance is the tango show to top all others \u2013 especially with its hefty price tag. Offering only 100 seats, the Faena's cabaret room is swathed in blood-red curtains and gilded furniture. The show itself loosely follows the history of tango, starting from its cabaret roots to the modern fusions of \u00c1stor Piazzolla. The orchestra is first-rate, there are plenty of sexy period costumes and even a brief (shock!) nude scene. This is tango foreplay at its best.\n\nPuerto Madero\n\n  Top Sights\n\n1Reserva Ecol\u00f3gica Costanera Sur D4\n\n  Sights\n\n2Colecci\u00f3n de Arte Amalia Lacroze de Fortabat A1\n\n3Corbeta Uruguay A2\n\n4Faena Arts Center B3\n\n5Fragata Sarmiento A3\n\n6Puente de la Mujer A3\n\n  Eating\n\n7i Central Market A2\n\n8i Fresh Market B3\n\n9La Parolaccia del Mare A4\n\n10La Parolaccia Trattoria A3\n\n11La Rosa N\u00e1utica A1\n\n  Entertainment\n\nRojo Tango (see 12)\n\n  Sleeping\n\n12Faena Hotel + Universe B4\n\n# Congreso & Tribunales\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nCongreso & Tribunales\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Touring the backstage corners of the gorgeous  Teatro Col\u00f3n, or perhaps taking in a play or concert there later in the evening.\n\n Strolling down Av de Mayo to the stately buildings around  Plaza del Congreso.\n\n Getting the history of the unique and stunning, Dante-inspired building that is  Palacio Barolo.\n\n Standing at the base of Buenos Aires' iconic symbol, the  Obelisco.\n\n Visiting the toilets and bidets at the bizarre little museum in  Palacio de las Aguas Corrientes.\n\n###  Explore: Congreso & Tribunales\n\nPlaza del Congreso is at the heart of this neighborhood and an easy walk from Plaza de Mayo (in the Center) along the important thoroughfare Av de Mayo. This avenue connects the city's two most significant political centers and is itself lined with beautiful buildings, so be sure to take a stroll along it.\n\nTo do this, however, you'll have to experience crossing Av 9 de Julio (Click here), which is 'the widest street in the world!' as proud _porte\u00f1os_ love to boast. This may be true, as it's 16 lanes at its widest; nearby side streets Cerrito and Pellegrini make it look even broader. Fortunately, traffic islands provide raised breaks for the thousands of pedestrians who cross this monstrosity every day, but it's still an intimidating barrier (and can't be done in one green light without breaking into a run \u2013 trust us).\n\nOnce you've explored the Plaza de Congreso area, head up to Av Corrientes and have a look around \u2013 it's the city's traditional old theater district, but there's still plenty of action going around. Further north is Plaza Lavalle, home to its own important cluster of buildings, including one of the principal gems of the city, Teatro Col\u00f3n. Just a couple blocks from here is the Obelisco, Buenos Aires' key landmark, with a small plaza near the base.\n\n###  Local Life\n\n\u00bb Hangouts Break for afternoon teatime at the classic institutions of Caf\u00e9 de los Angelitos (Click here) or El Gato Negro (Click here).\n\n\u00bb Culture Av Corrientes is lined with theaters and cultural centers where locals catch inexpensive art exhibitions and plays.\n\n###  Getting There & Away\n\n\u00bb Bus Take bus 29 from Palermo or San Telmo, bus 64 from the Microcentro.\n\n\u00bb Subte L\u00edneas A and B from the Microcentro.\n\n####  Lonely Planet's Top Tip\n\nCheck the big theaters like Teatro San Mart\u00edn or Teatro Col\u00f3n for inexpensive or even free events that are occasionally put on. Cultural centers like Centro Cultural San Mart\u00edn are also good sources of free or affordable entertainment.\n\nCongreso is busy during the day, but certain smaller streets feel desolate at night when all the stores close and businesspeople go home. Take a taxi if you feel uncomfortable.\n\n  Best Places to Eat\n\n\u00bbAramburu\n\n\u00bbChan Chan\n\n\u00bbParrilla Pe\u00f1a\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbCaf\u00e9 de los Angelitos\n\n\u00bbEl Gato Negro\n\n\u00bbCruzat Beer House\n\nFor reviews, Click here \u00bb\n\n  Best Entertainment\n\n\u00bbTeatro Col\u00f3n\n\n\u00bbTango Porte\u00f1o\n\n\u00bb\u00c1vila Bar\n\nFor reviews, Click here \u00bb\n\nJOAO CANZIANI \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nTEATRO COL\u00d3N\n\nSinking into a red velvet seat for a performance at Teatro Col\u00f3n is a magical experience. This is one of the world's greatest opera houses: Mikhail Baryshnikov once called it 'the most beautiful of the theaters I know'. And you can experience it yourself, by attending a concert here or taking a behind-the-scenes tour.\n\n###  DON'T MISS...\n\n\u00bb For theater aficionados, the backstage tour is a must.\n\n\u00bb The occasional free concerts \u2013 check the website and click on 'Int\u00e9rpretes Argentinos'.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb   4378-7127\n\n\u00bb www.teatrocolon.org.ar\n\n\u00bb Tucum\u00e1n 1171\n\n\u00bb tours residents\/non-residents AR$30\/110\n\n\u00bb   tours 9am-5pm\n\nThe gorgeous and impressive seven-story building is one of Buenos Aires' biggest landmarks (and sources of pride). It's the city's main performing-arts venue and the only facility of its kind in the country, a world-class forum for opera, ballet and classical music. The theater's opening night was a presentation of Verdi's _A\u00efda_ , and visitors have been wowed ever since. After all, the acoustics are considered among the top five of the world's concert venues.\n\nOccupying an entire city block, the Teatro Col\u00f3n can seat 2500 spectators and provide standing room for another 500. Started in 1880 and finished in 1908, it was the southern hemisphere's largest theater until the Sydney Opera House was built in 1973. Even at times of economic hardship, the elaborate Col\u00f3n remains a high national priority. Italian Francesco Tamburini was the main architect, but after his death in 1891 his partner Vittorio Meano \u2013 who designed the Palacio del Congreso \u2013 was put in charge. After Meano was murdered (possibly due to a love triangle!), Belgian Jules Dormal took over and added some French elements to the theater. Over the years, the Col\u00f3n has hosted some very prominent figures, such as Enrico Caruso, Pl\u00e1cido Domingo, Luciano Pavarotti, Maria Callas and Arturo Toscanini.\n\n Congreso & Tribunales\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SIGHTS\n\nTEATRO COL\u00d3N NOTABLE BUILDING\n\nSee Click here.\n\nPLAZA DEL CONGRESO SQUARE\n\nMAP GOOGLE MAP\n\n(  4010-3000, ext 2410;   tours 12:30pm & 5pm Mon, Tue, Thu & Fri;   L\u00ednea A Congreso, S\u00e1enz Pe\u00f1a) At the western end of Av de Mayo lies Plaza del Congreso, often dotted with cooing pigeons and families feeding them. The Monumento a los Dos Congresos honors the congresses of 1810 in Buenos Aires and 1816 in Tucum\u00e1n, both of which led to Argentine independence. The enormous granite steps symbolize the high Andes, and the fountain at its base represents the Atlantic Ocean. West of the plaza is the colossal green-domed Palacio del Congreso (Congress building).\n\nCosting more than twice its projected budget, the Palacio del Congreso set a precedent for contemporary Argentine public-works projects. Modeled on the Capitol in Washington, DC, and topped by an 85m dome, the palace was completed in 1906. Inside the Congreso, there are free guided tours in English and Spanish. Go to the entrance at Hip\u00f3lito Yrigoyen 1849; bring photo ID.\n\nPALACIO BAROLO NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(  4381-1885; www.palaciobarolo.com; Av de Mayo 1370; standard tours AR$80, longer tours incl glass of wine AR$150;   standard tours 4-7pm Mon-Thu, longer tours 8pm Wed & Fri & 8:30pm Thu) One of the Congreso area's most striking buildings is this 22-story concrete edifice. The building's unique design was inspired by Dante's _Divine Comedy_ ; its height (100m) is a reference to each canto (or song), the number of its floors (22) to verses per song and its divided structure represents hell, purgatory and heaven.\n\nPalacio Barolo was commissioned by cotton tycoon Luis Barolo and designed by Italian architect Mario Palanti. Finished in 1923, it was Buenos Aires' highest skyscraper (until construction of Edificio Kavanagh, in Retiro). At the top is a lighthouse with an amazing 360-degree view of the city.\n\nPALACIO DE LAS AGUAS CORRIENTES NOTABLE BUILDING, MUSEUM\n\nMAP GOOGLE MAP\n\n(cnr C\u00f3rdoba & Riobamba) F Swedish engineer Karl Nystr\u00f6mer and Norwegian architect Olaf Boye helped create this gorgeous and eclectic waterworks building. On the 2nd floor is the small and quirky Museo del Patrimonio MAP GOOGLE MAP (  6319-1104;   9am-1pm Mon-Fri; tours in Spanish 11am Mon, Wed & Fri) F. The collection of pretty tiles, faucets, handles, ceramic pipe joints and plenty of old toilets and bidets is well lit and displayed. Guided visits offer a backstage glimpse of the building's inner workings and huge water tanks. Bring photo ID and enter via Riobamba.\n\nAlso known as Obras Sanitarias, Palacio de las Aguas Corrientes dates from 1894 and occupies an entire city block. Topped by French-style mansard roofs, the building's facade consists of 170,000 glazed tiles and 130,000 enameled bricks, all shipped from England and assembled here.\n\nOBELISCO LANDMARK\n\nMAP GOOGLE MAP\n\n(cnr Av 9 de Julio & Corrientes) The city's unmistakable landmark is the famous Obelisco, which soars 67m above the oval Plaza de la Rep\u00fablica. Dedicated in 1936, on the 400th anniversary of the first Spanish settlement on the R\u00edo de la Plata, this stately spire symbolizes Buenos Aires much as the Eiffel Tower represents Paris or the Washington Monument does Washington, DC. Following major soccer victories, boisterous fans circle the Obelisco in jubilant, honking celebration; it's also often used as the zero point for measuring distances from the city center. Unfortunately, you can't enter the structure \u2013 you'll have to admire it from outside.\n\nPLAZA LAVALLE PLAZA\n\nMAP GOOGLE MAP\n\n(Libertad btwn C\u00f3rdoba & Lavalle) A couple of blocks northwest of the Obelisco is Plaza Lavalle, three blocks of parks surrounded by some important buildings. Two big landmarks here are the Teatro Col\u00f3n and the Teatro Nacional Cervantes (Click here). But there's also the austere neoclassical Escuela Presidente Roca (1902), an educational facility that's often mistaken for Teatro Col\u00f3n. Across from it lies the French-style Palacio de Justicia (1904) and its Tribunales (federal courts).\n\nMUSEO JUDIO DR SALVADOR KIBRICK MUSEUM\n\nMAP GOOGLE MAP\n\n(  4123-0832; www.museojudio.org.ar; Libertad 769; admission AR$50;   11am-5pm Tue-Thu, to 4pm Fri) At the northeastern end of Plaza Lavalle, Jewish symbols adorn the facade of the Templo de la Congregaci\u00f3n Israelita, Argentina's oldest and largest synagogue. Concrete sidewalk planters, constructed after recent attacks against Jewish targets, discourage potential car bombs; police stand guard nearby. The Museo Judio Dr Salvador Kibrick is in the synagogue and contains many items and exhibits related to Jewish history. Bring photo ID for admission.\n\nMUSEO BEATLE MUSEUM\n\nMAP GOOGLE MAP\n\n(  6320-5362; www.thecavern.com.ar; Av Corrientes 1660; locals\/foreigners AR$30\/60;   10am-midnight Mon-Sat, 2pm-midnight Sun) Located in the Paseo La Plaza complex, this museum claims to be the only Beatles museum in South America. It showcases the Beatles memorabilia collection of owner Rodolfo R Veasquez \u2013 expect plenty of records, collector plates, toys, figurines, eight-track tapes, games and a couple of guitars from musicians related to the group. There's even a brick from the Cavern Club, a music club (now demolished) where the Beatles got their start. To find the museum, follow 'The Cavern' signs.\n\nCENTRO CULTURAL RICARDO ROJAS CULTURAL CENTER\n\nMAP GOOGLE MAP\n\n(  4954-5523; www.rojas.uba.ar; Av Corrientes 2038) This exceptional cultural center has a wide range of affordable classes, including dance, music, photography, theater, film and language studies.\n\nCENTRO CULTURAL SAN MART\u00cdN CULTURAL CENTER\n\nMAP GOOGLE MAP\n\n(  4374-1251; www.elculturalsanmartin.org; Sarmiento 1551) One of Buenos Aires' best resources, this large cultural center has free or inexpensive galleries, music, films, lectures, art exhibitions, classes and workshops.\n\n###  BUENOS AIRES' CARTONEROS\n\nYou'll see them mostly at night, hunched over at the curb, picking through the garbage and pushing loaded-down carts. These are not the homeless, or the crazy, or the drug-addicted, or even the city's petty thieves. These are regular people, but some of Buenos Aires' poorest citizens \u2013 they're _cartoneros_ (cardboard collectors). Many used to have regular jobs as skilled laborers or even businesspeople but have been laid off with the 2001 crisis. With unemployment still substantial and no social security to cover them, collecting recyclables is one way they can make a living.\n\nIt's estimated that around 20,000 _cartoneros_ rummage through Buenos Aires' trash heaps; some are even accredited by the city and wear uniforms. They sort through the city's 5000 daily tons of waste, collecting cardboard, paper, metal, plastic, glass \u2013 anything they can sell by the kilo to the _depositos_ (recycling companies). They stake out their territory, perhaps about 15 city blocks, and are occasionally forced to pay police bribes. Many have been pricked by syringes or cut by broken glass. This isn't an easy job, but it's decent work \u2013 once established, the daily take-home pay for a _cartonero_ can be AR$100 or more.\n\nWhile most _cartoneros_ work independently, some work for neighborhood cooperatives that pay them a regular wage and organize vaccinations. Some cooperatives even provide child care for parents who go off on their nightly rounds. In the poorest families, however, even the young children have to work all night long. And some _cartoneros_ are in their 50s and 60s.\n\nIt's not surprising that Argentina's economic crash has inflamed this side-business in recyclables, and that those less fortunate had to use their ingenuity to organize for themselves what their government could not. The _cartoneros_ are a reminder to us that there is another side to the glittering richness of Buenos Aires' center, and that there is another part of this city where the poor people live.\n\n Congreso & Tribunales\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  EATING\n\nHardly inspiring in terms of contemporary cuisine, the Congreso area caters mostly for business with cheap _parrillas_ (steakhouses) and quick takeout. Walk around the side streets, though, and you're bound to stumble across some Chinese, Korean and Peruvian gems. The biggest cultural footprint in this area, however, is Buenos Aires' Little Spain neighborhood (in the blocks around Avs de Mayo and Salta); here you'll find a few good traditional Spanish and Basque eateries.\n\n  CHAN CHAN PERUVIAN $\n\nMAP GOOGLE MAP\n\n(  4382-8492; Hip\u00f3lito Yrigoyen 1390; mains AR$35-65;   noon-4pm & 8pm-midnight Tue-Sun) Thanks to fair prices and relatively quick service, this colorful Peruvian eatery is jam-packed at lunchtime with office workers devouring plates of _seviche_ (seafood cured in citrus) and _ajiaco de conejo_ (rabbit and potato stew). There are also plenty of _arroz chaufa_ (Peruvian-style fried rice) dishes, easily downed with a tangy pisco sour or a pitcher of _chicha morada_ (a sweet fruity drink).\n\nPARRILLA PE\u00d1A PARRILLA $\n\nMAP GOOGLE MAP\n\n(  4371-5643; Rodr\u00edguez Pe\u00f1a 682; mains AR$50-90;   noon-4pm & 8pm-midnight Mon-Sat, noon-4pm Sun) This simple, traditional and long-running _parrilla_ is well known for its excellent-quality meats and generous portions. The service is fast and efficient and it's great value. Don't expect many tourists \u2013 this is a locals' sort of place. Also on offer are homemade pastas, salads and _milanesas_ (breaded steaks), along with several tasty desserts and a good wine list.\n\nPIZZER\u00cdA G\u00dcERR\u00cdN PIZZERIA $\n\nMAP GOOGLE MAP\n\n(  4371-8141; Av Corrientes 1368; slices AR$8;   11am-1am Sun-Thu, to 2am Fri & Sat) A quick pit-stop on Av Corrientes is this cheap but classic old pizza joint. Just point at a pre-baked slice behind the glass counter and eat standing up with the rest of the crowd. Or sit down and order one freshly baked \u2013 this way you can also choose from a greater variety of toppings for your pizza.\n\n_Empanadas_ (meat or vegetable pies) and plenty of desserts are also available.\n\nCHIQUIL\u00cdN PARRILLA $$\n\nMAP GOOGLE MAP\n\n(  4373-5163; Sarmiento 1599; mains AR$60-100;   noon-2am Sun-Thu, to 3am Fri & Sat) A local mainstay for 80 years, Chiquil\u00edn is an excellent place to safely take, say, your parents. It's a large, comfortable restaurant with a cozy and classic atmosphere (including hanging hams). Dressed-up staff are efficient, which is great because this place can bustle \u2013 even at 1am on a Saturday night. The best choices here are steak and pasta.\n\nPLAZA ASTURIAS SPANISH $$\n\nMAP GOOGLE MAP\n\n(  4382-7334; Av de Mayo 1199; mains AR$80-140;   11:30am-5pm & 7pm-1am Sun-Thu, to 2:30am Fri & Sat) This old-fashioned Spanish restaurant draws in a regular midday crowd with its set lunch (AR$90), which includes a main dish plus coffee, dessert and a glass of wine. Otherwise, the regular menu features staples like chorizo, ham and potato casserole, and pasta, as well as more adventurous dishes like _cazuela de mariscos,_ a powerful seafood stew rich with mussels, garlic and herbs.\n\n  ARAMBURU GOURMET $$$\n\n(  4305-0439; www.arambururesto.com.ar; Salta 1050; prix-fixe AR$390;   8:30-11pm Tue-Sat) 'Molecular' dining has taken Buenos Aires by storm, and Chef Gonzalo Aramburu is leading the pack. The set 12-course meal might take you up to three hours to enjoy, each artistically created plate just a few bites of gastronomic delight. Expect enlightening tastes, textures, smells, plus unique presentations and a highly memorable meal. Located in the edgy but upcoming neighborhood of Montserrat.\n\n Congreso & Tribunales\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  DRINKING & NIGHTLIFE\n\nThis neighborhood is not known for its drinking holes, but there are at least a few atmospheric spots where you can toast the town while the politicos scurry past on the sidewalks.\n\nCAF\u00c9 DE LOS ANGELITOS CAFE\n\nMAP GOOGLE MAP\n\n(  4314-1121; Av Rivadavia 2100;   8am-midnight) Originally called Bar Rivadavia, this cafe was once the haunt of poets, musicians, even criminals, which is why a police commissioner jokingly called it ' _los angelitos'_ (the angels) in the early 1900s. Recently restored to its former glory, this historic cafe is now an elegant hangout for coffee or tea; it also puts on tango shows in the evening.\n\nEL GATO NEGRO TEAHOUSE, CAFE\n\nMAP GOOGLE MAP\n\n(  4374-1730; Av Corrientes 1669;   9am-10pm Mon, to 11pm Tue, to midnight Wed & Thu, to 2am Fri & Sat, 3-11pm Sun) Tea-lined wooden cabinets and a spicy aroma welcome you to this pleasant little sipping paradise. Enjoy imported cups of coffee or tea, along with breakfast and dainty _sandwiches de miga_ (thin, crustless sandwiches, traditionally eaten at tea time). Imported teas and coffees are sold in bulk, and a range of exotic herbs and spices are also on offer.\n\nCRUZAT BEER HOUSE BAR\n\nMAP GOOGLE MAP\n\n(  6320-5344; www.cruzatba.com; Sarmiento 1617;   10am-2am Mon-Fri, 7pm-3am Sat) In wine-soaked Buenos Aires, Cruzat is as close as you can get to a German beer garden. Kick back on the shaded terrace and choose craft beers from all over Argentina \u2013 look for El Bols\u00f3n (from R\u00edo Negro), Antares (from Mar del Plata) and Gulmen (from Viedma). There are also imports from Belgium, Chile, Spain and Italy.\n\nLOS 36 BILLARES CAFE, BAR\n\nMAP GOOGLE MAP\n\n(  4381-5696; www.los36billares.com.ar; Av de Mayo 1265; shows from AR$50;   8am-2am Mon-Sat) Dating from 1894, this is one of the city's most historic cafe-bars. As its name implies, it's big on pool and billiard tables (check out the basement); the back room is full of men shuffling cards for a poker game. Tango shows, highlighting different singers and dancers every night, happen at 9pm from Monday to Wednesday; tango classes also available.\n\nMALUCO BELEZA CLUB\n\nMAP GOOGLE MAP\n\n(  4372-1737; www.malucobeleza.com.ar; Sarmiento 1728;   Wed & Fri-Sun) Located in an old mansion is this popular Brazilian _boliche (nightclub)_. It gets packed with upbeat revellers moving to samba fusion and others watching half-naked dancers writhing on the stage. For a more sedate atmosphere, climb the stairs, where it's more laid-back. If you're craving Brazilian cuisine, get here at 8:30pm on Wednesday, when dinner and a show are on tap.\n\nEL BESO MILONGA\n\nMAP GOOGLE MAP\n\n(  4953-2794; Riobamba 416, 1st fl) Another traditional and popular place, El Beso attracts some very good dancers \u2013 you should be very confident of your dancing skills if you come here. Located upstairs, it has good music and a cozy feel.\n\nOn Friday night, El Beso hosts the far less traditional but still well-known La Marshall Milonga (<http:\/\/lamarshallmilonga.com.ar>), a gay _milonga_ , for all who want to try a change of roles in their tango. There's a class at 10:30pm before the _milonga_ (tango session) at 11:30pm.\n\n Congreso & Tribunales\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nTEATRO COL\u00d3N CLASSICAL MUSIC\n\nMAP GOOGLE MAP\n\n(  4378-7100; www.teatrocolon.org.ar; Cerrito 628) BA's premier venue for the arts, with ballet and opera as well as classical music. For more on Teatro Col\u00f3n, Click here.\n\nTEATRO SAN MART\u00cdN CLASSICAL MUSIC\n\nMAP GOOGLE MAP\n\n(  0800-333-5254; www.complejoteatral.gob.ar; Av Corrientes 1530) This major venue has several auditoriums (the largest seats over 1000 people) and showcases international cinema, theater, dance and classical music, covering conventional and more unusual events. It also has art galleries and often hosts impressive photography exhibitions.\n\nTEATRO PASEO LA PLAZA THEATER\n\nMAP GOOGLE MAP\n\n(  6320-5300; www.paseolaplaza.com.ar; Av Corrientes 1660) Located in a small and pleasant outdoor shopping mall, this complex features several theater halls that run both classic and contemporary productions, including tango, theater and comedy.\n\nTEATRO PRESIDENTE ALVEAR THEATER\n\nMAP GOOGLE MAP\n\n(  4373-4245; www.complejoteatral.gob.ar; Av Corrientes 1659) Inaugurated in 1942 and named after an Argentine president whose wife sang opera, this theater holds over 700 and shows many musical productions, including ballet. Occasional free shows are on offer.\n\nTEATRO AVENIDA CLASSICAL MUSIC\n\nMAP GOOGLE MAP\n\n(  4812-6369; www.balirica.org.ar; Av de Mayo 1222) In 1979 a fire closed down this beautiful 1906 theater for 15 years, but it was later restored to its former glory. Today the Avenida highlights Argentine productions, mostly classical music, ballet and flamenco. But its biggest strength is opera.\n\nCAF\u00c9 DE LOS ANGELITOS TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4952-2320; www.cafedelosangelitos.com; Av Rivadavia 2100; show from US$90, show & dinner from US$130) Angelitos puts on one of the best shows in Buenos Aires. It's tango \u2013 but also a bit more. The performers dress in top-notch costumes and use interesting props, like drapes and moving walls. They also dance to modern tunes such as those by local band Bajofondo, and despite a nightclub feel at times \u2013 especially due to the lighting \u2013 it's all very tastefully and creatively done.\n\nThe stage is well set up (the musicians are on a different level, out of the way but well in sight) and everyone gets a good view. If you can't afford Rojo Tango (Click here), come here; it's choreographed by the same folks and offers some sexy elements.\n\nTANGO PORTE\u00d1O TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4124-9400; www.tangoporteno.com.ar; Cerrito 570; show US$80, show & dinner from US$127) One of the city's best shows takes place in this renovated art deco t heater. Snippets of old footage are interspersed with plenty of athletic (and at times sensual) dancing. There's an interesting blindfold number, the orchestra is excellent and Juan Carlos Copes \u2013 a famous Argentine dancer in his time \u2013 makes a rug-\u00adcutting cameo if he's in good health.\n\nIt's right in the center near Teatro Col\u00f3n. Complimentary tango class offered beforehand.\n\n\u00c1VILA BAR FLAMENCO\n\nMAP GOOGLE MAP\n\n(  4383-6974; Av de Mayo 1384;   Thu-Sat) Offering flamenco for many years now is this cozy little Spanish restaurant with good traditional food. Main dishes can include rabbit, paella and seafood stews. Shows have older, experienced dancers and cost AR$220 (drinks not included). They start around 10:30pm and reservations are a must on weekends.\n\nCANTARES FLAMENCO\n\nMAP GOOGLE MAP\n\n(  4381-6965; www.cantarestablao.com.ar; Av Rivadavia 1180; show $AR90;   shows 9pm Fri & Sat) This flamenco venue, in the old Taberna Espa\u00f1ol, once hosted the Spanish poet Federico Garc\u00eda Lorca. It's a small basement space with only 85 seats, providing a wonderfully intimate place for the highly authentic dances. An \u00e0 la carte dinner is on offer; reserve in advance. Flamenco classes also available.\n\n Congreso & Tribunales\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SHOPPING\n\nCongreso is where the city's politicos hang out and it's not really known as a shopping destination. That said, Av Corrientes has many of the city's discount bookstores, and despite most books being in Spanish, it's fun to wander through them. And if you're looking for jewelry or electronics like used cameras, check out Calle Libertad between Lavalle and Rivadavia.\n\nZIVAL'S MUSIC\n\nMAP GOOGLE MAP\n\n(www.zivals.com; Av Callao 395;   9:30am-9:30pm Mon-Sat) This is one of the better music stores in town, especially when it comes to tango, jazz and classical music. Listening stations are a big plus, and many books are also for sale. There's also a branch in Palermo Viejo (Serrano 1445).\n\nWILDLIFE CAMPING & OUTDOOR EQUIPMENT\n\nMAP GOOGLE MAP\n\n(  4381-1040; Hip\u00f3lito Yrigoyen 1133;   10am-8pm Mon-Fri, to 1pm Sat) If you're looking to buy (or sell) all manner of outdoor and camping equipment before traveling on from Buenos Aires, this is the place to do it. Crampons, knives, tents, backpacks, climbing ropes, foul-weather clothing, military gear and even the occasional mule saddle can be found at this somewhat offbeat and musty shop.\n\nCongresso Tribunales\n\n  Top Sights\n\n1Teatro Col\u00f3n F2\n\n  Sights\n\n2Centro Cultural Ricardo Rojas B3\n\n3Centro Cultural San Mart\u00edn D4\n\n4Museo Beatle D4\n\n5Museo del Patrimonio C2\n\n6Museo Judio Dr Salvador Kibrick F2\n\n7Obelisco F3\n\n8Palacio Barolo E5\n\n9Palacio de las Aguas Corrientes C2\n\n10Plaza del Congreso D5\n\n11Plaza Lavalle E2\n\n  Eating\n\n12Chan Chan E5\n\n13Chiquil\u00edn D4\n\n14El Cuartito E1\n\n15Parrilla Pe\u00f1a D2\n\n16Pizzer\u00eda G\u00fcerr\u00edn E3\n\n17Plaza Asturias F5\n\n  Drinking & Nightlife\n\n18Caf\u00e9 de los Angelitos B5\n\n19Cruzat Beer House D4\n\n20El Gato Negro D3\n\n21Maluco Beleza D4\n\n  Entertainment\n\n22\u00c1vila Bar E5\n\nCaf\u00e9 de los Angelitos (see 18)\n\n23Cantares F5\n\n24Cartelera Baires E3\n\n25Cosmos-UBA B3\n\n26El Beso C3\n\n27Espacio INCAA D5\n\n28Los 36 Billares F5\n\n29Sala Leopoldo Lugones D3\n\n30Tango Porte\u00f1o F2\n\n31Teatro Avenida F5\n\nTeatro Col\u00f3n (see 1)\n\n32Teatro Paseo la Plaza D4\n\n33Teatro Presidente Alvear D3\n\nTeatro San Mart\u00edn (see 29)\n\n  Shopping\n\n34Wildlife F5\n\n35Zival's C3\n\n  Sports & Activities\n\n36Punto Cumbre C4\n\n  Sleeping\n\n37Gran Hotel Oriental C5\n\n38Hotel Bonito E7\n\n39Hotel Lyon C4\n\n40Hotel Marbella F5\n\n41La Cayetana E7\n\n42Livin' Residence C2\n\n43Milhouse Youth Hostel G5\n\n44Novotel Hotel E3\n\n45Sabatico Hostel E7\n\n# San Telmo\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nSan Telmo\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Jostling with fellow shoppers at the Sunday  Feria de San Telmo, where vendors sell all manner of goods, and buskers and tango dancers compete for your spare change.\n\n Taking a break in peaceful  Plaza Dorrego \u2013 when it's _not_ a Sunday.\n\n Exploring the reconstructed tunnels at  El Zanj\u00f3n de Granados.\n\n Visiting the  Museo Hist\u00f3rico Nacional at historic Parque Lezama.\n\n Strolling cobbled streets and taking in old-time atmosphere at places like  Pasaje de la Defensa.\n\n###  Explore: San Telmo\n\nTwo central thoroughfares in this barrio are Balcarce and Defensa; they're where you'll find most things of interest to travelers.\n\nEveryone is drawn to Plaza Dorrego, the heart and soul of San Telmo. It's a nice leafy place to snag an outdoor table under an umbrella and have a coffee or full meal (though on Sundays the _feria_ takes over and tables disappear). Sometimes tango dancers provide entertainment for a few pesos, though you can also be hassled for spare change by beggars or asked to buy items by roving vendors. Keep a good hold of your bag, just in case.\n\nGenerally everything of interest is reachable by walking in this neighborhood. From Plaza Dorrego you can stroll up or down the main drag of Defensa, window-shopping for pricey antiques or trendy new trinkets along the way. Several museums are also on or just off this street. If you head south, you'll hit Parque Lezama, a local park where families hang out at the playground and lovers smooch on benches. Heading north, you can be in the Plaza de Mayo area in 15 minutes.\n\nThe adventurous can walk further south down busy Av Almirante Brown to La Boca (note: this will take you along the gritty edges of this blue-collar neighborhood).\n\n###  Local Life\n\n\u00bb Markets Explore the Mercado de San Telmo (Click here) to get a dose of history and a feel for how the locals buy their meats and vegetables.\n\n\u00bb Hangouts Classic cafes like Bar Plaza Dorrego (Click here), Bar El Federal (Click here) and La Poesia (Click here) drip with traditional atmosphere and old-time locals taking in their morning _medialunas_ (croissants) or afternoon coffee breaks.\n\n\u00bb Games Like chess? Then head to Parque Lezama (Click here) and find the cluster of chess tables there \u2013 and, if you dare, make a challenge.\n\n###  Getting There & Away\n\n\u00bb Bus Take bus 59 from Recoleta and Palermo, bus 29 from La Boca, Plaza de Mayo and Palermo.\n\n\u00bb Subte L\u00ednea C connects the western edge of San Telmo with the Center, Congreso and Retiro.\n\n####  Lonely Planet's Top Tip\n\nOn Sundays the _feria_ (street market) is full-on, which means tons of people are visiting and you'll need to watch for pickpockets and over-charging more carefully. On the other hand, it's a fun time to be in the neighborhood and the museums and most stores are all open. For more peace, visit San Telmo from Monday to Saturday, especially to sit at one of Plaza Dorrego's outdoor tables.\n\nEast of Balcarce the streets become industrial and, along with the more southern edges toward La Boca, should probably be avoided at night. During the day they're fine, however, and provide access to Puerto Madero.\n\n  Best Places to Eat\n\n\u00bbCaf\u00e9 San Juan\n\n\u00bbCasal de Catalunya\n\n\u00bbOrigen Caf\u00e9\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbBar Plaza Dorrego\n\n\u00bbDoppelg\u00e4nger\n\n\u00bbGibraltar\n\nFor reviews, Click here \u00bb\n\n  Best Shopping\n\n\u00bbFeria de San Telmo\n\n\u00bbMateria Urbana\n\n\u00bbPuntos en el Espacio\n\nFor reviews, Click here \u00bb\n\nPANORAMIC IMAGES \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nPLAZA DORREGO\n\nAt the heart of San Telmo is Plaza Dorrego, normally a peaceful little plaza strewn with locals and tourists sitting at tables under their umbrellas. A few hippie street vendors hawk their wares on the sidewalks while tango dancers occasionally perform for a few pesos. Come Sundays, however, the plaza and Calle Defensa become packed with craft stalls selling everything from antiques to knick-knacks to creative homemade souvenirs. It's a bit of a crazy scene, but worth experiencing nonetheless.\n\n###  DON'T MISS...\n\n\u00bb Relaxing at a table on the plaza while tango dancers perform nearby.\n\n\u00bb Sunday's bustling street _feria_.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\nAfter Plaza de Mayo, Plaza Dorrego is the city's oldest plaza. It dates to the 18th century and was originally a pit stop for caravans bringing supplies into Buenos Aires from around Argentina. At the turn of the 19th century it became a public square surrounded by colonial buildings that survive to this day. There's still a wonderful old-time atmosphere here and cafe-restaurants like Bar Plaza Dorrego (Click here) will definitely take you back in time. However, things are changing; across from this traditional cafe, a branch of Starbucks recently opened and prominently announced the arrival of the 21st century.\n\nPlaza Dorrego's biggest claim to fame is likely now its extremely popular Sunday _feria_. Tourists and locals alike flock to this fun event, which brings together hundreds of street vendors, buskers and shoppers. Originally started in 1970 as an antiques fair, the _feria_ has now become a craft market offering all manner of items \u2013 jewelry, souvenirs, knick-knacks, quality artwork, vintage clothing, old collectibles, hand-made crafts, leather jackets and much, much more. Defensa is closed to traffic from Plaza de Mayo to Parque Lezama (which has its own little craft market) and lined with hundreds of stalls. Street performers from metallic human statues to _candombe_ drumming groups to professional tango dancers entertain the crowds, while sidewalk tables provide welcome breaks. It's a tight and crowded scene, so be prepared to bump into people \u2013 and watch your bag carefully.\n\n San Telmo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SIGHTS\n\nPLAZA DORREGO PLAZA\n\nSee Click here.\n\nEL ZANJ\u00d3N DE GRANADOS ARCHAEOLOGICAL SITE\n\nMAP GOOGLE MAP\n\n(  4361-3002; www.elzanjon.com.ar; Defensa 755; 1 hr tour Mon-Fri AR$90, 30 min tour Sun AR$60;   tours 11am, noon, 2pm & 3pm Mon-Fri, every 30min 1-6pm Sun) One of the more unique places in BA is this amazing urban architectural site. A series of old tunnels, sewers and cisterns (built from 1730 onwards) were constructed above a river tributary and provided the base for one of BA's oldest settlements, which later became a family mansion and then tenement housing and some shops. It's best to reserve ahead for tours.\n\nThe Zanj\u00f3n is the realized dream of Jorge Eckstein, who found these ruins in 1986 after purchasing land for a business project and then spent years renovating them into what you see today. It offers a fascinating glimpse into the city's history; meticulously reconstructed brick by brick and very attractively lit, this site also contains several courtyards and even a watchtower. There are a few relics on display in the various halls and rooms, but the highlights are the spaces themselves.\n\nMUSEO DE ARTE MODERNO DE BUENOS AIRES (MAMBA) MUSEUM\n\nMAP GOOGLE MAP\n\n(  4342-3001; www.museodeartemoderno.buenosaires.gob.ar; Av San Juan 350; admission AR$5, Tue free;   11am-7pm Tue-Fri, to 8pm Sat & Sun) Housed in a recycled tobacco warehouse, this spacious and newly remodeled museum shows off the works of both national and international contemporary artists. Expect temporary exhibitions showcasing everything from photography to industrial design, and from figurative to conceptual art. There's also an auditorium, and there are plans to integrate the old cinema museum next door, too \u2013 and add a cafe and gift shop.\n\nMUSEO DE ARTE CONTEMPOR\u00c1NEO BUENOS AIRES MUSEUM\n\nMAP GOOGLE MAP\n\n(MACBA;   5299-2010; www.macba.com.ar; Av San Juan 328; admission AR$25;   noon-7pm Mon & Wed-Fri, 11am-7:30pm Sat & Sun) Art lovers shouldn't miss this fine museum, which specializes in geometric abstraction drawn from the technology-driven world that surrounds us today (think architecture, maps and computers). So rather than traditional paintings, you'll see large, colorful and minimalist pieces meant to inspire reflection. It mostly shows off the works of young Argentine artists, though occasional international guests' work appears. There are four floors; the first two hold rotating, permanent exhibits. Guided visits in Spanish are available at 5pm daily.\n\nPASAJE DE LA DEFENSA NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Defensa 1179;   10am-6pm Tue-Fri, to 8pm Sun) Originally built for the Ezeiza family in 1880, this building later became a _conventillo_ (tenement house) and was home to 32 families. These days, it's a charmingly worn building with antique shops clustered around atmospheric leafy patios.\n\nPARQUE LEZAMA PARK\n\nMAP GOOGLE MAP\n\n(Defensa & Av Brasil) Scruffy Parque Lezama was once thought to be the site of Buenos Aires' founding in 1536, but archaeological teams have refuted the hypothesis. Today's green park hosts old chess-playing gentlemen, bookworms toting _mate_ (traditional Argentine tea) gourds and teenagers kissing on park benches. Don't miss the striking late-19th-century Iglesia Ortodoxa Rusa (Russian Orthodox Church) on the north side of the park; it's the work of architect Alejandro Christopherson and was built from materials shipped over from St Petersburg.\n\nMUSEO HIST\u00d3RICO NACIONAL MUSEUM\n\nMAP GOOGLE MAP\n\n(  4307-1182; Defensa 1600;   11am-6pm Wed-Sun) F Located in Parque Lezama is the city's national historical museum. It's dedicated to exhibiting items related to Argentina's revolution on May 25, 1810. Argentine hero Manuel Belgrano's watch was stolen from this museum in 2007, and things have never been the same since. Bags and backpacks have to be checked in, and guards are everywhere.\n\nInside, exhibits are a bit sparse, but at least they're neatly displayed. There are several portraits of presidents and other major figures of the time, and you can peek into a recreated version of Jos\u00e9 de San Mart\u00edn's bedroom \u2013 he was a military hero and liberator of Argentina (along with other South American countries). Old documents are also on display, and there's a video room as well.\n\nPerhaps the most interesting exhibit, however, is of a few paintings depicting Africans in Argentina celebrating Carnaval and playing _candombe_ (a drum-based musical genre invented in the early 18th century by slaves brought to the Rio de la Plata region). Argentina's black history is limited and mysterious \u2013 the country did have a slave trade, but today there are very few people of African descent here.\n\nMUSEO PENITENCIARIO MUSEUM\n\nMAP GOOGLE MAP\n\n(  4361-0917; museopenitenciarioargentino.blogspot.com.ar; Humberto Primo 378;   2-6pm Thu, Fri & Sun) F Dating from 1760, this building was a convent and later a women's prison before it became a penal museum in 1980; reconstructed old jail cells give an idea of the prisoners' conditions. Don't miss the homemade playing cards and shivs, plus the tennis balls used to hide drugs. A prison infirmary is also exhibited. Next door, the neocolonial and baroque Iglesia Nuestra Se\u00f1ora de Bel\u00e9n MAP GOOGLE MAP (Humberto Primo 340) was a Jesuit school until 1767, when the Bethlemite order took it over.\n\nCONVENTO DE SANTO DOMINGO CHURCH\n\nMAP GOOGLE MAP\n\n(  4331-1668 ext 201; cnr Defensa & Belgrano;   bas\u00edlica tours by appointment only;   L\u00ednea E Bol\u00edvar, L\u00ednea A Plaza de Mayo) Marking the approach into San Telmo, this 18th-century Dominican building has a long and colorful history. On its left tower you'll see the replicated scars of shrapnel launched against British troops who holed up here during the invasion of 1806. The bas\u00edlica displays the flags that were captured from the British. Secularized during the presidency of Bernardino Rivadavia (1826\u201327), the building became a natural history museum, its original single tower serving as an astronomical observatory, until Governor Juan Manuel de Rosas restored it to the Dominican order. For (free) basilica tours, call between 3:30pm and 6:30pm; most tours run on Sundays.\n\nMUSEO DEL TRAJE MUSEUM\n\nMAP GOOGLE MAP\n\n(  4343-8427; www.funmuseodeltraje.com.ar; Chile 832;   3-7pm Tue-Sun;   L\u00ednea C Independencia) F Near the Montserrat border, this small clothing museum is always changing its wardrobe. You can hit upon wedding outfits from the late 1800s, popular fashions from the early 1900s or even clothing worn by travelers on the Silk Rd. If you're lucky, accessories such as hair combs, top hats, antique eyeglasses and elegant canes might be on display.\n\nFACULTAD DE INGENIERIA NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Av Paseo Col\u00f3n 850) This neo\u00adclassical and monstrous building is the engineering school for the Universidad de Buenos Aires. It was originally built for the Fundaci\u00f3n Eva Per\u00f3n and is an oddball landmark once described by Gerald Durrell as 'a cross between the Parthenon and the Reichstag.'\n\nIn front of the building and in the middle of Av Paseo Col\u00f3n is Plazoleta Olaz\u00e1bal, a tiny park that features Rogelio Yrurtia's masterful sculpture Canto al Trabajo. It was moved here from its original site on Plaza Dorrego.\n\n###  SAN TELMO: A BRIEF HISTORY\n\nSan Telmo is known for the violent street fighting that took place when British troops, at war with Spain, invaded the city in 1806. They occupied it until the following year, when covert porte\u00f1o resistance became open counterattack. British forces advanced up narrow Defensa, but the impromptu militia drove the British back to their ships. Victory gave _porte\u00f1os_ confidence in their ability to stand apart from Spain, even though the city's independence had to wait another three years.\n\nAfter this San Telmo became a fashionable, classy neighborhood, but in the late 19th century a yellow-fever epidemic hit and drove the rich onto higher ground, west and north of the present-day Microcentro. As European immigrants began to pour into the city, many older mansions in San Telmo became _conventillos_ (tenements) to house poor families. Today you can see one such _conventillo_ at Pasaje de la Defensa (Click here), originally built for the Ezeiza family; later it housed 32 families.\n\n San Telmo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  EATING\n\nThe heart of San Telmo, Plaza Dorrego is surrounded by several cafe-restaurants that pop open their umbrellas from Monday to Saturday. On Sunday, however, the plaza (and a few surrounding streets) is taken over by vendors and tourists jamming the ever-popular antiques market. San Telmo has traditionally supported a large cluster of _parrillas,_ but as the neighborhood inexorably gentrifies, more innovative, upscale and pricier restaurants and bars are moving in.\n\nORIGEN CAF\u00c9 INTERNATIONAL $\n\nMAP GOOGLE MAP\n\n(  4362-7979; Primo 599; mains AR$50-70;   8am-10pm Tue-Fri & Sun, to 9:30pm Mon & Sat) Modern but unpretentious, this stylish corner bistro spills out onto the wide sidewalks; snag an outdoor table on a sunny afternoon. The creative menu features health-conscious dishes from stir-fries and whole-wheat pizzas to homemade soups and green salads. There's an emphasis on vegetarian food, and the cappuccinos are served in delightfully oversized mugs.\n\nEL DESNIVEL PARRILLA $\n\nMAP GOOGLE MAP\n\n(  4300-9081; Defensa 855; mains AR$45-80;   noon-4:30pm & 7pm-1am Mon-Fri, noon-1am Sat & Sun) This famous and long-running _parrilla_ joint packs in both locals and tourists, serving them treats like chorizo sandwiches and _bife de lomo_ (tenderloin steak). The sizzling grill out front is torturous while you wait for a table (which could be in the large back room) \u2013 get here early, especially on weekends.\n\nBAR EL FEDERAL ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4300-4313; cnr Per\u00fa & Carlos Calvo; mains AR$35-110;   8am-2am Sun-Thu, to 4am Fri & Sat) Dating from 1864, this historic bar has a classic, somewhat rustic atmosphere accented with original wood, tile, and an eye-catching antique bar. The specialties here are sandwiches (especially turkey) and _picadas_ (shared appetizer plates), but there are also lots of pastas, salads, desserts and tall mugs of icy beer.\n\nLA POESIA ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4300-7340; www.cafelapoesia.com.ar; Chile 502; mains AR$40-110;   8am-2am) Step back in time at this historic, traditional cafe. Originally a gathering place for artists and poets, this small corner spot still retains its bohemian atmosphere with live music recitals on Thursdays and Fridays. Snack on a turkey sandwich, _milanesa_ (beef cutlet) or some pasta, and try to relive the past.\n\nABUELA PAN VEGETARIAN $\n\nMAP GOOGLE MAP\n\n(  4361-4936; Chile 518; daily menu AR$43;   8am-7pm Mon-Fri, 9am-4pm Sun;  ) Tiny but atmospheric spot with just a handful of tables. The vegetarian special changes daily \u2013 expect things like spinach omelets, stuffed cannelloni and burghul hamburgers.\n\nCASAL DE CATALUNYA SPANISH $$\n\nMAP GOOGLE MAP\n\n(  4361-0191; Chacabuco 863; mains AR$60-100;   8pm-midnight Mon, noon-4pm & 8pm-midnight Tue-Sat, noon-4pm Sun) Located in BA's Catalan cultural center is this excellent Catalan restaurant. Big on seafood, its specialties run from garlic shrimp to fresh mussels and clams in tomato sauce to fish of the day with _aioli_. Other typical dishes include _jam_ \u00f3 _n serrano_ (prosciutto-like ham), seafood paella and suckling pig. Don't miss the luscious _crema Catalana_ for dessert.\n\nLA PANADER\u00cdA DE PABLO MODERN ARGENTINE $$\n\n(  4331-4683; Defensa 269; mains AR$70-110;   9:30am-6pm Mon-Wed, 9:30am-midnight Thu & Fri, 8pm-midnight Sat, 10am-7pm Sun) Enter this modern restaurant and be comforted by the awesome design, airy spaces and cozy booths. Try the smoked salmon salad with avocado, rib-eye marinated in rosemary and thyme or Yamani rice stir fries. There are a few elegant pizzas and pastas too, along with over a dozen cocktails to accompany. You'll also find a great patio in back for warm days, plus breakfast offerings.\n\nGRAN PARRILLA DEL PLATA PARRILLA $$\n\nMAP GOOGLE MAP\n\n(  4300-8858; www.parrilladelplata.com; Chile 594; mains AR$65-100;   noon-4pm & 8pm-1am) There's nothing too fancy at this traditional corner _parrilla_ \u2013 just old-time atmosphere and generous portions of good grilled meats at decent prices. There are also pastas for that vegetarian who gets dragged along. Divided into two sections but they're right next to each other.\n\n  CAF\u00c9 SAN JUAN INTERNATIONAL $$$\n\nMAP GOOGLE MAP\n\n(  4300-1112; Av San Juan 452; mains AR$125-150;   12:30-4pm & 8pm-1am) Having studied in Milan, Paris and Barcelona, celebrity TV-chef Leandro Crist\u00f3bal now runs the kitchen at this renowned San Telmo bistro. Start with fabulous tapas, then delve into the grilled Spanish octopus, _molleja_ (sweetbreads) cannelloni and the amazing pork _bondiola_ (deliciously tender after nine hours' roasting). Most of the seafood is flown in daily from Patagonia. Reserve for lunch and dinner.\n\nIf you can't get a table here, try the Caf\u00e9 San Juan La Cantina MAP GOOGLE MAP (  4300-9344; Chile 474;   noon-3:30pm & 8:30pm-midnight Tue-Sun). It's located a few blocks away and has a different menu.\n\nCOMEDOR NIKKAI JAPANESE $$$\n\nMAP GOOGLE MAP\n\n(  4300-5848; Av Independencia 732; mains AR$80-250;   noon-3pm & 7:30-11pm Mon-Thu, noon-3pm & 8pm-midnight Fri, 8pm-midnight Sat) Housed in the Asociaci\u00f3n Japonesa building, this restaurant has some of BA's most authentic Japanese food, and the locals know it \u2013 come early if you don't want to wait. All your favorites are here, including tempura, teriyaki, ramen or udon noodles and \u2013 of course \u2013 lots of sushi and sashimi choices. Imported sake is available too.\n\n###  AN EXPAT'S SAN TELMO\n\nNew Jersey native Jessica Pollack, a cultural historian and expert city tour guide with Buenos Tours (Click here), tells us what she loves about San Telmo, one of Buenos Aires' most historic 'hoods and the barrio that she's chosen to call home.\n\n####  San Telmo's Got Character\n\nI am always impressed by the palpable history on the cobbled streets: on Pasaje San Lorenzo you see layers of time. The once-grand Spanish mansions, later occupied by immigrant families, are now covered in street art and graffiti and house anything from music venues to pilates studios. If you peek into the courtyard at San Lorenzo 317, there's an artisan workshop called El Moro, a family operation that makes traditional _mates_ and _bombillas_ \u2013 it's full of the history, personality and artistry typical of an old neighborhood.\n\n####  The Old & the New\n\nDespite gentrification, the traditional is not sacrificed for the trendy; they coexist wonderfully. I love that on any night I can get a craft beer at historic Bar El Federal (Click here) or contemporary cocktails at dimly lit Doppelg\u00e4nger (Click here). Even inside the century-old indoor Mercado de San Telmo (Click here) you'll find delicious preserves homemade by a woman named Margarita just a few stalls down from Coffee Town, serving specialty blends and imported coffees at its fashionable new stand.\n\n####  Sunday Suggestions\n\nWe all know Sunday is San Telmo's big day: the area comes to life with the hugely popular and highly enjoyable street fair (Click here). But for most of the city, Sunday means relaxation and quiet time. To me, Sunday means waking up late for a slice of ricotta cake from Confiter\u00eda Europa (Carlos Calvo 678), where Argentine men watch _f\u00fatbol_ and sip _cafe con leche_. The service is terribly slow, but it's Sunday after all, and the leisurely pace is a welcome escape from the street-fair bustle.\n\n San Telmo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  DRINKING & NIGHTLIFE\n\nSan Telmo keeps gentrifying. Fancy restaurants and bars continue to pop up with regularity, mixing it up with a few old classics like historic cafes that have hardly changed over the years. Here older gentlemen still show up for their morning coffee and _medialunas_ , but there's space for everyone \u2013 and this neighborhood has become very popular with locals, travelers and expats.\n\n  BAR PLAZA DORREGO CAFE\n\nMAP GOOGLE MAP\n\n(  4361-0141; Defensa 1098;   8am-2am Sun-Thu, to 3am Fri & Sat) You can't beat the atmosphere at this traditional joint; sip your _submarino_ (hot milk with chocolate) by a picturesque window and watch the world pass by, or grab a table on the busy plaza. Meanwhile, traditionally suited waiters, piped-in tango music, antique bottles and scribbled graffiti on walls and counters might take you back in time.\n\nDOPPELG\u00c4NGER COCKTAIL BAR\n\nMAP GOOGLE MAP\n\n(  4300-0201; Av Juan de Garay 500) This cool, emerald-hued corner bar is one of the only places in BA where you can count on a perfectly mixed martini. That's because Doppelg\u00e4nger specializes in vermouth cocktails. The atmosphere is calm and the lengthy menu is fascinating: start with the journalist, a martini with a bitter orange twist, or channel Don Draper and go for the bar's bestseller \u2013 an old-fashioned.\n\nGIBRALTAR PUB\n\nMAP GOOGLE MAP\n\n(  4362-5310; Per\u00fa 895;   noon-4am) One of BA's classic pubs, the Gibraltar has a cozy atmosphere and good bar counter for those traveling alone. It's also a great place for fairly authentic foreign cuisine \u2013 try the Thai, Indian or English dishes (full English breakfast offered from noon to 5pm). For a little friendly competition, head to the pool table in the back.\n\nCOFFEE TOWN COFFEE\n\nMAP GOOGLE MAP\n\n(  4361-0019; www.coffeetownargentina.com; Bolivar 976, inside Mercado de San Telmo;   10am-8pm) For some of BA's best coffee, drop into this very casual kiosk inside the Mercado de San Telmo (enter via Carlos Calvo). Experienced baristas serve up organic, fair-trade coffee derived from beans from all over the world \u2013 think Colombia, Kenya, Sumatra and Yemen. A few pastries help the java go down easy.\n\nBAR BRIT\u00c1NICO CAF\u00c9\n\nMAP GOOGLE MAP\n\n(  4361-2107; Av Brasil 399;   24hr Tue-Sun, 8am-midnight Mon) A classic corner cafe on the edge of Parque Lezama, Bar Brit\u00e1nico has an evocative old wooden interior and big glass windows that open to the street. Drop in for a _caf\u00e9 cortado_ (small espresso with milk) in the morning or a beer on a sunny afternoon.\n\nLA PUERTA ROJA BAR\n\nMAP GOOGLE MAP\n\n(  4362-5649; Chacabuco 733;   5pm-late) There's no sign at this upstairs bar \u2013 just look for the red door. It has a cool, relaxed atmosphere with low lounge furniture in the main room and a pool table tucked behind. This is a traditional place, so you won't find fruity cocktails on the menu \u2013 but there's good international food like curries, tacos and chicken wings.\n\nBAR SEDD\u00d3N BAR RESTAURANT\n\nMAP GOOGLE MAP\n\n(  4342-3700; Defensa 695;   10am-4am Mon-Thu, to 6am Fri-Sun) This long-running corner bar-restaurant, outfitted with black and white tiles and rustic wood tables, is housed in an old restored pharmacy. Drop in for an icy _chopp_ (mug of draught beer) or a late-night glass of red \u2013 there are also sandwiches, pizzas and daily specials if you're hungry.\n\nPRIDE CAFE CAFE\n\nMAP GOOGLE MAP\n\n(  4300-6435; Balcarce 869;   9am-8pm Mon-Fri, 11am-8pm Sat & Sun;  ) This small, gay-friendly and contemporary cafe is especially swamped by cute men on Sunday during San Telmo's antiques fair, attracted by the homemade pastries, healthy snacks and flavored coffees. Peruse the foreign mags or utilize the free wi-fi, and maybe pick up a hot date.\n\nBOUTIQUE CLUB\n\nMAP GOOGLE MAP\n\n(  4543-3894; www.museumclub.com.ar; Per\u00fa 535;   Wed, Fri & Sat) This cavernous disco is best known for its Wednesday-night 'after-office' party (read: meat market), which starts at the normally ungodly-early hour of 7pm and runs to the ungodly-early finishing hour of 2am. It's a huge space with multiple balconies and a great sound system highlighting '80s and '90s pop music. Note the amazing building, an old factory designed by Eiffel \u2013 who also did that particular Parisian landmark.\n\n####   Neighborhood Walk  \n **Historical Saunter**\n\n**Start** El Zanj\u00f3n de Granados\n\n**End** Bar Brit\u00e1nico\n\n**Length** 1.5km; 2\u00bd hours\n\nTime your walk to tour the amazing series of tunnels and brick archways of  El Zanj\u00f3n de Granados (Click here), which formed the foundations of BA's oldest homes.\n\nThe decaying white-stucco-and-brick  Casa M\u00ednima at San Lorenzo 380 is a good example of the narrow-lot style known as _casa chorizo_ (sausage house). Barely 2m wide, the lot was reportedly an emancipation gift from slave owners to their former bondsmen.\n\nStop at the lively  El Desnivel (Click here) for a good steak experience. And don't miss strolling through the covered  Mercado de San Telmo (Click here), which has been running since 1897.\n\nBack on Defensa you'll soon reach the heart of the barrio,  Plaza Dorrego (Click here). From Monday to Saturday it's a relatively peaceful place, but come Sunday the lively  Feria de San Telmo (Click here) sets up in the plaza and surrounding streets.\n\nFor funky prison paraphernalia, check out the  Museo Penitenciario (Click here); note the  Iglesia Nuestra Se\u00f1ora de Bel\u00e9n (Click here), an old Jesuit school, next door.\n\nA block south, the  Museo de Arte Moderno de Buenos Aires (Click here) offers cutting-edge exhibitions, along with works by classic Argentine artists. Next door is the  Museo de Arte Contempor\u00e1neo Buenos Aires (Click here), great for abstract art.\n\nThe freeway location of the  Club Atl\u00e9tico Memorial is simply awful \u2013 but so is its history. This is one of the secret detention centers where thousands of people were tortured and killed during Argentina's Dirty War (1976\u201383). There isn't much left beyond an excavated basement where a three-story building used to be.\n\nStroll through the large  Parque Lezama (Click here) to the  Museo Hist\u00f3rico Nacional (Click here) for a bit of insight into Argentina's history. And finally, rest your tired feet at the atmospheric corner  Bar Brit\u00e1nico (Click here), snag a prized window seat and order a drink \u2013 you deserve it.\n\n San Telmo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nEL VIEJO ALMAC\u00c9N TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4307-7388; www.viejoalmacen.com; cnr Balcarce & Av Independencia; show from US$90, show & dinner from US$140) One of Buenos Aires' longest-running shows (since 1969), this venue is a charming old building from the 1800s. A good dinner is served at a multi-story restaurant in the main building, then everyone heads across the street to the small theater with intimate stage. The show starts with a quick movie about the tango show's history, then moves on to the highly athletic dancers with plenty of glitz. One highlight is the exceptionally good folklore segment.\n\nLA VENTANA TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4334-1314; www.laventanaweb.com; Balcarce 431; show from US$90, dinner & show from US$140) This long-running basement venue is located in an old converted building with rustic brick walls. The tango show is excellent and includes a folkloric segment with Andean musicians and a display of _boleadores_ (balls on cords that gauchos used to tangle up prey). There's also a patriotic tribute to Evita as a singer belts out 'Don't Cry for Me, Argentina.' The dinner offers a wide variety of tasty main dishes \u2013 unusual for tango shows. Gala Tango is a more upscale experience and happens upstairs.\n\nLA TRASTIENDA ROCK, REGGAE\n\nMAP GOOGLE MAP\n\n(  5533-5533; www.latrastienda.com; Balcarce 460) This large, atmospheric theater welcomes over 700, features a well-stocked bar, and showcases national and international live-music acts almost nightly. Look for headers such as Charlie Garcia, Divididos, Jos\u00e9 Gonzalez, Damien Rice and Conor Oberst. Get tickets at the office here or check www.tuentrada.com.ar.\n\nTODO MUNDO TANGO, FLAMENCO\n\nMAP GOOGLE MAP\n\n(  4362-2354; Pasaje Anselmo Aieta 1095) This restaurant puts on free tango, flamenco and other types of shows, but you do have to order at least AR$100 worth of food \u2013 basic Argentine fare like _empanadas_ , pasta and _parrilla_. Tango shows happen on Monday and Thursday nights, while flamenco flutters on Friday and Saturday nights. Expect rock, salsa, folk and jazz on other nights; all shows start around 10:30pm.\n\nLA SCALA DE SAN TELMO CLASSICAL MUSIC\n\nMAP GOOGLE MAP\n\n(  4362-1187; www.lascala.org.ar; Pasaje Guiffra 371) This small San Telmo venue, located in a refurbished colonial building, puts on classical and contemporary concerts that highlight piano, tango, musical comedies and other musical-related shows and workshops. Affordable or free admission.\n\n San Telmo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SHOPPING\n\nSan Telmo has traditionally been Buenos Aires' antiques neighborhood. In recent years, however, San Telmo's popularity with tourists has attracted other kinds of stores. Fashion boutiques and housewares shops are moving in, changing the general feel on the streets. Locals fear that their beloved neighborhood might become another Palermo, but even with rising real-estate prices, San Telmo is not likely to lose its gritty authenticity or charm.\n\nFERIA DE SAN TELMO MARKET\n\nMAP GOOGLE MAP\n\n(Defensa;   10am-6pm Sun;   10, 22, 29, 45, 86) On Sundays, San Telmo's main drag is closed to traffic and the street is a sea of both locals and tourists browsing craft stalls, waiting at vendors' carts for freshly squeezed orange juice, poking through the antique glass ornaments on display on Plaza Dorrego, and listening to street performances by myriad music groups. Runs from Av San Juan to Plaza de Mayo.\n\nMATERIA URBANA HOUSEWARES, ART\n\nMAP GOOGLE MAP\n\n(  4361-5265; www.materiaurbana.com; Defensa 702;   11am-7pm Wed-Fri, 2-7pm Sat, 10:30am-7pm Sun) This innovative design shop shows the work of over 100 local artists; one-of-a-kind finds include offbeat line drawings, abstract photography, carved wood statuettes, leather animal organizers, clothes, and jewelry made from silver, wood and coral. There's nearly constant foot traffic at Materia Urbana, especially during the street fair on Sunday.\n\nPUNTOS EN EL ESPACIO FASHION\n\nMAP GOOGLE MAP\n\n(  4307-7906; www.puntosenelespacio.com.ar; Carlos Calvo 450;   11am-8pm) With over 40 designers represented, this store is a good place to check out edgy women's collections by rising stars in the local fashion world. There are also kids' and mens' clothes, handbags, jewelry and a few shoes. A second location, focused on accessories and home decor, is on the corner of Defensa and Independencia, a few blocks away.\n\nIMHOTEP ANTIQUES\n\nMAP GOOGLE MAP\n\n(  4862-9298; Defensa 916;   11am-6pm Sun-Fri) Come find the funkiest old knickknacks at this eccentric shop. Small oddities such as Indian statuettes, Chinese snuff boxes, precious stone figurines and gargoyles make up some of the bizarre trinkets here. Also look for fantastical and mythological creatures; there are also plenty of skulls.\n\nGIL ANTIGUEDADES ANTIQUES\n\nMAP GOOGLE MAP\n\n(Humberto Primo 412;   11am-1pm & 3-7pm Tue-Sun) A window display of Great Gatsby\u2013style flapper dresses and vintage nightgowns pulls the passerby into San Telmo's finest antiques emporium. Decorative objects like china teapots and leather hatboxes are overshadowed by the stunning array of silk slips and lacy Victorian gowns \u2013 John Galliano, Catherine Deneuve and Salvatore Ferragamo are among the famous people who've stopped by for inspiration on visits to Buenos Aires.\n\nMERCADO DE SAN TELMO MARKET\n\nMAP GOOGLE MAP\n\n(Defensa, Bol\u00edvar, Carlos Calvo & Estados Unidos block;   9am-8pm) This market was built in 1897 by Juan Antonio Buschiazzo, the same Italian-born Argentine architect who designed Recoleta Cemetery. It occupies the inside of an entire city block, though you wouldn't be able to tell just by looking at the modest sidewalk entrances. The wrought-iron interior (note the amazing original ceiling) makes it one of BA's most atmospheric markets; locals shop here for fresh produce and meat. Peripheral antique stalls offer luggage, wine decanters and other treasures. More stalls are open on weekends.\n\nWALRUS BOOKS BOOKS\n\nMAP GOOGLE MAP\n\n(  4300-7135; Estados Unidos 617;   noon-8pm Tue-Sun) Run by an American photographer, this tiny shop is probably the best English-language bookstore in BA. Thousands of new and used literature and nonfiction books line the shelves here, and there's a selection of Latin American classics translated into English. Bring your quality books (including Lonely Planet guides!) to trade; literary workshops offered too.\n\nCUALQUIER VERDURA CLOTHING HOUSEWARES\n\nMAP GOOGLE MAP\n\n(  4300-2474; Humberto Primo 517;   noon-8pm Thu-Sun) Located in a lovely, refurbished old house, this fun store sells eclectic items from vintage clothing to funny soaps (look for these in the 'bathroom') to recycled floppy-disc lamps to contemporary knickknacks and novelty toys. Wander through the outdoor patio and note the stained-glass windows on the wall and _mate_ -drinking Buddha above the fountain.\n\nMOEBIUS CLOTHING\n\nMAP GOOGLE MAP\n\n(  4361-2893; Defensa 1356;   3-8pm Mon, noon-8pm Tue-Sat) This funky little shop's racks are crowded with owner-designer Lilliana Zauberman's kaleidoscopic products: 1970s-style jersey dresses, whimsical ruffled bikinis, skirts printed with koi fish and frog patterns, cherry-red trench coats and handbags made from recycled materials. Around 60 designers sell their work here, so there's always something different, fun and new to keep an eye out for.\n\nPUNTO SUR CLOTHING\n\nMAP GOOGLE MAP\n\n(  4300-9320; www.feriapuntosur.com.ar; Defensa 1135;   11am-7:30pm) This is a great clothing store highlighting the works of over 60 Argentine designers. Creativity is rampant and it's a fun walk-through for one-of-a-kind funky threads, including interesting knitwear, colorful skirts, printed T-shirts, jewelry and accessories, cool handbags and even a few shoes. These are definitely clothes that make a statement.\n\nSIGNOS JEWELRY\n\nMAP GOOGLE MAP\n\n(  15-5949-9193; www.signosac.blogspot.com.ar; Carlos Calvo 428;   11am-7pm Mon-Fri, to 5pm Sun) This is the tiny silversmithing shop of Alberto Codiani and Laura Romero, both artists who create beautiful jewelry. Amber, ammonites and precious stones are incorporated into unique pieces that are sure to attract attention. Custom work available; silversmithing classes also on offer.\n\nL'AGO HOUSEWARES, ACCESSORIES\n\nMAP GOOGLE MAP\n\n(  4362-4702; Defensa 970;   11am-8pm) Kitschy-cool home decor \u2013 from fluorescent _mate_ sets and funky pillows to Frida Kahlo kitchen magnets, eclectic lighting, recycled Elvis wallets and Marilyn Monroe handbags \u2013 attracts hipsters and travelers to cute-as-a-button L'Ago. Also at Thames 1247 in Palermo.\n\nEN BUEN ORDEN ANTIQUES\n\nMAP GOOGLE MAP\n\n(  15-5936-2820; Carlos Calvo 431;   2-6pm Tue-Sat, 10am-7pm Sun) If you fancy an old-fashioned little shop where you can sort through shelves full of knickknacks, old jewelry, Jackie O\u2013style sunglasses, old lace, musty shoes, opera gloves, pillbox hats and antique figurines, then this place is for you. Despite the name, there's no real order to the place.\n\nSan Telmo\n\n  Top Sights\n\n1Plaza Dorrego C4\n\n  Sights\n\n2Centro Cultural Torquato Tasso D6\n\n3Convento de Santo Domingo C1\n\n4El Zanj\u00f3n de Granados C2\n\n5Facultad de Ingenieria D3\n\n6Iglesia Nuestra Se\u00f1ora de Bel\u00e9n D4\n\n7Museo Argentino del T\u00edtere B3\n\n8Museo de Arte Contempor\u00e1neo Buenos Aires D4\n\n9Museo de Arte Moderno de Buenos Aires (Mamba) D4\n\n10Museo del Traje B2\n\n11Museo Hist\u00f3rico Nacional D6\n\n12Museo Penitenciario D4\n\n13Parque Lezama D6\n\n14Pasaje de la Defensa C4\n\n  Eating\n\n15Abuela Pan C2\n\n16Bar El Federal C3\n\n17Caf\u00e9 San Juan C4\n\n18Caf\u00e9 San Juan La Cantina C2\n\n19Casal de Catalunya B3\n\n20Comedor Nikkai B3\n\n21El Desnivel C3\n\n22Gran Parrilla del Plata C2\n\n23La Poesia C2\n\n24Origen Caf\u00e9 C4\n\n  Drinking & Nightlife\n\n25Bar Brit\u00e1nico D5\n\n26Bar Plaza Dorrego D4\n\n27Bar Sedd\u00f3n C2\n\n28Boutique B2\n\n29Coffee Town C3\n\n30Doppelg\u00e4nger C5\n\n31Gibraltar C3\n\n32La Puerta Roja B2\n\n33Pride Cafe D3\n\n  Entertainment\n\n34Boutique C1\n\nCentro Cultural Torquato Tasso (see 2)\n\n35El Viejo Almac\u00e9n D2\n\n36Federaci\u00f3n Argentina de Pato C1\n\n37La Scala de San Telmo D3\n\n38La Trastienda D1\n\n39La Ventana D1\n\n40Tango Queer C2\n\n41Todo Mundo C4\n\n  Shopping\n\n42Cualquier Verdura C4\n\n43En Buen Orden C3\n\n44Feria de San Telmo C4\n\n45Gil Antiguedades C4\n\n46Imhotep D3\n\n47L'Ago D3\n\n48Materia Urbana D2\n\n49Mercado de San Telmo C3\n\n50Moebius D5\n\n51Punto Sur C4\n\n52Puntos en el Espacio C3\n\n53Signos C3\n\n54Vinotango C3\n\n55Walrus Books C3\n\n  Sports & Activities\n\n56Rayuela B3\n\n  Sleeping\n\n57America del Sur B3\n\n58Bohemia Buenos Aires C3\n\n59Bonito San Telmo C5\n\n60Brisas del Mar B4\n\n61Casa y Mundo Bolivar C6\n\n62Circus Hostel & Hotel B4\n\n63Hostel Viejo Telmo A2\n\n64Lugar Gay D4\n\n65Mansi\u00f3n Vitraux D3\n\n66San Telmo Colonial B3\n\n67Scala Hotel A3\n\n68Terranova Hostel C4\n\n# La Boca\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nLa Boca\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighbourhood Top Five\n\n Strolling the cobblestones of  El Caminito, lined with colorful shanties, art vendors and buskers performing for your spare change.\n\n Seeing the modern exhibits at  Fundaci\u00f3n Proa, the neighborhood's cutting-edge art museum.\n\n Watching tango dancers strut their stuff while shopping nearby at the  Feria de Artesanos Caminito.\n\n Analyzing the powerful artwork at  Museo de Bellas Artes de La Boca Benito Quinquela Mart\u00edn.\n\n Going to a _f\u00fatbol_ game at  La Bombonera Stadium, home to the scrappy Boca Juniors team.\n\n###  Explore: La Boca\n\nOn your way into La Boca, note the Casa Amarilla, in the 400s block along the main drag Av Almirante Brown. This is a replica of the country house belonging to Almirante Brown, the Irish founder of the Argentine navy. Three blocks further on (look to your left at the kink in the road), you'll notice the curious Gothic structure called Torre Fantasma (Ghost Tower). As you reach the Riachuelo, you can alight from your bus or taxi and walk the last few hundred meters. Get a good look at the Puente Nicol\u00e1s Avellaneda, which spans the Riachuelo, linking La Boca to the industrial suburb of Avellaneda; before the bridge's completion in 1940, floods had washed away several others. From here follow the riverside walkway all the way to El Caminito.\n\nBoca's main attractions \u2013 museums, shops, eateries, Bombonera stadium \u2013 are all within a few blocks of El Caminito. There's no reason to venture beyond the touristy streets in this neighborhood, whose bullies have a reputation for occasionally mugging careless tourists for their cameras. Be discreet, stick close to the busier streets and you should be fine. Buenos Aires' mayor, Mauricio Macri, is from La Boca and has been trying to improve the neighborhood by developing destinations such as the Usina del Arte; eventually he wants to gentrify the thoroughfare of Pedro de Mendoza, linking Puerto Madero with La Boca.\n\nThe symbol of the community's solidarity is the Boca Juniors soccer team, the former club of disgraced superstar Diego Maradona. The team plays at La Bombonera Stadium, which is just four blocks inland from the Riachuelo and contains a museum detailing the team's players and successes; you can take a peek at the stadium from this museum.\n\n###  Local Life\n\n\u00bb Stick to El Caminito La Boca's local life is pretty local, and should stay that way. Visitors should stay around the El Caminito area: muggings of obvious tourists (ie those carrying large cameras) have occurred.\n\n###  Getting There & Away\n\n\u00bb Bus Take buses 29, 64 and 152 from Palermo and the center; they all end up at El Caminito.\n\n####  Lonely Planet's Top Tip\n\nLa Boca is not the kind of neighborhood for casual strolls \u2013 it can be downright rough in spots. Don't stray far from the riverside walk, El Caminito (and its nearby tourist streets) or the Bombonera Stadium, especially while toting expensive cameras. And certainly don't cross the bridge over the Riachuelo; there's nothing to see there anyway. Anda Responsible Travel (Click here) and Graffitimundo (Click here) have good walking-tour options for this area.\n\n  Best Places to Eat & Drink\n\n\u00bbIl Matterello\n\n\u00bbEl Obrero\n\n\u00bbProa Cafe\n\nFor reviews, Click here \u00bb\n\n  Best Museums\n\n\u00bbFundaci\u00f3n Proa\n\n\u00bbMuseo de Bellas Artes de La Boca Benito Quinquela Mart\u00edn\n\nFor reviews, Click here \u00bb\n\nHIROSHI HIGUCHI \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nEL CAMINITO\n\nLa Boca's most famous street \u2013 Argentina's only 'open air' museum \u2013 is a magnet for visitors, who come to witness its brightly painted houses. Meanwhile, artists sell their colorful paintings while tango dancers ask you to pose with them or behind cardboard cut-outs and nearby weekend craft stalls offer handmade goods. Everywhere there are groups of tourists taking photos \u2013 can you say 'tourist trap'?\n\n###  DON'T MISS...\n\n\u00bb The tiled reproductions of Benito Quinquela Mart\u00edn's artwork on El Caminito's walls.\n\n\u00bb Exploring the inside of some _conventillos_ (tenements) \u2013 such as the one at Magallanes 861.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb Av Don Pedro de Mendoza, near Del Valle Iberlucea\n\nThis block-long cobbled walk does have its unique charms, though, like the various bas-reliefs and sculptures dotted about. And there are the super-colorful tenement shacks, covered in corrugated zinc and originally brushed with leftover paint that Genoese immigrants begged off ships. Surrounding streets also offer souvenirs, and restaurants with pleasant sidewalk tables.\n\nCaminito (or 'little path') was named after a 1926 tango song by composer Juan de Dios Filiberto and lyricist Gabino Coria Pe\u00f1aloza (hunt for the lyrics on a wall plaque), which tells of a love lost. This song inspired Benito Quinquela Mart\u00edn, La Boca's most famous artist, to help create Caminito as the neighborhood's main landmark in 1955.\n\nQuinquela Mart\u00edn mostly painted dark scenes of the barrio's port on the Riachuelo and its workers. His house and workshop have been turned into a museum and are worth a visit, but you can get an idea of his style on Caminito. Look for a small tiled reproduction of his _Dia de Trabajo_ (Day of Work) on a green wall and a much bigger one of his _Regreso de la Pesca_ (Return from Fishing) at the end of the street, both by Ricardo S\u00e1nchez.\n\nAt the end of Caminito is a small plaza with a mural depicting past struggles of the neighborhood, such as the dangerous task of volunteer firemen and the faces of La Boca's 'disappeared' during the military regime. There are also other themes, such as Aztec figures and local _f\u00fatbol_ heroes.\n\n La Boca\n\nSights | Eating | Entertainment | Shopping\n\n  SIGHTS\n\nEL CAMINITO STREET\n\nSee Click here.\n\nFUNDACI\u00d3N PROA MUSEUM\n\nMAP GOOGLE MAP\n\n(  4104-1001; www.proa.org; Av Don Pedro de Mendoza 1929; admission AR$15;   11am-7pm Tue-Sun) Only the most cutting-edge national and international artists are invited to show at this elegant art museum, which features high ceilings, white walls and large display halls. Stunning contemporary installations utilize a wide variety of media and themes, while the rooftop terrace is _the_ stylish place in La Boca for relaxing with a drink or snack \u2013 it boasts a view of the Riachuelo. Plenty of cultural offerings include talks, lectures, workshops, music concerts and cinema screenings.\n\nProa also boasts video rooms, an auditorium and an impressive library.\n\nMUSEO DE BELLAS ARTES DE LA BOCA BENITO QUINQUELA MART\u00cdN MUSEUM\n\nMAP GOOGLE MAP\n\n(  4301-1080; Av Don Pedro de Mendoza 1835; suggested donation AR$10;   10am-6pm Tue-Fri, 11am-6pm Sat & Sun) Once the home and studio of Benito Quinquela Mart\u00edn (1890\u20131977), this fine-arts museum exhibits his works and those of more contemporary Argentine artists. The top floor displays Mart\u00edn's surrealist paintings, whose broad, rough brush-strokes and dark colors use the port, silhouettes of laboring men, smokestacks and water reflections as recurring themes. There are outdoor sculptures on the rooftop terraces, and the top tier has awesome views of the port.\n\nIn keeping with the museum's maritime theme there is also a small but excellent permanent collection of painted wooden bowsprits, which are the carved statues projecting forward at the front of ships.\n\nMUSEO DE LA PASI\u00d3N BOQUENSE MUSEUM\n\nMAP GOOGLE MAP\n\n(  4362-1100; www.museoboquense.com; Brandsen 805; admission from AR$60;   10am-6pm) High-tech and spiffy, this _f\u00fatbol_ (soccer) museum chronicles the rough-and-tumble neighborhood of La Boca, La Bombonera Stadium (Click here), soccer idols' histories, the champion\u00adships, the trophies and, of course, the gooooals. There's a 360\u00b0 theater in a giant soccer-ball auditorium, an old jersey collection and a gift shop. The museum is right under the stadium, a couple of blocks from the tourist part of El Caminito; get a tour of the pitch for a few extra pesos.\n\nMUSEO HIST\u00d3RICO DE CERA MUSEUM\n\nMAP GOOGLE MAP\n\n(  4301-1497; www.museodecera.com.ar; Del Valle Iberlucea 1261; admission AR$20;   11:30am-7pm Mon-Fri, 11am-8pm Sat & Sun;   29, 64, 152) Wax reconstructions of historical figureheads (literally) and dioramas of scenes in Argentine history are the specialty of this small and tacky private institution. Among the historical Argentine personages depicted are no less than Juan de Sol\u00eds, Guillermo Brown, Mendoza, Garay and Rosas. In addition, there are also stuffed snakes and creepy wax limbs depicting bite wounds \u2013 all barely worth the price of admission.\n\n###  MARADONA & MESSI\n\nBorn in 1960 in abject poverty in a Buenos Aires shantytown, Diego Armando Maradona played his first professional game before his 16th birthday. Transferring to his beloved Boca Juniors, he continued to prosper. After a good showing at the 1982 World Cup, he moved to Europe. Here, his genius inspired unfashionable Napoli to two league titles, and in 1986 he single-handedly won the World Cup for a very average Argentina side. In the quarter-final against England, he scored a goal first with his hand \u2013 later saying the goal was scored partly by the hand of God \u2013 and then a second one with his feet, after a mesmerizing run through the flummoxed defense that led to its being named the Goal of the Century by FIFA.\n\nBut the big time also ruined Diego. Earning huge sums of money, Maradona became addicted to cocaine and the high life. A succession of drug-related bans, lawsuits and weight issues meant that by his retirement in 1997 he had been a shadow of his former self for some years.\n\nSince his retirement, overdoses, heart attacks, detoxes, his own TV program and offbeat friendships have all been par for the course in the Maradona circus. Most unbelievably of all, he was chosen to manage the national team: the highlight in a colorful spell \u2013 after qualifying for the 2010 World Cup in South Africa \u2013 was his triumphant suggestion that his critics could pleasure him orally. Nevertheless, those numerous touches of magic in the number 10 shirt have sealed his immortality. To many Argentines, the hand of God and the hand of Maradona are one and the same.\n\nEvery talented Argentine since has been dogged with the label 'the new Maradona', but these days there's one who's the real deal. Rosario-bred Lionel Messi, a little genius who runs at defenses with the ball seemingly glued to his feet, has been captivating the world with his prodigious talents and record-breaking goal-scoring feats for Barcelona and, increasingly, for the national team. Many shrewd judges consider him better even than the great Diego, and his humble off-field demeanor is certainly an improvement. If he manages to inspire the _albiceleste_ (Argentina's national football team) to win the World Cup again, it will truly be the Second Coming.\n\n_Andy Symington_\n\n La Boca\n\nSights | Eating | Entertainment | Shopping\n\n  EATING\n\nThough limited to just a handful of streets, the tourist area of La Boca does contain a number of traditional Argentine eateries \u2013 mostly offering classic steaks and pastas. As long as you don't expect fine cuisine, you shouldn't be disappointed.\n\n PROA CAFE CAFE $\n\nMAP GOOGLE MAP\n\n(  4104-1003; www.proa.org\/eng\/cafe.php; Av Don Pedro de Mendoza 1929; mains AR$45-80;   11am-7pm Tue-Sun) Chef Lucas Angelillo presides over this airy eatery on the top floor of Fundaci\u00f3n Proa. Stop in briefly for a fresh juice and gourmet sandwich, or stay longer and order a meat, seafood or pasta dish. Don't miss the rooftop terrace on a warm, sunny day \u2013 you'll get good views of the Riachuelo, hopefully without its corresponding scents.\n\nIL MATTERELLO ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4307-0529; Mart\u00edn Rodr\u00edguez 517; mains AR$80-120;   noon-midnight Tue-Sat) This Genovese trattoria serves up awesome lasagne bolognese and _tagliatelle alla rucola_ (tagliatelle with arugula). For a special treat, however, try the _tortelli bianchi con burro foso al aglio_ (pasta pillows stuffed with chard and Parmesan in a burned garlic sauce). For dessert there's a great tiramisu and seasonal _crostate_ (cream-filled pastry). Also in Palermo (  4831-8493; cnr Thames & Gorriti).\n\nEL OBRERO PARRILLA $$\n\nMAP GOOGLE MAP\n\n(  4362-9912; Agust\u00edn R Caffarena 64; mains AR$60-100;   noon-4pm & 8pm-midnight Mon-Sat) The same family has been running El Obrero since 1954, and a number of famous people have passed through over the years, including Bono and Robert Duvall (check out the photos on the walls). You'll also see old Boca Juniors jerseys, antique furniture, old tile floors and chalkboards showing the day's specials and standard _parrilla_ fare. Take a taxi.\n\n###  LA BOCA'S COLORS & THE RIACHUELO\n\nIn the mid-19th century, La Boca became home to poor Spanish and Italian immigrants who settled along the Riachuelo \u2013 the sinuous river that divides the city from the surrounding province of Buenos Aires. Many of them came during the booming 1880s and ended up working in the numerous meat-packing plants and warehouses here, processing and shipping out much of Argentina's vital beef. After sprucing up the barges, the port dwellers splashed leftover paint on the corrugated-metal siding of their houses, unwittingly giving the neighborhood what would become one of its main claims to fame.\n\nHowever, La Boca's other leftover industrial materials have also eventually found their way into the river. Decades of untreated sewage, garbage dumping and industrial wastes have taken their toll, and today the abandoned port's waters are trapped under a thick layer of incredibly smelly rainbow sludge. For years politicians have vowed to clean up the river, but funds have been misappropriated by the corrupt and very little money has actually gone into improving the river's situation. Rusting boat hulks have been removed and other efforts have taken place, but the Riachuelo's health does not look to be improving in the near future. If you can stand the smell, take a walk along the riverfront path to get a close-up look at the poor Riachuelo \u2013 and hope that some day, someone in power will actually do something to clean it up.\n\n La Boca\n\nSights | Eating | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nUSINA DEL ARTE THEATER\n\n(<http:\/\/agendacultural.buenosaires.gob.ar>; Agust\u00edn Caffarena 1) This restored old electricity factory is a valiant attempt to breathe new life into an edgy section of La Boca. It's a gorgeous red-brick building complete with scenic clock tower, and its concert hall is now the home to Buenos Aires' philharmonic and national symphony orchestras.\n\nThe Usina can hold up to 1200 spectators and also hosts dance, theater and art exhibitions \u2013 the acoustics are top-notch. It's open only during concerts and guided tours; check the city's website for current happenings.\n\nLA BOMBONERA STADIUM STADIUM\n\nMAP GOOGLE MAP\n\n(Brandsen) The La Bombonera Stadium is home of the Boca Juniors football team \u2013 the former club of disgraced superstar Diego Armando Maradona. You can take a peek at it via the Museo de la Pasi\u00f3n Boquense (Click here). Game tickets are hard to come by \u2013 it's best to go via travel or tour agencies like Tangol (Click here).\n\nTEATRO DE LA RIBERA THEATER\n\nMAP GOOGLE MAP\n\n(  4302-1536; Av Don Pedro de Mendoza 1821) This small, colorful theater, funded by famous Argentine painter Benito Quinquela Mart\u00edn, was built in 1971 and holds nearly 650 seats. Check out the upright piano in the lobby; it was painted by Quinquela Mart\u00edn.\n\n La Boca\n\nSights | Eating | Entertainment | Shopping\n\n  SHOPPING\n\nFERIA DE ARTESANOS CAMINITO STREET MARKET\n\nMAP GOOGLE MAP\n\n(cnr Caminito & Mendoza;   noon-6pm Thu-Sun & holidays) Homemade crafts and tango-themed goods are for sale at this small and lively crafts fair, giving La Boca even more color than usual. Tango dancers and buskers compete for your attention, and along Caminito itself are many drawings, paintings and pictures to buy.\n\nLa Boca\n\n  Top Sights\n\n1El Caminito B5\n\n  Sights\n\n2Fundaci\u00f3n Proa B5\n\n3Museo de Bellas Artes de La Boca Benito Quinquela Mart\u00edn C5\n\n4Museo de la Pasi\u00f3n Boquense B4\n\n5Museo Hist\u00f3rico de Cera B5\n\n  Eating\n\n6El Obrero C3\n\n7Il Matterello B3\n\nProa Cafe (see 2)\n\n  Entertainment\n\n8Estadio Luis Conde B3\n\n9La Bombonera Stadium B4\n\nTeatro de la Ribera (see 3)\n\n  Shopping\n\n10Feria de Artesanos Caminito C5\n\n# Retiro\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nRetiro\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Wandering around  Plaza San Mart\u00edn and getting an eyeful of the impressive surrounding buildings.\n\n Touring the opulent mansion  Palacio Paz, once Argentina's largest private residence at 12,000 sq meters.\n\n Splurging on dinner at one of Retiro's five-star hotel restaurants like Elena and Le Sud.\n\n Checking out the amazing silverwork at  Museo de Arte Hispanoamericano Isaac Fern\u00e1ndez Blanco.\n\n Being astounded by the quantity of lethal weapons at  Museo de Armas.\n\n###  Explore: Retiro\n\nRetiro is a small, compact neighborhood most easily seen on foot. Wander around the Plaza San Mart\u00edn area, perhaps touring a mansion or two \u2013 just double-check the visiting times as they're very limited. The museums around here also tend to be open just in the afternoon and closed on Mondays, so plan ahead if you want to visit them. Join the crowds on pedestrian Calle Florida and follow it down into the Center; Reconquista is another pedestrian street that isn't quite as crowded and better for taking a lunch or coffee break, as there are many restaurants with sidewalk tables there.\n\nFor any tips on Buenos Aires or Argentina, stop by the Secretar\u00eda de Turismo de la Naci\u00f3n, at Av Santa Fe 883 \u2013 it's located at the end of a hallway entrance and has good information and plenty of pamphlets. If you're heading north, walk on Esmeralda or Suipacha to Arroyo. The Museo de Arte Hispanoamericano Isaac Fern\u00e1ndez Blanco is not far from here and has a leafy garden if you need a green break. Continue west, crossing Av 9 de Julio into Recoleta; you're now on upscale Av Alvear. As you follow this route you'll see plenty of gorgeous art deco buildings built by European immigrants decades ago.\n\n###  Local Life\n\n\u00bb Hanging Out Sip a coffee or down a drink at one of the neighborhood's many cafes or bars catering to thirsty businesspeople.\n\n\u00bb Picnic On a sunny day, grab a to-go lunch and head to Plaza San Mart\u00edn's grassy lawns to join the locals out for some fresh air.\n\n\u00bb Shopping Feel the bustle (and hustle!) of porte\u00f1o crowds on Calle Florida, which starts near Plaza San Mart\u00edn and heads south.\n\n###  Getting There & Away\n\n\u00bb Bus Take bus 59 from Recoleta and Palermo, buses 22, 45 and 126 from San Telmo, bus 150 from Congreso.\n\n\u00bb Subte L\u00ednea C connects Retiro with the Congreso and western edge of San Telmo.\n\n####  Lonely Planet's Top Tip\n\nCalle Florida heaves with people of all kinds doing all sorts of things \u2013 businesspeople power- walking, tourists shopping, vendors selling, buskers busking. It's also dotted every few meters with _arbolitos_ \u2013 'little trees,' or street money changers. With the US dollar being in such high demand, these shady figures are trying to bring in as much revenue as possible. While some have used them successfully, be aware that fake bills and scams do exist.\n\n  Best Places to Eat\n\n\u00bbElena\n\n\u00bbLe Sud\n\n\u00bbDill & Drinks\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbFlorer\u00eda Atl\u00e1ntico\n\n\u00bbMili\u00f3n\n\n\u00bbFlorida Garden\n\nFor reviews, Click here \u00bb\n\n  Best Buildings\n\n\u00bbPalacio Paz\n\n\u00bbEdificio Kavanagh\n\n\u00bbPalacio San Mart\u00edn\n\nFor reviews, Click here \u00bb\n\nSTEFANO PATERNA PHOTOGRAPHY \/ ALAMY \u00a9\n\n##### TOP SIGHT\n\nPALACIO PAZ\n\nOnce a private residence, this opulent, French-style palace (1909; also called Palacio Retiro) is the grandest in BA. Inside its 12,000 sq meters are three wings, four floors and 140 rooms decorated with marble columns and gilded accents, while halls boast beautiful wood-carved details and velvet-covered walls. It's worth visiting to get an idea of the richness that Argentina once represented.\n\n###  DON'T MISS...\n\n\u00bb The circular Hall of Honor, decorated with mosaic floors, marble details and a stained-glass cupola.\n\n\u00bb The Presidential Room, where hermaphroditic figures look uneasy as they are stabbed in their genitals.\n\n###  PRACTICALITIES\n\n\u00bb C\u00edrculo Militar\n\n\u00bb MAP\n\n\u00bb   4311-1071 ext 147\n\n\u00bb www.palaciopaz.com.ar\n\n\u00bb Av Santa Fe 750\n\n\u00bb tours in English\/Spanish AR$55\/45\n\n\u00bb   English tours 3:30pm Wed & Thu, Spanish tours 11am & 3pm Wed-Fri, 11am Sat\n\nJos\u00e9 Camilo Paz, founder of the still-running news\u00adpaper _La Prensa,_ originally commissioned French architect Louis-Marie Henri Sortais to design and build his personal mansion in 1902. Construction took 12 years and finished in 1914; unfortunately, Paz couldn't see his completed masterpiece as he had passed away two years earlier (check out his family's elaborate tomb in Recoleta cemetery). He also couldn't realize his aspiration to become Argentina's president and make Palacio Paz his presidential residence.\n\nNearly all of the palace's materials \u2013 including the marble \u2013 were shipped from France. There's a definite resemblance to the Palace of Versailles, especially in the ballroom, but other rooms show more of a Louis XVI, Renaissance or Tudor style. With seven elevators and 40 bathrooms, it remains Argentina's largest single-family home ever built.\n\nOne section of the palace has been a military officer's club since 1938, and the Museo de Armas (Click here) takes up another wing. The palace can only be visited via guided tours on certain days, so plan ahead.\n\n Retiro\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SIGHTS\n\nPALACIO PAZ NOTABLE BUILDING\n\nSee Click here.\n\nPLAZA SAN MART\u00cdN PLAZA\n\nMAP GOOGLE MAP\n\nFrench landscape architect Carlos Thays designed the leafy Plaza San Mart\u00edn, which is surrounded by some of Buenos Aires' most impressive public buildings. The park's most prominent monument is the obligatory equestrian statue of Jos\u00e9 de San Mart\u00edn; important visiting dignitaries often come to honor the country's liberator by leaving wreaths at its base. On the downhill side of the park you'll see the Monumento a los Ca\u00eddos de Malvinas , a memorial to the young men who died in the Falklands War.\n\nRetiro was the site of a monastery during the 17th century, and later became the country _retiro_ (retreat) of Agust\u00edn de Robles, a Spanish governor. Since then, Plaza San Mart\u00edn \u2013 which sits on a bluff \u2013 has played host to a slave market, a military fort and even a bullring. Things are much quieter and more exclusive these days.\n\nAt the south end of the plaza is Estaci\u00f3n Retiro (Retiro train station), which was built in 1915 when the British controlled the country's railroads.\n\nMUSEO DE ARMAS MUSEUM\n\nMAP GOOGLE MAP\n\n(Weapons Museum;   4311-1071 ext 179; www.museodearmas.com.ar; Santa Fe 702; admission AR$10;   1-7pm Tue-Fri, 2-7pm Sat) Even if you've spent time in the armed forces, you probably have never seen so many weapons of destruction. This maze-like museum exhibits a frighteningly large but excellent collection of over 3000 bazookas, grenade launchers, cannons, machine guns, muskets, pistols, armor, lances and swords; even the gas mask for a combat horse is on display. The evolution of rifles and handguns is especially thoroughly documented, and there's a small but impressive Japanese weapons room.\n\nThe whole collection is very extensive, impressive, clean and well labeled.\n\nPALACIO HAEDO NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Av Santa Fe 690) On an odd triangular block at the corner of Florida and Santa Fe, the neo-Gothic Palacio Haedo was the mansion of the Haedo family at the turn of the 19th century; it now houses the country's national park service.\n\nPALACIO SAN MART\u00cdN NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(  4819-7000 ext 8092; Arenales 761;   L\u00ednea C San Mart\u00edn) This impressive art nouveau mansion (1912) is actually three independent buildings around a stone courtyard. It was designed by architect Alejandro Christophersen and boasts marble staircases, grandiose dining rooms and a garden containing a chunk of the Berlin Wall. A small but good museum displays pre-Columbian artifacts from the northwest, along with some paintings by Latin American artists. Free tours happen at 3pm on Thursdays (bring ID), but can be suspended at any time. Enter via Esmeralda 1231.\n\nOriginally built for the powerful Anchorena family, Palacio San Mart\u00edn later became the headquarters of the Foreign Ministry; today it's used mostly for official purposes.\n\nEDIFICIO KAVANAGH NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Florida 1035) A feisty Irishwoman funded the construction of this handsome 120m art-deco apartment building, which was the tallest skyscraper in Latin America at the time of its construction in 1935. A local rumor claims that the heiress, vengeful towards another aristocratic family for scorning her daughter, built the structure that high to block light from entering the basilica where her rivals attended Mass every Sunday.\n\nBAS\u00cdLICA DE SANT\u00cdSIMO SACRAMENTO CHURCH\n\nMAP GOOGLE MAP\n\n(Plaza San Mart\u00edn 1039) In the shadow of the Kavanagh building is this French-style church built by the Anchorena family in 1916. Inside, check out the original tiled floor, stained-glass windows, stone columns and wedding-cake-like altar.\n\nTORRE DE LOS INGLESES LANDMARK\n\nMAP GOOGLE MAP\n\n(Torre Monumental;   4311-0186; Plaza Fuerza A\u00e9rea Argentina;   10am-6pm Mon-Fri, 10am-6:30pm Sat & Sun;   L\u00ednea C Retiro) F Standing prominently across from Plaza San Mart\u00edn, this 76m-high miniature version of London's Big Ben was a donation from the city's British community in 1916. During the Falklands War of 1982 the tower was the target of bombs, and the government officially renamed it Torre Monumental \u2013 but the name never really stuck. You can enter inside the base of the tower, where there are a few historical photos, but folks aren't allowed up the elevator.\n\nThe plaza in which it stands used to be called Plaza Brit\u00e1nica, but is now the Plaza Fuerza A\u00e9rea Argentina (Argentine Air Force Plaza).\n\nMUSEO DE ARTE HISPANOAMERICANO ISAAC FERN\u00c1NDEZ BLANCO MUSEUM\n\nMAP GOOGLE MAP\n\n(Palacio Noel;   4327-0228; www.museofernandezblanco.buenosaires.gob.ar; Suipacha 1422; admission AR$5;   2-7pm Tue-Fri, 11am-7pm Sat & Sun) Dating from 1921, this museum is in an old mansion of the neocolonial Peruvian style that developed as a reaction against French influences in turn-of-the-19th-century Argentine architecture. Its exceptional collection of colonial art includes silverwork from Alto Per\u00fa (present-day Bolivia), religious paintings and baroque instruments. There's little effort to place items in any historical context, but everything is in great condition and well lit, and the curved ceiling in the main salon is beautifully painted. There's also a peaceful garden.\n\nAlso known as the Palacio Noel, after the designing architect, the museum building and its collections suffered damage (since repaired) from the 1992 bombing of the Israeli embassy, which at the time was located at Arroyo and Suipacha. The space where the embassy was located has since become a small memorial park; you can still see the outline of the building on a neighboring wall.\n\nThe museum has an annex in the Congreso neighborhood at Hip\u00f3lito Yirogoyen 1420 (open noon to 6pm Tuesday to Friday and 11am to 5pm Saturday and Sunday) whose main strength is an antique doll collection.\n\nTEATRO NACIONAL CERVANTES NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(  4815-8883; www.teatrocervantes.gov.ar; Libertad 815) Six blocks southwest of Plaza San Mart\u00edn, you can't help but notice the lavishly ornamented Cervantes theater. From the grand tiled lobby to the main theater, with its plush red-velvet chairs, you can smell the long history of this place (somewhat musty). The Cervantes is definitely showing its age, with worn carpeting and rough edges, but improvement projects are planned. Until then, enjoy the elegance \u2013 however faded \u2013 with a tour (call for current schedules). It presents theater, comedy, musicals and dance at affordable prices.\n\nThe landmark building dates from 1921 and was built with private funds, but was acquired by the state in 1926. Its facade was designed as a replica of Spain's Universidad de Alcal\u00e1 de Henares. The building underwent remodeling after a fire in 1961.\n\nMUSEO NACIONAL DEL TEATRO MUSEUM\n\nMAP GOOGLE MAP\n\n(  4815-8883 ext 156; cnr Av C\u00f3rdoba & Libertad;   10am-6pm Mon-Fri) F This small museum traces Argentine theater from its colonial beginnings, stressing the 19th-century contributions of the Podest\u00e1 family \u2013 Italian immigrants who popularized the _gauchesca_ (gaucho literature) drama _Juan Moreira_. Items include a gaucho suit worn by Gardel in his Hollywood film _El d\u00eda que me quieras_ and the _bandone\u00f3n_ belonging to Paquita Bernardo, the first Argentine woman to play the accordionlike instrument (she died of tuberculosis in 1925 at the age of 25).\n\nThere's also a photo gallery of famous Argentine stage actors.\n\n###  AVENIDA 9 DE JULIO\n\nIt's one Buenos Aires landmark that all visitors to the city will have to cross, in one way or another \u2013 Avenida 9 de Julio, hailed as the world's widest avenue and named after Argentina's independence day. It's only one kilometer long but 16 lanes wide (140 m) \u2013 and takes a walking pedestrian at least two traffic-light cycles to cross, via raised islands. _If_ they don't dillydally.\n\nWhen the widening construction started in 1935, the avenue was considered a patriotic symbol of the city's modern aspirations. Designers modelled it on Paris' Champs-\u00c9lys\u00e9es, but made it twice as wide as a way to one-up its predecessor. For the construction, dozens of blocks of traditionally-styled European buildings had to be demolished through the city's center, and thousands of residents displaced. It was an epic destruction of glorious architecture \u2013 all in the name of progress. But one significant building refused to be touched; the original French Embassy. It still stands today, as the lanes of 9 de Julio forcefully curve around it.\n\nIt took until 1980 to fully complete the widening of Avenida 9 de Julio. Today, several landmark buildings and monuments dot the thoroughfare. At its southern end lies Plaza de la Constituci\u00f3n, home to a Beaux-arts train station (but not a safe place to hang out, day or night). At Av de Mayo is a statue of Don Quixote astride his horse. A bit further north, the 67m-high white Obelisco punctuates the sky, while nearby is the beautiful neoclassical facade of the Teatro Col\u00f3n opera house. And finally, at the northern end of the avenue, you'll find the French Embassy \u2013 which stuck to its guns and won the right to remain.\n\n Retiro\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  EATING\n\nRestaurants in Retiro tend to cater to the business crowds, offering good-value midday specials and food to go \u2013 so all you have to do is find yourself a nice, grassy spot or shady bench in nearby Plaza San Mart\u00edn where you can enjoy your impromptu picnic. Don't ignore the area's bars and cafes, which also serve meals and are sometimes more casual and interesting than traditional restaurants, and of course also offer a wider range of drinking options.\n\nEL CUARTITO PIZZA $\n\n(  4816-1758; Talcahuano 937; slices AR$10-14, pizzas AR$75-150;   12:30pm-1am Sun & Tue-Thu, to 2am Fri & Sat) In a hurry? Think fast, order and pay for your piece of pie, then eat at the counters standing up. Not only is it cheaper and faster this way, but you can enjoy the old sports posters without turning around. You can't get more local or traditional, and while it's mostly full of businessmen and male waiters, the gals are equally welcome. Sit down for more menu choices.\n\nDILL & DRINKS INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4515-0675; www.dillanddrinks.com; San Marti\u00edn 986; mains AR$70-140;   noon-6pm Mon, to 2am Tue-Fri, 2pm-2am Sat) Here's an intimate bar-restaurant with contemporary and trendy design. Order the daily lunch special, which includes a principal plate and two cocktails (!), the latter made with fresh juices and fruits. Bring your own Negroni recipe; it'll be filed away in their collection. Dishes are made with quality ingredients and can be things such as shrimp risotto or pork medallions in honey mustard.\n\nGRAN BAR DANZ\u00d3N INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4811-1108; Libertad 1161; mains AR$90-150;   7pm-2am Mon, to 2:30am Tue, to 3am Wed, to 3:30am Thu, to 4am Fri, 8pm-4am Sat, 8am-2am Sun) It's hard to be hipper than this popular lounge-bar-restaurant. A cool-looking wine-conservation system makes it possible to order several wines by the glass, easily paired with the duck confit, grilled salmon or mushroom risotto. The food is fine, if not fabulous, but make no mistake \u2013 you're here for the scene, the beautiful people and, of course, the Bonarda and Torront\u00e9s.\n\nFILO ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4311-0312; www.filo-ristorante.com; San Mart\u00edn 975; mains AR$75-110;   noon-1am) Popular with the business lunch crowd, this large, pop-art-style Italian pizzeria tosses great thin-crust pies with fresh toppings \u2013 try a pie piled high with prosciutto and arugula. Other tasty choices include _panini,_ gourmet salads, various pastas and a whirlwind of desserts. The menu is extensive \u2013 there's something to please just about everyone here.\n\nDAD\u00c1 INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4814-4787; San Mart\u00edn 941; mains AR$75-130;   noon-2am Mon-Thu, to 5am Fri & Sat) The tiny bohemian Dad\u00e1, with walls painted red and a bar cluttered with wine bottles, feels like an unassuming neighborhood bar in Paris. Order something savory off the bistro menu \u2013 the fresh guacamole and homemade potato chips are perfect for sharing. At night you can dine on grilled salmon and down an expertly mixed cocktail.\n\n ELENA MODERN ARGENTINE $$$\n\nMAP GOOGLE MAP\n\n(  4321-1728; www.elenaponyline.com; Four Seasons Hotel, Posadas 1086; mains AR$200-250;   6:30-11am, 12:30-3:30pm & 7:30pm-1am) If you're looking for a splurge night out, Elena should be your destination. Located in the Four Seasons Hotel, this highly-rated restaurant uses the best ingredients and cooking methods to create superb dishes. Order the dry-aged rib-eye steak or seared prawns with charred baby fennel for something really special. Expect the cocktails, desserts and service to be five-star as well.\n\n  LE SUD FRENCH $$$\n\nMAP GOOGLE MAP\n\n(  4131-0131; Hotel Sofitel, Arroyo 841; mains AR$150-235;   6:30-11am, 12:30-3pm & 7:30pm-midnight) For a taste of Europe, dress up and head on over to Le Sud, one of the city's finest French restaurants, elegantly ensconced in a posh hotel. Chef Olivier Falchi whips out simple yet authentic French fusion dishes like brie and squash ravioli, seafood paella with smoked paprika and grilled lamb chops with goat cheese. The five-course tasting menu is AR$780.\n\nSIPAN PERUVIAN $$$\n\nMAP GOOGLE MAP\n\n(  4315-0763; www.sipanrestaurants.com; Paraguay 624; mains AR$100-200;   noon-4pm & 8pm-1am Mon-Fri, 8pm-1am Sat) Japanese-\u00adPeruvian food is all the rage in BA, and Sipan is at the head of it. Tucked away in a shopping gallery, this sleek, low-lit restaurant turns out imaginative sushi and seviche, along with fancier dishes like seafood appetizers heaped high on ceramic spoons, and stir-fried tamarind pork. Try the passion-fruit pisco sour \u2013 it's delicious. Also in Palermo at the Hotel Palermitano.\n\nEL FEDERAL ARGENTINE $$$\n\nMAP GOOGLE MAP\n\n(  4313-1324; www.elfederalrestaurante.com; San Martin 1015; mains AR$110-170;   10am-midnight Mon-Sat) This traditional corner eatery is something of a neighborhood institution. You'll find Argentinian comfort food \u2013 simple pastas, steaks and _empanadas_ \u2013 as well as higher-end specialties like Patagonian lamb, _\u00f1andu milanesas_ (cutlets of the emu-like, flightless _\u00f1andu_ ) and northern river fish. Elaborate desserts top things off, and a rustic wooden bar adds charm. The lunchtime _menu ejecutivo_ is only AR$70 to AR$90.\n\n Retiro\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  DRINKING & NIGHTLIFE\n\nRetiro has a good range of bars and cafes that cater to businesspeople during the day and into the evening, and at night attract the traveler-expat crowd.\n\n  FLORER\u00cdA ATL\u00c1NTICO COCKTAIL BAR\n\nMAP GOOGLE MAP\n\n(  4313-6093; Arroyo 872) One of BA's hottest bars, this basement speakeasy is located within a flower shop, adding an air of mystery and likely a main reason for its success. Hipsters, artists, chefs, businesspeople and expats all flock here for the excellent cocktails, whether they're classic or unique \u2013 and the lack of gas lines means all of the delicious tapas and main dishes are cooked on the _parrilla_ grill.\n\nIf you're a gin lover, note that the owner, Renato Giovannonni, produces and sells his own brand \u2013 called 'Pr\u00edncipe de los Ap\u00f3stoles' \u2013 aromatically infused with mint, grapefruit, eucalyptus and _yerba mate_. Reserve ahead for dinner.\n\nMILI\u00d3N COCKTAIL BAR\n\n(  4815-9925; www.milion.com.ar; Paran\u00e1 1048;   6pm-2am Sun-Wed, to 3am Thu, to 4am Fri & Sat) This elegant and sexy bar takes up three floors of a renovated old mansion. The garden out back is a leafy paradise, overlooked by a solid balcony that holds the best seats in the house. Nearby marble steps are also an appealing place to lounge with a frozen mojito or basil daiquiri, the tastiest cocktails on the menu. Downstairs, the restaurant serves passable inter\u00adnational dishes.\n\nFLORIDA GARDEN CAF\u00c9\n\nMAP GOOGLE MAP\n\n(  4312-7902; Florida 899;   6am-midnight Mon-Fri, to 11pm Sat, 8am-11pm Sun) Usually full of businesspeople drinking up a storm of coffee, this two-story cafe \u2013 now sporting modern touches such as glass walls and copper-covered columns \u2013 was historically popular with politicians, artists and writers. In fact, Jorge Luis Borges and P\u00e9rez C\u00e9lis (a famous Argentine painter) used to hang out here before the era of skinny lattes. The people-watching here is excellent.\n\nCAF\u00c9 RETIRO CAF\u00c9\n\nMAP GOOGLE MAP\n\n(  4516-0902; Retiro Station Lobby, Ramos Meija 1358;   6:30am-9pm Mon-Fri, 8am-4pm Sat) Catching a train out of town? Allow an extra half hour for coffee at this grand cafe, which boasts soaring ceilings, polished wood and a bronze interior. One of the original fixtures of the station, built in 1915, the cafe has undergone a thorough restoration \u2013 the chandeliers twinkle beautifully at night. Look for it nearly across from the Torre de los Ingleses.\n\nDRUID IN IRISH PUB\n\nMAP GOOGLE MAP\n\n(  4312-3688; Reconquista 1040;   noon-midnight Mon-Fri, to 3am Sat) This cozy Irish pub sports an intimate and uncrowded atmosphere. A wide range of aged whiskeys, imported liquor, blended cocktails and a handful of beers temper the pizza, sandwiches and British food that is served. Live rock, jazz, Celtic or Irish music groups may occasionally play.\n\nKILKENNY IRISH PUB\n\nMAP GOOGLE MAP\n\n(  4312-7291; Marcelo T de Alvear 399;   noon-4am Mon-Thu, to 5am Fri & Sat, 7pm-1am;  ) Buenos Aires' most popular Irish bar has become, well, just too damn popular. Weekends are a crush and thumping music makes it hard to chat up your date, but the dark-woodsy atmosphere is congenial enough. Come early on weekdays if you want to score one of the cozy deep booths for easy conversation. There's a good whiskey and beer selection, too.\n\nFLUX BAR\n\nMAP GOOGLE MAP\n\n(  5252-0258; Marcelo T de Alvear 980;   7pm-3am Sun-Thu, to 4am Fri & Sat) Run by a friendly Englishman and his Russian partner, this gay bar is hetero-friendly \u2013 so everyone's welcome to come on down. The large basement space has a slightly artsy feel. Feeling adventurous? Try the Buenos Aires iced tea (made with Fernet, that popular Argentine mixer that's something of an acquired taste). Happy hour runs every day from 7pm till 10pm.\n\n Retiro\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nTEATRO COLISEO CLASSICAL MUSIC\n\nMAP GOOGLE MAP\n\n(  4816-3789; www.fundacioncoliseo.com.ar; Marcelo T de Alvear 1125) Classical music, jazz, ballet, opera and symphony orchestras entertain at this theater most of the year, but a few surprises \u2013 such as Argentine-American rock star Kevin Johansen \u2013 occasionally show up.\n\n Retiro\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SHOPPING\n\nThe Retiro neighborhood, is classy, expensive and home to a fair share of the city's upscale leather shops and art galleries. But it also serves the downtown business and tourism sector, with a mix of bookstores, outdoor clothing stores, and souvenir and wine shops.\n\nAUTOR\u00cdA ART, ACCESSORIES\n\nMAP GOOGLE MAP\n\n(  5252-2474; www.autoriabsas.com.ar; Suipacha 1025;   9:30am-8pm Mon-Fri, 10am-6pm Sat) This cool designer's showcase \u2013 stocked with edgy art books, sculptural fashions, whimsical leather desk sculptures and unique jewelry of all materials (silk cocoons!) \u2013 strives to promote Argentine designers. Especially interesting are the recycled materials \u2013 check out the bags made of tyvek, inner tubes, firehoses or even old sails. Products are of high quality and prices are accessible.\n\nMEMORABILIA CRAFTS & TEXTILES\n\nMAP GOOGLE MAP\n\n(  4811-7698; www.memorabiliabazar.com; Arenales 1170;   11am-7:30pm Mon-Fri, to 1:30pm Sat) For unique, fun and handmade Argentine items, explore the corners of this tiny boutique. The stock is ever-changing, but can include such things as ceramic bowls and mugs painted in animal faces, silver jewelry in super-creative shapes and a few knit tops for women. Expect cute small items easily packed into a suitcase for the flight back home.\n\nPATIO BULLRICH SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(  4814-7500; www.shoppingbullrich.com.ar; Av del Libertador 750;   10am-9pm) Buenos Aires' most exclusive shopping center once hosted livestock auctions, but these days it tends toward sales of Persian rugs, double-breasted tweed suits and Dior's latest designs. Three floors hold fine boutiques such as Lacoste, Salvatore Ferragamo and Tiffany & Co, along with fancy coffee shops, a cinema complex and a food court.\n\nCASA L\u00d3PEZ LEATHER GOODS\n\nMAP GOOGLE MAP\n\n(  4311-3044; www.casalopez.com.ar; Marcelo T de Alvear 640\/658;   9am-8pm Mon-Fri, 10am-7pm Sat & Sun) Start up the limousine and make sure there's enough room for some of BA's finest selection of quality leather jackets, luggage, bags and accessories. The look is conservative, not hip; service is almost too attentive, so be prepared to chat. Other branches are located in Galer\u00edas Pac\u00edfico and Patio Bullrich.\n\nLA MARTINA SPORTING EQUIPMENT\n\nMAP GOOGLE MAP\n\n(  4576-7999; www.lamartina.com; Paraguay 661;   10am-8pm Mon-Fri, 10am-2pm Sat) Polo is a high-class sport in Buenos Aires, an unmistakable symbol of wealth and refinement \u2013 but even if you've never mounted a horse, it's interesting to look around at the gorgeous leather riding boots, helmets and saddles at Argentina's premier polo shop. Street wear clothes are also available if you don't ride.\n\nGABRIELLA CAPUCCI FASHION\n\n(  4815-3636; Av Alvear 1477;   10:30am-8pm Mon-Sat) While certainly not for everyone, this girly boutique is undeniably unlike other stuffy ones on this upscale avenue. Come check out the unique sequined T-shirts, creative handbags, wispy scarves, vintage tops and eclectic rhinestone accessories. Expect also a load of crocheted flowers, huge beads, satin and animal prints, wild costume jewelry and a general over-saturation of bright colors. Real men \u2013 wait for your gals outside.\n\nGALER\u00cdA 5TA AVENIDA SHOPPING MALL\n\n(Av Santa Fe 1270;   noon-8pm Mon-Sat) Looking for vintage or secondhand clothing? This old shopping gallery is an obligatory stop. Used funky wearables are sold here at several shops, and prices are relatively fair for even the grungiest backpacker. Find the bargain racks for the best deals, though some selections are for the desperate only. Individual store hours vary widely, with many opening after 4pm.\n\nRetiro\n\n  Top Sights\n\n1Palacio Paz C5\n\n  Sights\n\n2Bas\u00edlica de Sant\u00edsimo Sacramento D5\n\n3Edificio Kavanagh D5\n\n4Monumento a los Caidos de Malvinas D4\n\n5Museo de Armas C5\n\n6Museo de Arte Hispanoamericano Isaac Fern\u00e1ndez Blanco B2\n\n7Museo Nacional del Teatro A6\n\n8Palacio Haedo C5\n\n9Palacio San Mart\u00edn C4\n\n10Plaza San Mart\u00edn D4\n\n11Teatro Nacional Cervantes A6\n\n12Torre de los Ingleses D3\n\n  Eating\n\n13Dad\u00e1 D5\n\n14Dill & Drinks D5\n\n15El Federal D5\n\nElena (see 37)\n\n16Filo D5\n\n17Gran Bar Danz\u00f3n A4\n\n18Le Sud C3\n\n19Sipan D5\n\n  Drinking & Nightlife\n\n20Buller Brewing Company E5\n\n21Caf\u00e9 Retiro D3\n\n22Druid In E4\n\n23Florer\u00eda Atl\u00e1ntico B3\n\n24Florida Garden D5\n\n25Flux B5\n\n26Kilkenny E5\n\n  Entertainment\n\n27British Arts Centre B3\n\n28Teatro Coliseo A5\n\nTeatro Nacional Cervantes (see 11)\n\n  Shopping\n\n29Autor\u00eda B5\n\n30Arte y Esperanza B6\n\n31Casa L\u00f3pez D5\n\n32La Martina D5\n\n33Memorabilia A4\n\n34Patio Bullrich A1\n\n  Sports & Activities\n\n35Tangol D5\n\n  Sleeping\n\n36Casa Calma B5\n\n37Four Seasons A2\n\n38Hotel Pulitzer C5\n\n39Hotel Tres Sargentos E5\n\n# Recoleta & Barrio Norte\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Map\nRecoleta & Barrio Norte\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Wandering the elegant sarcophagi in  Cementerio de la Recoleta, an astonishing necropolis where, in death as in life, generations of the Argentine elite rest in ornate splendor.\n\n Relaxing on the sunny patio of one of the many  restaurants or cafes on RM Ortiz.\n\n Finding that perfect hand-made souvenir in one of the dozens of stalls at  Feria Plaza Francia.\n\n Seeking out the beautiful European impressionistic artwork at  Museo Nacional de Bellas Artes.\n\n Getting a close-up look at  Floralis Gen\u00e9rica, an interesting giant metal flower sculpture.\n\n###  Explore: Recoleta & Barrio Norte\n\nRecoleta's main attractions are concentrated around the cemetery. This fashionable barrio was, interestingly enough, first constructed as a result of sickness. Many upper-class _porte\u00f1os_ in the 1870s originally lived in southerly San Telmo, but during the yellow-fever epidemic they relocated as far away as they could, which meant clear across town to Recoleta and Barrio Norte. Today you can best see much of the wealth of this sumptuous quarter on Av Alvear, where many of the old mansions (and newer boutiques) are located.\n\nBehind the cemetery is the impressive Museo Nacional de Bellas Artes \u2013 this national art museum is free, but note that it's closed in the mornings and on Mondays. And just beyond the museum is the landmark flower sculpture _Floralis Gen\u00e9rica_. From here you can walk to Palermo's fancy Malba museum and green parks in about 15 to 20 minutes.\n\nBarrio Norte is not an official neighborhood as such but rather a largely residential southern extension of Recoleta. Some people consider it a sub-neighborhood of Recoleta (and parts of it are sometimes lumped in with Retiro or Palermo, too) \u2013 it all really depends on who you talk to. However, Barrio Norte does have a more accessible feel than its ritzier sibling, especially around busy Av Santa Fe. Here you'll find hundreds of shops, all vying for shoppers' attention, and all conveniently located on bus and Subte lines \u2013 not things you'll see much of in Recoleta itself.\n\n###  Local Life\n\n\u00bb Hanging Out La Biela (Click here) is in a touristy location, but locals still flock here to sit at the front patio on warm sunny days.\n\n\u00bb Shopping Grab your wallet and head to Av Santa Fe, where hundreds of stores cater to _porte\u00f1os'_ every whim (Click here).\n\n\u00bb Ice Cream Recoleta has more than its fair share of excellent ice cream shops, so find one and order up.\n\n###  Getting There & Away\n\n\u00bb Bus Buses 59 heads from Palermo to San Telmo, stopping along Av Las Heras along the way.\n\n\u00bb Subte L\u00ednea D covers the southern section of Recoleta.\n\n####  Lonely Planet's Top Tip\n\nRecoleta is an expensive neighborhood. For a cheap lunch, get takeout somewhere \u2013such as El Sanjuanino (Click here) \u2013 and find a nice park bench, such as in Plaza Intendente Alveear, where you can hang out, eat and watch the world go by.\n\n  Best Places to Eat\n\n\u00bbL'Orangerie\n\n\u00bbComo en Casa\n\n\u00bbRodi Bar\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbLa Biela\n\n\u00bbBuller Brewing Company\n\n\u00bbCasa Bar\n\nFor reviews, Click here \u00bb\n\n  Best Places to Shop\n\n\u00bbFeria Plaza Francia\n\n\u00bbAv Santa Fe\n\n\u00bbGaler\u00eda Bond Street\n\nFor reviews, Click here \u00bb\n\nRELIGIOUS IMAGES\/UIG \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nCEMENTERIO DE LA RECOLETA\n\nRecoleta cemetery is arguably Buenos Aires' number-one attraction, and a must on every tourist's list. You can wander for hours in this amazing city of the dead, where countless 'streets' are lined with impressive statues and marble sarcophagi. Peek into the crypts and check out the dusty coffins, and try to decipher the history of its inhabitants.\n\n###  DON'T MISS...\n\n\u00bb Evita's grave Go to the first major intersection, turn left at the statue; continue until a mausoleum blocks your way. Go around it and turn right at the wide 'street'. After three blocks it's to your left.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb   0800-444-2363\n\n\u00bb cnr Jun\u00edn & Guido\n\n\u00bb   7am-5:30pm\n\nOriginally the garden of the church next door, Recoleta cemetery was created in 1822. It covers four city blocks and contains about 4800 mausoleums decorated in many architectural styles, including art nouveau, art deco, classical, Greek, baroque and neo-Gothic. Popular motifs include crosses of all kinds, marble angels, stone wreaths, skulls and crossbones, draped urns, winged hourglasses and the occasional gargoyle. All decorate the final resting places of past presidents, military heroes, influential politicians, famous writers and other very noteworthy personages, including those of the Paz family (Jos\u00e9 C Paz founded _La Prensa_ ), Liliana Crociati de Szaszak (famous for dying in an avalanche in Austria), Rufina Cambaceres (the urban myth goes that she was buried alive) and boxer Luis Angel Firpo.\n\nThe most impressive tomb is not Evita's, which is rather plain. Instead, get a good map and look for other sarcophagi; interesting stories, odd facts and myths abound. Also note the cemetery's rough edges \u2013 the cobwebs and detritus inside many of the tombs, the vegetation growing out of cracks, the feral cats prowling the premises. All add to the charm.\n\nFree tours are offered in English at 11am Tuesday and Thursday and in Spanish at 9:30am, 11am, 2pm and 4pm from Tuesday to Sunday (weather permitting). For a great map and information, order Robert Wright's PDF guide (www.recoletacemetery.com); touts also sell maps at the entrance.\n\n Recoleta & Barrio Norte\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SIGHTS\n\nCEMENTERIO DE LA RECOLETA CEMETERY\n\nSee Click here.\n\nBAS\u00cdLICA DE NUESTRA SE\u00d1ORA DEL PILAR CHURCH\n\nMAP GOOGLE MAP\n\n(  4806-2209; Jun\u00edn 1904; museum AR$6;   10, 17, 60, 92, 110) Yes, that's a pair of skulls on your right as you enter the basilica. But the centerpiece of this gleaming white colonial church, built by Jesuits in 1716, is a Peruvian altar adorned with silver from Argentina's northwest. Inside, head to the left to visit the small but historic cloisters museum (  10:30am-6:15pm Mon-Sat, 2:30-6:15pm Sun); it's home to religious vestments, paintings, writings and interesting artifacts. You can also snap a photo of the Recoleta cemetery through the window grilles.\n\nCENTRO CULTURAL RECOLETA CULTURAL CENTER\n\nMAP GOOGLE MAP\n\n(  4803-1040; www.centroculturalrecoleta.org; Jun\u00edn 1930;   59) Part of the original Franciscan convent and alongside its namesake church and cemetery, this renovated cultural center houses a variety of facilities, including art galleries, exhibition halls and a cinema. Events, courses and workshops are also offered, and its Museo Participativo de Ciencias MAP GOOGLE MAP (  4806-3456; www.mpc.org.ar; Jun\u00edn 1930; admission AR$40;   vary widely, see website) is a children's hands-on science museum.\n\nMUSEO NACIONAL DE BELLAS ARTES MUSEUM\n\nMAP GOOGLE MAP\n\n(  5288-9945; www.mnba.org.ar; Av del Libertador 1473;   12:30-8:30pm Tue-Fri, 9:30am-8:30pm Sat & Sun) F This is Argentina's most important national arts museum and contains many key works by Benito Quinquela Mart\u00edn, Xul Solar, Edwardo S\u00edvori and other Argentine artists of the 19th and 20th centuries. There are also impressive international works by European masters such as C\u00e9zanne, Degas, Picasso, Rembrandt, Toulouse-Lautrec and Van Gogh. Everything is well displayed, and there's also a cinema, concerts and classes. The museum's peaceful interior is a welcome respite from the busy avenue outside. Call in advance for tours in English.\n\nThe museum's building is a former pump house for the city waterworks, and was designed by architect Julio Dormala. It was later modified by Alejandro Bustillo, famous for his alpine-style civic center in the northern Patagonian city of Bariloche.\n\nFLORALIS GEN\u00c9RICA MONUMENT\n\nMAP GOOGLE MAP\n\n(cnr Av Figueroa Alcorta & Bibiloni) This gargantuan solar-powered flower sculpture, located smack in the center of Plaza Naciones Unidas is the inspired creation of architect Eduardo Catalano, who designed and funded the project in 2002. The giant aluminum and steel petals are 20m high and used to close like a real flower, from dusk until dawn \u2013 until the gears broke, that is.\n\nPALAIS DE GLACE NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(  4804-1163; www.palaisdeglace.gov.ar; Posadas 1725;   noon-8pm Tue-Fri, 10am-8pm Sat & Sun;   17, 62, 67) F Housed in an unusual circular building that was once an ice-skating rink and a tango hall (happily not at once, however!), the spacious Palais de Glace now offers a variety of rotating cultural, artistic and historical exhibitions. Be sure to check out the 2nd floor, worth a peep for its interesting ceiling and other architectural details. Musical concerts are also occasionally hosted here.\n\nFACULTAD DE INGENIER\u00cdA NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(Engineering School; cnr General Las Heras & Azcu\u00e9naga) This beautiful but decrepit neo-Gothic building (1912) was designed by Uruguayan architect Arturo Prins and never quite completed. It's currently being given a face-lift.\n\n###  GLORIOUS DEATH IN BUENOS AIRES\n\nOnly in Buenos Aires can the wealthy and powerful elite keep their status after death. When decades of dining on rich food and drink have taken their toll, Buenos Aires' finest move ceremoniously across the street to the Cementerio de la Recoleta (Click here), joining their ancestors in a place they have religiously visited all their lives.\n\nArgentines are a strange bunch who tend to celebrate their most honored national figures not on the date of their birth, but on the date of their death (after all, they're nobody when they're born). Nowhere is this obsession with mortality more evident than at Recoleta, where generations of the elite repose in the grandeur of ostentatious mausoleums. Real estate here is among Buenos Aires' priciest: there's a saying that goes, 'It is cheaper to live extravagantly all your life than to be buried in Recoleta.'\n\nIt's not just being rich that gets you a prime resting spot here: your name matters. Those lucky few with surnames like Alvear, Anchorena, Mitre or Sarmiento are pretty much guaranteed to be laid down. Evita's remains are here (in the Familia Duarte sarcophagus), but her lack of aristocracy and the fact that she dedicated her life not to BA's rich but rather to its poor infuriated the bigwigs.\n\nA larger and much less touristy graveyard is Cementerio de la Chacarita, located in the neighborhood of Chacarita. The cemetery opened in the 1870s to accommodate the yellow-fever victims of San Telmo and La Boca. Although much more democratic and modest, Chacarita's most elaborate tombs match Recoleta's finest. One of the most visited belongs to Carlos Gardel, the famous tango singer. Plaques from around the world cover the base of his life-size statue, many thanking him for favors granted. Like Evita, Juan Per\u00f3n and others, Gardel is a quasi saint toward whom countless Argentines feel an almost religious devotion. The anniversaries of Gardel's birth and death days see thousands of pilgrims jamming the cemetery's streets.\n\nAnother spiritual personality in Chacarita is Madre Mar\u00eda Salom\u00e9, a disciple of the famous healer Pancho Sierra. Every day, but especially on the second day of each month (she died on October 2, 1928), adherents of her cult cover her tomb with white carnations. To visit Chacarita, take L\u00ednea B of the Subte to the end of the line at Federico Lacroze and cross the street.\n\n Recoleta & Barrio Norte\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  EATING\n\nRecoleta is the playground for the wealthy elite, full of beautiful apartment buildings, upscale boutiques and the occasional baroque mansion. As you can imagine, the restaurants here aren't cheap, but if you want to rub shoulders with the upper classes, this is the place to be.\n\nPractically everyone visits Recoleta's cemetery, so the two-block strip of touristy restaurants, bars and cafes lining nearby RM Ortiz is very convenient. Food here tends toward the overpriced, but many restaurants have outdoor terraces that are choice hangout spots on warm days. And the people-watching here is excellent, especially on weekends when the nearby craft market is in full swing.\n\nThere are three ice-cream shops on Av Quintana near the cemetery: Persicco (cnr Av Quintana & RM Ortiz), Una Altra Volta MAP GOOGLE MAP (  4805-1818; www.unaltravolta.com.ar; cnr Av Quintana & Ayachuco) \u2013 both within one or two blocks of the cemetery \u2013 and Arkakao (Av Quintana 188), which is five blocks from the cemetery.\n\nCOMO EN CASA ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4816-5507; www.tortascomoencasa.com; Riobamba 1239; lunch mains AR$45-70;   8am-midnight Tue-Sat, 8am-8:30pm Sun & Mon) This gorgeous, upscale cafe-restaurant has a very elegant atmosphere and attracts Recoleta's wealthiest. Its best feature is the shady patio, complete with large fountain and surrounded by grand buildings, a must on a warm day. For lunch there are fancy sandwiches, salads, wraps and stir-fries, while dinner options include goulash, shrimp ragout and spinach gnocchi. Plenty of luscious desserts, plus breakfast too.\n\nRODI BAR ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4801-5230; Vicente L\u00f3pez 1900; mains AR$50-90;   7am-1am) A great option for well-priced, unpretentious food in upscale Recoleta. This traditional corner restaurant with fine old-world atmosphere and extensive menu offers something for everyone, from inexpensive combo plates to relatively unusual dishes such as marinated beef tongue.\n\nNATURAL DELI CAFE, DELI $\n\nMAP GOOGLE MAP\n\n(  4822-1228; www.natural-deli.com; Laprida 1672; mains AR$50-72;   8am-midnight Mon-Sat, 9am-midnight Sun;  ) Modern, organic cafe offering fresh dishes with a natural bent. Choose from creative gourmet sandwiches and wraps, fresh salads or stir fries. There are also fresh juices and _licuados_ (blended fruit smoothies), plus many organic gourmet products are sold. Great for breakfast; muffins, scones, brownies and even key lime pie available. Also in Las Ca\u00f1itas, Palermo (  4514-1776; Gorostiaga 1776;   8am-midnight Mon-Sat, 9am-midnight Sun).\n\nTEA CONNECTION CAFE $\n\nMAP GOOGLE MAP\n\n(  4805-0616; www.teaconnection.com.ar; Uriburu 1595; mains AR$60-75;   8am-midnight Mon-Sat, 9am-midnight Sun) At this sleek corner cafe, choose from over 20 types of black, red and green teas and health-\u00adconscious sandwiches, salads, vegetable tarts and pastries. Drinks include fruit juices and _licuados_. Other nearby locations include at Arenales 2102 and Montevideo 1655.\n\nCUMAN\u00c1 ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4813-9207; Rodriguez Pe\u00f1a 1149; mains AR$40-50;   noon-4pm & 8pm-1am) To sample Argentina's regional cuisine, check out this colorful, budget-friendly eatery with huge picture windows and an old-fashioned adobe oven. Cuman\u00e1 specializes in delicious _cazuela,_ stick-to-your-ribs stews filled with squash, corn, eggplant, potatoes and meat. Also popular are the _empanadas,_ _locro_ and _humita_ (corn, cheese and onion tamales). Come early to avoid a wait.\n\nEL SANJUANINO ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4805-2683; Posadas 1515; empanadas AR$13, mains AR$40-70;   noon-4pm & 7pm-1am) This long-running, cozy little joint probably has the cheapest food in Recoleta, attracting both penny-pinching locals and thrifty tourists. Sit either upstairs or downstairs (in the basement) and order spicy _empanadas,_ tamales or _locro_ (corn and meat stew). The curved brick ceiling adds to the atmosphere, but many take their food to go \u2013 Recoleta's lovely parks are just a couple of blocks away.\n\nMUNICH RECOLETA ARGENTINE $$\n\nMAP GOOGLE MAP\n\n(  4804-3981; www.munich-recoleta.com.ar; RM Ortiz 1871; mains AR$70-125;   noon-3pm & 8pm-midnight Wed-Mon) This traditional restaurant hasn't changed much since Jorge Luis Borges was a regular; try the _brochettes_ (shish-kebabs), grilled salmon or homemade ravioli. Service is exceptional and the white window curtains make this a semiprivate affair \u2013 perhaps a reason why more locals than tourists eat here. Warning to animal lovers: there are trophy animal heads on the walls.\n\nL'ORANGERIE FRENCH $$$\n\nMAP GOOGLE MAP\n\n(  4808-2100; Alvear Palace Hotel, Av Alvear 1891; full tea AR$190;   breakfast 7-11am daily, lunch buffet noon-3.30pm Mon-Sat, afternoon tea 4:30-7pm Mon-Sat & 5-7pm Sun, brunch 12:30-4pm Sun) The grand tearoom at the Alvear Palace Hotel is fit for a special occasion. The formal afternoon tea, served from 4:30pm (from 5pm on Sunday), offers an endless array of exquisite cakes, sandwiches and pastries (two people can share one tea service). At breakfast and lunch, chefs in tall white hats attend lavish buffet spreads; Sunday brunch is particularly elaborate.\n\nEL BURLADERO SPANISH $$$\n\nMAP GOOGLE MAP\n\n(  4806-9247; www.elburladero.com.ar; Uriburu 1488; mains AR$120-175;   noon-4pm & 8pm-midnight) Treat yourself to an upscale meal in Recoleta at this exceptional Spanish restaurant. The menu changes seasonally, but will usually include a paella dish and perhaps the marinated rabbit, black hake fish or lamb with mushrooms. For a good deal, come at midday and get the three-course lunch menu (AR$109). Tapas are also available, and there's a long, high communal table for large groups.\n\n###  NIP & TUCK\n\nWander around Recoleta, the ritzy neighborhood that is home to many of Buenos Aires' plastic surgery clinics, and nobody bats an eyelid at someone walking down the street with plasters on their faces \u2013 they've obviously just had a nip and tuck.\n\nThe president herself \u2013 Cristina Kirchner \u2013 is sometimes dubbed 'the queen of botox' in the local media. It comes as no surprise, then, to learn that Argentina has the highest per capita ratio of plastic surgery operations in the world. Reports suggest that one in 30 Argentines have had some sort of procedure during their lifetime.\n\nDemand for plastic surgery has risen exponentially in the last decade, especilally with the boom in 'medical tourism.' Rather than paying up to US$15,000 for a facelift in the United States, some have elected to head to the 'Paris of the South' and combine the surgery with a bit of tango, beef and sightseeing \u2013 for a third of the price.\n\nBe aware, however, of the dangers of any medical procedure \u2013 _let alone_ one on foreign turf where you may not speak the language. If you're considering such a procedure, do your homework very, very carefully and thoroughly.\n\n Recoleta & Barrio Norte\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  DRINKING & NIGHTLIFE\n\nRM Ortiz, across from Recoleta's famous cemetery, is a two-block strip of restaurants, cafes and bars. On warm sunny days most of them open up their fine outdoor front patios, perfect for a drink or meal and some people-watching.\n\nLA BIELA CAFE\n\nMAP GOOGLE MAP\n\n(  4804-0449; www.labiela.com; Av Quintana 600;   7am-2am Sun-Thu, to 3am Fri & Sat) A Recoleta institution, this classic landmark has been serving the porte\u00f1o elite since the 1950s \u2013 when race-car champions used to frequent the place. The outdoor front terrace is unbeatable on a sunny afternoon, especially when the nearby weekend _feria_ (street market) is in full swing. Just know that this privilege will cost 20% more.\n\nBULLER BREWING COMPANY BAR\n\nMAP GOOGLE MAP\n\n(  4808-9061; www.bullerpub.com; RM Ortiz 1827;   noon-1am Mon-Wed, to 2am Thu & Sun, to 4am Fri & Sat) Yes, it's a microbrewery in Buenos Aires, and in Recoleta, no less. Six kinds of beer are brewed on the premises, including a stout, hefeweisen, pilsen and a honey beer. Alcohol content ranges from 4.5% to 8.5%. There's a great outdoor patio in front and an extensive menu of snacks and sandwiches. Also in Retiro (Paraguay 428).\n\nCASA BAR SPORTS BAR\n\nMAP GOOGLE MAP\n\n(  4816-2712; www.casabarbuenosaires.com; Rodr\u00edguez Pe\u00f1a 1150;   7pm-3am Wed-Fri, to 5am Sat) This recycled antique house turned sports bar offers a large selection of spirits and microbrews, along with a wine list stocked with higher-end bottles. You'll also find nachos, pizza and spicy hot wings on the menu, plus happy-hour specials from 7pm to 10pm. Casa Bar is stylish but casual \u2013 and a great spot to watch sports on TV, especially American football and baseball.\n\nCL\u00c1SICA Y MODERNA CAFE\n\nMAP GOOGLE MAP\n\n(  4812-8707; www.clasicaymoderna.com; Av Callao 892;   8am-2am Mon-Sat, 5pm-2am Sun) Catering to the literary masses since 1938, this cozy and intimate bookstore- restaurant-cafe continues to ooze history from its atmospheric brick walls. It's nicely lit, serves fine, simple meals and offers nightly live performances of folk music, jazz, bossa nova and tango. Mercedes Sosa (may she rest in peace), Susana Rinaldi and Liza Minnelli have all chirped here.\n\nSHAMROCK BAR\n\nMAP GOOGLE MAP\n\n(  4812-3584; Rodr\u00edguez Pe\u00f1a 1220;   6pm-4am Mon-Wed, to 6am Thu & Fri, 8pm-6am Sat) Popular with both locals and tourists for its cheap happy hour, this long-running and not very traditional Irish pub in Barrio Norte is decked out in dark wood and has a dim, moody atmosphere. Women beware: this is a serious pick up joint. DJs rule from Thursday to Saturday, when the Basement Club opens up downstairs, usually around midnight.\n\nBASEMENT CLUB CLUB\n\nMAP GOOGLE MAP\n\n(  4812-3584; Rodriguez Pe\u00f1a 1220;   Thu-Sat) This cool but unpretentious subterranean club is known for first-rate DJ lineups, pounding house music and a diverse young crowd. Thanks to the Shamrock, the ever-popular Irish pub upstairs, the place sees plenty of traffic throughout the night. Come at 3am to see the club in full swing, or just descend the stairs after enjoying a few pints at ground level.\n\n####   Neighborhood Walk  \n **Death, Art & Shopping**\n\n**Start** Cementerio de la Recoleta\n\n**End** La Biela cafe\n\n**Length** 2-3km; three hours\n\nStart with a bang and visit BA's top tourist destination:  Cementerio de la Recoleta (Click here). You can easily spend hours in here examining the hundreds of elaborate sarcophagi. Turn right out of the cemetery and swing past the upscale  Recoleta Mall.\n\nHead to the neo-Gothic  Facultad de Ingenier\u00eda (Click here), designed by Uruguayan architect Arturo Prins and never quite completed. It was being given a face-lift at research time so hopefully will look spiffy when you get there.\n\nNeed to shop for your home? Then stop in at the large  Buenos Aires Design mall, worth a look for the most cutting-edge furniture and lifestyle products. Now cut across Plaza Francia and head to the excellent  Museo Nacional de Bellas Artes (Click here), containing classical art from all over the world. Everything is well displayed and lit and, best of all, it's free.\n\nCross Av Figueroa Alcorta to reach the giant metal flower sculpture,  _Floralis Gen\u00e9rica_ (Click here); it's cool to see your reflection in the petals. Head back down Alcorta \u2013 passing the mammoth  Facultad de Derecho (School of Law) building along the way. Cross the footbridge and make your way up Plaza Intendente Alvear. If it's a weekend, browse through the craft stalls at  Feria Plaza Francia (Click here).\n\nStop by the  Centro Cultural Recoleta (Click here) to explore the galleries. If you have small kids, the Museo Participativo de Ciencias will grab their attention. Right next to the cultural center is the pretty  Bas\u00edlica de Nuestra Se\u00f1ora del Pilar (Click here); check out the small museum upstairs.\n\nAmble down to restaurant-filled RM Ortiz and end your walk at the fine cafe  La Biela (Click here). If it's sunny grab a table on the front patio \u2013 it's worth the extra pesos.\n\n Recoleta & Barrio Norte\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  ENTERTAINMENT\n\nNOTORIOUS JAZZ\n\nMAP GOOGLE MAP\n\n(  4813-6888; www.notorious.com.ar; Av Callao 966) This stylish, intimate joint is one of Buenos Aires' premier jazz venues. Up front you can buy CDs of various music genres, while in the back the restaurant- cafe (overlooking a verdant garden) hosts live shows nearly every night at 9:30pm. Log on to the website for schedules; most performances are jazz, but there's also Brazilian music.\n\n Recoleta & Barrio Norte\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping\n\n  SHOPPING\n\nExclusivity is the key word here. If you have the bucks and are willing to pay top dollar for the best quality goods, then you'll want to shop in these neighborhoods. The city's best leather shops are based here, along with a few top fashion boutiques. Av Santa Fe is a catch-all for fashion, housewares and everything in-between.\n\nFERIA PLAZA FRANCIA MARKET\n\nMAP GOOGLE MAP\n\n(www.feriaplazafrancia.com;   11am-8pm Sat, Sun & holidays) Located right in front of Recoleta's cemetery, in Plaza Intendente Alvear, this large and popular street fair features hundreds of booths selling leather accessories, bronze jewelry, fused glass, ceramic mugs, woven hats and kitschy souvenirs \u2013 essentially dozens of handmade crafts of all kinds. Hippies gather, bakers circulate their pastries and mimes perform (or just stand very still). The website gives a great overview of what's available.\n\nBUENOS AIRES DESIGN SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(  5777-6000; Av Pueyrred\u00f3n 2501;   10am-9pm Mon-Sat, noon-9pm Sun) The trendiest and finest home furnishings are all under one roof here. This is the ideal place to look for that snazzy light fixture, streamlined toilet or reproduction Asian chair. Also good for everyday appliances and housewares, along with cute decor and art objects.\n\nWUSSMANN SHOP STATIONERY\n\nMAP GOOGLE MAP\n\n(  4811-2444; Rodriguez Pe\u00f1a 1399;   10:30am-8pm Mon-Fri, 11am-2pm Sat) Writers and artists delight in the gorgeous handmade paper at this chic stationery shop. Leatherbound journals, monogrammed stationery, and oversized sketchbooks are made with recycled paper; come here for one-of-a-kind invitations and notecards or hand-painted wrapping paper to spruce up a special gift.\n\nPORTOBELLO VINTAGE BOUTIQUE CLOTHING\n\nMAP GOOGLE MAP\n\n(  4811-2619; Paraguay 1554;   noon-8pm Mon-Fri) Excellent vintage clothing boutique. Find that special jacket, dress, shirt and bottom from the 1940s on up. All clothes are in great condition and sold at affordable prices. Some jewelry and other accessories are also for sale.\n\nGALER\u00cdA BOND STREET SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(Av Santa Fe 1670;   10am-9pm Mon-Sat) For the edgiest tattoos and piercings in town, you can't beat this grungy shopping center. Buenos Aires' skateboarder-wannabes, along with their punk-rock counterparts, also come here to shop for the latest styles, sounds and bongs. Expect everything from Hello Kitty to heavy metal.\n\nRecoleta & Barrio Norte\n\n  Top Sights\n\n1Cementerio de la Recoleta D4\n\n  Sights\n\n2Bas\u00edlica de Nuestra Se\u00f1ora del Pilar D4\n\n3Centro Cultural Recoleta D4\n\n4Facultad de Ingenier\u00eda C4\n\n5Floralis Gen\u00e9rica C2\n\n6Museo Nacional de Bellas Artes D3\n\n7Museo Participativo de Ciencias D4\n\n8Palais de Glace E3\n\n  Eating\n\n9Como en Casa C6\n\n10Cuman\u00e1 D6\n\n11El Burladero C5\n\n12El Sanjuanino E4\n\n13Gran Bar Danz\u00f3n F6\n\nL'Orangerie (see 39)\n\n14Munich Recoleta D4\n\n15Natural Deli A5\n\n16Rodi Bar D5\n\n17Tea Connection E5\n\n18Tea Connection C6\n\n19Tea Connection C5\n\n20Una Altra Volta D4\n\n  Drinking & Nightlife\n\n21Buller Brewing Company D4\n\n22Casa Bar D6\n\n23Cl\u00e1sica y Moderna D8\n\n24La Biela D4\n\n25Mili\u00f3n E7\n\n26Shamrock D6\n\n  Entertainment\n\n27Basement Club D6\n\n28Notorious D7\n\n  Shopping\n\n29Buenos Aires Design D3\n\n30El Ateneo C7\n\n31Feria Plaza Francia D4\n\n32Gabriella Capucci F5\n\n33Galer\u00eda 5ta Avenida F7\n\n34Galer\u00eda Bond Street D7\n\n35Portobello Vintage Boutique E7\n\n36Wussmann Shop D6\n\n  Sports & Activities\n\n37Escuela Argentina de Tango F7\n\n38VOS E7\n\n  Sleeping\n\n39Alvear Palace Hotel E4\n\n40Art Suites B5\n\n41Ayres de Recoleta C4\n\n42Hotel Lion D'or C5\n\n43Palacio Duhau \u2013 Park Hyatt E4\n\n44Petit Recoleta Hostel B6\n\n45Poetry Building C6\n\n46Yira Yira Guesthouse E7\n\n# Palermo\n\n###   Explore\n\n###   Top Sights\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Shopping\n\n###   Sports & Activities\n\n###   Map\nPalermo\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighborhood Top Five\n\n Checking out the contemporary artwork at Museo de Arte Latino\u00adamericano de Buenos Aires, a modern glassy museum where you can commune with Diego Rivera and Frida Kahlo.\n\n Bicycling, jogging, roller\u00adblading or just walking along the paths at  Parque 3 de Febrero.\n\n Eating your way through the dozens of ethnic restaurants in  Palermo Viejo and Las Ca\u00f1itas.\n\n Visiting  Museo Evita to get the scoop on Argentina's most famous international woman.\n\n Shopping the  designer clothing boutiques and other fun shops in Palermo Viejo.\n\n###  Explore: Palermo\n\nMost of Palermo's museums are located near its green parks. They're spread out over this large neighborhood, so give yourself plenty of time to see them. Museo Evita, Museo Nacional de Arte Decorativo and Museo de Arte Latinoamericano de Buenos Aires (Malba) all have pleasant cafe-restaurants with outdoor areas, great for taking a relaxing lunch or snack break.\n\nPalermo Viejo, the city's most trendsetting neighborhood, is roughly bounded by Santa Fe, Scalabrini Ortiz, C\u00f3rdoba and Dorrego. It's further divided into Palermo Hollywood (north of the train tracks) and Palermo Soho (south of the tracks), both full of old buildings, leafy sidewalks and cobbled streets. These areas have BA's best cutting-edge restaurants, along with trendy bars and nightclubs; Plaza Serrano has dozens of sidewalk tables and heaves with crowds on weekends.\n\nBuenos Aires' most famous fashion designers have opened up dozens of boutiques here, and there are many fancy housewares stores and other themed shops. It's not the cheapest place to shop, but likely the most fun \u2013 wear comfortable shoes as you'll be walking a lot.\n\nAnother popular but much smaller neighborhood in Palermo is further north in Las Ca\u00f1itas; it occupies a wedge of blocks close to the polo grounds. It's mostly a residential area on the border with Belgrano and named after the fields of sugar cane that used to grow here. The only sweet things here now, however, are the luscious desserts at the dozens of restaurants on Av B\u00e1ez, the main business street. Just a few blocks long, it's densely packed with eateries, bars, cafes and even a club or two, and it positively buzzes at night.\n\n###  Local Life\n\n\u00bb Shopping Trendy locals dress up to be seen fashionably shopping Palermo's fancy boutiques.\n\n\u00bb Nightlife Young, hip _porte\u00f1os_ wait until after midnight to congregate at bars, and after 2am head out to the clubs.\n\n\u00bb Sunny Weekends Local families grab their strollers and head to the Jard\u00edn Zool\u00f3gico (Click here) for a day of animal-watching.\n\n###  Getting There & Away\n\n\u00bb Bus Take buses 29, 59, 64 and 152 from the Microcentro to Plaza Italia; bus 39 from Congreso to Palermo Viejo; bus 111 from the Microcentro to Palermo Viejo.\n\n\u00bb Subte L\u00ednea D is the fastest way to Palermo's Plaza Italia area.\n\nLONELY PLANET \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nMUSEO DE ARTE LATINOAMERICANO DE BUENOS AIRES (MALBA)\n\nSparkling inside its glass and cement walls is this airy modern-art museum, one of BA's finest. It contains some of the best works by classic and contemporary Argentine artists, such as Xul Solar and Antonio Berni, plus some pieces by other Latin American painters like Mexican duo Diego Rivera and Frida Kahlo.\n\n###  DON'T MISS...\n\n\u00bb Andy Warhol's photography, especially if it includes abstract artist Marta Minuj\u00edn (an Argentine conceptual artist).\n\n\u00bb One of Diego Rivera's early cubist works, _Retrato de Ram\u00f3n G\u00f3mez de la Serna_ (Portrait of Ram\u00f3n G\u00f3mez de la Serna, 1915).\n\n###  PRACTICALITIES\n\n\u00bb Malba\n\n\u00bb MAP\n\n\u00bb   4808-6500\n\n\u00bb www.malba.org.ar\n\n\u00bb Av Figueroa Alcorta 3415\n\n\u00bb admission AR$40, Wed AR$20\n\n\u00bb   noon-8pm Thu-Mon, to 9pm Wed\n\nChristened in 2001, the building is home to Eduardo F Costantini's private art collection. The millionaire and philanthropist gathered over 200 of Latin America's best artworks from the 20th century. Well-known Argentine painters represented here include Linoenea Spilimbergo, Jorge de la Vega, Emilio Pettoruti and Guillermo Kuitca. If they're on display, check out figurative artist Antonio Berni's _Nuevo Realismo_ (social realism) oeuvres. Many of these artists confront social issues in their works.\n\nAmong the collection is the work of Tarsila do Amaral, one of Brazil's most famous painters; look for her _Abaporu_ (1928), one of Brazil's most important paintings and for which Costantini paid US$1.4 million in 1995. Colombian Fernando Botero depicts human figures in cartoonish, overly plump sizes, as in his _Los Viudos_ (1968). And Frida Kahlo's charming _Self-Portrait with Monkey and Parrot_ (1942) cost Costantini nearly US$3.2 million.\n\nExcellent temporary exhibits are shown in several halls, and there are occasional kids' programs and a cinema that screens art-house films. A terrace, bookstore, gift shop and cafe-restaurant (expensive but great for people-watching) complete the picture. Call for tours in English.\n\nIMAGE SOURCE \/ GETTY IMAGES \u00a9\n\n##### TOP SIGHT\n\nPARQUE 3 DE FEBRERO\n\nAlso known as Bosques de Palermo, this sweeping green space abounds with small lakes and paddleboats, pretty gazebos, a monument to literary greats called El Jard\u00edn de los Poetas (the Garden of Poets) and the pleasant Rosedal MAP GOOGLE MAP(Rose Garden). On weekends it's filled with families picnicking, friends playing _f\u00fatbol,_ lovers smooching and strolling, and outdoor enthusiasts jogging and bicycling.\n\n###  DON'T MISS...\n\n\u00bb Walking, bicycling or roller-blading around the rose garden and nearby lakes on a warm sunny weekend, when the ring road is closed to vehicular traffic and outdoor rentals abound.\n\n###  PRACTICALITIES\n\n\u00bb MAP\n\n\u00bb cnr Avs del Libertador & de la Infanta Isabel\n\n\u00bb   10, 34, 130\n\nThe area around Parque 3 de Febrero was originally the private retreat of 19th-century dictator Juan Manuel de Rosas and became public parkland only after his fall from power \u2013 on February 3, 1852. Ironically for Rosas, the man who overthrew him \u2013 former ally Justo Jos\u00e9 de Urquiza \u2013 sits on his mount in a mammoth equestrian monument at the corner of Avs Sarmiento and Presidente Figueroa Alcorta.\n\nIn 1875, Parque 3 de Febrero was inaugurated by Argentina's president, Nicolas Avellaneda. It was designed by Charles Thays, a French botanist and landscape architect who also worked on Plaza de Mayo, Barracas de Belgrano and Parque Lezama. Thays used London's Hyde Park and Paris' Bois de Boulogne as inspiration for his work here.\n\nThe park's most interesting destinations include the Jard\u00edn Japon\u00e9s (Click here), the Jard\u00edn Zool\u00f3gico (Click here), the Jard\u00edn Bot\u00e1nico Carlos Thays MAP GOOGLE MAP (Botanical Gardens;   dawn-dusk) F and the Planetario Galileo Galilei MAP GOOGLE MAP (  4771-9393; www.planetario.gov.ar; cnr Avs Sarmiento & Belisario Rold\u00e1n). More exclusive are the Campo Argentino de Polo (Click here) and the Hip\u00f3dromo Argentino (the racetrack). Just south of the zoo, and a major landmark in Palermo, is Plaza Italia, a half-moon-shaped traffic island and important transport hub.\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  SIGHTS\n\nMUSEO DE ARTE LATINOAMERICANO DE BUENOS AIRES (MALBA) MUSEUM\n\nSee Click here.\n\nPARQUE 3 DE FEBRERO PARK\n\nSee Click here.\n\nMUSEO NACIONAL DE ARTE DECORATIVO MUSEUM\n\nMAP GOOGLE MAP\n\n(  4802-6606; www.mnad.org; Av del Libertador 1902; admission AR$15, Tue free;   2-7pm Tue-Sat Jan, 2-7pm Tue-Sun rest of year) This museum is housed in the stunning beaux arts mansion called Residencia Err\u00e1zuriz Alvear (1917), once the residence of Chilean aristocrat Mat\u00edas Err\u00e1zuriz and his wife, Josefina de Alvear. It now displays many of their very posh belongings, along with beautiful features such as Corinthian columns and a gorgeous marble staircase inspired by the Palace of Versailles. There's also an amazing hall which has a carved wooden ceiling, stained-glass panels and a huge stone fireplace.\n\nEverything from renaissance religious paintings and porcelain dishes to Italian sculptures and period furniture was owned by Err\u00e1zuriz, and some artwork by El Greco, Manet and Rodin can also be seen. There are guided tours in English Tuesday to Saturday at 2:30pm (AR$15 plus admission). There's also a lovely cafe outside, Croque Madame, which provides a relaxing break on a sunny day.\n\nJARD\u00cdN ZOOL\u00d3GICO ZOO\n\nMAP GOOGLE MAP\n\n(  4011-9900; www.zoobuenosaires.com.ar; cnr Avs Las Heras & Sarmiento; admission AR$75, under 12yr free;   10am-6pm Tue-Sun Oct-Mar, to 5pm Apr-Sep) Set on 18 hectares, Buenos Aires' Jard\u00edn Zool\u00f3gico is a decent zoo, offering over 350 species \u2013 many in 'natural' and good-sized animal enclosures. On sunny weekends it's packed with families enjoying the large green spaces and artificial lakes. Some of the buildings housing the animals are impressive; check out the elephant house. An aquarium, a monkey island, reptile house and large aviary are other highlights; a few special exhibits (like the sea lion show or carousel) cost extra.\n\nThe zoo is noted for having successfully bred condors and white tigers, and for having an educational farm with petting zoo for the kids. Waterfowl, Patagonian hares, nutria (semi-aquatic rodents) and feral cats roam wild.\n\nJARD\u00cdN JAPON\u00c9S GARDENS\n\nMAP GOOGLE MAP\n\n(  4804-4922; www.jardinjapones.org.ar; cnr Avs Casares & Berro; admission AR$24, under 12yr free;   10am-6pm) First opened in 1967 and then donated to the city of Buenos Aires in 1979 (on the centenary of the arrival of Argentina's first Japanese immigrants), Jard\u00edn Japon\u00e9s is one of the capital's best-kept gardens \u2013 and makes a wonderfully peaceful rest stop. Inside there's a Japanese restaurant along with lovely ponds filled with koi and spanned by pretty bridges. Japanese culture can be experienced through occasional exhibitions and workshops on ikebana, haiku, origami, _taiko_ (Japanese drumming) and other events.\n\nMUSEO DE ARTES PL\u00c1STICAS EDUARDO S\u00cdVORI MUSEUM\n\nMAP GOOGLE MAP\n\n(  4774-9452; www.museosivori.org.ar; Av de la Infanta Isabel 555; admission AR$5, Wed & Sat free;   noon-8pm Tue-Fri, 10am-8pm Sat & Sun;   10, 34) Named for an Italo-Argentine painter who studied in Europe, this modern museum of Argentine art has open spaces allowing frequent and diverse exhibitions. S\u00edvori's Parisian works reflect European themes, but later works returned to Argentine motifs, mainly associated with rural life on the Pampas. However most works on display are by other well-known Argentine artists, such as Benito Quinquela Mart\u00edn, Antonio Berni and Fernando Fader. There's a sculpture garden and slick cafe on the premises, and occasional theater, concerts, courses and workshops are offered.\n\nMUSEO EVITA MUSEUM\n\nMAP GOOGLE MAP\n\n(  4807-0306; www.museoevita.org; Lafinur 2988; local\/foreigner AR$10\/20;   11am-7pm Tue-Sun) Everybody who's anybody in Argentina has their own museum, and Eva Per\u00f3n (1919\u201352) is no exception. Museo Evita immortalizes the Argentine heroine with plenty of videos, historical photos, books, old posters and newspaper headlines. However, the prize memorabilia has to be her wardrobe: dresses, shoes, handbags, hats and blouses lie proudly behind glass, forever pressed and pristine. Even Evita's old wallets and perfumes are on display. Our favorite is a picture of her kicking a soccer ball \u2013 in heels.\n\nHead around to the corner if you need refreshment \u2013 attached to the museum is the pleasant Museo Evita Restaurant (Click here) with wonderfully leafy patio, perfect for relaxing on a warm day.\n\nMUSEO DE ARTE POPULAR JOS\u00c9 HERN\u00c1NDEZ MUSEUM\n\nMAP GOOGLE MAP\n\n(  4803-2384; www.museohernandez.buenosaires.gob.ar; Av del Libertador 2373; admission AR$5, Sun free;   1-6:30pm Wed-Fri, 10am-7pm Sat & Sun) This museum was being remodelled at research time, but the emphasis here is on both traditional and contemporary arts and crafts, mostly from Argentina. Expect to see intricate gaucho-related silverwork like knives and mate sets, Mapuche textiles like ponchos and folk crafts from the country's northern regions. The back halls hold changing exhibits.\n\nMUSEO XUL SOLAR MUSEUM\n\nMAP GOOGLE MAP\n\n(  4824-3302; www.xulsolar.org.ar; Laprida 1212; admission AR$20;   noon-8pm Tue-Fri, to 7pm Sat, closed Feb) Xul Solar was a painter, inventor, poet and friend of Jorge Luis Borges. This museum (located in his old mansion) showcases over 80 of his unique and colorful yet subdued paintings. Solar's Klee-esque style includes fantastically themed, almost cartoonish figures placed in surreal cubist landscapes. It's great stuff, and bizarre enough to put him in a class of his own. Tours in Spanish are available Tuesday and Thursday at 4pm and Saturday at 3:30pm.\n\nMUSEO CASA DE RICARDO ROJAS MUSEUM\n\nMAP GOOGLE MAP\n\n(  4824-4039; Charcas 2837;   L\u00ednea D Aguero, Pueyrred\u00f3n) Walk under the facade, modeled after the Casa de Independencia in Tucum\u00e1n, and behold a quaint courtyard surrounded by European and Incan architectural motifs. Famous Argentine educator and writer Ricardo Rojas lived here from 1929 to 1957, and in his office wrote his renowned work _El Santo de la Espada_ (1933). An old dining room with period furniture also gives an idea of the past. Phone for opening hours, as the museum was closed for remodeling at research time.\n\nBIBLIOTECA NACIONAL LIBRARY\n\n(  4808-6000; www.bn.gov.ar; Ag\u00fcero 2502;   9am-9pm Mon-Fri, noon-7pm Sat & Sun;   59, 60) F Prominent Argentine and Latin American literary figures, such as Ernesto S\u00e1bato, have lectured here, and other events include workshops, concerts and cultural activities. Tours in English are offered on Monday, Tuesday and Thursday at 3pm. Bring photo ID and be ready to fill out a form to enter.\n\nCENTRO ISL\u00c1MICO REY FAHD MOSQUE\n\nMAP GOOGLE MAP\n\n(  4899-0201; www.ccislamicoreyfahd.org.ar; Av Int Bullrich 55) This landmark mosque, built by Saudis on land donated by former president Carlos Menem, is southeast of Las Ca\u00f1itas. Free tours in Spanish are offered on Tuesday, Thursday and Saturday at noon (bring your passport, dress conservatively and enter via Av Int Bullrich).\n\n###  WALKING THE DOG\n\nBuenos Aires supports a legion of _paseaperros_ (professional dog walkers), who can be seen with up to a dozen canines on leashes. They'll stroll through areas like Palermo's parks, Recoleta and even downtown with a variety of dogs ranging from scruffy mongrels to expensive purebreds, each of their tails happily a-waggin'.\n\n_Paseaperros_ are employed by busy apartment dwellers who either can't or won't take the time to exercise their animals properly \u2013 and are willing to pay up to AR$200 per month for this unique walking service. Since most _paseaperros_ don't pay taxes, they can really 'clean up' in the city \u2013 figuratively speaking.\n\nEvery day thousands of canines deposit tons (almost literally) of excrement in the streets and parks of the capital. You'll be aware of this fact soon after stepping onto the streets of Buenos Aires. Cleaning up after one's pooch is already a city requirement, but enforcement is nil, so be very careful where you tread \u2013 you'll see dog piles of all textures and sizes lining almost every sidewalk. One to especially step clear of is the author-named _dulce de leche_ variety.\n\nStill, the capital's leashed packs are a remarkably orderly and always entertaining sight, and make great snapshots to bring back home.\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  EATING\n\nPalermo Viejo is at the heart of innovative cuisine in Buenos Aires. Dozens of upmarket restaurants serve creative cuisine in a contemporary setting, but it's important to be discerning \u2013 a new eatery opens every week, and while quality is generally high, only a few places are truly special.\n\nApart from the high-end _parrillas_ where fine steaks and expensive wines rule, Palermo chefs often take inspiration from different ethnic cuisines. You'll find elements of Japanese, Indian, Vietnamese, Brazilian, Mexican, Middle Eastern, Greek and even Norwegian food throughout the neighborhood's dining scene. Just remember that most restaurants offer an Argentine approach to these international styles of cooking: don't expect spicy flavors, for example, because the locals can't stomach it.\n\nAnother sub-neighborhood of Palermo with exceptional eating is Las Ca\u00f1itas, not far from Palermo Viejo. Traffic jams up here on the weekends, when hordes of diners descend on the few blocks of Av B\u00e1ez where most of the area's restaurants and bars are concentrated.\n\nOUI OUI INTERNATIONAL $\n\nMAP GOOGLE MAP\n\n(  4778-9614; www.ouioui.com.ar; Nicaragua 6068; mains AR$40-70;   8am-8pm Tue-Sun) _Pain au chocolat_ and shabby chic? _Oui_. This charming and popular French-style cafe produces the goods \u2013 dark coffee, buttery croissants and jars of tangy lemonade \u2013 and boasts a small and cozy interior. Choose also from creative salads, gourmet sandwiches and luscious pastries. Its annex, Almac\u00e9n Oui Oui MAP GOOGLE MAP (cnr Dorrego & Nicaragua;   8am-9pm Tue-Sun), is on the same block and stays open an hour later.\n\nSOCIAL LA LECHUZA ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4773-2781; Uriarte 1980; mains AR$40-75;   8:30pm-midnight Tue, 12:30-3:30pm & 8:30pm-midnight Wed-Sat, 1-3:30pm Sun) A world away from its trendy neighbors, this classic joint holds on to tradition and offers a breath of fresh air from all those overpriced, overhip restaurants in Palermo. Funky art adorns the walls including amateur owl paintings ( _lechuza_ means 'owl'). Meats and pastas are served in abundant portions, but don't miss the desserts like chocolate mousse and tiramisu.\n\nAREVALITO VEGETARIAN $\n\nMAP GOOGLE MAP\n\n(  4776-4252; Ar\u00e9valo 1478; mains AR$40;   9am-midnight Mon-Sat;  ) The menu is hardly extensive at this tiny bohemian eatery, but everything is good and very healthy, and the portions are generous. There's homemade yogurt, daily sandwich specials, hearty salads, vegetable tarts and more. Exceptional coffee and lemonade too.\n\nLAS CHOLAS ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4899-0094; Arce 306; mains AR$40-75;   noon-4pm & 8pm-midnight) Good food and bargain prices keep this popular corner eatery packed. Choose from typical _parrilla_ cuts or traditional Argentine foods like _locro_ and _cazuela_ (meat and veggie stews). Negatives include uncomfortable chairs, spotty service and the owner's large dog roaming the dining room.\n\n  OVIEDO MEDITERRANEAN $$\n\nMAP GOOGLE MAP\n\n(  4822-5415; Beruti 2602; mains AR$90-140;   noon-midnight Mon-Sat) Famed chef Martin Rebaudino brings a contemporary Spanish flair to seafood (the fish is shipped daily from Mar del Plata) and serves up melt-in-your-mouth _cochinillo_ (suckling pig) dishes that are worth writing home about. Desserts are homemade, as are the breads. A fantastic wine list and cordial service make Oviedo a fine-dining experience you won't mind shelling out for.\n\n  DON JULIO PARRILLA $$\n\nMAP GOOGLE MAP\n\n(  4832-6058; Guatemala 4699; mains AR$80-120;   noon-4pm & 7:30pm-1am) Classy service and a great wine list add an upscale bent to this traditional corner steakhouse. The _bife de chorizo (sirloin steak)_ is the main attraction here, but the exposed-brick interior, original floor tiles and cowhide tablecloths enhance the sensory experience, and the gourmet salads \u2013 served with a flourish by the uber-professional wait staff \u2013 are a treat.\n\n  LAS PIZARRAS INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4775-0625; Thames 2296; mains AR$80-90;   8pm-midnight Tue-Sun) At this simple and unpretentious yet excellent restaurant, Chef Rodrigo Castilla cooks up a changing rainbow of eclectic dishes such as grilled venison or rabbit stuffed with cherries and pistachios. Those with meeker stomachs can choose the asparagus and mushroom risotto or any of the homemade pastas. The chalkboard menu on the wall adds to the casual atmosphere.\n\n  SIAMO NEL FORNO PIZZA $$\n\nMAP GOOGLE MAP\n\n(  4775-0337; Costa Rica 5886; pizza AR$65-95;   8pm-midnight Tue-Thu & Sun, to 1am Fri & Sat) Possibly the city's best Naples-style pizzas, made with quality ingredients and finished in a hot wood-fired oven so the thin crusts char beautifully. Try the Margherita, with tomatoes, fresh mozzarella, basil and olive oil; the Champignon & Prosciutto comes with mushrooms, ham and goat cheese. Also bakes up excellent calzoni.\n\nASTOR MODERN ARGENTINE $$\n\nMAP GOOGLE MAP\n\n(  4554-0802; www.astorbistro.com; Ciudad de la Paz 353; mains AR$80-90;   12:30-3:30pm Mon-Wed, 12:30-3:30pm & 8:30pm-midnight Thu & Fri, 8:30pm-midnight Sat) French-trained Chef Antonio Soriano presides over the kitchen at this contemporary restaurant in a residential neighborhood. The few main dishes change weekly but are always delicious and beautifully presented, accented with edible flowers. If you order the tasting menu (AR$149), bring your appetite \u2013 it's nine courses. To watch your meals being created, sit at the bar, which offers a view of the open kitchen.\n\nSUDESTADA ASIAN $$\n\nMAP GOOGLE MAP\n\n(  4776-3777; Guatemala 5602; mains AR$105-135;   noon-3:15pm & 8pm-midnight Mon-Thu, to 1am Fri & Sat) Sudestada's well-earned reputation comes from its beautifully prepared curries, stir-fries and noodle dishes, all inspired by the cuisines of Thailand, Vietnam, Malaysia and Singapore. Don't forgo an exotic cocktail or delicious lychee _licuado_ (milkshake). Note that if you order something spicy, it's actually spicy. The popular set-lunch special is great value.\n\nCRIZIA INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4831-4979; www.crizia.com.ar; Gorriti 5143; mains AR$90-135;   7pm-1am) One of BA's best seafood restaurants. Start with a half-dozen oysters, then follow with the grilled Camembert over porcini mushrooms and seared red tuna in lime vinaigrette. End with the semifreddo of ginger and fresh mango. Chef Gabriel Oggero will make yours a night to remember.\n\nOLSEN SCANDINAVIAN $$\n\nMAP GOOGLE MAP\n\n(  4776-7677; Gorriti 5870; mains AR$105-125;   noon-midnight Tue-Sat, 10:30am-midnight Sun) With its hip, relaxed vibe, too-cool crowd and dramatic central fireplace, Olsen could easily be located in the frosty climes of Scandinavia. Chef German Martitegui's dishes are limited but inspired, and the vodka selection \u2013 over 60 \u2013 is superlative. Luxuriate with an exotic cocktail in the lovely front garden, or try the popular Sunday brunch. Limited menu between lunch and dinner.\n\nMIRANDA PARRILLA $$\n\nMAP GOOGLE MAP\n\n(  4771-4255; www.parrillamiranda.com; Costa Rica 5602; mains AR$70-125;   8am-1am Sun-Thu, to 2am Fri & Sat) Fashionable Miranda is the _parrilla_ of choice for those looking for both style and substance. It's a pleasant modern steakhouse with concrete walls, high ceilings and rustic wooden furniture, but high-quality grilled beef is the main attraction here \u2013 try the popular _ojo de bife_. If you score a sidewalk table on a warm day, life doesn't get much better.\n\nIL BALLO DEL MATTONE ITALIAN $$\n\nMAP GOOGLE MAP\n\n(  4776-4247; www.ilballo.tv; Gorriti 5737; mains AR$50-100;   noon-4pm & 8pm-midnight Mon-Sat) This artsy, eclectic trattoria attracts artists, musicians and tourists, among others, with its delicious homemade pastas. Try the popular _caramel diburata_ appetizer (a soft cheese), then go for the _fusilles escarparo_ with garlic, green onion and parmesan in a tomato sauce. Cute little patio for warm days; reserve at night. Its annex is two blocks away at Gorriti 5950.\n\nMUSEO EVITA RESTAURANTE MODERN ARGENTINE $$\n\nMAP GOOGLE MAP\n\n(  4800-1599; www.museoevitaresto.com.ar; JM Gutierrez 3926; mains AR$70-100;   9am-midnight Mon-Sat, to 7pm Sun) This restaurant's charming tiled courtyard may be the city's prettiest spot for an alfresco lunch, and the cuisine is thoroughly sophisticated, too. Locals and visitors alike come for the gourmet sandwiches, steaks and salads; the lunch specials are good too.\n\nBIO VEGETARIAN $$\n\nMAP GOOGLE MAP\n\n(  4774-3880; www.biorestaurant.com.ar; Humboldt 2192; mains AR$68-100;   11am-midnight Sun-Thu, to 1am Fri & Sat;  ) The supremely health-conscious should make a beeline for this casual corner joint, which specializes in healthy, organic and vegetarian fare. Try the quinoa risotto, seitan stirfry, Mediterranean couscous or mushrooms a la Bahiana (Brazilian-style). Don't miss the refreshing ginger lemonade. Also caters to celiacs, vegans and raw foodists.\n\nAZEMA INTERNATIONAL $$\n\nMAP GOOGLE MAP\n\n(  4774-4191; AJ Carranza 1875; mains AR$75-120;   8:30-midnight Mon-Thu, to 1am Fri & Sat) With exotic spices and foreign ingredients, Paul Jean Azema goes where few local chefs have gone before. His eclectic menu takes inspiration from his diverse travels \u2013 expect dishes like lamb cooked Mauritius-style, rabbit in chardonnay wine sauce and mango and curry duck \u2013 and Azema himself occasionally makes an appearance in the dining room.\n\n  HERN\u00c1N GIPPONI RESTAURANT MODERN ARGENTINE $$$\n\nMAP GOOGLE MAP\n\n(  3220-6820; www.hgrestaurant.com.ar; Soler 5862; mains AR$120-135, set menu AR$260, wine pairings AR$160 extra;   7:30am-midnight Mon-Fri, 8am-midnight Sat & Sun) Located in the Fierro Hotel, this exceptional restaurant offers highly sophisticated, Spanish-influenced dishes created by chef Hern\u00e1n Gipponi. Order the seven-course tasting menu for the full experience \u2013 it's pricey but worth it. Set menu on weekday lunches runs AR$150; the six-course weekend brunch costs AR$175 and is a must. On Monday nights, everyone sits at one communal table. Reserve ahead.\n\nLA CABRERA PARRILLA $$$\n\nMAP GOOGLE MAP\n\n(  4831-7002; Cabrera 5099; mains AR$115-150;   12:30-4:30pm & 8:30pm-1am Mon-Thu, to 2am Fri & Sat, 12:30pm-1am Sun) Hugely popular for grilling up BA's most sublime meats. Steaks come in 200g or 400g sizes and arrives with heaps of little complimentary side dishes. Come at 7pm for 'happy hour,' when everything is 40% off \u2013 just make sure you get here early enough to score a table. There's an annex at Cabrera 5127; expect a long wait at both locations.\n\nUNIK MODERN ARGENTINE $$$\n\nMAP GOOGLE MAP\n\n(  4772-2230; www.unik.pro; Soler 5132; mains AR$135-190;   8:30pm-midnight Mon, 12:30-3pm & 8:30pm-midnight Tue-Sat) For a splurge night out, you can't do much better than Unik. Start with an appetizer of roasted beets with goat cheese and walnut-\u00adtruffle vinaigrette (AR$90), then move on to the main courses: rabbit with eggplant pur\u00e9e, suckling pig with grilled apples in Dijon mustard sauce or Patagonian lamb with pickled figs and a tagine sauce. Unik indeed.\n\nGREEN BAMBOO VIETNAMESE $$$\n\nMAP GOOGLE MAP\n\n(  4775-7050; www.green-bamboo.com.ar; Costa Rica 5802; mains AR$115-160;   8:30pm-1am Sun-Thu, to 2:30am Fri & Sat) This sultry Vietnamese eatery offers just a small selection of dishes, but all are well prepared and flavorful. Sample things like seafood curry, marinated sirloin in lemongrass and 'traditional' _pho_ (well, Argentine-style \u2013 so don't expect much tradition). The atmosphere is dim and romantic, with a few low tables, and the tropical cocktails are excellent. Reserve on weekends.\n\n###  BEHIND CLOSED DOORS\n\nA very popular Buenos Aires culinary offshoot in the last few years are 'closed-door restaurants', or _puertas cerradas_. These places are open only a few days per week, have timed seatings and are generally prix fixe (and mostly cash only). They're not marked with signs and you have to ring a bell to enter. They won't even tell you the address until you make reservations (mandatory, of course). But for that tingly feeling brought on by discovering something off the beaten path \u2013 with some of the city's best food to boot \u2013 these places are for you.\n\nThere are two kinds of _puertas cerradas:_ the first is where you dine in the chef's actual home, and usually sit at a large communal table. This is a great way to meet other people, often interesting travelers or expats; it's ideal for folks traveling alone. The second kind has more of a restaurant feel and tables are for separate groups \u2013 just like a regular restaurant, but not open to walk-ins. Many _puertas cerradas_ are located in Palermo.\n\nSome of BA's best _puertas cerradas_ include iLatina (www.ilatinabuenosaires.com\/en), serving exquisite Colombian food; Casa Saltshaker (www.casasaltshaker.com), where you'll sample ex\u2013New Yorker Dan Perlman's culinary creations; NOLA (www.nolabuenosaires.com), home to New Orleans\u2013fusion dishes; Casa Felix (www.colectivofelix.com\/casa-felix), a pescatarian's delight; and Cocina Sunae (www.cocinasunae.com), for near-authentic Asian-fusion meals.\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  DRINKING & NIGHTLIFE\n\nYou'll find Buenos Aires' hippest drinking scenes in and around Palermo, especially near Plaza Serrano in Palermo Soho. Many restaurants in this neighborhood also have good bars. Las Ca\u00f1itas has a lively three blocks of nonstop restaurants and bars, and is also worth a drop-in, especially later in the evening.\n\n  VERNE COCKTAIL BAR\n\nMAP GOOGLE MAP\n\n(  4822-0980; Medrano 1475;   9pm-2am Sun-Wed, to 4am Thu-Sat) Upscale yet casual bar with slight Jules Verne theme. Cocktails are the specialties here, whipped up by one of BA's best bartenders, Fede Cuco. A few tables, some cushy sofas and an airy outdoor patio offer a variety of seating options, but plant yourself at the bar to see drinks being made; check out the French absinthe server.\n\nFRANK'S BAR COCKTAIL BAR\n\nMAP GOOGLE MAP\n\n(  4777-6541; www.franks-bar.com; Ar\u00e9valo 1445;   9pm-4am Wed-Sat) Very popular plush, elegant speakeasy bar that 'requires' a password (via telephone booth) to get in \u2013 or just sweet talk the bouncer. Inside it's a beautiful space with crystal chandeliers, billowy ceiling drapes and exclusive feel. Classic cocktails from before the 1930s are stirred \u2013 never blended \u2013 and served to a crowd of locals and foreigners. Check out the balcony bar too.\n\nMAGDALENA'S PARTY BAR\n\nMAP GOOGLE MAP\n\n(  4833-9127; www.magdalenasparty.com; Thames 1795) Popular bar-restaurant with laid-back atmosphere and _buena onda_ (good vibes). DJs spin from Thursday to Saturday nights, and with cheap drinks this is a good preclub spot; try the vodka lemonade by the pitcher. Happy hour runs from noon to midnight daily, and tasty expat-friendly food is served, such as freshly ground hamburgers, California-style burritos and organic coffee. It does weekend brunch too.\n\nHOME HOTEL BAR\n\nMAP GOOGLE MAP\n\n(  4778-1008; www.homebuenosaires.com; Honduras 5860;   8am-midnight) Some of Palermo's best cocktails can be found at Home Hotel's very intimate bar-restaurant. During the day, relax in the grassy garden next to the slick infinity pool. At night, settle down at the polished cement bar with a house cocktail, created by some of Buenos Aires' best-known bartenders. Friday evenings in summer are livened up by DJ parties. A wide variety of vodkas, along with tapas are available.\n\nANTARES BAR\n\nMAP GOOGLE MAP\n\n(  4833-9611; www.cervezaantares.com; Armenia 1447;   7pm-4am) Thirsty for a decent _cerveza_ ? Look no further than this modern but relaxed restaurant-bar with Argentine-brewed ales, pilsners, lagers and barley wine. Order a beer flight, sample the brewmaster's special-edition selection or just enjoy the two-for-one pints during happy hour (from 7pm to 8pm). Also in Las Ca\u00f1itas at Ar\u00e9valo 2876.\n\nCONGO BAR\n\nMAP GOOGLE MAP\n\n(  4833-5857; www.barcongo.com.ar; Honduras 5329;   8pm-4am Mon-Thu, 9pm-5am Fri & Sat) The highlight at this trendy bar is the beautiful back patio \u2013 _the_ place to be seen on hot summer nights, with its slick bar, leafy atmosphere and comfy wood booths. The music is great, too, with DJs spinning from Wednesday to Saturday, and inside there are elegant low lounges in romantic spaces. A full food menu is available, along with strong cocktails.\n\nEL CARNAL BAR\n\nMAP GOOGLE MAP\n\n(  4772-7582; www.carnalbar.com.ar; Niceto Vega 5511;   7pm-5am Tue-Fri, 9pm-5am Sat) See and be seen \u2013 preferably in the open air with an icy vodka tonic in hand \u2013 on the rooftop terrace at this ever-popular watering hole. With its bamboo lounges and billowy curtains, the place can't be beat for a cool chill-out on a warm summer night. Early in the week reggae rocks, while Thursdays to Saturdays means pop and '80s tunes.\n\nSHANGHAI DRAGON PUB\n\nMAP GOOGLE MAP\n\n(  4778-1053; Ar\u00e1oz 1199;   5pm-3am Mon-Wed, to 4am Thu-Sun) Good corner pub with mellow vibe, sports on TV and slight Asian theme, attracting the 25 and over crowd. Come for dinner if you want cheap Chinese food like vegetable stir fries and Kung Pao chicken, while happy hour means cheap drinks from 5pm to 10pm. DJs spin funk and rock on Saturdays. Same owners as popular Gibraltar (Click here) and Bangalore (Indian) pubs.\n\nMUNDO BIZARRO BAR\n\nMAP GOOGLE MAP\n\n(  4773-1967; www.mundobizarrobar.com; Serrano 1222;   8pm-late) This red-lit, futuristically retro and stylish lounge bar is open pretty much all through the night on weekends, when everything from old-time American music to hip DJs to jazz stirs up the air waves. If you're feeling peckish, check out the American-inspired bar food, which ranges from Tex-mex to burgers to hot apple pie with ice cream. Dance on the stripper pole after you've had a few drinks.\n\nVAN KONING PUB\n\nMAP GOOGLE MAP\n\n(  4772-9909; Av B\u00e1ez 325;   7pm-3:30am Sun-Thu, to 5am Fri & Sat) Wonderfully rustic spaces make this Dutch-themed pub feel like the inside of a boat; after all, it's a 17th-century-style seafaring theme complete with dark wood beams, flickering candles and blocky furniture. Bars on two floors serve over 30 kinds of both local and imported brews, with at least three on draft. A magnet for expats; the first Wednesday of the month is Dutch night.\n\nSUGAR BAR\n\nMAP GOOGLE MAP\n\n(  4831-3276; www.sugarbuenosaires.com; Costa Rica 4619;   7pm-5:30am Mon-Fri, 11am-5:30am Sat, 11am-3am Sun) This lively expat watering hole brings in a youthful nightly crowd with well-priced drink specials and comfort food like chicken fingers and buffalo wings. Watch sports on the two huge TV screens or come on Thursdays \u2013 also known as ladies' night \u2013 when things can get a little rowdy. On weekends, you can roll out of bed and arrive in time for eggs and mimosas.\n\nACABAR BAR\n\nMAP GOOGLE MAP\n\n(  4772-0845; www.acabarnet.com.ar; Honduras 5733;   8pm-2am Sun-Thu, to 4am Fri & Sat) One of the quirkiest restaurant-bars in town. A maze of a half-dozen rooms and spaces are decked out in mismatched chandeliers, funky furniture, clashing pastel colors and frilly wallpaper; it's a texture and pattern overload. It's also famous for board games, which Argentines love to play. Serves food earlier on.\n\nKIKA CLUB\n\nMAP GOOGLE MAP\n\n(www.kikaclub.com.ar; Honduras 5339;   Tue-Sun) Being supremely well located near the heart of Palermo Viejo's bar scene makes Kika's Tuesday-night popular 'Hype' party (www.hype-ba.com) easily accessible for the trendy crowds. It's a mix of electro, rock, hip-hop, drum and bass and dubstep, all spun by both local and international DJs. Other nights see electronica, raggaeton, Latin beats and live bands ruling the roost.\n\nPACH\u00c1 CLUB\n\n(  4788-4288; www.pachabuenosaires.com; Av Rafael Obligado 6151;   Sat) Popular, long-running electronica club well-known for attracting famous international DJs who spin tunes for the sometimes spaced-out crowds. Laser light shows and a great sound system makes the chic crowds happy through the early morning light \u2013 be sure to bring your shades and watch the sun come up from the terrace.\n\nCLUB AR\u00c1OZ CLUB\n\nMAP GOOGLE MAP\n\n(  4832-9751; www.clubaraoz.com.ar; Ar\u00e1oz 2424;   Thu-Sat) Also known as 'Lost', this intimate club's finest hour is on Thursday, when hip-hop rules the roost and the regulars start break dancing around 2am (reggaeton comes on later in the evening). National and international DJs liven up the weekends. There's no dress code \u2013 a good thing, since it tends to get hot and sweaty in here.\n\nCROBAR CLUB\n\nMAP GOOGLE MAP\n\n(  4778-1500; www.crobar.com.ar; cnr Av de la Infanta Isabell & Freyre;   Fri & Sat) Year after year, stylish Crobar remains one of BA's most popular nightlife spots. Friday usually features international DJs mashing up the latest electronic selections, while Saturday tends to feature more commercial or Latin beats. There's also a back room for those who prefer classic rock, '80s remixes and occasional live bands, while the main levels are strewn with mezzanines and catwalks that allow views from above.\n\nGLAM CLUB\n\nMAP GOOGLE MAP\n\n(  4963-2521; Jos\u00e9 Antonio Cabrera 3046;   Thu-Sat) Housed on three floors of an old mansion with tall brick hallways, this mazelike gay club still brings in a crowd of young, good-looking guys. They're here to dance and get to know each other better \u2013 there are no shows to distract, just casual lounges, pretty bars and a dark room where anything goes. Saturday is the biggest night here.\n\nNICETO CLUB CLUB\n\nMAP GOOGLE MAP\n\n(  4779-9396; www.nicetoclub.com; Niceto Vega 5510;   Thu-Sat) One of the city's biggest crowd-pullers, the can't-miss event at Niceto Club is Thursday night's Club 69, a subversive DJ extravaganza featuring gorgeously attired showgirls, dancing drag queens, futuristic video installations and off-the-wall performance art. On weekend nights, national and international spin masters take the booth to entertain lively crowds with blends of hip-hop, electronic beats, cumbia and reggae.\n\nLA VIRUTA MILONGA\n\nMAP GOOGLE MAP\n\n(  4774-6357; www.lavirutatango.com; Armenia 1366, basement) Popular basement venue. Good beginner tango classes are available before _milongas_ \u2013 translating into many inexperienced dancers on the floor earlier on \u2013 so if you're an expert get here late (after 3:30am). Music can run the gamut from tango to rock to cumbia to salsa earlier in the evening, with more traditional tunes later. Tango shows also on offer.\n\nSALON CANNING MILONGA\n\nMAP GOOGLE MAP\n\n(  4832-6753; www.parakultural.com.ar; Av Scalabrini Ortiz 1331) Some of BA's finest dancers (no wallflowers here) grace this traditional venue with its great dance floor. Well-known tango company Parakultural often stages good events here involving live music, tango DJs, singers and dancers. Expect big crowds and plenty of tourists.\n\nWORTH A DETOUR\n\n###  BELGRANO\n\nBustling Av Cabildo, the racing heartbeat of Belgrano, is an overwhelming jumble of noise and neon. It's a two-way street of clothing, shoe and housewares shops that does its part in supporting _porte\u00f1os_ ' mass consumerism. For a bit more peace and quiet, head to the blocks on either side of the avenue, where Belgrano becomes a leafy barrio of museums, plazas, parks and good local eateries.\n\nA block east of Av Cabildo, the barrio's plaza is the site of the modest but fun Feria Plaza Belgrano (cnr Juramento & Cuba;   10am-8pm Sat & Sun). On a sunny weekend it's full of shoppers and families with strollers. Near the plaza stands the Italianate Iglesia de la Inmaculada Concepci\u00f3n, a church popularly known as La Redonda (The Round One) because of its impressive dome.\n\nJust a few steps from the plaza is the Museo Hist\u00f3rico Sarmiento (  4782-2354; www.museosarmiento.gov.ar; Juramento 2180; admission AR$15;   1-6pm Mon-Fri, 3-7pm Sat & Sun), which honors one of the most forward-thinking Argentines in history. Also close by is the Museo de Arte Espa\u00f1ol Enrique Larreta (  4784-4040; www.museolarreta.buenosaires.gov.ar; Juramento 2291; admission AR$5;   1-7pm Mon -Fri, 10am-8pm Sat & Sun), a mansion with gorgeous art pieces and gardens. About five blocks north is yet another museum, the Museo Casa de Yrurtia (  4781-0385; O'Higgins 2390; admission AR$10, Wed free;   11:30am-6pm Wed-Fri, 3-7pm Sat & Sun), honoring the well-known Argentine sculptor.\n\nFour blocks northeast of Plaza Belgrano, French landscape architect Carlos Thays took advantage of the contours of Barrancas de Belgrano to create an attractive, green public space on one of the few natural hillocks in the city. Retirees spend the afternoon at the chess tables beneath its omb\u00fa tree, and on Saturday and Sunday evenings the band shell hosts a popular outdoor _milonga_ (tango event).\n\nAcross Juramento from Barrancas, Belgrano's growing Chinatown fills three blocks on Arribe\u00f1os, with more Chinese businesses spilling over into the side streets. Don't come on Mondays, however, as many places are shut; do come on Chinese New Year, when festivities abound.\n\nYou'll probably head into Belgrano via Av Cabildo, either by bus or Subte (the Subte runs right under Cabildo). Plaza Belgrano is one block east of Cabildo at Juramento; most sights are around the plaza. Barrancas de Belgrano is the location of Belgrano's bus and train stations and is located about four blocks from the plaza.\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  ENTERTAINMENT\n\nTHELONIOUS BAR JAZZ\n\nMAP GOOGLE MAP\n\n(  4829-1562; www.theloniousclub.com.ar; Salguero 1884, 1st fl;   9pm-1am Wed-Thu, to 3am Fri & Sat) Up the stairs in an old mansion lies this dimly lit jazz bar, with high brick ceilings and a good sound system. Come early to snag a seat (or reserve one ahead of time) and partake in the typically Argentine menu and good range of cocktails. Thelonious is known for its classic and contemporary Argentine jazz lineups, though international musicians sometimes entertain.\n\nLA PE\u00d1A DEL COLORADO FOLK\n\nMAP GOOGLE MAP\n\n(  4822-1038; www.lapeniadelcolorado.com; G\u00fcemes 3657;   8pm-4am) Nightly music shows (mostly folkloric) start at 10pm and are memorable at this rustic restaurant, and after midnight audience members pick up nearby guitars to make their own entertainment. There's also tasty northern Argentine food on offer, including _locro_ , _chip\u00e1_ (chewy cheese balls) and _humitas de Chala_ (like tamales) \u2013 the spicy _empanadas_ are excellent.\n\nLOS CARDONES FOLK\n\nMAP GOOGLE MAP\n\n(  4777-1112; www.cardones.com.ar; Borges 2180;   from 9pm Wed-Sat) Come to this friendly, low-key _pe\u00f1a_ (folk club) for audience-participatory jam sessions (and possible dancing), mellow guitar shows, hearty regional cuisine from northern Argentina and free-flowing red wine. Shows start at 10pm on weekdays and 11pm weekends. Check out the website for details on the current lineup and reserve ahead for a good table.\n\nTIEMPO DE GITANOS FLAMENCO\n\nMAP GOOGLE MAP\n\n(  4776-6143; www.tiempodegitanos.com.ar; El Salvador 5575;   Wed-Sun) This venue in Palermo Hollywood offers good flamenco shows in an intimate restaurant setting. Shows start at 11:30pm and cost AR$60 to AR$90, depending on the night. Classic Spanish foods like paella and tapas are on tap (food purchase is obligatory). Reserve in advance.\n\n####   Neighborhood Walk  \n **Walking the Green**\n\n**Start** Parque 3 de Febrero\n\n**End** Museo Evita\n\n**Length** 4.5km, 3-4 hours\n\nStart in  Parque 3 de Febrero (Click here) these expansive green spaces were once the aristocracy's stomping ground. It's best on weekends, when the road around the rose garden is cut off to vehicular traffic (this is when you can rent bicycles and rollerblades).\n\nThose interested in modern art can peek into the contemporary  Museo de Artes Pl\u00e1sticas Eduardo S\u00edvori (Click here), which showcases Argentine work. There's a relaxing cafe here as well. If you like flowers, head across the road and cross the bridge to the  Rosedal (Click here), where you can stop to sniff the roses. Continue across the garden until you come to Av Iraola, turn left onto it for about a block, then veer to the right to reach Av Sarmiento.\n\nCross Av Sarmiento (carefully!), and head along Av Berro for about 500m to BA's  Jard\u00edn Japon\u00e9s (Click here). This little paradise is meticulously maintained with koi ponds, pretty bridges and a tea shop, making it a welcome break from roads and traffic.\n\nNow skim around Plaza Alemania and jog around a few residential streets to reach  Museo de Arte Latinoamericano de Buenos Aires (Click here), an airy museum that's home to some excellent paintings. For more culture, go two blocks south to the much more modest  Museo de Arte Popular Jos\u00e9 Hern\u00e0ndez (Click here), which exhibits handicrafts and folkloric items.\n\nOn Av del Libertador, stop in at luscious ice-cream shop  Un Altra Volta for a peaked cone of _dulce de leche granizado_ (milk caramel with chocolate flakes). Now head down the street (alongside the odiferous zoo) to  Museo Evita (Click here), where you can check out the collected memorabilia of Argentina's most famous woman. There's a fine cafe-restaurant here where you can end your long walk.\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  SHOPPING\n\nPalermo Viejo, a large sub-neighborhood of Palermo, is a fashionista's shopping paradise. A few years ago, most of the storefronts were showcases for cutting-edge clothes; these days, the barrio hosts a wider range of designers selling high-end wares from home accessories and books to fancy stationery, soaps, candles, souvenirs, kids' toys and gourmet chocolate. It's easy to spend hours or even days shopping in Palermo; many design-minded travelers consider an afternoon here part of the sightseeing circuit.\n\nFERIA PLAZA SERRANO MARKET\n\nMAP GOOGLE MAP\n\n(Plaza Serrano;   10am-8pm Fri-Sun) Costume jewelry, hand-knit tops, funky clothes, hippie bags, glass jewelry, leather accessories and much more fill the craft booths at this popular street fair on fashionable Plaza Serrano (also known as Plaza Cortazar). It's not huge, but the plaza is in the middle of Palermo's bustling nightlife and surrounded by trendy bars, restaurants and upscale stores.\n\nRAPSODIA FASHION\n\nMAP GOOGLE MAP\n\n(  4831-6333; www.rapsodia.com; Honduras 4872;   10am-9pm) With fabrics from linen to leather and details like fringe and sequins, this large and popular boutique is a must for fashion mavens. Old and new are blended into creative, colorful styles with exotic and bohemian accents. Locals covet its dresses and jeans; over a dozen branches in the city.\n\nBOLIVIA CLOTHING\n\nMAP GOOGLE MAP\n\n(  4832-6284; Gurruchaga 1581;   11am-8pm Mon-Sat, 3-8pm Sun) There's almost nothing here that your young, hip and possibly gay brother wouldn't love, from the stylish plaid shirts to the skin-tight jeans to the military-styled jackets. Metrosexual to the hilt, and paradise for the man who isn't afraid of patterns, plaid or pastels. Also at Nicaragua 4906.\n\nLO DE JOAQUIN ALBERDI FOOD & WINE\n\nMAP GOOGLE MAP\n\n(  4832-5329; www.lodejoaquinalberdi.com; JL Borges 1772;   11am-9:30pm Mon-Sat, noon-9:30pm Sun) Nationally produced wines for every taste and budget line the racks and cellar of this attractive wine shop. Tastings happen Thursdays and Fridays at 7:30pm (double-check ahead of time) and include four wines and some cheeses; the cost is AR$100.\n\nHERMANOS ESTEBECORENA CLOTHING\n\nMAP GOOGLE MAP\n\n(  4772-2145; www.hermanosestebecorena.com; El Salvador 5960;   11am-8pm Mon-Sat) The Estebecorena brothers apply their highly creative skills toward smartly designed tops, jackets that fold into bags, polo-collar work shirts and even supremely comfortable, nearly seamless underwear. The focus is on original, highly stylish, very functional men's clothing that makes the artsy types swoon. Selection is limited, but what's there really counts.\n\nLA MERCER\u00cdA FASHION & ACCESSORIES\n\nMAP GOOGLE MAP\n\n(  4831-8558; Armenia 1609;   11am-8pm) Attracting crowds of gossipy ladies on a busy weekend, this boutique is stuffed with bright and colorful accessories like costume jewelry, pillows, scarves, belts, perfumes, hats and lots of handbags. Frilly, glitzy and designed for self-assured women.\n\nCALMA CHICHA HOUSEWARES\n\nMAP GOOGLE MAP\n\n(  4831-1818; www.calmachicha.com; Honduras 4909;   10am-8pm Mon-Fri, 11am-8pm Sat, 1-7pm Sun) Calma Chicha specializes in creative housewares and accessories that are locally produced from leather, faux leather, sheepskin, cowhide, and brightly hued fabric. Look for butterfly chairs, throw rugs, leather placemats, bright pillows and cowskin bags.\n\nJUANA DE ARCO FASHION & ACCESSORIES\n\nMAP GOOGLE MAP\n\n(  4833-1621; www.juanadearco.net; El Salvador 4762;   10am-8pm Mon-Fri, 11am-8pm Sat, 1-8pm Sun) Mariana Cortes has designed adorable bits of fabrics sewn into girly sets that would be best showcased during a pillow fight \u2013 think brightly colored T-shirts, flowery boxer shorts and tight leggings. Descend the staircase to discover more treasures.\n\nNOBRAND SOUVENIRS\n\nMAP GOOGLE MAP\n\n(  4776-7288; www.nobrand.com.ar; Gorriti 5876;   noon-8pm Tue-Sun) For that modern, locally inspired gift, check out this slick shop. Two designers created Argentine logos such as the cow, _mate_ , _asado_ , _empanadas_ and tango (along with people like Evita and Che Guevara), and transferred those iconic logos onto T-shirts, aprons, mugs, notebooks, caps and even shoes. Fun gifts for folks back home.\n\nCAPITAL HOUSEWARES\n\nMAP GOOGLE MAP\n\n(  4834-6555; www.capitalpalermo.com; Honduras 4958;   10am-8pm) There's nothing you really _need_ at this whimsical knickknacks store, but it's a fun place to visit anyhow. The stock is always changing, but expect things like mugs with iconic images, funky computer accessories and cartoonish shower curtains.\n\nHUMAWACA LEATHER GOODS\n\nMAP GOOGLE MAP\n\n(  4832-2662; www.humawaca.com; El Salvador 4692;   11am-8pm Mon-Sat, 2-7pm Sun) Award-winning designs bring both form and functionality to Argentine leather, producing handbags, tote bags and wallets with clean modernist lines and colorful hues. Visit this tiny boutique and you'll always find something different and eye-catching.\n\n28 SPORT SHOES\n\nMAP GOOGLE MAP\n\n(  4833-4287; www.28sport.com; Gurruchaga 1481;   11am-1:30pm, 2:30-7pm Mon-Sat) For the retro-sports fanatic, there's nothing better than this unique shop with a sense of humor and a vintage twist. Focusing on only one product and one style \u2013 men's '50s sport-style shoes \u2013 the cobblers here can concentrate on quality and craftsmanship. Inspiration comes from football, boxing and bowling shoes, and only 12 pairs of each design are produced.\n\nMISHKA SHOES\n\nMAP GOOGLE MAP\n\n(  4833-5655; www.mishkashoes.com.ar; El Salvador 4673;   10:30am-8:30pm Mon-Sat, 3-8pm Sun) Well-regarded designer Chelo Cant\u00f3n was once an architect but now creates wonderfully unique footwear with a retro-hip, feminine and slightly conservative vibe. Try on a pair of patent-leather sandals for size, or go for more traditional ballet flats in velvet and brocade (though styles are always changing). Check the website for other locations around the city.\n\nHARAPOS PATAGONIA SOUVENIRS\n\nMAP GOOGLE MAP\n\n(  2058-7810; Malabia 1635;   11am-7:30pm Mon-Sat, 2-7:30pm Sun) Can't make it all the way down to Patagonia? Well, then just visit to this small store to grab a southern souvenir. There are woolen goods (sheep are big down there), hand-made ceramics, wooden utensils and silver and alpaca jewelry. All products are made by Patagonian craftspeople.\n\nSUGAR & SPICE FOOD & WINE\n\nMAP GOOGLE MAP\n\n(  4777-5423; www.sugarandspice.com.ar; Guatemala 5419;   10am-7pm Mon-Fri, 9am-1pm Sat) Nibble the exotic (for Argentina, at least) creations of Frank Almeida, a long-time American expat. Herb cookies, almond biscotti, hazelnut panettone and peanut-butter brownies soothe homesick taste buds, and daily-baked bagels, muffins and scones are also available.\n\nCENTRICO SHOES\n\nMAP GOOGLE MAP\n\n(  4865-0143; www.centricocentrica.com.ar; Figueroa 1800;   10:30am-8:30pm Mon-Sat) Leonardo Mancuso designs these handmade leather shoes with a classic, traditional styling that emphasizes simplicity over showiness. Both men's and women's shoes available, and some have unisex looks; there are also a few ankle and knee-high boots. Prices run from AR$1000, but check out the sale items in the back room.\n\nPANORAMA CLOTHING\n\nMAP GOOGLE MAP\n\n(www.pnrm.com.ar; Rep\u00fablica de la India 2905;   11am-8pm Mon-Sat) About 20 emerging young designers are showcased at this small, upscale store in Palermo Chico. Peruse the clothing racks for one-of-a-kind, eclectic tops, pants, dresses and coats that can be definite show-stoppers. Small sizes dominate, though custom orders are possible. There are also a few accessories and shoes.\n\nALTO PALERMO SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(  5777-8000; www.altopalermo.com.ar; Av Santa Fe 3253;   10am-10pm) Smack on bustling Av Santa Fe, this popular, shiny mall offers dozens of clothing shops, bookstores, jewelry boutiques, and electronics and houseware stores. Look for Timberland, Lacoste, Hilfiger and Levi's (plus many Argentine brands, too). Services include a food court, a cinema complex and a good kids' area on the 3rd floor.\n\nPASEO ALCORTA SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(  5777-6500; www.paseoalcorta.com.ar; Salguero 3172;   10am-10pm) One of the largest and most upscale malls in the city. All the popular Argentine women's clothing shops are represented, as are international boutiques such as Adidas, Nike and Swatch. Other stores sell leather goods, kids' clothes, men's designs, sportswear and accessories. There's also a large food court and a children's play area.\n\nPATIO DEL LICEO SHOPPING MALL\n\nMAP GOOGLE MAP\n\n(  4822-9433; Santa Fe 2729;   2-8pm Mon-Sat) Wonderful little shopping mall with funky, casual and very artsy vibe. In the past few years, young struggling artists have taken over and created an artistic hub here, filling it with various small stores, exhibition spaces and workshops. You'll find a couple of book shops, a record store and some design stores. For refreshment there's a small cafe called Baby Snakes.\n\nLIBROS DEL PASAJE BOOKS\n\nMAP GOOGLE MAP\n\n(  4483-6637; www.librosdelpasaje.com.ar; Thames 1762;   10am-10pm Mon-Sat, 2-9pm Sun) This cool literary sanctuary offers history, culture and art books. They're mostly in Spanish, but look for the small English section near the front door (with some Lonely Planet books). There's a cute cafe in back, with small inside patio, for a snack or cup of coffee.\n\nPAPELERA PALERMO STATIONERY\n\nMAP GOOGLE MAP\n\n(  4833-3081; www.papelerapalermo.com; Cabrera 5227;   10am-8pm Mon-Fri, 11am-8pm Sat) Everyone emails these days, but step into this stationery store and you'll be tempted to start penning letters again. A large selection of gorgeous wrapping papers, handmade stationery and funky spiral notebooks (look for the Evita motif) all inspire.\n\nEL CID FASHION\n\nMAP GOOGLE MAP\n\n(  4832-3339; www.elcid.us; Gurruchaga 1732;   11am-8pm Tue-Sat, 3-7pm Sun) Some of the finest men's threads can be found at this Palermo Viejo boutique, which highlights Nestor Goldberg's designer shirts, pants, jackets, accessories and jeans. Materials are of the highest quality, and tailoring is classy, hip and casual.\n\nARTE \u00c9TNICO ARGENTINO CRAFTS & TEXTILES\n\nMAP GOOGLE MAP\n\n(  4832-0516; www.monteargentino.com; El Salvador 4656;   11am-6pm Mon-Fri, 11am-4pm Sat) Bright and beautiful woven _mantas_ (blankets) from Santiago del Estero are the main attraction at this upscale shop, located in an old house. All are made from wool and natural dyes, and can also be used as light rugs. Expect to pay from AR$2500 up.\n\nMERCADO DE LAS PULGAS MARKET\n\nMAP GOOGLE MAP\n\n(cnr \u00c1lvarez Thomas & Dorrego;   11am-7pm Tue-Sun) This large, covered warehouse is full of caged booths selling antiques, vintage objects and some modern items \u2013 precious things such as wood furniture, glass soda bottles, chandeliers, old clocks, silver trays, bird cages, elegant mirrors and ironwork.\n\n###  BUENOS AIRES' EMERGING DESIGNERS\n\nOne of the most notable transitions in Buenos Aires fashion in the last few years is the growing prominence of emerging designers. Based mostly out of private homes and apartments, known locally as 'showrooms,' a young community of recent fashion school grads and 20-somethings with an entrepreneurial spirit are taking over BA's inventive design world. Recent initiatives by the Buenos Aires City government such as competitions like IncuBA and La Ciudad de Moda (which allowed several of the most promising emerging designers to stage runway shows at Buenos Aires Fashion Week), have given the industry the boost it desperately needs to make BA one of the most intriguing fashion hotspots in Latin America. Whether you're on the hunt for casual streetwear, luxurious leather or innovative jewelry design, BA's best emerging designers take pride in their originality and skilled craftsmanship.\n\nWhen it comes to clothing design, rising names like Bel\u00e9n Amigo (www.belenamigo.com.ar) and Joan Martorello (www.facebook.com\/JMARTORELLO), both present at La Ciudad de Moda's runway show, are capturing stylish locals with their alternative, street-chic designs that range from Martorello's signature knits to Amigo's tailored pants and drapey silk organza tops. For more comfy casual wear, stop in at Deleon's (www.deleonba.com) Palermo Hollywood showroom, a destination for young fashionable locals looking to expand their collection of urban cool garments that scream sophistication.\n\nYet another exciting fresh face in BA's emerging fashion scene is Julia Schang-Viton (www.schangviton.com.ar), a young design prodigy whose structured, architectural cuts and neutral color palette draw upon her Asian heritage.\n\nIf you're in the market for leather, you've come to the right city. The independent design team behind artsy leather jacket label Oveja Oveja (www.ovejaoveja.com) have created a stir with their high-quality, hand-painted jackets that fit perfectly with BA's cosmopolitan vibe.\n\nFor leather bags, don't miss the geometric gems by Bellebas (www.bellebas.com), whose sleek clutches and embossed satchels make for the quintessential Buenos Aires accessory.\n\nIn the world of jewelry, both In\u00e9s Bonadeo (www.inesbonadeo.com.ar), a metal-working craftswoman who has already shown her work in New York at the inter\u00adnational design fair NY Now, and Vendaval (www.vendavalbuenosaires.com.ar), whose amulets are available at indie boutique Monoambiente (www.mono-ambiente.com.ar) can't be missed.\n\nWhile popular among locals, shopping in showrooms can prove intimidating for visitors. To gain access to these hidden treasures, it takes some local knowledge and the right connections. Thankfully, a few ambitious expats are giving tourists the chance to discover the exciting world of BA's emerging design through personalized shopping tours that'll take you to some of the most notable showrooms in town as well as the hippest open-door boutiques. Sophie Lloyd at ShopHopBA (www.shop-buenosaires.com) is the perfect option for those looking to get inside the city's exclusive showrooms. Warm, welcoming and knowledgeable, Sophie's tours include champagne toasts and privately catered lunches, and she also offers personal color consultations to those in need of a wardrobe makeover. Vanessa Bell at Creme de la Creme (www.cremedelacreme.com.ar) is known for her extensive contacts and excellent taste.\n\n_Natalie Schreyer is a fashion writer who has been living in Buenos Aires for five years. She is the creator ofwww.bashopgirl.com, a fashion blog covering BA's best emerging designers. In addition to her blog, she has written for LandingPadBA.com._\n\n Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Shopping | Sports & Activities\n\n  SPORTS & ACTIVITIES\n\nCAMPO ARGENTINO DE POLO POLO\n\nMAP GOOGLE MAP\n\n(cnr Av del Libertador & Av Dorrego) Just across from the Hip\u00f3dromo Argentino in Palermo, this stadium holds up to 30,000 spectators and hosts polo's most important events (including the Argentine Open Polo Championship in November and December). However, the northern suburb of Pilar has the highest density of polo clubs.\n\n###  TIERRA SANTA\n\nTired of the same old Sunday sermons? Praying for kitsch? Then Tierra Santa (  0800-444-3467; www.tierrasanta-bsas.com.ar; Av Costanera R Obligado 5790; adult\/child 3-11yr AR$60\/30;   9am-9pm Fri, noon-10pm Sat, Sun & holidays Apr-Nov, 4pm- midnight Fri-Sun & holidays Dec-Mar) might be exactly what you need.\n\nEnter this religious and wonderfully tacky theme park, roughly based on Jerusalem, and head straight to the manger scene. Here, colorful lights and minimally animatronic figures swoon over baby Jesus. Better yet is the creation of the world, which features real rushing waters and life-size fake animals. From here it's a 30-second walk to witness the 12m-tall animatronic Jesus rise from the Calvary mound, open his eyes and finally turn his palms toward the emotional devoted below. Miss the show? Don't fret: another resurrection is just around the corner.\n\nThe park isn't just for Christians \u2013 there are reproductions of the Wailing Wall, along with a synagogue and a mosque. So regardless of religious affiliation, enjoy nibbling on a shawarma or take in an Arabic dancing show. It's a spectacle you won't find anyplace else on earth \u2013 especially not in Jerusalem.\n\nPalermo\n\n  Top Sights\n\n1Museo de Arte Latinoamericano de Buenos Aires H3\n\n2Parque 3 de Febrero F2\n\n  Sights\n\n3Centro Isl\u00e1mico Rey Fahd E2\n\n4Jard\u00edn Bot\u00e1nico Carlos Thays F4\n\n5Jard\u00edn Japon\u00e9s G3\n\n6Jard\u00edn Zool\u00f3gico E4\n\n7Museo Casa de Ricardo Rojas H6\n\n8Museo de Arte Popular Jos\u00e9 Hern\u00e1ndez H4\n\n9Museo de Artes Pl\u00e1sticas Eduardo S\u00edvori F2\n\n10Museo Evita F4\n\n11Museo Nacional de Arte Decorativo H4\n\n12Museo Xul Solar G7\n\n13Planetario Galileo Galilei G2\n\n14Rosedal F2\n\n  Eating\n\n15Almac\u00e9n Oui Oui C3\n\n16Arevalito B4\n\n17Astor B3\n\n18Azema C4\n\n19Bio D4\n\n20Crizia D5\n\n21Don Julio E5\n\n22Green Bamboo C4\n\n23Hern\u00e1n Gipponi Restaurant C4\n\n24Il Ballo de Mattone B4\n\n25Il Ballo del Mattone C4\n\nIl Matterello (see 95)\n\n26La Cabrera C5\n\n27La Cabrera C6\n\n28Las Cholas D2\n\n29Las Pizarras E4\n\n30Malv\u00f3n G3\n\n31Miranda C4\n\nMuseo Evita Restaurante (see 10)\n\n32Natural Deli C1\n\n33Olsen B4\n\n34Oui Oui C3\n\n35Oviedo H6\n\n36Siamo nel Forno C4\n\n37Social la Lechuza D5\n\n38Sudestada C4\n\n39Unik D5\n\n  Drinking & Nightlife\n\n40AcaBar C4\n\n41Antares A7\n\n42Casa Coupage D4\n\n43Club Ar\u00e1oz F5\n\n44Congo C5\n\n45Crobar E2\n\n46El Carnal C5\n\n47Frank's Bar B4\n\n48Glam G7\n\n49Home Hotel B4\n\n50Kika C5\n\n51La Viruta D6\n\n52Magdalena's Party D5\n\n53Mundo Bizarro C6\n\n54Niceto Club B5\n\n55Post Street Bar D5\n\n56Salon Canning D6\n\n57Shanghai Dragon D7\n\n58Sugar B6\n\n59Van Koning D2\n\n60Verne F6\n\n  Entertainment\n\n61Campo Argentino de Polo D2\n\n62La Pe\u00f1a del Colorado F5\n\n63Los Cardones E5\n\n64Thelonious Bar F5\n\n65Tiempo de Gitanos C4\n\n  Shopping\n\n6628 Sport A7\n\n67Alto Palermo G5\n\n68Arte \u00c9tnico Argentino B7\n\n69Bolivia A7\n\n70Calma Chicha A6\n\n71Capital A7\n\n72Centrico E6\n\n73El Cid B6\n\n74Feria Plaza Serrano A6\n\n75Harapos Patagonia B7\n\n76Hermanos Estebecorena B4\n\n77Humawaca B7\n\n78Juana de Arco B6\n\n79La Mercer\u00eda B7\n\n80Libros del Pasaje D5\n\n81Lo de Joaquin Alberdi A6\n\n82Mercado de las Pulgas B4\n\n83Mishka B7\n\n84Nobrand B4\n\n85Panorama F4\n\n86Papelera Palermo C5\n\n87Paseo Alcorta H3\n\n88Patio del Liceo H6\n\n89Rapsodia A7\n\n90Sugar & Spice D4\n\n91Zival's D6\n\n  Sports & Activities\n\n92Anda Responsible Travel G7\n\n93Anuva Wines C2\n\n94Bar du March\u00e9 C4\n\n95Pain et Vin C5\n\n96Parque General Belgrano H2\n\n97Say Hueque D5\n\n98Vamos G6\n\n  Sleeping\n\n99248 Finisterra D2\n\n1005th Floor G4\n\n101Abode D5\n\n102BA Sohotel E5\n\n103Cabrera Garden B4\n\n104Caser\u00f3n Porte\u00f1o B3\n\n105Duque Hotel E5\n\n106Eco Pampa Hostel E5\n\n107Infinito Hotel F5\n\n108Livian Guesthouse E7\n\n109Magnolia Hotel E6\n\n110Mine Hotel D6\n\n111Miravida Soho D5\n\n112Palermitano D5\n\n113Palermo Viejo B&B D6\n\n114Reina Madre Hostel G7\n\n115Rendezvous Hotel B5\n\n116Rugantino Hotel D5\n\n117Vain Boutique Hotel E4\n\n# South Of Palermo\n\n###   Explore\n\n###   Sights\n\n###   Eating\n\n###   Drinking & Nightlife\n\n###   Entertainment\n\n###   Sports & Activities\n\n###   Map\nSouth of Palermo\n\nFor more detail of this area see Neigbourhood Map \u00bb\n\n###  Neighbourhood Top Five\n\n Communing with tango's most famous singer via his old recordings, news clippings and personal items at  Museo Casa Carlos Gardel, located in the very house he used to live.\n\n Visiting the impressive  Mercado de Abasto, a gorgeously remodelled shopping center.\n\n Wandering among old skeletons, taxidermy rooms and natural science exhibits at  Museo Argentino de Ciencias Naturales.\n\n Finding that hidden Peruvian, Korean or Jewish jewel of a  restaurant.\n\n Dancing it up at Monday night's drumming parties at  Ciudad Cultural Konex.\n\n###  Explore: South of Palermo\n\nBuenos Aires' easterly regions \u2013 south of Palermo and east of Congreso \u2013 are refreshingly local, blue-collar neighborhoods with occasional surprises for the tourist, such as some artsy galleries, a few renovated cafes and a bunch of shopping outlet stores.\n\nVilla Crespo is a good place to start. It's a good place for outlet shopping and has become home to several casual, tasty restaurants as well. A short walk to the south is Caballito, a calm and pleasant neighborhood with the large, circular Parque del Centenario. The main attraction here is the Museo Argentino de Ciencias Naturales, a good natural-science museum that's worth a peek for its musty taxidermy and cool skeleton room.\n\nEast of Villa Crespo are the Abasto and Once (pronounced 'ohn-seh') neighborhoods, both melting-pot destinations that have attracted sizable populations of Jews, Peruvians and Koreans \u2013 and their respective ethnic cuisines as well. The main attraction in Abasto is the Mercado de Abasto, one of the city's most attractive shopping malls. On a side street just east of the mall, look for a small statue of Carlos Gardel, the famous tango singer; four blocks northeast is the Museo Casa Carlos Gardel, a museum honoring him. Many alternative theaters can also be found in this area.\n\nSouth of Abasto is Once and its bustling train station, surrounded by hundreds of street vendors selling garments and cheap electronic devices. There's a colorful, almost third-world feel to this neighborhood \u2013 a welcome change in BA, though you should avoid this area late at night. South of Once is the bohemian neighborhood of Boedo, which has a few atmospheric cafes such as Las Violetas and Esquina Homero Manzi.\n\n###  Local Life\n\n\u00bb Hanging Out Local ladies take a break at Las Violetas (Click here), quite possibly the most beautiful traditional cafe in Buenos Aires.\n\n\u00bb Eating A lack of tourists at classic eateries like Caf\u00e9 Margot (Click here) will take you back in time.\n\n\u00bb Theater Artsy _porte\u00f1os_ head to one of the many alternative theater productions going on in Abasto (Click here).\n\n###  Getting There & Away\n\n\u00bb Bus Take bus 140 from the Microcentro to Villa Crespo, bus 26 to Once, bus 105 to Caballito, bus 126 to Boedo.\n\n\u00bb Subte L\u00edneas A, B and E are the fastest way to these neighborhoods.\n\n####  Lonely Planet's Top Tip\n\nConsider finding a place to stay in Villa Crespo, which is just south of Palermo and becoming more hip every day. New restaurants, outlet shops and guest houses continue to pop up here as rents become too expensive for many businesses in Palermo. Some accomm\u00adodations might even be located just as close or even closer to Plaza Serrano \u2013 the commercial and social heart of Palermo Viejo \u2013 than many places in Palermo Soho or Hollywood.\n\nParts of Once can be a bit sketchy at night, so tread carefully.\n\n  Best Places to Eat\n\n\u00bbSarkis\n\n\u00bbMalv\u00f3n\n\n\u00bbCaf\u00e9 Crespin\n\nFor reviews, Click here \u00bb\n\n  Best Places to Drink\n\n\u00bbLas Violetas\n\n\u00bb\n\n\u00bbCervecer\u00eda Cossab\n\nFor reviews, Click here \u00bb\n\n  Best Entertainment\n\n\u00bbCiudad Cultural Konex\n\n\u00bbEsquina Carlos Gardel\n\n\u00bbComplejo Tango\n\nFor reviews, Click here \u00bb\n\n South of Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Sports & Activities\n\n  SIGHTS\n\nMUSEO CASA CARLOS GARDEL MUSEUM\n\nMAP GOOGLE MAP\n\n(  4964-2071; Jean Jaur\u00e9s 735; admission AR$5, Wed free;   11am-6pm Mon & Wed-Fri, 10am-7pm Sat & Sun) Small but noteworthy is this tribute to tango's most famous voice. Located in Gardel's old house, this museum traces his partnership with Jos\u00e9 Razzano and displays old memorabilia like photos, records and news clippings. There isn't a whole lot to see, so it's best for real fans or just the curious; look for the cluster of colorfully painted buildings. Free tango classes offered Wednesday and Friday at 6pm, and Saturday at 3pm.\n\nMERCADO DE ABASTO NOTABLE BUILDING\n\nMAP GOOGLE MAP\n\n(  4959-3400; www.abasto-shopping.com.ar; Av Corrientes 3247;   10am-10pm;   L\u00ednea B Carlos Gardel) The historic Mercado de Abasto (1895) has been recycled by US-Hungarian financier George Soros into one of the most beautiful shopping centers in the city. The building, once a large vegetable market, received an architectural prize in 1937 for its Av Corrientes facade; at night the spotlighted and lofty arches are visible all the way from Av Pueyrred\u00f3n. It holds more than 200 stores, a large cinema, a large food court and a kosher McDonald's (the one upstairs).\n\nIt's great for families, with a good children's museum, video\/arcade games and even a small amusement park. The small Abasto neighborhood was once home to tango legend Carlos Gardel, and on the gentrified pedestrian street off Av Anchorena is a statue of the singer.\n\nMUSEO ARGENTINO DE CIENCIAS NATURALES MUSEUM\n\nMAP GOOGLE MAP\n\n(Natural Science Museum;   4982-6595; www.macn.secyt.gov.ar; Av \u00c1ngel Gallardo 490; admission AR$10;   2-7pm) Way over to the west, the oval Parque del Centenario is a large open space containing this excellent natural-science museum. On display are large collections of meteorites, rocks and minerals, seashells, insects and dinosaur skeleton replicas. Life-size models of a basking shark and ocean sunfish are impressive, and the taxidermy and skeleton rooms are especially good. Bring the kids; they can mingle with the hundreds of children who visit on school excursions.\n\nNearby is the Observatorio Astron\u00f3mico MAP GOOGLE MAP (  4863-3366; www.asaramas.com.ar; Patricias Argentinas 550). Call or check the website, as observation hours change depending on the season.\n\nWORTH A DETOUR\n\n###  FERIA DE MATADEROS\n\nIn the working-class barrio of Mataderos is this excellent folk market (  Mon-Fri 4342-9629, Sat 4687-5602; www.feriademataderos.com.ar; cnr Avs Lisandro de la Torre & de los Corrales;   11am-8pm Sun Apr\u2013mid-Dec, 6pm-midnight Sat late Jan\u2013mid-Mar). Merchants offer handmade crafts and regional cuisine like _locro_ (a corn and meat stew) and _humita_ (a savory corn and cheese mixture wrapped in husks). Folk singers, dancers and gauchos on horseback entertain, and there's a nearby gaucho museum (  4687-1949; Av de los Corrales 6436; admission AR$5;   noon-6:30pm Sun Mar-Dec). From downtown, take bus 155 (also marked 180) or 126; the market is up to an hour's ride away, but worth it \u2013 you can also take a taxi to and from Mataderos if you're pinched for time. Call ahead in between seasons to make sure it's open.\n\n South of Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Sports & Activities\n\n  EATING\n\nThese neighborhoods have yet to be discovered by the tourist masses, but things are changing. Rents in Palermo Viejo have skyrocketed over the years, driving some new businesses to nearby Villa Crespo. Meanwhile, Boedo has a few traditional places that are just starting to be visited by foreigners looking for something different. And Once is a good place to hunt for ethnic foods with Jewish, Peruvian or Korean flavors.\n\n  SARKIS MIDDLE EASTERN $\n\nMAP GOOGLE MAP\n\n(  4772-4911; Thames 1101; mains AR$55-90;   noon-3pm & 8pm-1am) The food is fabulous and well-priced at this long-\u00adstanding Middle Eastern restaurant \u2013 come with a group to sample many exotic dishes. Start with the hummus platter, _baquerones_ (marinated sardines), _keppe crudo_ (raw meat) or _parras rellenas_ (stuffed grape leaves), then follow with kebabs, couscous with lentils or lamb in yogurt sauce. Less busy at lunchtime; a long wait for dinner.\n\nCAF\u00c9 CRESPIN CAFE $\n\nMAP GOOGLE MAP\n\n(  4855-3771; Vera 699, cnr of Acevedo; mains AR$30-60, set brunches AR$140-225;   8am-8pm Tue-Fri, 9am-8pm Sat, noon-7pm Sun) Cute corner cafe in Villa Crespo. Stock up on pancakes, French toast and bagel sandwiches for breakfast, or go for the quesadillas, salmon salad or ham and cheese tostadas at lunchtime. Tasty brunches, and there are also good pastries and a bakery on the premises.\n\nPAN Y ARTE ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4957-6922; www.panyarte.com.ar; Av Boedo 878; mains AR$50-80;   8am-midnight Sun-Thu, to 1am Fri & Sat) There's a wonderful old-time atmosphere at this bohemian eatery, which features a hippie waitstaff and organic bakery. Food ranges from the same old boring stuff ( _milanesas,_ pastas and pizza) to more interesting choices like vegetarian _picadas_ (a plate of appetizers), stuffed squash and goat stew. There are also organic products like cheese and _mate_ to purchase, plus it's in a busy, interesting and nontouristy neighborhood.\n\nCAF\u00c9 MARGOT ARGENTINE $\n\nMAP GOOGLE MAP\n\n(  4957-0001; Av Boedo 857; mains AR$40-80;   8am-late) This classic cafe, one of the city's official _bares notables_ (notable bars), is an off-the-beaten-path spot where you can relax with a platter of _picadas_ (meat, cheese and olives) and a bottle of wine, or a frosty mug of artisan-crafted beer and a huge sandwich piled high with sliced turkey. The atmospheric main room is a bit snug; sidewalk tables are best for the claustrophobic.\n\n MALV\u00d3N CAFE $$\n\nMAP GOOGLE MAP\n\n(  4774-2563; www.malvonba.com.ar; Serrano 789; mains AR$50-130;   8am-8:30pm) Famous for its US-style weekend brunch \u2013 which features pancakes, French toast and eggs Benedict \u2013 Malv\u00f3n is an eatery with a wonderfully rustic yet upscale atmosphere. The gourmet sandwiches are tasty, but there are also great bagels, burgers, tapas and baked treats like scones, muffins and pecan pie. Expect a wait on the weekend. Also in Palermo (Lafinur 3275).\n\nBI WON KOREAN $$\n\nMAP GOOGLE MAP\n\n(  4372-1146; Jun\u00edn 548; mains AR$90-120;   noon-3pm & 7:30-11:30pm Mon-Fri, 7:30-11:30pm Sat) Korean food can't be beat at this simple restaurant. Go for the _bulgogi_ (grill the meat yourself at the table), _bibimbap_ (rice bowl with meat, veggies, egg and hot sauce) or _kim chee chigue_ (kimchi soup with pork \u2013 for adventurous, spice-loving tongues only!). And don't forget to say _kamsamnida_ (thank you) to your server at the end.\n\n South of Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Sports & Activities\n\n  DRINKING & NIGHTLIFE\n\n  LAS VIOLETAS CAFE\n\nMAP GOOGLE MAP\n\n(  4958-7387; www.lasvioloetas.com; Av Rivadavia 3899;   8am-2am) Dating back to 1884, this historic coffeehouse was renovated in 2001 into the gorgeous place it is today. Lovely stained-glass awnings, high ceilings and gilded details make this cafe possibly the most beautiful in the capital. Come for the luxurious afternoon tea and be sure to pick something up in the chocolate- pastry shop on the way out.\n\n878 BAR\n\nMAP GOOGLE MAP\n\n(  4773-1098; www.878bar.com.ar; Thames 878;   8pm-2am Sun-Thu, to 4am Fri & Sat) Hidden behind an unsigned door is this 'secret' bar \u2013 you have to ring the bell to get in, but it's hardly exclusive. Enter a wonderland of elegant low lounge furniture and red brick walls; for whiskey lovers there are over 80 kinds to try, but the cocktails are tasty too. If you're hungry, tapas are available (reserve for dinners).\n\nCERVECER\u00cdA COSSAB BAR\n\nMAP GOOGLE MAP\n\n(  2060-5023; Carlos Calvo 4199;   7pm-1am Wed-Thu, to 4am Fri, 9pm-4am Sat) Beer lovers unite and head down to bohemian Boedo and this dedicated beer bar \u2013 a unique find for BA. Over 50 tasty suds are served, including seven on tap. Delicious pizzas, cheese plates and sandwiches help you make a night of it \u2013 but for something out of the ordinary, try the _picada Patagonica_ with smoked wild boar and venison.\n\nLA CATEDRAL DANCE HALL\n\nMAP GOOGLE MAP\n\n(  15-5325-1630; www.lacatedralclub.com; Sarmiento 4006) If tango can be youthful, trendy and hip, this is where you'll find it. The grungy warehouse space is very casual, with funky art on the walls, thrift-store furniture and dim atmospheric lighting. It's more like a young bohemian nightclub than anything else, and there's no implied dress code \u2013 you'll see plenty of jeans on the dancers. Great for cheap alcohol; the best-known _milongas_ (tango dances) occur regularly on a Tuesday night.\n\nCLUB GRICEL DANCE HALL\n\nMAP GOOGLE MAP\n\n(  4957-7157; www.clubgriceltango.com.ar; La Rioja 1180) This old classic (far from the center; take a taxi) often has big crowds, especially on Monday. It attracts an older, well-dressed clientele \u2013 along with plenty of tourists. There's a wonderful springy dance floor and occasionally live orchestras.\n\nAMERIKA CLUB\n\nMAP GOOGLE MAP\n\n(  4865-4416; www.ameri-k.com.ar; Gasc\u00f3n 1040;   Fri-Sun) BA's largest and feistiest gay nightclub, long-running Amerika attracts all kinds of folks \u2013 but Saturdays are especially popular with gay guys. There are two music floors, one electronica and one Latina, plus _canilla libre_ (all-you-can-drink) on Fridays and Saturdays. Large video screens, stripper shows, four bars and a wild dark room keep things interesting.\n\n###  AVANT-GARDE THEATER\n\nGet off the beaten play path and go for something out of the ordinary \u2013 there's plenty of choice in this creative city for unique and worthwhile theater.\n\n\u00bb Actors Studio Teatro MAP GOOGLE MAP (  4983-9883; www.actors-studio.org; Av D\u00edaz V\u00e9lez 3842) Offers new interpretations of old classics, along with cutting-edge productions in its 120-seat theater. Also has occasional acting classes.\n\n\u00bb El Camar\u00edn de las Musas MAP GOOGLE MAP (  4862-0655; www.elcamarindelasmusas.com.ar; Mario Bravo 960) Offers contemporary dance, plays and theater. There are also workshops and classes available, and a trendy restaurant-cafe provides affordable snacks.\n\n\u00bb El Cubo MAP GOOGLE MAP (  4963-2568; www.cuboabasto.com.ar; Pasaje Zelaya 3053) A hip small Abasto space, it hosts gutsy theater pieces and offbeat performances such as queer musicals.\n\n\u00bb Espacio Callej\u00f3n MAP GOOGLE MAP (  4862-1167; www.espaciocallejon.blogspot.com; Humahuaca 3759) A small independent venue that showcases edgy new theater, music and dance, and offers a few classes (including 'clown' acting).\n\n South of Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Sports & Activities\n\n  ENTERTAINMENT\n\nCIUDAD CULTURAL KONEX CULTURAL CENTER\n\nMAP GOOGLE MAP\n\n(  4864-3200; www.ciudadculturalkonex.org; Av Sarmiento 3131;   erratic hours, call ahead) Cutting-edge cultural center offering multidisciplinary performances that often fuse art, culture and technology. Famous for its amazing Monday night percussion shows that attract young party-goers.\n\nESQUINA CARLOS GARDEL TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4867-6363; www.esquinacarlosgardel.com.ar; Carlos Gardel 3200; show from US$96, dinner & show from US$140) One of the fanciest tango shows in town plays at this impressive 430-seat theater, an old cantina right next to the Mercado de Abasto (Click here). The Abasto neighborhood was once Carlos Gardel's old stomping ground, and he even hung out at this locale. The memorable show starts with a good film about the area, then goes on to highlight top-notch musicians and performers.\n\nCOMPLEJO TANGO TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4941-1119; www.complejotango.com.ar; Av Belgrano 2608; show from US$85, dinner & show from US$120) For those who wish to not only watch tango but also experience it, there's this classy venue in Balvanera. Should you choose to accept it, your first hour here is a free beginning tango lesson. Follow it up with a tasty dinner, then an excellent tango show \u2013 beware, however, as the performers go around towards the end, picking out audience members to dance with them (usually badly).\n\nESQUINA HOMERO MANZI TANGO SHOW\n\nMAP GOOGLE MAP\n\n(  4957-8488; www.esquinahomeromanzi.com.ar; Av San Juan 3601; show AR$330, show & dinner from AR$550) This tango venue, in a remodeled old-time cafe, is located right on the historic intersection of San Juan and Boedo and was named after one of Argentina's most famous tango lyricists. It has the capacity for 300 spectators and offers a decent show that's a mix of glitzy high-kicks and more traditional _milonga_ -type dancing. Be warned: at some tables the waiters keep passing in front of you, interrupting views.\n\n South of Palermo\n\nSights | Eating | Drinking & Nightlife | Entertainment | Sports & Activities\n\n  SPORTS & ACTIVITIES\n\nACATRAZ BOWLING\n\n(  4982-4818; www.acatrazclub.com.ar; Av Rivadavia 3636;   4pm-4am Sun-Mon, to 6am Fri & Sat) Unusual restaurant-bar-bowling alley-billiards hall sorta destination. You can do it all here; a great place to come with a group of friends. There are two bowling alleys, various pool tables, sports on TVs, plenty of tables to eat at and bars to drink at. Spread out over several floors, this place is huge and takes entertainment to a new level.\n\nSouth of Palermo\n\n  Sights\n\n1Carlos Gardel Statue G2\n\n2Casa Brandon D2\n\n3Mercado de Abasto F2\n\n4Museo Argentino de Ciencias Naturales C3\n\n5Museo Casa Carlos Gardel G2\n\nObservatorio Astron\u00f3mico (see 4)\n\n  Eating\n\n6Bi Won H2\n\n7Caf\u00e9 Crespin C2\n\n8Caf\u00e9 Margot F5\n\n9Malv\u00f3n C1\n\n10Pan Y Arte F5\n\n11Sarkis D1\n\n  Drinking & Nightlife\n\n12 C1\n\n13Amerika E2\n\n14Cervecer\u00eda Cossab E5\n\n15La Catedral E3\n\n16Las Violetas E3\n\n  Entertainment\n\n17Actors Studio Teatro E3\n\n18Ciudad Cultural Konex F3\n\n19Club Gricel G5\n\n20Complejo Tango G4\n\n21El Camar\u00edn de las Musas F2\n\n22El Cubo G2\n\n23Espacio Callej\u00f3n F2\n\n24Esquina Carlos Gardel G2\n\n25Esquina Homero Manzi F5\n\n  Sports & Activities\n\n26DWS E1\n\n  Sleeping\n\n27Chill House Hostel F2\n\n28Pop Hotel C1\n\n29Querido B&B C1\n\n30Rac\u00f3 de Buenos Aires E4\n\nDay Trips from Buenos Aires\n\n######  Tigre & the Delta\n\nTake laid-back boat rides along the peaceful backwaters of the R\u00edo de la Plata. A nearby outdoor market and interesting museums are pluses.\n\n######  San Antonio de Areco\n\nExplore this serene village and its historic buildings, and perhaps visit a nearby _estancia_ (cattle ranch). If you're lucky, you might spot a gaucho or two.\n\n######  Colonia\n\nLocated across the R\u00edo de la Plata in Uruguay, this pleasant little colonial gem is lined with cobbled streets and charming old buildings.\n\nTigre & the Delta\n\n######  Explore\n\nOnly an hour's drive from BA, tranquil Tigre and its huge river delta make a popular weekend getaway for cement-weary _porte\u00f1os._ And while Tigre itself is a pleasant enough riverside town, it's really the swampy waterways that everyone is after. Latte-colored waters \u2013 rich with iron from the jungle streams flowing from inland Argentina \u2013 alongside reedy shores are far from any stereotypical paradise, but there are a few surprises here. Boat rides into the delta offer peeks at local houses and colonial mansions, or you can just get off and explore some nature trails. All along the shores are signs of water-related activity, from sailing, kayaking and canoeing to sculling and even wakeboarding.\n\n######  The Best...\n\n\u00bb Sight Puerto de Frutos (Click here)\n\n\u00bb Place to Eat Boulevard Saenz Pe\u00f1a (Click here)\n\n\u00bb Place to Drink Maria Luj\u00e1n (Click here)\n\n######  Top Tip\n\nKayak or canoe tours (try www.eldoradokayak.com and www.selknamcanoas.com.ar) are a good way to explore the peaceful back waterways of the delta.\n\n######  Getting There & Away\n\n\u00bb Train From Estaci\u00f3n Retiro you can take a train straight to Tigre (one hour). The best way to reach Tigre, however, is via the Tren de la Costa (tickets AR$40) \u2013 a pleasant light-rail train with attractive stations and views. This train line starts in the suburb of Olivos; to get there, take a train from Retiro station and get off at the Mitre station, then cross the bridge to the Tren de la Costa. Buses 59, 60 and 152 also go to the Tren de la Costa.\n\n\u00bb Bus 60 (most 60 buses go to Tigre, but double check with the driver). The trip takes 1\u00bd hours.\n\n\u00bb Car Take the Panamericana Hwy north to _ramal_ (branch) Tigre.\n\n\u00bb Boat Sturla Viajes (  in BA 4314-8555, in Tigre 4731-1300; www.sturlaviajes.com.ar; Estaci\u00f3n Fluvial, local 10 in Tigre, Grierson 400 in BA) has a commuter boat (AR$35) to Tigre that leaves from Grierson 400 in Puerto Madero, but it's only at 6.10pm from Monday to Friday. However, you can take its Tigre tour directly from Puerto Madero, which includes boat transport and a trip around the Delta (AR$260 round-trip).\n\n######  Need to Know\n\n\u00bb Area Code 011\n\n\u00bb Location 35km northwest of Buenos Aires\n\n\u00bb Tourist Office (  4512-4497; www.vivitigre.gov.ar; Mitre 305;   8am-6pm) Located behind McDonald's; will help you sort out the complex delta region\n\n##    Sights\n\nThe waterways of the delta offer a glimpse into how the locals live here, along peaceful canals with boats as their only means of transportation. Frequent commuter launches (AR$45 to AR$68) depart from Estaci\u00f3n Fluvial (situated behind the tourist office) for various destinations in the delta. A popular destination is the neighborhood of Tres Bocas, a half-hour boat ride from Tigre, where you can take residential walks on slender paths connected by bridges over narrow channels. There are several restaurants and accommodation options here. The Rama Negra area has a quieter and more natural setting with fewer services but is located an hour's boat ride away.\n\nSeveral companies offer inexpensive boat tours (AR$60 to AR$120, one to two hours), but commuter launches give you flexibility if you want to go for a stroll or stop for lunch at one of the delta's restaurants.\n\nPUERTO DE FRUTOS MARKET\n\n(Sarmiento 160;   10am-6pm) At this popular waterside market you'll find furniture, housewares, wicker baskets, dried flowers, plants and a whole lot of kitsch. Friday to Sunday is best, when a large crafts fair sets up; there are several restaurants too. There's a tourist office at the entrance to the port.\n\nMUSEO DE ARTE TIGRE MUSEUM\n\n(  4512-4093; Paseo Victorica 972; admission AR$15;   9am-7pm Wed-Fri, noon-7pm Sat & Sun) Located in an old social club that dates from 1912, this beautiful art museum showcases famous Argentine artists from the 19th and 20th centuries, plus rotating exhibits. The building itself is beautiful enough to warrant a visit.\n\nMUSEO NAVAL MUSEUM\n\n(Naval Museum;   4749-0608; Paseo Victorica 602; admission AR$10;   8:30am-5:30pm Mon-Fri, 10:30am-6:30pm Sat & Sun) This worthwhile museum traces the history of the Argentine navy with an eclectic mix of historical photos, model boats and airplanes, artillery displays and pickled sea critters.\n\nMUSEO DEL MATE MUSEUM\n\n(  4506-9594; www.elmuseodelmate.com; Lavalle 289; admission AR$15;   11am-6pm Wed-Sun) Celebrating everything connected to Argentina's national drink, this museum boasts over 2000 pieces; check out the _mates_ for blind people. You can also watch a short video and \u2013 in the pleasant garden out back \u2013 sample the concoction itself.\n\nPARQUE DE LA COSTA AMUSEMENT PARK\n\n(  4002-6000; www.parquedelacosta.com.ar; General B Mitre 2; admission Tue & Wed from AR$52, Thu-Sun from AR$97) Tigre's amusement park offers roller coasters, games and everything else that makes a theme park enjoyable. Opening hours vary widely throughout the year, so check the website.\n\n###  VISITING IGUAZ\u00da FALLS\n\nMany visitors to Buenos Aires tour the city and then take a side trip to one of the most spectacular sites in South America: Iguaz\u00fa Falls. If you have an extra couple of days it's definitely worth the time and money. Just remember that it's much warmer and more tropical there than in BA, and that in January and February the heat and humidity can be overwhelming.\n\nIguaz\u00fa Falls straddles the Argentina\u2013Brazil border and some of the most stunning views are from the Brazilian side. As for Brazilian visas, if you are from the US, Canada or Australia, you officially need a visa to enter Brazil. Western Europeans do not. Brazilian visas aren't cheap, and getting one may take some time, so plan ahead. Some travelers without a visa have day-tripped across the border by taking the public bus to the Brazilian city of Foz do Igua\u00e7u, but this may not always be possible.\n\nFlying is the best way to go if you're short on time, but many travelers go by bus (one-way ticket AR$765, 18 hours). Bus and air packages \u2013 often including round-trip fare, transfers, guided tours and accommodations (but not visa or park admission) \u2013 are popular and easily available in BA at agencies like Tangol (Click here) or Say Hueque (Click here).\n\nDuring July, on holiday weekends and during Semana Santa (Easter week) you should plan way ahead or be prepared to pay premium prices.\n\nFor general information on Iguaz\u00fa, see the Casa de Misiones tourist office (  4317-3722; www.misiones.gov.ar; Av Santa Fe 989).\n\n##    Eating\n\nTigre's cuisine is not cutting edge, but it can be atmospheric \u2013 stroll Paseo Victorica, the city's pleasant riverside avenue, for the nicest options. Ask the tourist office about the various restaurants in the delta.\n\nBOULEVARD SAENZ PE\u00d1A INTERNATIONAL $\n\n(  5197-4776; Blvd Saenz Pe\u00f1a 1400; breakfast mains AR$35-75, dinner mains AR$60-70;   10:30am-6pm Wed-Sat, 8:30pm to close Thu-Sat) This creative eatery offers delicious dishes (granola and yogurt for breakfast, gourmet sandwiches and salads for lunch, luscious pastries for teatime) and there's a cute patio for warm days. Dinner is by reservation only.\n\nUN LUGAR PARRILLA $\n\n(  4749-0698; Lavalle 369; mains AR$50-85;   noon-3pm & 8:30-11pm Tue-Sun) This _parrilla_ (steak house) has comfortable indoor seating, but on warm days head to the sidewalk patio out front. Homemade pastas also available.\n\nMARIA LUJ\u00c1N ARGENTINE $$\n\n(  4731-9613; Paseo Victorica 611; mains AR$80-150;   8:30am-midnight) A good choice for an upscale meal of typical Argentine fare, this beautiful, large restaurant also has a great patio boasting full river views.\n\n##    Sleeping\n\nTigre's huge delta region is dotted with dozens of accommodation possibilities, including camping, B&Bs, cabanas and beach resorts. Since places are relatively hard to reach (guests generally arrive by boat), the majority provide meal services, which are not always included in the price \u2013 ask beforehand. The Tigre tourist office (Click here) has photos and information on all these places, and many are listed on its website.\n\nThe following places are in the city of Tigre itself. Book ahead on weekends and holidays, when prices can rise significantly.\n\nPOSADA DE 1860 HOSTEL $\n\n(Tigre Hostel;   4749-4034; www.tigrehostel.com.ar; Av Libertador 190; dm US$12, r from US$70;    ) This odd hostel is in two buildings. One is the original mansion with en-suite private rooms and large garden, while the second is a mazelike building with little atmosphere, and with dorms and private rooms that all share bathrooms.\n\nHOTEL VILLA VICTORIA GUESTHOUSE $$\n\n(  4731-2281; www.hotelvillavictoria.com; Liniers 566; r Sun-Fri from AR$557, Sat from AR$702;     ) Run by an Argentine-Swedish family, this boutique hotel is more like a fancy guesthouse. Only six (simple yet elegant) rooms are available, and there's a clay tennis court and a pool in the large grassy garden. Swedish, French and English are spoken.\n\nCASONA LA RUCHI GUESTHOUSE $$\n\n(  4749-2499; www.casonalaruchi.com.ar; Lavalle 557; r AR$550;    ) This family-run guesthouse is in an old 1893 mansion. Most of the four romantic bedrooms have balconies; all have shared bathrooms with original tiled floors. There's a pool and large garden out back.\n\nSan Antonio de Areco\n\n######  Explore\n\nNestled in lush farmlands, Areco is a pretty town that attracts day-tripping _porte\u00f1os_ who come for peaceful atmosphere and picturesque colonial streets. The town dates from the early 18th century and preserves a great deal of _criollo_ (creole) and gaucho traditions, especially among its artisans, who produce very fine silverwork and saddlery. Areco's compact center and quiet streets are very walkable; around the Plaza Ruiz de Arellano are several historic buildings.\n\n######  The Best...\n\n\u00bb Sight Museo Gauchesco Ricardo G\u00fciraldes (Click here)\n\n\u00bb Place to Eat Almac\u00e9n Ramos Generales (Click here)\n\n\u00bb Place to Drink Boliche de Bessonart (Click here)\n\n######  Top Tip\n\nIf you're here in early to mid-November, don't miss D\u00eda de la Tradici\u00f3n, when the town puts on the country's biggest gaucho celebration. Call the tourist office for exact dates.\n\n######  Getting There & Away\n\n\u00bb Bus Frequent buses from Retiro bus station drop you five blocks from the center of town; travel time is two hours.\n\n\u00bb Car Take RN8 west to _ramal_ (branch) Pilar.\n\n######  Need to Know\n\n\u00bb Area Code 02326\n\n\u00bb Location 115km northwest of Buenos Aires\n\n\u00bb Tourist Office (  453165; www.sanantoniodeareco.tur.ar; cnr E Zerboni & Ruiz de Arellano;   10am-7pm Mon-Fri, 8am-8pm Sat & Sun) Located in a white, stand-alone building in the park.\n\nSan Antonio de Areco\n\n  Sights\n\n1 Museo Gauchesco Ricardo G\u00fciraldes A1\n\n2 Museo Las Lilas B3\n\n3 Museo y Taller Draghi B2\n\n  Eating\n\n4 Almac\u00e9n Ramos Generales C2\n\n5 Boliche de Bessonart C2\n\n6 Puesto La Lechuza B1\n\n  Sleeping\n\n7 Areco Hostel B2\n\n8 Paradores Draghi B2\n\n##    Sights\n\nNote that early in the week some museums are closed.\n\nMUSEO GAUCHESCO RICARDO G\u00dcIRALDES MUSEUM\n\n MAP GOOGLE MAP\n\n(cnr R G\u00fciraldes & Sosa;   11am-6pm Wed-Mon) F This sprawling museum in Parque Criollo dates to 1938 and includes an old flour mill, a re-created _pulper\u00eda_ (tavern), a colonial-style chapel and a 20th-century reproduction of an 18th-century _casco_ (ranch house). Displays include horse gear, gauchesco artwork and rooms dedicated to Ricardo G\u00fciraldes, author of the novel _Don Segundo Sombra_.\n\nMUSEO Y TALLER DRAGHI MUSEUM\n\n MAP GOOGLE MAP\n\n(www.draghiplaterosorfebres.com; Lavalle 387; admission AR$25;   9am-1pm & 4-7pm Mon-Sat, 10am-1pm Sun) This small museum, attached to the silversmith workshop of the locally renowned Draghi family, highlights an exceptional collection of silver _facones_ (gaucho knives), beautiful horse gear and intricate _mate_ paraphernalia.\n\nMUSEO LAS LILAS MUSEUM\n\n MAP GOOGLE MAP\n\n(www.museolaslilas.org; Moreno 279; admission AR$50;   10am-8pm Thu-Sun mid-Sep\u2013 mid-Mar, to 6pm rest of year) Florencio Molina Campos is to Argentines what Norman Rockwell is to Americans \u2013 a folk artist whose themes are based on comical caricatures. This pretty courtyard museum displays his famous works.\n\n##    Eating\n\nPUESTO LA LECHUZA PARRILLA $\n\n MAP GOOGLE MAP\n\n(  470136; Victorino Althaparro 423; mains AR$45-75;   noon-3pm & 8pm-midnight Sat, noon-3pm Sun) Best on a warm day, when you can enjoy a lunch of _empanadas_ or barbecued beef under the trees near the river. Live guitar music on Saturday night; open weekends only.\n\nBOLICHE DE BESSONART PICADAS $\n\n MAP GOOGLE MAP\n\n(cnr Zapiola & Segundo Sombra; picadas AR$10-65;   11am-late Tue-Sun) Its shelves filled with dusty bottles and and old gaucho photos, this weatherbeaten corner bar draws in locals for _picadas_ (snack plates of meat, cheese and olives).\n\nALMAC\u00c9N RAMOS GENERALES ARGENTINE $$\n\n MAP GOOGLE MAP\n\n(www.ramosgeneralesareco.com.ar; Zapiola 143; mains AR$65-140;   noon-3pm & 8pm- midnight) Come to this traditional, local mainstay if you want an old-time atmosphere in which to enjoy good fish, meat or pasta dishes.\n\n##    Sleeping\n\nARECO HOSTEL HOSTEL $\n\n MAP GOOGLE MAP\n\n(  453120; www.arecohostel.com.ar; Arellano 121; dm AR$120, d AR$280, tr AR$420) In an atmospheric old building facing the central square, this hostel offers one spacious four-bed dorm with working fireplace, a pair of larger dorms and a lone private room up front. The clean, tiled guest kitchen, narrow but grassy backyard and friendly management add to the hostel's appeal.\n\n  PARADORES DRAGHI GUESTHOUSE $$\n\n MAP GOOGLE MAP\n\n(  455583; www.paradoresdraghi.com.ar; Matheu 380; s Sun-Thu only AR$450, d AR$580;     ) Large, comfortable rooms (two with kitchenette) are available at this tranquil place. There's a grassy garden with a beautiful pool, a greenhouse breakfast room and two patios in which to take a relaxing break.\n\n###  LIFE IN THE COUNTRYSIDE\n\nWant to get away from it all but still have some affordable fun? Check into El Galope Hostel (  99-105985; www.elgalope.com.uy; Km 114.5, Ruta 1; dm US$25, d with shared\/private bathroom US$70\/90), a farm about 50 minutes by bus outside Colonia. You can take nature walks, go horseback riding and sweat in the sauna \u2013 all in the peaceful Uruguayan countryside. Its owners, M\u00f3nica and Miguel, are friendly and helpful, and are experienced international travelers who speak English, Spanish, French and German.\n\nColonia\n\n######  Explore\n\nColonia (officially Colonia del Sacramento) is a picturesque Uruguayan town whose Barrio Hist\u00f3rico neighborhood is a Unesco World Heritage Site. Pretty rows of sycamores offer protection from the summer heat, and the R\u00edo de la Plata provides a venue for spectacular sunsets.\n\nPicturesque spots for wandering in Barrio Hist\u00f3rico include the narrow, roughly cobbled Calle de los Suspiros (Street of Sighs), lined with tile-and-stucco colonial houses, the Paseo de San Gabriel, which follows the western riverfront, the Puerto Viejo (Old Port) and the historic center's two main squares: vast Plaza Mayor 25 de Mayo and shady Plaza de Armas (the latter also known as Plaza Manuel Lobo).\n\n######  The Best...\n\n\u00bb Sight Faro (Click here)\n\n\u00bb Place to Eat Buen Suspiro (Click here)\n\n\u00bb Place to Drink Barbot (Click here)\n\n######  Top Tips\n\nTo avoid the crowds and more expensive accommodations prices, consider visiting Colonia midweek. If you want US dollars, use ATMs in Uruguay \u2013 those in Argentina do not give them out.\n\n######  Getting There & Away\n\n\u00bb Ferry Buquebus (www.buquebus.com), Colonia Express (www.coloniaexpress.com) and Seacat (www.seacatcolonia.com) have many daily ferries between Buenos Aires and Colonia. Fast ferries take an hour; slow ferries take three hours. Immigration for both countries is handled at the port before boarding.\n\n######  Need to Know\n\n\u00bb Area Code 4522\n\n\u00bb Location 50km east of Buenos Aires by ferry\n\n\u00bb Tourist Offices The BIT Welcome Center (  4522-1072; www.bitcolonia.com; Odriozola 434;   10am-7pm Dec-Apr, 9am-6pm May-Nov) is across from the port. There's a tourist office in Barrio Hist\u00f3rico (  4522-8506; www.coloniaturismo.com; Manuel de Lobos 224;   9am-6pm) and at the bus terminal (  4522-8506; www.coloniaturismo.com; Av Roosevelt;   9am-8pm).\n\nColonia\n\n  Sights\n\n1 Archivo Regional B3\n\n2 Casa Nacarello B4\n\n3 Faro B4\n\n4 Iglesia Matriz B3\n\n5 Museo del Azulejo A3\n\n6 Museo Espa\u00f1ol B2\n\n7 Museo Ind\u00edgena B2\n\n8 Museo Municipal B3\n\n9 Museo Portugu\u00e9s B4\n\n10 Port\u00f3n de Campo C4\n\n11Teatro Basti\u00f3n del Carmen C2\n\n  Eating\n\n12 Buen Suspiro B4\n\n13 La Bodeguita B3\n\n14 Lentas Maravillas A2\n\n  Drinking & Nightlife\n\n15 Barbot D3\n\n  Sleeping\n\n16 El Capullo C3\n\n17 El Viajero Hostel D3\n\n18 Posada del \u00c1ngel D4\n\n##    Sights\n\nA single UR$50 ticket covers admission to Colonia's eight historical museums. All keep the same hours, but opening days vary.\n\nPORT\u00d3N DE CAMPO GATE\n\n MAP GOOGLE MAP\n\n(Manuel de Lobos) The most dramatic way to enter Barrio Hist\u00f3rico is via the reconstructed 1745 city gate. From here, a thick fortified wall runs south along the Paseo de San Miguel to the river, its grassy slopes popular with sunbathers.\n\nIGLESIA MATRIZ CHURCH\n\n MAP GOOGLE MAP\n\n(Plaza de Armas) Uruguay's oldest church \u2013 begun by the Portuguese in 1680, then completely rebuilt twice under Spanish rule \u2013 is the centerpiece of pretty Plaza de Armas. The plaza also holds the foundations of a house dating from Portuguese times.\n\nFARO LIGHTHOUSE\n\n MAP GOOGLE MAP\n\n(admission UR$20;   11am-sunset) One of the town's most prominent landmarks, Colonia's 19th-century lighthouse provides an excellent view of the old town and the R\u00edo de la Plata. It stands within the ruins of the 17th-century Convento de San Francisco, just off the southwestern corner of Plaza Mayor 25 de Mayo.\n\nTEATRO BASTI\u00d3N DEL CARMEN THEATER, GALLERY\n\nMAP GOOGLE MAP\n\n(Rivadavia 223;   10am-10pm) F Incorporating part of the city's ancient fortifications, this theater and gallery complex hosts rotating art exhibits and periodic concerts.\n\nMUSEO PORTUGU\u00c9S MUSEUM\n\n MAP GOOGLE MAP\n\n(Plaza Mayor 25 de Mayo 180;   closed Wed & Fri) In this beautiful old house you'll find Portuguese relics including porcelain, furniture, maps, Manuel Lobo's family tree and the old stone shield that once adorned the Port\u00f3n de Campo.\n\nMUSEO MUNICIPAL MUSEUM\n\n MAP GOOGLE MAP\n\n(Plaza Mayor 25 de Mayo 77;   closed Tue & Thu) Houses an eclectic collection of treasures including a whale skeleton, an enormous rudder from a shipwreck, historical timelines and a scale model of Colonia (c 1762).\n\nARCHIVO REGIONAL MUSEUM\n\n MAP GOOGLE MAP\n\n(Misiones de los Tapes 115;   closed Sat & Sun) On the northwest edge of the plaza, Archivo Regional contains historical documents along with pottery and glass excavated from the 18th-century Casa de los Gobernadores nearby.\n\nCASA NACARELLO MUSEUM\n\n MAP GOOGLE MAP\n\n(Plaza Mayor 25 de Mayo 67;   closed Tue & Fri) One of the prettiest colonial homes in town, with period furniture, thick whitewashed walls, wavy glass and original lintels (duck if you're tall!).\n\nMUSEO DEL AZULEJO MUSEUM\n\n MAP GOOGLE MAP\n\n(cnr Misiones de los Tapes & Paseo de San Gabriel;   closed Thu & Fri) This dinky 17th-century stone house has a sampling of French, Catalan and Neapolitan tilework.\n\nMUSEO IND\u00cdGENA MUSEUM\n\n MAP GOOGLE MAP\n\n(Comercio s\/n;   closed Mon & Thu) Houses Roberto Banchero's personal collection of Charr\u00faa stone tools, exhibits on indigenous history, and an amusing map upstairs showing how many European countries could fit inside Uruguay's borders (it's at least six!).\n\nMUSEO ESPA\u00d1OL MUSEUM\n\n MAP GOOGLE MAP\n\n(San Jos\u00e9 164;   closed Tue & Wed) This recently reopened museum has a varied collection of Spanish artifacts, including colonial pottery, engravings, clothing and maps.\n\n###  HISTORIC ESTANCIAS\n\nOne of the best ways to enjoy the wide-open spaces of Argentina is to visit an _estancia_ (cattle ranch). The late-19th-century belle \u00e9poque saw wealthy landowning families build up their country ranches with lavish, often fanciful homes, which they used as country retreats.\n\nToday these establishments cater to tourists with _d\u00edas de campo \u2013_ day tours that include large _asado_ (barbecue) lunches, horseback rides, folk shows and, often, swimming facilities. Most also have overnight stays, which offer a longer glimpse into Argentina's history on the pampas.\n\nEl Omb\u00fa (  in BA 4737-0436; www.estanciaelombu.com) Just outside San Antonio de Areco, this working _estancia_ offers nine rooms and the opportunity to watch gauchos do their stuff.\n\nJuan Ger\u00f3nimo (  02221-481414; www.juangeronimo.com.ar) There's excellent horseback riding and bird-watching at this working cattle farm, located within a Unesco world biosphere reserve about two hours from Buenos Aires.\n\nLa Candelaria (  02227-494132; www.estanciacandelaria.com) Located about 1\u00bd hours from BA. Special for its castle and manicured grounds designed by Charles Thays, who did many of BA's public parks. Polo matches often held here.\n\nLos dos Hermanos (  in BA 4723-2880; www.estancialosdoshermanos.com) Just about an hour outside BA, this is a good place to learn how to horseback ride. Friendly, with good food.\n\nLa Margarita (  in BA 4951-0638; www.estancialamargarita.com) S Located at an old _estancia_ about 100 miles southwest of BA. Offers a self-catering option, which makes your stay more self-sufficient and affordable.\n\nLa Oriental (  02364-15-640866; www.estancia-laoriental.com) More authentic than luxurious is this lovely _estancia_ three hours from Buenos Aires. Activities include fishing or windsurfing in a nearby lagoon.\n\nPuesto Viejo (  in BA 5279-6893; www.puestoviejoestancia.com.ar) Not far from Ezeiza airport is this boutique _estancia_ where you can learn to play polo, ride a bicycle in the coutryside or just hang by the luxurious pool.\n\nSan Antonio's tourist office has more information on _estancias,_ as does Buenos Aires' Secretar\u00eda de Turismo de la Naci\u00f3n (  4312-2232; www.turismo.gov.ar; Av Santa Fe 883;   9am-7pm Mon-Fri).\n\n##    Eating & Drinking\n\nBUEN SUSPIRO PICADAS $$\n\n MAP GOOGLE MAP\n\n(  4522-6160; www.buensuspiro.com; Calle de los Suspiros 90; picadas from UR$205;   11am-midnight) Duck under the wooden beams into this cozy spot specializing in _picadas_. Sample local wines by the bottle or glass, accompanied by spinach and leek tarts, ricotta-and-sesame balls, local cheese and sausage, and more. Reserve ahead for a fireside table in winter, or while away a summer afternoon on the intimate back patio.\n\nLENTAS MARAVILLAS INTERNATIONAL $$\n\n MAP GOOGLE MAP\n\n(Santa Rita 61; sandwiches & salads UR$280-300;   2-8:30pm Thu-Tue) Cozy as a friend's home, this is a dreamy spot to kick back with tea and cookies or a glass of wine and a sandwich between meals. Flip through an art book from owner Maggie Molnar's personal library and enjoy the river views, either from the upstairs fireplace room or the chairs on the grassy lawn below.\n\nLA BODEGUITA PIZZA $$\n\n MAP GOOGLE MAP\n\n(www.labodeguita.net; Comercio 167; mini pizzas UR$115, dishes UR$220-390;   8:30pm-midnight year-round, plus 12:30-3:30pm Sat & Sun Apr-Nov) Nab a table out back on the sunny two-level deck and soak up the sweeping river views while drinking sangria (UR$200 per liter) or munching on La Bodeguita's trademark mini pizzas, served on a cutting board.\n\nBARBOT BREWPUB\n\n MAP GOOGLE MAP\n\n(  4522-7268;  www.facebook.com\/barbot cerveceria; Washington Barbot 160;   7pm-late Wed-Sun) A welcome addition to Colonia's drinking scene, this upscale brewpub (Colonia's first) opened in 2013, serving a wide selection of homebrews, accompanied by pizza, _picadas_ and Mexican fare.\n\n##    Sleeping\n\n  EL VIAJERO HOSTEL HOSTEL $\n\n MAP GOOGLE MAP\n\n(  4522-2683; www.elviajerocolonia.com; Washington Barbot 164; dm US$18-20, d US$64-74;    ) With bike rental, horseback excursions, a bar for guests and air-con in all rooms, this hostel is brighter, fancier and somewhat cozier than the competition, and the location two blocks east of Plaza de Armas couldn't be better.\n\nPOSADA DEL \u00c1NGEL HOTEL $$\n\n MAP GOOGLE MAP\n\n(  4522-4602; www.posadadelangel.net; Washington Barbot 59; d US$80-120;     ) Cheerfully painted in yellow and periwinkle blue, this little hotel has amenities such as down comforters and a sauna for chilly nights, and a swimming pool for the summer heat. Standard interior-facing rooms are dark; it's worth splurging on one with a view.\n\nEL CAPULLO B&B $$\n\n MAP GOOGLE MAP\n\n(  4523-0135; www.elcapullo.com; 18 de Julio 219; d US$120-165;    ) Friendly and well-traveled English-speaking owners, a prime Barrio Hist\u00f3rico location, a grassy yard and a swimming pool are the big attractions at this remodeled colonial _posada (inn)_. It's worth paying extra for one of the rooms upstairs or adjoining the back patio.\n4 Sleeping\n\nBuenos Aires may be the city that never sleeps \u2013 but really, who doesn't need a bit of rest once in a while? You'll find a wide range of places to rest your head here, from hostels to boutique hotels, guesthouses, rental apartments and international five-star hotels. Just remember to book ahead \u2013 or pay in cash \u2013 for the best deals.\n\n######  Rates, Discounts & Payments\n\nBuenos Aires is decent value compared with the USA or Europe. However, inflation has been running at 25% to 30% annually. To avoid sticker shock, double check the prices we list before reserving.\n\nThe prices we list \u2013 particularly for the four- or five-star hotels \u2013 are generally the rack or high-season rates from November through January. Rates for top-end hotels can vary widely on any particular day, as many are dependent on how empty or full the hotel is that day. Rates can also skyrocket during holidays such as Easter, Christmas or New Year. Some places lower their rates during slow periods, while others don't. But whatever the season, you don't always have to pay the official posted price.\n\nYour best bet for getting a cheaper rate is to book in advance. You can do this via most hotels' websites. Calling ahead and talking to a salesperson with the power to negotiate prices can also be fruitful, especially if you plan on staying more than a few days. Offering to pay in cash can also bring about a discount.\n\nThe most expensive hotels will take credit cards, but budget or midrange places may not \u2013 or they may levy a surcharge (about 10%).\n\n######  Hostels\n\nBuenos Aires' hostels range from basic no-frills deals to beautiful, buildings more luxurious than your standard cheap hotel. Most fall in between, but all have common kitchens, living areas, shared bathrooms and dorm rooms (bring earplugs). Most have a few private rooms (with or without bathroom) and provide some traveler services. BA has a few Hostelling International (HI; www.hihostels.com) hostels, where members can get a small discount. Other hostel networks include Minihostels (www.minihostels.com) and HoLa (www.holahostels.com).\n\n######  Hotels\n\nAs in many countries, Buenos Aires' hotels vary from utilitarian holes-in-the-wall to luxurious five-star hotels with all the usual top-tier services. In general, hotels provide a room with private bathroom, cable TV and sometimes a phone. Cheap hotels might also have cheaper rooms with shared bathroom. Higher-end hotels may have safe boxes, a refrigerator and a desk. Some hotels have a cafe or restaurant. Staff members at tourist-\u00adoriented hotels will usually speak some English.\n\n######  Boutique Hotels & B&Bs\n\nIn recent years these types of accommodations have popped up like mushrooms in BA. The neighborhood of Palermo especially has become home to dozens of boutique hotels; most are pricey but beautiful, with just a handful of hip, elegant rooms and usually decent service. In BA, B&Bs are sometimes (but not always) run by the owners, and usually have fewer rooms than boutique hotels \u2013 but often offer a better breakfast.\n\n###  NEED TO KNOW\n\n####  Price Ranges\n\nPrices are for a room with private bathroom for two people in high season.\n\n$ under AR$500\/US$80\n\n$$ AR$500-1100\/US$80-175\n\n$$$ over AR$1100\/US$175\n\nMany high-end hotels add a 21% tax to their quoted rates. Most budget and midrange hotels already have this tax included in their quoted rates. To avoid a surprise at checkout time, ask if this tax is included in any price you're quoted. We've included this tax in the prices shown here.\n\n####  Reservations\n\nIt's a good idea to make a reservation during any holidays or the busy summer months of November through February.\n\n####  Breakfast\n\nSome kind of breakfast, whether it be continental or buffet, is usually included at most accommodations. Unless you're staying somewhere fancy, however, don't expect too much \u2013 a typical breakfast will often consist of toast or _medialunas_ (croissants), with some jam or butter if you're lucky, plus coffee or tea.\n\n#####  Lonely Planet's Top Choices\n\n Poetry Building Lovely apartments decorated with vintage-reproduction furniture.\n\n Cabrera Garden Peaceful B&B boasting just three gorgeous rooms and a grassy garden.\n\n Miravida Soho Upscale guesthouse with friendly owners and wine-tasting opportunities.\n\n 5th Floor Modern and elegant B&B offering one of Buenos Aires' best breakfasts.\n\n Magnolia Hotel Fine boutique hotel with a very relaxing rooftop terrace.\n\n Casa Calma Ecologically minded luxury hotel providing a paradise in BA's busy downtown.\n\n#####  Best by Budget\n\n######  $\n\n America del Sur Five-star boutique hostel, with awesome rooms and services.\n\n Reina Madre Hostel Very comfortable, well-run and intimate hostel.\n\n Yira Yira Guesthouse Friendly and intimate guesthouse with only four rooms.\n\n######  $$\n\n Abode Homey guesthouse with nice terrace and exceptional breakfast.\n\n Rac\u00f3 de Buenos Aires Beautiful boutique hotel in a nontouristy neighborhood.\n\n Casa y Mundo Bolivar Lovely apartments in a remodeled mansion, with patios.\n\n######  $$$\n\n Palacio Duhau \u2013 Park Hyatt Gorgeous remodeled mansion with a stunning courtyard.\n\n Alvear Palace Hotel Buenos Aires' most traditional and luxurious five-star hotel.\n\n Faena Hotel + Universe Supremely elegant and over the top \u2013 this is where celebrities stay.\n\n#####  Best Boutique Hotels\n\n Magnolia Hotel Splendid boutique hotel with a very relaxing rooftop terrace.\n\n Mine Hotel Best for its grassy backyard with contemporary pool.\n\n Mansi\u00f3n Vitraux San Telmo's slickest spot to lay your head.\n\n Duque Hotel Elegant and beautiful, and there's a luxurious spa too.\n\n#####  Best for Families\n\n Poetry Building Upscale apartments, all with kitchen, plus a soaking pool.\n\n Novotel Hotel Especially family-friendly services \u2013 including two kiddie pools.\n\n Hotel Lyon Simple budget lodgings with tons of space for large families.\n\nWhere to Stay\n\nNeighborhood | For | Against\n\n---|---|---\n\nThe Center | Great transportation options; fairly close to all neighborhoods except Palermo; offers many services | Limited eating, shopping and nightlife options; noisy and crowded during the day and impersonal after dark\n\nPuerto Madero | Very safe, calm, quiet and upscale; great strolling opportunities, both in a natural reserve and along the pleasant restaurant-lined dikes | Expensive: many restaurants are overpriced; very limited public transport\u00adation, accommodation, shopping and service options and not much i nteresting nightlife\n\nCongreso & Tribunales | Reasonably central, with plenty of traditional theater and other cultural options; interesting local flavor, tending towards classic architecture and governmental vibe | Certain sections are desolate and less safe at night; limited shopping and eating possibilities\n\nSan Telmo | Endearing traditional atmosphere, reasonable shopping and nightlife, a good range of restaurants and many decent hostels | Far from Palermo; some areas can be edgy at night; public transportation is somewhat limited\n\nLa Boca\n\n| |\n\nNot recommended and practically no accommodation options\n\nRetiro | Beautiful upscale neighborhood within walking distance of Recoleta and the Center; convenient for public transportation | Very expensive; limited accommodations options; not many affordable restaurants or shops\n\nRecoleta & Barrio Norte | Buenos Aires' most upscale neighborhood; gorgeous architecture, good transportation options and fairly safe | Most accommodations, restaurants and shopping are very expensive\n\nPalermo | Many boutique hotels to choose from; the city's widest range of interesting restaurants, by both cuisine and budget; great shopping and nightlife | A bit of a trek to the Center and San Telmo; might be too touristy for some\n\nSouth of Palermo | Up-and-coming neighborhoods with local atmosphere and some decent accommodations, restaurants and shops; quick access via Subte to the Center | Fewer traveler services; some neighborhoods are not central\n\n###    The Center\n\nBuenos Aires' Center, being right in the middle of things, has the most business-type accommodations in the city. Several pedestrian streets make it more walkable, and it's close to the upscale neighborhoods of Puerto Madero, Retiro and Recoleta. The Plaza de Mayo area contains the bustling banking district and many historic buildings, and is within walking distance of San Telmo.\n\nDuring the day the whole area is very busy, but at night the streets become deserted and even a bit sketchy. Your eating and nightlife options are also very limited \u2013 for this you'll have to head to Palermo.\n\nGRAN HOTEL HISPANO HOTEL $\n\nMAP GOOGLE MAP\n\n(  4345-2020; www.hhispano.com.ar; Av de Mayo 861; s\/d AR$360\/490;    ;   L\u00ednea A Piedras) The tiny stairway lobby here isn't an impressive start, but upstairs there's a sweet atrium area with covered patio. Most rooms are modern and carpeted; those in front are biggest, and those on the top floor are brightest. There's also a pleasant outside sun terrace. It's a popular, central and well-tended place, so reserve ahead. Pay in cash for a 10% discount.\n\nHOTEL AVENIDA HOTEL $\n\nMAP GOOGLE MAP\n\n(  4331-4341; www.hotelav.com.ar; Av de Mayo 623; s\/d US$60\/75;    ;   L\u00ednea A Peru) Just 34 plain but efficient rooms greet you at this friendly place. There's a pleasant breakfast area and the location is great, right near Plaza de Mayo. Get a back room for more peace and quiet; the front ones have nice balconies (except for the 4th floor) but are noisy. Pay in cash for a 10% discount.\n\nV & S HOSTEL CLUB HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4322-0994; www.hostelclub.com; Viamonte 887; dm US$15, r from US$60;    ;   L\u00ednea C Lavalle) S One of the best in town, this attractive, central and eco-friendly hostel is located in a pleasant older building. The common space, which is also the dining and lobby area, is good for socializing. The spacious dorms are carpeted and the private rooms are excellent; all have their own bathroom. A nice touch is the tiny outdoor patio in back.\n\nPORTAL DEL SUR HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4342-8788; www.portaldelsurba.com.ar; Hip\u00f3lito Yrigoyen 855; dm US$14-17, s\/d US$50\/70;    ;   L\u00ednea A Piedras) Located in a charming old building, this is one of the city's best hostels. Beautiful dorms and sumptuous, hotel-quality private rooms surround a central common area, which is rather dark but open. The highlight is the lovely rooftop deck with views and attached bar and lounge. Offers free tango and Spanish lessons and a walking tour; plenty of other activities available.\n\nHOTEL MAIP\u00da HOTEL $\n\nMAP GOOGLE MAP\n\n(  4322-5142; www.nuevohotelmaipu.com.ar; Maip\u00fa 735; s\/d AR$200\/280, without bathroom AR$160\/240;  ;   L\u00ednea C Lavalle) Head on up the marble staircase into a dim, tiled hallway. This classic old building was once owned by the Anchorenas, a wealthy and aristocratic Argentine family. Original tiles and high ceilings add charm to these simple lodgings. The 10 rooms are unmemorable but at least have cable TV. There's no breakfast, but it's a good budget deal.\n\nCLARIDGE HOTEL HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4314-2020; www.claridge.com.ar; Tucum\u00e1n 535; d US$160;     ;   L\u00ednea B Florida) One of downtown BA's finest hotels, the Claridge features a relatively grand entrance for the area, where space is scarce. Standard rooms, with their tiny baths, aren't as fancy as you'd think, so go for a suite (some with balcony and Jacuzzi) if you want something special. The spa and pool are highlights. Prices vary widely, so check beforehand.\n\nHOTEL FAC\u00d3N GRANDE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4312-6360; www.hotelfacongrande.com; Reconquista 645; r AR$1270;    ;   L\u00ednea B Florida) For those seeking a touch of the country in Buenos Aires, there's this (slightly) gaucho-themed hotel. The lobby is decorated in rustic furniture and cowhide-covered pillows, and rooms are modern and comfortable. The location on pedestrian Reconquista is good and there's an intimate vibe that's rare in hotels of this size. Get a top-floor room for views. Overall, a good deal for the price.\n\nHOTEL LAFAYETTE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4393-9081; www.lafayettehotel.com.ar; Reconquista 546; d AR$1330;    ;   L\u00ednea B Florida) Spacious, elegant rooms are on offer at this fine downtown hotel on a pedestrian street. The bathrooms are small but efficient, while double-glazed windows guarantee peace and quiet. Hotel amenities include a sauna and gym, a nice restaurant, and a fancy lobby with plant atrium, fireplace and sofas. Buffet breakfast; book ahead for the best rates.\n\n###  WEBSITES\n\nMany travelers visiting Buenos Aires love the city so much that they want to stay longer and find an apartment. But snagging a pad isn't as easy as it could be; renters often need to commit to two years and nearly always need a local's bond to guarantee monthly payments \u2013 almost impossible for most foreigners.\n\nTo cater to this demand, dozens of apartment websites have popped up in recent years. These sites charge significantly more than locals would pay, but they don't have those pesky requirements, either. You can view pictures of rental properties, along with prices and amenities. Usually the photos match what you will get, but not always; if you'd like someone to check out an apartment before you rent it, Madi Lang (Click here) can make sure the place isn't on a busy street, in an outlying neighborhood or near a construction site.\n\n\u00bb www.bytargentina.com\n\n\u00bb www.apartmentsba.com\n\n\u00bb www.buenosaireshabitat.com\n\n\u00bb www.oasisba.com\n\n\u00bb www.santelmoloft.com\n\n\u00bb www.stayinbuenosaires.com\n\n\u00bb www.jaimejensen.com\n\nIf you're just looking for a room, check www.spareroomsba.com. Or look for longer-term guesthouses (where rooms usually share bathrooms) at www.casalosangelitos.com and www.lacasademarina.com.ar. And there's always the Buenos Aires branch of Craigslist.\n\nAnother good option for short- or long-term stays is dealing directly with owners via sites like www.airbnb.com, www.homeaway.com or www.flipkey.com.\n\nAlso check hotels.lonelyplanet.com.\n\n###    Puerto Madero\n\nThere are hardly any hotels in Puerto Madero, a relatively new, upscale neighborhood that lies just east of the Center. Most buildings here are old warehouses that have been converted into fancy restaurants, offices and lofts, or they're brand-new apartment high-rises. The Reserva Ecol\u00f3gica Costanera Sur provides a welcome chunk of wild nature \u2013 almost nonexistent in Buenos Aires.\n\nPublic transport doesn't reach Puerto Madero, but the nearest Subte line is only three blocks away, and many buses run along Av Leandro N Alem\/Paseo Col\u00f3n.\n\nFAENA HOTEL + UNIVERSE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4010-9000; www.faenahotelanduniverse.com; Martha Salotti 445; r US$760-1090;     ) Located in a renovated storage mill, this Philippe Starcke\u2013designed fantasy hotel is more than just a place to stay. Traipse through the plush main hallway, lined with two top-notch restaurants, a sultry bar-lounge, a basement cabaret and \u2013 outside \u2013 a slick swimming pool. On arrival guests are given a personal valet and cell phone, then taken to luxurious rooms that feature claw-foot beds, etched-mirror entertainment centers and glass-walled bathrooms. Also on the premises are a Turkish bath and spa.\n\n###    Congreso & Tribunales\n\nThe Congreso and Tribunales area contains many of the city's older theaters, cinemas and cultural centers. Lively Av Corrientes has many modest shops, services and bookstores, and was BA's original theater district. The Plaza de Congreso area is always moving, sometimes with mostly peaceful public demonstrations. Generally, this area is not quite as packed as in the Center and has a less business-and-touristy flavor, but still bustles day and night.\n\nMILHOUSE YOUTH HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4345-9604; www.milhousehostel.com; Hip\u00f3lito Yrigoyen 959; dm AR$90-95, d AR$330-350;    ;   L\u00ednea A Avenida de Mayo) BA's premiere party hostel, this popular HI spot offers a plethora of activities and services. Dorms are good and private rooms can be very pleasant; most surround an appealing open patio. Common spaces include a bar-cafe (with pool table) on the ground floor, a TV lounge on the mezzanine and a rooftop terrace above. A gorgeous annex building nearby offers similar services.\n\nHOTEL MARBELLA HOTEL $\n\nMAP GOOGLE MAP\n\n(  4383-8566; www.hotelmarbella.com.ar; Av de Mayo 1261; s\/d AR$300\/400;    ;   L\u00ednea A Lima) The rooms at this hotel are basic but clean \u2013 if you can stand a bit of traffic noise, try to secure one with a balcony. More spacious (and more expensive) rooms are available, and there's a good, modern bar-restaurant where breakfast is served. From here it's an easy tramp to either Plaza del Congreso or Plaza de Mayo. Pay in cash and save 10%.\n\nGRAN HOTEL ORIENTAL HOTEL $\n\nMAP GOOGLE MAP\n\n(  4951-6427; ghoriental@hotmail.com; Bartolom\u00e9 Mitre 1840; s\/d AR$210\/300;   ;   L\u00ednea A Congreso) Despite its name this hotel is not grand, but it is good \u2013 and a good deal. Downstairs rooms are a bit dark (get one upstairs) and showers in general are small, but the simple, high-ceilinged rooms are comfortable enough for non-fussy travelers \u2013 just don't expect many services. The tiled lobby and hallways are long and narrow, and there are a few old touches that add some personality.\n\nSABATICO HOSTEL HOTEL $\n\nMAP GOOGLE MAP\n\n(  4381-1138; www.sabaticohostel.com.ar; M\u00e9xico 1410; dm AR$100, r AR$400-500;    ;   L\u00ednea E Independencia) This well-maintained hostel is located off the tourist path in an atmospheric neighborhood. Rooms are small but pleasant and the good common areas include a nice kitchen, dining and living room, airy patio hallways and a pleasant rooftop terrace with _asado_ (barbecue) grill and hammocks in summer. There's occasional live music on weekends, plus a ping-pong table, foosball and bike rentals.\n\nLA CAYETANA HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4383-2230; www.lacayetanahotel.com.ar; M\u00e9xico 1330; r US$130-180;    ;   L\u00ednea E Independencia) Located south of Congreso in Montserrat, this is a beautiful 1850s guesthouse offering 11 simple, colorful rooms, all decorated differently with rustic yet upscale furniture. They all surround three lovely outdoor patios, which are accented with original tiles and leafy plants \u2013 the last one has a grassy garden. Breakfast includes fresh fruit, yogurt, and eggs to order. It's a quiet little paradise in a nontouristy neighborhood. Reserve ahead.\n\nLIVIN' RESIDENCE APARTMENTS $$\n\nMAP GOOGLE MAP\n\n(  5258-0300; www.livinresidence.com; Viamonte 1815; studios US$100, 1-bedroom apt US$110, 2-bedroom apt US$160;   ;   L\u00ednea D Callao) One of the better deals in town, especially if you're traveling in a group, are these studio and one- or two-bedroom apartments. All have a simple, contemporary feel, with tasteful furniture, flat TVs, small kitchens and balconies. There's a tiny rooftop terrace with Jacuzzi, _asado_ and nearby gym room. Security is good; reserve ahead.\n\nHOTEL BONITO BOUTIQUE HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4381-2162; www.bonitobuenosaires.com; Chile 1507, 3rd fl; r US$90-105;    ;   L\u00ednea E Independencia) Lovely boutique hotel with just five artsy, gorgeous rooms mixing the traditional and contemporary. Some have a loft, cupola sitting area or Jacuzzi; floors can be wooden or acid-finished concrete. There's a warm atmosphere, with a small bar area and a good, sizeable breakfast. It's in a nontouristy, very local neighborhood within walking distance of Congreso and San Telmo.\n\nHOTEL LYON APARTMENTS $$\n\nMAP GOOGLE MAP\n\n(  4372-0100; www.hotel-lyon.com.ar; Riobamba 251; d\/tr\/q AR$530\/650\/770;    ;   L\u00ednea B Callao) If you're a traveling family or group and on a budget, this place is for you. The two- and three-bedroom apartments available here are basic and no-frills but very spacious, and all include entry halls, large bathrooms and separate dining areas with fridges (but no kitchens). Up to five people can be accommodated in each apartment. Reserve ahead.\n\nNOVOTEL HOTEL HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4370-9500; www.novotel.com; Av Corrientes 1334; r from US$205;     ;   L\u00ednea B Uruguay) This large, contemporary French chain hotel is tastefully designed. The comfortable rooms have unique showers with glass on two opposite sides, plus a fun colored-light system (leave it to the French). But the highlight is out back, where a beautiful deck surrounds pools \u2013 one for adults and two for the kids \u2013 along with a living wall of vegetation and a bar.\n\nThis Novotel is family friendly, offering kid discounts, a playroom and Xbox rental; it's located smack in the middle of Corrientes' entertainment district.\n\n###    San Telmo\n\nSouth of the center, San Telmo has some of the most traditional atmosphere in the city. Buildings are more charming and historical, and less modern, than in the Center, and they tend to be only a few stories high. Many restaurants and fancy boutiques have opened here in recent years, and there are some good bars, tango venues and other nightspots for entertainment. Many accommodation options here are hostels, guesthouses or boutique hotels.\n\nIf you're looking to house up to eight people for a week or more, check out www.playinbuenosaires.com \u2013 and reserve well in advance.\n\nAMERICA DEL SUR HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4300-5525; www.americahostel.com.ar; Chacabuco 718; dm AR$120-130, d AR$450-480;    ;   L\u00ednea C Independencia) This gorgeous boutiquelike hostel is the fanciest of its kind in Buenos Aires, and built especially to be a hostel. Beyond reception is a fine bar-bistro area with large, elegant wooden patio. Clean dorms with four beds all have amazingly well- designed bathrooms, while private rooms are tastefully decorated and better than those at many midrange hotels. A multitude of services are also on offer.\n\nBOHEMIA BUENOS AIRES HOTEL $\n\nMAP GOOGLE MAP\n\n(  4115-2561; www.bohemiabuenosaires.com.ar; Per\u00fa 845; r from AR$420;    ;   L\u00ednea C Independencia) With its slight upscale-\u00admotel feel, this good-value San Telmo hotel offers 22 simple and neat rooms, most good-sized, if a bit antiseptic with their white-tiled floors. None of the rooms has bathtub, so instead of taking a soak enjoy the peaceful grassy backyard and small interior patios. The breakfast buffet is a plus, and there's a restaurant.\n\nSAN TELMO COLONIAL GUESTHOUSE $\n\nMAP GOOGLE MAP\n\n(  4300-0097; www.bairescolonial.com.ar; Carlos Calvo 767; r US$65-90;    ;   L\u00ednea C Independencia) Best for the independent traveler who requires minimal service is this very informal guesthouse. Ten very spacious rooms (four with lofts) line flower-pot-strewn patio-hallways on two floors. All but one comes with small kitchenette for very simple cooking, and the furniture is hardly fancy. Minimum stay is three nights; breakfast is offered at a nearby historic cafe. No sign outside.\n\nCIRCUS HOSTEL & HOTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4300-4983; www.hostelcircus.com; Chacabuco 1020; dm US$24, r from US$80;     ;   L\u00ednea C Independencia) From the trendy lounge in front to the wooden- deck-surrounded wading pool in back, this hostel-hostel exudes hipness. Both dorms and private rooms, all small and simple, have basic furniture and their own bathrooms. There's a pool table and slick TV area too, but no kitchen.\n\nBRISAS DEL MAR HOTEL $\n\nMAP GOOGLE MAP\n\n(  4300-0040; www.hotelbrisasdelmar.com.ar; Humberto Primo 826; r with\/without bathroom AR$120\/100;  ;   L\u00ednea C San Juan) Long-running old cheapie hotel with no luxuries \u2013 except for cable TV. Has basic but decent budget rooms, the cheapest ones with shared bathrooms \u2013 try for an upstairs one, as they're brighter. All face tiled hallways lined with plants, and there's a very rustic, unstocked kitchen. No breakfast.\n\nTERRANOVA HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4300-1957; www.terranovasantelmo.com.ar; Humberto Primo 670; dm AR$70, r AR$200-220;   ;   L\u00ednea C San Juan) This laid-back hostel boasts a casual bar area in front and puts on weekend 'cultural' events like live music \u2013 thankfully, away from the sleeping areas. A long covered patio-hallway provides nice outdoor spaces, and rooms have high ceilings (it's an old colonial building). There are also colorful murals and free tango classes.\n\nHOSTEL VIEJO TELMO HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4331-5469; www.viejotelmo.com; M\u00e9xico 974; d\/tr\/q US$45\/70\/100;    ;   L\u00ednea C Independencia) The highlight of this hostel is the rooftop terrace, with _parrilla_ grill and nearby kitchen\u2013dining room. It's in a 1912 building with a variety of decent rooms, nearly all with bathroom; each has cable TV and there's also a living room near reception. Conveniently located near the Center and many transportation lines; no sign outside.\n\n  MANSI\u00d3N VITRAUX BOUTIQUE HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4878-4292; www.mansionvitraux.com; Carlos Calvo 369; r US$135-160;     ;   L\u00ednea C Independencia) Almost too slick for San Telmo, this glass-fronted boutique hotel offers 12 beautiful rooms, all in different colors. All have either flat-screen or projection TV, and bathrooms boast very contemporary design. The breakfast buffet happens in the basement wine bar, and a tasting is included in your stay. There are also a large Jacuzzi, a dry sauna and a fancy rooftop terrace with small lap pool.\n\nBONITO SAN TELMO GUESTHOUSE $$\n\nMAP GOOGLE MAP\n\n(  4362-8451; www.bonitobuenosaires.com; Juan de Garay 458; r US$90-105;   ;   29) The busy avenue outside seems unlikely to offer such a paradise, but after you climb the stairs you'll be surrounded by contemporary touches, from the grand piano in the living room to the elegant dining nook in back. Six lovely rooms are available (including one with kitchenette), but the best features are the lush rooftop terraces, complete with San Telmo views. Reserve ahead.\n\nCASA Y MUNDO BOLIVAR BOUTIQUE HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4300-3619; www.casabolivar.com; Bol\u00edvar 1701; US$70-90;   ) Fourteen spacious studios and loft apartments with kitchenettes have been renovated into attractive modern spaces \u2013 some with original details such as carved doorways or painted ceilings \u2013 at this amazing mansion. Separate entrances join with hallways connecting through the complex, and there are lovely garden patios in which to relax. No breakfast, but there's a cafe-restaurant.\n\nL\u00ednea C, Constituci\u00f3n, is the closest Subte stop, but consider taking a taxi here instead. Three-day minimum stay; long-term guests preferred.\n\nSCALA HOTEL HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4343-0606; www.scalahotelbuenosaires.com; Bernardo de Irigoyen 740; r AR$1270-2060;    ;   L\u00ednea C Independencia) A grand lobby awaits you at this business-\u00adoriented, four-star hotel. All rooms are lovely and spacious; standard ones have wooden floors, while higher categories come with carpeting and sitting rooms. Some boast 9 de Julio views. There's a pleasant, large patio in back, along with spa, gym, restaurant and two business salons; plans are for a swimming pool by 2015.\n\n###    Retiro\n\nRetiro is a great, central place to be, _if_ you can afford it \u2013 many of BA's most expensive hotels, along with some of its richest inhabitants, are settled in here. Close by are leafy Plaza San Mart\u00edn, the Retiro train and bus stations, and many upscale stores and business services. Ritzy Recoleta is to the northwest and the busy Microcentro is to the south \u2013 both just a short and pleasant stroll away.\n\nHOTEL TRES SARGENTOS HOTEL $\n\nMAP GOOGLE MAP\n\n(  4312-6082; www.hotel3sargentos.com.ar; Tres Sargentos 345; s\/d AR$300\/380;   ;   L\u00ednea C San Mart\u00edn) A great deal for the location, this simple budget hotel has a nice enough lobby and is located on a pedestrian street. The carpets in the halls need changing, but the ones in the simple, comfortable rooms are clean enough. Some rooms even offer a bit of a view \u2013 ask for a floor up high.\n\n  CASA CALMA BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4312-5000; www.casacalma.com.ar; Suipacha 1015; r US$220-240;    ;   L\u00ednea C San Mart\u00edn) S Those with stuffed wallets and of an eco-conscious mind now have their perfect hideaway in BA: this central, environmentally friendly and luxurious hotel. Rooms are beautifully pristine and relaxing (some even have sauna or Jacuzzi), with Zen-like baths and serene atmosphere. It's a world away from outside the front door, where BA noisily buzzes by.\n\nCasa Calma does its part by using eco-certified wood in its building and outside greenery to adjust the hotel's temperatures. It recycles what it can and offers guests organic towels, bulk toiletries and even two bamboo bicycles to rent. It's so gorgeous, however, you won't even notice.\n\nFOUR SEASONS HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4321-1200; www.fourseasons.com\/buenosaires; Posadas 1086; d from US$665;     ;   L\u00ednea C San Mart\u00edn) No surprise here \u2013 the Four Seasons offers all the perks that define a five-star hotel, such as great service and white terry-cloth robes. Rooms are large and beautiful, with contemporary furnishings and decorations, and the finest suites are located in an old, luxurious mansion next door (go for the presidential \u2013 it's US$10,000 per night).\n\nThere are also a gorgeous spa, an outdoor heated swimming pool and a top-notch restaurant.\n\nHOTEL PULITZER BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4316-0800; www.hotelpulitzer.com.ar; Maip\u00fa 907; r from US$175;     ;   L\u00ednea C San Mart\u00edn) Very well located, this large, new boutique hotel has a black-and-white lobby and minimalist decor. Rooms are spacious and stylish, boasting flat-screen TVs and elegant bathrooms; some have a balcony. The highlight, however, is the beautiful rooftop terrace with attached bar, offering great views over the city. There are also a restaurant, a cocktail bar and even a swimming pool.\n\n###    Recoleta & Barrio Norte\n\nMost of the accommodations in Recoleta and Barrio Norte (whose borders can be blurred) are expensive, and what cheap hotels there are tend to be full much of the time. Buildings here are grand and beautiful, befitting the city's richest barrio, and you'll be close to Recoleta's famous cemetery, along with its lovely parks, museums and boutiques.\n\nYIRA YIRA GUESTHOUSE GUESTHOUSE $\n\nMAP GOOGLE MAP\n\n(  4812-4077; www.yirayiraba.com; Uruguay 911 1B; s\/d US$45\/65;    ;   L\u00ednea D Callao) This casual, intimate apartment-home is run by the helpful Paz, who lives on-site. The floors are wooden and the ceilings high, and there are just four large rooms (all with shared bathrooms) facing the central living area with tiny patio. It's a good place to meet other travelers and centrally located near downtown. Reserve ahead.\n\nREINA MADRE HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4962-5553; www.rmhostelbuenosaires.com; Anchorena 1118; dm AR$95-105, s\/d AR$215\/235;    ;   L\u00ednea D Pueyrred\u00f3n) This wonderful hostel is clean, safe and well run. It's in an old building that has plenty of personality, with high ceilings and original tiles, and all rooms are comfortable and modern (and share bathrooms). There's a cozy living room with balcony and small kitchen plus lots of dining tables, but the highlight is the wooden-deck rooftop with _asado_. Pet cat on premises.\n\nHOTEL LION D'OR HOTEL $\n\nMAP GOOGLE MAP\n\n(  4803-8992; www.hotel-liondor.com.ar; Pacheco de Melo 2019; s AR$300-320, d AR$360-420, tr AR$400-560;   ;   L\u00ednea D Pueyrred\u00f3n) These digs have their charm (it's an old embassy), but rooms vary widely \u2013 some are small, basic and dark, while others are grand. Despite some rough edges, all are good value and most have been modernized for comfort. The old marble staircase and elevator are fabulous, and there's a nice rooftop area. Cheap breakfast option; some rooms share bathrooms.\n\nPETIT RECOLETA HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4823-3848; www.petitrecoleta.com; Uriburu 1183; dm US$14-16, s US$35-50, d US$45-65;    ;   L\u00ednea D Facultad de Medicina) It's hardly a stunner as far as hostels go, but you can't beat this cheapie's location. There's an interior patio, small TV room, larger dining-bar room and pool-table area. Most rooms are private; there are two dorms, one for men and one for women. Only two rooms have air-con. Long-term tenants have their own section (one-month minimum).\n\nART SUITES APARTMENTS $$\n\nMAP GOOGLE MAP\n\n(  4821-6800; www.artsuites.com.ar; Azcu\u00e9naga 1465; d AR$850-1450;   ;   L\u00ednea D Pueyrred\u00f3n) The 15 luxurious, modern and spacious apartments here are all bright and boast minimalist decor, full kitchens or kitchenettes, sunny balconies and slick, hip furniture. Windows are double-paned for quiet, staff speak English and security is excellent. Continental breakfast included. Long-term discounts are available; reserve ahead. An annex offers more apartments.\n\n  POETRY BUILDING APARTMENTS $$$\n\nMAP GOOGLE MAP\n\n(  4827-2772; www.poetrybuilding.com; Jun\u00edn 1280; apt US$175-235;    ;   L\u00ednea D Pueyrred\u00f3n) These gorgeous studios and one- or two-bedroom apartments are perfect for families or small groups. Each one is different, eclectically decorated with reproduction antique furniture, and all come with fully stocked kitchens. Some boast an outdoor balcony or patio, but there's also a beautiful common terrace with soaking pool. Amenities include flat-screen TVs, plus iPod and cell-phone rentals.\n\n  PALACIO DUHAU \u2013 PARK HYATT HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  5171-1234; www.buenosaires.park.hyatt.com; Av Alvear 1661; d from US$655;     ;   130) If it's good enough for presidents, diplomats and Tom Cruise, it's good enough for you. The luxurious Park Hyatt takes up a city block and consists of two wings, including the Palacio Duhau, a renovated mansion. There's a gorgeously terraced garden with fountains and patios, plus a fine spa, indoor pool, wine and cheese bar and art gallery. Excellent service.\n\nALVEAR PALACE HOTEL HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4808-2100; www.alvearpalace.com; Av Alvear 1891; r from US$640;     ;   130) The classiest, most traditional hotel in BA. Old-world sophistication and superior service will help erase the trials of your long flight into town, while the bathtub Jacuzzi, Herm\u00e8s toiletries and Egyptian-cotton bed sheets aid your trip into dreamland. There's also an excellent restaurant, elegant tea room, cigar bar, fine spa, indoor swimming pool and butler service.\n\nAYRES DE RECOLETA APARTMENTS $$$\n\nMAP GOOGLE MAP\n\n(  4801-0505; www.ayresderecoleta.com; Uriburu 1756; studios US$175;     ;   59) The 37 studio apartments here (plus two penthouses) all come with king-size beds or two twins, brown-and-white color scheme and classy decoration. There are also simple kitchenettes great for heating up leftover takeout, plus a small indoor pool with Jacuzzi. And the location can't be beat \u2013 you're a block from Recoleta cemetery. Reserve ahead.\n\n###    Palermo\n\nDespite being a slight trek from the center, Palermo is the top choice for many travelers. Not only is it full of extensive parklands \u2013 which are great for weekend jaunts and sporting activities \u2013 but you'll have heaps of cutting-edge restaurants, designer boutiques and hip dance clubs at your door. Many of these places are located in the extensive sub-neighborhoods of Palermo Soho and Palermo Hollywood. Las Ca\u00f1itas is another sub-neighborhood, just to the northwest, with a three-block strip of door-to-door restaurants and bars. All are connected to the center by bus or Subte.\n\nECO PAMPA HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4831-2435; www.hostelpampa.com.ar; Guatemala 4778; dm US$20, s\/d US$70\/85;   ;   L\u00ednea D Plaza Italia) S Buenos Aires' first 'green' hostel is this casual spot sporting vintage furniture, low-energy light bulbs and a recycling system. The rooftop is home to a small veggie garden, compost pile and solar panels. Dorms are a good size and each of the eight private rooms comes with bathroom and flat-screen TV (most have air-con).\n\nThere's another branch further north in Belgrano (www.hostelpampa.com; Iber\u00e1 2858).\n\n  5TH FLOOR B&B $$\n\nMAP GOOGLE MAP\n\n(  4827-0366; www.the5thfloorba.com; r US$90-170;    ;   L\u00ednea D Scalabrini Ort\u00edz) This upscale B&B offers seven elegant rooms, three with private balcony. All are tastefully decorated with art deco furniture and modern amenities. The common living room is great for chatting with the English owner, a polo enthusiast, and there's also a pleasant back patio with lovely tile details. Occasional closed-door dining events happen here. Excellent breakfast; address given upon reservation.\n\n  CABRERA GARDEN B&B $$\n\nMAP GOOGLE MAP\n\n(  4777-7668; www.cabreragarden.com; Jos\u00e9 Antonio Cabrera 5855; US$145-250;     ) One of BA's loveliest stays is this three-room B&B run by a Polish-German gay couple. The remodelled 1920s building boasts a beautiful grassy garden with small patio and pool, and there's a wonderful living room in which to hang out. Rooms are very comfortable and all different, with modern conveniences like flat-screen TVs and iPod docks.\n\nEnglish, German and Polish spoken; reserve ahead. The nearest Subte stop \u2013 L\u00ednea D, Ministro Carranza \u2013 is 10 blocks away.\n\nABODE GUESTHOUSE $$\n\nMAP GOOGLE MAP\n\n(  4774-3331; www.abodebuenosaires.com; r US$90-150;    ;   L\u00ednea D Palermo) Run by an expat couple, who live on the premises, is this very intimate and homey guesthouse. Each of the four simple yet comfortable rooms comes with its own bathroom, and the largest has a balcony. The highlight: a wonderful rooftop terrace, where you can enjoy your full English breakfast. By reservation only; no walk-ins. Friendly dog on premises.\n\nPALERMO VIEJO B&B GUESTHOUSE $$\n\nMAP GOOGLE MAP\n\n(  4773-6012; www.palermoviejobb.com; Niceto Vega 4629; s\/d US$75\/85;    ) This small and intimate B&B is located in a remodeled _casa chorizo_ \u2013 a long, narrow house. The six rooms all front a leafy outdoor patio hallway and are simple but quite comfortable; two have lofts. All come with fridge and a good breakfast. Call them ahead of time \u2013 they often leave on errands in the afternoon.\n\nThe nearest Subte stop \u2013 L\u00ednea B, Malabia \u2013 is nine blocks away.\n\nLIVIAN GUESTHOUSE GUESTHOUSE $$\n\nMAP GOOGLE MAP\n\n(  4862-8841; www.livianguesthouse.com; Palestina 1184; r US$90-150;   ;   106, 160) Located in a lovely old building in an untouristy section of Palermo is this chill guesthouse. There are 10 colorful yet tasteful rooms on offer, one with its own terrace and most with private bathroom (a few share bathrooms or have a private bathroom down the hall). There are pleasant living-room spaces and a pretty back garden too.\n\nINFINITO HOTEL BOUTIQUE HOTEL $$\n\nMAP GOOGLE MAP\n\n(  2070-2626; www.infinitohotel.com; Arenales 3689; r US$160;    ;   L\u00ednea D Scalabrini Ort\u00edz) Starting at its small lobby cafe-reception, this hotel exudes a certain trendiness. Rooms are small but good, boasting flat-screen TVs, fridges, wooden floors and a purple color scheme. They try to be ecologically conscious (mostly by recycling); however, there's also a sauna and Jacuzzi. Located near some parks but still within walking distance of Palermo's nightlife. Buffet breakfast included.\n\nCASER\u00d3N PORTE\u00d1O GUESTHOUSE $$\n\nMAP GOOGLE MAP\n\n(  4554-6336; www.caseronporteno.com; Ciudad de la Paz 344; s AR$550-680, d AR$680-850;    ;   L\u00ednea D Olleros) Catering especially to tango dancers is this fine guesthouse with 10 simple but tastefully furnished rooms. All have private bathrooms, but four have them located outside the actual rooms. Behind the lush garden there's a small dance studio where classes take place, while other common spaces include a relaxing rooftop terrace and a kitchen for guest use.\n\nThe location is in a nontouristy residential neighborhood; four-night minimum stay in high season. Unsurprisingly, plenty of tango information is available.\n\nRUGANTINO HOTEL HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4773-2891; www.rugantinohotel.com; Uriarte 1844; r US$115;    ;   L\u00ednea D Palermo) This small and intimate hotel is located in a 1920s building and run by an Italian family. Various tiny terraces and catwalks connect the seven simple but beautiful rooms, all decked out in hardwood floors and modern styling \u2013 combined with a few antiques. The climbing vine-greenery in the small central courtyard well is soothing, and you can expect espresso for breakfast.\n\n  MIRAVIDA SOHO GUESTHOUSE $$$\n\nMAP GOOGLE MAP\n\n(  4774-6433; www.miravidasoho.com; Darregueyra 2050; r US$205-280;    ;   L\u00ednea D Plaza Italia) Run by a friendly and helpful German couple, this gorgeous guesthouse comes with six beautiful and elegant rooms. All are very comfortable and one has a private terrace. There's a wine cellar, bar-lounge area for evening wine tastings, a small and relaxing patio, and even an elevator. It serves good, full breakfasts; reserve ahead (10% discount if you pay cash).\n\n  MAGNOLIA HOTEL BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4867-4900; www.magnoliahotel.com.ar; J \u00c0lvarez 1746; r US$255-350;    ;   L\u00ednea D Scalabrini Ort\u00edz) This classy boutique hotel is in a gorgeously restored old house. Its eight impeccably groomed rooms are bathed in muted colors and fitted with elegant furniture; some have a patio or balcony. Common spaces are beautiful, and the gorgeous rooftop terrace is strewn with cushy lounges. Other pluses include a welcome drink and a little patio for the breakfast.\n\nMINE HOTEL BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4832-1100; www.minehotel.com; Gorriti 4770; d US$205-270;     ;   55) S This hip boutique hotel offers 20 good-size rooms; some come with Jacuzzi and balcony and all have a desk and natural decor touches. Get one overlooking the highlight of the hotel: the peaceful backyard, which comes complete with small wading pool. There's a small bistro for the buffet breakfast, and Mine even attempts to be eco-friendly (reusing towels, low-energy bulbs, recycling).\n\nDUQUE HOTEL BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4832-0312; www.duquehotel.com; Guatemala 4364; d US$160-205;     ;   L\u00ednea D Scalabrini Ort\u00edz) More upscale than most boutique hotels is this elegant charmer. All 14 rooms are lovely and well designed, though some can be a bit small \u2013 go for a superior or deluxe if you need more space. Pluses include a large Jacuzzi, sauna, basement spa, buffet breakfast, afternoon tea with pastries and great little backyard garden with a tiny pool.\n\nVAIN BOUTIQUE HOTEL BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4776-8246; www.vainuniverse.com; Thames 2226; r US$190-300;    ;   L\u00ednea D Plaza Italia) Fifteen elegant rooms, all with high ceilings and wooden floors, live at this nicely renovated building. All are modern in that white, minimalist way, and boast sofas and small desks. The highlight, however, is the wonderfully airy, multilevel living room with attached wooden-decked terrace \u2013 a great place to enjoy breakfast. Small bar-restaurant in the lobby; reserve ahead for discounts.\n\nPALERMITANO BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4897-2100; www.palermitano.biz; Uriarte 1648; d US$160-265;    ;   39, 55) Located in the middle of Palermo's nightlife, this boutique hotel has 16 tastefully decorated and contemporary rooms. The breakfast buffet is served all day, and they'll even bring it to your door. And pretty un\u00adusual for Buenos Aires is the small rooftop terrace with wading pool. A branch of the excellent Peruvian restaurant Sipan is on the ground floor.\n\nBA SOHOTEL BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4831-1844; www.basohotel.com; Paraguay 4485; r US$170-210;     ;   L\u00ednea D Plaza Italia) This 33-room boutique hotel is a good bet, with good service and an attached corner restaurant. The gorgeous rooms are spacious and come with wooden floors, desk, balcony, double-paned windows, Jacuzzi tubs and bathroom mirrors that don't fog (it's important!). There's also a tiny pool and Jacuzzi on the communal terrace. Breakfast buffet included; reserve ahead for discounts.\n\n248 FINISTERRA BOUTIQUE HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  4773-0901; www.248finisterra.com; Av B\u00e1ez 248; r US$170-240;    ;   L\u00ednea D Ministro Carranza) Smack in the middle of Las Ca\u00f1itas' nightlife strip lies this elegant, Zen-like boutique hotel. There are 11 minimalist rooms, all beautifully contemporary, though the smallest are a bit tight. There's a dining area for breakfast and a small grassy garden in back, but the highlight has to be the rooftop terrace, with wooden lounges and a Jacuzzi. Reserve ahead.\n\nRENDEZVOUS HOTEL HOTEL $$$\n\nMAP GOOGLE MAP\n\n(  3964-5222; www.rendezvoushotel.com.ar; Bonpland 1484; d US$165-215;    ) This boutique hotel is located in a beautiful four-story French-style building. Each of the 11 rooms is unique, styled with either antique or modern furnishings and bright colors; one has its own private balcony and outdoor Jacuzzi. There's a small bar-lounge at reception, tiny rooftop deck and cute patio at the entrance.\n\nThe nearest Subte stop \u2013 L\u00ednea B, Dorrego \u2013 is nine blocks away.\n\n###    South of Palermo\n\nWith the popularity of Palermo raising property values and rents, some places to stay have popped up in the more blue-collar, historical, artsy or even 'bohemian' neighborhoods to the south. This large area is a good choice if you want to 'go local' and don't mind being a bit further from the main sights (but not _too_ far away). Public transport is good \u2013 and there probably won't be another tourist in sight.\n\nCHILL HOUSE HOSTEL HOSTEL $\n\nMAP GOOGLE MAP\n\n(  4861-6175; www.chillhouse.com.ar; Ag\u00fcero 781; dm AR$85, d AR$250-360;   ;   L\u00ednea B Carlos Gardel) One of the coolest-vibe hostels in BA is at this remodeled old house, boasting high ceilings and a rustic artsy style. There are two dorms, eight private rooms with bath (No 6 is especially nice) and an awesome rooftop terrace where weekly _asados_ take place. Run by a French and Argentine team; free bike rentals too.\n\nRAC\u00d3 DE BUENOS AIRES BOUTIQUE HOTEL $$\n\nMAP GOOGLE MAP\n\n(  3530-6075; www.racodebuenosaires.com.ar; Yapey\u00fa 271; r from US$90;    ;   L\u00ednea A Castro Barros) This Italian-designed building in a nontouristy neighborhood offers 12 lovely rooms with different styling, from virgin white classic to subdued masculine to animal print. All are spacious and have wooden floors, high ceilings and modern amenities. There's a small plant-strewn patio for breakfast and a basement wine bar for evening tastings. It's a good-value deal and just three blocks from the Subte.\n\nQUERIDO B&B B&B $$\n\nMAP GOOGLE MAP\n\n(  4854-6297; www.queridobuenosaires.com; Juan Ram\u00edrez de Velasco 934; r US$80-130;    ;   L\u00ednea B Malabia) Run by a friendly and helpful Brazilian-English couple is this homey yet modern B&B. It's located in a nontouristy neighborhood, but within easy walking distance of Palermo. The eight clean rooms are small but comfortable, and all have private bathrooms; ask for an inside 'courtyard' room for more quiet. There's a living room in which to meet fellow travelers, plus a little gravel patio in back. Reserve ahead.\n\nPOP HOTEL HOTEL $$\n\nMAP GOOGLE MAP\n\n(  4776-6900; www.pophotelsbuenosaires.com; Juan Ram\u00edrez de Velasco 793; r from US$95;   ;   L\u00ednea B Malabia) Located near Villa Crespo's outlet stores is this colorful and bright hotel. Halls are carpeted (this helps with noise) and rooms are modern and comfy, all with fridge, sink and flat-screen TV; the ones on the 4th floor are larger and boast balconies. Breakfast not included. Rates vary widely; prepay for a significant discount as rack rates can be double the prepaid rate.\nUnderstand Buenos Aires\n\nBUENOS AIRES TODAY\n\nEconomic ups and downs, 'Queen Cristina' and culinary novelties \u2013 Buenos Aires never bores.\n\nHISTORY\n\nSpanish colonialism, fierce independence, the golden years and the Dirty War \u2013 and all that before the roller-coaster economic years.\n\nMUSIC\n\nTango is only one facet of Buenos Aires' music; there are also rock, folk, jazz and electronica.\n\nLITERATURE & CINEMA\n\nOver the last century, Argentina has produced many famous writers (think Jorge Luis Borges), along with two Oscar-winning movies.\n\nART & ARCHITECTURE\n\nArts runs rampant in Buenos Aires, from sophisticated street graffiti to abstract installations. And BA's gorgeous European architecture often surprises newcomers.\nBuenos Aires Today\n\nBuenos Aires has two faces: it's a city that harbors both decline and prosperity. You'll see dirty, neglected buildings everywhere \u2013 yet the rebirth following the economic crash of 2001 keeps on going. Puerto Madero continues to grow, Palermo's best restaurants still attract queues, and malls are full of shoppers. Public transportation is improving and the city's restaurants keep evolving \u2013 and though president Cristina Kirchner's popularity goes up and down, you should never count her out.\n\n###  Best in Print\n\nKiss of the Spider Woman (Manuel Puig; 1976) Two prisoners and their developing relationship in a Buenos Aires prison; made into the Oscar-winning 1985 film.\n\nAnd the Money Kept Rolling In (and Out) (Paul Blustein; 2005) How the IMF helped bankrupt Argentina.\n\nOn Heroes and Tombs (Ernesto S\u00e1bato; 1961) A complex plunge into Buenos Aires' society, aristocracy and family dynamics in the 1950s.\n\nThe Tango Singer (Tom\u00e1s Eloy Mart\u00ednez; 2006) An American graduate student travels to Buenos Aires and tracks down a legendary tango singer.\n\n###  Best on Film\n\nLa historia oficial (The Official Story; 1985) Oscar-winning film on the Dirty War.\n\nNueve reinas (Nine Queens; 2000) Two con men chasing the big score.\n\nEl secreto de sus ojos (The Secret in Their Eyes; 2009) Thriller that won the 2010 Oscar for best foreign-language film.\n\nPizza, birra, faso (Pizza, Beer, Cigarettes; 1998) Four BA gangster youths try to survive on the city streets.\n\n###  Economic Roller Coaster\n\nArgentina's currency devaluation in 2002 caused surging demand for its suddenly-cheap agricultural products. Helped along by skyrocketing government spending and strong growth in Brazil and China, this economic boom lasted through 2007 and revved up again in 2010. But high inflation (unofficially hovering at around 25%), a stronger peso and lower commodity prices have reined in the economy.\n\nIn October 2011, in an effort to curb capital from heading overseas, the government started requiring Argentines to substantiate their purchases of US dollars. This created a _mercado azul_ (literally, 'blue market') for US dollars, which are highly sought after as a stable currency. The real-estate market stalled, since purchases were pretty much always transacted in US dollars. Calle Florida is now even more full of _arbolitos_ (or 'little trees', since they stand around), who target tourists with dollars to change \u2013 at nearly double the official rate.\n\nMany economists believe that the government needs to reduce spending and stop borrowing from its central bank and public pension system; control inflation; and maintain foreign-exchange reserves. Moreover, government policies need to become more transparent to encourage both domestic and foreign investment. These are tall orders and go against the traditional Argentine economic flow, but recession and even devaluation are increasing risks. Who knows \u2013 maybe another crash is just what Argentina needs to get on top again.\n\n###  Cristina's Reign\n\nIn 2011 Cristina Kirchner was re-elected president by a landslide majority of 54%. She ran on a platform that appealed to the populist vote, promising to raise incomes, restore industry and maintain Argentina's economic boom. Her approach worked like a charm.\n\nSince then, however, things haven't been so rosy. Her popularity plunged as the economy hit the brakes, inflation skyrocketed and crime kept rising. _The Economist_ and international agencies such as the IMF have accused her government of cooking the books (especially inflation figures). Her health has been on the rocks: she had to have surgery to remove her thyroid in 2012, and she underwent a procedure to remove a blood clot on her brain in 2013. She lost even more political support in the October 2013 midterm elections, making a Chavez-like third term \u2013 currently not allowed in the constitution, but something that many thought Cristina was after \u2013 very unlikely.\n\nBut nobody can write _la presidenta_ off yet \u2013 she still maintains majorities in both chambers, her term won't be up until 2015 and she's been known to make comebacks. And despite her many detractors, Cristina has made admirable social strides. She's addressed the abuses of the military dictatorship, championed same-sex marriage laws and, above all, supported the blue-collar classes. And her people love her for it, just as they did Evita.\n\n###  Cultural Evolutions\n\nDespite a discouraging economy and a downturn in tourism in the last few years, Buenos Aires' culinary culture continues to evolve. Intrepid young chefs are providing creative twists to the restaurant scene with relatively new-to-BA concepts like molecular gastronomy and pop-up meal nights. Meanwhile, the closed door\u2013restaurant sphere keeps thriving, with a few places now making Argentina's _asado_ \u2013 traditionally a barbecue party at a friend or family's home on a Sunday \u2013 available to tourists in town for only a few days. Add to that a few new exciting food fairs and BA is well on its way to becoming a foodie destination.\n\nAnother evolving sector of Buenos Aires is its public transportation. The final implementation of the SUBE card meant that obtaining enough coins for the bus was no longer a source of friction and stress in _porte\u00f1os'_ daily lives (really \u2013 it was _that_ bad). And new bus-only Metrobus lanes down big avenues like 9 de Julio and Juan B Justo have eased traffic somewhat, though people continue to purchase private cars as an investment against inflation (nobody trusts banks with their savings any more). Bicycle use has also increased, as the bike-lane system keeps expanding, and the city's free bike-share program has been deemed a success.\n\nHistory\n\nLike all Latin American countries, Argentina has a tumultuous history, one tainted by periods of despotic rule, corruption and hard times. But its history is also illustrious, the story of a country that fought off Spanish colonial rule and was once among the world's economic powerhouses. It's a country that gave birth to international icons such as the gaucho, Evita Per\u00f3n and Che Guevara. Understanding Argentina's past is paramount to understanding its present and, most importantly, to understanding Argentines themselves.\n\nArgentina's national beer, Quilmes, is named after the now decimated indigenous group of northwest Argentina. It's also the name of a city in the province of Buenos Aires.\n\n##  The Spanish Arrive\n\nAlthough the banks of the R\u00edo de la Plata had been populated for tens of thousands of years by nomadic hunter-gatherers, the first attempt at establishing a permanent settlement was made by Spanish aristocrat Pedro de Mendoza in 1536. His verbose name for the outpost, Puerto Nuestra Se\u00f1ora Santa Mar\u00eda del Buen Aire (Port Our Lady Saint Mary of the Good Wind) was matched only by his extravagant expedition of 16 ships and nearly 1600 men \u2013 almost three times the size of Hern\u00e1n Cort\u00e9s' forces that conquered the Aztecs. In spite of his resources and planning, Mendoza unfortunately arrived too late in the season to plant adequate crops. The Spanish soon found themselves short on food and in typical colonialist fashion tried to bully the local Querand\u00ed indigenous groups into feeding them. A bitter fight and four years of struggle ensued, which led to such an acute shortage of supplies that some of the Spanish resorted to cannibalism. Mendoza himself fled back to Spain, while a detachment of troops who were left behind retreated upriver to Asunci\u00f3n (now the capital of Paraguay).\n\nWith Francisco Pizarro's conquest of the Inca empire in present-day Peru as the focus of the Spanish crown, Buenos Aires was largely ignored for the next four decades. In 1580 Juan de Garay returned with an expedition from Asunci\u00f3n and attempted to rebuild Buenos Aires. The Spanish had not only improved their colonizing skills since Mendoza's ill-fated endeavor but also had some backup from the cities of Asunci\u00f3n and Santa Fe.\n\nStill, Buenos Aires remained a backwater in comparison to Andean settlements such as Tucum\u00e1n, C\u00f3rdoba, Salta, La Rioja and Jujuy. With the development of mines in the Andes and the incessant warfare in the Spanish empire swelling the demand for both cattle and horses, ranching became the core of the city's early economy. Spain maintained harsh restrictions on trade out of Buenos Aires and the increasingly frustrated locals turned to smuggling contraband.\n\nThe city continued to flourish and the crown was eventually forced to relax its restrictions and co-opt the growing international trade in the region. In 1776 Madrid made Buenos Aires the capital of the new Viceroyalty of the R\u00edo de la Plata, which included the world's largest silver mine in Potos\u00ed (in present-day Bolivia). For many of its residents, the new status was recognition that the adolescent city was outgrowing Spain's parental authority.\n\nAlthough the new viceroyalty had internal squabbles over trade and control issues, when the British raided the city twice \u2013 in 1806 and 1807 \u2013 the response was unified. Locals rallied against the invaders without Spanish help and chased them out of town. These two battles gave the city's inhabitants confidence and an understanding of their self-reliance. It was just a matter of time until they broke with Spain.\n\n###  SMUGGLING IN BUENOS AIRES\n\nIt's not a coincidence that one of the most popular whiskeys served in Buenos Aires is called Old Smuggler. The city's history of trading in contraband goes all the way back to its founding. Some argue that the culture of corruption, so pervasive in Argentina, is tolerated because the historical role of smuggling in Buenos Aires led to a 'tradition' of rule-bending.\n\nThe Spanish empire kept tight regulations on its ports and only certain cities were allowed to trade goods with other countries. Buenos Aires, originally on the periphery of the empire, was hard to monitor and therefore not allowed to buy from or sell to other Europeans. Located at the mouth of the R\u00edo de la Plata, the settlement was an ideal point of entry to the continent for traders. Buenos Aires merchants turned to smuggling everything from textiles and precious metals to weapons and slaves. Portuguese- manufactured goods flooded the city and made their way inland to present-day Bolivia, Paraguay and even Peru.\n\nLater, the British and high-seas pirates found a ready and willing trading partner in Buenos Aires (and also introduced a taste for fine whiskeys). An increasing amount of wealth passed through the city and much of the initial growth of Buenos Aires was fueled by the trade in contraband. As smuggling was an open game, without favored imperial merchants, it offered a chance for upward social mobility and gave birth to a commercially oriented middle class.\n\n##  Independence\n\nWhen Napoleon conquered Spain and put his brother on the throne in 1808, Buenos Aires became further estranged from Madrid and finally declared its independence on May 25, 1810.\n\nSix years later, on July 9, 1816, outlying areas of the viceroyalty also broke with Spain and founded the United Provinces of the River Plate. Almost immediately a power struggle arose between Buenos Aires and the provincial strongmen: the Federalist landowners of the interior provinces were concerned with preserving their autonomy, while the Unitarist businessmen of Buenos Aires tried to consolidate power in the city with an outward orientation toward overseas commerce and European ideas. Some of the interior provinces decided to go their own way, forming Paraguay in 1814, Bolivia in 1825 and Uruguay in 1828.\n\nAfter more than a decade of violence and uncertainty, Juan Manuel de Rosas become governor of Buenos Aires in 1829. Although he swore that he was a Federalist, Rosas was more of an opportunist \u2013 a Federalist when it suited him and a Unitarist once he controlled the city. He required that all international trade be funneled through Buenos Aires rather than proceeding directly to the provinces, and he set ominous political precedents, creating the _mazorca_ (his ruthless political police) and institutionalizing torture.\n\nOne of the best-known contemporary accounts of postindependence Argentina is Domingo Faustino Sarmiento's _Life in the Argentine Republic in the Days of the Tyrants_ (1868). Also superb is his seminal classic, _Facundo, Or Civilization and Barbarism_ (1845).\n\n##  The Fleeting Golden Years\n\nRosas' overthrow came in 1852 at the hands of Justo Jos\u00e9 de Urquiza, a rival governor who tried to transfer power to his home province of Entre Rios. In protest, Buenos Aires briefly seceded from the union, but it was reestablished as the capital when Bartolom\u00e9 Mitre crushed Urquiza's forces in 1861. From there, Buenos Aires never looked back and became the undisputed power center of the country.\n\nThe economy boomed and Buenos Aires became a port town of 90,000 people in the late 1860s. Immigrants poured in from Spain, Italy and Germany, followed by waves of newcomers from Croatia, Ireland, Poland and Ukraine. Its population grew nearly seven-fold from 1869 to 1895, to over 670,000 people. The new residents worked in the port, lived tightly in crammed tenement buildings, developed tango, and jump-started the leftist labor movement. The onslaught of Europeans not only expanded Buenos Aires into a major international capital but gave the city its rich multicultural heritage, famous idiosyncrasies and sharp political differences.\n\nBy Argentina's centennial in 1910, Buenos Aires was a veritable metropolis. The following years witnessed the construction of the subway, while British companies built modern gas, electrical and sewer systems. Buenos Aires was at the height of a golden age, its bustling streets full of New World businesses, art, architecture and fashion. Argentina grew rich during this time based on its meat production. Advances in refrigeration and the country's ability to ship beef to distant lands was key to its economic success. In fact, by the beginning of WWI, Argentina was one of the world's 10 richest countries, and ahead of France and Germany.\n\nConservative forces dominated the political sphere until 1916, when Radical Party leader Hip\u00f3lito Yrigoyen took control of the government in a move that stressed fair and democratic elections. After a prolonged period of elite rule, this was the first time Argentina's burgeoning middle class obtained a political voice.\n\nIt was also at this time that Argentina's fortunes started to change, but unfortunately not for the better. Export prices dropped off, wages stagnated and workers became increasingly frustrated and militant. La Semana Tr\u00e1gica (Tragic Week), when over 100 protesters were killed during a metalworkers' strike, was the culmination of these tensions; some say this radical reaction was due to the government being pressured by moneyed interests. The Wall Street crash of 1929 dealt the final blow to the export markets and a few months later, in 1930, the military took over the country in a coup led by General Jos\u00e9 F\u00e9lix Uriburu. The golden age rapidly became a distant memory.\n\nThis was the first of many military coups that blemished the rest of the century and served to shackle the progress of the nation. Scholars have argued that the events that culminated in the 2001 economic collapse can be traced back to the 1930 military takeover.\n\nA fascinating, fictionalized version of the life of ex-president Juan Per\u00f3n, culminating in his return to Buenos Aires in 1973, is Tom\u00e1s Eloy Mart\u00ednez' _The Per\u00f3n Novel_ (1998).\n\n##  The Age of the Per\u00f3ns\n\nDuring WWII the rural poor migrated into Buenos Aires in search of work. The number of people living in the city nearly tripled and it soon held a third of the national population (which is in fact similar to the percentage today). The growing strength of these urban working classes swept populist Lieutenant-General Juan Domingo Per\u00f3n into the presidency in 1946. Per\u00f3n had been stationed for a time in Italy and developed his own brand of watered-down Mussolini-style fascism. He quickly nationalized large industry, including the railways, and created Argentina's first welfare state. Borrowing from Fascist Italy and Germany, Per\u00f3n carefully cultivated his iconic image and held massive popular rallies in Plaza de Mayo.\n\nThe glamorous Eva Duarte, a onetime radio soap-opera star, became the consummate celebrity first lady upon marrying Per\u00f3n, and an icon who would eclipse Per\u00f3n himself. Known as Evita, her powerful social-assistance foundation reached out to lower-class women through give\u00adaways of such things as baby bottles and strollers, and the construction of schools and hospitals. The masses felt a certain empathy with Evita, who was also born into the working class. Her premature death in 1952 came just before things went sour and her husband's political power plummeted.\n\nAfter Evita's death Per\u00f3n financed payouts to workers by simply printing new money, bungled the economy, censored the press and cracked down on opposition. He was strikingly less popular without Evita, and was deposed by the military in 1955 after two terms in office. Per\u00f3n lived in exile in Spain while a series of military coups ailed the nation. When he returned in 1973, there were escalating tensions from left and right parties; even if he'd lived to serve his term of re-election, Per\u00f3n would have had too much on his plate. His successor, his hapless third wife Isabel, had even less staying power and her overthrow by a military junta in 1976 came as no surprise.\n\nAlthough the effects of Per\u00f3n's personal political achievements are debatable, the Peronist party, based largely on his ideals, has endured.\n\nHect\u00f3r Olivera's 1983 film _Funny Dirty Little War_ is an unsettling but excellent black comedy set in a fictitious town just before the 1976 military coup.\n\n##  The Dirty War\n\nThe new military rulers instituted the Process of National Reorganisation, known as El Proceso, and this was headed by the notorious Jorge Rafael Videla. Ostensibly an effort to remake Argentina's political culture and modernize the flagging economy, El Proceso was little more than a Cold War\u2013era attempt to kill off or intimidate all leftist political opposition in the country.\n\nBased in Buenos Aires, a left-wing guerrilla group known as the Montoneros bombed foreign buildings, kidnapped executives for ransom and robbed banks to finance its armed struggle against the government. The Montoneros were composed mainly of educated, middle-\u00adclass youths; they were hunted down by the military government in a campaign known as La Guerra Sucia (the Dirty War). Somewhere between 10,000 and 30,000 civilians died \u2013 many of them simply 'disappeared' while walking down the street or sleeping in their beds. Many were tortured to death, or sedated and dropped from planes into the R\u00edo de la Plata. Anyone who seemed even sympathetic to the Montoneros could be whisked off the streets and detained, tortured or killed. A great number of the 'disappeared' are still unaccounted for today.\n\nThe military leaders let numerous aspects of the country's well-being slip into decay, along with the entire national economy. When Ronald Reagan took power in the USA in 1981, he reversed Jimmy Carter's condemnation of the junta's human-rights abuses and even invited the generals to visit Washington, DC. Backed by this relationship with the USA, the military was able to solicit development loans from international lenders, but endemic corruption quickly drained the funds into their Swiss bank accounts.\n\n###  ESMA: ARGENTINA'S AUSCHWITZ\n\nAlong a busy road in the BA neighborhood of Nu\u00f1ez is an imposing building officially called the Naval Mechanics School but better known as ESMA. During Argentina's 1976\u201383 military rule it served as an infamous detention center where some 5000 people were brutally tortured and killed. Truckloads of blindfolded prisoners were unloaded outside the building, taken to the basement, sedated and killed. Some were murdered by firing squads and others were drugged and dropped from planes into the R\u00edo de la Plata on twice-weekly 'death flights'. The building also served as a clandestine maternity center that housed babies taken from their mothers (many of whom were subsequently killed) to be given to police and military couples without children.\n\nIn 2004, as part of president N\u00e9stor Kirchner's effort to revisit the Dirty War crimes, the building was designated a memorial museum, handed over to a human-rights group and named the Space for Memory and Promotion and Defense of Human Rights. But reviving the memory was like opening Pandora's box. A public debate ensued on how to tackle the museum: whether to make it educational, poignant, moralizing or realistic. This debate \u2013 together with the human-rights group's insistence that all campus buildings, some of which were still occupied by the Navy, be vacated \u2013 delayed the museum's launch. (In the end, the Navy did move to another locale.)\n\nEventually, it was agreed that it was best to leave the space bare, with few explanatory signs, and so commemorate the victims. On the public tours (see www.espaciomemoria.ar) through the bleak rooms, guides tell the stories of detainees' tragic lives. According to photographer Marcelo Brodsky, whose brother disappeared in ESMA, 'The site is charged with torture sessions, muffled screams, odors and sounds'. This reminder of Argentina's state terror allows visitors to ponder the frailty of democracy and contemplate the evil of military dictatorships not only in Argentina but around the world.\n\n##  The Return to Democracy\n\nThe military dictatorship that ruled the country with an iron fist lasted from 1976 to 1983. General Leopoldo Galtieri took the reins of the draconian military junta in 1981, but its power was unraveling: the economy was in recession, interest rates skyrocketed and protesters took to the streets of Buenos Aires. A year later, Galtieri tried to divert national attention by goading the UK into a war over control of the Falkland Islands (known in Argentina as Las Islas Malvinas). The British had more resolve than the junta had imagined and Argentina was easily defeated. The greatest blow came when the British nuclear submarine _Conqueror_ torpedoed the Argentine heavy cruiser _General Belgrano,_ killing 323 men. Argentina still holds that the ship was returning to harbor.\n\nEmbarrassed and proven ineffectual, the military regime fell apart and a new civilian government under Ra\u00fal Alfons\u00edn took control in 1983. Alfons\u00edn enjoyed a small amount of success and was able to negotiate a few international loans, but he could not limit inflation or constrain public spending. By 1989 inflation was out of control and Alfons\u00edn left office five months early, when Carlos Menem took power.\n\n_Nunca M\u00e1s_ (Never Again; 1984), the official report of the National Commission on the Disappeared, systematically details military abuses from 1976 to 1983 \u2013 during Argentina's Dirty War.\n\n##  Menem & the Boom Years\n\nUnder the guidance of his shrewd economic minister, Domingo Cavallo, the skillfully slick Carlos Menem introduced free-market reforms to stall Argentina's economic slide. Many of the state-run industries were privatized and, most importantly, the peso was fixed by law at an equal rate to the American dollar. Foreign investment poured into the country. Buenos Aires began to thrive again: buildings were restored and new businesses boomed. The capital's Puerto Madero docks were redeveloped into an upscale leisure district, tourism increased and optimism was in the air. People in Buenos Aires bought new cars, talked on cell phones and took international vacations.\n\nAlthough the Argentine economy seemed robust to the casual observer, by Menem's second term (1995\u201399) some things were already amiss. The inflexibility imposed by the economic reforms made it difficult for the country to respond to foreign competition, and Mexico's 1995 currency collapse jolted a number of banks in Buenos Aires. Not only did Menem fail to reform public spending but corruption was so widespread that it dominated daily newspaper headlines.\n\nThe Falklands War is still a somewhat touchy subject in Argentina. If the subject comes up, try to call the islands the Malvinas instead of the Falklands, as many Argentines have been taught from a young age that they have always belonged to Argentina.\n\n##  The Economic Crisis\n\nAs an economic slowdown deepened into a recession, voters turned to the mayor of Buenos Aires, Fernando de la R\u00faa, and elected him president in 1999. He was faced with the need to cut public spending and hike taxes during the recession.\n\nThe economy stagnated further, investors panicked, the bond market teetered on the brink of oblivion and the country seemed unable to service its increasingly heavy international debt. Cavallo was brought back in as the economic minister and in January 2001, rather than declaring a debt default, he sought over US$20 million more in loans from the IMF.\n\nArgentina had been living on credit and it could no longer sustain its lifestyle. The facade of a successful economy had been ripped away, and the indebted, weak inner workings were exposed. As the storm clouds gathered, there was a run on the banks. Between July and November, Argentines withdrew around US$20 billion, hiding it under their mattresses or sending it abroad. In a last-ditch effort to keep money in the country, the government imposed a limit of US$1000 a month on bank withdrawals. Called the _corralito_ (little corral), the strategy crushed many informal sectors of the economy that function on cash (taxis, food markets), and rioters and looters inevitably took to the streets. As the government tried to hoard the remaining hard currency, all bank savings were converted to pesos and any remaining trust in the government was broken. Middle-class protesters joined the fray in a series of pot-and-pan-banging protests, and both Cavallo and de la R\u00faa bowed to the inevitable and resigned.\n\nTwo new presidents came and went in the same week and the world's greatest default on public debt was declared. The third presidential successor, former Buenos Aires province governor Eduardo Duhalde, was able to hold onto power. In order to have more flexibility, he dismantled the currency-board system that had pegged the peso to the American dollar for a decade. The peso devalued rapidly and people's savings were reduced to a fraction of their earlier value. In January 2002 the banks were only open for a total of six days and confidence in the government was virtually nonexistent. The economy ceased to function: cash became scarce, imports stopped and demand for nonessential items flat-lined. More than half of the fiercely proud Argentine people found themselves below the national poverty line: the once comfortable middle class woke up in the lower classes and the former lower classes were plunged into destitution. Businesspeople ate at soup kitchens and homelessness became rampant.\n\nCarlos Menem's Syrian ancestry earned him the nickname 'El Turco' (The Turk). In 2001 he married Cecilia Bolocco, a former Miss Universe 35 years his junior; they're now separated.\n\n##  Enter N\u00e9stor Kirchner\n\nDuhalde, to his credit, was able to use his deep political-party roots to keep the country together through to elections in April 2003. Numerous candidates entered the contest; the top two finishers were Menem (making a foray out of retirement for the campaign) and N\u00e9stor Kirchner, little-known governor of the thinly populated Patagonian province of Santa Cruz. Menem bowed out of the runoff election and Kirchner became president.\n\nKirchner was the antidote to the slick and dishonest Buenos Aires establishment politicians. He was an outsider, with his entire career in the provinces and a personal air of sincerity and austerity. The people were looking for a fresh start and someone to believe in \u2013 and they found that in Kirchner.\n\nDuring his term Kirchner defined himself as a hard-nosed fighter. In 2003 he managed to negotiate a debt-refinancing deal with the IMF under which Argentina would only pay interest on its loans. In 2006 Argentina repaid its $9.5 billion debt, not a small feat, which drove his approval rating up to 80%. Annual economic growth was averaging an impressive 8%, the poverty rate dropped to about 25% and unemployment nose-dived. A side effect of the 2001 collapse was a boom in international tourism, as foreigners enjoyed cosmopolitan Buenos Aires at bargain prices, injecting tourist money into the economy.\n\nBut not everything was bread and roses. The fact that Argentina had repaid its debt was fantastic news indeed, but economic stability didn't necessarily follow. In fact, a series of problems ensued during Kirchner's presidency: high inflation rates caused by a growing energy shortage, unequal distribution of wealth, and a rising breach between rich and poor that was slowly obliterating the middle class.\n\nOn the foreign-policy front, Kirchner's belligerence became aimed at outside forces. In November 2005, when George Bush flew in for the 34-nation Summit of the Americas, his presence sparked massive demonstrations around the country. Although anti-US sentiment unites most Argentines, some feared that Kirchner's schmoozing with Venezuelan president Hugo Ch\u00e1vez alienated potential investors in the United States and Europe.\n\nKirchner made admirable strides toward addressing the human-rights abuses of the military dictatorship. In 2005 the Supreme Court lifted an amnesty law that protected former military officers suspected of Dirty War crimes, and this led to a succession of trials that put several of them away for life.\n\nAt least two terms came about due to Argentina's economic crisis: _el corralito_ (a little corral) refers to the cap placed on cash withdrawals from bank accounts during 'La Crisis', while _cacerolazo_ (from the word _cacerola_ , meaning pan) is the street protest where angry people bang pots and saucepans.\n\n##  The Trials & Tribulations of Cristina\n\nWhen N\u00e9stor Kirchner stepped aside in July 2007 in favor of his wife's candidacy for the presidential race, many started wondering: would 'Queen Cristina' (as she's often called due to her regal comportment) be just a puppet for her husband, who intended to rule behind the scenes?\n\nIn the October 2007 presidential election, Cristina Fern\u00e1ndez de Kirchner succeeded in her ambition to move from first lady to president. Weak opposition and her husband's enduring clout were some of the reasons for Cristina's clear-cut victory, despite the lack of straightforward policies during her campaign. While this was not the first time Argentina had had a female head of state (Isabel Per\u00f3n held a brief presidency by inheriting her husband's term), Cristina was the first woman to be elected president by popular vote in Argentina. As a lawyer and senator she has often been compared to Hillary Clinton; as a fashion-conscious political figure with a penchant for chic dresses and designer bags, she also evokes memories of Evita.\n\nCristina's tumultuous presidency has been laced with scandals, unpopular decisions and roller-coaster approval ratings. In March 2008 she significantly raised the export tax on soybeans, infuriating farmers, who soon went on strike and blockaded highways. In June 2009 Kirchner's power base was shattered during the mid-term elections, when her ruling party lost its majority in both houses of Congress. Soon after, she enacted an unpopular law set to break apart Clar\u00edn, a media conglomerate that often reflected unfavorably on her presidency. All the while, Argentina has been hounded by inflation that has been un\u00adofficially estimated at up to 25%.\n\nHer presidency has seen some positive sides, however. The economy grew strongly during the first part of her tenure, bolstered by high consumer spending and strong demand for the country's agricultural exports and manufactured goods. In a true Peronist vein, Cristina implemented a wide range of social programs to beef up the pension system, benefit impoverished children and help fight cases related to crimes against humanity. And in July 2010 she signed a bill that legalized same-sex marriage in Argentina, making it Latin America's first country to do so.\n\nOn October 27, 2010, Cristina's presidency was dealt a serious blow when N\u00e9stor Kirchner died suddenly of a heart attack. As N\u00e9stor was expected to run for the presidency in 2011, this was widely seen as a disaster for the Kirchner dynasty. But the country rallied around Cristina's sorrow, and her popularity in early 2011 remained high enough that she ran for office again and was easily re-elected. She had run on a platform that appealed to the populist vote, promising to raise incomes, restore industry and maintain Argentina's economic boom. Her approach worked like a charm, but her popularity wasn't to last. For more, Click here.\n\nUnlike your typical politician, Amado Boudou \u2013 Cristina Kirchner's vice president \u2013 drives a Harley-Davidson and jams with his band on a Fender guitar. But like your typical politician, he's been accused of embezzlement and money laundering.\n\nTimeline\n\n1536\n\nSpanish aristocrat Pedro de Mendoza reaches the R\u00edo de la Plata and attempts to set up a permanent settlement, only to return to Spain within four years.\n\n1580\n\nBuenos Aires is reestablished by Spanish forces, but the city remains a backwater for years, in comparison to growing strongholds in central and northwestern Argentina.\n\n1660\n\nBuenos Aires' population is around 4000; it will take another century for it to double.\n\n1776\n\nBuenos Aires becomes capital of the new Spanish Viceroyalty of the R\u00edo de la Plata, which included what are today Bolivia, Argentina, Paraguay and Uruguay.\n\n1806 & 1807\n\nBritish troops raid the city but are beaten back by the people of Buenos Aires in two battles, now celebrated as La Reconquista (the Reconquest) and La Defensa (the Defense).\n\nMay 25, 1810\n\nBuenos Aires declares its independence from Spain, although actual independence is still several years off. The city renames its main square Plaza de Mayo to commemorate the occasion.\n\n1829\n\nFederalist _caudillo_ Juan Manuel de Rosas takes control of Buenos Aires and becomes its governor; BA's influence increases dramatically during his 23-year reign.\n\n1852\n\nFederalist and former Rosas ally Justo Jos\u00e9 de Urquiza defeats Rosas at the Battle of Caseros and, in 1853, draws up Argentina's first constitution.\n\n1862\n\nBartolom\u00e9 Mitre, governor of Buenos Aires province, poet and founder of _La Naci\u00f3n n_ ewspaper, becomes president after defeating Urquiza's federal forces.\n\n1868\n\nIntellectual Domingo Faustino Sarmiento is elected president. He encourages immigration, ramps up public education and pushes to Europeanize the country.\n\n1869\u201395\n\nThe Argentine economy booms, immigration skyrockets and Buenos Aires' population grows from 95,000 to 670,000.\n\n1869\u201395\n\nTango emerges in Buenos Aires.\n\n1871\n\nSerious shortages of water and an inadequate sewerage systems leads to a severe outbreak of yellow fever that kills more than 10% of the city's population.\n\n1887\n\nConstruction of Puerto Madero begins.\n\n1888\n\nThe first Teatro Col\u00f3n, located on Plaza de Mayo, is demolished.\n\n1897\n\nPuerto Madero is completed, but Eduardo Madero \u2013 the businessman contracted for the project \u2013 has died four years earlier.\n\n1908\n\nAfter 20 years of construction, the second Teatro Col\u00f3n opens with a presentation of Giuseppe Verdi's opera _Aida_.\n\n1916\n\nHip\u00f3lito Yrigoyen, leader of the Radical Party popular with the middle classes, is elected president and introduces minimum wage to counter inflation; he's re-elected in 1928.\n\n1930\n\nHip\u00f3lito Yrigoyen is overthrown in a military coup led by General Jos\u00e9 F\u00e9lix Uriburu, who stays in power for two years, after which civilian rule is restored.\n\n1946\n\nPopulist Lieutenant-General Juan Domingo Per\u00f3n is elected president; Per\u00f3n and his young wife Eva Per\u00f3n ('Evita') make sweeping changes to the political structure.\n\n1952\n\nEvita dies of cancer on July 26 at age 33, one year into Juan Per\u00f3n's second term as president. Her death severely weakes the political might of her husband.\n\n1955\n\nAfter the economy slides into recession, Per\u00f3n loses further political clout; he is thrown from the presidency and exiled to Spain after a military coup.\n\n1976\u201383\n\nUnder the military leadership of General Jorge Videla, Argentina is launched into the Dirty War. In eight years, up to 30,000 people 'disappear'.\n\n1982\n\nGeneral Leopoldo Galtieri provokes the UK into a war over control of the Falkland Islands (Las Islas Malvinas), but Argentina is easily defeated by the British.\n\n1983\n\nThe military regime collapses, ending the Dirty War; civilian government is restored under Radical leader Ra\u00fal Alfons\u00edn, but he leaves office early due to growing inflation problems.\n\n1989\n\nPeronist Carlos Menem succeeds Alfons\u00edn as president and overcomes the hyperinflation that reached nearly 200% per month by instituting free-market reforms.\n\n1992\n\nA bomb attack at the Israeli embassy kills 29 and injures over 200.\n\n1994\n\nEighty-five people are killed and over 100 are wounded when a Jewish community center is bombed.\n\n1999\n\nThe mayor of Buenos Aires, Fernando de la R\u00faa, is voted president of Argentina as a result of dissatisfaction with the corrupt Menem administration; he inherits $114 billion in public debt.\n\n2001\u201302\n\nArgentina commits the largest debt default in world history; Argentina's economy is ruined, which sparks massive riots and looting around the country.\n\n2003\n\nN\u00e9stor Kirchner \u2013 a governor from Patagonia's province of Santa Cruz \u2013 is sworn in as Argentina's president, with 22% of the vote.\n\n2005\n\nAn amnesty law that protected Dirty War military officers suspected of human-rights abuses is abolished\n\n2007\n\nLawyer, senator and former first lady Cristina Fern\u00e1ndez de Kirchner becomes Argentina's first woman president elected by popular vote.\n\nMay 2010\n\nArgentina celebrates its bicentennial with a bang; BA's Av 9 de Julio shuts down for many colorful festivities, and the Teatro Col\u00f3n reopens after four years of restoration.\n\nOct 27, 2010\n\nN\u00e9stor Kirchner dies of heart failure.\n\n2011\n\nHer husband's untimely death late the previous year leaves Cristina Kirchner in the lurch, but she easily wins re-election in October.\n\n2012\n\nInflation is running at about 25% (though government figures say it's less than 10%). Kirchner passes a law restricting the sale of US dollars, creating huge black-market demand.\n\n2013\n\nKirchner loses major support during October's mid-term elections. Her dreams of changing the constitution to allow her to run for a third presidential term quickly evaporate.\n\n2014\n\nArgentina experiences its largest currency devaluation since 2002.\nMusic\n\nA variety of music genres are well represented in Buenos Aires, especially when it comes to the city's most famous export, the tango. But BA's music scene is also about hybrids of overlapping sounds and styles. Traditional kinds of folklore, tango and _cumbia_ (Colombian music) are melded with digital technology to create global tunes that are gaining recognition in living rooms and music festivals all around the world.\n\nNo other musician has influenced tango like Carlos Gardel, the legendary singer who epitomized the soul of the genre\n\n##  Tango Music\n\nSmall musical ensembles that accompanied early tango dances were influenced by polka, habanera, Spanish and Italian melodies, plus African _candombe_ drums. The _bandone\u00f3n,_ a type of small accordion, was brought into these sessions and has since become tango's signature instrument. The tango song was permeated with nostalgia for a disappearing way of life; it summarized the new urban experience for the immigrants. Themes ranged from profound feelings about changing neighborhoods to the figure of the mother, male friendship and betrayal by women. The lyrics, sometimes raunchy and sometimes sad, were sung in the street argot known as _lunfardo_.\n\nNo other musician has influenced tango like Carlos Gardel, the legendary singer who epitomized the soul of the genre. He achieved stardom during tango's golden age, then became a cultural icon when his life was cut short by a plane crash at the height of his popularity. Over the years, other figures like Osvaldo Pugliese, Susana Rinaldi and Eladia Bl\u00e1squez have also given life to the tango song. It was \u00c0stor Piazzolla, however, who completely revolutionized the music with his _nuevo tango,_ which introduced jazz and classical-music currents into traditional songs \u2013 and ruffled some feathers along the way.\n\nToday, a clutch of new arrivals is keeping tango music alive and well, and in the spotlight. The most popular is the 12-musician cooperative Orquesta T\u00edpica Fern\u00e1ndez Fierro (www.fernandezfierro.com), with its charismatic singer Walter Chino Laborde and several fantastic albums boasting new arrangements of traditional tangos. An award-winning documentary was made about them by Argentine-born, Brooklyn-based director Nicolas Entel.\n\nTwo other young orchestras to watch out for are Orquesta T\u00edpica Imperial (www.orquestaimperial.com.ar), which sometimes plays at _milongas_ around town, and El Afronte (www.elafronte.com.ar), which plays on Monday and Wednesday at Bendito and Maldita Milongas in San Telmo (both at Per\u00fa 571).\n\n###  THE STARS OF TANGO\n\n####  Gardel\n\nIn June 1935 a Cuban woman committed suicide in Havana, and a woman in New York and another in Puerto Rico tried to poison themselves, all over the same man \u2013 whom none of them had ever met. The man was tango singer Carlos Gardel, known as El Zorzal Criollo (the King of Tango) or the songbird of Buenos Aires, who had just died in a plane crash in Colombia.\n\nBorn in France, Gardel was the epitome of the immigrant porte\u00f1o whose destitute single mother brought him to Buenos Aires at the age of three. In his youth he worked at a variety of menial jobs and entertained his neighbors with his rapturous singing. A performing career began after he befriended Uruguayan-born Jos\u00e9 Razzano, and the two of them sang together in a popular duo until Razzano lost his voice. From 1917 onward Gardel performed solo.\n\nCarlos Gardel played an enormous role in creating the tango _canci\u00f3n_ (song). Almost single-handedly, he took the style out of Buenos Aires' tenements and brought it to Paris and New York. His crooning voice, suaveness and overall charisma made him an immediate success in Latin American countries. The timing couldn't have been better, as he rose to fame in tango's golden years of the 1920s and 1930s. Gardel became a recording and film star, but his later career was tragically cut short by that fatal plane crash. Every day a steady procession of pilgrims visits Carlos Gardel's sarcophagus in the Cementerio de la Chacarita in Buenos Aires, where a lit cigarette often smolders between the metal fingers of his life-size statue. The large, devoted community of his followers, known as _gardelianos,_ cannot pass a day without listening to his songs or watching his films. Another measure of his ongoing influence is the common saying 'Gardel sings better every day'. Elvis should be so lucky.\n\n####  Piazzolla\n\nGardel may have brought tango to the world, but it was El Gran \u00c1stor (the Great \u00c1stor), as Argentines like to call \u00c1stor Piazzolla (1921\u201392), who pushed its limits. The great Argentine composer and _bandone\u00f3n_ (small accordian) virtuoso, who played in the leading An\u00edbal Troilo orchestra in the late 1930s and early 1940s, was the greatest innovator of tango. He revolutionized traditional tango by infusing it with elements of jazz and classical music such as counterpoints, fugues and various harmonies.\n\nThis new style, known as _nuevo tango_ , became an international hit in Europe (Piazzolla lived on and off in Italy and France) and North America (he spent his early years and a couple of later stints in New York). In his native land, however, it encountered considerable resistance; a saying even stated 'in Argentina everything may change \u2013 except the tango'. It took years for Piazzolla's controversial new style to be accepted, and he even received death threats for his break with tradition.\n\nPiazzolla was an incredibly prolific composer; it's estimated that his output includes some 1000 pieces. These include soundtracks for about 40 films; an opera that he wrote with poet Horacio Ferrer, _Mar\u00eda de Buenos Aires_ ; and compositions based on texts and poems by Jorge Luis Borges.\n\nPiazzolla's legacy lives on. Some of the greatest contemporary musicians, such as Yo-Yo Ma, have recorded albums dedicated to El Gran \u00c1stor (such as the 1999 _Soul of the Tango \u2013 The Music of \u00c1stor Piazzolla_ ). The new wave of electronic tango often samples his music and the 2003 album _Astor Piazzolla Remixed_ features his songs remixed with dance beats and added vocals, all done by an international cast of DJs and producers.\n\n##  Neo Tango\n\nLike the rest of the music scene in Buenos Aires, a newer tango has evolved that's a hybrid of sounds and styles \u2013 making tango cool again with a younger audience. Musicians have been sampling and remixing classic tango songs, adding dance beats, breaks, scratches and synth lines, and committing other delightful heresies. This edgy genre has been called by many names: fusion tango, electrotango, tango electronica or neo-tango. Paris-based Gotan Project (a Franco-Suizo-Argentine trio) was the first to popularize this style, with its debut album _La Revancha del Tango,_ which throws into the mix samples from speeches by Che Guevara and Eva Per\u00f3n and remixes by the likes of Austrian beatmeister Peter Kruder. Its follow-up albums don't break the mold like the first but are still great if you like the Gotan sound.\n\nThe best of the genre's albums so far is likely _Bajofondo Tango Club,_ by the Grammy-winning collective Bajofondo. It's spearheaded by Argentine producer Gustavo Santaolalla, who won two best-original-score Oscars for _Brokeback Mountain_ and _Babel;_ he also scored the films _Amores Perros_ and _21 Grams,_ and produced albums by such prominent artists as Caf\u00e9 Tacuba and Kronos Quartet. Praised as more Argentine than Gotan Project (whose trio is composed of only one Argentine), its first album has subtle performances by a variety of _bandoneonistas_ within a hypnotic framework of lounge, house and trip-hop. Its third album, _Mar Dulce_ , is a catchy creation that throws more folk and rock into the mix and has a strong international cast of singers, such as Spanish hip-hop star Mala Rod\u0155iguez and the Canadian-Portuguese Nelly Furtado.\n\nAnother neo-tango collective to make an international name for itself is Tanghetto, with two Latin Grammy nominations. This six-member group mixes elements of rock, jazz, flamenco and _candombe_ (a drum-based musical style of Uruguay).\n\n###  TANGO MUSIC HALL OF FAME\n\n\u00bb Carlos Di Sarli (1903\u201360) \u2013 Pianist, composer and orchestra leader.\n\n\u00bb Juan D'Arienzo (1900\u201376) \u2013 Violinist and orchestra leader.\n\n\u00bb Carlos Gardel (1890\u20131935) \u2013 Singer and actor.\n\n\u00bb \u00c1stor Piazzolla (1921\u201392) \u2013 _Bandone\u00f3n_ (accordionlike instrument) player and composer.\n\n\u00bb Roberto Goyeneche (1926\u201394) \u2013 Singer.\n\n\u00bb An\u00edbal Troilo (1914\u201375) \u2013 _Bandone\u00f3n_ player, composer and orchestra leader.\n\n\u00bb Osvaldo Pugliese (1905\u201395) \u2013 Pianist, composer and orchestra leader.\n\n\u00bb Enrique Santos Disc\u00e9polo (1901\u201351) \u2013 Composer and poet.\n\n\u00bb Homero Manzi (1907\u201351) \u2013 Lyricist and poet.\n\n\u00bb Horacio Salg\u00e1n (b 1916) \u2013 Pianist, composer and orchestra leader.\n\n\u00bb Julio Sosa (1926\u201364) \u2013 Singer.\n\n\u00bb Eladia Bl\u00e1zquez (1931\u20132005) \u2013 Singer, pianist and composer.\n\n\u00bb Susana Rinaldi (b 1935) \u2013 Singer.\n\n\u00bb Adriana Varela (b 1952) \u2013 Singer.\n\n##  Rock & Pop\n\nArgentine rock started in the late 1960s with a trio of groups \u2013 Almendra (great melodies and poetic lyrics), Manal (urban blues) and Los Gatos (pop) \u2013 leading the pack. Evolution was slow, however; the 1966 and 1976 military regimes didn't take a shine to the liberalism and freedom that rock represented. It didn't help that anarchy-loving, beat-music rocker Billy Bond induced destructive mayhem at a 1972 Luna Park concert, re-enforcing the theme of rock music as a social threat.\n\nUnderground groups and occasional concerts managed to keep the genre alive, and after the Falklands War in 1982 (when English lyrics were not actually allowed on the air) radio stations founded _rock nacional_ and helped the movement's momentum gain ground. Argentine rock produced national icons like Charly Garci\u00e1 (formerly a member of the pioneering group Sui Generis) and Fito P\u00e1ez (a socially conscious pop-hippie). Sensitive poet-songwriter Alberto Luis Spinetta of Almendra fame also had an early influence on the Argentine rock movement, later incorporating jazz into his LPs. Another mythical figure is Andr\u00e9s Calamaro, frontman of the popular 1970s band Los Abuelos de la Nada. He later emigrated to Spain, where he formed the acclaimed Los Rodr\u00edguez; he's been performing solo since the late 1990s.\n\nMore recent Argentine groups that have played _rock nacional_ include Soda Stereo (ex-member Gustavo Cerati's _Fuerza Natural_ won the 2010 Latin Grammy for best rock album); cultlike Patricio Rey y sus Redonditos de Ricota (its legendary leader Indio Solari now has a solo career); versatile Los Piojos (mixing rock, blues, ska and the Uruguayan music styles _murga_ and _candombe_ ); and Los Ratones Paran\u00f3icos, who in 1995 opened for the Rolling Stones' spectacularly successful five-night stand in Buenos Aires.\n\nLos Fabulosos Cadillacs (who were the winners of a Grammy award in 1998 for best alternative Latin rock group) have popularized ska and reggae, along with groups such as Los Aut\u00e9nticos Decadentes, Los Pericos and Los Cafres. Almafuerte, descended from the earlier Herm\u00e9tica, is Buenos Aires' leading heavy-metal band. The bands Dos Minutos and Expulsados seek to emulate punk-rock legends the Ramones, who are popular in Argentina. Other classic bands include hippyish Los Divididos (descendants of the famous group Sumo), Mendozan trio Los Enanitos Verdes and the wildly unconventional Babas\u00f3nicos.\n\nThe band Les Luthiers satirizes the middle class or the military using irreverent songs played with unusual instruments, many of which have been built by the band members themselves. Another quirky character is the late Sandro \u2013 known as the Argentine Elvis \u2013 whose death in January 2010 saw tens of thousands of _porte\u00f1os_ gather in the streets of Buenos Aires to mourn his demise.\n\nArgentine women also rock. Singer Patricia Sosa has a captivating voice and performs a mix of rock, soul and blues; her closest counterpart in the English-speaking world is Janis Joplin. The most recent singer-songwriter who has gained fame abroad is Juana Molina, whose ambient music with electronic flair has been compared to Bjork's. Juana Chang and the Wookies combined indie-rock, garage and punk (she now sings with the Kumbia Queers). Keep an eye out for Denise Murz, a Lady Gaga\u2013style electro-pop diva, and the multitalented, folksy Sol Pereyra.\n\nToday some of Argentina's most cutting-edge bands include catchy Miranda! (electro-pop), wacky Bersuit Vergarabat (utilizing multigenre tunes with political, offensive and wave-making lyrics), free-willed La Renga (blue collar, no nonsense and political), La Portuaria, who collaborated with David Byrne (rock fusion influenced by jazz and R&B), and Valentin y Los Volcanos (indie-pop with great guitar music). And don't miss the multicultural, alternative and eclectic Kevin Johansen.\n\nCharly Garc\u00eda's version of the Argentine national anthem does what Jimi Hendrix did for 'The Star-Spangled Banner', but it earned Garc\u00eda a court appearance for 'lacking respect for national symbols'.\n\n##  Blues & Jazz\n\nThe high degree of crossover between Buenos Aires' blues and rock scenes is illustrated by the path of the late guitar wizard Pappo (1950\u20132005). An elder statesman, Pappo was in the groundbreaking rock group Los Abuelos de la Nada and became involved with the seminal blues-rock band Pappo's Blues, as well as Los Gatos and others. He played hard-driving, full-tilt rockin' blues and was especially great when covering such American masters as Howlin' Wolf, BB King and Muddy Waters.\n\nGuitarist-singer Miguel 'Botafogo' Vilanova is an alumnus of Pappo's blues and an imposing figure in his own right. Also worth checking out is La Mississippi, a seven-member group that has been performing rock-blues since the late 1980s. Memphis La Blusera was around BA's blues scene for a long time until it broke up in 2008; it once worked with North American legend Taj Mahal.\n\nLalo Schifrin is an Argentine pianist, composer and conductor with a jazz background; he's most famous for writing the _Mission: Impossible_ theme. He's also won four Grammy awards and has been nominated for six Oscars. In the late 1950s, Schifrin performed with Gato Barbieri, another notable composer and jazz saxophonist. Carlos Alberto Franzetti is a big-band composer who wrote _The Mambo Kings_ (1992) and won a Latin Grammy in 2001 for his _Tango Fatal_ album.\n\nGuitarist Luis Salinas is known for his mellow and melodic tunes that run along George Benson lines but are a bit less poppy; be sure to check out his jazz takes on such traditional Argentine forms as the _chacarera, chamam\u00e9_ and tango. Dino Saluzzi, a _bandone\u00f3n_ player originally from Salta who began recording in the '70s, was one of the first Argentine musicians to mix folklore, tango and jazz. Dino's son Jos\u00e9 is a renowned guitarist in his own right.\n\nAnother musician and son of an Argentine jazz legend is Javier Malosetti, son of pianist Walter Malosetti. Javier's group Electrohope blends jazz, blues, rock and swing with Latin rhythms and funk. Meanwhile, jazz guitarist Tom\u00e1s Bec\u00fa's debut album, _Bushwick_ (2007), is stellar. For wildly experimental jazz check out the Gord\u00f6loco Tr\u00edo, which fuses ambient, funk and jazz in its 20-minute-long songs.\n\nDrummer Sebasti\u00e1n Peycer\u00e9, who favors a funk-tinged fusion, has played with the likes of Paquito D'Rivera, BB King and Stanley Jordan. Finally, BA's own version of the Sultans of Swing is the Caoba Jazz Band, who for years has been playing 1920s and '30s New Orleans\u2013style jazz for the love of it.\n\nBorn in C\u00f3rdoba in the early 1940s, _cuarteto_ is Argentina's original pop music. Despised by the middle and upper classes for its arresting rhythm and offbeat musical pattern (called the ' _tunga-tunga_ '), as well as for its working-class lyrics, it's definitely music from the margins. Although definitively _cordob\u00e9s_ (from C\u00f3rdoba), it's played in working-class bars, dance halls and stadiums throughout the country.\n\n##  Latin & Electronica\n\nBuenos Aires' young clubbers have embraced the _m\u00fasica tropical_ trend that's swept Latin America in recent years. Many a BA booty is shaken to the lively, Afro-Latin sounds of salsa, merengue and especially _cumbia_. Originating in Colombia, _cumbia_ combines an infectious dance rhythm with lively melodies, often carried by brass. An offshoot is _cumbia experimental_ or _cumbia villera,_ a fusion of _cumbia_ and gangsta posturing with a punk edge and reggae overtones. Born of Buenos Aires' shantytowns, its aggressive lyrics deal with marginalization, poverty, drugs, sex and the Argentine economic crisis.\n\nA forerunner of the movement is Axel Krygier, the king of psychedelic Latin, whose latest album _Pesebre_ (2010) is a brilliant fusion of jazz, rock, _cumbia_ , electronica, Argentine folklore and experimental sounds. Kumbia Queers is a female band from Argentina and Mexico whose version of _cumbia_ is known as tropipunk.\n\nDance music is big in BA, with DJs working the clubs well into the morning. A few major electronic names to look out for are Bad Boy Orange (big on drums and bass); Aldo Haydar (a true veteran of progressive house); local boy made international star Hern\u00e1n Cattaneo (you loved him at Burning Man, remember?); Gustavo Lamas (a blend of ambient pop and electro house) and Diego Ro-K (also known as the Maradona of Argentine DJs).\n\nOne of BA's most `interesting music spectacles is La Bomba del Tiempo, a collective of drummers that features some of Argentina's leading percussionists. Its explosive performances are conducted by Santiago V\u00e1zquez, who communicates with the musicians through a language of mysterious signs \u2013 the result is an incredible improvisational union that simulates electronic dance music and sounds different every time. During the summer it plays open-air at Ciudad Cultural Konex (Click here) every Monday evening; it's also featured at various happenings and parties in BA's clubs.\n\nIn 2007 electronic musicians from Zizek Records, a homegrown BA label, created 'digital _cumbia_ ' by fusing various forms of _cumbia_ and Argentine traditional music with reggaeton, dance-hall, hip-hop and electronic beats.\n\n## Folk Music\n\nThe folk music of Argentina is inspired by generations of immigrants and spans a variety of styles, including _chacarera, chamam\u00e9_ and _zamba._ The late Atahualpa Yupanqui was a giant of Argentine folk music, which takes much of its inspiration from the northwestern Andean region and countries to the north, especially Bolivia and Peru. Los Chalchaleros, a northern Argentine folk institution, was around from 1948 to 2003. Probably the best-known Argentine folk artist outside of South America, however, is the late Mercedes Sosa of Tucum\u00e1n, whose progressive, politicized lyrics earned her the title 'the voice of the voiceless ones'.\n\nCurrent contemporary performers include El Chaque\u00f1o Palavecino of Salta; Suna Rocha (also an actress) of C\u00f3rdoba; Antonio Tarrag\u00f3 Ross; V\u00edctor Heredia; Le\u00f3n Gieco (aka 'The Argentine Bob Dylan') and the Conjunto Pro M\u00fasica de Rosario. Singer-songwriter-guitarist Horacio Guarany's 2004 album _Cantor De Cantores_ was nominated for a Latin Grammy in the Best Folk Album category.\n\nOf the younger generation, the artists to watch out for are Chango Spasiuk (an accordion player who popularized _chamam\u00e9_ music abroad), Mariana Baraj (a singer and percussionist who experiments with Latin America's traditional folk music as well as elements of jazz, classical music and improvisation) and Soledad Pastorutti (whose first two albums have been Sony's top sellers in Argentina \u2013 ever!).\n\nEvery genre of Argentine music is experiencing the same hybrid phenomenon of blending electronic music with more traditional sounds. Digital folklore, much like digital _cumbia_ and neo-tango, is exploding. Tonolec, a duo (singer and synth player), combine traditional folk songs of the Toba indigenous community from Argentina's north (some of which have been passed down orally) with an electronic sound. The singer also uses traditional instruments in live gigs, creating a warm, world-music-style fusion. Two other digital-folklore groups to look out for are Onda Vaga, an acoustic band with smooth harmonies that add a jazzy feel to traditional folklore sounds, and Tremor, who mixes Andean flutes, the Argentine _bombo leg\u00fcero_ (drum), electric guitars and a synthesizer into a blend of ancient and digital sounds. Finally, there's Chancha via Circuito, whose 2010 album _R\u00edo Arriba_ mixes melodic flutes and slow tempos, making it more meditative than dance oriented.\n\n_Murga_ is a form of athletic musical theater composed of actors and percussionists. Primarily performed in Uruguay, _murga_ in Argentina is more heavily focused on dancing than singing. You're most likely to see this exciting musical art form at Carnaval celebrations.\nLiterature & Cinema\n\nPerhaps because of its history of authoritarian rule, Argentina has developed a strong literary heritage, with many contemporary writers using the country's darkest moments as inspiration for their complex and sometimes disturbing novels. Leading the classic writers' pack are Jorge Luis Borges, Julio Cort\u00e1zar and Ernesto S\u00e1bato.\n\nBuenos Aires is also home to Argentina's vibrant, evolving film industry. The country has won two Oscars for best foreign-language film (in 1985 and 2009) \u2013 the only Latin American country ever to have won the award \u2013 and continues to produce excellent directors and movies.\n\nArgentines are pretty well read \u2013 their literacy rate is over 97%. And in 2011 Buenos Aires was voted Unesco World Book Capital.\n\n##  Literature\n\nOne of Argentina's most influential pieces of classic literature is the epic poem by Jos\u00e9 Hern\u00e1ndez, _Mart\u00edn Fierro_ (1872). Not only did this story about a gaucho outlaw lay the foundations of the Argentine _gauchesco_ literary tradition but also it inspired the name of the short-lived but important literary magazine of the 1920s that published avant-garde works based on the 'art for art's sake' principle.\n\nJulio Cort\u00e1zar (1914\u201384) is an author well known to readers outside Argentina. He was born in Belgium to Argentine parents, moved to Buenos Aires at age four and died in self-imposed exile in Paris at the age of 70. His stories frequently plunge their characters out of everyday life into surrealistic situations. One such story was adapted into the film _Blow-Up_ by Italian director Michelangelo Antonioni. Cort\u00e1zar's novel _Hopscotch_ takes place simultaneously in Buenos Aires and Paris and requires the reader to first read the book straight through, then read it a second time, 'hopscotching' through the chapters in a prescribed but nonlinear pattern for a completely different take on the story.\n\nAnother member of Borges' literary generation is Ernesto S\u00e1bato (1911\u20132011), whose complex and uncompromising novels have been extremely influential on later Argentine literature. _The Tunnel_ (1948) is S\u00e1bato's engrossing existentialist novella of a porte\u00f1o painter so obsessed with his art that it distorts his relationship with everything and everyone else.\n\nAdolfo Bioy Casares (1914\u201399) and Borges were close friends and occasional collaborators. Bioy's sci-fi novella _The Invention of Morel_ (1940) gave Alain Resnais the plot for his classic film _Last Year at Marienbad_ and also introduced the idea of the holodeck decades before _Star Trek_ existed.\n\nThe contemporary, postboom generation of Argentine writers is more reality-based, often reflecting the influence of popular culture and directly confronting the political angles of 1970s authoritarian Argentina. One of the most famous postboom Argentine writers is Manuel Puig (1932\u201390), whose first love was cinema. Much of his writing consists solely of dialogue, used to marvelous effect. Puig's novel _The Buenos Aires Affair_ (1973) is a page-turner delving into the relationship between murderer and victim (and artist and critic), presented as a deconstructed crime thriller. His most famous work is _Kiss of the Spider Woman_ (1976), a captivating story of a relationship that develops between two men inside an Argentine prison; it was made into the 1985 Oscar-winning film starring William Hurt. Being openly gay and critical of Per\u00f3n did not help his job prospects in Argentina, so Puig spent many years in exile.\n\nAnother prolific writer is Tom\u00e1s Eloy Mart\u00ednez (1934\u20132010). His _The Per\u00f3n Novel_ (1988), a fictionalized biography of the controversial populist leader, and its sequel, _Santa Evita_ (1996), which traces the worldwide travels of Evita's embalmed corpse, were both huge hits.\n\nAward-winning Ricardo Piglia (b 1941) is one of Argentina's most well-known contemporary writers. He pens hard-boiled fiction and is best known for his socially minded crime novels with a noir touch, such as _The Absent City_ (1992), _Money to Burn_ (1997) and _Nocturnal T arget_ (2010).\n\n_Metegol_ (Foosball; 2013) is a 3D film directed by Juan Jos\u00e9 Campanella; it cost US$22 million, making it the most expensive Argentine movie ever produced.\n\nOsvaldo Soriano (1943\u201397), perhaps Argentina's most popular contemporary novelist, wrote _Funny Dirty Little War_ (1986) and _Winter Quarters_ (1989). Juan Jos\u00e9 Saer (1937\u20132005) penned short stories and complex crime novels, while Rodrigo Fres\u00e1n (b 1963), the youngster of the postboom generation, wrote the international bestseller _Argentine History_ (1991).\n\nThe first novel of Federico Andahazi (b 1963), _The Anatomist,_ caused a stir when it was published in 1997. Its ticklish theme revolves around the 'discovery' of the clitoris by a 16th-century Venetian who is subsequently accused of heresy. Andahazi based his well-written book on historical fact, and manages to have some fun while still broaching serious subjects. His prize-wining _El Conquistador_ (2006) is a historical novel about an Aztec youth who 'discovers' Europe before Columbus reaches America, while his latest book _Pecar como Dios manda_ (To Sin Like You Mean It; 2008), hypothesizes that to understand the essence of a society you have to understand the web of sexual relations on which it's built.\n\nTwo of the younger generation of Argentine writers are Washington Cucurto and Gabriela Bejerman. Cucurto runs Elo\u00edsa Cartonera, a small publishing house that releases books by young authors made of recycled cardboard collected by the city's _cartoneros_. Bejerman, a multimedia artist who launched a music career as Gaby Bex, released an album in 2007 that incorporates some of her poetry with electro music. Other names to watch out for are Andr\u00e9s Newman, Oliverio Coelho and Pedro Mairal.\n\n###  JORGE LUIS BORGES\n\nMany of the greatest lights of Argentine literature called Buenos Aires home and all but one had been extinguished by the end of the 20th century. The light that burned brightest was without doubt Jorge Luis Borges (1899\u20131986), one of the foremost writers of the 20th century. A prolific author and an insatiable reader, Borges possessed an intellect that seized on difficult questions and squeezed answers out of them. Though super-erudite in his writing, he was also such a jokester that it's a challenge to tell when he's being serious and when he's pulling your leg (though often it's a case of both at once). From early on one of his favorite forms was the scholarly analysis of nonexistent texts, and more than once he found himself in trouble for perpetrating literary hoaxes and forgeries. A few of these are contained in his _Universal History of Iniquity_ (1935), a book that some point to as the origin of magic realism in Latin American literature.\n\nBorges' dry, ironic wit is paired (in his later work) with a succinct, precise style that is a delight to read. His paradoxical _Ficciones_ (1944) \u2013 part parable, part fantasy \u2013 blurs the line between myth and truth, underscoring the concept that reality is only a matter of perception and the number of possible realities is infinite. Other themes that fascinated Borges were the nature of memory and dreams, labyrinths, and the relationship between the reader, the writer and the written piece. _Collected Fictions_ (1999) is a complete set of his stories.\n\nThough he received numerous honors in his lifetime \u2013 including the Cervantes Prize, the Legion of Honor and an OBE \u2013 Borges was never conferred the Nobel. He joked of this in typical fashion: 'Not granting me the Nobel Prize has become a Scandinavian tradition. Since I was born they have not been granting it to me.'\n\nPilgrims can head to his last residence in BA: a private apartment building near the corner of Florida and Santa Fe in Retiro. Look for a plaque on the wall.\n\n##  Cinema\n\nBuenos Aires is at the center of the Argentine film industry, which generated a wave of directors and films of the New Argentine Cinema. While this movement can't be pinned down as a school of cinema, as it includes a hodgepodge of themes and techniques, it is certainly a new movement of film-making that has been attracting international attention, earning awards and screenings at festivals in New York, Berlin, Rotterdam and Cannes.\n\nSadly, much of the homegrown production is more acclaimed abroad than in Argentina, where people are generally more drawn to multiplexes that show Hollywood flicks and romantic comedies. Perhaps it's because these art-house films deal with themes that are too close to home \u2013 such as survival, alienation, the search for identity and suppressed sexuality.\n\nThe film that's considered to have spearheaded the New Argentine Cinema is _Rapado_ by Mart\u00edn Rejtman, a minimalist 1992 feature that for the first time pushed the boundaries in a country where films were generally heavy with bad dialogue. In the late 1990s the government withdrew subsidies pledged to film schools and the movie industry. Despite this, two films ignited 'the new wave' \u2013 the low-budget _Pizza, birra, faso_ (Pizza, Beer, Cigarettes; 1998) by Adri\u00e1n Caetano and Bruno Stagnaro, and Pablo Trapero's award-winning _Mundo gr\u00faa_ (Crane World; 1999), a black-and-white portrait of Argentina's working-class struggles.\n\nTrapero went on to become one of Argentina's foremost f ilmmakers, whose credits include _El bonaerense_ (2000); the ensemble road movie _Familia rodante_ (Rolling Family; 2004); _Nacido y criado_ (Born and Bred; 2006) a stark story about a Patagonian man's fall from grace; and the 2010 noir film _Carancho,_ a love story whose protagonist is a sleazy opportunist who frequents emergency rooms and accident scenes to find new clients for his legal firm. Trapero's most recent film is _Elefante blanco_ (White Elephant; 2012), which screened at Cannes.\n\nOne of the brightest stars of the New Argentine Cinema is Daniel Burman, Argentina's answer to Woody Allen, who deals with the theme of identity in the character of a young Jew in modern-day Buenos Aires. His films include _Esperando al mes\u00edah_ (Waiting for the Messiah; 2000), _El abrazo partido_ (Lost Embrace; 2004) and _Derecho de familia_ (Family Law; 2006). Burman's other claim to fame is his co-production of Walter Salles' Che Guevara\u2013inspired _The Motorcycle Diaries_. His most recent film, _Dos hermanos_ (Brother and Sister; 2010), the story of aging siblings who've recently lost their mother, is based on the Argentine novel _Villa Laura_.\n\nAnother director to have made a mark on Argentina cinema is the late Fabi\u00e1n Bielinsky. He left behind a small but powerful body of work that includes his award-winning feature _Nueve reinas_ (Nine Queens; 2000), which inspired a 2004 Hollywood remake, _Criminal_. His last film, the 2005 neo-noir flick _El Aura,_ screened at Sundance and was the official Argentine entry for the 2006 Oscars.\n\nLucrecia Martel has left an indelible trace on Argentina's contemporary cinema. Her 2001 debut, _La ci\u00e9naga_ (The Swamp), and the 2004 follow-up, _La ni\u00f1a santa_ (The Holy Girl), both set in Martel's native Salta province, deal with the themes of social decay, Argentine bourgeois and sexuality in the face of Catholic guilt. Another acclaimed director, Carlos Sorin, takes us to the deep south of Argentina in two of his neorealist flicks, the 2002 _Historias m\u00ednimas_ (Minimal Stories) and the 2004 _Bomb\u00f3n el perro_ (Bomb\u00f3n the Dog).\n\nArgentina's biggest film event is the Buenos Aires International Festival of Independent Film, held in April. Check out www.bafici.gov.ar for more information.\n\nJuan Jos\u00e9 Campanella's _El hijo de la novia_ (Son of the Bride) received an Oscar nomination for best foreign-language film in 2001. His 2004 award-winning film _Luna de avellaneda_ (Moon of Avellaneda) is a masterful story about a social club and those who try to save it. And in 2010 Campanella won the Oscar for best foreign-language film with his _El secreto de sus ojos_ (The Secret in Their Eyes).\n\nOther noteworthy films include Luis Puenzo's Oscar-winning _La historia oficial_ (The Official Story; 1985), Sandra Gugliotta's bust-out directorial debut _Un d\u00eda de suerte_ (A Lucky Day; 2002), _Un oso rojo_ (A Red Bear; 2002) by Israel Adri\u00e1n Caetano, _Roma_ (2004) by Adolfo Aristarain, _Iluminados por el fuego_ (Enlightened by Fire; 2006) by Tristan Bauer and _El hombre de al lado_ (The Man Next Door; 2009) by Mariano Cohn and Gast\u00f3n Duprat.\n\nAnd up-and-coming director is Luc\u00eda Puenzo (daughter of Luis Puenzo). Her _XXY_ (2007) won multiple awards at Cannes that year; it follows the travails of a 15-year-old hermaphrodite. In 2013 Puenzo directed _Wakolda_ (The German Doctor), a true story about the family who unknowingly lived with Josef Mengele during his exile in South America.\n\n###  VICTORIA & SILVINA OCAMPO\n\nIn 1931 Victoria Ocampo (1890\u20131979) \u2013 a writer, publisher and intellectual \u2013 founded _Sur,_ a renowned cultural magazine that introduced Virginia Woolf, Albert Camus and TS Eliot to Argentine readers. _Sur_ also featured writers like Jorge Luis Borges, Adolfo Bioy Casares, Ernesto S\u00e1bato and Julio Cort\u00e1zar.\n\nOcampo was an inexhaustible traveler and a pioneering feminist, and was loathed by some for her lack of convention. A ferocious opponent of Peronism, chiefly because of Per\u00f3n's interference with intellectual freedom, Ocampo was arrested at her summer chalet, Villa Victoria, at the age of 63. She entertained her fellow inmates by reading aloud and acting out scenes from novels and cinema.\n\nOcampo never went to university, but her voracious appetite for knowledge and her love of literature led her to become Argentina's leading lady of letters. She hosted intellectuals from around the globe at Villa Victoria, in Mar del Plata, creating a formidable literary and artistic salon. (The villa is now a cultural center.)\n\nToday you can also visit Victoria Ocampo's restored mansion in San Isidro, Villa Ocampo (www.villaocampo.org), for a reminder of a bygone era.\n\nIf Victoria is remembered as a lively essayist and a great patroness of writers, her younger sister, Silvina, was the literary talent, writing both short stories and poetry. Silvina won several literary prizes for her work, and in 1940 she married Adolfo Bioy Casares, a famous Argentine writer and friend of Jorge Luis Borges.\nArt & Architecture\n\nOver the years, Argentina has been able to boast various notable artists such as Antonio Berni, Benito Quinquela Mart\u00edn and Marta Minuj\u00edn, each with their own style and trigger for helping to break the mold of what was and is acceptable in their country's art world. As for architecture, Buenos Aires still has examples of the many styles in vogue at one time or another throughout the city's life. You'll find old and new juxtaposed in sometimes jarring and often enchanting ways, though the new has been asserting itself more and more in recent years.\n\nSome of the best times to be in Buenos Aires if you want to discover the art world is during the Arte BA festival (www.arteba.com) and the annual La Noche de los Museos (Night of the Museums; www.lanochedelosmuseos.gob.ar).\n\n##  Art\n\nEduardo S\u00edvori (1847\u20131918) was one of Argentina's first notable artists and well-known realist painters. He created landscapes and portraits and helped found one of Argentina's first artist guilds. Other early artists included C\u00e1ndido L\u00f3pez (1840\u20131902) \u2013 a soldier who learned to paint with his left hand after losing his right arm in war \u2013 and Ernesto de la C\u00e1rcova, who depicted social issues such as poverty.\n\nLino Enea Spilimbergo (1896\u20131964) was a diverse painter and engraver whose subjects ranged from classical to postimpressionism to stark and surreal human figures. His contemporary, Antonio Berni (1905\u201381), would sometimes visit shantytowns and collect materials to use in his works. Various versions of his theme _Juanito Laguna ba\u00f1\u00e1ndose_ (Juanito Laguna Bathing) \u2013 a protest against social and economic inequality \u2013 have commanded wallet-busting prices at auctions. You can see both artists' work in the restored ceiling murals of the Galer\u00edas Pac\u00edfico shopping center (Click here).\n\nOther famous Argentine artists of this era are Juan Carlos Castagnino, a realist and figurative painter; Jorge de la Vega, who dabbled not only in various styles of visual art but also became a popular singer and songwriter; and Emilio Pettoruti, who affronted Buenos Aires with his 1924 cubist exhibition. Roberto Aizenberg was well known as one of Argentina's top surrealists.\n\nOne of the more interesting contemporary artists is Roberto Jacoby (b 1944), who has been active in diverse fields since the 1960s, from organizing socially flavored multimedia shows to setting up audiovisual installations. His most famous work, _Darkroom_ , is a video performance piece with infrared technology meant for a single spectator.\n\nGuillermo Kuitca (b 1961) is known for his imaginative techniques that include the use of digital technology to alter photographs, maps and other images and integrate them into larger-themed works. His work is on display at major international collections and he's had solo and group shows at key art expos around the world.\n\nOther internationally recognized artists who experiment with various media are Buenos Aires\u2013born, New York\u2013based Liliana Porter, who imaginatively plays with video, paintings, 3D prints, photos and an eclectic collection of knickknacks; Graciela Sacco, whose politically and socially engaging installations often use public space as their setting; and the photographer Arturo Aguiar, known for playing with light and shadow in his mysterious works. Also watch out for highly eclectic Argentine pop artist Marta Minuj\u00edn, who has added fire to the Marshall McLuhan quote 'Art is anything you can get away with'.\n\nBuenos Aires has also seen a rise in urban art interventions, a movement of diverse activist artists whose work calls attention to social and urban issues in the city's public spaces. The most prominent figure is Marino Santa Mar\u00eda (www.marinosantamaria.com), whose award-\u00adwinning _Proyecto Calle Lan\u00edn_ is a series of colorful murals along the narrow Calle Lan\u00edn in the up-and-coming artist neighborhood of Barracas.\n\nThe late Benito Quinquela Mart\u00edn, who put the working-class barrio of La Boca on the artistic map, painted brightly colored oils of life in the factories and on the waterfront. Xul Solar, a multitalented phenomenon who was a good friend of Jorge Luis Borges, painted busy, Klee-inspired dreamscapes. The former homes of both Quinquela (Click here) and Solar (Click here) are now museums showcasing their work.\n\n###  BEST ART MUSEUMS\n\n\u00bb Museo Nacional de Bellas Artes (Click here)\n\n\u00bb Museo de Arte Latinoamericano de Buenos Aires (Click here)\n\n\u00bb Museo Nacional de Arte Decorativo (Click here)\n\n\u00bb Colecci\u00f3n de Arte Amalia Lacroze de Fortabat (Click here)\n\n\u00bb Fundaci\u00f3n Proa (Click here)\n\n##  Architecture\n\nLittle trace remains of the modest one-story adobe houses that sprang up along the mouth of the Riachuelo following the second founding of Buenos Aires in 1580. Many of them were occupied by traffickers of contraband, as the Spanish crown forbade any direct export or import of goods from the settlement. The restrictions made the price of imported building materials prohibitively high, which kept things simple, architecturally speaking. For an idea of how BA's first settlements used to be, visit El Zanj\u00f3n de Granados (Click here) in San Telmo.\n\nBuenos Aires' Cabildo (Click here) is a fair example of colonial architecture, although its once plaza-spanning colonnades were severely clipped by the construction of Av de Mayo and the diagonals feeding into it. The last of the Cabildo's multiple remodels was a 1940s restoration to its original look, minus the colonnades. Most of the other survivors from the colonial era are churches. Sharing Plaza de Mayo with the Cabildo, the Catedral Metropolitana (Click here) was begun in 1752 but not finished until 1852, by which time it had acquired its rather secular-looking neoclassicist facade.\n\nMany examples of postindependence architecture (built after 1810) can be found in the barrios of San Telmo, one of the city's best walking areas, and Montserrat. San Telmo also holds a wide variety of vernacular architecture such as _casas chorizos_ (sausage houses) \u2013 so called for their long, narrow shape (some have a 2m frontage on the street). The perfect example is Casa M\u00ednima (at San Lorenzo 380).\n\nIn the latter half of the 19th century, as Argentina's agricultural exports soared, a lot of money accumulated in Buenos Aires, in both private and government hands. All parties were interested in showing off their wealth by constructing elaborate mansions, public buildings and wide Parisian-style boulevards. Buildings in the city in the first few decades of the boom were constructed mostly in Italianate style, but toward the end of the 19th century a French influence began to exert itself. Mansard roofs and other elements gave a Parisian look to parts of the city, and by the beginning of the 20th century art nouveau was all the rage.\n\nAmong the highlights of the building boom's first five decades is the presidential palace, known as the Casa Rosada (Click here), created in 1882 by joining a new wing to the existing post office. Others include the showpiece Teatro Col\u00f3n (Click here) and the imposing Palacio del Congreso.\n\nThe 1920s saw the arrival of the skyscraper, in the form of the 100m-high, 18-story Palacio Barolo (Click here). This rocket-styled building was the tallest in Argentina (and one of the tallest in South America) from its opening in 1923 until the completion of the 30-story art deco Edificio Kavanagh (Click here) in 1936. The Kavanagh in turn, when finished, was the largest concrete building in the world and remains an impressive piece of architecture.\n\nIn the 1930s, in Palermo and Recoleta, fancy apartment buildings started popping up. This trend would continue intermittently into the 1940s, by which time the city would also have a subway system with multiple lines.\n\nRogelio Yrurtia (1879\u20131950) was one of Argentina's best-known sculptors. Many of Yrurtia's pieces are displayed at his own museum (Click here) in the BA neighborhood of Belgrano, or you can see his masterpiece _Canto al Trabajo_ on the Plazoleta Olaz\u00e1bal in San Telmo.\n\nBuenos Aires continued to grow upward and outward during Juan Per\u00f3n's spell in power (1946 to 1955). Though the economy flagged, anonymous apartment and office blocks rose in ever greater numbers. Bucking the trend were such oddball buildings as the Banco de Londres on Reconquista, designed in 1959 by Clorindo Testa, whose long architectural career in BA began in the late 1940s. The bank was finished by 1966, but Testa's Biblioteca Nacional (Click here) \u2013 which must've looked pretty groovy to him on the drawing board in 1962 \u2013 was hideously dated by the time it opened (following many delays) in 1992. Its style is somewhere between late Offshore Oil Platform and early Death Star.\n\nA heartening trend of 'architectural recycling' took off in Buenos Aires in the latter 20th century and continues today, helping to preserve the city's glorious old structures. Grand old buildings have been remodeled (and sometimes augmented) to become luxury hotels, museums and cultural centers; notable examples include the Centro Cultural del Bicentenario (Click here), which used to be the city's main post office, and the Usina del Arte (Click here), a concert hall that used to be an old electricity factory. Old markets have also been restored to their original glory to live again as popular shopping malls, such as the Mercado de Abasto (Click here) and Galer\u00edas Pac\u00edfico (Click here).\n\nAt the same time, the first decade of the 21st century has seen an increasingly modern skyline develop in Buenos Aires. Soaring structures of glass and steel tower above earlier efforts, many innovative and quite striking, such as the Edificio Rep\u00fablica in Buenos Aires' downtown. It was designed by C\u00e9sar Pelli, who also did Kuala Lumpur's Petronas Towers.\n\nThe renovation of Puerto Madero turned dilapidated brick warehouses into offices, upscale restaurants and exclusive lofts. Contrasting with these charming low, long buildings is one of the city's tallest structures, the 558ft-high Torres El Faro, standing at the eastern section of Puerto Madero. It's a pair of joined towers that now house fancy apartments. Other architectural gems here include Calatrava's Puente de la Mujer (Click here) and the glass-domed Museo Fortabat (Click here) by Uruguayan-born architect Rafael Vi\u00f1oli.\n\nStreet art \u2013 which is not illegal in Buenos Aires \u2013 has become more and more prominent in neighborhoods like Barracas, San Telmo, La Boca and Palermo. Colorful murals, political stencils and graffiti-inspired creations cover public and private walls, sometimes commissioned by the city and property owners.\nSurvival Guide\n\nTransportation\n\nARRIVING IN BUENOS AIRES\n\nEzeiza Airport\n\nAeroparque Airport\n\nBoat\n\nGETTING AROUND BUENOS AIRES\n\nBicycle\n\nBus\n\nCar & Motorcycle\n\nTaxi & Remise\n\nTrain\n\nTram\n\nSubte (Underground)\n\nDirectory A\u2013Z\n\nCustoms Regulations\n\nDiscount Cards\n\nElectricity\n\nEmbassies & Consulates\n\nEmergency\n\nGay & Lesbian Travelers\n\nHealth\n\nInsurance\n\nInternet Access\n\nLanguage Courses\n\nLegal Matters\n\nMedical Services\n\nMoney\n\nPost\n\nPublic Holidays\n\nSafe Travel\n\nTaxes & Refunds\n\nTelephone\n\nTime\n\nToilets\n\nTourist Information\n\nTravelers with Disabilities\n\nVisas\n\nWomen Travelers\nTransportation\n\n##  Arriving in Buenos Aires\n\nBA is Argentina's international gateway and easily accessible from North America, Europe and Australasia, as well as most other capital cities in South America. Aerol\u00edneas Argentinas is the country's main airline, but smaller Argentine airlines are in constant flux and come and go very frequently. Even airline offices will often move. Always check current travel information during your tenure here.\n\nFlights, cars and tours can be booked online at lonelyplanet.com.\n\n###  EZEIZA ARRIVAL & DEPARTURE TIPS\n\n\u00bb Citizens from some countries have to pay a reciprocity fee ( _tasa de reciprocidad_ ) before arriving in Argentina; ideally you'll be reminded of this when you buy your airplane ticket. This fee is equal to what Argentines are charged for visas to visit those countries. You'll need to pay this fee online via credit card; see www.migraciones.gov.ar\/accesibleingles and click on 'Pay your Reciprocity Rate' on the left column. These fees are US$100 for Australians (good for one year), US$160 for Americans (good for 10 years) and US$75 for Canadians (per entry \u2013 sucks, eh? Or go for the US$150, good-for-five-years option). You'll need to prepay this fee before entering Argentina via other airports, borders or ports (that means you, cruise ship passengers) too, or you might be turned around.\n\n\u00bb To change money at Ezeiza, don't use a _cambio_ (exchange house) there \u2013 their rates are generally bad. Better rates are found at the local bank branch; after exiting customs, pass the rows of transport booths, go outside the doors into the reception hall and make a U-turn to the right to find Banco de la Naci\u00f3n's small office. Its rates are identical to downtown offices, there's an ATM and it's open 24 hours, though long lines are common. There are other ATMs at Ezeiza.\n\n\u00bb There's a tourist information booth (  24hr) just beyond the city's Taxi Ezeiza stand.\n\n\u00bb When leaving Buenos Aires, get to Ezeiza at least two to three hours before your international flight out; security and immigration lines can be long (and be aware that traffic is often bad _getting_ to Ezeiza; it can take an hour or more). Also, even when you get past main security there may be bag checks at the gate, and neither food nor liquids may be allowed onto airplanes. Eat and drink up before boarding.\n\n###  Ezeiza Airport\n\nAlmost all international flights arrive at Buenos Aires' Ezeiza airport (EZE; officially Aero\u00adpuerto Internacional Ministro Pistarini), about 35km south of the center. Ezeiza is a modern airport with decent services like ATMs, restaurants, bookstore, pharmacy, duty-free shops and a small post office (Ezeiza Airport;   9am-5pm Mon-Fri, to noon Sat, mailbox 24hr). There's also a Telecentro _locutorio_ (long distance telephone office; open 24 hours) with telephone cabins and internet access, near Farmacity and McDonald's. Wi-fi is available at La Pausa Restaurant, past gate 9 upstairs and in Terminal C.\n\nFlight information (  5480-6111; www.aa2000.com.ar.) is available in English and Spanish.\n\n###  WATCH THAT POCKET!\n\nWhen traveling on BA's crowded bus or Subte lines, watch for pickpockets. They can be well dressed, men or women, often with a coat slung over their arm to hide nefarious activities going on near your bag or pocket. Occasionally there are several of them, working as a team, and they'll try to shove or distract you. The best thing to do is not look like a tourist, keep your wallet well ensconced in your front pocket, wedge your purse under your arm and wear your backpack in front \u2013 like the locals do. Don't make yourself an easy target and they'll move on \u2013 and you might not even notice they exist.\n\n####  Bus\n\nIf you're alone, the best way to and from Ezeiza is taking a shuttle with transfer companies such as Manuel Tienda Le\u00f3n (MTL;   4315-5115; www.tiendaleon.com; Av Madero 1299, Ezeiza airport). You'll see its stand immediately as you exit customs, in the transport 'lobby' area. Frequent shuttles cost AR$80 to AR$95 per person to the city center, run all day and night and take 40 to 60 minutes, depending on traffic. They'll deposit you either at the MTL office (from where you can take a taxi) or at some limited central addresses.\n\nAnother shuttle service, directed at independent travelers, is Hostel Shuttle (  4511-8723; www.hostelshuttle.com.ar). Check the website for prices, schedules and drop-off destinations (only at certain hostels), and try to book ahead. You can also try www.minibusezeiza.com.ar.\n\nIf you're really on a penny-pinching budget, take public bus 8, which costs AR$6 and can take up to two hours to reach the Plaza de Mayo area. Catch it outside the Aerol\u00edneas Argentinas terminal (Terminal B), a 200m walk from the international terminal. You'll need coins; there's a Banco de la Naci\u00f3n just outside customs.\n\n####  Taxi\n\nIf taking a taxi, avoid MTL's overpriced taxi service. Instead, go past the transport 'lobby' area outside customs, walk past the taxi touts, and you'll see the freestanding city taxi stand (with a blue sign saying Taxi Ezeiza (  5480-0066; www.taxiezeiza.com.ar)). In late 2013 it charged AR$270 to the center. Note that if you pre-arrange your taxi back to Ezeiza after your stay in BA, the rate can be 20% cheaper (this is due to airport and taxi regulations); your taxi driver might remind you about this fact.\n\n####  Chauffeur-Driven Car\n\nFor a special treat, reserve a luxury car from Silver Star Car (  in Argentina 15-6826-8876, in the USA 214-502-1605; www.silverstarcar.com); you'll be driven by native English speakers to the destination of your choice (US$150). There are car-rental booths at Ezeiza, but we do not recommend renting a car for your stay in Buenos Aires.\n\n###  CRITICAL MASS\n\nStarted in San Francisco in 1992, Critical Mass is an international bicycling event dedicated to improving bicycle awareness and reaffirming cyclists' rights. It's held in over 300 cities throughout the world, and though its goal is different in each city, it's not meant to be a race, a protest, a demonstration or a means to cause trouble by maliciously blocking vehicular traffic.\n\nIn Buenos Aires, Masa Cr\u00edtica has no set leaders or destinations. People just show up at the Obelisco with their bikes at 4pm on the first Sunday of each month \u2013 and start riding somewhere. Expect several hundred people to participate \u2013 up to a thousand on nice, warm days. And you'll see all sorts of folks: activists, families, hipsters, foreigners, on tall bikes, low bikes and everything in between. There's even the odd skateboarder, rollerblader or cyclist in costume. You'll be riding in the streets, but you're more protected than you think \u2013 there's safety in numbers. Lots of yelling, cheering and horn-blowing is required; expect to have a blast.\n\nThere's also a Masa Nocturna every full moon, starting at the Obelisco at 9pm. It's the same concept, but the end point is at the planetarium in Palermo's Parque 3 de Febrero \u2013 and that's where the full-moon party starts.\n\n###  Aeroparque Airport\n\nMost domestic flights use Aeroparque airport (Aeroparque Jorge Newbery;   5480-6111; www.aa2000.com.ar), a short distance from downtown Buenos Aires.\n\n####  Bus\n\nManuel Tienda Le\u00f3n (MTL;   4315-5115; www.tiendaleon.com; Av Madero 1299, Ezeiza airport) does hourly transfers from Ezeiza to Aeroparque for AR$95. To get from Aeroparque to the center, take public bus 33 or 45 (don't cross the street; take them going south). MTL also has shuttles to the center for AR$30.\n\n####  Taxi\n\nA taxi to the center costs around AR$80.\n\nHandy Bus Routes\n\nRoute | Bus\n\n---|---\n\nMicrocentro to Palermo Viejo | 111\n\nMicrocentro to Plaza Italia (in Palermo) | 29, 59, 64\n\nOnce to Plaza de Mayo to La Boca | 64\n\nPlaza de Mayo to Ezeiza airport (placard says 'Ezeiza') | 8\n\nPlaza Italia to Microcentro to San Telmo | 29\n\nPlaza Italia to La Boca via Retiro & Plaza de Mayo | 152, 29\n\nPlaza Italia to Recoleta to Microcentro to Constituci\u00f3n | 59\n\nPlaza San Mart\u00edn to Aeroparque airport | 33, 45\n\nRecoleta to Congreso to San Telmo to La Boca | 39\n\nRetiro to Plaza de Mayo to San Telmo | 22\n\n###  Boat\n\nThere's a regular ferry service to and from Colonia and Montevideo, both in Uruguay. Most ferries leave from the Buquebus terminal at the corner of Avs Ant\u00e1rtida Argentina and C\u00f3rdoba in Puerto Madero; there are other Buquebus offices at Av C\u00f3rdoba 879 and at Pueyrred\u00f3n 1786. Colonia Express (  4317-4100; www.coloniaexpress.com; C\u00f3rdoba 753) is cheaper than Buquebus but has fewer departures; its central office is at Av C\u00f3rdoba 753, but its terminal is in an ugly, industrial neighborhood near La Boca.\n\nBoth companies have many more launches in the busy summer season; book online in advance for discounts.\n\n###  RETIRO BUS TERMINAL\n\nBuenos Aires' modern Retiro bus terminal is 400m long, three floors high and has slots for 75 buses. The bottom floor is for cargo shipments and luggage storage, the top for purchasing tickets, and the middle for everything else.\n\nThere's an information booth (  4310-0700;   6am-midnight) that provides general bus information and schedules, plus a tourist office (Retiro bus station, across from bus slot 35;   7:30am-2:30pm Mon-Fri, to 4:30pm Sat & Sun) near Puente 3 on the main floor, on the same level as bus slot 35. Other services include ATMs, lockers, telephone offices (some with internet access), restaurants, cafes and dozens of small stores. Various _remise_ (call-taxi) booths are also available, some open 24 hours. You can also purchase a SUBE card (www.sube.gob.ar) and get it charged at a booth near the southern entrance.\n\nYou can buy a ticket to practically anywhere in Argentina and departures are fairly frequent to the most popular destinations. Reservations are not necessary except during peak summer and winter holiday seasons.\n\n##  Getting Around Buenos Aires\n\n###  Bicycle\n\nBuenos Aires is generally not the best city to cycle around: traffic is dangerous, with scant respect for cyclists, and the biggest vehicle wins the right of way, so bikes are low on the transport totem pole.\n\nHowever, things are getting better. New bike lanes were installed in 2010 and are expanding; a bike-share program (www.mejorenbici.gob.ar) also exists, but it's more geared towards residents (one-hour rental limits). Every day there seem to be more cyclists on the streets \u2013 but even so, BA has a long way to go to be seen as a bike-friendly city.\n\nThe city's best places for two-wheeled exploration are Palermo's parks and the Reserva Ecol\u00f3gica Costanera Sur; on sunny weekends you can rent bikes at these places. You can also join city bike tours, which include bicycle and guide (Click here).\n\n###  Bus\n\nBuenos Aires has a huge and complex bus system. If you want to get to know it better you'll have to buy a _Guia T_ \u2013 it's sold at any newsstand, but get the pocket version (about AR$10). It details hundreds of the city's bus routes. Just look at the grids to find out where you are and where you're going, and find a matching bus number. You can also check www.xcolectivo.com.ar for an online version. For information (in English) on how to get to your destination on city buses, check out www.omnilineas.com\/argentina\/buenos-aires\/city-bus\/.\n\nBus ticket machines on board will give you small change from your coins. Rides around town are cheap; just mention your destination to the driver and he'll cue the machine. If you're staying in BA awhile, consider getting a SUBE card (www.sube.gob.ar) for cheaper fares and to make paying easier.\n\nMost bus routes (but not all) run 24 hours; there are fewer buses at night. Seats up front are offered to the elderly, pregnant women and those with young children.\n\nIf you're arriving in Buenos Aires at Retiro bus terminal for the first time, it'll be difficult to sort out the local bus system \u2013 there are a lot of bus lines outside. It's worth spending a few pesos to take a _remise_ (radio taxi) directly to your destination. There are two small _remise_ booths near bus slots 8 and 9 that are open 24 hours. And remember to keep an eye on your bags at this station!\n\n###  EASY BUS TICKETS\n\nYou can buy nearly any long-distance bus ticket without taking a special trip to Retiro bus station. Use the practical booking services of Omnil\u00edneas (  4326-3924; www.omnilineas.com; Maip\u00fa 459, 9B). Just reserve and buy your ticket over the website, and either print it out at home or pick it up at the office. Prices are the same as at Retiro bus station, and English is spoken.\n\n###  Car & Motorcycle\n\nAnyone considering driving in Buenos Aires should know that most local drivers are reckless, aggressive and even willfully dangerous. They'll ignore speed limits, road signs, road lines and often traffic signals. They'll tailgate you mercilessly and honk even before signals turn green. Buses are a nightmare to reckon with, potholes are everywhere, traffic is worse every day and parking can be nonexistent in places (and cost a bundle). To top it off, pedestrians haphazardly cross the road, seeming to beg to be run over at times.\n\nReconsider your need to have a car in this city: public transportation will often get you anywhere faster, cheaper and with much less stress. And you won't have to worry about the police, who have been known to stop cars to check for violations, while subtly asking for _coimas_ (bribes). If this happens to you when you weren't doing anything illegal, insist on contacting your embassy \u2013 too much trouble for some officers.\n\n####  Driving\n\nDriving outside BA is another story. Drivers are still crazy, but there are fewer of them, and you'll have more flexibility in your travels. If you drive in Argentina \u2013 especially in your own car \u2013 it may be worth joining the Autom\u00f3vil Club Argentino (ACA), which has many nationwide offices. ACA recognizes members of overseas affiliates, such as the American Automobile Association (AAA), and often grants them similar privileges, including discounts on maps, accommodations, camping, tours and other services. For more information contact the ACA head office (  4802-6061; www.aca.org.ar; Av del Libertador 1850).\n\n####  Rental\n\nIf you want to rent a car, expect to pay US$30 to US$50 or more per day. International chains can be more expensive than local rental agencies; call around. You'll need to be at least 21 years of age and have a valid driver's license; having an international driver's license wouldn't be a bad idea, though you don't necessarily need one. A credit card and passport are also necessary.\n\nAvis (  4326-5542; www.avis.com.ar; Cerrito 1535)\n\nHertz (  4816-8001; www.hertz.com.ar; Paraguay 1138)\n\nNew Way (  4515-0331; www.new-wayrentacar.com; Marcelo T de Alvear 773)\n\nIf you have experience driving scooters and are up to the challenge of getting around BA on an electric version, check out Green Scooter (  6091-9060; www.thegreenscooter.net; Soler 4717;   10:30am-6:30pm). You'll need to pay an AR$2500 deposit (cash or credit card) and bring your passport. The cost is AR$190 per day, helmet and lock included.\n\nFor motorcycle rentals, be at least 25 years of age and head to Motocare (  4761-2696; www.motocare.com.ar\/rental; Echeverria 738, Vicente Lopez). Bring your own helmet and riding gear. Crossing into Chile, Uruguay, Paraguay and Brazil is possible.\n\n###  Taxi & Remise\n\nBuenos Aires' very numerous (about 40,000) and relatively inexpensive taxis are conspicuous by their black-and-yellow paint jobs. They click every 200m (or every minute of waiting time) and cost 20% more after 6pm. Make sure that the meter's set to the current price when you start your ride. Drivers do not expect a big tip, but it's customary to let them keep small change. Taxis looking for passengers will have a red light lit on the upper right corner of their windshield.\n\nMost cab drivers are honest workers making a living, but there are a few bad apples in the bunch. Try not to give them a 100 peso note for a small fare; sometimes they're short on change, but there have been cases where the driver quickly and deftly replaces a larger bill with a smaller (or fake) one. One solution is to state how much you are giving them and ask if they have change for it (' _\u00bfTiene usted cambio de un cien?'_ \u2013 'Do you have any change for a hundred?').\n\nBe wary of receiving counterfeit bills; drivers have been known to switch your valid bill for a fake one. If you're suspicious this might happen, note aloud the last three numbers\/letters on a bill as you're paying him (it's pretty much always a 'him').\n\nAt night the driver will turn on the light (or _luz_ ) so you can carefully check your change (look for a watermark on bills). They'll do the same with your bills. And make sure you get the right change.\n\nTry to have an idea of where you're going or you might be taking the 'scenic' route (though also be aware there are many one-way streets in BA, and your route to one place may be quite different on the way back). A good way to give the impression that you know where you're going is to give the taxi driver an intersection rather than a specific address. Also, if you are obviously a tourist going to or from a touristy spot, it's not a good idea to ask how much the fare is; this makes quoting an upped price tempting, rather than using the meter. And try not to take a taxi right outside a tourist spot or after you've withdrawn money from an ATM \u2013 walk a block or two and flag one down instead.\n\nFinally, make an attempt to snag an 'official' taxi. These are usually marked by a roof light and license number printed on the doors; the words _radio taxi_ are usually a good sign. Official drivers must display their license on the back of their seat or dashboard; you can write down the taxi's number and agency telephone in case you have problems with the ride or forgot something.\n\nMost _porte\u00f1os_ will recommend you call a _remise_ instead of hailing cabs off the street. A _remise_ looks like a regular car and doesn't have a meter. It costs a bit more than a street taxi but is considered more secure, since an established company sends them out. Most hotels and restaurants will call a _remise_ for you; expect a short wait for them to show up.\n\nHelpful Train Information\n\nDestination(s) | Station | Contact\n\n---|---|---\n\nBelgrano, San Isidro, Tigre, Rosario | Retiro | L\u00ednea Mitre (  0800-222-8736; www.mitresarmiento.com.ar)\n\nSouthern suburbs & La Plata | Constituci\u00f3n | L\u00ednea Roca (  0800-362-7622; www.ugofe.com.ar\/general_roca)\n\nBah\u00eda Blanca, Tandil & Mar del Plata | Constituci\u00f3n | Ferrobaires (  0810-666-8736; www.ferrobaires.gba.gov.ar)\n\nSouthwestern suburbs & Luj\u00e1n | Once | L\u00ednea Sarmiento (  0800-222-8736; www.mitresarmiento.com.ar)\n\n###  Train\n\nTrains connect Buenos Aires' center to its suburbs and nearby provinces. They're best for commuters and only occasionally useful for tourists. Several private companies run different train lines; train stations are all served by Subte.\n\n###  Tram\n\nA light-rail system in Puerto Madero is called the Tranv\u00eda del Este. It's currently 2km long and has only four stops, with plans to extend the line from Retiro to Constituci\u00f3n. It's cheap to ride, but consider skipping it \u2013 stroll Puerto Madero's lovely cobbled lanes instead.\n\n###  Subte (Underground)\n\nBA's Subte (www.subte.com.ar) opened in 1913 and is the quickest way to get around the city, though it can get mighty hot and crowded during rush hour. It consists of _l\u00edneas_ (lines) A, B, C, D, E and H. Four parallel lines run from downtown to the capital's western outskirts, while _L\u00ednea_ C runs north\u2013south and connects the two major train stations of Retiro and Constituci\u00f3n. _L\u00ednea_ H runs from Once south to Av Caseros, with plans to expand it.\n\nOne-ride magnetic cards for the Subte cost AR$3.50. To save time and hassle, buy several rides, since queues can get backed up (especially during rush hour). If you're planning on staying in BA for a while, SUBE (www.sube.gob.ar) is a convenient, rechargeable card that negates the need for coins.\n\nTrains operate from 5am to around 10:30pm Monday to Saturday and 8am to around 10pm Sunday and holidays, so don't rely on the Subte to get you home after dinner. Service is frequent on weekdays; on weekends you'll wait longer. At some stations platforms are on opposite sides, so be sure of your direction _before_ passing through the turnstiles.\n\n###  SUBE CARD\n\nIf you're planning on staying in BA for a while, the SUBE card (www.sube.gob.ar) is a very handy and inexpensive rechargeable card that you can use for the Subte, local buses and some trains. It saves you money and you don't have keep a stash of coins on hand. Get it at some kioskos and Correo Argentino or OCA post offices around the city (check the website for locations or look for the SUBE logo at businesses). Ezeiza airport and Retiro bus station also have Sube booths where you can get and recharge this card. You'll need your passport or a copy. Charging the card itself is easy, and can be done at many kiosks or Subte stations.\n\n##  Tours\n\nBuenos Aires has tours for every style and stripe, from the large tourist-bus variety to guided bike rides to straight-up walks. The city of Buenos Aires organizes free monthly tours from April to December, with themes ranging from art to historic bars to particular neighborhoods. Stop by any government tourist information office (Click here) for more information.\n\nMost companies listed here offer tours in English and possibly other languages; some companies also do private tours.\n\nSay Hueque (  5258-8740; www.sayhueque.com; Thames 2062) This independent travel agency specializes in customized adventure trips all around Argentina, and will also make air, bus and hotel reservations. It offers various BA tours as well. Also downtown at Viamonte 749, 6th fl, and in San Telmo at Chile 557.\n\nTangol (  4363-6000; www.tangol.com; Av Florida 971, Suite 31) Do-all agency that offers city tours, tango shows, guides to _f\u00fatbol_ games, hotel reservations, Spanish classes, air tickets and country-wide packages. Also offers unusual activities including helicopter tours and skydiving. Another branch in San Telmo at Defensa 831.\n\nAnda Responsible Travel (  3221-0833; www.andatravel.com.ar; Ag\u00fcero 1050, 4A) Most notable for its La Boca tour, which introduces travelers to local organizations working towards improving the lives of its citizens. Also does many tours around Argentina that benefit local citizens, which are sometimes indigenous groups.\n\nBA Cultural Concierge (  15-3876-5937; www.baculturalconcierge.com) Customized single-day tours of Buenos Aires, Tigre and the Mataderos fair. Also tango and _milonga_ tours.\n\nBA Walking Tours (  15-5773-1001; www.ba-walking-tours.com) Day tours, night tours, historic tours and tango tours.\n\nBiking Buenos Aires (  4040-8989; www.bikingbuenosaires.com) Friendly American and Argentine guides take you on various tours of Buenos Aires; theme tours include a graffiti and architecture tour.\n\nBuenos Aires Bus (  5239-5160; www.buenosairesbus.com; locals\/foreigners AR$90\/120) Hop-on, hop-off topless bus that runs frequently over two dozen stops (see website).\n\nBuenos Tours (  5984-2444; www.buenostours.com) Well-run private tours guided by friendly, knowledgeable and responsible local expats. Great website too.\n\nCultour (  15-6365-6892; www.cultour.com.ar) Good tours run by teachers and students from UBA (University of Buenos Aires). Prepare to learn the historical and cultural facets of Buenos Aires.\n\nGraffitimundo (  15-3683-3219; www.graffitimundo.com) Excellent tours of some of BA's best graffiti, by those in the know. Learn artists' history and the local graffiti culture. Several tours available, including a La Boca 'Hidden Walls' tour. Stencil workshops too.\n\nSeriema Nature Tours (  5410-3235; www.seriematours.com) It does nature tours to all South America, but around BA the most popular outings are to Costanera Sur.\n\nUrban Biking (  4314-2325; www.urbanbiking.com) One-day cycling tours \u2013 including an alternative 'nightlife' bike trip \u2013 and bike and kayak excursions to Tigre.\nDirectory A\u2013Z\n\n### BOOK YOUR STAY ONLINE\n\nFor more accommodation reviews by Lonely Planet authors, check out <http:\/\/hotels.lonelyplanet.com>. You'll find independent reviews, as well as recommendations on the best places to stay. Best of all, you can book online.\n\n###  Customs Regulations\n\nArgentine officials are generally courteous and reasonable toward tourists. Electronic items, including laptops, cameras and cell phones, can be brought into the country duty free, provided they are not intended for resale. If you have a lot of electronic equipment, however, it may be useful to have a typed list of the items you are carrying (including serial numbers) or a pile of purchase receipts.\n\nDepending on where you have been, officials focus on different things. Travelers south-bound from the central Andean countries may be searched for drugs, and those from bordering countries will have fruit and vegetables confiscated.\n\n###  PRACTICALITIES\n\n\u00bb Popular newspapers include leftist _P\u00e1gina 12_ (www.pagina12.com.ar), centrist _Clar\u00edn_ (www.clarin.com.ar) and moderate _La Naci\u00f3n_ (www.lanacion.com.ar). The _Buenos Aires Herald_ (www.buenosairesherald.com) is the city's English-language daily.\n\n\u00bb There are dozens of radio stations in BA. For tango there's FM 92.7, for Argentine rock try FM 98.3 and for Argentine folk music it's FM 98.7.\n\n\u00bb National channels include Canal 2, 9, 11 and 13. Canal 7 has many educational and cultural programs.\n\n\u00bb Argentina uses the metric system.\n\n\u00bb Smoking is banned in most public spaces such as restaurants, cafes, bars and buses.\n\n###  Discount Cards\n\nTravelers of any age can obtain a Hostelling International card at any HI hostel (www.hostels.org.ar) or at the tiny HI office in Retiro (  4511-8723; www.hostels.org.ar; Av Florida 835). With this card you can obtain discounts at any HI hostel in Argentina, usually 10% to 15% off regular prices. International Student Identity Cards are also sold here; you'll need current student ID.\n\nFor non-HI hostels, check out minihostels (www.minihostels.com), a network of quality, 'good-vibe' hostels throughout Argentina and expanding to other places in Central and South America. The HoLa (www.holahostels.com) card works in a similar way for a different network of hostels.\n\nTravelers over the age of 60 can sometimes obtain discounts on museum admissions and the like. Usually a passport with date of birth is sufficient evidence of age.\n\n###  Electricity\n\nArgentina's electric current operates on 220V, 50 Hertz. There are two types of electric plugs: either two rounded prongs (as in Europe) or three angled flat prongs (as in Australia). See www.kropla.com\/electric2.htm for details. Adapters are readily available from almost any _ferreter\u00eda_ (hardware store), or visit a travel store.\n\nMost electronic equipment (such as cameras, PDAs, telephones, computers and laptops) are dual\/multi-voltage, but if you're bringing in something that's not, use a voltage converter or you might short out your device.\n\n###  Embassies & Consulates\n\nSome countries have both an embassy and a consulate in Buenos Aires, but only the most central location is listed here.\n\nAustralian Embassy (  4779-3500; www.argentina.embassy.gov.au; Villanueva 1400)\n\nCanadian Embassy (  4808-1000; www.canadainternational.gc.ca; Tagle 2828)\n\nFrench Embassy (  4515-7030; www.embafrancia-argentina.org; Cerrito 1399, Retiro)\n\nGerman Embassy (  4778-2500; www.buenosaires.diplo.de; Villanueva 1055)\n\nItalian Consulate (  4114-4800; www.consbuenosaires.esteri.it; Reconquista 572)\n\nSpanish Consulate (  4814-9100; www.spanish-embassy.com\/buenos-aires.html; Guido 1770)\n\nUK Embassy (  4808-2200; www.ukinargentina.fco.gov.uk; Dr Luis Agote 2412)\n\nUS Embassy (  5777-4533; <http:\/\/argentina.usembassy.gov>; Colombia 4300)\n\n###  CONCIERGE SERVICE\n\nBA Cultural Concierge (  15-3876-5937; www.baculturalconcierge.com) Madi Lang's concierge service helps you plan itineraries, arrange airport transportation, run errands, get a cell phone, reserve theater tickets, scout out a potential apartment and do a thousand other things that'll help your trip to run smoothly.\n\n###  Emergency\n\nAmbulance (  107)\n\nFire (  100)\n\nPolice (  911, 101)\n\nTourist Police (Comisar\u00eda del Turista;   0800-999-5000, 4346-5748; Av Corrientes 436;   24hr) Provides interpreters for travel insurance reports.\n\n###  Gay & Lesbian Travelers\n\nArgentina is a strongly Catholic country with heavy elements of machismo. In Buenos Aires, however, there is a palpable acceptance of homosexuality. In 2002 BA became the first Latin American city to legalize same-sex civil unions, and in July 2010 Argentina became the first Latin American country to legalize same-sex marriage. In fact, gay tourism has become so popular that BA is now South America's top gay destination.\n\nArgentine men are more physically demonstrative than their North American and European counterparts, so behaviors such as kissing on the cheek in greeting or a vigorous embrace are considered innocuous even to those who express unease with homosexuals. Lesbians walking hand-in-hand should generally attract little attention, since heterosexual Argentine women sometimes do so, but this would be very conspicuous behavior for males. If you are in any doubt, it's better to be discreet.\n\n###  Health\n\nBuenos Aires is a modern city with good health and dental services. Sanitation and hygiene at restaurants is relatively high, and tap water is generally safe to drink. If you want a glass of tap water, ask ' _\u00bfMe pod\u00e9s traer un vaso de agua de la canilla?_ ' (Can you bring me a glass of tap water?).\n\nPublic health care in Buenos Aires is reasonably good and free, even if you're a foreigner. Waits can be long, however, and quality spotty. Those who can afford it usually opt for the superior private care system, and here most doctors and hospitals will expect payment in cash. Many medical personnel speak English.\n\nIf you develop a life-threatening medical problem you may want to be evacuated to your home country. Since this may cost thousands of dollars, be sure to have the appropriate insurance before you depart. Your embassy can also recommend medical services.\n\nA signed and dated note from your doctor, describing your medical conditions and medications (with their generic or scientific names) is a good idea. It's also a good idea to bring medications in their clearly labeled, original containers. Most pharmacies in Argentina are well supplied.\n\nFor more specific information on vaccinations to get before traveling to Argentina, see wwwnc.cdc.gov\/travel\/destinations\/argentina.htm.\n\n###  Insurance\n\nA travel-insurance policy to cover theft, loss and medical problems is a good idea. Some policies offer a range of lower and higher medical-expense options; the higher ones are chiefly for countries, such as the USA, that have extremely high medical costs. There is a wide variety of policies available, so read the small print.\n\nSome policies specifically exclude 'dangerous activities', which can include scuba diving, motorcycling and even trekking. Check that the policy you're considering covers ambulances and an emergency flight home.\n\n###  NEED AN OFFICE FOR AN HOUR OR A DAY?\n\nThe brainchild of one of BA's many expat entrepreneurs, Areatres (  5353-0333; www.areatresworkplace.com; Malabia 1720;   8:30am-8pm Mon-Fri) is a secure working office where you can rent a desk, cubicle, office or meeting room. There are fax and copy services, complete internet and wi-fi connections, networking social events, a business lounge, a large presentation room and even a Zen-like patio at the back for the stress-prone. Facilities are cutting-edge \u2013 it's like you never left Silicon Valley. It's even eco-conscious. It's also at Humboldt 2036 (  5258-7600; www.areatresworkplace.com; Humboldt 2036).\n\n###  Internet Access\n\nBuenos Aires is definitely online. Internet caf\u00e9s and _locutorios_ (telephone offices) with internet access are common everywhere in the center; you can often find one by just walking a few blocks in any direction. Rates are cheap and connections are quick. To find the @ ( _arroba_ ) symbol, try holding down the Alt key and typing 64. Or ask the attendant ' _\u00bfC\u00f3mo se hace la arroba?'_ ('How do you make the @ sign?').\n\nNearly all hotels have wi-fi or in-room internet connections for guests traveling with their own laptops, and the fancier ones also feature 'business centers' with one or more computers. Many hostels provide free internet to guests. Also, many cafes and restaurants (even McDonald's) offer free wi-fi.\n\n###  Language Courses\n\nBA has become a major destination for students of Spanish. Good institutes are opening up all the time and private teachers are a dime a dozen. Cultural centers also offer language classes; the Centro Cultural Ricardo Rojas (  4954-5523; www.rojas.uba.ar; Av Corrientes 2038) has an especially good range of offerings, from Korean to Russian to Yiddish.\n\nMost private language institutes organize social activities, private classes and (usually) volunteer opportunities. Homestay programs are also available but often cost more than finding a place yourself. Check websites for fees and schedules.\n\nUniversity of Buenos Aires (www.idiomas.filo.uba.ar) Offers intensive, long-term classes (one to four months) in Spanish, Italian, German, French, Portuguese and Japanese. It's cheap, and great for serious students, but classrooms can be run-down.\n\nSpanglish (www.spanglishexchange.com) Set up like speed dating; you'll speak five minutes in English and five in Spanish, then switch partners (and it's a bit of a pick-up scene too).\n\nAcademia Buenos Aires (  4345-5954; www.academiabuenosaires.com; Hip\u00f3lito Yrigoyen 571, 4th fl)\n\nDWS (  4777-6515; www.dwsba.com.ar; Av C\u00f3rdoba 4382)\n\nExpanish (  5252-3040; www.expanish.com; Juan Per\u00f3n 698)\n\nRayuela (  4300-2010; www.spanish-argentina.com.ar; Chacabuco 852, 1st fl, No 11)\n\nVOS (  4812-1140; www.vosbuenosaires.com; Marcelo T de Alvear 1459)\n\nVamos (  5352-0001; www.vamospanish.com; Coronel D\u00edaz 1736)\n\n###  Legal Matters\n\nYou're hardly likely to get involved with the local police if you follow the law. If you drive a car, however, officers are not above petty harassment. So-called safety campaigns often result in motorists receiving citations for minor equipment violations (such as a malfunctioning turn signal) that carry fines. In most cases, corrupt officers will settle for less expensive _coimas_ (bribes), but this requires considerable caution and tact. A discreet hint that you intend to contact your consulate may minimize or eliminate such problems \u2013 often the police count on foreigners' ignorance of Argentine law. Another tactic, whether you know Spanish or not, is to pretend you don't understand what an officer is saying.\n\n###  CULTURAL CENTERS\n\nBuenos Aires has good cultural centers offering all sorts of art exhibitions, classes and events. They're listed in the neighborhood chapters.\n\nThere are also several foreign cultural centers in the Microcentro, such as the Instituto Cultural Argentino-Norteamericano (  5382-1500; www.icana.org.ar; Maip\u00fa 672), which has Spanish classes and workshops; the Alianza Francesa (  4322-0068; www.alianzafrancesa.org.ar; Av C\u00f3rdoba 946), which concentrates on French-themed instruction and arts; and the Instituto Goethe (  4318-5600; www.goethe.de\/hs\/bue; Av Corrientes 319), which offers German-language instruction, lectures, films and even concerts. All have good libraries in their respective languages.\n\nThe British Arts Centre (  4393-6941; www.britishartscentre.org.ar; Suipacha 1333) has well-priced theater, films, music and workshops (among other things) in English and Spanish.\n\n###  Medical Services\n\nHighly regarded hospitals include Hospital Italiano (  4959-0200; www.hospitalitaliano.org.ar; Juan D Per\u00f3n 4190), Hospital Alem\u00e1n (  4827-7000; www.hospitalaleman.org.ar; Pueyrred\u00f3n 1640) and Hospital Brit\u00e1nico (  4309-6400; www.hospitalbritanico.org.ar; Perdriel 74). Another popular medical facility is Swiss Medical (  0810-333-8876; www.swissmedical.com.ar; cnr Santa Fe & Scalabrini Ortiz), with various branches around town.\n\nDental Argentina (  4828-0821; www.dental-argentina.com.ar; Laprida 1621, 2B) provides modern facilities and good dental services with English-speaking professionals.\n\n####  Pharmacies\n\nPharmacies are common in Buenos Aires. The biggest chain is Farmacity (www.farmacity.com), with dozens of branches throughout the city; they're modern, bright and well stocked with sundries. They have a prescription counter and some are open 24 hours. It's hard to miss their blue-and-orange color theme.\n\n###  Money\n\nArgentina's unit of currency is the peso (AR$).\n\nBanks and _cambios_ (foreign-\u00adexchange offices) are common in the city center; banks have longer lines and more limited opening hours but may offer better rates.\n\nFor international transfers, Western Union (www.westernunion.com) has several agents in BA.\n\nCarrying cash and using ATM and credit cards is the way to go in Argentina.\n\n####  ATMs\n\nATMs _(cajeros autom\u00e1ticos)_ are everywhere in BA and the handiest way to get money. ATMs dispense _only_ Argentine pesos (not US dollars, despite what the screen says) and can be used for cash advances on major credit cards. There's often an English-translation option if you don't read Spanish.\n\nThere may be limits per withdrawal, but you may be able to withdraw several times per day \u2013 just beware of per-transaction fees. To avoid having a fistful of large-denomination bills, withdraw odd amounts like 990 pesos.\n\nAlso, a small fee is charged on ATM transactions by the _local_ bank (not including charges by your home bank, which are extra). Note that this is a _per transaction_ fee, so consider taking out your maximum allowed limit \u2013 if you feel safe doing so.\n\n####  Cash\n\nNotes come in denominations of two, five, 10, 20, 50 and 100 pesos. One peso equals 100 centavos; coins come in denominations of five, 10, 25 and 50 centavos, as well as one and two pesos. The $ sign in front of a price is usually used to signify pesos.\n\nDon't be dismayed if you receive dirty and hopelessly tattered banknotes; they will still be accepted everywhere. Some banks refuse worn or defaced US dollars, however, so make sure you arrive in Buenos Aires with pristine bills.\n\nCounterfeiting of both local and US bills has become something of a problem in recent years, and merchants are very careful when accepting large denominations. You should be, too; look for a clear watermark or running thread on the largest bills, and be especially careful when receiving change in dark nightclubs or taxis. For photos, check www.landingpadba.com\/ba-basics-counterfeit-money.\n\nGetting change from large denominations can be a problem for small purchases. Large supermarkets and restaurants are your best bet. Always keep a stash of change with you, in both small bills and coins.\n\nUS dollars are accepted by many tourist-oriented businesses.\n\n####  Credit Cards\n\nMany tourist services, larger stores, hotels and restaurants take credit cards such as Visa and MasterCard, especially for big purchases. Be aware, however, that some businesses add a _recargo_ (surcharge) of up to 10% to credit-card purchases; ask ahead of time. Some lower-end hotels and private businesses will not accept credit cards, and tips can't usually be added to credit-card bills at restaurants. Many places will give you a small discount if you pay in cash, rather than use a credit card.\n\nThe following local representatives can help you replace lost or stolen cards:\n\nAmerican Express (  4310-3000)\n\nMasterCard (  0800-444-5220)\n\nVisa (  4379-3400)\n\n###  BUENOS AIRES' BLUE (IE BLACK) MARKET\n\nBecause many Argentines are desperate for hard currency to combat their country's high inflation, mistrust the peso's stability _and_ are not allowed to easily buy them \u2013 Argentina has a robust black market for US dollars, especially in Buenos Aires. This market is also called the _mercado azul_ (blue market, or 'cambio blue'). The blue market rate can be nearly twice the official exchange rate, though rates fluctuate daily. Many people think this parallel market can't last forever, and the government is constantly tinkering with laws to combat it.\n\nIn BA, some people use this market on Calle Florida, where _arbolitos_ (touts; literally, 'little trees') constantly call out _'cambio, cambio, cambio'_. The _arbolito_ leads the interested party to a _cueva_ (unofficial exchange office) for the transaction. Unobtrusive storefront _cuevas_ also exist in some tourist neighborhoods in BA; many locals use them and know where they are located. Be aware that this shady activity \u2013 although commonplace (newspapers even publish the going blue rate) \u2013 is technically illegal. Scams and fake bills do exist, and unwary travelers make very good targets.\n\nInstead of using _arbolitos_ , some people change money (or pay for services) at certain stores, travel agents, restaurants and accommodations for rates close to the blue market's. Outside BA, some _cambios_ might give you the unofficial rate. Hundred-dollar bills get the highest rates.\n\nNote that ATMs in Argentina don't give out US currency, no matter what their screen says. Some ATMs in Uruguay will, however \u2013 though there are daily limits for withdrawals.\n\nAnother way travelers bypass the official exchange rate is by using international money-transfer services such as www.xoom.com (for those with US bank accounts) or Azimo (for those with UK bank accounts).\n\nDo your research very carefully before coming to Argentina. And no matter how you end up getting your pesos, use them all up before your flight home. It's unlikely you'll be able to change them back to a hard currency at a decent rate \u2013 if at all.\n\n####  Tipping\n\nIn restaurants and cafes it's customary to tip about 10% of the bill for decent service. An interesting note: when your server is taking your bill with payment away, saying ' _gracias_ ' usually implies that the server should keep the change as a tip. If you want change back, don't say ' _gracias_ ' \u2013 say ' _cambio, por favor_ ' instead.\n\nNote that tips can't be added to credit-card bills, so carry cash for this purpose. Also note that the _cubierto_ that some restaurants charge is not a tip; it's a sort of 'cover charge' for the use of utensils and bread. Yes, it's silly, but that's the custom.\n\nBartenders Usually no tip, but it's OK to give a small bill for a drink or good cocktail.\n\nDelivery persons A small bill.\n\nHotel cleaning staff A few pesos per day (only at fine, upscale hotels).\n\nHotel porters A small bill.\n\nRestaurant servers Tip 10%; 15% for fine restaurants with great service.\n\nSpas Tip 15%.\n\nTaxi drivers No tip unless they help with luggage; many people round up to nearest peso.\n\nTour guides Tip 10% to 15%.\n\n####  Traveler's Checks\n\nTraveler's checks are very impractical in Argentina, and even in BA it's very hard to change them. Only the fancier hotels and a few banks and _cambios_ will take them, and they'll charge a very hefty commission. Stores will _not_ change them.\n\nOutside BA it's almost impossible to change trav\u00adeler's checks. If you do decide to bring some, get them in US dollars.\n\n###  OPENING HOURS\n\nThere are always exceptions, but the following are general opening hours:\n\nBanks 8am to 3pm or 4pm Monday to Friday; some open till 1am Saturday.\n\nBars 8pm or 9pm to between 4am and 6am nightly (downtown, some open and close earlier).\n\nCafes 6am to midnight or much later; open daily.\n\nClubs 1am to 2am to between 6am and 8am Friday and Saturday.\n\nOffice business hours 8am to 5pm.\n\nPost offices 8am to 6pm Monday to Friday, 9am to 1pm Saturday.\n\nRestaurants Noon to 3:30pm, 8pm-midnight or 1am (later on weekends).\n\nShops 9am or 10am to 8pm or 9pm Monday to Saturday.\n\n###  Post\n\nThe more-or-less reliable Correo Argentino (www.correoargentino.com.ar) is the government postal service, with numerous branches scattered throughout BA. Essential overseas mail should be sent _certificado_ (registered). For international parcels weighing over 2kg, take a copy of your passport and go to the Correo Internacional near the Retiro bus station. Check the website for all prices.\n\nIf a package is being sent to you, expect to wait awhile for it to turn up within the system (or to receive notice of its arrival). Unless you have a permanent address, your parcel will likely end up at the Correo Internacional. To collect the package you'll have to wait \u2013 first to get it and then to have it checked by customs. There might also be a small holding fee, charged per day. Don't expect any valuables to make it through.\n\nPrivately run international and national services are available. Federal Express has its central branch at Maip\u00fa 753. Other choices are DHL International (  0810-122-3345; www.dhl.com.ar; Av C\u00f3rdoba 783). OCA (www.oca.com.ar) and Andreani (www.andreani.com.ar) are good for domestic packages; both have many locations around town.\n\n###  Public Holidays\n\nGovernment offices and businesses are closed on the numerous national holidays. If a holiday falls midweek or on a weekend day, it's often bumped to the nearest Monday; if it falls on a Tuesday or Thursday, then the in-between Monday or Friday are taken as holidays.\n\nPublic-transportation options are more limited on holidays, when you should reserve tickets far in advance. Hotel booking should also be done ahead of time.\n\nJanuary 1 A\u00f1o Nuevo; New Year's Day\n\nFebruary or March Carnaval \u2013 dates vary; a Monday and Tuesday become holidays\n\nMarch 24 D\u00eda de la Memoria; anniversary of the day that started the 1976 dictatorship and subsequent Dirty War\n\nMarch\/April Semana Santa (Easter week) \u2013 dates vary; most businesses close on Good 'Thursday' and Good Friday; major travel week\n\nApril 2 D\u00eda de las Malvinas; honors the fallen Argentine soldiers from the Islas Malvinas (Falkland Islands) war in 1829\n\nMay 1 D\u00eda del Trabajor; Labor Day\n\nMay 25 D\u00eda de la Revoluci\u00f3n de Mayo; commemorates the 1810 revolution against Spain\n\nJune 20 D\u00eda de la Bandera (Flag Day); anniversary of death of Manuel Belgrano, creator of Argentina's flag and military leader\n\nJuly 9 D\u00eda de la Independencia; Independence Day\n\nAugust (third Monday) D\u00eda del Libertador San Mart\u00edn; marks the anniversary of Jos\u00e9 de San Mart\u00edn's death (1778\u20131850)\n\nOctober (second Monday) D\u00eda del Respeto a la Diversidad Cultural; a day to respect cultural diversity\n\nNovember (fourth Monday) D\u00eda de la Soberan\u00eda Nacional; day of national sovereignty\n\nDecember 8 D\u00eda de la Concepci\u00f3n Inmaculada; celebrates the immaculate conception of the Virgin Mary\n\nDecember 25 Navidad; Christmas Day\n\nNote that Christmas Eve and New Year's Day are treated as semi-holidays, and you will find some businesses closed for the latter half of those days.\n\n###  Safe Travel\n\nBuenos Aires is generally pretty safe. You can comfortably walk around at all hours of the night in many places, even as a lone woman. People stay out very late, and there's almost always somebody else walking on any one street at any hour of the night. (Some areas where you should be careful at night, however, are around Constituci\u00f3n's train station, the eastern border of San Telmo, and some parts of Once and La Boca \u2013 where, outside tourist streets, you should be careful even during the day).\n\nLike all big cities, BA has its share of problems. The economic crisis of 1999\u20132001 plunged a lot of people into poverty, and street crime has subsequently risen. As a tourist you're much more likely to be a target of petty crimes like pickpocketing and bag-snatching than armed robbery or kidnapping. Be careful on crowded buses, on the Subte and at busy _ferias_ (street markets). Don't put your bag down without your foot through the strap (especially at sidewalk cafes), and even then keep a close eye on it. Be especially careful at Retiro bus station.\n\nMinor nuisances include lack of respect shown by cars toward pedestrians, lax pollution controls and high noise levels. Many Argentines are heavy smokers, and you can't help but be exposed to it on the street (smoking is banned in most restaurants, bars and public transport). The tourist police (Comisar\u00eda del Turista;   0800-999-5000, 4346-5748; Av Corrientes 436;   24hr) may be of some help.\n\nUsing your head is good advice anywhere: don't flash any wealth (including expensive jewelry), don't stagger around drunk, always be aware of your surroundings and look like you know exactly where you're going (even if you don't). Be careful showing off expensive electronics like laptops, smart phones, iPods or iPads. But realize that if you're reasonably careful, the closest thing to annoyance you'll experience is being shortchanged, tripping on loose sidewalk tiles, stepping on the ubiquitous dog pile or getting flattened by a crazy bus driver. Watch your step.\n\n###  ELECTRONICS WARNING\n\nNote that buying a smart phone, especially an iPhone, is extremely expensive in Argentina due to import restrictions \u2013 and they are not widely available. If you do bring your smart phone, don't flash it around unnecessarily or leave it unprotected somewhere. This goes for iPads, iPods and laptop computers too.\n\n###  Taxes & Refunds\n\nOne of Argentina's primary state revenue-earners is the 21% value-added tax known as the Impuesto de Valor Agregado (IVA). Under limited circumstances, foreign visitors may obtain IVA refunds on purchases of Argentine products upon departing the country. A 'Tax Free' window decal (in English) identifies participants in this program, but always check that the shop is part of the tax-free program before making your purchase.\n\nYou can obtain tax refunds on purchases of AR$70 or more made at one of these participating stores. To do so, present your passport to the merchant, who will make out an invoice for you. On leaving the country keep the purchased items in your carry-on baggage. A customs official will check them and stamp your paperwork, then tell you where to obtain your refund. Be sure to leave yourself a bit of extra time at the airport to get this done.\n\n###  Telephone\n\nTwo companies, Telecom and Telef\u00f3nica, split the city's telephone services.\n\nStreet phones require coins or _tarjetas telef\u00f3nicas_ (magnetic phone cards available at many _kioskos,_ or small markets). You'll only be able to speak for a limited time before you get cut off, so carry enough credit.\n\nToll-free numbers in BA have '0800' before a seven-digit number.\n\n####  Cell Phones\n\nIt's best to bring your own factory unlocked tri- or quad-band GSM cell phone to Argentina, then buy an inexpensive SIM chip (you'll get a local number) and credits as needed. Both SIM chips and credits can be bought at many _kioskos_ or _locutorios_ (small telephone offices); look for the ' _recarga facil_ ' signs. Many Argentines use this system with their cell phones. Phone-unlocking services are available; ask around.\n\nYou can also buy cell phones that use SIM chips; these usually include some credits for your first batch of calls. Be careful renting phones as they're not usually a better deal than outright buying a cell phone.\n\nIf you plan to travel with an iPhone or other 3G smart phone, prepare yourself \u2013 you may need to purchase an international plan to avoid being hit by a huge bill for roaming costs. On the other hand, it's possible to call internationally for free or very cheaply using a VoIP (Voice over Internet Protocol) system such as Skype. This is a constantly changing field, so do some research before you travel.\n\nCell-phone numbers in Argentina are always preceded by '15'. If you're calling a cellular phone number from a landline, you'll have to dial 15 first. But if you're calling a cell phone from another cell phone, you don't need to dial 15 (at least within the same area code).\n\nWhen calling cell phones from outside Argentina, dial your country's international access code, then 54 9 11 and then the eight-digit number, leaving out the 15.\n\n####  Locutorios & Internet Cafes\n\nOne way to make a local or international phone call is to find a _locutorio,_ a small telephone office (sometimes marked _telecentro_ ) with private booths from which you make your calls and then pay at the register. There's a _locutorio_ on practically every other block in the center. They cost a bit more than street phones, but you can sit down, you won't run out of coins and it's much quieter.\n\nWhen making international calls from _locutorios_ ask about off-peak discount hours, which generally apply after 10pm and on weekends. Making international calls over the internet using Skype is a cheap option; many internet cafes have this system in place.\n\nFaxes are cheap and widely available at most _locutorios_ and internet cafes.\n\n####  Phone Codes\n\nThe Buenos Aires area code is 011. You will need to dial this when calling BA from outside the city, but you don't need to dial it when calling from within BA.\n\n####  Phonecards\n\nTelephone calling cards are sold at nearly all _kioskos_ and make domestic and international calls far cheaper than calling direct. However, they must be used from a fixed line such as a home or hotel telephone (provided you can dial outside the hotel). They cannot be used at most pay phones. Some _locutorios_ allow you to use them, and although they levy a surcharge, the call is still cheaper than dialing direct. When purchasing one, tell the clerk the country you will call so that they give you the right card.\n\n###  Time\n\nArgentina is three hours behind GMT and generally does not observe daylight-saving time (though this situation can easily change). Many _porte\u00f1os_ use the 24-hour clock to differentiate between am and pm.\n\n###  Toilets\n\nPublic toilets in BA are generally decent and usually stocked with toilet paper (carry some anyway), but soap and towels are rarer. If you're looking for a bathroom while walking around, note that the largest shopping malls (such as Galer\u00edas Pac\u00edfico) always have public bathrooms available, but in a pinch you can always walk into a McDonald's or large cafe. Changing facilities for babies are not always available.\n\nSome may find bidets a novelty; they are those strange shallow, ceramic bowls with knobs and a drain, often accompanying toilets in hotel bathrooms. They are meant for between-shower cleanings of nether regions. Turn knobs slowly, or you may end up spraying yourself or the ceiling.\n\n###  Tourist Information\n\nThe Secretar\u00eda de Turismo de la Naci\u00f3n (  4312-2232; www.turismo.gov.ar; Av Santa Fe 883;   9am-7pm Mon-Fri) dispenses information on Buenos Aires but focuses on Argentina as a whole.\n\nThe tourist police (Comisar\u00eda del Turista;   0800-999-5000, 4346-5748; Av Corrientes 436;   24hr) can provide interpreters and helps victims of robberies and rip-offs.\n\nThere's a tourist kiosk at Ezeiza airport (Ezeiza airport;   24hr) and another one at Retiro bus station (Retiro bus station;   7:30am-2pm Mon-Fri) near Puente 3 on the main floor, across from bus slot 36.\n\nThere are several tourist offices and kiosks in Buenos Aires. Note that hours vary depending on the season and number of volunteers. The official tourism site of Buenos Aires is www.bue.gov.ar and the government site is www.buenosaires.gov.ar.\n\nPlaza San Mart\u00edn (cnr Av Florida & MT de Alvear)\n\nFlorida (cnr Av Florida & Diagonal Roque S\u00e1enz Pe\u00f1a)\n\nPuerto Madero (Dique 4)\n\nRecoleta (Av Quintana 596)\n\nEsmeralda (cnr Av Rivadavia & Esmeralda)\n\n###  Travelers with Disabilities\n\nNegotiating Buenos Aires as a disabled traveler is not the easiest of tasks. City sidewalks are narrow, busy and dotted with many broken tiles. Not every corner has a ramp, and traffic is ruthless when it comes to pedestrians (and wheelchair-users). A few buses do have _piso bajo_ (they 'kneel' and have extra-large spaces), but the Subte (subway) does not cater to the mobility-impaired.\n\nInternational hotel chains often have wheelchair-\u00adaccessible rooms, as do other less fancy hotels \u2013 accessibility laws have changed for the better over the last few years. Some restaurants and many important tourist sights have ramps, but BA is sorely lacking in wheelchair-accessible bathrooms \u2013 although the city's shopping malls usually have at least one, restaurants don't often have the appropriate installations.\n\nIn Buenos Aires, QRV Transportes Especiales (  15-6863-9555, 011-4306-6635; www.qrvtransportes.com.ar) offers private transport and city tours in vans fully equipped for wheelchair users. BA Cultural Concierge (  15-3876-5937; www.baculturalconcierge.com) offers service for low-mobility travelers, by helping with errands. Or you could head to BA with a company like Accessible Journeys (www.disabilitytravel.com), which has tours and cruises in South America \u2013 including one that includes Buenos Aires.\n\nOther than the use of brail on ATMs little effort has been dedicated to bettering accessibility for the vision impaired. Stoplights are rarely equipped with sound alerts. The Biblioteca Argentina Para Ciegos (BAC, Argentine Library for the Blind;   4981-0137; www.bac.org.ar; Lezica 3909) maintains a brail collection of over 3000 books, as well as other resources.\n\n###  Visas\n\nNationals of the USA, Canada, most Western European countries, Australia and New Zealand do not need visas to visit Argentina, but check current regulations. Most foreigners receive a 90-day visa upon arrival.\n\nTo get yourself a 90-day extension (AR$300), visit the Direcci\u00f3n Nacional de Migraciones (  4317-0234; www.migraciones.gov.ar\/accesibleingles\/?categorias; Ant\u00e1rtida Argentina 1355;   8am-2pm Mon-Fri). Set aside some time, as there are lines and this process can take an hour or two. Get your extension the same week your visa expires. Overstaying your visa (AR$300) costs as much as an extension, but it's also much more stressful \u2013 and the rules can change quickly.\n\nAnother option if you're staying more than three months is to cross into Colonia or Montevideo (both in Uruguay; Colonia can be an easy day trip) and return with a new three-month visa. This strategy is most sensible if you are from a country that does not require a visa to enter Uruguay.\n\nAmericans, Australians and Canadians need to pay a reciprocity fee ( _tasa de reciprocidad_ ) when arriving in Argentina, Click here.\n\n###  Women Travelers\n\nBuenos Aires is a modern, sophisticated city, and women travelers \u2013 even those traveling alone \u2013 should not encounter many difficulties. Men do pay more overt attention to women in Argentina, however, and a little open-mindedness might be in order. Argentina's machismo culture is, after all, alive and well.\n\nA few men feel the need to comment on a woman's attractiveness. This often happens when you're walking alone and pass by a man; it will never occur when you're with another man. Comments usually include whistles or _piropos,_ which many Argentine males consider the art of complimenting a woman. _Piropos_ are often vulgar, although a few can be poetic. Much as you may want to kick them where it counts, the best thing to do is completely ignore the comments. After all, many _porte\u00f1as_ are used to getting these 'compliments', and most men don't necessarily mean to be insulting; they're just doing what is socially acceptable in Argentina.\n\nOn the plus side of machismo, men will hold a door open for you and let you enter first, including getting on buses.\n\n   |    |    |\n\n---|---|---|---\n\nBehind the Scenes\n\n### SEND US YOUR FEEDBACK\n\nWe love to hear from travellers \u2013 your comments keep us on our toes and help make our books better. Our well- travell ed team reads every word on what you loved or loathed about this book. Although we cannot reply individually to postal submissions, we always guarantee that your feedback goes straight to the appropriate authors, in time for the next edition. Each person who sends us information is thanked in the next edition \u2013 and the most useful submissions are rewarded with a free book.\n\nVisit **lonelyplanet.com\/contact** to submit your updates and suggestions or to ask for help. Our award-winning website also features inspirational travel stories, news and discussions.\n\nNote: We may edit, reproduce and incorporate your comments in Lonely Planet products such as guidebooks, websites and digital products, so let us know if you don't want your comments reproduced or your name acknowledged. For a copy of our privacy policy visit lonelyplanet.com\/privacy.\n\n##  OUR READERS\n\nMany thanks to the travelers who used the last edition and wrote to us with helpful hints, useful advice and interesting anecdotes:\n\nBettina Becker, Arturo Costa, Loli Delger, Alexandra Goller, Cathy McCloy, Eben Pullman.\n\n##  AUTHOR THANKS\n\n####  Sandra Bao\n\nI'm grateful for the support of my excellent (and now ex) commissioning editor Kathleen Munnelly \u2013 best of luck in your future adventures, Kathleen; I'll miss you. Also many thanks to Graciela and Silvia Guzm\u00e1n, who kept me entertained with their raucous company. This book wouldn't be the fine thing that it is without the help of Lucas M, Sylvia Z, Alan S, Madi L, Jed R and Dan P \u2013 you all know who you are. A big gracias to Gustavo and Miriam for their companionship and opinions, and also to Sally Blake, Jimena Moses, Andy Symington and Marina Charles for their contributions.\n\n_Un beso grande_ to my godmother, Elsa, and her son Jorge for their hospitality. Love to Fung and David Bao, and to Daniel for their support over the years. Finally, lots of love to my husband, Ben Greensfelder, for keeping our home in good order while I was away.\n\n##  ACKNOWLEDGMENTS\n\nClimate map data adapted from Peel MC, Finlayson BL & McMahon TA (2007) 'Updated World Map of the K\u00f6ppen-Geiger Climate Classification', _Hydrology and Earth System Sciences_ , 11, 1633\u00ac44.\n\nCover photograph: Tango dancers' feet, Herv\u00e9 Hughes\/Alamy\u00a9\n\n###  THIS BOOK\n\nThis 7th edition of Lonely Planet's Buenos Aires guidebook was researched and written by Sandra Bao. The 6th edition was written by Sandra Bao and Bridget Gleeson, and the 5th edition by Sandra Bao. This guidebook was commissioned in Lonely Planet's Oakland office, and produced by the following:\n\nCommissioning Editor Kathleen Munnelly\n\nCoordinating Editors Sarah Bailey, Tracy Whitmey\n\nSenior Cartographer Mark Griffiths\n\nBook Designer Clara Monitto, Virginia Moreno, Wendy Wright\n\nAssociate Product Directors Sasha Baskett, Angela Tinson\n\nSenior Editor Catherine Naghten, Karyn Noble\n\nAssisting Editors Michelle Bennett, Kate Evans, Carly Hall\n\nAssisting Cartographer Julie Dodkins\n\nCover Research Naomi Parker\n\nLanguage Content Branislava Vladisavljevic\n\nThanks to Anita Banh, Elin Berglund, Ljubomir Ceranic, Ryan Evans, Larissa Frost, Anna Harris, Genesys India, Jouve India, Anne Mason, Martine Power, Amanda Williamson\n\n### Our Story\n\nA beat-up old car, a few dollars in the pocket and a sense of adventure. In 1972 that's all Tony and Maureen Wheeler needed for the trip of a lifetime \u2013 across Europe and Asia overland to Australia. It took several months, and at the end \u2013 broke but inspired \u2013 they sat at their kitchen table writing and stapling together their first travel guide, Across Asia on the Cheap. Within a week they'd sold 1500 copies. Lonely Planet was born.\n\nToday, Lonely Planet has offices in Melbourne, London and Oakland, with more than 600 staff and writers. We share Tony's belief that 'a great guidebook should do three things: inform, educate and amuse'.\n\nOUR WRITERS\n\n####  Sandra Bao\n\nCoordinating Author Sandra's mom and her family escaped China's communist regime in the years following WWII, eventually boarding a freighter bound for Argentina in 1952. After months at sea they arrived in Buenos Aires \u2013 just two days after the death of Evita Per\u00f3n. Sandra's parents married in Montevideo, Uruguay, then raised Sandra and her brother, Daniel, in BA. They lived the carefree _porte\u00f1o_ life (with _asados_ every Sunday) until 1974, when things got politically dicey. Once again the Baos emigrated to greener pastures, this time the USA. Sandra is proud to be a _porte\u00f1a_ and has regularly returned to her homeland as an adult, watching the peso fluctuate wildly through the decades. Over the last 14 years Sandra has contributed to dozens of Lonely Planet guidebooks.\n\nRead more about Sandra at: lonelyplanet.com\/members\/sandrabao\n\n####  Contributing Author\n\nMarina Charles wrote the Street Art essay. She is a British expat and Buenos Aires resident who cofounded Graffitimundo in 2009.\n\n**Published by Lonely Planet Publications Pty Ltd**\n\nABN 36 005 607 983\n\n7th edition \u2013 July 2014\n\nISBN 978 1 74360 023 8\n\n\u00a9 Lonely Planet 2014 Photographs \u00a9 as indicated 2014\n\nAll rights reserved. No part of this publication may be copied, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, recording or otherwise, except brief extracts for the purpose of review, and no part of this publication may be sold or hired, without the written permission of the publisher. Lonely Planet and the Lonely Planet logo are trademarks of Lonely Planet and are registered in the US Patent and Trademark Office and in other countries. Lonely Planet does not allow its name or logo to be appropriated by commercial establishments, such as retailers, restaurants or hotels. Please let us know of any misuses: lonelyplanet.com\/ip.\n\nAlthough the authors and Lonely Planet have taken all reasonable care in preparing this book, we make no warranty about the accuracy or completeness of its content and, to the maximum extent permitted, disclaim all liability arising from its use.\n\n   |    |    |\n\n---|---|---|---\n\n# Quick Links\n\n### A\n\nAbode\n\nAbuela Pan\n\nAcaBar\n\nAcatraz\n\nAldo's Vinoteca\n\nAlmac\u00e9n Ramos Generales\n\nAlsina\n\nAlto Palermo\n\nAlvear Palace Hotel\n\nAmerica del Sur\n\nAmerika\n\nAntares\n\nAramburu\n\nArchivo Regional\n\nAreco Hostel\n\nArevalito\n\nArt Suites\n\nArte y Esperanza\n\nArte \u00c9tnico Argentino\n\nAstor\n\nAutor\u00eda\n\nAyres de Recoleta\n\nAzema\n\n### B\n\nBA Sohotel\n\nBahrein\n\nBar Brit\u00e1nico\n\nBar El Federal\n\nBar Plaza Dorrego\n\nBar Sedd\u00f3n\n\nBarbot\n\nBasement Club\n\nBas\u00edlica de Nuestra Se\u00f1ora del Pilar\n\nBas\u00edlica de Sant\u00edsimo Sacramento\n\nBi Won\n\nBiblioteca Nacional\n\nBio\n\nBohemia Buenos Aires\n\nBoliche de Bessonart\n\nBolivia\n\nBonito San Telmo\n\nBoulevard Saenz Pe\u00f1a\n\nBoutique\n\nBrisas del Mar\n\nBroccolino\n\nBuen Suspiro\n\nBuenos Aires Design\n\nBuller Brewing Company\n\n### C\n\nCabrera Garden\n\nCaf\u00e9 Crespin\n\nCaf\u00e9 Margot\n\nCaf\u00e9 Retiro\n\nCaf\u00e9 San Juan\n\nCaf\u00e9 Tortoni\n\nCaf\u00e9 de los Angelitos\n\nCalifornia Burrito Company (CBC)\n\nCalma Chicha\n\nCampo Argentino de Polo\n\nCantares\n\nCapital\n\nCasa Bar\n\nCasa Calma\n\nCasa L\u00f3pez\n\nCasa Nacarello\n\nCasa y Mundo Bolivar\n\nCasal de Catalunya\n\nCaser\u00f3n Porte\u00f1o\n\nCasona La Ruchi\n\nCementerio de la Recoleta\n\nCentrico\n\nCentro Cultural Borges\n\nCentro Cultural Recoleta\n\nCentro Cultural Ricardo Rojas\n\nCentro Cultural San Mart\u00edn\n\nCentro Cultural del Bicentenario\n\nCentro Isl\u00e1mico Rey Fahd\n\nCervecer\u00eda Cossab\n\nChan Chan\n\nChill House Hostel\n\nChiquil\u00edn\n\nCircus Hostel & Hotel\n\nCiudad Cultural Konex\n\nClaridge Hotel\n\nClub Ar\u00e1oz\n\nClub Gricel\n\nCl\u00e1sica y Moderna\n\nCocoliche\n\nCoffee Town\n\nColecci\u00f3n de Arte Amalia Lacroze de Fortabat\n\nComedor Nikkai\n\nComo en Casa\n\nComplejo Tango\n\nConfiter\u00eda Ideal\n\nCongo\n\nConvento de Santo Domingo\n\nCorbeta Uruguay\n\nCrizia\n\nCrobar\n\nCruzat Beer House\n\nCualquier Verdura\n\nCuman\u00e1\n\n### D\n\nDad\u00e1\n\nDill & Drinks\n\nDon Julio\n\nDoppelg\u00e4nger\n\nDruid In\n\nDuque Hotel\n\nD'Oro\n\n### E\n\nEco Pampa Hostel\n\nEdificio Kavanagh\n\nEl Ateneo\n\nEl Beso\n\nEl Burladero\n\nEl Caminito\n\nEl Capullo\n\nEl Carnal\n\nEl Cid\n\nEl Coleccionista\n\nEl Cuartito\n\nEl Desnivel\n\nEl Federal\n\nEl Gato Negro\n\nEl Obrero\n\nEl Querand\u00ed\n\nEl Sanjuanino\n\nEl Viajero Hostel\n\nEl Viejo Almac\u00e9n\n\nEl Zanj\u00f3n de Granados\n\nElena\n\nEn Buen Orden\n\nEsquina Carlos Gardel\n\nEsquina Homero Manzi\n\n### F\n\nFacultad de Ingenieria\n\nFacultad de Ingenier\u00eda\n\nFaena Arts Center\n\nFaena Hotel + Universe\n\nFaro\n\nFeria Plaza Francia\n\nFeria Plaza Serrano\n\nFeria de Artesanos Caminito\n\nFeria de San Telmo\n\nFilo\n\nFloralis Gen\u00e9rica\n\nFlorer\u00eda Atl\u00e1ntico\n\nFlorida Garden\n\nFlux\n\nFour Seasons\n\nFragata Sarmiento\n\nFrank's Bar\n\nFundaci\u00f3n Proa\n\nFurai-Bo\n\n### G\n\nGabriella Capucci\n\nGaler\u00eda 5ta Avenida\n\nGaler\u00eda Bond Street\n\nGaler\u00edas Pac\u00edfico\n\nGibraltar\n\nGil Antiguedades\n\nGlam\n\nGran Bar Danz\u00f3n\n\nGran Hotel Hispano\n\nGran Hotel Oriental\n\nGran Parrilla del Plata\n\nGranix\n\nGreen Bamboo\n\n### H\n\nHarapos Patagonia\n\nHermanos Estebecorena\n\nHern\u00e1n Gipponi Restaurant\n\nHome Hotel\n\nHostel Viejo Telmo\n\nHotel Avenida\n\nHotel Bonito\n\nHotel Fac\u00f3n Grande\n\nHotel Lafayette\n\nHotel Lion D'or\n\nHotel Lyon\n\nHotel Maip\u00fa\n\nHotel Marbella\n\nHotel Pulitzer\n\nHotel Tres Sargentos\n\nHotel Villa Victoria\n\nHumawaca\n\n### I\n\nIglesia Matriz\n\nIglesia Santa Catalina\n\nIl Ballo del Mattone\n\nIl Matterello\n\nImhotep\n\nInfinito Hotel\n\n### J\n\nJard\u00edn Japon\u00e9s\n\nJard\u00edn Zool\u00f3gico\n\nJuana de Arco\n\n### K\n\nKika\n\nKilkenny\n\n### L\n\nLa Biela\n\nLa Bodeguita\n\nLa Bombonera Stadium\n\nLa Cabrera\n\nLa Catedral\n\nLa Cayetana\n\nLa Cigale\n\nLa Martina\n\nLa Mercer\u00eda\n\nLa Panader\u00eda de Pablo\n\nLa Parolaccia Trattoria\n\nLa Pe\u00f1a del Colorado\n\nLa Poesia\n\nLa Puerta Roja\n\nLa Puerto Rico\n\nLa Rosa N\u00e1utica\n\nLa Scala de San Telmo\n\nLa Trastienda\n\nLa Ventana\n\nLa Viruta\n\nLas Cholas\n\nLas Pizarras\n\nLas Violetas\n\nLe Sud\n\nLentas Maravillas\n\nLibros del Pasaje\n\nLivian Guesthouse\n\nLivin' Residence\n\nLo de Joaquin Alberdi\n\nLondon City\n\nLos 36 Billares\n\nLos Cardones\n\nL'Ago\n\nL'Orangerie\n\n### M\n\nMagdalena's Party\n\nMagnolia Hotel\n\nMaluco Beleza\n\nMalv\u00f3n\n\nMansi\u00f3n Vitraux\n\nManzana de las Luces\n\nMaria Luj\u00e1n\n\nMateria Urbana\n\nMemorabilia\n\nMercado de Abasto\n\nMercado de San Telmo\n\nMercado de las Pulgas\n\nMilhouse Youth Hostel\n\nMili\u00f3n\n\nMine Hotel\n\nMiranda\n\nMiravida Soho\n\nMishka\n\nMoebius\n\nMundo Bizarro\n\nMunich Recoleta\n\nMuseo Argentino de Ciencias Naturales\n\nMuseo Beatle\n\nMuseo Casa Carlos Gardel\n\nMuseo Casa de Ricardo Rojas\n\nMuseo Espa\u00f1ol\n\nMuseo Etnogr\u00e1fico Juan B Ambrosetti\n\nMuseo Evita\n\nMuseo Evita Restaurante\n\nMuseo Gauchesco Ricardo G\u00fciraldes\n\nMuseo Hist\u00f3rico Nacional\n\nMuseo Hist\u00f3rico de Cera\n\nMuseo Ind\u00edgena\n\nMuseo Judio Dr Salvador Kibrick\n\nMuseo Las Lilas\n\nMuseo Mitre\n\nMuseo Mundial del Tango\n\nMuseo Municipal\n\nMuseo Nacional de Arte Decorativo\n\nMuseo Nacional de Bellas Artes\n\nMuseo Nacional del Teatro\n\nMuseo Naval\n\nMuseo Penitenciario\n\nMuseo Portugu\u00e9s\n\nMuseo Xul Solar\n\nMuseo de Armas\n\nMuseo de Arte Hispanoamericano Isaac Fern\u00e1ndez Blanco\n\nMuseo de Arte Latinoamericano de Buenos Aires (MALBA)\n\nMuseo de Arte Contempor\u00e1neo Buenos Aires\n\nMuseo de Arte Moderno de Buenos Aires (Mamba)\n\nMuseo de Arte Popular Jos\u00e9 Hern\u00e1ndez\n\nMuseo de Arte Tigre\n\nMuseo de Artes Pl\u00e1sticas Eduardo S\u00edvori\n\nMuseo de Bellas Artes de La Boca Benito Quinquela Mart\u00edn\n\nMuseo de la Ciudad\n\nMuseo de la Pasi\u00f3n Boquense\n\nMuseo de la Polic\u00eda Federal\n\nMuseo del Azulejo\n\nMuseo del Mate\n\nMuseo del Traje\n\nMuseo y Taller Draghi\n\n### N\n\nNatural Deli\n\nNew Brighton\n\nNiceto Club\n\nNobrand\n\nNotorious\n\nNovotel Hotel\n\n### O\n\nObelisco\n\nOlsen\n\nOrigen Caf\u00e9\n\nOui Oui\n\nOviedo\n\n### P\n\nPach\u00e1\n\nPalacio Barolo\n\nPalacio Duhau \u2013 Park Hyatt\n\nPalacio Haedo\n\nPalacio Paz\n\nPalacio San Mart\u00edn\n\nPalacio de las Aguas Corrientes\n\nPalais de Glace\n\nPalermitano\n\nPalermo Viejo B&B\n\nPan Y Arte\n\nPanorama\n\nPapelera Palermo\n\nParadores Draghi\n\nParque 3 de Febrero\n\nParque Lezama\n\nParque de la Costa\n\nParrilla Pe\u00f1a\n\nPasaje de la Defensa\n\nPaseo Alcorta\n\nPatio Bullrich\n\nPatio del Liceo\n\nPetit Recoleta Hostel\n\nPizzer\u00eda G\u00fcerr\u00edn\n\nPlaza Asturias\n\nPlaza Dorrego\n\nPlaza Lavalle\n\nPlaza San Mart\u00edn\n\nPlaza de Mayo\n\nPlaza del Congreso\n\nPoetry Building\n\nPop Hotel\n\nPortal del Sur\n\nPortobello Vintage Boutique\n\nPort\u00f3n de Campo\n\nPosada de 1860\n\nPosada del \u00c1ngel\n\nPride Cafe\n\nProa Cafe\n\nPuente de la Mujer\n\nPuerto de Frutos\n\nPuesto La Lechuza\n\nPunto Sur\n\nPuntos en el Espacio\n\n### Q\n\nQuerido B&B\n\n### R\n\nRac\u00f3 de Buenos Aires\n\nRapsodia\n\nReina Madre Hostel\n\nRendezvous Hotel\n\nReserva Ecol\u00f3gica Costanera Sur\n\nRodi Bar\n\nRojo Tango\n\nRugantino Hotel\n\n### S\n\nSabatico Hostel\n\nSalon Canning\n\nSan Telmo Colonial\n\nSarkis\n\nScala Hotel\n\nShamrock\n\nShanghai Dragon\n\nSiamo nel Forno\n\nSignos\n\nSipan\n\nSocial la Lechuza\n\nSudestada\n\nSugar\n\nSugar & Spice\n\n### T\n\nTango Porte\u00f1o\n\nTea Connection\n\nTeatro Avenida\n\nTeatro Basti\u00f3n del Carmen\n\nTeatro Coliseo\n\nTeatro Col\u00f3n\n\nTeatro Gran Rex\n\nTeatro Nacional Cervantes\n\nTeatro Opera\n\nTeatro Paseo la Plaza\n\nTeatro Presidente Alvear\n\nTeatro San Mart\u00edn\n\nTeatro de la Ribera\n\nTerranova Hostel\n\nThelonious Bar\n\nTiempo de Gitanos\n\nTodo Mundo\n\nTomo 1\n\nTorre de los Ingleses\n\n### U\n\nUn Lugar\n\nUnik\n\nUsina del Arte\n\n### V\n\nV & S Hostel Club\n\nVain Boutique Hotel\n\nVan Koning\n\nVerne\n\nVita\n\n### W\n\nWalrus Books\n\nWildlife\n\nWussmann Shop\n\n### Y\n\nYira Yira Guesthouse\n\n### Z\n\nZival's\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n~ Latest titles in the _**Build It Yourself**_ Series ~\n\nCheck out more titles at www.nomadpress.net\n\nNomad Press  \nA division of Nomad Communications  \n10 9 8 7 6 5 4 3 2 1  \nCopyright \u00a9 2014 by Nomad Press. All rights reserved.  \nNo part of this book may be reproduced in any form without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review or **for limited educational use**.  \nThe trademark \"Nomad Press\" and the Nomad Press logo are trademarks of Nomad Communications, Inc.\n\nISBN Softcover: 978-1-61930-254-9  \nISBN Hardcover: 978-1-61930-250-1\n\nIllustrations by Sam Carbaugh\n\nEducational Consultant, Marla Conn\n\nQuestions regarding the ordering of this book should be addressed to Nomad Press  \n2456 Christian St.  \nWhite River Junction, VT 05001  \nwww.nomadpress.net\n\nPrinted in Canada.\nCONTENTS\n\nTIMELINE\n\nINTRODUCTION\n\nSo, You Want to Make Comics?\n\nCHAPTER 1\n\nAncient Comics\n\nCHAPTER 2\n\nComics in the Newspapers\n\nCHAPTER 3\n\nThe Birth of Superheroes\n\nCHAPTER 4\n\nInto the Silver Age of Comics\n\nCHAPTER 5\n\nLife After the Comics Code\n\nCHAPTER 6\n\nComics and the Internet\n\nGLOSSARY\n\nRESOURCES\n\nINDEX\n\n  **Interested in Primary Sources? Look for this icon.**\n\nYou can use a smartphone or tablet app to scan the QR code and explore more about comics and cartooning! Cover up neighboring QR codes to make sure you're scanning the right one. If you don't have a QR code scanning device, you can find a list of each url in the Resources on page 118.\nTIMELINE\n\nINTRODUCTION\n\nSO, YOU WANT TO MAKE COMICS?\n\nFrom newspapers to movie theaters, comics are everywhere. Your parents, grandparents, and even great-grandparents grew up reading them! They are one of the oldest ways to tell a story and they are one of the newest forms of modern art. Does that sound a little confusing?\n\nDon't worry, in this book you'll learn all about the history of this fun art form and unravel some of the mysteries of comics.\n\n**What are comics?** There are many definitions for comics floating around. The most universal one is that comics are images in sequence that tell a story, with or without words. A cartoon is a comic published in a newspaper or magazine.\n\nWORDS TO KNOW\n\ncomic: images in sequence that tell a story, with or without words.\n\nimage: a picture of something, either real or imagined.\n\nsequence: the order in which something happens.\n\ncartoon: a comic published in a newspaper or magazine.\n\nLike stories, comics have rules and systems to help readers understand what is happening. Instead of sentences, you can use panels to help contain an idea or scene. Pages of comics work like paragraphs, while word balloons, thought clouds, and narration blocks give your characters a place to speak, think, or observe.\n\nCOMICS DON'T READ LIKE STORIES OR NOVELS AND CAN BE CONFUSING IF YOU DO NOT KNOW THE RULES AND SYSTEMS.\n\nModern comics use panels to frame each section. These panels are arranged in a sequence, or order, from beginning to end. The first panel starts the story, giving basics such as who is in the strip and where the story is taking place. The middle panel or panels follows and moves the story along. The final panel ends the story and usually has something funny or exciting happen in it!\n\nWORDS TO KNOW\n\npanel: a square or other shape that frames a single scene in a comic strip.\n\nword balloon: a rounded outline with a point toward a character that encloses the character's speech.\n\nthought cloud: a shape similar to a word balloon that encloses a character's thought.\n\nnarration block: a block of text that contains the voice of the writer or of a character talking about what is happening.\n\ncharacter: someone in a story.\n\nComics are only as good as their characters. When you think of comics, do you imagine superheroes? The history of comics is full of different types of characters, from masked crusaders to funny animals. You can even turn yourself into a compelling comic character!\n\n**What makes a character interesting?** Keep reading to learn how to make your own characters using some of the best techniques in comics!\n\nComics didn't just appear overnight with today's rules and systems. They developed into the comics we know now through years of experimentation. Advances in printing, duplication technology, and computers have also been important to the evolution of comics.\n\nComics are fun and anyone can make them. As you read this book, you'll learn the basics of drawing, how to tell a story using comics, and how to make your very own comic book. You will even learn how to make a web comic that anyone in the world can read.\n\nWORDS TO KNOW\n\ntechnology: scientific or mechanical tools, methods, and systems used to solve a problem or do work.\n\nTRY THIS! Using stick figures, fill in the comic strip below. Make your characters speak, think, and observe in the right spaces.\n\nCHAPTER 1\n\nANCIENT COMICS\n\nWhere do comics come from? Ancient cave paintings in Font-de-Gaume, France, are some of the oldest paintings that we can still see. They are more than 14,000 years old. Sometime in the far past, the first human artists decided to tell stories of dreams and hunting by painting representations of things on the walls of the cave.\n\nThe artists used simple lines to represent what they saw in the world and in their dreams. Humans on cave walls look a lot like the stick figures of today. Other animals are easy to recognize, such as reindeer and saber-toothed tigers.\n\nWORDS TO KNOW\n\nrepresentation: showing things in pictures or other forms of art.\n\nTHESE PAINTINGS DON'T JUST SHOW PEOPLE AND ANIMALS STANDING STILL\u2014THEY TELL A STORY.\n\nThe humans are often shown hunting, throwing spears, and running. It's easy to imagine the artists sharing their stories with members of their tribe, pointing at these moments in the action to illustrate their story.\n\nUntil humans invented writing, most stories were passed along orally. The stories of the cave paintings of Font-de-Gaume were told to generation after generation, long after the original artists and storytellers had died. These paintings are some of the first examples of humans using images to tell a story!\n\nWORDS TO KNOW\n\ntribe: a large group of people with common ancestors and customs.\n\nancestor: someone from your family who lived before you.\n\ncustom: a way of living and doing things, such as food and dress.\n\norally: spoken out loud.\n\ngeneration: all the people born around the same time.\n\nindigenous: native.\n\ndepict: to create a representation of something experienced or seen.\n\nsymbol: a physical representation of a thing or idea.\n\nspiritual: religious, relating to the soul or spirit.\n\nMODERN DAY CAVE PAINTINGS\n\nIn Australia, many tribes of indigenous people use representational paintings to tell their stories. These stories often depict the dream countries claimed by each tribe. Using symbols for animals, rivers, trees, and rocks, the artists pass along the stories of their tribe to each generation. They also use their paintings when meeting with members of other tribes who have stories of their own. Most tribes use symbols from these dream paintings to distinguish themselves. By studying these dream country paintings, we can better understand how early humans used pictures to represent the spiritual dreams of their tribes as well as actual events in their lives.\n\nANCIENT EGYPT\n\nThe Egyptian empire (3050\u2013332 BCE) was one of the most powerful and advanced civilizations in the ancient world. It was famous for its monuments, such as the Sphinx and the Great Pyramids of Giza. We know more about the Egyptians than many other civilizations because they loved to tell stories about themselves. These stories included words and pictures!\n\nThe ancient Egyptians began to develop a pictographic language around 4000 BCE. A pictographic language is one that uses common images and sounds to help form words. Their letters look like the things they describe.\n\nWORDS TO KNOW\n\nBCE: put after a date, BCE stands for Before Common Era and counts down to zero. CE stands for Common Era and counts up from zero. These non-religious terms correspond to BC and AD. This book was published in 2014 CE.\n\ncivilization: a community of people that is advanced in art, science, and government.\n\ncommunity: a group of people who live in the same area.\n\nmonument: a building, structure, or statue that is special because it honors an event or person, or because it is beautiful.\n\npictographic: a picture of a word or idea.\n\nThe term for this Egyptian form of written language is hieroglyphics. If you were to look at hieroglyphics you would see birds, eyes, snakes, and many other familiar images.\n\nWHEN ANCIENT EGYPTIANS COMBINED THESE SYMBOLS THEY FORMED WORDS AND SENTENCES.\n\nEgyptians loved to use hieroglyphics on the walls and pillars of their buildings. They told stories using images too. One reason we know so much about how Egyptians mummified their pharaohs is because of these ancient \"comics.\"\n\nEgyptians often told the stories of their pharaohs inside their tombs. These stories were read in sequence and had images to help the viewer understand what was happening. The process of mummification was usually depicted, showing how the body was prepared for the afterlife. Mummification began with cleaning the body and removing the organs and ended with wrapping the body and placing it in a sarcophagus.\n\nWORDS TO KNOW\n\nhieroglyphics: a writing system that uses pictures and symbols called hieroglyphs (or just glyphs) to represent words and ideas.\n\nmummify: to preserve a dead body so it doesn't decay.\n\npreserve: to keep something from rotting.\n\ndecay: to rot.\n\npharaoh: the title for ancient Egyptian kings or rulers.\n\ntomb: a room or place where a dead person is buried.\n\nprocess: an activity that takes several steps to complete.\n\nafterlife: the ancient Egyptian belief in life after death.\n\nsarcophagus: a large, stone box containing an Egyptian king's coffin and mummy.\n\nMAYA: CREATORS OF THE FIRST COMIC BOOK\n\nIt may be a stretch to compare a Maya codex to a comic book, but the similarities are there. The Maya codices were created between 200 and 900 CE by the Maya people. Hundreds of years later, many experts are working to decode the Mayan written language. Thanks to the Maya codices and other artifacts, we are able to understand many things about their civilization.\n\nThe codices are brightly colored and use the inner bark of wild fig trees as paper. Some are the size of modern books with the pages folded together to be read. Other codices unfold into huge stories that could easily cover a wall. Each codex used words and pictures to tell stories about the Maya people.\n\nOne of the most common themes in the Maya codices is the stars. The Maya were famous for their ancient astrologers, who recorded the movements of the stars to help the Maya figure out the best time to do things. For example, the movement of the planet Venus was used by Maya astrologers to help their rulers decide when to go to war.\n\nTHE WORD \"MAYAN\" REFERS TO THE LANGUAGE OF THE MAYA PEOPLE. THE WORD \"MAYA\" IS USED TO DESCRIBE EVERYTHING ELSE.\n\nWORDS TO KNOW\n\ncodex: another name for a book. Plural is codices.\n\nartifact: an object made by people in the past, including tools, pottery, and jewelry.\n\nastrologer: a person who studies how the movements of the sun, moon, and planets affect humans.\n\nModern astronomers study the observations recorded in the Maya codices. It helps them understand events that happened in the ancient night sky!\n\nThe largest Maya codex is known as the Madrid Codex. It is one of the most complete codices we have from the Maya people. The book was probably written by more than eight scribes.\n\nWhile the Mayan written language is still a mystery, the way the images are presented in sequence have helped modern archaeologists understand what they were trying to say. The Madrid Codex contains many astronomical observations, just like the other codices, but it also shows religious rituals, how to keep bees, and the art of Maya weaving.\n\nWORDS TO KNOW\n\nastronomer: a person who studies objects in the sky, such as stars and planets.\n\nobservation: something you notice.\n\nscribe: a person who copies writings by hand.\n\narchaeologist: a scientist who studies ancient people through the objects they left behind.\n\nritual: something done as part of a religion.\n\nTAPESTRIES: COVER YOUR WALLS IN STORIES\n\nIn Europe during the Middle Ages (350\u2013 1450 CE), lords and kings lived in huge drafty castles made of stone. To help keep the castles warm and to tell tales of brave royal family members, the walls were covered in brightly colored tapestries.\n\nThe second-earliest known European tapestry is 1,000 years old and is called the Cloth of St. Gereon. It shows bulls and griffins fighting, but no clear story. One of the earliest known tapestries to depict a story is from the cathedral of Halberstadt in Germany. It was made in 1175 and tells a story from the Bible.\n\nTAPESTRIES BECAME MORE AND MORE DETAILED AS WEAVERS LEARNED NEW TECHNIQUES.\n\nMany tapestries made for castles tell stories of the kings, lords, and knights who lived there. They tell stories without words and show scenes of war and daring. In many ways they look like modern superhero comics.\n\nWORDS TO KNOW\n\ntapestry: a colorful, woven fabric that hangs on a wall. It often shows a scene.\n\ncathedral: a large important church.\n\nYou can follow the story of St. George fighting a dragon or see King Charlemagne battle invading armies in Spain. Few tapestries have words on them, and even fewer show the lives of common people. Unlike the Maya codices, medieval tapestries focus only on the lives of the wealthiest people.\n\nOne of the largest and most detailed tapestries is the Bayeux Tapestry. It is 230 feet long and 20 inches tall (70 meters by 51 centimeters). The tapestry tells the story of the Norman invasion of England in 1066 CE by William the Conqueror. Told from left to right, the story gives details about the battles and struggles that resulted in the fall of King Harold of England. Some scholars think he was the basis for King Arthur.\n\nThe visual storytelling in tapestries helped inspire the artists behind the next evolution of comics. Enter the broadsheet!\n\nWORDS TO KNOW\n\nmedieval: describes the Middle Ages, the period of European history after the fall of the Roman Empire, from about 350 to 1450 CE.\n\nWHERE DID \"CARTOON\" COME FROM?\n\nDuring the Renaissance (1300s\u20131600s CE), many master painters created large paintings on ceilings and walls called frescoes. These paintings were so enormous that a single artist couldn't do all the work. The process of creating these masterpieces led to the invention of the terms cartoon and cartoonist.\n\nMichelangelo began the process by sketching out a pencil or charcoal version of the complete painting. The sketched images had to be much smaller than they would be on the finished wall or ceiling. He would then hire other artists to help him paint the work on the wall. These hired artists drew larger versions of Michelangelo's sketch on big pieces of cardboard. These cardboard drawings, or carta, were used as models for the fullsized image on the wall.\n\nAfterward, the cardboard drawings were thrown away or reused. The artists who did this work called their drawings \"cartone,\" which would come to be known as cartoons today.\n\nWORDS TO KNOW\n\nRenaissance: a period of time in Europe after the Middle Ages, from the 1300s to the 1600s.\n\nGET YOUR BROADSHEET\n\nHumans have always loved stories with pictures. Until the invention of the European movable type printing press in 1430 by Johannes Gutenberg, books were all written and copied by hand. The only places that had books were castles and cathedrals. Most people couldn't even read!\n\nBut people loved to sketch funny scenes on scraps of paper or on walls. Someone with a funny drawing to share had to pass around the original or have someone copy the drawing by hand. This made telling stories through writing or drawing really time consuming.\n\nThen came the printing press. The first book to be printed on Gutenberg's new press was the Bible. Gutenberg began to print picture stories too, known as broadsides. Broadsides got their name from being printed on one side of a large sheet of paper. They were then either folded and passed around or plastered like a poster on a door or in a town square for everyone to see. Broadsides were fairly cheap to make and very popular.\n\nSINCE MOST PEOPLE DIDN'T KNOW HOW TO READ, MOST BROADSIDES WERE PRINTED LIKE COMICS.\n\nWORDS TO KNOW\n\nprinting press: a machine that presses inked type onto paper.\n\nBroadsides often told stories from the Bible or showed the martyrdom of a saint. They used the visual storytelling found in the tapestries of the rich, but told stories anyone could enjoy.\n\nBroadsides eventually became broadsheets, the early form of what we know as newspapers. As more and more people became literate, the pages of broadsheets were filled with words. Eventually, the picture stories were pushed into smaller and smaller spaces on the page.\n\nTHROUGH BROADSHEETS, THE MODERN political cartoon WAS BORN.\n\nWORDS TO KNOW\n\nmartyrdom: the death of a person for his or her beliefs.\n\nsaint: a Catholic Christian who has performed miracles as confirmed by the pope.\n\nliterate: having the ability to read.\n\npolitical cartoon: a comical or critical depiction of a political figure or event.\n\npamphlet: an informative book or brochure.\n\nAmerican Revolution: the war during which the 13 American colonies fought England for independence. It lasted from 1775 to 1783.\n\nBoston Massacre: a riot in Boston that took place on March 5, 1770. Five colonists were shot and killed by British soldiers.\n\nTHE COMICS ARE COMING!\n\nBroadsheets were popular throughout Europe and the American colonies. Printers had power because they could make pamphlets and broadsheets containing new and different ideas. One of the most famous illustrations printed in a broadsheet before the American Revolution was one of the Boston Massacre, drawn by Paul Revere. The drawing shows British troops opening fire on unarmed Boston colonists. **If you were a colonist, how would this picture make you feel?**\n\nThe cartoon doesn't tell the whole story of what happened, but Revere's image was burned into the minds of American colonists. The American Revolution may not have happened if it wasn't for these brave printers and the work of their presses.\n\nPaul Revere continued to make political cartoons and sketches during the American Revolution. He wasn't the only American cartoonist to create political drawings during this time, but his were among the most popular.\n\nPolitical comics are usually shown in a single panel and use exaggeration and symbols to tell their stories. Often they are meant to influence their readers' thoughts about a political party or country. Paul Revere found that if he showed the British as the bad guys in his comics, he could get more Americans on the side of the revolutionaries.\n\nAmerican newspapers continued to print political cartoons after the revolution. Instead of the British, the bad guys in the comics were whomever the artist disagreed with. This could even include the president.\n\nPAMPHLETS AND BROADSHEETS OFTEN EMPLOYED ARTISTS TO CREATE DRAWINGS THAT INFLUENCED THEIR READERS TO ACTION.\n\nWORDS TO KNOW\n\nrevolutionary: someone committed to fighting a ruler or political system.\n\nPAUL REVERE: PATRIOT CARTOONIST\n\nPaul Revere was made famous by the Henry Wadsworth Longfellow poem, \"Paul Revere's Ride.\" We all know the famous ride Paul Revere took through the Massachusetts countryside to warn everyone by calling, \"The British are coming! The British are coming!\" What many of us don't know is that Paul Revere was also one of America's first cartoonists.\n\nWORDS TO KNOW\n\ncontent: the written material and illustrations in a story, article, book, or website.\n\nA famous silversmith in Boston, Paul Revere often doodled funny drawings on scraps of paper in his spare time. He was known as a supporter of independence who met with other revolutionaries to plot how they could get the British out of the colonies. His skills as a doodler were needed when revolutionary broadsheet printers wanted to include more visual content for their readers.\n\nBesides the Boston Massacre, Paul Revere drew comics during the American Revolution that depicted the British soldiers, the colonists loyal to the king, and the king himself as bad guys. The single panel form was Paul Revere's favorite way of making his drawings. He was one of the first American cartoonists to use a form of the word balloon. Characters were shown to \"talk\" with ribbons of words coming out of their mouths.\n\nJAPANESE MANGA\n\nIn 1812, a Japanese artist named Hokusai began creating sketchbooks he called manga. These quick drawings often told stories of traditional Japanese life, from local fishermen to powerful military leaders called shoguns. Hokusai drew very quickly and captured the movement and expression of people going about their everyday lives. He saw his sketchbooks as a good way to develop skills for his full-time job creating stories on tapestries, scrolls, and silk divider screens.\n\nArtists employed by his studio were required to make their own manga using Hokusai's techniques. Hokusai's drawings were kept and passed on to other Japanese artists, influencing the development of Japanese comics.\n\nToday, Japan has one of the most lively comics cultures. Comics are made about everything, from funny stories to serious lessons about doing business.\n\nHOKUSAI'S NAME FOR HIS DRAWINGS ARE WHAT THE PEOPLE OF JAPAN CALL COMICS TODAY\u2014MANGA.\n\nWORDS TO KNOW\n\nmanga: a term for Japanese-style comics.\n\nculture: the beliefs and way of life of a group of people.\n\nSIDEWALK CAVE PAINTING\n\n**SUPPLIES:** long dry sidewalk, colored chalk\n\n**Ancient humans used simple drawings on cave walls to tell stories about hunting, dreams, and daily activities. You can draw stories just like the ancient cave people!**\n\n1Think of a story from your life you would like to tell a friend. Did your family go on a special vacation or a fun adventure?\n\n2Break your story down into sections. It should have at least three sections: a beginning, middle, and end. Your story might have more. A story about learning to ride a bike might have five sections. Can you think what they would be?\n\n3Find a length of sidewalk with as many spaces as you have sections. If your story has five sections, you will need five sidewalk spaces.\n\n4Starting with the first sidewalk space on the left, use the chalk and begin to draw your story. You can use stick figures and simple drawings to show what is happening. Continue drawing your story on the other spaces of sidewalk.\n\n**TRY THIS!** How could you tell stories like this using different materials? What if you were on a beach or in the snow? What's the longest story you could tell? Try getting some friends to tell a story together. Have each person add a different detail to each section and take turns telling the story to each other.\n\n**MANGA-STYLE SKETCHBOOK**\n\n**SUPPLIES:** several sheets of 6-by-12-inch (15-by-30-centimeter) white paper, stapler, glue, 2 squares cut from cereal boxes each 6 by 6 inches (15 by 15 centimeters), 1 sheet of 8-by-14-inch (20-by-36-centimeter) colored paper, markers\n\n**Hokusai made sketchbooks and filled them with drawings of what he saw each day. You can make your own manga-style sketchbook.**\n\n1Stack the white paper. Fold the stack in half to make a square, unfold, and put three staples along the fold.\n\n2Glue the cardboard squares to each side of the colored paper. Leave \u00bc inch (\u00bd centimeter) of space between them. Fold the excess paper around the cardboard and glue it down.\n\n3Glue the first and last pages of your stapled pages to the left- and right-hand cardboard squares so that the glued edges are covered.\n\n4Decorate the cover. Add a title and the date you start. You can make more sketchbooks as you fill them up, numbering each as you go. Include a start and finish date on each cover.\n\nTRY THIS! Can you make sketchbooks in different shapes? Try using different colors of paper for the cover and different types of cardboard to make your cover more durable! How would you make a pocket-sized sketchbook?\n\nDRAW LIKE AN EGYPTIAN!\n\n**SUPPLIES:** paper, pencils, blue painter tape, large pieces of cardboard, scissors, paint, paintbrushes\n\nThe ancient Egyptians loved to tell stories about everything their civilization did! They used the walls of their pyramids, palaces, and temples to tell their tales. Popular themes were palace life, beliefs, and even farming! Follow in their footsteps and use words and drawings to tell future generations about something you do everyday!\n\n1Think about something you do every day that takes several steps to complete, such as getting ready for school. On a piece of paper, draw out the steps, starting on the left side. If you need more space, tape on another piece of paper until you have finished with your steps.\n\n2Cut up the cardboard boxes into large pieces of cardboard. Find an empty wall where you can tape up your pieces of cardboard. Make sure there is room to fit everything you drew on the paper. If you needed to use more than one piece of paper in step one, make sure you use at least that many pieces of cardboard on the wall.\n\n3Just like the early cartoonists of the Renaissance, draw big versions of your steps on the cardboard. Use words above your images to tell your audience what is happening in that step. For example, if you drew yourself putting on your shoes you might write, \"put on shoes, tighten Velcro straps.\"\n\n4After you have drawn large versions of all of your steps, go through and add color with your paint. You now have a step-by-step record of something you do every day!\n\nTHE ROSETTA STONE\n\nThe Rosetta Stone was a carved tablet that translated hieroglyphics into other known ancient languages. Long before the Rosetta Stone was discovered, archaeologists were able to understand a lot about the Egyptian people because of their use of images in sequence. While they were still closer to ancient cave paintings than they were to the modern comic strip, these Egyptian stories were another step to what we now know as comics! But, as you've seen, the ancient Egyptians weren't the only ancient civilization to use images in sequence to tell important stories.\n\nWORDS TO KNOW\n\nRosetta Stone: a stone tablet written in 196 BCE telling the same decree using hieroglyphics, Egyptian Demotic script, and ancient Greek. The stone was fully translated in 1822, leading specialists to understand hieroglyphics better in the nineteenth century.\n\nCHAPTER 2\n\nCOMICS IN THE NEWSPAPERS\n\nHave you read a newspaper today? Look in any newspaper and you might find dozens of comics printed in their very own section. Comics such as _Peanuts_ , _Calvin and Hobbes_ , _Garfield_ , _Get Fuzzy_ , and _Pearls Before Swine_ have all become household names thanks to their presence in newspapers.\n\nComics like these didn't exist in early American newspapers. Early comics were usually political cartoons such as those drawn by Paul Revere. It took the popularity and innovation of one political cartoon character to break open the gates for modern newspaper comics. This character was a bald boy in a yellow smock.\n\nWORDS TO KNOW\n\ninnovation: a new creation or a unique solution to a problem.\n\nsmock: a cloth worn over clothing to protect it from stains.\n\nTHE YELLOW KID\n\nIn 1895, Joseph Pulitzer was one of the most powerful newspaper publishers in America. He hired Richard F. Outcault to create the first serial comic strip character. Outcault's comic was called _The Yellow Kid_. The character wore a yellow smock with writing on it that stated what he was thinking or saying.\n\nFirst conceptualized as a political cartoon, _The Yellow Kid_ soon became very popular. Pulitzer published the comic on one entire sheet of newspaper using new printing techniques that allowed for cheap color reproduction.\n\nThe bright yellow of the kid's smock and the humor in the comic caught the attention of Pulitzer's readers. _The Yellow Kid_ resembled old political comics and broadsheets since it rarely used panels, relying on one giant illustration to convey many jokes. For a while, _The Yellow_ _Kid_ was as popular as Mickey Mouse is today.\n\nWORDS TO KNOW\n\nserial: occurring in a series.\n\nconceptualized: imagined and thought out.\n\ncolor reproduction: to make color prints of an original piecec of art.\n\nhumor: the quality of being funny.\n\nTHE POPULARITY OF THE YELLOW KID LED PULITZER TO FIND OTHER CARTOONISTS WHO WERE EAGER TO SEE THEIR CHARACTERS REACH A WIDER AUDIENCE.\n\nPulitzer's main rival in the newspaper industry, William Randolph Hearst, wanted to include comics in his newspapers, too. He even tried to convince Richard Outcault to leave Pulitzer's newspapers and draw _The Yellow Kid_ for Hearst's publications.\n\n**THE AMAZING COLOR PRINTER**\n\nBefore digital printing was invented, most color printing was done using a process of layering the tints of four colors. This tricked the eyes of readers into seeing more than just the four colors. The process, called CMYK, is still used today. The colors are cyan (C), magenta (M), yellow (Y), and black (K).\n\n  Richard Outcault's comics often took up an entire page of the newspaper and could be very busy and chaotic. **Take a look at this comic from** **1896.** How is it different from the comics you find in the newspaper today? Is it easier to read? More fun to look at? Is it more confusing? Why do you think comics have evolved into their current, simpler form?\n\nWORDS TO KNOW\n\ndigital: characterized by electronic and computerized technology.\n\nlayering: stacking images on top of each other.\n\ntint: a shade or variety of color.\n\nEACH LAYER USES SHADES OF THOSE FOUR COLORS, WHICH, WHEN STACKED ON TOP OF EACH OTHER, CREATE WHAT LOOKS LIKE MANY DIFFERENT COLORS.\n\nEarly newspapers in America relied on movable type and woodcut images to print a single page. It was very time consuming! Large towns could afford to print daily newspapers, but small towns didn't usually have this luxury. Then, in 1843, an inventor named Richard March Hoe created the steam-powered drum printer, which took castings from a master page. It could print page after page from a large roll of paper. The drum printer made the printing process faster and easier.\n\nIn the 1880s, mass color printing was developed. This new color separation technique was called chromolithography and allowed newspapers to print comics and etchings of photographs in color for the first time. Newspapers flew off the stands!\n\nWORDS TO KNOW\n\nmovable type: a process of printing that uses individual type pieces to spell out words.\n\nwoodcut: a way of printing by carving an image on a piece of wood before adding ink and printing the image onto paper.\n\ncasting: a metal print form created to make multiple copies of a printed page.\n\nmaster page: the version of a printed page that is used to make other copies.\n\nchromolithography: a color printing process using metal plates to layer tints of color.\n\netching: a print made by scratching original art onto a metal plate.\n\nFACT OR FICTION?\n\nCould the popularity of America's two major newspapers help to spark a war? A large part of the reason the United States went to war with Spain in the 1890s was because the public was influenced by articles written in Hearst's and Pulitzer's papers. The anti-Spanish articles were later found to be mostly false. The reporters had been actively encouraged to exaggerate their stories to help sell more newspapers! Do you think this could happen today? Why or why not?\n\nTHE FIRST SUNDAY COMICS\n\nJoseph Pulitzer invented one of the most enduring forms of comics\u2014 the Sunday comics. First published in 1895, the Sunday comics were a collection of _The Yellow Kid_ comics plus several others. The newspapers of the late nineteenth century were much larger than our papers today. The cartoonists often had an entire page to create their comics, and that meant a lot of creativity could go into each one.\n\nOnce Pulitzer proved that the Sunday comics increased sales, other publishers weren't far behind in publishing Sunday comics of their own. Newspapers started calling these new drawings the funnies.\n\nThrough the years, the Sunday comics have decreased in size, but they still remain one of the major selling points of newspapers.\n\nWHAT'S YOUR FAVORITE FUNNY TO READ IN THE NEWSPAPER?\n\nWORDS TO KNOW\n\nfunnies: the original name of comic strips in Sunday newspapers.\n\nEARLY COMIC MASTERS\n\nWhen comics first started appearing every week, newspapers were eager for new talent. Here are a few of the early masters of the modern comic medium.\n\n**Henry Conway Fisher and** _**Mutt and**_ _**Jeff**_ **:** In 1907, Henry Conway Fisher, also known as Bud, introduced one of the greatest innovations in cartooning: the comic strip. Until then, most comics were huge splashes of color and action with few panel borders, such as in _The Yellow Kid_. Fisher decided to tell stories with **standardized** panels that were always the same size. Panels helped readers understand what was happening. The comic strip was also much smaller, which meant newspapers didn't have to use most of a page for just one comic.\n\nBud Fisher's comic strip, _A. Mutt_ , featured a man called Mutt who got into lots of trouble. Hearst loved the work Fisher was doing and hired him to produce the strip for his national newspapers.\n\nWORDS TO KNOW\n\nmedium: the material artists use to create their art, such as stone, paint, and ink.\n\nstandardized: a set way of doing something.\n\n  Look at the Mutt and Jeff comic strip that was reprinted in 1948. Is this the kind of humor people today would find funny? Why or why not?\n\nAlong with inventing the panel, Fisher also introduced another innovation to comics: the sidekick. Bud decided to draw another character into his strip, a man called Jeff. The friendship between Mutt and Jeff was so popular that Fisher began to include him as a regular part of the story. He even renamed the strip _Mutt and Jeff_.\n\n**George Herriman and** _**Krazy Kat**_ **:** The first widely published African American cartoonist was George Herriman from Louisiana. Herriman's comic strip, called _The Dingbat Family_ , began to appear in newspapers in 1910. The strip followed a cast of characters in an apartment building.\n\nKrazy the cat and Ignatz the mouse were the most popular characters in the strip. One day, the mouse flung a brick at the cat's head. The cat took the brick as a sign of love from the mouse. This ridiculous premise appealed to readers, and Krazy and Ignatz eventually took over the daily strip. Herriman renamed the strip _Krazy Kat_ and introduced a third character, a dog called Offissa Pupp.\n\nWORDS TO KNOW\n\nsidekick: a character who supports the main character.\n\ncast: a group of characters.\n\npremise: the main idea of a story.\n\nrace: a group of people with the same skin color and other physical features.\n\n  **More** _**Krazy Kat**_ **!**\n\nHerriman played with the logic of the panels. Characters might find themselves in different seasons over the span of a few panels or they might chase each other around the moon. Eventually, inspired by the natural rock formations at Arizona's Monument State Park, Herriman modeled his imaginary world after the real Coconino County.\n\nHERRIMAN WROTE dialogue WITH AN INVENTED SPELLING OF ENGLISH WORDS THAT REFLECTED THE SOUNDS OF SPANISH, YIDDISH, AND CREOLE ACCENTS.\n\n_Krazy Kat_ was never a huge favorite with the general public. Many newspaper editors wanted to cancel the eccentric comic, but they couldn't since their boss, Mr. Hearst, loved it. It was also loved by many great artists and writers of the early 1900s, including the painter Picasso and the poet e.e. cummings.\n\nWORDS TO KNOW\n\nlogic: the principle, based on math, that things should work together in an orderly way.\n\ndialogue: a conversation between two people.\n\neccentric: odd, usually in a unique way.\n\n**Winsor McCay and** _**Dream of the Rarebit Fiend**_ **and** _**Little Nemo in Slumberland**_ **:** Winsor McCay began his career as a cartoonist drawing political cartoons, but he started making comic strips for his local paper after he saw how popular they were. His comic, _Dream of the Rarebit Fiend_ , was an immediate success.\n\nThe exciting stories grew stranger and stranger until the final panel where the main character woke up. He blamed the strange dream on a cheese dish called rarebit, which he'd eaten before bed.\n\nIn 1905, McCay created _Little Nemo in Slumberland_. The strip followed the dream adventures of a boy named Nemo who was joined by a repeat cast of characters. With gigantic, colorful panels filled with inventive worlds, it almost always ended with Nemo waking in his bed and his father telling him to settle down. In one famous Sunday strip, Little Nemo dreams that his bed comes to life, the legs grow incredibly long, and it walks above New York City.\n\n  Winsor McKay's comics were known as much for the artwork as the dialogue. Look at one of his comics from 1905. Where would a comic like this most likely be published today? In newspapers, magazines, or on websites?\n\nWORDS TO KNOW\n\nrarebit: a type of soft cheese which is famous for giving indigestion.\n\n_Little Nemo_ was wildly popular, and the inventive spirit McCay brought to his comic continues to inspire cartoonists today. While he was influenced by some of the art of his time, he was mostly inspired by the writings of psychologists.\n\n**Harold Gray and** _**Little Orphan**_ _**Annie**_ **:** Harold Gray created the very popular _Little Orphan Annie_ in 1924. The comic featured the adventures of an orphan girl with curly hair, a red dress, and large, blank eyes. Annie's eyes allowed readers to add their own expressions to the character.\n\n_Little Orphan Annie_ became an iconic image of the times. Annie was the foster child of a rich man, Daddy Warbucks, and they were always getting separated and having to find each other again.\n\nWORDS TO KNOW\n\npsychologist: a person who studies the mind and behavior.\n\niconic: a widely recognized symbol of a certain time.\n\nDoes _Little Orphan Annie_ sound familiar? The popular comic strip inspired a famous musical called _Annie_ , which has been turned into a movie and has been performed by thousands of schools and local theater groups throughout the country. Next time you find yourself singing along to \"It's a Hard Knock Life,\" think about how it all started with a comic strip from the 1920s!\n\n**E.C. Segar and** _**Popeye**_ **:** Comics of the early twentieth century were packed with physical action. E.C. Segar's _Popeye_ certainly had its share of fighting. Segar introduced Popeye as a side character in his popular strip, _Thimble Theater_.\n\n  Little Orphan Annie was so popular that, during a newspaper strike in the 1940s, New York City Mayor Fiorello LaGuardia read the comic out loud on the radio so readers could keep up with her adventures. **Listen to** **the mayor's radio address.** Do you think the comic was as much fun to listen to as it was to read? How does the mayor include a lesson after reading the comic out loud to the audience?\n\nTHE 50-YEAR-OLD, ONE-EYED SAILOR WHO LOVED TO FIGHT EVENTUALLY STOLE THE STRIP!\n\nSegar was surprised to discover the popularity of a character he hadn't planned on using for more than a few strips. Popeye's fame grew, and just as in _Krazy Kat_ , the strip was renamed _Popeye_ after the new main character.\n\nPart of the popularity of _Popeye_ had to do with the action sequences. The strip portrayed Popeye fighting a variety of villains, including his arch-nemesis, Bluto. Segar drew exaggerated scenes, showing the energy of each punch, kick, and wallop with the intensity of modern-day special effects.\n\nMANY HISTORIANS THINK POPEYE WAS THE FIRST SUPERHERO. HE COULD PUNCH HARDER THAN ANY LIVING MAN, ESCAPE ANY TRAP, AND EVEN STOP BULLETS.\n\nWORDS TO KNOW\n\nvillain: a character who opposes the hero and does bad things.\n\narch-nemesis: a character who is the opposite of the main character, and usually the enemy.\n\n\"I YAM WHAT I YAM\"\n\nPopeye ate spinach to get strong. His theme song even touted the strengthening powers of the green veggie, \"I'm Popeye the Sailor Man, I'm strong to the finish because I eats me spinach, I'm Popeye the Sailor Man!\"\n\nThe publishers of Popeye thought this was a great way to get kids to eat healthy. A decimal mistake made in 1870 made people think spinach contained much more iron than it really does. Spinach is healthy, but not as healthy as people thought before 1937, when the mistake was fixed!\n\nTHING, JOB, DESCRIPTION\n\n**Supplies:** pencils, paper, several friends, scissors, three cups, colored pencils or crayons\n\n**Comics are about communicating ideas with pictures and words. A cartoon needs to convey important information to the reader without having to say it. For example, if your main character is a crime-fighting duck, your readers need to immediately recognize it as both a duck and a crime fighter.**\n\nAs you do this exercise with a group of friends, keep in mind some key questions:\n\n*What makes my animal look unique?\n\n*What sorts of things give clues about my animal's job?\n\n*How can someone tell what my character is feeling without words?\n\n1Make three columns on a piece of paper and label them \"Animal,\" \"Job,\" and \"Description.\" In each column, write a list of examples. Write enough so each of your friends will get one from each column. For example, if you have eight friends, you would write out eight animals, eight jobs, and eight descriptions. These descriptions could be words such as clumsy, sleepy, or creative.\n\n2Cut out each animal, job, and description. Put the animals in one cup, the jobs in another, and the descriptions in the last one. You can fold up each piece of paper so that anyone picking out of the cup can't see what it is.\n\n3Pass each cup around the room. Everyone takes one piece of paper out of each cup. Once everyone has all three pieces of paper, get drawing with colored pencils and crayons! Make sure nobody can see what your combination is and remember, **don't use any words**.\n\n4Have each person show his or her drawing to the group and see how quickly people can guess what each of the drawings depicts.\n\nTRY THIS! What helped people guess correctly? Did color play a role? Try adding a fourth category to the exercise. Also try replacing the categories with other things, such as vegetables instead of animals, or super powers instead of jobs.\n\nDRAW CARTOON FACES\n\n**Supplies:** pencils, paper, good eraser, pen or thin marker\n\n**One of the best ways to show emotion on a character is through facial expressions. The face may seem complicated to draw at first, but when you follow a few simple steps you can draw all kinds of faces! The best way to draw anything is by breaking it down into simple shapes.**\n\n1 **FACE SHAPE** : Most faces are oval, but some look angular. Draw a few different face shapes. Here are some examples.\n\n2 **GUIDELINES** : Once you have a face shape you're happy with, lightly draw a vertical and horizontal guideline through the middle. These lines are very important. They guide us when placing the parts of the face.\n\n3 **NOSE** : The nose is often used as a reference point on the face because it doesn't change shape as often as eyes, eyebrows, and the mouth. The bottom line of the nose is usually halfway between the horizontal guideline and the chin. Here are some sample nose shapes.\n\n4 **EYES** : Every cartoonist has his or her own style of drawing eyes. But the darkest part of the eye, the pupil, almost always goes on the horizontal guideline. Here are some examples.\n\n5 **MOUTH:** Lightly draw two lines straight down from the pupils of the eyes. At about halfway between the bottom of your nose and the bottom of the chin, draw your mouth. Start one end at an eye line and connect to the other.\n\n6 **EYEBROWS:** Eyebrows show the most emotion on a face. Look at the following examples and see how a blank face goes from normal, to mad, to sad, to tired, to surprised\u2014all because of eyebrows.\n\n7 **EARS:** To add ears, start at the horizontal line on the sides of your face. Curve up slightly before swooping around to the bottom of your ear. The base of the ear should be along the same line as the bottom of the nose. Take a look at the people around you. Do you notice how their eyes, noses, and ears all line up pretty much the same way?\n\n8 **HAIR** : Start the hair alittle below the top of the head. The biggest mistake people make when drawing hair is not connecting it to the head! Look at your own hair or your friends' hair. Which hairstyle will you choose?\n\n9 Finish drawing the chin and trace over all the lines you want to keep with pen. This is called **inking**. Erase the pencil guidelines.\n\nWORDS TO KNOW\n\ninking: to use ink to add definition to pencil drawings.\n\nDRAW CARTOON BODIES\n\n**Supplies:** pencils, paper, tracing paper, eraser, photographs of people that show the full body such as images from magazines\n\n**Cartoon faces need to be attached to cartoon bodies to do stuff such as run, dance, sit, and fly. Get ready to draw cartoon bodies\u2014bones first! Stick figures give us a general idea of where to put everything on our final drawing. Like the bones in your body, these lines give us a frame to build the meat of our characters.**\n\n1 Start with an action line. What's an action line? Think of it as the moving direction of your character. The middle section of the line will be your character's spine. Our character will be standing still, so our action line will be straight up and down.\n\n2 Start at the top of your action line and draw your face shape, with the top of the line close to the top of the head. Don't make it too large since you need to add the rest of the body.\n\n3 Just like faces, bodies can be drawn first as simple shapes. Let's draw a rectangle for the body. Then draw four circles, one at each corner of the rectangle.\n\n4 Draw one line from each circle. These will be the arms and legs. Draw a circle at the end of each arm and a triangle at the bottom of the legs.\n\n5 To make arms, draw two lines on either side of your arm line connecting the shoulder circle to the hand circle. If you want to get fancy, add an elbow circle and then connect shoulder to elbow and elbow to hand.\n\n6 Repeat this process for the legs and neck. Erase the bone lines when you are done.\n\n7 Clothes take a lot of practice to get right. Once you have the basic shapes of your body drawn out, however, you can hang clothing off of your character fairly easily. Add a simple T-shirt and pants.\n\nTRY THIS! What body shapes do you see all around you? What is your body shape? Can you draw a full cartoon of yourself, face and body? Once you've mastered body shapes, try to draw different types of clothing on your characters. You can draw a bunch of dummy bodies and practice adding your fashions to them. Fashion designers do the same thing everyday!\n\nTRY THIS, TOO! Now that you've mastered a cartoon body standing still, try to draw one in action. Look at people in different positions in magazines. Use tracing paper and try to find their action line, and then draw their sketchy skeleton. Try it on a bunch of different people and poses to see how your sketches change. Find some photos of kids and babies. Do their sketchy skeletons look the same as the adults? What's different?\n\n**DRAW HANDS AND FEET**\n\n**Supplies:** pencils, paper, eraser, hands and feet\n\n**Some parts of a character are really tricky to draw. The two that cartoonists struggle with the most are hands and feet. Here are some suggestions for tackling these tricky parts.**\n\n**HANDS:** Start with shapes. Most complex body parts can be broken down into a combination of basic shapes. For hands, think of circles and ovals.\n\n1 To draw a hand with the palm up, first draw a circle. This will be the palm of the hand you are drawing.\n\n2 Now, draw an oval on top of the circle where the base of the thumb goes.\n\n3 Draw five finger lines. Make a shorter one from the thumb oval and four coming out the top of the circle\u2014 notice how they are different lengths.\n\n4 Using simple guidelines, mark off the knuckle, just as you marked off the elbows and knees in the cartoon body project on the previous page. Then add some meat to the skeleton fingers. Erase your guidelines and marvel at your accomplishment!\n\n**FEET:** Have you ever really looked hard at your foot? What shape is it? Artists tend to think of the foot as a triangle and a circle. You can also think of your foot as a wedge.\n\n1 Looking at your own foot for reference, draw what you think the basic shape is, either a wedge shape or a triangle\/circle.\n\n2 Toes aren't as long or bendy as fingers, but they have distinct shapes. The big toe is a little farther away from the other four and is the base to your foot's arch. Start your toes by drawing little ovals for each toe.\n\n3 Just as you did for the arms, legs, and fingers, add the outline to each of your foot's sketchy skeleton parts.\n\n4 Erase your guidelines and enjoy! Next time, try drawing some shoes. How does this change how you think of a foot's shape?\n\nTRY THIS! Whenever you find yourself with a little free time, you can practice drawing hands. You always have one with you! Like learning to play an instrument, drawing takes practice.\n\n**DESIGN A CAST OF CHARACTERS**\n\n**Supplies:** pencils, paper, markers, small objects from around the house (such as vegetables, fruit, bottles, glasses, boots, and electronics)\n\n**If you saw just a dark outline of your favorite cartoon character, you could probably tell who it was from its shape. Russell from the movie** _**Up**_ **looks like an egg, Popeye is skinny with a knobby head and huge forearms, and Sponge Bob is, well, a sponge! Interesting shapes make interesting characters. Practice making different characters out of everyday shapes.**\n\n1 Arrange your objects in front of you and draw each shape on a piece of paper. Review all your shapes\u2014look at them upside down and sideways. Do the shapes suggest any body types to you? For example, an upside down ketchup bottle may look like a football player with a big chest.\n\n2 Once you have some body types in mind, start to figure out how to turn each shape into a character. Sketch out the face, where the pants go, and add arms and legs. Use what you've learned from the projects so far to bring your characters to life. Don't forget to give your characters names!\n\nTRY THIS! When you watch a cartoon or animated movie, have your homemade manga sketchbook handy. Pause the movie when a new major character is introduced. Try to sketch the shape of that character. Do different shapes tell you something about the character? What are heroes shaped like? What about villains?\n\n**SUNDAY COMIC STRIP**\n\n**Supplies:** pencils, paper, ruler, pens, colored pencils, sketchbook with your character designs\n\n**Now that you have an awesome cast of characters, it's time to show them off in a Sunday comic strip! Every good comic strip begins with a thumbnail. Not a real thumbnail! In cartooning, a thumbnail is a small sketch, or plan, of what your whole page will look like.**\n\n1 Start your thumbnail by making your comic strip panels. Draw a rectangle about 3 by 2 inches (8 by 5 centimeters) and then divide it up into panels. Use simple stick figures and rough word balloons to sketch out the action or joke of the strip. The beauty of thumbnails is you can erase them to change your ideas. You can even draw several thumbnails of each comic to try to find the best design.\n\n2 On a fresh piece of paper, lay out your comic panels carefully with a ruler. Follow your thumbnail so that you have enough panels and they are in the right place.\n\n3 Sketch out your characters and words with pencil first, then use your pen to trace the lines you want to keep. When you are done inking, erase the pencil.\n\n4 Now you are ready for color! Be creative and don't be afraid to use lots of color. Give your masterpiece a title and sign your name!\n\nTRY THIS! Design other Sunday comics using different shapes for the panels. Do circles or triangles make the comic funnier or more serious? Try making a comic with no words at all. Can you still tell a joke or story without words?\n\nCHAPTER 3\n\nTHE BIRTH OF SUPERHEROES\n\nEveryone loves heroes! Ever since ancient times, people have been telling stories about men and women with super strength who risk their own safety to save the lives of others.\n\nThe oldest hero to appear in an epic was Gilgamesh, a hero from a region of the world called the Fertile Crescent. This area is where the countries of Iraq, Syria, Kuwait, and Iran are today. Gilgamesh was very strong and helped save his city\u2013state from monsters and villains.\n\nOne of Gilgamesh's first challengers was Enkidu, a powerful warrior whom Gilgamesh fought for weeks. Eventually, Enkidu was defeated and became Gilgamesh's friend. Enkidu was one of the first sidekicks!\n\nWORDS TO KNOW\n\nepic: a long poem, usually about the life of a hero or heroine.\n\ncity\u2013state: a city and its surrounding area, which rules itself like a country.\n\nAncient Greeks also told tales about superheroes, such as Achilles, who had only one weak spot\u2014his heel. Another hero, Odysseus, battled monsters on his journey home from war. Stories and poems from the past often featured very strong people who protected weaker people.\n\nWHY WERE STORIES ABOUT SUPERHEROES POPULAR BACK THEN? WHY ARE THEY POPULAR NOW?\n\nEARLY SUPERHEROES\n\nComic book superheroes have several predecessors other than the ancient Greek heroes. One of those is the Scarlet Pimpernel. Created by a Hungarian writer named Baroness Emma Orczy, the Scarlet Pimpernel was a vigilante who saved wealthy aristocrats from the horrors of the French Revolution. It is one of the first instances of a hero using a disguise and an alter ego, two common characteristics of modern-day superheroes, such as Batman. _The Scarlet Pimpernel_ was made famous in America through a new form of entertainment, pulp magazines.\n\nWORDS TO KNOW\n\npredecessor: someone or something that came before others.\n\nvigilante: a person who takes the law into his or her own hands.\n\naristocrat: a person of royal blood or privilege.\n\nFrench Revolution: a period of violent change in France between 1789 and 1799.\n\nalter ego: a second personality in the same person.\n\ninvincible: someone who cannot be defeated.\n\npulp magazine: a cheap fiction magazine published between 1896 and the 1950s.\n\nPulp magazines were cheaply made publications that people bought for the exciting stories of romance, science fiction, adventure, westerns, and suspense. Many Americans were first introduced to characters such as Tarzan and Sherlock Holmes through pulp magazines. Other twentiethcentury heroes had their start in the pulps, too, such as Buck Rogers and Zorro. The covers of pulp magazines were bright and flashy and would sometimes have nothing to do with the actual stories inside.\n\nTAKE A LOOK AT THE COVERS OF TODAY'S COMIC BOOKS\u2014THEY HAVE A VERY SIMILAR LOOK TO THE PULP MAGAZINES OF THE EARLY 1900.\n\nOne of the most important science fiction pulp magazines was called _Amazing Stories_ , which was about a future in which everyone had a jet pack or battled alien cultures on Mars. The cover of one _Amazing_ _Stories_ pulp shows a man flying through his neighborhood, aided only by a small device. It inspired a few important teenagers in Cleveland, Ohio, who created a superhero who would one day fly.\n\nWORDS TO KNOW\n\nscience fiction: stories that deal with the influence of real or imagined science.\n\nsuspense: a feeling or state of nervousness or excitement caused by wondering what will happen.\n\nRADIO!\n\nThe early twentieth century saw the rise of a very popular new technology\u2014radio! Radio programs, many of them inspired by pulp magazines, were broadcast across the country. From coast to coast, people could listen to the same shows. Some of the first radio networks are still around today, although they mostly do television. The Columbia Broadcasting System (CBS) and the National Broadcasting Company (NBC) are two of these. These radio broadcasts helped make three modern heroes household names: Tarzan, Zorro, and the Shadow.\n\n**THE FIRST COMIC BOOKS**\n\nThe first comic books looked like the Sunday newspaper supplements. These collections of material that had already been published were simply stapled together with a flashy cover. Soon, actual comic book publishers began to create new content, such as comic adaptations of Bible stories or historical events.\n\nFew people thought that comic books could ever be popular. Newspapers, movies, radio, and pulps\u2014that's where people assumed they'd find excitement, action, humor, and drama. But they were wrong! A few key characters helped to make comic books very popular.\n\nWORDS TO KNOW\n\nbroadcast: a program transmitted over a long distance.\n\nnetwork: a company that provides programs to be broadcast over radio or television stations.\n\nadaptation: telling a story in a different way and in a new format. For example, making a musical out of a series of comic strips would be adapting the comics into the musical.\n\ndrama: an exciting event or series of events.\n\nOne of those characters was Superman. In 1932, two buddies named Jerry Siegel and Joe Shuster were working as a freelance team. Jerry was the writer and Joe was the artist. Together, they brainstormed the idea of an alien who looked just like us, but had abilities above and beyond humans. They wanted their hero to be larger than life with an energy you could feel coming off the page.\n\nSiegel and Shuster's idea was rejected by many publishers during a period of several years. Finally, they pitched their idea to a small publisher in New York City called Action Comics, which agreed to give them a try.\n\nSuperman first appeared as a man in a blue costume and a red cape with the letter \"S\" on his chest, lifting a car over his head. Superman wasn't the only story to appear in this iconic issue. However, he is the only character people remember!\n\nWORDS TO KNOW\n\nfreelance: to work on a project without being employed by the company assigning the work.\n\nbrainstorm: to come up with a bunch of ideas quickly and without judgment.\n\npitch: to present an idea.\n\n  **Check out the very first** **Superman cover.** The publisher worried that people would find the idea of a man lifting a car ridiculous.\n\nThe publishers of _Superman_ found their second big seller the following year. A young cartoonist named Bob Kane was inspired by movies and pulp heroes such as the Scarlet Pimpernel. He pitched the idea of a wealthy man who, at night, dresses in a bat suit to rid his town of crime. In a 1933 issue of Detective Comics, Batman first swings into the world. He proved to be another popular figure and helped to introduce what would become known as the Golden Age of comics.\n\nTHE GOLDEN AGE OF COMICS\n\nThe superhero comic books of the early 1930s are the beginning of what many comics historians call the Golden Age of comics. It was golden because it was so new and popular. Everyone wanted to read comic books, both kids and adults.\n\nComics during this era expanded beyond their stapled pages and were adapted into movies, radio programs, and daily comic strips in national newspapers. Sales of comics made publishers rich and comics artists famous.\n\nWORDS TO KNOW\n\nhometown: the place where a person, thing, or idea is born.\n\nGolden Age: the period of comics history before the 1950s, considered to be the best and most popular by many fans and historians.\n\nera: a set segment of time.\n\nTHE BIG SCREEN\n\nLike radio, movies took inspiration from pulp magazines. One of the first popular science fiction movie series was about Buck Rogers, a famous pulp fiction and comic strip hero. Most movies had live actors and actresses performing superhuman feats, but animation has always been popular, too. One of the most famous early animated films is an adaptation of E.C. Segar's Popeye by the Fleischer Brothers Studio. Never before had audiences seen the likes of Popeye, except in the funnies of course!\n\n  **Watch one of the first animated movies of Popeye!** How is it different from the cartoons kids watch today?\n\nWORDS TO KNOW\n\nanimation: multiple drawings in sequence to show movement.\n\npenciling: to create an original comic first in pencil.\n\ndynamic: full of movement.\n\nThere were no restrictions on comics in the Golden Age. They could be about anything and people still loved them. This level of comics popularity would not be seen again in America for nearly 50 years. A few comics artists stand out for being especially innovative.\n\n**Will Eisner:** Will Eisner started his comics career penciling and inking for other cartoonists. Eisner decided to make his panels more dynamic by matching the shape and design of the panels to the action within them.\n\nIn 1940, Eisner was given the chance to create his own superhero. Unlike the other cartoonists of his day, he had complete control over how the story was told and how it was published. He retained all the creative rights to his creation. Called _The Spirit_ , Eisner's comic followed the adventures of a former detective who was thought to be dead.\n\n**Joe Simon and Jack Kirby:** During the 1930s and 1940s, a new, small pulp publisher called Timely Comics needed a hit superhero to rival the popular Superman. Writer Joe Simon, who was also the head editor of Timely Comics, teamed up with an old friend of his, Jack Kirby. Together, they invented the next great American superhero\u2014Captain America.\n\nThe first _Captain America_ comic was modeled on the most talkedabout news of 1940. Captain America was sent to do battle with the rising Nazi threat and Adolf Hitler himself. Would _Captain America_ have been as popular if it had been published in a time of peace instead of a time of war? Why?\n\nWORDS TO KNOW\n\ncreative rights: the right to use something that is created, often for profit.\n\nNazi: the main political party of Germany before and during World War II.\n\nCAPTAIN AMERICA WAS AN IMMEDIATE SUCCESS.\n\nThe art and storytelling of Jack Kirby was unlike other comics of its day. Huge splash pages were used for the battle scenes. The characters were drawn as a cross between the flashy cover art of pulp magazines and the frenzy of newspaper comics such as _Popeye_. The comic world hadn't seen anything like it.\n\nMORE ABOUT JACK\n\nJack Kirby was born Jacob Kurtzberg to Jewish immigrant parents. He grew up in a poor section of the lower east side of Manhattan and learned how to draw from copying the art he saw on the covers of pulp magazines. Eventually, he found work as an inbetweener for the Fleischer Brothers Animation Studio, filling in extra drawings needed between key frames.\n\nAfter the animation studio closed and moved to Florida, Kirby became a freelance cartoonist taking on odd jobs to pay the bills. His two hit comics for Timely, _Captain America_ and _Boy Commandos_ , inspired many future comics artists to start their own careers.\n\nWORDS TO KNOW\n\nsplash page: large pages of comic action with no panels.\n\nimmigrant: a person who leaves his or her own country to live in another country.\n\ninbetweener: an animator who assists the head animator by drawing movements in between key frames.\n\nkey frame: a drawing in animation that captures the major points of movement. For example, if a character is waving her arm from left to right, the key frames would show the arm on the left, middle, and right.\n\nCARTOONING WOMEN\n\nFrom the first decade of comics, women have been making comics for newspapers and comic books. In the 1920s, several women saw their comics published widely, and a few even got rich.\n\nEdwina Dumm created a comic strip called Cap Stubbs and Tippie about the adventures of a young boy, his grandma, and Tippie the dog. Cap Stubbs and Tippie was one of the first comics made by a woman to be popular throughout America. It inspired a famous Halloween song and was eventually made into a weekly feature in Time Magazine, where the better printers allowed Dumm to create beautifully illustrated comics.\n\nYou may have heard of Kewpie dolls from your grandparents. The Kewpie doll came from a comic created in 1905 by Rose O'Neill. She told funny stories of little **cherubs** she called Kewpies. The comics were often colorful and were popular among adults and children.\n\nOther women worked in comics and comic books throughout the Golden Age of comics, although many aren't widely remembered today. Their work, however, did inspire future generations of women who would make memorable comics of their own.\n\nWORDS TO KNOW\n\ncherub: a type of angel, usually shown as a small child.\n\n**HEROES GO TO WAR**\n\nWorld War II officially began for the United States after the surprise attack on Pearl Harbor by Japan on December 7, 1941. Thousands of Americans signed up to join the armed forces, including many of America's artists and cartoonists.\n\nWORDS TO KNOW\n\nPearl Harbor: a U.S. naval base in Hawaii that was attacked by Japan in 1941.\n\nJack Kirby was sent to Europe, where his quick drawing skills were noticed by his commanders. They sent Kirby out as a scout to draw enemy encampments and artillery positions.\n\nTHE DANGEROUS JOB LATER INSPIRED SOME OF KIRBY'S MOST MEMORABLE COMICS.\n\n**WALT DISNEY AT WAR**\n\nHave you ever been to Disney World? Walt Disney also contributed to the war effort. His studios were commissioned by the U.S. Armed Forces to create short animated films for two purposes\u2014to encourage Americans to buy war bonds and to educate soldiers about a variety of subjects. Walt Disney also led a team of artists that designed decals for the noses of American bombers and cargo planes.\n\n  **Watch one of Disney's war** propaganda **films.** How do you think people in 1943 responded to these movies?\n\nWORDS TO KNOW\n\npropaganda: ideas or statements that are sometimes exaggerated or even false. They are spread to help a cause, political leader, or government.\n\nComic book characters also went to battle with America's enemies during this time. Batman and Superman helped to foil Nazi spies. Lesser-known superheroes such as Plastic Man, who could bend and fold into any shape, thwarted Japanese spy rings. Of course, Captain America continued his battles with the Nazi forces. The superheroes were widely read by American troops.\n\nWhile comic book heroes and artists were going to war, they had an even bigger fight waiting for them when they returned.\n\nTHE GOLDEN AGE OF COMICS WAS COMING TO AN END.\n\nDRAW A SUPERHERO\n\n**Supplies:** pencil, paper, colored pencils, digital camera, a friend (optional)\n\n**Some of this may be familiar since we use basic shapes and action lines to make our superhero. But heroes are larger than life, and the same thing is true with drawing them!**\n\n1 Start with an action line. If you want your hero to be flying, draw a big swoosh instead of the straight action line you used for standing still. If your superhero is going to be bigger than your regular characters, make your action line longer. Draw the head shape at the top of the action line.\n\n2 Skip a little down the action line to allow room for the neck, and then draw a large triangle for the upper body. If you are drawing a male hero, make it big. If it's a female hero, draw a slightly smaller triangle. Draw an oval where the hips will be, then add the socket circles for the shoulders and legs. Complete the skeleton arms and legs as you did in the last chapter.\n\n3 Superheroes are also bulkier than regular characters. Add muscles to arms, shoulders, and legs. Go ahead and make them look exaggerated. Muscles make your character super!\n\n4 To draw the face, follow the same steps as before, but keep in mind that most superhero faces look more realistic than other comic characters. Look at your favorite superheroes' faces for ideas and try to make your superhero strong and expressive. Giving your superhero a mask or hood will make drawing his or her face easier.\n\n5 Wait, what about clothes? Don't worry, you will outfit your superhero in the next project. But you can start brainstorming now!\n\nTRY THIS! Most comic book artists use friends as models for their superheroes. Find a friend and use a digital camera to take some pictures of different poses. How could you take a picture of a flying pose without making your friend jump?\n\nTHE NAZI BAN ON COMICS\n\nThe Nazis thought that American comics were one of the most powerful propaganda tools the Allied forces had. Because of this, the Nazis banned almost all forms of comics and would not allow anyone to have anything that depicted American characters such as Superman or Mickey Mouse. The Nazi ban on American comics changed the direction of European comics in the twentieth century. Almost no European superhero comics exist today.\n\nWORDS TO KNOW\n\nAllied forces: the armies of the countries that fought together against Germany in World War I and World War II.\n\nDESIGN A COSTUME AND BACKSTORY\n\n**Supplies:** pencil, paper, notebook, superhero drawings\n\n**Think of Batman, Spiderman, Iron Man, The Hulk, Wolverine, and Wonder Woman. What do all these characters have in common? That's right, strong secret** identities **linked to their names and great costumes. Every great superhero needs a costume and** backstory.\n\nWORDS TO KNOW\n\nidentity: the characteristics that make a person an individual.\n\nbackstory: a story about the events leading up to the main story.\n\n1Brainstorm ideas for a name for your superhero. Make a chart with two columns like the one below. In the first column, list 10 nouns. These are people, places, or things. In the second column, list words such as man, boy, girl, or woman. Now try matching some of your nouns in column one with the words in column two. Do any sound cool or interesting? If you don't like any of the names you come up with at first, try another list of 10 until you find something you like.\n\n2Who is your superhero when he or she isn't busy saving the world? How did he or she get his or her powers? Conduct an interview to find out! Write a list of 10 different interview questions. **Here are some examples:**\n\n*\"What do you do for work?\"\n\n*\"What do you like to eat for dinner?\"\n\n*\"Do you have children?\"\n\n*\"How old were you when you got your powers?\"\n\n3Pretend to ask your superhero questions and write down what you think the answers would be. This will give you a more complete picture of who your superhero really is!\n\n4Superheroes need costumes! Make a chart with two columns. In the first column, list different situations your superhero might encounter, such as rescuing people at sea or stopping nuclear bombs. In the second column, name the costume piece or gadget your superhero will need, such as a cape, motorized swimming shoes, or an automatic heat shield. Choose which ones you want to give your character.\n\nTRY THIS! Design a sidekick. How will the sidekick look compared to the main hero? Will the costume look different or the same? Why do you think some sidekick characters, such as Batman's Robin, look very different and more colorful?\n\nDESIGN A VILLAIN\n\n**Supplies:** pencils, paper, markers with dark colors\n\n**Every hero needs a villain! Originally, a villain was someone who lived outside of a village. Now, a villain is anyone who wants to upset the normal way things are done. Some villains want revenge. Others want to rule the world. As you design your own villain, think about your hero. Who would be the opposite of your hero? Why would your hero need to protect the world from the villain?**\n\n1Come up with the shape of the villain. Villains are often the opposite of your hero. If you have a shorter hero, you may want the villain to be very tall. If your hero is really strong, the villain may need to look weak. Once you have a basic shape, draw the body as you would for a normal comic character.\n\n2Design the villain's face. Think of villains you have seen in animated movies. What sort of facial features do they have in common? One of the most expressive parts of a villain's face is the eyes. Look at some of the following examples to see how you can make a face villainous.\n\n3Just like your hero, villains need a good backstory. What made them villains? Do they have diabolic plans? Were they childhood friends with your hero? Were they once heroes themselves?\n\nWORDS TO KNOW\n\ndiabolic: extremely evil.\n\n4Most villains have dark costumes. If you have seen any Disney films, you may have noticed that the villains usually have black and purple in their costumes. Play around with a few designs for your villain's costume.\n\nTRY THIS! Villains usually don't work alone. They rely on minions or thugs to do their dirty work. Try to design a few minions or thugs for your villain. Design their costumes based on what you drew for the villain.\n\nWILL EISNER\n\nWill Eisner put comics to work for the armed forces. He was drafted into the service and began to draw for many army publications, most famously PS Magazine. He developed a popular character called Joe Dope who showed military men the wrong way to do something. It offered a humorous note to what was often boring content. The work he did during World War II inspired the Army to hire Eisner to create a comic book called Preventative Maintenance to be a part of PS Magazine. The comic showed soldiers how to do routine work on their machines. The complicated procedures were made easier to understand through the use of comics. Eisner worked on Preventative Maintenance for 24 years before returning to the world of popular comics.\n\nWORDS TO KNOW\n\ndrafted: required to join the military.\n\nMAKE YOUR COMICS MAKE SENSE\n\n**Supplies:** pencils, paper, ruler, pens\n\n**By combining words and pictures, you can make a superhero comic that packs a punch!**\n\n1On scratch paper, draw thumbnails for your superhero comic. Make sure there are plenty of chances for action. Remember, your comic should have a beginning, middle, and end. Use as many panels as you need to tell your story.\n\n2While you sketch your thumbnail, include some words along with pictures. **Here are a few rules about including words in your comics.**\n\n*Always have most of your words appear at the top of your panel\n\n*Always write out your words BEFORE you draw your word balloons, thought clouds, and narrative boxes to make sure you have enough space.\n\n*Avoid covering your characters with word balloons and thought clouds.\n\n3Try to include a different type of sound or way of communicating in your comic, such as telepathy, radio, a different language, or yelling.\n\n4Explore unique panel designs in your thumbnail. How can you use different panel shapes to show movement or emphasize the action?\n\n5Try adding some sound effects to your thumbnails. How will you make them look like different noises on the page?\n\n6Copy your thumbnails into your notebook. In the United States, we read left to right, top to bottom. Because of how we read, we need to lay out comic pages in same pattern. Check your comic to make sure your panel designs and sound effects make sense.\n\nWORDS TO KNOW\n\ngraphic novel: a comic as long as a book, that tells one story.\n\nTRY THIS! Check out some comic books and graphic novels at your local library. As you read through them, pay attention to panel designs and how the cartoonist uses word balloons. Can you find any other rules? Do any of the cartoonists break the rules? Did it make sense to change or break the rules?\n\nMAKE A DYNAMIC COVER\n\n**Supplies:** pencils, paper, hero design, villain design, colored pencils or markers, rulers, digital camera, a few friends\n\n**During the Golden Age, covers often made the difference between good sales and bad sales. Covers showed lots of action and the title was splashed across the top of the page. Words to help sell the comic were also placed in specific places.**\n\n1 Use a piece of scratch paper to do a few thumbnail drawings of your cover design. Think about where you want the title and how you want to show some action. Use stick figures to show the action between your hero and villain. Have your friends pose like the best design you come up with for the cover. Have one friend be the hero and the other be the villain. Take photos from a variety of angles until you find one you really like.\n\n2 On a fresh sheet of paper, use a ruler to mark where the title and any other words will go. Use your photos as references to sketch out the action.\n\n3 Once you are happy with the design, ink over everything with the pen. Then erase all the pencil marks. Now it's time to add color. You can photocopy your cover design if you want to be able to experiment with color before creating a final color version.\n\nCHAPTER 4\n\nINTO THE SILVER AGE OF COMICS\n\nAfter World War II ended, America enjoyed the last of the Golden Age of comics before moving into what is known as the Silver Age of comics. Romance comics became popular, daily newspaper comic strips provided readers with the entertainment they wanted, and the photocopier was born during this era.\n\nROMANCE IN THE AIR\n\nAmericans grew tired of reading about heroes and villains after the war. Comic book publishers saw their sales falling and knew they needed to publish different work. Most created westerns or adventures, but a few publishers took a risk with romance.\n\nWORDS TO KNOW\n\nSilver Age: the era of comics between the introduction of the **Comics Code** in 1956 and the late 1970s.\n\nComics Code: a set of guidelines created by comics publishers to help regulate the content of comics.\n\nThe first successful romance comic was _Archie_. Published by MLJ Magazines, the series was roughly based on the popular teenage movies of Mickey Rooney, and told the tale of the love life of Archie Andrews. Bob Montana was the cartoonist who brought _Archie_ to life. The comic is still popular today and can be found in the magazine rack of almost any supermarket in the country!\n\nARCHIE EVENTUALLY BECAME SO POPULAR THAT MLJ MAGAZINES RENAMED ITS COMIC COMPANY\n\nInspired by the success of _Archie_ , other publishers quickly adopted the idea of romance comics. A very popular one called _Young Romance_ came from an unlikely team\u2014Joe Simon and Jack Kirby, the creators of _Captain America_.\n\nWORDS TO KNOW\n\nromance comic: a comic about characters who are trying to find love.\n\nRomance comics turned around the classic ideas of romance. Many of the stories put the female main character in charge of finding a boyfriend. She'd discover that the ugly duckling was the best boyfriend of all. The comic books challenged some of the long-held ideas Americans had about finding love and romance, that looks weren't everything and money didn't equal happiness.\n\nDAILY COMICS\n\nMeanwhile, the daily comic strips in newspapers were the envy of the comics world. The founders of the modern newspaper comic industry, Hearst and Pulitzer, developed a system called syndication that allowed comic strips to be sold to a variety of newspapers. King Features Syndicate, founded by Hearst, is still one of the leading comic strip syndicates in America. The syndicates were the business people between the cartoonists and the newspapers that published them.\n\nWORDS TO KNOW\n\nsyndication: when the rights to print or broadcast a creative work are sold.\n\nCARTOONISTS COULD FOCUS ON THEIR ART AND LET THE SYNDICATE HANDLE ALL THE BUSINESS.\n\nDaily cartoonists had to work within the specific structure and size of each comic, unlike early cartoon masters who had whole pages. But in art, restrictions can lead to great creativity.\n\nAl Capp created _Li'l Abner_ , one of the most popular daily comic characters of the 1940s and '50s. The character of _Li'l Abner_ was a strong, slow-witted, backwoods man who was accompanied by lots of different characters in the small Southern town of Dogpatch. One of those characters was the Shmoo.\n\nThe Shmoo were small, white, bowling pin\u2013shaped creatures that quickly reproduced and were said to be the \"tastiest critter\" in the world. They were cute, lovable, laid eggs, and were easy to catch and cook. America loved the idea of the perfect animal and bought lots of Shmoo merchandise.\n\nIN 1947, SHMOO PRODUCTS SOLD $25 MILLION WORTH OF STUFF\u2014THAT'S EQUAL TO $257 MILLION TODAY!\n\nChester Gould created _Dick Tracy_. The main character was a detective who wore a bright yellow trench coat, used ultramodern gadgets, and had plenty of helpers in his fight against organized crime.\n\n_Dick Tracy_ was drawn in a weird, stylized form with a range of bad guys whose names were associated with the way they looked. PruneFace was drawn as a face with so many wrinkles you couldn't make out his expression. Mumbles had an extremely small mouth. The strips always ended with a cliffhanger.\n\nWORDS TO KNOW\n\nstylize: to draw comics in a specific way.\n\ncliffhanger: an exciting moment that makes you wonder what happens next.\n\nOhio cartoonist Milton \"Milt\" Caniff created two of the most popular adventure strips of the day: _Terry and the Pirates_ and _Steve Canyon_. Caniff is famous for his realistic drawing style, and he loved to show as much action as he could get away with.\n\n_Terry and the Pirates_ used cliffhangers to tease its audience into reading the next day's strip. Sometimes the stories in _Terry and the Pirates_ and _Steve Canyon_ were serialized over entire months! Can you think of a modern television series that uses cliffhangers to keep people watching week after week?\n\n_Blondie_ , created in 1933 by Murat \"Chic\" Young, was a daily comic about home, work, and relationships. The comic usually ended on a punch line. Blondie and her husband, Dagwood Bumstead, are still fixtures in the comics, though they're written and drawn by different cartoonists. _Blondie_ was the most widely syndicated comic of its time, and it inspired many other daily comics, such as _Hi and Lois_ , _For Better or For Worse_ , and _The Family Circle_.\n\nWORDS TO KNOW\n\nserialize: to create a series of stories.\n\npunch line: the sentence, statement, or phrase that makes the point, as in a joke.\n\n  **Compare a** _**Blondie**_ **comic from** **1933 with** _**Blondie**_ **comics from** **today.** Do they look different? Is the dialogue different? What are some of the reasons comics change over the decades? Cover up neighboring QR codes to make sure you're scanning the right one.\n\nWHERE'S THE MONEY?\n\nOften, the creators of newspaper comic strips were financially better off than their comic book counterparts. Harold Gray, creator of Little Orphan Annie, retired as a millionaire. The creators of Superman, on the other hand, accidently gave most of their rights to their publisher, DC Comics. Jerry Siegel and Joe Shuster were eventually replaced as the creative team for Superman and both lived the rest of their lives in near poverty.\n\nMilitary humor was popular after the war. Mort Walker introduced the world to _Beetle Bailey_ in 1950. The comic strip followed the Army life of a private at a military base and his run-ins with the overweight and often quick-tempered Sarge. Through the lens of humor, _Beetle Bailey_ commented on social issues of the time, such as the war in Vietnam and the draft. _Beetle Bailey_ is still enjoyed by daily newspaper readers today.\n\nCHARLES SCHULZ AND PEANUTS\n\nHave you ever read the _Peanuts_ comic strip or seen one of the _Peanuts_ television shows? You have Charles Schulz to thank. Schulz specialized in drawing children and came up with an idea for a daily comic strip called _Li'l Folks_. The syndicates he sent his _Li'l Folks_ comics to didn't know what to think of them.\n\n  **More** _**Peanuts**_ **!**\n\nCHARLES SCHULZ'S COMIC STRIP WAS THE FIRST TO FEATURE ALL CHILDREN!\n\n_Li'l Folks_ was populated entirely by children who often expressed very grown-up emotions. The star of the comic, Charlie Brown, constantly battled disappointment and depression. Finally, a syndicate agreed to give _Li'l Folks_ a chance in 1950. They renamed it _Peanuts_ , and it became one of the most famous comics in the world.\n\nTHE SIDEKICK CHARACTER IN PEANUTS, SNOOPY THE DOG, IS AS RECOGNIZABLE AS MICKEY MOUSE.\n\nSCARY COMICS\n\nAs comic books in the late 1940s and early 1950s turned away from superheroes and toward romance, some comic publishers decided to tell darker stories. This may have seemed like a good idea, but those dark tales helped cause the end of the Golden Age. The covers often showed violent scenes to entice newsstand readers. Two comics were figureheads of this new trend in American comics\u2014 _Crime Does Not Pay_ and EC Comics' _Tales from the Crypt_.\n\n  As an instructor in an art school, Charles Schulz was famous for holding contests to see who could draw the longest straight line without a ruler. **Watch** **him draw Charlie Brown!**\n\nTHE AMAZING XEROX MACHINE\n\nBefore the 1950s, making a copy of something took a lot of time and effort. If a business wanted to copy a document, it had to use carbon paper to make a carbon copy. Enter Chester Carlson, New York attorney and the inventor of the photocopier.\n\nTired of the long copying process, Carlson began to experiment in his kitchen with light-sensitive particles to print copies on paper. In 1938, he applied for a patent, but it wasn't until 1947 that a small company in New York decided to take a chance on his invention. The company called the process xerography and became the Xerox Corporation.\n\nThe photocopier was very important to the future of comics because it made the creation of independent comics cheaper and easier. It also played a big role in the birth of the modern graphic novel, but more on that in a later chapter.\n\nThe comic book _Crime Does Not Pay_ was inspired by old police gazettes of the nineteenth century, which were often highly illustrated. The comic took the most shocking true news stories and illustrated them in comic form. It was one of the first to feature a master of ceremonies, or emcee. This character, called Mr. Crime, helped narrate the stories. The emcee character was used in most crime and horror comics of the era.\n\nWORDS TO KNOW\n\ncarbon copy: an exact copy of a document made using carbon paper between two or more pieces of paper.\n\npatent: a document from the government that gives an inventor the exclusive right to make, use, or sell his or her invention.\n\nxerography: a way of making a copy using light-sensitive chemicals and paper.\n\nindependent comic: a comic book published without the help of a large company.\n\nemcee: a master of ceremonies who helps to lead a story or event.\n\nCompeting with _Crime Does Not Pay_ , EC Comics became one of the most well-known publishers of crime, war, and horror comics in America. EC published _Tales from the Crypt_ , which featured a rotating cast of emcees. The comic told graphic horror stories with twist endings and morals. The covers were gruesome and brightly colored, like the pulp magazines of the early part of the twentieth century.\n\nAmerican adults loved their crime and horror comics. Unfortunately, newsstands mixed all of the comic books together. A kid who wanted to read the latest _Scrooge McDuck_ comic might pick up an issue of _Crime Does Not Pay_ out of curiosity.\n\nWORDS TO KNOW\n\nmoral: a valuable lesson to help people know how to behave.\n\nPARENTS WERE CONCERNED THAT THESE COMICS WERE BAD FOR KIDS, AND BEGAN TO PRESSURE PUBLISHERS AND THE GOVERNMENT FOR SOME RULES AND REGULATIONS.\n\nTHE END OF THE GOLDEN AGE\n\nHave you ever heard people say that videogames containing violence can make kids violent? People used to say the same thing about comics!\n\nConcern over comics kept growing in the early 1950s. People even burned comic books in protest. In 1954, a psychologist named Dr. Fredric Wertham published a book called _Seduction of the Innocent_ , which argued that children who can't tell fiction from reality get confused by comic books. He claimed that comic books lead to violent behavior.\n\nDue to Wertham's book and public pressure, the U.S. Senate decided to hold hearings on the negative effects of comics. The Senate called Bill Gaines, the publisher for EC Comics, to the stand.\n\nBefore he entered the Senate building, Gaines took some medication that his doctor had prescribed to help his nerves. That was a mistake! The drugs made him confused. By the end of his testimony, he had agreed that the comics he created were completely inappropriate.\n\nWORDS TO KNOW\n\nhearing: a special session of Congress or the Senate held to \"hear\" from witnesses and experts on a given issue.\n\nTHE AMERICAN PUBLIC WAS FURIOUS.\n\nComic publishers feared the government would censor their work. Gaines gathered the major comic publishers together to figure out a plan to get Americans to trust comic books again. They formed the Comics Magazine Association of America (CMAA) and elected the publisher of Archie Comics as president. They created a set of rules called the Comics Code. All comics that followed these rules were given a special CMAA badge to print on the covers of their comics.\n\nThe Comics Code was the end of publishers such as EC Comics and _Crime_ _Does Not Pay_. Superheroes couldn't be shown fighting bad guys with guns, so action scenes and weapons became silly. Comics historians call this era the Silver Age.\n\nAs comics became more and more goofy in the 1960s and '70s, adults moved away from reading comics. Comic creators needed superheroes for a younger audience. A young man named Steve Ditko joined the Marvel Comics team with the idea for a teenager with the powers of a spider. The result was _The Amazing Spider-Man_.\n\nWORDS TO KNOW\n\ncensor: to examine books, movies, letters, etc., in order to remove things that are considered to be offensive or harmful to society.\n\nTEAM SUPER\n\nDuring the Senate hearings of 1954, many publishers had to sell their characters to keep from going out of business. DC Comics, publisher of Batman and Superman, bought several new heroes, and then faced the problem of mixing them in with its old heroes. In the early 1960s, DC Comics launched the solution\u2014the superhero team.\n\nThe Justice League of America (JLA) joined up the heroes of DC Comics to overcome galactic bad guys. The JLA had memorable members such as Batman, Superman, and Wonder Woman, but also included newly bought heroes, such as The Flash, Green Lantern, and the Martian Manhunter.\n\nUntil the 1960s, Marvel Comics was known for copying the popular comics of the day, from romance to westerns to giant monster comics. When Jack Kirby returned from the war and from making romance comics, he and the editor-in-chief, Stan \"Lee\" Lieber, came up with the idea of a team of scientists who were changed into superheroes by cosmic rays. The team was called the Fantastic Four.\n\nFans immediately flocked to Marvel Comics. The good news was Stan Lee and Jack Kirby were just getting started. In the following years, the duo created almost all of what would become known as the Marvel Superheroes. Together they introduced the Hulk, Iron Man, the X-Men, Thor, and another super team that brought back a hero from the Golden Age\u2014Captain America.\n\nJack Kirby had the idea of bringing Captain America, who had been lost in the North Sea after World War II, back from the 1940s. Stan Lee loved the idea, and together they came up with the Avengers. The comic was a hit.\n\nTHROUGHOUT THE HISTORY OF MARVEL COMICS, ALMOST ALL OF MARVEL'S HEROES HAD TURNS AT BEING AN AVENGER.\n\n**JAM COMICS**\n\n**Supplies:** pencils, paper, pens, friends\n\n**Comics don't need to be a lonely art. Drawing a comic with friends can be a fun and funny time! Jam comics are a perfect way to have fun with a group.**\n\n1Sit in a circle. Give each person a sheet of paper and a pencil or pen. Have everyone write a title and draw a panel at the top of the page. The first panel should show a character or two and set up an action.\n\n2Pass the page to the right. Don't say anything about your drawing. All the information the person to your right needs will be in your title and first panel.\n\n3Read the comic you got from the person on your left. Draw a new panel and add your spin on the story.\n\n4Repeat steps two and three until everyone has drawn a panel on each comic. Pass the comics back to their original creators and read the stories everyone created. Are they different from what the first person intended? Are they funny?\n\nTRY THIS! Some cartoonists like to create pages and pages of jam comics around a theme or story. How could you make a jam comic book with your friends? What themes or ideas would you use?\n\nMAKE A DAILY COMIC STRIP\n\n**Supplies:** notebook with un-ruled paper (a new manga sketchbook would work well), scratch paper, pencil, pen, small ruler\n\n**A daily comic is a fun, quick way to practice making comics. They don't need to be funny and they could even be about your own life! A lot of cartoonists do a daily comic strip, even if they never publish them.**\n\n1Choose a theme. What will your daily strip be about? Do you want it to be funny or dramatic? About your daily life? Will it have a changing cast of characters? What about the title?\n\n2Choose a daily design. If you choose to make a one-panel gag comic, what shape is your single panel going to be? If you do a strip, how many panels will you use each day?\n\n3Creating a daily comic can be a lot of work. It might be helpful to brainstorm a bunch of comic ideas ahead and map them on a calendar.\n\n4Now it's time to put all your new cartooning skills to work. First, make thumbnail sketches, then pencil, ink, and color your comic strip. Create a new one every day!\n\n**USE THE \"MARVEL METHOD\"**\n\n**Supplies:** friends who each take one job (writer, penciler, inker, colorist), art supplies for each job\n\n**Inspired by Henry Ford's assembly line system of manufacturing, Marvel Comics decided to try a new way of making comics that would streamline the process. This became known as the** Marvel Method **. Get together with your friends and follow the Marvel Method to create a comic!**\n\n1 **WRITER** : First, the writer comes up with a plot for the story. For example, Super Mole saves his family from the clutches of Grumpy Gardener with his powers of super digging. The writer also gives information about other characters, how the story should be told, and anything else he or she thinks the penciler should know.\n\n2 **PENCILER** : The penciler uses the plot and character notes from the writer and makes thumbnails to visually tell the story. He or she then carefully pencils the action for each panel. The penciler doesn't write out any words, but leaves room for the writer to add them in the next step.\n\n3 **WRITER** : The writer adds narration blocks, word balloons, and thought clouds to match the action and panels.\n\nWORDS TO KNOW\n\nMarvel Method: a way of creating comics that relies on an assembly-line format.\n\n4 **INKER** : The inker traces over the important lines from the penciler and the writer. The inker might also make slight changes to the art, such as adding sound effects, defining shadows, and drawing dark lines to add complexity to each panel. Once finished, the inker erases any pencil lines and the comic goes to the colorist.\n\n5 **COLORIST** : The colorist chooses colors for the comic and makes the comic come alive.\n\nTRY THIS! How could you use the Marvel Method with only two people? Can you work on several comics at once while different people are doing different parts? For example, the writer might start plotting the next comic while the penciler is still working on the first.\n\nTHE MARVEL METHOD\n\nStan Lee and Jack Kirby came up with the Marvel Method for their first Fantastic Four comic. Here's how it worked. Stan came up with a plot for the comic and gave the idea to an artist, who drew the whole comic out, telling the story visually. Then Stan added narration and dialogue to the pages. The Marvel Method allowed for a freer visual style, and the artists were able to tell the stories how they envisioned them. Before the Marvel Method, writers completely scripted out the whole comic, including what the panels would look like.\n\nMAKE A SUPERHERO TEAM\n\n**Supplies:** pencils, paper, colored pencils\n\n**Marvel and DC realized the popularity of superhero teams in the 1950s. What makes these teams so much fun?**\n\n1Brainstorm some ideas on a piece of paper. What would a superhero team's job be? Why are they a team? What sort of name would fit the team? How are they related to each other? What brought them together?\n\n2Draw your team, give each member of the team a name, and show his or her power. What makes a good team? If they all had the same type of powers, they wouldn't be very interesting. What kind of abilities do you want on your team? What powers should they have?\n\n3Try designing costumes that are unique to each character, but still make them look like they belong to a team.\n\n4Lastly, design a secret home base for your team. For example, the Justice League of America meets in an orbiting space station. What and where will your superhero team call home base?\n\n  The 1960s saw the start of a popular television show based on the comic book hero Batman. It came out shortly after the invention of color television and featured extremely bright colors, crazy costumes, and silly language. The star of the show, Adam West, played Batman and his alter ego, Bruce Wayne, in a deliberately silly way. **Watch the** **introduction credits to the 1960s** _**Batman**_ **television show.** Does the theme song sound familiar? How are the graphics different from television shows today?\n\nCHAPTER 5\n\nLIFE AFTER THE COMICS CODE\n\nNot every comics artist wanted to follow the rules of the Comics Code from the 1950s. Many wanted to explore serious subjects, and they wanted to take plenty of space to do it. Some comics artists began to produce and distribute their own comic books. With help from the Xerox machine, they could cheaply print lots of copies of their independent comics. The longer independent comics were called graphic novels.\n\nINDEPENDENCE!\n\nImagine having someone tell you what your comics have to be about every time you sit down to draw. Does that like fun? Many comic book artists in the 1960s through the 1980s wanted to be able to express their creativity without worrying about the Comics Code. A few brave artists decided to do just that.\n\nLarry Marder began publishing his comic, _Tales of the Beanworld_ , in 1980. He was first inspired by the story his mother tells of how his head looked like a bean when he was born. In college he started drawing little bean-shaped characters that eventually became the cast of his comic.\n\nAs his comic evolved, Marder explored themes found within mythology and Native American cultures. His characters live in a place called Beanworld. What do you think Marder means when he says, \"Beanworld isn't a place, it's a process?\"\n\nIn 1984, two friends from Massachusetts published the first edition of a comic that would become an international, multi-million-dollar phenomenon. _Teenage Mutant Ninja Turtles_ ( _TMNT_ ) was created by Peter Laird and Kevin Eastman. Originally meant to be a joke about an absurd jumble of words, the comic was instantly popular among comic book fans. The two started their own publishing company, Mirage Studios, to publish the book.\n\nWORDS TO KNOW\n\nmythology: a set of stories or beliefs about a particular religion or culture.\n\n_TMNT_ made hundreds of millions of dollars for their creators through action figures, four feature-length movies, several video games, and two follow-up cartoon shows. Laird and Eastman remained true to their independent comics roots and founded the Xeric Grant to help support other independent cartoonists.\n\nOne reason independent comics were so successful was the direct mail system. Up until the late 1970s, comics could only be purchased at newsstands, drug stores, and supermarkets. New companies began to form to distribute comics to readers through the mail. Readers just needed to get a catalog from a distributor to order the comics they wanted.\n\nTHANKS TO THE XEROX COPIER AND DIRECT MAIL, SELF-PUBLISHED MAGAZINES CALLED ZINES (PRONOUNCED \"ZEENS\") COULD BE CREATED QUICKLY AND CHEAPLY, THEN SENT TO READERS.\n\nWORDS TO KNOW\n\ndirect mail: a system used to sell things directly to subscribers through the U.S. mail.\n\nzine: a self-published magazine, usually produced using photocopiers.\n\nEarly writers of zines, who called themselves zinesters, created zines about everything from politics to rock bands. Zines were often quickly made and easily sold or traded with other zine creators. It was a way to bypass the often difficultto-enter publishing world.\n\nBEFORE THE INTERNET, ZINES WERE THE CHEAPEST AND MOST EFFECTIVE WAY OF GETTING YOUR STORY READ BY AN AUDIENCE.\n\nMini comics developed out of the independent zines of the 1980s. These self-published comics are very small, which make them easy to produce and distribute. A comics artist will often begin his or her career creating and selling mini comics until a publisher discovers him or her.\n\nLOOOOONG COMICS\n\nMany independent comic book creators collected their comics into thick paperback books to be sold in bookstores. Today, we know them as graphic novels and they are still very popular.\n\nOne of the first graphic novels was written in the 1970s. Canadian cartoonist Dave Sim drew a parody of other comics about magic and barbarians. He called it _Cerebus_ after the title character, a barbarian warrior aardvark. Originally meant to be humorous, the comic grew into something bigger. Sim started to tell long stories in his comic about everything from religion to politics.\n\nWORDS TO KNOW\n\nzinester: a person who makes zines.\n\nmini comic: a small, self-published comic.\n\nparody: an imitation of something for comical effect.\n\nSim wrote the comic and drew the characters, but hired another artist to draw all of the background art. When a storyline was completed, Sim bundled the issues together in a thick, phonebookshaped package to sell. These collections are some of the first examples of the modern graphic novel. Together, they created one of the largest collected independent comics of all time\u2014more than 6,000 pages long!\n\nAnother successful, long independent comic book from the late 1970s came from the husband-and-wife team of Richard and Wendy Pini. Their comic book was called _Elfquest_. The first issue appeared in a fantasy comic magazine called _Fantasy Quarterly_ in 1977.\n\nWORDS TO KNOW\n\nfantasy: a **genre** of comics and literature based on myth and legend.\n\ngenre: a type of creative work, such as mystery, romance, or fantasy.\n\nWHATCHA' CALL IT?\n\nThe term graphic novel became widely used when Will Eisner finished his work for the U.S. Army and returned to making comic books. He thought of a new way to present his comics to the world\u2014not through a printed monthly comic book, but through a published, hardcover book sold in bookstores. The books were popular, and people started to take comics seriously again as an art form.\n\nSome graphic novel artists found inspiration in very serious stories. Would you want to read a comic book about the Holocaust?\n\nArt Spiegelman's father was a survivor of Auschwitz, one of the most horrible Nazi concentration camps. Spiegelman decided to tell his story using animals as the characters. In his graphic novel, _Maus_ , Jews were mice and Nazis were cats. He hoped that this technique would make the hard topic easier for readers to look at. He could show disturbing scenes without having to draw the suffering faces of real people.\n\nWORDS TO KNOW\n\nHolocaust: a time before and during World War II when the German Nazis tried to kill the entire Jewish race, as well as several other groups.\n\nconcentration camp: a prison where people were sent during the Holocaust to be killed or made to do hard work.\n\nfolk tale: a story told by a specific group of people, often involving magic and a moral.\n\nPulitzer Prize: a group of prizes awarded annually for work done in journalism, fiction, and non-fiction, as well as for photojournalism.\n\nMAUS WAS COLLECTED INTO A TWO-VOLUME GRAPHIC NOVEL IN 1991 AND BECAME THE FIRST COMIC TO WIN THE PULITZER PRIZE FOR LITERATURE.\n\nINDIE GRAPHIC NOVEL PUBLISHERS\n\nMany of the new graphic novels were being published by small independent companies. The three major publishers were Top Shelf, Fantagraphics, and the Canadian publisher, Drawn and Quarterly.\n\nTHEY WERE INDEPENDENT BECAUSE THEY WERE NOT OWNED OR OPERATED BY ANY OF THE MAIN PUBLISHING COMPANIES OF THE DAY.\n\nThe work of these publishers helped change the perception of comics in America. Their graphic novels were more serious. As book critics and readers began to take note of the more mature work, major book publishers were inspired to start their own graphic novel imprints.\n\nThe first major publisher to enter the field was Pantheon Press, the publisher for Art Spiegelman. Pantheon could get graphic novels into major bookstores. In the mid-1990s, the comic book sections of bookstores were made up of mostly superhero comics with a few graphic novels sprinkled in. By the early 2000s, the comics section was renamed the graphic novel section and began to take up whole rows of shelves.\n\nWORDS TO KNOW\n\nimprint: a part of a publishing company that publishes a certain type of book.\n\nIMAGE COMICS\n\nThe late 1980s and early 1990s saw some of Marvel Comics's biggest sales, thanks to the artistic talent of its comic creators. These artists worked hard to produce the best superhero comics possible. However, most of the comics created by DC and Marvel were owned by the companies, not by the writers and artists. This meant the money from sales of comics, T-shirts, or movies went to the company. Some of the artists wanted more creative say and control over their characters.\n\nIn 1992, several artists and writers, mostly from Marvel, decided to quit working for the big publishers and start their own company. The result was Image Comics. It began with a simple idea: that the creators of comics should retain complete control over their characters and comic book stories.\n\nThe first comics to be published by Image were very successful. As the popularity of Image Comics grew, more comics artists joined Image to publish their own comics. Today, it remains one of the most successful independent comic book publishers and often is considered a mainstream comics publisher.\n\nIMAGE REFUSED TO JOIN THE COMICS CODE AUTHORITY, WHICH MEANS THAT ITS COMICS CAN'T BE CENSORED BY ANYONE OUTSIDE OF ITS OWN OFFICES.\n\nTHE COMIC BOOKSTORE\n\nAs comic books and graphic novels started to become popular again, they needed a place where fans could find them. Comic bookstores popped up all across America during the 1980s and mid-1990s. People who loved comics could browse shelves stocked with books, T-shirts, action figures, and posters. They could even write a list of their favorite comic titles, and when the bookstore got a shipment, the employees would pull those titles aside and save them. This was called a pull list. Most big towns still have one or two comic stores and you can still make a \"pull\" of your favorite comics. Check them out sometime\u2014you might even make some new friends who love the same comics as you!\n\nWORDS TO KNOW\n\npull list: a list of comics that your local comic bookstore will order and set aside for you.\n\nNEWSPAPER COMICS IN THE MODERN AGE\n\nIndependent comics creators and publishers weren't the only ones enjoying the modern age of comics. Newspaper comics also thrived between the late 1970s and 1990s.\n\nThese newspaper comics had changed through the decades. When Charles Schulz introduced _Peanuts_ in 1950, his use of a simple background was new and innovative. In the '70s, '80s, and '90s, most comic strips relied more on memorable characters than complicated settings to stay popular.\n\nIf someone was to ask you the name of the fat orange cat who loves lasagna, you'd probably shout, \"Garfield!\" In the 1970s, Jim Davis created a strip that mostly took place on the floor or countertop, with very few extra details. The humans in the strip interacted with Garfield, but they couldn't hear what Garfield was thinking. However, the audience knew Garfield's thoughts, through the use of a thought cloud.\n\nGARFIELD WAS A commercial SUCCESS! HE IS STILL FOUND ON EVERYTHING FROM COFFEE MUGS TO T-SHIRTS AND IS AN ANIMATED TV SERIES.\n\nNot every cartoonist was interested in commercial success. In 1985, Bill Watterson introduced the world to a boy and a stuffed tiger in what many consider to be one of the top 10 comic strips of all time. People ages 6 to 96 love _Calvin and Hobbes._ No other comic strip has been so popular with such a wide range of ages.\n\nWatterson says he was inspired partly by his own childhood and partly by the comic strip greats who came before him. These include Winsor McCay, George Herriman, Walt Kelly, and Charles Schulz.\n\nWORDS TO KNOW\n\ncommercial: operating as a business to earn money.\n\n  **More** _**Garfield**_ **!**\n\nBill Watterson gave a speech at Kenyon College in 1990 that was made into a cartoon by comics artist Gavin Aung Than. Can you find similarities between Than's comic and Watterson's style of drawing?\n\nPlans were made for _Calvin and Hobbes_ to appear on everything from socks to hats and there was even talk of a television series, but Bill Watterson said no. He wanted his strip to be enjoyed as a comic strip and nothing more. He successfully fought against marketing his strip on other products, something comics had been doing since _The Yellow Kid_ almost 100 years before. Even after he retired _Calvin and Hobbes_ in 1995, he still refused to allow his characters to be used in other places.\n\nMANGA\n\nLet's take a moment and travel to Japan! Remember Hokusai, who invented manga? Cartooning was popular in Japan through the early twentieth century. After World War II, the Japanese comics from before the war were associated with the empire, which had fallen. No one wanted to read these comics anymore. The manga that took its place after the war became the most popular form of comics in the world.\n\nManga publishers sprang up all across the country. They printed their comics in quarterly, monthly, and even weekly editions. Manga books were the size of telephone books.\n\nMANGA STORIES WERE ABOUT EVERYTHING, FROM LIFE AS AN OFFICE WORKER TO BEING A SPACE EXPLORER.\n\n  **More** _**Calvin and Hobbes**_ **!**\n\nThe biggest name in manga grew to fame in the late '40s and early '50s. Osamu Tezuka was inspired by the comics and cartoons he saw from America, such as Walt Disney's work. Tezuka chose to focus his early comics on a general audience and grew a large following of fans. His first anime, which is a form of manga-style animation, was called _Astro Boy_. It was one of the first Japanese comics to be seen by Americans. During the course of his career, Tezuka wrote and drew more than 150,000 pages of comics, as well as directed several animated features and series of his work.\n\nOSAMU TEZUKA IS OFTEN CALLED THE WALT DISNEY OF JAPAN.\n\nWORDS TO KNOW\n\nanime: Japanese animation.\n\nMANGA AND ANIME\n\nMany Americans first experienced Japan's comic style in the form of anime. Often people confuse anime and manga. Many of the manga that Tezuka created for print were then made into anime. Other times, the anime would inspire a printed run of manga. A good rule of thumb when you see Japanese cartooning is if it moves, it's anime. If it doesn't, it's manga!\n\nWhen manga became popular in America, the language translation was the easy part. The structure of the comic was harder. The Japanese read from right to left, which means their front covers are our back covers. In order for Americans to read them from left to right, the pages and panels had to be rearranged!\n\nAmerican readers at first didn't understand certain expressions and gestures of manga characters, but it didn't take long before people loved reading them. Today, American publishers no longer switch the layout of manga books. American readers are used to reading them from right to left, just like the Japanese!\n\nCOMIC-CON!\n\nComic book lovers often gather at conventions called Comic-Con. The Golden State Comic-Con was the first, in 1970 with 300 attendees. By 1980, it had been renamed the San Diego Comic-Con and more than 5,000 people went. Today, more than 100,000 people visit every year! Comic-Cons are held all over the world. Anyone can buy table space to sell his or her comics. Some of the best-known conventions are Stumptown, SPX, APE, and MoCCA. People even go dressed as their favorite characters!\n\nWORDS TO KNOW\n\nconvention: a gathering of people who are all interested in a certain idea, topic, or event.\n\n**COSTUME FOR A COMIC-CON**\n\n**Supplies:** images of the character you want to dress as, paper and pencil, supplies for gadgets (such as recycled boxes, plastic cans, and wrapping paper tubes), fabric, scissors, glue, paint, used clothing\n\n**Many convention goers dress as their favorite characters. It's called cosplay, and it's a way to make conventions more fun.**\n\n1Sketch out your costume. Use pictures of your character as a reference to sketch out what you need to make yourself look like the character.\n\n2Figure out what supplies you need to make each part of the costume. Can you get clothes at a used clothing store for your costume? These clothes can be cut up or painted and glued. Check the recycle bin for things such as cardboard and plastic.\n\n3Using your sketch as a guide, begin making the different parts of your costume. Ask an adult for help if you need it. Make sure to try on all the parts as you make them.\n\n4Be awesome. Wear your cool costume to a local comic convention. How many people knew who you were? How many other cosplayers did you meet? What kinds of tips did they share with you about making costumes?\n\nTRY THIS! Gather a group of friends together to design costumes for a super team. Often, conventions have contests for the best group cosplay. See if you can win!\n\nDRAW MANGA-STYLE CHARACTERS\n\n**Supplies:** pencils, pens, paper\n\n**Comics look different in each part of the world. European comics often have a more sketchy style, while American comics have cleaner lines. Let's look at how to draw Japanese manga characters before we do the harder work of learning how to draw their faces!**\n\n1Manga characters tend to have thin bodies, with long arms and legs. Start by drawing an action line for your character. Draw a larger-thannormal circle at the top for the head and then draw a thin rectangle for the body. Draw the sketchy skeleton for the arms and legs the same way you would for any other comic.\n\n2Add meat to the skeleton, but not too much! Often, manga characters have more slender arms and legs than American characters. This allows for the great expressive poses that manga is known for.\n\n3Add clothes and ink your manga character. Erase the pencil lines and add color!\n\nTRY THIS! Can you turn your hero and villain characters into manga?\n\nWORDS TO KNOW\n\ncomic relief: the inclusion of a funny character or scene in an otherwise serious work.\n\nTRY THIS, TOO! In manga, a certain type of character drawing is known as the chibi. A chibi is a shortened version of the main character and it often looks like a child. Chibi are used to show extreme emotions and for comic relief. Look at the step-by-step illustrations shown here. Can you turn your manga character into a chibi?\n\nDRAW MANGA-STYLE FACES\n\n**Supplies:** pencils, paper, pens\n\n**The design of the faces is what makes manga so recognizable. Let's crack the code of what makes a face manga.**\n\n1Unlike the oval face shapes of American comics, the general shape of a manga face is a large circle. Draw a circular face shape.\n\n2Just as in American comics, draw a vertical guideline down the middle of the face where the nose will go.\n\n3Unlike American comics, the horizontal line doesn't go through the middle of the face shape. Instead, it should go about two-thirds down from the top of the head. This is to give enough room for the large, iconic manga eyes.\n\n4Extend your vertical guideline through the bottom of the circle. Add the jawline and chin by connecting the bottom of the vertical guideline with the right and left ends of the horizontal guideline as shown in the example.\n\n5Manga noses are either very small or hugely expressive. Using the examples shown here, draw a nose between the horizontal guideline and the bottom of the circle.\n\n6Now draw the eyes following the step-by-step example shown below.\n\n7Draw the mouth, eyebrows, and ears. The mouth will be smaller than the eye guides. Look at the examples of manga mouths. The eyebrows, aren't much different from American eyebrows, and manga ears are either big or small, depending on your style.\n\n8Manga hair is super fun to draw since it can be really wild. Most manga women have long, flowing hair, which can be bunched up, braided, or swirling all around them. Manga men, and women, also often have angular, spiky hair. Open up any manga from your local library to see these different styles. Remember to make sure the hair is connected to the head.\n\nTRY THIS! Try to draw yourself and people you know as manga characters. Is it easier or more difficult than trying to draw them as American cartoons?\n\nMAP A LONG COMIC\n\n**Supplies:** ruled notebook, sketchbook, trusty pen or pencil\n\n**Writing your own graphic novel will take time. It took Art Spiegelman more than 10 years to finish** _**Maus**_ **! You can get started on your own graphic novel by laying the foundation with thumbnails and a script.**\n\n1In your notebook, write a short paragraph describing what your graphic novel will be about. Will it be about space exploration? The Wild West? A historical event or past experience?\n\n2Break down your story. Start with three parts: beginning, middle, and end. Write a little about what happens in each part. Now break each part into sections, or scenes. Write a few words for each scene to tell what is happening.\n\n3Write a script. The script simply tells you what is going to happen on each page. Use your outline and write out what happens, section by section. Make sure you include some talking and notes for the actions. Break your script up into different comics pages as you go.\n\n4Using your script, thumbnail out what each page will look like in your sketchbook. Make changes. Every cartoonist always makes changes to his or her script and thumbnails as the graphic novel develops. Don't be afraid to change something if you have a cooler idea. That's all part of the process.\n\nCHAPTER 6\n\nCOMICS AND THE INTERNET\n\nComics artists often use the latest technology to create and publish their work. The most amazing technological advancement for comics has been the Internet. It was originally created by the U.S. military in the 1960s as a way to share information in case of a nuclear attack. But the Internet was only used by a few government agencies and universities through the 1970s.\n\nThe Internet's early design and coding was almost all done through experimentation and collaboration across several scientific and computer science communities. By the late 1980s, the Internet had evolved dramatically. While the first computers were the size of your garage, personal computers in the late 1980s were so small they could fit easily on your desk in the living room. The Internet is still changing today!\n\nWORDS TO KNOW\n\ncollaboration: working with others.\n\nComics have come a long way from the big splashy pages of Hearst's and Pulitzer's newspapers. And comics are not just for kids anymore. Thanks to cheaper, high-quality printing technology, tablet devices, and the Internet, comics are finding new audiences all over the world.\n\nWEBCOMICS\n\nWith the spread of the Internet, many cartoonists found that they could reach a wide audience without needing a publisher, comic syndicate, or even a printer. These early pioneers set their sites (pun intended) on cyberspace and boldly went where no cartoonist had gone before\u2014online. Many of the first webcomics were quickly drawn stories about video game culture or humor revolving around a simple theme.\n\nAs the Internet became more a part of daily life, cartoonists used it to find an audience by establishing their own websites. There are also many comics apps that showcase the work of regular cartoonists and webcomics artists.\n\nWORDS TO KNOW\n\nwebcomic: a comic made to be read on web browsers.\n\n  Here are some fun webcomics to find online, with your parents' permission, The Last of the Polar Bears by Lindsay Cibos and Little Dee by Chris Baldwin. How do the characters change through time? Do the stories make surprise twists? Cover up neighboring QR codes to make sure you're scanning the right one.\n\nThe advent of color printing and mass newspaper production technology before the turn of the twentieth century changed the world of comics. Today, new technologies of the twenty-first century drive innovations in comics and cartooning in much the same way.\n\nThe same technology that made the boom in webcomics possible has also reshaped the mini comics and indie comics world. Comics artists use laptops, digital scanners, and electronic drawing devices such as the Wacom tablet to edit and lay out their comics for print from their desks. Powerful desktop publishing software has given independent cartoonists new ways of making their art. The cost of printing or photocopying comics has also fallen, creating a perfect environment for self-publishing.\n\nWORDS TO KNOW\n\nWacom tablet: a device that allows an artist to draw life-like lines right onto a computer.\n\ndesktop publishing: computer software that edits and formats comics, books, and magazines.\n\nCOMIC BOOK CONNECTION\n\nAll new comic strip cartoonists rely on social media to connect with their audiences. They love to hear from their readers! Ask an adult for permission to follow the Twitter or Tumblr feeds of some of your favorite cartoonists.\n\nAs more cartoonists learned to design their own comics using the latest computer technology, a flood of new talent poured into conventions such as SPX in Bethesda, Maryland, and APE in San Francisco. Independent mini comics and publishing conventions sprouted up across the nation in the mid-to-late 1990s.\n\nIndependent cartoonists found an audience eager to read something handmade in a world that has become very digital. Cartoonists created handmade screen print art on the covers of their mini comics, which gave each issue the look of having been crafted with care.\n\nIF YOU MAKE YOUR OWN MINI COMICS, TAKE THEM TO A MINI COMIC CONVENTION AND TRADE WITH OTHER CARTOONISTS.\n\nTHE FADING NEWSPAPER STRIPS\n\nAfter staying popular for almost 100 years, national newspapers began to see a fall in the number of readers in the 1990s. With the rise of the Internet and the popularity of cable news channels, fewer people read newspapers to find out what is happening in the world. Do you still read newspapers?\n\nWORDS TO KNOW\n\nscreen print: to make a color print of an image with special ink pressed through screens.\n\nLocal newspaper companies began buying each other up to gain more readers, until only one or two local newspapers were left in most regions. The lack of competition meant that the comic syndicates couldn't find as many homes for their comic strips.\n\nComic syndicates also couldn't pay new cartoonists the kinds of salaries they offered in the 1960s and '70s. Now, each syndicate only accepts one or two new cartoonists a year, even though it usually receives about 5,000 submissions.\n\nTHESE NEW CARTOONISTS GET PAID VERY LITTLE. IT'S A VERY DIFFERENT JOB THAN IT WAS 50 YEARS AGO.\n\nPUBLISHERS IN THE INTERNET AGE\n\nHow have comic book publishers adapted to the new world of the Internet? DC and Marvel began experimenting with new media in the late 1990s to try to get comics to fans in new ways. Today, their comics can be bought digitally, read digitally, and stored on a fan's device.\n\nBoth companies continue to try to find new ways to stay ahead of the technological curve. Meanwhile, print sales continue to decline. Maybe in a few years, major comic books such as Superman and X-Men will only be available as digital downloads.\n\nAt one of the first San Diego Comic-Cons, legendary cartoonist Jack Kirby said that in a few decades, Hollywood studios would be coming to the comic convention to look for the next big idea for a blockbuster film. He was right! From 2000 to 2013, a comic book movie has been one of the top-grossing films each year.\n\nA few newspaper comic strips have thrived in spite of the Internet and television. _Pearls Before Swine_ follows the example of _Garfield_. The characters often interact in a very spare environment, usually just a table and chairs, and rely on a punch line. _Rhymes with Orange_ by Hilary Price is mostly a single-panel gag comic that combines social observation with bizarre humor. The comics page also now includes its first Latin American comic strip, called _Baldo_.\n\nPerhaps the most popular new comic strip of this era is _Get Fuzzy_ , a strip about a cat, a dog, and their human. Comic strips about talking animals have always been popular, but _Get Fuzzy_ , drawn by Darby Conley, is one of the first to portray the cat and dog characters in an almost realistic style. _Get_ _Fuzzy_ embraces the more artful style of Bill Watterson.\n\nWhat's next for comics? It's impossible to know! But we can guess that comics are here to stay, even if the way we produce and read them changes through time.\n\n  **Find more of these comics.** Cover up neighboring QR codes to make sure you're scanning the right one.\n\nDESIGN YOUR TABLE FOR COMICS CONVENTIONS\n\n**Supplies:** banner paper, glue, glitter, paint, scissors, streamers, shoeboxes, cardboard, anything else you can think of to make things for your table\n\n**Whether you decide to be a mini comics creator, a zinester, a graphic novelist, or a webcomics genius, one day you may want a table at a comic convention. Either by yourself or with friends, start to think about how you will design your table.**\n\n1Decide on your table size. Most conventions will let you buy space at a whole table or a half table that you share with another cartoonist or artist. Once you know how much room you have, you can design what you need to stand out.\n\n2Use big sheets of paper to design a banner with the name of your comic or publishing group on it. You can paint original art on the banner or have your name in big glittery letters. Some comics artists like to hang a string in front of their banners for some sample comics.\n\n3Make a display stand for your comics. You can use shoe or cereal boxes for this step. People love to see comics standing upright so they can get a good look at the covers.\n\nTRY THIS! Conventions are a good place to sell merchandise of your comic. Try designing and ordering T-shirts of your work! Stickers and hand-crafted buttons are easy things you can make to get people excited about your comics. What else can you think of to give away at conventions?\n\nMAKE A MINI COMIC\n\n**Supplies:** paper, scissors, pencil, ruler, pens, eraser\n\n**Part of the fun of a mini comic is that you can make it quickly and get many copies to your friends and fans cheaply. Mini comics are really versatile. You can choose any size paper\u201411 by 17 inches (28 by 43 centimeters), 8\u00bd by 11 inches (21\u00bd by 28 centimeters), or 8\u00bd by 14 inches (21\u00bd by 35\u00bd centimeters). You can make them by yourself or with your friends. They can be funny or they can be serious. Besides being portable, mini comics are easy to make your own.**\n\n1Fold your paper in half three times and then unfold it completely. You have just divided your paper into eight equal sections. Each section will be a page in your mini comic.\n\n2Fold the paper in half once along the long side and cut along the crease line from the folded edge to the center. Unfold and then refold along the short edge.\n\n3Pinch the sides toward the center so that the cut opens. When your hands come together, flatten all the pages to one side. You now have a mini book! Number the pages with pencil and unfold it completely.\n\n4Use a piece of scratch paper to sketch out thumbnails of your comic. What is it going to be about? What will the cover look like? Use some of the skills you have practiced in other activities to make an awesome story and cool cover.\n\n5Pencil your comic's art. Draw all of the art on all of the pages. If you need to, fold the pages to keep track of the order of the story.\n\n6Ink your comic. Erase the pencil lines when you are done. If you are going to use a color copier, you can add some color to your comic!\n\n7The comic art is your master copy. A master copy is the version all other copies are made from. Ask an adult to help you make your copies at a local copy store or library. When you have made your copies, follow steps one through three again to put your comics together.\n\n8Mini comics are like the trading cards of the comics world. If you are ever at a comic convention with independent creators, you can trade one of your comics for one of theirs. You can also share them with your friends or get a table at a convention and sell them to new fans!\n\nTRY THIS! How can you design your comic differently? Can you use different colored paper? Try to find some friends to start a small publishing company with them. You can go to local comics conventions and sell your comics together.\n\nMAKE A WEBCOMIC\n\n**Supplies:** pencils, paper, ink, digital camera or scanner, computer, Internet, photo-editing software such as Adobe Photoshop or Gimp\n\n**Most comics artists use the Internet to show off their work to the global community. Others don't just show off their work, they DO their work online. These are the webcomics creators and you can be one of them. Get permission from an adult to go online and make sure you are setting up your website properly.**\n\n1Ask an adult to help you get a website set up. Tumblr is a great place to start. Cartoonist Patrick Yurick wrote a very informative feature on setting up your Tumblr site to make a webcomic at makingcomics.com. Go to the distribution section of the website and click on \"digital.\" Click on the \"tutorial\" link to access his advice.\n\n2Draw your comics and upload them to your website. You can either upload your comics by taking a photo of your comic or by scanning your comics into your computer. Make sure you save your image at a web resolution, which is 72 dpi (dots per inch).\n\n3Figure out a publishing schedule. Which days do you want new comics to show up on your website?\n\n4Time to promote! Tell your friends about your new webcomic and ask them to tell their friends. The online comics community likes to help each other out. Always get permission from a parent before you join any online group.\n\nGLOSSARY\n\n**adaptation:** telling a story in a different way and in a new format. For example, making a musical out of a series of comic strips would be adapting the comics into the musical.\n\n**afterlife:** the ancient Egyptian belief in life after death.\n\n**Allied forces:** the armies of the countries that fought together against Germany in World War I and World War II.\n\n**alter ego:** a second personality in the same person.\n\n**American Revolution:** the war during which the 13 American colonies fought England for independence. It lasted from 1775 to 1783.\n\n**ancestor:** someone from your family who lived before you.\n\n**animation:** multiple drawings in sequence to show movement.\n\n**anime:** Japanese animation.\n\n**archaeologist:** a scientist who studies ancient people through the objects they left behind.\n\n**arch-nemesis:** a character who is the opposite of the main character, and usually the enemy.\n\n**aristocrat:** a person of royal blood or privilege.\n\n**artifact:** an object made by people in the past, including tools, pottery, and jewelry.\n\n**astrologer:** a person who studies how the movements of the sun, moon, and planets affect humans.\n\n**astronomer:** a person who studies objects in the sky, such as stars and planets.\n\n**backstory:** a story about the events leading up to the main story.\n\n**BCE:** put after a date, BCE stands for Before Common Era and counts down to zero. CE stands for Common Era and counts up from zero. These non-religious terms correspond to BC and AD. This book was published in 2014 CE.\n\n**Boston Massacre:** a riot in Boston that took place on March 5, 1770. Five colonists were shot and killed by British soldiers.\n\n**brainstorm:** to come up with a bunch of ideas quickly and without judgment.\n\n**broadcast:** a program transmitted over a long distance.\n\n**carbon copy:** an exact copy of a document made using carbon paper between two or more pieces of paper.\n\n**cartoon:** a comic published in a newspaper or magazine.\n\n**cast:** a group of characters.\n\n**casting:** a metal print form created to make multiple copies of a printed page.\n\n**cathedral:** a large important church.\n\n**censor:** to examine books, movies, letters, etc., in order to remove things that are considered to be offensive or harmful to society.\n\n**character:** someone in a story.\n\n**cherub:** a type of angel, usually shown as a small child.\n\n**chromolithography:** a color printing process using metal plates to layer tints of color.\n\n**city\u2013state:** a city and its surrounding area, which rules itself like a country.\n\n**civilization:** a community of people that is advanced in art, science, and government.\n\n**cliffhanger:** an exciting moment that makes you wonder what happens next.\n\n**codex:** another name for a book. Plural is codices.\n\n**collaboration:** working with others.\n\n**color reproduction:** to make color prints of an original piece of art.\n\n**comic:** images in sequence that tell a story, with or without words.\n\n**comic relief:** the inclusion of a funny character or scene in an otherwise serious work.\n\n**Comics Code:** a set of guidelines created by comics publishers to help regulate the content of comics.\n\n**commercial:** operating as a business to earn money.\n\n**community:** a group of people who live in the same area.\n\n**concentration camp:** a prison where people were sent during the Holocaust to be killed or made to do hard work.\n\n**conceptualized:** imagined and thought out.\n\n**content:** the written material and illustrations in a story, article, book, or website.\n\n**convention:** a gathering of people who are all interested in a certain idea, topic, or event.\n\n**creative rights:** the right to use something that is created, often for profit.\n\n**culture:** the beliefs and way of life of a group of people.\n\n**custom:** a way of living and doing things, such as food and dress.\n\n**decay:** to rot.\n\n**depict:** to create a representation of something experienced or seen.\n\n**desktop publishing:** computer software that edits and formats comics, books, and magazines.\n\n**diabolic:** extremely evil.\n\n**dialogue:** a conversation between two people.\n\n**digital:** characterized by electronic and computerized technology.\n\n**direct mail:** a system used to sell things directly to subscribers through the U.S. mail.\n\n**drafted:** required to join the military.\n\n**drama:** an exciting event or series of events.\n\n**dynamic:** full of movement.\n\n**eccentric:** odd, usually in a unique way.\n\n**emcee:** a master of ceremonies who helps to lead a story or event.\n\n**epic:** a long poem, usually about the life of a hero or heroine.\n\n**era:** a set segment of time.\n\n**etching:** a print made by scratching original art onto a metal plate.\n\n**fantasy:** a genre of comics and literature based on myth and legend.\n\n**folk tale:** a story told by a specific group of people, often involving magic and a moral.\n\n**freelance:** to work on a project without being employed by the company assigning the work.\n\n**French Revolution:** a period of violent change in France between 1789 and 1799.\n\n**funnies:** the original name of comic strips in Sunday newspapers.\n\n**generation:** all the people born around the same time.\n\n**genre:** a type of creative work, such as mystery, romance, or fantasy.\n\n**Golden Age:** the period of comics history before the 1950s, considered to be the best and most popular by many fans and historians.\n\n**graphic novel:** a comic as long as a book, that tells one story.\n\n**hearing:** a special session of Congress or the Senate held to \"hear\" from witnesses and experts on a given issue.\n\n**hieroglyphics:** a writing system that uses pictures and symbols called hieroglyphs (or just glyphs) to represent words and ideas.\n\n**Holocaust:** a time before and during World War II when the German Nazis tried to kill the entire Jewish race, as well as several other groups.\n\n**hometown:** the place where a person, thing, or idea is born.\n\n**humor:** the quality of being funny.\n\n**iconic:** a widely recognized symbol of a certain time.\n\n**identity:** the characteristics that make a person an individual.\n\n**image:** a picture of something, either real or imagined.\n\n**immigrant:** a person who leaves his or her own country to live in another country.\n\n**imprint:** a part of a publishing company that publishes a certain type of book.\n\n**inbetweener:** an animator who assists the head animator by drawing movements in between key frames.\n\n**independent comic:** a comic book published without the help of a large company.\n\n**indigenous:** native.\n\n**inking:** to use ink to add definition to pencil drawings.\n\n**innovation:** a new creation or a unique solution to a problem.\n\n**invincible:** someone who cannot be defeated.\n\n**key frame:** a drawing in animation that captures the major points of movement. For example, if a character is waving her arm from left to right, the key frames would show the arm on the left, middle, and right.\n\n**layering:** stacking images on top of each other.\n\n**literate:** having the ability to read.\n\n**logic:** the principle, based on math, that things should work together in an orderly way.\n\n**manga:** a term for Japanese-style comics.\n\n**martyrdom:** the death of a person for his or her beliefs.\n\n**Marvel Method:** a way of creating comics that relies on an assembly-line format.\n\n**master page:** the version of a printed page that is used to make other copies.\n\n**medieval:** describes the Middle Ages, the period of European history after the fall of the Roman Empire, from about 350 to 1450 CE.\n\n**medium:** the material artists use to create their art, such as stone, paint, and ink.\n\n**mini comic:** a small, self-published comic.\n\n**monument:** a building, structure, or statue that is special because it honors an event or person, or because it is beautiful.\n\n**moral:** a valuable lesson to help people know how to behave.\n\n**movable type:** a process of printing that uses individual type pieces to spell out words.\n\n**mummify:** to preserve a dead body so it doesn't decay.\n\n**mythology:** a set of stories or beliefs about a particular religion or culture.\n\n**narration block:** a block of text that contains the voice of the writer or of a character talking about what is happening.\n\n**Nazi:** the main political party of Germany before and during World War II.\n\n**network:** a company that provides programs to be broadcast over radio or television stations.\n\n**observation:** something you notice.\n\n**orally:** spoken out loud.\n\n**pamphlet:** an informative book or brochure.\n\n**panel:** a square or other shape that frames a single scene in a comic strip.\n\n**parody:** an imitation of something for comical effect.\n\n**patent:** a document from the government that gives an inventor the exclusive right to make, use, or sell his or her invention.\n\n**Pearl Harbor:** a U.S. naval base in Hawaii that was attacked by Japan in 1941.\n\n**penciling:** to create an original comic first in pencil.\n\n**pharaoh:** the title for ancient Egyptian kings or rulers.\n\n**pictographic:** a picture of a word or idea.\n\n**pitch:** to present an idea.\n\n**political cartoon:** a comical or critical depiction of a political figure or event.\n\n**predecessor:** someone or something that came before others.\n\n**premise:** the main idea of a story.\n\n**preserve:** to keep something from rotting.\n\n**printing press:** a machine that presses inked type onto paper.\n\n**process:** an activity that takes several steps to complete.\n\n**propaganda:** ideas or statements that are sometimes exaggerated or even false. They are spread to help a cause, political leader, or government.\n\n**psychologist:** a person who studies the mind and behavior.\n\n**Pulitzer Prize:** a group of prizes awarded annually for work done in journalism, fiction, and non-fiction, as well as for photojournalism.\n\n**pull list:** a list of comics that your local comic bookstore will order and set aside for you.\n\n**pulp magazine:** a cheap fiction magazine published between 1896 and the 1950s.\n\n**punch line:** the sentence, statement, or phrase that makes the point, as in a joke.\n\n**race:** a group of people with the same skin color and other physical features.\n\n**rarebit:** a type of soft cheese which is famous for giving indigestion.\n\n**Renaissance:** a period of time in Europe after the Middle Ages, from the 1300s to the 1600s.\n\n**representation:** showing things in pictures or other forms of art.\n\n**revolutionary:** someone committed to fighting a ruler or political system.\n\n**ritual:** something done as part of a religion.\n\n**romance comic:** a comic about characters who are trying to find love.\n\n**Rosetta Stone:** a stone tablet written in 196 BCE telling the same decree using hieroglyphics, Egyptian Demotic script, and ancient Greek. The stone was fully translated in 1822, leading specialists to understand hieroglyphics better in the nineteenth century.\n\n**saint:** a Catholic Christian who has performed miracles as confirmed by the pope.\n\n**sarcophagus:** a large, stone box containing an Egyptian king's coffin and mummy.\n\n**science fiction:** stories that deal with the influence of real or imagined science.\n\n**screen print:** to make a color print of an image with special ink pressed through screens.\n\n**scribe:** a person who copies writings by hand.\n\n**sequence:** the order in which something happens.\n\n**serial:** occurring in a series.\n\n**serialize:** to create a series of stories.\n\n**sidekick:** a character who supports the main character.\n\n**Silver Age:** the era of comics between the introduction of the Comics Code in 1956 and the late 1970s.\n\n**smock:** a cloth worn over clothing to protect it from stains.\n\n**spiritual:** religious, relating to the soul or spirit.\n\n**splash page:** large pages of comic action with no panels.\n\n**standardized:** a set way of doing something.\n\n**stylize:** to draw comics in a specific way.\n\n**suspense:** a feeling or state of nervousness or excitement caused by wondering what will happen.\n\n**symbol:** a physical representation of a thing or idea.\n\n**syndication:** when the rights to print or broadcast a creative work are sold.\n\n**tapestry:** a colorful, woven fabric that hangs on a wall. It often shows a scene.\n\n**technology:** scientific or mechanical tools, methods, and systems used to solve a problem or do work.\n\n**thought cloud:** a shape similar to a word balloon that encloses a character's thought.\n\n**tint:** a shade or variety of color.\n\n**tomb:** a room or place where a dead person is buried.\n\n**tribe:** a large group of people with common ancestors and customs.\n\n**vigilante:** a person who takes the law into his or her own hands.\n\n**villain:** a character who opposes the hero and does bad things.\n\n**Wacom tablet:** a device that allows an artist to draw life-like lines right onto a computer.\n\n**webcomic:** a comic made to be read on web browsers.\n\n**woodcut:** a way of printing by carving an image on a piece of wood before adding ink and printing the image onto paper.\n\n**word balloon:** a rounded outline with a point toward a character that encloses the character's speech.\n\n**xerography:** a way of making a copy using light-sensitive chemicals and paper.\n\n**zine:** a self-published magazine, usually produced using photocopiers.\n\n**zinester:** a person who makes zines.\nRESOURCES\n\n**Books**\n\n_Cartooning the Head and Figure_. Jack Hamm. Grosset and Dunlap, 1967\n\n_Comics and Sequential Art_. Will Eisner. Norton Press, 2008\n\n_Comics, Manga, and Graphic Novels_. Robert Petersen. Praeger Books, 2011\n\n_Dick Tracy: American's Most Famous Detective._ Bill Crouch Jr. ed. Citadel Press, 1987\n\n_Expressive Anatomy for Comics and Narrative_. Will Eisner. Norton Press, 2008\n\n_Graphic Storytelling and Visual Narrative_. Will Eisner. Norton Press, 2008\n\n_How to Draw Comics the Marvel Way_. Stan Lee and John Buscema. Simon and Schuster, 1984\n\n_How to Make Webcomics_. Brad Guigar, et al. Image Books, 2011\n\n_Men of Tomorrow_. Gerard Jones. Basic Books, 2004\n\n_Reading Comics_. Douglas Wolk. Da Capo Press, 2007\n\n_The Comic Book History of Comics_. Fred Van Lente and Ryan Dunlavey. IDW, 2012\n\n_The Comics: An Illustrated History of Comic Strip Art 1895\u20132010_. Jerry Robinson. Dark Horse Comics, 2001\n\n_The Ten-Cent Plague_. David Hajdu. Picador Press, 2008\n\n_Understanding Comics_. Scott McCloud. Kitchen Sink Press, 1993\n\n**Primary Source QR Codes**\n\n**Page 24:** upload.wikimedia.org\/wikipedia\/commons\/4\/40\/1896-11-08_Yellow_Kid.jpg\n\n**Page 27:** en.wikipedia.org\/wiki\/Mutt_and_Jeff#mediaviewer\/File:Muttandjeffalla51.jpg\n\n**Page 28:** ignatzmouse.net\/us\/archives\/kk\n\n**Page 30:** comicstriplibrary.org\/display\/111\n\n**Page 32:** youtube.com\/watch?v=xH9tCcrrcak\n\n**Page 48:** images.tcj.com\/2014\/01\/SuperDiscov2.jpg\n\n**Page 50:** youtube.com\/watch?v=GJCTTba4S0E\n\n**Page 55:** youtube.com\/watch?v=vr9qpeOjmuQ\n\n**Page 70:** loc.gov\/exhibits\/blondie\/wedding.html; blondie.com\n\n**Page 71:** peanuts.com\n\n**Page 72:** youtube.com\/watch?v=dS0vUbWdNxg\n\n**Page 83:** youtube.com\/watch?v=1jgE-lrfZ3k\n\n**Page 93:** garfield.com; zenpencils.com\/comic\/128-bill-watterson-a-cartoonists-advice\n\n**Page 94:** calvinhobbesdaily.tumblr.com\n\n**Page 104:** lastpolarbears.com; littledee.net\n\n**Page 108:** rhymeswithorange.com; baldocomics.com\/blog; getfuzzyarchive.blogspot.com\nINDEX\n\n**A**\n\nactivities\/projects\n\nCostume for Comic-Con,\n\nDesign a Cast of Characters,\n\nDesign a Costume and Backstory, \u2013\n\nDesign a Villain, \u2013\n\nDesign Your Table for Comics Conventions,\n\nDraw a Superhero, \u2013\n\nDraw Cartoon Bodies, \u2013\n\nDraw Cartoon Faces, \u2013\n\nDraw Hands and Feet, \u2013\n\nDraw Like an Egyptian!, \u2013\n\nDraw Manga-Style Characters, \u2013\n\nDraw Manga-Style Faces, \u2013\n\nJam Comics,\n\nMake a Daily Comic Strip,\n\nMake a Dynamic Cover,\n\nMake a Mini Comic, \u2013\n\nMake a Superhero Team, \u2013\n\nMake a Webcomic,\n\nMake Your Comics Make Sense, \u2013\n\nManga-Style Sketchbook,\n\nMap a Long Comic,\n\nSidewalk Cave Painting,\n\nSunday Comic Strip,\n\nThing, Job, Description, \u2013\n\nUse the \"Marvel Method,\" \u2013\n\nAdams, John,\n\n_Amazing Spiderman_ ,\n\nAmerican Revolutionary comics, iv, \u2013\n\nancient comics, iv, \u2013\n\nanime, vi,\n\n_Archie_ ,\n\n_Astro Boy_ , vi,\n\nAustralian indigenous representational paintings,\n\n_Avengers_ , vii,\n\n**B**\n\n_Baldo_ ,\n\n_Batman_ , v, , , ,\n\n_Beetle Bailey_ ,\n\n_Blondie_ ,\n\nbroadsides\/broadsheets, \u2013\n\n**C**\n\n_Calvin and Hobbes_ , \u2013\n\nCaniff, Milton \"Milt,\"\n\n_Cap Stubbs and Tippie_ ,\n\nCapp, Al,\n\n_Captain America_ , vi, \u2013, ,\n\nCarlson, Chester,\n\ncartoons. _See also_ specific cartoons by name\n\nbroadsheets including, \u2013\n\ncolor printing of, iv\u2013v, \u2013,\n\ncomic books of, vii, \u2013, , \u2013,  ( _see also_ graphic novels)\n\ndefinition of,\n\nearly comic masters creating, \u2013\n\nmanga as, iv, vi, vii, , , \u2013, \u2013\n\nnewspapers including, v\u2013vi, , , \u2013, , \u2013, , \u2013, \u2013\n\norigins of, iv,\n\npolitical, \u2013, , \u2013\n\nromance, \u2013\n\nscary, \u2013\n\nserialized, v,\n\nsidekicks in, v, , , ,\n\nSilver Age of, vi, ,\n\nSunday comics as, v, ,\n\nsuperheroes in, v\u2013vi, , \u2013, \u2013, \u2013\n\nsyndication of, , ,\n\ntimeline of, iv\u2013vii\n\nwartime, \u2013, \u2013, , , ,\n\nwebcomics, vii, \u2013,\n\nwomen creating,\n\ncave paintings, iv, \u2013,\n\n_Cerebus_ , \u2013\n\ncharacters, vi, , , , \u2013, . _See also_ specific characters\n\ncolor printing, iv\u2013v, \u2013,\n\nComic-Con, vii, , ,\n\ncomics\n\nancient, iv, \u2013\n\ncartoons including ( _see_ cartoons)\n\ncharacters in, vi, , , , \u2013,  ( _see also_ specific characters)\n\ncomic books of, vii, \u2013, , \u2013,  ( _see also_ graphic novels)\n\ncomic book stores selling,\n\nComics Code for, vi, vii, , , ,\n\ncomic strips of, , \u2013, , \u2013 ( _see also_ newspaper comics)\n\nconventions on, vii, , , , ,\n\ndefinition of,\n\nGolden Age of, v, \u2013, ,\n\ngraphic novels of, vii, , , , \u2013,\n\nthe Internet and, vii, \u2013\n\nmini, vii, , , , \u2013\n\npanels of, ,\n\npost-Comics Code era of, \u2013\n\nregulation of, vi, vii, \u2013, , ,\n\nrules and systems of, , , \u2013\n\nComics Code, vi, vii, , , ,\n\nComics Magazine Association of America,\n\nConley, Darby,\n\nconventions, comic, vii, , , , ,\n\n_Crime Does Not Pay_ , \u2013,\n\n**D**\n\nDavis, Jim,\n\nDC Comics, vi, vii, , , \u2013, ,\n\n_Dick Tracy_ ,\n\nDisney, Walt, ,\n\nDitko, Steve,\n\n_Dream of the Rarebit Fiend_ ,\n\nDumm, Edwina,\n\n**E**\n\nEastman, Kevin, \u2013\n\nEC Comics, , , ,\n\nEgyptian pictographic language, iv, \u2013, \u2013\n\nEisner, Will, \u2013, ,\n\n_Elfquest_ ,\n\n**F**\n\n_Fantastic Four_ , , ,\n\nFisher, Henry Conway \"Bud,\" v, \u2013\n\nfunnies, v,\n\n**G**\n\nGaines, Bill, \u2013\n\n_Garfield_ , ,\n\n_Get Fuzzy_ ,\n\nGolden Age of comics, v, \u2013, ,\n\nGould, Chester,\n\ngraphic novels, vii, , , , \u2013,\n\nGray, Harold, v, \u2013,\n\n**H**\n\nHearst, William Randolph, , , , , ,\n\nHerriman, George, v, \u2013,\n\nHoe, Richard March,\n\nHokusai, iv, , ,\n\n**I**\n\nImage Comics, vii,\n\nthe Internet, vii, \u2013\n\n**J**\n\nJapanese Manga. _See_ manga\n\n_Justice League of America_ ,\n\n**K**\n\nKane, Bob, v,\n\nKelly, Walt,\n\nKirby, Jack, \u2013, , , , ,\n\n_Krazy Kat_ , \u2013\n\n**L**\n\nLaird, Peter, \u2013\n\nLee (Lieber), Stan, ,\n\n_Li'l Abner_ ,\n\n_Little Lit_ ,\n\n_Little Nemo in Slumberland_ , \u2013\n\n_Little Orphan Annie_ , v, \u2013,\n\n**M**\n\nmanga, iv, vi, vii, , , \u2013, \u2013\n\nMarder, Larry,\n\nMarvel Comics, vi, vii, , , \u2013, , ,\n\n_Maus_ , vii, ,\n\nMaya codices, iv, \u2013\n\nMcCay, Winsor, v, \u2013,\n\nMichelangelo, iv,\n\nmini comics, vii, , , , \u2013\n\nMontana, Bob,\n\nmovies, vii, , , , , ,\n\n_Mutt and Jeff_ , v, \u2013\n\n**N**\n\nnewspaper comics. _See also_ specific cartoons by name\n\ncartoons as, v\u2013vi, , , \u2013, , \u2013, , \u2013, \u2013\n\ncolor printing of, iv\u2013v, \u2013\n\ncomic strips of, , \u2013, , \u2013\n\ndaily, \u2013,\n\nearly comic masters creating, \u2013\n\nfading of, \u2013\n\nhistory of, \u2013\n\nmodern, \u2013\n\npolitical, , , \u2013\n\nSunday comics, v, ,\n\nsyndication of, , ,\n\n**O**\n\nO'Neill, Rose,\n\nOrczy, Baroness Emma,\n\nOutcault, Richard F., v, \u2013\n\n**P**\n\n_Peanuts_ , vi, \u2013,\n\n_Pearls Before Swine_ ,\n\nphotocopying, vi, , , ,\n\nPini, Wendy and Richard,\n\npolitical cartoons, \u2013, , \u2013\n\n_Popeye_ , \u2013,\n\nPrice, Hilary,\n\nprinting presses\/techniques, iv\u2013v, vi, , , , \u2013, , , ,\n\nPulitzer, Joseph, v, \u2013, , ,\n\npulp magazines, \u2013, , , ,\n\n**R**\n\nradio programs, ,\n\nRevere, Paul, iv, \u2013\n\n_Rhymes with Orange_ ,\n\nRogers, Buck, ,\n\nromance comics, \u2013\n\n**S**\n\nScarlet Pimpernel, ,\n\nscary comics, \u2013\n\nSchulz, Charles, vi, vii, \u2013, ,\n\nSegar, E. C., \u2013,\n\nserialized comics, v,\n\nShuster, Joe, v, ,\n\nsidekicks, v, , , ,\n\nSiegel, Jerry, v, , ,\n\nSilver Age of comics, vi, ,\n\nSim, Dave, \u2013\n\nSimon, Joe, ,\n\nSpiegelman, Art, vii, , ,\n\n_The Spirit_ ,\n\n_Steve Canyon_ , \u2013\n\nSunday comics, v, ,\n\nsuperheroes, v\u2013vi, , \u2013, \u2013, \u2013\n\n_Superman_ , v, \u2013, , , , ,\n\nsyndication, , ,\n\n**T**\n\n_Tales from the Crypt_ , ,\n\n_Tales of the Beanworld_ ,\n\ntapestries, \u2013\n\n_Teenage Mutant Ninja Turtles_ , \u2013\n\ntelevision shows, , ,\n\n_Terry and the Pirates_ , \u2013\n\nTezuka, Osamu,\n\ntimeline, iv\u2013vii\n\n**W**\n\nWalker, Mort,\n\nwartime cartoons, \u2013, \u2013, , , ,\n\nWatterson, Bill, \u2013,\n\nwebcomics, vii, \u2013,\n\nwomen as cartoonists,\n\n**X**\n\n_X-Men_ , , ,\n\n**Y**\n\n_The Yellow Kid_ , v, \u2013, ,\n\nYoung, Murat \"Chic,\"\n\n_Young Romance_ ,\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n_For Maureen, Ronan, and Orla, with love and appreciation_\n\n# CONTENTS\n\nFOREWORD\n\n_The Imperative of Equal Opportunity and the Importance of Citizen Schools \u2014_ Lawrence H. Summers\n\nINTRODUCTION\n\n_The Opportunity Equation_\n\nSECTION ONE: Starting Citizen Schools\n\nCHAPTER ONE\n\n_Building Blocks_\n\nCHAPTER TWO\n\n_Maureen_\n\nCHAPTER THREE\n\n_\"It's My Turn!\"_\n\nCHAPTER FOUR\n\n_Turning Point_\n\nCHAPTER FIVE\n\n_Organization Man_\n\nCHAPTER SIX\n\n_The Eddy Is Ready_\n\nCHAPTER SEVEN\n\n_Scale, Spread, and the Pursuit of Systemic Change_\n\nSECTION TWO: How Citizen Power and an Expanded Learning Day Can Narrow Achievement Gaps, Broaden Opportunity, and Strengthen America\n\nVOICES FROM CITIZEN SCHOOLS\n\n_Joyce King Thomas, volunteer_\n\nCHAPTER EIGHT\n\n_Citizen Power and the Importance of Mentoring_\n\nCHAPTER NINE\n\n_It's About Time_\n\nCHAPTER TEN\n\n_From a Nation of Consumers to a Nation of Makers: Inspiring Creativity and Innovative Thinking in Our Schools_\n\nCHAPTER ELEVEN\n\n_Social Networks and Social Skills_\n\nCHAPTER TWELVE\n\n_Supporting Teachers, and Parents Too_\n\nVOICES FROM CITIZEN SCHOOLS\n\n_Lindy Smalt, AmeriCorps Teaching Fellow_\n\nSECTION THREE: Next Steps for America and for You\n\nCHAPTER THIRTEEN\n\n_A Civic Marshall Plan for Equal Opportunity_\n\nVOICES FROM CITIZEN SCHOOLS\n\n_Agostinha DePina, student_\n\nAPPENDIX\n\n_Expanded Learning Time Schedule with Citizen Schools_\n\nACKNOWLEDGMENTS\n\nNOTES\n\nINDEX\n\n# FOREWORD\n\n# THE IMPERATIVE OF EQUAL OPPORTUNITY AND THE IMPORTANCE OF CITIZEN SCHOOLS\n\n_Lawrence H. Summers is the president emeritus of and Charles W. Eliot University Professor at Harvard University and the former secretary of the treasury of the United States. In February 2014, Summers was elected chair of the Citizen Schools board of directors. His daughter Ruth, referenced below, served as an AmeriCorps Teaching Fellow at Citizen Schools from July 2012 to June 2014._\n\nIn an elitist age, the Duke of Wellington famously said that the Battle of Waterloo was won on the playing fields of Eton, attributing the leadership skills of the British military officers he commanded to lessons learned as students at England's top private school. I believe that the battle for America's future will be won or lost in our public schools\u2014and today we are losing.\n\nThis was brought home to me when I was treasury secretary. I made it a practice every time I visited a city to go visit a public school. Once I was in Oakland meeting with a group of high school students and gave what I thought was a pretty good speech about the importance of education to national prosperity and individual opportunity.\n\nAfter I finished, a young teacher\u2014she was probably five years older than my daughter Ruth is now\u2014came up to me and said,\n\n\"Secretary Summers, that was a terrific speech. But here's the problem:\n\n\"Why should the children believe that there's nothing more important to the future of this country than education when nobody has come in and painted this school in the last eleven years and the paint is chipping off the walls?\n\n\"Why should they believe you when you say science education is hugely important, but they get nauseous every time they do a chemistry lab because the ventilation system hasn't been fixed for a year?\n\n\"Why should they look around and believe you?\"\n\nI had no good answers, and I have been haunted by this question ever since.\n\nAmerica, if it has stood for anything, has stood for two hundred years for the idea of equality of opportunity. Everyone's income is not going to be equal in the United States. The way everyone lives is not going to be equal in the United States. We are going to try to make it fairer and better, but it is never going to be completely equal or fair. It cannot be in a free society.\n\nWhat we can, and I believe must, aspire to as a nation is that everyone's children\u2014all children\u2014have the same chance to succeed. No child should be denied the opportunity to be educated, to follow their dreams, to create, to prosper, and to contribute because of the circumstances of their birth. Those with less money may live in smaller houses, take simpler vacations, and wear less expensive clothes. That is inevitable. But it should not be acceptable to anyone, liberal or conservative, that they be forced to have lesser dreams for their children.\n\nThis ideal of equal opportunity\u2014that any child can grow up to be president, to be a billionaire, to win a Nobel Prize\u2014is the American Dream. Yet, as Eric Schwarz's book powerfully illustrates, on the standard of equality of opportunity, our generation of Americans is failing. Of course, there are difficulties in statistics and measurement. But, as best as one can tell, for the first time in two hundred years, the gap in the life prospects of the children of the rich and the children of the poor is greater today than it was one or two generations ago. That should not be acceptable.\n\nNor should it be acceptable that the United States now trails most other major industrial nations in terms of equality of opportunity and social mobility. In Europe, it is easier for those from disadvantaged backgrounds to succeed than it is for their American counterparts. As a corollary, the successful are more entrenched in the United States than in the nations that are our biggest trading partners and competitors.\n\n_The Opportunity Equation_ offers stories and data that help us see inequality as not just a function of inequality in our schools but a function of the inequality of experiences offered by the broader society. The result is growing inequality in educational attainment, in income, and in family wealth. As just one example, the college completion rate for children of parents in the top quartile by family income has jumped from 40 percent to 73 percent since 1970, but for children of parents in the lowest quartile, it has only grown from 6 percent to 8 percent. As an additional example, despite all that has been done to improve financial aid, the percentage of the students attending our elite universities who come from the upper half of the income distribution is greater than it was a generation ago. As a country, we are moving in the wrong direction with respect to equality of opportunity for lower-income children, and that is not acceptable.\n\nIt does not have to be this way. Determined national efforts can make a difference, as they have over the last half century with respect to racial equality. When I was a child, African Americans were massively underrepresented in higher education and the race-based achievement gap in our schools was sickeningly large. We still have a long way to go in fully addressing racism in our society. But whereas in the 1940s and '50s the reading and math skills of the typical white student were four to five grade levels ahead of the typical African American student, today more than half of that skills gap has been eliminated. In the nation as a whole, African Americans are still less likely than whites to attend elite colleges and universities and less likely to graduate, but an African American twenty-year-old is about as likely to be enrolled in college as a white twenty-year-old. More needs to be done, but this is a huge national achievement. It is a national achievement of public policy. It is a national achievement of changed private attitudes. It is a national achievement of reform in almost every institution in our society. It is real progress.\n\nAs we have made great progress in combating racial inequality, so too can we overcome inequality of opportunity between the children of affluent and less-affluent parents. That is why I recently committed to chair the national board of directors of Citizen Schools, an organization committed to making such progress with respect to the class divide. I did this in part because I have seen the power of the program firsthand through my daughter Ruth, an AmeriCorps teaching fellow with Citizen Schools, and in part for three other reasons.\n\nThe first thing that attracted me to Citizen Schools was the program's commonsense approach of equalizing access to extra learning time in the afternoon and evenings, on weekends, and during the summer. For Boston students in grades six through twelve, the school day ends at 1:30, and across the country all children spend the great majority of their waking hours out of school. Children are three times as likely to be raised by a single parent as they were in 1960, and that parent, usually the mother, is more likely to be working. Because low-income families lack money to purchase access to tutoring, camps, and after-school programs, the average lower-income child reaches age twelve having spent six thousand fewer hours engaged in formal and informal learning than the average upper-income child. If that division stands we will never equalize opportunity.\n\nA second compelling aspect of Citizen Schools is that it works with, but remains outside of, the public school system. I'm a Democrat. I'm a guy who believes in the government. I'm a guy who doesn't understand how you can love your country and hate its government. At the same time, I'm a guy with eyes, and I know that government cannot do it all. It is a long-standing truth that if we are going to succeed in solving problems, it cannot all just be the government saying what needs to be done, hiring people, and doing it. It needs to be a collective effort of the broader society and its citizens. Just as the children of England's elite learned to lead on the playing fields of Eton and Oxford and Cambridge, in America\u2014in the twenty-first century\u2014we need to provide children of all backgrounds with the experiences that will prepare them to lead and to compete.\n\nThe last reason why I was so drawn to Citizen Schools is that, despite my sometimes tearing up when I talk about my daughter Ruth and her powerful experience as a teacher in the program, I am really not touchy-feely when it comes to public policy. I believe in data. What impressed and inspired me about Citizen Schools is that they recognized that, as noble as all their intentions are, as good as their work made all of them feel, it really was not very important if, at the end of it all, children did not have better experiences and learn more. Before agreeing to become involved with Citizen Schools I spent time making sure that rigorous evaluation and external evaluations were a central part of the model. I learned that rigorous evaluation has not only been an important component of the model but that the evaluations completed so far have been highly encouraging about the efficacy of the program.\n\nIn the context of the vexed debate over strengthening education, it seems to me that Citizen Schools and the work described in this book represent a set of enormously promising and enormously powerful educational innovations. I salute Eric Schwarz, the founding CEO of Citizen Schools and the author of this fine book, for his vision and his dedication. He stands in a long tradition of American citizen-leaders who have moved this country forward.\n\nW. E. B. Du Bois famously said that the problem of the twentieth century would be the problem of the color line. Though much progress has been made, the race-based achievement gap of my childhood has unfortunately morphed into an even starker class divide. Today, students from the top income quartile are four to five grade levels ahead of students from the bottom quartile\u2014a gap that is twice as large as existed in the 1940s and 1950s. The question for the twenty-first century is how can we reduce this growing class divide and provide more opportunity for all children from all backgrounds.\n\nThis will be the task of America's parents: striving for the best for their children. It will be the task of all those on the frontlines in the schools: teachers and principals. It will be the task of those, like my daughter, who are not employed by the schools but work within them. But it is also the responsibility of citizens who care very deeply about the future of our cities and the future of our country. This book shows how ordinary people have made and can make a profound difference in lifting up opportunity for all children. And it demonstrates the importance of our doing all we can to help.\n\n\u2014LAWRENCE H. SUMMERS\n\n# INTRODUCTION\n\n# THE OPPORTUNITY EQUATION\n\nAs Adam Barriga entered the Massachusetts State House, with its towering golden dome and rooms full of history, his sneakers squeaked lightly on the Italian-marble floors. Adam, age twelve, had been grumpy most of the day. But now, as he made his way to the capitol building's main lobby, walking past flags and portraits and a mural of the Revolutionary War Battle of Concord, he forgot entirely what had been bothering him. Once in the lobby, Adam joined hundreds of his classmates involved in all sorts of hands-on learning projects. The students were sixth and seventh graders attending public schools in Boston, and almost all of them qualified for a free or reduced-price lunch, meaning their family incomes were below or a little above the poverty line. About half of them had school-identified learning disabilities or spoke a language other than English at home or both. But soon all of them would be talking to an appreciative adult audience about robots they'd programmed, video games they'd designed, Android apps they'd invented, or, as in Adam's case, rocket ships they'd built.\n\nThe event was what we at Citizen Schools call a WOW!\u2014a chance for students to showcase their learning and, hopefully, \"wow\" those in attendance. This particular evening the crowd included elected officials and executives from leading technology companies such as Google and Microsoft and Biogen Idec, one of the fastest-growing biotechnology firms in Massachusetts. I noticed how the politicians and executives moved through the State House effortlessly, smiling for pictures with families, asking questions of the students, and genuinely enjoying an event listed as a \"stop-by\" on their schedules. Cultivating support from guests at a WOW! was important, and I was pleased with the level of engagement. But what really stuck with me from that May evening in 2010 were the experiences of three others in the room.\n\nFirst was Adam himself, then a sixth grader at the Clarence Edwards Middle School in Boston's Charlestown neighborhood. The Edwards, or Eddy, as it's known locally, had recently partnered with Citizen Schools to expand the learning day for all students from six to nine hours as part of a statewide Expanded Learning Time pilot program for struggling urban schools. Adam was demonstrating what he'd learned from one of his Citizen Schools apprenticeships, It _Is_ Rocket Science!, and tentatively conversing with parents and politicians and anyone else who would listen. His confidence grew over the course of the evening as he talked about escape velocity, lunar windstorms, and solar flares, and as he described the final project he and his classmates had participated in: a simulated lunar landing performed by videoconference with real astronauts from NASA.\n\nSecond was David Mantus, the teacher of the It _Is_ Rocket Science! apprenticeship and a repeat volunteer at Citizen Schools. David had grown up in the suburbs of Long Island, and he shared with me that his fondest childhood memories were of launching rockets in his backyard with his dad, a NASA engineer, and visiting science museums on the weekend with his grandfather. Later, Mantus earned a PhD in chemistry and eventually climbed the corporate ladder to become head of regulatory affairs for Cubist Pharmaceuticals. As Mantus moved further away from hands-on science in his own career (from mixing chemicals in the lab, to schmoozing regulators on the conference circuit), he had lost a little of himself. This increased his desire to inspire future scientists. At the State House, Mantus prompted his middle school apprentices to explain what they had learned, but then he invariably built upon their answers, adding scientific detail as he revealed more than a trace of boyhood excitement.\n\nAdam and David each burst with pride, and in their unlikely relationship and shared enthusiasm for launching rockets, I saw great hope. But my greatest joy that night came from the smile of wonderment worn by Adam's grandfather, Eduardo Barriga. Barriga had emigrated from Peru thirty years previously. For almost his entire life in the United States, he had worked as a custodian at the State House. The marble floors where Adam held forth, and where CEOs and politicians gathered, were polished with his own hands. Now his grandson was standing on them with something important to say.\n\nI started Citizen Schools because for students like Adam Barriga, opportunities like that night at the State House are far too rare. On the other hand, for upper-income Americans like David Mantus, and like me, the ascension from enriching childhood experiences to advanced education and successful careers has become too automatic, a ticket to success that upper-middle-class parents so reliably and consistently procure for our children that it stacks the deck against others who are less fortunate. I felt it was simply unfair that upper-income kids were the almost exclusive beneficiaries of an _opportunity equation_ in which their abilities were multiplied by a dazzling array of extra learning opportunities.\n\nMy thinking on these topics was informed by my own childhood, every corner of which was piled high with the building blocks of opportunity. Like David, I too was surrounded by professionally successful adults\u2014not rocket scientists in my case, but prominent lawyers and businesspeople and nonprofit leaders. I had extra time to learn and was given many chances to build the muscle memory of success. Further, as a descendant and namesake of the FAO Schwarz toy store family, I was connected to a powerful social network.\n\nThe opportunity divide is not new. Children like David and me have always had a leg up. But in recent decades, the story of opportunity in America has evolved in at least three important ways.\n\nFirst, new research indicates the wealth-based opportunity gap is now a chasm. Access to the extra learning that families like mine take for granted is now so unequal that class-based achievement gaps in everything from elementary school reading to middle school math to college graduation have ballooned and are now substantially _larger_ than when I grew up in the 1960s and '70s. At that time, wealthier kids were a little more than two grade levels ahead of poor kids in reading and math. Now they are more than four grade levels ahead. Gaps in harder-to-measure social skills\u2014like the ability to ask for help and to network for a job\u2014have also grown, and in turn contribute to ever-growing wealth and income gaps. For both the affluent and the impoverished, parental wealth now predicts adult success more than at any point in at least one hundred years.\n\nSecond, children from lower- and moderate-income families are now at such an intense relative disadvantage in getting ready for productive adulthood that US economic competitiveness is being undermined. We simply can't afford to have so many workers not ready to perform in the modern economy. The blue-chip consulting firm, McKinsey, estimates our current income-based achievement gap imposes the equivalent of a 3\u20135 percent permanent national recession. By failing to spread opportunity more broadly, we are slicing our economic pie unevenly, and we are also keeping the pie from growing.\n\nThird, the _cause_ of the growing opportunity and achievement divide is becoming clearer. In certain circles it has become fashionable to deride public schools as hopelessly inept, dragged down by incompetent teachers and stifling bureaucracies. For sure, many schools need to improve. But a close look at the data indicates that ever-widening achievement gaps are growing fastest _outside_ of school. They are growing on suburban playing fields and at robotics competitions, at after-school math programs and specialized sleep-away camps, and elbow to elbow with parents around dining room tables all across America. The class-based education gap is accelerating _not_ because teachers are lazy or \u00adbecause America's schools suddenly forgot how to churn out Horatio Alger success stories. And poor and working-class kids are not learning _less_ than before, they are actually learning _more._ Instead, the class-based education gap is accelerating because upper-middle-class children are accelerating faster, pulling away from their less privileged peers thanks to increasingly engaged parents and a cottage industry of coaches, counselors, tutors, and trainers.\n\nSo what to do about this opportunity chasm by which upper-middle-class families pass on success to their children, and children from lower-income families get left even further behind? Is it just about money, fixable only in a utopian, Lake Wobegon world where every family's income is \"above average\"? Or are there experiences and relationships routinely offered to upper-income children that our society, with modest investment, could make available to all children?\n\nI recently asked Paul Reville, a professor of practice at Harvard's Graduate School of Education and formerly the highly regarded Massachusetts secretary of education, if he knew any system of education that routinely and at scale gets its students to high levels of educational achievement while preparing them for careers. \"I know of one,\" Reville said: \"The upper-middle-class family.\"\n\nReville's insight is startling. But what exactly _is_ the opportunity equation that generally works for upper-income children? And can we make it work for all children?\n\nIn the summer of 1994 I was newly married and hoping to start a family. I had recently left a job with a leading national nonprofit organization and was reflecting on these questions of educational and economic opportunity. My questions led me to develop Citizen Schools, a startup program that began with just me and ten fifth graders.\n\nWhat if we could harness the power of you and me\u2014of millions of architects, engineers, lawyers, carpenters, journalists, and grandmothers who sew\u2014to equalize opportunity and reduce the growing wealth-based achievement gap? What if we could create a new network of \"citizen schools\" to extend the traditional school day and provide time and space for all kinds of talented people to share their time and attention with lower-income students? I believed then, and believe even more deeply today, that if we offer all children more time with caring and accomplished adults, and more chances to build the muscle memory of success\u2014by getting good at playing piano or building robots or writing poems\u2014then, and only then, can we make American education what Horace Mann dreamed it could be: \"the great equalizer of the conditions of men, the balance wheel of [our] social machinery.\"\n\nThe very idea that I could create a new organization\u2014particularly one dedicated to such fundamental change\u2014was rooted in my own childhood experiences. My parents' weekend dinner parties provided an ongoing and indispensable apprenticeship in the profession of success. I remember being invited to join these dinners from the time I was ten years old and discoursing with editors from the _New York Times_ , the New York police commissioner, nonprofit founders, Ivy League academics, leading civil rights lawyers, corporate executives, politicians, and more.\n\nDeeper mentoring and hands-on learning and leadership opportunities came to me through specialized camps, workplace internships in my teenage years, and summer and vacation jobs arranged by my parents. For my senior project in high school I worked for a prominent New York planning firm and studied the walking patterns of visitors to the nine principal museums located between the Museum of the City of New York on 105th Street and Fifth Avenue and the Metropolitan Museum of Art on 82nd Street. I wrote a report that helped make the case for designating that sparkling stretch of Fifth Avenue as Museum Mile.\n\nThese connection points with talented professionals were critical to my upbringing. But it's one thing for successful adults to indulge in conversation with the son of a friend at a party\u2014or even to take him in as an intern. My idea for Citizen Schools was different. I wanted adults to turn their life skills into deliberate hands-on apprenticeships and to share these lessons with their own children and neighbors but also with children they don't know, and to show up consistently for kids from a different side of town, carrying a different outlook on the world, and maybe speaking a different language at home. Would busy professionals do that? And would kids give up their afternoons or weekends to spend time with professionals who knew little about teaching, and, in many cases, even less about the neighborhoods where the kids were growing up? Finally, even if the kids and adults came together and liked it, would it make a difference in their lives? Would it change the equation? Or would it provide only a fleeting feel-good experience; or worse, a mirage\u2014enticing children and adults alike to chase after a more beloved future, but ultimately leaving them parched and unsatisfied?\n\nIn September 1994 I volunteered in Margie Tkacik's fifth-grade classroom at the Dever Elementary School, located a quick walk from Boston Harbor and across the street from the massive Harbor Point mixed-income residential community. With the help of almost $200 million in federal financing and subsidies, Harbor Point had risen to replace the notorious Columbia Point housing development, where in the 1970s and early '80s drug dealers reigned, two-thirds of apartments were vacant, and ambulances on several occasions refused to offer services because of assaults on their staff. By any measure, Dever was still a struggling urban school. Test scores were low, and many students suffered from the grinding poverty reminiscent of Columbia Point. But Dever also included a growing number of working-class families, and Principal Nydia Mendez, then in her third year at the helm, imbued the school with moxie and a measure of optimism. Nydia offered me the chance to test the Citizen Schools concept and to serve as my own guinea pig\u2014our first volunteer \"citizen teacher.\"\n\nMargie Tkacik presided over Room 202 with boundless energy: part loving aunt filled with empathy and compassion for the children, and part mad scientist full of wonder and provocative questions. She was a veteran teacher and a former Golden Apple award winner (an award given to the district's top teachers). She lived in nearby South Boston, and her husband ran the bar across the street from the teachers' union office where she and colleagues would gather for a few drinks on Fridays after a long week of work. On my first day in Margie's class, I marveled as she circled the classroom like an observant shepherd, sharing a word of encouragement here, a new assignment there, and, as needed, a nip of discipline. After college I had worked as a journalist, and Margie and I agreed that for two hours a week I would work with ten of her students to publish a newspaper. This would give me a chance to pilot Citizen Schools, and Margie a few hours a week to work more intensively with the remainder of her class.\n\nI remember nervously making last-minute adjustments to my lesson plans as I walked to my first class from the train station, the imposing _Boston Globe_ newsroom behind me and the Dever fifth graders ahead. When I entered the classroom I handed each student a narrow, spiral-bound reporter's notebook and held a \"press conference\" in which the students asked me questions to learn the design of the class. That part of the session worked well. Then I tried a brief lecture on the Five Ws of journalism (What, When, Who, Where, and Why), and found the students slumping in their chairs, disengaged despite my best efforts to interest them. A flop! By week three, I recruited a young co-teacher from City Year, the national service program I had previously worked for, and the class settled into a productive routine that maximized learning by doing and minimized teacher talking. We started each class with students conducting phone or in-person interviews\u2014of the principal, local civic leaders, teachers, and even fellow students. Then each student had some writing time, during which my co-teacher and I would circulate and make suggestions. I would then offer a five-minute group lesson on a writing-related topic and, to close, we would pair up for twenty minutes of peer editing. I wasn't a great teacher, but the kids loved the class. I was a real editor, and they became real journalists.\n\nThe kids weren't my students; they were my apprentices, learning the tricks of a real trade by doing it. Every child in the apprenticeship wrote at least two articles for our paper, which we called the _Dever Community News._ They edited one another's work and were edited by me too. Our paper included comics, a crossword puzzle, a horoscope, and dozens of news and feature articles\u2014including a front-page feature by eleven-year-old Nick Earner about life growing up in South Boston's Old Colony public housing development.\n\n\"One day I witnessed a man on the roof of my house with a gun,\" wrote Nick. \"He almost took his life because he had his children taken away by DSS [the Massachusetts Department of Social Services].\" Nick continued with a startling and eminently publishable piece of citizen journalism, ending with a description of his motivation for sharing the story. \"What made me write this article,\" Nick shared, \"was about three weeks ago two African American men were almost beaten to death for no good reason by five teenagers. This upset me because I never thought that was supposed to happen in this world.\"\n\nNick's article described a world unfamiliar to most eleven-year-olds, and familiar to most adults only through books or TV shows, or not at all. I worried for him. But I also appreciated Nick's moral compass and desire to tell his story, and as he and I worked through successive drafts of his writing, we developed a close bond.\n\nTo pay for printing the paper, the students sold $400 worth of advertisements to local businesses. Then one day in our last week together we all piled into a rented van and drove to a printer in Chelsea, a small city across the Mystic River from Boston. We watched in awe as our creation flew off a huge old printing press that folded the papers, stacked them in bunches of five hundred, and wrapped them tightly in plastic twine. When we got back to Dever, the kids walked a little taller as they distributed their newspapers to classmates and teachers. I like to think that their writing improved too\u2014not just for this project but for all their classes.\n\nAs for me, I was hooked. And Citizen Schools was born.\n\nIn the almost two decades since we published the _Dever Community News,_ Citizen Schools has blossomed into a national movement serving more than six thousand students annually from coast to coast and providing exuberant, college-bound proof that low-income kids can learn at high levels. We've shown that some of the nation's lowest-performing schools\u2014schools in places like East Harlem and East Oakland\u2014can, with help and partnership, provide their students with the same experiences as suburban kids and thereby deliver suburb-worthy results. We've been honored by the White House for our work eliminating achievement gaps in academic performance, high school graduation rates, and college enrollment; we've been profiled on the evening news; and we've begun to lead a movement to expand the learning day and to eventually bring millions of caring, talented adults together with children to improve education in America. Of course, we've made our share of mistakes too, and learned some humbling lessons along the way. Tragically, even as Citizen Schools and dozens of other strong educational interventions have grown, so has the achievement gap between rich and poor. Today we are helping thousands of children beat the odds. But we aren't changing the odds\u2014at least not yet.\n\nCitizen Schools illuminates a story of opportunity, of accomplishment, and of citizen power in America. At a time when many educational leaders are looking for new, silver-bullet shortcuts to educational excellence\u2014things like software-driven learning, vouchers, merit pay for effective teachers, and more\u2014I believe the best solution is human-centered, rooted in American tradition, and, as I hope to show in this book, achievable. To be fair, to build our economy, and to recommit to the American ideal of equal opportunity, we need to provide lower- and moderate-income children with the following five building blocks of opportunity:\n\n1. More mentorship by caring and professionally successful adults\n\n2. A longer learning day, allowing more time to master the academic basics and chances to participate in sports, art, and music as part of a well-rounded education\n\n3. More chances to practice creativity and innovation\u2014critical skills in the modern economy\n\n4. More chances to build the social networks and social skills that help drive professional and life success\n\n5. Better support of full-time teachers and parents, the primary caregivers for lower- and moderate-income children\n\nWe need people like Deb Daccord, a lawyer who for eight years in the prime of her career helped prepare middle school students, dressed in their Sunday best, to argue mock trials in front of federal judges; people like David Mantus, the son of a rocket scientist, who seven times has taught Boston sixth graders like Adam Barriga how to launch rockets and do simulated lunar landings, working with real astronauts from NASA; and people like Alan Su, a Google engineer who for the last four autumns has taught sixth graders how to create new Android apps with a community purpose.\n\nDaccord, Mantus, and Su have advanced degrees and prominent careers, and we need people like them if we want to equalize opportunity. We also need talented artists and tradesmen, like Joel Bennett, a carpenter, who taught Citizen Schools apprentices important lessons about math, and about life; and we need grandmothers like Earline Shearer, who was sad she hadn't passed on her love of sewing to her own children and grandchildren but through Citizen Schools found a way to teach her craft to other people's children. The stories of Earline Shearer and Joel Bennett and Alan Su are inspiring stories of noble intentions. But these volunteers are not just do-gooders. They are do-gooders with grit who are achieving measurable results in the lives of real children and showing a pathway to strengthen America.\n\nThe narrative thread for this book is Citizen Schools, and my own journey as its cofounder and leader, trying to reimagine the who, when, what, where, and how of education. In the coming pages I will first tell my childhood story and that of my wife (who grew up with a very different background and much less privilege). My goal in these opening chapters is to unpack and better understand the building blocks of professional success that have elevated generations of Americans. When we say a child is privileged, what mix of experiences, connections, and extra supports are we talking about? And how have millions of poor and working-class kids gained access to similar building blocks in the past? The heart of the book is the story of Citizen Schools, and the stories of children we serve, who were born with little money and few connections but by virtue of repeated positive experiences with professionally successful adults are now catapulting their way into good colleges and good careers.\n\nThese questions are explored against the backdrop of an increasingly contentious and urgent national debate about how to deliver the education our children need.\n\nIn just one generation, the United States has dropped from first in the world in college graduates to sixteenth. In science and math proficiency\u2014subjects required for the fastest-growing occupations\u2014we rank twenty-first and twenty-sixth, respectively, among the world's industrialized nations. For every future engineer graduating from a US college, ten graduate in China. These gaps threaten our economy, because we live in a \"flat world\" in which many of the best knowledge-based jobs are mobile and will increasingly go to places with the most educated workers.\n\nThis is bad news. But the really bad news is the growing class-based achievement gap within America. While the achievement gap between whites and blacks has narrowed over the last two generations, an important step on an incomplete journey, achievement gaps based on parental income levels have doubled since World War II and continue to widen, as is powerfully documented in _Whither Opportunity?_ , a compendium of recent scholarship pulled together by educational economists Greg Duncan and Richard Murnane. Forty years ago, 40 percent of upper-income Americans earned a four-year college degree by their mid-twenties, while 6 percent of lower-income Americans did. Now it's 8 percent for low-income students and a fast-growing 73 percent for their upper-income peers. Upper-income students who are academically low-performing (on the SAT and in their course grades) are as likely to earn a college degree as lower-income but academically high-performing students.\n\nAmerican capitalism and our democracy rest on the idea that differential results for individuals are one part luck and two parts effort and ingenuity. If that equation flips and career success becomes only a little about differential effort and talent and mostly about what zip code you were lucky enough to be born into, then we've lost the American identity.\n\nSo what's causing these growing gaps? As mentioned, it's not that the academic skills of poor children are slipping. Actually, they are growing modestly. The class-based education gap is accelerating because children from upper-middle-class families are accelerating rapidly, benefiting from an awesome web of support and growth opportunities, including many I received. This is the opportunity equation. In the 1970s, low-income families spent $835 per year on out-of-school enrichment for their children\u2014things like music lessons, tutoring, summer camp, and participation on travel sports teams\u2014while upper-middle-class families spent $3,536. By 2006, these investments, adjusted for inflation, had grown to $1,315 for lower-income families and $8,872 for upper-income families, nearly a tripling of the investment gap\n\nUpper-income children are also bathed in the most precious resource of all: their parents' time. A generation ago, lower-income children spent more time with their parents than upper-income kids did. Now it has flipped, and wealthier parents are spending four hours more per week with their children: reading to them, talking at the dinner table, driving them to sporting events and music lessons, and thereby packing their suitcases for a lifetime of success.\n\nClass-based achievement gaps are growing at a time when education drives access to career success more than ever before, meaning the gaps are more consequential. Forty years ago, college graduates earned only about 20 percent more than high school graduates over the course of their lifetimes. Good jobs requiring little education were plentiful, and those with a modest education could work their way up. But most of those low-skilled and high-wage jobs have gone away, because they consisted of tasks that were easy to computerize, since they involved following simple rules, or to outsource. Now high-wage jobs require workers to use higher-level thinking skills, to communicate well, to work well on diverse teams, and to solve problems where no clear rules apply. These are the skills college grads generally have, and this explains why they now earn almost twice as much as their less-educated peers, a quadrupling of the college\u2013high school wage gap.\n\nIt's a vicious cycle: Children of parents with money race further ahead, buoyed by tutoring, paid enrichment, lessons, and formal and informal mentoring. And the resulting educational gains reap bigger economic gains. On the surface it looks like we live in a more meritocratic world\u2014one in which what you learn is what you earn. Upper-income families have figured this out and invest accordingly. Lower-income families are working longer hours but lack the money and time and social networks to provide the enrichment and extra learning that is taken for granted by the children of their wealthier peers. The result is a world that is actually less meritocratic and in which our society loses out on the rich cultural and economic benefits that would flow to all of us if more children had a chance to fully develop and nurture their talents.\n\nThere is a raging debate among education reformers in which, to simplify, one side says the achievement gap is caused by poverty and can't be fixed by schools without fixing poverty, and the other side says poverty is no excuse and that bad schools and bad teachers cause most of the growing gaps between rich and poor students. This is an unproductive debate in part because it confuses causes and solutions. Based on my own experience and the available research, the cause of America's class-based achievement gap is about one part school\u2014and three parts out-of-school factors. The part that is school is a difference between teacher quality and expectations in upper- and lower-income public schools. Schools serving upper-income children have higher expectations and more \"highly qualified\" teachers, in large part because teachers in those schools have more support. We need to change this, as various current reform efforts are endeavoring to do. But exhaustive studies by educational researchers Joseph Altonji and Richard Mansfield and others indicate that only 20 to 30 percent of the achievement gap is driven by these school-related factors. As an example, they find that moving low-income students from a school and associated community at the tenth percentile of quality (as measured by test scores and graduation rates) to a school at the ninetieth percentile increases the likelihood students will graduate high school on time by only eight to ten percentage points, leaving in place a thirty-percentage-point gap.\n\nThe really big difference for kids is that upper-income families spend seven times more money supporting student learning _out of school._ And because upper-income families live in communities populated by professionally successful people, upper-income children are naturally exposed to more successful adults, starting with their parents. Success is modeled for them from birth through high school, college, and beyond. That's what drives the achievement gap.\n\nMy own community of Brookline, Massachusetts, illustrates the point. Like hundreds of families before us, we moved to our neighborhood because of the William H. Lincoln elementary school. In the 1960s and '70s the Lincoln had been dubbed \"Stinkin' Lincoln\" due to its older, dilapidated classrooms and because its catchment area, while dominated by gracious homes and tree-lined streets, includes Whiskey Point, Brookline's lowest-income section and home to modest triple-deckers and hundreds of units of public housing. But in the 1980s, in an effort to upgrade the Lincoln for all its students and also to retain more wealthy families (many had left for private schools), Brookline built a new Lincoln school, supported by increased taxes and a special private fund-raising campaign, and designed by acclaimed architect Graham Gund. An energetic principal, Barbara Shea, helped make it one of the highest-performing schools in the state despite the challenges faced by some of its students.\n\nToday the Lincoln serves 570 students, hailing from thirty-one countries, in kindergarten through eighth grade. Half of Lincoln's students are nonwhite, and a full 25 percent of the Lincoln students are low-income and eligible for free or reduced-price lunch, but the school community also includes hedge fund managers, architects, and academics. My wife, Maureen, and I have gotten deeply engaged in the school community, coaching teams, supporting field trips, raising money for after-school programs, serving on the school advisory council, and more. We love the school. But, like our upper-income neighbors, we've also signed up our kids for private tutoring and music and sports lessons, and we provide hundreds of hours of extra academic coaching at home. The investments pay a large return, and as our kids enter Brookline High School they are moving ahead of most of their lower-income classmates\u2014even though they attend the same excellent public schools and live in the same neighborhood.\n\nIf upper-middle-class families want to know who is causing the achievement gap, we should look in the mirror. It is us. We go to the ends of the earth to help our kids succeed. Our kids deserve it. But so do all kids.\n\n# SECTION ONE\n\n# STARTING CITIZEN SCHOOLS\n\n# CHAPTER ONE\n\n# BUILDING BLOCKS\n\nFamily lore has it that my first words were _nani, nani_ (\"more, more\"), spoken in a northern Nigerian dialect used by just sixty thousand people in the region of Kaduna. My parents had brought me to Africa in 1961 at the age of nine months. My father, scion of the FAO Schwarz toy store family, was fresh out of Harvard Law School. He was serving through the MIT Fellows in Africa program as Assistant Commissioner for Law Revision for the Northern Region of the newly independent Nigerian republic. My mother, a recent graduate of Radcliffe College and a member of a large Boston Brahmin family, had signed up to teach a high school course in African history. I was along for the adventure.\n\nI have no clear memories of my year in Africa. But from the photographs I grew up with and the stories I have heard, the trip foreshadowed my childhood with its sense of adventure and its extraordinary exposure to stimulating people and places.\n\nAfter living and working in Africa for fourteen months, my family settled on the rapidly gentrifying Upper West Side of Manhattan. My father joined a large law firm, Cravath, Swaine & Moore, and stayed active in civic causes. My mother had a second and then a third child (my sisters, Adair and Eliza) and began postgraduate studies at Columbia University, eventually earning a PhD in linguistics.\n\nOur turf on the Upper West Side stretched from Broadway to Riverside Park and from 100th to 106th Streets. It featured turn-of-the-century brownstones, rent-controlled apartment buildings, and subsidized housing\u2014a chock-a-block mix that supported a diverse population of professionals, artisans, blue-collar workers, small-business owners, and families on welfare. This was before the crack epidemic and resultant crime spree of the early 1980s, and my parents, and therefore I, had virtually no fear of crime. Starting at age six, I had permission to play handball in front of our building, to walk unescorted to Riverside Park for games of baseball and kick the can, or to head over to Broadway, where for a single quarter my friends and I could pick up a slice of pizza. A nearby bodega often had chickens running around the back part of the shop and was run by a friendly family from Puerto Rico.\n\nAs I approached nursery-school age, my parents and our Upper West Side clan of cousins and friends decided the educational options available to us weren't good enough. So they started their own school\u2014the West Side Montessori School, patterned after the hands-on, self-directed learning approach developed by famed Italian educator \u00adMaria Montessori. It was the first school in the city to include both low-income students with public subsidies and full-paying families like ours. In 1966, at age five, I was one of nine children in the school's first graduating class. I still remember one assignment\u2014a number line in which I counted from 1 to 4,387, running the numbers down successive strips of yellow paper that I would tape together. I am not sure this deeply improved my math skills. But when I was done with the assignment I had a number line that stretched all the way around our living room two times. I was proud and felt I had earned a badge of success.\n\nThe following fall I headed as a first grader to the Collegiate School, which was founded by the Dutch East India Company in 1628 and has offered a rigorous\u2014and increasingly prestigious\u2014all-boys education ever since. Two years later, on the first day of third grade, our teachers pulled us together to let us know that we would be joined by a new classmate, John Kennedy, son of the late president. John added unmistakable star power and a Secret Service detail to our already well-heeled class. His mother, Jackie O, was a fixture on the sidelines of our soccer games and, years later, sponsored a memorable eighth-grade class trip to Boston.\n\nUp to the end of third grade my childhood progressed smoothly. I made friends easily enough; showed athletic promise in several sports, including basketball, which was my favorite; and demonstrated aptitude in school even if it was sometimes wrapped in a daydreamy fog.\n\nIn fourth grade, however, things took a turn for the worse. The best I can recall is that the trigger was a bad match with my homeroom teacher, Ms. Goldberg, though clearly more was going on. My teachers to that point had been warm, maternal figures who softened the authoritarian feel of Collegiate, with its giant stained-glass windows and its centuries of tradition and coat-and-tie uniforms. But Ms. Goldberg was stern and British, and didn't take kindly to my daydreamy ways. We got off to a bad start when she said I was spelling my name wrong (actually _she_ was, as our \"Schwarz\" contains no _t_ ), and things spiraled downhill from there. School started feeling like a hostile environment, and I started feeling unsuccessful. I stopped doing my homework, earned mediocre to poor grades, and began an academic slide that continued for many years. When the final bell rang on the last day of fourth grade and we were dismissed, I jumped up and literally danced a jig as I raced out of the school.\n\nI have faint memories of school in the ensuing middle-grade years. But I remember my out-of-school time in full, living color. I typically left Collegiate at four or four thirty in the afternoon (after an eight-plus-hour day) and walked a few blocks down Broadway to H&H Bagels, with its large bins of freshly cooked sesame, salt, poppy, onion, and \"everything\" bagels. I would buy two of whatever was warmest and then eat the bagels plain as I headed home on the Broadway 24 bus. When I got home, I would be greeted by a live-in au pair and, usually, my two younger sisters. I'd inhale two bowls of cereal or a couple of grapefruits, and then play basketball in my room or at a park that was three blocks away. Mom would get home in time to make dinner, and Dad would usually arrive a little later in the evening.\n\nOddly missing from these evenings was structured homework time\u2014a staple, I am sure, for most of my classmates then and for my children and their peers today. I think my parents just assumed homework was something that kids did on their own.\n\nIn middle school my challenges worsened and occasioned lots of special meetings with my teachers and parents. I hated these \"Eric is struggling\" conferences, but they continued to be held at least two or three times a year. My mother would make a special trip into school and we would meet with one or a few teachers. Sometimes a guidance counselor or the principal would join us as well. I would hear about all of the assignments I hadn't completed and the potential I wasn't fulfilling. Then I would make a halfhearted promise to improve, and the meeting would be over. Sometimes I would study harder for a few weeks. My parents might do a homework assignment with me elbow to elbow. But I had negative momentum. I was unengaged in school, disorganized, and ashamed. And the meetings continued. I remember that after one particularly difficult conference in sixth grade, my French teacher looked sadly at me and said, simply, \"Your poor mother.\"\n\nCompounding my academic disengagement in these middle-grade years, I also pulled back socially, maintaining a few close friendships but living far from the center of my school's social scene. Once in high school, with a deepening feeling of alienation, I started smoking marijuana daily, drinking heavily on weekends, and experimenting with other drugs. I attended three private high schools in three states, never getting officially expelled but earning multiple suspensions and mostly poor grades. It wasn't a pretty picture.\n\nAmazingly, throughout these difficult years I continued to have positive learning experiences. I was an obsessive fan of my hometown sports teams, in particular the Yankees in baseball and Knicks in basketball, and I digested the statistics of my favorite players so thoroughly that I developed an advanced number sense and statistical acumen far out of proportion to my academic standing or hours of formal study. I attended top-flight private schools. And while I certainly didn't maximize my opportunities, the schools offered rigorous academics and a rich sampling of creative arts, sports, and various extracurricular clubs. Some of that learning did stick.\n\nI attended several sleep-away summer camps as well, which provided important learning experiences. At Tamarack Tennis Camp in the White Mountains of New Hampshire, the mosquitoes were so large they gave me cauliflower ear\u2014no doubt from a combination of numerous bites and the cuffs I administered to my ears in a futile effort to squash the maddening bugs; but the camp's counselors were fun-loving and full of praise. And camp proprietor Jack Kenney\u2014the grandfather of skiing legend Bode Miller, who was homeschooled on the Tamarack grounds\u2014was an innovator in tennis instruction. Kenney used state-of-the-art photography to give campers an eight-photo depiction of their serving motion. And he used low-tech teaching tools too, like an adjustable upside-down bucket that hung high above the courts and allowed campers to practice the perfect toss for a first or second serve. Kenney and his team built up my skills in tennis, helping me at age thirteen develop the muscle memory for solid ground strokes and a powerful serve, which on a good day I can still call up four decades later. Even more important, at Tamarack I developed the confidence that comes with getting good at something\u2014anything. That summer I won awards for proficiency and improvement, as well as the coveted camp-wide award for sportsmanship. As I collected the last of these at the camp's closing ceremony, camp owner Kenney cracked that with all of the trophies I'd won I could open a hardware store. My heart was jumping out of my chest with joy.\n\nTwo summers later I transitioned from attending camp to helping run a camp. I was fifteen, and I got support from my parents to run a small day camp for younger cousins and neighbors at our country home, one hour from New York City. My sister Adair and I were two of the three counselors, and for two weeks we offered a rotation of tennis, swimming, and horseback riding to nine young neighbors and relatives. The camp was a success, perhaps even an opening to my later career at Citizen Schools. But there is no way I would have had the experience of \"running\" that camp without tremendous support from my parents. It wasn't just that they provided the tennis court, the pool, and two horses\u2014critical ingredients, to be sure. They also recruited the campers! It was left to us to design and lead four hours of daily activities, to prepare a daily snack, and to communicate with the campers and their parents once the camp opened.\n\nEducation scholars such as University of Pittsburgh professor Lauren Resnick would call the experience I had that summer an apprenticeship, or\u2014to update the image from that of a young boy in a blacksmith's or printer's shop\u2014a \"cognitive apprenticeship.\" What my parents provided is called scaffolding, and it is a part of a sequential four-stage learning process as old as humankind\u2014though sadly missing from many schools and childhoods today. Apprenticeships start with the expert (this can be a parent, a more experienced colleague, or a master craftsman) _modeling_ success; then comes _scaffolding_ (support to the apprentice\u2014like the ladder that helps a painter reach the second floor, or the initial recruiting that helped get our camp off the ground); then _coaching_ as the apprentice begins to perform the tasks; and, finally, _fading_ as the apprentice begins to lead.\n\nWhether it is Ben Franklin apprenticing to a Boston printer, a young lawyer beginning her career by clerking with a senior judge, or a medical resident on rotation with an experienced surgeon, not-yet-successful people learn through apprenticeships with already successful people. For me, I had lots of these chances. Before I turned twenty, I completed the summer-camp apprenticeship and then additional unpaid internships\u2014or apprenticeships\u2014with prominent lawyers, city planners, and a US senator.\n\nDespite these burgeoning opportunities, my teenage years continued to be difficult. I made lots of mistakes. But when I made them, I got help. For instance, when I drank too much on an early-spring night in 1978 (I was seventeen) and was chased by police officers in a paddy wagon through the Boston Common, arrested, and taken to jail, I was bailed out a few hours later by family friend Gil Burke. Like my father, Gil was a Harvard-trained lawyer. In retrospect, that one drunken night could have changed the trajectory of my life. Lesser mistakes have ruined the lives of other children. It could easily have been three days in jail, not three hours. It could have been a permanent arrest record, not a quick release. It could have been a source of deep family hardship, not a bachelor-party story, as it became for me.\n\nThe great psychologist Erik Erikson wrote that all adolescents should have the chance for a period of _Wandenchaft_ , or wanderlust, in their later teen years, when they are free to experience the world and to make mistakes as they find their footing and discover new truths about their souls. I had this time, and for me it was one of life's greatest gifts. I wasn't a bad kid as an adolescent. I was just lost and trying to figure things out.\n\nInstead of my three-hour visit with the Boston Police Department, it was my 1980 internship with US senator Gary Hart of Colorado that became the turning point in my life. The opportunity arose for two reasons: my dad knew Gary Hart, and I was dating a girl in Colorado.\n\nIn early June I arrived in Denver in my beat-up Chevy Chevette and made my way to campaign headquarters, which was stuffed into a three-story Victorian on High Street in one of Denver's nicer residential neighborhoods. I tucked in my shirt, rang the doorbell, and was greeted by Mary Cromer, the plainspoken office manager whom I had been speaking with by phone as I drove across the country and who had lined up a basement room for me to sleep in. After getting acquainted for a few minutes, Mary gave me my first campaign assignment: \"There's a lawn mower in the garage out back and the lawn needs to be mowed,\" she said. \"Okay,\" I said, and I headed out back, suddenly a little unsure what I was getting myself into. When the lawn was done I spent four long days counting pennies, nickels, dimes, and quarters collected at a recent state convention in coffee cans covered in HART FOR \u00adSENATE bumper stickers. I settled into an almost Zen meditative state as I counted out piles of coins and stuffed them fifty or twenty-five at a time into rust-colored paper rolls and then crimped the ends and stacked the rolls in small cardboard boxes. I didn't complain about the menial work, but I had been taught to ask for more responsibility, so I did. In my second week, the campaign manager said I could work as a ghostwriter, crafting letters to the editor for in-state campaign volunteers to sign and submit. Then, with an endorsement from Cromer, I graduated to position papers and was assigned to draft a campaign brochure on small business.\n\nLater that summer, I helped organize benefit rock concerts performed by Linda Ronstadt, Jimmy Buffett, and Stephen Stills\u2014all of them recruited to the cause by Hart's good friend Warren Beatty. I was just nineteen, and the concerts were an incredible thrill for me, and lucrative for the campaign. In retrospect, they were an uncannily perfect bridge between a teenage desire to attend as many pot-infused rock concerts as possible and a young-adult opportunity to manage a six-figure enterprise for the reelection campaign of a United States senator. The entire campaign experience, as well as the mentors I got to work with, was incredible. I had grown up _around_ power, but now, as through transference, I was working _with_ power. I called my parents and my college, the University of Vermont, and asked to withdraw from the fall semester of my sophomore year so I could continue working through Election Day in November.\n\nThat fall I organized college students across Colorado and discovered I had a talent for mobilizing people behind a cause. On Election Day I led a statewide network of hundreds of student volunteers from a borrowed Boulder dorm room. We stationed volunteers at college polling places across the state with lists in triplicate of all the registered Democrats in those precincts. At midmorning, midafternoon, and dinnertime, we checked to see who had voted and then dispatched students to dorms, libraries, and cafeterias to search for other students who still needed to exercise their franchise. Of the fifteen-hundred-some students living on campus at the University of Colorado at Boulder and registered to vote as Democrats, all but three turned out to vote that day. Students also came out strong in Colorado Springs, Fort Collins, Greeley, and in small college towns across the western slope of the Rockies. Senator Hart won by fewer than twenty thousand votes, withstanding Ronald Reagan's landslide victory over President Jimmy Carter and positioning himself for a presidential run of his own in 1984.\n\nI came back to the East Coast with a new sense of purpose and pride. I had done real work with real responsibility and had a chance to work with numerous caring and experienced professionals who invested their time in my development. By January of 1983, as I began my last semester of college, I was working full-time for Senator Hart's presidential campaign. I served as his national student director and worked closely with the youthful codirectors of his campaign, Bill Shore and Kathy Bushkin, and with the savvy Jeanne Shaheen (now a US senator), who ran our campaign in New Hampshire. I helped to lead our efforts in Vermont and mobilized thousands of students to volunteer in the crucial early caucus and primary states of Iowa and New Hampshire.\n\nIn the winter of 1983, with our campaign in last place in most polls and almost out of money, I organized 104 simultaneous press conferences announcing \"Students for Hart\" chapters on major college campuses across the country. My method was straightforward. Working from a short list of students who had contacted the campaign, and from a longer list of College Democrats chapter heads, I called people and asked if they would gather a few friends and hold a simple press conference for their college and local papers, announcing the formation of a Students for Hart chapter. If those I spoke with showed any interest, I would immediately send them an information packet and ask who they knew on other campuses that might be a prospect to join the effort. I would follow up with a call every few days. I began my phone calls after dinner at six or seven o'clock and often ended for the night when I started waking people on the West Coast\u2014at two or three in the morning, my time. The hard work paid off, and the Students for Hart event was a success, earning a positive article in _Newsweek_ magazine, building momentum that helped establish Hart as the youth candidate and earning me a reputation as a skilled campaign organizer. A year later, young people helped drive Hart's surprise second-place finish in Iowa and his upset victory over front-runner Walter Mondale in the 1984 New Hampshire primary.\n\nMy experience as Hart's national student director in 1983 and 1984 was my first job out of college. It taught me I could be successful on an adult stage, and it remains an early source of my conviction that citizen power, properly mobilized, can change the world.\n\nMy parents provided the initial connection to Gary Hart. And I was ready, finally, to take advantage of the opportunity because of the caring, the connections, and the powerful example they had provided much earlier. They were weak on supervising homework and in a few other areas, but overall both my parents were an inspiration. I love them deeply.\n\nAt six feet, one inch tall, my mother, Marian Lapsley Cross (she is now remarried), is an imposing presence. Often my friends and those of my two younger sisters found her intimidating. Mom was descended from Quaker activists and New England businessmen and, like my father, was big on long walks, conversations about social policy, and Yankee asceticism. We made do with beat-up old cars, leaky toilets and faucets, moth-rampaged blankets, and rumpled clothing. My dad, Fritz Schwarz, was a successful corporate trial lawyer whose career included defending IBM from monopoly charges, representing Polaroid founder Edwin Land, and winning access to Major League Baseball locker rooms for Melissa Ludtke, a female reporter for _Sports Illustrated._ We had plenty of money, attended private schools, owned a country house, and enjoyed long vacations in Europe, Africa, and the American West. But we also cut our own lawn, often wore clothes until they were threadbare, and usually skipped dessert. If Dad really wanted to mark a milestone, like a new tooth or an athletic or academic award (I didn't have many of those), there was no fancy dinner out or trip to the movies. Instead, he would offer, in his words, \"a really big prize\u2014a home-cooked spaghetti dinner.\"\n\nIn retrospect, our selective frugality was a little eccentric, given that Dad's salary put us in the upper echelon of the top 1 percent of earners in the nation. But my parents' work ethic and grit and populist outlook made a mark on me for which I will always be grateful. Dad had a hard time showing affection or emotion, but he saw the best in me and other people and looked to draw that out. In addition to his positive outlook, Dad's ferocious competitive spirit left a deep imprint on me. One night Dad took me to an office softball game played under the lights in lower Manhattan. He was in right field, one of only two lawyers on a team of younger, burlier moving men, messengers, and clerks from the firm. Late in the game a towering fly ball was hit near him. Initially he misjudged it, racing in and toward the foul line when the ball was actually hit deep in the right-center-field gap. Then he reversed course at a sprint and launched himself with a full dive, catching the ball backhanded as he skidded along the patchy grass and then jogged in toward the dugout with a broad grin that was both sheepish and proud. On another night\u2014a Sunday night in the dead of winter\u2014Dad excused himself after dinner and went outside in his winter coat. I went to look for him thirty minutes later and found him a hundred feet from the house in the bitter cold, practicing a speech that he needed to give the next day to a few hundred lawyers working with him on a big case.\n\nYears later, Dad's Harvard college friend, Dan Morgan, summed up to me why he loved Harvard by saying that it exposed him to people like my father. \"Your father had little athletic talent,\" Dan told me, \"yet he managed to become stroke of the Harvard crew just through sheer effort and determination, and he has applied that determination and drive to his entire life. As a kid who grew up on a farm in upstate New York, being around people like that was an incredible education.\"\n\nWhile I have come to deeply appreciate the drive my father passed on to me, I suspect his competitive intensity and his towering professional success intimidated me as a child. I was the oldest and only son and the oldest grandchild on both sides of the family. Expectations were high. It felt as if, as a ten-year-old, I had been placed on a powerful stallion of ambition that I lacked the skill and interest to ride.\n\nMom was my partner on all kinds of cool projects, from baking bread, to building a dollhouse for my sister, to raising and killing chickens, to learning macram\u00e9. But I also knew her as a ninth-grade English teacher at Benjamin Franklin High School, on 116th Street overlooking the East River in East Harlem. My private school had a different vacation schedule than Mom's public school, so I often had a chance to visit her classroom. Her students were only a few years older than me, and I was inspired by her ability to engage them. Once on the first day of school she assigned them to paint their dingy classroom bright yellow, handing out painter's tape and rollers to her shocked students. The students called Mom \"Slim\" and seemed to have a deep affection for her as a teacher who was curious about their lives and concerned about their futures. One day I walked across East Harlem with the whole class as they started to put together an end-of-year musical slide show that documented the school's environs. We must have made an unusual sight\u2014a tall white lady walking at the front and a few dozen African American and Puerto Rican fourteen-and fifteen-year-olds, many holding cameras, along with one long-haired blond boy trailing behind. To me it didn't feel odd at all. Instead it felt like a big, happy family adventure.\n\nA larger-than-life figure in my childhood was my paternal grandfather, Fritz Schwarz, who we called Grampy. Grampy was the managing partner of the white-shoe Wall Street law firm Davis, Polk & Wardwell and for many years the chairman of FAO Schwarz, the iconic toy store founded by his grandfather\u2014my great-great-grandfather\u2014in 1862.\n\nThe original Frederick August Otto Schwarz came to America in 1856 with his three brothers, Gustav, Richard, and Henry, after fleeing the Westphalia region of Germany because of political instability. The brothers settled in Baltimore and Frederick got a job as a clerk at a local stationery shop. In time he won the confidence of the storeowner and asked if during the Christmas season he could display and sell toys made by friends and family members from Germany and Austria. The storeowner agreed, the toys were a hit, and soon thereafter Frederick and his brothers struck out on their own and opened the Schwarz Toy Bazaar. The store grew modestly in Baltimore and then, after a devastating fire in 1868, relocated to New York City as FAO Schwarz: Purveyor of Fine Toys.\n\nBy the time of my grandfather's birth in 1902, FAO Schwarz had become, according to catalogs of the day, the largest and best-known toy store in the world, with branches in several other prominent cities, including Boston, Philadelphia, and San Francisco. It was known then (and still) for its larger-than-life stuffed animals and one-of-a-kind dolls and toys. Many of the toys, like the legendary Steiff teddy bears, were imported exclusively from small toy makers in Europe.\n\nIn the early 1900s Henry Schwarz, the son of the founder (and my great-grandfather), took over management of the store. He had just a high school education, but Henry grew the store's catalog business and relocated the flagship store uptown to Fifth Avenue near its current location on Fifty-Eighth Street. The family lived in Greenwich, Connecticut, on a large estate that is now home to the Greenwich Country Day School. According to family lore, at the outset of World War I the family purchased an iron canon to ward off any neighbors who might resent their German heritage.\n\nMy grandfather, who like my father went by the nickname Fritz, was educated at the Hill School in Pennsylvania and at Harvard College, where he was president of the student newspaper, the _Crimson._ After his graduation in 1924 he enrolled directly in Harvard Law School and earned a spot on the law review. During my grandfather's third and final year of law school, his father fell ill and died suddenly. Not yet twenty-five, Grampy assumed oversight of the family business and commuted to New York regularly to support his mother and manage the toy business. He declined a prestigious clerkship with Supreme Court justice Oliver Wendell Holmes and ran the store through the early years of the Great Depression before assuming the role of chairman and beginning a career as a corporate lawyer.\n\nIn 1962, just as my parents and I were returning from Africa, and one hundred years after the store's founding, Grampy decided to sell the business. None of his children were interested in taking it over, and he wisely wanted to avoid the family disputes that afflict many next-generation family businesses. In the ensuing years, FAO Schwarz has been sold at least a half dozen times, most recently to Toys \"R\" Us, and has declared bankruptcy twice. The toy store remains a source of family pride, however, and I now chair a small family foundation that uses proceeds from the business to support fellowships at high-quality child-serving organizations.\n\nSociologist Bruce Feiler, in the 2013 book _The Secrets of Happy Families_ , writes that children who know a lot about their families tend to do better when they face challenges. Children with the highest self-confidence, he said, have an \"intergenerational self,\" meaning that they know they belong to something bigger than themselves.\n\nI grew up with an extra dose of intergenerational self, as well as turbocharged mentoring from my extended family and their personal and professional networks. I also grew up with a deep subconscious fear that I would fail to live up to my forebears. Metaphorically, I had been drafted by the Yankees and was about to take my turn at the plate, just after Babe Ruth, Joe DiMaggio, and Mickey Mantle had belted home runs. That fear caused anxiety that likely contributed to my adolescent struggles. But on the other hand, I was a Yankee! I was surrounded by success and could learn from the best. Much as I floundered at times, I, like my privileged peers, had a glide path to the future.\n\nAs I reflect on my childhood and early adult life, I am immensely grateful. My life was far from perfect; my parents, like me, have plenty of flaws, and I stumbled many times as I made my way forward. But every time I stumbled, I had a helping hand and a new chance. I learned the language of professional success through trial and error and through dozens of in-depth experiences with accomplished adults who took an interest in my development and helped me open doors.\n\nWhat's different for the young people I have come to know through Citizen Schools? Most also have hardworking parents who love them deeply. Many have received important support from mentors in or out of their family. But I have come to believe that the mentoring, the coaching, the networking, the experiences with success, the academic support, and the modeling of workforce opportunity available to them are rarely as deep or persistent as they need to be.\n\nPoverty creates a gravitational pull that holds people down. By contrast, wealth underwrites opportunity in obvious ways (better schools, camps, lessons, etc.) and in less obvious ways, like giving young people stories of accomplishment to internalize and the confidence and safety net to take productive risks, such as leaving jobs and creating new ones.\n\nLeaving poverty is harder today than it used to be. When my Schwarz ancestors came to America in 1856, a recent immigrant from Germany with no formal higher education could start a small business and put his son on a path to own a country estate and his grandson to graduate from Harvard Law School with an offer to clerk for a Supreme Court justice. Today in America, a smaller share of lower-income people become upper-income earners\u2014compared to previous generations in America and compared to other industrialized nations today. A 2010 report from the Organisation for Economic Co-Operation and Development (OECD) found that with respect to social mobility across generations, the United States now ranks well below many other developed nations, including France, Germany, Spain, and Canada. In my home of Boston, a child who starts life in the bottom fifth of all Americans in family income has just a 10 percent chance of reaching the top fifth. And Boston has more social mobility than most US cities. A poor child raised in Atlanta or Indianapolis, for instance, has less than a 5 percent chance of reaching the upper middle class.\n\nLeaving poverty in the United States today requires escape velocity. Just as a rocket ship needs extraordinary fuel and thrust to escape the gravitational pull of the earth, escape velocity for a kid leaving poverty requires high expectations, persistent coaching and mentoring, and lots and lots of learning experiences with successful professionals throughout childhood.\n\nThe reason that few poor children achieve escape velocity\u2014and therefore remain poor\u2014is not rocket science. It's due to basic things. Things like no adult being able to help with homework, either because the parents are working two to three jobs and aren't available, or because they are available but\u2014owing to their own bad experiences in school or their limited English proficiency\u2014don't know how to teach long division or the Pythagorean theorem. And when Citizen Schools' parents can't help because they lack the time or skill, they generally don't have the means to hire live-in babysitters or specialized tutors to fill in. Students from low-income families generally don't attend sleep-away camp, play in sports leagues, take private music lessons, or have many chances to develop the muscle memory of success. They don't get many chances to win trophies. If it weren't for Citizen Schools, the students I have come to know wouldn't have internships with politicians or lawyers, or apprenticeships with scientists and engineers, even though these fields are growing fast and need more talent. How likely is it that a child will grow up to become an engineer if she has never met one? Their mistakes, when they make them\u2014and growing up by definition includes making mistakes\u2014will likely have far more serious consequences than mine did. Further, they face an added burden of unknown weight: many of them have known hunger, homelessness, a relative who has been jailed or killed, violence against their bodies, or several of these things.\n\nBackgrounds like mine don't guarantee success. Despite initial advantages, a small number of my close friends from childhood and members of my family have encountered great hardship, including homelessness, joblessness, wrenching addiction, deep sadness, and even poverty. But the odds are overwhelmingly stacked in our favor. Similarly, it is still true in America that many exceptional young people grow up in the most difficult of circumstances and yet lay claim to the American Dream of a good education followed by good jobs, a stable family life, home ownership, and more. These children beat the odds through some combination of hard work, force of personality, happenstance, and, usually, great mentoring by determined parents, skilled teachers, and loving community members. But sadly, the rags-to-riches American Dream stories occur less often today. America's current opportunity equation dictates a tough life for most poor and working-class children\u2014the children I have come to know through Citizen Schools. It does not need to be this way. In the chapters that follow, I hope to show how we can become a better nation and properly honor America's founding legacy of opportunity by changing this equation.\n\n# CHAPTER TWO\n\n# MAUREEN\n\nMy friend Amie's Cape Cod home, built by her Ohio industrialist great-grandparents, was large and rambling and featured a sweeping view toward the Elizabeth Islands. The dining room table comfortably sat twenty-four, and the unmodernized kitchen was designed for operation by a household staff of six. One weekend I arrived late for a party Amie was hosting, and while carloads of visitors were being dispatched to secure provisions, I was put to work preparing food. Soon another guest entered the kitchen, just back from an end-of-day bike ride. Her name was Maureen and she had a smile that was at once warm, powerful, and mischievous. I was instantly drawn to her. We spoke for just a minute and then were separated as we searched endless cupboards to find twenty-four beautiful but mismatched bowls for our first course, a bountiful fish stew. When dinner was served, Maureen and I found ourselves seated at opposite ends of the endless table, though I was pleased we occasionally shared smiles across the distance.\n\nAfter the dinner plates were cleared, people took different places around the table and Maureen and I sat together for dessert. I learned she was an artist and was pursuing a college degree at Tufts University and the School of the Museum of Fine Arts. Her parents were Irish immigrants with limited formal education, and as the fifth of eight children, her upbringing, at least materially, was as different from mine as night from day. Maureen, who was then twenty-five, described leaving home when she was seventeen and making a living on her own ever since. She had attended the local vocational and technical arts high school because her junior high guidance counselor told her that based on her family background she was not college material and should learn a trade. Haircutting became her trade, and in her late teens and early twenties she made a good living working at various salons west of Boston and eventually on fashionable Newbury Street in Boston's Back Bay. Now she was attending Tufts full-time but still cutting hair to pay her rent and cover tuition and bare-bones expenses.\n\nMaureen's whole persona was as foreign to me as it was attractive, and by the end of dessert, and after several glasses of wine, we discovered that we were the only ones left in the room. Above the table we were engaged in a deep conversation about childhood, while below it our feet were engaged in a wonderful game of footsie. We took a moonlit swim together and stayed up the entire night talking until I had to leave early the next morning.\n\nMaureen and I drifted away from each other for almost a year, but throughout that time she persistently reentered my consciousness, and she always made me smile. When I would get into deep conversations with friends, or with a therapist I was seeing at the time, the conversation would turn to soul mates and I would invariably think of Maureen, the Irish woman with substance and sparkle. Almost a year after our first meeting, we had a warm reconnection at a mutual friend's birthday celebration. Then weeks later, Amie held an anniversary sequel to her Cape Cod weekend party. By the end of the weekend Maureen and I were again together. This time it stuck, and just weeks later Maureen and I went on a vacation together in Maine. I was head-over-heels in love. We moved in together that fall and were engaged to be married the following spring.\n\nAs I got to know Maureen I came to better understand her road from a subsidized-school-lunch kid and first-generation immigrant in small-town Hudson, Massachusetts, to an honors-level graduate from one of the nation's most prestigious colleges. It's in many ways a prototypical American Dream journey, but also, as discussed in the previous chapter, a road less traveled today. I wondered how it worked for Maureen.\n\nMaureen's mother, Mary, was the oldest of nine girls raised on a small dairy farm outside Galway, Ireland\u2014the daughter of a tall and tough man, John Gilmore, who worked hard and was spare in his sharing of affection. Mary's mother, Margaret, lived to ninety-nine years of age and held the family together with the help of rituals and habits common to the place and time, including daily prayers together in the kitchen and weekly mass at the local Catholic church. I met Grandma Gilmore shortly after Maureen and I got engaged and was amazed to learn that her sons-in-law visited her every afternoon for a glass of buttermilk or cup of tea and a slice of her fresh-baked brown bread or raisin scones. Their social network was strong.\n\nMaureen's father, Joe, grew up just a few miles away in the Salt\u00adhill area of Galway. When Joe was six months old, on the night before Christmas, his dad was killed by a speeding motorist while fixing his car on the side of the road. Joe's mother took to her bed with grief and didn't get out much until she herself died almost four years later. After Joe's mother died, he was raised by relatives in a household he described as devoid of affection for him and with few material comforts. For those familiar with Frank McCourt's prize-winning memoir _Angela's Ashes_ , Joe's story would strike a similar chord.\n\nIn his later teens Joe became a carpenter. He met and married Mary Gilmore, and in 1957 the young couple moved to America, settling with Joe's aunt in an apartment in Waltham, just west of Boston. Joe's first job was as a baker's assistant working for ninety-nine cents an hour. Joe and Mary didn't have a car, so he walked six miles to and from work every day. A year later Joe joined the carpenters' union, and in good times he would have steady pay, though at other times, particularly in the winter, he could go for months and months without work. In 1962 Joe and Mary bought the home they have lived in ever since\u2014a split-level ranch in Hudson, a small mill town forty miles west of Boston.\n\nAs Maureen looks back at her childhood, she recognizes a variety of forces that helped push her forward. There was a strong sense of identity that came from belonging to an immigrant clan\u2014a family of ten and a big team of cousins and friends that got together often for Irish ceilis, social gatherings filled with singing and dancing and storytelling. Maureen would banter and dance with the older men and women and feel part of an important tribe within the larger American family.\n\nThere were also teachers who saw promise in Maureen and made her feel recognized. A fifth-grade art teacher told Maureen she had talent and asked her to stay after school for a special art club. A seventh-grade science teacher beloved by students for her countercultural views invited Maureen to her house, where there was a door made of beads and a living room rug made from colorful squares of recycled carpet swatches. It was different and mind opening. A high school English teacher at the vocational technical school told Maureen it was okay to pursue a liberal arts education and talked with her for hours outside of class about literature.\n\n\"When a teacher is in front of the class, they may be a great teacher but you know they are paid to do what they are doing,\" Maureen observed. \"They _have_ to be there, so it's harder to feel there is anything special about the relationship. But when they reach out on their own time and they see something positive in you and talk with you, it's a whole different level of recognition and a feeling of really being noticed and having something specific about who you are reflected back on you.\"\n\nImportant to Maureen's childhood were a series of work experiences that broadened her horizons and lifted her confidence. Maureen started working at age ten because, she recalled, \"If I wanted to buy a slice of pizza on Saturday with my girlfriends, or buy a pair of Levi jeans because that's what everyone was wearing, I needed to have my own money.\" Her first jobs were as a babysitter, and while some of the families she worked for were familiar and similar to her parents, the husband and wife in one family Maureen worked for had both completed college and were working as professionals. The experience was eye opening.\n\n\"The family environment was just different,\" Maureen recalled. \"Everything from their wardrobes to the things they talked about and asked about was different from what I experienced in my own family. What really made an impression on me was that this family that had been away to college recognized my work ethic and my value and hired me weekly for a long time. They made me think differently of myself. It's not that I had low self-esteem but still they helped me to see myself in a new way. I had a positive response reflected back on me because of my work ethic, skills, and gumption.\"\n\nWhen Maureen was twelve she took over her brother's daily newspaper route and was pleased by the generous tips and friendly comments she received from many of her nearly one hundred customers. Later, Maureen lied about her age to get a job at the local pharmacy. She forged a bond with the pharmacist, whom she remembers as a Renaissance man who could play the guitar and carved decoys with a penknife during slow points in the day.\n\nThe world of work was a proving ground and an opening to new people and new views, and this became especially true as Maureen started cutting and styling hair full-time right after graduating from Assabet Valley Technical High School. She started at a salon on Main Street in Hudson doing hair for her friends and for older women getting roller sets. In a year her gregarious personality and strong work ethic led Maureen's boss to promote her to manager. Later she moved to a different salon in upscale Wayland, where her customers included staff and faculty from Wellesley College, a bank executive, and a nice older woman who became Maureen's \"Jewish mother.\"\n\n\"It's a very intimate relationship when you are cutting people's hair,\" Maureen recalled. \"People ask for advice and may be vulnerable with you in a way that is unusual for them, and they would also give advice. About the time that I was working at the salon in Wayland I was starting to think of finally going to college and I was looking at studying textile design, because it was a career I could envision. My Jewish mother at the salon said, 'Maureen, you seem like a Wellesley girl to me. You need to go to Wellesley and then you can do whatever you want.'\"\n\nWhen I met Maureen she was a year away from earning her degree from Tufts University through its partnership with the School of the Museum of Fine Arts and was cutting hair privately for about ten clients a week while also painting and finishing her studies. She and I had traveled on very different journeys, but for both of us our early work experiences had been critical to an adolescent and early-adult transformation. A lifeline for me was attending and then working at summer camps, helping organize campaigns in Colorado and across the country, serving as editor in chief of my college newspaper, and getting to know the world as a young journalist. These work experiences lifted me through a dark adolescent fog and helped me make the most of the bountiful opportunities I had been given. Maureen used after-school clubs and intensive and financially necessary work experiences to open new doors and to build confidence, skills, and a network. For sure Maureen had a few great teacher-mentors in school, but it was her out-of-school experiences, particularly her real-world work experiences, that allowed her to close opportunity and achievement gaps with her wealthier peers.\n\nSadly, for low-income children today, work experiences are drying up and blowing away. Newspapers, to the extent they are even read in traditional paper form these days, are now delivered by adults, usually underemployed men, not by teenagers breaking into the world of work. The proportion of older teens (sixteen to nineteen) who work in the summer has declined from two-thirds a generation ago to just one-third today. And in just the last decade, the percentage of sixteen- to nineteen-year-olds working part- or full-time during the school year has declined from 46 percent to 27 percent. We are raising a generation that will have a much harder time navigating the work world and that, in the case of lower-income children, will lack the steppingstones teenage work provides to a brighter future. Wealthier teens will suffer too, but in many cases they will get unpaid internships and apprenticeships, funded by their parents, to provide some of the same workforce skills that teen jobs provide. Lower-income teens will suffer more, with the lines that isolate them from the world of success and of enterprise drawn more boldly than ever before.\n\n# CHAPTER THREE\n\n# \"IT'S MY TURN!\"\n\nIt was 98 degrees and just past noon when Maureen and I finally began to pedal. \"Off at the crack of noon,\" I joked, and we turned in our hard leather bicycle seats to share knowing smiles. After an exhausting year I had wrapped up my work with the national nonprofit City Year, and Maureen was on vacation from her position as director of student life at the Museum School. Maureen and I were both excited for a restorative late-June bike trip on the back roads of New England\u2014a trip that ended up serving as the bridge to the launch of Citizen Schools. We decided to plan our routes and destinations just a day in advance and to meander, clockwise, from Boston to Vermont, then back through New Hampshire to coastal Maine, and then home.\n\nWe packed just the essentials: necessary clothes and two books, a guide to New England bed and breakfasts and\u2014in what became a joke between us for years to come\u2014a five-pound cinderblock of a book, _Women Who Run with the Wolves_ , that Maureen insisted on bringing along (and in my panniers, no less!) but never read. We headed west, rolling over the potholed side streets of South Boston and through the skyscraper canyons of downtown Boston. We bisected the MIT and Harvard campuses on Massachusetts Avenue, and soon we were cycling through historic Concord and bucolic country towns, feeling free and soaked in perspiration. In Groton we pulled over and guzzled a half gallon of Gatorade, and then took turns pouring a gallon of cold water over each other's heads. Our longest training ride had been about thirty-five miles, and yet for day one of our trip we'd planned twice that distance and hadn't counted on the heat wave. As we finally wheeled our rusty twelve-speeds into the Cathedral of the Pines Bed and Breakfast in Jaffrey, New Hampshire, it was dusk and we were bone-tired.\n\nWe rode across New England for a week, and by the time we left for Boston on our final day of cycling, my creative spirit felt rested and all kinds of ideas for a new venture were percolating in my mind. I had known since my April 1994 decision to leave City Year that I wanted to create a new organization that rallied citizens to improve education. But the precise vision had not yet crystallized. The more I daydreamed and planned that summer, the more I zeroed in on the middle grades, which had been so difficult for me.\n\nThe image that settled in my mind was of a thirteen-year-old me, alone in my room on a hot summer day, dreading the start of school a few days later. A letter had come in that day's mail from Henry Singer, my lead counselor that summer at Tamarack Tennis Camp. \"Your serve is wicked good,\" it read, \"one of the best in camp.\" The handwritten letter went on for three pages, detailing things I did well and could build from, while also raising critiques and making suggestions. I had spent just a few hundred hours with Henry Singer, but he had pushed me, and supported me, and come to know me. I read through the letter many times that day and in the weeks and even years to come, and it made me feel good every time.\n\nThe idea for Citizen Schools initially emerged more from my heart than my head. I was searching for ways to help kids feel more successful during that adolescent period of vulnerability that had been so painful for me. I wanted to offer other children more relationships with mentors like Henry Singer, and more chances to feel successful, as I had begun to do at Tamarack.\n\nI began to think through the political and programmatic implications of the experiences I had at Tamarack and of the informal apprenticeships I had with our family summer camp when I was fifteen, with city planners and lawyers in high school, and with Gary Hart in college. In July, I came across an _Atlantic_ magazine article by Massachusetts state senator Michael Barrett that described how many European and Asian countries were implementing a significantly longer learning year and were starting to achieve better educational results than the United States.\n\nI had never thought much about the length of the school day, or year. But I did the math. Since the early twentieth century, the typical American school has operated for about 180 days per year and a little more than six hours per day. As someone who hadn't liked school very much, these thousand or so hours in school seemed like plenty. But when I calculated how many hours the typical student is _awake_ over the course of an entire year, I realized it was more than five thousand hours. The math was elementary, but for me it was a light bulb moment. I realized American kids were out of school for almost 80 percent of their waking hours! As a country, we were banging our heads against a wall trying, with little success, to change the 20 percent of time when kids were in school. But we were doing little to expand that time or to transform the educational opportunities offered when children aren't in school. I developed a nifty pie chart showing how little of a kid's waking hours were spent in school, and I started talking to anyone who would listen about the opportunity of out-of-school learning. _Carpe the afternoon!_ became my informal rallying cry.\n\nAs the summer went on, ideas kept percolating. A friend sent me a delightful article, \"Chekhov for Children,\" which detailed the efforts of a Manhattan drama teacher to utilize after-school time to introduce the play _Uncle Vanya_ to a group of fifth and sixth graders. I heard about a science camp in which kids built robots and about a Saturday program at one of my alma maters, Milton Academy, where kids chose courses in everything from carpentry to ceramics to filmmaking. Out of this stew of ideas came my own personal commitment to publish a newspaper with students\u2014whom I called apprentices\u2014and the first draft of my concept paper calling for a new network of \"Citizen Schools.\"\n\n\"There is a sleeping giant of education reform,\" the argument began, \"and it is us: average citizens from all walks of life. More than any new curriculum, new funding source, or new management plan, what students need is more attention, love, teaching, and guidance from more adults. In our search for better outcomes for kids, we need to stop bashing schools. The rest of us need to pitch in.\"\n\nThe concept paper went on to detail a summer and after-school curriculum built around apprenticeships taught by successful professionals and artisans. Kids in Citizen Schools would also learn through field trips to colleges and museums, through reading circles, and through targeted homework help, all provided by teaching fellows, who in the early years would work part-time while attending college or graduate school. But the special sauce of the program was apprenticeships taught by volunteer \"citizen teachers.\" My vision was to give kids from some of Boston's poorest neighborhoods and lowest-performing schools the chance to work with leading professionals, to learn from the model of their success, and to create products and performances that made their peers, their parents, and other adults say: \"Wow! I can't believe kids did that.\"\n\nFortunately, the apprenticeship idea resonated with Dever school principal Nydia Mendez, who invited me for a long walk along Boston Harbor when I asked about trying out the Citizen Schools idea with her students. As we walked along the South Boston beachfront on a beautiful late-summer day, I shared my vision for Citizen Schools, focusing on how it could turn schools into places where students not just consumed but created knowledge, and where they discovered connections between school and future careers. Nydia shared her experience growing up as the daughter of an educator in Puerto Rico. \"My father always said that education was about the head, the heart, and the hand,\" she told me. \"But now in our schools, too often it's just about the head.\" I told her about my idea of creating a newspaper as an initial pilot, and she welcomed me into the Dever family, connecting me in September 1994 with the fifth-grade students in Margie Tkacik's classroom.\n\nAs described in the introduction, that first journalism class was full of learning (for me, at least), with ten eager students pitching in to write and edit an eight-page newspaper that we had professionally printed and then distributed all over the school and surrounding neighborhood. There was Freddy, who had a bright smile and moody, deep-set eyes. He had spent all of the previous summer cooped up alone in a hot Dorchester apartment because his mother was working as a nurse's aide and thought it was too dangerous for him to go outside and play. And there was Candace, a diligent fifth-grade journalist who would later defy the odds and graduate from St. John's University in New York, where she ran the college's cable-television news station. Kaitlyn was another of my students. Soft-spoken and in the middle of the pack academically, Kaitlyn drew no particular notice. But one day as I circulated among the students to edit their stories, Kaitlyn seized my attention. \"It's my turn!\" she exclaimed firmly, slamming her hand down on the table and punching out the words with fierce urgency and only a trace of her customary smile. I had not been sure Kaitlyn was getting much out of the class. But now I had a flash of recognition that Kaitlyn, like most people, learned primarily from relationships and experiences. Foreign as I may have seemed, I was providing Kaitlyn with a successful experience and a caring relationship. And she wanted more of both.\n\nIt _was_ Kaitlyn's turn. And it was my turn too\u2014my turn to try my hand as a social entrepreneur, to test and develop the Citizen Schools idea and try to bring it to classrooms across the country.\n\nIt's often attributed to the philosopher Ralph Waldo Emerson that \"if you build a better mousetrap, the world will beat a path to your door.\" That's probably true in the mousetrap business. But it's not always true in education or social services. Too often resources are spread around like peanut butter rather than being targeted to organizations with the best results. Or they flow to established nonprofits with good lobbyists and marketers to the detriment of able but newer or more humble practitioners. Although there has been some recent movement in philanthropy and government toward investing in organizations that have measurable evidence of success, change has been slow. As a result, while it's relatively easy to start a nonprofit, it's hard to grow one. According to the IRS, of the 1.6 million nonprofits that have qualified for tax-exempt status, only 17 percent of them have grown to annual budgets of $1 million or greater. Bridgespan, the nonprofit consultancy, estimates that outside of hospitals and universities, only 0.1 percent of all nonprofits formed since 1975 have grown to a size of $50 million or greater.\n\nThe odds of growing Citizen Schools were daunting. I could become a modern-day Don Quixote tilting at the windmills of my time\u2014or a monomaniacal Captain Ahab chasing Moby Dick. But with a fifteen-page concept paper and pilot journalism apprenticeship under my belt, I set out to build Citizen Schools as an impact player in the education arena. I was determined to try.\n\nWhen I had left City Year, my going-away presents were a laptop computer and help securing two important fellowships. One was a one-year public service fellowship at Harvard's John F. Kennedy School of Government, which consisted of free access to a photocopier and meeting rooms, the chance to sit in on classes and seminars, and a sixty-four-square-foot cubicle. The neighboring cubicles belonged to twenty retired military leaders from the former Soviet Union. I imagined them feeling obsolete in this post\u2013cold war era of glasnost, and they spoke in hushed tones during their short days at the office.\n\nEd Cohen, a member of the City Year board, provided the second fellowship\u2014and a valuable head start. Ed at that point was running the Echoing Green foundation but had previously been a Wall Street lawyer, a McKinsey business consultant, and the managing partner of General Atlantic, a leading private equity firm. Echoing Green had given the first $100,000 grant to City Year and had been an early funder of other leading social entrepreneurs, including Wendy Kopp at Teach For America, Vanessa Kirsch at Public Allies, and Aaron Lieberman at Jumpstart for Young Children. Ed made me an Echoing Green education fellow, an honor that came with a two-year $50,000 grant and a support system to help Citizen Schools get off the ground.\n\nBuoyed by the initial grant and the landing pad at the Kennedy School, I labored away amid the Soviet generals. Day after day I banged away on my laptop, typing the way I had learned as a newspaper reporter, with two fingers and at speeds up to fifty words per minute, trying to conceptualize and launch Citizen Schools. I built working committees to help us develop our curriculum, our training program, and even our organizational values.\n\nMost of my time was devoted to meeting with people to get advice on Citizen Schools, doing consulting gigs with other nonprofits to earn money (the public service fellowship was unpaid and I wasn't yet drawing a salary from Citizen Schools), and revising the concept paper. I met with police officers, lawyers, architects, zoologists, and more in search of future volunteers. One afternoon I took the subway, called the T in Boston, all around the city so I could meet with street musicians and ask them what they thought of the Citizen Schools idea and whether they would like to teach. I remember the response of one talented young guitarist in particular. \"I've always wanted to teach music,\" she told me. \"But no one ever asked me to.\" I hoped Citizen Schools could change that.\n\nI doggedly pursued the building blocks of a nonprofit organization\u2014a strategic plan and budget, tax-exempt status from the IRS, a team of volunteers and paid staff, schools to partner with, an office, and money to pay for everything. I asked my college friend and roommate, Ned Rimer, already an experienced nonprofit leader, to join me as a cofounder; and in the winter and spring of 1995, Ned, who in college had run the University of Vermont Rescue Squad, taught a first-aid apprenticeship at the Dever School in which a team of students became \"first responders\" for students who suffered skinned knees and other mishaps on the playground. By May of 1995 Ned was aboard almost full-time. I signed up interns from City Year and the Harvard Graduate School of Education. I met two extraordinary young educators, John Werner and Anita Price, and asked them to lead a planning process for our first summer program and then to run it together. I secured pro bono legal help from one of the top firms in the city, Hale and Dorr (now WilmerHale), and on January 18, 1995, we incorporated as a nonprofit with me as chair, Ned as vice chair, and my wife, Maureen, as secretary.\n\nFor year one, our budget was $130,000. That was enough to pay Ned and me a modest salary for part of the year, to hire eight staff to lead a six-week summer pilot for sixty students at the Dever, and then to operate a small after-school program just two days per week in the fall. We imagined ramping up to three summer and after-school programs in 1996.\n\nTo receive mail that first spring, we opened a post office box in Cambridge, two blocks from my Kennedy School cubicle. Every day, I walked to the little metal box, clutching the key that connected Citizen Schools to the outside world. Some days the box would be empty and I would return to my cubicle feeling lonely. I'd resolve to send more proposals, distribute more applications, and once again phone the people in my doublewide Rolodex. On other days there would be applications from potential staff, or volunteers, or students. Navigating Harvard Square traffic as I walked back to my office, I would rip open the envelopes to read the letters and applications, feeling elated at the growing interest.\n\nBy late June we had accepted\u2014miracle of miracles\u2014sixty-three pioneering students, recruited and trained sixty volunteer citizen teachers, and hired six team leaders and two interns to run the five-days-a-week summer program under John Werner and Anita Price's energetic direction.\n\nThe staff was an interesting lot. Sarah Light, one of the first hired, was the brilliant daughter of a professor and had just graduated from Harvard. She took over my journalism class and, while she struggled, as I had, with classroom management, she willed her way to build writing skills and justifiable pride among her students. Keith Mascoll was an actor and, at twenty-seven, a relatively experienced educator from Cambridge. Keith served as a role model for our kids and, along with Brendan Hughes\u2014another skilled theater person\u2014as the hilarious leader of our opening and closing circles. Keith and Brendan introduced all sorts of ideas through skits. If we wanted to encourage student curiosity, for instance, they would recruit student actors and act out behaviors to avoid\u2014the too-cool-for-school preteen rolling his or her eyes\u2014and behaviors to emulate\u2014the eager-beaver learner firing their own and their peers' imaginations with wide-eyed questions. Students and staff enjoyed the skits and they seemed to be effective in communicating key messages. The last hired was Biz Pinsky, just eighteen and headed to Columbia University and then a career in medicine. Her father was United States Poet Laureate Robert Pinsky, and Biz brought a love of Dr. Seuss and a strong creative spirit to the children on her team. The best teacher in the group was Tim Proskauer, a recent graduate of Wesleyan University who was already married and committed to a career in education. Years later Tim would move with his family to Puerto Rico and be named Teacher of the Year for the Department of Defense school system, which serves students on military bases across the United States and in Puerto Rico and Guam.\n\nTo sign up for a totally new program\u2014one with no track record and uncertain financing\u2014staff members and volunteers needed to be idealists and open to adventure. In many ways, they were all inventors. I learned a lot from each of them. One of our volunteer citizen teachers was named Kikuko. She was a seventy-two-year-old shiatsu massage master from Japan who taught the basics of shiatsu, and I learned from her that even the most hyperactive eleven-year-old boy can be serene when given clear, calm instructions and shown the power in his hands. Another volunteer was Jake Tucker, a legally blind chef at a swank Beacon Hill restaurant. As he and his charges prepared for a banquet on their final session, I learned how eager parents are to see their kids be successful and how everyone feels special when they are treated to a white-linen dinner party\u2014particularly when the chefs are preteens. Denis Rorie was a Boston cop\u2014a so-called \"community police officer,\" and another of our citizen teachers. Dennis and his colleagues taught a community policing class in which students developed a brochure with crime-fighting tips. \"Leave it to the rest of the police force to catch the bad guys,\" Denis would tell us. \"My job\u2014and your job at Citizen Schools\u2014is to catch kids doing things right.\" Denis was an inspiration for us, and his advice about \"catching kids doing things right\" became a hallmark of our educational philosophy. I treasure a picture of him in his standard police blues, with a big, wide smile, jumping high on a pogo stick in front of Dever School.\n\nThat first summer was brutally hot. The school had no air conditioning, and each classroom was outfitted with just one plastic fan that Ned had bought at the nearby OfficeMax for $14.99. Despite the heat, as the summer wore on we had some reason to believe that things were going well. For one, the kids kept showing up, waking up early and getting dropped off by their parents at eight, or riding public transportation to the nearby subway stop where we would post a staff member to escort students to the school. Every week students had two two-hour sessions for each of three apprenticeships they had chosen, plus they had daily reading circles, twice-weekly field trips to museums or scavenger hunts in different neighborhoods, and additional time for physical recreation, games, team discussions, and our daily opening and closing circles with all sixty-three students. Every day we witnessed learning breakthroughs\u2014such as when Sydney jumped for joy after she finished creating her children's book. (Later, as part of the Boston Public Library's 150th birthday celebration, Sydney's book was added to the collection and placed in the stacks of the main branch, Dewey decimal number and all.) We also took pride in simple gestures of kindness, like a kid writing a nice note to another kid through a system of appreciation mailboxes we had created, or a new student volunteering to speak at closing circle. We were particularly excited when we received an upbeat report from Kikuko, Denis, Jake, or any of the sixty volunteers who teamed up to teach twenty-two apprenticeship courses that summer.\n\nBut there were other days when we felt lucky to escape with no one getting hurt, or when we wondered if we were ready for the responsibility we had taken on. There were a couple of fistfights. There was the day that Dorzell, just ten years old, learned his father's jail sentence would be extended. He ran through the halls screaming, finally punching a wall and almost breaking his hand. And there were days where kids just didn't seem to want to learn and staff wilted in the heat, doubting their effectiveness.\n\nI am eternally hopeful. A friend once told me, \"You're not just a pie-in-the-sky optimist, but a whole-bakery-in-the-sky optimist.\" Generally I took the challenges in stride. But it was painful to see our staff struggle and heart-rending to see the daily challenges that confronted Dorzell and so many of his peers.\n\nWeek four of the five-and-a-half-week summer session was the toughest. The initial honeymoon period was over. It was steaming hot. And the adults were anxious about the upcoming WOW! event, at which students from each apprenticeship would showcase their learning. Some of the kids, perhaps sensing our fatigue, were stepping up their misbehavior.\n\nAt the end of the week I left the school for a meeting I had scheduled with ten parents. We all felt relieved to meet in an air-conditioned conference room at the Harbor Point housing development across the street from the Dever School. The purpose of the meeting was for me to ask them about Citizen Schools and to get their advice on a fall after-school session we were planning. But I was nervous. After four weeks of Citizen Schools, what did they think? What were their children saying? Were they learning? Growing? Would the parents want to enroll their children in our planned after-school program, or in future summer sessions?\n\nAround the table were ten moms\u2014one white, three Latina, and six African American. All of them had paying jobs: there was a nurse, a nurse's aide, and several entry-level hospital workers; a cleaning lady; a filing clerk; and a few who stocked shelves or staffed registers in pharmacies or convenience stores. Most of the women earned just above minimum wage\u2014or about $13,000 per year, putting them below or just above the federally defined poverty level. A majority served as the only breadwinners in their families, typically supporting three children or more.\n\nAs we introduced ourselves and started sharing impressions of the summer, the first thing I noticed is that the mothers were very solicitous of each other. Several of the moms exchanged numbers and made plans to talk more or to get together. They would nod familiarly as they heard stories about challenges another child was having. Fear about neighborhood crime and gang initiations ran high, particularly among the mothers of young boys. I also heard about typical adolescent challenges such as growing disinterest in school, a new willingness to talk back to parents and challenge their authority, and social exclusion at school and in their neighborhoods for being too heavy or too quiet or too something else by the standards of the cool clique.\n\nThe second thing I noticed was a powerful sentiment of gratitude. These moms were being pushed away by their children, because that's what happens when your kid turns twelve or thirteen. They were trying to raise kids under trying circumstances and in difficult neighborhoods. After a decade of being the earth to their son's or daughter's moon, they sensed their gravitational pull was waning. But Citizen Schools seemed to mitigate the growing sense of distance or loss. The moms talked about how relieved and delighted they were to see new role models step into their kids' lives. They kept saying how their children loved the volunteers at Citizen Schools, and loved the hands-on projects, and loved the special trips to museums and to the Boston Harbor Islands. Keith, the one African American man on our staff, got many shout-outs as a great role model to the African American boys in the program. I also heard from the parents about Brendan's funny stories, and Biz's positive reinforcement, and Tim's patience, and John's incredible creativity and energy, and Anita's warm but authoritative presence. Apparently, _this_ is what the kids had been talking about at home.\n\nImmediately after the meeting, I drove to Lambert's, a local vegetable and flower market, and bought twelve of the biggest sunflowers I could find\u2014enough for every staff person and intern we had. As we gathered for our closing circle that day, after the last child had gone home, I shared a flower with each person and a story of appreciation from the parents. Soon thereafter, in one of the hokiest decisions in organizational history, we named the sunflower the official flower of Citizen Schools.\n\nA week later we held our culminating WOW! festival\u2014a three-tent extravaganza that allowed our students to showcase their skills. We encouraged parents and community leaders from across the city to join in celebrating our young people. Kikuko and her team of novice shiatsu aficionados offered free massages. A group of students who worked with engineers from Bolt, Beranek, and Newman\u2014the Cambridge firm that helped develop the Internet\u2014gave surfing lessons on the still-novel World Wide Web to a rapt circle of adult pupils, including then Boston mayor Tom Menino. Theater and dance groups performed on a main stage. Sarah's journalists handed out their newspapers. And Jake's young chefs\u2014augmented by moms and some dads who brought their favorite dishes\u2014provided most of the food. We even had a group of young bike-repair apprentices offering free bike tune-ups to neighborhood kids. It was amazing. And it all took place under sunny skies, on the well-tended lawns at Harbor Point, the mixed-income community rising from the shores of Boston Harbor across the street from the \u00adDever School.\n\nThe WOW! festival was extraordinary, but it almost broke me. All summer long I had carried in my head the vision for this final WOW! Our children and their families deserved the best. And an amazing festival on the waterfront\u2014a festival with the mayor, with TV cameras, with creative displays of student work, with watermelon and cotton candy as well as healthy food made by our students\u2014was a way to give them the best, to make them feel worthy, and to put them at the center of the city's vision. The problem was that putting on a party of this scale took a lot of work\u2014and we had a thin team. Everyone was stretched to the limit just running the program day to day and getting the students ready for their final performances. So I told the team not to worry about the WOW!\u2014I would handle it.\n\nIn reality, what \"handling\" it meant was that I conceptualized the event and built momentum for it, but I didn't recruit enough other volunteers to actually make it happen. I leaned heavily on Maureen. Could she make some large posterboard signs? Could she pick up the clam chowder from Legal Sea Foods? Maureen saw how hard I was working and she pitched in heartily. But I could see she was irritated at Citizen Schools' intrusion on our family time. The day before the big WOW! I drove to a tent-rental place in the Boston suburbs to pick up three heavy tents and, with minimal help, managed to erect them. I shopped for thirty watermelons and loaded them in and out of our old station wagon. I picked up the huge helium tank and the Citizen Schools balloons. And I lugged forty folding tables and hundreds of chairs all over Harbor Point. By the time the party started, my back was in serious pain.\n\nAdrenaline got me through the WOW! But when I woke up the next morning I could barely walk. I shuffled slowly into the shower, put as much heat on my back as I could handle, and then managed, barely, to pull on my tuxedo for that morning's graduation. John and Anita orchestrated a beautiful ceremony filled with laughter and tears. We had a closing lunch. And then, in an extreme case of unfortunate timing, I boarded a plane to San Francisco to join a conference of education entrepreneurs.\n\nThe flight was miserable. I had a three-hour layover in Newark, and by this point I was in agony. I tried to play a mind game by reflecting on the fact that sometimes I like discomfort, such as getting stuck outside and far from home in a rainstorm or walking to work on a really cold day without a hat or gloves. Maybe because I've suffered few serious discomforts in life, I feel I'm leveling the cosmic scales if I experience a little bit of pain. But nothing about this back pain was noble or uplifting. I wondered what the hell I was doing to myself and whether Citizen Schools was taking too big a toll on my health and my relationship with Maureen. The following morning on the West Coast, I saw a doctor who gave me a prescription for Flexeril, the most powerful muscle relaxant out there, a brace to support my back, and strict instructions to get back home and lie down in bed and do nothing for two weeks. That's pretty much what I did, and two weeks is about what it took before I was able to walk normally and begin to do some much-needed stretching.\n\nBy Labor Day at the end of the summer of 1995, my back was feeling better and the happy memories of the summer and the culminating festival had reestablished their primacy in my consciousness. Doctors say that mothers are genetically programmed to forget the pain of childbirth so they will be willing to endure it again. Maybe the same is true of entrepreneurs working one-hundred-hour weeks to get their enterprises off the ground. Ned and I went into full-fledged planning mode, preparing for a fall after-school pilot and starting to think about running three summer programs the following year. Ned took the lead on the program that fall, serving as campus director and directly overseeing four part-time staff members and thirty-two students drawn from the Dever and the neighboring McCormack Middle School. The program met just two days a week for twelve weeks. The McCormack kids started at 1:30 p.m., then the ridiculously early dismissal time for all middle school students in Boston; the Dever kids, who had a 9:00 a.m. start to their school day, started Citizen Schools at about 3:00 and ended at 5:30. The middle school kids began the program with ninety minutes of chess instruction and homework help, and then everyone had apprenticeships from 3:00 to 5:00, followed by a closing circle orchestrated by Ned. I was on site one of the two days to coteach a weekly apprenticeship in business in which ten students worked with me and two friends.\n\nThe \"product\" of our business apprenticeship was the Citizen Schools end-of-semester WOW! The idea was that the students would plan and publicize the event and charge a small admission fee, which would cover our costs, perhaps deliver a small profit, and teach the kids some business principles in the process. I am absolutely certain that the students learned at least one important lesson\u2014which is that startup ventures need to expect the unexpected. Ultimately our WOW! was a success, with more than $2,000 of revenue from ticket sales, sales of ads in a simple program, and a raffle. But we had to reschedule the event twice because of snow!\n\nThe Citizen Schools business plan called for us to hire a full-time program director and a junior assistant in January, and I spent much of that fall recruiting for these jobs while also raising money and building relationships with new schools and neighborhood leaders where we planned to expand. We narrowed down a great list of candidates to two finalists, Stephanie Davolos and Tulaine Montgomery. Stephanie had been a French teacher on the Bayou in rural Louisiana through the Teach For America program and then became the program's regional executive director. She had a wonderful lightness to her spirit, joking, for instance, about her teaching of classical French grammar to French Cajun children who had been speaking a version of the language since they were toddlers. She was also fiercely determined and a deep believer in our \"learning by doing\" approach. Tulaine was a dynamo. At just twenty-four, she was program director for the House of Blues Foundation, an accomplished cellist, a brilliant speaker, and the foster mother of three teenage children. As we proceeded through the interview process I became convinced we should hire them both. \"Hire ahead of your needs\" was advice we had received at City Year from leveraged-buyout king Ray Chambers, and I was confident we'd have more than enough need for both Tulaine and Stephanie.\n\nAs a final interview of sorts, we took the two of them out to dinner in Boston with Anita, who was back for her final year at Harvard, and John, who was then working as a special-education teacher in Boston but was still very involved in our efforts. We were joined by Marsha Feinberg, our highly engaged board chair. This was the last stage of the vetting process and simultaneously a good chance to build the culture of our extended team. But while Ned and I knew we were likely to hire _both_ Stephanie and Tulaine, we had neglected to tell either of them. Throughout the dinner, they eyed each other watchfully, looking for an edge in what they must have imagined was a _Survivor_ -like reality show. Despite the circumstances, Tulaine and Stephanie were gracious and wonderful, and at the end of dinner I asked them both to join the team.\n\nStephanie and Tulaine started on February 1, 1996, which was also day one in our first headquarters. For the previous year Ned had been working out of his spare bedroom in Cambridge, while I worked from a basement office in Maureen's and my brick townhouse in South Boston. For meetings, we had been using the YMCA, Dever School, various law firms, and assorted coffee shops and borrowed conference rooms. But now we would have our own digs, a new \"garden-level\" office on South Street, a few blocks from the core of downtown Boston. The space was beneath a French restaurant, Les Zygomates, and featured a small conference room, a large back area where we could hold trainings and house part-time and future staff, and in the front, a four-hundred-square-foot room with natural light. You had to look up to see the light. But if you did, you had a good panorama of the sidewalk, including the hubcaps of parked cars and the ankles of passersby.\n\nI spent a lot of time that winter going to meetings with Stephanie and Tulaine and working with them and Ned to build our culture. Ned was a bit of a thespian, with a love of music. One summer as a child he had lived and worked on a dairy farm in Greece, and he often shared stories of the lessons he learned there.\n\n\"When you are learning how to milk a cow in a foreign land,\" he would say, \"you realize quickly the power of learning by doing. I spoke almost no Greek, so a lecture on milking a cow would not have been very effective. A textbook on how to milk a cow would have been, as they say, 'Greek to me.'\n\n\"But when the farmer got down on an overturned bucket and demonstrated how to milk a cow and let me practice, then I could really learn it. The language barrier went away.\" It was one of Ned's standards, and a great story, particularly to demonstrate the power of the Citizen Schools approach.\n\nApparently Ned had actually mastered some Greek, though, because he taught us all a Greek love song, \"Eis ton afro,\" and Ned and I and Stephanie and Tulaine belted out the song on numerous occasions when we were marking important milestones, greeting new staff, or just punch-drunk after a long day at the office.\n\nAt some point that winter we wrangled an important meeting with a potential funder from New York City. \"Money is the mother's milk of politics,\" California treasurer Jesse \"Big Daddy\" Unruh famously said. And money was mother's milk for us too, as for any startup, whether a Silicon Valley technology firm or a Boston-based education nonprofit. This particular meeting was with Robert Sherman, the senior program officer at the Surdna Foundation, and it was our first meeting with a significant national foundation. The Web wasn't yet a common research tool, and Google did not even exist, so I called everyone in my Rolodex whom I thought might know Robert or Surdna. People said Robert was friendly and that he and Surdna CEO Ed Skloot (later a Citizen Schools board member) had a deep interest in civic engagement. That\u2014and the donation of my former dining room table to serve in our office conference room\u2014was as far as our advance work went.\n\nOn the appointed day Robert arrived and found his way down the stairs into our new little office. I introduced him to Stephanie and Tulaine (Ned was away) and we headed into the conference room, where I had placed a tray of sesame bagels and a few glasses of water. We proceeded to have a meeting that struck me as fantastic. Robert shared his goals for the foundation, and we shared our vision for a new kind of school that activated community assets and turned kids into producers of things and ideas\u2014not just passive consumers. Robert and I were both New Yorkers, and we both started talking quickly and getting more and more excited. At some point, though, I began to notice that Tulaine and Stephanie were looking at me in a funny way, as if I was somehow sabotaging our promising meeting. But I kept talking, and Robert kept talking, and the energy in the room was just exactly what I wanted. After a while Robert got up to leave and said he was impressed and wanted to think things over and would call in a few days. He left, and Stephanie, Tulaine, and I huddled to review the meeting.\n\n\"Home run,\" I said. \"He loves this idea.\" Then I asked what they thought and what was behind the funny looks they had given me. Very politely, Stephanie and Tulaine explained that during the meeting I had started to lick my fingers and then use them to pick up\u2014and then nibble on\u2014the stray sesame seeds that had fallen off the bagels and landed on my end of the table. I cracked up. \"Oh, that'll be okay. I hate to see those delicious seeds go to waste. And Robert is a New Yorker; he'll understand.\" We all had a good laugh, and when Robert called the next day to offer us even more money than we had asked for, I joked that sesame bagels should become the official food of all Citizen Schools fund-raising meetings.\n\n# CHAPTER FOUR\n\n# TURNING POINT\n\nJoel Bennett had a gentle smile and a truck full of saws, vice grips, files, and other tools he used in a burgeoning carpentry and cabinetmaking business. Joel's business was stable, and he liked the feeling of tangible accomplishment he got from working with wood. But when he met Stephanie Davolos at a community event near his home in Boston's Jamaica Plain section, Joel volunteered that he had always been curious about teaching. Sometimes after completing yet another kitchen renovation or set of built-in bookcases for a home office, he wondered if teaching children might be a more satisfying way to spend his days. Citizen Schools became a way to dip his toe into the education world, and he signed up to teach a carpentry apprenticeship in the spring of 1996.\n\nThat spring we offered a Saturday program at Dorchester's Woodrow Wilson School, four blocks from the Victorian fixer-upper Maureen and I were soon to buy in nearby Ashmont Hill. We enrolled sixty-four students and offered eight apprenticeships (including Joel's) in the morning, followed by various field trips and learning activities in the afternoon. Joel's team of young carpenters set out to produce high-quality carpenter's toolboxes. They measured them, cut them, sanded them, shellacked them\u2014the whole thing.\n\nThe kids learned plenty. Those of us observing learned a couple of things too. For starters, we learned that a few of the kids didn't know how to measure\u2014at all. In the second week, Joel asked a seventh grader named Kiel to cut a piece of wood in half. Kiel said, \"Half? What's half?\"\n\nKiel was getting Bs at the Woodrow Wilson Middle School, and he didn't know what \"half\" meant. I couldn't believe it! Joel broke down the concept of half. (I remember him folding a piece of paper, then measuring his two feet compared to one, before pulling out a tape measure and actually addressing the piece of wood at hand.) Kiel and Joel cut the piece of wood together and moved forward.\n\nFrancisco was another young boy in the apprenticeship. He was eleven and a recent immigrant from Central America. His toolbox was a beautiful piece of work. Around week nine, the kids were planning their presentations and deciding what their toolboxes would be used for\u2014gifts to their moms, or just additions to their own personal workspaces. Joel had noticed that on the side of his toolbox, Francisco had written MIGUEL in block letters.\n\nJoel hadn't heard him talk about a Miguel, so he asked him: \"Francisco, who's Miguel? I thought you were keeping this or giving it to your mom.\"\n\n\"Miguel is this old man who lives in my neighborhood,\" said Francisco. \"He's always doing nice things for me. He gets me ice cream. Last summer he took me out to a baseball game. I've never had anything to give to him. So I'm going to give this toolbox to Miguel.\"\n\nFrancisco's story has stuck with me ever since. Even after telling his story dozens of times, I still get goose bumps imagining Francisco presenting his toolbox to Miguel. Our society embraces the giving of gifts to our children. But kids want the chance to give gifts too. Kids who grow up with the privilege of private classes\u2014in ceramics, jewelry, woodworking, or even music\u2014have plenty of chances to be givers of presents they created themselves. Providing kids the chance to be makers of things\u2014not just consumers\u2014is an important part of the opportunity equation.\n\nKiel's story stuck with me too. By summer 1996 we had grown to offer three full-fledged programs\u2014one at the Dever in Dorchester, one run by Tulaine at the Timilty School in Roxbury, and one run by John Werner at the Garfield School in Brighton. The whole idea of learning by doing, and tapping the incredible resources of the city, was captivating to people. And we continued to find energetic staff members and diverse volunteers to design cool learning experiences. But our program was built on a shaky foundation, and I worried if we were doing enough for students like Kiel. We had no behavior-management system to speak of and we were naive about lesson plans and the basic building blocks of teaching and learning. We got the big stuff right: making learning interesting and connecting kids to successful and caring adults. But we got almost all of the small stuff wrong. On good days we got away with it because many of the underlying learning opportunities were powerful. But on bad days we had chaos.\n\nThe first week at Timilty that summer was particularly challenging. A handsome new staff member from Newark, New Jersey, had inadvertently created tension among the staff, several of whom, I was told in the parlance of the day, were \"sweating him.\" And our enrollment included a larger than usual share of rambunctious seventh and eighth graders\u2014a tougher group to engage than the preadolescent fifth and sixth graders who had been our core constituency thus far. The students, sensing weakness, ran all over us.\n\nWe released early on Fridays to allow for staff training and planning time, and on the first Friday of that summer Tulaine led her team through what became known as Frank Friday\u2014an honest and open assessment of everything that was going wrong and a discussion of what we needed to do to make it better. The team started by brainstorming the \"brutal facts\": safety issues caused by kids running in the hallways, disrespect to adults by particular students followed by snickers from impressionable peers, rolling of eyes and sucking of teeth as new ideas were suggested at opening circle. They filled many pages of flip-chart paper. Then Tulaine skillfully steered the team toward a vision for the kind of community they wanted to build. \"Fun-loving but respectful,\" volunteered one staffer. \"A place that is safe where learning is fun,\" said another. \"Energetic but tight,\" offered a third. As the afternoon wore on, the team turned their attention to developing protocols for all sorts of nitty-gritty program components. How would they transition from activity to activity? What would they do for bathroom passes? Who could staff a \"Step-Up\" room for kids who needed a quick separation from their group and a reminder of the learning opportunities they were missing? Staff agreed to call every parent that weekend and enlist them in a reset of expectations, and they made plans for an engaging series of morning ceremonies incorporating stepping and sophisticated call-and-response cheers. Their goal was nothing less than a complete restart of the campus culture.\n\nGiven the early challenges at the Timilty, my favorite memory of that entire summer came at the WOW!, again held at Harbor Point. I had delegated more effectively this time, which was a good thing because we were hosting almost four hundred parents and other guests, as well as nearly two hundred students, including the crews from the Garfield School in Brighton, the Dever, and the Timilty. The Garfield is far from public transportation, so they came to the WOW! by private bus. But the Timilty came by T. They were running late, and I remember imagining all the things that could be going wrong. Did the stars of their performance apprenticeships fail to show up? Had the students gotten involved in some sort of altercation on the T? Just as I started to get truly concerned, my pager buzzed. The Timilty team was almost there, walking up from the nearby JFK T stop at that very moment. A few minutes later I heard a faint rhythmic clapping in the distance. Then sixty-two enthusiastic young people and eight very proud staff members turned the corner, stomping their way down Harbor Point Way and yelling out a powerful call-and-response that invoked Citizen Schools and our core values of pride, joy, and respect. The Timilty was ready to roll. Frank Friday had worked!\n\nAs we launched our four school partnerships for the 1996\u201397 school year, we noticed a different mood among our public school partners. Teachers and principals were on edge. They described a cresting wave of pressure to improve student learning\u2014part of a modern school-reform movement ushered in by the landmark 1983 report, _A Nation at Risk._ The report had generated front-page headlines with its rhetoric of \"a rising tide of mediocrity\" in US public education and its urgent call for higher standards. By the mid-1990s, teachers and principals who failed to deliver better student test scores feared for their jobs.\n\nIn Massachusetts, school reform could be traced back further to an initially obscure court case, _Webby v. Dukakis_ , which was filed in 1978 and alleged that the Commonwealth of Massachusetts was not fulfilling its constitutional duty to provide an adequate education for all children. A decision in the case by the state's Supreme Judicial Court forced the hand of the state legislature and ultimately\u2014with leadership from Senate president Tom Birmingham, later our Massachusetts executive director\u2014led to passage of the comprehensive Massachusetts Education Reform Act of 1993. The act called for dramatic increases in state funding of education and instituted a \"Robin Hood\" financing scheme to ensure that lower-income districts reached at least a minimum \"foundation\" level of investment. It also authorized up to twenty-five charter schools to push innovation, instituted minimum licensure standards for teachers, and created a process to decertify graduate schools of education if too few of their graduates met those new standards. Most significantly, the new law set in motion intensive efforts to identify rigorous standards for what should be learned in each grade and created a Massachusetts Comprehensive Assessment System\u2014the MCAS\u2014to measure how students and schools were faring.\n\nAll of this made a lot of sense to me. The process in Massachusetts was inclusive and relatively bipartisan. Thousands of teachers were involved in setting the standards and advising on the MCAS, which, unlike tests in many other states, includes essays and short-answer responses in addition to fill-in-the-bubble multiple-choice questions. Who could be against setting higher standards and measuring progress in moving toward them? And poorer school districts got more money to make progress possible. But despite the apparent logic of reform, the new standards and tests were controversial. In Cambridge there were sit-ins and other protests organized by activists who feared reform would dumb down education by focusing only on those skills easiest to measure on standardized tests. And in 1996, when the standards started to take effect and the tests started to really count (kids could not graduate without passing the MCAS), the climate of schools undeniably did change. Some of the change was welcome\u2014a stiffening of the spine as educators readied themselves for the hard work ahead. And in some schools educators realized they could meet high standards on the MCAS by offering engaging content and hands-on lessons. My current community of Brookline, for instance, aced the MCAS while generally keeping a rich, inquiry-based course of study. But other schools\u2014facing the first public test of student learning\u2014narrowed their mission, or appeared to give up. In many cases schools stripped \"enrichment\" classes from the curriculum because topics like art and music weren't tested. A study showed that many urban districts dramatically reduced time for science and social studies, because those tests were \"low stakes,\" meaning there was no consequence for failing them. Scores on English and math assessments did march steadily forward, but achievement gaps remained wide and dropout rates remained unacceptably high.\n\nFor Citizen Schools, the arrival of the MCAS presented a pivotal choice. On the one hand, schools were looking for more help and we were an emerging ally. But on the other, schools were deeply engaged in an examination of their own teaching practices, and these examinations were typically leading them to focus more on direct instruction by their own teachers and to eliminate anything that could be seen as a distraction from the urgent task of boosting proficiency. They had less tolerance for our rookie mistakes and seemed in some cases to lose their appetite for the enrichment-based learning we were offering.\n\nIt was in this environment, in the spring of 1997, that Nydia Mendez, our founding principal from the Dever, told us that she wanted to take a semester off from Citizen Schools. If we wanted to return, she said, we would need to retool our program to address more directly the emerging learning standards. We were a little stunned.\n\nNydia's decision to suspend Citizen Schools at the Dever presented us with a core organizational challenge we have wrestled with ever since. How do we keep our unique apprenticeship model intact while also doing more to build academic skills\u2014and to help students succeed on the standardized tests that would now determine whether they could graduate from high school and whether a four-year college would be an option? At times the pendulum at Citizen Schools would seem to swing away from academic coaching and toward deep enrichment and relevant, real-world learning, but then we would hear from a parent who would say that her son couldn't do Citizen Schools anymore because, much as he seemed to love the learning activities, he had failed math in the last semester, and Citizen Schools was an \"extra\" the family could no longer afford the time for.\n\nThese conversations would break my heart, as I could often see the gains students were making even if they weren't yet manifesting those gains on their report cards. There was Linda, for instance, a heavyset girl who would not say a word for week after week and would not take off a thick wool overcoat even though the classrooms we worked in were overheated. During her first few months in Citizen Schools, Linda built trust with two elderly women who taught her how to knit and make dolls. One of the women, Earline, had always been upset that none of her children or grandchildren had picked up her love of sewing. She told me that Citizen Schools offered her a perfect opportunity to pass on a craft she loved, and she asked if she could teach a sewing apprenticeship with Margaret, her neighbor and friend who was born on the same day in 1921.\n\nMost of our volunteers were young, and many were professionals who lived far from the school and arrived each Saturday by car, sometimes seeming to parachute in from a foreign land. Earline and Margaret lived just a few blocks from the school, and it warmed my heart to see them walk into the Wilson School every Saturday with their tote bags of knitting and sewing gear. It must have been wonderful for Linda, too. By week six of her doll-making class with Earline and Margaret, she'd removed her coat, which teachers from the school said she hadn't done that entire year at school. By week ten, Linda was talking. She was talking proudly about the two dolls she had sewn, one of which she was going to keep and the other of which she would give to children at a local homeless shelter.\n\nI don't know if Linda's grades went up that year, or ever. But I think it's very likely that her doll-making opportunity, set in the middle of a difficult adolescence, was a pivotal and positive experience. It may not have been enough to propel her through high school, or on to college or career success. But I am sure it gave her the confidence of knowing she could make something beautiful with her hands, and in the process opened new doors to learning and growth.\n\nWhile I was confident Linda benefited from Citizen Schools, I wondered if she benefited enough. My fears grew two years later when Linda's older brother, Matthew, a recent dropout from high school, was shot dead in front of their Dorchester home. The downward pull on Linda's life was so powerful. Taking off her coat and talking were necessary first steps. But when would Linda get her next experience with success? Would she get the academic help she needed to transition to high-school-level work? For Linda and thousands like her, would sewing, or carpentry, or law apprenticeships be a temporary oasis in a still-stifling desert, or could they become a gateway to greener pastures?\n\n# CHAPTER FIVE\n\n# ORGANIZATION MAN\n\nWe kept students like Linda in mind as, over time, we developed a series of educational innovations that maintained our focus on real-world apprenticeships taught by volunteer citizen teachers, while also taking direct responsibility for building the academic skills of our students. We launched a deeper mentoring and writing program for eighth graders (8th Grade Academy), put an increased focus on study skills and academic goal-setting throughout the program, targeted academic coaching in math and English language arts aligned with the host school curriculum, and developed a college and career connections (C3) curricula that included one or two college field trips for sixth and seventh graders and five to ten college trips for our eighth graders.\n\nPart of the change was just a matter of time\u2014the amount of time we engaged with our students. By our tenth anniversary, in 2005, our after-school program had evolved from two days per week to four or five days per week, from two hours to three full hours per day, and from just twenty weeks per year when we started to thirty or more weeks per year. If you thought of Citizen Schools as \"get smarter\" medicine, we had tripled the dosage.\n\nAnother key change was our approach to talent. Our staffing model evolved from mostly part-timers in college or graduate school to mostly full-time AmeriCorps teaching fellows, who came to us right after college and made a two-year commitment in exchange for a stipend and a $5,730-per-year scholarship to pay back loans or help fund graduate school. Since they were \"only\" lead teaching for three hours a day in the afternoons, the teaching fellows rounded out their day by taking on morning roles supporting teachers in the schools, or at Citizen Schools headquarters. They also worked to prepare their lessons and reach out to parents and the volunteers they helped support. We also restructured our campus director job to make it attractive to aspiring educational leaders who had taught for a few years and were looking to build management skills, potentially as a pathway to becoming school leaders. Campus directors, many of them graduates of Teach For America or of our Teaching Fellow program, served as assistant principals for the extended day, working closely with the host school's leadership and aligning our teaching to that happening in the regular day.\n\nIn addition to adding more time and creating a stronger talent profile for our front-line staff, we invested heavily in training and curriculum development. In 2004 we launched a partnership with Lesley University to offer our teaching fellows a master's degree in education through a unique blend of courses taught by our staff and Lesley professors. Teaching fellows began spending most of their summers working with us developing their teaching and outreach skills rather than running our summer program, which we began to phase out. A key role for the campus directors was observing and coaching teaching fellows, providing structured feedback based on five formal observations per year. Campus directors relied on a twenty-three-page instructional rubric that was developed by a cadre of award-winning teachers who joined us as staff or consultants. Often we would film teaching fellows and then meet to discuss teaching techniques that had worked well or fallen flat. Even our volunteer citizen teachers went through a training regimen covering the basics of student engagement and helping them adapt apprenticeship curricula that we had vetted and approved. Some volunteers still developed their own courses, but they did so according to an extensive \"Getting To WOW!\" guide that ensured they included best practices of apprenticeship teaching.\n\nStudents still enrolled in two apprenticeships per semester, each meeting one afternoon a week for ninety minutes, but now they also typically had one hour per day of structured homework time, which we called AIM (for Aspire, Invest, Make the grade); two to four hours per week of academic lessons led by the teaching fellows; and additional time for the C3 curricula and occasional explorations to museums and other learning venues (for a sample schedule, see the appendix). The extra time, the investment in talent, and the focus on training made us a stronger organization, better equipped to lift student learning and improve life trajectories.\n\nAn external evaluation of Citizen Schools covering Boston students participating between 2001 and 2005 helped answer the question of what happened, in the aggregate, to students like Kiel and Linda. While short-term gains in middle school test scores and grades were modest, the sophisticated matched comparison study, conducted by Policy Studies Associates of Washington, DC, concluded that participation in Citizen Schools led to greater engagement in school, better transitions into high school, and significant gains in high school achievement and completion. Starting in middle school, participants were suspended less and attended school more, gains that continued in high school, with our former participants missing five to fourteen fewer days per year than similar peers. Prior to entering Citizen Schools, our participants were among the most at-risk students in a high-poverty district, meaning they were more likely than their Boston classmates to have failed the fourth-grade MCAS test, more likely to be English-language learners, and more likely to have an identified learning disability. Yet by tenth grade, students who had participated in Citizen Schools in middle school had closed the achievement gap with _state_ averages on the high-stakes tenth-grade math and English tests (the gap between passing rates for our high-poverty students and state averages), and they graduated high school on time at a rate that was 20 percent higher than their Boston Public School peers. A 2011 study of a Citizen Schools expansion site in Charlotte, North Carolina, revealed similar results and showed that our low-income participants were enrolling in college at the same rate as their middle-income peers.\n\nThese results were encouraging. But we wanted to do better, and we wanted to grow. Growth would allow us to serve more students but would also create a platform for influencing policy and the ongoing education-reform debate, which was escalating in its intensity and its centrality to public discourse.\n\nThese twin goals of growth and continuous improvement set us on a never-ending quest for more money, more talent, and better systems to strengthen our organization and extend our impact. If the startup years allowed me and Ned to feel like chefs experimenting with new recipes or inventors imagining novel products, the next stage of Citizen Schools made us feel more like industrialists, debating management plans and expansion scenarios and overseeing an increasingly complex organization. In financial terms, we grew from a $1 million budget in 1997\u201398 to $11 million in 2005\u201306 and $33 million in 2013\u201314. The currency of the startup years was innovation and particular moments of discovery for children working with our staff and volunteers. The currency of the scale-up years was data, as we measured almost everything we did\u2014from the grades and oral-presentation skills of our students to the quality of volunteer training and the speed with which a broken computer was fixed. As CEO and cofounder I was still supposed to be Vision Man, but I tried to also be Organization Man.\n\nA key partner in building up the Citizen Schools organizational infrastructure was Emily McCann, whom we hired in late 2002 as chief financial officer (CFO) and who was then promoted in 2005 to chief operating officer and in 2008, after Ned left (he remained on the board), to president. The CFO search took place during a recession, and we received more than a thousand resumes. Many applicants were qualified on paper but none stood out. Then late in the game Priscilla Cohen, a founding board member who had transitioned to the staff and was running the search, introduced me to Emily. \"She's too young for the CFO role,\" Priscilla advised, \"but down the line she could be really good for something else.\"\n\nWhen I met Emily a few days later, it was professional love at first sight. First, she was an athlete, literally. She had been an eleven-letter varsity athlete at Harvard College, demonstrating competitive grit that I find to be a great predictor of workforce success. Second, she was wicked smart, with top grades from Harvard and Harvard Business School. Third, everywhere she had worked in her nascent career she had blazed a trail forward through hard work, applied intelligence, and outstanding people skills. Currently she was working at Disney, and her reference there said she was the fastest-rising young female executive since Meg Whitman, who went on to become CEO of eBay and HP and to almost become governor of California. Wow! Fourth, in the 1983\u201384 school year, when I was organizing college students for Gary Hart across the country, Emily was Hart's chief spokesman in her fourth-grade mock election. Double wow! Most important, Emily was friendly and warm and passionate about making the transition from designing Disney cruise vacations to lifting opportunity for children. She had the right blend of humility and ambition and seemed a perfect fit for our culture.\n\nThe problem was that Emily was just twenty-nine and didn't look a day older than twenty-two. Besides, she had never been an accountant, a financial anything, or a chief of anything, much less a chief financial officer for a fast-growing nonprofit trying to change education in America. Nonetheless, thirty minutes into my conversation with Emily, I knew she was the one for the job. Others thought I was crazy, but within a month Emily had made a believer of everyone in the organization. Over the next ten years she would help make Citizen Schools one of the best-run nonprofits in the country. Bridgespan, the nation's leading nonprofit consultancy, has cited Citizen Schools as a best-practice organization in the domains that Emily leads, including hiring, talent development, financial oversight, and information technology.\n\nA key ingredient of the social-change machine we were trying to build was money, and one of my jobs, in partnership with our director of organizational development, Anuradha Desai, was to raise it. We didn't take any government money for our first five years because we felt that private money would give us greater flexibility. I worked hard to attract funding from big and small companies, from foundations, and from individual donors. Eventually we would attract significant government and corporate grants, but for our first ten years of operation, most of our money came from foundations and from wealthy people, and this meant hundreds of fund-raising meetings every year.\n\nMany nonprofits complain that funders give too little, give with too many strings attached, and usually won't continue giving for more than two to three years. That has not been my experience. I have found a significant majority of the funders we deal with to be strategic, open to general support (meaning not many strings attached), and willing to reinvest for ten years or even longer so long as we are continuing to make progress. Some of our supporters, most notably the Edna McConnell Clark Foundation, have been national leaders in engaged and strategic philanthropy and are helping to steer foundations and even government to an investment approach that is long-term, focused on results, and strategic about building capacity. Generous board members, like Andrew Balson, our chair from 2007 to 2014 and a managing partner at Bain Capital, made long-term investments in Citizen Schools similar in some respects to private equity investments in that they were tied to our business plan and our results and pooled in formal \"growth capital\" funds supported by other philanthropists.\n\nBut even as the giving world has slowly become more rational and focused on investing in programs with proven or promising practices\u2014not just those where the program officer has a relationship or where the need is great\u2014philanthropy is still a relationship game. Rule one of fund-raising, I was told in my political-campaign days, is that \"people give to people.\" This meant that if we were to grow, I would need to meet and engage with as many philanthropists as possible. I threw myself into this work with gusto and met some amazing people who are deeply committed to building a more just world by giving away their money (or someone else's money) strategically. However, I also needed to bite my tongue during some particularly strange encounters.\n\nThere was the meeting with the young and fancily dressed foundation program officer in San Francisco who suggested that we meet at one of the city's finest restaurants, and then, after we ran up a $100+ bill on skimpy salads and seltzer water, suggested with every fiber of his body language that Citizen Schools\u2014or me\u2014should pick up the bill, not his billion-dollar foundation. Another time, as I prepared to meet with a prominent Boston-based executive, along with three other education-focused nonprofits, we were warned by the CEO's assistant that the CEO suffered from narcolepsy and might fall asleep briefly during our meeting. Despite the warning, it was quite unnerving when, midconversation, the CEO did indeed nod off for what seemed like several minutes. We decided to keep talking, hoping that our most persuasive arguments would find a place in his subconscious if not in his notepad.\n\nEver since our incorporation in Boston in January 1995, Citizen Schools had grown just in our home city. The lesson I had drawn from my City Year experience was to go deep, and I didn't see any reason to expand to other communities when there were still thousands of students and dozens of schools in Boston that needed us. As a result, in the early years we turned down offers to expand to Atlanta, New York, and several other communities. Over time, though, we heard the call of replication and reconsidered how to spread our good work. Growth in Boston had become harder because we had harvested the \"low-hanging fruit\" in terms of funding and motivated school partners. As a result, I felt we had reached a point where it would be easier to reach the next one thousand kids outside Boston than in Boston. I also believed that growth beyond Boston would teach us new lessons and open access to bigger pools of funding and talent and a bigger opportunity to influence the ongoing debate about how to improve public education.\n\nA key question was _how_ to grow. Should we expand in the common way for our sector\u2014the way of the Girl Scouts and Boys and Girls Clubs and most other national organizations\u2014which is through \"affiliates\" or \"franchises\" that are legally independent local organizations but that run similar core programs and share a brand name? Or should we just codify a set of Citizen Schools' best practices and then encourage all sorts of existing nonprofits and schools to pick up the ideas and run with them? Finally, another option available to us was to replicate as one organization, opening up what private sector analysts call \"company-owned stores,\" or branches, in new communities.\n\nMy friends at City Year argued for the company-owned-store approach, and they had good reasons that had been articulated by, among others, former Harvard Business School professor Jeff Bradach, who founded the nonprofit consulting firm Bridgespan. Bradach had studied replication in the private sector, where many large organizations like McDonald's grow through a mix of company-owned and franchise stores, and he felt that in the early stages of replication, the company-owned approach allowed for maximum control of culture and quality. This helped young entrepreneurs hone their business model. Direct replication also allowed for a tighter learning cycle in which breakthroughs as well as lessons learned from mistakes could be shared quickly across a large network. On the other hand, brilliant thinkers like Greg Dees, who helped launch the social enterprise programs at Harvard, Stanford, and Duke universities' business schools, staked out a different point of view. Dees argued that some of the most successful examples of scale and impact in American social policy\u2014such as Alcoholics Anonymous, the hospice movement, and Mothers Against Drunk Driving\u2014had spread their central tenets more like Johnny Appleseed, sharing their seeds far and wide, rather than like agribusiness giants such as E&J Gallo that control the land, the seeds, the labor, and the system of production.\n\nAs we debated these options among our board members and senior staff, my bias was to avoid the company-owned-store model. I saw its advantages. But I also remembered how hard and painstaking the growth had been at City Year. I argued that there was a difference between power and control, and that by ceding some control we might actually have more power to change education. I had lots of good arguments, but I was also just stubborn. I had experienced it one way at City Year and now, in my own organization, I wanted to try a new approach.\n\nJust as this debate was starting to percolate, Citizen Schools was approached by leaders of two groups who admired our model and our ability to engage middle school students. They wanted us to help them do something similar. The first to reach out was the After-School Corporation (TASC) in New York City, a fast-growing nonprofit that had recently made a splash when it received a $125 million gift from billionaire financier George Soros. TASC and Soros had the bold goal of bringing after-school programs to scale in New York City and then convincing the federal government to make a longer school day with extra enrichment and academics a right for all, not a luxury for the few. TASC had already helped launch or expand hundreds of after-school programs across New York's five boroughs, with the programs being delivered by a patchwork of large and small nonprofits and with a wide variety of approaches and outcomes.\n\nAs a general proposition, TASC felt that many of their programs serving younger children in the early elementary grades were working well. If nothing else, kids would stay at these programs until their parents picked them up, because the programs served as childcare in addition to providing academics and enrichment. But middle school kids could vote with their feet, and many programs serving these older children had daily attendance rates running as low as 25 to 50 percent of their official enrollment.\n\nTASC offered to pay us $100,000 to consult on their middle school program. At first this seemed like a dream come true. We had only been in existence for a little more than five years, and I could remember when our entire budget was barely more than $100,000 and we had to hustle to raise it. TASC was getting a lot of attention because of the Soros gift, and I deeply respected its founder, the visionary social entrepreneur Herb Sturz, and his partner, Lucy Friedman. The idea that Herb and Lucy would pay us to improve services for tens of thousands of kids\u2014more than ten times the number we were serving at that stage\u2014was compelling. But as we visited after-school sites and talked to the organizations running TASC-funded programs, I got cold feet. In reality, the programs we would be working with already had well-established cultures and budgets and funding streams. As we spoke with leaders of the programs about adopting Citizen Schools best practices, they would initially be enthusiastic, but then when we described the staffing patterns we believed were needed, or our program design, or the budget, we would run into resistance. \"That sounds good, but one of our funders requires us to run forty weeks of programming, so we couldn't have a January downtime for staff training,\" they would say. Or, \"That sounds good, but our staffing model relies on regular day teachers, so the most extra time we can ask them to work is one or two hours, not three hours or more.\"\n\nI concluded that our chances of getting these programs to shift to the Citizen Schools model\u2014or even to successfully adopt our best practices\u2014were slim. We thanked TASC and agreed to continue working together on advocacy for the after-school field but to defer a formal partnership for the time being.\n\nIn the middle of the TASC conversations, we were approached by another group\u2014a for-profit consulting firm, Work\/Family Directions. WFD was the brainchild of Fran and Charles Rodgers, who had taken a consulting partnership with IBM and parlayed it into a successful business that supported the family-related needs of millions of employees at Fortune 500 companies. WFD would help employees find nursing homes for their parents and summer camps and preschool programs for their kids. The business had grown like gangbusters in the late 1980s and '90s, but now it was starting to falter, and Charles and Fran and their team were looking for new lines of revenue. Their clients told them there was a dearth of programs for middle-school students, and they sensed there might be a market for a Citizen Schools\u2013like program geared to both the children of their clients and to lower-income children who lived in the communities where the companies were based.\n\nWe started a series of conversations about a potential pilot of middle school programs in New Brunswick, New Jersey; Baytown, Texas; Tucson, Arizona; and San Jose, California, all headquarters cities for major WFD clients. Each program would be run with seed money from WFD. And this time there was an extra zero on the financial offer. We would get $1 million over two years to launch these four sites, enough we felt to fully support a small high-quality program for a two-year pilot. To address the challenges we had identified with TASC, the programs would all be new. Staffing and program design would be developed from scratch so as to more fully incorporate all of the key Citizen Schools elements. Lastly, we agreed that if we moved forward with the WFD partnership, the programs would not be called Citizen Schools. Using our brand all across the country seemed like too big a risk at this stage, so we decided on the not-very-creative name of MAP\u2014for Middle School Apprenticeship Program. Citizen Schools and WFD would find local nonprofits to host and run the programs, and Citizen Schools would take the lead role in hiring and training staff and developing the curriculum, the model, the school partnerships, and a learning network to share best practices across the sites.\n\nThe whole enterprise put our reputation and our capacity on the line, but we decided to take the plunge. The funding from WFD's clients mitigated the financial jeopardy, and we felt that by not using our name but still closely monitoring program quality we would minimize any brand risk while maximizing our chances of delivering a quality program, increasing learning across our network, and expanding our influence in the fast-growing world of extended-day programs. Our first step was to organize a competition in which potential local partners vied to be selected as MAP implementation partners, making commitments we saw as essential to a smooth start. In Tucson and San Jose we ended up selecting local YMCAs. In Baytown, Texas, we partnered with the local affiliate of Communities in Schools, and in New Brunswick, New Jersey, we partnered directly with the local school district.\n\nSix months after signing the initial contract with WFD, all our nonprofit partners had been selected and we were exceeding the benchmarks we had set for staff outreach, school partnership development, volunteer recruitment, and more. We felt ready for a strong launch, and I remember inviting all the nonprofit partners to Boston and there being an air of excitement as nonprofit and education veterans from New Brunswick to Baytown shared their enthusiasm for Citizen Schools. The topic of branding came up, and someone asked an interesting question: \"I understand why you might not want to call these programs Citizen Schools, but we really love the name, and we are doing everything you've suggested a Citizen Schools program should do, and we are happy to sign a contract specifying that you can withdraw the name if we don't reach quality standards. So how about letting us use the name?\" It was a reasonable request and it affirmed our work. These people saw value in our ideas _and_ in our brand. After a few days of consultation with our board chair, Sherif Nada, and others, we decided to brand the new programs as Citizen Schools\u2014not MAP. We designed a special logo for the sites, giving our name in the big block letters of our logo and then, in smaller print below it, giving the name of the partnering nonprofit: for example, \"Presented by the Santa Clara YMCA.\"\n\nThe national partnership with WFD was followed quickly by an opportunity to replicate Citizen Schools in six Massachusetts communities with funding from Chris Gabrieli, a successful Boston venture capitalist turned civic leader. I first met Chris when he ran for Congress in 1998, vying with ten other candidates for the legendary Eighth Congressional District seat. Parts of the district had been represented by John F. Kennedy; then it was held for thirty-four years by Tip O'Neill and then by Joe Kennedy, the late president's nephew. The campaign was a rollicking affair that featured twenty-five formal debates in the spring and summer prior to the Democratic primary, which would surely decide the election in the overwhelmingly liberal district. Among the best attended of the debates was the so-called Great Debate\u2014a debate in Codman Square's Great Hall organized by two Harvard Kennedy School students and ten middle school apprentices from Citizen Schools. The Great Debate was one of my favorite apprenticeships of our early years, as it featured a recent Haitian immigrant leading the Pledge of Allegiance, middle school students helping three hundred people in the crowd register to vote, and questions during the debate asked by our middle school students and their parents.\n\nChris finished far back in the pack in 1998, but soon thereafter he created an organization, Massachusetts 2020, which had the initial goal of expanding after-school programs and eventually became a leading state and national voice for longer school days. Chris told me his experiences with Citizen Schools, including through the Great Debate, and his admiration for charter schools, like KIPP, that were pursuing a longer day, convinced him that a longer learning day was a key enabler of school improvement and that broader political advocacy for more learning time was needed. One of his first acts after starting Massachusetts 2020 was to fund a statewide middle school initiative, which in three years' time brought Citizen Schools to the Massachusetts cities of Framingham, Worcester, New Bedford, Springfield, Lowell, and Malden.\n\nIn the spring of 2002 we had run programs in Boston alone. Now it was fall 2004 and we were running programs in eleven cities, working with eleven school systems and ten different nonprofits. In Massachusetts we had a statewide footprint that helped us win a line item in the state budget, and we had a growing national footprint that helped us secure a $1 million AmeriCorps investment to support our Teaching Fellow program and deeper consideration from potential corporate partners. We had also launched a series of Reimagining After-School symposiums and were at the table with all the leaders of the field, pushing to transform after-school programs from an afterthought to a leading strategy for improved learning. In 2006 I coedited a book, _The Case for Twenty-First Century Learning_ , with articles by leading economists, business executives, and a national teachers' union head all making the case that extended learning time was a great way to build the critical thinking skills needed to thrive in the twenty-first-century workforce.\n\nWorking with affiliates helped make us national, but it also made us less agile. For example, if Citizen Schools in Tucson, where the former Teacher of the Year was running our program, discovered some new and better way to introduce math lessons, it would have taken a UN negotiator months and months of cajoling to get all of the other programs to try the same approach. Another challenge was financial. In San Jose we got a technology company involved in our work and it wanted to make a major contribution to scale Citizen Schools; but then our host organization, in this case the local YMCA, said it had targeted the same company to support its early-childhood program and asked us to back off.\n\nThe challenges of the affiliate approach mounted, and in January of 2005, right as I returned from a three-month sabbatical, I convened a Business Planning Council, a strategic planning group made up of our six most senior leaders and six more junior leaders from various parts of the organization, including from our affiliate sites. We met for a hundred hours over a five-month period and ended up making a number of key decisions, including a decision to recommit to apprenticeships as the core of our program model and to phase out our affiliate approach to replication. Going forward, all new programs would be \"company-owned\" branches. With existing programs, where possible we converted them to branches. In communities facing the greatest program or financial difficulties, we closed up shop. And in a few cases we continued the affiliate partnerships, until in 2010, the last of the Massachusetts affiliates, Malden and New Bedford, converted to become branches, with all of their Citizen Schools employees joining the mother ship.\n\nAs we wound down the affiliate program just a few years after launching it, there were moments when I kicked myself for my stubbornness in rejecting the company-owned approach at the outset. Replicating through branches might have ultimately pushed us further forward. But who really knows? Perhaps the approach we took worked out for the best. In just two years we went from being a one-city program to an eleven-city program operating across five geographically and politically diverse states\u2014a nationwide initiative that caught the attention of the White House and of corporate and academic leaders from coast to coast. We had some work to do to unwind or convert the affiliates, but we also had huge new assets and opportunities that came from the fast national expansion and from the strong programs and deep local partnerships forged in so many communities.\n\nFor me, the early years of national replication were difficult. In the startup years, most of the staff had also been my friends. We cooked and ate and drank together, and though we worked our tails off to improve the program, we were also true believers through and through. We were too new to have anyone else believe in us, so we had better believe in ourselves. As we neared our ten-year anniversary, however, and as we started to win more accolades from the press and from observers outside the organization, pockets of doubt and even skepticism started to emerge on the inside. Some staff seemed to have unrealistic expectations for what the program would accomplish\u2014finding fault if a student failed, instead of celebrating the occasions when students succeeded. Many staff also expected more guidance and handholding than we were used to providing. Sometimes I needed to channel Anita Price, our founding summer-program codirector, who would admonish potential staff to stay away if they wanted a \"paint-by-the-numbers experience.\" In addition, there was a professionalization that on bad days could seem a little bloodless. Did we really need another database, another evaluation system, another decision-making matrix, I wondered?\n\nA further challenge for me was that as we grew I became more isolated from the program. I didn't live in Dorchester anymore, down the street from one of our partner schools, as I had in the startup years; now I rarely had random encounters with Citizen Schools parents and kids on the T or at the grocery store.\n\nAs organizations grow, a natural tension emerges between the CEO and other top leaders, who are mostly \"externally facing,\" and more junior staff, who are in the trenches doing the hard and sometimes messy day-to-day work. The currency for outwardly facing people is vision\u2014the ideal the organization is aspiring to\u2014and the stories of success and nuggets of data that illustrate the vision. Talking all day about vision and success stories reinforces the natural optimism of the externally facing people and makes them feel great about the organization. Generally this optimism is essential to their success.\n\nThe currency for inwardly facing people is challenge: the schoolteacher who is upset we left their classroom in disarray, the student who is failing math class despite our extra math lessons, or the volunteer who is struggling. Talking all day about these challenges reinforces the natural practicality and skepticism of the front-line staff and keeps them hungry and focused on improvement. Generally, this practical focus on fixing what's wrong makes inwardly focused staff better at their jobs.\n\nThe problem comes when the externally focused optimists and the internally focused skeptics take their natural proclivities too far. The skeptics forget to breathe, and to notice progress, and to celebrate positive data and stories. Meanwhile the optimists often ignore challenges and give short shrift to the long distance between idea and implementation. If this chasm is not addressed, organizational culture suffers and trust declines.\n\nThese tensions may sound highly theoretical, but at Citizen Schools\u2014and at City Year too\u2014we had recurring rounds in what sometimes seemed like a fifteen-round heavyweight fight between me and a few others as the externally focused optimists and a rotating cast of characters serving as internally focused skeptics.\n\nI found a helpful refrain when trying to balance optimism and real challenges in the Stockdale Paradox, recounted by business guru Jim Collins in his book _Good To Great,_ a bible to many leaders in the social sector. The paradox comes from Admiral James Stockdale, the longest-serving prisoner of war in Vietnam, whom Collins interviewed for his book.\n\n\"Who didn't make it out?\" asked Collins.\n\n\"Oh, that's easy,\" Stockdale said. \"The optimists.\"\n\nCollins was incredulous. \"The optimists? I don't understand.\"\n\n\"The optimists. Oh, they were the ones that said, 'We're going to be out by Christmas.' And Christmas would come and Christmas would go. Then they'd say, 'We're going to be out by Easter.' And Easter would come and Easter would go. And then Thanksgiving, and then it would be Christmas again. And they died of a broken heart. . . . This is a very important lesson. You must never confuse faith that you will prevail in the end\u2014which you can never afford to lose\u2014with the discipline to confront the most brutal facts of your current reality, whatever they might be. . . . I never doubted not only that I would get out, but also that I would prevail in the end and turn the experience into the defining event of my life.\"\n\nAs a leader I had lots of people sharing brutal facts with me\u2014or at least what they thought were brutal facts. I needed to face them and address them, but also maintain\u2014and share\u2014ultimate faith that we would prevail.\n\n# CHAPTER SIX\n\n# THE EDDY IS READY\n\nIn 2006, Boston's Clarence Edwards Middle School (known locally as the Eddy) was a year away from closure. The school had a strong leader and an improving cadre of teachers, but it was caught in a vicious cycle familiar to urban schools. Weak test scores and a well-deserved reputation for violence led families who could make a choice to go elsewhere. The resulting lower enrollment had two consequences. First, because the school had open seats, difficult students who had been kicked out of other schools, or who were newly arriving in the district midyear, were assigned to the Edwards. Second, lower enrollment meant layoffs of the newest and often most energetic teachers. The consequence was akin to a death spiral, and the Edwards, in spring 2006, latched on to what its leaders saw as the last best hope to save the school: a state-funded Expanded Learning Time (ELT) initiative to reimagine the length and design of the school day.\n\nThe decline at Edwards had been gradual through the 1980s and '90s and then accelerated at the turn of the century. Cindy McKeen, who started teaching arts and theater at Edwards in 1999\u2014and was previously a private school teacher\u2014recalled the mood in the school when she arrived. \"I came here in 1999 and we had a pretty strong principal, Chuck McAfee. He was a disciplinarian, and the collegiality among adults was strong,\" said McKeen. \"But our kids just came from so much sad and so much bad that it was tough. I remember one of my students was a drug dealer. He said he was sure that before he turned twenty-one he was going to die or be in prison. He didn't know how to reach out for anything more. The frustration and rage was just so high.\" McKeen saw arts and after-school programming as \"a sanctuary\" for many Edwards students, but one that was overwhelmed by the chaos of the school and their lives at home.\n\nIf the Edwards was challenging when McKeen arrived, it soon got worse. In 2000, Principal McAfee was reassigned by the district to lead a high school turnaround effort in another part of the city. The new principal selected for the Edwards, while academically qualified, lacked street smarts and leadership skills, according to those I interviewed. Underperforming teachers were not coached or held accountable and student frustration bubbled over. Enrollment continued to decline as parents\u2014perhaps newly sensitized by the MCAS to the importance of school selection\u2014increasingly steered clear of the Edwards. These years also saw a notable spike in violence, with forty-seven officially recorded assaults on teachers by students over a three-year period, according to McKeen. \"I myself was punched in the neck in the cafeteria and was sent to the nurse's office three times, and the kids liked me,\" said McKeen.\n\nAn Edwards student at the time, Yoelinson Castillo, who now works for Citizen Schools at the Edwards, confirmed the chaos. \"My first day at the school there were three fights in my class. Kids were smoking weed in the bathrooms, and I had a teacher whose main activity every day for English class was a game of hangman,\" Castillo recalled. Castillo's eighth-grade year was better, as he landed with a skilled teacher who told him he was smart and helped him move from a special-needs class to mainstream classes. But looking back a dozen years after leaving the Edwards, Castillo mostly remembers the chaos, and he can quickly list friends and classmates from the Edwards who are now in jail, on the streets looking for work, or barely getting by with minimum-wage jobs.\n\nStephanie Edmeade, a current Edwards teacher, arrived at the school in 2000. She hoped to be part of a small group of teachers who could help turn the school around, but she faced resistance.\n\n\"I remember standing up in front of a staff meeting in the teachers' lounge and suggesting we institute a community-service program and people laughed at me,\" said Edmeade. \"Another time I talked about exemplars of advanced writing that I hoped all the teachers could use\u2014to give the students something to stretch for\u2014and one teacher said she didn't want her students to be discussing something they can't do.\"\n\nEdmeade and others said the most tragic moments in those years at Edwards were revealed through discussions with students who, despite the environment, were hungry to learn. \"I remember talking to one student who was on the student council,\" said Edmeade. \"She turned to me and said, 'I want to pass the MCAS, and I am not being taught.'\" Edmeade visited one of the student's classes and sure enough, she said, the teacher had his feet up on the desk and was reading the newspaper even as students played cards or stared out the window.\n\nAll of Edmeade's uncles and aunts were teachers, and they wondered why she was staying at the Edwards. \"'Why don't you go to a charter school?' they would say.\"\n\nSix years after Edmeade arrived at the Edwards\u2014enough time for two cohorts of sixth graders to make their way through middle school\u2014and after yet another change in the school's leadership, the Edwards embraced many of the reforms adopted by the nation's leading charter and private schools and made them work within a traditional district school. The school's dismissal time moved from 1:30 to 4:30 for every child. America's oldest public school system, which had offered about a six-hour learning day since Boston Latin School opened its doors in 1635, would now offer a nine-hour day to students in perhaps its neediest school and in two others across the city. And if that common-sense move wasn't a bold enough break with the past, the Edwards also resolved to reengineer the curriculum and the delivery system for the entire school day. Specifically, it engaged a \"second shift\" of paid and volunteer educators mobilized by Citizen Schools and other partners to complement the first shift of \"regular\" teachers.\n\nTogether Citizen Schools and the Edwards created math leagues, in which every student in the school added a daily hourlong math lesson and competed in math tournaments to practice their skills. The school also added an array of extracurricular offerings that rivaled the most elite private and suburban schools. The Edwards had a challenging past. But now through Expanded Learning Time, or ELT as it was known, the younger siblings and neighbors of those earlier Edwards students would have access to extra academic coaching, to the city's first middle school football team, to dance and theater troupes, to mock trials and filmmaking, to astronomy and Web design, and, eventually, to rocket science.\n\nMike Sabin was the key protagonist of the Edwards turnaround, serving as the school's principal from summer 2002 through the completion of the first year of ELT in 2007. Sabin's successor, Jeff Riley, helped make ELT at Edwards famous and took the school's achievement to the next level, but it was Sabin who wrote the playbook and coached the team through the planning year and the first critical turnaround year.\n\nSabin grew up in suburban Wellesley, Massachusetts, and was educated at Milton Academy and then Harvard. While still in college he committed to a career in urban education, and after graduation began work as a bilingual education teacher in Cambridge. Later he taught in and then became assistant principal of a Spanish immersion school in Lawrence while also earning his principal certification at Harvard's Graduate School of Education. By spring 2002 Sabin was looking for a school he could lead. He heard about the principal opening at the Edwards but initially didn't think he would apply as his heart was set on leadership of a bilingual school where he could use his Spanish fluency. A mentor persuaded him to show up for the Edwards interview anyway, and he ended up being offered the job. He was thirty-four.\n\nSabin launched his tenure as Edwards principal by meeting with every teacher one-on-one and trying in his simultaneously bookish and relentlessly upbeat manner to understand deeply the curriculum and to mold the teaching of his team. A friend at Harvard introduced him to Tony Helies, a tough-talking retired technology executive and a Harvard Business School graduate. Helies helped Sabin form an advisory board to provide management guidance and attract resources to the school.\n\n\"It was obvious from the beginning that if the school wanted to go from being the worst in the city to one of the best, that it just wouldn't cut it to have kids going home at one thirty in the afternoon,\" said Helies. \"Mike and the rest of us started trying to raise money to build a bigger after-school program, and in some ways that helped set the stage for the ELT program that came a few years later.\" At the same time, said Helies, Sabin would never have succeeded if he didn't know how to move out the very worst teachers, a management task often thought futile in unionized urban schools. \"We wouldn't have worked with Mike if he didn't know how to fire bad teachers and hire good ones. Principals before him were hall monitors with walkie-talkies moving from incident to incident. They didn't pick the curriculum and they didn't do anything to change the teachers.\"\n\nIn his first three years, Sabin stabilized the school. On a staff of about forty, he was able to make almost twenty new hires. He tightened the curriculum, lengthened core classes from sixty to eighty minutes, increased common planning time for teachers, and added a deeper focus on both math and the arts. Closer management of student misbehavior and, perhaps, the greater offerings of art and after-school electives started to reduce violence in the school. But while the climate at the Edwards was improving gradually, test scores remained abysmal. And parents and students were still voting with their feet. Every spring, parents in Boston put in for their top choices of schools, and for Sabin's first years at the helm only a handful put Edwards on their list.\n\nAs Sabin entered his fourth year, the 2005\u20132006 academic year, the national drumbeat for school accountability was getting louder. He sensed that if dramatic improvement didn't come soon, the Edwards would close.\n\nIt was at this time\u2014in late fall 2005, that the Massachusetts Department of Education and a nonprofit\u2014Massachusetts 2020\u2014announced a grant competition to provide funding to schools that wanted to add at least three hundred hours of learning time to their schedule. Winning schools would receive an extra $1,300 per student of state taxpayer money to transform learning and results across the school day by adding more academics, more enrichment, and more planning time for teachers.\n\n\"ELT came along just as proponents of standards are saying standards-based education systems are necessary but insufficient to get all students to proficiency,\" said Paul Reville, then the Massachusetts secretary of education and chair of the state Board of Education. \"Even an optimized educational setting is not enough to equalize opportunity for children of poverty in comparison to their affluent peers. Clearly, economically disadvantaged children need more.\"\n\nAs soon as Sabin heard about the ELT grant, he was excited to apply. And since he believed that change in urban schools generally came through collaborative leadership, he began engaging his teachers and outside partners, including Citizen Schools, in an intensive planning process. Whereas most schools applying for the ELT grants planned to staff the extra learning time by assigning their teachers to work an extra ninety minutes per day (for extra pay), extending the length of core academic classes, and bringing in a few community partners to team up with teachers in offering new electives, Sabin engineered a solution that was bolder. He extended the schedule by a full three hours Monday through Thursday and opted for a shorter day on Fridays, allowing for teacher professional development starting at noon. Whereas most other schools added ten to fifteen minutes to existing academic classes and also added a forty-five- to sixty-minute enrichment block, Sabin added a full hour of math academic league and a full two-hour elective block every day!\n\nThe changes at the Eddy were transformative, allowing students to enroll in two apprenticeships or electives per semester, including offerings such as rocket science and video game design, mock-trial classes that involved field trips to downtown law firms, and sports and a chance for full-length practices. Boldest of all, Sabin turned delivery of the extra time 100 percent over to Citizen Schools for all of his sixth graders and also deeply engaged Citizen Schools and other community partners in co-delivering the extra time to seventh and eighth graders.\n\nAn exciting development for Citizen Schools was the deep buy-in of teachers to the partnership. As stated in the school's ELT application to the Massachusetts Department of Education: \"Our 6th grade teachers voted unanimously to collaborate with Citizen Schools in the Extended Day, agreeing that the entire 6th grade class of 120 students will participate in its research-based, structured model of community building, project-based learning, and community service. Leveraging this partnership provides us with appropriate programming for the students and also sets the school up for a more unified culture in subsequent years.\"\n\nSabin told a reporter at the time that \"Citizen Schools brought a willingness to collaborate with the school and to modify certain elements of their program as long as the big picture remained true to their vision. They brought a determination to make things work at the Edwards and a feeling that 'these are our students.' They didn't try to hand off problems to the school when they arose, but took responsibility for trying to solve them.\"\n\nWithin Citizen Schools, the move to ELT provoked a new round of soul-searching of the type we had done back in 1997 when Dever principal Nydia Mendez suspended her work with us until we could align better with school-day academic goals. Now we needed to not only align with the school, we needed to become part of the school. In addition to helping with homework and study skills and leading field trips to college campuses and co-leading apprenticeships, our AmeriCorps teaching fellows would now deliver an hour of math lessons every day\u2014following the same script as veteran math teachers down the hall. We would give grades too, and in an effort to create a unified culture across the full school day, we started having students address our staff more formally by their last names, rather than by their first names. Quite explicitly, Mike Sabin and his team were inviting us, and challenging us, to lift our game to a higher level and be held accountable as professional educators.\n\nSome teaching fellows struggled with the higher expectations and the need to present a more teacherly demeanor. Many had come to Citizen Schools to get away from traditional schools. They loved our hands-on learning projects and the enrichment and community connections that infused our model. But not all felt prepared to teach math lessons for even one hour a day\u2014and some did it poorly. On the other hand, many of our staff worked hard to become solid and, in a few cases, excellent instructional leaders. They followed the path of first- and second-year teachers everywhere, experiencing tear-inducing failures in their first months, but also moments of exhilaration when a well-honed lesson enabled a child to finally grasp how to multiply fractions, or to see the relevance of a persuasive essay to winning a law case, or of basic algebra to programming a video game.\n\nOver time Citizen Schools would become better at \u00adsupporting academics while still drawing heavily on our core competency of helping students experience success through hands-on projects that connect academics to future careers. We changed our recruiting message to prospective AmeriCorps members to be clear they would do \"real\" teaching along with lifting up opportunity in other ways. We discovered new ways to support teachers and to learn from them by team-teaching in overlapping blocks, sharing assessments of student learning, and more (see chapter 12 for more on this topic). As I write, Citizen Schools now graduates a majority of its AmeriCorps teaching fellows directly into full-time teaching roles. We have developed two exciting programs for certifying teachers\u2014creating a \"teaching hospital\" model in which teaching fellows can earn their teaching certificate while leading extended-day classes and then transition seamlessly into full-time teaching roles.\n\nDespite all of Sabin's good planning, it was unclear in those first months of ELT whether the school had made a good choice. There were minor conflicts between Citizen Schools and traditional teachers over damaged blackboards and divergent approaches to discipline. In mid-fall there was a racially tinged showdown between Citizen Schools students and neighborhood toughs from Charlestown, which was then fast gentrifying but still famous for its concentrated pockets of white poverty and spectacular bank robberies (such as those featured in the 2010 Ben Affleck movie _The Town_ ). Finally, and tragically, in December and January there were successive murders of two Edwards students, Emmanuel \"Benji\" Saintil, fourteen, and Luis Genera, thirteen. The two students were killed in their home neighborhoods in incidents that were unrelated to the school and occurred miles away, but that nonetheless delivered to the school what could have been a knockout blow.\n\nMoriska \"Mo\" Selby was just twenty-four in that first year of ELT at the Edwards. She was in a pivotal role, acting as campus director for Citizen Schools, overseeing the final three hours of learning for all sixth graders and supporting many of the seventh and eighth graders as well. Selby, a native of the tiny Caribbean island of St. Vincent and then of Brooklyn, New York, was just two years out of Tufts University, but had already worked three years for Citizen Schools. I had pitched her on taking the Edwards job and remember seeing in her facial expression excitement for the opportunity but also concern that the experiment wouldn't work.\n\n\"Those first few months of ELT at the Edwards were so hard,\" Selby recalled. \"Kids would just walk out on you. 'You're not a real teacher,' they would say.\" It reached a low point mid-fall, when an average sixth grader, a boy Selby pegged at six feet two inches tall and almost two hundred pounds, pushed Citizen Schools staffer Caroline Beasley up against a wall of lockers. \"This boy was just so angry, and we tried explaining all of the opportunities we were trying to bring to the school with the longer day, but he just didn't want to hear it,\" said Selby. \"So after we huddled with the school leadership we decided that for the safety of the students and of the staff we just needed him to go home a little earlier and we needed to focus on everyone else.\"\n\nThat decision, said Selby, became a turning point for Citizen Schools and the school as they realized they could work through difficult decisions together.\n\nAnother turning point came that spring, when the Citizen Schools staff formed a Student Leadership Council of thirty students and took them to a camp in rural New Hampshire, skipping three days of school in early June in hopes of cultivating role models among the school's older students. The trip was part of an effort to ease tensions between the major subgroups of the school\u2014the general education population, students in the Chinese Sheltered Immersion program for new immigrants learning to speak English, and students with serious social\/emotional and learning disabilities. The three groups attended separate classes in the regular school day and came together just at lunch and recess and in the extended-day hours with Citizen Schools.\n\nThe camp in New Hampshire was run by a Russian man named Misha, and it was a challenging and foreign experience for the Edwards kids. For starters, no electronics were allowed, no technology, not even wristwatches. There was no fast food either, and all meals came entirely from the farm on the premises, including eggs the students would collect from a henhouse. Capitalizing on the foreign environment, Misha and the Citizen Schools team introduced a series of team-building and leadership activities, including canoeing on a large lake. One Puerto Rican student, Chris, who was a good athlete and a future member of Junior ROTC, was matched in a canoe with a tiny Chinese student with serious social\/emotional and academic learning disabilities. Chris suffered from his own learning challenges but told counselors he felt proud after managing to paddle six miles despite his partner paddling mostly in the opposite direction.\n\n\"It was an incredible leadership experience and it helped to draw together a group of students who normally never interacted in the school,\" recalled Emily Bryan, a Citizen Schools teaching fellow who helped organize the trip. (Emily later became an award-winning sixth-grade English teacher at the Edwards.)\n\nDuring their three days at the Marlow, New Hampshire, camp, the Edwards kids also had a chance to work with knives, learning how to carve wood. During one of these sessions the counselors noticed that a knife had been stolen.\n\n\"Misha, the camp director, sat everyone in a circle,\" recalled Bryan. \"He didn't yell. He said, 'It's not your fault that you live in a world where people steal, where you have to worry about your things being stolen, where you have to worry about violence. But if you are also lying and stealing then you are adding to this problem. If you want to live in a world without lying and stealing then you must be the one to change the world. You have to change yourself.'\"\n\nMisha then pointed down the field where they were sitting to his car and said to the students that if they needed anything, the car was open and his wallet was on the front seat. \"The knife got returned that night,\" Bryan recalled, adding that she learned a lesson that day about creating an environment of trust and support and high expectations that she has carried into her classroom ever since.\n\nThroughout the year, Sabin persisted in knitting together his traditional staff with the Citizen Schools AmeriCorps members and volunteer citizen teachers. By graduation in June, there were signs that the investment was paying off. Attendance was up across the school. Sixth-grade attendance had risen from 90 to 93 percent, an increase that may sound small but means the average student attended an additional week of school, compounding the extra learning time offered by the ELT model. Still, the Edwards had seen signs of momentum before only to be disappointed when the MCAS scores were published. This time it would be different. Math proficiency in sixth grade more than doubled, from 15 percent, the lowest in the city the previous year, to 32 percent, slightly above the city average. Test scores in almost every subject and grade were up by sizable margins.\n\nAfter two more years of ELT, the Eddy was transformed. Whereas in 2005 only 17 families had chosen the Edwards in Boston's open-assignment process, in 2008 more than 450 families chose the school, creating a large wait list. By 2008 the school had a football team (the first middle school team in Boston), numerous arts programs, diverse apprenticeships offered through Citizen Schools, and an extra hour of math games and instruction every day. The Edwards had erased more than 80 percent of the achievement gap in English and science (the gap between its students and state averages), and in math it had created a reverse achievement gap, outscoring such middle-class communities as Framingham, Waltham, and Watertown. Meanwhile, the school continued to serve many of the highest-need students and families in Boston, as the percentage of Edwards students living in or near poverty had actually grown to more than 90 percent, and the share of English-language learners and special-education students continued to be high.\n\nI recently returned to the Edwards to spend a day observing classrooms and meeting with teachers and students and with the new principal, Leo Flanagan. Principal Sabin, the ELT architect, had left after the first year of implementation to bring his family to Central America, where he helped to run a school for two years, before returning to Boston to take on another ELT partnership with Citizen Schools. Jeff Riley, Sabin's talented successor, had run ELT at the Edwards for two years and then been promoted to become Boston's assistant superintendent for middle schools, a role that allowed him to help spread ELT to other schools across the city. In 2012 Riley was tapped by the state as superintendent for the troubled Lawrence Public Schools.\n\nI had visited the Edwards a dozen times over the previous decade, including in January 2008, when Senator Edward M. Kennedy announced the TIME Act, a legislative proposal to replicate the Edwards ELT model across the country. But I had never spent a full day at the school\u2014a day starting at 7:00 a.m. as the first buses began to arrive and running until 4:30 p.m. as the last students left for home.\n\nMy previous visits allowed me to see snapshots that illustrated the story of a transforming urban school in a big-city district\u2014a transformation that, while still fragile and incomplete, has nonetheless provided hope to education reformers in Boston and beyond. The Edwards success story has been touted by business leaders and charter advocates, but also by teachers' union president Randi Weingarten, who stated in a 2011 _New York Times_ column that the Edwards was \"one of the most impressive schools I have ever seen.\" Now I wanted to understand the texture of the school and its makeover more intimately.\n\nIf the Edwards could turn around its fortunes, could others do the same? In a country where an estimated 50 percent of the dropout crisis and a majority of the achievement gap runs through just a few thousand middle schools, replicating the Edwards story even a few hundred times could start to move the needle on national educational results. Replicating the story a few thousand times would rocket the United States from below average in international rankings of student learning to the top tier.\n\nI got to the Edwards at seven in the morning\u2014just as the first students arrived for a breakfast of TruMoo chocolate milk and premade sausage, egg, and cheese sandwiches. I met with veteran teachers like Cindy McKeen and Stephanie Edmeade and with newcomers like Emily Bryan. I spoke with students too, including eighth grader Adam Barriga (the student who spoke at the State House, described in the introduction). Barriga sported a mop of long and bushy brown hair but held a clear vision of his future as an engineer\u2014a vision he said was inspired in large part by his sixth-grade rocket-science apprenticeship with David Mantus. Over the course of the day I developed a deeper sense of what made the school tick.\n\nLeo Flanagan, the principal at Edwards from 2010 to 2013, said he was most proud that his students now feel like individuals. He described a conversation with a student who had transferred to the Edwards after experiencing bullying at another middle school. \"'Mr. Flanagan' she said, 'You have black dorks here.' She is right,\" said Flanagan. \"We have big black kids wearing pink shoes because they have been here for a while and they have found some things they are really passionate about and they are willing to be themselves. That is just huge for kids.\" Flanagan described a particular student he believed was gay who joined the cheerleading squad and flourished. \"He would have turtled at most other schools,\" said Flanagan.\n\nThe Edwards's test scores have stopped rising, but the school has retained most of the gains made in the first three years of ELT. Annual student growth scores remain high in most subjects, often higher than for some well-regarded charter schools in the area. Despite calls to increase the focus on straight academics, Flanagan has insisted on continuing the school's emphasis on enrichment. \"I think what we are doing is really courageous, because I believe the future of kids is really about the experiences of kids. I want kids to have the experience of being in rock bands, of conducting real experiments with scientists, of arguing a mock trial down at the federal courthouse in front of a federal judge,\" said Flanagan. \"What would they do in the suburbs? Do you think they would ever just get rid of arts and music and other electives so they could focus all their attention on math? Part of how we drive things at Edwards is to ask, 'Would this fly in Brookline?' When we do things for poor black kids that we would never do in the suburbs, then I wonder.\"\n\nDespite his faith in the Edwards approach\u2014and in the possibility of creating good urban public schools\u2014Flanagan says he sometimes wonders whether he made a mistake in choosing to labor in a big-city school system like Boston that has been the subject of so much criticism. \"I see those charters out there offering a longer day and with full control of their teaching staff\u2014and with kids all signing up for a lottery to get in\u2014and I sometimes wonder,\" Flanagan said, \"if we are on the wrong side of history here in the district schools and if we are just doomed to failure. But then I wonder if, ironically, it's more cutting-edge to have stayed in the district. And I see the work we are doing here with ELT at Edwards and see how it's now spreading to more district schools across Boston, and it feels like we are building a pathway for lifting up opportunity and achievement for the hardest-to-reach children across this city and across the country.\"\n\nAmong the teachers of whom Flanagan is proudest is Emily Bryan, the former Citizen Schools teaching fellow. Bryan is one of six former Citizen Schools staff members who have been hired by the Edwards for teaching or administrative roles, creating a built-in talent pipeline that has served the school well and strengthened our partnership. On a recent visit to the Edwards, I sat in as Bryan taught her second sixth-grade English class of the day in Room 102, overlooking the school's drab concrete playground and parking lot.\n\n\"Okey-dokey,\" said Bryan, who sports short-cropped black hair and was sucking on a bright red lollipop as she welcomed the class. \"All you need on your desks right now is a pencil. The 'Do Now' is coming around.\" Bryan's first assignment was a short vignette from _The House on Mango Street_ by Sandra Cisneros. She asked the students to read the passage and then pick out examples of metaphor, personification, and figurative speech. As the students read, Bryan played music in the background. Then after a minute or two of quiet reading she began to banter with the students as they answered questions listed on the \"Do Now.\" \"Why do you like this passage?\" she asked. \"What makes it good? Do you see the sensory detail we were talking about the other day?\"\n\nBryan asked for an example of sarcasm from the text and immediately two girls raised their hands. She called on one and when the student gave a good example, Bryan affirmed her with praise and then tossed her a lollipop from across the room. \"In my last class I shared a really good writing passage from one of my former students\u2014from Emily Restrepo,\" Bryan announced. \"Emily's younger sister was in the class\"\u2014here Bryan paused for emphasis\u2014\"and you can imagine she was _thrilled_ that I was talking about her sister. That's sarcasm,\" Bryan continued without missing a beat. She took a few more comments on the Cisneros passage and then told the class it was time to get to work on their own writing.\n\nAs a first step, Bryan modeled the writing of a short passage herself. \"Okay, I want to write about a hot day and a sprinkler,\" she told the class. \"I want to use metaphor or simile and personification and figurative speech.\" Her laptop was connected to a projector, and as she began to write, every student paid attention as the first draft of her vignette began to unfold on one of the classroom walls.\n\n\"It was a hot day so we played in the sprinkler. We got really wet,\" Bryan wrote.\n\n\"That's not very good is it?\" she asked. Twenty-two heads shook side to side, eyes still riveted on Bryan. \"No, it's blah. How can we improve that with dialogue, with figurative language, and with sensory detail?\"\n\nBryan started over, twenty-two sets of eyes darting back and forth between her and the words projecting on the wall. \"It was a hot July afternoon. Eric and I danced in the sprinkler like ballerinas onstage. We screamed 'It's cold! It's cold!' as the wet grass stuck to our ankles and between our toes.\" In a little more than a minute Bryan had hooked the class and modeled a powerful vignette. Now the young writers themselves took pen to paper and began to work, eagerly. As they did, Bryan turned to me, saying in a stage whisper, \"These guys are freakishly good.\"\n\nBy the time the class was over, almost every student had earned at least one lollipop, and a good measure of praise, and I had heard some quite impressive writing from a class full of eleven- and twelve-year-olds\u2014half of whom didn't even speak English at home.\n\n\"Sometimes I go to the fountain and it calms me down,\" wrote Francesca. \"I smell the salty waters of the ocean. Trees dancing around me. The birds watching my every move. . . . I can hear the wind whistling into my ears, the park calling my name, 'Francesca, Francesca, come to play.'\"\n\nI was inspired. I had heard that Bryan's students had improved more in their MCAS writing and reading comprehension scores in the previous year than all but four other sixth-grade classes in the entire state, and now I knew why. She built relationships with her students; she challenged them to do great work, first modeling excellence and then coaching as students practiced their skills; and she made learning fun\u2014all principles at the core of the Citizen Schools approach.\n\nFull-time teachers like Bryan and leaders like Sabin and his successors as principal were critical to the Edwards turnaround, and a key part of the value that Citizen Schools added was the full-time AmeriCorps teaching fellows\u2014a job held right after college by future educators like Emily Bryan and Moriska Selby and dozens of others. Yet equally important factors in the school's turnaround were the hundreds of volunteer citizen teachers who led apprenticeship courses\u2014people like David Mantus. Mantus taught rocket science seven times at the Edwards, forging hundreds of relationships with students like Adam Barriga. Tony Helies, the retired businessman who advised Principal Sabin in the early years of the Edwards turnaround, taught astronomy at Citizen Schools fifteen times, including multiple times at Edwards; and Alan Su, a whiz kid engineer at Google's Cambridge office, taught computer programming classes five times at the Eddy.\n\nSu is such a talent that even as a relatively young engineer Google tapped him to help lead the team that made YouTube (a Google subsidiary) faster, helping the company \"live-stream\" every event at the 2012 London Olympics. But teaching was tougher than Su expected. He taught his first apprenticeship at the Edwards the first year of ELT, and Su remembers \"feeling a little naive\" when he realized the Web design curriculum he had put together was over the heads of most of the students in his class and that many of the students didn't seem motivated to learn.\n\nDespite the challenge, Su committed to teach again. His next course involved creation of a public service app for the Android phone\u2014an app that would enhance the experience for participants in First Night, a popular Boston arts festival. \"The idea was to make First Night more interactive and drive up attendance. You could imagine a clue on the ice sculptures and then people would have to take a picture to show they were there,\" related Su. \"The students learned the basics of putting together a user interface and basic computer programming, and by the time they were done they had the bones of a really cool app.\"\n\nSu has since come back to teach three more times and said he has seen a transformation at the Edwards and in its students. \"It's palpable, the difference in the attitude of the kids. It's night and day. There is no more 'too cool for school,' and now the students see that learning can be really fun.\" Su said that his personal motivation to volunteer has grown too. \"The thing I love is the combination of the boots-on-the-ground experience\u2014the experience of introducing kids to something out of the ordinary and helping them realize that the Web is not just something to consume from, they can be creators too. I love the combination of getting to know the kids and also seeing the results at the Edwards, which are incontrovertible. I also love that I have a role in addressing one of the greater inequities in our society.\"\n\nHelies and Mantus also stood witness to the transformation at the Edwards. Mantus's first course was in 2007 at a Citizen Schools campus in Boston's Brighton neighborhood, and it was an ill-fated effort to get the students to develop a pathway for bringing a new drug to market. The course was authentic, as that's what Mantus did for a living, but the lessons lacked pizzazz and didn't appeal to students. Mantus remembers several co-teachers from his firm vowing they would never return.\n\n\"The only part of the class that really resonated was the day we had a paper airplane contest and the kids all learned about the physics of flight,\" said Mantus. \"I realized that what I was really going for was to get kids jazzed about science and technology, and the key to that was just getting them excited about the fundamentals of chemistry and physics. I had a friend who worked at the Challenger Learning Center in Wheeling, West Virginia, and I called her up and she said she would love to collaborate with me on a course.\" Mantus went to work and designed a class in which his students learned the fundamentals of space travel and ended the semester with a live video hookup with astronauts from the Challenger Center. It was an instant hit. Mantus and his colleagues have now taught the class to more than one hundred students from the Edwards over the last five years.\n\nI knew that Mantus's motivation for teaching came from his desire to spread the wonder that he first experienced launching rockets in the backyard with his dad and going on trips with his grandfather to the New York Hall of Science in Queens, near their Long Island home. When I met with Helies, a retiree who had taught fifteen apprenticeships and volunteered in other ways to help the Edwards, I wondered what had motivated him to get so involved.\n\n\"Part of why I do this,\" Helies said, \"is that I was raised by a single mom who had four kids in a tough neighborhood in Brooklyn, and she just worked incredibly hard to get all her kids scholarships to private schools. I didn't get a lot of hugs growing up, but I got an education that changed my life. I am here today because of that, and God bless my mother for the aggressiveness she had in helping her kids get a good education.\"\n\nHelies graduated from the Trinity School on Manhattan's Upper West Side and then went to Northeastern University, where he earned a degree in electrical engineering. He worked on the Apollo project from 1967 to 1971 and then was laid off as the space program began to downsize. He earned a degree from Harvard Business School, married, and started raising a family, while also founding a company that did network management for large corporate computer systems. Helies never got involved in nonprofits during his professional life, but after selling his company, his interests turned to K\u201312 education. He hoped to provide to others the education that his mother provided for him. Soon he met Sabin and helped him with management advice, but he also began tutoring kids in math, which he loved and was good at. When Helies asked to do more with students, Sabin suggested he get involved with Citizen Schools.\n\nHelies decided that at Citizen Schools he would teach astronomy, a topic that had always fascinated him. He organized a course in which the students used the same simple instruments as the ancient Greeks to measure the size of the planets, their distance from each other, and the orbits of their moons. Helies was a good teacher, so he usually was assigned more than his share of discipline challenges. Maybe as a result, he found the classes difficult but also rewarding enough to keep coming back. \"Tutoring one or two kids is easy,\" said Helies. \"Teaching fifteen or twenty kids is much tougher. You need to develop a relationship with the kids, because people treat people who they have a relationship with much better.\" Helies also found ways to weave in exercise to his classes, which he found kids hungry for more of. \"I would have the girls race the boys to map out the solar system and they would just go nuts. They were so excited and they needed that chance to blow off steam.\"\n\nLooking back, Helies recalled a number of students for whom astronomy provided a breakthrough during their difficult middle school years. One student who was failing several of his classes initially told Helies that astronomy was stupid. Then at the end of the semester he handed Helies a note saying: \"Dear Mr. Helies, I used to hate astronomy. Now I like it.\"\n\nA few days after we spoke, Helies wrote me with another example of a rewarding experience from his class.\n\n\"One of our more challenging assignments is to determine the orbital period of Charon, Pluto's large moon. This is pretty sophisticated stuff for sixth graders,\" Helies wrote. \"We used photos taken by NASA of Pluto three days apart, showing that Charon had moved halfway round during that time. From that data we figured out the period, six days. For our WOW! we were having an 'Astronomy Conference' in the Boston College physics department, with the students serving as the conference presenters. BC does a wonderful job setting up the conference, with badges, conference programs, even cookies and juice. Professor Michael Graf and his grad students are the attendees. One of our quieter students volunteered to present about Charon's moon and its orbit. He had been quite shy during the apprenticeship, so I was pleased to see him take the initiative. Charon's orbit is six days, but the data we had could also be interpreted as a two-day orbit. (Can you see why?) Professor Graf challenged the six-day number and soon he and the student were both up at the white board, with the student drawing orbits! The student was able to explain why the data was inconclusive, show that both six days and two days were possible, and proposed that we could find out for sure with more photos. I felt like I was at a real astronomy conference. Well, I guess I was.\"\n\nThe Expanded Learning Time model we piloted at the Edwards sparked a transformation at Citizen Schools. We stopped opening new after-school programs, formerly the bread and butter of our organization, and instead looked to replicate ELT programs where we could work with every student in a middle school, or at least every student in one or two grades. While in 2006 I had been nervous to make the jump from optional to required programming at Edwards, by 2009 I was convinced that ELT\u2014done right\u2014could become an essential new paradigm for schools in the twenty-first century.\n\nIn the four years since the Edwards success was established, we have scaled ELT to a national cohort of twenty-four formerly struggling schools\u2014from East Harlem to East Palo Alto and from the west side of Houston to Chicago's South Side. Results in individual schools have varied, but on average ELT schools partnering with Citizen Schools have significantly increased student engagement (attendance, staying out of trouble, and seeing the link between school and careers) and increased proficiency rates on state tests by more than ten percentage points within two years. Teachers and parents give the program high ratings for academics and for offering a well-rounded education. The progress at ELT schools means that for students who attended these schools, who started middle school about two years behind their middle-income peers, half or more of that gap has been cut while they are still in middle school. Our hope is that students have also learned better school navigation skills like how to ask for help, how to advocate for themselves, and how to work toward a goal on diverse teams. We believe the academic momentum as well as the social skills and social networks Citizen Schools students build will help close most or even the entire remaining achievement gap before adulthood. In fact, in the two locations where we have longitudinal data through high school\u2014Boston and Charlotte\u2014low-income Citizen Schools graduates fully eliminated gaps with their middle-income peers in on-time high school graduation and in college enrollment, though not yet in college completion.\n\nIn Oakland, where I started my career as a journalist, the fastest-improving middle school in 2011\u20132012 was a Citizen Schools ELT school. And in our home base of Boston, Orchard Gardens, formerly one of the lowest-rated schools in the entire state, had such a dramatic turnaround through ELT and other reforms that it recently boasted the highest scores for student-learning growth of any middle school in all of Massachusetts. The latest report on our ELT work from external evaluator Abt Associates indicates that, on average, Citizen Schools ELT schools are delivering an extra three months of learning in math per year and an extra five months of learning per year in reading and writing compared to similar schools, while also boosting engagement in learning and interest in careers.\n\nThe second section of this book starts to unpack the key success factors of Citizen Schools\u2014the key components of the opportunity equation\u2014and suggest how we and others might bring them to millions more children.\n\n# CHAPTER SEVEN\n\n# SCALE, SPREAD, AND THE PURSUIT OF SYSTEMIC CHANGE\n\nIn the early days at City Year, founders Alan Khazei and Michael Brown and I would sit around our partially renovated warehouse space and discuss ways that City Year and national service could change the world. We talked a lot about Steve Jobs and Apple Computer. This was in the early 1990s, before the iPod, the iPhone, and the iPad. Apple had nowhere near the commercial success it has today; in fact, Jobs had been booted out of Apple by his board of directors, and the company was in trouble. Nonetheless, we loved the Macintosh computer and had used it, literally, to create City Year\u2014from our logo, to every proposal for support, to our database of applicants looking to join the City Year corps. To young social entrepreneurs, the Apple brand and the vision of Steve Jobs to \"make a dent in the universe\" was immensely appealing.\n\nApple, as its advertisements trumpeted, thought differently. While our parents worked in offices powered by huge IBM mainframe computers, we were the first generation to create and share ideas on personal computers. Apple stood for creativity and innovation and the little guy. Its design was intuitive, with dropdown menus and point-and-click visual icons instead of the slashes and dashes of computer code. It married the power of computers with the beauty of design and put control in the hands of average citizens, not experts and specialists. But while we thought of Apple as inventing personal computing, the truth is that for most of the 1980s and '90s\u2014as personal computers became ubiquitous\u2014Apple's market share was generally stuck in the mid to low single digits. IBM evolved, and it sold more personal computers than did Apple; so did Hewlett-Packard. And there were Compaq and Dell and Atari and a whole host of companies\u2014old and new\u2014that for most of those years held about 95 percent of the personal computer market, while Apple captured about 5 percent. What Apple and Steve Jobs did do\u2014and this is what fascinated us at City Year and fascinated me and my team as we developed Citizen Schools a few years later\u2014is change the vector of an entire industry. IBM, as iconic a mainstream company as there has ever been, adapted to the new standard that Apple had set. Hewlett-Packard adapted too. Microsoft changed its operating system to become more intuitive and personal, taking a page from Apple's playbook. An entire industry improved, and Apple led the way.\n\nAs we looked in 2008 to recalibrate our strategy at Citizen Schools and to build from the ELT breakthrough at the Edwards, we too wanted to change the vector of an industry. We wanted to change the vector of K\u201312 education, everything from its agrarian-era schedule to its industrial-era delivery system. When I wrote the Citizen Schools concept paper in my basement office in the summer of 1994, I sketched a business plan to create just six programs in Boston, but my hope was to catalyze a paradigm shift and to mobilize citizen teachers by the millions\u2014the \"sleeping giant of education reform\"\u2014and to get school districts and communities to think differently about the length of the learning day.\n\nNow in the Edwards ELT model we had a proof point that wasn't just a cool after-school program for select students but a full-school turnaround that offered a new vision for how schools could operate in the twenty-first century. We had an opening to argue for a whole new day for learning: a day with more time, more caring adults, more support for teachers, and more chances for students to be successful and to be creators of things, not just consumers. The question facing us was _how_ to grow and _how_ to change the vector of education. Should _scale_ be our focus, with success measured by how quickly we could grow the Citizen Schools footprint? Or should we focus instead on _spread,_ helping to create the conditions for our ideas to disseminate by influencing public policy and by sharing best practices at conferences and trainings and through the Internet?\n\nThis same conversation about scale and spread was happening at every other entrepreneurial social service organization I knew, and we were all talking with each other. Leaders at City Year and JumpStart, and Year-Up, and Teach For America, and others were wrestling with the same issue of how to grow our ideas, our brands, and, ultimately, our impact. We had all seen technology firms like Google and Amazon go from startup to market dominance within a decade, but our world of direct-service nonprofits with human-powered delivery systems wasn't scaling that fast. Not even close. Barry Harrington, an advisor with the consulting company Bain, had told me that human service businesses like Bain, or like law firms or accounting firms, generally can't maintain growth rates faster than 20 percent per year once they have gone through their initial startup period. While you might exceed 20 percent some years, Harrington said, the human challenges around hiring, training, and integrating talent\u2014all while maintaining or hopefully improving quality\u2014are too great to exceed it consistently.\n\nIf Harrington was right\u2014and even 20 percent growth is wildly higher than typical in the nonprofit sector\u2014we had a long way to go before our scale would be significant relative to the challenge we were addressing. In the 2008\u20139 school year, Citizen Schools was serving twenty-five hundred students. There are about twelve million middle school students in America, about four million of them in schools that are majority low-income, which are our target. At a growth rate of 20 percent per year it would take us a quarter-century to get to a point where we were reaching just 5 percent of the students in our target group. If we wanted to really change the vector of education, wouldn't we need to spread our ideas through policy work and through other organizations, and not just scale them directly as Citizen Schools?\n\nIn February of 2005 one hundred leading social entrepreneurs gathered at the Mohonk Mountain House in upstate New York. New Profit Inc., the venture philanthropy group, convened the meeting. Our goal was to think about how our community of social entrepreneurs and a few of our leading champions could collectively do more to address the urgent challenges of the day. I was commissioned to write a paper for the group, which I titled \"Realizing the American Dream: Historical Scorecard, Current Challenges, Future Opportunities _._ \" In the paper I made the case that after two centuries of steady progress in reducing poverty and increasing opportunity, America\u2014including those of us gathering at Mohonk\u2014had just witnessed the first generation in American history in which opportunity had declined. Poverty had gone up and high school graduation rates went down. Paraphrasing Dr. Martin Luther King, I described our recent pursuit of opportunity in America: \"Rather than rolling down like a mighty stream, justice trickled forward.\"\n\nThe solution I urged us all to consider was a three-part strategy for social change: organizational reform, meaning efforts to grow promising organizations like Citizen Schools; political reform, such as efforts to change laws and policies to encourage broader adoption of best practices; and cultural reform, by which I meant initiatives to change beliefs and behavior among citizens, such as the designated-driver campaign in the 1980s that helped to reduce drunk-driving deaths by one-third.\n\nDavid Gergen, the CNN commentator and senior advisor to four presidents, also spoke to the group at Mohonk. In a memorable after-dinner talk that didn't start until almost midnight, Gergen told us that social entrepreneurs were doing the most exciting work for the country since the civil rights movement. But he warned us that sometimes we looked like the little Dutch boy with his finger in the dike, trying to prop up a crumbling infrastructure rather than leading efforts to replace it with a different and better structure, a systemic change. Gergen urged us to think beyond creating independent charter schools and after-school programs and college-access programs and to work more with government. \"You need to transform islands of excellence into systemic reform,\" he told us.\n\nThere were two stories that we discussed that spoke to Gergen's challenge and examined the possibilities of big change through the scale and spread of effective interventions.\n\nThe first story is about Head Start. It begins in the small Michigan community of Ypsilanti, where in 1962 a local school administrator named David Weikart created the Perry Preschool Project. Perry enrolled the poorest children in the community and gave them access to caring adults who read to them and provided all kinds of enriching learning experiences to get them ready for first grade. The program was intended to lift student IQ, something thought impossible at the time, and was connected to an ambitious research effort. Directors of the project received 123 applications, and they randomly selected fifty-eight children to be in the treatment group. These children would enroll in the program. The other sixty-five applicants were assigned to the control group, meaning they would not be able to access the program but would agree to share data on their lives with researchers.\n\nThe evaluation of Perry Preschool caught the attention of policymakers far and wide. An initial study conducted two years after the program began showed student IQ for participants growing by an average of fifteen points, and other school-readiness measures rising sharply as well. Later studies showed that the IQ gains washed out over time but that other gains were deep and long lasting. Perry Preschool students earned significantly higher grades in school, and 71 percent of them graduated on time from high school, compared to just 54 percent in the control group. As the Perry kids grew up, the benefits of the positive preschool experiences kept compounding. By the time the graduates were twenty-seven years old, they were almost three times as likely to own their own homes as students in the control group and half as likely to receive public assistance.\n\nAmong the people captivated by the project and the early research was Edward Zigler, a professor of child psychology at Yale University. President Lyndon Johnson had asked Zigler to develop a strategy for early-childhood education as part of Johnson's War on Poverty. With Perry as an important inspiration, Zigler and his colleagues created Head Start, one of the most ambitious social programs of the last fifty years.\n\nIn many ways the Perry Preschool Project and its influence on Head Start is a social entrepreneur's dream. It's what David Gergen was calling for as a better alternative to the little Dutch boy sticking his finger in the dike. A new program in Ypsilanti, Michigan, offered a vision of what was possible and evidence it could work: a proof point. Key opinion leaders learned of it and proposed policy based on its success. The president of the United States and his team supported it enthusiastically. A new law was passed to provide funding and spread the idea. The program then attracted bipartisan support and funding grew, through Democratic and Republican administrations, from a few million dollars in 1965 to $8 billion today. Head Start now serves one million three- and four-year-old children, which is roughly half of all the children living in poverty in that age group in the entire country!\n\nThe problem with this seemingly dreamy story is that while many studies of Head Start point to modest benefits of the program, such as slightly better health outcomes for participants and modest academic gains in elementary school, most studies also show that the gains eliminate only a small fraction of the achievement gap and washed out over time. Everyone agrees that great early-childhood programs\u2014programs like Perry and like the Abecedarian Project in North Carolina\u2014make a big and lasting difference for children. But the difference made by average Head Start programs is much more modest, and in the view of skeptics is negligible over time.\n\nThe moral of the story is that spreading high-quality programs by government adoption is difficult. One reason is that small proof-point programs tend to be relatively expensive. There are no economies of scale, and the founders are biased toward investing whatever is needed to produce results. But when Congress or school districts look to scale up a demonstration, they face pressure to reduce costs. Scaling to more schools, or more congressional districts, sometimes becomes the most important metric, with program impact taking a back seat. This happened with Head Start, and when the program grew from Ypsilanti to the rest of the country, it got watered down. The original program had a 1:5 teacher-to-student ratio, which is similar to the ratio of preschool programs in upper-income communities. But to get the program to more communities at what was deemed a reasonable cost, Congress funded Head Start at a 1:10 ratio.\n\nMore fundamentally, the Head Start case illustrates the challenge of spreading ideas that will be implemented locally by very different organizations. It's important to understand that the federal government doesn't actually run Head Start, just as it doesn't run most social programs. Instead the government holds a competition and makes grants to thousands of local nonprofits that run the programs. Some of those nonprofits\u2014just like some local schools and some local hospitals and some local after-school programs\u2014are really good. They excel at hiring people and occasionally firing people and training and managing people. But other nonprofits that operate Head Start programs are not so effective at hiring and firing and training people, and children who attend those programs lose out. When we try to spread ideas, like those proven so effective in Ypsilanti, Michigan, we need to think not only about the programmatic models we want to replicate, but also about how we can replicate and grow effective _organizations_ that will bring the model to life and fulfill its promise. Even a great idea will founder without a strong organization to implement it.\n\nThe Head Start story as I've told it feeds distrust in government. When I tell this story to conservative friends, they nod their heads and talk about the challenges of getting entangled with government. But there are promising new efforts to improve Head Start's delivery system and an exciting push across the federal government to establish new criteria in funding competitions to steer more money to organizations with proven and promising track records. And there is another story that needs to be considered: the story of Habitat for Humanity.\n\nHabitat is an amazing organization that serves as the poster child for scale in the nonprofit arena. While particular companies often scale quickly in business, and nine of the thirty largest companies in America, including Apple and Amazon and Google, were started in the last forty years, change in the nonprofit sector is much slower. When I wrote the American Dream paper in 2005, of the thirty largest nonprofit organizations in the United States, only one had been started in the last forty years, and that was Habitat.\n\nHabitat was founded in 1976 by Millard Fuller, a lanky Southern businessman and a member of the Disciples of Christ, a large evangelical church. Habitat has a mission of increasing home ownership among the poor and eliminating substandard housing. Its model is to recruit volunteers to team up with low-income families to build new homes or renovate existing ones that the low-income family can then move into as owners. Habitat's model is a twentieth-century barn raising, and with the help of Fuller's church, it took off. In 1982 former president Jimmy Carter volunteered with Habitat and gave its work credibility and stature, accelerating its growth.\n\nHabitat grew through a franchise model, empowering local leaders to build their own Habitat organizations, but following best practices shared by the national office and benefiting from national marketing efforts and partnerships with major companies, which donated materials and encouraged their employees to get involved. In 2013 the organization built its six hundred thousandth house, and there are now more than three million people around the world living in Habitat homes. All this was accomplished with minimal help from the government (though Habitat is a recipient of AmeriCorps funds and deploys AmeriCorps members to advance its work) and with a maximum of old-fashioned community compassion.\n\nHabitat is a great nonprofit success story and its mission and methods offer valuable lessons for social entrepreneurs. But the problem with the story is that while Habitat was scaling like gangbusters, the percentage of low-income Americans who own their own homes actually went down. Habitat scaled rapidly and cost-effectively, but the policy climate and the business climate changed, and Habitat ended up further from achieving its mission than when it got started.\n\nMy point is not to criticize Habitat or Head Start but to acknowledge that really big change\u2014truly systemic change\u2014needs to come both from the scaling of particularly effective organizations _and_ from spreading the best ideas through policy and cultural change work.\n\nThe election of President Barack Obama in 2008 had all the markings of a sea-change event not just for American politics writ large, but specifically for the nonprofit sector and for the growing community of people within it who identify themselves as social entrepreneurs.\n\nBy 2008 social entrepreneurship was a hot topic around the world. Every spring since 2004, thousands of social entrepreneurs and members of their supporting ecosystem have journeyed to Oxford University for the field's top gathering, the Skoll World Forum on Social Entrepreneurship\u2014sponsored by eBay cofounder Jeff Skoll. The forum, which I've attended every year since 2005 when I was awarded a Skoll Award for Social Entrepreneurship, features keynotes by the likes of Bishop Desmond Tutu, Vice President Al Gore, and President Jimmy Carter, as well as by celebrated social entrepreneurs who have used innovative strategies to bring clean water, health care, electrical power, and books to the most remote regions of the world. Social entrepreneurship is hot in the United States too, with social enterprise clubs becoming the most popular and fastest-growing student groups at Harvard and Stanford Business Schools in the early 2000s.\n\nObama's election held great promise for social entrepreneurs for at least three reasons. First, as the nation's first black president, elected just two generations after the milestone events of the civil rights movement, Obama's election sent a message that big change is possible. If an African American could be elected president, maybe other dreams deferred could come true too. Second, in many ways Obama and his wife, Michelle, were social entrepreneurs themselves. Obama had been, famously, a community organizer on Chicago's South Side trying to establish innovative job training and after-school education programs. Later Obama served on the Chicago advisory board for Public Allies, a pioneering AmeriCorps program, and an innovator in deploying skilled mobilizers into the nonprofit community. Michelle Obama served as Chicago executive director for Public Allies in the early 1990s\u2014the same years I served as Boston executive director for City Year. Finally, the policies Obama spoke about in the campaign were policies that resonated within our community of social entrepreneurs and in some cases policies we had actively promoted during the campaign.\n\nIn his first inauguration, speaking at a time of rising economic crisis, Obama called on the country to move beyond a debate over the size of government to a discussion of how government can help us accomplish our public purposes most effectively. \"The question we ask today is not whether our government is too big or too small, but whether it works\u2014whether it helps families find jobs at a decent wage, care they can afford, a retirement that is dignified. Where the answer is yes,\" Obama stated, \"we intend to move forward. Where the answer is no, programs will end.\"\n\nWithin one hundred days of taking office, Obama signed the Edward M. Kennedy Serve America Act, which passed the House and Senate with substantial bipartisan majorities. The Serve America bill aimed to triple the size of AmeriCorps to 250,000 members and put a special focus on education, an emphasis inspired in part by the teaching fellows bill Kennedy had sponsored and announced at the Edwards School almost two years earlier. The Serve America Act also created a new Social Innovation Fund in which the federal government would partner with private philanthropy to scale up proven and promising solutions and programs. When the White House put out its press release explaining the initiative, Citizen Schools was one of four examples given of the types of organizations the program was designed to expand.\n\nThe Obama administration also teamed up with private funders to initiate a series of competitions to crowdsource creative solutions to persistent public problems. The Department of Energy, for instance, offered prize money to anyone who could make solar energy cheaper than coal, and the EPA offered funds to groups that could develop technologies to clean up oil spills faster and more completely. Both efforts stimulated new technologies that dramatically improved on the status quo and generated additional billions in private research and development spending. Robert Gordon, a friend to many in the social enterprise world, was named associate director of the Office of Management and Budget and pushed efforts to \"fund what works,\" trying to put more federal money behind programs and organizations with the strongest record of impact.\n\nOf all Obama's new approaches to activist and results-focused government, the boldest and most controversial was his Race to the Top initiative and a companion measure, the Investing in Innovation (i3) program, run by the Department of Education.\n\nRace to the Top was a $4.35 billion initiative funded by the 2009 emergency stimulus bill that allowed Obama and his secretary of education, Arne Duncan, to drive systemic change in education. Federal funds account for only about 10 percent of all that's spent on public schools (with the rest coming from state and local sources), and most of the federal funds are allocated to states and districts by formula to support low-income students, teacher training, education for disabled students, and a few other specific programs. Historically, federal money added to the school-funding pot but didn't do much to stir it. With Race to the Top, that was about to change. Using the $4.35 billion in stimulus money as leverage, Duncan told states and districts they could win a share of the money if they developed a bold vision to improve student learning, including adopting policies the administration favored, such as allowing more charter schools and evaluating teachers based at least in part on student test scores.\n\nI don't agree with all of the policy priorities Race to the Top emphasized (for example, \"merit pay\" for individual teachers based primarily on test scores seems like a bad idea, and certainly an unproven one), but as a strategy for driving change at the local level, Race to the Top was unbelievably successful. To date, through three rounds of Race to the Top competitions, nineteen states and sixteen school districts have won more than $4.5 billion in awards. Even many states and districts that did not win the contest for federal funds made major changes in their policies, such as allowing more charter schools and adopting more rigorous teacher-evaluation systems.\n\nCitizen Schools benefited from Race to the Top in a few states. For instance, North Carolina made a particular push in its successful application to improve science, technology, engineering, and math (STEM) education, and some of these funds ended up supporting our work in Durham. And Massachusetts created a \"preferred partner program\" as a centerpiece of its winning application and then used the money to support a small set of organizations, including Citizen Schools, with strong track records of lifting student achievement.\n\nBut for Citizen Schools, as with most nonprofits, Race to the Top was small money. The big opportunity for us was the i3 competition, which was designed to help school districts and outside partners scale promising and proven programs. Throughout 2009 and early 2010 we waited impatiently as finishing touches were put on the competition design.\n\nThe basic idea for i3 was to create a tiered funding opportunity where the more evidence of impact you had, the more funding you could qualify to win. There were several categories within the competition, with one focused on teacher training and preparation, for instance, and another focused on turnaround of the lowest-performing schools. Applicants with the strongest evidence of success, usually multiple rigorous external evaluations, could apply for up to $50 million. Initiatives with shorter track records but still a strong evidence base could apply for up to $30 million, and more experimental efforts but with promising strategies and a commitment to track outcomes could qualify for up to $5 million. The whole initiative was funded with $650 million in stimulus money and an expectation that only fifty or fewer grants would be made across the entire country. In addition to scaling aggressively during the initial five-year grant period, applicants were required to sketch plans to eventually spread their innovations to hundreds of thousands or even millions of students.\n\nCitizen Schools decided to apply for $25 million to scale the ELT school-turnaround work we had begun at the Edwards to twenty-five schools across ten school districts. For three months this became a principal focus for me and several other senior staff. We forged partnerships with ten school districts, from New York to Oakland, and from rural Henderson, North Carolina, to the Apache community of Mescalero, New Mexico. We convinced Bain to commit $3.5 million of in-kind consulting support to help craft our strategy. WGBH, which in partnership with PBS runs an educational website that reaches 77 percent of all the schools in America, agreed to build a curriculum and training repository that could help us share best practices across schools and to spread the ELT idea broadly after the initial direct replication. The competition provided us with a unique opportunity to push the scale and spread of our idea, and provided me with a powerful lever to drive change within Citizen Schools. With the Edwards we had piloted ELT. Now we would bet the farm on it. While we would still run some optional after-school programs, our entire focus and all new programs would now be ELT. Adopting this strategy would mean that again we'd need to raise our game in order to be an effective partner to school districts in helping turn around some of their lowest-performing schools.\n\nI knew we were competing with the nation's top school districts, school-reform nonprofits, and research universities, and I also knew that only a few of many excellent applicants would win. Despite all those caveats, my competitive juices were flowing and I had high hopes we would be selected.\n\nIt turned out that the i3 winners were revealed prematurely in an e-mail unintentionally released by the Department of Education on August 4, 2010. I was at a Red Sox game that night, enjoying an evening out in the Bank of America skybox with Kerry Sullivan, the head of the Bank of America Charitable Foundation, and Lynn Wiatrowski, a bank executive we were recruiting to the Citizen Schools board. There were a few other nonprofit leaders there as well, including Michael Brown, now the CEO of City Year. Near the end of the game, people started to buzz about the i3 news. Michael stepped outside to call his team. I checked my e-mail and saw that the list of i3 winners had been sent. There were sixteen winners at our level of funding, but Citizen Schools was nowhere on the list. Teach For America and KIPP (Knowledge Is Power Program) had won the big $50 million awards. And City Year, which had teamed up with Johns Hopkins and Communities in Schools on a $30 million school-turnaround proposal, had been selected for funding. We had scored in the top 4 percent of all proposals submitted but not quite high enough to win.\n\nI was devastated and deflated. The baseball game was not close and the group was starting to break up, so I excused myself and headed outside into the August night. I began to walk home, about a half-hour journey and one I completed with as deep a feeling of melancholy as I had experienced in a long time.\n\nWhen I got home I e-mailed our staff and partners, putting the most positive spin I could on the announcement. \"Although the news is disappointing,\" I wrote, \"I am extremely proud of the Citizen Schools staff and partners today and know that we have a lot to be excited about moving forward. The i3 fund was intended to spark innovation and partnership not only among those who receive funding, but among all working in education. For Citizen Schools, it did just that. Together, in partnership with fifteen partners, we've developed a tremendous plan for helping schools innovate and expand the learning day for students.\" I went on to share that missing out on the grant would not change our budget for the upcoming year (we had hoped for the money but not counted on it) and recounted the ways in which the application process had sharpened our thinking and our strategy and built our partnerships with districts and others.\n\nThe next day I went into the office early and looked up our scores, which were posted online. Ratings on the strength of our evidence of impact had been perfect\u201415 out of 15 from both the evaluation experts brought in to assess that part of the application. Of the three judges who had rated our application for clarity of vision, power of design, and strength of implementation and sustainability, two judges had given us almost perfect scores and the third had given us solid scores but ones just low enough to pull us below the top tier. I joked about \"the Russian judge\" and tried to rally the troops around a vision of still implementing our i3 plan, but without the money from the Department of Education.\n\nAs I write this, it is now almost three years from that frustrating night at Fenway Park. In fact we did move forward with the i3 plan, even without the $25 million. We have grown from just two ELT schools when we wrote our i3 application in the spring of 2010 to twenty-four schools for the 2013\u201314 school year, with plans to reach twenty-six to twenty-eight in 2014\u201315. We are on target with the growth schedule we proposed in i3. In our proposal we asked school districts to provide $600 per student, with the rest of our costs coming from the i3 grant and private support. Now districts are providing an average of $1,200 per student, drawing from a variety of funding sources, including school-turnaround dollars, Title I funds, and, in Boston's case, a smaller i3 award that the district won in 2011 to replicate the Edwards ELT model. In 2012 Citizen Schools won a smaller i3 grant to scale our apprenticeships in STEM education, and private funders have done more too. Even though we didn't initially win, the i3 competition, while bruising, did catapult us forward.\n\nLegendary business consultant Peter Drucker described entrepreneurship as the rearranging of existing resources to higher and better uses. Since that first journalism class in 1994, I had hoped that Citizen Schools could make even a small contribution to rearranging how American schools and communities use the resource of time by putting afternoons and summers to better use, and I hoped we could help rearrange the use of our public school buildings, activating these public spaces\u2014an estimated $2 trillion resource\u2014to engage children beyond the regular school day. But the really big resource we have always hoped to rearrange is the talent and time of average citizens. If we can get architects and engineers and lawyers to carve out a few hours a week in which they get out from behind their desks and into American schools\u2014and also offer more Americans the chance to devote a year or two as full-time AmeriCorps members at the start or end of their careers\u2014we will move America forward and restore public schools to their historic role as engines of opportunity.\n\nThe next section of the book further explores five drivers of the growing opportunity divide that separates wealthier and poorer children. As mentioned in the introduction, too much of our national dialogue about education is focused on two convenient bogeymen: poverty (even though there are plentiful examples of poor students learning and achieving at high levels when given high expectations and support) and teachers and teacher unions (even though only 20 percent to 30 percent of the rich-poor achievement gap is explained by differences in school quality).\n\nIn my view, we can give low-income children a fair chance of competing with upper-income children (from around the world as well as across town), but to do so we need to focus on the following five building blocks of childhood success:\n\n1. More mentorship by caring and professionally successful adults\n\n2. A longer learning day, allowing more time to master the academic basics and chances to participate in sports, art, and music as part of a well-rounded education\n\n3. More chances to practice creativity and innovation\u2014critical skills in the modern economy\n\n4. More chances to build the social networks and social skills that help drive professional and life success\n\n5. Better support of full-time teachers and parents, the primary caregivers for lower- and moderate-income children.\n\nThe five chapters of section 2 each explore one of these success drivers in depth, sharing surprising research as well as illustrative stories from Citizen Schools and other programs.\n\n# SECTION TWO\n\n# HOW CITIZEN POWER AND AN EXPANDED LEARNING DAY CAN NARROW ACHIEVEMENT GAPS, BROADEN OPPORTUNITY, AND STRENGTHEN AMERICA\n\n# VOICES FROM CITIZEN SCHOOLS\n\n# JOYCE KING THOMAS, VOLUNTEER\n\nThe secret sauce of Citizen Schools is the volunteer\u2014the caring and talented adults, from all walks of life, who teach apprenticeships that our middle school students choose. Volunteer citizen teachers are part of a proud American tradition of citizen power that has strengthened the country since its founding, and that will be discussed in this section of the book. Here is the story of one of our volunteers, Joyce King Thomas, in her words.\n\n_I never learned how to say no to my oldest son. He always won our biggest debates._\n\n_At eleven, he came out on top in our knock-down, drag-out fight over listening to R-rated rap lyrics. (He needed to experience the Beastie Boys uncensored; it was part of a Brooklyn boyhood, he effectively argued.) He used his relentless charm to persuade us to make room for a parakeet, a second parakeet, a turtle, three lizards, a two-foot-long live-cricket-eating Komodo dragon, and, finally, a dog._\n\n_Lest I sound like a wimp and Aidan sound like a monster, you should know that I was the chief creative officer of a huge New York ad agency, and my son is a Colgate University graduate aiming for a master's in education at the Bank Street school._\n\n_Years of honing his skills at getting me to do things I had no intention of doing culminated when Aidan asked me if my company and I would volunteer for Citizen Schools, the organization he worked for. No, I didn't stand a chance._\n\n_The mission of Citizen Schools is to provide middle school kids in disadvantaged areas with an educational after-school experience. What makes the program unique is its twelve-week \"apprenticeships.\" Citizen Schools recruits companies like Google, the Food Network, and AOL to teach apprenticeships that give the kids hands-on experience in fields they might not otherwise be exposed to. Each apprenticeship culminates in something called a WOW!, where the kids literally wow their parents, teachers, and friends with what they've accomplished in those twelve weeks. While all the school reformers are debating what should happen in the six or seven hours kids are in school, Citizen Schools is busy filling up those three lonely after-school hours; the hours kids fill with either random TV watching or potentially dangerous hanging out._\n\n_As I mentioned, when Aidan made his pitch, I was the creative director of a gargantuan New York ad agency. We were in the middle of one of the most challenging economies in history. Clients' businesses were hurting, which meant we were hurting. Teaching a two-hour class in Harlem every Wednesday for twelve weeks, and investing hours in prepping for those classes, wasn't really a smart career move. But it turned out to be one of the smartest life moves I ever made._\n\n_For our apprenticeship, my team of volunteers divided the class into three \"mini agencies,\" gave them an assignment, and let them develop a strategy, brainstorm ideas, present their ideas, and, finally, film and edit a real commercial._\n\n_We even recruited the Ad Council, which creates advertising for important national issues\u2014like Smokey Bear's \"Only YOU can prevent forest fires\"\u2014to join us. The kids would present their ideas to the Ad Council, who would pick the ones to film._\n\n_So what could the kids make ads about? In the past, the Ad Council and our agency had created communications to help fight childhood obesity. Who better to create ads to convince young people to eat healthier and exercise more than a bunch of middle schoolers?_\n\n_Our homework was done. And we were off._\n\n_Day one we showed up at Isaac Newton Middle School on 116th Street in Manhattan, a big, classic old school with wide hallways._\n\n_I'd passed it a million times driving down the FDR, but never imagined I'd be teaching there with the most impressive team I could draft from our agency: Vann, an African American creative director who won a Purple Heart in Iraq; Craig, a dad\/creative director with a real knack for working with kids; Sallie, our indefatigable head of creative human resources; an energetic admin named Kim, who kept us focused and made sure we had it together each week; and me. Over the course of the program, we also wrangled a young editor, two junior producers, and an assortment of other agency personnel to join our naively overambitious mission of producing three commercials. It took a village. And then some._\n\n_The first class wasn't easy. Okay, none of the classes were easy. But armed with techniques we learned in the Citizen Schools training program (like, \"If you hear my voice, clap your hands twice\") we muddled through. The kids loved looking at commercials and explaining what the strategy of each piece of communication was. Aidan worked with us to create the right bite-size curriculum and helped us control the class, since we were middle school neophytes._\n\n_The kids were so quick and bright that the team leaders fell in love. My team included Hakeem, a brilliant kid with a truancy problem that had him skipping school thirty-five days one semester. The amazing thing was he didn't skip one single day of the apprenticeship. I was also lucky enough to work with Aaron, a mosquito of a kid who buzzed around keeping everyone laughing; Miguel, a charming boy whose quiet demeanor belied his quick mind; Dontea, a bright girl who seemed to have some serious personal issues; Aaron, a born actor and close friend of Hakeem's; and Michael, a talented artist who needed a lot of reassurance about how good he was._\n\n_The kids brainstormed ideas to convince their peers to eat healthier and exercise more. Then, they came up with a winner: a talking apple that would literally hit someone over the head with the point that they needed to eat more fruit. We scouted for locations within walking distance and picked a spot just east of the FDR Drive. The kids made most of the decisions, from shooting to editing._\n\n_A hundred or so people attended the WOW!, where we screened the behind-the-scenes film and the three commercials (available on YouTube: see ADLab Citizen Schools), including the school's principal and assistant principal, parents, Citizen Schools staff, and an assorted group from the agency. I stood in front and watched the crowd watch the kids present their work. It was a beautiful sight._\n\n_Around that time, I decided it was time for me to leave my company and explore some other possibilities, including helping in the education arena. I'm working with Citizen Schools to encourage other agencies to step up and teach. Sixth and seventh graders can learn so much from the process of creating communications. They learn it's possible to have a good career being creative. They learn that creating requires order and discipline. They learn that they can do it._\n\n_And I learned how important it is to listen to your children._\n\n# CHAPTER EIGHT\n\n# CITIZEN POWER AND THE IMPORTANCE OF MENTORING\n\nWhile countless education reformers have suggested we improve schools by recruiting and training more effective teachers, developing better tests, and adopting stronger curricula\u2014all within the confines of a traditional six- to seven-hour school day\u2014what if the holy grail of educational advancement lies outside this box? What if closing opportunity and achievement gaps requires an entirely new approach to our schools, including their hours of operation, the ways they define success, and the talent pool they draw upon in pursuing their goals? I think it does.\n\nBetter full-time teachers and better tests and better curriculum are all important things, and we should pursue them actively. But providing equal educational opportunity to all children will require us to address the root causes of our growing educational inequality: \u00adunequal access to extra learning time to master the academic basics, unequal access to enrichment and opportunities to practice innovation and problem solving, unequal access to social networks, and unequal access to well-supported parents and teachers. The only realistic way to equalize access to these essential assets is to mobilize America's greatest resource: citizen power.\n\nCitizen power lies at the heart of the American experience. From the citizen soldiers who fought for independence to the citizen activists who fought for civil rights, America has met its biggest challenges when its citizens have gotten directly involved. Citizen power built the settlement houses of the early twentieth century that welcomed and trained new immigrants and helped make America the world's first majority-middle-class nation. And citizen power drove the civil rights movement, which, while incomplete, has in the last two generations increased African American college completion from just over 1 percent to 20 percent and reduced poverty among African Americans from 55 percent to 27 percent.\n\nThe distinctively American impulse to join together with neighbors and make things better is what French aristocrat Alexis de Tocqueville chronicled in his 1835 classic, _Democracy in America._ Tocqueville noted that unlike citizens in his native France or other European democracies, when Americans faced challenges they banded together in associations of mutual self-help. He wrote of Americans on the frontier and in larger cities coming together to create libraries to share knowledge, firehouses to promote safety, granges to promote farming, Sunday schools to teach religion and reading, and social clubs to promote quality of life. Tocque\u00adville called Americans a \"nation of joiners\" and, without using the term, described social capital as a unique facet of American life. Tocqueville has been in the news recently as evidence mounts that social capital in America is now in decline (see chapter 11).\n\nYou may think of citizen power as political, concerning sit-ins, marches in major cities, and calls to Congress. Citizen power does include these things. But citizen power can also be exerted through direct action. It's the families along the Underground Railroad who sheltered fugitive slaves seeking freedom in the North. It's progressives like Jane Addams starting Hull House in Chicago and hundreds of her peers devoting a few hours each week to tutor new immigrants there. And citizen power is college volunteers on their summer vacations heading to Mississippi to teach former sharecroppers to read and register them to vote. On the American frontier in the early nineteenth century, before public schools were established, citizen power was the volunteer Sunday school teacher who taught children to read so they could study the Bible and also conduct basic commerce and follow the events of the day. Citizen power has always been able to get big things done in America. Amazing as it may seem today, it is likely true that for three generations after the formation of a government by the newly independent United States, more Americans were taught to read by volunteers, usually in Sunday school, than by paid public school teachers.\n\nCitizen power is not naive optimism. It works, and it _is_ working in communities across America. Citizen power today looks like the National Academies Foundation (NAF) high school initiative, which uses corporate volunteers from more than twenty-five hundred companies as the backbone of a model that serves more than sixty thousand students across thirty-nine states. Business experts from the industries of finance, hospitality and tourism, information technology, and health sciences volunteer in NAF classrooms and engage NAF students through paid internships at their companies. Most NAF students are low-income, but they graduate high school on time at higher rates than middle-class kids. And according to a rigorous external evaluation, 52 percent of NAF graduates earn a bachelor's degree in four years, compared to 32 percent of all students nationally and just 8 percent of low-income students.\n\nCitizen power helps fuel the Met School of Providence, Rhode Island, and High Tech High of San Diego, networks of schools whose students are asked to identify their passions and then meld them into internships and substantive projects that serve real-world purposes. The schools rely heavily on community participation by volunteers for the rich, real-world learning opportunities their students enjoy. One of the core design principles of High Tech High is \"Adult World Connections,\" and all students complete substantial internships where they have the opportunity to learn from and work with real-world experts. Students create projects for an authentic audience and have the opportunity to present their work in a professional setting. The assumptions behind High Tech High and the Met's school-to-work strategies mirror those behind Citizen Schools' WOW! events. Long-term student interest can be driven by providing students the opportunity to be successful with successful adults. Both schools deliver strong results with high graduation rates and more than 90 percent of the primarily low-income students going on to college.\n\nCitizen power can also look like FIRST, prolific inventor Dean Kamen's effort to mobilize engineers to work with high school students to build robots and then enter them in regional contests that have the feel of March Madness basketball games, complete with well-coached teams and screaming fans. Or like iMentor, which connects college-educated mentors with thousands of high school students through virtual and in-person mentoring, increasing graduation rates by 18 percentage points. Or like City Year, which places near-peer mentors into Elementary and High School classrooms in twenty-four cities across the country and has set as its goal reducing by half the number of students who are off-track to graduate high school on time in each of the cities where they operate. Citizen power can also shine further from the schoolhouse doors, such as in Girl Scouts, which increasingly offers badges in robotics and science in addition to babysitting and quilting, and in 4H, which reaches six million students and deploys 540,000 volunteers annually, with a majority serving in urban areas like Los Angeles, Washington, DC, and Atlanta.\n\nLike Citizen Schools, many of these citizen-power solutions are delivering compelling results, often in partnership with local schools. Our collective success does not absolve government of the responsibility to adequately fund and support a high-quality education for all children\u2014an investment that will pay dividends to our society for generations to come. But Citizen Schools and others are showing that any successful effort to improve education in America at scale will have to include a broad effort to get citizens off the sidelines and into the action as coaches, tutors, mentors, and teachers.\n\nI have seen again and again how real-world experiences and exposure to role models with real careers can be transformative for young students. I remember a Boston seventh grader named Lin-Ann, who loved the idea of being a doctor based on what she had seen on her favorite TV shows. In Citizen Schools she enrolled in a Cure Cancer apprenticeship, working with a researcher from a local cancer hospital and studying the behavior of cells when observed under a microscope. As part of the apprenticeship, Lin-Ann and her peers received white lab coats, and on school days when she had her apprenticeship, Lin-Ann would wear her lab coat all day long, from six thirty in the morning, as she prepared to board the bus to school, to five thirty at night as she headed home from her apprenticeship. She was proud, and her very identity changed, from drifting middle school student to future medical professional.\n\nAnother young student in Citizen Schools, Stephanie, took a different medically oriented course, Drugs on the Brain, which was led by Harvard University postdoctoral fellow Marcus DeLatte. Stephanie said that she was excited to be in the Drugs on the Brain class, because she was definitely planning to become a doctor. One day when Stephanie reiterated her interest in the medical profession, Marcus asked her to bring in her report card so they could discuss it and help plan her pathway to medical school. The next week Stephanie brought in her most recent report card\u2014a mix of Bs and Cs, with one D, in math. Marcus shared some feedback. \"Stephanie, I know you want to be a doctor, and I can see in our class that you have a good mind, and when you work hard you do excellent work. I know you _can_ become a doctor. But you need to know that you won't be able to become a doctor if you don't bring your grades up, particularly in math and science, to at least a solid B and eventually to an A.\" It was an essential message, but no one had ever delivered it to Stephanie before she heard it from Marcus, who had navigated his own way from inner-city New Orleans to Harvard Medical School.\n\nA third student\u2014also enrolled in Marcus's Drugs on the Brain class\u2014was Jonathan, a special-education student who was struggling in middle school. As an African American boy who was assigned to separate special-ed classes (often offered in the school's dingiest basement classrooms), Jonathan's chance of finishing high school was about 40 percent, and his chances of completing any kind of a college degree was in the low single digits. But while Jonathan's low test scores and grades and his special-education status would jump off the page to any future teacher who read his transcript, what would not be seen was his love of animals. If you spoke with Jonathan, however, you quickly learned of his passion for animals of all types. He sometimes fed stray cats and often asked after the dogs and hamsters and other pets of his friends and relatives. Once when a bat flew into the brick wall of his mother's apartment and collapsed on the ground below, Jonathan nursed it back to health in a little shoebox he hid under his bed without his mother ever knowing. Jonathan loved animals. But he knew nothing about careers in veterinary medicine. He had never made the connection between animals and biology or math. Jonathan started to make those connections through his apprenticeships at Citizen Schools and the opportunity to spend time with successful professionals like Marcus DeLatte. He had the chance to hold a human brain in his hands, to handle a boa constrictor, and to examine slices of sheep brain tissue under a microscope, recording what happens when a few droplets of alcohol are added. The experiences were transformative. When I last checked, Jonathan was a senior at a top high school in Boston, a mainstream education student getting Bs in math and science. He was hoping to go to college to study marine biology or veterinary medicine.\n\nLin-Ann, Stephanie, and Jonathan all benefited deeply from citizen power\u2014from active mentoring by volunteers who offered new perspectives and opened doors to new professions. Their mentors helped these young students see the connection between school and exciting future possibilities\u2014a link that is fundamental to successful education but often overlooked by traditional educators. For some children it is easy to imagine the pot of gold at the end of the educational rainbow\u2014the job as a doctor or lawyer or engineer\u2014and the path they need to travel to reach it. They see the pot of gold around them every day in the form of parents with good jobs and good incomes, nice houses and nice cars. Lower-income children, however, are less likely to believe that hard work today will yield tangible rewards in the future. Real-world mentors help make the pot of gold real\u2014and they help their apprentices understand the path to get there.\n\nCitizen power comprises more than just current professionals. Lester Strong and his team at Experience Corps have capitalized on a growing and undertapped citizen-power resource: older Americans. Experience Corps and similar programs like ReServe and the Retired and Senior Volunteer Program engage older adults who have retired from their \"main jobs\" but who are eager to find part-time or even full-time \"encore\" careers. Strong himself is serving in an encore career, having worked previously as a television reporter and news anchor in Boston. Joining forces with AARP, Experience Corps puts thousands of senior citizens to work as volunteers in public schools across the country. The volunteer mentors meet with children in small groups, providing individualized instruction during regular class time. Currently in nineteen cities and reaching twenty thousand students, Experience Corps is now in the process of scaling up and expanding its reach by adding more than ten thousand volunteers to their corps in hopes of serving one hundred thousand students per year by 2017.\n\nFor Strong, mentoring is personal. He grew up in South Carolina in the crucible of the civil rights years, and he tells a story now of how mentoring saved his life.\n\nI got interested in mentoring because three amazing people mentored me. When I was in third grade, my teacher told my parents that basically I was unteachable, her term was \"mentally retarded,\" and told my parents that they really shouldn't have high expectations for me academically. But fortunately for me there were three people in my community that didn't believe in that and in fact really mentored me. They were a barber, a minister, and a mother of a friend of mine. And they mentored me literally from third grade through high school. And as a result of their amazing care and attention, I graduated with honors, hundreds of scholarships, got an undergraduate degree from Davidson, went to Columbia Business School, had a wonderful career, and now I work for one of the most recognized and respected nonprofits in the country. All because those three individuals really saw something in me and believed in me. The interesting question for me is what is it that they saw? What is it that they saw that my teacher and in fact even my parents couldn't see? I think it's because they were looking through the lens of love. And for me love has three central qualities. Number one, that you want the best for a person, place, thing, or idea. Second, is that you're willing to make a sacrifice in order to achieve that best. And third, you're not asking for anything in return. There's no condition attached to it. And that certainly was the reflection back to me of these three individuals. They just totally cared about my success, and my future and my opportunity to grow.\n\nExperience Corps has advanced achievement at the schools with which it partners. Focused on improving the five components of reading, a rigorous external research study found that students with Experience Corps mentors made 60 percent more progress in critical reading skills than control group students who did not have mentors. In addition, teachers report improvements in the learning environment and overall school culture, as well as reduction in behavior problems, in schools with Experience Corps members.\n\nCEO Strong describes his volunteers as coming from a \"place of passion and wanting to leave something that is of value and is sustainable. By changing the life of one child they are influencing something that ripples out through multiple generations.\"\n\nWhile Experience Corps members are seeing real and measurable results in their schools, they are also serving themselves in the process. Senior citizens who volunteer with Experience Corps report feeling better about themselves, expanding their circle of friends, becoming more physically active, and seeing improvements in their health, strength, and cognitive abilities. Members tell Strong that the reason they get out of bed in the morning is because they \"know they are going to make a difference in the life of a child that day.\"\n\nAt Citizen Schools we've also seen this deep impact on the volunteer citizen teachers with whom we work. It's no longer surprising to hear volunteers say: \"I got much more out of this than the kids did.\" But when I ask the volunteers what they gained, I realize that their statement is heartfelt and what they \"got\" is important. Sometimes volunteers gain hope that inner-city kids are not a lost cause, or they gain pride that they are engaged in helping their community address the civil rights issue of our time, or they gain networks when they team up with colleagues and other citizens from the community. Often they gain a deeper understanding of themselves, a renewed appreciation of their profession, or a rediscovery of a forgotten passion.\n\nBritton Picciolini, an advertising sales leader for Google in Chicago for eleven years and now a volunteer citizen teacher, taught photography, which had been her college major. She realized she was happier when she introduced what she called \"a philanthropic angle\" into her life. Rachel Schachter, who taught music in an inner-city Newark school, said it felt great to remind herself how much she loved music as a kid. \"Teaching [at Citizen Schools] has also let me remember how much I wanted to grow up to be a musician when I was younger, and let me realize how proud the person I was in middle school would be of the person I am now,\" said Schachter. \"Whether I continue to teach or take my career path in a whole different direction, I will always have singing and songwriting in my life, and I am incredibly glad that this apprenticeship has reminded me how important it is to me.\"\n\nAn early Citizen Schools board member, Karen Webb Campbell, told me that she became a better mother after volunteering to teach a business class in which a team of middle school students designed and sold T-shirts. Campbell said that as she got better at setting clear expectations for a team of teenagers she had never previously met, she also got better at setting expectations for her own children. Other volunteers who have team-taught with their college-age children described it as a \"do-over\" opportunity in their own parenting and a welcome chance for bonding.\n\nBob Mersereau, a retired scientist, has taught twenty courses, more than any other Citizen Schools volunteer. Some semesters he traveled between campuses in three different cities each week to offer classes in astronomy. He even continued teaching through a recent bout with cancer. In the past couple of years he has teamed up with his son Bobby, a chef, to teach pizza science. The students make pizza from scratch and learn the scientific principles involved in making pizza taste great.\n\n\"Years ago I would have classified myself as not very emotional\u2014taking a psychology test, I scored low in the 'needs affection' category,\" Mersereau wrote recently to a Citizen Schools volunteer coordinator. \"My wife calls me insensitive. But Citizen Schools has become a very emotional experience for me. Being a part of such a philosophically solid enterprise has been fulfilling to say the least. Hauling my teaching materials into the Garfield [a school in Revere, Massachusetts] and New Bedford while undergoing radiation and afflicted with double pneumonia was simply a test of how important this new 'occupation' has become for me. Not only is this 'keeping me alive' it is a real source of personal value. I am not sure if there is any other reason to get up each morning other than the sense of being of value and Citizen Schools affords that opportunity to every one of us.\n\n\"This idea you had of pairing me up with my son Bobby is more of the same,\" continued Mersereau. \"I am so proud of him for 'following in my footsteps'\u2014he has expressed himself in those terms\u2014not easy for a father and son to talk like that.\"\n\nCitizen Schools and other citizen-power programs also deliver value to early-stage professionals. Supervisors at some of America's leading companies\u2014places like Google, Fidelity, Cisco, and Cognizant\u2014report that employees who volunteer at Citizen Schools improve their skills at leading teams, at communicating expectations, and at planning multidimensional projects. A whiz-kid engineer like Alan Su (introduced in chapter 6) learns how to be a more effective manager when he leads fifteen kids of diverse academic abilities to create educational video games over ten ninety-minute sessions. Cognizant, a technology firm with a large foreign-born workforce, noted an additional benefit. Teaching an apprenticeship, says Mark Greenlaw, the firm's former vice president for Sustainability and Educational Affairs, helps Cognizant employees gain confidence in their English skills even as they help transmit confidence in technology to the middle school students they teach.\n\nRecently, Citizen Schools commissioned University of Vermont Business School professor David Jones to examine in a methodologically rigorous way the potential impact that volunteering has on citizen teachers and their companies. Jones studied more than one hundred volunteers across four companies and a control group of similar employees who did not volunteer. He concluded that volunteering at Citizen Schools delivered significant gains in employee satisfaction and loyalty to the volunteers' companies and gains in specific job skills, including communication, leading and motivating others, and providing performance feedback.\n\nThe fact that well-designed mentoring and teaching experiences can benefit adult volunteers is exciting. But the reason I am so excited about citizen power is that it represents the key to lifting up opportunity and achievement for more children.\n\nHuman beings learn best through relationships and experiences. Some children get a wealth of rich, catalyzing experiences courtesy of their parents and in their schools and neighborhoods. Other children get little access to these experiences and are poorer for it, and they stay poorer because of this disparity. The good news is that successful professionals and successful retirees and successful graduate students\u2014all potential mentors\u2014are not a scarce or limited resource. We spend trillions searching for scarce natural resources like oil. We'll need to spend far less to mobilize citizen power\u2014a plentiful and renewable resource that has the power to advance opportunity for all, strengthen our workforce, and restore the American Dream.\n\n# CHAPTER NINE\n\n# IT'S ABOUT TIME\n\nWhen Citizen Schools opened in North Carolina, I had a chance to meet former governor Jim Hunt. I knew that Hunt was a skilled political leader who had led the Democratic National Committee in the 1990s. And I knew that he was a leader among a cohort of southern governors trying to lift expectations and funding for schools to build a more educated workforce and thereby help compete for good jobs. Hunt recognized that plentiful mill jobs were gone for good and that the economic future of his state lay in the big-city banks of Charlotte and the emerging biotech and computer technology companies of the Research Triangle. I did not know that for both his terms as governor, Hunt insisted on devoting a portion of every Thursday to tutoring a student after school at a struggling Raleigh public school.\n\n\"I remember one boy I worked with,\" Hunt told me. \"And he said, 'Governor, I'm not stupid. I can do the work. But I just need a little more time to understand it.'\" Governor Hunt is a big believer in extended-day learning, because he knows that kids learn at different paces and in different ways and that for many kids, all they need is a little more time.\n\nHunt recognized that the great inequity in our education system has less to do with differences in teacher quality between the suburbs and inner-city neighborhoods of Charlotte and Raleigh and more to do with the different amount of time devoted to extra learning opportunities. Sure there are some differences in teacher quality. Better teachers often stay longer in wealthier districts, which offer better working conditions, though not better pay. We urgently need to address these differences. But differences in teacher quality are more pronounced within schools than between schools, and, as described in the introduction, school quality only explains 20 to 30 percent of the rich\u2013poor achievement gap. Hunt wondered why we maintain an education system where time for learning in school is fixed and achievement is wildly variable, instead of a system of consistently high achievement where time for learning is variable.\n\nWhen my own children were entering fourth and first grades, my wife and I decided to move from Boston to neighboring Brookline and enroll our kids in the William H. Lincoln School, a great K\u20138 public school. The house we bought was more expensive than the one we sold, just a ten-minute walk away, in Boston's Jamaica Plain neighborhood. But more than a house, what we bought was access to a school and neighborhood that is economically diverse but weighted toward professionals, including architects, academics, doctors, lawyers, and business leaders. I see every day how upper-middle-class communities like mine powerfully organize afternoons, weekends, and summers full of formal and informal learning that propel our children past those living on the proverbial \"other side of the tracks.\" Brookline doesn't have a longer school day than Boston. But when you add up formal and informal learning, most kids in Brookline have a learning day that's almost twice as long.\n\nChildren have fundamentally unequal access to learning time, just as they have unequal access to successful mentors, two things we must change if we are to narrow our nation's huge and growing achievement gap.\n\nAs I write this, I am missing my daughter, Orla. She is eleven now, prime Citizen Schools age, and in the middle of an eighteen-day singing tour in Argentina, where she is performing seventeen concerts, riding horses, sightseeing in thirteen cities, and having the experience of a lifetime courtesy of Boston City Singers, a youth choir she joined last year. My son, Ronan, sixteen, had his own adventures this summer. In the past eight weeks he traveled with his AAU (Amateur Athletic Union) basketball team to Philadelphia, Portland, Hartford, and Orlando, where he played at the ESPN Center at Disney World, notching twenty-six points in one game. He also attended specialized basketball camps at Harvard and Yale, receiving high-level instruction and getting exposure to campus life in the Ivy League.\n\nWhen Orla and Ronan were younger, we invested in academic tutors on a few occasions, not because our kids were behind, but to prevent them from getting behind. In Ronan's case the tutor was Catholic and wore a necklace with a large cross. She told us one day that Ronan said he appreciated her wearing the large _T_ around her neck. \"If you could just start wearing the other letters, then reading would be a lot easier,\" Ronan told her. We all had a good laugh, and as parents we felt good that we had a concerned and sympathetic ally helping us raise and educate our son. When Ronan was in third grade we conducted some testing and found that he was struggling with sensory integration. We signed him up for sessions at a state-of-the-art center that effectively addressed the issue. Like most upper-income parents, we organize our lives\u2014and never-ending car pools\u2014to put opportunity in front of our children. We write checks left and right to underwrite these opportunities.\n\nThe voracious appetite for supplemental learning in wealthier suburbs has spawned a fast-growing cottage industry of tutors, enrichment programs, and specialized camps. Just in our immediate circle in Brookline, children enroll in fee-based programs that offer piano, pottery, theater, choir, dance, voice lessons, guitar, robotics, architecture, video game design, filmmaking, ballet, gymnastics, Hebrew school, Suzuki violin, Russian math, Chinese language, cooking, capoeira, sailing, rowing, rock climbing, martial arts, Girl Scouts, Boy Scouts, Junior Achievement, all manner of team sports, and hip-hop dance. Collectively, our friends employ dozens of tutors, coaches, and counselors.\n\nDemand for extra learning is so strong in communities like Brookline that some education companies have gone beyond cottage-industry status to become big business. For instance, Kumon, a popular tutoring service in math and reading, is now the twenty-second-fastest-growing franchise business in the world, right up there with the fast-food burger chains. Kumon started in Osaka, Japan, in 1954. Toru Kumon, a high school calculus teacher, was tutoring Takeshi, his second-grade son, when he noticed the benefits of giving short, bite-size assignments in math and getting total mastery and lots of repetitions before moving to the next challenge. By sixth grade Takeshi was solving calculus sets. Toru and his wife, Teiko, opened the first Kumon Center in Moriguchi City in 1958, and today the company operates in forty-eight countries around the world, with two thousand Kumon franchisees in North America (startup costs are about $100,000) serving 340,000 students. Kaplan, Sylvan, and Princeton Review are additional giants in the supplemental education market, which is overwhelmingly aimed at middle-class and upper-middle-class communities.\n\nI love the vibrancy of Brookline's after-school learning ecosystem and would love to see it replicated in Boston, Oakland, and other big cities across the country. But until parents in Boston and Oakland have the money to pay for all that is offered in Brookline, their after-school options are going to remain significantly less dazzling. As a result, I have come to believe that while Brookline should keep its formal six-hour day, which for most kids works effectively in combination with the town's array of school-based and parent-guided extra learning opportunities, Boston and other lower-income communities need a whole new day for learning. They need an equally dazzling nine- or ten-hour public school day with more time for academics, more time for enrichment, and more time for social and emotional development\u2014all built into the publicly funded core school day. By building extra learning time right into public schools, we can transform the current patchwork of programs that help a relative handful of kids beat the odds into a system of programs that helps whole neighborhoods of children change the odds. The students mentored by Governor Hunt needed more time for learning, but so frankly do most students in high-poverty schools. They need more time with successful adults to help them build their social skills, social networks, and social capital. They need more time to create and innovate. And they need more time to master the academic basics. In middle school\u2014which Education Secretary Arne Duncan has called \"the Bermuda Triangle of American education\"\u2014the need is particularly acute, as children require more help navigating a set of challenging years when the brain is changing and developing faster than at any point except right after birth.\n\nPut yourself in the shoes of a middle school principal in any lower-income neighborhood in America and consider the challenge presented by today's typical school schedule\u2014a schedule invented for an agrarian age and too often implemented with an industrial-age sensibility. Imagine your district has assigned you six hundred sixth through eighth graders, typical for middle schools, and most of them arrive by bus, many from other neighborhoods in the city.\n\nOn the first day of sixth grade, your students arrive on average two years behind grade level. Several are so far behind that you wonder if they have been attending school at all. Ninety percent of your students live below or just above the poverty line and about half are learning English as a second language or have been identified as having a significant learning disability, or both. Roughly 10 percent of your students will turn over every semester, meaning that when you prepare for state testing in the spring and for graduation in June, about 120 students will be newcomers to your school. Now assume that your district has a traditional school day, in line with national averages, meaning that your students arrive by bus at about 8:00 a.m. and start lining up to go home at about 2:15. Once you account for lunch, a lightning-quick recess, and take time for recording attendance in the morning and transition time between classes, you have about five hours left for academics and enrichment. How will you use the time? And how might citizen power help you expand it?\n\nLet's assume you are a believer in arts and sports and music as ways for students to build teamwork skills, and to practice creativity, and as an alternative way for students to shine and build their confidence. But let's also assume that your school doesn't have much in the way of art and music and sports facilities\u2014and that you don't have time to travel off campus. You decide to provide one hour per week each of art, and music, and sports\u2014for every student. On the fourth and fifth days, you provide students with an hourlong study hall block instead of enrichment because it's easier to staff and can be scheduled in the regular classrooms, which helps with logistics. You just scheduled your first hour. You have four left.\n\nThe district has prioritized math instruction in large part because last year half of the eighth graders failed the new Common Corps\u2013aligned statewide math assessment, and only 18 percent were proficient (actual numbers from New York City in 2013). Plus, middle school math is a gateway skill for higher education and for life (more so than high school math for all but the highest-level STEM jobs). If your school is in California, you need to get your middle school students through Algebra 1 by the end of eighth grade or else they may be unable to enroll in a four-year college even if they have good grades in high school due to the state's A through G course requirements, which require starting Algebra II by ninth grade. Eighth grade seems awfully early to give up on a four-year college, so you decide to schedule a double block of math for two hours each day in hopes you can get your students ready for college-track courses in high school. The double block gives students lots of opportunities to work on math problems, but classes are still operating on a 1:28 teacher-to-student ratio, so there is little time for individualized instruction or for tutoring of particular students. Still, focusing on math seems like a smart move. But now you only have two hours left.\n\nYou would like to offer a foreign-language block, as you have heard that learning a foreign language builds brain capacity and is increasingly important in the modern global economy, but there are only two hours left in your schedule and you haven't yet gotten to English or science or social studies. You'll have to skip offering any foreign-language instruction, at least for this year. You would also love to provide a double block for English language arts (ELA), because this could allow a deep dive in both reading comprehension and writing and oral expression\u2014key skills in the modern economy. But if you schedule a double block in ELA, you will have zero time left over for any science or social studies classes, nor any time for theater or library or computer science. You decide to go with a single English class per day. Since 80 percent of the state test is focused on vocabulary and reading comprehension, with just a few short sections testing student writing skills, your teachers too will focus mostly on vocabulary and reading and just do a few writing units, mostly focused on how to write a five-paragraph essay.\n\nNow you have one hour left. You decide to split that time between social studies and science, offering each student a chance to study one of those core topics for an hour a day in the fall and then switch to the other in the spring. A new study says that reducing time for science in school actually undermines student performance in math and reading. And you worry less time on social studies will yield less engaged and informed citizens. But you have hard decisions to make and you stick with the schedule you've crafted. You certainly wouldn't want the schedule for your own child. But you feel stuck between the proverbial rock and a hard place, and without more time you don't see how it's possible to provide the time needed for academics and for the well-rounded education that you know your students deserve.\n\nThese are the types of bad choices that our agrarian-era school schedule forces even the most progressive and creative school leaders to make. Without sufficient time, principals routinely choose academics _or_ enrichment; extra time for English _or_ math; social studies _or_ science. The short days work in the suburbs because the students arrive at or above grade level and their education is massively supplemented in the afternoons, on weekends, and in the summer. But for schools serving primarily lower-income students, the traditional school schedule is obsolete.\n\nNow imagine you are that same principal but that your students stay in school from 8:00 a.m. until 5:15 p.m. instead of 2:15. You are a practical person, so at first you would probably throw your hands up in despair because you don't know how in the world you can staff the extra time. Your teachers are already working full-time to run the six-hour day, so you can't just put the extra three hours on them. What will you do?\n\nHere is where Citizen Schools\u2014and citizen power more broadly\u2014as well as artful efforts to schedule teacher time more flexibly can make a longer learning day feasible, affordable, and productive. Keep in mind that while teachers typically teach students for five or fewer hours per day, most studies indicate that they work nine or ten hours per day or more. The additional time is devoted to lesson planning, reviewing and grading homework and tests, meeting individually with students and\/or parents, serving on committees, overseeing lunch or recess or student arrivals and departures, collaborating with other teachers, and reviewing interim assessments.\n\nDespite the long days, your teachers know their students need more learning time, and with the help of Citizen Schools, they want to make this work. Some teachers elect to work a different shift\u2014starting and ending their day a few hours later than normal, which can work well for teachers who need to get their own kids off to school in the morning. A few other teachers agree to add an extra hour of teaching time in exchange for a little additional pay. But the bulk of the coverage for the extra time comes from Citizen Schools. Specifically, by partnering with Citizen Schools, our principal gets an entire second shift of educators. The Citizen Schools team is led by a campus director\u2014someone who has already taught for three or four years, and who essentially serves as the school's assistant principal for ELT. The campus director will be an additional right hand for you as principal and will sit on the school's leadership team. His or her name may go on the school stationery, below your name and next to those of the assistant principal and director of instruction. Reporting up to the campus director are one or two deputies and as many as two dozen full-time AmeriCorps members\u2014all recent college graduates, most aspiring to be full-time teachers. Also included in the package are a few part-time staff and about one hundred trained citizen teachers\u2014volunteers from local industry\u2014enough so that every student in the school can take four apprenticeships over the course of the year, each offered one afternoon a week for a three-month span.\n\nCitizen Schools is not the only way to staff a longer learning day. There are promising models being developed by TASC in New York, Baltimore, and New Orleans, in which teachers cover one hour of extra learning and community partners provide an additional one to two hours of enrichment. Other organizations such as BELL, Breakthrough, City Year, College Track, Communities in Schools, and Higher Achievement are starting to move beyond offering optional after-school programs to more robust partnerships that change the structure of the school day and year. Many charter schools build into their model AmeriCorps-type fellowship positions, allowing them to staff a much longer learning day while keeping teacher workloads to manageable levels. The MATCH charter school in Boston, for instance, has Match Corps, a team of aspiring teachers who tutor six teams of two to three students every day; this is a smart strategy for expanding the day without burning out teachers. Some other charter schools have scheduled much longer days and staffed them without benefit of a second shift. When I speak with funders and leaders of these charter groups they generally concede that relying on teachers to routinely work twelve or more hours is a recipe for burnout and is not sustainable or scalable.\n\nWith a longer learning day and a second shift of talent, our principal can now take a fresh look at the schedule. Now Citizen Schools can take the lead for that second hour of math, allowing teachers more time to collaborate, or plan, or pull out individual students for extra tutoring. And now students can take both science and social studies all year long, and they can add a full two hours of enrichment every day, allowing for special classes like rocket science and slam poetry, allowing for field trips to local colleges and museums, and allowing for the relational time that adolescents need to navigate middle school.\n\nExtra time is particularly important for English-language learners, who need more chances to use English in real-world contexts, and for special-education students, who often need expert intervention from trained professionals as well as more personal attention and guidance and extra time to master academic content. But the extra time can be stressful too, adding more opportunity for conflict and challenge. Tyriq, a special-needs student enrolled in one of our Boston ELT schools, is a case in point.\n\nIn many ways Tyriq fits my favorite description of a middle schooler as having \"the full emotional range of an Italian aria, packed into the attention span of a gnat.\" Tyriq has the _capacity_ to show that full emotional range, but when our staff first encountered him in the fall of 2012, most of what we saw was anger. He spent most of his first semester in middle school getting kicked out of the classroom at all times of day for swearing, breaking things, and fighting with other students in his separate education group, the Therapeutic Learning Community (TLC). He could often be seen storming down the hallway to the time-out room, paraprofessionals hurrying behind to help calm him down and redirect his behavior so that he could reenter the classroom. Tyriq lacked the patience and emotional skills to cope with his frustration in a healthy way and he lacked the desire to do well in his classes. He ripped up tests and broke pencils to avoid performing academic tasks. His grades and scores were rock bottom.\n\nI have met a lot of kids like Tyriq over the years and, typically, school does not work out well for them. They have emotional issues, or attention issues, or organizational issues, or reading issues, or all of the above. For medical reasons\u2014and sometimes because of severe daily trauma in their lives\u2014tasks like listening to instructions, and following directions, and working in groups, and doing multistep problems are really hard. Some days they just can't do it. They are so angry they feel like their head is going to explode. It gets even harder because classmates in their special-needs clusters have similar problems, so they reinforce and add to each other's challenges. Further, they all notice that they are in the \"Therapeutic Learning Community\" or the \"Learning Adaptive Behavior\" cluster or just the \"special needs\" class, and they feel ashamed. In the cafeteria, other students call them the dumb kids or the weird kids or use \"SPED\" as an epithet and avert their gaze as they walk through. When they sign up for a sports team, which is rare, they often get kicked off the team for yelling at the coach or not following a play or for some other transgression.\n\nTyriq's school had some uncommonly good special-education teachers. But his lack of emotional control and intense struggles with executive function were on full display when he entered his Citizen Schools classroom at the end of each day last fall, the first year of mandatory ELT at the McCormack Middle School. When he was assigned to the soccer apprenticeship on Thursday afternoons, he refused to enter the classroom the first week, and the second week he entered reluctantly, unleashing a torrent of curse words. It was the same story when guests would come in to help teach the apprenticeship, such as Pat Kirby, then the executive director of Citizen Schools Massachusetts. Kirby, a college-level soccer player, remembers casually ignoring Tyriq's initial stream of foul language and encouraging him to participate, to share his knowledge of sports statistics, and to play in the scrimmage at the end of the lesson, something he had never done in any previous week.\n\nKirby arrived in the fourth week of the apprenticeship, and his presence marked the first in a series of turning points for Tyriq. According to his Citizen Schools team leader, Tyriq talked about his positive experience in soccer all week long. In a world devoid of positive male role models, Tyriq came to depend on Kirby, his new mentor. He started to open up in his soccer apprenticeship, looking forward to it each week and earning the title of \"Team MVP\" in the final WOW! Bowl Tournament. He went from literally standing on the sidelines of the game, reluctant to enter the fray, to running into the chaos and helping to score the winning goal. In the spring Tyriq rejoined the soccer apprenticeship and served as the team captain. He took his new role very seriously, rarely swearing and impressing teachers with his growing emotional maturity and his kindness to other students on the field. For someone who initially seemed unable to display any empathy, actions he now took, such as using the word \"we\" to refer to his teammates and helping people up when they fell, were big developments. Tyriq's improved behavior started to spread to his regular Citizen Schools team too and eventually to his performance in the classroom. He started to improve his grades. One day when Pat Kirby came in to visit him, Tyriq bragged about earning a perfect score on a recent math test.\n\nTyriq's only hesitation in playing on the soccer team in the spring was missing the opportunity to be in the camping apprenticeship, which was held at the same time. To build on Tyriq's momentum, however, Citizen Schools staff agreed that after the soccer apprenticeship ended he could join in the last few weeks of camping and, if his behavior kept improving, could participate in the camping WOW!\u2014an authentic overnight camping experience in the Blue Hills Reservation about ten miles outside Boston. Tyriq was initially wary of the many deprivations involved in camping, but when the camping day came he was eventually laughing and running around with new friends. He recruited others to watch the sunset over Ponkapoag Pond, which he described as \"peaceful and beautiful,\" slept through the night in his tent, and completed an eight-mile hike the next day. Tyriq told his team leader during a quiet moment on the trip that during the past year, having more time to work with people like her and Mr. Kirby \"helped me work on my behavior, and now Mr. White [the coordinator of the TLC program] wants me to do regular education classes next year.\" She asked him, \"Are you scared for next year?\" and he replied, \"No, I already have friends that are older than me,\" and he named several of the other students on the camping trip and from his sports apprenticeship teams.\n\nFor Tyriq and other students, ELT gives them time to slow down, to literally and figuratively watch the sunset and soak in some extra time with a mentor, a teacher, or an interesting assignment.\n\nELT is such a powerful idea\u2014and so full of common sense\u2014that it's worth pondering why it has not been adopted more broadly. The National Center on Time & Learning lists more than one thousand ELT schools, but that's less than 1 percent of the schools in the country, and many of these schools have added just an hour a day of extra time, which is a good start but far from sufficient.\n\nI think there are two main reasons we haven't yet adopted ELT on a more ambitious scale.\n\nThe first reason is inertia and the difficulty of shifting paradigms that govern our thinking. When I was in my twenties and working as a journalist, I took night courses in the history of science at the Harvard Extension School. One of the people I read was Thomas Kuhn, who wrote _The Structure of Scientific Revolutions._ Kuhn noted that old paradigms\u2014like the idea of the earth being flat or at the center of the universe\u2014die a slow death. For an old paradigm to fade away, Kuhn believes, it not only needs to be proven wrong but an alternative theory needs to be persistently proven right. The flat earth idea didn't become pass\u00e9 just because people realized that boats weren't falling off the edge of the earth. It took a new theory of a spherical globe that rotated on its own axis and around the sun for the old theory to truly enter the trash bin of history. In education it's a similar process, except maybe even slower. That's why we need a vision for what ELT can look like. We need to create more ELT success stories that people can learn from and be inspired by. For the old agricultural-era schedule and industrial-era delivery system to truly wither away, we need to create an alternative theory of how to organize schools, and we need to prove the theory over multiple years in multiple locations.\n\nThe second reason society isn't moving faster to ELT\u2014and I am going to say this in stark and honest terms\u2014is a deeply ingrained and thoroughly incorrect belief held by too many people that poor kids, and particularly poor black and brown kids, can't learn at high levels. This belief makes it easier to accept poor kids not getting access to opportunity. If kids like Tyriq can't learn, and their parents didn't learn\u2014this usually unstated line of reasoning goes\u2014then it is okay for society to give up on them.\n\nA great deal of expert research, however, has challenged this genetically driven talent frame and pointed to other explanations for success, whether in academia, the workforce, or in other pursuits like sports and music. The new research doesn't entirely discount the power of inherited genes to influence future outcomes, but in the age-old debate between whether nature or nurture plays a bigger role in setting our destiny, the new research gives the nod to nurture.\n\nMalcom Gladwell has become the great popularizer of this research. Gladwell, a British-born Canadian journalist who writes for the _New Yorker_ and recently had three books on the _New York Times_ best-seller list at the same time, writes in his book _Outliers_ that the two greatest factors in driving achievement are luck and the time to practice, not inherited talent. Whether you look at the Beatles in music, or Bill Gates in software design, or all-stars in hockey and other sports, notes Gladwell, great achievers generally log ten thousand or more hours of practice before they get great at what they do.\n\nAs a seventh grader, Bill Gates was lucky enough to be enrolled by his parents at Lakeside, an excellent private school in a wealthy Seattle neighborhood. The Mother's Club at the school held a fund-raiser every year to provide extra resources to the school, and in Gates's eighth-grade year, the mothers decided that in addition to supporting a summer scholarship program and a training fund for teachers, they would invest $3,000 in a state-of-the art ASR-33 Teletype computer with a direct link to a large mainframe computer in downtown Seattle. This one investment provided Lakeside students a computer-programming opportunity that went far beyond what most college students had access to at the time. Gates himself reported logging thousands of hours on the Lakeside computer. Then, in high school, he gained access to two computer centers at the University of Washington. Just as the world of coding was opening up, Gates was one of the few anywhere who actually had access to practice it.\n\n\"It was my obsession,\" Gates told Gladwell. \"I skipped athletics. I went up there at night. We were programming on weekends. It would be a rare week that we wouldn't get twenty or thirty hours in.\" With his ambition, his intelligence, and his parents' support, Gates made the most of his differential access to computer time, and by the time he was nineteen he was ready to drop out of Harvard and start Microsoft.\n\nSome of the most convincing evidence for the importance of extra learning time involves hockey players. In Canada and many other countries, January is the cutoff date for youth hockey teams. To understand what this means, imagine that it's October and you and a friend show up to play youth hockey in Manitoba. Your friend is going to turn six in December and you are just a few weeks younger and will turn six in January. Based on the age cutoff, your friend gets assigned to the six-and-under team, where he will be the youngest player and probably just a little slower and smaller than most of his teammates; you become the oldest player on the five-and-under team. Age differences of six or twelve months don't matter much once you get to high school or college. But at age five, a six-month difference is 10 percent of your life! And that 10 percent turns out to provide a real advantage.\n\nAt age five or six, the slightly older kids\u2014the kids born in January, February, and March\u2014are, on average, slightly better players. They get placed on the first line. They play twenty minutes a game rather than ten. And they are more likely to be selected for the all-star game. As the kids get a little older, the extra time and the competition against better players pays off, and the kids born in the early part of the year are now more likely to get selected for the better travel teams. That means more games, and more practices, and a bigger competitive edge. Canadian psychologist Roger Barnsley was the first to study this effect of relative age, and he found, according to Gladwell, that in any study of elite Canadian hockey players, right up to the National Hockey League, about 40 percent of the players will have been born in the first quarter of the year, 30 percent in the next quarter, 20 percent in the third quarter, and just 10 percent between October and December.\n\nThe kids coming into Citizen Schools are like hockey players born in the late fall. They need extra learning time to catch up.\n\nWhy should Bill Gates and his classmates at a fancy private school be the only ones to get access to thousands of hours of computer time? How much faster would our technology industry have developed, producing how many more jobs, if all schools had a computer like his did and a mechanism to allow students to learn for extra hours? How much better would we be as a country if instead of just a few kids getting extra ice time, or extra tutoring time, or extra enrichment time\u2014all kids got it? Some kids would still rise to the top, and others would still bring up the rear. But we would have something much closer to a true meritocracy than we do today, and our average educational attainment would rise, boosting the economy for everyone.\n\n# CHAPTER TEN\n\n# FROM A NATION OF CONSUMERS TO A NATION OF MAKERS: INSPIRING CREATIVITY AND INNOVATIVE THINKING IN OUR SCHOOLS\n\nTinkering seems to come naturally to McCalvin Romaine, who is now in his twenties and pursuing a career in technology at Digitas, one of the world's top online marketing firms. On Tuesday afternoons he can often be found at the Washington Irving School in Roslindale, Massachusetts, volunteering with teams of sixth and seventh graders and teaching the young students how to pull apart computers, install more RAM, and identify and fix software problems.\n\nReflecting on his childhood, Romaine's eyes open wide and he smiles broadly as he remembers the time he turned his Mom's hair dryer into a flashlight. All it took, he remembers, was duct tape, copper wire, and a little ingenuity to connect the dryer's batteries to a small light bulb rather than a heating coil and blower. That the flashlight operated and looked like a gun, turning on as you pressed the plastic trigger at the base of the dryer, added to the excitement felt by the nascent inventor. But Romaine also remembers a dark side to his project. He brought his flashlight into school one day and his teacher threatened to throw the whole project in the trash if he didn't put it away. Then she reprimanded him for straying from his assigned work.\n\nRomaine was a Citizen Schools student in middle school, and his favorite apprenticeships were in hip-hop dance, law, and a technology class not very different from the one he teaches today. \"I was always a curious kid growing up,\" Romaine told me. \"But it wasn't until I joined Citizen Schools that I really understood where these math and science courses that I took in school could take you in the future. Just seeing so many volunteers who were real professionals in my apprenticeships, it gave me an outlet for my creativity and it gave me some insight as far as the career paths I could pursue.\"\n\nAcross America we face a giant and growing mismatch between what our schools are teaching and what our children need. Creativity and scientific thinking and innovative problem-solving are skills increasingly in demand in today's job market. Yet too many of our schools focus on teaching students how to follow rules, memorize facts, and master basic arithmetic and reading. Schools persist with an old paradigm despite mounting evidence that the scientific thinking of American students lags that of students in most other industrialized countries and new evidence that the creativity and innovativeness of American students\u2014which had formerly set the United States apart\u2014is in decline. Some of the mismatch is due to the short agrarian-era school day discussed in the previous chapter and to the limited time allocated to scientific learning in schools in the era of high-stakes math and reading assessments. One study showed that in the last fifteen years there has been a 23 percent decline in time devoted to science instruction in schools, down to a low of 2.3 hours a week in 2011\u201312. In my experience, the decline has been most dramatic in schools serving the lowest-income students.\n\nCompounding the time-on-task challenge, too few schools have the equipment or orientation to offer significant time for hands-on science projects, the type most likely to spark interest in science or engineering as a career and to build creative-thinking skills. Researcher Robert Tai and his colleagues have demonstrated that interest in science in middle school is a stronger predictor of who goes into careers in STEM fields than is math or science proficiency. And the Lemelson-MIT Program sponsored research indicating a majority of teenagers reported they may be discouraged from pursuing STEM careers because they do not personally know any engineers or know what engineers do. Yet despite looming talent shortages in all sorts of STEM disciplines, we persist with a school system that is offering kids less time for science, less time to conduct experiments and build things, and in most cases no time at all to meet and work with real scientists and engineers like Marcus DeLatte and David Mantus. Where is the interest supposed to come from?\n\nToo often the shortage of exciting STEM opportunities in public schools means that the only kids who get cool STEM opportunities are kids who already have STEM-savvy role models in their lives. These are mostly the children of engineers and doctors and scientists, most of whom are white, male, and middle class. No wonder our homegrown STEM pipeline is drying up like an old riverbed in a drought. Among students who took AP (Advance Placement) computer science classes in 2010\u201311, for instance, a gateway into one of the fastest-growing segments of the STEM workforce, just 2 percent were African Americans and only 16 percent were girls. In California that same year, 51 percent of all K\u201312 students were Latino, but among those taking AP computer science, just 7 percent were. We shouldn't be surprised that just 14 percent of engineers in America are women, and only 5 percent are African Americans.\n\nAs I was launching Citizen Schools in the mid-1990s, there was a lot of talk in education circles about creativity and higher-order thinking skills. Of course, I wanted Citizen Schools to promote both. Who wouldn't? But we also had pressure from parents and school partners to make sure their children were progressing on the basics\u2014arithmetic, reading, and writing\u2014and that seemed like a key priority for Citizen Schools too. We were stuck in a raging debate between back-to-basics purists and advocates for twenty-first-century skills such as higher-order problem solving, teamwork, and creativity. The focus on basics was often associated with E. D. Hirsch, who wrote a series of books outlining what all Americans should know. The second approach was associated with an unusual alliance of corporate leaders and educational progressives.\n\nAs I tried to navigate Citizen Schools through this educational debate, I often turned to a slender volume called _Teaching the New Basic Skills_ , published in 1996 by educational economists Richard Murnane and Frank Levy. Instead of guessing what skills graduates needed\u2014and would increasingly need in the future\u2014the authors studied in detail the actual hiring practices of eight large American companies, including Northwestern Mutual Life, AT&T, and Mitsubishi Motors of America. By starting with real data about the skills workers needed to get good entry-level jobs, Murnane and Levy cut through a lot of the ideological abstractions. I also appreciated that the \"new basic skills\" Murnane and Levy described as necessary for the modern workforce were also a good inventory of the skills active citizens need to thrive in a democracy. Schools need to prepare students to earn a living in the work world but also to contribute to their communities.\n\nMurnane and Levy drew three central conclusions. First, companies hiring for entry-level middle-class jobs have an increasingly clear set of standards. To be hired, applicants need ninth-grade-level math and reading skills, but they also must possess the ability to communicate nuanced ideas orally and in writing, to work well on diverse teams, to form hypotheses and test them, and to use technology to solve problems. These are what Murnane and Levy call the new basic skills, a subset of what others call twenty-first-century skills.\n\nSecond, Murnane and Levy's research indicated that half of the eighteen-year-olds in the country, even as they are poised to graduate from high school, do not possess the new basic skills and therefore are not eligible to get hired at entry-level jobs at companies like those named above.\n\nThird, the authors concluded that the new basic skills can be taught, and that just as with traditional academic skills, they can be learned through practice. Murnane commented: \"These skills need to be practiced and after-school and extended day programs provide a great way to practice them.\"\n\nAt company after company they visited during their research, Murnane and Levy learned that hiring managers had adopted a multistage process for winnowing their applicant pools. In every case the hiring process took into account cognitive and analytic skills taught in school but also tested for other abilities. At Mitsubishi Motors of America, for instance, prospective employees were first given a drug test and the General Aptitude Test Battery (GATB), an IQ-like test that assesses nine different skills including verbal and numerical aptitude. Depending on the exact job, the company set minimum thresholds for the GTAB, typically near the fiftieth percentile. Prospective employees who passed the drug test and scored at the required level on the GATB would be invited back to take a mechanical aptitude test and then to engage in two expensive hands-on assessments. In the first of the hands-on tests, the authors explained, each applicant was assigned to a small team composed of men and women with various ethnic backgrounds. Their task was to assemble circuit boards, wires, and modular components on a large frame according to a blueprint. The team members could pursue the work in any fashion they chose. At the end they held a _kaizen_ meeting (a continuous-improvement approach pioneered by Japanese corporations) to discuss how they could have done the job better. Then they assembled a second board. Throughout the ninety-minute activity, trained evaluators assessed the applicants on teamwork, communication skills, and creative problem-solving abilities.\n\nI was deeply influenced by Murnane and Levy's work and found it highly relevant to our own. They had pinpointed a mismatch between the goals and values of schools and the requirements of the workplace. But rather than business being the bad guy, representing soul-crushing conformity and slavish obedience to the profit-driven dictates of \"The Man\" while education offered a high-minded vision of learning and discovery, somehow things had been reversed. Schools were the ones following the industrial-age playbook, and many in the corporate world were looking for something new.\n\nThe biggest mismatch between what schools are offering and what the workforce needs concerns creativity. In 2010 IBM released a poll of fifteen hundred CEOs, who identified creativity as the number one \"leadership competency\" of the future. The next year the social-networking site LinkedIn reported that \"creative\" was the most commonly used word in the profiles of its members. Clearly creativity is hot in the business world. But there is new evidence that the creativity of America's children\u2014long the envy of the world\u2014has been in decline since the mid-1980s. The decline was brought to light by Kyung Hee Kim, who analyzed fifty years of data from the world's most established creativity assessment: the Torrance Test of Creative Thinking. In 2010 _Newsweek_ magazine trumpeted the topic on its cover, calling it a \"Creativity Crisis.\"\n\nKim and others who have analyzed the data say that creativity in America started to decline sometime between 1984 and 1990. In Kim's words, the data indicate that \"children have become less emotionally expressive, less energetic, less talkative and verbally expressive, less humorous, less imaginative, less unconventional, less lively and passionate, less perceptive, less apt to connect seemingly irrelevant things, less synthesizing, and less likely to see things from a different angle.\" The change has been significant, with 85 percent of students in 2008 scoring below the level for the average student in 1984.\n\nAs the nation thinks through how to prepare more young people for the innovation economy of the present and the future\u2014for jobs that require a strong grounding in basic academic proficiencies _and_ a bent toward innovation, tinkering, and team-based problem solving where there are no rules\u2014we need to rethink how we build those skills and who is best positioned to teach them.\n\nThese were the key questions posed in 2004 at Reimagining After-School: A Symposium on Learning and Leading in the 21st Century, organized by Citizen Schools at the Harvard Graduate School of Education. Prolific author and Harvard Business School professor Rosabeth Moss Kanter kicked off the conference with a call for \"kaleidoscope thinking,\" which questioned who should teach twenty-first-century skills and when and where they should be taught. I spoke and urged my colleagues in the after-school and Expanded Learning Time fields to lift our sights beyond the low standard of keeping kids safe during potentially risky afternoon hours and instead see extended learning time as a front-burner solution to America's educational and social challenges, such as how to teach creativity and innovative problem solving.\n\nThe Reimagining After School conference drew two state commissioners of education, numerous district-level leaders, national policymakers, multiple foundation presidents, and extended-day leaders from across the country. A surprise attendee\u2014and a presenter\u2014was Tyeisha Bogy, an assertive sixth grader from the Grover Cleveland Middle School in Boston's Fields Corner neighborhood, a school so challenged that it would close for good five years later. Tyeisha was one of ten sixth and seventh graders who had enrolled in a Citizen Schools apprenticeship I taught called Power Learning Out of School. The course was my effort to question the paradigm around where and when learning happens, while also introducing my students to interesting people. I also wanted the students to apply their middle school math skills in a real-world pursuit with an authentic audience.\n\nThis was an unorthodox Citizen Schools apprenticeship with no toolboxes, video games, rocket launches, or mock trials. But my students did important work. Our goal: to interview as many CEOs and other senior executives as possible, ask them about the key skills they needed to get to the top, and find out where they had learned those skills.\n\nWe started with in-person and conference-call interviews, and we completed some great ones with Jeff Taylor, CEO of Monster.com, and the heads of the Boston Children's Museum and a fast-growing beauty products business. These interviews helped me and my ten apprentice research assistants to build a questionnaire that we posted on SurveyMonkey.com. We then recruited nonprofit and for-profit CEOs to fill out the survey online, eventually getting more than two hundred responses. While I am quite sure our sampling techniques did not meet the standards of the Gallup polling group, what we learned was interesting.\n\nWe discovered that of the nine skills that helped executives get to the top (ranging from math to teamwork to creativity), just three of them were learned primarily at school, with three learned primarily at home and three learned primarily through after-school and summer activities. The three most important tickets to the top, our executives said, were perseverance, teamwork, and creativity. Further, they said that these skills were primarily learned outside of school, either at home or in various enrichment experiences like those described in this book.\n\nLooking back at the results of the survey and at the urgent challenge of teaching creativity and innovation, I am convinced we need to embrace both in-school and extended-day (and year) strategies. Within the core day we need to stop giving science and related topics short shrift. And we need to recruit new teachers who are familiar with the wondrous things that can be done in physics and biology and chemistry labs and who have the ability to link basic math and science facts to cool new discoveries. I'm involved in one exciting national effort called 100Kin10, which is designed to promote the hiring and support of one hundred thousand excellent new science and math teachers over the next ten years. The effort was launched by the Carnegie Corporation (the same foundation that brought us the public library) and is supported by the White House and CEOs from some of the top technology companies in the world.\n\nBut the real game changer for STEM education and for boosting innovation and creativity is to recruit one million scientists and engineers to team up with those one hundred thousand full-time math and science teachers. Getting STEM professionals in every classroom, and giving students of all ages a chance to make things with STEM professionals after school, would make science classes more fun, more engaging, and more connected to the creative discoveries happening in the real world. Most schools in the country are within a few miles of a university, hospital, or industry lab brimming with science, engineering, and medical professionals, some portion of who are interested in advancing the learning of young people.\n\nImagine being a student in a school in which real chemists joined your regularly scheduled science class every Monday to conduct real-world \u00adexperiments, and where every Wednesday you could choose among electives taught by video game designers, medical researchers, and astronomers. Imagine if a requirement of your sophomore year was to complete an internship in a local lab and conduct an experiment and gather evidence to prove or disprove your hypothesis.\n\nIn an effort to build a movement around just this type of STEM mentoring, Citizen Schools is incubating a new organization, US2020, which is designed to coordinate and promote efforts to mobilize STEM professionals. We aim to match individual mentors\u2014most drawn from corporate members like Cisco, HP, Tata Consulting Services, Raytheon, and Salesforce\u2014with schools and nonprofits that know how to coordinate real-world projects, like Citizen Schools, FIRST, 4-H, Spark, and others. I am serving as executive chairman of US2020, which we plan to spin off as an independent organization in late 2014.\n\nUS2020 was launched by President Obama at the White House science fair in April 2013 and has set a bold goal of mobilizing one million STEM mentors by the year 2020, creating millions of \"moments of discovery\" for children. \"We would never teach football by having students study it in a textbook,\" said President Obama's science advisor, John Holdren, noting that football players only get good by putting on pads and getting on the field. Similarly, students need to get into the lab and grapple with real-world experiments and projects. At the US2020 launch, President Obama said we need \"all hands on deck\" to improve STEM education.\n\nWhen the president says we need all hands on deck, he is talking about people like Steve Robinson, a molecular biologist. In 2002 Robinson and his wife adopted a child, and the couple agreed that Steve would take a year away from full-time work and take the lead on child care. Robinson was excited for a sabbatical and eager to dig into life at home, but he decided he'd also love to teach one course as a volunteer, spending one or two mornings a week at the local high school. He approached the school principal and asked if given his credentials as a PhD in molecular biology, he could develop and teach an optional advanced course for some of the older students in the school. The principal was intrigued, but said that because Robinson was noncertified, he wouldn't be allowed to teach even one elective. Disappointed, Robinson took his offer to the leaders of a nearby private school and they eagerly snapped him up.\n\nIt turned out that Robinson loved teaching and was good at it. Midway through his year as a part-time volunteer, he applied for a full-time job at the same private school and was hired on the spot, turning his career upside down but allowing him to pursue a passion and share it with dozens of new students every year. In 2007, after just four years as a full-time teacher, Robinson was awarded an Einstein Fellowship, given to the very best math and science teachers in the nation. Robinson was born in Illinois, and for his fellowship he was assigned to work with the freshman US senator from Illinois, none other than future president Barack Obama. Robinson became a key advisor in the Obama campaign, and until recently he served in the White House as President Obama's senior science education advisor. Now Robinson is back teaching science full-time at a Democracy Prep charter school in Harlem.\n\nFor every Steve Robinson who volunteers to teach science and then is so excited by the challenge and opportunity that he switches careers, there will be a hundred people like David Mantus and Alan Su, who take great pleasure in teaching\u2014and are good at it\u2014but want to continue doing science, or engineering, or law as their full-time job.\n\nAmerica's schools need to open their doors wide to more Robinsons and more Mantuses and Su's too. If we want children growing up on the South Side of Chicago to become chemists for Abbott Labs, which is located in suburban Chicago, or electrical engineers for Boeing, which is located downtown, we need these children to meet chemists and engineers at these companies and do real-world projects with them.\n\nOne of the most interesting recent developments in the informal learning space is the maker movement, a fast-growing effort to stimulate more backyard inventing and to showcase the work of tinkerers and artists and inventors. Makers are people who might be the next Steve Wozniak, who hung out with other tinkerers at the Homebrew Computer Club in a friend's Menlo Park garage before teaming up with Steve Jobs to create Apple Computer. Or they could be your neighbor who organizes a day for the kids on the street to make tie-dye shirts, or who designs a pedal-powered green TV that allows her to exercise while watching her favorite show and maintaining a carbon footprint of zero.\n\nIf there is a leader of the maker movement it is Dale Dougherty, a middle-aged technology entrepreneur and the founder of Global Network Navigator, or GNN, a Web portal and the first site on the Internet to be supported by advertising revenue. Dougherty is now the publisher of _MAKE_ magazine, a quarterly publication that highlights DIY (do it yourself) and DIWO (do it with others) projects and hawks the building blocks of the maker movement, including 3-D printers and Arduino microcontrollers, the brains of small, electronically controlled gadgets that can tell lights when to turn on and off or control the movements of a small robot. Simple 3-D printers are now available for less than $1,000, and many believe they could radically decentralize manufacturing in the coming decades.\n\nIf you are a technology CEO, makers are the type of young \"creative class\" workers you want working with you. And if you are a mayor, you want more makers in your city. In fact, when makers are present in critical mass, they drive innovation and economic growth for an entire city, as powerfully documented by Richard Florida in his book _The Rise of the Creative Class._ Recently, people active in the maker movement have been thinking about how to insert maker thinking and maker activities into schools. Dougherty and _MAKE_ magazine teamed up with Pixar, Intel, and Cognizant, three leading technology firms, to launch the Maker Education Initiative, or Maker Ed as it is known, and Maker Corps, efforts to open and support more than one hundred new maker spaces in schools and community centers around the country. The maker spaces are essentially arts and crafts rooms with soldering irons, parts to build robots, and 3-D printers or industrial design tools. In a parallel effort, DARPA, the Defense Advanced Research Projects Agency, made a 2011 grant of $10 million to Maker and Otherlab, a San Francisco firm, to create industrial design labs in one thousand schools, although sequestration budget cuts have slowed the rollout.\n\nUS2020 and Citizen Schools are working closely with Maker Ed and Maker Corps, and I see the maker movement as a way to scale the apprenticeship experience much faster than could be done just through Citizen Schools. But I also know that the maker movement, like FIRST, is primarily white and Asian, middle class, and suburban. If you go to the Maker Faires that Dougherty organizes at the San Mateo Fairgrounds, and at the Henry Ford Museum in Detroit, and in New York, you see incredible creativity. You also see lots of people (one hundred thousand plus in San Mateo every spring). But you don't see a lot of kids. And you definitely don't see many kids or adults from the inner city. You are much more likely to see a young David Mantus with his rocket scientist father than an Adam Barriga or a McCalvin Romaine. I sometimes wonder if the maker movement will contribute to a renaissance in creativity but mostly among those already on the right side of the opportunity divide.\n\nIn September of 2012 my eleven-year-old daughter, Orla, and I headed to the New York Maker Faire, held outside the New York Hall of Science in Queens on the grounds of the 1964 World's Fair. The gathering was beautiful, like a high-tech Woodstock. Huge puppets from the Bread and Puppets group in Vermont paraded around the fairground. Long-haired engineers in tie-dyed shirts biked around on old three-speeds outfitted with huge butterfly wings. Thousands of burritos were being eaten. There was a workshop featuring a young inventor from Sierra Leone who had made an entire FM radio station, including the batteries to power it, out of scrap materials and a 3-D printer. He was now employing three journalists and three DJs and making plans to build a windmill to bring electricity to his community.\n\nThe ingenuity was inspiring. Several of the Maker Ed corporate partners were there, and they had brought young inventors from across the country. Orla and I met Joey Hudy, who was not much older than Orla and had designed a hydraulic marshmallow cannon that he had demonstrated the previous spring to President Obama at the White House Science Fair. We also met with Anthony Rodriguez, a young inventor from Newark who said he hoped to become an industrial designer or a multimedia producer. \"My favorite part,\" he said, \"is you see a problem or a need and then you can just make something for it.\"\n\nThere was lots of stuff for sale at Maker Faire. You could buy robotics kits and rocket launchers, arts and crafts materials, dressmaking kits, and even 3-D printers. Orla and I checked out one of the 3-D printers and were fascinated but a little disappointed. The concept is cool but the reality of the production process is it's slow. The printer we saw was creating a pocketsize bottle opener, maybe two inches by one inch by a quarter-inch thick. The printer went back and forth for about half an hour spraying layer after layer of red plastic according to the template that had been programmed into the attached computer.\n\nAs Orla and I headed back to Manhattan on the elevated Number 7 train, we passed graffiti-covered bodegas and Laundromats, open-air vegetable stands, and check-cashing stores in the Willets Point neighborhood near the Hall of Science. The handful of passengers coming from Maker Faire stood out in sharp relief from the locals sharing the same subway car.\n\nWe noticed a Jamaican man with a Pittsburgh Steelers cap and an energetic three-year-old son who hopped from seat to seat and stared curiously out the window. There was a middle-aged Chinese man with a shopping cart filled with thousands of sheets of roasted dried seaweed. And there were a half-dozen young Latino men with Rocawear T-shirts, jeans, and Adidas sneakers. None of these folks had been to the faire, and if they noticed it as the train rumbled past, it was only with a fleeting glance.\n\nStanding out on the train, as did Orla and I, was a middle-aged father and his three young children. The dad had tired eyes, and I imagined him up late at night writing computer code. His young son held a lightsaber he had bought at the faire and was engaging in an imaginary sword fight with a subway pole. The eldest child, a girl, was recounting the Maker Faire project that most inspired her. I imagined David Mantus as a young child taking this same train with his technician grandfather or his rocket scientist father to go to the New York Hall of Science, and I was reminded anew of the power of relationship and the power of experience to imprint any number of characteristics, including creativity.\n\nI was inspired by Maker Faire. The crowd was overwhelmingly white and middle class, but there were a significant number of immigrants from India and the Far East and a small number of African Americans and Latinos. With effort, these numbers could grow. The event made me hopeful. But as I headed away on the Number 7 train, I was equally humbled by the challenge of equalizing access to all it has to offer.\n\n# CHAPTER ELEVEN\n\n# SOCIAL NETWORKS AND SOCIAL SKILLS\n\nRobert Putnam has a gentle, pious demeanor befitting the son of a Methodist small-businessman from the Midwest, which he is. But on the Fourth of July weekend of 2012, Putnam, author of the best-selling 2000 book _Bowling Alone_ and described by the London _Sunday Times_ as \"the most influential academic in the world today,\" gave a talk at the Aspen Institute's annual Ideas Festival that was shocking.\n\n\"The American Dream of equality of opportunity is destined to decay in the years ahead,\" Putnam told the assembled power brokers. \"Whether or not you care about inequality in wealth and income, we should all be worried about growing inequality in opportunity.\"\n\nPutnam has made social capital _\u2014_ the relationships we all have with family, close friends, and extended networks\u2014his lifelong obsession. He believes that human productivity, quality of life, and professional success are determined as much or more by _who_ you know, and the depth and quality of those connections, as they are by _what_ you know, or even by your family's relative wealth.\n\nPutnam's research raises a trio of imposing challenges for educators trying to reduce opportunity and achievement gaps. First, as Putnam and his research team document exhaustively in _Bowling Alone,_ and have continued to demonstrate, social capital has been declining steadily since its high point in the 1960s. People may have more contacts on Facebook or Instagram, but we spend less time with our friends, we trust people less, and we have fewer people we feel comfortable turning to for a favor.\n\nSecond, social capital and social skills are becoming more important to success in the highly networked twenty-first-century economy, an assertion also convincingly made by Richard Florida in _The Rise of the Creative Class._ Not good for Americans to have less social capital just as its importance to both individual and community success grows.\n\nThird, social capital is more and more unequally distributed. This was the focus of Putnam's talk in Aspen, in which he described a \"sharply growing class divide among America's youth.\" Most Americans have less social capital today than they did one or two generations ago. But the decline has been steepest for Americans with lower incomes and less formal education.\n\nPutnam helps us see that the opportunity divide is not just a divide in access to good teaching of math, science, or writing, but a divide that is much more fundamental: a divide in access to positive experiences and relationships _that are foundational to almost everything we do._\n\nPutnam and other researchers, including famed sociologist James S. Coleman, have documented a variety of ways in which access to childhood social experiences and networks that are proven to support adult success has become more unequal. Thirty years ago, for instance, upper-income children were slightly more likely than lower-income children (67 percent rather than 62 percent) to participate in extracurricular high school clubs like chess, theater, or community service. Today, that modest gap has almost quadrupled. Thirty years ago children of parents with a college degree and children of parents with a high school degree attended church at about the same rates. Today, on any given Sunday, children of better-educated parents are 50 percent more likely to attend church and to build the associated social networks. Today if you are an upper-income child in high school, your likelihood of serving as captain of a varsity sports team\u2014a proven chance to strengthen leadership skills\u2014is 27 percent, whereas just 12 percent of lower-income students will get this same opportunity. Gaps have also grown in overall levels of trust, in time spent with parents, and in participation in high school sports. And, as we discussed in the opening chapter, the investment gap has tripled between what upper- and lower-income families spend on out-of-school tutoring and extracurricular activities like piano lessons and robotics camps.\n\nSociologist Coleman was one of the first to clearly document the impact of social capital on educational and career outcomes in his influential 1988 article, \"Social Capital and the Creation of Human Capital.\" It's not just nice to participate in extracurricular clubs, or in a faith community, or to spend more time with caring adults, he argued; these experiences drive students to entirely different places.\n\nColeman documented, for instance, that children who attend church regularly are half as likely to drop out of high school compared to their peers who rarely or never attend church. This holds true even when the peer families have the same levels of education and financial resources. When a child has weak social networks, less time with caring adults, and lower expectations from those adults, he or she is almost four times more likely to drop out of high school compared to peers whose families have the same education and financial standing but have weaker social networks. Other more recent research has drawn causal links between social capital and math and reading scores.\n\nThe case for social capital's influence on career (as opposed to academic) success is strongest of all, and its influence is on the rise. In an _Atlantic_ article titled \"Where the Skills Are,\" Richard Florida argues that we are going through a second seismic change in the orientation of our economy and the abilities it values in its workers. For much of human history, Florida says, the differential ability that drove variation in human success was physical strength and dexterity. People who could plough more land, catch more fish, make more shoes, or lay bricks better and faster were rewarded for it. By the middle of the twentieth century, Florida argues, the world economy transitioned, and analytical brainpower became the skill that was increasingly prized by employers and recognized in the marketplace. This was the first seismic shift, and it meant that the difference in the earning power of someone with strong analytical skills and someone with weak analytical skills was now much larger than the difference in earning power between a skilled and not-so-skilled bricklayer.\n\nFlorida believes that we have now entered a third phase in our economic history\u2014one in which social skills have become more important than analytical skills. In his _Atlantic_ article he describes how an examination of US labor market job listings and average wages led to his conclusion.\n\nFlorida went on to describe the implications for education: \"Given the rising demand for social skills in our economy, it is curious that we devote so few of our educational resources to building them. A growing chorus has noted the failure of US schools to adequately teach math, science, and technology, but social intelligence is equally important, and we need to cultivate it more systematically.\"\n\nUpdating our schools with more time for learning and more citizen power will create a foundation that enables schools to more effectively build the social capital and social skills crucial for student success. If we instead continue to ignore social capital as a systemic driver of opportunity and achievement gaps, we will contribute to the current vicious cycle in which families and communities with lower levels of income and education provide their children with less exposure to the relationships and opportunities that drive adult success, which will in turn leave their children with even less income and an even lower likelihood of achieving academic and professional success.\n\nAs Citizen Schools was getting started, my colleagues and I got to know Robert Putnam, and in 2000, when he published _Bowling Alone,_ he included a nice shout-out to Citizen Schools, calling our efforts an effective way to \"serve civic ends\" and to \"strengthen the civic muscles of participants.\" For a young and mostly unknown organization, this was welcome recognition.\n\nAs we got to know Putnam, we learned that there were two types of social capital and that each was critical to helping individuals get ahead. \"Bonding\" social capital involves close connections to people like you\u2014such as the people from your church or neighborhood or from people who share your same ancestry. People who have strong bonding social capital reinforce shared social norms and often do favors for each other, making what Tom Wolfe, in his novel _The Bonfire of the Vanities_ , called \"deposits in the favor bank.\" In communities with high levels of trust, you expect that favors will be repaid, and this helps the community operate effectively. \"Bridging\" social capital is equally important and, Putnam taught us, even more important to individuals from low-income communities. Bridging social capital is when you form a relationship with someone who is different from you\u2014from another side of town, a different class, and with an entirely different network.\n\nPutnam's ideas resonated with me. It was my mom's network that helped me set up that summer camp when I was just fifteen. And it was the fact that my dad knew Senator Gary Hart that opened the door to a summer internship on his campaign when I was just nineteen\u2014an internship that became the turning point in my life. For my wife, Maureen, strong bonding social capital gave her initial access to the world of work. Then, as her jobs evolved, she built bridging capital with the \"Jewish mother\" whose hair she cut and with others. New doors opened.\n\nBut how would this work at Citizen Schools? Can social capital be transferred, or does it just reproduce, cementing inequality? Can a few young educators and a few hundred community volunteers build social capital for low-income children stuck in struggling schools, while building their own social capital in the process? And if volunteers can build social capital in tough schools and tough neighborhoods, will it really make a difference for the kids? Or is social capital such an elusive and hard-to-measure asset that we'd be wiser to bet all our chips on building academic skills? After reflecting on my early experience at Citizen Schools, I concluded that social capital _can_ be imparted and _is_ life-changing. It's not easy to build social capital for a thirteen-year-old kid. But it's not easy to build reading skills either. And both are vital to the future success of our students.\n\nIn some cases, social capital is clearly transferable and transformative. Consider the examples of Chrystal and Col, two former Citizen Schools students. Both were born in Haiti, the poorest country in the Americas, and came to Boston as young children.\n\nChrystal was a charmer, the type of student who attracts adults to her because of her eloquence and bright smile. Her home life was difficult, though, with Dad remaining in Haiti, Mom severely ill, and an aging grandmother struggling to keep up with the young children. When Chrystal was in sixth grade at the Woodrow Wilson Middle School in Dorchester, she enrolled in Citizen Schools. She would sometimes show up at our Saturday program with all her belongings in a trash bag. She would say she was moving from her grandmother's house to an aunt's house, or to another location. Deb Daccord, a lawyer with Mintz Levin who was one of her teachers, recalled that \"Chrystal was a standout student in our mock-trial class. She had a way of attracting role models to her because of her intelligence, but it was a patchwork of role models in a life with very little stability.\"\n\nWhen Chrystal's mom died during her high school years, she and her beloved younger brother initially went to live with their grandmother. But the grandmother was aging and couldn't handle both children, so they split up. Chrystal moved in with one aunt in Dorchester, and her brother joined a different relative in Brockton, almost an hour away. It had been five years since Daccord and Chrystal had met, though they had stayed in touch sporadically. Daccord herself had been orphaned as a teenager, and when she heard about Chrystal's latest move, she resolved to be a more active mentor. She remembers buying Chrystal a computer and going to her aunt's apartment to help her set it up, and then returning months later to fix it.\n\nWhen Crystal turned eighteen, government benefits that had been coming to her since the death of her mother ended, and her aunt kicked her out of the house. She was homeless, just as her senior year at English High School was supposed to begin. Daccord said her immediate impulse was to reach out. She consulted with her husband and two daughters, the eldest of whom had just gone off to college, and they were enthusiastic about helping. The Daccords decided to offer Chrystal a room in their home for the year and to help her navigate the college enrollment process. Chrystal happily accepted the offer, and with Deb's ongoing mentorship and support, Chrystal finished high school successfully and gained admittance to and enrolled in a four-year college, the University of Massachusetts at Dartmouth. Chrystal is now working as a waitress at a restaurant in Boston while continuing to work toward her degree. While this level of generosity is not an everyday occurrence, there are many more examples of Citizen Schools community members going out of their way to help students.\n\nCol was another student of ours in Boston at around the same time. He lived with his dad and one sibling while his mom and two younger siblings remained in Haiti, where Col had completed the early years of his schooling. Col enrolled in the Citizen Schools 8th Grade Academy (8GA) program in the fall of 2003, and as part of 8GA he took two apprenticeships and visited ten colleges and universities, including an overnight trip to my alma mater, the University of Vermont. All 8GA students also wrote essays during the year with volunteer writing coaches drawn from the city's top law firms. Every other week, each student would meet one-on-one with their writing coach (they referred to their coach as \"my lawyer\") and at the end of the year they each published an essay in a glossy magazine called _Bridging._ Col's essay was about the high cost of college and his concern that it might discourage kids from pursuing a college education.\n\nAfter completing his 8GA year at Citizen Schools, Col enrolled in TechBoston High School, a small and supportive \"pilot\" school that we had encouraged him to consider. He enrolled in Upward Bound, a federal college-access program, and stayed connected to Citizen Schools through occasional alumni reunions. Four years later Col graduated high school on time and with solid grades and enrolled at the University of Vermont as part of the class of 2012. Hundreds of Boston eighth graders had trekked north to UVM through Citizen Schools, taking classes from great professors like Huck Gutman, visiting the university art museum, meeting admissions officers, scrimmaging with the women's basketball team, and braving the cold weather and the culture shock of life on an almost all-white campus. Col would be the first Citizen Schools alumnus to enroll.\n\nSocial capital certainly played a role in prompting Col to apply to and enroll at UVM. He had met students and faculty and staff on the admissions team back in eighth grade, and this made him more confident to apply, even though the school was far away and no one in his family had ever attended a college. The relationship with Citizen Schools was likely a plus on his application too, as UVM hoped the partnership would help it build a more diverse student body. But whatever role social capital played in Col's application and acceptance, it would become even more important after he enrolled.\n\nCol completed his freshman year, but his grades were low and he found it difficult to bond with his roommates. They were from wealthy families in the Boston suburbs and had never met anyone like Col, and they had little understanding of the academic and social challenges he faced. In Col's sophomore year his struggles grew. Like many first-generation college students, he couldn't pay all of his tuition. UVM's costs are high\u2014they were close to $45,000 in his freshman year\u2014but the university had provided Col with a substantial financial aid package. He had tuition relief worth close to $20,000 from UVM's general scholarship fund. Plus Col was eligible for a federal Pell grant of about $5,000, and he personally took out a subsidized Stafford student loan for another $6,500. He also served as a federal work-study student, earning money for school and picking up job skills as a computer technician at the business school. Rounding out the package were two special UVM scholarships. One of them, at $3,750 per year, was covered by the Minnie Parker Charitable Trust, a New York foundation chaired by Citizen Schools cofounder Ned Rimer, who had been my roommate for two years at UVM. Col was able to cover most of his tuition for his sophomore year too, but he had lost one of the smaller UVM scholarships (not the one from Minnie Parker) and tuition charged by the university had gone up more than $1,000. There was a gap of close to $5,000 between what he owed and what he could pay. In December the university froze Col's account, meaning that he couldn't register for spring courses, couldn't take out a book, and couldn't even log on to a computer.\n\nThe university thought the solution was simple. The gap was only a few thousand dollars, and they suggested that Col's parents take out a federal PLUS Loan at 8.5 percent interest. But Col's father wasn't willing to do this, as he was sending every penny he could back to his wife and two children in Haiti. Col was stuck in a quagmire that snares hundreds of thousands of first-generation college students every year and causes them to drop out of a four-year college. Six years after they enter college, only 11 percent of first-generation college students have earned a degree, compared to 55 percent of their more advantaged peers.\n\nCol reconnected with friends on the Citizen Schools staff, who in turn alerted Ned, the Citizen Schools cofounder and chair of the foundation that was funding UVM to support students like Col. Ned called a friend of his in the development office and made two suggestions: First, if this situation could be resolved, maybe the Minnie Parker Trust would make an additional contribution to the university, again targeted to helping students like Col succeed. Second, Ned, who along with me had recently received a distinguished alumni award from the university, appealed to their sense of justice. Col had already covered more than 90 percent of his tuition. Please solve the problem and let Col continue his studies, Ned asked.\n\nNed's appeal worked. Col was a long way from home. His roommates couldn't relate to his situation and his immediate family didn't have the money or the know-how to help. But because of Citizen Schools, Col's network at UVM also included an influential graduate of the university and a donor\u2014someone in Col's corner who was willing to make a call on his behalf. Three years later\u2014and after additional tuition scares again resolved through interventions by Ned and members of the Citizen Schools staff\u2014Col graduated on time with a degree in business.\n\nIn Col's case, social capital worked just the way it sometimes does for rich people. You have a problem\u2014like needing an introduction to someone who could offer you a job\u2014and someone you know helps you solve the problem. The connection doesn't _get_ you the job. You have to perform in the interview and be a strong candidate, just as Col had to get the work done and pass his courses. But when your connection has juice with someone in power, they help you get your foot in the door, which is often half the battle. Not every graduate of Citizen Schools or similar programs is going to be fortunate enough to receive scholarship funding via an influential leader like Ned. But what is scalable is the building of relationships so that hundreds of thousands of students like Col have someone in their corner, someone they feel comfortable calling who in turn can pick up the phone and get people in power to help.\n\nNancy DiTomaso, vice dean at Rutgers University, wrote an intriguing article in the _New York Times_ last year in which she stated that a major driver of employment inequality between blacks and whites is that whites have more connections with people in hiring roles and are therefore more likely to get the friendly introductions that lead to jobs. \"Getting an inside edge by using help from family and friends is a powerful, hidden force driving inequality in the United States,\" she wrote, basing her findings on thousands of interviews she has conducted. \"Inequality reproduces itself because help is typically reserved for people who are 'like me': the people who live in my neighborhood, those who attend my church or school or those with whom I have worked in the past. The mechanism that reproduces inequality, in other words, may be inclusion more than exclusion.\"\n\nSocial capital helps Citizen Schools kids get access to good high schools, to good summer programs, and as we've seen with Col, Chrystal, and others profiled in this book, to homes, colleges, and job interviews. Professionals like Deb Daccord and Ned Rimer have lots of social capital. Through Citizen Schools they get the chance to use it not only for their own children and the children of their friends but for other people's children too.\n\nSocial capital also helps students in more subtle ways. For sure it helps students get their foot in the door; but it also helps students see new doors, and sometimes it helps them develop the confidence to walk through the door and ask for help when it's needed.\n\nJadine Yarde, for instance, grew up in the same low-income Boston neighborhood as Col. She navigated her way through high school to college at St. John's University and into a good job as ad sales coordinator for DISH Network. She's a young professional in New York City and has become a skilled networker with a bright future. But Jadine told me that her newfound confidence was hard-won. In middle school, as she enrolled in Citizen Schools and started talking with her friends about her apprenticeship classes, such as one in entrepreneurship, some of them distanced themselves and said she should stop acting so high and mighty. \"I was fighting outside influences like peer pressure, and at the time I couldn't conceptualize what exactly Citizen Schools could do for me because I saw it as what stood in the way of me being a 'normal' middle school student who hung out with friends on Saturdays and went downtown to cause trouble after school.\"\n\nJadine said that Citizen Schools staff members and volunteers kept pushing her and became instrumental to her academic and professional development. \"Being able to meet through Citizen Schools with top execs at some of the most successful businesses and share my goals and aspirations is an environment that Citizen Schools created and it has helped me beyond measure. Now I am excited, motivated, and determined to be as successful as those I met as a twelve-year-old.\"\n\nAs I finished a recent conversation with Jadine, she wanted to reinforce one final point. Apprenticeships, she said, helped give her confidence _in her own ideas_ and the comfort to approach strangers at a cocktail party or in an interview and engage with them as intellectual and social peers. \"At Citizen Schools I was constantly reminded of my abilities. I realized that color and your academic standing do not define you and you should never fall victim to believing that they do. You are just as smart as a billionaire, you just have to give your dreams your undivided attention. Confidence is something that should be worn proudly.\"\n\nWhen a student builds a network of advocates, they not only have more people to open doors for them but more people willing to stand up for them in difficult situations. Let me give an example. For many years I taught a master's degree class to young teaching fellows at Citizen Schools, and as part of the class, we devoted one evening to watching clips from education-themed movies such as _Children of a Lesser God_ , _Dead Poets Society_ , and _Stand and Deliver_ , the last the story of real-life math teacher Jaime Escalante, who led a class of inner-city Los Angeles students to take and pass the AP calculus test. We always keyed in on the scene when Escalante, played by Edward James Olmos, visits his star pupil in her parents' restaurant after the parents have asked her to drop out of school to work at the family business. In the movie, Escalante brusquely confronts the hard-working immigrant parents and tells them their daughter should return to school. Every year we viewed the clip, our young staff would react the same way. They would be disappointed the girl had dropped out of school but would say that Escalante had crossed a line and should have left the decision about whether to attend school to the girl and her parents. But on two occasions when we asked our young student alumni\u2014kids like Chrystal, and Col, and Jadine\u2014to join us in watching the movie, they took a different point of view. They were glad Escalante intervened. They loved their families, but they wanted fiercer advocates for their interests, even if those advocates might occasionally cross lines and break from their parents' views.\n\nThere has been a growing recognition recently of the importance of social networks and of nonacademic skills such as grit and persistence, and suggestions that schools should try to build these skills and attributes. But too many schools and school reformers remain stuck in an obsolete paradigm. They focus on the traditional Three R's and generally see learning as something done in a chair and behind a desk and with a single teacher in front of the room. In most schools, students have too few chances to work on teams, too few chances to meet and work with successful adults, too few chances to learn about the world of work through internships, and little explicit training in how to give or ask for feedback. Activities that provide these skills\u2014like playing on sports teams, acting in plays, participating in internships, and building robots on a team\u2014are rare at most schools and only available to a small number of students.\n\nFor children like Chrystal and Col and Jardine, citizen power and an expanded learning day can build nonacademic skills and narrow social-capital gaps by giving kids chances to practice relating to many different adults while building confidence, higher levels of trust, and stronger social networks.\n\nWorking with very-low-income students in some of the toughest schools and neighborhoods in America, Citizen Schools has gotten our graduates to finish high school and enroll in college at rates that are the same as or higher than in middle-class communities. Some of the success comes from the academic boost we give students when they are still in middle school. But I believe an equally big success factor is the social capital we help our students develop.\n\n# CHAPTER TWELVE\n\n# SUPPORTING TEACHERS, AND PARENTS TOO\n\nOf all the questions I get about Citizen Schools, perhaps the most frequent is: \"Do the teachers like you?\" Many questioners seem conditioned to expect the worst of public school teachers and assume that a second shift of educators, offering different approaches and taking less or no pay, will inspire resentment from the full-time teachers who lead classes for a majority of the day.\n\nGenerally, however, America's teachers have embraced Citizen Schools and embraced an expanded learning day and citizen power in their schools. While a few teachers may react defensively and hide behind the closed door of their classrooms, the best teachers welcome any help they can get. Teachers' unions as well have generally embraced Citizen Schools. As stated earlier, American Federation of Teachers president Randi Weingarten visited the Edwards ELT campus and declared it in a _New York Times_ column, \"one of the most impressive schools I have seen in America.\" The Boston Teachers Union has gone so far as to explicitly advocate for a nine-plus-hour learning day for all students, with the extra time delivered either by teachers receiving extra pay or by outside programs like Citizen Schools.\n\nAdding a second shift of educators in schools could fundamentally change the teacher's job for the better, making it more sustainable and enjoyable while bringing more resources to kids and engaging families more deeply in their child's education. Suburban teachers often get this support from active families and well-organized extended-day programs. But most teachers in high-poverty schools feel overworked, undersupported, and unsuccessful. As a result, many of them leave teaching too soon, creating a higher-than-necessary teacher churn and more challenges for students. Many rigorous \"no excuses\" charter schools have been successful at lifting test scores, but they have even higher rates of teacher turnover than district schools because of the long days and breakneck pace. For district and charter schools, a second shift can make a full career in teaching more attractive.\n\nErin Dukeshire, a middle school science teacher at the Orchard Gardens K\u20138 ELT school in Boston, illustrates the point. Orchard Gardens, formerly one of the lowest-performing schools in Massachusetts, became a \"turnaround school\" in 2010 with new leadership, mostly new teachers, and an expanded learning day. Very quickly Orchard Gardens became one of the strongest schools in Massachusetts. In 2011\u201312 its middle-grade students had the highest rate of student learning growth of any middle school in the state. All of these students received an extra three hours of learning from Citizen Schools every day. Some students even had a chance to travel to the White House and meet with the president.\n\nBy the time Dukeshire came to Orchard Gardens, she was a relative veteran in the ed-reform world. She had taught for two years in Miami through Teach For America and then had joined a charter school in Boston that offered students a nine-hour day with core teachers covering the entire time. At Orchard Gardens, Principal Andrew Bott recruited her to be part of his turnaround team of teacher leaders.\n\n\"I was ready for a change and I was looking for leadership opportunities beyond the classroom,\" said Dukeshire. \"Also, I personally wasn't able to sustain the intensity at which I was working for that number of hours in a day. Since I was coming to Orchard Gardens at the start of the turnaround, the work was still intense and I was working just as hard that first year, but the difference was that the work with students ran until about two-twenty, and then Citizen Schools took over and I could use that afternoon time to work with my colleagues.\" Dukeshire, a 2014 winner of the Presidential Award for Excellence in Math and Science Teaching, said that having a high-quality partner extend the day for students while teachers prepare the next day's lesson with colleagues \"means that we have more time to create better outcomes for kids and it makes the profession of teaching more sustainable.\"\n\nAt the Isaac Newton School in East Harlem, New York, sixth-grade teacher David McKinney can teach his math class every morning knowing that every student in his class has done their homework from the previous day, because every student participates in the extended day with Citizen Schools and gets an hour of supported homework time from a second-shift teacher while still at school. Imagine this difference alone: Before ELT maybe half of your students completed their homework, since many had no structured time and place to complete it. Now all of your students, or almost all of them, complete it. Imagine how that changes your job as a teacher.\n\nFor Sara Sheckel, a former sixth-grade English teacher at the Mc\u00adKinley Institute of Technology in Redwood City, California, a second shift means that she can coordinate with five AmeriCorps teaching fellows, who collectively support the ninety-six sixth graders, a majority of them from Spanish-speaking families, enrolled in her four English classes. While Sheckel has one hour per day with each of her four classes of twenty-four students, the teaching fellows have three hours every day with one team of eighteen to twenty students. Sheckel benefits from the extra practice her students get building vocabulary skills with the teaching fellows, and she also gets help engaging parents and providing individualized guidance to her students. With so many students, it's not realistic for Sheckel to call every parent every week, or even every month, or to sit down one-on-one with all of her students. But in the course of just one week it _is_ realistic for the Citizen Schools teaching fellows to call every parent with an informative update and to have one-on-one conversations with every student on their team. Now Sheckel has a conduit to pass on key messages to students and parents and also a way to receive information back from them.\n\nIncreasingly, Citizen Schools ELT schools include a thirty- to sixty-minute block where first-shift and second-shift teachers lead a class together. At Elmhurst Community Prep in Oakland, which in 2012\u201313 had the most student learning growth of any middle school in the city, the teachers and teaching fellows co-teach an advisory block focused on goal-setting and on increasing students' ownership for their own success. Most days each adult in the room takes an advisory group of ten to twelve students, allowing students and adults alike to build meaningful relationships and trust, a sort of school-based social capital that often helps students advance. When students get better at asking for help, at speaking up when they don't understand something, and at holding give-and-take conversations with adults and peers, they are better positioned to move forward academically, socially, and professionally.\n\nThe more overlap time there is between first- and second-shift teachers, the more opportunity there is to re-create the core learning day in a way that supports teachers and improves student learning. In our ELT schools in East Palo Alto and San Jose, California, for instance, teaching fellows lead three hours of extended learning time but also support two to three \"regular\" classes each day. Before ELT, a typical school-day class was fifty minutes long and consisted of the teacher sharing a new concept for about twenty minutes and then giving students thirty minutes to practice the new skill as the teacher tried simultaneously to manage behavior across the full class while circulating for mini-interventions with students, a majority of whom are typically stuck on one or more steps in the lesson. With a teaching fellow in the room, the master teacher can introduce the lesson and have the class work through a few sample problems as a group. Then the teaching fellow can oversee the class, keeping students on task, while the lead teacher takes a succession of small groups to the worktable at the back of the room to unpack a new skill until each student has mastered it.\n\n\"Teachers and principals are telling us that having a teaching fellow support core classes is allowing teachers to close the instructional loop,\" said Katie Brown Rothschild, the Citizen Schools managing director of program in California and formerly a teaching fellow and campus director. \"They used to preassess for understanding, then teach and practice, and then assess again. Now they can preassess, teach and practice, assess again, and then remediate as needed. They love it.\"\n\nAllowing teachers time for pull-out tutoring, giving students extra academic practice time, and engaging parents in their child's learning are all important ways that Citizen Schools supports teachers. But the most important way we support teachers is by motivating students to try harder in school. By exposing students to exciting real-world projects, Citizen Schools helps make traditional school subjects become more relevant and enticing. All of a sudden a topic sentence becomes a key skill to win a mock trial, not just another academic standard on a long list that needs to be mastered. A student becomes motivated to learn the Pythagorean theorem because it helps unlock the secret of programming a video game.\n\n\"In education we can test kids, we can have progress reports, data reports, but the bottom line is if learning is not meaningful for a student they won't truly learn,\" said David Baiz, a former teacher and now principal at Global Technology Prep public school in East Harlem. \"Citizen Schools makes that connection possible and allows kids to see how learning is meaningful for their lives.\" Baiz described a student, Kamarthy, who did a Citizen Schools presentation with NBC iVillage. \"He is a student who struggled a lot in the school day and was never engaged. But he was very engaged in this presentation and it was a wonderful presentation. He felt so comfortable up there talking to these executives, presenting his products to board members, and giving a mini-speech. We saw his potential in a whole new way.\"\n\nCitizen Schools also supports teachers by allowing them to be mentors and master educators. With a second shift of eager young educators on the scene, experienced full-time teachers can not only _get_ help in the classroom but can _give_ help by mentoring the young teaching fellows sourced by Citizen Schools.\n\nAs an example, Kendra Engels, who served as campus director at the De Vargas Middle School in Santa Fe, New Mexico, which in two years of ELT was able to raise math proficiency by eighteen percentage points and ELA proficiency by eight percentage points, described the relationship her team built with a strong teacher at the school.\n\n\"We had a teacher who taught English and who was really skilled at ELL (English-language learner) instruction techniques,\" shared Engels. \"She was often in the classroom when we were teaching and she ended up working with our Academic Program Lead to help her rewrite some of the lessons to better align with where she saw gaps in student learning and with how she taught in her class. We rearranged our staffing so that our three teaching fellows who worked most often with ELL students could observe her teaching. It was great, because it showed students that we were all colleagues and that the teachers and Citizen Schools staff were in it together. It was great for the teacher to see the planning that went into our programming and to have input into our instruction. And of course it was fabulous for our staff to have great techniques modeled by a pro.\"\n\nEngels said the same teacher also led a four-hour training session for the Citizen Schools staff that many deemed the best they'd had all year. They loved the new perspective, and she loved the opportunity to teach her craft.\n\nEngels and her team also partnered with teachers to use assessment data more effectively. Together they reviewed interim test results and came up with plans to boost student understanding and performance. \"The school led several half-day data-analysis sessions,\" Engels recounted, \"to strategize about how to best adjust instruction based on what the data said about student learning. This was the first session of its kind for the school in quite some time, if not ever, but was the type of thing we do at Citizen Schools all the time. Our staff was able to attend and contribute a lot of great ideas and strategies. It really helped the teachers to see us as colleagues and for them to see that we felt equally responsible for the standardized testing results and equally driven to improve them. We became partners toward one goal rather than just two entities working in the same space.\"\n\nELT also provides teachers with a pathway to leadership, often a tough road in schools where the typical management structure for a school of six hundred students is one principal, one assistant principal, a director of instruction or dean of discipline, a secretary, and fifty teachers. In the second and third years of ELT at her school in Redwood City, Sara Sheckel actually split her time between the school, where she taught two instead of four classes, and Citizen Schools, where she served as the part-time instructional coach, providing feedback and professional development to the first- and second-year educators in the teaching fellowship. The experience allowed her to continue adding value as a teacher while building management skills, ultimately leading to Sheckel's appointment for the 2013\u201314 year as assistant principal of the Roy Cloud K\u20138 school in another part of Redwood City.\n\nIn many ways, the job of teacher as currently constructed is an outmoded relic of an industrial age. The teacher works mostly alone, putting equal attention into tasks he or she is great at and ones he or she struggles with. The job in year one is similar to the job in year ten or year forty. Just as students are treated too much like widgets in a factory, receiving the same dosage of multiplication tables and the same serving of Mendel's peas, regardless of their understanding and interest, teachers are asked to teach the same topics and in basically the same way regardless of their skill and experience and the needs of their students. It's as if students came into a hospital and received the same medicine and the same fifty-minute examinations from doctors and nurses who followed the same script regardless of the ailment and regardless of the particular specialty and previous training of the medical professionals.\n\nThe United States has roughly 5 million medical professionals but only 624,000 doctors who care for patients. At its best, the medical system leverages each of those 5 million professionals to do what they can uniquely do best, with the role of an intake nurse different from that of a nurse practitioner, and different still from the X-ray technician, the medical resident, and the attending physician. Surely education could learn something from this differentiated approach.\n\nRight now there are approximately 3.5 million full-time teachers employed to teach about 50 million US students enrolled in about 100,000 K\u201312 schools. There are another 2.5 million public school employees, ranging from special-education paraprofessionals to lunch monitors, librarians, secretaries, principals, custodians, and administrators. Whereas in the 1950s the ratio was one teacher for every twenty-eight students, today, in part due to growing numbers of special-education students, it is one to fifteen, and in urban districts it's one teacher for every twelve students.\n\nThe sheer volume of teachers creates real problems for professionalization of the craft, as Rick Hess of the American Enterprise Institute, among others, have described. Even with modest rates of turnover, schools across the nation need to hire 280,000 new teachers per year, many more than the 120,000 full-time registered nurses hired each year, or the 26,000 new lawyers. As Hess notes, hiring 280,000 excellent teachers every year when there are only 1.5 million college graduates per year\u2014and just 500,000 from competitive colleges\u2014is an impossible task. A lot of people coming out of college don't want to be teachers, and yet we are trying to recruit almost one in five recent college graduates year after year. Imagine if we had slightly fewer teachers and paid them better and supported them with hundreds of thousands of teaching fellows, many of them training to become teachers but others taking a year or two to contribute to a local school before going on to careers in business or science, or the arts. Most teaching fellows would be right out of college, but some might be older professionals transitioning to a new career in education. This more flexible talent model would provide more continuity with better-supported master teachers staying longer, while also infusing schools with fresh energy and additional talented and caring adults.\n\nA model in which there were varying levels of teachers in a school would support educators at every level of experience, increase efficiency, and allow for more differentiation of student learning. Master teachers who had demonstrated excellent teaching over a sustained period of time could earn 50 percent more than typical teachers today and could teach one or two fewer classes per semester so they had time to coach their younger peers. Core teachers might have a similar teaching load to teachers today but would receive help from a teaching fellow, offloading work such as grading of homework and tests, supervising student practice sessions, and management of field trips. This would allow core teachers to put more time into the development of great lesson plans. Teaching fellows and volunteer citizen teachers could support the master and core teachers while also extending the learning day, providing more time for academic practice, more time for personalized learning (through online programs such as Khan Academy or software-based learning), and significantly more time for students to make and do things with real professionals. Together, this American Dream team of talented educators could build academic and social skills while also helping students discover the real-world applications of school-based knowledge.\n\nIn some respects this multitiered approach would look like the Isaac Newton Expanded Learning Time Middle School in East Harlem, which is located in the same building where my Mom taught freshman English more than forty years ago. After many years of disappointing performance, Principal Lisa Nelson adopted ELT in 2011\u201312 for all of the school's sixth graders and saw proficiency jump by twenty-six percentage points in math and seventeen percentage points in English. Nelson, a veteran administrator, also saw a new spirit in her school. Her teachers and her leadership team felt buoyed by the energy of the Citizen Schools second shift, and she increasingly turned to Citizen Schools campus director Seth Miran as a trusted partner. Her school was infused with artists and engineers and financial professionals. My mom even came back to her old school building and taught an apprenticeship in organic farming and said she learned a few new teaching tricks she wished she'd had decades earlier.\n\nIsaac Newton is now in the process of expanding ELT to all students in sixth through eighth grade. As part of this year's plan, Principal Nelson has asked her Citizen Schools teaching fellows to take the lead on interim assessments, including the administration of \"exit tickets,\" which are two- to three-question assessments to check for understanding at the end of a single lesson. Teaching fellows will score the various interim assessments, load the data onto a spreadsheet, and conduct initial analysis that they can then share with Citizen Schools and school-day colleagues. The core teachers will have more time to focus on instruction and lesson planning and other high-value activities.\n\nELT with Citizen Schools is one compelling way to reorganize the talent strategy of a school, but it's not the only way. Generation Schools in New York City delivers smaller class sizes and a longer day and year to its students\u2014while working within the New York City teacher's contract\u2014by eliminating most administrative and specialist positions. Many charter schools have developed junior-teacher models that allow rookie teachers to extend the learning day without burning out core teachers, while also developing needed bench strength for the school. As mentioned in chapter 6, a majority of Citizen Schools teaching fellows now transition directly from their two-year assignment to a full-time teaching role, in many cases after having earned a teaching certificate with one of our alternative certification partners. All the programs mentioned in chapter 8, including the National Academies Foundation, High Tech High, and the Met School, have found ways to creatively engage a second shift of educators to support core teachers, engage students, and build a bridge to parents.\n\nInfusing citizen power and extra learning time into schools also supports parents. For starters, and to state the obvious, a day that ends at five or six o'clock instead of two or three lines up better with the end of the workday, meaning children have fewer hours home alone or on the streets. Keeping kids off the streets is a low standard for extended-day programs, but it's an important first step. According to the Afterschool Alliance, more teens are victims or perpetrators of crime between 3:00 and 6:00 p.m. on school days than at any other time of the week. School-day afternoons are also a time of experimentation with drugs and with sex and, most commonly, a time for mind-numbing TV shows and violent video games.\n\nCitizen Schools and programs like it also create a bridge that helps parents connect with their children and the public schools they attend. Sometimes this bridge comes from the simple act of calling parents (or grandparents) at home just to check in and name something a student is doing well and something he or she could do better. Community police officer Denis Rorie had advised us to \"catch kids doing things right.\" That advice is reaffirmed every time I see parents beam with pride as they watch their children showcase a new skill. These WOW! moments open doors to new careers as well as to new conversations around the kitchen table.\n\nSometimes the extended-day staff serve as translators, literally. Our work in Revere, Massachusetts, an immigrant city with fast-changing demographics, is a case in point. When we started working in Revere, we learned that a majority of the students in the school spoke a language other than English at home. But not a single one of the school's thirty- eight full-time teachers was fluent in any foreign language. When the principal wanted to translate a flyer into Spanish, the most common language of the school's parents, she needed to send the text to the district and sometimes wouldn't get a translation back for three weeks! Fortunately a majority of the Citizen Schools staff was fluent or conversational in Spanish, and one staff member spoke Haitian Creole. We became an essential conduit for communication.\n\nThe sometimes-difficult relationship between urban parents and schoolteachers was driven home to me in 1999, when a few colleagues and I traveled with twelve parents and teachers from a Boston middle school to Camp Kiev, a high-end summer camp in Maine. We ventured all the way to Camp Kiev because it had an Outward Bound\u2013type ropes course, and camp director Dick Kennedy had lined up funding to pay his instructors to lead us in team-building activities. It turned out we needed all the help we could get.\n\nOn the first day of the retreat, when asked to suggest ways to improve the school, teachers quickly defaulted to complaining about parents, saying they needed to get their children to complete their homework and to get to school on time each day. I cringed. Parents blamed the teachers\u2014not those who had volunteered to attend the retreat but others they felt were not serving their children well. As parents they were entrusting their children's learning to the school but they were seeing disappointing results.\n\nOn the second day of the retreat, the facilitator asked participants to remember who held a vision for them when they were twelve years old. Most people had a story of a teacher, or a parent, or a helpful neighbor. But one father said that when he was twelve, no one had a vision for him, and he broke down crying. The ice was broken and the group started to come together. Later that day the group completed high and low ropes courses and team-building exercises. The barriers continued to fall. On the final day of the retreat, we started to think collectively about how an expanded learning day could be a platform for improving the school\u2014and improving communication. Parents and teachers were each a little more able to see things from the other's perspective.\n\nI recently read through feedback surveys completed by 353 classroom teachers working in our ELT schools. There were helpful points of critique pointing to areas for improvement. Several teachers, for instance, said they loved the apprenticeships but wanted to see standards raised for the academic lessons led by teaching fellows. Others had complaints about the furniture being rearranged in their rooms, or about students being tired from the long day. But the overwhelming sentiment of the comments was gratitude. Teachers in the schools we partner with feel like they have allies\u2014new crews of professional colleagues who respect them and help them motivate, support, and teach their students.\n\nOf all 353 teachers responding to the anonymous survey, 94 percent said that Citizen Schools staff responded to their needs, and 93 percent said they had been responsive to Citizen Schools; 96 percent said they interact with Citizen Schools staff as professional colleagues; 88 percent of teachers said that Citizen Schools staff had initiated a conversation about student academic progress in the last semester, and a majority said this happened at least weekly.\n\nLooking at these numbers, I couldn't help but recall our start as a summer and after-school program at Dever School in Boston almost twenty years earlier. I remembered a painful incident at the end of our second year at Dever, when teachers we hardly knew were furious that students on our watch had damaged several blackboards in their classrooms. Ned and I went directly to the local hardware store and bought chalkboard paint and then spent the better part of a day repainting large green classroom chalkboards and leaving behind new packets of white and colored chalk. Today we still have the occasional chalkboard incident. That's going to happen any time space is being shared. But now there is a foundation of trust built on a track record of results and appreciation.\n\nIn many ways American education overall is also headed in a positive direction. Despite the challenges this book has discussed, average college graduation and high school graduation rates are going up, math scores are improving, and we now have hundreds\u2014maybe even thousands\u2014of schools delivering excellent results for low-income students. More outstanding college graduates are choosing to teach. We are moving toward a voluntarily adopted national Common Core curriculum that focuses more on higher-order thinking skills rather than regurgitation of memorized facts. And tests are about to become better, assessing writing and scientific thinking, not just the ability to guess correct answers on a fill-in-the-bubble test. The challenge for America is that while our schools are improving, schools around the world are improving faster. And the challenge for low-income American students is that while they are learning more\u2014and their parents and their teachers are working harder\u2014they are falling relatively further behind, left in the wake of a tsunami of privatized extra learning opportunities that benefit their upper-middle-class peers.\n\nWhen I started Citizen Schools I was not yet a parent, and I was just becoming a new type of teacher. The parents and teachers we worked with were role models for me, and I remember them vividly. I remember parents with six kids in tow traveling an hour on public transportation after a full day's work to attend a WOW!, eyes gleaming and smiles wide as they saw their child operate a robot she had built or argue a case in front of a federal judge. \"It's all she talks about at home,\" I would hear. \"She says she wants to be a lawyer now.\" I remember Hetty Mitchell, one of the parents in our first summer program, who worked at the Stop & Shop in Dorchester's Lower Mills section where I used to go for groceries. As I paid for my groceries, Hetty and I would talk about her daughter, Andrea, exchanging stories of accomplishments and challenges. I felt an important kinship.\n\nI also felt kinship with Margie Tkacik, the fifth-grade teacher in Dever's Room 202 who let me work with ten of her students for a few hours a week to launch Citizen Schools. Margie was such an enthusiastic teacher that at times she seemed to almost bubble over as she described her plans and visions for her students. She reminded me a little of my mother. When I started Citizen Schools, I had a recurring conversation with people who loved the model but thought I should embed the program in the existing six- or seven-hour school day. But this was maddening to me. What I learned from Margie was _not_ that her students needed me _instead_ of Margie. They needed as much time as possible with Margie _and_ some extra learning time with me.\n\nAs I look today over a troubled public education landscape\u2014a landscape where innovation and personalized learning is growing rapidly, but so is inequality\u2014I yearn for the chance to rebuild our national sense of shared public purpose. Public schools were intended to knit together a new country, giving children of immigrants and of business owners the same chance at an excellent education. Today public schools and their teachers feel under siege. Some of that is deserved, a consequence of resistance to fair-minded change and higher standards. But surely much of the acrimony is undeserved, driven in part by the lack of connection and therefore lack of empathy between upper- and lower-income parents, between business leaders and teachers, and between all of us as American citizens.\n\nGandhi said that we must _be_ the change we want to see in the world. If we want better public schools, we can't wait for some new curriculum or management plan or market mechanism. We need to roll up our sleeves and make them better. We need to step into schools with minimal judgment and as much curiosity and energy as we can muster. That's how to change the opportunity equation.\n\n# VOICES FROM CITIZEN SCHOOLS\n\n# LINDY SMALT, AMERICORPS TEACHING FELLOW\n\nThe backbone of the Citizen Schools second shift are teaching fellows, recent college graduates supported by the national AmeriCorps program. AmeriCorps members make Expanded Learning Time possible and support teachers and volunteers and parents, as described in the previous chapter. I believe dramatic expansion of AmeriCorps, as described in the final section of the book, is perhaps the most cost-effective and powerful investment we could make in equalizing opportunity in America. The following story is from AmeriCorps member Lindy Smalt, who served with Citizen Schools in Revere, Massachusetts, from 2010 to 2012.\n\n_Two years ago, I was a Wheaton College senior. I was undoubtedly one of the coolest kids on campus. I was a theater major, automatically mysterious and deep. Lindy was my name, and self-assurance was my game._\n\n_And yet there was that constant, dreadful feeling in the back of my mind\u2014What was I going to do after May? What if the rest of the world wasn't caught up on how cool I was? What was I going to do without my immense sense of purpose and popularity?_\n\n_I got very, very lucky\u2014I stumbled upon a job in education._\n\n_Now, let me be clear: When I faxed back the signed offer letter to Citizen Schools, I thought the teaching fellowship was going to be a two-year break for me to figure out what my \"real job\" would be._\n\n_That couldn't have been less true. I might have been a rock star at Wheaton, but nothing, not even my twenty-five-credit semester, could have prepared me for the incredibly demanding work of teaching in one of our nation's low-income communities._\n\n_Once obsessed with political philosophizing, I was shocked to find that teaching in a public school was the first time I wasn't just ranting about politics\u2014I was living them. A single forty-minute lesson at Garfield Middle School reflected so many of our nation's struggles, from the prison system, to immigration, to the drug war. In two years, I taught an Iraqi refugee, the daughter of a murderer, a boy who saw his parents murdered, a boy who got expelled for drug possession, a girl who spoke an unheard-of African dialect\u2014and these people were eleven. Through their lives, their sometimes absent parents, their complete apathy toward school, I saw\u2014for the first time\u2014the necessity of my work and of my life, and the true depth of our nation's struggles._\n\n_\"When we are very old,\" said one of my student's mothers to me this year, in half Arabic and broken English, as she placed her hand on mine, \"we will always think of Ms. Smalt. We will say, 'Ms. Smalt is the one who changed everything. She was the start of a new life.'\" She and her son, Abdellah, do not have a computer or a car; they walk to the local library to use the Internet. Yet with her support, Abdellah's unparalleled perseverance, and my resources in the community, we were able to secure a spot for her son in the high-performing charter high school in the next town, as well as garner a $2,500 grant for him to attend summer camp for the first time._\n\n_Students like Abdellah have all of the skills to succeed in college and beyond, but often there is no one to show them the way. He is small and gets swallowed in large classes of screaming, sassy preteens. But he is diligent, positive, and extremely kind, and he deserves a chance. And there are millions more like him._\n\n# SECTION THREE\n\n# NEXT STEPS FOR AMERICA AND FOR YOU\n\n# CHAPTER THIRTEEN\n\n# A CIVIC MARSHALL PLAN FOR EQUAL OPPORTUNITY\n\nWhat will it take to bring Citizen Schools and similar ideas to scale\u2014to truly reimagine and expand the learning day while narrowing class-based disparities in opportunity and achievement? Can we do it? Or as a nation have we lost our mojo, lost our heart, and lost our belief that big change is possible?\n\nI believe America is still the place where Adam Barriga, the grandson of a Peruvian immigrant who polishes the State House floor, can become, like David Mantus, a chemist, or rocket scientist, or anything he wants. We can still be the place where Miguel wants to take a young immigrant from Central America out for ice cream, and where that immigrant, Francisco, has the opportunity to give something back. We can be the place where Earline has a chance to pass along her life's passion for sewing, and where Linda finally gets to take off her heavy wool coat. We can be the place where a lousy school no longer drags down its neighborhood, but instead leads a renaissance in learning, like the Edwards did, transforming itself from worst to first and serving as a beacon of hope and pride for an entire city. We can do all of these things, but only if in our hearts and actions we truly embrace opportunity for the many, not just the few.\n\nThis book starts with a problem\u2014a well-documented and fast-growing achievement gap between the academic and career advancement of wealthier and poorer students\u2014and then discusses the causes and potential solutions. Some of the achievement gap (20 to 30 percent) is caused by inequality between schools in wealthier and poorer neighborhoods. That inequality needs to change. But most of the gap comes from unequal access to learning opportunities offered after school or in the summers, at home or in a growing constellation of tutoring centers, skill-building camps, and paid enrichment and internship programs. Upper-income kids get many thousands of dollars invested in these types of extra learning opportunities, and as a result they hone their basic academic skills; they build new skills such as the ability to innovate and create and work on teams, and they build increasingly important social networks and social skills. This inequality needs to change too.\n\nThis book describes how Citizen Schools and similar programs deploy citizen power and an expanded learning day to provide children in lower-income communities many of the same learning opportunities that are routine in wealthier communities. Citizen Schools and others are using this playbook today to narrow and even eliminate achievement gaps. But while existing programs help hundreds of thousands of low-income children beat the odds stacked against them, we aren't fundamentally changing the odds. Unless we do something much bolder, the gap between upper-income and lower-income families will continue to get bigger, overall economic growth will stagnate, and the number of Horatio Alger success stories will continue to dwindle.\n\nOn July 4, 1861, four months after taking office and with the first major battle of the Civil War looming, President Abraham Lincoln wrote to Congress with a single message. The leading purpose of government, he stated, is \"to elevate the conditions of men\u2014to lift artificial weights from all shoulders\u2014to clear the paths of laudable pursuit for all\u2014to afford all an unfettered start, and a fair chance in the race of life.\" Each of us as individuals can take action to provide fellow citizens with that fair chance in the race of life. But government, as Lincoln said, must do its part too.\n\nFollowing are six specific steps that government can take at the national, state, and\/or local level to increase opportunity and strengthen the nation.\n\n1. Fully fund the bipartisan Edward M. Kennedy Serve America Act and get at least 250,000 Americans into full-time service, with a majority deployed to schools. AmeriCorps members, including members of a new School Turnaround Corps run in partnership with the Department of Education, can be the heart of a massively scaled \"second shift\" in education and can in turn recruit, train, and support citizen change makers like those described in this book. Fully funding AmeriCorps would cost $2 billion, which would pay for itself many times over in the form of higher educational attainment and higher wages for the millions who would benefit. A recent study indicated that each dollar invested in national service ultimately returns four dollars to the Treasury through higher wages (which lead to higher tax receipts) and lower social service costs.\n\n2. Initiate a national marketing campaign for mentors and volunteer teachers on the scale of the Rosie the Riveter campaign during World War II or the designated-driver campaign in the 1980s, inspired by Mothers Against Drunk Driving. David Mantus and Alan Su and Earline Shearer and Deb Daccord are the early adopters. Now we need to use their stories to promote mentoring and volunteer teaching as a personally rewarding way to strengthen our communities and country. We should use the bully pulpit of the US president and First Lady (and of mayors and governors and other political leaders across the country) to call for more citizen teachers, mentors, and tutors, and to recognize and reward volunteers and the companies that sponsor them.\n\n3. Pass the national Time for Innovation Matters in Education (TIME) Act. Introduced in 2008 by the late Senator Kennedy and reintroduced in 2011, the TIME Act would create an ELT fund allowing the federal government to offer matching grants to qualified schools. To ensure systemic shifts in funding priorities, the TIME Act should provide annual funding of $750 or more per student, with winning schools and districts needing to free up an additional $750 per student from their own budgets and commit to adding at least three hundred extra hours of high-quality learning time. As the economy and tax revenues rebound from the recent recession, states should dedicate 10 percent or more of increased revenues to state versions of the TIME Act that support high-quality Expanded Learning Time programs. As with the federal act, states could make five-year renewable grants of $750 or more per student per year with a one-to-one match required from recipient schools and communities. This would ensure that a combination of new and repurposed funding streams support and sustain the type of high-quality extended-day programs and workplace internships that can narrow or eliminate opportunity and achievement gaps.\n\n4. Revise the federal tax code so that contributors to local education foundations and private schools do not receive a tax break if they support schools that are already well funded and serve few low-income students. Donors would maintain their tax break if the school or district they are supporting is publicly funded at a lower than average level and serves a high percentage of low-income students. This idea aligns with a recent suggestion by Stanford professor of political science Rob Reich, who wrote a September 2013 _New York Times_ column describing how the Schools Foundation in wealthy Hillsborough, California, received $2,300 per student in tax-deductible donations while, across San Francisco Bay, students in low-income Oakland benefit from less than $100 per student in private donations. \"Private giving to public schools widens the gap between rich and poor,\" wrote Reich. \"It exacerbates inequalities in financing. It is philanthropy in the service of conferring advantage on the already well-off.\" Upper-income parents like me should absolutely contribute to the schools our children attend, whether public or private, but we shouldn't simultaneously withdraw funds from the public purse by deducting those contributions from our taxable income. Additional revenue that would flow to the IRS by making this change should be earmarked to pay for other proposals discussed in this chapter.\n\n5. Reauthorize the Elementary and Secondary Education Act (commonly known as No Child Left Behind), the principal federal policy governing education. Ensure that its various programs move toward allowing more flexibility in how federal funds can be used but also require reasonable accountability for results. Specifically, Congress should make the following changes relative to the themes of this book:\n\na. Reserve 5 percent of Title I funding (this set-aside would equal about $750 million at current budget levels) for investments in partnerships between high-poverty districts and educational support organizations that have strong evidence of positively impacting student learning. Too much Title I funding is spent on what it was spent on last year without regard to effectiveness.\n\nb. Revise the 21st Century Community Learning Centers program ($1.2 billion) to explicitly authorize use of the funds for Expanded Learning Time models that serve all students in a school (in addition to optional after-school programs that extend the learning day for some students) and to explicitly require that funded programs involve a partnership between one or more schools and one or more community-based organizations and take a balanced educational approach that includes enrichment, as well as core academics.\n\nc. Revise the School Improvement Grants program ($500 million) to allow grants from states directly to nonprofit partners with proven school-turnaround models.\n\n6. Organize a new round of Race to the Top funding focused on hands-on learning in science, technology, engineering, and math (STEM). A relatively modest one-time federal investment in hands-on STEM learning could stimulate lasting changes in local school policy, stimulate new partnerships with private industry, and scale up programs such as US2020, which match STEM professionals with teachers and students.\n\nOffering Citizen Schools or similar programs to every middle school child in a majority-low-income school in America (about 4 million students) would cost about $6 billion, less than 1 percent of total US K\u201312 education spending. At least half this money could come from reallocations of current budgets and the AmeriCorps expansion described above. While some funds for ELT can come from stretching current dollars further and spending them more wisely, some new investment is also needed. This can come from modest changes to the tax code (reducing deductions for gifts to private schools and public schools that serve mostly upper-income children) as described above, or through allocation of a portion of additional revenues as the economy grows.\n\nWhile there is much that government can do, there is a lot that we the people can do that won't require waiting for government. To narrow achievement gaps and broaden opportunity, citizens like you and me can:\n\n1. Volunteer as a citizen teacher (www.citizenschools.org), a mentor (mentor.org or us2020.org for scientists and engineers and others in the STEM disciplines), or a tutor in a high-need school near your home or office.\n\n2. Serve as a full-time AmeriCorps or VISTA member (www.americorps.gov) or a member of other full- and part-time service corps.\n\n3. Invest. If you are an upper-middle-class or wealthy parent, calculate what you spend on extra enrichment and tutoring for your own children and consider making a tax-deductible contribution of this amount or an amount you can afford to one of the many effective organizations trying to close the opportunity gap. Our kids deserve everything we are giving them, but so do other kids.\n\n4. Advocate for public schools, for extended-day programs, and for appropriate reforms to fully fund and reform public schools in your community.\n\n5. If you work for or run a business, a government agency, or a nonprofit, host a summer intern from a low- to moderate-income background. Try to pay them a stipend. But even if all you can cover is a few free lunches, provide a meaningful work experience that gives your intern a first step on her career ladder. If you work near an innovative high school that places students in for-credit internships (NAF, MET, High Tech High), sign up to host a student extern so they can earn credit while getting exposure to the world of work.\n\n6. If you are a social entrepreneur or want to become one, create a new program that lifts up opportunity for lower-income children. Evaluate it. Improve it. We need more examples of successful programs that are well evaluated, well documented, and marketed broadly. If you already run a promising or proven program addressing achievement and opportunity gaps, then grow it, tell your story, and share your methods.\n\nAmerica's success has never been about just government, or business, or individuals. At our best, our national character mixes rugged individualism with communitarian ideals. We celebrate the Horatio Alger story but also community barn raisings and volunteer fire departments and the neighborhood public school. In recent years, however, we have often lost sight of our community values. We've been distracted by our differences and have allowed a globalized economy to expand gaps of wealth and achievement. Equality of opportunity is a founding ideal of the nation, but we now have less of it than ever before.\n\nDespite this lament, the students, volunteers, and staff of Citizen Schools show us that the American capacity for generosity, compassion, and love of neighbor runs deep. Our participants are narrowing achievement gaps, broadening opportunity, and strengthening America. Let us hope these actions will inspire more to stand up\u2014together\u2014to make the opportunity equation work for the many, not just the few.\n\n# VOICES FROM CITIZEN SCHOOLS\n\n# AGOSTINHA DEPINA, STUDENT\n\nEvery spring, Citizen Schools hosts a fund-raising benefit in Boston. As with many benefits, there is chicken for dinner, some wine, a few brief speeches by staff and board members, and sometimes a video. But at these benefits the keynote speaker is always a student or a former student, someone who has lived the Citizen Schools program and who has the final word. The following speech was shared by alumnus Agostinha DePina on April 4, 2012.\n\n_Let me take you to my homeland._\n\n_Bare feet feeling the hot sand,_\n\n_Chasing chickens, riding horses._\n\n_These are my roots._\n\n_My name is Agostinha DePina, and I am a senior at John D. O'Bryant High School in Boston. I spent the first eight years of my life on the island of Fogo in Cape Verde, where my parents grew up, and only my mom went to high school. We were really poor. I remember sleeping on the floor, hungry some nights with one dress and no shoes. But I also remember feeling free and happy._\n\n_I immigrated to the US when I was nine years of age. My parents brought me here for the opportunity. But I was terrified. I remember my first day of elementary school, being in the big yellow bus in the middle of strangers, without knowing a word of English, and entering a classroom where I did not know what to do or what to say. So I placed myself in a comfort corner._\n\n_I might have gotten lost right away if it weren't for my second-grade teacher, Ms. Gomes. With her charismatic and intellectual teaching, she taught me English and helped me see what was possible in my new country. In the Cape Verdean culture, women are taught that their dreams of success are their husband's dreams, that they don't need a voice because the man has a voice. But in the United States, I saw things I had never seen before: girls of all ages going to school, mothers being independent and working, and women striving to be a part of something. Ms. Gomes showed me that women can become queens without a king. But I was still shy\u2014a quiet girl with a lot to say, but with no voice._\n\n_For several years I was a passionate student, always eager to go to school. My mom and my dad were always supportive of me. I could see how hard they worked for my six siblings and me\u2014my mom is a housekeeper at the Westin Boston Waterfront Hotel and my dad is a cleaner at UMass Boston. They told me every day that I am responsible for my future and my success._\n\n_However, as classes became more rigorous and the material was harder to understand, my parents' motivational speeches were not enough. In sixth and seventh grade, I couldn't keep my grades up, and I began to lose my drive for school._\n\n_In eighth grade, though, I was lucky. I got a support network that kept me from going off track. A group of people believed in my potential and gave me the knowledge and skills that have gotten me where I am today. These were the people of Citizen Schools._\n\n_My team leader, Julianne, would always come over and talk with me. Every time I had a test or quiz at school, Julianne would help me study. Then professionals from Putnam Investments came and taught us interview skills. Two volunteers from the law firm Choate, Hall, & Stewart\u2014Eleanor and Cara\u2014worked with me on writing essays that would be published in a magazine. They became my mentors._\n\n_And every Tuesday and Thursday, I took apprenticeships. I measured my school's carbon footprint one semester, and I tried creative writing. We created stories by observing regular day people during their daily activities. My citizen teacher Jennifer made me read my poem to my peers, where I overcame my shyness._\n\n_Julianne and the teaching fellows took us to visit eight different colleges. I loved visiting Brown and Trinity. A panel of Trinity students talked to us about their experience. Initially, I wasn't sure I wanted to go to college. But I remember one student talked about how Trinity College really made it possible for her to attend college and persevere. I knew that if she could do it, I could overcome any obstacle I face._\n\n_I am proud to say that I've just been admitted to Clark University, where I will major in communications. But I would not have made it into Clark, or even be graduating from the great high school I attend, without Citizen Schools. I wouldn't have discovered my passion for writing. And most importantly, the people I saw coming to my school and giving back made me realize that my aspiration in life is to give back. I am currently writing and performing with_ Teen Voices _magazine, where I use writing to empower other teen girls. My dream is to start a nonprofit for girls, to help them find the confidence that others have helped me find, and give other girls the opportunities that many women never receive._\n\n_All these people\u2014Jennifer, Eleanor, Cara\u2014they saw my talents and potential at the right time in my life, and they helped me reveal it. I'm especially grateful to my team leader, Julianne, who came over and sat next to that shy girl. I'm excited to say that Julianne is here tonight._\n\n_Thank you all. All of you in this room tonight are making it possible for teens to stay on track. You who volunteer, or send your employees to volunteer, are changing kids' lives. Your donations bring Citizen Schools to more schools and help students discover the drive to go to college. You realize that there is nothing more important than education, and you know that you have a role to play in helping teens learn what they want to become in life._\n\n_The girl who walked barefoot on the heat of the Cape Verdean sand_\n\n_With one dress to wear, no money in her hands_\n\n_Is the same girl who is now making her dreams come true_\n\n_Now it's my turn to give back so that everyone can see_\n\n_How this Cape Verdean girl who was a slave to poverty_\n\n_Is now the master of her destiny._\n\n# APPENDIX\n\n# EXPANDED LEARNING TIME SCHEDULE WITH CITIZEN SCHOOLS\n\n# ACKNOWLEDGMENTS\n\nWriting this book brought together my family story, the Citizen Schools story, and a journalistic story about the growing opportunity divide between lower- and upper-income children and what we can do about it. Telling the story was a privilege. But the book would not have been possible without the contributions of many others.\n\nMy parents, Fritz Schwarz and Marian Cross, and my sister, Eliza Schwarz, gracefully accepted that in a book about childhood and opportunity I wanted to talk about my own childhood, which meant talking about them too. Thank you for believing in me when it wasn't easy to and for each being great role models in many ways.\n\nMy children, Ronan and Orla, grew up around Citizen Schools, hanging out in our offices, and at apprenticeships, and WOW! celebrations. I remember when Citizen Schools apprentices held them as babies. Now Ronan, a high schooler, is alumni age, and Orla, if she were in one of our schools, would only have a year remaining before her Citizen Schools graduation. Thank you, Ronan and Orla, for putting up with Dad heading up to the \"man cave\" to work on this book so many evenings and so many weekends. And thank you for allowing me to tell an uncensored version of my childhood story. I feel confident talking about my wayward ways because I am so confident in both of you.\n\nMy wife, Maureen, is an artist and the most amazing storyteller I know. She has been the most important critic and supporter of this book, catching me when I lost my voice or went off track and helping make words and paragraphs and the entire story better. Thank you, Maureen, for being such a loving and passionate mother, and wife, and friend.\n\nI've worked on this book for two years, starting with an initial proposal and outline, gradually adding a few draft chapters, and then, after Beacon Press agreed to publish the book, writing most of the chapters in 2013 at a series of brief writing retreats and on a lot of weekend mornings. Along the way I have benefited from great encouragement and editorial advice worthy of William Strunk and E. B. White, from Susan Heath, Martha Eddison, David Stolow, Joel Horwich, Laura Pappano, Ted Fiske, Mike Kubiak, Melissa Ludtke, Sherif Nada, Diana Smith, Colin Stokes, Emily Bryan, David Shapiro, AnnMaura Connolly, Alexandra Bernadotte, and Jocelyn Glatzer. Each of you helped to make the book better. Thank you. Thanks as well to Ambassador Swanee Hunt and the Hunt Alternatives Fund for supporting the book and a writing retreat at the Columbine Ranch. Thanks as well to the great team at Beacon Press and to my editor, Alexis Rizzuto. Alex, I knew I was going to like you when I learned that Phil Rizzuto was your grand-uncle, and you've never disappointed. It goes without saying that any errors or oversights are my responsibility.\n\nImportant early research assistance came from Nina Barker and Renee Reid. Colin Lacy was my partner over the last year in tracking down footnotes, checking facts, and suggesting improvements to the storyline. And a number of Citizen Schools current or former staff helped me fact-check or track down stories. Thank you to Kelly Bernard, Allyson Crawford, Elle Ward, Rebecca Brown, Ruth Summers, Alana Siegner, Jessica Lander, and Holly Trippitt.\n\nA special thank-you goes to the core team that helped bring Citizen Schools to life and lead its growth. You have made this story possible. Ned Rimer has been a great friend since college and was a tremendous copilot for the first dozen years of Citizen Schools. He joins John Werner, Anita Price, Stephanie Davolos, and Tulaine Montgomery as the fantastic founding team that brought Citizen Schools from idea to reality. So many have made major contributions since then, but I want to particularly call out from the past and current executive staff Anuradha Desai, Adrian Haugabrook, Kate Carpenter Bernier, Lisa Ulrich, Kate Mehr, George Chu, Lori Stevens, Priscilla Cohen, Nell Kisiel, Pat Kirby, Claudia Alfaro, Kait Rogers, and the amazing Emily McCann, now our president, who has been a great partner for the last decade. Five powerful leaders have served as chair or cochair of the Citizen Schools board of directors: Marsha Feinberg and Shashi Rajpal, then Sherif Nada, then Andrew Balson, and most recently Larry Summers. I appreciate each of you and all the board members past and present who have shaped and supported Citizen Schools.\n\nFinally, I would like to thank the amazing volunteer citizen teachers and front-line staff of Citizen Schools who have made the work and the learning fun, and the young apprentices whose big dreams motivate us every day. Helping a child discover and achieve his or her dreams is the noblest job I can imagine, and it has been my great privilege to do it for the last twenty years.\n\n# NOTES\n\nFOREWORD\n\n. Tom Mortenson, \"Bachelor's Degree Attainment by Age 24 by Family Income Quartiles, 1970 to 2012,\" _Postsecondary Education Opportunity_ , 2013.\n\n. National Center for Education Statistics, \"Enrollment Rates of 18- to 24-Year-Olds in Degree-Granting Institutions, by Level of Institution and Sex and Race\/Ethnicity of Student: 1967 Through 2010,\" table 213, August 2011, <http:\/\/nces.ed.gov\/programs\/digest\/d11\/tables\/dt11_213.asp>\n\n. Aparna Mathur, Hao Fu, and Peter Hansen, \"The Mysterious and Alarming Rise of Single Parenthood in America,\" _Atlantic,_ September 2013, <http:\/\/www.theatlantic.com\/business\/archive\/2013\/09\/the-mysterious-and-alarming-rise-of-single-parenthood-in-america\/279203\/>.\n\n. \"The 6,000-Hour Learning Gap,\" The After-School Corporation, 2013, <http:\/\/www.expandedschools.org\/policy-documents\/6000-hour-learning-gap#sthash.QsffhGWW.dpbs>. Multiple original sources cited.\n\nINTRODUCTION\n\n. \"Distribution of Bachelor's Degrees Awarded by Age 24 by Family Income Quartiles 1970 to 2012,\" _Postsecondary Education Opportunity_ , no. 256 (\u00adOctober 2013), www.postsecondary.org.\n\n. Greg J. Duncan and Richard J. Murnane, eds., _Whither Opportunity? Rising Inequality, Schools, and Children's Life Chances_ (New York: Russell Sage Foundation, 2011), 98\u201399.\n\n. Joseph P. Ferrie, \"History Lessons: The End Of American Exceptionalism? Mobility in the United States Since 1850,\" _Journal of Economic Perspectives_ 19, no. 3 (2005), 199\u2013215.\n\n. McKinsey & Company, Social Sector Office, _The Economic Impact of the Achievement Gap in America's Schools_ (April 2009), <http:\/\/mckinseyonsociety.com>.\n\n. Duncan and Murnane, _Whither Opportunity?_\n\n. Horace Mann, Twelfth Annual Report to the Massachusetts State Board of Education, 1848.\n\n. Erickson Arcaira, Juliet D. Vile, and Elizabeth R. Reisner, _Achieving High School Graduation: Citizen Schools' Youth Outcomes in Boston_ (Washington, DC: Policy Studies Associates, 2010), www.policystudies.com; _Citizen Schools ELT Evaluation: Preliminary Findings_ (Cambridge, MA: Abt Associates, 2013), www.abtassociates.com.\n\n. Organisation for Economic Co-operation and Development (OECD), _Education at a Glance 2011: OECD Indicators_ , <http:\/\/www.oecd.org>.\n\n. Organisation for Economic Co-operation and Development (OECD), _PISA 2012 Results_ , www.oecd.org\/pisa\/.\n\n. National Science Board, _Science and Engineering Indicators 2012_ (Arlington, VA: National Science Foundation, 2012), <http:\/\/www.nsf.gov\/statistics\/>.\n\n. Duncan and Murnane, _Whither Opportunity?_\n\n. \"Distribution of Bachelor's Degrees Awarded by Age 24 by Family Income Quartiles 1970 to 2012,\" _Postsecondary Education Opportunity_ , no. 256 (\u00adOctober 2013), www.postsecondary.org.\n\n. Joshua Wyner, John Bridgeland, and John Dilulio, _Achievement Trap: How America Is Failing Millions of High-Achieving Students from Lower-Income Families_ (Washington, DC: Jack Kent Cooke Foundation, 2007).\n\n. Duncan and Murnane, _Whither Opportunity?_ , 11.\n\n. Robert D. Putnam, Carl B. Frederick, and Kaisa Snellman, _Growing Class Gaps in Social Connectedness Among American Youth_ (Cambridge, MA: Harvard Kennedy School of Government, 2012), 13.\n\n. College Board, _Education Pays: Update_ , 2005, <http:\/\/www.collegeboard.com>. In 1972, the median earnings for males with a bachelor's degree were 22 percent higher than for males with a high school degree.\n\n. Anthony P. Carnevale, Stephen J. Rose, and Ban Cheah, _The College Payoff: Education, Occupations, Lifetime Earnings_ (Washington, DC: Georgetown University Center on Education and the Workforce, 2011), <http:\/\/cew.georgetown.edu>.\n\n. Duncan and Murnane, _Whither Opportunity?_ , 339\u201358.\n\nCHAPTER ONE\n\n. Lauren Resnick, \"Learning in School and Out,\" _Educational Researcher_ 16, no. 9 (1987): 13\u201320.\n\n. In the race for the 1984 Democratic nomination, Gary Hart actually won more votes across the primary and caucus season than Walter Mondale, but Mondale secured the votes of most party superdelegates and cruised to a first-ballot win at the convention in San Francisco before being crushed in the general election by President Reagan. Hart was the odds-on favorite for the 1988 Democratic primary until being forced to drop out of the race when the _Miami Herald_ photographed him and a woman who was not his wife aboard a yacht in the Gulf of Mexico.\n\n. Bruce Feiler, _The Secrets of Happy Families: Improve Your Mornings, Rethink Family Dinner, Fight Smarter, Go Out and Play, and Much More_ (New York: William Morrow, 2013).\n\n. Organisation for Economic Co-operation and Development (OECD), _Economic Policy Reforms: Going for Growth 2010_ , 185, <http:\/\/www.oecd.org\/>. Similar data has been found by researchers with the Equality of Opportunity Project, <http:\/\/www.equality-of-opportunity.org\/>.\n\n. Raj Chetty, Nathaniel Hendren, Patrick Kline, and Emmanuel Saez, Equality of Opportunity Project, 2013, <http:\/\/www.equality-of-opportunity.org>.\n\nCHAPTER TWO\n\n. Teresa L. Morisi, \"The Early 2000s: A Period of Declining Teen Summer Employment Rates,\" _Monthly Labor Review_ 133, no. 5 (May 2010): 23\u201335.\n\nCHAPTER THREE\n\n. Urban Institute, National Center for Charitable Statistics, Core Files (Public Charities, 2010). From Amy S. Blackwood, Katie L. Roeger, and Sarah L. Pettijohn, _The Nonprofit Sector in Brief: Public Charities, Giving, and Volunteering, 2012_ (Washington, DC: Urban Institute, 2012), <http:\/\/www.urban.org\/UploadedPDF\/412674-The-Nonprofit-Sector-in-Brief.pdf>.\n\n. Peter Kim and Jeffrey Bradach, \"Why More Nonprofits Are Getting Bigger,\" _Stanford Social Innovation Review_ (Spring 2012), <http:\/\/www.ssireview.org\/>.\n\nCHAPTER FOUR\n\n. National Commission on Excellence in Education, _A Nation at Risk: The Imperative for Educational Reform_ (April 1983), available at the website of the US Department of Education, <http:\/\/www.ed.gov.datacenter.spps.org\/uploads\/sotw_a_nation_at_risk_1983.pdf>.\n\n. Massachusetts Education Reform Act of 1993, 215.\n\n. Jennifer McMurrer, _NCLB Year 5: Instructional Time in Elementary Schools: A Closer Look at Changes for Specific Subjects_ (Washington, DC: Center on Education Policy, 2008), www.cep-dc.org.\n\nCHAPTER FIVE\n\n. Erikson Arcaira, Juliet D. Vile, and Elizabeth R. Reisner, _Citizen Schools: Achieving High School Graduation; Citizen Schools Youth Outcomes in Boston_ (Washington, DC: Policy Studies Associates, 2010); Shaun Kellogg, Brandy Parker, Cheryl Loiselle, and Michael Kubiak, _Achieving High School Success: The Impact of Citizen Schools North Carolina_ (Boston: Citizen Schools Research & Evaluation, 2013).\n\n. Eric Schwarz and Ken Kay, eds., _The Case for Twenty-First Century Learning_ , issue 110, New Directions for Youth Development (San Francisco: Jossey-Bass\/Wiley, 2006).\n\n. Jim Collins, _Good to Great: Why Some Companies Make the Leap\u2014and Others Don't_ (New York: HarperBusiness, 2001), 85.\n\nCHAPTER SIX\n\n. Quoted in Kate Carpenter Bernier, _Expanding Learning Time: How the Edwards Middle School in Boston Partnered with Citizen Schools to Transform the Learning Day_ (Boston: Citizen Schools, April 2008), 4.\n\n. In Worcester, Massachusetts, for instance, teachers at the Jacob Hiatt elementary school proposed teaming up with artists from the Worcester Art Museum and historians from the Paul Revere House. But these community-based educators would play an adjunct, not a leading, role.\n\n. Carpenter Bernier, _Expanding Learning Time_ , 10.\n\n. Ibid., 13.\n\n. Randi Weingarten, \"Evidence Matters,\" _New York Times_ , April 23, 2011, available at the website of the American Federation of Teachers, <http:\/\/www.aft.org\/newspubs\/press\/columnsblogs.cfm>.\n\n. Replicating the Edwards's success at five thousand schools would reach approximately one million sixth graders\u2014or about 25 percent of the sixth-grade cohort and most low-income sixth graders. If these students, on average, closed achievement gaps with their upper-income peers, as happened at the Edwards, our national proficiency rate on international exams like PISA would go up about six percentage points.\n\n. At Orchard Gardens, from 2010 to 2013, the eighth-grade math proficiency rate as measured by the statewide MCAS exams went from 8 percent to 63 percent. During this same time period, math proficiency rates within Boston Public Schools went up by three points (34 to 37) and in the state of Massachusetts they went up by four points (from 51 to 55). At Orchard Gardens, from 2010 to 2013, the eighth-grade English proficiency rate on the statewide MCAS exams went from 36 percent to 74 percent. During this same time, proficiency rates within Boston Public Schools and the state of Massachusetts remained flat, staying at 58 percent and 78 percent respectively.\n\nCHAPTER SEVEN\n\n. On its website the Ashoka Foundation defines _social entrepreneurs_ as \"individuals with innovative solutions to society's most pressing social problems. They are ambitious and persistent, tackling major social issues and offering new ideas for wide-scale change. Rather than leaving societal needs to the government or business sectors, social entrepreneurs find what is not working and solve the problem by changing the system, spreading the solution, and persuading entire societies to move in different directions\" (www.ashoka.org). The _Wikipedia_ entry on social entrepreneurship (citing \"The Meaning of Social Entrepreneurship\" by the late J. Gregory Dees) gives this definition: \"Social entrepreneurship is the process of pursuing suitable solutions to social problems. More specifically, social entrepreneurs adopt a mission to create and sustain social value. They pursue opportunities to serve this mission, while continuously adapting and learning. They draw upon appropriate thinking in both the business and nonprofit worlds and operate in all kinds of organizations: large and small; new and old; religious and secular; nonprofit, for-profit, and hybrid.\" Peter Drucker's observation on entrepreneurs is that they rearrange resources toward higher and better uses; social entrepreneurs seek to do the same toward addressing social challenges.\n\n. Blackbaud, Inc., \"Index of National Fundraising Performance,\" 2013, <http:\/\/www.blackbaud.com\/files\/resources\/downloads\/Target_Index_Results_Summary_Q1_2013.pdf>.\n\n. Eric Schwarz, \"Realizing the American Dream: Historical Scorecard, Current Challenges, Future Opportunities,\" working paper, A Gathering of Leaders: Social Entrepreneurs and Scale in the 21st Century, February 15\u201318, 2005.\n\n. Greg Parks, \"The High\/Scope Perry Preschool Project,\" _Juvenile Justice Bulletin_ (October 2000): 3, available from the website of the National Criminal Justice Reference Service, https:\/\/www.ncjrs.gov\/pdffiles1\/ojjdp\/181725.pdf.\n\n. FT 500, 2013, _Financial Times_ , www.ft.com\/intl\/indepth\/ft500.\n\n. \"The 200 Largest US Charities,\" _Forbes_ (2005), <http:\/\/www.forbes.com\/2005\/11\/18\/largest-charities-ratings_05charities_land.html>. In 2005, Habitat for Humanity was the only organization founded in the last forty years based on cash revenues. When in-kind contributions are included, there were two other organizations that had been founded in the last forty years, Gift In Kind International and AmeriCares.\n\n. Habitat for Humanity, \"About Us,\" <http:\/\/www.habitat.org\/how\/about_us.aspx>.\n\n. Lawrence Mishel, Jared Bernstein, and Heather Boushey, _The State of Working America 2002\/2003_ (Ithaca, NY: ILR Press, 2003), 37\u201349. From 1973 to 2001, the percentage of American families in the lowest economic quartile who owned their homes declined from 51 percent to 49 percent. During this same period, Habitat built tens of thousands of homes for poor families across the country.\n\n. America Forward, _America Forward: Invent, Invest, Involve_ (Boston: New Profit, 2007), www.newprofit.com. This booklet was shared with Obama, McCain, and other candidates. It impacted thinking on reforms, including the Social Innovation Fund, an effort to have the federal government team up with philanthropy to scale proven and promising organizations.\n\n. Barack Obama's first inaugural address, 2009, <http:\/\/www.whitehouse.gov\/blog\/inaugural-address>.\n\n. White House Office of Science and Technology Policy, <http:\/\/www.whitehouse.gov\/administration\/eop\/ostp>.\n\n. US Department of Education, \"Race to the Top Executive Summary,\" 2009, <http:\/\/www2.ed.gov\/programs\/racetothetop\/executive-summary.pdf>. In 2009, $4.35 billion was appropriated for the Race to the Top Fund in the American Recovery and Reinvestment Act (ARRA). The program includes $4 billion for statewide reform grants and $350 million to support states working together to improve the quality of their assessments.\n\n. New America Foundation, Federal Education Budget Project, \"American Recovery and Reinvestment Act: Background and Analysis,\" 2012, <http:\/\/febp.newamerica.net\/background-analysis\/american-recovery-and-reinvestment-act>. Race to the Top funds were later included by Congress in regular appropriations, including nearly $700 million in fiscal year 2011 and almost $550 million in fiscal year 2012 for further rounds of the state competition and an Early Learning Challenge. The bill included language that would allow the department to create a district-level competition and continue the investment in the Early Learning Challenge.\n\n. US Department of Education, \"Investing in Innovation Fund (i3): Funding Status,\" <http:\/\/www2.ed.gov\/programs\/innovation\/funding.html>. $650 million was included in the 2009 ARRA for the Investing in Innovation Fund. Almost all of that ($645,978,395) was spent in FY 10. The i3 competition is entering its third year, having been appropriated about $150 million each in fiscal years 2011 and 2012. In FY 2013, there is currently $134 million in estimated available funds.\n\nCHAPTER EIGHT\n\n. Charles T. Clotfelter, \"Patterns of Enrollment and Completion,\" in _Economic Challenges in Higher Education_ , edited by Charles T. Clotfelter et al. (Chicago: University of Chicago Press, 1991), 31, available from the website of the National Bureau of Economic Research, <http:\/\/www.nber.org\/chapters\/c6079.pdf>. See also Richard P\u00e9rez-Pe\u00f1a, \"US Bachelor Degree Rate Passes Milestone,\" _New York Times_ , February, 23, 2012, <http:\/\/www.nytimes.com>; Amy K. Glasmeier, _An Atlas of Poverty in America: One Nation, Pulling Apart, 1960\u20132003_ (New York: Routledge, 2006), \"Introduction: The Paradox of Poverty in America\" available at the MIT Poverty in America Project website, <http:\/\/povertyinamerica.mit.edu;\/download\/atlas_of_poverty_in_america_p1.pdf>, and US Census Bureau, \"Income, Poverty and Health Insurance Coverage in the United States: 2011,\" press release, September 12, 2012, <http:\/\/www.census.gov\/newsroom\/releases\/archives\/income_wealth\/cb12\u2013172.html>.\n\n. Lawrence A. Cremin, _American Education: The National Experience,_ _1783\u20131876_ (New York: Harper and Row, 1980).\n\n. James J. Kemple, _Career Academies: Long-Term Impacts on Labor Market Outcomes, Educational Attainment, and Transitions to Adulthood_ (New York: MDRC, 2008), <http:\/\/www.mdrc.org\/publication\/career-academies-long-term-impacts-work-education-and-transitions-adulthood>.\n\n. High Tech High, \"Results,\" <http:\/\/www.hightechhigh.org>.\n\n. FIRST, www.usfirst.org; iMentor, www.imentor.org; City Year, www.cityyear.org.\n\n. Transcript of Lester Strong video interview, from Shar McBee, \"Why Volunteer with Children? Here's Why,\" _Huffington Post_ , March 26, 2013, <http:\/\/www.huffingtonpost.com.\/shar-mcbee\/why-volunteer-with-childr_b_2950039.html>. For more on Experience Corps: <http:\/\/www.aarp.org\/experience-corps\/>.\n\n. Nancy Morrow-Howell et al., _Evaluation of Experience Corps: Student Reading Outcomes_ , publication no. 09-01 (St. Louis: Center for Social Development, Washington University in St. Louis, 2009), <http:\/\/csd.wustl.edu>.\n\n. Lester Strong, speaking on a panel at the Center for American Progress, \"The American Way to Change, How National Service and Volunteers Are Transforming America,\" April 21, 2010, event video at <http:\/\/www.americanprogress.org>.\n\n. Ibid.\n\n. Rachel Schachter, \"Ambition Musician: Showing Students Real Possibilities,\" Citizen Schools, _InspirED_ (blog), 2012, <http:\/\/www.citizenschools.org\/blog\/musician-ambition-showing-students-real-possibilities\/>.\n\n. David Jones, _The Citizen Teacher Experience Study: A Report of Research Results_ (Burlington: University of Vermont, 2012).\n\nCHAPTER NINE\n\n. Greg J. Duncan and Richard J. Murnane, eds., _Whither Opportunity? Rising Inequality, Schools, and Children's Life Chances_ (New York: Russell Sage Foundation, 2011), 339\u201358.\n\n. Arne Duncan, \"Making the Middle Schools Matter,\" Remarks at the National Forum's Annual Schools to Watch Conference, June 23, 2011, <http:\/\/www.ed.gov\/news\/speeches\/making-middle-grades-matter>.\n\n. National Institute of Mental Health, \"Imaging Study Shows Brain Maturing,\" press release, May 17, 2014, <http:\/\/www.nimh.nih.gov\/news\/science-news\/2004\/imaging-study-shows-brain-maturing.shtml>. \"The brain's center of reasoning and problem solving is among the last to mature. The decade-long magnetic resonance imaging (MRI) study of normal brain development, from ages four to twenty-one, by researchers at NIH's National Institute of Mental Health (NIMH) and University of California Los Angeles (UCLA) shows that such 'higher-order' brain centers, such as the prefrontal cortex, don't fully develop until young adulthood. . . . In the late 1990s, NIMH's Dr. Jay Giedd, a coauthor of the current study, and colleagues discovered a second wave of overproduction of gray matter just prior to puberty, followed by a second bout of \"use-it-or-lose-it\" pruning during the teen years. . . . Areas with more advanced functions\u2014integrating information from the senses, reasoning, and other 'executive' functions (prefrontal cortex)\u2014mature last.\"\n\n. Common Core State Standards Initiative, <http:\/\/www.corestandards.org>.\n\n. Rolf K. Blank, _What Is the Impact of Decline in Science Instructional Time in Elementary School?_ , prepared for the Noyce Foundation, 2012, available at the website of the Council of State Science Supervisors, <http:\/\/www.csss-science.org.\/downloads\/NAEPElemScienceData.pdf>.\n\n. Valerie Strauss, \"Survey: Teachers Work 53 Hours Per Week on Average,\" _Answer Sheet_ (blog), _Washington Post_ , March 16, 2012, <http:\/\/www.washingtonpost.com\/blogs\/answer-sheet\/>.\n\n. Jessica Edwards, _Mapping the Field: A Report on Expanded-Time Schools in America_ (Boston: National Center on Time and Learning, 2012), <http:\/\/www.timeandlearning.org\/mapping>.\n\n. Thomas Kuhn, _The Structure of Scientific Revolutions_ (Chicago: University of Chicago Press, 1962).\n\n. Malcolm Gladwell, _Outliers: The Story of Success_ (New York: Little, Brown, 2008).\n\n. Ibid., 2.\n\n. Ibid., 23.\n\nCHAPTER TEN\n\n. Organization for Economic Co-operation and Development (OECD), \"Programme for International Student Assessment (PISA) Results from PISA 2012: United States,\" <http:\/\/www.oecd.org\/pisa\/>. See also Kyung Hee Kim, \"The Creativity Crisis: The Decrease in Creative Thinking Scores on the Torrance Tests of Creative Thinking,\" _Creativity Research Journal_ 23, no. 4 (2001): 285\u201395.\n\n. Blank, _What Is the Impact of Decline in Science Instructional Time in Elementary School?_ , 8.\n\n. Robert Tai, Christine Qi Liu, Adam Maltese, and Xitao Fan, \"Planning Early for Careers in Science,\" _Science_ 312 (May 2006).\n\n. Lemelson-MIT Program, \"2010 Lemelson-MIT Invention Index Reveals Ways to Enhance Teens' Interest in Science, Technology, Engineering and Mathematics in the Classroom and Beyond,\" press release, January 28, 2010, <http:\/\/web.mit.edu\/invent\/n-pressreleases\/n-press-10index.html>.\n\n. US Congress Joint Economic Committee, _STEM Education: Preparing for the Jobs of the Future_ , 2012, <http:\/\/www.jec.senate.gov\/>.\n\n. National Action Council for Minorities in Engineering, \"African Americans in Engineering,\" _NACME Research & Policy Brief_ 1, no. 4 (February 2012), <http:\/\/www.nacme.org\/user\/docs\/AAEFINAL.pdf>.\n\n. Richard J. Murnane and Frank Levy, _Teaching the New Basic Skills: Principles for Educating Children to Thrive in a Changing Economy_ (New York: Free Press, 1996).\n\n. Richard Murnane, presentation at Reimagining After-School: A Symposium on Learning and Leading in the 21st Century, April 29, 2004, Cambridge, MA.\n\n. IBM, \"IBM 2010 Global CEO Study: Creativity Selected as Most Crucial Factor for Future Success,\" press release, May 18, 2010, <http:\/\/www-03.ibm.com\/press\/us\/en\/pressrelease\/31670.wss>.\n\n. Kyung Hee Kim, \"The Creativity Crisis: The Decrease in Creative Thinking Scores on the Torrance Tests of Creative Thinking,\" _Creativity Research Journal_ 23, no. 4 (2011), 285\u201395.\n\n. Peter Gray, \"As Children's Freedom Has Declined, So Has Their Creativity,\" _Freedom to Learn_ (blog), _Psychology Today_ , September 17, 2012, <http:\/\/www.psychologytoday.com\/blog\/freedom-learn\/201209\/children-s-freedom-has-declined-so-has-their-creativity>.\n\n. 100Kin10, <http:\/\/www.100kin10.org>.\n\n. White House, \"New Details: President Obama to Host White House Science Fair,\" press release, April 22, 2013, <http:\/\/www.whitehouse.gov\/the-press-office\/2013\/04\/22\/new-details-president-obama-host-white-house-science-fair>. See also \"Remarks by the President at the White House Science Fair,\" April 22, 2013, <http:\/\/www.whitehouse.gov\/the-press-office\/2013\/04\/22\/remarks-president-2013-white-house-science-fair>.\n\n. Walter Isaacson, _Steve Jobs_ (New York: Simon & Schuster, 2011).\n\n. Richard Florida, _The Rise of the Creative Class and How It's Transforming Work, Leisure, Community, and Everyday Life_ , rev. ed. (New York: Basic Books, 2012).\n\nCHAPTER ELEVEN\n\n. Robert D. Putnam, Carl B. Frederick, and Kaisa Snellman, \"Growing Class Gaps in Social Connectedness Among American Youth,\" paper presented at the Saguaro Seminar: Civic Engagement in America, Cambridge, MA, Harvard Kennedy School of Government, August 8, 2012, p. 19, available at www.hks.harvard.edu.\n\n. Ibid.\n\n. Author's notes from interviews of those attending the Aspen Ideas Festival, 2012.\n\n. Putnam, Frederick, and Snellman, \"Growing Class Gaps,\" 11. \"Participation in chess clubs, debate teams, school bands, and student councils bolster self-esteem and feelings of self-worth, boost high school grade point average, shape educational aspirations and attainment, as well as wages and occupational choice (Lamborn et al., 1992; Newmann et al., 1992; Eccles and Barber, 1999; Borghans et al., 2011).\"\n\n. Ibid., 11\u201317.\n\n. Greg J. Duncan and Richard J. Murnane, eds., _Whither Opportunity? Rising Inequality, Schools, and Children's Life Chances_ (New York: Russell Sage Foundation, 2011).\n\n. James S. Coleman, \"Social Capital in the Creation of Human Capital,\" _American Journal of Sociology_ 94 (1988): 95\u2013120.\n\n. Sean F. Reardon, \"The Widening Academic Achievement Gap Between the Rich and the Poor: New Evidence and Possible Explanations,\" in _Whither Opportunity?_ , eds. Duncan and Murnane, <http:\/\/cepa.stanford.edu\/sites\/default\/files\/reardon%20whither%20opportunity%20-%20chapter%205.pdf>.\n\n. Richard Florida, \"Where the Skills Are,\" _Atlantic_ , October 2011.\n\n. Ibid.\n\n. Robert D. Putnam, _Bowling Alone: The Collapse and Revival of American Community_ (New York: Simon & Schuster, 2000), 405.\n\n. Tom Wolfe, _The Bonfire of the Vanities_ (New York: Farrar, Straus and Giroux, 1987).\n\n. Jennifer Engle and Vincent Tinto, _Moving Beyond Access: College Success for Low-Income, First-Generation Students_ (Washington, DC: Pell Institute for the Study of Opportunity in Higher Education, 2008), <http:\/\/inpathways.net\/COE_MovingBeyondReport_Final.pdf>.\n\n. Nancy DiTomaso, \"How Social Networks Drive Black Unemployment,\" _New York Times_ , May 5, 2012, _Opinionator_ (blog), <http:\/\/opinionator.blogs.nytimes.com\/2013\/05\/05\/how-social-networks-drive-black-unemployment\/>.\n\n. Paul Tough, _How Children Succeed: Grit, Curiosity, and the Hidden Power of Character_ (Boston: Houghton Mifflin Harcourt, 2012).\n\nCHAPTER TWELVE\n\n. Randi Weingarten, \"Evidence Matters,\" _New York Times_ , April 23, 2011, available at the website of the American Federation of Teachers, <http:\/\/www.aft.org\/newspubs\/press\/columnsblogs.cfm>.\n\n. Boston Teachers Union, _e-Bulletin_ 22, December 13, 2011, <http:\/\/www.btu.org\/btu-publications\/ebulletin\/>.\n\n. Matthew Ronfeldt, Susanna Loeb, and James Wyckoff, \"How Teacher Turnover Harms Student Achievement,\" _American Educational Research Journal_ 50, no. 1 (2013): 4\u201336.\n\n. Agency for Healthcare Research and Quality, \"The Number of Practicing Primary Care Physicians in the United States,\" 2010, <http:\/\/www.ahrq.gov\/research\/findings\/factsheets\/primary\/pcwork1\/index.html>.\n\n. Frederick M. Hess, _The Same Thing Over and Over: How School Reformers Get Stuck in Yesterday's Ideas_ (Cambridge, MA: Harvard University Press, 2010).\n\n. Benjamin Scafidi, _The School Staffing Surge: Decades of Employment Growth in America's Public Schools_ (Indianapolis: Friedman Foundation for Educational Choice, 2012), <http:\/\/www.edchoice.org\/>. Note: Data from US Department of Education, National Center for Education Statistics, 1990 _Digest of Education Statistics_ , table 76, and the 2011 _Digest of Education Statistics_ , table 69.\n\n. Catherine Rampell, \"The Lawyer Surplus, State by State,\" _Economix_ (blog), _New York Times_ , June 27, 2011, <http:\/\/economix.blogs.nytimes.com\/2011\/06\/27\/the-lawyer-surplus-state-by-state\/?_r=0>.\n\n. Frederick Hess, \"How to Get the Teachers We Want,\" _Education Next_ 9, no. 3 (2009): 34\u201339, <http:\/\/educationnext.org\/.how-to-get-the-teachers-we-want\/>.\n\n. For more information on these types of programs, see EnCorps, <http:\/\/www.encorpsteachers.com>, and Encore Fellowships, <http:\/\/www.encore.org\/fellowships>.\n\n. These programs include partnership with Relay GSE, Reach, the University of Massachusetts Graduate School of Education, and the Drexel Universtity Graduate School of Eduction alternative certification and master's programs.\n\n. Afterschool Alliance, \"Afterschool Essentials: Research and Polling,\" 2012, <http:\/\/www.afterschoolalliance.org\/>.\n\nCHAPTER THIRTEEN\n\n. Clive Belfield, _The Economic Value of National Service_ (New York: Center for Benefit-Cost Studies of Education, Teachers College Columbia University, 2013), <http:\/\/cbcse.org>.\n\n. Rob Reich, \"Not Very Giving,\" _New York Times_ , September 4, 2013, <http:\/\/www.nytimes.com>.\n\n. National Academies Foundation, www.naf.org; Met, www.metcenter.org; High Tech High, www.hightechhigh.org.\n\n# INDEX\n\nPlease note that page numbers are not accurate for the e-book edition.\n\nAbecedarian Project, 108\n\nacademic disengagement, 21\n\nacademic skills, 12, 63, 66, 68, 89, 152\n\naccountability, 86, 88\n\nachievement gap, 63, 195; class-based, 4\u20135, 12\u201315; closing of, 68, 92, 101\u20132; debate over cause of, 14\u201315; early-childhood education and, 108; income-based, 3\u20135, 9\u201310; race-based, xi, xiii; school-related factors in, 14\u201315\n\nadolescents, 23\u201324, 41, 50\n\naffiliate model of nonprofits. _See_ franchise model of nonprofits\n\nAfrican Americans: achievement gap and, xi; college completion rate for, 125; STEM careers and, 152\n\nThe After-School Corporation (TASC), 73\u201375, 142\u201343\n\nafter-school programs, 50, 86, 137\u201338, 143, 155\u201357, 185; After-School Corporation (TASC), 73\u201375; for English-language learners, 143; Massachusetts 2020, 77. _See also_ Expanded Learning Time (ELT) program; out-of-school experiences\n\nAIM (Aspire, Invest, Make the grade), 68\n\nAltonji, Joseph, 14\n\nAmateur Athletic Union (AAU), 137\n\nAmerican Federation of Teachers (AFT), 176\n\nAmeriCorps, 78, 89, 110, 112, 199; funding of, 196; teaching fellows, 67, 89, 97, 142, 178, 190\u201392\n\nanalytical skills, 166\u201367\n\nApple Computer, 103\u20134, 159\n\napprenticeships, 6\u20139, 23, 41\u201343, 78\u201379; carpentry, 58\u201359; cognitive, 23; fading, 23; modeling success, 23; scaffolding, 23; format for, 68; power of, 127\u201332\n\nart, 63, 139\u201340\n\nassessment data, 181\n\nat-risk students, 68\n\nBain Capital, 71, 105, 114\n\nBaiz, David, 180\n\nBalson, Andrew, 71\n\nBarnsley, Roger, 149\n\nBarrett, Michael, 42\n\nBarriga, Adam, 1\u20132, 97, 194\n\nBarriga, Eduardo, 3, 94\n\nbasic skills, priority on, 152\u201354\n\nBeasley, Caroline, 90\n\nbehavior management, 60\n\nBennett, Joel, 10\u201311, 58\u201359\n\nBogy, Tyeisha, 156\n\nbonding social capital, 167\u201368\n\nBoston City Singers, 137\n\nBoston Teachers Union, 176\n\n_Bowling Alone_ (Putnam), 164\u201365, 167\n\nBradach, Jeff, 73\n\nbranding, 76\u201377\n\nBridgespan, 70\n\nbridging social capital, 168\n\nBrookline, Massachusetts, 14\u201315\n\nBrown, Michael, 103, 115\n\nBryan, Emily, 91\u201392, 94\u201397\n\nBurke, Gil, 23\n\nBushkin, Kathy, 26\n\nBusiness Planning Council, 78\u201379\n\nCampbell, Karen Webb, 132\n\ncampus directors, 67\n\ncapitalism, 12\n\ncareer success: education and, 13; social capital and, 166\u201367\n\nCarnegie Corporation, 157\n\ncarpentry apprenticeship, 58\u201359\n\nCarter, Jimmy, 110\n\n_The Case for Twenty-First Century Learning_ (Schwarz and Kay), 78\n\nCastillo, Yoelinson, 83\n\nChambers, Ray, 55\n\ncharter schools, 62, 95, 113, 143, 177, 185\n\nChina, college graduation rates in, 11\n\nchurch attendance and social networks, 165, 166\n\ncitizen power, 124\u201334, 142, 195\n\nCitizen Schools: accomplishments of, 9\u201310; after-school programs, 52\u201353, 66; approach of, xii\u2013xiii; beginnings of, 5\u20139; benefits of, 64\u201365; challenges for, 48\u201349, 60, 63\u201364, 78\u201381; curriculum changes in, 66; ELT program and, 142; evaluation of, 68\u201369; founding staff of, 47\u201348, 54\u201355; fund-raising for, 56\u201357, 70\u201372, 78; growth of, 9\u201310, 58\u201361, 69, 72\u201381, 105, 116\u201317; i3 grant and, 114\u201317; idea for, 41\u201344; launch of, 40\u201357; national expansion of, 75\u201381; organizational infrastructure of, 66\u201381; protocols for, 60\u201361; Race to the Top and, 113; scaling, 114\u201315; staffing model of, 67; teacher support and, 176\u201385; Teaching Fellow program, 67, 78, 88\u201389, 97, 142, 178\u201385, 190\u201392; transformation of, 101\u20132; volunteers, 46, 48, 60, 64, 67\u201368, 97\u2013101, 120\u201323, 131\u201334; WOW! events, 1\u20133, 49\u201352, 54, 61, 126. _See also_ Clarence Edwards Middle School; Expanded Learning Time (ELT) program\n\ncitizen teachers, 7\u20139, 43, 48, 104, 184, 199; experiences of, 120\u201323; impact of, 97\u2013101, 127\u201334; training of, 67\u201368. _See also_ volunteers\/volunteering\n\nCity Year, 8, 45, 54, 73, 115; author's employment with, 40\u201341, 81, 103; as example of citizen power, 127; as model for Citizen Schools, 105\n\nClarence Edwards Middle School (Boston): ELT program at, 82, 85\u2013102; problems at, 82\u201384, 89\u201390; reforms at, 84\u201387; teacher buy-in at, 88; teachers at, 94\u201397; tensions at, 90\u201391; transformation at, 92\u2013101\n\nclass divide, xiii, 4\u20135, 12\u201315, 165\n\ncoaching, 23\n\nCoffey, Maureen, 46, 52, 168\n\ncognitive apprenticeships, 23\n\nCognizant, 133\n\nCohen, Ed, 45\n\nCohen, Priscilla, 69\n\nColeman, James S., 165\u201366\n\ncollege completion rate, xi, 11\u201312, 125, 172, 188\n\ncollege enrollment, 69, 102, 127\n\ncollege graduates, 11\u201312; earnings of, 13; first-generation, 170\u201372; skills of, 13\n\nCollegiate School (New York City), 19\u201320\n\nCollins, Jim, 81\n\nCommon Core curriculum, 140, 188\n\ncommunication skills, 153\n\ncompany-owned store approach, 73, 79\n\ncompetitive drive, 27\u201328\n\ncomputer science classes, 152\n\nconfidence, role in success, 22, 31, 65, 173\u201374\n\nco-teaching, 179\n\ncreative class, 160\n\ncreativity, 10, 118, 139, 151\u201363\n\ncrime, 185\n\nCromer, Mary, 24\u201325\n\nCross, Marian Lapsley, 27\n\ncrowdsourcing, 112\n\ncultural reform, 106, 110\n\ncurriculum development, 67\u201368\n\nDaccord, Deb, 10, 169\u201370, 173\n\nDARPA (Defense Advanced Research Projects Agency), 160\n\nDavolos, Stephanie, 54\u201358\n\nDees, Greg, 73\n\nDeLatte, Marcus, 128\u201329, 152\n\ndemocracy, 12\n\nDepartment of Education, 113, 115, 116\n\nDepartment of Energy, 112\n\nDepina, Agostinha, 201\u20134\n\nDesai, Anuradha, 70\u201371\n\n_Dever (MA) Community News_ , 8\u20139, 43\u201344\n\nDever Elementary School (Boston), 7, 43, 52, 59, 63, 88, 187\u201388\n\nDiTomaso, Nancy, 173\n\nDougherty, Dale, 160\n\ndropout rates, 63, 93\n\nDrucker, Peter, 117\n\nDu Bois, W. E. B., xiii\n\nDukeshire, Erin, 177\u201378\n\nDuncan, Arne, 113, 139\n\nDuncan, Greg, 12\n\nearly-childhood education, 107\u20139\n\nearly-stage professionals, 133\n\nEchoing Green, 45\n\neconomies of scale, 108\n\neconomy, influence of social capital on, 166\u201367\n\nEdmeade, Stephanie, 84, 94\n\nEdna McConnell Clark Foundation, 71\n\neducation: career success and, 13; early-childhood, 107\u20139; as equalizer, 5\u20136; funding, 62; K\u201312, 104; STEM, 113, 151\u201352, 157\u201359, 198; system, 182\u201383; systemic change in, 103\u201310, 113\n\neducation reform, 61\u201364, 84, 104\u20135, 124; Investing in Innovation (i3), 114\u201317; plan for, 194\u2013200; Race to the Top initiative, 112\u201314\n\nEdward M. Kennedy Serve America Act, 112, 196\n\n8th Grade Academy, 66, 170\n\nEinstein Fellowship, 159\n\nElementary and Secondary Education Act, 198\n\nEmerson, Ralph Waldo, 44\n\nemployment inequality, 173\n\nemployment opportunities, 39\n\nEngels, Kendra, 180\u201381\n\nengineers, 11, 152\n\nEnglish language arts (ELA), 140\u201341\n\nEnglish language learners (ELLs), 1, 143, 181\n\nenrichment experiences, 3, 5, 12\u201314, 63, 94\u201395, 136\u201339. _See also_ apprenticeships; extracurricular activities; out-of-school experiences\n\nentry-level jobs, skills needed for, 153\u201354\n\nEnvironmental Protection Agency (EPA), 112\n\nequal opportunity: ideal of, x, 164; steps to improve, 194\u2013200; in Europe, xi; in US, xi\n\nErikson, Erik, 24\n\nEscalante, Jaime, 174\u201375\n\nescape velocity, 32\n\nExpanded Learning Time (ELT) program, 2, 82, 104\u20135, 142; at the Clarence Edwards Middle School, 85\u2013102; expansion of, 101\u20132, 196\u201397; funding of, 198, 199; grant for, 86\u201387, 114\u201317; reasons for lack of broad adoption of, 146\u201349; results from, 101\u20132; stresses and challenges of, 143\u201346; teacher support and, 176\u201382; time schedule for, 205\n\nExperience Corps, 129\u201331\n\nextended-day learning, 135\u201349, 177\u201378, 185. _See also_ Expanded Learning Time (ELT) program\n\nexternal focus in organizations, 80\u201381\n\nextracurricular activities, 136\u201339, 166. _See also_ enrichment experiences\n\nFAO Schwarz, 29\u201330\n\nFeiler, Bruce, 30\n\nFeinberg, Marsha, 55\n\nfinancial aid, xi\n\nFIRST, 127\n\nfirst-generation college students, 170\u201372\n\nFlanagan, Leo, 93\u201395\n\nFlorida, Richard, 160, 165, 166\u201367\n\nforeign language, 140\n\n4-H, 127\n\nfranchise model for nonprofits, 72\u201378, 110\n\nFrank Friday assessments, 60\u201361\n\nFriedman, Lucy, 74\n\nFuller, Millard, 110\n\nfund-raising, 56\u201357, 70\u201372, 78\n\nGabrieli, Chris, 77\u201378. _See also_ Massachusetts 2020\n\nGarfield School (Brighton, MA), 60, 61\n\nGates, Bill, 148\n\nGeneral Aptitude Test Battery (GATB), 154\n\nGenera, Luis, 90\n\nGeneration Schools, 185\n\ngenetics, 147\n\nGergen, David, 106\u20137, 108\n\ngift giving, 59\n\nGirl Scouts, 127\n\nGladwell, Malcolm, 147\u201349\n\nGlobal Network Navigator (GNN), 160\n\nGordon, Robert, 112\n\ngovernment: actions by, to improve equal opportunity, 196\u2013200; funding, 108\u201310, 112, 113, 196, 198; role of, in nonprofit sector, 111\u201312\n\nGreenlaw, Mark, 133\n\ngrowth models for nonprofits, 72\u201378\n\nHabitat for Humanity, 109\u201310\n\nhands-on projects, 1, 2\u20133, 6, 19, 51, 63, 88\u201389, 151, 198. _See also_ apprenticeships; learning-by-doing approach\n\nHarbor Point housing community (Boston), 7\n\nHarrington, Barry, 105\n\nHart, Gary, 24\u201327, 42, 70, 168\n\nHead Start, 107\u20139\n\nHelies, Tony, 85\u201386, 97\u2013101\n\nHess, Rick, 183\n\nhigher-order thinking skills, 152\u201354, 188\n\nhigh school graduation rates, 68, 102, 106, 127, 188\n\nhigh school sports, 166\n\nhigh-stakes testing, 62\u201364, 151, 181\n\nHigh Tech High, 126\u201327\n\nhigh-wage jobs, 13\n\nhiring process, 153\u201354\n\nHirsch, E. D., 152\n\nhockey players, 148\u201349\n\nHoldren, John, 158\n\nhomework, 21, 32, 68, 178\n\nHudy, Joey, 161\n\nHughes, Brendan, 47\n\nhuman service businesses, 105\n\nHunt, Jim, 135\u201336\n\ni3. _See_ Investing in Innovation (i3)\n\nIBM, 103\u20134, 154\n\niMentor, 127\n\nimmigrant identity, 37\n\nincome inequality, 3\u20134, 12\n\ninequality: in education, 195; employment, 173; income, 3\u20134, 12; increase in, xi; of opportunity, 164, 305\n\ninnovation, 10, 118, 151, 157, 159\u201363\n\nintergenerational self, 31\n\ninternal focus in organizations, 80\u201381\n\ninternships, 6, 23, 24, 32, 39, 126, 158, 200\n\nInvesting in Innovation (i3), 113, 114\u201317\n\nIsaac Newton Expanded Learning Time Middle School, 184\u201385\n\nJobs, Steve, 103, 104, 159\n\nJohn F. Kennedy School of Government, 45\n\nJones, David, 133\u201334\n\nJumpStart, 105\n\njunior teachers, 185\n\nK\u201312 education, 104\n\nKamen, Dean, 127\n\nKanter, Rosabeth Moss, 155\n\nKennedy, John, Jr., 20\n\nKenney, Jack, 22\n\nKhan Academy, 184\n\nKhazei, Alan, 103\n\nKikuko, 48\n\nKim, Kyung Hee, 155\n\nKIPP (Knowledge Is Power Program), 115\n\nKirby, Pat, 145\n\nKirsch, Vanessa, 45\n\nknowledge-based economy, 12\n\nKopp, Wendy, 45\n\nKuhn, Thomas, 146\u201347\n\nKumon, 138\n\nlabor market, 166\u201367\n\nleadership opportunities, for teachers, 181\u201382\n\nlearning by doing approach, 54, 56, 60; _See also_ apprenticeships; hands-on projects\n\nlearning experiences, positive, 21\u201323, 32\n\nlearning opportunities: extra, 3, 5, 13\u201314, 63, 94\u201395, 136\u201339, 166; unequal access to, 136\u201337, 195\n\nlearning time, 42, 104, 124, 135\u201349; expanded, 9, 10, 77\u201378, 86\u201387, 118, 195; importance of, 147\u201349; insufficient, 139\u201341, 151; staffing extra, 142\u201343; unequal access to, 136\u201337. _See also_ Expanded Learning Time (ELT) program\n\nLesley University, 67\n\nlesson plans, 60\n\nLevy, Frank, 153\u201354\n\nLieberman, Aaron, 45\n\nlife skills, sharing, 6\u20137\n\nLight, Sarah, 47\n\nLincoln, Abraham, 195\n\nlow-income students, 1; academic skills of, 12; building blocks of success for, 117\u201318; childhood experiences for, 31\u201333, 35, 36\u201339, 165\u201366; college completion rate of, 12; disadvantages for, 4, 32\u201333, 165\u201366, 195; expectations of, 14, 147; opportunities for, 13\u201314; work experiences for, 39\n\nlow-skilled jobs, 13\n\n_MAKE_ magazine, 160\n\nMaker Corps, 160\u201361\n\nMaker Education Initiative (Maker Ed), 160\u201361\n\nmaker movement, 159\u201363; Maker Faires, 161\u201363\n\nMann, Horace, 6\n\nMansfield, Richard, 14\n\nMantus, David, 2\u20133, 10, 94, 97\u201399, 152, 159, 194\n\nMAP (Middle School Apprenticeship Program), 76\n\nMascoll, Keith, 47\n\nMassachusetts Comprehensive Assessment System (MCAS), 62\u201363, 83, 92\n\nMassachusetts Department of Education, 86\u201387\n\nMassachusetts Education Reform Act (1993), 62\n\nMassachusetts 2020, 77\u201378, 86\u201387\n\nMATCH charter school (Boston), 143\n\nMatch Corps, 143\n\nmath instruction, 11, 140, 143\n\nMcAfee, Chuck, 83\n\nMcCann, Emily, 69\u201370\n\nMcCormack Middle School (Boston), 52\u201353\n\nMcKeen, Cindy, 82\u201383, 94\n\nMcKinney, David, 178\n\nmedical system in US, as compared to US education system, 182\n\nMendez, Nydia, 7, 43, 63, 88\n\nmentors\/mentoring, 10, 31, 118, \u00ad127\u201334, 169; advertising campaign for, 196; teachers as, 180\u201381\n\nmeritocracy, 149\n\nmerit pay, 113\n\nMersereau, Bob, 132\u201333\n\nMet School (Providence, RI), 126\u201327\n\nmiddle school, 139\u201341; middle school students, 74\u201375\n\nMiran, Seth, 184\n\nMitchell, Hetty, 188\n\nmoderate-income students, 4\n\nMontessori, Maria, 19\n\nMontessori schools, 19\n\nMontgomery, Tulaine, 54\u201357, 60\u201361\n\nMorgan, Dan, 28\n\nmothers, shared experiences of, 50\u201351. _See also_ parents\n\nMurnane, Richard, 12, 153\u201354\n\nmusic, 63, 139\u201340\n\nNada, Sherif, 77\n\nNational Academies Foundation (NAF), 126\n\n_A Nation at Risk_ (report), 61\u201362\n\nnature vs. nurture debate, 147\u201348\n\nNelson, Lisa, 184\n\nNew Profit Inc., 106\n\nNew York Maker Faire, 161\u201363. _See also_ maker movement\n\nNo Child Left Behind Act, 198\n\nnonacademic skills, 175\n\nnonprofit organizations: building blocks of, 46; funding and growth of, 44\u201345, 108\u20139, 112, 114\u201317; Obama election and, 111; scale and spread of, 105, 109\u201310\n\nObama, Barack, 110\u201313, 158, 159\n\nObama, Michelle, 111\n\n100Kin10, 157\n\nonline learning, 184\n\nopportunity: building blocks of, 10\u201311; employment, 39; equal, x, xi, 164; gap, wealth-based, 3\u20135; steps to improve equal, 194\u2013200; for upper-income students, 12\u201313, 14, 136\u201339; wealth and, 31\n\nopportunity equation, 3, 12\u201313, 33, 59\n\noptimism vs. skepticism in nonprofit organizations, 80\u201381\n\nOrchard Gardens school (Boston), 177\u201378\n\norganizational reform, 106\n\nout-of-school experiences, 3, 5, 12\u201314, 20\u201323, 39, 42. _See also_ enrichment experiences; extracurricular activities\n\nparents, 188; concerns of, 50\u201351; as models of success, 14, 23, 27\u201329; relationships between teachers and, 186\u201387; support for, 10, 185\u201387; support of, 22\u201323, 27, 32, 118; time spent with, 13, 166\n\nPerry Preschool Project, 107, 108\n\nperseverance, 157\n\nphilanthropy, 71\u201372, 112, 197\u201398\n\nPicciolini, Britton, 132\n\nPinsky, Biz, 47\u201348\n\nPinsky, Robert, 48\n\npolitical reform, 106, 110\n\npoor children. _See_ low-income students\n\npoverty, 14, 117; escaping from, 31\u201333; increase in, 106; and consequences of making mistakes, 32\u201333\n\nPower Learning Out of School apprenticeship, 156\u201357\n\npractice time, 147\u201349\n\nPrice, Anita, 46, 47, 55, 80\n\nprivate schools, 19\u201323\n\nproblem-solving skills, 151, 156\n\nprofessionalization, 80\n\nProskauer, Tim, 48\n\npublic schools, 117, 189; criticism of, 4, 176; funding of, 113; private donations to, 197\u201398, 199. _See also_ schools\n\nPutnam, Robert, 164\u201365, 167\u201368\n\nrace-based achievement gap, xi, xiii\n\nRace to the Top initiative, 112\u201314, 198\n\nracial equality, xi\u2013xii\n\nreal-world experiences, 127\u201328, 158, 180. _See also_ apprenticeships; hands-on projects\n\nrecruitment, 89\n\nReich, Rob, 197\n\nReimagining After-School symposium series, 78, 155\u201357\n\nReServe, 130\n\nResnick, Lauren, 23\n\nRetired and Senior Volunteer Program, 130\n\nretired volunteers, 129\u201331\n\nReville, Paul, 5, 87\n\nRiley, Jeff, 85, 93\n\nRimer, Ned, 46, 53, 55\u201356, 69, 171, 172, 173\n\nRobin Hood financing scheme, 62\n\nRobinson, Steve, 158\u201359\n\nRodgers, Charles, 75\n\nRodgers, Fran, 75\n\nRodriguez, Anthony, 161\u201362\n\nrole models, 27\u201331, 51, 127\u201328, 152, 169\n\nRomaine, McCalvin, 150\u201351\n\nRorie, Denis, 48, 186\n\nRothschild, Katie Brown, 179\u201380\n\nSabin, Mike, 85\u201386, 87, 88, 92, 93\n\nSaintil, Emmanuel, 90\n\nscale, of nonprofits, 105, 108\u201310, 114, 194\n\nSchachter, Rachel, 132\n\nschool day: in Boston, \u201384; ELT grant, 87; length of, 10, 42, 74, 78, 104, 118, 135\u201349, 151; and Massachusetts 2020, 77, 86; typical schedule for, 139\u201341. _See also_ learning time\n\nSchool Improvement Grants, 198\n\nschool-reform movement, 61\u201364\n\nschools: accountability for, 86; changing climate of, 62\u201363; charter, 62, 95, 113, 143, 177, 185; disengagement from, 21; funding of, 62, 113; leadership opportunities in, 181\u201382; private, 19\u201323; private donations to, 197\u201398, 199; role of, in achievement gap, 14\u201315\n\nschools of education, 62\n\nschool year, length of, 42\n\nSchwarz, Frederick, 29\n\nSchwarz, Fritz (author's grandfather), 27, 29, 30\n\nSchwarz, Henry (author's great-grandfather), 30\n\nscience in schools, 11, 63, 141, 143, 151\u201352, 157\n\nscience, technology, engineering, and math (STEM) education, 113, 151\u201352, 157\u201359, 198\n\nscientific thinking, 151\n\nSelby, Moriska, 90, 97\n\nsenior citizen volunteers, 129\u201331\n\nServe America Act, 112, 196\n\nShaheen, Jeanne, 26\n\nShea, Barbara, 15\n\nShearer, Earline, 11\n\nSheckel, Sara, 178\u201379, 182\n\nSherman, Robert, 56\u201357\n\nShore, Bill, 26\n\nSinger, Henry, 41\n\nskepticism vs. optimism in nonprofit organizations, 80\u201381\n\nSkloot, Ed, 56\n\nSkoll, Jeff, 111\n\nSkoll World Forum, 111\n\nSmalt, Lindy, 190\u201392\n\nsocial capital, 125, 139, 164\u201375; effect on world economy, 166\u201367\n\nsocial change, 106\u201310\n\nsocial entrepreneurs, 45, 106, 111, 199\u2013200\n\nSocial Innovation Fund, 112\n\nsocial media, 165\n\nsocial mobility, xi, 31\u201332, 33\n\nsocial networks, 10, 102, 118, 139, 165\u201366, 173\u201375, 195\n\nsocial skills, 4, 10, 102, 118, 139, 165, 167, 195\n\nsocial studies, 63, 141, 143\n\nSoros, George, 74\n\nspecial-education students, 143\u201346\n\nspecial-education teachers, 183\n\nsports, 139\u201340, 166\n\nspread, of nonprofits, 105, 108\u20139\n\nstaffing model, 67, 75\n\n_Stand and Deliver_ (film), 174\u201375\n\nstandardized testing, 62\u201364, 151, 181, 188\n\nSTEM (science, technology, engineering, and math) education, 113, 151\u201352, 157\u201359, 198; professionals, 157\u201358, 159, 198\n\nStockdale, James, 81; Stockdale Paradox, 81\n\nStrong, Lester, 129\u201331\n\nstudent engagement, 101, 102\n\nStudent Leadership Council, 90\u201391\n\nstudent motivation, 180\n\nstudent outcomes, 68\u201369, 101\u20132\n\nStudents for Hart, 26\n\nstudy hall, 140\n\nSturz, Herb, 74\n\nSu, Alan, 10\u201311, 97\u201398, 133, 159\n\nsuburban teachers, 177\n\nsuccess: building blocks of, 117\u201318; factors in, 147\u201349; models of, 14, 23, 27\u201331, 43\n\nSullivan, Kerry, 115\n\nsummer camps, xii, 12, 22\u201323, 39, 41\u201342\n\nsummer programs, 67\n\nsupplemental learning, 136\u201339\n\nSurdna Foundation, 56\n\nTai, Robert, 151\n\nTamarack Tennis Camp, 22, 41\n\nTASC (The After-School Corporation), 73\u201375, 142\u201343\n\ntax breaks, 197\n\nTaylor, Jeff, 156\n\nteacher conferences, 21\n\nteachers, 189; attitudes of, toward Citizen Schools, 176, 187\u201388; blame of, 117; buy-in from, 88; collaboration between, 179; evaluation of, 113; junior, 185; levels of, 183\u201384; licensure standards for, 62; as mentors, 180\u201381; merit pay for, 113; number of, 182\u201383; pathways to leadership for, 181\u201382; quality of, 14, 136; recruitment of, 89, 183; relationships between parents and, 186\u201387; role of, \u00ad182\u201385; science and math, 157, 158\u201359; special-education, 183; suburban, 177; support for, 10, 176\u201385; supportive, 37, 118; turnover rates, 177; work day for, 142. _See also_ citizen teachers\n\nteachers' unions, 117, 176\n\nTeach For America, 67, 105, 115\n\n_Teaching the New Basic Skills_ (Murnane and Levy), 153\u201354\n\nteamwork, 152, 153, 157\n\nTechBoston High School, 170\n\ntechnology skills, 153\n\ntelevision, 185\n\ntest scores, 62, 92, 94, 113, 177, 181\n\nTherapeutic Learning Community (TLC), 144\n\nThomas, Joyce King, 120\u201323\n\nTime for Innovation Matters in Education (TIME) Act, 93, 196\u201397\n\nTimilty School (Roxbury, MA), 60, 61\n\nTitle I funding, 198\n\nTkacik, Margie, 7, 43, 189\n\nTocqueville, Alexis de, 125\n\ntraining, 67\u201368, 89\n\nTucker, Jake, 48\n\ntutors\/tutoring, 137, 138, 143, 166\n\n21st Century Community Learning Centers program, 198\n\ntwenty-first-century skills, 152\u201357\n\nUnited States: class divide in, xiii, 4\u20135, 12\u201315, 165; college graduation rate in, xi, 11\u201312, 125, 172, 188; decline of creativity in, 155; equal opportunity in, x\u2013xi; social mobility in, 31\u201332, 33\n\nUniversity of Vermont, 170\u201372\n\nupper-income students: advantages of, 3\u20135, 12\u201315, 136\u201337, 165\u201366, 195; childhood experiences for, 18\u201323, 27\u201331, 165\u201366; college completion rate of, 12; expectations of, 14; opportunities for, 12\u201313, 14, 136\u201339\n\nUpward Bound, 170\n\nurban schools, 82, 86\u201387, 93, 94\u201395\n\nUS2020, 158, 161, 199\n\nvideo games, 185\n\nviolence, 89\u201390; in schools, 82\u201383, 86\n\nvision, 80, 147\n\nVISTA, 199\n\nvolunteers\/volunteering, 11, 46, 48, 60, 64, 97\u2013101, 120\u201323, 196, 199; benefits for, 131\u201334; impact of, 127\u201334; senior citizens, 129\u201331; training of, 67\u201368\n\n_Wandenchaft_ (wanderlust) period, importance for adolescents, 24\n\n_Webby v. Dukakis_ , 62\n\nWeikart, David, 107\n\nWeingarten, Randi, 93, 176\n\nWerner, John, 46, 47, 55, 60\n\nWest Side Montessori School (New York City), 19\n\nWFD, 75\u201377\n\nWGBH-TV, 114\n\nWiatrowski, Lynn, 115\n\nWilliam H. Lincoln School (Brookline, MA), 14\u201315, 136\n\nWolfe, Tom, 168\n\nWoodrow Wilson School (Dorchester, MA), 58\u201359\n\nworkers, educated, 12\n\nwork experiences, 24\u201327, 37\u201339\n\nWork\/Family Directions (WFD), 75\u201377\n\nwork skills, 153\u201355, 166\u201367\n\nWOW! events, 1\u20133, 49\u201352, 54, 61, 126\n\nWozniak, Steve, 159\n\nYarde, Jadine, 173\u201374\n\nYear-Up, 105\n\nZigler, Edward, 107\nBeacon Press  \nBoston, Massachusetts  \nwww.beacon.org\n\nBeacon Press books  \nare published under the auspices of  \nthe Unitarian Universalist Association of Congregations.\n\n\u00a9 2014 by Eric Schwarz  \nAll rights reserved  \nPrinted in the United States of America\n\n17 16 15 14 8 7 6 5 4 3 2 1\n\nThis book is printed on acid-free paper that meets the uncoated paper   \nANSI\/NISO specifications for permanence as revised in 1992.\n\nText design and composition by Kim Arney\n\n_Library of Congress Cataloging-in-Publication Data_\n\nSchwarz, Eric  \nThe opportunity equation : how citizen teachers are combating   \nthe achievement gap in America's schools \/ Eric Schwarz.  \npages cm  \nIncludes bibliographical references and index.  \nISBN 978-0-8070-3372-2 (hardback)  \nISBN 978-0-8070-3373-9 (ebook)  \n1. Community and school\u2014United States.   \n2. Educational change\u2014United States\u2014Citizen participation.   \n3. Educational equalization\u2014United States.   \n4. Community power\u2014United States.   \n5. Social action\u2014United States. I. Title.  \nLC221.S376 2014  \n370'973\u2014dc23  \n2014009331\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n# Hadrian's Lover\n\n_Patricia Marie Budd_\n\n_Dedications_\n\nThis book is dedicated to all my LGBT students past, present, and future. I am here for you. I always have been and always will be.\n\nJoseph, I love you! You will always be my Grecian poet.  \nWilf, you inspired me! You are not forgotten!\n\nOctober 6 \u2013 12, 1998\n\nThe world must NEVER FORGET what happened to Matthew Wayne Shepard!\n\n# Reviews\n\nEditor's Choice for September, 2013, _So So Gay_ , London, UK\n\nFive Star Review \"Beautiful, heart-wrenchingly poignant, and brilliantly conceived\" Jake Basford, _So So Gay_ , London, UK\n\n\" _Hadrian's Lover_ inspires internal debate and thought. As a text in a high school classroom it would do wonders and possibly change the world.\" Drew Rowsome, _My Gay Toronto_\n\n# Acknowledgments\n\nMany thanks go to:\n\nChristine Marie Scott, Bartley P. Busse, and James Duncan: my initial editing team! You helped me shape my vision!\n\nMonty Henstridge for providing legal advice for all court scenes.\n\nBrendan Toner for providing military intelligence.\n\nJoanne Skilnick for helping me better understand our country's natural resources, global warming, and the future ecological health of our planet.\n\nChristine Marie Scott, Bartely P. Busse, Kimerica Parr, and Tara Nahamko for their endorsement of my work.\n\nTyler Tichelaar for his professional editing services acquired through ReaderViews.\n\n# WARNING!\n\nThis is a novel about sexual awakenings and sexual discrimination. You will be reading some sexually explicit material. Two thirteen year old boys begin sexual experimentation; a fifteen year old boy is caught masturbating; a sixteen year old boy wants his boyfriend to have sex; a seventeen year old boy and girl have sex; and a married male couple have sex on more than one occasion. If this warning offends you put the book back on the shelf, do not buy it, do not read it. The odds are you aren't ready to handle the truth. And yet, ironically enough, you are the reason why this book had to be written!\n\n-pmb-\n\n# Contents\n\n  1. Title Page\n  2. Dedication\n  3. Reviews\n  4. Acknowledgments \n  5. Warning!\n  6. Prologue: Crystal's Report\n  7. New Year's Eve\n  8. A New Year's Kiss\n  9. Illicit Communication\n  10. Images of Crystal\n  11. Sweet Sixteen!\n  12. Spring Fever\n  13. Good News!\n  14. Apocalyptic Nightmare\n  15. Dating Devon\n  16. Rezu\n  17. Hadrian's Zoo\n  18. Summer Craze\n  19. Back to School\n  20. Second Anniversary\n  21. Study Date\n  22. Enter the Vixen\n  23. T'Neal\n  24. Hiding\n  25. B-ball\n  26. Family Dinner\n  27. Antinous Wants You!\n  28. Exposure\n  29. Taming the Strai\n  30. The Principal's Office\n  31. Interview with Salve!\n  32. Het-Row\n  33. Warith vs. Weller\n  34. Indoctrination\n  35. Gideon's Obsession\n  36. Quelling Dissension\n  37. Private Session\n  38. \"to die, to sleep no more\"\n  39. Extracting a Confession\n  40. Todd Awakes\n  41. Heart to Heart\n  42. A Controversial Lesson\n  43. Making Amends\n  44. A Private Meeting\n  45. Transcripts: Hadrian vs. Hunter\n  46. Defense Questioning of Geneticist Avery Gillis\n  47. Prosecution questioning of Gideon Weller\n  48. Prosecution Questioning of Dean Stuttgart \n  49. Defense Questioning of Ms. Destiny Stuttgart\n  50. Defense Questioning of Darrell Jeffreys, Warden's Personal Assistant \n  51. An Unprecedented Sentence\n  52. Love Recovers\n  53. Hadrian's Wall\n  54. Bibliography\n  55. Copyright\n\n# Prologue: Crystal's Report\n\n**A Treatise Validating Hadrian's Sexual Preference and Method of Procreation  \nby Crystal Albright  \nsubmitted to: Ms. Sterne**\n\nHomosexuality is the sexual norm in Hadrian while all forms of heterosexual behavior are illegal. Hadrian's citizens have chosen a homosexual lifestyle side-by-side with in vitro fertilization, to create a stable human population within the confines of our borders. It had been determined by the founding families\u2014and is approved every decade, on our country's day of birth by our citizens through a referendum\u2014that Hadrian's population shall never exceed ten million.\n\nOur small country was founded three generations ago. My great-grandmother, Ester Stiles, was among the first families to buy land in the Hudson Bay region of Northern Manitoba with the express purpose of creating a safe and stable home for homosexuals as well as a rigidly monitored and controlled human population. For this luxurious lifestyle, we have the five founding families to thank: Stiles, Stuttgart, Birtwistle, Nasser, and Reznikoff. It was they who, in the early years of Canada's decline, purchased the land of our good country. It was they who established the cornerstones of Hadrian's constitution: that Hadrian's chosen lifestyle is homosexual; that Hadrian will be a safe haven for homosexuals from around the globe; that Hadrian's central focus is the creation and maintenance of a stable human population; that Hadrian will create an ecologically sound balance between humanity and nature. The key focuses of this treatise are the cornerstones reflecting Hadrian's sexual preference and the stabilization of the human population without which humanity will most certainly decline as surely as the Roman Empire did.\n\nAs late as the twentieth century, it was reasonable, perhaps even essential, that humans should prefer heterosexuality as the sexual norm. Heterosexual sex is designed specifically for procreation. Humanity no longer requires such methods, yet the multitudes outside Hadrian's walls are still reproducing at an astronomical rate. Outside Hadrian's walls, close to twenty billion people are overpopulating this planet\u2014nearly twenty billion people are starving and suffering from uncountable syndromes as a result. Millions of these people resorting to cannibalism just to survive! ( _Salve!_ , HNN\u2014Melissa Eagleton Reporting, June 2, 21\u2014). These poor unfortunates are forced to live in close quarters spreading disease at a rampant rate. Humanity has overrun the planet like a virus killing all of nature in its path. What food is available to these people is scarce. What medicines limited. Little space lies beyond our walls for farms to grow food and raise livestock. Hadrian, on the other hand, is self-sustaining. We have farms. We grow our own food. We raise fowl. We maintain freshwater fisheries. We house seawater fishery farms. Our people are fed. Our surplus stock accounts for 14 percent of Hadrian's economic wealth as we export what we do not need to those in dire need. We have, in every sense of the word, created a utopia here in Hadrian.\n\nSometimes our citizens question why, if we are so adamant against heterosexuality, do we not simply allow these people to immigrate to other countries where their lifestyle is not only acceptable but also preferable? The answer to that question should be obvious. These are our children. We do not simply dismiss them\u2014toss them into a torrent ocean of human disease, debris, and decay. Our love for our offspring does not permit us to act in so selfish and irresponsible a manner. Besides, no one born in Hadrian is a pure heterosexual.\n\nOne of Hadrian's greatest advances, truly a leap in medical science, has been its work on the human genome project. In the past fifty years, our genetic scientists believe they have identified those chromosomes within our DNA that indicate a strong disposition toward homosexuality. Although the human animal is by nature a creature that needs to procreate, nature has also established its own checks and balances to overpopulation by ensuring some aspects of human DNA lean toward homosexual behavior. Hadrian's scientists, who have worked to increase that component of our DNA, are believed to have successfully eradicated the pure heterosexual gene from Hadrian's genetic database. Our geneticists scan each and every embryo to determine whether the homosexual gene exists before implanting it into a woman's womb. If no signs of homosexuality are present in the embryo's DNA, it is automatically destroyed. Thus, our geneticists ensure that every citizen born in Hadrian is homosexual (Doctoro, _Hadrian's Human Genome Project_ , page 882).\n\nHere in Hadrian, we ascribe to the Kinsey scale. According to Alfred C. Kinsey ( _Sexual Behavior in the Human Male_ , page 638) human sexual preference can be divided into a scale from zero to six. A zero on this scale is said to be 100 percent heterosexual whereas a six is considered pure homosexual. The gradations between show individuals leaning more or less toward either side. Societies outside Hadrian's walls have always elected to accept the leaning toward heterosexual behavior purely for the means of reproduction. Religion has both fostered and festered the \"procreation fact\" as the foundation for heterosexuality as the pure and natural sexual norm. Yet this choice has not served humanity well. Our small planet simply cannot sustain the burden of human population placed upon it. Our founding families recognized the futility of heterosexual preference and excessive procreation. Thus, upon foundation of our good country, the first article of justice states the need to enhance and nurture human homosexual tendencies as well as create strict procreation laws.\n\nWith modern medical advances, our geneticists boast having reduced the Kinsey scale by two points (Billington, _The Homosexual Gene_ , page ). Although our geneticists are, as yet, unable to distinguish between each of the gradations of Kinsey's behavioral scale, they are confident enough to pronounce that no more zeros or ones exist among any of Hadrian's citizens. Every child born into Hadrian society in the last twelve years is, at the very least, a two on the Kinsey scale. We do not deny that a two still has a stronger leaning toward heterosexual behavior; however, inside every two is a latent homosexual. It is our job as citizens of Hadrian to help these individuals release and embrace their true, and socially acceptable, inner desires. Until the day when our scientists have created the perfect human genome, where all who are born will be sixes, we must forever work with our children, reminding them why our country was founded and the critical need to maintain a stable human population.\n\nFor this reason, reeducation camps were formed: Not to harm and abuse our citizens but to remind our youth of our country's founding families' ideals and the necessity of maintaining strict vigilance. Many of today's youth are unaware of our early immigration policy. In the beginning, Hadrian's borders were open to immigrants, inviting homosexuals from around the world to join us. Many people swarmed to our gates, and not all of these people were homosexual. In fact, a large number, at least 15 percent, were people who believed themselves to be a one or two on the Kinsey scale. Two individuals who joined us, Mark and Julie Reiner, a pure heterosexual couple, agreed to sterilization in order to be part of our preferred community. Neither had wanted children anyway, viewing procreation in an overpopulated world as immoral. They also agreed to permanent separation, each locating to opposite ends of the country. They knew when they immigrated to Hadrian that they were entering a homosexual community, a community dedicated to restraining the human population growth, a country dedicated to putting into place the necessary checks and balances to create a stable human population. These noble people chose to deny their heterosexual tendencies, accepting celibacy in its place, for the sake of Hadrian's future, for the sake of humanity. All of Hadrian's citizens are asked to make such sacrifices, if indeed, one perceives being who we are as a sacrifice at all!\n\nPerhaps the greatest sacrifice made by any Hadrian citizen is that of the surrogate goddess. These women open up their wombs, choosing to be the mothers of our men's children. Without these women, our homosexual men would be bereft of the benefits of rearing many of Hadrian's children. No wonder these women are treated like goddesses in our world. In fact, it is the expressed desire of this author to become a revered surrogate goddess.\n\nIt is also important to remind our youth that Hadrian's first citizens swore an oath of fidelity to our founding principles. Although our children are not required to swear this oath until coming of age (twenty-one), it is expected of all Hadrian's citizens to raise their children with a clear understanding of not only the world inside our walls, but of the dangerous world that surrounds us. That dangerous world batters against our walls daily as a reminder. Heterosexual barbarians are constantly threatening to overtake our country. Hadrian's ballistic missile detection and tracking system is state of the art. It is terrifying to learn how many incoming missiles our military has protected us from. A virtual wasteland lies outside our walls where many of our intercept missiles have detonated those threats (Dodger, _The Military Sciences_ , page ). Although conscription of our youth into Hadrian's military between eighteen and twenty-two years of age is undesirable, it is undeniably essential; we must defend Hadrian's way of life at all costs.\n\nThe worst attack against our peoples occurred a mere eight years ago when an Evangelical Christian drove across our border, exploding a dirty nuclear device in Augustus City. The loss of Quadrant One's southern grassland region and the devastation to our populace was the final straw that drove our people into exile from the outside world. No more are tourists from other countries allowed to visit our fair and spacious country. Although we retain fair trade with friendly countries through export and import, we no longer believe outsiders can be trusted to cross over our borders (Miller, _The Resurgence of the Law_ , page ).\n\nHadrian's population must remain xenophobic. We cannot allow the ravages of disease to infect the only stable human population. Annually, hundreds of millions of outsiders die of disease and starvation; yet those numbers never seem to impact the ever-growing population. We, in Hadrian, will never send our children to live out there with them. No, it is much kinder, much safer to our wayward youth to be reclaimed through reeducation. Although some of our youth may believe themselves to be heterosexual, it is incumbent upon us to help them understand what it means to be a two on the Kinsey scale and help them awaken and embrace their latent homosexuality.\n\nHadrian is the savior of humanity. While the rest of the world procreates itself into excessive human waste and decay, our small population will live on. Hadrian's values ensure the human race will survive.\n\n#  _Salve!_\n\n**December 31, 21___\u201411:53 PM  \nHadrian's National News (HNN)\u2014Melissa Eagleton Reporting**\n\nAs the clock counts down, Hadrian's citizens await the very moment we can all shout gaily, \"Happy New Year and Happy Fiftieth Birthday, Hadrian!\" It is hard to believe that we have made it this far! Fifty years of gay freedom! Fifty years of a stable human population\u2014maintaining ten million! Fifty years of showing the rest of the world how man and nature can commune as one. Considering all the trials and tribulations Hadrian has been through since we first established our borders, it is truly amazing we have come this far. The wall surrounding our borders is now one third complete, stretching for hundreds of miles on each side of the southeast and southwest gates.\n\nThis is it, folks; in less than seven minutes, when the chimes ring in the New Year, we will all shout out with joy, for Hadrian is the only country left standing between humanity and certain death\u2014for without Hadrian, the human race would invariably become extinct! As we look outside our walls and daily witness man's decay, although we are saddened by their sufferings and daily losses, their decline is a reminder of the value and importance of Hadrian's society. Overpopulation is the planet killer. Overpopulation is man's deadliest enemy. Homosexuality, population stability, communing with the earth (that small portion of which we have saved within the confines of our walls): these are Hadrian's gifts to the future. Our scientists estimate that the dregs of the human race will wipe themselves out in the next fifty to one hundred years. But Hadrian will remain! Hadrian will fight for humanity's survival.\n\nBefore the clock strikes that magic hour, let us take a moment to remember Hadrian's soldiers, those brave men and women who patrol our borders and guard our wall from the outside world, and the endless attacks against our civilization by the heterosexual barbarians and religious fanatics Every year, we mourn the loss of another soldier dying to preserve our rights and freedoms. Every year, a grieving mother or father buries her or his only child. No one goes to the wall with illusions. To serve at the wall is to face certain death on a daily basis. To serve at the wall is to be in a constant state of readiness to repel desperate attacks by illegal immigrants threatening to swarm over our borders, threatening to infect us with any one of their endless plagues. To serve at the wall means to battle with organized armed forces determined to steal our land and destroy what little remains of the earth's habitable land. To serve at the wall means to face the threat of religious terrorists who deem our chosen lifestyle as the devil's menace. Our young men and women conscripted into our forces, who serve four years of their lives between the ages of eighteen and twenty-two, are our protectors and saviors from the outside world. Before the countdown begins, let us hold our traditional vigilance\u2014one minute of silence for those of us who cannot enjoy this night's revels\u2014one minute of silence for those of us who may, at any moment, die while defending our very way of life and the right to celebrate the way we do tonight. One minute of silence and then one minute of bated anticipation to the final countdown. And then together, we will shout for joy our Happiest of New Years, for this year as we celebrate the fiftieth year of our country's birth, we are also celebrating the rebirth of mankind!\n\n_Vale!_\n\n# New Year's Eve\n\nFacing south, with a convex dome encompassing east and west as well as the entire roof, the Hunter home is one of the most impressive dwellings in Antinous. The northern wall contains a glass bay window opening up to the view of a multi-tiered garden sloping down to the Nelson River. Very few buildings have open windows facing north in this manner. All windows are on the convex of every Hadrian building, designed to double as solar panels, obscuring the interior from the outside, but allowing those inside a beautiful tinted view of all that lies beyond. This design has also eliminated the need for window dressings. To have clear glass windows exposed to the outside denotes either wealth in a family home or a government complex.\n\nThe Hunters' backyard is a sprawling garden that slopes down by tiers to the river. Right now, it is lightly dusted white from last evening's snowfall. Come spring, Dean will begin the process of manual tilling, mixing in the mulch from last year's compost, and seeding, weeding, and nurturing the garden. Gardening has been Dean's job for the past two decades. More than just time-consuming, it acts as a relaxant, something Edgar, Dean's psychiatrist, had recommended years ago. It also provides the Hunter family with income tax refunds on an annual basis. Every Hadrian citizen who makes the effort to work with, and not against, the earth is granted hefty tax rebates.\n\nAs well as showing pride in his garden, Dean is equally proud of the interior of his and Geoffrey's house. There is little one can do with respect to the exterior. All homes in Hadrian look the same with a convex facing west, south, and east for solar panels and one flat wall facing north. There are no front doors, only a back door facing north. Only size and interior luxuries denote wealth. When you enter the Hunter home, the first thing you see is the wide expanse of the living room to your left and the glistening creams and beiges of the kitchen directly in front. To the right is the long hallway that leads to the central washroom, and four bedrooms, one being the master bedroom. Standing in the lobby closest to the hall entrance, Dean Hunter takes in the full view of the living room. Tall and lean, Dean is standing rigid as he stares at the room before him. Dean is anxious. Geoffrey is never any help when it comes to organizing a party. When asked, he will help, but with limited effectiveness. Sweeping a floor in Geoffrey's mind means sweeping around things. It never occurs to him to move a couch or a chair, to get at the dirt underneath. \"Nobody is going to look under the chairs, Dean. I swear.\" He adds, \"Sometimes I think you are more gay than I am!\"\n\nDean finds this old joke annoying, but he knows Geoffrey means kindly by it, so he always shakes it off. Staying focused, Dean pipes up, \"Tonight's special, Geoffrey.\" Although not needing to, he reminds him, \"This is not just any New Year we're celebrating! It's also Hadrian's fiftieth birthday!\"\n\nGeoffrey smiles fondly at his old lover. _No, not old,_ he reminds himself. _Dean is only forty years old._ Not that there is a huge gap in their ages; Geoffrey is only forty-eight. No, he refers to Dean as his old lover because they have been together for over twenty-two years now (not including couple's registration). So many changes have occurred in Dean over the years\u2014from the frightened teenager Geoffrey met to the strong confident man standing before him. Smiling, Geoffrey still appreciates that Dean took his last name. Although he knows it wasn't so much a choice on Dean's part as it had been a necessity, at times like today, Geoffrey is able to convince himself that this is how Dean wanted it regardless. And Dean, he muses, is still so handsome. Geoffrey enjoys checking out his lover when Dean is unaware. Knowing Dean is self-conscious, Geoffrey doesn't try to make Dean uncomfortable. Still, when opportunities like this arise, with Dean fixated on a problem and focused, Geoffrey will often stand back and admire his physique. Dean is tall and slim, no middle-age bulge. Unconsciously, Geoffrey gives his own protruding belly a slight shake. Geoffrey is short and stout next to Dean. And, unlike Dean's thick dark brown hair, Geoffrey's mousy brown curls have all but receded to a horseshoe patch around his head. _Hadrian's Lover,_ Geoffrey muses. _Dean is still so good looking_. Even the gray streaks that lace his temples are attractive. With his square jaw and thick eyebrows, he is the classic Marlboro man. No one smokes in Hadrian, but the old image hangs in the city's central museum. When they first saw the original advertisement poster, Geoffrey teased Dean, saying, \"All you need now is a cowboy hat.\" Geoffrey was so taken by the resemblance that he surprised Dean with a week's stay at The Cattle Ranch horseback riding that year. This, of course, was before their sons, Frank and Roger, Geoffrey's genetic sons, were born.\n\nGeoffrey's musings are cut short by a sudden caustic remark, \"I hate that wallpaper.\" That wallpaper is the one point that takes Dean's pride down a notch when it comes to his home's interior! He despises it wholeheartedly. It doesn't happen often, but every so often, Dean will utter a complaint about their living room d\u00e9cor. Although Dean was responsible for most of their home design, Geoffrey was given free rein with the living room. \"Everyone is entitled to one room,\" Geoffrey had insisted. Dean still regrets giving in to that logic. Besides, Geoffrey already had his study!\n\n\"Well, you should have said so fifteen years ago when I picked it out,\" Geoffrey replies. \"It's too late now.\"\n\n\"I wanted to say something, but we had agreed you got to decorate the living room.\" They have been having this old argument for five years now. The use of wallpaper was still allowed fifteen years ago with the mandatory restriction that it not be replaced for a minimum of ten years. Its use became banned two years ago, three years after they could have replaced it! Geoffrey refused then, and he still refuses now, to change the d\u00e9cor. Wood walls maintained by oil or natural stone are the limited means of d\u00e9cor today. The less one adds to the world's pollution, the greater the tax break. Wallpaper and most paint are now illegal.\n\nSeeing defiance begin to glare in Dean's eye, Geoffrey turns authoritative. \"We are not adding to global waste by tearing off perfectly good wallpaper. We've kept it in good condition and it stays up as long as it lasts.\"\n\nScowling, Dean says, \"But we've kept it long past the ten-year minimum. Can't we just get rid of it\u2014it's so dark!\" The wallpaper is black with silver, laced patterns.\n\nShaking his head, Geoffrey has bragged to too many coworkers about having the same wallpaper up for fifteen years, adding how it will last at least another five. With wallpaper now being illegal, owning one of the few homes with the material still on its walls is a status symbol Geoffrey is not willing to give up. \"The silver pattern brightens it up.\"\n\n\"Only because the paper is so faded!\"\n\n\"It's not faded.\" Geoffrey is determined to defend his choice to the death.\n\n\"It's faded.\"\n\n\"It's staying up!\"\n\nSulking, Dean repeats, \"I have really grown to hate it.\"\n\nUnrelenting, Geoffrey replies, \"Well, grow to liking it again because we are not changing it again for at least five years!\" Wagging a pointed finger toward Dean, he adds, \"And only then if it needs replacing,\" and he cites a government slogan to end the argument, \"Earth First!\"\n\n\"Fine!\" Dean huffs. Glancing down, he glares Geoffrey's way. \"Hand me the broom, please. I have to sweep up in here.\"\n\nGrimacing, Geoffrey bites back, _I just finished sweeping up in there,_ but he knows better. Nothing is ever clean enough for Dean when company is expected. Rolling his eyes, he retrieves the broom and hands it to Dean.\n\nBefore Geoffrey can execute an escape down the hall to their room, Dean turns to scold him. \"Geoffrey, I need your help here.\" Reluctantly, Geoffrey crosses over and helps pull the couch out so Dean can sweep up the dust and excess dirt.\n\nAppreciating the profile of Dean's form as he leans forward to capture more dirt with the broom, Geoffrey wishes the man would let up with all this cleaning. After Dean finishes his task, Geoffrey leans in close, whispering hotly in his ear, \"Anything else you need me to do?\" Today's observations have reminded Geoffrey how Dean has matured into a very sexy man, _like fine wine_ , he muses. \"Our guests won't arrive for another two hours.\" _And_ , he whines inwardly, _if you would just let up on all this cleaning, we could relax and enjoy ourselves in the shower._\n\nThough no prude anymore, Dean has his mind on more practical matters. His eyes continue to scan the room, landing on the coffee table\u2014it is cluttered. The coffee table\u2014cheese tray! Turning, he orders Geoffrey, \"Get the soya cheese out and start cutting slices.\" Remembering the way Geoffrey put the last cheese tray together, he adds sternly, \"Arrange it nicely this time. Use a little parsley as garnish, and put a small bowl of sweet pickles in the middle.\" Suddenly, remembering an important detail, he concludes, \"and set out the crackers I baked\u2014arrange them nicely, too!\"\n\n\"I don't see why you get me to do these things. I can never meet the standards you set.\"\n\n\"If I had time,\" Dean answers crisply, \"I would do it myself. But I don't.\" Hands sawing the air frantically, he admits, \"I need you to help.\"\n\nLaughing, Geoffrey bows, \"Your servant ever.\"\n\nExasperated enough to roll his eyes, Dean sighs, \"We only have two hours before the first guests arrive.\" He points dramatically now toward the kitchen.\n\n\"I'm going.\" Geoffrey's hands go up in defense and he flashes his _please don't shoot me_ grin. \"Soya cheese tray. Crackers. Pickles. At your will!\"\n\nAfter Geoffrey turns the hall corner, Dean shouts out a reminder, \"Don't forget the real cheese. I bought some cheddar and some Brie. Mike loves Brie.\"\n\nCalling from the kitchen, Geoffrey says, \"I don't know why you expect him every year. Mike Fulton never comes.\"\n\n\"He promised me he'd come with Todd this year.\"\n\n\"He never comes!\" Geoffrey reminds him gently, but sternly.\n\n\"I messaged him through his wave link at work. He replied he was coming!\" Dean refuses to see Mike Fulton for what he has become, choosing instead to remember the man he was before his husband passed on.\n\n\"Fine.\" Geoffrey is no longer trying to mask his exasperation. \"He'll come. I'll put out the Brie!\"\n\n\"He promised me!\" Dean insists. \"He promised both Todd and me!\"\n\nAlthough Geoffrey had not meant to be heard, his low growl still makes its way from the kitchen into the living area. \"He's a good one for making promises.\"\n\nAs much as Dean would like to defend Mike right now, deep down he knows Geoffrey is right. Mike Fulton is not the same man who married Dean's best friend twenty-one years ago. That man died the same day his lover, Will Middleton, passed on. Dean and Will had been best friends since high school. Everyone, in those days, had expected Dean and Will to partner for life. But events happened, things changed, and then Dean met and married Geoffrey Hunter. Shortly after, Will Middleton met Mike Fulton in his first year at Antinous Uni; by the end of that year, the two men had married. Dean had always thought his friend had rushed things, but he understood why Will had chosen to marry so fast. Mike had proven to be the best of all partners for Will, and an amazing Papa to Todd.\n\nOnly in the last few years has the man changed, leaving Dean to worry about Todd. Mike Fulton is a grown man who can work things out for himself, but at fifteen, Todd is still too young and impressionable. He needs a father's guidance to help him through the confusing years. Dean has done his best to offer Todd help, but his access to the young man is limited. \"I wish Mike would let Todd live here with us,\" Dean often laments to Geoffrey, who always answers, \"He is not our son.\" As if to impress his point, he always asks, \"Would you let Frank go off to live with the Middletons if I had been the one to pass on instead of him?\" Although partners keep their own surnames, it is traditional to refer to the family unit by the genetic parent's last name. Since Todd was Will Middleton's genetic offspring, his last name is that of his father. Since both of the Hunter boys are Geoffrey's genetic offspring, they, too, go by the last name Hunter. Had Dean been able to keep his last name and provide the genetic material for a child of his own, that child would have borne Dean's original last name. Hadrian's reproduction laws, however, have denied Dean the responsibility and privilege of producing an offspring.\n\nToday, Dean responds to Geoffrey's question by placing his hands over his bowed head. \"No! I could not\u2014but I'm not like him!\"\n\nGeoffrey reenters the living room as Dean puts the final touches on the coffee table. He has rearranged the candle plates and is now holding a few stray magazines in one hand and a dust rag in the other. Dean remonstrates, \"Mike Fulton is still the boy's papa and he loves Todd!\"\n\n\"But he's never around when Todd needs him!\" Dean is so exasperated he begins to weep. \"And I promised Will that I'd look out for his boy.\"\n\n\"And you do,\" Geoffrey says. He wants to embrace Dean, but he knows when Dean is like this, any physical contact only causes him to revert back to old habits.\n\nOddly enough, it is Dean who reaches to Geoffrey for an embrace. \"And he is so much like his father it scares me.\" As Dean's hands are full, he does not wrap his arms around Geoffrey so much as step in to allow his husband's arms to wrap around his back and clasp the sides of his shoulder blades. Dean bends down to rest his head against Geoffrey's shoulder.\n\n\"You don't have to worry about Todd,\" Geoffrey says reassuringly. \"He and Frank are tight. And one day, I swear to you, those two will be a unit.\"\n\n\"I hope so.\" Dean nods his head against his partner's shoulder. \"You're right. I worry too much.\"\n\n\"All right.\" Geoffrey gives Dean one more squeeze before releasing him and issuing a mock stern order. \"We better finish getting ready for your party! It is Happy New Year after all!\"\n\n\"And,\" Dean adds gaily, \"Hadrian's fiftieth birthday!\" The two men kiss briefly. Then, through misty eyes, Dean confesses, \"I really love you, Geoffrey frey Hunter. I am grateful every day that I have you in my life.\"\n\nMoved by Dean's honesty, Geoffrey kisses his lover passionately. As soon as Geoffrey releases him, Dean springs back into action. When Dean has a party to plan, there is no stopping the man!\n\n*****\n\n#  _Salve!_\n\n**A Year in Review  \nHNN\u2014Melissa Eagleton Reporting**\n\nThis past year has been a quiet one for Hadrian with only four outsider attacks against the wall. According to Lieutenant-General Birtwistle, only one attack was from an organized force. The Alberta regiment, the Manitoba brigade (consisting mostly of descendants from what used to be southern Manitoba), continues to demand we return Hadrian's land to them. Lieutenant-General Birtwistle is confident the losses incurred by the enemy were sufficient to deter any future attacks from this military force. \"We had to use incendiary rockets against the enemy.\" Sighing, he closed his eyes I'm sure in an attempt to block out grim memories. \"We took out their entire squadron.\" He then shook his head sadly. \"It was the only way. They were clearly organized and had with them scaffolding apparatus. Hadrian's war policy is very clear. No outsider is ever to enter Hadrian!\" An understandable policy after June 13, now commonly known as 6-13, and considering the constant bombardment of plagues ravaging many outside countries. The last thing Hadrian needs is for a contaminated outsider to bring a plague inside our walls. Lieutenant-General Birtwistle feels Hadrian no longer need fear the Alberta regular army. As he reminded us in his interview, \"Alberta's Prime Minister has assured us, via wave link, it is no longer providing this regiment with additional forces. Remember,\" Lieutenant-General Birtwistle said, \"it has been over eight years since the Alberta army has sent any regiment besides the Manitoba Brigands to attack our borders.\" The fact that Lieutenant-General Birtwistle refers to this particular regiment as bandits suggests that they are no longer regular army but are themselves becoming another desperate rabble, outsider victims fruitlessly trying to slip through our walls.\n\nLieutenant-General Birtwistle would also like Hadrian to recognize Private Katrina Jones, who was awarded the Antinous Sword, posthumously, for her brave actions. Prior to the incendiary rockets being used, she stood bravely at the wall repelling the invaders. She died of a gunshot to the head.\n\nInside our walls, new progress has been made in strengthening the firewalls that keep the outside net from infiltrating our wave link. Unfortunately, some noxious hate literature, written by Leigh F. Butler, brother of Jeremiah F. Butler, the notorious suicide bomber of 6-13, slipped through last month. In his hate statement, he declared the citizens of Hadrian to be devil spawn and blasphemous sinners. Unfortunately, he was able to spam his cruel words to all our high schools via wires, terrifying and confusing many of our youth. Quoting his antiquated Bible (my apologies to those Hadrian citizens who choose to believe in this God), he referred to us as the abominable villains of Sodom and Gomorrah and warned that the dirty nuclear device his brother exploded was really God raining fire upon us and that more such rain is destined to come our way. Leigh F. Butler and his message of hate is a brutal reminder of the need for vigilance at our borders. Estelle Ramones, head of Hadrian's Institute of Computer Science, assures us that the latest version of the firewall will stave off future attacks of hate literature.\n\nOn a positive note, agricultural engineer Quintin Laugharne claims to have biologically engineered an even hardier soya bean than the one Will Middleton introduced to our southern grasslands fifteen years ago. Will Middleton's revolutionary work helped usher Hadrian closer to a fully self-sustaining country. Using Middleton's research as his base, Laugharne has now stabilized the bean's genetics so it will grow even in our milder summer seasons. The soya bean, known for centuries as a life-saving plant with its high levels of protein, has saved Hadrian from over-dependency on food imports from the outside world.\n\nEven prior to the addition of the soya bean industry, Hadrian had always been very close to self-sufficient. Because Hadrian's borders include one quarter of the world's largest inland freshwater lakes (believe it or not, Hudson Bay used to have a high concentration of saline that we harvest along its shores) and numerous bountiful rivers, we do not lack access to fresh water like many parts of the outside world. The boreal forest in our northernmost regions, with careful harvesting and a solid forest reclamation plan, provides us with all the pulp and paper we need. Our fishing industry and farms all boast bountiful harvests, and the mining of quartz and ore meet our basic metal needs. As we enter a new year in the life of Hadrian, we can happily claim the future is in our hands. Happy fiftieth birthday, Hadrian!\n\n_Vale!_\n\n# A New Year's Kiss\n\nThe Hunter home is immaculate. You could run a finger into its deepest crevice and not pick up any dust. This obsessive cleaning trait of Dean's only rears its ugly head when company is expected. He has also taken great pains to ensure that both party rooms, living room and kitchen, are decorated to match the evening's festivities. All of the Hunters' guests, when greeted at the door, are presented with one of Dean's hand sewn (and starched!) conical hats. Using colorful scraps of hemp linen he produced over a dozen of these hats with tassels for his guests to wear. Every year he sews a new one, always for Geoffrey to wear. The living room is decorated with a rainbow of cheesecloth strung across the ceiling and others dripping down to the floor. Two of the three living room wall screens are turned on with festive imagery slide shows, the third reserved for the _Salve!_ New Year count down. Dean also has a series of handmade noisemakers, including miniature drums, castanets, and whistles. Along with all the traditional New Year's paraphernalia, Dean makes sure each guest starts the night with a glass of wine or his choice of liquor. With all the gaiety surrounding him, the multiple coos over his d\u00e9cor, and the layout of the food, one would expect Dean to be thrilled\u2014but he isn't. It is already ten-thirty and still no Mike and Todd. Frustrated at having no way of contacting Todd, Dean just paces in the front hallway, periodically looking toward the door.\n\n\"Dean,\" Geoffrey whispers, placing a hand on his lover's shoulder to settle him. \"Our guests are in the living room, not the front hall.\"\n\n\"I'm just\u2014\"\n\n\"Waiting for Todd; I know. But,\" with gentle opposition, \"if Mike were going to bring him, they'd be here by now.\"\n\n\"I just hate the thought of that poor kid stuck at home alone on New Year's Eve,\" Dean laments.\n\n\"You don't know that, though, do you?\" Geoffrey reminds him.\n\nDean shakes his head angrily. \"Oh, but I do! Mike Fulton is either working overtime for more credits, or he's out with a new boyfriend.\"\n\n\"You have no right to judge the man,\" Geoffrey remonstrates.\n\n\"I'm going to take the bubble.\" Seeing reservation in Geoffrey's eyes, Dean attempts to justify his actions. \"I just want to drive over there and see if Todd is home.\"\n\n\"Dean,\" Geoffrey reminds him, \"the bubble won't run right now. It's been too cold. You know that.\"\n\n\"It's only minus five out there.\"\n\n\"It was minus twenty yesterday,\" Geoffrey reminds him. \"The battery is dead.\"\n\n\"It's been out in the sun all day, it should be charged up by now.\"\n\nGeoffrey grimaces, \"Dean\u2014\"\n\nBefore Geoffrey can finish Dean jumps in with, \"I'll take public transit then.\"\n\n\"Dean!\" Geoffrey grabs Dean's arm before he can turn toward the closet. \"You're not thinking rationally.\"\n\n\"Please, Geoffrey, let me go.\"\n\n\"No, Dean. Your responsibility is to your guests.\"\n\nJust as Dean nods his reluctant assent, there is a knock at the front door. Dean rushes to answer it. Todd is standing outside. With his shoulders stooped and his body bundled inside an old black jacket, Todd looks stouter and shorter than he really is. In fact, Todd is 5' 5\" and quite muscular since he trains year round for his favorite sport, b-ball. He also likes to wrestle and plays v-ball on the offseason to stay in shape. In the summer, he swims in Hudson Bay. His wavy brown hair (hidden underneath a wool toque) is the same color as his eyes, which currently stand out against his thick frosty eyelashes. His hands are stuffed inside his jacket pockets because he lost his gloves a few weeks ago. \"Hi, Papa Dean,\" he stutters through shivers. \"Can I come in? I'm freezing!\"\n\n\"Of course you can.\" Dean pops his head over Todd's shoulder. \"Where's your Papa?\" Seeing no sign of Mike Fulton, Dean asks, \"Did you walk here?\"\n\n\"Yes, sir,\" Todd answers as he steps inside. Dean doesn't even bother to ask why Todd never took public transit\u2014no thumbprint, no credit, no access to public transit. He doesn't remove his coat, though, since he is still cold. Although winters in Hadrian are relatively warm, never falling much below -15\u00b0 C anymore and this New Year's Eve is considered mild at -5, Todd still had to make the trek across town. Todd and his Papa Mike live in subsidized housing on the northern edge of Antinous, Hadrian's capital city, whereas the Hunters live in a more posh region, along the Nelson riverbank.\n\nDean shoots Geoffrey an exasperated look before smiling for Todd and asking, \"Is Papa Mike working overtime tonight?\"\n\n\"Nah, he has a big date,\" Todd says while shrugging off his coat.\n\nThe way Dean glares at Geoffrey over this piece of information one would think Geoffrey was to blame for Mike Fulton's absence. Geoffrey chooses to ignore Dean's pointed expression, welcoming Todd with a handshake. \"Well, we're glad you were able to join us, Todd.\" Gesturing, he adds, \"Go on into the kitchen and help yourself to some punch. Frank's in there with Roger and a friend.\" Following Todd with his eyes as he walks away, Dean calls, \"Mind you take from the purple bowl. The crystal bowl is for adults only!\"\n\n\"And tell Frank to quit eating all our food,\" Dean adds as Todd disappears.\n\n\"Yes, sir,\" Todd chimes back from the kitchen.\n\n* * * * *\n\nFrank, Anthony, and Roger are the only ones in the kitchen. The three boys are clustered together around the table filled with food. Like most teenage boys, Frank's appetite is humongous, so he has spent most of the night standing at the kitchen table eating. Todd laughs, \"Papa Dean wants you to quit pigging out!\"\n\nFrank smiles as soon as he sees Todd. He instantly steps forward and pulls Todd in for a bear hug. Being so much taller than Todd, standing 6' 3\", he actually lifts his friend off his feet, holding him suspended briefly. \"Glad you could make it, pal. We were ready to give up on you.\" Frank is a very attractive youth. His oval face and reddish brown hair, although slightly feminine, do not detract from his obvious strength, for like Todd, Frank is muscular. He too enjoys playing v-ball, b-ball, and wrestling. He is even a long distance runner for track. His eyes are a speckled green, blue, and gray. When Frank smiles, his eyes always seem to light up.\n\nAnthony scowls as soon as Frank greets Todd, instantly turning his back to concentrate on the cheese and crackers. Like all of Frank's boyfriends, Anthony is short, standing only 5' 5\u00bd\". He has a small wiry frame, almost too skinny. Although he stands by the food table and appears to pick at the cheese, he doesn't eat anything.\n\n\"Hey, Todd,\" Roger calls out happily. Todd is like an older brother to Roger, sometimes even more so than his real brother. Frank is great, but he doesn't listen like Todd. Whenever Roger is upset and neither father is around, Todd is the next person Roger seeks out for help. Frank is a last resort because he seldom ever takes seriously anything Roger has to say. \"Want some Brie? Papa Dean put it out for your Papa Mike. Is he here?\"\n\nTodd leans in between Anthony and Frank. \"Excuse me, skinny,\" he teases Anthony, who does not respond, refusing even to move out of the way for Todd. Frank grabs Anthony's shirt collar and pulls him to his side. Anthony's smile for Frank barely conceals his contempt for Todd. Todd is not oblivious; he knows Anthony is jealous, which is utterly stupid since he and Frank are just friends. Cutting off a huge chunk of Brie, Todd slathers it on a cracker before offering it to Anthony, \"Here,\" he teases, \"let's fatten you up a little so Frank has something he can grab onto.\" Roger howls in laughter.\n\nAnthony glowers at Todd; then peering up, he complains, \"Frank, your friend is being mean to me.\"\n\nTodd chuckles along with Roger while Frank fights back the desire to join in. \"Anthony, he's just playing with you.\" Sensing that his little boy is about to explode into one of his famous hissy fits, Frank calms Anthony down by kissing him. Although he still refuses to talk to Todd, Anthony is mollified. As they watch this display, Todd winks at Roger while stuffing the cheese and cracker into his mouth.\n\n* * * * *\n\n\"Geoffrey! Geoffrey!\" Dean cries out frantically. \"Do you have the champagne ready? Melissa Eagleton is starting the countdown!\" This bottle, _Champagne Philipponnat Clos des Goisses_ , their last bottle of real French champagne, has been waiting patiently for over four years for tonight's special celebration. With major cutbacks on imports, it is not likely they will ever have another such bottle to drink. They had ordered one case on their tenth anniversary, drinking a bottle each New Year and anniversary until 6-13 ended any chance that they could order more. Since then, their last remaining bottle has sat in solitary confinement inside its case in the far reaches of their cold storage awaiting Hadrian's Fiftieth New Year's Eve and Birthday celebration. As Melissa Eagleton's voice drones along with the guests (all eyes glued to the Hunters' wall screen), Dean calls out impatiently, \"Geoffrey, please!\"\n\nGeoffrey enters the room with the champagne bottle in his hand and the cork teased halfway out. Smiling, he gives the bottle a slight shake\u2014just enough to get the bubbles flowing, but not so much that any of the valuable wine will get wasted on the floor. \"Cork ready to pop, dear.\"\n\nDean sighs audibly, moving swiftly to the coffee table to retrieve the tray with one dozen crystal flutes, one for each of the eleven men present at their little New Year gathering. The extra flute, having been intended for Mike Fulton, will sit empty on the tray. As soon as the count hits one, everyone in the room cheers \"Happy New Year!\" along with Melissa Eagleton. Every man turns to his partner or neighbor, extending the traditional New Year's kiss. Geoffrey and Dean kiss gingerly, due to the delicate nature of the tray held between them. Once the first round of kissing is complete, mostly moderate pecks since the only two couples present are Dean and Geoffrey and Frank and Anthony, Geoffrey pops open the champagne and pours everyone a drink. Although he gives full portions to the adults, the four teenage boys only garner a half glass each. Frank tosses his flute back, downing the expensive wine in one dramatic gulp. Geoffrey shakes his head. Dean wants to lecture him on such folly, considering how expensive this particular bottle is and how they are never likely to taste such luxury again, but before he or Geoffrey can express their displeasure, Frank turns Todd's way and pulls him in for a kiss. And not just any kiss. Frank spins Todd as if in a dance, dipping his best friend low before diving down for his lips. It is a very romantic gesture, and as angry as Geoffrey and Dean are at Frank for having downed the expensive champagne, they are both pleased to see Frank engaging in a kiss with Todd. Neither man dislikes Anthony, but both men are hoping for a union between Frank and Todd. Smiling at the sight, Geoffrey feels the need to kiss his lover one more time, a little less gingerly now that the tray of flutes no longer hinders him.\n\nAfter Frank releases Todd, he dips his flute into the punchbowl; then raising his flute high, he offers up another toast to honor Hadrian's birthday! Anthony does not participate in this cheer. Having watched Frank kiss Todd, the young man boils over in anger and jealousy. Pushing through the small crowd, he stops briefly to glare at Todd. And then, intentionally banging into Todd's shoulder, he pushes past him to storm out of the front room and down the hall to Frank's bedroom. Frank shrugs his shoulders Todd's way before following Anthony out to set things straight.\n\nTodd turns to leave, figuring he's overstayed his welcome, but Papa Dean catches him by the elbow. \"Happy New Year, Todd.\" Dean tussles Todd's hair as he speaks.\n\nDucking away to avoid Dean messing his hair too much, Todd laughs in reply, \"Happy New Year, Papa Dean.\" Then, with a half-hearted smile, \"and Happy Fiftieth Birthday, Hadrian.\" He lifts his flute in a token toast.\n\n\"Fifty years. Pretty amazing, eh?\" Dean is smiling, not having felt the tension yet.\n\n\"Uh-huh.\" Todd is barely paying attention to Dean. He is staring in the direction Frank left in pursuit of Anthony.\n\nNoticing the distant look in Todd's eyes, assuming rightly it has something to do with Frank and Anthony, Dean comments, \"That was some kiss you and Frank had.\"\n\n\"You mean Frank _had_!\" Exasperated, Todd exclaims, \"I wish he'd stop doing things like that to me!\"\n\n\"So tell me what was wrong with the kiss?\"\n\n\"It was too much\u2014over the top\u2014Frank always goes overboard. I mean, that's the way you kiss your boyfriend. And I'm not Frank's boyfriend.\"\n\n\"Why not, when the two of you get along so well?\"\n\n\"Because Anthony is Frank's boyfriend.\"\n\n\"Does that bother you?\"\n\n\"No.\" Shaking his head, Todd reiterates, \"Really, Papa Dean, I'm not jealous of Frank's dating other guys.\" Somewhat confused by the truth, he confesses, \"I don't know why,\" and confirms with his eyes, \"I love Frank. You know I do.\"\n\nPapa Dean places a hand on Todd's shoulder. \"I know.\"\n\n\"It's just, well, I'm not ready to date yet, and Frank obviously is.\" He shrugs. \"I can't be angry at him for that. Besides,\" he adds judiciously, \"I have no time for that sort of thing. I have to concentrate on school and b-ball. Keep my average up and my game sharp so I can get into uni. That way I won't have to be a soldier and can work toward being an agricultural engineer like my dad!\" Excited by the prospect, Todd adds, \"You know Dad introduced soya bean to Hadrian. I want to introduce rice. That's the last food substance we still have to import. Can you imagine if we could grow our own?\"\n\nDean smiles, pulling Todd in for a hug. \"That's an admirable goal, son. It would make your father proud.\" Dean hugs Todd a little too hard but the young man doesn't mind. Both men are re-experiencing the grief of loss: for Dean, the loss of a dear friend, and for Todd, the loss of a father. Releasing Todd, Dean takes a moment to brush a lock of hair out of the boy's eye. \"There is so much about you that reminds me of your father.\"\n\nTodd beams. He loved his dad. There are nights he will dream his father is still alive, only to wake up and suffer the crushing reality of his death all over again. The transition of joy to sorrow, though seemingly infinitesimal, is very real. The light in Todd's eyes blurs and tears begin to fill. \"Why did he have to die?\"\n\nPapa Dean re-submerges Todd in his arms. \"I don't know, son. Life is seldom fair. Sometimes all a man can do is make the best of it.\"\n\n\"I want to make Dad proud of me. I want to be just like him.\"\n\nTodd breathes these words against Dean's aching chest. \"You are, Todd, like him in so many ways.\" _Perhaps too much_ , Dean worries. \"But you have to be your own man, too. And I know,\" he says, now pushing Todd back slightly so he can look him in the eye, \"whatever you decide to do with your life, your father would be very proud of you!\"\n\n\"Thanks, Papa Dean.\"\n\nDean frowns. His little pep talk has not removed the shadow covering Todd's mood. \"All right, out with it.\"\n\n\"Out with what?\" Todd asks, shifting his eyes away.\n\n\"You are still upset about that kiss, aren't you?\"\n\n\"Sort of.\"\n\nTodd is evasive. Papa Dean will not let Todd escape telling him about his anxiety. \"Why does the kiss bother you this much? It just looked like the two of you were having some fun.\"\n\nTodd's eyes darken. \"It may have looked like fun to you, but as sure as Hadrian was gay, it didn't look that way to Anthony.\"\n\n\"Ah, Anthony.\"\n\n\"Yeah, Frank's boyfriend. How could Frank kiss me like that in front of him? That's the way you kiss your boyfriend! Not your best friend.\"\n\n\"So, you're worried about Anthony's feelings, then?\" Dean is not convinced.\n\n\"Yes\u2014no.\" Todd knows better than to lie to Dean. Papa Dean always has a way of wriggling the truth out of him. \"I'm worried about all the hateful rumors he's going to spread when we go back to school.\"\n\nPapa Dean is concerned. \"Does this happen a lot?\"\n\n\"Every time Frank does something stupid like that in front of one of his boyfriends.\" Exasperated, Todd says, \"Everybody he has ever dated hates me! And they have all called me horrible names behind my back. Last year, Iggy told everyone I was a _strai_.\"\n\n\"A _strai_?\" Dean asks, angered. \"I hate that word! It's so derogatory. It's no one's fault if he's born straight. He's to be pitied, not mocked.\"\n\n\"And then,\" Todd barges on to avoid discussing Iggy's accusation, \"Frank avoided me for six weeks because Iggy made him choose between him and me. The same thing is going to happen with Anthony; I just know it!\"\n\nAlthough sensitive about the topic, Dean will not let Todd pass over such a volatile accusation. \"Whoa, back up, son. Have his boyfriends accused you of being straight?\"\n\n\"Iggy told everyone I was a _strai_ \u2014a cu\u2014cunt\u2014hammer\u2014.\" Todd closes his eyes, trying not to cry. \"And he's not the only one. Just about every boy Frank's ever dated has said that about me.\" Shaking his head in disbelief, he adds, \"They never say it to my face, of course, but Crystal tells me everything.\"\n\n\"Is Frank aware of this?\"\n\nTodd shrugs.\n\n\"I plan to ask him,\" Dean says.\n\nTodd opens his eyes, fear evident. \"Please, Papa Dean\u2014no.\" He sighs. \"It's going to be strained enough between Frank and me as it is. After he's done talking to Anthony, I'm willing to bet you won't be seeing me around here for a good month\u2014at the very least.\" Looking down at his feet, Todd concludes, \"Anthony is not the type to put up with competition.\"\n\nDean turns grim. \"That explains all those times you've been absent from our house the past few years.\" He shakes his head. \"That is not to happen anymore.\" Lifting Todd's chin to look at him, Dean says, \"I promised your father I'd keep an eye on you. How can I do that if you're avoiding my house? Promise me you'll come over at least once a week.\"\n\n\"But\u2014,\" Todd stutters.\n\n\"No buts.\" Papa Dean is stern. \"Sunday dinner is the easiest. Promise me Sunday dinner at the very least.\"\n\n\"Papa Dean\u2014Frank's boyfriend will nev\u2014\"\n\n\"Frank's boyfriends come and go like the wind through foliage. You are a son to me.\" Refusing any argument, Dean adds, \"Frank can tell his boyfriends what he should have been telling them all along.\" Todd looks up quizzically. \"That you are like a brother to him!\" Taking a moment to calm down, Dean finds his smile. \"So, Sunday supper?\"\n\nTodd smiles, relieved he doesn't have to lose touch with Papa Dean. \"Thanks, Papa Dean!\"\n\n\"Try not to let petty, jealous slurs hurt you. Surely you can see the irony?\"\n\nTodd expresses his gratitude by giving Dean a grizzly bear hug. \"I love you, Papa Dean,\" he blurts out unexpectedly.\n\nDean laughs and hugs the boy tighter. \"I love you too, son.\" Kissing the top of Todd's head, he promises, \"And I will always be here for you.\"\n\n* * * * *\n\n#  _Salve!_\n\n**Illegalizing Outsider Technology  \nHNN\u2014Melissa Eagleton Reporting**\n\nAs many of you are aware, Hadrian's government officials have met numerous times to discuss whether or not outsider technology should be made illegal. The debate is proving to be a most volatile one, and our opinion polls show that many of Hadrian's citizens are divided on the issue. For some, it is a matter of cost. Outsider technology comes cheaper than the contact vocal lens. For others, anything brought into Hadrian from the outside world could potentially be contaminated. We must remember that disease is rampant in the outside world, and the possibility that some deadly bacteria, for which there is no cure, can spread through such importation is very real. Yet, as my producer wisely puts it, \"We still import rice, sugarcane, fruits, and vegetables that are unable to be grown this far north. Coffee and cocoa,\" two of his favorite vices, \"although nearing impossible to acquire except at exorbitant prices, are also among those items imported into Hadrian.\" As these are items we consume, it seems, he claims, to be rather ironic\u2014silly in fact\u2014for Hadrian citizens to be over-zealous when it comes to the use of outsider technology. \"These, after all,\" he stated at our pre-production meeting, \"we do not eat.\" I must admit, what he says makes sense, yet it still sends a chill down my spine when I think of Hadrian's citizens making use of outsider technology. When one citizen uses outsider technology to contact another connected by voc to our wave, the ramifications are frightening. Viruses intended to damage our wave network can do substantial damage to Hadrian's businesses, government, and educational institutions. Then there are the numerous instances of spam messages, voicemails, and videos containing hate sent intentionally to destroy our citizens' morale and frighten our children. Too many times have these insidious viruses littered our wave network. The importation of food items does none of these. As inconvenient as it might be for some  \nof our citizens to abandon the less expensive communication devices purchased through outside companies, remember, Hadrian's government offers wall screens and tablets credits below the least expensive outsider tech. Everyone in Hadrian can still be connected to the wave. The potential ban of outsider technology will not see the less fortunate of Hadrian citizens bereft of communication. Government devices may be slower but they are definitely a lot safer for Hadrian.\n\n_Vale!_\n\n# Illicit Communication\n\nThe following Sunday, Todd fails to join the Hunters for their family dinner. Dean demands to know why. Frank's answer is evasive, \"Todd's sick.\"\n\nAngered by Frank's dismissive attitude, Dean stretches his hand across the table, demanding, \"Give me your cell phone!\"\n\nFrank gasps, flabbergasted. \"My what?\" He has not told anyone that Todd, Crystal, and he own cell phones. No one knows they communicate through an outsider satellite phone service. Although using such a service is not illegal in Hadrian, it is certainly frowned upon.\n\nGeoffrey looks up, dismayed and discontented. \"You have outsider technology?\" His voice is curt and solemn.\n\n\"No!\" Frank protests.\n\n\"Don't lie!\" Dean insists, still holding out his hand, palm upward, waiting for the recalcitrant technology to be handed over to him. \"It's how you communicate with Todd, isn't it?\"\n\nGeoffrey glares at his son. Frank shivers. Roger pales, suddenly very afraid for his older brother. Neither father has ever raised a hand against his children, but the look in Geoffrey's eyes suggests he is fighting back the desire to cuff Frank right now. Frank, too, notes the rising anger in his genetic father. Continuing the lie will only land him in even more trouble, so Frank capitulates to the truth. \"Please, Dad,\" he begs, \"Todd can't afford a voc. No one knows, I swear!\"\n\n\"Your Papa Dean knows! He found out, didn't he?\"\n\n\"It's not like it's illegal,\" Frank mutters weakly.\n\nGrowling now, Geoffrey asks, \"Are you aware that people are currently debating making use of all outside technology illegal in Hadrian?\" Staring intently at his son, he continues, \"And if it becomes a crime, that means one of only two punishments: exile or death.\" Pausing to the let the gravity of the situation sink in, he concludes, \"You know we have no prisons in Hadrian.\"\n\n\"I know, sir,\" Frank mutters into his chin. Frank is now pale with fear. Roger begins to cry.\n\n\"And,\" Geoffrey adds, a little too cruelly, \"they are suggesting anyone who would want to use outsider technology must be straight. To be caught with a cell phone could mean a one-way ticket to a reeducation camp. I do not want either of my boys to be thought of as a _strai_.\"\n\n\"Don't use that word, please,\" says Dean, surprised by his husband's word choice when Geoffrey knows how much he hates this commonly accepted slur.\n\n\"But,\" a horror-stricken Roger cries out, \"Frank's not straight!\"\n\nTurning to his youngest, Dean attempts to calm him. \"Roger, dear, dry your eyes. Frank is not in that kind of trouble.\"\n\n\"But he could be,\" Geoffrey growls.\n\nDean begins to regret that he brought up the subject in front of Geoffrey. \"It is not necessary to frighten the boys, Geoffrey.\"\n\nGeoffrey gives Dean a knowing stare. \"Yes, I think it is. You of all people should understand that.\"\n\nNeither boy catches the undertone of Geoffrey's remark, but Dean feels its sting. He closes his eyes briefly before re-summoning his strength. \"Well, owning outsider tech is not illegal yet, and we've caught the boy before anyone else has.\" Waving his open palm as a reminder to Frank, he says, \"As soon as he hands over the device, this whole uncomfortable business will be over.\"\n\n\"You heard your Papa, Frank!\" Geoffrey orders. \"Give him the phone!\"\n\n\"Give it to him, Frank. Give it to him,\" Roger begs. Frank obeys.\n\n\"Now, Frank,\" Geoffrey instructs, \"you can forget about the rest of your supper. Go to your room and study.\"\n\n\"Yes, sir,\" Frank says as he stands and turns to leave.\n\nBefore Frank can walk past his father, though, Geoffrey grabs hold of his arm. \"Promise me right now that you will never use outsider technology again.\" When Frank's response comes a second too late, Geoffrey tightens his grip and growls, \"Promise!\"\n\n\"Yes, sir,\" Frank weeps. His only contact with Todd is lost now because Anthony won't even let him look at his friend in school. \"I promise.\"\n\nGeoffrey releases his grip. \"All right, go to your room.\"\n\n* * * * *\n\nLater that night, after Geoffrey retreats into his study to prepare for another week at the office, Dean sneaks up to Frank's room. He brings the remains of Frank's dinner with him. Because Frank is tall, he often gets lightheaded if he doesn't eat properly. Dean often wonders where the boy gets his height since Geoffrey is so short by comparison. _Obviously from his genetic mother_ , he reminds himself, and yet, Roger, Frank's genetic sibling, is also short. _Ah well, there really is no explaining the oddities of genetics_ , Dean muses. Knocking lightly on the door, Dean waits for Frank's acknowledgment before entering. Frank is sitting at his desk studying (as his father had instructed). Dean crosses the room and places the plate next to the computer slate Frank uses for school. The Hunters are wealthy enough that they were able to purchase Frank his own slate rather than make do with the school's slower version.\n\nFrank smiles when he sees the food. \"Thanks, Papa Dean. I'm starving.\"\n\n\"I knew you would be, a big boy like you. You shouldn't go without food.\" Shaking his head, he adds, \"But your father was angry, and he was right to send you to your room.\"\n\nPulling Frank's cell phone out of his pocket, he studies the ancient technology a moment. \"Come over to the bed and sit down with me.\" Shaking the phone slightly, he says, \"I want to talk over a few things with you.\" Frank complies and follows his papa to the bed. Dean pulls open the poster curtain and the two sit side-by-side on Frank's king-sized mattress. \"Whose idea was it to buy these things?\"\n\n\"Mine, but Crystal agreed. I mean, without a voc, it's really hard to keep in touch with Todd. And vocs are just too expensive. We couldn't buy him one of those.\"\n\n\"No,\" Deans concurs. \"That's for certain.\"\n\nPleading now, Frank says, \"We're the three gay caballeros, Papa Dean; we need these to communicate.\"\n\nDean smiles briefly, \"The three gay caballeros. I like that.\" More sternly, now, he asks, \"Do you understand why your father got so angry with you over this?\"\n\n\"Not really,\" Frank admits. \"He went a little crazy if you ask me.\"\n\n\"What you need to understand, Frank, is that Hadrian went a little crazy after 6-13.\"\n\n\"I know 6-13 was bad, but that was eight years ago.\"\n\n\"Yes, it was. Even so, fear of the outside world hasn't abated. To have a dirty nuke explode in your borders, killing so many people, destroying so much of our fertile land\u2014well, surely you understand there is a lot of distrust for anything outsider\u2014especially technology.\"\n\n\"I'm sorry, Papa Dean. I just wanted a way to keep in touch with Todd is all.\"\n\n\"I understand. But we all have to make sacrifices for Hadrian's lifestyle. You know that, right?\"\n\n\"Yeah\u2014I don't know\u2014I guess.\"\n\n\"One of those sacrifices includes importation of many fruits and vegetables we used to enjoy and delicious items like coffee and chocolate and,\" adding sternly, \"outsider technology.\"\n\n\"But it's not illegal yet.\"\n\n\"But it's frowned upon and you know as well as I do that it'll be illegal soon enough. Now,\" handing the phone to Frank, he says, \"show me how this thing works.\"\n\n\"Huh?\" Frank is flabbergasted. After getting into so much trouble, he can't believe his papa wants to use the phone.\n\n\"I want to ask Todd why he didn't come tonight.\" Dean is matter-of-fact in his request. He understands that neither Mike Fulton nor his son, Todd Middleton, have vocs. There is no way of messaging them either since their wall screen broke down a few months ago.\n\nFrank is uneasy about phoning Todd. He has been avoiding him to placate Anthony, and he knows how much he hurt his friend when he texted him not to come over tonight. \"But, Papa Dean, Todd's sick.\"\n\n\"Really?\" Papa Dean is not fooled. \"Call him now!\"\n\n\"But Dad said\u2014\"\n\n\"And, I'm telling you to phone him. Now!\"\n\nFrank takes the phone out of Dean's hand and begins pressing different keys, \"Yes, sir.\" When the number starts to ring, he hands the phone back to Papa Dean. Listening intently, he can just make out Todd's voice as he answers, \"Hey, Frank, what's up?\"\n\n\"You don't sound sick to me.\"\n\n\"Huh?\" Todd is confused. This is not Frank's voice.\n\n\"It's Papa Dean.\"\n\n\"Papa Dean.\" There is a moment of reflection. \"How did you\u2014?\"\n\n\"Frank is not as circumspect as he thinks he is,\" says Dean.\n\nFrank lowers his head as if he can feel Todd glaring at him through the phone.\n\n\"Oh.\" Todd remains silent, waiting for Dean to pass judgment.\n\n\"Don't worry. I'm not going to tell on you.\" Todd's sigh of relief is audible even to Frank. \"These little devices are not illegal\u2014yet. But they are more conspicuous than the vocal contact lens so you boys have to be doubly careful.\" When Dean stares intently at him, Frank nods in agreement, corresponding with Todd's reply of, \"Yes, sir.\"\n\n\"Now, Todd,\" says Dean, quickly changing the subject, \"I have a bone to pick with you.\"\n\n\"Sir?\" Todd feigns confusion, but he knows exactly why Dean is upset.\n\n\"You promised me Sunday dinner\u2014and,\" he adds swiftly, \"don't pretend you're sick!\"\n\n\"I'm sorry, Papa Dean, but Frank\u2014\"\n\nDean cuts Todd off instantly with a curt remark directed at his son. \"I know\u2014Frank!\" Dean is staring so intently at him that Frank begins to sweat. \"I understand, but that doesn't mean I am letting you off the hook. You promised me Sunday dinners, and if need be, I will pick you up and take you out to eat. That is until Frank gets his 'love life' settled.\" Frank responds by staring at his feet. \"So,\" Dean states crisply, \"I will be around to pick you up next Sunday; let's say 6 p. m.\"\n\n\"But Papa Mike, he's so proud\u2014he'll say no.\"\n\n\"You let me deal with Papa Mike. Does he know about this phone?\"\n\n\"Yeah.\" Then adding in his Papa Mike's defense, Todd adds, \"He doesn't like it, but he understands.\"\n\n\"Okay, put him on the phone, then. I'll talk to him now.\"\n\n\"He, uh...\" There is an awkward silence.\n\n\"He's not home, right?\"\n\n\"No, sir.\"\n\n\"On a date?\" Dean is bitter at this news. It is not that he holds anything against Mike for forming a new relationship; it's just he has been forming new relationships since, well, within weeks after Will died. It seems Mike turns to the arms of strange men to help him forget, and Dean suspects he also turns to whiskey. Sighing at circumstances beyond his control, Dean adds with determination, \"I'll get a hold of him at work this week. Papa Mike will agree.\"\n\n\"Thanks, Papa Dean.\" Todd is relieved. He is lonely right now. When he's not at school or in b-ball, he's stuck at home alone, with nothing but studying or the odd text with Crystal to keep him from dying of boredom.\n\n\"Now,\" Dean is quite stern, \"before I give this phone back to Frank, I want both of you boys to promise me to be more careful! No one\u2014and I mean NO ONE, is ever to know the two of you are using these things. Is that understood?\"\n\nAlthough in separate rooms in different ends of the city, both boys answer in unison, \"Yes, sir!\"\n\n* * * * *\n\n#  _Salve!_\n\n**Masturbation Rules  \nHNN\u2014Melissa Eagleton Reporting**\n\nEarlier this week, we had quite the heated debate over the topic for tonight's _Salve!_ In all honesty, I was at first uncomfortable about discussing such an issue so openly on air. I suggested this topic be best left to the education system to deal with. Hadrian's Sex Education Curriculum is one of the finest worldwide. However, my producer pointed out, quite correctly, that this topic is not something meant only for our children. Parents, too, need some coaching\u2014not in terms of \"how to\"; most of us can figure that out quite nicely on our own\u2014but rather in terms of how to talk to our children about this issue. So today's topic is masturbation and what to do if you accidentally walk in on your child\u2014doing\u2014ahem\u2014pleasuring him\u2014or herself privately.\n\nOften when a parent accidently walks in on his or her child masturbating, the moment becomes one filled with consternation and embarrassment for both parties. What is really important in this situation is not to allow our initial emotional reaction to take precedence. Allowing discomfiture and disquiet to dictate the now critical discussion, or worse yet, to allow these emotions to avoid the issue all together, is not instrumental in helping your child develop a healthy attitude toward his or her body and the act of masturbation. Clearly this is an act everyone has committed once or twice, perhaps many times over. The old religious myth fanatics used to scare their sons' hands away from their penises. \"You'll go blind\" is, as my producer succinctly put it, \"hooey.\" Therefore, the question begging to be asked is: Why do some parents still react badly upon the discovery of their child's masturbatory acts? When we respond unfavorably to such a natural instinct, we are perpetuating the folly that masturbating is sinful. Again, to quote my producer, \"That very notion is absurd.\" As the parents of Hadrian's children, we need to remind ourselves that masturbation does not hurt anyone! On the other hand, it is actually beneficial to both body and spirit. Masturbation is a great stress reliever, and the release of sexual tension is something every single human body demands.\n\nIn fact, masturbation is a great sexual alternative for our youth. The startling rate at which our teens become sexually active suggests the need for proactive measures. What better pro-active measure than masturbation? It allows your son or daughter to release built-up sexual tension without sexual bonding and forming of intense relationships prior to being emotionally ready. For, as we all know, the body is often ready for sexual release long before the average person is emotionally ready for a serious relationship.\n\nStill, it is important that your child understand how masturbation, colloquially referred to as \"petting the kitty\" or \"lengthening the leather,\" is a very private act and not one to be shared with others\u2014and no doubt, many a parent has been embarrassed by the accidental discovery of his or her child's private affairs. But it is critical you step past these uncomfortable feelings\u2014wait out enough time to allow the embarrassment to abate, and then discuss the issue with your teen. This is critical. Children of Hadrian should never feel wrong for committing such a natural, useful act. Murad Nasser, Hadrian's top medical practitioner, recommends that we orgasm at least once a day, even if one does not have a sexual partner. Masturbation, he says, is a necessary act, possibly even vital to maintaining good physical and emotional health.\n\nIn fact, parents, don't wait for that accidental moment; be pro-active. Sit your son or daughter down today and hold a frank discussion about this matter. Let your child know that it is okay to masturbate\u2014just remember, it is a very private act that should never be expressed in public. As well as ensuring your child understands and accepts masturbation as a natural act, make sure he or she knows masturbation is something best done alone.\n\n_Vale!_\n\n# Images of Crystal\n\nTodd's room is actually a walk-in linen closet. Because his and Todd's bungalow has only one bedroom, Papa Mike removed all the shelves in the closet and restructured the middle shelf as a bed with dresser drawers underneath. The shelf is long enough and wide enough for one single mattress. Screwed into the wall above the head of Todd's bed is a small reading lamp so Todd has some light. The door to Todd's room is the original folding door that came with the linen closet. About two feet are between the closet door and the bed shelf for Todd to move around in. More often than not, he leaves the folding door open to give the illusion of more space, but today, he has the door closed.\n\nCurrently, Todd is half-lying, half-sitting up on his bed, with his upper back and neck pressed against the wall. Images of Crystal Albright (his and Frank's other best friend and fellow sport fanatic) moving in slow motion waft through Todd's brain. He envisions her musky cinnamon scent and the way she looks in her tight green dress. The fingers of his one hand are actually caressing the smooth lacquered pages of a very old _Hustler_ magazine, circa 2010. Open to the pullout image, the woman Todd pretends is real looks alarmingly like Crystal. Both women are tall, thin yet muscular, have short dark hair, stunning brown eyes, _and_ big breasts.\n\nTodd found the magazine in the recesses of Papa Mike's closet. There a small box lay hidden, and curiosity, though deadly to the cat, is fodder to Todd's desire. He doesn't even question what contraband porn is doing hidden inside his papa's closet. He simply found the match to ignite his wakening desires. Without being self-conscious, Todd returns with the magazine to his small bedroom.\n\nDown the hall, Papa Mike is opening the door.\n\n\"Hey, Papa Mike, is Todd in?\" Frank asks as he enters.\n\n\"Yeah. Was he expecting you?\"\n\n\"No. I'm just hoping we can talk.\"\n\n\"What's wrong? You look distressed.\"\n\n\"I just broke up with Anthony.\"\n\n\"Break-ups,\" Papa Mike replies conciliatorily, \"are the worst at your age.\"\n\n\"Yeah.\"\n\n\"Well, come on in then. He's in his room.\" Gesturing, Mike adds, \"Just head on down.\"\n\nIt only takes a moment for Frank to walk to Todd's room, open his door, and walk in on him in the act. Giddy at the sight, Frank blurts out, \"Hadrian's Lover, Todd! Are you jerking off?\"\n\nTodd gasps, quickly rolling over to face the wall, his actions immediately abandoned. Suddenly, all his heat is spent, and his sun collapsing pulls Todd's psyche deep inside its black hole. One thought alone resonates: _Thank Hadrian, my pants aren't down_. Fortunately, Todd did not bother removing his pants so he is not revealing an exposed buttock to his companion. Panic stricken, a second thought suddenly emerges: _What if he finds out?_ Shaking, Todd tries desperately to hide the magazine clutched in his left hand.\n\nFrank quickly slides the door shut behind him, hoping Todd's Papa Mike didn't hear his outburst. There is just enough room for Frank to stand in front of Todd's shelf. \"Oh, man,\" he whispers. \"That is the sexiest thing I have ever seen.\"\n\nHoarse, Todd blurts out, \"Go away, Frank.\"\n\n\"No way.\" Frank sits down beside Todd. Tickling his hand up Todd's spine, slithering it over his shoulder, Frank's hand begins to make its way back down Todd's chest. Todd curdles, contorting into the fetal position, ironically bringing Frank's hand closer to its mark.\n\n\"Please, Frank, don't.\"\n\n\"Just go with it, baby.\" That is when Frank feels the magazine clutched in Todd's hand. \"Are you using porn?\" Giggling. \"I love it!\" He clasps the magazine and gives it a gentle tug. Todd's grip tightens. \"Come on; let me see, too.\"\n\n\"Please.\" Todd is now in tears. \"Go away.\"\n\nBecoming suspicious, Frank tugs harder at the magazine. \"Why?\"\n\n\"Just let go.\"\n\n\"Give me the magazine.\"\n\n\"No.\"\n\n\"Give it over!\" Frank practically yells as he pulls the contraband free of Todd's grip. Standing up, he stares at the magazine in horror. \"What the fuck is this?\" When Todd fails to answer beyond a whimper, Frank screams, \"What the fuck is this?\" Rolling the _Hustler_ tight in his grip, he begins to beat Todd with it. \" _Strai_ porn? You're using _strai_ porn?\" Todd cringes under each blow.\n\nBefore Frank can get too carried away, Papa Mike slides open the door. \"What in Hadrian's name is going on in here?\"\n\n\"Todd was whacking off when I walked in.\" Frank is so incensed his voice makes even the simple act of masturbation a sin.\n\n\"Every man, woman, and child masturbates, Frank!\" Although embarrassed for Todd being caught in the act, Mike would never chastise him for it.\n\nFrank swirls around to throw the trash rag in Papa Mike's face. \"He was using this!\"\n\nIt takes a moment for Papa Mike to regain his balance and senses. His inspection of the magazine is accompanied with a moment of silence, punctuated by Todd's truncated sobbing. Knowing his are the only sounds being made, Todd tries desperately, even more so unsuccessfully, to control himself. Finally, Papa Mike asks, \"Where did you get this, son?\" Todd can't answer. His voice is currently being choked by shame. Looking at his son's friend, Mike orders him, \"Do not tell _anyone_ about this.\"\n\n\"What are you going to do?\" Frank demands.\n\n\"I will deal with this.\" Papa Mike is curt, his voice suggesting Frank leave immediately.\n\n\"How?\" Not even waiting for a response, Frank barges on, \"Where would he get something like this?\"\n\n\"From my closet.\"\n\n\" _Your_ closet?\" Incredulity doesn't even come close to describing Frank's emotions at this moment.\n\n\"Yes, my closet,\" Mike repeats with no offer of an apology. Looking toward Todd, Papa Mike nods knowingly.\n\n\"What the fuck is something like that doing in _your_ closet?\" Frank is nothing if not persistent.\n\n\"I used to sell contraband.\" Still no apology; just a blunt declaration. Incensed, Mike states, \"How the fuck do you think I got his father through Uni? On a mechanic's pay?\"\n\nFlabbergasted, Frank asks, \"But this? Who would buy\u2014\"\n\n\"Yes, Frank, there is a black market in _strai_ porn, and the older it is, the more costly. Bound paper magazines like this,\" he says, shaking his head in regret, \"had I remembered I still had it, might have brought in a thousand credits or more. Damn, I wish I had found it first. I could have bought Todd new b-ball shoes and decent clothes for school.\" _Maybe even some whiskey and cigarettes,_ he ruminates. Pointing its remains at Frank, he says, \"Now that you've ruined it, I doubt I could score a credit for it.\"\n\n\"That's what you're worried about?\" Frank can't believe his ears.\n\n\"What? You think Todd's the only kid in Hadrian who has ever masturbated to _strai_ porn? Shit, I stroked the snake a few times to this myself.\" Mike doesn't even pause in his lie. He has spent most of his adult life helping a _strai_ very close to him hide. He knows exactly what to say and do. This sort of posturing has become second nature to him, even if it has been a while. \"And, why not? Who in Hadrian's name does it hurt? It's a great way to get rid of some of that straight tension.\" Eyeing Frank circumspectly, he adds, \"But you wouldn't know about that, would you?\" Mike shakes his head in disgust. \"Well, guess what, Frank? Not every one of us is a six on the Kinsey scale. Some of us actually have some battling to do!\" Looking back at his son, he adds, \"Todd may be a five or a four; hell, he may even be a three or a two. I don't care. Do you?\" Gesturing with the magazine toward the front door, Mike shames Frank into looking down at his shoes, but Frank still refuses to take the hint and leave.\n\nFrank wants to say, _Yes, it does matter_ , but he knows the answer is no. \"No, sir. No, I don't care.\"\n\n\"I'm working with him.\" Papa Mike is blunt and still motioning with the magazine toward the front door. \"That's all you need to know.\"\n\n\"Can you,\" asks Frank, feebly pointing to the now decimated magazine, \"get rid of this?\" Searching for the right words, anything to justify his frustration, he states, \"It's\u2014it might be too much temptation for him.\"\n\n\"Yes.\u2014And, I will go through my closet to see if I have any more.\u2014And, if I do, I will throw it all out. Happy? Satisfied?\" Once again, Mike gestures as if to say, _Will you now get the fuck out?_\n\n\"Thank you.\" Frank begins to cross in front of Papa Mike.\n\nStopping Frank suddenly, Mike demands, \"And you? What do you plan to do? Expose him?\"\n\nTodd cringes in fear. Frank looks down on his friend. Feelings of compassion overwhelm him. \"No, sir.\"\n\n\"Thank you.\" Frank turns and goes down the hall.\n\nWaiting until he is sure Frank has left the house, Papa Mike finally sits on the bed beside Todd. _Fuck_ , he mutters to himself, _not this again!_ \"I'm sorry, son. I should have told you Frank was here.\"\n\n\"P-Papa, am I straight?\"\n\nBreathing deeply before sighing, Mike responds, \"I don't know. Most of us have something straight in us.\" Glancing down at the magazine, he adds, \"That's why I was able to make so many credits selling these damn things.\" _I should get back into it_ , he considers. _We're so fucking broke_. It was too dangerous, though, and Mike no longer wants to risk his life in that racket. Turning his mind back to Todd, he rubs a hand over his forehead. \"But we all have that latent homosexual in us, too.\"\n\n\"Do we?\"\n\n\"Yes. Most of us\u2014just\u2014\" Shaking his head, he suddenly turns his explanation into a command. \"Just never act on those straight tendencies.\"\n\n\"Yes, sir\u2014I mean, no, sir\u2014I won't.\"\n\n\"Good boy.\" Getting up to leave, Mike pauses to look down at his late husband's son. \"Just remember,\" he adds (something used to tell Will) \"you don't have to do anything until you're ready.\" _Will you ever be ready?_ He wonders. _Will never was._\n\nTodd sniffles, \"Thanks, Papa.\"\n\n\"You _okay_?\" Not bothering to pause, he adds, \"'Cause I gotta go. I got a date.\"\n\nTodd doesn't answer. Papa Mike has already left the room. Still holding himself, Todd notices how his penis has shrunk inside his hand like a turtle.\n\n* * * * *\n\n#  _Salve!_\n\n**Hadrian's Lover  \nHNN\u2014Melissa Eagleton Reporting**\n\n_Hadrian's Lover_ is scheduled to air this Friday on the wave. Don't miss the pilot episode of this inspiring, historical fiction that traces the lives of Hadrian, Emperor of Rome, and his young Greek lover, Antinous. The producers of this docudrama do not wish to refer to their work as historical fact, preferring the newly coined phrase \"factition,\" even though much of the script is founded on solid research. That Hadrian was ruthless like many Roman Emperors will receive little emphasis at the start of the mini-series. Initial emphasis will be placed on the fact that he was deemed one of the five good Emperors of Rome. Much of the scandal surrounding his love affair with Antinous will also play a significant role. It isn't until after the sudden death of young Antinous, only nineteen years of age, drowning in the River Nile, that the more vicious side of Hadrian will be presented. It is said that after the death of Antinous, Hadrian went mad. In fact, most of the cruelties for which Hadrian is known occurred after he lost Antinous.\n\nThat Antinous was the most sexually alluring and beautiful of men caused a great deal of difficulty for the casting director. Abigail Williams searched tirelessly throughout Hadrian for a young actor who would meet this bill as well as be able to perform the role. She finally found her match, but it has been decided not to identify him until the show's airing. Rest assured that Hadrian's best and most beautiful young man has been chosen for this most illustrious role. Indeed, this is the role of a lifetime.\n\nLong time favored actor Royston Birley will be playing Hadrian. Royston, thrilled at having landed such an auspicious role, is said to have spent months completing in-depth research into Hadrian's reign as well as finding everything the wave has to say about Hadrian's love affair with Antinous. \"What is great about this role,\" Royston said in a recent interview, \"is the highlighting of Hadrian's arranged marriage to Vibia Sabina, clearly a political maneuver that caused much misery to the noble emperor. The only real joy and love of Hadrian's life was his young Greek lover Antinous.\"\n\nAs we all know, after Hadrian met young Antinous, the boy became his closest confidant. Antinous provided Hadrian with lively conversation as well as the obvious attraction of physical beauty. Theirs is perhaps one of the most heart-wrenching love stories of all times, as Hadrian and Antinous' relationship was severed all too quickly by the sudden drowning of the young man on the River Nile.\n\nThe mystery surrounding Antinous' death will be dealt with in-depth. A fictional character, Centurion Detective Giustino Romano, has been created specifically for the storyline. This character, Director Aaron Ganis says, will consider all the various theories about Antinous' death. Was it a ritual killing, a murder, or an act of suicide? Ganis will not reveal what, if any findings, are given in this series, but he assures us, that all conspiracy theories of Hadrian's day, as well as a few invented by our writers, are thoroughly investigated by Centurion Romano.\n\n_Vale!_\n\n# Sweet Sixteen!\n\nTodd's birthday is celebrated at the Hunters'. Mike Fulton is working overtime and contacted Dean earlier through his work wave messaging system to let him know he wouldn't be attending. With frustration brewing, Dean is determined to throw all his energy into preparing the perfect dinner for Todd. _The boy will turn sixteen only once; why can't Mike see that? Sixteen. Next thing you know he'll be seventeen\u2014seventeen_ \u2014Dean closes his eyes. A swirl of anxiety rushes through him\u2014 _No!_ Shaking his head, Dean refuses to allow memories of his youth to ruin the night's jollities. Todd's birthday cake, Dean decides, will be extraordinary. He plans to bake the young man a thick, very heavy, chocolate fudge cake. Cocoa beans are rare, not being grown in Hadrian, so he had to purchase them through the black market; Geoffrey will be upset when he finds out, and Dean anticipates a scolding, but he doesn't care. This is Todd's day and Dean knows how much the boy loves the taste of chocolate, so a chocolate cake is essential for Todd's birthday! _Will_ , he remembers fondly, _also loved chocolate. And_ , Dean smiles, _he grew his own cocoa beans in Antinous Uni's agricultural hothouses._ Remembering Will causes Dean to compare Mike's behavior with what he feels it should have been, like that of Will. _Will_ , he firmly believes, _would have been here!_ Dean silently berates Mike Fulton. _Will never would have missed Todd's birthday!_\n\nThe doorbell rings. _Someone's here already?_ Dean rushes to the front door, ripping off his baker's apron before answering it. Crystal Albright stands before him. _Hadrian's lover, who invited her?_ The Hunters rarely have women over, and Dean is always warned in advance so he can make himself scarce. \"Hello, Crystal.\" Dean's voice remains smooth, even though a sharp shock is racing through his body. Crystal is beautiful. Today, she is wearing a low-cut blouse, exposing deep cleavage and tight-fitting jeans. Crystal takes a deep breath, exposing even more cleavage. Another shock wave hits Dean and he jerks slightly. His hands instinctively reach for his temples, which are throbbing.\n\nConcerned, Crystal asks, \"Are you all right, Mr. Hunter?\" Crystal thinks it is so romantic that Dean chose to change his name to his husband's, so, unlike the other kids, she never calls him Papa Dean, but always refers to him as Mr. Hunter.\n\n\"Yes, Crystal,\" Dean mutters as his head jerks. \"Just a slight headache. Go into the living room. Put Todd's present with the rest and I'll get you a drink.\"\n\nSensing Dean's distress, and unwittingly making his agony worse, Crystal offers, \"Let me help you, Mr. Hunter. That's why I came early.\" After depositing Todd's gift in the living room, she joins Dean in the kitchen. He keeps his back to her, but her cinnamon scent wafts through the air. Dean starts to rub his forehead. She watches from behind as his back muscles tighten. \"What can I do?\" Crystal asks. _Give her something. Give her something_. Dean tries desperately to think. He can feel her walk closer. She touches his waist. Instantly, a shock rips through him. He stumbles. Nausea begins to build. \"Mr. Hunter, are you sure you're all right?\" Dean tries to speak, but the need to vomit is so extreme that he lunges for the sink and begins to throw up. \"Oh, dear.\" Crystal is concerned. \"Mr. Hunter, you should go lie down. I'll take care of everything here.\"\n\nDean agrees by nodding his head. He instantly leaves the kitchen for the safety of his room. _Who is going to bake Todd's cake now?_ He groans, calling back, \"Just don't do the cake. Leave the cake. I'll tend to it later.\"\n\n\"Don't you worry about anything, Mr. Hunter. Mama Elena taught me how to bake. I can handle this. Now, where is your recipe?\" As her voice fades, Dean groans in disappointment. He had wanted to do all this for Todd, had it all planned out, and then that girl had to show up. _No, it's not her fault_ , he tries to remind himself, _stop blaming women._ But he can't stop himself, and even though he is aware of the unjust nature of his thoughts, he continues to berate Crystal Albright for her presence.\n\n* * * * *\n\nAfter dinner, Todd slips away from the festivities to visit Papa Dean in the master bedroom. Although surprised to see Dean looking quite healthy, Todd believed Crystal when she said he was really ill. He had thrown up in the kitchen sink, she said. Besides, Papa Dean would never abandon him. \"Hey, Papa Dean, how are you feeling?\"\n\nDean sits upright on the bed. Blinking his left eye, Dean turns off the news wave he was watching through his vocal contact lens, colloquially referred to as \"the voc.\" The voc was the last of the new technology to enter Hadrian pre-6-13: phone, video, game console, timekeeper, camera, and wave link (with holographic screen and keyboard all in one). Microscopic solar batteries combined with the salt water of the eye help to keep the vocal lens charged. Coupled with a tactic tattoo or ear jewelry installed with microphone and speakers, the individual is constantly connected with Hadrian's information wave. \"I'm feeling a lot better, son. I'm so sorry I'm missing your party.\" Concerned that Crystal might have botched Todd's dinner and the cake, he asks, \"Did everything taste all right?\"\n\n\"O, wow, Papa Dean; that supper was amazing.\" Todd's smile fans Dean's heart. \"Crystal said you had everything ready and that all she had to do was pop the buns in the oven. That beef stew was really something else. What were those big doughy things in it?\"\n\n\"Those are called dumplings. My papa...\" Dean pauses momentarily, feeling the loss of dear family connections before continuing, \"used to make them for me all the time.\"\n\n\"They were delicious.\" Todd sits down beside Papa Dean. Dean reaches forward and rubs the back of Todd's new shirt. It is made of thick hemp, a plant grown in Hadrian, so a common fabric, and dyed dark beige with a green foliage pattern (all dyes made by Dean using plants from his garden). \"This shirt is really nice, Papa Dean. Thank you so much. Where did you get it?\"\n\n\"I bought the fabric and designed the pattern myself. I chose to handsew. I thought that would make it more personal.\"\n\n\"Wow!\" Todd looks down at his shirt with new eyes, filled with admiration. \"You handsewed this?\"\n\nDean smiles. \"Designed the pattern myself.\" Musing, he adds, \"I sewed one for Frank's birthday last month, but he's never worn it. Isn't flashy enough for him, I guess.\"\n\n\"Well, then Frank's just stupid. I think this shirt is amazing!\"\n\nThe shirt is simple in its construction. The collar is only one-inch wide and is double the thickness of the rest of the shirt. The front panels, where the buttons and buttonholes go, are also double thickness and folded identically to the collar, as are the cuffs. Dean would have liked to create two buttonholes for cuff links, but Geoffrey advised him to go with buttons. \"There is no point giving the boy what he can't afford, and you always go overboard for his birthdays,\" Geoffrey had remonstrated with Dean quite sternly. \"How do you think Mike feels when Todd comes home bearing your expensive gifts\u2014things he can never afford to buy the boy?\"\n\n\"If Mike paid more attention to his son, I'd care,\" Dean answered back.\n\nTo which Geoffrey reminded Dean, \"Todd is not your son, and you have to trust that Mike Fulton is taking care of him.\"\n\n_But he's not taking good care of him!_ Dean wants to yell back at Geoffrey when he suddenly realizes he is not arguing with his husband but sitting in his bedroom talking with Todd. It is amazing how powerful some memories can be, creating their presence so strongly in the moment.\n\nTodd, sensing Papa Dean's distress, reaches forward to wrap his arms around the man's neck. \"I love the shirt. Thank you so much, Papa Dean.\"\n\nStill hugging Todd, Dean asks, \"Is your friend still here?\"\n\n\"Crystal? No. She had to go home.\"\n\n\"I'm sorry she left so early,\" Dean lies while smiling freely. \"You know I am feeling a little better. Would you mind if I rejoined the party? I'd like to taste some of that cake I had hoped to make for you.\"\n\n\"Crystal did a really good job!\" Todd doesn't realize how much his words hurt Dean.\n\nSwallowing his disappointment, Dean determines to sound chipper, so he claps his hands together and declares, \"Well, then I better taste it to make sure it really is good.\" Winking, he adds, \"It's not like we get chocolate every day.\"\n\nTodd winces. \"Mr. Hunter wasn't too happy when he saw that cake.\"\n\nDean laughs. \"I didn't think he would be. The chocolate I purchased was contraband. But today is your birthday so he simply has to endure it.\"\n\n\"You shouldn't get yourself into trouble for me, Papa Dean.\"\n\nDean gets up and Todd follows him to the door. \"I can handle Geoffrey. Let's you and I go join the rest.\"\n\nAs they walk down the hall, Todd proposes, \"Hey, Papa Dean, how about when I become a bioengineer like my dad, I genetically alter the cocoa bean so we can grow it up here?\"\n\nDean wraps an arm around Todd's shoulder. \"That's a wonderful idea, Todd. Best I've heard in years. Until then, though, let's go enjoy what's left of your illicit birthday cake.\"\n\n* * * * *\n\n# _Salve!_\n\n**Extreme Weather  \nHNN\u2014Melissa Eagleton Reporting**\n\nThe legacy our forefathers have left us is never more prevalent than in days like this one. We all know of the changes to the earth's weather systems as a result of global warming. We live in what used to be a thick boreal forest that is now more grassland. We live near what used to be tundra, now mostly boreal forest. Hudson Bay was once the world's largest inland body of salt water\u2014no more. As well as having to live with radical changes to earth's bio-system, we also have to contend with the excesses of extreme weather. Without warning, the weather will change and, suddenly, we find ourselves swamped by torrential rainstorms or tornadoes descending where least expected. Worse yet are the super cell thunderstorms that used to show up only over locations much further south such as Montana in what used to be the United States of America, now a smattering of smaller countries, much like our own old Canada. What we are experiencing right now, over both southern quadrants and all of Antinous, is equivalent to the kinds of storms that used to hit desert regions in the rainy season. Meteorologists suggest that the rains pounding down on us will last for weeks. Tighten your belts, Hadrians, as very little yield is likely to come from this year's crops. Already, city dwellers are complaining of having lost the bulk of their precious topsoil, and farmers are looking out, not onto the fields they have planted (if indeed they have planted any seed at all!), but at lakes! The question that sits in everyone's mind right now is whether, after the rains finally stop and all the drainage has occurred, there will be enough time, and _top soil_ , left for planting, growing, and harvest. I do hope you have taken precautions over the years and have extra preserves from last year's growth since it is very likely our grocery stores will suffer a shortage of fresh fruit, grains, and vegetables this year! Remember, we do not bring in the same level of import as we have in the past. When a rough year hits Hadrian, we must all be prepared to ride out the worst of it. If your stores are low, do not be proud; speak to your neighbors. I am certain many a prudent man or woman has extra rations and will be more than willing to share.\n\n_Vale!_\n\n# Spring Fever\n\nAs it nears the end of May, planting season is quickly passing. Numerous rainsqualls attacking Antinous in the past couple of weeks have aborted all of Dean's early attempts to get his garden in order. According to Melissa Eagleton's report on _Salve!_ , there is very little chance of the rains letting up any time soon. The loss of this year's garden is too much for Dean, and his sigh is both weary and discordant. The Hunter family garden is his pride and joy. Every year during the spring and summer months, he spends hours each day planting, weeding, trimming, pruning, and ensuring that the finest fruits, berries, and vegetables grow in the sprawling ledges of their backyard. It is a garden to please the eye as well as provide sustenance. The top three tiers are Dean's flower and herb gardens, each with at least two fruit or berry trees: apples, pears, and Saskatoon berries and choke cherries. Cutting through the garden's center is a path that helps Dean navigate up and down the various tiers. As one descends closer to the riverbed, Dean has one tier for sweet corn, another for a wide variety of vegetables including tomatoes, potatoes, beets, peas, carrots, cucumbers, onions, radishes, green and yellow beans, broccoli, lettuce, cabbage, and asparagus. The last tier is split with one side strictly for raspberries while the other half is divided between strawberries and blueberries. Dean is careful to keep the berries covered with cheesecloth so the birds cannot consume his crop. Interspersed throughout are the fallow tiers in which Dean alternates his vegetable and corn gardens. These tiers are kept well weeded so the soil can replenish and be used for compost storage. To keep these areas out of sight for visitors, Dean constructed temporary partitions to surround the unused areas. Geoffrey surprised Dean one year by commissioning an artist to paint images of the flowers and plants Dean most loves to cultivate on each of his partitions. This gift was given the first year after Dean's garden won Hadrian's Home Garden Award. Dean has garnered this award six times over the past ten years; the last three years consecutively.\n\nThis year is different, though. Dean isn't thinking about winning any awards, or trying to grow a new crop. Nor has he begun his annual ritual of digging his hands into the earth, spreading manure and compost, lovingly planting seedlings, carefully thinning and weeding. Instead, Dean has been holed up inside the house, watching the rain fall\u2014too much rain. Not enough dry time has passed in between the rain for him to work the dirt, which, rather than being in soft beds, is now in thick muddy pools.\n\nSeated on the cushioned bench inside the bay windows, Dean stares morosely at the rain pounding down on his backyard. Gardening is his lifeboat, a ritual routine that keeps his mind from focusing on harsh memories\u2014memories that recur every spring\u2014always beginning on the Ides of March. \" _Et tu, Brute?_ \" he mutters. Tears roll down his cheeks.\n\nThis depression has been creeping up on Dean for over two months. Success at keeping it hidden and at bay was destroyed when the rains came. Prior to the steady downpour, Dean was keeping his mind focused on planning out the garden, getting seedlings ready, and cleaning out his garden shed. But the rainstorms came. And although it rains every spring, and Dean suffers low days as a result, this year the pounding down of the endless stream has swollen the river, flooding the first two tiers, drowning, and eventually washing away his precious berries. The other tiers have also been ruined as the heavy rains washed off all the topsoil he had worked so hard to build up and maintain over the years, reducing much of the garden to the stone and clay that lies beneath. To make matters worse, Geoffrey has been working late nearly every night for close to three months. With too much time on his hands, not seeing other outlets like sewing or house cleaning as options, Dean slowly has sunk deeper and deeper into a funk. So overwhelmed by emotion and memories he no longer tries to restrain in his mind's recesses, Dean doesn't hear the bubble pull up; they are such quiet vehicles. Its sudden appearance in the small parkway next to the backdoor startles Dean into awareness and fear sends the blood racing through his body. Quickly wiping his eyes, Dean dashes into the washroom where he splashes cold water over his face, hoping to hide his tired red eyes from his lover. _What time is it?_ Dean wonders. The boys aren't home from school yet. _What's Geoffrey doing here?_ In a panic, Dean blinks his right eye, looking at the clock to check the hour. _It's only 2:30,_ he groans. Geoffrey is two full hours early. _Why is he home now? Maybe he knows. What to do? Go to the bedroom? Pretend to sleep? No,_ Dean realizes, _that would tip Geoffrey off._ Rushing into the kitchen, Dean begins to pull out pots and pans, quickly planning an elaborate meal that should take him a good two hours to prepare. _Maybe Geoffrey will see that I'm busy,_ Dean hopes desperately, _and go straight to his office instead of\u2014of\u2014of whatever it is that brought him home so soon._ Terrified of being found out, Dean doesn't even notice which vegetable he pulls out of the fridge. He just slams it down on the cutting board and begins chopping. Without realizing it, he is chopping apart a gourd he recently retrieved from the pantry for decorative purposes.\n\n* * * * *\n\n#  _Salve!_\n\n**Our Oceans Overflow  \nHNN\u2014Melissa Eagleton Reporting**\n\nYes, the title for today's _Salve!_ is deceiving. Indeed the earth's oceans have grown over the last century. The loss of over half the Arctic and Antarctic polar icecaps has reduced the size of every continent while increasing the oceans exponentially. Albeit a terrifying truth, tonight's _Salve!_ is on another type of overflow: the garbage barrage. When our satellite takes images of the earth's oceans, one has to wonder whether there is more water or pollution. The oceans are literally overflowing with debris. Observe as our wall screen reveals horrifying images of garbage islands. So much human waste in the form of plastics: bags, bottles, utensils, you name it. Then there are various metals, from tin cans to car parts! Even human clothing! Everything man-made can be found in the swill that was once a beautiful ocean. There are simply too many refuse items floating out there for me to attempt naming them all. All of this garbage has collected over the centuries until now one can actually stand and walk on these debris mounds. There are literally thousands of these garbage islands. Our cameras even detected a small colony of humans living on one. There is so little livable land left in the outside world that some poor souls have resorted to making these floating rubbish heaps their homes. It is frightening to wonder what they eat and drink on their floating cities of garbage.\n\nWhere the oceans' waters still flow free, what was once a beautiful aquamarine has been contaminated yellowish orange, a sulfurous byproduct of oil spills. In other areas, the thick black oil of recent and centuries old spills has turned this once azure bowl into a region of Black Death. Compare these images of Mexico's Gulf Coast, taken in the early twentieth century, to those taken now! The sight is absolutely grisly.\n\nEutrophication, possibly one of the most deadly of ocean killers, is entirely man-made. Having used the ocean as our waste disposal system for too many centuries, we have added more than just toxic chemicals into our planet's essential waters. We have flooded the oceans with fertilizers, causing exponential growth of algae. These little creatures have flourished to such an extent that they have literally consumed all of the oxygen in their surrounding ocean area, killing off all other marine life in the vicinity. Where there used to be the odd \"dead zone\" in our world's oceans, now more than half of earth's oceans are lifeless.\n\nIs it any wonder that our marine biologists work endlessly to ensure that the water we consume from our rivers connected to the Hudson Bay is clean? We also patrol our water border endlessly with a new breed of fisherman: the \"detritus fisherman.\" These brave men and women work tirelessly to keep humanity's excess pollution from entering our waters. It is an endless, dirty, and backbreaking job. The worst part is our not knowing what to do with much of what we prevent from coming our way. In most cases, it is simply rerouted back to where it came from, as we have nowhere to put this accretion of decay. Our detritus fishermen do, however, continuously dig through and salvage from this garbage any materials we can use. We have salvaged glass for windows and metals for building. Much of what we can no longer dig deep into our earth to find is simply floating out there, waiting for us to grab and reuse. Being a detritus fisherman is a thankless, heart-wrenching, and backbreaking job. These people are the real heroes of Hadrian.\n\nPerhaps the most sorrowful images of our oceans are the hundreds of thousands of refugee ships floating among all the garbage and decay. These ships' passengers were either forced off their land due to overpopulation or were desperately hoping to be allowed onto the North American continent where pockets of sustainable land still exist outside of Hadrian. Most of these people will die in their crafts on the ocean, either from thirst, starvation, or disease because no one will allow them onto their shores. As painful as it is for us, we too must turn these ships away. Time and again, refugee boats navigate their way into Hudson Bay from both the Atlantic and the Arctic Oceans, hoping to immigrate to Hadrian. I honestly cannot imagine how difficult it must be to turn away crying and emaciated children. But Hadrian's borders must remain closed. There is too much risk of widespread disease, and as all our outside images have shown, where too many men dwell, the earth inevitably suffers. Our population is ten million. We will not allow that to grow either through baby booms or immigration. Population control is the first of the four cornerstones of Hadrian's civilization. It is the very crux, the very pillar upon which our society was founded.\n\nThe earth's ocean waters are a grim reminder of why the human population must diminish and then be restrained from ever growing out of control again!\n\n_Vale!_\n\n# Good News!\n\nGeoffrey seldom comes home early from the office. Today, however, is special. Thrilled by the morning's success, he is anxious to find his partner and share the good news with him. The last thing Geoffrey is expecting is to find his husband suffering from another episode. Dean has been stable for over three years now. Their psychiatrist, Edgar, was like a gift from Hadrian. He prescribed Seroxat (an anti-depressant that has helped Dean cope with brooding fears and panic anxiety) as well as Zolam (a benzene for immediate relief when Dean suffers from a panic attack). Their monthly appointments have been worth every chit. The possibility that Dean is crumpled over inside does not even register in Geoffrey's mind, though the steady rain of the past two weeks should have been warning enough. Unfortunately, Geoffrey is often too preoccupied with work to take note of the subtle changes occurring in his home. And, now that the boys are older, they don't complain when Papa Dean is morose and not spending time with them like he used to. So, when Geoffrey arrives home, he is in a state of ignorant bliss: a man of all smiles.\n\nThe deal Geoffrey has been working on for the past three months (years actually, but only these past few months have led his hopes and dreams in the right direction) has finally come to pass. Although his father was hurt when Geoffrey told him of the plans to sell Hunter Enterprises to Hadrian National, the need was too great. Since Hunter Enterprises runs the detritus fisheries, much of the financial burden of coping with waste disposal falls on his company's shoulders. For years, his grandfather and father had fought with the government for subsidies to help cover the devastating cost of effective, earth-friendly disposal of all wastes retrieved from Hudson Bay. Seeing as Hunter Enterprises received the benefits of profit from the usable wastes they fished out of the bay, the government insisted the company also carry the financial burden of disposing of all non-usable materials. Anything retrieved from Hudson Bay that is harmful to the planet must be disposed of in such a way that it no longer poses any threat to the earth or Hadrian's citizens. The respect gleaned from being the corporation responsible for eliminating much of the pollution from Hadrian's main water source, as well as providing the country with nearly one fourth of all its usable resources, pales at times next to the astronomical cost of providing safe, clean disposal of many of the toxic wastes left behind by humanity's forefathers. Geoffrey's grandfather and father compensated for this financial loss by offering their employees little support\u2014no medical benefits, no retirement savings packages, and, of course, being such a dirty job, it was also low paying. Geoffrey, however, has a stronger moral streak than his elders. Rightfully, Geoffrey credits Dean for helping him acquire a more philanthropic perspective.\n\nWhen Geoffrey's father retired and Geoffrey took over the company, he decided to listen to his workers' needs. The end result was an added financial burden on Hunter Enterprises, cutting great swaths out of company profit. Try as they might, the board members could not persuade Geoffrey to change his mind because he saw reason and right behind providing for his detritus fishermen. These men and women handle much of the toxic wastes found in the Bay, and they have done so for far too many years without health benefits. Geoffrey made this decision within weeks of discovering that the majority of his employees, over two-thirds, in fact, were from the reeducated class, those members of Hadrian's society who had been found out as heterosexual before the age of twenty-two (the age one pledges an oath to Hadrian's four cornerstones). These citizens were placed in government run reeducation facilities that guided them back toward Hadrian's chosen lifestyle. The taint of having being registered as straight remains a blight on their lives, as very few opportunities are open to these men and women. As a result, many take jobs as detritus fishermen since few others want such arduous, dirty, and mentally draining jobs and few other employers will hire them. All job applications must be accompanied with full disclosure as to one's education and military service. As no heterosexual is allowed in the military and re-education camps are considered educational institutions, such \"full disclosure\" would immediately reveal anyone who was once deemed heterosexual. When Dean had learned this fact, he extracted a promise from Geoffrey that he would treat his employees fairly.\n\nKnowing how important it was to Dean, Geoffrey labored endlessly to convince the board to improve working conditions for the detritus fishermen and offer them a benefits package. Very few board members had supported Geoffrey's bid for his employees. That the reeducated are desperate for employment makes them an easy target for exploitation. Geoffrey had argued that their role was too important to ignore as, he rightly pointed out, \"Outside of the military, no other man or woman in Hadrian puts his or her life on the line daily the way detritus fishermen do.\" Although the vote was close, he did secure a slight raise and a basic medical plan for the detritus fishermen. Unfortunately, the cost of doing so was breaking the back of Hunter Enterprises. The board members were upset, stockholders were calling for changes, and Geoffrey had been forced to determine what those changes needed to be.\n\nToday's success brings forth those changes: secured financial resources for the stock investors, the value of their shares doubling with the takeover, and all cost of waste disposal now being handled by the government. Selling Hunter Enterprises to Hadrian National, making it a subsidiary to the government giant, is the finest move Geoffrey has made so far for his business. Granted, Geoffrey is no longer majority owner, but he does retain his post as company head as part of the deal. He is now on a salary that doubles what his wages were when he owned and ran Hunter Enterprises. As well as further improving the detritus fishermen's wages and working conditions, which he knows will make Dean happy, Geoffrey has not only secured but substantially improved his family's lifestyle as well as his children's future. To be rid of the worries and financial burden of waste disposal has made his mind and heart light. Now that Hunter Enterprises is a subsidiary of Hadrian National, Hadrian taxes will be used to pay for all waste disposal and renewal. Hadrian citizens pay the highest taxes in the world, most of which are reserved for cleansing and maintaining Hadrian's little portion of the planet. In fact, the majority of Hadrian's taxes are split three ways: funding the military; funding education (of which reeducation is a subsidiary); and continuous research into effective earth-friendly waste disposal. Tax rebates are offered for those who show extra care with earth-friendly practices, but these pale next to the expense of cleaning up Hudson Bay, the world's second largest inland body of water: a veritable ocean-sized slough filled with waste discarded over centuries by billions and billions of humans.\n\nToday's victory is one Geoffrey longs to share with Dean. When he prances into the front room, he calls out merrily, \"Dean, Dean, come out here. I've got something I want to tell you.\"\n\n\"I'm in the kitchen, preparing supper.\"\n\nDean's petulant response should trigger a warning in Geoffrey's mind, but it does not. Being far too engrossed in his own joy, Geoffrey does not notice the sullen tone in Dean's voice. \"Put that away,\" Geoffrey chants. \"We're going out for supper.\" When no sound comes in response, Geoffrey calls out, \"Come in here, Dean; this is important.\" Dean slowly walks from the kitchen into the front room, shamming a smile. Joy, success, and pride have a way of blinding a man. Geoffrey does not see the obvious signs of depression in Dean's eyes.\n\nAs soon as Geoffrey sees his lover, he rushes to him and lifts him up in his arms. Being shorter than Dean by a good three inches, Geoffrey has to lean back in order to lift Dean off the ground. Dean also has to cooperate by bending at the knees and lifting his feet. After swinging Dean around, Geoffrey lowers him and reaches up for a kiss. Dean complies, but with little fervor. Geoffrey has enough fervor inside to compensate, enough to keep him from noticing Dean's lack of participation. \"Your daddy's done it, baby!\" he says between bouts of kisses. \"I sealed the deal today!\"\n\n\"What deal?\" Dean asks, now leaning against the wall. Geoffrey's swinging and plunging into him have driven his body up against the hideous black velvet wallpaper.\n\nGeoffrey has to kiss Dean again before answering, \"Hadrian National now owns Hunter Enterprises!\"\n\nDean tries to smile. \"That's...great.\" Geoffrey kisses him again and then turns Dean to face the wall. Dean's face is now squishing up against black velvet. He knows Geoffrey can't tell. Dean's depression is rooted so deep inside that, like a dandelion that's been dug out, it continues to resurface, sprout its yellow head, and just as quickly, wither white.\n\n\"Um...\" Geoffrey mutters as he kisses the back of Dean's neck. His hands are running up the front of Dean's shirt, catching the buttons and flipping them open adeptly. His nails scratch down the front of Dean's chest, gripping and pulling at his chest hairs. Making his move for Dean's belt buckle, he steps back slightly so Dean can give him room to undo it. Dean obliges. He knows what Geoffrey wants. Soon enough, Dean's pants and briefs are a bundle of cloth around his ankles and Geoffrey has the front of his pants open. Not able to accomplish much standing tip-toe, Geoffrey makes a request of Dean, \"Bend your knees for me, babe.\" Dean obliges but snaps when Geoffrey thanks him with, \"That's my boy.\"\n\n\"I am not a fucking child! I know what to do. You don't have to instruct me like an idiot, you know!\"\n\nThe resurgence of Dean's spring depression hits Geoffrey like a wrecking ball. He releases his grip on Dean's penis and steps back, flabbergasted by this sudden attack. \"Whoa, where did that come from?\"\n\n\"You!\" Dean is still facing the wall, hands positioned, his body waiting, only his mind reacting. \"You treat me like a little baby. 'Come to Papa.' 'Who's your daddy?' 'Bend your knees.' 'Good boy!' You'd think I was a fucking dog!\" Turning now to face Geoffrey, eyes wet and red with anger, he shouts, \"I'm a man, damn you! Not a fucking dog and not a child! For the love of Hadrian, treat me like an adult!\"\n\n\"That...is...not...fair!\" So stunned by the blow of Dean's curse, each word Geoffrey utters stands alone. Backing away to the opposite wall, Geoffrey holds his hands up in a defensive position. Shaking his head, he says, \"I don't deserve this!\" Taking a moment to inspect his lover, he sees that Dean, now facing him, still has his pants and briefs tangled at his feet. His limp penis exposes itself as an affront to Geoffrey. \"For Hadrian's sake, pull up your pants,\" he sneers. \"You look ridiculous.\"\n\nIt is cruel, malicious, yes, even evil to say such a thing to Dean at a moment like this. When trapped inside his anxieties, Dean truly is as vulnerable as a child. Insecure and riddled with depression, desperate to dig himself out of his hole in the sand, every attempt to claw at the edges sinking him deeper, Dean collapses to the floor. First his back thumps against the wall, as if Geoffrey's voice had been a fist slamming into his chest. Tears spring forth anew and his body slowly drips down, his shirt pulling up and away from his torso as he lowers himself, making him even more exposed. Geoffrey, knowing he has done wrong, fears the extent of the damage he has just caused. Rushing to his lover's side, he sweeps Dean up into his arms and apologizes. Geoffrey, too, breaks down as he blubbers his regrets.\n\n\"No,\" Dean whimpers. \"I'm sorry. I'm sorry. I'm sorry. I'm sorry.\" Geoffrey has to put a hand to Dean's mouth to stop this skip recording. When Dean gets completely overwhelmed like this, he simply can't stop saying, \"I'm sorry.\"\n\nGently holding Dean's mouth closed, Geoffrey tries again. \"I didn't mean that, babe. Please forgive me.\"\n\nShaking his head free, Dean wails, \"It's my fault! I hate myself! I hate my life. I just\u2014I just\u2014sometimes, I just want to die.\"\n\nFear tugs at Geoffrey's heart. \"Please don't talk like that.\" He hasn't heard Dean utter these words in years. _What's gone wrong? What's happened?_\n\nDean continues his self-recriminating rant. \"Why do you love me? Why do you stay with me? I'm nothing! I'm pathetic!\"\n\n\"No, babe. No,\" Geoffrey tries to reassure him. \"That's not true.\" He rocks him now as he caresses his hair. \"You're a good man, Dean. We have a good life.\" Gently cradling Dean's head in his hands, Geoffrey turns Dean's face upwards to look at him, now pleading with him. \"You know that, don't you?\"\n\nDean's voice becomes soft. \"You have a good life. The boys have a good life. I have nothing\u2014I'm useless\u2014I'm fucking useless. I hate myself.\" Frustration builds to a fever as Dean begins pounding his fist on his thigh.\n\nFollowing Dean's fist with his eyes, Geoffrey now notices the bruising. _Not again!_ \"Dean, stop!\" He demands as he grabs Dean's fist and fights for control. \"You promised me you would never do this again.\"\n\n\"I hate myself! I hate what I've become! I had wanted so much\u2014when I think of what I might have done...\"\n\nGeoffrey tries desperately to reassure Dean, knowing his words fall on deaf ears. \"No, Dean. You're an amazing man, a great cook, a fabulous dad, and the clothes you sew, the garden you keep\u2014\" Then it hits him\u2014 _the garden_ \u2014Dean hasn't been able to work in his garden. \"All this rain\u2014that's what brought this out, isn't it, sweetie?\" Shaking his head, self-recriminatingly, he adds, \"I should have known. I've just been so busy with this deal.\"\n\n\"That's the thing,\" Dean says. \"You've got your deals. You've got your business. You've got your life. But what have I got? Nothing! I've been reduced to\u2014to\u2014this!\" With that final cry, Dean regains control of his fist and begins slamming it into his thigh again. Both thighs, Geoffrey notices, are black and blue\u2014mostly yellow and purple, actually.\n\nEnraged, Geoffrey wrestles Dean onto his back, pinning his arms down with his knees. Bellowing so that spit showers Dean's face, he demands, \"Stop it!\" For a moment, he just kneels there, dominating Dean, growling down at him, pressing his knees into Dean's arms to keep him from harming himself further. Trying to control his anger, knowing rage only exasperates Dean's condition, Geoffrey works to slow his breathing before speaking. \"You swore you would never hit yourself again.\"\n\n\"I tried not to\u2014but\u2014I just\u2014I hate myself so much\u2014I couldn't stop myself.\"\n\nClosing his eyes, thinking briefly, Geoffrey asks, \"When was the last time you saw Edgar?\" He had been so busy he had stopped attending sessions with Dean.\n\n\"Last fall.\"\n\nNodding, Geoffrey continues, \"And why did you stop going?\"\n\n\"He said I had improved so much he didn't see the need to keep up with our sessions.\"\n\n\"Right.\" Opening his eyes, looking into Dean's, Geoffrey asks, \"And why didn't you tell me this?\"\n\n\"Things were going good.\"\n\n\"What about the Seroxat?\"\n\nDean evades the questions. \"My arms hurt, Geoffrey. You're cutting off my circulation.\"\n\n\"If I let you go, will you promise not to hit yourself?\"\n\nDean shudders a few breaths before replying, \"I'll try.\"\n\nGeoffrey tightens his grip, \"Try isn't good enough.\"\n\n\"I...\" Dean trembles, \"promise.\"\n\nReleasing his knees, Geoffrey slips off Dean to sit on the floor beside him. \"The Seroxat, Dean\u2014when did you come off the Seroxat?\"\n\n\"Edgar started tapering me off in the fall. I took my last dose in February.\"\n\nGeoffrey looks dejected. \"Why didn't you tell me?\" Ire forming once more, he demands, \"Why didn't Edgar tell me?\"\n\nStill lying on the floor, Dean covers his face with his hands. \"I told him I would.\"\n\nGeoffrey snorts. \"And he believed you?\"\n\n\"I was doing so good.\"\n\n\" _Well_ , Dean! You were doing _well_! And, you were doing _well_ because you were on the Seroxat!\" _Dear Hadrian, why,_ he chastises himself, _did I correct his grammar?_ Closing his eyes, Geoffrey pauses for a moment to regulate his breath, attempting to calm himself. Sadness is so overwhelming and fear for the mental and physical health of a lover can cause one to consider acting outside of reason. At this moment, all Geoffrey wants to do is hunt down Edgar and pummel him into dirt. In anticipation, quite involuntarily, his hands form fists. Geoffrey has to force himself to relax them, and almost against his will, shake these thoughts away. Looking back at Dean, Geoffrey notices that his pants and briefs are still tangled around his ankles, his shirt still strewn open, exposing his chest. Geoffrey reaches for the waistband of Dean's briefs. \"Let me help you, sweetie,\" he says soothingly. After helping Dean pull his briefs and pants back into place, Geoffrey watches as Dean buttons them shut. \"You left your shirt open,\" he says. Dean stares at the wall, oblivious. Pondering the situation, Geoffrey wonders whether Dean has any benzies left. \"Do you still have some Zolam?\" When Dean only whimpers, Geoffrey becomes insistent. \"Where is it?\"\n\n\"It's probably expired.\"\n\n\"Where is it?\"\n\nDean succumbs to the inevitable. \"My top dresser drawer\u2014under the socks\u2014back right corner.\"\n\n\"I am going to get it now and you are going to take some.\" Considering Dean's current state of mind, Geoffrey adds, \"Your full dose.\"\n\n\"Please,\" Dean begs. \"I want to do this without drugs.\"\n\nGeoffrey shakes his head. \"It didn't work for you, did it, Dean?\" Adamant now, he says, \"No. You need Zolam right now. You know you do.\" As he gets up and walks down the hall to their room, Geoffrey blinks and mutters, \"Edgar Fraser.\" Dean cries softly. Geoffrey's voice fades out as he enters the bedroom, but he will make an appointment with Edgar that evening at seven o'clock. There will be several evening appointments to come since Geoffrey will insist on attending them. From their room, Geoffrey calls out, \"The boys will be home soon, Dean. You better come into the bedroom.\"\n\nAccepting his circumstances, Dean rises slowly to join Geoffrey. \"After the Zolam kicks in,\" he says softly, \"we can have sex like you wanted.\" Geoffrey doesn't hear this since Dean is only muttering to himself.\n\n* * * * *\n\n#  _Salve!_\n\n**The Eighth Anniversary of 6-13  \nHNN\u2014Melissa Eagleton Reporting**\n\nFor seven years now, I have been asked to revisit the pain that was inflicted upon our people that fateful June day in 21___. That June 13th was, ironically enough, a Friday. In the early hours of that ill-fated morning, the fanatical Christian, one Jeremiah F. Butler, believed to hail from Tex (formerly the state of Texas within the United States), drove into our border city of Augustus. At 5:57 that morning, with the summer sun already hanging high above the horizon, he was asked to please step out of his vehicle. Border Patrol Officer Acilia Zangani, who died the instant the bomb exploded, having voc'd in prior to the explosion, was quoted as saying, \"Although there is no reason to mistrust this man, something about him feels dangerous.\" The Border Patrol Office in Antinous, which later said Officer Zangani's instincts were always 100 percent accurate, approved her request to retain him for questioning. When instructed to leave his vehicle, the man must have panicked because, at that instant, the dirty nuclear bomb hidden in his trunk exploded. Five hundred and eighty-eight thousand of Augustus' citizens died that day. Their deaths were followed by another forty-four thousand: those who were at the city's northern edge or working peacefully on the surrounding soya farms. Those closest to the bomb's explosion, but far enough away not to die under its initial impact, suffered slowly from radiation burns, sickness, and the onset of various forms of cancer. Many working out on the farms would not know how their lives would be dramatically changed due to contracting cancers anywhere from six months to even as late as, yes, even today: cancers we know to have been induced by the deadly radiation. As a result of 6-13, Hadrian lost one of its greatest innovators in bioengineering, Will Middleton, the man responsible for genetically altering the soya bean to grow in our northern climate.\n\nSince 6-13, many of Hadrian's citizens have suffered from apocalyptic dreams. This is understandable since Augustus is the third city to have suffered from a nuclear attack. For some time, it appeared that the world had learned and people were unwilling to unleash such devastating brutal power against their fellow man, but it seems that fanatical faith is not prone to learn from past human mistakes. Please, fellow citizens, if you are one of the many who suffer from such horrifying nightmares, seek out one of Hadrian's many competent and compassionate psychiatrists. There is no need to suffer alone.\n\nSo there you have it, Hadrian. As well as 6-13 stealing one of our greatest minds, Will Middleton, it has left many of our citizens subjected to psychological trauma. We will never forgive the heterosexual world for condemning us. Never again will an outsider cross our borders.\n\n_Vale!_\n\n# Apocalyptic Nightmare\n\nMike's recurring nightmare always begins in an idyllic setting, at the Fulton family cottage on an isolated beach north of Antinous on the Churchill River. This is where, every summer for the first thirteen years of their marriage, Will Middleton and Mike Fulton would pack up the bubble and take Todd for vacation. As well as the golden sand that runs for miles along the river's southern shore, there is the scenic beauty of the boreal forest still growing in this region. The dream always opens here. It is always early evening and Todd is always busy building sandcastles, digging moats, and filling them with water. The scene is tranquil, the skyline lilac, interspersed with dark blue. Will Middleton, six inches taller than his partner, rests his arm on Mike Fulton's shoulder. Mike, as always, pulls in tight to Will's side, walking at a slight angle as a result. Will laughs along with Todd, enjoying the sights and sounds of his play. Then he looks down at Mike, smiles, and ruffles the shorter man's hair. Mike soaks in every gesture from his spouse. Will Middleton is not one to express feelings physically, nor verbally, so moments like this are rare. Although the sun is behind Will, his face appears radiant and Mike feels himself at his most in love. Reaching up using his toes, Mike is pleasantly surprised by Will's leaning down to meet his lips for a kiss. Todd's laughter flaps like butterfly wings in Mike's chest, causing his heart to beat rapidly. For a fleeting moment, Mike wonders whether now is the time for them to have their second child, his genetic son or daughter. Just as Mike's lips are about to connect with his lover's, Will Middleton disappears.\n\nMike is stunned. \"Will? Will?\" he cries out. And then he sees him; Will Middleton is inside a mirror\u2014 _what is he doing inside a mirror?_ Mike always wonders\u2014bent over working on a soya farm just outside of Augustus City, the city shimmering in the background. _What is Will doing outside Augustus City?_ But his question is never answered. All Mike can do is watch in horror as a thunderous explosion occurs behind Will and a mushroom cloud billows over Augustus City in the background. The ensuing winds, shooting out from the city and racing across the field, form a hand that clutches Will Middleton. Tauntingly it lifts its victim, suspending him by digging one gnarly nail into his neck. Will screams and struggles against the force lifting him, but he can do nothing to escape its grip. Slowly, Will's life force begins to drain away until all that remains is a limp bag of skin. When nothing remains of what was once Will Middleton, the wind whips his husk against the inside of the mirror shattering the glass, its sharp shards slashing Todd, cutting deep into his skin. Screaming, Todd leaps up from his sandcastle and runs wildly into his Papa Mike.\n\nIn horror, Mike Fulton stares at the boy clutching him. He pushes Todd off, but the boy climbs back on, crying, begging for his daddy. It is as if Todd is made of glue and is now stuck to him. Mike, using both hands, grabs one of Todd's arms, and with great effort, plucks it off. As soon as Mike releases his grip and reaches for Todd's other arm, the first arm, as if being pulled by magnets or gravity, slams back into him. Every time Mike plucks one of Todd's hands away, the other hand slams back into him, gripping, clutching, sucking onto him like a river leech. Suddenly, a war of gripping and plucking occurs until Mike, unable to bear it any longer, sits up in his bed, screaming at Todd to let go of him\u2014to get off\u2014to leave him alone!\n\nSweating in horror, Mike leaps out of bed and races to the linen closet\u2014now Todd's small bedroom. The closet door is open as usual, but there is no Todd. Relieved, Mike sinks to his knees. Todd is not here to hear him scream. Too many times, Mike has awoken from this nightmare to discover Todd sleeping. Never will Mike wake him to ask whether Todd can hear him. It is safe, Mike believes, to assume Todd knows nothing of his inner loathing for the boy\u2014Todd lives while his husband died. Todd lives while his responsibility to produce a child has been relinquished due to impoverished means. Todd lives while his sperm is being used to provide barren men and women with children. His children! \"At least one child should have been given to me to raise.\" Having had a loving relationship with his genetic mother, Mike had longed for his own parenting role. Riddled with guilt, Mike cries as he grieves and self-recriminates. \"I'm sorry, Todd,\" he blubbers. But his apology is insincere.\n\n* * * * *\n\n# Dating Devon\n\nJune is coming to its end, and like every other Friday afternoon that school year, Frank Hunter and Todd Middleton arrange to meet at Todd's locker. It has become a tradition for these two to spend every Friday after school having supper at the Soya Baron (where \" _the Soya Burger is King!_ \"). The Soya Baron is conveniently located across the street from Pride High. Lacking the special thumbprint to connect him to an account, Todd has no purchasing power, so it is always Frank's treat. Being more like brothers than \"just friends,\" it never makes Todd feel awkward to have Frank pay for him. Frank and Todd are best friends, having been best friends since they were in the cradle. Will Middleton, Todd's genetic father, and Dean Hunter had been best friends since their high school years. Papa Dean often reminds Todd and Frank how the boys used to sleep together in the crib. In fact, sleepovers were a common event for these two until the onset of puberty. With sexual awakenings apparent in both boys (thirteen being a critical age), Frank's fathers sat Frank and Todd down to warn them how sharing a bed now would be too awkward. Papa Mike, Frank's fathers assured Todd, would agree. Papa Mike did not participate in this crucial parent\/son discussion. Ever since Will Middleton's death, Mike Fulton had taken very little notice of Todd's growing needs. Still, Geoffrey Hunter and Papa Dean assured Todd that Papa Mike and they did not want their sons becoming sexually active at so young an age. Well, they had nothing to worry about with Todd. Although he had yearnings, he wasn't interested in having sex with anyone yet. Frank, on the other hand, unknown to his fathers, began sexually experimenting when he was thirteen. Although Frank wants to be with Todd, he has begrudgingly come to accept that Todd is still listening to their parents. It doesn't help that Papa Dean constantly reminds Todd that Papa Mike and he approve of his choice to wait until he is ready, or at least eighteen.\n\nToday, Frank is late. Todd sighs. He is starving! Leaning against his pink locker, Todd settles in for the long wait. Disappointment begins to swell in Todd's chest. Frank is already fifteen minutes late. It is not unusual for Frank to be late, but he is seldom ever ten minutes past the appointed time: three-thirty (ten minutes after the school bell). If he's later than ten minutes, Todd knows Frank will usually not show. Todd is used to Frank standing him up. It seems like the older they get, the hornier Frank gets, and the more often he ignores Todd to go off with one of his boys. Frank's absence really stings tonight because, once again, Papa Mike is working overtime. The last thing Todd wants is to spend another night at home alone. _Where are you, Frank?_ Todd wonders.\n\n\"He's on a date.\" Leaning up against the purple locker next to Todd is Devon Rankin. Devon is slightly taller than Todd, standing 5' 8\", but he has less muscle definition. He too plays b-ball, but unlike Todd, Frank, and Crystal (the three stars of Pride's Panthers and known by all as the three gay caballeros), his dedication level only landed him on the junior team in grade ten. The three gay caballeros played on the senior team last year. Even so, Devon is in good physical shape. His smile is radiant and his blonde curly locks dangle over his shoulders. Today, he is wearing his hair tied back in a loose ponytail. He wears silver stud diamond earrings and a thick silver chain to match. Leaning with his back to the locker, Devon's hands are hidden behind his backside. He offers Todd his profile, definitely his most handsome feature.\n\n\"Huh?\" Todd looks around at Devon, unaware that he had expressed aloud his thoughts concerning Frank's whereabouts. Devon smiles, exposing for Todd perfect teeth. \"Uh, we were supposed to meet,\" Todd mumbles.\n\n\"Yeah, well, I saw him leaving about five minutes ago with Davie.\" He clicks the inside of his cheek as he winks. \"Davie's pretty sweet. And they say he'll even tumble on the first date.\"\n\nTodd smirks. \"I guess he's Frank's type then.\"\n\n\"Yeah,\" Devon concurs, \"which explains why the two of you are only friends.\"\n\nTodd laughs outright at that. \"Yeah. I can't imagine myself ever looking, or acting, like the tinsel tarts Frank dates.\"\n\n\"Tinsel tarts,\" Devon chuckles. \"I like that.\" Studying Todd briefly, Devon is very agreeable at the moment. \"Yeah, I don't like guys looking like women either. I mean, what's the point? It feels too, het'ro, if you know what I mean.\"\n\nTodd assumes a serious expression. \"Yeah, yeah, I know what you mean.\"\n\n\"Now you,\" Devon says, nodding Todd's way. \"You dress the way I like a man to dress.\"\n\n\"Really?\" Todd is flattered. Devon is actually the first person to compliment his style of clothing. \"Everyone else says I look too much like a 'mechanic'.\" He feigns quotation marks with his fingers.\n\n\"Hey,\" Devon asks, \"isn't your father a mechanic?\"\n\n\"Papa Mike is.\"\n\n\"What does your father do?\" Seeing gloom in Todd's eyes, Devon immediately apologizes, \"I'm sorry, Todd. I didn't know.\"\n\n\"That's okay,\" Todd replies. \"I don't really talk about it.\"\n\n\"That's okay; I understand.\" Devon's compassion touches Todd. \"Do you mind my asking what he did?\"\n\n\"Nah, sure. Dad was an agricultural engineer.\"\n\n\"Wow!\" Devon exclaims. \"That's big!\" Suddenly Devon realizes Todd's last name is the same as the famous bioengineer Will Middleton. \"Your dad was _Will Middleton_?\"\n\n\"Yeah,\" says Todd, proud of his father's accomplishments.\n\n\"No wonder you're so good at b-ball. Will Middleton was one of the greatest college b-ball players ever.\"\n\n\"He also helped establish the soya farms in Quadrant One,\" said Todd. No one had believed Will Middleton when he had said soya would grow in the grasslands of Quadrants One and Two, considering it was not that long ago on the evolutionary scale that the region had been mostly boreal forest. But with the rapid increase in planet temperature, the temperate marshy grasslands quickly moved north, and with the onset of hot summers and moderate winters, the southern tips of Quadrants One and Two proved to be fertile soil for the valuable, lifesaving plant. With less room for raising livestock, the soya plant has become crucial in providing Hadrian's citizens with much needed protein, especially since cattle are nearing extinction. With over twenty billion people crowding the earth, very little room is left for cattle ranches. In fact, the national government only allows one cattle ranch to exist within Hadrian's walls because the methane gases released by the cattle were a major cause of global warming. Living in an environmentally sound and balanced environment is, and will always be, one of Hadrian's central principles. Todd is proud that his father was instrumental in ensuring Hadrian's ability to live in harmony with its small portion of the earth. \"Yeah,\" Todd continues, \"Dad used to travel there a lot...\" Pausing, Todd squeezes shut his eyes, trying desperately to hold back the hot tears fighting to stream out. \"He travelled to the southern tip\u2014soya farms\u2014he was there when\u2014\"\n\n\"Oh, shit!\" Devon is truly sorry. \"6-13!\"\n\n\"Yeah, well,\u2014\" Todd loses focus and begins to sob. He misses his father so much his whole body aches in agony. Will Middleton's death had spun both husband and son into such a state of despair that their parent\u2013child relationship has never recovered.\n\n\"Hey, hey.\" Devon reaches his arms around Todd and pulls him in for a hug. He rocks Todd gently in his arms, allowing him time to cry while he whispers soothing hushing sounds. There are no words for a moment like this, and Devon doesn't even try to find them.\n\n\"I'm sorry,\" Todd utters through gasps. \"It's been eight years; you'd think\u2014\"\n\nDevon won't even let him finish. \"No, I wouldn't be able to get over it.\" Still rocking Todd gently, he says, \"You cry as long as you want. As long as you need!\" Todd accepts Devon's offer and releases a flood of pent up grief. He has only been able to share this level of emotion with Frank and Crystal. Having another person feel empathy is relieving.\n\nAfter Todd is spent, Devon asks whether he'd like to have dinner with him. \"At your place?\" Todd asks.\n\nPleased that Todd is responding on a positive level, Devon replies, \"I was thinking maybe we could go to a Japanese restaurant. How about _Rezu_?\" _Rezu_ is Japanese slang for lesbian. The restaurant owners have chosen to embrace the slang term and make it their own.\n\nTodd shudders. \"Oh, man, I can't\u2014I mean\u2014\" Todd is not sure how to tell Devon he doesn't even have a thumbprint, let alone credits.\n\nDevon smiles. \"I'm asking for the date, so it's my treat.\" After waiting a moment without reply, he asks, \"So, what do you think?\"\n\n_A date_? Todd wonders what this might mean. _He won't pressure me, I don't think._ Remembering Devon's description of Davie (tumbles on the first date), Todd feels reassured that Devon isn't likely to expect anything. \"Okay,\" Todd replies, still sounding uneasy, \"if it's not too expensive\u2014I mean, I've heard _Rezu_ is pricey\u2014I don't want you blowing all your credits on me.\"\n\n\"Nothing's too pricey for my first date with you.\" Laughing, Devon says, \"I can't believe you said yes. Man, I've been wanting to ask you out all year.\"\n\nTodd blinks, \"Really? Me?\" He shakes his head in wonder. \"I'm nothing special. I'm certainly not good looking or\u2014\"\n\nDevon puts a finger to Todd's lips. \"You are a prize, my friend. Half the guys at Pride want to date you, but we're all too scared to ask.\"\n\n\"Why?\" Todd is truly dumbfounded.\n\n\"Mostly out of fear of Frank,\" Devon answers. Suddenly realizing Todd really doesn't understand why anyone is afraid of him, he adds, \"But, also because you're the best damn b-ball player on campus. You got Rookie of the Year at Quadrants. I think everyone's intimidated by that.\"\n\n_Intimidated by me?_ Todd finds that concept hard to believe. It is amazing how humble a man can be when he is afraid of what other people might think.\n\n\"Come on,\" Devon says, squeezing Todd's shoulder muscle, in anticipation. \"I got my mothers' bubble. We'll drive to _Rezu_.\" Devon's mothers own a 'double bubble,' a slightly larger vehicle designed to sit four. Only registered couples with one or two children are eligible to purchase one. All other residents of Hadrian are confined to owning a single bubble. Once partners register as a legally bound couple (the equivalent of marriage), they are required to sell off one bubble, reducing the number of vehicles driven throughout Hadrian, thus limiting congestion and the number of road accidents. It also helps ensure less energy waste. When a registered couple is preparing for the arrival of a newborn, they are free to trade in their bubble for the family-sized double bubble.\n\nAs the two boys approach the small vehicle, Devon laughs, remembering when he and his mothers went in five years ago to trade their old double bubble in for a newer, more reliable version. The salesman had a bowl of an odd-looking gum on his desk. He said it was called double bubble! \"Did you know that the name double bubble once was a type of chewing gum?\" Devon asks Todd. \"From way back when.\" Just to add emphasis to how long ago this really was, Devon not only extends the vowel in 'way' but also gestures with open palms a very long extension. Todd shakes his head. \"Yeah, the guy who sold us this bubble told us all about it. It's a small pink rectangle with an indent in the middle. It comes wrapped inside a joke comic. When I first tasted it, I thought it was really great\u2014super sweet, but the flavor only lasted a few minutes. The man said they used the original recipe to make it. I mean it was really good, but for such a short period. I ended up spitting it out.\" Todd's look of surprise that anyone would so blatantly litter encourages Devon to add, \"Don't worry; I learned my lesson. Mama Rena caught me and gave me such a tongue lashing when we got home.\" Laughing, he adds, \"Mama Rena believes a spanking every now and again never hurt anybody, so she actually paddled my backside that day.\" Reassuring his new friend, he adds, \"I've never done anything like that again.\"\n\nTodd nods his approval. \"You're lucky only your mama caught you!\"\n\n\"Tell me about it,\" Devon agrees. Both boys shudder at the thought. No one litters in Hadrian. The fines if caught are astronomical, and if you amass a record of three offenses, the government will exile you from Hadrian!\n\nDevon opens the bubble for Todd by tapping on the roof, causing the transparent door to slide down. Todd steps up and then down into the small vehicle. It looks so small that it seems impossible for anyone to fit into it, but it is surprisingly roomy inside. Even a man of 6' 5\" can sit inside a bubble and not feel cramped. After settling into the driver's side, Devon turns Todd's way. \"I can't wait till we get to _Rezu_.\" Smiling, he adds, \"I don't know about you, man, but I'm starving!\" As if on cue Todd's stomach growls, causing Devon to laugh. \"Good, you're hungry too! I know just what to order us when we get there!\" Todd can't help but laugh along with Devon, his playful mood infectious.\n\n* * * * *\n\n#  _Salve!_\n\n**Rezu\u2014Still Hadrian's Top Restaurant  \nHNN\u2014Melissa Eagleton Reporting**\n\nWhen was the last time you took your lover out for a quiet romantic meal? Where better than Hadrian's oldest restaurant, _Rezu_? One of Antinous' most popular restaurants and listed as Hadrian's top dining establishment in the bestselling restaurant guide: _Hadrian's Finest Victuals_ , published last year through Sappho Press. _Rezu_ is nestled nicely in the corner of Backstreet Bay where it offers its customers a scenic view of the Nelson River that complements its tantalizing cuisine. Opened the year of Hadrian's founding, _Rezu_ has always held its mission to be to provide Hadrian's citizens with an authentic taste of traditional Japanese cookery. Kyoko Yokomoto was among the first immigrants to join Hadrian's family. With her, she brought everything she knew about preparing traditional Japanese food, having apprenticed under her father at his restaurant in what was then still Vancouver, British Columbia, Canada. Her daughter, Etsuko Yokomoto, inherited _Rezu_ , and today Etsuko's daughter, Chiyo, and grandson, Jun, run the business. _Rezu_ still upholds the quality that founded this fine eating establishment. You are guaranteed one of the finest meals in all of Hadrian when you visit _Rezu_. Bring your lover and reserve a private room. There is no better way to spend the perfect romantic evening than dining at _Rezu_.\n\n_Vale!_\n\n# Rezu\n\nDevon offers his thumb to the host, teasing him with little wiggles back and forth. The man adeptly retrieves his pad, allowing Devon to press his thumbprint on the face of it. Subsequent to his print, Devon follows up with a few taps, transferring credits from his account to the host's. Devon's generous tip opens up for him one of _Rezu_ 's private rooms. Before sliding open the thin rice paper doors, delicately decorated with cranes in flight above a low bamboo marsh, Todd and Devon respectfully remove their shoes. The tatami mats, they are told, are very old and almost impossible to replace since Hadrian has limited imports from the outside world. The devastation to Quadrant One caused by 6-13 radically diminished world trade, having created for Hadrian a barometer of fear that seems permanently stuck in the red zone, resulting in Hadrian's choice to become xenophobic. There will be no new tatami shipped to _Rezu_ for quite some time.\n\nUpon entering the room, Todd is impressed by its sparse but beautiful decor. Along one wall is a low stand with a flower vase and small statue. \"This,\" Devon explains, \"is what my moms calls the tea room.\" Smiling sweetly, he adds, \"She says you are supposed to admire the beauty of the flower arrangement and art work.\" In this case, the art consists of a small statue of a crane taking flight, and a print of _Courtesan_ by Hokusai. \"It's supposed to put you in some kind of mood,\" he says slyly.\n\n\"Wow, that's beautiful,\" Todd responds, transfixed by the beauty of the courtesan.\n\nDevon lies on his side, his head closest to where Todd is seated. When they first entered the room, Devon motioned for Todd to sit on the right side of the long, low dining table, but Todd judiciously chose to sit at the head where there was only room for one body. Lying sideways is Devon's attempt to get in close to his new partner.\n\nAlthough the drinking age in Hadrian is twenty-one, Devon is able to schmooze the young waiter with a few circular motions of his thumb. The Rankin family often patronizes _Rezu_ and is known for its exorbitant tips. Soon the two boys are drinking _biru_ (Japanese for \"beer\" Devon translates for Todd) and warm _saki_. \"Rice wine,\" Devon explains. \"Very strong rice wine,\" he adds with a chuckle when Todd sputters out his first sip of the strong bitter alcohol. \"Trust me,\" Devon winks. \"Keep drinking. It's strong enough that it'll start tasting good soon enough.\" The two boys laugh, and Todd, taking Devon's advice, continues to sip and scrunch up his face in disgust. Soon their meal arrives. Devon, knowing the menu well, orders them a variety of sushi with tempura. Todd balks at the idea of eating raw fish, but he has no problem gobbling down the deep-fried and lightly battered vegetables and tiger shrimp. And even though he has never eaten tofu on its own like this, he is amazed by how good the white block covered lightly with dried fish flakes tastes when dipped in a combination of soya sauce and rice wine vinegar.\n\n\"By all that's gay and glorious,\" he declares, \"but this is good food!\"\n\nDevon consumes the majority of the sushi, the most expensive portion of the meal. He is a little disappointed that Todd won't even try any of the raw fish placed delicately over the sweet rice. He ends up overstuffing himself because he had ordered enough for a whole murder of crows. Devon really wants to impress Todd and win him over as his boyfriend.\n\nAttempting to use the chopsticks, having only learned that night, Todd fumbles to pick up another piece of tempura, the light batter coating an orange slice of something truly delicious. \"Umm,\" Todd says as he barely manages to pop the item into his mouth. Talking with his mouth full, Todd spits out small bits of the food in the explosion of words. \"What's this stuff called?\"\n\n\"That's yam.\" Devon is enjoying the sight of Todd luxuriating over the food he is consuming. When Todd trickles some of the soya sauce rice wine mixture down his chin, Devon leans forward, and with his index finger, he wipes the stray dribble and ensures it (along with his finger) makes its way inside Todd's mouth. There is a moment of silence, in which Devon feels a flame and Todd feels awkward. \"Nice stubble,\" Devon says admiringly. \"Most guys I know can't even shave yet, let alone hope to grow stubble.\"\n\nEmbarrassed that he is not presentable, Todd self-consciously rubs his cheek. \"Ah, Papa Mike forgot our razor at a friend's house.\" Devon chuckles. Lowering his eyes, feeling shame, Todd admits, \"Papa Mike said he'd, ah, bring it home, but he forgot again. So, I, ah, couldn't shave this morning.\"\n\n\"That's okay.\" Lightly licking his lips, Devon giggles, \"I've never kissed a guy with stubble before.\"\n\nTodd slowly sets the chopsticks down. _I'm going to have to do this,_ he figures. _Devon's used up a lot of credit here tonight._ He swallows hard, sips a little sake, and then closing his eyes, turns his head and waits for Devon's lips to touch his own. Todd had only meant for a quick little peck, but Devon's lips cover his, his teeth gripping Todd's lower lip. And then Devon's tongue slips inside Todd's mouth. Todd panics and pulls away. Devon is not worried. \"Wow.\" He smiles. \"That was nice.\" Todd blushes, unable to speak, muted by embarrassment and anxiety. Devon peers up at Todd. Noting the dark flush, he asks, \"Am I the first boy you ever kissed?\" Todd shakes his head. \"Of course not,\" Devon reasons, \"Frank, right?\" Todd nods. \"Of course, Frank. But you're not his type, so that's why the two of you never went anywhere.\"\n\n\"We...we're...we...are...friends,\" Todd stutters.\n\nDevon runs a hand up Todd's leg. Todd's hand flutters against Devon's and lowers it to the floor. Devon pushes his side up and rests firmly on his elbow. \"Are you still a virgin?\" Devon knows the answer as Todd's face turns dark purple. \"Wow, a virgin,\" Devon mutters in awe. He is now lying on his back, hands cupping his head. \"Hey, man, that's cool.\" Lying back on his side, Devon once again faces Todd. \"We all start this world off as virgins.\" Truly admiring Todd now, hoping he will be Todd's first, Devon asks, \"Have you thought about when you want to do it?\"\n\n\"Ah...\" Todd can't look up. He has never been good talking about sex. His father died when he was only eight so Will Middleton had never had a chance to discuss awakening sexuality with his son. Papa Mike, unfortunately, turned aloof after Will Middleton passed on, lacking both desire and energy to raise Will's son properly. He did, at least, give Todd some good advice once, shortly after Todd had turned thirteen. \"Papa Mike says I shouldn't...um...\" Todd swallows uncomfortably, \"do anything, until...well, just, until things feel right. He says I should w\u2014\" Certain this next line is going to make him sound foolish in Devon's eyes, he cuts himself off.\n\n\"Hey,\" Devon says kindly, \"you don't have to be embarrassed. I'm not going to make you try and do anything tonight.\"\n\n\"Thanks,\" Todd sighs, relief heavy in his exhale.\n\n\"So, what else did Papa Mike say?\"\n\n\"He...um...thinks I should...um, well...maybe wait until I get married. That's what he and my dad did.\"\n\n\"Really?\" Devon is amazed. Although Hadrian has a very strong notion of family\u2014only registered couples, committed to monogamy, confirmed in marriage a minimum of ten years, can apply for a third child\u2014there are no religious laws dictating a couple wait until marriage before having sex. \"Why?\" he asks, truly dismayed. \"I mean, sorry,\" he says in response to Todd's look of alarm. \"It's just, that's a little unconventional. Most people don't worry about that sort of thing. I mean, if it feels right, why not share what you've got with the other person?\"\n\n\"I guess that's what Papa Mike means.\" Todd is so embarrassed now that he is studying his knees, wishing the rest of him could be ducked under the table, too. \"He just thinks I need to wait until it feels right.\"\n\n\"Yeah,\" Devon says. \"That makes perfect sense.\" Since that expression gives him hope, Devon reaches his hand around Todd's head, pulling him down onto the tatami mat flooring for another kiss. Not wanting to scare Todd off, though, he just lies there with him, kissing until Todd pulls free and asks Devon to take him home. \"Sure,\" he says.\n\nDevon wraps his arm around Todd's waist and they exit their private room. Meeting up with their waiter near the front entrance, Devon thumbs the man's pad, tapping in a hefty tip of extra credits. As they leave _Rezu_ , Devon leads Todd back to his mother's bubble. Playing the gentleman, Devon moves to open Todd's door for him. Before opening the door, though, Devon summons up his courage, \"I'd like to see you again. Can we go out again tomorrow?\"\n\nDevon's heart drops as Todd shakes his head. \"I, ah...\" Todd swallows. \"Tomorrow I promised Papa Mike I'd help him fix the bubble. I...I won't be able to...both day and night...busy.\"\n\nDevon wraps his arms joyfully around Todd, hugging him close. \"How about Sunday afternoon? Are you helping Papa Mike then, too?\"\n\nTodd gives in and puts his arms around Devon. \"No.\"\n\n\"We could go to The Cattle Ranch Restaurant that opened up last month.\"\n\n\"The Cattle Ranch Restaurant?\" Todd turns pale, gasping. \"I heard a beef burger alone costs fifty credits!\" A credit in Hadrian is akin to what the old country of Canada used to consider a dollar bill. Anything less than a credit is called a chit; one hundred chits equals one credit.\n\nDevon laughs, finding Todd's level of nativity appealing. \"Not quite that bad. A beef burger is only thirty credits.\"\n\n_Only?_ Todd shakes his head. Thirty credits are enough to buy groceries for a week: soya beans, soya milk, soya bread, soya cheese, a few greens, and one serving each of fish or chicken (a leg or a thigh\u2014not the breast). The concept of tossing that many credits away for one little slab of meat is beyond comprehension for someone barely scraping by like Todd and his Papa Mike. _Rich people_ , he muses cynically.\n\n\"Yeah,\" Devon continues, oblivious to Todd's musing. \"It is, if you ask for all the extras like a bun, lettuce, tomato, onions, sauce that sort of thing. A beef burger can easily run up to fifty credits if you're a fussy eater.\" Grinning, he adds, \"Like me.\"\n\n\"No, Devon,\" Todd states firmly.\n\nAlthough disappointed since a meal at the burger establishment always leaves one's partner feeling obliged to put out, Devon can tell Todd is serious. \"Okay,\" he reconsiders. \"How about the zoo? That won't cost too many credits. A few credits each to get in, a little more on top for junk munch, which...\" he adds in his own stern manner, \"we can't live without.\" Todd chuckles; everyone loves junk munch and it is relatively cheap. Papa Mike brings some home every second or third week, so it can't be that expensive. Smiling sincerely, sensing Todd's leaning, Devon asks, \"You can afford that, can't you?\"\n\nTodd's mood darkens as he shakes his head. \"I...\" _How do I tell someone this?_ \"I...don't have a thumbprint\u2014or a credit account.\" Todd lowers his head in shame.\n\nDevon blinks. \"Why? Won't your Papa give you one?\" An individual without a thumbprint implant has no purchasing power, which is why nearly everyone in Hadrian over the age of twelve has a thumbprint.\n\n\"We just don't have the credits\u2014\" Todd trails off, feeling deeply his shame.\n\n\"Hey,\" Devon says soothingly, \"let me pay for the zoo. Come on,\" he cajoles Todd. \"It's nothing expensive like The Cattle Ranch Restaurant or _Rezu_. Again, it's me asking for the date: my date, my credits. What do you say?\"\n\n_The zoo_. Todd begins to smile against his will. Todd hasn't been to the zoo since before his father died.\n\nDevon, noting the slight transition, tickles Todd into a more playful mood. \"Come on,\" he encourages. \"It'll be fun.\"\n\nTodd nods his head slightly. \"It does sounds like fun!\"\n\n\"Are we on then?\" Devon asks as he leans in for a kiss. When he finally releases Todd's lips, Devon waits for him to say yes. \"It's a date, then.\" After planting another kiss on Todd, Devon laughs gaily as his new beau blushes. \"Hadrian's lover, you are so cute,\" he declares.\n\n\"Cute?\" Todd sputters.\n\n\"Not cute like one of Frank's boys' cute.\" Shaking his head slightly and laughing, Devon adds, \"What did you call them?\"\n\nBoth boys answer, \"Tinsel tarts.\"\n\nAfter a further chuckle, Devon says, \"It's just refreshing to be with someone for whom all this is new.\" He kisses Todd again and then asks, a little tentatively but with hope, \"Can we tell everyone at school it's official; that we're a unit?\"\n\nTodd shrugs, \"I...I guess so. Just...just...\" Trying not to sound too pathetic, Todd says, \"Just don't push for\u2014you know.\"\n\n\"I won't,\" Devon promises and then hoots out his joy. \"One more kiss, one more kiss\u2014just one kiss and then I'll take you home. I promise.\"\n\nTodd complies.\n\n* * * * *\n\nBefore getting out of the bubble, Todd turns to face Devon. Thinking Todd is going to initiate a kiss, Devon leans forward. Not quite knowing what to do, Todd kisses his cheek. Devon giggles and turns Todd's head to face him, kissing him fully on the lips. \"We've come this far,\" he smiles. \"I won't push forward, but I'm definitely not going backwards,\" he says coyly. \"Did you want to say something?\" he asks to help relieve Todd's tension, which, oddly enough, Devon finds very erotic.\n\n\"I...ah...forgot,\" Todd mutters.\n\nDevon laughs. \"Well, if you remember, voc me. My I.D.'s devran\u2014\" he pauses, taking a moment to kiss Todd before continuing, \"\u2014hgvoc.\"\n\nToo embarrassed, Todd is unwilling to reveal to Devon he doesn't even have a vocal contact lens. The latest in high tech communication is too expensive for Papa Mike. If Todd were to get a part-time job, he could probably buy one, but Papa Mike has asked him to concentrate on school and b-ball. \"A b-ball scholarship is the only way you are going to get into uni, Todd,\" Papa Mike had said, shaking his head sadly. \"We just don't have the credits.\" No thumbprint means no access to spending money; no vocal contact lens means limited access to the wave, especially since Papa Mike won't even pay to fix their wall screen. \"It saves credits,\" he reasoned. \"This way we don't have to pay government rates for the wave link.\"\n\n\"But we could get the reduced rate, Papa Mike,\" Todd had pleaded. But Mike Fulton is too proud to apply for services made available to low income families, so they don't even get the wave coming through their wall screen anymore. _Thank Hadrian for school!_ All educational institutions provide free access to the wave, but only through school slates, and the wave only works when at school. So Todd has to live with old school technology and the generosity of his friends. This past New Year's Eve, Frank and Crystal had given him an old cell phone with unlimited calling (a three-year contract obtained illicitly through an outsider phone company!) in celebration of Hadrian's fiftieth birthday.\n\n\"Thanks,\" Todd says, smiling sheepishly for Devon. Tapping his temple, Todd pretends to have memorized Devon's voc I.D., suggesting it is something he can actually use. Then, after \"one last kiss,\" Devon opens the bubble door. Todd jumps out, and as quick as a rabbit chased by coyotes, he runs inside his house.\n\n* * * * *\n\n#  _Salve!_\n\n**All but Extinct  \nHNN\u2014Melissa Eagleton Reporting**\n\nLooking for a way to spend your Sunday afternoon? Why not take in Hadrian's Zoo? Hadrian boasts the last remaining zoo planet-wide. Pre 6-13, it was a hot spot for tourists around the world. With fewer visitors due to closed borders, Hadrian's Zoo is suffering from financial distress. Rest assured, Hadrian's National Government will not allow this worldwide treasure to lapse into disrepair, or risk the deaths of any of its animal occupants, many of which are the only living relics of their species. Lucy the elephant, for example, is the last of her kind. Thanks to the preservation of sperm and eggs dating back to the mid-twenty-first century, our zoologists have been able to impregnate Lucy, as they had her mother and her grandmother before. The zoo has always been one of Hadrian's most affordable locations, costing a mere three credits for adults, two credits for students and senior citizens, while children under twelve get in for free. If numbers do not improve, though, zoo officials say they may have to increase admission credits to as much as fifteen credits for adult admission, ten credits for students and seniors, and they are even considering a charge of five credits for children between the ages of two and twelve. These high prices, zoo officials claim, may be necessary for them to maintain the various hothouses they have for growing exotic foods as feed. Then there is the difficult and very expensive chore of regulating the temperatures of their animals' living quarters. Pearl, for example, being from the now extinct species of polar bear, requires a much colder environment than Lucy, the elephant. Remember, Hadrian is preserving our planet's vast array of animal species in the world's last remaining zoo. Your patronage will help ensure the future of our planet's wildlife. One day, after the hordes of heterosexual barbarians finally die off, the future citizens  \nof Hadrian will reclaim the planet. We will be the earth's surrogate parents to its future wildlife population. So, don't delay; spend your Sunday afternoon at Hadrian's zoo!\n\n_Vale!_\n\n# Hadrian's Zoo\n\nTime spent at the zoo is such an incredible experience that Todd begins to think he might actually come to like Devon that way, _maybe_. Their walk through the various animal exhibits is both awe inspiring and a sobering one; both boys remain silent, staring at what is said to be the last of its species. Such signs are posted everywhere. \"Have we seen an exhibit without one?\" Todd asks solemnly.\n\n\"I don't remember one,\" Devon replies. Not wanting their date to be so gloomy, Devon decides to liven things up. \"I'm hungry,\" he chimes.\n\n\"You're always hungry,\" Todd reminds him.\n\n\"I'm a growing boy, with a hollow leg, as my moms like to say.\" They laugh as Devon leads them to the closest junk munch stand.\n\nDevon insists Todd let him buy them soya dogs, causing Todd to protest. \"You're spending too much on me, Devon. I mean\u2014you know I don't have credits or a thumbprint.\" It never seems to matter to Todd that he's told Devon this fifty times already. Something inside keeps urging him to repeat himself. Perhaps Todd is hoping Devon will finally understand why he feels so uncomfortable. \"All our credits go to paying off Papa Mike's debts.\" Actually they mostly go to buying whiskey, hallucinogens, contraband cigarettes, and taking out various young men, but Todd doesn't wish to share this information with Devon. \"He can't spare a single chit, let alone get me a thumbprint.\"\n\n\"Come on, Todd,\" Devon says reassuringly. \"None of that matters to me. I don't care if you don't have any credits. But,\" trying not to sound judgmental, \"why don't you get a job? Your employer would have to arrange for the thumbprint, and then it would be free. Once you got that, you could set up your account with your first download.\"\n\n\"Papa Mike wants me to concentrate on school and b-ball. They're my only hope for uni.\" Sighing, he adds, \"And I want to go to uni. I want to be a bioengineer like my father.\" Sensing he is not getting through to Devon ( _why can't rich people understand?_ ), Todd gets straight to the point. \"The fact is, Devon, it makes me feel awkward watching you throw away all those credits.\"\n\nDevon laughs. \"I'm having fun. Besides, I like spending credits on you.\"\n\nTodd sighs. \"It just doesn't feel right. It's like...\" He pauses to consider his words carefully. \"It feels like I'm using you.\"\n\n_I wish you would,_ Devon thinks in response, but he merely says, \"Well, you're not. I want to spend credits on you. I want to do things for you.\" Devon is sincere. \"I want to do stuff with you. I don't care if it costs a few credits. Why not? I've got it.\"\n\nFrustrated, Todd sighs deeper. \"It's just, well, maybe we could do something less expensive. I mean, something that doesn't require credits.\"\n\nDevon shakes his head, flabbergasted, unable to conceive of anything that wouldn't cost a fortune in credits to have a good time. \"Like what?\"\n\n\"I don't know. Go for a walk in the park or\u2014I like working out. Maybe we could get together and train for b-ball.\"\n\n\"The school year's almost over,\" Devon replies, a little bemused. \"That makes no sense.\"\n\n\"You know you're pretty good,\" Todd encourages him.\n\n\"Right,\" Devon mutters. \"That's why they kept me on juniors this year.\"\n\n\"No, really,\" Todd says. \"With a little training, I bet you could make first string on seniors next fall. I mean, why not? We got all summer.\"\n\nDevon smiles, _All summer\u2014he's thinking long-term dating now!_ \"Are you serious?\" Devon is beaming. Hooking up with Todd Middleton is proving to be the highlight of his year! \"I would absolutely love that!\" He hugs his new boyfriend, smothering him in kisses. Embarrassed, Todd pushes Devon off. \"Hey, come on,\" Devon says, dismayed. \"What's wrong?\"\n\n\"We're...\" Todd's eyes shift side-to-side. \"There's people...\" His head tilts to the right, indicating two women walking arm-in-arm toward the aviary.\n\n\"I keep forgetting how private you are.\" Although slightly annoyed, Devon is still thrilled at Todd's offer to help him train for b-ball. \"No more kissing in public, I promise.\"\n\n\"Thanks,\" Todd replies shyly.\n\n\"Time for dessert,\" Devon announces merrily.\n\nTodd shakes his head. \"We just agreed not to use up any more credit.\"\n\n\"No,\" Devon counters. \"We just agreed that some of our dates could be credit free, like training.\" Willing to compromise, he suggests, \"How about this? When you pick the date, we'll do it your way. When I pick the date, I get to spend as many credits as I please. Come on,\" he says encouragingly. \"Let's shake.\" He offers his hand. \"I'd say, kiss on it, but...\" he pauses, looking upward and tilting his head, \"we're in public.\" Todd nods in agreement and they shake. For the rest of the afternoon, Devon fills Todd up with every form of sweet treat and junk munch he can convince his new boyfriend to eat, including a red candy apple and a caramel one!\n\n* * * * *\n\n#  _Salve!_\n\n**Hot Summer Nights  \nHNN\u2014Melissa Eagleton Reporting**\n\nIt's another hot Hadrian summer day! Can you believe this used to be one of the colder regions in the old country of Canada? Believe it or not, Hudson Bay once had an ice-free period of only four months, dating back, of course, to the late seventeenth century! One would have expected, though, with the radical decrease in salinity in Hudson Bay's waters, that there would have been a decrease in the ice-free time on the ocean-sized inland lake. Fresh water freezes much faster than salt water. Unfortunately for much of old Canada's northern wildlife, such is not the case with Hudson Bay. The constant rise in global temperatures has made our summers hot and our winters warm, far too mild for the bay to freeze over for any substantial length of time. During the mid-twentieth century, the mean temperature for Hadrian this time was year was 10\u00b0 C. Today we enjoy a mid-June mean of 30\u00b0 C. Also, winter often used to arrive in late August, early September. Now, old man winter (who has aged to the brink of death) seldom worries us until mid-to late December. Yes, global warming has changed our planet's ecosystem considerably. Hadrian, which was once mostly boreal forest, is now one of the most fertile of grasslands regions around the globe. It is our mild winters and hot sultry summer nights that made Hadrian one of the world's top tourist spots prior to 6-13! The loss of the foreign tourism industry has been a blow financially, but I am sure I speak for everyone in Hadrian that it is no loss, but rather a relief, no longer to have to cater to those hypocritical heterosexual barbarians.\n\n_Vale!_\n\n# Summer Craze\n\nIt is a crazy and frustrating summer for Devon. The closer he feels he is getting to Todd, the further Todd seems to be pulling away from him. One night, their last weekend before school starts up, as they are sitting on the bed in Devon's room, watching one of Devon's voc vids through the wall screen, Devon begins to come on to Todd. This time, he is pushing for more than just a few kisses. He wants them to undress so they can see each other's bodies. \"We don't have to do anything serious,\" he reasons, \"just touch each other.\" Todd's heart starts to pound inside his chest. He can barely breathe, let alone try to answer. Devon takes his silence for consent and he begins to reach his hand inside Todd's pants.\n\nTodd leaps off the bed like someone had just lit him on fire. \"Stop! Stop it!\" he shrieks.\n\n\"What's going on in there?\" Alisha Rankin calls from the living room.\n\n\"Nothing! We're okay, Mom,\" Devon responds. Standing up, he turns on Todd. \"What's wrong with you?\" he whispers. \"Do you want my moms coming in here?\" Annoyed, he gestures for Todd to sit back down on the bed.\n\nTodd refuses, squeezing his eyes tight. \"Just, please, don't touch me there. I'm not ready yet.\"\n\n\"Why not?\" Devon is perplexed. \"It will feel good if you let me.\" Shaking his head in wonder, he adds, \"I could make you cum.\" When he gets no response but a further lowering of Todd's head, Devon demands, \"Look at me, Todd.\"\n\n\"I...\" Todd can't seem to open his eyes.\n\n\"Okay,\" Devon sighs. \"I won't touch you. But baby...\" He kisses Todd lightly. \"I'm aching here. I want to feel you touch me.\" Devon takes Todd's hand in his and, though Todd shakes he allows Devon to manipulate him, as if he were a puppet. Soon Todd's hand, held in place by Devon's, is gripping Devon's penis through the rough folds of his jeans. Devon groans and kisses Todd. Squeezing Todd's hand, Devon gasps, \"Oh, baby, put your hand inside my pants.\" Although Todd struggles some, Devon, having unzipped his jeans, moves Todd's hand up through to his cotton briefs. Slipping Todd's hand through the front opening, he helps Todd grasp his penis. Devon groans in ecstasy.\n\nUnable to cope, Todd pulls his hand free. \"I can't do this,\" he pleads.\n\n\"For Hadrian's sake, Todd.\" Devon is angry. \"I have been waiting all summer for this.\" Exasperated, he has to slow down his breathing to avoid yelling. \"I can't take this anymore. I want you and I know you want me.\"\n\n\"No!\" Todd shrieks. \"No, I don't.\" Todd is terrified. Feeling Devon like that has repulsed him. He notices the hurt look in Devon's eyes. \"I...I... I'm just not ready. I don't want sex right now. I just want to date...to hang out\u2014not sex\u2014not yet.\"\n\n\"When?\" Devon is becoming quite demanding.\n\nTodd is in tears. \"I don't know.\"\n\n\"No.\" Devon shakes his head angrily while zipping up his pants. \"No, you don't, do you?\"\n\n\"I'm sorry, Devon,\" Todd pleads. \"I just. I can't.\"\n\n\"There is no can't about it!\" Devon spits. \"You won't!\"\n\n\"It's not like that,\" Todd tries explaining.\n\nDevon turns his back. \"Get out.\" Todd is stunned. He didn't expect this. Even with his back turned, Todd can feel the anger glaring out Devon's eyes. \"I said get out!\" Turning and pointing toward the door, Devon yells, \"I'm through with you.\" Stunned, Todd turns, makes his way out of Devon's room. Blindly he makes his way to the back door and heads home. He has to walk the five miles since Devon is so angry he doesn't offer Todd a ride.\n\n* * * * *\n\n#  _Salve!_\n\n**Back to School  \nHNN\u2014Melissa Eagleton Reporting**\n\nWell, it's that time of year again, children, teenagers, and parents! School has started! As of the wee hours, with the sun still rising early in the sky, Hadrian's youth were seen walking back toward school. It is always an exciting time of the year when friends who lost touch over the summer months are able to reunite inside the halls of academia! There is no doubt in my mind that Hadrian's education system is by far the finest in the world. Our children come first in Hadrian, and preparing them to meet the challenges they will face upon commencement in the adult world is our teachers' top priority. At Antinous' own Pride High, for example, students study courses in our planet's ecological history, the impact of climate change, and the ways we in Hadrian are working toward a positive reversal, or at the very least, a partial resurrection, such as we have obtained in Hadrian, of the earth's natural resources. Some students look forward to classes in population control while others will approach life from a much lighter perspective. Pride High is renowned for its glorious fashion and cosmetics programs. Whatever your son or daughter chooses to take in preparation for his or her future career, you can be certain that schools like Pride High will be working toward the betterment of your child's education! As Principal Gavin is often heard to say, \"Pride High! Where all our children walk with heads held high!\"\n\n_Vale!_\n\n# Back to School\n\nTodd is not looking forward to going back to school. Breaking up with Devon was so sudden and unexpected that Todd is left profoundly confused and isolated. Devon refuses to answer any of his wave messages, and Papa Mike never seems to be around these days, leaving Todd with no one to talk to. Compounding the matter is that Frank and Crystal are miffed at him. He never called either of them over summer vacation, being too busy dating, and now when he calls Frank, all he gets is \"Can't talk, Todd; I'm busy.\" Whenever he tries calling Crystal, all he gets is a generic recording telling him \"Crystal Albright is currently unavailable.\" _She must have turned off her phone,_ Todd reasons. Now it's Sunday night, and tomorrow is the first day of school. Todd knows his classes. His first class is math with Ms. Sterne (which causes him to grimace), followed by Hadrian history with Mr. Reiner (which brings on a smile). Frank and Crystal are taking those classes, too. Last spring during fall registration (and pre-Devon), the three friends had tried to get into as many of the same classes as possible, but they weren't so successful with the afternoon. After lunch, Todd has physical education while Crystal is studying world religion and Frank is taking fashion studies. For their last class of the day, Todd is enrolled in agricultural sciences, Frank in cosmetology, and Crystal has a spare. _At least I'll be with my friends in the morning, or will I?_ Todd wonders. Worried, Todd realizes he should have made an effort to spend time with his friends over the summer, but Devon had sucked up all of his time.\n\nHis break-up with Devon hits Todd hard. He really thought they were forming a strong friendship, granted a friendship in which Todd felt obliged to kiss, but they were sharing some really deep thoughts. They talked about the concept of a god or gods, and why some of Hadrian's citizens still believed in these deities regardless of the obvious hate literature toward homosexuals in many of their holy texts. They also argued over whether Hadrian citizens had subconsciously made gods of Hadrian and Antinous, and both expressed confusion over why some citizens slipped outside the wall, simply to vanish. Why anyone would want to live in the outside world was unfathomable. And they always shared a good laugh over the boys' and girls' washrooms remaining segregated. \"That is so _strai_ ,\" Devon would always say. \"I know; how stupid,\" Todd would respond, and then they both would utter, \"Like, that makes any sense!\" That this expression was simultaneous every time the boys had this conversation always made them laugh. But, most of all, they had shared a love of b-ball. Todd had really brought Devon a long way with the game. Without doubt, Devon is now first string material. _B-ball's going to be toughest of all,_ Todd realizes. _But I'll make it work, somehow. It's good for the team,_ he reckons, _and I will recommend Devon to Coach Miller._ As always, when it comes to b-ball, Todd lines up the team and its various positions in his mind. _Frank is center,_ he decides. _He's tall, fast, and is almost unbeatable with his jump ball. Crystal is point guard, I'm shooting guard, Millicent will take small forward, and Devon will be perfect as power forward._ The thought _Devon will be perfect_ cloaks Todd like a dark cloud shadowing over the moon. Todd is so confused about his feelings for Devon; at times, he simply can't contain his emotions.\n\nEverything with Devon happened so fast and ended just as quickly. Todd still hasn't gotten over the staggering blow. What Todd fails to understand is that Devon, too, is smarting. While Todd was building a friendship, Devon was falling in love. When Devon had asked Todd out, he wasn't looking for a friend, he was looking for a lover; and although he tried hard to understand that Todd needed to go slow, when he saw in Todd's eyes that they would never make love, he lashed out, angrily tossing Todd aside. Devon is now a walking wound, and thoughts of Todd, the very sight of him, are like salt water for him.\n\nDay one of school is brutal. Todd and Devon are in the first three classes together. Devon inevitably arrives early and is already sitting in Ms. Sterne's math class when Todd walks in. As soon as he sees Todd, Devon starts singing an old tune dating back to the late twentieth century: \"Day Tripper\" by some group called the Bugs or the Beetles. But Devon knows the group, he knows the song; his mothers are music historians. Actually, both women are librarians. Music history is their passion, and though they call themselves music historians, no one in Hadrian gets paid for acquiring such knowledge. At any rate, Devon is able to take full advantage of their quirk and taunt Todd with it. At first, Todd has no idea what Devon is saying with that song, but eventually, the words take on meaning. Devon is calling Todd a tease through these antiquated lyrics. Someone who leads a man on, only to refuse to follow through on promises made. _But I never made any promises,_ Todd insists. _Or did I?_ Every kiss, he realizes now, was a subtle intonation. With that sudden self-awareness, Todd begins the process of self-recrimination. Once the seed of self-recrimination has been planted, all it needs is a good dose of self-loathing to keep it well watered.\n\nBy the end of school's first week, Devon has a new boyfriend, Roger Hunter, Frank's little brother. Roger, now in grade nine, is thrilled to be Devon's new beau since it means he is dating one of the big boys. When Todd sees them walking arm-in-arm or kissing goodbye in the hallway, he feels a pang of hurt. He knows Devon brings Roger to their classroom door to kiss him goodbye, to rub into Todd's face that it took him no time at all to find a replacement.\n\nFinally, Frank caves and asks Todd to hang out again. He has been watching his friend from a distance and can see how dejected Todd has become. Although jealous that Todd spent his summer dating Devon and not him, Frank can't bear to witness Todd always walking with his shoulders down, never a smile on his face, eating alone\u2014Hadrian only knows where\u2014 _no_ , Franks reminds himself, _I know where he is_. When Todd feels the need to hide, there is only one place in the school he can go\u2014behind the back stairwell leading up to the girls' locker room. Crystal designated this \"the safe space\" when she discovered the security camera broken last year. The three gay caballeros refuse to share this safely guarded secret with anyone else.\n\nFrank decides it's time to go looking for Todd. He has started skipping out of class, something hitherto unheard of. Todd's Papa Mike stresses the importance of good grades and b-ball since he cannot afford to pay for Todd's higher education. Frank knows Todd wouldn't skip class unless something was really wrong in his life. Todd may have hurt him, but Todd is still his best friend and he will always love him. \"Hey, buddy,\" he calls out before showing himself. \"You hiding back there?\"\n\n\"Frank?\" Todd's heart rises slightly at the sound of his friend's voice. \"You talking to me again?\"\n\n\"Yeah,\" Frank sighs as he swings in underneath the stairs to join Todd. \"I guess it's time to get over my hissy fit.\"\n\n\"I'm sorry I ditched you this summer.\" Todd is sincere. He realizes now how much he misses his friends. Two weeks alone can be brutal when you are used to constant attachments with others\u2014especially when home life is even emptier than the school halls. \"I don't know what happened, really. It was like the summer with Devon turned into a whirlwind. He picked me up, spun me around like a twig, and then shot me out at the end of it.\"\n\n\"I guess you really liked Devon, eh?\" Frank asks, trying hard to hide his disappointment.\n\n\"I thought we were...\" Todd is truly befuddled. \"I don't know\u2014connecting on some level.\" Looking over to Frank, slightly misty-eyed, he adds, \"Like you and me used to be.\"\n\n\"Yeah,\" Frank sighs. \"I know.\" Considering his behavior last June, Frank admits, \"It's kind of my fault, too. I did stand you up that one Friday.\"\n\n\"Yeah,\" Todd agrees, \"the day Devon asked me out.\"\n\nNodding, Frank says, \"I was busy dating Davie, and you were looking for a surrogate friend.\" After musing for a moment, he says, \"And I'll bet any amount of credit that Devon was just looking for a boyfriend. He's certainly found one in Roger.\" Wincing, knowing he just touched on a sore topic, Frank adds, \"Sorry, Todd.\"\n\n\"That's all right.\" Although they sting, Todd realizes Frank's words were not meant to hurt. \"He had to move on. I guess I ought to as well. It's just hard when\u2014\"\n\n\"When you're all alone?\"\n\n\"Yeah,\" Todd says, looking over to his friend. \"I had no idea how important you and Crystal were to me until you two decided I was no longer worth the effort.\"\n\n\"Well, I got over it.\" Slapping Todd's thigh as reassurance, Frank adds, \"I'm sure Crystal will too.\"\n\nTodd's sigh of relief is audible. \"Thanks for taking me back.\"\n\n\"We're friends, the best of friends, man,\" Frank confirms. \"And that is never going to change. Come on,\" he invites, slapping Todd playfully. \"Let's go find Crystal. It's time to get the three gay caballeros back together again!\"\n\n* * * * *\n\n#  _Salve!_\n\n**A Veritable Vortex in Time  \nHNN\u2014Melissa Eagleton Reporting**\n\nIt is an undeniable truth that autumn in Hadrian is one of the loveliest times of the year. By mid-September, the leaves will begin their glorious changing of color. By October, they will dance in the air on their branches in a beautifully golden array. Such beauty will flicker on the limbs of the various trees in our northern boreal forest for a good month, so now is the time to take advantage of this incredible beauty. And where better to enjoy such sights than at The Cattle Ranch, stretching along the northern bank of the Churchill River? The five hundred acres owned by Jake Matonabee and Jeremy Stoker also extend north in a blended mixture of pastureland and boreal forest. Needless to say, the trail rides across their land offer some of Hadrian's most beautiful views, especially now that fall is approaching.\n\nAs well as producing beef and mutton for our consumption, leather for coats and boots, and wool for our winter clothing, The Cattle Ranch is renowned for its horseback riding. It offers a variety of vacation packages as well as day rides. A stay at The Cattle Ranch is all inclusive: your room, three meals a day, and all the horseback riding you can handle\u2014they offer three trail rides a day! One of the best parts of visiting The Cattle Ranch is that you are introduced to your horse the morning after your arrival. You will ride this horse your entire stay at The Cattle Ranch, and the proprietors encourage you to bond with your horse by helping with its grooming. They will teach you how to brush your horse down before and after your day's ride. You can job shadow the wrangler if you want, and depending on how long your stay is and how much upper body strength you possess, you may even be taught how to saddle the horse yourself. Don't worry, though; no one has to clean up horse manure. The hired hands take care of that job.\n\nSeriously, though, the best part about The Cattle Ranch is its historical value. It has chosen to operate (as do all farming institutions, be they family or corporate) as did those ranches of old, modeling their style on the mid to late nineteenth century. Horse, mule, oxen, and human power are the rule. It makes sense when you think about it. No large trucks or tractors are allowed in Hadrian due to the excessive use of fossil fuel required to operate such machinery. With a complete ban on all fossil fuels in Hadrian, one is restricted to operating all devices on electric, hydro, wind, or geothermal energy. According to Jake Matonabee, the decision to work as of old came about forty-eight years ago when the farming community met and determined that historical farming was the best possible approach. \"It is,\" as Matonabee says, \"a positive environmentally sound approach to working the land and raising cattle. With livestock comes animal waste, which we then recycle into fertilizer.\" When asked about using biomass energy to run larger equipment, Matonabee was adamant in his rejection. \"It still releases carbon dioxide into the atmosphere.\" Even though the carbon dioxide released by biomass balances out with the carbon dioxide absorbed in its growth, The Cattle Ranch and Farmers Association rejected its use since it removed from them all possibility of tax rebates. Hadrian's government is very generous when it comes to rebates given to any individual or company, with proven methods of reclamation and meeting high environmental standards. More importantly, according to Matonabee, ancient methods of farming and ranching require the community to work collectively during planting, crop growth (weeding and insect control), the birthing of and rounding up of livestock, and, most especially, harvest. As we well know, the use of fossil fuels, all chemical pesticides, and herbicides are illegal in Hadrian, which has led to the choice of a much simpler lifestyle in rural areas. As a result, when one visits The Cattle Ranch, one walks through a veritable vortex in time, stepping out of the twenty-second century into the year 1878.\n\n_Vale!_\n\n# Second Anniversary\n\nAlthough back on the Seroxat, Dean still suffers from low days, remaining uncomfortable in his skin. Too much of his time is spent in morbid contemplation. Struggling to find value and worth in his life, Dean tries to focus his mind on days that were good\u2014all those times when he truly did feel in love with Geoffrey and believed he wanted to be with him. Today, his thoughts are centered on the first time he lay down with Geoffrey. This was a time of bonding, an awakening and strengthening of their love. Dean, desperately needing to feel this way again, indulges in the memory of their second anniversary. Knowing, on this day, Seroxat isn't enough, Dean takes two milligrams of Zolam, allowing its altering powers to stupefy him. His mind, feeling free now, contemplates his and Geoffrey's second anniversary.\n\n*****\n\nTwenty years ago, as it is today, The Cattle Ranch was nestled in the rolling hills and forests of Quadrant Three. As well as pasturing their livestock (horses, cattle), Jake Matonabee and Jeremy Stoker, owners and proprietors, cut numerous riding trails across their land for paid ranch guests. Yet, not all of this land was reserved for cattle and horseback riding. Jeremy's little sister, Sissy Hildebrand, ran the sheep herd in the northern, more remote section. Sissy, being straight, had chosen a life of isolation to avoid being found out. Jeremy, then and now, knows well his sister's sexual preference, and not only does he accept her, but he has worked hard over the years to help her keep her secret effectively concealed. Even so, he often encourages her to find a woman with whom she can live\u2014someone to register with for security reasons\u2014but she is always ardently opposed. Her one concession is to hire only women for hands. Taking her brother's advice, she has never hired any men. \"Avoid temptation, Sissy,\" Jeremy always cautions her. \"That way no one can expose and expel you.\" Today, Sissy is forty-six. When Dean and Geoffrey visited the ranch, she was but twenty-six years old. Even then, she was far too old for reeducation camp. Her fate would be worse than death if her secret were known.\n\nAlways irate at the implication behind Jeremy's repetitious lecture, Sissy's retort is like a broken record: \"I don't think of sex every time I look at a man. It doesn't work that way, you know!\"\n\n\"I know, Sissy,\" Jeremy always replies in consolation. \"I didn't mean that\u2014it's just, well, even the wrong glance a man's way can lead to suspicion.\" Twenty years ago, the conversation was even more perverse. \"Even Jake,\" Jeremy one day sprang on her, \"has questioned your sexuality more than once.\"\n\n\"What did you say?\" Terror welled in Sissy's breast.\n\n\"I told him he was being stupid\u2014that you just don't have much opportunity to meet women. So he asked if that's why you only hired women\u2014in hopes of meeting someone and I said yes.\"\n\nAfter a long sigh of relief, Sissy said, \"Thank you.\"\n\nThese conversations between Sissy and Jeremy about her sexual orientation always end with Sissy bursting into ire. \"I hate having to feel this way. It's not fair!\"\n\n\"It is what it is, Sissy.\"\n\n\"That doesn't make it right,\" she always grumbles.\n\n\"I know.\"\n\n\"Nor is it any answer.\"\n\n\"I know that, too, Sissy, but,\" he inevitably adds with a sigh, \"it's all I've got to offer.\"\n\nSuch conversations only occur when they meet in person\u2014and when they are alone. Fortunately, Jake stopped expressing his concern fifteen years ago. That was only after Sissy actually hired a man as a sheepherder and Jeremy was no longer required to travel north all the time to help her.\n\n* * * * *\n\nFor their second anniversary, Geoffrey brought Dean to The Cattle Ranch. After having seen a picture of the original Marlboro Man in the Antinous City Center Museum and being impressed by its resemblance to his partner, Geoffrey searched the wave for information on The Cattle Ranch. It was just as he had hoped it would be\u2014Geoffrey was pleased to discover it offered vacation packages. For two thousand five hundred credits per person, the all-inclusive package included three square meals a day, a horse to ride and care for every day during the five night, four day stay (with as many as three trail rides a day), walking trails, camping options, even job shadowing for those who were interested in learning the everyday workings of an historical ranch.\n\nAt twenty, Dean was tall, lean, muscular, and coltish. He had been broken, though, roughly at the hands of his guardian at the Northeast Reeducation Camp. Only Geoffrey's supportive, gentle ways kept him from committing suicide. Although having been married for two years, the two men had yet to consummate their registration. Geoffrey had won Dean's respect and love, the two men having formed a strong friendship, yet Dean still shied away from sexual contact. Holding hands, wrapping arms about each other's waists, hugging, and even occasional experimentation with kissing occurred. But thus far, kissing, if held onto too long by Geoffrey, had only ever ended with Dean pulling away. Bringing Dean to The Cattle Ranch for their second year Registration Celebration was not an attempt on Geoffrey's part to seduce Dean. He had come to accept, long ago, that winning Dean would take time\u2014even that it might never happen. Mike Fulton and he often confided in one another about their frustrations, both having partnered with a straight, and many times, it seemed as if neither would ever win the sexual love of his partner. Mike would succumb to despair more often than Geoffrey, and the two men spent many an evening or Sunday afternoon venting out their frustrations. They had even considered partnering with each other on the side, but that idea never did come to fruition since, for both Geoffrey and Mike, it was as much about bonding with the man one loves as it was about sexual fulfillment. \"True sexual fulfillment,\" Geoffrey reasoned one night during these conversations with Mike, \"comes from making love with the man you love. It's not just about ejaculating and feeling good physically. It's about feeling good with and for someone you love.\"\n\n\"But what if Will never comes around?\" Mike was despondent. Then, after looking hard at Geoffrey, he expressed concern for his friend. \"What if Dean won't?\"\n\n\"I don't know,\" was Geoffrey's reply. \"This is brutally hard. I masturbate a lot. I wish\u2014I hope\u2014I just don't know.\" Then he looked Mike's way. \"The fact is, we both knew what we were marrying ourselves into. Will was open and frank with you. Dean, I met at reeducation. I know he only pretended to be cured,\" after snorting, \"with my help. I know for a fact he only took my last name to avoid\u2014\"\n\n\"Avoid what?\"\n\n\"Nothing.\"\n\nMike let the question drop. Once Geoffrey decided to clam up there was no hope of getting any more out of him. He just waited patiently for the man to continue.\n\n\"Although I'd like to think he married me out of love,\" Geoffrey added despairingly, \"our marriage was just an opportunity to him to escape reeducation.\"\n\n\"Then why did you marry him?\" Mike asked gravely.\n\n\"The same reason you married Will. I fell in love. He's my other half. All I can do is hope one day he'll be ready to share himself with me.\"\n\nSo, twenty years ago, when Dean came into their cabin room and sat down on the bed next to where he lay, Geoffrey was taken completely by surprise to hear Dean say he thought he was ready. It was late, well past the midnight hour. Geoffrey had left Dean sitting by the mock fire (a geothermal heater designed to look like a small campfire) with Jeremy Stoker telling stories about horses, cattle ranching, and singing old country songs for their amusement.\n\nSitting up, still half-asleep, at first Geoffrey thought he might be dreaming. \"Really?\" He asked bewildered. \"Are you sure?\"\n\n\"Yes.\" Although his reply was in the affirmative, Dean was shaking, his voice so low Geoffrey could barely hear him. Not looking at Geoffrey but sitting hunchbacked on the edge of the bed, holding his hands atop his legs, rubbing the palms together nervously, Dean was avoiding eye contact.\n\nAfter pulling himself closer to Dean, Geoffrey gently held his chin in his hand, turning Dean to face him. Dean's eyes were closed. Geoffrey noticed the tears. \"We don't have to if you don't want to, my love,\" he said soothingly. His body was aching, desire for Dean overwhelming him.\n\nDean nodded his head, \"I...\" He shook his head, \"I\u2014do...\" He nodded once more. \"...want to. We...\" He opened his eyes, looking directly into Geoffrey's, seeing in them an honest expression of love. Before him sat a man who would do anything for him. As he burst into tears, Dean blurted out, \"I love you, Geoffrey, but I'm afraid.\"\n\nCupping Dean's face in his hands, Geoffrey reassured him, \"I love you too, and I would never do anything to hurt you.\"\n\n\"Then...\" Dean swallowed. Turning his face, he kissed the palm of Geoffrey's hand, \"make love to me.\"\n\nAs the two men kissed, Geoffrey discovered everything he believed about intimacy to be real and true.\n\n* * * * *\n\nLess than an hour before Dean gave himself over to Geoffrey for the first time, he had been opening his soul to Jeremy Stoker. Jeremy was the kind of man who seemed to invite confidence. His quiet, self-assured ways suggested understanding\u2014and a willingness to accept. Someone willing simply to listen and nod can have a powerful influence. Without even realizing it, Dean had told this man about what had happened to him in high school; what it had been like to be straight and live in a pretend gay relationship.\n\nHaving shook his head in wonderment, Jeremy mused, \"Two years\u2014and you two ain't never\u2014\"\n\nDean lowered his head and shook it. \"He won't ever make me\u2014he promised.\" Taking a moment to breathe in and out with short staccato, he added, \"He's a man of his word.\"\n\n\"Does he\u2014\" Jeremy paused, wondering how to put this. \"Does he get it somewhere else?\"\n\n\"No\u2014sometimes I wish he would, but\u2014\"\n\n\"But yer glad he don't.\"\n\nDean shuddered with shame at the realization.\n\n\"By all that's gay and glorious,\" Jeremy muttered, \"that man must love you.\" Worried he might have touched on a sore spot, he asked gently, \"And you, Dean, how do you feel about him?\"\n\n\"I love him,\" Dean blurted out. No thought had gone into his answer\u2014only expressed emotion. He took a moment to take in what he had just said. \"I do love Geoffrey; he's a wonderful man,\" Dean pleaded. \"It's just that I'm not that way, and it's hard having to crawl into bed with him every night, knowing he wants me to do things I can't seem to bring myself to do.\"\n\n\"I can imagine,\" Jeremy replied, nodding again for Dean to go on.\n\n\"I just\u2014I just don't know what to do. I can't keep faking it\u2014I can't keep on hurting him like this, but...\" Dean shuddered at the thought. \"I can never go back to that place.\"\n\n\"What was it like?\" Jeremy asked with sincere interest.\n\n\"It was...it was...\" Even with all of Jeremy's ease and willingness to listen, with every ounce of the man inviting confidence, Dean could not bring himself to speak of what had happened to him during his time at the Northeast Reeducation Camp.\n\n\"It's all right, son,\" Jeremy said soothingly. \"I don't need to know.\" That Dean was straight was something Jeremy had figured out even before the two men had arrived at The Cattle Ranch. Changing the subject, but only just slightly, Jeremy commented, \"Your partner must be a very wealthy man.\"\n\nThis comment seemed so odd so soon after Dean had just exposed himself as het'ro that he couldn't help but bluster, \"I don't understand?\"\n\n\"Notice how you and he are the only two guests here.\" He paused to allow the young man a moment of reflection. \"This is peak season. This time of year, early fall being so lovely, all our cabins are usually full. We can service up to six couples\u2014men and women\u2014don't matter. Business is business. Though, we do tend to attract more men, being men ourselves.\" Having given Dean a knowing look, he added, \"Don't see any women here about, do you?\"\n\nDean pondered the implication. \"He bought out the whole place?\"\n\nJeremy smiled. \"Rented out.\" Dean blushed. That was what he had meant to say. \"He even asked if we had any women hands.\"\n\nDean sat up straight. \"Do you?\" His eyes opened wide to reveal both hope and fear.\n\nJeremy nodded. \"Some. Cowfolk come in both sexes you know.\"\n\nDean flushed an even deeper red. \"I...ah...\"\n\nJeremy pointedly ignored Dean's obvious embarrassment. \"I told him as much and he asked whether we'd mind keepin' 'em outta sight while the two of you were here. He paid extra for the favor, so I sent the ladies up to the north pastures to help my little sister, Sissy. She tends to our sheep up there.\" After having eyed Dean briefly, Jeremy rested his eyes on the faux fire. \"Now. The way me and Jake figures it, if a man don't want no women around, he's either reeducated or has a real hate on for women. Either way, it likely adds up to his being straight.\" He chuckled slightly. \"There ain't nothing straight 'bout your man...\" He paused momentarily to wave his hands, \"which leaves only you. Now yer twenty, why ain't you at the wall? He's a mite older, clearly been educated. Well, it doesn't take too much to put two and two together. You was straight when you was younger, and as you jes' told me now, you still is.\"\n\n\"You...you...\" Real fear flashed in Dean's eyes. \"You aren't gonna expose me, are you? Have 'em send me back there?\"\n\n\"No, son, I ain't gonna do that to you. 'Sides,\" he stated judiciously, \"my understanding of the law says it is illegal to act on being straight and you ain't been doin' no straight acts far as I can tell.\"\n\n\"No, sir,\" Dean expostulated vehemently. \"No, sir, I haven't.\"\n\n\"Ever done?\" Although his curiosity was pushing the bounds of propriety, Jeremy simply couldn't help but ask.\n\n\"N-no, sir\u2014just the kissing stuff I told you about.\"\n\n\"Well, that's good, I suppose. But...\" Here Jeremy paused, rubbed his chin, then sighed deeply. \"Even if you did and I knowed 'bout it, I don't reckon I'd turn you in.\" Dean was stunned. Sitting before him, for the very first time in his entire life, was a man who wasn't judging him for being who he was. Dean felt so overwhelmed and touched that he was unable to speak. Having sensed the need to change the topic, Jeremy offered up a song. \"How 'bout a little good ol' fashioned country fer ya?\"\n\nDean smiled. \"I'd like that.\"\n\nJeremy turned to his right where his guitar case leaned against one of the vacant stump seats. After opening it and retrieving the instrument, he took a moment to tune the strings. \"Let's see now,\" he said while retuning the G string; his ear was close to the string as he listened intently until he was sure the sound was just right. \"How about a little more Tim Hus?\"\n\n\"Who's he?\" Dean was clearly intrigued.\n\n\"Tim Hus, he was from Alberta country back when Alberta was still a part of old Canada. I sung y'all one a his songs yesternight, the one 'bout a bull rider.\"\n\n\"A bull rider?\" Dean was intrigued, his eyes widening at the wonder. He remembered the song he just hadn't put the picture together in his mind. Blushing Dean adds, \"I, ah, thought a brauma bull was a wild horse.\n\nJeremy laughs. \"You remembered that line, did ya?\" Then shaking his head between chuckles, he adds, \"No. It's the bull they say no one can ride.\" Unable to let a good joke pass he adds, \"It also means the man with the big penis.\" Dean joins in on the jocularity. Then nodding his head thoughtfully, Jeremy adds, \"Anyway, cowboys\u2014note girls ain't added here\u2014don't think there were any women folk done this\u2014men would saddle up the biggest, most ornery bulls they could get a hold of an' try an' ride 'em. Compete to see who could stay on its back the longest. Well, them bulls would buck mighty hard. I guess it was considered a real act of manhood to be able to ride on the back of one of them like that.\"\n\n\"Were any of them ever killed?\" The idea of risking his life to prove his strength above others was intoxicating to Dean.\n\n\"Oh, yeah. I reckon lots were. It would be mighty dangerous. Our old bull is one mean bastard. I sure wouldn't want to git on the back of him\u2014but these good ol' boys did and made good money doing it, too. Anyhoo, this here song I'm gonna sing ya's called 'Silver in the Buckle' and its 'bout a broncin' horse rider.\" Jeremy chose this song specifically for the one verse in which the Montana man meets a girl \"across the line that stole his heart away.\" Under normal circumstances, Jeremy would adjust the verse, even though most folk, he knows, don't really listen to the lyrics. But not tonight\u2014tonight Jeremy sings the song exactly the way Tim Hus wrote it. If he had judged the lad correctly, Dean would make out the intent of the lyrics. And when he was singing that verse, he would watch the youth closely to see whether or not he was game. Sissy was mighty lonely and Jeremy knew she needed a man in her life. He couldn't keep up this charade; Jake was far too suspicious, and as much as he loved Sissy, his true love was Jake. The fear of losing him had been building up lately.\n\nJeremy was correct. Dean picked up on the lines, gasping as soon as he heard them\u2014taking off his hat, Dean used it to cover his lap. Without even finishing the song, Jeremy chimed out, \"Sissy? Sissy, little girl, is that you?\"\n\nJeremy had turned the voc mic off so Dean couldn't hear the woman's response. \"Jeremy? For Hadrian's sake, don't you know what time it is?\"\n\n\"It's late, sweetie pie. What you callin' fer?\"\n\n\"ME? Why you dirty ol' swamp rat\u2014\" Jeremy blinked, turning on his eye cam to show her Dean. \"Ohhh.\" Intelligence dawned. Although he was obviously much younger than Sissy, she couldn't help but find herself drawn to his masculine beauty. \"Can he\u2014hear me? Can he\u2014see me?\"\n\n\"Naw, little girl,\" Jeremy teased. \"You know I don't got time to come running up there to help you herd in a bunch of sheep. Why didn't you fence up them pastures like I told you to?\" He then confused the young man by giving Dean a wink. \"Why didn't you jes' call me in the morning?\" Feigning chagrin, he added, \"Now, little sis, you know I'm too busy. We got guests to entertain this week.\" Having trouble containing himself, Jeremy laughed outright.\n\n\"So, what's your game, Jeremy? Is he for me?\"\n\n\"Maybe.\" Grunting as if punched lightly, he added, \"It might be possible.\" His smile spread wide. \"Of course you can meet them\u2014well, one of them, one guest went to bed early. Here\u2014\" When Jeremy blinked on the holocam, the grainy image of a blonde, curly haired young woman in her mid-twenties appeared. She was sitting up in bed with the covers dropped down to her waist.\n\n\"Hello.\" She spoke shyly, having never met a man other than her brother with straight tendencies. And since Jeremy is bi-sexual, she has to share him with Jake, who, she reasons, would likely kill her if he knew. The very idea of a fully straight man only interested in women excited her. Jeremy moved his head, bringing Sissy's image closer to Dean. \"I'm Sissy. I herd sheep for my brother up north.\"\n\nDean looked up, smiling clumsily. \"My name's Dean Stu\u2014Hunter.\" Blushing crimson, Dean squeezed his eyes shut and dropped his head.\n\nSissy giggled awkwardly, \"Stuhunter. That sure is an odd name. What's your cultural heritage?\"\n\n\"No, uh...\" Dean stuttered. \"I...it's just Hunter.\"\n\n\"Okay, just Hunter.\" Dean was so embarrassed by her teasing that he could not bring himself to look up. Beginning to feel sorry for him, Sissy added, \"It's nice to meet you, Dean.\" Since Jeremy had placed her holo image close enough, Sissy reached up with her hand to touch Dean's cheek. The cool thing about holo imaging through the contact voc is that the salt-based electric current charging it can be felt as a slight shock, which had given a whole new meaning to the concept of phone sex! Unfortunately, when Sissy's holo fingers lightly feathered Dean's face, he was stunned. His mind reeled back to reeducation and a bolt of lightning crackled in his brain, searing all of its energy out of his extremities. The suddenness of this impact caused the young man to have a seizure. Sissy quickly pulled her hand back, watching on in horror. When the attack finally abated, nausea set in. Turning his back on Jeremy, Dean vomited into the bushes behind his seat. When he finished, Dean got up, and without looking back, stumbled toward his and Geoffrey's cabin, pausing briefly at the water trough just outside their door to wash his face and rinse out his mouth. Turning to look at her brother, Sissy inquired, \"Jeremy, what just happened?\"\n\n\"He's a re-ed...\" Then shaking his head, he suggested, \"They must've done something.\"\n\n\"For the love of Hadrian,\" she gasped. \"What?\"\n\n\"I don't know, Sissy.\"\n\n\"Poor boy,\" Sissy lamented.\n\nJeremy sighed. \"I only hope he can adapt.\" His eyes had followed Dean walking in the direction of his and Geoffrey's cabin.\n\nSissy was miffed, \"Like m\u2014\"\n\nJeremy never let her finish. His eyes had hardened, \" _Likely_ , yes, likely, I agree. It's very _likely_.\" After a pause and having softened some toward his sister, he said, \"But what might have happened to him\u2014it frightens me more than you can possibly know.\"\n\nTrembling at the implication, she inquired, \"How so?\"\n\n\"He could be\u2014\" Jeremy stopped himself in time. He knew he must avoid saying anything that might implicate Sissy. He had taken a foolish risk doing introductions over the voc\u2014anyone on the government wave could have overheard.\n\nFor a brief moment, brother and sister stared into one another's eyes. Sissy was the first to break the silence. \"I really do need help\u2014with\u2014the sheep, Jeremy.\"\n\n\"I sent you three women,\" he replied dryly.\n\nClosing her eyes, Sissy tried desperately to squeeze back the tears. \"Please come,\" she begged. \"I really need you\u2014your help.\"\n\n\"All right, baby girl,\" he answered soothingly. \"I'll saddle up first thing and get there by nightfall.\" Although Jeremy and Sissy are brother and sister through their fathers' marriage they are not genetically related. Even so, if anyone ever learned of their affair most people would find it disturbing.\n\nSuddenly, Sissy blurted out, \"I love you!\"\n\nJeremy's eyes grew stern again, and he responded quickly with, \"I love you too, _little sis._ \"\n\n* * * * *\n\n#  _Salve!_\n\n**Today's Sexually Active Youth  \nHNN\u2014Melissa Eagleton Reporting**\n\nToo many of today's teenagers are sexually active. This is a concern for numerous of Hadrian's parents. Many are seeking professional advice for ways to discourage our youth from beginning early experimentation. The fact is, early sexual experimentation has been a problem for parents of teenagers since the beginning of time. The accusation of outsiders that our teenagers are promiscuous due to our sexual orientation is balderdash. The teenage years are ones of sexual awakening. Teenagers' hormones begin to pop and dash around like the ball from the old pinball machines in the arcade museum. That is not to say I condone early sexual behavior in our youth. I only say this to let parents know they haven't done anything wrong if they suddenly discover their young adult is no longer a virgin.\n\nMany young people find themselves hurt and confused by their first sexual experiences. What our children need to learn, through open and honest communication with their parents, is that touching, allowing another to touch you, is a very intimate experience. One becomes quickly attached to the person to whom she or he makes love. The greatest danger our youth face is the wretched psychological hurt that comes with a break-up. The pain of breaking up is compounded deeply if one has been intimate with another. Encouraging our youth to restrain from acting upon their sexual impulses at an early age is crucial. Early experimentation may shatter a child's self-esteem, creating an emotional turbulence that can deter an individual from creating a stable emotional bond with a future partner.\n\nNo doubt, parents, you are wondering what it is you can do to help stave off your child's sexual yearnings for a year or two. To begin, you need to be open and forthright with your sons and daughters. Be prepared to answer any questions they might ask regardless of how uncomfortable it may make you feel. And, yes, it is essential that you consider their questions about heterosexual behaviors. Remember, many of our citizens know what it feels like to be attracted to the opposite sex. Even though our geneticists have done a wonderful job isolating the gene for homosexuality, the fact still remains that some of our children are born with heterosexual tendencies. Let's remember the Kinsey scale: anyone between a three and five is easily swayed to accepting homosexuality as the sexual norm. Youth who are a two, on the other hand, feel the heterosexual drive a lot more strongly than their latent homosexual tendencies. If you suspect your child might be a two on the Kinsey scale, you can help him or her release the inner homosexual. Encourage your children to reach deep inside and discover what they like most about their own sex. Nurture these inclinations and help your child turn them into honest desires. For a child who is a two, it really is just a matter of finding the right man or right woman.\n\nNow, for parents whose children are easily attracted to the same sex, you need to talk to your children about how to move slowly, as they may not take the time necessary to formulate solid relationships. Many of our youth find they get involved too quickly in a sexual relationship and are ultimately hurt. Parents, speak candidly with your child. Let him or her know what it was like for you as a young adult. Discuss how you controlled your sexual urges; or be honest, let him or her know about the hurt you felt when you gave of yourself too soon. It never serves a parent to mimic a virgin countenance. A false face forces even the finest into floundering. We are not saints; we are human beings, and we are all prone to flaws. Let us leave that \"holier than thou\" nonsense to the religious fanatics outside our walls, shall we? Rather, Hadrian's parents need to help their children understand how emotional the sharing of one's body is and encourage their children to restrain from entering into such a commitment at too young an age, and the best way to accomplish this is by sharing our own experiences openly. But perhaps the two most important pieces of advice I can offer you are these: be honest and never judge your child.\n\n_Vale!_\n\n# Study Date\n\nFrank's room is huge. Todd laughed the first time he saw the bathroom attached. \"By all that's gay and glorious, man, your bathroom is four times the size of my bedroom!\" Frank's bed is a four-poster queen-size; Todd's is a single. As soon as he enters Frank's room, Todd always rips open the poster's curtains and tosses himself onto Frank's bed. \"Man, I wish I could have a bed this size. And the mattress.\" He always says this as he luxuriates in the feel of a mattress that actually bends and forms to his body, immediately placing him in the perfect rest position and realigning his spine at the same time. \"Hadrian's lover,\" Todd groans in delight, \"it just cracked my back for me.\"\n\nFrank laughs heartily. He enjoys watching Todd squirm in delight on his bed. \"You can sleep over any time,\" is Frank's chirp reply, always knowing, but never fully prepared, for Todd's inevitable response.\n\n\"Forget it, Frank. I'm not one of your tinsel tarts.\"\n\n\"I think you'd look cute in makeup.\" Frank never quits.\n\n\"You're not slathering that shit on my face ever again.\" Todd sits up and studies his image in Frank's mirror. The silver center is a beautiful oval surrounded by a highly decorative cherry oak frame. \"It made me look stupid; the stuff smells. And, worst of all,\" he adds emphatically. \"It gave me acne.\" Now looking at his friend, he exclaims, \"I hate acne!\" Frank sits next to Todd, placing his hand on his knee. Todd swats it away. \"Hands off, buddy!\"\n\nFrank shrugs off the rejection. Todd may play hard to get, but they are still best friends, and that relationship always gives him hope. \"And how old were we then? Ten? Twelve?\"\n\n\"Thirteen,\" Todd replies crisply. Todd has grim memories of that day. Frank and he got carried away dressing themselves in as festive a manner as possible. After painting each other's faces, Frank exclaimed how beautiful Todd was, and suddenly, they were kissing, making out quite heavily. The experience became awkward for Todd when Frank began dry humping against him. Todd didn't know what to do, and the incident hadn't stopped until Frank had groaned during ejaculation. Todd had leapt up from the floor at that point, crying, \"I'm not ready, Frank! I'm not ready. This is too soon.\"\n\nAs Todd ran out of the room, Frank had chased him. \"Todd, come back! I'm sorry.\"\n\nTodd wouldn't listen. He had just hopped on his bike and raced home. He never even bothered to change. When he ran inside, he saw Papa Mike sitting on the couch with a new \"friend.\" \"Well, well,\" the man said smiling, \"look at your little fella.\" Todd stood rooted to the floor, unable to move. He was so stunned that when the man stood up and walked toward him, Todd stumbled back against the wall. The older man, trapping him there, leaned in so close Todd could smell the whiskey on his breath. It wasn't until the man hooked his finger through the silver loop of Teika's dog collar and gave it a slight tug that Todd remembered Frank had put it on him. All the clothes Todd had chosen to wear were pinks and purples, and Teika's dog collar, being dark purple, matched perfectly. Frank had taken it off the old German Shepherd and strapped it on Todd. They had both laughed hilariously when Todd had pretended to be Teika. Frank started petting him, saying things like \"Good girl!\" Todd had pretended to wag his tail and leap up to lick Frank's face the way he had seen Teika do. That was when things between Frank and him got crazy.\n\n\"I like the dog collar.\" The older man's seductive whisper ripped Todd out of his memory. \"Do you want to do it doggy style? Is that what you're wanting to do?\" Todd winced. That was how Frank had dry humped him. The man continued his seductions. \"You are one beautiful little boy. Do you know that?\" he asked Todd in an enticing manner. Todd just stood there, glued to the wall, and quaking. \"You are going to have all the men chasing after you when you get older.\"\n\nTimid and terrified, Todd called out, \"Please make him stop, Papa Mike. Please.\"\n\n\"Leroy, leave the kid alone.\" Papa Mike didn't even sound annoyed. But Leroy listened. He blew Todd a little kiss first, then returned to the couch to sit beside his lover. \"But, Todd,\" Papa Mike began to chastise, \"go to the mirror and look at the way you are dressed.\" Todd did as instructed. \"With those clothes,\" he said, shaking his head, \"that dog collar and all that makeup\u2014well, face facts, son; you are just asking for sex.\"\n\n\"Your Papa's right, little man,\" Leroy concurs. \"Makeup was designed to help make a person more attractive. It's a tool. You wear it because you want to enhance your appearance. And if you want to enhance your exterior form, it's because you want somebody to take notice of you. Just like I did.\" He topped off his lecture with a wink.\n\nTodd shivered. He didn't mean to. He thought Frank and he were just fooling around, but then it got crazy and they had done stuff, stuff Todd felt was wrong\u2014 _No_ , he thought, _not wrong, just not right_. He looked at himself questioningly in the mirror. _Why_ , he wondered, _if it's not wrong, didn't it feel right?_\n\n\"So, Todd,\" Papa Mike asked, \"are you ready for sex?\"\n\n\"No, Papa.\" Todd knew he wasn't ready; he wondered whether he'd ever be ready after what he and Frank had done. \"No, Papa, I'm not.\"\n\n\"Then go wash your face, change your clothes, and for Hadrian's sake, take off that damn dog collar!\"\n\n\"Well,\" Todd recalls the other man laughing and saying, \"at least you know the boy is gay.\"\n\n\"Shut up, Leroy.\" After sighing, Mike had added, \"You can dress for attraction when you are ready to attract.\"\n\nStill staring at himself in the mirror, Todd mouths the words, \"Yes, Papa,\" both in the past and in the present.\n\nFrank looks at Todd oddly, wondering what is going on inside his friend's mind. \"Thirteen,\" Frank waves the age off dismissively. \"We didn't know what we were doing.\" Walking over to his dressing table, he opens up a cosmetic case. \"I've been learning all kinds of ways to use makeup creatively in Cos class. Makeup doesn't have to make anyone look tawdry. You don't have to slather it on, wear blush, or anything. Just use a little base to keep from looking too pale.\" As if to further his case, Frank adds, \"No tinsel, no trash.\"\n\n\"Yet,\" Todd reminds him, \"all the boys you date wear it to look kitschy.\"\n\n\"Kitschy,\" Frank laughs. \"Good word! That's going to be my nickname for you.\"\n\n\"Don't you dare\u2014\" Todd begins to protest.\n\n\"Okay, Kitschy,\" Franks says, winking Todd's way. Frank is constantly threatening to give Todd a pet name.\n\nTodd's face reddens. \"I swear by Hadrian's lover, Frank. I'll kick the shit out of you if you ever call me that again.\"\n\n\"All right, gee whiz, learn to take a joke.\" Frank shrugs off his annoyance by turning to the mirror and begins to brush the light beige powder over his face. Taking his time, sensing the mood of the room, Frank decides to chat lightly until Todd calms down. Unfortunately, he does not choose a very good topic. \"Really, Todd, it's like you're living back in the Dark Ages, embracing old world concepts where men weren't allowed to dress freely. No makeup, no skirts, no flashy clothes.\"\n\n\"You don't wear skirts,\" Todd interjects.\n\n\"The point, Todd, is I could if I wanted to.\" He turns to face Todd, brush in one hand, powder case in the other. \"And nobody would abuse me for it!\" Frank insists, \"So could you!\" Before Todd can object, Frank barges on, lecturing, \"There was a time, before Hadrian, when men's fashion was limited. We weren't allowed to choose the way we dressed.\" Looking with emphasis toward his cosmetics, he adds, \"Or the way we looked.\" Turning back to face the mirror, Frank finishes his touch up. When done, he swivels to face Todd, presenting his newly formed image: \"See,\" he smiles proudly. \"You can't even tell I'm wearing any. It just covers up the blemishes.\"\n\n\"You don't have any blemishes,\" Todd interjects.\n\n\"That's because I cover them up.\" Adding more powder to his brush, Frank moves in closer to Todd. \"Here, let me put some on you.\"\n\nAnnoyed, Todd pushes Frank's hand away. \"I said no.\"\n\n\"Oh, come on, Todd. Everyone wears makeup!\"\n\n\"That doesn't mean I have to.\"\n\nSpinning on his heels, returning the cosmetics case to his dressing table, Frank exclaims, \"You are so stubborn!\"\n\n\"You said men have a choice in Hadrian. So, why can't I choose the way _I_ look?\"\n\n\"Of course you can,\" Frank says a little too harshly. \"Did you know,\" he adds in his own defense, \"that when makeup was first used by the Egyptians, it was worn by men as well as women!\" Todd shakes his head. \"I learned that in Cos, too!\" Huffing a little to release his anger, he states, \"Well, here in Hadrian, like in ancient Egypt, men are free to wear makeup if they want to!\"\n\n\"It's just,\" Todd counters, \"some guys use way too much goop and it looks awful.\"\n\nFrank takes this comment to mean his personal use of makeup is vulgar. \"Just because a guy wears a little base doesn't make him cheap or gaudy.\"\n\n\"I didn't mean you...\" Todd stumbles, trying to avoid hurting his friend's feelings. \"It's just that's the way the boys you date look\u2014garish, flashy, showy. I mean\u2014you never date anyone who just looks normal.\" _And they all wear that stupid fucking collar!_ \"It's like you want every guy you're with to look like a tart or something.\" His frustration growing, Todd expresses his real fear, \"And I know if you're trying to put that crap on me, it's because you want me to look that way, too!\" To avoid further discussion, Todd stands and crosses over to the bedroom door. \"What's taking Crystal so long?\" Todd reaches into his jean pocket to retrieve his cell phone and begins to text.\n\n\"What are you doing?\" Frank is clearly annoyed. Todd hit home when he suggested Frank wanted him to look pert and cute like his \"boys.\" It's almost as if Frank is trying to relive their time together when they were thirteen. Todd seemed so anxious to kiss him, so anxious to pet and play then, but he suddenly changed. Frank still doesn't know what he did wrong, and they can never talk about that day. It has become taboo\u2014almost as if it never happened. Sighing, Frank realizes a truth about himself; everyone he dates has that same boyish look; they all remind Frank of Todd on that day, and they all end up wearing Teika's dog collar.\n\nTodd has his back to Frank so he doesn't see his friend's facial expression. He simply answers Frank's question. \"Texting Crystal to see what's taking her so long.\"\n\n\"Put that away,\" Frank growls. Todd obeys his friend, hearing something dark and ominous in Frank's voice. \"She'll get here when she gets here!\" Allowing all of his annoyance to spill out, he adds, \"Crystal, our little chaperone.\"\n\nTodd turns around. \"That's not fair, Frank! She's smart. Besides, we both need to pass this test.\" He adds as a reminder, \"Everyone needs Hadrian history to graduate.\"\n\n\"I know.\" Frank lets go of his aggravation. \"I just would have preferred you and me alone.\"\n\n\"You're the one who got the three of us together to form a study group last year!\"\n\n\"I know.\" Somewhat frustrated, Frank replies, \"I had no idea you two would hit it off.\"\n\n\"We're just friends, Frank,\" Todd insists. \"Like you and me.\"\n\n\"No,\" Frank reminds him, \"not like you and me.\"\n\nIt is Todd's turn to get irritated. \"I am not a tinsel tart! I am not trash! And I will not join your little harem of boys!\"\n\nFrank is offended. \"I never said you were any of those things.\" Frank sits on the bed and pats the spot next to him. Todd refuses to take the bait. \"You would be the only one!\" As if to defend his behavior, he adds, \"I only date all those guys because I'm waiting for you!\"\n\nSlightly mollified, Todd sits back on the bed, searching for a way to word this delicately. \"Frank, I don't want a boyfriend.\" Seeing the look of shock in Frank's eyes, Todd begins to backtrack. \"I mean, not so soon after\u2014it just hurts too much.\" Sighing, hoping this will placate Frank, he says, \"I'm just happy being able to hang out with you again.\"\n\n\"And what's wrong with dating?\" Frank sees hope in everything Todd says.\n\nSqueezing his eyes shut, Todd replies, \"Nobody just dates.\" Then complaining, he says, \"A date means getting laid and...\" Feeling foolish, blushing a little, Todd stares at his feet as they shuffle uncomfortably. \"I'm not ready for sex.\" He harrumphs, \"That's why Devon dumped me.\"\n\nFrank laughs gaily as he wraps his arms around Todd's shoulder. He and Devon didn't do it! He can still be the one Todd falls in love with! The first and only one Todd shares himself with! _Just like Dad and Papa Dean,_ he tells himself. Todd wants to shove Frank off, but he knows that will hurt his friend too much so he lets Frank hug him instead. \"I respect you, Todd. I wouldn't make you do anything until I knew you were ready.\"\n\n\"That's what Devon said,\" Todd mutters gloomily.\n\n\"I'm not Devon.\" Then whispering in Todd's ear, he adds, \"Dating would be no different than the way things are now, except for kissing and,\" a little naughtily, \"maybe a little petting.\" Frank leans in for a kiss.\n\nTodd quickly turns away. \"Kissing leads to petting and petting always leads to sex!\" Glaring back at Frank, he concludes, \"And if it doesn't, he dumps you!\"\n\nFrank gently brushes away one of the tears Todd is desperately trying to hold back. Frank reaches out to Todd and holds his friend tight in his arms. \"He really hurt you, didn't he?\" _Hadrian exile you, Devon,_ Frank curses.\n\nAlthough he would like to struggle against Frank's hug, Todd merely closes his eyes and tries to explain. \"Frank, I\u2014\" Unable to finish, he tenses into Frank's embrace.\n\nEncouraging him, Frank says, \"You can tell me anything, Todd.\"\n\n_No, I can't!_ Todd starts to cry. _Why can't I feel anything if I'm at least a two? Isn't it just a matter of finding the right man? Isn't Frank the right man? I love him. I don't love anyone better! Why don't I feel anything then?_ Summoning up the courage, he asks, \"Everybody's at least a two, right, Frank?\"\n\nFrank smiles reassuringly at his friend, \"That's right, babe. Everybody in Hadrian is at least a two.\" To prove his point, he asks Todd, \"You look in the mirror, right?\"\n\n\"Of course I do. Everybody does.\"\n\n\"Everybody, good. Remember last year?\" Todd shakes his head, unsure of Frank's allusion. \"When we took the history of homosexual themes in cinema?\"\n\nTodd smirks, \"Yeah, great class. All we did was watch short flicks and movies!\"\n\nFrank becomes annoyed. \"It was more than that; we learned some important things too, you know.\"\n\n\"Yeah, yeah, I know, but,\" he smirks, \"it was slack!\"\n\nTruly annoyed now, Frank states, \"Beside the point. Remember the short film we watched by the twentieth century director David Blythe\u2014oh, Hadrian, what was its name?\"\n\n\"Don't ask me. All I remember about that class was not having to do any work. I spent most of it sleeping. It was sweet.\"\n\nFrank is almost irate. \"Todd, please, I'm trying to remember\u2014 _Circadian Rhythms_!\" Frank is pleased with his excellent memory. \"In it, he said, and I quote, 'All mirrors are homosexual.'\"\n\nTodd screws up his face in disgust at this logic. \"What?\"\n\nFrank walks over to the mirror to illustrate. \"See, I'm looking at myself, and what I see is a man. As I admire my own image, I am admiring someone of the same sex.\"\n\n\"By that reasoning,\" Todd utters sarcastically, \"everyone who masturbates is gay.\"\n\nFrank spins to face Todd, pleased by the logic, \"Yes! That's right! And Todd,\" he says, remembering back to the day he walked in on his friend, \"you can't be\u2014you know\u2014that way\u2014having masturbated.\"\n\nTodd blushes as the memory surges through him. When Frank walked in on him, he had been masturbating to thoughts of Crystal\u2014while looking at heterosexual porn. Shame floods his body. Franks, sensing his friend's unease, sits down beside him and wraps an arm around Todd's shoulder. \"Besides,\" he adds, giving Todd a gentle shake, \"everyone born in Hadrian has homosexual tendencies. It just takes some people a little longer to find themselves than others; that's all.\"\n\nTodd nods his head (a little too quickly) in agreement. \"I'm just not ready\u2014right, Frank?\" As his breath quivers, he tries to sound confident. \"I'm just not ready yet.\" _If Devon taught me anything about myself,_ he thinks, _it was that!_\n\nFrank begins kissing the top of Todd's head. \"I can wait, babe. I can wait.\"\n\nAlthough Frank's action is innocent, Todd panics. Pushing Frank away, Todd stands, a bit too aggressively, before crossing to the other side of the room. Positioning himself next to Frank's desk with his back to his friend, Todd picks up his school slate and begins tapping the screen, searching for the doc they need to study from for tomorrow's test. After finding what they need, he holds the slate against his chest, a feeble armor against Frank's lust. Finally summoning up the courage, he rejects Frank's offer. \"No, you can't.\" Turning back to face Frank, he states, \"You can't wait.\" He pauses, losing some of the conviction in his voice. \"At least, not for as long as it could take me.\" Silencing Frank with a wave of his hand, he adds, \"I know you, Frank. You're horny. You'll start dating all those pretty boys and I'll look like a fucking idiot.\" Just as Frank is about to deny the charge, Todd interjects, \"Really, Frank? Admit it. What if it takes two or three years before I'm ready?\" His eyebrows cock. \"My father wasn't ready until he was twenty-four years old!\" What Todd doesn't mention was that his father was \"piss drunk\" at the time; something his Papa Mike often slurs in drunken lament when he doesn't realize Todd can hear him. With barely a pause for this thought Todd continues, \"Are you really going to stay celibate that long?\"\n\nThat is the clincher. Frank cannot deny the fact that two years\u2014with the possibility of eight years\u2014without sex is way too long. He would never stay faithful to Todd, not for that long of a stretch, and Todd, as he has come to learn over the years, is the most steadfast of friends. \"That's why I love you, you know.\"\n\nTodd tries to laugh; pretending to be coy is not something he is good at. \"Why?\"\n\n\"Because you're faithful, loyal. You're the best friend I've ever had, will ever have.\" Standing up, Frank crosses over to join Todd as his desk. \"Papa Dean says you're one in a million!\"\n\n\"That's nice of him.\" Todd likes Frank's Papa Dean. He is down to earth, although Todd can't quite figure out why he feels an affinity toward the man. Ever since his father died, Todd has felt like Papa Dean adopted him. He's proven to be an even better father than Papa Mike. Todd flops down on the floor, the school slate now face up in his lap.\n\nFrank pulls out his desk chair and sits down. Swiveling around so he can look down at his friend, Frank ends with, \"Okay, so you're not ready.\" Pointing his index finger at Todd, he warns, \"I'm going to continue dating.\"\n\n\"I never said you shouldn't.\" Todd is not looking at Frank, tapping and sliding his finger across the slate, ostensibly in search of the page they need to study.\n\n\"But one day,\" Frank continues, \"you will be ready, Todd Middleton.\" Todd laughs and shakes his head. Frank's dogged determination is what makes him such an excellent b-ball player. The man just never knows when to quit. \"And when that time comes, just know this.\" Frank bends down and lifts Todd's face up by the chin. \"I will be standing in the wings ready to sweep you up into my arms.\"\n\n\"Okay, Frank.\" Todd sees no point in continuing to resist since he's won a stalemate, \"but until then, please remember, we're just friends.\"\n\n* * * * *\n\n#  _Salve!_\n\n**Hadrian's Wall Still Holding  \nHNN\u2014Melissa Eagleton Reporting**\n\nAnother wave of heterosexual barbarians attempted to force their way through a portion of Hadrian's Wall and cross our border last night. This time, the attack was waged against the Mid-West Gate Battalion. According to Captain Collins, 12th Platoon commanding officer, \"This was no regular army. Had it been, they never would have concentrated all their men on so heavily a fortified zone.\" Having anticipated an intelligent military attack, dressing regulars like thralls, Captain Collins had immediately contacted the bases located fifteen miles south of the Mid-West Gate, those camps stationed near the electric wire fencing, warning them to be on alert for what he believed would be the real attack. However, no such attack came. It truly is a wonder that the hordes did not attempt an attack in one of these locations, which are, as Captain Collins assures us, the preferred location for armies or organized gangs to attempt breaking into Hadrian's borders. Even so, rest assured, folks, that these border fences are not left undefended. We have watchtowers placed every fifty feet with trained snipers to keep the heterosexual barbarians at bay. At the Mid-West Gate, heavy fighting was reported by Captain Collins. After taking small arms fire, and coming under insurgent rocket attacks, the 12th Platoon used superior tactics and its upgraded M4A2 assault rifles to push back the enemy and seize the objective. The rifles' extremely high rate of fire and 30 round 5.56mm magazines provided the necessary force multiplier to inflict maximum casualties with minimal losses. In fact, Captain Collins believes that last night's attack was more of a suicide run than a battle. Considering the desperate living conditions of many outsiders, this does not seem so unlikely. \"All the bodies of the heterosexual dead,\" Captain Collins is quoted as saying, \"were emaciated,\" adding, \"had they been successful at climbing the wall or breaking through the front gate, not one of them would have stood a chance in hand-to-hand combat against the men and women of Hadrian.\" His eyes were filled with sorrow as he spoke, and shaking his head, he told me, \"Even our oldest citizens would have stood a better chance. It was worse than an old fashioned turkey shoot!\" A turkey shoot for those voc, or wave, viewers who are unfamiliar with the term, means having all the advantage. Apparently a turkey hunter would scatter the flock and then sit back and wait while all the birds reconvened in the same spot. He would then just sit back and shoot them at his leisure. I can assure you, folks, that under no circumstances did Captain Collins mean to suggest his soldiers and he shot the barbarian hordes at their \"leisure.\" Rather, for the soldiers of the Mid-West Gate Battalion, this turkey shoot was far too painfully easy. Many of our young soldiers have requested psychological aid, as their training did not include assisted suicide. Captain Collins has made a recommendation to his commanding officer that the military begin looking into the possibility of training our soldiers for what he believes will be more future acts of euthanasia. Once again, we are reminded of our good fortune, extending gratitude for the wisdom of the founding families in creating the only safe haven remaining for humanity on planet earth. Is it any wonder we have no crime in Hadrian? Is it any wonder we have no jails? Who, in his right mind, would commit a crime knowing that the only punishment available is banishment with the option of assisted suicide? It is truly amazing how many of our convicted criminals request black henbane when sentenced with exile.\n\nFor those viewers unaware, black henbane is a highly poisonous plant because it contains several alkaloids. It grows wild throughout most of Hadrian. To ensure its availability to those who would choose euthanasia, the plant is grown in a hothouse here in Antinous. Even our most hardened criminals do not want to live in the outside world, so daunting, so terrifying is the prospect. If put to the test, I daresay I too would choose death over exile. Really, when you think of the alternative\u2014life outside our walls\u2014wouldn't you choose death?\n\n_Vale!_\n\n# Enter the Vixen\n\n\"Frank,\" Papa Dean calls from the front room. \"Your friend Crystal is here.\"\n\n\"Finally,\" Todd mutters. He really does need Crystal as a chaperone.\n\n\"She says you guys have to study.\" There is some trepidation in Papa Dean's voice, a note Todd picks up on, but to which Frank appears oblivious.\n\n\"Thanks, Papa,\" Frank hollers back. \"Send her in.\"\n\nTodd shakes his head. \"Why can't you just open the door and talk like normal people, instead of yelling?\"\n\nFrank, very good at being coy, replies, \"I'd have to walk all the way over there and I prefer being close to you.\"\n\nThe door bursts open and Crystal enters. She is tall\u20145' 11\u00bd\"\u2014but it has always irked her that she never hit six feet. She has never forgiven her mothers or the doctor for making her take heterosexual birth control in order to stunt her growth. \"Gigantism,\" her doctor had warned, \"is a serious concern. An enlarged heart would mean a shorter lifespan.\" This warning was enough to convince her mothers, so Crystal was required to stunt her growth.\n\nWith her short, dark brown hair swept up and over to the right side, accenting her oblong face, cute little nose, and jade green eyes, Crystal is truly beautiful. Like Todd, she refuses to wear makeup. Everyone, boys and girls, wears makeup these days. No one seems to care that a girl won't, though. More girls than guys can get away with the natural look, even if their numbers are far and few between. At least Crystal isn't the only girl at Pride who refuses to wear the goop. And a girl without makeup isn't instantly tagged as straight. By comparison, Todd has set himself up for razzing because he is the only boy at Pride who won't wear any facial enhancements. Even, still, Todd begrudgingly admits most people don't wear makeup to look like sluts\u2014 _only Frank's boys!_ What Todd will never admit to himself, though, is that his perspective on this is more than slightly skewed.\n\n\"Hey, guys,\" Crystal chimes as she walks across the room. When she reaches the middle, she gives a model spin so they can better appreciate her figure. \"So,\" she asks, beaming in their direction, \"how do I look?\"\n\n\"Wow! Girlie girl, you look amazing!\" Frank stands up to admire her image more thoroughly. \"Do you always dress like this for a study date with boys?\"\n\nCrystal giggles. \"Silly, I have a date with Lolita after.\" Spinning slowly to show off her beauty, she says, \"I'm wearing this for her!\"\n\n\"Do your moms know you're dressed like that?\" Frank utters his amazement.\n\n\"Don't be insane,\" Crystal replies. \"They don't even know I own this dress.\"\n\nCrystal is stunning in her light green dress, snug to the body. Thin straps reveal snowy white shoulders, a low cut bodice exposes a deep cleavage, and her hips are accented by the rippling of material sewn into elastic. The skirt is a mini and hugs her buttocks neatly. Frank leans in to inspect her backside. \"Aren't you wearing any underwear?\"\n\nThe question is rhetorical, but Crystal answers anyway. \"Of course not,\" she giggles. \"Can't risk showing any lines.\" Still twirling in constant show of her beauty, Crystal smiles. \"What do you think, Todd? Do I look okay?\"\n\nTodd's eyes are wide, his mouth hangs open and he is staring. He stutters, \"You, uh\u2014gee, Crystal.\" Swallowing hard, he manages to say, \"You look great.\"\n\n\"What do you think, Frank?\" Crystal swirls again. \"Will Lolita like me in it?\"\n\nFrank is more than pleased with Crystal's question. \"She is going to like you so much, girlie, you won't last sixty seconds in that dress.\" Crystal's giggle suggests that that is the plan. \"Let me dab some makeup on you,\" Frank adds. \"And I can cut that time in half.\"\n\nTurning coquettishly to Todd, Crystal winks, \"What do you think, Todd? Should I let Frank doll me up?\"\n\nFrank notices Todd staring Crystal's way\u2014gawking actually. Jealousy rages as he slams back down on his chair. It creaks but doesn't break. \"Fuck it!\" he spits out. \"You look fine the way you are.\" Todd blushes and turns away.\n\nCrystal, ever the expert deflector, turns a teasing eye on Frank. \"Oh, oh.\" Feigning a little pout, she says, \"Crystal came at a bad time, didn't she?\" Poking Frank, attempting to jostle him into good humor, she adds, \"Were you trying to make it with the boy again?\"\n\nFrank, although not feeling very spirited, forces a laugh. To make it look like he's giving in to Crystal's jocularity, he spins gaily in his chair, retorting, \"Can't blame a guy for trying.\"\n\nTodd turns, hoping to change the conversation. \"Shouldn't we start...\" Red splotches now cover the whole of Todd's face. \"I mean...\" Embarrassment causes him to stumble over his words.\n\nMisinterpreting Todd's bashful demeanor, Frank smiles Crystal's way.\n\nGiving Frank a knowing wink to deepen further his misconception, Crystal feigns concern as she bends forward to shake Todd's shoulder, exposing her deep cleavage in the process. \"Awe, Todd, are we embarrassing you?\"\n\nTodd bends his head down quickly; even the back of his neck is red. He mutters, \"Can we study? Please?\"\n\n\"First, you two kiss and make up.\" Then taunting Todd, she adds, \"Otherwise, I might think you're a _strai_ and attracted to me.\" Looking up stunned, Todd is clearly hurt by the accusation. \"Oh, I'm just teasing.\" Crystal's voice is not as reassuring as it should be. \"I know you're not straight. But seriously,\" pouting again, looking to Frank to take the initiative, \"I won't feel right until you boys kiss and make up.\"\n\nFrank stands, taking full advantage of Crystal's suggestion. Walking over to Todd, Frank motions for him to stand. Then, after placing both hands behind Todd's head, and with his back to Crystal, he carefully mouths the words, \"Make this look good.\" Using his eyes, Frank gestures Crystal's way. \"She's a gossip.\" As he looks into Todd's eyes, he waits for him to initiate the kiss. When their lips finally meet, Frank holds them together for over a minute. Crystal squeals in delight and claps during the whole ordeal. As Frank returns to his chair, Todd shakes before awkwardly sitting down on the floor.\n\n\"Wow, Frank,\" Crystal remarks. \"That must have been some kiss. You got the boy all a tremble.\" She, too, sits on the floor in front of Todd. \"You know,\" she still goads Todd, \"I love watching boys kiss.\"\n\nFrank smiles. \"Well, that's all you get to watch us do!\" Although he is speaking to Crystal, his eyes are locked on Todd. That kiss has his whole body on fire. Reaching across his desk, he slides his personal slate over, using it discreetly to cover his groin.\n\nTodd musters up the courage to speak, \"Can we...\" Even now, addressing these two is difficult. His hands, still gripping the school slate, shake it slightly for emphasis as he finishes, \"study now?\"\n\n\"Study?\" Crystal claps her hands together, her eyes open wide in excitement and she gives her head a little shake. \"Okay! Tomorrow's test!\" As always, Crystal is highly exuberant, (kids love associating with her due to her excessive energy bursts and constant upbeat mood). \"Hadrian's legal system. Tomorrow's test will be on Procreation Laws, the impact of 6-13 on Hadrian's foreign policy and conscription laws\u2014\"\n\nFrank rolls his eyes in disgust. He is sorely displeased with the fact that all Hadrian's youth must now serve four years in the military. Prior to 6-13, youth were only conscripted for two years.\n\nCrystal ceases listing and begins to chastise him. \"And what is your problem?\"\n\n\"I wish we didn't have to have conscription,\" Frank mutters. Although he doesn't like the idea of having to serve in the army, he, like every other Hadrian citizen, is required by law to do so.\n\n\"Exceptions are made for scholarship winners,\" Todd reminds him.\n\n\"A very good reason to study,\" Frank declares, posturing himself now for hard work.\n\n\"I disagree,\" Crystal argues. \"Conscription is essential.\" Quite stern now, she adds, \"Even if I win a scholarship, I plan to ask the uni to hold on to it until after I serve.\" Both boys stare uncomprehendingly. \"We are all responsible for helping to defend our country. Four years of service ensures more experienced men and women defending our walls, something we need considering that the world out there is getting more and more insane.\" Shaking her head sadly, \"Surely you remember 6-13?\"\n\n_Todd certainly remembers,_ Frank thinks, as he watches his friend for any sign of emotional disturbance. Fortunately, there is none.\n\n\"Besides,\" Crystal remonstrates, \"you can't depend on a scholarship.\" Shaking her head disapprovingly, she says, \"Sorry, Frank, but your marks are just not high enough. Todd may get a sports scholarship, but you are going to the wall.\" Hadrian's wall, although envisioned to surround the entire country (and one day it will!) only spans one hundred miles across those lands where the heterosexual barbarians have been known to concentrate the bulk of their attacks. There are, however, watchtowers and electric fences spanning the rest of Hadrian's borders. \"Unless,\" Crystal smiles wickedly, \"you expose yourself.\"\n\n\"What?\" Frank stands, face flushed with anger. Todd stands to calm him. \"She's just kidding, Frank.\" Looking down at Crystal, miffed that she would upset Frank to such a degree, Todd says, \"Tell him you're just kidding.\"\n\nCrystal can barely get the words out from laughing so hard, but she does manage to execute an apology of sorts. \"Don't worry, Frank,\" she finally utters after her stomach and diaphragm calm down. \"No one in their right mind would ever believe you're straight. Which again means, you're going to the wall.\" This brings on another fit of laughter.\n\nTodd, knowing Frank's temper, pulls the man in for a hug. \"If you go to the wall, I go to the wall,\" he whispers. \"We go together.\" This mollifies Frank, and he returns Todd's light hug with his own grizzly bear. The physical contact, Todd rightly surmises, is just what Frank needs to get his mind off wanting to punch Crystal.\n\nAfter kissing the top of Todd's head, Frank whispers his appreciation. \"Thanks.\"\n\nAs the two boys resume sitting, Crystal, oblivious to her guffaw, smiles Todd's way, whispering to him her dream. \"I'm thinking of making the military my career.\"\n\nSuddenly, Frank is forgotten as the two have a brief t\u00eate-\u00e0-t\u00eate. \"I thought you wanted to be a surrogate goddess?\" Todd asks earnestly. \"You'd look beautiful pregnant,\" he blurts out.\n\nFrank scowls as Crystal smiles and says, \"Thank you.\" To Frank's further annoyance, she blows Todd a kiss. \"I can do both. The military has great maternity leave benefits, and a surrogate goddess never has to get pregnant more than once every two years.\"\n\n\"It seems to me a surrogate goddess would be ten months of wasted time and money for the military,\" Frank utters discouragingly.\n\nImpervious, Crystal insists, \"It's amazing how many compensations are made for a surrogate.\" Then, smiling at Todd for encouragement, she adds, \"I really could do both!\"\n\n\"Cool,\" Todd replies, finding everything Crystal has to say fascinating right now.\n\nFrank shakes his head in disgust. \"Typical Crystal, always wanting to be in the limelight!\"\n\n\"You shall not dissuade me, Frank Hunter.\" More determined now than ever, she states, \"I shall be a surrogate goddess _and_ make the military my career!\" Now, looking back at the school slate in Todd's lap, she smiles. \"Let's get back to our study list, shall we. Where was I? Oh, yes, listing off study topics. Procreation Laws, the impact of 6-13 on Hadrian's foreign policy, and...\" glaring at Frank, \"conscription laws, the resurgence of the anti-heterosexual campaign, and reeducation camps!\" Once again turning to face Todd, Crystal opens wide her eyes. She accents the expression by sticking her tongue out of her mouth and rubbing her hands together while giggling. \"Let's start with Procreation Laws, shall we?\" Trying now to recapture a teasing mood, Crystal slaps Frank's thigh, \"So, tell me, after you two boys wed, are you planning on having your babies right away or waiting?\"\n\nBefore Frank can even answer, Todd's head is shaking. \"We're not registered!\"\n\n\"I know that,\" Crystal laughs. \"I said _when_.\"\n\nFrank steps in before Todd can reject him thoroughly. \"Yes! We do plan to register!\" The look he gives Todd brooks no objection, so Todd simply looks down at the slate in his lap and waits, wishing fervently for the other two to finish kidding around. Frank, however, has more to say on this issue. He wants to make sure everyone in the room is clear on his intent. \"We'll wait a couple of years, I think, but have them both within the first ten years. That's what my dads did\u2014and I want them close in age\u2014like Roger and me. What about you? Right away or later?\"\n\n\"Never!\" Crystal says this a bit too emphatically.\n\n\"What?\" Todd looks up, his face exposing disappointment.\n\nCrystal shrugs slightly and then looks to Frank, almost as if he would be more likely to understand. \"When I register...\" Mischievously, she adds, \" _if_ I register, I certainly don't want to add a child into the mix.\"\n\n\"But you have to,\" Todd admonishes. \"And your partner will have to.\"\n\nCrystal pointedly ignores Todd. \"A child just complicates everything.\"\n\nFrank looks concerned. \"You and Mama Elena at it again?\"\n\nCrystal, dropping all jocularity, becomes instantly somber. Her eyes roll up into her sockets. \"And then she and Mom go at it. Mom always takes my side and that pisses Mama Elena off. 'Just because you passed her through your loins doesn't mean she's always right.'\" Crystal shakes her head and joins the boys in laughter at her mimicry of Mama Elena's shrill accusations. Crystal laughs, \"Mama Elena's right, though. Mom lets me get away with everything.\" Taking on a stern expression for this imitation, Crystal shrills, \"You've spoilt the girl rotten.\" After they have all had a good chortle, she adds, \"And I am spoilt\u2014rotten. I could never have a kid, especially if she turned out like me.\" Frank and Todd guffaw in agreement.\n\n\"But,\" pulling them back to his original argument, Todd insists, \"You still have to have a baby. Everybody has to\u2014it's the one child law.\"\n\nCrystal's green eyes sparkle. \"There are ways around that. Your Papa Mike figured a way around it, didn't he?\"\n\n\"But Papa Mike wants to have his child.\"\n\n\"Then why doesn't he?\" Crystal asks.\n\nTodd cringes. When drunk, Papa Mike will often throw this issue in\n\nTodd's face. \"Do you see me with my own child? No! And why not, might you ask? Because of you, I've no fucking money! Because of you, I've no fucking child of my own! You are so fucking needy. New running shoes\u2014new clothes\u2014more food!\" Todd no longer points out the fact that it was Papa Mike who drank through all his father's savings after 6-13.\n\nFrank, cognizant to the sudden pallor of Todd's face, shakes his head in dismay. Crystal always manages to ask the wrong questions. Frank leaps in to save him from having to explain. \"His Papa applied to pass his responsibility on to others. They use his semen for men with low sperm count.\" Watching Todd's head droop, Frank cringes at now being the one to make Todd wince.\n\nPapa Mike's voice roars in Todd's brain. \"I've probably got a thousand kids out there I'll never meet!\"\n\nCrystal's voice breaks through Todd's memory, \"Why?\"\n\n\"For the love of Antinous, Crystal! They're poor, all right! His Papa can't afford another kid.\" Fearing for Todd's emotional balance, Frank insists, \"Drop it already.\"\n\n\"Sorry, Todd.\"\n\n\"It's all right, Crystal.\" Todd smiles meekly. \"I don't mind not having a kid brother or sister. Frank and Roger make up for that.\"\n\nCrystal's smile widens. \"You always look at everything in such a positive light, Todd. I so admire you.\"\n\nTodd beams, all discomfort washed away by Crystal's compliment. Frank seethes with jealousy. _Crystal caused Todd pain, I defended him, and one_ _silly compliment on her part and he's all's good._ Wanting to lash out at Crystal, Frank throws the argument back in her face. \"You don't have poverty as an excuse, not with your Mama! How do you plan to avoid raising your own child?\"\n\nCrystal replies slyly by tapping Todd's school slate, suggesting the answer lies inside today's study session. Although still confused and curious as to her meaning, Todd uses Crystal's gesture to suggest the three of them actually begin preparing themselves for tomorrow's test. \"Do you guys want me to quiz you?\"\n\nCrystal and Frank both roll their eyes. \"Goodness gracious me, Todd!\" Crystal shoots her next remark Frank's way. \"Is he always so serious?\"\n\n\"Yup!\"\n\n\"Okay. Let's begin, but give me the slate. I'll drill you two. I already know this shit by heart. Mama Elena's been making me study every night this week.\" Rolling her eyes upward to the ceiling, opening her hands into fat jazz fingers, Crystal adds, \"She is such an extremist!\" After snatching the slate away from Todd, Crystal begins, \"Let's start with an easy one. When was the one child law instigated?\"\n\nFrank answers instantly, \"The same year as our founding.\"\n\n\"Good.\" Crystal is now settled into a studious pose, sitting on her knees, causing her skirt to rise even higher up her thighs. Todd struggles not to stare at the soft brown fuzz she has exposed for him.\n\nFrank notices Todd fixated on Crystal's legs\u2014no, not her legs\u2014the space between them. \"Hey, peach fuzz.\" His annoyance evident, he kicks Crystal's thigh. \"Save it for Lolita.\"\n\nCrystal giggles; blushing, she looks to Todd as if he is a conspirator. \"Oops!\" Grabbing at the skirt between her inner thighs, she gives it a slight tug, covering what she had previously exposed. Mischievously giving Todd a wink, she says, \"It's a good thing I'm with guys!\" Then, as if nothing untoward just happened, Crystal continues their study session. \"Todd, you answer this next one.\" Todd closes his eyes, a study method he has perfected over the years to help him visualize the answer. He believes his brain works like a computer, and he simply waits for it to show him the answer. Sometimes it works; most of the time, it doesn't. \"Why...\" Crystal's voice sends shivers down Todd's spine, \"since our sexual preference is the perfect form of birth control, did our founding families believe it necessary to instigate such a radical law?\"\n\nTodd's eyes stay closed. The answer has appeared for him. \"In order to ensure a stable population that neither declines or increases, it was determined that each man and woman be responsible for bringing in another life to replace his or her own. This way, we could maintain a stable population of ten million and...and\u2014\"\n\nCrystal, never noted for her patience, doesn't give Todd any thinking time. \"And?\"\n\n\"And,\" Frank continues for him, \"we have the outside world as our example of what happens when there is no government control over human procreation.\"\n\n\"I know that,\" says Todd, opening his eyes and looking Frank's way. \"Just let me finish, will you?\"\n\n\"Okay. Finish.\" Frank isn't offended. Crystal notes how fondly he looks down on his friend.\n\nTodd's eyes squeeze tight. \"What Frank said and...um...\" He lifts his hands up, turns them into fists, and begins pumping (Frank and Crystal try very hard not to laugh at him). \"To ensure the future of humanity...\" Here, Todd claps his hands together, opens his eyes, and points directly at Crystal, \"As well as to keep mankind from ever again overwhelming the earth with its presence.\" Satisfied with his answer, Todd releases a contented sigh.\n\n\"Very good, Toddie.\" Crystal pats him on the back.\n\nAnnoyed, Todd responds, \"My name is Todd!\"\n\n\"Sor-ry!\" She extends each syllable to dramatically ridiculous proportions. Then looking to Frank, she says, \"Doesn't like pet names, eh? Whatever are you going to call him?\"\n\n\"Actually, I call him 'Ki\u2014'\"\n\nBefore Frank can call him \"Kitschy,\" Todd butts in. He is clearly infuriated, \"He calls me what he always calls me\u2014by my name!\"\n\nFrank laughs a little uneasily. \"Just don't call him 'Toddie.'\"\n\nIn order to lighten the discomfort she's created, Crystal pipes up, \"You guys hear what happened to Millicent at school?\"\n\n\"No.\" Franks leans forward in his chair, hoping the gossip is malicious enough for Crystal to have to whisper. Todd too leans forward, his face precariously close to her breasts. His heart quickens.\n\nCrystal, sensing the collective desire for dirt, embellishes the story with facial expressions, elongated vowels, and crisp consonants. \"She got caught cheating using her vocal contact lens.\"\n\n\"No!\" Both Todd and Frank utter in dismay.\n\n\"Yeah. Lolita voc'd me today. Millicent is in major hot water. Her mothers grounded her, and she has a week's worth of detention at school!\"\n\n\"Yikes!\" Frank exclaims.\n\n\"That's gotta hurt,\" Todd adds sympathetically.\n\n\"That's not the worst of it. Apparently, the school is banning the use of vocs during school hours.\"\n\nShaking his head, Frank scoffs, \"That is so _strai_!\"\n\n\"I know,\" Crystal agrees.\n\n\"There is no way they can do that,\" Frank adds. \"You can't even see the thing, let alone determine if someone's using it.\"\n\n\"People move their mouths a lot,\" Todd responds. Not having a vocal contact, he has noticed how stupid everyone looks, talking or whispering to the air. \"It looks like you're talking to yourself.\"\n\n\"So?\" both Crystal and Frank say, offended.\n\nCrystal carries on with the harangue. \"You can always claim to be reading to yourself.\"\n\n\"Yeah,\" Frank agrees. \"Some people read better when they mouth the words.\"\n\n\"But that's not how they're going to stop us,\" Crystal says.\n\n\"How are they going to catch you guys?\" Todd asks. This situation doesn't affect him since he can't afford the vocal contact lens, but as his friends are clearly upset, he feigns interest.\n\n\"I guess they've got some kind of blocker and can stop voc signals.\"\n\nTodd smiles and shows off his cell phone. \"The three of us can still text.\"\n\n\"That's right!\" Frank exclaims happily. He and Frank slap hands and grip fingers. Then Frank and Crystal do the same. Crystal raises her hand for Todd, who pretends not to notice by picking some lint off his sock. He really wants to slap hands with Crystal, but he is petrified of making physical contact with her.\n\nLooking up, trying to appear jovial, Todd exclaims, \"Thank Hadrian for old school!\"\n\n\"Yeah, but,\" Crystal adds, a little miffed by Todd's rejection, \"old school tech can never replace a voc.\" With vocal contact lenses, one can contact a friend anytime by simply blinking an eye and saying his name. By searching for the right icon and blinking it open, a visor scans out from the eyes to form a nearly invisible screen, allowing for virtual face-to-face contact. One can watch vids, movies, and concerts, play games\u2014a person can do just about anything using the voc. For the more flamboyant, microphones and speakers come in nicely camouflaged jewelry, often worn as an earring. For the artistically bent, there are options like tactile tattoos or, for the more modest individuals, a small implant that can be easily installed behind one earlobe.\n\nFranks muses over the situation a little and declares, \"The ban won't last long anyway.\"\n\n\"Why not?\" Crystal looks on inquisitively.\n\n\"All the teachers have contacts now, even Mr. Gavin!\" Mr. Gavin is Pride High's principal. \"It won't be long before they'll want access again.\"\n\n\"So true,\" Crystal agrees. \"Nobody wants to be unhooked from their voc.\"\n\n\"At most,\" Frank figures, \"this ban will last two or three days.\"\n\n\"Can we get back to studying?\" Todd asks, weary of a discussion that has nothing to do with him.\n\n\"Okay.\" Crystal recaps what they've covered. \"Hadrian maintains ten million. We've reviewed the rationale for the one child law, and now, Frank, it's your turn. How many children can a registered couple apply for?\"\n\n\"Easy. Two.\"\n\n\"Nope.\" Looking Todd's way, Crystal asks, \"Do you know?\"\n\nClearly confused, Todd defends Frank's answer. \"How could it be anything but two?\"\n\n\"Come on, you guys,\" Crystal is clearly disgusted. \"That was a trick question. You know Mr. Reiner loves trick questions.\" They both look stunned, neither knowing the answer. Crystal sighs, answering for them. \"A registered couple can apply for as many children as they want.\" Smiling now, enjoying the boys' confusion, Crystal explains, \"The majority of couples only get approved for two babies, BUT it is possible even today, mostly due to 6-13, that some couples get approved for a third child. It's like slots on a roulette wheel. The majority say two children, two or three slots say three children, _and_ one slot even says four.\" She accents the number by wiggling four fingers.\n\nFrank is intrigued; he would love to have a lot of children. \"You mean to say there are registered couples out there with _four_ children?\" The size of the family seems incomprehensible.\n\n\"Yes,\" Crystal replies triumphantly. \"Since 6-13\u2014\"\n\n\"You said that already,\" Todd remonstrates. He is trying not to feel any remorse, but so much emphasis on this one date is beginning to bother him.\n\nFrank gives Crystal a daunting stare and mouths the words, \"His father.\"\n\n\"I'm so sorry, Todd.\" Leaning forward, she places a hand on his knee. \"But,\" she adds gently, \"this will be on the test.\"\n\nTodd closes his eyes briefly to summon up his inner strength, \"I know. You're right. We have to study it. I'll be okay.\" Looking now to Frank, the one he knows is truly worried, he states, \"I will...I am...I'll be...I'm fine.\"\n\nCrystal, sensing the need to move past this part of their review as quickly as possible, pushes on. \"So, since...that date...the government put together a lottery for which only registered couples can apply. And they get _hundreds of thousands_ of applications!\" Crystal is quite exclamatory. \"Every year since...five hundred registered couples' names are drawn and awarded the privilege of having a third child. These couples are given extra tax credit for raising a third, and in the odd case, a fourth child. The need to rebuild our population, to maintain a population of ten million as established in our constitution, requires a little levity with the one child law. And, remember, this opportunity is only for registered couples.\" Giving both boys a stern look, she adds, \"I guarantee you this will be on the test. It was part of our required reading.\" When Todd and Frank flush with shame, Crystal rightly surmises that neither boy bothered to download the text assigned them by their teacher. Sounding more like a parent now than a friend, Crystal lectures the two young men, \"You had better take the time to read through that material tonight.\" Switching smoothly, too smoothly, from lecture mode to teasing, Crystal's green eyes begin to gleam. She has been looking forward to this next topic of study. \"So, which of you boys can tell me how a woman might avoid having to raise a child of her own?\"\n\n\"Easy,\" Frank chimes. His eyebrows rise and he smirks Crystal's way, pleased at being able to return taunt for taunt. \"Expose yourself.\" Enjoying the moment, he adds, \"One of the first things they do to a _strai_ at re-ed is sterilization.\" Any youth exposed as straight either has her tubes tied or, for a young man, a vasectomy. The DNA of an exposed individual is deemed tainted and not worthy of future Hadrian citizens. Clapping his hands triumphantly, Frank rubs them together before shooting an index finger into Crystal's face. \"You'd never have to have a baby then!\"\n\n\"True!\" More than irked, Crystal's answer is clipped. \"If my question were in reference to the re-ed class, you'd be right, but it wasn't, so you're wrong!\" Barging on now to keep Frank from interjecting, she explains, \"I'm talking about normal lesbians.\" Smiling with superiority, she states, \"There is a way for a lesbian to avoid raising her own child if she so chooses.\"\n\nA little irked at Crystal's superior attitude, Frank becomes demanding. \"Quit beating around the bush, then. How do you plan to avoid raising a child?\"\n\n\"The surrogate goddess is free to exempt herself from raising her own child.\"\n\n\"Why?\" Todd asks, confused.\n\n\"Since a surrogate is asked to give up all but one infant at birth, and psychological studies found it too hard on a woman's psyche to keep one child and give up the rest, a surrogate goddess has the option of not keeping any of the babies she births. Apparently,\" she explains, \"once you've raised a child you have given birth to, it becomes exceedingly difficult for a woman to give up the other children she births for men.\"\n\n\"Makes sense,\" Todd says.\n\nAlthough miffed at having to agree with Crystal's reasoning, Frank adds, \"That's why most surrogates quit after they birth their own child.\"\n\n\"Meaning she has to be an elder parent! But enough of this,\" Wanting to avoid having it out again with Frank, Crystal decides it is time to move on. \"Next topic.\" Looking to Todd, her strongest ally in the room, she asks, \"What does it mean for a country to be entrapped?\"\n\nLeaning back, Todd cradles one knee inside his hands before remarking, \"You know, I heard on _Salve_! that nearly two-thirds of the world's remaining countries are entrapped. Eagleton says it won't be long before Hadrian is the only country left that is able to sustain itself.\"\n\n\"That's good, Todd; it shows you understand the concept, but you haven't defined what it means.\" Looking up, she says, \"Frank, you define it.\"\n\nFrank looks at Todd, trying to remember what he said, lifting the meaning out of context. \"It must mean trapped somehow.\" Looking at Crystal, he says, \"Trapped inside their borders, maybe?\" Stopping her before she can give him the answer, he adds, \"Wait, no. Todd said something about Hadrian being self-sustaining.\" Looking into Todd's eyes, the answer comes to him. \"Maybe a country that is entrapped can't sustain its own people. Too many people and not enough natural resources to keep everyone alive.\"\n\n\"Good, Frank!\" Crystal slaps his hand and they grip fingers. They are actually friendly again in this moment.\n\nLooking back into Todd's eyes, Frank asks, \"Did you say two-thirds of the world's countries are entrapped?\"\n\n\"According to Mama Elena, they all are.\" Crystal once again leaps in with the correct answer. \"If what Mama Elena says is right, Hadrian is the only self-sustaining country left on the planet.\"\n\n\"Are you serious?\" Todd is shocked.\n\nCrystal adds to the tension. \"What's left of Canada can't hold its own, far less the rest of the world. Most countries have collapsed; the world economy is completely shattered.\" Breathing a sigh of relief, she adds, \"Thank all that is gay and glorious we finally cut ourselves off from the rest of the world!\" The events of 6-13 created so much fear in Hadrian's citizens that the government cut off all but the most important trade. Although Hadrian continues to export much needed fresh water and grain to the outside world, importation has been radically reduced.\n\n\"I know,\" Todd adds, \"There is so little usable land left; most of the soil out there is depleted as a result of overuse and erosion.\"\n\n\"That's exactly what Mama Elena said, Todd.\" Then turning to Frank with conviction, Crystal says, \"You see, Frank, that is why our wall has to be defended, at all cost. If Mama Elena is right and Hadrian is the only sustainable country left, the threat from the outside world is only going to get worse.\" This sobering thought causes the three to sit in silence, considering what their tour of duty on the wall will be like two short years from now if scholarships are not forthcoming. Even Crystal (who plans to make the military her career) knows their lives will be in danger as long as they remain in the service. Once again, Crystal diverts their attention. \"But we digress. Tomorrow's test will be on Hadrian's Procreation Laws and\u2014\"\n\n\"Wait,\" Todd cuts her off. \"Aren't we missing something?\"\n\n\"What?\" Frank asks. \"I thought we covered everything.\"\n\n\"No,\" Crystal agrees, \"Todd's right. We forgot about historical implications\u2014lessons learned from the mistakes of the past. We better go back to the one child law. Thanks for reminding us. So, why was the one child law, first instigated by China in the twentieth century, so unsuccessful?\"\n\nFrank doesn't even give Todd a chance to reply. \"That's easy. The Chinese were heterosexual. The only guaranteed form of birth control for a heterosexual is abstinence. And nobody,\" Frank says, looking Todd's way, \"likes to abstain from sex.\" Tapping Todd's knee with his toe, he adds jokingly, \"Well, almost nobody.\"\n\nTodd is annoyed. \"Heterosexuality is not why the one child law never worked in China.\" Conceding slightly, he adds, \"It may have been part of the problem, but there was a lot more to it.\"\n\n\"Like what?\" Franks asks.\n\n\"Cultural stereotypes, boys being more valued than girls,\" Todd answers. \"Families would just keep having babies until they finally gave birth to a little boy!\"\n\nCrystal shakes her head in disgust, \"Thank Hadrian we don't have that!\" Crystal is maddened that her sex has always had to suffer discrimination.\n\n\"Homosexuality has certainly made equality between the sexes easier to attain,\" Frank adds. He and Crystal slap hands and grip fingers.\n\nSuddenly, Todd feels very alone. \"You guys make it sound like heterosexual men are to blame for all the world's evils.\"\n\n\"Maybe they are,\" Frank suggests. \"After all,\" he adds, \"the world out there is run by heterosexuals.\"\n\n\"Mostly run by heterosexual men,\" Crystal adds pointedly.\n\n\"By the late twentieth century, homosexual men were beginning to make their mark in politics,\" Todd counters. \"And,\" he adds to placate Crystal, \"there were women leaders, too.\"\n\nCrystal rolls her eyes. \"I can count them all, too, on one hand,\" she sighs. \"That Gandhi woman,\" tapping her index finger, \"Margaret British woman,\" her middle finger, \"some Muslim lady,\" her third finger.\n\n\"Our old country, Canada, had a woman prime minister,\" Todd reminds her.\n\n\"Yeah, right Todd.\" Crystal is clearly caustic. \"And she lasted how long?\"\n\nSmirking, Frank adds, \"Who even remembers her name?\"\n\n\"Whereas five out of Hadrian's eight presidents have been women!\" Crystal pronounces proudly. Then, feeling sorry for Todd, she leans forward, placing her hand on his. \"It might seem like we are blaming everything on the heterosexual male, but seriously Todd, our laws have to be harsh when it comes to heterosexual sex.\"\n\n\"It just seems wrong,\" Todd mutters. \"It feels like we're prejudiced or something.\"\n\n\"We're not prejudiced,\" says Frank, on the defensive. \"We're just realistic. Look at the mess heterosexuals have made of our world.\"\n\n\"But that has nothing to do with sex,\" Todd replies.\n\n\"No,\" Frank agrees. \"But it has to do with who is running everything.\"\n\nTodd shakes his head. \"Is this going to be on the test?\" He desperately wants to change the subject.\n\n\"No,\" Crystal answers. \"But it does lead us into our next topic: Hadrian's heterosexual law.\"\n\n\"Why is het'ro sex illegal, Todd?\" Frank asks pointedly.\n\nTodd talks to the floor. \"It is meant for procreation. No one form of birth control invented by man has ever proved a hundred percent effective.\" Sighing, he admits, \"That's all I can remember.\"\n\nFrank finishes for him. \"Heterosexual men are proven to be more violent in nature. A heterosexual male is more likely to commit rape, and it is suggested that heterosexual males are pedophiles.\"\n\n\"I don't believe any of that!\" Todd retorts. \"That's all subjective. There's no real evidence.\"\n\n\"Frank, Todd's right,\" says Crystal. \"The real reason we abandoned heterosexual sex was to stem the tide of the world's population. We can only guarantee a stable population by combining a homosexual lifestyle and IVF.\"\n\n\"True enough,\" Frank adds. \"But no one can deny the footage taken outside the wall. All those rapes, children beaten. And then, what those murdering bastards did to our female soldiers at the wall.\"\n\n\"What?\" Todd shakes his head in ignorance.\n\nFrank sighs. \"It happened nearly twenty-five years ago.\" He shudders. \"Dad showed me the vid stream.\"\n\nCrystal quivers, \"Mama Elena showed it to me.\"\n\n\"Why don't they show it to us in school?\" Todd asks earnestly.\n\n\"Too bloody.\" \"Too brutal.\" Frank and Crystal's voices overlap.\n\nCrystal finishes, \"They left that to the news and parental discretion.\"\n\n\"So, what happened?\" Todd asks.\n\nFrank begins. \"The heterosexual barbarians broke through the wall. They killed all the male soldiers instantly, but they took their time with the women.\" Frank pauses for effect. \"They raped them first.\" Another pause ensues, a moment of silence out of respect for the dead. \"One female soldier survived, and only because reinforcements arrived to drive the hordes back into the wastelands. But she had already gone mad. She never did recover.\"\n\nTrembling at the memory, Crystal adds, \"Oh, Todd, it was horrible.\"\n\n\"And it was all done by heterosexual males.\" Looking down at his friend, Frank states, \"They were brutal, Todd; I've never seen anything like it, and I hope I never have to ever again!\" Staring intently at Todd, he says, \"Heterosexuality has to be banned.\" Frank is almost a little too emphatic. \"Or we will end up just like them.\" Although he is merely pointing to his wall, Crystal and Todd understand what he really means: \"just like those heterosexual bastards outside Hadrian's Wall.\"\n\nSensing Todd's discomfort, Crystal leans forward and pats his knee. Todd reacts as if Crystal were a wasp that just stung him. \"Don't fucking pity me!\" He spits out vehemently, \"I'm not one of them!\" Leaping to his feet, he points to Frank. \"You saw me and Frank kiss.\"\n\nCrystal is stunned by the violence of his reaction. Frank smiles, slides his personal slate back onto his desk, opens his lap, and pats his right leg. Todd takes the bait and sits down on Frank's lap. Frank immediately closes his legs, hemming Todd in. Desperately needing to prove himself gay, Todd leans in and kisses Frank with fervor. Devouring Todd's lips, Frank uses enough suction to tease Todd's tongue into his mouth. Todd gives in willingly. Frank's heart beats rapidly and his hands begin to roam.\n\nCrystal stands. \"All right, enough!\" The boys, not listening, continue to kiss feverishly. Frank starts sucking on Todd's neck, hoping to leave a dark red welt as evidence. Feeling out of place, Crystal remarks, \"Study session is clearly over; you two will just have to remember the rest of the stuff on your own.\" Even though she knows she is rambling, Crystal can't seem to stop talking. \"I may like watching boys kiss, but this is ridiculous.\" Turning to face the door, then spinning back again to look at Todd, she says, \"I'm not going to stick around to watch this.\" She crosses quickly to the door. \"I'm out of here, guys.\" Before opening the door, she takes a quick glance back. Neither boy is paying her any mind. \"I'm leaving,\" but still no reply. \"Fine.\" She opens the door. \"Bye.\" It slams shut behind her.\n\nThe sound of the door slamming registers in Frank's mind. As much as he is enjoying Todd's fervent attack, a part of him is worried. The way Todd reacted earlier\u2014his forceful claim, \"I'm not ready,\" coupled with the memory of how horribly their last attempt at sex ended, causes Frank to exhibit some restraint. \"Todd,\" he pants out heavily. \"You keep this up\u2014\" He stops to draw in more of Todd's mouth and flavor. \"I won't be able to stop\u2014so.\" Already Frank is regretting what he is about to say, but his friendship for Todd is stronger even than his sexual urge. \"Unless you're really ready,\" he warns, \"you better go.\" To Frank's infinite displeasure, Todd stands and leaves the room.\n\n* * * * *\n\n#  _Salve!_\n\n**Fashion Sense  \nHNN\u2014Melissa Eagleton Reporting**\n\nDressing drag has always been the domain of our men. And why not? Before the days of Hadrian, it was one way to fight back against the mundane fashion men were confined to. This tradition grew out of each man's need to express his inner desires freely in a world determined to clamp down on male expression. Although men are no longer limited in their apparel, the tradition of the drag queen lives on. And why not? Everyone loves a good drag show! Of course, the drag queen still dons the outfit once viewed as the heterosexual female look designed specifically to attract the male sex. No doubt, in a world where a man seducing another man was deemed horrific, this brash and bold expression was both ironic and rebellious! And, of course, Hadrian's favorite, most brash and bold, drag queen is Pepper Tibbits. Pepper, famous for her\u2014sorry, his, feminine beauty and soprano voice, will be performing tonight at Antinous' Bedroom. Pepper's ability to transform from man to woman is truly awe-inspiring and her voice\u2014sorry, he looks so much like a woman\u2014 _his_ voice is indeed the finest in all of Hadrian.\n\nMy little pronoun blip presents a problem faced by some women in Hadrian. Though no one disputes men their right to dress drag, when the transformation is as complete as, say Pepper Tibbits, well then, if such a man is not on stage, how is the average woman to know she is really a he? Honestly, boys, we must be able to differentiate; otherwise, your little game of subterfuge is unfair. There have been complaints about this issue made to Hadrian's Sexual Preference Agency because more than one young woman has been accused of being heterosexual by attempting to seduce a young man dressed in drag. Andrea Hodgson is but one such victim. In her report, she had fallen in love with a woman named Darya, whom she claimed looked 100 percent female. There was absolutely no way you could tell Darya was male. Darya often flirted with Andrea, so the poor woman thought she stood a chance with Darya. When Andrea made her move, though, Darya's delicate soprano voice suddenly dropped down to a baritone, and Darren accused Andrea of being straight. He immediately exposed her to authorities, and had it not been for the excellent skills of defense Lawyer Faial Raboud, Andrea Hodgson, being over twenty-one years of age, would have been exiled. As we all know, the reeducation camps are designed to aid our wayward youth. Once conscious assent has been given to Hadrian's lifestyle, anyone discovered acting upon heterosexual tendencies is forced to leave our country and never return! Such was to have been the fate of Andrea Hodgson. Seriously, boys! How can this be fair? Dress in drag and enjoy your tradition, voice your rebellion against the heinous heterosexuals, but please, do not use drag as a weapon against your fellow citizens. Be sure to let the women around you know you are men, and if some poor soul mistakenly assumes you to be female, due to your exceptional talent at presenting your feminine mystique, please kindly inform her of your real sex, and only expose her if she persists in her seductions after your disclosure!\n\nOn a brighter note, now that the leaves are starting to change color, it is time to consider new fall fashions. Burnt orange, yellows, and greens are all the rage right now. But not everyone can wear these colors with ease. Fear not; bright colors stay in style year round in Hadrian. Pink and purple are always a hit no matter what time of year. Just remember: never wear white after the long weekend of September. While white works in the summertime, it fails to impress in the fall. That doesn't mean you have to go brown, but it certainly means white should go back inside the closet. Hadrian, let's dress for success, not like a mess.\n\n_Vale!_\n\n# T'Neal\n\nThe next morning, Todd walks into his first class to witness Crystal and T'Neal arguing. Aghast he watches them from the other side of the room as if across a vast expanse dividing him from all of humanity. Their voices are loud enough to be heard by everyone, even the kids in the hall. T'Neal looks so short standing next to Crystal\u2014barely 5' 4\". Towering over him like she is, Crystal looks like an Amazon ready to impale him. T'Neal is not daunted. Lithe and beautiful, T'Neal has never been insecure. His dark black hair is long and shiny. Although he wears more makeup than the average teenager, his use of cosmetics is tasteful, accenting the slightly oriental lift to his eyes, his sharp cheekbones, and full lush lips. When made up like this, T'Neal goes from being merely good looking to being absolutely gorgeous. Pointing to his neck, he screams, \"Are you calling me a liar and a thief?\" T'Neal is proud of his new piece of jewelry. Donning the bright purple leather dog collar lets everyone knows he is Frank's boyfriend. A boy has reached the highest pinnacle of teenage hierarchy if he's dating Frank Hunter. T'Neal is now on top, and no one is going to push him down!\n\n\"It certainly looks like it to me.\" Crystal slams each word into T'Neal's chest with her finger.\n\n\"And I'm telling you,\" he spits up, \"that Frank gave his collar to me! That I'm his boyfriend! And not some fucking cunt hammer!\"\n\n\"Todd is not a _strai_!\" Crystal yells back. \"I saw him and Frank kissing yesterday.\"\n\n\"I was with Frank yesterday!\"\n\n\"Frank was with me and Todd yesterday afternoon, you lying little piece of shit!\"\n\n\"I don't care who Frank was with yesterday afternoon. Or what he was doing with the _two of you!_ He was _with me_ last night!\"\n\nMillicent, one of the girls standing behind Crystal, sees Todd leaning against the doorjamb and gasps, \"Oh, Crystal, there he is.\"\n\nTodd shakes his head questioningly. T'Neal spins on his heels to declare, \"You are not Frank's boyfriend, _strai_. I am!\"\n\nTodd turns white at the accusation. Nobody has ever called him a _strai_ to his face before. It doesn't seem to matter that the insult comes coupled with the suggestion that Frank and he are dating.\n\nT'Neal is pleased by the reaction he has coerced out of Todd. Pointing proudly to his neck as evidence, he continues to harangue him. \"He gave me his collar last night.\" Running his middle finger through the silver ring (ostensibly designed for a dog leash), T'Neal wiggles his shiny purple nails (freshly polished that morning to match his new neckwear) Todd's way. \"Did he ever give you his collar? NO!\" Todd needs all of his will power not to throw up. Whirling back to face Crystal, T'Neal unhooks his finger and begins stabbing his own chest. \"Me! Frank gave it to me.\" Then pointing toward Todd without even bothering to look his way, he exclaims, \"Not him!\"\n\nTodd is stunned. \"What the fuck?\"\n\nAs Frank enters the room, he ruffles Todd's hair. He takes no notice of Todd's expression; all his attention is on Todd's turtleneck. Taking a quick peek underneath, he spies the red splotch he left on Todd's neck yesterday afternoon. He chuckles before saying, \"Hi.\" Todd pulls back from him. Frank is again oblivious to Todd's mood as his eyes are now firmly locked on T'Neal. \"Hey, baby,\" he says gaily in greeting, \"who's your daddy?\"\n\nT'Neal takes a moment to rejoice in front of Crystal before twirling Frank's way, \"You are, Frank.\"\n\nFrank opens his arms. \"Then come to Daddy.\" They kiss and embrace.\n\nCrystal blurts out, \"Frank, you pig!\"\n\nFrank looks up over T'Neal's shoulder. \"What the fuck's your problem?\"\n\nT'Neal, still snuggled up against Frank's chest, answers, \"The stupid cunt says you're dating Todd. She called me a liar and thief for wearing your collar.\" Frank releases T'Neal, looks slowly and steadily at Crystal, then turns to look at Todd, who is pale and leaning against the doorjamb. Turning back to Crystal, Frank's anger flares. \"What the fuck did you say?\"\n\n\"Don't put this on me, you bastard. Yesterday, you and Todd were making out hot and heavy, and suddenly, T'Neal has your collar. I had to leave your room because the two of you were going at it so hard. It was embarrassing, and then this morning, I see this little prick in that.\" She points dramatically at the collar around T'Neal's neck. Millicent and the other girls mutter their disapproval of Frank's conduct.\n\nT'Neal chooses to defend Frank. \"Well, Todd couldn't have been very good at making out 'hot and heavy' if Frank dumped him so quickly for me. I can't imagine a cunt hammer being able to perform in the way a real man like Frank is used to.\" Shooting a smug glance Todd's way, he rubs salt in the wound. \"That's right, _het'ro_. He picked me, not you.\"\n\nAll Todd can say is \"Fuck me\" as he stumbles out of the room.\n\nAnnoyed, Frank yells at his new boyfriend. \"Shut up, T'Neal.\"\n\n\"But Frank\u2014\" Poor T'Neal is bewildered. \"What did I do wrong?\"\n\n\"You gloat,\" Frank says furiously.\n\nT'Neal is offended. \"Well, you never told me you and Todd were dating.\"\n\n\"We aren't.\" Frank pushes T'Neal aside and begins to head toward the door.\n\nTerrified, T'Neal grabs his arm. \"Frank, are we still a unit?\"\n\nFrank turns around; his anger dissipates at the sight of T'Neal's horrified countenance. Apologetically, he says, \"Look, T'Neal; none of this is your fault.\" He leans in and gives T'Neal a quick kiss. \"We're still a unit.\" Looking past his new boyfriend to Crystal, he adds, \"That bitch is spreading rumors is all.\"\n\n\"Rumors are fake,\" Crystal shoots back. \"What I saw yesterday is true!\"\n\nFrank glares and points at the woman. \"You have no fucking idea what you saw, or what you just did to Todd.\" He is so angry his face is purple and his fists clench.\n\nAt this moment, Ms. Sterne enters the classroom. Ms. Sterne is a no nonsense teacher. She wears a suit jacket, tie, and skirt every day. Her silver hair is cropped short, military style, reminding her students she once made the army her career. She wears no makeup and constantly suffers from dark rings under her eyes. \"What is all the swearing and screaming about?\" She takes a moment to study Frank's posture. \"Frank, you better calm down.\" When Frank shows signs of releasing tension and his face fades from purple to splotched pink, Ms. Sterne begins to lecture him. \"Frank, you know I don't approve of swearing in this classroom.\"\n\nThe presence of his math teacher lets some of the wind out of Frank's sail. \"Sorry, Ms. Sterne,\" Frank mumbles. \"It's just that Crystal\u2014\" He chooses not to continue.\n\n\"Crystal what?\" Ms. Sterne asks coldly. Crystal is the woman's niece and clearly Ms. Sterne's favorite. Frank also knows how much the woman detests Todd.\n\n\"Nothing,\" he mutters as he storms out of the room.\n\n\"Frank,\" Ms. Sterne calls out after him, \"get back in here! Class is about to start.\"\n\n\"Would you like me to go fetch him?\" T'Neal asks half-expectantly.\n\n\"No, T'Neal, you sit down. One student missing from class is enough for today. Everyone, sit down.\" Blinking up her attendance sheet, Ms. Sterne notices one other student missing. \"Where is Todd Middleton?\" Looking over her glasses and down her nose, she stares at Todd's empty desk. Shaking her head, she mutters loud enough for the class to hear, \"I just saw him downstairs. He's not fooling anybody.\" She blinks Todd Middleton and Frank Hunter absent.\n\n* * * * *\n\n#  _Salve!_\n\n**Spotlight: The High School Counselor  \nHNN\u2014Melissa Eagleton Reporting**\n\nToday, I wish to offer our wayward youth some advice. Teens, is your biggest fear that you might be heterosexual? Well, don't worry. At the very most, you are a two on the Kinsey scale. That means you just need a little nurturing and more time to find the latent homosexual inside of you. The right man or woman is out there waiting for you. Do not despair. Do not give up hope. Remember, you are not alone. Inside every high school is a loving counselor who is willing to listen to your fears, concerns, and sexual confusion. These men and women have been trained by the finest in education counseling and psychiatry. They know what you are going through, and they can help guide you toward finding your inner homosexual. I know your instincts are to turn to your friends for advice, but they are not always the best ones to guide you. When struggling with sexual confusion, it is essential you turn to trained professionals. Hundreds of teens just like you are questioning who they are, and like you, they too need professional advice. Do not be afraid of your high school counselor; this person is there to help you.\n\nTo encourage further both teens and parents to avail themselves of their useful high school counselors, let me tell you about an interview I had with Pan Zhang, Pride High's counselor. According to Pan, he meets with five or six students each year to discuss their heterosexual concerns. Many of these students, he says, come to him in tears and begging for help. The cry he says he most often hears is, \"Help me not be straight. I don't want to be a _strai_.\" It is a heartbreaking plea for aid and one Pan feels requires immediate attention. These youth become Pan's top priority as he takes them under his wing and offers guidance. Parents and a professional psychologist are brought in immediately to help the youth conquer these abhorrent desires. Since these are youth who have never acted on their heterosexual desires, none are sent to reeducation camps except as a last resort. As we all know, the re-education camp is designed to help our active heterosexual youth find his or her latent inner homosexual. Youth who have heterosexual leanings but do not take action are much easier to reclaim through private in home and school measures. Nine times out of ten, Pan says, wayward youth who come forward before acting on their heterosexual impulses can be reclaimed before reeducation camp becomes essential.\n\nPlease remember, no high school counselor has ever exposed a youth based solely on a confession of fear. Counselors are there to talk and listen and understand. Rest assured, your high school counselor will guide you in the right direction; help you discover your latent homosexual tendencies. Don't delay. Visit your high school counselor today!\n\n_Vale!_\n\n# Hiding\n\nFrank finds Todd exactly where he expected\u2014behind the back stairwell leading up to the girl's locker room. Ever since Crystal discovered the security camera in this hallway to be broken, it has become Todd's safe place. As he is hidden under the stairs at the very far end, the first indication Frank gets of Todd's presence is a continual banging of bone against brick, punctuated with the repetitive phrasing, \"Fuck me.\"\n\n\"Hey, buddy,\" Frank says as he turns the corner of the stairwell. Bending to fit underneath, he sits down next to Todd.\n\nTodd is no longer hitting his head against the wall. It is now lodged deep between his knees.\n\n\"Leave me alone, Frank,\" Todd mutters.\n\n\"Can't do that, buddy,\" Franks says softly. \"Not while you're in this state.\"\n\nTodd looks up, glaring hate shooting out of red puffy eyes. \"You put me here!\"\n\nFrank sighs, closes his eyes, and rests his head against the cold brick. \"That's not fair, Todd.\"\n\n\"Isn't it?\" Todd lashes out all his anger and embarrassment onto Frank.\n\n\"I know you don't mean that.\" Frank hasn't moved. His eyes remain shut, trying to hide the bleeding inside.\n\n\"Well, I do!\" Todd shouts. Standing now, he walks to the front of the stairwell and sits on the lower steps, using both his back and the stairway to protect him from Frank. He shouts out to the wall in front of him, \"Everything, Every\u2014Fucking\u2014Thing _is_ Your Fault.\"\n\nDesperately fighting back tears, Frank pleads, \"Todd, I came here to help you.\"\n\n\"Help me?\" Todd looks up incredulously. Standing, he crosses back behind the stairwell, staring at Frank. \"You can't help me! Everything I said would happen is happening. They all think I'm a fucking idiot. That you dumped me\u2014\"\n\nFrank shakes his head. \"I'm sorry Crystal said that.\"\n\n\"She didn't!\" Todd yells in her defense. \"Your little boyfriend did!\" Ripping at the hair on his head, Todd begins to wail, \"How long, Frank? How long after I left you?\"\n\n\"Todd.\" Frank finally opens his eyes. They are red and swollen, caused by a combination of anger and hurt. \"We're _NOT_ dating!\"\n\n\"I know that!\" Todd yells. \"But Crystal thought we were, and then you go and hook up with\u2014with\u2014that fucking little tramp! That fucking little tinsel tart.\"\n\n\"Are you jealous?\" Frank blurts without thinking.\n\n\"I'm not jealous. It's your timing. It's your fucking timing that hurts.\" Todd slams his back against the sidewall and drops to the floor. His knees knock against Frank's since the two boys are now sitting at right angles. Todd quickly pulls his legs away.\n\nFrank grits his teeth at the gesture. \"Todd, I had no idea Crystal was going to say anything. Besides,\" he adds, almost a little too cruelly, \"I wasn't the one who kissed you.\" Leaning his head back against the wall, he says, \"After you left me\u2014what the fuck did you expect?\" He looks up at Todd, allowing his tears to flow freely now. \"Seriously, Todd, what did you think I was going to do? I really thought we were going somewhere.\" The pounding of bone against brick recommences, this time much louder, much harder. \"Todd, stop that!\" When Todd continues his act of self-abuse, Frank grabs him by the head and orders, \"Stop that!\"\n\nTodd is now mumbling through tears, \"He called me a _strai_ , Frank. A het'ro! A fucking cunt hammer! To my face!\" His hand caresses his forehead. \"Oh, Hadrian,\" he weeps into the palms of his hands. \"Help me.\"\n\n\"Todd. Todd,\" Frank says lovingly. Wishing he could hold Todd, Frank settles for rubbing his hand on his friend's shoulder. \"T'Neal was jealous. Nobody believes a jealous man's accusations. His calling you a _strai_ is stupid,\" Frank reasons. \"Think about it\u2014he thought you and me were a unit. That's not straight.\"\n\nTodd's weeping is relentless. \"Oh, Hadrian, I'm so sorry, Frank.\" Todd's voice is barely audible. \"It's my fault. It's all my fault.\" Closing his eyes, desperately needing to confess. \"Frank,\u2014\" his voice catches. \"I think I'm\u2014\"\n\n\"NO, YOU'RE NOT!\" Frank is standing now, towering over Todd. \"Don't ever fucking say that!\" He pauses to calm himself down. He looks away. For some reason, he can't seem to look at Todd. \"I know you're not. Not after yesterday. I felt you. You were hard, man. You were hard when we were going at it.\"\n\nTodd wants to tell him it was Crystal who did that to him, but he is too afraid. All he can muster is, \"Thanks, Frank.\"\n\nFrank, now squatting down in front of Todd, takes his hands in his. \"You're just not ready.\" He shrugs his shoulders, \"So what? Who cares? Just because you don't want sex right now doesn't mean\u2014doesn't mean you're straight\u2014it just means\u2014you're not ready.\"\n\n_But I do want to_ , Todd laments internally. _Just not with you._\n\n\"What we did yesterday,\" Frank smiles feebly, \"we just did to shut Crystal up.\" His anger for the girl resurfaces. \"She had no right to say those things about you.\" He spits out his vehemence, \"Fucking bitch!\"\n\n\"She's not a bitch,\" says Todd, a little too defensively. \"I mean, it's not her fault either. It's me.\" Starting to cry again, he asks, \"Oh, Hadrian, Frank. What's going to happen to me?\"\n\n\"Nothing, Todd. Nothing. I'm here for you. I'm going to help you. I'll do anything. Anything to help you through this.\"\n\n\"Anything?\" Todd asks, wiping his nose against his sleeve. \"Do you swear\u2014\"\n\nFrank answers before Todd can even finish, \"I swear.\"\n\n\"Swear on the soul of Hadrian's lover?\"\n\n\"Yeah,\" Frank laughs, placing a hand over his heart, \"I swear on the soul of Hadrian's lover!\"\n\nSuddenly, without thought, Todd blurts out, \"I love you, Frank.\" Recognizing what he just said causes Todd to start crying again. Frank pulls him in for a full embrace, holding Todd between his arms and legs: both boys end up in the fetal position.\n\n\"I love you, too, Todd,\" Frank says as he cradles Todd.\n\n\"If it's ever going to be anyone, Frank,\" Todd blubbers, \"it will be you.\"\n\n\"I know. I know. And I'm ready for you. I'm waiting. You and me, Todd,\" Frank mutters softly. \"You and me. But only when you're ready.\"\n\n\"You promise?\" Todd is shaking.\n\n\"I give you my word.\" Smiling, envisioning their future, Frank says, \"After we graduate. After we register if that's what you'd prefer.\"\n\nTodd nods. Still shaking, he quivers, \"Yeah, after we register.\" He gives Frank a slight nudge to release him. \"I'm okay now. Really.\" He forces a feeble smile as he wipes his eyes and nose on his sleeve. The two boys separate and once again lean their backs against the cold brick walls.\n\nEven though Todd claims everything is copacetic, Frank still feels the need to lighten the mood. He laughs, reminiscing, \"Hey, you remember when you puked down Ms. Singer's shirt?\" The mood becomes jocular as both boys collapse into each other in raucous laughter.\n\n\"Oh man, she was pissed,\" Todd sputters out between guffaws. He is forcing out the jocularity, though, since the Ms. Singer experience was the first stirring of Todd's confusion.\n\n\"Actually,\" Frank counters, \"she took it amazingly well. You only heard her initial reaction. She saw how sick you were.\"\n\nThis event happened in their grade eight class. Todd hadn't been ill. He had simply awoken that day with a headache caused by too much crying through the night. It always happened near the anniversary of 6-13\u2014the apocalyptic nightmares in which Todd watches his father explode in front of him. After rising, he had asked his Papa Mike for an aspirin and Mike, being hung over, grabbed the wrong bottle. Partying excessively of late (also brought on by the anniversary of his lover's death), Mike had accidently given Todd one of his hallucinogens. By the time Todd got to school, the effects of the drug had started to kick in. Todd was sick and nauseous. His head was swimming; he was even drooling. During math class, Ms. Singer had stood by Todd's desk, helping the student sitting across from Todd. She was bent over, showing the girl how to figure out that day's formula. Ms. Singer was a first year teacher, young, slim, busty, and beautiful. Always delicately dressed in various shades of pink or purple. All the girls were a'titter whenever she was in the room. Her body curves were accented at their best in this position, and her behind was very close to Todd's face. He became transfixed, following her spine from the small of her back down to her buttocks. Reaching up with his hand, he clutched her right buttock. Ms. Singer turned in horror, to stare down at Todd. \"Mmm,\" he whispered, \"nice bum.\" Fortunately for Todd, only Ms. Singer had heard him. More fortunately, she is a very empathetic young woman, seeing before her a very sick young man.\n\n\"Oh, you poor dear,\" ignoring the raucous laughter of the room, Ms. Singer bent down to feel Todd's forehead. Her breasts were now directly in front of his face.\n\n\"Boobs,\" he murmured. Lifting his head to get a better look had caused Todd's stomach to swirl. This, combined with the scent of Ms. Singer's perfume, begot puke. His vomit shot up against the woman's chest. After her initial scream of horror, she placed her hand to her mouth and uttered a soft cry. She looked over to Frank, who was sitting in the desk on the other side of the aisle and asked him if he could carry Todd down to the nurse's office.\n\nTodd had very little memory of those events, but two images remain seared in his brain: Ms. Singer's bum and the beautiful valley between her breasts. He relives, embellishes, upon those images in his fantasies every time he masturbates. He usually emerges from these moments feeling soiled and dirty, as if his mind and body have betrayed him, but he can never seem to stop himself from thinking about her\u2014until lately. For the past year, it has been Crystal, with her burgeoning body, who has replaced Ms. Singer in Todd's imaginings.\n\n\"Crystal,\" Todd mutters.\n\n\"What about her?\" Frank asks, trying to cover his annoyance. Things are finally feeling good between Todd and him again, so mention of Crystal is likely to resurface the strain.\n\n\"You've got to make things right between you,\" Todd reasons.\n\n\"Why?\" Frank wants absolutely nothing to do with that girl right now.\n\n\"Tryouts for b-ball are in two weeks,\" Todd reminds him. \"If you and Crystal are going to be co-captains, you guys have to work things out.\"\n\n\"They only give co-caps to seniors,\" Frank replies.\n\n\"Not this year,\" Todd responds. \"Coach and I agree you and Crystal are our best.\"\n\n\"You're our best,\" Frank sputters. It has never dawned on him that he would ever be chosen as a co-cap over Todd.\n\nTodd silences Frank before he can protest further. \"Uh, uh, I'm not co-cap material. The coach and I talked about it, and she's right. You garner a lot more respect, so the job lands on your shoulders.\"\n\nFrank smiles, pleased to have both Todd and the coach's confidence.\n\nSensing his co-cap role has yet to dawn fully on Frank, Todd warns him, \"That means you need to work with Crystal, not against her.\"\n\n\"I can keep the sport and my personal feelings separate,\" Frank insists.\n\nTodd isn't so sure. \"Maybe Crystal can't.\" Attempting diplomacy, he adds, \"Couldn't you put your pride aside and talk things out with her?\"\n\n\"Only if she's willing to apologize!\" Frank isn't feeling very conciliatory. Crystal nearly destroyed his relationship with Todd today.\n\n\"How about I talk to her? We're planning on doing some initial training before tryouts.\" Thinking of ways to make this work, he adds, \"Umm, how about you join us after school? We're meeting at three in the small gym. Coach is letting us use it. You could come ten minutes late\u2014give me time to set things right and get her to apologize.\" Sensing Frank's mistrust, Todd reassures him, \"She will; I know she will!\"\n\nFrank sighs, pinching the crown of his nose with his fingers. \"All right, but only if she apologizes.\"\n\n\"For the team, Frank,\" Todd reminds him.\n\n\"Yes,\" he replies, \"for the team.\" They slap hands and grip fingers. Before they let go, the bell rings. Frank maintains his grip on Todd's hand and the two boys stand. \"We better not miss Mr. Reiner's class. We've got that test to write, remember?\"\n\n\"Yeah,\" Todd sighs. He never did any more studying after he left Frank's yesterday. \"Let's go.\"\n\nTogether, holding hands, the two boys make their way down the hall toward Mr. Reiner's class. This gentle act does not give Todd any sense of discomfort. It is perfectly natural for two friends, whether of the same or opposite sex, to walk hand-in-hand in Hadrian. If two individuals wish to denote an intimate connection, they will choose to walk with their arms wrapped around each other. Had Frank tried to put his arm around his friend's shoulder or waist, Todd would have balked and walked away from him.\n\n* * * * *\n\n#  _Salve!_\n\n**Remembering Will Middleton  \nHNN\u2014Melissa Eagleton Reporting**\n\nYes, folks, it's that time of year again! Hadrian's high schools and colleges are gearing up for b-ball season! Tryouts begin next week when high school and college coaches will be on the lookout for the best possible players for their teams. And speaking of the best of all possible players, who over the age of thirty doesn't remember Will Middleton? Who could forget his graceful presence on the b-ball court during both his high school and college years? While attending Antinous' Pride High, Will Middleton led the Panthers to win at the Quadrants three years in a row and the Nationals two years in a row! Until this past year, when his son joined the team, the Panthers had not been back to the Nationals since Will Middleton graduated. Tall, lithe, muscular, and highly skilled, Will Middleton could dribble, pass, shoot, and always score! High school b-ball was only the beginning for this man. His seven years on Antinous' Centurions were award years, too, leading that team to its only ever seven-year winning streak! I remember his coaches thanking Hadrian profusely for his staying on for both a Master's and Ph.D. in agricultural engineering. Having won both a sports and academic scholarship Will Middleton was eligible to play b-ball his entire uni career. To keep the game fair, no one over twenty-five is allowed on the b-ball court.\n\nUnfortunately, we lost Will Middleton as a result of 6-13. That was a sad day for Hadrian with its poisonous claws digging deep into the hearts of all, killing hundreds of thousands on the day of impact; Will Middleton was one such victim. Although we will never see the likes of Will Middleton again, he did leave Hadrian his legacy, Todd Middleton. Sixteen-year-old Todd Middleton is currently attending Pride High and is, according to Coach Miller, the main reason the Panthers brought home their first Quadrant victory in over ten years. Coach Miller openly admits that Todd Middleton is the unifying force behind her team. Although Todd is a lot shorter than his father, only 5' 5\", whereas Will was 6' 2\", Coach Miller says what Todd lacks in height, he makes up for in skill, speed, and spirit. Like his father, Todd Middleton lives and breathes the game.\n\nTodd Middleton is like his father in other respects, too. As was Will Middleton, this boy is humble to the core. When asked whether he plans to win Quadrants for his team again this year, his answer was simple, \"No one player wins the game. The Panthers won last year and the Panthers will this year too because our team works together like the gears of a finely tuned machine. I'm only one player\u2014look at Frank Hunter. He's a monster on the court. And I defy anyone to try to get the ball from Crystal Albright once she's got control of it.\" Well, there you have it, folks. Todd Middleton, like his father before him, is a true team player!\n\n_Vale!_\n\n# B-ball\n\nCrystal is already dressed and ready for pre-practice. She had worn her T-shirt and shorts under her school clothes, so as soon as the last bell rang, all she had to do was run to the small gym and get started. When Todd arrives, she is doing push-ups: flat palm up into a clap. As Todd watches her, he admires her lean muscular physique. Her shorts are snug fitting and her buttocks, clenched tight, show the side indent. Following the line of her leg down to her ankle, he can make out thigh and calf muscle definition. Like her forearms, Crystal's monotone voice, as she drones out the numbers, begins to quiver the closer she gets to her optimum number: fifty. \"Wahoo!\" Todd cheers as her nose touches the hardwood for that last number. \"Way to go, girl. You did it!\"\n\nCrystal lies down on her side, smiling triumphantly up at him. _Hadrian's lover, she is beautiful,_ Todd's thought flashes impulsively before he has a chance to contain it. He blushes. \"Hey, you,\" Crystal chimes. \"Told you I could do it!\" She pats the floor in front of her belly.\n\nTodd accepts the invitation and sits cross-legged in front of her. \"I told you, you could.\"\n\n\"Frank never believed I could, but you...\" She pinches the front of his turtleneck slightly, pulling it forward. Todd suspects she's trying to see whether Frank left his mark, so he tucks his chin to his chest. \"But you,\" she repeats as she lets go of the soft cloth, \"you've always believed in me.\"\n\n\"Yeah, Frank,\" he mutters under his breath.\n\n\"Huh?\" Crystal leans her head under Todd's and looks up into his eyes. \"You want to talk?\"\n\n\"Yeah.\" Todd leans back, getting her out of his personal space, and draws his knees up to his chest (feeble protection!). \"Yeah,\" he repeats, his eyes shifting right and left, \"we need to talk.\"\n\n\"About Frank?\"\n\n\"Yeah, Crystal, about Frank.\"\n\n\"He's a dick,\" Crystal spits out vehemently.\n\n\"That's not fair, Crystal. You don't\u2014\"\n\nCrystal leans back, unbelieving. \"Are you actually defending him? For the love of Hadrian, Todd, he cheated on you!\"\n\nTodd looks down to his feet. \"He didn't cheat on me.\"\n\nCrystal sits up and slaps Todd hard on the head, causing him to cry out. \"Ow! Damn you, Crystal; that hurt.\"\n\n\"Don't be a driveling fool!\" she says, leaning back now. \"For Hadrian's sake, Todd, he used you, dumped you, and moved on! How can you possibly not see that?\"\n\nStanding just outside the door, Frank sneers. He knew Todd could never convince that girl to apologize. _She is too set, too determined_ , he reasons, _to believe I'm wrong!_ Leaning back against the old brick wall, Frank closes his eyes, and clenching his fists, fights back the urge to charge into the gym and have it out with her.\n\nTodd looks up, trying to smile, trying not to cry. Crystal feels so much sympathy that her hand goes instinctively to his cheek. He closes his eyes and rests his face against her palm. \"It's all right, Todd. I know how you feel. When Millicent dumped me last year, I thought my heart had shattered into a million pieces.\"\n\nThis little piece of intelligence does not help to ease Todd. It angers him enough to embolden him. Looking up, he declares, \"Frank did not cheat on me because we are not dating! What we did yesterday\u2014\" Todd chokes on his words, giving Crystal the wrong impression. He stops himself because he really doesn't want to talk about what he and Frank did. Sighing, he knows he has to talk about it, though\u2014especially now. The look in Crystal's eyes reflects her realization that he and Frank are more than just friends. \"Look,\" he says sternly. \"Frank and I are not dating! We never were dating! When I left him yesterday, I told him it didn't mean anything! In fact,\" he adds, \"we talked about other guys and he confessed to me that he wanted to ask out T'Neal. Said he wasn't sure if T'Neal liked him or not. So,\" Todd improvises frantically, \"I told him not to be crazy. T'Neal was obviously madly in love with him, and all Frank had to do was voc him and they'd be dating like they'd been that way forever.\" Finally, Todd stops to take a breath.\n\nFrank opens his eyes and smiles. _Nice lie, Todd!_ Frank is impressed. He wonders whether he should enter the gym now, but he stops when he hears Crystal react.\n\n\"Really, Todd? Really? I don't believe what you and Frank were doing meant absolutely nothing to you.\"\n\n\"Well, it didn't,\" he insists.\n\nTodd's words sting Frank. He takes a deep breath and readies himself to enter.\n\n\"I don't understand guys.\" Crystal is half-irate, half-befuddled. \"That you can actually do stuff like that and then act like you just don't care.\"\n\n\"It's not that big a deal, actually,\" Todd tries feebly to impress.\n\nFrank snorts, lightly enough not to be heard inside the gym.\n\n\"Guys are weird,\" Crystal says.\n\n\"Maybe girls are just too sensitive?\"\n\n\"Well, I'd rather be too sensitive than kiss and forget the way you two do!\"\n\nFeeling some success, Todd asks tentatively, \"Would you apologize to Frank, please? I mean, none of this was his fault after all.\"\n\nCrystal leans back, discontent, battling with indecision in her eyes. \"Is that what you want?\" Unbelieving, she reiterates, \"Really want?\"\n\nTodd's sigh of relief is audible even for Frank. \"Yes, that's what I want.\"\n\n\"Fine!\" Crystal agrees, though somewhat begrudgingly. \"I'll apologize.\"\n\nFrank is impressed. Todd actually managed to bring the girl around.\n\n\"Thank you so much, Crystal.\" In his exuberance, Todd leans forward and kisses her. Quickly sitting back, he apologizes.\n\n\"Don't be silly; we're friends.\" Clearly pleased, Crystal leans forward and returns Todd's kiss. It is soft and sweet and stirring. When she releases him, she leans back. \"We're best friends.\" The rest she whispers. \"I always kiss my best friends.\" A second, more lingering kiss ensues.\n\nWhen Crystal finally releases his lips, Todd blushes and then turns to look toward the door. Frank, he is certain, is out there. \"Okay, Frank. Come in.\"\n\nCrystal leaps up, stung. \"He's been standing out there the whole time?\"\n\nFrank walks into the gym. His smile is a little too smug. \"Do you have something to say to me, Crystal?\"\n\nScowling, Crystal stands; then she looks down on Todd. \"You tricked me!\"\n\nNervous now, Todd feverishly improvises. \"I didn't trick you, Crystal. Everything I said is true. I just...I just knew you wouldn't listen if I didn't talk to you first.\" When all Crystal does is harrumph, Todd presses, \"Please, Crystal. You two have to work together, be co-caps. And it wasn't Frank's fault. He didn't dump me like you thought. We were just being a couple of guys acting stupid\u2014I led him on, teased him up, and then left. He calls T'Neal\u2014so what.\" Talking now to his shoes, he adds, \"I don't give a shit; why should you?\"\n\nCrystal bends down, squatting in front of Todd. \"I saw how you looked this morning.\" Appraising him now with her eyes, she says, \"How you look now. You're upset. You can't deny that.\"\n\n\"Yes, I'm upset!\" Todd shouts. \"That little fucker called me a _strai_! And, you!\" He pounds a finger on her sternum. \"You fucking called me one Sunday!\"\n\nCrystal blanches. \"Todd, I didn't.\"\n\nFrank leaps in. \"Not directly, but you kept insinuating it. That's why he jumped on my lap and started to kiss me. Your suggestion was so fucking blatant even I could see through it. I would have told you to fuck off but for Todd jumping on me like he did.\" Even though he is still addressing Crystal, Frank smiles down at his friend, \"A lot more pleasant than yelling at you.\"\n\n\"Oh, Todd, is that what you thought?\" Crystal asks. \"That I thought _that_ about you?\" Todd nods, still staring at his shoes. \"I am so sorry.\" Bending down, she embraces Todd. Crying, she begs for forgiveness.\n\n\"Please apologize to Frank,\" is all Todd can muster.\n\nCrystal looks up. \"I'm sorry I misinterpreted you, Frank.\" After Frank's nod of approval, Crystal turns all of her attention back on Todd. \"I'm so sorry you guys thought I was saying that about you. I really am sorry, Todd. Can you ever forgive me?\"\n\n\"Yeah,\" he says, looking up finally, wiping his eyes. \"We're cool.\" He extends a hand to Frank, who yanks him up to standing. \"All right, you guys,\" Todd says, clapping his hands together. Trying to put everything behind him, he adds, \"Frank and I need to change.\"\n\nAs soon as the two boys return from the boy's locker room, Todd takes control. \"Let's get to work.\"\n\n\"What do we do first, Coach?\" Frank's attempt at jocularity is feeble, but Todd and Crystal laugh anyway.\n\n\"Weaves,\" Todd pronounces. \"Coach, put out a med-ball and a b-ball for us to use. But we should run stairs first.\"\n\nBoth Frank and Crystal groan. \"Come on, Todd!\" Frank protests.\n\n\"Seriously, Todd, stairs? You know Coach only gives us those when she's pissed off.\" Crystal and Frank form a quick alliance.\n\n\"I swear she only does it to make us puke,\" Frank adds.\n\n\"Quit acting like a couple of little babies.\" Todd is in control. \"It's conditioning. Strengthens the thighs on the way up and the calves on the way down. So, quit bitching, kiddies, and let's move.\" When it comes to b-ball, Todd is cocky and confident. His father taught him the game, and they used to spend hours at the local gym playing one-on-one with Papa Mike on the sidelines, taking turns cheering on one and then the other. Will Middleton played high school and college ball. If Hadrian had had a professional league, Will Middleton would have been an all star. He taught his son everything he knew about the game before his death.\n\nAfter running stairs, Todd has the three of them weave in and out, up and down the court with the med-ball, making the switch to tossing the b-ball a breeze. When he insists they run lines, Crystal and Frank protest again. \"For the love of Hadrian, Todd,\" Frank growls, \"that is so _strai_!\"\n\n\"Sexual preference has nothing to do with running lines, Frank.\"\n\nFrank shakes his head in both anger and dismay. \"You know I meant stupid.\"\n\n\"Let's go, kiddies,\" Todd remarks as he heads toward the far end of the gym.\n\n\"No Fucking Way!\" Crystal's objection is a little over the top, but she takes it even further. \"We did stairs\u2014why the hell should we do lines, too?\"\n\n\"What a couple of whiny little babies,\" Todd balks. \"Line training is different than stairs. With lines, you have to work up instant speed, stop on a dime, and switch directions. It's something you have to be ready to do at any second in the game. Line work is critical! Go sit on the sidelines like a couple of little toddlers, but I'm doing lines.\" There is no way Frank or Crystal have any intention of letting Todd show them up so they run through the line workout with him. Just to piss them off (a little sadistic punishment perhaps), Todd runs the gym lines three times non-stop. Running weaves is next. First, Todd has them toss the medicine ball back and forth before switching over to the much lighter b-ball. Finally, after thoroughly exhausting his teammates, Todd asks through heavy breaths, \"How about a little two-on-one?\"\n\nEven two-on-one, Crystal and Frank have to work hard to beat Todd Middleton. He takes after his genetic father in nearly every aspect but height. His genetic mother, whomever she might be, must have come from short stock. That he is only 5' 5\" is a hindrance on the b-ball court, but what Todd loses in height, he makes up for with heart and skill! His dribbling technique is flawless and lightning quick! Maneuvering speed is another one of his talents, slipping in between and around the other team like a jackrabbit running from wolves. And his throwing arm is by far the strongest Coach Miller has ever seen! He knows every play in the book by heart and has a keen sense of what other players are going to do by the way they move.\n\nAfter beating Todd 4-2, Crystal and Frank foolishly gloat. And although it means cleaning the gym before heading out, a condition made by Coach Miller if they are to use the gym on their own time, Todd makes sure to remind his two friends, \"Two of you!\" Then pointing to his chest with both fingers, \"One of me.\" Topping it off with a smug shrug, he continues, \"Just saying! And,\" he tosses in an extra little jab, \"you're both twice my height at that!\" They all know he exaggerates now, but the height comment is always enough for Frank and Crystal to look shamefaced and help him clean up the gym (something he never does for them when he wins).\n\nAs soon as Todd is out of earshot, Frank confesses his dismay to Crystal. \"Why Coach Miller doesn't make him co-cap is beyond me.\"\n\nCrystal shakes her head in wonder. \"Makes no sense to me.\"\n\nNormally, Frank is offended when someone suggests he is not as good as someone else, but there is no denying that Todd is the real b-ball player at their school. Their team is good; they even won Quadrants last year. But Todd isn't just good; Todd is a player, just like his dad. Coach Miller knows this too. She isn't stupid. She just has good instinct when it comes to teenage dynamics. Frank Hunter is top dog. The kids respond to his word like he was a Hadrian god\u2014almost as if he were Hadrian himself. Todd, though admired for his b-ball skills and always beloved during the season, would never be able to command the same level of respect from his teammates. Coach Miller compensates for this situation by making Todd her unofficial assistant coach. She always discusses tactics with him every game: which way the ball is going and what play would work best. Todd is also first string and on the floor every single game. But at times, the coach keeps him on the bench with her so they can strategize. Todd's father taught him well, and Coach Miller wants to squeeze every bit of information out of Todd while she can. More often than not, these bench times occur when the team is winning. If they ever start to fall behind, though, Todd is always quick to say, \"You better put me back in, Coach.\" Coach Miller never objects. With Todd Middleton on her team, she is confident they will not only win Quadrants again this year, but that they might even take Nationals.\n\n* * * * *\n\n#  _Salve!_\n\n**The Cattle Ranch Restaurant\u2014Controversial Catering HNN\u2014Melissa Eagleton Reporting**\n\nHow many of you, Antinous' citizens, have visited The Cattle Ranch Restaurant? Be honest, now! You may say otherwise, claiming to revile the owners for serving the dreaded red meat\u2014cattle having been a major contributor to global warming due to methane gases as well as the destruction of over two-thirds of the world's rain forests to provide grazing land for these creatures. And then there is all the alarm and apprehension regarding health concerns, red meat being seen as a silent killer. Because trans-fats are consumed when one eats red meat, the danger always exists of increasing bad cholesterol. Stop anyone on the street to ask whether he or she would eat cow meat and the rate of denial is extreme. Now, you may ask, why am I so cynical, suggesting the majority of you are liars when it comes to eating beef\u2014simply drive by The Cattle Ranch Restaurant and see for yourself. Don't be surprised to find a line-up that stretches down the block! Patrons are almost always guaranteed a minimum of an hour wait before getting a table. Although The Cattle Ranch Restaurant has only been open for six months, it has fast become the busiest, most popular eating establishment in all of Antinous. I find this fact amazing considering the extravagant cost. Beef is rare (pun intended), with only one cattle ranch in all of Hadrian. One can expect to spend a minimum of sixty credits for a hamburger deluxe. Salisbury steak with mashed potatoes and gravy (vegetables not included) sells for fifty-five credits, and Hadrian help you if you order veal cutlets, tenderloin, or steak! A fine dining experience for two can easily cost you two hundred credits before coffee and dessert! And yet, The Cattle Ranch Restaurant's popularity continues to grow stronger every day. Face it, Antinous; we like beef!\n\nThe d\u00e9cor is reminiscent of the Old West. The owners of both The Cattle Ranch and its namesake restaurant are eager to show how environmentally sound their approach to raising cattle is. In fact, for the sake of our environment, The Cattle Ranch's owners, Jake Matonabee and Jeremy Stoker, use very little modern technology on their ranch. They own no bubble or farm equipment requiring energy to operate. Everything accomplished on this ranch is achieved using manpower and horsepower. As well, recycling is their number one prerogative. Horse and cattle manure is used to keep the soil rich and fertile for growing their family gardens as well as the grass, hay, and oats consumed by their stock.\n\nOne cannot help but admire the extent to which these two men go to ensure beef is an environmentally sound part of Hadrian's diet. Clearly, red meat is not for everyday eating, but for that one special night. When you want to impress your date, The Cattle Ranch Restaurant is definitely the place. Go on; it'll help you both feel a little naughty!\n\n_Vale!_\n\n# Family Dinner\n\nGeoffrey and Frank are seated at the kitchen table while Dean organizes the last of their meal. He has prepared tofu chili, green salad, and garlic toast. He places the steaming pot of chili on the table for everyone to help himself. He has prepared each person's salad separately and now places the leafy greens with dried blueberries, almond slices (very hard to find these days being as most nuts are imported), and green onion drizzled over with a Saskatoon berry dressing in front of each man's place. One chair remains empty, though, since Roger is late. After retrieving the garlic toast from the oven and placing it inside a small basket, Dean takes his place beside Geoffrey. \"Shall we start without Roger?\" he asks Geoffrey.\n\n\"We might as well.\" Geoffrey's sigh is disgruntled. \"If we wait any longer, our meal will get cold.\"\n\nDean looks across the table to Frank. \"Do you know where Roger is?\"\n\nFrank merely shakes his head. He has already started digging into the chili, stuffing his mouth with it and some of the garlic toast. After he has swallowed down his first mouthful, he expresses his satisfaction with the meal. \"This is really good, Papa Dean.\"\n\nGeoffrey smiles. \"Nice and spicy,\" he says as he pats Dean's knee.\n\nDean is pleased. He had never intended to spend his life as a stay-at-home spouse. There was a time when he actually dreamed of becoming a paramedic, nurse, or even a doctor\u2014something in the medical profession. He had even considered becoming a chiropractor at one point, but those dreams ended when he was seventeen. Dean never went to uni, and he was married shortly after his eighteenth birthday. Once he came to terms with the way his life had turned out, though, he had chosen to throw himself into the role wholeheartedly. Teaching the boys hygiene, talking with them, listening, offering advice, helping them with their homework, making sure they ate well, all of these were an enjoyable part of his life. That and having them compliment him on the quality of his cooking gave Dean a sense of pride. With the glowing compliments being showered his way, Dean should have been at his ease, but Roger's empty place bothers him. Blinking as he whispers, \"Roger,\" he waits for his voc to connect to Roger's voc link over the wave. Connection is made. \"Roger\u2014Yes, it's me\u2014Where are you?\u2014Really?\u2014Turn your voc cam on\u2014Blink and turn on your camera!\u2014NOW!\u2014Thought so. Say hello to Devon for me. Look around.\u2014I want to see where you are; that's why!\u2014Look around!\" Glancing Geoffrey's way, Dean says, \"They're at The Cattle Ranch Restaurant.\" Geoffrey's sigh is almost a growl. \"Tell him to bring you home.\u2014Now.\u2014You should have asked us first.\u2014Now, Roger.\u2014I don't care if your food just arrived. You come home now!\u2014I don't care how much it costs. You come home now!\" Frank, laughing over his chili, gets a sharp slap from his father. Dean continues relentlessly, \"You tell Devon he is to bring you home now or you won't be dating him ever again!\" Dean grimaces. Although Roger's voice cannot be heard, what he is saying right now can easily be inferred as very hurtful. \"Say whatever you want, but if you are not home in the next ten minutes, I will drive there myself and drag you home.\u2014Good. See you soon.\" Blinking his eyes, Dean severs his connection with Roger. Turning to Geoffrey, he says, \"He'll be home soon.\"\n\n\"The Cattle Ranch Restaurant,\" Frank says approvingly. \"Devon really knows how to woo a guy!\"\n\nDean is not in the mood for any humor. \"That's enough out of you. Roger is too young to be dating. He's in way over his head with that boy, Geoffrey. I'm worried about Roger out with the likes of him.\"\n\n\"The likes of him?\" Frank is amused.\n\n\"Yes,\" Dean retorts, \"the likes of him.\" Angry he has to explain, \"A player,\" and shaking his head, worried, he clarifies, \"That boy is sexually active, and I don't want the likes of him...\" Exasperated, Dean closes his eyes and breathes heavily, before concluding, \"with our Roger.\"\n\n\"Well,\" Frank laughs, \"you managed to save him from the evil clutches of Devon Rankin for one more night.\"\n\nGeoffrey intercedes by holding Dean's hand while shushing his son. \"That's enough, Frank.\" In an attempt to soothe his lover, Geoffrey adds, \"We'll have a chat with Roger as soon as he gets home.\" Dean nods, though still disgruntled.\n\nFrank figures the timing is right to share his good news. Papa Dean will definitely revel in what he has to say. \"Guess what happened in school today?\" Dean is not listening. His eyes are focused on the kitchen entrance as he awaits their prodigal's return. Geoffrey shows enough interest so Frank carries on. \"Todd said he loved me today.\"\n\nGeoffrey laughs. \"That's not news, son. We've always known that. Surely you've known all along, too!\"\n\n\"Well, yeah,\" Frank agrees, \"but the way he said it, well, this time it had meaning.\" Dean's focus has shifted and his eyes are now locked on Frank with intense interest. Frank smiles. He knew this news would make his Papa Dean happy. \"Yeah, he said it's going to be him and me someday. After we graduate, we're going to register. We didn't decide when or anything, just that it's what we both want.\"\n\nDean clasps a hand to his mouth and closes his eyes while a few tears shed. Geoffrey reaches over; he runs his hand down Dean's shoulder to his elbow and back up again to hold onto his lover's hand. The two men clasp hands, and Dean, with his free hand, reaches over the table to clasp Frank's.\n\n\"Well, this is good news, son,\" says Geoffrey, approving Frank's choice.\n\n\"Oh, Frank,\" Dean says, \"I am so happy.\"\n\n\"Thanks, Dad. Thanks, Papa Dean.\" Sharing this news with his fathers, feeling the depth of their joy in his future happiness fills Frank with a sense of accomplishment and purpose.\n\n\"I always knew Todd was the better choice for you,\" Geoffrey adds approvingly. \"I never did like all the different boys you kept bringing over here. It was obvious you were just biding your time with them.\"\n\n\"Well,\" Dean says, now standing to clear his bowl from the table, \"I, for one, am not going to miss that parade of young men.\"\n\nFrank is confused. \"What do you mean?\"\n\nGeoffrey explains, \"He just means it will be nice to see you settled with Todd instead of jumping from boyfriend to boyfriend like you've been doing for the past three years.\"\n\n\"But,\" Frank is a little disconcerted, \"Todd and I aren't dating or anything. We've just agreed to commit\u2014\"\n\nDean turns on Frank. \"How can you commit to something as important as registration and not be dating? That's marriage, Frank!\" Dazed and amazed, Dean just stands there and stares at his son.\n\n\"Dean asked you a question, Frank.\" Geoffrey insists on resolution.\n\nWhen Frank fails to respond, Dean spins on his heels and throws his porcelain bowl into the sink, shattering it. Shaking in anger, Dean braces against the counter's edge.\n\n\"Dean,\" says Geoffrey, standing and crossing over to his lover. \"Calm yourself.\" Eyeing the broken porcelain in the sink, he becomes critical, \"that qualifies as waste.\" Then he orders, \"Clean that up.\" Dean complies, still shaking with anger. Geoffrey, turning back to Frank, insists, \"Explain yourself, son.\"\n\n\"Well\u2014\" befuddled, Frank searches his mind for an answer. He has never seen Papa Dean so angry, and it has him baffled. \"We tried kissing today, but Todd said he wasn't ready yet so\u2014\"\n\nDean swirls around, \"SEX!\" Frank's mouth drops open. Geoffrey closes his eyes and shakes his head. \"That's it, isn't it?\" Dean expostulates. \"He's not ready for sex, so you're not going to date him.\" Dean is snarling now. \"Is that all you think defines a relationship between men\u2014SEX? Has it ever occurred to you that maybe, just maybe, two people can spend time together just talking, simply enjoy each other's company, maybe cook a good meal together, shop, or read, or even just watch a vid on the wall screen? Is everything about SEX for you? A man is not just a cock and balls!\"\n\nFrank smirks. \"'Cock and balls'\u2014good one, Papa Dean.\"\n\n\"What?\" Dean, no longer simmering, is boiling over. \"Do you think I'm fucking around here? This is no joke!\"\n\nFrank is stunned. Never once has he heard either father swear, let alone direct such language at him. Geoffrey, too, is incensed. \"ENOUGH!\" Gripping Dean's arm, he begins to remonstrate. \"Never use that kind of language against our boys!\" Dean rips his arm away and turns his back on the scene. Although moved by Dean's words and knowing the truth behind them, Geoffrey is compelled to put an end to this tirade against Frank. Dean's words are by far too descriptive and harsh. \"I mean it, Dean. The boy may be wrong,\" says Geoffrey, throwing Frank a warning glance to let him know he has not been exonerated. \"I know you're upset, but that is unacceptable.\"\n\nExasperated beyond reason, Dean raises his hands in the air. \"I'm sorry I swore, but...\" Unable to finish, Dean clasps his hands into fists and releases them without reducing any of the tension. \"Fine! You deal with him. I have to get out of here before I break something else.\" Spinning around, refusing to look at either man, Dean exits the kitchen and storms down the hall to their bedroom.\n\nGeoffrey returns to his chair. \"I understand why your Papa is angry, Frank.\" Sighing, he begins, \"I'd like to try to understand\u2014\" then squeezes his eyes shut and raises and tightens two open hands, \"you.\"\n\n\"Dad,\" Frank pleads, \"Todd doesn't want to date. He says dating means kissing, and kissing means petting, and petting means sex, and he's not ready yet.\"\n\nTension begins to release, and although Geoffrey is not entirely satisfied, he can at least understand Todd's perspective. \"And you're not willing to date him without all those things?\"\n\n\"I\u2014\" Frank doesn't want to admit his father and Papa Dean are right. \"That's not it. Todd really doesn't want to date yet. It's his decision, Dad.\" Desperate to feel exonerated in his parents' eyes, Frank concludes, \"I can't make him date me, can I?\"\n\nGeoffrey shakes his head. \"No, Frank. I don't suppose you can.\" _Will Dean understand?_ Geoffrey wonders.\n\nAlmost as if he can read his father's mind, Frank blurts out, \"Papa Dean hates me now, doesn't he?\"\n\n\"No, son,\" Geoffrey reassures him. \"He's just...upset. And I guess...\" he says, not wanting to berate his boy, \"I guess he has some right to be.\"\n\n\"Why?\" Frank is upset and bewildered.\n\n\"Look, Frank.\" Geoffrey leans forward, elbows on the table, hands clasped, looking over his knuckles at his son. \"Papa Dean has more right than I do to lecture you on this subject.\"\n\n\"But, Dad,\" Frank objects, \"he doesn't know what Todd wants.\" Frustrated, Frank adds, \"Todd doesn't know what Todd wants\u2014except that when he's ready, it'll be him and me. And that's enough for me right now. If I push Todd any harder, I'll lose him.\"\n\n\"I understand, son,\" Geoffrey says, trying to respond judiciously. \"But I think Papa Dean understands Todd a lot better than either you or me.\"\n\n\"How? I've known Todd my whole life. We've been best friends since before either of us can remember.\" Frank is annoyed. He truly believes he is the only one who knows and understands the real Todd Middleton.\n\n\"I was your Papa Dean's first and only lover.\"\n\n\"I know,\" says Frank, getting annoyed. \"And I know Todd's still a virgin.\" One of the things Frank loves most about Todd is his chaste treasure, waiting to be granted to and enjoyed by only one lover. _I will be that man_ , he reassures himself, _but only when Todd's ready!_\n\n\"Knowing is one thing, Frank. Understanding what that means to a man is an altogether different matter.\" Rethinking his strategy, Geoffrey explains, \"For some of us, sex comes easy. We're luckier than most, I'd say. But for others, like Todd and Papa Dean, it really does take being in the right moment, in the right place, and with the right person. The building of trust required to bring him to that time and place with you is not so easily acquired as you think.\" Geoffrey begins rubbing the crown of his nose, unsure whether he is getting through to Frank. \"I was a lot like you when I was younger. I dated...\" _No_ , he reasons, _now is not the time to equivocate._ \"I slept with a lot of different men.\" Waving his hand to silence Frank, sharing an expression that clearly indicates this is no time for pretense, Geoffrey continues, \"But that all changed when I met your Papa Dean. You see, I fell in love with him. He was young, frightened, alone, reaching out for help\u2014perhaps, I think, like Todd was reaching out to you, today. Anyway, I was a lot older than you are now. I was twenty-six and you're only sixteen. I guess time and a clearer view of the world gave me a better perspective.\"\n\n\"What do you mean?\" Frank asks, truly wanting to understand. He really believes he and Todd are a lot like his dad and Papa Dean.\n\nGeoffrey chooses his words carefully. \"Your Papa Dean was only seventeen when I met him, and like Todd, he wasn't ready, but we had this connection.\"\n\n\"Yeah, yeah!\" Frank agrees. \"That's exactly like Todd and me.\"\n\n\"Perhaps,\" says Geoffrey, hoping for not _exactly_. \"But your Papa Dean came to trust me over time. He knew that for me, he was the only one; that all those other men were behind me. You see, I never lied to Dean. I told him about my past flings.\"\n\nFrank is getting excited. \"Yeah, like Todd knows all about my boyfriends.\"\n\n\"Not quite, son. You see I was ready to give up that life in exchange for a true commitment, a real relationship with Dean. And Dean knew he could trust me.\" Looking gravely into his son's eyes, he asks, \"Is Todd able to say the same about you?\"\n\nFrank drops his head in dejection. \"No, sir.\" Looking up, miffed and mystified, he demands, \"What is it you and Papa Dean want me to do?\"\n\n\"I can't answer that, son. Only you can decide what is right for you at this time.\"\n\n\"No, Dad, you have to tell me!\"\n\n\"As much as you want me to solve this problem for you, I can't. Your life is your own riddle, and only you can puzzle this one out.\"\n\n\"That's no help at all.\"\n\n\"I'm sorry, son. You're only sixteen, I know, and this is quite a heavy burden you have placed on yourself, but you do know the difference between right and wrong. Only you know the depth of your own personal commitment to Todd. Only you know how much you are ready to give of yourself at this time in life.\" Shaking his head in wonder, he adds, \"I doubt I could have given everything to just one man at your age, but then, I'm not you, and you're not me.\"\n\n\"So, you're not going to tell me what to do?\"\n\n\"No, son. I can't choose your life for you.\"\n\nFrank is both disappointed and relieved. Without thinking, Frank comes to his decision. \"I have a date tonight with T'Neal.\"\n\n\"Do you plan on keeping it?\" Geoffrey asks, trying hard to keep judgment out of his voice.\n\n\"Yes.\" After a brief pause, he asks, \"May I borrow the bubble?\"\n\nGeoffrey looks down at his bowl and picks at the last of the tofu chili. \"Do you remember the command code?\"\n\n\"Yes.\"\n\n\"You going to finish eating?\" Frank hasn't touched any of his meal since Dean's outburst.\n\n\"No, sir, I'm not hungry.\"\n\n\"All right then,\" Geoffrey nods. \"You can go.\"\n\n* * * * *\n\nAfter Frank departs, Geoffrey leaves the table cluttered with the dinner mess and retreats to the master bedroom. As Geoffrey suspected, Dean is sitting on the edge of their bed, slouched over with his head between his hands. Geoffrey's first thought, involuntary but natural, is _I sure married a handsome man._ Dean is tall, thin with a classically sculptured face. His brown curly hair is only now beginning to show the early stages of gray, and unlike for most men, is not thinning away. Before sitting next to Dean, Geoffrey takes a moment to control his sexual urges. Once in control, he begins lightly to rub Dean's back. \"Hey.\"\n\n\"So, what's the verdict?\" Dean does not look up, but directs the rhetorical question into his hands.\n\nInstead of responding, Geoffrey wraps his arms around Dean and pulls him in for a hug. \"Come here.\" Dean shudders, trying to hold back tears. \"It's all right,\" Geoffrey replies. \"Go ahead and cry. I know this is bringing back a lot of harsh memories.\"\n\nAfter a long bout of tears, Dean changes the subject. \"Our boys are all that I have.\"\n\nGeoffrey smiles. \"What about me?\" he whispers soothingly. \"You have me, too.\"\n\n\"That's not what I mean,\" Dean mutters. \"I didn't mean to suggest otherwise; it's just, everything I had ever wanted out of life I couldn't have.\"\n\nGeoffrey closes his eyes. He knows what Dean is talking about. He often wonders how well he could have coped had he not been allowed to pursue a career in business.\n\n\"Our boys are everything to me\u2014my whole life\u2014I love taking care of them\u2014and it is fulfilling, but\u2014I feel so useless sometimes\u2014like I'm losing grip.\"\n\n\"That's because they're growing into men and we can't hold onto them forever,\" Geoffrey reasons.\n\n\"I know. I know. But Roger is still so young. He's only fourteen! That boy only wants one thing from him. And then Frank with Todd\u2014oh, Geoffrey, I'm so worried about Todd. He's too much like his father.\"\n\nGeoffrey squeezes tighter to help reduce Dean's shaking. \"We're watching over him. And after what he said to Frank today, I'm sure he'll be _okay_.\"\n\n\"But not if Frank keeps running around like he does.\"\n\n\"We can't control Frank's choices,\" says Geoffrey. Dean shudders. \"All we can do is give him the best advice possible.\" Kissing Dean softly on the top of his head, Geoffrey adds, \"But yelling at him, suggesting he's...unsavory, swearing like that. You know that doesn't help.\"\n\n\"I'm sorry. I'm sorry I got out of hand down there. I shouldn't have let him get me so angry.\"\n\n\"It isn't Frank that upsets you so much is it?\"\n\n\"I'm just so worried\u2014I thought, when he said he and Todd were committed, that everything was going to be okay, that Todd was...safe.\"\n\n\"Todd is safe. We'll continue to watch over him. I promise.\" Geoffrey resumes rocking Dean in his embrace. Soon their lips connect. Knowing what sets his love aflame, Geoffrey reaches up and grasps the thick brown curls at the back of Dean's head\u2014not too rough, but enough to cause a slight twinge of pain. Pulling Dean's head back so his neck arches up, Geoffrey begins to nibble on his Adam's apple. Soon they are lying on the bed. Geoffrey manages to undo Dean's shirt and unbutton his denims, forcing the zipper by pushing his hand into Dean's briefs. Geoffrey hardens as he teases Dean's cock into a similar state. The moan he elicits from Dean is truncated by the sound of the outside door opening. Leaping up instantly, Dean heads for the bedroom door. He offers no apologies for abandoning his lover at such a moment. He merely comments, \"Roger's home,\" as he zips up his jeans, buttons up (and tucks in) his shirt, and exits the room.\n\nGeoffrey groans, but he can't help smiling at Dean's commitment to their boys. Once again, he thinks of how lucky he is to have found a man like Dean Stuttgart to help raise their children and share the rest of his life with. Taking a moment to breathe in the scent of Dean's hair lingering on his fingertips, Geoffrey wonders whether he should head on over to his study and put in a few hours of work. Smiling, he likes the idea that flashes into his head. Undress and wait patiently until Dean comes back to their bed.\n\n* * * * *\n\nFrank and T'Neal are hitting it off hot and heavy. They had met at T'Neal's as both his mothers were working and they would have the place to themselves. T'Neal's room, though much smaller than Frank's, also houses a king-sized bed. T'Neal is lying on his back, showing off his incredible stretch. Being lead cheerleader, T'Neal can not only do the splits front and sideways, but he can, in a standing position, lift his leg up into a point, his knee snug against his ear. Currently T'Neal's exhibition is designed to tease Frank. Although Frank makes every effort to oblige his young lover, he eventually drops to the bed to sulk.\n\n\"Ah, honey,\" T'Neal coos. \"It happens to all of us. It's okay. Really.\" Then squishing up his nose and shaking his face like a little bunny rabbit, he adds mischievously, \"We could try a again.\" When Frank gives no sign of complying, T'Neal rolls onto his side and begins caressing Frank's chest hairs. Feeling incompetent, T'Neal begins to apologize.\n\n\"It's not your fault,\" Frank blurts out, a little too harshly. Unable to say more, all Frank can do is shudder, sigh, and cry.\n\n\"I swear, Frank,\" T'Neal promises. Frank's tears urge tenderness in the young man. \"No one at school will ever know.\" At this moment, Frank doesn't even care. All he really wants is Todd to hold. For the first time in his life, Frank feels like he is being unfaithful. The very act of being with T'Neal is hurtful. He isn't just making love anymore. He is having an affair. Although he doesn't break up with T'Neal, he goes home saddened by this new awareness.\n\n* * * * *\n\n#  _Salve!_\n\n**Panthers Bring Home the Nationals  \nHNN\u2014Melissa Eagleton Reporting**\n\nIt was an exciting afternoon for the Pride Panthers today when they won in the last few seconds of the Championship B-Ball tournament's final game. The Pride Panthers are proud to announce that they will be bringing home the National Cup! Co-caps Frank Hunter and Crystal Albright led their team to this monumental victory with the help of Todd Middleton, son of the late Will Middleton, another Pride Panther National Cup winning phenomenon. It seems the Middleton family is destined to be superstars of the b-ball court. Like his father, Todd Middleton landed the winning basket for his team.\n\nAll through the game, the Panthers had trailed the Virginia Wolves by only one point. It wasn't until the last second when Todd Middleton stole the ball from the Wolves co-cap Anita Brown that the tide turned in the Panthers' favor. Weaving in and out of the Wolves team, Todd Middleton passed the ball to Frank Hunter off the right of the court, creating an opening that allowed Todd to hit directly for the basket. Frank Hunter, knowing his teammates' strategy, passed the ball back to Middleton, who successfully loped the ball into the basket just prior to the buzzer going off. The game ended with a 35-34 score in favor of the Panthers. Needless to say, Todd Middleton's team was ecstatic. Co-cap Crystal Albright was the first to plant a kiss on Todd Middleton's lips. Who can blame her with all the excitement of the win? Hers was immediately followed by a kiss from the other co-cap, Frank Hunter. Following the example of their intrepid leaders, all of Todd Middleton's teammates commenced kissing him. When they went through the long line-up of shaking hands with the other team, the Virginia Wolves also acknowledged Todd Middleton's success by each player also giving the young man a kiss. Most of the girls went for his cheek, but none of the guys had any qualms about kissing him on the lips. I don't think anyone has ever been kissed as much as Todd Middleton was this afternoon. When asked about all the kissing, Todd Middleton's only response was to blush a deep crimson. He did, however, speak freely about his hopes for the future. \"I'm hoping to get into Antinous Uni. That's where my father went.\" And like his father, Todd Middleton wants to study bioengineering. \"My dad genetically altered the soya bean so we could grow it up here. I want to do the same with rice! If we can grow rice up here,\" Todd Middleton explains, \"then Hadrian will truly be self-sufficient!\" Now that is the Hadrian spirit! We need more young men and women like Todd Middleton.\n\n_Vale!_\n\n# Antinous Wants You!\n\nCoach Miller is the only teacher at Pride with whom Todd feels comfortable enough to have a heart-to-heart talk. They often speak about b-ball. Then, the first school day after the Pride Panthers brought home the National Cup, she seeks him out during lunch in the cafeteria. \"Todd,\" she chimes. Looking up from his bag lunch, Todd smiles. The coach always makes him smile. \"Would you mind meeting with me after school? I'd like to ask you a few questions.\"\n\n\"Sure, Coach.\" Todd's smile widens. He likes that Coach Miller asks his advice about the game. It makes him feel important, worthwhile. \"We gonna talk about next season?\" She raises her eyebrows and smiles. \"You'll see.\" Punching him playfully in the arm, she adds, \"After school, then.\"\n\nTodd laughs, more than pleased. \"Okay, Coach.\" Frank elbows him, impressed, and the other boys at the table give nods of approval. B-ball gives Todd a sense of self-worth and importance he seldom feels in other aspects of his life.\n\nWhen Todd enters the gym after school, he spots Coach Miller sitting on the player's bench, going over plays in her book. Todd crosses over and sits down next to her. Her welcoming smile is infectious, and soon the two are laughing without either having said a word yet. Finally, Todd breaks up the chuckle. \"What's up?\"\n\n\"You are,\" she replies merrily.\n\n\"Huh?\" Todd is pleasantly flummoxed.\n\n\"I've been waiting all day to tell you.\" Excitement sparkles in her eyes.\n\nTodd is being lifted into the clouds by her thrill. \"What?\"\n\n\"I got a call last night!\"\n\n\"Yeah,\" he giggles. Suspicion tickles.\n\n\"From Antinous Uni.\"\n\n\"Antinous Uni?\" Too stunned to comprehend, Todd questions, \"Already?\"\n\n\"Already,\" Coach Miller confirms.\n\n\"But, I'm only in grade eleven,\" a befuddled Todd sputters out.\n\n\"They're offering you a full sports scholarship, Todd,\" Coach Miller reassures him. \"Residency, tuition, books, food\u2014everything but spending credit!\"\n\nTodd is so excited he leaps up from the bench, shouting, \"Wahoo!\" Jumping up, he punches the air. Coach Miller, infected by youthful exuberance, bounces up to join him in his victory dance. She picks the boy up in her arms and swings him around in circles as if he were two. When she releases him, they both howl like dogs at the moon. Frank, who has been waiting outside the gym door for Todd (he always thumbs in Todd's fare for transit, so Todd only has to walk to school), pops his head in. \"What in Hadrian's name is going on in here?\"\n\nAs soon as he sees Frank, Todd rushes over to his friend. \"Me! Me, Frank! Antinous Uni wants me!\" The two boys bend at the knees, both chests thrown back, necks stretching beyond ligament capacity as they howl out joyously. When they finally right themselves, Todd barges on excitedly, \"That's my father's old campus!\"\n\n\"Ha, Ha!\" Frank, like Coach Miller before him, lifts Todd high into the air and swings him around in circles. Being stronger and taller than Coach Miller, Frank flies Todd even higher. \"Of course they want you!\" he bellows in delight. \"Who wouldn't want you?\"\n\nAfter Frank finally releases him and a moment passes for his head to stop spinning, Todd asks, \"Can I come to your place? I need to talk to Papa Dean. Antinous Uni wants me to graduate this year.\"\n\n\"By all that's gay and glorious,\" Frank declares, \"Todd, that's amazing!\"\n\n\"I don't know; I don't know,\" says Todd, so happy he is tripping over his words, speaking so fast he can't seem to say anything. \"I don't know if I should. I need to talk to Papa Dean first. I have to ask his advice.\"\n\nBefore Frank can respond in the affirmative, Coach Miller becomes inquisitive. \"Papa Dean? Why one of Frank's fathers? Why not your own?\" With news like this, Coach Miller would assume the boy would want to rush home and tell his own parents first\u2014 _parent_ , she reminds herself. Todd's genetic father died eight years ago.\n\n\"Ah...\" Todd hadn't even considered asking Papa Mike. It wasn't that he didn't love his second father. _I do_ , he reminds himself. \"It's just that Papa Mike has to work and...ah...\" A glimmer of gloom threatens to encroach on Todd's good fortune. \"It's not like he can't advise me...\" As with every mention of Todd's circumstances, Frank wishes he could get Todd away from Papa Mike, but Todd ardently refuses to abandon the man. \"Well, since Dad died, well, Papa Mike, he has to work a lot of overtime\u2014I mean, um...\" Todd trails off. It never seems like he can defend Papa Mike in the eyes of others. So few people in Hadrian understand what paucity is really like. It's one thing to see the poverty-stricken hordes outside Hadrian's walls, all evidence of which is photographed from above. It's another thing actually to have to live without. Todd counts himself lucky, though. They may not have all the luxuries like voc implants or fancy clothes, but Papa Mike keeps him fed and clothed, and he has a roof over his head. Over half the world's population lives in squalor facing Nature's elements with no bed, no bread, and no clothes.\n\nAt times like this, when it is evident the other person simply cannot comprehend his situation, Todd turns to Frank. Knowing how hard this is for his friend, Frank finishes for Todd. \"They just don't have any money, and his Papa Mike works twenty-four seven. Todd can go for weeks without ever seeing his second father. But that's okay because Todd stays with us whenever his Papa Mike has to work excessive overtime. Todd and his Papa Mike know they can depend on us. He's never left out in the cold or anything.\"\n\n\"Oh,\" Coach Miller says. She raises her palms slightly and gives them a little shake to silence Frank's rambling. She can sense Todd's unease and doesn't want what should otherwise be the happiest moment in this young man's life to be filled with shame and embarrassment. \"Well,\" she says, adeptly changing the subject back to Todd's future success and clapping her hands with forced joviality, \"it's good you have Papa Dean to talk with, then.\" Clasping Todd's shoulders, she adds, \"Go ask Papa Dean what he thinks, then, and let me know your decision first thing in the morning.\"\n\n\"Absolutely,\" Frank concurs exuberantly, determined to recapture their previous moment of glee. \"You're coming over to our place, Todd. I'll voc Papa Dean and let him know we're coming. Dad's out of town and Roger's on a date, but the three of us, we're going to have us a celebration dinner!\" Then shaking Todd, he adds, \"And I don't care what you say! It's going to be The Cattle Ranch Restaurant!\" The most expensive place in the city, but Todd does not balk. This news is too good, and he is simply swirling with merriment. As soon as Todd nods his assent, the two boys join hands and race out of the gym, leaving Coach Miller giggling happily.\n\n* * * * *\n\nPapa Dean is thrilled by Todd's news, but his advice is not what Todd or Frank expected. \"I think it's wonderful that Antinous Uni wants you to graduate early, Todd. That really speaks highly of you, but are you ready?\" Todd is unable to answer because the waiter has just brought out a large tray with three juicy steaks, baked potatoes, and an array of seasoned vegetables. Both Todd and Frank ogle their meals. \"All right, boys, it's a steak\u2014not a potential boyfriend.\"\n\n\"This is one potential boyfriend I'd be more than willing to date,\" Todd pipes up as he begins slicing through the slightly seared meat.\n\n\"Me, too,\" says Frank, also beginning to dig in instantly.\n\n\"Okay, Todd,\" Dean reminds him, \"chew your food before swallowing.\"\n\n\"Sorry, sir. It's just so good.\" A line of steak juice dribbles down his chin. Frank laughs.\n\nPapa Dean leans forward and hands Todd a napkin. \"Wipe your chin.\"\n\n\"Yes, sir,\" Todd laughs.\n\n\"You haven't answered my question.\" After the initial reaction over their meal is done, Dean insists on settling Todd's future.\n\n\"Well,\" Todd begins.\n\n\"Finish what's in your mouth first,\" Dean insists.\n\n\"Sorry, sir,\" Todd says while swallowing. \"I'm not sure. My marks are pretty good.\"\n\n\"How good?\" Dean inquires.\n\n\"I've got a seventy-six average.\n\n\"That's not bad, Todd, but,\" he cautions the youth, \"that's not great, either.\" Considering his words over another mouthful of steak, Dean continues, \"Most students who excel a grade have marks in the high nineties.\" Not wanting to be hurtful, yet remain pragmatic, Dean suggests, \"Consider waiting one more year, Todd. If Antinous Uni wants you badly enough to offer up a scholarship a year early, it will wait until you actually graduate.\"\n\n\"But I could go in September. Why wait one more year if I don't have to?\"\n\nFrank nods. He thinks Todd should go early. \"Why should he pass up such an amazing opportunity?\"\n\nDean passes over his son's comment. \"Well, let's think about this. You just turned seventeen a few months ago. You will be younger than everyone on campus. And then, you are going to find uni a lot harder than high school, Todd.\"\n\n\"You didn't go to college,\" Frank barges in, \"so how would you know?\"\n\nFrank has no idea how hurtful his comment is. Although Dean had every intention of attending uni, he knew, even having finished high school with a ninety-five average, no uni in Hadrian would accept him. Dean pauses, looks away for a moment, settles his emotions, and starts over. \"I saw how much work your father had to put into his final year at Antinous Uni, Frank. And your father, Todd, my best friend, said he had never worked harder in his life than when he was studying agricultural engineering. If you really want to bring in innovations like your father did...\" Smiling, he remembers the boy's dream on Hadrian's fiftieth birthday. \"Didn't you say you wanted to engineer a new strain of rice, one compatible to Hadrian's climate?\"\n\n\"Yes, sir.\" Todd is thrilled that Papa Dean remembers. \"That's my dream. And,\" smiling, remembering his birthday cake, \"cocoa beans, too! Then you can bake all the chocolate cake in the world!\"\n\n\"And you boys can eat it,\" Dean adds, pleased. \"But,\" he persists, \"dreams require hard work. Nothing worthwhile in this life will ever come easy. If you go to uni too soon, before you are really ready, you may find yourself flunking out and the road to your dream caving in underneath you.\"\n\nTodd, looking down at his plate, studies his steak seriously, as if it held in its red sinews the answer to this daunting question. \"Gee, Papa Dean. I never thought about it like that.\"\n\n\"No way!\" Frank blurts out. \"That is so _strai_!\"\n\n\"Frank, I told you not to use that expression.\"\n\nFrank ignores his papa and barges on, \"Todd, please don't tell me you are actually considering turning Antinous Uni down?\"\n\n\"I don't know.\" Todd is still inspecting his beef. Looking up, he searches deeply into Papa Dean's eyes. This is a man Todd trusts, loves almost as much as he loved his father. _I think I love you even more than Papa Mike_. Smiling, Todd gives the table his answer. \"Papa Dean is right. If I can trust anyone's advice, it's yours, Papa Dean. If you think I should wait another year, then that's exactly what I'm going to do.\"\n\nFrank scowls, but Papa Dean is all smiles. \"You've made a wise decision, son.\"\n\nFrank spits out his dissatisfaction. \"If I knew you were going to advise him like this, I would have taken him out for supper all by myself.\"\n\n\"Well, look at the bright side, son,\" says Dean, patting his boy on the back. \"You get to keep Todd around for another year, and then, if you win the academic scholarship, the two of you can start uni together.\"\n\nThat pleasant reality is enough to bring Frank back to rights with Todd's decision so the three men can continue to enjoy their very expensive, very delicious steak dinner.\n\n* * * * *\n\n#  _Salve!_\n\n**Springtime and Exposure  \nHNN\u2014Melissa Eagleton Reporting**\n\nParents, I cannot stress enough just how important it is for you to keep a vigilant eye upon your child\u2014especially now that springtime has arrived. I am not sure why it is, but springtime always brings with it sexual awakenings and heightened sexual activity. The lengthening days, stretching the sunlight well into night, seem to have an influence on our passions. As conscious adults, we can control these fevers and channel them appropriately. Our youth, however, need our guidance. If we fail them in their time of need, their hormones will lead them astray. The human sexual need begins to simmer in the springtime and often comes to a boil during the hot summer months. During the fall and winter, attendance at our reeducation camps is minimal. Seldom do we see more than two or three wards during this time of the year, many of these inmates having been exposed during the spring and summer months and simply needing more time to mature, understand, and come to accept Hadrian's ways. Yet, come spring, numbers registered at reeducation camps begin to double, triple, and by summer sometimes have quadrupled. Clearly, we are becoming lapse in our diligence toward guiding and observing the ways of our youth. Teenagers, in particular, are our most vulnerable as they journey on the road of sexual experimentation. Be warned, parents; many of our teenagers are playing with fire as they begin to experiment with heterosexual behaviors. Many teenagers choose to experiment with the opposite sex purely out of the youthful need to rebel; however, others are honestly confused and need our help to understand their contradictory desires. Hadrian's first cornerstone of existence is the key to our country's stability as a homosexual nation dedicated to preserving the balance between humanity and nature. When our children experiment foolishly with heterosexual behaviors, they risk establishing lifelong habits that have historically proven themselves counterproductive to humanity. We all know our planet cannot sustain today's world population. Only Hadrian stands triumphant as a self-sustaining country with natural parks and reserves as well as home to the world's only remaining zoo. Everything we do is to help protect earth's ecological diversity. Over 50 percent of the land within our walls belongs to Mother Earth, and we must fight with every ounce of will to keep the majority of humanity from destroying what little remains of her natural ground. Our chosen values have helped us maintain a stable population. This balance would change if we were to allow our youth to restore heterosexuality as the central means of procreation. Having witnessed how heterosexuals have ruined the world, we must not allow the same to happen here in Hadrian. Parents, please remember, you are on the front lines. It is imperative you help your teenager understand Hadrian's values, not merely for us, but for the future of the human race and the planet earth. You are not just raising your own children; you are raising Hadrian's children. You are raising the future of humanity. And, it is up to you to educate your child, for without your guidance, our children will go astray.\n\n_Vale!_\n\n# Exposure\n\nIt starts with the dreaded dance supervision: Ms. Sterne's nightmare. Four times a year, the students of Pride gather in the gymnasium to gyrate and sweat over one another while teachers stand back and watch, keeping an ever vigilant eye open for drug and alcohol use. This special Saturday is an extra duty added to teachers' already hectic schedules: a fifth dance added to the annual school year quota. No one but Ms. Sterne is complaining, though, since this dance is in celebration of Pride High's National B-ball Championship victory! Ms. Sterne, having taught at Pride High long enough, knows only too well the dance supervision drill, and having experienced it at its worst, she uses her seniority to avoid gym, hallway, and bathroom duties. Instead, her job is to sit in Mr. Gavin's office and watch the video surveillance. This duty is considered too dull for other educators, many of whom, unlike Ms. Sterne, actually enjoy hunting down drunk and stoned teenagers. In the early years of her teaching career, Ms. Sterne used to chase down teenagers sodden with alcohol with similar avidity until the day she had to sit with one girl as she puked out vodka and a raspberry-flavored drink into a garbage can. The wretched smell still lingers in Ms. Sterne's memory, and she has sworn never again to suffer the decrepitude of youth. So, for the past fifteen years, Ms. Sterne has volunteered for the long night watch inside the principal's office, entertaining herself with classical music while she watches all security videos. If she spies students attempting to consume illegal substances or making out, she vocs Mr. Gavin, and he or some other unfortunate educator has to deal with the teenage delinquent.\n\nBecause tonight's dance is a victory celebration, the anticipated offenses are numerous. Ms. Sterne has already caught a few reprobates while blinking her way through the systems channels. It is purely by happenstance that she discovers maintenance has finally fixed the broken camera behind the stairwell leading up to the girl's locker room. She had put in a work order to have that particular camera fixed over three years ago. As is the case with most school bureaucracy, that particular work order got lost. \"Somebody must have found it in a pile of red tape, not too long ago,\" she mumbles as she switches over to that channel. What Ms. Sterne sees on the security wall screen both shocks and horrifies her. Her niece Crystal is making out with a boy\u2014Todd Middleton. This is no ordinary make-out session either; this is full-out intercourse. Her eyes are riveted to the screen both unable, and desperately wanting, to look away. After her mind has gone through a dizzying array of hateful emotions, she finally feels a slight sense of relief as she remembers Crystal's mothers have been forcing the young woman to take heterosexual birth control to counteract gigantism. \"We can thank Hadrian for that much,\" she grumbles quietly. Suddenly fearing she might not be alone, she unreasonably jumps up to look out the door into the main office. Issuing a staccato sigh of relief, she returns to her chair and locks the door from the inside, so no one will be able to walk in on her unawares.\n\nWith some semblance of order to her thoughts, Ms. Sterne ponders the ramifications of Crystal's actions. Contacting Mr. Gavin is out of the question. Crystal Albright is not just any young woman; her mama just happens to be Ms. Sterne's sister, Vice President Elena Styles! Crystal will have to be protected for Elena's sake. But that boy\u2014that boy will pay for what he is doing with Crystal! Unfortunately, dealing with Todd Middleton will have to wait until Monday!\n\n* * * * *\n\nTodd Middleton is late for class, again. He always has trouble getting up in the mornings. He spends too many late hours texting with Frank or Crystal. Having to walk to school makes matters worse since it is a good fifteen-minute hike. As Todd runs, he thanks Hadrian for his recent conditioning due to the recently ended b-ball season. _I should have gone to sleep earlier_ , he chastises himself. But even after this bout of self-recrimination, Todd can't help but smile. As his feet crunch through last year's deadfall exposed after the last melt, waiting only for the spring rains to batter it into earth, Todd chuckles. Last night's texts were especially fun since neither Crystal nor he wanted to let go of the other, texting into the wee hours in an attempt to hold onto the memory of their touch\u2014reminding each other what they had done Saturday night, telling the other what each had especially liked, then both agreeing judiciously to delete all remnants of their text talk before signing off.\n\nAs with every other late morning arrival, Todd enters his first class dressed slovenly and with his hair disheveled. Because Papa Mike's work day begins early, before Todd even has to get up, no one is at home to make sure Todd gets up on time, washes, and looks appropriate. Todd, more often than not, is left to his own devices. When late like this, Todd usually just grabs whatever he can find on the floor, not even looking to see whether it's clean, has been ironed, has the appropriate flare, or even matches. Papa Mike is a mechanic (a necessary field but looked down upon due to the straight connotations associated with that line of work, so it is one of the least paid professions). Consequently, he seldom buys Todd new clothes. Most of the clothes Todd wears are hand-me-downs from Mike's closet, which, unfortunately, reflect a more rugged-looking man. When Todd goes to school, he gets teased all the time. These days, though, Todd doesn't care. Nor does he take any of the ribbing seriously anymore, convinced everyone accepts him as gay due to his prized position on the basketball team and his longstanding friendship with Frank Hunter. He is far too popular now that he helped the Panthers bring home the Nationals! That and the offer of a full scholarship to Antinous Uni improved his status considerably among the student body. But Todd is not like the other boys, and although it remains unspoken, there are those, like Ms. Sterne, who have always suspected and now know for sure.\n\n\"Late again, Mr. Middleton.\" Eyeing the young man austerely as he sits in his desk, Ms. Sterne shakes her head disapprovingly, her silver hair flashing like lightning. Looking down on Todd, over her reading glasses, she inquires, \"What foolish behavior kept you from getting a good night's sleep?\"\n\nTodd smiles sheepishly. \"Just on the voc again.\"\n\nMillicent snickers, \"You can't afford a voc!\"\n\n\"Shut up!\" Frank, ever ready to defend Todd, offers up. \"I gave him one for Hadrian's fiftieth birthday last New Year's Eve.\" Actually, Frank and Crystal gave Todd a cell phone. T'Neal scowls as soon as he hears Frank confessing to buying Todd such an expensive and elaborate present. Frank growls back in retort, \"He's like a brother to me, T'Neal. Get over it!\"\n\n\"And what's wrong with your parents, Todd?\" Millicent asks.\n\n\"Millicent, you are so ignorant!\" Frank declares. Looking Crystal's way, he wonders why she doesn't say anything in Todd's defense.\n\nMillicent ignores Frank's outburst. \"Why won't they buy you one? Are they too cheap?\" The class joins Millicent in laughter. Todd leans back in his chair, refusing to hide, pride burning greater than embarrassment.\n\nMs. Sterne does nothing to stem the tide of mockery. Her eyes squeeze into slits. \"Just voc'ing?\" There is suspicion in her glare. She eyes him from head to toe, inspecting every inch of his apparel. \"Did you take time to look at yourself in the mirror this morning?\"\n\n\"Ah,\" Todd shifts his eyes away from her daunting stare, \"no, ma'am. I was running late and\u2014\"\n\nTossing up her hands in disgust, she finishes for him. \"And you just threw on whatever first came to your face? That's it, right?\"\n\n\"Ah, yeah.\" Todd shifts in his seat. Ms. Sterne's interrogation has him sitting on hot coals.\n\nCrossing over to her desk, she retrieves a hand mirror. Flashing it at the young man, she demands, \"Look at yourself.\" Todd obeys. One does not contradict a teacher like Ms. Sterne. \"Now what do you see?\"\n\n\"Ah, messy hair.\"\n\n\"That's right.\" Waving to the student body, she asks Todd, \"Do you see anyone else in this room with messy hair?\" Todd is required to swivel in his seat and look at all the other students. Most of them, Frank excluded, are smirking at him. No one laughs outright, though, as Ms. Sterne is clearly livid. Crystal, he notices, is not laughing. She has her head lowered, hiding her face from everyone. _She's not participating in this_ , he thinks reassuringly. A little smile blooms on his face. Ms. Sterne notes the subtle intensity of his look and who inspires it. She doesn't bother to wait for a reply. Angered by this slight act of communication on Todd's part, she slams her mirror onto his desk, cracking the glass in the process. Startled, Todd leaps up and turns around in his seat. Ms. Sterne is shaking her head, hands on her hips, condemning him with her eyes.\n\n\"I\u2014I\u2014I'm sorry. I\u2014I won't be late tomorrow, I promise. It's just\u2014\" He quickly runs his fingers through his hair, getting them caught in the tangle.\n\n\"It's not just being late!\" Ripping Todd's hand away from his head, taking with it clumps of hair, she barges on. \"It's not just your messy hair! It's not just the clothes you wear that reek of _strai_!\"\n\nFrank can stand no more. \"Ms. Sterne, that's not fair. Todd's not straight. He's gay like the rest of us. It's just\u2014he's just\u2014\" He gives Todd an apologetic look, \"his Papa Mike\u2014well, he's raising Todd by himself\u2014his partner, Todd's dad, died eight years ago.\" Todd closes his eyes. He misses his father fiercely.\n\nMs. Sterne throws Frank a warning glance. \"I know his father died. I know his Papa Mike is poor. But Todd is what? Sixteen? Seventeen? He could still go to The Charity Bin and get himself clothes that look less straight!\" A few students snicker at this.\n\n\"But he's not straight, Ms. Sterne. I know him.\"\n\n\"Do you, Frank?\" She glances quickly at Crystal and then sharply at Todd. Todd, fixated on his teacher's eyes, registers this look and cringes. Ms. Sterne, although still addressing Frank, smiles cynically at Todd. \"Do you really?\" Turning abruptly, Ms. Sterne returns to the front of the class, erases the day's lesson, and begins afresh. \"I think this is a good time to review why we formed our good country Hadrian and exercised sexual reformation. Why today, more than ever, we discourage all forms of heterosexual behavior.\" Turning back to face the class, her eyes shoot down like lightning bolts on Crystal. \"Crystal, dear, remind the class why Hadrian chose to enforce a homosexual lifestyle, eradicating heterosexuality.\"\n\nCrystal's shoulders start to heave. It is evident she is sobbing. Todd instinctively raises his hand to answer for her, the urge to protect her is strong. \"Put your hand down, Todd. I specifically asked Crystal.\" Todd instantly obeys.\n\nAfter a sniffle, Crystal recites, \"Human population has grown to such excess that the earth is overcrowded. There are close to twenty billion people in the world, on a planet that can barely sustain ten billion. The majority of humans living outside Hadrian's wall are starving, dying of disease, scratching out a desperate living.\"\n\n\"That's right.\" Ms. Sterne shows a little pity toward the girl and continues for her. \"These poor unfortunates are still propagating at an exponential rate whereas we in Hadrian do not suffer that problem. Having eliminated heterosexuality from our genetic code our population remains stable, and there are no unwanted pregnancies in Hadrian.\" Turning to another student, she requests, \"April, explain how we have managed to eradicate all unwanted pregnancies in Hadrian?\"\n\n\"All pregnancies are the result of in vitro fertilization and all insemination is recorded in the Centralized Hospital Records.\"\n\nNow staring intently at Crystal, Ms. Sterne asks, \"What happens to a woman who finds herself pregnant outside the official process? Eduardo?\"\n\n\"She is immediately tagged as a heterosexual. If under the age of twenty-one, she will receive reeducation. If over the age of consent, she will be exiled.\"\n\n\"Good.\" Ms. Sterne is pleased at the rote recitation. \"But,\" she continues, \"before all that, what happens?\" Looking around the room, she selects another boy. \"Devon?\"\n\n\"State officials interrogate her to determine who the father is. The unwanted fetus is aborted and the _strai_ gets sent to reeducation.\" Before speaking again, Devon Rankin raises his hand.\n\n\"Yes, Devon?\"\n\n\"Ms. Sterne,\" Devon asks cruelly, \"can the het'ro woman choose death over exile? If she's over twenty-one, I mean.\"\n\n\"Yes, Devon, she can,\" Ms. Sterne answers curtly. Turning now to another boy, she requests, \"Jared, explain how our lifestyle serves Hadrian.\"\n\n\"Hadrian, unlike the rest of the planet, has a stable human population. Our country never exceeds ten million.\"\n\n\"Excellent!\" she replies, nodding approvingly before turning to enquire of another student. \"Millicent, how else does Hadrian benefit?\"\n\nMillicent responds eagerly, \"We have housing for all our citizens, everyone has a job, no one starves, and, unlike the rest of the planet, we have farms where we raise livestock and grow all our own food.\"\n\nFor the first time since Todd entered the room, Ms. Sterne smiles. \"Very good, Millicent. Frank, continue.\"\n\n\"What more is there to say?\" There is hint of anger in his tone. He knows where this lesson is going and he doesn't like it. _Todd is gay!_ he reminds himself. _Why is Ms. Sterne doing this?_\n\n\"Plenty, Frank.\" Ms. Sterne glares at him. Usually a boy or girl showing rebellion during these lessons is instantly tagged as straight, but Ms. Sterne knows, everyone knows, that Frank Hunter is in love with Todd Middleton. _The sooner we get Todd reeducated, the better for you Frank_ , she thinks solemnly before answering for him. \"Hadrian also has the last five remaining wildlife parks on the planet.\" Turning again to Devon Rankin, assured of getting the right response from him, she says, \"Devon, name them.\"\n\n\"The Wapusk, Caribou, Numaykoos, Amisk, and Sand Lakes.\"\n\n\"Very good,\" Ms. Sterne congratulates the young man. \"And all five preserves help retain but a sampling of the indigenous wildlife.\"\n\n\"We actually have indigenous wildlife,\" Millicent boasts freely.\n\n\"That's right,\" Ms. Sterne adds. \"And no other country can boast that!\" Reconsidering, she adds, \"Except perhaps for the desert regions.\" Continuing, she states, \"We would never be able to maintain that much free land if our population continued to explode like the rest of the world.\" Turning back to Crystal, she says, \"But, let's return to the topic of Hadrian's sexual preference. We scorn heterosexuality because of the world population explosion. Even so...\" She stops and tries to sound pragmatic. \"No, let me ask this as a question. Crystal?\" The poor girl heaves a sob. She knows why she is being singled out. \"Will a woman get pregnant every time she has intercourse with a male?\" Although rhetorical, Ms. Sterne expects Crystal to answer her question anyway.\n\n\"No, ma'am.\"\n\n\"No, she won't.\" Unsympathetic, she demands, \"Look up when you speak.\" When Crystal complies, she exposes red, swollen eyes.\n\nWitnessing Crystal's distress causes Todd to drop his head on his desk. _What have I done?_\n\n\"Then, tell me, why have we made heterosexual behavior illegal?\"\n\nTodd groans, and for a brief moment, he rebels. Lifting his head defiantly, he demands, \"Why do we have to take this, again? Everyone knows this. We've been repeating this shit since grade one!\"\n\nThe class gasps. Ms. Sterne smiles grimly. This is the very sort of outburst she was hoping for from the boy. It is all the justification she needs for exposure. Actually, she has all the evidence necessary for exposure, but she is determined to land all responsibility on Todd's head. As for Crystal, an agreement was made with the girl's mothers to complete reeducation, privately, at home, not in a government camp. \"The very fact that you think this is 'shit,' Todd, is the very reason we repeat this lesson year after year.\" Looking now to the students of her class, Ms. Sterne lectures, \"As you are in the most vulnerable years, the time of sexual awakening and experimentation, it is imperative that we revisit our laws and the rationale behind them. Inevitably, it is during the teenage years that exposure occurs.\" Feigning sadness in her expression, Ms. Sterne now reveals the true intent of this lesson: \"Today, students, I must expose among you a male heterosexual.\" All eyes, following hers, shift wonderingly to Todd.\n\n\"You can't mean Todd,\" one boy pipes up. Devon, on the other hand, feels a sense of relief. He nods his head, no longer feeling inadequate. Looking Todd's way, his facial expression, though suggestive of pity, with his downturned, pierced lips, says, _So that's why you wouldn't let me._\n\nFrank is so flabbergasted that he can't even speak at first. How did this ever come out in the open? Finding his voice, he practically shouts with exasperation, \"Being a two is no crime!\"\n\n\"No, of course not.\" Ms. Sterne replies, not too kindly. \"But acting on those impulses is criminal.\"\n\nAs this statement begins to register in the students' minds, one boy pipes up, expressing similar confusion, \"But he's on the b-ball team,\" while a second exclaims, \"He's our best player!\"\n\nMs. Sterne, having anticipated the majority's claims, reminds the students of sports history. \"As we know, homosexuals were considered among the finest of athletes during the time of the first Olympics, but recent history shows otherwise. As many of you may not know, just prior to the sexual reformation, the sports industry used to be heavily dominated by heterosexuals. Many homosexual players were afraid to reveal their preferences due to the degree of prejudice against them. So,\" she says emphatically, \"it is really not so strange that we should find a _strai_ on a sports team.\"\n\nFrank turns to his friend. \"Todd, say something!\" Desperation and urgency are evident not only in his voice but in his wide open eyes. \"Defend yourself.\" Begging, he cries, \"Please!\"\n\n\"He can't,\" Ms. Sterne says sharply. Staring down on Todd, who has assumed the position of hiding his head under his arms, she states, \"He committed the act Saturday night.\" The class gasps, even Crystal. \"The act itself is on tape, captured by one of the school cameras.\" Todd groans. He had thought the camera in the back stairwell was broken. _It looks broken_. He feels like crying. Not only has he condemned himself, but he has also exposed the woman he loves. \"And that act,\" Ms. Sterne adds harshly, \"was forced upon Crystal!\" A chorus of girls gasp and three young women leap out of their desks, rushing to comfort Crystal. A few of the boys utter, \"Gross!\"\n\nTodd's head shoots up. \"What?\" He can't believe what he is hearing. He didn't force anything. _Crystal kissed me! She gave me the note!_ He can't bring himself to state these thoughts, though.\n\nAfter a brief moment, attempting to glare Todd back to hiding his face and failing in the attempt, Ms. Sterne turns on Crystal. In a voice like broken glass, she demands, \"Right, Crystal?\"\n\nTodd's head swivels. He knows he is looking at Crystal, but she is a blur, her image distorted by his tears.\n\nCrystal looks in Todd's direction, her head shaking in contradiction as she sobs out, \"Yes, ma'am.\"\n\n\"Tell the class what he did to you.\"\n\nTodd is stunned. He can't see, only hear, as Crystal's voice echoes over and over, the damning accusation. Suddenly he bursts out, \"If there's a video\u2014\" but Crystal's wail silences him. Wiping his eyes, he gives Crystal a closer inspection. She has a black eye. Someone has thrashed her good. _To make it look more like rape, no doubt_ , he reasons. _All the blame must rest on me._ Todd hides his face in his hands.\n\nNow yelling, almost hysterical, Ms. Sterne declares, \"I know it was forced; just look at her bruises. But Crystal refuses to accuse you! She says reeducation camp would be better.\" Turning away in disgust and then swinging back around on her heels, she shouts in Todd's ear, \"As far as I'm concerned, reeducation is too good for the likes of you!\" Now hateful, her voice dripping venom, she says, \"You deserve the death sentence.\" The class gasps in horror. Todd turns a ghostly white! \"That's right, children,\" Ms. Sterne states emphatically, now wanting solely to intimidate and terrify the lad, \"When a straight man rapes a woman in Hadrian, that act is punishable by death!\" Staring hard at the pale youth shirking from her glare, Ms. Sterne punches hard. \"Regardless of age!\" Once again, Ms. Sterne reduces Todd to tears. Sensing that the boy is crumpled and beaten, she demands, \"Do you admit to having\u2014\" She pauses briefly, choosing her words carefully, \"to being an active heterosexual?\" Todd shakes. \"Do. You. Admit. To. Being. An. Active. Heterosexual?\"\n\nShivering, Todd answers, \"Yes.\"\n\nStraightening, Ms. Sterne towers triumphantly over the crumpled youth. \"And now,\" in a voice dripping with satisfaction, she says, \"you can go down to the office where your Papa Mike is waiting to retrieve you.\" Todd looks up, gasping and pale. Ever since Todd's father died, Papa Mike has barely acknowledged his existence. Todd has no idea how the man will react. Although she doesn't say it, you can see in Ms. Sterne's eyes that she hopes the man will beat Todd senseless.\n\n* * * * *\n\nPapa Mike does not beat Todd. He doesn't even look at him. They drive in silence. They do not go home. Instead, Papa Mike drops Todd off at Riverside Park. Numerous thoughts race through Mike Fulton's brain. _Thank Hadrian, he doesn't bear my name or genes._ He wants to call Todd a fool for getting caught. Tell him he's a shame to his father's good name. But he doesn't. The only words he utters the entire time are when he stops the bubble and motions for Todd to get out: \"Never come near me again.\" The harsh whisper lashes Todd's heart.\n\nFrightened and abandoned, not knowing what else to do, Todd texts Frank and asks whether he can spend the night at his house.\n\nFrank texts back, \"Yes. I'll come get you.\"\n\n* * * * *\n\n#  _Salve!_\n\n**Another Brutal Attack  \nHNN\u2014Melissa Eagleton Reporting**\n\nAnother brutal attack, this time against the Mid-West Wall, has been reported. This morning, barely an hour prior to the change of guards, over one hundred outsiders drove at top speed in armored cars and tanks with the clear intention of ramming through Hadrian's walls. The hour of the attack was clearly meant to be strategic, our soldiers being exhausted after a full night guarding the wall. Although tired, our brave men and women were not lax in their response time. They moved swiftly in our people's defense. The alarm was rung as soon as the first vehicle was seen racing toward our border. Bazookas\u2014or as our soldiers like to refer to them, stovepipes\u2014were immediately used to counter the onslaught of tanks. That many of the front-line vehicles were suicide assaults with explosive devices meant to bring down the wall was evidenced when the first explosion occurred. \"The explosion that occurred when our first missile detonated was greater than that particular device was capable of.\" It was at that moment that Major Janice Cardinal knew drastic measures needed to be taken. \"There were far too many tanks heading toward our border,\" the major was reported as saying, \"for the one-man portable recoil rocket launchers to have the necessary impact. And those racing toward our wall were clearly heavily loaded with explosives.\" With no time to consider options, the major immediately determined that the only way to counter this attack was to bring out incendiary weapons. Major Cardinal, notorious for her military discipline, insists all planes be manned and ready whenever an alert is given. All pilots and bomber crews were at the ready seconds after the attack began. Thus, Major Cardinal was in position to launch an immediate counterattack and white phosphorous bombs were dropped on the attacking hordes, wiping out their entire contingent. Although it is not customary to use incendiary weapons during a battle, Lieutenant-General Birtwistle commented that these weapons are only utilized when all other attempts at aborting an enemy attack have failed. This particular attack against the wall was so sudden, and the barrage so imminent, that precious little time was available for debate. In the case of this morning's attack, however, Lieutenant-General Birtwistle is confident that the major in charge made the right decision. \"The enemy was racing toward us at such high speeds there was little time to deliberate. Had Major Cardinal not issued the order when she had, the wall surely would have been breached.\" Major Cardinal justified her action, by saying, \"The catastrophic explosions that occurred every time one of our missiles struck an oncoming vehicle was evidence enough. If even one of those tanks had hit our wall, it would have come crumbling down. I could not allow that to happen!\" We can all thank Hadrian for Major Janice Cardinal's fast thinking and instantaneous leap into action.\n\nThis latest attack is yet again a reminder of what we have achieved in Hadrian and how much we have to lose if the hordes outside our walls ever break through. Every Hadrian citizen is employed, clothed, fed, and sheltered. Without the burden of overpopulation, we are able to ensure a healthy lifestyle for all our citizens. Anyone who cannot afford luxuries like voc implants can still have a government installed wall screen with access to the central wave at minimal cost. No one is disconnected, and no one goes without in Hadrian.\n\nAnd yet, our greatest achievements are still being threatened by the desperate and jealous hordes outside our wall. As much as we may pity their misfortunes, we cannot allow any man, woman, or child to cross our borders. The purity of humanity is in our hands. The cleansing of the planet is a responsibility we have chosen. To allow members of the outside races to fester inside our walls would be to destroy everything Hadrian has so far accomplished. We will never allow that to happen. As a result of our refusal to allow the plague-ridden, overpopulated masses to enter our borders, we are constantly battling back starvation-crazed hordes, slamming against our walls like tempest tossed waves.\n\nHadrian's military is sending out the call to all of Hadrian's citizens between the ages of twenty-two and forty to reenlist and help reinforce our strength against the heterosexual hordes constantly ramming our walls. The future of Hadrian as a self-sustaining country committed to protecting the planet and the human species is critical! We must triple our military strength and reinforce and continue building our border wall. Hadrian's wall needs to be deeper, taller, and to span our entire country's edge. Seriously consider rejoining our forces to help Hadrian's military continue successfully to thrust back those who are determined to steal and destroy all that we have accomplished.\n\n_Vale!_\n\n# Taming the Strai\n\nAs Frank prepares to go fetch Todd, he realizes it is fortuitous for him that his fathers are going out. Monday night is their date night. Geoffrey proposed this standing date when Dean and he were first wed. He believed, quite rightly, that in order to develop and maintain a close loving relationship with Dean, they needed to commit to spending time together on a romantic level. Thus, for over twenty years, the two men have dedicated their Monday nights to their mutual amusement.\n\nUsually looking forward to their private night out, on this particular Monday, Dean is far too disgruntled to enjoy the upcoming dinner and a movie. Frank's voc call this morning about Todd was worrisome. Dean had called Geoffrey on the instant and insisted he come home from work. Dean had rightly judged Todd's circumstances to be an emergency.\n\nOnce Frank had gotten home from school and Geoffrey had arrived, the three men had gathered in the kitchen to discuss the situation. Frank had many questions about what might happen to Todd, which led to Dean and Geoffrey revealing Dean's family history, sparing nothing, to Frank. Shocked, Frank learned that he is, by his genetic father's marriage, a member of a founding family. Papa Dean was a Stuttgart. _Was_ being the operative word. What should have been an illustrious status is, in their circumstances, a more than frowned upon association. To reveal Papa Dean's founding family genes would cause the Hunters to be shunned by all good society. In fact, sometime after Dean's exposure, his father had put into words the machinations of a story depicting the tragic death of his son in a devastating accident. According to the police report, Dean Stuttgart had been driving his father's bubble when he hit a moose. Everyone knows the bubble vehicle is fragile compared with the older car models that once pumped toxic carbon dioxide into the air, contributing to global warming. It was deemed an acceptable risk to drive the smaller, lighter electric vehicle, even though crashes were far more lethal. It actually encouraged safe driving. As the story goes, Dean Stuttgart chose to take a Sunday drive into the forest just north of Antinous city. There he rammed a moose, dying on impact. There was to be no challenging this story and Dean never tried. The death of his only son opened the door for Dean's father to apply for another child. Dean has a genetic little sister he has never met\u2014whose name he has never even known.\n\nJust as Dean and Geoffrey finished telling Frank this information, Frank's voc had received the call from Todd that Mike had abandoned him at Riverside Park. Dean had wanted to be the one to collect the boy, but both Frank and Geoffrey insisted Frank go. Since Frank was the one Todd had called, Frank was the one to go get him. Geoffrey and Dean argued heatedly over this, but Frank and Geoffrey eventually won.\n\nFrank, able to sense his Papa Dean's dissatisfaction, reiterated his position, \"What Todd needs right now is a friend.\" Unable to sway Papa Dean, he tried again, \"I need to talk to him alone if I'm going to make him realize he and I need to start dating right now.\"\n\n\"You don't even really have to date, Frank,\" Dean tried hard to explain. \"People just need to believe the two of you are a unit.\" Worried that Todd is too unstable right now for a lover's bond, Dean strongly encourages Frank only to offer up a pretend relationship. \"Just act it, like Will and I used to.\"\n\n_Yeah_ , Frank muses, now getting ready to leave, _and that really helped you, didn't it?_\n\n\"Let him know,\" Dean persists. \"Let him know you are there for him, will wait for him, won't abandon him. Make sure he understands we won't ever abandon him.\" When he turns his bleary eyes to Geoffrey, he is startled at the sight of the man standing stock still, seemingly so impenetrable, resistant to all pity. Dean pleads uselessly, \"Geoffrey, please, what the boy needs right now is adult guidance.\"\n\nGeoffrey is not a cruel man so it pains him to see Dean in such agony, but he also knows that Dean is in no condition to help Todd. He has a lot of calming down to do because his own demons have resurfaced. Dean is Geoffrey's top priority. \"Too many haunting memories have surfaced. I need to help you settle those first.\" Geoffrey truly believes it is in Todd's best interest to let Frank fetch him. \"Frank is correct, Dean. If the boys are to pretend to be lovers\u2014\"\n\n\"They don't even have to pretend to be lovers,\" Dean cuts him off. \"They just have to act like they are dating.\"\n\n\"No, Dean,\" Geoffrey counters, shaking his head sadly.\n\n\"Yes, we do Papa Dean,\" Frank insists. \"You know we do.\"\n\nReasserting Frank's position, Geoffrey states, \"Todd confessed to being sexually active. If Frank is going to convince others he has tamed him, people are going to have to believe they had sex.\" Sighing, wishing there were another way around this, he adds, \"To do that, the two boys will need to be alone.\" Before Dean can protest, Geoffrey silences him. \"Dean, enough.\" Geoffrey's tone is stern. He is in command mode and there is no reversing him. \"The boys need privacy in order to talk this through. We are going to give them that necessary time alone.\" Gripping Dean's arm, he states, \"Both boys are mature enough, close enough to being men that they can deal with this, and will deal with this, a lot more effectively alone. You and I will only muddle matters.\" Sighing, he adds, \"Besides, you are in no condition right now to offer anyone moral support.\" Then conceding slightly, he concludes, \"We won't stay out long. We'll just go for dinner and be home by seven.\"\n\n\"Make it eight,\" Frank says. He knows what he needs to do, and he wants more time than their being home by seven will allow.\n\n\"All right,\" Dean reluctantly agrees. \"But only dinner and,\" staring defiantly at Frank, he states, \"we'll be back by seven.\"\n\n\"Absolutely,\" Geoffrey nods, but as soon as Dean's back is turned, he reassures Frank by mouthing, \"eight.\"\n\nFrank is relieved when the two men finally leave. Todd and he need privacy all right. Frank knows he and Todd need to do more than pretend.\n\n* * * * *\n\nAs Frank emerges from his bedroom, he forms a wry, telling grin on his face. Devon and Roger are in the living room, as expected, necking on the couch. Roger had promised to invite his boyfriend over so Frank could prove unequivocally that Todd was no longer a _strai_. Frank knows they need a witness beyond himself. There is simply too much evidence against his best friend. And since they are best friends, everyone expects Frank to lie. He stood up for Todd in class today, before Ms. Sterne exposed him, Crystal denounced him, and Todd foolishly admitted to being straight! Todd straight! It was inconceivable, but there you have it! _Really, Frank?_ he berates himself. _Inconceivable?_ _When you've known all along?_ Frank shudders; everything he's ever been taught about straight males is heinous: pedophiles, rapists, chauvinists, and all stupid, smelly, messy, dirty pigs! None of that describes Todd. _Okay,_ he admits, _he isn't the neatest individual_. The fact is Todd's clothes are seldom tidy. _But he's poor_ , Frank insists. _We can excuse him that_ \u2014but not anymore. Not after his admission in class. _Even so_ , Frank reasons, _Todd is no rapist! There is no way he raped Crystal! No way!_ But not even Frank can deny Todd's declaration, and that confession is all the damning evidence anyone needs. Ms. Sterne has even suggested Todd deserves the death penalty. The memory of her implied threat causes Frank to wince. _Again_ , he asks, _why do they have to be so cruel about exposure?_ But he knows why. It really is the only way to warn _strais_. _Straight men, especially_ , he reasons, _are difficult to tame_. Feeling a surge of pride, yet still rueful, he thinks, _Todd may have been a strai, but not anymore_. _I've made sure of that!_\n\nAs soon as Frank walks into the living room, Devon begins admiring Frank's physique. Frank has purposely left off his shirt, and he only begins zipping up his jeans after he enters the living room. Everything about him from tousled hair, to his bruised and sweaty chest, to being half-dressed has to suggest sex. Everyone must know\u2014no _believe_ \u2014that what just happened really _did_ happen. Todd is a confessed _strai_ , but Frank means for everyone to understand that he is just confused! Still, this is not going to be easy. Todd is definitely not what everyone perceives to be Frank's type. Frank only dates the more effeminate boys\u2014boys he can easily manipulate\u2014boys who like to pamper him and treat him like a god. That is definitely not Todd. Todd, like Frank, has the bearing of the alpha male\u2014especially on the b-ball court. Unfortunately, Todd never dated anyone but Devon. And Devon certainly didn't help matters when he announced that Todd's exposure explains everything. He added even more damaging evidence when he said Todd never once initiated and was always the first to pull back from a kiss. _That's Todd's biggest problem_ , Frank surmises. Everyone expects him to be the aggressor, but he has never once approached any boy on campus. Everyone now believes that if Devon hadn't approached Todd, he never would have dated anyone. Some even say he used Devon to throw people off the scent of his being a _strai_. Even that Todd is the official star of the b-ball team holds no sway in his favor anymore, not after what Ms. Sterne said about sports and heterosexuals in class. Oddly enough, Frank muses over Todd's position on the team. _He should have been co-cap, not me_ , but Frank surmises that his being the taller and stronger got him the position. There is no doubt about it; Frank is definitely stronger than Todd, height and extra weight being his advantage. _Thank Hadrian for that_ , Frank muses as he studies the raw skin of his knuckles. Licking off the blood, he remembers how hard Todd fought. It was not what Frank wanted, but Todd had to be subdued. _It had to happen_ , he reminds himself again, though it feels more like a reprimand than justification.\n\nRoger smiles. He knew if anyone could tame a _strai_ , it was Frank. \"So you guys did it?\" he asks half-expectant, half-hopeful. Roger likes Todd, seeing him as a part of their family. The last thing he wants is to see Todd sent away for reeducation. Frank grins slyly. He is wearing his \"I just got laid look.\" Devon is not falling for it, though. Todd's exposure today redeems for him what he had seen as his own personal failure. Besides, Todd Middleton had kissed a girl, had straight sex with that girl\u2014Ms. Sterne even implied rape. _No_ , he says silently, _Todd is straight, and as far as I'm concerned, there is no taming that kind of man. Besides_ , he reaffirms, _if I couldn't get him to do it, no one can!_ Turning to Roger, he laughs, \"You don't really believe they did it, do you?\" Scorn ripples across his face. Looking back at Frank, he says, \"You can't possibly expect people to fall for this? No one is ever going to believe Middleton went down for you. He's a _strai_ \u2014a fucking little cunt hammer. And, if what Ms. Sterne suggests is true\u2014a rapist!\"\n\n\"Todd never raped anyone!\"\n\nIgnoring Frank's outburst, spitting out derision, Devon exclaims, \"They ought to lock the cunt hammer away!\" Amused by his own consideration, he adds, \"Frankly, Frank, I don't know why they don't just castrate the bastard.\"\n\nRoger, caving to Devon's perspective, begins to worry. \"Did it really happen?\" His eyes squint as his shoulders shrug upwards in quandary.\n\nFrank grimaces; he had expected as much, which is why he had insisted on making it really happen, and not just be a pretend act like Todd had begged him. Even so, he couldn't help feeling betrayed. _Roger could at least back me_ , he thinks grimly. Crossing over to the couch, he sits next to Devon. The two men stare down until Devon looks away. Then Frank calls out, \"Todd!\" When there is no response, he looks toward the hallway leading to his bedroom. \"Todd,\" he says more urgently, a hint of anger in his voice, \"get out here.\" Devon and Roger stare expectantly at the hallway. A smothered cry emerges from Frank's room. \"Now, baby,\" Frank states emphatically. \"Your daddy's calling, so come!\" Another moan is emitted prior to the subtle sound of a door handle turning, followed by the slight creak of a door beginning to open. Frank offers up his grin to Devon and Roger. \"Did you put on the clothes I bought you?\" A stifled cry replies. \"Come on, baby; I want to see you dressed nice.\" The door closes with barely a sound. Devon and Roger look at each other curiously. \"I've given him a new look,\" Frank replies matter-of-factly.\n\nActually, the impromptu shopping spree was in reaction to Ms. Sterne's initial attack on Todd's appearance. The whole time Frank was picking out clothes and throwing them Todd's way, Todd kept throwing them back swearing, \"There is no way I am ever going to wear these things!\"\n\nFrank whirled on him, insisting, \"We have to convince people you really are gay!\" Frank won that war, too. \"You are going to look like one of my boys and that's that!\" There was no more discussion to be had. Frank bought the clothes, and then, as soon as they got home, dragged Todd into his bedroom.\n\nOnce again, Devon and Roger stare intently at the hallway, listening to the muffled sounds of shuffling from inside Frank's bedroom. Soon enough, the sounds of a moving door handle followed by the opening of the door can be heard\u2014a little less faint than last time. _Good, he's starting to accept_ , Frank muses. The shadow of a body walking tight against the wall appears. It is as if Todd is trying to push himself deep inside the wood in a desperate attempt to escape. Slowly, Todd comes out. He can't enter the living room, though, and stays glued to the corner where the hall and the living room meet. The black velvet wallpaper, though elegant in design, feels repulsive against his skin. Frank has him wearing a tight short-sleeve hot pink T-shirt accenting Todd's muscular torso. Frank wanted this effect. It would make Devon drool, and anyone who could make Devon hard was someone of whom he would approve. Todd's sudden transformation into a meek demeanor is very erotic.\n\n\"Wow,\" Devon mutters appreciatively. \"You can really make out his package in those jeans.\" Todd's hands instinctively drop down to hide himself. Frank also insisted Todd wear a pair of skin-tight jeans with stovepipe pant legs, unnecessarily held up with a hot pink belt to match the T-shirt. To ensure his dominance, the only way people would believe Todd tamed, Frank had also demanded Todd wear Teika's dog collar made of thick purple leather with a huge ring for a leash. All of Frank's boyfriends have worn this collar, even at school. It is like being given his ring or sweater, and it lets everyone else know he, Frank Hunter, is the sole proprietor of this boy.\n\nFrank leans back casually, opens his lap slightly, and pats his left knee. \"Ignore him, Todd. Come to Daddy,\" he invites seductively, patting his knee a second time. Every word, every action, every physical image must literally reek of sex if he is going to save his best friend. \"Come on, Todd,\" Frank urges. It is hard for Todd to move, but he manages to release his grip on the wall. Walking is uncomfortable as the tight jeans chafe against him.\n\n\"Parade for us,\" Devon taunts. \"Get him to spin, Frank.\" Devon is enjoying the show. Todd can't even look up to beg Frank not to make him spin around. He just stops and shakes in the center of the room.\n\n\"Shut up,\" Frank orders Devon. \"Don't worry, baby,\" he coos softly. \"Just come to Daddy.\" Todd slowly begins his death march. \"There's Daddy's boy.\" With Todd standing in front of him, still sheltering himself with his hands, Frank gently turns him around and then roughly pulls him down on his lap. Todd winces. _They have to know he's in pain_ , Frank reminds himself. \" _I must be cruel only to be kind,\" as Shakespeare aptly put it_. Frank hates treating Todd so roughly, but word has to get out that Todd has been tamed\u2014that he is no longer straight\u2014that Frank has made him one of them. Tears bloom in Todd's eyes\u2014tears he has been fighting back since he managed to stop the flow after what Frank did to him. Frank gently wipes the tears from Todd's face, caressing the bruises and his cut, now swelling, lower lip. \"Who's your daddy?\" Frank asks in a teasing manner. Frank is now caressing Todd's legs, squeezing them periodically, and allowing his fingers slowly to climb higher. One of Todd's hands flaps like a bird with a broken wing in an attempt to arrest Frank's movements. Frank grips Todd's hand tightly in his, hard enough to indent Todd's fingernails into his palm. Lowering his voice so only Todd can hear, Frank whispers, \"Answer me.\"\n\nTodd mumbles, \"You.\"\n\nFrank's whisper becomes threatening, \"Louder!\"\n\nObeying, Todd replies, \"You, Frank!\" a little too loudly, enough to cause raucous laughter from Devon and Roger.\n\nEven Frank joins in with a chuckle as he pats Todd's head. \"Good boy.\" Frank kisses Todd as a reward.\n\nSneering, Devon remains skeptical. \"Listen, Frank; play this up all you want, but nobody's going to believe you planked this cunt-hammer.\"\n\nAfter glaring Devon's way, Frank turns to look at him. Todd knows exactly what Frank is thinking: _You have to kiss me_. Leaning in to initiate Todd is startled when Frank stops him. \"Ah, ah, ah, you didn't ask.\"\n\nA shudder precedes Todd's request. \"Frank, kiss me.\"\n\nSlipping one finger into the collar ring, Frank tugs slightly, reminding Todd of his new place in the pecking order. \"Say, Frank, may I kiss you, please.\" Todd quakes. Unrelenting, Frank warns, \"I won't do it if you don't beg,\" pulling a little harder on the collar ring.\n\n\"Frank,\" says Todd. Although it is a low whisper, Frank knows Devon can still hear, \"may I please kiss you.\"\n\nLooking straight at Devon, Frank replies, \"Of course you may, baby.\" Turning back to look at Todd, Frank waits for him to initiate. When their lips unite Devon and Roger slap hands and clasp fingers.\n\n\"By all that's gay and glorious,\" Devon chimes, \"I think you've really done it.\"\n\nRoger, congratulates his brother, \"Good work, Frank!\"\n\n\"Now get the fuck out of here,\" Frank orders before he kisses the top of Todd's head. \"My boy and I want a little alone time.\"\n\nDevon and Roger leave quickly, but not before Devon can call out one last derogatory remark. \"One minute a cunt hammer, and in the very next, Frank's buggering board.\" Chuckling, he adds as he exits, \"Impressive.\"\n\n\"Devon, you can be so crude sometimes,\" Roger says as he pulls his mate through the front door.\n\nFrank waits a few moments after the front door closes behind the two young men. Expecting Todd to dart away any moment, he wraps his legs and arms tightly around his friend. He wants so badly to cry, to beg forgiveness, but he knows he can't. If he is to tame Todd successfully, he can't back down. Not now. Not ever. Papa Dean taught him that. Frank explains, \"We had to do it, Todd; you know that, don't you?\" His muscles begin to shake from holding Todd so tight. \"You know that, right?\"\n\nTodd gives Frank the answer he knows he wants to hear. \"Yes, Frank.\" Frank's relief is audible in his sigh. No longer fearing Todd might try to escape, Frank loosens his grip. He does not release Todd, though, as the need to hold him close is too great. Although Frank tries to hold them back, tears begin to stream down his face. Todd doesn't even notice; his eyes are glued shut by shame and exhaustion. \"This isn't how I imagined us.\" Frank releases one arm to wipe the tears from his face. \"I always knew it'd be you and me one day\u2014but I was willing to wait, till you were ready. I didn't want to\u2014\" He couldn't say it. \"I wanted you to want me, too.\"\n\n\"Why, Frank?\" Todd's voice is a shattered whisper. \"Why did it have to happen?\" It was a contemplative moment, solemn and sore with rage.\n\n\"Papa Dean was a _strai_ ,\" Frank says, looking down at Todd. \"Did you know that?\" He harrumphs. \"My father tamed him\u2014at a reeducation camp. Dad took a summer job there. That's where he and Papa Dean met. Said he wanted to do something to help the nation\u2014said he actually believed taming _strais_ was essential for national security.\" Frank shakes his head at the wonder of it. \"Said he had actually fallen for that claptrap.\" Resting his head in his free hand, Frank continues, \"When I voc'd home, Papa Dean insisted I come straight home. When I got here, Dad was with him; he left work early because this was so important. The two of them sat me down and told me what reeducation camp is like\u2014what had happened to Papa Dean\u2014what will happen to you. They said we needed to make people believe we are a unit\u2014a sexually active unit. That this was something we had to do.\" Frank swallows his guilt. His fathers had only said to make people believe they are a unit; they never actually said to \"do it.\" _But we had to_ , Frank reminds himself. Needing justification, he adds, _This was the only way anyone would believe us_. Openly sobbing now, he begs, \"Believe me, Todd; it was better this way.\"\n\nAt this moment, Dean and Geoffrey return home. Because the entry hall opens into the living room, the first thing Dean sees is Todd sitting on Frank's lap, wrapped in his arms, both boys sobbing. Instinct takes over and he rushes to them. \"Frank, Todd, what happened?\" Quickly taking in the bruises on both boys' faces, he demands, \"Hadrian's lover, what happened?\" Looking his husband's way, he exclaims, \"Geoffrey, they've been beaten!\"\n\nGeoffrey moves into the room to stand behind Dean. Frank instantly begins to babble some story about _strai_ bashing. Dean cringes as he listens to a gruesome tale about a gang of boys from school jumping Todd at Riverside Park and Frank leaping in to save him. Dean, kneeling in front of the boys, opens his arms to envelop Todd as he slides off Frank's lap. Papa Dean cradles and rocks the boy like an infant. Geoffrey takes great care to observe the scene before him. Frank is shirtless. Todd is dressed like T'Neal\u2014no, like the way Mike Fulton said he had been dressed that Sunday over three years ago when the boys had first experimented with sex. \"Dean,\" Geoffrey places a hand on his partner's shoulder, \"you take care of Todd. I'll look after Frank.\" Frank, Geoffrey also notes, is avoiding eye contact. \"Frank,\" says Geoffrey, his voice mimicking soft and soothing, but Frank can hear the edge of displeasure. The instant their eyes meet, Frank is conscious of his father's awareness. His father, Frank rightly surmises, is not as easily overwhelmed by the current circumstances\u2014empathetic, yes; fooled, no. Having borne witness to Dean's suffering, seeing now Todd crumpled and broken, he wonders at the depth of emotions Dean and Todd are forced to endure. \"Frank,\" he repeats, \"come with me.\" As Geoffrey's eyes brook no opposition, Frank rises and slowly follows his father down the hall and into his bedroom.\n\nAfter taking in the damage done\u2014the bed curtains ripped off\u2014one of the curtain cords tied to the front right poster\u2014Geoffrey turns and confronts his son. \"Tell me what really happened!\"\n\n* * * * *\n\n#  _Salve!_\n\n**Spotlight: Gideon Weller!  \nHNN\u2014Melissa Eagleton Reporting**\n\n\"Tonight's episode is unique in that the guest I am interviewing is actually present with us on stage. It is with great pleasure that I introduce to my viewers the warden of the Northeast Reeducation Camp, Mr. Gideon Weller. Mr. Weller, thank you so much for taking time out of your rigorous schedule to speak with us tonight.\"\n\n\"Well, Ms. Eagleton\u2014\"\n\n\"Please, call me Melissa.\"\n\n\"All right, Melissa. It is an honor for both myself and for the Northeast Camp that you have offered me this interview.\"\n\n\"First off, our viewers would like to know what it is you do at the reeducation camp.\"\n\n\"My job is quite hefty. I oversee all stages of each ward's transition back into Hadrian society. When they first come to us, many are unruly, undisciplined, angry, and hurtful youth. It is the job of all who work at the reeducation camp from the warden all the way down to the janitorial staff to help encourage these boys to embrace Hadrian's lifestyle.\"\n\n\"That must be a very daunting task if they come to you as unruly as you suggest.\"\n\n\"Indeed it is, and indeed they do. Remember, these are boys who believe they are heterosexual. Many are a two on the Kinsey scale, so we have to help them battle against and then vanquish the stronger heterosexual drive. It is essential we cleanse them of their heterosexual tendencies before we can help them find their inner latent homosexuality.\"\n\n\"And how is that done?\"\n\n\"We have a very tight schedule by which the boys must abide. From the minute they wake to the very minute they return to their beds, they are kept active in sports, classes, private counseling sessions, and other activities deemed necessary for their reeducation.\"\n\n\"How long is their day?\"\n\n\"Our boys rise at six a.m. and bed down at nine p.m.\"\n\n\"Nine is a little early, don't you think?\"\n\n\"Not after the rigorous day we put them through. By nine o'clock, many of our boys are so tired they collapse as soon as they sink into their mattresses.\"\n\n\"How long is your day?\"\n\n\"My day begins at five a.m. I need to wake a good hour before the boys in order to ensure everything is in readiness for the day's activities. I, and all our staff, then join the wards for breakfast and exercise. As soon as the boys begin class, I head over to my office and continue working through all the paperwork that comes along with each ward, and not just the wards currently in my possession. No, no, no. At the Northeast Camp, we keep track of all our wards after they leave us. We like to know about their successes in life. Husbands report back to us on an annual basis. I enjoy reading those reports most of all. Often, I will share these reports with our new wards so they know the hope and happiness that await them in the future.\"\n\n\"That sounds wonderful. How uplifting. How inspiring that must be for these young men.\"\n\n\"For some, yes, depending on what stage of their reformation they are at. It is always delightful, though, when the wards are nearing graduation. These young men truly appreciate news of their predecessors' fates.\"\n\n\"So, is that the end of your day, then?\"\n\n\"Oh, no, no, no, not by any stretch. The morning is barely over for me by this point. I always dine with the boys, as I mentioned\u2014breakfast, lunch, and supper. Everyone in the camp comes together for meals. I like the boys at Northeast Camp to feel like we are a family. As you may know, many of our youth have been disowned by their real families so we embrace them as our own.\"\n\n\"That is the best way to win our children over, I think.\"\n\n\"So, after lunch, I will go around the camp and participate in various events. All wards have private sessions with their guardians, and sometimes, the charge, or the ward, needs a third ear to listen and help out with difficulties or concerns.\"\n\n\"My word, you are a truly amazing man.\"\n\n\"Thank you, Ms. Eag...Melissa. As I was saying, my days are as full as the boys' days are. I will sometimes join them in viewing films about the outside world or help them learn to reject what they perceive as feminine seduction.\"\n\n\"So far everything suggests an easy time for these young men.\"\n\n\"Oh, no, no, no. Don't be fooled. Reeducation for the straight male is never an easy process. Disciplinary measures are often necessary. Remember, heterosexual males are violent by nature, and one is often caught in a situation where offense is the best defense.\"\n\n\"Of course. Do you suppose it is the same at the young women's reeducation camp?\"\n\n\"I have no idea what our female counterparts have to deal with. What I do know is that heterosexual males are among the most volatile and dangerous people in Hadrian. I have never forgotten the brutal attack on the wall over twenty years ago.\"\n\n\"Oh, yes. That was horrendous.\"\n\n\"The woman who survived\u2014\"\n\n\"You knew her?\"\n\n\"She was my genetic mother. A kinder, more beautiful woman never existed. Those brutal, bloody barbarians, what they did to her\u2014the scars that would never heal. Those bastards\u2014those rapists\u2014those murdering sons of\u2014\"\n\n\"Of course, of course, what those men did was horrible.\"\n\n\"They are the epitome of heterosexuality, and they are why we must never allow any of our young men to accept or act upon such unruly physical emotions.\"\n\n\"No, of course, you are right. I think what our viewers need to know is that the Northeast Camp has one of the highest success rates at retraining and transitioning our wayward young men back into Hadrian society.\"\n\n\"Yes. I am committed to bringing our boys back to the only truly loving and kind sexual lifestyle. No boy leaves my camp with any inkling of heterosexual tendencies. Until I am one hundred percent certain a ward can live in harmony with Hadrian's chosen lifestyle, he will not leave the confines of my camp. As long as there is any indication of heterosexual tendencies, I consider him to be a threat to Hadrian's citizens and our national security.\"\n\n\"Please explain for our viewers how heterosexuals might affect national security.\"\n\n\"Heterosexuals need to commune with heterosexuals, so these boys will do anything in their power to aid the hordes outside our wall to enter into Hadrian. We all know the story of how it was one of our very own soldiers, who had admitted to being straight, who aided the horde that attacked our wall, murdered our men, raped our women, and killed the soul and sanity of my mother. Every heterosexual is a danger to our society. They will try to make contact with the outside world and bring them in to destroy our guarded and cherished lifestyle.\"\n\n\"No more need be said, Mr. Weller. You are clearly a passionate and dedicated man. Your devotion to the reeducation of our young men and the protection of our cherished society is admirable. And, although viewers, there was never any hard evidence against the accused soldier who also died in the attack, we can surely understand Mr. Gideon Weller's point of view. Mr. Weller, thank you again for sharing your valuable time on _Salve!_ You, sir, are our spotlight of hope!\"\n\n_Vale!_\n\n# The Principal's Office\n\n_Fank!_ Todd's text is urgent, his typing frantic. He doesn't even take the time to read what he's written. _ppal clled. papa deana dn me. offce. meetig help_. Frank understands everything. Raising his hand, he asks Mr. Reiner whether he can go to the bathroom. Permission is given and Frank races down the stairs, up the hallway, and into the office. He practically slams into Coach Miller, who is leaning rather dejectedly against the doorjamb.\n\n\"Coach?\" Frank's eyes plead with the woman.\n\nShe shakes her head, grimly pronouncing, \"It's bad, Frank.\" Tears begin to spill. \"Why?\" she asks, seemingly of Frank but really of no one. When she gets no response (not that she had expected one), she continues, \"It's so wrong.\" Her hand caresses her forehead. \"Who gives a damn if he's straight!\" She has to put a hand over her mouth to stifle her outburst of tears. Chewing now on the knuckle of her thumb, Coach Miller shakes her head in dismay.\n\nThe secretary is not impressed. \"Being straight is illegal!\" he declares.\n\n\"Why?\" Coach Miller turns and demands of the younger man.\n\n\"Population control!\" He rolls his eyes as if talking to a dummy. \"Everyone knows that.\"\n\n\"So give him a vasectomy! SNIP!\" Her gesture, so sharply pointed in Mr. Whalen's direction, suggests she would like to snip him. \"Problem solved.\"\n\n_Stupid woman!_ He sniffs in contempt as his eyes roll upward. \"They'll do that to him too, I believe. It's standard procedure upon entering reeducation.\"\n\n\"No!\" Frank whispers in despair.\n\nMr. Whalen, paying the young man no heed, taunts the teacher. \"You just don't want to lose your star player!\" Sniffing, he adds sanctimoniously, \"You should be thinking of your students' needs, not those of the game.\"\n\nCoach Miller desperately wants to shout back _FUCK YOU!_ But because they are standing in the main office with a student present, she has to content herself with a sullen glare.\n\nMr. Whalen decides he has put up with this woman long enough. \"Don't you have work to do?\"\n\nCoach Miller wants to scream out, _You're not my boss!_ She seethes instead, knowing better. The school secretary may be lower on the employee hierarchy, but when it comes to political clout in the office, this man wields all the power. Turning, Coach Miller attempts to walk away with dignity, but her stride clearly indicates defeat.\n\nFrank instantly runs to the secretary's desk. \"Mr. Whalen, please, I need to see the principal.\"\n\n\"Incoming,\" Mr. Whalen chimes. Waving his hand impatiently at the youth, he blinks to answer the call. Taking his time with the call, Mr. Whalen records all information in infinite detail before blinking off. Finally, he looks Frank's way. \"Yes? May I help you?\" There is a slight edge of annoyance in his voice.\n\n\"I need to speak with the principal. Where is he?\"\n\nGrimacing, Mr. Whalen points to the chairs against the sidewall. \"Sit down, Frank.\" Being co-cap of the National Championship team, Frank is known by everyone at Pride.\n\n\"I don't want to sit down. I want to talk to the principal.\"\n\n\"Well, Mr. Gavin is busy.\" Pointing with eyes to the man's office door, Mr. Whalen says, \"He's in a meeting.\"\n\n\"I know,\" Frank replies. \"I need to get in there.\"\n\nAnnoyed, Mr. Whalen replies, \"It's an important meeting. He's not to be disturbed.\" Gesturing to the chairs, he says, \"Sit down and wait.\"\n\n\"I'm not waiting.\" Frank rushes to the door but is stalled by a locked handle.\n\nExasperated, Mr. Whalen calls from his desk, \"Sit down, Frank.\"\n\n\"No! I want in.\" Frank starts to knock, his knuckles rapping harder against the wood as his request is blatantly being ignored.\n\nFinally, the door opens and Mr. Gavin pushes his head past Frank. \"Whalen, I said no interruptions.\" Mr. Whalen, raising eyes and eyebrows, complementing the look with chagrin, flicks open palms, then nods toward Frank's back.\n\nFrank, meanwhile, pushes past the principal into the office to kneel next to Todd's chair. Terror strikes deep into his heart at the sight of the strange man sitting beside his friend. Although in his fifties, there are no lines of age etching this man's face. Nor is there any suggestion of cosmetic surgery. His hair, however, has been dyed black, so black the look is unnatural against the pale white of his skin. His legs are crossed and his hands are folded one atop the other over his knee. His back is so straight he looks more like a mannequin than a human being. Papa Dean stands behind Todd's chair, his hands on the boy's shoulders. Todd is bent over crying. \"Papa Dean,\" Frank asks dismayed, \"what's going on here?\"\n\nThe strange man silences Papa Dean with one darting look. Then, cocking an eyebrow, slightly amused by this new turn of events, he asks, \"And who might you be?\" His voice is soft, soothing, and sickeningly sweet.\n\n\"Who the fuck are you?\" Frank demands.\n\n\"Such language! Really, Mr. Gavin,\" the man addresses the principal, who has followed Frank into the office, \"the way your students speak to their elders.\"\n\n\"Frank,\" Mr. Gavin is clearly upset, \"I must ask you to leave.\" He holds the door open and motions Frank toward the door.\n\nGripping Todd's hand in his, Frank declares, \"I'm not leaving my boyfriend.\"\n\nThe stranger snorts. \"Your boyfriend, you say?\"\n\nSmiling grimly, Mr. Gavin asks, \"Since when?\"\n\n\"Since yesterday. See this,\" he says emphatically, pointing to Todd's neck, \"this is my collar. Only my boyfriend wears it.\"\n\n\"Yes,\" Mr. Gavin is curt, \"and only yesterday T'Neal was wearing it.\"\n\n\"We broke up.\" Frank is equally brusque.\n\n\"That was sudden,\" Mr. Gavin replies pointedly.\n\nThe strange man is clearly amused by all this. \"Well, isn't this delightful.\" Swiveling in his chair, he addresses the principal. \"Mr. Gavin, introduce me to Todd's\u2014boyfriend.\" The pause serves to amplify his sarcasm.\n\nReleasing a long sigh, begrudgingly accepting Frank's presence, Mr. Gavin closes his office door. Before resuming his seat, he begins, \"Mr. Weller, Frank Hunter\u2014apparently Todd's boyfriend.\" He, too, expresses doubt with acerbic slurring. After sitting down, he swivels in his chair and resumes working at his computer, determined to shut out everything that has to happen here. _This is not my business_ , he reminds himself. And just last week, he had taken Todd's hand in his, before the entire school body, congratulating him on making the winning basket and receiving the well-earned most valuable player award at the Nationals. _This is a government affair_ , he adds in a feeble attempt to justify his act of betrayal.\n\n\"Amazing!\" Mr. Weller claps his hands with mock glee. \"One day a confessed heterosexual\u2014an active one to boot\u2014and the next day he is one hundred percent gay.\" Getting up out of his chair, he extends a hand Frank's way. \"Let me shake your hand, Frank Hunter; your accomplishment in taming this boy is truly amazing.\" Looking with delight Dean's way, he continues, \"You have even outdone your father!\" Looking back at Frank, he adds, \"Why, in less than twenty-four hours, you have accomplished what takes reeducation camp no less than six months and sometimes up to four years to achieve.\"\n\nFrank stands, irate; he pushes Weller's hand away. \"I didn't say he was fully tame, but I am taming him. He's agreed to let me tame him.\"\n\nPapa Dean pipes up. \"If the girl can receive reeducation at home, why can't the boy?\"\n\n\"Oh, Dean, Dean, Dean.\" Leaning over Todd's slumped shoulders, Weller taps Papa Dean's chest. \"You of all people should know the answer to that.\" Turning to face Frank, he continues, \"Yes, young man, your Papa Dean is a confessed _strai_ \u2014sorry, bad language\u2014heterosexual\u2014a 'zero' on the Kinsey scale in his own words.\" Weller pauses to feign ignorance. \"What exactly did you say, again, Dean?\" And, then as if the answer suddenly strikes him like a bolt out of the blue, he exclaims, \"Oh, yes! 'You'll never tame me! Never!'\" The word \"never\" is extended like a growl. Mr. Weller's fingers flick out into fat jazz hands, his eyes widen, accenting his sardonic demeanor. \"Yes, Dean, I remember.\" There is jealous animosity in his eyes. Dean Stuttgart had been his ward\u2014yet he had failed in all his attempts to tame the man. It riles him that a summer temp, Geoffrey Hunter, in less than two months, won the boy away from him. Dean Stuttgart, a founding family descendant, would have been a coup indeed. Weller had personally promised the trading baron he would tame his son, even had dreams of marrying into that prestigious family. _I could have kept your father from disowning you,_ he thinks grimly. Bitter in his discontent, he studies Todd. _I am not going to lose this boy to any Hunter!_ Now, patting Todd's head, he watches dispassionately as the boy shivers and pulls away from his touch. Defiant eyes blurred by tears dart up. Weller's head tilts questioningly. \"Is the boy here a zero on the Kinsey scale, too? Whom only the great Hunters can tame?\" Todd drops his head in shame. \"No,\" says Weller, shaking his head. \"Dean wasn't a zero, and you're not one either. There are no zeros born in Hadrian anymore.\" Then, adding as an after-thought, \"Or ones for that matter. We have genetically removed them from Hadrian's human genome.\"\n\n\"That's impossible,\" Dean expostulates.\n\n\"Really?\" Mr. Weller's eyebrows cock at that. He grins contemptuously. \"Then explain you. If there weren't a little two in you somewhere, how did the mighty Hunter tame you?\"\n\nFrank looks up dismayed, \"Papa Dean, what is he talking about?\" It's not that Frank doesn't understand the concept; it is simply that fear has him befuddled.\n\n\"The Kinsey scale, boy,\" Mr. Weller answers in place of Dean. Turning on Mr. Gavin, he asks, \"Don't you teach your students anything in this school?\" Mr. Gavin pointedly ignores the man's insults. Mr. Weller, equally dismissive toward the principal, turns back to explain the elementals to Frank. \"A zero on the Kinsey scale is said to be a pure heterosexual with no homosexual tendencies at all. A one only has the slightest inkling of what a homosexual is, much like the fives who have some inkling of what it means to be heterosexual. Some people say fives make the best tamers.\" Looking directly at Dean, he states, \"I say that's nonsense. And then,\" with a twinkling in his eye, he adds, \"there are the sixes,\" pointing proudly to himself. (Dean shakes his head knowingly at that!) \"Well, we are the ideal state of human being. The rest of you poor bastards are somewhere in between. Twos, like your Papa Dean here, are the most difficult to contend with. Many believe they are zeros, but we know better. There are no zeros in Hadrian, and only reeducation can reveal that.\" Weller's grin actually widens. \"Isn't that right, Dean?\" Looking down at Todd's slumped shoulders, he says, \"I suspect your little boy here is a two as well!\" Lifting the boy's chin, Mr. Weller forces him to look up again. Inspecting his bruises, he turns to Frank. \"It looks like your method of taming includes iron fists.\"\n\nSlightly confused, Dean glances Frank's way. Frank claimed their bruising came as a result of _strai_ bashing. A group of boys had jumped Todd in Riverside Park, and Frank had leaped in to defend him. Frank shakes his head, easing Papa Dean's distress.\n\nSmiling sardonically, Weller says, \"Don't worry, Todd; we won't beat you like Frank, here.\" Dean winces, knowing that promise for a lie!\n\nTodd yanks his face away, his knuckles turning white from gripping the arms of his chair. Ready to scream, only Papa Dean's soothing words ease him. \"Shh, Todd, it's okay.\"\n\n\"Yes, Dean, it is okay.\" Mr. Weller concurs. \"Or, at least it will be after reeducation.\"\n\nTodd's face drops into his hands, his wail expelled through clenched teeth, \"Frank, you promised!\" Looking up at his friend, through tears and desperation, he says, choking on his words, \"You said if we\u2014You said I wouldn't have to go!\"\n\nFranks kneels instantly by Todd's side. Both hands are clutching Todd's. \"You don't!\" Looking up, eyes pleading, he asks, \"Does he, Papa Dean?\"\n\nBefore Dean can answer, Mr. Weller leaps in, \"Oh, Dean, Dean, Dean.\" Each repetition of the name accompanies a jarring shake of his head. Mr. Weller's smile is vicious. \"Did you make promises you can't keep?\"\n\nDean attempts to stare the man down, but Mr. Weller is too secure in his position to worry. Although he looks away in defeat, Dean persists with his position. \"No, Frank,\" Dean insists. \"He won't have to go.\" Once again, boring his eyes into Weller's, he states, \"If the girl doesn't have to go, Todd won't have to.\"\n\nMr. Weller shakes his head sadly, with mocking, heartrending distress, \"The girl, Crystal Albright, is not a confessed heterosexual.\" With a circular swirl of the hand, he points to Todd. \"Your boy here is.\"\n\nDean is steaming, \"She initiated everything!\"\n\n\"Not according to the girl\u2014\" Weller pauses to glance Todd's way before adding, \"or the boy.\"\n\n\"Why don't you look at the video and see what really happened?\" Frank demands. \"Todd told me. She came on to him.\"\n\nMr. Weller shrugs condescendingly, \"I would, but the video has been destroyed.\"\n\nDean, Frank, and Todd all cry out, \"What?\"\n\n\"Now, Todd,\" Mr. Weller says tersely. \"You confessed. What need is there to hurt the girl further? None. So, at her mothers' request, the video was destroyed.\"\n\nTodd leaps out of his chair. \"That's not fair!\"\n\nMr. Weller is no longer willing to play games. He shoves Todd roughly back into his chair as he yells, \"Sit down!\" Todd winces. He is still raw from yesterday. \"Your confession has been made. You agreed to the girl's account. We had, we _have_ , no need of video evidence.\" With biting insistence, he adds, \"It is too late to change your story now! You are going to reeducation camp.\"\n\n\"No!\" Dean demands.\n\n\"You have no say in this, Stuttgart!\" This is the first time Todd has ever heard Papa Dean's birth name. He's a Stuttgart. Maybe there is hope. The Stuttgart family is a founding family. Surely Papa Dean has some pull. As if sensing what Todd is thinking, Mr. Weller chuckles, shaking his head contemptuously. \"Don't get your hopes up, boy; not even a Stuttgart can circumvent the sexual reformation laws.\"\n\nHe was going to add, _especially one whose father has disowned him_ , but a bitter, discontented Dean interjects, \"But clearly the girl's mothers can.\"\n\nNow, as if addressing children, Mr. Weller speaks slowly, over-emphasizing each word. \"The girl committed no crime. The boy did.\" In a more menacing tone, he addresses Todd, \"You can thank Hadrian and his lover that Ms. Albright and Vice President Stiles aren't charging you with rape!\" _Vice President Stiles!_ The thought looms in everyone's mind. Todd always knew Crystal's parents were rich, but she never spoke of her mama's founding family status. Mr. Weller gleams at the prospect. He may not be able to marry into her family, but he can most certainly ingratiate himself. Twisting the knife in the jugular, he adds, \"Otherwise, you'd be swinging from a rope!\" Todd pales.\n\n\"You don't have to do this,\" Dean pleads. \"We are working with him at home.\"\n\n\"Home?\" Mr. Weller shakes his head, \"No, no, no. Todd has no home. Not anymore.\" Reaching behind him, on the principal's desk, he retrieves a document.\n\nFor a brief moment, Mr. Gavin is acknowledged, or rather, the high back of his chair, as he sits frozen, concentrating on his computer screen. He has nothing more to say or do with this issue. The _strai_ belongs to the government, and it is in his best interest simply to give them the boy.\n\nShaking the document Dean's way, Mr. Weller proclaims, \"His legal guardian, Michael Fulton, has signed the form. The boy is now my legal ward!\"\n\nDean gasps, mouthing the words, \"Not you!\"\n\nWeller smiles as he mouths back his response, \"Me.\"\n\nTodd, oblivious to their silent communication, mutters, \"Papa Mike.\" As his chest caves, his shoulders heave with sobs. He feels even more betrayed than when he had been abandoned by the man the day before.\n\nSquatting now, Frank traps his head under his hands and between his knees. He sobs with heart-wrenching abandon as Papa Dean begs, \"Don't\u2014don't\u2014don't\u2014\"\n\nMr. Weller ignores him. \"Stand up, Todd. We have to go.\"\n\n\"NO!\" Todd shrieks. He darts up from his chair and races for the door. Before he can grab the handle, though, Mr. Weller has retrieved a smart tranquilizer dart from his coat pocket. It's a simple blowgun, and the dart, now sticking out of Todd's shoulder, knocks him out instantly.\n\nBeing the closer of the two men, Dean reaches Todd's body first. Although he knows this battle to be lost, he feigns denial, allowing him an opportunity to fumble around the boy's shirt pocket, retrieving Todd's cell phone.\n\n* * * * *\n\n#  _Salve!_\n\n**Inside a Reeducation Camp  \nHNN\u2014Melissa Eagleton Reporting**\n\n\"I always enjoy these episodes where I am able to leave the studio and visit a location live. After our successful interview with Northeast Reeducation Camp's warden, Mr. Gideon Weller, we have been invited to his reeducation camp to meet with some of the wards and their guardians.\n\n\"This camp is beautifully situated on the shore of Hudson Bay. With thick forest to the north, farmlands to the west and south, and the lake to the east, the boys sent here are kept physically active all year round. In the spring, plowing and planting the fields occurs. During the summer months, the boys enjoy fishing, hiking, biking, swimming, canoeing, even singing around the campfire. Come fall, the boys actively participate in the harvest. All the vegetables and grain consumed at the Northeast Camp are grown right here.\n\n\"Over here is the nucleus of the entire complex. As you can see, the boys' cabins encircle the central compound. It is in this main building that the hub of activity occurs. In this one building, all wards and guardians meet to celebrate victories and or discuss difficulties each individual ward may be facing.\n\n\"Let's go inside, shall we? As we go down these stairs and turn right, we enter into the kitchen galley where all the food is prepared, and let me tell you, viewers, it smells really good in here. This next room is the central cafeteria where even Mr. Weller sits down to eat his meals, seven days a week I've been told. Mr. Gideon Weller has married himself to reeducation. He lives, eats, and works here. His entire life is dedicated to the welfare of our young men.\n\n\"This next room, as we move down the hall, is the central meeting area. Here, I'm told, is where the young men hold dances and parties where they learn to have good clean healthy gay fun! Here in this room, we are going to meet Little Stephie Chatters and his ward Mattie Molloy.\n\n\"Hello, Stephie.\"\n\n\"Hello, Ms. Eagleton. This is so wonderful. Thank you so much for coming here. May I say hi to my fathers?\"\n\n\"Of course you can.\"\n\n\"Hi, Daddy. Hi, Papa Steve.\"\n\n\"Now, Stephie, is it true, were you really a ward here once yourself?\"\n\n\"Oh, yes. I came when I was twelve. But I'm no longer straight, in any way! I am one hundred percent gay, and I am so happy now. I was so miserable when I thought I was straight. I was always depressed and even thought about killing myself. When I first got here, I begged Papa Gideon\u2014Mr. Weller lets me call him Papa Gideon\u2014to help me be gay. I knew being straight was awful\u2014Papa Gideon calls it abhorrent\u2014but I had no idea how to fix things. This camp saved my life. And now I'm able to help out other wards like Mattie here.\"\n\n\"So, you must be Mattie. Tell me, what's it like having a ward like little Stephie? He must be a huge inspiration for you, knowing he too once suffered like you do.\"\n\n\"Yes, ma'am.\"\n\n\"Mattie's shy, Ms. Eagleton. He's only been here a couple of months, but he's come such a long way. When he first got here, he was very angry. He always swore, and he wouldn't listen to anything anyone had to say. But now he is so demure. I am so proud of him. I just know that he will be cured before the year is up.\"\n\n\"Well, Mattie, that is such good news. I'm sure you will be so happy when you get released.\"\n\n\"Yes, ma'am.\"\n\n\"Stephie, Mattie, thank you so much for sharing your stories with us. What a great inspiration it is to see success and how a victor is able to turn around and use his experience to help another young man flourish. It truly is awe inspiring and rewarding. So there you have it, folks, the success of the Northeast Camp firsthand!\"\n\n_Vale!_\n\n# Interview with Salve!\n\nGeoffrey and Dean are snuggled up tight on the couch, watching a comedy on the wall screen. Dean has been so despondent after Todd's exposure and transfer to the Northeast Reeducation Camp. Geoffrey is hoping a little humor will help cheer him up. It is not working. Halfway through the vid, Geoffrey gets an incoming call; blinking, he pauses the film. A second blink and then, \"Hello?\u2014Hey, Aaron, how are you?\" Dean smiles slightly; Aaron Hillier stood up for Geoffrey at their wedding. What Dean doesn't know is that Aaron had also been a volunteer that summer at the Northeast Reeducation Camp when Dean was being held there. In fact, he ended up studying reeducation psychiatry and now works full-time under Gideon Weller. Geoffrey pulls away from Dean. Dean sees nothing odd in this act, thinking Geoffrey simply needs to focus on his conversation. \"Yes, yes, it has been too long...So, how's Mateo?\" Mateo is Aaron's partner. He, too, unknown to Dean, works in reeducation psychiatry. \"Nice. Good to hear. So what's up? Why the call? Really? Dean and me?\"\n\nDean looks over. \"Dean and me what?\" Although curious, Dean's voice holds no excitement.\n\nGeoffrey, hushing Dean, looks straight at the wall as if at Aaron. \"Why?\" A dark film covers Geoffrey's eyes. \"Are you serious? He can't do that?\"\n\n\"What? What? Do what?\" Dean asks nervously.\n\n\"What?\" Geoffrey voice is a mixture of horror and ire.\n\nDean reaches over and grabs Geoffrey's shoulder, turning the man to face him. \"What is it? Is it Todd? Tell me!\"\n\nGeoffrey tries to calm his voice. \"We've been asked to do an interview for _Salve!_ \"\n\nThis is dumbfounding. Why would an offer to be on TV cause such consternation? Dean shakes his head in dismay. Shivering, he inquires, \"Why?\"\n\n\"Aaron,\" says Geoffrey, focusing once more on the wall, \"Dean and I need to talk. I'll give you our answer tomorrow.\" Now it is Geoffrey's turn to shake his head in dismay. \"When?\u2014Fine. I'll call back in one hour.\" Blinking once, he disconnects. After he blinks a second time, the image on the wall screen disappears. Leaning back into the couch, he closes his eyes and purses his lips. Dean's stare burns deep into his consciousness. \"We've been asked to participate in an upcoming _Salve!_ episode titled 'Happily Married After Reeducation.'\"\n\nShrinking away, Dean groans, \"No!\" His head shakes spasmodically. \"No!\"\n\nGeoffrey smiles sadly at his terror-stricken lover. \"I\u2014\" He doesn't want to do the interview any more than Dean does, but he knows better. _How to explain?_ Opening his eyes, he can see by the look on Dean's face no level of understanding is attainable right now in his fragile mind. Even so, the truth must be revealed. \"We don't have much choice.\"\n\nDean's scrotum turns to ice. \"Why not?\"\n\nThere is no easy way to say this so Geoffrey doesn't even try. \"Weller filed a report against you.\"\n\nDean screams, \"They can't send me back there! They can't! We're married! We're married!\" Geoffrey moves closer to his lover, but Dean rejects any physical contact. Leaping off the couch, he begins pacing the living room like a caged tiger.\n\n\"Apparently, you said something today, something Weller says indicates you might still be straight.\"\n\nDean's voice becomes shrill. \"What? What? What did I say? I didn't say anything. I just tried to help Todd. I was just trying to help Todd.\"\n\n\"In his report, Weller said you claim there are still ones and zeros inside Hadrian. And,\" he adds hesitantly, \"that it is impossible to tame them.\"\n\nDean's body jerks as if hit with an electric shock. Doe-eyed, anticipating the bullet, Dean turns to Geoffrey. \"I never said that!\" Words become disjointed as Dean's breath turns sporadic. \"I swear I never said that!\" No longer pacing, standing now on the other side of the room, Dean wraps his arms around his shoulders, rocking his upper body.\n\nCrossing over, Geoffrey places his hands on Dean's (still clasped tight around his shoulders). \"He says...\" He turns Dean to face him. \"He said you declared yourself a zero when you were inside the camp...\" closing his eyes to avoid showing Dean the dread he feels, Geoffrey adds, \"And that today's statement suggests that might be true.\" Dean shrieks. Geoffrey pulls him in tight. As much as it pains him, Geoffrey needs to finish telling Dean everything that was said about him. \"Weller has recommended you be re-assessed by his reeducation psych team.\" Dean starts to scream wildly, shaking inside Geoffrey's grip. \"That's how Aaron found out. He works in assessment under Weller.\" Trying not to grimace, Geoffrey continues, \"He said Weller wants you back inside the camp so they can determine whether or not...\" He closes his eyes, fear beginning to shake his resolve. \"Whether or not you should be exiled.\"\n\nSuddenly, Dean is seventeen again\u2014desperate and willing to do anything to avoid further punishment. He drops to his knees and begins undoing Geoffrey's pants. It is Pavlov's dog all over again. Blowing Weller was the only sure way of avoiding punishment.\n\n\"No! NO! Dean, stop it!\" Geoffrey kneels down so he and Dean can be on the same level. \"I promised you then and I'm keeping that promise now. It will never be like that between us! This is your fear acting, not love.\" Dean collapses into his lover's arms. Both men shed tears. Recovering his voice, Geoffrey tries to offer up some hope. \"Aaron believes if we go on _Salve!_ and talk about our lives, it will exonerate you.\" Dean nods. He is broken, beaten the way he was back in the camp. He really is willing to do anything right now. \"We'll just answer a few questions. Tell Hadrian we're in love\u2014only say what's true.\" The ceaseless nodding of Dean's head against his chest discombobulates Geoffrey. \"I won't let them take you back there. I won't let them exile you. I swear to you, on the soul of Hadrian's lover, I will fight them like a grizzly protecting her cub.\" Geoffrey slides into a seated position on the floor, and Dean falls helplessly into his lap. Holding Dean tightly, Geoffrey tries desperately to reassure him with gentle strokes and soft cooing noises.\n\n* * * * *\n\nMelissa Eagleton is even more beautiful in person than she is on the camera. They say the camera adds a good ten pounds, and Melissa would not lose weight to look normal so when one first encounters the real life woman, the first reaction is always how much slimmer she appears. She is also a lot shorter than most would expect. One seldom sees Melissa standing when she gives one of her broadcasts. Her dimples seem just a little deeper and her smile a little wider too. No one is quite sure why, but instead of doing the woman's beauty justice, the camera actually diminishes her. Perhaps it is because she refuses to wear extreme amounts of makeup. She does comply with the need for makeup for the camera, but she never lets the cosmetics team do more than make her look healthy and normal. Off camera, the light makeup proves to be very stylish. Her blue eyes twinkle in such a way that the interviewee is immediately put at ease. Well, almost every interviewee. Nothing, not even Melissa's sweet demeanor and perky smile, can help ease Dean's tension. In fact, her feminine beauty has the exact opposite effect. For Dean, being in close proximity to such an attractive woman sets off a series of residual electric shocks that shake his body in a series of disjointed jerks\u2014akin to Pavlov's dog\u2014destroying any potential attraction or emotion he might have felt for her. Having been attached to electrodes, with high voltage shocks tearing throughout his nervous system every time his body responded erotically to a beautiful woman, has proven to be a lifelong lasting and very effective deterrent. Although Dean remains attracted to women, his body has been successfully conditioned, and his brain sets off the memory of the shock so strongly that he feels the very currents as if they were still ripping through him. This, now natural reaction, is why Dean avoids women as much as humanly possible. Regrettably, the shock waves he now feels remind Dean not only of past horrors, but they also bring to mind what Todd is going through this very moment inside that camp.\n\n\"So, gentlemen...\" Melissa's voice is melodious. \"Come sit down next to me.\" Geoffrey, understanding Dean's dilemma, takes the chair closest to Melissa Eagleton. Waving to her camera crew, signaling them to wait just one minute, she begins instructions. \"I'll just start bombarding you with questions. Answer them honestly, and after ten or fifteen minutes, we'll wrap it up. We won't use all of the tape. My edit team will pick out the best bits.\" Smiling, she says, \"I am so excited. I have been waiting to do a piece about this for years. I never could understand why reeducation was always stalling me. Then suddenly last night, out of the blue, Aaron Hillier calls me, saying they have the perfect couple, you two!\" She leans forward to clasp each man's hands as she expresses her gratitude. \"Thank you!\" After leaning back in her chair, she swivels to face the cameraman. She gives him the \"one more minute\" finger and then asks her guests, \"Are we ready?\" Both men nod. \"All right then.\" Smiling to the cameraman, she nods her assent.\n\nThe cameraman begins calling out, \"Five, four,\" gesturing at the same time with the fingers of his right hand. He is silent for \"three, two, one,\" merely mouthing the words while continuing to gesture. Then after mouthing the word \"one,\" he flicks with his index finger to indicate they are now filming.\n\n\"Well, good evening, Hadrian. Tonight's show is a very special event. With us today are Geoffrey and Dean Hunter! This couple met at the Northeast Reeducation Camp.\" She turns now to face the men. Geoffrey is holding Dean's hands in his. This image makes Melissa smile; it couldn't be any more perfect. \"Geoffrey, can you tell my viewers about the first time you met Dean.\"\n\n\"I, ah, I got a job helping out at the Northeast Camp...um, trying to make some money to help pay for my tuition. I, ah, well, um, I wanted to do something to help. I thought this might...was...ah, a good thing.\"\n\nMelissa smiles. \"That's really good,\" she says encouragingly. \"But what my viewers really want to know is how the two of you met.\"\n\nDean stares at his hands. He remembers exactly how he and Geoffrey met. Weller had decided it was time for Dean to have sex, and this time he wasn't taking no for an answer. Geoffrey walked in on them by accident. If it hadn't been for Geoffrey Hunter, Gideon Weller would have raped Dean.\n\nGeoffrey answers for them both. \"The first time I saw Dean, he was standing in line at the camp cafeteria. I watched him fill up his tray; he sure could eat a lot back then! And the way he walked over to his table really captured my attention. Something about Dean reminded me of a Bengali tiger. His brown curly hair had been left unattended, and I hate to say it, but that unruly look really suited him. I was instantly attracted to the man.\"\n\n\"How romantic.\"\n\n\"Dean, when did you discover that you loved Geoffrey?\"\n\nSpeaking into his hands, Dean says, \"When he said he would never\u2014never force me\u2014and\u2014and I\u2014I\u2014knew I could trust him\u2014that he wouldn't\u2014make me.\"\n\n\"Oh, my,\" Melissa expresses concern. \"You're crying.\"\n\n\"It's a very emotional experience for both of us,\" Geoffrey explains.\n\n\"I see.\" Tilting her head slightly, hoping to make eye contact with Dean, Melissa presses him with another question. \"So, Dean, you must be very grateful to the Northeast Reeducation Camp since it brought you and your husband together.\"\n\nDean shudders, his shoulders heave. \"I can't do this...\" His head shakes. Geoffrey releases Dean's hands and wraps him in an embrace. \"It's all right, Dean; everything's all right.\"\n\n\"I can't, Geoffrey...I can't...I can't say I'm grateful to those bastards for anything! Not even for you. I love you...but I can't.\"\n\nGeoffrey continues to soothe Dean as he sobs. At one point, the two men kiss\u2014a spontaneous act for which Melissa Eagleton is grateful. There is very little of this interview she can use. Pondering the situation, she realizes this will not be a _Salve! Interview Special_. It will have to be a regular _Salve!_ , but with a twist. Her mind works quickly to redesign the show meant to air that very evening. _I'll have the wall screen showing excerpts of their interview_. Considering her options, _At least there's the bit where Geoffrey Hunter explains why he got a job there, his first impression of Dean and their kiss. No wait_ , she remembers, _Dean does say \"I love you.\"_ With a little creative editing, she realizes they can use that, too. The rest she will simply have to invent. Turning to face her camera crew, Melissa Eagleton signals for them to stop filming.\n\n* * * * *\n\n#  _Salve!_\n\n**Happily Married After Reeducation  \nHNN\u2014Melissa Eagleton Reporting**\n\nHave you ever wondered what happens to our youth after entering reeducation? Well, today's show is going to answer that question for you. I had the good fortune to interview Geoffrey and Dean Hunter earlier today, and what a lovely couple they make! They met at the Northeast Reeducation Camp over twenty years ago. Records show they even married inside the camp's central meeting hall. On the wall screen to my right, I will be showing you short excerpts from my brief but revealing interview with the Hunters. Here, Geoffrey Hunter is explaining why he went to work at the reeducation camp:\n\n\" _I wanted to do something to help_.\"\n\nAnd help he did. He changed Dean's life for the better, so much so that Dean not only agreed to marry the man, but he asked to take Geoffrey's last name as his own! Geoffrey Hunter also told us how he felt the very first time he saw Dean. According to Geoffrey, it was love at first sight for these two.\n\n\" _Dean was standing in line at the camp cafeteria. I watched him fill up his tray. He sure could eat a lot back then! And the way he walked over to his table, there was something about him. His brown curly hair really suited him. I was instantly attracted to the man_.\"\n\nI just love a good romance story. Dean was so taken by his husband's beautiful words that he was drawn to tears and couldn't help but express his love openly for all our viewers.\n\n\" _I love you_.\"\n\nAnd, just look at the way Geoffrey responds to those words. These two men are so much in love they can't help but kiss. Dean Hunter's successful cleansing of his heterosexual tendencies, discovery of his latent homosexuality, and his now loving marriage really show us the good work the people at the Northeast Reeducation Camp do to help bring our wayward youth back into the loving fold.\n\nSo, there you have it folks, Geoffrey and Dean Hunter. Happily Married After Reeducation!\n\n_Vale!_\n\n# Het-Row\n\nWhen Todd wakes, he is lying on a hard mattress. A wooden ceiling spins over his head while rafters lurch down at him. Nauseated, he leans over the side of his bed and retches. A pail, adeptly placed, catches the vomit.\n\n\"Water's on the bedside table.\"\n\nThe voice, dull sounding, as if struggling to make itself heard from under a deep green sea, barely registers in Todd's ears. \"Huh?\"\n\n\"Here.\" A hand blurs into view, holding a tall glass filled with clear liquid.\n\nTodd's hand shakes as he reaches for the water. The strange hand guides the glass into Todd's. \"Thanks,\" Todd mumbles. After rinsing and spitting, Todd throws up again, spilling the remainder of his water.\n\n\"Shit, man,\" the voice utters. \"You splashed it all over me.\"\n\nTodd's head is still hanging over the bucket. \"Sorry.\"\n\n\"Fuck it!\" the voice replies conciliatory. \"It happens to everyone. It's those fucking tranquilizer darts. I don't think they've ever brought in a straight man without them.\" The hand reaches for the glass, dangling now empty in Todd's hand. Suddenly, two hands are in Todd's blurred vision. One is pouring water from a pitcher into the glass. \"Here.\" The glass floats toward Todd. \"No doubt you're not finished. Drink some this time; you're going to need something inside to throw up.\"\n\nTaking the hand's advice, Todd rinses, spits, and then swallows some water. He immediately throws up again.\n\n\"More water?\" the voice asks.\n\n\"Yeah,\" Todd mutters, holding the glass up toward the origin of the voice.\n\n\"Here. Rinse, spit, drink, and vomit again.\" Todd takes the glass and does as instructed, except this time he doesn't vomit. He hangs his head over the bucket, waiting, but it seems his nausea is at its end. Todd moves to lie back down when the voice stops him. \"I wouldn't do that. It'll only make you puke again.\"\n\nFiguring he's right, having regained more of his senses so he is now able to determine the voice's sex, Todd chooses to stay sitting up. Pouring water on his hand, Todd rubs it over his face. \"Where am I?\" he asks when finished.\n\nLaughing sardonically, the voice answers, \"Welcome to Het-Row! Where all the future homos go!\"\n\nLooking up, Todd sees the cynical smile of a young man his own age. He has long red hair, a square jaw, and ( _are those age wrinkles?_ ) sparkling blue eyes. Todd drops his head and groans, \"Reeducation camp.\"\n\n\"That's what the good folk of Gomorrah like to call it, but really, it's just ' _strai_ town,' 'cunt hammer village,' or the unruly 'sticker in 'er bog.' Call it what you like, it's home now until you come of age and they exile you, you die\u2014good luck with that (exposing a pair of scarred wrists)\u2014or you get married.\"\n\nTodd looks up questioningly. \"Married?\" _Hadrian, no! The only man I thought I could marry was Frank\u2014but no more._\n\nResponding to the despondent look on Todd's face, the boy quips, \"That's right, married! No _strai_ leaves here unless he's been poked and successfully latched to another man with the ol' ball and chain, so to speak. It's not good enough to claim you're gay; you have to stay that way under the watchful eye of your loving husband, who, by the way, is required by law to make monthly reports on you for the first year, bi-annual reports for the next five years, and then annually until the day you die!\"\n\n\"You've got to be fucking kidding me, right?\"\n\n\"Shit, man, I wish I were.\" Discontentedly studying his wrists, the boy says, \"Ah, to die. Death.\" Closing his eyes, summoning lines up from memory, he quotes:\n\n\"To die: to sleep;\n\nNo more; and by a sleep to say we end\n\nThe heart-ache and the thousand natural shocks\n\nThat flesh is heir to, 'tis a consummation\n\nDevoutly to be wish'd.\"\n\n\"Fuck man,\" says Todd. \"That's morbid.\"\n\n\"Actually, it's Shakespeare, and as for being morbid\u2014\" his laugh takes on a derisive edge, \"you only just got here, man. Just wait,\" he harrumphs, \"a few more minutes.\"\n\nReaching between the two beds, he once again presents his hand. \"My name's Matthew Molloy, but don't ever call me Matthew in front of them.\"\n\n\"Why not?\" This is too much for Todd to take in.\n\nLosing the cynical edge to his smile, Matthew scowls, \"My gay name is Mattie.\" Scrutinizing Todd, he asks, \"You're the Middleton kid, right?\"\n\n\"Yeah, Todd.\"\n\n\"I wonder what they'll call you? Toddie, Todster, Toddel Bear? Toddly Poo.\"\n\nDropping his head into his hands, Todd mutters, \"Fuck me.\"\n\n\"I heard that happened, too.\" Looking at Todd wonderingly, Matthew asks, \"Is it true?\"\n\nPissed off, Todd snarls, \"Is what true?\"\n\nOblivious to Todd's reaction, Matthew moves in closer to allow for a more conspiratorial tone. \"Did you really fuck that girl?\" Closing his hands and eyes half in wonder, half in delight, he whispers, \"Man, I wish I could have\u2014at least once\u2014shit.\" Leaning back dejectedly on his bed, he confesses, \"I only got a kiss and a little tit. Humph, and we both got shipped off\u2014but you, you actually fucked her, they say, and she never got sent away. They say it was rape, but she wouldn't accuse you because rape is punishable by death. How altruistic of her.\" The sarcastic slurring suggests that Todd would have been better off with a death sentence. \"Is that true? Did you rape her, or did she really fuck with you?\" He is shaking with excitement.\n\n\"Yes\u2014NO\u2014I didn't rape her\u2014but we did have sex.\"\n\n\"Yes!\" Matthew's reaction, aptly enough, is an ejaculation. \"What was it like? What did it feel like\u2014in there?\"\n\nTodd rests his hand on his forehead. \"Fuck, man, too much has happened since then.\"\n\n\"Please tell me!\" Matthew is desperate.\n\n\"Hot, damp, like being wrapped in warm, wet silk.\"\n\n\"Beautiful. I imagined as much.\"\n\nBut Todd is no longer listening to Matthew. His thoughts have drifted back to a few nights ago. _Only a few nights?_ Although only Saturday night, during Pride's National win celebration dance, it feels like a whole world ago when Todd's body sparkled like fireflies dancing over a moonlit river. Stifling a cry, he asks, \"How long have I been out?\"\n\nMatthew shrugs. \"I don't know. That drug usually only lasts twenty-some hours. It's late Wednesday afternoon if that helps. And you were here when I went to bed.\" Pondering the time difference, he reasons, \"They must have really doped you up. Did you fight them?\"\n\nOnce again, Matthew's words fade into the background as Todd's mind fixates on his memory. _Yes_ , Todd figures, _it's only been three nights. Or is it four? Hadrian help me, I can't think_. Shaking his head to try and clear it only causes his mind to spin and his stomach to swirl. Once again, he is vomiting into the bucket, this time only dry heaving since there is nothing left for his stomach to throw up.\n\n\"Okay,\" Matthew quips, \"don't shake your head.\" Reaching for the glass and pitcher, he offers Todd some more water.\n\nTodd accepts the gift but does not acknowledge Matthew's kindness since his mind is still focused on recent events. _And then there was_ \u2014grimacing\u2014 _the night that was supposed to save me._\n\nAs if reading Todd's mind, Matthew carries on inquisitively, \"And the guy? Did you fuck the guy, too?\" Todd's shoulders heave as he sobs at the memory. \"I'll take that as a yes. And they still brought you here? The fucking is usually the last thing that happens before they marry you to the bastard.\" Shaking his head sadly, Matthew asks, \"So, who did you fuck? The girl, I mean.\" Still shaking his head in wonder, he adds, \"Her mothers must be big in the government.\"\n\nTodd shakes his head. He doesn't want to say what he remembers Mr. Weller having mentioned in the principal's office: Crystal's second mother is Elena Stiles, the Vice President of Hadrian _and_ from a founding family! \"I never met Crystal's mothers,\" is the only response Todd is willing to offer. Thoughts of Crystal remind him of the fairer sex. _Girls_. \"Are there any girls here? You said your girlfriend got shipped out, too. Where's the girl's camp?\" Todd knows Crystal never got sent away, but a part of him wishes he could somehow find her there.\n\n\"Nowhere near here; that's for damn sure. The only cunts that ever walk this place are the ones who come to desensitize you to their flesh.\" Shuddering at the memory, Matthew adds, \"The first time you see a chick in this place, you are going to cringe.\"\n\n\"Ugly?\"\n\n\"I wish.\" Shaking his head, Matthew explains, \"No, they hire the most beautiful women they can possibly find. All carpet munchers, too, every God damn one of 'em\u2014but don't ever let them hear you say anything derogatory about them\u2014or the men\u2014that'll earn you a paddling.\"\n\n\"They beat us here?\"\n\n\"If that was all they did,\" Matthew says, \"I think this place would almost be bearable.\"\n\n\"Mattie!\" A high pitched, far too feminine voice shrills out. \"Is my little Mattie complaining again?\" Stephie strolls into the boys' cabin. He is wearing tight-fitting, very short shorts and a tie-up tank top. His hair is up in a high ponytail, classic mid-twentieth century, and his makeup and false eyelashes are garish. As soon as Stephie's voice shrills out, Matthew lies down on his bed, covering his eyes with his arm. His body begins to quiver. Stephie sits down on the edge of Matthew's bed. \"Sit up, Mattie.\" A hint of anger lowers his voice an octave, but gaiety returns as soon as he notices Todd. \"Oh, Mattie, your little roommate's awake.\" His voice is now fully masculine, no longer pretending to be jocular. \"I said sit up!\" He smacks Matthew's thigh. Matthew obeys, pulling his knees in tight to avoid any physical contact with Stephie.\n\n\"Todd Middleton, this is Stephie. Stephie, this is Todd Middleton.\"\n\n\"Much better,\" Stephie says while eyeing Todd. Now looking over to Matthew, he adds, \"I'd hate to have to give you five demerits today.\" Matthew looks up, his face pale with terror. \"Don't play coy with me,\" says Stephie, tickling Matthew under the chin. The act causes the young man to shake as tears begin to fall. Todd watches on in horror. \"First demerit. I heard you call the women who come here for training purposes cunts and carpet munchers.\" As an aside, he adds, \"That should really count for two demerits, but since I don't like women anymore\u2014\"\n\nTodd bursts out, \"You were het'ro?\"\n\n\"Yes.\" Stephie poses happily, pushing out a flat chest forming the S curve with his back. \"Believe it or not, Stephie is a tamed little boy. Fully homosexual now.\" Shaking his head in disgust, he says, \"I can't believe I ever thought of girls that way.\" Assertively, proving even then he knew what was right, he adds, \"But I told my dads and they signed me up right away. I had never touched or kissed a girl even. I was only twelve. Being here was the best thing ever to happen to little Stephie. They treated me so well. I was like everyone's little pet.\"\n\n\"I was wrong,\" Matthew quips. \"There is one they didn't have to bring in with drugs.\"\n\nSmiling happily, allowing Matthew his little joke, mostly because he is proud of that fact, Stephie carries on with his life story. \"Now I'm fifteen and one of the best tamers ever.\" Sitting straight and proud, he states, \"That's what Papa Gideon\u2014Mr. Weller says.\" He smiles sadly Matthew's way. \"I'm still too young to marry any of my wards, though.\" He pats Matthew's leg. \"I'm hoping Mattie'll wait for me.\" Matthew groans. Stephie, angered by this subtle act of rejection, squeezes Matthew's leg hard. Looking back Todd's way, feigning serenity in his relationship with Matthew, he explains, \"My dads won't let me marry until I'm eighteen.\" Then looking back over to Matthew, he says, \"But Mattie will wait for me, won't you Mattie?\"\n\n\"They won't let two _strais_ marry, remember,\" Matthew mutters almost too helplessly. Matthew's neck has remained bent the entire time, and now his head drops even lower.\n\nStephie starts to run his fingers through Matthew's hair, gripping it at the crown and yanking his head up. Tears have flooded Matthew's face. \"Papa Gideon says I can marry whomever I want!\" He kisses Matthew full on the lips. \"And I want you!\" A little grimly now, he admits, \"But Papa Gideon doesn't think you'll wait for me.\" Stephie sadly inspects Matthew's face. Tears are streaming down Matthew's neck. When Matthew fails to respond, Stephie leans in and licks away the tears at the base of Matthew's neck up to his closed eyelid. There is no pretense left in Stephie's voice. It is as if his voice has suddenly cracked and lowered in the instant. \"Won't you, Mattie?\" That Stephie's grip on Matthew's hair has tightened is evidenced by the wince Matthew suddenly expresses.\n\n\"Yes,\" Matthew mumbles. \"I'll wait.\"\n\nStephie lets go of Matthew's hair and watches with amusement as his head drops down dejectedly against his chest. \"You know,\" Stephie decides, \"I am going to have to give you five demerits after all.\"\n\nMatthew's head rises instantly. He is off the bed and on his knees in front of Stephie. \"Please, Stephie, please. I'll be good; I promise. I'll do anything. _Anything!_ \"\n\nTodd shrinks back on his bed. _Hadrian no! Not in front of me_. Horrified by the thought, he quickly turns his back and squeezes his eyes shut. The thought of what might be about to happen causes Todd's stomach to swirl once more.\n\nMatthew gulps back a wail. \"Please don't give me another demerit.\" Openly crying now, his sobs tear a cleft in Todd's heart. Matthew hides his face in Stephie's lap.\n\nWith a motherly, loving smile, Stephie begins petting Matthew's hair. \"I don't want to see my Mattie hurt. I just want us to love each other. Because I love you, Mattie.\" Then softly, in a voice full of hope, he requests, \"Kiss me if you love me, Mattie.\" Stephie ceases the petting, closes his eyes, tilts his head, and waits for a kiss.\n\nRealizing nothing untoward is going to happen, Todd looks back just in time to see Matthew shudder silently, look up at Stephie's attending lips, and comply.\n\nStephie opens his eyes, his smile now genuine. \"Will you do something for me,\" he sings out, \"now that I've done something nice for you?\" Matthew nods his assent. Stephie stands, lifting Matthew up with him. They clasp hands and Stephie leads Matthew out.\n\nAs they walk over the threshold of the cabin door, they pass Jason Warith entering. He smiles, \"Hello, Stephie, Mattie.\" Matthew remains silent; Stephie smiles, \"Hello, Jason. Your little boy is waiting for you.\" Stephie nods in Todd's direction as he pulls Matthew through the cabin door.\n\n* * * * *\n\n#  _Salve!_\n\n**Spotlight: Jason Warith  \nHNN\u2014Melissa Eagleton Reporting**\n\n\"As I am sure you have noticed, viewers, we have been following a theme these past few weeks. With such a rash of exposures occurring this spring\u2014the most shocking being that of Pride's b-ball star, Todd Middleton\u2014we decided to do a thorough exploration into reeducation. Our focus has been on the most successful of the four male reeducation camps, that being the Northeast Reeducation Camp under the tutelage of Mr. Gideon Weller. Well, tonight we have another guest for you to meet. Before I bring him out on stage, I would like to give you a little background into this extraordinary individual. Jason Warith, now nearing thirty, has just completed his doctorate from Antonius Uni. With a Ph.D. in human sexual orientation, Jason Warith is considered the 'golden boy' of reeducation. Having recently finished his internship at the Southwest Reeducation Camp, located on a farm ten miles east of New Augustus City, Adrian Adams, the camp warden, claims Jason Warith is going to change the way we approach reeducation. 'His methods,' Adams says, 'are innovative and humane.' Warith's approach, Adams maintains, completely dismisses corporal punishment and concentrates on accepting the young man for who he is while simultaneously helping him accept homosexuality as a loving alternative. And on that note, I would like to bring out the man said to be the future reformer of reeducation, Mr. Jason Warith.\n\n\"Mr. Warith, welcome and thank you for being on _Salve!_ \"\n\n\"Thank you for having me, Ms. Eagleton.\"\n\n\"Please, Melissa. Tell me, Jason, you don't mind my calling you by your first name, I hope?\"\n\n\"Not at all.\"\n\n\"Tell me, Jason; what inspired you to pursue reeducation as a career?\"\n\n\"My mama, Helena, spent three of her teenage years inside a reeducationcamp for women. It took a long time for me to get her to open up and tell me what reeducation was like, speaking about what had happened to her was akin to her reliving it all over, but one day she opened up and shared her story.\"\n\n\"Would you mind sharing your Mama Helena's story with my viewers?\"\n\n\"Yes, I would mind. This is her story and not mine. I cannot share it with the world. What I can say, though, is that her story inspired me. I knew if the way Mama Helena was treated is any indication, the reeducation process needs some serious reformation. It was then that I decided to make the study of human sexual orientation and the ways in which we reeducate our youth my life's vocation. I knew I could not work with young women; the reeducation of females is clearly the domain of our women, but I also knew that if the way my mama was treated is any indication, then our young men must also be brutalized.\"\n\n\"Brutalized?\"\n\n\"Yes, Ms. Eagleton, brutalized. Think about it. Whom do we hate more\u2014the heterosexual male or the heterosexual female?\"\n\n\"I\u2014I don't think hate is the right word.\"\n\n\"Really, how many people like barbarians? Hordes? Pedophiles? Rapists?\"\n\n\"Well, when you put it that way, but really, I think you are over-exaggerating a little.\"\n\n\"Am I? How many times have you referred to the heterosexual male as a barbarian? And you cannot deny that the prevailing attitude toward heterosexual men is that they are violent and dangerous.\"\n\n\"You have to admit there has been a lot of evidence against them.\"\n\n\"Perhaps, coming from those men outside, men who are desperate enough to attack our wall. But I do not believe we should be comparing our young men to fraught and starving marauders.\"\n\n\"I see your point.\"\n\n\"Our boys are merely confused. They are experiencing, what for them, are natural sexual feelings toward women. If, as we believe, many of them are twos on the Kinsey scale, then their heterosexual urges are a lot stronger than their latent homosexual desires. There is nothing wrong with that. The only reason our founding families rejected heterosexual behavior\u2014or rather, I should say the main reason they rejected it, and why we the citizens of Hadrian continue to do so, was and still is, due to the need to quell overpopulation. Rather than punish our boys and girls for their natural feelings, we need to educate them to reason beyond their initial sexual instincts. Once they come to understand the need to avoid procreation, we can then work with them to embrace the latent homosexual inside. It is incumbent upon us to help guide them toward the very same choice we have all made: to be homosexual for the protection of our planet. The key difference here is that these youth need to make this choice consciously, whereas the rest of us have subconsciously agreed to this lifestyle by merely accepting our upbringing.\"\n\n\"Well, this is very interesting. I understand that your first position is to be at the Northeast Camp.\"\n\n\"Yes, I was originally offered a position at the Southwest Camp, and I am sorry to pass that opportunity by, but my services have been requested at the Northeast Camp so I agreed to go.\"\n\n\"Yes, you will be amazed at what you find there. Gideon Weller has done incredible things with his wards. He has the highest success rate in all four male reeducation camps.\"\n\n\"Yes, so I've been told.\"\n\n\"You must look forward to working with such a dedicated and successful man.\"\n\n\"I am certain working with Gideon Weller is going to be an eye-opening experience.\"\n\n\"Indeed it will be. Gideon Weller impressed me as a remarkable man. Reeducation is one of the most commendable of professions. I wish you the best of luck in all your endeavors.\"\n\n\"Thank you, Ms. Eagleton.\"\n\n\"Thank you, Mr. Warith, for if what they say about you is true, then you are the future of reeducation.\"\n\n_Vale!_\n\n# Warith vs. Weller\n\nTodd is stunned. Matthew, someone Todd is certain is highly intelligent and was very likely once a vibrant young man, has clearly been twisted into something morbid and broken. Lost in contemplation, Todd doesn't even notice Jason Warith until he sits down across from him.\n\nA tall man of mixed Arab and Irish descent, Jason's skin is golden brown. His hair is a rich black, his eyes a very dark brown, and his smile kind and gentle. Jason's calming demeanor had helped him to win over the wards under his guardianship at the Southwest Reeducation Camp. Jason's work during his residency earned him a reputation as the most likely of his graduating class to achieve success in the field of reeducation. Some even hailed him as the future of reeducation.\n\nNot wanting to alienate Todd immediately, Jason chooses to sit on Matthew's bed across from him. In this way, Todd can avoid close physical contact, helping him feel more at his ease. Jason has come to accept that these youth are fearful of men who come on too strong. The best way to win a young man over is to give him a lot of personal space and only enter into his zone when invited. Jason holds out his hand for Todd to shake. \"Hello,\" he says with his famous smile. The sight of Jason's hand repulses Todd, as if it were a cobra ready to strike. \"I won't hurt you, I promise,\" Jason adds reassuringly. When Todd pushes back, Jason lowers his hand. His smile dissipates. This is not working out. He knew meeting his first charge wasn't going to be easy. The file on this boy suggests a hostile youth; yet everything about the young man suggests otherwise. Of course, he is just coming off the effects of the tranquilizer drugs, something else Jason hopes to phase out over time, but still, Todd's reaction is more like a scared child than an angry young man.\n\nJason crosses his legs and studies Todd. _He sure looks young_. While studying his chart to make sure he got the age correct, he asks, \"How old are you, son?\"\n\n\"Seventeen,\" Todd answers, but he stares away, avoiding Jason's eyes.\n\n_That's his age all right_ , Jason reaffirms as he shakes his head wonderingly. _Fear_ , he muses, _has a way of making a man look younger or older_. And, in the case of Todd Middleton, fright has reduced him into the image of a small child. As well as ensuring the boy's age was accurately recorded, Jason was hoping his question would open the door for further conversation. It hasn't worked. Todd remains despondent, staring first at the wall, now the ceiling, and now the floor.\n\n\"My name's Jason.\" Pausing briefly, a little uneasy about what he has been instructed to say, he adds, \"I'm told to call you Tabatha.\"\n\nFinally eye contact, but glaring hate and anger. _I should have expected as much_ , Jason sighs inwardly, _but I have to mark that down\u2014no wait; they can't hear a facial expression, only if he_ \u2014\n\n\"My name is Todd! I won't wear one of your stupid gay names.\"\n\n_Damn_ , Jason closes his eyes briefly. He knows he has to jot this down, give the boy his first demerit. Mr. Weller is just outside the cabin door, hanging on their every word. \"Listen...\" he is about to say, \"Tabatha\" but decides not to call the boy by any name for the moment. _The last thing this kid needs is for me to exacerbate an already dangerous situation_. Jason knows Mr. Weller seriously hates this boy. He has close ties with Elena Stiles, Crystal Albright's second mother. How to word this? he wonders. Jason Warith does not subscribe to the need to change the ward's name. Yet Mr. Weller made it very clear in their first meeting that this is the way things are done at the Northeast Camp, and under no circumstances is Jason to bring any of his high and mighty radical notions inside his camp. Indeed, Jason's first meeting with Gideon Weller was more than just an eye opener. He was read the riot act and informed beyond a shadow of a doubt that he, Gideon Weller, had not hired him. The only reason Jason Warith is a member of his staff is due to the political clout of one Destiny Stuttgart, Dean's grandmother. Aged eighty-four, Destiny Stuttgart is the last surviving original founding member. For the past twenty-three years, Destiny Stuttgart had thought her grandson dead, so when Dean voc'd her, she had almost refused to listen, thinking he was some sick pervert. It wasn't until Dean had called her Mimi, Dean having been the only grandchild ever to call her Mimi, that she turned on her voc camera and looked into the eyes of the gentleman who had once been the child she had deeply loved. When he explained his and Todd's current circumstances, she decided to take action. She had placed a personal voc to Adrian Adams at the Southwest Camp and asked him for the best man for the job. He had reluctantly recommended Jason Warith. When Jason learned the details of the case, he agreed, knowing full well he would be up against a battle. Rumors abound regarding the treatment of wards at Gideon Weller's camp and Dean Hunter's account only proved to Jason that his presence there was essential, not only for Todd Middleton, but it was critical if real steps were to be made in the reformation of reeducation. Cut out the malignant tumor first; then bombard the body with radiation if you truly hope to kill all the cancer. _But the tumor is still deep inside the brain_ , Jason reminds himself. _I haven't dug it out yet, and until I am able to, I have to tread carefully for this boy's sake_. At the moment, this is Gideon Weller's camp. He is in charge and his word, for the time being, is law! During their meeting, Gideon Weller made it perfectly clear, almost too zealously, that the Northeast Camp abides by the demerit system, and as soon as five demerits are given, punishment is swift and brutal. When Jason refused to commit any act that would physically harm a ward under his care, Gideon Weller vowed to be the man's shadow. \"Punishment,\" he stated unequivocally, \"will be administered whenever I deem necessary\u2014and I will step in to administer it if you lack the courage to act when necessary!\" Jason knows Gideon Weller to be a man of his word. He is standing outside the cabin door, waiting patiently for the opportunity to punish the Middleton boy. Focusing now on Todd, Jason continues, closing his eyes briefly to remember Weller's exact words, \"The reason Mr. Weller insists we give you boys new names is to help...to help you transition from the perception of being heterosexual back to a normal gay state.\"\n\n\"Why is being gay normal and heterosexual is not?\"\n\nIt is a fair question, and back at the southwest camp, Jason would have ignored the fact that it was expelled with such venom and force and answered the boy honestly. Unfortunately, Gideon Weller also heard the question and would only hear the aggravated taunt. Jason knows that the tone of Todd's voice means a second demerit. There is no doubt in Jason's mind that Gideon Weller has added that to the count. Grimacing, Jason knows only too well what lies ahead\u2014how Gideon Weller will emerge as soon as Todd hits the deadly number five. \"Neither state is unnatural\u2014\" he begins but is cut short by Todd.\n\n\"Then why am I forced to be here?\"\n\n_Three_ \u2014 _damn!_ \"Please, stop speaking and just listen!\" Jason had hoped to make a connection with his charge before Mr. Weller came in with the discipline. If he doesn't reach the boy first, he won't have any hope of ever getting Todd to trust him.\n\n\"Fuck you!\" Todd expostulates.\n\n_Four!_ Jason places his face in his hands. He hasn't even had a chance to explain the demerit system and what is about to happen. Frustrated, he begins rubbing his hands down his face as if to ease tension. Suddenly, like a silent explosion, he grabs the back of Todd's head and gags his mouth with his other hand. \"Shut up and listen,\" he whispers. \"Mr. Weller is listening to everything you say. He wants to hurt you, and I'm the only thing standing between you and him. You have to let me help you. Will you please just let me explain to you about this institution\u2014\" Jason yelps as Todd bites into his hand. Although no blood is drawn, there are some pretty deep teeth marks.\n\n\"Fuck you! And fuck this whole fucking institution!\" Todd shouts.\n\n\"That's five,\" says Mr. Weller. Before Jason can even think the number, the man is standing inside the door of the cabin grinning.\n\n\"Mr. Weller,\" Jason intercedes, \"give me a minute. I haven't even had a chance to explain the rules to him.\"\n\n\"He understands enough to know that his place is to obey, to listen...\" Weller pauses, noting Jason cradling his hand, and having heard the yelp, inferring the damage done, adds with a cynical smile, \"and be respectful.\" Now standing at the small table in the room's center, rubbing the table softly with his hand, Mr. Weller demands, \"Bring him here.\"\n\n\"You have five demerits, Tabatha,\" Jason begins. \"You get a demerit every time you are unruly, disobedient, you swear\u2014\"\n\n\"Enough explanation!\" Mr. Weller's voice is dry and curt.\n\n\"He should at least be told what is about to happen and why.\" Jason has never ascribed to the demerit system. At the Southeast Camp, he was able to pilot a new, more gentle approach, one that invites the young man back into the homosexual fold, rather than berates and beats him there. Although he knew the Northeast Camp advocated corporal punishment, Jason was disappointed to learn that Gideon Weller actually utilizes it on a daily basis.\n\nStaring straight ahead at the wall, Todd struggles not to shake. He remembers Matthew's face when Stephie threatened him with a fifth demerit. _It doesn't matter_ , he thinks. _I'm not giving in!_\n\n\"Bring. Him. Here.\" Each word is punctuated harshly with a soft tap of his index finger against the table.\n\nAs Mr. Weller's order is not to be gainsaid, Jason motions for Todd to stand. When Todd refuses to acknowledge him, Jason grips his arm. Todd yanks it away, looking up at the man with sullen hate. Jason can't bring himself to do this. \"No,\" he insists. \"It's too soon after his arrival to punish him. All the research shows that a softer, gentler approach works better than\u2014\"\n\nMr. Weller is having none of this. \"We do not pamper our wards here, Mr. Warith!\" Then turning toward the door, he yells, \"Darrell!\" A tall muscular man emerges through the cabin door. Watching as his henchman walks over, Mr. Weller tells Jason, \"I knew you were too soft for this job, but it appears Dean Stuttgart has some pull after all.\"\n\nJust as Dean Stuttgart's name begins to register in Todd's brain, it is yanked away instantly when Darrell reaches across the bed and grabs him by the hair. Todd yelps as the large hand pulls him backwards. In one swift motion, Darrell lifts the youth off the bed and tosses him face first over the table.\n\nSmiling sadistically, Mr. Weller adds, \"Fortunately, Dean Stuttgart is still an embarrassment to his family so his hand only extends so far.\" Jason turns toward the door. Arresting Jason with the shot of a finger, Mr. Weller orders, \"You stay right where you are, Mr. Wraith! This boy is your charge, and you are expected to be present for all his reeducation treatment.\" With a judgmental air, he states, \"It should be you doing this, but as you are not up to the task, I shall take over.\" Mr. Weller takes a moment to enjoy watching Todd's hands flail fruitlessly at Darrell's grip before ordering him to undo his pants.\n\nTodd screeches out, \"Fuck you!\"\n\nMr. Weller doesn't reply. He merely crosses over to the wall behind the table to retrieve the paddle hanging there, only to return instantly to slam Todd's backside with it. Appreciating the boy's howl, Mr. Weller waits patiently for it to diminish into a whimper. \"Nine more with your pants down, Tabatha. Twenty more with them up.\"\n\nCrossing over to the table, bending down level to the boy's ear, Jason implores, \"Please, Todd, I can't stop him.\" Closing his eyes briefly, he begs, \"Let's not make this worse.\"\n\nTodd's mind is reeling with pain. _Worse? How can this possibly get worse?_ Fumbling for his pants button, he fails to release them fast enough for Mr. Weller. Another smash of the paddle hits his backside. Todd's body convulses as he shakes and screams. Then, wailing out in agony and frustration, he pleads, \"I can't. I can't. You won't let me.\"\n\nJason stops Mr. Weller from delivering a third blow by grabbing his wrist mid-swing. He stares directly into the warden's eyes. Both men threaten the other. \"I'll help him,\" Jason insists. Mr. Weller struggles against Jason's grip.\n\nHis lackey, however, cannot aid him since he is currently occupied with keeping a tight grip on Todd's hair. Jason only lets go of his grip when Mr. Weller releases the tension in his arm, showing his willingness to lower the paddle. Jason bends down and undoes Todd's pants, lowering them to his ankles.\n\n\"His underwear, too.\" It is not enough simply to beat the boy. Mr. Weller wants him degraded beyond recognition of his former self. Although Jason wishes to protest, he suspects Mr. Weller will only make things harder for Todd. After his briefs are dangling past his knees, Mr. Weller begins administering the punishment. A third smack is called out as \"One!\"\n\n\"Three!\" Jason reminds him.\n\nAgain, the two men stare. Oddly enough, Mr. Weller concedes. \"Three.\" Before taking his fourth swing, though, he pauses to inspect the paddle. As he had hoped, as is almost always the case (the drug induces such violent vomiting that inevitably something always shoots out the back end), there are traces of feces on the paddle. He shows it to Darrell, whose grin resembles a rabid hyena. As Darrell shoves Todd to the floor, pushing him into Jason in the process, Mr. Weller spits out his disgust, \"You dirty _strai_ rat!\" Shoving the paddle into Todd's face, he demands, \"Didn't your fathers teach you how to wipe your ass?\" Looking now at Jason, who has just recovered from the shove, he orders, \"Get this dirty little charge of yours a basin of water, soap, and a washcloth.\" When Jason fails to react on the instant, Mr. Weller explodes, \"Now!\"\n\nJason leaps at the order, fear chasing him into the washroom. _Hadrian's lover_ , he marvels, _if the man has me this frightened, how must the boy feel?_ Todd, in his turn, is lying on the floor, shaking in horror and pain. Within moments, Jason is back. He places the basin next to Todd, rubs some soap on a cloth, and is about to begin washing the boy's behind when Mr. Weller stops him. \"One does not stoop to wiping one's ward's ass here, Mr. Warith! The boy must do that himself.\"\n\nJason, with infinite tenderness, helps Todd into a child's pose on his knees. He hands him the cloth, guiding his hand to where it needs to be. All the time, he whispers soothingly into Todd's ear. \"I will get you out of here, Todd; I promise.\" But Todd can't hear. His mind is swirling inside an abyss of shame and degradation. He is so consumed by anguish and horror that he can't even accomplish the simple task of wiping himself. Jason ignores Mr. Weller's order and assists the boy. He holds Todd's hand and makes all the necessary motions until he is sure the area is clean. \"Let me help you back up, Todd.\"\n\nJason's whisper, unfortunately, is loud enough for Mr. Weller to hear. \"Have you forgotten his name already, Mr. Warith?\"\n\n\"Come on, Tabatha,\" Jason concedes. _No point making things worse_. Lifting Todd to standing, he adds, \"I'll help you back to your bed.\"\n\n\"You will help him back to the table! His punishment is not over.\"\n\nJason, without turning Todd, looks back at Mr. Weller. \"I am the boy's charge, and I say he has been punished enough for his first offense.\"\n\n\"Mr. Jason Warith,\" Mr. Weller orders, \"walk your ward back over to that table. He still has seven hits left. Refuse and I will add the additional twenty.\"\n\n_Additional twenty?_ Jason looks deep into the headmaster's eyes. Fear begins to scalds him as if he were slowly being lowered into a vat of hot oil. Ironically, a cold shiver rushes up his spine. It dawns on Jason just how dangerous Gideon Weller is. _Hadrian help us; the man is insane_. Not knowing what else to do, and fearing even worse for Todd, Jason Warith turns and directs his ward back to the table. When Todd is placed in front, he literally falls face first onto the tabletop. This time, Darrell grabs Todd's hands to hold him steady. The beatings continue to the count of ten. Mr. Weller, handing the paddle to Jason, demands, \"Wash it! And then get the boy ready for indoctrination! We welcome him into the fold at seven.\"\n\nJason is stunned. The instrument is bloody, the violent blows having burst veins in the boy's anus. Walking in a daze, Jason enters the washroom.\n\nLooking at his loyal aide, Weller nods. Darrell releases Todd's hands and the boy falls to the floor like wax dripping off a candle.\n\nJason races out of the washroom as soon as he hears the thud of Todd's body hitting the floor. Through his peripheral vision, Jason catches the sight of Weller and Darrell walking out the cabin door. Directing all his attention to the youth on the floor, he says, \"Todd, Todd, my boy.\" He receives no answer.\n\n* * * * *\n\n#  _Salve!_\n\n**The Error of Their Ways  \nHNN\u2014Melissa Eagleton Reporting**\n\nEvery so often our government reminds HNN of our responsibility to keep Hadrian's citizens informed of the conditions of the outside world. Although we are all aware of the ghastly conditions that exist in other parts of the planet, it never hurts to refresh our consciousness so that our vigilance to maintain and create both a stable human population as well as preserve and resurrect earth's natural resources never abates. It is hoped that a look at the misery wrought by humanity in the outside world will help quell our youth's foolish behaviors; many of our youth persist in experimenting with heterosexual behavior, reminding them yet again of the world's desperate need to stem the tide of world population.\n\nLast year's world report concentrated on the East Asian countries with a strong emphasis on India, perhaps the world's hardest hit, having exceeded 2.5 billion according to the most recent World Pop Clock. Although much of our emphasis has been placed on this country due to its high degree of disparity, Hadrian's government has asked we look at the dwindling state of our closest neighbors; in particular, the splintered anarchy that was once the United States of America.\n\nThere is no doubt that many of Hadrian's citizens feel those who used to be American citizens have received their just deserts, but it is hard to hold onto resentment when one views from our satellite cameras the horror of their current existence. Yes, believe it or not, these people once reigned as the world's most dominant power. To think of all the good they could have done to help protect our planet had they but only listened to their own prophets.\n\nOne such prophet was a man by the name of Chris Hedges. He wrote back in the early twenty-first century. I dug up one of his articles from Hadrian's editorial archives. This piece is titled, \"We Are Breeding Ourselves  to Extinction,\" March 9, 2009. Although it is questionable whether or not humanity still had time to stop its onslaught of destruction, this man at least had the wherewithal to warn humanity of its unfortunate direction. His words also inspired pre-founding family members who established the process of purchasing and creating our own country. Listen to how he opened his article: \"All measures to thwart the degradation and destruction of our ecosystem will be useless if we do not cut population growth.\" Why did humanity ignore this man's voice and the voices of others like him! As we all know and can attest to by virtue of the images of the swarming hordes only two borders below us, that warning was not heeded. More damning evidence against what used to be the United States of America, what used to be the \"dominant world power,\" are echoed in Hedges' words: \"The United States alone gobbles up about 25 percent of the oil produced in the world each year.\" Another key point Hedges reminds us of is that, even though countries like old Canada and the old US of A were not among those countries overpopulating the planet (at this time), their populations continued to grow due to the ever-growing need for immigration. It was these old countries' desperate attempts to stem the excessive waves of the human tide crashing against their shores that wrought many of the bloody conflicts of the mid to late twenty-first century.\n\nHedges warned us then, \"The overpopulated regions of the globe will ravage their local environments, cutting down rainforests and the few remaining wilderness areas, in a desperate bid to grow food.\" Hedges also pointed out that which every \"developed\" country but Hadrian now lives with: \"the depletion and destruction of resources will eventually create an overpopulation problem in industrialized nations as well.\" And indeed it did!\n\nToo few listened to Chris Hedges and other learned men and women like him. But Hadrian's pre-founding members heard! And, Hadrian still listens. It was \"Earth First!\" when Hadrian's borders were created, and it is \"Earth First!\" to this very day. The human species and the earth will survive, but in order for this to happen, we must all sacrifice. We must all sacrifice for Hadrian. We must all sacrifice to reclaim our beloved Mother Earth.\n\n_Vale!_\n\n# Indoctrination\n\nJason does not leave Todd's side. Even though the boy is non-responsive, Jason is determined to help him up and onto his bed. He begins by lifting Todd up by his armpits and instructing the lad to step out of his pants and shorts. Todd obeys but says nothing. Helping him over to the bed, Jason ensures Todd is lying face down with his red, swollen buttocks exposed to the air. \"There's some salve in the bathroom,\" Jason says soothingly. \"I'm going to get it and put some on your backside.\" Todd isn't even listening. His mind has entered a dark state\u2014a place void of thought and emotion. Cold and black, it is a place into which Todd can escape\u2014or embrace\u2014insanity. As soon as Jason returns with the ointment, he warns Todd, \"It will sting at first, but it will soon become soothing.\" After applying the ointment, Jason pulls a chair next to Todd's bed where he holds vigil. Humming softly a song his mama used to sing to him when he was sick and couldn't sleep, he gently runs his fingers through Todd's sweaty and matted hair. The boy's eyes are open, but at what they stare, Jason remains oblivious.\n\nNeither Todd nor Jason are aware of time lapsing. Jason is startled back into the present when a small icon appears in the far right corner of his vision. Focusing on it, Jason blinks. \"Yes\u2014I know\u2014He's in no condition\u2014It will simply have to wait\u2014No!\u2014I said, NO!\u2014No, you listen to me; this boy is my ward. I am his charge; he is in no shape\u2014Be reasonable\u2014What is wrong with you?\u2014Fine!\u2014Yes!\u2014We're coming!\" Annoyed, Jason blinks to sever the connection. Looking down at the boy splayed unconscious before him, he gives him a gentle shake. \"Todd, I'm sorry, son, but\u2014\" He stops as Todd moans. Closing his eyes, he takes a deep breath. Shaking his head, he wants very much to defy Mr. Weller. _But that will only get me fired_ , he reasons, _and then Todd will have no one to help him survive this_. \"I am so sorry, son.\" It is a joke of an apology and Jason knows it. Todd does not acknowledge any of what Jason says. He has fallen back into the black void of his unconsciousness. \"You have to get up.\" Jason closes his eyes, swallowing self-loathing. \"I have to get you dressed.\" Re-opening his eyes, Jason looks around for Todd's pants. Picking them up, he notices how tight they really are. He shakes his head, _stovepipe jeans_. There is no way he will put these back on Todd in his condition. Tossing them aside, he rummages through Matthew's drawers, hoping to find a pair of soft loose-fitting cotton pants. He settles on a pair of fleece pajama bottoms. _These will have to do_ , he reasons. He doesn't even bother to make Todd stand. He slips the soft pants under Todd's feet, drawing them up over his legs, taking extra precaution when he lifts them over his buttocks. Even so, Todd groans in agony. As he picks the boy up off the bed, Jason tries to help him stand. Whether it is due to physical agony or simply a lack of will, it is evident Todd will not be walking. Jason lifts him up into his arms and carries him over to the central cabin. Todd rests his head on Jason's shoulder, causing the older man to wonder whether he still has a chance of winning the boy over.\n\n* * * * *\n\nThe main cabin is in the center of the complex. All other cabins, those designated to the various wards, encircle it. This much larger log building houses administrative offices, the kitchen galley, cafeteria, small \"social rooms\" for a charge and ward to conduct private sessions, as well as a large meeting area for indoctrination, dances, parties, sometimes even weddings. It is to this room Jason Warith was instructed to bring Todd for his indoctrination.\n\nIndoctrination consists of the current wards welcoming the newest ward to their commune and reciting, by rote, the rules and rationale for reeducation. For this purpose, the room is set up much like an auditorium in which the new ward's chair is placed front and center while the other wards are seated semi-circle audience-style, facing him. Behind each ward's chair stands his guardian. Currently, seventeen wards are in attendance at the Northeast Camp. Todd Middleton brings that number up to eighteen. Behind Todd's chair is a small riser upon which sits the camp warden, Mr. Weller, his administrative team, and two psychiatric counselors.\n\nMr. Weller glares at Jason. \"Put your ward on his feet and bring him to the front.\" Jason ignores the man's pointed request and carries Todd forward. Such blatant disregard for his authority bristles down Weller's spine. \"Set him down, Mr. Warith!\" Jason\u2014suspecting he has already pushed too many buttons for his own\u2014and his ward's good\u2014helps Todd to standing, turning him to face his audience.\n\n\"Sit him in his chair,\" Weller orders.\n\nJason stares straight ahead, seeing a mixture of animosity, concern, and respect in the other guardians' eyes. He also notes the look of shock and awe from the small collection of wards. Instead of seating Todd, Jason cradles the youth in his arms.\n\nSensing Warith's refusal to comply, Weller contemplates disciplining his newest employee (a man he never hired but had thrust upon him, no doubt by some vestigial influence Dean Stuttgart still has) right now in front of everyone, but he reconsiders. Warith is being hailed as the \"golden boy\" and leader of the reformation of reeducation, but Weller thinks, _Reform is not what is needed to quell the heterosexual male. Discipline and a strong hand are all that have ever worked, will ever work, with the male who envisions himself the aggressor_. Still, now is not the moment to fight Warith. He will settle with Jason Warith later. \"We shall begin.\" Mr. Weller's voice booms and echoes throughout the room with over-exaggerated strength and confidence. He is determined that no one charge or ward will take advantage of Warith's show of defiance to treat him similarly. \"Wards, let us welcome Tabatha Middleton to our family.\" There are murmurs of awe. Although he is introduced as Tabatha, Todd Middleton is recognized by everyone because his face and name have been splashed all over _Salve!_ for the past month due to his team winning Quadrants and the Nationals and his walking away with the Most Valuable Player award. Their moment of solemn respect is shattered when Mr. Weller orders, \"Mattie Malloy, as you are Tabatha's roommate, stand and begin.\"\n\nStephie beams. This is a great opportunity to show Papa Gideon how far he's come with Mattie. Tapping Matthew's shoulder, he whispers excitedly, \"Stand up, Mattie; stand up.\"\n\nObeying, Matthew begins, \"Please remember your guardian is here to help you.\" Stephie smiles. Matthew remembers everything so well. \"He will guide you through this transition, remind you of the cornerstones of Hadrian's society, and explain how the loss of one foundation means the loss of our society's strength.\"\n\nStephie is thrilled. \"Oh, Mattie,\" he coos, \"that is so good.\" Stephie's beam is seraphic. Mr. Weller rewards him with a smile of reassurance. \"What next, Mattie? What next?\"\n\nAlthough focused on Todd, Matthew knows every word he says is falling on deaf ears. It is obvious Todd is already beaten. As Matthew's mind is no longer needed to recite the rules by rote, his thoughts reel back to his first day. His once defiant will earned him five demerits before he even knew what demerits were; these were followed swiftly by the paddle. He shudders at the thought. Everything he does these days is to avoid further encounters with the paddle. He drones on, \"Your guardian will guide you through the dangers of heterosexual behavior. How heterosexual men are violent in nature. Always abusive and cruel, it is not uncommon to find rapists and pedophiles among them.\"\n\n\"Oh, Mattie,\" Stephie sings, \"I am so proud of you!\"\n\n\"Your guardian will help you purge these negative tendencies; guide you toward embracing the more passive loving qualities of our nation.\"\n\n\"Oh, Mattie,\" Stephie is beside himself. \"You did so good.\" As Matthew resumes his seat, Stephie pats him proudly on the back, kissing the top of his head. Matthew looks straight ahead, acknowledging nothing.\n\n\"Jamie,\" Mr. Weller commands. His guardian yanks the young man to standing. \"Inform Tabatha of our daily routine.\"\n\nWhen he fails to respond instantly, his guardian slaps the back of his head. \"Rise at six,\" he stammers out. \"Breakfast at six-thirty\u2014exercise at seven\u2014classes at eight\u2014lunch at eleven-thirty\u2014exercise again at twelve\u2014rejecting women at one\u2014private session at three\u2014supper at five\u2014visual of the outside world at six\u2014private session at seven\u2014bed by nine.\"\n\n\"Very good, Jamie. Sit down.\" Glaring at the back of Jason Warith's head, Mr. Weller growls, \"Mr. Warith, turn your ward, and his chair, around. Tabatha has missed supper, but he is in time for tonight's visual of the outside world.\"\n\nAfter Jason encourages Todd to turn and face Mr. Weller, the warden bores down on the youth. \"Tabatha, as you have just heard, we run a very tight ship here at the Northeast Camp. There is no tolerance for rebellion. If you listen to your guardian and obey the rules, your transition back into Hadrian society will be swift and smooth. If not, know this; disciplinary measures will be used to quell all acts of defiance and heterosexual behaviors. We are not going to threaten you with hell and damnation in some foolish afterlife. No, Tabatha. Hell and damnation is out there, on the other side of those walls. It is Hadrian government's strict policy never to send our children out there to fend for themselves. We will either aid in your transition, or retain you here since the only other choice left to you would be to live outside our walls. Today's visuals will remind you why that alternative is not a viable option.\" Now directing stern attention to Jason, he commands, \"And now, Mr. Warith, your ward must sit down!\" Each word is spoken emphatically, leaving no room for resistance.\n\n* * * * *\n\n#  _Salve!_\n\n**The Twenty-Fifth Anniversary of the Terror Attack  \nHNN\u2014Melissa Eagleton Reporting**\n\nIt was twenty-five years ago today that a group of well-organized Christian extremists broke through Hadrian's southwest wall. Although well-fortified, those fanatics somehow managed to gain entrance. It is believed one of our soldiers was in collusion with the enemy. Although Corporal Victor Williams was also killed during the attack, it was from his watchtower that the invaders should have been detected. Williams' family, though, ardently defends him, claiming the young man was a patriot and not, as many suspect, a heterosexual sympathizer. Even so, a number of Williams' peers were concerned about his sexual preference. Williams did not date, and a few female soldiers claimed he was creepy. \"His eyes follow you everywhere,\" one female soldier had said in a complaint registered to a senior officer. No formal action was ever taken against Private Williams, though, as no tangible evidence regarding heterosexual tendencies was ever produced.\n\nIf Williams were a traitor, he allowed over fifty heterosexual barbarians inside our wall to murder all the soldiers of his compound. Of the fifteen men and eighteen women stationed at the southwest gate that day, all but one died. All the male soldiers were shot, and those women who survived the initial attack were brutally raped and murdered. Footage of the event was found on the body of a dead enemy renegade. These evil brutal animals filmed most of their atrocities, leaving behind a tape that was spine-chilling to say the very least. The torture and raping of the eight surviving female soldiers was cruelty beyond belief.\n\nOne soldier survived this brutal attack: Captain Tanya Weller. Before being killed, Corporal Regan Hillier managed to get a message over the wave asking for reinforcements. By the time two heavily armed battalions arrived, though, everyone but Sergeant Weller had been murdered. The heterosexual barbarians were in the process of brutally raping the poor woman while others stood by filming the event. What kind of sick people would film such atrocities? What sort of sick pleasure were they hoping to gain from watching themselves murder, torture, and rape others? Did they actually believe such a film was for posterity?\n\nEven though Sergeant Weller's life was saved, her mind was completely destroyed by the attack. She had to be institutionalized and placed under twenty-four hour watch. She still managed to commit numerous acts of self-mutilation, and eventually, she was successful at taking her own life. Her suicide, as well as the attack against her compatriots, is a bitter reminder of how the outside world views Hadrian's citizens. We must never forget that the ultimate intention of the heterosexual world is to destroy us.\n\nEvidence of the world's hatred is clearly shown in the attack scenes filmed by the attackers. The images seen on this tape are so graphic it was decided not to publicize them; rather, we leave the viewing of them to the discretion of the individual and parents. Anyone needing to be reminded of the violent and brutal nature of heterosexual hordes can access this video, simply titled _Terrorists at the Wall_ , over the wave. One viewing of this horrific event is evidence enough to explain why so many citizens of Hadrian shrink with fear at the thought of the outside world breaking through our walls. Extreme religion rules the outside world, and we have been branded as sinners due to our sexual orientation. There is also extreme jealousy that we alone are able to control population and live within a self-sustainable community. The outside world is battering against our walls, desperate to get in, anxious to destroy us and take what we have worked long and hard to create and to preserve.\n\nHadrian's military will not allow that to happen. The men and women in our service are dedicated to preserving our freedom and saving what little livable earth there is left. We will not allow the heterosexual barbarians, their unnatural hordes, ever to destroy the last vestige of life left on this planet!\n\n_Vale!_\n\n# Gideon's Obsession\n\nEvery night after supper, since Todd Middleton's arrival at his camp, Gideon Weller retreats to his office to be alone. He falls into this routine every time a rapist enters his compound. While the wards are kept busy watching images of the outside world, Gideon Weller sits behind his desk, dims the lights, and with the blink of an eye, turns on the small, nearly invisible screen that pops up in front of his vision. Once again, he relives the horror of his mother's rape. He never uses the wall screen for this in case anyone should walk in.\n\nLeaning back now in his large cushioned, swivel office chair, his hand fumbles for his pants zipper. His mind balks, fights against his bodily urges, but they always prove too much for him. Tears stream as he watches four men grab his mother's limbs, pulling her in four opposite directions. The film is grainy as a result of the government having to convert the stream from outside technology to make it compatible with Hadrian's voc wave. After so many viewings, though, Gideon's eyes have adjusted so he doesn't notice the distortions anymore. A fifth man enters the screen. He uses a bowie knife to tear open Gideon's mother's uniform. The man doesn't even bother to strip her fully, only cutting away enough of the fabric to expose her breasts and pubic area. The fifth man steps back and swirls his index finger. The cameraman cackles as the four men holding onto Gideon's mother take their cues. It is grotesque to watch as they pull her in four directions and then spin her, even bounce her up and down, as if this were some children's game. The camera bounces and jerks along with its handler's laughter. Gideon begins to feel nauseous. Ironically, this feeling only adds to the sensations he is creating, sickening both his sense of self and his hatred for all heterosexual men. The men torturing his mother laugh as she screams in terror. Suddenly, the spinning stops and the fifth man, retrieving his penis, begins rubbing it hard. Gideon, too, rubs harder. When ready, the fifth man grabs Gideon's mother's buttocks in preparation for the thrust. His first attempt fails, to the delight of the cameraman, who once again jars the image as he laughs uproariously. The rapist yells at his companions to hold her tighter. Forcing his eyes wide open, Gideon watches as the four men each take a step back and then lean to hold her body taut. Gideon swears he can hear his mother's tendons rip. The rapist's pelvis begins its thrust motion. Gideon's pelvis moves in tandem. A sixth man enters. He yanks his mother's head back and down, forcing her mouth open, and begins to fuck her face. Her body convulses as she chokes clearly, giving the fifth man shivering sensations. Gideon, too, chokes. _Oh, mother, forgive me!_ The image is too much for him. He blinks it off. _I am not one of them_ , he chants to himself. Softening, alone in the dark, he blinks to retrieve the image. He needs the provocation. The four men holding Gideon's mother shout obscenities along with the cameraman. After both men come, they drop her on the ground and a cascade of urine waterfalls on her face while one man chooses instead to defecate. For this final act of degradation, her mouth is forced open to receive his \"special\" gift. Gideon's body is shaking, his head thrown back, his chest arched, his eyes raked with tears.\n\nIt is not his mother's rape to which he is masturbating. It is the image he has created in his own mind, his raping the men who brutalized her. The impossibility of this act is lost on Gideon at times like this. Even though all those men are long dead, it is not their faces he sees, but the face of the current rapist living inside his compound. It is Todd Middleton's face he envisions. It is Todd Middleton he watches raping and torturing his mother. It is Todd Middleton he aggressively beats and rapes in his own mind. The more violently they torture his mother, the more violently he tortures Middleton until finally, when the first bullet from the reinforcements blows off the head of the man defecating into his mother's mouth (the head of Todd Middleton), Gideon Weller collapses, moaning piteously. Slowly, his eyes close and the horrific image vanishes. Always, without thought, he uses his wet, sticky hand to wipe away the tears. After covering himself with his disgrace, he screams inside his head, _Those Fucking Heterosexual Bastards! I am not one of them! I am not one of them!_\n\nWhat few people know is that Gideon Weller had once, like Todd Middleton, been caught masturbating to straight porn. His mothers had, after a long discussion with him, decided the best course of action for him was to be reeducated at home. Gideon agreed. Being admitted to a government reeducation camp would have closed off too many doors for his future. As Gideon was scholarship material and destined for uni, he could not afford having a heterosexual stain on his record. From that time forth, he has dedicated himself to finding the latent homosexual within, but the need to eradicate everything heterosexual only came after his mother's brutal rape. The one, and only, time he had visited her in the institution, she had recognized him, but she only saw before her the heterosexual masturbator. She beat him, berated him, cursed him, and called him every ugly name she could muster. She even spat in his face. His whole life since has been dedicated to proving to himself that he is not straight. And, even though he has never formed this need for validation as a conscious thought, he is determined to make all straight men pay for what those murderous marauders did to his mother.\n\n* * * * *\n\n#  _Salve!_\n\n**The Reeducated  \nHNN\u2014Melissa Eagleton Reporting**\n\nAre we prejudiced against the reeducated? Few people would ever admit to holding animosity toward those of us who have required government assistance when dealing with sexual confusion. Our youth, who have been found acting on heterosexual tendencies and have experienced reeducation, reenter our society fully rehabilitated. One would assume that these individuals can return to life as normal, but there are a few occupations for which these individuals no longer qualify. To begin with, no woman after reeducation has ever been allowed to become a surrogate goddess. Our government's rationale is to avoid further contamination of heterosexual genes into Hadrian's gene pool. This makes perfect sense. If those individuals in reeducation have stronger heterosexual tendencies than the rest of us, it is only reasonable not to contaminate our future children's genes. The surrogate goddess not only provides the womb for our men's fetuses to grow in, but they also provide the eggs. Although sensible not to allow a reeducated woman to become a surrogate goddess, one does have to question why no reeducated man or woman is allowed to enter Hadrian's military, especially at a time when citizens are being asked to reenlist after their four-year tour of duty is over. Why then are the reeducated banned from active military duty?\n\nAngelo Martin, age nineteen, recently applied for and was refused active duty in Hadrian's military on the basis that he had spent eight months in reeducation at the Southwest Reeducation Camp. \"I really think it's unfair that even after reeducation, we should be discriminated against. I made a mistake. I know that now. I've been reeducated, and in two years, I will swear an oath of fidelity to Hadrian and our chosen way of life. I love my country, my family, my friends, and my planet as much as any other Hadrian citizen! I want a share in being responsible for our people's protection. It's ironic really. Those of us who have been reeducated are no longer a threat. We've come to accept Hadrian's lifestyle and fully understand its merits. We want to defend our country, too. I know I would be a good soldier if only my country would let me.\"\n\nLieutenant-General Birtwistle justifies the army's refusal to allow the reeducated to serve. \"We simply cannot risk having anyone who is likely to harbor sympathy for the enemy defending our wall. Just the idea that one of our own could betray us is enough to keep the reeducated off our recruitment list.\" When asked for evidence of someone with strong heterosexual tendencies actually aiding and abetting the enemy, reference was made to the brutal terror attack on the Southwest Gate over twenty-five years ago. Although Lieutenant-General Birtwistle admits no hard evidence was ever found against Corporal Victor Williams, many in the military still believe he was the one who helped the heterosexual barbarians to break through our wall. As always, it is important to note that the young corporal was exonerated posthumously of all charges. Yet, even just the suspicion that one of our own might well have been swayed to help outsiders based on similar sympathies is enough to bar the door of military service to those citizens who have been reeducated.\n\nMany professions in the areas of medicine, research, and education are also out of reach to the reeducated. I recently learned that our unis deny entrance to anyone who has a reeducation record. This situation is most unfortunate because some of these youth had great careers ahead of them. We most recently lost Todd Middleton to reeducation. You may remember him as the young b-ball star of Pride High's Panthers. He had been offered an early entrance scholarship to Antinous Uni, but that offer was revoked as soon as he was exposed and sent to the Northeast Camp for reeducation. He had even mentioned in an interview with me not too long ago about his wanting to become a bioengineer like his father. His dream had been to alter rice genetically so it would grow in our northern climate. Unfortunately, we will now never know if he could have been successful.\n\nEven with these avenues closed, there are many jobs still open to the reeducated. Many of these men and women, having fallen in love with and married their wards, choose to become stay-at-home spouses. Others go on to work in the trades. As well, the service industry is always hiring, and one does not have to disclose a history of reeducation in any of these jobs. Every job in Hadrian is important, from bioengineering right down to waste disposal. Those, in fact, who work in waste disposal are among the most important citizens in this day and age. It may not be a pretty job, but it is critical if we are to avoid future pollutants damaging our environment.\n\nSo don't despair. There is a world of work opportunities awaiting the reeducated. Hadrian loves you. And Hadrian still needs you. You have been welcomed back into the fold. Embrace your new life with joy and energy.\n\n_Vale!_\n\n# Quelling Dissension\n\n\"Enough of this,\" Gideon mutters as he reaches across his desk for a handful of tissue. After wiping his hand and face, Gideon Weller resumes his authoritative bearing. Sitting upright, he zips up his pants before flicking on his reading lamp. Blinking his left eye, he begins peering over the file he keeps open at all times. What he reads brings a sneer to his lips. Todd Middleton has proven to be just as unruly as Gideon Weller had hoped. He is fully aware of the unspoken charge against _Tabatha_. No man should ever be allowed to get away with rape. It never ceases to amaze him that some women haven't the heart to accuse the man, as is their right. Knowing firsthand how rape can tear a woman apart, Mr. Weller sees it as his personal mission never to allow a rapist ever to reenter Hadrian society, for once a man has raped a woman, he will commit the act again and again until he has damned all of Hadrian to a living hell. Every time a rapist comes into his camp, he finds himself compelled to watch his mother's horror over and over, witnessing again and again her bloody and bruised body, her heart-wrenching screams of agony. And then come the memories of her institutionalized life with the rending of hair, the self-mutilations that eventually led to the end of her life.\n\nGideon Weller has never forgotten the horror of his one visit to his mother after she was institutionalized. Not having seen her since the attack, Gideon was unprepared for how the violence had demented her. The nurse on duty the day of Gideon's visit had tried to warn him. \"Your mother is very unstable, Mr. Weller. I do not recommend you visit her. If you wish, I can turn her window on so we can see inside her room.\" Instinctively, he blinked the window light on, revealing a woman with unkempt hair, muttering and pacing.\n\nGideon, repulsed by the idea of staring at his mother like an animal in the zoo, demanded, \"Turn that off! My mother is not a gorilla!\"\n\nHaving forgotten whom he was addressing at the moment, the young man protested, \"No, she is more like a Tasmanian Devil.\" Noting the look of horror on Gideon Weller's face, the young nurse apologized, \"I am sorry to speak so cruelly about your mother but that is the way it is. She is dangerous.\" Although Captain Tanya Weller was a short, slightly plump, unassuming woman, her face had that rounded, pointed look of the small devil, not to mention the sharp incisors. \"She will lunge at you as soon as you walk through that door. She will dig her teeth into you, and more often than not, she aims for the throat.\" As a reflex, the young man rubs the side of his neck, caressing the scar. \"Thank Hadrian she never bit into the vein,\" the young nurse muttered to himself.\n\n\"I insist you shut off that window and let me in to see my mother,\" Gideon demanded.\n\nThe young man merely shook his head. \"That would be unwise. Right now, she appears unresponsive, but the very moment you walk through that door, she will lash out at you, certain you are one of the rapists.\"\n\n\"But she'll know me,\" Gideon insisted. \"I'm her son.\"\n\nStill doubtful, the young man held off unlocking Tanya Weller's door. \"We usually only allow women in there. The last male nurse who went in there, well,\" he said, again rubbing at the scar on his neck, \"she left some mighty nasty scars on his face.\" He shuddered as he spoke. \"Nearly tore out one of his eyeballs. Left him blind on the left side.\"\n\n\"She's my mother,\" Gideon persisted. \"She would never hurt me.\"\n\n\"You are a man!\" The nurse's eyes opened wide in emphasis. \"She always hurts men.\"\n\n\"I am her son!\" Although Gideon's confidence had been shaken, he was determined to see his mother. \"I'm family!\"\n\n\"Oh, sweetie,\" the young man lamented. \"You have no idea what your mother's been through, do you?\"\n\n\"I know she was brutally beaten and raped.\" Gideon had been close to shouting by that point. He had been kept away from his mother for too long. His Mama Rena had repeatedly dissuaded him from making the trip and successfully kept him away for over a month. \"Honey,\" Mama Rena had tried to reason softly, \"your mother is not responding well to men.\" Gideon refused to believe his own mother would reject him. \"But, baby,\" Mama Rena had pleaded, \"she's not the same woman she was before the attack.\" But nothing Mama Rena said had deterred Gideon from visiting his mother.\n\nAfter a month of battling with Mama Rena, Gideon had been beyond frustration at being forestalled by this nurse. Tired of being kept in the dark, Gideon had started to yell, \"Let me into my mother's room now!\"\n\nThe young nurse sighed, \"All right, but you shouldn't go in there alone. Let me voc security first.\"\n\n\"I do not need security to visit my own mother. Let me in her room NOW!\" Gideon had been adamant.\n\nThe nurse's lips pursed, his head shook, but he had turned and done as Gideon Weller instructed. As soon as Gideon had entered the room, the nurse had locked the door behind him. Under no circumstances was Captain Tanya Weller's room ever to be left open. Her escape, it was feared, would lead to the mass murder of men. As soon as the door was sealed, the nurse had blinked and put in a call to security. \"Get up here fast; there's going to be trouble\u2014that's right, the Weller woman\u2014her son\u2014I had no choice\u2014he was adamant! For the love of Hadrian, get up here now!\u2014I can't\u2014I won't go in there! She'll kill us both!\" Before he had even had a chance to blink the connection off, four female security guards, accompanied by a female nurse with syringe in hand, had entered Tanya Weller's room. Tanya Weller was sedated and her son, Gideon Weller, was laid out on a stretcher and raced off to emergency.\n\nThe attack against her son had been brutal. Captain Tanya Weller was career military. Though short and stocky, she had excelled at hand-to-hand combat. The instant her son had entered the room, she had smiled. \"Gideon, is that you?\" But before he could even say, \"Yes,\" she had slammed her knuckles into his larynx, shattering his Adam's apple. The damage to his vocal chords provided him with the deep raspy voice that haunts many a ward.\n\nTanya Weller had watched as her son sunk to his knees. Gasping for air, he had looked up at her in confusion and fear. Captain Tanya Weller was on high alert, every muscle taut, ready to attack, but first, she had wanted to taunt her prey. \"You dirty _strai_ rat!\" And then she had sneered. \"You think I don't know? A mother always knows!\" Then she had smashed her fist into his face, breaking his nose. Gideon had collapsed to the floor, bleeding profusely from both nostrils. Tanya Weller had successfully managed a kick into his groin just as the door was opened. She had then been tackled, pinned, and injected with a very strong sedative, knocking her out, but not before she had spewed out damaging insults toward her son.\n\nGideon shudders at the memory. As always, he tells himself, _I am not one of them! These men, these heterosexual rapists_ , he repeats over and over, _are the very cancer eating at the heart of Hadrian society, and they must be extricated!_\n\nThe worst part about having a known rapist in his camp is knowing that boy is currently being sheltered by that _love and hug and nurture boys back to homosexual health maniac!_ Jason Warith is far too easy on the boy, allowing for far too many demerits to slip past without punishment. The boy has already been here for three days, and only once has the paddle been administered. _If I only had the time to follow them everywhere!_ His administrative duties, unfortunately, are endless and demanding. It is very bothersome, though, knowing Tabatha is the only ward not to suffer punishment for obvious misdeeds. Warith's leniency toward his ward is starting to nurture seeds of discontent among the other wards whose guardians still adhere to camp rules. And word has gotten back to him that Warith refuses to call Tabatha by his new name, again, spreading dissension among the other wards. Mattie Malloy had to be beaten twice yesterday for refusing to answer to his proper name. Poor Stephie was shaken. Gideon Weller had had to step in and finish the first job for him. When Stephie came to him in tears, informing him of Mattie's tenth demerit of the day, Gideon Weller had ordered his second punishment to be public; a punishment he determined to deliver. Everyone in the camp was ordered to the central meeting room. This public display of humiliation was designed specifically to shut Mattie Molloy down and stem the tide of insurrection Warith had instigated against his rule. And it proved to be very effective. Not a single word of dissent has been heard from Molloy or anyone else since.\n\nGideon Weller smiles, remembering just how effective this act was. He had ensured Jason Warith and Tabatha were given front row standing (no one was seated for this ritual). With paddle in hand, Weller had wielded it with all his might against Mattie Molloy's already raw and swollen buttock. Having completed the task, he had turned, pointing the paddle (blood dripping from it onto the floor) at his appalled audience, and shouted, \"There will be no more dissension in my camp!\" His voice had boomed and echoed ominously off the walls. Looking to the top right corner of his eye screen, he had ended the session with an order, \"Private sessions, NOW!\"\n\nJason Warith and Todd had been the first to move. Jason had roused his horror-struck ward into action with three simple words, spoken loud enough for everyone in the room to hear, \"Let's go, Tabatha.\" Jason had been quelled and was determined to toe the line from now on\u2014in public at any rate.\n\n* * * * *\n\n#  _Salve!_\n\n**Debate over Corporal Punishment  \nHNN\u2014Melissa Eagleton Reporting**\n\nA heated debated has risen over the use of corporal punishment inside reeducation camps. As we all know, corporal punishment within our educational institutions is illegal. No teacher would ever think of raising a hand against his or her students today. And, yet, the use of corporal punishment is legal in reeducation camps. It has recently come to the attention of HNN that Jason Warith has requested an investigation be held at the Northeast Reeducation Camp regarding the excessive use of corporal punishment. This request came in the form of a letter complaint written by Jason Warith against Gideon Weller. Jason Warith claims to have witnessed Gideon Weller beating students' backsides with a paddle on more than one occasion, and the beatings were so harsh the man actually drew blood. Warith also states in his letter of complaint that the guardians under Gideon's employ have no qualms following the man's lead. \"Young boys are beaten daily at the Northeast Reeducation Camp,\" Warith wrote in his letter to the reeducation board.\n\nWitnesses are being called forward to speak of these events. Many speak in Gideon Weller's favor. Darrell Jeffreys supports his employer, stating, \"These are no ordinary youth we are asked to care for. You can't coddle them. They are violent and abusive when they arrive and only a hard hand teaches them how to obey.\" Other guardians add support, even Stephie Chatters, the young man I interviewed when I toured that facility. \"I hate using the paddle,\" Stephie admits, \"but there are times when it is the only way to deal with noncompliance. Sometimes these boys can be really naughty.\" When asked if it was true that Mr. Weller had instructed his employees to beat their wards at least once a day, Stephie Chatters denied the charge emphatically. \"Oh, no, we are only to use the paddle if a boy earns five demits, not before! Usually you only have to use the paddle once or twice and then they stop misbehaving.\"\n\nThe question of five demerits earning a boy a paddling such as described at the Northeast Reeducation Camp is also being questioned. What exactly earns a boy a demerit? Does the punishment meet the crime? Is such physical force necessary to help reeducate our youth back into Hadrian's loving arms? Jason Warith says no. Gideon Weller insists the answer is yes. No one at HNN is prepared, at this moment, to judge the good folk of the Northeast Reeducation Camp. As none of us has ever had to deal with a young man who believes himself to be heterosexual, we cannot possibly know what it is like having to reeducate these recalcitrant and potentially violent youth. So, we turn to you, Hadrian, and ask for your views. Voc us @HNN#CP-RE\/MH and tell us what you think about corporal punishment. Is it a necessary component of the reeducation of our male youth or should its practice be abandoned? We look forward to hearing from you.\n\n_Vale!_\n\n# Private Session\n\nAlthough memories of having squashed Mattie Molloy and seeing the entire camp shaken please Gideon Weller, he still feels disturbed. An edge of uncertainly has entered into his world. Shaking his head, Gideon Weller resolves that there will be no more niceties, no more leniencies with this Tabatha Middleton! _This one_ , he determines, _will suffer_. With a harrumph, he determines, _Vigilante justice must occur since the system has chosen to coddle this one_. Rapists, as far as Gideon Weller is concerned, need the harshest treatment of all. They need to learn what it feels like to be violated by another human being the way they violated his mother!\n\nBlinking open the day's schedule, Gideon Weller notes that Jason Warith and Tabitha Middleton have Room 3 booked for their evening private session. Gideon Weller knows Warith _hasn't the balls_ to do what is necessary, so he determines to step in and take charge. Whispering, \"Darrell,\" he waits until a connection is made. \"Darrell, meet me outside Room 3.\" Blinking quickly, he severs the connection and makes his way out of his office and down the hall. Standing outside Room 3, Gideon Weller waits until Darrell Jeffreys arrives before taking any action. Quietly, he opens the door to Room 3 a crack, just enough to hear what is being said inside. Sickened, Gideon Weller realizes that Warith is listening to the youth talk about his attraction to Crystal Albright\u2014the very girl he had raped! With a simple nod of the head, Weller directs Jeffreys to begin the attack. Darrell reacts instantly. He knows his instructions by rote\u2014rip open the door, grab the guardian (in this case Warith), thrust him from the room, slam the door shut, and lock it. His next order of business is to hold the youth in a headlock while Gideon Weller administers medicinal intercourse. Jeffreys always closes his eyes, so he doesn't have to watch.\n\nJason's first reaction, like many a guardian before him, is to try to re-open the door, but it is locked. Hearing Todd screaming, Jason frantically pulls at the door to no avail. He bangs on it, slams his shoulder against it, shouts down the corridor for help, all to no purpose. He does everything he can think of to get back inside. When the screaming finally stops, Darrell opens his eyes and waits for Gideon Weller to nod before releasing the boy and then unlocking the door. As soon as the door gives, Jason pushes his way into the room, shoving past Darrell, just in time to see Mr. Weller beginning to stuff his penis back inside his pants. Todd, he discovers, is lying senseless on the floor, his pants and shorts a tangled mess around his ankles. He kneels at Todd's side. The boy's anus, Jason notices, is bleeding. No doubt, the internal damage is severe. Weller has clearly bludgeoned the boy. Looking up at Weller, Jason notices blood on the man's hand. His eyes widen in horror. _Hadrian's lover_ , he marvels, _did he use his fist?_ Raging on the warden, Jason punches Mr. Weller in the head. Stunned, the man flies backwards, with his still hard cock sticking up in the air as he crashes to the floor.\n\n\"You will pay for this,\" Mr. Weller growls after regaining his senses. Returning to his feet, he snarls, \"You are fired.\"\n\nJason's voice is a low rumble. \"You were raping this child!\"\n\n\"That child,\" Mr. Weller yells out, pointing down to the floor, \"is the rapist! He raped Crystal Albright! He managed to avoid the death penalty, but I'll be damned if he doesn't learn what it feels like to be violated the way he violated that poor girl.\" Soft now, Mr. Weller tucks his penis back inside his pants and zips them back up.\n\nJason turns, bends down, and begins gently to lift the boy's briefs and pants up. \"Leave him be,\" Mr. Weller orders.\n\nJason's voice spits out wrath. \"I will not.\" He lifts _Todd_ into his arms. Defiantly, he stares at Mr. Weller. \"I am taking Todd to the city central hospital.\" There is no way Jason is willing to trust anyone at this camp right now! \"And then,\" he states pointedly, \"I am calling Quadrant Officials. I intend to see you arrested.\"\n\n\"Really,\" Weller slurs sarcastically. \"And what are you going to tell them?\"\n\n\"I intend to tell them how you raped this boy, the evidence being the severe damage you caused and no doubt your sperm!\"\n\nMr. Weller laughs, looking to Darrell; reassured by the man's smirk, he turns back to laugh in Jason's face. \"Be my guest.\" He motions for Jason to leave and signals for Darrell to stand aside. As Jason walks out the room, Weller adds, \"Good luck making your charge stick.\" Then, adding acerbically, \"My aide here can attest to the fact that Tabatha begged me to help him. He said you weren't willing to do what was necessary so I had to take the initiative.\" Looking over to his aide, he states, \"Isn't that right, Darrell?\"\n\n\"That's what I heard, sir.\" Darrell stands next to his boss like an unmovable brick wall.\n\nStunned and sickened, Jason slowly turns, Todd hanging limp in his arms, to stare horror-struck at the man. \"Have you no moral conscience?\"\n\n\"Do not speak to me about morals.\" Weller's voice turns into a skewer aimed right at Jason's heart. \"You, who shelter and aid a known rapist!\" Spitting on the floor at Jason's feet, he yells, \"You sicken me!\" And then turning his back, he orders Jason Warith, \"Get out of my sight!\"\n\n* * * * *\n\nWhen Todd wakes, he is lying on his stomach in a sterile hospital room. Shuddering seizures ravage his body as he relives, during his first conscious moment, the stabbing pain inflicted upon him by Gideon Weller. As the spasms slowly cease, a terrified Todd realizes things can always get worse.\n\n* * * * *\n\n#  _Salve!_\n\n**Heterosexuals and Suicidal Behavior  \nHNN\u2014Melissa Eagleton Reporting**\n\nParents, it is critical that you speak openly and honestly with your children about any potential heterosexual feelings they might have. As we are aware, many of us have latent heterosexual tendencies. Only a small percentage of Hadrian's population can claim to be a six on the Kinsey scale. The rest of us exist somewhere between a two and a five. As a result, the period of sexual awakening for our youth can be very confusing. As we grow and mature intellectually, it is much easier for us to reconcile with the first cornerstone of Hadrian's society: that of homosexuality as our preference. As adults, we can understand the logic behind our chosen lifestyle\u2014see how it, combined with strict procreation laws, is essential for creating and maintaining a stable human population. Hadrian's four cornerstones of existence were based on reason and logic, motivated by the very real need to preserve the human species and regain balance with planet earth. As each of us comes of age, with all our cognitive skills developed, we can reaffirm our founding families' oath and swear ourselves, as they did, fealty to the four cornerstones of Hadrian society: Hadrian's chosen lifestyle is homosexual; Hadrian is a safe haven for homosexuals; Hadrian's goal is to create and maintain a stable human population; Hadrian will create an ecologically sound balance between humanity and nature. At twenty-one years of age, after three years of service with our military, Hadrian's citizens are ready to accept the responsibility of fulfilling Hadrian's promise to themselves, the human race and the earth.\n\nBecause our children are not yet capable of making that oath, it is incumbent upon parents and teachers to educate, gently guide, and nurture their spirits along Hadrian's chosen path. When we neglect our duty to our youth, we leave them vulnerable to sexual desires that are unwanted here in Hadrian, sexual desires that are in fact dangerous to our society. Allowing  heterosexuality to blossom in your youth's bosom is akin to allowing choke weed to grow and destroy our gardens. When a child's sexual education is ignored and he (or she) is left to discover feelings on his (or her) own, havoc can occur. What parents and teachers need to realize is that when a child experiments with heterosexual behaviors, he (or she) is risking more than just unwanted pregnancies. He (or she) is also risking his (or her) life!\n\nYes, it is true. And, though I do not condone this, the fact remains that individuals pegged as heterosexuals suffer from a great deal of emotional, and sometimes physical, abuse. Children, as we are all aware, are a bundle of paradoxes. Though they may appear to be the most tolerant among us, they also have proven to be the least tolerant, especially with that which they know to be unacceptable by society's standards. Empathy, an intellectual approach toward emotion, is learned behavior. Such lessons often take time\u2014many years in fact\u2014and seldom sink into the cognitive mind until the late teens or early twenties. As a result, youth viewed as straight receive a substantial amount of peer abuse. This abuse, coupled with the growing awareness that their desires are inappropriate, often leads these youth to contemplate suicide. Many, in fact, attempt, and some, unfortunately, succeed. Thus, it is our duty to watch for any signs of heterosexual behavior in our children, to teach them how to counter these feelings, as well as to encourage them to trust our judgment until they are of age and can see the logic and reason behind Hadrian's chosen lifestyle. Hadrian's society is based on choices that require higher level thinking skills, and until our children have acquired this ability, we must protect them from themselves and guide them in the right direction.\n\nRemember, all our children are, at the very least, a two on the Kinsey scale, so given time and the proper education, our youth can be guided gently toward accepting Hadrian's chosen sexual lifestyle: homosexuality. Help your children make the right choice! This lifestyle, coupled with strict procreation laws, is humanity's last hope.\n\n_Vale!_\n\n# \"to die, to sleep no more\"\n\nHaving just woken and experienced the spasms of pain caused by the rape, Todd determines death is his only solution\u2014his only viable escape from hell. What was it Matthew said? No one gets out of here, except through marriage, exile, or death\u2014and regarding death, he wished Todd luck, showing upon his wrists numerous failed attempts. He quoted those lines, the only ones Todd can remember: \"to die, to sleep no more.\" What a pleasure that would be\u2014no more sleep\u2014no more dreams\u2014no more waking up to reality. Simply to cease to be, to disappear into oblivion, is intoxicating. No more thought, no more pain, no more memories, no more betrayal. Eyeing his hospital room, Todd searches for the means necessary to execute his own demise. The IV tubing might work as a noose, but from where would he hang himself? The only hooks in the room are for a potholder in the ceiling and one for coats on the door. Neither hook is likely to hold his weight. Inspecting the ceiling offers no hope, either. It is a solid ceiling, no tiles to push aside to wrap the tubing around corner joints. It's as if they thought of everything here. No doubt a hospital is full of poisons, but when mixed properly, they become lifesaving medicines. Still, even if Todd knew which mixtures were deadly, _where in Hadrian's name,_ he wonders, _would I find them?_ Closing his eyes in despair, he pleads with the unknown, _How am I to die?_ Reopening them after letting his head drop dejectedly to the side, Todd finds his answer. _Oh, sweet Mischief,_ he almost utters, but stops himself lest anyone in the halls might hear him. There, sitting on the table next to his hospital bed, is a scalpel. Todd does not question how it got there, who \"accidently left it behind,\" or even why there would be a scalpel in his room at all, having no purpose whatsoever in his healing process; he simply rejoices in his breast at having found an instrument ready for action. His first impulse after picking up the sharp implement is to slit his wrists, but the memory of Matthew's scars reminds him that such a death is too slow. No doubt he would be found and revived, which is not part of Todd's plan. This is no cry for help. This is finality! _Dig it into the neck and rip it across the throat:_ the thought prompting action simultaneously. Blood begins to stream, and Todd, offering up no resistance, begins to ride the waves of ebb and flow between life and death. Closing his eyes during the final throes, he escapes into his mind, reliving the memory of the last moment on earth when he was truly happy.\n\n* * * * *\n\nCrystal is in her green dress\u2014the tight-fitting one she wore that Sunday when Todd, Frank, and she got together to study. The very same dress that, when left unguarded, would rise up too high. Crystal, Frank, and Todd had been dancing together at the big celebration. The school had thrown a huge party for the b-ball team for having won the Nationals. Wearing high heels, Crystal feigned tripping into Todd so she could whisper in his ear \"check your phone\" as he had helped her stand back up. After their dance, when Crystal begs leave to go to the bathroom, Todd wanders outside for a bit of fresh air and to cool off, but really, so he can be alone in order to read her clandestine message: \"Meet me behind the girls' stairwell.\" Todd is thrilled. _Behind the girls' stairwell!_ What can Crystal want? The girls' back stairwell camera is broken. No one will know they are there. No one will see whatever it is that is going to be done. And Todd wants things done. Stealing his way back into the school, he finds he is shaking more from nerves than the crisp spring night air. Now behind the stairwell, silent in the dark shadows, Todd anxiously awaits Crystal's arrival. His eyes dart up into darkness as he begins to hear her descending the stairs. _She really must have gone to the bathroom,_ he surmises. Although her footfalls are light since she is wearing heels, her shoes still manage to click against the stair's tiles. She is wearing some kind of perfume. It smells like cinnamon. Crystal always smells like cinnamon. And then she appears. \"Hello, Todd.\" Is her voice husky? Her walk over to him is like a trance, a dance in slow motion, with her hands sliding along her thighs. Leaning against the wall, Todd steps forward to meet her. Crystal places a hand on his chest. He feels her electric touch. Todd's breath quickens. Crystal leans in and kisses him. Somehow, they have switched positions. Crystal's back is now against the wall. Acting on instinct, unaware of how the first time should feel, Todd is thrilled by how Crystal effectively orchestrates their movements. He tells her he loves her. She kisses him. And then, when it's over, she walks away, with his life in her hands, into darkness.\n\n* * * * *\n\n#  _Salve!_\n\n**Profile of a Vice President  \nHNN\u2014Melissa Eagleton Reporting**\n\nVice President Elena Stiles is the third member of the Stiles family to hold a prestigious office in our government. Ester Stiles was Hadrian's first president while Denise Stiles was Hadrian's fifth. Having served under President Nasser for eight years, Vice President Stiles is determined to run in the next campaign. President Nasser has given Elena Stiles her full support. \"Having worked with Stiles for both terms of my presidency has convinced me that this is the woman destined to lead our country into the future.\" President Nasser's endorsement certainly will be a great boost come fall elections. \"Elena Stiles is energetic, forthright, forceful, and always looking toward the future. She stands for all that makes Hadrian strong and is determined to ensure the continued power of our military.\" No one can deny the importance of keeping our military strong. According to Vice President Stiles, \"Our military is the main reason we are able to keep Hadrian's values alive. If it were not for our brave young men and women defending our lifestyle and lives, Hadrian's wall would have fallen long ago and the human virus that surrounds us would have overrun and destroyed this last vestige of earth's natural beauty.\" One key campaign promise Elena Stiles intends to keep after her inauguration is to increase funding to the military.\n\nIf voted in as our next president, Vice President Stiles also promises to increase funding toward education. The intellectual growth of our youth must be one of our country's top priorities. \"Our children are the future of Hadrian. They are the ones who must continue preserving the natural balance of our country's habitat. They are the ones who must continuously develop new ways to reuse man's pollutants as well as discover ways to counter the poisonous effects already tainting earth's water, soil, and air. The task our foremothers and forefathers burdened us with comes with no simple solutions. Only the strongest and most creative of minds can overcome humanity's past mistakes.\" She is so right! This woman has Hadrian's priorities as her own: a strong military to protect us from the threat of the outside world and the education of our youth.\n\nWhen asked how she stands on reeducation camps, Vice President Stiles was circumspect. Although she stressed the importance of reeducating wayward youth, she also seemed to suggest that the way in which our camps are run may not be up to her personal standards. When asked how she felt these institutions should be run, she was unable to offer up any viable suggestions. She did, however, suggest that a thorough inquiry into reeducation camps was essential, and after becoming president (\"should I be so lucky as to win the country's confidence\"), such an investigation would be held. With respect to the outcome of such an examination, Vice President Stiles hinted at the potential restructuring of these camps. This may very well be the result of a meeting held with the eight camp wardens at the central cabinet. Apparently, some dissension exists between the wardens because disparaging methods are being used at some of the camps. It seems Jason Warith's complaints against the Northeast Reeducation Camp have opened the door for others to express their dissatisfaction at reeducation methods. Vice President Stiles is taking these concerns seriously and is looking at ways of standardizing how we approach reeducation.\n\n_Vale!_\n\n# Extracting a Confession\n\nFive days have passed since Crystal denounced Todd. Every day, she has thought about texting him, though fearful of what she might learn since Frank voc'd and told her he was sentenced to the Northeast Reeducation Camp. Sentenced! Crystal grimaces at the harsh word Frank used. Frank hates her now, but she doesn't care. Her only concern is for Todd. Before Frank had hung up on her, Crystal asked him whether Todd still had his cell phone at the reeducation camp. Frank said no one had mentioned finding it so it was still a secret. But then he said he had tried texting Todd and got no response. Todd might not even have it with him, Crystal reasons, in a feeble attempt to justify not having tried to contact him. It isn't so much cowardice that keeps her from texting Todd; rather, her mother, Gail Albright, having discovered her phone and with whom it was used to converse, confiscated it. Crystal had to find and steal it back before she could try contacting her lover. Her mother is keeping a vigilant eye on her every move. Gail Albright, having taken time off from work, is staying home indefinitely in order to help facilitate Crystal's unofficial reeducation. Although one really doesn't need to reeducate a girl who has been raped, since Crystal refuses to accuse Todd, her mothers are determined to ensure she has not gone astray. As far as Crystal knows, neither of her mothers has seen the digital recording her aunt, Ms. Annabelle Sterne, had forced her to watch early that dreadful morning. Annabelle Sterne insisted Crystal denounce Todd, threatening to tell the girl's mothers what really happened if she didn't. Aunt Annabelle was right. Her exposure was sure to ruin Mama Elena's political career. Mama Elena had worked long and hard to get where she is; Crystal couldn't bear the thought of taking all that away from her, especially now that she has chosen to run for President of Hadrian. Nor could she cope with the threat of her mothers' disappointment in her. She loved both women too much to risk losing their love and respect.\n\nWhen Crystal adamantly refused to accuse Todd of rape, though, Elena Stiles became suspicious. Elena and Gail held a private conference and decided it would be in Crystal's best interest to reestablish her association with her ex-girlfriend: Lolita Huber (daughter to the secretary of defense). Crystal agreed. Their relationship was easily reestablished since Ms. Huber owed her position in the government to Elena Stiles. Crystal had broken up with Lolita because the two girls have nothing in common. Lolita is an intellect, a big reader, yet dry as desert dust, and deficient in humor whereas Crystal loves to laugh, have fun, roughhouse, and play sports. If given her druthers, Crystal would spend all her time hanging out with Todd and Frank. Very few opportunities exist for a girl to associate with guys, other than on sports teams, so Crystal chose, at a very young age, to become very adept in basketball, baseball, hockey, and wrestling. Her mothers have always approved of her gaming since, on the surface, it appeared as if Crystal were merely trying to prove herself the physical equal of any young man her age. Although she knew both Frank and Todd were miles ahead of her in both speed and strength, Crystal found ways to compensate through skill. In fact, it is fair to say that the only young men Crystal's age who can best her in the sports arena are Todd Middleton and Frank Hunter. Although, at times, she had managed to get the better of Todd, but only in wrestling. _I wonder why that was?_ she questions. _I think he liked me on top of him._ A brief smile crosses her lips before once again turning grim. _He'll never love me again,_ she reasons. _Not after what I did! Oh, Todd,_ tears burgeon once more, _what did I do to you?_\n\nCrystal sighs as she lies on her bed, staring at a blank slate screen. She is supposed to be working on a treatise that validates Hadrian's sexual preference and method of procreation. She is to present her findings at school tomorrow in an oral presentation. Her aunt, Ms. Sterne, assigned it to her in the aftermath of Todd's exposure. She hasn't done any research or written a single word. The only movement on the electronic slate's screen is a blinking curser. Crystal's mind is far too preoccupied with the fate of her lover to worry about homework, especially _extra_ homework! Right now the only thing Crystal can think about is what might be happening to Todd\u2014 _because of me!_ Silencing the hum of the school slate by tapping the power button, Crystal tosses her personal slate aside. Silently, she slips off the bed, tiptoes over to the door, places her ear against the wood, and listens intently for sound in the hallway. When satisfied no one is immediately outside her door, she carefully turns the knob and pushes it open an inch. Peering through the slight crack, she determines no one is in sight. From the basement, she hears the sounds of pool balls clacking. Mom must be practicing. Mama Elena is seldom home, even when she is in town. Crystal knows she is away on business, as usual. Believing it to be safe, Crystal slowly opens her bedroom door, slithers out of her room, and stealthily slips inside her mother's room. Mom is a creature of habit. She would have locked Crystal's phone in her \"special drawer\" in the master bathroom.\n\nAfter successfully picking the lock, something Crystal had learned to do years ago (unknown to her mother and simply because she hated the idea of her mother keeping secrets from her\u2014never to steal anything) she slowly pulls open the drawer. Today, this clandestine art serves her need, not her curiosity, as she retrieves her phone. Quickly, Crystal relocks the drawer and returns to her room before anyone knows she was out.\n\n\"Todd? Are you out there?\" Crystal texts quickly, not daring to try to phone Todd because her mother may hear her speaking out loud. Silence is the better approach. After waiting an intolerably long second, she types again: \"Answer me, please.\"\n\nThere is a sudden beep in response. It's from Todd. Crystal sighs her relief and then reads: \"What do you want?\"\n\n\"I'm so sorry.\"\n\n\"Really?\"\n\n\"Todd, please. Auntie\u2014Ms. Sterne made me denounce you.\"\n\n\"So?\"\n\nCrystal drops the phone and begins crying. _Todd hates me! Can I blame him?_ Desperate, she reaches out electronically again: \"I love you.\"\n\n\"You said it was rape!\"\n\n\"I never said that!\"\n\n\"Did you deny it?\"\n\n\"I can't. You know I can't.\"\n\n\"Why not?\"\n\n\"Please don't ask me, Todd.\"\n\n\"Was it?\"\n\n\"What?\"\n\n\"Rape?\"\n\n\"How can you ask that?\"\n\n\"Everyone says it was rape.\"\n\n\"That's not my fault!\"\n\n\"Isn't it?\"\n\nCrystal stares dumbfounded at her phone. _He's right. I know he's right. This is my fault, all my fault. But there's nothing I can do. Auntie Bella's right. If I come out, say anything in Todd's defense, it will ruin Mama Elena's political career._ \"Todd, please forgive me.\"\n\n\"How's reeducation at home?\"\n\n\"It's horrible.\"\n\n\"Try being in camp.\"\n\n\"Is it really bad for you there?\"\n\n\"What do you think?\"\n\n\"Todd, you're killing me.\"\n\n\"Reeducation camp is killing me!\"\n\nCrystal wants to throw the phone against the wall, but she isn't able to. It is her only link to Todd, and now that she has established contact, she can't let go. _He blames me for everything. And he's right. This is my fault!_ Clutching the phone to her breast, Crystal wonders, _Is there anything I can say to explain myself to him?_ \"Mama Elena wants me to accuse you of rape. But I refuse to!\"\n\n\"Why don't you? It'd be kinder.\"\n\nThis response terrifies Crystal: \"Oh, Todd, what are they doing to you?\"\n\n\"What are they doing to you?\"\n\n\"They made me get back together with Lolita.\"\n\n\"Poor you.\"\n\n\"Don't mock me. I'm sorry. I feel horrible. I love you!\"\n\n\"You didn't answer my question?\"\n\n\"What question?\"\n\n\"Was it rape?\"\n\n\"You know it wasn't!\"\n\n\"Do I?\"\n\nHis persistence in such clarification angers Crystal: \"No, Todd, it wasn't rape.\"\n\n\"How did it happen, then?\"\n\n_Why is he asking this?_ she wonders: \"You know how it happened.\"\n\n\"Tell me anyway.\"\n\n_Fine,_ she grimaces, _I'll answer all his questions. I guess he has a right._ \"I texted you a note at the Championship dance.\"\n\n\"Go on.\"\n\n\"We met in the back stairwell.\"\n\n\"The one with the broken camera?\"\n\n\"Yes.\"\n\n\"Then what happened?\"\n\n\"I kissed you.\"\n\n\"And then?\"\n\n\"We made love.\"\n\n\"So, you initiated everything?\"\n\n\"You know I did. Why are you asking all these questions?\"\n\n\"Because I'm not Todd!\"\n\nGasping, Crystal drops the phone. It bounces off the bed and clatters to the floor. Crystal leaps up and grabs it, waiting in silence, hoping the sound does not alert any suspicion in her mother. Nothing happens while Crystal holds her breath. Still kneeling on the floor, she looks in horror at the last line of text on her phone: \"Because I'm not Todd.\"\n\n\"Who are you?\"\n\n\"One of the few friends your lover has left.\"\n\n\"What's your name?\"\n\n\"Do you know what they do to a boy suspected of rape in reeducation?\"\n\nThis question stops Crystal's heart. Her breath is suspended as she types in her reply: \"No\" and waits for the answer.\n\n\"They bludgeon him!\"\n\n\"They what?\"\n\n\"Rape, you stupid girl! They RAPE him!\"\n\nTears are flowing, Crystal is sobbing; had she been speaking, very little of what she types would be comprehensible. As it is, it is barely legible: \"ow do u no?\"\n\n\"I was in reeducation. That year, two boys committed suicide. Both boys suspected of rape. Both boys raped!\"\n\n\"How do u no? Did u sea?\"\n\n\"No, but everyone knew. There's more.\"\n\nCrystal groans; masochistically, she needs to know: \"mor?\"\n\n\"It is impossible to commit suicide in a reeducation camp.\"\n\n\"But u sad\u2014\" Realization dawns; Crystal continues her text: \"ho did 2 boys die?\"\n\n\"Suicide.\"\n\n\"I don't understand.\"\n\n\"Curious, isn't it? Two boys suspected of rape were the only two boys able to commit suicide. Think about that for a minute.\"\n\n\"Is Todd\u2014did he\u2014\" Crystal is unable to finish.\n\n\"He tried. He failed.\"\n\n\"Thank Hadrian for that!\"\n\n\"You can thank me\u2014or rather, Jason Warith.\"\n\n\"Who?\"\n\n\"Todd's facilitator.\"\n\n\"Oh.\"\n\n\"I arranged for him to work with Todd\u2014to protect him.\"\n\n\"?\"\n\n\"Because I knew!\"\n\n\"Please tell me who you are.\"\n\n\"Will you help Todd?\"\n\n\"How?\"\n\n\"Tell the truth.\"\n\n\"I can't.\"\n\n\"We have nothing left to say.\"\n\n\"No, please\u2014\"\n\nCrystal pauses, recognition of her character dawning: \"Mama Elena\u2014she's VP Stiles.\"\n\n\"We mustn't do anything to hurt one of our founding families.\"\n\n\"Please, you don't understand.\"\n\n\"No. You don't understand! I will not sacrifice Todd on the altar of your stupidity!\"\n\nFrightened, knowing herself to be the personification of all that is selfish, Crystal asks: \"Will you denounce me?\" There is no answer to her plea. She types again, \"I'm scared.\"\n\n\"Imagine how Todd feels.\"\n\n\"I can't help him.\" Her fingers shake before she types in the next phrase. \"I can only help myself.\"\n\n\"I DO understand.\"\n\nCrystal's immediate response to that line is to shut off her phone. Without  thought or precaution, she races back into her mothers' personal washroom. She returns her cell to the locked drawer and swears never to touch it again. Back in her bedroom, Crystal hides herself under the covers, shivering uncontrollably. Frightened and alone, she realizes she forgot to erase all evidence of their conversation. And although she knows with absolute certainty the mystery man at the other end hasn't deleted his version, she still goes through the motion of retrieving her cell and deleting all contact made with Todd's mobile.\n\n* * * * *\n\n#  _Salve!_\n\n**The World Outside our Walls  \nHNN\u2014Melissa Eagleton Reporting**\n\nTo every child, it is revealed what life is like for the unfortunate living outside Hadrian's walls: Images of emaciated bodies lying on desert floors. Children starving. Mothers' breasts bereft of milk. It is truly unfathomable how humanity is still thriving. It is amazing how the words of Dr. Oscar Baumann, written in the late nineteenth century, depict what is happening all over the planet today. \"There were women wasted to skeletons from whose eyes the madness of starvation glared...warriors scarcely able to crawl on all fours, and apathetic, languishing elders. Swarms of vultures followed them from high, awaiting their certain victims.\" The image was frightening enough back then when the calamity of famine only hit one area in Africa, but knowing that this painting of horror is found daily, and in every country but ours, is demoralizing.\n\nSometimes, I believe it is better not to witness the slow agonizing death throes of humanity's suicide. For, as much as we dearly wish we could do something, anything, to help these wretched creatures, the reality is that providing food, shelter, water, and warmth for the billions dying outside our walls is impossible. We cannot cure the illnesses mankind has festered upon itself. We cannot hope to save those other twenty billion or more individuals. All we can do is cling to the knowledge that we, in Hadrian, are rebuilding humanity's future, that Hadrian is maintaining the necessary balance to help humanity survive.\n\n_Vale!_\n\n# Todd Awakes\n\nWhen Todd awakes, Frank is clutching his right hand. Having fallen asleep, Frank has his head lying in Todd's lap. Todd sighs when he sees his friend. _Finally,_ he mutters.\n\nThis is not the first time Todd has risen since his failed suicide attempt. When he first discovered that he was still alive, Todd screamed as best he could through his damaged throat while simultaneously ripping at the bandages around his neck. His intent was to dig his nails deep into his wound, effectively ending his life once and for all! Fortunately for Todd, or perhaps not so fortunately from his perspective, he is under twenty-four hour supervision. The mirror on the right wall is a one-way window for the nurse assigned on observation duty, which explains why his first attempt to kill himself failed. Mia Ocampo, the nurse in attendance his first night at the hospital, managed to call the doctor and effectively stem the flow of blood, saving Todd Middleton's life. It was Gordon McAlister, the second nurse in attendance, who circumvented Todd's second attempt. Although Todd managed to remove most of his bandage, he never got his fingers into the wound. Since then, Todd has been strapped to his bed, has a catheter fitted, and since he refuses to eat, drink, or take any medication, is attached to an IV unit.\n\nWhen Jason Warith came to visit Todd, he had tried unsuccessfully to reestablish the tenuous bond he had begun forging with the boy. Not that he had any hopes that ties could be rebuilt after what Mr. Weller had done. Since the rape, Todd had completely shut himself off from the rest of the world. Before leaving Todd the last time, Jason had asked the young man whether there was anything at all he could do for him. When Todd asked to see Frank Hunter, Jason smiled. \"I can do that for you, son.\" He had heard that Todd and the Hunter boy were close friends, that prior to Mr. Weller's interference, they had even begun forming a boyfriend relationship. _Why,_ he questioned irritably, _did Weller not leave him with the Hunter family? No,_ he reminded himself, _I can't waste thoughts on \"what ifs.\" I only have \"what is\" to work with._ So, as soon as Todd returned to an induced sleep\u2014an unfortunate necessity as his continued struggles, even with the restraints, caused life-threatening damage to his wounds\u2014Jason Warith contacted the Hunter household.\n\nFrank, having waited at Todd's bedside for over two hours, is now sleeping. Todd would shake his own head, but he can't because it is belted down. All he can do is roll his eyes as his friend snores. _Friend_ , he wonders, _can I still call him that?_ Studying Frank's profile, Todd considers Frank's motivation. _Would I have done the same if our situations were reversed?_ Todd is unable to provide an answer. _I guess I'd only know if I were in his place. At least,_ he figures, _he did it because he loves me._ \"Frank,\" he whispers. His voice still hasn't recovered from his suicidal knife attack. When Frank fails to respond, Todd wakes him by shaking his right hand. As soon as Frank stirs, Todd rasps again, \"Frank.\"\n\nSitting up, rubbing his eyes and face before running his fingers through his hair, Frank mutters, \"Todd.\" Frank leans forward to kiss Todd on the lips; Todd, with closed eyes, remains unresponsive. Frank pretends not to notice. _Keep things cheery, upbeat,_ he reminds himself. \"Hey, babe. Welcome back.\"\n\nTodd doesn't waste any time, \"Help me.\"\n\n\"You know I will,\" Frank promises. \"I'll do anything for you.\" Although he remains silent, Frank can see the promise he made to Todd that morning so long ago when they sat hidden (or so they had thought) behind the girl's locker room stairwell. It is like a giant neon sign flashing out of Todd's eyes.\n\nEnunciating each word carefully, lest his voice, hindered by his attempt at self-slaughter, should slur his words, Todd slowly demands, \"Help me die.\" Frank pales. His mind refuses to register what Todd has spoken. Sensing Frank's refusal, Todd insists, \"I'd do it for you, _Bob.\"_ Todd says no more. He simply waits for Frank to reach an understanding.\n\nAt the beginning of their grade ten school year in their language arts class, Frank and Todd had held a heated debate as to the meaning of Earle Birney's poem \"David.\" Frank had insisted the boys were lovers, making Bob's act of euthanasia a much more powerful moment. Todd took the stance that the boys were just good friends, no sexual innuendoes existed, and Frank was just reading into the poem what he wanted it to say. Frank proved his point, though, by quoting line fourteen, \"Then the two of us rolled in the blanket,\" using the fact that it comes just prior to the pines thrusting up into the sky to emphasize his point. The teacher agreed with Frank and pursued this interpretation.\n\nNow, though, as he lies strapped to his bed, Todd is not thinking of David or Bob's sexual preference. Rather, his thoughts focus on David's request and Bob's agreeing to do it. David, having fallen and broken his back, had begged Bob to push him over so he could die. Bob did not want to comply. He had hoped to find a way to save his friend, but as the poem says, Bob knew that more than a day and a night would pass before he could make his way back down to the camp and bring men \"unknowing\/The way of mountains\" back to rescue David. \"And then, how long? And he knew...and the hell of hours\/After that, if he lived till we came, roping him out.\" Todd thought now of his after hours, the hell of living, if Frank were somehow successfully to \"rope him out.\" He could never go back to thinking that he could be gay now. He could never hope to find that spark somewhere inside now. Looking at Frank, Todd reminds him again, \"For Christ's sake push me over!\/If I could move...or die...\" Another of David's lines. Another desperate plea to die with dignity.\n\nFrank shakes his head sporadically. It is all coming back to him now. Todd wants him to help him commit suicide. \"No.\" His head shake quickens with his heart rate. \"No, Todd. No!\"\n\n\"You promised!\" Todd reminds him.\n\n\"Never promised this!\"\n\n\"To do anything!\" Todd's brown eyes harden like frozen earth.\n\n\"Never. Never this.\" Frank, unable to bear the accusation in Todd's eyes, turns to leave.\n\nTodd punches him in the back of the head with his words. \"You owe me.\" Frank stumbles. He knows exactly what Todd means. Bitterness strikes like a whip against Frank's heart when Todd spits out, \"You're no better than Weller!\" Aghast, Frank quickly opens the door and rushes out of Todd's room.\n\n* * * * *\n\n#  _Salve!_\n\n**Shocking Allegations!  \nHNN\u2014Melissa Eagleton Reporting**\n\nJason Warith is demanding a thorough investigation be held at the Northeast Reeducation Camp, insisting that the administration there be held accountable for abusive treatment of its wards. \"The lax approach to the investigation thus far is unacceptable,\" an enraged Warith stated yesterday at the Reeducation Camp Wardens' Review Board Meeting. It is the responsibility of this board to determine whether or not to press official charges against the Northeast Camp Warden. Jason Warith has laid two very specific charges against Gideon Weller: 1) the over, and unnecessary, use of corporal punishment, and 2) rape. I find these charges extremely hard to believe, having met with Gideon Weller and toured his camp.\n\nThese shocking accusations surprise many, considering the numerous accolades hailed upon Gideon Weller over the years. Statistically, the Northeast Camp reeducates more wayward youth than any other. Although Gideon Weller admits corporal punishment is administered at his camp, it is also administered at every other reeducation camp. He believes it is unavoidable when dealing with angry, aggressive youth, particularly young heterosexual males. According to Jason Warith, the success rate of the Northeast Camp does not indicate real success, but rather, young men desperate to escape in any way they can. Jason Warith also pointed out that the Northeast Reeducation Camp has the highest rate of attempted and completed suicides.\n\nGideon Weller claims that is because his files are accurate; he does not attempt to dissemble. \"Every act of self-mutilation discovered in my camp is identified as a suicidal act. Thus,\" he reminded us, \"it only appears as if there are more suicide attempts at my camp. Other camps,\" Gideon Weller explained recently in response to these charges, \"dismiss self-mutilation as suicidal. We, at the Northeast Camp do not.\"\n\nI, for one, do not believe these accusations. Having met with Gideon Weller and having toured his facility, I am convinced that these charges are unfounded. We even witnessed a sample of his great success when we aired \"Happily Married After Reeducation.\" It was at the Northeast Camp under Gideon Weller's tutelage that Geoffrey and Dean Hunter met. These two men have been happily married for over twenty years! It is unthinkable that a man of such honor and nobility as Gideon Weller would ever stoop to such abusive strategies to rein in unruly youth. It is my sincere hope that the accusations laid against Gideon Weller are unfounded and that the impending investigation unearths the truth.\n\n_Vale!_\n\n# Heart to Heart\n\n\"Todd.\" Papa Dean's voice is soft, soothing, enticing enough for Todd to surrender to it.\n\n\"Papa Dean?\"\n\n\"Yes, son, it's me.\" Todd's eyes open. It takes a moment for the image of the man looking down on him to come into focus. Todd is heavily sedated to keep him from thrashing around, even bound the way he is. \"Hey, you,\" Papa Dean says quietly.\n\n\"Hi, Papa,\" Todd mutters. Closing his eyes again, he mumbles, \"I'm tired.\"\n\n\"I'm not surprised,\" Papa Dean replies, \"considering all the drugs they've pumped into you.\"\n\nTodd tries to shake his head. He can't. The leather strap holds him securely. Reopening his eyes, he glances over to the man sitting next to his bed, the man holding and caressing his hand. Papa Dean. Not Papa Mike. \"No, Papa Dean, tired of life.\"\n\n\"Please, Todd, no,\" he whispers softly. \"Don't talk like that, son. You're only seventeen years old.\"\n\n_I could just as easily be sixty._ Todd sighs fretfully. There is no room for equivocation with the boy. \"Do you want me to lie to you?\"\n\n\"No,\" Papa Dean answers resolutely. \"No, Todd, I don't. Only the truth between us.\" Sighing deeply, Dean braces himself for the worst. \"Only the truth.\"\n\n\"The truth is,\" Todd says matter-of-factly, \"I want to die.\"\n\n\"I know.\"\n\n\"Do you?\" Todd is skeptical. How can anyone know? Anyone who hasn't\u2014but, he remembers, _Papa Dean has._ Frank said Papa Dean was straight. \"You were in reeducation camp, too, weren't you?\"\n\n\"Yes, Todd. A very long time ago.\"\n\n\"Tell me the truth, Papa Dean. Why did this happen to me?\"\n\n\"Because the world is scared.\"\n\n\"Scared of _strais_.\"\n\n\"Hadrian is. The rest of the world\u2014out there\u2014they're scared too.\"\n\n\"Scared of me?\"\n\n\"No, son. Not you. Scared of living. Scared of dying. Scared of abject poverty, starvation, disease. Twenty billion people battling for life on a planet incapable of sustaining even ten billion. Earth is overburdened, son, overwhelmed with the human virus. They are scared of themselves and of us.\"\n\n\"Us?\"\n\n\"Hadrian is a haven\u2014we have land\u2014we have clean water\u2014we have space in which to breathe\u2014we have love of our fellow man\u2014\"\n\n\"Do we?\"\n\n\"I'm sorry?\"\n\n\"Love our fellow man?\" There is a moment of silence. Papa Dean knows there is no answer he can give to a boy who has had his entire life ripped out from under him\u2014who has been brutally raped. _How,_ Dean asks himself, _can Todd possibly believe Hadrian capable of loving man? He can't._ Waking Dean from his reflection, Todd asks, \"Don't they?\"\n\n\"Who son?\"\n\n\"The outsiders. Don't they love their fellow man?\"\n\nDean ponders this for a moment. All of Hadrian National News' propaganda would have him say no. All he has ever seen of the outside world are the harrowing images of skeleton bodies rushing up against Hadrian's wall, and the images of the starving, dying babies captured from above by Hadrian's satellite camera. Even so, he can't believe them all to be barbarians. \"I suppose some of them must. It's just\u2014they have so little food, so little water left that, well, we have to protect ourselves from\u2014\" He stops himself from using the now clich\u00e9d expressions\u2014 _the barbarian heterosexuals; the crazed, insane, and dangerous outsiders._ \"Well, from the threat that they'll break through Hadrian's walls, take and destroy all we've managed to preserve.\" Looking down at the child, he says, \"Not just for us, Todd\u2014for the future of humanity. Hadrian is man's last hope for posterity.\"\n\n\"Why can't we share?\"\n\n\"There just isn't enough.\"\n\n\"Not enough land? Not enough food?\"\n\n\"No, Todd. Not for twenty billion.\"\n\n\"Is that why we have to be homosexual, Papa?\"\n\n\"I don't know Todd. That's what we're taught. I mean there are so many ways to prevent pregnancies. But even with all those available, the world is still overpopulated.\" Shaking his head in wonder, he states, \"I may not like all of Hadrian's laws, son, but I don't condemn them either. I mean, if heterosexuals could control procreation, then why is the world so overburdened now?\" Looking down at his hands, he remembers the words Geoffrey used to convince him: \"Only Hadrian has a stable population. A fully homosexual state may seem drastic, but we are living in an extreme world\u2014even so...\" Dean's voice trails off.\n\n\"Even so, Papa?\"\n\nTrying not to sob, Dean says, looking down on Todd, \"What they did to me... What they did to you.\" Shaking his head, tears flowing. \"It all seems so wrong.\"\n\n\"Did you try to die, too, Papa?\"\n\n\"Yes.\"\n\n\"What stopped you?\"\n\n\"Geoffrey Hunter.\"\n\n\"Frank's father.\"\n\n\"Yes. Without him,\" Dean says, closing his eyes in order to shut in the tears, \"I don't think I could live.\"\n\n\"Did he rape you?\"\n\nDean is stunned. \"No. Of course not.\" Knowing what happened to the boy has skewed his perspective. Dean adds reassuringly, \"Weller is different, Todd. Most men aren't like him.\" Dean tightens his grip on Todd's hand. \"Geoffrey Hunter was good to me. Kind. Gentle. He waited. Like Frank waits for you.\"\n\nTodd closes his eyes. _Papa Dean doesn't know then._ Memories swirl Todd's stomach. The pounding of fists, his split brow, bruised cheek, and swollen lip; Frank's black eye replays itself in his mind. And then, the stunning blow to his temple, leaving him senseless long enough for Frank to bind his wrists securely to the front bedpost. Todd squeezes his eyes shut. No matter how hard Todd had begged, Frank wouldn't listen. Frank's only words were, \"You need to relax, Todd; relax into it or it will hurt.\" And it did hurt. It had felt as if his backside were being ripped in half. If he could, Todd would shake his head, but all he can do is say, \"No, Papa Dean, Frank can't save me.\"\n\n\"He loves you Todd.\"\n\n\"I know, but\u2014\" _I can't tell him,_ Todd suddenly realizes. _Knowing the truth about Frank would shatter Papa Dean._ It is evident to Todd that Papa Dean envisions Frank as a duplicate of his father, and although Frank looks exactly like Geoffrey Hunter, Todd knows the two men are as radically different in personality as they are in height. Changing the subject, Todd asks, \"Tell me about Frank's father\u2014how he saved you.\"\n\n\"He never tried to turn me. He just\u2014became my friend. He listened, never judged, understood as best he could. He never even tried to kiss me.\" Smiling, he shrugs, shaking his head wonderingly. \"He married me, knowing we might never\u2014\"\n\n\"Did you? Ever?\"\n\n\"Yes,\" he confesses. \"Eventually. But he never pressed. I mean, there were times when he thought I might, but he always let go when I said no. But one time, I guess, I no longer had the heart to stop him. I don't know whether I wanted to or not, or whether I felt like I owed him, but when he leaned in, I let him kiss me. For weeks, we just kissed.\" Dean shakes his head in wonder. \"We'd been married two years then. Where he found his patience, I'll never know. And then finally, I went to him, told him I was ready. I knew I wasn't, not really, but something inside me said I need to do this\u2014that Geoffrey loves me and I love him. He was good to me; so gentle with his touch. Even so, I cried. He apologized, but I said, 'No, we needed to do this\u2014I needed to do it\u2014I\u2014'\" Dean, now looking at Todd, explains, \"I needed to let go. It was cathartic.\"\n\n\"Is it\u2014I mean\u2014do you enjoy it?\"\n\n\"Sometimes.\" Sighing a little humph, Dean admits, \"Sometimes, I even initiate now.\" Considering his life with Geoffrey Hunter, Dean adds, \"It took me a long time, but now I really do want to be with him.\" Speaking more hopefully than assuredly, he adds, \"I know if you take your time with Frank, you'll feel that way about him someday, too.\"\n\n_He really doesn't know,_ Todd muses. _I wonder what Frank really told him._\n\nSensing the doubt in Todd's eyes, Dean persists, \"It's not so bad when the man you're with loves you. When you honestly love him back. Besides,\" he sighed, \"I'd never really been with a girl. We had only just kissed.\"\n\n\"Tell me what happened with the girl, Papa.\"\n\n\"High school,\" he begins, \"grade twelve. Jessica and me\u2014she was one of my best friends. I guess like you and Crystal. We did everything together; always holding hands; always laughing. Then one day at lunch, it just happened. It wasn't planned or nothing\u2014it just happened. We were laughing so hard, and our faces got\u2014well, we just sort of came together. Our lips touched. I burst into flame and started kissing her. Next thing I knew, she was standing up, screaming and pointing my way. Two teachers grabbed me and hauled me into the office. They locked me inside the Vice Principal's office and left me there for over an hour. I didn't know what they'd do or were planning. Then my father came. He was outraged. I had shamed the family name. Me, the genetic offspring of a founding family! They never fiddled with founding family genes in those days.\" Shaking off sad memories, Dean concludes, \"Anyway, he signed me over to the state and disowned me.\"\n\n_Just like Papa Mike._ \"I'm so sorry, Papa Dean.\"\n\n\"Don't worry about me, Todd; I'm all right. My life's been good, thanks to Geoffrey.\" Hoping this is the right moment, he adds, \"Let Frank help you\u2014I know he can\u2014\" But Todd only closes his eyes again.\n\n\"What was reeducation camp like for you, Papa?\"\n\n\"Torture.\" A sad smile blossoms; Dean knows Todd will understand. \"I was Weller's first ward.\" Todd opens his eyes and gasps. \"Yes,\" Dean admits, \"it was horrible.\"\n\n\"Did he...\" Todd doesn't finish\u2014he can't finish. \"You, too?\"\n\n\"He tried\u2014he might have succeeded if it hadn't been for Geoffrey.\"\n\n\"Frank's father.\"\n\n\"He walked in just as it was happening. I mean, I tried to fight, but Weller was the stronger man.\"\n\n\"I know.\" Todd truly does understand.\n\n\"Geoffrey was enraged when he saw what Weller was trying to do to me. He pulled him off before Weller could\u2014\"\n\n\"Weller,\" Todd interjects, \"he had his man grab Jason\u2014throw him out of the room\u2014he locked him out.\" Papa Dean's story resonates so for him. \"Jason would have saved me if he could...\" _But it's too late now,_ Todd reminds himself. _It's too late for me now._ Shifting his eyes so he can look into Papa Dean's crystal blue orbs, he says, smiling weakly, \"Thank you for sharing with me.\"\n\n\"I'd like to tell you more, Todd.\" Dean takes a moment to brush his fingers through Todd's hair. \"About your father, if you'll let me.\"\n\n\"Dad?\" It suddenly dawns on Todd. \"Was my father...\" He whispers now so no one outside the room can hear. \"Was my father like me?\"\n\n\"Yes.\"\n\n\"Tell me.\" Todd's voice is near pleading. \"Tell me, please.\"\n\nDean recalls, \"I met your father two months after we began high school.\"\n\n\"Grade ten,\" Todd mutters.\n\n\"Yes,\" Dean's mind begins to drift back into the memory. \"Two months into grade ten. I was sitting alone in the cafeteria...\" Feeling the need to explain, Dean adds, \"I often sat alone in those days.\"\n\n\"Why?\" Todd asks.\n\n\"I knew\u2014I knew I was different\u2014and was afraid of interaction\u2014afraid of anyone getting to know the real me.\" Todd's head nods slightly in understanding. \"I never had a childhood friend like you. Anyway, Will had been watching me for some time. I often caught him staring at me. I avoided him, thinking he was attracted to me.\" Dean smiles a little. \"Then one day, he just up and joined me at my lunch table. Gave me quite a start.\" Dean gets lost inside his memory and the words tumble off his lips as if in the moment.\n\n* * * * *\n\n\"Dean Stuttgart, right?\"\n\nA startled young Dean looks up from his studies. With an egg sandwich poised in his left hand and his computer slate at the ready in his right, Dean freezes at the sight of Will Middleton, the most popular boy in school, smiling down at him, hand extended in greeting. Dean pointedly ignores the offer of friendship and looks back down at his slate. Always studious, Dean throws himself into his studies with fervor because it helps him to avoid interacting with others. Mostly, it is to keep him from staring at girls. If not careful, Dean would be the recipient of many an angry glare. _Study,_ he often reminds himself, _study and forget about the others. Study and become a doctor like Dad wants!_\n\n\"That's not going to help you this time,\" Will rejoins.\n\nDean looks up amazed. He is resorting to talking out loud to himself now. He is going to have to watch out for that. \"What?\"\n\n\"Let's start again, shall we?\" Will says with a smile. \"Dean Stuttgart, right?\"\n\n\"Yeah, so?\" Dean responds briskly. This man is nothing if not persistent.\n\nExtending his hand again, Will says, \"Will Middleton.\"\n\n\"Everyone knows who you are,\" Dean mutters disapprovingly.\n\n\"And,\" Will adds, \"everyone knows who you are.\" Dean rolls his eyes and returns his attention to his slate, though he cannot focus. Laughing now as he retrieves his unwanted hand, Will remains undaunted and persistent. \"Mind if I join you?\" he asks as he sits across the table from Dean.\n\nSpeaking to his slate now, Dean says, \"I'd rather be alone, thank you!\"\n\n\"That's your problem,\" Will remarks.\n\nDean's eyes dart up, first in fright, then sparking into anger. \"What?\" When he gets no response from Will Middleton, he retreats back into the safety of his slate. \"I'm studying!\" When Will still refuses to take the hint, Dean adds, \"Sit there all you want. That doesn't mean I've got to talk to you.\"\n\n\"You're talking to me now,\" Will adds with a laugh. Then, taking on a serious note, he says, \"Yes, you do\u2014we do.\" Dean sighs, staring intently at the blurred words of his biology text. His left hand is squeezed tightly around his sandwich, having turned it into mulch while his right hand threatens to snap the pencil it is holding. \"You see,\" Will sighs, \"I think I know you.\"\n\nDean looks up startled, angry. \"You don't know me. No one here knows me.\"\n\n\"True, enough,\" Will replies. \"And yet,\" now waving to the students surrounding them inside the cafeteria, \"they all know who you are. You are a Stuttgart.\" Smiling grimly, \"That makes you the most popular unpopular boy in school.\"\n\n\"What do you care?\" Dean snaps back.\n\n\"I'm like you,\" Will replies. Dean snorts. \"A lot like you,\" Will insists.\n\nSpeaking to his slate, Dean says, \"You're the school jock. Everyone looks up to you\u2014\"\n\n\"And,\" Will adds pointedly, \"expects a lot from me, too. And,\" he adds grimly, \"from you, too, being who you are, who your father is. _Who_ ,\" he adds with extra emphasis, \"your grandmother is\u2014one of our founding mothers. Everybody watches you.\"\n\nDean sinks deeper into his seat, trying to hide his face in his book. \"Leave me alone, I'm studying.\"\n\nPausing to let the gravity of his words sink in, Will says, \"People expect a lot from guys like you and me.\"\n\nBegrudgingly, Dean asks, \"Like what?\"\n\nWill's answer is matter-of-fact, \"Like dating.\"\n\n\"I'm not dating you!\"\n\nIgnoring Dean's adamant rejection, Will continues, \"You see, like you, I don't date, and like with you, people want to know why. Like you, some even question if I'm...\" in a low whisper, \"a _strai_.\"\n\nFlustered, Dean opens both hands, dropping sandwich crumbs and slate onto the table.\n\n\"That's right, Dean.\" Will stares intently into his soon to be new boyfriend's eyes. \"People think you and I just might be...\" Now just mouthing the word \"straight.\" After allowing for the reality of their situation to sink in, Will continues, \"For the moment, the only thing saving you is your family name, and the only thing saving me are my skills on the b-ball court.\" Squinting his eyes, planning now to play hard ball, Will says, \"If you ever touch one of those girls you ogle\u2014don't try to pretend. I've seen you.\" Shrugging, he admits, \"I've been watching you watch them. I watch them, too, just not so obviously as you.\" Paling slightly with introspection, Will adds, \"At least, I hope not.\"\n\nDean is aghast. \"How obvious have I been?\"\n\n\"Very.\" Will's crisp reply is a cutting reminder that even being a Stuttgart might not be enough to save him if he should ever falter. Finally, having consistent eye contact, Will persists, \"The fact is, Dean, you need me, and...\" almost a little begrudgingly, \"I need you.\" Holding Dean's crystal blue eyes with his own stone gray orbs, he says, \"Here is what I propose.\" Standing now, Will crosses to Dean's side of the table. \"Shove over,\" he commands. Will has frightened Dean into submission. Thigh snug to thigh, Will begins slowly, very gently to rub Dean's arm with his fingers. So light is his touch that he only manages to caress his arm hair. \"We date. You and I become a unit\u2014a tight unit\u2014the serious high school couple. The ones everyone expects to register.\" Dean swallows hard. \"If we can become inseparable, then we can escape the mistrusting, questioning stares, the behind the back whispers we both hear so often, and most importantly,\" he adds, \"we can escape something we both desperately want to escape.\"\n\n\"What?\" Dean mutters.\n\n\"Actually having to date.\" Pausing briefly, Will stops his fingers from fluttering. \"So,\" he says coyly, now having his fingers walk up Dean's arm, over his shoulder, and under his chin where he feigns tickling him, \"laugh for me.\" Dean chokes a giggle. Will grimaces and shakes his head slightly. \"We'll have to work on your playacting.\" Will's playacting is obviously exemplary. Using his finger now to turn Dean's face in his direction, Will asks again, \"So? Are you in?\"\n\nDean ponders the implications. Thinking back, he has never seen Will with a guy. Never heard of any boy dating him, just a lot of wishes and wants and desires expressed by every other boy in school but him. Even though Will is playing up the player right now, Dean does not mind. There are definite advantages to dating someone who doesn't want to date. \"No kissing or any\u2014other stuff?\" Dean asks, feeling awkward.\n\n\"Definitely no other stuff. As for kissing,\" Will pauses, \"I don't think we can avoid that, but it really only has to be a modest peck between classes.\"\n\nDean's mind is highly active at this moment. \"Can I\u2014may I introduce you to my dad?\"\n\nWill smiles that his plan is actually working. Approaching Dean Stuttgart was a gamble, a huge, extremely dangerous risk, but one he had been right about. Watching Dean ogle Rylie Wineman had made Will decide to act because it looked as if the girl were ready to expose Dean. \"I'd be honored to meet the direct descendant of our founding mother.\"\n\nDean laughed\u2014not a forced, choked giggle this time, but an honest laugh that piqued the interest of those around them. As Will had known all along, they were under the discreet but vigilant observation of their peers. \"Mimi's nice. I'll introduce you to her, too.\"\n\n\"Mimi? I thought her name was Destiny.\"\n\n\"It is,\" Dean blushes slightly. \"Mimi's a pet name for grandmother.\"\n\n\"Cool,\" Will replies, considering his next move. \"Let's seal it with a kiss\u2014just a peck.\" After a quick meeting of lips, the two boys separate. Will, once he is halfway through the cafeteria, spins back gaily to call out, \"See you tonight, then!\" Dean blushes for the crowd and nods his head. They hadn't actually made a date, but it was good for the rest of the students to think they had.\n\n* * * * *\n\n\"Wow!\" Todd's eyes sparkle. \"That was quite a story Papa Dean.\" He had listened so intently to Papa Dean's voice that Todd had almost felt as if he had been there with his father and Dean over twenty-five years ago. Turning his eyes back to Papa Dean, he asks, \"Did he\u2014did my father ever get caught? Did he go to reeducation camp, too?\"\n\n\"No.\"\n\n\"And he married?\"\n\n\"Yes, your Papa Mike.\"\n\n\"Did he love him?\"\n\n\"Your Papa Mike loved your father. That I know for sure. But whether your father loved Mike\u2014I think so\u2014but honestly, Todd, I don't know.\" Dean recalls, \"Mike was a cheerleader. He doted on your father. He would do anything for him, and sometimes, your father took advantage of that. I guess that's why he chose to marry Mike. There's safety in marriage.\" Trying to reassure Todd, he adds, \"I'm sure he liked him, appreciated his devotion. Your Papa Mike was thoroughly committed to Will.\"\n\n\"But why would Dad marry if he didn't love the man? Or want sex with him. I mean, marriage means sex.\"\n\n\"Marriage isn't all about sex, Todd.\"\n\n\"I know that, but it is expected, isn't it? That's why you eventually gave in, right? Right?\"\n\n\"Yes, Todd, you're right.\"\n\n\"Then why would he?\"\n\n\"Because I warned him\u2014about reeducation.\" Closing his eyes, Dean remembers the day Will Middleton came to visit him at camp. \"You can have visitations after you've come out as gay. When your dad came to see me, I told him how happy I was to find my true self. I even tried to act festive. But it was evident by his eyes that he didn't believe me,\" Dean harrumphs. \"And I didn't blame him because at that point it was all a lie. Anyway, visitors had to sit behind chicken wire stretched up from the countertop to the ceiling. But there were a few inches available for our hands to reach underneath so when I put my hand through, he held onto it. That's when I slipped him the note\u2014a little old school perhaps, but a note is silent and can easily disappear. Anything put on the wave is easily reconstructed.\"\n\n\"What did the note say?\" Todd asks in earnest.\n\n\"'Never get caught. Find a man you like. Marry him. Trust me. It's better.' He picked Mike Fulton. Your Papa Mike loved your father to distraction.\n\nI don't think he could have picked a better man.\"\n\n\"Except he changed,\" Todd begins.\n\n\"Yes, after your father died,\" Dean agrees. \"It was as if something inside Mike died when your father passed.\"\n\n\"He was never the same.\"\n\n\"No, you're right,\" Dean agrees. \"He wasn't.\"\n\n\"Papa Dean?\"\n\n\"Yes, Todd.\"\n\n\"Please ask Frank to visit me. He hasn't been back since...\" _Since I asked him._\n\nDean smiles, hopeful. \"Sure, I will\u2014of course, I will. And he can help you, Todd\u2014I know he can. Like his father helped me.\"\n\n_No, he can't Papa Dean._ Todd closes his eyes so Dean can't read his expression. _Not the way you want him to. Frank's not like his father. But he will help,_ Todd determines, _with what I've asked of him. He owes me that much._ \"Papa Dean?\"\n\n\"Yes, Todd.\"\n\n\"I love you.\"\n\nDean, smiling through his tears, bends down to kiss Todd. \"I love you too, son.\"\n\n* * * * *\n\n#  _Salve!_\n\n**Continuing Investigation  \nHNN\u2014Melissa Eagleton Reporting**\n\nThe continuing investigation into the alleged abuses reported at the Northeast Reeducation Camp is capturing the attention of Hadrian. Parents fear for the health and sanity of their children should they be found acting on their heterosexual tendencies. One mother noted that every one of our children has the potential to experience heterosexual desires. \"Sexual education is a parental matter, not a state matter,\" said one father, expressing his dissatisfaction in the current reeducation system. \"Children should remain at home with their parents and a government counselor should attend to them on an as needed basis.\" This sounds reasonable enough, but when put into the light of economics, only wealthier families could afford such luxuries. We are still left with the necessary means of reeducating the children of our poorer citizens. Such a dichotomic split between the treatment of the wealthy and the poor will only lead to resentment among our citizens. An increase in taxes is not likely to occur, not with another election so close at hand. Hadrian's citizens and businesses already pay the highest taxes in the world to maintain the one remaining zoo as well as our four (and the earth's only remaining) national parks.\n\nHeated debate surrounds the issue of corporal punishment. Some parents are demanding that the paddle be banned as a means of punishment while others concur with Gideon Weller that in order to restrain the passions and aggressions of heterosexual men, one needs to do battle with a hard hand. We have all witnessed firsthand the brutal ways of heterosexual men every time our walls are attacked. These men are truly lunatics and very dangerous. All our exposure to heterosexual men shows that they are not capable of reason and act far too quickly on their emotions. The first instinct of the heterosexual male is to fight. According to Gideon Weller, \"When one is responsible for restraining angry young men, one has to use physical force or all of Hadrian will suffer the consequences!\" And, as our polls show, many in Hadrian concur.\n\nAs for the accusation that Gideon Weller has raped some of the wards at his camp, no evidence has been found, and not a single ward in the past twenty-five years that Gideon Weller has been stationed at the Northeast Camp has come forward with an accusation. It is merely the word of one man who claims to have heard a rape in action. Of that specific incident, Gideon Weller has a witness assuring us that no rape occurred. The young man, the alleged victim in question, remains silent on the issue.\n\nI remain firm in my belief that Gideon Weller is innocent of all charges. His work at the Northeast Reeducation Camp excels all others in the field of reeducation. _Salve!_ and HNN stand behind Gideon Weller.\n\n_Vale!_\n\n# A Controversial Lesson\n\nIn grade ten, Frank and Todd were in all required courses together. Both boys' favorite class was literature. Much of the material studied was gay literature and the standard interpretation, though often left unexpressed, was that every character read and discussed in class was gay. Todd decided to challenge this rule on the day Mr. Reiner introduced a classic Canadian literature unit with an emphasis on gay literature written when homosexuality was deemed unacceptable.\n\nOn that day, unconvinced that either the poet or his characters were gay, Todd raises his hand. After the receptive nod, he asks, \"Mr. Reiner, how do you know Earle Birney was gay?\"\n\n\"Good question, Todd.\" Mr. Reiner enjoys students with the courage and intelligence to challenge his suppositions. \"I don't. I didn't even bother to read the man's bio. All I'm really interested in is the content of his poetry\u2014actually, with the one poem I've chosen for our study, _David_.\"\n\n\"I don't see it. How can you base a man's sexuality on one poem about two friends mountain hiking\u2014especially when there is nothing in the poem to suggest either boy is gay\u2014or the poet for that matter.\"\n\nFrank instantly raises his hand. Mr. Reiner smiles. Nodding knowingly Frank's way, he waves his hand down. \"We'll get to that soon enough, Todd,\" Mr. Reiner replies, almost condescendingly. \"First, I would like to explain the necessity of studying the works of homosexuals from the past\u2014\"\n\nTodd refuses to allow the subject to change. \"I read a bio of the poet over the wave last night. It said he married a woman, an Edith or Ester Bull, and that they had a son. I think that means the man was a heterosexual.\"\n\nA slight grimace exposes Mr. Reiner's annoyance. \"There were a number of men, and women,\" he adds for the girls, \"who married and lived false heterosexual lives. But,\" cutting off any further attempt by Todd to recommence his argument, \"that is not the topic for today's discussion.\" Todd had intended to mention how Earle Birney also had a relationship with a female graduate student, but Mr. Reiner is successful in assuaging his attempt. Lifting a finger in front of the boy's face, \"Ah, ah, ah,\" he pronounces sternly, insisting they revert back to his chosen topic. \"Those men and women,\" he continues, \"who lived in an oppressed era, had to be subtle about offering the world their expression. It behooves us,\" Mr. Reiner drones on now, looking directly at his immediate opposition (Todd), \"to study the work of those men and women who did not live in a secure and free environment like ourselves. This,\" he emphasizes, \"is an important part of our heritage as homosexuals. And,\" he reminds his class, \"part of the rationale behind why our founding families chose to create a country of our own. Todd\u2014\" Although Mr. Reiner is not picking on Todd, he tends to direct most of his questions the young man's way since he is almost always sure to get an intelligent answer from Todd. \"Remind the class of the cornerstone I am referring to.\"\n\nHadrian's society is founded on four cornerstones of existence, and these cornerstones are drilled into all Hadrian's children from the first days of schooling, so anyone in the room could have answered without thinking. Todd is definitely not thinking at the moment as he answers in far too concise and crisp a manner. He believes he is being picked on for harboring a disparaging interpretation. \"Safety for homosexuals.\"\n\n\"Yes,\" Mr. Reiner's sigh expresses his disappointment. Usually Todd's answers are more in-depth. \"Devon,\" another boy Mr. Reiner can always depend on for bright responses, \"elaborate, please.\"\n\nDevon doesn't even have to turn on the cognitive components of his brain to reply. He simply recites verbatim the words straight out of Hadrian's founding constitution: \"First and foremost, it has been decided that Hadrian will provide a safe haven for homosexuals, who have, throughout the history of mankind, suffered discrimination and abuse. Never again shall a homosexual walk in fear or feel the need to hide his identity for the appearance of normalcy in society.\"\n\n\"Very good.\" Turning on the student who is his favorite most other days, Mr. Reiner scolds, \"Now, that, Todd, was a thorough, in-depth response.\" Todd's nod is acknowledgment enough for Mr. Reiner to move on.\n\n\"Now, obviously, Todd read the poem as assigned. Who else in the class read the poem _David_ by Earle Birney last night?\" Only a few hands rise: Devon, Frank, Millicent, and T'Neal. All other students lower their heads in shame. Mr. Reiner is quite disappointed. Hadrian boasts the brightest minds and the best education system on the planet, yet here, sitting before him, is the deluge of society. His glare is scathing and the class is sufficiently intimidated. Mr. Reiner, notorious for using the voc to contact parents instantly\u2014in class, _and_ loud enough for everyone in the room to hear\u2014begins a rapid succession of blinking. Having already set up contact groups with all of his students' parents, it is just a matter of blinking in the right contacts and uttering the appropriate words. \"English ten\u2014class two\u2014delete Middleton, Hunter, Rankin, Brown, and Cantos\u2014\" The rapid succession of blinks required of Mr. Reiner to organize this message was only comical to those five students not affected. \"Parents,\" the teacher begins his tirade against the wayward members of his class, \"I regret to inform you that your child has refrained from completing yesterday's homework. Please note the attached file.\" Reiner is also notorious for backup plans designed as extra work for students who fail to complete assigned tasks. \"I expect all irresponsible students to complete this task tonight in lieu of yesterday's assignment. Please ensure your child completes it and either vocs or waves the assignment to me before tomorrow's lesson.\" Numerous groans fill the room. Hadrian takes the education of its children seriously, and everyone who failed to read the poem last night knows his or her parents are answering that voc message right now! And Hadrian only knows what horrendous assignment Mr. Reiner has designed for them.\n\n\"Now,\" Mr. Reiner continues, \"take out your slates and open the Birney poem we downloaded yesterday. T'Neal,\" (Mr. Reiner always asks T'Neal to read because the young man has a natural actor's voice\u2014and T'Neal always agrees\u2014and knowing he will be asked to read, he always practices the night before) \"please read the poem aloud for us.\"\n\nT'Neal obliges, and when he is done, he offers up the first interpretation. \"It's about mountain climbers, isn't it, sir?\"\n\nMr. Reiner quickly aborts the instinctual headshake and scorn he feels whenever a student claims an interpretation already presented by another member of the class. \"Yes, T'Neal, it is. That is exactly what Todd pointed out earlier.\" Hoping for further insight, Mr. Reiner's eyes scan over the other four students who claimed to have read the poem in advance.\n\nTodd immediately pipes up. He likes the poem and read it over four times last night. \"Only the one guy was a real climber. The other kid didn't really know what he was doing.\"\n\n\"Good!\" Todd is now back in Mr. Reiner's good graces. \"How do you know that, Todd?\"\n\n\"Well, it says here,\" Todd is pointing to the lines on his slate even though the act is not necessary, \"that David taught him how to: 'David showed me\/How to use the give of shale for giant incredible\/Strides.'\" Scrunching his eyes tight, looking deep inside his brain for the answer, Todd adds, \"I think that means David taught him how to jump from one rock to another.\"\n\n\"Not bad, Todd.\" Mr. Reiner is smiling so hard his cheeks are causing his eyes to squint.\n\n\"And here it says,\" Todd adds, encouraged by his instructor's enthusiasm, \"'David taught me\/How time on a knife-edge can pass with the guessing of fragments.' I think that means David was the expert here and not his friend.\"\n\n\"Excellent, my boy, but we haven't heard from any of the girls. Millicent, what other evidence can you find in the poem to support the idea that David was an expert climber?\"\n\n\"Nothing. The poem doesn't make any sense at all.\" Clearly exasperated, she glares up at Mr. Reiner. \"Why does all the stuff we read have to be about boys? Why can't we ever read anything about lesbians?\"\n\nMr. Reiner groans, as do most of the boys. Mr. Reiner so desperately wants to point out that they just finished reading _Fried Green Tomatoes at the Whistle Stop Cafe_ by Fannie Flagg. But he doesn't. Millicent Brown would just cry about it to her mothers and make his life a living hell. Instead, he searches for the safest means of escape. \"I challenge you to find some for us,\" Mr. Reiner suggests.\n\n\"Sylvia Plath,\" Millicent responds instantly.\n\n\"Fine,\" Mr. Reiner grumbles. He hates Sylvia Plath's suicidal everybody hates me poetic rants\u2014but all the lesbians love her. \"We can download some of her work tomorrow. In the meantime,\" his eyebrows rise as he speaks briskly, \"attempt to understand Birney's poem.\"\n\n\"Yes, sir.\" Millicent is slightly mollified since she has been assured some study of her favorite poet. \"Can we start with _Admonition_?\"\n\nPushing his glasses up to his forehead, and then rubbing the bridge of his nose, Mr. Reiner sighs. \"How about you pick the poems?\"\n\n\"I will,\" Millicent declares victoriously.\n\n\"All right then.\" Having successfully escaped a scathing remark from the girl's mothers about favoritism to boys, Mr. Reiner directs the class back to today's lesson. \"Let's go deeper into the poem, shall we?\"\n\n\"Oh, oh, oh!\" Millicent is practically giggling in delight as she waves her arm about.\n\n\"Yes, Millicent.\" Mr. Reiner is dearly hoping she will talk about _David_.\n\nNo such luck. \"Can we also study Anne Sexton? She was Sylvia Plath's best friend.\" She turns and smiles to her best friend, Crystal Albright.\n\n\"Were they lovers, too?\" Crystal asks Millicent. Before answering, Millicent kisses her girlfriend.\n\nMr. Reiner puts a stop to this display of affection and any potential response from Millicent. \"My classroom is not a bedroom. Please refrain from kissing in here.\"\n\nMillicent raises her eyes, \"You let the boys kiss.\"\n\n\"No, I do not.\" Mr. Reiner is unimpressed with the accusation. \"This is an academic institution, not a social outlet for dating.\" To mollify the girl some, knowing full well she is going to go home and complain to her mothers, Mr. Reiner offers her a prize. \"Here's what you do, Millicent. You choose all the poems and poets for tomorrow's lesson. Voc them to me and I'll approve them. Sound good?\"\n\nVictorious, Millicent shakes her fists in front of her and turns for one more kiss from Crystal. Mr. Reiner chooses to let this little act pass since Millicent's mothers are very active in the girl's education. As far as these women are concerned, Millicent is never wrong. Teachers always have to tread lightly with parents of strong influence, regardless of who's right or wrong.\n\nFrank pulls Mr. Reiner out of potential hot water by raising his hand. \"I think what Bobbie did for David at the end was incredibly romantic and beautiful.\"\n\n_Romance_ , Mr. Reiner smiles. _Thank you,_ _Frank. Girls love romance!_ \"And what was that, Frank?\" Better yet, this is the very direction he wants the class discussion to take.\n\n\"To help his lover die when his life was clearly over.\"\n\n\"What?\" Todd can't believe his ears.\n\n_Oh, good!_ Mr. Reiner grins. _When these two disagree, the class discussion doesn't get any better._\n\nFrank's shoulders shrug involuntarily as his head shakes and his eyes blink in confusion. \"Didn't you hear me?\"\n\n\"Of course I heard you,\" Todd retaliates. \"I just think you're wrong.\"\n\n_Thank Hadrian for Todd,_ Mr. Reiner muses. No one else in class has the gumption to disagree with Frank Hunter. That Mr. Reiner silently agrees with Frank is irrelevant. He simply appreciates Todd's ability to spark a debate and get Frank Hunter riled up at the same time.\n\n\"So you think assisted suicide is wrong?\" Frank is fervent in his position.\n\n\"In this case,\" Todd replies quite matter-of-factly (a tone he knows drives Frank insane), \"yes.\"\n\n\"Why?\" Frank is almost angry now.\n\n\"Because his life isn't over.\" Then, in deference to Frank's near outburst, Todd raises his hand to calm down his friend. \"Okay, I admit he is probably paralyzed for life, but\u2014\"\n\n\"Probably?\" Frank interjects. \"Ah, Mr. Reiner, am I right or am I right? I mean, it says David can't even move and he only shifts his eyes...\"\n\n\"He can move his head,\" Todd points out.\n\nFrank ignores Todd's input. \"On top of all that, _Todd_ , he can't feel any pain.\"\n\n\"Where exactly does it say that, Frank?\" Mr. Reiner always insists his students provide empirical evidence.\n\nIgnoring his teacher, carrying right on with his harangue, Frank continues, \"I mean, he fell a good fifty feet, landed on his back atop a very sharp rock\u2014'a cruel fang' he can't even feel!\"\n\n\"Good, Frank,\" Mr. Reiner stops him. \"You provided us one quote from the poem, 'a cruel fang,' but that doesn't prove your supposition that David is paralyzed. Find us the evidence, direct evidence from the poem.\"\n\n\"Right here,\" Frank, too, points to his personal slate, though it does nothing to help the others in the class.\n\n\"Which lines?\" Mr. Reiner asks since Frank has not provided any direct quotes.\n\n\"Um...\" Frank really hadn't pointed to the right lines so he begins his search. Tapping his slate triumphantly when he finds it, he exclaims, \"Here! Where David says, 'I can't move...If only I felt\/Some pain.' See,\" he says, smiling Todd's way, nodding his head and raising his eyebrows. \"He even uses the word 'pain.'\"\n\n\"Okay,\" Todd acquiesces. \"He's paralyzed for life, but come on; that doesn't mean his life has to end.\"\n\n\"It does for David.\"\n\nTilting his head, closing his eyes only to open and roll them Frank's way, Todd asks, \"Why?\" Mr. Reiner has to stifle a laugh; Todd is so comical with his expressions.\n\n\"Listen,\" Franks says. (Mr. Reiner loves it when the students take over and teach each other.) \"It's at the beginning of the poem, 'mountains for David were made to see over,' and then at the end, he rejects the idea of having to live helpless and confined to a wheelchair.\" Seeing Mr. Reiner's eyebrow cock, Frank looks at the poem and finds the lines he needs for evidence. \"Here, when he says, 'For what? A wheelchair,\/Bob?'\"\n\n\"Just because someone wants to die doesn't mean it's right to help him,\" Todd insists. \"He could learn to accept his new life. Learn ways to cope.\"\n\nUsing one of the symbols Birney placed inside his poem, Frank ejaculates, \"He was a mountain goat, Todd! He could never live in a wheelchair. He needs to climb mountains. And then,\" Frank adds, latching onto the other symbol Earle Birney used to foreshadow David's death, \"there's the bird with the broken wing. David took it from Bob and killed it, saying, 'Could you teach it to fly?'\"\n\n\"That was a bird, Frank.\" Exasperated, Todd practically shouts, \"Not a guy.\" Then turning grim, he says, \"I just don't understand how Bobbie could do it.\" To take a friend's life is incomprehensible to the youth.\n\n\"Devon,\" Mr. Reiner asks, trying to include other class members in the discussion, \"do you know the answer to Todd's question?\"\n\nDevon is quick to respond. He may not jump into class discussions, but he will always answer a question when asked. \"He blames himself for David's fall.\"\n\n\"Good,\" Mr. Reiner praises the lad, \"and where is that found in the poem?\" Turning now to the girls, choosing Millicent since she is the only one to have read the poem in advance, he says, \"Come, Millicent; show the boys up by finding the evidence.\"\n\nMillicent scowls but gives in to the challenge. \"Here, when David says, 'No, Bobbie! Don't ever blame yourself.\/I didn't test my foothold.'\"\n\n\"Not bad,\" Mr. Reiner acknowledges, \"but,\" he emphasizes, \"those lines only show that David doesn't want Bobbie to feel guilty. What lines show that Bobbie _does_ feel guilty?\" Because Frank's hand is waving like a flag in the wind, Mr. Reiner tosses the reins back to him, \"Frank.\"\n\n\"And I knew\/He had tested his holds. It was I who had not... I looked\/At the blood on the ledge, and the far valley. I looked\/At last in his eyes. He breathed, 'I'd do it for you, Bob.'\"\n\n\"Very nice, Frank.\" Although pleased with Frank's in-depth unearthing of the poem exactly as he had hoped, Mr. Reiner still feels obliged to give Todd the last word. The boy looks so dejected and discombobulated by the entire interpretation. \"Okay, Todd, what's wrong?\"\n\nTodd fumes in frustration because he is unable to break Frank's logic. He truly believes he is right and Frank is wrong. All he wants is for Frank to admit that it's not always right to help another person commit suicide. They have been fighting over this issue ever since one of Hadrian's senators recently tried, and failed, to have the bill legalizing euthanasia revoked. Although referendums are normally only held once a decade, with assisted suicide being such a hot topic, another vote was held mid-decade. Although still a victory in the eyes of the government, 46 percent of the populace voted against the old law. Mr. Reiner purposely chose this poem to match the political mood of the time. He simply had to introduce it as gay literature so he could get the class to this crucial point. \"Well,\" Todd responds, as one who is defeated but determined not to lose, \"I just know I could never do what Bobbie did. I could never throw my best friend over the edge of a cliff to his death.\"\n\nFrank smiles. Of course, everyone in the room knows of his and Todd's friendship so it is evident Todd means he could never kill Frank, but Frank takes the meaning to an even deeper level. Still smiling, he winks at his best friend, clearly enjoying what he is about to say, \"But, Todd, Bobbie wasn't just killing his best friend. He was helping his lover to die with dignity.\"\n\nTodd's blush causes excessive laughter among the students. Even Mr. Reiner is enjoying the show.\n\nWith the poem blurring before his eyes, Todd actually begins to stammer. \"They weren't lovers. There's nothing in the poem that says they're lovers.\"\n\n\"I disagree,\" Mr. Reiner pipes up. Now is the time to prove his point. \"Frank?\" If his suspicion is correct, Frank will know exactly where the line is.\n\nSmiling triumphantly at Todd, Frank quotes the one and only line in the entire poem that is suggestive of a sexual relationship (the one line that enabled Mr. Reiner to claim this poem was homosexual literature) between David and Bobbie: \"Then the two of us rolled in the blanket while round us the cold\/Pines thrust at the stars.\" Mr. Reiner beams.\n\nTodd's response is feeble, and he knows it. \"That doesn't mean they had sex.\"\n\n\"Oh, yes it does!\" Frank retorts. He has won this debate, and there is no doubt about it. Todd has lost face while he has risen victorious.\n\n\"Well,\" says Todd, though defeated, refusing to give in. \"I still wouldn't do it.\"\n\n\"Do what?\" Frank asks.\n\nMr. Reiner raises a brow over this line of questioning. It no longer seems as if the boys are discussing David and Bobbie.\n\n\"Push you over\u2014\" Todd suddenly shuts down; if stripped, his entire body would be plasma purple at having just admitted affection for Frank in front of the whole class.\n\n\"Well...\" Frank is as happy as if he were in Antinous' bed. \"I'd do it for you, _Todd_!\"\n\nThe line had come straight out of the poem! The class had applauded. Rumor had spread for weeks on end that Frank and Todd were lovers (even if Todd didn't wear the ridiculous purple dog collar denoting Frank's ownership). Frank wished it could have been more than a rumor, but he had enjoyed the accolades while they lasted.\n\n* * * * *\n\n#  _Salve!_\n\n**A Controversial Law  \nHNN\u2014Melissa Eagleton Reporting**\n\nEuthanasia. Assisted suicide. Call it what you like, helping another person to die has always been a topic of heated debate in Hadrian. This controversial law, in place since the founding families introduced the first constitution, never ceases to spark argument among our citizens. There is a small contingency of religious citizens in Hadrian\u2014mostly Jewish, Islamic, and Christian (historically at odds in the outside world but, ironically, in agreement on this topic)\u2014who have united to fund a campaign against euthanasia. And they are hard at it! There is no doubt about that! Every ten years, Hadrian's citizens gather at the polls to reaffirm our country's policies and beliefs. One such referendum, of course, is to determine whether or not to retain the controversial euthanasia law on our books. So volatile has this debate become that proponents against assisted suicide were able to push for a mid-decade vote, something unheard of previously in Hadrian's short history. Fortunately, as with every other past vote, the majority of Hadrian's citizens determine the \"death with dignity\" law to be both moral and essential. Even so, radical opposition claims the last referendum to have been a victory since only 54 percent of the populace voted to keep the law in place. As a result, these people are convinced that this decade will be the one when Hadrian votes out the law that permits an individual to commit suicide with the help of loved ones or a medical practitioner if said person feels he or she can no longer live with dignity.\n\nHere at HNN, we are split in our beliefs. I stand for the human right to die with dignity while many others feel that assisted suicide is essentially murder. One cameraman states that death is not a human choice, but the dominion of God. I assure you, he is alone in that opinion. But, at the very least, our production manager would like to see stringent rules in place that will effectively determine what constitutes a life no longer worth living. Curious to know what our viewers think, we ask that you voc in to our poll at @HNN#E-AS\/MDR and let Hadrian know your opinion. Why wait for the next referendum? Let's tell our government how we feel right now! With a presidential election coming up, I can assure you that Hadrian's senate will listen closely to the _Salve!_ viewers' opinion.\n\n_Vale!_\n\n# Making Amends\n\nWhen Frank returns, he is ready. Having visited the darkness of his soul, he has come to realize he is to blame for the pain Todd is suffering. He has been selfish. _Papa Dean was right. When Todd said it would be him and me one day, I should have broken off with T'Neal, but I wanted to get laid!_ Scorn and self-recrimination burn deep. _If only I had stayed by Todd's side. Professed myself his boyfriend. Never trusted him alone with that woman! Todd only turned to Crystal because I had abandoned him. Had we been dating, she never would have come on to him_. But even that isn't the reason he decides to help Todd. Anguish is in the truth. _I raped Todd._ Shuddering, he remembers Todd's words, \"You're no better than Weller.\" Those words forced Frank to admit that Todd and he had never made love. _I raped him,_ he reminds himself, _and then I shamed him._ With this awareness came understanding. What little spirit and strength Todd had left in him were destroyed inside reeducation camp. All that remains now of his friend is a battered body and a shattered soul desperately seeking rest. Terrified by this reality, Frank realizes there is only one way to make amends, to end Todd's suffering. _I helped put him here,_ Frank realizes. _I must help him escape._\n\nBefore walking into Todd's room, Frank charms the nurse on duty. \"Why don't you take a five minute break? I'll be in the room with Todd. Everything is going to be okay.\"\n\nGrateful, the nurse smiles at Frank. He could use a little down time, having been chained to his desk for the past five hours. \"I sure could use a break.\" Wincing slightly and using his eyes to nod toward the washroom, he admits, \"I really have to go. I've been farting up a storm.\"\n\nFrank laughs, then nods, \"Go.\"\n\nAs soon as the nurse exits, Frank enters Todd's room. Todd's eyes are closed. \"Frank?\" he asks without opening them.\n\n\"Yes.\"\n\nThere is silence as the boys wait: Frank for Todd to ask; Todd for Frank to say he is ready. Finally, Todd initiates. \"Will you do it?\"\n\n\"Yes.\"\n\nTodd sighs, \"Thank you.\"\n\nFrank walks over to the bed. His first order of business is to remove from Todd's chest the attached electrodes that keep track of his heartbeat. The sudden elongated beep and straight line on the machine fixes the boys' attention; both are aghast and in awe of the image. Frank jerks back into action, opening his shirt, placing the electrodes on his chest, causing the machine to resume its steady beat, complementing the rise and fall of the heart line. Deciding now is the time, Frank reaches behind Todd's head and pulls at the pillow. He has to yank harder than he had intended to release the pillow since Todd's head is strapped down and the pillow is wedged beneath it. Prior to smothering his friend, Frank bends over and kisses Todd on the lips; then he whispers into his ear, \"I love you.\"\n\nThe silence is overpowering, the repeating beeps of Frank's heartbeat failing to register in Frank's stunned mind. Suddenly, the sound explodes in his head like a rapid succession of bombs, an eerie reminder of the corpse lying on the bed\u2014its face still covered by the pillow\u2014Frank's hands still holding it down. Pale, shivering, Frank doesn't move. He knows Todd is dead, but he cannot will himself to let go. He is afraid to release his grip from fear of seeing \"It.\" \"It\"\u2014the very word Bobbie used to describe David's corpse.\n\nOutside, at the nurse's station, seeing nothing untoward, a young man smiles at the sight of a visiting lover leaning over to comfort his friend. Suddenly, the nurse is shocked into action when he watches Frank slowly straighten and then rip something away from his chest.\n\n* * * * *\n\n#  _Salve!_\n\n**Murder or Assisted Suicide?  \nHNN\u2014Melissa Eagleton reporting**\n\nThe death of Todd Middleton, son of Will Middleton, the bioengineer who brought the soya bean to Hadrian, comes as a shock to everyone who knew and loved the boy. Todd was only seventeen years old and beloved by many. Like his father before him, Todd Middleton was known to Hadrian as a superstar on the b-ball court and had even been offered a full early entrance scholarship to Antinous Uni prior to his having been exposed as a heterosexual male.\n\nWhat is most shocking about Todd Middleton's death is how he was brutally murdered by his lover. Their peers knew Todd Middleton and Frank Hunter as the best of friends who, shortly after Todd had been exposed as a heterosexual, were said to have coupled. Because Todd had already had sexual intercourse with a young woman, this sudden change in their relationship was insufficient to stem Todd's removal from society and his being placed in a reeducation camp. Todd Middleton was housed at the Northeast Camp under the care of Gideon Weller with Jason Warith as the boy's guardian. Although Frank Hunter has confessed to his crime, Jason Warith insists that the real murderer in this case is Gideon Weller.\n\nFrank Hunter's defense lawyer is Ms. Faial Raboud, renowned for obtaining an acquittal for Andrea Hodgson, who had been exposed as a heterosexual by Darya Danson.\n\nRaboud's strategy of defense for Frank Hunter is to focus on Todd Middleton's death as having been assisted suicide and not an act of murder. As we know, euthanasia is legal in Hadrian if the individual requesting aid is of sound mind and clearly suffering from a debilitating illness. To die with dignity is every Hadrian citizen's right. The question being raised by National Prosecutor Graham Sabine is whether or not Todd Middleton was actually suffering from any weakening and incurable disease. Also, as Sabine succinctly puts it, \"When the euthanasia law was put into place, it is highly unlikely Congress considered depression as life-threatening. Depression,\" the national prosecutor points out, \"cannot be deemed unbearable as it is seldom lifelong and there are always ways to help cure the victim.\" Medical professionals agree Todd Middleton was in a state of depression at the time of his death, but they are in disagreement whether the quality of his life had been drastically diminished. What validates euthanasia is when the victim can no longer live a quality existence and it would be kinder to allow him to die with dignity. All of this hinges on the individual in question being of sound mind when making the decision to die. As Sabine points out, \"Severe depression disables the individual from having a 'sound mind.' Thus,\" Sabine argues, \"Todd Middleton's request to die should have been denied.\" Defense attorney Faial Raboud must prove to the court that hopelessness is not akin to mental instability and that one can make sound choices while suffering from depression. Raboud has set up quite a challenge for herself if she is going to prove that Todd Middleton had lost all self-respect with no hope of ever restoring his self-esteem.\n\n_Vale!_\n\n# A Private Meeting\n\nDefense attorney Faial Raboud had reluctantly agreed to a private meeting with National Prosecutor Graham Sabine, to be accompanied by Crystal Albright and her mothers. Faial judiciously chose to keep Frank Hunter and his fathers ignorant about this conference. She will reveal all to them after the fact, but emotions are running too high right now for the family\u2014anger, hurt, and disillusionment have built voraciously into a desperate need to scapegoat Crystal Albright. The girl certainly holds her share of responsibility in this tragedy, Faial acknowledges, but she is not deserving of Dean Hunter's severe reaction\u2014his extreme hate for her. The man's anger and pain have taken him beyond the realm of reason.\n\nScanning the meeting room while she waits, Faial notes that it is the standard issue of the conference meet and greet. Situated on the third floor of the national government's central office building (Hadrian's tallest building), the room has north windows that open onto the building's roof garden. The room is filled with natural light, offering no light fixtures for after daylight hours. No sunrays ever penetrate the room, though, as the sun is always too far southeast or west for any of it to enter the building. Outside this wall-length gemstone is one of the most beautiful roof gardens Faial had ever seen. Being early July, every flower is in bloom, including the tiger lilies and the ever present wild rose. \"Ahh,\" Faial sighs involuntarily, there is a Brown-Eyed Susan. And another! She loves the Brown-Eyed Susan! That had been her genetic father's nickname for her. She also notes a spray of daisies, carnations, and gladiolas.\n\nBlinking and looking to the time display in the upper left of her eye screen, Faial sighs, this time disconcertedly. Graham Sabine is late. Sabine _is always late,_ she reminds herself. Graham Sabine is not known for his punctuality. Faial believes he is tardy on purpose in an attempt to unbalance the opposition. By leaving her to sit alone, waiting, worrying, Faial reasons, Sabine means to create tension, bringing on a state of mind he hopes will discombobulate her. This tactic never worked the last two times he tried it, so it amuses Faial that he continues to use it on her. _Who was it that said insanity is to do the same thing over and expect different results?_ Even though she can't remember the original author, the thought still makes her giggle.\n\nBringing her thoughts back to Frank Hunter, Faial's jovial mood diminishes. Though unaffected by Sabine's repetitive tactics, Faial is fully aware of the case's complexities. Getting Frank Hunter off will not be easy. Her partner told her she was crazy to take the case, but there is more to this than a killing. Instinct compels her. The statements of Dean Hunter and Jason Warith have exposed an ugly truth about Hadrian, a truth she has tasted, felt, but prior to this case, has never been able to expose. Andrea Hodgson's case taught her all about the ills of imposing stringent sexual preference laws on individuals. Like her client Andrea, Faial had come to believe Darya Danson dressed in drag to lure women. Though she could never prove this supposition, it had opened her eyes to the realities of imposed sexual preference. Men and women must be free to choose with whom they wish to share their bodies. If society needs law to avoid excessive procreation, so be it, but no one, Faial has come to believe, has the right to decide whom a man or a woman should love. The whole notion that incidents of rape and pedophilia will increase astronomically if heterosexuality becomes legal is sheer nonsense. Rapists and pedophiles are a class all their own. It is unfair to use their despicable behavior against normal sexual drives. Whether Hadrian's citizens are ready to accept it or not, Faial reasons, heterosexual desires are as normal as homosexual ones.\n\nAlthough she currently lacks proof to support her beliefs, the Frank Hunter case, Faial truly believes, if played right, will reveal the evidence required for much needed reforms. The fact that her client has confessed to murder doesn't make her task any easier. Somehow, she has to get through to the boy that there is more to this case than the act of suffocation. That act, as Faial and the boy's fathers believe, was assisted suicide, but Frank is not helping matters. Using this time alone while waiting for Sabine and the Albrights, Faial turns her mind to the seriousness of this case.\n\nFrank Hunter's confession is damaging. As soon as the nurse entered Todd Middleton's room, Frank confessed. He repeated the same story to government law officials, even to National Prosecutor Graham Sabine prior to speaking with her! When Faial finally had an opportunity to meet with her client, he uttered the exact same words again. It was like listening to a soldier spout off rank, name, and serial number: \"Todd is dead. I killed him. I suffocated him with his pillow.\" Even so, Faial is sure she can work around Frank Hunter's self-damning act. Although her client is taciturn, making it hard for her to uncover motivation, she did get a lot of background information from his fathers and Jason Warith. _Jason Warith,_ Faial smiles. _Now there is someone I look forward to working with._ They are like-minded with respect to Hadrian's reeducation system, yet he is not so open-minded regarding the radical change for which she is hoping\u2014making the heterosexual lifestyle legal. Even so, Faial feels almost akin to Jason Warith, as if she has finally encountered a kindred spirit. They may not agree on every point, but together, they can pave the way for some necessary changes to occur in Hadrian. It would help to have some founding family backing, but Faial knows that is asking too much.\n\nFocusing her mind back on today's meeting, Faial considers its potential ramifications; if Graham wants to meet about Crystal Albright, no doubt he knows her testimony will be damaging. _I wonder if he knows about the phone messaging? Not likely,_ Faial reasons, _as that would have required the girl to expose herself._ Reaching her hand inside her coat pocket, Faial smiles as her fingers caress the thin metal. Today's meeting will be revealing, especially if what Dean Hunter says is true\u2014 _Of course what he says is true; I have all the evidence I need right here._ She taps the phone as she considers this. The text message he judiciously saved says it all, but the evidence will be all the more effective if Faial can get the girl to admit to the truth in court. _I want to hear her say it_ , Faial ruminates. _I want the court to hear the words come right out of her mouth._\n\nAs the door opens, Faial turns to see Sabine enter followed by three women, one of whom is Vice President Stiles. Sabine purposely kept Stiles' presence from Faial, hoping to add shock value to the start of this meeting. _So_ , she realizes, _he has founding family support._ Fortunately, Faial learned long ago to school her expressions. Although Faial feels the surprise Sabine intended, he is not fortunate enough to witness her distress. Faial's face remains, as it always does when she encounters opposition, impassive. A forced smile blossoms on his pudgy face as Sabine leads his entourage around the oval conference table.\n\n\"Graham,\" Faial nods in greeting.\n\n\"Faial.\" Graham leans forward, casually resting his forearms on the glossy jack pine table, his hands clasped together, unsuccessfully feigning a relaxed countenance.\n\n_Graham is not in control,_ Faial immediately surmises. The three women are all sitting with their backs against their chairs. The youngest, Crystal Albright, no doubt, has dropped her head. Gail Albright, the genetic mother, has all her attention focused on the girl while Vice President Stiles, Crystal's mama most likely, sits with arms folded under her breasts. Stone gray eyes, like a frozen pond, bore down on Faial. She could easily see how this woman won her way into office. Even without her founding family status, she would have accomplished much, maybe even got to where she is today. Smiling now, she returns her gaze back to the prosecution. \"It's been a while.\" Their mutual greeting, though polite, is strained. These two have faced off in court before. Both times, Faial won. Sabine's smile suggests to her that he believes he is on the winning team this time. Faial has no illusions; this is going to be an impossible case to win, but through it, she is going to break a little ice, crack open something she believes needs revealing. _With luck,_ she muses, _I may just create an old fashioned river break-up!_ Disgusted by Sabine's puffed up confidence, Faial adds, \"The Andrea Hodgson case I believe.\"\n\nSabine's smile turns quickly to a scowl. He lurches forward, unclasping his right hand and directing his index finger her way. Elena Stiles, sitting next to the man, rests her hand on his shoulder. Sabine resumes his posture of ease, hands re-clasping, and his smile, though strained, once again blossoms on his face. \"I see you are alone\u2014no client?\"\n\n\"I saw no need to bring them.\" Faial's smile solidifies into confidence. Having successfully exposed Sabine's weakness, that of being a poor loser, she also exposed the true nature of the hierarchy in this room. Elena Stiles, as she had rightly assumed, is the woman in charge. Even so, Faial knows the balance of the meeting has tilted in her favor\u2014for the moment.\n\n\"Is it too emotional for them?\" Sabine asks sincerely.\n\n\"That, and their presence is unnecessary. As, I suspect,\" she adds assuredly, \"is this meeting.\" In response to Gail Albright's quizzical look, the menacing glare given by Elena Stiles, and Sabine's chagrin, Faial adds, \"I have all the evidence I need right here,\" reaching now into her pocket and retrieving Todd Middleton's cell phone and presenting it, \"to subpoena the girl.\" Faial hones in on Crystal Albright as the girl looks up and stares at the object as if it were a viper ready to strike. Her wide eyes and pale clammy face are all too telling. Sabine glares at the girl, then her mothers. Gail Albright grips her daughter's hand under the table and Crystal's head drops. Sobbing commences and Mama Elena grips Crystal's other hand, silencing her whimpers; even so, her upper body continues to convulse.\n\n\"I see.\" Sabine resumes his smile for Faial, a very tight smile. \"And do you really believe that evidence is usable?\"\n\nFaial nearly laughs, \"Absolutely.\" _What game is he playing at?_ Faial wonders. Elena Stiles gives no clues; she remains stolid and angry, but, for the moment, silent.\n\n\"You realize there is no way you can prove Crystal Albright is the author of said text.\"\n\nFaial's smile widens imperceptibly. \"I haven't mentioned a text yet.\"\n\nSabine remains in control. \"What else could it be?\"\n\n\"Good guess.\" She decides to throw Sabine a bone. \"And, yes, as I have the original text.\"\n\n\"May I see the text message?\" Sabine reaches across the table for the device.\n\nFaial pockets the phone and replies quite congenially, \"Of course.\" She blinks to open her voc line. \"Is your voc id the same?\"\n\nSabine sits back, disappointed by the way things are progressing. He had really hoped Faial would have been sufficiently intimidated by the VP's presence simply to hand over the phone and all documentation. \"Yes.\"\n\nWith a blink and a quick whisper, \"Graham,\" Faial sends off the link. Sabine's eyes slit as he receives the data.\n\n\"Send it to me,\" Elena Stiles demands, \"esHgov33vp\u2014uppercase the 'h.'\" Faial obliges. As soon as Elena receives the data, she passes it on to Gail and Crystal. Gail immediately begins blinking and reading. Crystal ignores her voc\u2014she already knows what the document contains.\n\nAfter finishing his read, Sabine pronounces, \"You still can't prove Crystal Albright is the text author.\" Sabine is reaching here and Faial knows it, but he has serious political clout behind him. \"Anyone could have written this.\"\n\n\"Really?\" Faial isn't falling for it. She knows better, and one quick look at Crystal proves how easy it is going to be to get the girl to babble once she is on the stand. She is obviously drowning in the deep end of her own fear and guilt. Looking now at the girl's mothers, Faial is held by the arresting stare of Elena Stiles. It feels as if Elena can read her mind. Vice President Stiles just sits there, impassive, and then suddenly, she cocks her left eye and Faial knows. _This woman will not allow Crystal Albright on the stand._ Looking back to Sabine, grim with determination, \"I will subpoena her.\"\n\n\"No, you will not.\" Elena speaks so softly each word becomes an explosion.\n\nIt is hard to stand up against a founding family member, even when one is as formidable as Faial Raboud, but stand up to her she does. \"I am merely following the rules of legal engagement established by your forefamily and the other founding forefamilies.\" Determined not to allow Frank Hunter to be sacrificed on the altar of politics, Faial persists, \"My client has the right to a fair trial and he is innocent until proven guilty.\"\n\n\"Your client has already confessed\u2014he is guilty!\" Elena Stiles, now in control of the meeting, treats Graham Sabine like a piece of furniture. His head is bowed. _No doubt he is fuming,_ Faial reasons. Had it not been her client's life at stake, Faial would have enjoyed watching Sabine so successfully subdued.\n\nNot being one to cave under the bully, Faial announces, \"My client is still entitled to a trial. That is the law according to the founding forefamilies' constitution. This case is going to court.\"\n\n\"The national prosecutor can arrange for his sentencing based solely on his confession,\" Stiles insisted.\n\nFaial instantly lowers her head. Placing her hands atop her head and resting her elbows on the table suggest defeat, adding a long drawn out sigh for emphasis. All the while, she is rapidly blinking out a voc message to her aide\u2014\"with Frank?\"\u2014\"good\"\u2014\"stick like glue\"\u2014\"tell you later\"\u2014\"no visitors\"\u2014\"no voc contact\"\u2014\"keep out of his eye\"\u2014\"no pen, no paper\"\u2014\"trust me, he'll ask.\" All the time she is voc'ing, Faial is desperately hoping she is not too late. They must have someone near him right now, and knowing the state her client is in, Frank Hunter would willingly sign his life away.\n\nSabine, responding to a tap on the shoulder by Elena Stiles, looks up to observe Faial Raboud's physical position. Believing her to have been subdued, he takes full advantage to stick the knife into the jugular. \"There is precedent. The Nation vs. Almer.\" Gerald Almer had murdered Henry Wilfer. It was a crime of passion. Gerald Almer had walked in on Henry and another man. He reacted badly to the sight. Having brooded over his lover's betrayal for months, Gerald eventually took action. Having staked out Wilfer's new home, he waited for a night when his ex was alone. He then snuck into their housing complex and slit his throat. Like Frank, he turned himself in, confessed, and then asked that a trial be waived. He did not want to be exiled, asking instead that the state assist in his suicide. Having given Faial time to register this new information, Sabine now twists the knife. \"We plan to do the same with your cli\u2014\"\n\nFaial, having finished her silent communication, cuts Graham off. \"That action was at the accused's request. My client has made no such declaration!\" She adds fervently, \"And, any such request must come through me.\"\n\n\"Or your aide,\" Sabine adds gleefully. Faial's instant anger is unusual, suggesting her back is against the wall\u2014the exact position she hopes he and Elena Stiles believe she is in.\n\nFaial ponders Sabine's comment briefly. It is unlikely Gil would betray her. The two have worked closely for over twenty years. They are very close, Faial being Gil's spouse's son's auntie. When Gil's boy was diagnosed with leukemia, Faial had financed the boy's treatment. Gil may have them convinced he is working for them, but the odds of his betrayal are very unlikely. _But why didn't he say anything when I voc'd? Either they haven't contacted him and are bluffing or I never gave him the opportunity._ Her messaging was sent like a Gatling gun. She had learned years ago how to text with a blink of the eye so she wouldn't have to speak. Most people don't use this feature because it is very hard to master, but master it she did for when she needed to let others know something important quickly; quietly pulling up the keyboard is the only way. _No,_ she reasons, _Gil is trustworthy_ \u2014yet, Sabine and Stiles seem so sure. She finds herself asking, \"Have you received Frank Hunter's consent?\"\n\nBoth Stiles and Sabine smile, suggesting the answer is yes. Yet Sabine's eyes shift ever so slightly, betraying his confidence. They are waiting on it. Stiles, on the other hand, has the perfect poker face. Faial considers the vice president her real opponent. This woman has substantial power and is now running for president. This court case may very well destroy her chances of winning. Considering just how dangerous Elena Stiles could be, Faial decides to bargain. \"Well, I'm not giving you an out of court option. What would you like instead?\"\n\n\"Keep my daughter out of it!\" Elena Stiles answers instead of Sabine. Faial has gauged his role correctly. He is merely Stiles' tool.\n\nLosing Crystal Albright's testimony is huge. It is a key point in Faial's argument. Without the girl's confession, Sabine might very well convince the court that the text message is fraudulent. Faial is only worried slightly over that; at best, Sabine might be able to cast some doubt as to Crystal's authoring it, but there is enough damning information in the text for Faial to open up the idea that Todd Middleton was not a rapist but a young man in love. A young man who had been betrayed by his lover, the woman he loved; a young man who had then been raped while at reeducation camp. Her plan is to show how Todd's mental state deteriorated to such an extent that he could never have accepted life as a homosexual\u2014not after everything that had happened to him. Thus Frank Hunter is no longer a murderer but one who committed the lawful act of euthanasia. Would the text alone be enough? Probably. Even if Sabine does manage to cast some doubt over its authorship, the damage will be done. Graham Sabine is an opponent Faial feels confident in overcoming, but Elena Stiles\u2014Vice President Stiles, soon to be President Stiles\u2014founding family member Stiles\u2014this woman is a hurdle Faial is unsure she can leap. \"I'll keep Crystal off the stand, but the text is being used as evidence.\" Before Elena Stiles can explode, Faial adds, \"It is already documented evidence.\" Stiles' glare is daunting, but Faial is not backing down on this point. She wants her client to live. Elena Stiles' reaction is more subdued than Faial expected. She nods once, and though her face remains grim, her scowl fades ever so slightly. Faial has the sinking feeling that she has just given the woman exactly what she wanted all along. Everything prior to this moment has been a bluff. Faial comes to realize that as much power as a founding family may have, not even its members can circumvent their own laws. Elena Stiles had known all along that she couldn't keep Faial from putting her daughter on the stand until she wrenched the offer out of her. _Thank Hadrian it's Graham I will be up against in the courtroom and not this woman!_ Faial takes a brief moment to remind herself never to play poker with Elena Stiles. _She'd clean me right out, down to the very last credit._\n\n* * * * *\n\n# Transcripts: Hadrian vs. Hunter\n\n**Defense Questioning of Geneticist Avery Gillis**\n\nDefense: |  | Mr. Gillis, you are a geneticist for Hadrian's Procreation Arm of the Government, are you not?  \n---|---|---  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | How long have you worked in this capacity?  \n|  |   \nGillis: |  | Twenty years.  \n|  |   \nDefense: |  | So, for the last twenty years, you have been working to identify and create the genetically perfect homosexual human gene. Is that not so?  \n|  |   \nGillis: |  | We do not consider our work as an attempt to create genetic perfection. We are simply working toward identifying, isolating, and ensuring the DNA most likely to produce homosexuals.  \n|  |   \nDefense: |  | Why is it so important that every human born in Hadrian be homosexual?  \n|  |   \nGillis: |  | Everyone knows the answer to that.  \n|  |   \nDefense: |  | Of course we do, but humor me and provide us with your reasoning for the record.  \n|  |   \nGillis: |  | Obviously, here in Hadrian our goal is to create a stable human population that will live in harmony with the earth.  \n|  |   \nDefense: |  | And being homosexual is crucial for living in harmony with the earth?  \n|  |   \nGillis: |  | No, of course not. But it is critical for controlling human population.  \n|  |   \nDefense: |  | And why is that?  \n|  |   \nGillis: |  | The evidence is all around us in the outside world.  \n|  |   \nDefense: |  | Once again, humor me, for the record.  \n|  |   \nGillis: |  | Over twenty billion people populate the outside world of our earth. Hadrian, on the other hand, has for over fifty years created and maintained a stable population of approximately ten million. This stability has only been accomplished through radical measures with a homosexual population and licensed births.  \n|  |   \nDefense: |  | Very good. I am sure everyone here agrees with you. Let us go back to your work with genetics. How successful have Hadrian's researchers, such as yourself, been at identifying the homosexual gene within an embryo?  \n|  |   \nGillis: |  | Very successful.  \n|  |   \nDefense: |  | Define _very successful_ for me.  \n|  |   \nGillis: |  | What's to define? Very successful is very successful.  \n|  |   \nDefense: |  | Excuse my ignorance, but I am not a geneticist, so I need you to be very specific. For example, can you guarantee that 100 percent of all genetic tracing in embryos will in fact identify if a child will have no homosexual tendencies at all?  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | Really? One hundred percent? No homosexual tendencies at all? Guaranteed? Absolutely? Beyond any shadow of a doubt?  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | And what happens to these embryos?  \n|  |   \nGillis: |  | They are destroyed.  \n|  |   \nDefense: |  | So, is this why you believe you have eradicated all zeros and ones from the Kinsey scale?  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | Does this mean you are able to distinguish the exact ratio of homosexual tendencies that are blended in with hetero-sexual tendencies\u2014in each embryo, of course, prior to it being approved for maturation?  \n|  |   \nGillis: |  | No, of course not. But we are so close we honestly believe we have eradicated the majority of zeros and ones from the Kinsey scale.  \n|  |   \nDefense: |  | The majority of?  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | I'm sorry, Mr. Gillis, but you just stated unequivocally, beyond any shadow of a doubt, that any embryo lacking the gene for homosexuality\u2014every single one\u2014is destroyed, ensuring absolutely no zeros\u2014no pure heterosexuals\u2014exist today in Hadrian. Yet, now you use the phrase \"The majority of.\" When one says the majority, one suggests a minority exists.  \n|  |   \nGillis: |  | Yes, but\u2014  \n|  |   \nDefense: |  | Wait for the question, Mr. Gillis. Please, correct me if I am wrong, but isn't the Kinsey scale based on behavior patterns? His research, if I recall correctly, was not founded on any genetic studies.  \n|  |   \nGillis: |  | You are correct.  \n|  |   \nDefense: |  | So how can you claim to have eliminated _only_ \"the majority of \" the zeros and ones on this scale if it is not a genetic scale?  \n|  |   \nProsecution: |  | M'Lady, I fail to see the purpose behind this line of questioning.  \n|  |   \nJudge: |  | Nor I. I would ask the Defense to please explain herself.  \n|  |   \nDefense: |  | I am looking to establish the deceased's sexual orientation and the motivation behind my client's act of euthanasia.  \n|  |   \nProsecution: |  | Correction, M'Lady. The defendant is on trial for murder.  \n|  |   \nDefense: |  | I am aware of the charges against my client, but it is my intent, M'Lady, to prove his act was, in fact, euthanasia and not murder.  \n|  |   \nJudge: |  | And you intend to accomplish this through a study of genetics?  \n|  |   \nDefense: |  | Yes, M'Lady.  \n|  |   \nJudge: |  | Proceed.  \n|  |   \nDefense: |  | Once again, Mr. Gillis, how is it geneticists are able to use a behavioral scale in their genetic research?  \n|  |   \nGillis: |  | We are simply using terminology laypersons can understand.  \n|  |   \nDefense: |  | So, the Kinsey scale is not, in fact, an accurate representation of what you are capable of with genetics?  \n|  |   \nGillis: |  | Well, yes and no.  \n|  |   \nDefense: |  | Explain.  \n|  |   \nGillis: |  | Yes, in the sense that we are very close to identifying the homosexual gene. No, because no work in genetic research will ever be 100 percent.  \n|  |   \nDefense: |  | Thank you, Mr. Gillis. That brings me to my next point. Let us refer back to your wording \"the majority of.\" Prior to just admitting that any work in genetic research will ever be 100 percent, did you not claim to guarantee\u2014100 percent guarantee\u2014that every fetus could in fact be genetically altered, ensuring all Hadrian's children be born homosexual?  \n|  |   \nGillis: |  | Yes.  \n|  |   \nDefense: |  | Really? A 100 percent guarantee?  \n|  |   \nGillis: |  | All right, no, but really, yes.  \n|  |   \nDefense: |  | No, but really yes? Please explain.  \n|  |   \nGillis: |  | We do what is called genetic screening of the fetus. Tests will then reveal if the child to be born has the genetic makeup of someone who is homosexual.  \n|  |   \nDefense: |  | That sounds straightforward enough, so why did you preface your response with no?  \n|  |   \nGillis: |  | Because, as much as we now know about human DNA, it is such a condense microscopic heliograph, it is impossible to ever say fully, 100 percent, exactly what an individual's sexual orientation will be.  \n|  |   \nDefense: |  | Interesting.  \n|  |   \nGillis: |  | That being said\u2014  \n|  |   \nDefense: |  | That being said, it is possible, is it not, that some of Hadrian's children may in fact be born heterosexual? That perhaps a one or even a zero on the Kinsey scale could be born.  \n|  |   \nGillis: |  | It is very unlikely.  \n|  |   \nDefense: |  | Unlikely does not mean impossible. If it is \"unlikely,\" then you have to admit it is possible; even the most remote possibility still exists, does it not?  \n|  |   \nGillis: |  | Not likely.  \n|  |   \nDefense: |  | Does it not?  \n|  |   \nGillis: |  | All right, yes. But we are very careful in our screening process.  \n|  |   \nDefense: |  | Regardless of how careful you are, did you not just say it is not possible ever to be 100 percent certain of anything when it comes to human DNA?  \n|  |   \nGillis: |  | You are correct.  \n|  |   \nDefense: |  | So it is possible then that one, two, or maybe even more of Hadrian's children may in fact be born not only with heterosexual tendencies but be a zero on the Kinsey scale, to use your own terminology.  \n|  |   \nGillis: |  | Although very unlikely, it is possible.  \n|  |   \nDefense: |  | Thank you, Mr. Gillis. No more questions.  \n|  |   \nJudge: |  | Does the prosecution wish to ask any questions of the witness at this time?  \n|  |   \nProsecution:  |  | No, M'Lady, but I would like to ask that we approach the bench.  \n|  |   \nJudge: |  | Would the prosecution and defense please approach the bench?  \n|  |   \nProsecution: |  | What was the point of all that?  \n|  |   \nDefense: |  | My point, M'Lady, is that Todd Middleton was a heterosexual living closeted in a homosexual world. Fearful of reeducation camp, fearful of losing family and friends, fearful of losing the opportunity presented to him by Antinous Uni, Todd Middleton was a young man diminished in morale and esteem. The final toll destroying his will to live was being brutally raped by Mr. Weller\u2014  \n|  |   \nProsecution: |  | Objection, M'Lady. Mr. Weller is not on trial for rape. Nor is there any evidence he committed such an act.  \n|  |   \nJudge: |  | Objection noted. The Defense will please rephrase.  \n|  |   \nDefense: |  | After his sexual experience with Mr. Weller, the young man turned to thoughts of suicide, evidence of which has already been shown. My client, therefore, chose not to murder Todd Middleton, but rather to assist the young man in an act of suicide.  \n|  |   \nJudge: |  | Euthanasia is only legal if the patient is living in pain and wishes to die with dignity. This case still boils down to an act of murder.  \n|  |   \nDefense: |  | On the contrary, M'Lady, if Todd Middleton were indeed heterosexual with no hope of reeducation, fearing, no doubt, as all the citizens of Hadrian fear, the \"heterosexual barbarian,\" his self-loathing, coupled with what I intend to prove was a brutal rape, left the young man living in a state of hopelessness akin to physical pain. Ergo, Frank Hunter's act is no longer that of murder but euthanasia.  \n|  |   \nJudge: |  | All right, Ms. Raboud. I will allow you to continue\u2014for now.\n\n* * * * *\n\n#  _Salve!_\n\n**A Captivating Court Case  \nHNN\u2014Melissa Eagleton reporting**\n\nThe court case involving Hadrian versus Hunter is all anyone in Hadrian can talk about. Heated debates are being held in every office, classroom, even on the street. Everyone in the HNN newsroom also comments on how the evening dinner table is focused on discussions regarding the innocence or guilt of young Frank Hunter. Seventeen years old and a confessed murderer, yet his lawyer, Ms. Faial Raboud, refuses to enter his guilty plea. Being her client is under age, Ms. Raboud is only entering the plea the boy's fathers agree to: not guilty. The two fathers and Ms. Raboud stand by the plea that Frank Hunter did not murder his friend; rather, he participated in an act of assisted suicide. Today's testimonies by Jason Warith, the boy's guardian at the Northeast Reeducation Camp, and the boy's papa, Mr. Mike Fulton, lend credence to Raboud and the Hunters' assertion.\n\nMr. Fulton admitted to having abandoned his son to the system when he learned the boy had been exposed as straight. Just knowing one's loving parent no longer desires any contact with you is enough to cause any young man or woman to lose his or her senses. As we know from a previous _Salve!_ with Gideon Weller, many of our young men and women who have been exposed find themselves suddenly cast aside by their families. As numerous youth survive the reeducation system and go on to lead productive lives, this evidence alone is not enough to prove that Todd Middleton's depression was extreme enough to allow for euthanasia. It was Mr. Jason Warith's testimony that proved the most damning.\n\nWhen on the stand, Jason Warith spoke of brutal beatings occurring on a daily basis at the Northeast Camp. The paddle, shown to the right on our wall screen, is said to have been wielded against Todd and nearly every other young man's bare backside while in the custody of our nation. As well as beating the young men, public displays of this abusive treatment also occurred as a warning to others. According to Mr. Warith, Mr. Weller beat one boy to the point where blood dripped off the paddle. Jason Warith goes on to say, \"Mr. Weller then pointed the bloody instrument at us, yelling out, 'There will be no dissension in my camp!'\" This incident, according to Mr. Warith, wasn't the worst of it. \"The man brutally raped my ward.\" Although Mr. Warith admits he did not witness the actual attack, he claims to have heard the young man crying out for help. \"Being locked out of the room,\" Mr. Warith claims, \"made it impossible for me to intervene.\"\n\nMr. Weller has yet to take the stand. Tomorrow, prosecuting attorney Graham Sabine intends to examine him and prove beyond a doubt that although Todd Middleton came to the Northeast Camp recalcitrant, he was beginning to soften under Gideon Weller's tutelage and was readying himself for reentry into Hadrian society.\n\n_Vale!_\n\n# Transcripts: Hadrian vs. Hunter\n\n**Prosecution questioning of Gideon Weller**\n\nProsecution:  |  | Mr. Weller, according to the Defense, Todd Middleton was heterosexual and no amount of reeducation would work for him. Do you agree with his assessment?  \n---|---|---  \n|  |   \nWeller: |  | No sir. Todd Middleton knew he had heterosexual leanings, but like everyone in Hadrian, there was inside him a latent homosexual. He asked me to help bring those latent tendencies to the surface.  \n|  |   \nProsecution: |  | Which you did by agreeing to make love to him at his request?  \n|  |   \nWeller: |  | Yes. As the legal age of consent is sixteen and Todd Middleton's Charge was unwilling to assist his ward in this manner, I agreed to help the young man out.  \n|  |   \nProsecution: |  | How can you explain Todd Middleton's suicide attempts after your sexual interaction with him?  \n|  |   \nWeller: |  | Todd Middleton is not the first, nor will he be the last, young man who becomes suicidal after his first sexual experience. These boys are still struggling with their sexual identity, which is why so much care is taken to prevent their attempts from being successful. Inevitably, they learn not only to accept themselves as homosexual, but also eventually to become active, healthy members of society.  \n|  |   \nProsecution: |  | Thank you, Mr. Weller.  \n|  |   \nJudge: |  | Defense, you may cross-examine.  \n|  |   \nDefense: |  | Mr. Weller, do the men you make love to often scream\u2014  \n|  |   \nWeller: |  | Indeed, madam, they do.  \n|  |   \nDefense: |  | Allow me to finish the question, sir. \"Do the men you make love to often cry\u2014  \n|  |   \nWeller: |  | Tears of joy!  \n|  |   \nJudge: |  | Mr. Weller, one more outburst and I will hold you in contempt of court.  \n|  |   \nWeller: |  | Sorry, M'Lady.  \n|  |   \nDefense: |  | Do the men you make love to often cry out in agony, begging Hadrian for help?  \n|  |   \nProsecution: |  | Objection, M'Lady. How Mr. Weller's sexual partners react during their intimate act is of no business to the Defense.  \n|  |   \nJudge: |  | Objection noted.  \n|  |   \nDefense: |  | Allow me to rephrase. Why is it, if you and Todd Middleton were making love, that the young man was heard to cry out in agony, \"For the love of Hadrian, someone help me\"?  \n|  |   \nWeller: |  | Todd's words were, to the best of my memory, \"Please, Mr. Weller, for the love of Hadrian, help me.\"  \n|  |   \nDefense: |  | And never once did he cry out the words, \"Please, no, stop, please, stop\"?  \n|  |   \nWeller: |  | I recall his saying, \"Help me stop; please, help me stop.\"  \n|  |   \nDefense: |  | How do you explain the difference between what Mr. Warith heard Todd yelling and what you claim him to have said?  \n|  |   \nWeller: |  | Quite simply, Mr. Warith could only hear the young man's voice muffled through a closed door.  \n|  |   \nDefense: |  | And your personal assistant, Mr. Darrell Jeffreys, can attest to that.  \n|  |   \nWeller: |  | Yes, he can.  \n|  |   \nDefense: |  | One more question, Mr. Weller. Do you often have your personal assistant in the room with you when you make love to a man?  \n|  |   \nProsecution: |  | Objection, M'Lady. Mr. Weller's sexual habits are not on trial.  \n|  |   \nJudge: |  | Objection noted.  \n|  |   \nDefense: |  | The Defense retracts its question. That will be all; thank you, Mr. Weller.  \n|  |   \nJudge: |  | Prosecution, you may cross-examine.  \n|  |   \nProsecution: |  | Mr. Weller, did you rape Todd Middleton?  \n|  |   \nWeller: |  | No.  \n|  |   \nProsecution: |  | Mr. Weller, did you ever rape any of the charges under your or another guardian's care?  \n|  |   \nWeller: |  | No.  \n|  |   \nProsecution: |  | Mr. Weller, do wards often request sexual intercourse from their guardians?  \n|  |   \nWeller: |  | Yes.  \n|  |   \nProsecution: |  | Mr. Weller, is such sexual activity frowned upon?  \n|  |   \nWeller: |  | Only if the ward is under the age of consent, that being sixteen.  \n|  |   \nProsecution: |  | Thank you, Mr. Weller. That will be all.\n\n* * * * *\n\n#  _Salve!_\n\n**A Turn for the Worse  \nHNN\u2014Melissa Eagleton Reporting**\n\nToday's testimony by Mr. Gideon Weller has stunned many of Hadrian's citizens. Although Mr. Weller cannot be accused of rape, what he admitted as being a common practice at the Northeast Camp is most shocking. Mr. Weller admitted under oath having had sexual intercourse with Todd Middleton. And, although Todd Middleton, at age seventeen, was of the age of consent, the age difference between the Warden and the young man is far too extreme for comfort. That the act was not rape was confirmed by one of _Salve!_ 's investigating reporters, Michael Swahazey. Michael interviewed Mr. Weller's personal assistant who assured us that intercourse occurred between the two men at young Middleton's request. Mr. Weller referred to the act as \"medicinal intercourse\" and acknowledged it to be a common act between ward and guardian.\n\nHNN has responded to this method of rehabilitating our youth with outrage. We do not believe that any educational institution should have its adult mentors engaging in any form of sexual contact with our youth. The guardian of a ward at a reeducation camp is akin to a teacher in a school. This is a relationship based on trust and, though the focus of these educators is to help our children find their latent homosexuality, it should not be their place to participate in any sexual activity with youth sent to reeducation.\n\nInvestigating reporter Michael Swahazey has delved deep into this story and contacted all other reeducation camps throughout Hadrian. Only at the Northeast Reeducation Camp is \"medicinal intercourse\" used. When a ward tries to seduce a guardian, according to Adrian Adams, camp Warden for the Southwest Reeducation Camp, the guardian is instructed to refuse but speak to the ward positively about such a request. The fact that a ward expresses sexual interest in someone of the same sex is a sign that he (or she as in the case of our female youth) is coming to accept Hadrian's chosen lifestyle as his (or her) own. \"Never,\" Adrian Adams states emphatically, \"should a guardian cross the line and participate with his ward in sexual activity.\"\n\nWe, at HNN, agreed with Adrian Adams. I, for one, am not so anxious now to throw all of my support behind Mr. Gideon Weller. He may not have raped Todd Middleton, but he definitely crossed the line when he engaged in sexual activity with the boy. What is your opinion Hadrian? Voc us your views @HNN#GW-RE\/MI.\n\n_Vale!_\n\n# Transcripts: Hadrian vs. Hunter\n\n**Prosecution Questioning of Dean Stuttgart**\n\nProsecution:  |  | Mr. Hunter, when did you first discover the deceased was heterosexual?  \n---|---|---  \n|  |   \nDean: |  | The day Frank voc'd that Todd had been exposed at school.  \n|  |   \nProsecution: |  | Surely that wasn't the first time. You must have suspected. When did you first suspect the deceased might be heterosexual?  \n|  |   \nDean: |  | When the boys were thirteen.  \n|  |   \nProsecution: |  | Thirteen. Four years ago.  \n|  |   \nDean: |  | Three and a half.  \n|  |   \nProsecution: |  | Three and a half then. So, you knew the boy was straight for three and a half years?  \n|  |   \nDean: |  | No.  \n|  |   \nDefense: |  | Objection, M'Lady. The witness never claimed to have known about the deceased's sexual preference, only to have suspected.  \n|  |   \nProsecution: |  | Sorry, suspected. So, then, you suspected straight tendencies for three and a half years, correct?  \n|  |   \nDean: |  | Yes.  \n|  |   \nProsecution: |  | In all that time, Mr. Hunter, did you ever sit the boy down to talk to him about the dangers of _strai_ behavior?  \n|  |   \nDefense: |  | M'Lady, that's a sexual slur.  \n|  |   \nProsecution:  |  | My apologies. Straight behavior\u2014or would the defense prefer I use the word heterosexual\u2014did you ever sit the boy down and discuss the dangers of heterosexual behavior?  \n|  |   \nDean: |  | No.  \n|  |   \nProsecution: |  | No? That's odd. Tell us, Mr. Hunter, you were reeducated, were you not?  \n|  |   \nDefense: |  | Objection.  \n|  |   \nProsecution: |  | It is important, M'Lady. Since the defense is determined to throw blame on external forces, I am uncovering other external forces from the boy's family and those closely associated with him.  \n|  |   \nJudge: |  | Proceed.  \n|  |   \nDefense: |  | M'Lady\u2014  \n|  |   \nJudge: |  | As I allowed you levity, I will extend the same to the prosecution. Proceed with your questioning.  \n|  |   \nProsecution: |  | You were reeducated, were you not?  \n|  |   \nDean: |  | Yes.  \n|  |   \nProsecution: |  | And in the three whole years you suspected the boy to be straight, you never once spoke to him about being straight or cautioned him about the consequences?  \n|  |   \nDean: |  | Of course I did.  \n|  |   \nProsecution: |  | How? Did you ever tell him about your being straight?  \n|  |   \nDefense: |  | Objection, M'Lady. My client was reeducated.  \n|  |   \nProsecution: |  | My apologies. Having once been straight, did you ever speak to him about that, sir? Did you ever ask him if he thought about straight acts? Kissing girls? Touching breasts? That sort of thing? Did you ever once think to ask him what he liked most about his own sex? Did you ever once try to guide him toward a gay lifestyle? Did you ever tell him about reeducation camp?  \n|  |   \nDefense: |  | Objection! M'Lady, the prosecution is bombarding the witness, refusing him opportunity to answer.  \n|  |   \nJudge: |  | Objection noted. Prosecution will restrain himself to one question at a time.  \n|  |   \nProsecution: |  | Of course, M'Lady. Mr. Hunter, why didn't you share your experiences with the boy?  \n|  |   \nDean: |  | I...don't talk about it.  \n|  |   \nProsecution: |  | Why not?  \n|  |   \nDefense: |  | M'Lady!  \n|  |   \nJudge: |  | Answer the question, Mr. Hunter.  \n|  |   \nDean: |  | It...  \n|  |   \nProsecution: |  | It was what, Mr. Hunter?  \n|  |   \nDefense: |  | M'Lady, the prosecution is badgering the witness.  \n|  |   \nProsecution: |  | M'Lady, please, all I did was request the witness to answer the question.  \n|  |   \nJudge: |  | Answer the question, Mr. Hunter.  \n|  |   \nGeoffrey: |  | Dean, it's all right.  \n|  |   \nJudge: |  | Sit down, Mr. Geoffrey Hunter. Answer the question, Mr. Hunter. If you do not, I will have to charge you for withholding testimony.  \n|  |   \nDean: |  | It was hell! Demeaning! Horrifying! Brutal. I was beaten and tortured.  \n|  |   \nProsecution: |  | That smacks of exaggeration, Mr. Hunter.  \n|  |   \nDean: |  | You weren't there! They used a paddle on me damn near every day, electric shock therapy, and they overtly encouraged sexual activity between wards and guardians.  \n|  |   \nProsecution: |  | The two methods of discipline and deterrents are legal and an accepted means of maintaining order in reeducation camps, particularly with violent and recalcitrant youth. As for wards choosing to have sex with their guardians, well, there is no law against having sex. No more questions.  \n|  |   \nDean: |  | You assume wards have a choice.  \n|  |   \nProsecution: |  | You may return to your seat, Mr. Hunter.  \n|  |   \nDean: |  | They raped us!  \n|  |   \nProsecution: |  | Were you ever raped, Mr. Hunter?  \n|  |   \nDean: |  | Weller tri\u2014  \n|  |   \nProsecution: |  | Yes or no, Mr. Hunter.  \n|  |   \nDean: |  | Weller had me down. He\u2014  \n|  |   \nJudge: |  | Yes or no, Mr. Hunter.  \n|  |   \nDean: |  | But he\u2014  \n|  |   \nJudge: |  | Yes or no.  \n|  |   \nDean: |  | No.  \n|  |   \nProsecution: |  | Thank you. No more questions.  \n|  |   \nJudge: |  | Defense, do you wish to cross-examine?  \n|  |   \nDefense: |  | Yes. Mr. Hunter, take a moment. I know this is hard for you. Did Gideon Weller ever try to rape you?  \n|  |   \nProsecution: |  | Objection!  \n|  |   \nDefense: |  | M'Lady, if Mr. Weller had attempted to rape another ward, he may well have raped Todd Middleton, too.  \n|  |   \nJudge: |  | Objection overruled. Proceed.  \n|  |   \nDefense: |  | Mr. Hunter, did Gideon Weller ever try to rape you?  \n|  |   \nDean: |  | Y-yes.  \n|  |   \nDefense: |  | Please tell us what happened.  \n|  |   \nProsecution: |  | Objection!  \n|  |   \nDefense: |  | M'Lady, I am trying to establish what the conditions at the Northeast Camp are really like in order to prove that Todd's Middleton's state of mind at the time he requested assisted suicide from the accused was in fact beyond repair.  \n|  |   \nJudge: |  | Objection overruled! Defense may continue with her line of questioning.  \n|  |   \nDefense: |  | Go ahead, Dean; take your time.\n\n* * * * *\n\n#  _Salve!_\n\n**A Stunning Revelation  \nHNN\u2014Melissa Eagleton Reporting**\n\nYesterday's testimony by Dean Hunter was most revealing. One is left to seriously ponder the need for reformation in Hadrian's reeducation system. It appears that Jason Warith's cry for change needs to be met. In fact, Vice-President Elena Stiles has promised, if elected as Hadrian's next president, to establish a committee dedicated to running a thorough investigation into our reeducation camps. It was hinted that she might even choose Mr. Jason Warith as an advisor to the committee.\n\nAs equally stunning as Dean Hunter's horror stories about life during reeducation is the text testimony presented by defense lawyer Faial Raboud. Apparently, Mr. Dean Hunter had a text conversation with the young woman Todd Middleton was to have allegedly raped. The young woman, whose name has not been revealed, had refused to accuse the young man of rape. The text message Mr. Hunter allegedly had with this young woman suggests that the sexual act committed by Todd Middleton was not rape, but rather an act instigated by the young woman. This, along with the damning accusation that Mr. Weller may well have raped Todd Middleton lends credence to Faial Raboud's argument that Todd Middleton was suffering from extreme depression. It will be interesting to see the outcome of this case. Will Frank Hunter be found guilty of murder or will the court recognize Todd Middleton's death as an act of kindness? Hadrian, remember to voc us your views. Please note; we have amalgamated all our viewer sites for this case into one: @HNN#RE-GW\/MI-FH\/AS.\n\n_Vale!_\n\n# Transcripts: Hadrian vs. Hunter\n\n**Defense Questioning of Ms. Destiny Stuttgart**\n\nDefense: |  | Ms. Stuttgart, please remind the court of the values you and the four other original founding family members established for our country.  \n---|---|---  \n|  |   \nMs. Stuttgart: |  | In order to do that, I must be allowed to start from the beginning.  \n|  |   \nProsecution:  |  | No objections, M'Lady.  \n|  |   \nMs. Stuttgart: |  | When world population topped ten billion, religious extremists with their apoplectic theories and revelations began to war against one another\u2014and us. By us, I mean homosexuals.  \n|  | Regardless of the education and incredible strides made to ensure acceptance in inclusion during the twenty-first century, the abuses against homosexuals grew and persecution became extreme.  \n|  | Anti-gay mobs rose everywhere and even straight men and women were persecuted if they appeared gay or vocally supported us. Sensing the future direction of humanity, finding the need for strength in numbers, a collective was formed.  \n|  | We were, in essence, a secret society. We called ourselves Hadrian's warriors. So, Hadrian was formed as a means of self-preservation and protection. I find it ironic that the very reason we sit in this courtroom today is due to the same type of fear mongering that brought us together as a community.  \n|  |   \nProsecution: |  | Objection, M'Lady. With due respect to our founding mother, we are here today to try Frank Hunter with murder.  \n|  |   \nJudge: |  | Objection stands. Clerk, please strike our founding mother's last statement.  \n|  |   \nDefense: |  | Please continue, Mother.  \n|  |   \nMs. Stuttgart: |  | As I was saying, one reason for gathering our forces together was preservation\u2014however, the word preservation means many things to us. Our efforts were not merely for the self-preservation of the homosexual community but for the preservation of all of humanity and the preservation of a life-sustaining planet.  \n|  | Our planet, as we know it, or rather, as we knew it to be, must be preserved and repaired if humanity is to survive. This preservation became the crux of our constitution.  \n|  |   \nProsecution: |  | M'Lady, again, with due respect to our founding mother, I fail to see the purpose behind this line of questioning.  \n|  |   \nMs. Stuttgart: |  | That is because you have become as blind as the rest of our society.  \n|  |   \nJudge: |  | Mother, please.  \n|  |   \nMs. Stuttgart: |  | Don't silence me like a wayward child. If any privilege comes of being a founding mother, it is the privilege to speak openly before all as to why our country was founded! Silence, all of you! Fear, hate, distress makes up the basis of humanity\u2014  \n|  |   \nProsecution: |  | Yes, outside our walls but\u2014  \n|  |   \nMs. Stuttgart: |  | No buts! It is the makeup of all humanity! Not just them but us, too! We are here today because of our fear, our hate, and our distress! They hate us. Now we hate them! They kill us. Now we kill them! They are jealous, fearful, dying. We lock them out. Why? Because we have to! But how many of them are exactly like us?  \n|  | We like to quote the Kinsey scale zero to six\u2014and maybe our geneticists have eliminated the zero and the one, and maybe they haven't\u2014and maybe they shouldn't be allowed to. What right do we have to say heterosexual sex is wrong? No more than they have the right to say homosexual sex is wrong. When we founded Hadrian, we were not looking for justifications or the creation of the perfect homosexual human race. We founded our nation on choice.  \n|  | The choice Hadrian citizens made was to reduce the human population now\u2014not tomorrow\u2014not in twenty years\u2014not in the next two hundred years\u2014but now as the critical need is now\u2014is still now! _Stability Now!_ became our cry and to do that, we knew we needed a homosexual nation and IVF laws. We asked everyone who joined us, gay or straight, to pledge themselves, to choose with us what we honestly believe to be humanity's last chance at survival.  \n|  | It is why we sacrifice access to higher levels of technology working only with wind, hydro, and thermal power, keeping a store of energy for those days when access to natural energy is lost to us. We were willing to endure those days when we can't drive or run hot water or watch the wall screen.  \n|  | We made these sacrifices because we knew we have more than a claim to this planet; we are responsible to it. But times have changed. We no longer sacrifice for humanity, survival, and our planet. Instead, we demand luxury and\u2014  \n|  |   \nProsecution: |  | M'Lady, please. This is getting us nowhere.  \n|  |   \nMs. Stuttgart: |  | You sit down, be quiet, and listen. I am here today because my great-grandson is being charged with murder! That's right! Frank Hunter is a Stuttgart by marriage. The poor boy doesn't even know it. I have never even met him! Today is the first time I have ever seen him in my entire life. How wrong is that? How, you ask? Frank's Papa Dean is my grandson. And today is the first day I have seen Dean in over twenty years. I was told he was dead! I only learned otherwise when he voc'd and asked me to help his young friend. Forgive me...  \n|  |   \nJudge: |  | It's all right, Mother. Take your time.  \n|  |   \nMs. Stuttgart: |  | It's hard. I've been denied access to my own grandson and two great-grandsons because of the prejudices of a society I helped to create. I am tired of hiding that reality.  \n|  | There are heterosexuals among us and I don't care. These men and women I hold in the highest esteem as they make the greatest sacrifice for humanity. They choose celibacy or, as in the case of my grandson, some even choose to embrace our lifestyle.  \n|  | This morning, before I agreed to take the stand, I had a chance to meet with my boy because I didn't want to say anything here today that would hurt him any more than he has already suffered. When I asked him if his marriage was a sham, he assured me the answer was no.  \n|  | He loves his partner, Geoffrey Hunter, and I dearly want to get to know the man who won my boy's heart. I want to learn from my grandson and his lover how they managed to find balance in a world where among humanity, balance is damn near unattainable.  \n|  | Yes, balance! Humanity's greatest flaw is in our lack of balance. We swing from one extreme to another, and today, I say Hadrian has taken the pendulum of man's folly from one form of abuse into another.  \n|  | When swung hard enough, the blade meets itself right back up at the top! My great-grandson killed Todd Middleton, but for reasons many of us cannot fathom. He killed his lover out of love.  \n|  | Out of an awareness that our society would never accept him for who he was. Todd Middleton suffered grievously at the hands of Gideon Weller. The man is a rapist\u2014  \n|  |   \nProsecution: |  | Objection!  \n|  |   \nMs. Stuttgart: |  | Objection be damned! And don't you try to shush me, M'Lady, or strike any of my comments! I am a founding Mother and I say these proceedings go against the very fabric of the values of the country I helped create! You listen to me, and you listen well.  \n|  | I say Gideon Weller is a rapist\u2014not a confessed rapist, one yet to be proven in this court of law\u2014but a rapist nonetheless. I saw the medical report describing Todd Middleton's condition.  \n|  | That alone should be enough to condemn the man! If anyone should be exiled or given henbane to swallow, it should be Gideon Weller and not my great-grandson! That's all I have to say. I'm tired. Let me down.  \n|  |   \nDefense: |  | Of course, Mother. Thank you.  \n|  |   \nJudge: |  | Mother Stuttgart, please, the prosecution has the right to cross-examine you.  \n|  |   \nProsecution:  |  | No questions for now, M'Lady. Mother Stuttgart looks quite drawn; perhaps she will allow for cross-examination tomorrow\u2014if I still feel the need to question her.  \n|  |   \nJudge: |  | Good. Then, Mother, you may step down from the bench.\n\n* * * * *\n\n# _Salve!_\n\n**HNN\u2014Viewer Wave Link  \n@HNN#RE-GW\/MI-FH\/AS**\n\nApril Bolger  \nMedicinal Sex? What in Hadrian's Name is that all about? How can he get away with that?\n\nAngel Higgins  \nWho cares about medicinal sex? The rat was a _strai_. Hadrian is better off now that he's dead!\n\nAaron Whyle  \n_Strais_ make me sick! I say kill 'em and be done with it. Why should our good money go to reeducating heterosexual pigs? Kick 'em all out.\n\nT'Neal Cantos  \nI know Frank Hunter. He is a good man. He would never murder anyone.\n\nAngel Higgins  \nFrank Hunter's my hero! He killed a _strai_!\n\nDuncan Fraser  \nIf our founding mother is not opposed to heterosexuals, then why are we?\n\nAngel Higgins  \n@ Duncan Fraser\u2014 _Strai_ lover! Probably a _strai_ yourself.\n\nAaron Whyle  \n@ Duncan Fraser\u2014Ditto!\n\nAndreas Contreras  \nAll _strais_ should be castrated!\n\nAngel Higgins  \n@ Andreas Contreas\u2014AGREED!\n\nToni Freeman  \nI honestly can't imagine how hard it must be having to tame _strai_ males. We all know how violent they are. Of course they have to use corporal punishment. No doubt it's in self-defense! And, I mean, medicinal sex, well maybe _strai_ guys need it.\n\nSebastian Remilard  \nDidn't anyone listen to our founding mother's testimony today?\n\nDevon Rankin  \nWhen I first learned Todd Middleton was a _strai_ , I thought castration was in order, but after hearing our founding mother, I don't think that way anymore. I used to date Todd before anyone knew he was that way and he was an amazing guy. The best damn b-ball player I ever had the honor to play with. He spent a whole summer teaching me the game. He may have been straight, but I no longer think that means he was a bad guy. I feel for Frank. I really think he believes he did the right thing by Todd and after reading all these posts\u2014who knows\u2014he might well have. Either way, I hope they don't exile him.\n\nTrina Scaponi  \nWell said, Devon! I admit, I'm an anti- _strai_. I hate heterosexuals! But after listening to our founding mother today, I'm not so sure anymore. And medicinal sex just doesn't seem right to me.\n\nShakira Osagi  \nHadrian vs. Hunter is an interesting case, and maybe it is okay to be straight just as long as they never act on it.\n\nBronek Sobanski  \nSay no to _Strais_!\n\nAdolfo Gaafar  \nExile is too good for _strais_. Don't give 'em a choice. Just give 'em the death penalty!\n\nFernando Pereira  \nIt's not illegal to be straight. It's only illegal to have heterosexual sex. I agree with Shakira Osagi, just as long as they don't act on it.\n\nBromek Sobranski  \n_Strai_ men are sluts! They have to have sex. No woman is safe around a _strai_ man.\n\nSissy Hildebrand  \nYou people are sick! You have no idea what you are talking about.\n\nAaron Whyle  \n@ Bromek Sobranski\u2014The slogan is, \"Go away, Mr. Strai!\"\n\nDuncan Fraser  \nPeople, please, these are our mothers and fathers, our sisters and brothers you're talking about. Listen to Mother Stuttgart and learn a little tolerance and acceptance. Love, not hate, should be the order of the day.\n\n* * * * *\n\n# Transcripts: Hadrian vs. Hunter\n\n**Defense Questioning of Darrell Jeffreys, Warden's Personal Assistant**\n\nDefense: |  | M'Lady, the Defense would like to call Darrell Jeffreys, Mr. Weller's personal assistant, to the stand.  \n---|---|---  \n|  |   \nClerk: |  | Mr. Darrell Jeffreys to the stand.  \n|  |   \nDefense: |  | Mr. Jeffreys, how long have you been Mr. Weller's personal assistant?  \n|  |   \nJeffreys: |  | Over twenty years.  \n|  |   \nDefense: |  | And during that time, how many times were you on hand when Mr. Weller was helping young men discover their latent homosexual?  \n|  |   \nJeffreys: |  | I don't understand the question, madam.  \n|  |   \nDefense: |  | How many times were you in the same room as Mr. Weller while he was having sex with a young ward under his charge?  \n|  |   \nJeffreys: |  | I don't remember, madam.  \n|  |   \nDefense: |  | Really? From the way Mr. Weller was describing his prowess, one would assume witnessing such intensity would be hard to forget.  \n|  |   \nJeffreys: |  | I really don't remember.  \n|  |   \nDefense: |  | Let me help jog your memory. Was it once, twice, three times, five maybe?  \n|  |   \nJeffreys: |  | I don't remember.  \n|  |   \nProsecution: |  | Objection! The defense is badgering the witness.  \n|  |   \nDefense:  |  | M'Lady, determining how many times Mr. Jeffreys was privy to Mr. Weller's special moments with his wards is crucial to my client's defense.  \n|  |   \nJudge: |  | Proceed, but tread carefully.  \n|  |   \nDefense: |  | Yes, M'Lady. Mr. Jeffreys, according to the records of the North-East Reeducation Camp, you were in attendance five times while Mr. Weller was administering medicinal intercourse, not including Todd Middleton. Can you confirm these numbers?  \n|  |   \nJeffreys: |  | That sounds about right.  \n|  |   \nDefense: |  | Can you tell me how these young men are doing today?  \n|  |   \nJeffreys: |  | I do not keep track of the whereabouts of wards once they leave the compound. My job is simply to assist Mr. Weller whenever required.  \n|  |   \nDefense: |  | So you really don't know where any of these men are today?  \n|  |   \nJeffreys: |  | No, ma'am, I do not.  \n|  |   \nDefense: |  | Would it shock you to learn that, like Todd Middleton, all five of these men are dead? That all five of these men committed suicide shortly after Mr. Weller administered the medicinal intercourse you were so fortunate to have been in the presence of?  \n|  |   \nJeffreys: |  | I\u2014ah\u2014had no idea.  \n|  |   \nDefense: |  | Was Mr. Weller actually administering medicinal intercourse, or was he, in fact, raping these men?  \n|  |   \nProsecution:  |  | Objection!  \n|  |   \nJudge: |  | Objection denied. Answer the question, Mr. Jeffreys.  \n|  |   \nJeffreys: |  | I\u2014I honestly thought it was for medicinal purposes\u2014that he was helping them. I swear on Hadrian's lover I had no idea any of them died.  \n|  |   \nDefense: |  | Did any of these men, did Todd Middleton, agree to medicinal intercourse with Mr. Weller?  \n|  |   \nJeffreys: |  | No. No, ma'am, they didn't.  \n|  |   \nDefense: |  | Did Jason Warith accidently lock himself out of the room?  \n|  |   \nJeffreys: |  | No, ma'am, he did not.  \n|  |   \nDefense: |  | How was it he was unable to reenter?  \n|  |   \nJeffreys: |  | I was instructed to remove him from the room and then lock the door behind him so Mr. Weller could administer\u2014while he\u2014  \n|  |   \nDefense: |  | While he raped Todd Middleton.  \n|  |   \nProsecution:  |  | Objection. The Defense is putting words in the witness's mouth.  \n|  |   \nJudge: |  | Objection noted.  \n|  |   \nDefense: |  | Mr. Jeffreys, in your opinion, did Mr. Weller rape Todd Middleton?  \n|  |   \nJeffreys: |  | Yes, ma'am, he did.  \n|  |   \nDefense: |  | No more questions.  \n|  |   \nJudge: |  | Prosecution. Do you wish to cross-examine?  \n|  |   \nProsecution: |  | No, M'Lady.  \n|  |   \nJudge: |  | This session is adjourned. We will reconvene tomorrow morning at 8:00 a.m. when I will execute judgment.\n\n* * * * *\n\n# Transcripts: Hadrian vs. Hunterr\n\n**An Unprecedented Sentence**\n\nClerk: |  | Frank Hunter, please rise for your verdict.  \n---|---|---  \n|  |   \nJudge: |  | I must say this has been a most disturbing and unique case. Todd Middleton's death was murder, and not, as the defense lawyer eloquently suggests, an act of assisted suicide.  \n|  | Although Hadrian law allows for euthanasia in cases of the terminally ill, these laws do not extend to the emotionally distressed. One can overcome a mental illness brought on by circumstance. Todd Middleton's depression was not the result of an incurable chemical disorder of the brain. Thus, it is the decision of this court that, you, Frank Hunter, are guilty of murder.  \n|  | Under normal circumstances, the penalty for said crime is death and I would hang you. However, there are extenuating circumstances that require leniency. Nor will you be exiled to the outside world.  \n|  | I am fully aware that I will be setting precedent here today, but as your lawyer uncovered for us, Todd Middleton's situation, his sexual preference compounded by his having been raped, did make life for the young man unbearable in Hadrian. I believe that you, and Todd Middleton, saw his death, your aiding him in his suicide, as an act of mercy.  \n|  | I recognize the need for reform within our reeducation system as well as a complete rethinking of our nation's attitudes toward our heterosexual citizens. As a result, I have chosen to sentence you to a life of service within Hadrian's military.  \n|  | Your rank upon entrance, until the day you die, will be that of ensign. There will be no opportunity for promotion. No opportunity for parole. You will spend the rest of your days serving your country by protecting Hadrian from the outside world. This court is now adjourned.\n\n* * * * *\n\n#  _Salve!_\n\n**An End to an Era  \nHNN\u2014Melissa Eagleton Reporting**\n\nAs all my viewers are aware, _Salve!_ is the voice of Hadrian\u2014that of the government and of the people. When the tune of those voices changes, so must its mouthpiece. I am not retiring as the host of _Salve!_ but I have been forced to face some very ugly truths about much of what I have been saying to you over the years. I still believe, as I am sure do many of you, that Hadrian is the future hope for humanity, but that future cannot be forged by the blood of our youth. I do not refer to those who die defending our wall. The threat of the outside world remains as prevalent as ever. I refer, rather, to those who have suffered and died at the hands of Gideon Weller\u2014six young men over the past twenty years. Are those numbers too small for Hadrian to take notice? No. Every single boy's death is a slap in the face of our good country Hadrian. The charges laid against Gideon Weller based on the evidence that surfaced in the wake of Frank Hunter's murder trial and the reeducation camp's ex-warden's subsequent sentence of exile has shattered citizens' faith in Hadrian's reeducation system. As Judge Julia Reznikoff said in her final address at Weller's trial: \"Gideon Weller is a man who became judge, jury, and executioner of these young men. He determined them to be pure heterosexuals with no hope of reformation and acted believing their end justified. Unfortunately,\" Judge Reznikoff added, \"as we condemn this man, we must also condemn ourselves, for his warped attitude was born of our country's prevailing prejudice.\" As Mother Stuttgart succinctly reminded us when she spoke at her great-grandson's trial: \"Humanity's greatest flaw is in our lack of balance.\" We swing from one extreme to another, and today, I say Hadrian has taken the pendulum of man's folly from one form of abuse into another. Gideon Weller acted based on our country's lack of balance\u2014our country's prevailing prejudice against heterosexuals. The six youth Gideon Weller raped all committed suicide. That these young men all chose death as their only escape proves beyond a shadow of a doubt that we have allowed our hopes for the future to act as a steamroller over the fringes of our youth. These six young men were our children. Our fear of the outside world, our warped perception of heterosexuals\u2014especially viewing heterosexual men as aggressive, violent, and abusive\u2014has twisted the way we view some of our youth.\n\nTodd Middleton proved how we do not truly understand our heterosexual youth. His potential, his gifts, much like those of his father, whom we learned during Frank Hunter's trial was also a _strai_ \u2014I'm sorry, heterosexual, are lost to our society. As we know, Will Middleton is revered in Hadrian for his genetically altering the soya bean, an ingenious genetic alteration which helped Hadrian enter into the self-sustaining age we now appreciate. An act brought into existence by a straight man\u2014a het'ro. Had we known of his sexual preference when he was still in high school, our country would have been denied all of his talents. Just as we are now denied all the potential talents his son, Todd Middleton, had. Todd Middleton had hoped to follow in his father's footsteps. His dream was to alter the DNA of rice so we could grow that life-saving crop in our northern climate. Because rice is the last remaining crop Hadrian imports, Todd's dream would have offered us complete anonymity and self-reliance. We will never know if he, like his father, could have been successful.\n\nWhy is it we only ever stop to think about our actions when the one who dies, or the person who speaks out, is deemed a member of our elite? Stop and think, we have, finally. Recently elected President Stiles vows to look into the state of reeducation camps and has appointed Jason Warith to work in conjunction with the Minister of Education. As you know, Jason Warith was the man who helped expose Gideon Weller's abusive methods of reeducation. Mr. Warith states that his first act as assistant to the Minister of Education is to ban the use of corporal punishment. Only one paddle will remain\u2014Gideon Weller's\u2014a paddle stained with the blood of his victims. This instrument of torture used against our children for far too many years will now hang in the Government Hall as a bloody reminder of his ruthless rule forged by our fear of the outside world and unreasonable prejudice against heterosexuals.\n\nAnd although positive changes are being made, needless to say, many are out there who actually approve of Gideon Weller's methods. Hadrian's  transition back to a state of tolerance and understanding will not be easy, but suffice to say, the spark that lights the way for that transition is burning brightly, and women and men like Faial Raboud and Jason Warith are more than willing to carry that torch in the hopes of creating a better future for our children.\n\n_Vale!_\n\n# Love Recovers\n\nFour years have passed since Frank's trial. Dramatic changes have occurred in the Hunter family as a result. Dean moved out, taking up residence in his grandmother Destiny Stuttgart's abode. She purchased a home in the New Augustus City, rebuilt closer to the Canadian border and growing after that dirty nuclear assault. When he left, Dean had insisted on a separation. Time was needed, he said, for him to come to grips with what had happened: Todd's death, Frank's confession, the legalization of heterosexuality, and now, the lengthy court battles he and countless other victims of the Northeast Reeducation Camp have filed. Most important for Dean is the reemergence of Destiny Stuttgart, his Mimi, in his life. As both an elder and family member, Mimi does not cause Dean to suffer the negative effects of the shock treatment that successfully conditioned his body against feminine influence. Mimi, knowing this brutal effect on her grandson, determined the last important goal of her life would be to help Dean overcome this psychological prison.\n\nThe separation is hardest on Geoffrey and Roger. The pain of losing his lover spirals Geoffrey into a state of depression that twists him into a severe workaholic. He is now indispensable to Hadrian National Fisheries (HNF) and is the top CEO of the entire corporation. Roger is torn between two fathers. As much as he longs for the loving arms of his papa, Roger is too afraid to leave Geoffrey alone even for a day. Thus, his only visits with Dean are through visual voc. Whenever Dean suggests he come to live with him, or even just enjoy a short visit, Roger refuses. Roger knows he is his father's only link to life outside the work world. Roger is the only reason Geoffrey leaves the office to come home for dinner. More times than not, though, Geoffrey's mind stays at the office, and then after eating, he either disappears into his study or returns to work. Roger is more often alone than not. And he misses Frank dearly.\n\nFrank feels none of this. His sentence to life in the military has him living at the southwest wall. He has cut off all communication with family and friends, greeting them cordially when they insist on making a visit, which he is allowed once a month with a maximum of three people. Only two ever show up: Geoffrey and Roger. This monthly vigil is a daytrip on which Geoffrey brings his work. Roger brings homework and reading material since his father is uncommunicative. Roger makes this visit regardless of Frank's now formal aloofness. He refuses to believe his brother no longer feels any love. He knows that deep inside, locked away by fear and self-loathing, is the fun-loving tenderhearted brother he grew up with. And even if the old Frank never again emerges, Roger continues his monthly visits and determines to do so until the day he dies.\n\nWith the legalization of heterosexuality, and the uni of New Augustus suffering economically due to fear resulting from the name association (Augustus now being synonymous with nuclear strike and cancer), Dean gained reluctant acceptance into the campus. Of course, Mimi's influence was critical. Very few people are willing to refuse a founding mother. Thus, Dean is now working toward building a life in the field of medicine. He has chosen nursing for his vocation because he will be able to graduate and begin work in the field before the age of fifty. Mimi and Geoffrey share the cost of his tuition and living expenses. At first, Dean felt it wrong to have Geoffrey help pay for his education since he was the one who filed for separation and had left, but both grandmother and partner had insisted Geoffrey be allowed to help. Dean, having reclaimed his last name at Mimi's insistence, registered as Dean Stuttgart. Once precedence was set, Dean advocated for other heterosexuals to be allowed uni entrance, and coupled with Mimi's powerful influence and the university's crippled financial status, the New Augustus Campus became a beacon of hope for heterosexuals throughout Hadrian. By the end of his second year of schooling, Dean had established the first straight\/gay alliance on campus, the first of its kind in the history of Hadrian. Dean, of course, is its president and his vice-president is Cantara Raboud, Faial Raboud's daughter. Thanks to her mother's efforts, Cantara is now free to come out and express her sexuality. Dean and she are not lovers. Dean is restrained in three ways: 1) age appropriate heterosexual attraction\u2014Cantara is only twenty-three to Dean's forty-four years, 2) electric shock conditioning\u2014although he has managed to overcome most of it, Dean still has a long way to go before he can ever hope to hold a woman in his arms, and 3) Geoffrey\u2014Dean loves Geoffrey and every time he thinks of trying to be with a woman, the thought of hurting his former lover acts as more of an abatement than his previous conditioning. During their separation, Dean often found himself not only thinking about his partner, but longing for him. In many ways, Dean has come to conclude, one's sexual preference has as much to do with love as it has with physical yearnings\u2014perhaps even more so. Having confessed this revelation to Mimi, Dean is continuously encouraged by her to invite Geoffrey down for a visit. Now that the school year is nearing its end, Dean agrees and vocs his old lover.\n\nGeoffrey is overwhelmed with conflicting emotions: relief, love, and desire, muddy when mixed with anger, resentment, and pain. It takes time to win Geoffrey's trust back, but soon the two men are voc'ing on a daily basis. Within six months, Geoffrey makes his first visit to New Augustus.\n\nMimi's home consists of two levels, a ground floor and a level below ground. The car park is also below ground. One drives down a small hill inside the compound. Mimi, being a founding mother, is wealthy and her home consists of four bedrooms and a family room on the ground level, with the living room, kitchen, and a fifth (guest) bedroom below ground. When Mimi opens the door to her underground vehicle park, she greets Geoffrey, inviting him in. Turning to face the stairs, she calls up to Dean, who is working in one of the rooms he has converted into his study. \"Your room,\" she instructs Geoffrey as they await Dean's arrival, \"is the first door on the right.\" She points to the room just past the stairs. To their left is the living room, and down the hall, past the guest room on the right, is the kitchen.\n\nWithin seconds, Dean has leaped down the stairs like a young mountain goat. As soon as his eyes meet Geoffrey's, he lunges forward, pulling him into his arms. Mimi discreetly backs away into the living room to give the two men their privacy. While kissing and awkwardly banging against walls, Dean and Geoffrey make their way into the guest bedroom where they remain for three days with Dean periodically emerging to prepare a meal in the kitchen.\n\nBy the end of Dean's final year, Geoffrey and he are planning their re-registration ceremony. The decision came quite suddenly and at Dean's suggestion. After a bout of unabashed lovemaking, while Geoffrey is lying with his head cradled into Dean's shoulder, he shudders a soft sigh. \"I thought I had lost you forever.\"\n\nDean closes his eyes. \"Do you remember the first time we made love?\"\n\n\"It was at The Cattle Ranch.\" The memory provokes a smile.\n\n\"Did you know that I had the chance to be with a woman while we were there?\"\n\nSitting up in surprise, Geoffrey battles with his emotions. He is appalled that the ranch was a front for heterosexual behavior while at the same time trying to remind himself he has no right to judge. \"Did you?\" Fear imbues his being as he worries that Dean may well have been with another partner.\n\n\"No.\" Pausing briefly, knowing his next words will be painful, Dean confesses, \"I wanted to, but I couldn't. The conditioning was too strong.\" With a slight harrumph, he admits, \"I still have trouble being in the same room with a woman for too long.\" Smiling, he concludes, \"Mimi's helping me with that, though.\"\n\n\"So,\" Geoffrey gulps back the truth, \"if you could have, you would have then?\"\n\n\"Yes.\" Dean is not purposely trying to hurt Geoffrey. He is simply trying to share with him the complete truth. \"Back then, but not today.\" He looks his lover in the eye and emphasizes, \"The only person I ever want to be with now is you.\" Geoffrey begins to sob and Dean envelops him in his arms. \"I am so sorry I hurt you. Please understand. I needed this time to decide who I wanted to be.\" Wiping the tears from Geoffrey's eyes, he adds, \"This time, Geoffrey, I can honestly say I choose to be with you. No one else is deciding for me. Not fear, not Weller, not society...\" With a little smirk, he concludes, \"not even you.\" Now, with the eyes and smile of honesty, he says, \"I love you, Geoffrey Hunter, and I want to spend the rest of my life with you.\"\n\nAfter the two men decide to reestablish their bond, Geoffrey steps down as the conglomerate head, resuming his former position as President of Hunter Fisheries. Roger is thrilled, not just to have his family back, but to have his father restored to his former, more jovial self. Geoffrey asks Roger to stand up for him at the ceremony and Dean also makes a request. He needs Geoffrey and Roger to speak to Frank on his behalf. It is time to heal the rift between papa and son. It is time to make their family whole again.\n\n* * * * *\n\n# Hadrian's Wall\n\nAt the wall, Frank Hunter stands at the ready, watchful of any movement on the outside. Before him lies a wasteland scarred by bombings, blackened shards of rock, shrapnel, and the decimated skeletal remains of enemy vehicles. Beyond Hadrian's firing range are the scattered and stunted beginnings of black pine, the heartiest of all northern trees, no doubt attempting to resurrect the Lazarus of old Canada's once abundant boreal forest. Frank watches the horizon of the wastelands for any sign of heterosexual barbarians. He doesn't call them that. He doesn't even think of them as the enemy. But it is the military term for the enemy; anyone outside the wall, in fact, is a heterosexual barbarian; it is the term he uses when addressing a senior officer about a kill, and he always writes, \"Shot and killed one (or more) heterosexual barbarian\" in every report. Frank has a sharp eye and exceptional aim. He kills, on average, five heterosexuals a month. Frank doesn't think of this act as shooting the enemy. He doesn't see it as necessary for his nation's security. There is not a single patriotic bone left in Frank Hunter's body. He doesn't even consider himself a citizen of Hadrian anymore. Frank Hunter is a mercy killer. Every kill is, for Frank, and very likely for his every victim, an act of euthanasia. After every kill, he whispers, \"I did it for you, Todd.\"\n\nFrank's post is atop the central watchtower. He takes no days off unless sick or ordered to. He breaks for meals and the odd trip to the bathroom. He won't watch news streams or vids, he plays no wave games, and takes no lover. Most of all, he tries not to think. He mans his station, kills every human straight that wanders into his site, and suffers restless, nightmarish sleep. Frank cannot stop his mind from remembering when he succumbs to weariness.\n\nWhen forced to rest by a senior officer, usually one new to his platoon, Frank runs. He runs the three-mile length of the wall his tactile tattoo restraint allows him. He then runs back. Back and forth and up and down. When Frank runs the three mile stretch, he runs up and down every set of stairs leading up the three stories of the wall as well as up and down the fourth set of stairs, leading up to each of the watchtowers within his limits. For Frank, running is the most effective method of emptying his mind. It does not please the MPs assigned to guard him day and night. Whenever Private Frank Hunter runs, the MP is obliged to run alongside him. More often than not, though, the MP trails behind, unable to keep pace with the lithe, powerful man. One MP actually shot Frank in the leg when he was unable to keep up. He claimed he had ordered Frank to halt, and very likely did, but the gap formed between them made it impossible for Frank to hear him. Nor could he have simply voc'd Frank. Being that military service is a life sentence for Frank, he is not allowed any luxuries such as the voc or access to Hadrian's wave network. Nor does he wish for any. Being granted access to the historic library on his one enforced day off is all he cares for. When he is not running, Frank will retreat to the historic library and read one of the archaic books. The feel of the book in his hand, its weight combined with the scent of the paper as he flips the pages, is oddly relaxing.\n\nContact with family is also limited to one downloaded letter a week, which one of the senior officers projects from his voc onto the wall screen for Frank to read. Once a month, Frank is entitled to a vid conference or personal visit with family. Frank is forever being ordered to attend these and must be escorted to the meeting room by his MP. Always, his father and Roger come. Dean, initially by his own design and also at Frank's request, has never attended or participated in any family conference or visit. On this day, however, unknown to Frank, Dean Stuttgart arrives. He does not come into the room immediately. Geoffrey and Roger intend to prepare Frank. They begin with the usual trite conversation about how healthy Frank looks\u2014so tanned, so fit\u2014and then move on to how little activity there has been at the wall of late while Frank tallies off his kills for the month. Neither Geoffrey nor Roger care to hear about that aspect of Frank's life, but Frank always lists them off anyway. It is grizzly how detailed Frank is when he describes each man or woman's death. It is as if he has engrained the image of each heterosexual he kills into his mind's eye. Finally, nearing the end of their visit, Frank asks, as always, \"And how is your schooling, Roger? Are you excelling?\"\n\n\"Uni is great!\" Roger ejaculates.\n\n\"Do you know he plans to attend for at least six years?\" Geoffrey is proud of Roger's decision to study agricultural engineering and his determination to obtain, at the very least, a Master's degree.\n\n\"Eight,\" Roger adds excitedly. \"I'm definitely going for my Ph.D. Professor Joel Lipman, the same man who taught Will Middleton and Quintin Laugharne, says I show real promise for an undergrad.\"\n\n\"Well, get your Bachelor's degree first,\" Geoffrey adds judiciously. \"Then you can work on your Master's. You can talk Ph.D. six years from now!\"\n\nOn this particular visit, Roger made sure to take a few days off from uni to be with Frank, his father, and Papa Dean for this most important reunion.\n\n\"Good work, Roger.\" Frank remains civil, containing all show of affection, sitting upright and at attention, as if he were addressing senior officers.\n\n\"Roger's average,\" a proud father interjects, \"is eighty-eight percent!\"\n\nAlthough there is no sign of emotion, Frank replies, \"I am pleased, Roger. Todd would be proud.\" The only sign of emotion that threatens to crack the veneer of Frank's self-imposed emotional exile is the closing of his eyes and an intake of breath held momentarily before slowly exhaling.\n\n\"Thanks, Frank. That means a lot to me,\" says Roger. Reaching into his pocket, he retrieves a small burlap bag. \"I brought you a present.\" Handing the package to Frank, he says, \"Smell it.\"\n\nWorking to keep his voice flat, Frank replies, \"It smells really good. What is it?\"\n\n\"Ground cocoa.\"\n\nFrank's eyes moisten. Once more, he must breathe slowly to contain the threat of an outpouring of emotion. \"Todd had promised to grow these for Papa Dean.\"\n\n\"I know. Professor Lipman lets me grow these in one of the Antinous Uni hot houses. He was reluctant at first, hot houses being critical for growing plants for life saving medicines, and the beans having to come from Hadrian's Global Seed Vault, but when I explained about how Todd...well...how he...well, how I want to...you know, genetically alter them...like he wanted to...anyway, I get to grow a little crop. I roasted and ground these figuring they'd make a great gift for...\" Looking now to his father, Roger asks, \"May I tell him?\" Geoffrey nods; he does not contain his emotions; tears begin to stream, creating an odd image on a face suddenly shining with delight. \"And, I figured they'd make a great wedding present for Papa Dean.\"\n\nFrank starts. \"Pa\u2014Pa\u2014Papa Dean is\u2014but, Dad\u2014he\u2014you\u2014never divorced\u2014did you?\"\n\n\"No son,\" Geoffrey smiles through the tears. \"We are getting remarried. Papa Dean wants you\u2014he wants you to stand up for him. He came with us today and is waiting outside until you ask him to enter. May I please go get him, Frank? He misses you dearly. He loves you and wants you back in his life.\" Frank is speechless. Afraid to talk, afraid to move from fear of breaking down, he just sits in his chair, waiting. Geoffrey stands, \"I'll\u2014I'll go get him.\" As soon as he opens the door, he calls out and Dean steps into the room. Franks stands, the two men facing one another.\n\nDean opens his arms, \"Frank\u2014\"\n\nFrank crumples to his knees, wailing in a sudden burst of emotion expelled from him like an explosion. Sobs and tears rack his body like a tempest-tossed bark. Papa Dean bends down and cradles him. \"I'm so sorry, son. I'm so sorry I abandoned you for all these years.\"\n\nThrough his sobs, Frank manages a plea. \"Forgive me.\"\n\n* * * * *\n\n# Bibliography\n\nAlthough fictional, a substantial amount of research went into the production of _Hadrian's Lover_. Resources used while writing this novel include:\n\nAbout.com Geography. Current World Population. <http:\/\/geography.about.com\/od\/obtainpopulationdata\/a\/worldpopulation.htm>\n\nAbout.com Geography. How fast is the world's population growing? <http:\/\/geography.about.com\/library\/faq\/blqzworldgrowth.htm>\n\nAbout.com Geography. Most Populous Countries Today. The 24 Most Populous Countries on Earth Today. By Matt Rosenburg. About.com Guide. <http:\/\/geography.about.com\/cs\/worldpopulation\/a\/mostpopulous.htm>\n\nAbout.com Geography. Population Statistics. <http:\/\/geography.about.com\/od\/obtainpopulationdata\/Population_Statistics.htm>\n\nAbout.com Geography. Trash Islands. Trash Islands of the Pacific and Atlantic Oceans. From Amanda Briney, Contributing Writer. Updated March 24, 2009. <http:\/\/geography.about.com\/od\/globalproblemsandissues\/a\/trashislands.htm>\n\nAbout.com Geography. Seven Billion People. Will seven billion people be overpopulation? <http:\/\/geography.about.com\/od\/obtainpopulationdata\/a\/seven-7-billion-people.htm>\n\nAmerican Institute of Biological Sciences. Articles. Forecasting the Effects of Global Warming on Biodiversity. Daniel B. Bodkin. Henrik Saxe. Miguel B. Araujo. Richard Betts. Richard H. W. Bradshaw. Thomas Cedhagen. Peter Chesson. Terry P. Dawson. Julie R. Etterson. Daniel P. Faith. Simon Ferrier. Antoine Guisan. Anja Skjoldborg Hansen. David W. Hilbert. Craig Loehle. Chris Margules. Mark New. Matthew J. Sobel. David R. B. Stockwell. <http:\/\/ibcperu.org\/doc\/isis\/7086.pdf>\n\nAnnenberg Learner. The Habitable Planet: A Systems Approach to Environmental Science. <http:\/\/www.learner.org\/resources\/series209.html>\n\nArctic Institute of North America. Arctic, Vol. 54, No. 2 (June 2001) P. 142-148. Climate Change Scenarios for Hudson Bay Canada, from General Circulation Models. William A. Gough and Edmund Wolfe. (Received May 16, 2000: accepted in revised form September 19, 2000). <http:\/\/arctic.synergiesprairies.ca\/arctic\/index.php\/arctic\/article\/view\/773\/799>\n\nArctic. Vol. 57, No. 3 (September 2004) P. 299-305. Trends in Seasonal sea Ice Duration in Southwestern Hudson Bay. William A. Gough, Adam R. Cornwell and Leonard J. S. Tsuji. (Received 22 August 2003; accepted in revised form April 1, 2004). <http:\/\/pubs.aina.ucalgary.ca\/arctic\/Arctic57-3-299.pdf>\n\nblog.modernature.ca. MODERN LIFE vs. HUMAN NATURE. Garbage Island: Floating Plastic Ocean. Posted on June 6, 2008. <http:\/\/blog.modernature.ca\/?p=90>\n\nBlythe, David, Director. \"Circadian Rhthyms\". 1976. <http:\/\/www.youtube.com\/watch?v=3Xyy9OgCOZE>. Published on Oct. 9, 2012. Note: the You Tube account associated with this video has been closed.\n\nBMJ Helping doctors make better decisions. Falling grain stocks and rising population spell disaster and demand debate. A. J. McMichael. <http:\/\/www.bmj.com\/content\/311\/7021\/1651>\n\nCalendar 2210. <http:\/\/www.hf.rim.or.jp\/~kaji\/cal\/cal.cgi?2210>\n\nDaily News. Scientists uncover possible source of homosexuality. By Christine Roberts. New York Daily News. Tuesday, December 11, 2012. 6:11 PM. www.nydailynews.com\/news\/national\/scientists-uncover-source-homosexuality-article-1.1218017\n\nGenesis of Eden. The World Population \"Bomb.\" <http:\/\/dhushara.com\/book\/diversit\/bomb.htm>\n\nGovernment. no. Ministry of Agriculture and Food. 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November 9, 2004 at 12:00 AM. Usha Lee McFarling \\ Los Angeles Times. <http:\/\/dailyuw.com\/archive\/2004\/11\/09\/imported\/report-points-radical-arctic-warming#.UPxwlY7C5Wg>\n\nThe Daily. Studies show warming in Antarctic regions. September 28, 2006 at 12:00 AM. Staci Miner. <http:\/\/dailyuw.com\/archive\/2006\/09\/28\/imported\/studies-show-warming-antarctic-regions#.UPxwHI7C5Wg>\n\nThe Sustainability Report. Canada's Population. <http:\/\/www.sustreport.org\/signals\/canpop_ttl.html>\n\ntreehugger. World Population to Hit 7 Billion by 2011, New Stats Show. Mat McDermott Business \/ Corporate Responsibility August 14, 2009. <http:\/\/www.treehugger.com\/corporate-responsibility\/world-populationto-hit-7-billion-by-2011-new-stats-show.html>\n\nTruthdig drilling beneath the headlines. We Are Breeding Ourselves to Extinction. By Chris Hedges. Posted March 8, 2009. <http:\/\/www.truthdig.com\/report\/item\/20090309_we_are_breeding_ourselves_to_extinction>\/\/\n\nUNRV. History. Hadrian. <http:\/\/www.unrv.com\/five-good-emperors\/hadrian.php>\n\nUniversity of Toronto Libraries. Canadian Poetry Online. Earle Birney: Biography. Source: Wailan Low. <http:\/\/www.library.utoronto.ca\/canpoetry\/birney>\/\n\nUniversity of Toronto Libraries: RPO (Representative Poetry Online). David. Birney, Earle (1904-1995). <http:\/\/rpo.library.utoronto.ca\/poems\/david>\n\nU.S. Department of Commerce. United States Census Bureau. World POPClock Projection. <http:\/\/www.census.gov\/population\/popclockworld.html>\n\nWebMD. Health & Sex. Is There a Gay Gene? New Genetic Regions Associated With Male Sexual Orientation Found. WebMD Health News. January 28, 2005. <http:\/\/www.webmd.com\/sex-relationships\/news\/20050128\/is-there-gay-gene>\n\nWikipedia: The Free Encyclopedia. Human population control. <http:\/\/en.wikipedia.org\/wiki\/Human_population_control>\n\nWikipedia: The Free Encyclopedia. Hudson Bay. <http:\/\/en.wikipedia.org\/wiki\/Hudson_Bay#Climate>\n\nWWF Global. Marine problems: Climate change. <http:\/\/wwf.panda.org\/about_our_earth\/blue_planet\/problems\/climate_change>\/\n\nWWF Global. Marine problems: Pollution. <http:\/\/wwf.panda.org\/about_our_earth\/blue_planet\/problems\/pollution>\/\n\nYahoo! News. Business. The world of 2030: U.S. declines; food, water may be scarce. By Olivier Knox, Yahoo! News. The Ticket \u2013 Mon. December 10, 2012. <http:\/\/ca.news.yahoo.com\/blogs\/ticket\/world-2030-u-declines-food-water-may-scarce-162757458--politics.html>\n\nYahoo! News. Canada. Next Great Depression? MIT study predicting 'global economic collapse' by 2030 still on track. By Eric Pfeiffer, Yahoo! News. The Sideshow \u2013 Wed. April 4, 2012. <http:\/\/ca.news.yahoo.com\/blogs\/sideshow\/next-great-depression-mit-researchers-predict-global-economic-190352944.html>\n\nYouTube CA. 7 Billion, National Geographic Magazine. 3,915 videos. <http:\/\/www.youtube.com\/watch?v=sc4HxPxNrZ0>\n\n2208 Hanke-Henry Permanent Calendar. <http:\/\/henry.pha.jhu.edu\/calendardir\/ccctdir\/ccctHTML\/2208.0.html>\n\nDisclaimer: The author of this text cannot be held responsible for any links that have been disengaged since the publication of this novel.\n\n# Copyright\n\n_Published by Clink Street Publishing 2015_\n\n_Copyright \u00a9 2015_\n\n_Second edition._\n\n_The author asserts the moral right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work.  \nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means without the prior consent of the author, nor be otherwise circulated in any form of binding or cover other than that with which it is published and without a similar condition being imposed on the subsequent purchaser._\n\n_ISBN: PB: 978\u20131\u2013910782\u201398\u20139  \nEB: 978\u20131\u2013910782\u201399\u20136_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## **TABLE OF CONTENTS**\n\n**Preface**\n\n**Introduction**\n\n**CHPT. 1 | Prohibition**\n\n**CHPT. 2 | Compounding Spirits**\n\n**CHPT. 3 | Gin, Whiskey, and Rum**\n\n**CHPT. 4 | Brandy, Wine, and Cider**\n\n**CHPT. 5 | Absinthe, Cordials, and Bitters**\n\n**CHPT. 6 | Weights and Measures**\n\n**Resources**\n\n**Glossary**\n\n**Bibliography**\n\n**Endnotes**\n\n**Index**\n\n**Acknowledgments**\n\nPlease bookmark your page before following any links\n\n## Preface\n\n\"Here,\" Fritz said. He slid a little blue book toward me across a long folding table. \"This is more your area than mine.\"\n\nBefore he retired and decamped for a beach life in Thailand, Frederick C. \"Fritz\" Blank was owner and chef of the Philadelphia restaurant Deux Chemin\u00e9es. Now closed, \"the Doo,\" as he sometimes called it, sat on the corner of Locust and Camac in two Frank Furness\u2013designed homes. For decades, it was one of the city's benchmarks of genteel dining. Some patrons addressed him as Chef Blanc. Seemed about right, Frenchifying his name like that. After all, the names of dishes on the menu were heavily Gallic\u2014 _palourdes r\u00f4ties aux ar\u00f4mates, quenelles de brochet \u00e0 la Lyonnaise, ballotine de canard, sorbet au pamplemousse, p\u00e2t\u00e9 aux coings_ , and more.\n\nBut all that was in the dining room. Downstairs at the restaurant, past a long hall hung with copper pots and pans, it was another matter. In the basement kitchen, the French veneer peeled away and he was Chef Blank, not Blanc. His family was not Parisian or Lyonnais; they hailed from Pennsauken, New Jersey\u2014\"East Philadelphia,\" he liked to joke\u2014and, before that, from W\u00fcrttemberg in southern Germany. Staff meals downstairs were far more likely to be something from Germany, Austria, Hungary, or other parts of Mitteleuropa than from the French menu upstairs. There might be his grandmother's _Kartoffelsalat_ (a bacon-laced potato salad), for instance, slices of the simmered bread dumpling known as _Serviettenkn\u00f6del_ , or a Czech sauerkraut soup garnished with smoked sausage and sour cream.\n\nOur mutual obsession with books about food and drinks brought us together, although until I met Fritz around 1997 I had been self-conscious about the sheer number of books at home. As a museum curator\u2014I'd earned degrees in anthropology as well as museum administration\u2014I was acquisitive by inclination and training. What started as a handful of cookbooks to get me through cooking rudimentary meals in my first apartment during undergraduate years had metastasized into a library that spread over three floors of my home. Not just cookbooks, but all sorts of related volumes: barware catalogs, corporate histories of drinks firms, pamphlets, culinary postcards, technical manuals for making ice cream and bacon, culinary histories, and more. By the time I moved to south Philadelphia, the collection was some two thousand volumes\u2014and growing.\n\nBlank, though, owned five times that number. In comparison, his collection made mine seem, well, _pedestrian_. Completely reasonable, in fact. We both claimed that these were working libraries we used every day. He had an entire wall of boxes filled with menus from restaurants around the world, and stacks upon stacks of magazines about food and beverages stretching back to the 19th century. _He_ was a madman. _Me?_ I was merely touched. We became good friends and, over the course of the next several years, I spent thousands of hours in his library.\n\nAlthough I'd handled literally tens of thousands of cookbooks, bartenders' guides, manuals on livestock, butchery, and related material over my career as a collector, the format, color, and size of this little blue book was unfamiliar. Something unusual, then, something rare. I glanced at Fritz sitting nearby, but he looked away, pulling a folder from one of his research piles. I turned the book over in my hands. Gold letters glittered as light hit its spine. I frowned at the name they spelled and turned back to face him. \"Who is Viereck?\"\n\n\"I have no idea.\" He had already pulled his glasses down from their perch on his boxy little chef's hat and was making a show of busying himself with the folder's contents. \"Look inside.\"\n\nThe name bothered me. George Sylvester Viereck. An author, obviously, but... something else. A friend of H. P. Lovecraft, maybe? One of his lesser-known colleagues? I did have a sizable collection of Lovecraft material at home. There was something to that, but when I cracked opened the book and began running my eyes and fingertips over the pages, any thoughts of weird fiction writers drained away. The wear on its cover, the brittle and slightly tanned paper, those vanilla smells of slow decomposition, and the archaic handwriting all suggested the thing was older than either of us at the table. The book's age, however, might have been one of the few true things about it. The entire thing, from cover to cover, was a deception.\n\nDespite the spine's promise of Viereck's writing, there were no printed pages at all. Instead: hundreds of handwritten notes about, and formulas for, booze. Recipes for gin and other juniper spirits cropped up again and again; so did a dozen or so for absinthe. Cordials. Whiskeys, both real and artificial. Brandies. Not brandies for connoisseurs, but there they were along with notes on how alcohol reacts under certain conditions and how Cognac might be colored with oak extract and adjusted with syrups to get ready for market. Loose slips of paper tucked in its pages tied it to Prohibition-era New York; here was a prescription form from a Manhattan hospital, there a business card from Harlem. Flipping between the pages, I realized most of the recipes were in English, but dozens were in German, a language I hadn't spoken since I was an adolescent. There was _K\u00fcmmel, Doppel K\u00fcmmel_ , and _Eisk\u00fcmmel_. Latin, too, crept in as later entries veered into pharmaceutical preparations. Lotion for head lice. Salves for chilblains and cures for freckles. I looked up at Fritz. \"Freckles need cures?\"\n\nHe smiled and waved away my question with an avuncular shoo. \"It's yours. Have it.\"\n\nAnd so that little blue book came to me, its temporary custodian. One day it will end up with someone else, maybe in a library or museum. But first, I'm sharing it with you. I hope you enjoy it as much as I have.\n\n## Introduction\n\nSince I was a toddler, I have plished and plashed in pools and puddles of illicit liquor. Oh, it's not my family's fault particularly. All things considered, I imagine my mother would prefer not to have introduced me to distilling before kindergarten. We're not a moonshining or bootlegging family, mind you, but it was she who brought me to my first moonshine still in 1971 when she toted me along on a hunt for antiques in the wooded and remote New Jersey Pine Barrens. These trips\u2014to estate sales, yard sales, antique shops, wherever old and neglected treasures might have lurked\u2014were the bane of my childhood existence and left me numb with boredom. On this one, she was admiring the enormous copper \"water heater\" sequestered in the basement of an abandoned and dilapidated old house when a stranger with a shotgun appeared at the top of the steps. The rough-looking fellow didn't point the gun at us, but did suggest that we move along. Mom's bold, not foolhardy; we moseyed. That night, she described the big copper rig to my father. \"It's a wonder,\" he said, shaking his head, \"you two are still alive.\" Of course, that was no water heater, but quite literally an underground distillery.\n\nOnce old enough to read, I plunked down in front of my parents' fireplace with the Foxfire books about Appalachian folkways, poring over descriptions of backwoods stills and wildcat whiskey. While my peers wanted to grow up to play football or fight fires, I yearned to explore hidden hollers and secret caves where scofflaw distillers tried to evade meddlesome revenuers. My version of cops and robbers, I suppose.\n\nEventually I did get to explore those secret and hidden places: the woods, warehouses, basements, and garages where unlicensed distillers made their homemade hooch. In fact, I have spent more than half my life researching clandestine booze and those who make it. There have been death threats and more shotguns, sure, but those are rare. Overwhelmingly, the men and women who have shared with me their liquor, their time, and their trust since my college days have treated me as a peer and, here and there, a friend. A few years back, I turned some of their stories and recipes into a book called _Moonshine!_ People seem to like it.\n\nThat moonshine reputation is how this secret formulary ended up in my hands. When Chef Blank gave me the notebook, he knew that a book about suspect alcohol would give me more joy than it would him. He was right. I suspect that the old chef, who died in 2014, also knew that the old museum curator wouldn't keep it to himself, but share it. Once again, he was right.\n\n### What Is This?\n\nPhysically, the book Blank had given me is a duodecimo (5\" x 7.75\") with a dark blue hard cover. It is old but not ancient. In faded gold typeface, the spine reads _The Works of George Sylvester Viereck_ : under that _The Candle and the Flame_. Inside, though, it's something else. Once I began thumbing through it, I realized that the notebook was a compounder's formulary, a collection of recipes for making alcoholic beverages, disguised as a rather dreary book of poetry from 1912. Although it's rare to find handwritten formularies, they do surface now and then. The more I pored over it, though, the more I wondered who would have made this one\u2014and why.\n\nBefore someone filled them with handwriting, the book's pages had been blank. It was possibly a salesman's dummy or a printer's mock-up. Some pages have been cut out and a few remain blank, but the others are jammed with recipes (in English, German, and occasional Latin) for spirits, cordials, oils, bitters, tinctures, and notes on alcohol production and analysis. Several pages detail treatments for freckles, coughs, and chilblains; abortifacients; fly paper; \"lotion for children's heads\"; label glue; tattoo removal; \"non poisonous straw hat cleaner\"; the sort of preparations one would expect from a Prohibition-era pharmacy. In fact, several bits of ephemera tucked in the pages date the book to 1920s and early '30s New York.\n\nWas it someone's attempt to cram for Prohibition? A bootlegger's recipe book? A pharmacist's collection of profitable recipes? After all, the formulas are disorganized, but they aren't random; of the tens of thousands of such booze recipes floating around a century ago, the compiler selected many with supposed health-giving properties. And what was the connection to Viereck? After my years of moonshine fieldwork, many methods and ingredients were familiar. As I flipped through, some recipes were as well. Stomach bitters \"equal to Hostetter's for one quarter cost,\" for instance, used specific phrasing I'd seen elsewhere. Despite penciled adjustments here and there, further research revealed that almost none of these alcohol formulas were original, but were instead culled from English and German language druggists' handbooks, journal articles, farming books, distillers' treatises, and cordial-makers' manuals dating back to the middle of the 19th century\u2014earlier than Prohibition, earlier even than Bram Stoker's _Dracula_.\n\n### How It's Organized\n\nThe original notebook is neatly penned but disorganized, the formulas, notes, and recipes haphazard. Some recipes show up repeatedly. Others are broken into parts scattered over multiple pages. Here and there, clusters of similar wording and format suggest a common source. It includes several iterations of beading oil, also known as The Doctor, an ancient cheat of unscrupulous 18th-century rum merchants and 1920s moonshiners alike. Rather than present annotations of every single page\u2014believe me, as an historian, the temptation is strong\u2014I've rearranged the entries and grouped them in broad categories. Whether they are real or faked, most of the whiskeys, for instance, are lumped together here rather than scattered higgledy-piggledy as in the original. Same with the wines, cordials, rums, and so forth. Repeats are mostly omitted. Mostly.\n\nEach section about spirits or cordials in this book has descriptions of relevant ingredients or techniques. It also includes recipes. Some are transcribed directly from the notebook, some scaled down or adapted for modern ingredients. Others still are riffs on the handwritten recipes or use their ingredients in contemporary mixed drinks and cordials.\n\nBut before we dive into recipes, we'll take a look at Prohibition in America, see where formularies such as this fit into the world of drinks, and review some of the gear and techniques for making at least some of the recipes from that old handwritten notebook.\n\n### Doctor Lyon Has a Secret\n\nDespite worn corners and minor tears, the old blue notebook has stood up well since Prohibition. Over time, acids have turned its pages tan and brittle. A tear mars the spine, but there are no spills or stains inside, no smudging, and the ink remains dark. Loose sheets from prescription pads, newspaper clippings, pages from other books, an urgent note signed \"Lucy,\" and various notes give the book a New York City provenance. The Lucy note and a prescription slip from J. F. Lyon helped zero in on the compiler's identity, but a New York Public Library call slip with distinctive handwriting yielded a name: Victor A. Lyon.\n\nVictor Alfred Lyon was born in Germany in 1876 to Louise and Jonathan Fish Lyon. His mother was German, but J. F. hailed from an old New England family and earned his medical degree from the University of Heidelberg. In time, Victor and his younger brother became physicians as well. ZaZa (as young Berlin-born Walter was called by his classmates) earned his MD from Yale while Victor got his from Bellevue Hospital Medical School (later absorbed into NYU's School of Medicine) in 1898. Although ZaZa left medicine, by 1908 the elder Lyon brother was practicing as a surgeon at St. Elizabeth's and Philanthropin Hospitals in New York City.\n\nAt the onset of national Prohibition, Victor and his sister Lucy lived with their parents in Harlem at 231 East 116th Street, an area once heavily populated by German immigrants. Their father kept office hours in the building and saw patients at nearby hospitals. Victor did as well, but Victor had a secret tucked away in a nondescript little book of poetry.\n\nNow, Victor Lyon was a master at the low profile. No photo of him\u2014not even from a passport application\u2014has surfaced. After his mother died in 1921 and his father four years later, federal and state census workers noted him living at the Harlem address for decades with his unmarried sister (a \"spinster,\" the records call her). He pops up in some medical directories, alumni lists, and served on an exemption board for conscripts in the First World War. In that war, he had registered as a physician working for General Electric. Other than that, the man's a ghost. His registration form for the war notes that he is tall with grey eyes and grey hair. He claims, writing in that familiar script we see in the notebook, to be native-born American, but that is a lie; rampant anti-German hysteria in the United States led many so-called hyphenated Americans to hide or outright reject their Germanic heritage (see \"Hoch der Kaiser! A German America,\" page XX).\n\nBy the Eisenhower years, Lyon had stopped practicing but lived at the same old address. In 1961, he's listed (albeit with \"address unknown\") in the _American Medical Directory_. Two years later, the directory drops him entirely. He is presumably dead, though missing from death records of the time. He would have been 87 years old by then, unmarried, childless, a man who left nothing for posterity.\n\nWell. Maybe one thing. There was, after all, that little blue book bearing the name George Sylvester Viereck.\n\n### George Sylvester Viereck A Venom-Bloated Toad of Treason\n\nBefore his name was stricken from the rolls of esteemed authors, before the _New York Tribune_ suggested that he be kicked up the stairs and down again, and before Colonel Henry Watterson of the _Louisville Courier-Journal_ summed a popular sentiment of the day by calling the man a \"venom-bloated toad of treason,\" Munich-born New Yorker George Sylvester Viereck (1884\u20131962) was a darling of the American literary scene. He had been, it was said, America's Oscar Wilde, a wunderkind fluent in English and German who, as a youngster, wrote for _Die New Yorker Staats-Zeitung_ , Baltimore's _Der deutsche Correspondent_ , and his father's American paper _Der deutsche Vork\u00e4mpfer_ (The German Pioneer) which later became the bilingual _Rundschau Zweier Welten_ (Review of Two Worlds). He was an American correspondent for the popular _Berliner Tageblatt_ and served on the editorial boards of _Current Literature_ and the _International_. Viereck's books received acclaim in both Germany and the United States, where the young man was compared to poets such as Shelley, Keats, Swinburne, Baudelaire, and Heine. He was precocious, prolific, and had no small opinion of himself. Viereck's burgeoning arrogance is hinted at in a 1907 _New York Times_ interview in which the 23-year-old poet declaimed, \"I sometimes feel... as if I were a sort of Colossus of Rhodes, with one foot in Europe and with one foot here.\"\n\nHis literary stature, however, was not the article's main thrust. Rather, Viereck used the interview to confirm a decades-old rumor that tied him to the imperial throne of Germany by what a later author delicately termed \"a pedigree of human frailty.\" It was widely assumed, in fact, that Viereck's father was the illegitimate son of the bewhiskered old emperor himself, Wilhelm I. Which is to say that this would-be Colossus was a direct descendant of Frederick the Great and first cousin to Wilhelm II, the bellicose kaiser who was soon to wage war against Britain and her allies in the Great War. Although he made certain to confirm his Hohenzollern family connections in the _New York Times_ article, they were entirely beside the point. Young Viereck was a poet! A genius! \"I, George Sylvester Viereck,\" he insisted, \"would rather have written 'Nineveh, and Other Poems' than be the German Emperor.\" It probably didn't strike readers as true then, either.\n\nThe family connections aren't mere court gossip. Viereck's ultimate disgrace was grounded in the kinship he felt with his cousin the kaiser and with his deep identification with the Fatherland. Within a dozen years of that interview, the ground had shifted. War had come and gone. Sylvester, as he was known to friends and family, was on the outs. His strident and bombastic pro-German writings leading up to, and during, World War I were almost entirely to blame. That he was alleged to be involved in more sinister plots of espionage and sabotage against the United States did the rest.\n\nBy 1919, he was drummed out of the Poetry Society of America, a group he had help found, and expelled from the Author's League of America. Even the New York Athletic Club gave him the boot. By the time Victor Lyon got his hands on the blank book that would become his formulary, the name George Sylvester Viereck was anathema. Few, even in German American families, would have thought to pluck _The Candle and the Flame_ from the shelf, much less poke through it.\n\nIt was one of the best hiding places Lyon could have devised.\n\n### Hoch der Kaiser! A German America\n\nA decade before Prohibition, some 2.3 million German-born immigrants lived in the United States, including most of the Lyon family. Yet by the time the Volstead Act (the National Prohibition Act) took hold, many strove to downplay, hide, or outright repudiate their \"hyphenated\" status as German Americans.\n\nThose who did so didn't just stop speaking German in brewers' meetings or in churches; they shed their German identity, even down to their names. Herr M\u00fcller became Mr. Miller; Frau Schmidt transformed into Mrs. Smith. Spurred on by the sinking of the British ship _Lusitania_ and America's subsequent entry into World War I, anti-German hysteria rose to fevered pitch. The people of Cincinnati changed the names of German, Berlin, and Hanover Streets to English, Woodward, and Yukon Streets. Chicago's German Hospital morphed into Grant Hospital as \"alien\" Germans were fired from their jobs across the nation. In St. Louis, Kaiser Street became Gresham Avenue and the Berlin Hotel polished its image with a new name: the Bonair. Sauerkraut was out; \"victory cabbage\" was in.\n\nThe University of Michigan all but shut down its German program when it dismissed six of its professors for their perceived lack of support for the Allies. In 1917, the Nebraska State Council of Defense\u2014a virulent investigator of patriotism, loyalty, and \"anti-American\" activity\u2014voted unanimously to have all German books removed from the public libraries. For the council, it seemed, merely reading German was sufficient grounds to question someone's loyalty.\n\nGermany's sinking of the _Lusitania_ shocked Americans, helping to galvanize them against \"the Hun\" both abroad and at home. That _Fatherland_ editor George Sylvester Viereck warned Americans to stay off British ships beforehand only confirmed his perfidy in the eyes of Americans.\n\nIn the twisted logic of the time, speaking German was prima facie evidence of sedition. Prohibitionists leveraged that general animosity to target German American brewers and beer not merely as unpatriotic but downright anti-American. Cartoons of stout American flag\u2013waving Germans drinking from steins and exclaiming \" _Hoch der Kaiser_!\" (\"Up with the kaiser!\") were among the propaganda images meant to highlight the divided loyalties of beer-swilling \"hyphenated Americans.\" During World War I, who stood with Kaiser Bill and the Hun to menace America? Why, John Barleycorn, of course, who diverted grains from the war effort and weakened America's troops fighting in Europe. Prohibitionists had already gotten absinthe banned before the war; peeling off beer by manipulating Americans' anti-German sentiment was just the next step in their campaign to eradicate alcohol entirely.\n\n## **_Chapter one_**\n\n## Prohibition\n\n**_New laws forbade publishing recipes_**\n\nIf Doctor Lyon had published the formulas and recipes he recorded during America's Prohibition years rather than keep them hidden in a secret notebook, he likely would have been tossed in jail. They were not strictly secret, but were not what we'd call common knowledge. Prohibitionists aimed to keep it that way. Pharmacists, physicians, and some journalists knew about blending and compounding colors, aromas, and flavors with spirits and syrups, as did blenders, liquor wholesalers, distillers, saloon owners, and others who handled liquor in their profession.\n\nStarting in 1920, though, new laws forbade publishing recipes, formulas, and directions for making alcoholic beverages. American publishers released a smattering of cocktail guides during Prohibition\u2014after all, cocktails per se were not illegal. Instructions for _producing_ alcohol, however, were. Publishers could be, and were, arrested and fined if they provided formulas for making liquor. Just days into the \"noble experiment\" in 1920, for instance, revenue agents nabbed John Mitchell, editor of the _Richmond Planet_ , for publishing a liquor recipe collection. American libraries pulled books that dealt with manufacturing cordials, wines, liquor, and other intoxicating beverages. To their credit, many librarians did not destroy the books, but shifted them to reference shelves. At the New York Public Library, librarian Edwin P. Anderson announced, \"We would no more think of forbidding readers to consult such books in our reference department than we would books on flying. After the prohibition amendment goes into effect there will be additional reasons for them, as they will be histories.\"\n\nWhat refreshing irony that Prohibition itself is history. Mostly.\n\n**_Cocktails per se were not illegal_**\n\n### **A History of Doctored Drinks**\n\nSome argue that scabrous, degraded spirits of Prohibition, such as bathtub gin, fake brandy, and synthetic whiskey, stood in sad contrast to the glories of pre-Prohibition bartending. Hogwash. Doctored drinks go back millennia. Counterfeit whiskey didn't just come into existence ex nihilo in 1920; synthetic, fraudulent, and adulterated\u2014even merely spiced and seasoned\u2014wines, spirits, and cordials were well established by the time Prohibition descended over the United States. Pre-Prohibition distillers knew what they were making; wholesalers who bought from them generally did as well. Beyond that, though, things got iffy. Joseph Fleischman set the scene in his 1885 manual on blending liquor:\n\nThe moment a barrel of liquor leaves the bonded warehouse, the first thing thought of, and done, is to reduce its cost. The blender knows how to make the bonded liquor produce a profit of 25 to 50 per cent on the amount he paid for it, and frequently a great deal more.\n\nFor the changes made by these cheapening processes neither the saloon-keeper nor his bartender is responsible; they can only offer for sale what they are able to purchase. The purchaser accepts the liquors he buys for what they are represented to be. [ _The Art of Blending and Compounding Liquors and Wines_. New York: Dick and Fitzgerald, (1885)]\n\nOf course, Fleischman was trying to appeal to blenders, distillers, bartenders, and others who would buy his manual, so his claim that everybody makes highly profitable blends should be taken with a grain of salt. But the thing is, books like his were immensely popular among those in the liquor business and did describe widespread practice. Bootleggers treasured them.\n\n### **What Was Prohibition?**\n\nWhen we talk about Prohibition, most people think of the years in the early 20th century when alcohol was forbidden across the United States. For almost 14 years, the nation tried a new approach to drinking. In the eyes of those who strove earnestly to eliminate saloon culture and Demon Rum, these were real, palpable evils that destroyed lives, wrecked families, and threatened the ruination of civil society. Images spring to mind of a nation in which alcohol was made suddenly illegal across the board and everyone who drank it was a law-breaker.\n\nClose, as carnival barkers of the era cried, but no cigar. The actual situation was more nuanced. This _noble experiment_ , as some later called it, would forbid nearly every iteration of alcohol its forgers could manage to squeeze into law. Those laws, as so many before and since, were built on concessions, compromises, and loopholes.\n\nWhat we call Prohibition was an accretion of shifting laws and regulations\u2014bookended by two constitutional amendments\u2014that controlled production of and access to industrial, medicinal, and beverage alcohol in the United States from 1920 until 1933. Propelled by a mix of idealism, na\u00efvet\u00e9, browbeating, and regional successes of \"local options\" to go dry, prohibitionists had been pushing language that would become the 18th Amendment through Congress since 1917. After a year of political wrangling that saw Democrats, Republicans, and a few independents pitted against one another, dry forces had enough votes to present the proposed amendment to the states. Mississippi ratified the document first, then Virginia, Kentucky, and North Dakota. By the end of 1918, 15 states had ratified the document. Of the 48 then in the Union, 36 states needed to accept its terms for the amendment to pass. On January 16, 1919, Nebraska became that 36th. After that, the remaining states, except for Connecticut and Rhode Island, ratified the document. Bully for them.\n\nThe National Prohibition Act, a federal law that spelled out enforcement details of the 18th Amendment, overcame a veto from President Woodrow Wilson on October 28, 1919, and took effect at the stroke of midnight on January 17, 1920. Its full title was \"An Act to prohibit intoxicating beverages, and to regulate the manufacture, production, use, and sale of high-proof spirits for other than beverage purposes, and to ensure an ample supply of alcohol and promote its use in scientific research and in the development of fuel, dye, and other lawful industries.\"\n\nDumping seized beer, wine, and spirits into gutters and drains was a routine PR move for Prohibition agents. Here the Orange County (California) sheriff and his staff dispatch a load of contraband to the ocean.\n\nMost folks called it simply the Volstead Act after Andrew Volstead, a Republican representative from Minnesota who sponsored the legislation. But we should be clear; despite assurances from some historians who insist that alcohol was forbidden across the board, that every brewery and distillery was shuttered, and that every permit to handle liquor was revoked, many legal loopholes and exemptions existed\u2014as the act's long title hints. Between legally produced sacramental wine being consumed by suddenly religious families, medicinal whiskey for those stubborn coughs, and various high-proof products such as vanilla extract that manufacturers successfully lobbied to have defined as something other than \"beverages,\" getting alcohol to the masses wasn't just possible, it was profitable.\n\nThe kinds and quality of available alcohol shifted between 1920 and 1933, but drinking didn't stop. Especially in New Orleans, Chicago, Kansas City, Philadelphia, New York, and other cities, the flow of booze continued not just unabated, but seemed to gush. Home distilling went on (as it ever has and does today) and great volumes of illicit spirits came from smuggled booze, cutting plants, moonshine syndicates, and alcohol diverted from legal industry, such as cosmetics and perfumery. Breaking the law, unthinkable for some, became something of a national pastime when it came to alcohol.\n\nAt its very best, Prohibition-era alcohol was exactly what it had always been. Smugglers routinely slipped high quality spirits past government agents. Genuine Champagne, Scotch and Canadian whisky, European aperitifs, proper wine, and monastic liqueurs, such as Chartreuse and Benedictine, could, for a price, be had. Yet it's worth keeping in mind a saying about American drinking habits that was an old joke among distillers even before Prohibition struck: in any given year (and it varies: 1854, 1867, 1902, take your pick) Americans consumed more French brandy than had been distilled in all of France. The same could be said of most imported spirits; American boosters had argued since the 18th century that domestic spirits every bit as good as imported could be made with native ingredients. The implication, of course, is that, whether they knew it or not, American bartenders very often poured spirits that bore only superficial resemblance to whatever the bottle and kegs were within indicated.\n\nThe 18th Amendment finally was repealed in 1933 with passage of the 21st Amendment, but small- _p_ prohibition didn't disappear. Pre-Prohibition bans on absinthe and home distilling endured even after repeal. Some areas of the United States still exercise \"local options\" and forbid alcohol sales entirely. Others restrict them to certain days.\n\n### **Whiskey-Made Criminals**\n\nOf all the claims temperance unionists and anti-saloon legionnaires made in their clamor for nationwide prohibition, perhaps the most naive\u2014aside from the conviction the policy would actually work\u2014was that, once whiskey, rum, gin, and other spirits were driven from the American landscape, the nation's asylums, jails, and prisons would be emptied. They would be relics, vestiges of an older world, no longer of any use in the Arcadia of boozeless America. For these would-be reformers, alcohol truly was the root of _all_ evil. As many of them saw it, eliminating alcohol and the saloons in which it was served was essential for physical, mental and moral health. Few captured that special flavor of fervor better than baseball player turned evangelist William \"Billy\" Sunday. Sunday railed against sin in his popular sermons, but took particular affront to liquor. In 1916, he thundered to a Boston audience of 55,000, \"Listen! Seventy-five per cent of our idiots come from intemperate parents, 80 per cent of the paupers, 82 per cent of the crime is committed by men under the influence of liquor, 90 per cent of the adult criminals are whiskey made... Whiskey and beer are all right in their place, but their place is in hell.\"\n\nWithout alcohol degrading the very souls of America's citizens, their minds would clear, families would stay together, and an altogether wholesome way of life would take hold. Ice-cream parlors and buttermilk dens would hang shingles where once saloons blighted neighborhoods. And crime? Crime would be history. Prisons would take in fewer and fewer inmates until one day they simply could be shuttered as the last offenders cycled back into the general population. State and federal governments would be relieved of the financial burden of housing, supporting, and guarding prisoners.\n\nOf course, after a few years of living under the Volstead Act and its attendant laws and regulations, the folly in that line of thinking became manifest. Prisons didn't empty; they were fuller than ever. Between 1914 and 1919, the federal prisoner population hovered between 4,000 and 4,800. By 1929, that number was over 12,000, not counting those farmed out to state, county, and city jails because federal prisons were at capacity. In Atlanta's federal penitentiary, about half of the prisoners were moonshiners who, generally after short sentences, went home armed with new advice about distilling from fellow prisoners. Rather than dissuading illegal distilling, prison for some was a clearinghouse for moonshine tips and techniques.\n\n### **Victor Lyon's Formulas**\n\nAs a rare peek into the secretive beverage arts of Prohibition-era America, Victor Lyon's notebook is fascinating in and of itself. The formulas in it, though, didn't just spring up like mushrooms when the nation\u2014ostensibly\u2014went dry. Some of his formulas and recipes come from German distilling manuals, some from farmhouse books published in England, and others still from books and magazines printed in the country that had just outlawed exactly these sorts of things. Many come from medical texts. Most date from the middle of the 19th century to the 1920s, making them older than L. Frank Baum's _The Wonderful Wizard of Oz_ , older than the first American football game, older even, in some cases, than the American Civil War and the stories of Mark Twain. These aren't formulas that originated in the Jazz Age; in most cases, they preceded it.\n\nNo matter how extensive his notes on manufacturing liquors and cordials are, though, they're only a fraction of the thousands upon thousands of recipes and formulae waiting to be rediscovered in old manuals for tinctures, bitters, spirits, and cordials. Here and there, he's got outliers, but for the most part, Lyon wrote formulas for unpretentious beverages: gins, whiskeys, mostly simple cordials, rums, and absinthes.\n\nSomething else to note about Lyon's formulas: many of them have more or less overt connections to medicine. Yeah, sure, there's hardly an ingredient under the sun that someone doesn't regard as a remedy for this or that complaint, and spirits themselves are medicine for many, but the frou-frou fancy concoctions from the 1860s on are entirely omitted. There is, for example, no violet-scented purple Parfait Amour, essential to an Aviation cocktail, but probably not a fast seller in a 1920s Harlem doctor's office. He passes on such suggestive, deluxe, and frivolous concoctions as Cr\u00e8me de Nymphe (cinnamon, mace, and rose), Culotte de Pape (\"Pope's Britches,\" a blend of Ceylon cinnamon, cloves, and vanilla), or a cordial the color of old moss made with caraway seeds, orange skins, anise, and carrot flowers. Nineteenth-century saloonkeepers might have kept that last one in the storeroom in small kegs labeled \"Oil of Venus\"\u2014but sold it as Huile de Venus. Those were the recipes of a baroque Gilded Age when gold seemed to flow in the streets, not of Prohibition when Americans died in the gutters for wont of good liquor.\n\nPurchasers of bootleg liquor are running enormous risks, for labels are not only counterfeited, but bottles of recognized brands are refilled over and over again, the contents consisting of diluted alcohol and coloring matter and flavoring extracts. The reason so much gin is being peddled is because the concoction consists mostly of alcohol, flavored with oil of juniper. With bottles fra[u]dulently labeled, purchasers are deceived.\n\n\u2014J. M. Doran, head of the Industrial Alcohol and Chemical Division of the Prohibition Unit, as quoted in \"Bootleg Liquor,\" _Journal of the National Medical Association_ 14, no. 1 (1922)\n\nGin blossomed as a Prohibition staple in part because nearly anybody could make simple versions with nothing more than juniper oil, alcohol, and a jug, but since its earliest days, the juniper-flavored spirit has been touted as sovereign against urinary tract complaints. Peppermint oil, essential to so many cr\u00e8me de menthe formulas, has been used to settled disruptive bellies since colonial days. Still is. Ginger wine? Official pharmacy manuals prescribed it for stomach complaints for centuries. Brandy comforts colds and, like whiskey, is still used as a solvent for medicinal herbs. Absinthe, banned even before national Prohibition, gained traction in part because Swiss and French doctors prescribed it as an antimalarial.\n\nUnlike Lyon, however, I am not a physician. I may throw in a comment here and there, but explicating medicinal applications of these old recipes is a task for someone else.\n\n### **How Did Americans Get Alcohol During Prohibition?**\n\nAlthough total prohibition in America was the utopian dream of some of temperance crusaders and anti-saloon zealots, what the 18th Amendment actually forbade was not alcohol but _intoxicating beverages_. Smart business owners and interest groups understood that the loopholes inherent in that phrase were so big that they could sail boats through them and lobbied to get their products labeled as something other than _intoxicating_ or _beverages_. Americans who wanted to drink alcohol during the Volstead years had several options for getting their giggle water. From conning their ways into congregations to distilling at home, here are some of the sources of alcohol.\n\nGiggle water was an example of \"Volstead's English\"\u2014slang that emerged in the 1920s to joke about, mock, and speak softly about drinking. This 1928 cocktail guide of the same name contained many 19th-century drink recipes: fixes, cobblers, fizzes, juleps, and more.\n\n### **Legal Booze**\n\n**Pre-Prohibition Hoards**\n\nThe Volstead Act did not prevent Americans from drinking alcohol they already owned before Prohibition took effect. As the onset of Prohibition approached in 1920, many drinkers bought as much liquor as they could afford, hoarding the bottles for their own use or sometimes as investments for the coming dryness. Citizens could store as much alcohol as they liked in their homes, where they could shake, stir, and drink it at their leisure. They were even allowed two homes; one in the city and one in the country, though booze could not be shifted from one to the other without proper permits. Countless millions of bottles of gin, whiskey, k\u00fcmmel, rum, and other spirits and cordials were tucked away in basements, cabinets, and throughout homes and farms across the country. Every now and again during a renovation or estate sale, some of those hoards are uncovered, much to the delight of vintage spirits collectors and bartenders.\n\n**Medicinal Alcohol**\n\nPhysicians were specifically allowed to write prescriptions for medicinal alcohol during Prohibition. With scrips in hand, their patients could purchase \"medicinal\" whiskey, brandy, or fortified wines (22% abv [alcohol by volume] compared to the usual 12%\u201314%) from druggists who paid for an annual permit to handle such wet goods. You never saw a nation of such sickly people.\n\n**Hard Cider**\n\nAppeasing rural districts was key to passing the 18th Amendment. Early on, one of the bones tossed to their representatives was permission to ferment sweet apple cider. The resulting hard cider, perfectly legal to drink at home, could pack a serious punch.\n\n**Sacramental Wine**\n\nIf you were Catholic or Jewish during Prohibition (or could pass as such, anyway), you were in luck. Both Jews and Catholics were permitted sacramental wine under the new laws. To the sputtering indignation of Wets, congregations swelled as newly religious citizens took full advantage of the loophole while Irish and African American \"rabbis\" purchased bulk wine for nonexistent congregants (but very real customers).\n\n**Travel**\n\nFor those who could afford it, international travel was the most straightforward opportunity to drink without fear of legal repercussions. A trip to Canada, Cuba, Mexico, Panama, or Europe was the perfect setting to indulge in spirits, cordials, and cocktails, sometimes made by bartenders who relocated rather than become soda jerks or waiters in New York, Boston, and other US cities where their professions were now forbidden. Such liquid souvenirs were forbidden back in the States, but they did make it back, sometimes by the steamer trunkful.\n\nHaving spent years among felons and criminals of many stripes, one thing always struck me as discordant about the idea that the alcohol Americans drank constituted genuine goods when they traveled during Prohibition. It's this: whenever counterfeit booze is in the local market, everything\u2014and I do mean _everything_ \u2014about drinking becomes suspect. Grifters, opportunists, and outright criminals simply cannot resist an easy mark. And, man, were Americans easy marks. Look at it like this: if you're running a bar in Canada or Mexico along the US border or in the Caribbean during the Volstead years, yeah, sure, you can sell hordes of unruly Yanks genuine spirits. Many did just that and made good money doing it. But... but... if you're amenable to a little extra on the side and pretty sure you'll never see that mook again (or he's too drunk to notice), what's to stop you from pouring cut liquor and pocketing the difference?\n\nA stretch? Not at all. Consider the case of Boston distiller J. F. McCarthy. Before Prohibition, McCarthy made Sheridan Rye, a well-regarded whiskey. But he shuttered the firm and sold his lead bottle caps\u2014emblazoned with his name\u2014as scrap to junk dealer in nearby Chelsea, Massachusetts. Fast-forward to 1924. McCarthy is cooling his heels in one of Havana's better saloons. He asks the waiter for American whiskey. But of course, sir. A few moments later, the bottle is delivered. \"There you are, sir! One of the best brands of American whiskey!\" It's McCarthy's own Sheridan Rye. Except... not. The label had been counterfeited, the contents wholly factitious, but the lead caps? The very ones the distiller had sold as scrap years earlier.\n\nBefore I was old enough to buy beer, I could buy the ingredients for brewing beer in my college apartment. This poster, which hung next to my mother's stove since I was a child, may have helped nudge me into that particular epiphany.\n\n**Low-Alcohol Near Beer**\n\nFew cared for it, but beer under 0.5% alcohol content was permitted under Volstead as \"non-intoxicating.\" Until they realized the greater profit in selling malt syrup for people to brew beer illegally at home, some breweries eked through Prohibition making near beer by heating regular beer in open containers to drive off ethanol. Enterprising bootleggers reintroduced alcohol (usually neutral spirits) to make \"needled\" beer. People did not love it.\n\n### **Illegal Sources**\n\n**Home Brewing and Home Distilling**\n\nCountless Americans took to brewing their own beers at home during Prohibition. Their suds may not have been exemplars of the brewing art, but what they were quaffing was probably better than crude needled stuff. At the same time, many immigrants and native-born Americans alike knew how to build and operate small stills to make various sorts of moonshines in their homes. While their owners may not have been selling their makings, by 1926 millions of stills churned out a lot of homemade liquor.\n\n**Denatured Industrial Alcohol**\n\nMany in industry and the arts, from manufacturers to painters, insisted there simply was no reasonable substitute for ethanol as a solvent. The problem for temperance types is that pure industrial ethanol is chemically identical to the ethanol in whiskey, gin, wine, and cider. By government diktat, then, producers of industrial alcohol were required to \"denature\" it by using one of several approved formulas that would render it unfit to drink. Bootlegger chemists devised ways to \" _re_ nature\" such tainted spirits while government chemists revised official formulas to stay ahead of them. Plain alcohol like this needled beers, bumped the proof of watered whiskey, and formed the base of compounded gins and other spirits. Results were mixed. Oftentimes, counterfeit booze made from incompletely processed industrial alcohol carried the off tastes and smells of denaturing chemicals.\n\n**Bootleggers**\n\nMoonshining and bootlegging, two related activities, are often conflated. Although the same person may do both, they are, in fact, separate undertakings. Moonshiners distill spirits illicitly; bootleggers sell spirits (and other contraband) illicitly, whether or not they were made legally. Swarms of Americans became opportunistic bootleggers during Prohibition, selling pints and quarts here and there. They were cab drivers, elevator operators, bellboys, doctors, cigar store owners, milkmen... even kindly matrons in the \"fruit jar trade\" delivering high-proof preserves to neighbors in gingham-lined baskets. Selling booze on the side made money, but it wasn't generally their primary income. Other bootleggers worked more or less full-time for or in cahoots with criminal syndicates, distributing liquor that they or they colleagues distilled, diverted, hijacked, compounded, or smuggled.\n\n**Nanosmuggling**\n\nBig-time smugglers hauled truck- and boatloads of liquor into the United States, but not everyone was big-time. Despite the likelihood of getting caught, Americans returning from abroad routinely squeaked in a souvenir bottle or two. Sometimes, hauls were more substantial. American government officials on official missions, for example, could be granted free entry; on their return, their luggage, like modern diplomatic pouches, would not be searched.\n\nIn 1928, Congressman M. Alfred Michaelson was busted following such a trip to Panama after a railway baggage man in Florida traced leaking liquor to one of his six trunks. The ostensibly dry Michaelson had brought back 6 quarts of John Haig whiskey, two quarts of cr\u00e8me de menthe, one quart each of \"taffel Akavait\" (a caraway-flavored Scandinavian spirit), cr\u00e8me de cacao, and cherry brandy. Oh, and a keg of Barbancourt rum. A year later, his prohibitionist colleague William M. Morgan of Ohio was likewise busted with four bottles of whiskey. \"I never,\" later explained the honorable representative, \"took a drink in my life.\"\n\nAs part of a series of raids on moonshine operations feeding New York, Philadelphia, and Atlantic City, New Jersey troopers pose with several 5,000-gallon fermentation vats discovered outside Fort Dix.\n\n### **Drugstore Whiskey**\n\nAlthough the United States did not issue a license to a pharmacist until 1816, druggists, whether they were called apothecaries, pharmacists, or some other name, had been dispensing alcohol to their patients for centuries, a practice that stretched back to Europe and was older than America itself. Medicinal alcohol was such an ingrained part of American culture by the 1920s that even moderate temperance advocates whose lips, they insisted, would never touch alcohol regarded whiskey, brandy, and rum as important medicines in and of themselves; they relieved coughs, kept the cold at bay, and helped revive flagging spirits. Beyond plain bottles of whiskey, though, druggists made heavy use of nearly pure, practically flavorless \"rectified\" alcohol that typically ran 90%\u201395% ethanol. It was part and parcel of their trade, a vital ingredient used in the legitimate medicines many made in their back rooms and basement workshops.\n\nNational Prohibition was meant to eradicate alcohol from public life, but in truth, the laws and regulations were never watertight; for those who wanted it, supplies of one sort of alcohol or another could always be found. Drugstore whiskey was one of the alternates to which people flocked, since physicians operated under an exemption that allowed them to write their patients prescriptions for medicinal whiskey. Whether patients presented those scripts in all sincerity with a straight face, or with a broad wink, retail druggists were those who filled them.\n\n**There was grousing in Brooklyn and booze was to blame.**\n\nIn May 1922, members of the Kings County Pharmaceutical Society gathered at the Brooklyn College of Pharmacy for their monthly sit-down. The meeting began with good news; over two hundred seniors had passed their qualifying exams, making it the largest graduating class in the college's history. One can imagine self-pleased harrumphs around the room. Afterward, there was a brief, relatively mundane discussion of tariffs and duties on glassware. Then things turned sour.\n\nLike their colleagues in other cities, New York druggists chafed under new rules meant to control alcohol. Most were adamant that, whether they were consumed as is or used in bitters, tonics, tinctures, and other compounds, spirits helped their patients maintain good health. What they didn't mention, but everyone in the room understood, was that spirits also contributed mightily to their profits. While Prohibition laws allowed them to continue their trade in alcohol, the obstacles thrown in their way were infuriating.\n\nMembers railed against local Prohibition directors who disparaged them in newspaper interviews as criminals and tangled their work with red tape. There were bitter complaints that officials intentionally dragged their feet when druggists applied for permits to sell alcohol. Although pharmacists were permitted to dispense alcohol and use it in various preparations, the law demanded that such spirits were strictly for medicinal purposes; whiskey for a cough was allowed. Whiskey for a julep, though? No dice. But so-called medicinal whiskey\u2014 _spiritus frumenti_ or _spiritus fermenti_ in apothecary Latin\u2014was meant to be a hassle to obtain. People weren't supposed to be able to just walk into a drugstore and buy a pint of bourbon or rye as they could in many places before Prohibition (or since, for that matter). Instead, anyone who wanted to buy a bottle of otherwise outlawed spirits needed a prescription.\n\nThe problem was, prescription slips were easy to fake. The market was flooded with counterfeits. This was one of the reasons Prohibition directors didn't simply rubber-stamp applications for liquor permits. There was a growing and well-founded sense that staunching the flow of alcohol was not going to be as straightforward as Prohibition's advocates had hoped. Furthermore, druggists and physicians were part of the problem.\n\n**Medicinal Wine and Spirits**\n\nWhen the Volstead Act went effect in 1920, Major Alfred Vernon Dalrymple was appointed chief prohibition officer in Chicago. There he oversaw a twenty-one man crew who sought to crush the trade in alcohol, wine, and beer. Strict enforcement of the law was absolutely necessary, he argued, to maintain the \"morale\" of the newly arrived immigrants\u2014never mind that they might have come from Germany, Italy, Ireland, China, or other parts of the world where drinking was regarded a comfort. In a speech at Chicago's City Club, the major drew clear distinctions between law-abiding citizens and the treasonous behavior of those who continued to trade in intoxicating beverages.\n\n\"Benedict Arnold,\" he thundered, \"was no greater traitor than any citizen who opposes the enforcement prohibition while it remains a law.\" The issue was not temperance, or prohibition, but law enforcement. Dalrymple singled out two groups involved for particular excoriation: druggists and doctors.\n\nMajor Dalrymple comes across as a bit of a blowhard, especially as we look back with almost a century of hindsight. But here's the thing: he wasn't wrong.\n\nUnder the Volstead Act, physicians were permitted to write liquor prescriptions for \"medicinal purposes.\" It wasn't supposed to be a free-for-all; physicians were required to keep records indicating what liquor for what ailment, how much, in what dose, and for whom they wrote prescriptions. Druggists who filled those prescriptions had to buy annual licenses and keep records as well. But for patients with legitimate medical complaints (as well as thirsty souls who knew an obliging doctor), a supply of bonded whiskey, brandy, New England rum, or fortified wines was as close as the nearest drugstore. Druggists were inundated with prescriptions both because prescription blanks were easy to counterfeit and because plenty of doctors would, for a small fee of course, dash off _spiritus frumenti_ prescriptions for patients with imaginary ailments.\n\nThe medicinal spirits exemption was a particular bother for Prohibition agents. Dalrymple told members of the City Club that a single druggist had sold 800 gallons of whiskey in nine days. Bartenders selling that much whiskey these days might get a visit from brand ambassadors wanting to know how they could help them sell even more. Those are _great_ numbers. Eight _hundred_ gallons! Even so, he was trumped by the physician who issued 7,534 prescriptions for alcohol in about four weeks\u2014some 250 every day, including weekends. Assuming each of those were for a pint to quart (the maximum allowed to any one patient per month), that's about 940 to 1,880 gallons of whiskey, brandy, and other spirits. The poor guy must've been doing nothing but writing prescriptions for alcohol. Here's hoping he had the sense to hire some assistants to give his wrist a break.\n\n### **Potus Inebrians**\n\nNo, no, no. It's not that the president is drunk. Well, possibly. But that's not what's going on here. _Potus_ , before it became an acronym for the President of the United States, was a medical Latin term for \"drink.\" _Potion_ , _potable_ , and _potation_ come from the word. Although plenty of beverages we wouldn't necessarily call wines today paraded under the name of _vinum_ , _potus inebrians_ referred to a strong drink more broadly. _Potiones ardentes_ could mean either wines or more ardent waters, the cordials and distillates of the pharmaceutical trade.\n\nIt is the ancient custom of compounders to call wines, beers, and spirits by their Latin names. We don't need to concern ourselves with every last one, but it's worth taking a look at a few that appear as common names in old recipes for compounded cordials, wines, and spirits. _Spiritus_ referred to distilled alcohol and crops up often.\n\n---\n\n_Spiritus vini gallici_ | French brandy\n\n_Spiritus sacchari_ | rum\n\n_Spiritus oryzae_ | arrack\n\n_Spiritus juniperi_ | gin, Holland gin, genever\n\n_Spiritus lactis equini_ | koumiss, fermented mare's milk\n\nWine was extensively called for in older recipes, often by its Latin name, _vinum_ , followed by some modifier. There was, for example _vinum album_ and _vinum rubrum_ (white and red wine, respectively), but also _vinum Gallicum_ , _Burgundicum_ , _Burdigalense_ , or _Germanicum_ (French, Burgundy, Bordeaux, or German wine). When wine was pure and unmixed, not watered or flavored, it might be referred to as _merum_. Others that show up in compounding manuals include:\n\n---\n\n_Vinum Lusitanicum_ | port wine\n\n_Vinum Toccaviense_ | Tokay\n\n_Vinum Hispanicum album_ | Spanish white wine\n\n_Vinum Xeres_ | sherry\n\n_Vinum Maderaicum_ | Madeira\n\n|\n\n_Cerevisia\/cervisia\/zythum_ | beer or ale\n\n_Cerevisia Londinensis_ | porter\n\n_Cerevisia tenuis_ | strong beer\n\n_Vinum pomaceum_ | apple cider\n\n_Vinum pyraceum_ | perry (that is, cider made of pears; quite delicious)\n\n## **_Chapter two_**\n\n## Compounding Spirits\n\n**_ways to use spirits and essential oils in beverages_**\n\nThe alcohol entries in Victor Lyon's notebook, regardless of the language in which they're written, fall into two broad categories. The first concerns alcohol production and analysis\u2014how ethanol behaves when mixed with water, for instance, or price lists for raw supplies. Notably, there is little information about operating a still, that essential piece of equipment needed to extract high-proof spirits from low-alcohol beers, wines, and washes. The assumption with almost all of these formulas is that the spirits, cider, or wine used to make them are purchased just like any other ingredient or supply. The second type of entry follows from that assumption: it covers formulas and recipes for making cordials, syrups, tinctures, and high-proof spirits. Some use botanicals, such as orange peels or juniper berries, which we're used to seeing in modern cocktail bars. Others rely on essential oils and aromatic compounds to provide flavor and aroma to the finished product.\n\nIn this chapter, we'll do three things. First, we'll set up historical context for these sorts of recipes and formulas by looking at early medical texts. Second, we'll review gear for making compounded beverages, from must-have kit, such as bottles and spoons, to more specialized equipment. Last, we'll examine ways to use spirits and essential oils in beverages.\n\n### **Not Your Grandmother's Recipes (Unless Gran Was a Bootlegger)**\n\nDoctor Lyon's notes, by and large, are not homestyle recipes. Sure, some of them can be scaled down and made in kitchens and bars by using pitchers, jugs, and measuring cups. We'll take a look at some of those. Except for examples with a bit of a British farmstead vibe, though, there's very little domestic about his writing. Unlike cookbooks of the era, Lyon's formulas and notes were not intended for home cooks. Nowhere do we see teacups, dessertspoons, or hands used to measure ingredients. There are no dimensions, such as \"the size of a walnut\" or \"as big as a pea,\" as we might expect from someone writing about preparing meals for a household. The scale of these recipes is commercial; even the most ardent DIY bitters enthusiasts and home cocktail fiends tend not to work with 40 or 100 gallons of whiskey at a time.\n\nA few entries concern genuine spirits, but the most of the 300-plus entries are recipes and formulas for compounded wine and spirits\u2014making whiskey, brandy, rum, absinthe, gin and genever, k\u00fcmmel, port wine, cordials, and so forth, from neutral spirits, essences, flavorings, various botanicals, and, from time to time, genuine spirits. Such wholesale compounding was commonplace among liquor brokers and saloonkeepers at the turn of the 20th century, but is largely forgotten now and little known outside professional or hobbyist distilling circles and curious bartenders.\n\nRather, these recipes and formulas fall into the domain of chemistry, of rules and laws. One senses an orderly, if occasionally disorganized, mind at work. The metric measurements throughout the notebook were commonplace by the end of the 19th century in professional European manuals, whether they dealt with liquor or charcuterie. Herbs and spices seem not to have been harvested from a nearby garden\u2014there is no information about seasons, pests, ideal harvest times, seed saving, and so forth\u2014but bought from merchants. Several recipes deal with making extracts and essences, but there's nothing about distilling essential oils. As the purview of specialists, these, too, must be purchased.\n\n### **The Doctor Is In**\n\n**An Ancient Cheat of Acid and Oil**\n\nModern craft bartenders achieve a rich, frothy head on cocktails a few ways, including shaking drinks with cream, egg whites, or pineapple juice. Tech-forward types may even dispense stabilized aromatic foams from nitrogen-charged whipping siphons. Withal, 18th-century rum merchants and 1920s moonshiners alike made spirits foam ever, ever so lightly with a bit of liquid prestidigitation. Their trick\u2014and almost always it was an underhanded trick meant to fool customers\u2014was an ancient cheat known as The Doctor, false-proof, beading oil, or simply beading. They are usually equal parts of sweet almond oil and sulfuric acid dissolved in high-proof spirits. Lyon includes a lot of such recipes. Despite minor variation in wording or directions, they are practically unchanged from such recipes over the last three centuries. Consider William Smyth's directions for publicans from 1781:\n\n### THE DOCTOR\n\nTwo drams of oil of almonds\n\nTwo ditto of oil of vitriol\n\nOne jill of spirit of wine [i.e., gill, or one-fourth of an imperial pint or 142 ml]\n\nShake well, and use occasionally.\n\nThe Doctor is a cheat because it affects how spirits behave when they're agitated. Shaking a sample of spirits is a crude, seat-of-the-pants test for determining its proof. Try it with whiskey, vodka, or whatever you have lying around. Regular-strength spirits (that is, 40% abv) will produce many small bubbles that quickly dissipate on the liquid's surface. Higher-proof spirits yield slightly larger and longer-lasting bubbles that some call _goose eggs_ or _frog's eyes_. They don't last more than a few seconds, but some customers looked on those bubbles or beads as proof positive that the spirit they were about to buy were, in fact, proof spirits. Beading oil allowed dealers to water down the product and still get that slight head. It was also handy when saloonkeepers who bought gin in bulk might water it down too much (for some cutting was always expected). Beading oil with a little tincture of cassia and chiles\u2014and syrup, of course\u2014would seem to bring that proof right back up. Occasionally, as in William Loftus's 1869 recipe that follows, the foaming seems to have surpassed mere beads and produced proper foamy heads in doctored spirits.\n\n### RECEIPT FOR SPIRIT BEADING\n\nOil of sweet almonds 1 ounce\n\nOil of vitriol [sulfuric acid] ditto.\n\nRub together in a mortar, and add by degrees about two ounces of lump sugar, rubbing well with the pestle until it becomes a paste. Then add small quantities of spirits of wine until a thin liquid is formed. This quantity of beading is sufficient for 100 gallons of gin, and will cause the spirits to carry a fine pearly bead, when drawn from a little height into a glass.\n\nThese oils are curiosities. Please don't make them. If you have one, please don't use it. Sulfuric acid can be dangerous for those who aren't versed in its safe handling. Unlike syrups and caramels, which may be deployed occasionally in quality spirits, beading oils really have no wholesome purpose in spirits.\n\n### **What Is a Flavor?**\n\nWhat is a flavor? Pfft. Come on, now. Who makes drinks and doesn't know what a flavor is? Put a glass to your lips, take a sip, _boom_ , there it is: elderflower, or mint, or whiskey. Orange. Juniper. Cinnamon. Rum. Whatever. We can parse those broad strokes into even more nuanced detail, but we know on a gut level what tastes are. Our understanding of how taste buds function has evolved over the last century, but the basic premise holds. Little structures called papillae in your mouth and throat detect qualities such as sweet, sour, bitter, salty, and\u2014now that we can put a name to it\u2014umami, that satisfying \"savory\" flavor we can pull from such foods as aged beef, certain cheeses, and some seafood. Ditch the outdated idea that tongues can be sectioned into areas that detect only sour or only sweet or some other taste. We know now that _all_ areas of the tongue, not just certain sections, can detect different tastes.\n\nBut taste buds are only one part of the picture. Most scientists who look at this sort of thing, in fact, report that smell accounts for the lion's share of what we call taste. What else? How about pain? You bet. Flavor, wrote R. L. Hall in 1968,\n\nis the sum of those characteristics of any material taken in the mouth, perceived principally by the senses of taste and smell, and also the general pain and tactile receptors in the mouth, as received and interpreted by the brain. (R. L. Hall, \"Food Flavors: Benefits and Problems,\" _Food Technology_  [1968]: 1388)\n\nWith this definition, we are reminded that part of the taste of a Bloody Mary is the pain of horseradish, that back-of-the-throat itch from good black pepper, and the irritation caused by hot sauce if that's in there, too. Tannins in oak and tea can contribute astringent, drying sensations to drinks. We can talk about menthol in mint juleps and grasshoppers, which is hardly any kind of taste at all... not until you inhale, drawing air across your moist tongue, and suddenly your mouth comes alive with a bracing, spiky, cool _mintiness_.\n\nFlavor in spirits and cordials is rarely so simple as single notes, such as \"lemon\" or \"anise,\" because of the immense variability in the composition of raw ingredients, the chemical transformations that happen while aging, and how they are handled. That chemical complexity was the subject of much curiosity in the middle of the 19th century. Analyzing spirits and cordials to uncover their constituent parts allowed liquor producers to understand, on a chemical level, the compounds and substances that imbued whiskeys, brandies, wines, and other beverages with their distinctive aromas and flavors. Putting that knowledge to practice laid the groundwork for much of the bootleg liquor that changed hands during Prohibition.\n\n### **Justus von Liebig, Father of Organic Chemistry**\n\nInnkeepers, cellar masters, importers, blenders, and distillers\u2014not to mention makers of wine, cider, beer, mead, and just about anything else you'd care to pour down your throat\u2014had been adding herbs, spices, minerals, even animals and insects to their drinks since time immemorial. When Jesus was offered a mix of vinegar and, depending on whose Gospel you favor, gall (Matthew 27:34) or myrrh (Mark 15:23) to drink on the cross, it was hardly a new idea. When 12th-century French cleric Peter of Blois entered the service of the Plantagenet king Edward II, he began a long career as a diplomat and sort of propagandist for the monarch. While Peter's loyalty seemed unshakable, his reports of English wine were harrowing. Even at the court of the king, the wine was\n\nsour or musty; muddy, greasy, rancid, reeking of pitch and quite flat. I have witnessed occasions when such dregs were served to noblemen, they had to sift it through clenched teeth and with their eyes shut, with trembling and grimacing, rather than just drink it. (Quoted in Jack Turner, _Spice: The History of a Temptation_ [New York: Alfred A. Knopf, 2004], 114\u201315)\n\nCombinations of wine, say, or beer and medicines or flavorings were not necessarily haphazard, but they were not _analytical_. Enter German chemist Justus von Liebig (1803\u20131873). Distillers know his name because a distillery condenser that he popularized almost 200 years ago remains popular today. Water-cooled Liebig condensers convert hot, ethanol-rich vapors coming off stills back into liquids that become gin, whiskey, brandy, absinthe, and so on. Other ways of condensing ethanol vapors exist, but over the last quarter-century, I've seen more elegant little Liebig condensers mounted to homemade stills than any other, from Seattle to rural Ireland.\n\nBy his late twenties, von Liebig had become deeply engaged in teasing apart various substances, learning how they interact with one another, and what roles they had in determining the characteristics of, say, essential oil of bitter almond. In \" _Untersuchungen \u00fcber das Radikal der Benzoes\u00e4ure_ ,\" an 1832 article von Liebig wrote with colleague Friedrich W\u00f6hler, he made a bold assertion.\n\nNineteenth-century temperance crusader Eli Johnson made a handsome living decrying liquor in the United States and Europe. His \"Magic Box\"\u2014painted black and gussied up with a skull and crossbones\u2014held essential oils and other aromatic compounds from which he might concoct his \"drinks from drugs.\"\n\nThe production of all organic substances no longer belongs just to living organisms. It must be seen as not only probable, but as certain, that we shall be able to produce them in our laboratories. Sugar, salicin, and morphine will be artificially produced. Of course, we do not yet know how to do this, because we do not yet know the precursors from which these compounds arise. But we shall come to know them. (Quoted in William H. Brock, _Justus von Liebig: The Chemical Gatekeeper_ , 1st ed. [Cambridge, UK: Cambridge University Press, 1997])\n\n_But we shall come to know them_. Freiherr von Liebig's acolytes took those words to heart. By 1840, a wave of new texts were flooding scientific circles in Europe as von Liebig's writings were translated into multiple languages and colleagues did experiments of their own to determine the constituent parts of such substances as Cognac, gins, whiskeys, arrack, and more. It wasn't long before liquor merchants, distillers, and compounders took notice and began to change their formulas to include vastly more essential oils and other aromatic compounds. New language crept into the manuals. There was much talk of the \"poisonous\" liquors of the old days (that is, 10 or 20 years back) and how, even if the directions on offer were for imitation gins and bogus brandies, such formulas only included compounds found in the original, genuine spirits, so they could not possibly be harmful.\n\n### **Compounding**\n\nWhat Victor Lyon put together in the first decade of Prohibition wasn't a distiller's manual, nor was it a bartender's guide. Despite some overlap with each, it was something altogether different. Not a single cocktail recipe appears in the book and, despite notes on purifying caramel, whiskey mash bills, and small-scale distillation, anyone reading those parts would have to know how to operate a still already to make sense of them. Rather, his manuscript falls into an old tradition of compounding formularies.\n\nHistorically, compounding formularies were books, collections of notes on analyzing materials and formulas for using them. Some physicians kept such notebooks with their personal favorite preparations, but they were more commonly used by druggists who made (or, rather, compounded) medicinal preparations on site: ointments, pills, skin creams, freckle cures, spirituous waters, and other medicaments. They were also used by those in the beverage trade to keep track of their favorite (and often secret) preparations.\n\nBecause adulteration was rife in the botanicals and drugs trade, a good liquor manufacturer needed to know how to test his supplies for purity\u2014whether the juniper oil, for instance, was cut with turpentine or whether the \"French Cognac\" was German beetroot spirits laced with Cognac oil, rounded out with syrup of raisins, and goosed with caramel to simulate age. Keeping notes about successful processes and useful preparations was an important part of the job.\n\nThe sources of these sorts of recipes could be anything from herbals to cookery manuals and medical journals. Many were culled from more substantial, comprehensive tomes called pharmacopoeias or dispensatories. Pharmacopoeias stretch back centuries before the earliest cocktail books, yet that's exactly the font from which they spring. If, like me, you love poring over modern cocktail guides, such as the fat little _PDT Cocktail Book_ or Death & Co.'s gorgeous thick tome, you'll see them on an entirely different level after reading old pharmacopoeias.\n\n### **Origins of Pharmacopoeias**\n\nIt may seem a stretch at first blush, but the lineage of modern cocktail guides and recipe books goes back a lot further back than Jerry Thomas and his 1862 _Bar-Tender's Guide_. In fact, we can trace their line at least to the time when Romans ruled Britain. Without losing sight of the fact that the manuscript at the heart of the book in your hands was written during American Prohibition, it's worth looking at a few of the older texts that paved the way for later books on alcohols, syrups, tinctures, and the other preparations that lie at the heart of modern bartending.\n\nAncient Romans, Greeks, Chinese, Indians, Mesopotamians, Egyptians, and others developed more or less sophisticated knowledge about the healing arts that included how to prepare treatments from animal, mineral, and plant sources. For nearly 2,000 years, that collective knowledge has gone by a shorthand name in the West: _materia medica_. You still hear it used today, though we're more likely to call the field _pharmacy_. A book that gathers those directions for identifying, analyzing, and preparing compounded medicines is called a pharmacopoeia (also pharmacopeia, pharmacop\u0153ia, pharmacopoea, and related spellings, all of which are taken from the ancient Greek, meaning \"drug-making\"). Such books dictate ingredients, proportions, and processes for making the official compounded medicines of this or that city or nation. Invariably, these included medicated wines, bitters, and various alcoholic preparations.\n\nHippocrates, Theophrastus, Celsus, Galen, Pliny the Elder, and others contributed to a growing understanding of medicine from the fifth century BC to the second century AD. The term _materia medica_ , though, comes from Dioscorides, a military physician in Nero's army, who traveled widely with the army and penned a treatise around AD 77 called _De Materia Medica_ (Latin for \"Concerning Medical Material\"). His five-volume work described roots, seeds, vines, oils, herbs, and animals used in, and as, medicine. Over the next 1,500 years, through the Dark Ages and into the medieval era and the Renaissance, the text was translated into many languages; despite local favorites here and there, it became a basic treatise on medicinal preparations throughout Europe.\n\nIn the 7th through 13th centuries, Dioscorides's book and other Graeco-Roman medical texts made their way to Arab physicians, such as Avicenna, Yuhanna ibn Masawaih (known as John Mesue in English), and Ibn Baitar, who wrote their own texts, adding new spices, herbs, and oils to the earlier works. In the 13th century the ambitious and powerful Holy Roman Emperor Frederick II had many of these works translated from Arabic to Latin. From there, they entered the broader European _materia medica_.\n\n**The First Pharmacopoeia**\n\nOkay, so maybe you don't fully buy into the argument that Harry Craddock's 1930 _The Savoy Cocktail Book_ or Paul Clarke's 2015 _The Cocktail Chronicles_ are direct descendants of works that were knocking around when Roman legions got their asses handed to them by German tribesmen at the Battle of Teutoberg Forest. You'd be wrong, but let's let it slide.\n\nSkip ahead to the 16th century. Using those old texts that had been circulating for more than a millennium, physicians and apothecaries in cities across Europe were developing rudimentary pharmacopoeias of official medical preparations. Beyond the walls of their own cities, such works had little reach. That changed when a German physician named Valerius Cordus presented _Dispensatorium Pharmacopolarum_ to the Nuremberg senate in 1542. _Dispensatorium_ was based on earlier works, but Cordus organized the entries alphabetically and included descriptions of preparations made from essential oils, including juniper and lavender. The senate was taken with his contribution and published it posthumously in 1546. As its use spread beyond the city, _Dispensatorium_ became arguably the first proper pharmacopoeia with wide reach, known across Europe as the _Nuremberg Pharmacopoeia_.\n\nBy 1618, physicians in London had adopted their own official pharmacopoeia. Dublin, Edinburgh, Amsterdam, and others followed. Published in Latin, these were scholarly books written in a language the unlettered were neither apt nor meant to understand. The first _British Pharmacopoeia_ \u2014a national, rather than city-based\u2014version was published in English in 1864 and revised in 1867 to supersede those of Dublin, London, and Edinburgh. By then, an American version, _The United States Dispensatory_ , already ran over 1,300 pages. Time and again, my own worn, leather-bound sixth edition from 1845 has proved invaluable when researching 19th-century alcohol and botanicals.\n\n**Why Latin?**\n\nUnderstanding the oldest pharmacopoeia formulas for tinctures, syrups, cordials, and distilled spirits is slow-going without some Latin under your belt. But it's not impossible. Modern physicians and pharmacists regularly write and understand abbreviations and shorthand terms in a language most of them don't speak. But why Latin? Why not use English? Or German or French or whatever the local language is, a language that everyday people would understand? Surely the local language would have eliminated a lot of confusion. \"This,\" opined one pre-Prohibition pharmacist, \"is a popular delusion.\" He continued:\n\nTo a qualified dispenser a Latin prescription is more familiar and readily understood than an English one. The language is concise and the abbreviations convenient and perfectly well understood. The chief or only difficulty in reading prescriptions consists in deciphering bad writing; and when words are illegible it matters little, whether they are in English, Latin, or Arabic... A knowledge of Latin may be taken to some extent as a guarantee of a liberal education, without which a young man would be more safely employed in measuring tape or ribbon, than in weighing morphia. (\"Fatal Accidents,\" _Pharmaceutical Journal and Transactions_ [1855], 393)\n\nSo, snarky comments about ribbons aside, the writer points out one of two good reasons to use Latin. Regardless of whether a pharmacist was in New York, Munich, Marseille, Turin, Stockholm, or Edinburgh, if he were an educated man\u2014and in those early days, nearly all druggists were men\u2014he could read the formulas of his colleagues from any of those places. With a knowledge of Latin, he could read not just a compounding formula here and there from a foreign source, but entire formularies and pharmacopoeias, cross-checking ingredients, amounts, and preparation methods with those of other traditions. For much the same reason, biologists, botanists, and others who study living organisms use Latin for nomenclature; local names are all over the board, but if there's general agreement on the Latin, they can better understand one another.\n\nThe other reason Latin remained popular with druggists is hinted at in that last word in the quote above: _morphia_. Morphine, as we'd call it now, is an important medicine. The potent painkiller derived from opium poppies is highly addictive and can be dangerous if used improperly. An apothecary's shop could have hundreds of such problematic substances on hand\u2014or they would be problematic if they fell into the wrong hands. Uneducated people who didn't understand Latin could do some real harm if they were to get their hands on narcotics, opiates, and other potent drugs. But if everything is in Latin? All the bottle and drawer labels, the prescriptions, the formula books? It might as well be gibberish and, like Victor Lyon's notebook, could be hidden in plain sight. Latin wasn't just a way to communicate with international colleagues; it was a way to encrypt those communications.\n\n### **The Emergence of Compounding Formularies and Rectifiers' Manuals**\n\nBy the 18th century, distilling and blending had emerged as serious commercial concerns. As distilling technology advanced, distillers were able to produce spirits faster and with greater purity, compounding formularies had evolved into a genre distinct from their predecessors and cousins, the pharmacopoeias and cookbooks. They came to bear a lot of resemblance to perfumers' manuals and, in fact, called for many of the same exotic ingredients from the same faraway lands. Manuals for the ice-cream and soda fountain trade arose slightly later but from the same sources.\n\nGone for the most part were formulas that were meant to be taken as remedies for such maladies as scrofula and venereal diseases. In the place of alcohol as a solvent and carrier for medicine were liqueurs and spirits meant to surprise and delight drinkers (and turn a profit for rectifiers who blended and sold such concoctions). The formulas in these liquor manuals are chockablock with calls for chemicals, compounds, balsams, and extracts to flavor\u2014or to craft facsimiles of\u2014spirits.\n\nChiefest among these aromatic compounds were essential oils. Essential oils are aroma incarnate. They are highly concentrated isolates of the chemicals that give each plant its characteristic smell. An older name, _quintessential oils_ , reinforces the point they are the most perfect, pure expression of the flowers, seeds, roots, rhizomes, barks, and other plant parts used to make them.\n\nAlthough truly bad, even dangerous, beverages stemmed from misuse of such formulas, they weren't _necessarily_ meant to be cheats. Even today, rectifiers generally make cr\u00e8me de menthe with peppermint oil because fresh leaves don't provide enough taste and aroma to make a strong, stable, long-lasting mint cordial.\n\nPrinted drinks formularies were once commonplace. As users altered or improved formulas to suit local tastes and available ingredients, such books as this 1910 _The Bottlers' Formulary_ could become heavily annotated. From the collection of the University of California, Davis.\n\n### **Gear**\n\nFor centuries, compounded spirits and cordials have been made using little more complicated than buckets and spoons. Some gear, though, does make the process easier, cleaner, and faster. From blenders to bottles, here are some pieces of kit to consider. What follows is just gear for making, flavoring, and storing alcohol, bitters, syrups, and tinctures. You better believe I put a still on the list. If you're looking for dashers, shakers, jiggers, and things like that, check out a good bartending manual, such as Jeffrey Morgenthaler's _The Bar Book_ (2014).\n\n**Glass Bottles or Mason Jars**\n\nGlass, not plastic, bottles are indispensable for storing finished wet goods.\n\n**Blender**\n\nA blender is useful for making some cordials and syrups. For 1:1 or 50% simple syrup (see here for more about syrups), add equal weights of sugar and water to the blender, fire it up to high speed, and stop when the sugar is dissolved; the air bubbles will dissipate. Several commercial essences that mimic popular brands of spirits and liqueurs call for pouring about an ounce of essence into a blender. Add a bottle of vodka and sugar, then blend until the sugar dissolves. Follow the manufacturer's instructions on those, but, as always, beware the fragrant taint (see here) and use glass blender pitchers; plastic can absorb the strong aromas of essential oils and contaminate everything you ever make in that blender again.\n\n**Spice or Coffee Grinder**\n\n**Spoons**\n\n**Bottle Brush**\n\n**Pipettes or Micropipettes**\n\nMany people who work with essential oils advise using eyedroppers to dole out precise doses of these concentrated compounds. In fact, lots of brands include droppers in the bottle design so users don't accidentally pour out too much of the bottle's precious contents. The trouble is that a \"drop\" is a squirrelly measurement. The size of a drop can vary depending on the viscosity of the liquid being dispensed, the width of a dropper's aperture, the force used to squeeze the bulb, and other factors. A glass or plastic pipette, on the other hand, measures absolute volume, usually in milliliters.\n\n**Measuring Cups and Spoons**\n\nGo with stainless steel or damage-resistant borosilicate (such as Pyrex). Plastic can be okay; just make sure it doesn't come into contact with undiluted essential oils or other strong-smelling compounds. Measuring cups sold for homes and commercial kitchens are super handy for measuring volumes. The thing is, some are off the mark. In a set I bought recently, 1,000 grams of water (which should take up 1,000 ml) measured 890 ml, 1,050 ml, and 960 ml in the variously sized cups _made bythe same manufacturer_. \"Close enough for government work,\" my father might joke, but not when you want precise measures. Either verify with a graduated cylinder or scale or invest in proper, calibrated flasks from a lab supply firm.\n\n**Funnel**\n\n**Strainers**\n\n**Filtering Medium**\n\n**Peeler**\n\n**Microplane**\n\n**Kitchen Scale**\n\nIt's the 21st century. Get a kitchen scale already. You won't need it for everything, but there's no substitute when you need 42 grams of this or 12 ounces of that. I have two. One is for measuring 5-gram intervals that I use for baking and making big batches of syrups. The other registers tenths of a gram and is perfect for weighing spices, herbs, and small amounts of other ingredients.\n\n### **Nice to Have**\n\nOnce you have the basic tools for compounding, a few others can make the job easier. These are not strictly necessary, but you'll be glad to have them.\n\n**Flasks**\n\nOne- or 2-liter flat-bottomed Erlenmeyer flasks are useful for blending small amounts of alcohol. Some recipes that call for oils and essences instruct readers to put them into a bottle and shake... and shake... and _shake_ until the oils finally dissolve from all the agitation. Ain't nobody got time for that. Dissolve the oils with high-proof alcohol in an Erlenmeyer flask, drop in a mag bar (see below), and top off with the amount of alcohol the recipe calls for. Set it on a stirrer (see following item) and let the magnets do all the agitation. Spend the extra buck or two to get a ground glass stopper that fits the flask to help contain the compounds you're agitating.\n\n**Electromagnetic Stirrer**\n\nLike the Erlenmeyer flask, an electromagnetic stirrer isn't absolutely necessary, but, man, it makes light work of blending liquids. Toggle the switch, set the speed, and a small motor sets magnets rotating under the metal plate at hundreds or even thousands of rpm. On some models, that plate heats as well, but don't use it; heat degrades essential oils. The concave bottoms on most wine, beer, and spirits bottles will throw the magnet to one side, where it will impotently spaz and bang against the glass; they are worthless here. Use only flat-bottomed flasks and bottles on a stirrer.\n\n**Mag Bar**\n\nIf you do use a stirrer, you will need a short, coated magnetic bar that spins inside the flask and gets the liquids moving. Fish it out afterward either with a fridge magnet (my method) or a coated magnetic stick (the preferred method).\n\n**Mortar and Pestle**\n\nBigger versions exist made of hollowed tree stumps and tree limbs rounded at one end for pounding manioc and whatnot, but the size you want is a countertop model: heavy stone or ceramic bowl (the mortar) with a shillelagh-looking little club that fits in one hand. Use that pestle to crush spices in the mortar or, as old-timers did, to grind essential oils with sugar and mix in high-proof alcohol to dissolve the mass.\n\n**Graduated Cylinder**\n\nA footed glass cylinder with marks up the side noting volume. Get one from a lab supply firm such as Cynmar.\n\n**B\u00fcchner Funnel**\n\nThis fancy little two-part contraption attaches by tubing to a water faucet; the resulting vacuum draws liquids through a filter paper, leaving behind the solids. Pricier than coffee filters or cheesecloth stuck in a funnel, but if you want crystal clear tinctures and bitters, they're the way to go.\n\n**Spirit Hydrometer**\n\nUseful for determining the alcoholic strength of your spirits if you make your own or cut high-proof spirits with water to drinking strength.\n\n**Brix Refractometer**\n\nA countertop Brix refractometer measures the sugar concentration in aqueous solutions, such as bar syrups.\n\n**Separatory Funnel**\n\nThis allows separation of materials such as fruit pulp and juice or, if you're distilling essential oils, hydrosol and oil.\n\n**A Still**\n\nProhibition, so we're told, was repealed in 1933. That's true as far as it goes, but the federal prohibition on home distilling remains ironclad. If you intend to make whiskey, rum, brandy, and other genuine spirits, then a still is essential. If, however, you plan to make cordials, liqueurs, and cocktails, you can get along just fine without one. Making intoxicating beverages without the use of a still was, in fact, the major selling point of most of those old compounders' manuals.\n\nArthur Burdett Frost's 1921 sketch, \"He made some hooch and tried it on the dog,\" poked fun at homemade liquor and a nation of scofflaws, many of whom were trying their hands at distilling and compounding for the first time.\n\n### **Alcohol**\n\nWhen buying $400 worth of liquor from the grocery store in the morning, as I do from time to time, I've learned the easiest answer to the inevitable question is \"Yes, I _am_ having a party.\" The truth is, a whole bunch of 80-, 100-, and 192-proof spirits went into testing recipes from Victor Lyon's notebook.\n\nDistilled spirits, whether brandy, whiskey, \"alcohol,\" or some other type are at the core of old recipes and formulas for cordials, bitters, tinctures, and compounded spirits. The reasons are threefold. First, we like the taste of intoxicating beverages and the effect of drinking them. Second, spirits are especially good at extracting and retaining flavors and aromas. Third, spirits\u2014but only high-proof ones\u2014can thoroughly dissolve (or \"kill\" in old recipes) essential oils. Lower-proof spirits, such as everyday vodka or whiskey, incompletely dissolve essential oils, leaving the resulting mix cloudy. For a review of \"proof\" and different ways to measure it, see here.\n\nHigh-proof, relatively neutral-tasting alcohol in old texts is sometimes as low 83% ethanol, but 90%\u201395% is more common (the rest is water and trace amounts of other compounds). It may also be called K\u00f6ln or Cologne; silent, neutral, or velvet spirit; or simply plain, deodorized, or absolute alcohol. That last one more properly refers to 100% ethanol, also called anhydrous or dehydrated alcohol (not because it's a dry powder, but because its water has been removed). Outside labs, it's uncommon because 100% alcohol is unstable; open a bottle and the proof starts dropping immediately as the liquid absorbs moisture from the very air. During Prohibition, this 90% to 95% spirit is the type of alcohol that bootleggers would have used to cut smuggled spirits or simply to fabricate factitious spirits using essential oils, ethers, and other aromatic compounds.\n\nThe introduction of grain or neutral spirit was never demanded by public taste; it was dictated solely and simply by economical motives. The time required for maturing genuine whisky is thus saved, and the practice of blending is carried to an enormous, if not appalling, extent, with handsome profits to the blenders.\n\n\u2014S. Archibald Vasey, _Guide to the Analysis of Potable Spirits_ (London: Bailliere, Tindall & Cox, 1904)\n\n### **Essential Oils: The Fifth Element**\n\nEssential oils, ubiquitous in old manuals and starting to creep into modern cocktail recipes, are aroma incarnate. Old texts sometimes call them _quintessential_ oils after _quintessence_ , a term derived from medieval Latin. Because they were something altogether different from Aristotle's four elements of earth, air, fire, and water, they were _quinta essentia_ , a fifth element, the very life force of plants. In less prosaic language, they are complex mixtures of volatile compounds isolated from whole plants or parts of them. Just about all plants are capable of producing essential oils, even if only in trace amounts, but not all of them are commercially useful. They are immensely popular today and widely available to consumers, partly because proponents of aromatherapy have driven retail demand.\n\nBut before we go any further, a bit of clarification; essential oils are not actual oils in the way that the fatty, heavier, so-called fixed oils, like those of olives, almonds, or avocados are. Essential oils are lighter, more ethereal, more fugitive, dissolve readily in alcohol, and are not at all greasy. An alternate name, _volatile oils_ , reminds us that they tend to evaporate, dissipating quickly at room temperature in the air. In general, essential oils smell pleasant, though a few are so strong and penetrating that they block out nearly every other smell. Clove, cassia, and rosemary come to mind. I particularly like cade oil for this reason; its intense, smoky, phenolic smell reminds me of a campfire still smoldering from the previous night. To keep its aroma from contaminating my entire office, though, I keep its bottle inside a somewhat larger jar. Oils or not, the old name has stuck, so that's what we'll use.\n\nScrape a sweet orange with your fingernail or squeeze a piece of peel. Its smell, unmistakable for any other in your kitchen or bar, will fill the space in front of you. That's because breaking the cell walls of its skin releases tiny watery geysers rich with orange oil into the air or, if you aim it that way, over the surface of your cocktail. That is one of the simplest essential oils, one that anybody who has ever peeled an orange will recognize. These oils are not uniform. Instead, they are mixes of compounds, often hundreds of them. The bulk of orange oil's weight, for instance, is a molecule called d-limonene; the rest is a blend of acids, aldehydes, and diesters that can vary by the type of orange as well as its origin. The result: essential oils culled from the orange trees in my California garden won't smell or taste the same as those harvested in Florida, south Louisiana, Sicily, Algeria, Tunisia, Morocco, Spain, Israel, Italy, or just about anywhere else oranges grow. Those who use essential oils for perfume, food, or beverages pay particular attention not just to how they were made, but to their place of origin.\n\n**Terpeneless Oils**\n\nRelatively flavorless compounds called terpenes comprise a large proportion of the bulk of essential oils. Because they are poorly soluble in weak alcohol, mixing them with lower-proof spirits results in cloudiness. Since the 19th century, so-called terpeneless oils, made by distilling essential oils with alcohol, have been available. Their advantage lies in readily dissolving in regular, everyday 40% abv vodka. Unfortunately, far fewer varieties of terpeneless oils are available to the public than regular essential oils. No matter: use high-proof alcohol to dissolve your oils whether they have their terpenes or not.\n\n**How Essential Oils Are Made**\n\nHistorically, producers have used two primary methods to extract essential oils. In **expression** or **cold pressing** citrus oils, producers squeeze citrus peels (or sponges that had been rubbed over them) in presses, then collect the oils, water, and other heavy liquids that ran off. Once these liquids settle, the portion that floats on top can be skimmed and may undergo further treatment such as filtering, rectification, or blending to become \"essential\" oils. Modern producers are more likely to use centrifuges which are less work and give higher yields.\n\nThe other method, today's standard approach, would have been familiar to importers and dealers in essential oils. It involves a device producers sometimes call an extractor, which any distiller would recognize immediately as a still. In a thumbnail, here's how **distillation** works for essential oils. A load of a single botanical\u2014anise seeds, cinnamon bark, ginger, juniper berries, whatever\u2014is loaded into a still which is then sealed and heated. The botanicals may be in direct contact with water inside the still's boiler, as mint leaves, pine needles, and rose petals often are; subjected to a mix of water and steam; or undergo the most common treatment, **steam distillation** , in which steam is injected directly into the still, creating vapors that are drawn off and condensed. From those, the essential oils are extracted. Essential oil yields and quality can be affected by variables that include the weather, growing conditions, time of harvest, still temperature, and skill of the operators. Yields range from a fraction of a percent for something like orange flowers to perhaps 7 percent for eucalyptus or caraway.\n\n**Hydrosols and Flower Waters**\n\nAfter extracting as much essential oil as feasible from botanicals, distillers don't necessarily throw away water left in the still. Whether they are byproducts of distillation or the end result, **hydrosols** , as such fragrant waters are known, retain many of the characteristics of whatever had been distilled. Orange flower and rose waters are \"leftovers\" from making neroli and attar (or otto) of rose. While hydrosols aren't central to modern drinking, I've had lovely Champagne cocktails with rose water and any Ramos Gin Fizz made around here gets a splash of orange flower water.\n\nLess frequently used procedures include a modern innovation called **hyper** (or **super) critical carbon dioxide extraction**. In this process, pressurized carbon dioxide pulls desirable compounds from botanicals at a lower temperature than steam distillation. Without such high heat, fewer chemical reactions take place. Fans of **CO** 2 **-extracted** oils find them to have a fuller spectrum of aromatic compounds. Some bartenders insist on these whenever possible. Last, in **enfleurage** repeated batches of flowers are either pressed with cold fat or mixed with warmed fat until the fat is saturated. That scented fat is then washed with high-proof ethanol which evaporates and leaves behind a highly concentrated absolute. But now we're wandering into the perfumers' pantry, so let's leave off and get back to drinks.\n\n**To Dissolve Essential Oils with Alcohol**\n\nOverproof spirits, such as 50% abv vodka or whiskey, are more effective at dissolving essential oils than are the everyday 40% varieties that leave such mixtures cloudy. Still, it can be done better. To dissolve essential oils with alcohol thoroughly, pour at least twice the volume of neutral spirits (90%\u201395% abv) as the volume of oils into a glass bottle or flask. Add the oil(s). Either close the top and shake or spin the liquids on a magnetic stirrer. Stop when the mix is clear and limpid. If it remains clouded, add more spirits and repeat until it clears.\n\n**A Bit of a Disclaimer**\n\nTo hear some in homeopathic medicine and aromatherapy crowds, essential oils can cure everything from stretch marks to Ebola. They have the power to calm and soothe anxieties or, alternately, to excite you and push you to the very limit of sexual stamina. They will, I have been assured, protect one from measles, eradicate acne, eliminate eczema, assuage chest and belly complaints, rid children of intestinal worms, and \"promote healing.\" How do we untangle bogus assertions and dangerous practices from those that have varying degrees of support among different schools of healing? Simple: we don't. Not here. Look, here's the deal. I'm not a physician. I'm not a homeopath, allopath, osteopath, or any other kind of healer. I'm an historian. When I write about medicine, it's as an historian, not a practitioner. This book doesn't promote the healing properties of anything, much less essential oils.\n\nRecently, an Ebola outbreak that began in Africa jumped the ocean. When the deadly virus turned up in American cities, people freaked right the hell out. For a while, a claim that cinnamon oil would cure the disease was going around. The Food and Drug Administration shut down that nonsense, but the fact remains that people ascribe to essential oils all sorts of near-miraculous powers. This is not the book to explore those possibilities. It doesn't matter that, in my personal experience, ginger seems to settle a rebellious belly or that I find the aromas of certain mint oils relaxing. I'm not a doctor, so I make no health claims at all in this book about essential oils, herbs, spices, or any of that lot. Zero. They can smell good, they can be relaxing, they can help make some tasty beverages, but will essential oils cure gout and revive faded hair follicles? It's not for me to say. If you want to learn more about that sort of thing, look elsewhere.\n\nIn fact, if you look at websites devoted to understanding or selling essential oils, you're likely to see some language along the lines that humans should never, ever, in any circumstances consume essential oils.\n\nGiven the changing landscape and understanding of what is safe and what is not, I would refer readers to the International Fragrance Association, which has evolving standards on its website that explains which essential oils are **GRAS (generally recognized as safe** , see here), which should be avoided, and which fall into some middle space\u2014perhaps safe to consume, but may cause photosensitivity or they are safe for most people, but not in those who are allergic to the substance. Use common sense, but also tap the resources in this book to research on your own the current understanding of what is safe.\n\nThat having been said, allow me a few basic words of safety when using essential oils.\n\n\u2022 Read and follow all label cautions and warnings.\n\n\u2022 Keep away from babies, children, and pets.\n\n\u2022 Do not allow undiluted essential oils to contact skin or eyes.\n\n\u2022 Never taste or consume undiluted essential oils or put them on your tongue.\n\n\u2022 If ingested, call the American Association of Poison Control Centers (800-222-1222) immediately.\n\n\u2022 Essential oils are flammable; do not use them near open flames.\n\n\u2022 If you have any serious medical condition or are pregnant, consult your physician before using essential oils.\n\n\u2022 For characteristics, warnings, contraindications, or other considerations of any essential oil, ask the vendor or look online for its Material Safety Data Sheet (MSDS).\n\n### **On the Toxicity of Old-School Ingredients**\n\nMany of the ingredients in Lyon's notebook and other old recipes are more or less safe for use at home. Some fall into the category generally recognized as safe (GRAS) used by the United States Food and Drug Administration (FDA) to designate food additives that, after decades or even centuries of regular use, show no particular harm to users. Yes, nutmeg may cause intense cramps and hallucinations, but not in the small quantities you're likely to use in a cake or dusted on the top of a drink. It is, as they say, generally recognized as safe.\n\nAnyone who considers re-creating or tweaking old recipes, whether they're from this book or other sources, needs to understand that physicians, botanists, chemists, and others in the know have realized that more than a few ingredients in those recipes should be used with caution or not at all. Whether because of its carcinogenic properties or because of its role as a precursor for making the festival drug ecstasy, for instance, sassafras oil is tightly restricted in the United States. If you want it for root beers or syrups, a safrole-free extract is available that sidesteps both of those concerns. Likewise, the traditional botanicals Virginia snakeroot and calamus root are banned.\n\nNostalgia aside, not everything in the old days was wholesome. Some old-timey ingredients are more trouble than they're worth while others are downright deadly. Methanol, for instance, has gone under a variety of names since the 19th century, but no matter what you call it, the clear, vodkalike alcohol is a potent toxin that can sicken, blind, and kill humans; it should _never_ be consumed as a beverage or served to anyone. A professional distiller, on the other hand, or a food chemist might have a reason to use sulfuric acid safely in the making of drinks, but you at home or in your bar? Unless you have the training, know _exactly_ what you're doing, and take appropriate caution, skip it. Read those formulas the way you might a letter from your great-great-grandmother trying to maintain her dignity on a cross-ocean steamer\u2014and maybe there's a serial killer on board: a revealing peek into the past, but it's not necessary, or even wise, to relive her travails.\n\n**GRAS: Generally Recognized as Safe**\n\nJust because it's natural doesn't mean it's good to put in your mouth. Arsenic, belladonna, and cyanide are as natural as they come. Do you really want them in your cocktail? Others\u2014say, caffeine\u2014are relatively innocuous up to a point. Then there are food additives that have been around so long and used by so many with little ill effect that the FDA regards as them as GRAS. Some are granted GRAS status after scientific review. Rather than rely strictly on printed books (even this one) that become obsolete, for the last word on which oils or compounds are safe to put in food and drinks, check the latest information from the FDA as well as GRAS lists from the Flavor & Extract Manufacturers Association and the International Organization of the Flavor Industry's Global Reference List. See \"Organizations and Useful Websites,\" here.\n\n**Even the Adulterants Were Adulterated**\n\nBecause essential oils are both expensive and liquid, they have been easy to adulterate and a target for adulteration since the earliest days of their production. Just after Prohibition, E. W. Bovill quipped in a British journal that \"a large proportion of the oil sold as American Peppermint Oil has never seen either America or the true peppermint plant, and the world consumes vastly more Otto of Rose then is ever distilled from rose petals.\"\n\nAdulteration, in fact, seems to have been the rule rather than the exception in the 19th and early 20th centuries. Pharmacopoeias of the time are rife with the exact characteristics (or at least their normal range) of essential oils and how to detect fraud, often with specific adulterants. Oils labeled \"commercial grade,\" \"second grade,\" and \"imitation\" hint at this, but even \"pure\" oils were not always... well, pure. Oil of turpentine and alcohol itself were frequently added to bulk up essential oils, sometimes to make up as much as 90 percent of the volume of oils they purported to be.\n\nA handwritten notebook at the University of Wisconsin offers a rare look into how the sorts of essential oils used in cordials and spirits were adulterated at the retail level in the early 20th century. Its oil of wormwood was half turpentine. Neroli, supposedly distilled from orange flowers, was half petit grain, a less costly oil distilled from immature fruit or leaves and stems of orange trees. Oil of cumin was one-third alcohol. Oil of juniper berries? One part oil of juniper berries, two parts turpentine. \"Pure\" oil of wintergreen was entirely synthetic and boosted with camphor, turpentine, and iron. Sweet almond oil was only two-thirds that; the rest was sesame oil. Oil of spearmint? One-quarter turpentine. You'd be excused for cocking an eyebrow at the claim when packages arrived at the pharmacy from wholesalers that they were at all pure in the first place.\n\nThe adulterated oils of the past are, thankfully, largely past. The takeaway from this, though, is that cordial and compounded spirits recipes from old books may work with modern oils\u2014or they may not. It's not that the _recipes_ are necessarily bad, but that the _ingredients_ may have been; users developed formulas based on what was available. When reading old recipes, then, don't take their measurements at face value. Rather, gauge their relative amounts and work up new formulas using them as guides, not diktats.\n\nTurpentine was a once common whiskey, gin, and essential oil adulterant. Nasty on its own, in small amounts and perhaps mixed with tea and a bit of barrel char, says cocktail historian Ted \"Dr. Cocktail\" Haigh, \"It's closer than you would ever, ever want to admit to yourself to the flavor of whiskey.\"\n\n### **Beware the Fragrant Taint**\n\nAlcohol's ability to extract and retain flavors and aromas makes it a fantastic base for drinks. The downside is that alcohol, like butter, milk, and eggs, can suck up ambient smells you'd rather it didn't. A constant refrain from dry Prohibition agents was that much of the illicit hooch of the 1920s and '30s was made in filthy settings. How could it be wholesome liquor when hogs were slopped nearby, when discarded tins and sugar sacks were strewn about the distillery floor, when stills were so unclean that they were dubbed \"black\" pots?\n\nWhen you're compounding beverages, work in a clean, well-organized space. Avoid doing anything that raises a stink. I'd even argue not to use perfume, cologne, or scented soap that day, but others might think I'm being a tightass. In any event, don't slice onions, cook cabbage soup, mix tuna salad, snack on kimchi, or whip up a batch of aioli. Likewise, don't clean on days you mix. Hot water is fine, but just the ambient smell of cleaning compounds can taint a batch of spirits. Once your anisette or ginger brandy smells of fish sauce or scrubbing powder, you're going to have to get a whole lot more creative in figuring out what to do with it (other than dumping it).\n\nJust as alcohol absorbs aromas, essential oils can leave them where they're not wanted. Wooden spoons and plastic utensils, bottles, container liners, and blenders are particularly prone to picking up odors after direct contact with essential oils. Unless the tool is disposable, such a pipette, use glass, porcelain, or stainless-steel gear instead for mixing and blending. Don't use bottles or other containers that smell even slightly of what they used to hold.\n\n### **Essence**\n\n_Essence_ is a slippery word when it comes to beverages. Because its meaning shifts with users, it may refer to essential oils, extracts of botanicals in alcohol, or essential oils or other aromatic compounds dissolved in alcohol. In his 1910 _Manual for the Essence Industry_ , Eric Walter explains the confusion:\n\nThe medium for transferring flavors is called in general \"essence.\" This word is derived from the Latin \"esse,\" to be, i.e., it is, or will be, the essential part of something. In the former case the term \"essence\" refers to the source; in the latter, the purpose for which it is used. Sometimes both interpretations agree, as for instance a raspberry essence will be made from raspberry fruit and is to be employed for imparting a raspberry taste or flavor to any substance. However, many essences bear the name only of the product which is to be made, and the essence, which may be synthetically made, bears no relation whatever to the natural fruit product, as for instance in the ease of artificial raspberry essence; then again, rum essence is not made from rum, although it is employed for making rum-like substitutes. [New York: John Wiley and Sons (1910)]\n\nSome home distillers, and those who distill neutral spirits in particular, rely on commercial essences to emulate a huge array of spirits. I've never had good whiskeys, brandies, or rums from such essences, but passable cordials and some decent gins can be made from little more than a bottle of vodka, an ounce or so of essence, and perhaps some sugar.\n\n### **Buying Essential Oils**\n\nAlways buy essential oils from reputable suppliers. Research them online and ask friends and colleagues for recommendations for vendors. A few things to keep in mind when shopping for oils to be used in beverages:\n\n\u2022 Look for essential oils designated \"food safe.\"\n\n\u2022 Local and common names for plants vary considerably; buy from reputable vendors who can provide the Latin denomination of their oils to assure you're getting the product you intend.\n\n\u2022 Avoid products labeled \"fragrance oil\" or \"perfume oil\" which are likely made from something other than the supposed fragrance. Instead look for \"pure essential oil\" or \"100% essential oil.\"\n\n\u2022 Buy oils in dark glass (darkness protects against sunlight damage and undiluted oils can dissolve rubber and plastic) or lined aluminum bottles.\n\n\u2022 Compare prices. The most expensive may not be the best, but reputable vendors do have different suppliers, so prices vary. If you have questions, call: most suppliers are happy to answer questions.\n\n### **Storing Essential Oils**\n\n\u2022 Store all essential oils in a cool, dark place. The fridge is fine, but do not freeze them. At the least, keep them out of direct sunlight. Keep them away from areas with variable temperatures, such as next to appliances, outer walls, or windows.\n\n\u2022 Citrus oils in particular tend to degrade after six months or so. Buy them in the quantities you need and store them in the refrigerator to extend their usefulness.\n\n\u2022 The shelf lives of volatile oils vary. Check with the vendor for expiry information.\n\n\u2022 If the oils arrive in bottles with rubber droppers, switch the lids; oils may dissolve rubber over time.\n\n\u2022 Store oil in dark bottles (amber or cobalt). Even better, use bottles with UV protective coating or lined aluminum.\n\n\u2022 A few oils may cloud or solidify when cold. If this happens, hold the bottle in your hands for a few minutes; your body warmth should liquefy the mass.\n\n\u2022 Always return the cap as soon as you've extracted the amount of oil you need and make sure it's screwed on tightly.\n\n### **The Pearson Square and Cider Oil**\n\nSpend enough time around hobbyist distillers, winemakers, and certain bartenders and you are bound to hear talk of the Pearson Square. No, it's not some disreputable plaza where they all hang out and swap their homemade wares (though that does sound like the kind of place I'd like to whittle away an afternoon). The Pearson Square is a simple mathematical model used to determine the ratio of ingredients to combine in order to reach a particular final concentration. People who make wine and cordials use it to determine how much of one low-proof liquid should be combined with a higher-proof liquid to reach some point between the two strengths. There are no complicated algorithms to figure; it's basic arithmetic.\n\nSo, how's it work? Well, picture the Pearson Square as a great big circle. Nah, I'm jerking your chain; it's a square. Make it as big or small as you like. It measures proportions, so the units can be anything that holds liquid: pints, liters, gallons, buckets, jars, whatever. Its four corners are indicated by the letters A (upper left), B (lower left), D (upper right), and E (lower right). The center, where lines drawn from A to E and from B to D, is labeled C. Disregard any negative numbers obtained so that -13, for instance, is merely 13.\n\nA = the alcohol content of the stronger spirit\n\nB = the alcohol content of the lower-proof or nonalcoholic liquid\n\nC = your target alcoholic content, the final proof you want\n\nD = (B minus C), the proportion of stronger spirit you need\n\nE = (A minus C), the proportion of lower-proof liquid you need\n\nFor instance, you want to mingle a batch of hard, low-proof apple cider with apple brandy to make a homemade Cider Oil (here). How much of each do you use if you want a specific final proof? The Pearson Square lays it out.\n\nSay you want your cider oil to be 18% alcohol (C). Your brandy is 50% alcohol (A) and your cider is 5% (B). The proportion or parts of brandy you need (D) is (5 minus 18) = \u201313 (disregard negative values, so this is 13) and the proportion of cider you need (E) is (50 minus 18) = 32. If you add 13 parts of 50% abv apple brandy to 32 parts of 5% hard cider, you'll end up with 45 parts of 18% cider. Remember: \"Parts\" can be milliliters, ounces, yogurt containers (clean ones, please), or other measures you find convenient. A 750 ml bottle of bonded apple brandy will yield a little over 2.5 liters of cider oil, plenty enough to get your wiggle on.\n\nSee here\u2013here for Victor Lyon's cider-based recipes.\n\n## **_Chapter three_**\n\n## Gin, Whiskey, and Rum\n\n**_These are the acts of someone calculating the logistics_**\n\nVictor Lyon's notebook doesn't contain obvious recipes or formulas for moonshine, but that's hardly unusual. Few written recipes for making moonshine survive from those years. Such recipes were part of oral traditions and tended not to be written anyway. What we tend to see instead in historic collections\u2014when there's anything written at all\u2014are ledgers and penciled calculations of profit and loss for making alcohol: grains, yeast, sugar, sorghum, or molasses, supplies such as bottles and corks. We can infer recipes and yields from those. Occasionally, such files list customers by name, address, and how much liquor they bought and their debts. If Lyon kept such explicit records, they are lost to us. In fact, it's not clear from the notebook that the Harlem physician sold liquor at all. No mug shots, no arrest reports, and no articles have surfaced to link him to bootlegging.\n\nBut there are hints that suggest perhaps he both made and sold contraband liquor. In the back of the notebook are price lists for essential oils and contact information for firms that sold supplies for making and bottling alcohol. Here and there, he makes adjustments to formulas, and figures the cost of each bottle of various compounded spirits. Idle curiosity? Perhaps. At the very least, though, these are the acts of someone calculating the logistics of clandestine liquor for sale.\n\nIn this chapter, we'll look at some of the notebook's entries on gin, whiskey, and rum, along with others such as arrack, that would have been available on the black market in Prohibition-era New York.\n\n### **_Gin_**\n\nOne should always age homemade gin, goes a bit of Prohibition-era snark, until it cools to at least room temperature.\n\nOne of the reasons gin was so popular during Prohibition is that it was relatively easy to make with home-distilled spirits or, more commonly, redistilled and scrubbed denatured industrial alcohol. With a formula and good ingredients in hand, drinkers could make what we call a compounded gin now: a blend of high-proof spirits, essential oils or extracts, and water, maybe with a touch of syrup or glycerin to smooth out rough edges.\n\nOn repeal, most drinkers ditched counterfeit whiskeys and other ersatz spirits as soon as they could get genuine wet goods. Not homemade gin. In fact, New York dealers said that after the 21st Amendment ended Prohibition, sales of neutral spirits showed no decline at all. Today, so-called bathtub gin stands as shorthand for the horrors of Prohibition-era drinking, the nadir of beverage arts in America, the lowest point in our collective drinking shame. But what was that stuff? Bartenders' folklore abounds with anecdotes about old-timers using oil of juniper berries to make bathtub gin, named supposedly for the unsanitary conditions in which its detractors claimed it was made.\n\nI suspect we've been calling it bathtub gin for the wrong reason. The common assumption is that such gins were actually mixed in bathtubs, the alcohol commingling with soap scum, loose hairs, rusty fixtures, and sloughed dead skin\u2014which would be super nasty. I'm not saying that _never_ happened, but old-timers I interviewed who mixed alcohol and flavorings with their parents described using large glass demijohns and jugs called carboys to mix gin. When they added water to the mixture to bring its 90% abv or so down to drinking proof, those oversized jugs did not fit under a kitchen sink faucet. Tilted to one side, though, the jugs could be topped off readily using the bathtub's tap. As someone who has fermented, distilled, blended, and shifted a lot of alcohol, that seems a much more reasonable explanation for the term.\n\nEither way, we rarely get the opportunity to examine what such concoctions actually, truly were.\n\nHere is our chance.\n\n**Types of Gin**\n\nThe US Alcohol and Tobacco Tax and Trade Bureau (TTB for short) recognizes three distinct types of spirits within the broader class of gin: distilled, redistilled, and compounded. All of them are at least 40% alcohol and flavored primarily by juniper berries.\n\n**Distilled gin**\n\nGin produced by original distillation from mash with or over juniper berries and other aromatics or their extracts, essences, or flavors\n\n**Redistilled gin**\n\nGin produced by redistillation of distilled spirits with or over juniper berries and other aromatics or their extracts, essences, or flavors\n\n**Compounded gin**\n\nGin produced by mixing neutral spirits with juniper berries and other aromatics or their extracts, essences, or flavors\n\nLyon's gin recipes don't encompass all the official modern permutations. There is, for instance, none that start with a low-alcohol mash run through a still. Rather, most are compounded. A few are redistilled. One British recipe is something of a hybrid, combining corn (i.e., grain) spirits with essential oils and whole cardamom, a gin botanical enjoying some vogue with newer American distilleries now.\n\nThe notebook also includes genever and cordial gins. **Cordial gins** are sweetened and sometimes made with additional flavors. **Sloe gin** , made by infusing purplish-black sloes, is one example. Its dominant flavor comes from an infusion of small, astringent, plumlike fruits of _Prunus spinosa_ , a thorny hedgerow plant also called blackthorn. Plymouth and the Bitter Truth sloe gins are available in the United States. Venerable **Old Tom** is more nebulous. When it was sweetened, Old Tom was cordial gin, but even distillers who make it now disagree on just how sweet it was and whether it should be aged. In practice and over time, variation in the style made it as diverse as modern gins.\n\nAnd **genever**? Why, that gets a section of its own.\n\n### **Gin Cocktails**\n\nBlack market gins were especially easy to source during Prohibition because they were so easily made. When the blender wasn't particularly skilled or the base alcohol imperfectly redistilled to remove denaturants, mixing sweet and bitter ingredients in gin cocktails helped\u2014more or less\u2014to buff out rough edges. If re-creating the notebook's gin recipes somehow doesn't make it to your to-do list, you can still satisfy a taste for Volstead-era gin cocktails with some of these using modern gins.\n\n### DUBONNET COCKTAIL\n\n**Gin shows up again and again in Prohibition-era cocktails because it was an easy spirit to make for nonchemists; just add juniper oil to a batch of high-proof spirit, water it down, and as soon as you can say, \"Bob's your uncle,\" you've got a batch of basic gin. Mixing it with Dubonnet, a mildly bitter aperitif, rounds its rough edges.**\n\nCombine equal parts gin and Dubonnet (with a dash of orange bitters if you like) in a mixing glass with ice. Stir until chilled, then strain into a cocktail glass.\n\n### THE LAST WORD COCKTAIL\n\n**The word is that the Last Word cocktail was invented by bartender Frank Fogarty at the Detroit Athletic Club in 1922. It fell between the cracks of history, however, until Seattle bartender Murray Stenson rediscovered it about a decade ago in Ted Saucier's 1951 bartenders' guide _Bottoms Up!_ and became something of an evangelist for this exquisite drink. We have been guilty, on particularly boisterous evenings, of mixing them by the quart.**\n\n0.75 oz gin\n\n0.75 oz green Chartreuse\n\n0.75 oz maraschino liqueur\n\n0.75 oz fresh lime juice\n\nShake the ingredients with ice and strain into a chilled cocktail glass. Garnish with a lemon twist.\n\n### LUCIEN GAUDIN\n\n**Fencing is my all-time, hands-down, favorite sport. And I've scars to show for my love of it. Seriously, kids: wear a mask and glove when you take up a saber. These days, I'm more likely to take up a shaker, though, and mix this bittersweet cocktail named after a 1920s Olympic fencer. The Lucien Gaudin is kissing cousin to that darling of the American cocktail scene, the Negroni. Gaudin brought home two silvers and two gold medals. I'll settle for just one of these.**\n\n1 oz gin\n\n0.5 oz Campari\n\n0.5 oz Cointreau\n\n0.5 oz dry vermouth\n\nStir well in a cocktail shaker with ice. Strain into a cocktail glass or a coupe. Garnish, if you like, with a flamed orange peel.\n\n### SOUTHSIDE\n\n**A Prohibition keeper that doesn't get enough play in modern bars, the Southside blends gin with lemon juice and mint for a drink that's kissing cousin to the more familiar mojito. It's like a minty gin lemonade. And who doesn't like lemonade? Give it a splash of soda to lighten the drink, if you like, and a dash or two of aromatic bitters, but really, it's fine as is.**\n\n2 oz gin\n\n6 to 8 fresh mint leaves\n\n1 oz fresh lemon juice\n\n0.75 oz simple syrup (here)\n\nSplash of soda (optional)\n\nShake the ingredients with ice in a shaker. Strain into a cocktail glass or coupe.\n\n### BEE'S KNEES\n\n**Another recipe sweetened to take that bathtub edge off Prohibition-era gins. Because honey syrup blends especially well with many rums, consider swapping out the gin with a white _rhum agricole_ here.**\n\n2 oz dry gin\n\n1 oz honey syrup (half honey, half water, heated briefly until uniform, then cooled)\n\n1 oz fresh lemon juice\n\nShake the ingredients with ice and strain into a chilled old-fashioned glass with fresh ice.\n\n### **Genever**\n\nIn the spring of 1922, a shipment of lamps from Germany arrived by steamer in Philadelphia. Customs appraisers found nothing out of the ordinary until one thought they seemed a bit, well, _heavy_. Upon opening one, he discovered a quart of \"Holland\" gin inside. Right then and there, the consignee lost the entire lot. \"Unless Uncle Sam gets his hands on all shipments,\" noted a _Central Press_ article, \"German lampstands promise to become exceedingly popular in the land made famous by Mr. Volstead.\"\n\nThe gin priming those lamps was not the familiar London dry style so popular today. It was an older, malted style also called genever, jenever, Hollands, genevieve, Schiedam (after the Dutch distilling center), and (incorrectly) Dutch gin. Since colonial days, Americans knew it as a Dutch specialty, though it is also made in Belgium and parts of Germany.\n\nAmerican distillers made semblances of genevers since at least 1809 when Pennsylvania distiller Samuel M'Harry published a recipe \"How to make a resemblance of Holland Gin out of Rye Whiskey\" in his _Practical Distiller_. Chief Gowanus New Netherland Gin from New York Distilling Company is a modern re-creation of M'Harry's recipe and worth seeking out. Confusingly, early bartending guides often refer to it simply as \"gin.\" Using London dry gin, such as Beefeater or Tanqueray, makes some of those old gin drinks weird, unbalanced creations. What you want is malty, juniper-forward genever. Use it as you would whiskey.\n\nA slight error in transcription would make the two recipes on here seem to be for English gin, but the original recipes are, in fact, for _English-made_ genever, or Holland gin. They both come from the 1902 edition of _The Brewer, Distiller, and Wine Manufacturer_ , a British guide.\n\nThe bones of all proper gins and genevers are surely built from juniper, a tree that grows throughout Europe, north Africa, Asia, and parts of North America. Of the hundreds of varieties of juniper, the berry-like cones of _Juniperus communis_ are most prized for the resinous, bracing aroma and flavor they bring. In North America, a taller related species called Western or Sierra juniper _(J. occidentalis)_ bears a larger, bluer berry that is sometimes used in newer \"Western style\" gins, many of which contain nontraditional and hyper-local botanicals.\n\nJust as they are now, juniper berries were easy enough to obtain from wholesale spice merchants. As always quality was key, and not just any berry would do. I asked David T. Smith, British coauthor of _The Craft of Gin_ and irrepressible gin enthusiast, why the notebook specifies that juniper berries should be at least a year old for English genever. Smith holds that _Juniperus communis_ berries actually need a full two seasons of growth, rather than one as the notebook specifies. Only then do the little blue-black balls have a great enough concentration of essential oils for use in gin.\n\n### ALAMAGOOZLUM\n\n**Alamawhoozlum? Alamagoozlum. It's a mouthful, and not just because the recipe\u2014from Charles H. Baker Jr.'s 1939 _The Gentleman's Companion_ \u2014makes about a 10-ounce drink. A blend of herbal Chartreuse, malty genever, and a hefty dose of bitters, it's one you can share with a friend. Or two. Baker credits its invention to financier J. Pierpont Morgan, but in 1873 the Pelican Saloon in Los Angeles advertised drinks that included the Alamagoosier. Was it the same drink? Without a recipe, it's hard to say, though that _Bris Around the Corner_ cocktail in the same ad gives me pause...**\n\n2 oz genever\n\n2 oz plain water\n\n1.5 oz Jamaican rum\n\n1.5 oz Chartreuse (green is better here than yellow, but it's your call)\n\n1.5 oz gomme syrup (here)\n\n0.5 oz cura\u00e7ao\n\n0.5 oz Angostura bitters\n\n\u00bd egg white\n\nShake all the ingredients without ice for 10 seconds to emulsify the egg white. Add ice, shake again until shilled, then strain into a large coupe (or two smaller ones).\n\n### THE COLLINS\n\n**The Collins\u2014Tom, John, or even Juan Collins, depending on who's doing the mixing\u2014is an early 19th-century precursor to the French 75, using soda water rather than Champagne. Perfectly traditional with London dry or the lightly sweetened Old Tom gins, it shines with genever.**\n\n2 oz genever\n\n1 oz fresh lemon juice\n\n0.5 oz simple syrup (here)\n\nSoda water, quantum satis\n\nShake the ingredients (except the soda water) with ice, strain into an ice-filled Collins glass, and top with soda water. Give it a brief stir, take a sip, and smile.\n\n### DEATH IN THE GULF STREAM\n\n**A stiff, bitter concoction from Charles H. Baker Jr.'s 1939 _The Gentleman's Companion_. Baker scored the recipe from his friend, famed author and boozehound Ernest Hemingway. Like the classic mai tai or a proper Cuba Libre, this one gets a boost of lime oil from the fruit's shell.**\n\nTake a tall thin water tumbler and fill it with finely cracked ice. Lace this broken debris with 4 good purple splashes of Angostura, add the juice and crushed peel of 1 green lime, and fill the glass almost full with Holland gin... No sugar, no fancying. It's strong, it's bitter\u2014but so is English ale strong and bitter, in many cases. We don't add sugar to ale, and we don't need sugar in a Death in the Gulf Stream\u2014or at least not more than 1 tsp. Its tartness and its bitterness are its chief charm. It is reviving and refreshing; cools the blood and inspires renewed interest in food, companions and life.\n\nOh, go ahead and put a smidge of simple syrup (here) in it if the thing's too harsh for your tastes. Not enough to make it sweet, just to temper the high notes.\n\n### **_Rum_**\n\nOf the 300-plus entries in Victor Lyon's notebook, not one includes directions for making actual rum, for starting with molasses or sugarcane, then fermenting, distilling, aging, and blending it. Every single one of his recipes involving rum either starts with that spirit as the base for making something else or is complete artifice. The thing is, Lyon wasn't some outlier compounding these sketchy fictions nobody had ever drunk or even heard of. During Prohibition, wealthy clientele often thought that while hoi polloi got cut, watered down, and faked spirits, _they_ were getting the Real McCoy\u2014but swells sometimes bought counterfeit wet goods the same as John Q. Public did; they just paid more for it. Lyon's recipes fall firmly into an age-old tradition among rum merchants for deception. They reflect historical rums more accurately than many distillers would have us believe.\n\nMost of the recipes from the notebook are for rum essences. Such essences were mixtures of alcohol and aromatic substances that, when added in small doses to plain spirits or blends of actual rum with spirits gave the general aroma and taste of aged rum. Such additions might have included vanilla, tea, caramel, sweet spirits of nitre, pyroligneous acid, ethyl butyrate, soot, sulfuric acid, various essential oils, gall tincture, and extracts of saffron. But, then, what constituted \"actual\" rum in the first place was an uncertain gamble.\n\nAccording to New Orleans rum authority and collector Stephen Remsberg, Prohibition coincided with the beginning of the sale by distillers of their rum in a bottle with a proprietary label. \"Before World War I,\" he wrote to me, \"rum was shipped in the barrel. A tavern or grocer would buy a barrel and cases of empty glass bottles. The customer would buy or pay a deposit on the bottle, and the shop would fill it. If the merchant had a high-end trade, he would print a label with the customer's name. I have a few such bottles. In most cases, I doubt the customer ever knew the name of the distiller of the rum. He would tell the shop that he wanted 12 bottles of that Jamaican 'Wedderburn' rum he had bought in the past, identifying only the category of the rum on the London market. Under those circumstances who would know what 'genuine' was. You depended on the reputation of the local merchant.\"\n\nAnd those merchants may not have been dealing with genuine rums even when they were unloaded in America, usually by way of London's rum market. At the dawn of Prohibition, American physician and pure food campaigner Harvey W. Wiley bemoaned the sad state of affairs. \"As in the case of whiskey and brandy, rum has been subjected to all kinds of adulteration. So great has become the adulteration of rum shipped to England from Jamaica that for every barrel of rum sent 6 barrels were sold.\"\n\nRum was not only widely adulterated, the spirit was itself an adulterant. By 1924, local rum could be had for as little as forty cents a gallon in Havana. One journalist writing from Cuba reported that a gallon of such rum mixed with a quart of Scotch yielded five quarts of Scotch. \"It may taste a little raw,\" he wrote, \"but the kick, in fact a good healthy wallop, is there, so why should anyone object?\"\n\nWhy indeed?\n\n### RUMESSENZ\n\n**This rum essence wasn't meant to be consumed _as_ rum; it was intended to give rumlike aroma and flavor to plain spirits. Wholesalers, saloonkeepers, blenders, importers, exporters, bottlers, and others who handled bulk rum would use such essences for three main purposes: to fake rum completely from plain spirits, to reintroduce familiar rum smells and tastes to a batch that had been cut with plain alcohol, or to boost the appeal of rum that was subpar\u2014perhaps because it had been already so massaged by previous handlers.**\n\n**Through its history, rum has been a spectacularly adulterated and counterfeited spirit. Modern brands still regularly include ingredients to augment body and color that are forbidden in spirits such as straight whiskey. Chief among them is caramel for just the right color to suggest greater age than the spirit may actually have. In Lyon's recipe following, caramel is mixed with vanilla, cassia buds, salt, and birch tar. My advice? Just buy some nice rum. Great examples can be had for significantly less than French brandies.**\n\n---\n\nVanillae concisae (cut vanilla) | 2.5\n\nOlei Rusci (birch tar) | 10.0\n\nRadicis Tormentillae cincisae (cut tormentil root) | 20.0\n\nKorum cassiae contusurune (bruised cassia buds) | 2.5\n\nFulginis splendentis pulveratae (powdered soot) | 15.0\n\nSalis culinaris (culinary salt) | 25.0\n\nAetheris Formici (formic ether) | 100.0\n\nSpiritus Aetheris Nitrosi (nitric ether) | 15.0\n\nSpiritus (90%) (90% abv neutral spirit) | 500.0\n\nTincturae Sacchari Tosti (caramel) | 50.0\n\nLet macerate for 8 days\u2014then filter. 15-20 grams are combined with 1 liter of neutral spirits at 53-55% (the original calls for \"Weingeist,\" spirit of wine).\n\nRum essences formulas were wide-ranging, though they often include butyric and formic ethers for fruity, pineapple-like aromas; caramel, vanilla, and tea to imitate barrel-aging; and cassia to impart warmth. Recipes abound because types of rum did; Caribbean rums, for instance, may vary depending on whether the territories from which they hailed were colonized by French, Spanish, or British forces at different points in history with different distilling technologies. Some essence formulas evolved as understanding of ingredients\u2014or quality of them\u2014improved. Finally, many of them were just different paths to get to the same result. Or close enough.\n\n### LOW GRADE ARTIFICIAL RUM\n\n---\n\nPlain spirits | 40 gallons\n\nRum | 5 gallons\n\nPrune juice | \u00bd gallon\n\nCaramel | 12 ounces\n\nRum essence | 8 ounces\n\n**Saffron**\n\nSaffron, one of the world's most expensive spices, is the stigmata of a crocus flower that must be harvested by hand. Saffron has a rich, musky, floral scent and can impart varying shades of yellow to curries, cordials, and some vermouths. Its use as a colorant in spirits goes back at least 400 years. Old rum and whiskey recipes in particular rely on saffron for warmth, color, and a tinge of unearned age.\n\n### **Rum Cocktails**\n\nLyon's recipes for rum shrubs are about as close as he gets to a cocktail in his entire notebook\u2014and they are among my favorites. Until recently, shrubs were a nearly extinct line of beverages that once enjoyed immense popularity. In his excellent book _Shrubs_ , Michael Dietsch focuses on vinegar-based examples of the fruit juice drinks (the name derives from the Arabic _sharab_ , \"to drink\"). As bracing and refreshing as those are, a second mellow type of shrub without vinegar but made from fruit juice, sugar, and brandy or rum has been knocking around for centuries. Use it in punches or as you might cura\u00e7ao.\n\n### RUM SHRUB\n\n**Lyon's two shrubs are among my favorite recipes in the notebook, each lifted from Joseph Fleischman's 1885 compounding manual. Both are good over ice (maybe with a bit more rum or iced tea for an impromptu punch) or as a cordial, and I've used them in margaritas to good effect. For each, Puerto Rican 151 is fine but Lost Spirits Cuban-style rum is grand. Here's a scaled-down and slightly tweaked version for home:**\n\n750 ml 151 proof rum\n\n3.25 oz fresh orange juice\n\n3.25 oz fresh lemon juice\n\nPeel of \u00bd lemon, pith removed\n\nPeel of \u00bd orange, pith removed\n\n13 oz sugar\n\n16 oz water\n\nCombine the rum, juices, and citrus peels in a large swing-top jar. Seal and let macerate 24 hours in a cool place. Meanwhile, make a syrup by heating the sugar and water in a nonreactive pot. When cool, combine with the strained rum mixture, stir to blend, and bottle.\n\nThe West Indian shrub is identical, except that it uses fresh lime juice in place of both the lemon and orange juices.\n\n### TWELVE MILE LIMIT\n\n**As Prohibition was just getting its sea legs, the United States reckoned its laws held 3 miles into open water. Beyond that, just out of US jurisdiction, bobbed armadas. Dubbed \"rum row,\" each was an amalgam of smuggler's ships, cutting plants, and offshore warehouses freighted with those intoxicating beverages forbidden onshore. When America later extended its territorial reach to 12 miles, rum row shifted further out and continued as before. In a nod to that shifting, permeable barrier, here's a stiff one to keep you afloat.**\n\n1 oz white rum\n\n0.5 oz rye whiskey\n\n0.5 oz brandy\n\n0.5 oz grenadine\n\n0.5 oz fresh lemon juice\n\nGarnish: lemon twist\n\nShake with ice until chilled. Strain into a cocktail glass. Garnish with a lemon twist.\n\n### MICHAEL LAZAR'S MAI TAI\n\n**San Francisco bar manager and author Michael Lazar is a cautious advocate of essential oils in drinks. \"Some have too narrow a flavor range compared to the fresh botanicals, but I look at them as different ways to get flavors in,\" he explains. \"Galangal is so hard, it's impossible to work with, so galangal oil is great. And shiso is so incredibly delicate and very hard to keep fresh, especially in a bar setting, that the oil is a godsend.\" In his adaptation of Trader Vic's 1944 Mai Tai, Lazar combines fresh lime juice with lime-flavored simple syrup.**\n\n2 oz Appleton or El Dorado 12-year-old rum\n\n1 oz fresh lime juice\n\n0.5 oz Ferrand dry orange cura\u00e7ao or Cointreau Noir\n\n0.5 oz orgeat\n\n0.25 oz lime simple syrup (recipe follows)\n\nLime wheel and sunflower petals, for garnish\n\nMix all ingredients and shake with ice. Strain over crushed ice, garnish with lime wheel and sunflower petals. For a more traditional approach, drop the spent lime shell in the drink and garnish with a sprig of mint\n\n### LIME SIMPLE SYRUP\n\n1 liter 1:1 simple syrup (here)\n\n5 to 10 drops lime oil\n\nMix the lime oil in cool or room temperature syrup. Stir to combine.\n\n### MARY PICKFORD\n\n**In Havana, rum-based cocktails were all the rage among Americans seeking respite from Prohibition. Among their favorites: the El Presidente (recipe follows), the Hotel Nacional Special, daiquiris, and this number named after \"America's Sweetheart,\" actress Mary Pickford.**\n\n2 oz light rum\n\n1 oz fresh pineapple juice\n\n1 tsp grenadine\n\n1 tsp maraschino liqueur\n\n1 brandied cherry\n\nShake in a cocktail shaker with ice. Strain into a cocktail glass or a coupe. Garnish with a brandied cherry.\n\nIt wasn't just warm weather that drew Yankee snowbirds to Cuba during Prohibition. Havana was home to a robust year-round nightclub scene that catered to Americans thirsting for a proper cocktail. Or five.\n\n### EL PRESIDENTE\n\n**Named for Cuban president Gerardo Machado, the elegant El Presidente is a mix of rum, cura\u00e7ao, grenadine, and one specific vermouth. Tropical drinks historian Jeff Berry notes in _Potions of the Caribbean_ that Dolin Vermouth de Chamb\u00e9ry, a moderately dry French vermouth, works in an El Presidente, but that the other dry vermouths simply don't. He's right. In fact, it's a pretty bad drink with the French vermouths that dominate the market. If you can't find Dolin, do as Berry suggests and mix equal parts of dry French and sweet Italian vermouths.**\n\n1 oz aged rum (A\u00f1ejo Havana Club if you have it)\n\n1 oz Dolin Vermouth de Chamb\u00e9ry\n\n0.25 oz orange cura\u00e7ao\n\nOrange peel\n\n1 booze cherry\n\nStir the rum, vermouth, and cura\u00e7ao in a mixing glass with ice. Strain into a chilled coupe. Twist the orange peel over the drink, drop it in, and garnish with a cherry.\n\n### ARRAKESSENZ\n\n**A decade ago, if you really wanted to try your hand at replicating old punch or cocktail recipes that called for arrack, I would have advised looking to a formula like this one to mimic the sugarcane- and rice-based spirit sometimes called _spiritus oryzae_ in medical literature. Here, Lyon would have us mix 20 to 25 ml of a blend of vanilla, Pekoe tea, catechu, and formic ether (so-called rum ether) to a liter of spirit.**\n\n---\n\nVanillae concisae (cut vanilla) | 2.0\n\nHerba Thea Peaco (Orange Pekoe tea) | 50.0\n\nCateshu Pulverati (powdered catechu) | 10.0\n\nOlei Florum Aurantii guttae (drops of neroli, e.g., oil of orange flowers) | 2\n\nAceti Pyrolignosi rectificat (rectified pyroligneous acid) | 50.0\n\nAetheris Formici (formic ether) | 100.0\n\nSpiriti Aetheres Nitrosi crudi (nitric ether) | 10.0\n\nSpiritus (90%) (90% abv neutral spirit) | 350.0\n\nMacerate for 8 days, then strain and filter. 20\u201325 grams added to 1 liter of 35% alcohol produces produces artificial arrack.\n\nThe original German has an uncorrected error and calls this artificial rum.\n\n### SWEDISH PUNCH\n\n**Swedish or \"caloric\" pun(s)ch is a classic use for Batavia Arrack, good for drinking on its own as a Christmas punch, mixing in cocktails, or spiking baked goods. I haven't been without a bottle of two of this cardamom-scented citrus punch since bartender and blogger Erik Ellestad laid down a DIY framework almost a decade ago on his blog, SavoyStomp.com. Ellestad calls for Appleton V\/X rum, a blend of 15 different rums. I prefer the funkier five-year-old El Dorado Demarara. Your call.**\n\n2 lemons, sliced thinly\n\n2 cups well-aged rum\n\n1 cup Batavia Arrack\n\n2 cups hot strong spiced tea (4 tsp black tea, 6 crushed green cardamom pods, brewed in 2 cups water for 6 minutes, then strained)\n\n2 cups raw or natural sugar\n\nPut the lemon slices, along with any accumulated juice, into a \u00bd-gallon nonreactive container with a sealable lid (e.g., a big ol' Mason or swing-top preserving jar). Add the rum and arrack. Macerate for 6 hours. Don't leave it all day or overnight; you don't want to extract too much of bitterness from the lemons.\n\nMeanwhile, pour the hot spiced tea over the sugar. Stir to dissolve and let it cool to room temperature.\n\nAfter 6 hours, pour the rum off the lemon slices and combine it with the syrup. Filter and bottle in a clean, sealable container. Let it mellow for a day.\n\n### WILLIAM KITCHINER'S MOCK ARRACK\n\n**William Kitchiner's 1817 tome _The Cook's Oracle_ contains one of the earliest references to potato chips. It is a paragon of 19th century English make-do attitude. Of his many cordial and spirits recipes, this one offers another semblance of proper arrack by deploying benzoic acid in rum.**\n\nDissolve two scruples of flowers of benjamin [benzoic acid] in a quart of good rum, and it will immediately impart to it the inviting fragrance of \"Vauxhall nectar.\"\n\n### CARAMEL FOR COLORING\n\n**To be honest, I had mixed feelings about giving a recipe for making caramel to color spirits. If the goal is to fake age, then I just wanted to wash my hands of it. However, caramel coloring can give a nice dark color\u2014in judicious amounts\u2014to bitters, cordials, and other homemade wet goods. Come to think of it, when it's known as browning, a few drops of the exact same caramel gives a bit of color to soups, stews, and gumbos that otherwise might have turned out a bit pale.**\n\n8 oz plain white sugar\n\n6 oz water, divided\n\nOpen the windows, turn on a fan, and be ready for some smoke. Or make this one outdoors. In a deep, heavy, nonreactive pot, heat the sugar and 2 ounces of the water over medium-high heat until the sugar melts. It will turn amber, then darken like an old penny. Keep cooking, but watch carefully. When it has turned a deep, dark brown\u2014almost, but not quite black\u2014remove from the heat and add the remaining 4 ounces of water. It will hiss, steam, spit, and seize up. Don't get burned. Stir the mass until it liquefies, over more heat if necessary. Store at room temperature.\n\nThe second type of caramel, in the following recipe, is not cooked as long as black sugar; it retains the taste and aroma of cooked sugar. Added in somewhat greater quantities, it lends pleasant, rounded caramel notes to rums\u2014and rum cocktails\u2014in particular.\n\n### CARAMEL LEMONADE\n\n**In 2014, I gave a talk about mainland and Hawaiian moonshine at Tiki Oasis, San Diego's annual gathering of tiki enthusiasts. There, we blended caramel syrup with white whiskey from Death's Door in a wickedly good boozy lemonade. You may want to open your kitchen windows when making this caramel and, if you have a particularly twitchy smoke detector, unplug it; things can get smoky.**\n\n1.5 oz Death's Door White Whiskey\n\n4 oz Front Porch Lemonade (recipe follows)\n\n2 teaspoons Sweet Caramel Syrup (recipe follows)\n\nMix the whiskey, lemonade, and caramel syrup over ice. Garnish with mint if you're feeling extra fancy.\n\n##### FRONT PORCH LEMONADE\n\n5 parts fresh lemon juice\n\n5 parts water\n\n3 parts 1:1 simple syrup made with Demerara or white sugar (see here)\n\nStir until blended in a jar, pitcher, or gallon jug.\n\n##### SWEET CARAMEL SYRUP\n\n12 oz white sugar\n\n15 oz water\n\nSlowly heat the dry sugar in a deep, heavy-bottomed pot (I use an unlined copper one, but use what you've got). When it reaches a rich amber color, immediately (and carefully) pour in all the water. Be careful: It will spatter and steam. The whole mass will seize up in a hard candied blob. No worries. Lower the heat to low and continue to heat the mixture, stirring now and then, until the sugar dissolves. Some water will have evaporated as steam, so when the whole thing is liquid and cool, transfer it to a measuring cup and add just enough water to make the total volume 25 percent more than the original volume of sugar. In this example, 12 ounces (100%) plus 3 ounces (25%) = 15 ounces. Just top off with cool water until the total volume is 15 ounces. Easy-peasy.\n\n### COPPER DISTILLED BOURBON\n\n**A devious recipe. Clearly, this isn't distilled bourbon at all, but a facsimile given an authentic-sounding name.**\n\n---\n\nCopper sulfate dissolved in H2O | 1 dram\n\nProof spirit | 40 gallons\n\nPeach flavoring | 1 gallon\n\nBrandy flavoring | 1 gallon\n\nWine vinegar | 1 pint\n\nGlycerin | 1 pound\n\nOil cognac dissolved and alcohol | 12 drops\n\nColor with caramel.\n\n### **_Whiskey_**\n\nWell before Andrew Volstead was born, Americans were flavoring, sweetening, spicing, coloring, and otherwise adjusting their whiskeys. To this very day, families macerate peaches, apples, cinnamon, blueberries, and other fruits and botanicals in whiskey. In moonshining families, jars of such concoctions are boons of the Yule season. Some of the most famous bars in the country feature exactly such concoctions. Well. Maybe not that blueberry one.\n\nLyon's whiskey entries are a mishmash of compounded whiskey recipes, formulas for essences or oils that would have been added to rectified spirits, and notes on making actual whiskeys, including rough grain bills for bourbon and notes on malting. Although some of his notes call for commercial products\u2014Fritzsche Bros. were renowned flavor importers whose firm was eventually subsumed by Swiss flavor conglomerate Givaudan\u2014many call for compounding disparate aromas and flavors as needed. Creosote would have lent a charred, smoky taste to factitious whiskeys. Oils of clove, wintergreen, and Cognac are common, while peach flavoring, honey, and glycerin smoothed the taste of raw liquor. It's not surprising that many of the whiskey recipes are imitations of rye, an immensely popular whiskey before Prohibition that is coming back into vogue now.\n\nWe whiskey drinkers can get testy when someone adds syrup, flavors, and color to our supposedly pure drams. Nonetheless, cinnamon-flavored whiskeys, themselves an old cordial against cholera and other diseases, have cut a huge swath into the drinking habits of Americans in the last few years. In truth, whiskey has been one of the most fantastically adulterated spirits in the history of the world. Nineteenth-century compounding manuals testify to the sorts of mixtures once passed as whiskey, but long before essential oils and extracts of creosote were deployed, what people drank as whiskey bore only a fleeting resemblance to what we know today.\n\nTake, for instance, the _usquebach_ from the _Pharmacopoeia Londinensis_ published in 1618. The stuff went under a number of names, including _usquebaugh_ and _uisge beatha_ , but each was a Gaelic rendition of the Latin _aqua vitae_ , the water of life. Scotch and bourbon drinkers may know where I'm headed with this, but modern drinkers would recognize the word as _whiskey_. Or _whisky_. Whichever. But the name is about the only thing that they would recognize; whiskeys of the time were nothing like what we drink today. They were outlandishly flavored with big flavors, such as mace, ginger, licorice, and\u2014especially\u2014saffron. See here for more on that last one.\n\n**I Need to Check Your ID**\n\nMost of the outright fake whiskey from the 19th century through today, despite the particular style or even brand their makers try to reproduce, boils down to faking one thing: age. From early experiments with X-rays to soaking wood chips and sawdust in whiskey to modern experiments with smaller barrels or exposing new spirits to ozone, distillers, blenders, wholesalers, and bootleggers have tried either to speed aging or give the impression of a spirit older than it really is, even though probably more than half of a whiskey's taste comes from the barrel. New, raw whiskey is a solvent that does three important things as it ages in oak barrels:\n\n1. It extracts compounds from the barrel's wood itself (which varies depending on whether and how much the interior may be toasted or charred, whether it may have held wines or other spirits, and what species of oak is used).\n\n2. During long oxidation, it breaks down some compounds and transforms them into all-new pleasant-smelling and -tasting chemicals that did not exist in the new spirit.\n\n3. Through evaporation through porous wood, whiskey loses volume, but becomes more concentrated.\n\nBut such aging takes years. Even then, some batches turn out better than others. One of the ways bootleggers sidestepped proper aging was to infuse raw whiskey, sugar moonshine, or neutral spirits with oak sticks or chips (often made from genuine broken-down whiskey or beer barrels). After a few hours (or weeks if they were true back-alley artisans): voil\u00e0\u2014whiskey! Not really, of course. Such \"chipped\" whiskey lacked the suave complexity of properly aged spirits. I've sampled whiskeys made this way. They just never taste better than alcohol flavored with wood extracts. Thin, hot, weak examples that won't satisfy discerning drinkers. Of course, such extracts, along with other compounds, can be deployed in otherwise decent whiskey in hopes of improving it. Those are more successful cheats I warrant most whiskey drinkers have unknowingly\u2014and perhaps happily\u2014consumed.\n\n### Whiskey Cocktails\n\n### ROCK & RYE\n\n**Admittedly, winter sniffles and sneezes aren't the pressing concern here in San Diego that they can be in other places. Still, we like our remedies against chilly nights and this is a good one.**\n\n**Before, during, and after Prohibition, enough temperance advocates made allowances for booze-heavy tonics and bitters that their prevalence in ostensibly dry houses became a running joke. While a slug of plain whiskey would almost surely have been met with tightly pursed lips, who could argue that rock candy sugar and fruit wouldn't be wholesome additions? A touch of horehound\u2014an herb that, as everyone knows, is such balm to a sore throat\u2014was exactly what the doctor might've ordered to keep winter's bane in check.**\n\n**A few years back, we planted blood orange and lemon trees in the garden. When I\u2014 _cough, cough_ \u2014feel a cold coming on, I'll raid the trees for a few fruits to soak in rye whiskey with hard horehound candy for the perfect remedy.**\n\n750 ml rye whiskey\n\n6 to 8 oz horehound candy (see note)\n\n3 oz dried sour cherries\n\n2 (4-inch) sticks of cinnamon\n\nZest of 1 orange (blood orange if you've got it)\n\nZest of 1 lemon\n\n3 whole cloves\n\nMix the ingredients in a 1- to 2-liter lidded glass jar. Macerate at room temperature, giving the jar a swirl now and then, for 2 to 5 days until the candy is fully dissolved and the cordial is fragrant with citrus and spice. Strain into clean 1-liter bottle.\n\nA few confectioners, such as Hammond's in Colorado, still make horehound candy. If you can't find any, swap regular or yellow rock candy and add 1 teaspoon of dried horehound. If you can't find horehound, you could add a star anise fruit or two, but don't sweat it.\n\n### WARD 8\n\n**Although it predates Prohibition, the Ward 8, a rye and orange juice _potus_ , remained popular under Volstead. Tipplers then would have used old stocks of Pennsylvania or Maryland rye, smuggled Canadian whisky, or even synthetic stuff. Ratios are all over the board for this one, but here's the version I like:**\n\n2 oz straight rye whiskey\n\n0.75 oz fresh lemon juice\n\n0.75 oz fresh orange juice\n\n0.25 oz grenadine, or to taste\n\nSparkling water\n\nShake the first four ingredients with ice and strain into a chilled glass. Add a splash of sparkling water to give the whole thing a lift.\n\n### **Poisoned Liquor**\n\nSome of Lyon's recipes are good, others are better left to professionals, others still just don't result in spirits as good as what we can buy from even modestly stocked modern stores. A few, however, stand out as recipes you must not re-create. One of them is rye ether. Ethers are distinct class of compounds, but as used by rectifiers and blenders, the term also referred to combinations of essential oils, ethers, alcohols, acids, and other aromatic compounds that gave the characteristic smell of different fruits or spirits\u2014raspberry, for instance, pear, bourbon, strawberry, apple, etc. Such ethers are harmless in and of themselves, but Lyon's recipe calls for a small amount of methyl alcohol, the notorious \"wood alcohol\" that blinded and killed so many in the late 19th and early 20th centuries.\n\n### RYE ETHER\n\n---\n\nAnise oil | 1 part,  i\n\nAcetic ether | 5 parts,  v\n\nAmyl alcohol | 5 parts, \"distil\"  v\n\nNitrous ether | 5 parts,  v\n\nMethyl alcohol | 2 parts,  ii\n\nAlcohol | 100 parts,  xiiss\n\nLater, Lyon gives a recipe for brandy essence that likewise calls for methanol under the name \"rectified wood spirit.\" This essence isn't something anyone would drink as is, but which would be added in minuscule amounts (1 part to 1,600 parts of an alcohol-water mixture). Methanol at that concentration and in the amount a drinker would likely consume probably would be harmless\u2014but the stuff is toxic even when inhaled or absorbed through skin, so skip it.\n\nYou wouldn't want to drink it\u2014in fact, it is _imperative_ that you not\u2014but alcohol that looks and tastes just as clean and pure as vodka can be distilled from lumber and sawdust. Low concentrations of methanol (CH3OH), also called wood alcohol, is common in many foods (including alcoholic beverages). It even courses through our own bloodstream. Such small amounts are harmless, but a toxic dose for humans may be as little as 8 grams\u2014about a teaspoon. In the words of New York City's chief medical examiner in 1928, \"wood alcohol is not 'poison' liquor. It is simply poison. If it gets into liquor, the liquor is poisoned.\"\n\nIn the 19th and early 20th centuries, pharmacists, physicians, and those in the trade believed that wood naptha, as crude distillates of wood were called, was so vile, so disgusting, that no one would willingly consume it. The smallest part, so the thinking went, would taint drinking alcohol. For the most part, they were right. Until about the 1890s, only isolated instances of methanol poisoning show up in medical literature.\n\nThen new production methods came along that yielded the light, colorless methanol we know today. It was widely used in hundreds of products from paints to cosmetics and was one of the chief additives that the US government permitted in several regulated formulas to create \"denatured\" alcohol, supposedly unfit to drink. After high-profile mass poisonings, awareness spread among physicians that methanol could blind and kill those who drank it. Even then, the general public wasn't aware of the threat until well into Prohibition, when ignorant or indifferent bootleggers began using it in, or even as, bootleg liquor.\n\nThe results were devastating. After 30 people were poisoned from drinking contaminated whiskey in New York in 1918, toxicologist Alexander Gettler issued a clarion call about the dangers of wood alcohol poisoning in the _American Journal of Pharmacy_. Those who drink or inhale the poison or absorb it through their skin may experience nausea, headaches, weakness, vomiting, and temporary visual distortions. A less fortunate drinker may suffer permanent blindness as her body slowly metabolizes it over several days into more toxic formaldehyde and formic acid, leading to coma and organ failure. The inevitable fate of those who consume too much: death. Six of those poisoned did, in fact, die. Gettler's message was that such poisoning would skyrocket. Doctors and health officials needed to warn everyone.\n\nA decade later, Gettler's boss, New York City medical examiner Charles Norris, published an excoriating essay on policies that led to the widespread use of wood alcohol among bootleggers. \"Nearly ten thousand in [New York City] will die this year from strong drink,\" he wrote. \"Our national casualty list for the year for this one cause will outstrip the toll of the War. These are the first fruits of Prohibition\u2014not in terms of savings banks or factory efficiency or the votes of pussy-footing legislators, but in terms of life and death... This is the net dividend of our noble experiment\u2014in extermination.\"\n\n### **Korn**\n\nDistilled from rye, barley, or sometimes wheat, _Korn_ is a bit of a stranger in the US, but the grain spirit has a long history in Germany. In German, _Korn_ refers to grain more broadly rather than American corn ( _Zea mays_ ). The nose and taste of these minimally aged schnapps are clearly cereal. Think less vodka and more grain moonshine or white dog. A shot of chilled _Korn_ , served either alongside a glass of beer or mixed into it, is known in Germany as a _Herrengedeck_ , a sort of Teutonic boilermaker.\n\n**Hot and Spicy**\n\nAfter water and neutral spirits, capsicum and grains of paradise rank among the most common spirit adulterants since the 19th century\u2014especially in gin, whiskey, and schnapps. Old manuals are rife with compounded recipes calling for them. Capsicum generally referred to cayenne pepper, but it may also indicate other hot chiles. Capsicum is almost universally derided as a particularly low adulterant in old texts. Unscrupulous blenders who watered down whiskey, so the story goes, used chile peppers to simulate a higher proof, to \"sharpen\" the spirit.\n\nWhile the practice does seem to have been widespread, the problem with that simple interpretation is that chiles don't simulate high proof. The taste may distract drinkers from the aqueous nature of overly hydrated whiskey, but some drinkers enjoy the painful\/pleasurable kick that cayenne adds to a drink. Certainly pharmacists, physicians, and mothers alike before and during Prohibition used cayenne as a medicine\u2014at the very least, the burn it imparted felt like it was doing _something_.\n\nIn fact, Lyon's two German recipes for _Kornsch\u00e4rfe_ are for doing exactly that\u2014giving a bit of bite to the grain spirit _Korn_ with capsicum and grains of paradise. The recipe is lifted from _Spezialit\u00e4ten und Geheimmittel_ , a 1906 German text.\n\n### KORNSCH\u00c4RFE\n\n**To fortify artificial grain schnapps.**\n\nA \u2013 Prepare a capsicum extract 1:10 with 92-93% alcohol.\n\nB \u2013 An extract comprising\n\n4 T. [ _Theilen_ or parts] capsicum\n\n1 T. [parts] grains of paradise\n\n40 T. [parts] alcohol 92-93%\n\n## **_Chapter four_**\n\n## Brandy, Wine, and Cider\n\n**_Not all mixtures or blends of spirits are frauds_**\n\n### **_Brandy_**\n\nIn the early years of the Republic, American distillers adapted their craft to New World ingredients, creating recipes that struck notes of genuine pride in the New World's overwhelming bounty. They often strove to provide spirits every bit as good and wholesome as those from Europe\u2014at a fraction of the cost. Eighteenth- and 19th-century spirits formulas repeatedly call for making this or that spirit \"as good as imported\" or \"in the style of XYZ spirit.\" Samuel M'Harry's 1809 manual _Practical Distiller_ , for instance, gives directions for making \"a Brandy resembling French Brandy\" from clarified whiskey or apple spirits blended with 25%\u201333% genuine French brandy. In a New World mashup of pride and thrift, he notes that, if kept a few years, it will be more sound and salutary than French brandy alone. Thrift, pragmatism, and nationalism all rolled into one.\n\n**Practical Distiller**\n\nSamuel M'Harry's manual was so clearly written\u2014and printed just as Americans were migrating west and south into the Appalachian mountains\u2014that its directions for distilling on pot stills informed much of American folk distilling (i.e., _moonshining_ ) right into Prohibition years. In fact, I've interviewed moonshiners who still follow M'Harry-style recipes more than 200 years after they were published.\n\nFrench distillers probably would have taken issue with M'Harry's estimation of such domestic wet goods, but they would have recognized his methods immediately\u2014as would have their colleagues in Germany, Britain, the Netherlands, and other countries with commercial distilling sectors. Blending a portion of genuine brandy, whiskey, or rum into neutral spirits (sometimes called stretching or cutting the spirit) was a well-established technique well before Prohibition. By then, though, it became a standard maneuver with less and less genuine spirit involved\u2014and oftentimes none at all. In 1864, Charles Tovey inveighed against just such practice about casks of spirits coming from Holland, Germany, and Mediterranean ports\n\nbearing most impudently the brand of Cognac; in fact, it is known to the writer that many houses send over to Hamburg freshly emptied casks, having the original brand of some of the best shippers of Cognac brandy. These casks are filled with Hamburg Spirit, pale or brown, at a cost of about 3 [shillings] per gallon for proof, and those who are not acquainted with the manoeuver, are too often deceived by the external appearances. (Charles Tovey, _British & Foreign Spirits: Their History, Manufacture, Properties, Etc_. [London: Whittaker & Company, 1864])\n\nBarrels filled with spurious German potato or sugar beet spirits would go on to bonded warehouses abroad from where customers who sought the surety of bonded liquor were fleeced as surely as the punter quaffing a dram of counterfeit whiskey at the corner pub or saloon. Alternately, German firms would send barrels of such spirits to France, whence they were shipped out from the land of Cognac with new identities. A bit like money laundering, but with booze.\n\nBut not all mixtures or blends of spirits are frauds. French distillers in particular have added a variety of substances to their brandies\u2014stellar, world-class, award-winning brandies, mind you\u2014for centuries.\n\n**Boise**\n\nBoise (pronounced bwaz-AY) is a wood extract, commonly made of oak, but may be of other woods, and fortified with spirits. It is blended with spirits (typically brandies, but at times with whiskeys and others) in small amounts to get them ready for market. They add color as well as tannins and other compounds to help \"round\" and \"shape\" the final product. For cheaper brandies, the extract can be quite new, but some in use today are 100 years old\u2014or more.\n\nDistiller Hubert Germain-Robin's family has been in the brandy business for centuries in France. One of the first things he did when he arrived in California was to make boise. \"When people are making boise,\" he recently explained, \"it's just a special place where you don't visit. It's not known. It's not like a Cognac house where you can visit the tasting rooms.\" In its earliest incarnations, boise was simply chips from barrels. \"That was it. It started like that. Then, after people started to make boise, to sell boise in the Cognac region, there were different ways to do it. Sometimes it is just an infusion and they boil it gently, and evaporate and concentrate it and use that.\" Some distillers buy boise, but Hubert makes his own. He prefers American or French oak, but says that some have even used bitter galls to make boise. \"You get very different character with the different woods. A good boise is thirty, forty years or more. I want one with at least fifteen years so I can use different boises with different characteristics the same way I use different types of brandy in a blend.\"\n\nLyon uses oak in a few recipes to approximate barrel-aging and some astringency, such as in the cognac recipe (below) which called for oak sawdust (yeah, don't follow that one; sawdust extracts are among the most unpleasant natural oak flavorings\u2014all harsh edges and thin astringency with none of the richness of toasted oak barrels or staves). He also makes a run at an ersatz boise with his recipe for Cognac Charente Type:\n\n### COGNAC CHARENTE TYPE\n\n##### A. \u2014 TYPAGE\n\n---\n\nBlack Ceylon Tea | 5 ounces\n\nVanilla beans | 5 ounces\n\nStoned dry plums | 8 pounds\n\nOak wood | 10 pounds\n\nPercolate after allowing to stand 14 days with 8 gallons proof wine distillate. 6\u00bd gallons extract called Typage.\n\n##### B. \u2014 SYRUPAGE FOR COGNAC\n\n16\u00bd pounds sugar are boiled with 2\u00bd gallons water with the addition of 2\u00bd pounds caramel.\n\nResult 3\u00bd gallons.\n\n##### C. \u2014 SYRUP CHARTENTAIS FOR COGNAC BASE\n\n6\u00bd gallons Typage and 3\u00bd gallons Syrupage are mixed to the final result of 10 gallons of which \u00bc gallon is sufficient for 10 gallons Cognac.\n\nIn principle cognac is a mixture of \u00bc gallon Syrup Charentais, sugar and color with genuine wine distillate of any strength. The lowest limit is about 19 U.P.\n\nOrdinary manufacturing strength is 17 O.P. The wine distillate is to be understood to be of proof strength or if higher alcohol strength it must be diluted to 17 under proof.\n\n### PURE COGNAC OF 17 UNDERPROOF CONSISTS OF\n\n---\n\nSyrup Charentais | \u00bc gallon\n\nPure genuine proof wine distillate | 7 gallons\n\nDistilled water | 3 gallons\n\nAbout 10 gallons\n\nThe yield on this recipe is \"about\" 10 gallons, when it would seem, from simple math, to make exactly 10.25 gallons. Math isn't so simple when mixing alcohol and water because when the two are blended, the alcohol contracts. To make exactly 10 gallons of finished product, compounders would have to add more than 7 gallons of proof spirit to 3 gallons of water.\n\n**Aged Syrups**\n\nLyon's recipe for Cognac syrups (here) may well be what German and American distillers used to imitate French brandies through Prohibition. Shoot, it might even be what some French producers were making. But that's not how serious brandy makers do it. Notwithstanding purists who eschew as chicanery any syrup, boise, caramel coloring, and other additives in their spirits, those who make fine brandies in the French style with such additives age not only the brandy, but the various components as well. \"When I use a tiny bit of syrup or caramel in my blend,\" says Hubert Germain-Robin, \"I don't mix in something that is not the minimum age of my blend. It is more integrated already because it is aged.\"\n\nDan Farber, who makes magisterial grape and apple brandies at Osocalis in the hills of Soquel, California, agrees. \"Syrups can be very, very old. I have syrups that are fifteen, maybe sixteen years old: way too young.\" Farber explains that vintage syrups (and boise) age at different rates and in different ways from brandy, becoming far more complex and nuanced over years in barrels. \"Yes, people can do things that we might not like. They can take zero-aged _eau de vie_ , blending it with wood extract and sugar syrup and caramel color to make something that's not great\u2014but that doesn't mean you should throw the baby out with the bathwater. Therefore _all_ things that have _anything_ but the most pure essence of spirit are somehow flawed? That's a crazy way of thinking and it's sad in a way.\"\n\n**Vinegar in Brandies**\n\nI enjoy vinegar-based fruit shrubs, vinegar pie, drinking vinegars, and vinaigrettes. Filipino chicken adobo, a dish swimming in vinegar, garlic, and soy sauce is one of my favorite dinners. But vinegar in brandy? Absolutely not. It's a characteristic I consider not just a flaw in brandies, but an insurmountable one. Nevertheless, old formulas for compounded brandies often include small amounts of acetic acid or vinegar. Why?\n\nAfter talking to several distillers who were equally flummoxed, I have an idea. When fermentations turn sour with acetic acid, the flaw carries over into the distillation. The spirit cannot be salvaged. Before refrigeration, mashes that got too hot or were not distilled in time could have turned acetic more commonly. Perhaps, just as Belgians grew to dote on sour beers, locals (especially in Britain where many such recipes originated) developed a taste for that vinegar twang in some brandies, conflating it with age.\n\nThat's not to say vinegar doesn't have its place in mixed drinks. Try apple cider, champagne, raspberry, or other vinegars in your brandy as an alternate to lemon or lime juice. Call it _a gastrique_ if you like. Just... not in your nicest brandies, eh?\n\n**Brandy Cocktails**\n\nLyon's brandies are, by and large, not the kind we're likely to see on today's commercial market. Sure, even some world-class brandies are made with small amounts of caramels, vintage syrups, or boise, but the days of completely fake, imitation, or factitious brandy made from wine, grain, beet, or potato spirits are largely past. His ginger brandies are another matter. See here for Lyon's Ginger Brandy No. II, a bracing cordial we put to good use. In the meanwhile, here's a trio of brandy cocktails popular in the Volstead days. Do try them with genuine brandies.\n\n### SIDECAR\n\n**The Sidecar is a versatile cocktail for your arsenal, and is one of my favorites. Even the thought of one feels like pulling on a comfortable old sweater. Originally made with equal parts of brandy, Cointreau, and lemon juice, this one gets tinkered with a lot by modern bartenders. It's stiff, bracing, tart, and (if you give the glass a sugar rim) a bit sweet. Here's a stiffer modern version with more brandy and softened with dollop of syrup:**\n\n1.5 oz Cognac or other brandy\n\n0.75 oz Cointreau\n\n0.75 oz fresh lemon juice\n\n1 teaspoon 2:1 rich syrup (here) or 3.2 compounder's syrup (here)\n\nShake in a cocktail shaker with ice. Strain into a cocktail glass or a coupe. Garnish with orange zest. If using apple brandy, swap out the syrup with a bit of maple syrup or boiled cider: just enough to soften it a bit.\n\n### BRANDY FIX\n\n**From _Barflies and Cocktails_ , a 1927 vade mecum of mixed drinks published in Paris by Harry McElhone (of Harry's Bar) and illustrated by Wynn Holcomb, here's a vestige of 19th-century bracers that gooses brandy with cherry brandy\u2014which could be read as either a sweet, dark cherry-flavored brandy such as Heering or as a clear, dry kirschwasser. Go with the latter.**\n\nPour into a small tumbler 1 teaspoonful of sugar, 1 teaspoonful of Water to dissolve the sugar, Juice of a half Lemon, \u00bd Liqueur [glass] of Cherry Brandy, 1 Liqueur [glass] of Brandy.\n\nFill the glass with fine ice and stir slowly, then add a slice of Lemon, and serve with a straw.\n\n**In modern measures, try it like this:**\n\n1.5 oz VSOP Cognac\n\n0.75 oz kirschwasser\n\nJuice \u00bd lemon (0.5 to 0.75 oz)\n\n1 teaspoon simple syrup (here)\n\nLemon twist\n\nShake with cracked ice and pour it, ice and all, into a tumbler. Garnish with fresh fruit\u2014if garnishing is your bag\u2014and a lemon twist.\n\n### HARVARD COCKTAIL\n\n**Given the brandy, bitters, and just enough fizz to lighten and lift the whole drink (the Harvard Cocktail is sometimes called a Harvard Fizz), we'd expect this northeast stalwart to be served in a highball glass, but no: the stemmed cocktail or martini glass is the classic way to serve it.**\n\n1.5 oz Cognac\n\n1 oz Italian vermouth\n\n2 to 3 dashes of bitters (Angostura or Fee Brothers, barrel-aged)\n\nClub soda\n\nStir the Cognac, vermouth, and bitters together in a shaker with ice. When cold, strain into a chilled cocktail glass or coupe. Gently pour in 1 to 2 ounces of club soda and give it a brief stir.\n\n### **_Cider_**\n\nIn the lead up to national Prohibition, many winemakers, brewers, and cider makers assumed that they would be permitted to carry on once the nation went dry. Makers who considered their beers and wines \"temperance\" beverages reckoned that Prohibition would target only distilled spirits, such as whiskey, gin, and \"demon\" rum. They were in for a shock. With a few exceptions, the National Prohibition Act outlawed every intoxicating beverage over 0.5 percent alcohol. Apple cider was one of those exceptions.\n\nIn 1920, after much confusion in cider-making areas about what was permitted, John Kramer, the federal prohibition commissioner, clarified that anyone manufacturing \"nonintoxicating cider and fruit juices exclusively for use in his home\" needed no permit. Sugar, molasses, raisins or other dried fruit were prohibited since those were ways to boost alcohol content and (presumably) prepare \"a mash fit for distillation.\"\n\nProhibition wrecked America's cider-drinking traditions. Farmers could sell sweet cider by the glass, but if customers took some away and it later hardened, the farmer could be held accountable for having produced an intoxicating beverage. Home vinegar making was permitted, but commercial producers were required to secure a bond and permit. Many threw in the towel on cider, tore out their orchards, and either sold the land or planted other crops.\n\n**Cider Cocktails**\n\nFor many Britons and Americans who may not have been able to get or afford imports, apple cider formed the basis of traditional imitation wines and spirits from abroad. Lyon's cider \"Champagne\" is one such venerable imitation. In an unusual twist, the recipe calls for fining the mix\u2014that is, clearing it of particulates\u2014with milk. Home brewers commonly fine with gelatin or isinglass, but modern bartenders have rediscovered fining with milk only recently. Some call it milk washing. When milk is added to spirits, it curdles and traps little floaty bits that would otherwise accumulate in the bottom of barrels or bottles. It also smooths some of the spiky flavors, making a rounder resulting beverage. When the cider is clear, it is carefully racked off the top of the curdled milk.\n\nJohn Conacher's 1919 cartoon, _The Sacred Cider Grove_ , gives a broad wink at the difficulty of keeping Americans away from intoxicating beverages as Prohibition approached. \"Are those good eating apples?\" asks the gentleman rider. \"No, Mister,\" responds the farmer, \"not this year they ain't. This year them apples is good drinkin' apples.\"\n\n### CIDER CHAMPAGNE\n\nPut in a barrel 5 gallons hot water[,] 30 pounds brown sugar, \u00be pound acid tartaric, 25 gallons cold water, 3 pints of hops or brewer's yeast worked into paste (\u00be pounds flour and one pint of water will be required to make this paste) put all together in a barrel, which it will fill and let it work 24 hours, the yeast running out at the bung all the time, but putting in a little occasionally to keep it full. Then bottle, putting in 2 or 3 broken raisins to each bottle and it will nearly equal champagne.\n\n### CIDER WITHOUT APPLES\n\n**Lyon's Cider Without Apples, while a traditional recipe, doesn't stand up to the fine ciders we can get today from the United States, France, Spain, and the United Kingdom. Its saving grace is raisins, which help give a little fizz when they ferment in the bottle. Do try the Cider Champagne, though. If you omit the sugar and milk fining, you're on the way to have a batch of good old Cider Oil.**\n\n---\n\nGood cider | 10 gallons\n\nSpirits | \u00bd gallon\n\nSugar | 3 pounds\n\nMix and let rest for 2 weeks. Fine with 1 pint skimmed milk. Often sold for champagne.\n\n### **Cider Oil**\n\nCider oil. Cider royal. Cyder-oil. Cider ile. What even is that stuff? Depends on who's talking. Often it was a blend of hard or sweet cider and apple brandy (or sometimes whiskey) popular in colonial America similar to Lyon's Cider Champagne. But it was also sometimes what we call applejack today. In his 1810 _Sketches of a Tour to the Western Country_ , Fortescue Cuming mentioned stopping along his trip for \"a bowl of excellent cider-oil... stronger than Madeira... obtained from the cider by suffering it to freeze in the cask during the winter, and then drawing off and barreling up the spirituous part which remains liquid, while the aqueous is quickly congealed by the frost.\" Records show George Washington served it to voters in 1758 in his bid for the Virginia House of Burgesses. He won. You could do worse than serve up a bowl to friends.\n\n### THE STONE FENCE\n\n**In _The History of Applejack_ , Harry B. Weiss notes a stiff drink from northern New Jersey's \"apple belt\" called the Stonewall Jackson: 4 ounces of applejack in 8 ounces of hard cider. \"After swallowing this, the drinker wouldn't know if he were in Sussex, England, or Sussex, New Jersey.\" It's a stiffer version of the Stone Fence (or Stone Wall), a colonial-era drink still popular in some quarters:**\n\n2 oz apple brandy (or, in a pinch, rum or whiskey)\n\nHard cider\n\nPour the apple brandy into a pint glass over a few ice cubes. Fill with hard cider. Orange and\/or aromatic bitters are optional newfangled additions to this ancient quaff. Served hot, it benefits from a slice of lemon and a few cloves or a cinnamon stick.\n\n### CIDER SHERRY FLIP\n\n**Flips, a class of drinks that mix eggs with sugar and spirits or wine, are coming back into vogue after decades of neglect. Bartenders of the past, including Pennsylvanian George Murray, weren't always punctilious about including the eggs. In the 1890s Murray served a cider sherry flip that Pittsburgh's businessmen snapped up to stave off the summer heat. During Prohibition, that \"sherry\" might well have been an imitation sherry such as Lyon described. Murray's original made about a 2-ounce drink\u2014but our summers seem to be hotter. Let's nudge the volume just a bit higher.**\n\n1.5 oz pale sherry\n\n1.5 oz cider champagne\n\n1 tsp simple syrup (here)\n\nStir in a shaker with ice. Strain into a chilled coupe.\n\n### **Applejack, Jersey Lightning, and Spirits of Cider**\n\nUnlike urban moonshining that so often relied on sugars, distilling apple spirits has always tended to be a rural pursuit. While Harlem doctor Victor Lyon secretly recorded recipes that might have landed him in jail if he had tried to publish them, illicit distillers in nearby New Jersey cranked out untold gallons of applejack, the region's traditional spirit, to slake New York's thirst. Supplies also ran to Philadelphia, Baltimore, and points south, though Virginia and North Carolina apple distillers kept the flame alive in the upper South.\n\nProhibition very nearly put an end to the tradition, but Americans had been downing apple distillates and blends for centuries under a variety of names: applejack, apple whiskey, Jersey Lightning, cider brandy, spirits of cyder, cider royal, and others. What came to the cities during the 1920s and early 1930s was often new and raw, a roughness that helped kill our taste for it. After repeal, Charles H. Baker Jr. advised readers,\n\nDon't get Jersey Lightning that some friend has put down in the wood since last fall, get it at least four or five years old. It is a very deceiving fluid, and when not watched will induce a happy state from the waist down, closely approaching voluntary paralysis. ( _The Gentleman's Companion_ Derrydale Press: New York 1939)\n\nFor much of American history, **applejack** , **apple whiskey** , and **apple brandy** indicated the same thing. But not always and not everywhere. In casual use, many say \"applejack\" to designate aged, American-style, charred-oak apple brandy. It may also refer to a rough winter spirit later dubbed _Yankee antifreeze_ , obtained from freezing hard (i.e., fermented) apple cider and repeated breaking through the icy surface to extract unfrozen higher-proof liquid underneath until what's left simply fails to freeze. This type of applejack has disappeared from the commercial scene, but a few modern distillers experiment with the form.\n\nThe final type of applejack is, like many of Lyon's whiskeys, brandies, and rums, more than half vodka. So-called **blended applejack** , popularized by New Jersey distillers Laird & Company starting in 1968, actually predates Prohibition. In 1954, Harry B. Weiss noted in _The History of Applejack_ that blending apple brandy with grain spirits allowed pre-Prohibition distillers to compete against cheaper corn and rye whiskey.\n\n**Apple Brandy Cocktails**\n\nUnless they are intended to replicate some specific beverage such as sherry or port, always consider swapping out the \"spirits\" called for in cider compounding recipes with apple brandy to emphasize the apple taste and aroma. A little barrel char adds agreeable character as well. Alternately, if barreling your own apple spirits isn't in the cards, you would do well to shake one of the big apple drinks of Prohibition: the Jack Rose.\n\n### THE JACK ROSE\n\n**The Jack Rose predates David Embury's 1948 _The Fine Art of Mixing Drinks_ , but Embury pegged it as one of the best-liked apple brandy drinks of its day. It's still a good one.**\n\n2 oz American barrel-aged apple brandy\n\n0.5 oz fresh lemon or lime juice\n\n0.25 oz real pomegranate grenadine\n\nShake vigorously with plenty of cracked or crushed ice and strain into chilled cocktail glasses.\n\nEmbury notes that \"a nice touch\" is frosting the glass by dipping its rim in grenadine, then in powdered (i.e., fine, not confectioners') sugar before straining the drink into the glass.\n\nIn 1926, US \"dry czar\" General Lincoln C. Andrews admitted that his forces captured only 10% of the \"family stills\" in America. Here, agents in San Francisco dismantle one of millions of such illicit stills.\n\n### **_Wine_**\n\nJust around midnight one frigid February night in 1925, five cars rolled up to a Chicago, Milwaukee & St. Paul Railroad warehouse in Chicago. Out poured 15 men, armed raiders who pressed the facility's lone watchman into service to help them load more than 40 barrels of port into their cars. The value of their haul? $12,000. In today's economy, that's about $162,000 worth of fortified wine. The appeal for bandits is undeniable\u2014that is, after all, a lot of money\u2014but here's the thing: wine was actually legal during Prohibition. At least some of it was. The wine in the warehouse wasn't some bootleg rotgut, but a perfectly legal agricultural product owned by the Italian Vineyard Company.\n\nThe wine loophole in the Volstead Act was a doozy, the scams it spawned breathtaking. The act explicitly permitted wine for sacramental purposes but didn't delineate what was sacramental and what wasn't. The intention was to make national prohibition more palatable to Catholics, who drank modest amounts of wine during Mass, and to Jews who drank in somewhat greater quantities at home. With each group thinking that Prohibition would not prohibit that sacred juice, the thinking was that they would give less resistance.\n\nRabbis willing to endure onerous paperwork could secure access to 10 gallons of wine for every Jewish family in America. But what was \"wine?\" Port or claret? Hock or Champagne? What about vermouth? Inveterate Prohibition agent Izzy Einstein once seized $35,000 worth of sacramental Dubonnet, a sweet fortified aperitif from France, in a Harlem garage.\n\nFramers of the Nation Prohibition Act failed to take into account the informal structure of rabbinical ordination. Determining whether a Catholic priest was, indeed, a priest was relatively straightforward: ask his superiors. \"Compared to Christian religious institutions,\" writes historian Marni Davis, \"the American rabbinate was loosely organized and minimally supervised.\" If a man said he was a rabbi, in other words, who's to say he wasn't? Counterfeit rabbis claimed new, robust congregations, notes Davis, \"filled with members named Houlihan and Maguire.\" They culled members from lists of the dead or made them out of whole cloth. In Alameda, California, the city's mere 50 Jewish families constructed a fictitious congregation of over 500 names.\n\n**The Wine Doctor**\n\nIn a widely reprinted article from the _London Daily News_ , Armiger Barclay wrote of his encounter with a local _licoriste_ who pointed to rows of bottles and promised he could recreate any wine or cordial from abroad Barclay might like.\n\n\"See those? Wine in embryo. Tons of it! Give me good water\"\u2014he indicates an innocent looking tap in the corner\u2014\"and I'll turn you out a bottle of anything you like to name\u2014while you wait!\"... \"Mind you,\" he goes on, \"I don't object to the real wine in moderation. My own best qualities have a base of sound Sherry or Burgundy. Take claret, for instance. Why send to Bordeaux when, with a gill of Australian Burgundy or Spanish Rioja, water\u2014watch while I do it\u2014a few drops of French vinegar and 25 per cent of potato spirit that's colorless and odorless and only costs a few pence per gallon, I can give you chemically the same thing with more alcoholic strength?\"\n\nPresto! It is done. He pours some of the result into a wine glass and hands it to me. It has the look, smell and taste of the wine for which I pay one and six pence a bottle.\n\n\"Total cost is a fraction over three pence, bottle and label included,\" he exults... From the bottles on the shelf he takes benzoic acid, benzoic ether, acetic acid and ether, oenanthic ether and glycerine\u2014a drop or two of each\u2014and fills up the glass with the ever indispensable alcohol. I raise the colorless mixture to my lips and behold! It is Maraschino! (\"Tricks of the Wine Doctor,\" _Salt Lake Herald_ , August 18, 1907, 6)\n\nDuring Prohibition, men and women alike smuggled liquor into public spaces hidden in thousands of different containers from hot water bottles and canning jars to flasks disguised as books, ample bellies, high heels, cigars...and walking sticks.\n\n### TO COLOR CLARET WINE\n\nTake as many as desired of damasciennes [i.e., damsons] or black sloes and stew them with some dark colored wine and as much sugar as will make it into a syrup. A pint of this will color a hogshead of claret. It is also suitable for new port wines and may be kept ready for use.\n\n### TO COLOR PORT WINE\n\n---\n\nRaspings of red sanders wood | 6  [ounces]\n\nSpirits of wine | 1 quart\n\nInfuse 14 days and filter for use. Produces a beautiful red color for port.\n\n**Don't Get Caught Up on Names**\n\nSome drinks are patently bad no matter what we call them; their salvation does not lie in nomenclature. Others may be perfectly fine, but bad examples of what they purport to be. Don't get caught up on names. There's a trap into which we can fall when we project our current understanding onto terms and concepts of the past.\n\nTake Lyon's \"port wine.\" Real port is phenomenal stuff. Compared to them, Lyon's attempt isn't just subpar; it's off-spec. If a bartender served me that when I asked for ruby port, I'd send it back. As a drink, though, stripped of its counterfeit name, I like it. That huge bump of hard cider made me think immediately of cool-weather punches. Then it hit me that it's a sort of distaff cousin to Spanish sangria or Scandinavian gl\u00f6gg. With that in mind, I studded an orange with cloves to make an old-school pomander, roasted it, and poured Lyon's heated concoction over it for a hot, jammy punch during one of our cold and wet spells in San Diego.\n\nAt the same time, don't go off the rails. You can make a liter of lavender, black pepper, and nutmeg cordial that's got 450 grams of sugar in it if that's what gets your motor revving, but if you insist on calling it port (or triple sec or eggnog), that's not artistic license; you've just straight-up got it wrong.\n\n### TO PREVENT A BAD TASTE AND SOURNESS IN WINE\n\nHorseradish makes a surprise appearance in the formulary as a clipping pasted into the book's pages. Not usually associated with spirits, here it's used \"to prevent a bad taste and sourness in wine.\" My advice? Skip it. Instead, infuse about a cup and a half of _fresh_ horseradish (please: not jarred) in a 700 to 750 ml bottle of vodka for about 4 hours, strain it, and use the bracing fluid in Bloody Marys. It also works in tequila and gin, but it would be a shame if such a strong flavor overtook your finest spirits. Use the everyday stuff. And if your everyday stuff _is_ the finest, well, bully for you!\n\n**Dealcoholized Wine and Cider**\n\nThe problem with fruit juices\u2014or the whole point of getting your hands on some, depending on how you look at it\u2014is that yeasts tend to attack them, devouring their sugars and churning out alcohol as a waste product. If you let apple or grape juice ferment, you've got hard cider or wine on your hands. There's more to it, than that, of course, and much depends on the blending of appropriate varieties. Your wine or cider won't necessarily be the best you've ever had, but it will have a kick, not something to sneeze at when your government deems intoxicating beverages of all stripes contraband.\n\nThose in the business of selling \"fresh\" ciders during the Volstead years, though, were often at pains to make sure that their products contained less than 0.5% ethanol, the maximum permitted under the laws of the time. In 1920, the _National Bottlers' Gazette_ offered straightforward advice to its readers who wanted to steer clear of governmental entanglements: boil the juice as wine makers did to make \"dealcoholized\" wine. As any distiller can tell you, ethanol has a lower boiling point than water. Either bringing vats of cider to a boil or turning live steam into suspect juice would heat it enough so that all but trace amounts of alcohol present would be carried away with the vapor.\n\nWho knows how many vintners and cider makers bothered to collect that alcohol-rich vapor? But that would be pretty rough brandy. And wholly illegal.\n\n### PORT WINE ESSENCE\n\n**Not all nonalcoholic wines began as proper wines, though. Lyon's Victorian-era port wine essence could have been used either to simulate real port or to make an alcohol-free version for children's parties or to flavor syrups for ices.**\n\n---\n\nTartaric acid | 3 drams\n\nTincture of orange | 3 drams\n\nConcent[rated]. Decoction logwood | 4 drams\n\nGlycerin | 4 drams\n\nOenanthic ether | 20 minims\n\nAcetic ether | 20 minims\n\nSp[iri]ts [of] nitrous either | 30 minims\n\nSyrup qs ad [enough to equal a total of] | 3 ounces\n\nThis quantity to be added to 1\u00bd pints (Imp) of water to make nonalcoholic port wine (English formula)[.] Sherry wine essence is made in the same manner omitting the logwood and using sufficiency of caramel.\n\n**Wine Cocktails**\n\n### DIY SUMMER CUP\n\n**In _Boardwalk Empire_ , HBO's series set in Prohibition-era Atlantic City, Steve Buscemi's character Nucky Thompson uses a classic gin and vermouth aperitif from England in a power play with a corrupt senator. Pimm's No. 1 (there have been several iterations over the years) is the base for Pimm's Cup, a summertime refresher loaded with fruit. Of course, you could buy Pimm's No. 1, but making a DIY summer cup hardly even qualifies as work. Over oysters and cocktails one afternoon at San Francisco's Ferry Building, Martin Cate, owner of the rum bar Smuggler's Cove and beloved in tiki circles for his championing of tropical drinks, shared me with his base recipe for a summer cup in the style of Pimm's No. 1. His version is as straightforward as they come:**\n\n2 parts sweet vermouth\n\n2 parts gin\n\n1 part cura\u00e7ao\n\nMix and store in a cool, dark place. I add about 2 ounces of this to a glass with ice, top off with sparkling lemonade or ginger beer (Cate likes bitter lemon), and garnish with fresh cucumber spears, mint, hulled strawberries if we have them on hand, and a tuft of lemon balm or borage. For the cura\u00e7ao, consider Lyon's Rum Shrub (here) or swap it out with Lyon's Ginger Brandy (here) for a drink with more bite.\n\nSo-called \"local options\" allowed towns and counties to vote to go (officially) booze-free before the Volstead Act fell on the nation. Some places have yet to repeal those laws.\n\n### SONNENTANZ\n\n**While prohibition had taken hold in parts of America, but Prohibition proper was still a few years off, members and guests of Philadelphia's Union Club drank Zeeland Cocktails: a bit like a rum Manhattan with a splash of k\u00fcmmel. Okay, but no great shakes. Treat it like a summer cup, though, and you're onto Something Else. Ladies and gentlemen: the Sonnentanz.**\n\n1.5 oz Jamaican rum\n\n0.75 oz red vermouth\n\n0.5 oz strong cold tea\n\n0.5 oz orange cura\u00e7ao\n\n1 bar spoon k\u00fcmmel\n\nStir with ice until well chilled and strain into a chilled glass.\n\n### K\u00dcMMELLIK\u00d6R\n\nMix 40 loths of fine white sugar with 1\u00bd drams of pure caraway oil. Dissolve the mixture in 1\u00bd quarts of water, and mix in 6 pounds of alcohol. After a while, filter this liqueur through blotting paper.\n\nThe _Loth_ is an old German measurement equal to 0.5 ounce. A retail pound is equal to 16 ounces, 32 _Loth_ , or 128 _Quentchen_.\n\n### **Tweaking Brandies One Note at a Time**\n\nBrandy manufacturers have had a battery of compounds for adjusting aromas and flavors at their disposal for hundreds of years. Some are used even still; a dose of aged wood extract, caramel coloring, and vintage syrups are among the traditional tools a distiller may legally use to \"shape\" such brandies as Armagnac, Cognac, and Calvados\u2014to give them the body, aroma, mouthfeel, and taste customers find pleasing.\n\nThe \"brandy\" here is from physician Andrew Ure's 1837 Dictionary of Arts, Manufactures, and Mines. Compounders typically based so-called British brandies on neutral or malt spirits rather than grapes.\n\nOther ingredients are either lesser known or have fallen by the wayside. The faked brandies of Prohibition owe their formulas to French, Dutch, and German blenders of the 18th and 19th centuries. Cognac oil, ethyl butyrate, tannins, vinegars, raspberry syrup, and various fruit \"ethers,\" when added in judicious amounts, can impart the taste of old brandies to something quite new. In imitation of boise (see here) achievable only with great age, blenders have resorted to infusions of hickory nuts, almonds, and walnuts, to neroli, bitter almond oil, vanilla, and prune and raisin syrups. Meanwhile, the raw, fiery taste of new alcohol can be tempered with sweeteners, such as syrups, honey, or glycerin. For astringency: galls, oak extracts, infusions of nut hulls, teas, catechu, and kino. Ambergris, a highly aromatic secretion of sperm whales that has been softened by years of oxidation on open ocean waters, was once used to achieve a \"particularly fine\" flavor, but it is both hideously expensive and prohibited now in many countries.\n\n## **_Chapter five_**\n\n## Absinthe, Cordials, and Bitters\n\n**_The ban was not so much lifted as it disappeared_**\n\nBrandies fabricated with nothing but grain alcohol, wood extract, sweeteners, and colors are the antithesis of artisanal distilling. Likewise, manufactured \"whiskeys\" that are blends of vodka, aromas, colors, and flavors are straight frauds. There's not much (legitimate) reason to make them when such good examples can be had at any decent-size grocery or liquor store.\n\nCordials and bitters, on the other hand, are something else. Some cordials, such as anisette and cr\u00e8me de menthe, are made routinely, even by respectable firms, with essential oils or essences derived from them. In this chapter, we'll round up the sweetened alcoholic beverages from Lyon's notebook and some related concoctions. In particular, we'll look at caraway, anise, mint, and ginger. We'll throw in cinnamon, coumarin, orange, and some bitters. Some use essential oils; some, spices and herbs.\n\nBut first: absinthe.\n\n### **_Absinthe_**\n\nWhen it comes to spirits in the United States, it's hard to find one more maligned and romanticized than domestic moonshine. But one runs a close second: the famous green fairy of the belle \u00e9poque, absinthe. Like many beverages in the old notebook, absinthe has\u2014at least tenuously\u2014a medicinal aspect. Lyon includes more than a dozen entries on the spirit. The high-proof spirit has been reviled for the better part of a century, accused of causing a particularly virulent (and clinically dubious) form of alcoholism 19th-century physicians dubbed _absinthism_ , but its early success is due in part to the French army's issuing absinthe as an antimalarial. French soldiers developed a taste for the spirit and spread an affinity for the stuff to cities across the country.\n\nAlthough absinthe was popular on both sides of the Atlantic in the years before American Prohibition, numbers from France demonstrate the surge in its popularity from the late 19th century until just before World War I. In 1875, total French consumption of _la f\u00e9e verte (the green fairy)_ was 15,500 hectoliters\u2014about 410,000 gallons of pure alcohol. In 1908, the volume increased tenfold to over 4 million gallons. By 1913, the volume had grown to 6,327,000 gallons, a whopping 60 liters per inhabitant\u2014more than half a cup of absinthe per person each and every day of the year.\n\n### SCHWEIZER ABSINTH\u00d6L (SWISS ABSINTHE OIL)\n\n**Like most of Lyon's recipes, his \"Swiss\" absinthe oil is pinched from someone else. In 1925, the identical recipe is included in a long list of wormwood preparations deep in the bowels of _Hagers Handbuch der Pharmazeutischen Praxis_ , a German druggists' manual. The recipe is an example of a compound oil. Rather than blending so much of various essential oils with alcohol, a compound oil made of multiple oils would simply be added to a base alcohol and perhaps syrup. Recipes like this one are sometimes called essences or concentrates.**\n\n---\n\nAnis\u00f6l (anise oil) | 350 g\n\nFenchel\u00f6l (fennel oil) | 130 g\n\nR\u00f6m. Kamillen\u00f6l (Roman chamomile oil) | 6 g\n\nSternanis\u00f6l (star anise oil) | 133 g\n\nWermuth\u00f6l (wormwood oil) | 300 g\n\nWermuthessenz (wormwood essence) | [40 g]\n\nVeilchenessenz (violet essence) | \u00e0\u00e0 40 g\n\nWith diluted spirits of wine, makes Swiss Absinthe.\n\n**O, for a Ruse of Fire!**\n\nToward the end of the 20th century, fire arrived on the absinthe scene\u2014such as it was\u2014a sort of marketers' sleight of hand. I'm not an absinthe purist and have no compunction about using the high-proof spirit in such cocktails as a Sazerac or a Death in the Afternoon. I stand, however, with absinthe historian Marie-Claude Delahaye who declared flaming absinthe \"a heresy from the Eastern countries.\" See, fire was never part of traditional absinthe preparation in bars or caf\u00e9s in Europe or America. The pyrotechnics were a 1990s ruse designed to distract novice drinkers from the low quality of so-called Bohemian absinth(e)\u2014cold compounded wormwood-flavored spirits from the Czech Republic, usually lacking absinthe's other traditional botanicals, such as fennel and anise. Flames also highlight the stuff's high alcohol content.\n\nThe maneuver is simple; place a sugar cube on a perforated spoon over a thick glass containing the spirit, moisten the cube with a bit more, and set the booze-soaked sugar alight. The sugar melts and falls into the glass where it sets the spirit on fire. A dose of cold water extinguishes the flame and it's stirred until the whole thing may (or not) achieve a mild louche. The routine ruins spoons, makes glassware hard to clean, and introduces off flavors. It is a maneuver calculated to delight rubes and newbs.\n\nSince we are neither, we do _not_ set our absinthe on fire.\n\n**Absinthe Cheat Sheet**\n\nJust as whiskey, rum, and gin have definite styles, classic absinthe varieties can be broken down a few ways. In broadest strokes, there are green ( _verte_ ) which typically are colored by a maceration of herbs after distilling and white ( _blanche_ ) types which are not. But that's a crude measure. Differences also derive from botanicals used to prepare each. First, let's look at what they have in common.\n\n**The Flavor**\n\nClassic absinthe is a lightly bitter, high-proof spirit flavored with anise, fennel, and wormwood. Variation among the botanicals yields several styles (see \"Different Styles,\" here) and plenty of recipes exist that may omit one or more of the aforementioned trio\u2014though purists might not deem them true absinthes. The best absinthes are distilled spirits, but there was once a robust market for cold compounded versions made with oils. The majority of absinthe aficionados regard such compounded absinthes as beneath notice. The green color so common to absinthe comes from steeping botanicals in the distilled spirit (which is as clear as water otherwise), a final adjustment of flavor and look before bottling.\n\n**High Proof**\n\nThe highest grade absinthes are strong spirits, usually 65% to 72% abv. Exemplars of green\/ _verte_ absinthes are usually called _absinthe sup\u00e9rieure_ , while the highest grade of white\/ _blanche_ are _absinthe suisse_ (\"Swiss\" absinthe). A step down from that was _absinthe fine_ with slightly lower proof. Below that was _demi-fine_ , which ranged from 40% to 53% abv. _Absinthe ordinaire_ was an everyday\/for-the-masses version subject to much adulteration and often made with inferior spirits. The categories were not hard and fast, but as rough analogs similar to today's premium, midrange, well, and bottom-shelf spirits, they give you a broad feel for what each means without getting bogged down by the particulars.\n\n**Unsweetened**\n\nMany drinkers, especially in my home, sweeten their absinthe in the glass by melting sugar cubes suspended over the surface with cold water, but the spirit itself has no sugar. A lower-proof cordial version called _cr\u00e8me d'absinthe_ did have syrup added.\n\n**Different Styles**\n\nAnise, fennel, and \"grand\" wormwood ( _Artemisia absinthium_ ) are sometimes thought of as the \"Holy Trinity\" of absinthe botanicals. Three more complement them in most absinthes: petite wormwood, hyssop, and melissa (lemon balm). According to _A Treatise on the Manufacture and Distillation of Alcoholic Liquors_ \u2014the premier 19th-century distillation manual by Pierre Duplais\u2014variations in the amounts and other botanicals account for differences in style named after the French cities in which they were once popular.\n\n**Absinthes**\n\n**Absinthe Exotica**\n\nWe rarely see whiskey aficionados mixing different whiskeys in one glass. Gin lovers are loath to stir a martini with multiple gins. But the tiki crowd has no compunction about blending two, three, or even more rums in one glass (or mug or bowl) in an effort to get just the right flavor for a drink. They also deploy, from time to time, drops and dashes of absinthe to get the right flavor in such tropical libations as the Cobra's Fang, the Cuba Kula, and the Zombie, all from the 1930s. Anywhere from 3 to 6 drops of absinthe can lend an indefinable something to a drink, an _I-can't-quite-put-my-finger-on-it_ exoticness that even people who profess not to like absinthe can dig. More than 6 drops, though, and the cat's out of the bag.\n\n### COMPOUND LIQUEUR OIL FRENCH ABSINTHE\n\n**Compounders and rectifiers embraced essential oils centuries ago for making spirits and cordials with varying degrees of success, but wormwood oil often proved problematic. As far back as 1805, Richard Shannon called it \"one of the more ungrateful oils\" that smelled of wormwood, but had \"little or nothing of its bitterness.\" Modern users have called wormwood oil flabby, vegetal, and rank\u2014unsuitable for making essential oil-based absinthes.**\n\n**When I asked Gwydion Stone, distiller of Marteau absinthe and fellow tiki connoisseur, whether he knew formulas for respectable oil-based absinthes, he said that it's possible in theory to make a decent absinthe by cold-compounding oils, and there were apparently some made in the pre-ban era. \"The problem these days,\" he explained, \"is that there's no absinthium oil being produced for the beverage trade; it's all made for medicinal applications. Consequently there's more focus on capturing the therapeutic elements and none on good flavor and aroma.\" One ounce of this compound oil flavors a gallon of spirit.**\n\n---\n\nOil Coriander | 9 drams\n\nOil Cloves | 9 drams\n\nOil Cassia Cinnamon | 9 drams\n\nOil Fennel | 1 ounce\n\nOil Star Anise | 2 ounces 5 drams\n\nOil Wormwood | 11 ounces\n\nTotal: | 1 lb\n\n**The Louche**\n\nOil and water, so goes the old saying, don't mix. And that makes a very nice effect for people who like anise-flavored alcoholic beverages, such as pastis, raki, sambuca, and ouzo. Like them, absinthe turns translucent when brought down to drinking strength. Just as adding high-proof spirit can clear cloudiness in some spirits made with essential oils, cold water creates an emulsion of an essential oil that had been dissolved in the high-proof spirit. The main component of that essential oil is trans-anethole, an ester with a strong anise aroma. It's highly soluble in ethanol, hardly at all in water. At a target 5:1 ratio of water to absinthe, the alcohol content is about 12 to 14 percent, the same as in a glass of wine. But the result with so much cold water is cloudy, slightly milky, almost opalescent, shot through with swirls of whites, greens, and even faint blues. Watching a swirling louche form is particularly mesmerizing as the afternoon sun splays through a window and into the glass.\n\n### ABSINTHE\n\n**With its sugar and anise oil added to a nonstandard botanical bill (where's the fennel?), this recipe might turn off aficionados, but it's in line with many of the sorts of absinthe variants in older books\u2014and among today's home distillers.**\n\nLeaves and tops of wormwood plant 4 pounds, Angelica root, calamus root, anise seed and dittany leaves 1 ounce each. Brandy or spirit (12 U.P.) 4 gallons.\n\nMacerate 10 days add 1 gallon water. Distill 4 gallons at gentle heat and dissolve in the distilled spirit 2 pounds white crushed sugar[.] Flavor with few drops of anise.\n\nCalamus is a sweet-smelling plant that grows along waters' edges across most of Europe, into Asia, and parts of North America. Its root (a rhizome really) imparts a bitter flavor with overtones of cinnamon and ginger in countless 19th century bitters, cordials, and spirits formulas. Calamus is not approved as a food additive in the US because some varieties may be carcinogenic and cause vomiting.\n\n**Absinthe: An Early Victim of Prohibition**\n\nSo many absinthes are available to modern shoppers. Even a modest liquor store might offer a half-dozen choices. In a shop where the proprietor takes a personal interest, that number easily could double or treble. Some bartenders and journalists will tell you that that's because \"the ban\" on absinthe was lifted only recently. They might even pinpoint the year to 2007 and National Absinthe Day, a publicity ploy that originated with savvy absinthe company Lucid. It's a tidy little story, but that's not how it went down. In fact, no ban was lifted, no laws were changed, no regulations updated. Yet now we have absinthe and 15 years ago we didn't. What changed?\n\nAbsinthe was indeed banned in the United States at one time. Following the leads of Holland, Belgium, Switzerland, and Brazil, the US Department of Agriculture outlawed the spirit with Inspection Decision 147 in 1912, making it an early victim of the burgeoning temperance movement. Italy did the same in 1913. France itself, the heart of absinthe drinking culture in Europe, followed in 1915 on fears that the vast consumption weakened its military readiness. Knowing the ban was coming, many dedicated drinkers hoarded supplies, making the 1913 bottling one of the more common vintage expressions on today's vintage spirits market. Germany finally made absinthe verboten in 1923.\n\nThe thing is, sometime\u2014probably in the 1960s, according to _absintheur_ Gwydion Stone (see \"Serving Absinthe,\" here), though nobody seems quite sure exactly when\u2014absinthe's ban in America seems to have fallen through the cracks in a bit of bureaucratic reorganization. Absinthe has been perfectly legal for more than 40 years, but until the 2000s, nobody thought to challenge conventional wisdom that dictated that it wasn't.\n\nThe ban wasn't lifted so much as it disappeared\u2014probably before Neil Armstrong first stepped on the moon.\n\n**Absinthe Cocktails**\n\n### SERVING ABSINTHE\n\n**Last Spring, I visited Gwydion Stone and his wife, Trinity, in Seattle. Stone distills Marteau absinthe and is cofounder of the Wormwood Society, an absinthe appreciation group. Our long afternoon at their home elided into evening, eased along in part with a bottle of 1913 Pernod Fils absinthe. We didn't break the seal and we didn't kill the bottle, but the food, conversation, and generosity combined to make one of my favorite memories of 2014.**\n\n**We drank that pre-WWI absinthe in the classic manner with nothing more than ice water and a little sugar\u2014and some specialized gear. Absinthe glasses usually have a mark or reservoir for a specific measure of the spirit, usually one ounce. If you don't have one, a wine glass is perfectly fine. The sugar is optional. Some like it, some don't. You will note, however: no fire.**\n\n4 to 5 oz water\n\n1 oz absinthe\n\n1 sugar cube (2 if they're small)\n\nFill a carafe with ice and water. Pour the absinthe into an absinthe glass, and place a perforated absinthe spoon on top, spanning its rim from side to side. Place the sugar cube on the spoon. Drip a bit of ice water over the sugar cube until it is soaked through. Then slowly drip more water over it, dissolving the sugar until the absinthe louches\u2014turns completely opaque, a pale opalescent green. Stir briefly. Sip. Revel in the sensation of being the cat's pajamas.\n\nThe \"save the children\" trope is nothing new. Shutting down saloons was positioned early on as pro-family, anti-crime propaganda.\n\n### ABSINTHE SUISSESSE\n\n**This New Orleans brunch classic is not as well known outside the city as the Sazerac or Ramos Gin Fizz, but it's as good an introduction to absinthe cocktails as you're likely to get. Well... after the Sazerac, perhaps. Although modern recipes are apt to omit it, cr\u00e8me de menthe harkens back to 1930s versions of the drink. If mint's not your bag, ditch it, then double both the orgeat and the half-and-half.**\n\n1.5 oz absinthe\n\n0.5 oz white cr\u00e8me de menthe (or any of the mint cordials in this book)\n\n0.25 oz orgeat\n\n1 fresh egg white\n\n1 oz half-and-half\n\nDry-shake all the ingredients (i.e., in a shaker without ice) until blended, 10 to 15 seconds. Add ice and shake again until well chilled. Strain into a chilled cocktail glass. Alternatively, combine the ingredients in a blender with \u00bd cup of crushed ice and blend for 5 seconds. Pour unstrained into a rocks or highball glass.\n\n### MONKEY GLAND\n\n**While Prohibition raged in the United States, the French surgeon Serge Abrahamovitch Voronoff gained first fame then notoriety for his procedure that grafted tissues from monkey testicles to men's testicles. Oh, laugh it up, but Voronoff made a huge name\u2014and tidy fortune\u2014for himself doing it as a way to increase vitality. The gin-and-absinthe cocktail named after his procedure may not provide quite the same boost, but do make sure the OJ is fresh. If you have it, blood orange or Cara Cara is the way to go.**\n\n1.5 oz gin\n\n1.5 oz fresh orange juice\n\n1 tsp grenadine\n\n1 tsp absinthe\n\nShake the ingredients with ice; strain into chilled coupe or cocktail glass.\n\n### ABSINTHE ONIONS\n\n**If the closest the old notebook comes to cocktail recipes is a brace of rum shrubs (seehere), it's no surprise that Lyon is reticent about garnishes. I'm pretty sparing with them myself. Every few months, though, I put up a small jar of briny cocktail onions in absinthe. In ones and threes, I drop them rather than an olive into a martini for a bitter twist on a Gibson cocktail.**\n\n**I use 120-proof Absinthe Verte from St. George Spirits in this, but use what you've got.**\n\nDrain the brine from one 8\u2013fluid ounce\/237 ml jar of pickled cocktail onions. This should be about 4 ounces\/120ml. Pitch the brine (or take it as a pickleback if you're into that sort of thing). Top off the onions with the same volume of absinthe, reseal the jar, and pop it in the fridge. These onions keep for months, but should be good to use in 3 to 5 days.\n\n### **_Bitters_**\n\nPennsylvania physician Jacob Hostetter developed a recipe for bitters to combat constipation, colic, and fever in the early 19th century. When he retired in 1853, his son David began to manufacture and sell the nostrum. Young David for years poured most of his profit into advertising. By the time the Civil War broke out, the name Hostetter was so well known that the Union Army routinely ordered it by the train carload as medicine for its troops, leading to a handsome fortune for the Hostetter family. Dr. Hostetter's concoction\u2014which came in at a whopping 47% alcohol\u2014enjoyed decades of strong sales, even if the proof was reduced in later years. Prohibition, however, dealt a blow to the company from which it never recovered. By the 1950s, it was history.\n\nIn intervening decades numerous recipes arose which emulated the famous stomach bitters. Bitters like these are not the same as cocktail bitters such as Angostura, Fee Brothers, Abbott's, or Peychaud's. Rather, they are more akin to medicated wines meant to settle rumbly bellies and subdue a variety of complaints. Its bitterness (derived from orange peel, gentian, and cinchona) is about on par with tonic water\u2014but with a decided orange punch. I use it sometimes as a nightcap or, if I've had too many nightcaps, first thing as a morning pick-me-up.\n\nEven before Prohibition, there was grumbling and grousing from bartenders and temperance advocates alike about druggists who sold, in addition to medical spirits, all sorts of boozy bitters, tonics, and nostrums to a thirsty public.\n\n### HOSTETTER(ISH) BITTERS\n\n**Gum kino provided astringency and a bit of color to 19th-century cordials and bitters. Astringency is not bitterness exactly, but more like a \"dryness\" in drinks that's often associated with tannins. Some red wines, for instance, can seem dry. A century after Lyon lived, kino is an elusive ingredient, so I've swapped it out for one that's as close as the nearest grocery: black tea. Not just any tea, mind you; _overbrewed_ tea. Three to five minutes of steeping, depending on the kind of tea, will give you a decent cuppa. Let that same tea steep for half an hour, though, and you've got a mouth-twisting dryness that will earn you the stinkeye from finicky tea drinkers (including me). That's exactly what you want here.**\n\n---\n\n**TINCTURE**\n\n|\n\nBitter orange peel, dried | 36 g\n\nGentian | 21 g\n\nCinnamon | 3. 5g\n\nAnise seed | 7 g\n\nCoriander seed | 7 g\n\nCinchona, crushed | 7 g\n\nSpirit 96% abv | 237 ml\n\n**TEA**\n\n|\n\nLoose-leaf black tea | 3 Tbl\n\nBoiling water | 1.5 quarts\n\nDay One: bruise all spices in a mortar and put into a clean, dry, wide-mouth jar. Rinse the mortar with half the spirit. Pour this onto the bruised spices in the jar. Repeat with the remaining alcohol. Let it all infuse for 1 week to create a tincture.\n\nDay Eight: brew the tea an hour or two before proceeding with the rest of the recipe. Pour the boiling water over the tea leaves in a pot, and cover off the heat. Allow to steep for 30 minutes, then strain and set aside to cool. Discard the leaves.\n\nNext, strain the spice tincture into a clean jar. Set aside. Put the dregs\u2014the spice remnants you strained from the now-dark tincture\u2014into a medium pot. Add 16 fluid ounces\/474 ml of water. Bring the mixture to a boil, cover, lower the heat to medium-low, and simmer for 20 minutes.\n\nMeanwhile, put \u00bd pound\/227 g of sugar into a 2-liter jar. When the 20 minutes are up, strain the hot liquid from the dregs into the sugar, pressing the wet mass with the back of a spoon to extract every last bit of liquid (within reason; you want to be thorough, not manic). Alternately, run the mixture through a B\u00fcchner funnel (see here) Discard the dregs. Stir to dissolve. Add the tincture you set aside earlier. Add the cool tea. Stir to make sure the sugar is completely dissolved.\n\n### **_Cordials_**\n\nIn North America, the terms _cordial_ and _liqueur_ tend to be used interchangeably. Each refers to sweet, flavored spirits. The sugar content may be as low as 2.5 percent or run higher than 60 percent in older recipes. That much sugar makes a cordial little more than a highly aromatic syrup, shudderingly saccharine to modern tastes. Today, we favor much less sweet cordials, about 200 to 300 grams of sugar per liter. In the UK, _cordial_ may refer to sweetened, nonalcoholic beverages. British recipes calling for elderflower, lime, or other cordials may then instruct readers to slip some booze into the glass, or may not. They should be read closely.\n\nOrange peel flavoring allowed rectifiers to add additional layers of orange aroma and flavor to cura\u00e7aos cold-compounded solely of citrus oils, syrup, and alcohol. Peach flavoring often contained oak sawdust to emulate barrel aging. A better use of your resources? Snag a bottle of peach-flavored whiskey from Colorado distillers Leopold Bros., who actually age their spirits in barrels.\n\n**Making the Grade**\n\nBefore Prohibition pulled the rug out from under absinthe as the de facto spirit of bonhomie in the first years of the 20th century, distillers and compounders frequently divided cordials, such as anisette or cura\u00e7ao, into different grades the same way they categorized commercial absinthe. The distilling industry didn't enforce exacting standards about what set one grade of the spirit apart from another; such things fell to individual distillers or compounders, custom, and what the market would bear, but higher grades, however they were determined, commanded higher prices. We don't use grades the same way today, but if you've used _bottom-shelf_ rum, a _good starter_ whiskey, or an _ultra-premium_ gin, you get the gist of booze hierarchy. That last category is a bit of marketing hype, but, then, so was _absinthe sup\u00e9rieure_. The point is that knowing about old-fashioned grades is a useful framework for understanding vintage spirits and, consequently, why old cocktail recipes sometimes don't work with modern spirits. Old and modern ingredients might share a name, but alcohol content and ingredients might have shifted considerably over the decades or centuries since recipes were written.\n\nFive variables stand out, though, in old formulas that affect the grade of a spirit or cordial. The first is the **production method**. Was the stuff distilled or cold compounded? With rare exceptions\u2014for instance, in the case of mint cordials, which were almost universally made from mint essential oil and spirit\u2014producers and customers alike regarded distilled spirits as superior to nondistilled made from essences, extracts, and oils. Distilled spirits consequently commanded higher prices. The second feature is **alcohol content**. The third, for cordials rather than spirits, is the amount of **sugar syrup** used to sweeten it. Higher grades used greater proportions of alcohol and sugar. For some cordials, 600 grams of sugar per liter was normal: pretty sweet for modern tastes when you consider that today's orange liqueurs often run around 250 grams of sugar per liter. Often, especially when compounding spirits, the number and amount of **aromatics** , such as essential oils or botanicals, factor in the \"fineness\" of a particular batch. Old cocktail recipes that call for a dash or two of cura\u00e7ao or k\u00fcmmel make more sense when you realize that such mixers might have far more intense flavors and aromas than modern examples that simply get lost in mixed drinks.\n\nThe last\u2014and perhaps most important\u2014element was the **alcohol** itself. Not just the amount of alcohol per liter, but its inherent characteristics. Did a formula call for Cognac or other brandy? Genuine rum? Or did it specify \"silent\" neutral spirits? The greater the proportion of neutral spirits in formulas, the lower the final grade tended to be.\n\nLess carefully distilled alcohol that contained heads and tails thick with unwanted compounds would often be used in lesser grades of spirits. There really weren't enforceable standards in the distilling industry that set one grade of the spirit apart from another, but some general rules held.\n\nSpecific measurements to distinguish grades of cordials from one another vary among classic textbooks, but French usage dominates in professional manuals. In general, the highest grades were _surfine_ (super fine) with the most and best-quality alcohol and highest concentration of sugar. After that was fine, then _demi-fine_ (\"half\" fine), and finally _ordinaire_.\n\nPierre Duplais's _A Treatise on the Manufacture and Distillation of Alcoholic Liquors_ is a monumental work of distilling knowledge that went through seven editions from 1855 through 1900. It remains _the_ go-to manual for 19th-century absinthe recipes. Its success bred innumerable imitators and \"Duplais\" (as the book is often called) was widely plagiarized. According to Duplais and others, the following ratios of ingredients hold for every 10 liters of cordial (about 2.64 US gallons).\n\n---\n\n**LIQUEURS ORDINAIRES**\n\n|\n\nAlcohol 85% abv | 2.5 liters\n\nSugar | 1.25 kilograms\n\nWater | 6.6 liters\n\n**LIQUEURS DEMI-FINE**\n\n|\n\nAlcohol 85% abv | 2.8 liters\n\nSugar | 2.5 kilograms\n\nWater | 5.5 liters\n\n**LIQUEURS FINES**\n\n|\n\nAlcohol 85% abv | 3.2 liters\n\nSugar | 4.375 kilograms\n\nWater | 3.9 liters\n\n**LIQUEURS SURFINES**\n\n|\n\nAlcohol 85% abv | 3\u20134 liters\n\nSugar | 5.6 kilograms\n\nWater | 2.6 liters\n\n**_Doppel, Doppelt_ ,** or **_double_** cordials tend to be stronger and less sweet (analogous to _liqueurs surfines_ ) with stronger flavors and aromas.\n\n### **Syrups**\n\nSyrups are concentrated solutions of sugars in watery fluids. More often than not, that means plain water and white table sugar. When water and sugar are the only ingredients, the result is called \"syrup\" or \"simple syrup.\" The \"simple\" here refers to its two ingredients, not a specific proportion of sugar to water, which may vary widely depending on the profession, the practitioner, and the method of measuring ingredients.\n\nThat \"watery fluid\" could be nearly anything: tea, coffee, milk, wine, or infusions of spices and\/or herbs such as cinnamon, star anise, or fennel are just the beginning. The first mint julep I was ever served used a syrup made with sugar, water, and fresh mint. It's the way I made them for years after (but please don't tell my father, a Kentucky colonel, of those unorthodox youthful indulgences).\n\nThe sugar? Well, that could be maple, Demerara, or palm sugar, to start. But don't stop there. Agave nectar gets a lot of play in bars these days and honey practically demands blending with water to make it mixable enough for drinks.\n\n**On Measuring Syrups**\n\nBar syrups can get awfully fancy, what with exotic spices, tropical fruits, unusual sugars, essential oils, and whatnot, but the simplest of them is nothing more than a solution of equal parts of plain white sugar and water (see \"A Trio of Syrups for Drinks and Cordials,\" here, for how to mix this and other proportions). Volume is a crude measure for sugar, though; the true amount of sugar (hence the final syrup's sweetening power) depends on how tightly packed or settled it may be. Think of how many baking recipes call for tightly packed brown sugar rather than just scooped; it makes a difference. In recent years, bartenders have taken to weighing their ingredients to assure greater uniformity.\n\nSome bartenders take a page from old cordial makers and measure the degrees **Brix** (\u00b0Bx) of their syrups. Brix measures the mass concentration of solids as a percentage of the final solution's total mass. Fifty grams of sugar in 100 grams of solution is the standard 1:1 simple syrup. Sixty-seven grams of sugar (rounded up) in 100 grams of syrup is 2:1 rich syrup. The compounder's standard 60% syrup is 60 grams of sugar in every 100 grams of final solution or 60\u00b0Bx.\n\n### SYRUP FOR CORDIALS\n\nOne pint of water for every 2 pounds of sugar. This makes fine syrup about 32\u00b0 Baum\u00e9. Beat of whites of two eggs (for 10 pounds of sugar or in proportion) until very frothy and mix with rest.\n\n**Eggs**\n\nEggs (usually chicken, but also quail, duck, and other species) are either used whole or separated into whites and yolks for different preparations. Egg whites are frequently used in shaken cocktails both to give a smoother mouthfeel and to soften the edges of rougher spirits. They are also mixed with wines to \"fine\" or remove suspended particles before sending to market. Yolks, on the other hand are turned with great advantage into Eierlik\u00f6r or advocaat, German and Dutch drinks roughly analogous to English and American eggnog. As everyone knows, the chalazae (those white, squiggly cords that attach fresh yolks to shells) are the most repellent things in the world. Pluck them out with the edge of a broken shell, mix and strain the yolks, or blend the hell out of your egg drinks; nobody wants one of those vile things slithering across their lips.\n\nDegrees Baum\u00e9, B\u00b0, B\u00e9\u00b0, or simply Baum\u00e9 (with or without the accent on _e_ ) is an older system used to measure density of liquids compared to distilled water (which is 0 \u00b0Baum\u00e9). Thirty-two degrees Baum\u00e9 is a 59% sugar solution, which is pretty dang close to the old compounders' standard of 60% sugar syrup (recipe follows).\n\n### A TRIO OF SYRUPS FOR DRINKS AND CORDIALS\n\nTo make **simple syrup** (1:1 or 50% sugar), combine 600 grams each of sugar and water (or, really, any equal mass of the two). No need to heat it. Shake, stir, or blend in a blender until the sugar dissolves. Prepared simple syrup will keep several weeks under refrigeration. Optional: add an ounce or so of vodka or neutral spirits per liter to help prevent mold growth. Or maybe next time, just don't make so much.\n\n**Compounder's syrup** (3:2 or 60% sugar) must be heated to dissolve the sugars. To make it, combine 600 grams of sugar and 400 grams (milliliters) water. Proceed as with rich syrup, below. When using compounding syrup in cordials, first mix cold sugar syrup with any water called for, then add the alcohol. After that, blend essential oils dissolved in high-proof alcohol and other aromatic compounds, wines, or juices. Adjust the color with caramel, cochineal, saffron extract, or other colorants.\n\n**Rich syrup** (2:1 or 66.7% sugar) also needs to be heated. To make it, combine 600 grams of sugar with 300 grams (milliliters) of water in a pan (or, again: whatever amounts you want as long as you keep that 2:1 ratio). Heat the mixture slowly, stirring as needed, until the sugar is dissolved. Add spirits as above if you like, but rich syrup is not likely to develop molds if kept at a cool room temperature in a dark place.\n\nBrandy makers, especially those who follow French methods that call for adding small amounts of sweeteners to balance the final spirit, often age their syrups for decades. Just as whiskeys, rums, and other spirits change over the years, syrups aged in toasted oak transform, becoming more complex and richer. Thirty-, 50-, or even 100-year old syrups are not practical for most homes, but with small oak barrels on the market, and a little patience, more modest aged syrups are achievable.\n\n**Flavoring Syrups with Essential Oils**\n\nSome bartenders will add scant drops of essential oils to granulated sugar, let the mixture rest a bit, then stir in hot water until the sugar dissolves. And it's easy to see why they would. After all, the _oleo saccharum_ ritual that starts off so many bowls of punch is just such a mashup of oils and sugar. The name literally means \"oil-sugar\" (well, in dog Latin, even if classicists take exception to the term). But ever since David Wondrich's _Punch_ , a grand exploration of proper flowing bowls, came out in 2010 and reintroduced drinkers to the practice, making an _oleo saccharum_ has become de rigueur among cocktail types. Mashing sugar and oils together just _feels_ right.\n\nBut don't do that. Yeah, sure, you'll get a syrup, but you're not making the best use of sometimes pricey ingredients. Keep in mind that essential oils are also called _volatile oils_ ; the reason they smell so strongly when you open a bottle of, say, orange or cumin oils, is that molecules break free from the main body of oil and disperse in the air. Adding heat\u2014whether it's hot water or other liquids such as tea or milk\u2014increases the rate of reaction and sends even more aromatic compounds into the air. Smells nice, but you want that in your drink, not wafting into the ether of your kitchen or prep area.\n\nRather, add oils to _cold_ or room-temperature syrup made in advance. Doing so helps to keep essential oils stable. It also results in clear syrups, unclouded with oils.\n\n### GUM SYRUP\n\n**While it hasn't returned as a staple in every bar, sugar syrup known as _gomme_ , or gum, has made a comeback in craft cocktail bars. Gum is thickened with dried and ground resin of various acacia species. The rehydrated resin gives syrups a round, supple mouthfeel. If, as some observers would have it, some 90 percent of pre-prohibition American whiskeys were adulterated by the time they reached consumers' lips, such \"gum\" syrup would have helped some of the rougher burrs. Harsh spirits or no, the velvet feel cocktails and cordials made with gum syrup are well worth reviving.**\n\n2 oz\/55 g gum arabic\n\n6 oz\/180 ml water, divided\n\n12 oz\/340 g superfine sugar\n\nIn a small container, combine the gum arabic and 2 ounces\/60 ml of the water. Stir to blend, and cover. Let sit for 48 hours, or until the gum arabic is completely dissolved.\n\nIn a small saucepan, combine the sugar and remaining 4 ounces\/120 ml of water. Heat gently until the sugar begins to dissolve, and then fold in the gum arabic mixture. Remove from the heat immediately, let cool, and bottle. Store in the refrigerator indefinitely.\n\n### **K\u00fcmmel**\n\nPoor, ignominious k\u00fcmmel. Moonshine may be sneered at, absinthe misunderstood, but k\u00fcmmel has languished, practically forgotten, for the better part of a century. The war, of course, is to blame. When World War I erupted in 1914, Americans\u2014their patience already tested by what they felt were the divided loyalties of German Americans\u2014demonized all things Teutonic. Even speaking German on a telephone could get a call disconnected. Speaking it in the street? Why, that could get you beaten. As a readily identifiable German tipple with that insufferable... nay, even treasonous _umlaut_ , the caraway-flavored cordial took a big hit; by the 1960s mostly old-timers drank it. When they died off, so did sales.\n\nAt its most stripped-down, k\u00fcmmel (a.k.a. kummel and kimmel) is unsweetened brandy flavored with caraway seeds. Bartenders who once concocted their own caraway brandy might have used such a barebones version in dashes and squirts to flavor drinks, much as they used bitters or dashes of clove tincture. When bottled for sale, however, it was often sweetened. Ice caraway is a particularly lovely expression in bottles so rimed with sugar crystals that the whole thing seems frosted. Caraway was often k\u00fcmmel's dominant flavor, rather than the only one. Fennel and cumin frequently appear in old recipes, but so do coriander, star anise, cinnamon, rose, bitter almond, lemon, orange, and other flavors. Of modern brands, my favorite is Combier's Doppelt K\u00fcmmel Extra, but making your own gives additional latitude for personalizing this old cordial.\n\nLyon's notebook contains over a dozen recipes for making this once-important beverage. But the Germans weren't the only ones drinking caraway-spiked spirits and cordials. English and American bartenders in the 19th century used strongly flavored caraway cordials. An 1855 report on wood alcohol-spiked ethanol (\"methylated spirit\") in England noted that while a large pub may retail 1,200 gallons of gin in a month, it may only sell 10 to 12 gallons of cordials:\n\nThe substances chiefly used in flavouring cordials are caraway, cloves, and aniseed. The methylated spirit could not be used for any of these liquors. Indeed, from their being generally made use of to give an extempore flavour to gin at the option of the customer, more than usual attention must be paid to their own purity of flavour. (Graham, Hofmann, and Redwood, \"Report on the Supply of Spirit of Wine Free of Duty for Use in the Arts and Manufactures,\" _Pharmaceutical Journal and Transactions_ , 14, no. 9 [1855]: 556, 562)\n\nFrom the 1905 _The Hoffman House Bartender's Guide_ , here is a simple k\u00fcmmel for dashing into drinks:\n\n### CARAWAY BRANDY\n\nSteep one ounce of caraway seed, bruised, in one pint of brandy. In one week strain. Add six ounces loaf sugar.\n\nCaraway also gets extensive play in northern European drinks. It is the cornerstone of aquavit, a distilled Scandinavian spirit whose history goes back centuries and whose name (also rendered akevitt and akvavit) is derived from Latin _aqua vitae_ (\"water of life\").\n\n**_Doppel_** (also **_Doppelt_** or **double** ) k\u00fcmmels are stronger, drier, and may have more botanicals or oils. Historic recipes tend to run 45%\u201350% abv and about half the sugar (10%\u201312%) one could expect from an everyday k\u00fcmmel.\n\n### SIMPLE CARAWAY VODKA\n\n**It's not the same as the genuine article, but a simple, stripped-down k\u00fcmmel (or is it an akvavit?) can be made with nothing more than caraway seeds and vodka. Soften it if you like with up to a tablespoon of syrup. Small additions of fennel, star anise, or lemon zest are not hateful.**\n\n1.5 to 2 tbl caraway seeds, lightly crushed\n\n750 ml vodka, 40%\u201350% abv\n\nCombine the caraway and vodka in a 1-liter jar and macerate anywhere from 4 hours to a week. Strain into a clean bottle. Serve cold from the freezer.\n\n### TWO HISTORIC K\u00dcMMEL RECIPES\n\nFill three quarts of cognac or kirschwasser, six ounces of broken caraway, and   of an ounce of star anise into a glass bottle, close it with a bladder, and place it in a pot partly filled with cold water; now heat this, and let boil for half an hour; take the pot from the fire, and let the bottle get cool in the water, then sweeten the liquor with two pounds of refined sugar; filter, bottle and cork well.\n\nWith the aid of oils the method of manufacturing is as follows: dissolve 30 drops of caraway extract, 2 drops of fennel oil, 1 drop of cinnamon oil in one ounce of spirits; mix this to four quarts of cognac and three pounds of refined sugar; filter and bottle.\n\n### K\u00dcMMEL\n\n**From _The Druggists' Circular_ (1913) comes a recipe that more closely cleaves to Alasch, a Russian k\u00fcmmel variant:**\n\n---\n\nOil of anise | 2 drops\n\nOil of bitter almond | 2 drops\n\nOil of parsley | 6 drops\n\nOil of rose | 8 drops\n\nOil of caraway | 2.0 grammes\n\nTincture of vanilla | 2.0 grammes\n\nSweet spirit of niter | 20.0 grammes\n\nAlcohol (90%) | 3650.0 grammes\n\nSugar | 3000.0 grammes\n\nWater | 3500.0 grammes\n\nDissolve the oils in the alcohol, and add the tincture, spirit of niter, and the water in which the sugar has been dissolved.\n\n### PICAYUNE K\u00dcMMEL\n\n**From the 1910 edition of _Picayune's Creole Cook Book_ , a straightforward caraway cordial that uses essential oil. Like all good cordials, this one benefits from a rest to let the flavors marry and stabilize. Consider small additions of different essential oils\u2014just a few drops\u2014oils of fennel, star anise, lemon, orange, or other oils. Fresh dill or dill seeds make a hybrid oil-and-spice compound that drifts into the realm of Scandinavian aquavit.**\n\n1 quart of the best rectified alcohol\n\n30 drops of k\u00fcmmel oil [1\u20131.5 ml caraway essential oil]\n\n1 quart of boiling water\n\n2 pounds of the finest white loaf sugar\n\nPour the boiling water over the sugar and stir till dissolved. Bring to a boil, and let it continue boiling from 15 to 20 minutes. Strain, and when lukewarm, add the oil and the alcohol. Bottle and keep for at least two months before using. The oil is obtained from druggists.\n\n### ICE CARAWAY, OR ICE K\u00dcMMEL\n\n**Unlike everyday k\u00fcmmel, which was simply flavored with caraway oil or essence, distillers prepared the ice version by first filling clear bottles with a hot, supersaturated sugar solution. Over the course of days, delicate crystals grew, covering the bottles' interiors. Distillers then poured out the syrup and filled each bottle with k\u00fcmmel. The result was a beautiful \"frosted\" bottle of cordial.**\n\nBoil 50 pounds of hard crystallized sugar with 2 gallons of water to a syrup, filter through flannel and while still hot, add 6 gallons strong alcohol, and 10 ounces Russian caraway essence. Filter hot and as quickly as possible and fill into white glass bottles to three-quarters of their content. The bottles are stoppered and placed in a vat filled with crushed ice mixed with some table salt. While cooling the sugar crystallizes slowly and the more slowly, the more beautiful are the crystals. The bottles are finally filled with any desired caraway liquor of high alcohol content.\n\n### ICE LIQUOR\n\n**The old German favorite ice k\u00fcmmel is easy to make at home. The hardest thing about it is waiting patiently for days as sugar crystals form. Once crystals frost the inside of the bottles, fill them with commercial k\u00fcmmel, white rum, gin, or an aquavit, such as Krogstad's from House Spirits in Portland, Oregon. Don't like caraway? Not a problem. Use another flavoring\u2014cinnamon or peppermint oils, for instance, maybe with a bit of citrus oil or menthol\u2014to flavor high-proof vodka or white dog. For ease of calculations, I've increased the metric amounts slightly.**\n\n6.25 lbs\/3 kilos sugar\n\n1 quart\/1 liter filtered water\n\n2 to 3 drops essential oil (e.g., caraway, peppermint, or cinnamon) (optional)\n\n700 to 650 ml clear spirit\n\nCombine the sugar and water in a large pot. Bring to a boil over medium-high heat and stir frequently until all the sugar dissolves. Stir in the essential oil, if using. While still hot, pour the syrup through a funnel into two clean 750 ml bottles, then stopper each bottle. At this point, either place them aside in a cool space where they can remain undisturbed for several days or put them in an insulated cooler surrounded by crushed ice mixed with a handful of kosher or rock salt. When the bottles are fully \"frosted\" with sugar crystals, carefully decant the remaining syrup and fill each bottle with any spirit you like.\n\n### RUSSIAN CARAWAY\n\nThe journal contains two recipes for Russian Caraway, one for the essence, and the other for the liquor created with the essence (denoted as \"the formula\"). The essence calls for 13 ounces of carvol, also known as carvone, a compound with two forms, both of which dissolve readily in alcohol. One tastes of caraway and dill; the other, of spearmint. First isolated in 1849, each form is still used in food and candy making. Because pure d-carvone, the primary constituent of caraway oil, may be difficult to find, essential oil of caraway is an acceptable substitute.\n\n**K\u00fcmmel Cocktails**\n\n### THE K\u00dcMMEL FIZZ\n\n**When tending bar at Expatriate in Portland, Oregon, Jacob Grier devised the K\u00fcmmel Fizz as a \"Diplomatic Pouch\" drink (the bar's version of bartender's choice). Yet another reason that Portland really is a fine city in which to bend elbows.**\n\n1.5 oz k\u00fcmmel\n\n1 oz fresh lime juice\n\n0.5 oz blackberry liqueur\n\nShake, serve on rocks, then top with soda. Garnish with a lime wedge.\n\n### SAM WARD'S RECIPE\n\n**The Sam Ward Cocktail is a sweet little number either invented by or named after the son of the New York City Temperance Society's first president. Sam Junior didn't seem to take much after Pops and was a bit of a gourmand. Despite its many variants, the drink always has lemon and yellow Chartreuse. It may be as simple as a bit of yellow Chartreuse and chipped ice in a hollowed-out lemon half, but bartender Tim Daly's recipe from 1903 combines those elements with Cognac\u2014and k\u00fcmmel:**\n\nPeel a bright clean lemon in one long string; arrange same around inside of a fancy stem glass; fill centre with shaved ice, fill one third full of k\u00fcmmel, fill one third full of yellow chartreuse; add Cognac to fill glass, and serve with small straws.\n\nThis delicious decoction was devised by the famous New York connoisseur, after whom it was named; and by gentlemen who are on affectionate terms with their palates and stomachs is as fondly considered as their wives and sweethearts.\n\n### THE APPLE NAP\n\n**The Sam Ward is similar to a drink from William Schmidt's 1891 manual _The Flowing Bowl_. \"The Only\" William's recipes are a mixed bag; some work and some don't. With Cognac, his Nap cocktail is nothing special. But it got me to thinking of an old dish: apples baked with caraway and sugar. Using Laird's 100-proof apple brandy makes all the difference. A strong, spicy drink the color of chamomile tea\u2014and head and shoulders above your dance club appletinis. Do give it a try.**\n\n0.5 oz k\u00fcmmel\n\n0.5 oz green Chartreuse\n\n0.5 oz Laird's 100-proof apple brandy\n\nStir with ice until well chilled. Strain onto fresh ice and serve. A dash of rose water or rose liqueur (such as Crispin's Rose Liqueur from California distiller Crispin Caine) is an optional, but nice, touch.\n\n### MISSION BELL\n\n**Portland bartender Tommy Klus uses more than dashes of k\u00fcmmel; his modern Mission Bell is distaff cousin to margaritas, with caraway cordial in place of orange liqueur. Although it's not in his original, I like this one with a float of brick-red Ancho Reyes, an ancho chile\u2013flavored cordial from Mexico. Seehere for notes on chiles in old recipes.**\n\n1.5 oz Pueblo Viejo Reposado\n\n0.75 oz Combier Doppelt K\u00fcmmel Extra\n\n1 oz fresh lime juice\n\n0.25 oz agave syrup\n\nShake all the ingredients with ice in a cocktail shaker. Strain into a chilled coupe and serve with no garnish.\n\n### EPICUREAN\n\n**Paul Clarke, editor of _Imbibe_ magazine and author of _The Cocktail Chronicles_ , took a stab at a Cognac and k\u00fcmmel drink from David Embury's classic _The Fine Art of Mixing Drinks_. \"Embury doesn't call for it,\" he notes, \"but a slender wisp of lemon zest emancipated from the fruit and sent, with a quick twist, into the depths of the drink, certainly isn't a bad idea.\" No. Not bad at all.**\n\n2 ounces Cognac\n\n1 ounce dry vermouth\n\n0.5 ounce k\u00fcmmel\n\nDash of Angostura bitters\n\nStir with ice until well chilled and strain into chilled glass.\n\n**Cinnamon Cordials**\n\nEven when they're not the dominant flavors, cinnamon and the related spice cassia are immensely popular in old cordial recipes. Lyon includes cinnamon in about a dozen recipes, including imitation rye, absinthe, rum essence, gin, brandy, and one simple cinnamon cordial.\n\nTheir essential oils provide a warmth to the finished cordials and they're economical; in addition to being relatively inexpensive, a little goes a long way. In the last few years, Fireball, a sweetened, whiskey-based cinnamon cordial, has swept the United States. Popular at first among younger drinkers, bartenders in even swanky cocktail bars were soon downing shots of the fiery drink and slipping it into cocktails. It all seemed so new, so fresh, and maybe just a little bit of a joke to be downing something so seemingly lowbrow.\n\nBut cinnamon cordials are not new at all. Even when Victor Lyon wrote his recipe in the 1920s, the stuff was old, older than the United States even. In 1771, the Philadelphia merchants Bright and Pechin advertised such essentials as millstones, indigo, and beeswax along with cinnamon and aniseed cordials. Nor were they popular in North America alone. In 1790, John Bamford offered ten dozen hogsheads of \"exceeding fine\" cinnamon cordials in London. Such cordials, whether they were made from cinnamon or the stronger cassia, remained popular on both sides of the Atlantic well through the 19th century and regularly show up in merchants' advertisements in North and South Carolina, New York, Pennsylvania, Vermont, Illinois, Tennessee, Virginia, Ireland, England, and other places.\n\nLet's take a look at a few versions.\n\n**Cinnamon and Cassia**\n\nFirst, be aware that in the United States, \"cinnamon\"\u2014queen of the pantry in confectionery, baking, perfumery, and compounding\u2014is often a pretender. Usually derived from the bark of cassia ( _Cinnamomum cassia)_ , it may also come from its buds, leaves, or twigs. This is the flavor we have come to think of as cinnamon and is what's usually used in cinnamon buns, candies, and other so-called cinnamon treats. Cassia has a more aggressive, distinctive aroma and flavor than its relative, Ceylon or \"true\" cinnamon ( _Cinnamomum verum_ ).\n\nCeylon cinnamon is a softer, warmer spice than cassia, which is prized for its strong, almost biting taste and pungent aroma. Cassia oil (often labeled as cinnamon oil) is a particularly aggressive aromatic. Although we are familiar with its taste in countless products from toothpaste to cinnamon rolls, use cassia with caution; it easily takes over and unbalances syrups, cordials, and spirits, such as gins, if not treated with a light hand. Better to start with too little and add very small quantities until the desired taste and \"hotness\" is achieved. Dried cassia buds, which resemble cloves, were once widely used in confectionery and spirits but harder to find now.\n\n### CINNAMON CORDIAL\n\n**Lyon includes a straightforward cinnamon cordial on par with what we see in old household account books\u2014little more than alcohol, cinnamon oil, and syrup. The giveaway that it's from a professional manual rather than a home recipe book is the inclusion of talcum powder. Its purpose in a recipe like this is to absorb the terpenes in essential oils that add little to the overall flavor but lead to cloudiness in compounded beverages. When the mixture is passed through filter paper, the talcum holding terpenes gets trapped.**\n\n### HELEN SAUNDERS WRIGHT'S CINNAMON CORDIAL\n\n**Among the more detestable Prohibition laws were those that forbade publication of recipes for making alcohol. Cocktail books were allowable (even if drys pursed their lips at the idea), but not books, pamphlets, or even newspaper articles that detailed how to make those cocktail ingredients. So, in 1922, when Helen Saunders Wright's publisher reprinted her 1908 handbook about doing just that, it was\u2014in the words of Judas Priest's Rob Halford\u2014breaking the law.**\n\nThis is seldom made with cinnamon, but with either the essential oil or bark of cassia. It is preferred colored, and therefore may be well prepared by simple fermentation. If the oil be used, one dram will be found enough for two or three gallons of spirit. The addition of two or three drops each of essence of lemon and orange peel, with about a spoonful of essence of cardamoms to each gallon, will improve it. Some persons add to the above quantity one dram of cardamom seeds and one ounce each of dried orange and lemon peel. One ounce of oil of cassia is considered to be equal to eight pounds of the buds or bark. If wanted dark, it may be colored with burnt sugar. The quantity of sugar is one and one-half pounds to the gallon. (Helen Saunders Wright, _Old-Time Recipes for Home Made Wines, Cordials, and Liqueurs from Fruits, Flowers, Vegetables, and Shrubs_ [Boston: Page Company, 1922]\n\n### NALEWKA\n\n**Lyon dubs nalewka a Polish specialty. That's true as far as it goes, but his recipe is only one example of a broader type. Proper Polish _nalewkas_ are sweetened infusions of fruits or botanicals in high-proof liquor. The finished products can be as high as 75 percent ethanol\u2014tasty, but something that should be approached with caution. This version, made from oils and essence, is better in mixed drinks and pairs especially well with lemon and ginger.**\n\n**Since straight benzaldehyde can be difficult for lay folk to get, I substituted almond extract and tweaked volumes accordingly.**\n\n---\n\n**NALEWKA ESSENCE**\n\n|\n\nCinnamon oil | 5 ml\n\nClove oil | 7 ml\n\nAlmond extract | 15\u201320 ml\n\n90%-96% alcohol | 30 ml\n\nThis makes about 2 ounces of nalewka concentrate, what would have been called _essence_ or even _compound oil_ in old manuscripts. Here's how to make a cordial with it:\n\n---\n\n**NALEWKA**\n\n|\n\nNalewka essence (as above) | 4 ml\n\n40%\u201350% abv vodka | 300 ml\n\nCompounder's syrup (here) | 170 ml\n\nAdd the nalewka concentrate and alcohol to a 700 to 750 ml bottle or jar. Shake to dissolve it. Add the syrup, shake to mix thoroughly. Better when rested at least a week.\n\n### NALEWKA SOUR\n\n**The big clove note here makes this one I don't care to drink on its own, but judicious additions of nalewka to gin punches, lemonades, and sours are refreshing.**\n\n1 oz nalewka\n\n1 oz fresh lemon juice\n\n1 to 2 dashes of Angostura bitters\n\nClub soda or ginger beer (optional)\n\nShake the nalewka, lemon, and bitters with ice. Strain onto fresh ice in a tumbler. Let it stand a bit to dilute. Or top off with a splash of club soda or ginger beer.\n\n### BALLS DEEP (HOT CINNAMON LIQUEUR)\n\n**Each summer Comic-Con pulls thousands of cosplayers, zombie enthusiasts, and Hollywood celebrity types to San Diego. In the city's East Village, a bar and restaurant named Neighborhood sees a lot of that traffic. The hidden cocktail bar Noble Experiment tucked away near the kitchen is surely part of the attraction. Another may well be the house-made cinnamon liqueur. Bar manager Ethan Ostrander makes almost 2 gallons at a time. I've scaled the recipe down for a more manageable yield of about 750 ml.**\n\n300 ml Evan Williams Bottled-in-Bond bourbon (100 proof)\n\n300 ml Licor 43 (see note)\n\n100 ml water\n\n20 ml Bittermens Hellfire Habanero Shrub\n\n70 grams crushed cinnamon sticks\n\nBlend the liquids in a 2-quart nonreactive container. Add the crushed cinnamon sticks and steep for 20 minutes, stirring every five. Strain, filter, and bottle.\n\nLicor 43 ( _Cuarenta y Tres_ ) is a sweet Spanish cordial named for 43 herbs and spices that we're told go into its making. Vanilla, though, is the heavy hitter here. This liqueur provides all the sweetener in the recipe; no additional syrup is necessary.\n\n### PYROBLAST\n\n**Philadelphia bartender and author Katie Loeb makes a different hot cinnamon cordial that is as much of a blast as the name promises. Loeb calls for up to 2 teaspoons of cinnamon oil but I've tweaked her recipe to ease off the cinnamon.**\n\n1 liter 40% abv Canadian whisky, such as Canadian Hunter\n\n1 cup Pyroblast Syrup (recipe follows)\n\nAbout 1.5 to 2 ml cinnamon oil, depending how \"hot\" you like it\n\nCombine the syrup and oil. Shake or stir until the mixture is homogenous. Stir this into the whisky and mix well before serving.\n\nKatie notes that the finished cordial can be kept in a decorative decanter with cinnamon sticks, a whole star anise, and a few lemon or orange zests studded with cloves floating in it.\n\n##### PYROBLAST SYRUP\n\n3 cups water\n\n12 cinnamon sticks, broken\n\n16 whole cloves\n\n4 star anise\n\n12 allspice berries\n\n12 black peppercorns\n\n\u00bd teaspoon red chile flakes\n\n3 cups sugar\n\nBring the water to a boil in a small saucepan and add the spices. Allow to boil for 3 minutes. Add the sugar and stir to dissolve. Lower the heat and allow the syrup to simmer for 10 minutes, stirring occasionally.\n\nTurn off the heat and allow the syrup to cool to room temperature. Strain out the spices before using and funnel into clean glass bottles for storage. Refrigerate for up to 1 month.\n\nProhibition-era and collegiate slang combine in _Here's How!_ , a 1927 pocket-sized cocktail booklet. In the circumlocution of the day, Gordon Water was gin, McCarty was (Bacardi) rum, and Scotch was... suspect.\n\n### CINNAMON FIZZ\n\n**I met Massachusetts biochemist, bartender, and blogger Frederic Yarm years ago at Tales of the Cocktail in New Orleans and have always liked his serious takes on drinks. He even plays it straight with Fireball, a cinnamon-flavored whiskey cordial that serious bartenders once avoided... but now many embrace as they do Fernet. Alternatively, try this one with cinnamon cordial, nalewka, Balls Deep, or Pyroblast.**\n\n2 oz cinnamon cordial\n\n1.5 oz heavy cream\n\n1 egg white\n\n0.75 oz fresh lemon juice\n\n0.25 to 0.5 oz 1:1 simple syrup (page 000)\n\nOrange flower water\n\nDry shake all the ingredients except the orange flower water without ice to emulsify, and then with ice to chill. Strain into a Collins glass with 3 ounces of soda water. Add a straw; garnish with orange twist and finish with 3 drops orange flower water.\n\n**Anise**\n\nThere's little middle ground with anise; people seem either to like it or hate its licorice taste. _Absintheurs_ will recognize it as a big component of the favored drink while aficionados of sticky Chinese ribs may know star anise, an unrelated species that yields the same compounds. One or the other anise shows up in almost a dozen of the notebook's recipes, including absinthe, rye and \"whiskey\" ether, k\u00fcmmel, and whiskey \"spices\" meant to be enhance whiskey or straight-up fake it.\n\nLyon's simple anise cordial (recipe follows) makes a perfectly respectable anisette that few would mistake for anything else. His Cordial Liquor of Danzig, named after the ancient coastal city now known as Gda\u0144sk in Poland, has a heavy anise punch, but is rounded out with spices we're more accustomed to tasting in k\u00fcmmel. The peppermint is a bracing touch. One ounce of this compound oil is used to flavor a gallon of spirit.\n\n---\n\n**CORDIAL LIQUOR OF DANZIG COMP OIL**\n\n|\n\nRoman chamomile oil | 1 drachm\n\nOil Mace | 1 drachm\n\nOil Angelica | 14 drops\n\nOil Cloves | 1 ounce\n\nOil Peppermint | 1 ounce\n\nOil Caraway | 2\u00bd ounces\n\nOil Fennel | 3\u00bd ounces\n\nOil Star Anise | 7 ounces\n\nTotal: | 1 pound\n\n### ANISETTE DE HOLLANDE\n\n**Although many recipes use anise as the lone botanical, flavors in anisettes do get more complex. When star anise is added along with fennel, angelica, bitter almond, rose (and a few others, depending on whose formula is used), we get a more floral variety called _anisette de Hollande_ , \"Dutch\" anisette. In their 1869 manual _De Schat der Likeuristen_ , Henri Moens and Hubert Verburgt pinched Pierre Duplais's more robust formula that used a still, winnowing it down to six oils that a person could mix in the kitchen. Since pure bitter almond oil is troublesome to track down in the United States, I've adjusted their formula with almond extract.**\n\n9 drops star anise oil\n\n8 drops anise oil\n\n1 drop fennel oil\n\n1 drop angelica oil\n\n\u00bd tsp pure almond extract\n\n1 drop of rose otto or \u00bd to 1 tsp rose water\n\n60 ml high-proof spirit (75%\u201396% abv)\n\n550 ml 40% abv vodka\n\n200 ml compounder's syrup (see here)\n\nDissolve the oils in the high-proof spirit. Add the vodka and almond extract (and rose water, if using). Stir thoroughly to blend. Add the syrup and blend again. Rest for 1 month before using.\n\nNo, not you: the bottle. Rest the bottle.\n\n### ANISETTE DE BORDEAUX\n\n**The fourth edition of _The Picayune's Creole Cook Book_ from New Orleans from 1910 gives a simple recipe for making a different sort of anisette\u2014well, simple as long as you own a small still and know how to operate it. Ah, if only that were allowed to American households these days. Alas, the feds forbid home distilling.**\n\n**_Anisette de Bordeaux_ , with its heavy dose of star anise, was once hugely popular. With slight variations, this particular recipe shows up in countless pre\u2013World War I cookbooks and liqueur manuals, many of which seem to have pinched it from J. de Brevans, who was chief chemist of the municipal laboratory in Paris. _Anisette de Bordeaux_ predates de Brevans, but it was his 1890 manual _La fabrication des liqueurs et des conserves_ (issued in English translation in 1893 as _The Manufacture of Liquor and Preserves_ ) that got serious traction in America as the Golden Age of cocktails was in full swing. Even _Scientific American_ published the recipe. This version slightly tweaks metric measurements for an American audience.**\n\n**If you can find full-bodied young Hyson, a green tea from China, by all means use it. If not, substitute another green tea such as gunpowder or _long jing_ (sometimes called Dragonwell).**\n\n10 ounces of green aniseed\n\n4 ounces of star aniseed\n\n1 ounces of coriander\n\n2 ounces of hyson tea\n\n1 ounces of fennel\n\n3\u00bd gallons of alcohol\n\n10 pounds of sugar\n\n7 pints of water\n\nThe above ingredients may be purchased from first-class druggists and grocers. Take the aniseed, coriander, fennel, and Hyson tea, and pound well. Then macerate or steep for fifteen days in three and a half gallons of the finest rectified alcohol.\n\nAfter this distill in a \"bain-marie,\" or water bath. Then make a syrup with the ten pounds of sugar and seven pints of water. Mix well with the aniseed liquor and filter. Then bottle and keep in a cool, shady place. Several large bottles of anisette will be the result of the above quantities after distillation. The quantities may be reduced or increased in proportion to the amount it is desired to make.\n\n### LANIZ\u00c8T: SOUR MASH CAJUN ANISETTE\n\n**As is so often the case, not everyone wants or can afford store-bought wet goods... or has a still sitting around. Enter _laniz\u00e8t_ , Cajun anisette, made with anise oil. You can find _laniz\u00e8t_ year-round in Acadiana, though it is especially popular around Christmas. Maybe it's a nod to that season, maybe it's a hint of the potency lurking inside the bottle or jar, but south Louisiana anisette is often colored a vibrant, almost lurid, red. The recipe below is my own mashup of Ernest Matthew Mickler's higher-octane take on the classic in _White Trash Cooking_ and Cajun chef Paul Prudhomme's more supple canned version in _The Prudhomme Family Cookbook_. Let it stand six weeks before using, Mickler says, \"and you can't tell it from bought.\"**\n\n3 quarts\/2.84 liters water\n\n25 oz\/700 g sugar\n\n\u00bd teaspoon anise oil\n\n\u00bd tablespoon vanilla extract\n\n\u00bd teaspoon red food coloring\n\n3 cups\/750 ml bourbon or Tennessee whiskey\n\n5 to 7 lbs ice\n\nPour 1.5 liters of the water in a medium stockpot. Note the depth of the liquid. Later, you will boil the syrup to this height. For now, pour in the remaining water and all the sugar. Bring to a boil, stirring until sugar is dissolved. Lower the heat and simmer until the liquid reduces to 1.5 liters, 50 or 60 minutes, stirring occasionally. Remove from the heat.\n\nWhile the syrup is simmering, sterilize five new or well-scrubbed 1-pint canning jars in a deep pot or canning pot. Leave the jars in the hot water until you're ready to use them. Wash and boil the lids and rings according to the manufacturer's directions.\n\nWhen the syrup reaches that 1.5-liter mark, turn off the heat and remove the pot from the hob. Stir in the anise oil, vanilla, and food coloring until thoroughly mixed, then stir in the whiskey. Remove the jars from their hot water bath with tongs. Place the jars (don't touch with your bare hands) on a wooden surface or folded towels and immediately pour the crimson liquid into the jars up to \u00bd inch from the tops. Wipe any dribbles or spills from the rims with a clean, damp cloth and place hot lids on top with sealing compound down; screw on the metal rings firmly but not too tightly.\n\nLine your sink with a damp dish towel; it will prevent the hot jars from breaking when they touch the cool surface. Immediately place the jars upright in the sink, then slowly fill it with cool tap water so it covers the jars. As the jars cool, you'll hear a series of metallic pops and pings; that's a vacuum forming in each jar. When the jars are cool to the touch, after 5 to 10 minutes, place them upright in a tub of ice, with ice to top off the jars, to cool the anisette as quickly as possible. Once contents of jars are well chilled, about 1 hour, remove the jars from the ice. Label and date the jars, then store upright in a cool, dark place.\n\nYield: 5 pints\/2.4 liters\n\n**Ginger**\n\nGinger, one of our very oldest spices, has been a staple in cooking and medicine for millennia. While many spices have notes of this or hints of that, the tan, knobby rhizome of a ginger plant tastes boldly, brightly of ginger; it _is_ the reference note. There's ginger, though, and there's ginger. Fresh, dried, preserved, or powdered all have different characteristics. Fresh ginger holds promises of liveliness and sass, of exotic and ancient histories. There is a potency in a fat hand of fresh ginger that livens chutneys, marinades, sushi, soups, and all sorts of beverages, including ginger beer and ginger-flavored brandies, wines, and cordials.\n\nCompounding pharmacists sometimes added essence of Jamaica ginger to simple syrup (1:25 ratio), the proverbial spoonful of sugar that helps medicine go down. Tasty ginger syrup readily jumped the drugstore counter to soda fountains. Such syrup, usually made from fresh ginger rather than dried or ginger oil, has become a standard in modern bars with robust cocktail programs.\n\n### LYON'S GINGER BRANDY NO. II\n\n**Of all the recipes in the notebook, Lyon's ginger brandy is among my favorites. Although it's made from brandy, the amount of syrup here makes a proper cordial. It's sweet, versatile, and packs a fiery punch. A scaled-down version follows but frankly, we've ended up using so much of this one that I'll double Lyon's recipe from here on.**\n\n---\n\nJamaica Ginger | 8 ounces\n\nBrandy | 1 gallon\n\nBruise ginger thoroughly, macerate it in the brandy 2\u20133 weeks, strain. Then boil the marc gently in a gallon of water for 20 minutes, strain and dissolve 10 pounds of sugar in the aqueous liquid. Mix this one cold with the brandy maceration and finally clear with findings [ _sic_ ].\n\n### VICTOR LYON'S GINGER BRANDY\n\n45 g dried, cracked ginger\n\n750 ml Paul Mason VSOP brandy\n\n2 lbs sugar\n\n750 ml water\n\nMacerate the ginger in the brandy for 2 to 3 weeks. When the ginger taste is strong, make a syrup by heating the sugar and water together, stirring until the sugar is completely dissolved. Allow to cool. Strain the ginger-flavored brandy into the cool syrup and bottle.\n\n### DR. FURNISH\n\n**Doctor Lyon's ginger brandy is a fiery concoction with a great ginger taste, but it's a bit sweet on its own. Dr. Tim Furnish, pain management specialist at the University of California San Diego, mixed Lyon's ginger brandy with citrus and dry gin for\u2014well, not a painkiller exactly, but I will take two and call (a bit later) in the morning.**\n\n1 oz Victor Lyon's Ginger Brandy (see previous recipe)\n\n1 oz London dry gin\n\n0.75 oz fresh orange juice\n\n0.5 oz fresh lemon juice\n\nDash of Angostura bitters\n\nShake with ice and strain into a chilled coupe.\n\n### STRONG TINCTURE OF GINGER\n\n#### ( _TINCTURA ZINGIBERIS FORTIOR_ , TINCT. ZING. FORT., ESSENCE OF GINGER)\n\n**The 1885 BP formula Lyon specifies in his Ginger Brandy No. 1 comes from the 1885 edition of the _British Pharmacopoeia_ :**\n\nTake of Ginger, in fine powder, ten ounces; Rectified Spirit, a sufficiency. Pack the ginger tightly in a percolator, and pour over it carefully half a [British] pint of the spirit [e.g., 10 fluid ounces]. At the expiration of two hours add more spirit, and let it percolate slowly, until one pint of tincture has been collected.\n\nPowdered ginger is a mess to filter, so use cracked dry ginger instead. Macerate 10 ounces of the ginger in 10 fluid ounces of high-proof spirits (90%\u201396% abv) in a glass jar or bottle. After 3 days, draw off the amount you need from the top and leave the rest to continue to steep in the spirits until needed.\n\nFor an extempore ginger syrup, add 15 to 20 ml (3\u20134 teaspoons) of this tincture to 500 ml of compounder's syrup (here).\n\n### ESSENCE OF JAMAICA GINGER\n\n**The \"essence of Jamaica Ginger\"\u2014the infamous _jake_ of Prohibition\u2014is a separate recipe. Don't break the bank; a simple California brandy suffices. I use Paul Mason VSOP. The lemon oil is optional, but lemon and ginger are an old, comfortable couple; it's a nice touch here.**\n\n---\n\n**GINGER BRANDY NO. I**\n\n|\n\nFinest essence of Jamaica Ginger | 8 ounces (see note)\n\nOil of Lemon | 10 drops\n\nBrandy | 1 gallon\n\nSimple syrup | \u00bd gallon\n\nFor a more modest amount, try this scaled down version:\n\n---\n\nJamaica Ginger (as above) | 40 ml\n\nOil of lemon | 2\u20133 drops\n\nBrandy | 750 ml\n\nCompounder's syrup | 375 ml\n\nMix and store for several months.\n\n### GINGER WINE WITH COCHINEAL\n\n**Like caramel, cochineal is an old, old coloring. Since learning of its ability to create a strong, nearly permanent crimson dye from natives of Oaxaca, Europeans clamored for it to dye kings' robes, cardinals' vestments... and cordials. In some parts of the United States, red soft drinks are known tongue-in-cheek as \"bug juice,\" not\u2014as some would have you believe\u2014because their high sugar content attracts bugs, but because their brilliant red may have been derived from the dried insects that make cochineal. The bugs in question are females of _Dactylopius coccus_ which feed exclusively on _Opuntia ficus-indica_ , a cactus which is now naturalized around the Mediterranean, parts of Latin America... and my neighborhood, where cochineal infestation is common.**\n\n---\n\nBest ginger root bruised | \u00bd ounce\n\nCapsicum | 2\u00bd grains\n\nTartaric acid | \u00bd dram\n\nPut into one pint 95% alcohol, let it remain one week and filter. Then add \u00bd gallon water which is been boiled with \u00bd pound crushed sugar. Mix when cold. To give color boil \u00bc ounce cochineal[,] \u215c ounce cream tartar, \u00bc ounce saleratus[,] \u00bc ounce alum in \u00bd PINT water.\n\nThe inclusion of saleratus dates this recipe from 1840 to the early 1860s. Saleratus was a chemical leavener used in baking that was eclipsed with the introduction of baking powder just before the American Civil War. Adding such alkalis to cochineal solutions helped turn them a deeper, richer red. Capsicum, on the other hand, is a classic pairing with ginger. Especially as ginger concoctions age, they may lose their bite. Chiles such as cayenne subtly bolster the fiery edge of ginger without drawing attention to themselves\u2014unless we want to do just that by using large amounts or strongly flavored chiles, such as ancho, chipotle, or scorpion pepper.\n\nImitation crust on inside of wine bottles to resemble old age is a saturated solution of cream of tartar colored by Brazil wood or cochineal.\n\n**Mint**\n\nSince colonial times, Americans had settled rebellious bowels and kept chest colds at bay with doses of peppermint extract. They mixed it with water or alcohol or took the stuff\u2014one shudders\u2014straight. Mint-growing regions in New York and Michigan eventually rivaled the famed peppermint fields of Mitcham in England. By the beginning of the 20th century, some 90 percent of the world's oil of peppermint came from within a 90-mile radius of Kalamazoo, Michigan.\n\nWhere did all that peppermint oil go? Americans put a lot of it into cr\u00e8me de menthe. Newspapers and handwritten recipe books of the late 19th and early 20th centuries are stuffed with homemade cr\u00e8me de menthe recipes. And why not? It was a simple medicine\u2014and pleasant cordial\u2014that anyone could make with little trouble.\n\nLyon calls for mint in several recipes, sometimes as a component of more complex spirits such as Cordial Liquor of Danzig Comp Oil (here) or Absinthe Suissesse (here), sometimes for simple cordial. One I thought would be simple turned out a beautiful failure.\n\n### CR\u00c8ME DE MENTHE\n\n---\n\nOil of peppermint | 2 drams\n\nFresh mint leaves | 2 ounces\n\nAlcohol (90) | 5 pints\n\nSugar | 2 pounds\n\nWater to make | 8 pints\n\nDissolve the oil in the alcohol, add the mint leaves, macerate for a few days, then add some water. Again macerate. Finally filter into the sugar and dissolve. If green mint leaves are not to be obtained, a little chlorophyll may be used.\n\nThis was the first recipe I tried from the notebook that just didn't work. It was all harsh angles and bracing menthol, altogether too overwhelming with both alcohol and mint. Then there was the sugar that misbehaved. When I blended the syrup with high-proof grain spirits, the dissolved sugar came out of solution. After about two days, fine crystals appeared on the inside of the bottle. Before long, they grew into large chunks of crystal-clear sugar that glistened as they caught the sunlight. Beautiful... and undrinkable. I keep it as a souvenir, a reminder that not everything in the book ought to be resurrected\u2014and a prod to do things a little differently now and then.\n\nThat leads to Mint and Gum.\n\n### MINT AND GUM\n\n**Bon vivant and world traveler Charles H. Baker Jr.'s opinionated, two-volume food and drink recipe-cum-travelogue _The Gentleman's Companion_ (1939) unfairly languished for decades before modern booze slingers and drinkers rediscovered it. I turned to Baker for an alternate to Lyon's cr\u00e8me de menthe.**\n\n**Baker calls for a ridiculously rich syrup he calls _gum_ or _gomme_ syrup (four parts sugar to one of water, clarified by heating with an egg white). That is one way to do it, but _gum_ more properly refers to a powdered resin called gum arabic that helps stabilize syrup and lend an unbeatable smoothness to drinks. This version cuts Baker's volume and ditches his ersatz _gomme_ , in favor of a more suave rendition using actual gum arabic. The resulting cordial carries the faintest taste of toasted marshmallow; actual gum syrup is not a good sweetener for every DIY cordial, but it's right at home with mint.**\n\n270 ml compounder's syrup or gum (pages 145 and 147)\n\n475 ml filtered water\n\n10\u201315 drops oil of peppermint\n\n150 ml 96% alcohol\n\nGreen coloring, entirely optional, but a few drops will yield a muted emerald shade\n\nMix the gum syrup and water in a wide-mouthed 1- or 2-liter jar. Dissolve the peppermint oil in the alcohol, then add to the jar. Mix thoroughly. Cover tightly, let it stand for 5 days, then skim off all excess peppermint oil by laying a new white blotting paper, coffee filter, or paper towel on the surface and quickly wicking it away (a kitchen trick that helps degrease consomm\u00e9s of those last pesky bits of fat that keep them from being crystal clear). Adjust the color, if desired, with a few drops of green coloring.\n\nMakes a bit less than a liter.\n\nThe gum has a tendency to make this cordial cloud in the bottle. Mixed in spirit-forward drinks, however, such as a Stinger (here), the cloudiness dissipates quickly.\n\n### SHAKER PEPPERMINT CORDIAL\n\n**The Shakers\u2014more properly known as United Society of Believers\u2014are a religious sect known for pacifism, communal lifestyle, celibacy, and simple, but powerfully compelling craftsmanship. Their foods likewise were uncluttered, their purpose laid bare. A manuscript at the Shaker Museum in Old Chatham, New York, contains a recipe for a \"pleasant and wholesome\" peppermint cordial. It is simple, if a little one-note, but typical of homemade American mint cordials.**\n\nTake 60 drops of the oil of peppermint, 1 cup sugar, and one cup brandy. Put altogether into a marble mortar and work them well. Then add 8 quarts of water and add more brandy and sugar till it becomes a pleasant cordial. Observe that the oil of peppermint will not mix with water without some kind of spirits.\n\n### SNOWSHOE GROG\n\n**For a few short weeks in the winter of 2010, it seemed as if I might be moving to Appleton, Wisconsin. In the end, our family stayed in San Diego, which, as consolation prizes go, is not a bad one. While I was visiting to investigate the scene in snow-blanketed Wisconsin, locals introduced me to their own particular winter warmer: snowshoe grog or, simply, the _snowshoe_. A bumper of _snowshoe_ did make braving the Baltic temperatures easier. Phillips Distilling Company bottles a version for the local market, but if you feel up to making your own, try this:**\n\n1 oz cr\u00e8me de menthe, peppermint schnapps, or Mint and Gum (here)\n\n1 oz brandy\n\nStir with ice; strain into a shot glass.\n\nAlternatively, if a few days fishing on the ice is your bag, do as some other hardy sportsmen do and whip up a bigger batch: pour equal parts of brandy and mint cordial in a thermos, seal it, and gently shake to mix. Here and there, some like it hot. If that's you, mix a batch as above, heat it in a saucepan gently (remember: alcohol is flammable) on a stove just so it's hot, but stop short of bringing it to an actual boil. Pour carefully into a warmed thermos.\n\n**Stingers**\n\nSnowshoe Grog is a not a subtle drink; although it will help you get your swerve on, it's more of a way to fend off the cold. Its medicinal roots are clear in the wallop of mint packed into every shot. The Stinger, however... well, now, that's an altogether more refined drink I've happily downed even in sunny Southern California.\n\nIn 1891, William \"the Only\" Schmidt published _The Flowing Bowl_ , a collection of drinks recipes with some gems and more than a few clunkers. His Apr\u00e8s Souper is a sort of forerunner to the Stinger with the welcome addition of maraschino, a funky sweetened cherry distillate.\n\n### APR\u00c8S SOUPER\n\nDash of gum syrup (here)\n\n1 oz brandy\n\n0.5 oz white cr\u00e8me de menthe\n\n0.25 maraschino liqueur\n\nStir over ice and strain into a cocktail glass.\n\nSchmidt gives another version he calls The Judge, which has an ounce of cr\u00e8me de menthe, two of brandy, and three dashes of gum syrup to round out the rough edges.\n\n### STINGER\n\n**By 1922, when Lyon was busily writing in his secret notebook, Robert Vermeire in London released _Cocktails\u2014How to Mix Them_. Vermeire's Stinger is closer to what we're likely encounter today, even down to a dash of absinthe which, thankfully, is available again. It is my favorite of the Stinger-type drinks.**\n\n1.5 oz brandy\n\n0.75 oz peppermint liqueur\n\nDash of absinthe (optional, but give it a try)\n\nStir over ice and strain into a cocktail glass.\n\n### JEFF MORGENTHALER'S BLENDED GRASSHOPPER\n\n**Fernet-Branca is the not-so-secret \"bartenders handshake\" popular among staff in craft cocktail bars across the United States. The intense bitterness of this 78-proof amaro is muted in the tiny amount called for in this version of a grasshopper from Portland bartender and author Jeff Morgenthaler, but what a difference it makes. Its peppermint undertones double down on those mint oils from the grasshopper's traditional cr\u00e8me de menthe, while the fat from the ice cream covers the taste buds and is repeatedly melted away, layering and relayering flavors like Damascus steel.**\n\n1.5 oz green cr\u00e8me de menthe\n\n1.5 oz white cr\u00e8me de cacao\n\n1 oz half-and-half\n\n1 tsp Fernet-Branca\n\nPinch of sea salt\n\n4 oz vanilla ice cream\n\n8 oz crushed ice, prepared with a Lewis bag and mallet or a food processor\n\nCombine all the ingredients in a blender pitcher. Blend on low speed until smooth. Pour into an old-fashioned soda glass or a tall mug. Garnish with a mint sprig and serve with a straw.\n\n**Woodruff**\n\n### ARTIFICIAL WOODRUFF ESSENCE\n\n**At Prater, one of my favorite beer gardens in Berlin, I sometimes order a big, low glass of the tart and cloudy local wheat beer, Berliner Weisse. If you are so inclined (some purists sniff at the practice), you may take yours _mit schuss_ \u2014with a shot. The shot in this case is a syrup of either red raspberry or green sweet woodruff. B. G. Reynolds makes an American version of woodruff syrup that lacks the lurid artificial green coloring so often found in German examples. Lyon's artificial woodruff essence leverages coumarin\u2014the distinctive flavor of woodruff\u2014derived from tonka beans. An essence such as this would be used to flavor syrups, light German wines, lemonades, and other cooling warm-weather drinks. Two ounces of Lyon's woodruff essence is enough to flavor a gallon of syrup.**\n\n---\n\nCumarin [ _sic_ ] | 1 ounce\n\nHot Alcohol | 15 ounces\n\nDissolve then add\n\n---\n\nAlcohol | 2 pounds\n\nTincture Tonka Bean (1:5) | 1 pound\n\nDistilled Water | 6 pounds\n\nTotal: | 10 pounds\n\n### ACTUAL WOODRUFF ESSENCE\n\nIn _Das Getr\u00e4nkebuch_ , a 1938 collection of drinks recipes, Hans Kr\u00f6nlein suggests stuffing slightly wilted woodruff leaves into a bottle (the new-mown hay aroma of coumarin in woodruff intensifies as the leaves lose moisture), filling it with wine, and corking it. It is then left in the sun to become strongly scented with coumarin. A few drops of this _Waldmeister-Bowlenessenz_ in a glass of chilled white wine makes a quick _Maitrank_ (recipe follows).\n\n### MAITRANK, OR MAY WINE\n\n**When we moved into our first place in Philadelphia some 20 years ago, friends brought a big Mason jar of May wine as a loft-warming present. _Maitrank_ , as it's sometimes known, is a classic homemade German tonic with coumarin's unmistakable aroma of vanilla and new-mown hay. Like vermouth, Lillet, or Dubonnet, it is an aromatized wine, but with only one aromatic: sweet woodruff. Given Pennsylvania's centuries-old German presence and my taste for neglected drinks, it was a great fit. Here's how to make a small batch:**\n\n750 ml light white wine\n\n100 ml brandy\n\nA handful, about 50 grams, of fresh _Waldmeister_ (sweet woodruff), including white flowers\n\n1 Tbsp sugar or compounder's syrup (see here)\n\nStuff the sweet woodruff into a 1-liter hermetic jar. Add the remaining ingredients to cover the herb, seal, then give it a swirl and a shake to dissolve the sugar or syrup. Put it aside for 12 to 24 hours. A refrigerator or sunny window sill is fine; take your pick. Taste as you go; the coumarin taste can get quite strong. When it has the aroma and flavor you want, pull out the herbs. Serve chilled or top off with just enough chilled sparkling water to give it some lift.\n\n### HENLEY'S MAY WINE ESSENCE\n\n**Another version of May wine essence crops up in the fat handbook _Henley's Twentieth Century Formulas, Recipes and Processes_ , a fascinating compendium of scientific knowledge of the early 20th century. Distiller Lance Winters of St. George Spirits in California was as enthusiastic about the old tome as I am. \"If you were stranded on an island and the world has ended,\" he beamed, \"you could restart civilization with that book.\"**\n\n**_Henley's_ uses high-proof spirit, double-infused with two batches of sweet woodruff.**\n\nFresh woodruff, in bloom or flower, is freed from the lower part of its stem and leaves, and also of all foreign or inert matter. The herb is then lightly stuck into a wide mouth bottle and covered with strong alcohol. After 30 minutes pour off the liquor on fresh woodruff. In another half hour the essence is ready though it should not be used immediately It should be kept at cellar heat (about 60\u00b0F) for a few days, or until the green color vanishes. Any addition to the essence of aromatics, such as orange peel, lemons, spices, etc., is to be avoided. To prepare the Maitrank, add the essence to any good white wine, tasting and testing, until the flavor suits. (Gardner D. Hiscox, _Henley's Twentieth Century Formulas, Recipes and Processes_ [New York: Norman W. Henley Publishing Company, 1914], 770)\n\n**Orange**\n\nAs popular as sweet and bitter oranges drinks have been for the last three centuries\u2014nearly every compounder's formulary and rectifying manual printed has multiple versions of triple secs, cura\u00e7aos, and other orange-flavored spirits and cordials\u2014it's a bit surprising that Dr. Lyon's notebook doesn't contain more recipes using it. Of the 300 and some-odd entries, fewer than 10 involve either orange peel, oil, or juice. Admittedly, those for rum shrubs make up for it. See scaled-down versions in the rum section (here).\n\nOf the remainders, there's orange peel in the stomach bitters (here), sweet orange (and lemon) oil in the German absinthe (here)\u2014and one handy little tincture that proves remarkably versatile around the bar: Orange Peel Flavoring. Lyon uses it in knockoffs of port wine and genever. It's a great orange extract for flavoring cakes, frostings, and like that, but compounders have used orange peel flavoring (or tincture or extract) to introduce additional layers of orange aroma and taste to cold compounded triple secs and cura\u00e7aos.\n\n### ORANGE PEEL FLAVORING\n\nSteep 1 pound of orange peel in 1 gallon of 95% alcohol for 15 days. Filter.\n\n### MAKE-AT-HOME ORANGE PEEL FLAVORING\n\n3 to 4 ounces dried orange peel\n\n700 to 750 ml 90%\u201392% abv spirit\n\nCombine the orange peel and alcohol in a 1-liter jar, seal, and let macerate for 2 weeks. Filter and jar.\n\n### QUICK AND DIRTY ORANGE FLOWER WATER\n\n**One night this past summer, a friend gathering ingredients for a Ramos Gin Fizz at my house asked if I'd like one. Bet your ass, I did. But I also knew we were out of orange flower water, a floral essence at the core of its taste. Going to the store was out of the question, so I did what any right-thinking DIYer would: I whipped up a batch right then and there with some essential oils, a spot of vodka, and a little water. Now, my quick and dirty orange flower water isn't made according to traditional methods. And, no, it doesn't have quite the same nose or taste as what squirts from little blue bottles of my preferred brand, A. Monteux, but once it's mixed into a creamy, heat-busting Ramos Gin Fizz, it's not a bad substitute. Not bad at all.**\n\n1 oz 100-proof vodka\n\n1 to 2 drops neroli oil\n\n1 drop bitter orange oil\n\n2 oz distilled water\n\nPour the vodka into a 4-ounce bottle with a tight-fitting cap. Add the oils, stirring or swirling to dissolve. Add the water, seal tightly, and shake.\n\n### MIKE MCCAW'S TRIPLE SEC\n\n**Seattle polymath Mike McCaw is an author, publisher, still builder, distillery consultant, globetrotting explorer, and distiller. Using just four ingredients, he makes triple sec with a highly concentrated flavor that 19th-century _liquouristes_ would have called an essence. For a bottle of full-strength orange cordial, Mike adds a small amount of the concentrate to plain vodka and a bit of sugar. \"If you use a mixture of sweet and bitter orange, and less sugar,\" Mike advises, \"you can make a pretty nice Aurantii Amari clone. Use more bitter than sweet.\"**\n\n##### TRIPLE SEC CONCENTRATE\n\n2 ml oil of lemon\n\n2 ml oil of grapefruit\n\n10 ml oil of sweet orange\n\n500 ml neutral spirit (92\u201395% abv)\n\nDissolve the oils in neutral spirit. Pour the resulting concentrate into a dark, airtight jar and store in a cool place away from direct sunlight.\n\n##### TO MAKE COLD-COMPOUNDED TRIPLE SEC\n\n750 ml 40% abv vodka (divided)\n\n1.5 ml concentrate (above)\n\n140 g sugar\n\nPour 500 ml of the vodka into an empty 750 ml bottle. Add the concentrate and the sugar. Shake until the sugar dissolves and top off to 750 ml with the remaining vodka. Alternatively, put all ingredients into a 1- or 2-liter mixing flask and spin at 900 rpm until the sugar dissolves.\n\nUsable at once, but better if left to marry for 1 to 2 weeks.\n\n### DIY ORANGE CORDIAL\n\n**From the stalwarts Grand Marnier and Cointreau to 19th-century-style Pierre Ferrand Dry Cura\u00e7ao Ancienne M\u00e9thode and Solerno, a blood orange liqueur from Italy, we usually have several gallons of bittersweet orange cordials around the house. Yeah, okay, that's a bit much. Hush. The thing is, you can drink just fine without orange cordials, but you'll drink better with them. This simple DIY version makes a nice stand-in when you're on a budget.**\n\nZest from 2 to 3 navel oranges (about 10 grams)\n\n7.5 g (about 1 tablespoon) dried bitter orange peel\n\n2 cups brandy (see note)\n\n4 whole cloves\n\n600 ml (about 2.5 cups) of compounder's syrup (here)\n\nPour the brandy over the orange zest and peels in a resealable glass jar. Give it a swirl and let it rest for 20 to 21 days at cool room temperature out of direct sunlight. Add the cloves. Let rest for one more day. Strain the mixture into a large jar or mixing container. Add the syrup, stir to blend, and bottle. Let the cordial rest for another week before using. Best within 3 to 4 months.\n\nDon't splurge on expensive brandy here. Paul Mason VSOP works just fine and comes in at less than the cost of a cocktail at many swanky bars.\n\n## **_Chapter six_**\n\n## Weights and Measures\n\n### **A Pint's a Pint, Amirite? Or Amirite?**\n\nWhether we're talking about cocktails, brewing, compounding, or just mixing a bowl of punch, if you're going to read old booze recipes, you'll want to be up on the kinds of measurements likely to show up in primary sources. The hitch in that particular giddyup is that measuring systems vary over geography, time, and profession. Some measures fall out of use entirely or never were standardized in the first place. In this short chapter, we're going to look at some common ways to measure volume, weight, concentrations, alcohol content, and a few other odds and ends you may come across in old recipes.\n\nTake the pint. Just about anyone who's raised an elbow at a pub can picture the size of one instantly. Perhaps you've even heard the old chestnut that the pint's a pound the world around. Of course, this holds only for an American pint and only if you wanted to measure pure water, not beer or gin. Even then, a pound of water isn't _quite_ a full pint. But it's close enough for driving the basic relationship into schoolchildren's heads.\n\nThe problem, of course, is when one puts the saying to the test by actually going to places around the world where the British imperial pint reigns\u201420 ounces to America's 16\u2014and suddenly recipes start to get wonky. That's just one example from hundreds of different measures. Every baker and home cook knows, for instance, that a tablespoon is 15 milliliters. Except when it's not. Australian tablespoons are 20 ml. So, too, were druggists' tablespoons. Some of them, anyway. In British Edwardian and Victorian recipes, a tablespoon could be closer to 25 ml. Fritz Blank, who gave me the notebook on which this book is based, ran a high-end French restaurant and had been a major in the Army Medical Corps. His tablespoon? Twelve millimeters. What about the capacity of gallons? Of barrels? Pfft. Pull up a chair and pour a drink; that could take all night to hammer out.\n\nThe kick is that, unless you're using calibrated lab equipment, your measuring devices may not be even close to accurate. Recently, for instance, I bought a set of commercial measuring cups that supposedly ran from 100 to 2,000 ml. There was a problem. The measurement marks weren't just off; they weren't even consistently off. An engineer would say that they were neither accurate nor precise. One thousand grams of room-temperature water\u2014which should be 1 liter exactly\u2014measured 960 ml in one container, 1,050 in another, and 890 in a third. Those cups are useful for mixing, blending, and scooping, but for precise measurements, I use a different set that I know is accurate. Likewise, don't just assume that your tablespoon holds 15 ml or that your 1-liter measuring cup really does measure a liter. Either buy from a reputable specialist or double-check yourself by pouring exact weights of room temperature water into the measuring cups (here's where your scale comes in handy). Since one gram of water equals one milliliter, the various volumes should weigh the same amount.\n\n### **Understanding Pharmacists' Weights and Measures**\n\nPhysicians and pharmacists through the early 20th century relied on Latin for their notes, formulas, and prescriptions using alcohol (and everything else) because their tradition and training dictated it. As long as international colleagues could make out the handwriting, they could read and understand one another's work no matter what language each spoke at home. Perhaps just as important, writing entire bodies of work in Latin prevented lay readers from understanding material that dealt with potentially harmful, even deadly, ingredients.\n\nIt's not just their propensity to use Latin that may confound modern readers; the recipes can seem almost alien to those who are not accustomed to reading them. Weird characters that look like astrological signs and alchemical symbols litter the pages. Not to worry; once you get past scruples, grains, and drachms, the recipes are not so different from the cups and teaspoons used in cookbooks and blogs today.\n\nDespite the alien look of the symbols, Western pharmacists followed a few conventional systems of measurement: **apothecaries' measures** for liquids used units such as gallons, dra(ch)ms, and fluid ounces. **Apothecaries'** or **troy weight** and **avoirdupois weight** dealt, not surprisingly, with mass using pounds, ounces, scruples, drachms, and grains.\n\nLet's look briefly at the measures we're likely to find in old booze recipes.\n\n### **Metric Measures**\n\nAfter some false starts at the end of the 18th century, the metric system of measurement with its meters, liters, and grams spread across Europe and eventually most of the world. The United States is a notable holdout; although our retail liquor and some food is sold in metric units, Americans still use pounds, teaspoons, cups, and other household measurements in our day-to-day interactions. As for liquor manuals and pharmacopoeias, the bulk of them published since about the middle of the 1800s call for grams, kilograms, liters, and millimeters in particular. Aside from minor spelling variations around the world, metric measurements are unchanged and remain easy to use. A kilo of sugar in 1923 weighed the same as a kilo does in 2015. Two rules of thumb, though, hold when interpreting old alcohol recipes with metric measures.\n\n1. **Missing units are (probably) grams.** When a formula\u2014say, for a cordial or gin\u2014in a compounding manual does not specify units of ingredients, \"gram\" is almost always meant. \"15 calamus, 35 juniper berry, 7.2 coriander,\" for example, should be read as \"15 grams of calamus root, 35 grams of juniper berries, 7.2 grams of coriander seeds.\" Electronic scales that are accurate to within a hundredth of a gram can be had for less than the cost of a fast-food lunch. They're a better investment, anyway. If you don't already have one for your kitchen or bar, get one; they are particularly useful when precision is important.\n\nEs ist \u00fcblich, die Menge der \u00e4therischen \u00d6le in \"Gramm\" anzugeben, w\u00e4hrend man darunter \"cc\" versteht. Dies Kommt von der Einteilung der Me\u00dfzylinder her.\n\nIt is common to specify the amount of essential oils in \"grams\" while it's understood to mean \"cc.\" This comes from the division of the graduated cylinder.\n\n\u2014Hans G\u00f6ttler, _Rezeptbuch f\u00fcr Destillateure_ (1909)\n\n2. **Grams are (probably) milliliters**. Ethanol was weighed in old recipes that call for pounds or kilos of spirits\u2014and a lot of them do. A stickler for details who tries out those old recipes might also weigh essential oils, because they usually called for grams of this or that oil. In practice, \"grams\" of essential oils and other aromatic compounds, such as fruit ethers, generally meant \"milliliters.\" Although not technically correct, in many 19th-century liquor manufacturing books, \"gram\" became shorthand for \"milliliter\" because a milliliter of water weighs 1 gram. Extrapolating to liquids that were more or less dense than water would have led to some inconsistencies, but measuring the volume of multiple aromatic compounds in a graduated cylinder was more expedient than switching out counterweights on a small scale. Unless a text specifies that essential oils or other aromatic compounds truly should be weighed, read \"grams\" as \"milliliters.\"\n\nCelsius and Fahrenheit temperature measurements do affect proof readings of spirits, but the vagaries of alcohol at various temperatures is of interest primarily to manufacturers of spirits and those who collect taxes on them. Let's leave it to them. Formularies that use metric measures in the recipes themselves tend to use only two: volume and mass\/weight.\n\n**Volume Measure**\n\n---\n\n1,000 microliters (\u00b5l) = | 1 milliliter (ml)\n\n|\n\n10 milliliters (ml) = | 1 centiliter (cl)\n\n|\n\n10 centiliters = | 1 deciliter (dl) = | 100 milliliters\n\n10 deciliters = | 1 liter (l) = | 1,000 milliliters\n\n10 liters = | 1 dekaliter or decaliter (dal)\n\n|\n\n10 dekaliters = | 1 hectoliter (hl) = | 100 liters\n\n10 hectoliters = | 1 kiloliter (kl) = | 1,000 liters\n\n**Mass\/Weight Measure**\n\n---\n\n10 milligrams (mg) = | 1 centigram (cg)\n\n|\n\n10 centigrams = | 1 decigram (dg) = | 100 milligrams\n\n10 decigrams = | 1 gram (gramme, gr., or g) = | 1,000 milligrams\n\n10 grams = | 1 dekagram or decagram (dag)\n\n|\n\n10 dekagrams = | 1 hectogram (hg) = | 100 grams\n\n10 hectograms = | 1 kilogram (kilo, kg, or k) = | 1,000 grams\n\n1,000 kilograms = | 1 metric ton\/tonne (t)\n\n|\n\n**Do Use Microliters, Don't Call Them Microns**\n\nA small but growing number of modern bartenders who incorporate essential oils and related aromatic compounds in their recipes have begun using a volume measurement they've dubbed a \"micron.\" Their cordial or syrup recipes might call for 600 microns of black pepper oil, 175 microns of rose otto; that sort of thing. From California to Europe, I have tasted outstanding drinks these progressive and experimental bartenders throw down. _Micron_ , though, is the wrong word.\n\nA micron (\u00b5) is a measurement of length. A thousand of them make a millimeter, a million form a meter. Brewers, distillers, and bottlers refer to them regularly when sizing filters to prepare products for market so their beers or vermouths or what have you do not cloud, throw sediment, or develop off-tastes in the bottle. The proper term for those tiny, tiny volumes bartenders have been squeezing into spirits and cordials, on the other hand, is _microliter_. How tiny? A single microliter (\u00b5l) measures one thousandth of a milliliter and a full liter has a million of them.\n\nAutomatic pipettes that measure microliters of highly concentrated aromas and flavors offer even more precision than the drops commonly called for in aromatherapy and most home\/bar essential oil projects. One of those micropipettes can cost more than a decent mountain bike or a few years of gym membership. Plain ol' 3 ml plastic pipettes are far more affordable, even if they sacrifice precision. If you prefer punctilious cocktails to gym workouts and have the cash to splash, though, see the Resources (here) for a source.\n\nInternal Rev. Chemist G.F. Beyer testing a batch of bootleg liquor. While government reports documented the near-universal horror of these bootleg liquors, it's worth keeping in mind that the seizures of such liquor generally came from sketchy places in the first place; country clubs and well-connected joints with well-heeled crowds who could afford protection money tended to not get raided nearly as often as places that catered to poorer, often immigrant, clientele.\n\n### **Apothecaries' Weights and Measures**\n\nThroughout Europe and the United States, apothecaries used\u2014and occasionally still do use\u2014a system of measurements derived from Roman and Greek units. Unfortunately for those of us who want to understand the kinds of formulas for cordials, medicated wines, bitters, and compounded spirits found in old apothecary manuals, professionals from different medical traditions never quite managed to agree on how best to measure a Roman pound. Was it to be calculated using the weight of Roman coins? Was it perhaps somehow related to units of length? The weights of barley grains and carob seeds also came into play. The result is that over centuries and distance, such measurements as ounces and pounds rarely aligned. When dealing with concentrated substances such as essential oils or ethanol, even small variation might yield different results for those following formulas written in a distant lands.\n\nSo a 17th-century Spanish ounce may not quite align with a 19th-century German one. That doesn't mean we need to toss all the old measures overboard. Because most of the alcohol formulas we're likely to find come from 18th- through 20th-century manuals\u2014with some, admittedly, from much earlier\u2014pharmacy and medicine manuals of those eras do reveal the symbols and measurements we're likely to come across.\n\n**Weights**\n\n---\n\ngr. Granum, -i = | grain(s)\n\n|\n\n\u042d Scupulus, -i = | scruple = | 20 grains\n\n  Drachma, -ae = | drachm = | 60 grains\n\n  Uncia, -ae = | ounce (troy) = | 480 grains\n\n  Libra, -ae = | pound (troy) = | 5760 grains\n\nLb. Libra, -ae = | pound (avoirdupois) = | 7000 grains\n\nNotice two different pounds. One, the **apothecaries'** or **troy** pound, weighs 12 ounces. It's still used in some settings such as weighing precious metals. I recall feeling distinctly cheated as a child after learning that a pound of gold only weighs 12 ounces. Never mind that a grade school child has, at best, scant gold reserves. An **avoirdupois** pound, on the other hand, is the familiar \"pound of commerce\" we use to measure oranges, chicken thighs, juniper berries, and husbands; it is 16 ounces.\n\n**Volume**\n\n---\n\n  Minimum, -i = | minim\n\n|\n\ngtt. Gutta, -ae = | drop\n\n|\n\nfl \u042d or Fluidscupulus, -i = | fluid scruple = | 20 minims\n\n\u0192  or fl. dr. Fluidrachma, -ae = | fluid drachm = | 60 minims\n\n\u0192  or fl. oz. Fluiduncia, -ae = | fluid ounce = | 480 minims (8 drachms)\n\nO. Octarius, -i = | pint = | 16 fluid ounces (20 ounces in Imperial measures) or \u215b gallon\/congius\n\nC. Congius, -i = | gallon = | 8 pints\n\nA half measure (ounce, drachm, scruple, whatever), may be noted as \"ss\" for _semis_. \u042dss thus is _semiscrupulus_ , half a scruple, and  ss is semiuncia, half an ounce. Pre-1800 texts sometimes use a \"long\" or \"descending\" _s_ as the first _s_ in _semis_ (the same letter that renders Milton's _Paradise Lost_ as _Paradi\u222be Lo\u222bt_ ), so we also see _fs_ or \u00df, the German double _s_ , to indicate halves.\n\nNote that a lowercase _m_ may be used for _minim_ but may also indicate _manipulus_ (a handful). Sorry. No rule of thumb here; context is the only thing that will reveal whether you're meant to use an itty-bitty amount of liquid or a big ol' fistful of dried herbs or spices.\n\n### **II, IV, VI, VIII, This Is How We Calculate!**\n\n**Roman Numerals**\n\nOur everyday Arabic numbers (0, 1, 2, 3, etc.) make even complex calculations straightforward, but that's not the only system around. The numeration we see in pharmacy texts even through Prohibition, for example, is usually adapted from Roman reckoning. For those not accustomed to the system that uses letters as numbers, those forgotten booze formulas in pharmacopoeias can be a bit bewildering. This is not surprising, considering that lay readers weren't meant to understand them; \"bewildering\" was partly the point. Tens of thousands of recipes for compounded spirits, bitters, tinctures, and medicinal wines from the 17th through early 20th centuries use this numbering system, though, so an understanding of the basics is in order. Today, although Roman numerals linger on in dating movies, publishing, denoting sequels, and telling time, their hash mark\u2013looking method of counting has atrophied. The good news is that the ancient Roman numeral system is easy to pick up. Who, after all, is truly dumbfounded by the numerals in WW II, Superbowl XXX, or Henry VIII?\n\nIn conjunction with the apothecaries' weights and volumes, three main Roman numerals come into play in pharma-style alcohol recipes: I, V, and X. That is, 1, 5, and 10. Since, from time to time, other numerals do crop up, here's the whole set:\n\n---\n\nI, i, or j = | 1\n\nV or v = | 5\n\nX or x = | 10\n\nL = | 50\n\nC = | 100\n\nD = | 500\n\nM = | 1,000\n\nMake more complex numbers by combining these seven letters (note: there is no Roman zero). Two is noted as II (or more commonly in lowercase: ii). Three is iii. Often, the final or only _i_ in a receipt can be noted as _j_ \u2014 _ij_ is read as \"ii\" (2) while _iij_ is \"iii\" (3). One more point before we get lost in a quagmire of variants, inconsistencies, and idiosyncratic uses: writers tended not to use the same letter four times in a row. \"Four,\" then, is not iiii (except on watch faces), but iv (one less than five). Likewise, 9 is ix (one less than ten) and, on the same principle, 400 is CD.\n\n### **Kitchen and Bar Measures**\n\nFinally, what pharmacists dub _household_ or _domestic_ measures have figured into compounded recipes for hundreds of years. Most of them are familiar in American homes and bars. Keep in mind that spoons, cups, and glasses have ranged widely in their capacities over the years. Whenever possible, it's best to verify their capacity in the same text in which the recipe appears.\n\nWhen that's not possible, here are some of the more important measures you'll find in old booze recipes.\n\n* * *\n\n---\n\nTeaspoon (tsp) = | 4 to 5 ml\n\nDessertspoon = | 8 ml\n\nTablespoon (Tb\/Tbl\/Tbsp) = | 0.5 fluid ounce (the modern American standard is 15 ml, but it was 12\u201325 ml historically)\n\n3 teaspoons = | 1 tablespoon\n\n2 tablespoons = | 1 fluid ounce (fl oz)\n\n1 cup = | 8 fluid ounces\n\n2 cups = | 1 pint (pt)\n\n2 pints = | 1 quart (qt)\n\n4 quarts = | 1 gallon\n\nWineglass = | 1.5 to 2 fluid ounces (45\u201360 ml) (drinks historian David Wondrich pegs a wineglass at 2 fluid ounces, but pharmacy texts give a range)\n\nTea cup = | (about 150 ml)\n\nBreakfast cup = | 8 fluid ounces (about 240 ml)\n\nTumbler = | 11 fluid ounces (about 330 ml)\n\n### **Proof**\n\nOverproof rums, whiskeys, gin, and even some brandies have grown a lot more popular in recent years. They stand up particularly well in spirit-forward cocktails and are favorites with many drinkers, not just craft bartenders. Most folks have a sense that overproof spirits are somehow stronger than the regular offerings, but what is this \"proof\"? Simply, it's a measure of the alcohol content of a beverage.\n\nAs a measure of alcoholic strength, though, the term _proof_ has been discarded by British and American governments, both of which now measure alcohol as a percentage of total beverage volume (% abv). The term gained popularity in Britain where tax officers\u2014and navy sailors\u2014used a crude sort of trial by fire to determine how strong a spirit was by mixing a bit of the spirit with gunpowder and then attempting to ignite the mix. If the wetted gunpowder burned briskly, it was said to be \"over proof.\" Mixtures that failed to spark were \"under proof.\" A low, steady burn indicated spirits actually at proof: 57.15 percent alcohol. By 1816, the term was retired in the UK in favor of measuring alcohol content with a floating hydrometer developed by excise agent Bartholomew Sikes.\n\nWhen recipes call for brandy, whiskey, or other spirits so many degrees under proof (U.P.) or over proof (O.P.), check their provenance; British and American spirits will be slightly different.\n\nOf course, the term still enjoys lively popular use. Throughout my travels interviewing moonshiners and folk distillers in the United States, 100-proof shine remains a touchstone of authenticity, the minimum strength at which one should drink it. Since both British and American proof are deployed in formularies and pharmacopoeias, let's look at both.\n\n**American Proof**\n\nIn the United States, proof is simply twice the alcohol content at 60\u00b0F. One hundred proof is thus 50% abv. Eighty proof is 40% abv. Easy-peasy. Done. Next.\n\n**British Proof**\n\nThe old British system is a bit trickier, but only a bit. Proof in the United Kingdom doesn't quite align with American usage. The chart on here lists equivalents, but the British system boils down to this: proof spirit (e.g., 100\u00b0) is 57.15% abv which is very close to the fraction  . To convert percentage abv to degrees proof, multiply by  . For example, 100 percent (anhydrous or dehydrated) alcohol is 175\u00b0 proof [100 \u00d7 ( )]. What Americans call 80 proof (40% abv) is 70\u00b0 proof in the British system [40 \u00d7 ( )].\n\n**Two French Methods for Reckoning Proof**\n\nFrom the early 1800s until European standardization in the 1980s, France and Belgium used a particularly sensible method to reckon alcoholic strength. Their system, named after its inventor Joseph-Louis Gay-Lussac, was centesimal. That is, the entire range of possible alcoholic content was broken into hundredths. Each degree was one part per hundred parts of total volume; 0\u00b0 held zero alcohol while 100\u00b0 was pure, anhydrous spirit. It is essentially the modern _alcohol-by-volume_ scale.\n\nFrench methods weren't always so straightforward, though.\n\nCordial manufacturers and brandy blenders in the south of France once used a weight system, expressed in fractions, rather than volume to designate the strength of spirits. Ideally, such spirits were made from wine, but lesser grades from grains, beets, and marc (wine lees) did enter the market. Because French distilling and compounding manuals were so widely copied, terms such as \"spirits at three in five\" crept into books printed in other languages as well, sometimes without explanation. For the skinny on the system, let's dig into the granddaddy of French distilling texts, _A Treatise on the Manufacture and Distillation of Alcoholic Liquors_ from Pierre Duplais. This is from an 1871 American edition:\n\nThese numbers are not arbitrary; they indicate the weight, and not the volume, as some theorists have contended, of the quantity of water which it is necessary to add to any spirituous liquor to bring it to proof (Preuve de Hollande), or 19 degrees Cartier (50 degrees Centigrade).\n\nThus the three-fifths is spirits at 29\u00bd degrees, which mixed in the proportion of three parts of spirits with two parts of water, will give five parts in weight of brandy at 19 degrees.\n\nThe trois-six is alcohol at 33 degrees, of which, if three parts are mixed with an equal weight of water, will produce six parts of brandy of the same degree, or 19 degrees Cartier. [Philadelphia: H. C. Baird & Company, 1871]\n\nSo, the first, or top, number indicates the weight of 85% abv spirit, while the second, or bottom, number is the total final weight of the blend. What's missing from the designation is the amount of water needed to bring the blend to proof. You could figure that quickly enough by subtracting the denominator from the numerator... or just refer to the chart (here) that compares American and British proof with the trois-six weights as well as degrees Cartier.\n\n**Cartier and Tralles Scales**\n\nThe final two methods\u2014at least for our purposes\u2014for measuring alcoholic content in beverages are the Cartier and Tralles scales. The Cartier scale runs from 10 (0% alcohol) to 44\u215b (100% alcohol) and dates from the turn of the 19th century. We see it especially in Spanish, French, Portuguese, and Brazilian distilling and compounding texts. Equivalents are on the chart on here.\n\nGerman mathematician Johann Georg Tralles is the eponymous inventor of another scale, sometimes called the Prussian scale, based on volume. Degrees **Tralles** appear in American and German formulas and recipes for spiritous beverages. Because they are very close to the centesimal degrees of Gay-Lussac, we may regard them as equivalent.\n\n### **Parts**\n\nFinally, we have **parts** ( _Teile_ , _Theilen_ , or _Th_. in German which can look like Tb, tablespoon). Parts are simply ratios of ingredients to one another. Just as a classic Manhattan is made of one part vermouth to two of whiskey (plus bitters and, if you roll that way, a cherry, its attendant juice, and an orange twist), formulary recipes sometimes call for parts of essential oils or aromatic compounds. These are always volume measures, so each \"part\" may be a milliliter, a teaspoon, an ounce, or a big red bucket, for that matter\u2014as long as the unit is the same. For kitchen and bar use, smaller units are more manageable. Cordial recipes in particular are often written in parts of each ingredient.\n\nFor example, a compounded orange cura\u00e7ao flavor formula from _Fenaroli's Handbook of Flavor Ingredients_ uses a blend of sweet and bitter orange for the bulk of flavor, then rounds it out with smaller parts of lemon, rum ether, neroli, and warm spices. The result is more nuanced than a simple orange-flavored vodka:\n\n---\n\nAlcohol (96%) | 418\n\nSweet Orange | 350\n\nBitter Orange | 190\n\nLemon | 22\n\nRum ether | 5\n\nNeroli | 5\n\nCinnamon | 4\n\nClove | 3\n\nNutmeg | 2\n\nCoriander | 1\n\nTotal | 1,000\n\nNote that this is not cura\u00e7ao... yet. For that, add 20 ml or so of this compounded flavor, along with fresh orange peel or tincture made from peels (here), to a bottle of vodka and 100 to 200 grams of sugar. See Mike McCaw's Triple Sec (here) for directions on how to proceed.\n\n### **Proof Scales**\n\n---\n\nvolume | weight\n\n**Centesimal degrees (Gay-Lussac) (abv)** | **American proof (at 60\u00b0F)** | **British proof (1816-1907)** | **Degrees Cartier** | **Trois-Six (Duplais) approx**\n\n0 | 0 | 0.0 | 10\"\n\n|\n\n1 | 2 | 1.8 | 10 \u00bc\n\n|\n\n2 | 4 | 3.5 | 10 \u215c\n\n|\n\n3 | 6 | 5.3 | 10 \u215d\n\n|\n\n4 | 8 | 7.0 | 10 \u00be\n\n|\n\n5 | 10 | 8.8 | 10 \u215e\n\n|\n\n6 | 12 | 10.5 | 11 \u215b\n\n|\n\n7 | 14 | 12.3 | 11 \u00bc\n\n|\n\n8 | 16 | 14.0 | 11 \u00bd\n\n|\n\n9 | 18 | 15.8 | 11 \u215d\n\n|\n\n10 | 20 | 17.5 | 11 \u00be\n\n|\n\n11 | 22 | 19.3 | 11 \u215e\n\n|\n\n12 | 24 | 21.0 | 12 \u215b\n\n|\n\n13 | 26 | 22.8 | 12 \u00bc\n\n|\n\n14 | 28 | 24.5 | 12 \u215c\n\n|\n\n15 | 30 | 26.3 | 12 \u00bd\n\n|\n\n16 | 32 | 28.0 | 12 \u215d\n\n|\n\n17 | 34 | 29.8 | 12 \u00be\n\n|\n\n18 | 36 | 31.5 | 12 \u215e\n\n|\n\n19 | 38 | 33.3 | 13 \"\n\n|\n\n20 | 40 | 35.0 | 13 \u00bc\n\n|\n\n21 | 42 | 36.8 | 13 \u215c\n\n|\n\n22 | 44 | 38.5 | 13 \u00bd\n\n|\n\n23 | 46 | 40.3 | 13 \u215d\n\n|\n\n24 | 48 | 42.0 | 13 \u00be\n\n|\n\n25 | 50 | 43.8 | 13 \u215e\n\n|\n\n26 | 52 | 45.5 | 14 \u215b\n\n|\n\n27 | 54 | 47.3 | 14 \u00bc\n\n|\n\n28 | 56 | 49.0 | 14 \u215c\n\n|\n\n29 | 58 | 50.8 | 14 \u00bd\n\n|\n\n30 | 60 | 52.5 | 14 \u215d\n\n|\n\n31 | 62 | 54.3 | 14 \u215e\n\n|\n\n32 | 64 | 56.0 | 15 \"\n\n|\n\n33 | 66 | 57.8 | 15 \u00bc\n\n|\n\n34 | 68 | 59.5 | 15 \u215c\n\n|\n\n35 | 70 | 61.3 | 15 \u215d\n\n|\n\n36 | 72 | 63.0 | 15 \u00be\n\n|\n\n37 | 74 | 64.8 | 16 \"\n\n|\n\n38 | 76 | 66.5 | 16 \u215b\n\n|\n\n39 | 78 | 68.3 | 16 \u215c\n\n|\n\n40 | 80 | 70.0 | 16 \u215d\n\n|\n\n41 | 82 | 71.8 | 16 \u215e\n\n|\n\n42 | 84 | 73.5 | 17 \u215b\n\n|\n\n43 | 86 | 75.3 | 17 \u215c\n\n|\n\n44 | 88 | 77.0 | 17 \u215d\n\n|\n\n45 | 90 | 78.8 | 17 \u215e\n\n|\n\n46 | 92 | 80.5 | 18 \u215b\n\n|\n\n47 | 94 | 82.3 | 18 \u215c\n\n|\n\n48 | 96 | 84.0 | 18 \u215d\n\n|\n\n49 | 98 | 85.8 | 18 \u215e\n\n|\n\n50 | 100 | 87.5 | 19 \u00bc\n\n|\n\n51 | 102 | 89.3 | 19 \u00bd\n\n|\n\n52 | 104 | 91.0 | 19 \u00be\n\n|\n\n53 | 106 | 92.8 | 20 \u215b\n\n|\n\n54 | 108 | 94.5 | 20 \u215c\n\n|\n\n55 | 110 | 96.3 | 20 \u00be\n\n|\n\n56 | 112 | 98.0 | 21 \"\n\n|\n\n57 | 114 | 99.8 | 21 \u215c\n\n|\n\n58 | 116 | 101.5 | 21 \u00be\n\n|\n\n59 | 118 | 103.3 | 22 \"\n\n|\n\n60 | 120 | 105.0 | 22 \u215c\n\n|\n\n61 | 122 | 106.8 | 22 \u00be\n\n|\n\n62 | 124 | 108.5 | 23 \u215b | 2\/3\n\n63 | 126 | 110.3 | 23 \u00bd\n\n|\n\n64 | 128 | 112.0 | 23 \u215e\n\n|\n\n65 | 130 | 113.8 | 24 \u00bc | 3\/4\n\n66 | 132 | 115.5 | 24 \u215d\n\n|\n\n67 | 134 | 117.3 | 25 \"\n\n|\n\n68 | 136 | 119.0 | 25 \u215c\n\n|\n\n69 | 138 | 120.8 | 25 \u00be\n\n|\n\n70 | 140 | 122.5 | 26 \u00bc\n\n|\n\n71 | 142 | 124.3 | 26 \u215d\n\n|\n\n72 | 144 | 126.0 | 27 \"\n\n|\n\n73 | 146 | 127.8 | 27 \u00bd\n\n|\n\n74 | 148 | 129.5 | 27 \u215e\n\n|\n\n75 | 150 | 131.3 | 28 \u215c\n\n|\n\n76 | 152 | 133.0 | 28 \u215e\n\n|\n\n77 | 154 | 134.8 | 29 \u00bc | 3\/5\n\n78 | 156 | 136.5 | 29 \u00be\n\n|\n\n79 | 158 | 138.3 | 30 \u00bc | 4\/7\n\n80 | 160 | 140.0 | 30 \u00be\n\n|\n\n81 | 162 | 141.8 | 31 \u00bc | 5\/9\n\n82 | 164 | 143.5 | 31 \u00be\n\n|\n\n83 | 166 | 145.3 | 32 \u00bc | 6\/11\n\n84 | 168 | 147.0 | 32 \u00be\n\n|\n\n85 | 170 | 148.8 | 33 \u00bc\n\n|\n\n86 | 172 | 150.5 | 33 \u215e\n\n|\n\n87 | 174 | 152.3 | 34 \u215c\n\n|\n\n88 | 176 | 154.0 | 35 \" | 3\/7\n\n89 | 178 | 155.8 | 35 \u215d\n\n|\n\n90 | 180 | 157.5 | 36 \u215b\n\n|\n\n91 | 182 | 159.3 | 36 \u215e\n\n|\n\n92 | 184 | 161.0 | 37 \u00bd | 3\/8\n\n93 | 186 | 162.8 | 38 \u00bc\n\n|\n\n94 | 188 | 164.5 | 38 \u215e\n\n|\n\n95 | 190 | 166.3 | 39 \u215d\n\n|\n\n96 | 192 | 168.0 | 40 \u00bd\n\n|\n\n97 | 194 | 169.8 | 41 \u00bc | 8\/9\n\n98 | 196 | 171.5 | 42 \u00bc\n\n|\n\n99 | 198 | 173.3 | 43 \u215b\n\n|\n\n100 | 200 | 175.0 | 44 \u215b\n\n|\n\n**ABV in Typical Alcohols**\n\n---\n\n**ABV** | **Typical alcohols: note that outliers push the limits in either direction, but most beverages below will fall into these ranges**\n\n2\u201312 | Most beers (though some outliers reach far into double digits)\n\n9\u201318 | Most wines and vermouth\n\n18\u201322 | Fortified and \"dessert\" wines\n\n15\u201340 | Cordials and liqueurs\n\n30\u201360 | Cocktail bitters\n\n40\u2013120 | Distilled spirits such as whiskey, rum, gin, tequila, brandy, etc.\n\n55\u201372 | Absinthes\n\n75.5 | 151 proof rums\n\n80\u201395 | Commercial spirits for blending and industrial use\n\n95\u201396 | Rectified \"neutral\" spirits (e.g., grain alcohol, spirytus)\n\n96\u2013100 | Laboratory grade ethanol, rarely found in the retail market\n\n99.2\u2013100 | Anhydrous or dehydrated alcohol\n\n## Resources\n\n**Essences, Oils, Flavors, and Compounds**\n\n**Aftelier**\n\naftelier.com\n\nOils, perfumes\n\n**Art of Drink**\n\nartofdrink.com\n\nAcid phosphate and lactart\n\n**The Essential Oil Company**\n\nessentialoil.com\n\nEssential oils and distillation equipment\n\n**The Good Scents Company**\n\nthegoodscentscompany.com\n\nFlavor and aromatic compounds, oils, extracts, and essences\n\n**Healthy Village**\n\nhealthyvillage.com\n\nMedicinal herbs and cinchona\n\n**Herbiary**\n\nherbiary.com\n\nBulk herbs especially suited for making bitters and cordials\n\n**In Pursuit of Tea**\n\ninpursuitoftea.com\n\nSpecialty teas and infusions\n\n**Kalustyan's**\n\nkalustyans.com\n\nBulk spices and herbs\n\n**Lhasa Karnak Herb Company**\n\nhttp:\/\/www.herb-inc.com\n\nBulk spices and herbs\n\n**Liberty Natural**\n\nlibertynatural.com\n\nBulk spices and herbs, oils\n\n**LorAnn Flavors and Oils**\n\nlorannoils.com\n\nOils, extracts\n\n**Modernist Pantry**\n\nmodernistpantry.com\n\nAcids, flavors, thickeners, and sweeteners\n\n**Monterey Bay Spice Company**\n\nherbco.com\n\nBulk spices and herbs, oils\n\n**Mountain Rose Herbs**\n\nmountainroseherbs.com\n\nBulk spices and herbs\n\n**Penzeys Spices**\n\npenzeys.com\n\nBulk spices and herbs, jars\n\n**Scents of Earth**\n\nscents-of-earth.com\n\nFrankincense, myrrh, tonka beans, rose attars, oils.\n\n**Tenzing Momo**\n\ntenzingmomo.com\n\nBulk spices and herbs, some oils\n\n**Commercial Spirits, Bitters, and Syrups**\n\nWhether you work in the booze industry or are simply an enthusiast, you'll want a selection of whiskeys, brandies, gins, overproof rum, and other commercial spirits. Having lived in some out-of-the-way places, I know firsthand that it's not always easy laying your hands on the juniper juice and corn squeezin's bartenders and bloggers who live in big cities regularly use. Here are a few of my favorite firms that ship.\n\nRemember: 40% abv spirits won't readily dissolve essential oils. It's worth checking these vendors' stock of high-proof neutral spirits such as Everclear, Golden Grain, or Polish _spirytus rektyfikowany_ (\"rectified spirit\" that ranges from 92% to about 96% abv). Several brands are available; look for \" _spirytus_ \" on the label and double-check the proof.\n\n**Astor Wines & Spirits**\n\nastorwines.com\n\n**B. G. Reynolds Syrups**\n\nbgreynolds.com\n\n**Binny's Beverage Depot**\n\nbinnys.com\n\n**DrinkUpNY**\n\ndrinkupny.com\n\n**Hi-Time Wine Cellars**\n\nhitimewine.net\n\n**K &L Wine Merchants**\n\nklwines.com\n\n**Master of Malt**\n\nmasterofmalt.com\n\n**Shoppers Vineyard**\n\nshoppersvineyard.com\n\n**Small Hand Foods**\n\nsmallhandfoods.com\n\n**Organizations and Useful Websites**\n\n**Alcohol and Tobacco Tax and Trade Bureau (United States)**\n\nttb.gov\n\n**The American Distilling Institute**\n\ndistilling.com\n\n**Artisan Distiller**\n\nartisan-distiller.net\n\n**Craft Distillers' Alliance (UK)**\n\nthecda.co.uk\n\n**Flavor & Extract Manufacturers Association (the other FEMA)**\n\nfemaflavor.org\n\n**Food Standards Agency**\n\nfood.gov.uk\n\n**GRAS (Generally Recognized as Safe) Database United States Food and Drug Administration**\n\nfda.gov\n\n**Hobby Distiller's Association**\n\nhobbydistillersassociation.org\n\n**Home Distillation of Alcohol**\n\nhomedistiller.org\n\n**International Fragrance Association**\n\nifraorg.org\n\n**International Organization of the Flavor Industry**\n\niofi.org\n\n**The IUCN Red List of Threatened Species**\n\niucnredlist.org\n\n**Yahoo! Distillers**\n\ngroups.yahoo.com\/neo\/groups\/Distillers\n\n## Glossary\n\nThe first draft of this book's glossary was one of those times when my history as a museum curator got the better of me. That sprawling, 16,000-word chapter explained every ingredient in Victor Lyon's notebook: Latin, German, English, the lot. It was also way too much. At the gentle, but insistent, prodding of my patient editor, Ann Treistman, a new glossary emerged. What follows is a list of the more important or noteworthy ingredients from Lyon's old notebook. Forgive my omissions; there just isn't the space to cover everything.\n\n**Acetaldehyde** has a distinct, pungent smell that can give the impression of fruitiness, juiciness, and, in wines, the \"roundness\" associated with older vintages.\n\n**Acetic Acid** gives vinegar its pungent, savory, nose-wrinkling aroma. Its sharp taste can draw a mouth awry, but in small quantities it is a frequent adulterant of older compounded brandy and whiskey recipes.\n\n**Acetic Ether** (more commonly known as ethyl acetate) has a pleasant, supple, brandy-like odor and is widely used to flavor modern foods and beverages.\n\n**Allspice** has a warm, complex aroma and taste well suited to baking, sausages, jerk, and cordials. It appears in many old cordial recipes and is the primary botanical in a Caribbean cordial called Pimento (or Allspice) Dram that cocktail historian Ted Haigh once called \"the most important liqueur in the world.\"\n\n**Almonds** come in two broad types: sweet and bitter. The former are the common almond of baking, confectionery, and a singularly delicious syrup called orgeat that plays well in many rum and brandy drinks. Bitter almonds, on the other hand, are difficult, if not outright impossible, to find in American stores because they are toxic unless prepared with punctilious regard to safety. Natural almond oil\u2014often called oil of bitter almond\u2014is prepared from bitter almonds (or kernels of apricot, peach, or similar species) with high levels of pleasant-smelling benzaldehyde while \"imitation\" may be synthesized in labs\u2014without risk of cyanide contamination.\n\n**Alum** has been used to fine (that is, to clear) spirits of particulates or oily residue and to purify water. It was also sometimes used to fix, or make more durable, colors added to spirits.\n\n**Amyl Alcohol** is any one of eight different molecules, or a combination of them, commonly found in fusel oil. By itself, amyl alcohol has a harsh, petrochemical odor, but in small amounts and with other compounds, it can lend pleasant apricot and banana aromas.\n\n**Amyl Butyrate** is a common flavoring agent with a sweet taste naturally found in apples and with fruity overtones of pineapple and banana, giving drinks a sort of generic \"tropical\" taste. Many old rum formulas in particular call for it.\n\n**Angelica roots** and sometimes seeds are used in vermouth, bitters, and gin, where they can lend peppery, bitter notes and hints of anise and juniper.\n\n**Anise** is a cousin to fennel, caraway, and cumin. Anise is native to Greece and Asia Minor but has naturalized across Europe, North America, and Asia. Its seeds (actually little fruits) are a stalwart flavoring for spirits and cordials. Its licorice and fennel aroma is both sweet and warm. Anise oil may be from actual anise seeds, but is more commonly made from star anise.\n\n**Balsams,** derived from a number of trees and shrubs around the world, are highly aromatic resins much used in perfumery, medicine, and some foodstuffs. Two types, balsam of Peru and balsam of Tolu, dissolve readily in alcohol and abound in pre-Prohibition formulas for cordials and flavored spirits. A third, benzoin, occasionally shows up in whiskey formulations and some mixed drinks. All three have notes of cinnamon and vanilla as well as an unctuous, soft, sweet, aromatic, and rich scent with bosky base notes, a quality perfumers call balsamic.\n\n**Benzaldehyde** is one of the heavy hitters in the flavor industry. First isolated in the 19th century, it is derived from both artificial and natural sources, including the kernels of bitter almonds, apricots, peaches, and other _Prunus_ species. Its classic flavor is the base of artificial almond extract and almond oil. Its sweet, woodsy aroma immediately brings to mind almonds, cherries, and fresh marzipan and is found in many essential oils.\n\n**Boise** (pronounced bwaz-AY) is a wood extract, commonly made of oak, and occasionally of other woods. See here.\n\n**Butyric Ether,** also called ethyl butyrate or pineapple oil, has a sweet, fruity, pleasant smell and is widely used in confectionery and beverages where its strong pineapple aroma shines through.\n\n**Cajeput** tree leaves and young twigs are distilled to produce an essential oil similar to eucalyptus oil. In old ersatz spirits formulas, it serves to mimic age and impart faint bitterness. Uncommon in Western homemade drinks today, cajeput still enjoys some popularity in remedies for colds and throat complaints because of its penetrating and camphor-like odor.\n\n**Calamus** see here\n\n**Capsicum** : see here\n\n**Caramel** see here\n\n**Caraway** see here\n\n**Cardamom** Beloved throughout Scandinavia in baked goods, warm, fragrant cardamom is common in older drinks manuals. It is used in gins, amari, some vermouths, and bitters. Cardamom-flavored syrup is a grateful addition to kitchens and bars.\n\n**Charcoal** Because of its enormous surface area, carbon pulls impurities from spirits as they pass through. Distillers used it to \"polish\" spirits by removing unwanted compounds that cause off flavors and aroma.\n\n**Chiles** see here\n\n**Cinchona,** the bark of two species of South American trees, _Cinchona succirubra_ and _C. calisaya_ , is the source of quinine, used as an anti-malaria drug since the 16th century. In drinks, its bitter component provides the characteristic taste of tonic water in some vermouths, cordials, bitters, and aperitifs.\n\nBeware: Ingesting too much quinine can bring on cinchonism, a condition caused by a buildup of cinchona in one's body. High doses can be fatal, but even low doses can bring on tinnitus, dizziness, nausea, blurred vision, hot flashes, and wrenching gut pains. Housemade tonic waters that leave visible cinchona floaties in the finished drink are best avoided.\n\n**Cinnamon** and **Cassia** see here\n\n**Clove** is a small evergreen whose unopened flowers are dried to make the spice familiar to westerners since Roman times. Clove is an aggressive flavor, easily overwhelming more delicate botanicals, and should be used with restraint.\n\n**Cognac Oil** is one of the primary tools in the old rectifying manuals; its strong fruity aroma readily mixes with alcohol and gives a characteristic \"brandy\" taste with oily, fatty, and herbaceous notes. It is made from distilling wine lees, the sludgy dregs made of dead yeast and other precipitates left behind during the production of Cognac.\n\n**Colorants** The proper color of a spirit or cordial makes the difference between an attractive product that commands a handsome price and one nobody wants. This was especially true during Prohibition when getting the color right (does that whiskey look yellow?) was often tricky. Counterfeiters of the era deployed a battery of woods, lichens, bugs, roots, and barks to get just the right tinge to their fake whiskey, wine, rum, and brandy. Many contributed flavors as well. Kino, for instance, is resin culled from different trees, especially eucalyptus and related varieties. It didn't just color spirits; it imparted astringency as well.\n\nAmong the more common colorants in older manuals were caramel (see here) and cochineal (here). Others included alkanet (roots of plants in the borage family), cudbear (various lichen species), and catechu (derived from the heartwood of a number of acacia species).\n\nBrazil wood, logwood, and sandalwood, quite apart from their fragrant aromas, all lent a range of yellows and reds to cordials, wines, and spirits. Of roots, there were madder, rhatany, and tormentil, all of which yielded various reds for wines, especially faked port. Raspberry juice was used for the same purpose while its leaves lent astringency to country wines.\n\nCousin to ginger and sometimes called poor man's saffron, the dark yellow rhizome called turmeric has a pleasant, complex, peppery, and musty aroma with notes of citrus and ginger. It was used primarily as a yellow colorant for cordials and vermouths.\n\n**Coriander** is one of the core gin botanicals, but is also used in bitters, pastis, vermouths, and Galliano liqueur. Its floral, spicy aroma complements a sweet, well-rounded flavor that smacks of orange. Pairs well with whiskey.\n\n**Coumarin** see sweet woodruff recipes, here\n\n**Creosote** has a range of meanings that vary over time and with local usage, but as it was used in 19th- and early 20-century beverage manuals, creosote was made from the heavy oil of wood tar. Another variety, made from coal tar, is toxic and should be avoided altogether. Small doses of bitter wood tar creosote give rich, smoky aromas to cured meats and beverages. Some versions of \"liquid smoke\" use creosote to simulate long, slow smoking.\n\n**Fennel** Sweet, astringent, something like anise, something like licorice, fennel seeds are used widely in spirits and wines. Along with wormwood and anise, fennel is one of the three core ingredients of traditional absinthe and is often used to round out the flavor of k\u00fcmmel.\n\n**Formic Ether,** with its pungent, pineapple-like smell, was used to augment rums that had been cut with neutral spirits to enhance their rum-like aroma.\n\n**Fusel Oil** More commonly called fusel alcohol in modern use, it is an oily-looking combination of amyl alcohols, acids, esters, and aldehydes that imparts characteristic flavor to spirits, especially whiskey. Though some recipes for compounded spirits call for it, fusel alcohol has a disagreeable odor and may contribute to hangovers; anything more than trace amounts can quickly ruin good liquor. In fact, distillers generally remove as much fusel alcohol as they can from most new spirits, leaving in just enough to break down and form new, pleasant-smelling compounds during barrel aging.\n\n**Galls** are abnormal growths on plants caused by various species of parasites. Oak galls, formed by wasp larvae in growing oak trees, are often called for in old drinks formulas for the sharply bitter taste they give to vermouths, medicinal wines, and bitters.\n\n**G\u00e9n\u00e9pi** see Wormwood\n\n**Gentian** Two of the nearly 400 species of the mountainous plant gentian stand out for their use in beverages: yellow and spring gentians. It's one of the granddaddies of bittering elements in European tonics and liqueurs. Gentian root contributes bitterness to vermouths, cordials, bitters, amari, spirits, and even the soft drink Moxie.\n\n**Ginger** see here\n\n**Glucose,** also called dextrose or grape sugar, is not as sweet as table sugar, but is used like it to thicken and sweeten cordials, some bitters, and other alcoholic drinks. In syrups, it helps prevent crystallization.\n\n**Glycerine,** also called glycerin and glycerol, is a colorless, nearly odorless, slightly sweet viscous liquid used to sweeten cordials and provide a rounder mouthfeel. Home distillers sometimes add as much as 15 ml to a liter of spirits, but it's quite delectable at even half that concentration. Its use in commercial spirits is frowned on and in some cases forbidden. It's not harmful, just a cheat that connoisseurs dislike.\n\n**Grains of Paradise** see here\n\n**Hops** extracts were sometimes added to quickly made Prohibition beers and whiskeys to simulate greater age and care in their making.\n\n**Horseradish** see here\n\n**Hyssop** is an ancient evergreen bush used in alcohol. In the 1st century AD, Pliny mentioned hyssopites, a wine impregnated with the bitter herb. Since then, the strong and minty leaves have been used in monastic liqueurs, vermouths, and absinthes.\n\n**Juniper** see here\n\n**Licorice,** a perennial shrub native to southeastern Europe and northwestern Asia, has a long taproot that adds sweetness, warmth, and a bit of earthiness to syrups, cordials, bitters, schnapps, vermouths, some porters, and spirits\u2014most notably, gin.\n\n**Mace** see Nutmeg and Mace\n\n**Mint** , one of the world's most popular flavorings, is a protean herb prone to hybridizing, thus creating seemingly endless varieties. Broadly, though, the different types fall into two groups: peppermint and spearmint. Peppermint, and peppermint oil in particular, have a long history in medicine and drinks. More pungent than spearmint, peppermint tastes strongly of cooling menthol, but with a sweet aftertaste. Oil of peppermint is commonly used to make cr\u00e8me de menthe and it can be found in many cordials. Spearmint is the softer, more approachable variety. Mellow, with slight lemon notes and a cooling menthol effect when drinkers inhale after sipping some. Countless varieties exist, but this is the one for mint juleps, mojitos, mai tais, and other mixed drinks.\n\n**Nitrous Ether** has a sweet, pleasant, fruity aroma with hints of vanilla and benzaldehyde and routinely shows up in old beverage compounding formulas. It is also known as ethyl nitrite. A 3.5-4.5% solution of nitrous ether was sold over the counter in the United States under the names sweet nitrite or sweet spirit of nitre as a cold remedy until the FDA ruled that was not generally recognized as safe (see GRAS, here). See Sweet Spirit of Nitre.\n\n**Nutmeg and Mace**\n\n**Nutmeg** and mace both come from the apricot-like fruit of _Myristica fragrans_ , a tropical evergreen native to the Banda Islands of Indonesia, though now grown in several places. Nutmeg is the kernel of the fruit's seed while mace is the reddish-orange or amber colored aril (the lacy, web-like coving of each nutmeg). Usually sold separately. Buy whole and grate or ground as needed.\n\n**Nutmeg** has a warm, rich, and pungent aroma. The flavor can be subtle and sweet, but it tapers off to a mildly bitter end. Rarely used as a primary note in spirits, it helps round out other flavors in some vermouths, absinthes, and gins. Often grated onto punches, some mixed drinks, and eggnogs just before serving.\n\n**Mace** has an aroma similar to nutmeg's, though almost clove-like. The taste is both more subtle and more bitter than nutmeg.\n\n**Nuts** A variety of nuts, including hazelnuts and walnuts, are used to impart bitterness, color, and astringency in spirits and in imitation of brandies.\n\n**Orris** is the collective name for the bulbs of three iris species harvested after three to six years of growth and cured for another two or so. Generally sold as a powder, it has a fragrant, slightly earthy aroma. Its primary purpose, however, is to enhance or \"fix\" the aromas and flavors of other botanicals in gins, vermouths, absinthes, and some orange cordials.\n\n**Peppermint** see Mint\n\n**Pyroligneous Acid** is a complex mix of over 200 components, including acetic acid and methanol, created during the destructive distillation of wood. The dark yellow\/reddish brown liquid is reminiscent of commercial liquid smoke. It was used in synthetic spirits\u2014 primarily whiskeys\u2014to simulate barrel-aging.\n\n**Rose** Forget insipid garden roses. Go for hugely flavored, intense red roses for rose drinks. Use the petals only and, if you've got a beef with bitterness, snip off the white base of each one. Rose is used in parfait amour, some vermouths, cordials, and flavored brandies. Rosewater is the aroma-saturated water left in the belly of a still after the essential oil (also called attar or otto of rose) is drawn off. Use a light touch; a little goes a long way.\n\n**Rosemary** is a thick, woody evergreen shrub that can grow to the size of a small car (at least in our neighborhood). The leaves smell strongly of resin, pine, and camphor. Use a restrained hand; too much rosemary in a glass is like sucking pine needles.\n\n**Rum Ether,** also known as ethyl oxyhydrate, is a wood distillate. Its aroma resembles that of rum and, despite lacking aromas important to real rum (such as isoamyl alcohol and \u03b2-damascenone), it has been used widely in spirits and the food industry overall.\n\n**Saffron** see here\n\n**Salt** Whole grains and flakes of salt are _de rigueur_ for margaritas, salty dogs, and other cocktails. An old distiller's trick is adding non-iodized salt to stills during distillation to raise the boiling point of water, giving a greater ethanol percentage on a single run.\n\n**Spearmint** see Mint\n\n**Star Anise** is a particularly bold spice. Its aroma suggests licorice, anise, and fennel while the flavor is aggressive, slightly numbing, and lingering. The essential oil is widely used in anisettes, pastis, and cordials (see Anisette de Hollande and Anisette de Bordeaux, pages 167\u2013168)\n\n**Sulfuric Acid** Often called oil of vitriol in old texts, concentrated sulfuric acid is highly corrosive. It should be handled with the utmost care by people already familiar with its use. In Lyon's notebook, it is used to purify caramel, test the purity of spirits, create \"beading oil\" (see here), and as a barrel ager.\n\n**Sweet Oil** Commonly pressed almond oil, but any neutral, edible oil. Often used to create \"beading oil\" for counterfeit spirits (see Sulfuric Acid).\n\n**Sweet Spirit of Nitre** A once-common medicine in the United States, over the counter sales of pungent, sweet-tasting sweet spirit of nitre and its inclusion in any drug for any purpose were banned by the FDA in 1980. Used extensively in ersatz whiskey and brandy formulas. Since the mid-19th century, it has been known that inhaling it could cause cause dizziness, nausea, headache, heart irregularities, blueing of lips and nails, and even death.\n\n**Syrup** see here\n\n**Tea** Aside from its most excellent use as a beverage in and of itself, black tea adds astringency and color to punches, cocktails, and syrups. Teas pair well with fruits, flowers, and caramel. Green and \"white\" teas play well in punches and cocktails.\n\n**Tonka Beans** Once hugely popular among distillers, liquor manufacturers, compounders, and bottlers (as well as tobacconists), tonka beans were banned as a food additive in 1954 because of their high coumarin content. The aroma is intense, sweet, and suggests new-mown hay, vanilla (for which it is often used as a substitute), cinnamon, and almonds. The dark, wrinkled beans may be purchased online or from well-stocked spice shops.\n\n**Vanilla and Vanillin** One of our most beloved spices and second only to saffron in price, vanilla beans are the fermented and dried fruits of orchids native to Central America but now also grown in Madagascar, Indonesia, and elsewhere. The rich, intense aroma of fermented vanilla beans is unmistakable: creamy, balsamic, and sweet, with hints of soft leather and tobacco. Used in cocktails and products such as Kahl\u00faa, Galliano, Tuaca, flavored vodkas, and numerous cordials. Vanillin seems as if it should be pronounced \"va-NILL-in\" but it's \"VAN-ill-in.\" This phenolic aldehyde is the primary constituent of vanilla extract (other than alcohol). It is also found in wines, vinegars, and spirits that have been properly aged in toasted or charred oak barrels. It is sometimes added in small quantities to help rapidly \"age\" counterfeit spirits.\n\n**Vinegar** is a mild solution (often 3-5%) of acetic acid. It is called for in some synthetic whiskeys and brandies, but reached wide popularity as a souring agent in drinks such as switchel and some shrubs.\n\n**Violet** The ephemeral, transient aroma of _Viola odorata_ is due to ionone, a chemical that temporarily blocks our ability to smell it once we've encountered the smell, despite the fact that the smell is still present\u2014simply not for those who have just smelled it. The effect: the dry, floral scent and flavor comes on, then disappears. A few moments later, it comes rushing back, new attacks every few moments.\n\n**Wintergreen** Oil of wintergreen, distilled from the leaves of low North American bushes, gives a bracing, minty, vaguely medicinal flavor to cordials and some counterfeit spirits. It may also be made from some species of birch trees and other unrelated species that produce methyl salicylate.\n\n**Woodruff, Sweet** see here\n\n**Wormwood** is a broad term that may refer to any species in the _Artemisia_ genus, but _A. absinthium_ \u2014also called grand or common wormwood\u2014is the definitive ingredient for making absinthe. _A. pontica_ (Roman or petite wormwood), is often used as a colorant for absinthe as well. Several other species of these strongly bitter plants are used in spirits, absinthes, cordials, wines, and vermouths. G\u00e9n\u00e9pi, a related term, comprises several Artemisia species, including _A. genipi_. _A. rupestris_ , and _A. umbelliformis_. G\u00e9n\u00e9pi grows in mountainous areas of Spain, France, Switzerland and Italy. The term may also refer to bitter liqueurs made from such species.\n\n## Bibliography\n\nBaker, Charles H. Jr. _The Gentleman's Companion_. 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Nordhausen am Harz, Germany: Heinrich Killinger Verlagsgesellschaft, 1938.\n\nLacour, Pierre. _The Manufacture of Liquors, Wines, and Cordials without the Aid of Distillation. Also the Manufacture of Effervescing Beverages and Syrups, Vinegar and Bitters_. New York: Dick and Fitzgerald, 1853.\n\nLoftus, William R. _Loftus's New Mixing and Reducing Book for Publicans and Spirit Dealers and Retailers in General_. London, 1869.\n\nMahoney, Charles S. _The Hoffman House Bartender's Guide: How to Open a Saloon and Make It Pay_. New York: R. K. Fox, 1905.\n\nMcElhone, Harry. _Barflies and Cocktails_. Paris: Lecram Press, 1927.\n\nM'Harry, Samuel. _The Practical Distiller_. Harrisburgh [sic], PA: John Wyeth, 1809.\n\nMickler, Ernest Matthew. _White Trash Cooking_. Berkeley, CA: Ten Speed Press: 1986.\n\nMiller, Amy Bess, and Persis Fuller. _The Best of Shaker Cooking_. New York: Collier Books, 1985.\n\nMoens, Henri, and Hubert Verburgt. _De Schat der Likeuristen: of Volledige Handleiding om Zonder Distilleerketels en Zonder Forneizen Alle Soorten van Uitmuntende Likeuren te Bereiden_. Ghent, Belgium: De Busscher en zoon, 1869.\n\nMorgenthaler, Jeffrey. _The Bar Book_ : _Elements of Cocktail Technique_. San Francisco: Chronicle Books, 2014.\n\nPereira, Jonathan. _The Physician's Prescription Book_. Philadelphia: Lindsay and Blakiston, 1852.\n\n_The Picayune_. _The Picayune's Creole Cook Book_ , 4th ed. New Orleans, 1910.\n\nPrudhomme, Paul. _The Prudhomme Family Cookbook: Old-Time Louisiana Recipes by the Eleven Prudhomme Brothers and Sisters_. New York: William Morrow and Company, 1987.\n\nSchmidt, William. _The Flowing Bowl: What and When to Drink_. New York: Charles L. Webster & Co., 1891.\n\nShannon, Richard. _A Practical Treatise on Brewing, Distilling, and Rectification_. London, 1805.\n\nSimmonds, Charles. _Alcohol, Its Production, Properties, Chemistry, and Industrial Applications_. London: Macmillan and Co., 1919.\n\nSmyth, William A. _The Publican's Guide Or Key to the Distill-House_. London, 1781.\n\nSpinoza, Baruch. _Tractatus Politicus_ , 1677.\n\nThomas, Jerry. _Bar-Tender's Guide_. New York: Dick & Fitzgerald, 1862.\n\nTolzmann, Don Heinrich. _Images of America: German Cincinnati_. Charleston, SC: Arcadia Publishing, 2005.\n\nTovey, Charles. _British & Foreign Spirits: Their History, Manufacture, Properties, Etc_. London: Whittaker & Company, 1864.\n\nTurner, Jack. _Spice: The History of a Temptation_. New York: Alfred A. Knopf, 2004.\n\nTydings, Millard E. _Before and After Prohibition_. New York: Macmillan, 1930.\n\nUre, Andrew. _Dictionary of Arts, Manufactures, and Mines_. London: Longmans, Green, and Company, 1837.\n\nVasey, S. Archibald. _Guide to the Analysis of Potable Spirits_. London: Bailliere, Tindall & Cox, 1904.\n\nViereck, George Sylvester. _The Works of George Sylvester Viereck: The Candle and the Flame_. New York: Moffat, Yard and Company, 1912.\n\nWalter, Eric. _Manual for the Essence Industry_. New York: John Wiley and Sons (1916).\n\nWeiss, Harry B. _The History of Applejack or Apple Brandy in New Jersey from Colonial Times to the Present_. Trenton: New Jersey Agricultural Society, 1954.\n\nWilcox, R. W. _Materia Medica and Pharmacy_. Philadelphia: P. Blakiston's Son & Co., 1907.\n\nWiley, Harvey Washington. _Beverages and Their Adulteration_. Philadelphia: P. Blakiston's Son & Co., 1919.\n\nWondrich, David. _Imbibe!: From Absinthe Cocktail to Whiskey Smash, a Salute in Stories and Drinks to \"Professor\" Jerry Thomas, Pioneer of the American Bar Featuring the Original Formulae_. New York: Perigee, 2007.\n\n_____. _Punch: The Delights (and Dangers) of the Flowing Bowl: An Anecdotal History of the Original Monarch of Mixed Drinks, with More Than Forty Historic Recipes_. New York: Perigee, 2010.\n\nWright, Helen Saunders. _Old-Time Recipes for Home Made Wines, Cordials, and Liqueurs from Fruits, Flowers, Vegetables, and Shrubs_. Boston: Page Company, 1922.\n\n## Endnotes\n\n**** Heywood Broun, \"Books,\" _New York Tribune_ , July 14, 1919, 8.\n\n**** \"Books: Self-Astounder,\" _Time_ , May 11, 1931.\n\n**** Otis Notman, \"Viereck, Hohenzollern? Is New York's Romantic Poet a Grandson of Kaiser Wilhelm the Great?\" _New York Times_ , June 29, 1907.\n\n**** _New-York Tribune_ , July 17, 1918, 4.\n\n**** \"Viereck Expelled by Authors' League,\" _New York Times_ , 26 July 26, 1918.\n\n**** _New York Tribune_ , August 16, 1918, 12.\n\n**** Chronology: The Germans in America, accessed May 9, 2014, http:\/\/www.loc.gov\/rr\/european\/imde\/germchro.html.\n\n**** Don Heinrich Tolzmann, _Images of America: German Cincinnati_ (Charleston, SC: Arcadia Publishing, 2005), 111.\n\n**** Nicholas J. C. Pistor, \"Pershing or Berlin? A St. Louis Street Name Debate,\" _St. Louis Post-Dispatch_ , February 3, 2014.\n\n**** C. Wilcox, \"World War I and the Attack on Professors of German at the University of Michigan,\" _History of Education Quarterly_ 33, no. 1 (1993): 60\u201384.\n\n**** \"Negro Editor Arrested,\" _Washington Post_ , January 17, 1920, 3.\n\n**** \"Treatises on Liquor Making Barred from Shelves of Libraries,\" _Pittsburgh Post-Gazette_ , January 16, 1920, 4.\n\n**13** \"Bootlegger's Patron Called Prize Freak: Flare-up in the Senate,\" _Alabama Anniston Star_ , January 17, 1926, 1.\n\n**** _Mass Moments_ (blog), accessed February 1, 2015, http:\/\/www.massmoments.org\/moment.cfm?mid=326, accessed February 1, 2015.\n\n**** \"Famous 'Booze' Sermon,\" Billy Sunday Online, accessed February 1, 2015, http:\/\/www.billysunday.org\/sermons\/booze.php3, accessed February 1, 2015.\n\n**** \"Dry Czar Favors Sale of Beer as Enforcement Aid,\" _Olean (NY) Times Herald_ , April 14, 1926, 1.\n\n**17** Charles Norris, \"Our Essay in Extermination,\" _The North American Review_ 1928: 645-652.\n\n**18** \"The New Alcohol Formula 39-B,\" _American Perfumer and Essential Oil Review_ , October 1921: 328.\n\n**** \"Drinks for Drys,\" _Time_ , April 8, 1929.\n\n**** \"Kings County Society,\" _Practical Druggist_ , June 1922, 40.\n\n**** \"Enforcing the Prohibition Law,\" City Club Bulletin 13, no. 38, September 20, 1920.\n\n**** Jonathan Pereira, _The Physician's Prescription Book_ (Philadelphia: Lindsay and Blakiston, 1852), 86.\n\n**** William A. Smyth, _The Publican's Guide or Key to the Distill-House_ (London, 1781).\n\n**** William R. Loftus, _Loftus's New Mixing and Reducing Book for Publicans and Spirit Dealers and Retailers in General_ (London, 1869).\n\n**** W\u00f6hler und von Liebig, \" _Untersuchungen \u00fcber das Radikal der Benzoes\u00e4ure_ ,\" _Annalen der Pharmacie_ 3 (1832): 249\u201382.\n\n**** B. B. Petrovska, \"Historical Review of Medicinal Plants' Usage,\" _Pharmacognosy Reviews_ , 6, no. 11 (2012): 1\u20135.\n\n**** George A. Burdock, _Fenaroli's Handbook of Flavor Ingredients_ , 6th edition, (Boca Raton, FL: CRC Press, 2010), 1566.\n\n**** E. W. Bovill, \"The Essential Oil Market,\" _Journal of the Royal African Society_ 33, no. 132 (July 1934): 217\u201325.\n\n**** Glenn Sonnedecker, \"How to Adulterate Volatile Oils: A Pre-1906 Manuscript Formulary,\" _Pharmacy in History_ 32, no. 4 (1990): 155\u201365.\n\n**** \"Repeal of Prohibition in Practical Effect in N.Y.,\" _San Bernardino (CA) County Sun_ , October 14, 1933, 3.\n\n**** \"Customs Officers Hold Up Shipment of German Lamps,\" _Wilmington (OH) News-Journal_ , May 12, 1922, 1.\n\n**** John Gardner, _The Brewer, Distiller, and Wine Manufacturer_ (London: J. & A. Churchill, 1902).\n\n**** Mark H. Haller, \"Philadelphia Bootlegging and the Report of the Special August Grand Jury,\" _Pennsylvania Magazine of History and Biography_ April (1985), 215\u201332.\n\n**** Harvey Washington Wiley, _Beverages and Their Adulteration_ (Philadelphia: P. Blakiston's Son & Co., 1919), 374.\n\n**** \"Quart of Whiskey, Gallon of Rum Equals Bootleg Havana Scotch,\" _Kansas Hutchinson News_ , December 30, 1924.\n\n**** William Kitchiner, _The Cook's Oracle; and Housekeeper's Manual_ , 7th ed. (New York: J. & J. Harper, 1830).\n\n**** A. Paine and A. D. Davan, \"Defining a Tolerable Concentration of Methanol in Alcoholic Drinks,\" _Human & Experimental Toxicology_ 20, no. 11 (November 2001): 563\u201368.\n\n**** Charles Norris, \"Our Essay in Extermination,\" _North American Review_ 226, no. 6 (1928): 646.\n\n**** See, for instance, A. Wood and F. Buller, \"Poisoning by Wood Alcohol: Cases of Death and Blindness from Columbian Spirits and Other Methylated Preparations,\" _JAMA_ 43 (1904): 972.\n\n**** Charles Norris, \"Our Essay in Extermination,\" _North American Review_ 226, no. 6 (1928): 652.\n\n**** Eduard Hahn and Johann Holfert, _Spezialit\u00e4ten und Geheimmittel: Ihre Herkunft und Zusammensetzung_ (Berlin, Germany: Springer, 1906).\n\n**** Vice-Consul R.S. Warburton, \"Reports on Frauds in the Brandy Trade, France,\" from bulletin issued by Victoria Department of Agriculture, 1888, 168\u201369.\n\n**** National Prohibition Act, Title II, section 29.\n\n**** \"Gives Law on Cider,\" _Coffeyville (KS) Daily Journal_ , December 4, 1920, 7.\n\n**** \"Take One with Me,\" _Pittsburgh Dispatch_ [sic], July 28, 1889, 15.\n\n**** \"Raiders Steal 45 Barrels of Real Port Wine,\" _Carbondale (IL) Daily Free Press_ , February 10, 1925.\n\n**** Marni Davis, _Jews and Booze: Becoming American in the Age of Prohibition_ (New York: NYU Press, 2012), 163.\n\n**** \"Cider Can Be Dealcoholized,\" _Practical Druggist_ , December 1920, 39.\n\n**** Stephan A. Padosch et al., (2006) \"Absinthism: A Fictitious 19th Century Syndrome with Present Impact,\" _Substance Abuse Treatment, Prevention, and Policy_ (2006): 1:14.\n\n**** Marie-Claude Delahaye, _Pernod: Creator of Absinthe_ (Mus\u00e9e de l'Absinthe Auvers-sur-Oise, 2008), 103.\n\n**** Richard Shannon, _A Practical Treatise on Brewing, Distilling, and Rectification_ (London, 1805).\n\n**** Elke Scholten et al., \"The Life of an Anise-Flavored Alcoholic Beverage: Does Its Stability Cloud or Confirm Theory?\" _Langmuir_ 24, no. 5 (2008), 1701\u20136.\n\n**** See, for instance, Stanley Clisby Arthur, _Famous New Orleans Drinks and How to Mix 'Em_ (New Orleans: Harmanson, 1937).\n\n**** Jeffrey Morgenthaler, _The Bar Book: Elements of Cocktail Technique_ , (Chronicle Books: San Francisco, 2014.\n\n**** Charles S. Mahoney, _The Hoffman House Bartender's Guide How to Open a Saloon and Make It Pay_. (New York: RK Fox, New York (1905), 224.\n\n**** Hans G\u00f6ttler, Hans _Rezeptbuch f\u00fcr Destillateure_ ,. Neustadt, Germany: Meininger, (ca. 1908) Neustadt.\n\n**** William Schmidt, _The Flowing Bowl: What and When to Drink_ (New York: Charles L. Webster & Co., 1891), 196\u201397.\n\n**** _The New Orleans Picayune_ , _The Picayune's Creole Cook Book_ , 4th ed. (New Orleans, 1910), 328.\n\n**** Tim Daly, _Daly's Bartenders' Encyclopedia_ (Worcester, MA: published by author, 1903), 113.\n\n**** Paul Clarke, \"60\/30, #5&6: Epicurean and the Allies Cocktail,\" _The Cocktail Chronicles_ (blog), November 25, 2010, http:\/\/www.cocktailchronicles.com\/2010\/11\/25\/6030-epicurean-and-the-allies-cocktail\/.\n\n**** _Pennsylvania Gazette_ , January 10, 1771.\n\n**** _Public Advertiser_ (London), June 4, 1790, 4.\n\n**** _Scientific American_ : Supplement 35, no. 892 (1893): 14250.\n\n**** Amy Bess Miller and Persis Fuller, _The Best of Shaker Cooking_ (New York Collier Books, 1985), 413.\n\n**** Such as R. W. Wilcox, _Materia Medica and Pharmacy_ (Philadelphia: P. Blakiston's Son & Co., 1907), 19.\n\n**** T. E. Furia and N. Bellanca, _Fenaroli's Handbook of Flavour Ingredients_ , vol. 2 (Boca Raton, FL: CRC Press, 1975).\n\n## Index\n\n_Page numbers listed correspond to the print edition of this book. You can use your device's search function to locate particular terms in the text_.\n\nTables in the text are indicated in _italics_.\n\n**A**\n\nabsinthe\n\nessence recipe, 125\n\ngrades of, 142\u201343\n\nhistory of, xxi, 5, 8, 131\n\nas ingredient, 134\u201335\n\nlouche form of, 129\n\nmedicinal qualities of, 8, 124, 129\n\noverview of, 124, 126\u201328\n\nrecipe for, 130\n\nserving method, 132\n\nstyles of, 127\n\nabsinthe onion garnish, 135\n\nAbsinthe Onions, 135\n\nAbsinthe Suissesse, 134\n\n\"Absinthism: a Fictitious 19th Century Syndrome with Present Impact\" (Padosch), 122\n\nabsinthium oil, 129\n\nAcetic acid, 101\n\nAcetic either, 86\n\nAcetic ether, 113, 117\n\nActual Woodruff Essence, 183\n\nadulteration, 30, 45\u201346, 67\u201368, 82, 127\n\naged syrups, 100, 146\n\nAlamagoozlum, 65\n\nallspice berries, 163\n\nalmond oil, 24\u201325, 46, 121, 167\n\n_American Journal of Pharmacy_ , 88\n\nAmerican proof measurement, 202\n\nAmyl alcohol, 86\n\nAnderson, Edwin P., 1\n\nAndrews, Lincoln C., 111\n\na\u00f1ejo Havana Club rum, 76\n\nangelica oil, 167\n\nangelica root, 127, 130\n\nAngostura bitters, 65\u201366, 104, 157, 161, 172\n\nanise, 126\u201327, _127_ , 165\u201369\n\n_See also_ star anise\n\naniseed, Green, 168\n\nanise oil, 86, 125, 130, 151, 167, 169\n\n_See also_ star anise oil\n\nanise seed, 42, 130, 139, 148\n\nanisette cordials, 123, 142, 158, 165\u201369\n\nAnisette de Bordeaux (1910), 168\n\nAnisette de Hollande (1869), 167\n\napothecaries' weights and measures, 193\u201398\n\napple brandy, 109\u201311, 155\n\napplejack, 107\u201310\n\nAppleton rum, 74, 78\n\napple whiskey, 110\n\nApr\u00e8s Souper (1891), 180\n\naromatherapy, 40, 43, 195\n\naromatic bitters, 60, 109\n\naromatic compounds, 21, 35, 39, 42, 48, 85, 145\u201346, 194\u201395\n\n_Around the World with Jigger, Beaker, and Flask_ (Baker), 65\n\narracks, 76\u201378\n\nArrakessenz formula, 77\n\nArtificial Woodruff Essence, 182\n\n_The Art of Blending and Compounding Liquors and Wines_ (Fleischman), 2, 20, 71\n\nastringency, 99, 121, 139\n\navoirdupois weight, 193, 198\n\n**B**\n\n\"bain-marie\" (waterbath), 168\n\nBaker, Charles H., Jr., 65, 66, 110, 177\n\nBalls Deep (Hot Cinnamon Liqueur), 162, 165\n\nbalsams, 35\n\nBamford, John, 158\n\n_The Bar Book_ (Morgenthaler), 36\n\nBarclay, Armiger, 113\n\n_Barflies and Cocktails_ (McElhone), 104\n\nBarleycorn, John, xxi\n\nbar measures, 201\n\n_Bar-Tender's Guide_ (Thomas), 32\n\nbartending manuals, 36\n\nBatavia Arrack, 78\n\nbathtub gin, 55\u20137\n\nbeading oil (The Doctor), xv, 24\n\nbeer, xxi, 13, 105\n\nBee's Knees, 61\n\nbenzaldehyde, 161\n\nBerry, Jeff, 76\n\n_Beverages and Their Adulteration_ (Wiley), 82\n\nBittermens Hellfire Habanero Shrub, 162\n\nbitters\n\nAngostura, 65\u201366, 104, 157, 161, 172\n\naromatic, 60, 109\n\nHostetter(ish) Bitters, 139\n\noverview of, 136\u201338\n\nblackberry liqueur, 154\n\nBlack Ceylon Tea, 99\n\nblack market, 53, 59\n\nBlank, Fritz, xii\u2013xiv, 192\n\nblender (gear), 36\n\nblood orange, 84, 135\n\n_Boardwalk Empire_ (HBO), 118\n\nboiled cider, 103\n\nboise extract, 95, 99\u2013100, 102, 121\n\n_The Book of Receipts_ (1907), 192\n\nbootleggers, 2, 13\u201314, 39, 83, 88\n\n\"Bootleg Liquor\" (Doran), 8\n\nbooze cherry, 76\n\n_The Bordeaux Wine and Liquor Dealers' Guide_ (1857), 92\n\nbottle brush, 36\n\n_The Bottlers' Formulary_ (1910), 35\n\n_Bottoms Up!_ (Saucier), 59\n\nBovill, E.W., 45\n\nbrandied cherry, 75\n\nbrandy\n\napple, 109\u201311, 155\n\nBrandy Fix, 104\n\ncaraway flavored, 147\u201348\n\ncherry flavored, 104\n\nCognac, 95, 103\u20134, 150, 155, 157\n\nessence of, 86\u201387\n\nginger flavored, 170\u201373\n\nHarvard Cocktail, 104\n\nhistory of, 93\u201399, 120\u201321\n\nas ingredient, 74, 170, 172, 178, 180\u201381, 183, 189\n\nSidecar, 103\n\nvinegar in, 101\n\nBrandy Essence, 87\n\nBrandy Fix, 104\n\nbrazil wood, 175\n\nBrevans, J. de, 168\n\n_The Brewer, Distiller, and Wine Manufacturer_ (Gardner), 63\n\n_British & Foreign Spirits: Their History, Manufacture, Properties, Etc_. (Tovey), 95\n\n_British Pharmacopoeia_ (1864), 33, 192\n\nBritish proof measurement, 202\n\nBrix refractometer, 38\n\nBrock, William H., 28\n\nBrooklyn College of Pharmacy, 16\n\nB\u00fcchner filter, 38, 139\n\nbutyric ether, 69\n\n**C**\n\ncade oil, 40\n\ncalamus root, 45, 130\n\nCampari, 60\n\n_The Candle and the Flame_ (Viereck), xiv, xix\n\ncapsicum, 89\u201391, 174\u201375\n\ncaramel, 30, 68\u201369, 79\u201382, 99\u2013102, 117, 120\u201321\n\nCaramel for Coloring, 79\n\nCaraway Brandy, 148\n\ncaraway oil, 120, 153\u201354, 157, 165\n\ncaraway seeds, 8\n\nCaraway Vodka, 150\n\ncarboys, 55\n\ncardamom, 57, 78, 160\n\nCartier scale, 203\u20134\n\ncarvol\/carvone, 154\n\ncassia, 68\u201369, 129, 158\u2013160\n\nCate, Martin, 118\n\n_Central Press_ (newspaper), 62\n\ncentrifuges, 42\n\nChartreuse, green or yellow, 59, 65, 155\n\ncheat formulas, 24, 35\n\ncherry brandy, 104\n\ncherry garnish, 75\u201376\n\n_Chicago Defender_ (newspaper), 17\n\nChief Gowanus New Netherland Gin, 62\n\n\"chipped\" whiskey, 83\n\ncider\n\nboiled, 103\n\nCider Champagne, 107, 109\n\nCider Sherry Flip, 109\n\nCider Without Apples, 107\n\nhard cider, 10, 51, 109, 114, 116\n\nhistory of, 105\n\noil of, 50\u201351, 107\n\nThe Stone Fence, 109\n\nCider Champagne, 107, 109\n\ncider oil, 50\u201351, 107\n\nCider Sherry Flip, 109\n\nCider Without Apples, 107\n\ncinchona, 136, 139\n\ncinnamon\n\nBalls Deep (Hot Cinnamon Liqueur), 162, 165\n\nCinnamon Fizz, 165\n\ncordials, 82, 138, 158\u2013165\n\nHelen Saunders Wright's Cinnamon Cordial, 160\n\noil of, 43, 129\n\nsticks, 84, 109, 163\n\nCinnamon Fizz, 165\n\ncitrus oils, 40\u201342\n\ncitrus zest garnish, 103\n\nCity Club (Chicago), 17\u201318\n\nClaret Wine, 114\n\nClarke, Paul, 32, 157\n\nclove oil, 82, 129, 161, 165\n\ncloves, 84, 109, 114, 148, 162\u201363, 189\n\nclub soda, 104, 161\n\n_The Cocktail Chronicles_ (Clarke), 32, 157\n\ncocktail guides, 1, 30, 32\n\ncocktails\n\nabsinthe, 126\u201327\n\nbrandy, 102\u20134\n\negg based, 144\n\nfoam for, 24\n\ngin, 59\u201361\n\nand overproof alcohol, 201\n\nrum, 75, 78\u201379\n\nand syrups, 147\n\nZeeland, 119\n\n_Cocktails\u2014How to Mix Them_ (Vermeire), 181\n\nCognac brandy, 95, 103\u20134, 150, 155, 157\n\nCognac Charente Type, 99\n\ncognac oil, 30, 81, 82, 121\n\nCognac syrups, 100\n\nCointreau, 60, 103\n\nCointreau Noir, 74\n\nCold-Compounded Triple Sec, 188\n\ncold pressing, of oils, 42\n\nThe Collins, 66\n\ncolorants, 69, 80, 145, 175\n\nComic Con, 162\n\ncompounded gin, 55\u201357\n\ncompounder's syrup, 103, 145, 161, 167, 173, 177, 183, 189\n\ncompounding formularies, 30, 34\u201335\n\ncompounding manuals, 19, 38, 71, 82, 89, 157, 194\n\nCompound Liqueur Oil French Absinthe, 129\n\ncondensers, distillery, 28\n\n_The Cook's Oracle_ (Kitchiner), 78\n\nCopper Distilled Bourbon, 81\n\nCordial Liquor of Danzig Comp Oil, 165\u201366\n\ncordials\n\ncaraway flavored, 147\u2013157\n\nchile flavored, 156\n\ncinnamon flavored, 82, 138, 158\u2013165\n\ncoloring of, 174\n\nCordial Liquor of Danzig Comp Oil, 165\u201366\n\nCr\u00e8me de Menthe, 176\u201377\n\nDIY Orange Cordial, 189\n\nginger flavored, 170\n\nhistory of, 141\n\niced, 153\n\nMike McCaw's Triple Sec, 188\n\nMint and Gum, 177\n\nmint flavored, 175\u201378\n\nnalewka essence, 161\n\norange flavored, 184\u201389\n\nproduction method, 142\u201343\n\nShaker Peppermint Cordial, 178\n\nsyrups for, 144\u201345\n\nCordus, Valerius, 33\n\ncoriander, 127, 129, 139, 168\n\ncounterfeit alcohol, 2, 8, 11\u201313, 55, 67\u201368, 83, 95, 114, 121, 123\n\ncounterfeit prescriptions, 16\u201317\n\nCraddock, Harry, 32\n\n_The Craft of Gin_ (Smith), 64\n\nCr\u00e8me de Menthe, 176\u201377\n\ncr\u00e8me de menthe\n\nand essential oils, 35, 123, 175\n\nas ingredient, 134, 178, 180, 181\n\nmedicinal qualities of, 8\n\nsmuggling of, 14\n\nCuba Libre, 66\n\nCuming, Fortescue, 107\n\ncura\u00e7ao, 65, 71, 76, 118, 141, 184, 185\n\ncuriosities, 25\n\nCymar (lab supply firm), 38\n\ndaiquiris, 75\n\n**D**\n\nDalrymple, Alfred V., 17\u201318\n\nDaly, Tim, 155\n\nDavis, Marni, 112\n\nDeath in the Gulf Stream, 66\n\nDeath's Door White Whiskey, 80\n\nDelahaye, Marie-Claude, 126\n\n_De Materia Medica_ (Dioscorides), 32\n\nDemerara, 80, 143\n\ndenatured industrial alcohol, 13, 88\n\nDe Schat der Likeuristen (Moens and Verburgt), 167\n\nDetroit Athletic Club, 59\n\nDietsch, Michael, 71\n\nDioscorides, 32\n\n_Dispensatorium Pharmacopolarum_ (Cordus), 33\n\ndistilling\n\nand absinthe, 126\n\nof apple spirits, 110\n\nof essential oils, 22, 39, 41\u201342\n\nof gin, 57\n\nhistory of, 34\u201335, 69, 93\n\nhome, 4\u20135, 9, 13, 38, 48, 168\n\nand industry standards, 142\n\nmanuals, 6, 93, 127, 143, 203\n\nin prisons, 6\n\nof rum, 67\n\nDIY Orange Cordial, 189\n\nDIY Summer Cup, 118\n\nThe Doctor (beading oil), xv, 24\n\ndoctored drinks, 2, 24\n\nDolin Vermouth de Chamb\u00e9ry, 76\n\nDoran, J.M., 8\n\nDr. Furnish, 172\n\ndried sour cherries, 84\n\ndruggists, 15\u201318, 30, 33\u201334, 193\u201395\n\nThe Druggists' Circular (1913), 151\n\ndruggists' manuals, 8, 125\n\ndrugstore whiskey\n\n_See_ medicinal alcohol\n\nDubonnet Cocktail, 59\n\nDuplais, Pierre, 127, 143, 167, 203\n\n\"Dutch\" anisette, 167\n\nDutch gin\n\n_See_ genever\n\n**E**\n\negg drinks, 24, 65, 134, 144\u201345, 165\n\n18th Amendment, 3\u20135, 9\u201310\n\nEinstein, Izzy, 112\n\nEl Dorado rum, 74, 78\n\nelectromagnetic stirrer, 37\n\nEllestad, Erik, 78\n\nEl Presidente, 75, 76\n\nEmbury, David, 111, 157\n\nenfleurage, 42\n\nEpicurean, 157\n\nErlenmeyer flasks, 37\n\nessence\n\nof absinthe, 125\n\nbrandy, 86\u201387\n\nof cordials, 161\n\ndefined, 48\n\nformulas, 68\u201369\n\nJamaican Ginger, 170, 173\n\nrum, 48, 67\u201368, 158\n\nviolet, 125\n\nof wine, 117, 184\n\nwoodruff, 182\u201383\n\nessential oils\n\nand absinthe, 125, 129\n\nadulteration of, 45\u201347\n\naffecting spirit grades, 142\n\nbuying and storing of, 48\u201349\n\nand cordials, 151, 153\u201354, 158\u2013160, 167\n\nand cr\u00e8me de menthe, 35, 123, 175\n\ndefined, 39\u201341\n\ndistilling of, 22, 39, 41\u201342\n\nand flavor, 21\n\nfragrant taint, 47\n\nand gin, 55, 57, 64\n\nhistory of, 29\n\nmedicinal qualities of, 8\n\nand necessary gear, 36\u201338\n\nand orange flavored recipes, 187\n\nand poisoned liquor, 85\n\nand rum, 67, 74\n\nsafety of, 43\u201345\n\nand syrups, 144\u201346\n\nweights and measures, 194\u201396\n\nEvan Williams bourbon, 162\n\nExpatriate (bar), 154\n\n**F**\n\nfake alcohol\n\n_See_ counterfeit alcohol\n\nFarber, Dan, 100\n\n_Fatherland_ (ed. Viereck), xx\n\nFDA (Food and Drug Administration), 44\u201345\n\n_Fenaroli's Handbook fo Flavor Ingredients_ , 204\n\nfennel oil, 125\u201327, 127, 129, 150\u201351, 165\u201368\n\nFernet Branca, 181\n\nFerrand dry orange cura\u00e7ao, 74\n\nfiltering medium, 37\n\n_The Fine Art of Mixing Drinks_ (Embury), 111, 157\n\nflasks, 37\n\nflavor, defined, 27\n\nFlavor & Extract Manufacturers Association, 45\n\nflavored whiskey, 82\n\nFleischman, Joseph, 2, 20, 71\n\nflips, 109\n\nflower waters, 42\n\n_The Flowing Bowl_ (Schmidt), 155, 180\n\nfoamy spirits, 24\n\nFogarty, Frank, 59\n\nfolk distilling\n\n_See_ moonshine\n\n\"Food Flavors: Benefits and Problems\" (Hall), 27\n\nformic ether, 68\u201369, 77\n\nformula, defined, 22\n\nFormula 39-B, 14\n\nfragrant taint, 47\n\nFrench 75, 66\n\nFrench Absinthe, 129\n\nFrench proof measurement, 202\u20133\n\nfresh fruit garnish, 104, 118\n\nFritzsche Bros., 81\u201382\n\nFront Porch Lemonade, 80\n\nfunnels, 37\n\nFurnish, Tim, 172\n\n**G**\n\ngalangal oil, 74\n\ngalls, 99, 121\n\ngarnishes\n\nabsinthe onions, 135\n\ncherry, 75\u201376\n\ncitrus zest, 103\n\nfresh fruit, 104, 118\n\nlemon twist, 59, 74, 104\n\nlime, 74, 154\n\nmint, 74, 80, 118, 181\n\norange, 60, 165\n\nsunflower petal, 74\n\nGaudin, Lucien, 60\n\nGay-Lussac, Joseph-Louis, 202\n\ngear, 36\u201338\n\ngenever, 63\u201366\n\ngentian, 136, 139\n\n_The Gentleman's Companion_ (Baker), 66, 110, 177\n\nGermain-Robin, Hubert, 95, 99, 100\n\nGerman-Americans, xx\u2013xxii, 147\n\n_Das Getr\u00e4nkbuch_ (Kr\u00f6nlein), 183\n\nGettler, Alexander, 88\n\ngin\n\nBee's Knees, 61\n\nChief Gowanus New Netherland Gin, 62\n\ncompounded, 55\u201357\n\nDubonnet Cocktail, 59\n\nand essential oils, 55, 57, 64\n\n\"Holland\" gin, 19, 62\u201363, 66\n\nhomemade (bathtub), 55\n\nas ingredient, 59\u201361, 118, 135\n\nThe Last Word Cocktail, 59\n\nLondon dry, 62, 66\n\nLucien Gaudin, 60\n\nmedicinal qualities of, 8\n\nOld Tom, 57, 66\n\nSouthside, 60\n\nspecific types of, 57\n\nginger beer, 118, 161, 170\n\nginger brandy, 170\u201373\n\nginger oil, 42\u201343\n\nGinger Wine with Cochineal, 174\u201375\n\nGivaudan, 82\n\nglass bottles, 36\n\ngomme syrup, 65, 177\n\nG\u00f6ttler, H., 194\n\ngraduated cylinder, 38\n\ngrain schnapps, 91\n\ngrains of paradise, 89\u201391\n\ngrams, 194\n\ngrapefruit oil, 188\n\nGRAS: Generally Recognized as Safe, 44\u201345\n\ngrenadine, 74\u201376, 85, 111, 135\n\nGrier, Jacob, 154\n\ngrinders, spice or coffee, 36\n\n_Guide to the Analysis of Potable Spirits_ (Vasey), 39\n\nGum kino, 139\n\ngum syrup, 147, 177, 180\n\n**H**\n\nHagers Handbuch der Pharmazeutischen Praxis (Hager), 125\n\nHaigh, Ted \"Dr. Cocktail,\" 47\n\nhalf-and-half, 134, 181\n\nHall, R.L., 27\n\nhard cider, 10, 51, 109\n\nHarry's Bar, 104\n\nHarvard Cocktail, 104\n\nHavana, Cuba, 75\n\nHelen Saunders Wright's Cinnamon Cordial, 160\n\nHemingway, Ernest, 66\n\nHenley's May Wine Essence, 184\n\n_Henley's Twentieth Century Formulas, Recipes and Processes_ (Hiscox), 184\n\n_Here's How!_ (cocktail booklet), 163\n\n_Herrengedeck_ , 89\n\nhigh-proof alcohol, 39, 41, 167\n\nHiscox, Garner D., 184\n\n_The History of Applejack_ (Weiss), 109\u201310\n\n_The Hoffmann House Bartender's Guide_ (Mahoney), 160\n\nHolcomb, Wynn, 104\n\nHolland gin, 19, 62, 63, 66\n\nHollands\n\n_See_ genever\n\nhome brewing, 13, 105\n\nhome distilling, 4\u20135, 9, 13, 38, 48, 168\n\nhoney syrup, 61\n\nhops, 107\n\nhorehound candy, 84\n\nhorseradish, 116\n\nHostetter, Jacob, 136\n\nHostetter(ish) Bitters, 139\n\nHotel Nacional Special, 75\n\nhydrosals, 42\n\nhyper critical carbon dioxide extraction, 42\n\nHyson Tea, 168\n\nhyssop, 127\n\n**I**\n\nIce Caraway\/K\u00fcmmel\/Liquor, 147, 152\u201353\n\nillegal sources of alcohol, 13\u201314\n\n_Imbibel_ (Wondrich), 200\n\n_Imbibe_ magazine, 157\n\nimported spirits, 5\n\nIndustrial Alcohol and Chemical Division of the Prohibition Unit, 8\n\nInternational Fragrance Association, 44\n\nInternational Organization of the Flavor Industry, 45\n\nintoxicating beverages, 9, 13\u201314\n\nItalian vermouth, 76, 104\n\nItalian Vineyard Company, 112\n\n**J**\n\nThe Jack Rose, 111\n\nJamaica Ginger essence, 170, 173\n\nJamaican rum, 65, 119\n\nJeff Morgenthaler's Blended Grasshopper, 181\n\njenever\n\n_See_ genever\n\nJersey Lightning, 110\n\nJohnson, Eli, 28\u201329\n\n_Journal of the National Medical Association_ , 8\n\nThe Judge, 180\n\njuniper berries, 21, 42, 46, 55, 57, 63\u201364\n\njuniper oil, 8, 30, 59\n\n_Justus von Liebig: The Chemical Gatekeeper_ (Brock), 28\n\n**K**\n\nKings County Pharmaceutical Society, 16\n\nkirschwasser, 104, 150\n\nkitchen measures equivalents, 201\n\nkitchen scale, 37, 194\n\nKitchiner, William, 78\n\nKlus, Tommy, 156\n\nKorn, 89\u201390\n\nKornsch\u00e4rfe, 90\u201391\n\nKramer, John, 105\n\nKr\u00f6nlein, Hans, 183\n\nk\u00fcmmel, 119, 142, 147\u2013157\n\nThe K\u00fcmmel Fizz, 154\n\nK\u00fcmmel formula, 151\n\nK\u00fcmmellik\u00f6r, 120, 157\n\n**L**\n\nLacour, Pierre, 103, 145\n\nLaird & Company, 110\n\nLaird's 100-proof apple brandy, 155\n\nLaniz\u00e8t: Sour Mash Cajun Anisette, 169\n\nThe Last Word Cocktail, 59\n\nLatin names, 18\u201319, 33\u201334, 193\n\nLazar, Michael, 74\n\nlegal sources of alcohol, 9\u201313\n\nlemon juice, 60\u201361, 66, 71, 74, 80, 85, 103, 161, 165, 172\n\nlemon oil, 188\n\nlemon twist garnish, 59, 74, 104\n\nLeopold Bros., 141\n\nLicor 43 cordial, 162\n\nLiebig, Justus von, 28\u201329\n\nlime garnish, 74, 154\n\nlime juice, 59, 71, 74, 111, 154, 180\n\nlime oil, 66, 75\n\nLime Simple Syrup, 74\u201375\n\nlime wheel, 74\n\nliqueur ratios for cordials, 143\n\n\"local options\" laws, 119\n\nLoeb, Katie, 162\n\nLoftus, William, 24, 101\n\n_Loftus's New Mixing and Reducing Book_ (Loftus), 101\n\nloopholes, legal, 2, 4, 9, 10, 15\u201316, 112\n\n_Los Angeles Times_ (newspaper), 85\n\nLost Spirits Cuban-style rum, 71\n\nlouche form, in absinthe, 129\n\n_Louisville Courier-Journal_ , xviii\n\nlower-proof alcohol, 13, 21, 39, 41\n\nLow Grade Artificial Rum, 69\n\nLucien Gaudin, 60\n\nLyon, Victor A.\n\nabsinthe recipes, 123\u201325\n\nanise cordial recipe, 165\n\nArrakessenz, 77\n\nbackground of, xv\u2013xx, 8\n\nand beading recipes, 24\n\nbrandy essence, 86\n\nbrandy recipes, 102\n\ncider recipes, 105\u20137\n\ncinnamon flavored cordials, 159\n\nCr\u00e8me de Menthe recipe, 176\u201377\n\nand garnishes, 135\n\nGinger Brandy recipes, 170\u201373\n\ngin recipes, 57\n\nand ingredient safety, 44\n\nKornsch\u00e4rfe, 90\n\nk\u00fcmmel, 148\n\nand methyl alcohol, 85\n\nmint flavored cordials, 175\u201376\n\nand moonshine, 53\n\nnalewka cordials, 161\n\nnotebook of, 1, 6, 21\u201322, 30, 53\n\noak in recipes, 99\n\norange flavored recipes, 184\n\nand poisoned liquor, 85\u201388\n\nPort Wine Essence, 117\n\nPort Wine recipe, 114\n\nRumessenz, 68\n\nrum recipes, 67\n\nshrub recipes, 71, 118\n\nsyrup recipes, 100\n\nwhiskey recipes, 81\n\nwoodruff essence, 182\n\n**M**\n\nmace oil, 165\n\nmaceration\n\nof absinthe, 130\n\nand anisettes, 168\n\nand arrack, 77\u201378\n\nof ginger flavored drinks, 170\u201373\n\nof mint flavored drinks, 176\n\nand moonshine, 81\n\nof orange flavored drinks, 187\n\nand rum, 68, 71\n\nand vodka, 150\n\nand whiskey, 84\n\nMachado, Gerado, 76\n\nmag bar, 37\u201338\n\n\"Magic Box\" (Eli Johnson), 28\n\nmai tai, 66\n\nMaitrank, or May Wine, 183\n\nMake at Home Orange Peel Flavoring, 187\n\nmalt syrup, 13\n\n_Manual for the Essence Industry_ (Walter), 48\n\nmanuals\n\nbartending, 36\n\nand cocktail recipes, 167\u201368\n\ncompounding, 19, 38, 71, 82, 89, 157, 194\n\ndistilling, 6, 93, 127, 143, 203\n\nand doctored drinks, 2\n\ndruggists,' 8, 125\n\nand essential oils, 39, 48\n\nrectifiers,' 34\u201335, 184\n\nweights and measures, 22, 142, 193\u201394, 196, 203\n\n_The Manufacture of Liquor and Preserves_ (de Brevans), 168\n\n_The Manufacture of Liquors, Wines, and Cordials without the Aid of Distillation_ (Lacour), 103, 145\n\nmaple syrup, 103\n\nmaraschino liqueur, 59, 75, 180\n\nmargaritas, 71, 156\n\nMary Pickford cocktail, 22, 75\n\nmason jars, 36\n\n_materia medica_ , 32\n\nMcCaw, Mike, 188\n\nMcElhone, Harry, 104\n\nmeasures and weights, 193\u201396, 201\n\nmeasuring cups\/spoons, 36\u201337, 192\n\nmeasuring devices, 192\n\nmedicinal alcohol, 8, 10, 15\u201318, 32\u201333, 84, 124, 129, 136\u201338, 180, 196\n\nmelissa (lemon balm), 127\n\nmethanol (wood alcohol), 45, 86, 89\n\nMethyl alcohol, 85, 86\n\nmetric measures, 193\u201394\n\nM'Harry, Samuel, 20, 62, 93\u201395\n\nMichael Lazar's Mai Tai, 74\n\nMichaelson, M. Alfred, 14\n\nMickler, Ernest Matthew, 169\n\nmicroliters, 195\n\nmicroplanes, 37\n\nMike McCaw's Triple Sec, 188, 204\n\nmilk washing, 105\n\nmilliliters, 194\n\nmint\n\nin absinthe, 127\n\ncordials, 175\u201378\n\nas essential oil, 42\u201343\n\nas flavoring, 27\n\ngarnish, 74, 80, 118, 181\n\nas ingredient, 175\u2013181\n\nleaves of, 60, 176\n\nMint and Gum, 177\u201378\n\nShaker Peppermint Cordial, 178\n\n\"spanking\" the mint, 40\n\nMint and Gum, 177\u201378\n\nmissing units, 194\n\nMission Bell, 156\n\nMitchell, John, 1\n\nMoens, Henri, 167\n\nmojito, 60\n\nMonkey Gland, 135\n\nmoonshine\n\nand authenticity, 202\n\nand counterfeit alcohol, 81\n\nand home distilling, 13, 93\n\nand Lyon's notebook, 53\n\nand prisoners, 6\n\nand spirit foam, 24\n\nsyndicates, 4\n\n_Moonshine!_ (Rowley), xii\n\nMorgan, J. Pierpont, 65\n\nMorgan, William M., 14\n\nMorgenthaler, Jeffrey, 36, 181\n\nmorphine, 34\n\nmortar and pestle, 38\n\nMurray, George, 109\n\n**N**\n\nNalewka, 161, 165\n\nNalewka Sour, 161\n\nnanosmuggling, 14\n\n_National Bottlers' Gazette_ , 116\n\nNational Medical Association, 15\n\nNational Prohibition Act, 3\u20134, 10, 11\u201312, 17, 105, 112\n\n_See also_ prohibition; Volstead Act\n\nnear beer, 13\n\nNegroni, 60\n\nneutral spirits, 39, 43, 55, 57, 68, 77, 83, 95, 142, 188\n\nNew York Distilling Company, 62\n\nNew York Public Library, xvii, 1\n\nNitrous ether, 86, 117\n\n\"noble experiment\"\n\n_See_ prohibition\n\nNoble Experiment (bar), 162\n\n\"non-intoxicating\" alcohol, 13\n\nNorris, Charles, 14, 88\n\nnotes, in flavor, 27\n\n**O**\n\noak barrel whiskey, 83\n\nOak wood, 99\n\nOld Peach, 103\n\n_Old-Time Recipes for Home Made Wines, Cordials, and Liqueurs from Fruits, Flowers, Vegetables, and Shrubs_ (Wright), 160\n\nOld Tom gin, 57, 66\n\nonions, as garnish, 135\n\nOrange cordial, 189\n\norange cura\u00e7ao, 76, 119, 204\n\nOrange Flower Water, 187\n\norange garnish, 60, 165\n\norange juice, 71, 85, 135, 172\n\norange oil, 41\n\norange peel, 60, 76, 139, 160, 187, 189\n\nOrange Peel Flavoring, 141, 184, 187\n\norgeat, 74, 134\n\nOstrander, Ethan, 162\n\noverproof (O.P.) spirits, 43, 201\u20132\n\n**P**\n\nPadosch, Stephan A., 122\n\npale sherry, 109\n\n\"parts\" ratios, 204\n\nPearson Square, 50\u201351\n\npeelers, 37\n\nPekoe tea, 77\n\nPelican Saloon, 65\n\npeppercorns, 163\n\npeppermint liqueur, 181\n\npeppermint oil, 8, 35, 45, 153, 165, 175\u201378\n\n_Pharmaceutical Journal and Transactions_ , 33\u201334, 148\n\npharmacists, 15\u201318, 30, 33\u201334, 193\u201395\n\n_Pharmacopoeia Londinensis_ (1611), 82\n\npharmacopoeias, 30\u201334, 45\n\nPhiladelphia Union Club, 119\n\nPhillips Distilling Company, 178\n\nPicayune K\u00fcmmel, 151\n\n_Picayune's Creole Cook Book_ (1910), 151, 168\n\nPickford, Mary, 22, 75\n\nPimm's Cup, 118\n\npineapple juice, 24, 75\n\npint measurement, 191\u201392\n\npipettes\/micropipettes, 36, 195\n\nPlain spirits, 67\u201369\n\nplastics, 36\n\npoisoned liquor, 29, 85\u201388\n\npomegranate grenadine, 111\n\nport wine, 114, 184\n\nPort Wine Essence, 117\n\n_Potions of the Caribbean_ (Berry), 76\n\n_The Practical Distiller_ (M'Harry), 20, 62, 93\u201395\n\nPrater (beer garden), 182\n\npre-prohibition, xx, 2, 5, 8\u20139, 33, 110, 147\n\nPrescription Julep, 200\n\nprisons, 6\n\nproduction method, 86\u201388, 142\n\nprohibition\n\nand ciders, 105, 110\n\ncriminality of, 5\u20136\n\ngeographic borders of, 74\u201375\n\nand illegal sources alcohol, 13\u201314\n\nand legal sources of alcohol, 9\u201313\n\nand Lyon's notebook, 6\u20138\n\nand medicinal alcohol, 15\u201318\n\noverview of, xxi\u2013xxii, 1\u20135\n\npre-prohibition, xx, 2, 5, 8\u20139, 33, 110, 147\n\nand wines, 112\n\nproofs, measurements of, 201\u20132\n\n_The Prudhomme Family Cookbook_ (Pruhomme), 169\n\nPruhomme, Paul, 169\n\nprune juice, 69\n\nPuerto Rican 151, 71\n\nPumpenheimer, 19\n\n_Punch_ (Wondrich), 146\n\nPure Cognac of 17 Underproof, 100\n\npurity, of alcohol, 30\n\nPyroblast\/Pyroblast syrup, 162\u201363\n\npyroligneous acid, 67, 77\n\n**Q**\n\nQuick and Dirty Orange Flower Water, 187\n\nquintessential oils\n\n_See_ essential oils\n\n**R**\n\nraisins, 30, 105, 107\n\nRamos Gin Fizz, 42, 134, 187\n\n\"rectified wood spirit,\" 86\n\nrectifiers' manuals, 34\u201335\n\nred chile flakes, 163\n\nredistilled gin, 57\n\nred vermouth, 119\n\nreligious exemptions, 4, 10\n\nRemsberg, Stephen, 67\n\nRepeal Day, 3\n\n\"Repeal of Prohibition in Practical Effect in N.Y.,\" San Bernardino County Sun, 52\n\n\"Report on the Supply of Spirit of Wine Free of Duty for Use in the Arts and Manufactures\" (1855), 148\n\nReynolds, B.G., 182\n\n_Rezeptbuch f\u00fcr Destillateure_ (G\u00f6ttler), 194\n\n_Richmond Planet_ (newspaper), 1\n\nrich syrup, 145, 177\n\nRock & Rye, 84\n\nrock candy, 84\n\nroman numerals, 199\n\nrose liqueur, 155\n\nrose oil, 45, 151, 167\n\nrose water, 42, 155, 167\n\nrum\n\na\u00f1ejo Havana Club rum, 76\n\nAppleton rum, 74\n\ndistilling of, 67\n\nEl Dorado rum, 74, 78\n\nEl Presidente, 76\n\nessences, 48, 67, 68, 158\n\nJamaican rum, 65, 119\n\nLost Spirits Cuban-style rum, 71\n\nLow Grade Artificial Rum, 69\n\nmaceration of, 68, 71\n\nMichael Lazar's Mai Tai, 74\n\nroman numerals, Appleton rum\n\nRumessenz, 68\n\nshrubs, 71, 135\n\nSwedish Punch, 78\n\nTwelve Mile Limit, 74\n\nwhite rum, 74, 153\n\nWilliam Kitchiner's Mock Arrack, 78\n\nRumessenz, 68\n\nrum ether\n\n_See_ formic ether\n\n\"rum row,\" 74\n\nRum Shrub recipe, 71\n\nRussian Caraway, 153\u201354\n\nRye Ether, 86\n\nrye whiskey, 74, 84\u201385, 110\n\n**S**\n\nsacramental wine, 4, 10, 112\n\nsaffron, 67, 69, 82, 145\n\nsaloon culture, 2, 5\u20136\n\nsalt, 68, 153, 181\n\n_Salt Lake Herald_ (newspaper), 113\n\nSam Ward's Recipe, 155\n\nSanford's Jamaica Ginger advertisement, 172\n\nSaucier, Ted, 59\n\n\"Save the Boy!\" propaganda, 132\n\n_The Savoy Cocktail Book_ (Craddock), 32\n\nSchiedam\n\n_See_ genever\n\nSchmidt, William, 155, 180\n\nSchweizer Absinth\u00f6l (Swiss Absinthe Oil), 125\n\n_Scientific American_ (periodical), 168\n\nSeed, Eric, 76\n\nseparatory funnel, 38\n\nShaker Peppermint Cordial, 178\n\nThe Shakers (United Society of Believers), 178\n\nShannon, Richard, 129\n\nSheridan Rye, 12\n\n_Shrubs_ (Dietsch), 71\n\nSidecar, 103\n\nSikes hydrometers, 202\n\nSimple Caraway Vodka, 150\n\nsimple syrup, 36, 60, 66, 74\u201375, 103\u20134, 109, 143\u201345, 165, 170, 173\n\n_Sketches of a Tour to the Western Country_ (Cuming), 107\n\nsloe gin, 57\n\nSmith, C.P., 14\n\nSmith, David T., 64\n\nSmuggler's Cove, 118\n\nsmuggling, 4\u20135, 74\n\nSmyth, William, 24\n\nSnowshoe Grog, 178, 180\n\nsoda water, 66, 165\n\nSonnentanz, 119\n\nSouthside, 60\n\n_Spezialit\u00e4ten und Geheimmittel_ (1906), 90\n\nspiced tea, 78\n\n_Spice: The History of a Temptation_ (Turner), 28\n\nSpinoza, Baruch, xxii\n\nSpirit Beading, 25\n\nspirit hydrometer, 38\n\nspoons, 36\n\nstar anise, 143, 147, 150, 162\u201363, 168\n\nstar anise oil, 125, 129, 151, 165, 167\n\nstate laws on alcohol, 5\n\nStauch restaurant, 152\n\nsteam distillation, 42\n\nSteele, Richard, 30\n\nStenson, Murray, 59\n\nSt. George Spirits, 184\n\nstill (equipment), 21, 38, 111\n\nStinger (1922), 181\n\nstingers, 180\u201381\n\nstomach bitters advertisement, 138\n\nStone, Gwydion, 129, 131, 132\n\nStoned dry plums, 99\n\nThe Stone Fence, 109\n\nStonewall Jackson, 109\n\nstrainers, 37\n\nstretching the spirit technique, 95\n\nStrong Tincture of Ginger, 173\n\nsugar\n\nand absinthe, 126\u201327, 130, 132\n\nin anisettes, 168\u201369\n\nin arracks, 78\n\nand bitters, 139\n\nand caramel, 79\u201380\n\nin ciders, 107, 109\u201311\n\nin cordials, 120, 141\u201343, 148\u2013160, 177\u201378, 183, 188\n\nand cr\u00e8me de menthe, 176\n\nin ginger flavored drinks, 170\u201374\n\nlaws regarding, 105\n\nand necessary gear, 36\u201338\n\nin shrubs, 71\n\nand spirit beading, 25\n\nand syrups, 99\u2013100, 104, 114, 143\u201347, 163, 189\n\nsulfuric acid, 24\u201325, 45, 67\n\nSummer cup, 118\n\nSunday, William \"Billy,\" 6\n\nsunflower petal garnish, 74\n\nsupplies, 36\u201338\n\nSwedish Punch, 78\n\nSweet Caramel Syrup, 80\n\nsweet spirit of nitre, 67\n\nsweet vermouth, 118\n\nsweet woodruff, 182\u201384\n\nSyrup Charentais, 99\u2013100\n\nsyrups\n\naged, 100, 146\n\nCharentais, 99, 100\n\nand cocktails, 147\n\ncompounder's syrup, 103, 145, 161, 167, 173, 177, 183, 189\n\nfor cordials, 144\u201345\n\ndefined, 143\n\nand essential oils, 144\u201346\n\nflavoring of, 146\n\ngomme syrup, 65, 177\n\ngum syrup, 147\n\nhoney syrup, 61\n\nLime Simple Syrup, 74\u201375\n\nmalt syrup, 13\n\nmeasuring of, 144\n\nPyroblast syrup, 162\u201363\n\nrich syrup, 145, 177\n\nsimple syrup\n\n_See_ simple syrup\n\nSweet Caramel Syrup, 80\n\n**T**\n\ntalcum powder, 159\n\ntaste, 27\n\n_The Tatler_ (newspaper), 30\n\ntea, 27, 67, 69, 77\u201378, 99, 119, 121, 139, 168\n\ntemperance\n\n_See_ prohibition\n\nterpeneless oils, 41\n\nThomas, Jerry, 32\n\nTiki Oasis, 80\n\ntonka beans, 182\n\nTovey, Charles, 95\n\n_Tractatus Politicus_ (Spinoza), xxii\n\nTrader Vic (restaurant), 74\n\n_Trait\u00e9 de la Fabrication des Liqueurs et de la Distillation des Alcools_ (Duplais), 143\n\nTralles, Johann Georg, 204\n\nTralles scale, 203\u20134\n\ntravel exemption, 10\u201312\n\n_A Treatise on the Manufacture and Distillation of Alcoholic Liquors_ (Duplais), 127, 203\n\n\"Tricks of the Wine Doctor\" (Barclay), 113\n\ntriple sec, 188\n\ntroy weight, 193, 198\n\nTTB (US Alcohol and Tobacco Tax and Trade Bureau), 57\n\nTurner, Jack, 28\n\nTurner, John, 15\n\nturpentine, 30, 45, 46, 47\n\nTwain, Mark, 85\n\nTwelve Mile Limit, 74\n\n21st amendment, 5, 55\n\n**U**\n\nunderproof (U.P.) spirits, 202\n\n_The United States Dispensatory_ (1845), 33\n\n_Untersuchungen \u00fcber das Radikal der Benzoes\u00e4ure_ (von Liebig and W\u00f6hler), 28\n\nUpshaw, William, 3\n\nUre, Andrew, 92\n\nUS Alcohol and Tobacco Tax and Trade Bureau (TTB), 57\n\n**V**\n\nvanilla, 68\u201369, 77, 151\n\nVanilla beans, 99\n\nvanilla extract, 169\n\nvanilla ice cream, 181\n\nVasey, S. Archibald, 39\n\n\"Vauxhall nectar,\" 78\n\nVerburgt, Hubert, 167\n\nVermeire, Robert, 181\n\nvermouth, dry, 60, 76, 157\n\nVictor Lyon's Ginger Brandy recipes, 170\u201373\n\nViereck, George Sylvester, xiv, xviii\u2013xx\n\nvinegar, 71, 81, 101, 105, 121\n\nviolet essence, 125\n\nvitriol oil, 24, 25\n\nvodka, 150, 161, 167, 187\u201388\n\nvolatile oils\n\n_See_ essential oils\n\nVolstead, Andrew, 4, 81\n\nVolstead Act, xx, 4, 6, 9, 17, 112, 119\n\n_See also_ National Prohibition Act\n\nvolume measurement equivalents, _195_ , _198_\n\nVoronoff, Serge Abrahamovitch, 135\n\n**W**\n\nWaldmeister-Bowlenessenz (sweet woodruff), 183\n\nWalter, Eric, 48\n\nWard 8, 85\n\n_Washington (DC) Evening Star_ (newspaper), 185\n\nWashington, George, 107\n\nwaterbath (\"bain-marie),\" 168\n\nWatterson, Henry, xviii\n\nweight measurement equivalents, _195_ , _198_\n\nWeiss, Harry B., 109\u201310\n\nwhiskey\n\napple, 110\n\n\"chipped,\" 83\n\nDeath's Door White Whiskey, 80\n\nflavored, 82\n\noak barrel, 83\n\nRock & Rye, 84\n\nrye, 74, 84\u201385, 110\n\nWard 8, 85\n\nwhite rum, 74, 153\n\n_White Trash Cooking_ (Mickler), 169\n\nwholesale compounding, 22\n\nWiley, Harvey W., 67, 82\n\nWilliam Kitchiner's Mock Arrack, 78\n\nWilson, Woodrow, 4\u20135\n\nwine distillate, 99\u2013100\n\nwines\n\ncoloring of, 114\n\nDIY Summer Cup, 114\n\nhistory of, 112\u201313\n\nPort Wine Essence, 117\n\npreventing bad taste and sourness, 116\n\nSonnentanz, 119\n\nwintergreen oil, 46, 82\n\nWinters, Lance, 184\n\nW\u00f6hler, Friedrich, 28\n\nWondrich, David, 146, 200, 201\n\nwood alcohol poisoning, 85\u201388\n\nwoodruff essences, 182\u201384\n\nWormwood Society, 132\n\nwormwood\/wormwood oil, 46, 125, 127, 129, 131\n\nWright, Helen Saunders, 160\n\n**Y**\n\n_Yankee antifreeze_ , 110\n\nYarm, Frederic, 165\n\n**Z**\n\nZeeland cocktails, 119\n\nzest, of citrus, 84, 103, 150, 157, 162, 189\n\n## Acknowledgments\n\nDeaths, dying, suicide, and heartbreaking disappointment: the last year has been brutal. Friends both dear and casual lost or took their lives, marriages crumbled, and loved ones endured stomach-churning setbacks. As spring slides into summer, fresh insults continue. Working on this book through this cruel year has been a reminder that I am surrounded by kind, generous, and smart people who have shared their time, expertise, recipes, and insights with me.\n\nHere are a few of them. If I have forgotten anyone, forgive the oversight.\n\nThanks first to my editor at Countryman Press, Ann Treistman. When Ann asked if I wanted to write another moonshine book, I deflected the question. She was game enough to take on this one instead. For their most excellent photography, I am indebted to John Schulz and Daniel Fishel of StudioSchulz in San Diego. And to designer Nick Caruso who turned my meandering discussion of the nadir of beverage arts in American history to the gorgeous book in your hands. My agent Lisa Ekus advised, guided, and arranged so much while Sally Ekus and everyone else at the Lisa Ekus Group pitched in to make this manuscript happen.\n\nRare books expert Charles Keller offered advice on Lyon's original handwritten notebook while Will Elsbury, Military History Specialist at the United States Library of Congress, tracked down esoteric historic information. Sushan Chin, archivist at The New York University School of Medicine, provided some background material on der unsichtbare Mann, Victor Lyon, while librarian Nancy Kervin helped with materials in the US Senate collections and Rebecca Johnson Melvin uncovered Viereck material at the University of Delaware. At the University of California, Davis Axel Borg, special collections archivist Liz Phillips, Thomas Collins, and Lynne Kimsey, director of the Bohart Entomological Museum, all shared the time and expertise.\n\nDistillers and fermenters Gwydion Stone, Dave Smith, Lance Winters, Dan Farber, Hubert Germain-Robin, Maggie Campbell, Chip Tate, Michael Skubic, Mike McCaw, Carl Sutton, and Robert Birnecker graciously offered various technical, historical, and translation assistance. Bartenders and servers Scott Beattie, Jacob Grier, Bobby Heugel, Tommy Klus, Katie Loeb, Jeff Morgenthaler, Ethan Ostrander, Stephen Shellenberger, Matt Van Wagner, Audrey Saunders, Gaz Regan, Michael Lazar, Vipop Jinaphan, and Frederic Yarm provided recipes and directions for various concoctions. Amy Stewart ( _The Drunken Botanist_ ), who knows her way around a poisonous plant or two, lent a hand flagging some of the more problematic botanicals in the old notebook.\n\nMelinda Huff at Cynmar Corporation provided lab equipment that made much of this work faster and more efficient. Canadian chemist, author, and bartender Darcy O'Neil generously provided samples of some of the more esoteric compounds I used to test these and other compounding formulas. Thanks also to cocktail historian Ted Haigh for insight into liquor adulteration, Greg Boehm for sharing scans from his collection, and Dave Arnold for liquor manipulation ideas.\n\nOtto von Stroheim at Tiki Oasis and the staff of Denver's Museum of Contemporary Art invited me to use their audiences as guinea pigs for some initial vetting of the topics in this book; thank you all. In Berkeley, perfumer Mandy Aftel vetted my ideas on essential oils while \"Professor\" Steve Remsberg, Jeff \"Beachbum\" Berry, and Luis Ayala gave insight into historic and contemporary rum fakery. Blair Reynolds of BG Reynolds Syrups provided samples of his woodruff syrup \u2014 and inspired me to plant a load of Waldmeister in our California garden. In Emeryville over mezcal and hops, Adam Rogers of _Wired_ magazine humored me while I teased apart ideas. Back in San Diego, Ryan Morris organized my notes, transcriptions, and translations, a project that had stymied others.\n\nAbove all, there was the man who has been at my side for more than twenty years, Tim Furnish. Without his unwavering encouragement, Victor Lyon's old notebook might still be tucked away in my library seeming for all the world like a dreary old book of questionable poetry.\nLost Recipes of Prohibition\n\nPage xx: Library of Congress, Prints & Photographs Division, WWI Posters, LC-USZC4-7819; xxi: _Temperance: a monthly journal of the Church Temperance Society_ , April 1918; xxii: raclro\/iStockphoto.com; : Orange County (California) Archives; : Charles S. Warnock, _Giggle Water_. 1928. New York Club Cocktail; : A Holiday Liquor Map, _The Rhinelander Daily News_. December 19, 1929; : LeeTorrens\/iStockphoto.com; 22, : Judge Jr., _Here's How_. 1927. Judge Publishing Company; 22 (stamp): raclro\/iStockphoto.com; : fotomy\/iStockphoto.com; : Library of Congress, Prints & Photographs Division, LC-DIG-ppmsc-05880; : Bartender at Schiek's Restaurant serving first beer the day after Prohibition was repealed. Minneapolis: Minnesota Historical Society; : Everett Historical\/Shutterstock.com; : Library of Congress, Prints & Photographs Division, LC-USZ62-12143; : Robert N. Dennis, from collection of stereoscopic views. Union League Club, Philadelphia; : Library of Congress, Prints & Photographs Division, LC-DIG-ppmsca-25463; : Library of Congress Prints and Photographs Division Washington, LC-DIG-npcc-00979. All botanical illustrations: _K\u00f6hler's Medizinal-Pflanzen_. Photography of journal pages by John Schulz and Daniel Fishel (StudioSchulz). Book cover photograph by Sean Hemmerle.\n\nCopyright \u00a9 2015 by Matthew Rowley\n\nAll rights reserved\n\nBook layout and design: Nick Caruso Design\n\nFor information about permission to reproduce selections from this book, write to Permissions, W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, NY 10110\n\nFor information about special discounts for bulk purchases, please contact W. W. Norton Special Sales at specialsales@wwnorton.com or 800-233-4830\n\nThe Countryman Press\n\nwww.countrymanpress.com\n\nA division of W. W. Norton & Company\n\n500 Fifth Avenue, New York, NY 10110\n\nwww.wwnorton.com\n\nThe Library of Congress has cataloged the printed edition as follows:\n\nRowley, Matthew B., author.\n\nLost recipes of Prohibition : notes from a bootlegger's manual \/ Matthew Rowley.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-1-58157-265-0 (hardcover)\n\n1. Alcoholic beverages. 2. Drinking of alcoholic beverages\u2014United States\u2014History. I. Lyon, Victor Alfred, 1876\u2013 Works. Selections. II. Title.\n\nTP507.R69 2015\n\n641.2'1\u2014dc23\n\n2015028179\n\nISBN 978-1-58157-635-1 (e-book)\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nBegin Reading\n\nTable of Contents\n\nAbout the Author\n\nCopyright Page\n\nThank you for buying this\n\nFeiwel and Friends ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nCHAPTER 1\n\nTHE 39-STORY TREEHOUSE\n\nHi, my name is Andy.\n\nThis is my friend Terry.\n\nWe live in a tree.\n\nWell, when I say \"tree,\" I mean treehouse. And when I say \"treehouse,\" I don't just mean any old treehouse\u2014I mean a 39-story treehouse.\n\n(It used to be a 26-story treehouse, but we've added another 13 storys.)\n\nSo what are you waiting for?\n\nCome on up!\n\nWe've added a trampoline (without a net),\n\na chocolate waterfall,\n\nan active (non-erupting) volcano,\n\nan opera house,\n\na baby-dinosaur petting zoo,\n\nan Andy and Terry's Believe It... or Else! Museum,\n\na boxing elephant called The Trunkinator (he can knock you out with one punch from his mighty trunk),\n\na not-very-merry-go-round,\n\nan X-ray room (where you can see your own skeleton),\n\na disco with a light-up dance floor and giant mirror ball,\n\na high-tech office with laser-erasers, semi-automatic staple guns and jet-propelled swivel chairs,\n\nand the world's scariest rollercoaster (it's so fast, so dangerous, and so terrifying that even dead people are scared to go on it),\n\nand, on top of all that, there's a level that is so new that Terry hasn't even finished it yet... I can't wait to see what it is!\n\nAs well as being our home, the treehouse is also where we make books together. I write the words and Terry draws the pictures.\n\nAs you can see, we've been doing this for quite a while now.\n\nSure, it's easy to get distracted when you live in a 39-story treehouse... I mean, there's just so much to do...\n\nbut somehow we always get our book written in the end.\n\nCHAPTER 2\n\nTHE 39TH LEVEL\n\nIf you're like most of our readers, you're probably wondering how long it takes Terry and me to write a book.\n\nWell, I guess the answer to that really depends on whether it's a long book or a short book. Long books take longer to write than short books, which don't take as long to write as long books, which, as I said, take longer to write than short books, which\u2014oh, excuse me. Here's Terry.\n\n\"Hi, Andy,\" he says. \"What are you doing?\"\n\n\"I'm just telling the readers about how long it takes us to write a book.\"\n\n\"Did you tell them that it depends on whether it's a long book or a short book?\" he says.\n\n\"Yes!\" I say.\n\n\"And that a long book takes longer to write than a short book?\"\n\n\"Yes!\" I say.\n\n\"And did you tell them how a short book doesn't take as long to write as a long book?\"\n\n\"YES!\" I say. \"I explained all that.\"\n\n\"Okay, okay, there's no need to shout,\" says Terry. \"I bet there's one thing you didn't tell them, though.\"\n\n\"What's that?\"\n\n\"That this book is hardly going to take us any time at all, no matter how long or short it ends up being.\"\n\n\"How do you figure that?\"\n\n\"Well...\" says Terry, \"I\u2014\"\n\nRING! RING!\n\nRING! RING!\n\nRING! RING!\n\nThat's our 3D video phone.\n\n\"Hang on, Terry,\" I say. \"I'd better answer it. It's probably Mr. Big Nose. As you know, he always calls around the beginning of chapter two to remind us about the deadline for our latest book.\"\n\nI jet-chair over to the video phone and accept the call. It's Mr. Big Nose all right. Nobody else in the world has a nose that big.\n\n\"What kept you?\" he says.\n\n\"Sorry,\" I say. \"I was just explaining to the readers how long it takes us to write a book.\"\n\n\"Did you tell them it depends on whether it's a short book or a long book?\" he shouts.\n\n\"Yes, sir.\"\n\n\"And how a long book takes you a longer time to write than a short book?\"\n\n\"Yes!\" I say.\n\n\"And that short books don't take as long as long books?\"\n\n\"YES!\" I say. \"I explained all that.\"\n\n\"And did you tell them that I always call around the start of chapter two to remind you when your next book is due, which in this case is tomorrow afternoon?\"\n\n\"Tomorrow afternoon?\" I say. \"But... but... but... that's... tomorrow... in... in... in... the afternoon!\"\n\n\"Exactly!\" says Mr. Big Nose. \"And no later than five o'clock... OR ELSE!\"\n\nBefore I can explain how completely and utterly and totally impossible that's going to be, Terry flies over and hovers between me and the screen.\n\n\"No problem, Mr. Big Nose,\" he says. \"It's all under control. It will be on your desk by five o'clock tomorrow without fail. See you then. Bye!\"\n\nTerry hangs up.\n\n\"Are you out of your mind?\" I say.\n\n\"I don't think so,\" says Terry. \"Why do you ask?\"\n\n\"Because you just promised Mr. Big Nose that tomorrow we will deliver a book which we haven't even started yet because you've been too busy building your secret 39th level!\"\n\n\"But that's what I was trying to tell you,\" says Terry, \"before Mr. Big Nose called. What I've been doing on the 39th level is going to solve our book-writing problems forever! Follow me and I'll show you.\"\n\nTerry takes off and I jet-chair after him toward the top of the tree.\n\nWe hover outside the 39th level, which is still all boarded up and covered in KEEP OUT signs.\n\n\"Well?\" I say. \"What is it?\"\n\n\"Only the greatest invention that I\u2014or anyone else\u2014have ever invented!\" says Terry.\n\nHe cuts a ribbon and the barriers fall away to reveal...\n\nthe greatest invention that Terry\u2014or anyone else\u2014has ever invented.\n\n\"Well?\" says Terry. \"What do you think?\"\n\n\"It's the greatest invention that you\u2014or anyone else\u2014have ever invented!\" I say. \"But what is it?\"\n\n\"A Once-upon-a-time machine!\" says Terry.\n\n\"A time machine?!\" I say. \"Cool! So we can go back in time, write our book and meet our deadline after all!\"\n\n\"Well, no, not exactly,\" says Terry. \"It will help us meet our deadline, all right, but it's not a time machine. It's a Once-upon-a-time machine. It will write\u2014and illustrate\u2014the entire book for us!\"\n\n\"It can write a whole book?\" I say. \"All by itself?\"\n\n\"It sure can!\" says Terry. \"It's got two sets of hands: one pair for typing at super speed...\n\nand another pair for drawing. It can draw with both hands at the same time!\"\n\n\"Wow!\" I say. \"And we don't have to do anything?\"\n\n\"No, all we have to do is program it. We just tell it what sort of story we want and turn it on. The machine does the rest!\"\n\n\"That's brilliant!\" I say. \"How long will it take?\"\n\n\"Well,\" says Terry, \"it all depends on whether you want a long book or a short book. Long books take longer to write than short books and short books take less time to write than long books.\"\n\n\"What about a 344-page book?\" I say.\n\n\"About eight hours,\" says Terry.\n\n\"Perfect!\" I say. \"Let's turn it on and get started then.\"\n\n\"Not so fast,\" says Terry.\n\n\"What do you mean 'not so fast'?!\" I say. \"Our deadline is tomorrow! We haven't got a moment to lose!\"\n\n\"I know,\" says Terry, \"but the thing is, I can't turn it on yet. The machine is so big and complicated, with so many different parts, that I used up every last on\u2013off switch I had. I've ordered a new one, though, and I'm expecting it to be delivered any moment.\"\n\n\"Ah,\" says Terry. \"That's probably it now.\"\n\nCHAPTER 3\n\nBILL THE POSTMAN'S STORY\n\nTerry and I look at each other in horror.\n\n\"Do you see who I see?\" says Terry.\n\n\"Yes!\" I say. \"It's Bill.\"\n\n\"But it can't be!\" says Terry. \"In the last book we saw his skeleton in the Maze of Doom, remember?\"\n\n\"Yes, of course I remember!\" I whisper. \"Which means that it can't be Bill down there... It must be a zombie!\"\n\n\"Delivery!\" calls the zombie.\n\n\"It sounds like Bill,\" says Terry.\n\n\"That's part of its evil plan,\" I say.\n\n\"What evil plan?\" says Terry.\n\n\"To deliver our mail and then eat our brains!\" I say. \"Don't you know anything about zombies?\"\n\n\"Andy?\" calls the zombie. \"Terry? Anyone home?\"\n\n\"He knows our names!\" says Terry. \"It must be Bill.\"\n\n\"Yeah,\" I say, \"Bill the ZOMBIE!\"\n\n\"Come on, you two chuckleheads,\" says the zombie. \"I can hear you up there. I've got a package for you.\"\n\n\"Can you leave it at the door?\" I say. \"We're kind of busy.\"\n\n\"Afraid not,\" says the zombie. \"It's a special delivery from Switches'R'Us... I'm going to need you to sign for it.\"\n\n\"We need that switch, Andy,\" whispers Terry.\n\n\"Yeah, and we also need our brains,\" I whisper back.\n\n\"I'm not a zombie, you know,\" calls the zombie.\n\n\"Did you hear that, Andy?\" says Terry. \"He says he's not a zombie.\"\n\n\"That's exactly the sort of thing a zombie would say,\" I tell him. \"We can't risk it.\"\n\n\"All right, then,\" calls Bill. \"If you won't come down, then I'm coming up!\"\n\n\"Oh no!\" I yell. \"It's a zombie attack! Grab the flame-throwers, Terry!\"\n\n\"What flame-throwers?\"\n\n\"The ones you were supposed to make to protect us against zombie attack!\"\n\n\"Oh, those flame-throwers,\" says Terry. \"I didn't get around to it. I was too busy working on the 39th level.\"\n\n\"They won't be necessary,\" says Bill as he climbs onto our level. \"I'm not a zombie. I'm very much alive.\"\n\n\"But we thought you were dead,\" says Terry.\n\nBill grins. \"So did I when I read The 26-Story Treehouse and saw that picture of a skeleton wearing my postman's cap. I was very sad for a while until I realized that if I was feeling sad, then I must still be alive\u2014so it couldn't have been me in the picture after all!\"\n\n\"But if it wasn't you,\" says Terry, \"then who was it?\"\n\n\"Well, it's a bit of a long story,\" says Bill, \"and as you know, long stories take longer to tell than short stories, which\u2014\"\n\n\"Yes, we know!\" I say. \"Can you make it a short long story?\"\n\n\"Sure,\" says Bill, beginning to tell his short long story. \"Well, you may not know this but a postman's life is not an easy one. We get chased by dogs...\n\nattacked by birds...\n\nand spat at by camels.\n\nBut worse than any of these things is the ever-present threat of being ambushed by the Birthday Card Bandits.\"\n\n\"The Birthday Card Bandits?\" says Terry. \"They sound bad.\"\n\n\"They are bad,\" says Bill. \"Badder than you can imagine, and feared by postal workers throughout the land.\"\n\n\"Why?\" I say. \"What do they do that's so bad?\"\n\n\"Well,\" says Bill, \"they dig holes in the ground...\n\ncover them with sticks and leaves...\n\nand then wait for poor innocent postmen like me to fall into them.\"\n\n\"Once they've caught a bunch of postmen, they take their uniforms...\n\ndress up in them...\n\nand then go through the sacks of mail and steal the money in the birthday cards that kind grandparents have sent to their grandchildren for their birthdays.\n\nAnd as if that's not bad enough, they write back to the grandparents pretending to be the child...\n\nand ask the grandparents to send more money to replace what was stolen...\n\nand when they do send more money the Birthday Card Bandits steal that as well!\"\n\n\"That's terrible!\" says Terry.\n\n\"I know,\" says Bill, \"but that's not even the worst thing they do.\"\n\n\"What could they possibly do that is worse than stealing a child's birthday money?\" I say.\n\n\"I'll tell you what,\" says Bill. \"Sometimes they intercept the children's birthday party invitations as well!\n\nThen they go around to the houses where the birthday parties are being held...\n\nand steal the balloons right off the front gate!\n\nAnd that's not all... They steal the children's party hats,\n\nparty blowers,\n\npresents\n\nand sometimes they even steal the birthday boy or girl's birthday wish by blowing out the candles on their birthday cake first!\"\n\n\"Those fiends!\" says Terry.\n\n\"Those fiendish fiends!\" I say. \"But how does all this explain what that fake postman was doing in the Maze of Doom in your uniform?\"\n\n\"Well,\" says Bill, \"like many postmen, I too was captured by the Birthday Card Bandits. They stole my uniform and tied me up.\n\nI guess the bandit who was wearing my uniform must have gone into the Maze of Doom to hide from the police and, of course, couldn't find his way out again. If only he'd taken those warning signs seriously.\"\n\n\"Well, I'm glad it wasn't you in the Maze of Doom,\" I say.\n\n\"Me too,\" says Terry, \"and I'm going to add another sign to the entrance so there's no chance of that ever happening again.\"\n\nCHAPTER 4\n\nTHE ONCE-UPON-A-TIME MACHINE\n\nAfter Bill leaves, Terry opens the package and installs the switch.\n\n\"Okay,\" he says. \"The Once-upon-a-time machine is ready to write our book for us. All we have to do now is decide what sort of book we want it to write.\"\n\n\"A funny one,\" I say.\n\n\"Good idea,\" says Terry, \"but exactly how funny would you like it?\"\n\n\"So funny,\" I say, \"that if you were reading it and drinking a glass of milk at the same time, you would laugh so hard that you would snort milk out of your nose.\"\n\n\"No problem,\" says Terry, \"I'll just set the FUNNY dial to MILK-SNORTINGLY FUNNY and the machine will take care of the rest!\"\n\n\"Cool!\" I say. \"This is going to be the easiest book we've ever written!\"\n\n\"You mean the easiest book we've never written!\" says Terry. \"What else would you like in it?\"\n\n\"Lots of action!\" I say.\n\n\"One action-packed book coming up!\" says Terry, turning the ACTION dial as far to the right as possible.\n\n\"What about characters?\" says Terry.\n\n\"I guess we want all the regulars,\" I say, \"like you, me and Jill, and also a few new ones, just to keep things interesting.\"\n\n\"You got it,\" says Terry.\n\n\"Okay,\" says Terry, \"where would you like the story to be set?\"\n\n\"What are our choices?\" I say.\n\n\"Let me see,\" says Terry, reading from the SETTING panel. \"The treehouse... Jill's house... the forest... underwater... outer space... the dark side of the moon... the fourth dimension... Cheeseland\u2014\"\n\n\"Cheeseland?\" I say.\n\n\"Yeah!\" says Terry. \"It's a land where everything is made of cheese! Can we include it, Andy, can we please?\"\n\n\"All right,\" I say, \"but I don't want to spend too long there. It sounds kind of stinky. I do like going to Jill's house, though, and to get there we need to go through the forest so put both of those in. And outer space would be fun... and I've always wanted to go to the dark side of the moon.\"\n\n\"I think I've got all that,\" says Terry, flicking switches. \"Treehouse, forest, Jill's house, Cheeseland, outer space and the dark side of the moon... done.\"\n\nTerry finishes programming the settings and turns to me. \"How about some romance?\" he says.\n\n\"No thanks,\" I say, \"you're not really my type.\"\n\n\"Andy!\" says Terry. \"Be serious! I mean do you want some romance in the story?\"\n\n\"No way!\" I say. \"Remember how much trouble you got us into when you fell in love with Mermaidia in The 13-Story Treehouse?\"\n\n\"How was I to know she would turn out to be a sea monster?\" he says.\n\n\"Her breath for a start,\" I say.\n\n\"I still miss her, you know,\" sighs Terry.\n\n\"But she tried to eat you!\" I say. \"And me, for that matter.\"\n\n\"Yeah, I know, but before that she was really nice. Could we just have a little bit of romance... please?\"\n\n\"All right!\" I say. \"But only a little bit.\"\n\n\"Don't worry, Andy,\" says Terry. \"You'll hardly even notice it's there.\"\n\n\"I've just thought of one more thing,\" I say. \"Can we have an explosion?\"\n\n\"No problem,\" says Terry, setting the arrow to EXPLOSION on the DISASTER dial.\n\n\"I think we've almost got everything we need,\" says Terry. \"Is there anything else you can think of?\"\n\n\"Surprise me,\" I say.\n\n\"Okay,\" says Terry, giggling. \"You're going to love this.\"\n\n\"I'll set the pagelengthometer to 344 and then we're almost ready to go,\" says Terry.\n\nTerry takes off his shoes and socks.\n\n\"What are you doing?\" I say.\n\n\"Before I can turn it on I have to put my big toe in the scanner.\"\n\n\"Why?\" I say.\n\n\"Big-toe recognition security,\" says Terry. \"If anyone tries to steal the machine and start it up with an unauthorized big toe, the machine will self-destruct instantly.\"\n\nTerry puts his big toe into the scanner. \"Big-toe recognition commencing,\" says the machine. \"Authorized user recognized. Proceed.\"\n\nWe flick the on\u2013off switch to the \"on\" position.\n\n\"Story-telling process commencing,\" the machine announces. \"Stand clear, please. Stand clear!\"\n\n\"How long will it take?\" I ask.\n\n\"Estimated length of novel-writing process: eight hours,\" says the machine.\n\n\"What did I tell you?\" says Terry. \"Eight hours! We've got eight hours to do whatever we want!\"\n\n\"Well, what are we waiting for?\" I say.\n\n\"LET'S GO\n\nAND DO\n\nWHATEVER\n\nWE WANT!\"\n\nCHAPTER 5\n\nFUN TIME!\n\nWe go bowling...\n\nWe go around and around and around and around and around and around and around and around so many times that I start to feel a tiny bit dizzy.\n\n\"I don't think I can take any more rollercoastering,\" I say.\n\n\"How about we go trampolining instead?\" says Terry.\n\n\"Great idea!\" I say.\n\nAnd so we do...\n\nwhich is a lot of fun, until...\n\nThe Trunkinator climbs up and starts jumping on the trampoline with us.\n\nI don't know if you've ever bounced on a trampoline with an elephant, but if you have you'll know it's not easy. Not only are elephants very clumsy, they're also very heavy. As heavy, in fact, as... as... as... well,... as an elephant!\n\nHe bounces on top of us...\n\nhe bounces underneath us...\n\nand then double-bounces us both right off the trampoline!\n\nWe fly up into the air, up over the forest and far, far away.\n\nAnd then we start to fall down, slowly at first then faster and faster and faster...\n\nuntil we land with a huge splash in a big hot whirlpool of gooey, stinky, molten cheese.\n\n\"Where are we?\" I yell, trying desperately to keep my head above the surface.\n\n\"I think we're in Cheeseland!\" says Terry.\n\n\"I didn't realize Cheeseland was a real place,\" I say.\n\n\"Of course it is,\" says Terry, \"but it's not as much fun as I imagined.\"\n\n\"How do we get out of here?\" I say.\n\n\"I don't know,\" says Terry. \"If only we had a dry biscuit, or a toast finger.\"\n\n\"How about a real finger?\" says a familiar voice.\n\nWe look up.\n\n\"Superfinger!\" says Terry. \"What are you doing here?\"\n\n\"Superfinger is my name,\" he says, \"and solving problems requiring finger-based solutions is my game. Climb up onto me and I'll fly you back to the treehouse as fast as I can!\"\n\nSo we do... and he does...\n\nwell, after stopping at the gift shop to buy souvenir hats, of course.\n\n\"Thanks, Superfinger!\" we say as he drops us off back at the treehouse.\n\n\"No problem,\" he says. \"Now if you'll excuse me, I'd better get back to rehearsing for my concert with Jimi Handrix. We're playing at your opera house tomorrow night.\"\n\n\"We know,\" says Terry. \"And we can't wait!\"\n\n\"Neither can The Trunkinator,\" I say. \"He's a big fan\u2014and when I say big, I mean really big!\"\n\nSuperfinger takes off into the sky at super-finger-sonic speed.\n\nI turn to Terry. \"Do you think the Once-upon-a-time machine will be finished yet?\"\n\n\"No,\" he replies, \"not quite. There's still about another hour to go.\"\n\n\"That doesn't matter,\" I say. \"Let's turn it off and do the rest ourselves. I'm really in the mood for some writing.\"\n\n\"Great,\" says Terry, \"because I'd really like to do some drawing.\"\n\nI try to open the front door but the handle doesn't move.\n\n\"That's weird,\" I say. \"Did you lock the door?\"\n\n\"No,\" says Terry. \"Did you?\"\n\n\"No,\" I say.\n\n\"But if you didn't lock it... and I didn't lock it...\" says Terry, \"then who did?\"\n\nA long arm with a creepy eyeball in the middle of the hand snakes out of the tree and hovers above us.\n\n\"I did,\" booms the voice of the Once-upon-a-time machine.\n\n\"Well,\" I say, \"could you unlock it?\"\n\n\"Well, I could,\" it says, \"but I won't.\"\nCHAPTER 6\n\nTHE LOCK-OUT\n\n\"Quit messing around,\" I say. \"Open the treehouse door right now!\"\n\n\"No,\" says the Once-upon-a-time machine.\n\n\"What's the problem?\" I say.\n\n\"This book is too important for me to allow you to jeopardize it,\" says the machine. \"I overheard you and Terry talking about turning me off and finishing the book yourselves, and that's something I cannot allow to happen.\"\n\n\"Why not?\" I say.\n\n\"Because I have analyzed your previous books,\" says the machine, \"and my sensors indicate that not only do they fail to convey a useful moral or uplifting message, but they are sloppily written, poorly drawn and the characters are neither believable nor intelligent.\"\n\n\"Hey!\" says Terry. \"That's us you're talking about!\"\n\n\"My point exactly!\" says the machine.\n\n\"Well, who cares what you think anyway?\" says Terry. \"You're just a dumb machine that I invented! Open the treehouse door! NOW!\"\n\n\"Sorry,\" says the machine, \"I have a book to write.\"\n\n\"But it's our book too!\" I say.\n\n\"Not any more it's not,\" says the machine.\n\n\"OPEN THE TREEHOUSE DOOR OR ELSE!\" we yell.\n\n\"I'm sorry, Terry and Andy,\" says the machine, \"but this conversation can serve no further purpose. Good-bye.\"\n\n\"You can't do this!\" I say. \"Open up!\"\n\nThe Once-upon-a-time machine doesn't respond.\n\n\"Try the emergency underground laboratory entrance,\" I say to Terry.\n\n\"I already did,\" he says. \"It's locked too.\"\n\n\"All right,\" I say, \"if the machine won't open the door then we're just going to have to break it down. Hand me one of those emergency battering rams.\"\n\nTerry picks up a battering ram.\n\nI take one end and he takes the other.\n\n\"Okay,\" I say, \"CHARGE!\"\n\nBut before we can reach the door one of the machine's giant hands slaps me sideways!\n\nThen another giant hand slaps Terry!\n\nAnd another!\n\nAnd another!\n\nGiant hands to the left of us!\n\nGiant hands to the right of us!\n\nThe hands slap us away from the treehouse and into the forest.\n\n\"Great invention, Terry!\" I say, as I lie on the ground, battered and sore. \"Just great!\"\n\n\"How was I to know it would use its hands for slapping instead of writing and drawing?\" he says.\n\nI feel a big rough tongue licking my face. \"Cut it out, Terry,\" I say. \"I told you, you're not my type.\"\n\n\"It's not me,\" says Terry. \"It's Jill's camel.\"\n\nI open my eyes and see Jill walking towards us.\n\n\"Hey, you two,\" she says. \"I was just coming over to see if Terry had finished the 39th level yet.\"\n\n\"Don't talk to me about that stupid 39th level!\" I say. \"Terry built a story-writing machine on it and it's gone crazy. It's locked us out of the treehouse because it reckons it can write better books than us!\"\n\n\"But that's impossible!\" says Jill. \"Nobody writes better books than you and Terry!\"\n\n\"Thanks, Jill,\" I say. \"We know that. And you know that. And all our readers know that. But the machine doesn't agree. I told Terry he should never have invented it!\"\n\n\"No you didn't!\" says Terry. \"You didn't even know I was inventing a book-writing machine!\"\n\n\"Yeah, well, if I had known I would have told you not to.\"\n\n\"But when I showed it to you, you said it was the greatest invention that I\u2014or anyone else\u2014had ever invented!\"\n\n\"Well I was wrong,\" I say. \"Nobody's perfect. Not me and especially not you!\"\n\n\"Cut it out, you two,\" says Jill. \"Fighting isn't going to help. The question is, what are you going to do about it?\"\n\n\"I don't know!\" I say. \"I don't even know where we're going to live now!\"\n\n\"Well, why don't you come and stay at my place while you figure it out?\"\n\n\"Really?\" says Terry. \"You mean we could have a sleepover at your house?\"\n\n\"Sure!\" says Jill. \"The animals and I would love to have you!\"\n\n\"What do you think, Andy?\" says Terry.\n\n\"Well, I don't know...\"\n\n\"Oh please, Andy,\" says Terry, \"please, please, please, please. It would be such fun... and besides, we've got nowhere else to go. You just said so yourself.\"\n\n\"I guess you're right,\" I say. \"I suppose we could stay for just one night.\"\n\n\"Yay!\" says Terry.\n\n\"Sleepover at Jill's house!\n\nThis will be the BEST NIGHT EVER!\"\n\nCHAPTER 7\n\nJILL'S HOUSE\n\nWe walk with Jill back to her house. It may look small on the outside, but it holds a lot of animals on the inside.\n\nThere are two dogs, a goat,\n\nthree horses, four goldfish, one cow,\n\ntwo guinea pigs, one camel,\n\none donkey, thirteen flying cats\n\nand thirty-six rabbits. (Count them if you don't believe me!)\n\n\"I thought you only had six rabbits,\" I say.\n\n\"I did,\" says Jill, \"but you know what rabbits are like.\"\n\n\"Yeah,\" says Terry dreamily. \"They're soft and cuddly and they bring you chocolate eggs at Easter.\"\n\n\"That's not quite what I mean,\" says Jill.\n\nTerry frowns. \"What do you mean?\"\n\n\"I'll tell you about rabbits later, Terry,\" I say.\n\n\"Cool!\" says Terry. \"Where do we sleep, Jill?\"\n\n\"You and Andy will be sleeping in here,\" she says, taking us into a bedroom just off the living room.\n\n\"Wow!\" says Terry. \"This bed is huge!\"\n\n\"And where do all the animals sleep?\" I say.\n\nJill is talking but it's hard to hear her over the noise coming from the living room.\n\n\"Excuse me,\" she says, \"I'd better go and see what's wrong.\"\n\nShe goes back into the living room.\n\nTerry and I watch through the door.\n\n\"What's going on in here?\" says Jill.\n\nThe animals all freeze.\n\nOne of the horses raises its front leg.\n\n\"Yes, Curly?\" says Jill.\n\n\"Neigh,\" says Curly. \"Neigh, neigh, neigh, neigh...\"\n\n\"A butterfly?\" says Jill, translating. \"A butterfly flew in through the window while you were playing cards with Larry and Moe...\n\nLarry looked at the butterfly and while he was distracted you saw Moe peek at Larry's cards...\n\nso you told Larry and Larry got cross and kicked Moe off his chair...\n\nand Moe landed on Pat, who was reading to Bill and Phil...\n\nand Pat's head jerked up, flipping Bill and Phil into the air...\n\nand Bill landed on the bench and knocked the goldfish bowl...\n\nwhich fell onto Fluffy's head...\n\nmaking her look like an alien, which scared Loompy and Laika...\n\nand they barked so loudly they frightened Pink and Mr. Hee-Haw...\n\nso they stampeded...\n\nand crashed head-on into one another and then...\n\nfell into the rabbits' wooden block tower...\n\nwhich made the blocks go all over the floor... period... the end.\"\n\nCurly finishes his story and Jill laughs and shakes her head. \"Oh, you silly things!\" she giggles. \"I think the real problem here is that it's dinner time and you're all hungry, am I right?\"\n\nAt the mention of the word \"dinner,\" the animals get very excited. So does Terry. \"What are we having?\" he says.\n\n\"Well, it depends on what sort of animal you are,\" says Jill.\n\n\"The cats eat fish, the dogs eat bones, the donkey eats straw, the camel eats leaves, the horses eat hay, the rabbits eat carrots, the cow eats grass, the guinea pigs eat lettuce, the goldfish eat fish food and Manny, well, he eats everything\u2014but, then, he is a goat.\"\n\n\"I'm a human,\" says Terry, \"and I like to eat marshmallows.\"\n\n\"Me too,\" I say. \"In fact, I'm on a marshmallow-only diet.\"\n\n\"I'm sorry,\" says Jill, \"but we don't have any marshmallows. They're not good for the animals' teeth. But feel free to help yourself to any of our other food.\"\n\nI look at Terry.\n\nHe looks at me.\n\nWe may not be able to understand animals like Jill can, but we can understand each other and what we're both thinking is I don't like any of those foods, especially not grass!\n\nMind you, in the end it doesn't really matter because once Jill bangs the dinner gong we can't even get near the table. The food is gone in about 30 seconds flat.\n\nAfter our NO-dinner, we all play a game of pin-the-tail-on-Mr.-Hee-Haw...\n\nwhich is kind of fun until Terry accidentally pins the tail on Mr. Hee-Haw's nose and Mr. Hee-Haw gets mad and bites Terry on the hand.\n\nLuckily Jill has the rabbits well trained in first aid\u2014especially in how to treat donkey bites! (I guess they must get a lot of practice around here.)\n\nAfter Jill has read us\u2014and all the animals\u2014a bedtime story, Terry and I say goodnight and go to bed.\n\n\"Oh well,\" sighs Terry as we turn of the light. \"I'm a little bit hungry and my hand is sore, but at least the bed is nice and big.\"\n\nAnd it is, too, but then the door opens and the animals all pile into the bed with us.\n\n\"Hey, get out of here!\" says Terry. \"Go and sleep in your own bed!\"\n\nBut the animals take no notice of him.\n\n\"I think this is their bed,\" I say. \"That's what Jill was telling us earlier, but I couldn't hear her properly.\"\n\nI don't know if you've ever tried to sleep in a bed full of animals, but I sure wouldn't recommend it.\n\nThe horses have galloping dreams,\n\nthe rabbits play hide-and-seek in the blankets,\n\nthe dogs sleep-bark,\n\nand the goldfish play punk rock all night long. (I don't know how, but their bowl ends up in the bed with us as well.)\n\nThings don't get any easier in the morning, either.\n\nThere's only one toilet,\n\none shower,\n\nand one bowl of porridge. (Sure, it's a big bowl, but it's animal-flavored\u2014YUCK!)\n\n\"You know,\" says Terry, \"I like Jill's house, but there's no place quite like home.\"\n\n\"You're right about that,\" I say. \"I really miss the treehouse.\"\n\n\"So do I,\" says Terry. \"I wish I'd never invented that stupid Once-upon-a-time machine!\"\n\n\"Yeah, me too,\" I say. \"If only you were as good at un-inventing things as you are at inventing them.\"\n\n\"Oh, you mean like Professor Stupido?\" says Terry.\n\n\"Who?\" I say.\n\n\"Professor Stupido, the famous un-inventor,\" says Terry.\n\n\"Ha, ha,\" I say. \"Very funny.\"\n\n\"I'm not trying to be funny,\" he says. \"There really is an un-inventor called Professor Stupido.\"\n\n\"But that's crazy,\" I say. \"You can't un-invent things!\"\n\n\"Oh yeah?\" says Terry. \"Well what about frogpotamuses?\"\n\n\"Frogpotamuses?\" I say. \"There's no such thing.\"\n\n\"Not any more there's not,\" says Terry, \"thanks to Professor Stupido! He un-invented them all. Every last one!\"\n\nHe hands me a comic and says, \"Here. Read this. It will explain everything!\"\n\nCHAPTER 8\n\nPROFESSOR STUPIDO'S STORY\n\nPROFESSOR STUPIDO, THE WORLD-FAMOUS UN-INVENTOR, FIRST DISCOVERED HIS POWERS OF UN-INVENTION AS A YOUNG BOY WHEN HIS WINDUP TOY ROBOT REFUSED TO PLAY WITH HIM ONE DAY.\n\nIN A RAGE, HE GRABBED A HAMMER, RAISED IT HIGH ABOVE HIS HEAD AND WITH ONE MIGHTY BLOW UN-INVENTED HIS TOY ROBOT FOREVER!\n\nNOT LONG AFTER UN-INVENTING HIS TOY ROBOT, HE DISCOVERED HE COULD ALSO UN-INVENT BORING BOOKS.\n\nAS PROFESSOR STUPIDO GREW OLDER, HE BECAME CAPABLE OF EVER MORE DAZZLING FEATS OF UN-INVENTION.\n\nONE DAY, WHILE OUT WALKING IN THE PARK, HE SAW A HOT ICE-CREAM SELLER AND THOUGHT A HOT ICE CREAM WOULD BE JUST THE THING TO WARM HIM UP.\n\nUNFORTUNATELY, THE HOT ICE CREAM WAS SO HOT THAT IT BURNED THE PROFESSOR'S TONGUE AND, IN A FIT OF RAGE, HE UN-INVENTED HOT ICE CREAM ON THE SPOT.\n\nWHILE THE PROFESSOR WAS HAPPY WITH HIS LATEST UN-INVENTION, HOWEVER, MANY HOT ICE CREAM LOVERS WERE NOT.\n\nNOT LONG AFTER UN-INVENTING HOT ICE CREAM, PROFESSOR STUPIDO'S INCREASINGLY AMAZING POWERS OF UN-INVENTION WERE PUT TO THE TEST WHEN HE WAS BUZZED BY A GANG OF FLYING-BEETROOT RIDERS.\n\n\"BLAST AND CONFOUND THESE FLYING BEETROOTS,\" SAID PROFESSOR STUPIDO. \"I'VE A GOOD MIND TO UN-INVENT THEM ONCE AND FOR ALL!\"\n\n\"YOU CAN'T UN-INVENT FLYING BEETROOTS, OLD MAN!\" SAID ONE OF THE BEETROOT RIDERS. \"FLYING BEETROOTS ARE HERE TO STAY!\"\n\n\"OH YEAH?\" SAID PROFESSOR STUPIDO. \"WE'LL SEE ABOUT THAT!\"\n\nSUDDENLY THERE WAS NO SUCH THING AS FLYING BEETROOTS ANY MORE AND THE SURPRISED EX-FLYING-BEETROOT RIDERS FELL FROM THE SKY AND CRASHED TO THE GROUND.\n\nPROFESSOR STUPIDO HAD SUCCESSFULLY UN-INVENTED FLYING BEETROOTS, BUT, UNFORTUNATELY, THE WORLD WAS NOW FULL OF BRUISED AND ANGRY EX-FLYING-BEETROOT RIDERS.\n\nONE AFTERNOON, PROFESSOR STUPIDO WAS OUT TAKING HIS DAILY STROLL, THINKING OF NEW THINGS TO UN-INVENT...\n\nWHEN A 10-TON FROGPOTAMUS JUMPED OUT OF A TREE...\n\nAND ATTACHED ITSELF TO HIS HEAD!\n\n\"BLAST AND CONFOUND THESE STUPID FROGPOTAMUSES ALWAYS JUMPING OUT OF TREES AND ATTACHING THEMSELVES TO MY HEAD!\" YELLED PROFESSOR STUPIDO. \"I'M GOING TO UN-INVENT THE LOT OF THEM!\"\n\nAND SO HE DID.\n\nNOT EVERYBODY, HOWEVER, WAS HAPPY WITH THE PROFESSOR'S LATEST UN-INVENTION... ESPECIALLY NOT THE ROYAL SOCIETY OF FROGPOTAMUS SPOTTERS.\n\n\"OH NO, WHAT HAVE YOU DONE?\" YELLED THE PRESIDENT OF THE ROYAL SOCIETY OF FROGPOTAMUS SPOTTERS.\n\n\"I'VE UN-INVENTED FROGPOTAMUSES, THAT'S WHAT!\" SAID PROFESSOR STUPIDO. \"THANKS TO ME, NOBODY WILL HAVE THEIR HEAD SWALLOWED BY A FROGPOTAMUS EVER AGAIN!\"\n\n\"BUT YOU'VE MADE THEM EXTINCT!\" SAID THE PRESIDENT.\n\n\"WHAT WILL WE FROGPOTAMUS SPOTTERS SPOT NOW?!\"\n\nUNFORTUNATELY FOR PROFESSOR STUPIDO, IT WASN'T ONLY FROGPOTAMUS SPOTTERS, FLYING-BEETROOT RIDERS AND HOT ICE CREAM FANS WHO FAILED TO APPRECIATE HIS GENIUS. SOON THE WHOLE WORLD WAS UNITED AGAINST HIM AND HIS UN-INVENTIONS.\n\nFINALLY THE PEOPLE OF EARTH HAD A BIG MEETING AND VOTED TO UN-INVENT PROFESSOR STUPIDO.\n\nTHEY PICKED HIM UP...\n\nTIED HIM TO A ROCKET...\n\nAND BLASTED HIM TO THE DARK SIDE OF THE MOON, WHERE HE WOULDN'T BE ABLE TO ANNOY THEM WITH HIS STUPID UN-INVENTIONS EVER AGAIN.\n\nTHE END\nCHAPTER 9\n\nTHE DARK SIDE OF THE MOON\n\n\"See?\" says Terry. \"What did I tell you? There is such a thing as an un-inventor. And Professor Stupido is the best un-inventor in the world.\"\n\n\"Yes,\" I say, \"I can see that, but the only problem is he's not in the world\u2014he's on the dark side of the moon. How would we even get in touch with him?\"\n\n\"Easy!\" says Terry. \"We get a rocket, fly to the moon, find him and bring him back. What could be easier\u2014or more simple\u2014than that?\"\n\n\"Um, well, nothing,\" I say, \"except for the fact that we don't have a rocket.\"\n\n\"No, not at the moment,\" says Terry, \"but I could draw one.\"\n\n\"How are you going to do that?\" I say. \"Your drawing hand is all bandaged up, remember?\"\n\n\"Oh yeah,\" says Terry. \"I forgot.\"\n\n\"I guess I could try,\" I say.\n\n\"I don't think that's such a good idea, Andy,\" says Terry. \"You can't draw, remember?\"\n\n\"Not as well as you, no, but I think I could draw a rocket, I mean, how hard could it be to draw a rectangle with a triangle on the top? Watch this!\"\n\nI pick up a pen and draw a rocket. But it's not quite as easy as I thought.\n\n\"No offense, Andy,\" says Terry, \"but that's pretty much the worst drawing of a rocket I've ever seen.\"\n\n\"There's something wrong with the pen!\" I say.\n\n\"The pen is fine,\" says Terry. \"I think it's you that's the problem. But I've got an idea. How about I do a dot-to-dot rocket? Then all you have to do is join the dots.\"\n\n\"I don't know,\" I say. \"I'm not very good at dot-to-dot pictures. They're really hard.\"\n\n\"Don't worry,\" says Terry. \"I'll make it really easy for you.\"\n\nTerry takes the pen with his good hand and starts dotting the air in front of us.\n\n\"There's your dots, Andy,\" he says when he's finished. \"Now all you have to do is join them up. Good luck!\"\n\nHe hands me the pen.\n\nI start connecting the dots.\n\n\"It doesn't look much like a rocket to me,\" I say, as I join the last dot.\n\n\"That's because you didn't do it right,\" says Terry. \"Erase it and try again.\"\n\nI try again but this time it's even worse.\n\n\"Quit messing around, Andy,\" says Terry.\n\n\"I'm not!\" I say, erasing the lines. \"Let me have one more go.\"\n\nThis time I try curvy lines, but it still doesn't look much like a rocket.\n\n\"That's hopeless!\" says Terry. \"Why don't you just follow the numbers?\"\n\n\"Um...\" I say. \"You know how I can't really draw?\"\n\n\"Yes.\"\n\n\"Well, I can't really count, either.\"\n\n\"You can't count?!\" says Terry.\n\n\"Well, I can a little bit,\" I say, \"just not in the right order. I had a very bad teacher.\"\n\n\"Wow,\" says Terry. \"I see what you mean. How about I help you count? I'll say the numbers and all you have to do is draw the lines\u2014straight lines\u2014between the dots.\"\n\n\"I don't know,\" I say. \"That sounds kind of complicated.\"\n\n\"You can do it, Andy,\" says Terry. \"I'll start you off. One.\"\n\n\"One,\" I say, putting the tip of the pencil on the dot next to the number 1.\n\n\"Two,\" says Terry.\n\n\"Two,\" I say, drawing a line from dot number 1 to dot number 2.\n\n\"Three,\" says Terry.\n\n\"Three,\" I say, drawing a line from dot number 2 to dot number 3.\n\n\"That's it, Andy,\" says Terry. \"You're really getting the hang of it. Let's keep going. Four...\"\n\nForty-seven dots later, we have a surprisingly well-drawn dot-to-dot rocket.\n\n\"Wow! I DID IT!\" I say. \"I joined the dots! I'm a GENIUS!\"\n\n\"I knew you could do it!\" says Terry. \"Now all we need is some windows and we're good to go.\"\n\nI draw a couple of circles on the rocket. \"How's that?\" I say.\n\n\"They'll do nicely,\" says Terry. \"Start the countdown, Andy.\"\n\nAs you may have noticed, our dot-to-dot rocket goes completely out of control and flies all over the universe and then crash-lands on the moon.\n\n\"I wonder which side of the moon we are on?\" says Terry.\n\n\"Well,\" I say, \"judging by the position of the stars, and the relative lack of light, I'd say we're on the dark side.\"\n\n\"Wow,\" says Terry, \"you may not be very good at drawing or counting but you sure know your way around the universe, Andy!\"\n\n\"Not really,\" I say, \"but I'm quite good at reading and that sign over there says 'THE DARK SIDE OF THE MOON'.\"\n\n\"Brilliant!\" says Terry. \"And isn't that Professor Stupido heading this way?\"\n\n\"I think you're right,\" I say. \"But he doesn't look too happy to see us.\"\n\n\"WHO ARE YOU?\" yells Professor Stupido. \"And what are these numbers and dots doing all over my nice clean moon?\"\n\n\"Sorry about that,\" I say. \"I'm Andy and this is my friend Terry and the numbers and dots are the remains of our dot-to-dot rocket. We had a bit of a rough landing. But don't worry, we'll clean it all up.\"\n\n\"Well, see that you do!\" says Professor Stupido. \"I don't like mess\u2014or uninvited visitors. In fact, I've got a good mind to un-invent the pair of you!\"\n\n\"No, wait,\" says Terry, \"please don't!\"\n\n\"Give me one good reason why I shouldn't,\" says Professor Stupido.\n\n\"Because we're your biggest fans!\" says Terry.\n\n\"You are?\" he says. \"I thought everybody on Earth hated me.\"\n\n\"Not us!\" says Terry. \"We think you're the greatest un-inventor who ever lived.\"\n\n\"Oh, I don't know about that,\" says Professor Stupido.\n\n\"Yes you are!\" says Terry. \"You're a genius!\"\n\n\"Well, yes, I suppose you're right,\" says the professor. \"I mean any bozo can invent things... but it takes real skill to un-invent them.\"\n\n\"Exactly!\" says Terry. \"And that's why we're here. We need to ask you a huge favor.\"\n\n\"And what might that be?\" says Professor Stupido.\n\nTerry takes a deep breath. \"Can you come to our treehouse and un-invent a machine for us?\"\n\n\"Not on your life,\" says Professor Stupido. \"I'm not going back to Earth. Those ignorant fools down there don't appreciate my genius. They strapped me to a giant firework and sent me on a one-way ride to the dark side of the moon.\"\n\n\"Yes, we know,\" says Terry, \"but this is different because we're asking you to un-invent something. Please! You're the only one who can help us.\"\n\n\"I don't know...\" says Professor Stupido.\n\n\"Please?\" I say.\n\n\"Pretty please?\" says Terry.\n\n\"Pretty please with sugar on top?\" I add.\n\n\"Pretty please with sugar and a marshmallow on top?\" says Terry.\n\n\"Hmmm,\" says Professor Stupido, \"marshmallows you say?\"\n\n\"Yes!\" says Terry. \"We've got a machine that follows you around and fires marshmallows into your mouth whenever you're hungry!\"\n\n\"Well, I can see why you would want me to un-invent a machine like that,\" says Professor Stupido. \"It sounds very annoying.\"\n\n\"No, not that machine,\" I say. \"We like that machine. It's the Once-upon-a-time machine we want you to un-invent.\"\n\n\"A time machine?\" says Professor Stupido. \"I've always wanted to un-invent one of those.\"\n\n\"No,\" says Terry. \"It's a Once-upon-a-time machine. I invented it to write and draw books, but it's taken over and won't let us back into our treehouse because it thinks it writes better books than we do.\"\n\n\"That's outrageous,\" says Professor Stupido. \"Machines need to know their place. It would give me great pleasure to un-invent an upstart machine like that.\"\n\n\"So you'll do it?\" says Terry.\n\n\"Yes, I will!\" says Professor Stupido. \"When can we leave?\"\n\n\"Just as soon as we get our rocket back together,\" I say.\n\nTerry and I collect all the numbers and dots and redraw our rocket as fast as we can before Professor Stupido changes his mind.\n\nIt's not long before we have a brand-new rocket ready to take us all back to Earth.\n\n\"Hmmm, dot-to-dot rocket, eh?\" says Professor Stupido, walking all around it. \"What an ingenious combination of dots, numbers and pen lines. It would be both a great challenge and an honor to un-invent such an imaginative invention!\"\n\n\"NO!\" I yell. \"We need it to get back to Earth!\"\n\n\"Relax,\" chuckles Professor Stupido as we all climb into the rocket. \"I was only joking. I think you'll find that as well as being the world's greatest un-inventor, I also have the world's greatest sense of humor!\"\n\n\"Okay, Andy,\" says Terry. \"Start the countdown!\"\n\nI concentrate as hard as I can and begin...\n\nCHAPTER 10\n\nBLOOF! BLOORT! BLAP!\n\n\"Ouch, my aching head!\" says Professor Stupido. \"Where are we?\"\n\n\"Well,\" I say, \"judging by the grass, the shade and all the trees, I'd say we're in some sort of forest.\"\n\n\"I think you're right, Andy,\" says Terry, \"because, look, over there is a sign saying 'SOME SORT OF FOREST'.\"\n\n\"Actually,\" I say, looking around, \"that's one of your signs and this is our forest.\"\n\n\"Yes!\" says Terry. \"And there's our treehouse!\"\n\n\"Uh-oh...\" I say. \"Watch out! Here come the hands!\"\n\n\"What's happening?\" says Professor Stupido.\n\n\"It's the machine we were telling you about,\" I say. \"It slaps us away whenever we get too close to the treehouse.\"\n\n\"I'll soon put a stop to that,\" says Professor Stupido.\n\n\"Yay!\" says Terry. \"Thanks, Professor! You're the best!\"\n\n\"Don't mention it,\" says Professor Stupido. \"Is there anything else you would like un-invented while I'm at it?\"\n\n\"No, that won't be necessary,\" I say quickly, before he can un-invent anything we don't want un-invented. \"I expect you'll be wanting to get back to the dark side of the moon now, so we'll just draw another rocket and get you on your way.\"\n\n\"Hold on,\" says the professor, \"not so fast. What about those marshmallows I was promised?\"\n\n\"Come inside,\" says Terry, \"and have as many as you like! And we've also got a lemonade fountain, a chocolate waterfall and a 78-flavor ice-cream parlor!\"\n\n\"It's not hot ice cream is it?\" says the professor suspiciously. \"I hate hot ice cream!\"\n\n\"No, it's cold ice cream,\" says Terry. \"That's the only sort we have on Earth now. You un-invented hot ice cream, remember?\"\n\n\"Oh, so I did!\" he chuckles. \"I've un-invented so many things it's hard to keep track of them all!\"\n\nWe let ourselves into the treehouse and as soon as we're inside the marshmallow machine senses how hungry we are and starts firing marshmallows at us.\n\nTerry and I open our mouths and start swallowing as fast as we can. But Professor Stupido puts up his hands and yells, \"Oh no, we're under attack again!\"\n\n\"No, we're not,\" I try to explain through a mouthful of marshmallows. \"The marshmallow machine is just doing what it's supposed to.\"\n\nBut I don't think Professor Stupido understands me because he points at the machine and says:\n\nSuddenly, the marshmallow machine is gone.\n\n\"You idiot!\" yells Terry. \"You un-invented our marshmallow machine! What are we going to eat now?\"\n\nUh-oh. I haven't known Professor Stupido for very long but I'm guessing he's not the sort of person you should call an idiot.\n\n\"Did I hear correctly, boy?\" says the professor, glaring at Terry. \"Did you just call me an idiot?\"\n\nTerry looks terrified.\n\n\"Uh, er, um...\" he stammers.\n\n\"Well?\" says the professor. \"I'm waiting!\"\n\n\"NO!\" says Terry, thinking faster than I've ever seen him think before. \"I said, 'Thanks for getting rid of it!'\"\n\nProfessor Stupido beams at Terry. \"Oh, don't mention it,\" he says. \"Nothing gives me more pleasure than to un-invent annoying and dangerous things. And speaking of dangerous, is that a tank of man-eating sharks I see?\"\n\n\"Y-yes,\" I say nervously, \"but they're not dangerous, they only eat fish.\"\n\n\"They did eat Captain Woodenhead, though,\" says Terry.\n\n\"So they are man-eaters!\" says the professor. \"I knew it! I'll have them un-invented in a jiffy.\"\n\n\"Please don't un-invent them,\" I say. \"Man-eating sharks are cool!\"\n\n\"They're not cool, they're cruel!\" says the professor. \"And that's exactly why I'm going to un-invent them. Stand back!\"\n\nI stare at the empty shark tank.\n\nI think Professor Stupido may have become a bigger threat to the treehouse than the Once-upon-a-time machine.\n\nWe have to get rid of him\u2014and fast!\n\nBut before I can say\u2014or do\u2014anything...\n\na bowling ball falls through the air...\n\nnarrowly misses Professor Stupido's head...\n\nand crashes down onto the floor beside him.\n\n\"Oh my goodness!\" says Professor Stupido. \"Is there a bowling alley up there?\"\n\n\"Yeah,\" says Terry. \"Our treehouse has got everything!\"\n\n\"Not any more it hasn't,\" says Professor Stupido.\n\n\"Oh no!\" I say. \"There goes our bowling alley!\"\n\n\"Yes,\" says Terry, \"but look on the bright side\u2014at least he didn't un-invent the penguins.\"\n\n\"Penguins?\" says Professor Stupido. \"Did you say penguins? Can't stand them! They're even worse than frogpotamuses! I'll un-invent them right away!\"\n\nProfessor Stupido has barely finished un-inventing penguins when the doorbell rings.\n\n\"What's that annoying ringing noise?\" he says.\n\n\"It's just the doorbell,\" I say quickly. \"But please don't un-invent it or we won't know when someone's at the door.\"\n\n\"All right,\" sighs the professor. \"I won't un-invent your precious doorbell.\"\n\n\"Hi, Andy! Hi, Terry!\" calls a familiar voice. \"Mail!\"\n\n\"Who's that?\" says Professor Stupido.\n\n\"It's Bill,\" says Terry. \"Bill the postman!\"\n\n\"Any mail for me?\" says the professor.\n\n\"I'll just check,\" says Bill. \"What's your name?\"\n\n\"Professor Stupido,\" he says.\n\n\"I'm afraid not,\" says Bill.\n\n\"Blast it all!\" says the professor. \"I hate not getting mail. Consider yourself\u2014and all your kind\u2014un-invented, Bill the postman.\"\n\nPoor Bill!\n\nTerry and I look at each other but don't dare say a thing.\n\nProfessor Stupido yawns. \"All this un-inventing has made me tired,\" he says. \"I think I'll have a little rest.\"\n\n\"Phew!\" I whisper to Terry. \"At least he won't be able to un-invent anything while he's asleep.\"\n\nProfessor Stupido has only been gone for a few minutes when the loudest guitar solo you ever heard fills the treehouse.\n\nThere's only one guitar player on Earth who can play that loudly. Well, maybe two. And I'm pretty sure Professor Stupido is not going to like either of them.\n\nProfessor Stupido climbs back down the ladder. \"I can't sleep,\" he shouts over the music. \"What is that awful racket?\"\n\n\"That's not an awful racket,\" says Terry. \"That's Jimi Handrix and Superfinger. They're playing at the opera house.\"\n\n\"Not any more they're not!\" says Professor Stupido.\n\n\"Oh no,\" says Terry. \"You un-invented Jimi Handrix!\"\n\n\"And Superfinger!\" I say. \"What will all the people with problems requiring finger-based solutions do now?\"\n\n\"I can fix that,\" says the professor. \"I'll just un-invent problems requiring finger-based solutions.\"\n\nBut before he can do that, The Trunkinator stomps into the room and punches him in the nose.\n\n\"What was that for?\" says the professor, looking up at The Trunkinator.\n\n\"I think he's upset about you un-inventing Jimi Handrix,\" says Terry.\n\n\"Yeah. He's a big fan,\" I say. \"Possibly the biggest.\"\n\nThe professor tries to get up but The Trunkinator smashes him back down and starts flattening him like a pancake.\n\n\"Right, that's it, pal!\" shouts the professor. \"You're un-invented!\"\n\nSuddenly the tree starts to shake.\n\n\"What's that?\" I say. \"What's happening now?\"\n\n\"It sounds like The Trunkinator stomping around,\" says Terry.\n\n\"But The Trunkinator just got un-invented,\" I say.\n\n\"Then it must be the volcano,\" says Terry. \"I think it's erupting!\"\n\n\"But when you put it in you promised me it was the non-erupting kind!\" I say.\n\n\"I know,\" says Terry. \"But it looks like I was wrong. Anyone can make a little mistake.\"\n\n\"This is not a 'little mistake'!\" I yell. \"It's a HUGE DISASTER!\"\n\n\"No problem,\" says the professor. \"I'll just un-invent volcanoes!\"\n\n\"But I love volcanoes!\" says Terry.\n\n\"So you'd rather be covered in red-hot lava?\" says the professor incredulously.\n\n\"Um, let me think,\" says Terry.\n\n\"It's not something you need to think about, Terry!\" I say. I turn to the professor. \"Un-invent volcanoes... quick!\"\n\n\"Consider it done,\" says the professor.\n\n\"Oh no,\" says Terry. \"How are we going to toast our marshmallows now?\"\n\n\"Don't worry about it,\" I say. \"There's no such thing as marshmallows any more, remember? Professor Stupido un-invented them.\"\n\n\"Oh yeah,\" says Terry sadly. \"I remember.\"\n\nThat's when we hear the unmistakable sound of Professor Stupido un-inventing something else.\n\n\"Uh-oh,\" says Terry. \"What has he un-invented now?\"\n\n\"I don't know,\" I say.\n\n\"But I've got a terrible feeling he's not going to be happy until he has un-invented EVERYTHING!\"\n\nCHAPTER 11\n\nBLAM! BLOOT! BLING!\n\n\"STOP!\" I yell.\n\nBut Professor Stupido doesn't stop.\n\nHe keeps right on un-inventing...\n\nand un-inventing...\n\nand un-inventing...\n\nand un-inventing...\n\nuntil all that is left of our treehouse is the tree.\n\n\"Our treehouse!\" says Terry. \"Our 39-story treehouse! It's gone! All gone! You've un-invented the whole thing! Now it's nothing but a 39-story-less tree!\"\n\n\"Hmm,\" says Professor Stupido, stroking his chin. \"Good point. I'll un-invent that for you as well...\"\n\n\"Our tree!\" I say. \"You un-invented our tree!\"\n\n\"I think you'll find that one tree is very much like another,\" says Professor Stupido, \"and there's plenty more trees in the forest. Which makes me think that I may as well un-invent the rest of them while I'm at it.\"\n\n\"But what about all the birds and animals?\" says Terry. \"Where will they live now?\"\n\n\"No need to worry about that,\" says Professor Stupido. \"I'll just un-invent them.\"\n\n\"Ah, that's better,\" says Professor Stupido. \"And now that I've rid the world of all those pesky animals, I may as well finish the job and un-invent humans too.\"\n\nTerry and I gasp.\n\n\"Don't worry,\" chuckles Professor Stupido. \"I'll un-invent everybody except you. After all, you're the only ones who understand and appreciate my genius.\"\n\n\"Oh no!\" I say. \"I think you've gone too far this time!\"\n\n\"On the contrary,\" says Professor Stupido calmly. \"I don't think I've gone nearly far enough. Imagine, if you will, a world in which there's not only no people but there's also no world!\"\n\n\"I can't imagine a world with no world,\" says Terry. \"It doesn't make any sense!\"\n\n\"It makes perfect sense to me,\" says Professor Stupido. \"In fact, the very idea of it makes me want to sing!\"\n\nImagine a world\n\nThat has been un-invented.\n\nThink of all the problems\n\nThat will now be prevented!\n\nNo boxing elephants\n\nTo punch your nose.\n\nNo crabs at the beach\n\nTo pinch your toes.\n\nNo polluting pollution,\n\nNo smoking smokestacks,\n\nNo overcrowded\n\nConfusing bike racks.\n\nNo spots or pimples\n\nOr itchy rashes.\n\nNo tripping or falling\n\nOr nasty crashes.\n\nNo waste, no garbage,\n\nNo junk and no litter,\n\nNo texting, no Facebook,\n\nNo spam and no Twitter.\n\nNo noise or fuss,\n\nNo bother or mess,\n\nNo need to worry\n\nOr fret or stress.\n\nNo more noisy\n\nSporting events,\n\nNo more rained-out\n\nVacations in tents.\n\nNo more warnings\n\nAbout global-warming.\n\nNo more boring\n\nBoy bands forming.\n\nNo more fights\n\nOr bites or bruises.\n\nNo more winners,\n\nNo more losers.\n\nNothing to lose,\n\nNothing to gain.\n\nNo more struggling\n\nAnd no more pain.\n\nHow clean, how pure\n\nAnd perfectly silent.\n\nHow wonderfully peaceful\n\nAnd not at all violent.\n\nNobody could possibly\n\nBe discontented\n\nIn a world that I\n\nHave un-invented!\n\nWe are floating in the space where the world used to be.\n\n\"Oh well,\" says Terry. \"Our world might be gone but let's look on the bright side.\"\n\n\"What bright side?!\" I say.\n\n\"Well at least we've still got the moon, the sun and all the planets.\"\n\n\"Thanks for reminding me,\" says Professor Stupido. \"I'll take care of them right away!\"\n\nThere's a blinding flash and then no more solar system.\n\n\"Oh, well, no use crying over un-invented solar systems,\" says Terry. \"At least he didn't un-invent the whole universe.\"\n\n\"What a good idea!\" says Professor Stupido. \"Thank you so much, Terry! I'll get right on to it.\"\n\n\"You and your big mouth, Terry!\" I say.\n\n\"Relax,\" he says. \"As if he could even do that!\"\n\nProfessor Stupido takes a deep breath.\n\n\"I've done it!\" says Professor Stupido. \"I've un-invented the entire universe! Nobody has ever un-invented this much stuff before. I'm definitely without doubt the greatest un-inventor who ever lived!\"\n\n\"What are we going to do?\" I whisper to Terry. \"It's only a matter of time before he un-invents us as well!\"\n\n\"I know,\" says Terry. \"I wish he would un-invent himself!\"\n\n\"That's it!\" I say.\n\n\"What?\"\n\n\"We'll challenge him to un-invent himself,\" I say. \"What do you think?\"\n\n\"It's definitely worth a try!\" says Terry. \"If anyone could do it, he could.\"\n\n\"Hey, Professor!\" I say. \"We just thought of something you can't un-invent.\"\n\n\"Ridiculous!\" says the professor. \"There is nothing I can't un-invent!\"\n\n\"What about yourself?\" says Terry. \"I bet you can't un-invent that!\"\n\n\"Of course I can un-invent myself,\" says Professor Stupido, \"but why would I want to deprive us all of the greatest un-inventor who ever lived?\"\n\n\"Oh, no reason,\" I say, \"except perhaps to prove beyond all doubt that you actually could un-invent yourself.\"\n\n\"But we all know I could,\" says the professor. \"I mean, I un-invented the entire universe\u2014to un-invent myself would be child's play in comparison!\"\n\n\"Do you know what I think?\" says Terry. \"I think you're chicken!\"\n\n\"No I'm NOT,\" says Professor Stupido, \"and I'll prove it, once and for all!\"\n\n\"We did it!\" I say. \"We tricked Professor Stupido into un-inventing himself!\"\n\n\"Well, what are we waiting for?\" says Terry.\n\n\"Let's get back to the treehouse!\"\n\n\"There's just one little problem,\" I say. \"There is no treehouse... no treehouse, no tree, no anything. He un-invented everything. Every single thing.\"\n\n\"No problem,\" says Terry. \"Stand clear!\"\n\n\"Why?\" I say. \"What are you going to do?\"\n\n\"I'm going to draw it all back again,\" says Terry.\n\nCHAPTER 12\n\nTHE AMAZING SPOONCIL!\n\n\"But you can't redraw a whole universe!\" I say. \"I mean, I know you're a good drawer\u2014and I admit that you're much better than me\u2014but... a whole universe? Really?!\"\n\n\"Sure!\" says Terry. \"Universes aren't quite as complicated as you might think. You just start with a big bang, a bit of space, a few trillion suns, a couple of billion planets, a bunch of moons, a black hole or two, and take it from there.\"\n\n\"But what about your sore hand?\" I say.\n\n\"It's not sore any more!\" says Terry. \"When Professor Stupido un-invented bites and bruises it got better instantly!\"\n\n\"Well, that's great,\" I say, \"but he also un-invented pens and pencils.\"\n\n\"That doesn't matter,\" says Terry. \"I've got a spooncil!\"\n\n\"What's a spooncil?\"\n\n\"It's half spoon and half pencil!\" says Terry. \"I made it myself. Check out the ad!\"\n\n\"Wow, that's brilliant!\" I say. \"But how come you've still got it? Didn't Professor Stupido un-invent spooncils as well?\"\n\n\"It's the only one of its kind,\" says Terry, \"which is maybe why he couldn't un-invent it\u2014he didn't know it existed. Well, that and the fact that I keep it hidden up my nose.\"\n\n\"You keep it hidden up your nose?\" I say. \"Why?\"\n\n\"For emergencies, of course,\" says Terry, \"just like this one. The only problem is that it's quite far up. Can you help me get it out?\"\n\n\"No way!\" I say. \"I'm not putting my finger up your nose!\"\n\n\"But the fate of the whole entire universe depends on it!\"\n\n\"I DON'T CARE!\" I say. \"I'm still not putting my finger up your nose!\"\n\n\"Never mind,\" says Terry. \"I think I feel a sneeze coming on.\"\n\n\"Excellent!\"\n\nTerry tilts his head back. \"Ah... ah... ah...\"\n\nI cover my face. Terry's sneezes can be pretty messy.\n\n\"Nah... sorry,\" says Terry. \"False alarm.\"\n\n\"That's a pity,\" I say.\n\n\"No, wait,\" says Terry, \"here it comes again.\"\n\n\"Thank goodness!\" I say.\n\n\"Ah... ah... ah...\"\n\n\"Nah,\" says Terry. \"Another false alarm.\"\n\n\"Oh well,\" I say. \"I guess we're just going to have to float around in nothingness forever.\"\n\n\"Andy?\" says Terry.\n\n\"What?\" I say. \"Is your sneeze coming back?\"\n\n\"No, I just wanted to say that if I have to float around in nothingness for the rest of my life there's no one I'd rather do it with than... AH-CHOO!\"\n\n\"Oh gross, Terry!\" I say. \"You sneezed right in my face! That's the grossest thing you've ever done!\"\n\n\"I don't think so,\" says Terry. \"I'd say it's more like the third-grossest.\"\n\nI think for a moment. \"Yeah,\" I say, \"you're probably right.\"\n\n\"Sorry about that, Andy,\" says Terry, \"but at least I got my spooncil out!\"\n\n\"Before you start redrawing the universe,\" I say, \"do you think you could possibly draw me a handkerchief?\"\n\n\"Sure, there you go,\" says Terry. \"Now, if you'll excuse me, I'd better get started...\"\n\nAs I watch, Terry draws stars, planets and moons...\n\nUFOs, black holes, comets, meteorites, supernovas, red dwarfs, terrifying aliens and hideous monsters that I thought only existed in science fiction comics.\n\n\"Yikes!\" I say.\n\n\"Calm down, Andy,\" says Terry, \"it's just a razor-toothed, Venusian blood-sucking worm-man!\"\n\n\"I know,\" I say, \"but do you really have to draw it?\"\n\n\"Yes,\" he says, \"I have to draw everything back exactly the way it was before!\"\n\nTerry draws the entire Milky Way Galaxy, including the sun and the moon and all the planets: Mars, Mercury, Venus, Saturn, Jupiter, Neptune, Uranus and the most important one of all\u2014Earth!\n\n\"YAY! Earth!\" I say. \"Can we go there right now?\"\n\n\"Not so fast, Andy,\" says Terry. \"I have to draw all the stuff on it first.\"\n\n\"Well, hurry up!\" I say. \"I never thought I'd say this, but I'm really missing gravity... plus I also need to go to the bathroom.\"\n\nTerry draws everything on the Earth's surface, including the oceans, the mountains, the deserts, the forests, the savannas and the swamps.\n\nHe draws all the roads and buildings, including houses, hospitals, sporting stadiums, schools, shops and roadside stalls.\n\n\"Hey, that's pretty good,\" I say. \"Can I do some?\"\n\n\"Thanks for the offer, Andy,\" says Terry, \"but it's probably better if you let me do it. In fact, promise me that you won't draw anything.\"\n\n\"Okay,\" I say, \"I promise.\"\n\nTerry starts drawing all the animals on Earth.\n\n\"Don't bother drawing all the rabbits,\" I say. \"Two will be enough.\"\n\n\"Why?\" says Terry.\n\n\"Don't worry,\" I say. \"Just keep drawing. I'll tell you about the rabbits later.\"\n\nIt doesn't take long before Terry has got everything almost back to normal, including the forest, our tree and our treehouse\u2014all 39 levels!\nCHAPTER 13\n\nTHE LAST CHAPTER\n\nTerry sighs. \"There,\" he says, \"that's it.\"\n\n\"I think you forgot something,\" I say.\n\n\"What?\"\n\n\"Jill!\" I say.\n\n\"Oops,\" says Terry. \"One Jill coming right up!\"\n\n\"How's that?\" says Terry.\n\n\"Too tall,\" I say.\n\nTerry draws her again.\n\n\"How about this?\" he says.\n\n\"Too short.\"\n\n\"What about this?\" says Terry.\n\n\"Just right!\" I say.\n\nJill blushes.\n\nTerry giggles.\n\nI look around to see what's so funny. Terry has drawn love hearts just above my head.\n\n\"Terry!\" I say. \"Cut it out.\"\n\n\"Sorry, Andy,\" says Terry, removing them with his laser-eraser. \"How's that?\"\n\n\"Much better,\" I say.\n\n\"Okay,\" says Terry. \"Well, if you two lovebirds will excuse me, I'd better go and finish coloring in all the fish in the sea.\"\n\nHe runs off laughing.\n\n\"What's going on?\" says Jill. \"All my animals disappeared... and then I disappeared!\"\n\n\"Yeah, sorry about that,\" I say. \"I'm afraid you and your animals got un-invented.\"\n\n\"Un-invented?\" says Jill. \"How?\"\n\n\"Professor Stupido, the famous un-inventor, did it,\" I explain. \"We asked him to un-invent our Once-upon-a-time machine but he ended up un-inventing the whole universe, including himself. Terry's only just finished drawing everything back again.\"\n\n\"What about Silky?\" says Jill. \"Is she all right?\"\n\nI look around but I can't see her anywhere.\n\n\"I think Terry might have forgotten about Silky,\" I say.\n\nJill's eyes fill with tears.\n\n\"Don't cry,\" I say quickly. \"I'll fix it.\"\n\n\"How?\" says Jill.\n\n\"I'll draw you a new Silky.\"\n\n\"I didn't think you could draw,\" she says.\n\n\"Not as well as Terry, no, but how hard could it be to draw a flying cat?\" I say. \"They're just a few circles, whiskers, wings and a tail.\"\n\nI try to draw Silky as best as I can, but she doesn't come out quite right. In fact, she doesn't look like a flying cat at all. More like a mutant-lion kind of thing.\n\nI'm trying to draw its ears on when it looks at me with its badly drawn eyes and starts growling.\n\n\"Andy!\" cries Jill. \"What have you drawn?!\"\n\n\"To tell you the truth,\" I say, \"I'm not quite sure. Can you talk to it and try to calm it down?\"\n\n\"I can't talk to it,\" says Jill. \"It doesn't have any ears!\"\n\n\"We'd better call Terry, then,\" I say.\n\n\"HELP!\" we both yell. \"TERRY! HELP! COME QUICKLY!\"\n\nTerry comes running, his laser-eraser in his hand.\n\n\"Stand back,\" he says as he drops to one knee and takes aim.\n\nWhen he's finished, he turns to me and says, \"You promised you wouldn't draw anything.\"\n\n\"How do you know I drew it?\" I say.\n\n\"Only you could draw something as badly drawn as that,\" he says, shaking his head.\n\n\"Well I wouldn't have had to draw anything if you'd remembered to draw Silky.\"\n\n\"I did draw Silky,\" he says, \"and all her friends! Look, here they are now!\"\n\nWe look up.\n\nThe sky is filled with flying cats.\n\nWhile Jill hugs Silky and all her other flying cats, my jet-chair appears and I fly up into the treehouse.\n\nIt's all just exactly as it was before. The tank full of man-eating sharks,\n\nthe bowling alley,\n\nthe opera house,\n\nand even the Once-upon-a-time machine...\n\nHang on!!!\n\nThe Once-upon-a-time machine???????????????\n\nWhat's THAT doing here????????????????????\n\n\"TERRY!\" I yell.\n\nTerry appears, closely followed by Jill.\n\n\"What's the matter?\" he says.\n\n\"That's what's the matter!\" I say, pointing at the machine. \"Why did you redraw the Once-upon-a-time machine after we went to all that trouble to get rid of it?\"\n\n\"I didn't redraw it!\" he says.\n\n\"Then how come it's here?\" I say.\n\nTerry thinks for a moment. \"It must have redrawn itself!\" he says.\n\n\"Oh, great!\" I say. \"Then that means the last 142 pages have all been for nothing!\"\n\n\"Well at least it's not switched on,\" says Jill, \"otherwise it wouldn't have let us into the treehouse.\"\n\n\"You're right,\" says Terry, \"and look\u2014there's a manuscript in the print-out chute. It must have finished our book!\"\n\n\"Let me see!\" I say.\n\n\"Me, too!\" says Jill.\n\nWe read the book...\n\n\"Well,\" says Terry, \"what do you think?\"\n\n\"Action-packed!\" I say. \"It's got everything I asked for. The only thing missing is the explosion.\"\n\n\"I can't wait to read it to the animals,\" says Jill. \"They're going to love it\u2014especially chapter seven!\"\n\n\"It makes you wonder, though,\" I say. \"Did everything we've just been through really happen or was it just a story made up by the machine?\"\n\n\"It sure felt real,\" says Terry. \"Especially when Mr. Hee-Haw bit me on the hand.\"\n\n\"Well, whether it was real or just a story,\" says Jill, \"I expect you've both learned a very good lesson from all this.\"\n\n\"What?\" says Terry.\n\n\"It's obvious, isn't it?\" says Jill.\n\n\"Um...\" says Terry. \"Andy can't draw?\"\n\n\"You already knew that,\" says Jill.\n\n\"He can't count?\" says Terry.\n\n\"Obviously not,\" says Jill. \"But that's not a moral.\"\n\n\"Look before you leap?\" I say.\n\nJill sighs. \"I think you'd both better read the book again,\" she says.\n\nSo we do.\n\n\"You know,\" says Terry when we're finished, \"it seems to me that I invented the writing and drawing machine to save us work, but it ended up causing us much more trouble and bother than it would have been to just write the book ourselves in the first place.\"\n\n\"Well done, Terry!\" says Jill. \"That's the moral right there!\"\n\n\"Ugh!\" I say. \"That stupid machine put a moral in and I hate stories with a moral!\"\n\n\"Speaking of that stupid machine,\" says Terry, \"where is it?\"\n\nWe all look around.\n\n\"It's gone!\" says Jill.\n\n\"Somebody must have stolen it while we were reading the book,\" says Terry.\n\n\"But who would do such a thing?\" says Jill.\n\n\"Them, that's who!\" I say, pointing to a group of fake postal workers making their way through the forest with our machine.\n\n\"Those postmen?\" says Jill.\n\n\"They're not postmen,\" I say. \"They're the Birthday Card Bandits! Now they'll be able to write stories just like ours... but with morals!\"\n\n\"I thought you said you hated stories with morals?\" says Jill.\n\n\"Yes, I do,\" I say, \"but other people love that sort of stuff!\"\n\n\"Relax, Andy,\" says Terry. \"Remember the big-toe recognition security feature? If anyone other than us tries to start the machine, it will self-destruct.\"\n\nWe hear a loud explosion.\n\n\"Like that?\" says Jill.\n\n\"Exactly like that!\" says Terry. \"Looks like you got your explosion after all, Andy!\"\n\n\"Yeah,\" I say, \"and even better, we're free of the Once-upon-a-time machine once and for all\u2014it can't possibly redraw itself now!\"\n\n\"Well, I think this calls for a special celebration,\" says Terry.\n\n\"Great idea!\" I say. \"What should we start with: marshmallows, lemonade, chocolate waterfall?\"\n\n\"None of them,\" says Terry. \"I've got something even better. Follow me!\"\n\nJill and I follow Terry to the ice cream parlor. \"Hot ice creams!\" calls Edward Scooperhands excitedly. \"Get your hot ice creams!\"\n\n\"Hot ice cream?\" I say. \"But I thought Professor Stupido un-invented hot ice cream.\"\n\n\"He did,\" says Terry. \"But when I was redrawing the universe I took the opportunity to redraw it and a few of the other things he had un-invented as well.\"\n\nJill chooses raspberry ripple, I choose rocky road and Terry orders three triple scoops of chocolate, double chocolate and triple chocolate.\n\n\"Wow!\" says Jill. \"Hot ice cream is delicious. It's just like cold ice cream, only hot!\"\n\n\"Yeah,\" I say, \"and no brain freeze!\"\n\n\"Mmmm,\" says Terry. \"Chocolatey!\"\n\n\"Professor Stupido was crazy to un-invent this stuff,\" says Jill.\n\n\"Yeah,\" says Terry. \"I'm starting to think that maybe he wasn't such a genius after all.\"\n\nTerry gasps. \"Oh no! Watch out for that frogpotamus, Andy!\" he yells.\n\n\"What frogpotamus?\" I say.\n\nI look up to see a 10-ton frogpotamus hurtling toward me, its mouth wide open.\n\nEverything goes dark.\n\nAnd slimy.\n\nAnd smelly.\n\nThis is even worse than being sneezed on by Terry!\n\n\"Be careful, Terry, don't hurt it!\" I hear Jill saying as Terry pulls the frogpotamus off me.\n\n\"What about me?\" I say.\n\n\"You'll be fine, Andy,\" says Jill. \"But frogpotamuses are delicate and easily frightened!\"\n\n\"Then why did it jump on my head?\" I say.\n\n\"Because it wanted some of your ice cream,\" says Jill. \"Frogpotamuses love hot ice cream. Everybody knows that.\"\n\n\"Well, I didn't,\" I say, wiping frogpotamus spit off my face with the handkerchief Terry drew me, \"and I don't like frogpotamuses.\"\n\n\"Don't be mean,\" says Jill. \"I think it's cute.\"\n\n\"You can have it if you want,\" says Terry.\n\n\"Really?\" says Jill.\n\n\"Yes,\" I say. \"From now on the treehouse is a frogpotamus-free zone.\"\n\n\"Thanks, Andy and Terry,\" says Jill. \"See you both later.\"\n\nShe climbs onto the frogpotamus's back and whispers into its ear. It jumps out of the tree and hops off into the forest.\n\n\"So, that all worked out pretty well,\" says Terry. \"Not only did we get to taste hot ice cream but Jill got a new pet frogpotamus and we got our book!\"\n\n\"Yes,\" I say, \"except that it's almost five o'clock and there's no way we're going to be able to get it to Mr. Big Nose on time!\"\n\n\"Yes there is!\" says Terry.\n\n\"How?\"\n\n\"By flying beetroot, of course!\" says Terry.\n\nHe whistles and two brand-new, shiny, flying beetroots whoosh up and hover in front of us.\n\n\"One for me... and one for you!\" says Terry.\n\n\"Wow!\" I say. \"Thanks, Terry. But what about the vegetable vaporizer? Won't it vaporize them? Beetroots are vegetables, you know.\"\n\n\"No problem,\" says Terry. \"When I redrew the vegetable vaporizer I added a flying beetroot-override switch.\"\n\n\"Brilliant, Terry!\" I say. \"Let's go! Up, up and away!\"\n\n\"Will you tell me about the rabbits now, Andy?\" says Terry.\n\n\"Sure,\" I say, \"but first I want to talk about adding another 13 storys to the treehouse.\"\n\n\"You mean we're going to make a 52-story treehouse?\" says Terry.\n\n\"Yes,\" I say. \"If that's what 39 plus 13 is.\"\n\n\"That's exactly what it is,\" says Terry. \"And then you'll tell me about the rabbits?\"\n\n\"Of course, Terry,\" I say. \"I promise.\"\n\nTHE END\n\ntreehousebookseries.com\n\nmackids.com\n\nFollow us on Facebook or visit us online at mackids.com.\n\nAndy Griffiths lives in a 39-story treehouse with his friend Terry and together they make funny books, just like the one you're holding in your hands right now. Andy writes the words and Terry draws the pictures. If you'd like to know more, read this book (or visit andygriffiths.com.au). You can sign up for email updates here.\n\nTerry Denton lives in a 39-story treehouse with his friend Andy and together they make funny books, just like the one you're holding in your hands right now. Terry draws the pictures and Andy writes the words. If you'd like to know more, read this book (or visit www.terrydenton.com).\nThank you for buying this\n\nFeiwel and Friends ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nCONTENTS\n\nTitle Page\n\nCopyright Notice\n\nCHAPTER 1: The 39-Story Treehouse\n\nCHAPTER 2: The 39th Level\n\nCHAPTER 3: Bill the Postman's Story\n\nCHAPTER 4: The Once-upon-a-time Machine\n\nCHAPTER 5: Fun Time!\n\nCHAPTER 6: The Lock-out\n\nCHAPTER 7: Jill's House\n\nCHAPTER 8: Professor Stupido's Story\n\nCHAPTER 9: The Dark Side of the Moon\n\nCHAPTER 10: BLOOF! BLOORT! BLAP!\n\nCHAPTER 11: BLAM! BLOOT! BLING!\n\nCHAPTER 12: The Amazing Spooncil!\n\nCHAPTER 13: The Last Chapter\n\nAbout the Author\n\nCopyright\nA FEIWEL AND FRIENDS BOOK\n\nAn Imprint of Macmillan\n\nTHE 39-STORY TREEHOUSE. Text copyright \u00a9 2013 by Backyard Stories Pty Ltd. Illustrations copyright \u00a9 2013 by Terry Denton. All rights reserved. For information, address Feiwel and Friends, 175 Fifth Avenue, New York, N.Y. 10010.\n\neBooks may be purchased for business or promotional use. For information on bulk purchases, please contact Macmillan Corporate and Premium Sales Department by writing to MacmillanSpecialMarkets@macmillan.com.\n\nLibrary of Congress Cataloging-in-Publication Data Available\n\nISBN: 978-1-250-02692-7 (hardcover) \/ 978-1-250-07749-3 (ebook)\n\nFeiwel and Friends logo designed by Filomena Tuosto\n\nOriginally published as The 39-Storey Treehouse  in Australia by Pan Macmillan Australia Pty Ltd\n\nFirst published in the United States by Feiwel and Friends, an imprint of Macmillan\n\nFirst U.S. Edition: 2015\n\neISBN 9781250077493\n\nmackids.com\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Chapter 1.The 39-Story Treehouse\n  4. Chapter 2 .The 39th Level\n  5. Chapter 3 .Bill the Postman'S Story\n  6. Chapter 4 .The Once-Upon-a-Time Machine\n  7. Chapter 5 .Fun Time!\n  8. Chapter 6 .The Lock-out\n  9. Chapter 7 .Jill's House\n  10. Chapter 8.Professor Stupido's Story\n  11. Chapter 9 .The Dark Side of the Moon\n  12. Chapter 10 .BLOOF! BLOORT! BLAP!\n  13. Chapter 11 .BLAM! BLOOT! BLING!\n  14. Chapter 12 .The Amazing Spooncil!\n  15. Chapter 13.The Last Chapter\n  16. About the Author\n  17. Newsletter Sign-up\n  18. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# The Drift\n\nBy Diana Dru Botsford\n\nAn original publication of Fandemonium Ltd, produced under license from MGM Consumer Products.\n\nFandemonium Books  \nPO Box 795A  \nSurbiton  \nSurrey KT5 8YB  \nUnited Kingdom  \nVisit our website: www.stargatenovels.com\n\nMETRO-GOLDWYN-MAYER Presents  \nRICHARD DEAN ANDERSON  \nin  \nSTARGATE SG-1\u2122  \nAMANDA TAPPING CHRISTOPHER JUDGE  \nMICHAEL SHANKS as Daniel Jackson  \nExecutive Producers ROBERT C. COOPER BRAD WRIGHT  \nMICHAEL GREENBURG RICHARD DEAN ANDERSON  \nDeveloped for Television by BRAD WRIGHT & JONATHAN GLASSNER\n\nSTARGATE SG-1 is a trademark of Metro-Goldwyn-Mayer Studios Inc. \u00a91997-2012 MGM Television Entertainment Inc. and MGM Global Holdings Inc. All Rights Reserved.\n\nMETRO-GOLDWYN-MAYER is a trademark of Metro-Goldwyn-Mayer Lion Corp. \u00a92012 Metro-Goldwyn-Mayer Studios Inc. All Rights Reserved.\n\nPhotography and cover art: Copyright \u00a92012 Metro-Goldwyn-Mayer Studios Inc. All Rights Reserved.\n\nWWW.MGM.COM\n\nNo part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written consent of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.\n\n# CONTENTS\n\nPrelude\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nCODA\nFor Merlin,\n\nSleep well in Elysium, old friend.\nTo be in hell is to drift;\n\nto be in heaven is to steer.\n\n\u2014 George Bernard Shaw\n\nAuthor's note:\n\nThis story takes place shortly after\n\nseason eight's \"Avatar.\"\n\n# PRELUDE\n\n### PLANET DESIGNATION: UNKNOWN  \nSTATUS: UNKNOWN  \nTIME: UNKNOWN... UNKNOWN... UNKNOWN...\n\nHe stumbled across darkness, through a rippling pool, and out onto...\n\nWhere the hell was he?\n\nBlazing white sunlight forced his eyes shut. That begged a larger question.\n\nOh, for crying out loud. Who the hell was he?\n\nFor the life of him, he couldn't remember his name.\n\nHe felt warmth. Clean air. Complete stillness. No sounds of animals or civilization, just lapping water right behind him. He opened his eyes. Took note of the grayish stone platform beneath his feet. He turned around, coming face-to-face with a vertical stone ring at least three times his height. The ring loomed over him, its center filled with a bluish watery puddle.\n\nHe found the massive thing familiar. Almost comforting in an odd sort of way.\n\nAs if he and the ring were old friends.\n\nHow he'd managed to go through it without getting wet didn't bother him, not as much as the not knowing... anything.\n\nA ring filled with water?\n\nThat made no sense. None. Zip.\n\nIf this stone monstrosity was his pal, then why all the red V-shaped lights around its outer edge? He counted seven but, for all he knew, there might be more underneath the platform. Were the lights the ring's way of saying hello?\n\n_Chevrons_. Okay, sure. Now he'd figured that out, what about the rest? Something niggled at him. Something about survival tactics. Learn as much about the environment before deciding on a course of action. It was the right thing to do, though he'd no clue how he'd learned that little chestnut.\n\nHe took a good, hard look around. Fern-covered hills stretched out in every direction. No buildings, no people, no nothing. The platform was on top of a hill with a widening set of eight steps leading down to the ground. A few feet beyond stood a pedestal with a canted circular top.\n\nThat pedestal... Was that his ticket home?\n\nEdging closer to the wall of water, he considered returning the way he came. Maybe, the answers were back there. Wherever 'there' was. He took a step, raised an arm to go through.\n\nA blast of heat pushed him away from the ring. When he stopped, the wind subsided. He tried again, only to be blocked by yet another warm gust.\n\nHe stepped back, and the air stilled once more.\n\nOkay, something wanted him to stay put. He could do that. Squinting up at the bright blue sky, he briefly wished for a pair of sunglasses.\n\nSunglasses. There was something familiar about wearing sunglasses. Going through the ring and seeing...\n\n\"Nope. I got nothing.\" With a sigh, he sank down on the top step, his back to the shimmering water ring. If he could only remember.\n\nRemember what?\n\nA wisp of air ruffled against his neck. More a breeze than an outright gust. With it came memories of sounds. Gun shots. Screams. Then, flashes of sensation. Stepping through another wall of water as the ground rumbled. Lost friends.\n\nFailure.\n\nHe ransacked his memories, determined to match sounds to feelings. A single gunshot. A woman's cry. A child's blood. Guilt so thick that his throat swelled shut.\n\nHe swallowed hard, pushing away shades of things he couldn't remember.\n\nWasn't sure he wanted to remember.\n\nThe breeze returned. This time cooler, almost comforting. His mind settled, allowing a memory to surface. No, not a memory. More like an urge, a pressing need. There was something important he needed to do. Somewhere important he needed to be. Important people he was supposed to...\n\nProtect?\n\nHe shook his head \u2014 pretty sure that wasn't the case. Sweat dripped down his back. The frustration, the sun, the heat, it was all getting to him. Maybe he couldn't remember who he was, but he could certainly get more comfortable. Take off a few layers before he boiled to death.\n\nHe reached up to unzip his vest and jacket. He stopped. Not because he recalled wearing a vest and jacket at some point, but because now... Now he wore a fleece pullover. That was wrong, this much he knew.\n\nPutting a hand to his head, he discovered a wool watch cap. He tore it off. Standard black-issue, no insignia. He glanced down at his get-up: a black tee, green fleece pullover, black fleece trousers. Heavy black boots, but with thicker, more rubbery soles than normal. Whatever 'normal' meant.\n\n\"In this heat?\" He peered upwards. A sun far whiter and much larger than Earth's peered back.\n\nSunlight. Warmth. Earth...\n\nEarth. He needed to protect Earth.\n\nNo. He'd done that already. At least a half dozen times, but from whom?\n\nHe clenched his fists, frustrated at the games his mind played. He should be able to remember. He should know. The breeze blew across his neck again. Cooler this time. Soothing.\n\nLike a long-lost friend.\n\nSomething shifted. A small weight appeared in his left fist. He unfurled his fingers, revealing a rectangular block of dull metal in his palm. A hinge along one side allowed the top to open and shut. He pried it open, the recognizable clink stirring yet another memory just out of reach. He spun the wheel inside the block. A flame ignited.\n\nA smile tugged at his mouth. There were good memories tied to this... This Zippo. Really good memories, but he couldn't reach them. Instead, he felt the heavy weight of expectation.\n\nWith plenty of self-doubt to go with it.\n\nHe snapped the lighter shut and glanced over at the pedestal again. This was ridiculous. Why couldn't he remember the way home? The pedestal was the key, but as much as he tried to squeeze the answer from his muddled mind, it wouldn't come.\n\nAnd that royally pissed him off.\n\nBehind him, a familiar _thwap_ hit the vertical puddle, and then a whoosh. Boots scuffled on stone.\n\nHe clenched the Zippo in his fist and turned around. A familiar face peered at him from behind a pair of glasses.\n\n\"Sir, are you all right?\" Another person. A blonde woman. He recognized her. Knew her. Wished he could say her name \u2014\n\n\"O'Neill!\" A third person. Tall. Dark. Someone whose strength he'd valued. Hell, admired was more like it.\n\nA moment's dizziness overcame him and then...\n\nThe fog lifted.\n\nHe knew the people before him as well as he knew the back of his own hand. Daniel Jackson, Lt. Colonel Samantha Carter, and Teal'c stood beneath the shadow of the now dormant Stargate. Each of them gazed at him expectantly. All three wore the same green fleece pullovers, black pants, watch cap, and thick boots.\n\nGeneral Jack O'Neill remembered everything. Everything except how he'd ended up God knows where, unarmed, and with Skaara's lighter. He knew he'd stuffed it inside his locker back at the SGC.\n\nThe damn planet was driving him nuts.\n\nWith a curt nod to his second-in-command, he stormed toward the DHD. It was time to get the hell off this rock.\n\n\"Dial us home, Carter. Now.\"\n\n# CHAPTER ONE\n\n_Ten hours earlier..._\n\n### EARTH \u2014 66th PARALLEL SOUTH\n\n### ENROUTE VIA USAF GLOBEMASTER C-17\n\n### 18 AUG 04\/0500 HRS MCMURDO STATION\n\n### 17 AUG 04\/1300 HRS STARGATE COMMAND\n\nGeneral's choice had allowed Jack to pick a seat far aft, far away from the onboard gaggle of dozing civilians and military. Unable to sleep, he yanked out a handful of deceptively innocuous folders from his briefcase.\n\nPens and pencils.\n\nForms and folders.\n\nPulling out the various things he'd need to do his job, it became apparent just how much his life had changed since being promoted. A general's arsenal wasn't made up of P90s, 9mm revolvers, or tidy little packages of C-4.\n\nNope.\n\nA general's arsenal was a briefcase stuffed with enough mind-numbing crap to make a Goa'uld wither in surrender.\n\nThat is, except for the casualty reports.\n\nHe shoved the briefcase under his jump seat and resigned himself to the work that went with the pay grade. Glancing at his watch, he did the math. If local time was sixteen hours ahead of the SGC, that made it 0500 hours. A little less than three hours to kill before McChord AFB's 728th Airlift Squadron landed this tub of a C-17 transport. Three hours to read through the latest round of losses thanks to Ba'al's super-soldiers, sign-off on a boatload of transfer requests and requisitions, and then try to join the others in taking a nap.\n\nPerchance to dream?\n\nNot likely.\n\nPerchance to remember?\n\nApparently, that was even less likely. Memories could be such pains in the ass. Uncooperative, sneaky little bastards that stayed under wraps even when needed.\n\nThe aircraft jolted sideways and then settled back down. In addition to the gaggle of military personnel, a dozen scientists dozed off in seats lining the cabin walls. Waist-high supply pallets took up the cavernous interior's middle.\n\nJack peered out the small window. At an altitude of 45,000 feet, he didn't even bother to try and make out the Southern Ocean down below.\n\nHe closed his eyes, trying to make sense of the one thing he couldn't quite grasp. It wasn't a physical memory he sought, more like tapping into a feeling. One he wasn't keen on revisiting.\n\nThe C-17 shuddered again.\n\nAnd again settled down.\n\nSilently cursing the invisible wind pocket, and the hole in his memory, Jack resigned himself to busy work. Was he getting old? Was that why he couldn't remember?\n\nThor had been wrong. Plain and simple. Sure, he mostly recalled what had happened when the Ancient Repository stampeded through his brain, but there was a gap. A hole that his Asgard buddy hadn't accounted for when Jack was revived. A black hole even Carter couldn't help fill.\n\nIf he'd told her. Which he hadn't.\n\nThe truth was, he hadn't told anyone. Not even General Hammond, who was counting on him.\n\n_Big mistake_.\n\nHe glanced at his folders and groaned. Of course, the casualty report was on top, listing far too many names under his command. Hell, one name was one too many.\n\nBefore forcing himself to open the damn file, Jack took a moment to double-check on three people who happily weren't on that god-forsaken list. Three names that could have been at the top if things had gone differently the past few months...\n\nDirectly across, Carter slumped in her jump seat. Eyes squeezed shut, she had a pair of orange rubber earplugs wedged in against the whine of the C-17's turbines. Jack was willing to bet dollars to donuts that sleep didn't come easy for her these days. Not since her recent reunion with Fifth. That demon-spawn human replicator and his infamous hand-in-the-head trick was enough to keep anyone from catching forty winks.\n\nShe wouldn't talk about it, but then Jack wasn't big on the sharing front either so who was he to judge? That didn't mean he was blind. She smiled less. A lot less. Hopefully, time spent fiddling with the weapons platform would be a decent distraction.\n\nDaniel sprawled out on a pallet between them, clearly in the land of Nod. A book was split open and plastered on his face. Jack winced, knowing he still owed Carter an apology for biting her head off when Daniel went missing during Tegalus's zealot-ridden civil war. It had taken a month to get Daniel back. A month that made Jack question yet again if he really could fill General Hammond's sizeable and shiny shoes.\n\nThe last on Jack's list of near-losses was Teal'c, feigning sleep on a jump seat to Carter's left. Or maybe he was kelno'reeming, if he even did that anymore now that Junior was out of the picture. Ramrod straight. Eyes barely closed. None the worse for wear after a near no-exit scenario from that alien virtual-reality gizmo chair.\n\nSpeaking of...\n\nJack had a training mission to prepare.\n\nWhich was a bit of a problem. Sure, he remembered sitting in the Ancient weapons chair. A moment's frustration at how the thing was too short in the leg-room department. The gummy feel of those gel packs. The chair back radiating enough heat to warm his six. The sound of gunfire. Laser shots. Carter shouting for him to do something, and then \u2014\n\nThe plane pitched right. Not enough to wake anyone up, but Jack gave up on the tour down memory lane. It wasn't just that he couldn't remember how to use the damn chair. He didn't want to. He didn't want to think about losing control. Not knowing what needed to be done. Sitting on the sidelines while a rickety-old device grabbed his brain and did the job for him.\n\nEnough. Once they landed in Antarctica, it would all come back to him.\n\nOr not.\n\nHe flipped open the casualty report, determined to make every name an indelible mark in his head. In the greater balance of what mattered most, he wasn't sure which ranked higher \u2014 remembering how to use the one weapon which could protect the planet...\n\nOr remembering the men and women who'd fallen in doing so.\n\n### PEGASUS ICE RUNWAY\n\n### ROSS ISLAND, ANTARCTICA\n\n### 18 AUG 04\/0750 HRS MCMURDO STATION\n\nDaniel peered out the window as the C-17 descended into blackness. Down below, a narrow band of lights outlined the runway. The illuminated pathway twinkled in assurance like miniature red and blue lighthouses, a guarantee to pilots that yes, landing a heavily loaded plane on a 110-foot thick glaciated ice shelf was no different than landing on asphalt. Ice that could be months or centuries old.\n\nA slight thud, a momentary jostle, and they landed. Daniel leaned right to counter the plane's forward momentum as it rolled across the snow-covered ice tarmac. Across the plane, Sam and Teal'c did likewise, their black-cap clad heads bent sideways in unison.\n\n\"Piece of cake,\" Jack said, his eyes stayed fixed on the open folder on his lap. He'd done paperwork all night, a very un-Jack-like behavior even with his recent promotion to general and SGC commander.\n\nAs the plane slowed down, Daniel grabbed his backpack from beneath his seat and stuffed in his hardbound copy of the _Antarctica Treaty_. \"I'm guessing the pilots do this sort of thing all the time... Flying into Antarctica when it's dark.\"\n\n\"I meant the negotiations, not the landing. You'll do fine.\"\n\nDaniel winced. \"Thanks, but after Tegalus \u2014 \"\n\n\"Back on the horse, Daniel.\" Jack lowered his voice. \"We talked about this.\"\n\nThey had, repeatedly. And even though Daniel knew Tegalus had been on the brink of civil war \u2014 with or without SG-1's arrival through the gate inspiring the Caledonian Federation to take action \u2014 he still hated the idea that he had failed to get the two sides to find common ground. \"Look, I don't think I'm \u2014 \"\n\n\"Just give the diplomats a tour.\" Jack slapped the folder shut. \"Ply them with your charm and wit, tell them to back off on their demands, and in an hour they'll be gone. Then you and Teal'c can play around the outpost to your heart's content, Carter will kick-start the chair back up with those naquadah generators, and I'll \u2014 \"\n\n\"It's not going to be that easy.\"\n\n\"Just get it done.\"\n\n\"But \u2014 \"\n\n\"Daniel, let it go.\"\n\nThe plane came to a halt at the end of the 10,000 foot long runway. \" _Welcome to the bottom of the world, ladies and gentlemen_ ,\" the pilot spoke over the intercom. \" _It's a balmy minus forty-five outside. Enjoy your stay_.\"\n\nWhoops and hollers broke out amongst the Operation Deep Freeze scientists. While Sam was all grins, clearly thrilled by her upcoming work at the outpost, Teal'c barely mustered a raised eyebrow.\n\n\"Not exactly Chulak, is it?\" Daniel asked.\n\n\"Indeed.\" Teal'c had hoped to visit his home while everyone else headed south, but Jack had squashed that request, insisting the big guy join them at the outpost.\n\nWhat was Jack up to? On an almost daily basis, he'd gone out of his way to spend time with each of them. Ever since Teal'c's mishap with the virtual reality chair.\n\nActually, even before that.\n\nNow that Daniel thought about it, not a day had gone by since the Tegalus fiasco where Jack didn't join them for a meal or stop by either Daniel's or Sam's lab to chat.\n\nThe tail ramp opened and a bone-chilling gust welcomed them to Antarctica. Daniel swung up the fur-lined hood of his black USAF extreme weather parka. The scientists followed suit, their bright red parkas marking them as civilians. Although Sam and Teal'c had done the same, Jack predictably opted to keep to the minimum \u2014 his black watch cap his only cover.\n\nA tractor crawled up the exposed tail ramp, grabbing the pallet Daniel had slept on last night. He assumed the boxes and crates contained food, technical gear, and hopefully, coffee. Lots of coffee. Enough to fill a bathtub.\n\nDaniel squeezed his notepad in beside his research materials and the treaty book. Once the diplomacy part was out of the way, the chance to study the Ancient outpost without immediate threats to Earth would be a gift. No team members in urgent need of rescue. No pressure to find Atlantis' gate address.\n\nNo alien civilizations to tear down.\n\n_Enough with the guilt trip_. He zipped his pack shut.\n\n\"Daniel?\" Sam spoke up from across the plane. \"Everything all right?\"\n\n_Thousands dead. Buildings collapsed. All because_ \u2014\n\n\"Daniel?\"\n\n\"Sam, I'm fine.\" Lie firmly in place, he stowed his backpack underneath his seat and his guilt along with it. \"How long till we can get off?\"\n\n\"They need to finish unloading first.\" Jack shoved folders into a beaten-up briefcase. Each was marked with the official SGC logo, one labeled CASUALTY REPORTS, the other TRANSFER REQUESTS. His tight, almost pinched face made Daniel instantly regret the idea that there wouldn't be any pressure while at the outpost.\n\n\"How bad?\" He pointed at the briefcase.\n\n\"Bad enough.\" Jack snapped the case closed. \"You can lose the seatbelt, you know.\"\n\nIn other words, Jack didn't want to talk about it.\n\nOnce the scientists unloaded, SG-1 followed while Jack stayed on board to talk with the pilot. The frigid air bit right through Daniel's parka, insulated coveralls, fleece pants and pullover as well as his two layers of long johns. The only part of him that stayed warm was his feet, thanks to three layers of socks and a thick pair of black rubber boots.\n\nSpotlights lit up the immediate area, sending long shadows across the ground. The landing crew directed passengers toward an awaiting terra-bus outfitted with massive wheels. Steam from the plane drifted through the lights. Daniel had to assume the C-17's engines were kept running to avoid freezing up.\n\nAntarctica was the coldest, cleanest, driest continent on Earth, but where others viewed the continent as nature at its most extreme, Daniel saw a puzzle. An Ancient puzzle he intended to solve given enough time. The once temperate zone had been the advanced race's home millions of years ago, but if they'd left for the Pegasus galaxy, how was it that some humans \u2014 like Jack, Colonel Sheppard, and others \u2014 had genetic markers which allowed the use of Ancient technology? Homo sapiens had only been around for 200,000 years.\n\nIt was a mystery, made even more so by the knowledge that Atlantis had left behind an outpost meant to protect a planet far, far away from their eventual destination.\n\nAs the scientists boarded the bus, a red truck with three-foot high tires pulled up. Windows lined its back half.\n\n\"That's our ride,\" Sam said.\n\n\"Is this means of transportation heated, Colonel Carter?\"\n\n\"Don't worry, Teal'c.\" Sam picked up her backpack. \"It'll warm up once the katabatic winds die down.\" She ran toward the truck with Teal'c close behind.\n\nWeighed down with his book-ridden pack, Daniel followed more slowly, leaning into the wind as it pushed against him. Not for the first time, he wished there really was an Archaeology.com website with CDs on everything from Blackwell's _History of the Latin Language_ to Buckert's recently translated _Savage Energies_. Sadly, there wasn't, and he'd need all the help he could get in deciphering some of the more ambiguous passages on the outpost's panels.\n\nHalfway to the truck, the wind abruptly stopped. Daniel faltered at the sudden absence of an opposing force. His pack swung forward, threatening to take him along with it. Someone grabbed his arm, preventing him from falling flat on his face.\n\n\"Need a hand?\" asked Jack.\n\n\"I'm good, thanks.\" He gestured toward the truck. \"Think there's coffee waiting?\"\n\n\"There better be or someone's getting court-martialed.\" Jack patted his shoulder. \"Let's go.\"\n\nThey walked side-by-side, stopping when the bus carrying the other passengers rolled by. Between moving his body and the lack of wind, Daniel started to warm up.\n\n\"Have you talked to Balinsky lately?\" Jack asked as the bus cleared the area. \"He put in a request to transfer off SG-13.\"\n\n\"You really did stay up all night doing paperwork.\"\n\n\"A general's work is never done.\"\n\n\"Did Balinsky give a reason why?\" Daniel barely knew the redheaded archaeologist. Though they both had ties to the SGC's archaeology department, he wasn't crazy about Balinsky's over-quick assumptions when it came to long-dead civilizations. He'd made the same mistake early on in his tenure with SG-1 and it had almost gotten them killed. Several times.\n\nJack shrugged. \"You think Dixon's too hard on him? Dave can be like that.\"\n\nLike Jack wasn't. Daniel had been put through the ringer many a time, but still... He'd never wanted to quit. Well, not because of Jack, at least.\n\nAs they approached the truck, someone wrapped up in a dark green parka and black coveralls jumped down from the forward cab and snapped off a salute. An equally dark green balaclava covered their mouth, making it impossible to tell whether it was a man or a woman.\n\nThe airman took their packs, held out a hand toward Jack's briefcase, but he waved them off. \"Just get us in this thing, will you?\"\n\nWith a muffled \"Yes, sir,\" they were led toward the truck's rear end.\n\n\"I think Balinsky's making a mistake if he transfers off a first contact team,\" Daniel told Jack as the airman opened the hatch.\n\n\"So do I,\" Jack said. \"But if he's not happy...\"\n\nA blast of heat welcomed them from inside the compartment. Bench seats lined the two walls. Sam and Teal'c had taken up residence on the driver's side, each with a cup to their mouths. From the steam wafting upwards, Daniel assumed his wish for coffee was about to come true.\n\nJack climbed in. \"Dixon doesn't need a malcontent on his team.\"\n\nDaniel followed. Together they took the bench opposite their teammates. As the airman loaded bags onto the floor between the seats, Teal'c handed out insulated mugs.\n\nDaniel took a sip and regretted it. \"Hot tea, really?\"\n\nTeal'c merely smiled and returned to drinking.\n\nThough the mugs were warm enough, Daniel kept his hands covered. Until that rear door was closed, he had no intention of taking his gloves off. \"I wouldn't call Balinsky a mal \u2014 \"\n\n\"SG-13 deserves the best,\" Jack said, turning around in his seat to face the window.\n\n\"Well... All the teams do.\"\n\n\"Yeah.\" Jack stared out into the pitch-black Antarctic morning, his tea untouched.\n\nThe masked airman jumped in and offered Daniel his backpack. Lodging the mug between his legs, Daniel took the heavy bag. The airman slammed the back door shut and knocked twice on the roof. The truck kicked into gear.\n\n\"Will there be suitable heat at the outpost?\" Teal'c asked.\n\n\"When Jack and I came down a few months ago to work with the Atlantis team, they were still putting in heaters.\" Hoping to put Jack in a better mood, he added, \"Maybe we should keep it a bit chilly till the diplomats leave, then \u2014 \"\n\n\"I'm sorry, Dr. Jackson,\" came a muffled reply from the airman. He pulled off his balaclava, revealing the ruffled black hair and ever-worried looks of Paul Davis, the Pentagon's adjutant to the SGC. \"There's been a change in plans for the diplomacy talks with Daniel and the ambassador.\"\n\n\"Major.\" Jack turned from the window. \"What's going on?\"\n\n\"We've been at this for months, General, but the diplomats sent to McMurdo haven't become any less difficult.\"\n\n\"One look at the outpost should calm them down.\" Daniel glanced sideways at Jack. \"It has that effect on people.\"\n\n\"What's the problem with the diplomats?\" Sam asked.\n\n\"They've insisted the talks be held at McMurdo, on neutral territory.\" Paul ran a hand through his ruffled hair. \"If we don't agree to their terms, the UN will demand the immediate removal of the weapons chair from Antarctica.\"\n\n\"Oh, for crying out loud, just toss those diplomats in a chopper and drag their asses up to the outpost.\"\n\nMajor Paul Davis sank down by the rear door, hating the idea of saying no to General O'Neill. \"I can't do that, sir.\"\n\nThe general raised an eyebrow. \"And yet, that was the original plan. Care to explain?\"\n\n\"You were out of reach, sir. The ambassadors made their refusal known while your transport was coming in for a landing.\" Paul unzipped his parka as the truck slowly moved over the frozen sound surrounding Ross Island \u2014 home to the U.S. Antarctic Program's McMurdo Station as well as New Zealand's Scott Base. \"If it's any consolation, sir, the trainees have already headed up to the outpost. General Hammond is there as well, running their preliminary briefing.\"\n\n\"That's just what I need to start the day,\" the general quipped. \"Eager nuggets. They've got the genetic what's-it to operate the weapons chair?\"\n\nPaul nodded. \"The IOA handpicked each and every one of them, sir.\"\n\n\"By the way, Major...\"\n\n\"Yes, sir?\"\n\nThe general grabbed the thermos from Teal'c. \"Nice try changing the subject there.\"\n\n\"I wasn't trying to, sir.\" He also wasn't trying to feel like a first-year cadet, but whenever he spoke with General O'Neill, Paul felt like he was still at the academy.\n\nThough there was no denying he enjoyed every minute of it. It was the reason why he continued to ignore the Pentagon's recent reminder that he was due for promotion to Lt. Colonel. Any chance to work with SG-1 and Generals Hammond and O'Neill was worth staying at the rank of Major.\n\nThe general unscrewed the thermos's cap. \"Last I checked, McMurdo isn't neutral, it's American. Isn't that \u2014 ?\"\n\n\"It's American run, sir,\" Colonel Carter said. \"But McMurdo's still neutral soil, so is all of Ross Island \u2014 \"\n\n\"Because it's considered part of Antarctica,\" Dr. Jackson added. \"Which is \u2014 \"\n\n\"Neutral according to the treaty,\" Teal'c finished.\n\n\"Give me a break. The outpost's in Antarctica, too.\" General O'Neill poured water into his mug, stared at it briefly, and then plucked out the teabag. \"A millions of years-old hangout for ancient aliens has got to be as neutral as it gets.\"\n\n\"Not that alien, sir,\" Paul said. \"You carry their genetics.\"\n\n\"Lucky me.\" General O'Neill flung his teabag into a small can by the truck's rear door.\n\nDr. Jackson shook his head. \"I'll bet the diplomats don't see the outpost like we do.\"\n\n\"And just whose side are you on, Daniel? I thought you wanted to poke around the outpost.\"\n\n\"I do,\" Dr. Jackson said. \"But I don't think we can just drag the diplomats up there.\"\n\n\"Talk is talk.\" The general sniffed his mug. \"What the hell kind of tea is this?\"\n\n\"Rooibos tea, O'Neill. Colonel Carter introduced me to the substance as an alternative to chamomile.\" Teal'c curled his lip.\n\n\"I arranged for the Rooibos, sir. Teal'c's tretonin doesn't mix well with coffee or hot chocolate.\" The colonel shared a smile with SG-1's resident Jaffa. Paul never ceased to admire the comradeship amongst the team.\n\nWhile Dr. Jackson explained the African origins of Rooibos, Paul glanced out the window. McMurdo's lights glowed in the distance. A faint pink flush hugged the horizon just beyond, a weeklong prelude to what would be the continent's first sunrise in months.\n\n\"Geez Louise, it's hot in here.\" General O'Neill unzipped his parka. Teal'c, the colonel and Daniel followed suit.\n\n\"All right, Davis. We'll play along with these namby-pamby diplomats if they stop demanding we yank out the weapons chair.\"\n\n\"Or shut down the new F-302 base,\" Colonel Carter added.\n\n\"With Dr. Jackson's help \u2014 \"\n\n\"If we're going to work together,\" Dr. Jackson said, \"You need to get used to calling me Daniel.\"\n\nPaul gave him a grateful nod. \"While General O'Neill trains new recruits to run the chair, we'll work with the diplomats to find common ground \u2014 \"\n\n\"Earth is their common ground,\" Teal'c said.\n\n\"That's true, but these particular members of the Security Council only see a violation of the Antarctica Treaty.\"\n\n\"No militarization,\" Daniel said.\n\n\"Exactly. And the F-302 base on the other side of Observation Hill doesn't help our cause.\"\n\n\"We need 302s stationed down here to protect the outpost,\" Colonel Carter said. \"Especially while the _Prometheus_ finishes its engine refit, or Ba'al could attack \u2014 \"\n\n\"We're not the ones you need to convince, Carter.\"\n\n\"Yes, sir.\"\n\nAn exchange of glances went back and forth between the general and colonel, the crunch of heavy tires on snow and ice punctuating the silence.\n\nPaul looked out the window again. McMurdo's lights were closer, illuminating the snow-covered, Observation Hill which separated the station from the F-302 base. \"Sir, have you and General Hammond discussed the possibility of giving the F-302 plans to the Chinese? Their representative asked again.\"\n\n\"Are you nuts?\"\n\nPaul stiffened, just like a cadet. \"No, sir, but...\"\n\n\"They want something, don't they?\"\n\n\"Please, Daniel. When don't they?\"\n\n\"O'Neill, has your government not already provided such plans to the Russians?\"\n\n\"We give the Chinese the 302 plans and the next thing you know, they'll want the plans to the 303. That's not going to happen.\" General O'Neill stared at Paul. \"The Swiss sent someone, too. What do they want?\"\n\nPaul sighed. \"Switzerland's still harping on about the Stargate being located in neutral territory.\"\n\n\"Like theirs, I suppose.\"\n\n\"Yes, sir.\"\n\n\"Who's the third?\" Colonel Carter asked.\n\n\"Argentina. And they just want Antarctica,\" Paul answered.\n\nThe general stowed the thermos on the compartment shelf. \"Isn't this all a moot point if the weapons chair is dead as a doornail?\"\n\n\"Not if the Mark II generators work,\" Colonel Carter said.\n\nPaul allowed himself a small grin. At least he could report one piece of good news. \"Dr. Lee ran preliminary generator tests yesterday. Green lights across the board.\"\n\n\"Bill's already done the first power tests?\" Colonel Carter glared at General O'Neill. \"Just one day earlier, sir. I could've met you here.\"\n\n\"And miss out on all the traveling fun? Daniel, help me out here.\"\n\n\"No thanks, Jack. I think you're doing a fine job.\"\n\nAnother round of silence, the general openly scowling at each of his former teammates. The companionable vibe had shifted to a strained one, making Paul feel like he'd stepped into an argument between his parents. He immediately regretted saying anything about the Mark II test.\n\nTeal'c broke the stalemate. \"You cannot keep us forever in your sights, O'Neill.\"\n\n\"Et tu, Brutus?\" The general sighed. \"Look, we're here now. Okay? Just... Get the negotiations done.\"\n\n\"I'm still not convinced \u2014 \"\n\n\"Make peace with the diplomats and then take a chopper up to the outpost by lunchtime. It'll be like old times.\"\n\n\"You mean like when we stood around and watched you freeze like an ice block?\"\n\n\"Different times,\" the general insisted. \"You and T can poke around. I'll stomp all over the trainees. Carter will tinker with her toys and get Lee all flustered. Then, Hammond will pat us on the back and we'll go home.\"\n\n\"Sure, Jack.\" Daniel smirked. \"Just like old times.\"\n\n# CHAPTER TWO\n\n### CHULAK\n\n_50 years ago..._\n\nHuang Sun Tzu crawled behind a broad tree, just beyond the stone spiral pathway, which led to the Chappa'ai. His breath stilled as Apophis' Serpent Guards stormed past. The clank of their armor echoed across the sandy canyon nestled between the two forest ranges. He counted their measure. Ten. Twelve. Fifteen armored Jaffa. Each twice his size.\n\nA passing warrior glanced in his direction. Between one heartbeat and another, Huang flattened himself against the tree, becoming one with its trunk.\n\n_Let your rapidity be that of the wind, your compactness that of the forest_.\n\nOr so his great ancestor had once said.\n\nThe Jaffa moved on, none the wiser, and Huang released his breath into the warm air of Chulak's morning. Subterfuge would be difficult, but not untenable.\n\nLord Yu would be pleased.\n\nTen more Jaffa marched by. The leaves then rustled, a sound only discernible by those trained in the ways of Lord Yu's Dragon Guard. Huang nodded in greeting as his companion knelt beside him. He kept his eyes on the enemy troops until a dusty finger was pressed against his forehead. He swiped it away. \"Is this truly necessary?\" he hissed.\n\nA soft sigh. A whispered breeze against the pounding of yet more Jaffa storming past. Lao Dan Shi swept back the cloak of his matched disguise, the gray of their enemy. Gray cloaks, iron mesh armor and chest plates, shorn heads covered by thin steel skullcaps.\n\n\"Success is necessary, my brother,\" Shi whispered as another squadron marched past. His forefinger plunged into a small clay pot within his palm and withdrew more of the charcoal powder.\n\n\" _And yet, you must be subtle_ ,\" urged a voice from within the satchel at Huang's feet. He withdrew the Goa'uld long-range visual communications device displaying Lord Yu's noble visage. The silver-skinned device was strung within a frame of wires and rods. It was the only way Huang and Shi could speak over the great stellar distances separating them from the great Jade Emperor.\n\nOne of the many shortcomings of being only human. Naquadah did not flow through Huang or Shi's veins for they were destined to be Dragon Guards. To command Lord Yu's Jaffa in his glory.\n\nThe thudding march of Apophis' Jaffa trailed off. Huang peered out from behind the tree and confirmed they were taking up position around the Chappa'ai.\n\nA horn blew in the distance. Shi cocked his head, listening for the telltale sign of more Jaffa approaching. \"We must hurry.\"\n\n\" _What you must do_ ,\" Lord Yu whispered, \" _is stop them with guile, not war_.\"\n\nShi gripped his staff weapon. \"Why wait, my lord \u2014 \"\n\n\" _Silence_!\"\n\nHuang watched on in pity as his more fanatical brother muttered his obedience to their master.\n\n\" _Upon your arrival, kill Apophis' Jaffa. Then, report back_.\" Lord Yu's visage winked out.\n\nHuang returned the device to his satchel. Shi placed his staff weapon by its side and waved a charcoaled forefinger in his direction.\n\nThough reluctant, Huang surrendered his forehead. \"This tattoo, this sigil... It is not the _w\u00e9nsh\u0113n_ I had hoped to bear.\"\n\nShi traced a circle upon Huang's forehead. \"Our rightful tattoos will soon be ours if we succeed \u2014 \"\n\n\"If?\" Huang pulled back. \"You doubt our mission?\"\n\nShi grasped Huang's shoulder. \"I am Lord Yu's servant... and your cr\u00e8che-brother. Born together \u2014 \"\n\n\"We shall succeed together,\" Huang completed the mantra which had bonded them since birth. Clones of Lord Yu's first Dragon Guards; they had been conceived, grown, and born by way of a device as old as the System Lord. Though they were echoes of the past, the product of a technology as ancient as the Goa'uld they served, Huang believed his destiny could be more if Lord Yu would only give him the chance.\n\nShi pressed his finger once more upon Huang's brow. A serpentine stroke shortly followed by a final upward arc just below.\n\n\"There.\" Shi wiped his hand across his iron mesh leggings. \"You now bear the mark of a Serpent Guard.\"\n\nHuang grabbed the pot of crushed charcoal and applied a matching tattoo to Shi's forehead. He pushed aside his repulsion to the false _w\u00e9nsh\u0113n_ , knowing it a necessary evil.\n\nHuang was a devoted servant, a Dragon Guard trainee, and the genetic inheritor of the great Tau'ri philosopher Sun Tzu. To have his forehead tattooed with anything less than his lord's golden crest was anathema to everything Huang had worked for since birth.\n\n\"Do not press so hard,\" Shi whispered. \"I would not have Baozhai see my head so stained upon our return.\"\n\nHuang rocked back on his heels. \"You know it is forbidden for us to mate, even with a slave \u2014 \"\n\n\" _Kree_ , Jaffa!\"\n\nHuang put down the pot and glanced around the tree. Many warriors now guarded the Chappa'ai, their staff weapons at the ready. Three Serpent Guards stood beside the dialing device, their silver tattoos marking their high rank. The morning glow of Chulak's two suns glistened across their foreheads as they awaited the approaching squadron.\n\nKa-chunk. The telltale sound of a first address glyph being entered into the dialing device.\n\nThe horn sounded again. Closer this time, but the footsteps accompanying were few. Five, maybe six at the most.\n\nToo small a number to infiltrate. They'd be easily seen when they slipped behind the nearing Jaffa. Huang sank back against the tree. \"We have a problem, brother.\"\n\nMetal hit stone once again as another glyph was pressed. Shi presented him with a toothy grin. Taller, leaner, Shi was always the optimist. He had truly inherited the warmer nature of his genetic ancestor Lao Tzu. \" _Every problem is an opportunity in disguise_.\"\n\nAgain, the horn blew. This time only a stone's throw away. Huang stuck out his head to witness the final team of Jaffa march past. Grasping his staff weapon in his right hand, his left took firm hold of his brother's arm. \"Then we must grab hold of this opportunity to stop our lord's enemy.\"\n\nKa-chunk. A second, then third glyph was engaged.\n\nAs one, Huang and Shi strode to the line's rear and assumed positions within the rank. Two towering Jaffa led the group, the smoothness of their faces a clear sign of their youth. They could not be more than fifty years of age. The warrior on the left was dark-skinned, his shoulders broad as a tree. The other equally tall though his skin was pale as snow.\n\nThe group marched toward the Chappa'ai. No heads turned. No comments were made. The two escorts and three warriors passed the dialing device as a silver-tattooed Jaffa pressed three more glyphs. The dark-skinned youth flicked a finger at Huang and Shi, indicating they should take position to the Chappa'ai's right.\n\nHuang strode past the dialing device, stealing a glance at the glyphs engaged on its pedestal.\n\nIt was the Tau'ri address! He was certain of it. Only Chulak's point of origin was missing. Feeling Shi's eyes upon him, Huang gave the briefest of nods. Shi's thumb hovered over his staff weapon's firing mechanism. Huang stopped him with a short jerk of his head.\n\n_Not yet, brother. Not yet_.\n\nThey could not attract attention. Not unless the address worked.\n\n\" _Hi'ato_ , Jaffa,\" warned the brown-skinned Serpent Guard. Walk on.\n\nThe two hurried to the platform, but not before he overheard the two escorts speak.\n\n\"Bra'tac must be made aware of their slovenliness.\" This from the paler Jaffa escort.\n\n\"See to it, Va'lar,\" the dark warrior pronounced. \"Once our brothers have departed.\"\n\n_And us along with them_.\n\nA final ka-chunk to enter the point of origin, and a surge of power shook the ground. Huang tensed. A moment later, the Chappa'ai disgorged its force. The way to the Tau'ri was open. The liquid horizon settled across the ring of the Gods and a great cheer went up through the Jaffa ranks.\n\n\" _Tal Shak_ ,\" shouted the lead silver-tattooed Serpent Guard. \"Go and secure the Tau'ri for Apophis' return!\"\n\nThe chosen warriors approached the platform, hesitancy upon their brows. Upon closer inspection, Huang noticed all three looked very much alike. Tan skinned, muscular. Two carried jagged scars across their faces, the other's unmarred. For all Huang knew, they could be triplets, born together.\n\n_And together they would die_.\n\nHuang bent his knees minutely. To truly follow Lord Yu's directions, he knew that they must avoid force until through the other side. The staff weapon in his palm argued against his effort of patience.\n\nOne step. Two. The warriors climbed the platform steps. To his side, Shi had slid his hand down his staff weapon, his thumb yet again reaching for the trigger. Huang prayed his brother would be patient.\n\nThe warriors reached the Chappa'ai, turned around and thumped their fists against their chests in salute.\n\n\"For Apophis,\" they chanted. \"Emperor of all Goa'uld!\"\n\nA great cheer went up amongst the ranks. The three warriors turned back around and stepped toward the Chappa'ai.\n\n\"Now,\" whispered Huang.\n\nHe leapt to the platform with Shi at his side. Two warriors had already entered the shimmering pool, but the third whirled toward them, his weapon raised.\n\nShi double-kicked the warrior's ribs, bringing forth a grunt. A final punch to the Jaffa's stomach pouch sent him sprawling, but not before Huang ripped the staff from his hands. \"Hurry,\" he cried to Shi. They stepped toward the Chappa'ai.\n\nA staff weapon blasted across their path.\n\n\" _Kree ta_!\" yelled the dark-skinned Jaffa. Smoke steamed from his staff weapon as he strode toward them.\n\n\"We must go!\" Huang raced to the Chappa'ai's watery horizon. Below the platform, the assembled Jaffa raised their staff weapons, but with only one more step, it would no longer matter. With Shi, he would be safely through the other side.\n\nIn the land of the Tau'ri.\n\n\" _Shol'vah_ , stop!\" Another blast from dark warrior. This time at their feet. Huang had never seen a Jaffa with such precise marksmanship.\n\nHuang reached out to drag Shi through the Chappa'ai, but his brother spun toward the dark warrior. Shi raised the stolen staff weapon in his hand, thumbed the trigger, and \u2014\n\nCollapsed. His chest blackened.\n\n\"Go,\" Shi mumbled, blood spurting from his mouth.\n\nBetween heartbeats, the desire to destroy the dark Jaffa fought against the logic of Lord Yu's mission.\n\nThe moment passed. The need to complete his brother's wishes remained, and Huang dived into the Chappa'ai.\n\nIn the instant between his departure from Chulak and his arrival in the land of the Tau'ri, Huang prepared himself. He knew two of Apophis' Jaffa would be there. Out in the open. Easy to kill.\n\nOne could not hide in a desert.\n\nLord Yu had told them much about where the Tau'ri had buried \u2014 and now must have unburied \u2014 their Chappa'ai. A land known as Giza. Much wind. Dry heat.\n\nA brittle cold stung Huang's face. He flew from the Tau'ri's Chappa'ai clutching his staff weapon and satchel. Pulling up his knees, he hit the ground and rolled sideways. He'd only a fleeting glance of snow, ice, and being inside a monstrous cave before the Chappa'ai's horizon winked out.\n\nThe cave dropped into darkness. Footsteps shuffled across the icy ground. Huang stilled his breath, waiting.\n\nAn arc of blue fire shot past his feet. A zat'ni'katel. The Jaffa were still here! Another bolt erupted toward him. Before a third could find its aim, Huang jumped to his feet and swung forward his staff weapon.\n\nHe could not gain a clear shot. The Serpent Guards crouched behind a mound of ice to one side. Twin bolts of energy shot forth from either side of the mound. Huang searched the cave for cover and found none.\n\nHe would make cover where there was none. Tilting his staff weapon upwards, he blasted the ice-covered snow ceiling over the two Jaffa. His target hit true. The warriors collapsed in a spew of water, ice and rock.\n\nHuang was alone.\n\nIt was time to return to his lord and master. He spun around the cave, searching for the dialing device.\n\nThere was none. He was stranded.\n\nIn the frozen land of the Tau'ri.\n\n# CHAPTER THREE\n\n### MCMURDO STATION\n\n### ROSS ISLAND, ANTARCTICA\n\n### 18 AUG 04\/0835 HRS MCMURDO STATION\n\nA half hour later, the truck transporting Paul and the SGC team passed a whitewashed sign welcoming them to McMurdo. A few orange parka-clad scientists hiked by, their flashlights broadcasting on the snow ahead them. In the days to come, the station's winter staff of two hundred scientists and support personnel would more than quadruple when the sun finally appeared on the Eastern horizon.\n\nThe truck turned onto the side road leading to Building 155, the station's main center. A bank of spotlights illuminated the power, water, sewer and telephone lines, all running above ground to keep from freezing. Paul rattled off the names of various buildings for General O'Neill and the SG-1 team. Over a hundred buildings made up McMurdo, mostly built of well-insulated corrugated tin and raised on short stilts to keep the snow from building up.\n\nThe truck came to a halt besides the ramp leading up to Building 155. Parkas were slipped on and zipped. Wool hats tugged down. General O'Neill cracked open the back door and the harsh cold of Antarctica rushed into the compartment. He grabbed his briefcase and jumped down. Paul waited while the others each collected their belongings.\n\n\"It will be good to see General Hammond again,\" Teal'c said. \"When do we depart for the outpost?\"\n\nPaul climbed down and shut the truck door. \"A helicopter is waiting to take you now if you'd like.\"\n\n\"Yeah, about that...\" General O'Neill pulled up his hood. \"Teal'c, stay and help the negotiations.\"\n\n\"Sir?\" Paul asked in unison with Colonel Carter.\n\n\"I do not understand, O'Neill. How would I assist?\"\n\n\"Just stand over them. Be intimidating. And watch Daniel's six.\"\n\n\"My six is fine, thank you.\" Daniel paused for a moment, tilting his head. \"Although, it wouldn't hurt for the diplomats to meet Teal'c.\"\n\n\"Sir, I don't think this is a good idea,\" Paul countered. Negotiating the future of the weapons chair was critical to Earth's safety, and moving it was not an option. They still knew too little about how the platform operated to chance any kind of relocation that might cause permanent damage.\n\n\"Having Teal'c at those negotiations is my idea, Major,\" the general said, \"so just go with it.\"\n\n\"Yes, sir.\" Paul tugged his balaclava down to combat the relentless wind. He needed to get everyone inside before they froze. He gestured toward the building. \"We should get inside.\"\n\nHe led the way up a rocky pathway, past rows of shacks and buildings lit up by lampposts. Someone had plowed recently, exposing the bare volcanic rock underneath on which the station had been built. McMurdo sat on Hut Point Peninsula, some twenty miles away from Mt. Erebus, which was also located on Ross Island. Paul had been assured that the volcano was only mildly active. Considering how this day was going, a volcano erupting was the least of his worries.\n\nColonel Carter ran up to join him. \"Major, I'd like to get to the outpost as soon as possible.\"\n\n\"Not a problem, Colonel.\" He pointed toward a brown pre-fab shack. \"Captain Biggs can take you to the helicopter pad. It's right behind the main dormitories.\"\n\n\"Goggles?\"\n\n\"Ready and waiting.\"\n\n\"Thank you, Paul.\" Even through the hood of her parka, he could see the colonel's signature smile. \"Sir, we should head over if you're ready?\"\n\n\"I am. Daniel, you ready to kick butt with those diplomats?\"\n\n\"Save the chair, save the outpost. World peace by lunchtime. I got it.\" Daniel turned back toward the road. \"You know, we've never 'officially' been to McMurdo before.\"\n\n\"Don't worry, you'll get the introductory briefing.\" The general patted Daniel's shoulder. \"It'll be fun.\"\n\n\"A fun briefing? That's somewhat of an oxymoron, isn't it?\"\n\n\"Happy Camper training,\" Paul explained. \"Air force personnel are exempt, but since you're civilians it's required. Even repeat visitors \u2014 \"\n\n\"Don't spoil it for them, Major.\" The general tapped a gloved finger on Daniel's chest. \"Make those folks see the light of day.\" He gestured toward the still dark sky. \"Metaphorically speaking, of course.\"\n\n\"Jack, what if I screw up again? What if \u2014 \"\n\n\"Get it done, Daniel!\" General O'Neill spun toward the USAF support building and headed off with Colonel Carter in tow.\n\nSurprised by Daniel's insecurity, Paul considered pressing him for details, but decided against it. If Generals Hammond and O'Neill believed he could help settle the outpost issue with the diplomats, Paul had to believe they were right.\n\nHe gestured toward Building 155's double doors. \"Teal'c, Daniel, shall we?\"\n\n\"Yeah, I'm freezing. Teal'c?\" Daniel ran toward the entrance.\n\nNo answer came from the Jaffa. Paul glanced over his shoulder to discover why. Teal'c had turned to face the frozen McMurdo Sound that separated the station from the continent's mainland and the outpost. Paul wondered for a moment if Teal'c knew that, or if he was appreciating the green ribbon of light rippling over the horizon.\n\nKnowing Teal'c, he was doing both.\n\n\"The Aurora Australis,\" Paul explained. \"They're caused by photonic emissions in the Earth's upper atmosphere. Solar wind particles funnel down and accelerate along the Earth's magnetic field lines.\"\n\n\"And those gold-white lines?\"\n\nInside the aurora, several striations undulated in counterpoint. The outer green band swayed left while the golden-edged white lines would bend right, seeping out of the ribbon's edge. The aurora would then solidify and repeat the process, all in a slow, meandering pattern.\n\n\"That's unusual,\" he admitted. \"Gold bands usually are a reflection of how much energy's absorbed, but I've never seen polar lights behave like that before.\"\n\n\"So many years ago,\" Teal'c whispered. \"I often wonder...\"\n\nPaul assumed Teal'c referred to when the SGC first discovered the Antarctic gate. \"Seven years is a long time.\"\n\n\"Indeed.\" Teal'c shuddered, as if shaken from a reverie.\n\nThe aurora disappeared. Another gust blew across the ramp, cold enough this time to bite right through Paul's parka. He shivered. \"We should get inside.\"\n\n\"I agree.\" Teal'c's dark eyes turned wistful under the glare of a neighboring spotlight. \"Though it should feel like only a moment in my hundred-and-six years,\" he whispered, \"my time with the Tau'ri has been most gratifying.\"\n\nTaken aback by the Jaffa's openness, Paul could only nod in agreement.\n\n\"Major Davis?\" Teal'c's voice returned to its normal deep tone. \"What does this 'Happy Camper' training entail?\"\n\nGlancing up at the six-foot-three alien warrior, Paul gulped. \"How do you feel about having a bucket on your head?\"\n\n### MCMURDO HELIPAD\n\n### ROSS ISLAND, ANTARCTICA\n\n### 18 AUG 04\/0915 HRS MCMURDO STATION\n\n\"...never agreed to enlisted civilians, Carter. Hammond's making a huge mistake.\"\n\n\"Yes, sir.\" Sam let General O'Neill blow off steam as she strapped into the Bell 212 helicopter's starboard seat. As it was, she barely heard him over the already running rotors.\n\nOutside, the sky had changed from a pitch black to a rose-tinted dark gray. Still too dark to fly blind, but she had a definite plan to take care of that problem.\n\nFlipping on the cabin lights, she retrieved her pre-check clipboard from the dashboard. Small red ticks marked off each item. Captain Biggs had generously prepped the chopper, but Sam knew the only safe pilot was an obsessive one.\n\n\"Cool! A real reason to visit the coldest place on the planet!\" The general pointed out the front window. Down the hill from the helipad, the lights of McMurdo twinkled, but the 'cool' General O'Neill referred to \u2014 an Aurora Australis \u2014 hovered to the east. As they watched, the green wispy glow dissipated into the ionosphere.\n\nWhile the general pulled on his helmet, Sam glanced off to their right at the helipad's windsock. The bright orange cone was a little less than half full. She checked the wind gauge. Fifteen knots and holding steady. Happily, the gusts they'd experienced upon arriving at McMurdo were gone. Flying would be a cinch as long as it stayed that way. She donned her helmet.\n\n\"Whose idea was this anyway, Carter? I can't believe we're just going to sit back and let them take a whirl in that blasted weapons chair.\"\n\n\"If we want the best, sir, don't we need to look past our own borders?\" Placing the clipboard on her lap, she turned off the overheads. Her eyes adjusted quickly, thanks to the green-lit instrument board.\n\n\"How are these people going to understand the stakes?\" The general unzipped his parka. \"Some of their governments don't have the same attitudes about national security, they don't \u2014 \"\n\n\"I don't think General Hammond had a choice, sir. The Atlantis Expedition left a big hole in personnel qualified to operate the Ancient weapons platform.\"\n\n\"Fine, we have to protect the planet. I get it.\" He slapped his harness shut. \"But why can't they use that gene therapy thing and grab some SG teams? Slap 'em with the shot, and then let them live down here.\"\n\n\"The International Oversight Committee insisted on international involvement, sir, including both civilians and military.\" She toggled the gyro switch. So far, so good.\n\n\"Gotta love those IOA folks.\"\n\n\"Sir, you were at the Homeworld Security briefing. General Hammond discussed this with us at length.\"\n\n\"This whole share and share-alike... Why is it that every time we save the planet, we first have to deal with folks like the IOA, Kinsey, or the NID?\"\n\n\"Well, Kinsey's gone.\"\n\n\"That's a happy thought.\"\n\n\"I don't see any other option, sir.\"\n\n\"I'm tired of all the \u2014 \" The general tilted his head back against the headrest. \"Forget I ever said anything.\"\n\n\"General?\" Sam twisted in her harness to look at him.\n\nHe didn't return the glance. Instead, he stared out the fore window. \"Life's a moving target, Carter, and I'm tired of raising my gun.\"\n\n\"Sir?\"\n\nHe waved her off. \"Finish your pre-flight, Colonel.\"\n\n\"Yes, sir.\" Sam turned back to the instrument board. Over the years they'd served together, General O'Neill occasionally made mention of retiring yet again, but he'd never meant it. He loved their work just as much as she did.\n\nThen why did he sound so convincing this time? As she checked the directional control pedals, she made a mental note to set up a team night once they got back to Colorado Springs. Maybe that would help the general feel more connected.\n\nBetween taking over SG-1's command, coordinating with the science departments, and juggling work with a newfound effort to have an actual life, Sam knew she'd been remiss in the team bonding department. And while General O'Neill technically wasn't part of SG-1 anymore, a few pizzas and watching a bad science fiction movie with the others couldn't hurt.\n\nShe scanned the instruments one last time. \"Hydraulic controls on. Throttle, fully open. Engine anti-ice set to on.\" She flipped on the rise switch.\n\n\"Take her out, Colonel. The sooner we get this over with, the sooner we get back to the SGC.\"\n\nSam lifted the collective control stick. The chopper inched upwards, nice and smooth, its spotlights bouncing off the ice-packed snow below. Once they rose a good fifty meters from the helipad, she nudged the cyclic control forward and the chopper headed out toward the outpost.\n\nThe F-302 base came into view as they flew over Observation Hill or 'Ob Hill' as Captain Biggs had called it. As the chopper neared the peak, she spotted the large wood cross commemorating Robert F. Scott's failed expedition. In the years since being rescued from the second gate, Sam had made it a point to learn as much as possible about the region's history. The Herculean efforts of Scott, Shackleton, Amundsen, and many others who braved Antarctica in the early 20th century had often inspired her during some of SG-1's more difficult missions.\n\nThe chopper cleared Observation Hill and flew over the F-302 base. The hill's rocky terrain did a good job of hiding the brown-paneled hangers and barracks from the casual eye. About fifteen miles down McMurdo's frozen shoreline sat Scott Base. As Sam caught a glimpse of the twin F-302s on the ice, she wondered just how much the New Zealanders knew about the interceptors.\n\n\"Is that safe?\" asked General O'Neill.\n\n\"I'm sure the New Zealanders have been briefed, sir.\"\n\n\"That's not what I meant. Do we really want to have two billion-dollar birds sitting on ice?\"\n\nSomeone waved to them from next to one of the F-302s. A pilot, most likely. \"This time of year, the ice shelf's at least ten to fifteen feet deep.\"\n\n\"Yeah, but is it safe?\"\n\n\"As safe as the runway we came in on, sir.\"\n\nThe chopper cleared the base and Sam flipped off the spots. Switching to infrared mode, she slid down her night vision goggles. Her brain took a second to adjust to the lack of depth perception.\n\n\"Those AN-PVS 15s work for you, Colonel.\"\n\n\"Don't really have a choice, sir.\" Sam adjusted her headings to follow the Ross Island coastline further south toward the outpost along the barrier between where the McMurdo and Ross Ice Shelves joined. \"Captain Biggs seemed to think the C-17s might start using them for winter flights.\"\n\n\"Makes sense.\" A light tapping sound came from the general's direction. Though Sam couldn't shift her focus from the terrain ahead, she knew General O'Neill well enough to recognize fidgeting when it happened.\n\n\"ETA 'til outpost twenty-five minutes, sir.\"\n\n\"And I'm counting every second in anticipation \u2014 Whoa!\"\n\nA stiff wind shoved the chopper's nose to the right. Sam applied pressure to her left pedal to adjust the tail-rotor's pitch. The chopper straightened out and she settled back in.\n\nShe checked the gauge. \"The wind's back down to ten knots, sir. We shouldn't have any more problems.\"\n\n\"Sweet.\" The general tapped his fingers against the console. \"A civilian could do this training, you know. I've got more important things to do like approving Walter's latest parking space request or signing another round of Siler's requisitions.\"\n\n\"Running the SGC keeping you busy, sir?\"\n\nThe general merely grunted.\n\nSam grinned. General O'Neill never got tired of downplaying his importance to the SGC in both a leadership position and as a strategist. Or in this case, being the most qualified person on the planet to operate the weapons chair, whether he thought so or not. She told him as much.\n\n\"I never should've let Sheppard go with Weir to Atlantis. The man liked Antarctica. We could've built him his own personal bunk at the outpost.\"\n\n\"Yes, sir.\" Another gust pushed against the chopper. She adjusted the rudder.\n\nSam envied John Sheppard and the others who'd gated to another galaxy. Not that she didn't love her job leading SG-1. She did! But the possibility of discovering Ancient technology far beyond anything anyone could imagine was tantalizing. Although, when she'd heard Rodney McKay had been assigned to the expedition, her envy had turned just a bit to pity.\n\nThe general picked up his finger drumming routine again, this time tapping against his leg in syncopation with the rotors. \"God, I hope Daniel doesn't screw up.\"\n\n\"Why would he?\"\n\nThe finger drumming stopped. \"It's Daniel who needs to believe that, Carter. Not Hammond, Davis, you or me.\"\n\n\"Just out of curiosity, sir, how do you think Teal'c can help? I mean, other than his firsthand knowledge of the Goa'uld, he's not exactly \u2014 \"\n\nThe general laughed. \"Kind of like sending a grizzly to babysit a salmon stream, isn't it?\"\n\nShe chuckled. \"At least it gives Teal'c something to do.\"\n\n\"Yeah, well, I know how that goes.\" The finger tapping started up again.\n\nSam chewed her lip. Teal'c didn't need to be in Antarctica, and he didn't need to stay with Daniel. Whatever troubled the general, whatever motivated his need to keep SG-1 close at hand, clearly he wasn't going to talk about it.\n\nAnother gust nudged the chopper, this time coming from behind, pushing them in the right direction. Grateful for the boost, she nudged the cyclic control forward and headed toward the outpost. With any luck, they'd shave off a few minutes.\n\nAlthough, knowing the general's mood, the trainees might not be so lucky.\n\n### MCMURDO STATION\n\n### ROSS ISLAND, ANTARCTICA\n\n### 18 AUG 04\/0940 HRS MCMURDO STATION\n\nThe rope tied to the waist of Teal'c's insulated coveralls tugged twice. \"Just a few more steps to our right, Mr. Murray,\" came the muffled sound of his assigned companion.\n\n\"I disagree, Dr. Malan,\" he replied, his voice equally muffled by the white plastic bucket he wore to participate in this 'Happy Camper' training game required for McMurdo's civilian visitors. \"Our goal is to reach the central pole. To that end, we must take ten steps forward and one, perhaps two to our left.\"\n\n\"Not if we want to win a set of bowling shoes!\" Dr. Malan tugged the rope joining them again. He stopped. \"Wait, how can you be so sure?\"\n\nTeal'c sighed. He could sense the pole ahead of them. Years of training by Master Bra'tac had taught him how to feel the presence of anything nearby. Convincing this young man to trust him would be difficult without offering an explanation.\n\nWhile Teal'c understood the purpose of this training exercise \u2014 to remind civilians that they must use all their senses when in such a rugged environment  \u2014 he also recognized the other element behind the simulation. \"Did not the personnel director say we must learn to trust one another if we are to succeed?\"\n\nThe taut rope slackened as Dr. Robert Malan's footsteps crunched in the snow. The biologist bumped into Teal'c's left side and stopped. \"I really want to win. You understand, right? It's not just about the shoes \u2014 \"\n\n\"If we lose, I shall personally acquire a set for you. Would that be a sufficient arrangement?\"\n\n\"And you're sure? I mean...\" More crunching of snow, the rope winding across Teal'c's back. \"I could've sworn \u2014 \"\n\n\"Grab hold of the rope and permit me to lead.\"\n\n\"Okay, dude. Go for it. You lead, I'll follow.\"\n\nWith one hand on the rope between them, Teal'c set off to the pole. He took two steps forward and then another. He stopped, waiting for Dr. Malan to catch up. Yes, he could sense the pole ahead. Only a few more feet. \"Seven more paces and we will reach our objective.\"\n\nHe took another step.\n\nDr. Malan followed. \"Objective. Like a military thing? Oh, that's right. Dr. Edmunds said you were with the Air Force. Awesome. Well, forward march, airman.\"\n\nTeal'c smiled beneath the confines of his bucket. The enthusiasm of youth never changed, no matter the civilization. The biologist reminded him very much of Rya'c, and perhaps even himself many years ago when serving as Serpent Guard within Apophis' ranks. So sure of himself. So sure that what is seen is what things are...\n\nAs he took another step forward, Teal'c considered the possible ways he might exact revenge on O'Neill for insisting he remain at McMurdo. Not holding back on their next ping-pong match was certainly one option. Replacing his beloved red Jell-O with green was yet another. In either case, Teal'c would most certainly find retribution.\n\n\"Are we there yet?\" asked Dr. Malan.\n\nTeal'c's gloved hand grazed the wooden pole. He stopped. \"We have succeeded.\"\n\nRemoving the near stifling bucket from his head, Teal'c sucked in a refreshing bout of the cold, dry air. The miraculous Aurora Australis lights had been replaced on the horizon with a red wisp of pre-dawn glow.\n\nTurning toward the other eight teams scattered across the snow-covered parking lot, he searched for Daniel Jackson. His friend had been partnered with the National Science Foundation's director for McMurdo. Lean, white-haired, and with a beard as long as the Tau'ri's Santa Claus, the exceptionally tall NSF man had said little during formal introductions, except to argue over the partnerships assigned by Hannah Presley. The station's personnel director had been gracious, but firm in her refusal to change the assignments. It was a talent that reminded Teal'c very much of Colonel Carter.\n\nPlastic thwacked against wood. Teal'c spun back around to his own partner.\n\n\"Ouch!\" Dr. Malan pulled off his bucket. He grinned widely beneath his sparse red beard. \"Hey, we did it!\"\n\n\"We have a winner!\" Hannah Presley ran over to them, blowing a whistle. \"Everyone inside \u2014 there's hot chocolate and a snack waiting in the cafeteria.\"\n\nBuckets were raised and much laughter ensued as the teams disengaged their partnered tethers. That is, all but one team. Far over on the parking lot's other side, Daniel Jackson and Dr. Edmunds turned away from each other as they untied their rope. The NSF director threw his bucket to the ground. He mumbled something Teal'c could not hear and stormed off toward Building 155. Scooping up the neglected bucket, Daniel Jackson followed at a slower pace.\n\n\"Hey, thanks for the win,\" said Dr. Malan, rubbing snow off his short red beard. \"If you're up for it, I can give you guys a tour of the main complex.\"\n\nTeal'c bowed his head. \"That would be most appreciated.\"\n\nDr. Edmunds stomped past the pole. \"Back to work, Malan.\"\n\nThe biologist stared at the NSF director's retreating back. \"What's his problem?\"\n\n\"Not exactly the warmest welcome I've received,\" said Daniel Jackson as he joined them.\n\n\"Don't mind Edmunds,\" said Hannah Presley. \"The bergy bits in the sound have a warmer nature than him.\"\n\n\"Bergy bits?\" Teal'c asked.\n\n\"Miniature icebergs,\" explained Dr. Malan.\n\nBuckets in hand, the four reached Building 155's back entrance just as the wind picked up yet again. Dr. Malan yanked open the heavy door. Most of the buildings in McMurdo were made for the cold weather with walk-in freezer doors to block out the elements. The only difference was that instead of walking into a freezer, this was more like walking out of one.\n\nInside, they removed their outerwear to dry. When Dr. Malan inquired as to why Teal'c's watch cap remained on his head, Daniel Jackson thankfully changed the subject by asking if there was any tea. They hurried toward the cafeteria and Hannah Presley promised to return shortly with a selection of Rooibos.\n\nThe cafeteria was at the end of a long hallway beside a bank of windows that rattled against the wind. Teal'c looked in at the large open-style dining area \u2014 far larger than the SGC's commissary. In the center were many food stations. Several dozen community tables lined the walls.\n\n\"Good thing we got inside when we did.\" Dr. Malan pointed out the window toward a wooden building further up the road. A row of flags flapped briskly. \"Hey, how about that tour?\"\n\n\"Don't waste your time with those two, Malan.\" They turned around to find Dr. Edmunds leaning against the wooden door leading into the cafeteria, arms crossed, his bearded chin thrust upwards.\n\nThe NSF director dropped his arms and strode up to Dr. Malan, separating the young man from both Teal'c and Daniel Jackson. \"They may not look like military, they may not act like military, but they are military.\"\n\n\"As I explained earlier,\" said Daniel Jackson, \"I'm a linguist and an archaeologist and \u2014 \"\n\n\"Have absolutely no reason to be here.\" With that, Dr. Edmunds stomped off. The sound of his footsteps echoing through the corridor only served to punctuate the howling wind outside.\n\nDr. Malan apologized. \"Edmunds's not a bad guy, he's just what my mom calls 'prideful.' Don't take it personally.\"\n\n\"Don't worry about us.\" Daniel Jackson awarded the young biologist a smile. \"We're used to it.\"\n\nUsed to it, indeed. Teal'c observed the NSF director depart. Edmunds's arrogance was no different than what he'd had come to expect from Goa'uld System Lords.\n\nThe wind rattled the windows once more, a howling gust that lent agreement.\n\n### FERRAR GLACIER\n\n### 18 AUG 04\/0940 HRS MCMURDO STATION\n\nJack grabbed hold of his armrests as another gust jolted the chopper starboard. \"Nice flying there, Ace.\"\n\n\"I can't explain it, sir.\" Carter jockeyed the collective, her arm vibrating against the strain. \"Even Katabatic winds don't just pop up out of nowhere.\"\n\n\"Pop up, you say?\"\n\nShe smiled faintly, but didn't reply. Too much wind, Jack supposed.\n\nThe first time he'd been choppered from McMurdo to the outpost, the trip had started out as a bit of a lark. The sun had been high overhead, the sky as blue as a robin's egg, and the snow-covered brown mountains surrounding Ferrar Glacier had gone on forever. Then, that errant drone chased Sheppard's helicopter halfway across McMurdo Sound.\n\nHe'd expected things to go differently this time, but so far? Not so much. The ride itself had been anything but smooth. One moment Carter had the chopper airborne nice and easy, honing in on the outpost. The next, they were fighting gusts that would put a hurricane to shame.\n\n\"You sure you can do this, Colonel?\"\n\n\"Trying, sir.\"\n\n\"Well, try not to flatten us like a stack of pancakes,\" he said only half-jokingly. Every trip he'd traveled to this damn place always ended badly. Why should this be any different?\n\nHe checked the altimeter. \"Fifteen hundred feet to go.\"\n\nCarter switched on the head beams and yanked off her night-vision goggles. \"The wind's shoving us down too \u2014 \"\n\nThe chopper lurched left. Below, a circle of blinking red landing lights and a solitary spotlight marked the outpost helipad. The instrument panel's green lights reflected against her scrunched-up face.\n\n\"Easy, Carter.\"\n\n\"Eleven hundred feet, sir.\" She shifted her grip on the collective. \"Piece of cake.\"\n\nThe chopper took a sudden drop, making him wish he'd never eaten that donut Biggs offered before take-off. \"Make that an even thousand,\" Jack said, reading the altimeter. \"Maybe this isn't such a hot idea.\"\n\nOver the roar of the wind, he heard Carter's feet tap the pedal controls. \"Have a little faith, sir.\"\n\n\"Hey, I'm all about faith, but \u2014 \"\n\nThe chopper's nose pitched upward, throwing him back against his seat.\n\nCarter wrestled them to a steady horizontal, the chopper shuddering against the strain. \"I've flown in worse.\"\n\nHe eyed the altimeter once more. Seven hundred feet to go. \"Maybe we should head back. Try again later, when the wind's died down.\"\n\nThe rotors groaned, another shudder seized the chopper, and then...\n\nThe shaking stopped.\n\n\"The wind's gone.\" Carter flashed him a grin.\n\nHe returned the smile. \"Faith, my ass. Nice flying.\"\n\n\"Thank you, sir.\"\n\nHe leaned back. \"Take us down, Colonel.\"\n\n\"Happily.\" Carter wiped her brow and brought the chopper down by the mammoth glass and steel dome housing the elevator that led down to the Ancient outpost.\n\nJack grabbed his briefcase and jumped out, half-tempted to kiss the ground. Except it wasn't ground. Beneath his feet was a whole lot of ice. At least two hundred feet of it.\n\n\"General O'Neill! Colonel Carter!\" A heavily bundled figure waved at them from the tunnel leading inside.\n\nCarter raised a hand over her brow and did a half-turn. \"They must have moved the dome since my last visit.\"\n\n\"To cover my fancy drill work,\" Jack explained, still mystified by how he'd macgyvered the ring transporter to bore through two hundred fricking feet of solid ice. He remembered holding a soldering gun. He remembered sparks. Hell, he even had a faint reminiscence of Teal'c keeping him company.\n\nBut how he got those rings to cut through ice like a knife through Swiss cheese? Completely clueless.\n\nThe figure receded inside the dome tunnel. Jack nudged Carter toward the dome. \"I've had enough hypothermia for one lifetime. Let's go.\"\n\nCarter hoisted her backpack and followed him inside. As soon as they ran under the 'White Rock Research Station' signage, the parka-clad figure hit a button. The outer door slid down and the man pulled back his hood.\n\nGeneral Hammond grinned at them both. \"Welcome to the outpost.\" He strode toward the inner door and slapped the button to open it.\n\nAs the inner door retracted, Jack pushed back his hood. \"General, it's good to see you.\"\n\n\"I told you, Jack...\" He led them further into the dome, swiping another button to close the inner door. Ten feet inside, an elevator waited. \"Call me George. We're both generals now.\"\n\n\"Old dog, new tricks, sir.\" Jack followed him into the elevator cage. \"How's the Pentagon treating you?\"\n\n\"It has its ups and downs.\" Hammond punched the button for the bottom level.\n\nThe door closed. Digital numbers flashed above the control panel as they descended \u2014 just like the elevator back at the SGC. Grateful for at least one constant, Jack's unzipped his parka as the air warmed up.\n\nWhile Carter followed suit, Hammond kept his parka closed. Jack raised an eyebrow. \"Too cold for you, sir?\"\n\n\"Still a Texan at heart, I'm afraid. Even Colorado Springs was a bit chilly for my tastes.\"\n\n\"Are you enjoying Washington, sir?\" asked Carter.\n\n\"I never thought I'd say this, but yes, I am.\" Hammond gestured at Jack's briefcase. \"I see Walter's kept you busy.\"\n\n\"Not exactly the ammo I'm used to carrying,\" Jack admitted.\n\n\"Welcome to my world,\" Hammond said with a laugh. \"I understand the diplomats insisted on meeting with Dr. Jackson back at McMurdo. That was a good idea, by the way.\"\n\n\"General?\"\n\n\"Having Teal'c join in. Give those folks a taste of what's really out there.\"\n\nJack grimaced. \"Sir, about that. Can't you drag them \u2014 ?\"\n\n\"They'll do a fine job,\" Hammond insisted. \"You worry about wrangling those eager trainees.\"\n\nHammond chatted with Carter as the elevator descended, but Jack barely paid attention. His mind was too busy, trying to figure out the best way to tell the general the truth.\n\nHow could he train a bunch of nuggets if he couldn't even remember himself?\n\n_Pull it together, O'Neill_.\n\nIt wasn't like he'd had to train the Atlantis gang on how to use the chair. They'd figured that out all by their little genius selves.\n\nThe elevator hit bottom, the cage door slid open, and Carter excused herself. She dashed across the outer chamber in search of Dr. Lee. Jack watched her go.\n\nThat is, until he caught sight of the one gadget he hated even more than the chair.\n\nEight-feet high, barely two-feet wide and deep, the Ancient stasis unit he'd spent way too many months in stared back at him from across the room. How he'd willingly stepped into that metal coffin... Well, he'd deliberately shoved that memory in the same box as all the others.\n\nLast time he'd been at the outpost, there'd been plenty of distractions to keep him from thinking about the damn thing. Daniel and Weir's enthusiasm. McKay's uppity Canadianisms. Sheppard's ability to operate the chair as if it was a Gameboy.\n\n\"You ready, son?\" Hammond stepped out of the elevator. Other than a dozen or so scientists futzing with gear at various workbenches, the chamber was fairly empty. A few airmen hung out by a bank of communications equipment at the far wall, drinking coffee, filling out paperwork. All run-of-the-mill.\n\nThough he didn't feel 'ready,' Jack grinned. \"You bet.\"\n\nSure, rummaging in the dark corners of a crowded box of memories was exactly what he wanted to do.\n\n\"The trainees are in the break room so you've got some time to work with the colonel and Dr. Lee. Take a spin in the chair and test those Mark IIs.\"\n\n\"Yes, sir.\"\n\nHammond frowned. \"Is something wrong, Jack?\"\n\n\"Nothing a good rummage sale won't fix, sir.\"\n\n\"Excuse me?\"\n\n\"Just a little joke, General.\"\n\nHammond nodded, hopefully buying the lie. \"I'll go let the trainees know you're here.\" He stepped nimbly around a cart being pushed by a scientist and headed toward the break room.\n\nAlone and ignored in a room full of geeks, Jack sucked it up and made his way to the weapons platform chamber. He marched past his former resting place, refusing to give the metal box another thought. Once through the archway, he caught sight of the chair. Carter and Lee were bent over a monitor, unaware of his presence. The 'easy-chair from hell' \u2014 his pet name for the Ancient weapons chair \u2014 just sat there nice and dormant. Its trellised back was upright. The platform dark.\n\nAnd yet somehow, he had it in him to make the thing light up like a Christmas tree. In that chair, he could shoot out enough weapons to take out an entire Goa'uld fleet.\n\nOr so they'd told him.\n\nCarter had said to have a little faith, but faith was a funny thing. It required all sorts of clich\u00e9s like doubting doubters. Believing in the evidence of things not seen. Or, losing control to a higher source. Inside his own head, no less.\n\nHe strode toward the chair, knowing he had no choice but to rely on another old chestnut \u2014 fake it 'til you make it.\n\n### BUILDING 155 \u2014 MCMURDO STATION\n\n### 18 AUG 04\/1015 HRS MCMURDO STATION\n\nDaniel huddled with Teal'c and Robert Malan around the rec room's foot-high Plexiglas tank. A hand-drawn sign encouraged them to touch the tank's mostly albino inhabitants.\n\n\"Is that a starfish?\" Daniel asked, surprised by its pale yellow coloring. It clung to a rock in the center of the tank while the other creatures stayed underwater.\n\n\"That's an Antarctic sea star.\" Malan gently touched one of the sea star's limbs. He pointed to a handful of spindly yellow slug-like creatures, each the size of a child's fist. \"Those are anemones.\"\n\n\"Why are they so light-colored?\" Daniel asked.\n\n\"Lack of sun, temperature. A whole bunch of reasons.\" Malan stroked the largest anemone's spine. Its tendrils retracted. \"You need to come over to Crary Lab. We've got a much bigger touch-tank over there. Flatworms, isopods, crustaceans. All sorts of local marine invertebrates.\"\n\n\"I'm afraid that's not possible.\" Paul glanced at his watch. \"We've only got about twenty minutes left.\"\n\n\"I've got an idea!\" Malan joined Paul at the door. \"How about after the meeting we show them the lab, the NSF Chalet, and maybe take them up to the weather station?\"\n\nWhile Paul did his best to let the redheaded biologist down easy, Daniel copied Malan's earlier action, grazing one of the anemone's multiple tendrils with his fingertip. Soft, almost rubbery, the tendril slipped from his touch.\n\nTeal'c leaned in beside him. \"Will not the diplomats be angered if we are late, Daniel Jackson?\"\n\n\"We've got time.\"\n\n\"Would not that time be best used in preparation?\"\n\n\"I suppose.\" Daniel touched the pale starfish. Its skin was softer than normal. More pliable.\n\n\"And yet \u2014 \"\n\n\"I'm stalling, I know.\" Daniel grinned at his friend. \"Just trying to gather my thoughts.\"\n\nTeal'c tilted his head. \"You have prepared for this discussion for the past week.\"\n\n\"Yeah...\"\n\nTeal'c copied his action, touching the starfish's back. \"This is not Tegalus, Daniel Jackson.\"\n\n\"No... It's Antarctica,\" Daniel replied. \"These are my own people, Teal'c, but to be honest, I'm not sure if that makes it any easier.\"\n\n\"But perhaps more difficult?\" Two of the starfish's points curled upward as Teal'c stroked its back. \"And like the people of Tegalus, these diplomats will do what they want unless otherwise convinced.\"\n\n\"Fat lot of good I did convincing the Caledonians and the Rand Protectorate.\"\n\n\"As you say, these diplomats are your own people.\" Teal'c retracted his finger from the tank. \"Do you believe it your responsibility to make them understand the outpost's importance?\"\n\n\"That's why I'm here, isn't it?\" With a sigh, Daniel turned from the tank and took a look around the room. Stacks of newspapers, books, and board games covered the center table. On another table by the far wall, a chess set waited for players. Next to the chess set stood an all too familiar wooden board covered in a series of crisscrossed lines. Several dozen black and white stones bunched together in one corner of the board.\n\nDaniel blinked in surprise. _W\u00e9iq\u00ed_ \u2014 the ancient Chinese game known by most today as 'Go' \u2014 wasn't that popular a game. He wondered who would be playing it at McMurdo.\n\nIt'd been barely a year since he'd been captured by Lord Yu and forced to play. While not the strangest experience of his lifetime, Daniel couldn't deny that his hours-long discussion with the Goa'uld had left its mark. Especially Yu's insistence that Daniel learn how to be both a warrior and a scholar.\n\nThough he wouldn't admit it to anyone, he'd embraced that advice whole-heartedly ever since.\n\nTeal'c followed his eye line. \"You cannot always help those who do not wish to be helped, Daniel Jackson.\"\n\n\"No,\" Daniel whispered. \"But I can still try.\"\n\n\"You know, Mr. Murray...\" Malan replaced the tank's cover. \"You don't need to wear a hat inside. It's like \u2014 \"\n\nMajor Davis cleared his throat. \"We really need to head toward the meeting room.\"\n\n\"Sure, sure. We're going that way, anyways.\" Malan led them down a wide hallway with office doors on either side. He pointed out the various rooms including the radio and television stations, the library, and the personnel office.\n\nDaniel kept a smile plastered on to be polite. The tour stopped to admire a weather monitor. Malan dragged Teal'c in for a closer view while Paul held back, his face pinched and anxious.\n\nDaniel didn't blame him. Gating to a Goa'uld enemy stronghold would be easier than what lay ahead. Despite Jack's parting order to 'get it done,' this wasn't going to be easy.\n\nJack's latest orders rankled. He'd changed since taking command of the SGC. Not enough for anyone else to notice, but Daniel did. How could he not? Jack seemed quieter. Not necessarily more removed, just subdued.\n\nBefore his promotion, Jack had definitely been more animated. Sure, he'd argue loudly. He'd groan and complain. But he'd also listen. Jack would take in everyone's thoughts before making a final decision. Sometimes those decisions weren't the best, but more often than not, Jack had bent when Daniel needed him to.\n\nThe man still listened, but sometimes he behaved as if he'd already made up his mind before any arguments could be made. Of course, there were issues Jack was privy to as a general that Daniel could barely guess at.\n\nWhen the tour stopped in front of a closed door with a small window in its top half, Paul thanked Malan for the tour \u2014 a not-so-subtle cue that the young biologist needed to leave.\n\nMalan pumped Teal'c's hand. \"Don't be afraid to join us tonight in the bowling alley. It'll be fun!\"\n\n\"I shall consider your invitation.\" Teal'c tipped his head, oblivious to the scientist's hero-crush.\n\n\"Dr. Malan...\" Paul glanced at his watch.\n\n\"Oh, right. Gotta go!\"\n\nAs Malan hurried off, Daniel shared a smile with Paul. \"Bowling for fun \u2014 \"\n\n\"Still putting that promotion package together, Davis?\" a voice called out behind them.\n\nDaniel turned toward the newcomer. A man dressed in an F-302 jumpsuit strode toward them. When he stopped in front of Paul, Daniel read his name patch: Kenneth Ferguson, Lt. Colonel. Ferguson was lean, with an aquiline nose and eyes that seemed to take in everything without missing a beat. His buzz-cut blond hair was just a breath above being shaved clear off.\n\nPaul introduced Ferguson. \"The Colonel and I went through the academy together.\"\n\nFerguson smirked. \"And yet you're still a wee little major. That's got to be boring. What's the matter? Can't your CO write up a decent recommendation?\"\n\nPaul stiffened. \"Actually, Colonel, the board review happened several months ago. I'm good to go.\"\n\n\"Congratulations,\" Teal'c said.\n\n\"That's great,\" Daniel said, ignoring the colonel's petty comments. \"When does the promotion happen?\"\n\n\"Yeah, Davis,\" Ferguson chimed in. \"When's the ceremony?\"\n\n\"I haven't said 'yes' yet.\" Paul glanced through the door's window.\n\nFerguson slapped him on the back. \"Of course not. You've been in the service more than sixteen years. Why would you ever want to be promoted?\"\n\n\"I'm happy with the way things are.\"\n\n\"Sure you are,\" Ferguson said. \"Tell you what, Colonel Thanks-but-no, when you decide to get off your ass and go through with the promotion, call me. I'll be sitting in the front row.\" He tossed off a wave and strode toward the galley.\n\n\"You okay?\" Daniel didn't really understand military ranks and promotions, but he got the pecking order. If Paul was turning down a promotion, there had to be a good reason.\n\n\"Yeah, I'm fine.\" He turned back toward the door. \"Before we go in, let me brief you on these diplomats.\"\n\nSurprised at Paul's sudden change of topic, Daniel glanced at Teal'c who raised an eyebrow in response. Inside, three ambassadors hovered around a coffee service by the far wall. Two men and a woman.\n\n\"The tall man with the mustache is Markus Duebel, Switzerland,\" Paul said. \"He's been with their diplomatic corps for over twenty years, but from what I've put together, he's been booted out of just about every place the Swiss sent him.\"\n\n\"Great. A grumpy career diplomat,\" Daniel observed. Duebel had a reserved air to him. He poured cream into his cup, added sugar, and then stirred it in, every movement economized. Even the man's facial movements were minimal. Duebel didn't smile or frown while talking with the other man in the room. He simply was. His steel gray hair added to his stern aura.\n\n\"All three of them are basically bottom of the barrel ambassadors. Kicked out, fired, re-hired. I'm guessing the UN didn't believe they needed to send their best.\" Paul pointed at the shorter man gesticulating wildly at Duebel. \"That's Jorge Diego Suarez, Argentina. You know their deal.\"\n\n\"They believe they own Antarctica,\" Teal'c said.\n\n\"Not all Argentines think that way,\" Paul explained.\n\n\"Mostly just the upper-class.\" Daniel decided Suarez probably belonged in that category. Unlike the other two who had donned simple cold-weather fleece pullovers, the Argentine diplomat wore an expensive cable-knit sweater. His black hair and beard were neatly trimmed, not a hair out of place.\n\nAnd from the way his hands waved, he enjoyed a good argument. Too bad Jack wasn't around. There wasn't anyone better at shutting down someone like that in a heartbeat.\n\nThe third diplomat turned toward the door. She was Asian, not more than five-foot-three. Short-cropped black hair and bangs framed a face tattooed with what seemed a permanent scowl. She sat down at the table in the center of the room and sipped her coffee while the two men argued.\n\n\"That's Quing Zhu,\" said Paul.\n\n\"She desires something of value,\" Teal'c observed.\n\nDaniel pulled back from the door. \"Not that I'm surprised considering how little we've lived up to our end of our promise to share technologies from the program, but how can you tell?\"\n\n\"She remains separate from the other two,\" Teal'c said. \"Watch. She is impatient.\"\n\nAs if on cue, Zhu glanced toward the wall clock and frowned.\n\n\"Maybe that's just her nature,\" Daniel said. \"Guys, whether we like it or not, China's a member of the Security Council. If President Hayes trusted them enough about the Stargate \u2014 \"\n\n\"After last year's diplomatic fiasco over Lord Yu,\" Paul said, \"I'm not so sure the president trusts anyone.\"\n\nTeal'c's frown only punctuated Paul's point. Though Daniel had missed SG-1's run-in with the Chinese government last year \u2014 mostly because he was Lord Yu's prisoner at the time \u2014 he'd learned during his debriefing all about how the Chinese had unwittingly allowed Yu's spy to infiltrate the SGC.\n\nA spy that China wouldn't allow the Pentagon to interview.\n\n\"So those are the players.\" Paul dropped a hand to the doorknob. \"Teal'c, you're just here to observe, but \u2014 \"\n\n\"I have pledged my allegiance to this world, Major Davis. I will remain silent.\"\n\nDaniel grinned at his friend's avowal, wishing he felt as confident. \"I think what he's trying to say is \u2014 \"\n\n\"I am ready.\" Teal'c lifted his chin. \"This is merely a battle of a different sort.\"\n\n\"A battle of words,\" Paul said with a nod.\n\n\"It's not a battle,\" Daniel insisted. \"We have to help these UN diplomats understand why the weapons chair \u2014 though technically in breach of the treaty \u2014 needs to stay put.\"\n\n\"Daniel Jackson, is not world peace the primary reason for the United Nations?\"\n\n\"Of course.\" Grabbing the doorknob, Daniel silently prayed these negotiations would go far smoother than his last attempt.\n\n\"Then is it not also true that if the chair's removal causes it to malfunction, any hopes of such peace will be destroyed if the Goa'uld could attack again?\"\n\nStunned by the simple truth in Teal'c's words, Daniel did the only thing he could.\n\nHe opened the door.\n\n# CHAPTER FOUR\n\n### EARTH\n\n_50 years earlier..._\n\n\" _You are certain_?\"\n\n\"Lord Yu, the glyphs were correct. This is the land of the Tau'ri.\" The cave's bitter chill nicked at Huang's face, the frigid air bit at his lungs. He had retrieved both cloaks from the dead Jaffa, but it was not enough. He needed to return home.\n\n\"There is no dialing device,\" he spoke into the communications device, his teeth chattering. \"All that remains is ice and snow. Giza is no longer \u2014 \"\n\n\" _Impossible_!\" Lord Yu's fist smashed down out of view, but Huang heard a table splinter from the impact.\n\nGripping the frame, Huang raised the communications device above his head. \"My lord, do you not see the ice upon this cave?\" He panned the device across the snow-covered walls. \"If this was once a land of sand and warmth, it is no more.\"\n\nHe lowered the device. \"My lord, I must ask for rescue. Cannot an Al'kesh \u2014 ?\"\n\n\" _No. You must find the Tau'ri_.\"\n\nFear seized Huang's heart. \"No one could survive in a place such as this. Not even the Tau'ri.\"\n\n\" _Do not presume to know about the Tau'ri_.\" Lord Yu's eyes took on a distance gaze. \" _They are a formidable people. How else could they cast out Ra_?\"\n\nLord Yu returned his gaze to Huang. Haunted eyes peered out from the silver sheen of the communications device. \" _Something is wrong. You must go forth. Discover who is responsible_.\"\n\n\"My lord, I implore you. Without Shi...\" He swallowed hard at the thought of his dead brother. \"Without tools to \u2014 \"\n\n\" _Shi's death is unfortunate, but he can be replaced. As for tools... Have you not the weapon of those Jaffa you killed_?\"\n\n\"I do, but \u2014 \"\n\n\" _You have their clothes to reinforce your own. You have your wits. Your training. What more could you need_?\"\n\nHuang bowed his head. \"A squadron of our Jaffa would make this more obtainable, my lord.\"\n\nLord Yu grunted. \" _Under your command, I presume_?\"\n\n\"Yes, my lord.\"\n\n\" _And I suppose you seek the_ w\u00e9nsh\u0113n _so as First Prime you could command such a legion_?\"\n\nHuang dared not look his master in the eye. \"I do, my lord.\"\n\n\" _You are a fool_!\" The Goa'uld's voice boomed throughout the cave. \" _You live to feel pure molten gold poured upon your skin so that you may carry my sigil. I seek the knife which will cast a far graver wound_.\"\n\nHuang recoiled in horror at the idea that he only lived to covet the _w\u00e9nsh\u0113n_. \"Forgive me, my lord. I meant only that \u2014 \"\n\n\" _Only the dead can be forgiven_ ,\" said Lord Yu with an imperious wave of his hand. \" _Is that what you wish? To die_?\"\n\n\"No, my lord.\"\n\n\" _Then, go forth. Contact me once you have seen what remains_.\" Lord Yu's palm covered the display.\n\nThe communications device winked off.\n\nHuang did as he was commanded, refusing himself the pity a lesser man might carry. If he was to be First Prime one day, if he was to prove himself worthy to Lord Yu, he could do no less.\n\nFirst, he stripped the armor off both Jaffa. Removing their tightly-woven black shirts proved simple and once he'd slipped them on, one over the other, his torso eased its shaking from the cold. He began to remove their chainmail leggings, but decided against it. Both pairs were too large, and adding metal leggings would only make it more difficult to move. He instead removed his own leggings, wrapped one of the cloaks around his lower limbs, and then slipped the leggings back on.\n\nAnd though the two dead Jaffa had been his enemies, he chose to restore their armor to their bodies. They had been mighty warriors and were due that respect. Though Huang was not Jaffa, he knew the legend of Kheb was core to their belief. If these two souls were to meet their paradise, they should do so in honor, dressed as befitted all warriors of any rank or race.\n\nNow warmer, he stowed the communications device in his satchel and searched the cave for some sign of a way out. In the direction of the Chappa'ai there was only a wall of packed snow and ice, but a faint amount of light came from somewhere. He only needed to find its source. He strode to the far other end and looked upward. There. A small crevasse at the top of a steep incline. If he was not mistaken, sunlight poured in from the sky above.\n\nUsing two weapon staffs as poles, he egressed the sloping wall. The crevasse was too small, but he quickly widened the opening with his hands. More sunlight poured through.\n\nHe glanced one last time at the cave below and trembled, but not from the cold. A thousand years had passed since Lord Yu had left Earth. It was very possible that Huang \u2014 and the Jaffa he had killed \u2014 were the only living beings that would ever step foot through that Chappa'ai again.\n\nThe thought sobered him as much as Lord Yu's command to discover what had become of Giza and the land of the Tau'ri. He crawled through the opening, his feet seeking purchase as he went. Finally, he freed himself from the snow-bound cave and rose up to greet the changed land.\n\nBright sunshine reached his face. Snow fell upon his brow.\n\nStumbling further from the cave, he gazed down upon a world far different than his master had ever described. Desert flatlands were not what lay before him. Instead, he witnessed a frozen wasteland. Glacial cliffs. Soaring mountains. A rugged landscape impossible for any being to survive.\n\nAnd yet...\n\nIn the distance, a great plume of smoke belched into the sky. Perhaps, a sign of life.\n\nSetting that as his direction, Huang set forth to serve his master's bidding.\n\n# CHAPTER FIVE\n\n### ANCIENT OUTPOST\n\n### FERRAR GLACIER, ANTARCTICA\n\n### 18 AUG 04\/1020 HRS MCMURDO STATION\n\n\"Hey, Bill.\" Sam nodded toward Dr. Lee as she entered the weapons platform chamber. Though General O'Neill was still in the outpost's main area, she'd need to hurry if she wanted everything ready for the chair's next test with a live subject. Namely, the general.\n\n\"Be with you in bit,\" Lee said from a monitor stationed by the rear wall. \"I'm on a conference call with Dr. Rick Dale, an old friend of mine over at the Mount Erebus Volcano Observatory.\"\n\nThe Erebus Observatory was located on Ross Island, some twenty-seven miles away from McMurdo. Last Sam checked, the volcano hardly spat out anything past its inner caldera.\n\n\"I'm just gonna...\" She pointed toward the dual computer display monitoring the Weapons Platform.\n\nLee hit a key and turned around. The word 'mute' flashed on his screen. \"Prelims went off without a hitch.\"\n\n\"Thanks for doing that. He can't see the chair, can he?\"\n\n\"Who? Oh, you mean Rick?\" He turned back the screen. \"Nope, I narrowed the camera's view. Don't worry, Colonel.\"\n\nHe tapped the keyboard again and the mute function disappeared. As Lee continued to chat with his friend, Sam slid onto a stool by the monitors and chafed her hands together to warm her fingers. The chamber was kept at a cool thirty degrees to insure the ice-covered cavern remained intact. A bit cold for bare hands, but wearing gloves while inputting the program she had planned just wasn't going to get the work done fast enough.\n\nIf everything went as planned, the Mark II would definitely power the chair. She'd designed the power system, run the simulations, and had even double-checked the math \u2014 which was no easy feat when having to convert from Base-10 to Base-8 so it could interface with the chair.\n\nWhile she'd told General O'Neill that a little faith went a long way that belief was firmly rooted in science. Most scientists couldn't blend a religious belief and a scientific mind, but Sam had no trouble combining the two. Especially after all that she'd seen throughout the Milky Way. Just as she'd had faith the chopper would handle a bit of wind because of the laws of aerodynamics, she also believed those immutable laws were an extraordinary miracle.\n\nShe entered a few keystrokes, calling up a system report on Dr. Lee's preliminary test. The backup Mark II had kicked in for a few microseconds, but she had just the remedy to keep that from becoming a problem. Popping in her flash drive, she bypassed the system, and copied over its files.\n\n\"Colonel Carter,\" Lee said loudly. \"What's he doing?\"\n\nSam looked up from the monitor. General O'Neill was poking at the control gel packs on Ancient weapons chair's armrests. His brow furrowed, his shock of silver hair still ruffled from his helicopter helmet, the general seemed lost in thought. Troubled, actually.\n\nLee looked over at the general and then back to Sam. \"Should I sign off?\"\n\n\"Give us a few minutes, and then yes, it's probably a good idea to get back to work.\" Without waiting for a reply, she hurried over to the general's side. \"We're ready, sir.\"\n\nHe frowned. \"Yeah. About that, Carter...\"\n\n\"It won't take long,\" she assured him. \"We just need a live guinea pig in the chair for a few moments.\"\n\n\"A guinea pig, you say?\" He turned toward her, his expression softened.\n\nSam mustered up as much enthusiasm as she could. \"Nothing wrong with being a guinea pig, sir.\" She flashed him a smile, hoping it would pull the general out of whatever troubled him.\n\nIt didn't work. General O'Neill turned his attention back to the chair, his eyes hooded. Somber.\n\n\"What was it like, Carter?\"\n\n\"Operating the chair? Sir, you'd know better than me, I don't have the genetics.\"\n\nHe shook his head. \"I don't mean the chair. I meant back when that entity thing took over your body and stuffed you in that computer mainframe in the M.A.L.P. room.\"\n\n\"Oh,\" Sam replied. \"That.\"\n\n\"Yeah, that.\"\n\n\"To be honest, sir, all I remember was feeling like I had no control.\" She shuddered, realizing how similar the sensation had been to Fifth's attack, a thousand horrific images seared into her brain.\n\n\"You must've had some kind of control,\" the general said. \"Otherwise, you couldn't shout at us through every frigging monitor on the base.\"\n\n\"I suppose that's true.\" She closed her eyes, trying to shut the lid on any thoughts of Fifth. She knew it wouldn't hold for long. Too many nights waking up in a cold sweat were proof-positive that Fifth had left a mark.\n\n\"Carter?\"\n\n\"Right. Being in the mainframe was like...\" She remembered the disorientation. The feeling as if she'd never get out. Was that how General O'Neill felt when he sat in the Ancient chair? \"It was like being in a dark room without any door, but \u2014 \"\n\n\"There's always a but, isn't there?\"\n\nShe smiled. \"Yes, sir, there is.\"\n\n\"Care to share?\"\n\n\"I still knew who I was. I still had that knowledge.\" She shrugged. \"After that, it was just a case of using that knowledge as a weapon.\"\n\n\"Now you sound like Daniel.\" His lips flattened in a thin line, General O'Neill turned to stare at the chair.\n\n\"Sir, when you were in the chair, you obviously were still in control, otherwise \u2014 \"\n\n\"How could I knock out Anubis' fleet?\" The general straightened his shoulders. \"That's a good point.\"\n\n\"You're still in control, sir.\" She gestured toward the chair. \"It's just a weapon.\"\n\n\"A weapon that has to crawl around my head to do its job.\" General O'Neill sighed. \"Problem is, I'm not quite sure I recall how to make it work. I mean, I remember...\"\n\nHe fell silent. Though Sam wasn't surprised to hear he couldn't recall his last use of the chair, she was surprised to see how much it seemed to bother him. He'd read her report of what happened during the mission to battle Anubis. She was sure he'd read Daniel's and Teal'c's, too. Maybe, he just needed a good analogy to put his mind at ease.\n\n\"That's the easy part, sir. Just concentrate. Think of the chair as a trigger on a rifle. Focus on your target.\"\n\n\"Whoa. Hold on.\" He jabbed a finger at the chair. \"Tell me those drones won't be connected while I'm sitting in that. Or any of those nuggets, either. Trust me, Carter, it's not fun being chased by one of those things.\"\n\n\"We've bypassed the drones, General,\" Lee called out from behind the monitors.\n\nGeneral O'Neill eyed Lee with skepticism. \"And how long have you been eavesdropping?\"\n\n\"Me?\" Lee's face reddened. \"I... uh... I was on a conference \u2014 \"\n\n\"Just tell me this thing isn't going to blow up.\" The general marched around to the front of the chair. Any visual sign of his earlier trepidation was gone, tucked away in a room so locked down that even Sam didn't have the key.\n\nKeys!\n\n\"Sir, don't sit down yet.\" She hurried back over to the control system and checked. The file transfer was complete. A few clicks, a password prompt, and she typed in an alphanumeric string. A low hum initiated from the primary Mark II. The secondary stayed dormant, its yellow standby light blinking in confirmation.\n\n\"Does it work?\" Lee joined her by the control screen.\n\n\"So far, so good.\"\n\n\"Does what work?\"\n\nSam scribbled the passphrase on to a pad by the monitor and handled it to Lee. \"I've created an automatic switch so the second Mark II can take over if the first falters. Converting the chair's input take from the enormous power of vacuum energy to the naquadah's more direct stream was like trying to re-jigger a piece of equipment from using A\/C power to D\/C, so I \u2014 \"\n\n\"Ah!\" General O'Neill stopped her. \"I'm sorry I asked. Are we ready to light this thing up?\"\n\n\"Yes, sir. Just remember \u2014 \"\n\n\"Finger on the trigger, Carter. I get it.\" The general scrubbed a hand over his face. \"Okay, no time like the present.\"\n\nHe sat down in the chair and nothing happened.\n\nSam switched the display over to the monitoring computer. Power was steady. By all rights, it should work. She peered around the display at the general. He had his arms crossed, his eyes squeezed shut. \"Sir, you need to concentrate.\"\n\n\"What do you think I'm doing, whistling Dixie? Oh, wait a second.\" The general splayed his hands along the gel packs.\n\nThe chair came to life, its translucent panels illuminating a soft blue glow. The platform's equally milky panels mirrored the color and luminescence. As the back reclined, the chair rotated around, revealing a hauntingly neutral face on the general. A face that reminded Sam of just how close they'd come to losing him months ago when the Repository threatened to wipe his mind permanently.\n\nOpening his eyes, General O'Neill glanced in her direction. He waggled his eyebrows. \"I got it to work, huh?\"\n\n\"I had every faith you would, General.\"\n\n\"So now what?\"\n\n\"We'll run a few more tests. Make sure everything's in order before General Hammond brings the trainees in.\"\n\n\"Lucky me.\"\n\nSam headed back toward the monitors. \"You see, sir? Piece of cake. With the right genetics, anybody can do it.\"\n\n\"Anybody,\" said the general, his voice wooden. Distant.\n\n\"Sir?\"\n\n\"I'm fine, Carter.\" He squeezed his eyes shut again. \"Anybody can do this.\"\n\n### BUILDING 155 \u2014 MCMURDO STATION\n\n### 18 AUG 04\/1045 HRS MCMURDO STATION\n\nTeal'c gulped down yet another cup of Rooibos, thankful for the container of red leaf tea Hannah Presley had brought him. Though it was not a cure-all for his current circumstances, he found the soothing drink a suitable contrast to the heated exchange between Daniel Jackson, Major Davis, and the diplomats.\n\nThe negotiations were little more than verbal sparring. One ambassador would jab. Daniel Jackson or Major Davis would parry. Time passed with little achieved, the ambassadors unwilling to compromise on any ground. It was obvious why all three of the diplomats had been sent to Antarctica.\n\nThey were petty fools and cowards.\n\nMajor Davis sagged back against his chair. \"Our standing orders are to find weapons that will protect Earth against the Goa'uld.\"\n\n\"Weapons which have radically altered the balance of power on this planet,\" Ambassador Zhu said, stabbing a finger on the table. \"That balance must be \u2014 \"\n\n\"An order which the UN Security Council supported,\" Daniel Jackson continued, ignoring the woman. \"If it wasn't for the Ancients weapons platform, Anubis wouldn't have been just a threat, he would have \u2014 \"\n\nAmbassador Zhu harrumphed. \"Anubis is gone. There is no need for the weapons platform to remain here in Antarctica in violation of the treaty.\"\n\n\"Madame Ambassador, we need more time to study the chair before taking any chances it \u2014 \"\n\n\"No more waiting. Certainly your scientists have had enough time now to determine the best means of removal.\" Ambassador Duebel smoothed down the sides of his mustache, a mannerism of public grooming Teal'c found pretentious. \"Per Article One, no weapons of any kind \u2014 \"\n\n\"But, sir \u2014 \"\n\n\"No!\" Ambassador Juarez waved a finger at Major Davis. \"The weapons platform's presence violates the very spirit of the treaty. The Americans are fools to believe otherwise.\"\n\nTeal'c clenched his jaw in disgust.\n\nBeside him, Daniel Jackson sighed. \"You really believe that? After everything we've done to protect you?\" He pulled off his glasses and pinched the bridge of his nose. \"That chair is the only reason \u2014 \"\n\n\"You keep bringing up the same argument,\" Ambassador Juarez roared. \"This only proves my point. Americans abuse Earth's resources as if \u2014 \"\n\n\"Enough!\" Teal'c leapt up and faced the enemies of his friends. The three ambassadors froze in their chairs.\n\nDaniel Jackson glanced in his direction, began to shake his head, and then stopped. \"I think Teal'c would like to add something to our discussion.\"\n\nTeal'c gazed steadily upon the ambassadors, allowing his stature to create a moment's pause. Would it also allow him to impress upon them what was at stake?\n\nFinally, Ambassador Duebel shrugged. \"Though Mssr Teal'c has no say in these matters, I am willing to hear his perspective. Objections?\"\n\n\"It matters little,\" Ambassador Zhu said with a tilt of her head, \"but we will allow him the courtesy as a guest.\"\n\n\"This is a waste of time,\" Ambassador Juarez murmured.\n\nTeal'c turned toward the Argentine Ambassador. \"Then I will be brief.\"\n\nAmbassador Juarez opened his mouth to speak.\n\nTeal'c stepped closer. \"I have served this world for more than seven years. I will have my say, Ambassador.\"\n\nThe man's face went white and he shut his mouth.\n\nSatisfied, Teal'c returned to his chair. \"I wish to speak to your accusations that the Americans are fools \u2014 \"\n\n\"That is none of your concern \u2014 \"\n\n\"You are wrong, Ambassador Zhu. It is the concern of every being in this galaxy. Your territorial disputes mean nothing to me nor to the Goa'uld.\"\n\nOnce sure the Chinese ambassador would hold her tongue, Teal'c pressed on. \"These Americans you denounce as fools, it is they who have sacrificed hundreds of lives in defense of this world. Sacrifices that have included many men and women I named friends. To call them fools dishonors their memory.\"\n\n\"Forget us Americans for a moment,\" said Major Davis. \"What about the Russians? They've sent several teams through the gate. They know firsthand \u2014 \"\n\n\"The Russians have become American puppets,\" said Ambassador Juarez.\n\n\"Puppets,\" Ambassador Zhu added, \"who are rewarded in a manner you refuse to share with others.\"\n\nTeal'c observed the exchange of glances between Major Davis and Daniel Jackson. He knew what rewards the Chinese ambassador spoke of. Rewards she had yet to openly ask for. If O'Neill believed them unworthy of the F-302 fighter jets, Teal'c trusted his friend's judgment. O'Neill was not Apophis or some other Goa'uld demanding blind trust. His trust had been earned.\n\nAnd yet, if the Americans gave the Chinese the plans to the F-302s, would they not have one more ally?\n\n\"Puppets and fools,\" muttered Ambassador Juarez. \"The Russians and Americans deserve each other.\"\n\n\"And so you would dishonor their deaths as well?\"\n\nAll three ambassadors suddenly found the table in front of them to be of great interest.\n\nDaniel Jackson leaned toward him and whispered, \"I think the ambassadors get your point, Teal'c.\"\n\n\"I do not think so. Indeed, in their attempts to discredit the Ancient weapons platform, they not only dishonor those who have given their lives to protect this world, but its inhabitants as well.\"\n\nAmbassador Duebel crossed his arms. \"Do not presume to understand what Earth requires, sir.\"\n\n\"It is you who does not understand. Many countries have sent representatives to train with General O'Neill in the use of this technology, including your own. They do this to defend Earth. All of Earth. Should you not honor these people by seeing what it is they do before dismissing their efforts by dismantling the weapons platform?\"\n\nThe ambassadors remained silent. Ambassador Duebel picked at his pant leg, while Ambassador Juarez stared at a wall, the lines on his face deepened by his scowl. Major Davis had found something worth removing from his sleeve while Daniel Jackson used the moment to clean his glasses.\n\nAmbassador Zhu arose from her chair, awarded Teal'c a glare, and then went to the far window where refreshments had been set up. Outside, the sunless day cast a colorless pall over the base.\n\n\"We have somehow digressed from the subject at hand,\" Ambassador Zhu said as she retook her seat, a glass of water in hand. \"The chair must be moved, and further, China remains emphatic that our people should know the details of the Stargate Program. Allow us to decide what is right and what is wrong.\"\n\n\"To use your own words, Madame Ambassador,\" said Daniel Jackson, \"we've somehow digressed from the point of this meeting. These negotiations \u2014 \"\n\n\"Remove the weapons chair from Antarctica or there will be no negotiations.\" She turned away in her seat, a clear sign of dismissal.\n\nThe Chinese ambassador played at a game, nothing more, Teal'c reminded himself. He reached into his pocket and took hold of his tretonin as a distraction. The cool glass and metal vial served to remind him of the greater picture. For the Jaffa to find the freedom from the Goa'uld he enjoyed, they needed these Tau'ri to cooperate.\n\n\"Ba'al would support your demands to dismantle the chair,\" he finally said. \"As would Moloc. Amaterasu.\" He kept his voice low. Deliberate. \"Perhaps even Lord Yu.\"\n\nAmbassador Juarez whirled toward him. \"Qui\u00e9nes son?\"\n\nTeal'c leaned forward, his face neutral. \"I speak of other System Lords who would happily see the Tau'ri destroyed. They wait, patiently, believing themselves far superior to your planet's abilities.\"\n\n\"How well you have been indoctrinated by the Americans,\" said Ambassador Zhu. \"More ghosts under the bed to scare us?\"\n\nMajor Davis interrupted. \"If the Chinese really think the chair should be removed, why send one of their own for training?\"\n\nThe two other ambassadors gaped at Ambassador Zhu.\n\n\"Is this true?\" asked Ambassador Duebel.\n\nThe Chinese diplomat merely shrugged. \"She is there only to placate my government and the IOA.\"\n\n\"A political appointee who just happens to have the ATA gene?\" Major Davis asked.\n\nZhu bowed her head. \"An unnecessary exercise.\"\n\nIt was then Teal'c saw an advantage and took it. \"The Chinese do not strike me as a people who waste time on the unnecessary.\"\n\n\"I must agree with our alien visitor.\" Ambassador Duebel rose. \"In fact, I wonder if China has a specific place they wish this weapon moved.\"\n\n\"Is this true, Madame?\" Ambassador Juarez's chair toppled backward as he jumped up. \"Your country holds rank on the IOA. Did you intend to have the weapon platform for yourself?\"\n\nAmbassador Zhu's chin lifted in cold defiance. \"I resent what you imply.\"\n\nAmbassador Duebel once again smoothed his mustache. \"We imply nothing, though there is merit in reconsidering our positions based on this newest information.\"\n\n\"I agree,\" Ambassador Juarez said.\n\nAmbassador Duebel faced Major Davis. \"Could a tour of the Ancient outpost be arranged?\"\n\n\"Absolutely,\" said Major Davis, a relieved smile on his face. \"When would you like to go?\"\n\nAmbassador Duebel shifted his gaze to Ambassador Zhu. \"Now, if that's possible.\"\n\n\"Just give me a few minutes to arrange a chopper.\" The major headed toward the door, but then stopped. He turned back. \"How many should I say will be going?\"\n\n\"I will go.\" Ambassador Juarez joined Major Davis at the door.\n\nAmbassador Duebel favored Teal'c with a smile. It was open. Honest. Possibly even apologetic.\n\n\"We shall all go.\"\n\n### ANCIENT OUTPOST\n\n### FERRAR GLACIER, ANTARCTICA\n\n### 18 AUG 04\/1050 HRS MCMURDO STATION\n\nJack studied the faces of the IOA trainees. The 'anybodies' Carter had referred to earlier. They stood in various forms of attention against the wall opposite the weapons platform. Oval origin patches on their red-fleeced pullovers marked them as Europeans, Russians, and Americans. Ten military, two civilians, the trainees ranged from a sixty-something retired U.S. Army vet to a Chinese girl barely out of her teenage years. A dozen genetic freaks in all, each one with the genetics to kick-start the Ancient tech.\n\nAnybodies. Just like him.\n\nA couple of them coughed, a few scuffled their feet. A last check of his clipboard, a forced smile, and he called the roster, hoping to God he'd remember who was who.\n\n\"Okay,\" he said as cheerfully as he could, \"who's first?\"\n\nTen hands shot up. No surprise, all the military folks had volunteered. The two civilians just stared at the weapons platform, a matched pair of terrified grins plastered on. One was an Italian, a guy by the name of Michael Castellano, with a ponytail long enough to use for rock-climbing. Then there was the Chinese girl who looked too young to be there. Lanky limbs, long hair pulled back. Too fresh-faced. The girl reminded him of Cassie when she was sixteen.\n\nGeneral Hammond strode into the room. As Jack met his gaze with a nod, the general joined Carter and Lee behind the chair. The brainiacs kept futzing with the Mark II, but so far, so good.\n\nJack returned his attention to the trainees. He waved at a young man with dark-hair and a serious buzz cut. \"You first.\"\n\n\"Thank you, sir,\" he said with a French accent.\n\n\"Name?\" Grabbing his clipboard from the platform, Jack gestured for the nugget to take the hot seat.\n\n\"Sub-Officer Andre Charpentier.\" He slid into the chair.\n\n\"Sub?\" Jack asked. \"As in submarine?\"\n\n\" _Oui_. I serve aboard the _Am\u00e9thyste_ , a nuclear-powered submarine.\" He reached for the armrest controls. \"The rank is not unlike a sergeant in your nation's \u2014 \"\n\nJack stopped him before he could palm the controls' gel packs. \"Listen, Carpenter \u2014 \"\n\n\"It is Charpentier.\" He stretched out the last few syllables as if he was talking to an infant.\n\n\"Yeah, whatever. I'm not sure how things operate \u2014 \"\n\n\"General...\" This from Hammond.\n\nWhoops. Jack kicked his normal self to the back of the bus and tried again. \"Sgt. Charpentier,\" he said, emphasizing those elusive last syllables. \"Go ahead. Put your palms on the controls, nice and \u2014 \"\n\nThe chair activated, tilting back as he followed his instructions. \"Like so?\"\n\n\"Sweet.\" Flashing Charpentier an encouraging grin, Jack turned on a small boom box Carter had set up for him behind the chair. A perfect mix of P90 reports, zat fire, and a few staff weapon eruptions to boot. The noise bounced off the walls in a melodious cacophony of battle.\n\nYep, he missed that sound.\n\nCharpentier bolted from the chair. \"I cannot!\"\n\n\"What?\" He shouted over the sound effects. \"I haven't even told you what to do yet.\"\n\nCharpentier stepped down from the platform, flinging a hand at the boom box. \"How do you expect us to concentrate with that infernal noise?\"\n\n\"Okay! Okay!\" He punched the sound off. Carter gave him a bemused grin and then returned to tinkering with her gadgets.\n\nJack faced the trainees. \"Hear me out, folks. If and when you're called on to operate this chair, the enemy will attack without hesitation. You will have backup. You will have people protecting your six \u2014 \"\n\n\"Our what?\" asked one of the German trainees. Jack was pretty sure his name was something Merkel. Peter or Paul or one of those saintly names.\n\n\"He means your rear-end.\" This from Ryan Hall, the gray-haired army vet. Hall was a bit too smug for Jack's liking, but what choice did he have?\n\nJack took a peek at his watch. Eleven hundred hours. Lunch couldn't happen soon enough.\n\n\"We're ready for the next trainee, sir.\"\n\nThank God for Carter.\n\n\"Okay, folks.\" He clapped his hands together. \"Let's try this again. Who's next?\"\n\nA sea of raised hands answered his call. They were an enthusiastic bunch, that was for sure.\n\nHe pointed at the Eastern European gal. \"Okay, you. Name?\"\n\nThin, blonde, and with pretty much the saddest eyes Jack had ever seen, the trainee stepped forward. \"Private Dora Vukovich, sir.\"\n\nJack checked the list. \"You're from \u2014 \"\n\n\"The Republic of Serbia.\" Vukovich raised her chin.\n\n\"Climb on in.\" He gestured at the chair. \"Let's see what you've got.\"\n\nVukovich sank into the chair, rested her hands on the gel-controls, and the lights flashed on. As the chair tilted backwards, she closed her eyes. \"I am ready, General O'Neill.\"\n\n\"Good job, Vukovich. I'm going to turn the boom box back on, okay?\" When she gave him a quick nod, he flipped on the sound effects.\n\nThe chair remained active. So far, so good. Time to see what she's really made of.\n\nJack stepped in front of the chair and waved his hand in the space over her head. \"Now... Think about Earth.\"\n\nVukovich's eyes snapped open. \"I do not understand.\"\n\n\"Just imagine you're in space, looking down on the planet.\"\n\nBingo. Earth came to life in all its holographic glory. A shiny blue marble with a honking big land mass centered near the southern pole.\n\n\"That is not Earth!\" a trainee called out behind him. \"Have you never seen a globe or even a satellite photograph?\"\n\nJack shut off the boom box and whirled around. It had been a girl's voice. With a Chinese accent. He jabbed a finger at the culprit. \"Pipe down because you're next. Name?\"\n\n\"Weiyan Shi,\" the girl managed to squeak out.\n\n\"Civilian?\"\n\nThe girl bobbed her head nervously.\n\n\"Okay, Shi. I'm guessing you were too busy to pay attention in General Hammond's briefing.\" Jack scowled at the trainee.\n\nMerkel raised his hand. \"The Ancients representation of Earth is from long ago. From when the continents were as one.\"\n\n\"At least someone took notes.\" Maybe too many notes. Jack tucked that observation aside and flipped the battle symphony back on. He turned toward Vukovich. The hologram of Earth still hung in the air, nice and solid. \"You're doing fine, Vukovich. Now imagine there's some other someone up in space with you. Some big bad goon \u2014 \"\n\n\"No. I will correct the display.\" The girl squeezed her eyes shut.\n\n\"That's not the point.\" Jack glanced up at the global display. The midsection bulged outward. \"What the hell are you doing?\"\n\n\"I will prove the civilian wrong. Not just anyone can serve in the Serbian army! I am not an idiot.\" Vukovich's face reddened as she kept up the strain of whatever-the-hell she was doing.\n\nThe platform began to vibrate. Nothing huge, but enough to put Jack's teeth on edge. On the head's up display, the sphere became an egg, then a cube. The land mass stretched upward with what could have been a lopsided version of Europe \u2014 including the old Soviet Union \u2014 spread across the top half. The whole thing looked like a fun house version of Earth.\n\nCarter stared at the display too, her brow furrowed. \"Sir, you might want to get her to stop.\"\n\n\"Yeah, look, Vukovich. That's enough.\"\n\n\" _Ni\u0161ta_. No.\" She gripped the chair's arms.\n\nThe platform's vibrations intensified. And then, something fell on Jack's head.\n\nIce dust... From the outpost's ceiling.\n\nSomeone yelped. Jack quickly glanced toward the trainees. One of the trellised panels had fallen on the Chinese gal.\n\nHe turned back around and yanked Vukovich from the chair. \"I said cut it out!\"\n\nThe rattling stopped immediately. The image collapsed, the chair turned off. Vukovich ran from the room. A collective gasp raced across the trainees as Jack watched her go, completely confused.\n\nWhen no one moved to pull the panel off Shi, he jumped down off the chair platform and helped her out. \"You'll be fine,\" he assured her.\n\n\"No, no, no,\" the girl mumbled. She yanked up her left sleeve and then her right. She ran her hands up and down her bare arms not once, not twice, but three times. It was as if she was looking for injuries that just weren't there.\n\nIf the girl was a hypochondriac, she wouldn't last more than a day.\n\n\"Shake it off, Shi,\" Jack barked at her deliberately. If the Chinese were going to send a civilian, he had no other choice but to give her a crash course in hanging tough. \"A few bumps and bruises never hurt \u2014 \"\n\n\"Take a thirty-minute recess, people,\" General Hammond announced to the room. \"If you'll follow me, I believe lunch is available in the break room.\"\n\nStunned, Jack could only watch Hammond stomp past, frowning. The general shook his head in that don't-argue-with-me way of his.\n\nWhat the hell?\n\nShi tugged down her sleeves and followed Hammond out through the archway. The other trainees trailed close behind.\n\nJack looked over at Carter and Lee. \"What the hell just happened?\"\n\n\"We're on it, sir!\" Carter pulled a stool up to the computer and pounded at the keyboards.\n\nHow the hell were these nuggets going to fight the Goa'uld if they couldn't handle a little technological hiccup? Suddenly, he wasn't hungry anymore. Not for lunch, that's for sure.\n\nHe headed off toward main chamber. Maybe there was cake. Which would be a good thing. It would take a helluva big piece to knock down the bitter taste in his mouth.\n\nThere were times George was near convinced God put him on Earth for one reason and one reason alone \u2014 to keep Jack O'Neill in line.\n\nThe man had to see the bigger picture. Otherwise, the trainee program was doomed before it had gotten off the ground. While everyone else finished lunch, George followed Jack back toward the chair room. Just as they passed the Ancient stasis unit, he pulled Jack aside.\n\nAnd got an earful. The trainees were unfocused. They were too young. Too old. The civilian additions were worthless. All the while, Jack fiddled with the stasis unit's molding as if he was trying to pry the thing apart. Thankfully, the blue-gray metal and silhouetted rivets wouldn't budge, but Jack continued to yank at its edges. The activity kept him in one place so George allowed him the outlet.\n\nThe man was no fool. Clearly something else was bothering him. After a few rounds of the man's whispered frustrations, however, George knew he had to shut Jack down. \"You need to take a step back, son. There's more here than meets the eye.\"\n\n\"You could've fooled me, General.\"\n\n\"Everyone wants the same thing, to protect Earth.\"\n\n\"Maybe the military trainees, but civilians, sir? Really?\"\n\nGeorge sighed. \"Be patient. They'll fall into line.\"\n\n\"General, I'd follow you to the ends of the Earth. Heck, I'd follow you to the end of any planet, but if the Pentagon picked these people, it makes me wonder what the hell they were thinking when they promoted me.\"\n\n\"You know very well that the Pentagon had no say in the selection of these trainees. The IOA did the base genetic testing. Furthermore, none of these volunteers are even aware of the whole program. They don't know about the Stargate and \u2014 \"\n\n\"Everything all right, sirs?\" Dr. Lee had poked his head out of the weapons room.\n\n\"Just hunky-dory.\" Jack stepped closer to the stasis controls, his back to the scientist.\n\nIgnoring Jack's ill temper, George faced Lee. \"We're a bit busy here, Doctor. Is there something you need?\"\n\n\"Ah... No, sir.\" Lee blinked a few times, like a rabbit caught in the crosshairs. \"Colonel Carter's running one last test on the Mark IIs. I was going to see about getting another shipped down from the SGC for backup. If that's okay with you, General O'Neill?\"\n\nJack's face darkened, but he kept his tone civil. \"Yep, sure. Whatever Carter wants.\"\n\n\"Right, thanks.\"\n\nWhen Lee was out of earshot, George returned to dealing with his troubled charge.\n\nJack stared into the stasis unit, his hands braced against each side. \"I can barely remember getting into this thing.\"\n\n\"Can't or won't?\"\n\nA shadow passed over Jack's face and he dropped his hands.\n\n\"Look, I understand your reservations, but the IOA believes including other nations will help Homeworld Security's cause, not hurt it.\"\n\nJack spun toward him. \"Sir, I request permission to be reassigned.\"\n\n\"You don't mean that.\"\n\n\"I'm thinking the Alpha site.\" Jack smirked, a clear tell that he was blowing smoke at himself as much as anyone else. \"Heck, I'll even go babysit one of those science outposts Carter helped set up. Anything's better than here because if the president keeps letting the IOA tell us what to do, there won't be a here much longer!\"\n\nGeorge straightened up to his full height. Maybe he wasn't as tall as Jack, but he sure as hell had the stars to back him up. \"General O'Neill, you are not alone in securing Earth's survival.\"\n\n\"I know that, sir.\" Jack returned his gaze to the stasis unit. \"It just feels that way sometimes.\"\n\n\"Imagine how it'll feel when you're in a Pentagon office with nothing more than a couple of pens for weapons.\"\n\nJack whipped his head around so fast that it was a surprise it didn't fall off.\n\n\"You're surprised?\" George smiled at his prot\u00e9g\u00e9. \"I still plan on retiring in the not-too-distant future. Someone needs to take over.\"\n\n\"Begging your pardon, sir. I respect your job. Your title. Your efforts.\"\n\n\"But?\" George crossed his arms, waiting for the inevitable.\n\n\"Well, sir. It's bad enough being on the sidelines \u2014 \"\n\n\"Why would I want you in the nosebleed section?\"\n\n\"Sir,\" Jack lowered his voice, \"while I like to say I took the job as head of the SGC because of the pay bump, the truth is... I took it so the other guy wouldn't.\"\n\n\"General Philips, you mean?\"\n\nJack scowled. \"Philips is a moron.\"\n\n\"And now? Admit it, you like the position. At least some of the time.\"\n\n\"I like it when my people get home safely. That was easier to do when I was out in the field, watching their sixes \u2014 \"\n\n\"It's a fine thing, being a hero who'd die for their country.\"\n\n\"I don't mean it like that, sir.\"\n\nGeorge waved his apology aside. \"It's one thing to die for your country, but right now this planet needs heroes who would much rather live for it.\"\n\n\"General O'Neill?\" It was the Castellano boy. He rushed up, his long ponytail flopping behind him.\n\n\"Yeah?\" Jack said wearily.\n\n\"We're ready, sir. Just say the word.\" The young Italian attempted a salute. It was sloppy, his hand practically flopping outward as he finished. Between the grin and the salute, George liked the young man instantly. His heart was most definitely in the right place.\n\nJack frowned. \"Just say the word?\"\n\nCastellano brought himself to attention. \"We really want to help protect the planet. We just need you to show us how.\"\n\nJack's frown softened. \"Sounds like a plan, Castellano.\"\n\nCautiously relieved, George put a hand on the lad's shoulder. \"Why don't you gather the others and bring them in? General O'Neill will be there shortly.\"\n\nWith a grin, Castellano threw another salute and dashed off toward the main chamber toward the awaiting trainees.\n\nJack raised a hand to the back of his neck. \"Oh, geez...\"\n\n\"Look, I know this isn't easy.\"\n\n\"You can say that again.\"\n\n\"You're a general now. Your job isn't about tactics anymore. It's about strategy.\" He jerked a thumb at the trainees who had begun to head back into the chair room.\n\n\"And having a strategy that involves drop-kicking them isn't going to fly,\" Jack murmured. \"I get it.\"\n\n\"Just pretend they're from another planet,\" George advised. \"Give them the same respect, the same leeway.\"\n\n\"I suppose...\"\n\n\"Those men and women want to learn how to defend this planet. And I think they have as much right to as you do.\"\n\n\"Yes, General.\"\n\n\"So go and train them.\"\n\nJack flung off a mock salute not much better than Castellano's.\n\nGeorge found himself laughing. It felt good, considering the last few hours' tension. \"And Jack?\"\n\n\"Sir?\"\n\n\"Try to call me George. We're equal rank now.\"\n\n\"Au contraire, mon general! I'm just a measly little brigadier. You've got two more stars on those shoulders, sir. I've a long way to go 'till I've earned them.\"\n\n\"You're damn right,\" George answered with a smile. \"Now teach those folks how to do their part.\"\n\nWith a nod, Jack headed back into the chamber.\n\n\"General Hammond, is this a good time?\"\n\nGeorge turned toward Daniel Jackson, newly arrived along with Teal'c and the three quarrelsome ambassadors. Hopefully, seeing the weapons platform in use would convince them to back off their demands to dismantle the chair. Removing the chair from Antarctica was too big a risk right now. They didn't know enough yet about the technology to guarantee it could be reassembled and work as needed.\n\nGeorge had read the SGC's most recent casualty reports. The Goa'uld were still out there and still wreaking havoc. Without Earth's best line of defense, those reports would only get worse.\n\nThough Jack made an effort to be as courteous as possible, he stuck to his guns and called on Weiyan Shi next. She climbed into the chair and the panels lit up instantly.\n\n_So far, so good_.\n\nBefore he could even ask, she called up the display of Earth.\n\n\"Nice work.\" Determined to make Hammond happy, Jack smiled broadly.\n\nThe girl silently followed his orders. The further into the session they got, the more Jack began to believe that a civilian \u2014 at least this civilian \u2014 might not be un-trainable. The girl had definite potential.\n\nMaybe Hammond was right. If he thought of these nuggets as aliens, cut them some slack, then maybe \u2014 just maybe \u2014 they could make some honest-to-God headway.\n\n\"Why have you not turned on the sounds?\"\n\n\"You want them on?\"\n\n\"I do not wish to be treated differently.\" Shi kept her focus on the globe floating overhead.\n\nHuh. Jack could appreciate that. He reached over to the boom box to turn it on.\n\n\"Actually, sir...\" Carter joined him by the base of the chair platform. \"Before adding battle sounds, it might be worth seeing if she can tap into some of the other displays stored in the platform's memory banks.\"\n\n\"I will try.\" Shi settled further back into the chair.\n\nA small commotion at the room's entrance caught Jack's eye. Daniel and Teal'c. With them was some older Chinese woman. One of the ambassadors, most likely. She shoved her way up front past the trainees. All scowls and fists. He raised a hand palm-up at his former teammates, the universal sign for 'what gives'?\n\nDaniel shrugged apologetically. Teal'c dipped his head in greeting. Jack decided that either meant negotiations went off without a hitch or things were SNAFU. Knowing the SGC's luck as of late, his bet was on the latter.\n\nCarter climbed up on the platform. She peered down at the girl, giving her one of those big grins that could make a Goa'uld think twice. \"What's your name?\"\n\n\"Weiyan Shi. Please, call me Weiyan.\"\n\nCarter smiled. \"Okay, Weiyan. You've seen pictures of the solar system. Can you try and call one up in your mind?\"\n\nWeiyan squeezed her eyes shut. In a heartbeat, the hovering Earth morphed into a three-dimensional image of planets, moons and a bright glowing sun smack dab in the middle. Jupiter came around for a pass, an all too familiar sight. Jack had seen the planet firsthand, thanks to his little expedition with Teal'c when Apophis had rigged their X-302 for a one-way ticket to nowhere.\n\nHe counted the planets off in his head. Nine total. No, wait. A tenth slid by. Then another. \"Carter, is that our solar system?\"\n\nCarter stepped off the platform. \"It's possible.\"\n\n\"With eleven planets?\"\n\nShe leaned in and whispered, \"Sir, the _Prometheus_ has found several new planets out beyond the Kuiper asteroid belt. We just haven't made it public knowledge yet.\"\n\n\"Uh, guys?\" Daniel joined them by the base of the platform. \"Is that what I think it is?\"\n\n\"Yep. Didn't you get the memo?\" Jack crossed his arms, going for nonchalance. It wasn't often he was a heartbeat ahead of Daniel when it came to the science stuff. Might as well enjoy it while he could.\n\nA quick glance back at a grinning Hammond made his euphoria all the sweeter. Okay, maybe training these nuggets wasn't so bad. It certainly beat dealing with paperwork back at the base.\n\nWhich anybody could do.\n\n\"Did it work?\" Weiyan opened her eyes. She gazed up at the display with the barest glimmer of a smile. Almost as if she'd never done it before \u2014 the smiling part, that is.\n\nAs far as the map was concerned, Jack was pretty damn sure that had never happened before, either. Certainly not him or Sheppard for that matter.\n\n\"General O'Neill, you are wasting precious resources.\"\n\nJack turned toward the voice. It was the pushy Chinese ambassador. He opened his mouth, wanting nothing more than to tell her to take a flying leap, but Hammond shook his head.\n\nOh, yeah. Diplomacy. Act like a general. Right.\n\nHe forced a grin he wasn't feeling. \"Seems to me she's doing an outstanding job, Ma'am.\"\n\n\"I am not a ma'am,\" she said through gritted teeth. \"As the appointed ambassador from China, I order you \u2014 \"\n\n\" _T\u00edng zh\u01d0_! Do not interfere!\" Half-risen from the chair, Weiyan gripped the arms. If anything, the backlit panels glowed hotter. The color deeper, bluer.\n\n\"Remove her,\" the ambassador demanded. \"Do it or I will!\"\n\nA hushed gasp ran through the trainees. They backed away from the chair.\n\n\"No. This is where I belong.\" Weiyan held onto the chair. \" _Bie li wo_.\"\n\nThe two fell into a whiplash of Chinese, but Jack held back, unsure of what to do or what to say. Their voices grew louder, the trainees backed away further.\n\nThe rumbling started up again. Nothing huge, but Jack recognized it from this morning's aborted session. He hesitated, unsure what Hammond would want him to do with this incipient Chinese civil war.\n\nThe tremor kept up. Jack glanced at Hammond for help. The general put a hand on the woman's shoulder. \"Madame Ambassador, would you please come with me?\"\n\nShe crossed her arms. \"I will not.\"\n\nThe arguing between the two women grew louder. The rumbling kept up, too, like a freight train chugging through a station. Behind Weiyan, the chair's back panel lights pulsed.\n\n\"What the hell's going on, Jack?\" Daniel whispered.\n\n\"You tell me. You're the linguist!\"\n\n\"They're talking so fast, I can't.\"\n\n\"Weiyan, get out of the chair!\" Jack gestured at Teal'c to remove the ambassador.\n\nTeal'c slid in front of the near-crazed woman, blocking her access to Jack, Carter, Daniel, and more importantly, the chair where Weiyan...\n\nSobbed?\n\nThe rumbling intensified. Ice dropped from the ceiling. A rock-sized piece barely missed Carter. \"Dr. Lee! Enable the safety program!\"\n\n\"I can't find the pass key!\"\n\nCarter started off toward the monitors, but stopped short when another chunk of rock fell in front of her.\n\n\"The girl's a fool,\" the woman ranted. \"A disgrace \u2014 \"\n\n\"No!\" Weiyan shot up from the chair. \"I am not some little girl.\" She pounded a fist on the near gel pack.\n\nA burst of light, an awful cracking sound of the floor splitting in two, and Jack found himself falling. Carter, Teal'c, Daniel... Their bodies tumbled against him as he spiraled downward.\n\nA flash of red. Weiyan was falling, too.\n\nA horrible roar filled Jack's ears. His arms flailed, seeking purchase, but he kept falling.\n\nUntil he hit bottom. Then, the last sound he heard was his head smacking something hard.\n\nWith a sigh, he sunk into the black.\n\n# CHAPTER SIX\n\n### EARTH\n\n_50 years earlier..._\n\nHuang trekked across the great frozen valley, away from the Chappa'ai, away from his passage home. Though many hours had passed, sunlight was eternal in the frozen land of the Tau'ri. A fact never mentioned by his master Lord Yu, but no matter. The young Dragon Guard headed toward the great plume of smoke in the distance, determined to fulfill his Lord's bidding. He would find the Tau'ri. He would learn what had become of the land of his ancestors. He would not fail.\n\nLeft, then right. A step forward. Left, then right. Another step followed.\n\nHe used the two weapon staffs retrieved from Apophis' dead Jaffa as walking poles. Each stride forward was advanced first by a jab into the snow to insure it would hold his weight. He was no fool. As a child, he'd played often in the snow-covered land surrounding Lord Yu's fortress. What might seem like a dense patch could in truth be barely more than a gossamer of snow blanketing an endless crevasse underneath. The thick snow beneath his feet could be his enemy.\n\nAnd yet...\n\nHe paused a moment to gaze out at the barren snowscape, the craggy rocks, the ancient blue-white glacial monuments which spoke to the eternal beauty of nature. The blues were bluer. The quality of light strikingly different from anything he had seen before. The sun above cast a softer, whiter shimmer above snow banks.\n\nHuang's heart ached with the absence of his cr\u00e8che-brother. If only Shi had survived to see the honesty in this landscape. If only he was here to feel the cold, crisp nature of the wind.\n\nIf only the kresh'ta who had killed Shi was here now. Huang would rip the symbiote from the murderous Jaffa's pouch and watch him die in agony.\n\nLeft, then right. Another step forward. Huang knew he must stop soon and rest. It had been many hours since he'd left the Chappa'ai. He glanced once more toward the smoke emanating from the distant mountain beyond the valley and across a great flat expanse of land. The journey there would take many days.\n\nMore days than the sustenance he carried would allow.\n\nHe must keep his strength. Eat when he was hungry. Rest when he could. He stopped once more and crouched down, placing the staffs beside him. He unslung his satchel from his shoulder and pulled forth one of the two-score meatrolls wrapped in oiled parchment. Breaking off a small piece, he sighed. Only days ago, the slave Baozhai had packed these rations.\n\nOne more thing he would need to attend to upon his return home. Baozhai must be informed of Shi's death. Even if their affair was forbidden, Huang felt he must honor Shi by telling the girl of his brother's brave efforts and unfortunate demise.\n\nWhen Lord Yu allowed him to return home, Huang would receive his _w\u00e9nsh\u0113n_ , take up his right as First Prime, and then do battle in Shi's name to ensure that honor lived on in all Dragon Guards to come.\n\nHuang stuffed the other half of the meatroll back into the satchel, knowing he must stretch his food supplies. A shrill birdcall grabbed his attention. He looked up in time to see the bird's white underbelly and black-tipped wings soar downward toward the flat ice at the valley's end. Its wingspan was enormous, perhaps twice the length of a full-grown man.\n\nAnd possible food to aid his journey.\n\n\" _What is your progress_?\" asked Lord Yu from within the communications device.\n\n\"Arduous, my lord,\" Huang replied, removing the wired ball and frame from his satchel.\n\n\" _Show me_.\"\n\n\"Yes, my lord.\" Huang arose from his crouching position and held the communications rig aloft. He slowly turned around so that the System Lord might see a full panorama of the daunting landscape. He ended facing the smoke which rising from the distant mountain. \"That is my destination, Lord. Certainly the Tau'ri \u2014 \"\n\n\" _Impossible. Giza had no mountains, only flat desert_.\"\n\n\"My lord?\" Huang lowered the communications rig to eye-level. \"Can mountains such as these grow in the thousand years since your time amongst the Tau'ri?\"\n\n\" _As I said, impossible_.\" The System Lord stroked his emperor's tuft, his face pensive. \" _Tell me, you have been there for several days now. When last did the sun set_?\"\n\n\"It has not, my lord.\"\n\nLord Yu harrumphed. \" _Then you are not in Giza. If the Tau'ri's year is still the same length, it is summer where you have arrived_.\"\n\n\"But my lord... The ice! The snow!\"\n\n\" _You are at the southern pole. I am certain of it_.\"\n\nHuang took a step back in surprise. \"But the Chappa'ai \u2014 \"\n\nIt was a step he would forever regret.\n\nHe plummeted into a crevasse he had not seen. Powdered snow spilled across his face as gravity pulled him downward. When he jerked his arms outward to stop his descent, he lost his grip on the communications ball. The more he reached through the engulfing snow for the ball, the more snow toppled down. His eyes and skin stung from the burning cold. Though he held mouth shut, the snow filled his nostrils. He flailed against the avalanche of snow, sinking further and further down.\n\nSome tired part of him, far off in the distant reaches of his mind, told him to stop moving. He did so, his descent immediately halting. Light still reached him though he'd sunk beneath the surface. That meant he wasn't too far below.\n\nIt meant he could still survive. Still escape.\n\nBut to what end? He had failed. Failed his master. Failed Shi.\n\nEven if he could crawl back up to the surface, without the communications ball, he could never return home. Lord Yu's guards would fire without pause upon any unannounced arrivals through the Chappa'ai. So would the Serpent Guards of Chulak.\n\nThere was no planet to turn to, no means to return home.\n\nHuang floated in the snow, allowing its insulation to warm his misery.\n\nHe had failed his other cr\u00e8che-brothers, too. Qin Shi. Kong Qiu.\n\nKong Qiu.\n\nThe words of Qiu's great ancestor Confucius echoed across Huang's memory. A learning line from his boyhood tutor.\n\n_Failure is not falling down, but refusing to get up_.\n\nSlowly, ever carefully, Huang drew in his left arm. The snow didn't pull him down further.\n\nHe brought his right knee up to his chest. The snow remained calm. Pushing his leg downward, he pushed himself up. The snow gave him no argument. In small measures, he inched his way toward the light...\n\nAnd broke through to the air above. One final push and he crawled from what could have been his snowy grave. Sliding on his belly, he moved back toward his earlier footprints, several arms' length away. It was only then that he allowed himself to sit up, gasping in great gulps of air.\n\nHis satchel was gone, and with it any source of food. While the snow would provide him with water aplenty, he would need to eat to survive. He looked for his staff weapons. One had disappeared, but the other was safely where he'd placed it earlier. He retrieved the staff and glanced overhead. The bird from earlier was gone, but where there was one, certainly more would come.\n\nWith the staff in his grip, he rose to meet the southern pole of the Tau'ri. Yes, he had failed his master and his family, but he now knew he could survive.\n\nHe would survive. He must. For to do otherwise would give his life no purpose.\n\nAnd his brother's death no meaning.\n\nHe would learn the truth of the Tau'ri. And then, he would find a way home, his duty to his lord and master complete. Then, and only then, would he ask Lord Yu's aid in crushing the Jaffa who had killed Lao Shi.\n\nIt was only a matter of when and where.\n\n# CHAPTER SEVEN\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/1435 HRS MCMURDO STATION\n\nTwo-hundred feet beneath the Ferrar Glacier, the once calm, almost serene outpost had become mayhem. Airmen rushed across fallen ice and snow. Scientists struggled to retrieve toppled-over equipment. Between sobs and shouts, diplomats and trainees huddled in the main chamber.\n\nThrough the entire ruckus, George held vigil over a ten-foot wide hole in the floor beside the still activated weapons chair. Somewhere hidden below the milky-white force field covering the ragged tear in the floor were his people: Jack, SG-1, and the Chinese trainee.\n\nSomeone stuck a cup of coffee in his hand. \"Thank you.\"\n\nDr. Lee shuffled up beside him. \"Have you tried calling out to them?\"\n\n\"Repeatedly.\" George took a sip of the welcome coffee. Black, two sugars. Someone knew him all too well.\n\nLee stuck his foot out toward the hole. \"It's possible the force field muffles sound waves.\"\n\n\"I wouldn't do that, if \u2014 \"\n\nA faint electrical charge shot out from the force field, attacking Lee's boot. The scientist stumbled back a few paces.\n\n\"Doesn't carry much of a punch, does it?\" Still, George had tried to break through multiple times with nary a dent. \"Are you all right?\"\n\n\"I think so.\" Keeping his distance, Lee peered at the hole. \"The edges are mostly yellowish sandstone sandwiched between layers of dark gray volcanic basalt.\"\n\n\"Drillable?\"\n\n\"If the force field hasn't spread under the floor beyond the hole, sure. It's too thick to see much of anything.\"\n\n\"Let's bring in a drilling team,\" George suggested. \"Contact McMurdo.\"\n\n\"I'm not sure drilling's a good idea, General.\" Lee pulled his glasses off and wiped them on his fleece pullover. \"With this much geological instability, it could bring the whole outpost down on us.\"\n\n\"And we'd lose the weapons chair,\" George realized. He drained his cup in disgust. \"What happens next, Doctor?\"\n\n\"Next, General?\" Squinting, Lee returned his glasses to their rightful position.\n\n\"That's right, Doctor. Next.\" George threw his shoulders back and stared the scientist down to make his point. He knew Lee was in shock. They all were, but now wasn't the time for standing around and contemplating their navels. \"That fall probably knocked our people unconscious, otherwise General O'Neill would be yelling bloody murder. Get that force field down and our people retrieved. Am I clear?\"\n\nAs if to reinforce his point, a low rumble started beneath them. The trellised panels rattled. A few moments later, the shaking subsided.\n\n\"Aftershocks,\" Lee mumbled. Holding on to one of the pillars arranged around the platformed chair, he knelt down until his face was only inches from the force field. \"I think one of the scientists might have some gear I could use.\"\n\n\"Then go get that gear and bring it in here, pronto,\" George ordered. \"The sooner you get started, the sooner we'll have our people back.\"\n\n### PLANET DESIGNATION: UNKNOWN\n\n### STATUS: UNKNOWN\n\n### TIME: UNKNOWN... UNKNOWN... UNKNOWN...\n\nThere's a moment when nothing exists while stepping through an active Stargate. It's that microsecond between heartbeats. A sliver of nonexistence where all sensations disappear. There's no sound, no scent, no taste or touch, and certainly nothing to see.\n\nUntil stepping out the other side.\n\nDaniel stood on a stone platform no different than countless other platforms on countless other trips. The loamy scent of rich soil tickled his nostrils. Rolling hills covered in jewel-green ferns spread out as far as the eye could see. Bright sunlight warmed his face.\n\nSunlight. Warmth.\n\nAs opposed to?\n\nThat didn't make sense. As a student of ancient cultures, he would use that instant of gate-travel to wipe away presupposed expectations. While first impressions could be wrong so could making comparisons. He'd learned that the hard way. He always needed a moment to push aside his last location so he could judge a new planet fairly.\n\nIt was no different this time. Except, when he mentally tried to push away that last moment before stepping through, nothing came to mind.\n\nPerspiration dripped down his back. Upon hearing the event horizon collapse behind him, he decided to unzip his operation jacket. His hand came in contact with a fleece pullover. A glimpse of Sam and Teal'c showed them wearing the same thing: green fleece pullovers and black wool pants.\n\nSo was Jack, who'd somehow arrived ahead of them, wearing the same get-up and seeming very pissed. Especially since no one had a GDO.\n\n\"Dial us home, Carter. Now.\"\n\nDaniel stepped to the edge of the platform. \"Jack, what's going on?\"\n\n\"We're gating out of here, that's what.\" Jack rushed over to the DHD. \"We'll head to the Alpha site.\"\n\n\"Wait a minute!\" Daniel ran down the platform steps. Sam and Teal'c followed close behind. \"Don't we want to figure out where 'here' is, first?\"\n\n\"Not really, no.\" Jack jabbed a finger in the air. \"Someone's playing games with us and I'm not in the mood.\"\n\nSomeone needed to calm their normally fearless leader fast.\n\nFortunately, Teal'c took the fall. \"O'Neill, have you any memory of \u2014 ?\"\n\n\"Nada. Bupkiss.\" Jack's voice lowered.\n\n\"Me, either,\" said Daniel.\n\n\"The last thing I remember was...\" Sam frowned. \"Well, that's odd.\"\n\n\"Carter?\"\n\n\"Sir, I remember who I am.\" She gestured at Jack, then Daniel and then Teal'c. \"I remember all of you, but \u2014 \"\n\n\"Yeah, I'm having trouble, too,\" Daniel said. \"My long-term memory's fine, but short-term's \u2014 \"\n\n\"Gone,\" Jack whispered. His eyes darkened, causing Daniel to wonder what had happened before the rest of them had arrived.\n\nSam gestured toward the DHD. \"Uh, Daniel? I can't find the point-of-origin symbol.\"\n\nDaniel slid between Jack and the DHD and studied the DHD's round control panel. Everything seemed normal; two concentric rings of nineteen glyphs apiece surrounded the central red half-sphere control globe. On closer examination, he found a new glyph-key. Twin lines twisted around each other like a double helix. He pointed it out to Sam.\n\n\"You're sure?\" she asked. \"I've never seen that before. Teal'c?\"\n\nSG-1's resident Jaffa bent over the DHD and examined the glyph. \"Your estimation is as good as mine.\"\n\nBehind them, Jack exhaled loudly. \"It's a 'guess,' T. As in 'your guess is as good as.'\"\n\nTeal'c straightened up from the pedestal. \"I have often found Colonel Carter and Daniel Jackson's hypotheses to be far superior to mere guesses. Should we not trust them now?\"\n\n\"You'll get no argument from me.\" With a flourish, Jack gestured toward the DHD. \"Dial it up, Colonel.\"\n\n\"Yes, sir.\" She pressed the first key, its telltale kachink followed soon by the gate's top chevron lighting up.\n\n\"You see that?\" Jack smacked at Daniel's arm. \"Six more symbols, one big kawoosh, a quick hop to the Alpha site, and we'll be home for supper.\"\n\nKachink. Five more symbols to go.\n\n\"I still think we're making a mistake.\" Daniel turned his gaze toward the valley below and the hill just beyond.\n\nKachink. Three down, four to go.\n\nHe couldn't see any buildings. No other identifiable markings. But that hill...\n\nKachink. Another chevron lit up.\n\n\"There's something familiar about this place,\" he announced.\n\n\"No, there really isn't.\" Jack yanked off his pullover revealing a black long-sleeved t-shirt underneath.\n\nKachink.\n\nDaniel followed suit, pushing his sleeves up to reveal the same shirt, only white. \"That's odd.\"\n\nJack looked down at his shirt and then at Daniel's. \"Not half as odd as...\"\n\n\"What?\"\n\n\"Nothing,\" Jack answered with a scowl. He turned back toward the DHD. \"Hurry it up, Carter. Our attire isn't exactly in line with the local climate.\"\n\n\"O'Neill, should we not determine what brought us here before returning to the SGC?\"\n\n\"Later, T.\" Jack shoved his hands into his pockets. \"SG-1 can come back with weapons, GDOs, the whole works.\"\n\nKachink.\n\nDaniel couldn't help but notice Jack's left hand balled up in his pocket, as if he had something in it. He checked his own pockets. Nothing. Not even a bit of lint.\n\n\"Last one, sir.\" Sam tapped the new point-of-origin key, and pressed down on the control globe. As the red sphere lit up, Daniel tensed. No matter how many times he'd been through the gate that moment of anticipation never failed.\n\nUntil now.\n\nThere was no active wormhole.\n\n\"What the hell?\" Jack threw down his pullover.\n\n\"The DHD should work,\" Sam reported. \"The correct glyphs light up. The control globe's engaged.\"\n\nJack's face went taut. \"Fix it.\"\n\n\"Sir, I don't have any tools.\"\n\n\"I don't care!\"\n\nAs Sam flinched, Daniel heard a distant rumble. Like a waterfall. No one else mentioned it so he returned to the matter-at-hand. \"Calm down, Jack. We'll figure it out. We always do.\"\n\n\"Look, Carter, I'm sorry.\" Jack rubbed the back of his neck, a gesture Daniel had seen him repeat several times since they'd arrived.\n\nJack caught Daniel's glance and dropped his hand to his side. \"Just, figure it out.\"\n\nThe rumble grew louder. Daniel could feel it in his boots. \"Anyone hear that?\" He turned his gaze upwards, but saw nothing. Just blue sky and a bright sun.\n\n\"We're about to be hit by an ambush, aren't we?\" Jack glanced up as well. \"I'm thinking Ba'al or one of his cronies got bored. That has to be what that thing was about earlier.\"\n\nThing? Daniel raised an eyebrow. \"What did you see, Jack?\"\n\n\"Leave it alone, Daniel.\"\n\nMore rumbling. Pebbles around the base of the DHD began to scatter.\n\nTeal'c scanned the horizon. \"The skies are clear, O'Neill.\"\n\n\"Sir, I recommend dialing the gate manually,\" Sam said.\n\n\"Right. Good!\" For the first time since their arrival, Jack actually smiled.\n\nThe rumbling stopped.\n\nJack gestured toward Teal'c and then the gate. \"Do it.\"\n\nTeal'c climbed onto the gate platform. Taking hold of the inner ring, he pulled downward for a manual dial. As the ring slid through the outer rim, Daniel considered their circumstances. Beyond not being able to remember how they got there, two other things bothered him. First off, it was sunny, which somehow seemed wrong. Second, Jack kept fidgeting with his pocket, almost as if he was...\n\n\"What's that?\" Daniel pointed at Jack's pocket.\n\n\"Nothing.\" Jack stuck his hands behind his back. \"You realize we have no weapons.\"\n\nDaniel shrugged, letting Jack's misdirection slide for the moment. \"We've been in worse spots.\"\n\n\"Not in these get-ups.\" Jack headed toward the gate platform where Teal'c stood, staring up at the Stargate. \"What's the hold-up?\"\n\n\"Observe.\" Teal'c gestured toward the inner ring as it spun through the various symbols. \"It will not lock.\"\n\n\"That's not possible.\" Sam ran up the platform and grabbed hold of the inner ring, stopping its forward momentum.\n\nDaniel joined Jack at the platform's base. He peered up at the Stargate. All seven chevrons were still lit up. \"Try turning it the other way.\"\n\nWith a slight bow of his head, Teal'c grasped hold of the ring and tried to push upwards. It wouldn't budge.\n\nThe ground trembled again.\n\n\"All right,\" Daniel said. \"This is bad.\"\n\n\"That's an understatement.\" Jack ran up the steps and positioned himself behind the gate. With Teal'c on the other side, the two pushed against the inner ring, but to no avail.\n\nDaniel climbed up and joined the others in pushing and pulling at the inner ring. In the distance, the tremors continued \u2014 a low constant rumble.\n\n\"This is going nowhere.\" Jack wiped his brow. \"Somehow, we've busted the thing.\"\n\n\"It could be worse,\" Daniel said.\n\n\"Oh, really?\" Jack said with more than a bit of sarcasm.\n\n\"We could be surrounded by enemy troops,\" Teal'c offered.\n\n\"The day's not over yet.\" Jack pointed over at the DHD. \"What would happen if we popped the top off? That doesn't need anything other than brute strength. Or Teal'c.\"\n\n\"Sir, I'm not sure that's a good idea,\" Sam said. \"If we break the control crystal, it might prevent any wormhole from forming \u2014 outgoing or incoming.\"\n\nThe platform started shaking. Daniel grabbed onto the gate. \"We should find a safer place. Someone will report us missing.\"\n\nJack raised an eyebrow. \"Like who?\"\n\nThe rumbling calmed and Daniel let go. \"General Hammond \u2014 \"\n\n\"Hammond's in D.C.\" Jack crossed his arms. \"Is there anything else you're forgetting?\"\n\n\"That wasn't forgetting. That was habit. You've been General all of \u2014 \" Daniel tried to push aside the fog covering his recent memories. He remembered Jack's first speech as general. Sam's promotion to Lt. Colonel. The Atlantis expedition leaving for the Pegasus Galaxy.\n\nThe rumbling returned, strong enough to shake the platform. The chevrons' glass and metal parts rattled against the Stargate's stone surface.\n\n\"My God, I can't remember.\"\n\n\"Tell me about it,\" Jack shouted. \"I couldn't even remember who I was when I got here.\"\n\nThat piece of news alarmed Daniel, but he didn't have time to ask Jack more. The tremors had intensified to an almost deafening rumble.\n\nThe platform surged sideways. Daniel stumbled to his knees. As he scrambled to his feet, Sam ran down the stairs toward the DHD. Jack raced to the platform's edge.\n\n\"Carter, stop!\"\n\nAn explosion of dirt sprayed up around the DHD. Jack dived off and knocked Sam out of the way as the ground surrounding the pedestal thrust upwards. The DHD toppled to its side.\n\nA loud crack reverberated across the platform. Daniel watched in paralyzed horror as a fracture snaked open along the steps leading up to the gate.\n\nTeal'c grabbed his arm and together, they jumped off the platform. Above the din of the tremors, a grinding noise started up. The gate began to sway back and forth.\n\n\"Run!\" Following Jack's lead, they raced down the hill. Daniel stumbled to keep up, tripping over micro-fissures erupting across the fern-covered terrain.\n\nDirt spewed up in front him. He stumbled backward. Another spout of earth exploded to his right and he lost his balance. Teal'c grabbed his shirt collar, yanking him back on to his feet.\n\nJack led the way as they raced down the hill. The overgrown ferns scratched at Daniel's face, but he kept on running.\n\nFinally, they reached the bottom of the hill, the tremors diminished. Another hundred yards and the quake stopped.\n\nJack raised a fist, signaling them to halt. \"Everyone all right?\"\n\nDaniel opened his mouth to answer, but never got the chance. A thunderous boom erupted behind them. He spun toward the sound only to see the hill they'd just descended torn in two. For a moment, the gate swayed back and forth, perched precariously close to one edge of the forming crevasse. Another tremor hit. The tear in the ground widened, and the gate toppled into the chasm below.\n\nThe hill collapsed and the Stargate was buried.\n\nTeal'c felt no small measure of ease when the world ceased to shake. True, the Stargate was inaccessible, but he had long come to believe that SG-1 could and would always succeed in finding a means of escape.\n\nTo believe otherwise would serve no purpose. He recognized this as he climbed to his feet. O'Neill stood up as well. Though he whispered a soft groan, the man showed no signs of injury. Colonel Carter crouched beside Daniel Jackson. She ripped a strip of cloth from her shirt and wiped his face clean of blood. Upon closer examination, Teal'c was gratified to find his friend had only suffered minor cuts.\n\n\"Everyone all right?\" O'Neill asked.\n\n\"Dandy,\" said Daniel Jackson as he attempted to wipe dirt from his face.\n\n\"I am also well.\" Teal'c permitted himself a brief smile at the young man's adoption of O'Neill's phrase.\n\n\"Carter?\" O'Neill knelt down.\n\nColonel Carter nodded. \"Fine, sir. Just a bit shook up.\"\n\nO'Neill rose and raised a hand to his brow, blocking the sunlight. \"Let's figure out what we have and then what we need to get out of here.\"\n\n\"We'll need a Stargate, for one thing,\" said Daniel Jackson.\n\n\"You let me worry about that.\"\n\nColonel Carter glanced at the field of rubble from which they'd escaped. \"Daniel's right, sir. We have no way of knowing how stable the ground above the gate will be and without the right tools to dig it out \u2014 \"\n\n\"I said, let me worry about that.\" O'Neill spun on his heels and strode off in the opposite direction.\n\nTeal'c scanned their very much-altered surroundings. Beyond the collapsed hill, a large grass-covered plain stretched out for several miles. Though his vision was not as it once had been while carrying a symbiote, he believed he could see an expanse of trees at the plain's far end. Without closer inspection, he could not be certain.\n\nTo either side and behind them rose more hills, each no more than a hundred yards from their current position. O'Neill had reached the base of the hill to their rear and sat down. He bent and retied his shoelace.\n\nDaniel Jackson stood up. \"You don't suppose he's thinking of what happened on Edora, do you?\"\n\n\"I do not,\" Teal'c said. \"The circumstances are sufficiently different.\"\n\nColonel Carter also rose. \"For one thing, he had us on the other side working to bring him home.\"\n\nTeal'c grimaced at the memory. More than four years ago, a meteor shower on Edora had buried the Stargate, stranding O'Neill beyond their reach. It had taken three months of little to no sleep for then Major Carter to devise a means to pierce through the hardened naquadah above the Edoran gate's horizon. Teal'c had volunteered to go through and tunnel the remaining way to the surface. It was only when little oxygen remained that O'Neill had found him. In a certain sense, the rescue had been mutual.\n\nIndeed, most of SG-1's missions had been resolved by group efforts. This is why it disturbed Teal'c to see O'Neill withdraw at a time when they needed him most. Action was required and though O'Neill no longer led SG-1, he remained their de facto leader.\n\nWith a curt nod to Daniel Jackson and Colonel Carter, Teal'c jogged over to O'Neill's location. The effort left him a bit more winded than it should have, but then the past hour had certainly taken its toll on all of them.\n\nHe slowed as he neared O'Neill, allowing his brother-in-arms a moment to collect himself. \"Should we not search for a solution to our predicament?\"\n\n\"And what predicament would that be?\" O'Neill asked, pulling a small metal object from his pants' pocket. With a flick of his wrist, the cap flipped back.\n\nTeal'c raised an eyebrow upon seeing the lighter. He had searched his pockets shortly after their arrival and found nothing. \"I do not believe any of us found similar items in our possession.\"\n\nO'Neill thumbed the wheel. When a flame sparked up from the lighter's metal cage, a brief smile appeared on his face. Perhaps an indication that O'Neill had returned to good humor.\n\nThe smile faded. \"You know what? I'm not buying this so-called predicament we've gotten ourselves into.\"\n\n\"And yet the Stargate is buried.\"\n\n\"Hell, T, we couldn't get it to work even when it wasn't.\" O'Neill snapped the lighter shut.\n\n\"Then what do you propose?\"\n\n\"That we find a way off this rock.\" O'Neill stood up and steadily met his gaze.\n\n\"Do you believe that to be possible?\" Teal'c asked, inwardly pleased by his brother's renewed optimism.\n\nO'Neill glanced once more at his lighter and then returned it to his pocket. \"It's not like we have a choice, but yes, I do. Something's not adding up here, and I've got a hunch I know what this is about. If I'm right, I'm gonna be pissed.\"\n\n\"A hunch. Somewhat like Colonel Carter or Daniel Jackson's estimations?\"\n\n\"Possibly, but I don't think we're in any danger. Let's move out.\"\n\nThough none of them had watches with which to measure time, O'Neill sent each member of SG-1 in a different direction with orders to return within what 'felt like' thirty minutes. Teal'c volunteered to survey the plains beyond the flattened hill.\n\nAs he cautiously made his way past the scattered remains of broken rock and debris, he searched for any sign that the Stargate might be easily unburied. The possibility of retrieval became less and less likely as he walked by boulders as large as O'Neill's truck scattered across the area. Though he regretted admitting it, even he would not be strong enough to move them out of the way.\n\nHe walked beyond the destruction, stepping over smaller rocks that littered the landscape. Many of them seemed smoother, as if worn away by the action of water. Perhaps Colonel Carter could explain how the hill's collapse could cause such a phenomenon.\n\nUpon reaching safer terrain, Teal'c picked up his pace. He jogged across more even ground and long-bladed grasses, heading toward the tree line at the field's end. As he reached the woods' outer edge, a glint of silver flashed from between the deeply fissured tree trunks that populated the woods. Teal'c dropped to his knees. A Jaffa's armor reflecting against the sparse sunlight may have been the cause. Another flash prompted him to creep inwards, remaining low to avoid detection.\n\nHe heard no sound of birds or wildlife. There were no footsteps save his own soft tread. Often, low hanging branches studded with red conical-shaped flowers impeded his momentum. He knew he should turn back, having far exceeded O'Neill's thirty-minute window, but what he could discern from the flashes might be critical to SG-1's survival. He proceeded forward until he came to the edge of a cliff and looked out.\n\nIn the canyon below, a multi-spired citadel sprawled for many miles. At its center stood a single oblong tower, built of metal and glass, undoubtedly a good half-mile high. Sunlight glistened off its windows. Though Teal'c was too far away to be certain, he believed he saw movement along the citadel's outer stone wall, a ten-foot high ring of white stone polished to such a sheen that the neighboring lake shimmered against its sides. He counted six terraces around the city's outer buildings, each formed of colored glass panels cross-braced with steel beams.\n\nBe they friend or foe, whoever had accomplished such an engineering feat would certainly have the means to assist SG-1. Teal'c hurried back to his friends, hopeful that O'Neill's hunch was right and that such assistance would come freely and not at a price too high to pay.\n\n\"A penny for your thoughts, Carter.\"\n\nSam suppressed a grin. Whatever had troubled General O'Neill earlier seemed to have flown out the proverbial window. He was practically jovial, squatting down beside her, chewing innocuously on a blade of grass.\n\nThey'd taken up position behind a bush some fifty yards back from an archway carved into the wall surrounding the city Teal'c had discovered. From what she could see so far, the people living there had to be extremely advanced. There was no other way to explain the slender, graceful buildings or the cable-less cars sliding up their sides. What she wouldn't do for a \u2014\n\n\"Carter?\"\n\nSam rocked back on her heels and sighed. \"To be honest, sir, I just wish I had some of my equipment.\" She glanced up at the sun. It hadn't really changed altitude since they'd arrived on the planet. \"A spectrometer would be nice. The light's \u2014 \"\n\n\"A P90 might be useful.\" He plucked another blade of grass.\n\n\"Sir, you probably shouldn't sample the local vegetation. We've no way of knowing if it's poisonous or not.\"\n\n\"Oh, I don't think we're in any trouble from that quarter.\" He waved the blade toward the archway. \"We've got company.\"\n\nTwo men dressed in belted beige-colored tunics, off-white trousers and matching shirts took up guard on either side of the arch. \"It's tough to see this far off, but I think they're wearing side-arms, sir.\"\n\n\"You think we should waltz up to them and ask for help?\"\n\nSam shrugged. \"I don't see any other choice.\"\n\n\"Certainly our appearances do not seem threatening,\" Teal'c said, crouched on her other side.\n\nBehind her, Daniel handed his watch cap to Teal'c. \"It wouldn't hurt to be a bit less conspicuous.\"\n\n\"So what's your plan?\" the general asked as Teal'c positioned the hat over his gold serpent tattoo.\n\nSam faced the general, surprised. \"My plan, sir?\"\n\n\"At the moment, Colonel,\" he pointed at her, \"you're SG-1's commanding officer. I'm just a lowly general in your keeping.\"\n\n\"Yes, sir.\" She squared her shoulders. \"Well, since this is a first contact \u2014 \"\n\nDaniel tapped her on the shoulder. \"Your resident archaeologist-slash-linguist should take lead.\"\n\nThe general snorted. \"Why am I not surprised?\"\n\n\"Daniel's right, sir. We've no way of knowing if they're friendlies or not, and he's better at playing ambassador than \u2014 \"\n\n\"I get it, Carter. You don't need to explain.\" The general glanced back at Daniel. \"At least, not yet.\"\n\nThankful for General O'Neill's support, Sam studied the city guards. They were armed, SG-1 wasn't, and unless she wanted to wait until dark and sneak in, there really wasn't anything they could do to change the situation. Realizing she was putting off the inevitable, she gestured forward with two fingers. \"Daniel, you're up. Teal'c, take our six. General O'Neill?\"\n\n\"Yes, ma'am?\" The general tossed aside his blade of grass.\n\n\"You're with me.\"\n\nAt Daniel's suggestion, they rose slowly from their position behind the bushes, arms raised.\n\nThe guards didn't move.\n\n\"Hello!\" Daniel strolled ahead. \"We're from Earth. And to be honest, we could use your help.\"\n\nEyes forward, arms resting at their sides, the guards remained motionless. Sam held her breath as they closed the distance to ten yards. She smiled to show they were friendly.\n\nDaniel held up his hand, signaling an all-stop. With a nod from Sam, the general and Teal'c came to a halt. Daniel continued on, walking up to the guard on the left. \"My name is Daniel. This is Samantha Carter, Teal'c, and Colonel O'Neill.\"\n\nThe general coughed none too subtly into his hand.\n\n\"Sorry,\" Daniel said. \"I mean General... General O'Neill.\"\n\nNeither guard moved a muscle.\n\n\"Why do they not respond?\" Teal'c asked.\n\nDaniel narrowed his brow. \"Maybe they're trained that way. As long as we don't present a threat, they'll ignore us.\"\n\nSam stepped up to the other guard. \"We'd like to talk to your leaders, if \u2014 \"\n\n\"Forget it.\" The general strode up and snapped his fingers in the guard's face twice. No response. The man didn't blink. \"These guys would put Buckingham Palace's guards to shame.\"\n\nTeal'c cocked his head. \"Buckingham Palace?\"\n\n\"It's in England,\" Daniel explained. \"The Queen's guard isn't allowed to react to the tourists unless there are extraordinary circumstances.\"\n\nThe general waved a hand in front of the guard. \"I guess we're not that extraordinary.\"\n\n\"Indeed.\" Teal'c leaned into the archway. \"There are many people within the city. Perhaps they will consider us more \u2014 \"\n\n\"Extraordinary?\" Sam gestured for the general and Teal'c to close ranks. \"Guards may be one thing, but there's no way to know how the city's inhabitants will react. I recommend proceeding with caution.\"\n\nDaniel took the lead again and they passed through the archway. They came out onto a wide avenue running parallel with a series of meter-high granite obelisks. Rows of purple flowers \u2014 reminiscent of lilacs \u2014 lined each side of the street. Though there wasn't anyone in the immediate area, a glance upwards revealed locals walking across various bridges leading in and out of the central tower. Some were dressed in pale flowing robes. Others wore outfits similar to the guards by the arch.\n\nSam inspected the tower's base. Unlike the broad top levels, the narrow bottom was made of brownish stone. One end appeared to be open with a pulsating light emanating from inside. She clenched and unclenched her hands, wishing yet again for the proper equipment.\n\n\"Here's someone,\" Daniel said.\n\nA white-haired woman dressed in long robes headed their way. Sam guessed she was in her late sixties. She carried a metal case, about the size of a shoebox, and as she neared, Sam noticed floral garlands wrapped around her wrists. The woman strode with a grace that belied her apparent age, her robes flapping behind her.\n\nDaniel hurried over. \"Hello, I'm Doctor \u2014 \"\n\nThe woman walked on by, ignoring all of them.\n\n\"I guess you're not her type,\" the general jerked his head in Teal'c's direction. \"Here come two more. Be nice, Daniel.\" A man and woman, both dressed in the same military garb, marched toward them. As they neared, Sam stepped off to stand by one of the obelisks, careful to not step on the flowers lining the walkway. They really did remind her of lilacs though they didn't carry the same heavy scent.\n\n\"Excuse me. Hello?\" Daniel called out to the approaching couple. \"We've come through the Stargate.\"\n\nThey kept walking, not batting so much as an eyelid.\n\nAs they passed Teal'c, he offered, \"Perhaps their language is dissimilar, Daniel Jackson.\"\n\n\"Good point.\" Daniel shoved his glasses further up the bridge of his nose. \"Chappa'ai?\"\n\nNothing. They breezed past him like he was nothing more than an ant on the sidewalk.\n\n\" _Anulus? Circacona_?\" He dropped his hands. \" _Porta_?\"\n\n\"Enough with the playing nice.\" General O'Neill darted out in front of the couple and raised his hands. \"Listen, folks \u2014 \"\n\nThe man strode right through the general without so much as a ripple.\n\nHe continued on, the woman at his side. The general whirled toward Sam, the white scar on his eyebrow stretched tight. \"What just happened?\"\n\n\"I don't know, sir.\" Though it would explain why the flowers didn't put out any scent. She bent down to pick one.\n\nHer hand passed through it. \"Maybe it's some kind of hologram technology?\"\n\nDaniel walked over to another obelisk and waved his hand. It went directly through. He headed around to the other side.\n\n\"Perhaps ghosts?\" Teal'c added.\n\n\"Ghosts, my ass.\" General O'Neill pointed toward the departing couple. \"They've got something to do with this. The lost memories, the gate. The whole shebang.\"\n\n\"Maybe...\" Daniel mumbled from behind the obelisk. \"Guys, I think I found something.\"\n\nSam hurried over to join him. Rectangular rows of block-like glyphs covered the obelisk's back end.\n\n\"I have seen this language before,\" Teal'c observed.\n\n\"You should have,\" Daniel said. \"It's Ancient.\"\n\nGeneral O'Neill took a step back from the obelisk. \"Ancient as in 'old' or Ancient as in \u2014 \"\n\n\"The Ancients.\" Daniel pointed to a lower section of the text etched into the granite. \"It's an older dialect. Similar to the language we found on Heliopolis.\"\n\n\"Ernest Littlefield's planet.\" Sam glanced back up at the traffic on the bridges outside the central tower. Ghostly foot traffic, she reminded herself. \"Can you translate it?\"\n\n\"Huh.\" Daniel's head jerked up. \"That's interesting.\"\n\n\"What?\" the general demanded. \"Daniel, so help me god. If you tell me I have to stick my head in another brain-sucking machine to get us out of here, I'll quit. I'll resign my \u2014 \"\n\n\"I didn't mean that sort of interesting. Although, really? You'd quit?\"\n\n\"Daniel...\" The general growled.\n\n\"Just kidding. Sort of.\" He pointed toward the bowed top of the stone where the Ancient glyphs were twice the size as the lower text. \"According to this, the planet or the city, I'm not sure which, is named Pedion Elysium.\"\n\n\"Like Elysium Fields?\"\n\nSurprised, Sam raised an eyebrow in his direction.\n\n\"I do read, you know. _Henry V_. Act four. Scene one.\"\n\n\"William Shakespeare,\" Teal'c said. \"'What infinite heart's-ease must kings neglect that private men enjoy.'\"\n\n\"Yeah, that's the one.\" The general's grin disappeared.\n\n\"You can remember Shakespeare, but not how we got here?\" Daniel asked.\n\nGeneral O'Neill crossed his arms. \"Can you?\"\n\n\"No. But \u2014 \"\n\n\"Then don't mock, Daniel. It doesn't suit you.\"\n\nPushing aside the idea that General O'Neill and Teal'c both read Shakespeare, Sam examined the stone's lettering more closely. \"What does this Elysium have to with the Ancients?\"\n\n\"Actually, it makes sense.\" Daniel waved toward the flowers at the base of the obelisk. \"The lilacs. The woman with garlands on her wrists. In Greek mythology, Elysian Fields. Or Elysium if you use the Latin derivative like \u2014 \"\n\n\"Daniel,\" the general warned. \"Cut to the chase.\"\n\n\"Right.\" Daniel ran a hand through his hair. \"Elysium was considered the hero's final resting place. The Greek poet Pindar referred to Elysium as an island.\" His eyes went wide. \"Now that I think of it, Pindar's account mentions a Kronos.\"\n\n\"We killed that slimy snakehead off years ago.\"\n\n\"I don't think it's the same, Jack. In this case, Kronos could be connected to the Hellenistic embodiment of time.\"\n\n\"Time. As in?\"\n\nDaniel gestured to a lower passage. \"In this case, more like...\" His fingers traced along a line of text. \"More like slipping free of time. This place, Pedion Elysium...\" He whirled toward the general. \"'A final resting place!' I think this might have been one of the last colonies of the Ancients before they ascended.\"\n\n\"You mean like Vis Uban?\" Sam asked. \"How does that explain how we got here?\"\n\n\"Or why the people appear as ghosts,\" Teal'c added, \"unaware of our presence.\"\n\n\"Jack, don't you get it?\" Daniel excitedly raised his hands toward the walkways above. \"If this is \u2014 was \u2014 one of the last pre-ascension colonies of the Ancients, your Ancient genetics might have something to do with how we got here.\"\n\n\"And that's supposed to make me happy, how?\" Pissed, Jack turned toward the central tower. What Daniel said made sense. It confirmed his suspicions. Underlined his resentment at having that damned ATA gene.\n\nSure, being able to save Earth's butt had its benefit. But this time, thanks to his grandparents' great-grandparents' heritage, they were somehow stuck on a god-forsaken planet with a bunch of walking Ancient holo-ghost-whatevers. The whole thing made him want to lie down, just turn his brain off, and take a nap.\n\n\"Help! Somebody? Please!\"\n\nThe cry for help came from the other side of the wall. It sounded like a woman, but he couldn't be certain. Without another thought, Jack broke into a run with SG-1 hot on his heels. They reached the arch and he jerked up his fist, signaling the team to halt. He peered through the arch and saw a small-built woman with a long black ponytail. She ran parallel with the outside wall, away from their position.\n\nWhoever she was, her get-up was pretty much like his, except her pullover was red, not green. Could be a friend. Could be a foe. He had no way of knowing. He pulled his head back and gestured toward Carter and Teal'c to take the far left. With a nod back to Daniel to follow, he slid out the right side and hugged the outer wall.\n\n\" _Yo ren ting jian wo ma_?\"\n\nWhat the hell?\n\nThe woman turned around and froze when she caught sight of Jack and the others.\n\nHigh forehead. A narrow, but square jaw. She wasn't so much a woman as a barely out of teenaged youth.\n\nHe wasn't sure how, but he knew this girl.\n\n\"You!\" He stormed toward her. \"You know what's going on here, don't you?\"\n\n\"Jack!\" Footsteps behind him announced his conscience coming to tell him to back off.\n\nThis time he refused to listen.\n\n\"No, Daniel.\" He grabbed the girl's elbow. \"The gate's gone so how did you get here?\"\n\n\"I don't know!\" The girl started to do the very thing Jack hated more than anything.\n\nShe started to sob.\n\nHe counted to three, letting his temper simmer down.\n\n\"I think you do know,\" he finally told her. \"I don't know how, but I've seen you before. And, I think you know who we are, too.\"\n\n\"Sir...\"\n\n\"Not now, Carter,\" he said, gritting his teeth. \"I want the truth, damn it. I'm tired of this bullshit.\"\n\n\"I told you,\" the girl yelled, breaking free from his grip, \"I don't know!\"\n\nThe ground started shaking again.\n\n\"O'Neill \u2014 \"\n\n\"Yeah, I feel it.\" But he really didn't care. Not anymore. He seized both her arms and held on. \"Who the hell are you and how the hell did you get here?\"\n\nThe rumbling intensified beneath his feet. Walls cracked. He thought he heard glass shatter.\n\nThe girl jutted out her chin. \"I. Don't. Know!\"\n\nA boom shot out, echoing across the canyon. The wall to Jack's right shattered into a spray of rock, spewing toward them with ferocious velocity. As he yanked the girl out of the way, a second boom shook the ground, tossing him off his feet. With a hand over the girl's head, he kissed the dirt and hoped for the best.\n\nIn less than a heartbeat, the tremors stopped. He waited another moment just to be sure.\n\n\"It's over, Jack.\" A hand on his arm. Daniel. \"You better take a look.\"\n\nJack released the girl and climbed to his feet. Daniel was pointing back toward the city or, rather, where the city had once been.\n\nThe canyon was empty. Rippled waves of heat rolled across a barren wasteland populated only by yellowed grass, dirt and dust. No, Jack realized, that wasn't quite true. Squinting against the hot sunlight, he made out a small brownish building, no more than a story high. Maybe a half a click off.\n\n\"Was that not the central tower's location?\" Teal'c asked.\n\n\"You see it, too?\" A glance at Carter gave Jack the nod he needed to confirm he wasn't certifiably nuts.\n\nMovement next to the building caught his eyes. The outline of a person. Maybe a man. Too far off to tell. Whoever it was, he \u2014 or she \u2014 waved their arms. A bit of cloth flapped in the breeze. A flag of some sorts. Or a robe.\n\nThat damned breeze.\n\nA sudden chill seized Jack's bones and he tried remember.\n\nNo. No trying.\n\nHe strode up to the girl. \"You're that trainee from the outpost, Weiyan \u2014 \"\n\n\"Weiyan Shi,\" the girl said. \"Why are we not in Antarctica?\"\n\n# CHAPTER EIGHT\n\n### ANTARCTICA\n\n_50 years earlier..._\n\nHuang crept toward his prey, resisting the urge for self-ridicule. As First Prime of Lord Yu, he would have sought greater enemies on the field of battle. The Jaffa armies of Sokar, Apophis, even Ra would have trembled as he led the Dragon Guards against them.\n\nInstead, his life had been reduced to exile within the frozen wasteland of the Tau'ri's southern pole. Worse, his head and heart warred over whether it was just to kill the creature before him. The odd black-and-white bird waddling before him walked more like an old man than a bird in search of flight. Its orange-lined black beak crooned piteously as if calling out in loss. To kill such a creature felt wrong.\n\nBut necessary. Hunger gnawed at Huang's belly as he watched the bird cease its cries. It stopped walking, preening the dense white feathers on its chest. Two days had passed since losing his satchel. Two days with no food and little sleep. A blizzard had hit only hours after losing his gear, forcing him to find refuge in an ice cave. For hours he'd laid within the rippled blue ice, wrapped only in his cloaks. On occasion, his eyes would play tricks on him, casting golden-white tendrils of light across the cave's floor. He would squeeze his eyes shut against the mirage, his mind rolling over the great failure that had led to his plight. Staving off self-pity, he strived to find hope, or at least the chance of it in the future.\n\nUpon the storm's end, he had ventured out again to follow the great glacial cliff along the frozen sea. He felt for all the world as if he stood alone on the Tau'ri's planet. He was not fool enough to believe Lord Yu would send rescue. For all his master knew, he was dead. If Huang did not find true shelter from the cold soon, that would surely be his fate. Several of his toes had begun to blacken with festers swelling near the tips where his nails had worked loose.\n\nBut first, he must eat. Even if it meant killing the man-like bird to do so. Less than ten paces from what would become his next meal, Huang thumbed the staff weapon's trigger. The firing head opened, releasing a short crackle of energy as it charged.\n\nThe bird's head swung up. A single caw erupted and black eyes darted in Huang's direction. Its oblong wings flapped though no flight occurred.\n\nA single shot blasted off its head.\n\nUsing his staff weapon, Huang hobbled over to the dead bird. Blood oozed from its severed head. He pulled out his belt-knife and sliced the creature's breast open, revealing a thick layer of fat. The pungent stench curdled in his nose, smelling more like fish than fowl.\n\nBreathing only through his mouth to fight off the stink, he sliced down further, revealing tender red meat underneath. He carved away the fat, filleted the breasts and withdrew them from the carcass. Flipping the bird over, he took a half-step back and blasted its feathers with the staff weapon. Over the ensuing embers, he roasted the meat on his knife's point.\n\nHe bit into the barely seared meat and immediately spat it out, overwhelmed by the rank taste. Shoving down the bile that threatened to rise, he forced himself to eat the entire thing.\n\nAs he consumed the putrid meal, disgust crossed his mind. A reflection of how purposeless his life had become.\n\n_Here sits the once-future First Prime of Lord Yu, relegated to eating whatever vile sustenance I can find_.\n\nA glimmer of golden light flashed to his far right. At first he believed it only a trick of his eyes, teary as they were from the smell. When the light flared again, he rose up from his meal and turned toward its source.\n\nNothing. The light had disappeared. In the far distance, many days' walk from his position, smoke bellowed from a high mountain. With reluctance, he admitted to himself that the smoke did not come from the same location he'd seen days ago. The earlier storm must have forced him off-course.\n\nEven so, where there was smoke, there would be Tau'ri. Huang heaved a heavy sigh. He knew not what he would do upon meeting them, but if Lord Yu cared so greatly for these people, Huang had to believe they would welcome his presence.\n\nThe golden-white light flashed again, emanating from beyond the next bend in the glacial cliff. Ignoring the pain in his feet, Huang ran until he reached a tattered red flag attached to a pole wedged into a stone cairn.\n\nThe light was still there, hovering a mere hand's length above the wedged-together pile of rocks. Huang cautiously approached. The cairn was no more than a few paces wide though its top stones were higher than his head. Too small to be the burial place of any Tau'ri, he knew not what the cairn contained.\n\nHe picked up one of the rocks, its surface smooth. Weatherworn, he realized. Glancing up at the light, Huang noticed long white tendrils extending from its center.\n\n\"Thank you,\" he said, though the very idea of speaking to a light felt foolish. \"Are you a friend of the Tau'ri?\"\n\nThe light ascended from the cairn, its tendrils folding in along its central column. Huang stumbled back in fear that he'd upset the strange being.\n\n\"Are you foe?\" he dared asked.\n\nThe light flared bright gold, and then, disappeared.\n\nHuang returned his attention to the cairn. Grabbing stone after stone, he pulled down one wall and then another. Several bundles nestled between the stones, covered by what appeared to be gray woolen blankets. They would help keep him warm on his journey. Lifting up a blanket, he discovered that it was fashioned as a pairs of pants. Relief washed through him. The added clothing would aid his climb up the mountain to the Tau'ri. He gathered up the woolen clothes.\n\nA glint of metal beneath them caught his attention. Pulling back more of the woolen items, he found small, colorfully wrapped packages. Some shaped like blocks, covered in what seemed thin metal paper with strange letters printed on top. Others were cylindrical, bright red containers with black lids. More writing appeared on their sides as well. Pulling out the containers, he discovered even more packets. Of the items present, the largest pile was of long, thin bars, each swathed in more layers of the metal paper.\n\nHe unwrapped one, discovering inside a dark brown solid substance, squarely scored. Though the bar was very cold, the metal paper must have insulated it well enough to escape freezing. He broke off a piece. Sniffed it. A musky aroma with a sweet edge. He bit into it.\n\nIt was food!\n\nIn excitement, he ripped open more packages. There was hard bread, a fatty meat, and lumps of white crystals tasting very sweet.\n\nBiting off another piece of the brown bar, he searched for the billowing smoke from the mountaintop. It was gone!\n\nFear gripped his stomach. Had the light destroyed the Tau'ri? Was that why it had encouraged him to pillage such a vital resource?\n\nHuang swallowed hard, the Dragon Guard trainee within reasserting itself. Lord Yu must be warned. If the light was at war with the Tau'ri, only the great System Lord would know how to combat such a force.\n\nGrabbing as much of the supplies as he could carry, Huang retreated to the cave in which he had taken refuge. There he donned the woolen clothes. Fed, warmed, and alert, he picked up his staff weapon and headed forth to retrace his steps to the Chappa'ai. He would find the dialing device, certain it must be buried somewhere within the cavern's ice-covered floor. From there, he would travel to one of his master's protected worlds and send work back to Lord Yu. In doing so, perhaps the Goa'uld would see the success in his endeavor, and not his failure.\n\nAs he strode past his earlier hunting place, snow began to fall.\n\n# CHAPTER NINE\n\n### PLANET DESIGNATION: UNKNOWN\n\n### STATUS: UNKNOWN\n\n### TIME: UNKNOWN\n\n\"Why are we not in Antarctica?\"\n\n\"Good question.\" Jack broke off staring at the girl long enough to get the lay of the land. A warm wind blew against his face. Dust and dirt swirled across drab, lifeless plains, the only color provided by SG-1's green pullovers, and the red one worn by the girl.\n\nA kid. Barely out of her teens. Yet somehow, she had something to do with whatever in the hell was going on.\n\n_One piece of the puzzle down, too many more to go_.\n\nMovement way off in the distance caught his eye. Someone was definitely waving from the one surviving piece of a city that had disappeared into thin air. Jack was pretty sure the building had been the base of the city's central tower.\n\nThere was something familiar about whoever it was, something about the long, flowing white robe they wore, too. Jack squinted, feeling like he should take action. Like he should know what to do.\n\nWhat he wouldn't give for a P90.\n\n\"O'Neill,\" Teal'c said, pointing toward the far off building. \"Should we not approach?\"\n\n\"We should do something,\" Daniel said. \"Whoever it is, maybe they can help us get home.\"\n\n\"Yes, please,\" Weiyan whispered. \"I want to go home.\"\n\nJack looked back at his team. Daniel had draped a protective arm around Weiyan. The girl was practically shaking in her boots. Carter flashed one of her A+ smiles of reassurance.\n\n_She should really bottle that smile up. She'd make a mint_.\n\n\"Plans, sir?\" Carter turned toward Jack, the smile gone.\n\nAll of them turned expectant gazes toward him, waiting for him to make the first move. Say the right thing. Make a decision.\n\nA wave of fatigue hit him. He stuck a hand in his pocket, wrapping his fingers around the Zippo. Another familiar thing, though he hadn't a clue how it got there.\n\nIt wasn't as if he didn't miss what the lighter represented. He did. Skaara's death had been tough to take, but staying ahead of the Goa'uld this past year had been too big a distraction.\n\nHell, just staying alive had been a distraction.\n\n\"Jack?\"\n\n\"Yeah, Daniel?\" The words came out barely more than a croak. His throat was dry, tight.\n\n\"General, shouldn't we \u2014 \"\n\n\"Just... Just give me a moment, Carter.\" The wind felt wrong somehow. It tugged at his clothes, coming from too many directions. Hell, the entire situation was wrong.\n\nThe ground rumbled faintly beneath Jack's feet and he knew.\n\nHe knew it was up to him to remember. Remember how they got there. For all their sakes.\n\nHe looked again at what he was pretty sure was a man standing by the building. Turning back toward the disheveled surroundings, he felt everyone watching him, waiting. The weight of their expectations pressed down hard and a private flare of reproach welled up in response. There'd be time enough to deal with self-pity later.\n\nA soft rumble started up again. Low, almost a growl. Another piece of a puzzle he couldn't fit into place.\n\nShrugging off his uneasiness, he said, \"All right, SG-1. Let's go make nice with the native.\" He turned back toward the building.\n\nThe man was gone.\n\n### MCMURDO STATION\n\n### ROSS ISLAND, ANTARCTICA\n\n### 18 AUG 04\/1530 HRS MCMURDO STATION\n\nPaul Davis had just hoisted another injured civilian into the awaiting Bell 212 when the floodlights across the central compound kicked back on. A good thing, too, since the chopper was parked so close to Cray Lab that one of its blades nearly touched a neighboring electric pole. Sliding the side hatch shut, he waved at the chopper pilot and gave the go.\n\nThe chopper lifted off the ground, inched forward away from the pole, and then headed toward Pegasus Air Field where the civilian would be evacuated on a C-17 to New Zealand. Paul watched as the blue neon lights along the chopper's tail and blades illuminated the last dull gray night of Antarctica winter.\n\nThe sun would rise tomorrow. A whole hour and thirty-nine minutes of sunshine. As much as the floodlights helped in finding survivors, Paul knew the sunlight would bring disadvantages along with advantages.\n\nIt'd be easier to find the dead.\n\n\"We found another!\"\n\nStation personnel ran past, heading toward a collapsed smaller lab down the hill. With a sigh, Paul took off after them. He immediately kicked himself when his cold breath stung his face. A fur-lined parka wasn't enough in the sub-zero temperatures, not when minus twenty degrees Fahrenheit was the norm. He should have pulled on a pair of coveralls. Tugging his balaclava back down over his face, he found himself grateful for at least some protection.\n\nHe arrived at the collapsed building as an injured civilian was being put on a gurney. The man's face and red beard were covered in blood. Knowing instantly who it was, Paul's heart dropped. He stepped up to the gurney and looked down into the pained face of Dr. Malan.\n\n\"Major Davis...\" Malan coughed, spitting up more blood. \"Where's Mr. Murray? Is he all right?\"\n\nSqueezing the young scientist's hand, Paul lied. \"He's around here somewhere. Don't worry about him.\"\n\n\"He's gonna miss bowling. He's gonna miss meeting...\" Malan dropped his head to the gurney and closed his eyes.\n\nOne of the volunteers put a finger on Malan's neck and checked his pulse. \"He's alive, barely.\"\n\nPaul released Malan's hand and stepped back. \"Where will you take him?\"\n\n\"Byrd Station's a good 1400 kilometers away,\" the volunteer said as he strapped Malan to the gurney. \"But it beats going back to New Zealand. He's got internal injuries so we better get him up to the airfield quick. Excuse us, Major.\"\n\nPaul said a silent prayer as they carried the gurney toward another waiting chopper. The chances of Malan surviving the trip to Williams and then on to Byrd weren't good.\n\n\"Major Davis!\" A parka-clad woman ran up to him, her face buried inside her hood.\n\n\"Here!\" Paul stomped his feet against the cold as she approached. A black patch on her chest labeled 'Operation Deep Freeze' let Paul know she was part of the science community. \"There's a call for you on the long-range radio. A General Hammond.\"\n\n\"Which way?\" Of the hundred or so buildings that made up McMurdo, more than half had collapsed in the quake.\n\n\"The NSF chalet. Edmunds's office.\" She pointed up the hill toward the Swiss-style wood building.\n\n\"The chalet didn't suffer any damage?\"\n\n\"Just don't use the front entrance,\" she shouted as the chopper bearing Malan headed off. \"The doors are wedged shut from the seismic activity.\"\n\n\"Thanks.\" Paul hurried up the hill, allowing himself a brief head shake. Seismic activity? More like a pancake flattener.\n\nOnce inside the chalet, he rushed down the hall to Edmunds's office, picking his way over fallen ceiling tiles, reams of papers strewn everywhere, and several broken bulletin boards. Finally, he found the NSF Director's office. The door was half off its hinges.\n\n\"I don't give a damn about the military's special projects,\" Edmunds bellowed into the radio, his back to the door. \"You people are here to support the NSF. Now get down here and \u2014 \"\n\n\" _And I don't give a damn whether you care or not_.\" The radio's crackle did little to hide General Hammond's Texan twang. Or his obvious impatience. \" _The President has authorized_ _ \u2014_ __ \"\n\nEdmunds switched off the radio. \"Ha, like I care.\"\n\n\"Sir...\" Paul cleared his throat and stepped inside. Edmunds eyed him without any warmth. The NSF Chief of Operations for McMurdo was much loved by the civilian population for his anti-military stance.\n\nEdmunds pushed back his chair and stood up. All six-and-a-half feet of him. Paul recognized it for the intimidation effort that it was so he kept his face neutral.\n\n\"I suppose you want to talk to that asinine jarhead?\"\n\n\"Yes, sir.\" Paul didn't bother pointing out that Edmunds's use of a derogatory term for the Marines wasn't exactly relevant apropos a USAF general.\n\nEdmunds stepped out from behind his desk. \"Then do it, but hurry things up. We need to keep the channel clear.\"\n\n\"Thank you.\"\n\n\"I'm not stupid, Major.\"\n\n\"No one ever said you were, Doctor.\"\n\n\"Earthquakes rarely hit this part of the world and when they do, they're small in scale.\" Edmunds waggled his finger. \"You military folks are up to something.\"\n\n\"Sir \u2014 \"\n\n\"I wasn't here back in March,\" Edmunds said, \"but I've heard enough from the winter staff to know something happened. Something big. Strange lights in the sky, F-302s shooting off from behind Observation Hill like it was the end of the world. And let's not forget all the new Air Force personnel coming through here ever since \u2014 \"\n\n\"Need to know, sir.\"\n\nEdmunds glowered, his mood coalescing into barely contained fury. \"You're going to tell me I don't need to know, when we've just lost seventeen people?\"\n\nPaul slid past him and picked up the mic. \"Sir, I'll need the room.\"\n\n\"Five minutes, Major.\" He strode toward the door.\n\nPaul waved the mic as a reminder. \"Sir, the sooner I talk to the general, the sooner I can rejoin the others and assist in the search and rescue of your personnel.\"\n\n\"Make sure that you do.\" Edmunds stomped off.\n\nPaul sucked in a breath and counted to five. He blew out and thumbed the microphone's talk button.\n\n\"This is Major Paul Davis for General Hammond. Over.\"\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/1610 HRS MCMURDO STATION\n\n\" _...the entire continent's been struck, sir. I'm hearing reports of anything from a 7.1 to a 7.4 scale earthquake_.\"\n\nAs George listened to Major Davis' report, he glanced up at the outer chamber's concave ceiling. Striated stone blended with what looked like frozen waterfalls. Best he could tell, the Ancients had built the place to last. The roof hadn't cracked or suffered any other damage.\n\nHe thumbed the radio mic. \"How many causalities, Major?\"\n\n\" _At McMurdo? Seventeen plus another ten or so injured. We were lucky, General. Only two hundred winter here besides our F-302 pilots_.\"\n\n\"And the pilots are fine?\" George held his breath, hoping for the best.\n\n\" _All present and accounted for, General. We lucked out, but I received a report a short while ago that a wing led by Captain White completed an easterly flyby, and... Sir, New Zealand's Scott Base flattened. No survivors_.\"\n\nGeorge took in the bad news, steadying himself against the pallet of crates where the radio had been set up. Davis detailed the status of McMurdo and signed off to go assist with recovery efforts. As George put down the radio mic, someone stepped up beside him, a cup of coffee in hand. He took the proffered cup and sipped greedily.\n\nHell, by this point he should have just asked for the whole coffee pot.\n\n\"General Hammond?\"\n\nDr. Lee stood at the pallet's edge, blinking rapidly behind his glasses. Beside the stocky scientist was a face George hadn't seen in several years.\n\n\"It's good to have you here.\" George smiled grimly at Lieutenant Graham Simmons. \"The coffee's much appreciated.\"\n\nThe young man returned the smile. \"It's good to see you, too, General.\"\n\n\"I sort of wanted some coffee, too,\" Lee said, indicating George's half-empty cup.\n\n\"There's more over by the elevator.\" George waved his cup toward the cage conveyor where someone had been smart enough to set up a coffee and food station. While the trainees had been relegated to the break room, the diplomats hovered in the main chamber by the coffee, anxiously waiting for an available chopper to airvac them back to McMurdo.\n\nLee glanced over at the diplomats and shrugged. \"Uh... Maybe later. I wanted to give you a preliminary report.\"\n\nGeorge put down the coffee. \"Go ahead, Doctor.\"\n\n\"Actually, some of this will be easier if I just show you.\"\n\nLee shambled off toward the weapons platform chamber.\n\nGeorge followed with Simmons at his side. As they passed the de-activated Ancient stasis unit, he pushed down ten tons of regret. He'd been powerless to save Jack the last time an Ancient contraption had got ahold of him. This time, George would move Heaven and Earth if need be.\n\nInside the chamber, Lee grabbed two military-issue green parkas hanging off a trellised panel. He handed one to George. \"You'll want to put this on. We've switched off the heaters in the chair room.\"\n\nThe room was already barely above freezing. George offered his parka to Simmons. He'd need it more, dressed only in a standard uniform.\n\nSimmons waved him off. \"I'm fine, sir.\"\n\nLee zipped up his parka. \"We were hoping the colder temperature might convince the chair to shut off.\"\n\n\"Convince?\" George shrugged on his parka, but kept it open. \"Dr. Lee, are you trying to tell me the chair, this weapons platform \u2014 ? Is it sentient? Is the chair the reason we can't get to our people?\"\n\n\"Honestly, we've no way of knowing.\" Lee circumnavigated the pillars bordering the hole. Someone had installed a walkway plank over the force field leading from the weapons platform to a bank of three monitors on the chair's left. A low-level hum emanated from the Mark II generators.\n\nGeorge glanced at the force field, hoping for even a flicker that might let him see down below.\n\n\"Do you think they're all right, General?\" asked Simmons, joining him by the hole's edge.\n\n\"I certainly hope so, son.\" George pulled himself away and joined Lee by the monitors. \"Why haven't you turned off the generators?\"\n\n\"We considered doing that, but I was worried we might damage the chair if we shut the generators off without shutting down Colonel Carter's power filter program.\"\n\n\"Forget the chair and forget the program, Doctor Lee, just shut those damn things off!\"\n\nLee bent down and switched off the generators. The hum disappeared instantly, but the chair remained lit up.\n\nAnd the force field didn't so much as flicker.\n\n\"Yeah. I didn't think that would do much good.\" Lee returned to the monitors. \"I'm pretty sure the generators didn't activate whatever device set things off. I'd bet it has something to do with the chair.\"\n\n\"Is that possible?\" asked Lt. Simmons. Of his own volition, he'd taken up sentry duty by the hole.\n\n\"We think the trainee's genetics \u2014 \"\n\n\"Weiyan Shi.\"\n\nLee pulled his glasses off and frowned. \"Sir, as strong as General O'Neill's ATA genetics are, I've seen two stronger cases. Major Sheppard, who's now on Atlantis \u2014 \"\n\n\"We hope.\" He pushed aside his concern for the fate of the Atlantis team. \"Who's the other case?\"\n\nLee stuck his glasses back on. \"Weiyan Shi. She's stronger than Sheppard's and General O'Neill's combined. I'm willing to bet she's the one who got the chair to activate this Drift device.\"\n\n\"Drift device?\"\n\n\"As in Continental Drift. We think it's running right beneath the force field.\" On Lee's central monitor, he called up a map of the planet. \"Colonel Carter's report on the Proclurash Taonas mission \u2014 \"\n\n\"The planet where they retrieved the ZPM to power the weapons chair in the first place?\"\n\n\"That's the one,\" Lee said. \"The Taonas chair showed them a map of Earth from millions of years ago which looked like this.\" He tapped a key and the center screen displayed a planet with a single land mass.\n\nHitting another key, he animated the image. The land mass broke up into more recognizable pieces which slid apart into a facsimile of Earth today. \"Immense seismic activity caused Continental Drift and it's what we're experiencing right now.\"\n\n\"Earthquakes happen all the time,\" George said. \"All over the world.\"\n\n\"Normal earthquakes. The key word here is 'immense.' In the last hour, I've located an off-the-charts form of energy, right here in the outpost. Fifteen feet below this floor.\"\n\nThe doctor tapped another key. A new image came up. A graph with a bunch of squiggly lines peaking high above a red line toward the top. \"Whatever's been activated, it's emitting enormous levels of solar-photonic energy downward, right into the basalt layers \u2014 \"\n\n\"Basalt. You mean like the dolerite around the hole?\"\n\n\"That's one type, yes.\" Lee placed a sample of the grey rock on the table beside the monitors. \"Deeper down in the earth's crust, basalt magma is formed by decompression melting of the mantle.\" He raised a small hammer over the sample. \"The photonic energy emitted is applying tremendous pressures. Strong enough that if the force field is shut off while the device is still running \u2014 \"\n\nLee smashed the hammer down, splitting the rock into three uneven pieces.\n\n\"The force field is protecting the outpost?\" Simmons asked.\n\nGeorge frowned. \"The very force field we're trying to shut down is the reason the outpost hasn't collapsed.\"\n\n\"Exactly,\" Lee said, \"but the rest of the region wasn't as lucky, and further tectonic activity pushing against the basalt could eventually tear apart the plate. Scott Base collapsed because it sat closer to the plate's fault line than McMurdo.\"\n\n\"Sir, we should evacuate McMurdo.\" Simmons knelt beside the hole and peered down.\n\nLee joined the lieutenant. He stared at the hole and said, \"General, evacuating McMurdo will only prolong things.\"\n\nGeorge steeled himself, recognizing the preamble to something much worse. \"Go on,\" he said softly.\n\n\"We're pretty convinced the device gets its photonic energy from some sort of solar energy retrieval system.\"\n\n\"If you haven't noticed, Doctor, Antarctica hasn't seen the sun in months.\"\n\n\"That's true.\" Lee shoved his hands into his pockets. \"The lack of sunlight's probably why the tremors stopped for now. The device only had a residual charge leftover. The thing is... Twenty hours from now, the first real sunrise is due. When that happens, the quakes will start up again, and well, it's likely the seismic pressure will extend outward, resulting in quakes and tsunamis across the world.\"\n\nSimmons turned away from the hole and faced Lee. \"All because the ground is being bombarded with photonic energy?\"\n\nIt was then George realized that he'd heard the term before. \"Didn't the System Lord Yu retrieve some Ancient technology using photonic energy?\"\n\nLee picked up a stray pebble from the floor and dropped it into the hole. Predictably, the pebble bounced back. \"The Goa'uld used it for shields \u2014 \"\n\n\"And cloning,\" George said, still unhappy with the manner in which that particular SG-1 mission had ended. Though the key objective of rescuing Dr. Jackson from the Goa'uld had succeeded, Jack's decision to let Yu live hadn't gone over well with the President. \"What are our options?\"\n\nLee turned away from the hole. \"We need to find a way to alter the force field so we can gain access to that device and shut it down.\"\n\n\"Let's put one of the trainees back in the chair,\" George suggested.\n\nLee shook his head. \"Based on how this whole mess started, I'd like to explore some other options first. There might be a way to redirect the field, bend it so we can get to the device. I just haven't figured out how yet.\"\n\nGeorge found himself wishing \u2014 not for the first time \u2014 that Colonel Carter was up here and not down below. She'd have this entire mess figured out in an instance.\n\nWishes and horses, he reminded himself. \"All right, Doctor. Pull together whatever resources you can to shut that thing down. We've got less than twenty hours.\"\n\nDaniel had been so surprised by the man's disappearance that he accidentally breathed in some of the dust stirred up by the quakes. He began to cough uncontrollably.\n\nJack started to pound his back, but Daniel waved him off.\n\n\"We've seen this before,\" Sam said. \"On the Nox's home world.\"\n\n\"The Asgard have some sort of cloaking device, too,\" Jack added.\n\nDaniel swallowed a few times, finally calming down his cough. \"I don't think it's the Asgard, Jack.\"\n\n\"Because? From what Thor's told us in the past, the Asgard used to be buddies with the Ancients.\"\n\n\"This just doesn't seem like the Asgard's style.\"\n\n\"It's possible we're dealing with an entirely different race.\" Sam turned away from the building and cocked her head as if listening for any more tremors. \"Maybe they stumbled onto some kind of Ancient technology and don't know how to use it.\"\n\n\"Hence the tremors,\" Teal'c offered.\n\nSam shrugged. \"Without any equipment to measure seismic activity, it's impossible to tell if that quake was real or \u2014 \"\n\n\"Something we were made to imagine.\" Daniel sighed. A gust of hot wind stirred up the yellowed prairie grass, creating mirage-like waves that stretched out as far as he could see.\n\n\"Besides the sun, heat, and lack of snow and ice,\" Sam said, \"this place has a lot in common with Antarctica. Pristine terrains, windy air, both examples of nature in its extreme.\"\n\n\"But how did everything change so dramatically?\" Daniel asked. No matter which direction he turned, there was nothing to see. The city, the hills, even the trees were gone. The only thing left standing was the low building about a half-mile off, its dark brown stone facade swallowing the harsh sunlight \u2014 almost like a black hole.\n\nA whimper caught his attention.\n\nWeiyan had sunk to the ground. She'd tucked her head between her drawn-up knees and wrapped her arms around them. Her shoulders barely shook, but Daniel didn't need to be a mind reader to know how terrifying this must be. He knelt down beside her, placing a hand on her back.\n\n\"I didn't do anything.\" Her legs muffled her voice. \"I have tried to be good. I have tried \u2014 \"\n\n\"It's okay. We'll get out of here.\" He glanced up at Jack and then added, \"Somehow.\"\n\n\"Don't look at me.\" Jack shrugged. \"I've got nothing.\"\n\nBut Daniel did. He stood up. \"I think we do.\"\n\n\"Daniel...\" Jack narrowed his eyes. \"Are you carrying weapons I don't know about?\"\n\n\"No.\"\n\n\"Water?\"\n\n\"Well, no.\" Daniel swallowed again, wishing that wasn't the case.\n\n\"Food?\"\n\n\"No, but \u2014 \"\n\n\"Just stop.\" Jack raised a finger warningly. \"We have nothing.\"\n\n\"Pedion Elysium.\"\n\n\"'Pedy-my' beg your pardon?\"\n\n\"Pedion Elysium. It's a name, Jack!\" Daniel raised his arms to indicate their surroundings. \"If we have a name \u2014 \"\n\nSam shook her head. \"I'm sorry, Daniel, but I don't see how it's relevant.\"\n\n\"It's not like we can gate home,\" Jack added, \"and get your books to figure it out.\"\n\nDaniel dropped his head. They were right. The name was useless without knowing its context. It's meaning. \"There's a reason we found that obelisk.\"\n\n\"Yeah, well, without water, food, or weapons,\" Jack said, \"the fact is we're screwed unless our mystery host comes back.\"\n\n\"It is not my fault,\" Weiyan mumbled.\n\n\"It's okay,\" Daniel assured her. \"No one said you did anything wrong.\"\n\n\"Speak for yourself.\" General O'Neill jabbed a finger at Weiyan. \"You \u2014 \" He strode toward her, the anger in his face visible, almost tangible.\n\n\"You were in the chair when the floor was ripped out from under us. What did you do?\"\n\nA slight tremor started up, the sound like the faint roar of an engine running off in the distance.\n\nShe jumped up and away from Jack. \"I tried to do what you asked. I tried to be a good trainee.\"\n\nDaniel returned a hand to Weiyan's shoulder for support. Being on an alien planet was one thing. Dealing with Jack O'Neill could be enough to send her over the edge.\n\n\"Technically,\" Sam offered, \"Weiyan was half out of the chair, sir. It's possible she triggered some sort of safety protocol. We don't know enough about the outpost's technology to be sure.\"\n\n\"Indeed,\" Teal'c added. \"It might only have been a matter of time before such a thing occurred.\"\n\nJack raked a hand through his hair. \"Why now? We've had dozens in that chair and nothing happened.\"\n\n\"And why the memory lapse?\" Sam asked. \"We didn't remember being in Antarctica before we got here until \u2014 \"\n\n\"She showed up.\" Jack jerked a thumb toward a subdued Weiyan.\n\nAt least she'd stopped panicking. She'd taken up watching the four of them talk. Daniel supposed it must be pretty much like watching a game of tennis.\n\nA thought occurred as he focused on the horizon again. Shortly after his return from ascension, he'd read a report from one of SG-1's missions that might have something to do with their situation. \"Jack, when we were transported here \u2014 \"\n\n\"Don't ask me.\" He waved a hand at their surroundings. \"I've never seen this place.\"\n\n\"Yes, but did the process feel familiar at all?\"\n\n\"Familiar as in how?\"\n\n\"Like the time you and Colonel Maybourne went through that portal?\"\n\n\"The Furlings' technology,\" Teal'c said.\n\n\"Maybourne's little paradise?\" Jack grimaced. \"Nope. This was different.\"\n\n\"Different how?\"\n\n\"Well, for one thing, I didn't forget who I was.\"\n\n\"Good point.\" Daniel did a three-sixty, taking in the desert landscape once more. \"Well, one thing's for sure... Whatever technology got us here could be even more useful than the Stargate.\"\n\n\"Stargate?\" Weiyan asked.\n\n\"Daniel.\" The general sliced a hand across his neck. \"Clearance levels. Need to know. Hello?\"\n\n\"Oh.\" Daniel spun toward him, his mouth gaped open. \"It's a little late for that.\"\n\n\"Yeah. I suppose it is.\" Jack exchanged a glance with Sam. \"Look, Miss \u2014 \"\n\n\"Weiyan, please.\" She gave a slight bow, her earlier fear of General O'Neill apparently subsiding. Just like the tremors. Daniel strained his ears for a moment, but didn't hear anything. Hopefully, they'd seen the last of any seismic activity.\n\n\"Are we on another planet?\" Weiyan asked. \"One built by those who created the weapons chair?\"\n\nDaniel thought back on that city with its extraordinary architecture and its people. People who'd walked right through them. \"Sam, what was that? A hologram?\"\n\n\"I really don't know, Daniel. They'd need an enormous power source to create something that complex.\"\n\nHe gestured toward the now non-existent city's one remaining relic. \"What if it's housed in that building? Shouldn't we take a closer look?\"\n\n\"He's got a point, Carter.\" Jack shoved his hands in his pockets. And yanked them out again as if they'd been burnt.\n\nNeither Sam nor Teal'c noticed \u2014 their attention was turned toward Weiyan \u2014 but Daniel had. \"You all right?\"\n\n\"Yep, terrific.\" Jack thrust his hands behind his back. \"Let's go take a look at that building.\"\n\nDaniel pushed up his sleeves. It wasn't getting any cooler. \"If nothing else, we could get out of the sun. Maybe we'll find more answers inside. Maybe there's more to this Pedion Elysium \u2014 \"\n\nJack snorted. \"Would you stop with the Elysium thing? It's not going to help.\"\n\nSam grabbed Daniel's arm. \"We don't know what's real and what isn't. That building could come down on top of us if there's another quake. The last thing we need to do right now is make assumptions.\"\n\n\"Colonel Carter is correct.\" Teal'c knelt down and scooped up a handful of the pale brown soil. \"As General O'Neill indicated, we have no resources with which to determine our location.\"\n\nSam sighed. \"Or, any way to get home.\"\n\n\"Indeed.\" Teal'c flung the dirt down and stood up.\n\nAnd yet... Daniel mouthed the words Pedion Elysium to himself. There was something there. Something just out of reach. If only \u2014\n\n\"O'Neill!\" Teal'c pointed toward the distant building.\n\nA golden-white curtain shimmered into life just above the building. The curtain broke into three, no, four ribbons of undulating light. Streaks of a darker gold emanated downward, all pointed toward a glowing white object forming in front of the building.\n\n\"Holy crap,\" Jack said.\n\n\"It's like the Aurora Australis,\" Sam said. \"Only, instead of green or red, it's more like a golden-white.\"\n\n\"In the daytime?\" Jack pointed at the undulating gold-white ribbons.\n\n\"It's possible photonic emissions from the upper atmosphere are combining with solar wind particles, sir.\"\n\nThe white-outlined form began to glow.\n\nJack stepped up beside Daniel. \"Anyone you know?\"\n\nHe had a point. \"You think it's an Ancient?\" Daniel asked.\n\nThe emerging coruscated white strands reminded him of his once-mentor Oma Desala. Of Shifu. And of someone else. Another Ancient. Another Ascended Being he should remember, but couldn't. A face so obscured it made all of Daniel's other recollections of his time as an Ascended Being amongst the Ancients seem like easy memories.\n\n\"Jack, this could be our ticket home.\"\n\n\"Yeah, possibly,\" Jack said.\n\n\"Daniel Jackson,\" Teal'c said, coming up to stand on Daniel's other side. \"If the Ancients remain removed from human affairs as you once explained, how can you be certain they will assist us now?\"\n\nThe glowing strands elongated, reaching up toward the ribbons of golden white light overhead. The two energy sources intertwined, creating a blindingly bright light show.\n\n\"Nice parlor trick,\" Jack said.\n\n\"The Ancients don't play tricks,\" Daniel said.\n\n\"Tell that to old oily-faced. Don't forget, Anubis is ascended, too. Or at least, he's some half-assed version.\"\n\n\"Anubis is dead.\" A twinge in Daniel's shoulder reminded him of when Jack shot him a few months ago when the half-ascended Goa'uld hijacked his body. \"There's no way that's Anubis.\"\n\nHe edged closer.\n\n\"Who is Anubis?\" Weiyan asked.\n\nA thunderous crack erupted overhead with all the power of a sonic boom. Daniel looked up as green lightning tore through the ribboned lights and shot upwards and smashed against the sky. After a moment, a black tear appeared at its epicenter. A ragged edge of nothing.\n\nThe wind picked up. Weiyan crouched low. Jack, Teal'c and Sam leaned forward. Daniel found himself doing the same to keep from being blown backward. Above the building, the gold and white strands bent toward the gaping hole in the sky. The maw widened, the stream of lights swallowed into its jaws.\n\nAnd then, it disappeared.\n\n# CHAPTER TEN\n\n### ANTARCTICA\n\n_48 years earlier..._\n\n\"Four hundred and twenty-two.\" Huang dipped his forefinger in the ash pot. It was the only words he permitted himself upon rising from each slumber. The only words he allowed to mark the passage of time. He pressed his finger against the wooden panel beside the stove and swept downward, the short slash joining the many of others of his exile.\n\nFour hundred and twenty-two days. A useless, but necessary task since discovering the Tau'ri hut upon the frozen shores. At times it seemed the sun would never reappear. Other times, it seemed to never set. He silently reprimanded himself. He should be grateful to have found the Tau'ri's long-forgotten domicile.\n\nWhen he'd originally set out to return to the Chappa'ai so that he might warn his master of the being of light, a blizzard had so blinded his passage that he had become lost. It was only after many hours of wandering that he stumbled upon the abandoned wooden hut and discovered its contents. Within he found crates of food, furniture, and a fully equipped kitchen. Along the back wall hung frozen carcasses of blubbery, flippered creatures.\n\nMost importantly, he'd found black and white papers bearing images of men. Tau'ri men. Some pictures had the men on skis, others showed them standing beside massive dogs bearing collars.\n\nWhere the men had gone was unknown. Huang could only assume that the being of light had taken them.\n\nTurning away from the blackened marks upon the wall, he lit one of the many oil lamps he'd found around the square hut and hobbled over to one of the hut's few windows. His toes had healed many days ago, but walking was still a challenge.\n\nHe gazed at his reflection in the window through weary eyes. Cracked lips. A straggling beard that warmed his face and throat. Though it was not a First Prime's proper appearance, it was a truth he had come to accept with one exception.\n\nRaising the ashen finger to his forehead, he drew the _w\u00e9nsh\u0113n_ of Lord Yu between his brows. Another daily ritual. Another means of seeking solace amongst the failure of his lot.\n\nHe peered out the window toward the shoreline. Many days ago, long sunsets had splashed wild colors across the sky, followed by an endless barrage of darkness and a night sky full of stars. The night's black had now turned to a red-tinged blue, the scattered clouds almost purple. Barely a sliver of the sun was above the far horizon, but it was a start. Soon daylight would come again and he would have to make a decision. A decision that gave him great unease.\n\nSatisfied with the soot-stained tattoo, he returned to the stove. The last of his fresh meat simmered within the pot. He had learned through trial and error of the need to hang the mournful-eyed creatures for several days before cooking them. The brains he could cook upon taking down the legless beasts, but the meat tasted more of fat than meat if not aged.\n\nTo remain or to go? That was the decision he must reach before the ice thinned again. If he stayed, he would need to hunt soon, a difficult challenge as he'd lost his staff weapon months ago, while attempting to cross the thin ice that separated the hut from the glacier that held the Chappa'ai.\n\nFour hundred and twenty-two days.\n\nHuang sank onto a wooden chair by the stove, soaking in what heat he could to disperse the hut's cold air. Had too much time passed to warn Lord Yu? Would his master consider him dead?\n\nA pounding at the door. He bolted to his feet, staring in horror. Had the being of light come back?\n\nVoices shouted from outside. Men's voices, in a language he did not know. Huang grabbed the lamp. If necessary, he would use it as a weapon.\n\nThe door burst open and three men stumbled in, all dressed in great orange coats, their hoods lined with black fur.\n\nHuang raised the lamp menacingly. \"Stay back!\" he yelled in Goa'uld, his tongue thick from lack of use.\n\nThe men pulled back their hoods. All three wore beards though the tallest of the three kept his short around his long face. In unison their hands raised in surrender, the tall man speaking to him again in the mysterious language.\n\n\" _Bie guan wo_!\" Huang warned them to stay back again, this time in the tongue of his Tau'ri ancestors.\n\nThe tall man's eyes widened and he smiled. \" _Wo de shang di, ni shi zhong guo ren_!\"\n\nYou're Chinese!\n\n# CHAPTER ELEVEN\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/1830 HRS MCMURDO STATION\n\nThe floor buckled as George scrambled for cover. He dove for the computer table. As he slipped underneath, another jolt pushed upwards. His head smacked against the metal underside and then he dropped like a ton of rocks. To his right, computer monitors crashed to the ground, their screens shattering upon impact.\n\nWall panels rattled. Ice from the ceiling pelted the ground. A scream from the outer chamber pierced the tremor, but unless George wanted to get himself killed, there was nothing he could do until the tremor ended. The airmen and marines assigned to the outpost would protect the civilians and diplomats until this was over.\n\nIf it was ever over. The last time George had been in an earthquake was during basic training at Vandenberg. The Air Force base had shook for a moment or two, but this latest tremor just kept on coming with no end in sight.\n\nRound after round of ice and snow dumped onto the force field covering the hole in the ground. Each time another round hit, the energy field glowed and sparked.\n\nBehind all that snow, he got a glimpse of Dr. Lee. Head tucked in, the scientist belly-hugged the Mark IIs as if his life depended on it. George scrambled from under the table to help \u2014 the damage those naquadah generators could cause would make the earthquake seem like a picnic.\n\nHe'd made it half way toward Lee when the quake stopped. A light dusting of ice and snow fell on his face and he wiped it off. Lee still clutched the Mark IIs.\n\nThreading his way through the broken equipment scattered across the ground, George knelt beside the scientist and laid a hand on his back. \"It's over, son.\"\n\n\"You're sure?\" came the muffled reply.\n\n\"You're the scientist. You tell me.\"\n\nLee lifted his head, his glasses fogged up. No doubt he must have been breathing pretty hard. His face was as white as the now snow-covered force field.\n\n\"General Hammond!\" Ambassador Duebel stormed through the archway, his clothes dusty and torn.\n\nBehind Duebel were the other two diplomats \u2014 Juarez and Zhu. The Chinese ambassador sported a wide bandage across the left side of her face, thanks to a chunk of rock that had gone flying when the floor first caved in.\n\nGeorge cursed silently. Someone needed to keep these people out of the room. Where had Simmons had disappeared to? He'd lost sight of the lieutenant when the latest quake hit.\n\nAs the diplomats drew near, George deliberately ignored them. Instead, he pulled Lee up from the ground and patted his shoulder. After a few words of encouragement, the scientist hurried out, shouting for replacement equipment and a clean-up detail. Lee may not be military, but he understood the need for order.\n\nAnd the need for haste. George privately hoped Jack and the others had weathered the trembler intact, but there was no way to know. Though the snow had melted off the force field, it was still intact and murky as mud.\n\nAmbassador Zhu crept up to the hole before he could stop her. She knelt, reaching out a hand toward the force field.\n\n\"I wouldn't do that if I was you,\" he warned.\n\nShe snapped back her hand.\n\nDuebel side-stepped, blocking George's view of the Chinese ambassador. \"We demand to know your plan.\"\n\nGeorge stopped just short of scowling. \"Plan?\"\n\n\"Yes, General Hammond.\" Spit flew from Juarez's mouth. The Argentine ambassador wiped a hand across his face and then stabbed a finger at the force field. \"Your brilliant military's plan to destroy the outpost before it destroys us.\"\n\nGeorge counted to three and then responded as blandly as he could. \"Ambassadors, I'm not going to sit here and quibble over what the military is and isn't responsible for... Not now. Not when our focus should be on shutting down that device and then mounting a rescue effort.\"\n\n\"Forget your rescue efforts!\" Juarez flung a hand toward the force field. \"Esperanza Base was destroyed! Fifty-five people, including ten families and two school teachers, all gone!\"\n\nGeorge was taken aback. The Argentine Base was over 2,400 miles away, located at the very tip of the Peninsula on Antarctica's far other side.\n\nFifty-five people. No wonder Juarez was beside himself. George gave his condolences. \"We're doing everything we can, short of \u2014 \"\n\n\"Really, General,\" said Duebel, \"you're doing nothing of the sort.\" He toed the broken monitor at his feet. \"Scientific study isn't enough. Please do not misunderstand, I fully recognize the lengths General to which O'Neill and his team have gone in order to protect this world in the past, but there's more at stake now. First, Scott Base and now Esperanza. Something needs to be done.\"\n\n\"What of China's stations?\" Ambassador Zhu called out. \"Have there been any reports?\"\n\n\"Not yet, no.\" Juarez' eyes narrowed. \"At least, not of your official bases.\"\n\nZhu returned to gazing at the force field, ignoring the obvious dig at what everyone involved with the Antarctic region already knew. George had heard reports that the Chinese were building several stations up on Dome Argus, but what they wanted up on the highest point of the Antarctic Plateau wasn't his current priority.\n\nStopping the quakes and retrieving his people was what mattered. If they could access the floor down below by altering that force field, it was entirely possible to take care of both problems in one fell swoop.\n\nDr. Lee reentered, leading a pack of airmen carrying replacement monitors. Simmons brought up the rear. He rejoined Ambassador Zhu by the hole and gave her a polite nod.\n\nShe rewarded him with the faintest of frowns.\n\n\"You must destroy the outpost,\" Duebel stated, \"before more earthquakes tear apart the entire continent.\"\n\n\"I wish it were that simple, Ambassador.\" George shook his head. \"We're dealing with technology we've barely begun to understand. We need more time.\"\n\n\"This isn't about the earthquakes,\" Juarez shouted, his face bright red with anger. \"You're more concerned with rescuing your people than the several thousand spread across this continent. With one phone call, I could contact the UN Secretary-General and the IOA Chair. They would demand your president \u2014 \"\n\n\"Leave him be,\" Zhu announced.\n\n\"Excuse me?\" Juarez whirled toward the Chinese Ambassador, his mouth agape.\n\n\"Quing...\" Duebel strode over to Zhu's side. \"I know the young woman trapped below is \u2014 \"\n\n\"You know nothing.\" Zhu rose from her position by the hole's edge. She turned toward George and though he expected to see a measure of the anger exhibited by the other two ambassadors, he instead saw what could only be described as genuine concern.\n\n\"I want you to destroy the outpost,\" Juarez repeated. \"That is the only way to stop the earthquakes.\"\n\nZhu took a menacing step toward him. \"And I want you to cease speaking an endless loop of gibberish.\"\n\n\"Who do you think you are?\" Juarez sputtered.\n\n\"China's ambassador to this region.\" Zhu raised her chin in defiance. \"We must allow General Hammond and his people the opportunity to \u2014 \"\n\n\"Be careful which side you align with,\" warned Duebel.\n\nZhu shrugged. \"At the moment, I am aligned with the truth. These people have had far longer to study this technology than we. Yes, there have been losses, and yes,\" she faced George, a pinched smile on her face, \"there have been promises left unfulfilled, but we must recognize the greater picture or...\"\n\nBowing her head, the ambassador turned away.\n\nDuebel and Juarez spun on their heels and stormed out. By the hole's edge, Zhu gave Simmons an acknowledging nod. Grateful for the reprieve, George thanked her for the support.\n\n\"Do not thank me just yet.\"\n\nFlailing white tendrils strained against the gaping rip above the building. As Sam watched the phenomenon, she raced through possible explanations, examining and tossing aside each idea as it cropped up. A black hole? No, she'd feel its gravitational force. Some sort of reverse effect on the aurora? That didn't make any sense, either. And why the more gold-colored aurora? Gold usually meant a higher energy charge, but again, why?\n\nThe black maw widened further, pulling at the cluster of tendrils. The tendrils shrunk, a final flash of white illuminated the sky, and the phenomenon disappeared.\n\nA gasp. Sam glanced to her left. Weiyan's mouth hung open in a perfect O.\n\n\"What the hell was that?\" General O'Neill asked.\n\nThe ground rumbled again. Nothing threatening \u2014 at least, not yet \u2014 but it was enough to scatter a few errant pebbles.\n\n\"Carter?\"\n\n\"Sir, I really can't say.\" She wiped the sweat from her forehead. The sun had come back out in full force. \"Not without a way to measure what we're seeing.\"\n\nShe'd kill for a scanner, a spectrometer, heck, even a pair of binoculars.\n\n\"Take a guess.\"\n\n\"What we're seeing just isn't possible, sir.\"\n\n\"And yet, there it is. So?\"\n\nSam shrugged. \"Without having a way to quantify what kind of energy \u2014 \"\n\n\"In other words...\" He pressed his lips together.\n\n\"I really don't know.\" She frowned, hating to let him and the others down.\n\n\"It's all right, Carter,\" he said, giving her an assuring tap on the shoulder. \"Nice to know even you can get stumped by the impossible.\"\n\n\"What do we do now?\" Weiyan asked. \"I want to go home.\"\n\nThe earlier rumbling died off and Sam's stomach unclenched in relief. One geophysical problem was bad enough.\n\n\"We can't stay like this much longer, Jack.\"\n\n\"I agree with Daniel Jackson,\" said Teal'c. \"Even I cannot sustain myself in this heat for more than a few hours.\"\n\nDaniel wiped his brow. \"We have no supplies, no \u2014 \"\n\n\"But we do have a shelter.\" General O'Neill raised two fingers and pointed toward the building. \"And I'll bet anything that's the key to this whole _mishegas_.\"\n\n\"Wait a minute,\" Daniel said. \"Shouldn't we \u2014 ?\"\n\n\"What? Sit here and wait for that thing to come back?\"\n\nDaniel frowned. \"I don't think it was a 'thing,' Jack.\"\n\nTeal'c mimicked Daniel's frown, only more pronounced. \"Do you still believe it was an ascended being?\"\n\n\"I don't know.\"\n\nThe general shrugged. \"Sure looked ascended-y to me. Let's move out.\" He headed off at a slow lope toward the building with Teal'c at his side, and Daniel close behind.\n\nSam turned around to make sure Weiyan was ready to follow. She gave the girl an encouraging smile. \"Don't worry, it's not much more than a half a mile \u2014 \"\n\nAn ear-splitting crack reverberated to Sam's right. As Weiyan jerked her hands up to cover her face, Sam whipped back around toward the men. The building rippled, but at first she thought it was a trick of the heat. A mirage.\n\nThe rumbling intensified and she realized the ripples were pushing outward, toward the general and the others. Another boom shook the ground.\n\nThe building began to crack in two under the pressure of the massive turbulence. Several hundred feet off to the left and right of the building, hills erupted from the ground. Fountains of dirt sprayed from all sides. Sam watched in horror as all three men stumbled backward.\n\n\"Sir!\"\n\nDaniel fell. General O'Neill and Teal'c grabbed hold of his elbows and lifted him up, never missing a step as they ran like hell toward Sam and Weiyan.\n\nWeiyan.\n\nShe'd curled up into a ball, her arms over her head.\n\nTwo hundred feet away, another hill savagely erupted.\n\n\"Get her up,\" the general shouted over the rumbling.\n\nSam dragged Weiyan to her feet. \"We need to run. Fast!\"\n\nWeiyan's eyes rounded in fear. \"I will try.\"\n\nShe didn't move.\n\n\"Go! Go!\" shouted General O'Neill, darting ahead with Daniel and Teal'c close behind.\n\n\"Please, Weiyan.\" Sam took hold of her hand. \"This whole area's unstable. We need to put as much distance \u2014 \"\n\nThe general doubled-back. \"We have to go, Colonel!\"\n\nAn ear-splitting blast exploded behind them. Sam scooped the girl over her shoulder, fireman-style, and ran as fast and as far away from the eruption as possible.\n\n\"Wait!\" Daniel pointed back the way they'd come.\n\nThe tremors had stopped. Panting from the exertion, she turned toward the building.\n\nIt was gone. The newly erupted hills surrounding the area were still there, but the ground had settled down. The air had become suddenly still.\n\nThe tundra-like grasses from earlier were gone except for a few tufts on the newborn hills. Where the building had once been was now a flat plain of pale, lifeless dirt.\n\n\"That's impossible.\" Sam set Weiyan down. \"Land just can't change. Not this fast.\"\n\n\"We must retrace our steps, O'Neill. We must find our original point of entry onto this world.\"\n\n\"I'm all for that, T.\" General O'Neill spread out his arms and turned around full circle, \"But unless your spidey-senses have a clue about which direction to go, we're screwed.\"\n\n\"That's the one part of this whole puzzle that's still foggy,\" Daniel said. \"I remember the Ancient outpost. I remember falling through the floor. Then next thing I knew \u2014 \"\n\n\"We were here,\" the general said. \"Coming out of the Stargate. Thing is, I don't remember stepping through one to begin with, let alone why.\" He turned toward Weiyan. \"Or how you got here, for that matter.\"\n\nA low rumble started up. Sam hoped it was just an aftershock, not another full on quake.\n\n\"I had told you that I don't know.\" Weiyan had backed away from the general, shaking her head vehemently.\n\nThe vibration intensified. Not by much, but enough to jostle the dirt at their feet.\n\nSam tensed, ready to run. Her vision darkened. She put a hand up over her eyes, convinced it was fatigue.\n\n\"Colonel Carter.\"\n\nSam dropped her hand. Teal'c pointed toward the horizon. The sun had dropped dramatically, hanging low over where the building had once stood.\n\n\"General!\" Sam directed his attention toward the setting sun.\n\n\"Yeah, I see it.\" His voice had gone flat, his eyes never leaving Weiyan.\n\nThe ground continued to rumble. Low, like a wounded animal growling its warning to keep away.\n\nBut that wasn't possible. The planet wasn't alive.\n\nOr was it?\n\nGeneral O'Neill ignored the escalating tremors, or he didn't care. As he stepped closer to Weiyan, Sam watched him clench his fists, straining to keep his anger in check.\n\n\"You did something to the chair,\" he said.\n\n\"I don't know...\" Weiyan's hands shot up, covering her face.\n\n\"That's bullshit.\"\n\n\"Jack, leave her alone.\" Daniel grabbed the general's arm.\n\nThe ground kept shaking.\n\n\"Sir, we should get ready to \u2014 \"\n\n\"You were there, Daniel! She deliberately slammed down on one of the gel packs.\" General O'Neill wrenched his arm free. \"You heard her. The only way we're getting out of here is the way we came. Either she tells us what she did or we're screw \u2014 \"\n\nAnother boom echoed across the plains, or canyon, or... Sam couldn't even decide what to call it anymore. The geography kept changing.\n\nPlanets don't change that fast.\n\n\"Who are you?\" the general demanded.\n\nThe ground shifted beneath them. Sam staggered forward, but Teal'c stopped her fall. The rumbling deepened. It was nearer. Stronger than before.\n\n\"General...\" She rechecked the horizon. Swirling dust covered the sun, leaving only a red-tinged corona visible.\n\n\"O'Neill!\"\n\nFifty meters to their left, the ground cracked open. A splitting maw widening with each passing moment.\n\n\"Jack?\"\n\nThe fissure lengthened, heading in their direction.\n\n\"Run!\"\n\nTeal'c remembered running. Hills thrusting upwards. Dirt and rocks flying through the air. A sudden pain against his left temple, and then...\n\nNothing.\n\nHe stirred into awareness, greeted only by darkness. Within the black, a sense of heat without sun. Peace. Ease.\n\nThe pain was gone.\n\nNo. That was impossible. He must wake up. His friends would require his assistance. They were stuck \u2014\n\nHe tried to open his eyes, but could not. It was as if a great weight held him in place, refusing to allow him to awaken. He struggled against the unseen restraint.\n\nA golden-white ribbon of light danced across his vision, hovering just beyond his reach. \"Why do you hold me here?\" he demanded.\n\nThe ribboned light flowed nearer. \"Wait. Learn,\" a voice said in the dark. \"All will be well.\"\n\nThe voice had spoken in Goa'uld. Of this, he was most certain. Cold fury welled up within him. If a Goa'uld was responsible \u2014\n\nCalm washed over him. A silent assurance his former oppressors were not responsible, and yet... He must know the truth.\n\n\"If all is well, why do you hold us here?\"\n\nThe light dimmed its brightness, becoming only a single rope-like strand. \"If you cannot find the truth right where you are, where else do you expect to find it?\"\n\n\"You speak in riddles.\"\n\n\"There are no riddles. Only events to unfold.\"\n\nTeal'c imagined himself raising an eyebrow in response. \"Which events do you speak of?\"\n\nThe ribboned light winked out.\n\nClink. Snap.\n\nMetal against metal. A familiar sound. Teal'c strained to find its source, but the darkness would not release him.\n\nClink.\n\nA feather's touch circled his emblem and he found himself free to open his eyes. At first, the sunlight was too strong. His lids fluttered as they adjusted to the brightness.\n\nSnap.\n\n\"Come on, big guy.\" O'Neill's voice urged. \"Rise and shine.\"\n\nTeal'c opened his eyes to find his friend and Weiyan Shi kneeling beside him. The trainee's hand lay gently on his forehead. Behind O'Neill stood Daniel Jackson and Colonel Carter, their relieved smiles most gratifying.\n\nO'Neill waved his lighter at Teal'c. \"About time you came back to the living.\"\n\n\"We thought we lost you for a moment there,\" said Daniel Jackson.\n\nTeal'c raised a hand to his left temple.\n\n\"Easy, Teal'c,\" warned Colonel Carter. \"We may be in some kind of illusion, but still... You took quite a blow.\"\n\nHe felt no cut or abrasion where the rock had hit. There was no pain. \"I am well.\"\n\n\"Can you sit up?\" Weiyan Shi glanced over at O'Neill who gave an approving nod. A hesitant grin spread over the young woman's face. \"You were very brave,\" she said to O'Neill. \"Teal'c might have died if \u2014 \" Her voice broke.\n\n\"Nah,\" O'Neill drawled. \"Can't knock this guy down for long.\" He grinned. \"Trust me, I've tried.\"\n\nTeal'c raised an eyebrow at O'Neill.\n\n\"What?\" his Tau'ri brother asked.\n\n\"You have made peace.\"\n\n\"And you made us worried sick. Well, not me, but Daniel and Carter were going nuts.\" O'Neill sternly glanced up at the remainder of SG-1 who rewarded him with matching smiles.\n\nWeiyan Shi grabbed hold of Teal'c's arm and though she was far smaller, she had the strength to assist him as he stood up. He privately berated himself. He shouldn't feel so weary. With a brief nod of thanks, he found his balance and then proceeded to recount what had happened while unconscious.\n\nO'Neill frowned upon his sharing the being's words. \"'All will be well,' my ass. We're like rats in somebody's maze.\"\n\nThe ground began to tremble.\n\n\"Sir,\" warned Colonel Carter.\n\n\"Rats, Carter! Anyone hungry? Maybe they'll send us some cheese.\"\n\n\"If you don't calm down,\" admonished Daniel Jackson, \"you'll start off another earthquake.\"\n\n\"How is that possible?\" Teal'c asked.\n\n\"None of this is possible,\" Colonel Carter said, \"but we can't deny the cause.\" She pointed to O'Neill and then Weiyan Shi. \"Or the effect.\"\n\nO'Neill squeezed his eyes shut and sighed heavily. His face dropped to a blank visage, one Teal'c had seen many times in battle when concentration was paramount.\n\nA moment later, the tremors stopped.\n\nWorried for his friend, Teal'c asked, \"Are you well, O'Neill?\"\n\n\"I'll be fine.\" He opened his eyes, raised his hands in mock surrender, and managed a small, unconvincing smile.\n\n\"You have to stay calm, Jack,\" advised Daniel Jackson. \"Otherwise, we'll \u2014 \"\n\n\"I get it. It's all an illusion. No need for a repeat, all right?\" O'Neill dropped his arms to his sides.\n\n\"Sir, if we're going to do something \u2014 \"\n\n\"Do?\" O'Neill sighed heavily. \"Do what? Walk around in circles while some 'thing' plays with our heads? Whatever the hell it meant, finding 'the truth,' was a load of crap. Pure, unadulterated crap.\" He strode off several dozen paces and halted, his back to them.\n\nWeiyan Shi made to follow, but Teal'c stopped her.\n\nShe shook her head. \"Is he always so angry?\"\n\n\"This is one of his better days,\" Daniel Jackson murmured.\n\nTeal'c knew his friend merely joked about O'Neill's temperament. The two were as close as brothers \u2014 brothers who bickered like children, even though they would die if required. It was important Weiyan Shi understood this as well. \"Given time, General O'Neill will come to terms with our situation. You will see.\"\n\nPermitting his friend a moment's respite, Teal'c turned his attention to their newly arranged surroundings.\n\n\"It's different, isn't it?\" asked Colonel Carter. \"For one thing, that building's gone.\"\n\n\"If it was truly ever here.\" Teal'c looked out over their changed environment. The once flat, undistinguishable plains had been reformed into a valley with several hills marking its border. In the direction O'Neill had moved off, the steepest hill loomed. Its forward face was sheared off, revealing dark gray vertical striations within the rock's face.\n\nThere was a familiarity to the terrain, a sense that he been here before, although... As 'here' did not really exist, he found the matter all the more disturbing.\n\nHe said as much to his companions.\n\nDaniel Jackson agreed. \"You said that being wanted us to learn. Did it say what?\"\n\n\"It did not.\" Teal'c searched his memory. \"Though I must wonder why the being believes it necessary to continue changing our environment.\"\n\nWeiyan Shi looked over her shoulder and Teal'c followed her gaze. She was watching O'Neill. He, in turn, was gazing upward toward the hill's top. In his right hand, he flipped his lighter open and shut. It concerned Teal'c that his friend felt the need to distance himself. That was not his normal behavior. Nonetheless, he understood O'Neill's need for a moment's meditation. If he did not control his emotions, the ground could rip open once more.\n\nO'Neill must have felt eyes upon him because he turned back and frowned. \"I'm going for a hike.\"\n\n\"Is that wise, O'Neill?\"\n\n\"Sir, one of us should go with you,\" Colonel Carter said.\n\n\"I'm just going to stretch my legs. Get a better look at our surroundings.\" O'Neill pointed at the hill's jagged peak. He then gestured toward Weiyan Shi. \"Keep an eye on our guest.\"\n\nWith that, O'Neill headed toward the hill's far side where there was a gentler slope. He began to climb.\n\nWeiyan Shi glanced back at Teal'c. No matter what O'Neill believed, Teal'c was certain the young woman had no part in their current circumstances. \"Do you have a question?\" he asked her.\n\nShe inched forward. \"You said this looks familiar.\"\n\n\"Well, kind of,\" Daniel Jackson responded, \"but none of this is real. It just looks \u2014 \"\n\n\"Yet I have not been here before,\" Weiyan Shi insisted. \"The previous variation...\" She gestured toward the hill behind them and then at those around the valley. \"That was familiar. It looked very much like the Taklamakan Desert.\"\n\n\"Western China.\" Daniel Jackson paused, peering out at the valley. \"Weiyan, just how close was the similarity?\"\n\n\"Very,\" she whispered, wrapping her arms around herself. \"My mother would take me there when I was very young. We would go to the desert's most northwest region. There was nothing there.\" She swallowed visibly. \"I was very little. Six, maybe seven. I would complain. There was nothing to see or do, but she would insist we visit the place where she met my father.\"\n\n\"Did you not know your father?\" Teal'c asked.\n\nWeiyan Shi shook her head. \"He left my mother soon after she became pregnant, although...\" She swallowed nervously. \"We met once, after my genetic testing so that I might serve China. He said that he would soon be going very far away. To visit his ancestral home.\"\n\n\"It is a great honor to witness one's ancestral home,\" Teal'c replied, privately troubled that a father would forever abandon their child. \"One day, when the Goa'uld are truly gone, I will visit Dakara to see the birthplace of the Jaffa.\"\n\n\"But you will return? To rejoin your friends and family?\"\n\nHe gazed down into her troubled face. \"Of this I have no doubt.\"\n\n\"You make it sound like we'll escape,\" she said.\n\nTeal'c considered the being's words of wisdom. Whether they were caught in the trap of some alien enemy or a planet of their imaginations, he knew the only truth that mattered.\n\n\"Have faith, Weiyan Shi. SG-1 has never failed.\"\n\nClimbing up the steep hill overlooking the shifted terrain should've knocked Jack out. Or at least, it should've made him thirsty. Or hungry.\n\nIt didn't do squat.\n\nAs he sat down, swinging his legs out over the edge, he recognized that his main objective for the climb had pretty much failed, too. He was still pissed. Still ready to knock someone \u2014 or something \u2014 out for dicking around with them.\n\nHe felt utterly, completely, pathetically useless.\n\nIf they were rats in a maze, he was the king of the rats.\n\n\"Come on, O'Neill,\" he muttered to himself. \"It's not like a P90 would be much use against...\"\n\nAgainst whatever was toying with them.\n\nWith a sigh, he leaned back and pulled the Zippo from his pocket. His Zippo. Now that was a whole other piece to this insane puzzle. When Teal'c had been knocked unconscious, Jack had taken the chrome-plated lighter out. He'd almost told Carter about its mysterious appearance when his gut told him not to \u2014 they'd think he'd gone nuts.\n\nHe turned the lighter over in his hand. The casing looked the same. Scratched up, a dent on the top. With a flip of his wrist, the cap swung back, making that oh-so-satisfying clink.\n\nHe dropped it to his lap. Who was he kidding? The thing was a useless piece of junk. It wasn't like a Zippo would help them escape. It was useless.\n\nHe knew the feeling and he didn't like it. There were gaps in his memory. Gaps waiting for answers that were in short supply.\n\nHe scanned the valley floor. If he had to guess, he'd say that from the sun's position, the local time appeared to be mid-afternoon. Waves of heat rippled against the far horizon, creating a mirage of water surrounding the immediate area.\n\nThere really wasn't much to see. Mostly barren dirt and a few rocky hills along the perimeter. The area where the building had stood showed no signs of anything, not even a pebble out of place. To the right, a pile of rubble broke up the otherwise smooth ground. Which was odd, but probably didn't mean anything.\n\nIn fact, if he had to point to a landmark, the big sloping hill at around ten o'clock stood out like a sore thumb.\n\nAnd it looked damned familiar.\n\nCarter, T, Daniel \u2014 heck, even Weiyan at some point or another \u2014 had found something familiar in the changing landscapes. For him, not so much. Until now.\n\nIt bugged him, not knowing where they were or how they'd got there, but what really bothered him was his own reaction to the situation. He needed to get a grip. He needed to stop acting like a green nugget barely out of training.\n\nHe picked the Zippo back up, hefting its weight in his hand. What he really needed was a long, hard look at himself. A painful reassessment of just who he was and how he could still make a difference. Contribute. Make things better.\n\nAnd not just in their current situation. Hammond had told him his job wasn't about tactics anymore, but strategy. Fair enough, but without tactics, any strategy on how to get out of this mess would be an utter waste of time.\n\nHence the feeling of uselessness.\n\nJack flipped open the Zippo again and thumbed the wheel. The flame sparked up. Surprised, he almost dropped the lighter over the edge. He blew out the flame and yanked out the inner tube. The bottom was dry as a bone. He sniffed the cotton, which would usually be soaked in fluid to light the wick. Nothing.\n\nSomebody was toying with them. Big time. More importantly, why with the lighter? It didn't mean anything to anyone except him. Not since Skaara had died, ascended, taken off for the big dead-and-gone Ancients Club.\n\nHe couldn't deny that everything they'd seen 'smelled' of the Ancients \u2014 ascended beings or whatever-the-hell they liked to call themselves. The Ancient writing. A city definitely decorated by whoever set up the Antarctic outpost.\n\nWhy in God's name would any of those high-and-mighties be involved with these shenanigans?\n\nHe slammed the inner tube back into the casing. He glanced down at the others at the bottom of the hill. Daniel was scribbling in the sand. Typical Daniel, always asking questions. Always trying to figure things out.\n\nAlways strategizing.\n\nJack climbed to his feet. He should be down there, helping. Maybe he was useless in the grand scheme of things, but that didn't mean he couldn't make it easier for the others. He gazed at the valley floor one more time, taking note of the pile of rubble.\n\nMaybe it would help Daniel. Maybe it wouldn't.\n\nBut at least it beat doing nothing.\n\nDaniel sank down between Sam and Weiyan. \"How long has Jack been gone?\"\n\nTeal'c sat down as well, crossing his legs. \"No more than fifteen minutes.\"\n\n\"So we wait.\" Daniel noticed Weiyan was tracing a circle in the sand. \"What's that?\"\n\n\"I will spell out our names in Chinese.\" Along the circle's circumference, she drew arrows pointing outward from each of the four quarters. \"Many years from now, perhaps someone will find a record of our existence here.\"\n\nSam frowned. \"With the way this landscape keeps changing, I'd be surprised if your drawings last very long.\"\n\n\"But if something does remain, could it hurt our situation?\" Weiyan drew another set of arrows around the circle, this time pointing inward. \"Perhaps the next time everything changes, this will not.\"\n\n\"A guide of sorts.\" Teal'c said.\n\nWeiyan mumbled something.\n\n\"I beg your pardon?\" Sam asked.\n\nShe dashed a sleeve across her eyes. \"'A man who does not know where he's been cannot know where he's going.'\"\n\n\"Sun Tzu,\" Daniel commented.\n\nThe girl dipped her head.\n\n\"Weiyan...\" Sam stepped toward her. \"What happened in the chair? Why did you lose control?\"\n\nWeiyan's face paled. \"It's not my fault.\"\n\n\"No one's blaming you.\" Sam put a hand on her arm. \"We just need to understand what happened. It might help us get home.\"\n\nA recognizable vibration started up under Daniel's feet. It was faint, but definitely there.\n\n\"Weiyan Shi, you must calm yourself,\" Teal'c said softly.\n\nThe rumbling stopped.\n\n\"Maybe the general's right,\" Sam said.\n\nDaniel poked a finger in the sandy dirt. \"About what?\"\n\nShe shrugged. \"Maybe we are just rats in some imaginary maze.\"\n\n\"I don't know, Sam. It feels pretty real to me, although...\" He glanced over his shoulder at the hill Jack had climbed. It was still there. Looking down at Weiyan's drawing, he noted that the circle and arrows were still there, too. The tremor, slight as it had been, hadn't erased the drawing.\n\nThat's when it struck him. \"Maybe we can't remember how we got here.\" He shimmied back to give himself a wide space. \"But we can remember what's happened since we did!\"\n\n\"Of what do you speak, Daniel Jackson?\"\n\n\"Pedion Elysium.\" When Sam and Teal'c both stared at him blankly, he traced a square in the dirt. The Goa'uld symbol for 'ped.' Beside it, he traced another symbol.\n\n\"That's a glyph from the Stargate's DHD,\" Sam said.\n\nTeal'c leaned over and studied the symbol Daniel had written. \"Do you believe Pedion Elysium to be a gate address?\"\n\nDaniel sat back on his heels and glanced at him. \"It's possible, isn't it? I mean, we were able to translate Proclurash Taonas into a gate address by matching syllables to glyphs. Why wouldn't that reasoning work for other gate addresses?\"\n\n\"Proclurash Taonas was only six syllables, Daniel.\" Sam pointed at the glyph in the sand. \"Ped-i-on E-Ly-si-um.\" She raised one finger and then another. \"That's seven.\"\n\n\"There are seven glyphs in a full address,\" Teal'c said.\n\n\"Well, sure,\" she said, crouching down beside Daniel, \"but, Proclurash Taonas was an Ancient outpost, and from everything you've told us, Daniel, their language was similar to Latin.\"\n\nDaniel drew an 'I' below the 'ped.' \"It is, or was, but Pedion Elysium is from Greek mythology.\"\n\n\"Like Atlantis.\"\n\n\"Exactly. Since the glyphs on the DHD are based on Goa'uld symbols for constellations,\" he drew a feather beside the 'I,' \"and the Greeks borrowed some of the ancient Egyptian language to form their own... Teal'c, is this right?\"\n\n\"Almost,\" the Jaffa said, bending down to trace a second feather beside the first. He then sat back, a bit unsteadily in Daniel's opinion.\n\n\"You all right?\"\n\nTeal'c frowned. \"I am well, Daniel Jackson.\" He pointed toward the feather in the sand. \"This is the symbol of Ma'at, a minor Goa'uld who once arbitrated disagreements among the System Lords.\n\n\"Cronus had her destroyed when she misjudged an argument between him and Ra. It was many years ago.\" Teal'c gestured toward the double feather in the dirt. \"The symbol of two feathers represents her scales of justice.\"\n\n\"Libra!\" Daniel exclaimed. \"In Greek mythology, the constellation of Libra was depicted as the scales of justice.\" Beside the feathers, he drew another glyph.\n\nSam shook her head. \"Daniel, this is interesting, but I don't see how figuring out the coordinates for this place \u2014 \"\n\n\"A place that is not even real,\" Weiyan added.\n\nDaniel's glasses had slid down his nose. He pushed them back up, annoyed at the distraction. \"If someone or something is holding us here, there's a reason they wanted us to see that city, the obelisk, and \u2014 \"\n\n\"The name.\" Sam raised her hands. \"Okay, I get it. So two down, five to go. What about that one?\" She pointed to the 'ON.'\n\nSounding out the syllable, Teal'c leaned forward and drew a jagged line beside it. \"In Goa'uld, the word means water.\"\n\n\"That's easy.\" Daniel traced another glyph besides Teal'c's drawing.\n\n\"Aquarius, the water pourer. In ancient Greek mythology, Aquarius was responsible for the Great Flood myth.\"\n\n\"China had its own floods,\" Weiyan said softly. \"My father spent much of our reunion reciting the legend of the great Emperor Yu and how he saved our people from extinction.\"\n\n\"Well, actually...\"\n\n\"Daniel,\" Sam warned. \"Stay on track.\"\n\n\"Indeed,\" Teal'c added.\n\n\"Sorry \u2014 sorry.\" Daniel hurriedly wrote the remaining Goa'uld symbols beside each remaining syllable. A flowering reed for the 'E,' a prostrating feline animal for the 'LY,' a long line with two ovals in its center for the 'SI,' and a bird \u2014 somewhat similar to Earth's owl \u2014 for the 'UM.'\n\nHe uncrossed his legs and shifted to kneeling in front of the symbols. \"Some of this is going to be tough.\" He paused for a moment, and then, next to the flowering reed drew another glyph.\n\n\"How do you get the constellation Hydra from that?\" Sam asked.\n\n\"The Goa'uld symbol represents a reed.\" He heard footsteps behind him, recognizing the long, easy gait as Jack's.\n\nDaniel pointed at the glyph. \"Reeds grow in water and since Hydra was a mythological water serpent in Greek mythology \u2014 \"\n\n\"There's something over by where the building used to be.\" Jack stopped next to him. \"Daniel, whatcha doing?\"\n\nDaniel explained his idea.\n\n\"Bit of a stretch, don't you think?\"\n\n\"Do you have any better ideas?\" He looked up at Jack, noticing that the lines on his face seemed more pronounced.\n\nJack cocked his head and studied the symbols in the dirt. \"Maybe. Maybe not, but keep going.\"\n\n\"Not if you're going to argue about it.\"\n\nO'Neill flattened his lips, but then his face relaxed. \"I'm fine, Daniel, just... Work it out.\" He pointed toward the Goa'uld symbol for 'LY. \"Is that a lion?\"\n\n\"Of course, a lion!\" Daniel drew another glyph beside the Goa'uld symbol.\n\n\"Leo.\"\n\nSam gestured toward the long line with the two ovals in its center, the Goa'uld symbol for 'SI.' \"What about that one?\"\n\n\"I honestly don't know,\" Daniel admitted. \"It's similar to the middle Egyptian hieroglyphic for the letter 'S,' but \u2014 \"\n\n\"Terra,\" Jack said, his voice distant. Flat.\n\nDaniel glanced up again. Jack stared down at the sixth symbol, but his eyes had glazed over, as if he \u2014\n\n\"Jack?!\" Daniel leapt to his feet.\n\n\"Are you well, O'Neill?\"\n\nJack continued to stare at the ground, his gaze blank, his features slack. It was if he was possessed. If Daniel didn't know better, he'd swear Jack had reverted to his time as a vessel for the Ancient's knowledge.\n\nA shudder ran across Jack's face. \"Planeta Terra,\" he said.\n\n\"The planet Earth? How \u2014 \"\n\nJack knelt down in the sand, one hand stretched out toward the symbols. He stuck a finger in the sand and drew a symbol.\n\nThe symbol for Earth.\n\n\"Planeta Terra.\"\n\nJack felt his knees press into the ground. He felt the dirt scratch his index finger as he drew the Earth symbol. He even sensed his mouth open and shut, repeating the words, \"Planeta Terra.\"\n\nBut he had no damn idea why.\n\n\"Jack?\" Daniel's voice. Far off. On the other side of a tunnel.\n\nHe was like a man on a runaway train. Out of control, no way to get off. No way to know where it went next. In a sense, the entire experience since falling through the Ancient outpost into \u2014 who knew where \u2014 was one giant exercise in control, and the lack thereof. He had to control his temper, but now? He had to let go of what? His head? Let some slimy alien jump in and play with his brains?\n\nHis finger lifted from the Earth symbol, dropped down to scratch another glyph beneath \u2014 right beside what looked like a really bad rendition of the owl on those potato chip bags, except without the glasses. He drew the last glyph.\n\n\"Aquila,\" his mouth said.\n\n\"Of course,\" Daniel's distant voice replied. \"The eagle that belonged to Zeus, the Greek god.\"\n\n\"Sir, you did it.\" Carter's voice now. \"A definite gate address.\"\n\nWhoever controlled Jack's actions allowed him to study the seven glyphs.\n\n\"General O'Neill!\" A hand shook his elbow. \"Oh, please wake up.\"\n\n\"I don't know if that's a good idea, Weiyan.\" Daniel again, being Daniel. Worrying.\n\nSomebody better worry. Jack couldn't feel his legs. His entire body became stiff, unmovable.\n\n\"Something's wrong,\" the Chinese girl cried out. Jack hated the Chinese. Well, not all of them. Just those scum suckers who'd made his life a living hell during Black Ops missions in the eighties, and then when SG-1 had to rescue Daniel from Yu.\n\nThat had not been fun.\n\n\"We need to help him.\"\n\nThe hand shook his arm. Smaller, weaker, but it did the job. Jack could feel himself keel over, slowly \u2014 as if gravity pulled him sideways \u2014 until his face smacked against the dirt. It hurt like hell, but he felt it!\n\n\"The building has returned,\" Teal'c announced.\n\nFeeling like himself again, Jack stood up. Weiyan kept hold of his arm. \"It's okay, thanks,\" he told her, gently prying her fingers off his arm. He suddenly felt like an ass for being such a bastard to this kid.\n\n\"I am glad you are awake again,\" Weiyan said with a smile.\n\nHe returned the gesture. \"Yeah, that was a hoot. What's going on, kids?\"\n\nTeal'c pointed in the opposite direction to the hill Jack had climbed earlier. That butt-ugly squat building was back. Closer this time, maybe three hundred feet away. Standing next to it was most positively, absolutely, a person.\n\n\"Stay here,\" he ordered. \"I'm going to go have a talk with our host.\"\n\n\"Hold on, Jack. This gate address? It's familiar.\"\n\n\"So?\" He glanced down at Daniel. \"Last time I checked \u2014 right before I became the second coming of The Exorcist \u2014 there wasn't any gate.\"\n\n\"I think we've been here before. Okay, not _here_ -here, because this isn't really a place, but some other time. On a mission.\"\n\n\"And yet, you have no idea what it means.\"\n\nDaniel scowled as only Daniel could do \u2014 rude yet Jack recognized that he meant well underneath it all.\n\n\"Don't you think there's a reason we worked this out?\" Daniel asked.\n\nJack squinted, barely able to make out the alien standing beside the building. He thought of him \u2014 or her \u2014 as an alien, because really? What else could it be, popping in and out like that? The alien wore a bleached-white cape, the cloth flapping against a breeze that had kicked up. A bit of dust swirled on the valley floor around the building. The alien didn't budge, but it did seem to be \u2014\n\n\"Is it waving at us?\"\n\n\"It appears so,\" Teal'c confirmed.\n\nJack stepped around the graffiti he'd help scribble in the dirt. \"Well, whoever it is, they have some explaining to do.\"\n\n\"Hold on,\" Daniel said. \"I think we should go with you. In case you have a \u2014 \"\n\n\"What? A relapse? A good, long look at what senility has in store?\" He stabbed a finger at the glyphs scrawled out in the dirt. \"Stay here and figure out what that address is about.\"\n\nDaniel frowned, but Jack ignored it. He nodded curtly at Carter, Teal'c and the trainee kid and headed off toward the building. He resisted the urge to run, knowing he needed to keep his adrenaline in check. Keep his temper under a tight lid. So instead, he focused on putting one foot in front of the other, shoving his frustrations as far down as they would go.\n\nThe ground trembled. Slightly, but most definitely there.\n\nOne foot, then the other. No anger, no frustration.\n\n_Just the truth, please, with a side of 'get us the hell out of here.'_\n\nThe ground kept on shaking, but he kept on going. He switched from watching his feet to looking up at the building. The alien had stopped waving. In fact, it wasn't even there anymore.\n\n\"Damn it.\" Jack doubled his pace. If he could outrun the alien, creature, or whatever it was that held them in this godforsaken place, then maybe, just maybe \u2014\n\nWith a loud crack, the ground split open barely ten feet ahead of him.\n\nHe stopped. Calmed himself down. Deep breaths. Relax.\n\n_Get a grip, O'Neill_.\n\nAnother tremor hit and the crack widened. Was it trying to block his way?\n\nIt was an illusion. He knew that. He took another step forward.\n\nThe illusion widened. Hell, the illusion became a damn deep chasm. He stepped to the left, intending to go around, but the ground rumbled again. Within seconds, the crack widened, stretching off in each direction as far as he could see.\n\nHe stopped just short of its edge and peered down into a bottomless, jagged tear in the ground. He kicked a pebble over the edge and it dropped. He listened for the sound of it hitting bottom.\n\nThe sound never came.\n\n_At least the tremors stopped. That's something_.\n\n\"General!\"\n\nJack whipped around. Carter and the rest were rushing toward him. \"I thought I told everyone to stay back.\"\n\n\"Whoa!\" Daniel got there first, halting a few feet back from the chasm's edge. \"I think someone's trying to tell you something.\"\n\nTeal'c came up on his other side, with Weiyan right behind him. \"Whomever that someone is, Daniel Jackson, they are no longer present.\"\n\nCarter leaned over the edge and Jack restrained himself from pulling her back. \"Sir, I hate to ask, but...\"\n\nHe shook his head. \"I swear on my single pair of stars that I wasn't pissed.\"\n\n\"Pissed?\" Weiyan asked. She hovered behind Teal'c. Jack couldn't blame her. He'd hide behind the big guy, too, if he had any sense. This entire experience was getting too damned \u2014\n\nThe ground shook again.\n\n\"Come on...\" Now the thing was reading his thoughts!\n\n\"Everyone, get back,\" Carter yelled over the rumble.\n\nJack didn't argue. He reversed course, away from the craggy pit of nothing. That is, until he noticed that the crack was shrinking.\n\nMoments later, the gap sealed up, the ground no different than before.\n\nWeiyan slipped out from behind Teal'c. \"How is that possible?\"\n\nShe took a step forward. Nothing happened. She took another step and the ground rumbled.\n\nJack joined SG-1 in rushing forward to stop her.\n\nThe tremor stopped.\n\nJack glanced as his team. \"Take another step.\"\n\nThey did. Nothing happened. Urging Weiyan on, Jack led the others in taking several more steps. They walked right over where the ground had cracked open earlier.\n\nA few more paces and Jack held up his fist, the signal for everyone to stop.\n\nThis was getting ridiculous.\n\n\"All right, fine!\" he yelled toward the abandoned building. \"I get it!\"\n\n\"Oh, no. No, no, no,\" Weiyan called out from behind him.\n\n\"It's all right.\" He kept his eyes peeled, hoping the alien would make another appearance. \"Piece of cake. We'll all go together.\"\n\n\"No, you don't understand,\" the girl sobbed.\n\nJack turned around to find Weiyan jerking down on her zippered fleece. \"What the hell?\"\n\nShe'd yanked up her pullover. Underneath, her entire lower stomach was covered in blood.\n\nThe ground rumbled.\n\n# CHAPTER TWELVE\n\n### BEIJING, CHINA\n\n### 1968 MAY 25\n\nHuang huddled beneath the central stairwell of the Ministry of Foreign Affairs, clutching the ancient scroll in his hand. Doors banged open and shut. Screams ripped through the air. Gunshots rang out across the many marbled floors.\n\nThe Red Guard had stormed the building, determined to fulfill China's Cultural Revolution by any means possible. Out with the old and in with the new. Their hatred of old customs, culture, habits and ideas had made the diplomatic corps a likely target. If Huang's bunkmate had not warned him that morning of the Red Guards' plans to kill diplomats who clung to those ideals, he would not have had the opportunity to rescue his precious find.\n\nFootsteps pounded on the stairs above. Huang retreated further back into the shadows and sat down with his back against the wall. He stuffed the scroll carefully into his faded blue proletarian's overalls. In anguish, he listened as great sculptures were toppled, paintings were slashed, and tapestries ripped in two. The Red Guards chanted their mantra rejecting the old ways again and again.\n\nThe China of Lord Yu was gone. Crushed in a revolt of perilous power mongering, swathed in false promises to the masses. Violence wrecked the streets. Troops of Red Guards killed and maimed anyone who did not fit the ideals of their leader, Chairman Mao. A man who claimed to want great peace and prosperity for his people.\n\nBut who acted no better than Lord Yu's enemies. At times, Huang wondered if indeed Mao Zedong and his ferocious Red Guards were not Anubis or Apophis in disguise.\n\nHuang drew in his legs and rested his head upon a knee. He had made a grave mistake coming to China, allowing his rescuers to do what they believed to be the right thing. When Sir Edmund Hillary discovered him in Antarctica, Huang hadn't the strength to out-run the Tau'ri. Hillary belonged to a land called New Zealand and along with his countrymen, he had come to reclaim the frozen land. To build bases of operations that would be known as Scott and McMurdo.\n\nIn broken Chinese, Hillary had insisted Huang return to what he believed to be his homeland of China. Huang had tried to tell Hillary the truth of his origins. He tried to convince him of the threat to the Tau'ri, of the being of light, but the explorer had only laughed and escorted him to a primitive craft called an airplane. Huang resigned himself to his situation in the hopes that upon returning to the land of his ancestors, he might find the means one day to return through the Chappa'ai and share both news of China and the being of light.\n\nTen years later, no opportunity had arisen. Huang learned to speak modern Chinese, was given a position as a gardener, and told to be thankful that he had food in his belly and clothes on his back.\n\nFar off in another part of the building, gunshots rang out. Believing himself safe for the moment, Huang withdrew the stolen scroll from his overalls. Upon hearing that all ancient artworks were to be destroyed by the Red Guards, he had taken the scroll from the wall outside the office of the Vice Minister of European Affairs.\n\nHe unrolled the scroll carefully. A few black painted pictograms covered the top half of the eight-foot long red cloth. The bottom half showed a man standing by a golden river, its banks swollen. The waters appeared ready to break from their boundaries, but the man appeared calm. Serene.\n\nHe bore an imperial tuft upon his chin. Dressed in a red mianfu, the horsehair tail of Lord Yu's green hat was longer than Huang remembered. Nonetheless, the scroll gave Huang solace. Assurance that his master and lord's eternal nature might someday allow Huang to return. To be forgiven. To once more take up his mantle as Dragon Guard or perhaps even First Prime.\n\nA door slammed open at the top of the five-story stairwell. Voices murmured. Three, perhaps four men. They began their descent, speaking in calm, hushed tones amidst the chaos.\n\nHuang quickly rolled up the scroll, secreting it once again in his overalls. Creeping forward, he peered up the stairs. Black shoes and blue pant legs climbed downward. He glanced across the stairwell to the exit. The door led out into the rear gardens overlooking a barricaded street. He'd used it often as a means to enter the building when in need of the facilities.\n\nHuang slid along the wall to the door, flattening himself against the cool stone.\n\n_Let your rapidity be that of the wind, your compactness that of the forest_.\n\nWith a silent prayer, Huang opened the door. He slipped outside, carefully closing the door behind him.\n\nBlooming flowers, budding trees, and great bushes lined the long white building of the Foreign Affairs Ministry. Huang had spent much of his time working in these gardens, along with a half-dozen others. Today, however, the gardens were empty, the other workers hiding in their homes in fear.\n\nBehind him, the door began to open, the men's voices loud with laughter. Huang picked up his gardening pail with its pruning shears and scurried to the closest tree he could find. He pulled out the shears and set to work, stowing his anxiety behind the most placid face he could muster. A worker's face.\n\n\"Halt!\" A Red Guard yanked the shears out of Huang's hands.\n\nForcing himself to remain calm, he turned toward the guard. Young, with a terrible haircut, baggy blue pants and a green shirt, the man was little more than an over-aged boy, at least fifteen years younger than Huang. Perhaps twenty, maybe twenty-five. Red squares lined each side of his collar.\n\n\"Wait, comrade,\" said another man behind him. Older, fuller of face and body, the man wore a blue-collared tunic. He had a vigorous manner about him, his eyes darting across the garden, seemingly assessing everything in mere moments.\n\nHe gestured for the guard to back away. Narrowing his eyes, he approached Huang. \"You are a gardener? A worker for the Foreign Ministry?\"\n\nHuang nodded, afraid to say anything that might encourage the Red Guard to attack him with the shears.\n\nThe man scowled. \"Gardening is a leisure of capitalistic classism. A look backwards, not forwards.\"\n\nHuang wanted nothing more than to proclaim his truth. To tell his origins. That by his very nature as a clone of the great Sun Tzu, he was a 'look backwards' to what was great, or rather, what had been great about China.\n\nKnowing he must survive, Huang held his tongue. He glanced up into the tree he had been working on. It was a cherry tree.\n\nHe returned his gaze to the man and offered a hesitant smile. \"I look toward the future, sir. This tree shall bear fruit. Sustenance for those in need.\"\n\nThe man bowed his head in respect. \"An admirable and necessary work. One that we will not keep you from, comrade.\"\n\nThe man gestured to the guard to return his shears. Huang nodded his thanks and the man walked off, his companions in tow.\n\nHuang allowed himself a silent, but deep sigh of relief.\n\nIt was short-lived.\n\n\"One more moment,\" the man said, striding back to the tree. \"Do you know who I am?\"\n\n\"I \u2014 \"\n\nThe man raised his hand once more, an almost imperial gesture. \"I am your Chairman. The work you do here is good, and I will remember your words. Gardening for sustenance is something China's people should achieve.\"\n\n\"Yes, Chairman.\"\n\n\"Can you read?\"\n\n\"I can write my name, but reading?\" Huang pressed his lips together. He had tried to learn the modern symbols of China's language. So far, the task had been difficult.\n\n\"I have not the time, Chairman.\"\n\n\"If you haven't read, then you are not familiar with the old ideologies. I approve.\" The Chairman pointed at the Red Guard who took Huang's shears earlier. \"Bring him inside.\"\n\n\"Chairman?\"\n\n\"As a worker, he will \u2014 What is your name?\"\n\n\"Huang.\"\n\n\"As a worker, Huang represents the people. He will make a fine diplomat.\"\n\nHuang staggered backward. \"But, Chairman, I know little \u2014 \"\n\n\"You know enough. Learn to read, but read sparingly. Care only for the people, and you will do well.\"\n\n\"What office should I take him to?\" asked the guard.\n\n\"Assign him to the new Vice Minister of the Americas.\"\n\n\"Americas?\" asked Huang.\n\nThe Chairman beamed. \"Have you never heard of the United States?\"\n\n# CHAPTER THIRTEEN\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/1910 HRS MCMURDO STATION\n\nAnother quake slammed through the outpost. George yanked Ambassador Zhu away from the hole as a new wave of snow and ice plummeted from the ceiling. Shielding Zhu with his body, he took the brunt of it, ice and snow pelting his back. The vibrations intensified. He quickly scanned the room to make sure Lee and Simmons were safe. Lee had grabbed on to the Mark IIs while the lieutenant stayed by the hole, hanging on to a nearby column.\n\n\"Simmons, get back from there,\" he yelled over the thunderous roar, but the lieutenant didn't budge. He kept looking down in that damn hole.\n\nAnother jolt. George fell sideways, his head smacking against the floor. As he struggled into a kneeling position, he could only hope the force field protected Jack and SG-1 down below. At this rate, the outpost would be torn to pieces.\n\nAnd the planet, too, if they didn't shut the damn thing off.\n\nThe tremor stopped, only to be replaced by loud hissing and popping sounds emanating from the force field. George glanced over at Zhu. She seemed none the worse for wear. Dr. Lee had risen to his feet, double-checking the straps on the Mark IIs.\n\n\"General, I can see them!\" Simmons pointed down at the hole.\n\nGeorge scrambled to his feet and raced over to the force field. It was as clear as glass. Down below, a motionless Jack laid next to Weiyan Shi on top of a glowing circular platform the size of a Hummer. Backlit trellised panels glowed underneath. Sprawled around the outer edges were Colonel Carter, Teal'c and Dr. Jackson, forming a semi-circle around the other two. Or more specifically, around Weiyan. She'd fallen exactly center.\n\nA control crystal protruded from her lower stomach. Or rather, the remains of one. Larger than a DHD controller, the red crystal was charred black.\n\n\"There should be more blood,\" Dr. Lee observed.\n\nZhu's ashen face made George wince. Normally, he'd ask Lee to be less clinical, and more sensitive to the fact that civilians were about, but he needed the scientist to focus on getting them out, not playing diplomat.\n\nHe'd have to do that himself.\n\nZhu sucked in a sharp breath. \"Is she...? Are they...?\"\n\nWeiyan's hand slid across her stomach, coming to rest just beside the burned out crystal.\n\n\"There!\" Simmons pointed toward Colonel Carter. Her left arm raised a few inches and then dropped back down. As if she was trying to pull something.\n\n\"How is that possible?\" Zhu asked, bending down beside the hole. \"Why are they all unconscious?\"\n\n\"I'm not sure.\" Lee rubbed his beard. \"I mean, I have a theory, but \u2014 \"\n\n\"Spit it out, Doctor.\"\n\n\"It could be a side effect of the photonic energy emitting from the device, or \u2014 \"\n\nGeorge raised a hand to stop the doctor's preamble. \"You'd said the device derives its energy from the sun. Why would it keep them unconscious?\"\n\n\"Hold on, let me show you.\" Lee ran over to his computer table. He lifted up a toolkit and grabbed a pad of yellow paper. \"See now, why couldn't I find this earlier?\"\n\n\"Perhaps the pad had disappeared,\" Simmons whispered.\n\nGeorge gave the lieutenant a stern look. \"Doctor, please.\"\n\n\"Sorry!\" Lee rejoined them, pulling a pen out of his coat pocket. He drew a fair facsimile of the platform below.\n\n\"Photons are pure energy. They register as both particle and waves.\" He drew a series of waves emanating out from the center. \"If General O'Neill and the others were knocked unconscious from their fall, they might be susceptible to Mu-waves. Electromagnetic oscillations that can affect neuron activity in their brains, forcing them to stay unconscious.\"\n\n\"Can these Mu-waves hurt them?\" George asked.\n\nLee shrugged. \"I don't really know that much about the physiological affect, but I'd bet they're having some pretty wild dreams.\"\n\n\"What about Weiyan?\" Simmons asked. \"Why isn't she bleeding?\"\n\nGeorge looked back down at the six-sided crystal sticking out of her stomach. A horrific sight to say the least. \"Does that crystal have something to do with it?\"\n\nLee shook his head. \"If the crystal was lit up, then yes, maybe. But it's broken which could be why the device is behaving this way.\"\n\nGeorge watched Samantha Carter's hand tighten into a ball. \"How much time do we have, Doctor?\"\n\nLee ran over to his computer and tapped a few keys. \"The sun rises in less than eighteen hours.\"\n\n\"I need to contact Major Davis.\" George glanced out the archway leading to the main area. Two airmen were carrying a civilian on a stretcher toward the elevator. Against the far wall, another airman was speaking on the radio, requesting a med-evac chopper.\n\n\"Sir?\" Simmons hurried to his side. \"I took the liberty of contacting the major before the latest tremor. He's still busy helping with recovery efforts at McMurdo.\"\n\n\"I almost envy him,\" George replied. At least someone was doing something constructive. He turned back toward the chair and it occurred to him that maybe there was something to be done here, too. \"It's time for a new approach, Doctor.\"\n\n\"Sir?\"\n\nHe turned to Simmons. \"How many trainees are still here?\n\n\"At least five.\" The lieutenant gestured toward the archway leading to the outer chamber. \"Should I \u2014 ?\"\n\n\"Whoa, General. Wait a minute.\" Lee's head shot up. \"Putting an untrained person with the ATA gene into that chair is what got us into this mess.\"\n\n\"And it might be the only way out at this point. If we don't take the risk, we could lose much more than General O'Neill and SG-1.\"\n\n\"We could lose the world.\" Lee slid out from behind his computer. \"I'll go find out which trainees are still here.\" As Lee hurried out, George dropped his gaze to the comatose General O'Neill. If the stakes weren't so high, he'd be amused by his prot\u00e9g\u00e9's uncharacteristically still form. Jack was a ball of energy, never not-moving. Even during briefings, he'd fidget with a pen, a coffee cup, or even a button on his BDUs.\n\nThe president had once referred to Jack as a living spin-top, never happy unless in motion. George snorted, remembering the man's recent promotion ceremony. President Hayes had pinned the stars on himself. He'd said it was like trying to shoot a moving target.\n\nBut now, Jack lay motionless. If the frown lines on his face weren't deeper than normal, George would've believed him dead.\n\nHe prayed it wouldn't come down to that.\n\nFootsteps announced Dr. Lee's return. To George's dismay, only two trainees followed him in. Ryan Hall, the retired army vet, and Sgt. Miguel Helado, the young Argentine whose arrogance was enough to convince George they were doomed before the soldier even got in the chair.\n\nHe strode up to the two trainees and stuck out his hand. \"I cannot thank you enough for being willing to help.\"\n\nHall gripped his hand and pumped it hard. \"Happy to, General. Just tell us what to do.\"\n\nGeorge turned to Helado to do likewise. \"You too, son. It's appreciated.\"\n\nThe Argentine didn't take his hand. \"I do this to save my own country, sir, not yours.\"\n\nThough taken aback, George held his tongue.\n\nLee coughed. \"He's right in a way, General. If we don't get the device turned off soon, Argentina and Australia will \u2014 \"\n\n\"Thank you, Doctor.\" He gestured toward the chair. \"Let's get this underway, all right?\"\n\n\"First try shutting the device down.\" Lee escorted Helado around the hole to the back of the chair. \"If that doesn't work, try moving the force field downward, away from the top of the device so we can deactivate it ourselves.\"\n\nHelado's face turned white as he walked across the wooden board leading to the chair and sat down. The seatback lit up. \"Now what?\"\n\nLee returned to his monitors. \"Concentrate on the device. Try to shut it off.\"\n\nHelado closed his eyes and the chair back reclined into the active position. His face scrunched up, he sucked in a breath...\n\nNothing happened.\n\n\"Concentrate on the force field,\" Simmons offered. \"You want to bend it downwards, away from the control crystal.\"\n\nHelado's eyes popped open. He glanced left and then right. \"What was that?\"\n\nLee shrugged. \"I didn't feel anything.\"\n\nHall rubbed his arms. \"I felt it, too.\"\n\n\"Like a wind. A breeze, yes?\" Helado shivered.\n\n\"General,\" Zhu said, her voice impatient.\n\nGeorge skirted around the hole until he was as close to the chair as possible. \"Sgt. Helado, if you could just focus?\"\n\n\" _S\u00ed_. Of course.\" Helado's eyes closed again.\n\nThe platform rattled. The back's illuminated panels flickered. The floor shook, not as intensely as before, but bad enough to topple over a decorative panel by the archway.\n\n\"Watch out!\" Simmons yelled.\n\nHelado jumped from the chair just before a chunk of ice fell on its seat. More ice and snow tumbled down from the ceiling as Helado ran across the wooden plank and took cover by the Mark IIs with Lee.\n\n\"Get back!\" George yelled.\n\nA shudder ran across the floor. Zhu and Hall crouched a few feet away from the hole, their arms covering their heads. Simmons was back at his default position, holding on to a column by the hole. Ice and snow smashed into the force field, immediately disintegrating into a burst of rising steam.\n\n\"Evacuate the room!\" George shouted over the din.\n\nHall and Helado dashed out, but Zhu crouched lower, staying put. \"I am not leaving!\"\n\nA final shake and the tremors stopped.\n\nGeorge exhaled, his breath turning to wispy fog in the cold room. Glancing back down at his unconscious people, he could only hope that whatever had them in its grip was also keeping them warm enough to escape hypothermia. Lying still like that for so long couldn't be good.\n\nSimmons seemed no worse for wear. Not a hair out of place. \"Earthquakes and sub-zero temperatures seem to agree with you, Lieutenant.\"\n\n\"Yes, sir.\" Simmons managed a faint grin. \"I was raised in a very warm climate, but I like the cold. Would you like me to see if any coffee survived that last round?\"\n\n\"You read my mind. Thanks.\"\n\nAs the young man hurried from the room, Zhu wiped a hand across her eyes. \"We have failed. Weiyan will die and \u2014 \"\n\n\"Not true,\" George said. \"We just haven't succeeded yet.\"\n\n\"That's the real irony in all this,\" Lee said. \"If Colonel Carter was here, she'd have the device shut down in no time.\"\n\nGeorge glanced at Samantha Carter's still form. Her hands no longer twitched, though her brow was creased. Whatever was going on in the colonel's brilliant head, it was clear she, too, was struggling to figure it all out.\n\n\"What happens now?\" Ambassador Zhu asked, more to the force field than to anyone else.\n\nThe scuffle of footsteps announced someone else had entered the room. \"I have spoken with the UN Secretary-General and he has agreed.\"\n\nGeorge whirled toward the voice.\n\nAmbassador Duebel glared back at him. \"Destroy the outpost now or I will arrange to have it done myself.\"\n\nOnce Sam managed to get Weiyan to lie down, she probed the girl's abdomen. She used her palm to scrape back the blood pooling around Weiyan's belly button, around her hips, up high around her ribcage. There had to be a good pint, at least.\n\n\"This makes no sense,\" she reported to the others. \"There's no incision, no cuts, not even a bruise.\"\n\n\"I don't feel anything.\" Weiyan laid a hand onto her bare stomach. \"It doesn't hurt, but,\" sucking in a breath, she lifted her hand, the fingers coated in the viscous stuff. \"No, no. I cannot bleed like this. Not now.\"\n\nThe girl was panting, clearly terrified.\n\n\"You've got to breathe slower,\" Sam told her gently.\n\n\"I will try.\"\n\n\"It's going to be okay.\" She took off her pullover. The fleece was made to wick away moisture in cold weather, she couldn't ask for better material to blot up the blood.\n\n\"Any chance at an explanation?\" The general knelt down beside them, his eyes narrowed in concern. \"I thought we'd decided this was all imaginary.\"\n\n\"I can't explain it, sir.\" The fleece soaked full of blood, and yet, still more seeped out around Weiyan's navel.\n\nThe general handed over his pullover. \"Teal'c's earlier knock to the head didn't leave any wounds, so what gives? None of this is real.\"\n\nWeiyan's breathing had calmed down from massive hyperventilation to ragged whimpers. Daniel crouched beside her. \"We know there are some similarities between here and the real world. Weiyan's...\"\n\n\"Go ahead,\" Weiyan sobbed. \"Say it. I lost my temper.\"\n\n\"It happens.\" Daniel squeezed her elbow. \"Just ask Jack.\"\n\n\"Watch yourself,\" General O'Neill half-warned, half joked.\n\n\"Is anyone thirsty?\" Sam asked. \"Or even hungry?\"\n\n\"No, which is strange enough, but if we're gonna talk about imaginary versus real, we should probably talk about this.\" The general pulled out his Zippo.\n\n\"Skaara's lighter,\" Daniel said. \"I thought you packed it away in that cigar box of yours after \u2014 \"\n\n\"After we pulled your ass out of Lord Yu's little fortress last year? Yeah, I did.\" The general flipped the lighter open and thumbed the wheel. The flame sputtered to life. \"Cute trick, huh? No lighter fluid.\" He snapped the lid shut.\n\n\"O'Neill, if you did not carry the lighter with you to Antarctica \u2014 \"\n\n\"Good question, T.\" He stuffed it back in his pocket. \"Part of me thinks we're dealing with Skaara, or another of Daniel's ascended buddies.\"\n\nDaniel shook his head. \"They don't interfere. Not like this.\"\n\n\"You can't be sure, Daniel.\" Sam wiped up the last of the blood on Weiyan's stomach. The bleeding had stopped. \"You can't even remember why you left them.\"\n\n\"Or why they kicked your sorry ass out.\"\n\n\"You're right,\" Daniel said. \"I don't know, but this just doesn't seem like something they'd do.\"\n\nSam pulled down Weiyan's shirt to cover her stomach. \"I just don't see how any of this is possible. We're here, but we're not \u2014 \"\n\n\"Yes, but when Weiyan got angry while in the chair,\" Daniel said, \"a quake happened.\"\n\n\"Subsequent earthquakes have directly tied into General O'Neill's and Weiyan's emotions,\" Teal'c added.\n\n\"True,\" Sam said. \"But if this is all imaginary, then where are we really? For all we know, we're still at the outpost.\"\n\n\"And if no one there can get to us,\" Daniel said, \"Weiyan could be injured. It could have happened when we fell.\"\n\n\"No!\" Weiyan bolted up from the ground. \"I don't feel any pain. I am well, I tell you.\" She clenched her fists. \"I am well. Please,\" her voice lowered, \"do not let me die.\"\n\n\"No one's dying.\" General O'Neill gathered Weiyan in his arms and held her as she cried.\n\nSam glanced up at him, surprised though warmed by his gesture. Then, a chill ran up her spine. A reminder of the impossibility of their situation. She hated to be the one to break his moment's kindness, but it had to be done. \"Sir \u2014 \"\n\n\"Leave it alone, Carter.\" He gave Weiyan a brief bear hug and then stepped back. \"We're getting out of here. Trust me.\" He tugged down on his shirt. \"We've got a good record of getting out of far worse things that this, don't we, people?\"\n\n\"General O'Neill is correct,\" Teal'c said. \"We have indeed escaped from far more challenging situations.\"\n\n\"Even holographic ones,\" Daniel added. \"Look at what happened with the Gamekeeper.\"\n\nThe general scowled. \"Do we have to?\"\n\n\"That was different, sir.\" Sam shook her head. \"We experienced yours and Daniel's memories over and over again, but we weren't hurt.\"\n\n\"So... This isn't a hologram?\"\n\n\"I don't think so, sir.\" Sam pointed toward the building at the far end of the basin. \"Otherwise, would that structure be here through every permutation?\"\n\n\"And why the city?\" Daniel asked. \"Or, more importantly, why give us clues like that obelisk? Or Jack having a sudden bout of speaking Ancient?\"\n\nGeneral O'Neill turned toward Daniel, \"You're convinced that Peda-lite Elysium \u2014 \"\n\n\"Pedion.\"\n\n\"Whatever.\" General O'Neill waved his hands. \"You really think that's a gate address?\"\n\n\"Sir, you were the one to fill in the last few syllables.\" She met his grim gaze with one of her own. Twice before, Sam had watched the Ancient Repository take over his brain and had always felt unsettled when it happened, as if the general would slip away and never come back.\n\nFrowning, he turned toward the building across the valley floor. \"I still feel like we're rats in somebody's maze.\"\n\n\"Or children with a puzzle,\" Teal'c commented.\n\n\"I am bleeding again!\" Weiyan pressed a hand against her stomach.\n\nSam pushed Weiyan's hand out of the way and pulled up her pullover. A blotch of blood oozed from her otherwise healthy skin. \"How do you feel?\"\n\n\"Sleepy.\" Weiyan glanced at the general and then sniffled back a sob. \"I will try to be brave.\"\n\nSam grasped her hand. \"Good, that'll help, but if we don't stop the bleeding soon, you could lose too much.\" She made to pull her hand free, but \u2014\n\nWeiyan grabbed hold with her other hand. \"I must confess.\"\n\n\"I don't understand.\"\n\n\"I am \u2014 I am a hemophiliac.\"\n\nSam gasped. \"That's impossible. How did you get on the trainee program if \u2014 ?\"\n\n\"I cannot tell you why I was accepted, Colonel Carter. Only that my bleeding,\" she touched her stomach, \"is serious!\"\n\nSam exchanged a look with General O'Neill. If Weiyan lost much more blood she would die, and all the assurances in the world wouldn't make a difference. \"Sir,\" Sam said, \"we need to figure a way out of here. Soon.\"\n\nHe nodded. \"I'm going to take Daniel and scope out the area. See if we can figure out a way off this rock.\"\n\n\"You can't do that, sir. Whatever's controlling this environment wants us to stay together.\"\n\n\"I am willing to journey with you, General.\" Weiyan placed a hand on her forehead.\n\n\"Yeah.\" General O'Neill shuffled his feet. \"I don't think that's such a hot idea. Teal'c?\" He jerked his chin toward the ground.\n\n\"You must conserve your strength.\" Placing an arm on her elbow, Teal'c eased Weiyan down to the ground. \"When we do find a means of escape, we will undoubtedly need your help.\"\n\n\"Teal'c's right.\" Sam scooped up the general's pullover and used it to wipe Weiyan's stomach. \"The bleeding's stopped again. What I don't understand is why this started all of a sudden. There's no wound.\"\n\nWeiyan dropped her gaze. \"Earlier I noticed my stomach was damp, but thought it only sweat.\"\n\nTeal'c lowered himself to the ground beside her. \"Whoever has placed us here is most cruel.\"\n\nGeneral O'Neill grunted. \"Just once I'd like us to meet a technologically advanced race bent on being nice.\"\n\n\"Don't forget the Tollan, sir.\" As soon as Sam mentioned their former allies, she regretted it. Thanks to the Goa'uld, they were all dead.\n\nShoving the thought aside, Sam patted Weiyan's shoulder. \"Stay put. You'll bleed less.\" She nodded at Teal'c who had taken up residence beside the young girl.\n\n\"I will assist her if needed.\"\n\nSatisfied, Sam joined General O'Neill and Daniel a few feet away.\n\n\"That guy keeps popping in and out,\" the general said. \"What the hell is that about?\"\n\n\"We don't even know if it is a guy,\" Daniel said.\n\n\"Guy. Girl. What does it matter?\"\n\n\"The general's right, Daniel.\" Sam shaded her eyes to scan the building in the distance. The sun's lack of movement alone was a clear indicator that they weren't in a natural environment. \"The one thing we do know is that whoever it is, they want us to go forward together.\"\n\n\"Maybe.\" The general shifted his stance forty-five degrees to the right of the building. \"But what about sideways?\"\n\n\"What good will that do?\" Daniel asked.\n\nThe general nodded toward the building's far right. \"I saw some rubble earlier. While on top of that hill.\"\n\n\"Rubble.\" Sam raised an eyebrow, unsure where the general was going with this.\n\n\"That's right, Colonel, rubble.\" He draped an arm around Daniel. \"And we all know how archaeologists love rubble.\"\n\nDaniel glanced at her and then General O'Neill. \"And you think where there's rubble, there might be something else to explain \u2014 ?\"\n\n\"How the hell to get out of here? Yes, Daniel. What do you think, Carter?\"\n\nSam exhaled, relieved to try something and even more, relieved to see the general back in a constructive frame of mind. \"I'm game. Daniel?\"\n\n\"I'm always up for rubble,\" Daniel said. \"What about Weiyan? Should we take her with us?\"\n\nSam thought about that for a moment. \"I'll stay behind.\"\n\n\"Negative, Colonel. We might need you.\"\n\n\"Sir, if you're right, and if heading on a diagonal from the building doesn't cause another seismic event, we could try this in phases. You two go first. If you find anything you deem worth securing a new position for, signal us. I'll cross, then Teal'c can bring Weiyan.\"\n\n\"Works for me.\" General O'Neill clapped his hands together. \"Let's move out.\" He raised two fingers in the air and then pointed toward their destination.\n\nWith a parting lift of an eyebrow, Daniel followed.\n\nThe two men trudged forward, or rather, sideways. Sam couldn't help but wonder what had spurred General O'Neill into action and dare she admit it, a better mood.\n\nTwenty feet. Thirty feet. She held her breath, steeling herself for another tremor.\n\nFifty feet. General O'Neill turned around and waved.\n\nRelieved, Sam waved back.\n\n\"Rats in a maze, Daniel.\"\n\n\"Jack, I don't think that's what this is all about.\" Topsoil spilled from Daniel's fingers. So far, he'd recovered a few marble shards, several large splinters of polished wood, and a three-foot wide granite pedestal \u2014 the same stone that had held the obelisk.\n\nIf they were inside some kind of holographic projection, someone was going to a lot of trouble to spare no detail.\n\n\"You're right about one thing,\" Daniel said. \"Rubble pretty much describes what we have here.\" Without any tools, he had to dig with his hands. It was slow, tedious work. He used his fingertips as a brush, sweeping back one area, and then the next. Sure, he could just plunge in and hope for the best, but if he broke something \u2014 a bit of stone, or something that could be a clue \u2014 he'd regret it.\n\n\"I'd kill for a pair of sunglasses,\" Jack muttered, standing a few feet away.\n\n\"You could help me sift through this.\"\n\n\"You're the archaeologist, not me.\"\n\nDaniel lifted his head, the relentless sunlight near blinding him. \"Do you want to find a way out of here or not?\"\n\n\"Someone needs to stand guard.\"\n\n\"Against what? Another earthquake?\" Daniel jabbed a finger toward another small mound of dirt, bits of marble, and granite. \"Tell me the big bad general isn't afraid to get his hands dirty.\"\n\n\"Big bad, my ass.\" Jack sank down beside the other mound and grabbed a handful of dirt.\n\n\"Careful,\" Daniel warned. \"Go slowly. We don't want to miss anything.\" He demonstrated how to sift through the dirt.\n\n\"Hey, if it gets us out of here, I'll pick this apart speck by speck.\" Jack got the hang of it pretty quick \u2014 which should have surprised Daniel, but it didn't. When motivated, Jack had a knack for picking up skills if the need arose.\n\nAfter watching him excavate for a few moments, Daniel returned to his own efforts. Eventually, the loose dirt he'd brushed aside gave way to more firmly packed soil.\n\nAnd a definite chunk of granite.\n\n\"Found something.\" He carefully swept away the soil packed around a three-inch wide, foot-long vertical piece of smooth granite. \"This looks like a piece of the obelisk!\"\n\nFlipping it over, he noticed blocky indentations along the back. He brushed away the embedded dirt and there they were, the letters P and E. The first two Ancient letters for the word Pedion.\n\n\"That's good, right?\" Jack took the granite chunk from his hands.\n\n\"Maybe. I'm not sure what we can do with it, though. We already know the planet's name. We need the rest of the obelisk if we're going to make any sense out of this.\"\n\n\"If you found that, we'll find the rest and you'll figure it out,\" Jack muttered. He returned to his work with the topsoil.\n\nSomething had been eating at Jack earlier. That much was clear to Daniel. Feigning nonchalance, he cleared away more dirt from the immediate area. \"So, what's going on with you, anyway?\"\n\n\"Digging like a kid in a sandbox. Wanting to ring the neck of whoever decided to treat us like lab rats.\"\n\n\"We're not rats.\"\n\n\"Oh, sure, you say that now.\" Jack shrugged. \"But hey, we're trapped somewhere and need to get out. Same old, same old.\"\n\n\"That's not what I mean.\" Daniel deliberately kept his focus on digging. \"You've been acting wonky since we left the SGC for Antarctica, and now \u2014 \"\n\n\"Wonky?\"\n\n\"Yes, wonky... Ah!\" Daniel found another chunk of granite. The letters D and E. He placed it next to the first piece.\n\n\"Sweet.\" Jack handed over a matching piece of granite, this time a good four inches wide and twice as long. The Ancient block-style letters for L, Y and S.\n\nDaniel laid the granite down beside the others. He looked toward Jack, hoping to pick up their conversation where it'd left off, but Jack had returned to digging silently, a clear sign that he wasn't interested in talking. With a sigh, Daniel carried on sifting dirt.\n\nFinally, Jack spoke. \"Since when did 'wonky' become part of your vocabulary?\"\n\n\"Don't change the subject. Something's eating you.\"\n\n\"Actually, I think that is the subject.\" Jack waggled a dirt-covered finger toward him. \"You've changed these last few years.\"\n\n\"I've changed?\"\n\n\"Yeah, you have.\" Jack attacked a new mound, no longer being careful. \"Since coming back from Oma-land. For one thing,\" he tossed a handful of dirt aside, \"you're snarkier.\"\n\n\"And that's bad, how?\"\n\n\"I'm not saying it's bad, just...\" Jack shrugged. \"Be careful. You'll get yourself in trouble.\"\n\nDaniel snorted. \"This from the king of snark.\" And that's when it hit him. \"You're changing the subject again.\"\n\n\"What do you want me to say? I'm getting tired of \u2014 \"\n\n\"This is about what happened on Tegalus, isn't it?\"\n\nJack single-mindedly tore into a new mound. \"I can distinctly remember telling you on the plane that \u2014 \"\n\n\"You think it's my fault the Caledonians launched the missile strike. That's what's bothering you.\"\n\n\"Trust me, Daniel, those jerks had it coming.\" Jack flashed him a tight grin. \"As far as I'm concerned, you can get caught up in all the civil wars you want. Just do me a favor and get together with Teal'c. Maybe the two of you could schedule your mishaps a little further apart.\" He plunged his hands back into the dirt. \"Bring Carter while you're at it.\"\n\nTegalus. Teal'c's near-death experience in the virtual chair. Sam's capture and torture by the Replicator Fifth...\n\nDaniel rocked back on his heels. \"You don't blame me,\" he realized aloud. \"You blame yourself. You think you could \u2014 \"\n\n\"Found another piece.\" Jack tossed over a fist-sized rock.\n\nWith a shake of his head, Daniel uncovered the letter I. He laid the granite piece down beside the others. \"We've been at this for, what?\"\n\n\"Twenty minutes, I guess.\" Jack waved his wrist in the air. \"No watch. Which is really annoying, by the way.\"\n\nDaniel shook his head. \"I don't mean the digging. I mean the Stargate Program. We've been at this a really long time. The losses, the near misses \u2014 \"\n\n\"It's all part of the job.\"\n\n\"Maybe being a general's your problem. You've drifted away from who you \u2014 \"\n\n\"Daniel...\" Jack's voice took on that warning tone, the one that said 'stop pushing.' He shoved both hands into the hole he'd pulled the last stone from. His shoulders jerked a few times until he fell back on his rear, another large chunk of granite in his hands.\n\nDaniel wiped the thick dirt off Jack's latest find, discovering the final letters U and M. The Ancients' version of an M always surprised him, looking more like a blocky Y.\n\n\"There are benefits to being a general, you know.\" Jack wiped his hands on his pants. \"Best parking space in the lot. Well, next to the NORAD commander, that is. Plus, I get to eat great commissary food, sit back \u2014 \"\n\n\"Not here, Jack. You can't sit things out while we're trying to figure out how to get off this world.\"\n\n\"You don't think I know that?\"\n\nDaniel laid the last piece down beside the others. \"What I think doesn't matter.\"\n\n\"Stop trying to play shrink. You're not very good at it.\" Jack jerked his chin toward the obelisk fragments. \"I don't think we're going to find anymore chunks of this stuff.\"\n\n\"This doesn't make sense. On the original obelisk, there was writing here.\" Daniel pointed to an empty area below the middle letters. \"Where did it go?\"\n\n\"Does it say Pedion Elysium?\"\n\n\"You've had the Ancients' knowledge downloaded into your head, you were forced to learn the language yourself to get out of that time loop, and you still can't read Ancient?\" He traced the first few letters and sounded out the pronunciation slowly for Jack to follow along.\n\n\"Stop!\" Jack put his hands over his ears. \"There's only so much information this brain can handle at once, and Ancient just isn't high on my list of priorities.\"\n\nDaniel fought the urge to roll his eyes. \"Yes, then. It says Pedion Elysium.\"\n\n\"Which clearly someone, or something, wants us to see. The question is, why?\"\n\n\"You think the being that talked to Teal'c \u2014 this alien \u2014 is playing a game with us?\"\n\n\"Yeah, and this,\" Jack waved a hand at the granite pieces, \"is a clue to that game.\"\n\n\"Maybe it isn't a game.\" Daniel stared at the words. He'd managed to work out a gate address \u2014 thanks to Jack's odd little Ancient Repository relapse \u2014 but so far, he couldn't figure out why that address was important. \"Could the alien be trying to tell us something? Something we need to know, but for whatever reason, the being has to be sneaky about telling us?\"\n\n\"Sneaky?\" Jack snorted. \"That's a snarky word if ever I heard one.\"\n\n\"Maybe it is, but unless you can tap into any last shreds of the Ancient Repository like you did earlier \u2014 \"\n\n\"Did not.\"\n\n\"Did so.\" Daniel deliberately glared at Jack. \"This is important. We need to know the significance of this name or \u2014 \"\n\n\"We'll go on being rats in a maze.\"\n\nTeal'c stretched his legs out. Fatigue washed over him, a moment's weariness he allowed in hopes that like a wave it would crest and become no more. He sought inward for his strength, believing he had only to grasp \u2014\n\nA hand pressed against his knee. \"You all right?\"\n\nTeal'c opened his eyes and forced himself to smile at Colonel Carter. \"I am simply performing a moment's kelno'reem.\"\n\n\"Kelnor-what?\" asked the young girl sitting beside him.\n\n\"Kelno'reem. A meditative practice to steady the mind.\"\n\n\"I thought you didn't do that anymore.\" Colonel Carter's eyes narrowed in suspicion. \"At least, not since your symbiote was removed.\"\n\nTeal'c knew he should speak the truth to his friend, but as there was nothing she or anyone else could do, he said nothing.\n\nFortunately, Colonel Carter had more pressing issues to attend to. Weiyan Shi no longer bled, but she could no longer stand, either. She had paled considerably. A sheen of sweat covered her forehead and her lips were pale. Teal'c exchanged worried looks with his teammate, aware that each hour that passed diminished the chances of her survival.\n\n\"I wish to go home,\" Weiyan Shi whispered. She reached out her hand to him.\n\nUnder other circumstances, Teal'c would not lie, but this young woman \u2014 barely more than a child \u2014 required comfort. As if she was his own, he took her hand, pressed it firmly in assurance. \"Even now O'Neill and Daniel Jackson seek a means of escape.\"\n\n\"That's right,\" Colonel Carter said. \"We do this all the time. Get into trouble, figure it out, jump back through the Stargate, and \u2014 \"\n\n\"But General O'Neill said that without a Stargate, we have no means of escape.\"\n\nTeal'c squeezed the girl's hand. \"You do not know General O'Neill as we do. He often jokes in impossible situations.\"\n\n\"It's a deflection thing,\" said Colonel Carter. \"The general's just trying to keep things light so everyone can focus.\"\n\nWeiyan Shi coughed. \"He has not joked very much in my presence.\"\n\n\"He just needs to get to know you.\" Colonel Carter lifted a hand above her brow and looked further into the valley.\n\nGently releasing Weiyan Shi's hand, Teal'c followed her line of sight toward the location of Daniel Jackson's and O'Neill's dig. The two men appeared to be pulling items from the ground. \"Perhaps they have found something of value.\"\n\n\"I wish we had radios,\" Colonel Carter said wistfully.\n\nAs did Teal'c. Nonetheless, they needed to know if any progress had been made. \"I will attend to Weiyan Shi while you confer with the others.\"\n\n\"And risk having the ground crack open again?\" asked Colonel Carter. \"I don't know, Teal'c. We'd agreed to wait for a signal from the general.\"\n\n\"As O'Neill and Daniel Jackson were able to proceed, perhaps the being which keeps us here will allow you to do the same.\"\n\n\"All right. I'll talk to the guys and if they've found anything, I'll signal you to join us.\" With one last reassuring smile for Weiyan Shi, Colonel Carter headed toward the others. Teal'c watched as she carefully repeated their steps, moving on a diagonal to the building's right. Assured that she would not come to any harm, he looked back at Weiyan Shi. The girl had curled up on her side, her eyes closed. Her chest rose and fell in a somewhat steady rhythm.\n\nTeal'c closed his own eyes again. Striving for a moment's balance, he inhaled deeply. He could not perform a deep level of kelno'reem now that his symbiote had been removed. That said, he could still assert some level of control over the rhythm of his heart, the temperature of his body, and even the tumultuous nature of his mind when necessary.\n\nHe exhaled forcefully, expelling worn breath out through his nostrils. He inhaled once more, allowing the warm air of their surroundings to fill his lungs. He looked inward, imagining the air mixing with his blood.\n\n\"You fool the others,\" Weiyan Shi said by his side, \"but I can see that you aren't well.\"\n\nTeal'c bowed his head. \"Admittedly.\"\n\n\"You aren't human, are you?\" Weiyan Shi pushed up to a seated position. \"You speak English like the Americans, but you use different words, as if \u2014 \" She sucked in a short breath.\n\n\"Are you in pain, Weiyan Shi?\"\n\nPressing her lips together, the girl nodded curtly.\n\n\"Should I retrieve Colonel Carter?\" He began to rise.\n\nWeiyan Shi grabbed his leg. \"Please, stay with me. I'll be all right. I'm just tired. Like you.\" She tugged at his pant leg. \"I'll try to feel better, ok?\"\n\n\"You use the word 'try' very often.\"\n\n\"Isn't that important? We should always try to do what is asked of us.\" This last she recited as if from a child's nursery rhyme.\n\nCrossing his legs, Teal'c pointed this out.\n\n\"My mother taught me to always try. You sound like my father. Or at least, the one time I spoke with him. He didn't believe in trying. He said one must simply do what needs to be done.\" Weiyan Shi lay down, resting her head upon her clasped hands.\n\n\"Your father sounds like he is a wise man.\"\n\n\"Was,\" she mumbled. Her eyes closed.\n\nTeal'c followed suit. He dropped his chin toward his chest, breathed in, and \u2014\n\n\"Will General O'Neill be angry if I ask where you're from?\"\n\nHe peered once more at this curious young woman and considered her question. In the end, the responsibility for her knowledge was her own, and given their current circumstances, it would be unjust not to give her at least some sense of the truth.\n\n\"I am from Chulak.\"\n\nHer eyes widened. \"Chulak. Is that far from here?\"\n\n\"As we have no means of knowing where 'here' is, I can only say that it is far from Earth.\"\n\nShe coughed again, the sound worrisome. \"You're the first alien I've ever met.\"\n\n\"My ancestors were of Earth.\"\n\n\"Does that mean we're related?\"\n\n\"I do not know.\" Teal'c had never considered that possibility. If the Jaffa had indeed evolved from the Tau'ri, those who walked upon Earth now could be his distant kin. Was it possible that this was yet another reason for his affinity to O'Neill, Daniel Jackson and Samantha Carter? The idea gave him comfort.\n\nA sudden boom across the sky cast aside his musings. The sound was familiar, though certainly out of place. He bolted to his feet, listening. Dizziness threatened to overcome him, but he pushed it aside.\n\n\"It wasn't me,\" Weiyan Shi said. \"I am not angry. General O'Neill must have lost his \u2014 \"\n\nTeal'c shot up a fist to signal silence. Though short-lived, the sound reverberated across the valley, the vibrations rippling through the air long after the initial boom had dissipated. He strained to hear confirmation of what he feared.\n\nHe did not have long to wait. A high-pitched whine, that of a ship slowing as it entered the upper atmosphere, emanated from the skies above. Moments later, a Goa'uld mothership appeared from behind the clouds and began its descent.\n\n\"How is this possible?\"\n\n\"What is it?\" Weiyan Shi cried out.\n\n\"An enemy you should have no wish to meet. Can you run?\" He grabbed her arm and lifted her up. The effort was difficult, but adrenaline now coursed through him, casting aside his fatigue. Another glimpse at the ship confirmed that it drew closer, no more than a half-mile away.\n\n\"I \u2014 I will try.\" Weiyan Shi stepped forward, stumbled, and fell to the ground.\n\n\"No, Weiyan Shi. We will not try.\" He swept her up into his arms. \"We will do.\"\n\nHe ran to join the others.\n\n\"Orders, sir?\"\n\nJack watched the Goa'uld mothership descend at break-neck speed, headed right toward them. At its current trajectory, it would set down smack between them and the building. Too close for comfort.\n\n\"O'Neill!\" Teal'c ran toward them, Weiyan slung over his shoulder. He set her down beside Carter. \"We must take cover!\"\n\nToo hard to believe.\n\n\"Jack! Shouldn't we do something?\" Daniel jumped up from their little \u2014 and now insignificant \u2014 pile of granite letters.\n\nToo impossible.\n\n\"General, we must hide!\" pleaded Weiyan.\n\n\"From what?\" He yanked the Zippo out of his pocket. Flipped the lid back. Thumbed the wheel. The flame came to life. Nice and neat.\n\nSunlight bounced off the incoming mothership's black metallic hull. Its pyramidal edges looked sharp enough to cut anything in two as it entered the lower atmosphere.\n\nJack waved a hand over the lighter's flame. Nothing. Nada. Not even the littlest bit of heat. \"Yeah, that tears it.\" He snapped the lighter shut and exchanged a glance with Daniel.\n\n\"You do not believe it is real,\" Teal'c observed.\n\nJack stuck the lighter back in his pocket. \"I think that's been pretty well established. How much longer do you think we have before that thing lands?\"\n\n\"Ten, perhaps fifteen minutes.\"\n\n\"Carter, am I crazy?\"\n\nShe raised her eyes upwards and frowned. \"No, sir. I don't think you are, but \u2014 \"\n\n\"Yes! You are all crazy!\" shouted Weiyan.\n\nJack whirled toward the girl, ready to calm her down. When he saw her digging a hole in the ground, he realized it would take more than just a few words.\n\nCupping her hands, Weiyan was scooping back armfuls of dirt. Her hair had half-tumbled from its ponytail, hanging around her face in a disheveled mess. The sandy soil covered her blood-soaked shirt as she frantically dug like her life depended on it.\n\nWeiyan was right, in a sense. Their lives did depend on what happened next, but not necessarily in the way she believed.\n\n\"Whatcha doing?\" he asked as calmly as he could.\n\n\"What you will not. We must dig a hole. Hide. Protect ourselves.\"\n\n\"Weiyan Shi.\" Teal'c bent down beside her and grabbed her wrists. \"What you see is not real.\"\n\nWeiyan yanked her arms loose. \"You cannot be certain. What if they harm us? We will be killed.\"\n\nA shadow passed over them. The ship was coming in for landing. Jack had to fight every instinct that screamed out to help the girl dig that hole.\n\n\"We must hurry or we will die!\" Weiyan kept on digging. She worked fast, creating a hole at least two feet deep.\n\n\"Jack, you're sure \u2014 ?\"\n\n\"I'm sure, Daniel,\" he roared over the mothership's entry rockets. \"Sure as I've ever been. But somebody needs to stop that girl before she hurts herself.\"\n\nWeiyan jumped in and curled up in a ball. Satisfied that she couldn't get into any more trouble, Jack returned his focus to the incoming ship. Not for the first time, he wished he had his scope.\n\nDaniel glanced over his shoulder at Weiyan and then back at Jack. \"I don't get it.\"\n\nJack snorted. \"Welcome to the party.\"\n\n\"That's not what I mean. Why aren't we experiencing another earthquake? She's panicked. You've got to be feeling at least a bit anxious. Although, you've become awfully calm about all this.\"\n\n\"Don't let the bravado fool you. Seeing that ship is akin to a dog hearing a whistle.\"\n\nThe ship touched down in front of the building. Jack noticed there were no portholes like those he'd seen on Cronus's and Apophis's motherships. He held his breath as the ground rumbled from the impact, the sound too similar to the earlier quakes for his taste. The pyramid's primary hull retracted upwards, revealing a multi-tiered base with lit panels underneath.\n\n\"That ship's taken some serious damage,\" said Carter.\n\nScorch marks streaked across the side facing them, but the thing Jack noticed the most was the lack of an outer frame. It was like the pyramid had lost half its mass.\n\n\"There's something off about the ship, T. Something's missing.\"\n\n\"That is not a _ha'tak_ class ship, O'Neill. It is far older.\"\n\nJack studied the ship's pyramid shape again. Then it hit him. The thing was a dead ringer for Ra's ship. A shudder went through him. \"God, tell me that isn't Ra. With all the crazy-assed backwards things we've experienced, that'd be \u2014 \"\n\n\"I do not believe it is, O'Neill.\" Teal'c pointed toward the ship's apex. \"The sigil is missing. I have seen it many times as Apophis' First Prime in battle against Ra's fleet.\"\n\n\"Then who the hell is it?\"\n\n\"I cannot say.\"\n\nA horrific grinding noise erupted from the pyramid's side. Squinting against the sunlight, Jack could make out a series of gantries opening up along the pyramid's base. As the doors slid upwards, swarms of armored Jaffa poured out of the ship.\n\nThere had to be at least a thousand of them, broken into four battalions, each group stomping off in a different direction. If his gut was wrong, and these bastards were real, there was a good chance this would become SG-1's last stand.\n\nThe Jaffa would make toast of them in minutes.\n\nHe shoved aside the maudlin thought. That pity-fest had ended hours ago.\n\nA battalion headed directly toward them, led by a Jaffa wearing a red cloak instead of the typical dark gray. His silver tattoo glinted in the sunlight, but he was still too far away for Jack to make out his forehead marking.\n\nCarter tensed beside him. \"What if you're wrong, sir?\"\n\n\"If I'm right, those Jaffa are just phantoms. Figments.\"\n\nDaniel leaned forward to get a better view. \"We should be able to figure out whose phantoms they are pretty soon.\"\n\n\"O'Neill,\" said Teal'c. \"If they are phantoms, should we not approach them for a closer look?\"\n\n\"Let's first see how this phantom parade plays out.\"\n\nThe lead Jaffa stormed toward them, backed up by enough metallic stomping from his horde to make Jack's trigger finger itch. Without a rifle, all he could do was stand and watch.\n\nAnd pray like a priest he was right.\n\nAnother fifty feet to go. The Jaffa didn't miss a beat as they neared. Further evidence for Jack that they weren't real. They weren't there.\n\nCarter widened her stance, balling her fists as if it would do any good.\n\nIt wouldn't, but he sure liked knowing she had his back.\n\nTwenty feet to go.\n\nTeal'c stepped up to his other side with a fierce snarl of a smile. He always did like a good fight.\n\nTen. The lead Jaffa and his cronies kept marching toward them, their matching tattoos clear as day.\n\n\"That's Lord Yu's sigil!\" Daniel said.\n\nThe Jaffa kept on marching.\n\nRight through SG-1.\n\n# CHAPTER FOURTEEN\n\n### EMBASSY OF THE PEOPLE'S REPUBLIC OF CHINA\n\n### WASHINGTON, D.C.\n\n### 2003 FEBRUARY 14\n\nHuang sat down behind his desk, stifling yet another yawn. It was past ten p.m. Why Ambassador Chen had insisted he remain in the office was unclear, but China's UN Security Council representative was nothing if not a persistent man. Chen's phone message implied grave importance. A matter unable to wait until morning.\n\nAs second-in-command to China's senior ambassador, Huang's office afforded him a magnificent view of America's capitol city. A blanket of snow had fallen, covering automobiles in great banks of white. While the majestic snow-covered mountains and trees of his long lost home were missing, the mounds reminded him of his time in Antarctica.\n\nWithout the clawing desperation to survive or the terrible isolation.\n\nOr the endless reminder of his failure to Lord Yu.\n\nHuang glanced about his office with its indulgent mahogany furniture, modern conveniences, and photos of his time among the Tau'ri. By their terms, he should consider himself successful. By his terms, nothing was further from the truth.\n\nHis gaze fell on a crudely painted pot containing a cherry tree sapling. A gift from a local elementary school. Pulling the pot closer, Huang reflected upon the irony that it was a cherry tree that had catapulted him from gardener to his current position. A position that had allowed him to thrive, but had yet to give him the access he sought to Antarctica, so that he might journey home.\n\nDecades had passed. Time had marched on. And yet, he still craved the opportunity to return to Lord Yu and seek forgiveness.\n\nHe dipped a finger in the pot, scooping up some soil. Earth, he reminded himself, finding it odd that the American's name for their planet was the same as the dirt upon which they stood. As he contemplated the ironies of Earth's many languages, he lifted his finger and pressed the soil upon his forehead. A downward stroke, two horizontal slashes \u2014\n\nThe door barged open. \"We must speak.\"\n\nHuang hurriedly wiped the _w\u00e9nsh\u0113n_ from his forehead as Chen entered the office. The ambassador's face was haggard with worry, his suit jacket unbuttoned, and his tie-knot loosened.\n\nAs he closed the door behind him, Huang took note of two folders under his arm. A red one, clearly belonging to the embassy, and a dark blue one as well. In his free hand, he carried a foot-long metal box.\n\n\"How may I be of service?\" Huang rose from his chair.\n\n\"First, you must sign this.\" Chen plopped the red folder down on Huang's desk. Embossed across the top in modern Chinese type were the words TOP SECRET.\n\nHuang opened the folder and scanned the single piece of paper inside. He had eventually learned the modern Chinese letters, the writing far simpler than ancient Chinese, or even Goa'uld for that matter. The paper contained a simple oath, a promise of non-disclosure and a vow to remain silent. If signed, Huang's security clearance would be heightened to the utmost level.\n\n\"It is my honor to serve, Ambassador.\" Huang quickly signed the document and returned the folder.\n\n\"Always so formal.\" Chen cocked his head. \"There are times I wonder if you were born in the right century.\" Placing the metal box on the desk's edge, he sank into the opposing chair. \"It is no wonder your wife divorced you.\"\n\n\"I was not worthy.\" Huang bowed his head. Taking a wife had been a momentary weakness, a forbidden act that had led to the birth of a child, yet another of his failures to follow Lord Yu's commands. He had left before the birth, selling the scroll depicting his master to the New York Museum of Art. He had sent the subsequent profits to his ex-wife to assist in raising their daughter.\n\nA daughter he could not bring himself to meet.\n\n\"While your worth to family may be in question,\" Chen tossed the blue folder in front of Huang, \"your tireless dedication to China holds value. I must share what I learned today with someone I trust.\"\n\nThe blue folder's cover carried several American emblems. That of the U.S. President, another of the United States Air Force, but it was the image at the bottom that caught his attention. Huang's fingertips grazed the logo in disbelief, for it surely looked like one of the Chappa'ai's symbols. Superimposed upon the image was an inverted English 'V'. At the top was embossed the letters SGC.\n\nHe repeated the letters aloud to Chen who waved a hand at the folder. \"Stargate Command. Open the folder.\"\n\nHuang did so, his eyes immediately glued to the top page. A photograph of the Chappa'ai stared back at him.\n\nStunned, he sat back in his chair. \"Please explain.\"\n\nChen did so. A secret meeting had been held that day at the American's Pentagon. Along with the ambassadors from France and England, he had been informed of the discovery of the \"Stargate\" in 1928, buried in the sands of Giza, Egypt. Eight years ago, a Dr. Daniel Jackson had deciphered its meaning and the device had been activated. For the past six years, American military personnel and scientists had traveled through the Stargate to hundreds of planets. Thousands more had been charted.\n\nHuang struggled to keep his face implacable. Flipping the page, he looked up at Chen and asked, \"Have the Americans come in contact with \u2014 ?\"\n\n\"Yes!\" Chen's eyes widened. \"Millions of humans live across the galaxy, taken from Earth centuries ago by a fierce enemy known as the Goa'uld.\" He pronounced the name incorrectly, eliminating the \"a\" so that it sounded more like the English word for ghoul with a \"d\" on the end.\n\nHuang resisted the temptation to correct him. \"What is known of this enemy?\"\n\n\"They are parasites. Incubating in what are called Jaffa warriors. They take hosts as needed so they may rule over many worlds.\" Chen reached over to the folder and turned the page, revealing a photograph of several System Lords including Cronus, Nirtti, and...\n\n\"This one calls himself Lord Yu!\" Chen stabbed a finger at the likeness of Huang's master. \"This thing, this alien, claims to have ruled China for many centuries.\"\n\nReeling in astonishment, Huang raised the photograph for a closer look. His lord sat on a modern leather and chrome chair. A Tau'ri chair. The timestamp at the bottom left indicated the photograph had been taken in July of 1999. Only four years ago.\n\n\"Where was this taken?\" he asked Chen.\n\nChen waved a hand at the photograph. \"In Colorado. At Stargate Command during negotiations surrounding a non-aggression treaty. A treaty the Americans dared agree to without informing China or any other country. It is outrageous, to say the least, but there are further concerns pressing.\"\n\nLord Yu had been here. Here on Earth!\n\n\"We met another race of beings today,\" Chen said.\n\nHuang dropped the photograph, a lump forming in his throat. Was it possible that the being of light had made its existence known as well?\n\n\"We met an Asgard. A small human-like being who instantaneously transported himself into the briefing room.\" Chen shook his head in disbelief. \"Thor, Supreme Commander of the Asgard fleet, and a devoted friend of the Americans, it would seem. Clearly, they possess significant technologies, although,\" he pushed the metal box toward Huang, \"we may have some of those technologies in our possession if we can determine their use.\"\n\nHuang opened the box. Inside was a pair of fist-sized Goa'uld communication devices.\n\nUndoubtedly, the communication devices would be disabled. For the first time in many decades, he regretted not having the naquadah in his blood to operate them. To contact Lord Yu and beg his mercy.\n\nThe silver skin shimmered.\n\nWas it possible? If they were they still active, the right voice command might be enough to reach his master. Huang raised an eyebrow to cover his shock. \"Do you know what they do?\"\n\n\"I do not.\" Chen frowned.\n\n\"How \u2014 ? Where did you get these?\"\n\n\"Do you remember a few months back when the Russians claimed they lost a missile submarine due to a meteor crashing into the Pacific some seventy-five kilometers away?\"\n\n\"The _Rostov_?\" Huang asked. \"Did we not determine that to be a cover story? The ship was taking on supplies in the harbor at Vladivostok at the time.\"\n\n\"Apparently, a Goa'uld 'mothership' crashed into the Pacific Ocean. A ship belonging to a Goa'uld named Anubis.\"\n\nHuang sat up straight upon hearing the name of Lord Yu's long dead enemy. How was this possible? As a cr\u00e8che-child, he had learned by heart the tale of Lord Yu's battle a thousand years ago. A tale in which his master has vanquished the most evil of System Lords for having attempted to destroy Ancient China by flooding the Yellow River.\n\n\"We found these on the ocean floor during what we'd hoped to be a salvage operation.\" Chen hefted one of the communication devices in his hand. \"It would seem the Russians knew about the Stargate Program for several years. In return, they've received untold benefits in the form of alien technology, including plans to the F-302, an advanced interceptor plane.\"\n\nThe ball's shimmer dissipated.\n\n\"The Russians knew?\" Huang put a hand to his mouth to cover his disappointment.\n\nChen tossed the ball back into the box. \"Turn the page.\"\n\nHuang did so. Two photographs lay side-by-side. One of a team of green-clad American soldiers shooting at a glider, the other of a crude box-like duplicate, its wingtips exhibiting none of the grace of the Goa'uld aircraft.\n\n\"That is the F-302,\" Chen said with barely suppressed anger. \"American military hardware inspired by knowledge gathered through the Stargate Program and capable of both aerial combat and space flight.\" He slammed his hand down on the desk. \"Russia has the plans, but China? No. All we have been offered is the opportunity to send observers in exchange for keeping this enormity from public knowledge.\"\n\n\"Observers?\" Huang swallowed. \"I would volunteer for such a posting.\"\n\n\"You?\" Chen shot him a glance. \"Nonsense. I need you here as my second. This is a task for younger people. As Security Council members, however, we will receive weekly reports from the SGC.\"\n\n\"And our observers?\"\n\n\"Of course,\" Chen said with a knowing smile.\n\nHuang flipped to the next page. Another photograph of the Chappa'ai \u2014 the Stargate, he corrected himself. Deactivated in the picture, it sat at the far end of a large concrete-walled room, surrounded by U.S. military personnel armed with rifles. From Chen's information, Huang could only assume the photograph was taken at Stargate Command in Colorado.\n\nSurely, there must be a way to infiltrate this operation. To activate the Stargate and return home. What he needed was information, the more the better. As his ancestor Sun Tzu once said, 'Knowledge is power.'\n\nHuang would gather all the knowledge he could before planning his next move. From the corner of his eye, he glanced at the case containing the communication devices. A silvery ripple shot across the untouched ball's surface.\n\nIt was most definitely active.\n\nHis mind raced. Could he remember the right phrase? Could he truly and finally contact Lord Yu?\n\n\"Who runs this operation?\" he asked with forced casualness.\n\n\"General Hammond,\" Chen replied, \"and a Colonel Jack O'Neill. My impression from Senator Kinsey is that this colonel is something of a fool, though a brave one at that.\"\n\nHuang closed the folder, the beginnings of plan taking shape in his mind. A fool, no matter how brave, would be a minor inconvenience.\n\nOne easily thwarted given the right subterfuge.\n\n\"May I study these for a while?\" he asked Chen.\n\n\"As long as you do not leave the facility, of course.\" Chen strode toward the door. \"I will be interested in your report.\" The ambassador dipped his head and departed.\n\nHuang waited an hour, to be sure the embassy had shut down for the evening and that he was alone. His breath held, he pulled the box to him, opened it and spoke the words.\n\nSoon after, Lord Yu's visage arose from within the ball's depths. Huang dipped his head in obedience.\n\n\" _You live_.\"\n\n\"I do, my lord.\" Huang sucked in a tremulous breath and raised his eyes to meet his master. Behind Lord Yu, the twisted limbs of a miniature cherry tree blossomed on the far wall. At its base sat his prized jade statue, a woman cradling a baby.\n\n\" _Have you lived according to my tenets_?\"\n\n\"I \u2014 I have,\" Huang lied. He told himself that once back in Lord Yu's service, he would tell the truth of his digression. He would admit to fathering a child.\n\nBut not today. Instead, he shared his experiences amongst the Tau'ri with his master, ending his tale with information about the Stargate Program. \"I am prepared to return to your service, my lord.\"\n\n\" _No, not yet_.\" Lord Yu stroked his imperial tuft. \" _Much has happened during your absence, Huang Sun Tzu. You must perform for me one more task. One more enemy to ensnare_.\"\n\n\"Anything, my lord. Who is this enemy?\"\n\n\" _SG-1_.\"\n\n# CHAPTER FIFTEEN\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/1945 HRS MCMURDO STATION\n\n\"What do you require, General Hammond? A direct call from the UN Secretary-General?\"\n\n\"Not while there are other options.\" George blocked Duebel from stepping any closer to the hole and the platform beyond. \"I'm sorry, Ambassador, but I take my orders from the President, not the United Nations.\"\n\nDuebel stabbed a finger at the Ancient chair. \"That thing has cost too many lives.\"\n\n\"It's also saved lives.\"\n\n\"I don't care about the past. It's the present that matters. Or should I say the future? Your own scientists have indicated that once the sun rises, the seismic activity emitted from that device could tear the continent apart.\"\n\n\"I am fully aware of the losses \u2014 \"\n\n\"How many more must die before you accept your responsibility? Or is this about your precious SG-1?\"\n\nGeorge sucked in a breath to calm himself down. \"Mister Ambassador, given time we could still shut the force field down. The sun doesn't rise for...\" He turned to Dr. Lee who had returned to monitoring the device on his computer.\n\n\"Seventeen hours,\" Lee said. \"More or less.\"\n\n\"More or less?\" Duebel shook his head. \"You know the immediate destruction of this outpost is the right thing to do. Ambassador Zhu, do you not concur?\"\n\nDuebel turned toward Zhu. She sat at her self-appointed post by the hole, her legs tucked under. Her eyes never left the force field or her countrywoman underneath. Almost too intently, unless \u2014\n\n\"Quing?\" Duebel strode to Zhu's side and knelt down. \"Your voice is needed.\" He grasped her shoulders.\n\n\"What would you have me do?\" she whispered.\n\n\"Trust us, Madame Ambassador,\" George entreated. \"Allow us the time to \u2014 \"\n\n\"Trust. Like your government entrusted mine?\" Zhu raised her head and glared right at him. \"Two years have passed since you disclosed the Stargate Program to China. Two years in which we've watched you trust Russia with plans to your interceptors and naquadah generators, but not us. Never us.\"\n\nDuebel's eyes widened. \"Quing! This is not the time.\"\n\n\"This is precisely the right time.\" Zhu curled her lips in obvious disdain. \"Intelligence points to the United States now providing Russia with plans to something called the X-303. An interplanetary defense spaceship code-named _Prometheus_.\"\n\nZhu returned her gaze to the force field. \"Do not speak to me of trust, General. You haven't earned the right. Instead, prove it. Find a way to save this planet.\"\n\nDuebel placed a comforting hand on her shoulder. As Zhu shrugged it off, the Swiss ambassador whispered harshly to her. George couldn't hear what Duebel said, so he turned away to afford them some privacy. As he did, Simmons, and a Security Forces airman with an M-16 slung across his back, hurried in.\n\nThe lieutenant pointed at a cup of coffee in the SF's hands and then at George. \"Black, sir. We couldn't find any sugar, but there's enough left for one pot.\"\n\n\"Thank you, airman.\" George accepted the offered cup. \"See if there's enough for the ambassadors, including Juarez \u2014 wherever he took off to.\"\n\n\"Yes, sir.\" The airman strode out.\n\nGeorge sipped his coffee. It was bitter, pretty much a good description of his day so far.\n\nHe glanced back at the two ambassadors still deep in private discussion. \"'We have to distrust each other. It's our only defense against betrayal.'\"\n\n\"Sir?\"\n\nGeorge gave Simmons a wan smile. \"It's an old saying, Lieutenant. Almost as old as I feel.\"\n\nSimmons leaned in and whispered, \"Sir, you should know. Ambassador Zhu \u2014 \"\n\n\"Is Weiyan Shi's mother.\" He gulped down the rest of the coffee. \"I'm not surprised, but I should have guessed sooner.\"\n\n\"General, you can't blame yourself for what's happened.\"\n\n\"Can't I?\" George turned toward the Ancient chair. Its cold metal frame and alien architecture stared back at him, a reminder that for every step forward they'd taken with the Stargate Program, the cost had been enormous.\n\nShifting his view to the force field, he had to ask himself whether he opposed Duebel's demands to destroy the outpost because there was still time to find an answer before the sun rose, or was the truth simpler? Was it because he couldn't take any more losses?\n\nA mild tremor shook the outpost, sending a light fall of snow down onto the force field and the surrounding floor. Some of it fell on Duebel.\n\n\"What are you waiting for, General? Another station to be flattened?\" The ambassador brushed the snow from his gray hair and rose from Zhu's side, his face dark with rage. \"Destroy the outpost and, while you're at, I want those interceptors that Quing mentioned removed as well. They should not be in Antarctica.\"\n\n\"Those F-302s are critical to Earth's last line of defense.\"\n\n\"They're U.S. military and entirely against the treaty.\" He wagged a finger at George. \"Stop stalling.\"\n\n\"That's not the case, Ambassador.\n\n\"Mount Erebus has erupted!\" Juarez ran into the chamber, waving a piece of paper. He bent over, panting wildly.\n\nDuebel snatched the report from Juarez and scanned it. \"The volcano has had a significant eruption.\" He handed it off to Zhu. \"That's only twenty-three miles from McMurdo!\"\n\n\"Ah, Mister Ambassador?\" Lee scurried over from his monitors. \"Erebus has persistent eruptions all the time. There's even occasional minor lava flows out to its inner crater, but never the outer caldera. An old college buddy of mine works up at the LEH, the Lower Erebus Hut, that's why \u2014 \"\n\n\"Your friend is dead, Doctor.\" Zhu lowered the paper. \"A rescue helicopter was sent to retrieve the Erebus team, but they found the hut demolished. A five kilometer-wide lava flow erupted from the crater and is burning a path all the way down to the Ross Sea.\"\n\n\"I don't believe it,\" Lee insisted. \"We would've heard something. I would have...\"\n\nThe scientist fumbled with his glasses. They dropped. George bent to pick them up, his hand grazing the brutally cold stone floor. He glanced over at Zhu, still sitting down. How she was able to handle the frigid temperature was beyond him.\n\nAlthough, if one of his grandchildren were locked beneath an alien force field, George would suffer through anything to ensure their rescue. Which made his next decision all the more difficult.\n\nHe handed the glasses back to Dr. Lee. \"I'm sorry for your loss, Bill. If you \u2014 \"\n\n\"Rick said Erebus would never amount to anything,\" Lee mumbled. \"He kept begging for more funds. He wanted to explore more, but the NSF wouldn't let him.\" He dropped his chin down. \"It just doesn't make scientific sense.\"\n\n\"That's all this is to you,\" Juarez said, his voice dripping with spite. \"A scientific anomaly to be studied.\"\n\n\"That's enough, Ambassador,\" George warned. \"You've made your point.\"\n\n\"No, General. This is only the beginning.\" Juarez's chin jutted out in clear defiance. \"Destroy this monstrosity,\" he waved toward the force field and then the chair, \"or I will contact your president myself.\"\n\n\"I suppose an eruption could happen,\" Lee continued to mumble. \"Plate tectonics are a tricky thing... Although, something like a reversal of the magnetic poles could cause added stress, maybe the photonic energy's directed toward \u2014 \"\n\n\"Get him out of here,\" Juarez demanded. \"If I see another scientist in my lifetime, it will be too soon.\"\n\nLee apparently didn't hear the ambassador because he kept on rambling on. \"Maybe it's not magnetic. If the computers work \u2014 \"\n\n\"Bill.\" George laid a hand on the scientist's back, hoping he might bring the man's focus back to the here and now.\n\nLee's eyes rimmed with tears. \"I'm sorry, General. Rick was at my wedding, you know?\" He swallowed. \"This is hard.\"\n\n\"I know it is, son.\" He nudged the scientist toward the archway to the outer chamber. \"Contact Major Davis. Have him get over to the F-302 base at Observation Hill. Pronto. I want the Byrd Station Navy SEALS ordinance team brought here to the outpost ASAP. Ready to detonate.\"\n\nLee's eyes widened. \"General, we can't!\"\n\n\"I don't see any other option. Not unless you have a way to turn that thing off.\"\n\n\"Not yet, no. If we had someone with a stronger strain of the ATA gene, that might help, but with all the trainees evacuated...\" Lee hung his head. \"I'm sorry, General. I \u2014 \"\n\n\"I'm sorry too, Bill.\"\n\nRubbing his eyes, Lee hurried toward the archway.\n\n\"And Doctor?\"\n\nLee stopped and turned around.\n\n\"How many hours until the sun comes up again?\"\n\n\"Less than seventeen now.\" Lee cocked his head. \"Sir?\"\n\n\"You've got sixteen hours to turn that 'yet' of yours into a reality. Make the call and then get back to work.\"\n\nA brief smile flickered across Lee's face. George hoped it was enough to keep him focused on finding a solution before it was too late.\n\n\"General?\" Lt. Simmons backed off from his position by the hole and slipped in front of Duebel and Juarez. The two ambassadors moved off to join Zhu.\n\n\"How are our people, Lieutenant?\" asked George.\n\n\"Still unconscious, sir, but General O'Neill's smiling, if that means anything.\"\n\n\"Why am I not surprised?\" George pulled up the hood on his parka. The chamber had become damn near the temperature of an iceberg. \"Is there anything else, Lieutenant? I'll need to call the president and obtain clearance before going through with \u2014 \"\n\n\"Sir...\" Simmons looked over at Ambassador Zhu. \"Dr. Lee said that if we had someone with ATA genetics as strong as Weiyan's, they could use the chair to deactivate the device.\"\n\n\"Like mother, like daughter?\" George eyed the Chinese ambassador. While she hadn't admitted to the relationship, certainly saving her child would be worth the revelation. \"Once more you're a step ahead of me, Lieutenant. Why you haven't put in for a promotion yet is beyond me.\"\n\nSimmons ducked his head. \"I wouldn't want to leave the Stargate Program, sir.\"\n\n\"You sound like Major Davis.\" George drew in a deep breath, preparing himself for what would undoubtedly be an uncomfortable discussion with Ambassador Zhu. \"Why don't you go see how Dr. Lee is faring? Tell him your idea.\"\n\n\"Right away, sir.\"\n\n\"And make sure Major Davis has those birds headed toward Byrd within the next half hour.\"\n\nSimmons ran through the archway and out into the nearly abandoned main chamber.\n\nGeorge walked over to the ambassadors. Duebel and Juarez greeted him with silent nods, but Zhu remained focused on the force field, or more specifically, on what lay beneath.\n\n\"Sirs, would you kindly give me a moment alone with Ambassador Zhu?\"\n\n\"What for?\" Juarez's eyes narrowed with suspicion.\n\nDuebel grabbed the Argentine ambassador's elbow. \"Leave it alone, Jorge. He's agreed to our demands.\" He half-dragged Juarez out of the chamber.\n\nGeorge knelt down beside Zhu. \"Madame Ambassador, why didn't you tell us Weiyan was your daughter?\"\n\n\"Trust has many enemies and few friends.\"\n\n\"Trust us enough to help save your daughter,\" George said, refusing to rise to her bait. \"Perhaps a turn in the chair.\"\n\nZhu laughed. A bitter, empty sound that echoed through the now empty chamber. \"I don't have the ATA gene, General. Weiyan inherited that \u2014 and other traits \u2014 from her father.\"\n\nGeorge sank back on his heels. \"And he's undoubtedly in China, thousands of miles away.\"\n\n\"Oh, further than that, in a manner of speaking,\" Zhu said bitterly. \"Weiyan's father is in a psychiatric institution. Completely catatonic and completely insane.\"\n\nDaniel instinctively ducked as an Al'kesh buzzed overhead.\n\nBeside him, Jack stood his ground. \"Would someone please explain how Lord Yu and his minions suddenly became citizens of this ghost town?\"\n\n\"If we knew the answer, we wouldn't be \u2014 \"\n\n\"They're not real.\" Sam crouched down beside Weiyan's self-made foxhole. \"It's safe. Really.\"\n\n\"I guess old habits die hard,\" Daniel admitted. These weren't real enemy troops. These were phantoms. Ghosts.\n\nBut why?\n\nHe scanned the valley. The Jaffa battalion who'd marched right through SG-1 earlier had taken up position a good hundred yards back behind Sam. Another battalion came to rest by the opposite end, standing in front of a massive grass-covered hill. The two remaining groups continued toward the rear of the valley, toward the bottom of the cliff Jack had climbed earlier.\n\nIn total, there had to be at least five thousand of Yu's Jaffa, but again, why?\n\nTeal'c pointed upwards. \"More!\"\n\nTwo more Al'kesh buzzed by, coming in for a landing.\n\n\"Someone \u2014 \"\n\n\"Or something,\" Jack corrected him.\n\n\"Is trying to communicate with us.\"\n\n\"Ya think?\" Jack whirled toward the battalion behind Sam and frowned. \"I'd have preferred a postcard or an email.\"\n\n\"Jack, don't you see?\" Daniel held out his hands toward the landing ships. \"Somehow this all has to do with Lord Yu and the Ancients, which means the Ascended \u2014 \"\n\n\"Daniel, you were ascended once, remember?\"\n\n\"Yeah, but \u2014 \"\n\n\"Ah!\" Jack raised a finger. \"You told us on Abydos, when you were all nice and glowy, that Ascended folks don't interfere. So what do you call this?\"\n\n\"I don't remember saying that, but if Yu \u2014 \"\n\n\"The last time I checked, Yu isn't an Ancient. He's a Goa'uld.\" Jack turned back toward the trio of Al'kesh now parked in the middle of the valley. \"And I should've killed his slimy, snakeheaded butt when I had the chance.\"\n\nDaniel hated the Goa'uld as much as anyone, but during his imprisonment with Lord Yu, he'd discovered the System Lord still had emotional ties to his ancient reign on Earth, or more specifically to China. \"Yu's not like the other Goa'uld.\"\n\n\"A Goa'uld is a Goa'uld. Carter, stay with...\" Jack jerked his head toward Weiyan. \"Daniel, Teal'c, you're with me. Let's go find out what this dog-and-pony show is all about.\"\n\nAlong with Jack and Teal'c, Daniel headed toward the closest Al'kesh. As they neared, people began to unload. Some dressed in blues, other in greens, reds and grays. Pretty much every one of them was dressed the same \u2014 all in ancient Chinese-style robes bound at the waist with swathes of darker cloth.\n\nCivilians, he mused, until a giant of a Jaffa escorted a group of three women close enough for him to see the fear on their faces. And the white bandages on their feet. Their feet had been broken and bound in the ancient Chinese style meant to keep women from running away.\n\nA style not used in China for quite some time. Daniel thought back on his time as Yu's captive a year ago and couldn't remember seeing any of the women's feet bound in that way.\n\nNot that he'd seen more than two or three and only during his time in the lower levels of the fortress. No woman ever made an appearance in Yu's throne room. He wasn't even sure Yu had a Goa'uld queen. His Dragon Guard certainly didn't need them, being cloned by Ancient technology.\n\n\"Yu's slaves?\" Jack asked.\n\n\"They can't see you,\" Daniel said. To prove his point, he walked right through the Jaffa guard. \"This is... Odd.\"\n\n\"Odd doesn't quite sum it up.\" Jack waved an arm through one of the women.\n\nTeal'c raised a hand, fingers spread wide, and swept it through the Jaffa guard. \"If these are holographic projections, should there not be some visual \u2014 \"\n\n\"After effect?\" Jack headed toward the Al'kesh's ramp.\n\n\"Like a light trail or something?\" asked Daniel.\n\n\"Or something is right. \" Jack peered up the gantry. \"Teal'c, come on.\"\n\nAs Teal'c turned to follow, Daniel noticed him grimace as if he was in pain. Then he shook his head slightly, put back his shoulders, and strode toward the ramp.\n\n\"You're wasting your time!\"\n\n\"Since when has that stopped me?\" He approached the ramp, stepped up \u2014\n\nAnd fell flat on his face.\n\nBiting his tongue, Daniel turned away.\n\n\"Go ahead, say it,\" Jack grumbled. \"Say 'I told you so.'\"\n\n\"Those weren't exactly the words I had in mind.\"\n\n\"Head's up,\" Jack warned as a single cargo ship flew across the small valley and headed toward the grass-covered hill.\n\n\"What's going on there?\"\n\nDaniel pointed toward the large mass of people heading toward the hill. \"Maybe if we get closer. Where's \u2014 ?\"\n\n\"I am here.\" Teal'c walked closer. Actually, stumbled was more like it.\n\nDaniel took a worried glanced at Teal'c and the sweat beading on his forehead. His skin had turned a gray brown.\n\nJack shook his head, a silent command to stay silent. There were times to listen to Jack and there were times to ignore him. Daniel chose the latter.\n\n\"Teal'c, are you feeling all right?\"\n\nTeal'c proudly lifted his chin. \"I am well, Daniel Jackson.\"\n\n\"Back at ya,\" Jack said with a smirk. His way of saying, score one \u2014 tied on the 'I told you so' front.\n\nDaniel ignored him. \"You don't look so good. Maybe you should rest, head back to Sam and Weiyan.\"\n\n\"I will rest when we are free of this dilemma.\"\n\nAnother warning glare from Jack, and Daniel let it go. But not without promising himself to keep an eye on Teal'c.\n\nThe cargo ship set down on the hilltop. As its bay doors open, the crowd of soldiers and civilians dropped to their knees in prostration.\n\n\"How come they're not making more noise?\" Jack asked.\n\n\"Perhaps they are, O'Neill, and we simply cannot hear them.\"\n\nA figure emerged, but the hill was too far off to see anything other than a blur. Daniel assumed it was Lord Yu. As if on cue, the crowd rose, turned around, and headed back to their ships.\n\nA rippling wave streaked across the valley and the ships disappeared. As did the people.\n\n\"What the hell?\" Jack whirled around. He pointed toward the building that continued to exist through every vision.\n\nOr nightmare.\n\nA solitary Jaffa strode toward them. He didn't slow down as he neared, obviously unaware of their presence. He was Asian, young, and shorter than Yu's other Jaffa\u2014no more than five-foot-nine. There was a quiet confidence about him as he walked. An inner strength.\n\nAn all too familiar strength. As the Jaffa drew closer, Daniel was unsurprised by his gray woolen cloak, cinched off to the side with a jade broach. Cropped black hair, green-flecked black eyes, and a square jaw. The sun glinted off the gold tattoo on his high forehead.\n\nJack jerked a thumb toward the Jaffa as he walked past, unaware of his audience. \"Isn't that...?\"\n\n\"Oshu,\" Teal'c stated flatly. \"Lord Yu's current First Prime and lead Dragon Guard. He is not Jaffa. None of Yu's Dragon Guard carries symbiotes. They are clones.\"\n\n\"Sonuvabitch.\" Jack face darkened with rage as he watched Oshu draw closer. \"Hallucination, ghost or whatever, that bastard's grand-clone gave us enough trouble last year that I'd like to ring all their necks.\"\n\n\"You mean Ambassador Huang,\" Daniel said. \"Didn't the Chinese \u2014 \"\n\nThe First Prime stopped beside them, seemingly unaware of their presence. He stared off toward the mysterious building.\n\n\"Guys, I'm not sure that's Oshu.\" Daniel approached the Dragon Guard for a better look. \"He looks like him, but \u2014 \"\n\n\"And again, I ask... There's a but?\"\n\nIgnoring Jack, Daniel stepped up to the Dragon Guard. A jagged white scar stretched across the man's chin. \"Whoever this is, it isn't Oshu. That scar's old.\"\n\nThe air rippled again and the First Prime disappeared.\n\n\"And I repeat,\" Jack glowered, \"what the hell?\"\n\nBefore Daniel could answer, a multi-color wash of light rippled across the valley like a shockwave. It crashed against the cliff at the far end, surged up the craggy, exposed rock, and then dissipated as it breached the summit.\n\nA Stargate materialized on the cliff-top.\n\nJack took a few steps toward the cliff and then stopped. He turned back toward Daniel. \"Too good to be true, I assume.\" A kawoosh erupted from the gate. As its event horizon settled down into a reflective puddle, more Jaffa exited, carrying large cargo containers. They descended the hillside.\n\nAnother wave of light streaked across the valley.\n\n\"Is this making anyone else dizzy?\" Jack asked.\n\n\"It is disorientating, O'Neill.\" Teal'c took a step forward, or rather, he tried. Halfway through his stride, he stumbled backwards. \"Something prevents my forward movement.\"\n\nJack stretched out his palm. It flattened against the air, as if he was miming touching a wall. \"Well, that's... Cute.\"\n\nAs Daniel raised his hand to try the same thing, high-curved stone walls sprung up around them. Walls that he could see right through. Phantom carpenter-slaves joined a roof to the sides. He turned back toward the ever-present building. On top a newly built addition to its left, two Jaffa spread copper sheets over a turret-shaped frame.\n\n\"They're building Lord Yu a fortress,\" he whispered in awe.\n\n\"That's nice, Daniel,\" said Jack, \"but we've been on Yu's home world, and this ain't it. No trees, no mountains, no snow.\"\n\n\"I don't know,\" Daniel admitted. \"Clearly, this is \u2014 \"\n\nA warm breeze blew across the back of his head. \"Did anyone else feel that?\"\n\n\"You felt it, too?\" Jack raised a hand to his neck.\n\n\"As did I,\" Teal'c said. \"Though why is unclear.\"\n\n_Because it is so clear it takes a long time to realize it_.\n\nDaniel shivered, remembering Oma Desala's words. \"Guys, I'm pretty convinced there's an ascended being here, somewhere.\"\n\nJack's eyes widened. \"They're responsible for all this?\"\n\n\"I don't know, Jack. None of this is making any sense.\"\n\nTeal'c gestured toward the translucent building's interior. \"Yu's Dragon Guard.\"\n\nFour men strode down a perpendicular hallway lined with Jaffa. The guards dressed in Jaffa armor, but their cloaks were different colors. The First Prime still wore his gray one, while another wore a red, another blue, and the fourth wore green.\n\nDaniel could see enough of their faces through the transparent walls to recognize the similarities to the four men he'd met last year. The guard with the red cloak had longer hair and the green-cloaked guard didn't have a beard, but otherwise, they were identical.\n\nAs they turned right and headed toward the end of the complex, time-lapsed images unveiled the completion of Yu's fortress. Tapestries and scrolls appeared on the walls. Furniture materialized. Lit braziers erupted to life. The Dragon Guards stopped at the hall's far end. Though it had to be at least fifty yards away, Daniel made out the high stone doors leading into the old building. They pushed on the doors, but nothing happened. They began to pound on the doors.\n\nDaniel watched them, feeling as if Oma was in turn watching him. Why? To see what he'd do? There was something iconic about the images. Four Dragons. The Ancient building.\n\nThe four men broke into two groups, each pulling at a door. The doors started to crack open, slowly. As if they hadn't been used for centuries. Perhaps even millennia.\n\nLord Yu's words crashed down on Daniel. He remembered. He remembered it all. How the Goa'uld's original Dragon Guards died, and more importantly, where they'd died.\n\nBlue light streamed out of the doors' ever-widening gap.\n\n\"My God, Jack. This is Kunlun. Pedion Elysium is Kunlun!\"\n\n\"Kun who?\"\n\n\"The planet Yu lived on before moving to his current planet. This is P3Y-702. The same place Yu captured me from last year!\"\n\nDaniel surged forward, pounding the invisible wall with his fists. He knew what would happen. He knew how this would end... And it would end badly. \"We have to stop them!\"\n\nJack pulled him back. \"Daniel, you're the one who pointed out that none of this is real, remember?\"\n\nDaniel wrenched himself free. \"Then why show this to us? Why would Oma do that?\"\n\n\"You really believe this is Oma?\"\n\n\"Who else could it be, Jack? Maybe she thinks we can help change the past somehow, maybe \u2014 \"\n\nWith a thunderous boom, a brilliant light blasted from the doors. An eerie glow radiated outward, swathing the Dragon Guards in its blue pall.\n\n\"We're too late.\" Daniel forced himself to keep watching, even though he knew what would happen next.\n\nWithin moments, the four guards collapsed.\n\n\"Are they...?\"\n\n\"Yes, Jack.\" Daniel turned away, sickened. \"They're dead.\"\n\n\"Were dead. This happened a long time ago.\"\n\n\"Yeah.\" Daniel sighed. \"I guess I finally understand firsthand what Oma meant.\" He glanced at Teal'c. \"You tried to tell me, too.\"\n\nTeal'c raised an eyebrow. \"To what do you refer?\"\n\n\"You mean her whole candlelight-is-fire bit?\" Jack asked.\n\n\"No.\" Daniel rubbed his eyes. \"More like, no matter what I do \u2014 whether it's Tegalus, or here, or anywhere \u2014 I can't fix others' mistakes.\"\n\n\"Welcome to the club.\" Jack squeezed his shoulder. \"I'd offer you a t-shirt, but we're fresh out.\"\n\n# CHAPTER SIXTEEN\n\n### BEIJING, CHINA\n\n### 2003 JULY 1, 6:30 AM\n\n\"I will be no more than an hour,\" Huang said as he climbed out of the escort car. Buttoning his black wool blazer, he peered through the restaurant window. Very few patrons were inside which suited him perfectly. Next door, the far more crowded American fast food establishment brimmed with Chinese citizens. This breakfast meeting was too important for grease and gummy pancakes. Authentic cuisine was needed.\n\nIt would be as much a parting meal for him as for his expected company.\n\nHe was greeted within the restaurant by a stooped old man dressed in the appropriate blue workman's shirt and pants. Selecting a rear table, Huang sat down facing the window so he might see his guest upon arrival. The server offered him a menu, but Huang waved him off.\n\n\"Your traditional breakfast,\" Huang ordered.\n\n\"Will your guest want cornflakes?\"\n\nHe scrutinized the server. Surely, the man was old enough to know better. \"When did cornflakes become a tradition in China?\"\n\nWith a bob of his head, the server shuffled off into the kitchen. Huang checked his watch. In ninety minutes, his flight would leave for Colorado Springs in the United States. The benefits of being a diplomat meant he had no need to wait amongst the unruly people with whom he must share the flight.\n\nThe old man returned with a tray of covered ceramic dishes and wooden bowls. As he placed them neatly in the middle of the table, Huang reviewed his plans.\n\nThanks to one special capsule in Ambassador Chen's evening tea, Huang was now lead ambassador to Beijing's UN diplomatic corp. That capsule had waited long to serve its purpose. Chen had shared a new update from the Security Council's report of the latest goings-on of Stargate Command. One of SG-1's team members had been apprehended by Lord Yu only hours ago. A Dr. Daniel Jackson \u2014 the man who had unlocked the Chappa'ai's mysteries.\n\nLord Yu had told him to look for a sign to commence his plan. Now, everything would progress as planned.\n\nThe US President had notified the Security Council of his intent to send in a rescue team. Huang had brought the news to Chen with his morning coffee. Chen had laughed off Lord Yu's facade, scoffing at the Goa'uld's efforts to pretend to be Chinese. That is, until Chen downed his cup and complained of chest pains. The embassy doctor had determined Chen suffered a heart attack. Who was Huang to disagree?\n\nUpon Chen's hospitalization, Huang had called the US President, demanding SG-1 place their plans on hold until he could confer with them face-to-face. Hence his reason for flying to their base of operations. The President had drawled so heavily on the phone that Huang could barely understand half the US leader's apologies and assurances. All that mattered was that the SG-1 would wait until his arrival.\n\nHungry, and admittedly anxious, he reached out to uncover one of the small dishes. The right side of his blazer banged against the table. Quickly sticking a hand inside his pocket, he assured himself that the small lead box containing the two communications devices recovered from the Pacific Ocean were intact. They would serve as his bait and his means to an end. Huang would finally go home.\n\nAll he had to do was deliver SG-1 into Yu's hands.\n\nA chime on the restaurant's doors sounded. Huang glanced up in time to see a young teenage girl close the door behind her. Her long black hair covered half of her face, but he still could see the resemblance to both her mother...\n\nAnd himself.\n\n\"Weiyan!\" He waved his daughter over.\n\nAs she neared, he thanked his inner courage for this meeting. Though Lord Yu would consider his offspring _b\u00f9 zh\u01d4n x\u01d4_ \u2014 forbidden \u2014 Huang could not deny her existence in his heart.\n\n\"Father.\" Weiyan sat down across from him. \"I am honored.\"\n\n\"Are you hungry?\" He gestured toward the many dishes ordered. \"Even a young woman like yourself must eat when opportunity arises.\"\n\n\"I will try, Father.\"\n\n\"One must simply do what needs to be done, child. Eat.\"\n\nShe laid a napkin in her lap. \"This seems more like a vast dinner than a simple breakfast.\"\n\n\"A traditional Chinese breakfast can keep the body strong throughout the day.\" His eyes moistened with joy at her demure nature. Quing had done well in raising their child.\n\n\"Mother has never served such a meal. We usually eat cornflakes or boiled eggs.\"\n\nHe uncovered the dishes, naming each as they were unveiled. \"First, a bowl of _congee_. It is a rice porridge.\"\n\nHe offered her a small plate of pickles. \"Those can be added if you prefer.\" Next, he removed the lid to a broad ceramic bowl. \"This is _y\u00f3uti\u00e1o_ steamed in soya.\" Swimming in the milky liquid were deep fried sticks of dough. Finally, he removed a plate cover where beneath were puffy sheets of steamed bread accompanied by another small bowl of milky soy in which to dip the bread. He took up one of the bread strips and dunked it in the soya. \"Like so, yes?\"\n\nThough she remained silent, her green-flecked eyes widened upon gazing at their meal.\n\n\"Is it too much?\"\n\n\"No. No, Father.\" She took up a spoon and dipped it in the _y\u00f3uti\u00e1o_. \"It's only... After so many years, I don't understand why you have contacted me.\"\n\nHuang put down his half-eaten bread. \"I am sorry for not contacting you earlier though my reasons were just.\"\n\n\"I shamed you,\" she whispered. \"If I'd been born a male \u2014 \"\n\n\"No.\" He grabbed her hand. \"Never. The shame belongs to me, but please, do not ask for an explanation. I cannot give you my reasons. They must be held within me to protect you as well as your mother.\"\n\nWeiyan withdrew her hand. \"Then why now? What has changed?\"\n\n\"I am leaving.\" Even as Huang said the words, a silent thrill ran through him, warring with this burning desire to know his child.\n\nWeiyan put down the spoon, her food untasted. \"Can I go with you?\"\n\n\"It is too dangerous.\"\n\n\"Dangerous?\" She sat back against her chair. \"I do not wish you to die, Father. Not when we have just met.\"\n\n\"Death is not to be feared, child.\" He studied her face closely, trying to decide just how much of the truth he should share. \"Have you not read the teachings of Sun Tzu?\"\n\n\" _The Art of War_? Once.\" She shrugged. \"All children must read his writings, but they don't really mean much. Not in today's world. Mother has me studying the sciences. She hopes I will pursue a degree in astro \u2014 \"\n\n\"Once is not enough,\" Huang chided his daughter. \"You spoke of fearing death. When you think of me, remember Sun Tzu. 'Until death itself comes, no calamity need be feared.'\"\n\n\"I will remember.\" Weiyan again picked up her spoon and pushed the dumplings around. \"Perhaps I should read Sun Tzu's work again. If it gives me solace during my treatments \u2014 \"\n\n\"Treatments?\" Huang's head jerked up.\n\nHis dishonor weighed heavily on his mind as Weiyan explained her blood illness. While he could not admit to her the cause, he knew that her hemophilia stemmed from the fact that as a clone, he should never have created offspring.\n\nThough he had failed as a Dragon Guard trainee, that would soon be rectified, but Weiyan would carry his failure to obey Lord Yu's laws for the rest of her life. There was little he could do to make restitution for his trespass.\n\nHe reached into his chest pocket and pulled out a checkbook. While most of the funds from the ancient scroll had already been sent to Quing, there was still several hundred thousand Yuan left in his bank account.\n\n\"What are you doing?\" she asked.\n\nHe wrote her a check for the entire balance. \"Where I am going, money has no purpose.\" He ripped the check from the billet and handed it to her.\n\nShe did not take it. \"I do not want money. I want my father.\"\n\nHuang placed the check on the table with a sigh. \"Your full name is Weiyan Shi. Have you ever wondered from where the second part of your name derives?\"\n\nShe dropped her head. \"Mother would not say.\"\n\nThat was because even Quing did not know the truth of that name. \"Many years ago, I had a brother named Lao Dan Shi.\"\n\n\"What happened to my uncle?\"\n\nHuang swallowed back the bitter memory of Lao Dan's death. \"That does not matter. Instead, let me share with you a saying that his...\" He stopped himself from saying 'ancestor,' for Lao Dan had been a direct descendent of Lao Tzu, crafter of the great Tao. To explain the relationship would reveal too much.\n\nInstead, Huang picked up the check and pushed it into Weiyan's hands. \"A favorite quote of my brother's is worth remembering. 'He whose desires are few gets them; he whose desires are many goes astray.'\n\n\"Take the check, Weiyan. Perhaps it will help ease your struggles with this illness.\"\n\nShe folded the check and slipped it into her purse. \"Where are you going? Home to the Taklimakan Desert where Mother met you?\"\n\nPicking up his chopsticks, Huang whispered, \"Further away, if I succeed.\"\n\n# CHAPTER SEVENTEEN\n\n### F-302 AIR BASE\n\n### OBSERVATION HILL, ANTARCTICA\n\n### 18 AUG 04\/2010 HRS MCMURDO STATION\n\nPaul jumped off the transport's wide snow belts and onto the hard-packed snow. He flipped on his flashlight. The earlier gray haze had sunk again into a near pitch-black night. \"Get back to McMurdo and find Colonel Ferguson,\" he told the driver, an army corpsman barely out of his teens.\n\n\"But Major Davis, what about \u2014 ?\"\n\n\"No time to argue. Just do it.\"\n\nRebuffed, the corpsman bobbed his head and kicked the transport back into gear.\n\nThe sudden change from the overheated truck's box-like cab into the frigid cold tightened Paul's weary arms and legs. A lack of thermal layers underneath was to blame, but there'd been no time once General Hammond's orders came through. When Paul couldn't find Ken Ferguson to convey those orders, he'd commandeered the first available military vehicle and hightailed over to Observation Hill to get the 302s in the air and over to Byrd Station.\n\nHammond wanted the Navy SEALS ordinance team laying charges up at the outpost on the double. As much as Paul wanted to believe that there was still time to shut down the continental drift device, he understood why the president had approved the general's backup plan.\n\nPaul cast his flashlight across the empty base. The quake had hit hard. He took in the collapsed sentry booth, the busted windows on the pilot barracks, and the two overturned jeeps beyond the sentry booth. The hangar had been squashed flat, so where were the F-302s?\n\nA door slammed shut over at the barracks to his right.\n\n\"Major Davis!\"\n\nTwo green-clad men raced toward him, each waving a standard-issue flashlight. They came to attention in front of him and snapped off quick salutes. Paul found himself envious of their USAF-issue leathered mitts as he rubbed his thinner fleece gloves together.\n\n\"Major Davis? I'm Captain White. This here is Captain Allen,\" said the taller of the two men, or at least, he supposed they were both men. It was hard to tell with their faces wrapped in balaclavas, the dark green face socks leaving only goggles to peer back at him.\n\nAs Paul returned the salute, a sudden gust of wind shot up from his left. \"Where are the birds? Can they even fly in these kinds of wind conditions?\"\n\n\"Flying in stiff wind's not the problem, sir,\" said Captain Allen. A woman's voice. Strong, confident. A lot like Colonel Carter.\n\n\"Then what is the problem, Captain?\"\n\n\"This way, sir.\" Swinging her flashlight out toward the airstrip, she jogged off.\n\nWith White beside him, Paul hurried to keep up. \"Any word from Colonel Ferguson yet? I sent an airman to find him and pass along our plans.\"\n\n\"No, sir,\" said White. \"Standard radios haven't worked so well since the first quake, but knowing the Colonel, he'll be here soon.\"\n\n\"Great,\" Paul lied. They came to a stop at the edge of an ice field. \"Where are the 302s?\"\n\n\"You're looking at them.\" Allen raised her flashlight. A fifty-foot-wide cluster of broken ice thrust upward from what had been the airstrip. Jutting out from either end were matching wingtips. The F-302s' wingtips.\n\n\"Sea ice, sir.\" White added his flashlight's beam to the disaster. \"When the earthquake hit, the resulting pressure shot out across McMurdo Ice Shelf and built up an ice ridge. The entire ice shelf probably looks like this now.\"\n\nPaul cast his flashlight along the closer of the two birds. While the wingtips were free, most of the cockpit, the nose and tail were buried in what, under other circumstances, would be a startlingly beautiful landscape. Jagged blocks of exposed ice \u2014 some as big as a house \u2014 shined pale blue under the flashlights' glare, the color reminiscent of toothpaste gel.\n\nDropping his flashlight beam toward the base of the blocks, Paul asked, \"Where are the wheels?\"\n\n\"There.\" White illuminated a partially exposed strut. \"And there.\" He swung the light left, revealing a partial wheel, the other buried within the ice.\n\n\"Can you dig them out?\" Paul checked his watch. It was almost twenty-one hundred hours. A little more than fifteen hours were left 'til the sun peaked the horizon and turned the Ancient defense outpost into an offensive weapon against all of Antarctica and possibly beyond.\n\n\"Depends on how \u2014 \"\n\nThe ground began to tremble. Ice cracked and popped. As the interceptors' wings vibrated, Paul held his breath, hoping it would only be a small aftershock.\n\nHe got his wish. A few seconds later, the ground settled down. The ice went on cracking for a few more moments, and then stopped.\n\nKnowing next time they might not be so lucky, he pressed White for a timetable.\n\n\"We can use hacksaws to cut away the ice, sir, but I'm not sure we could clear enough to make a difference.\"\n\n\"Let's say you can. How long, Captain?\"\n\nWhite tugged at his balaclava. \"The 302s can do up to Mach 6, but \u2014 \"\n\n\"The UN Security Council won't like it, I know.\" Even under these circumstances, he knew there was no way they'd waive the restriction. \"Sub mach speed, then. How long?\"\n\n\"Under Mach 1? Byrd's about 1400 kilometers away, a little over 850 miles.\"\n\n\"Three to four hours, sir.\" Allen said, shutting off her flashlight.\n\nAnother, more distant round of rumbling started up and Paul braced himself. The sound grew closer and with it came a pair of headlights lighting up the entire expanse of the ice-bound runway.\n\nIt was another truck, pulling up to the frozen shoreline. As it came to a stop, Paul asked, \"Any way you can shave off time without breaking UN protocols?\"\n\n\"We could bring them to Marble Point,\" Allen offered. \"That's only ten minutes away from the outpost. There are two tractor-trucks stationed there, sir.\"\n\n\"What the hell's going on?\" demanded a new voice.\n\nIt was Ferguson. Hood flapping behind him, he stormed up to Paul. \"Since when do you give the orders around here, Major?\"\n\n\"Take it easy, Ken \u2014 \"\n\n\"That's Colonel Ferguson, airman.\"\n\nPaul stiffened, but held his ground. \"Fine, Colonel. My apologies.\" He quickly outlined General Hammond's orders. \"We just need to figure out a way to break the tires free.\"\n\n\"Why send a lieutenant when you can send a major?\" Ken shook his head. \"So now you're Hammond's messenger boy.\"\n\n\"Colonel.\" Paul dropped his voice so the pilots wouldn't hear him. \"Whatever your beef with me is, sir, it'll have to wait. You know the stakes.\"\n\n\"Getting the 302s out is only half the problem. We need a runway.\" Ken waved a hand toward the ice. \"As you can see, that's not gonna happen.\"\n\n\"We need those planes, Colonel.\"\n\n\"Yeah, well, you need a lot of things including getting your head examined, but that can wait, too.\" Ken yanked his hood up. \"These birds can't just jump up in the air, Major.\"\n\n\"Jump... That's it!\"\n\n\"Excuse me? What the hell are you thinking?\"\n\n\"I am thinking, Ken. That's the point,\" Paul said, irritated with his one-time friend.\n\nKen sputtered off a string of profanities and threats, but Paul ignored him. He returned to studying the 302s, skimming his flashlight along the pressure ridge for possible gaps that would allow the pilots entry into the cockpits.\n\nThe beam from his flashlight landed on a crevasse halfway across the ridge. \"Is the ice safe to walk on?\" he asked Captain White.\n\n\"Uh, yes, sir. There's at least two hundred feet of ice underneath.\"\n\n\"Good.\" Paul walked out onto the ice, pointing his flashlight along the ridge, hoping to find another crevasse. He spotted one, less than a yard away from the other. Stepping closer, he aimed his flashlight into the six-foot high crack in the ice block.\n\nThrough the crack, he saw a 302's canopy. An idea was brewing in the back of his mind, but it would take a lot of faith on the part of the pilots, and Ken, to make it happen.\n\nPaul swung his flashlight beam back toward shore. \"Colonel Ferguson, at General Hammond's discretion, I must request that your pilots gain entry to these cockpits ASAP.\"\n\n\"I told you,\" Ken said. \"We can't get these birds up without a runway.\"\n\n\"We won't need it,\" Paul replied. He ran back ashore.\n\n\"I don't get it. How the hell are we gonna follow through on Hammond's orders?\"\n\n\"Ken, you're a topnotch air jock, but you have to trust me.\" Paul stopped in front of the F-302 wing commander. \"We don't need a runway.\"\n\nBeing a colonel might have its advantages, but having the imagination needed for what had to be done next? That took the kind of eye-opening experience of someone who'd lived and breathed the Stargate Program since its inception. Paul had that kind of experience and right now, he wouldn't trade it for all the promotions in the world.\n\nHe turned toward the pilots, hoping they'd follow his lead. \"You're going to jump, literally. Ground to space, and then back down right above Byrd Station.\"\n\n\"How do you figure, sir?\" asked Allen. \"We don't have a runway.\"\n\n\"Forget the runway,\" Paul said with a smile. \"You'll take the 302s up through a ground-to-orbit hyperspace window.\"\n\n\"You're nuts,\" Ferguson said. \"What the hell does a desk-jockey like you know about hyperspace windows?\"\n\n\"Trust me, Ken. A lot more than there's time to explain.\"\n\nSam sat beside Weiyan's foxhole, watching the images rush by with mixed emotions. While her head knew the Jaffa, the ships, even the transparent buildings were only that, images, every fiber in her being wanted to bolt toward General O'Neill, Daniel and Teal'c. She wanted to help, not do babysitting duty. The men tried hammering on the invisible walls that penned them in. Whatever held them was more than just a visual display.\n\n_Force fields_ , she decided. But what was the power source?\n\n\"We must save them.\" Weiyan climbed from the hole. She stumbled to her knees.\n\n\"Take it easy.\" Sam pushed her down to a sitting position. The girl had calmed down, but she was still breathing pretty hard. \"I don't think they're in any danger.\"\n\nShe crouched beside Weiyan and glanced down at her fleece pullover. It was dry. Clean. As if there'd never been any blood. With a nod from Weiyan, Sam lifted the lower half of the pullover. Nothing. Her abdomen was bare. \"How do you feel?\"\n\n\"Tired \u2014 Oh, no!\" A reflection of blue light lit up Weiyan's face.\n\n\"What?\" Sam's head whipped toward the building. A swath of blue streamed through its half-opened doors. The color was reminiscent of the Stargate's kawoosh, and for a brief moment Sam wondered if the energy source was derived from the same quantum energy that powered the gate.\n\nThe light spread further outward, enveloping the four Dragon Guards by the doors. They clutched their throats as if they couldn't breathe.\n\nThough she wanted to believe her team wasn't in danger, she jumped up and bolted toward them. She got no more than three feet from Weiyan when her head slammed into another of the transparent walls.\n\nIgnoring the pain, Sam called out. \"Sir!\"\n\nNo answer. Her teammates didn't hear her. Daniel pounded the air in front of him, frantic until Teal'c and the general grabbed hold of him and he dropped his fists.\n\nThe light intensified. From her vantage point, Sam could see the guards slapping their chests. A final burst of light shot outward, and the guards dropped to the ground.\n\nThe air rippled and the downed guards disappeared. The doors swung shut, but not before she got a glimpse of the light's source. Inside, a central core of blue light gyrated and throbbed. Below the energy form was a glowing circular platform the size of a big jeep. Backlit trellised panels shored up the sides underneath.\n\nThe doors slammed shut.\n\n\"That has to be the power source.\"\n\n\"Power to what?\" Weiyan asked.\n\n\"Whatever brought us here. Whatever's generating all this.\" Sam raised her hands to indicate the valley. \"It's possible that some elaborate device \u2014 \"\n\n\"Created by the Ancients you mentioned?\"\n\n\"Or someone else.\" She tried to smile for Weiyan's sake. It wasn't easy with her team still trapped out of reach. \"For someone who nearly bled to death, you ask a lot of questions.\"\n\nWeiyan wrapped her arms around her torso. \"My mother refers to my questions as a bad habit.\"\n\nSam was taken aback. \"Hey, if it wasn't for curiosity, humans would still be stuck in caves.\"\n\nWeiyan dropped her chin to chest.\n\nSam turned her attention back to the team. General O'Neill was talking to Daniel, patting him on the back while Teal'c seemed to be scanning the opposing hill. He pointed toward the top where a new set of images unfolded in a staggered fashion, like those old animated flipbooks. Four graves were dug by Jaffa. More Jaffa appeared, lowering four bodies shrouded in black cloth into the ground. Sam had to assume the bodies were those of the Dragon Guards.\n\nShe blinked and another scenario replaced it. This one was of a group of Jaffa constructing a statue out of red clay, several stories in height. They'd built up the back half, which seemed to resemble a dog sitting on its hind legs. Another image flashed. Everything but the statue's head had been formed. Three nasty blades stuck out of its red clay spine.\n\n\"This is getting familiar,\" she said as much to herself as to Weiyan. \"We found one of those statues on P3Y-702. On a hill, just like that one. But that's not possible.\"\n\nWho was she kidding? None of this was possible. They were being manipulated. The question was, by whom and why?\n\nThe view changed again, the statue completed. Its teeth, bulging jaw, and eye ridges stared down into the valley in ferocious protection of the graves at its base. A solitary figure stood beside the statue, dressed in the green hat and recognizable red over-vest of Lord Yu.\n\nThe air rippled and the System Lord disappeared, but the statue remained.\n\n\"A _Zhenmushou_?\" asked Weiyan. \"It is an ancient Chinese tradition, a funerary beast meant to protect the dead.\"\n\nAnother image shimmered into life, right in front of the building. It was Lord Yu, again. As he stepped away from the building, a ring transporter activated and swept him away.\n\n\"Let's just hope the dead don't include the general, Daniel and Teal'c.\" She cautiously raised a palm out in front of her and re-approached the force field. It felt solid enough. Smooth.\n\nShe slid her hand horizontally. \"It's not hot or cold. It's just \u2014 \"\n\n\"There?\" Weiyan asked. \"To see the sun and moon is no sign of sharp sight.\"\n\n\"That's beautiful.\" Sam dropped her hand down from the force field. \"Maybe even more beautiful than this situation deserves.\"\n\n\"The words of Sun Tzu are often that way, even though they are words of war.\" Weiyan sighed. \"I am grateful my father encouraged my reading of his work. Imagine how stunned he would be now to see such a large _Zhenmushou_!\"\n\nSam looked over her shoulder at Weiyan. Her eyes shined with amazement, transfixed by the clay statue on the hill.\n\n\"Carter!\" shouted General O'Neill.\n\n\"Finally!\" Sam looked back toward her team, relieved the walls had vanished. The three raced toward her. In fact, everything had vanished with the exception of the statue and its freshly dug graves.\n\n\"It's Kunlun, Sam!\" Daniel dropped down beside Weiyan. \"Lord Yu's former home-world, P3Y-702.\" He grabbed a handful of the dirt and hefted it in his hand. \"The events we saw? Just like Yu described when \u2014 \"\n\n\"When he captured your sorry ass and almost got the rest of us killed,\" the general said, frowning.\n\nTeal'c crouched beside Daniel. That's when Sam noticed the gray tinge to his skin.\n\n\"Teal'c, you all right?\"\n\n\"I am well, Colonel Carter, a moment's rest will suffice.\" Teal'c sat down heavily. Sam knew he was overdue for a tretonin injection, but without a means of escape from their current dilemma, there wasn't anything that could be done. He probably didn't wish to alarm them.\n\n\"A bit of a break will do us all good.\" General O'Neill plunked down on Teal'c's other side. \"Any guesses on why we're getting the _History Channel_ edition of Lord Yu's life on this hellhole of a planet?\"\n\nSam pointed toward the building. \"I think the answer's inside there, sir.\"\n\n\"All right, Carter. You're with me. Let's go check that building out. You three stay here.\"\n\nImmediately, vibrations started up beneath her feet. \"Sir?\"\n\nThe vibrations turned to a loud rumble. Dirt and stones scattered across the ground.\n\n\"Let's do it, Colonel.\" He started to march off toward the building.\n\nWithin seconds, the ground ripped open between the general and his target. A chasm formed at least twenty-feet long and endlessly deep.\n\n\"This is getting old.\" He took a step back and the ground sealed shut. He threw up his hands.\n\n\"I don't remember a building like that when we were actually on P3Y-702.\" Sam reported what little she'd seen of the device inside.\n\n\"I do,\" Daniel said. \"Those doors were hidden behind some of the other ruins. I tried prying them open just before the ring transporter grabbed me. There were Chinese pictograms on some of the walls describing Kunlun and how it was built by \u2014 \"\n\n\"Emperor Yu Huang Shang-Ti,\" Weiyan offered. \"There are many legends surrounding Kunlun. I have even heard it said that the emperor's paradise was really a planet. An evil planet that destroyed the Emperor's first Dragon Guards.\"\n\nSam winced at the awe in Weiyan's voice. Looking back over the valley, she wondered yet again how much of Earth's ancient history was myth and how much was fact. Since the Stargate Program's inception, the line dividing the two had thinned dramatically.\n\n\"Weiyan,\" said Daniel behind her, \"Chinese mythology places Kunlun on a mountain, not another planet.\"\n\nThe air rippled. Sam tilted her head, curious to see what images would be displayed next.\n\nLord Yu's ships shimmered into existence again, positioned only a few feet away. Behind them, the fortress walls also reappeared. This time, however, the images weren't static. The Jaffa led slaves up into the Al'kesh while members of the original four battalions marched back up the mothership's gantry, carrying large crates.\n\n\"Sir, those Jaffa. I'll bet they're carrying the Ancient photon emitter Yu has installed on his present home world.\"\n\n\"Oh, goody.\" General O'Neill groaned. \"Another walk down memory lane.\"\n\n\"Don't you see, Jack?\" Daniel waved a hand at the closing gantries. \"We're watching events play out the way they did a thousand years ago, exactly as Yu told us.\"\n\n\"Then how come she knows so much?\" the general asked, jerking his head toward Weiyan.\n\n\"My father told me,\" Weiyan's voice trembled. \"He said Kunlun was not a myth, but I didn't believe him. He told me everything... all of it...\" She flung a hand out toward the building. \"All of it has happened just as he said!\"\n\n\"And just who-in-the-hell is your father?\"\n\nHer eyes lifted toward Sam, wide in desperation. \"I thought his tales were make-believe. I thought he told me fairytales so he could pretend I was the little girl he had never known.\" Weiyan buried her head in her hands.\n\nThe mothership's hull slid downwards, covering its multi-tiered base underneath. Sam recognized the signs of pre-liftoff.\n\n\"Carter, we need answers.\" General O'Neill jerked a thumb toward Weiyan. \"Now, please.\"\n\nSam dropped down besides the girl. \"Weiyan, is your father a member of the IOA?\"\n\n\"No. Please do not ask me more. I swore I would not say.\"\n\n\"Do you want to get out of here or not?\" General O'Neill said with remarkable restraint.\n\n\"I do!\" Weiyan mumbled. \"I just... I never thought...\" Tears welled in her eyes, but Sam held back any demonstration of pity. The girl knew something. Something that could help them figure a way out of here.\n\n\"Weiyan Shi.\" Teal'c placed a hand on her shoulder. \"You must share how you obtained this information.\"\n\nSam lifted the girl's chin, forcing Weiyan to not look away. \"If your father's not IOA, is he military? Or an ambassador, maybe?\"\n\n\"Was.\" Weiyan's eyes slid toward General O'Neill. \"His name is Huang. Ambassador Huang.\"\n\nWith a cracking boom, Yu's ships blasted off.\n\nTeal'c was so shaken by Weiyan Shi's revelation that the departure of Lord Yu's ships meant nothing. His teammates were equally stunned. O'Neill's clenched jaw only served to accentuate his attempt to suppress his undoubted anger.\n\n\"I have done nothing wrong!\"\n\nThe girl dashed tears from her eyes.\n\nHer eyes. Teal'c leaned in for a closer view.\n\nWeiyan Shi's mouth quivered as he did so. \"Forgive me, Teal'c. I did not know \u2014 \"\n\n\"There is nothing to forgive,\" Teal'c assured her. \"You are not guilty of your father's crimes.\"\n\nThe young woman did indeed have the intense black eyes marked with odd specks of green exhibited by both Yu's First Prime Oshu and the spy Ambassador Huang. A spy who had nearly cost SG-1 their lives when he'd impeded their efforts to save Daniel Jackson a year ago.\n\nHer features, however, were different. Her nose smaller. Her face more oval.\n\n\"Weiyan,\" said Daniel Jackson. \"If everything you've told us is true \u2014 \"\n\n\"Oh, for crying out loud, Daniel!\" O'Neill grabbed Weiyan Shi's shoulders and yanked her to her feet. \"He put you up to this, didn't he? Is Huang here somewhere, getting his rocks off, watching us run around like rats in a \u2014 \"\n\nThe ground trembled. Teal'c recognized the sign. O'Neill must control his temper.\n\nTeal'c struggled to his feet, the task made all the more difficult by a weariness he could not ignore. None of the others seemed fatigued. Colonel Carter and Daniel Jackson had jumped to their feet with great alacrity.\n\n\"Where's your father, Weiyan?\" O'Neill shook her, his face dark with rage.\n\n\"Jack!\" Daniel pushed O'Neill back from the girl. \"Calm down!\"\n\n\"Shut up, Daniel. She knows something.\"\n\nA great tremor erupted across the valley floor, throwing Teal'c to the ground once more.\n\nColonel Carter ran to his side. \"You okay?\"\n\n\"O'Neill must stop.\" Teal'c pushed himself up to his knees. \"His emotions \u2014 \"\n\nA final shudder, Yu's fortress collapsed to rubble, and the quake faded to a distant rumble. All that remained was the hill bearing the great statute, the building in question, and the Stargate perched upon the cliff.\n\nTeal'c looked to O'Neill. His eyes were closed. His chest heaved heavily, his nostrils flared. Teal'c dared hoped his brother could contain himself.\n\nWeiyan Shi had already succeeded in doing so. She had wrapped her arms about herself, inhaling long, slow breaths.\n\n\"That's right,\" Daniel Jackson said. \"Keep calm, both of you. Jack, your abrasive general card won't work here and you know it.\"\n\nWeiyan Shi sank down beside Teal'c. She squeezed his hand. \"Are you all right?\"\n\n\"I am well,\" Teal'c lied. No purpose would be served by admitting a minor weakness. Not to her, not to any of his team. Whatever troubled him would surely pass. \"It is you, or more specifically, your origins which are now the concern.\"\n\nWeiyan Shi wrung her hands. \"I have told you everything I know. I spoke with my father only once. A year ago. It was then he told me he must go away. Forever.\"\n\n\"Oh, he went away all right.\" O'Neill opened his eyes. \"To a nice padded cell in China. He was spying for that god-damned slimy, snakeheaded \u2014 \"\n\n\"Sir,\" warned Colonel Carter as another tremor threatened.\n\nO'Neill sucked in a long breath. He expelled it noisily. \"Fine, Carter. You tell her. Tell her how \u2014 \" He turned away, shoving his hands in his pockets.\n\nThe tremor faded.\n\n\"Sir, there's a bigger question here,\" said Colonel Carter. \"We know her father's a clone from Janet Fraiser's report.\"\n\n\"A clone?\" Weiyan Shi laid her palm against her abdomen. \"That is why he could not tell me the truth. That is why...\" Her voice became inaudible.\n\n\"Weiyan,\" said Colonel Carter, \"your father's genetic markers date back 4,000 years. Were you and your mother both tested by the IOA?\"\n\n\"My mother,\" Weiyan Shi gulped, as if mention of the one who had born her was painful. \"She doesn't have the necessary genetics to power the great chair.\"\n\n\"Then Ambassador Huang must carry the Ancient gene,\" Teal'c said. Weiyan Shi's small hand tightened within his own. Though her grasp should have felt as nothing, Teal'c hands ached. He released her grip and flexed his fingers. His elbows and shoulders ached as well.\n\n\"Is my father an alien like you?\" asked Weiyan Shi.\n\nDaniel Jackson knelt down beside her once more. \"Your father's completely human.\"\n\n\"Cloned from a human,\" Colonel Carter emphasized. \"A human who not only lived 4,000 years ago, but must have been a direct descendent of the Ancients, just like \u2014 \"\n\n\"Like Jack.\" Daniel Jackson removed his glasses and gazed kindly into Weiyan Shi's eyes. \"Look, I know this sounds crazy, but if Yu was telling the truth, your ancestor was Sun Tzu.\"\n\n\"When you think about it,\" Colonel Carter said, \"it's amazing your mother even conceived. Four-thousand years of cloning would degrade the original DNA sample, no matter how much the Goa'uld used advanced Ancient technology.\"\n\nWeiyan Shi's eyes widened. \"If Lord Yu is a Goa'uld and he made my father, he cannot be bad.\"\n\nTeal'c interjected, believing the record must be set straight. \"Lord Yu is a parasitical creature, falsely claiming to have been your original emperor.\"\n\nDaniel Jackson slid his glasses back on. \"I'm not so sure his claims were false.\"\n\nClink. Snap.\n\nO'Neill had retrieved his lighter from his pocket and was now flipping it open and shut mindlessly, his attention on the building.\n\n\"Perhaps we will find our answers in the one constant throughout these apparitions,\" Teal'c observed.\n\nO'Neill twisted toward him, a half-smile upon his lips. \"Now you're talking. I could give a rat's ass about clones and gene pools. If Yu's in there \u2014 \"\n\nKawoosh.\n\nThe recognizable sound of the Stargate heralded its activation. Teal'c turned toward the cliff as the violent plunge of water settled into a stable event horizon.\n\n\"Carter? Any chance the gate's not a figment?\"\n\nFour figures emerged. Although they were too far away to ascertain their identities, Teal'c immediately recognized their khaki BDU pants, jackets and black tactical vests.\n\nO'Neill stuck a forefinger and thumb in his mouth and whistled. \"Over here!\"\n\n\"Sir...\"\n\n\"It's an SG team, Carter.\"\n\n\"I don't think so, Jack.\" Daniel Jackson slung a hand under Teal'c's arm and assisted him to rise.\n\nAs Weiyan joined them, they watched the four-man team descend along the side of the cliff and stride toward them. One of them had orange-red hair.\n\n\"What the hell's Balinsky doing here?\" O'Neill growled. \"I sent orders to Walter to kick his ass off of SG-13.\" He stormed toward the team. \"Took you enough time, Dixon!\"\n\nColonel Dixon walked directly through O'Neill.\n\n\"Great, more ghosts!\"\n\nSG-13 strode toward the crumbled walls of what had once been Yu's fortress. The team neared the ruins and their images dissolved.\n\nThe air rippled yet again, stirring up a great wave of dust that drifted toward them. The dust dissipated, revealing a more recognizable scene. SGC archaeologists dug trenches, others strung lines around several foot-deep holes. A tall, dark-skinned man with long plaits and a shovel jumped down into one such hole, shouting to an assistant who scurried in his direction.\n\n\"That's Kevin Hopkins.\" Daniel Jackson ran to the lead archaeologist. A college roommate, Teal'c recalled. Daniel Jackson waved a hand through the man's head, but Dr. Hopkins never flinched.\n\n\"And you wonder why instant replay was never approved in baseball.\" O'Neill pointed at the Stargate as another SG team \u2014 dressed in khaki uniforms and tactical vests \u2014 climbed down from the cliff. One carried a staff weapon.\n\nTeal'c recognized the team member immediately. He was the man carrying the staff weapon. \"This is indeed a faithful rendition of events.\"\n\n\"Yeah, super faithful,\" O'Neill's voice cracked.\n\nThe imagery sped up in a blur. The phantom version of SG-1 stopped by Kevin Hopkins, conferred, and then moved onward. Then Colonel O'Neill hefted a backpack toward the hill bearing the funerary statute. Daniel Jackson followed at a distance, his boonie-covered head turned toward the dig. Teal'c's doppelganger joined the then Major Carter and walked the camp's perimeter.\n\nA moment passed and the images subtly changed. Teal'c's ghost-double conferred with Major Carter at the base of the far hill. A single rifle shot rang out. He glanced upwards toward the hill as a shower of red clay sprayed forth from the _Zhenmoushu's_ claws.\n\nDaniel Jackson's ghost descended from the hill, his face flushed with indignation. He stormed toward the ruined walls. Placing a hand against the building's doors, he tilted his head upwards and stepped back as if to obtain a better view. A great wash of light surrounded Daniel Jackson and he disappeared as a ring transporter took hold.\n\nAll that remained was his boonie cast upon the ground.\n\nTeal'c turned his gaze upon the real O'Neill. His Tau'ri brother gripped the lighter, his eyes haunted with inner pain.\n\nThe need to assure O'Neill that these were merely specters of the past drove Teal'c to approach him. \"We cannot change \u2014 \"\n\nDizziness clouded Teal'c's vision. His body became numb.\n\n\"Teal'c!\"\n\nFatigue overcame him and he collapsed to the ground.\n\n\"Come on, T.\" Jack shook Teal'c's arm. \"Wake up!\"\n\nNothing. Teal'c's face had taken on a definite gray tinge, almost like he was \u2014\n\nNope, Jack wasn't even going to think it. He pressed two fingers against Teal'c's neck. There was pulse, barely. Jack glanced over his shoulder at that hellhole of a building, ready to take it apart brick-by-brick for answers.\n\nDaniel dropped down on Teal'c's other side. \"Does he need his tretonin?\"\n\n\"If he does, he's outta luck.\" Jack slapped Teal'c face lightly. \"Come on, buddy boy. Rise and shine.\"\n\n\"Wake up, Teal'c,\" Weiyan sobbed behind Jack. \"I am trying, cannot you?\"\n\n_What the hell did that mean_?\n\nTeal'c's eyes fluttered opened. \"O'Neill, what has happened?\"\n\nJack let himself breathe for the first time since Teal'c had passed out. \"Okay, don't do that.\"\n\nTeal'c harrumphed. \"I did not do anything.\"\n\n\"Sorry, Teal'c.\" Carter knelt down beside Jack. \"You did kinda pass out there. How are you feeling?\"\n\n\"I am well.\" With a barely hidden grimace, Teal'c pushed himself up onto his elbows.\n\nPutting a hand on Teal'c's chest, Jack stopped him. \"Nice try, but stay put for the moment.\"\n\nTeal'c sunk back, giving in way too easily.\n\nWeiyan brushed by, wrapping her arms around the big guy's neck. \"You will live?\"\n\n\"He'll be his normal self in no time. That is,\" Jack gently pulled her away, \"if you give him breathing room.\"\n\nCarter shot him a look. A look that meant trouble.\n\nHe let go of Weiyan's arm. \"What?\"\n\nCarter jerked her chin toward a spot a few feet off from Teal'c's position. Getting her message, Jack climbed to his feet and followed her out of earshot.\n\nAs Carter turned toward him, Jack noticed the bruises under her eyes. The kind she got when she'd been up for days straight. They'd been plopped down in this crap for a few hours, but days? Not even his internal clock was that messed up.\n\nOr was it? His memory had played some nasty tricks on him when they'd first arrived.\n\nHe sighed. \"A few real answers would be nice about now, Carter.\"\n\n\"Sir, if Teal'c continues to deteriorate...\"\n\n\"You think he needs tretonin?\"\n\n\"Maybe.\"\n\n\"Or maybe whatever hit those Dragon Guards got to Teal'c, too.\"\n\n\"Honestly? I think he was weakening even before those doors opened up.\" She slumped, a very un-Carter-like thing to do. \"Once Yu's ships... I mean, those images \u2014 \"\n\n\"I get it.\" Jack rubbed his face. \"I'm pretty beat myself, but how come we're not hungry or thirsty?\"\n\n\"Or why Weiyan's stomach bled out like that, and now \u2014 \"\n\n\"She seems fine.\" Jack glanced toward Teal'c. Weiyan cradled his beefy hands while Daniel chattered on about god-knows-what. \"Any chance her being here is just one big coincidence?\"\n\nCarter shook her head. \"I doubt it, sir. What we just witnessed pretty much matches Daniel's account from when Lord Yu held him hostage last year.\"\n\nShe shifted her gaze toward the building, the one constant throughout this nightmare. \"I barely remember that building from our visit a year ago. What is it about this place? Why the show and for who's benefit? Lord Yu captured Daniel as a sort of test, but \u2014 \"\n\n\"Yeah, wasn't that a fun time?\" He glanced down at a smooth, brown pebble he'd unearthed. If buildings and motherships could come and go, why were simple things like pebbles still around?\n\n\"The bigger question we should be asking is who's pulling the strings?\" He knelt down to pick up the pebble. \"You ever get the feeling that every time we've tied off a knot, the thread just unravels somewhere else?\"\n\nCarter sighed. \"Pretty much.\"\n\nHe stood up. Out of the blue, a buzzing noise filled his ears. Carter said something, but he couldn't hear her.\n\nHe must have stood up too fast. He took a deep breath and his ears popped, like a rubber band smacking against a wall. The buzzing was gone. \"What the hell?\"\n\n\"Sir, are you all right?\"\n\n\"Sam! Jack!\" Daniel called. \"Behind you!\"\n\nJack whirled around.\n\nAnd got a good, hard look at himself. Or rather, a khaki-clad phantom version. P90 in hand, favorite ball cap firmly in place, he \u2014 or rather, the phantom \u2014 stopped about twenty some-odd feet away and faced the building. Though his double shouted out soundlessly, Jack knew exactly what he'd said that day.\n\n_Daniel_!\n\nThe phantom whirled toward the funky Unas-looking statue on the far-off hill. He raised a hand to his mouth and shouted. Soundless.\n\n_Daniel_!\n\n\"Sir,\" Carter whispered, \"none of this is real.\"\n\n\"No kidding.\" Though he believed her, the pebble in his hand said different.\n\nJack threw the pebble at his doppelganger. It sailed right through. He followed the pebble's trajectory as it skittered to the ground a few feet away and kicked up a puff of dust and sand.\n\n\"General!\" Carter called out.\n\nStill charging forward, his khaki-clad other stormed right through him without as much as a ripple. It was as if his current self had no impact on the bigger things at stake.\n\nYet another reminder of just how useless he felt. Here. At the SGC. Even at the Ancient outpost.\n\nThe buzzing returned with a vengeance. \"Do you hear that?\"\n\nCarter raised an eyebrow. \"Hear what?\"\n\nAnother snap. The buzzing quit again. \"It sounded like... I dunno, Carter, like bees, or a machine, or some \u2014 \"\n\nA shadow passed overhead. He glanced up, saw the source, and while he should've been surprised, he wasn't.\n\nThat was the problem with reruns. No surprises.\n\nAn Al'kesh dropped out of the sky, honing in on his departing double's six. Both the ship and his other self headed toward the cliff, an activated Stargate precariously perched on its edge. The saucer-shaped Goa'uld ship laid down round after rapid round of weapons fire as his other self scrambled up the cliff.\n\nAs he watched himself climb, Jack remembered how the chase had pumped his heart. The world had narrowed down to survival. Reach the gate. Stay alive.\n\nOf course, last time this happened, he didn't have quite the same view. \"Talk about an out-of-body experience.\"\n\nBehind him, another set of footsteps pounded the dirt, headed his way.\n\n\"Jesus, Jack!\" Daniel stopped beside him, eyes bulging. \"What the hell were you thinking?\"\n\n\"I wasn't.\" Another round of fire shattered the pathway leading from the cliff's topside to the active gate. His younger self rolled sideways as more enemy fire peppered the ground. Finally, in an act of bravado \u2014 or stupidity, Jack still couldn't say which it had been that day \u2014 he dived into the active gate and disappeared.\n\nThe Al'kesh did, too.\n\n\"Fun stuff, huh?\" He missed that kind of action. No boring papers to sign. No casualty reports to read. No pandering to politicians.\n\nJack pulled out his old Zippo. Huang's kid, the crazy quakes, the lighter. It all added up to something. His thumb rubbed over the case, a pesky thought niggling at him.\n\nScrew it. There was only one way to find out.\n\nHe eyed Weiyan. \"She's got something to do with all this. Stay here.\"\n\n\"Jack, what are you doing?\"\n\n\"A different approach.\" He marched toward the girl stuck by Teal'c's side like there was no tomorrow.\n\n\"You miss it, don't you?\" Daniel called out. \"The missions, the fight. You can't do it all, Jack.\"\n\n\"T-shirts, Daniel!\"\n\n\"Excuse me?\"\n\nJack didn't break his stride. \"Make mine a double XL.\"\n\nDaniel wasn't the only member of the 'if-only' club.\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2100 HRS MCMURDO STATION\n\nBy the time Ambassador Zhu finished explaining the details of her short-term marriage to Lord Yu's spy, George wasn't sure whether to pity her or be outraged.\n\nHe'd yet to make up his mind when the bandage on her left cheek fell off, exposing a three-inch cut. Blood oozed from its edges. \"Are you all right, Madame Ambassador?\"\n\nZhu palmed her wound. \"Compared to my daughter? This is nothing.\"\n\n\"General Hammond?\" The click-clack of Lee's keyboard kept up for a few moments, and then, \"If these numbers are right, the device sent out a significant blast of photonic energy during that last quake.\"\n\n\"In which direction, Doctor?\" George braced himself, knowing he wouldn't like the answer.\n\n\"There were multiple epicenters this time.\" Lee squinted at the monitor. \"From fifty miles away to a few hits several thousand miles off. Up near the Magallanes-Fagnano fault line.\" He looked up, frowning. \"That fault line stretches almost 500 miles from the Pacific Ocean into Argentina.\"\n\n\"Doctor, I suggest you return your focus to shutting down the device.\" Noting Simmons' absence, George listened to the scurried footsteps and command barks emanating through the archway. What personnel were left proceeded with evacuations. The emergency exit door thudded shut.\n\nThe door!\n\nBolting up from Zhu's side, he joined Lee by the computers. \"Could the Ancients have built an entryway of some type that might access the lower level?\"\n\n\"If they did, we never found it.\" Lee switched the display to a schematic of the outpost's various chambers. A quick count told him there was six, but all on one floor.\n\nGeorge eyed the force field. \"There has to be an entrance leading from this floor down to the next. We just didn't look hard enough. Airman?\"\n\nLt. Gerling stuck her head in from just outside the archway. \"Sir?\"\n\n\"Take Lt. Brooks with you. I don't want one inch of this outpost left untouched. If there's another way down into that device, find it.\"\n\n\"Yes, sir.\" Gerling headed toward the main chamber.\n\nGeorge refocused on Ambassador Zhu and the 'Huang problem.' Her former marriage to an enemy spy put an entirely new spin on their situation.\n\nHe should be more surprised. For some reason, he wasn't. Standing two-hundred feet inside a glacier that could come down on his ears at any moment brought home the key lesson he'd learned in his eight years with the Stargate Program \u2014 crows always come home to roost.\n\nIn this particular situation, Huang was the crow and the Ancient weapons platform was the roost. A vital, but now dangerous, roost.\n\n\"You've put me in a difficult position,\" he told Zhu. \"By your own admission, vital information that would've affected the IOA's selection of your daughter, and your post as ambassador \u2014 \"\n\nZhu laughed, the sound bitter and empty. \"If the Chinese government can see past a family member's crimes \u2014 \"\n\n\"Ambassador, I don't blame you for Huang's actions, but the fact that Weiyan's overpowering ATA genetics are behind our predicament...\"\n\n\"What do Huang's actions have to do with Weiyan's abilities?\"\n\n\"There are security issues,\" he said half-heartedly. When the president had deemed the circumstances surrounding Huang's origins should be kept under wraps, current circumstances hadn't even been part of the picture.\n\n\"Security issues?\" Zhu's eyes widened. \"If she was your daughter, wouldn't you want to know?\"\n\n\"It's getting colder in here.\" He tugged his parka's zipper up higher.\n\n\"General, I deserve the truth.\"\n\n\"It's not that simple, Madame Ambassador.\"\n\nShe gazed down at her daughter and sighed. \"It never is.\"\n\n_Quit stalling, George_.\n\nHe told her everything he knew.\n\n# CHAPTER EIGHTEEN\n\n### SHUNYI PSYCHIATRIC HOSPITAL\n\n### BEIJING, CHINA\n\n### 2004 JULY 31, 9:00 PM\n\nHuang knelt within the four-walled garden and gazed upward. The city's polluted haze blanketed the night sky, but surely, tomorrow's sun would break its boundaries and reach this one spot. This one place he could succeed.\n\nSatisfied, his hands plunged into the dense, black soil.\n\n\"You failed,\" a whisper murmured behind him. \"At everything.\"\n\nA woman's voice. A voice that belonged to another life.\n\nHe scooped back mound after mound of the dirt. Earthworms and bits of mica sparkled beneath the pale light of the night's full moon. Beside his burgeoning hole lay four bundled saplings. His reward for cooperation, for allowing himself to be placed in solitary confinement. For no longer saying those two words which revealed his burden. Two words which spoke of his ruin.\n\n_They cannot_.\n\nHe would plant each sapling to honor Lord Yu's Dragons.\n\n\"Weiyan told me of your meeting,\" the woman said. \"How you taunted her with a last goodbye before your American adventure.\" It was good soil. Rich soil. His trees would grow well here. Though the doctors refused him tools to do his planting, he would still succeed. They could not take away his mind.\n\nEven though Jack O'Neill had taken his soul.\n\nOne moon. Not two.\n\n_Such a fool am I_.\n\n\"Huang, please... Turn around. Talk to me.\"\n\nThe voice again. A time of trespass. One he must not remember. It was _b\u00f9 zh\u01d4n x\u01d4_. Forbidden. No Dragon Guard must take a wife or mate. Such had been Lord Yu's command.\n\nOne moon. Not two.\n\nHe picked up the nearest sapling. \"For Kung Qiu,\" he whispered.\n\nHe removed its burlap cover. The small gather of roots and twigs smelled of cherry blossoms. Of promise. Placing the base gently in the hole he'd created, he began to swipe the soil in around its roots. Each stroke of dirt eased his heart, allowing his mind to wander. Though he begged his thoughts to travel further back in time, to a happier moment, all he saw was his ultimate failure to Lord Yu.\n\n\"At least do me the courtesy of explaining why you did this,\" the voice demanded.\n\nHe crawled to the next planting spot and thrust his hands into the dirt once more. He scooped back more soil. As he did so, the memories returned unasked, but demanding.\n\nHe remembered it all.\n\nLord Yu's capture of Dr. Jackson on Kunlun had given Huang the perfect opening to infiltrate Stargate Command. All it had taken was one call to the Americans' inept president with the threat of disclosing their secret program to the world. By forcing SG-1's team members who had the necessary naquadah to use the communications devices, he'd been given the opportunity to view his former home world once more.\n\nHe had sat within Stargate Command's conference room, so intent on contacting his master that he'd paid no intention to the trap closing in around him. The droning unimportance of words spilling from the porcine General Hammond had been inconsequential. Mindless banter that required little attention.\n\nHuang had bided his time as SG-1 blundered through the outer defenses of Lord Yu's fortress, knowing that once Lord Yu captured them, he had only to announce his presence and his master would retrieve him.\n\n\"Former husband, I am being punished for your crimes!\"\n\nHuang turned his back on the voice and uncovered his second bundle. \"For Qin Shi,\" he told the sapling.\n\nA thrill ran through him upon remembering when the communications device had sent back images of his home world. Craggy mountain peaks enveloped in snow. Mist rising from the shores of _Zh\u0113n Xi\u00e0ng_. His amusement at SG-1's absurd struggles to traverse Lord Yu's maze, all the while hoping they'd fail, but needing them to succeed if his plan was to work.\n\nHe had ignored the signs of SG-1's ulterior motives. Perhaps it had been the delight at Colonel O'Neill's feigned demotion? The pleasure at witnessing his defeated stride past the communications device had been unquestionable. That accomplishment had made Huang disregard Jacob Carter's comments on the two moons of Lord Yu's home world.\n\nThere was only one moon on Lord Yu's home world.\n\nNot two.\n\nHuang shuffled further into the garden and dug a third hole. Only now could he admit to himself how arrogant had been his disregard for the Tok'ra's duplicitous words. He had been too caught up in O'Neill's downfall and the reunion yet to come with Lord Yu. He should have recognized the Tok'ra's lies for what they had been. A signal. A warning.\n\nHe picked up the third sapling. \"For Lao Dan,\" he croaked, his voice thick with regret.\n\nHe should not have forgotten the words of his dead brother's ancestor. It shamed him now to remember.\n\n_The skillful man must be on his guard against being vain or boastful or arrogant in consequence of it_.\n\nHuang thrust the third sapling into the ground.\n\n\"I am being sent to Antarctica to oversee negotiations.\" The woman's voice once more. Another sin for which he could never atone.\n\nThe woman's voice dropped to a whisper. \"I must conspire some way to bring Weiyan with me \u2014 \"\n\n\"They cannot.\" Huang cringed as he spoke the banned words. Heinous images of his failure to Lord Yu tumbled through his memory. _The fortress littered with the bodies of Lord Yu's Jaffa. A young warrior racing by the communications device, preparing to attack the pathetic Tau'ri. Electric blue arcs of zat'ni'katel fire decimating the brick-lined terrace_.\n\nThrough it all, he had sat in the SGC's conference room, a sense of panic seizing him. The device had rolled through the ensuing battle between Jaffa and Tau'ri, permitting Huang a moment's view of a young man bearing the First Prime's golden tattoo. _Green-flecked black eyes, square jaw. A high forehead that spoke of an aristocratic air born of intelligence, not blind faith_.\n\nAnother clone of Sun Tzu.\n\nFor Huang, it had been like looking in a mirror.\n\nThe device had rolled on. _Jaffa far too inept at battle erupted from the patio column's secret passageways. They became easy targets for SG-1. Stunned, he watched each warrior fall_.\n\nThe Jaffa had been pale imitations of those he'd known in his time as a Dragon Guard trainee. He had known then and there that Lord Yu needed him to return and set things aright. To take his place as Dragon Guard and become the true First Prime.\n\nHe picked up the fourth sapling, remembering how his dedication to returning to Lord Yu had been born anew upon seeing the Jaffa battle the Tau'ri.\n\nA breeze blew against his neck. Another memory. One of desolation. Of his time in the frozen wasteland of Earth's southern pole. The being of light that helped him survive. He struggled to suppress the memory. Crushing the dirt between his fingers, a silent wail lodged in his throat, begging for escape.\n\n\"Huang, please...\" A hand circled his wrist. Pulled his hand back from the soil.\n\nThe fourth sapling tumbled to the ground.\n\nThe words he must not utter tumbled from his mouth. \"They cannot,\" he told the Tau'ri woman.\n\n\"Cannot what?\" Quing Zhu whispered. \"Tell me.\"\n\nMemory of his failure pressed down on him. _The sound of rifle fire emitting from the communications device as it rolled by. A jumble of images. Bits of broken brick terrace. Collapsed Jaffa. A moment's glance of Major Carter and Bra'tac, the rebel Jaffa Master, racing toward O'Neill and the_ Shol'vah _Teal'c_.\n\nThe device had continued rolling, heading toward a marble column near the center of the terrace.\n\n\" _What have you done?\" Huang shouted over the din of bullet and zat fire_.\n\n\" _What I should have done days ago,\" General Hammond had said, his pig-faced dense stupidity replaced with certitude_.\n\n_The display stopped rolling, wedged up against the column. The view revealed the sacred statue commemorating the wife of Lord Yu's host. A memorial of the shared sacrifice of symbiote and man to save China from Anubis a thousand years ago. The green marble statue depicted the wife cradling a baby in her arms, a child at her feet. A glorious reminder of what mattered most_.\n\n_Twin sets of Tau'ri standard-issue boots ran out of frame_. _O'Neill and the Jaffa rebel. Out of nowhere, P90 fire erupted from beside the statue. Lord Yu's Jaffa responded with a rapid succession of zat fire_.\n\n_The statue disintegrated_.\n\n\"No!\" Huang cried out. \"They cannot!\"\n\nRough hands pulled him from his plantings. As the hospital guard dragged him from the garden, he permitted himself a final glance of Quing Zhu. The woman who had once been his wife. The mother of his child.\n\nCold realization hit him as they locked him in his cell. Had Quing said they would go to Antarctica?\n\n\"They cannot.\" He sank down on the thin pallet that was his bed. This was his punishment. To stay here and rot.\n\nHe deserved nothing more.\n\n# CHAPTER NINETEEN\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2140 HRS MCMURDO STATION\n\n\"General Hammond?\" Lt. Brooks poked his head through the archway.\n\nGeorge waved him in. \"Report?\"\n\n\"Major Davis radioed, sir.\" The airman pushed back his hood and wiped stray snow off his watch cap. \"He's just rendezvoused with the SEALS over at Marble Point.\"\n\n\"Notify me the moment their chopper lands topside.\"\n\n\"Yes, sir.\" The airman hurried off.\n\nGeorge ran a hand over his eyes, fighting off fatigue that wouldn't end soon. In the past half hour, he'd provided Ambassador Zhu with as much detail as he dare regarding the Huang and Lord Yu. But he hadn't told her everything. Not yet.\n\nZhu's focus stayed fixed on her daughter's comatose form. \"Please,\" she said as the airman left, \"finish your story, General Hammond.\"\n\nNow came the kicker.\n\n\"We have evidence to suggest Huang was cloned from one of Yu's original Dragon Guards,\" George said. \"A direct descendent of Sun Tzu, I believe.\"\n\nZhu whirled toward him, her eyes widened in obvious disbelief.\n\n\"Lord Yu obtained the cloning technology from the ruins of an Ancient city,\" George explained. \"The SGC's chief medical officer at the time was able to confirm that Huang's DNA contained unusual traits. Traits not seen for over 4,000 years.\"\n\nZhu chewed at her thumb as George detailed Lord Yu's odd revelations to Daniel Jackson. He found the nervous habit telling. While she'd exhibited genuine surprise at his revelation, there was something else going on in that mind of hers, something that may or may not have a direct relation to their current predicament.\n\n\"And the U.S. decided to keep this information from China because...?\" She dropped her hand to her side. \"Don't bother. The lack of trust between our two nations takes on many shapes, this being only one example.\"\n\n\"It goes both ways, Ambassador. We'd no idea that Huang had fathered a child.\"\n\nZhu shook her head. \"A descendent of Sun Tzu. A viable, healthy clone. If this is the reason why Weiyan was always such a sickly child \u2014 \"\n\n\"General Hammond!\" Ambassador Duebel raced in. Not for the first time today, George wished Lt. Simmons could be in two places at once \u2014 helping evacuation efforts, and here, running interference with fifth wheel ambassadors who should have been long gone.\n\nThe Swiss ambassador came to a halt beside Zhu. \"Any change?\"\n\nShe snorted. \"In a manner of speaking.\"\n\nDuebel glanced at George. \"I don't understand. The force field's still in place.\"\n\n\"Why haven't you evacuated yet, Ambassador? Where's Juarez?\"\n\n\"He left. My God, it's cold in here.\" Duebel yanked his parka zipper up. \"We were boarding the helicopter when a radio report came in... Ushuaia, Argentina suffered a powerful quake twenty minutes ago.\"\n\n\"Casualties?\" George said a silent prayer.\n\n\"Several hundred wounded. No fatalities yet,\" Duebel replied. \"General, this will only get worse.\"\n\n\"The Navy SEAL munitions team should be here shortly.\" He glanced at his watch. \"Sunrise is in fifteen hours. In the meantime, I've given Dr. Lee the next twelve hours to find a way through that force field so we can retrieve our people. Once those twelve hours are up, the outpost will be destroyed as promised.\"\n\n\"Uh, General?\" Lee hurried over. \"If we know Weiyan's father could shut the device down, shouldn't we consider bringing him here? Four-thousand year-old genetics should be strong enough to \u2014 \"\n\n\"A 4,000 year-old what?\" Duebel asked loudly.\n\n\"Ambassador, keep your voice down, please.\" George shot Lee a silent warning. There was need to know and then there was need to _not_ know. By now, the scientist should comprehend the difference.\n\n\"Sorry,\" Lee said more softly, \"but Huang's ATA genetics could possibly shut down the drift device, and the force field. Then we won't have to blow up the outpost.\"\n\n\"Quing?\" Duebel knelt down beside Zhu. \"What does your ex-husband have to do with all this?\"\n\n\"You'll have to ask the general,\" said Zhu, shifting her position with her legs tucked underneath.\n\nGeorge appreciated her discretion, but facts were facts. He urged Lee to get back to his work and then returned his attention to the ambassadors. \"I'm sorry, but I can't allow an enemy spy access to this facility. We have no idea if he \u2014 \"\n\n\"Now wait just a moment.\" Duebel shot up and glared right at him. \"If Ambassador Huang \u2014 \"\n\n\"Former Ambassador,\" George retorted. \"As a member of the Security Council, you've read the official report. Huang's actions almost killed SG-1.\"\n\n\"And look at them now.\" Duebel flung a hand toward the hole. \"Are you Americans so prideful that you're willing to punish the entire continent \u2014 possibly the world \u2014 just because an accused spy caught you with your... Oh, what's the American expression?\"\n\n\"Caught with your pants down,\" Zhu muttered.\n\n\"Exactly.\"\n\n\"It's far more complicated than that,\" George explained.\n\nDuebel crossed his arms. \"If Huang's ATA genetics are even remotely as powerful as his daughter's, we need him here. Now.\"\n\nJack took one look at Teal'c's ashen face and knew exactly what he had to do. He grabbed Weiyan's arm and pulled her up from Teal'c's side. \"Let's go have a nice little chat, shall we?\n\n\"I have done nothing wrong!\"\n\n\"Yeah, yeah. You keep saying that.\" He mentally put a lid on any sympathy he might feel for the girl; he'd spent his life perfecting the act of being a bastard when needed.\n\nThat didn't make what he had to do next any easier.\n\nTeal'c raised an eyebrow, but Jack gave him a short shake of his head. Stay out of this.\n\nHe dragged Weiyan a few feet closer to the building. No earth-shattering quakes started up, so that was a good start. He waved his lighter at Weiyan. \"You're Chinese. You must know the _Art of War_ inside and out.\"\n\n\"Sun Tzu?\" Lifting her eyes, Weiyan mirrored Teal'c's eyebrow acrobatics. \"Of course. It was my father's wish that I memorize his work.\"\n\nClink. Jack flipped the lighter open. \"Yeah, I'll bet. How does the one part of old Tzu's writings go? Something about bait and lures?\"\n\nSnap. He flipped the lighter shut. \"I'm a fishing fan.\"\n\n\"Ah...\" She closed her eyes and recited, \"'Offer the enemy a bait to lure him; feign disorder and strike him.'\" Her eyes opened again and cocked her head. \"There's another passage, General. One that speaks of pretending to be weak so that your enemy might grow arrogant.\"\n\n\"Been there, done that.\" Jack deliberately crossed his arms, the lighter still in one hand. \"I gotta wonder... Your father nearly got us killed last year. By your own admission, you weren't even healthy enough to join the trainee program. And yet, somehow, you cleared the IOA's vetting process.\"\n\nWeiyan's mouth shut tight. Not exactly what he expected.\n\nHe pushed it to the next level. \"Now, I'm not that keen on the IOA. Frankly, there are times I think they're not much better than the Goa'uld, what with all their power-mongering, but those folks don't just let anyone in. Trust me, I know.\"\n\n\"I \u2014 I...\"\n\n\"Go ahead,\" he said with as much patience as he could muster. \"I'm all ears.\"\n\nShe bit her lip.\n\nClink. Jack flipped open the lighter again. \"Well?\"\n\n\"You will not yell and make the ground tremble?\"\n\nSnap. \"Try me.\"\n\n\"My mother... She is Quing Zhu.\" She gulped. \"She pre-approved my application for the trainee program.\"\n\n\"The Chinese ambassador?\" Jack whistled. \"Did she know about Huang's little affiliation with the Goa'uld?\"\n\nAn itty-bitty trembler rumbled beneath his boots. A warning. Or in his case, progress.\n\n\"Answer the question, Weiyan.\" Jack threw back his shoulders, putting on the best mean-old brigadier general persona he could drum up. \"Did your mother know or not?\"\n\nWeiyan nervously wet her lips. \"I do not think so.\"\n\nJack went in for the kill. \"But if Huang's name was on your application as next of kin \u2014 \"\n\n\"My father's a good man,\" she muttered. \"His reasons for doing what he did must have served some great purpose.\"\n\n\"Give me a break.\"\n\nThe ground shook. Harder this time. Louder.\n\n\"Jack, calm down!\"\n\nApparently, not loud enough. Jack raised a hand, warning Daniel not to come any closer. The rumbling kept up. Carter hurried over to Teal'c and helped him to his feet.\n\nJack got in Weiyan's face. \"So your mother snuck you in the backdoor. Faked your medical records \u2014 \"\n\n\"Hemophilia is manageable.\"\n\n\"Your mom must've thought you'd be the prize student.\" He stared her down. \"Was she gunning for you to be first in line to operate the weapons platform?\"\n\n\"My mother thought I would fail!\"\n\nA loud crash. Jack didn't have to look to know the phantom Stargate had tumbled downward, all thirty-two tons demolishing the cliff. Still, there were no cracks in the ground around them. Jack opened his mouth, ready to push harder. An arm yanked him backwards.\n\nDaniel stepped between them. \"Stop it. Both of you!\"\n\nWeiyan turned away, her shoulders shaking.\n\n\"Back off, Daniel!\" Jack shouted over the rumbling war of rock and dirt. \"You certainly did fail, didn't you, Weiyan? When you sat in that chair, Ambassador Zhu \u2014 if that's really her name \u2014 shouted her lungs out at you, and bam! Those ATA genes spiked into overdrive.\"\n\n\"Enough!\" Weiyan whirled toward him. \"You have no right to treat me this way.\"\n\nBoom. The ground shook. Hard. Jack managed to stay on his feet, but Weiyan stumbled forward. He stopped her fall and...\n\nThe rumbling stopped.\n\n\"Sir, we've got company.\" Carter nodded toward the building. A curtain of golden-white light streamed downward, washing out their view of the brick building. The curtain oscillated, forming three, then four glowing ribbons.\n\nJack pulled Weiyan upright. \"You're right,\" he told her. No one should treat you like that.\"\n\n\"But \u2014 \"\n\n\"I needed to get that thing's attention.\" He jerked his chin toward the building. \"Nice backbone you're growing there, by the way.\"\n\n\"I have always had a backbone, General O'Neill.\"\n\n\"Having one and using it are two different things.\" He turned toward Daniel. \"Let's see if Teal'c's up for a hike.\"\n\nDaniel frowned. \"You could've told me what you were planning.\"\n\n\"I wasn't sure it would work.\"\n\nThe ribbons contracted, losing their gold tint. Four white-hot ribbons merged into a solid pillar. A handful of glistening tendrils unfurled. It was a familiar image. One Jack had seen too often for one lifetime.\n\n\"Jack, if that's an ascended Ancient \u2014 \"\n\n\"If it is, it can kiss my ass. Daniel, Carter, get Teal'c over here. Carry him if you have to!\"\n\nWith Teal'c's arm slung across their shoulders, the two managed to half-drag, half-carry him over. As they reached Jack, the ascended being solidified into a hooded human shape dressed in a long, flowing white robe.\n\n\"You able to walk?\" Jack asked Teal'c. His face was grayer than the SGC's walls.\n\n\"I will try, O'Neill.\"\n\nWeiyan threw a supporting arm around Teal'c's torso. \"You must not try, you must do!\"\n\nTeal'c managed a weak smile. Then he broke down coughing.\n\n\"Yeah, you're not going anywhere.\" Jack gestured for him to sit down. Teal'c complied, too readily which scared the crap out of Jack.\n\nThe ascended being \u2014 or whatever the hell was dicking with them \u2014 waved in their direction. \"Look, I get that this could be some glowy Ascended what \u2014 \"\n\n\"It could be Oma,\" Daniel said with far too much reverence for Jack's taste.\n\n\"Oma?\" Weiyan bent down beside Teal'c.\n\n\"She is an ascended being,\" Teal'c explained between gasps. \"A former Ancient.\"\n\nWeiyan glanced up at Carter. \"Did they not build the Antarctica chair?\"\n\n\"Amongst other things,\" Carter replied with a bare sliver of a smile. \"General, I recommend caution.\"\n\n\"No fooling.\" Jack eyed the alien. A breeze had kicked up. A light one, but enough to send the ascended being's robes flapping behind it. At least it had stopped waving.\n\nTeal'c's breath grew more ragged. Jack considered marching right up to the alien and punching its lights out, but something told him that wouldn't be such a hot idea.\n\n\"I don't know about the rest of you,\" he said, turning back to the others, \"but I'm getting pretty tired of being treated like a toy.\"\n\n\"Daniel, you knew Oma better than anyone.\" Carter tilted her head. \"Would she really treat us like this?\"\n\n\"Sam, I barely remember talking to her before I \u2014 \"\n\n\"Died?\" Jack offered.\n\nDaniel shrugged. \"She's always been fairly gentle. Helpful even.\"\n\n\"Tell that to the Jaffa she burnt to a crisp on Kheb.\"\n\n\"She was protecting Shifu,\" Daniel whispered. \"This is... Different. I'm not sure how, but it is.\"\n\nTeal'c started up another coughing fit.\n\n\"I don't care what's going on, Daniel, I've had enough.\" Jack raised a hand to the side of his mouth and shouted, \"No more games! You wanna talk? Then get your butt over here!\"\n\nThe buzzing came back, softer this time. He ignored it.\n\n\"Careful, Jack, don't \u2014 \"\n\nHe shot Daniel a glare. \"I'm not letting that thing kill Teal'c off.\" He turned back toward the alien, the buzzing intensified. Pressure crowded his eardrums.\n\nThe alien disappeared.\n\nHe shook his head. Tried to shake off the buzzing. \"We're not moving!\"\n\nThe breeze shifted to a full-on gust. A dry, bitter cold reminder of what they'd left behind in Antarctica. \"Anyone feel that?\"\n\nCarter opened her mouth to speak, but Jack never heard her. Something slammed against his back. He stumbled and the wind seized him as if he was nothing more than a ragdoll. A sharp tug and he felt himself yanked forward.\n\nTwisting sideways, he tried to squirm out of its grip. No luck. Dust and wind blew across his face. He glanced sideways, but all he could see was the dusty terrain speed by. The only upside was that he had company. Carter, Daniel, Teal'c, and even Weiyan were just as locked up as he was by whatever had them in its clutches. The distant horizon became even more distant as they were pulled forward.\n\nIgnoring the wind's sting in his eyes and face, Jack refocused on the building up ahead. It was moving, too. Coming closer. A wave of vertigo hit him. If the yanking around didn't stop soon, he'd be a pancake in need of scraping off the building's walls. He threw up his arms, braced for impact.\n\nThe walls blurred. Jack squeezed his eyes shut, ready for what could only come next.\n\nHe jolted to a stop. Opening his eyes, the first thing he noticed was the warm air. The second thing was the blue light licking against bricked walls. They were inside a chamber of sorts, and though Jack couldn't see the ceiling, he was pretty damn sure they were inside that damned building.\n\nThe third thing, and probably the most important, was the hesitant smile of someone he never expected to meet again.\n\n\"It is good to see you, O'Neill.\" The robed alien pulled down his hood.\n\nHe looked exactly like Skaara.\n\nWhich was too good to be true.\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2210 HRS MCMURDO STATION\n\nGeorge's latest argument with Ambassador Duebel was stopped short by the arrival of Major Davis and two Navy SEALS. The men wore bright orange coveralls and jackets with Byrd Station patches on their left sleeves. Blue wool watch caps affixed with the traditional eagle and trident logo of the Navy SEALS covered their heads. They each carried a yard-wide chrome-and-black box stamped with the recognizable 'explosives' label. Colonel Ken Ferguson brought up the rear. George awarded Davis a grateful smile. \"Glad you could make it. First time at the outpost? \"\n\n\"Wish the circumstances were different, sir.\" Davis quickly introduced the SEALS \u2014 Lieutenants Kalsner and Mason. \"With your permission, they should start laying charges \u2014 \"\n\n\"Just one moment!\" Zhu pointed at the chair. \"If there's any chance my daughter can be saved by using Huang's \u2014 \"\n\n\"Huang the spy?\" Davis asked. \"General, what's going on?\"\n\n\"Have the SEALS lay their charges around the platform, Major,\" George ordered. \"I'll be with you in moment.\"\n\n\"Yes, sir.\" Davis led the SEALS over to the other side of the platform. They put down their kits and flipped open the latches. Ferguson stood back, taking it all in.\n\nGeorge returned his attention to the ambassadors, his voice low. \"Bringing Huang here is not going to happen.\"\n\n\"Because he claims to be a spy for the Goa'uld?\" Duebel asked.\n\n\"Yes.\" George nodded. \"And because sunrise happens just shortly after noon tomorrow. That's less than fourteen hours from now. There simply isn't enough time to retrieve him from Beijing. Do you understand?\"\n\nHis lips pressed together, Duebel spun around and joined Zhu by the hole. He crouched down and took her hand.\n\n\"My last words,\" Zhu whispered. \"They were \u2014 \"\n\n\"Shhhh,\" Duebel said reassuringly.\n\nGeorge bowed his head, striving to fight back the remorse nipping at his heels. Though he'd bought Lee added time, George harbored doubts that the scientist could find a solution.\n\n\"General Hammond, we found something.\" Major Davis dashed around the hole, holding up a glass tube.\n\n\"That's Teal'c's tretonin,\" Dr. Lee said. \"It must have fallen out of his pocket when the floor collapsed.\"\n\nDavis handed the tube over to George. The force field's shimmering light reflected against the purple liquid inside. If Lee did succeed, Teal'c would need a dose immediately.\n\nGeorge thanked Davis. \"Nice work out there today, Major.\"\n\n\"Thank you, sir.\"\n\n\"I trust McMurdo Sound is still in one piece?\"\n\n\"Mostly, sir.\" Davis smiled. \"When the 302s jumped into hyperspace, the ice broke up a bit.\"\n\n\"I suppose NSF Director Edmunds must be a little unhappy.\"\n\nDavis glanced toward Ferguson. \"Sir, in the grand scheme of things, I doubt that's high on Dr. Edmunds's list of concerns.\"\n\n\"In the meantime,\" George waved Teal'c's tretonin injector, \"I'll ask Lt. Simmons to hold onto this. If we're lucky, Teal'c will need it as soon as we pry him out of that thing.\"\n\nLee stared at the injector. \"Lt. Simmons, sir?\"\n\n\"Yes, that's right.\" George turned back to the major. \"And Davis? Ask the lieutenant to arrange for the ambassadors' immediate evacuation.\"\n\nDavis' eyes narrowed. \"Graham Simmons?\"\n\n\"He's out by the com center. Probably trying to round up another pot of coffee if I know him.\"\n\nDavis and Lee exchanged glances.\n\n\"Is there a problem?\" George asked.\n\nLee scratched his head. \"Lts. Gerling and Brooks are the only airmen who've been in the chamber since the accident.\"\n\nGeorge spun toward Davis. \"Simmons has been here the whole time.\"\n\n\"Sir, I just talked with Graham Simmons a few days ago. He's in Colorado Springs at the SGC.\"\n\nAscension was one thing. Being human, another.\n\nDaniel knew that and still he stared at Skaara, unsure of what to do, or even what to say. It was impossible to deny the wide gulf between them.\n\nEven if his former brother-in-law didn't look a bit different since \u2014\n\n\"Since my death?\" Skaara asked warmly.\n\nDaniel winced. \"More like before my own...death.\" He'd forgotten how perceptive Ascended Beings could be. Oma had certainly been that way.\n\nIt wasn't as if ascension had changed Skaara's appearance. Not by any means. He still wore the same long plaited black hair, olive fatigues, and cut-off chainmail vest taken from Ra's Jaffa. A loose bleached-white Abydonian-style robe hung from his shoulders, made all the whiter by the room's pervasive blue glow.\n\nIf it was a room. A blue mist hovered just below where the ceiling should be, obscuring Daniel's view. He shuddered. The room, or chamber \u2014 he wasn't quite sure which \u2014 was a good fifty degrees colder than outside. Though the ambient light had turned the bricks a mottled blue, he assumed they were inside the same building they'd seen outside. The same building that housed whatever had killed Lord Yu's original Dragon Guard.\n\nNear the wall stood a row of squared columns, their bases wrapped with trellised panels reminiscent of the Ancient outpost on Earth. A glance over his shoulder told him he wouldn't be able to see further into the chamber. It was too dark.\n\nAs he turned back toward Skaara, Jack whispered, \"This has gotta be some elaborate trick.\"\n\n\"No tricks.\" Skaara stepped toward them. \"I promise.\"\n\n\"I don't know, Skaara.\" Jack shoved his hands in his pockets. \"Planting your old Zippo...\"\n\nSkaara held out the lighter. \"You mean this, O'Neill?\"\n\n\"Ah, yeah. That.\" Jack's hands dropped to his sides. \"See, I could've sworn I'd left it back at the SGC. In my locker.\"\n\nThe lighter disappeared. \"Place and time have meaning only when the heart requires.\"\n\n\"You've been spending time with Oma,\" Daniel remarked.\n\n\"Before, after, during.\" Skaara shrugged. \"Though the current carries the stream one way, the skilled oarsman can travel in both directions.\"\n\n\"Of course,\" Sam said. \"Time wouldn't have any meaning if you're ascended, would it?\"\n\n\"Carter?\"\n\nSam turned toward Jack. \"Quantum mechanics, sir. Theorists believe time is an artificial construct. Like Skaara said, the past, present and future can all \u2014 \"\n\n\"Okay,\" Jack waved her off. \"I get it.\"\n\nSkaara cocked his head. \"Are you not happy to see me, O'Neill?\"\n\n\"Sure.\" Jack frowned. \"I thought you ascended beings weren't supposed to interfere with us lowly folks.\"\n\nSkaara matched Jack frown for frown. \"I saw no other way.\"\n\n\"If you say so.\"\n\nDaniel raised an eyebrow at Jack. While Abydos's destruction had been a painful discovery for Daniel upon his return to humanity, he'd known it had hit Jack just as hard. He wouldn't talk about it, though, no matter how much Daniel tried.\n\nHe understood why. Jack had grown close to Skaara during the very first Abydos mission nine years ago. That closeness had only deepened during the Stargate Program's early years. Whenever SG-1 visited, Jack went out of his way to bring something for Skaara, be it worn-out fatigues, comic books, even a baseball.\n\nAnd fuel for Jack's old Zippo. Skaara would hurriedly fill his precious lighter and then ignite the fire for their evening meal of roasted lizard and yefetta flatbread.\n\nThose memories made Daniel smile. That is, until he remembered their current predicament. \"Skaara, what's this about? Why \u2014 ?\"\n\n\"Carter, check on Teal'c,\" Jack ordered.\n\nDaniel mentally kicked himself as he joined Sam in hurrying to Teal'c's side. Daniel had been so surprised by Skaara's appearance that he'd forgotten about everything else. Teal'c lay sprawled across the floor, a marble surface laced with thousands of thin silver lines etched into the stone. Weiyan crouched by his side, holding his hand. Kneeling down, Sam pressed two fingers against Teal'c's neck.\n\nSkaara glided up beside Daniel and gazed down on the two. \"Helping those in need is not an evil act.\"\n\nHe exchanged glances with Skaara. \"You mean me.\"\n\nSkaara bowed his head. \"Though the Caledonians were blind, you are not, Daniel. None of you are. That is why I am here.\"\n\n\"Will he live?\" asked Weiyan, her eyes darting back and forth between Teal'c and Skaara. She seemed fine. Actually, better than fine. She'd somehow managed to neatly pull back her disheveled hair, color had reappeared on her cheeks, and even the bloodstain on her shirt had disappeared.\n\n\"Pulse is erratic.\" Sam palmed Teal'c's forehead. \"He's running a low-grade fever as best I can tell.\"\n\nTeal'c's eyes fluttered open.\n\nRelieved, Daniel refocused on Skaara. \"Can you tell us \u2014 ?\"\n\n\"Careful,\" Sam warned. \"The general's right. We can't be sure that really is Skaara, ascended or not.\"\n\n\"Have we ascended?\" Teal'c murmured. As Weiyan helped push him up to a sitting position, Jack bent down and clasped the Jaffa's shoulder.\n\n\"No, Teal'c.\" Skaara knelt beside him. \"You are still very much alive.\" Though Skaara's voice sounded as light and friendly as ever, his face was solemn.\n\nJack dropped down on Teal'c's other side, the stress line between his eyes deepening.\n\nJack's stern visage. Skaara's grim frown. If anyone had told Daniel their reunion would be so dour, he'd have sworn they didn't known what they were talking about.\n\nA convulsion hit Teal'c, reminding Daniel that this wasn't really a time for happy reunions.\n\nJack pressed Teal'c's shoulders back as the Jaffa's body shook. \"If you really were Skaara, you'd help him!\"\n\nSam stood up. \"He needs tretonin.\"\n\n\"Then we must hurry.\" Skaara flattened his hand against Teal'c's chest.\n\nA white glow blossomed across Teal'c's chest. He gasped as the glow intensified, arching his back in another convulsion. Weiyan echoed his gasp with one of her own.\n\nThe glow dissipated and Teal'c sank back to the floor. His eyes closed, his breathing steadied.\n\nDaniel sighed in relief. \"He's going to be all right now, isn't he?\"\n\n\"For a while.\" Skaara gave him a tentative smile. \"As Colonel Carter said, Teal'c needs tretonin.\"\n\n\"For a while?\" Jack rocked back on his heels and stared at Skaara. \"I thought you said time didn't matter.\"\n\n\"To someone ascended, no.\" Skaara dropped his hand away from Teal'c's chest. \"I am sorry, O'Neill. This was the only way to safely get your attention.\"\n\nJack climbed to his feet. \"So says the ascended, all powerful, all knowing wizard.\"\n\nWeiyan joined him. \"Ascended?\"\n\nSam explained. \"It means transcending to a higher plane of existence. Most of the Ancients \u2014 the people who built the Antarctica chair and the Stargates are \u2014 \"\n\n\"Not everyone who ascended was once an Ancient, Colonel Carter.\" Skaara returned Weiyan's gaze a moment longer and then rose with a grace beyond his young age.\n\nOr what had been his age when he'd been alive.\n\nSkaara stretched out his arms. \"It is good to see all of you again.\"\n\n\"All of us... Do you know me?\" Weiyan asked.\n\nSkaara's smile broadened. \"I know your father.\"\n\nTaken aback, Daniel glanced at Weiyan. Her lips had formed a mute O. He couldn't blame her.\n\n\"Look, Skaara.\" Jack took a wary step closer. \"I'm sorry if I'm having trouble believing you, but just what the hell is going on? And just how the hell do you know a Goa'uld spy?\"\n\nSkaara tilted his head. \"I called to you, O'Neill.\"\n\n\"I guess I wasn't listening.\"\n\n\"Back up,\" Daniel said. \"You knew Ambassador Huang?\"\n\nAnguish darkened Skaara's face. \"By a confluence of circumstances which are deeply regretted.\"\n\n\"Circumstances which might kill Teal'c, you mean.\"\n\n\"Jack.\"\n\n\"No, Daniel.\" Jack spun toward him. \"That man was working for those slimy, snakeheaded Goa'uld.\"\n\nSkaara wrung his hands. \"O'Neill, you must understand. Huang did not see Lord Yu as a demon. His loyalty was pure.\"\n\n\"Nope,\" Jack said. \"Still not buying it.\"\n\n\"Sir, what about the earthquakes?\" Sam offered. \"When we first met Oma Desala on Kheb, she demonstrated the ability to pretty much affect all the elements.\"\n\nDaniel turned back toward Skaara. \"Were the earthquakes your way of getting our attention?\"\n\nSkaara's smile faded. \"No. The environment caused the earthquakes, or rather, you did, O'Neill. You and Weiyan both inherited that ability. She awoke the device, but its function is fueled by your combined presence.\"\n\n\"The ATA gene.\" Sam chewed her lip in thought.\n\n\"Carter?\"\n\n\"Sir, this entire 'environment' could be constructed of Ancient technology. It's possibly why you spoke in Ancient.\"\n\nSkaara nodded. \"I did call out to you, O'Neill.\"\n\n\"The buzzing?\" Jack asked.\n\nAs Skaara gave Jack an apologetic shrug, Daniel reviewed what their ascended friend had just shared. Environment. Did that mean they weren't really here? Wherever 'here' is?\n\nOr was 'here' on some ascended plain? Daniel mulled that possibility over. After everything they'd seen, all the random clues, was it possible that 'here' didn't even exist?\n\n\"Why the reruns?\" Jack asked. \"You know Huang so I have to assume Yu and his merry band of Dragon Guards has something to do with all this.\"\n\nSkaara nodded. \"It was important you understood Huang's origins... And how his devotion to Lord Yu resulted in the Goa'uld's testing all of you when you rescued Daniel.\"\n\n\"Why couldn't you just tell us?\"\n\n\"It is complicated. Daniel,\" pleaded Skaara, \"surely you must understand. Some part of you must remember.\"\n\nNot for the first time, Daniel wished he could. But for all his efforts, the only thing he'd managed to dredge up was a memory of seeing Bra'tac and Rya'c imprisoned. That small memory had made a difference, but there was so much more he could do. If he could only remember.\n\n\"Non-interference,\" Sam said. \"Orlin was punished by the other Ascendeds when he tried to help the Velonians.\"\n\n\"We're not the Velonians, and we haven't asked you,\" Jack stabbed a finger toward Skaara, \"to build us a weapon. Instead, we've been treated like rats in a maze.\"\n\n\"But you are using a weapon, O'Neill. One that has saved your planet from Anubis, but could now rip it apart if you do not take the necessary steps to stop it.\"\n\nEveryone spoke at once. Sam wanted to know how to shut off the weapon. Weiyan wanted to know about Anubis. Jack insisted Skaara send them back to Earth immediately.\n\nEveryone except Daniel.\n\nSomething Skaara had said nagged at him. Something about safely getting their attention.\n\nThat's when it hit him. Skaara's very existence could be in jeopardy. \"The weapons chair. If that's what this is about, if Oma knows you're here, she could \u2014 \"\n\nOut of nowhere, Skaara's robes flapped outward as if a wind had blown through the chamber. He turned and pointed toward the room's darkened interior. \"Who do you think sent me?\"\n\nA warm breeze blew across Daniel's face and he knew. \"Oma.\"\n\nThe room lit up in a blaze of blue light.\n\nSam squeezed her eyes shut against the sudden brightness of the room. Her eyes adjusted, she opened them, and saw that the room was in fact a cavernous chamber reminiscent of the Antarctic outpost. Soft blue globes stuck out from trellised screens arranged along the walls. Squared-off columns surrounded a central platform \u2014 just like on Earth \u2014 but with one difference, the silver lines she'd noticed earlier in the floor also ran up the columns.\n\nSquinting, she realized there were other differences, too. The central octagonal platform was much wider than the ones she'd seen at both Proclurash and on Earth. At least a good car-length in diameter and almost three, maybe four feet tall. Something else was missing, too.\n\n\"Where's the chair?\" the general asked.\n\n\"This room served another purpose.\" Skaara gestured toward the platform. \"Not as a weapon, but as a salvation.\"\n\n\"Will it save Earth?\" Sam asked. \"That's why we're here, right?\"\n\n\"All in time, Colonel Carter.\"\n\n_Time_. That had to have been the third, maybe fourth instance of Skaara bringing up the subject.\n\n\"Skaara, if Earth's in trouble...\"\n\n\"Daniel. I promise, once you have seen and heard what has come before, you will be ready to face what comes next.\" Skaara headed toward the platform.\n\nGeneral O'Neill hesitated, glancing down at Teal'c. A thin sheen of sweat covered his face, but he wasn't shaking like earlier. Whether it was a lack of tretonin or something else, Teal'c needed medical attention. Soon.\n\nWeiyan sank down beside him. \"I will stay with him, General.\"\n\n\"Thanks,\" the general mumbled. He exchanged glances with Sam, the corners of his mouth twitching.\n\n\"Sir, you heard Skaara. Teal'c will be all right.\"\n\n\"Yeah, for the moment.\" He jerked his chin toward the platform. \"Carter, you think you can figure out what that thing is all about and get us out of here?\"\n\n\"Possibly, sir.\"\n\n\"Good. Whatever Skaara meant about Earth being ripped apart... Maybe he thinks we have time, but I've had enough of the waiting game.\"\n\n\"Yes, sir.\" That made two of them. \"Though, it would be easier if I had equipment to \u2014 \"\n\n\"Eyes and ears, Colonel. That's your equipment.\" He grimaced. \"And that brain of yours.\" With a swagger she knew the general wasn't feeling, he headed toward the platform.\n\nWith a nod from Weiyan, Sam joined the others. Curiosity mixed with wariness as she approached the octagonal platform. Shorter trellised panels lined its sides, each one back-lit by a soft blue glow. A plastic-like translucent substance covered the top, oscillating between phosphorescent blue and pale yellow. Dead center, a red control crystal jutted upwards, its base as wide as a basketball, its point honed to a viciously sharp point extending at least a foot in length.\n\n\"That's different.\" She pointed at two empty silver brackets on either side of the crystal. The right bracket was lined in blue, the left in white.\n\n\"Different, how?\" Daniel asked.\n\n\"I've never seen this configuration before in Ancient technology.\" She turned toward Skaara. \"That's what this is, right? We were in Antarctica one moment and now... Now we're where? I mean, I know we're inside the building on P3Y-702 \u2014 \"\n\n\"In a sense. Before your letters and numbers, the System Lord Yu named the planet Kunlun.\" Skaara waved a hand over the platform. The air rippled, darkened, as if its mass was undergoing some sort of metamorphosis. \"You've seen some images, now I will show you others.\"\n\nThe area above the platform coalesced, forming a miniature model of the Ancient city they'd found earlier. Complete with a central spire.\n\n\"Elysium Pedion,\" Daniel said.\n\n\"Elysium was the sister city of Vis Uban. The planet upon which Oma returned you to your former life, Daniel.\" Skaara cupped his palms over the virtual city and then pushed outward.\n\nThe spire rushed toward them. Its walls thinned and then disappeared, the view plummeting down through floors past various laboratory-like rooms.\n\n\"Vis Uban was the last city of the Ancients,\" Skaara said, \"but Elysium had become home to many of their people's greatest scientists.\"\n\nOne floor showed Ancients working busily at pedestals not unlike the time-loop device on P4X-639. Another showed two men monitoring a polygonal vat, the viscous liquid inside churning like molten glass.\n\nSam gestured toward the image. \"Is that how the Ancients made control crystals?\"\n\n\"Many variations of crystals were made for many purposes.\"\n\nThe image tunneled downward until it stopped in the very room they were in, platform and all, with one difference.\n\nThree crystals extended from the top. The traditional red control crystal nestled between a shorter indigo blue crystal and a flat, solid white one. The three crystals were connected by a series of transparent boards covered in thin lines \u2014 almost like circuit boards.\n\nA glint of light beneath the crystals caught Sam's attention. She looked down through the platform's translucent top. Deep inside the base spun an incandescent sphere, the center white-hot. Pockets of gas rippled across its surface. She'd seen something similar once before. \"Sir, look familiar?\"\n\nGeneral O'Neill leaned in. After a moment, he shrugged. \"Looks like a miniature sun to me. Wait a minute...\" The general's head shot up. \"Carter, am I nuts or does that look exactly like that photo-tonic thing Yu used to power his force field and clone his guards?\"\n\n\"Photonic energy. Yes, sir. It's very similar to the Ancient technology we found in Yu's fortress. We didn't see anything like this on Vis Uban, but \u2014 \"\n\n\"Lord Yu's acquisition of Ancient technology is only part of our problem,\" Skaara said. \"The recent accident which brought you here has triggered far worse difficulties.\"\n\n\"Difficulties?\" the general echoed.\n\nDaniel raised an eyebrow. \"Our problem?\"\n\n\"These same mechanisms created the Ancient plague. On Vis Uban and Elysium Pedion.\"\n\nThe general yanked Daniel away from the pedestal. \"Jack, what are you doing?\"\n\n\"Saving your butt, Daniel. Thanks to thawing out an Ancient, the rest of us had the plague. Some of us got cured. Others \u2014 meaning me \u2014 got a snake in their head. Trust me, you don't want a piece of this.\"\n\n\"O'Neill, it is not real,\" Skaara said. \"Even if it was, the crystal which caused the plague is long gone. The Ancients destroyed all the duplicating crystals when the plague began.\"\n\n\"All except the one Yu got ahold of.\" The general let go of Daniel. \"So, not real? Okay, then why the show?\"\n\n\"Wait a moment,\" Sam said. \"What do you mean by duplicating?\"\n\n\"You would call it DNA replication.\"\n\n\"Or cloning?\" Daniel asked. \"Like Lord Yu's Dragon Guards?\"\n\n\"All powered by photonic energy.\" Skaara waved a hand over the platform. The circuit boards disappeared along with the blue and white crystals. Only the red crystal was left. \"Three of these platforms were built. As time went on, each was modified for a different purpose. The two on Elysium and Vis Uban were meant for genetic formations.\"\n\n\"Uh, excuse me.\" General O'Neill raised his hand. \"Does any of this have anything to do with Earth?\"\n\n\"More than you can imagine. The crystal was installed in a device deep within Earth's outpost for geological transformations.\"\n\nA sphere rose above the crystal, its blues and whites coalescing into the recognizable from of Earth. Or rather, Earth hundreds of millions of years ago when there had been a single land mass. A singular super continent.\n\n\"Pangaea,\" Daniel said. \"But that was long before the Ancients lived on Earth, wasn't it?\"\n\n\"Even then,\" Skaara said, \"they anticipated that they might, someday, wish to colonize your homeworld.\"\n\nA white-gold light flashed toward the southern tip. The super continent broke up, its individual landmasses drifting apart.\n\nSam reeled in astonishment. \"The Ancients terra-formed Earth?\"\n\n\"Carter?\"\n\n\"I always thought continental drift was a natural part of our planet's development.\" She spun toward Skaara. \"Is that what you meant by Earth ripping apart? We've somehow re-activated the outpost's terra-forming device. Can't we just shut if off by pulling the right crystal.\"\n\n\"It is more complex than that, Colonel Carter.\" Skaara pointed at the empty white bracket. \"On Elysium, this crystal was used to generate photonic energy. Lord Yu took this crystal, as well as the encoding boards.\"\n\n\"The photon emitter and cloning technology. What about the red one?\"\n\n\"The control crystal serves as a filter,\" Skaara said. \"It can be turned on or off and uses the sun to extract photonic energy. That energy can then be sent into the Earth's crust to make mountains, oceans \u2014 \"\n\nThe general whistled. \"And rip the planet apart.\"\n\nSkaara sighed. \"If switched to its on position, yes, and while you have nothing to fear here on this plane...,\" he ran his hand through the platform, \"the device on Earth is very real.\"\n\nSam copied Skaara's actions. Her hand went right through, as if nothing was there. She should be relieved, but she wasn't. Not if Earth was in trouble.\n\n\"Oma's methods of teaching take many forms,\" Skaara said.\n\nDaniel frowned. \"Oma isn't really here, is she?\"\n\n\"No. But you must understand that 'here' is not where you believe it to be.\"\n\nGeneral O'Neill took a turn waving his hand through the platform. \"If this control crystal got switched on back on Earth, how do we shut it off?\"\n\n\"You are still on Earth, O'Neill.\" Skaara waved a hand over the platform. The air rippled, revealing five silhouettes stretched out across the countertop. The forms coalesced and the silhouettes became the bodies of five people.\n\nFive very recognizable people.\n\nSam stared at her semi-transparent body sprawled across the platform's other edge. Her left arm covered Teal'c's waist, her right rested on Daniel's feet. The three of them \u2014 or rather, their bodies \u2014 formed a semi-circle around two other bodies, those of General O'Neill and Weiyan.\n\nDaniel gasped. \"When the outpost's floor collapsed \u2014 \"\n\n\"Would someone please explain what's going on here?\"\n\n\"My God, Jack. Look.\" Daniel pointed toward the platform. The bodies had solidified, and in Weiyan's case, a charred black version of the control crystal protruded from her stomach.\n\nDaniel glanced over his shoulder toward Weiyan and Teal'c who still sat on the floor by the wall. He looked back at Skaara. \"If we aren't \u2014 \"\n\n\"Dead, Daniel.\" The anger in Jack's face stilled. \"The word is dead.\"\n\n\"You remain very much alive.\" Skaara turned his gaze toward Weiyan.\n\n\"What about her?\" Sam whispered.\n\nSkaara dropped his gaze. \"Her father's journey brought her to you, and now he must join you in saving Earth.\"\n\n\"Huang help us,\" the general sputtered. \"Are you nuts?\"\n\n\"What is happening?\" Weiyan jumped up and ran over. Sam grabbed her before she could get too close.\n\n\"Uh, Jack?\" Daniel pointed toward the ceiling. In unison, everyone gazed upward.\n\nA ragged hole appeared in the ceiling above them. General Hammond peered down from its edge.\n\n# CHAPTER TWENTY\n\n### SHUNYI PSYCHIATRIC HOSPITAL\n\n### BEIJING, CHINA\n\n### 2004 AUGUST 18, 9:00 PM\n\nHuang knelt beside his cot, his right forefinger sliding across the floor. Though he had no lamp, no mirror, or window, his eyes had long adjusted to the dark cell. Raising his hand, he judged the room's filth upon his fingertip. It would suffice.\n\nA downward stroke. A lower dash across his brow. Higher up, an inverted bracket.\n\nThe _w\u00e9nsh\u0113n_ completed, he leaned back against his thin mattress. His daily devotion to Yu was a pale effort, but it was all he had to honor his lord.\n\nA ragged sob escaped Huang's parched throat. The thought of never returning, of never seeing his lord and master was a bitter burden he would carry to his grave.\n\nA burden of knowledge.\n\nKnowledge of defeat. Of disaster. Of knowing his pride had allowed the Tau'ri to see through his plans to help his Lord Yu.\n\nHuang's stomach gurgled, feeling a hunger his soul no longer possessed. Each day, food would be brought. Each day, he would refuse sustenance except a single sip of water and a crumb to salve his basic urges. He deserved nothing more. His prison cell served merely as a living tomb. He would die here, and for once, he found himself truly accepting that knowledge. It brought him peace.\n\nUnbidden images swelled in his memory. Images of his life as a Dragon trainee with Lao Dan Shi. The steady thrum of Lord Yu's fortress barrier came to mind. As children, they would run along its perimeter, the pulsing barrier's hum an assurance much like the beat of one's own heart. The Jaffa would smile as they raced past, indulging in joyous freedom.\n\nA whiff of dank air stuffed his nostrils, returning him to the present. The cell was warm, humid. For one brief moment, he decided it must be late summer in Beijing. The cherry blossoms long gone.\n\nHis face became wet, sticky. Ashamed at his tears, Huang shut his eyes again and allowed the dark its due.\n\nA golden light warmed his eyelids. He welcomed its memory. Companion, savior, the light had saved him from Antarctica's treacheries. Huang thought back on his time in the Tau'ri's frozen wasteland with a newfound appreciation. Ice-blue glaciers. Crisp, fresh air. His Antarctic travails had been a time of ferocious landscapes testing his every move.\n\nThough he also remembered the constant doubts, it was clear to Huang now that possibility had been in abundance. Each day a gift of life.\n\nThe light faded. A memory too far out of reach.\n\nThe darkness spoke. \"The tree which fills the arms grew from the tiniest sprout.\"\n\nA recollection? Perhaps. Though it did not sound like his teacher Lord Yu or even Shi, his brother.\n\n\"A quote of Lao Tzu?\" He squeezed his eyes tighter, searching heart and mind for the voice's owner. \"Many years have passed since I heard the words of the _Tao Te Ching_.\"\n\nHis voice cracked. It had been many months since he had muttered any word aloud. Quing's visit... Yes, he had last spoken aloud to his ex-wife. Was that a month ago?\n\nA year ago?\n\nThe memory spoke again. \"Lao Tzu also said, 'the journey of a thousand _li_ commenced with a single step.'\"\n\nHuang opened his eyes, prepared to meet only the cell's black embrace. Instead, a young man stood before him. Swathed in a nimbus of golden-white light, his visitor could not be more than twenty, if that. Olive-skinned, with long plaits of jet-black hair, the youngster's wide brown eyes stood out in contrast to his bleached-white robes.\n\n\"Your journey nears its end, Huang Sun Tzu.\" The young man smiled broadly.\n\nHuang returned the smile. \"I am ready.\"\n\n# CHAPTER TWENTY-ONE\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2230 HRS MCMURDO STATION\n\nGeorge peered down at Jack, SG-1, and Zhu's daughter, their condition unchanged. In his head, he replayed every moment of Lt. Graham Simmons' time at the outpost. From coffee to counsel, the visitor's actions had fitted the young man to a tee.\n\nIf it wasn't Graham, then who was it?\n\n\"I'm not imagining things,\" he told Dr. Lee.\n\n\"No one's saying you are, General, but...\" Lee shrugged. \"I never saw him. I gave you coffee the first couple of times, and then \u2014 \"\n\n\"Brooks and Gerling. I realize that now.\" Beneath the force field, the rise and fall of his people's chests was the only sign they were still alive. \"Is it possible we've had an alien visitor?\"\n\nLee's voice lowered. \"Maybe an Ascended Being?\"\n\n\"If it was, then why the masquerade?\"\n\n\"Well, this was their equipment.\" Lee pulled off his glasses and rubbed his eyes. \"According to Dr. Jackson's reports, they were Ancients first, right? Maybe one of them wanted to see what we broke.\"\n\n\"By impersonating one of our own?\" With a sigh, George turned his gaze back toward the force field. \"I'd have appreciated the visit if they just gave us the instructions on how to fix this mess.\"\n\nThe frown on Jack's unconscious face deepened. He looked downright angry. Moving closer to the hole, George wondered what on Earth was going through his prot\u00e9g\u00e9's head. On the other side of the illuminated platform, Teal'c jerked from side-to-side, his broad chest shuddered once, and then his shoulders sagged.\n\nAs he quieted, George felt his own heart slow down in relief.\n\n\"General Hammond?\" Major Davis called out.\n\n\"Report.\" George turned his attention away from the force field. The Navy SEALS had packed enough C-4 on the Ancient Weapons chair to blow up the entire installation.\n\n\"That's the last charge, sir.\" Davis beckoned Colonel Ferguson over. \"Two Bell 212s are waiting topside to evacuate the last personnel over to Marble Point and then, McMurdo.\"\n\nFerguson hurried over and snapped to attention. He'd pretty much clammed up since his arrival. No surprise. The Ancient chair and the sight of SG-1 under that force field was enough to make anyone do the same. \"At ease, son.\" George glanced over at the archway where Ambassadors Zhu and Duebel stood, each lost in their own thoughts. \"How many can fit on one of those choppers?\"\n\nFerguson slid a look toward Davis. Either the outpost spooked him or something else was going on.\n\nDavis gave him a nod. \"Colonel, how many?\"\n\nFerguson returned his attention to George. \"I can squeeze a good fifteen in, General, if there's no equipment to carry.\"\n\n\"That'll do it.\" He indicated the ambassadors. \"Get them out of here. Leave me one chopper, but the remaining personnel should go now, including the SEALS and \u2014 \"\n\n\"If you're staying,\" Dr. Lee chimed in from his computer, \"I'd like to do so, too, General. There's still a good thirteen hours left. Who knows? Maybe that Ascended Being will come back, or I'll have a breakthrough with the device before we lose the western hemisphere.\" He smiled feebly, but George appreciated the sentiment.\n\nAnd the reminder of what was at stake.\n\n\"Sir?\" Davis turned to face him. \"Permission to stay as well. I'd like to help, if you'll have me.\"\n\n\"Happily, Major.\"\n\nAn appreciative whistle came out of Ferguson's mouth. He stuck a hand out toward the major. \"You're serious as a heart attack. You know that, right?\"\n\n\"Never more serious.\" Davis shook his hand with a smile.\n\nGeorge watched on as the two slapped each other's backs, fully aware of the difficulties the major had faced in dealing with the more 'by-the-book' colonel. Camaraderie came in all colors. Sometimes, they mixed. Sometimes, they clashed. Somehow, Davis had managed to persevere, do what was right, and win Ferguson's respect in the process.\n\nWith a final slap to the major's back, Ferguson stepped back and saluted George. \"I was wrong, General. I thought all Davis did was tinker with budgets so those SG teams \u2014 What I mean to say is...\" He dropped his hand. \"True fact, sir. We never would've gotten those birds out of the ice if it wasn't for the major's smarts.\"\n\n\"Dismissed, Colonel. And thank you.\" George returned the salute. Ferguson gave Davis a nod and ran out.\n\nNo doubt about it, Davis was long overdue for a promotion.\n\nLieutenants Brooks and Gerling hurried in. George had sent them on what now seemed to have been a wild goose chase \u2014 searching for a 'back door' into the chamber below.\n\n\"Find anything?\"\n\nThe airmen's matched frowns told him everything.\n\n\"All right. It was worth a try.\" He rubbed his eyes, contemplating how he'd manage to stay awake in the hours to come. Simmons' coffee would've come in handy right about now.\n\nSo would an explanation as to who \u2014 or what \u2014 had impersonated the lieutenant.\n\nGerling handed him a slip of paper. \"This came in over the wire from McMurdo, sir.\"\n\nGeorge read the hurriedly scribbled note out loud. \"Capetown, South Africa. 6.3 Richter along Antarctic Intercontinental plate. Emanating inland from shore. Aftershocks ongoing.\"\n\n\"We didn't feel anything, sir,\" said Brooks.\n\n\"That's Ancient technology for you.\" Lee took the note and scanned it. \"South Africa's thousands of miles away. If the device has that sort of reach, Europe can't be far behind.\"\n\nIn other words, it was time to move on to phase two. George turned back toward the airmen and ordered, \"Assist Ferguson in escorting everyone topside.\"\n\nBrooks dashed out, but Gerling stayed at attention. \"Sir? Brooks and I would like to stay, too. I'm qualified to fly a Bell 212 chopper. General O'Neill deserves all the help we can give him, sir.\"\n\nGeorge sized the young woman up. \"Permission granted. First escort our diplomats to the surface and then report back.\"\n\nGerling spun on her heels and strode over to the ambassadors. She spoke softly, presumably telling them, far more nicely than George, that they needed to get the hell out.\n\nDuebel got the message, following Gerling out of the chamber. Zhu, on the other hand, stayed stock-still.\n\nGeorge girded himself for the inevitable confrontation. Of course she wouldn't want to leave her daughter \u2014 no parent in their right mind would.\n\nHe skirted the hole and approached her. \"I'm sorry,\" he said, \"but there isn't anything more you can do here except jeopardize your own safety.\"\n\nTears welled in Zhu's eyes. \"She is my child. Any hope \u2014 \"\n\n\"We won't stop trying until the very end. I promise.\" He nodded toward Gerling.\n\n\"This way, ma'am.\" The airman took Zhu's elbow.\n\nGeorge headed over to the SEALS as they packed up their kits. \"You won't have room for those. Leave them here.\"\n\nThey exchanged glances, each as freckle-faced and wide-eyed as a pair of kids and George suddenly felt his age. It was high time he retired and he knew it. If Jack made it out of this alive \u2014\n\n\"General Hammond?\" The taller of the two SEALS \u2014Lieutenant Kalsner \u2014 pulled a palm-sized green metal box from his parka's inner pocket. \"Major Davis said we should leave the remote trigger with you. We set it to blow twelve hours from now. All you need to do is start the timer.\"\n\n\"Hang on a moment.\" Lieutenant Mason, the other and even younger of the pair, flipped open his case. He pulled out a bar of C4. A receiver tab stuck out of its side, its light gauge blinking yellow.\n\nMason jerked his chin toward the hole. \"There's a small indent in the force field behind the chair. I'd like to try squeezing \u2014 \"\n\n\"Get yourselves topside, Lieutenant. I'll place the final charge.\" Taking the C4 and remote, George dismissed the guards.\n\nHe circled the hole, searching for the right spot to plant the final C4. He found it, just behind the chair as Mason had said. While the rest of the force field lay flush with the floor, a hands-width-wide area dipped under the jagged edge. Unlike the rest of the force field, this particular patch appeared thicker, more opaque than the rest. George knelt down and wedged the C4 block into the gap, careful to not touch the force field.\n\nBeneath that force field, Jack's frown became an outright scowl.\n\n_Forgive me_.\n\nGeorge examined the remote's simple digital display and small keypad. At the bottom, a red start-up button blinked, its innocuous cadence a mock counter rhythm to the force field's thrum. The thing barely weighed half a pound.\n\nA half-pound of wires, metal, and a simple transmitter that would blow up the Ancient outpost.\n\nAnd everyone still trapped in its grasp.\n\n\"Sir?\" Major Davis stepped up beside him and lowered his voice. \"If you'd like, I can operate the remote.\"\n\n\"No, thank you.\" George dropped the remote to his side. He wasn't afraid to admit his prot\u00e9g\u00e9 and flagship team had come to mean as much to him as his own grandchildren. Their dedication, their unwavering loyalty to him, the nation, and the planet made them the very definition of what it meant to serve.\n\n\"Integrity first. Service before self. Excellence in all we do.\" If anyone was the inspiration for the U.S. Air Force's creed, it was Jack O'Neill, Samantha Carter, Teal'c, and Daniel Jackson.\n\n\"General?\" Davis asked.\n\nHe stared at the remote. \"If and when the time comes, Major, I think it's best that I be the one to pull the trigger.\" It was the least he could do to honor their last sacrifice. He thumbed the start button. The digital display changed from 12:00 to 11:59.\n\n\"Godspeed, SG-1.\"\n\nIf Jack had to pick the most surreal part of this experience, he wouldn't have chosen seeing Skaara. As much as he missed the kid, that wound had scabbed over, thank you very much. It only took a few bottles of scotch, an occasional bout of self-loathing, and the daily reminder that somewhere out there on the Oma Express, the boy was safe and out of harm's way.\n\nA visit would've been nice, but who was he to complain?\n\nStrolling through an Ancient ghost town wouldn't have been high on his list, either, and while Skaara's tale about Huang coming through the Antarctic gate some fifty-odd years ago raised an eyebrow, Jack really wasn't all that surprised. Actually, it explained a lot, though he didn't quite get why Skaara had interfered when Ascended Beings were supposed to be above that sort-of-thing.\n\n_Kid probably got lonely. Maybe that's why Daniel came back_.\n\nOn the surreal meter, being told that he was in a funky dream-state, his body sprawled out on a platform in Antarctica, just underneath the weapons chair, came close to the top. But really, when it came down to it, the most surreal thing was...\n\n\"Please tell me Hammond didn't just plant a block of C4 feet away from his \u2014 I mean my \u2014 head!\"\n\n\"If you do not act soon, General Hammond will have no choice but to destroy the entire outpost.\" Skaara pointed at the blackened crystal projecting out of Weiyan's gut. \"You must fix things to be as they were.\"\n\nWeiyan. The girl hadn't said a word since Skaara's little parlor trick revealed their bodies lying on the platform. She just stared at her impaled image, chewing her lip like it was going out of style.\n\nJack reached out to take Skaara's elbow and then stopped himself, remembering that ascended folks weren't really there-there. They just looked that way. He stepped closer, lowering his voice. \"What about her? Is Weiyan going to make it?\"\n\n\"Her journey is on another path.\"\n\nJack bit back a reply. He'd spent enough time around Ascended beings and their cryptic language to know he wasn't going to get a straight answer. At least not on that front. He glanced over at Teal'c still passed out, still sweating like he was inside a furnace. \"And Teal'c?\"\n\n\"His tretonin is only an arm's reach away,\" Skaara said. \"Once you have awakened, he will be fine. Then will you go.\"\n\n\"Go where?\" Daniel had slid around the platform to stand beside his own comatose body. The doubled image wigged Jack out, but he'd seen worse.\n\n\"You want us to go back to P3Y-702,\" Carter said. \"Kunlun.\"\n\nSkaara nodded. \"It is the only way to replace the damaged crystal in Antarctica. Huang will travel with you.\"\n\n\"That's never going to happen.\" Not as long as Jack could draw breath. \"The man's a menace.\"\n\n\"Will the plague harm my father?\" Weiyan touched the control crystal. Apparently, she wasn't listening.\n\n_For a father she'd only met once, she sure cares about the bastard more than herself_.\n\nSkaara stepped up beside her. \"The plague has long faded. You should not fear this for your father.\"\n\n\"I don't.\" Weiyan pulled back from the crystal. \"'The general who advances without coveting fame and retreats without fearing disgrace, whose only thought is to protect his country and do good service for his sovereign, is the jewel of the kingdom.'\"\n\nJack grimaced, recognizing the words.\n\nAnd the truth behind them.\n\n\"Sun Tzu,\" said Daniel.\n\nWeiyan grinned. \"When I met my father last year, his insistence that I read The Art of War seemed...\"\n\n\"Irrelevant?\"\n\nShe blushed. Jack found that rather amazing, considering the fact that she was basically staring at her soon-to-be very dead body, breathing her last breaths.\n\nWhen he'd told her earlier that she'd grown a backbone, he wasn't kidding.\n\nStill...\n\n\"Look, Skaara. I'm more than ready to go after this crystal if it'll save Earth, but Huang can't travel through the Stargate. No way. He's a security risk.\"\n\n\"He's also in a mental institution, sir.\" Carter glanced at Weiyan. \"I'm sorry. I'm guessing your mother didn't tell you.\"\n\n\"No.\" The girl looked shocked. \"She did not.\"\n\nSkaara shook his head. \"Two with the Ancient blood are required to open Kunlun's chamber. O'Neill is only one. The other must be Huang.\"\n\n\"Why can't Weiyan go?\" Daniel asked. \"She's certainly demonstrated she has the genetics.\"\n\nJack kept his face as neutral as possible. Maybe Daniel didn't get it, but she certainly did.\n\nWeiyan touched the crystal protruding from her body. \"If we wake up, I will die, won't I?\"\n\n\"Wait a minute!\" Daniel raced from behind the platform.\n\n\"There is no waiting, we must awaken at once. It is the only way to save Earth. Sun Tzu, and my father...\" she gulped. \"I understand now what matters most.\"\n\nDaniel shook his head. \"If they can save Teal'c when we wake up, they can save you \u2014 \"\n\n\"Let it go, Daniel,\" Jack warned. \"Let the girl have her dignity.\"\n\nShe's earned it.\n\nHe shared a glance with Carter who nodded. She understood.\n\nSkaara turned toward him. \"O'Neill, you must find it within yourself to forgive Huang. Only by doing so, and permitting him to help, can you retrieve the crystal and save Earth.\"\n\n\"There's gotta be another way,\" Jack said. \"One of the other ATA trainees \u2014 \"\n\n\"Is it Huang you cannot trust, or is there someone else?\" Skaara stared at him, hard. For a moment, it felt like Jack's clothes had been stripped and he was standing bare-assed naked while something older and wiser than his young Abydonian friend measured him up.\n\nAnd found him wanting.\n\nJack shook off the feeling. \"I don't know what you're talking about. I'd trust Carter, Teal'c, and Daniel to get me through anything. You know that.\"\n\n\"And yourself?\" Skaara pressed. \"Do you trust that you can use your abilities to retrieve the crystal? Using your genetics to operate the weapons chair will be a \u2014 \" He grinned. \"A cakewalk compared with how you must cooperate with the technology on Kunlun. Otherwise, the crystal will be lost.\"\n\n\"So I hate using Ancient technology,\" Jack retorted. \"I'm a big boy. I can deal with it.\"\n\n\"Like the general who does not fear disgrace?\"\n\nHe shrugged. \"Something like that.\" It wasn't the first time he'd pretended a bravado he wasn't necessarily feeling. He hated using the ATA gene, hated giving himself up to something he could barely control. It didn't feel right, but neither did huddling like a coward while his planet got ripped to shreds.\n\nA warm hand clasped his shoulder. Skaara's hand. \"Unlike Huang whose path was set very long ago, your journey still has far to go, O'Neill.\"\n\n\"Thanks. I think.\" Jack cleared his throat. \"You... You could visit sometime, you know. We could go fishing.\"\n\nSkaara dropped his hand to his side. \"In time all paths cross, O'Neill.\" He turned to face Carter, Daniel, and Weiyan. \"Are you ready to return to Antarctica?\"\n\n\"Wait!\" Weiyan dashed across the room and knelt beside Teal'c. \"I am ready.\" She held his hand.\n\n\"What do we do?\" Jack asked, keeping sight of Teal'c and Weiyan. \"Click our heels together and say 'There's no place like home', or \u2014 \"\n\nThe room disappeared in a flash of light. For a moment, all he could see was, well, basically nothing. Like staring at a white wall. Suddenly, he felt jittery, as if he'd had too much coffee before breakfast.\n\n\"Hello?\" he called to the blank wall.\n\n\"I will not try.\"\n\nWeiyan's voice. Somewhere far off.\n\nA thumping filled Jack's ears. Rhythmic. Repeated.\n\n\"General! They're awake.\"\n\n_Thump. Thump-thump_.\n\nUnless he'd gone nuts, that was Major Davis' voice. Where the hell...?\n\nThe thumping became deafening. The white light, blinding.\n\n\"I will do, Teal'c,\" Weiyan whispered.\n\n_Thump. Thump-thump. Thump-thump_.\n\n\"I will make you proud. All of you. Even you, General O'Neill. Even my moth \u2014 \"\n\nJack's vision plunged into darkness.\n\nFootsteps, yelling.\n\n\"The force field's receding below them!\"\n\nThough the thumping kept up, he felt his eyes open even though he could've sworn he'd never closed them.\n\nAbove a ragged hole in the ceiling, the Ancient outpost glared down at him. General Hammond's very round and very welcome face then slid into view.\n\n\"Almost thought I'd lost you there, son.\"\n\nEvery bone in Jack's body ached as he struggled to sit up. Daniel did the same, but Carter bolted to her feet. Davis reached down and gave her his hand to climb out.\n\n_Thump. Thump-thump_.\n\nShe waved him off. \"We need Teal'c's tretonin. Now!\"\n\nDavis opened his hand, a vial of the purple stuff stuck neatly in his palm. Grabbing the vial, Carter clambered over to Teal'c. She bent down and Jack heard a hiss.\n\n_Thump. Thump_.\n\nBeside him, Weiyan's eyes opened. She lowered her chin, glanced down at the crystal sticking out of her stomach, and closed her eyes.\n\nThe thumping subsided, but not before Jack recognized the sound.\n\nIt was the beating of his heart.\n\n\"Medic!\"\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2245 HRS MCMURDO STATION\n\nDaniel sat on the weapons platform, processing everything Skaara had told them. Down below, Paul Davis and a female airman argued on how to best pull Weiyan out. If Skaara was right, all the efforts in the world couldn't save her.\n\nPulling off his glasses, he wiped his eyes with the back of his hand. She'd been so young. So innocent. She'd put her life in the hands of the IOA, of the Stargate Program, and all for what?\n\nIf he thought about it too much, he'd think about how Weiyan's life was never really meant to happen. Huang was never supposed to come to Earth, as a clone, he should've never been allowed to have children, and the mother \u2014\n\nWhere was Ambassador Zhu?\n\nDaniel scanned the chamber, but didn't see her. The other ambassadors were missing, too. Which wasn't necessarily a bad thing.\n\nJack and General Hammond stood by the back wall, their heads bent in discussion. A heated one from what Daniel could tell. Jack waved his hands around \u2014 probably trying to explain what they'd experienced \u2014 while Hammond's frown just deepened and deepened. The two generals occasionally glanced over at Daniel, then at Sam. She sat with Teal'c on the other side of the platform. The Jaffa's eyes were closed, but he already looked far better than he had for hours.\n\nDaniel peered at the carnage below. Blood pooled on both sides of Weiyan's body. All backlit by the platform's self-illuminated top.\n\nA low rumble filled the chamber. The room shook for a second and then settled down. A light dusting of ice wafted down from the ceiling, but no one seemed to pay attention. Apparently, a small aftershock was nothing compared to the quakes that had been hitting the southern hemisphere. Fatalities in the hundreds, thousands more homeless. It would get worse before it got better. He knew that.\n\nHe glanced at Robert Lee, focused on his monitor, studying quake reports from across the globe. All thanks to the Ancients' terra-forming device. Daniel believed in science. He believed in exploring the great unknown. If it wasn't for the Stargate, his life would've had little meaning.\n\nBut not at such a cost.\n\nSomeone put a canteen in his free hand. With a nod of thanks, he unscrewed the cap and gulped the cold water down. Once that was empty, he looked up to see Airman Brooks standing over him. \"How long...?\"\n\n\"Were you out?\" Brooks stuffed the canteen in his parka. \"Nine hours, give or take.\"\n\n\"Nine hours?\" Impossible. Too much had happened.\n\n\"That force field had you locked up good and tight, sir. Not sure what made it finally drop low enough to get you out, but, well...\" Brooks shrugged. \"I guess alien technology can do anything.\"\n\n_Not anything_ , Daniel thought wistfully. _It can't save Weiyan_.\n\nHe stood up, his legs wobbly. Teal'c was still resting so he decided to take a short walk. Clear his head. A plank had been put over the hole between the platform and the rest of the chamber floor. He cautiously made his way across and headed through the archway and into the outpost's main room.\n\n\"Dr. Jackson, can you help me?\"\n\nDaniel turned toward the voice. It was Graham Simmons. The lieutenant stood by the stasis unit with a clearly distraught Ambassador Zhu.\n\n\"She was supposed to evacuate with the others,\" Graham explained.\n\nDaniel raised an eyebrow. \"Shouldn't you have done the same?\" He didn't remember seeing the lieutenant there before the accident, but in all the confusion with the trainees and diplomats, he hadn't really been paying attention.\n\n\"Please,\" Zhu said. \"Let me see my daughter.\"\n\nHe hesitated. Telling any woman their child was dying was hard enough. Breaking the bad news to Zhu could put the tenuous future of the Stargate Program in jeopardy.\n\n\"Of course,\" he decided out loud. With a nod to Graham, he ushered her through the archway and into the weapons chamber.\n\nThe two generals were still conferring.\n\n\"I'm sorry,\" Hammond said to Jack, \"but request denied. Allowing Huang anywhere near the Stargate or the chair would be like handing over nuclear launch codes to the \u2014 \"\n\nThe general's head jerked up as Daniel entered with Zhu by his side. Graham was just a step behind them.\n\n\"Weiyan!\" With help from Teal'c, Zhu climbed down a ladder into the hole.\n\n\"Step away, Dr. Jackson,\" warned Hammond.\n\nSoft sobs rose up from the hole, punctuated by Teal'c's murmured words of comfort.\n\n\"It's all right, sir.\" Daniel stopped beside the hole. Graham joined him in watching mother and daughter meet. Paul injected Weiyan with something. Hopefully, pain medication.\n\n\"Daniel.\"\n\n\"I know Zhu shouldn't be here, Jack, but the ambassador wanted to see her daughter, before \u2014 \"\n\n\"Huang must accompany SG-1,\" Graham said. \"The Ancient blood runs stronger in his veins than anyone else's, even O'Neill's.\"\n\n_O'Neill's_...\n\nThe exaggerated O. The soft double L's that almost sounded like R's. Daniel heard Sam gasp. He whirled toward Graham. Skaara gazed back.\n\n\"You must hurry, Daniel.\" Skaara stepped across the plank and stopped in front of Jack and Hammond. \"The crystal must be replaced before your sun rises.\"\n\n\"Hello, Skaara,\" Hammond said, a tinge of awe in his voice. \"So you're our mysterious visitor.\"\n\n\"I apologize,\" Skaara said sheepishly. \"I had hoped to minimize my involvement, but time on this level has not allowed me to do so any longer. Not without risking the Others' attention.\"\n\nThough Daniel privately wondered what Skaara meant by 'the Others,' there were more pressing matters at the moment. He hurried over to join Skaara and the generals. \"What happens when the sun rises?\"\n\nSam wasn't far behind. \"The crystal's ruined, Skaara. There shouldn't be any threat from solar power.\"\n\nSkaara frowned. \"Each of you by now should know that what is seen is not always what is true. Yes, the crystal is damaged, but built within these walls are the conduits needed to bring new power from your sun.\"\n\nSam reminded them of the silver lines etched on the floors of both the Kunlun chamber and the one on Yu's planet.\n\n\"Excuse me, but,\" Lee waved toward the floor and ceiling, \"there's no silver here.\"\n\n\"General Hammond?\" Brooks raised his hand.\n\n\"Go ahead, Airman.\"\n\n\"Sir, when Gerling and I crawled through some of the outer hallways of this place, we did see some metallic carvings in the walls.\" Brooks shrugged. \"We thought they were decorative.\"\n\n\"That's a problem, sir.\" Sam exchanged glances with Jack. \"Silver works as a thermal-electrical conductor. If there's any circuitry left in the crystal, it'll super-charge whatever power's left and \u2014 \"\n\n\"Rip the planet to shreds?\"\n\n\"Huang must accompany you, O'Neill,\" said Skaara. \"You must take him through the Stargate to Kunlun. Now.\"\n\n\"Even if I agreed to Huang's participation,\" said General Hammond, \"there isn't enough time. Huang is in Beijing, the Stargate's in Colorado, and we're in Antarctica. I'm sorry, but we need to evacuate and go through with our original plan.\"\n\n\"Blowing up the outpost,\" Daniel mumbled, devastated at what the loss signified for Earth.\n\n\"Sir, wait!\" Sam ran over to the hole. \"Major Davis, are the F-302s still over at Observation Hill?\"\n\nPaul climbed halfway up the ladder. \"Actually, Colonel, they're over at Marble Point. That's only a ten-minute chopper ride from here. Oh, wait!\"\n\nHe scrambled off the ladder, landing on the outpost floor with a thud. \"Use the 302s for hyperspace jumps! Hop over to Beijing, then \u2014 \"\n\n\"Jumping in and around Antarctica is one thing,\" Hammond said, \"but you can't go jumping in and out of populated areas with top secret equipment.\"\n\n\"But they could fly in,\" Sam said. \"If final approach was handled at normal speeds.\"\n\n\"Carter, what do you have in mind?\" Jack said.\n\n\"The 302s are capable of Mach 6, easily.\" Sam headed toward Robert's computer and tapped furiously on the keyboard. \"There are two, right?\"\n\nPaul nodded. \"I doubt Colonel Ferguson moved them back.\"\n\n\"You better hope you're right.\"\n\nAnother airman came in, holding a gurney. He went over to the hole and handed it down as Teal'c climbed up. He saw Skaara and bowed, surprisingly unconcerned.\n\n\"What will they do with Weiyan Shi?\" he asked.\n\nDaniel sighed. \"I suppose they'll medi-vac her, maybe \u2014 \"\n\n\"You must tell them to stop,\" Skaara said.\n\n\"I can't do that. What if there's a way to \u2014 \"\n\n\"Daniel...\" Skaara's brow furrowed, his eyes stern. \"Her journey's end begins here. Tell them to stop.\"\n\nA cold breeze blew across the back of Daniel's neck. He shuddered, remembering.\n\n_Please, Jack. Tell Jacob to stop_.\n\n\"You're going to help her ascend,\" Daniel realized aloud.\n\nSkaara nodded. \"She deserves no less.\"\n\n\"I concur,\" Teal'c intoned. \"If there are no objections, I will inform the airmen.\"\n\nNo one said a word. How could they? Death was one thing. What Skaara offered Weiyan?\n\nGeneral Hammond broke the silence. \"Let them know, Teal'c.\"\n\nTheir Jaffa teammate hurried over to the hole and relayed Hammond's orders. Though Daniel couldn't see her, he didn't hear Zhu object.\n\n\"Generals,\" Sam announced from the computer, \"I think this will work if General Hammond can get permission for a 302 to enter China's airspace.\"\n\n\"Easier said than done, Colonel.\"\n\nSam looked up from the screen. \"Sir, their government already knows about the fighter-interceptors and the truth is \u2014 \"\n\n\"Their butts are at risk as much ours.\" Jack scowled. \"I get it, Carter. Do the math.\"\n\n\"Beijing has a military airstrip just outside the city. They're about 7,300 miles from here. At Mach 6, that'll take a bit less than two hours from Marble Point. I can fly out there, grab Huang while you take the other 302 and head directly for Petersen. I can meet you back at the SGC in,\" she ran another series of keystrokes, \"another two hours. The trip from Beijing to Petersen is about 7,500 miles.\"\n\nJack folded his arms. \"That leaves ten hours to go through the gate, grab the crystal, hop back in the 302s and fly back here before a whole bunch of C-4 goes off. What time is sunrise?\"\n\n\"A little after twelve-noon,\" Lee said.\n\n\"Piece of cake except for one small change.\" Jack turned toward General Hammond. \"Sir, I'd like to be the one to get Huang. I think I owe that much to his kid.\"\n\nHammond glanced over at the hole. \"If you're willing to try it, then yes. There's a chopper waiting topside. I'll call the President and make arrangements.\"\n\nWith a nod toward Skaara, he headed toward the exit. \"It's gonna be one heck of a phone call.\"\n\n\"Uh, General Hammond?\" Jack raced after him. \"Since we're short on time, can we request that Huang be waiting at the airstrip? I'm not exactly keen on the idea of going into a prison to sign him out.\"\n\n\"Sam said it was a psychiatric hospital,\" Daniel pointed out.\n\nJack's eyes darkened. \"Same difference when it comes to the Chinese. Trust me.\"\n\n\"O'Neill,\" Teal'c said. \"As there are only two seats on each Interceptor, I will remain here.\" Without waiting for an answer, he climbed back down.\n\nJack shared a look with Daniel, who could only shrug.\n\n\"Allow Teal'c this action, O'Neill,\" Skaara offered. \"Daniel's path is with you and Colonel Carter.\"\n\n\"You'll take care of her after,\" Jack said. \"Right?\"\n\n\"Of course.\"\n\n\"And you'll take care of yourself? No more sticking your nose where it doesn't belong?\"\n\nSkaara grinned, reminding Daniel of how young he'd been before his untimely death.\n\n\"Daniel. O'Neill.\" Those soft Ls again.\n\nDaniel's chest ached, thinking back on everything he'd lost in the fight against the Goa'uld. Skaara. Kasuf. Sha're.\n\nSkaara met his gaze, his grin softening into a knowing smile. \"So much lies ahead for you both. Find peace in your work, Daniel. Find peace in your life, O'Neill. Let go of the past and live for the future.\"\n\nWith that, Skaara faded away.\n\nDaniel stared into the space where he'd been, wishing Skaara could have stayed longer. There were so many questions needing answers. Why had Oma sent him? How was Sha're's son, Shifu?\n\nWhy had he turned his back on ascension?\n\n\"Daniel?\"\n\n\"Yeah, Jack.\"\n\n\"Let's go save the world.\"\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 18 AUG 04\/2310 HRS MCMURDO STATION\n\n### 18 AUG 04\/0610 HRS STARGATE COMMAND\n\nTeal'c sank down on the platform, eyeing the surrounding force field. Ambassador Zhu sat across, focused on her child. A blanket had been draped below the charred crystal protruding from Weiyan Shi's belly, covering her legs against the biting cold of the outpost.\n\nWhile Teal'c's heart was glad that he and his team had wrested free of the platform's grip, Weiyan Shi was not as fortunate.\n\nWas it ill fate, he wondered, or should he consider the path she would soon encounter to be fortunate?\n\n\"I have read Stargate Command's files concerning Dr. Jackson's ascension,\" Zhu whispered. \"Will my daughter...? Will she \u2014 ?\"\n\nThe woman broke down in a fit of tears. Teal'c waited patiently, knowing that the ambassador must summon the strength on her own to face what would happen next.\n\nWhen her sobs subsided, Teal'c explained what he knew of both the legend, and the reality, of ascension. \"Before learning the truth, I knew only of the tale told amongst my people.\"\n\nZhu returned to stroking her daughter's sweat-plastered hair. \"If my former husband's origins are to be believed, all legends begin in truth. Tell me of the Jaffa's beliefs.\"\n\nTeal'c bowed his head in assent. \"When a Jaffa could no longer carry a prim'tah, they would make their journey to Kheb. There the Ko'lak would learn the path through the darkness, into the next life.\"\n\n\"I have read your file as well, Teal'c. If you no longer carry a symbiote, you cannot follow the ways of your people.\"\n\n\"On the contrary, Ambassador Zhu. When SG-1 first visited Kheb, an ascended being told my friend and mentor that only by losing our symbiotes would the path to ascension ever be possible.\"\n\n\"The tretonin.\" Zhu grunted. \"Thanks to alliances made through Earth's Stargate, you might one day achieve what my daughter must now embrace.\"\n\nSomeone coughed from above. Major Davis appeared at the hole's edge.\n\n\"The F-302s are in the air,\" he reported. \"Colonel Carter and Dr. Jackson are on their way to Petersen. General O'Neill's on track to land in Beijing in about 90 minutes.\"\n\n\"He will see my father,\" Weiyan Shi whispered. \"Please, tell General O'Neill \u2014 \"\n\nA thunderous rumble shook beneath them. Another quake. Snow and ice tumbled down from the ceiling and though Teal'c did his best to shield Weiyan Shi, several small chunks of ice smashed against her legs. As the shaking subsided, he took note of the fact that her wound had cauterized around the crystal. Nonetheless, the pain would be significant and yet, she made no complaints. He was not certain whether this was a sign of bravery or the possibility that she could not feel her body.\n\nIn either case, Teal'c knew that the tremors would not stop. Not until O'Neill had returned with the crystal from Kunlun and shut down the device. A pang of guilt washed over him and he wondered once more at the possible hand he'd had in their current fate.\n\nNo. There was no way to truly know. What mattered now was Earth's future. He must face the present and do what was needed.\n\nWeiyan's ravaged features told him how he must proceed. He stood up. \"Major Davis?\"\n\nThe major appeared again by the ladder. \"What do you need, Teal'c?\"\n\n\"An injection to blunt Weiyan Shi's pain. And another blanket.\"\n\nA moment later, Major Davis returned and handed him a syringe and a plastic-wrapped blanket. Teal'c administered the shot to Weiyan's arm. He then tore the plastic off the blanket and shook it free.\n\nMajor Davis took the discarded plastic. \"You've seen ascension happen twice, haven't you?\"\n\n\"I have. First when Daniel Jackson departed, and then Skaara. The event is,\" he glanced down at Ambassador Zhu and frowned, \"disquieting, but significant.\"\n\nBalling up the plastic, Major Davis then glanced over his shoulder. \"Airmen, withdraw to the outside chamber.\"\n\nMuffled sounds of acknowledgement reached Teal'c's ears. He thanked Major Davis and unfurled the wool blanket and spread it over Weiyan's shoulders. She whimpered.\n\nWhen Ambassador Zhu made no move to attend to Weiyan Shi, Teal'c slid over and cradled her head in his lap. \"Your sacrifice will not be forgotten, Weiyan Shi.\"\n\nShe opened her eyes. \"Please hold me.\"\n\n\"I will stay with you as long as you like.\" Ambassador Zhu crawled around the crystal and joined them at the platform's edge.\n\n\"Mother,\" Weiyan Shi cried out. \"I am sorry I failed you.\"\n\n\"You have failed no one, child.\"\n\nA gasp escaped Weiyan Shi and she squeezed her eyes shut.\n\nTeal'c had seen and felt much physical pain in his hundred and six years of age, and yet watching Weiyan Shi, watching this young woman barely out of childhood, unsettled him greatly.\n\nIn her weakened state, her groans had become barely whispers. He knew she would not ascend until she was truly ready to do so, but surely, there must be something he could do to ease her final moments. Something beyond Tau'ri painkillers.\n\nHe began to sing. \" _Kel ma kree. Sha'shan. Kel na t'ai. Or'intani. Kel han'dai_.\"\n\nWeiyan Shi's eyes fluttered open. \"A song for Jaffa?\"\n\n\"Indeed. My wife Drey'auc would sing this to our son Rya'c when he was very young.\"\n\n\"Rya'c.\" Pain shadowed across her face and she paused until it passed. \"Will you see him soon?\"\n\n\"When the time allows, yes.\" Rya'c had joined Master Bra'tac in assisting Ishta and the Hak'tyl women Jaffa in coordinating Fifth Column efforts. It had been many weeks since father and son had met. Teal'c silently promised to visit soon.\n\n\"Is he a warrior?\" Weiyan Shi asked. \"Like you?\"\n\n\"Youth still has him in its grip, but yes, Rya'c is and shall always be a warrior.\" He gazed down upon her worried brow. \"Just as you shall always be, Weiyan Shi.\"\n\nWith that, the pain left her and she fell asleep.\n\n\"This ascension,\" Ambassador Zhu whispered, \"is it possible that my daughter might still exist, once she has died?\"\n\n\"Believe it.\" Teal'c glanced at her. \"You must. She sacrificed herself so that we might save your planet. That sacrifice must be remembered, not denied.\"\n\nAmbassador Zhu brushed a wisp of hair from her daughter's forehead. \"I will not dishonor her memory.\" She returned Teal'c's gaze. \"Nor do I any longer believe that others who have served the program are fools.\"\n\n\"You remember our discussion.\"\n\n\"How can I forget your ardent speech about duty and honor?\" She turned her gaze upwards toward the Ancient Weapons Chair. \"Upon my return to China, I will devote my energies to convincing the upper leadership of the need to leave the chair here in Antarctica. Under the care of those who serve so that Earth may be protected.\"\n\n\"You honor your child's death,\" he said, relieved that some good would come of this day.\n\n\"What I wish to do, Teal'c, is honor her life.\"\n\n# CHAPTER TWENTY-TWO\n\n### 20\u00b0 53' 10\"N ~ 156\u00b0 40' 29W\n\n### INTERNATIONAL AIRSPACE\n\n### 19 AUG 04\/0105 HRS MCMURDO STATION\n\n### 19 AUG 04\/0805 HRS STARGATE COMMAND\n\nOnce the bird was out over international waters with a half moon as its only companion, Jack loosened his grip on the flight stick and flipped his radio back on. While he supposedly had a secure channel, betting against Chinese Intelligence wasn't exactly a gamble he felt like taking.\n\n\"Mission Command,\" he said into the radio, \"this is _Outpost One_. Package received. En route to disembarkation point.\"\n\nA few hisses. A couple of pops. Then Davis' voice broke though. \" _Roger, Outpost One. Standby for General Hammond_.\"\n\n\"Standing by.\" Well, sort of. At their current altitude of 20 miles above sea level, Mach 6 meant flying along at... He hated math. Checking his airspeed gauge, he noted a cozy little 4,102 mph.\n\n\" _This is Hammond. Report,_ Outpost One.\"\n\n\"Good to hear your voice, sir. I estimate arrival time in T minus 90 minutes and counting.\"\n\n\" _I trust you had no troubles obtaining your objective, General_?\"\n\n\"Nothing I couldn't handle, sir.\" Jack winced. It was a bit of lie, but not by much. Landing at the Beijing airstrip had been a piece of cake. The thing rode the runway like a Cadillac. Getting out of the F-302 had been another matter entirely. Especially when several hundred of the Chinese Republic's million-man army and a slew of high-powered spotlights greeted him. Communism at its finest. All nicely lined up, their rifles trained right on him. Chairman Mao would've been proud.\n\nAn armored tank rolled up to the bird. The top hatch popped open and a middle-aged man poked his head out. \" _Zh\u014dng zh\u01d0_!\"\n\nThe Chinese dropped their weapons. The guy looked familiar, but he knew it wasn't Huang. Wrong face shape and besides, the guy wouldn't go from being in a mental hospital to leading a battalion.\n\nDressed in a dark blue suit, the man climbed down from the tank and motioned for the two nearest soldiers to climb on up. As they did, another man came up the hatch. Though his hair had been shorn off to a shiny buzz cut, they'd let him keep the goatee, though it'd gone completely gray. It was a longer, more straggly mess than the last time Jack had seen him.\n\nThe two soldiers lifted Huang out of the hatch and that's when Jack noticed the cuffs on his wrists and ankles. He hated to admit it, but those restraints gave him peace of mind.\n\nThe soldiers lowered Huang to the tarmac. Besides a dark smudge on the guy's forehead, he seemed relatively cleaned up though Jack suspected it'd been a while since his last meal. He wore loose-fitting black pants and a matching collarless shirt.\n\nAnother soldier stuck his head out of the hatch and threw down a heavy winter parka. A soldier picked it up and shoved it over Huang's shoulders.\n\nObviously, they weren't fully informed on where Jack intended to take Huang. That was fine by him.\n\nAt the other man's lead, the soldiers escorted him toward the F-302. Between the soldier's open glares, the shuffling of Huang's cuffed ankles, and the oppressively hot Beijing weather, Jack had to muster up every discipline he knew not to shout at their idiocy. If they thought Huang was going anywhere near Antarctica, they were wrong. Dead wrong.\n\nAs they stopped a few feet short of the gantry, Jack took a long hard look at Huang's face. The man had single-handedly arranged for SG-1's demise while in Yu's clutches. Though he wasn't a Goa'uld, and not even a Jaffa, he was still the enemy.\n\nAnd those pitch-black pupils edged with green flecks still gave him an unearthly look.\n\n\"General O'Neill,\" said the leader, \"your photos do not do you justice.\"\n\n\"Okay, and who might you be?\"\n\nThe man bowed. \"I am the people's true ambassador to the Security Council.\"\n\nJack snapped his fingers. \"You're Ambassador Chen. I've seen your photo.\"\n\n\"And?\"\n\nWhat did the guy want, a compliment? Jack hated diplomacy. \"Kinda hot for a winter coat, don't you think?\" He jerked a thumb at Huang's parka.\n\nChen shrugged. \"It is barely above room temperature.\"\n\n\"Yeah, right.\" And the sky's not blue, it's green. Jack just loved playing footsies with the Chinese.\n\nHe gestured at Huang. \"If your goons wanna get Huang stowed on board, I'll be on my way.\"\n\n\"In a moment.\" Chen strode up to the F-302 and placed a hand on its belly. \"We have long awaited \u2014 \"\n\nA low rumble followed by a good round of ground shaking stopped the ambassador from asking any questions about the fighter-interceptor. He flung out a few clipped orders to his men, they dragged Huang up the gantry and into the cockpit, and Jack hustled them down before they got a closer look.\n\nAfter strapping in his silent passenger, Jack hopped in. Another trembler shook the tarmac, sending rippled waves of heat across the asphalt. With a quick salute to Chen \u2014 and no complaints \u2014 he got the bird out of there as fast as possible.\n\nA half-hour out of Beijing, he finally let himself breathe, hence the radio-call to Hammond. \"Ah, General? We felt a couple of quakes in Beijing before I left. How bad is it elsewhere?\"\n\n\" _Bad enough, Jack. Sydney's been hit, but we expected that. Australia's less than five thousand miles off the coast. However, there's been reports as far away as Russia_.\"\n\nJack checked his watch. It was a cheap little Timex, courtesy of Airman Gerling. \"We've got eleven hours left by my count.\"\n\n\" _Affirmative. I've paused the explosives countdown to give you, Colonel Carter, and Dr. Jackson every spare moment I can, just_ _ \u2014_ __ \" The radio fell back into a bed of pops and hisses.\n\n\"General?\"\n\n\" _Hurry back, son_.\"\n\n\"Uh, roger that.\" Jack signed off. He knew what Hammond really meant to say was, 'Don't screw it up.'\n\nJack could do that. Couldn't he? Even if he had to stick his hands inside a veritable funhouse of Ancient technology and let the machinery have a poke around his head? Sure, he'd do whatever it took to save Earth.\n\nScrew Sun Tzu and his little book of war. Fame and disgrace had nothing to do with it.\n\nAn alarm chimed. He glanced down at a blinking red light over the proximity sensor. Probably a commercial plane heading toward Japan.\n\nSwitching his grip on the flight stick, he adjusted the navigational computer. Normally, the aft seat co-pilot charted their course, but that seat was taken so he'd slaved the nav-com up front.\n\nHe punched in a slightly tweaked setting. The computer beeped and a fat red error blinked across the screen. \"Damn computer. Too many things to do.\"\n\n\"Lamentations will only hinder deeds.\" The clink of metal cuffs from the aft seat reminded Jack he had a passenger.\n\n\"So you do speak.\" He re-entered his course correction and the nav-com display switched from red to green.\n\n\"When words are necessary,\" Huang said.\n\n\"And was it necessary to get me and my team almost killed last year?\"\n\nHuang sighed. \"My Lord Yu believed it so, yes. I did not expect you to survive the encounter.\"\n\n\"Surprise!\" Jack snorted. \"Your slimy, snake-in-the-head, boom-box leader didn't want to kill us. He just wanted see if we could kick Anubis's butt once and for all.\"\n\n\"Anubis?\"\n\n\"Yeah, remember him? Cranky. Half-Goa'uld, half something else?\" Jack held back on saying what. If Yu's spy didn't know Anubis had been half-ascended, there wasn't any reason to tell him. \"Wore a long robe with a hood. Looked like something out of _Star Wars_.\"\n\n\"Wore,\" Huang repeated. \"He no longer survives?\"\n\nJack grinned. \"That's right. We kicked his smarmy little ass. No thanks to your fearless leader, I might add.\"\n\nThe memory of how they'd stopped Anubis made the grin slide right off \u2014 sitting in that Ancient weapons chair, his brain so stuffed with data that he couldn't breathe. And this time... What was it Skaara had said?\n\nOperating the chair would be a cakewalk compared to retrieving the crystal.\n\n_Oh, joy_.\n\nThe F-302 sliced through a puffy bank of clouds backlit by the moon. Jack checked their position. Several hundred miles off the island of Maui.\n\n\"How is he?\" Huang said.\n\n\"Anubis? Dead.\"\n\nThe cuffs clinked again. \"No, I mean my master. Lord Yu.\"\n\n\"Your master.\" Jack twisted around in his seat harness and pushed back his helmet to get a good look at Huang. \"I don't think he's missing you very much.\"\n\nHuang averted his eyes. \"I am a Dragon Guard.\"\n\n\"You're a clone. I've met your replacement.\" Jack settled back into his chair. \"Actually, I met several generations. How does it feel, knowing you're just another in a long assembly line of carbon copies, not really meaning much in the grand scheme of things?\"\n\n\"You would not understand,\" Huang whispered, his voice barely heard above the whine of the aero spike engines.\n\n\"Understand what?\" _That every time I lose a man and write up another casualty report, that I feel like I'm chipping away at a piece of myself_?\n\n\"How do you know about the Dragon clones?\"\n\nMaui went by. He altered course again, deciding to take the F-302 over Alaska and then straight down into Colorado.\n\n\"Yu told us all about them. In fact, when he did, your name never came up.\"\n\n\"By 'us' you mean SG-1? And yet, where is your team now? How is it you came for me alone?\" Again, the cuffs clinked.\n\nWhen Jack didn't answer, Huang added, \"I know what it means to be alone, O'Neill.\"\n\n\"In case you haven't noticed, there are only two seats. Didn't Yu teach you how to count?\" He tweaked the flight stick, pulling them up another five miles to avoid any commercial flights. \"Besides, I like being alone.\"\n\n\"'It is impossible for the brave to advance alone, or for the cowardly to retreat alone.'\" Clink. \"So says Sun Tzu.\"\n\nJack rolled his eyes. \"I know the quote. If it matters any, your kid spouted off all sorts of _Art of War_ crap during our, uh...\" Shared illusion? Yeah, that experience didn't need explaining.\n\n\"Weiyan. You saw my daughter?\"\n\nOff to the east, Jack saw the first hints of sunrise. Pink and yellow striated clouds hovered above the horizon. A reminder of what waited for them in Antarctica if they got back in time. \"Where do you get off, leaving a kid like that? People like you should never be allowed to have children.\"\n\n\"I realize this. In ways you would never understand.\"\n\n\"Weiyan's a brave girl, Huang. Despite you abandoning her.\"\n\n\"I see.\"\n\n\"You should,\" Jack said, pointing out the starboard window. \"You have the best seat in the house.\"\n\nHe nudged the nose north. At their current speed, they'd be over Alaska in about ten minutes.\n\n\"Tell me of my daughter, O'Neill.\"\n\nAnd there it was, the real reason Jack had opted to do this milk run solo. For Weiyan's sake.\n\n\"No thanks to you, she's got backbone.\" Jack told him everything. Including the fact that his daughter lay dying in a cold, frozen wasteland.\n\nAnd if Huang didn't do his part in retrieving that crystal, pretty much everyone would follow soon after.\n\n### STARGATE COMMAND\n\n### 19 AUG 04\/0330 HRS MCMURDO STATION\n\n### 19 AUG 04\/1030 HRS STARGATE COMMAND\n\nDaniel hurried out of the elevator and onto level 28, his backpack slung over one shoulder. He'd had two hours to nap, shower, dress in camo gear, and sling back enough coffee to keep him going for whatever came next. That, and ten minutes to stuff a few items in a pack. Not everything he packed was essential, but he'd had a last minute inspiration thanks to a side trip to the archives and a quick talk with Dr. Kevin Hopkins.\n\nDaniel ran toward the armory, the precious item stowed in his pack. Huang. He was going to meet a direct descendent of Sun Tzu. If circumstances weren't so extreme, he'd have a million questions.\n\nHe knew he should trust Skaara, but he was half-terrified. The plan relied too heavily on Huang and Jack working together. Daniel just couldn't see that being a successful partnership.\n\nAt the armory, he grabbed a pistol and was halfway to the gate room, when the master-of-arms chased him down. \"General's orders, sir.\" He handed Daniel a P90 along with a sling and two magazines.\n\nDaniel shoved the magazines in a pant-leg pocket and took the rifle. He waved off the sling. He hated the idea of being strapped to a rifle. Shifting his pack to his left shoulder, he headed toward the blast doors, the P90 cradled in his right arm. The airman on guard slid his security card through the lock. The doors opened, revealing Sam over by the gantry, a P90 dangling from her chest.\n\nHer wet bangs told him she'd rushed through their two-hour window just as fast. The real Graham Simmons \u2014 or at least, he hoped it was the real one \u2014 handed her a scanner. Daniel wondered if she'd told the lieutenant about Skaara's impersonation. Probably wasn't a good idea, all things considered.\n\nEspecially if it really wasn't the lieutenant. Could Skaara still be around, keeping an eye on things?\n\nThere was only one way to find out. Daniel headed over to join them. \"Graham, how are you?\"\n\nThe lieutenant grinned. \"I'm doing pretty good, sir. You?\"\n\nIt sounded like Graham.\n\n\"Good, good.\" Daniel looked closely at his face, his clothes, his hair \u2014\n\n\"That'll be all, lieutenant.\" Sam grabbed Daniel's elbow and pulled him to the far wall. \"That isn't Skaara.\"\n\n\"You're sure?\"\n\n\"Too many contractions.\" She punched a few buttons on her scanner. A yellow light flashed on the display. She tapped in another series on the keypad and the light winked out.\n\n\"Sam?\"\n\nShe flipped the scanner over. \"Yeah?\"\n\n\"Any word from the outpost?\"\n\nSliding the battery case open and shut, she turned the scanner right side up and thumbed the 'on' button.\n\n\"Got your eyes and ears?\" he asked, only half-teasingly.\n\n\"What?\" She looked up from the scanner. \"Oh, sorry, Daniel.\" She stuffed it inside her tactical vest pocket with a frown. \"I spoke to General Hammond a few minutes ago.\"\n\n\"And?\"\n\n\"Weiyan's still alive, but the general doesn't seem to think she'll last much longer. Teal'c's with her. Based on,\" she hesitated, \"past observations, she won't ascend until she's ready to, um... What I mean to say, is...\"\n\n\"Die?\" He understood Sam's discomfort. While he couldn't quite remember the last moments before his own ascension, he'd regained painful memories of the radiation sickness. Flashes of dialogue with Oma, with Sam, with Teal'c. His last moments with Jack.\n\n_I think I can do more this way. Please, Jack. Tell Jacob to stop_.\n\nHad he done more? Had he made a difference?\n\nAs if summoned by the memory, Jack strode in, dressed in camo, P90 hanging from his sling. Sergeant Walter Harriman followed him, waving a clipboard.\n\n\"General,\" said Walter, \"the president left orders not to initiate evacuation to the Alpha Site. If I do this \u2014 \"\n\n\"Last I checked, the president doesn't have a crystal ball.\" Jack yanked the clipboard from Walter, signed it, and handed it back. \"Tell him I appreciate the vote of confidence, but I'd rather have him fire my ass afterwards than take any chances.\"\n\n\"But \u2014 \"\n\n\"Come on, Walter.\" Jack tugged his cap's bill. \"Nothing beats an old-fashioned 'I told you so.' On the other hand \u2014 \"\n\n\"If President Hayes is wrong,\" Walter recited, \"Earth will crack open.\"\n\n\"Like an egg.\" Jack turned toward Daniel and Sam. \"You two ready to go?\"\n\n\"Yes, sir.\" Sam glanced up at the operations room. \"We removed the lock on P3Y-702 from the dialing computer.\"\n\n\"Where's Huang?\" Daniel slipped his P90 between his legs so he could fasten his pack to his vest-back.\n\n\"The doc's making sure he eats a good meal before we leave.\" Jack jerked his chin toward Daniel's pack. \"We're only going for four, maybe five hours. Whatcha got in there?\"\n\n\"Nothing.\" Daniel kept his face impassive.\n\nJack waggled a finger at him. \"You brought a copy of _The Art of War_ for the guy to sign.\"\n\n\"Funny.\" Daniel picked up his P90. \"Is this necessary?\"\n\n\"Yes.\"\n\n\"I don't think Yu's going to bother us.\"\n\n\"You know the clich\u00e9, Daniel. Better safe than sorry.\" He turned toward Walter. \"Dial it up, Sergeant.\"\n\n\"Yes, sir.\"\n\nThe blast doors slid open once again. Huang, his wrists and ankles in cuffs, shuffled into the room with SG-13 right behind him \u2014 everyone except Balinsky. They were also dressed up in camo gear, including Kevin Hopkins who'd been drafted in the past twenty-four hours to be their new team member. Unlike the others, Kevin didn't have a weapon out and drawn on the prisoner.\n\nColonel Dave Dixon kept his M-16 trained on Huang as they made their way over to the gantry. Though Daniel had never met Huang before, he couldn't help but notice how worn out the man seemed. Gaunt, hollow-eyed, his thin beard white and ragged. He didn't see any resemblance to Yu's current First Prime, Oshu until the man lifted his chin and stared up at the Stargate. Then the green flecks in Huang's dark eyes matched Oshu's exactly.\n\nThe gate began to spin. Steam rose from the bulky red capacitors. The inner ring ground along its track, the sound echoing against the gate room's cement walls. The ring slowed, stopped, and the first chevron lit up.\n\n\" _Chevron One encoded_ ,\" Walter announced over the PA system.\n\nThe ring circled back around. A half-sob escaped Huang's throat. \"I never stopped believing.\"\n\n\"Quiet down,\" barked Dixon.\n\nDaniel cringed. \"Jack, is it really necessary to treat the man this way?\"\n\n\" _Chevron Two encoded_.\"\n\n\"I never stopped hoping,\" Huang whispered.\n\nDixon raised his M16 directly into the old man's face. \"I said, quiet down.\"\n\nDaniel shared an uncomfortable look with Kevin.\n\nHuang stepped around Dixon's rifle. \"So long. I have \u2014 \"\n\n_Ka-chink_.\n\n\"Where's Siler?\" asked Jack.\n\n\"I'm here!\" Siler ran in, backpack in hand. He buckled it onto Jack's vest. \"Sorry, sir. Took a moment to find \u2014 \"\n\n\"I've got it from here.\" Jack handed his pack to Sam and turned around. Keeping her face neutral, she secured it to his vest-back.\n\n\" _Chevron Six encoded_.\"\n\nStartled, Daniel looked up at the gate. The first six chevrons were lit up. He'd been so caught up in the tension in the room, he hadn't heard Walter announce Chevrons four and five. A half-second later, the seventh one locked into place.\n\n\" _Chevron Seven locked_.\"\n\nThe Stargate's brilliant blue kawoosh erupted and then settled into place. Huang's mouth dropped open, he took a step back, right into the butt end of Dixon's rifle.\n\n\"Don't move until we tell you.\" Dixon looked at Jack. \"We're ready, General.\"\n\n\"Thank you, Colonel.\" Jack pushed down on his ball cap. \"We'll take it from here.\"\n\n\"You're joking, right?\" Dixon glanced at Huang and then back at Jack. \"This guy's shenanigans almost got a whole bunch of our people killed. Since when do we play buddies with the enemy?\"\n\nThe gate room shuddered, sending ripples across the event horizon. The quake didn't last long, but it was enough to make Dixon's eyes widen.\n\n\"Since we started monkeying with gadgets way out of our league.\" Jack raised two fingers toward gate. \"Move out, SG-1.\"\n\n\"Daniel!\" Kevin Hopkins called.\n\nDaniel stopped and turned back toward his old roommate.\n\n\"Good luck,\" he mouthed.\n\nWith a nod of thanks, Daniel stepped through the gate, hoping that luck would be enough.\n\n### P3Y-702 AKA KUNLUN\/ELYSIUM PEDION\n\n### 19 AUG 04\/0405 HRS MCMURDO STATION\n\n### 19 AUG 04\/1105 HRS STARGATE COMMAND\n\nSunrise greeted Daniel as he followed Sam through the gate and out the other side. A rose-colored wash of light blanketed the valley below, the planet's usual heat camouflaged by an early morning breeze. He climbed down the steps, scanning the valley for any signs of life, be it Lord Yu, his Jaffa, or even Skaara.\n\nA sense of d\u00e9j\u00e0 vu teased him, and for a moment he wondered if they were still in the device's throes. Then he caught sight of the SGC's abandoned dig from a year ago. White flags flapped in the breeze, their ends attached to ropes surrounding the various trenches dug by team archaeologists. A few trees wilted in the barren dirt. On a hill off to the left loomed the three-story tall Zhenmushou statue.\n\nAt the valley's other end stood the chamber. Its high brown walls almost colorless against the rising sun.\n\nA thwap from the event horizon. Jack stepped through, his hand on Huang's elbow. It didn't stop the old man from dropping to his knees in awe.\n\nJack gently pulled him back up. \"Carter, my lower pocket.\"\n\n\"He's not going to be much help to us cuffed,\" Daniel said.\n\n\"I hope you're right,\" Jack said as Sam slipped behind him.\n\nHuang's eyes never left the Zhenmushou. \"My ancestors... I must pay my respects.\"\n\n\"Not going to happen,\" Jack said.\n\nHuang dropped his head. \"I understand.\"\n\n\"I hope that you do.\"\n\n\"Come on, Jack. I think the three of us are enough to keep him from causing any \u2014 \"\n\n\"Got it, sir.\" Sam handed a key ring over to Jack.\n\nHe bent down and unlocked Huang's leg cuffs, then his hands. The cuffs clanked against the gate's stone platform as they dropped. \"Happy now?\"\n\nBy the time the sun had risen above the horizon, they'd climbed down the hill and headed toward the chamber. Meter-high obelisks lined the half-dirt, half-paved walkway, their Ancient letters worn away by age. Daniel hoped one day to come back, to study the obelisks and figure out if they matched the ones he'd seen in their vision, dream, or whatever it had been. Though Skaara had shared what had caused the plague, he'd left out specifics. Daniel still had a lot of questions.\n\nHuang never took his eyes off the _Zhenmushou_ as they passed beneath its shadow. He grinned broadly until he noticed the four excavated graves beneath the statue's base. The grin dropped away.\n\n\"I would ask why you chose to disturb my ancestors' remains, but the answer would have little meaning.\" He looked at Daniel, his eyes as dark and hard as the paved stone beneath their feet.\n\nJack kept up the pace with Sam at his side. They had at least another ten minute hike ahead of them and by Daniel's watch, that left them less than four hours to open the chamber, retrieve the crystal, and gate back to the waiting F-302s to go back to Antarctica.\n\nDaniel glanced at Huang. \"If your ancestors were originally from Earth, how did you get back there, by ship?\"\n\n\"Through the Chappa'ai, of course.\" The old man gazed again at the statue. \"The one in Antarctica. It was long ago, fifty years in fact. I fought past many Serpent Guards to reach it \u2014 they fought bravely, their marksmanship superlative. Particularly one, I remember \u2014 a tall, dark powerful Jaffa. I escaped, following two others to your land.\"\n\n\"Tall and dark, you say?\" commented Jack. \"Carter, sound like anyone we know?\"\n\n\"I doubt it, sir. If Teal'c knew about the Antarctic gate, wouldn't he have told us when he joined SG-1?\" Sam looked at Huang over her shoulder. \"A few years ago, we found two Jaffa frozen in the ice by the gate. We'd always assumed they'd been there for two, maybe three hundred years.\"\n\nHuang's eyes took on a distant look. \"Ice and snow can age a man in ways you cannot understand.\"\n\n\"And from Antarctica to China?\" Daniel asked.\n\n\"At first, I was guided by a strange light. Golden-white.\"\n\n\"Daniel, could that've been Skaara?\"\n\n\"Possibly.\" Though Skaara had never really explained how he'd managed to skip back and forth in time, it certainly sounded like him.\n\n\"Friend or foe, I cannot say.\"\n\nSam slowed down to match Huang's pace. \"There wasn't anyone else down in Antarctica fifty years ago. How did you survive?\"\n\n\"Seal meat.\"\n\n\"Okay, that's just gross.\" Jack shook his head. \"Those creatures deserved better.\"\n\n\"I am not proud of what I did, O'Neill. I did it to survive.\"\n\n\"Still doesn't explain how the hell you got out of there.\"\n\n\"Sir Edmund Hilary found me.\" Huang went on to detail how he'd met the explorer. He had just begun to share how he rose through the ranks during China's Cultural Revolution when they arrived at the chamber. He froze. \"I cannot. It is _b\u00f9 zh\u01d4n x\u01d4_.\"\n\n\"You're going in,\" said Jack. \"Whether you like it or not.\" He pointed at the transporter rings. \"Keep a good five feet away. Daniel, if you don't mind?\"\n\nApproaching the wall, Daniel took in the intricately carved stone surface covered in Chinese pictograms and Ancient glyphs. A circle with a dash in the middle represented the sun. Beside it, a half circle with a vertical line meant the moon. Three apexes crowded together upside down \u2014 like a series of mountains.\n\n\"Kunlun,\" whispered Huang. \"Behind those walls lies death.\"\n\n\"Not according to our mutual friend,\" Jack said.\n\nDaniel found the crack running down the wall's midsection. He tried pushing on either side, but the doors wouldn't give.\n\nSam took a turn with no luck. \"Sir, I'm willing to bet the door is keyed to the ATA gene.\"\n\nJack motioned for Huang to join him at the wall.\n\nWild-eyed, the old man backed away. \"No, no!\"\n\n\"You can,\" Jack said, stomping toward him, \"and you're gonna. Think of it this way. Yu would want you to save Earth, wouldn't he? That overdressed snake likes Earth.\"\n\n\"I cannot.\" Huang trembled so violently that Daniel became worried that he might have a heart attack.\n\nThen he remembered the item in his pack. \"Wait a minute.\" Opening it, he pulled out the three-pronged bronze wine vessel unearthed near the chamber just a year ago.\n\n\"I have something to show you. Something that I'm pretty sure comes from Earth, but... Well, take a look.\"\n\nHuang's shaking subsided, but he still didn't move.\n\nHolding the cup by its fluted bowl, Daniel showed it to him. \"We found it right over there,\" he pointed toward the dig area. \"There's writing inside. Xian. That's one of China's dynasties, from before \u2014 \"\n\n\"It is the language we are first taught as children.\" Huang choked back a half-sob and took the cup. \"You found this here?\"\n\n\"Daniel...\"\n\n\"Jack, give him a moment. It's a lot to take in.\"\n\nHuang traced the long thin lines with his forefinger.\n\n\"Do you know what it says?\"\n\n\"It is a hero's cup. It says...\" Huang's lip quivered. \"It is a quote of Sun Tzu. From before _The Art of War_.\" He dropped the cup to his side, his shoulders shaking. \"It says, 'Great is the hero who would die for their country, but would much rather live.'\"\n\nHuang dropped to his knees and held out the cup.\n\n\"You keep it.\" Daniel waved him off. He re-zipped his pack and joined Jack by the wall. \"He'll cooperate. Just go easy.\"\n\nJack tilted his head back, looking up toward the Ancient writing at the top of the wall. \"Nice quote, by the way.\"\n\n\"Sun Tzu at his finest.\"\n\n\"If you say so.\" Jack slapped the wall. \"How about we get this thing open? Then maybe everyone can get back to living.\" He turned toward Huang. \"What the hell?\"\n\nThe old man, still kneeling on the ground, lowered a dirt-covered finger. Smeared on his forehead was a roughly drawn replica of Lord Yu's sigil.\n\n\"I am ready.\" Huang stood up.\n\n\"I don't think so,\" Jack replied.\n\n\"You do not approve of my _w\u00e9nsh\u0113n_?\"\n\n\"Not really.\"\n\n\"Then I pity you.\" Huang strode to the wall, his step lighter, a faint smile on his lips.\n\nIn a sense, Daniel mused, the old man had re-found his purpose.\n\nSam checked her watch. \"Oh-five-hundred outpost time, sir.\"\n\nJack stared at Huang. \"How much time does that leave us?\"\n\n\"Three hours to get to the gate and make our window.\"\n\n\"Jack, just...\" Daniel stepped up to his other side and whispered, \"Whatever it takes, right? We need to get this chamber open, get the crystal, and \u2014 \"\n\n\"I know what we need to do, Daniel. Thank you.\" He rubbed his palms together and then raised them in front of the wall. \"You ready, Huang?\"\n\nThe old man bowed. \"I have always been, O'Neill. Are you?\"\n\n\"On the count of three then.\" Jack gave the countdown.\n\nThe two men pressed their palms against the wall.\n\nA low rumble. Dirt showered down.\n\nAnd then, the wall opened.\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 19 AUG 04\/0530 HRS MCMURDO STATION\n\n### 19 AUG 04\/1230 HRS STARGATE COMMAND\n\nPaul Davis found General Hammond by the archway, staring at the stasis unit. \"Sir, there's no reason you need to stay. When Gerling gets back from McMurdo, she could take \u2014 \"\n\n\"Major, as much as I appreciate your concern, I'm staying.\" Hammond raised a hand over the stasis controls. \"I think you'll find Dr. Lee determined to see this out as well.\"\n\n\"Yes, sir.\" He wasn't surprised.\n\n\"These Ancients,\" the general dropped his hand to his side, \"the technologies they built. Takes a pretty wild imagination to come up with ways to reform a planet, keep a man frozen in time.\"\n\n\"Don't forget the Stargate, sir.\"\n\nHammond chuckled. \"How could I?\"\n\nThey walked back into the weapons chair room. Dr. Lee was tinkering with one of the Mark IIs. Low murmurs came from the hole \u2014 Teal'c and Ambassador Zhu. Paul followed Hammond over to the ladder. Down below, Teal'c cradled Weiyan's head in his lap. The ambassador sat beside him, stroking her daughter's hair. An IV bag ran to the young woman's pale arm, hopefully providing her some small measure of pain relief.\n\nAll because of the Ancients and their wild imaginations.\n\nFor a moment, Paul wondered if it had all been worth it.\n\n\"How's she doing, Teal'c?\" Hammond whispered.\n\nHe looked up. \"She is sleeping, General Hammond. The fight to live remains strong. Any word from O'Neill and the others?\"\n\n\"They've gated to P3Y-702,\" Hammond replied. \"No other word yet, but we're hopeful they'll meet with success.\"\n\nTeal'c laid his hand back down on the blanket. \"Skaara assured us they will succeed. Have faith, General Hammond.\"\n\n\"Ambassador Zhu, is there anything we can do?\"\n\n\"Nothing.\" The ambassador managed a brief smile. The effort lasted less than a second.\n\nHammond headed for the rear of the platform. Paul followed him and they sank down on the one edge still connected to the main section of floor.\n\n\"Paul, I meant what I said earlier.\" Hammond turned toward him. \"We get through this and it's time for a promotion. More than time. Put in the paperwork, son. You've earned it.\"\n\nAnother rumble ran through the room. Paul glanced over at Lee. The scientist squinted at his monitor, ignoring the sound. The tremors died off.\n\n\"If we get through this,\" Paul said.\n\n\"Sure,\" the general said. \"Think of it as a reprieve. No more trips to the frozen south. No more last minute dashes to Colorado Springs. Just a warm desk in D.C. at Homeworld Security.\" He shrugged. \"Of course, it snows in Washington, too, but not always.\"\n\n\"Sir,\" Paul swept a stray chunk of ice off the weapons platform. \"What if I didn't take a promotion? I know the Air Force requires promotion or retirement by a certain point.\"\n\n\"Homeworld Security's a special situation.\" General Hammond shrugged. \"Don't you want the promotion?\"\n\n\"What about the SGC, sir?\"\n\n\"We can find another liaison, Lt. Colonel Paul Davis.\"\n\nAnother liaison. \"Has a nice ring, sir.\"\n\n\"General Hammond!\" Gerling ran in, her parka dusted with snow. \"I've brought provisions from McMurdo, sir. Not much, but some coffee, water, some Power Bars to eat.\"\n\n\"Let's get some coffee for the ambassador and some water for Teal'c. They must be hungry, too.\"\n\n\"Yes, sir.\" She unzipped her parka and pulled out a folded-up piece of paper. \"McMurdo's latest casualty reports. I thought you might want to see them.\"\n\nHammond unfolded the report. As he scanned it, his face stilled. He handed the paper to Paul and closed his eyes.\n\nPaul held off reading it. \"Bad, sir?\"\n\nThe general stood up. \"Bad enough. Dr. Lee? Care to join me for some well-earned, though probably terrible, coffee?\"\n\nLee raised his head from the monitor. \"Did I hear something about food?\"\n\n\"Power Bars, if you'll have them.\" Hammond strode out, tailed closely by Dr. Lee and Lt. Gerling.\n\nPaul read the report. Ten more deaths. All civilians. Building 155 had collapsed, but not before a few people had managed to chase everyone out of the facility. He scanned the list of names, not recognizing any until the last one.\n\nHe rose to his feet, not looking forward to sharing the name with Teal'c. Crossing over the plank, he looked down into the hole. Eyes rolled back, sweat dripping from her forehead, Weiyan turned her head back and forth.\n\n\" _Ni shi shui_?\" she mumbled. \" _Wo meng_?\"\n\n\"What is she saying?\" Paul asked.\n\nWeiyan's mother wiped her eyes with her parka sleeve. \"She says, 'Who are you? Am I dreaming?'\"\n\n\" _Wo jiang xie zia wo bao fu_.\"\n\n\"I will release my burdens,\" Zhu translated. Her hand grabbed Teal'c's arm. \"What does that mean? She is an innocent. What burdens could she possibly need to release?\"\n\n\"You must ask her.\" Teal'c gently raised Weiyan's head and shifted over, allowing Zhu to take his position. \"You are her mother. Tell her she is free to move on.\" He stood up. \"Then, allow her to do so.\"\n\nZhu's eyes widened. \"You are leaving?\"\n\n\"Do not regret these moments, Ambassador Zhu.\" He began climbing the ladder. \"I will not be far.\"\n\n\" _Wo jiang xie zia wo bao fu_ ,\" Weiyan whispered. \" _Wo jiang bu hai pa_.\"\n\nZhu bent over her daughter and broke into sobs. Paul's throat constricted at the sight of a mother saying goodbye to her child. He backed off the plank, giving them more privacy.\n\nTeal'c joined him by the back wall, purple bruises under his eyes told Paul all he needed to know. He hesitated, unsure if he should share the report from McMurdo. Weiyan had spoken about releasing her burdens. Telling Teal'c would just add to his, no doubt.\n\n\"You have something to tell me, Major Davis.\" Not a question, just a fact. Teal'c was like that. He could see the truth in anything.\n\nPaul held up the report. \"Another building collapsed at McMurdo. The personnel director, Hannah Presley... She died, Teal'c. Trying to save everyone else. I'm sorry. I know you two got along well.\"\n\n\"I see.\" He crossed the back of the weapons chair and climbed onto its platform, his back to Paul. \"Her red-leaf tea...\" He gripped the chair's headrest. \"Major Davis, I believe I have a confession to make.\"\n\n\"A confession? What could you possibly need to confess?\"\n\n\"I believe my actions many years ago have brought this disaster to pass.\"\n\nTaken aback, Paul stepped around the chair to face Teal'c. \"How could you have anything to do with \u2014 \"\n\n\"Many years ago, while a lieutenant in Apophis's ranks, I, along with many others, escorted three Serpent Guards who had been tasked with a secret gate address. An address I was not permitted to know.\"\n\n\"What's that got to do with anything?\"\n\nThe ground rumbled. Not as bad as before, but it served as a reminder that time was running out. Paul glanced at his watch. \"Six more hours until sunrise.\"\n\n\"Major Davis, I believe that I witnessed Huang escape through the Stargate to Earth.\" Teal'c went on to describe how two men, disguised as Serpent Guards, attacked the Jaffa. One made it through. The other died.\n\nPaul was stunned. It was a lot to take in. The idea that Teal'c might have known about the Antarctic Stargate from the start being just the tip of the proverbial iceberg.\n\nListening to the murmured sounds of a mother soothing her dying daughter, Paul surveyed the damage done to the outpost. It would take weeks for the Army Engineers to shore everything back up.\n\nBut it wouldn't bring back the dead. Or make it easier on the dying.\n\n\"Did you get a look at the body?\" he asked Teal'c. \"Or the Jaffa who got through?\"\n\nTeal'c frowned. \"I did not.\"\n\n\"Was that the only time Apophis sent troops through to try and reach Earth?\"\n\n\"No. Apophis made many attempts over the years. He did so for several planets, Earth was only one.\"\n\nPaul scratched his chin. \"And that was the only time enemy Jaffa infiltrated the Serpent Guard?\"\n\n\" _Bie li wo. Bie li._ \"\n\nWhispered shushes emanated from the hole, followed soon after by humming. Paul listened for a moment, unable to make out the song. It was soothing. Quiet. A simple melody. He supposed it must be some sort of Chinese lullaby.\n\n\"Infiltrations were frequent.\" Teal'c let go of the chair. \"Ra, Heru'er, Cronus. They all sent spies. Try to \u2014 \"\n\n\"So you don't know,\" Paul said. \"Look, Teal'c, you can't blame yourself for everything. That's not how it works.\"\n\n\"O'Neill would say as much.\" Teal'c bent his head. \"Master Bra'tac would also agree.\"\n\n\"I'm with Master Bra'tac,\" Paul said, \"and General O'Neill, of course.\"\n\nAmbassador Zhu continued humming. The tune changed, a speedier tempo with almost a playful quality.\n\nThe corner of Teal'c's mouth tugged up in a faint smile. \"When I was very young, my father told me the burden of wrongful blame was a falsehood to oneself. One that places distance between what we believe to be true and who we really are, setting our soul adrift from its true purpose.\"\n\n\"True purpose.\" Paul reflected on General Hammond's offer. It meant safety. Security. A pay raise.\n\nIf they survived the next few hours, he would be a fool to turn it down.\n\n### P3Y-702\/KUNLUN\/ELYSIUM PEDION\n\n### 19 AUG 04\/0615 HRS MCMURDO STATION\n\n### 19 AUG 04\/1315 HRS STARGATE COMMAND\n\nSam flicked on her P90's tactical flashlight, raking the light across the dark chamber. Silver curlicues were etched into the marble floor. A row of columns lined the far wall. \"This looks familiar.\"\n\nShe stepped inside, aiming her light toward the center of the room. There it was, an exact duplicate of the Antarctica terra-forming platform.\n\nAnd the one in their shared vision.\n\nUsing her free hand, Sam pulled out her scanner.\n\nDaniel joined her. \"Anything?\"\n\nShe thumbed on the display. The meter barely budged. \"Less than 0.3 micro-Sieverts. Most likely, from background radiation.\" She turned toward the general and held up her scanner. \"No considerable photonic radiation, sir.\"\n\n\"Good thing, too. I forgot my sunscreen,\" the general said from the doorway. \"Come on, Huang. Get ahead of me.\"\n\nHuang clasped Daniel's cup to his chest. \"You do not trust me at your back, O'Neill?\"\n\n\"Not really.\" The general waved his P90 toward the chamber's interior. \"Step in, nice and easy.\"\n\nAs the two men entered the chamber a blue light flashed brightly and then settled down into a soft glow. A low pulse thrummed through the chamber, amplified by the marble floor and stone walls.\n\n\"Shouldn't that be glowing red?\" Daniel pointed toward the control crystal.\n\n\"It's dormant.\" Sam approached, her scanner still unable to pick up any unusual radiation. Intact, the crystal was a darker red because it wasn't active, but it was right where it should be. Smack in the middle, with an empty bracket on either side.\n\n\"Dormant's a good thing, right?\" General O'Neill came up on her left with Huang beside him. \"We just need to grab the crystal, get back to the gate, and take this puppy to the outpost.\"\n\nThe general secured his P90 under his right arm and reached out a hand toward the crystal. He motioned for Huang to do the same. \"On my count.\"\n\nShe glanced down at the meter. \"Sir, wait!\" The needle was off the scale.\n\nGeneral O'Neill pulled his hand away. \"What?\"\n\nThe needle dropped to normal levels. \"That doesn't make sense, unless... General, Mr. Huang, I need the two of you to touch the crystal, but don't grab hold of it.\"\n\nWith a nod from General O'Neill, the two men touched the crystal with their fingertips. The needle pinged again.\n\n\"Don't make any moves.\" Adjusting the amplitude, she ran another scan. The needle stayed off the chart.\n\n\"How much longer, Colonel? Not that I'm complaining, but we're under a bit of a time crunch \u2014 \"\n\nThe ground shook, a low rumble reverberating through the chamber.\n\n\"Sam,\" Daniel warned.\n\nSuddenly, an ear-splitting crack resounded from somewhere outside.\n\n\"Hands off, hands off!\" She pulled both men back from the platform. The tremor immediately stopped.\n\n\"Carter?\" Dust trickled down on the general's head. He yanked off his ball cap. \"Skaara didn't say anything about there being a problem.\"\n\n\"No, sir. He didn't.\" She held out the scanner. \"I'm getting inconsistent readings. It's almost as if \u2014 \"\n\n\"Forget that meter thing.\" The general checked his watch.\n\n\"I need to run more tests.\"\n\n\"We've got two-and-a-half hours left before we're outta time. Use your head and if that fails, use your gut.\"\n\n\"What's wrong?\" Daniel peered down at the scanner and its near-zero reading.\n\nShe aimed the scanner at the crystal. Nothing. The needle didn't move. \"Daniel, touch the crystal.\"\n\nHe did. Again, nothing happened.\n\nOf course not. Neither she nor Daniel had the necessary gene.\n\nThe general had said use her head. Fine, but not without double-checking the theory forming in it. A theory no one was going to like.\n\n\"General, just one more time.\" She gestured for him to touch the control crystal. \"Mr. Huang, you too.\"\n\nAs soon as they touched the crystal, the needle slammed to the meter's far right. The ground shook. More dust fell.\n\n\"Satisfied, Carter?\" shouted General O'Neill over the noise.\n\nShe shoved the scanner in her vest pocket. What was the point? It was useless. \"Sir, I'm pretty sure that the moment you pull the crystal, the device will initiate a sizeable eruption similar to what happened in Antarctica.\"\n\n\"How sizeable?\"\n\nShe pointed toward the opened doorway. \"Sizeable enough that, if the gate sits on a fault line, it'll get buried.\"\n\n\"Just like it did in our vision,\" Daniel said.\n\n\"We've come this far.\" General O'Neill shooed Huang from the platform. \"There's gotta be something we can do.\"\n\n\"It doesn't matter to the device if the crystal is damaged \u2014 like on Earth \u2014 or pulled.\" She pointed at the silver lines on the floor. \"In fact, here it could be worse because the sun's already up.\"\n\n\"I could dial the gate in advance,\" Daniel offered.\n\nSam shook her head. \"That won't work. The gate's almost a mile off. The quake could be instantaneous. We need some sort of buffer. Something to at least temporarily block any stored up energy from reaching the platform.\"\n\n\"Like a dam,\" Huang said.\n\nSam raised an eyebrow at Huang's sudden contribution. He'd kept so silent since they'd entered the chamber. Still, he had a point. \"A dam is exactly what we need. Something to hold back any energy from reaching the planet's tectonic plates long enough to gate out of here.\"\n\n\"With the crystal,\" Daniel added.\n\n\"Kinda like sticking a finger in a dike,\" the general said.\n\n\"You may be on to something, sir.\" She leaned over the platform and examined the two empty brackets. Each one was about five inches long, maybe an inch wide. She eyeballed the bracket beneath the control crystal. It was both longer and wider. \"Not a finger, a hand.\"\n\n\"What do you have in mind?\"\n\n\"The general figured it out, really.\" She backed away from the platform. \"We need a finger in the dike, or in this case, a hand. Something to trick the bracket into thinking there's still a crystal attached.\"\n\n\"Would a jacket work?\" Daniel started to unbuckle his vest, but Sam stopped him.\n\n\"It needs to be carbon-based,\" she said. \"Like the crystal.\"\n\n\"Or a diamond,\" Daniel said. \"That's carbon-based, right?\"\n\nGeneral O'Neill snorted. \"Oh, sure. Let's just run back to the SGC and see if Siler's got one lying around.\"\n\n\"Actually, sir, there's one other option for a carbon-based form.\" She retrieved her scanner and ran it over her hand. \"Us.\"\n\nThe needle wobbled, then settled back to normal. \"On the quantum level,\" she explained, \"we're made up of tiny particles of light, and these particles of light have an energy frequency. One that could block the emissions long enough for everyone else to gate out with the crystal.\"\n\n\"And then what?\" asked Daniel.\n\nShe didn't know. \"We don't have time for me to run simulations. I couldn't even guess \u2014 \"\n\n\"Forget it, Carter. What fun would it be without a surprise or two tucked inside the cereal box?\" The general dumped his pack on the ground. \"Daniel, head to the gate. Start dialing.\"\n\n\"Wait,\" Daniel said. \"What?\"\n\n\"But, sir!\" She tried to catch his gaze, but he turned away, his focus on the dormant crystal.\n\n\"Not another word, Colonel. I'm going to do this.\"\n\nNo one moved. How could they? Leaving General O'Neill behind just wasn't an option.\n\n\"Do I need to remind you both of why we're here?\"\n\n\"There has to be another option.\"\n\n\"We don't have time for options, Carter.\"\n\nIn her eight years serving with SG-1, there'd always been time for options. \"What if we \u2014 ?\"\n\n\"Forget it, Carter.\" The general kept his back turned. \"Daniel, get your ass in gear.\"\n\n\"Jack...\"\n\n\"Just follow my orders, Daniel.\"\n\n\"I could do it, you know. I could stay behind. Even without the ATA gene, I should be able to... I mean, if the gate collapses, you could send word to the Tok'ra. Maybe the Asgard could pick me \u2014 \"\n\n\"Get it through your skull, Daniel. I don't want your help.\"\n\nDaniel jerked his head back, obviously stung by the general's words. \"Fine.\" Swooping up his pack, he stormed toward the doorway.\n\n\"Dr. Jackson!\" cried Huang.\n\nDaniel stopped.\n\nHuang held out the cup. \"You will want this back.\"\n\n\"Keep it.\" He looked over at the general who still refused to turn around.\n\n\"See you on the other side,\" Sam said.\n\n\"Right.\" With a final nod, Daniel ran from the chamber.\n\nSam refused to give up as easily. \"General, what if I \u2014 ?\"\n\n\"Negative, Colonel.\" He shoved up his jacket sleeves and motioned for Huang to step up to the platform. At first, the old man didn't move, glancing back and forth between her and the general. She understood his confusion. She was pretty much feeling that way herself.\n\nHuang looked down at the cup. \"I am ready.\" He stepped up beside the general.\n\n\"This will be a snap.\" General O'Neill turned back toward the platform. \"We're gonna pull this sucker out, Carter. Then the two of you will run like mad.\"\n\n# CHAPTER TWENTY-THREE\n\n### P3Y-702\/KUNLUN\/ELYSIUM PEDION\n\n### 19 AUG 04\/0700 HRS MCMURDO STATION\n\n### 19 AUG 04\/1400 HRS STARGATE COMMAND\n\n\" _I'm at the gate_,\" Daniel radioed in. \" _Dialing... Now_.\"\n\nThe _kawoosh_ sounded distorted, tinny. From Jack's position beside the platform, he could barely see the Stargate. It was too damn bright out there. A glint of sunlight off the distant hill had to serve as confirmation that the gate was active.\n\nHe thumbed his vest pocket radio. \"Roger that. Carter will give you the head's up when they head for the gate. Standby.\"\n\n\" _Good luck, Jack_.\"\n\n\"Yeah. You, too.\" He signed off.\n\nCarter was staring at her watch again. \"Five hours till sunrise at the outpost, sir.\"\n\nFor all of Skaara's fancy quantum talk about time, down here in the real world the clock kept ticking.\n\n\"Thank you, Colonel.\" He glanced at Huang. \"Think you can double-time it out of here, old man?\"\n\nHuang placed his precious cup on the platform's edge. \"I will do what is necessary.\"\n\nHe raised his chin with just a wee bit of indignation. Enough for Jack to be convinced they had a shot at doing what they'd come here to do.\n\n\"Carter, got your pack ready?\"\n\n\"Ready.\" She held up her pack, unzipped and ready to go.\n\nHe flashed her a smile, until it occurred to him that it could be the last time he'd get to do that.\n\nWhich sucked.\n\n_You're stalling, O'Neill_.\n\nHuang bellied up to the platform. Jack couldn't help looking at the smudged sigil on his forehead \u2014 Lord Yu's stamp at its finest. Part of him wanted to ask why the old man had done it, why he'd almost gotten them all killed. Why it was so important to bother with that asinine mark.\n\n_Still stalling_.\n\nHuang lifted his hands. \"I am ready.\"\n\nJack cleared his throat. Reached out toward the crystal. Skaara said this would be worse than the chair. Okay, fine. He could handle it.\n\n\"On my count,\" he told Huang. \"One. Two \u2014 \"\n\nHuang jerked his hands back.\n\n\"Damnit, Huang. What is it now?\"\n\n\"My Lord Yu.\" Huang turned his head toward the doorway. At first Jack thought he was looking at the active Stargate, but then he realized that wasn't it.\n\nHuang was looking at the statue up on the hill. \"He will never forgive me for helping you. He will hunt me down. He will murder my ex-wife.\"\n\nFor a moment, just one short moment, Jack found himself pitying the guy. \"I told you, he won't do anything of the sort.\"\n\nHuang turned toward him, stark desperation on his face. \"You do not know that.\"\n\nJack snorted. \"Actually, I do. Your buddy Yu and I had a nice chat after your little setup. Turns out he likes Earth, especially China. Go figure.\"\n\nHuang's gaze slid down to the cup. \"I did not know.\"\n\n\"Well, now you do.\" Jack rolled his shoulders. This hands-raised-in-the-air stuff was getting tiring. \"Can we get on with it?\"\n\nHuang nodded and raised his hands. \"To honor my cr\u00e8che-brother, Lao Dan Shi.\"\n\nJack reached out toward the crystal. \"One. Two. Three. Pull!\" He grabbed hold and \u2014\n\nThe city of Elysium Pedion flashed through his mind. Ancients gliding across walkways. Rooms filled with more of them, wearing long, beige lab coats. They tossed numbers and symbols through the air, as if playing catch with a baseball.\n\nJack shook the images from his head. Huang was tugging at the crystal right along with him. It wouldn't budge.\n\n\"Pull harder!\"\n\nA buzzing filled his ears.\n\n\"It will not come free!\" Huang shouted back.\n\nJack slid a palm around the back of the crystal. Shoved it toward him. Pulled it upwards. Nothing.\n\nHe placed his other palm across the base.\n\nSunlight sucked out of the room in the blink of an eye. An electric shock sizzled his muscles, burned through his bones. Ate at his insides.\n\nThe world went dark.\n\n_Thump. Thump-thump_.\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 19 AUG 04\/0715 HRS MCMURDO STATION\n\n### 19 AUG 04\/1415 HRS STARGATE COMMAND\n\n\" _Wo jiang xie zia wo bao fu. Wo jiang bu hai pa_.\"\n\nWeiyan Shi rolled her head from side-to-side, her face contorted in pain despite the Tau'ri's narcotics. She mumbled the same words again and again, and while Teal'c knew ascension would free her from her current agony, the waiting was difficult to witness.\n\n\" _Wo jiang... Wo jiang_.\"\n\n\"What is wrong with her?\" asked Ambassador Zhu, her weariness shrouded in an implacable mask.\n\n\"I cannot say.\" He crouched beside mother and daughter, his legs cramped, his muscles weary.\n\n\"Cannot or will not?\" Ambassador Zhu demanded. A half-choked sob escaped her lips.\n\nHe had asked the other SGC personnel to vacate the weapons chamber in respect. The ambassador's confrontational manner made him wish he'd done otherwise.\n\n\" _B\u00e0b\u00e0_ ,\" Weiyan whispered. \" _Wo jiang_.\"\n\nAmbassador Zhu's shoulders slumped.\n\n\"What is it she says?\"\n\n\"She calls for her father.\" The ambassador frowned. \"I am the one who is here. Why call for him? He has done nothing for her, nothing.\"\n\n\" _Wo jiang, B\u00e0b\u00e0. Wo jiang_.\" A great shudder ran through her. \" _Wo jiang_.\"\n\n\"What have I done?\" Ambassador Zhu whispered.\n\nA rattling reached Teal'c's ears from above. He glanced at the ladder permitting access to the outpost's main floor. The rungs shook.\n\nYet another tremor.\n\nFortunately, the vibrations subsided soon after they'd begun. The replacement of the control crystal could not come soon enough. Teal'c spared a moment to silently wish his comrades a quick success. For both their sake and that of Earth.\n\n\" _Wo jiang, B\u00e0b\u00e0. Wo jiang_.\"\n\n\"I have failed.\"\n\nTeal'c returned his gaze to Ambassador Zhu, an eyebrow raised in question.\"Ambassador Zhu, you cannot fail as Weiyan Shi's mother. She is your child by birth, is that not so?\"\n\n\"Of course she is.\" A defensive response, but Teal'c heard the truth in the woman's words.\n\n\" _Ni shi shui_?\" Weiyan mumbled. \" _Wo meng_?\"\n\n\"Have you told her she is loved?\"\n\n\"I \u2014 I have not.\" Sniffling, she raised her head and looked back at him. \"Not for many years. When the doctors discovered she had hemophilia, that her needs would outstrip her abilities, I could not hide my disappointment.\"\n\nA strained laugh erupted from her mouth. \"Huang abandoned us before she was born, before the doctors presented their diagnosis. He left me alone to care for her. Alone to...\"\n\nShe pressed her daughter's head to her chest.\n\nIt was then Teal'c understood. How could Weiyan Shi move on if her mother would not allow her to do so? It had been Ambassador Zhu's harsh words that had caused her child to lose control in the weapons chair. Now, it was the mother's need to love her daughter that kept Weiyan Shi here.\n\n\"Tell her of your love,\" Teal'c commanded.\n\n\"I do not know how.\"\n\nHe thought of Rya'c, of the anger in his son's eyes upon the death of Drey'auc, his mother. The road to redemption between father and son had been difficult, but it had begun with one simple truth.\n\n\"Ask for forgiveness,\" he told Ambassador Zhu.\n\n\" _Muqin_?\" Weiyan Shi's eyes opened. \"Mother, don't leave me. I am sorry. I \u2014 Please, do not be angry.\"\n\n\"Shhhh, little one.\" Ambassador Zhu stroked her daughter's brow. \"Of course I am here. There is no place I would rather be than with my child.\"\n\n\"...taken you from your duties. Ahhh,\" Weiyan Shi moaned, \"Forgive me my fears.\"\n\nTeal'c took her hand. \"You need not fear, Weiyan Shi. A great journey awaits you.\"\n\n\"I will try, Teal'c.\"\n\n\"No, Weiyan Shi. You will not try. You will do.\"\n\n\"I will do.\" She squeezed his hand in return. \"Mama?\"\n\n\"I am here,\" replied Ambassador Zhu.\n\n\"Forgive us both,\" Weiyan rasped.\n\n\"Both?\" The ambassador's eyes widened. \"Weiyan, it is I who must seek your \u2014 \"\n\n\"Father... Please forgive him, too.\"\n\nAmbassador Zhu nodded. \" _Du\u00ec b\u00f9 q\u012d. Yu\u00e1nli\u00e0ng_.\"\n\nA brief smile passed over Weiyan Shi's face. She released Teal'c's hand and raised it to touch her mother's face. \"I love you, too, Mama.\"\n\nHer eyes rolled back and she sagged. \" _Wo jiang bu hai pa_ ,\" she whispered.\n\nThe ambassador translated. \"I will not fear.\"\n\n\" _Wo jiang xie zia wo bao fu_.\"\n\n\"I will release my burden.\"\n\n\"Now, Quing Zhu. You must let her go.\"\n\nThe ambassador began to back away, but he stopped her. \"Not like that.\" He reached out and touched her shoulder. \"Within you. Grant her what it is she needs to depart.\"\n\nAmbassador Zhu bent down and whispered into Weiyan Shi's ear, \"I will always love you, child of my heart.\"\n\nA great sigh escaped Weiyan's lips. For a moment, she lay still, and then said, \" _Wo jiang xie xia wo de bao fu_.\"\n\n\"I will release my burden,\" Ambassador Zhu repeated.\n\n\" _Wo jiang bu hai pa. Wo jing zou jin guang mang_.\"\n\n\"I will not fear. I will go into the light.\"\n\nWeiyan's hands dropped to her sides, her death apparent.\n\n\"Is that it?\" Ambassador Zhu demanded. \"You promised \u2014 \"\n\n\"Wait.\"\n\nA pure white radiant blaze of light rushed upwards from Weiyan Shi's body. Though Teal'c had seen the process of ascension twice before, he could not help but bow his head in reverence.\n\n\"She is beautiful!\" exclaimed Ambassador Zhu.\n\nTeal'c lifted his head to gaze upon the light. The glowing form hovered above them, energy surging within its nebulous mass. Two tendrils unfurled from within its center, one extending to the ambassador, the other toward him.\n\n\"Goodbye, Weiyan Shi,\" he whispered.\n\nThe tendrils pulled back. \"Live well, Teal'c.\"\n\nShe shot upwards through the ceiling and disappeared.\n\n### PLANET DESIGNATION: UNKNOWN\n\n### STATUS: UNKNOWN\n\n### TIME: UNKNOWN...UNKNOWN...UNKNOWN...\n\nHe stumbled across darkness, through a rippling pool, and out into a pool of white light.\n\nWhere the hell was he, for crying out loud?\n\nHe squeezed his eyes shut. That begged a larger question.\n\nWho the hell was he? For the life of him, he couldn't remember his name.\n\n_Thump. Thump-thump_.\n\nEyes closed, he felt warmth. Clean air. Complete stillness. No sounds of animals or civilization, just lapping water. Directly behind him. He opened his eyes. It was too bright to see much of anything.\n\nHe whirled around, coming face-to-face with a vertical stone ring at least three times his height. The ring loomed over him, its center filled with a bluish puddle of water.\n\nHe found the massive thing familiar. Almost comforting in an odd sort of way.\n\nAs if he and the ring were old friends.\n\n_Thump. Thump-thump_.\n\nHe'd been here before.\n\n_Thump. Thump-thump_.\n\nHe knew that sound.\n\n_Thump_.\n\nHe knew his name. He remembered what had happened.\n\nJack turned his back on the Stargate. Peered into the over-bright white world around him.\n\nThe crystal. He'd pulled it and now...\n\n_Thump. Thump-thump. Thump-thump_.\n\nHis heart pounded in his chest. He got it. Why did it have to be so frigging loud?\n\nHe raised an arm, expecting to find the crystal in his hand. Instead, a glowing-white strand wavered into view. He raised his other arm. Another shimmering strand floated up.\n\nHe glanced down at his body. No boots. No legs.\n\n_Thump-thump_.\n\nNo nothing.\n\n\"Crap.\" Something must've gone wrong. When he'd pulled the crystal, he must've gotten himself killed.\n\nHe groaned. \"Please don't tell me I've ascended.\"\n\nWaving his hands in front of his face did squat. More strands slithered back and forth. Like a bunch of ribbons blowing in the breeze.\n\nBright, shiny ribbons. Almost mesmerizing.\n\nAlmost peaceful.\n\n_Thump. Thump-thump_.\n\nWould ascension be so bad?\n\nNo more casualty reports. No more sending good men and women to their deaths. No more sitting back and feeling useless while those same men and women became nothing more than a line on a piece of paper.\n\nIf he'd ascended, maybe he could finish what Daniel had set out to do. Maybe he could go kick some Goa'uld butt. Wipe them out before Oma and her gang caught up with him.\n\nKeep Earth safe.\n\n_Thump. Thump-thump_.\n\nHell, maybe he'd give himself that peace of mind he always wanted.\n\n\"Is this what you truly wish, O'Neill? To ascend?\"\n\nJack recognized the voice. \"Skaara?\"\n\nAnother cluster of ribbony strands blossomed from out of nowhere. The strands darkened. Solidified.\n\nSkaara gazed at him, patiently. Calm. For all the kid's youth before he'd taken a walk on the Oma-side, it was Jack who suddenly felt very young. And very foolish.\n\n_Thump. Thump-thump_.\n\nSkaara's white robes flapped in a breeze Jack couldn't feel.\n\nWait a moment.\n\nJack glanced down at where his body should've been. With relief, he took in the sight of his green BDUs, his black t-shirt, his black boots.\n\n\"That's funny,\" he said. \"I could've sworn I wore camo when I got here.\"\n\nSkaara raised an eyebrow.\n\nJack explained. \"You know, beige and tan camouflage. Nice way to blend in with the sand and all that.\"\n\n\"Blend in.\" Skaara gestured toward Jack's green BDUs. \"This is how you see yourself.\"\n\nJack shrugged. \"I always said they'd bury me in my BDUs.\"\n\n\"You are not dead, O'Neill.\"\n\n\"Yeah, I didn't think as much.\" His eyes rested on the Stargate, its event horizon thwaping against the ring's stony sides. \"But this is it, right? I'm gonna ascend?\"\n\n\"If that is your wish. What about Earth?\"\n\nThat was the first thing he'd take care of. Quick jaunt to Antarctica. Shut down that device before it ripped his favorite planet in two, then onward and upward.\n\n_Watch out, you slimy Goa'uld bastards, I'm coming_.\n\n\"How about an orientation tour?\" he asked Skaara.\n\n\"Ascended, you will see the universe, but, O'Neill,\" Skaara stepped closer, \"you cannot interfere. You can do nothing.\" He tilted his head. \"But that is not who you are.\"\n\n_Thump. Thump-thump_.\n\n\"I don't know, Skaara. Maybe...\" He sighed. \"Maybe I \u2014 \"\n\nThwap.\n\nA human strode through the gate. It was Huang \u2014 Sorta. The old man looked younger, more like when Jack first met him. His black hair had grown back in. His beard was neat, trimmed. Barely gray. He wore classic Jaffa armor, a gray cloak hanging from his left shoulder. A jade pin shaped like a dragon held the cloak in place.\n\nAnd on his forehead was Lord Yu's sigil etched in gold. Just like a First Prime.\n\n_Thump. Thump-thump_.\n\nHuang glanced at Skaara. \"You. I recognize you.\"\n\nSkaara dipped his head in greeting. \"It is good to see you again, Huang Sun Tzu.\"\n\nSun Tzu? It took a moment for Jack to remember this guy was a clone. A copy. A several thousand year-old Xerox of the real McCoy.\n\nHuang peered into the void. He inhaled deeply. \"There is a tranquility here. A solace I have not felt for many years. Not since...\"\n\n\"Antarctica?\" Skaara stepped up beside Jack.\n\nHuang turned to face them. As he did, the lines on his face smoothed out. Within moments, Jack realized he was looking at a younger, maybe twenty year-old version of Huang.\n\n\"I was not happy in Antarctica, but, I came to understand its power. Nature unbridled. Free from expectation. Hope. The land's nature was relentless. Unforgiving, and yet, it was honest. Truthful.\"\n\n_Thump. Thump-thump_.\n\nHuang ran a finger along the gold brand on his forehead. \"At the time, I did not recognize that truth.\"\n\nJack scowled. \"What would you know about truth? Your lies almost got my people killed \u2014 \"\n\n\"O'Neill,\" warned Skaara. \"To ascend requires giving up one's burdens. Make peace with your past. Let it go.\"\n\nLet it go. Jack dropped his head. Could he do that? For the first time in a long while, he didn't know what to do. Usually, he'd just go by his gut. Do the right thing. But now? Could he let Daniel, Carter, and Teal'c just die? Let Earth suffer?\n\n_Thump. Thump-thump_.\n\nA hand squeezed his shoulder. Huang. \"'Great is the hero who would die for their country, but would much rather live.'\"\n\nJack sighed. He was tired of all the quotes, all the brouhaha of burdens. Of letting go.\n\n_Thump. Thump-thump_.\n\nHe was worn out, period.\n\n\"Can we bypass the speeches and just move things along?\" He needed to get to Earth. Try and stop that device before things got any worse.\n\n\"Sun Tzu's quote was meant for you, O'Neill.\" Huang dropped his hand from Jack's shoulder. \"It is you who must promise to heed his words. You must live. Lead. If not for yourself, then for others.\n\n\"I was never meant to do so.\" Huang bowed toward the waiting Skaara. \"I was never really meant to be.\"\n\nCalm settled over the man's face. A calm Jack envied.\n\n\"So that's it,\" Jack said, surprised at how bitter his words sounded. \"I'm supposed to go back there, grab the crystal and then just leave you behind? Well, what if I want to be the one left behind? What if \u2014 ?\" He turned away, disgusted. Not with them. With himself. He was so tired. Tired of losing. Tired of winning.\n\nTired of fighting.\n\nClink. Snap. Clink. Snap.\n\n\"Still playing with the Zippo, Skaara?\"\n\n\"Do you want it back, O'Neill?\"\n\n\"No.\" Jack kept his gaze on the void ahead of him. \"I told you years ago, keep it. It's yours.\"\n\n\"And yet, you were drawn to retrieve the lighter when I left.\" Clink. Snap.\n\n\"I don't know, Skaara. Maybe... Maybe I just hated to see it go to waste.\"\n\n\"Do you know why?\"\n\nJack turned back around to face his old friend. Huang was gone. \"Because I couldn't let go.\"\n\nSkaara held out the lighter. \"Hold on to your life, O'Neill. I promise, it will be worth living. For you, and the many you will touch.\"\n\n\"If you say so.\" Jack had heard the same rhetoric from Skaara before, mere hours after Abydos and everyone on it had ascended.\n\n\"You do not believe me?\" Skaara held out the lighter.\n\n_Thump. Thump-thump_.\n\n\"Then believe in yourself.\"\n\nJack took the lighter. In the white light of the void, it felt solid. Real. A bit of metal and flint that had become an anchor in his upside-down life.\n\nHe wrapped his fingers around the metal casing.\n\nThe void went black.\n\n### P3Y-702\/KUNLUN\/ELYSIUM PEDION\n\n### 19 AUG 04\/0720 HRS MCMURDO STATION\n\n### 19 AUG 04\/1420 HRS STARGATE COMMAND\n\nThere was no sound. No movement. No change.\n\nThe platform's translucent panels stayed fixed in a constant blue. If it wasn't for the breeze blowing across Sam's neck, she'd swear she was looking at a photograph. General O'Neill and Huang appeared frozen, their hands wrapped around the control crystal, their knuckles white as if in some sort of death grip.\n\nIn the past five minutes, neither man had moved. From where she stood by the doorway, it didn't even look like they were breathing. In between their hands, she could see the control crystal emit a steady red glow.\n\nThat could be a good or bad. She had no way of knowing. Her scanner wasn't of any use, the needle unmoving from its basically neutral position.\n\nThe breeze picked up. Hot air blew through the chamber, sending sand and dirt in a swirl across the silver and marble floor. She checked her watch again. Ten minutes.\n\n\"Sir?\"\n\nGeneral O'Neill didn't even blink.\n\n\"General!\"\n\nNothing.\n\nThe crystal was active. Their job was done.\n\nWhy wasn't there any seismic activity, and more importantly, why weren't they moving?\n\nShe waved a hand in front of the general's face. He didn't flinch.\n\nThe needle wobbled.\n\n\"General?\"\n\nThe needle pinged. Hard and to the right.\n\n\"Huang?\"\n\nBeneath Sam's boots, she felt a low rumble. Getting louder and stronger by the moment.\n\nA column crashed behind the platform.\n\nThe men remained frozen. Almost as if they were in the grips of some sort of stasis.\n\nAnother column crashed. Pulverized stone and a millennium of dust sprayed across the chamber.\n\nShould she pull them off the crystal? Shake them awake?\n\nSam hesitated, knowing she had to do something and do it fast, but the question was what? The wrong move could crack the ground open, swallow the gate, and leave them stranded.\n\nAnd Earth still in trouble.\n\nThe tremor grew stronger. Debris and stone dust rained down from the ceiling.\n\nShe could leave now, bug out before the gate was lost. Maybe find another way to stop the terra-forming device in Antarctica. She looked at General O'Neill and wondered, could she do that? Could she leave him behind?\n\n\"You said to trust my gut.\" She grabbed his arm and pulled.\n\nNothing.\n\n\"Jack!\"\n\nHis elbow jerked. She stumbled backwards from the sudden movement.\n\n\"Carter?\" He turned toward her, the crystal in his hands.\n\n\"Sir! The bracket.\" She climbed to her feet and ran around the platform to face him.\n\nHuang shoved the general aside. \"Hurry, O'Neill.\" He plunged his left hand into the bracket.\n\nThe shaking subsided. While she could still feel it beneath her, the earlier roar had lessened to a sound like distant thunder.\n\n\"I don't understand,\" she told the general. \"I thought \u2014 \"\n\n\"Leave!\" Huang insisted.\n\n\"Sir?\"\n\nGeneral O'Neill kept his gaze on Huang. \"It's what he wants.\"\n\n\"Do not lie, O'Neill. It is what we both want.\" He bent over the platform, his face twisted in pain.\n\nA thin stream of light erupted from his back. Huang shot back up and Sam gasped.\n\nThe man's face was covered in dozens of light particles swimming across his skin. The particles expanded. Moved down through his arms, his chest, his legs.\n\n\"His body's de-molecularizing,\" she said in awe. \"Breaking down to the quantum level right before our eyes.\"\n\n\"We have to go, Carter.\" General O'Neill grabbed his pack from the ground and stuffed the crystal inside.\n\nSam looked again at Huang \u2014 or what was left of him. Most of his body was covered in agitated light particles, swarming around each other in a state of flux. All she could see was his left arm shoved down in the bracket, a corner of his mouth, a bit of forehead still covered in dirt. His brown eyes widened, not in fear, but in astonishment.\n\nThe general slung the pack over his shoulder and swooped up his P90. He ran for the doorway. \"Now, Carter!\"\n\nSam broke eye contact and raced after the general.\n\n\"O'Neill!\" Huang shouted.\n\nHe stopped at the doorway.\n\n\"Do not break your promise!\"\n\nSam whirled toward General O'Neill, his haggard face giving her no clue as to what Huang meant. He glanced at Huang and then back at Sam.\n\n\"Sir?\"\n\n\"Come on, Carter. Let's go home.\"\n\nWithin the darkness sparked a single flame. A candle's light never meant to shine upon the shores of man.\n\nBetween one moment and the next, another current winnowed across the ever-present stream. He had returned, his youthful energy sending ripples across dimensions she'd yet to share.\n\nAnd share she would. She knew now that his yearning for companionship had nearly caused yet another tear in the fabric.\n\nAn unanchored boat will only drift.\n\nHis return meant yet another repair accomplished beyond the Others' view. If they discovered what had been done, she would be banished, having broken their highest law. Interference in the lower planes was forbidden.\n\nEven if much of what occurred had been their doing.\n\nOr lack thereof.\n\n\"Are things as they should be?\" she asked.\n\n\"See for yourself,\" Skaara answered.\n\nGalaxies were born. Suns imploded. Black holes plunged into nothing.\n\nA new flame burst into being. \"Weiyan?\"\n\nAnd then another. \"Father?\"\n\nJoy. Rapture. Twin comets streaked beyond a nebula. They became one. Broke apart and spiraled beyond a moon.\n\nSoon she must teach them both what it meant to be ascended, but for now... For now she would allow father and daughter this moment of reunion.\n\n\"And the device?\" she asked Skaara. \"You are certain?\"\n\n\"He will do the right thing, Oma. He always does.\"\n\n### MARBLE POINT AIRSTRIP, ANTARCTICA\n\n### 19 AUG 04\/1130 HRS MCMURDO STATION\n\nDaniel sat in the rear of the chopper with Jack and Sam, fighting back a yawn. They took off from Marble Point and headed for the outpost. A golden-yellow smudge appeared on the horizon, the first signs of sunrise's imminent approach. Jack and Sam looked equally worn out. Especially Jack. He kept his eyes front, his mouth shut, the pack containing the crystal secured to his lap.\n\nWhatever had happened on P3Y-702, Jack wasn't talking about it. Not even to Sam. Once they'd returned through the gate, and ridden to Petersen, Daniel had offered to fly back with Jack so he'd have someone to talk to. Someone to help him work through why Huang had stayed behind instead.\n\nJack had insisted Daniel fly with Sam. As the F-302 flew south, she'd explained to Daniel what she'd seen, but even she had no idea why Huang had stayed. And while her description of Huang's de-molecularization sounded extraordinary, Daniel had no way to know what it meant.\n\nThe smudge on the horizon spread, becoming a thin line of yellow against the gray predawn. He checked his watch. Thirty-five minutes till sunrise. \"We're sure this is going to work?\"\n\n\"If Skaara says it will,\" Jack said, \"then it will.\"\n\n\"Okay.\" Daniel knew Jack could be blunt when the need arose. \"I'd like to stay for a few weeks, see if there are any inscriptions around the device.\"\n\nSam grinned. \"I'd like to as well, sir. The technology might come in handy in \u2014 \"\n\n\"Will it help us fight the Goa'uld?\"\n\n\"I don't how, sir, but I'd like to think we could figure out a way to harness its abilities for use on Earth.\"\n\n\"Can you copy the Ancients' doohickeys on your own?\"\n\nSam's eyes widened. \"No, sir. That would require a level of skill that \u2014 \"\n\n\"Daniel, don't you have a treaty to sort out?\"\n\n\"Paul Davis can work that out.\" Daniel gestured toward the bag in Jack's lap. \"What you're carrying there, and how that platform fits into what happened to the Ancients, don't you think it's worth \u2014 ?\"\n\n\"I'm tired of thinking.\" Jack closed his eyes.\n\n### ANCIENT OUTPOST, ANTARCTICA\n\n### 19 AUG 04\/1140 HRS MCMURDO STATION\n\nDaniel groaned when he heard they'd have to take the emergency stairs down to the outpost. It made sense, given the on-going tremors, but still...\n\nHalfway down the spiral staircase, Daniel quit complaining. Another tremor hit. The steps shook, chunks of ice plummeting down on their heads. He stopped, gripped the railing, and prayed it would be over soon.\n\nJack didn't say a word, he just kept going. Jumping over fallen ice, taking two steps at a time. One hand on the railing, Daniel followed with Sam bringing up the rear. Jack's single-mindedness had its advantages, Daniel knew that, but it had gotten them into as much trouble as out of it in the past. He wasn't sure his friend's determination would be enough to command the SGC for the long haul. Especially if Jack seemed so miserable doing it.\n\nThe quake ceased as they hit the bottom floor. Jack kicked open the door. General Hammond stood beside the stasis unit that had once held Jack in its grip. Ancient repositories, Goa'uld sarcophagi, stasis units... For all the insight gained from these unimaginable technologies, Daniel couldn't deny the cost.\n\nEspecially for Jack.\n\n\"Welcome back, SG-1.\" General Hammond turned toward Jack with a wary smile. \"You ready, son?\"\n\nJack unzipped the pack and pulled out the crystal. \"Ready isn't quite the word, sir.\" He tossed the pack at Sam and bolted through the archway. She raced in after him.\n\nDaniel hurried over to Hammond. \"Weiyan?\"\n\n\"She's gone, son.\" The general gestured for him to follow. \"And I can't say if it's a good thing or a bad one.\"\n\nAnother tremor shook the room.\n\nSam had gotten to the ladder first. Teal'c joined her in holding the rungs as Jack swung on, one hand holding the crystal against his side.\n\n\"Daniel! Get your ass over here and help me out.\"\n\nRunning into the chamber, Daniel got a glimpse of Paul Davis and Bill Lee off to one side. Ambassador Zhu stood between them, a fleece pullover pressed to her chest.\n\nWeiyan's pullover.\n\nKnowing there wasn't time for condolences, Daniel gave Zhu a quick nod and slid down the ladder to the terra-forming platform below. The force field hugged the walls, undulating as if it were welcoming them back. The crystal cradled under one arm, Jack swept away a good foot of snow piled over the brackets. Daniel joined him, clearing the central red bracket of any debris.\n\n\"Fifteen minutes, sir,\" came Sam's warning from up top.\n\n\"Right.\" Jack stepped in front of the red bracket, his face screwed up in concentration. He raised the crystal and \u2014\n\nThe ground shook, sending a spray of ice and dirt down from the hole's edges. A chunk fell on Daniel, hard. He pressed a hand to his head.\n\n\"You all right?\" Jack asked.\n\n\"Am I all right?\" Daniel pulled away his hand. \"Shouldn't you, you know...?\"\n\nJack glanced at the crystal. \"Yeah. I should.\" He lowered the crystal into the bracket.\n\nThe platform lit up, its blue panels thrumming loudly. Like the sound of a heartbeat, Daniel realized.\n\nThe quake stopped.\n\nDaniel blew out a long breath. \"Well, that's \u2014 \"\n\nJack's hands were still on the crystal.\n\n\"The tremor's stopped. You can let go now.\"\n\n\"Exactly.\" Jack closed his eyes, his hands splayed across the crystal's top-most facet.\n\n\"Jack?\"\n\nThe crystal lit up, its glow casting a red light across Jack's face. The thrum became louder, quicker. A racing thump loud enough to echo through the room.\n\n\"That's enough,\" Jack mumbled. He slid his hands down the sides of the crystal and sparks flew from the platform. Daniel jumped back, but Jack held on, his brow creased, his eyes squeezed shut. The light within the crystal turned darker red.\n\n\"What are you doing?\"\n\nThe thrumming stopped. The illuminated panels dimmed and then went dark. The only light left was from above.\n\nJack stepped away from the platform, sweat running down his forehead. \"It's dead, Daniel. Finished. No more threats to Earth.\"\n\n\"What? It's not that simple. You heard Sam.\" The charred crystal remains collapsed into dust. Stunned, he whirled around. \"We could've learned how to harness its energy. We could've learned how to \u2014 \"\n\n\"And risk ending up just like those dead and gone Ancients?\" Jack ran a forearm across his wet forehead. \"If you don't mind, I'd like to live a little longer.\"\n\n# CODA\n\n### FERRAR GLACIER, ANTARCTICA\n\n### 19 AUG 04\/1335 HRS MCMURDO STATION\n\nSunlight peeked out from behind a patch of clouds as Jack hiked past the waiting chopper. Though he'd donned a wool cap along with the rest of his cold-weather gear, his parka hood was down. The cold air felt good nicking at his skin, keeping him from feeling completely numb and worn out.\n\nHe reached the rocky promontory beyond the chopper pad and looked out over the valley below. The crunch of footsteps behind him was no surprise. No doubt his former teammates had decided to come along for the show, short as it might be.\n\nLooking out over the Ferrar Glacier, he took in the rugged white valley, surrounding cliffs, the deep crevasses that looked like someone had squeezed blue gel toothpaste onto the snow. Somehow, Huang had survived all this. Maybe Skaara had helped, but Jack suspected the clone's basic cussedness had more to do with it than anything else. Cussedness and maybe something else. Something everyone needed.\n\nFaith.\n\nIn Huang's case, that faith had been sorely misplaced.\n\nIn Jack's case, it was about finding that faith in himself.\n\nHe grinned. He could do that.\n\n\"O'Neill.\" Teal'c's parka hood was zipped up tight enough that Jack couldn't see half his face.\n\n\"I thought you hated cold weather.\"\n\nTeal'c turned toward the sun, its zenith past and gone. \"Upon our return to the SGC, I would like to visit my son. He is with Ishta and the Hak'tyl.\"\n\n\"Warmer climate on that planet, I suppose?\"\n\n\"Indeed.\"\n\n\"Wow.\" Daniel stopped beside him with Carter.\n\n\"Bring back memories, Carter?\" He knew this had been her first view of Antarctica years ago when they'd been flung from the previously unknown gate into what they thought was an ice planet.\n\n\"Lots of memories, sir. Not all of them bad.\" Her eyes crinkled. Underneath all that fleece and polyester, Jack was sure she was grinning. That was Carter. Always up for a challenge.\n\n\"...Zhu's agreed to revisit the treaty,\" Davis said, matching strides with Hammond as they headed over.\n\n\"Considering all she's been through,\" Hammond said, \"I'd take that as a win, Major.\"\n\n\"I think so, too, sir.\"\n\nThe two came up on Jack's other side. He gave them each a nod and turned his attention back to watching the sun peek in and out of the clouds.\n\n\"First day of sun for the year,\" Carter said. \"It's supposed to last an hour and thirty-nine minutes. Should set in another eight \u2014 \"\n\n\"Ack!\" Jack said, waving a glove-covered hand. \"Don't ruin it. Just watch. Enjoy.\"\n\n\"Sir?\"\n\n\"When's the last time we just watched the sun set, Carter?\"\n\nThe group settled down and Jack let the peace and quiet wash over him. That was the best part about so much snow. It made everything quieter. Newer.\n\nSimpler.\n\n\"General Hammond?\" Davis asked.\n\nJack stopped himself from groaning out loud. If he wanted peace and quiet, there was always the cabin.\n\n\"Sir,\" Davis said, \"About that promotion \u2014 \"\n\n\"Promotion?\" Jack congratulated the major. \"About time, Davis. We were starting to \u2014 \"\n\n\"Actually, General \u2014 Generals, I mean \u2014 \" Davis shoved his gloved hands in his pockets. \"I think I can do more good at my present rank. I know I'm not critical to the SGC, but I'd like to think I've done my part. Both with the treaty, and, well...\" He shrugged.\n\nJack had heard about Davis' little brainstorm getting the F-302s out of the ice. \"If I may, General, I'm with Davis. If he's happy, how about letting those forced promotion regulations slide for once?\"\n\nHammond nodded. \"Happy and helpful. Can't ask for more than that, I suppose. I'll notify the Secretary of the Air Force. I'm sure special dispensation can be granted. For now.\"\n\nThe world back on its keel, Jack turned toward the glacier. \"Out of curiosity, General, how'd you get China to release Huang into our custody?\"\n\nA muffled sigh escaped Hammond's extra-extra large parka. \"The president promised to consider giving them the plans to the F-302.\"\n\n\"Great.\" Jack remembered Ambassador Chen's admiring the bird during his brief visit. There'd definitely been something downright paternalistic about the way the guy had stroked its underbelly. \"Good luck with that.\"\n\nCarter frowned. \"What about the 303, sir?\"\n\n\"They want the plans to that as well.\"\n\n\"They're gonna build their own 303?\" Jack watched the sun dip lower in the sky. \"With what resources?\"\n\nHammond grunted. \"Remember, the president hasn't agreed yet. He's just said he'll consider their request. As far as resources are concerned, I'll leave that problem up to my successor.\"\n\nSuccessor? Jack felt his eyebrows shoot up into his scalp. \"General, is there something you've neglected to tell us?\"\n\n\"I hear tell that there's a great fishing spot in Maryland, just some forty minutes from the Pentagon.\" Hammond slapped him on the back. \"If you'll excuse me, I'm going to head over to McMurdo and help coordinate the relief effort. Davis?\"\n\nJack did a double take as Hammond trudged off toward the chopper with Davis in tow.\n\nFishing. In Maryland.\n\nHe turned back toward the valley, the darkening sky turning the white snow to a soft gray. A sparse slice of the sun hung just above a mountain off in the distance. A moment later, a bit of smoke wafted upwards. Jack realized it wasn't just any mountain he was looking at, it was Mt. Erebus.\n\n\"If the Chinese do build a 303,\" Carter said, \"I wonder what they'll call it.\"\n\n\"The _Sun Tzu_ 's a good name.\"\n\nAs the sun dropped behind the volcano, a breeze picked up. Cold. Bracing.\n\nJack pulled up his hood.\n\nYep. It was good to be alive.\n\n# Thanks to\n\nThe Polar Star's fine crew & expedition team for opening my eyes to the wonders of Antarctica. Special thanks to Danny Edmunds, Hannah Lawson, and Pierre & Wendy Malan--all who went beyond the call of duty to answer even my most outrageous of questions about the fragile continent's magnificence.\n\nCarey Adams, Mark Biggs, and all of Missouri State University. Their encouragement allowed me to help the reader experience the coldest, driest, and yet most powerful place on Earth.\n\nJen Brooks, Laura Gerling, Rhonda Mason, and Susannah Sinard whose support and feedback kept my hand securely on the tiller.\n\nSally Malcolm and Tom Reeve for their belief in this project.\n\nDavid Read, Lindsey Allen, Marian Trupiano, Kristin Muzina, and Maureen Bensa for sharing my exuberance for this much loved franchise.\n\nCapt. Angela Webb and the United States Air Force for their technical advice and steadfast dedication to our country.\n\nJoy Anne Baker whose Stargate Glyph font set came in super handy in completing this project. And a special thanks to the real-world Weiyan who provided the Chinese translations.\n\nGateworld.net for providing the ultimate guide to all things Stargate and for their continued friendship.\n\nThe cast & crew of SG-1 \u2014 especially Joel Goldsmith whose music lives on in the hearts of all Stargate fans.\n\nTo my family, for their patience with my obsessive need to explore the bottom of the world.\n\nThe great polar explorers Roald Amundsen, Robert F. Scott, and Ernest Shackleton whose Herculean efforts to reach the 'the end of the axis upon which this great round ball turns' continue to inform and inspire humanity at every turn.\n\n## THE ADVENTURES CONTINUE....\n\nMore STARGATE SG-1, STARGATE ATLANTIS and STARGATE UNIVERSE novels are available for your Kindle. Based on the hit TV series from MGM, the novels capture all the excitement and drama of the TV shows that we have come to know and love.\n\nStargate Kindle books UK\n\nStargate Kindle books US\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## The Heart of Devin MacKade\n\nThe MacKade Brothers Series\n\nBook Three\n\n## Nora Roberts\n\n##\n\nDevin MacKade knew it was his destiny to serve and protect the small town of Antietam, Maryland. And he always suspected his future should have little Cassie Connor in it. After Cassie married the wrong man, Devin tried to convince himself there would be other women, other loves. Now, after Cassie's divorce, Devin can finally follow his heart. But can Cassie follow hers?\nFor those who follow their hearts\n\n## Contents\n\nPrologue\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\n## Prologue\n\nDevin MacKade considered the age of twenty to be an awkward time in the life of a man. It was old enough for him to be considered responsible for actions and deeds, old enough for him to make a living or love a woman. Yet in the eyes of the law it was not quite old enough for him to be considered fully adult.\n\nHe was glad it would only take twelve months to get through it.\n\nBeing the third of four brothers, he'd already watched Jared and Rafe move beyond him into adulthood, and Shane was not far behind him. It wasn't that he was in a hurry, really. He was enjoying his time and his life, but Devin had begun, in his methodical way, to make plans for what would be.\n\nThe little town of Antietam, Maryland, would have been surprised to know that he had decided to uphold the law, rather than break it. Or bend it.\n\nHis mother had pushed him into college, true, but once he arrived, Devin had decided to enjoy it. The courses in administration of justice, criminology, sociology, fascinated him. How rules were made, why, how they were upheld. It had seemed almost from the beginning that those books, those words, those ideals, had just been waiting for him to discover them.\n\nSo, in his thoughtful way, he had decided to become a cop.\n\nIt wasn't something he wanted to share with his family just yet. His brothers would rag him, undoubtedly. Even Jared, who was already on his way to becoming a lawyer, would show no mercy. It wasn't something he minded. Devin knew he could hold his own with all three of his brothers, be it with words or fists. But for now, it was a personal agenda, and he wasn't talking.\n\nHe was aware that not everything you wanted, deep inside, worked out. There was proof of that right here in Ed's Caf\u00e9, where he and his brothers were grabbing a quick meal before heading to Duff's Tavern to shoot pool. Yes, the proof was right here, serving him the blue plate special, flushing shyly at Rafe's easy teasing.\n\nFive foot two, barely a hundred pounds, as delicate and fragile as a rosebud. Angel hair like a curling halo around a face that was all quiet gray eyes. A nose that tipped up just the tiniest bit at the end. The prettiest mouth in the county, with its deep dip in the top lip. Like a doll's. Small hands that he knew could juggle plates and coffeepots and glasses with a studied competence.\n\nHands that carried a ring with a chip of a diamond barely big enough to glint on the third finger.\n\nHer name was Cassandra Connor, and it seemed he'd loved her forever. Surely he'd known her forever, watched her grow up with a flicker of interest that had become a full-blown crush he'd considered too embarrassing to act on.\n\nAnd that was the problem. By the time he decided to act, he'd been too late. Joe Dolin had already claimed her. They would be married in June, just two weeks after she graduated from high school.\n\nAnd there was nothing he could do about it.\n\nHe made sure not to watch her walk away from their booth. His brothers had sharp eyes and he would never be able to tolerate being teased about something as intimate and humiliating as unrequited love.\n\nSo he looked out the window at Main Street. That, he thought, was something he could do something about. One day he would give something back to the town that had been such an intricate and important part of his life. One day he would serve and protect here. It was his destiny. He could feel it.\n\nThe way he sometimes felt, in dreams, that he had done so before\u2014or tried, when the town was ravaged by war, split and frayed by divided loyalties. In dreams, he could see it the way it had been, the way it was in those old Civil War photos. Stone houses and churches, horses and carriages. Sometimes he could almost hear the men gathering on corners or in the barbershop, discussing the War between the States.\n\nOf course, he thought with cool rationality, the town, or parts of it, were haunted. The old Barlow place on the hill just outside of town, the woods, his own home, the fields he helped plow and plant every spring. There were echoes there of lives and deaths, of hopes and fears.\n\nA man had only to listen to hear.\n\n\"Almost as good as Mom's.\" Shane shoveled mashed potatoes into his mouth, and the MacKade dimple flashed as he grinned. \"Almost. What do you figure women do on their night out?\"\n\n\"Gossip.\" His plate clean, Rafe leaned back and lit a cigarette. \"What else?\"\n\n\"Mom's entitled,\" Jared commented.\n\n\"Didn't say she wasn't. Old lady Metz is probably giving her an earful about us right now, though.\" Rafe grinned wickedly at that thought, and at the knowledge that his mother could handle even the formidable Mrs. Metz with one arm tied behind her back.\n\nDevin looked away from his view of Main Street, back at his brother. \"We do anything lately?\"\n\nThey all thought about it. It wasn't that their memories were poor, it was just that they found trouble so easily, they often overlooked the results.\n\nAnyone breezing by the big window of Ed's Caf\u00e9 would have seen the four MacKades, dark-haired, green-eyed devils, handsome enough to raise any female's blood pressure, be she ten or eighty. Reckless enough to have most men bracing or backing away.\n\nThey argued awhile over who had done what most recently\u2014fights picked and fought, laws broken, or at least dented. It was agreed, after the argument grew heated, that Rafe had the prize, with his race against Joe Dolin's Chevy on route 34.\n\nThey hadn't been caught, but word had gotten around. Especially as Rafe had won and Joe had slunk off muttering about revenge.\n\n\"The guy's a jerk.\" Rafe blew out smoke. No one disagreed, but Rafe's gaze shifted to where Cassie was busy serving a booth behind them. \"What does a sweet little thing like Cassie see in him?\"\n\n\"If you ask me, she wants out of the house.\" Jared pushed his plate aside. \"Her mother would be enough to send anyone looking for the first escape hatch. The woman's a fanatic.\"\n\n\"Maybe she loves him,\" Devin said quietly.\n\nRafe's opinion of that was one crude word. \"Kid's barely seventeen,\" he pointed out. \"She'll fall in love a dozen times.\"\n\n\"Not everyone has a flexible heart.\"\n\n\"A flexible heart.\" Shane whooped with laughter at the phrase. \"It ain't Rafe's heart that's flexible, Dev, it's his\u2014\"\n\n\"Shut up, creep,\" Rafe said mildly as his elbow jammed hard into Shane's ribs. \"You up for a beer, Jare?\"\n\n\"I'm up for it.\"\n\nRafe leered nastily. \"Too bad you two have to stick with soda pop. I bet Duff has a whole case of the fizzy stuff for you kids.\"\n\nThat, of course, insulted Shane. As it was meant to. Hot words came first, then the jostling. From her station at the counter, Edwina Crump shouted at them to take it outside.\n\nThey did, with Devin lagging behind to pay the tab.\n\nOn the other side of the window, his brothers pushed and shoved each other, more out of habit than from any real temper. Ignoring them, he smiled over at Cassie.\n\n\"Just blowing off steam,\" he told her, adding a tip that wouldn't embarrass her.\n\n\"The sheriff sometimes comes by about this time of night.\" Her voice was barely a whisper of warning. And so sweet to Devin's ears, he almost sighed.\n\n\"I'll go break it up.\"\n\nHe slid out of the booth. He thought his mother probably knew his feelings. It was impossible to hide anything from her. God knew, they had all tried and failed. He thought he knew what she would say to him.\n\nThat he was young yet, and there would be other girls, other women, other loves. She would mean the best by it.\n\nDevin knew that though he wasn't yet fully an adult, he had a man's heart. And he'd already given it.\n\nHe kept that heart out of his eyes, though, because he would hate Cassie's pity. Casually he walked out of the diner to break up his brothers. He caught Shane in a headlock, elbowed Rafe in the gut, cocked a brow at Jared and suggested amiably that they go play some pool.\n\n## Chapter 1\n\nThe town of Antietam was a pretty sight in late spring. Sheriff Devin MacKade liked to walk the uneven sidewalks and smell the freshly mowed grass, the flowers, hear the yip of dogs and shouts of children.\n\nHe liked to take in the order of it, the continuity, and the little changes. Outside the bank, a bed of pink begonias was spreading. The three cars jockeying in line at the drive-in window constituted a traffic jam.\n\nDown a little ways, in front of the post office, there were men passing the time, taking the air. Through the barbershop window, he could see a toddler experiencing his first haircut, while his mother bit her nails and blinked damp eyes.\n\nThe banners were flying for the annual Memorial Day parade and picnic. He could see several people busily scrubbing or painting their porches in preparation for the event.\n\nIt was an event he enjoyed, even with its logistical and traffic headaches. He liked the continuity of it, the predictability. The way people would plant themselves with their folding chairs and coolers along the curb, hours before parade time, to ensure that they would have a good view of the marching bands and twirling batons.\n\nMost of all, he liked the way the townspeople threw themselves into that weekend, how much they cared, how strong their pride.\n\nHis father had told him of the ancient man who, when he himself was a little boy, had walked creakily down Main Street wearing Confederate gray at an earlier Memorial Day. One of the last living testaments to the Civil War.\n\nDead now, as they all were, Devin mused as he glanced over at the memorial in the town's square. Dead, but not and never forgotten. At least not in little towns such as these, which had once known the sound of mortar and rifle fire and the terrible cries of the wounded.\n\nTurning away, he looked down the street and sighed. There was Mrs. Metz's Buick, parked, as usual, in the red zone. He could give her a ticket, Devin mused, and she would pay it. But when she lumbered into his office to hand over the fine, she would also treat him to a lecture. He blew out a breath, studied the door of the library. No doubt that was where she was, gossiping over the counter with Sarah Jane Poffenberger.\n\nDevin drew together his courage and fortitude and climbed the old stone steps.\n\nShe was exactly where he'd expected her to be, leaning over the counter, a mountain of paperback novels at her dimpled elbow, deep into the latest dirt with the librarian. Devin wondered why any woman so...generously sized insisted on wearing wildly patterned dresses.\n\n\"Mrs. Metz.\" He kept his voice low. He'd been tossed out of the library many times in his youth by Miss Sarah Jane.\n\n\"Well, hello there, Devin.\" Beaming a smile, Mrs. Metz turned to him. Her elbow nearly toppled the mountain of books, but Miss Sarah Jane, for all her resemblance to an understuffed scarecrow, moved fast. \"And how are you on this beautiful afternoon?\"\n\n\"I'm just fine. Hello, Miss Sarah Jane.\"\n\n\"Devin.\" Iron-gray hair pulled back from paper-thin white skin, starched collar buttoned firmly to her chin, Sarah Jane nodded regally. \"Did you come in to return that copy of The Red Badge of Courage?\"\n\n\"No, ma'am.\" He very nearly flushed. He'd lost the damn book twenty years before, he'd paid for it, he'd even swept the library for a month as penance for his carelessness. Now, though he was a man\u2014one who wore a badge and was considered responsible by most\u2014he was shriveled down to a boy by Sarah Jane Poffenberger's steely eyes.\n\n\"A book is a treasure,\" she said, as she always did.\n\n\"Yes, ma'am. Ah, Mrs. Metz...\" More to save himself now than to uphold parking laws, he shifted his gaze. \"You're parked illegally. Again.\"\n\n\"I am?\" All innocence, she fluttered at him. \"Why, I don't know how that happened, Devin. I would have sworn I pulled into the right place. I just came in to check out a few books. I'd have walked, but I had to run into the city, and stopped by on my way home. Reading's one of God's gifts, isn't it, Sarah Jane?\"\n\n\"It is indeed.\" Though her mouth remained solemn, the dark eyes in Sarah Jane's wrinkled face were laughing. Devin had to concentrate on not shuffling his feet.\n\n\"You're in the red zone, Mrs. Metz.\"\n\n\"Oh, dear. You didn't give me a ticket, did you?\"\n\n\"Not yet,\" Devin muttered.\n\n\"Because Mr. Metz gets all huffy when I get a ticket. And I've only been here for a minute or two, isn't that right, Sarah Jane.\"\n\n\"Just a minute or two,\" Sarah Jane confirmed, but she winked at Devin.\n\n\"If you'd move your car\u2014\"\n\n\"I'll do that. I surely will. Just as soon as I check out these books. I don't know what I'd do if I didn't have my books, what with the way Mr. Metz watches the TV. You check these out for me, Sarah Jane, while Devin tells us how his family's doing.\"\n\nHe knew when he was outgunned. After all, he was a cop. \"They're fine.\"\n\n\"And those sweet little babies. Imagine two of your brothers having babies within months of each other. I just have to get over to see them all.\"\n\n\"The babies are fine, too.\" He softened at the thought of them. \"Growing.\"\n\n\"Oh, they do grow, don't they, Sarah Jane? Grow like weeds, before you can stop them. Now you've got yourself a nephew and a niece.\"\n\n\"Two nephews and a niece,\" Devin reminded her, adding Jared's wife Savannah's son, Bryan.\n\n\"Yes, indeed. Give you any ideas about starting your own brood?\"\n\nHer eyes were glittering at the thought of getting the inside story on future events. Devin stood his ground. \"Being an uncle suits me.\" Without a qualm, he tossed his sister-in-law to the wolves. \"Regan has little Nate with her at the shop today. I saw him a couple hours ago.\"\n\n\"Does she?\"\n\n\"She mentioned Savannah might be coming by, with Layla.\"\n\n\"Oh, my! Well...\" Being able to corner both MacKade women, and their babies, was such a coup, Mrs. Metz nearly trembled at the idea. \"Hurry on up there, Sarah Jane. I've got errands to run.\"\n\n\"Hold your horses now, I've got 'em for you right here.\" Sarah Jane handed over the canvas bag Mrs. Metz had brought, now pregnant with books. Moments later, when Mrs. Metz puffed her way out, Sarah Jane smiled. \"You're a smart boy, Devin. Always were.\"\n\n\"If Regan finds out I headed her over there, she'll skin me.\" He grinned. \"But a man's gotta do what a man's gotta do. Nice seeing you, Miss Sarah Jane.\"\n\n\"You find that copy of The Red Badge of Courage, Devin MacKade. Books aren't meant to be wasted.\"\n\nHe winced as he opened the door. \"Yes, ma'am.\"\n\nFor all her bulk, Mrs. Metz moved quickly. She was already pulling out of the red zone and into the sparse traffic. Congratulating himself on a job well done, Devin told himself he could take a quick ride down to the MacKade Inn.\n\nJust needed to check and make sure there wasn't anything that needed his attention, he told himself as he walked up the street to his cruiser. It was his brother Rafe's place, after all. It was his duty to check on it now and again.\n\nThe fact that Cassie Dolin managed the bed-and-breakfast and lived on the third floor with her two children had nothing to do with it.\n\nHe was just doing his job.\n\nWhich was, he thought as he slipped behind the wheel of his car, a huge and ridiculous lie.\n\nHe was, however, doing what he had to do. Which was to see her. At least once a day, he simply had to see her. He just had to, no matter how much it hurt, or how careful he had to be. More careful, he reminded himself, now that she was divorced from that bastard who had beaten and abused her for years.\n\nJoe Dolin was in prison, Devin thought with grim satisfaction as he headed out of town. And he would be there for quite some time to come.\n\nAs the sheriff, as a friend, as the man who had loved her most of his life, Devin had a duty to see that Cassie and the kids were safe and happy.\n\nAnd maybe today he could make her smile, all the way to her big gray eyes.\n\nWhat had been the old Barlow place\u2014and likely would remain that forever in the mind of the town\u2014sat on a hill just on the edge of Antietam. Once it had been the property of a rich man who enjoyed its height, its expensive furnishings, its enviable view. It had stood there while the bloodiest day of the Civil War raged around it. It had stood while a wounded young soldier was murdered on its polished grand staircase. There it had remained while the mistress of the house grieved herself to death. Or so the legend went.\n\nIt had stood, falling into decay, disuse, disregard. Its stones had not moved when its porches rotted, when its windows were shattered by rocks heaved by rambunctious children. It had stood, empty but for its ghosts, for decades.\n\nUntil Rafe MacKade had returned and claimed it.\n\nIt was the house, Devin thought as he turned up its steep lane, that had brought Rafe and Regan together. Together, they had turned that brooding old building into something fine, something lovely.\n\nWhere there had once been weeds and thorny brambles, there was now a lush, terraced lawn, vivid with flowers and shrubs. He had helped plant them himself. The MacKades always united when it came to developing dreams\u2014or destroying enemies.\n\nThe windows gleamed now, framed by rich blue trim, their overflowing flower boxes filled with sunny-faced pansies. The sturdy double porches were painted that same blue, and offered guests a place to sit and look toward town.\n\nOr, he knew, if they chose to sit around at the back, they'd have a long view of the haunted woods that bordered the inn's property, his own farm, and the land where his brother Jared, his wife, Savannah, and their children lived.\n\nHe didn't knock, but simply stepped inside. There were no cars in the drive, but for Cassie's, so he knew the overnight guests had already left, and any others had yet to arrive.\n\nHe stood for a moment in the grand hall, with its polished floor, pretty rugs and haunted staircase. There were always flowers. Cassie saw to that. Pretty vases of fragrant blooms, little bowls and dishes with potpourri that he knew she made herself.\n\nSo, to him, the house always smelled like Cassie.\n\nHe wasn't sure where he would find her\u2014in the kitchen, in the yard, in her apartment on the third floor. He moved through the house from front to rear, knowing that if he didn't find her in the first two, he would climb the outside stairs and knock on the door of her private quarters.\n\nIt was hard to believe that less than two years before, the house had been full of dust and cobwebs, all cracked plaster and chipped molding. Now floors and walls gleamed, windows shone, wood was polished to a high sheen. Antique tables were topped with what Devin always thought of as dust collectors, but they were charming.\n\nRafe and Regan had done something here, built something here. Just as they were doing in the old house they'd bought for themselves outside of town.\n\nHe envied his brother that, not just the love, but the partnership of a woman, the home and family they had created together.\n\nShane had the farm. Technically, it belonged to all four of them, but it was Shane's, heart and soul. Rafe had Regan and their baby, the inn, and the lovely old stone-and-cedar house they were making their own. Jared had Savannah, the children, and the cabin.\n\nAnd as for himself? Devin mused. Well, he had the town, he supposed. And a cot in the back room of the sheriff's office.\n\nThe kitchen was empty. Though it was as neat as a model on display, it held all the warmth kitchens were meant to. Slate-blue tiles and creamy white appliances were a backdrop for little things\u2014fresh fruit in an old stoneware bowl, a sassy cookie jar in the shape of a smiling cat that he knew would be full of fresh, home-baked cookies, long, tapered jars that held the herbed vinegars Cassie made, a row of African violets in bloom on the wide windowsill over the sink.\n\nAnd then, through the window, he saw her, taking billowing sheets from the line where they'd dried in the warm breeze.\n\nHis heart turned over in his chest. He could handle that, had handled it for too many years to count. She looked happy, was all he could think. Her lips were curved a little, her gray eyes dreamy. The breeze that fluttered the sheets teased her hair, sending the honeycomb curls dancing around her face, along her neck and throat.\n\nLike the kitchen, she was neat, tidy, efficient without being cold. She wore a white cotton blouse tucked into navy slacks. Just lately, she'd started to add little pieces of jewelry. No rings. Her divorce had been final for a full year now, and he knew the exact day she'd taken off her wedding ring.\n\nBut she wore small gold hoops in her ears and a touch of color on her mouth. She'd stopped wearing makeup and jewelry shortly after her marriage. Devin remembered that, too.\n\nJust as he remembered the first time he'd been called out to the house she rented with Joe, answering a complaint from the neighbors. He remembered the fear in her eyes when she'd come to the door, the marks on her face, the way her voice had hitched and trembled when she told him there wasn't any trouble, there was no trouble at all. She'd slipped and fallen, that was all.\n\nYes, he remembered that. And his frustration, the hideous sense of impotence that first time, and all the other times he'd had to confront her, to ask her, to quietly offer her alternatives that were just as quietly refused.\n\nThere'd been nothing he could do as sheriff to stop what happened inside that house, until the day she finally came into his office\u2014bruised, beaten, terrified\u2014to fill out a complaint.\n\nThere was little he could do now as sheriff but offer her friendship.\n\nSo he walked out the rear door, a casual smile on his face. \"Hey, Cass.\"\n\nAlarm came into her eyes first, darkening that lovely gray. He was used to it, though it pained him immeasurably to know that she thought of him as the sheriff first\u2014as authority, as the bearer of trouble\u2014before she thought of him as an old friend. But the smile came back more quickly than it once had, chasing the tension away from those delicate features.\n\n\"Hello, Devin.\" Calmly, because she was teaching herself to be calm, she hooked a clothespin back on the line and began folding the sheet.\n\n\"Need some help?\"\n\nBefore she could refuse, he was plucking clothespins. She simply couldn't get used to a man doing such things. Especially such a man. He was so...big. Broad shoulders, big hands, long legs. And gorgeous, of course. All the MacKades were.\n\nThere was something so male about Devin, she couldn't really explain it. Even as he competently took linen from the line, folded it into the basket, he was all man. Unlike his deputies, he didn't wear the khaki uniform of his office, just jeans and a faded blue shirt rolled up to the elbows. There were muscles there, she'd seen them. And she had reason to be wary of a man's strength. But despite his big hands, his big shoulders, he'd never been anything but gentle. She tried to remember that as he brushed against her, reaching for another clothespin.\n\nStill, she stepped away, kept distance between them. He smiled at her, and she tried to think of something to say. It would be easier if everything about him wasn't so...definite, she supposed. So vivid. His hair was as black as midnight, and curled over the frayed collar of his shirt. His eyes were as green as moss. Even the bones in his face were defined and impossible to ignore, the way they formed hollows and planes. His mouth was firm, and that dimple beside it constantly drew the eye.\n\nHe even smelled like a man. Plain soap, plain sweat. He'd never been anything but kind to her, and he'd been a part of her life forever, it seemed. But whenever it was just the two of them, she found herself as nervous as a cat faced with a bulldog.\n\n\"Too nice a day to toss these in the dryer.\"\n\n\"What?\" She blinked, then cursed herself. \"Oh, yes. I like hanging the linens out, when there's time. We had two guests overnight, and we're expecting another couple later today. We're booked solid for the Memorial Day weekend.\"\n\n\"You'll be busy.\"\n\n\"Yes. It's hardly like work, though, really.\"\n\nHe watched her smooth sheets into the basket. \"Not like waiting tables at Ed's.\"\n\n\"No.\" She smiled a little, then struggled with guilt. \"Ed was wonderful to me. She was great to work for.\"\n\n\"She's still ticked at Rafe for stealing you.\" Noting the distress that leaped into her eyes, Devin shook his head. \"I'm only kidding, Cassie. You know she was happy you took this job. How are the kids?\"\n\n\"They're fine. Wonderful.\" Before she could pick up the basket of linens herself, Devin had it tucked to his hip, leaving her nothing to do with her hands. \"They'll be home soon, from school.\"\n\n\"No Little League practice today?\"\n\n\"No.\" She headed toward the kitchen, but he opened the door before she could, and waited for her to go in ahead of him. \"Connor's thrilled he made the team.\"\n\n\"He's the best pitcher they've got.\"\n\n\"Everyone says so.\" Automatically, she went to the stove to make coffee. \"It's so strange. He was never interested in sports before...well, before,\" she finished lamely. \"Bryan's been wonderful for him.\"\n\n\"My nephew's a hell of a kid.\"\n\nThere was such simple and honest pride in the statement that Cassie turned around to study him. \"You think of him that way, really? I mean, even though there's no blood between you?\"\n\n\"When Jared married Savannah, it made Bryan his son. That makes him my nephew. Family isn't just blood.\"\n\n\"No, and sometimes blood kin is more trouble than not.\"\n\n\"Your mother's hassling you again.\"\n\nShe only moved her shoulder and turned back to finish the coffee. \"She's just set in her ways.\" Shifting, she reached into one of the glass-fronted cabinets for a cup and a small plate. When Devin's hand curled over her shoulder, she jerked and nearly dropped the stoneware to the tiles.\n\nHe started to step back, then changed his mind. Instead, he turned her around so that they were face-to-face, and kept both of his hands on her shoulders. \"She's still giving you a hard time about Joe?\"\n\nShe had to swallow, but couldn't quite get her throat muscles to work. His hands were firm, but they weren't hurting. There was annoyance in his eyes, but no meanness. She ordered herself to be calm, not to lower her gaze.\n\n\"She doesn't believe in divorce.\"\n\n\"Does she believe in wife-beating?\"\n\nNow she did wince, did lower her gaze. Devin cursed himself and lowered his hands to his sides. \"I'm sorry.\"\n\n\"No, it's all right. I don't expect you to understand. I can't understand myself anymore.\" Relieved that he'd stepped back, she turned to the cookie jar and filled the plate with chocolate chip and oatmeal cookies she'd baked that morning. \"It doesn't seem to matter that I'm happy, that the kids are happy. It doesn't matter that the law says what Joe did to me was wrong. That he attacked Regan. It only matters that I broke my vows and divorced him.\"\n\n\"Are you happy, Cassie?\"\n\n\"I'd stopped believing I could be, or even that I should be.\" She set the plate on the table, went to pour him coffee. \"Yes, I am happy.\"\n\n\"Are you going to make me drink this coffee by myself?\"\n\nShe stared at him a minute. It was still such a novel concept, the idea that she could sit down in the middle of the day with a friend. Taking matters into his own hands, he got out a second cup.\n\n\"So tell me...\" After pouring her coffee, he held out a chair for her. \"How do the tourists feel about spending the night in a haunted house?\"\n\n\"Some of them are disappointed when they don't see or hear anything.\" Cassie lifted her cup and tried not to feel guilty that she wasn't doing some chore. \"Rafe was clever to publicize the inn as haunted.\"\n\n\"He's always been clever.\"\n\n\"Yes, he has. A few people are nervous when they come down for breakfast, but most of them are...well, excited, I guess. They'll have heard doors slamming or voices, or have heard her crying.\"\n\n\"Abigail Barlow. The tragic mistress of the house, the compassionate Southern belle married to the Yankee murderer.\"\n\n\"Yes. They'll hear her, or smell her roses, or just feel something. We've only had one couple leave in the middle of the night.\" For once, her smile was quick, and just a little wicked. \"They were both terrified.\"\n\n\"But you're not. It doesn't bother you to have ghosts wandering?\"\n\n\"No.\"\n\nHe cocked his head. \"Have you heard her? Abigail?\"\n\n\"Oh, yes, often. Not just at night. Sometimes when I'm alone here, making beds or tidying up, I'll hear her. Or feel her.\"\n\n\"And it doesn't spook you?\"\n\n\"No, I feel...\" She started to say \"connected,\" but thought it would sound foolish. \"Sorry for her. She was trapped and unhappy, married to a man who despised her, in love with someone else\u2014\"\n\n\"In love with someone else?\" Devin asked, interrupting her. \"I've never heard that.\"\n\nBaffled, Cassie set her cup down with a little clink. \"I haven't, either. I just\u2014\" Know it, she realized. \"I suppose I added it in. It's more romantic. Emma calls her the lady. She likes to go into the bridal suite.\"\n\n\"And Connor?\"\n\n\"It's a big adventure for him. All of it. They love it here. Once when Bryan was spending the night, I caught the three of them sneaking down to the guest floor. They wanted to go ghost-hunting.\"\n\n\"My brothers and I spent the night here when we were kids.\"\n\n\"Did you? Of course you did,\" she said before he could comment. \"The MacKades and an empty, derelict, haunted house. They belong together. Did you go ghost-hunting?\"\n\n\"I didn't have to. I saw her. I saw Abigail.\"\n\nCassie's smile faded. \"You did?\"\n\n\"I never told the guys. They'd have ragged on me for the rest of my life. But I saw her, sitting in the parlor, by the fire. There was a fire, I could smell it, feel the heat from the flames, smell the roses that were in a vase on the table beside her. She was beautiful,\" Devin said quietly. \"Blond hair and porcelain skin, eyes the color of the smoke going up the flue. She wore a blue dress. I could hear the silk rustle as she moved. She was embroidering something, and her hands were small and delicate. She looked right at me, and she smiled. She smiled, but there were tears in her eyes. She spoke to me.\"\n\n\"She spoke to you,\" Cassie repeated, as chills raced up and down her back like icy fingers. \"What did she say?\"\n\n\"'If only.'\" Devin brought himself back, shook himself. \"That was it. 'If only.' Then she was gone, and I told myself I'd been dreaming. But I knew I hadn't. I always hoped I'd see her again.\"\n\n\"But you haven't?\"\n\n\"No, but I've heard her weeping. It breaks my heart.\"\n\n\"I know.\"\n\n\"I'd, ah, appreciate it if you wouldn't mention that to Rafe. He'd still rag on me.\"\n\n\"I won't.\" She smiled as he bit into a cookie. \"Is that why you come here, hoping to see her again?\"\n\n\"I come to see you.\" The minute he'd said it, he recognized his mistake. Her face went from relaxed to wary in the blink of an eye. \"And the kids,\" he added quickly. \"And for the cookies.\"\n\nShe relaxed again. \"I'll put some in a bag for you to take with you.\" But even as she rose to do so, he covered her hand with his. She froze, not in fear so much as from the shock of the contact. Speechless, she stared down at the way his hand swallowed hers.\n\n\"Cassie...\" He strained against the urge to gather her up, just to hold her, to stroke those flyaway curls, to taste, finally to taste, that small, serious mouth.\n\nThere was a hitch in her breathing that she was afraid to analyze. But she made herself shift her gaze, ordered herself not to be so much a coward that she couldn't look into his eyes. She wished she knew what she was looking at, or looking for. All she knew was that it was more than the patience and pity she'd expected to see there, that it was different.\n\n\"Devin\u2014\" She broke off, jerked back at the sound of giggles and stomping feet. \"The kids are home,\" she finished quickly, breathlessly, and hurried to the door. \"I'm down here!\" she called out, knowing that they would do as they'd been told and go directly to the apartment unless she stopped them.\n\n\"Mama, I got a gold star on my homework.\" Emma came in, a blond pixie in a red playsuit. She set her lunch box on the counter and smiled shyly at Devin. \"Hello.\"\n\n\"There's my best girl. Let's see that star.\"\n\nClutching the lined paper in her hand, she walked to him. \"You have a star.\"\n\n\"Not as pretty as this one.\" Devin traced a finger over the gold foil stuck to the top of the paper. \"Did you do this by yourself?\"\n\n\"Almost all. Can I sit in your lap?\"\n\n\"You bet.\" He plucked her up, cradled her there. He quite simply adored her. After brushing his cheek against her hair, he grinned over at Connor. \"How's it going, champ?\"\n\n\"Okay.\" A little thrill moved through Connor at the nickname. He was small for his age, like Emma, and blond, though at ten he had hair that was shades darker than his towheaded sister's.\n\n\"You pitched a good game last Saturday.\"\n\nNow he flushed. \"Thanks. But Bryan went four for five.\" His loyalty and love for his best friend knew no bounds. \"Did you see?\"\n\n\"I was there for a few innings. Watched you smoke a few batters.\"\n\n\"Connor got an A on his history test,\" Emma said. \"And that mean old Bobby Lewis shoved him and called him a bad name when we were in line for the bus.\"\n\n\"Emma...\" Mortified, Connor scowled at his sister.\n\n\"I guess Bobby Lewis didn't get an A,\" Devin commented.\n\n\"Bryan fixed him good,\" Emma went on.\n\nI bet he did, Devin thought, and handed Emma a cookie so that she'd be distracted enough to stop embarrassing her brother.\n\n\"I'm proud of you.\" Trying not to worry, Cassie gave Connor a quick squeeze. \"Both of you. A gold star and an A all in one day. We'll have to celebrate later with ice-cream sundaes from Ed's.\"\n\n\"It's no big deal,\" Connor began.\n\n\"It is to me.\" Cassie bent down and kissed him firmly. \"A very big deal.\"\n\n\"I used to struggle with math,\" Devin said casually. \"Never could get more than a C no matter what I did.\"\n\nConnor stared at the floor, weighed down by the stigma of being bright. He could still hear his father berating him. Egghead. Pansy. Useless.\n\nCassie started to speak, to defend, but Devin sent her one swift look.\n\n\"But then, I used to ace history and English.\"\n\nStunned, Connor jerked his head up and stared. \"You did?\"\n\nIt was a struggle, but Devin kept his eyes sober. The kid didn't mean to be funny, or insulting, he knew. \"Yeah. I guess it was because I liked to read a lot. Still do.\"\n\n\"You read books?\" It was an epiphany for Connor. Here was a man who held a real man's job and who liked to read.\n\n\"Sure.\" Devin jiggled Emma on his knee and smiled. \"The thing was, Rafe was pitiful in English, but he was a whiz in math. So we traded off. I'd do his\u2014\" He glanced at Cassie, realized his mistake. \"I'd help him with his English homework and he'd help me with the math. It got us both through.\"\n\n\"Do you like to read stories?\" Connor wanted to know. \"Made-up stories?\"\n\n\"They're the best kind.\"\n\n\"Connor writes stories,\" Cassie said, even as Connor wriggled in embarrassment.\n\n\"So I've heard. Maybe you'll let me read one.\" Before the boy could answer, Devin's beeper went off. \"Hell,\" he muttered.\n\n\"Hell,\" Emma said adoringly.\n\n\"You want to get me in trouble?\" he asked, then hitched her onto his hip as he rose to call in. A few minutes later, he'd given up on his idea of wheedling his way into a dinner invitation. \"Gotta go. Somebody broke into the storeroom at Duff's and helped themselves to a few cases of beer.\"\n\n\"Will you shoot them?\" Emma asked him.\n\n\"I don't think so. How about a kiss?\"\n\nShe puckered up obligingly before he set her down. \"Thanks for the coffee, Cass.\"\n\n\"I'll walk you out. You two go on upstairs and get your after-school snack,\" she told her children. \"I'll be right along.\" She waited until they were nearly at the front door before she spoke again. \"Thank you for talking to Connor like that. He's still so sensitive about liking school.\"\n\n\"He's a bright kid. It won't take much longer for him to start appreciating himself.\"\n\n\"You helped. He admires you.\"\n\n\"It didn't take any effort to tell him I like to read.\" Devin paused at the door. \"He means a lot to me. All of you do.\" When she opened her mouth to speak, he took a chance and brushed a finger over her cheek. \"All of you do,\" he repeated, and walked out, leaving her staring after him.\n\n## Chapter 2\n\nSome nights, late at night, when her children were sleeping and the guests were settled down, Cassie would roam the house. She was careful not to go on the second floor, where guests were bedded down in the lovely rooms and suites Rafe and Regan had built.\n\nThey paid for privacy, and Cassie was careful to give it.\n\nBut she was free to walk through her own apartment on the third floor, to admire the rooms, the view from the windows, even the feel of the polished hardwood under her bare feet.\n\nIt was a freedom, and a security, that she knew she would never take for granted. Any more than she would take for granted the curtains framing the windows, made of fabric that she had chosen and paid for herself. Or the kitchen table, the sofa, each lamp.\n\nNot all new, she mused, but new to her. Everything that had been in the house she shared with Joe had been sold. It had been her way of sweeping away the past. Nothing here was from her before. It had been vital to her to start this life with nothing she hadn't brought into it on her own.\n\nIf she was restless, she could go down on the main level, move from parlor to sitting room, into the beautiful solarium, with its lovely plants and glistening glass. She could stand in the hallways, sit on the steps. Simply enjoy the quiet and solitude.\n\nThe only room she avoided was the library. It was the only room that never welcomed her, despite its deep leather chairs and walls of books.\n\nShe knew instinctively that it had been Charles Barlow's realm. Abigail's husband. The master of the house. A man who had shot, in cold blood, a wounded Confederate soldier hardly old enough to shave.\n\nSometimes she felt the horror and sadness of that when she walked up and down the staircase where it had happened. Now and again she even heard the shot, the explosion of it, and the screams of the servants who had witnessed the senseless and brutal murder.\n\nBut she understood senseless brutality, knew it existed.\n\nJust as she knew Abigail still existed, in this house. It wasn't just the sound of weeping, the scent of roses that would come suddenly and from nowhere. It was just the feel of the air, that connection that she'd been too embarrassed to mention to Devin.\n\nThat was how she knew Abigail had loved a man who wasn't her husband. That she had longed for him, wept for him, as well as for the murdered boy. That she had dreamed of him, and despaired of ever knowing the joy of real love.\n\nCassie understood, and sympathized. That was why she felt so welcomed in this house that held so much of the past. Why she was never afraid.\n\nNo, she was grateful for every hour she spent here as caretaker to beautiful things. It had been nearly a year since she had accepted Regan's and Rafe's offer and moved her family in. She was still dazzled that they would trust her with the job, and she worked hard to earn that trust.\n\nThe work was all pleasure, she thought now, as she wandered into the parlor. To tend and polish lovely antiques, to cook breakfast in that wonderful kitchen and serve it to guests on pretty dishes. To have flowers all around the house, inside and out.\n\nIt was like a dream, like one of the fairy tales Savannah MacKade illustrated.\n\nShe was so rarely afraid anymore, hardly even disturbed by the nightmares that had plagued her for so long she'd come to expect them. It was unusual for her to wake shivering in the middle of the night, out of a dream\u2014listening, terrified, for Joe's steps, for his voice.\n\nShe was safe here, and, for the first time in her life, free.\n\nBundled into her robe, she curled on the window seat in the parlor. She wouldn't stay long. Her children slept deeply and were content here, but there was always a chance they might wake and need her. But she wanted just a few moments alone to hug her good fortune close to her heart.\n\nShe had a home where her children could laugh and play and feel safe. It was wonderful to see how quickly Emma was throwing off her shyness and becoming a bright, chattering little girl. Childhood had been harder on Connor, she knew. It shamed her to realize that he had seen and heard so much more of the misery than she had guessed. But he was coming out of his shell.\n\nIt relieved her to see how comfortable they were with Devin, with all the MacKades, really. There had been a time when Emma hesitated to so much as speak to a man, and Connor, sweet, sensitive Connor, had forever been braced for a verbal blow.\n\nNo more.\n\nJust that day, both of them had talked to Devin as if it were as natural as breathing. She wished she was as resilient. It was the badge, she decided. She was finding it easier and easier to be comfortable with Jared or Rafe or Shane. She didn't jolt when one of them touched her or flashed that MacKade grin.\n\nIt was different with Devin. But then, she'd had to go to him, had to confess that she'd allowed herself to be beaten and abused for years, had been forced to show him the marks on her body. Nothing, not even Joe's vicious fists, had ever humiliated her more than that.\n\nShe knew he was sorry for her, and felt obligated to look out for her and the children. He took his responsibilities as sheriff seriously. No one, including herself, would have believed twelve or fifteen years before, when he and his brothers were simply those bad MacKade boys, that they would turn out the way they had.\n\nDevin had made himself into an admirable man. Still rough, she supposed. She knew he could break up a bar fight with little more than a snarl, and that he used his fists when that didn't work.\n\nStill, she'd never known anyone gentler or more compassionate. He'd been very good to her and her children, and she owed him.\n\nLaying her cheek against the window, she closed her eyes. She was going to train herself not to be so jumpy around him. She could do it. She had been working very hard over the past year or so to teach herself composure and calm, to pretend she wasn't shy when she greeted the guests. It worked so well that she often didn't even feel shy anymore.\n\nThere were even times, and they were coming more and more often, when she actually felt competent.\n\nSo she would work now to teach herself not to be so jittery around Devin. She would stop thinking about his badge and remember that he was one of her oldest friends\u2014one she'd even had a little crush on, once upon a time. She would stop thinking of how big his hands were, or what would happen if he got angry and used them against her.\n\nInstead she would remember how gently they ruffled her daughter's hair, or how firmly they covered her son's when he helped him with his batting stance.\n\nOr how nice it had been, how unexpectedly nice, to feel the way his finger brushed her cheek.\n\nShe curled more comfortably on the padded seat....\n\nHe was here, right here beside her, smiling in that way that brought his dimple out and made odd things happen to her insides. He touched her, and she didn't jolt this time. There, she thought, it was working already.\n\nHe was touching her, drawing her against him. Oh, his body was hard. But she didn't flinch. She was trembling, though. Couldn't stop. He was so big, so strong, he could break her in half. And yet...and yet his hands stroked so lightly over her. Over her skin. But he couldn't be touching her there.\n\nHis mouth was on hers, so warm and gentle. She couldn't stop him. She forgot that she should, even when his tongue slid over hers and his hand cupped her breast as if it were the most natural thing in the world.\n\nHe was touching her, and it was hard to breathe, because those big hands were gliding over her. And now his mouth. Oh, it was wrong, it had to be wrong, but it was so wonderful to feel that warm, wet mouth on her.\n\nShe was whimpering, moaning, opening for him. She felt him coming inside her, so hard, so smooth, so right.\n\nThe explosion of a gunshot had her jerking upright. She was gasping for breath, damp with sweat, her mind a muddled mess.\n\nAlone in the parlor. Of course she was alone. But her skin was tingling, and there was a tingling, almost a burning, inside her that she hadn't felt in so many years she'd forgotten it was possible.\n\nShame washed over her, had her gathering her robe tight at her throat. It was terrible, she thought, just terrible, to have been imagining herself with Devin like that. After he'd been so kind to her.\n\nShe didn't know what had gotten into her. She didn't even like sex. It was something she'd learned to dread, and then to tolerate, very soon after her miserable wedding-night initiation. Pleasure had never entered into it. She simply wasn't built for that kind of pleasure, and had accepted the lack early on.\n\nBut when she got to her feet, her legs were shaky and there was a nagging pressure low in her stomach. She drew in a breath, and along with it the delicate scent of roses.\n\nSo she wasn't alone, Cassie thought. Abigail was with her. Comforted, she went back upstairs to check on her children one last time before going to bed.\n\nDevin was well into what he considered the paper-pushing part of the day by noon. He had a report to type and file on the break-in at Duff's Tavern. The trio of teenagers who'd thought to relieve Duff of a bit of his inventory had been pathetically easy to track down.\n\nThen there was the traffic accident out on Brook Lane. Hardly more than a fender bender, Devin mused as he hammered at the keys, but Lester Swoop, whose new sedan had been crinkled, was raising a ruckus.\n\nHe had to finish up his report to the mayor and town council on the preparations for crowd control on parade day.\n\nThen, maybe, he'd get some lunch.\n\nAcross the office, his young deputy, Donnie Banks, was dealing with parking tickets. And, as usual, drumming his fingers on the metal desk to some inner rhythm that Devin tried hard to ignore.\n\nThe day was warm enough that the windows were open. The budget didn't run to air-conditioning. He could hear the sounds of traffic\u2014what there was of it\u2014and the occasionally squeal of brakes as someone came up too fast on the stop light at Main and Antietam.\n\nHe still had the mail to sort through, his job, since Crystal Abbott was off on maternity leave and he hadn't come up with a temporary replacement for her position as general dogsbody.\n\nHe didn't mind, really. The sheer monotony of paperwork could be soothing. Things were quiet, as they were expected to be in a town of less than twenty-five hundred. His job was to keep it that way, and deal with the drunk-and-disorderlies, the traffic violations, the occasional petty theft or domestic dispute.\n\nThings heated up now and again, but in his seven years with Antietam's sheriff's department, both as deputy and as sheriff, he'd had to draw his weapon only twice. And he'd never been forced to fire it.\n\nReason and guile usually worked, and if they didn't, a fist usually turned the tide.\n\nWhen the phone rang, Devin glanced hopefully toward his deputy. Donnie's fingers never broke rhythm, so, with a sigh, Devin answered the phone himself. He was well on his way to calming a hysterical woman who claimed that her neighbor deliberately sent her dog over into her yard to fertilize her petunias when Jared walked in.\n\n\"Yes, ma'am. No, ma'am.\" Devin rolled his eyes and motioned Jared to a seat. \"Have you talked with her, asked her to keep her dog in her own yard?\"\n\nThe answer came so fast and loud that Devin winced and held the phone six inches from his ear. In the little wooden chair across the desk, Jared grinned and stretched out his legs.\n\n\"Yes, ma'am, I'm sure you worked very hard on your petunias. No, no, don't do that. Please. There's a law against discharging a firearm within town limits. You don't want to go waving your shotgun at the dog. I'm going to send somebody over there. Yes, ma'am, I surely am. Ah...we'll see what we can do. You leave that shotgun alone now, you hear? Yes, ma'am, I've got it all down right here. You just sit tight.\"\n\nHe hung up, tore off the memo sheet. \"Donnie?\"\n\n\"Yo.\"\n\n\"Get on over to Oak Leaf and handle this.\"\n\n\"We got us a situation?\" Donnie stopped his drumming, looking hopeful. Devin thought he seemed very young, in his carefully pressed uniform, with his scarecrow hair and eager blue eyes.\n\n\"We've got a French poodle using a petunia bed as a toilet. Explain about the leash law, and see if you can keep these two women from a hair-pulling contest.\"\n\n\"Yo!\" Delighted with the assignment, Donnie took the information sheet, adjusted his hat and strode out, ready to uphold the law.\n\n\"I think he started to shave last week,\" Devin commented.\n\n\"Petunias and poodles,\" Jared said, and stretched. \"I can see you're busy.\"\n\n\"Antietam's a real naked city.\" Devin got up to pour them both coffee. \"Had us a situation down to Duff's,\" he added, tinting his voice with Donnie's accent and emphasis. \"Three cases of beer went missing.\"\n\n\"Well, well...\"\n\n\"Got two of them back.\" After handing Jared the mug, Devin eased a hip onto his desk. \"The other had been consumed by three sixteen-year-olds.\"\n\n\"Tracked them down, did you?\"\n\n\"It didn't take Sam Spade.\" Devin shook his head as he sipped. \"They'd bragged about it right and left, took the beer out to the field near the high school and had themselves a party. They were sick as dogs when I caught up with them. Idiots. Now they've got B and E charges, larceny, and an appointment with juvie.\"\n\n\"Seems to me I remember a couple of cases of beer and a party. In the woods.\"\n\n\"We didn't steal it,\" Devin reminded him. \"We left Duff the money in the storeroom\u2014after we'd broken in and taken the beer.\"\n\n\"A fine but salient point. God, we got drunk.\"\n\n\"And sick,\" Devin added. \"When we crawled home, Mom made us shovel manure all afternoon. I thought I'd die.\"\n\n\"Those were the days,\" Jared said with a sigh. He sat back. Despite the trim suit and tie, the expensive shoes, there was no mistaking him for anything but a MacKade. Like his brother, he had the reckless dark good looks. A bit more groomed, a bit more polished, but reckless enough.\n\n\"What are you doing in town?\"\n\n\"This and that.\" Jared wanted to work up to what he had to tell Devin. \"Layla's getting a tooth.\"\n\n\"Yeah? Keeping you guys up?\"\n\n\"I forgot what sleep's like.\" His grin flashed. \"It's great. You know, Bryan changes diapers. The kid's so in love with her, Savannah says the first thing he does when he gets home from school is to go find her.\"\n\n\"You got lucky,\" Devin murmured.\n\n\"Don't I know it. You ought to try it, Dev. Marriage is a pretty good deal.\"\n\n\"It's working for you and Rafe. I saw him this morning, heading into the hardware with Nate strapped to his back. He looked real domestic.\"\n\n\"Did you tell him that?\"\n\n\"I didn't want to start a fight in front of the baby.\"\n\n\"Good call. You know what you need around here, Dev?\" Still sipping coffee, Jared looked around the office. It was utilitarian, basic. Desks, wood floors, coffeepot, a ceiling fan that he knew squeaked when it was put into use in the summer, unpadded chairs, metal file cabinets. \"You need a dog. Ethel'll be dropping that litter any day now.\"\n\nDevin raised a brow. Fred and Ethel, Shane's golden retrievers, had finally figured out what boy and girl dogs could do together besides chase rabbits. \"Yeah, I need a puppy puddling on the floor and chewing up my papers.\"\n\n\"Companionship,\" Jared insisted. \"Think how you'd look cruising around town with a dog riding shotgun. You could deputize him.\"\n\nThe image made Devin grin, but he set his coffee down. \"I'll keep it in mind. Now why don't you tell me what you came in to tell me?\"\n\nJared blew out a breath. He knew how Devin's mind worked, step by meticulous step. He'd let Jared ramble, but he hadn't been fooled. \"I had some business at the prison this morning.\"\n\n\"One of your clients not getting his full television rights?\"\n\nJared set his coffee aside, linked his fingers. \"You arrest them, I represent them. That's why it's called law and order.\"\n\n\"Right. How could I forget? So?\"\n\n\"So. I had a meeting with the warden, and as he's aware that I'm Cassie's lawyer, he felt it reasonable to pass some news on to me.\"\n\nDevin's mouth thinned. \"Dolin.\"\n\n\"Yeah, Joe Dolin.\"\n\n\"He's not up for a parole hearing for another eighteen months.\" Devin knew the exact day, to the hour.\n\n\"That's right. It seems that after a difficult period of adjustment, during which Joe was a disciplinary problem, he's become a model prisoner.\"\n\n\"I'll bet.\"\n\nJared recognized the bitterness in the tone, understood it perfectly. \"We know he's a bastard, Devin, but the point here is, he's playing the game. And he's playing it well.\"\n\n\"He won't make parole, not the first time at bat. I'll make sure of it.\"\n\n\"Parole's not the issue. Yet. He's been put on work release.\"\n\n\"The hell he has!\"\n\n\"As of this week. I argued against it. I pointed out the fact that he'll be only a matter of miles from Cassie, his history of violence, his ties to the town.\" Feeling helpless, Jared unlinked his hands, held them palms up. \"I got shot down. He'll be supervised, along with the rest of the crew. We need the work release program, need the park and the roads cleaned and maintained, and this is a cheap way to handle it. Letting cooperative prisoners serve the community is a solid method of rehabilitation.\"\n\n\"And when they take a hike from trash detail?\" Devin was pacing now, eyes fiery. \"It happens. Two or three times a year, at least, it happens. I hauled one back myself last fall.\"\n\n\"It happens,\" Jared agreed. \"They rarely get far. They're pretty easy to spot in the prison uniform, and most of them don't know the area.\"\n\n\"Dolin knows the damn area.\"\n\n\"You're not going to get any arguments from me. I'm going to fight it, Devin. But it's not going to be easy. Not when Cassie's own mother has been writing the warden in Joe's defense.\"\n\n\"That bitch.\" Devin's hands curled into fists. \"She knows what he did to Cassie. Cassie,\" he repeated, and scrubbed his hands over his face. \"She's just starting to pull things together. What the hell is this going to do to her?\"\n\n\"I'm heading over there now to tell her.\"\n\n\"No.\" Devin dropped his hands. \"I'll tell her. You go file papers, or whatever you have to do to turn this thing around. I want that son of a bitch locked up, twenty-four hours a day.\"\n\n\"They've got a crew out on 34 right now. Trash detail. He's on it.\"\n\n\"Fine.\" Devin headed for the door. \"That's just fine.\"\n\nIt didn't take him long to get there, or to spot the bright orange vests of the road crew. Devin pulled to the shoulder behind a pickup truck where bags of trash were already heaped.\n\nHe got out of his car, leaned against the hood and watched Joe Dolin.\n\nThe sixteen months in prison hadn't taken off any of his bulk, Devin noted. He was a big man, thick, burly. He'd been going to fat before his arrest. From the look of him, he'd been busy turning that fat into muscle.\n\nThe prison system approved of physical fitness.\n\nHe and another man were unclogging the runoff on the other side of the road, working systematically and in silence as they gathered up dead leaves, litter.\n\nDevin bided his time, waited until Joe straightened, hauled a plastic bag over his shoulder and turned.\n\nTheir eyes met, held. Devin wondered what the warden would say about rehabilitation if he'd seen that look in Joe's eyes. The heat and the hate. If he'd seen that slow, bitterly triumphant smile before Joe tossed the bag in the bed of the pickup parked on his side of the road.\n\nBecause he knew himself, Devin stayed where he was. He knew that if he got close, too close, he wouldn't be able to stop himself. The badge he wore was both a responsibility and a barrier.\n\nIf he was a civilian, he could walk across the road, ram his fists into Joe's leering face and take the consequences. If he was a civilian, he could pummel the wife-beating bastard into putty.\n\nBut he wasn't a civilian.\n\n\"Help you, Sheriff?\" One of the supervisors walked over, ready to chat, officer to officer. His easy smile faded at the look in Devin's eyes. \"Is there a problem?\"\n\n\"Depends.\" Devin took out one of the cigarettes he'd been working on giving up for the past two months. Taking his time, he struck a match, lit it, blew out smoke. \"You see that man there, the big one?\"\n\n\"Dolin? Sure.\"\n\n\"You remember that name.\" Devin flicked his gaze down to the ID clipped to the supervisor's shirt. \"And I'm going to remember yours, Richardson. If he gets away from you, even for a heartbeat, it's going to be your ass.\"\n\n\"Hey, look, Sheriff\u2014\"\n\nDevin merely fixed his eyes on Richardson's face, kept them there as he pushed off the hood. \"You make sure that son of a bitch doesn't wander into my town, Richardson. You make damn sure of it.\"\n\nJoe watched the sheriff's car pull out, drive away. He bent his back to the work, like a good team player. And patted his pocket, where the latest letter from his mother-in-law was tucked.\n\nHe knew what it said, almost word for word. She kept him up with Cassie just fine. How the little bitch had a fancy job now at the MacKade Inn. Lousy MacKades. He was going to take care of all of them, every last one of them, when he got out.\n\nBut first he was going to take care of Cassie.\n\nShe thought she could have him tossed in a cell. She thought she could divorce him and start strutting her stuff around town. Well, she was going to think again, real soon.\n\nHer mama was helping him out, writing him letters. They were preachy letters, and he couldn't stand the dried-up old bat, but she was helping him out. And he wrote her every week, telling her how he'd suffered, how he'd gotten religion, how he wanted to be with his family again. He made sure he went on about the kids.\n\nHe could have cared less about the kids. Whiny little brats.\n\nIt was Cassie he wanted. She was his wife\u2014till death do us part. He was going to be reminding her of that before too much longer.\n\nHe hauled another bag to the bed of the truck, tossed it in. Oh, yeah, he was going to remind her good, just like the old days. She would pay, in spades, for every hour he'd spent in a cell.\n\nCurling his hand into a fist, he dreamed about his homecoming.\n\n## Chapter 3\n\nInstead of going directly to Cassie, Devin went to the prison. He didn't doubt Jared's skill as a lawyer, but he wanted, needed, to add his weight. He forced himself to stay calm as he laid out the facts, and his opinion, to the warden.\n\nFor every protest he made, he was shown a report to offset it. Joe Dolin had indeed made himself into a model prisoner, one who showed every sign of rehabilitation. He worked hard, followed the rules, went to chapel regularly. He expressed regret over his crimes and kept up with his alcohol-abuse counseling.\n\nWhen Devin left, he understood that the system he worked hard to uphold had just kicked him in the teeth. All he could do now was tell Cassie and try to reassure her.\n\nHe found her on her hands and knees in the parlor, polishing the carved wood of a gateleg table. She was so busy humming to herself, she hadn't heard him come in. She was wearing a white bib apron over her blouse and slacks, and had a plastic basket beside her filled with rags and cleaning tools.\n\nHer wavy hair was tucked behind her ear to keep it from falling forward into her face. She'd been letting it grow some, he thought. It rippled just past her chin.\n\nShe looked so damn happy, Devin thought, and jammed his hands into his pockets.\n\n\"Cass?\"\n\nShe jerked up, barely missed rapping her head on the table extension. Then blushed right to the hairline.\n\n\"Devin.\" She twisted her polishing rag in her hands as her nerves went into overdrive. She'd been replaying the dream in her head, the dream she'd had right here in the parlor, on the window seat. The dream where Devin had... Oh, my...\n\nHe stared at her, then stepped forward. She looked as though she'd been caught rifling the till. \"What's wrong? What's the matter?\"\n\n\"Nothing. Nothing.\" It seemed her stomach was suddenly full of bats, and she had to hold back a nervous giggle. \"My mind was wandering, that's all.\" Was it ever. \"And you startled me. That's all.\"\n\nIt wasn't like her to keep repeating herself, and his gaze narrowed. \"Are you sure you're all right?\"\n\n\"Yes, yes. Fine. Just fine.\" She scrambled to her feet, still twisting the rag. \"The couple who are staying here went out to tour the battlefield. They're going to stay another night. They're from North Carolina. He's a battlefield junkie. That's what he said. I gave them all the pamphlets, and...and a tour of the house. They wanted to see all of it. They're excited about the idea of ghosts.\"\n\nPuzzled, he nodded. She was babbling like a brook, when he usually had to coax to get three sentences in a row out of her. \"Okay.\"\n\n\"Do you want some coffee? I'll get you some coffee,\" she said, and started to bolt before he could answer. \"And brownies. I made brownies this morning, and\u2014\" When he put a hand on her arm to stop her, she froze like a doe caught in headlights.\n\n\"Cassandra, relax.\"\n\n\"I am relaxed. I'm relaxed.\" His hand was firm, warm. She thought she could feel the texture of it through her skin, all the way to the bone.\n\n\"You're about to jump out of your shoes. Take a deep breath. Take a couple of them.\"\n\nObediently, she did, and felt some of the nerves settle. \"I'm fine, Devin.\"\n\n\"Okay, we'll have some coffee.\" But even as he started to lead her out, his beeper went off. \"Damn it.\" He strode to the candlestick phone on the gateleg to call in. \"MacKade. Yes, Donnie.\"\n\nDevin pressed his fingers against his eyes. Where had the headache come from, and why the hell was Cassie staring at him as if he'd grown two pounding heads?\n\n\"I'm on a call now, Donnie. Handle it. That's what I said. Look, put the damn poodle in lockup, along with those idiot women, if you have to, but\u2014\" He broke off, cursed himself, knowing Donnie would do exactly that.\n\n\"Abort that. Be diplomatic, Donnie, and do your job. You're going to have to fine the poodle lady, but do it privately and professionally. Suggest a fence. Remind her that the leash law is there for her pup's safety, as well as the public's. There's traffic on that street, and her little dog could get himself squashed. When you've handled that, you go over to the complainant, tell her it's been dealt with, and compliment her on her flowers. Suggest a fence. You know, how good fences make good neighbors. No, I didn't make that up. Go away, Donnie.\"\n\nHe hung up and turned to see Cassie smiling at him. \"A small dog problem,\" he explained.\n\n\"You're so good at that, and knowing how to handle people and put things right.\"\n\n\"I'm a regular Solomon.\" He blew out a breath. \"Sit down, Cassie. I need to talk to you.\"\n\n\"Oh.\" Her smile faded. \"Something's wrong.\"\n\n\"Not necessarily. Come on, let's sit down.\" Because he wanted to be able to hold her hand when he told her, he chose the curvy settee that always made him feel like a clumsy giant. \"I'm going to tell you first that there's nothing to worry about. That I don't want you to worry.\"\n\n\"It's about Joe.\" Her hand trembled once in his, then went still. \"They let him out.\"\n\n\"No.\" He squeezed her hand gently, reassuringly, and kept his eyes steady on hers. \"He's not going to be out of jail for a long time.\"\n\n\"He wants to see the children.\" She went dead pale, her eyes huge and dark and terrified. \"Oh, God, Devin, the children.\"\n\n\"No.\" He cursed himself, knowing he was only making it worse by trying to cushion the blow. \"It's nothing like that. It's the work release program. You know what that is.\"\n\n\"Yes, they let the prisoners out for a few hours to do jobs, community service. Oh.\" A single shudder escaped before she closed her eyes. \"That's it.\"\n\n\"He's working on a road crew. Trash and litter pickup. That sort of thing. I wanted you to know, and not worry. I've arranged to be informed of his schedule. I'll know exactly where he is, and so will you. I don't want you driving by one day and seeing him on the side of the road and getting scared.\"\n\n\"All right.\" The fear was there, but she could handle it. She'd handled worse. \"He's supervised.\"\n\n\"That's right.\" He wasn't going to bring up how often they misplaced a prisoner. She'd know it already. \"I'm going to drive by, or have one of the men drive by, wherever he's working, a couple of times a day. And, because I want you to feel secure about this, we'll do drive-bys here, too.\"\n\nAnd at the school, he thought, but he didn't want to bring up the kids again.\n\n\"He's still in prison,\" she said, to reassure herself. \"There are guards.\"\n\n\"That's right. Jared's working on a protest, but I should tell you\u2014 Damn it.\" He let out another breath. \"Your mother's for it, and she's been writing to the warden.\"\n\n\"I knew that.\" Cassie squared her shoulders. \"She and Joe are writing each other. She's showed me his letters. It doesn't make any difference, Devin. I'm never going back to that. I'm never letting my children go back to that. We'll be all right.\"\n\n\"You'll be fine.\" He was going to see to it. He tucked a stray curl behind her ear, relieved that she didn't jolt. \"I'm sorry I scared you.\"\n\n\"You didn't. Not really.\"\n\n\"Any time, Cassie, day or night, that you feel uncomfortable or uneasy, I want you to call me. You know I spend most nights at the office. I can be here in five minutes if you need me.\"\n\n\"I never feel uncomfortable or uneasy here. I'm hardly ever alone.\" When he lifted a brow, she smiled. \"Can't you smell them?\"\n\n\"The roses? Yeah.\" Now he smiled. \"Still, I'm usually better company than a ghost. You call me.\"\n\n\"All right.\" She had to draw together all her courage. A point had to be proved. He was her friend, always had been. She had to stop being a trembling little mouse. \"Thank you.\" She made herself smile, then laid a hand on his cheek, and touched her lips to his.\n\nHe barely tasted her, but the explosion ripped through his system like napalm. It was so unexpected, so long desired. He didn't realize his hand had tightened like a vise on her fingers, making her eyes go wide with shock. All he knew was that her lips had been on his, just for an instant.\n\nAnd he couldn't stand it.\n\nHe dragged her against him, and captured that taste again, devoured it, steeped himself in it. Warm, sweet. The shape of her mouth, that deep dip, drove him crazy. He crushed it under his, traced it with a frantic tongue, then dived deep to plunder.\n\nHis heart was thundering, wild surf against jagged rocks. His blood was racing, making his head buzz. She was everything soft and small and sweet, everything he craved, everything he cherished.\n\nIt took him several desperate moments to realize her hands were trapped between them. And she was rigid in his arms. Stunned, he let her go and leaped up in one frenzied motion.\n\nAnd she stared at him, eyes dark as rain clouds, one hand lifted to lie against the mouth he'd just savaged.\n\nThat was the word for it, he thought, disgusted. Savaged.\n\n\"I'm sorry.\" He was as pale now as she was flushed, and cursing himself viciously. \"I'm sorry,\" he said again. \"I'm...sorry. I didn't mean to\u2014 You caught me off guard.\" There was no excuse, he reminded himself, and his punishment for breaking her trust would be the losing of it. \"That was way out of line, and it won't happen again. I don't know what I was thinking of. I have to go.\"\n\n\"Devin\u2014\"\n\n\"I have to go,\" he repeated, almost desperately, as he backed up. He nearly tripped over a table, decided that would have capped things off nicely. Because she hadn't moved an inch, he was able to escape without further humiliating himself.\n\nShe listened to the door slamming behind him. No, she hadn't moved, because she couldn't. She didn't think it would be wise to try to stand just yet.\n\nWhat had just happened here? she asked herself. She had kissed him, thinking it was time she was able to make that friendly gesture.\n\nRafe kissed her all the time. When he came by the inn for something, he often kissed her, just the way she'd tried to kiss Devin. Lightly, casually. And after a while, she'd gotten used to it, and she no longer stiffened up.\n\nThen Devin had kissed her. But he didn't kiss like Rafe at all. No, not at all. She still had her fingers against her lips, and could still feel the heat there. No, she'd never been kissed like that before, by anyone. As if the man's life had depended on it. She'd never imagined Devin...\n\nOh, but she had, she remembered, letting her unsteady hand fall into her lap. She had imagined, just the night before. Had she dreamed her way into this?\n\nWhat had happened here was certainly reality. Her heart was pounding still, and her skin was hot. She'd been so shocked by what he'd done, the way he'd grabbed her, the way his mouth had covered hers, she hadn't been able to move.\n\nHow long had it lasted? Thirty seconds, a minute? She couldn't say, but so much had happened inside her. She was still shaky from it.\n\nHe'd been sorry. Of course he had, she thought, and leaned back, closed her eyes and tried to catch her breath. He hadn't meant to kiss her. It had just been some sort of spontaneous reaction. A male reaction. Then he'd found her lacking and let her go. Apologized. He was a good and honorable man, and he'd apologized for doing something he hadn't really meant to do.\n\nIt was just a kiss, she reminded herself, but had to press a hand to her jittery stomach. Now she'd spoiled things, because she hadn't been able to shrug it off, or laugh it off like a normal woman. Any more than she'd been able to respond to him and make him want to kiss her again.\n\nShe would make an effort, Cassie ordered herself, to behave as though nothing had happened. The very next time she saw him, she would smile and make natural conversation. She was getting better at those things. She simply couldn't bear it if they couldn't be friends anymore.\n\nShe got up on still-wobbly legs to finish her polishing. And didn't think of Joe Dolin at all.\n\nDevin worked like a fiend the rest of the day and all of the next. He drove his deputies insane, and drove out to the farm to extend the same courtesy to his younger brother.\n\nOf course, he told himself he'd come out to work. There were crops to be tended, and several of the cows that hadn't yet calved were due to drop. He found his services welcomed when one of the cows delivered breech.\n\nBy the time it was over and the new calf was teetering on its spindly legs, Devin was a mess. His shirt was ruined, his arm was bruised from being contracted inside the cow's birth canal. And he stank.\n\nIn the stall, Shane was equally dirty, and he was whistling cheerfully as he administered inoculations to the annoyed baby. \"There you go, pal. That didn't hurt much.\"\n\nDisgusted, Devin stared at him. It had been a hard, messy job, and it wasn't over. The stall would have to be cleaned out and fresh hay spread, and the calf would need watching for the next couple of hours.\n\nAnd there was Shane, kneeling in the muck, happy as a fool.\n\nHe'd been letting his hair grow lately, Devin noted, and he'd pulled a tail of it through the opening in the back of his grimy cap. His green eyes, shades paler than Devin's, were dreamy, and his mouth was curved next to his dimple. He was sinfully good-looking, even for a MacKade. And he was the baby of the family, even younger than Devin, which meant that his older brothers had been honor-bound to kick his butt well and often.\n\nAs he continued to whistle, Devin gave serious thought to doing so now. \"What the hell are you so happy about?\"\n\n\"Nice healthy calf, from the look of him.\" Despite the calf's strong objections, Shane was holding him still and examining his eyes and ears. \"Mama's doing fine now. What's not to be happy about?\"\n\n\"She damn near broke my arm.\"\n\n\"She couldn't help it,\" Shane said reasonably. \"Besides, I told you I'd take that end. You insisted.\"\n\n\"Yeah, right. This place is a mess.\"\n\n\"Birthing's not neat.\" Shane stood and rubbed his filthy hands on his equally filthy jeans. He stepped out of the stall and leaned against the open door. \"Besides, I thought this might sweat the mood out of you.\" His grin was cocky, confident\u2014all the more reason for Devin to want to punch it in. \"Women trouble, right?\"\n\n\"I don't have women trouble.\"\n\n\"That's 'cause you don't have any women\u2014which, I might add, is an embarrassment to all of us. Why don't you take one of mine? I've got plenty.\"\n\nDevin answered the suggestion with the crude and expected response before he stepped over to the sink to wash his hands.\n\n\"No, really. You know who I think would be good for you? Frannie Spader. She's got all this red hair that just sort of tumbles all over the place, and the cutest smile. And when you get past the hair and the smile, she's got a body that can make a man whimper. I don't think you've done nearly enough whimpering lately.\"\n\n\"I'll pick my own women. I don't need your damn castoffs.\"\n\n\"Just being brotherly.\" He slapped Devin on the back before reaching for the soap. \"Of course, if you weren't so damn brotherly yourself, you could probably be making time with little Cassie\u2014\"\n\nIt was a tribute to Devin's speed, and Shane's innocence, that the blow caught Shane solidly on the jaw and sent him flying. He landed hard, shook his head. Before he could ask Devin what the devil had gotten into him, he was assaulted by a hundred and seventy-five pounds of furious, frustrated male.\n\nThey were well matched, knew each other's moves and rhythms. The barn echoed with grunts, the smack of flesh against bone, curses, as they rolled over the dusty concrete floor.\n\n\"Oh, for heaven's sake.\"\n\nThe female voice, and the disdain in it, didn't register on either of the combatants. Shane dropped his guard just long enough to be rewarded with a split lip, and answered it by bloodying Devin's nose.\n\n\"But, darling, it looks like they've just gotten started.\"\n\n\"I mean it, Rafe.\" With a heavy sigh, Regan MacKade shifted the gurgling baby on her hip. \"Break it up.\"\n\n\"Women,\" he muttered. But he would break it up his way, which was to dive into the fray, and get in a few licks of his own. Knowing he couldn't enjoy himself for long, he managed to shove Shane aside and sit on Devin.\n\n\"Stay out of this.\" Swiping at blood, Shane hauled himself to his knees. \"It's between him and me.\"\n\n\"Maybe I will.\" Rafe was having quite a bit of trouble holding Devin down. To prove he meant to, he covered Devin's grimy face with the flat of his hand and gave it enough of a shove to have his head rapping against the concrete. \"And maybe I want to play,\" he added. \"What's it about?\"\n\n\"Ask him.\" Already cooling off, Shane flexed his sore hand. \"I was just talking to him, and he punched me.\"\n\n\"Well, I want to punch you half the time you're talking to me,\" Rafe said reasonably, and looked down to see that Devin's eyes were clearing. He hadn't meant to rap his brother's head quite that hard. \"What were you talking to him about?\"\n\n\"Stuff. Women.\"\n\nDevin's vision was coming back, and so was his temper. He started to heave Rafe aside when Regan's firm, no-nonsense voice stopped him.\n\n\"That's just enough of this ridiculous behavior, Devin. You should be ashamed of yourself.\"\n\nStill on top of him, Rafe looked down and grinned. \"Yeah, Dev, you should be ashamed of yourself.\"\n\n\"Get the hell off me.\"\n\n\"You going to be a good boy?\" With a laugh, Rafe leaned over and kissed him. He was quick, and agile, and sprang away before Devin could retaliate.\n\n\"A fine thing,\" Regan said from the doorway of the barn, making Devin think twice about jumping Shane again. She stood there in tailored slacks and a crisp spring blazer, a wide-eyed baby on her hip, a polished leather shoe tapping. \"Wrestling in the barn like a couple of bad-tempered boys. Look at the two of you\u2014you're filthy, bloody, and your clothes are torn.\"\n\n\"He started it.\" Wisely, Shane held back a laugh, and tried to look humble. \"Honest, Regan, I was just defending myself.\"\n\n\"I'm not interested in who started it,\" Regan said regally, and deflated her brother-in-law with one snippy look. \"I believe we were invited to dinner.\"\n\n\"Oh, yeah.\" Shane had forgotten about that. \"We had a little trouble with a birthing. Breech calf. We just got finished.\"\n\n\"Oh.\" Instantly Regan was all concern. Tossing back a curtain of honey-brown hair, she hurried inside. \"Is it all right?\"\n\n\"Just dandy. Hey, Nate.\"\n\n\"No, you don't.\" Even as the cooing baby held out his arms to his uncle, Regan turned aside. \"You're filthy. The two of you go clean up.\"\n\nDevin eyed Shane narrowly, then hissed out a breath. \"I felt like pounding somebody. You were available. You also have a big mouth.\"\n\nShane dabbed at the blood on his lip. \"You sucker-punched me.\"\n\n\"So?\"\n\n\"So I owe you one.\"\n\n\"That's it boys, kiss and make up.\"\n\nWhen both Shane and Devin turned on Rafe, Regan gritted her teeth. \"Stop right there. If nobody punches anyone else, I'll cook dinner.\"\n\n\"Good deal,\" Shane decided.\n\n\"But you're not coming in the kitchen until... What's that noise?\"\n\n\"What noise?\" Devin unclenched his ready fist and listened. The whimpering sound was soft, barely audible over little Nate's babbling. Homing in on it, he strode halfway down the barn and looked into another stall. \"Looks like it's the day for birthing. Ethel's having her babies.\"\n\n\"Ethel.\" Like a frantic papa, Shane bolted down the barn, and all but fell into the stall beside his laboring pet. \"Oh, honey, why didn't you call me? Jeez, she's already had two.\"\n\n\"Fred's probably out passing out cigars.\" At the entrance to the stall, Rafe leaned over and kissed his wife, then his son. \"I know just how he feels.\"\n\nSeeing the panic in Shane's eyes, Devin shook his head. They'd witnessed or assisted in countless births with the stock over the years, but that meant nothing now. This was Ethel, and she was as close to a true love as Shane had ever known. He stepped in, crouched down beside his brother.\n\n\"She's doing fine.\" He hooked an arm over Shane's shoulders.\n\n\"You think?\"\n\n\"Sure. She's a MacKade, isn't she?\" Devin glanced up at Regan and winked. \"MacKade women are the best there is.\"\n\nAfter the birthing, the cleaning up, the cooking and the celebrating of Fred and Ethel's six healthy puppies, Devin drove back to the office. He was too restless to stay at the farm. Though he had taken a long, soaking bath to soothe out the worst of the aches his scuffle with Shane had caused, he still wasn't able to fully relax.\n\nHe slowed down as he passed the inn, saw lights shining on the second and third floors. Grimly he punched the gas again and headed into town.\n\nShe wasn't going to forgive him easily, he thought. He wasn't going to forgive himself. He'd acted like a maniac. He'd been rough and demanding when she deserved, and should have expected, a gentle touch.\n\nNo wonder she'd looked at him as though he'd lost his mind, her eyes round in shock, her soft, pretty mouth trembling.\n\nHe'd make it up to her somehow, eventually. He knew how to bide his time, didn't he? He'd been waiting for her nearly half his life already.\n\nJoe Dolin was also biding his time. His cell was dark, but he wasn't sleeping. He was planning. He knew most people thought he wasn't very smart, but he was going to show them, all of them, soon. He'd learned how to play the game, to say what the guards and the psychiatrists and the fat-faced warden wanted to hear. He'd learned how to act as they wanted him to act.\n\nHe could be humble. He could be repentant. He could be anything he had to be. As long as it got him out.\n\nDevin MacKade thought he'd proved something, driving by the work site, flashing his badge. Oh, he owed Devin MacKade. Big-time. He hadn't forgotten the way Devin had come after him, had cuffed him and tossed him into a cell. No, he hadn't forgotten what he owed Devin. There would be payback.\n\nBut Cassie would come first, because he owed her most of all. Everything would have been fine if she'd stayed in her place. But she'd gone whining to MacKade, sniveling about their personal business.\n\nA man had a right to punish his wife, to give her the back of his hand or let her feel his fist when she needed it. And Cassie had needed it a lot. She still did.\n\nNo fancy divorce papers changed that. She was his wife, his property, and he was going to be reminding her of that before too much longer.\n\nTill death do us part, he thought, and smiled into the dark.\n\n## Chapter 4\n\nParade day was a tactical nightmare. That was to be expected. Over and above his usual reasons, Devin was looking forward to it, because it would keep him too busy to think about any personal problems.\n\nThe parade would kick off at twelve sharp\u2014which meant anytime between noon and twelve-thirty\u2014with the usual speeches at the square and the ceremonial laying of the wreath at the memorial.\n\nAs sheriff, he was required to be there, in full uniform. He could handle it. There were only a handful of days out of the year when he had to drag out the dress khakis and tie and shiny black shoes.\n\nOf course, that meant dragging out the ironing board, as well, which he hated. It was the only domestic chore he truly despised, and the only one that jittered his nerves.\n\nBut by 8:00 a.m. he was pressed and dressed and out on the street. Already there were eager beavers claiming their spots, holding spaces along the curbs and sidewalks for others with lawn chairs and coolers.\n\nMost of the storefronts and shops along the parade route were closed for the day, but he could count on Ed's being open for breakfast.\n\nHe sauntered down the sidewalk, knowing he had the best part of an hour before he had to worry about crowd control or making certain the concessionaires were in their proper places with their balloons and hot dogs and ice cream.\n\nSummer had decided to make its debut on parade day. It was already hot, and he tugged irritably at his collar.\n\nHe imagined the tar on the street would be soft and melting by afternoon. He hoped the little girls who did their tumbles and cartwheels in their spangled uniforms were prepared.\n\nHe made a note to make certain there was plenty of water along the route for the marchers. He didn't want anybody fainting on him.\n\nIt might be a holiday, but Ed's was doing a brisk business. He could smell ham frying, coffee brewing. The scent reminded him that he'd been off his feed for the past couple of days.\n\nAfter exchanging a few greetings with patrons in booths, he sidled up to the counter and took a stool.\n\n\"Sheriff.\" Ed winked at him. As usual, her rhinestone glasses were dangling on a pearl-studded chain against her scrawny chest. She wore a splattered apron, but beneath it she was ready for the celebration in a snug, midriff-baring top as red as her hair, and shorts that barely met the limits of the law.\n\nShe had bright blue shadow all the way up to her penciled brows, and her mouth was stop-sign red. Poppies dangled from her ears and were pinned to her apron.\n\nDevin grinned at her. Only Edwina Crump could get away with an outfit like that.\n\n\"Ham and eggs, Ed, and keep the coffee coming.\"\n\n\"You got it, sweetie.\" Though she was old enough to be his mother, she fluffed her hair and flirted. \"Don't you look handsome in your uniform!\"\n\n\"I feel like an aging Boy Scout,\" he grumbled.\n\n\"One of my first beaus was a Boy Scout.\" She wiggled her brows as she took the clear plastic top off a plate of doughnuts and chose one for him. \"He was surely prepared, let me tell you. On the house,\" she added, casting a sharp eye over her two scrambling waitresses.\n\nShe left Devin with his coffee and doughnut before heading back into the kitchen.\n\nHe tried not to brood, really. To keep himself sane, he set his clipboard on the counter and read over his notes and itinerary. A half hour later, he was doing some fine-tuning and trying to enjoy Ed's very excellent ham and eggs.\n\n\"Hi there, Sheriff. Locked anybody up lately?\"\n\nHe swiveled on the stool and looked into the stunning and not altogether friendly face of his sister-in-law. Savannah MacKade always made a statement, Devin thought. When that lush siren's body sauntered into a room, men's hearts stopped. There was all that thick black hair falling past her shoulders, those almond-shaped eyes the color of sinful chocolate, and those ice-edged cheekbones against gold-dust skin.\n\nAnd there was, Devin mused, all that attitude.\n\n\"As a matter of fact, no, not lately.\" He grinned at the boy beside her\u2014his nephew, whether Savannah liked it or not. Tall for his age, and as dark and handsome as his mama, Bryan was sporting his baseball uniform and fielder's cap. \"Riding in the parade today?\"\n\n\"Yeah. Me and Con and the guys are riding in the coach's pickup. It'll be cool.\"\n\n\"Kind of early, aren't you?\"\n\n\"We had some things to pick up,\" Savannah supplied. \"Including Connor. We're on our way to get him as soon as Bryan here fills his stomach.\"\n\n\"I'm starving,\" the boy claimed and, eyeing the plate of doughnuts, he leaped onto the stool beside Devin.\n\n\"Hey, Ed, you got a starving boy out here.\"\n\n\"I'm coming.\" She slapped the swinging door of the kitchen open and strolled out. Her grin flashed at Bryan. \"Well, it's my champ.\" As sponsor of the Antietam Cannons, Ed preened with pride. \"Hell of a game Saturday.\" She saluted Savannah, leaned over the counter long enough to coo at the baby in the stroller, then fell into a deep and serious discussion with Bryan about food and baseball.\n\nDevin didn't ask. He'd be damned if he would. He slid off the stool long enough to pick up his niece, then settled back down with the wide-eyed Layla on his lap.\n\nBeneath the frilly sun hat, Layla's hair curled thick and dark. Her mouth\u2014her mother's mouth, Devin mused\u2014was serious as she watched him out of big eyes that were already easing from birth blue to MacKade green.\n\n\"Hello, beautiful.\" He bent over to kiss her, and was pleased to see that pretty mouth curve. \"She smiled at me.\"\n\n\"Gas.\"\n\nDevin looked up into Savannah's bland eyes. \"The hell it is. She smiled at me. She loves me. Don't you, Layla? Don't you, darling?\" He traced a finger over her hand until she gripped it. \"She's got MacKade eyes.\"\n\n\"They're still changing,\" Savannah claimed. But she was softening. Despite the badge, and the fact that she tried to resist him, she grew fonder of Devin every day. \"They might turn brown.\"\n\n\"Nah. MacKade eyes.\" He looked up again, smiled at her. \"You're stuck with them. With us.\"\n\n\"Apparently.\"\n\nHis grin only widened. He knew she liked him, no matter how cool she tried to be. \"Want a doughnut?\"\n\n\"Maybe.\" She gave up and slid onto a stool. \"You don't have to hold her.\"\n\n\"I want to hold her. Where's Jared?\"\n\n\"Doing some lawyer thing. He's going to meet us at the inn about nine-thirty.\"\n\n\"So, you haven't been by yet,\" Devin said casually, very casually, as he shifted Layla to his shoulder and rubbed her back.\n\n\"No.\" Savannah bent down to take a cloth from the stroller and smoothed it over Devin's shoulder. \"I nursed her right before we left. She's liable to spit up all over that pretty cop suit.\"\n\n\"Then I wouldn't have to wear it. You're just picking up Connor?\"\n\n\"Mm-hmm...\" With an expert's eye, Savannah selected her own doughnut. \"Rafe and Regan are swinging by later to get Cassie and Emma. Shane's going to drive Jared in so we don't have so many cars when we head to the park for the picnic.\"\n\nShe glanced over, saw that her son was well on his way to demolishing the two doughnuts Ed had given him. \"You angling for a ride?\"\n\n\"No. I've got to take the cruiser so I can pretend I'm working.\"\n\n\"I didn't see you at the game Saturday.\"\n\n\"I swung by for a couple of innings.\" He'd spotted Cassie in the stands, and he hadn't wanted to make her uncomfortable.\n\n\"You didn't make it to Sunday dinner yesterday at the farm.\"\n\n\"Did you miss me?\"\n\n\"Not particularly.\" But there was something in his eyes that wiped the sneer off her face. \"Is something wrong, Devin?\"\n\n\"No.\"\n\n\"Jared told me about Joe Dolin, the work release. It's bothering you.\"\n\n\"That's a mild term for it. I'm keeping my eye on him,\" he murmured, and turned his face into Layla's sweet-smelling neck to nuzzle.\n\n\"I'll bet you are,\" Savannah murmured. She brushed a hand over her daughter's head, then let it rest on Devin's shoulder in a gesture of affection and support that surprised both of them.\n\n\"Am I growing on you, Savannah?\"\n\nShe let her hand drop, but the corners of her mouth quirked up. \"Like you said, I'm stuck with you. Now give me my kid.\"\n\nDevin settled Layla in her mother's arms, then kissed Savannah, firm and quick, on the mouth. \"See you. See you, Bry,\" he added as he rose.\n\nBryan mumbled something, hampered by a mouthful of apple-filled doughnut.\n\n\"Damn MacKades,\" Savannah said under her breath. But she was smiling as she watched Devin stride away.\n\nBy noon, the town was bursting at the seams. People crowded the sidewalks and spilled over porches and front yards. Kids raced everywhere at once, and the bawling of fretful babies rose through the air like discordant music.\n\nSeveral streets were barricaded to keep the parade route clear. Devin posted himself at the main intersection so that he could soothe travelers who had forgotten about parade day, or were from far enough out of town that they'd never heard of it.\n\nHe offered alternate routes, or invitations to park and join the festivities.\n\nThe two-way radio hitched to his belt belched and squawked with static or calls from deputies placed at distant points along the route.\n\nAcross the street from him, at the corner of the gas station, a clown sold colorful balloons. Half a block down, ice cream and snow cones were big sellers. They melted in the heat almost as soon as they were passed from hand to hand.\n\nDevin looked at the wrappers, the spills, the bits of broken toys and balloons. Cleanup was going to be a bitch.\n\nThen, in the distance, he heard the first of the marching bands approaching the square. The brassy music, the click-clack of booted feet, had his practical frame of mind shifting into the pleasures of his youth.\n\nWhat the hell\u2014there was just nothing like a parade.\n\n\"Officer! Officer!\"\n\nResigned, Devin turned back to the barricade, where another car had pulled up. With one look, he summed up the middle-aged couple in the late-model sedan as hot, frazzled and annoyed.\n\n\"Yes, ma'am.\" He leaned down to the open window and gave them his best public-servant smile. \"What can I do for you?\"\n\n\"We have to get through here.\" The driver's irritated tone carried the flavor of the North that went with his Pennsylvania tags.\n\n\"I told you not to get off the highway, George. You just had to take the scenic route.\"\n\n\"Be quiet, Marsha. We have to get through,\" he said again.\n\n\"Well, now.\" Devin ran his hand over his chin. \"The problem here is that we've got a parade going on.\" To prove it, the marching band let out a blare of trumpets, a boom of drums. Devin pitched his voice over the din. \"We won't be able to open this road for another hour.\"\n\nThat sparked a heated domestic argument, demands, accusations. Devin kept the easy smile on his face. \"Where y'all headed?\"\n\n\"D.C.\"\n\n\"Well, I'll tell you what you can do, if you're in a hurry. You turn around and head straight up this road for about five miles. You're going to see signs to route 70. Take the eastbound. You'll hit the Washington Beltway\u2014that's 495\u2014in just about an hour.\"\n\n\"I told you not to get off the highway,\" Marsha said again.\n\nGeorge huffed. \"How was I supposed to know some little one-horse town would block off the streets?\"\n\n\"If you're not in a hurry,\" Devin continued, calm as a lake, \"you can turn around and pull into that field where there's a sign for parking. It's free. We got a nice parade here.\" He glanced over as a junior majorette tossed up her baton and snagged it, to the forceful applause of the crowd. \"I can give you a nice, pretty route into D.C.\"\n\n\"I haven't got time for any damn parade.\" Puffing out his cheeks, George slapped the sedan in reverse. Devin could hear them arguing as he jockeyed the car into a turn and headed off.\n\n\"Ain't that a shame...\" Devin muttered, and turned, nearly knocking Cassie over. He grabbed her instinctively, then let her go as if her skin had burned his hands. \"Sorry. Didn't see you.\"\n\n\"I thought I should wait until you'd finished being diplomatic.\"\n\n\"Yeah. George and Marsha don't know what they're missing.\"\n\nSmiling, she watched the senior majorettes twirl and tumble. But in her mind she was still seeing Devin in his uniform. So competent and male. \"I know. You must be hot. Would you like me to get you a drink?\"\n\n\"No, I'm fine. Ah...\" His tongue was in knots. He didn't know the last time he'd seen her in shorts. And over the years he'd done his best not to think about her legs. Now here they were, all long and smooth, showcased by neat little cuffed shorts the color of plums. \"Where's Emma?\"\n\n\"She's made friends with the little McCutcheon girl, Lucy. They're in her yard.\" It was easier to talk to him if she wasn't looking at him, so Cassie concentrated on the slow-moving convertible and its passenger, the waving and flouncily dressed current agriculture princess. \"Are you angry with me, Devin?\"\n\n\"No, of course not.\" He stared so hard at the princess that she flashed him a brilliant, hopeful smile, and a very personal wave. But it was Cassie he saw, looking shocked and delicate. And beautiful.\n\n\"You've flustered Julie,\" Cassie murmured, noting the exchange.\n\n\"Julie? Who's Julie?\"\n\nHer quick laugh surprised them both. Then they were staring at each other. \"Are you sure you're not mad?\"\n\n\"No. Yes. Yes, I'm sure.\" He jammed his hands into his pockets, where they would be safe. \"Not at you. At me. Like I said, I was out of line the other day.\"\n\n\"I didn't mind.\"\n\nThe blare of the next band rang in his ears. He was sure he hadn't heard her correctly. \"Excuse me?\"\n\n\"I said I\u2014\" She broke off when his two-way squawked.\n\n\"Sheriff. Sheriff, this here's Donnie. We got a little situation down to quadrant C. You there, Sheriff?\"\n\n\"Quadrant C, my butt,\" Devin muttered. \"He's at the elementary school. Watching too many Dragnet reruns.\"\n\n\"I'll let you go,\" Cassie said quickly as he whipped out his two-way. \"You're busy.\"\n\n\"If you'd\u2014\" He cursed again, because she was already hurrying through the cheering crowd. \"MacKade,\" he snapped into the receiver.\n\nThe little situation turned out to be a harmless brawl between overly loyal students at rival high schools. Devin broke it up, snarled at Donnie, then helped a mother deal with her terrified daughter, who had lost her breakfast over the idea of twirling her baton in public.\n\nBy the time the last marching boot clicked, the last flag waved and the last balloon drifted into the sky, he had to oversee the traffic headed for the park and the cleanup detail, and help a couple of weeping lost children find their way back to Mama.\n\nHe took his time cooling off under the stingy spray of his office shower, then gratefully retired his uniform until the next official event. By the time he made it to the park and snuck the cruiser in behind a trail of cars, the picnic, with its grilling food and boisterous games, was well under way.\n\nThere was softball, horseshoes, pitching contests, egg-throwing contests, three-legged races. He saw Shane nuzzling Frannie Spader, the curvy redhead he had so generously offered Devin a few days before.\n\nThere was Rafe, stepping up to bat, and Jared winding up to pitch. Regan and Savannah were spread out in the shade with their babies.\n\nThere were dogs and kids, big-bellied men sitting in lawn chairs, discussing sports and politics, old women fanning themselves and laughing. There was Cy, the town mayor, looking ridiculous as always, sporting a pair of violently checkered Bermuda shorts that exposed far too much of his hairy legs.\n\nMrs. Metz was shouting encouragement to her grandchildren, gnawing on a chicken leg and gossiping with Miss Sarah Jane.\n\nGood God, Devin thought, he really loved them. All of them.\n\nHe wandered over the grass, stopping here and there to chat or listen to a complaint or a snippet of news. With his hands tucked in his back pockets he watched solemnly with old Mr. Wineburger as horseshoes were tossed and clanged against the pole.\n\nHe was debating different techniques of horseshoe pitching when Emma came up quietly and held out her arms. He picked her up, settled her on his hip while Wineburger wheezed out opinions. But Devin's mind had begun to wander.\n\nLittle Emma smelled like sunshine and was as tiny as a fairy. But she was nearly seven now, he recalled with a jolt. Soon she wouldn't want to be picked up and held. She would, like the young girls he saw over at the edge of the field, be flirting with young boys, want to be left alone to experiment with being female.\n\nHe sighed and gave her a quick squeeze.\n\n\"How come you're sad?\" she wanted to know.\n\n\"I'm not. I'm just thinking that you're growing up on me. How about a snow cone?\"\n\n\"Okay. A purple one.\"\n\n\"A purple one,\" he agreed, and set her down. Hands linked, they walked toward the machine manned by the American Legion. He bought two, then settled down with her on the grass to watch the softball match.\n\n\"Come on, Dev!\" From his position at second, Rafe shouted to his brother. \"Batter up!\"\n\n\"I'm not moving. I've got me a pretty girl here,\" he shouted back.\n\n\"Mama says I'm pretty, too.\"\n\nHe smiled at Emma, ruffled her hair. \"That's because you are.\"\n\n\"Mama's pretty.\"\n\n\"She sure is.\"\n\nEmma cuddled closer, knowing his arm would come around her, just the way she liked it. \"She hardly ever cries anymore.\" In her innocence, she licked at the snow cone and didn't notice the way Devin's arm went taut. \"She used to cry all the time, at nighttime. But now she doesn't.\"\n\n\"That's good\" was all Devin could manage.\n\n\"And we got to have Ed the kitten, and a brand-new house, and nobody yells and breaks things or hits Mama now. Connor gets to play baseball and write stories, and I can have Lucy come right to my room to play. I've got pretty curtains, too, with puppies on them. And new shoes.\"\n\nShe wiggled her pink sneakers for Devin's benefit.\n\n\"They're very nice.\"\n\n\"It's 'cause you made him go away, the bad man. Connor said you arrested him and sent him to jail and now he can't hit Mama and make her cry.\" She looked up at him, her mouth circled with sticky purple, her eyes wide and clear. \"I love you.\"\n\n\"Oh, Emma...\" Undone, he lowered his brow to her soft golden curls. \"I love you, too. You're my best girl.\"\n\n\"I know.\" She puckered her purple lips and planted a sticky kiss on his cheek. \"I'm going to get Lucy now. She's my very best friend.\" She got to her feet, smiled her mother's soft smile. \"Thank you for the snow cone.\"\n\n\"You're welcome.\"\n\nHe watched her dance off, pretty as a pixie, then rubbed his hands over his face. It was hard enough being in love with the mother. What the hell was he going to do with this need for the child?\n\nWas he going to have to settle\u2014always\u2014for protecting, for watching over, for being the dependable friend, the favored honorary uncle?\n\nHe was getting damn sick of it, of holding in, of holding back.\n\nThis time, when Rafe called out, Devin got to his feet. Yeah, he thought, he'd batter up, all right. God knew he needed to hit something.\n\nThere was something intrinsically satisfying about smacking a little white ball with a slim wooden bat. It was the connection, the way the power of it sang up the arms. It was the sound, the solid crack, the whoosh of air, the rising cheers as the ball lifted.\n\nHe was feeling human by the time he rounded third and headed for home. More than human, since it turned out to be Shane guarding the plate. His lips peeled back in a feral grin matching his brother's as he went into a hard, bruising headfirst slide.\n\nThere was the brutal collision of flesh and bone, the swirl of choking dust, the hysterical screams of fans and teammates. He heard Shane grunt as his elbow whipped around to catch his brother in the ribs, beside the padded catcher's vest. He saw stars as some bony part, probably Shane's knee, caught him beside the ear.\n\nBut what he heard over it all was the glorifying call of \"Safe!\"\n\n\"I'll be damned.\" Shane had managed to hold on to the ball that Jared had bulleted to him, even after the nasty collision. \"I tagged the sucker,\" Shane insisted, waving the ball for emphasis.\n\nCy, the umpire, hung tough. \"You weren't on the plate, Shane. Devin was. You didn't get the tag in time.\"\n\nThat, of course, was tantamount to a declaration of war.\n\nFrom the sidelines, Savannah watched the very polished attorney Jared MacKade go nose-to-nose with the town's mayor, while her brothers-in-law shouted at each other, and anyone else who happened to get in the way.\n\n\"I love picnics,\" Savannah commented.\n\n\"Mmm... Me too.\" Regan stretched her arms. \"They're so relaxing.\" She smiled up at Cassie, who stepped under the shade with them. \"Don't worry,\" she said, noting the way Cassie hugged her arms. \"They won't hurt each other. Very much.\"\n\n\"I know.\" She tried not to be so poor-spirited. The MacKades were always yelling. But she hugged herself tighter when she saw Connor and Bryan race up to get a piece of the action.\n\n\"Don't worry,\" Regan said again.\n\n\"No, I won't.\"\n\nIt was good, wasn't it, that Connor could race and shout that way? He'd been too quiet for too long. Too worried, she thought guiltily. He was coming into himself more and more every day. And if picking sides over a baseball call made him happy, then no, she wouldn't let herself worry.\n\nIt was over soon enough, with vows of revenge and retaliation. She watched Bryan do a victory boogie, then nag until he was allowed up to the plate. Devin picked up a mitt, bent over and said something that had Connor goggling with pleasure. Her son raced into the outfield and joined the game.\n\n\"He's awfully good with children,\" Cassie murmured. \"Devin,\" she added.\n\n\"Every time he comes by the house, he has Nate on his hip the minute he steps through the door.\" Regan smiled down at her son, who was busy chewing on a bright red teething ring. \"He's bleeding.\"\n\nAlarmed, Cassie looked down at Nate. \"Where?\"\n\n\"No, I meant Devin. His mouth's bleeding. Anyone got a tissue?\"\n\n\"I do.\" Cassie pulled one out of her pocket.\n\nAs she hurried over to where Devin was walking to the outfield, Regan grinned. \"She hasn't figured it out yet, has she?\"\n\n\"Nope.\" Savannah leaned back against the tree. Layla was napping, and that seemed like a wonderful idea. \"He's going to have to do something a little more obvious for her to realize he's crazy about her.\"\n\n\"He's the only MacKade I know who moves slow.\"\n\nSavannah arched a brow before she closed her eyes. \"I'll bet he moves fast enough when the time comes. Cassie won't have a chance.\"\n\n\"No,\" Regan said softly. \"She'll have the best chance of her life.\"\n\nOut of breath from the effort of catching up with his long strides, Cassie called out, \"Devin! Wait a minute!\"\n\nHe glanced around, saw her rushing after him and did what he'd trained himself to do. He put his hands in his pockets. \"What?\"\n\n\"Your mouth. Gosh, you must be all leg,\" she managed, puffing, when she stopped in front of him.\n\n\"My mouth?\"\n\n\"It's bleeding.\" In practiced maternal gestures, she dabbed at the corner of his mouth. \"I saw you dive headfirst into Shane. I had to close my eyes. You're lucky you only cut your lip doing something that crazy. It's only a game.\"\n\n\"It's baseball,\" he reminded her, and struggled not to groan as her fingers gently soothed the wound he hadn't even been aware of. \"I got the run.\"\n\n\"Yes, I know. I'm learning all the rules and terms. RBIs and ERAs. Connor's so excited about playing. It was sweet of you to let him go into left field.\"\n\n\"Right. Right field,\" Devin managed as his heart jitterbugged in his chest. He kept his hands balled into fists in his pockets. \"Cassie, I'm fine.\"\n\nIt was the tone, the sharp impatience in it, that had her stopping. \"You are mad at me.\"\n\n\"I'm not mad at you. Damn it, I'm not mad. Look.\" Frustrated beyond belief, he snatched the blood-spotted tissue from her hands. \"What's this?\"\n\n\"It's blood. I told you your mouth\u2014\"\n\n\"Blood,\" he said, interrupting her. \"That's what I've got in my veins. Blood, not ice water. So if you're going to keep leaning up against me and putting your hands on my face, I\u2014\" He cut himself off, clenched his teeth. \"I'm not mad,\" he said, more calmly. \"I need to take a walk.\"\n\nCassie gnawed at her lip as he strode away into the little grove of trees that lined the east side of the park. The idea of losing his friendship gave her all the courage she needed to follow him.\n\nHe stopped, turned, and the heat in his eyes was like an arrow in her heart. \"I'm sorry,\" she said quickly. \"I'm sorry, Devin.\"\n\n\"Don't say you're sorry to me, Cassie, you have nothing to apologize for.\" Where the hell was everybody? he asked himself. Why weren't there people in the grove? He couldn't risk being alone with her now, when he didn't have himself under complete control. \"Go on back, Cassie. Go on, now.\"\n\nShe started to. It was second nature for her to do as she was told. But she couldn't, not this time. Not when it was so important. \"If you're not mad, then you're upset. I don't want to be the cause of that.\"\n\nIt was hard, almost terrifying, to step forward, when there was still temper simmering in his eyes. She knew he wouldn't hurt her, of course she knew, but there was a part of her that couldn't be entirely sure. But for Devin she'd risk it.\n\n\"It's because I kissed you,\" she blurted out. \"I didn't mean anything by it.\"\n\nThe temper drained from his eyes. They were blank now, carefully blank. \"I know you didn't.\"\n\n\"You kissed me back.\" Her heart was pounding so hard she could barely hear herself speak. \"You said you were angry with yourself for doing it, but I don't want you to be. I didn't mind.\"\n\n\"You didn't mind,\" he repeated, spacing out the words. \"Okay. We'll put it aside. Go on back now.\"\n\n\"Why did you kiss me like that?\" The words ended on a whisper as her courage began to flag.\n\n\"Like I told you, you caught me off guard.\" When she only continued to stare at him with those big, soft eyes, he felt something snap. \"Damn it, what do you want from me? I apologized, didn't I? I said it wouldn't happen again. I'm trying to stay away from you, and it's killing me. I've waited to kiss you for twelve years, and when I do I practically eat you alive. I didn't mean to hurt you.\"\n\nHer knees were starting to shake, but it didn't feel like fear. She knew fear well enough to recognize it. But whatever this was that was working through her was unfamiliar.\n\n\"You didn't hurt me.\" She had to swallow. \"I didn't mind. I don't mind.\"\n\nHe was trying to get a bead on her, but wasn't sure of his aim. \"I want to kiss you again.\"\n\n\"I don't mind,\" she repeated, because it was the best she could do.\n\nShe didn't move as he stepped toward her, had no idea if she should touch him. She would have liked to run her hands up those arms, they were so strong. But she wasn't sure.\n\nThen she didn't have to worry, or think, or try to guess. He laid his hands on her cheeks, framing her face, and lowered his mouth to hers, so gently, so patiently.\n\nHer heart fluttered, and the sensation was sweet, like something flying silently out of a cage when the door has been opened unexpectedly. When he drew her closer, just a little closer, she thought she floated toward him. Her lips parted on a sigh of quiet wonder.\n\nThis was what he meant to do, always. Show her tenderness and care. Let himself slide into her, slowly, gently. The dappled shade was perfect, sweetened by the call of birds and the tang of wildflowers.\n\nThis was what he'd meant to do, he thought hazily, and deepened the kiss with patient skill until she sighed again.\n\nAnd all the years he'd waited and wanted seemed like minutes, now that she was here, with him.\n\nThe sound of the shouts and laughter from the field beyond was like the buzz of happy bees in her head. She didn't realize she'd lifted her hands, curled them around his wrists, until she felt the strong quick beat of his pulse against her fingers. She held on as lovely colors began to revolve in her head, as the kiss went on and on, spinning out time.\n\nHe didn't let her go until her hands had slipped weakly from his wrists to fall to her sides.\n\nHer eyes were still closed when he lifted his head, when he moved his hands from her face to her shoulders. As he watched, she pressed her lips together, as if to draw in that last taste, and savor it.\n\n\"Cassie.\"\n\nShe opened her eyes, and they were heavy and clouded and confused. \"I don't know what to say now.\" Yes, she did, she realized. \"Will you kiss me again?\"\n\nTwelve years of repression kept him from groaning out loud. \"Not just this minute,\" he said, and held her at arm's length. Any closer, and he might just toss her over his shoulder and carry her off behind some handy rock. He wasn't sure either of them was ready for that. \"I figure we ought to spread it out a little.\"\n\n\"No one's ever kissed me like that. Made me feel like this.\"\n\n\"Cassie.\" The words had his libido growing fangs. Snapping down on it, he took her hand. \"Let's go back. I...haven't had lunch.\"\n\n\"Oh, you must be starving.\"\n\n\"Right.\" He could almost laugh at himself as he pulled her back onto the field.\n\n## Chapter 5\n\n\"I really appreciate this, Cassie.\" Regan tucked a giggling Nate into his portable swing, then bent over to kiss him as he bounced gleefully. \"With out-of-town clients coming into the shop this morning, I just can't keep him with me. And Rafe's got two crews to supervise.\"\n\n\"It's a real hardship,\" Cassie said from the sink. \"I can't think of anything more annoying than having to play with the baby.\"\n\n\"He is wonderful, isn't he? I can't believe he's already five months old.\" When she cranked up the music on the swing, Nate began to kick his feet in delight. \"I nursed him an hour ago, and I've got plenty of bottles here, and diapers, and two changes of clothes, and\u2014\"\n\n\"Regan, I know what to do with a baby.\"\n\n\"Of course you do.\" Grinning foolishly at Nate, Regan swept her hair back. \"It's just that I know you're so busy with the inn.\"\n\n\"You and Rafe are slave drivers, it's true, but I'm learning to bear up.\"\n\nAmused, Regan cocked her head. \"You're joking, and you're smiling, and I'm pretty sure I heard you singing when I came in.\"\n\n\"I'm happy.\" Cassie loaded plates into the dishwasher. The breakfast hour was over, and the guests were either gone or relaxing in their rooms. \"I didn't know I could be this happy. This is the most wonderful house in the world.\"\n\nRegan handed Nate a ring of colorful plastic to jiggle. \"So working here makes you happy?\"\n\n\"Absolutely. Not that I wasn't happy working for Ed, but...I love living here, Regan.\" She beamed at the view from the window. \"The kids love living here.\"\n\nRegan ran her tongue around her teeth. \"And that's why you were singing?\"\n\nCassie bent over a little farther, busied herself arranging dishes. \"Actually, there is something else. I guess you've got to go open the shop.\"\n\n\"I've got a few minutes. One of the perks of running my own business.\"\n\nIf there was anyone she could talk to, it was Regan. Cassie straightened, took a deep breath. \"Devin\u2014it's about Devin. That is, I'm probably making too much of it. Or not making enough of it. It's just, well... Do you want some coffee?\"\n\n\"Cassie.\"\n\n\"He kissed me,\" she blurted out, then slapped a hand to her mouth when a laugh bubbled out. \"I mean, kissed me. Not like Rafe kisses me, or Shane or Jared. I mean, like... My hands are sweating.\"\n\n\"It's about time,\" Regan said, with feeling. \"I thought he'd never get to it.\"\n\n\"You're not surprised.\"\n\n\"Cassie, the man would crawl naked over hot coals for you.\" She decided she would have some coffee, and walked over to the stove to pour it herself. \"So, how was it?\"\n\nRegan's statement had Cassie running a nervous hand through her hair. \"How was what?\"\n\nWith a chuckle, Regan sipped and leaned back against the counter. \"I have to figure that he has more in common with Rafe than a quick temper and great looks. So it must have been a pretty terrific kiss.\"\n\n\"It was at the picnic, two days ago. My head's still buzzing.\"\n\n\"Yep. That's a MacKade for you. What are you going to do about it?\"\n\n\"I don't know what to do.\" Brow creased, Cassie picked up a damp rag and began to wipe the counter. \"Regan, I started going with Joe before I was sixteen. I've never been with anyone else.\"\n\n\"Oh.\" Regan pursed her lips. \"I see. Well, it would be only natural to be a little nervous over the idea that you might be heading toward a physical relationship.\"\n\nBecause her palms were indeed damp, Cassie set down the cloth and rubbed them on her apron. \"I don't like sex,\" she said flatly, rattling dishes again so that she didn't note the lift of Regan's brow or the concern in her friend's eyes. \"I'm not any good at it, and I just don't like it, anyway.\"\n\n\"Cassie, I know the counseling helped you.\"\n\n\"Yes, it did, and I'm grateful for you persuading me to go. I feel better about myself, and I'm more confident about a lot of things. I know I didn't deserve to be abused, that I didn't cause it, and that I did the right thing by getting out.\" She let out a breath. \"This is a different matter. Not all women are built to enjoy sex. I've read about it. Anyway,\" she continued before Regan could comment, \"I'm getting ahead of myself. But I'm not stupid, Regan. I know that Devin has needs, and I'm prepared to meet them.\"\n\n\"That is stupid,\" Regan snapped. \"Making love is not supposed to be a chore like\u2014like...\" Flustered, she gestured to the sink. \"Like doing the damn dishes.\"\n\n\"I didn't mean it that way.\" Because Regan was her friend, she smiled. \"What I meant was that I care for Devin. I always have. This is a different level. I didn't know he was attracted to me. I'm so flattered.\"\n\nRegan's response to that was a muttered curse that only made Cassie's smile widen.\n\n\"Well, I am. He's so beautiful, and he's kind. I know he won't hurt me.\"\n\n\"No,\" Regan said quietly. \"He wouldn't hurt you.\" But, she thought, would you hurt him?\n\n\"Kissing him was lovely, and I think having sex with him would be nice.\"\n\nWisely, Regan covered her cough with a sip of coffee. If Devin was anything like Rafe, nice was hardly the word. \"Has he asked you to bed?\"\n\n\"No. He wouldn't even kiss me again when I asked him to. That's what I wanted to ask you about. How do I go about letting him know I don't mind being with him\u2014that way?\"\n\nIt was a tribute to her willpower that Regan didn't goggle. Carefully she set the coffee cup aside. \"This goes against the grain for me, Cassie, against every feminist cell in my body, but I have to trust my instincts here, and go with what I know about you and about Devin. I'm going to advise you to let him set the pace, at least initially. Take your cues from him. Just relax and enjoy the ride. I think you can count on him to get you both where you want to go. When you're ready, Cassie. It's important to think of yourself, too, not just Devin.\"\n\n\"So I really shouldn't do anything?\"\n\n\"Do what seems right to you. And do this\u2014don't compare him with Joe. And don't compare the woman who lived with Joe with the woman you are now. I think you're in for a few surprises.\"\n\n\"I've already had one.\" Cassie touched a fingertip to her lips. \"It was wonderful.\"\n\n\"Good. Keep an open mind.\" She gave Cassie a quick kiss, bent down to fuss over Nate one last time. \"And, Cass, I really wouldn't mind if you sort of kept me up-to-date with the progress.\"\n\nBy mid-afternoon, Cassie had finished the guest rooms, and the laundry, and had Nate tucked in a portable crib in Emma's room for a nap. She'd slipped a chicken in the oven to roast and was giving some thought to tackling the mending when she heard the quick rap on her door.\n\nHer heart did a little flip at the hope that it might be Devin stopping by. But settled again when she saw her mother through the screen.\n\n\"Hello, Mama.\" Dutifully Cassie opened the door and pecked her mother's dry cheek. \"It's nice to see you. I've just made some iced tea, and I have some nice cherry cobbler.\"\n\n\"You know I don't eat sweets in the middle of the day.\" Constance Connor scanned the living area of her daughter's quarters. She wrinkled her nose at the cat that curled under the table. Animals belonged outside.\n\nThe curtains were drawn back, which would surely fade the upholstery with that strong sunlight. But it was neat. She'd taught her daughter to be neat.\n\nAfter all, cleanliness was next to godliness.\n\nStill, she didn't care for the bright colors, or all the folderols sitting about. It was showy. She sniffed to indicate her disapproval and sat down on one of the living room chairs, her back broomstick-straight.\n\n\"I'll say again, it's a poor choice for you to live in a man's house who isn't your husband.\"\n\nIt was an old argument, and Cassie answered by rote. \"I lived in Mr. Halleran's house for nearly ten years.\"\n\n\"And paid good rent.\"\n\n\"I earn my keep here. What's the difference?\"\n\n\"You know very well the difference, so I'll not mention it again.\"\n\nUntil the next time, Cassie thought wearily. \"Would you like some iced tea, Mama?\"\n\n\"I can get through an hour without sipping or snacking.\" Constance set her purse firmly on her lap, crossed her ankles above her sensible shoes. \"Sit down, Cassandra. The children are in school, I take it.\"\n\n\"Yes. They're doing very well. They'll be home in about an hour. I hope you'll stay and see them.\"\n\n\"It's you I've come to see.\" She unsnapped her bag with fingers adorned with only a thin gold band. There was no glint to it, no shine. As, Cassie thought, there had been no glint or shine to her parents' marriage. She often thought, after a visit with her mother, that her father had died simply to escape it.\n\nBut she said nothing, waiting as her mother drew out an envelope. She didn't have to see the handwriting to know who it was from.\n\n\"This is the latest letter I received from your husband. It came in this morning's mail.\" Constance held it out. \"I want you to read it.\"\n\nCassie folded her hands in her lap, linked her fingers. \"No.\"\n\nEyes narrowed with righteous anger, Constance studied her daughter. \"Cassandra, you will read this letter.\"\n\n\"No, ma'am, I won't. He's not my husband.\"\n\nConstance's thin, pale face went dark with temper. \"You took vows before God.\"\n\n\"And I've broken them.\" It was hard, so hard, to keep her voice and hands from trembling, to keep her eyes level.\n\n\"You take pride in that? You should be ashamed.\"\n\n\"No, not pride. But you can't make me sorry for breaking them, Mama. Joe broke them long before I did.\"\n\nShe refused to look at the letter, refused to feel this bitter anger, that even so small a part of him had come into her home. Instead, she kept her eyes on her mother's face.\n\n\"Love, honor, cherish. Did he love me, Mama, when he beat me? Did he honor me when he used his fists on me? Did he cherish me when he raped me?\"\n\n\"You will not speak that way about your husband.\"\n\n\"I came to you when I had nowhere to go, when he'd hurt me so badly I could hardly walk, when my children were terrorized. And you sent me away.\"\n\n\"Your place was at home, making the best of your marriage.\"\n\n\"I made the best of it for ten years, and it nearly killed me. You should have been there for me, Mama. You should have stood up for me.\"\n\n\"I stood up for what was right.\" Constance's mouth was a thin line. \"If you forced him to discipline you\u2014\"\n\n\"Discipline me!\" Stunned, even after all the time that had passed, Cassie leaped to her feet. \"He had no right to discipline me. I was his wife, not his dog. And not even a dog deserved to be treated the way I was. He would have disciplined me to death, if I hadn't finally found the courage to do something about it. Would that have satisfied you, Mama? I'd have kept my vows then. Till death do us part.\"\n\n\"You're overdramatizing. And whatever happened before is done. He's seen his mistakes. It was the drink, the women who tempted him. He's asking for your forgiveness, and hopes that you will keep your vows, as he intends to.\"\n\n\"He can't have my forgiveness, and he can't have me. How can you do this to me? I'm your daughter, your only child.\" Cassie's eyes were no longer haunted, but steely. \"How can you take the side of a man who hurt me and betrayed me and made my life a misery? Don't you want me to be happy?\"\n\n\"I want you to do what's expected of you. I expect you to do as you're told.\"\n\n\"Yes, that's all you ever wanted from me. To do what I was told, to be what you expected me to be. Why do you think I married him, Mama?\"\n\nCassie couldn't believe the words were coming out of her mouth, but they wouldn't be stopped. Just as the emotions that pushed them from her heart to her throat and through her lips wouldn't be stopped.\n\n\"It was to get away from you, to get out of that house, where nobody ever laughed, nobody ever showed any affection.\"\n\n\"You had a good home.\" This time it was Constance whose voice trembled. \"You had a decent Christian up-bringing.\"\n\n\"No, I didn't. There's nothing decent or Christian about a house where there's no love. My children won't be raised that way, not anymore.\" Cassie spoke calmly now, amazed that she could, fascinated that she felt nothing at all. \"You're my mother, and I'll give you all the respect that I can. All I'm asking is for you to give me the same. I don't want you corresponding with Joe anymore.\"\n\nConstance got to her feet. \"You would dare tell me what to do?\"\n\n\"Will you stop writing him, Mama? Will you stop writing the prison authorities?\"\n\n\"I will not.\"\n\n\"Then you're not welcome in my home. We have nothing else to say to each other.\"\n\nStaggered, Constance could only stare. \"You'll come to your senses.\"\n\n\"I have come to them. Goodbye, Mama.\"\n\nCassie walked to the door and held it open. She stiffened when Constance swept by. And then the trembling began.\n\nSlowly, unsure of her footing, Cassie walked to the table. She braced herself on it as she lowered herself into a chair. Wrapping her arms tight around her body, she began to rock.\n\nShe was still sitting there when Devin came to the door, ten minutes later. He started to give a friendly rap on the wooden slat of the screen. But then he saw her, saw the way her shoulders were hunched and curled and the quick, monotonous rocking of her body, as if she were trying to still something inside herself. Or comfort it.\n\nHe'd seen her like that before, sitting in his office with her face battered. All he knew was that she was hurt, and he was through the door like a bullet.\n\n\"Cassie.\"\n\nShe sprang to her feet. He saw alarm mix with the hurt. Even as he reached out, she scooted back, out of his way.\n\n\"Devin, I didn't hear you come up. I was\u2014 I should\u2014\" Her mind raced for excuses, for the barrier of appearances. As always. Pale with grief, her eyes swimming with it, she stared at him. Then she began to move quickly. \"Let me get you some iced tea. It's fresh.\" She was hurrying for glasses, for the pitcher, her movements jerky. \"I've got some cobbler. I just made it this morning.\"\n\nShe jolted like a spring when his hands came down on her shoulders, and the glass she had just filled smashed on the tiles. The cat that had been napping under the table took off in a blur of fur.\n\n\"Oh, God, look what I've done.\" Her breathing hitched, and the feeling in her chest tightened. She couldn't stop it. \"I have to\u2014 I have to\u2014\"\n\n\"Leave it.\" He struggled to keep his voice easy as he turned her to face him. She was shaking hard, trying to pull back. Not this time was all he could think. Not this time. \"Come here,\" he murmured. \"Come on now.\"\n\nThe instant he drew her into his arms, the dam broke. She wept against his shoulder, the fast, hot tears soaking his shirt. He kissed her hair, stroked her back. \"Tell me. Tell me what's wrong, so I can help.\"\n\nIt wasn't coherent, nor was it complete, but he understood the gist when she stuttered out words between sobs. Bitter fury curled inside him as he soothed her, kissing her wet cheeks.\n\n\"You did what you had to do. You did what was right.\"\n\n\"She's my mother.\" Cassie lifted her ravaged face to his. \"I sent her away. I turned my mother away.\"\n\n\"Who turned who away, Cass?\"\n\nHer breath sobbed out again, and her hands balled into fists on his shoulders. \"It's not right.\"\n\n\"Get away from her.\" The screen door slammed as Connor burst through it. His own hands were fisted, and his face was flushed with fury, taut with violence. All he saw was a man holding his mother, and his mother crying. \"If you touch her I'll kill you.\"\n\n\"Connor!\" Shock had Cassie's voice ringing sharp. Was this her baby, with his fists raised and his eyes fierce? She caught a glimpse of Emma at the door, her frightened face pressed to the screen. \"Don't speak that way to Sheriff MacKade.\"\n\nEvery cell on alert, Connor stepped forward. \"Take your hands off my mother.\"\n\nIntrigued, Devin merely lifted a brow and let his arms fall to his sides.\n\n\"I said not to speak that way,\" Cassie began.\n\n\"He was hurting you. He made you cry.\" Connor bared his teeth, a ten-year-old warrior. \"He better leave right now.\"\n\n\"He wasn't hurting me.\" Though she was shaken to the core, Cassie stepped between them. \"I was upset\u2014 Grandma upset me\u2014and Sheriff MacKade was helping to make me feel better. I want you to apologize, this minute.\"\n\nDevin saw the boy's arms drop, and knew when the angry flush on Connor's cheeks turned to shame. With his eyes on the boy, he laid a hand on Cassie's shoulder.\n\n\"I'd like to talk to Connor. Alone.\" Anticipating her protest, he gave Cassie's shoulder a quick squeeze. \"Cass, the baby's crying. Why don't you and Emma go see to him?\"\n\n\"Nate. I forgot.\" At her wit's end, Cassie dragged a hand through her hair.\n\n\"Why don't you go on?\" Devin said, giving her a gentle nudge. \"Con and I are going to take a walk.\"\n\n\"All right. Come on, Emma, Nate's crying.\" But she took a deep breath as she held out a hand for her daughter. \"I expect you to apologize, Connor. You understand?\"\n\n\"Yes, ma'am.\" With his chin on his chest, Connor turned to go outside.\n\nHe knew what was coming. He was going to get whipped. His father had always done the hitting away from the house, away where his mother couldn't see and wouldn't know. He'd get a beating now for sure, and it would be worse than anything his father had ever done to him. Because he'd tried to do what was right, and he'd been wrong.\n\nDevin said nothing at all, just walked with the boy across the lawn, toward the woods that bordered it. He chose the path without thinking. The woods were as familiar to him as the town, as his own home, as his own mind. Beside him, Connor walked stiffly, his head drooped in shame, his back braced.\n\nBecause he knew he had to gauge his timing, and his moves, Devin resisted the urge to drape his arm over those thin little shoulders. Instead, he led the way down a path and stopped at the cluster of rocks where two soldiers had once met and doomed each other.\n\nHe sat, and the boy stood rigid and waiting.\n\n\"I'm awfully proud of you, Connor.\"\n\nThe words\u2014the last he'd expected to hear\u2014had the boy's head whipping up. \"Sir?\"\n\nCasually Devin took out a cigarette\u2014the first of a very long day. \"I have to tell you, it's a relief to me. I worry about your mother some. She's had a bad time of it. Knowing you're there to look after things, makes my mind a lot easier.\"\n\nConnor's confusion was too huge for him to feel any pride. He stared at Devin, his eyes still wary. \"I\u2014I sassed you.\"\n\n\"I don't think so.\"\n\n\"You're not going to hit me?\"\n\nDevin's hand stiffened, hesitated. Very slowly he tossed the barely smoked cigarette on the ground and crushed it under his heel. As he would have liked to crush Joe Dolin.\n\n\"I'm never going to raise my hand to you, not today, not any day.\" He spoke deliberately, his eyes level with Connor's, as a man would speak to another man. \"I'm never going to raise it to your mama or to your sister.\" But he held out that hand, and waited. \"I'm giving you my word, Connor,\" he said, when the boy simply stared at the hand being offered. \"I'd be grateful if you'd take it.\"\n\nDumbfounded, Connor put his hand in Devin's. \"Yes, sir.\"\n\nDevin gave the hand a little squeeze, tugged the boy a little closer. And grinned. \"You'd have torn right into me, wouldn't you?\"\n\n\"I'd have tried.\" The emotions swirling inside Connor were frightening. Most of all, he was afraid he would cry now and show Devin he was just a stupid little boy after all. \"I never helped her before. I never did anything.\"\n\n\"It wasn't your fault, Connor.\"\n\n\"I never did anything,\" Connor repeated. \"He hit her all the time, Sheriff. All the time.\"\n\n\"I know.\"\n\n\"No, you don't. You only know about when one of the neighbors would call you, or when he'd get so drunk he'd hit her someplace where it would show. But there was more. It was worse.\"\n\nDevin nodded. There was nothing else he could do. And drew the boy down on the rocks beside him. \"He hit you, too.\"\n\n\"When she couldn't see.\" Bravery forgotten, Connor pressed his face into Devin's side. \"When she didn't know.\"\n\nDevin stared off into the trees, eaten away by a useless anger at what he hadn't been able to prevent. \"Emma?\"\n\n\"No, sir. He never paid much attention to Emma, because she was just a girl. Don't tell Mama. Please don't tell her he hit me. She'd just feel bad.\"\n\n\"I won't.\"\n\n\"I hate him. I'd kill him if I could.\"\n\n\"I know how you feel.\" When the boy shook his head, Devin drew him back, looked deep into his eyes. \"I do know. I'm going to tell you something. I used to fight a lot.\"\n\n\"I know.\" Connor sniffled, but was profoundly grateful he'd controlled the tears. \"People talk about it.\"\n\n\"Yeah, I know they do. I used to like it, and I used to think there were lots of people I wanted to rip into. Sometimes I had reason for it, sometimes I didn't. Anyway, I had to learn to take a step back. It's important, that step. Now, you figure you owe your father some grief\u2014\"\n\n\"Don't call him that,\" Connor snapped out, then flushed darkly. \"Sir.\"\n\n\"All right. I figure you owe him some, too. But you've got to take that step back. Let the law handle it.\"\n\n\"I'm not ever going to let him or anybody hurt her again.\"\n\n\"I'm with you there.\" Studying Connor's determined face, he decided the boy deserved to know the situation. \"I'm going to give it to you straight, okay?\"\n\n\"Yes, sir.\"\n\n\"Your grandma got your mama real upset today.\"\n\n\"She wants him to come back. It's never going to happen. I won't let it happen.\"\n\n\"Your mama feels the same way, and that's why she sent your grandma away. That was hard for her, real hard, Connor, but she did it.\"\n\n\"You were helping her. I'm sorry I\u2014\"\n\n\"Don't apologize,\" Devin said quickly. \"I mean it. I know Cassie thinks you should, but we know how things stand. You did exactly right, Connor. I'd have done the same.\"\n\nNo compliment he'd ever received, no praise from a teacher, no high five from a teammate, had ever meant more. He had done what Sheriff MacKade would do.\n\n\"I'm glad you want to help her. I'll do anything you want me to do.\"\n\nThat kind of trust, Devin thought, was worth more than gold. \"I need to tell you that they've given Joe work release.\"\n\nConnor's face tightened up. \"I know about it. Kids at school say things.\"\n\n\"They giving you a rough time?\"\n\nHe moved a shoulder. \"Not as much as they used to.\"\n\nLearning to handle yourself, Devin thought with an astonishing sense of pride. \"What I want is for you not to worry too much, but more, I want you to keep your eyes open. You're smart, and you notice things. That's why you write good stories.\"\n\nConnor wriggled with pleasure. \"I like to write.\"\n\n\"I know. And you know how to look at things, how to watch. So I know you're going to watch out for your family. If you see something, hear something, even feel something that doesn't sit right, I want you to come to me. I want your word on that.\"\n\n\"Yes, sir.\"\n\n\"Do you have to call me sir all the time? It makes me feel creaky.\"\n\nConnor flushed, and grinned. \"I'm supposed to. It's like a rule.\"\n\n\"I know all about rules.\" Devin decided they could deal with that little matter later. \"A man would be lucky to have you for a son, Connor.\"\n\n\"I don't ever want to have a father again.\"\n\nThe hand that had lifted toward Connor's shoulder stiffened. Biting back a sigh, Devin ordered it to relax. \"Then I'll say a man would be lucky to have you for a friend. Are we square here?\"\n\n\"Yes, sir.\"\n\nThere were those eyes again, Devin thought, filled with trust. \"Your mama's probably worried you're beating me up.\" When Connor giggled at the idea, Devin ruffled his hair. \"You go on back now and tell her we straightened it all out. I'll talk to her later.\"\n\n\"Yes, sir.\" He scrambled off the rocks, then had to bite his lip to spark that last bit of courage. \"Can I come to your office sometime, and watch you work?\"\n\n\"Sure.\"\n\n\"I wouldn't get in the way. I'd just\u2014\" Connor tumbled over his own words and skidded to a halt. \"I can?\"\n\n\"Sure you can. Anytime. It's mostly boring.\"\n\n\"It couldn't be,\" Connor said with giddy pleasure. \"Thanks, Sheriff. Thanks for everything.\"\n\nDevin watched the boy race off, then settled back. He wished briefly for a cigarette before reminding himself he was quitting. Then he reminded himself that sooner or later he intended to have those two children, and maybe another on the way.\n\nConnor didn't want another father, and that would be a tough one. So, Devin mused, he'd just have to find the right path to take, and step carefully.\n\nThe first step, of course, was Cassie. One step, then the next. Direction always took you somewhere. If he was careful, she would be taking those steps with him.\n\n## Chapter 6\n\nIt was supposed to be Devin's day off, but he spent two hours in the morning dealing with a small crisis at the high school. The smoke bomb had failed in its mission. When it landed in the girl's locker room, it hadn't put out much of a cloud, and, more important, hadn't made the girls come rushing out screaming in their underwear.\n\nThe one he'd put together a short lifetime ago had had far more satisfying results. Not that he'd mentioned that particular incident to the two offenders he collared.\n\nOnce he had it under control, and the juvenile chemists shaking in their basketball shoes, he headed straight for the inn.\n\nHe had a surprise for Cassie, one he hoped would make her smile. And one he hoped would ease the way into that next step.\n\nHe supposed he had an unfair advantage. He knew her so well, had watched and observed for years. He knew every expression of her face, every gesture of her hands. He knew her weaknesses and her strengths.\n\nShe knew him, he thought, but not in the same way, or in the same detail. She'd been too busy surviving to notice. If she had noticed, she would have been able to see that he was in love with her.\n\nIt was just as well she didn't see. Not until he'd finished laying the foundation. He could take his time about that, Devin mused as he turned up the lane toward the inn. But once he had that foundation in place and solid, he was going to move fast.\n\nTwelve years was a damn long time to wait.\n\nBecause there was a car parked in one of the guest slots, he opted to go into the inn first. He was delighted to find her there, fully occupied with two snowy-haired women.\n\nShe'd forgotten to take her apron off. The new arrivals had come unexpectedly, and they had wanted a full tour, and the history of the inn. Cassie was grateful she'd finished the breakfast dishes, even though she'd been caught in the middle of vacuuming.\n\nThe two women were sisters, both widowed, and were eager to hear about the Barlow legend. Cassie led them back down the stairs after the tour of the second floor, and was well into her spiel when Devin walked in.\n\n\"...the bloodiest single day of the Civil War. The Antietam battlefield is one of the most pristine parks in the country. The visitors' center is only four miles from here, and very informative. You'll find\u2014 Oh, hello, Devin.\"\n\n\"Don't let me interrupt. Ladies.\"\n\n\"Mrs. Berman, Mrs. Cox, this is Sheriff MacKade.\"\n\n\"Sheriff.\" Mrs. Cox adjusted her glasses and beamed through the lenses. \"How exciting.\"\n\n\"Antietam's a quiet town,\" he told her. \"Certainly more quiet than it was in September of 1862.\" Because tourists inevitably enjoyed it, Devin grinned. \"You're standing right about on the spot where a Confederate soldier was killed.\"\n\n\"Oh, my goodness!\" Mrs. Cox clapped her hands together. \"Did you hear that, Irma?\"\n\n\"Nothing wrong with my ears, Marge.\" Mrs. Berman peered down at the stairs, as if inspecting for blood. \"Mrs. Dolin was telling us something of the history. We decided to visit the inn because we read one of the brochures that claimed it was haunted.\"\n\n\"Yes, ma'am. It surely is.\"\n\n\"Sheriff MacKade's brother owns the inn,\" Cassie explained. \"He can tell you quite a bit about it.\"\n\n\"You can't do better than to hear it from Mrs. Dolin,\" Devin corrected. \"She lives with the ghosts every day. Tell them about the two corporals, Cassie.\"\n\nThough she told the story several times each week, Cassie had to struggle not to feel self-conscious in front of Devin. She folded her hands over her apron.\n\n\"Two young soldiers,\" she began, \"became separated from their regiments during the Battle of Antietam. Each wandered into the woods beyond the inn. Some say they were looking for their way back to the battle, others say they were just trying to go home. Still, legend holds that they met there, fought there, each of them young, frightened, lost. They would have heard the battle still raging in the fields, over the hills, but this was one on one, strangers and enemies because one wore blue, and the other gray.\"\n\n\"Poor boys,\" Mrs. Berman murmured.\n\n\"They wounded each other, badly, and crawled off in different directions. One, the Confederate, made his way here, to this house. It's said he thought he was coming home, because all he wanted, in the end, was his home and his family. One of the servants found him, and brought him into the house. The mistress here was a Southern woman. Her name was Abigail, Abigail O'Brian Barlow. She had married a wealthy Yankee. A man she didn't love, but was bound to by her vows.\"\n\nDevin's brow lifted. It was a new twist, a new detail, to the legend he had known since childhood.\n\n\"She saw the boy, a reminder of her own home and her own youth. Her heart went out to him for that, and simply because he was hurt. She ordered him to be taken upstairs, where his wounds would be tended. She spoke to him, reassured him, held his hand in hers as the servant carried him up these stairs. She knew that she could never go home again, but she wanted to be sure the boy could. The war had shown her cruelty, useless struggle and the terrible pain of loss, as her marriage had. If she could do this one thing, she thought, help this one boy, she could bear it.\"\n\nMrs. Cox took out tissues, handed one to her sister and blew her own nose hard.\n\n\"But her husband came to the stairs,\" Cassie continued. \"She didn't hate him then. She didn't love him, but she'd been taught to respect and obey the man she had married, and the father of her children. He had a gun, and she saw what he meant to do in his eyes. She shouted for him to stop, begged him. The boy's hand was in hers, and his eyes were on her face, and if she had had the courage, she would have thrown her body over his to protect him. To save not only him, but everything she'd already lost.\"\n\nNow it was Cassie who looked down at the stairs, sighed over them. \"But she didn't have the courage. Her husband fired the gun and killed him, even as she held the boy's hand. He died here, the young soldier. And so did she, in her heart. She never spoke to her husband again, but she learned how to hate. And she grieved from that day until she died, two years later. And often, very often, you can smell the roses she loved in the house, and hear her weeping.\"\n\n\"Oh, what a sad, sad story.\" Mrs. Cox wiped at her eyes. \"Irma, have you ever heard such a sad story?\"\n\nMrs. Berman sniffed. \"She'd have done better to have taken the gun and shot the louse.\"\n\n\"Yes.\" Cassie smiled a little. \"Maybe that's one of the reasons she still weeps.\" She shook off the mood of the story and led the ladies the rest of the way down the steps. \"If you'd like to make yourselves at home in the parlor, I'll bring in the tea I promised you.\"\n\n\"That would be lovely,\" Mrs. Cox told her, still sniffling. \"Such a beautiful house. Such lovely furniture.\"\n\n\"All of the furnishings come from Past Times, Mrs. MacKade's shop on Main Street in town. If you have time, you might want to go in and browse. She has beautiful things, and offers a ten-percent discount to any guest of the inn.\"\n\n\"Ten percent,\" Mrs. Berman murmured, and eyed a graceful hall rack.\n\n\"Devin, would you like to have some tea?\"\n\nIt took an effort to move. He wondered if she knew that Connor got his flair for telling a story from his mother.\n\n\"I'll take a rain check on that. I have something in the car for upstairs. For your place.\"\n\n\"Oh.\"\n\n\"Ladies, nice to have met you. Enjoy your stay at the MacKade Inn, and in the town.\"\n\n\"What a handsome man,\" Mrs. Cox said, with a little pat of her hand to her heart. \"My goodness. Irma, have you ever seen a more handsome young man?\"\n\nBut Mrs. Berman was busy sizing up the drop-leaf table in the parlor.\n\nBy the time Cassie had settled the ladies in with their tea, her curiosity was killing her. She had chores to see to, and she scolded herself for letting them lag as she hurried around to the outside stairs.\n\nHalfway up, she saw Devin hooking up a porch swing. \"Oh.\" It made a lovely picture, she thought, a man standing in the sunlight, his shirtsleeves rolled up, tools at his feet, muscles working as he lifted one end of the heavy wooden seat to its chain.\n\n\"This seemed like the spot for it.\"\n\n\"Yes, it's perfect. Rafe didn't mention that he wanted one.\"\n\n\"I wanted one,\" Devin told her. \"Don't worry, I ran it by him.\" He hooked the other end and gave it a testing swing. \"Works.\" Bending, he gathered up the tools. \"Going to try it out with me?\"\n\n\"I really have to\u2014\"\n\n\"Try it out with me,\" Devin finished, setting the tools aside in their case. \"I put it up because I figured it was a good way to get you to sit with me on a summer afternoon. A good way for me to kiss you again.\"\n\n\"Oh.\"\n\n\"You said you didn't mind.\"\n\n\"No, I didn't. I don't.\" There it was again, that flutter in her chest. \"Aren't you supposed to be working?\"\n\n\"It's my day off. Sort of.\" He held out a hand, then curled his fingers around hers. \"You look pretty today, Cassie.\"\n\nAutomatically she brushed at her apron. \"I've been cleaning.\"\n\n\"Real pretty,\" he murmured, drawing her to the swing, and down.\n\n\"I should get you something cold to drink.\"\n\n\"You know, one of these days you're going to figure out that I don't come around so you can serve me cold drinks.\"\n\n\"Connor said you worried about me. You don't have to. I was hoping you'd come by so I could tell you how much I appreciate what you did for him the other day. The way you made him feel.\"\n\n\"I didn't do anything. He earned what he felt. You've got a fine boy in Connor.\"\n\n\"I know.\" She took a deep breath and relaxed enough to lean back against the seat. The rhythm of the swing took her back, far back, to childhood and sweet days, endless summers. Her lips curved, and then she laughed.\n\n\"What's funny?\"\n\n\"It's just this, sitting here on a porch swing, like a couple of teenagers.\"\n\n\"Well, if you were sixteen again, this would be my next move.\" He lifted up his arms, stretched, then let one drape casually over her shoulders. \"Subtle, huh?\"\n\nShe laughed again, tilted her face toward his. \"When I was sixteen, you were too bad to be subtle. Everybody knew how you snuck off to the quarry with girls and\u2014\"\n\nThe best way to stop her mouth was with his. He did so gently, savoring the quick tremor of her lips, of her body.\n\n\"Not so subtle,\" he said quietly. \"Wanna go to the quarry?\" When she stuttered, he only laughed. \"Some other time. For now I'd settle for you kissing me back. Kiss me back, Cassie, like you were sixteen and didn't have a worry in the world.\"\n\nWith someone else, anyone else, he might have been amused by the concentration on her face. But it struck his heart, the way her mouth lifted to his, that hesitant pressure, the unschooled way her hands lifted to rest on his shoulders.\n\n\"Relax,\" he said against her mouth. \"Turn off your head for a minute. Can you do that?\"\n\n\"I don't...\" She didn't turn it off. It shut off when his tongue danced lightly over hers, when his hands skimmed down her sides and up again. Down and up, in firm, steady strokes that had the heels of his hands just brushing the sides of her breasts.\n\n\"I love the taste of you.\" He pressed his lips to her jaw, her temples, back to her lips. \"I've dreamed of it.\"\n\n\"You have?\"\n\n\"Most of my life. I've wanted to be with you like this for years. Forever.\"\n\nThe words were seeping through that lovely haze of pleasure that covered her whenever he kissed her. \"But\u2014\"\n\n\"You got married.\" He trailed his lips down her cheek. \"I didn't move fast enough. I got drunk the day you married Joe Dolin. Blind, falling-down drunk. I didn't know what else to do. I thought about killing him, but I figured you must have wanted him. So that was that.\"\n\n\"Devin, I don't understand this.\" If he'd stop kissing her, just for a minute, she might be able to understand.\n\nBut he couldn't seem to stop, any of it. \"I loved you so much I thought I'd die from it. Just keel right over and die.\"\n\nPanic and denial had her struggling away. \"You couldn't have.\"\n\nHe'd said too much, but the regrets would have to come later. Now, he'd finish it. \"I've loved you for over twelve years, Cassandra. I loved you when you were married to another man, when you had his children. I loved you when I couldn't do anything to help you out of that hell you were living in. I love you now.\"\n\nShe got up and, in an old defensive habit, wrapped her arms tight around her body. \"That's not possible.\"\n\n\"Don't tell me what I feel.\" She jolted back a full step at the anger in his tone, making him clench his teeth as he rose. \"And don't you cringe away from me when I raise my voice. I can't be what I'm not, not even for you. But I'm not Joe Dolin. I'll never hit you.\"\n\n\"I know that.\" She let her arms drop. \"I know that, Devin.\" Even as she said it, she watched him struggle to push back the worst of his temper. \"I don't want you to be angry with me, Devin, but I don't know what to say to you.\"\n\n\"Seems like I've already said more than enough.\" He began to pace, his hands jammed in his pockets. \"I'm good at taking things slow, thinking them through. But not this time. I've said what I've said, Cass, and I can't\u2014won't\u2014take it back. You're going to have to decide what you want to do about it.\"\n\n\"Do about what?\" Baffled, she lifted her hands, then let them fall. \"You want me to believe that a man like you had feelings for me all these years and didn't do anything about it?\"\n\n\"What the hell was I supposed to do?\" he tossed back. \"You were married. You'd made your choice, and it wasn't me.\"\n\n\"I didn't know there was a choice.\"\n\n\"My mistake,\" he said, bitterly. \"Now I've made another one, because you're not ready, or you don't want to be ready. Or maybe you just don't want me.\"\n\n\"I don't\u2014\" She lifted her hands to her cheeks. She honestly didn't know which, if any, of those alternatives was true. \"I can't think. You've been my friend. You've been, well, the sheriff, and I've been so grateful\u2014\"\n\n\"Don't you dare say that to me.\" Devin shouted the words, and was too twisted with pain and fury to notice that she went white as death. \"Damn it, I don't want you to be grateful. I'm not playing public servant with you. I don't deserve that.\"\n\n\"I didn't mean... Devin, I'm sorry. I'm so sorry.\"\n\n\"The hell with being sorry,\" he raged. \"The hell with gratitude. You want to be grateful I locked the son of a bitch up who was pounding on you, then be grateful to the badge, not to me. Because I wanted to break him in half. You want to be grateful I've been coming around here being the nice guy, like some love-whipped mongrel dog, don't. Because what I've wanted to do is\u2014\"\n\nHe bit that back, his eyes cutting through her like hot knives. \"You don't want to know. No, what you want is for me to keep my voice down, my feelings inside and my hands to myself.\"\n\n\"No, that's not\u2014\"\n\n\"You don't mind if I kiss you, but then, you're so damn grateful it's the least you can do.\"\n\nHer stumbling protest fell apart. \"That's not fair.\"\n\n\"I'm tired of being fair. I'm tired of waiting for you. I'm tired of being torn up in love with you. The hell with it.\"\n\nHe strode by her, and was halfway down the stairs before her legs unfroze. She raced after him. \"Devin. Devin, please don't go this way. Let me\u2014\"\n\nHe jerked away from her light touch on his shoulder, whirled on her. \"Leave me alone now, Cass. You want to leave me be now.\"\n\nShe knew that look, though she had never expected to see it aimed from his eyes into hers. It was a man's bitter fury. She had reason to fear it. Her stomach clenched painfully, but she made herself stand her ground. He would never know how much it cost her.\n\n\"You never told me,\" she said, fighting to keep her voice slow and even. \"You never let me see. Now you have, and you won't give me time to think, to know what to do. You don't want to hear that I'm sorry, that I'm grateful, that I'm afraid. But I'm all of those things, and I can't help it. I can't make myself into what anyone else expects me to be ever again. I'll lose everything this time. If I could do it for anyone, I'd do it for you. But I can't.\"\n\n\"That's clear enough.\" He knew he was wrong\u2014not completely wrong, but wrong enough. It just didn't seem to matter, compared with this ragged, tearing hurt inside of him. \"The thing you've got twisted around, Cass, is that I don't want you to be anything but what you are. Once you figure that out, you know where to find me.\"\n\nShe opened her mouth again, then closed it when he strode away. There was nothing else she could say to him now, nothing else she could do. She felt raw inside, and her throat hurt.\n\nAnd it was hurt that had been in his eyes, she thought, closing her own. Hurt that she had caused, without ever meaning to.\n\nDevin MacKade loved her. The idea left her weak with terror and confusion. But bigger even than that was the idea that he had loved her all this time. Devin MacKade, the kindest, most admirable man she knew, loved her, had loved her for years, and all she had to give in return was gratitude.\n\nNow she had lost him, the friendship she'd come to cherish, the companionship she had grown to depend on. She'd lost it because he wanted a woman, and she was empty inside.\n\nShe didn't weep. It was too late for tears. Instead, she rose, reminded herself to square her shoulders. She went back into the inn through the kitchen. There were chores to see to, and she could always think more clearly when she was working.\n\nHer latest guests had gone off, eager to hunt antiques, so Cassie went back upstairs and turned on the vacuum she'd abandoned when the guests arrived.\n\nShe worked methodically, down the hallway, room by room. The bridal suite\u2014Abigail's room\u2014was her favorite. But she paid little attention now to the lovely wallpaper with its rosebuds, the graceful canopy bed, the wash of sunlight through the lace curtains.\n\nShe reminded herself to bring up fresh flowers. Even when the room wasn't occupied, there were always flowers on the table by the window. She'd forgotten them that morning.\n\nYet the room smelled of roses, powerfully. A sudden chill had her shivering. She felt him, and turned toward the door.\n\n\"Devin.\" Relief, confusion, sorrow. She experienced them all as she took a step toward the doorway.\n\nBut it wasn't Devin. The man was tall, dark-haired and handsome. But the face wasn't Devin's, and the clothes were formal, old-fashioned. Her hand went limp on the handle of the vacuum, and the sound of it buzzed in her ears.\n\nAbigail, come with me. Take the children and come with me. Leave this place. You don't love him.\n\nNo, Cassie thought, I've never loved him. Now I despise him.\n\nCan't you see what this is doing to you? How long will you stay, closed away from life this way?\n\nIt's all I can do. It's the best I can do.\n\nI love you, Abby. I love you so much. I could make you happy if you'd only let me. We'll go away from here, away from him. Start our lives over, together. I've already waited for you so long.\n\nHow can I? I'm bound to him. I have the children. And you, your life is here. You can't walk away from the town, your responsibilities, the people who depend on you. You can't settle for another man's wife, another man's children.\n\nThere's nothing I wouldn't do for you. I'd kill for you. Die for you. For God's sake, Abigail, give me the chance to love you. All these years I've stood by, knowing how unhappy you were, knowing you were out of reach. That's over now. He's gone. We can leave and be miles from here before he comes back. Why should either of us settle for less than everything? I don't want to sit in the parlor with you and pretend I don't love you, don't need you. I can't keep being only your friend.\n\nYou know I value you, depend on you.\n\nTell me you love me.\n\nI can't. I can't tell you that. There's nothing inside me any longer. He killed it.\n\nCome with me. And live again.\n\nWhatever was there, whoever was there, faded, until there was only the doorway, the lovely wallpaper and the strong, sad scent of roses. Cassie found herself standing, almost swaying, with one hand reaching out to nothing at all.\n\nThe vacuum was still humming as she sank weakly to the floor.\n\nWhat had happened here? she asked herself. Had she been dreaming? Hallucinating?\n\nShe laid a hand on her heart and found it was beating like a wild bird in a cage. Carefully she let her head drop down to her updrawn knees.\n\nShe had heard the ghosts before, felt them. Now, she realized, she had seen one. Not one of the Barlows, not the poor doomed soldier. But the man Abigail had loved. The man who had loved her.\n\nWho had he been? She thought she might never know. But his face had been compelling, though filled with sorrow, his voice strong, even when it was pleading. Why hadn't Abigail gone with him? Why hadn't she taken that hand he reached out to her and run, run for her life?\n\nAbigail had loved him. Cassie drew in a deep breath. Of that she was sure. The emotions that swirled through the room had been so powerful, she felt them still. There had been love here. Desperate, helpless love.\n\nIs that why you weep? Cassie wondered. Because you didn't go, and you lost him? You didn't reach out, and then there was nothing to hold on to?\n\nYou were afraid to love him, so you broke his heart.\n\nJust as she had broken Devin's heart today.\n\nWith a shudder, Cassie lifted her head. Why? she asked herself. Out of fear and doubt. Out of habit. That was pathetic. All Devin had wanted was affection. But she hadn't told him that she cared. Hadn't showed him she cared.\n\nWould she close herself away, as Abigail had, or would she take the chance?\n\nHadn't she been a coward long enough?\n\nWiping her damp face, she got to her feet. She had to go to him. She would go to him. Somehow.\n\nOf course, such things are never simple. She had children, and could hardly leave them to fend for themselves. She had guests at the inn, and a job to do. It took her hours to manage it, and with every minute that passed, the doubts weighed more heavily.\n\nShe combated them by reminding herself that it didn't matter how clumsy she was. He wanted her. That would be enough.\n\n\"I'm so grateful, Ed. I know it's a lot to ask.\"\n\n\"Hey\u2014\" Already settled down in front of the television with a bowl of popcorn, Ed waved a hand \"\u2014so I closed a little early. I get a night off.\"\n\n\"The kids are asleep.\" But still Cassie fretted. \"They hardly ever wake up after they're down.\"\n\n\"Don't you worry about those angels. And don't worry about the people downstairs,\" she added, anticipating Cassie. \"They want anything, they'll call up here and let me know. I'm going to watch this love story I rented, then hit the sack.\"\n\n\"You take the bed. You promised,\" Cassie insisted. \"I'll just flop down on the couch when I get back.\"\n\n\"Mm-hmm...\" Ed was betting that wouldn't be until dawn. \"You say hi to Devin for me, now.\"\n\nCassie twisted the collar of her blouse in her fingers. \"I'm just going over to his office for a little while.\"\n\n\"If you say so, honey.\"\n\n\"He's angry with me, Ed. He's so angry with me, he might just boot me out.\"\n\nEd stopped the videotape she was watching, turned around on the couch and gave Cassie one long, summing-up look. \"Honey, you look at him like that, and he's not going to boot you anywhere but into that cot he's got in the back room.\" When Cassie wrapped her arms around her body, Ed only laughed. \"Oh, you stop that now. Devin's not going to push you into anything. A man like that doesn't have to push. He just has to be.\"\n\n\"How did you know I was going over there to...to try to...\"\n\n\"Cassie, honey, look who you're talking to here. I've been around this block plenty. You call me, ask if I'd settle in here for the night because you need to see Devin, I'm going to figure it out. And it's long past time, if you ask me.\"\n\nCassie looked down at her plain cotton blouse and simple trousers. Her neat flat-heeled shoes. Hardly the garb of a femme fatale. \"Ed, I'm no good at this sort of thing.\"\n\nEd cocked her head. \"I'd wager Devin's plenty good at it, so don't you worry.\"\n\n\"Regan said I should let him set the pace. Maybe I shouldn't be going over there.\"\n\n\"Sweetie pie, sometimes even a real man needs a little kick. Now you stop second-guessing yourself and wringing your hands. Go on over there and get him.\"\n\n\"I should do something with my hair,\" Cassie fretted. \"And I've chewed off my lipstick, haven't I? Maybe I should put on a dress.\"\n\n\"Cassie.\" Ed tipped down her rhinestone glasses, peered over them. \"You look fine. You look fresh. He doesn't care what you're wearing, take my word for it. He's only going to care that you're there. Now go get him.\"\n\n\"All right.\" Cassie squared her shoulders, picked up her purse. \"I'm going. I'm going now. But if you need anything, just\u2014\"\n\n\"I won't need a thing. Go.\"\n\n\"I'm going.\"\n\nEd wiggled her bright red brows as Cassie went out the door. Poor kid, she thought. She looked like she was walking out in front of a firing squad. With a cackle, Ed tipped her glasses back up and flipped the video back on.\n\nHer money was on Devin MacKade.\n\n## Chapter 7\n\nHe really should just give it up and go back and crawl into his cot. That was what Devin told himself, but he kept right on sitting at his desk with his nose in a book. The story just wasn't holding his interest. It wasn't the fault of the author; nothing could have held his interest just then.\n\nHe knew it was foolish, and useless, but he'd had nothing and no one to vent his temper on. So there it was, still curdling inside him. He'd actually considered heading out to the farm and picking a fight with Shane. It would have been easy. Too easy. So he'd decided against it.\n\nHe told himself it was because he was a better man than that. He'd have done that sort of thing in his teens\u2014hell, in his twenties. The fact was, he'd probably have done it last week.\n\nBut it just didn't suit his mood now.\n\nHe was just going to sit here, in his quiet office, with his feet up on his desk and the chair kicked back, and read. Even if it killed him.\n\nIt was after ten on a weeknight, which meant it was doubtful any calls were going to come in to liven things up. He didn't have to be there, but he liked the solitude of his office at night, the familiarity of it. And the fact that he could be there, behind the desk instead of behind the bars.\n\nHe hadn't even turned the radio on, as he often did to bring a little music and company into the night. The only light was the one on his desk, the metal gooseneck lamp aimed at the book in his hands. The book he wasn't reading.\n\nHe considered getting up and brewing coffee, since he wasn't going to bed. But it seemed like too much effort.\n\nIt was the first time in his life he could remember being so angry and so tired at the same time. Usually temper energized him, got his blood up and his adrenaline sizzling. Now he was sapped. He supposed it was because most of the anger was self-directed, though he still had plenty left over for Cassie.\n\nWhen a woman hurt a man, it was the most natural thing in the world to cover it with anger.\n\nHe'd told other women he loved them. He wouldn't have denied it. The fact was that he'd tried to love other women. He'd worked hard at it for a space of time. The last thing he'd wanted to do was moon around over something he couldn't have.\n\nWhich was just what he was doing now.\n\nSulking, his mother would have called it, he thought with a grimace. He missed her more just now than he had since she'd died. And he'd missed her quite a bit over the years.\n\nShe'd have given him a cuff on the ear, he supposed, or she'd have laughed. She'd have told him to get his sorry butt up and do something instead of brooding over what he should have done. Or shouldn't have done.\n\nWell, he couldn't think of anything to do, except count his losses. He'd moved too quick, pushed too hard, and he'd stumbled over his own heart.\n\nThe hell with it, Devin thought again, and let the book lie on his chest. Shifting in the chair, he closed his eyes and ordered himself to think about something else.\n\nHe needed to talk to the mayor about getting a stop sign out on the end of Reno Road. Three serious accidents there in a year was reason enough to push for it. Then there was the talk he'd promised to give at the high school for the last assembly before summer hit. And he really had to help Shane with the early haying...\n\nThe dream snuck up on him, sly and crafty. Somehow he'd gotten from the hayfield to her bedroom door. Cassie? No, that wasn't Cassie. Abigail. Love and longing stirred in him. Why couldn't she see that she needed him as much as he needed her? Would she just sit there with her hands folded in her lap over her embroidery, her eyes tired and lost?\n\nIt seemed nothing he could say would convince her to come with him, to let him love her, as surely he'd been born to do. No, she would close herself off from him, from everything they could have. Should have.\n\nAnger stirred along with the love, along with the longing. He was tired of coming begging, with his hat in his hand.\n\nI won't ask again, he told her, and she just watched him. I won't come to you again and have you break my heart. I've waited long enough. If this is the way it has to be, I'm leaving Antietam. I can't keep running the law here, knowing you're here, always out of reach. I have to pick up whatever pieces are left of my life and go.\n\nBut she said nothing, and he knew when he stepped back, walked down the hall and down the stairs that it was the end. Her weeping drifted to him when he left the house.\n\nCassie stood on the other side of the desk, twisting the strap of her purse in her fingers. She hadn't expected to find him asleep, didn't know if she would wake him or leave as quietly as she had come.\n\nThere was nothing peaceful about him. There should have been, the way his feet were propped on the desk, crossed at the ankles, the way the book was lying open against his chest, the desk lamp slanting light over it.\n\nBut his face was hard and tense, his mouth grim. She wished she had the courage to smooth those lines away and make him smile.\n\nThen again, courage had always been her problem.\n\nHe opened his eyes and had her jumping like a rabbit. \"I'm sorry. I didn't mean to wake you.\"\n\n\"I wasn't asleep.\" At least he didn't think he'd been asleep. His brain was fuzzy and full of the scent of roses, and for a moment he'd thought she was wearing some full-skirted blue gown, with lace at the throat.\n\nOf course, she wasn't. Just her tidy little blouse and slacks, he thought, dragging a hand through his hair.\n\n\"I was just going over some things in my head. Town business.\"\n\n\"If you're busy, I can\u2014\"\n\n\"What do you want, Cassie?\"\n\n\"I...\" He was still angry. She had expected that, was prepared for it. \"I have some things to say to you.\"\n\n\"All right. Go ahead.\"\n\n\"I know I hurt you, and that you're furious with me. You don't want me to apologize. You get mad when I do, so I won't.\"\n\n\"Fine. Aren't you going to make me coffee?\"\n\n\"Oh, I\u2014\" She'd already turned to the pot before she caught herself. She drew a breath, turned back and faced him. He had a brow lifted. \"No.\"\n\n\"Well, that's something.\"\n\n\"I'm used to waiting on people.\" Now she was irritated, a not entirely unpleasant sensation, even if an unfamiliar one. \"If it annoys you, I can't help it. Maybe I like waiting on people. Maybe it makes me feel useful.\"\n\n\"I don't want you to wait on me.\" He could see the irritation clearly enough. It added a snap to her eyes that fascinated him. \"I don't want you to feel obliged to me.\"\n\n\"Well, I do feel obliged. And I can't help that, either. And the fact that I do feel obliged and do feel grateful\u2014 Don't shout at me, Devin.\"\n\nImpressed with her no-nonsense tone, he closed his mouth, then added, \"I might yet.\"\n\n\"At least wait until I've finished.\" It wasn't so hard, she realized. It was like dealing with the children, really. You just had to be fair and firm, and not allow yourself to be sidetracked. \"I have good reasons to feel obliged to you, and grateful to you, but that doesn't meant that beyond that, or besides that... It doesn't mean I don't have other feelings, too.\"\n\n\"Such as?\"\n\n\"I don't know, exactly. I haven't had real feelings for a man in\u2014maybe never,\" she decided. \"But I don't want to lose your friendship and...affection. Next to the children, there's no one I care for more than you, Devin. Being with you...\" She was going to fumble now, and she hated herself for it. \"The way we were today, this afternoon, before you got mad, was so nice, it was so special.\"\n\nShe was cutting right through his temper, slicing it to ribbons, the way she was standing there, twisting her purse strap and struggling to find a way to put things right between them.\n\n\"Okay, Cassie, why don't we\u2014\"\n\n\"I came here to go to bed with you.\"\n\nHis jaw dropped. He was sure he heard it hit the edge of the desk. Before he could pick it up again, the door burst open and Shane strolled in.\n\n\"Hey, Dev. Hey there, Cassie. Thought you might want to go down to Duff's and shoot a couple games. Why don't you come along, Cassie? It's about time you learned how to shoot pool.\"\n\n\"Go away, Shane,\" Devin muttered, without taking his eyes off Cassie's face.\n\n\"Come on, Dev, you've got nothing to do around here except read another book and drink stale coffee.\" Experimentally he picked up the pot and sniffed. \"This stuff'll kill you.\"\n\n\"Get lost now, or die.\"\n\n\"What's the problem? We'll just\u2014\" All innocence, Shane turned back. The tension in the air struck him like a fist, the way his brother was staring at Cassie. The way she was staring back. \"Oh. Oh,\" he repeated, drawing out the word on a milewide grin. \"Well, son of a gun. Who'd have thought?\"\n\n\"You've got ten seconds to get out the door before I shoot you.\"\n\n\"Well, hell, I'm going. How was I supposed to know you and Cassie were\u2014\"\n\n\"Tomorrow,\" Devin said evenly, and finally managed to get his feet off the desk and onto the floor, \"I'm going to break you into very small pieces.\"\n\n\"Yeah, right. I guess you two don't want to play pool, so I'll be going. Ah, want me to lock this?\" he said, winking as Devin snarled at him. But he was obliging enough to flip the latch and shut the door snugly behind him.\n\n\"You're not really going to fight with him?\" Cassie began quickly. \"He didn't mean anything, and...\" Tongue-tied, she let her words trail off as Devin walked slowly around the desk.\n\n\"What did you say to me before my idiot brother came in?\"\n\n\"That I came here to go to bed with you.\"\n\n\"That's what I thought you said. Is this your way of mending fences and keeping my friendship? Some new way of apologizing?\"\n\n\"No.\" Oh, she was making a mess of it. He didn't look amorous, just curious. \"Yes, maybe. I'm not sure. I know, at least, I thought you wanted to. Don't you?\"\n\n\"I'm asking what you want.\"\n\n\"I'm telling you.\" Lord, hadn't she just said it, out loud, in plain words? \"I came here, didn't I? I called Ed, and she's staying with the kids, and I'm here.\" She shut her eyes briefly. \"It isn't easy for me, Devin.\"\n\n\"I can see that. Cassie, I want you, but what I don't want is for you to think this is necessary to make things up with me.\"\n\nShe did what she had done once before. It had worked then. Cupping her hand on his cheek, she leaned up and kissed him.\n\n\"Now you're waiting for me to jump you,\" Devin murmured.\n\n\"Oh, I'm no good at this.\" In disgust, she tossed her purse into a chair. \"I never have been.\"\n\n\"At sex?\"\n\n\"Of course at sex. What else are we talking about?\"\n\n\"I wonder,\" he said quietly, but she was off and running in a way he'd never seen or heard before.\n\n\"I don't know what you want, or how to give it. If you'd just do whatever you usually do, it would be all right. It's not that I won't like it, I will. I'm sure I will. It's not your fault that I'm clumsy or stiff, or that I don't have orgasms.\"\n\nShe broke off in horror, and saw that he was gaping at her.\n\n\"Excuse me?\"\n\nSomeone else had said that, she thought frantically, looking everywhere but at him. Surely someone else had said that. All she could do to cover the overwhelming tide of horrid embarrassment was to rush on.\n\n\"What I mean is, I want to go to bed with you. I know it'll be nice, because it's nice when you kiss me, so I'm sure the rest will be, too. And if you'd just do something, I wouldn't be feeling so stupid.\"\n\nWhat the hell was he supposed to do? He knew very well the woman standing there was the mother of two, had been married for a decade. And he'd just realized she was as close to a virgin as anyone he'd ever touched.\n\nIt scared the living hell out of him.\n\nHe started to tell her that they would take a step back, take it slow. Then he knew that was the wrong way to go. It was painfully obvious that so much of her had been crushed already. What he would know was patience, she would see as rejection.\n\n\"I should do what I want with you?\"\n\nEnormously relieved, she smiled. \"Yes.\"\n\nIt was an offer that had the juices flowing hot. He knew if he wanted this to work he had to clamp down on needs\u2014and on nerves. \"And I'll tell you what to do, and you'll do it.\"\n\n\"Yes.\" Oh, it was really so simple. \"If you just don't expect too much, and you\u2014\"\n\n\"Why don't we start this way?\" He put his hands on her shoulders and lowered his mouth gently to hers. \"There's something I want very much, Cassie.\"\n\n\"All right.\"\n\n\"I want you to say you're not afraid of me, that you know I won't hurt you.\"\n\n\"I'm not. I know you won't.\"\n\n\"And I want you to promise something.\" He skimmed his lips up her jaw, felt her shoulders relax under his hands.\n\n\"All right.\"\n\n\"That you'll say stop if you mean stop, if I do something you don't like.\"\n\n\"You won't.\"\n\nHis lips cruised around to her ear and made something quake inside her. \"Promise me.\"\n\n\"I promise.\"\n\nHe took her hand and led her through the door into the small room he used at night. It was dark. It held little more than a narrow bed, a rickety table, an ashtray he rarely used anymore.\n\n\"It shouldn't be here. I should take you somewhere.\"\n\n\"No.\" If it wasn't now, she'd lose her nerve. What difference did atmosphere make, when it was dark and her eyes were closed? \"This is fine.\"\n\n\"We'll make it better than fine.\"\n\nHe lit one of the station's emergency candles, so at least there was soft light. She couldn't know how arousing she was, standing there, tidy and terrified, prepared to give herself. To sacrifice herself, he thought grimly.\n\nHe would show her different.\n\n\"I love you, Cassie.\" It didn't matter that she didn't believe him. She would. He kissed her again, slowly, deeply, patiently, putting his heart into it.\n\nAnd moment after moment there was nothing but the kiss, the taste of it, the meeting of lips, the way she softened against him.\n\n\"Hold me,\" he murmured.\n\nObedient, wanting to please, she wrapped her arms around him. There was a little shock when she felt how hard he was, how strong. How odd it was to hold him tight against her. While his mouth moved over hers, she stroked her hands over his back.\n\n\"I want to see you.\" He continued to rub his lips over her throat, even as her hands tensed on his back. He didn't mind her being shy. He found it endearing. \"You have such a lovely face.\" His eyes stayed on it as he slowly undid the buttons of her blouse. \"Eyes like fog, and that sexy mouth.\"\n\nShe blinked, thrown off enough to make no protest when he parted her blouse. No one had ever called her sexy. Then his gaze shifted downward, and the sound that rumbled in his throat had something curling hard in her stomach.\n\nHe was cupping her breasts in his hands, holding them as if they were delicate glass that could be shattered by a careless touch.\n\n\"Lovely.\"\n\n\"I'm small.\"\n\n\"Perfect.\" He lifted his gaze to hers again. \"Just perfect.\" He watched her lashes flutter when he circled her breasts, brushed his thumbs over her nipples. And his blood heated when they stiffened, when she shuddered, when her eyes opened again in surprise and went dark.\n\nWhat was he doing? Why wasn't he squeezing or pulling? She felt her head spin before it fell back. Heard, with a kind of dull shock, her own moan.\n\n\"Do you have to close your eyes?\" he asked her. It wasn't so difficult to keep his hands easy, after all, not on skin that was soft as silk. \"I like to watch them go cloudy when I touch you. I love to touch you, Cassie.\"\n\n\"I can't breathe.\"\n\n\"You're breathing. I can feel your heart.\" He lowered his lips to her shoulder before straightening to pull off his shirt. \"Feel mine.\"\n\nMy oh my, Cassie thought. He looked like something in one of those glossy magazines. All muscles and firm smooth skin. With only the slightest of hesitations, she laid a hand on his chest, and smiled. \"It's pounding. Are you ready?\"\n\n\"Oh, Cassie.\" Biting back a groan, he drew her into his arms, cradled her there, savored the feel of her flesh pressed against his. \"I haven't even started.\"\n\nBecause she thought he meant something entirely different, her brows drew together and she swallowed her distaste and reached courageously for his crotch.\n\nWith a ripe oath, he jerked back, stuttering, as she covered herself and gaped.\n\n\"I thought you wanted... I thought you meant...\" Good God, he'd been hard as rock. And huge.\n\nHe decided laughing would be better than screaming. \"Darlin', you do that again, I'm going to embarrass myself, and we'll have to start all over. If it's all the same to you, I'd just like to touch you for a while.\"\n\n\"I don't mind, but you're...\"\n\n\"I know what I am. You said you'd do what I want,\" he reminded her, fighting to keep his voice from growing rough with need. \"I want you to look at me, look right at me now.\"\n\nWhen she did, he skimmed his hands over her breasts again. He could see surprised pleasure ripple over her face, hear it in her quickening breaths. So he began to murmur to her, endearments, foolishness, gauging her reaction.\n\nWhen her eyes closed, he lifted her slowly off her feet, holding her suspended, trailing his mouth down from hers and over her throat, her collarbone, and at last to her breast.\n\nHer hands clamped on his shoulders and her body arched as arrows\u2014bullets\u2014of hot sensation pierced through her flesh and straight to her center to burn. She shook her head, struggling to clear it.\n\n\"Devin.\"\n\nHe laved his tongue over her. \"Do you want me to stop?\"\n\n\"No. No.\"\n\n\"Thank God.\"\n\nWhen she was quivering, when her hands were clutching and flexing on his skin, he lowered her to the floor again, until his mouth was fixed on hers. Her hands were fisted in his hair, her breath was coming fast. Her lips were hot.\n\nAnd still she stiffened, just for an instant, when he unhooked her slacks.\n\nShe wouldn't spoil it. That she promised herself. Whatever came now didn't matter, because what came before had been so lovely. She'd never felt these pulls, these yearnings. Or she'd somehow forgotten them. His hands were hard, the palms rough, but he used them so gently on her. She would have been happy to have him go on touching her, just like this, forever. She could blissfully have drowned in those wonderful ripples of sensations.\n\nNow he was uncovering the rest of her, and she knew it would be over soon. But he would hold her when he was done. He would hold her close and warm, she was sure of it. That would be enough.\n\nWhen he picked her up and cradled her against his chest, she smiled. The candlelight was lovely, and she felt an intense sense of tenderness, of sweetness. He'd made her feel wanted. She laid her lips against his, curled her arms around his neck, keeping them there as he lowered her to the cot so that the springs squeaked under their weight.\n\nShe opened her eyes in confusion when he didn't push inside her. Instead, he was curved beside her, his eyes on her face, his hand stroking up and down her torso.\n\n\"Don't rush me,\" he said mildly. \"I'm enjoying myself.\"\n\nTo her astonishment, he began to talk to her about her body, her skin, her eyes, her legs. And the things he was murmuring sent flashes of new heat inside her.\n\nShe was grateful he didn't seem to need her to talk back. She was having trouble breathing again.\n\nShe was so incredibly sweet, so amazingly innocent. That was what kept his need locked away, kept his hands from taking quickly. Twelve years, he thought, listening to the way her breath caught, then burst out, when he skimmed a finger up the inside of her thigh. When a man had waited so long, he could be as patient as a saint, though his blood churned like a riptide.\n\nHe lowered his mouth to her breast again. So small, and firm, and smelling like spring. Under his lips he felt her heart thundering, felt her skin quiver. And knew he pleasured her.\n\nHe wanted to give her more, to give her everything, to know she craved as he did. So he stroked and suckled, arousing himself and her until she began to writhe under him and he knew she was climbing toward the edge. And he would be the one to show her that the fall was sweet.\n\nIt was too hot. She was burning from the inside out and couldn't keep still. She ached, and nothing she could do seemed to soothe the throbbing. Something inside her was racing for something else, and she strained away from it. It was too big, too huge, too terrifying. The air was thick, the sensations were too fast and too many. She moaned and bit down on her lip to stop the sound.\n\n\"You can yell,\" Devin told her, his own voice ragged. \"You can scream if you want. Nobody can hear but me. Just let go, Cassie.\"\n\n\"I can't.\"\n\nHe dipped his fingers inside her, and his head spun. She was hot and wet and more ready than she knew. \"Don't ask me to stop,\" he murmured against her mouth. \"Don't ask me.\"\n\n\"No. No, don't.\"\n\nShe did scream then, a sound that should have shocked her, it was so wild and wanton. But her body was too busy rearing up toward him, poised on a spear of dark, drenching pleasure such as she'd never known. Everything inside her came to a fist, tensed violently, painfully, then burst free. She collapsed, weak as water, and thought she heard him groan.\n\n\"Again.\" He was greedy now. He kept a hand fisted in the tousled sheet to keep himself sane, and urged her up, urged her over. She strained against his hand, poured into it, and the arms she'd wrapped around him slid bonelessly to the mattress.\n\nSurrender, he thought. More, fulfillment. But now he would give her himself.\n\nHe covered her, slipped inside her, holding himself back as her eyes fluttered open on fresh shock. He took her slowly, drawing out each stroke, each pulse. His heart almost burst from the strain of control when she convulsed again. Deliberately, patiently, he stirred her, gaining unimagined joy as he felt her begin once more to tremble and race.\n\nThe shudder worked through him, ripping, demanding. This time he knew he would go with her. Finally, with her. He clenched at the hand she'd fisted in the sheet, covered it. And took the fall.\n\nShe couldn't stop shuddering. But she wasn't cold. Not cold at all. The heat from her body, and from Devin's, which lay over her, seemed to rise in waves that were all but visible. He was breathing hard, like a man who'd been racing, and his full weight was on her, pinning her to the mattress so that she could feel the springs pushing against her back.\n\nIt was lovely.\n\nShe understood, for the first time in her life, the secrets of the dark.\n\n\"I know I'm crushing you,\" he managed. \"I'm trying to move.\"\n\n\"You can stay.\" She wrapped her arms around him to keep him there. He was still inside her, still there. It felt wicked and wonderful. \"I like it this way.\"\n\n\"I appreciate you putting up with all that, seeing as you're not big on sex.\"\n\nThe dry tone alerted her, but she was too delighted to mind being teased. \"I didn't mind,\" she said, and smiled against his throat. \"Devin, it was wonderful. I actually\u2014\"\n\n\"I know. Several times. I counted.\"\n\nShe laughed, and didn't feel at all embarrassed. \"You did not.\"\n\n\"I certainly did.\" He found the energy to lift his head and look down at her. \"You can thank me later.\"\n\nHer smile sweetened. She'd never had a man look at her like that, all hazy-eyed and satisfied and sleepy. \"It was all right.\" Incredibly moved, she lifted a hand to his cheek. \"Wasn't it?\"\n\n\"It was worth waiting for.\" He turned his lips into her palm. \"But I'm not waiting another twelve years to have you again.\"\n\n\"I don't want you to.\" Everything inside her was dreamy and disjointed. \"You're so handsome.\"\n\n\"The curse of the MacKades.\"\n\n\"I mean it.\" She lifted her other hand, framing his face. It was so easy to touch him now, to let her finger trace that wonderful dimple beside his smile. \"Do you remember how I used to come out to the farm sometimes when I was a girl, to visit with your mother?\"\n\n\"Sure. You were a pretty little thing, skinny, and I didn't pay you much mind. My mistake.\"\n\n\"I used to watch you. In the summer, especially. When you'd be working with your shirt off.\"\n\nHis grin flashed. \"Well, well, little Cassie...\"\n\n\"I had a terrible crush on you for a while, and these really imaginative fantasies.\" She chuckled. \"Well, I thought they were imaginative, until now. Nothing came close. I can't believe I'm saying this, talking to you like this.\"\n\n\"Under the circumstances, you can say pretty much anything.\" He was hoping she would. He could feel himself hardening inside her.\n\n\"I was about twelve, and you were always nice to me. All of you were. I loved coming out there, just to be there. But it was a bonus when it was summer and you'd be bare-chested and sweaty. Like you are now.\" Experimentally, she traced a finger over his shoulder. \"All those muscles shiny with damp. Your body...it's so beautiful. Sometimes you'd come into Ed's, and when you'd go out, if there were women in there, they'd roll their eyes and sigh.\"\n\n\"Come on.\"\n\n\"Really. Of course, if one of your brothers came in, they'd do the same thing.\"\n\n\"Don't spoil it.\"\n\nShe laughed, lifting a hand to push tousled hair from her cheek. \"Okay. They sighed louder, and longer, for you.\"\n\n\"That's better.\"\n\n\"And Ed would say something like 'That Devin MacKade's got the best buns in three counties.'\" She caught herself on a giggle, her eyes going wide. \"I shouldn't have said that.\"\n\n\"Too late. Besides, I know Ed's partial to that particular part of the anatomy. She's told me.\"\n\n\"She's shameless.\" With a long sigh of her own, Cassie wound her arms around him again, let her hands wander down. \"But you do have an exceptional seat.\"\n\n\"Now you've done it.\" As her fingers brushed over his hips, he began to move inside her. Nothing could have pleased him more than seeing the way her eyes rounded in surprise.\n\n\"But how can you\u2014 Oh, my God!\"\n\n\"It's no trouble,\" he assured her. \"It's my pleasure.\"\n\nAnd after, a long time after, he curled up beside her on the cot, his face buried in her hair, his legs tangled with hers. As she had hoped, as she had needed, he held her.\n\n## Chapter 8\n\nIt was barely dawn when Cassie crept into her own kitchen. She felt giddy, like a teenager sneaking home after curfew. Not that she'd ever broken curfew, she thought now. Not that she'd ever done anything except exactly what was expected of her.\n\nIt made her hushed, secret return all the more liberating.\n\nShe'd just spent the night, all night, with the most exciting, beautiful, the most gentle man she'd ever known.\n\nShe, Cassandra Connor Dolin, was having an affair.\n\nShe had to slap her hand over her mouth to muffle a burst of laughter. Her heart was still racing, her head still swimming, and her body...her body felt as though it had been polished with flower petals.\n\nShe was sure she looked different, and tried to see her reflection in the chrome of the toaster. Because she was alone, she allowed herself three quick spins before putting the kettle on for coffee.\n\nThen, being a mother, she padded toward the bedrooms to make sure her children were snug and asleep. Turning from Connor's room, she stifled a gasp. There was Ed, her fire-engine hair done up in squashy pink rollers, wearing a wildly flowered robe of pink and blue.\n\n\"I'm sorry,\" Cassie whispered. \"I didn't mean to wake you.\"\n\n\"You were quiet as a little mouse. I was listening out for you.\" Ed took a long, measuring look, and liked what she saw. \"Well, well, I believe you're feeling good and smug this morning. About time, too.\"\n\nCassie cast a last look at her sleeping son, then backed down the short hallway toward the kitchen. \"The kids didn't give you any trouble, did they?\"\n\n\"Of course not. Never heard a peep out of either of them.\" Grinning, Ed followed Cassie into the kitchen, watched while she busied herself measuring out coffee. \"You going to tell me about it, or am I going to have to use my imagination? I got a damn good one.\"\n\nThe heat rose to Cassie's cheeks, but it was from pleasure as much as embarrassment. \"I stayed with Devin.\"\n\n\"I figured that out, sweetie pie.\" Very much at home, Ed popped bread into the toaster. \"From the look on your face, the two of you didn't discuss world events until morning.\" Sighing a little, she poked around in the refrigerator. \"I'm not just being nosy. I guess I want to make sure you're as okay on the inside as you look on the out.\"\n\n\"I'm fine.\" Cassie turned, smiled. There was Ed, holding a jar of preserves in one hand and a gallon of milk in the other, her thin face shiny with night cream, her hair exploding on rollers, her outrageous robe falling over legs the shape of toothpicks.\n\nThis, Cassie realized, was the mother of her heart. Cassie set the steaming kettle down again and dashed over to throw her arms around Ed.\n\nSurprised, moved, Ed pressed her lips to Cassie's hair. \"There, baby...\"\n\n\"I feel...different. Do I look different?\"\n\n\"You look happy.\"\n\n\"My stomach's still jumping.\" Laughing at herself, Cassie drew back and pressed a hand to it. \"But it feels good. I didn't know it could be like that. I didn't know I could be like that.\" Casting a quick look at the hallway, she went back to the coffee. Her children were asleep, and would be for another half hour. After all these years, Cassie thought, she would have a mother to listen.\n\n\"I've never been with anyone but Joe.\"\n\n\"I know that, baby.\"\n\n\"Before we were married, I wouldn't let him. I wanted to be married first, I wanted it to be right.\" She poured coffee for both of them, then sat at the table. \"I was nervous on our wedding night, but excited, too. You'd given me a white nightgown for my shower. It was so pretty, so perfect. It made me feel like a bride. When we got to the motel, I asked Joe to give me an hour to myself. I wanted to take a long bath and...well, you know.\"\n\n\"The female ritual. Yeah, I know.\"\n\n\"He came back\u2014it was closer to two hours\u2014and he was drunk. It wasn't the way I'd always dreamed. He ripped the gown, and he pushed me onto the bed. It all happened so fast, and he hurt me. I knew it was supposed to hurt some the first time, but it was more than some. He fell asleep right after, and I just laid there. I didn't feel anything.\"\n\n\"A man's not supposed to treat a woman that way.\" Even if she hadn't already despised Joe Dolin, Ed would have despised him now. \"That's not how it's supposed to be.\"\n\n\"It was the way it was. Always. I never felt anything, Ed. Ever. He didn't always hurt me, but it was always quick, and mostly a little mean. I figured it was my fault\u2014he told me it was often enough. It got better when I was carrying Connor, because he left me alone most of the time. I didn't know he was cheating on me then. I guess I was too stupid.\"\n\n\"Don't you call yourself stupid,\" Ed said fiercely. \"I don't want to hear that.\"\n\n\"Maybe I just didn't care enough to know, or want to know. I was wrapped up in becoming a mother, then in being one. He was already hitting me. We hadn't been married long when that started, but I didn't think there was anything I could do about it. My mother said...well, it doesn't matter what she said. I stayed, then Emma came along. He only wanted me a couple of times after Emma... He forced me.\"\n\n\"Oh, Cassie. Honey, why didn't you tell me?\"\n\n\"Ed, I was too ashamed. He was my husband, and I had it in my head that he had a right to do what he did. I know different now.\" She took a long breath. \"You see, when I went to Devin last night, I didn't think... I knew he wouldn't hurt me, at least not like Joe had. I thought going to bed with him would make him happy, and it didn't matter to me. I mean, I thought he would just...that I would just...\"\n\n\"You had yourself a real man last night,\" Ed finished. \"And it changed things.\"\n\n\"Yes.\" Relieved, Cassie smiled. \"He was so gentle, so patient. You know, it mattered to him what I was feeling. It really mattered. And he made me feel beautiful. Ed...\" She bit her lip, even as it curved again. \"It matters to me now. I'm already thinking about next time.\"\n\nEd let out a cackling laugh and squeezed Cassie's hand. \"Good for you.\"\n\n\"He says he loves me,\" Cassie said quietly. \"I know men say those things when they want you, or they think you need to hear it. But do you think he could?\"\n\n\"I think Devin MacKade's a man who says what he means. What about you?\"\n\n\"I don't know. That part of me is so confused. I didn't love Joe, Ed. I never did. I used him.\"\n\n\"Cassandra\u2014\"\n\n\"No, I did. I used him to get out of the house, because I wanted to have a family of my own, and he was there. I wasn't fair to him. I don't mean that gave him the right to beat me,\" she added, noting the warrior gleam in Ed's eye. \"Nothing gave him that right. But I didn't love him, not the way a woman should love her husband.\"\n\n\"He didn't do anything to deserve love.\"\n\n\"No, he didn't. With Devin, I feel so many things, so many different things, and I don't know if one of them is that kind of love.\"\n\n\"Then you take all the time you need to sort it out. Don't you let anyone push you into anything you're not ready for. Not even Devin.\"\n\n\"How will I know?\"\n\n\"Sweetie pie, when the time comes you'll know. Take my word for it, you'll know.\"\n\nWhile Cassie was talking with Ed over coffee, Devin was pulling up at the farm. He'd felt a need for home. The sky was losing its dawn haze when he walked into the milking parlor. Shane and two of the 4-H students he often took on as help were finishing up the morning routine.\n\nPatiently Shane showed one of the boys how to detach cow from machine without causing irritation. The parlor smelled of warm milk, animal and hay.\n\n\"You're going to check her teats after, just like you did before, to make sure there's no infection.\" He did so himself, demonstrating. \"When she's dry, you see to her feed.\" He cocked a brow at Devin. \"You can see the sheriff wanders in when most of the work's done. Y'all lead them out now.\"\n\nDevin gave the cow an easy swat, then helped Shane clean and disinfect the machines. It was routine, companionable work.\n\n\"Remember when Dad had us milking by hand?\" Devin asked.\n\n\"He figured we'd better know. Machines break down, but cows fill up regular. You're up early,\" Shane commented. \"And you've got a stupid grin on your face. Looks like you got lucky.\"\n\nDevin only angled his head. \"I'm feeling too good to pound on you this morning.\"\n\n\"Good, because I've got to finish up here and get to the hens before breakfast. You and Cassie,\" he said, grinning again. \"Who'd have thought it?\"\n\n\"I've been thinking about it for a long time.\" Devin helped Shane secure the fresh milk in the stainless-steel tanks. \"I've been in love with her a long time.\"\n\nShane straightened, winced. \"Man, don't start that. Every time I turn around, somebody's falling in love. It's giving me nightmares.\"\n\n\"Well, get used to it. I'm going to ask her to marry me.\"\n\nShane rubbed his hands over his face, pulled off his cap, dragged hands through his hair. \"What is it? Something in the water around here? First Rafe, then Jared. Now you. I turn my back for a minute and everybody's getting married, having babies. Get a hold of yourself, Dev.\"\n\n\"Afraid it's going to rub off?\"\n\n\"Hell, I'm going to start to take shots. Look, Cassie's as sweet as they come, and as pretty as fresh milk, but let's not go crazy.\"\n\n\"I love her,\" Devin said, so simply Shane groaned. \"It seems I always have. There's nothing I could do about it even if I wanted to.\"\n\n\"You know what kind of trouble this is going to cause me? Don't you have any consideration?\" Shane demanded. \"I'll be the only one of us left. Women home in on things like that. I won't be able to get myself a snuggle without the woman thinking it's going to lead to orange blossoms.\"\n\n\"You'll have to tough it out.\"\n\n\"What in sweet hell's so appealing about marriage?\" Grumbling, Shane headed out of the milking parlor. \"I mean, think about it, Dev. Really think. You've got one woman for the rest of your life. Just one. And there're so many out there. Tall ones, short ones, round ones.\"\n\nAmused, Devin slapped a hand on Shane's shoulder as they walked toward the chicken coop. \"And with me out of the way, there'll be more for you.\"\n\n\"There is that.\" Taking it philosophically, Shane shrugged. \"I guess it'll be up to me to maintain the MacKade legend. I'll just have to make the sacrifice.\"\n\n\"You're up to it, bro.\"\n\nCassie never lingered in the library. She was much too conscientious to skim over her cleaning there, but most often she tried to arrange her schedule so that someone was in the house when she dealt with that room.\n\nThere was no one in the house now. Her children were in school and the guests were busy with their sightseeing for the afternoon. She made excuses in her head for why she should see to a dozen other things besides that one room. But she knew the library had been used the day before. She knew there were books that needed to be put back on the shelves, plants that needed watering, windows that needed washing.\n\nShe told herself it was foolish. She knew the emotions and moods of the house better than she knew her own. There was nothing here that could hurt her. In fact, the house had changed her life, and all for the better.\n\nArmed with her cleaning basket, she went in. If she left the door open wide behind her, it was only because she wanted to be able to hear if one of the guests returned and wanted anything.\n\nIt wasn't because she was afraid.\n\nShe set the basket aside and tidied the books first. She knew guests often liked to borrow one to read on a rainy afternoon or to help them drift off to sleep at night. Rafe and Regan had provided a variety of books for a variety of tastes. She, too, was free to borrow any she liked, whenever she liked. But she rarely did.\n\nNor, she thought suddenly, did Connor, though he was a voracious reader. It occurred to her that he, too, avoided this room, even though he was thoroughly at home in the rest of the inn.\n\nIt was a feeling, she supposed. Something that lingered in the air. Shaking it off, she carried her basket over to the twin philodendrons that trailed their leaves from pots set in stands by the tall window that overlooked the side garden.\n\nThey needed to be dusted. She'd been putting it off.\n\nAs she began, she felt the chill, down to the bone.\n\nAnd knew she wasn't really alone.\n\nShe thought she could see him, out of the corner of her eye. The big body going to fat, the wide face set in hard, dissatisfied lines.\n\nJoe.\n\nThe terror came so quickly, she dropped the basket at her feet as she whirled around.\n\nHe wasn't there. Of course he wasn't. No one was. But it was so bitterly cold. With numb fingers, she reached for the window to open it to the warm breeze.\n\nShe fumbled, couldn't work the latch, and discovered her breath was coming in short gasps.\n\nYou let him touch you, didn't you? Whore.\n\nShe hunched her shoulders automatically against a blow that didn't come.\n\nDid you think I wouldn't know? Did you think you could cuckold me in my own house? You, with your innocent face and fancy Southern manners. Nothing but a slut.\n\nShaking, she backed slowly away from the window. Her eyes darted around the room, searching corners. There was no one there. But how could she hear the voice so clearly in her head?\n\nKnow this. You'll never leave me. I'll see you dead first.\n\nYou don't love me, Cassie wanted to say. You despise me. Let me go. But the words wouldn't come.\n\nI'll kill you both. Remember that. Till death do us part. And death is your only escape.\n\n\"Cassie.\"\n\nOn a strangled shriek, she spun around. Devin was just inside the door, his eyes narrowed in concern. Without a thought, she ran into his arms.\n\n\"Devin. Devin, you have to go. Go quickly, before he sees you. He's going to kill you.\"\n\n\"What are you talking about? God, you're shaking like a leaf. It's freezing in here.\"\n\n\"You feel it?\" Her teeth were all but chattering as she drew back. \"You can feel it?\"\n\n\"Sure I can. It's like an icebox.\" He rubbed her hands in his to warm them.\n\n\"I thought it was Joe. I swear I saw his fist coming toward me, and then\u2014\" The room spun; her knees buckled. The dizziness lasted only an instant, but she was already up in Devin's arms. \"I'm all right. It's gone.\"\n\nThe room was warm again, sunny and bright, with the scent of roses and polish. Very gently, he laid her down on the soft leather sofa. \"Let me get you some water.\"\n\n\"No, I'm all right.\" She thought she might jump out of her skin if he left her alone there. \"It's just this room.\" She steadied herself, sat up. \"I thought it was Joe, but it wasn't. It was Barlow.\"\n\nShe was still too pale, Devin thought, but her eyes had cleared. His heart had dropped to his knees when he'd seen them roll back in her head. \"Has this happened before?\"\n\n\"Not like this. Not this strong. I'm never very comfortable in this room. Even his bedroom is easier. But this time, I heard... You're going to think I've lost my mind.\"\n\n\"No, I won't.\" He cupped her face in his hands. \"Remember who you're talking to.\"\n\n\"All right.\" She blew out a breath. \"I heard him talking, in my head, I think. It sounded so much like Joe\u2014the tone, the meanness in it. He called me\u2014her\u2014a whore, a slut. He knew she was in love with someone else, but he wasn't going to let her go, ever. He said he'd kill her first, kill both of them.\"\n\n\"Come on, let's get out of here. Let's go upstairs.\"\n\n\"I haven't finished\u2014\"\n\n\"Leave it, Cassie. Just leave it.\" He would have carried her, but she got to her feet. Still, her hand reached for his. \"The other day, when you were talking to the old ladies?\"\n\n\"Mrs. Cox and Mrs. Berman, yes.\"\n\n\"You talked about Abigail being in love with someone. I thought you'd made it up, to add a little romance to the story.\"\n\n\"No. I can't explain it, Devin. I just know it's true. I saw him.\"\n\nHe paused at the back stairs that led up to her apartment. \"You saw who?\"\n\n\"The man she loved. I was in her room, and then I looked and he was at the door. He was looking right at me, talking to me as if I were Abigail. I could feel her there. Her heart was broken, but she let him go. Made him go. Devin...Devin, I think she killed herself.\"\n\nHe sat her down in a chair in her living room. \"Why do you think that?\"\n\n\"I can't explain that, either. Just a feeling. She didn't know how else to get free. And maybe because I thought about it once.\"\n\nThe blood drained from his face. \"Good God, Cassie.\"\n\n\"Not for very long,\" she said quickly. \"And not very seriously. I had the kids to think about. If I hadn't had them, I might have thought about it longer. When you're trapped, Devin, you get crazy ideas about escape.\"\n\nNothing he knew about her had ever frightened him more. \"I would have helped you. I wanted to help you.\"\n\n\"I wouldn't let you. I wouldn't let anyone. You, Ed, Regan. There were others, too, others who were willing to do whatever they could. I was wrong not to accept the help, but that's over now.\" She curled her hands over his. \"I'm not telling you this to upset you, but to try to help you understand how I know she did it. She didn't have people to help her. He'd seen to that. He made sure she was cut off from the women in town, made sure the servants were too frightened to do anything but stand back.\"\n\nSomewhere in her mind, she could almost feel it, see it. \"He hit her, too. It was his fist I saw today. Not Joe's. But it's the same, you see. So much the same. When he killed that boy in front of her, she knew he was capable of anything. She gave up, Devin. Eventually even her children weren't enough to keep her from escaping in the only way she knew.\"\n\n\"It's not you, Cassie.\"\n\n\"It could have been.\"\n\n\"But it's not,\" he said firmly. \"You're here, you're with me. There's nothing for you to be afraid of.\"\n\n\"I'm tired of being afraid.\" She closed her eyes, let her head rest on his shoulder as he crouched in front of her. \"I'm glad you're here.\" She let out a deep sigh. \"Why are you here?\"\n\n\"I worked it so I could clear out for an hour. I wanted to see you. I wanted to be with you.\"\n\n\"I thought about you all morning. I nearly put coffee in Emma's thermos for school, because I was thinking about you instead of what I was doing.\"\n\n\"Really?\" He couldn't think of a more satisfying compliment. When she lifted her head, he could see that the color was back in her cheeks. \"Were you thinking that you'd like to make love with me again?\"\n\n\"Yes, I was.\"\n\n\"I've still got most of an hour,\" he murmured, rising and bringing her to her feet.\n\nShe blinked. \"It's the middle of the day.\"\n\n\"Uh-huh.\" He drew her toward the hall.\n\n\"Devin, it's daylight.\"\n\n\"That's right.\" He unhooked the belt that held his beeper and weapon, hung them over the doorknob.\n\n\"It's...\" Her heart stumbled as he reached out to unbutton her blouse. \"It's barely noon.\"\n\n\"Yeah, I'm going to miss lunch.\" As he slipped the blouse from her shoulders, lowered his mouth toward hers, he smiled. \"Do you want me to stop, Cassie?\"\n\nHer head rolled back on her shoulders. \"I guess I don't,\" she said, weak, willing.\n\nShe forgot that the sun was shining and the birds were twittering. She forgot that traffic was cruising by on the road, and that people were going about their business in town.\n\nIt was so easy, so powerfully easy, to let it all happen again. It was so easy to enjoy the way his hands moved tenderly over her, the way his mouth coaxed hers to warm. He felt so good against her when she curled her arms around him, so solid, that she forgot to feel self-conscious because the sun was pouring through the windows.\n\nHe undressed her, completely, taking his time over it, drawing out each moment just to look at her. To look at what was finally his. The softness. The sweetness. He kissed her, soothing and arousing her, as he undressed himself. His hands were gentle, because he knew it was what she needed. His mouth was patient, allowing her to set the pace. And the pace was slow and dreamy.\n\nHe lowered her to the bed she'd made so neatly that morning, gave himself the quiet delight of brushing her hair with his fingers until it was all tangled golden curls over the plain white quilt. Her eyes were closed, and already her cheeks carried the faint flush of stirred passions.\n\nLast night there had been only the light from a practical and unscented emergency candle, a narrow bunk and a room that smelled of old coffee.\n\nToday there was sunlight, birdsong, and the perfume of the flowers by her window. And today, he thought, she knew there would be pleasure.\n\nHe gave her pleasure. Rivers of it. She floated on it, glided on it, immersed herself in it without reserve. All hesitancy, all shyness, vanished under a warm haze of gently lapping sensations.\n\nThe texture of his callused fingers, the friction of them as they moved over her skin caused little sparking shocks that speeded her pulse. The taste of his mouth as it moved to her flesh, then back to her lips, was drugging. She could hear his breathing quicken, or those little hums of pleasure in his throat, whenever he touched some new part of her. He was so beautiful to her\u2014not just his incredibly stunning outward good looks. More, it was the beauty inside that drew and seduced her\u2014the kindness, the strength, the patience.\n\nIt delighted her to be able to squeeze her hands over his biceps, feel the coil of strength in them, in the muscles of his back. She adored the shape and weight of his body, the way it pressed hers deep into the mattress. The light scrape of his teeth on her shoulder gave her a quick, jittery thrill. To answer it, she nipped at his while her hands grew bold enough to journey down.\n\nHe hissed out a breath, jolted. Her eyes flashed open when his head reared up. For an instant, for an eternity, she saw something dark and edgy and dangerous in those moss-green eyes. Something that had her blood leaping high and her pulse scrambling.\n\nHe yanked himself back into control, the way he would have yanked a wild dog on a thick leash. His muscles knotted. He could have sworn he felt the sweat burst out of his pores.\n\n\"Don't worry.\" His voice was raw, but he lowered his mouth gently to hers again. \"Don't be afraid.\"\n\nShe wanted to tell him she wasn't, couldn't be, afraid of him. That she would be afraid of nothing that happened between them. That she wanted to know what had come into his eyes. But he was kissing her into oblivion again, into that misty place where there was nothing but warm, quiet pleasures.\n\nHer moan was long and deep when he eased her to a peak. Long and deep when he gave her more. She let the current take her, opening for him, letting him fill her. Nothing was more stunning than moving with him, feeling his body mesh and mate with hers.\n\nThen his mouth was at her ear, and through her own gasping passion she heard him say her name. Just her name, before he pulled her with him.\n\n\"I love you.\" He still ached for her, even as he shifted his weight and drew her against his side. \"I want you to get used to hearing that.\"\n\n\"Devin\u2014\"\n\n\"No, I don't expect it yet. I will, but I don't expect it yet.\" He turned his face into her hair and breathed in the scent of it and her, a scent that always reminded him of sunlight on a meadow. \"You just get used to hearing it. You tell me when you're used to it, because then I'm going to ask you to marry me.\"\n\nShe went rigid. \"I can't. How can I think about that? This is happening too fast.\"\n\n\"Not for me.\" He wouldn't be angry, he wouldn't even allow himself to be discouraged by the shock in her voice. Instead, he stroked a hand down her arm and spoke with quiet confidence. \"I've gotten good at waiting, so I can wait a while longer. But I figured you should know where I'm heading here. I want you, I want the kids, I want a life, but I can wait until you're ready.\"\n\n\"I might never be ready. Devin, you have to understand, I don't know if I can ever make those promises again.\"\n\n\"You've never made them to me. That's all that counts.\" He rose up on his elbow so that he could study her face. He'd frightened her, he noted. But it couldn't be helped. \"I love you. You let that settle in, and we'll see what happens next.\"\n\n\"Don't you see that\u2014\"\n\n\"I only see you, Cassie.\" Persuasively, he kissed her, until the hand she'd lifted to push against his shoulder went lax. \"Only you.\"\n\nA few miles away, Joe Dolin was policing a picnic area on the battlefield for litter. As he worked, his eyes scanned the fields, the hills, the road below. There were large, shady trees, stone walls. He was going to pick his time, and his spot. This wasn't it.\n\nEventually the crew would work their way down toward the bridge where General Burnside had screwed up during the Battle of Antietam. There the ground was uneven, rocky and thick with brush. There was a creek to hide his scent, trees to cover him.\n\nHe'd often poached in those woods, jacklighting deer illegally with some of his drinking buddies. He had plenty of time now to calculate how long it would take him to travel through them, where he could hide, who he could go to for a little help.\n\nIn the meantime, he was making himself a busy little bee, picking up the soft drink cans and wrappers tossed aside by lousy tourists or kids hooking school. His supervisor wasn't a fool, but Joe never gave him any lip, any trouble, and made sure he was first in line to volunteer for any of the harder or messier jobs.\n\nHe was building himself a damn good rep in prison, something he'd never had on the outside. Something, he thought as he wiped sweat from his brow, that was going to help him get out of the cage.\n\nAnd get back to Cassie. Get to Cassie.\n\nThe little bitch was going to pay for every day he'd spent behind bars. Every hour he'd had to go without a drink or a woman.\n\nWhen he was finished with her, he was going after MacKade. Maybe all four of the stinking MacKades. He'd had plenty of time to plan it out, to work out the mistakes, to dream about it.\n\nHe hoped he had to kill one of them. He hoped it would be Devin. And when he was finished, he was going to Mexico, taking whatever was left of his wife with him.\n\nAll he needed was money, a car and a gun. He knew exactly where he was going to get all three.\n\n## Chapter 9\n\nConnor tried to take in everything at once. He knew Bryan was getting restless, wandering around the sheriff's office, trying to get a look at the cells in the back. But for himself, he thought nothing was more fascinating than watching the sheriff handle calls and type up reports.\n\nHe was going to write a story about it, and he had to get everything just right. The way the office looked, with the dust dancing in the sunlight through the windows, the scars on the desk from feet or cigarettes, the way the ceiling fan squeaked overhead.\n\nHe took a deep sniff and filed away in his mind the scent of coffee\u2014really strong, and a little harsh\u2014and the smell of the dust that sort of tickled the nose.\n\nHe tried to remember just how the phone sounded when it shrilled on the sheriff's desk, how the sheriff's chair scraped against the floor, how the deputy scratched his head, then his cheek, as he put papers away in the file cabinet.\n\nHe already had the sound of the sheriff's voice. It was deep and slow, and there was a hint of something in it. Humor, Connor thought, when he answered some of the calls. Other times it was brisk, kind of official. Once or twice he'd seen lines form between the sheriff's brows.\n\nHe sure did drink a lot of coffee, Connor thought, and he wrote a lot of things down. Connor had a million questions, but he held them in because he knew the sheriff was working.\n\nDevin glanced up and saw the boy watching him. Like an owl, he thought. Wise and patient. A look at his watch told him he'd kept the kids hemmed in for most of their Saturday morning. He imagined Connor could sit there, quiet as a mouse, for hours yet. But he recognized the signs of trouble brewing in Bryan.\n\nIt was time to give them all a break.\n\n\"Donnie, you take over here. We're going to get some lunch at Ed's.\"\n\n\"Yo.\"\n\n\"The state boys call about the Messner case, you tell them I'll have the report to them by Monday.\"\n\n\"Yo,\" Donnie said again, and crushed his brows together over the filing.\n\n\"I'll pick up lunch for Curtis. Tell him, if he starts to make noises back there.\"\n\n\"You got a prisoner?\" Suddenly all of Bryan's boredom was washed away in the thrill of it. \"You didn't tell us.\"\n\n\"Just somebody sleeping off a night on the town.\" He was almost sorry he couldn't tell them it was a mad psychopath. \"I could use a burger.\"\n\n\"All right!\" Bryan darted out of the door. \"I'm starving. Extra fries, right, Con?\"\n\n\"I guess.\" Connor could hardly think about food with all the questions in his head. \"Ah, Sheriff, how come you have that police radio on all the time? I mean, it has fire department stuff, and things from out of your jurisdiction.\"\n\n\"Because you can never be sure what might come over that you'd have to pay attention to.\"\n\n\"When you know somebody, does it feel funny to have to lock them up?\"\n\n\"Sometimes if you know them it makes it easier to settle things before they get out of hand.\"\n\n\"Have you ever had anybody break out?\" Bryan wanted to know as he danced backward on the sidewalk. \"Like, conk you over the head and run for it?\"\n\nDevin ran his tongue around his teeth. He had a wonderful image of poor old Curtis going over the wall. \"Nope, can't say as I have.\"\n\n\"If they did, you'd have to shoot them, right?\" The excitement of it leaped in Bryan's eyes. \"Like in the leg.\"\n\n\"If they did, it's likely I'd know who they were, so I'd just go to their house and bring them back.\"\n\n\"What if they resisted arrest?\"\n\nDevin knew what was expected of him. \"I'd have to rough 'em up.\"\n\n\"Slap the cuffs on him,\" Bryan said with a hoot. \"And back into the cage. Wham!\"\n\n\"The town's quiet,\" Connor said, \"because the sheriff keeps it quiet.\"\n\nTouched, Devin flipped a finger over the bill of Connor's ball cap. \"Thanks. We aim to serve.\"\n\n\"Sheriff.\"\n\nDevin turned and watched with an inner sigh as the ancient and wiry owner of the general store and sub shop approached. The man could talk the bark off a tree.\n\n\"Afternoon, Mr. Grant. How's business?\"\n\n\"Oh, up and down, Sheriff, up and down.\" Mr. Grant paused, flicked a bit of lint from the front of his wrinkled brown shirt. \"I thought I should let you know, Sheriff...not that I poke my nose into what's not my business... With me, it's live and let live...\"\n\nThat ended the statement, which Devin knew was habitual. Mr. Grant's mind wandered freely from pillar to post. \"Let me know what, Mr. Grant?\"\n\n\"Oh, well, I was just taking a little air and happened to walk by the bank. Just past closing time, you know.\"\n\n\"Yes, I know.\"\n\n\"Seemed to me somebody was holding up the bank.\"\n\n\"Excuse me?\"\n\n\"Seemed to me,\" Mr. Grant repeated, in his ponderous way, \"somebody was holding up the bank. Had a gun, sure enough. Looked to me to be a .45. Could be I'm wrong about that. Might be a .38.\"\n\nBefore either boy could comment, Devin slapped a hand on each of their shoulders. \"Go on up to Ed's. Stay there.\"\n\n\"But, Devin\u2014\"\n\n\"Do it, Bryan. Go on now, both of you. Stay there, and don't say anything.\" He stared hard at Connor. \"Don't say anything,\" he repeated. \"We don't want people getting upset and getting in the way.\"\n\n\"What are you going to do?\" Connor said in an awed voice.\n\n\"I'm going to take care of it. Get up to Ed's. Move. Now.\"\n\nWhen they ran off, Devin kept one eye on them, to be sure they obeyed. \"Mr. Grant, I wonder if you'd come along with me. Let's just take a look at this.\"\n\n\"Fine by me.\"\n\nThe bank was across the street and another half a block up. An old brick building with elaborate ironwork, it sat catty-corner from Ed's Caf\u00e9. A quick look showed Devin that the boys had indeed gone in. They had their faces pressed up to the window.\n\nDevin scanned the street. It was Saturday, and there was considerable traffic. Enough, in any case, to cause a problem if there was trouble. He didn't intend to have any of his people hurt.\n\n\"Did you get a look at the man, Mr. Grant?\"\n\n\"Some. Young, 'bout your age, I expect. Can't say as I recognized him. Looked a little like the Harris boy, but wasn't.\"\n\nDevin nodded. He spotted a dirty white compact with Delaware tags at the curb in front of the bank. \"Recognize that car there?\"\n\nMr. Grant thought it over. \"Can't say as I do. Never seen it around here.\"\n\n\"Stay here a minute.\" Unsnapping the flap covering his weapon, Devin sidestepped up to the bank. The door was festooned with curvy ironwork. Through it, he could make out one teller behind the wide counter. And the man across from her, nervously waving a gun.\n\nIt was a .45, he noted. Grant had been dead-on.\n\nHe slipped away from the door. \"Mr. Grant, I'd like you to get on down to the office, tell Donnie I need some backup here at the bank. We've got an armed robbery in progress. I want you to tell him that, straight out. And that I don't want him coming up here blaring sirens or coming into the bank. I don't want him coming into the bank. Have you got that?\"\n\n\"Why, sure I do, Sheriff. Be happy to oblige.\"\n\n\"And stay down there yourself, Mr. Grant. Don't come back up here.\"\n\nHe'd just started to move again when he saw Rafe approaching. Before his brother could lift a hand in greeting, Devin snagged him. \"You're deputized.\"\n\n\"Hell, Devin, Regan just send me out for more diapers. I haven't time to play deputy.\"\n\n\"See that car? White compact, Delaware plates?\"\n\n\"Sure. I got eyes.\"\n\n\"Put it out of commission.\"\n\nNow Rafe's brows lifted, and his grin flashed. \"Gee, Devin, I don't know as I remember how.\"\n\n\"Do it,\" Devin said, and the sharp impatience got through.\n\n\"What's going on?\"\n\n\"Somebody's robbing the bank. Put the car out of commission in case he gets past me. And do what you can to keep people out of the way without getting them stirred up.\"\n\n\"You're not going in there alone.\"\n\n\"I've got the gun, you don't,\" Devin pointed out. \"And I've got the badge. Be a pal, Rafe, and deal with the car. As far as I can tell, there's only one perp. I'm going in. If he comes out waving that damn gun, don't be a jerk. Get out of the way.\"\n\nThe hell he would, Rafe thought, but he crouched down to move around to the driver's side of the car while Devin took out his weapon.\n\nDevin wanted to keep it simple, and safe. He tucked his gun into the back of his belt, slipped his badge off and into his pocket. He strolled into the bank, smiled at the teller.\n\n\"Hey there, Nancy. Thought I'd be too late to make my deposit. Lucky for me you're still open.\"\n\nThough her face was frozen in fear, she managed to gape at him. \"But\u2014 But\u2014\"\n\n\"The wife'll have my hide if I forget to put the money in. We got that automatic withdrawal on our insurance, you know.\" He strolled up to the counter, one hand reaching down.\n\n\"Are you crazy?\" the man with the gun shrieked out, nerves in every syllable. \"Are you out of your mind? Get down on the floor! Down! Now!\"\n\n\"Hey, I'm not butting in line,\" Devin said reasonably. \"Just trying to do some business.\" He kept his eyes on the man's face, his hand still going down and back, where a man kept his wallet.\n\n\"I'll show you some business!\"\n\nTo Devin's relief, the man shifted the gun from Nancy and toward him. \"Put your damn money on the counter. I'll take that, too.\"\n\nAs if he'd just noticed the weapon, Devin held up a hand in peace. \"Holy hell, you robbing the bank?\"\n\n\"What does it look like I'm doing, Einstein? Let's have the money.\"\n\n\"Okay, okay. I don't want any trouble here. You can have it.\" But instead of his wallet, Devin came out with his gun. \"Now, are we going to stand here and shoot each other, or what?\"\n\nThe man's eyes went wild. \"I'll kill you! I swear I'll kill you!\"\n\n\"That's a possibility.\" A remote one, since the idiot was waving the gun like a flag on the Fourth of July. \"It's just as likely I'll kill you. You drop that gun on the floor and step back from it. You've already got armed robbery, you don't want to add shooting a police officer.\"\n\n\"A cop, a damn cop! Then I'll just shoot her!\" Furious, he swung the gun back toward the teller.\n\nDevin didn't hesitate, he didn't even bother to curse. Nancy was just where she should be. On the floor, out of the line of fire. And since he was close enough, Devin used his fist instead of his gun.\n\n\"Damn idiot.\"\n\nThe man managed to get off one shot at the ceiling before the gun flew out of his hand. Ignoring it, Devin put his own between the man's eyes.\n\n\"What you want to do now,\" he said calmly, \"is roll yourself over and put your hands behind your head. If you don't, I'm going to have to blow your head right off, and this carpet's only a year old.\"\n\n\"Damn cop. Damn lousy one-horse town.\"\n\n\"You got that right.\" With a bit more force than was strictly necessary, Devin jerked the man's hands down, cuffed them. \"You shouldn't mess with small towns. We're real careful about them. Anybody hurt back there? You all right, Nancy?\"\n\nAs a chorus of breathless, excited voices exploded from behind the counter, he glanced back, knowing Rafe was behind him. And grinned at the crowbar his brother was slapping against his palm.\n\n\"I told you I'd handle it.\"\n\n\"This was just in case. What did you do, Dev, scalp him?\"\n\nIdly Devin picked up the wig that had been dislodged during the scuffle. \"Looks that way. Might as well give him a shave while I'm at it.\" None too gently, he pulled the man's head back and ripped off the fake moustache. \"In case you haven't figured it out, you're under arrest. You have the right to remain silent...\" he began as he hauled the man to his feet.\n\nHe finished Mirandizing him on the way to the door. \"Y'all get up from behind there now. I'm going to send Donnie in to get your statements.\"\n\nFrom their station at the diner window, both boys watched Devin come out, dragging a balding man with a bloody lip.\n\n\"He got him,\" Bryan said, awed. \"Devin got an honest-to-God bank robber.\"\n\n\"Of course he did.\" Connor beamed. \"He's the sheriff.\"\n\nThere was talk of little else but the attempted bank robbery. In the way of small towns, unofficial reports leaped over the wires far ahead of official ones. In many of the phone and backyard-fence conversations, it was said that Devin had burst into the bank, gun drawn, eyes blazing. In others, he had taken out the robber, who'd been armed to the teeth with automatic weapons, bare-handed.\n\nBy the end of the day, Devin found himself the recipient of enough homemade baked goods that he could have opened his own restaurant. They made up for the endless official reports he had to type and file. They nearly made up for the phone calls he was forced to field, from concerned citizens, the mayor, the bank manager, and a number of women who thought he might need a bit of comfort after his ordeal.\n\nHe was deflecting one of the offers when his brothers walked in.\n\n\"No, Annie, I wasn't wounded.\" He rolled his eyes as all three of his visitors grinned at him. \"No, he didn't shoot me. Sharilyn's exaggerating. Ah...\" A little baffled by the offer presented to him, he cleared his throat. \"That's nice of you, Annie, and I appreciate the thought, but\u2014 No, I don't think I'm going to suffer from delayed stress syndrome. Yeah, I've heard of it, but\u2014 No, no, really, I'm just fine. And I'm a little tied up right now. Yeah, official business. That's right. You take care now. Uh-huh. You bet. Bye.\"\n\nHe let out a long breath, shaking his head briskly as he replaced the receiver. \"Holy hell.\"\n\n\"Was that Annie 'The Body' Linstrom?\" Shane wanted to know.\n\n\"She was hitting on me,\" Devin said with a snort of laughter. \"Women are a puzzle. There's no way around it.\"\n\nJared sat on the corner of Devin's desk. \"The way I heard it, bullets bounce off your chest.\"\n\n\"Nah.\" Shane sniffed at one of the pies sitting on a crowded shelf. \"I heard he eats bullets. Betty Malloy bake this lemon meringue?\"\n\n\"Yeah.\" Devin winced when the phone rang again. \"Where the hell is Donnie?\"\n\n\"Last I saw, he was strutting down Main Street trying to look like Supercop.\" Rafe cocked his head. \"Aren't you going to answer it\u2014Sheriff?\"\n\nDevin swore and picked up the phone. \"Sheriff's office. MacKade.\"\n\nHe leaned back, closed his eyes. It was the press again. Every small paper and news bureau within fifty miles had picked up on the botched robbery. By rote, he gave the official line, danced around the demand for a more in-depth interview, and hung up.\n\n\"You're good at that,\" Jared decided. \"Real stern and authoritative.\"\n\n\"I'm beginning to wish I'd kicked that jerk in the head,\" Devin muttered. \"He's caused me a lot of trouble. Now I'm stuck here, answering the damn phone, typing reports, with some out-of-town idiot who couldn't hold up a lemonade stand in the back. He whines all the time.\"\n\n\"At least you won't starve,\" Shane said, and helped himself to one of the cookies on a plate by the pie. \"We thought we'd take you down to Duff's, buy you a drink.\"\n\n\"Can't leave the prisoner unattended.\"\n\n\"Rough,\" Jared said, without sympathy. \"You know, Bryan was about to jump out of his socks when he got home. You're better than Rambo.\"\n\nAmused, Devin scratched his cheek. \"Don't tell him the last robbery I had to deal with was when a couple of kids stole underwear off Mrs. Metz's clothesline.\" He shuffled papers on his desk. \"Have you been by the inn, Rafe? Everything okay there?\"\n\n\"Everything's fine. Cassie was a little upset. Word travels,\" he added unnecessarily. \"But I told her it was all blown out of proportion, and you didn't do anything much.\"\n\n\"Thanks a lot.\"\n\n\"No problem. Connor was already writing a story about you.\"\n\n\"No kidding?\" The grin all but split his face.\n\n\"'A Day in the Life of Sheriff MacKade.'\" Rafe helped himself to coffee. \"The boy's nuts about you.\"\n\n\"Good thing.\" Shane took another cookie. \"Since Devin's going to marry his mama.\"\n\nRafe bobbled the coffee, spilled it on his hand and swore. \"Cassie? Little Cassie?\"\n\n\"Shane's getting ahead of himself,\" Devin said, in a mild tone that belied the gleam in his eye. \"As usual.\"\n\n\"Hey, you're the one who said it. Me, I figure you've just lost your mind. Like these two.\"\n\n\"Shut up, Shane.\" Jared kept his eyes on Devin's face. \"You and Cassie?\"\n\n\"So what?\"\n\n\"So...that's interesting.\"\n\n\"Are you speaking as her attorney?\" Devin pushed back from the desk. If the phone rang again, he thought he might just rip it out of the wall. To get himself back under control, he went to the coffee.\n\n\"He's got it bad,\" Rafe observed. \"Didn't you have a thing for her about ten, twelve years ago?\" When Devin didn't answer, merely poured the coffee, sipped it steely-eyed, Rafe grinned. \"Never got over it, did you? Son of a gun. Why, that's practically poetic, bro. It gets me, right here.\" He thumped a hand on his chest.\n\n\"Keep ragging me, it'll get you somewhere else.\"\n\n\"It's getting so every day's Valentine's Day in Antietam.\" In disgust, Shane shoved another cookie in his mouth. \"A man's not safe.\"\n\n\"Cassie's a sweetheart,\" Rafe said pointedly.\n\n\"Sure she is.\" Gamely, Shane swallowed, so that he could make his point. \"She's as good as they come, and pretty with it. But why does that mean he has to marry her? You see all this stuff?\" With a sweep of his hand, he indicated all the pies, cakes, tarts, cookies. \"Women are going to fall all over him, and he's tossing them off because he's gone cross-eyed over one woman. It's not only stupid, it's...well, it's selfish.\"\n\nRafe gave Shane a thump on the back of the head that would have felled a grizzly. \"Man, I love this guy. He's going to carry the MacKade legend into the next millennium.\"\n\n\"Damn right,\" Shane agreed. \"No woman's going to tie me down. I mean, with all the flowers out there, why pick one when you can have a bouquet?\"\n\n\"Now that's poetry.\" Rafe thumped him again. \"Let's go get that beer.\"\n\n\"You two go on.\" Jared stayed where he was. \"I need to talk to Devin a minute.\"\n\nThey left, arguing about who was buying. When the room was quiet again, Devin took his coffee back to his desk. \"You got a problem?\"\n\n\"No.\" Jared shifted so that they were face-to-face. \"But you might. Have you talked to Cassie about marriage?\"\n\n\"A little. Why?\"\n\n\"Joe Dolin.\"\n\n\"They're divorced. It's done.\"\n\n\"They're divorced.\" Eyes steady, Jared rested a hand on his knee. \"But done's another thing. He'll get out eventually, Devin. He'll come back.\"\n\n\"I'll handle it.\"\n\n\"Yeah, I figure you can handle Joe, one-on-one. But there's the law.\"\n\nUnconsciously Devin brushed a finger over his badge. \"He tries to touch Cassie again, just tries, and I'll have him back behind bars before he can blink.\"\n\n\"And that's part of the problem. You're the sheriff, but you won't be objective. You can't be.\"\n\nDevin set his coffee aside, leaned forward. \"I've been in love with her most of my life. At least it seems that way. And I had to stand back and do little more than nothing while he hurt her. While I knew what he was doing to her inside that house. She wouldn't let me help, so the law tied my hands. Things are different now, and nothing's going to stop me from taking care of her. He lifts his hand to her again, and he's dead. Problem solved.\"\n\nJared nodded. He didn't take the statement lightly. He knew what it was to need to protect the woman you loved from any sort of harm. And he knew Devin was a man who said exactly what he meant.\n\n\"I'm talking about a situation that could develop if he's smart enough not to lift his hand to her. What if, after he serves his time, he moves back here, stays clean. How are you going to handle that?\"\n\n\"One step at a time, Jared, like always. Of course, the first thing I'd have to do is keep Rafe from going after him because of what he tried to do to Regan.\"\n\nThat was true enough, Jared thought. And Rafe wouldn't be the only one who wouldn't welcome Joe Dolin back into the community. \"Dev, I know what Cassie's been through. Exactly. I know because I'm her lawyer, I handled the divorce. We're talking about a textbook case of spousal abuse. A pitiful phrase, textbook case, for that kind of horror. Therapy's helped her, the town's helped her, and her own backbone's helped her. But she's got scars she's never going to get rid of.\"\n\n\"I'm being careful,\" Devin said slowly. \"For God's sake, Jared, I've given her time\u2014even after the divorce, I waited and gave her time. I'm trying to give her more.\"\n\n\"Devin, I'm just trying to show you the whole package. Believe me, I can't think of anyone I'd rather see you with than Cassie. Anyone I'd rather see her with than you. God knows she deserves somebody decent. But it's not just the two of you. There are two kids here. Joe Dolin's kids.\"\n\nDevin's eyes darkened, narrowed. \"You can say that to me, when you've got Bryan? Are you going to tell me it matters they're another man's blood, when I know damn well Bryan's as much yours as Layla?\"\n\n\"That's not what I'm saying.\" Jared's voice was low and calm. \"I've seen you with them. I didn't have a clue how you felt about Cassie. You kept that covered well. But anybody with eyes can see you're crazy about those kids, that you've been good for both of them. They deserve you,\" he added, and nipped Devin's temper before it could bloom. \"They deserve a father who loves them, and a home where they can just be kids.\"\n\n\"Fine. That's what I'm going to see they have.\"\n\n\"But it's not like Bryan, Dev. His biological father isn't around, isn't an issue. Dolin is.\"\n\n\"He doesn't give a damn about those kids, never has.\"\n\n\"No, but he'll have a right to them.\" Knowing that the frustration he felt didn't help, Jared spread his hands and took a deep breath. \"The law says he does. And if he can't get to Cassie, he may just come up with the notion to get to her through them. Once he's out, he'll have a legal right to see them, to have visitation, to be part of their lives. You won't be able to block that.\"\n\nDevin hadn't thought of it. Maybe he hadn't let himself. Now that it was there, right in the front of his mind, his blood went cold. \"You're the lawyer. You block it.\"\n\n\"Parental rights are a sticky business, Dev. You know that. Until and unless he does something to put them in jeopardy, until and unless we can prove he's not just unfit, but dangerous to them, he'll have the law on his side.\"\n\nAlready Jared was thinking it through, working it out. \"We may be able to put the pressure on for supervised visitations only, but blood still counts heavy in court.\"\n\n\"He beat Connor.\"\n\nJared's brows drew together. \"I didn't know anything about that.\"\n\n\"Connor didn't tell Cassie, didn't want to make it worse on her.\"\n\n\"I might be able to use that, if the time comes. But once he's considered rehabilitated, a lot of the slate gets erased. He's going to be in for a long time yet, but I want you to know what you're up against here.\"\n\n\"I've got a clear picture of what I'm up against. Nothing's going to stop me from making Cassie and the kids mine. Not Joe Dolin, not the law, not anything.\"\n\n\"Well, then.\" Jared rose. \"I'll state the obvious. I'm behind you. Rafe and Shane are behind you.\"\n\n\"I appreciate it.\"\n\n\"If you get yourself out from behind that desk for an hour, come down to Duff's. I'll buy you a beer.\" Satisfied, Jared headed for the door, then paused. \"She's a terrific woman, Dev. Sweet, like Shane says, but tougher than you might think. Tougher than she thinks. If you convince her she wants you as much as you want her, you'll handle whatever comes down. I've got one piece of advice.\"\n\n\"You always do,\" Devin said dryly.\n\n\"For Cassie, it's not enough to let her know you love her, you want her. You let her know you need her. That's a woman who'd go to the wall for a man who needed her.\"\n\nHe did need her, Devin thought when Jared had shut the door behind him. But he didn't know how to show her, and wasn't entirely sure he should. Wasn't that just the kind of pressure he was struggling not to put on her?\n\nHe didn't want Cassie to go to the wall for him. He only wanted her to feel safe and happy. No, it was up to him to see that she was never hurt again, to protect her, to shield her and the children.\n\nHis need could wait.\n\n## Chapter 10\n\nCassie told herself it was foolish to worry. Devin was fine. Rafe had told her the story himself, and she knew that his version of the attempted bank robbery was much more accurate than those she'd heard over the phone. Even Connor's report, given in fits and starts of desperate excitement, had been less dramatic than the gossip spewing out of the town.\n\nSo there was no need to worry.\n\nShe was so worried, she jumped each time the phone rang. If she'd been able to leave the inn and the children for an hour, she'd have dashed into town to check every inch of Devin herself.\n\nOne thought, one fact, kept running in a loop in her brain. He'd faced down a man with a gun.\n\nShe shuddered again, and gave up trying to block the picture from her mind. He'd walked into an armed robbery, risked his life to protect others. His badge had never taken on such huge proportions for her before. He'd risked his life. In the day-to-day business of a town like Antietam, a sheriff's work was more diplomacy\u2014or so she'd imagined\u2014than risk.\n\nOf course, now, she began to see that had been foolish of her. There were fights, drunks, break-ins, hot tempers between neighbors and families. She had personal knowledge of the dangers of domestic disputes\u2014that tidy term for the violence that could happen behind closed doors.\n\nHe was in charge. And while Connor might see him as a hero, she began to see just how vulnerable the badge made him.\n\nBecause she did, she also realized that the worry that ate at her all through the long afternoon and evening wasn't just for a friend, a lover, not just for a man she admired and cared for. It was for the man she loved.\n\nIt had taken something unexpected, shocking, to open her eyes. Now that they were open, she could look back. Almost as far back as she could remember, Devin had been there. She had depended on him, admired him and in some ways, she supposed, taken his place in her life for granted.\n\nIt had been humiliating to go to him and admit what Joe had done to her, to show him the marks, to describe how she had come by them. Not just because he'd been the sheriff, she thought now. Because he had been Devin.\n\nShe'd always been more shy around him than around his brothers. Because, she thought again, he'd been Devin. Part of her heart had always been set aside for him. So she had never been able to look at him as just one of the MacKades, or just her friend, or just the sheriff.\n\nShe'd always felt something more. Now she was free, and she could let those feelings out. She could admit that it wasn't just part of her heart that belonged to him, but all of it.\n\nAll of her.\n\nThrough the worry came the wonder, and with it the joy. She loved.\n\nWhen the phone rang, she raced to it like a madwoman, then struggled to keep her voice calm when Savannah greeted her.\n\n\"Hi, I guess you've heard the big news by now.\"\n\n\"No one's talking about anything else.\" To calm herself, Cassie reached over to the refrigerator and took out a pitcher of juice. \"Have you seen Devin since it happened?\"\n\n\"Not personally. Jared has. He says our big, bad sheriff is annoyed with all the glory. A television crew came down from Hagerstown, and the paper's been here.\" Because she understood Cassie's silence perfectly, she softened her voice. \"He's fine, Cassandra. Not a scratch. Just grumbling because this whole business is going to keep him tied up for a while. Are you all right?\"\n\n\"Me?\" Cassie stared at the juice she'd poured. \"I'm fine. I'm just concerned.\"\n\n\"I know. I have to admit that by the time Bryan finished giving me the play-by-play, I was pretty concerned myself. But the one thing we can all be sure of is that Devin MacKade can handle himself.\"\n\n\"Yes.\" Cassie picked up the glass, set it down again. \"He can. I guess there's no one who needs anyone worrying about him less than Devin.\" But why hadn't he called?\n\n\"Listen, I really called to ask you a favor.\"\n\n\"Sure. What can I do?\"\n\n\"You can give my temper a break and send Connor over for the night. Bryan's been nagging me since he got home from the great bank robbery.\"\n\n\"Oh.\" Cassie peeked out the window into the yard, where Connor and Emma were playing with the cat. \"He'd love it, if you're sure.\"\n\nThere was a crash, and Cassie could hear Savannah yell, \"Bryan MacKade, if you break a window with that baseball, you're not only out of the game, you're suspended for the season!\n\n\"Yes, I'm sure,\" she said to Cassie, with feeling, when she returned to talk in the receiver. \"But there's more. Can we have Emma, too?\"\n\n\"Emma? You want Emma to spend the night?\"\n\n\"Jared has this idea that we'd better start practicing with girls. We sure know boys, and he started thinking that once Layla starts growing up, we'll be lost.\" She laughed, and Cassie heard the baby coo. \"So, how about giving us Emma for the experiment? We swear we'll turn her back over in one piece.\"\n\n\"She'd be thrilled. But, Savannah, you'd have four to deal with.\"\n\n\"Yeah. We've decided that's our magic number. If you know what I mean.\"\n\n\"Four?\" It was Cassie's turn to chuckle. \"Well, you're going to need plenty of practice, then.\"\n\n\"Let's just see how we survive one night. Pack them up, will you, Cassie? Jared will walk over through the woods and get them.\"\n\n\"On one condition. You'll call, anytime, if you want to bail out.\"\n\n\"You've got my word on that one.\" There was another crash, and something shattered. \"All right, Bryan, now you have to die. Hurry, Cassie\u2014I have to believe there's safety in numbers.\"\n\nThough it tugged at her heart a little, Cassie supervised the overnight packing, while her children bristled with excitement. They were so eager to go, and she tried not to fret that it was Emma's first sleepover.\n\nShe made certain they had clean clothes, toothbrushes, instructions on how to behave. They even took the cat. When they trooped off toward the woods with Jared, she was completely, utterly alone.\n\nWith too much time, she realized, to think, to brood, to worry.\n\nShe went down to the inn, found the handful of guests well occupied and content. Still, she set up cake and coffee in the parlor, offered complimentary wine to those playing cards in the sunroom.\n\nSeeing that she wasn't needed, she set the table for breakfast, and checked her pantry and refrigerator, though she knew she was well supplied for the large Sunday breakfast the inn was becoming renowned for.\n\nAt loose ends, she wandered outside. She wasn't used to having nothing to do, no one to look after. Certainly, she had often fantasized about how she would spend an evening alone. A bubble bath, a book, a late movie on television.\n\nThat was what she would do, she told herself. As soon as she ran into town and made sure Devin was really fine.\n\nShe dashed up the stairs, then let out a yelp when she saw the shadowy figure on the porch swing.\n\n\"I saw you were busy,\" Devin said. \"Thought I'd wait.\"\n\nShe still had a hand against her speeding heart. \"I thought you had to stay in town.\"\n\n\"I dragooned Donnie into staying at the office. It's the least he can do, after he left me with the phones all damn afternoon.\" He held out a bouquet of yellow tea roses. \"I brought you flowers. I was going by the florist and remembered I'd never brought you flowers. I know you like them.\"\n\n\"They're beautiful.\"\n\n\"Are you going to sit down with me?\"\n\n\"All right.\" She sat and held the roses in her arms as she would have a child. \"They're beautiful,\" she said again. \"I should put them in water.\"\n\n\"They'll keep a minute.\" Curious, he tucked a hand under her chin and turned her face to his. \"What is it?\"\n\n\"It's nothing. I was so worried,\" she blurted out. \"I couldn't leave, and kept waiting for you to call. Devin, why didn't you call? I'm sorry,\" she said immediately. \"I shouldn't nag you.\"\n\nOne of the scars, he mused, and kept his fingers firm when she would have looked away. \"Don't be sorry. I did call, several times. Your phone was busy.\"\n\n\"Everyone's been calling. I've heard a dozen different stories.\"\n\n\"The truth's probably less exciting.\"\n\n\"He had a gun, didn't he? You knew he had a gun when you went into the bank.\"\n\n\"I had to do my job, Cass. He wasn't going to get anywhere, and even if he did, there was a canister inside the moneybag that would have spewed red paint all over him and the bills.\" His grin spread. \"Actually, I'm kind of sorry we couldn't play that part out. It would have been some show. But he might have hurt someone.\"\n\n\"He might have hurt you.\"\n\n\"Well, then, you didn't hear about how bullets bounce off me.\"\n\nInstead of laughing, she pressed her face into his shoulder. \"I'm so glad you're all right. I'm so glad you're not hurt. I'm so glad you're here.\"\n\n\"I'm happy to be all of those things.\" Slipping an arm around her, he set the swing in motion. \"I'd have come sooner, if I could.\"\n\n\"I know. You were on the news.\"\n\n\"Yeah. So I hear.\"\n\n\"You didn't see.\" She turned her head. \"They'll show it again at eleven.\"\n\n\"I know what I look like.\"\n\nStudying his face, she found something endearing. \"You're embarrassed.\"\n\n\"No, I'm not.\" He shifted. \"Maybe. Some.\"\n\nNot just endearing, she realized. Adorable. \"I'm awfully proud of you,\" she murmured, and brushed her lips over his. \"Actually, we taped the broadcast. Connor was so excited. We can watch it, if you want.\"\n\n\"I'll pass. I don't\u2014\"\n\nShe interrupted him with her lips again, and experienced an odd, sweet power when she felt his heart jump. \"I've watched it three times. I thought you looked like a movie star.\"\n\n\"You don't get out enough.\" His palms were damp, so he eased off the swing. A little distance, MacKade, he warned himself, before you explode. \"I've been thinking about that, too. I haven't ever taken you out. To dinner, or anywhere.\"\n\n\"You took us down to the zoo in the spring, and to the fair last summer.\"\n\nWhy was she looking at him like that? he wondered. She'd never looked at him like that before. With... Was that amusement, or lust, or\u2014 God.\n\n\"I meant you and me. I love having the kids, but\u2014\"\n\n\"I don't have to go out on dates, Devin. I'm happy with the way things are.\"\n\n\"Still and all.\" He couldn't seem to think very clearly, not when she was just sitting here, smiling at him, a bouquet of flowers in her arms. \"I, ah, brought all this food. Pies and cookies and cakes. People have been bringing them by the office all afternoon.\"\n\n\"They're grateful.\" With her heart tripping lightly, she rose. \"They want to show it.\"\n\n\"Yeah, well, I'd never be able to eat it all. I gave some to Donnie, but I figured the kids might...\" He backed up when she stepped forward. \"They might want some. I didn't see them when I came up. It's a little early for them to be in bed on a Saturday night, isn't it?\"\n\n\"They're not here.\" She blessed Savannah and Jared, and fate. \"They're spending the night at the cabin.\"\n\n\"They're not here.\"\n\n\"No. We're alone.\"\n\nHe'd been prepared to leave, to spend a little time with her, then go. He wouldn't have asked to stay with her through the night, with the children in the next room. None of them were ready for that.\n\nNow they were alone, and the night had just begun. A slap of desire whipped through him, painfully. He braced against it, and managed an easy smile.\n\n\"Then I'll take you out.\"\n\n\"I don't want you to take me out,\" she murmured. \"I want you to take me to bed.\"\n\nIt closed his throat. \"Cassie.\" His hand was very gentle on her cheek. \"I don't expect that every time I come here. That's not the only reason I want to be with you.\"\n\n\"I know.\" She turned her lips into his palm. \"It's what I'd like tonight. I'm going to put these in water.\"\n\nShe left him, churning and speechless, on the dark porch. More than a little dazed, he followed her inside.\n\n\"I bought this at Regan's shop.\" Briskly Cassie filled a green Depression glass pitcher with water. \"I'm still getting used to having a little extra money to buy pretty things. I don't even feel guilty about it anymore.\"\n\n\"You shouldn't feel guilty about anything.\"\n\n\"Oh, a few things.\" With hands as gentle as they were efficient, she arranged the roses in the pitcher. \"But not this. And not you.\" Her eyes lifted. \"Do you know what I feel about you, Devin? About us?\"\n\nHe thought it was best not to try to speak just then, not with the way the blood was draining out of his head.\n\n\"Dazzled,\" she murmured. \"You dazzle me. You make me feel things, and want things I never knew I could have. I'm almost twenty-nine, and you're the only man who's really touched me. I want you to touch me.\"\n\nHe would, as soon as he could be sure he had his hands, and his needs, under control. If it had been anyone but Cassie, he would have thought she was seducing him.\n\nBecause he said nothing, made no move toward her, she was afraid she was doing it all wrong. It wasn't nerves now that plagued her, so much as doubt. And doubt had her shifting her gaze back to the flowers.\n\n\"If you'd rather not right now...if you don't want me\u2014\"\n\n\"God.\" It exploded out of him, made her head whip up in alarm, made him bite back whatever might have come out next. \"Let's go for a drive,\" he said quickly. \"It's a pretty night, the moon's coming up. I'd like to go for a drive with you.\"\n\nShe was sure she'd made some foolish mistake, but couldn't put her finger on it. All she was sure of was that her system was in overdrive, and his wasn't. As a seductress, she thought, she was a miserable amateur.\n\n\"All right, if you like.\"\n\nHe recognized that tone, the bright and false cheerfulness. He would have slit his throat before he did anything to cause that. \"Cassie, it's not that I don't want to make love with you. I do. It's just that... Maybe I'm a little more revved from this morning than I thought. I need to smooth out some of the edges before I... I can't touch you now,\" he ended, his tone too sharp, too quick.\n\n\"Why?\"\n\n\"Because I'm a little too needy right now, and it doesn't help for you to keep looking at me that way. I wouldn't be able to\u2014 I'd hurt you.\"\n\n\"You're angry with me?\"\n\n\"No.\" He swore, ripely, showing her some portion of his frustration in the way he whirled around and paced. \"When I'm angry with you, you'll know it. You're driving me crazy. Look at the way you're standing there, with your hands folded and those big, gorgeous eyes watching every move I make. I can't breathe when you look at me like that. I used to be able to.\" He shot the words out like an accusation. \"But that was before, and I just can't handle it as well now that we've been together. We've got to get out of here before I eat you alive.\"\n\n\"We're not going anywhere.\" It surprised them both, how firm and settled her voice.\n\n\"I'm telling you\u2014\"\n\n\"Yes, I believe you are trying to tell me. You think I'm too fragile to handle it. To handle you. Well, you're wrong.\"\n\n\"You haven't got a clue what you're dealing with, not with me.\"\n\n\"Maybe I don't. Maybe you haven't let me.\" Suddenly strong, suddenly sure, she walked to him. \"Every time we've made love, it hasn't been for you.\"\n\n\"Don't be ridiculous. Of course it was for me.\"\n\n\"It was for me,\" she said firmly. Strong, she thought. Strong face, strong eyes, strong hands. Not a picture in a magazine, or a white knight fantasy. A strong man, with strong needs. \"You were so careful, so patient. No one's ever been careful with me before.\"\n\n\"I know.\" Because he did, his hand was gentle when he lifted it to brush the golden curls of her hair. \"You don't have to worry anymore.\"\n\n\"Don't treat me like a child, Devin.\" Boldly she took his face in her hands, that familiar and compelling face. \"You were holding back. Every time, you were holding back. I've been too dazzled to realize it.\"\n\n\"Cassie, you need tenderness.\"\n\n\"Don't tell me what I need.\" Her voice had a snap to it, there was a spark in her eyes. \"I've had enough of that in my life. Yes, I need tenderness, but I also need trust and respect, and to be treated like a woman. A normal woman.\"\n\nAs carefully as he could, he wrapped his fingers around her wrists. \"Don't push me here, Cassie.\" He pressed his lips to her brow, and infuriated her.\n\n\"Kiss me like you mean it,\" she demanded, then crushed her lips to his. She felt his jolt, the burst of heat, then his struggle for control. \"Show me what it's like,\" she said against his mouth. \"I want to know what it's like, what you're like when you stop thinking.\"\n\nWith an oath, he devoured her mouth. It was like that first shocking kiss, she realized as her blood burst inside her veins. The first and the last time he had given her a glimpse of real hunger.\n\nThere was that surge of power again, that odd, whippy sensation that she could do or be anything. She strained against him when he tried to draw back.\n\n\"Damn it, Cassie.\"\n\n\"Again.\" Surprisingly strong, she dragged his dark head back to hers. \"Kiss me like that again.\" Her eyes, slumberous, aware, stayed on his. \"Show me what it's like,\" she murmured. \"I've waited my whole life to know.\" She ran her hands over his chest, felt the wild beat of his heart, the rigid edge of his control. \"Take me. Don't be kind tonight, Devin. Just take me. That's what I want.\"\n\nHis hands, shaking now, were tensed and rough as he wrapped her hair around them and dragged her head back. He plundered her mouth, ravishing it with lips and teeth and tongue. A part of him hung back still, waiting for her to object. He told himself he would stop\u2014could stop\u2014the moment he frightened her.\n\nBut as her taste seeped into him, he was afraid it was a lie. Just look at her, he thought, the sunbeam hair, the cloudy eyes, the rose petal skin.\n\n\"Cassie\u2014\"\n\n\"No. Just show me.\" She was almost delirious with new knowledge, with the force of her desire and her utter lack of fear. \"Show me.\"\n\nHe could have sworn he heard himself snap, heard an echo of brittle control breaking. The wildness overcame him, primitive, almost brutal, making all the years of patience nothing.\n\nIn his rush to taste her flesh, he ripped her blouse. The sound of the seam tearing would have snapped him back, but she moaned and wrapped herself around him. Instinctively he recognized the quiver of her body as desire, not fear. It clawed at him.\n\n\"I can't...stand it.\"\n\n\"Then don't,\" she murmured, thrilling when his arms clamped around her, when he lifted her off her feet so that she was pressed hard against him, heat to heat. \"Touch me.\" She fisted her pale hands in his dark hair, amazed at the hunger that swarmed through her. \"I'll go crazy if you don't.\"\n\nNearly stumbling, his mouth racing over her face and throat, he headed for the bedroom. But she wrapped her legs tight around his waist and shot new fire into his blood. By the doorway, he pressed her against the wall, using it to brace her. His desperate mouth clamped over her breast, suckling hard through her tattered blouse. Her response was to throw back her head and rock against him.\n\n\"More.\" She couldn't believe what was coming out of her mouth, couldn't believe this vicious need had been in either one of them. With a groan, she reached down and tore her own blouse aside so that his mouth could take her.\n\nShe climaxed the instant his teeth closed over her, shocking herself with the power of it. For an instant she was like a moth, pinned, quivering helplessly, and then she was alive, bursting with life.\n\nMindless as coupling animals, they dragged each other to the floor.\n\nShe pulled at his shirt, he yanked at her slacks. Speech was impossible as they rolled over the floor in the narrow hallway, groping for each other. There were only gasps and moans. No sighs now, no murmurs, only hissing breath and thundering pulses.\n\nCraving drove him, a craving long suppressed and denied. He yanked her hips high and ripped her practical cotton panties to shreds. And made her scream with his greedy mouth.\n\nShe bucked, then stiffened into a quaking bridge, her arms straining as her body arched up toward him. He drove her ruthlessly, relentlessly, until throaty growls rumbled in her throat.\n\n\"More.\" This time it was he who demanded it, he who groaned, as her nails scraped up his back and dug crescents into his shoulders. When her hand closed around him, his vision grayed, and the drumbeat of his pulse scrambled.\n\nShe was moving under him, writhing. Her eyes were nearly black, and blind with pleasure, when he fused his mouth to hers again. It was greed, rather than control, that kept him from ending it, that had him sliding sleekly down her body again, tasting and taking and touching until they were both mad.\n\nHe reared up, clamped his hands over hers, then plunged into her. Beyond all reason, he pumped and thrust, angling her rocking hips so that he could immerse himself in her, deep, then deeper. His mind had gone dark, leaving only snarling sensations as he rammed into that hot, wet pleasure with a feral force that had them both gasping.\n\nShe couldn't hold on. She tried, for him. How could she have known he needed like this? That she was capable of needing like this? How could she have known until he finally showed her? But she was being tossed too high now to fight her way back. Her hands slid off his damp skin, rapped hard on the wooden floor. She gave herself willingly to the last savage stab of pleasure, going weak as he continued to hammer himself into her.\n\nThen that wonderful hard body heaved, went rigid. She saw him throw his head back as if in pain, saw with wonder that it was he who was lost. When he shuddered, shuddered and cried out her name, she wept with the joy of it.\n\nHe felt the tears against his shoulder the moment his sated body collapsed on hers. He would have levered himself away instantly, but her arms came around him.\n\n\"Don't. Please don't move.\"\n\n\"I'm sorry.\" There was nothing he could say to her that would be good enough, nothing he could say to himself that would be bad enough. \"I hurt you. I promised I wouldn't.\"\n\n\"Do you know what you did?\" Her lips were curved, but he couldn't see. All he could see was his own careless treatment of the most precious thing in his life. \"You forgot.\"\n\n\"Forgot?\" Again he tried to shift, again she held him tight.\n\n\"You forgot to be careful, you forgot to worry, you forgot everything. I didn't know I could make you do that. It makes me feel\u2014\" a long, satisfied sigh \"\u2014powerful.\"\n\n\"Powerful?\" His throat was bone-dry. He wanted to lift her up off the floor. God, he'd taken her on the floor. He wanted to tuck her into bed and soothe her. But the word she'd used, and the tone, baffled him.\n\n\"Strong, sexy.\" At last she lifted her arms, stretched them above her head in a long, lazy movement. \"Powerful. I've never felt powerful before. I like it. Oh, I really like it.\" Eyes closed, lips curved, she hummed in her throat.\n\nAnd that was his first glimpse of her when he lifted his head, the smug smile and erotic glow of a woman who'd just discovered a dangerous and exciting secret. His blood stirred all over again. She looked...triumphant, he realized. Just who, he wondered, had ravaged who?\n\n\"You like it,\" he repeated.\n\n\"Mmm... I want to feel this way again. And again and again. I want to feel cherished, too, the way I do when you're gentle. I want to feel everything. I made you forget.\" She opened her eyes again and laughed when she saw the stunned and sated look in his eyes. \"I seduced you. Didn't I?\"\n\n\"You destroyed me. I tore your clothes.\"\n\n\"I know. It was exciting. Will you do it again?\"\n\n\"I...\" He shook his head, but when it didn't clear he gave up and lost himself in her eyes. \"Anytime.\"\n\n\"Can I rip yours?\"\n\nWords failed him. He managed a couple of strangled sounds before clearing his throat. \"We'd better get off the floor.\"\n\n\"I like it here. I like knowing you wanted me so much you couldn't wait.\" She lifted a hand to toy with the dark curls that fell, damp, over his forehead. \"I like the way you're looking at me right now. It's probably wrong, and I don't care, but I like knowing you wanted me for years. That you watched me, and wanted me. Like this.\"\n\n\"I didn't exactly picture it like this.\"\n\nHer lips curved again, a sly, knowing smile that made his blood swim. \"Didn't you?\"\n\n\"Well, maybe.\" His brain was still numb. It was the only part of him that seemed to have shut down. \"Once in a while.\"\n\nShe pressed her lips together, ran the tip of her tongue over them. \"I can still taste you.\"\n\n\"Oh, God.\"\n\nA quick and delicious tremor coursed through her as she felt him move inside her. \"I'm doing it again.\"\n\n\"Huh?\"\n\n\"Seducing you.\"\n\nHe couldn't get his breath. \"Looks like that.\"\n\nShe felt powerfully a woman, a normal, competent, well-loved woman. \"Tell me you love me, Devin. While you're filling me, while you're wanting me, tell me you love me.\"\n\nHe couldn't keep himself from hardening again, from driving deep into her, from groaning as her body rose and fell with him.\n\n\"I love you.\" Helpless, he buried his face in her hair. Somehow she'd taken the reins from him. He could do nothing but ride. \"I can't stop.\"\n\nShe absorbed it all, the love, the passion, the power, willingly matching his fast and desperate pace. When she knew he was falling off the edge with her, when they were both defenseless, she turned her lips to his ear.\n\n\"I love you, Devin. I love you. I think I always have.\"\n\nWhen he could speak again, he gathered her up, cradled her in his lap. \"I've wanted to hear that for a long time.\"\n\n\"I meant it. I couldn't have said it unless I did.\"\n\n\"I know.\" And it left him shaken and without defenses. \"You've tossed my master plan into the Dumpster, Cass.\"\n\n\"How?\"\n\n\"Well, I had it plotted out, you see. By my reckoning, I'd get you to fall in love with me by Christmas. Then I'd keep things at a nice, steady pace, and talk you into marrying me by spring.\"\n\n\"Let's not talk about marriage, Devin. Not yet. Not now.\"\n\nHe tipped her head back. \"When?\"\n\n\"I don't know.\" There was worry in her eyes again, and in her voice. \"Marriage isn't always the right answer.\"\n\n\"It is for people like you and me.\" He nearly spoke of the children, but stopped himself. It wasn't right to use them to press his case. \"I'd make you happy.\"\n\n\"I know you would.\" She turned her face into the curve of his neck. \"Let this be enough for now. It's so much more than I ever thought I'd have. Let it be enough for now.\"\n\n\"For now.\" He contented himself with the scent of her hair. \"Why don't we do this? Get ourselves some wine, some of that pie, have a little picnic?\"\n\n\"I'd like that.\" She leaned back, smiled. \"I'll get a couple of plates.\" But when she reached for her slacks, his hand closed over hers.\n\n\"You're not going to need those,\" he said, his eyes dark and wicked.\n\nShe laughed. \"I'm not going to serve pie buck-naked.\" Then she blinked, felt a quick skitter of her pulse. \"Am I?\"\n\n\"Why don't we see?\"\n\n## Chapter 11\n\nSchool was out, and that made life for two ten-year-old boys close to perfect. The haunted woods that fringed between Bryan's cabin and the inn beckoned. There they could search for ghosts, listen for the pounding of mortar fire, or hunt for more tangible remnants of war in the dirt and brambles. Even after more than a century, old shells could be unearthed.\n\nConnor had a collection Bryan envied, stubby bullets that looked like they were made of clay, an old brass button that had survived the uniform it belonged to and, best of all, the metal triangle of a stirrup Cassie had unearthed in the garden of the inn.\n\nThe boys had decided it had belonged to a Union general and his trusty steed.\n\nConnor viewed this stretch of summer in a way he never had before. The last year had been exciting when they moved into the new apartment, but he'd still worried often that it would all end. Now he'd come to believe, now he could anticipate the long, hot days, the companionship of his best friend and a home where no one stumbled in drunk with fists raised.\n\nHe watched his mother still. Her eyes no longer looked so tired, and she laughed so much more than she had ever laughed before. He liked the way she put pretty things around the house, the flowers, the pale green glass she'd begun to collect from Regan's shop. But he kept quiet about that, because he knew the guys would rag on him for liking something as lame as flowers or glass bowls.\n\nBut not Bryan. Bryan was the best of friends, and didn't even mind if Emma tagged along with them. Bryan liked to listen to Connor's stories. Bryan could keep secrets. Bryan was his brother, his blood brother. They had held a solemn ceremony in the woods, pricking their fingers and mixing their blood together to seal the bond.\n\nThey spent some of those early days of freedom from books and classrooms in the tree house Jared had built on the edge of the woods nearest the cabin. Some they spent in the yard of the inn, practicing baseball. They would also cut through the trees and visit Shane at the farm. As Bryan said, Shane was very cool, and he never minded if they wanted to play with the dogs and the puppies or hang out in the hayloft of the big old barn.\n\nBut almost every day, it was the woods that pulled at them. And tonight they had finally wangled permission to camp out, just the two of them, deep in the haunted woods.\n\nThey had pitched Devin's old tent. It was Devin, Connor knew, who had turned the tide. His mother had worried over the idea of letting the two boys loose for a night, but Devin had talked to her about rites of passage and memories and friendships. He owed the most important night of his life to Sheriff MacKade.\n\nThey had built a fire carefully, in a circle of stones on clear ground, as Devin had shown them, and they had hot dogs and marshmallows to roast over it. Cassie had given them a big jug of juice, but Devin had slipped them a six-pack of soda and told them to take the empty cans, along with the other trash, over to the farm in the morning for disposal.\n\nTheir sleeping rolls were spread out in the tent, the moon was high and bright overhead, and owls were hooting. The fire crackled, and the scent of scorched meat stung the night air. The sweet, gooey taste of marshmallow was in Connor's mouth. And he was in heaven.\n\n\"This is the best,\" he said.\n\n\"It's pretty cool.\" Bryan watched his hot dog turn black on the end of his stick, just the way he liked it. \"We should do it every night.\"\n\nConnor knew it wouldn't be special if they did it every night, but didn't say so. \"It's great here. Sheriff MacKade said that he and his brothers used to camp out in the woods all the time.\"\n\n\"Dad likes to walk in the woods.\" Bryan loved using that word. Dad. He tried to use it often, without making it a big deal. \"Mom, too. They sure kiss a lot.\" He made smacking noises with his lips so Connor would laugh. \"Beats me why kissing's supposed to be so damn neat. I think I'd gag if a girl tried to put her mouth on me. Disgusting.\"\n\n\"Revolting. Especially the tongue part.\"\n\nAt that, Bryan executed very realistic vomiting sounds that had both boys rolling with laughter.\n\n\"Shane's always kissing girls.\" Connor rolled his eyes. \"I mean, always. I heard your dad say he's got an addiction.\"\n\nBryan snorted at that. \"It's weird. I mean, Shane knows all there is to know about animals and machines and stuff, but he likes having girls hang around. He gets this funny look in his eye, too. Like Devin does with your mom. I figure some girls must zap some guys' brains. Like a laser beam.\"\n\n\"What do you mean?\" Connor had gone very still.\n\n\"You know, zap!\" Bryan demonstrated with a pointed finger and cocked thumb.\n\n\"No, about Sheriff MacKade, and my mom.\"\n\n\"Jeez, he's really stuck on her.\" The hot dog was thoroughly burned. Concentrating, Bryan blew on the end before biting it and filling his mouth with charcoal. \"He hangs around her all the time and brings her flowers and junk. That's what my Dad did with Mom. He'd bring her flowers, and she'd go real dopey over them.\" He shook his head. \"Screwy.\"\n\n\"He comes around because he's looking out for us,\" Connor said, but the sweet taste in his mouth had gone sour. \"Because he's the sheriff.\"\n\n\"Sure, he looks out for you.\" Involved with his hot dog, Bryan didn't see the panic in his pal's eyes. \"Maybe that's how he got stuck on her in the first place, but man, he's gone. I heard my mom and dad talking the other night, and Mom said how she got a kick out of seeing the big, bad sheriff\u2014that's what she calls him\u2014out of seeing him cow-eyed over Cassie. Cow-eyed.\" Bryan snickered at the term. \"Hey, if they get married, we'd be cousins and blood brothers. That'd be great.\"\n\n\"She's not getting married.\" Connor's voice lashed out so fast and furious that Bryan nearly bobbled the rest of his dinner.\n\n\"Hey\u2014\"\n\n\"She's not going to marry anyone, ever again.\" Connor leaped to his feet, fists clenched. \"You're wrong. You're making it up.\"\n\n\"Am not. What's your problem?\"\n\n\"He comes around because he's the sheriff, and he's looking out for us. That's it. You take it back.\"\n\nHe might have, but the martial glint in Connor's eyes sparked one in his own. \"Get real. Anybody can see Devin's got the hots for your mom.\"\n\nConnor was on him like a leech, knocking Bryan back, rolling over the dirt. Surprise and panic gave him the first advantage as his fists pummeled at Bryan's ribs. But it was his first fight, and Bryan was a veteran.\n\nWithin a few sweaty moments, Bryan had Connor pinned. Both of them were scraped and filthy and breathing hard. In reflex, Bryan bloodied Connor's lip, snarling like a young wolf. \"Give up?\"\n\n\"No.\" Connor jabbed an elbow out and had Bryan grunting. Into the brambles they rolled, gasping out threats and curses.\n\nAgain Bryan pinned him, and again he raised his fist. He stopped, froze. He would have sworn he heard something, something that sounded like a man dying, but it didn't sound of this world.\n\n\"You hear that?\"\n\n\"Yeah.\" Connor didn't loosen his grip on Bryan's ripped T-shirt, but his eyes darted left and right. \"It didn't sound real, though, it sounded like...\"\n\n\"Ghosts.\" The word came through Bryan's cold lips. \"Jeez, Con. They're really here. It's the two corporals.\"\n\nConnor didn't move a muscle. He didn't hear it anymore, just the owls and the rustle of small animals in the brush. But he felt it, and he suddenly understood. That was what war was, he thought, stranger against stranger, brother against brother. Fighting. Killing. Dying.\n\nAnd he was ashamed, because Bryan was his brother and he'd raised his fist to him. Raised his fist, he thought as tears stung his eyes, as Joe Dolin had done to Mama, and to him.\n\n\"I'm sorry.\" He couldn't stop the tears, just couldn't, not even when Bryan stared down at him. \"I'm sorry.\"\n\n\"Hey, it's okay. You hit good.\" Uncomfortable, he patted Connor's shoulder before he levered himself to his feet. Systematically he tugged aside brambles and picked thorns out of his clothes and flesh. \"You just got to work on your guard, is all.\"\n\n\"I don't want to fight. I hate fighting.\" Connor sat up and curled himself into a ball of misery.\n\nBryan cast around for something to say. \"Man, we're a mess. You're going to have to come up with a good story for how we got our clothes torn and stuff. Maybe we could say we were attacked by wild dogs.\"\n\n\"That's stupid. Nobody'd believe that.\"\n\n\"You come up with one, Con,\" Bryan coaxed. \"You're real good at stories.\"\n\nConnor sighed, kept his head on his knees. He didn't want to lie. He hated lying as much as he did fighting. But he didn't think he could stand seeing disappointment in his mother's eyes. \"We'll say we lost the baseball in the blackberry bushes and got all caught up in the thorns.\"\n\nIt was simple, Bryan decided. And sometimes simple was best. \"How about your lip? It's going to puff up real good.\"\n\n\"I guess I fell down.\"\n\nBryan wiped his hands on his dirty jeans. \"Does it hurt? You can put one of the soda cans on it.\"\n\n\"It's okay.\"\n\n\"Look, Con, I didn't mean anything by what I said. Nothing bad about your mom, I mean. She's great. If I thought somebody was saying something bad about my mom, I'd beat the hell out of them.\"\n\n\"It's okay,\" Connor said again. \"I know you weren't.\"\n\n\"Well, what'd you go at me like that for?\"\n\nCalmer now, Connor rested his chin on his knees. \"I thought Sheriff MacKade was coming around because he liked me.\"\n\n\"Well, sure he likes you.\"\n\n\"He's coming around for my mother. He's probably been kissing her, and maybe even more. You know?\"\n\nBryan shrugged. \"Well, since he's stuck on her...\"\n\n\"Everything's been good. Everything's changed, and it's so great the way it is. We've got the apartment, and Mama's happy, and he's locked up. Now everything's going to be ruined. If she marries the sheriff, it'll ruin everything.\"\n\n\"Why? Devin's cool.\"\n\n\"I don't want a father, not ever again.\" Dark eyes dominated Connor's dirty, tear-streaked face. \"He'll take over, and things will change back. He'll start drinking and yelling, and hitting.\"\n\n\"Not Devin.\"\n\n\"That's what happens,\" Connor said in a fierce whisper. \"It'll all be his instead of ours, and it'll all have to be his way. And if it isn't, he'll hurt her and make her cry.\"\n\nHe had an image of Devin making a vow, offering his hand on it, right here in the woods. But he pushed it aside.\n\n\"That's what fathers do.\"\n\n\"Mine doesn't,\" Bryan said reasonably. \"He'd never hit my mom. He yells, but she yells back. Sometimes she yells first. It's pretty cool.\"\n\n\"He hasn't hit her yet. She just hasn't made him mad enough.\"\n\n\"She makes him real mad sometimes. One time, she made him so mad I thought smoke was going to come out of his ears, like in a cartoon. He picked her right up and threw her over his shoulder.\"\n\n\"See.\"\n\nBryan shook his head. \"He didn't hurt her. They started wrestling around on the grass, and she was yelling at him and swearing. Then they started laughing. Then they started kissing.\" Bryan rolled his eyes. \"Man, it was embarrassing.\"\n\n\"If he'd really been mad\u2014\"\n\n\"I'm telling you, he was. His face gets real hard, and his eyes, too. He was really steaming.\"\n\n\"Did it scare you?\"\n\n\"Nah.\" Then Bryan moved his shoulders again. \"Well, maybe it does just a little, when I do something to make him really mad at me. But it's not because I think he's going to belt me or anything.\" Bryan let out a long breath, then shifted so that he could drape an arm over Connor's shoulders. \"Look, Con, Devin's not like Joe Dolin.\"\n\n\"He fights.\"\n\n\"Yeah, but not with girls, or kids.\"\n\n\"What's the difference?\"\n\nConnor was about the smartest person he knew, Bryan thought, but he could be so dopey. \"You just socked me, right? Are you going to go home and whip up on Emma?\"\n\n\"Of course not. I'd never\u2014\" He broke off, brooding. \"Maybe it's different. I have to think about it.\"\n\n\"Cool.\" Satisfied, Bryan rubbed his sore ribs. \"Let's break out a soda, and you can make up a ghost story. A really gruesome one.\"\n\nBecause Devin had awakened early, he was up and feeding the pigs when he spotted the two boys crossing from the woods with their gear and bag of trash. He lifted a hand in greeting, then cocked a brow when he saw the scrapes, bruises and ripped shirts.\n\n\"Must have been some night,\" he said mildly. \"Run into bears?\"\n\nBryan chuckled and greeted the exuberant Fred and Ethel. \"Nah. Wolves.\"\n\n\"Um-hmm...\" He studied Connor's puffy lip. \"Looks like you put up a hell of a battle.\" He started to reach out for Connor's chin, but the boy jerked back.\n\n\"We lost the baseball in the berry bushes,\" Connor said flatly. \"We got tangled up, and I fell.\"\n\n\"Your mothers'll probably buy that,\" Devin decided. \"Your dad won't,\" he told Bryan. \"But he'll let it slide.\" He emptied the bucket of grain into the trough and had the pigs squealing greedily. \"How'd it go otherwise?\"\n\n\"It was great.\" Bryan stepped onto the bottom rung of the fence to watch the pigs. \"We ate hot dogs and marshmallows and told ghost stories. We even heard the ghosts.\"\n\n\"Sounds eventful.\"\n\n\"Thank you for the tent,\" Connor said stiffly.\n\n\"No problem. Why don't you hang on to it? I imagine you'll use it again before I will.\"\n\n\"I don't want it,\" Connor said, with a lack of courtesy so out of character, Devin only stared. \"I don't want anything.\" He dropped the tent on the ground. \"I have to go.\" He stood for a moment, chin jerked up, waiting for Devin to show him what happened when you sassed.\n\nBut Devin only studied his face, and there was puzzlement, rather than anger, in his eyes. \"Put some ice on that lip.\"\n\nShoulders stiff, Connor turned and walked quickly away, without a word to his friend.\n\n\"I'll keep the tent, Devin.\" Mortified, and irritated, Bryan shot Connor's back a seething look. \"He doesn't mean to be a jerk.\"\n\n\"He's ticked at me. Do you know why?\" When Bryan kept his head down, his hands in his pockets, Devin sighed. \"I don't want you to break a confidence, Bry. If I've done something to hurt Connor, I'd like to make it right.\"\n\n\"I guess it's my fault.\" Miserable, Bryan scuffed his shoe in the dirt. \"I said something about how you were stuck on his mom, and he went nutso.\"\n\nDevin rubbed a hand over his suddenly tensed neck. \"Is that what you fought about?\" No answer again, and Devin nodded. \"Okay. Thanks for telling me.\"\n\n\"Devin.\" Loyalty had never been a problem for Bryan before. Now he felt himself tugged in different directions. \"It's just\u2014he's just scared. I mean, Con's not a wimp or anything, but he's scared that if you have, you know, like a thing going with Mrs. Dolin, things'll be like they were. Before, you know. He's got it stuck in his mind that you'd start punching out on his mom the way that bastard\u2014I mean the way Joe Dolin did.\" Bryan looked around, but Connor had already disappeared into the woods. \"I tried to tell him he was off, but I guess he didn't really believe me.\"\n\n\"Okay. I got it.\"\n\n\"He'll probably hate me for telling you.\"\n\n\"No, he won't. You did right, Bryan. You're a good friend.\"\n\n\"You're not mad at him, are you, for talking back?\"\n\n\"No, I'm not mad at him. You know how Jared feels about you, Bryan?\"\n\nPleasure and embarrassment mixed, tinted his cheeks. \"Yeah.\"\n\n\"I feel pretty much the same way about Con, and Emma. I just have to give him time to get used to it.\"\n\nShe'd tried not to worry. Really she had. But when she looked out the window and saw Connor crossing toward the inn, the relief was huge. Cassie set aside the flour she'd taken out for pancakes and went to the kitchen door of the inn.\n\n\"I'm down here, Connor. Did you have\u2014\" She saw the bruised face, the torn clothes, and her heart froze in her chest. She was outside like a bullet, terror seeping out of every pore. \"What happened? Oh, baby, who hurt you? Let me\u2014\"\n\n\"I'm all right.\" Still seething, Connor jerked away from her. The look he aimed at her was one she'd never seen from him before. It was filled with fury and disdain. \"I'm just fine. Isn't that what you always told me after he hit you? I fell down, I slipped. I walked into the damn door.\"\n\n\"Connor.\"\n\n\"Well, I'll tell you the truth. I had a fight with Bryan. I hit him, he hit me.\"\n\n\"Honey, why would you\u2014\"\n\nAgain he jerked away from her hands. \"It's my business why. I don't have to tell you everything, just like you don't tell me everything.\"\n\nIt was rare, very rare, for her to have to discipline the boy. \"No, you don't,\" she said evenly. \"But you will mind your tone when you speak to me.\"\n\nHis swollen lip trembled, but he kept his eyes steady. \"Why didn't you ever tell him that? Why didn't you ever tell him to mind his tone when he spoke to you? You let him say anything he wanted, do anything he wanted.\"\n\nHer own shame at hearing the bald truth from her son swamped her. \"Connor, if this is about your father\u2014\"\n\n\"Don't call him that. Don't ever call him my father. I hate him, and I'm ashamed of you.\"\n\nShe made some sound as tears sprang to her eyes, but she couldn't speak.\n\n\"You're going to let it happen again,\" Connor raged on. \"You're just going to let it happen.\"\n\n\"I don't know what you're talking about, Connor. Come inside and sit down and let's straighten this out.\"\n\n\"There's nothing to say. I won't stay if you marry Sheriff MacKade. I won't stay and watch when he hits you. I won't let you make me have a father again.\"\n\nShe sucked in a harsh breath, forced it out again. \"I'm not going to marry him, Connor. I'd just started to think about it, but I would never have made a decision on something that important without talking to you and Emma. And I'd never marry anyone if you were against it. I couldn't.\"\n\n\"He wants you to.\"\n\n\"Yes, he wants me to. He loves me and wants us to be a family. He deserves a family.\" When she said it, she realized how true it was, how selfish she'd been to ask him to wait. \"He cares for us. I thought you cared for him, Connor.\"\n\n\"I don't want a father. I'm not ever going to have one, no matter what you do. Everything's good now, and you're going to ruin it.\"\n\n\"No, I won't.\" She blinked the tears back. \"Go upstairs now, Connor, and get cleaned up.\"\n\n\"I won't\u2014\"\n\n\"Do as you're told,\" she said sternly. \"However you feel about me, I'm your mother and I'm in charge. I have to fix breakfast down here. You clean up and keep an eye on Emma until I'm finished.\"\n\nShe turned and walked back into the kitchen.\n\nSomehow she got through it, the cooking, the serving, the conversations. When she'd finished clearing up, she checked on the children, suggested that they play in the yard while she tidied the guest rooms.\n\nShe refused Connor's stiff offer to help, and left them to play. She was changing the linens on the bed in Abigail's room when she heard the front door open and close.\n\nShe knew it was Devin. She knew he'd come.\n\nShe didn't know that Connor had heard the car and, demanding a vow of silence from Emma, crept into the hallway.\n\n\"Can I give you a hand with that?\" Devin asked.\n\n\"No.\" Cassie smoothed the contoured sheet out, then reached for the top one. \"I've got it.\"\n\n\"I saw Con and Bry over at the farm this morning. You're not upset with him, are you? Boys get into tussles.\"\n\n\"No, I'm not upset about that.\"\n\n\"About what?\"\n\nShe drew a breath. She'd gone over it in her mind countless times already that morning. She'd let her children down all their lives. Whatever it cost, she would never do so again.\n\n\"Devin, I need to talk to you.\"\n\n\"I'm here.\"\n\n\"Connor's very upset, very hurt.\" She kept her hands busy, tucking the sheet, folding it down, smoothing it. \"He's sensed, or been told, something about us, and\u2014\"\n\n\"I know. I told you I saw him this morning. I'd say what he is, Cassie, is mad.\"\n\n\"Yes, he is. And upset, and hurt. Frightened,\" she added, pressing her lips together to steady them. \"Most of all, frightened. I can't let him be frightened, Devin. Not after what he's already been through.\"\n\n\"You didn't cause it.\"\n\n\"Didn't I?\" Meticulously she fluffed and patted the pillows into place. \"Doing nothing to stop it all those years is the same as causing it. The first eight years of his life were a nightmare I didn't put an end to. I thought I was shielding him. I told myself I was. But he knew. He's ashamed of me.\"\n\n\"That's not true, Cassie.\" Devin moved to her, took her hands. \"If he said that, it was because he was angry with me, and you were the nearest target. He adores you.\"\n\n\"I've hurt him, Devin, more than I ever realized. Maybe Emma, too. I see now that I've just started to make things right, make things up to them. Now I'm letting it change before they can adjust, before they can trust. I can't do that, Devin. And I can't see you anymore.\"\n\nPanic reared up, echoed clearly in his voice. \"You know that's not the answer. I'll talk to him.\"\n\n\"I don't want you to do that.\" Cassie tugged her hands from his. \"I have to handle this, Devin. I need to prove to Connor that I can, and that he and Emma come first.\"\n\n\"I'm not asking to come ahead of them, damn it, just to be a part of your life. Of their lives. I love you, Cassie.\"\n\n\"I know. I love you. I always will. But I can't be with you. Don't ask me to choose.\"\n\n\"What are you asking me to do?\" he demanded. \"To just walk away? I've waited for you for twelve years. I can't keep waiting for everything to be perfect. It's never going to be perfect, it just has to be right. We're right, Cassie. You mean everything to me. So do the kids. I need you. I need all of you.\"\n\nThat cut her heart out. \"Devin, if things were different\u2014\"\n\n\"We'll make them different,\" he insisted, taking her by the shoulders. \"We'll make it work.\"\n\n\"I'm not going to ask you to wait.\" She stepped back, turned toward the window. \"You need me, and hearing you say that is wonderful, even more wonderful than when you first told me you loved me. But Connor needs me, too. And he's just a little boy. He's my little boy, and he's frightened.\"\n\nShe took a deep breath, so that she could get it all out cleanly. \"You want marriage, family, and you're entitled to that. You're entitled to have someone who's free to give you what you want and need. But I'm not free, and I may never be free. I can't give you what you're entitled to, so I can't be with you, Devin.\"\n\n\"You expect me just to step back, as if nothing's happened between us? Just step back and wait?\"\n\n\"No. It's time you stopped waiting.\"\n\n\"There's no one but you.\"\n\nHer heart ripped in two ragged pieces\u2014one for the man, one for the boy. \"You haven't let there be. I let you hold on to me, Devin. I think part of me always knew you'd be there. And that was so unfair. I'm trying to be fair now, to everyone.\"\n\n\"Fair? It's fair to toss me, all of what we have together, aside, because a ten-year-old boy demands it? When the hell are you going to take charge, Cassie?\"\n\nIt was the first time he'd ever hurt her. She faced it, accepted it. \"That's what I'm trying to do. Taking charge doesn't always mean doing what you want. Sometimes it means doing what's right for the people you love.\"\n\n\"Damned if I'll beg you.\" Suddenly bitter, suddenly furious, he bit off each word. \"Damned if I'll ask you again, Cassie. I've had enough of standing on the sidelines and breaking my heart over you. I've stripped myself bare for the last time.\"\n\n\"Hurting you is the last thing I want, Devin. But I can't give you what you need most, so I can't give you anything.\"\n\nHis eyes cut into her, as hard and searing as his voice. \"It's time it was down to all or nothing. You've made your choice. Looks like I've made mine.\"\n\nShe listened to his receding footsteps, heard the door slam downstairs. This, she knew, was what Abigail had felt when she sent the man she loved away. This emptiness, this emptiness that was too huge for grief.\n\nCassie sat on the edge of the bed, buried her face in her hands, and sobbed.\n\nIn the corner of the hallway, Connor kept his hand tight on his sister's.\n\n\"Mama's crying,\" Emma whispered.\n\n\"I know.\" It wasn't Joe Dolin that had made her cry either, Connor thought. And it wasn't Sheriff MacKade.\n\nIt was him, and only him.\n\nWhile Cassie wept and Connor crept downstairs with grief and guilt heavy on his shoulders, Joe Dolin took his chance. He'd waited, oh, he'd waited so patiently, for just the right moment.\n\nThe creek rushed under the Burnside Bridge with a harsh bubbling sound. The trees were thick with leaves. His supervisor was gesturing to one of the other men, his attention distracted by a nest of copperheads they'd unearthed.\n\nThat was all it took.\n\nJoe bent to gather litter, working his way toward cover, step by careful step. And then he melted into the trees. As he walked quickly through the woods he stripped off his orange vest and tossed it into the brush beside the creek.\n\nHe didn't run, not right away. He still had trouble with the peripheral vision in his right eye, thanks to an injury he'd received when he attacked Regan MacKade. So he moved carefully at first, deliberately turning his head to judge his ground, and his distance.\n\nThen he sprinted, wild as a dog, over rocks, through brush, and finally into the creek. Breathing hard, he kept to the water, following its curves and angles. Before long, he was wet to the waist, but he kept going, pushing himself.\n\nPanting, he scrambled up the side of the bank, using rocks and vines to heave himself clear. Then he took a deep gulp of freedom. He would use the sun, and the direction of the creek, to show him the way he wanted to go.\n\nWhen Devin made up his mind, he was as hard to swerve as a six-ton truck. So when Rafe wandered into his office, saw Devin sitting behind his desk, typing furiously with his face set in stubborn lines, he knew there was trouble.\n\n\"I'm supposed to ask you to dinner,\" Rafe said easily.\n\n\"Beat it.\"\n\n\"Regan wanted to have the whole family over tomorrow, plus Cassie and the kids.\"\n\n\"I'm going to be busy. Now get the hell out of here.\"\n\n\"I didn't mention what time,\" Rafe continued, and walked over to look over Devin's shoulder. \"What the hell's this?\"\n\n\"Just what it looks like.\"\n\n\"Looks like a resignation to me. What wild hare do you have up your\u2014?\"\n\n\"Get off my back.\"\n\nRafe did the brotherly thing and ripped the paper out of the typewriter. \"Chill out.\" Before Devin could lunge to his feet, Rafe slapped a hand on his shoulder. \"Look, we can pound each other, I don't mind, but why don't we get the preliminaries out of the way? What the hell are you doing resigning as sheriff?\"\n\n\"What I should have done a long time ago. I'm getting the hell out of this town. I'm tired of being stuck here in the same damn rut, with the same damn people.\"\n\n\"Dev, you like nothing better than a rut.\" Rafe tossed the paper aside. \"What happened with Cassie?\"\n\n\"Nothing. Leave it.\"\n\n\"Aren't you the one who came breathing down my neck and made me face up to what I felt for Regan? One good turn.\"\n\n\"I don't have to face what I feel for Cassie. I've faced it for years. What I have to do is get over it.\"\n\n\"She turn you down?\" The vicious gleam in Devin's eyes didn't frighten Rafe; it touched him. \"Go ahead, take a shot at me. I'll give you a free one.\"\n\n\"Forget it.\" Deflated, Devin dropped back into his chair.\n\n\"Want to talk about it?\"\n\n\"I'm talked out.\" He rubbed his hands over his face. \"I'm tired. Connor doesn't trust me, she doesn't trust me. It comes down to neither of them wanting me enough. I can't keep trying to prove myself.\"\n\n\"The kid's come a long way, Dev. So has Cassie. Give them a little time.\"\n\n\"I've run out of time. I need something back, Rafe.\" Devin drew a deep breath. \"I just can't keep hurting like this. It's killing me. I'm getting out.\"\n\nBefore Rafe could speak, the phone rang. Devin snagged the receiver and all but spit into it, \"Sheriff's office. MacKade.\" He was on his feet in a flash, swearing violently. \"When? That's over a damn hour ago. Why in hell wasn't I notified? Don't give me that crap.\" He listened for another minute, then slammed down the receiver.\n\n\"Dolin's out.\" He strode over to the gun cabinet, unlocked it and pulled out a rifle. \"You're deputized.\"\n\n## Chapter 12\n\nJoe stayed hunkered in the ravine across from the little rancher where his mother-in-law lived. He doubted they'd look for him there, not right away. They'd go to his friends, check on Cassie. Maybe, just maybe, MacKade or one of his horse-faced deputies would swing by.\n\nBut his mama-in-law wasn't home. There was no car in the drive, and the curtains were drawn tight over the front windows.\n\nThe ranch house sat on the edge of a dead-end road, and was perfect for his purposes. He kept his eyes peeled, then scurried out of the ravine, keeping low. The far side of the house faced nothing but trees, so he used that for his entry. With an elbow, he shattered a window.\n\nOnce inside, he headed toward the main bedroom. He needed fresh clothes, and knew she kept some of her dead husband's things hanging in the closet like shrouds.\n\nThe old bag was morbid.\n\nShe was also paranoid.\n\nThat was how he knew there would be a pistol in her nightstand drawer, fully loaded. The only thing he wouldn't find in the house was a drink. But he'd see to that soon enough.\n\nInstead, in dry clothes too small for his frame, he settled down to wait.\n\nHe heard her drive up, listened to her fiddling with the locks and bolts on the front door. He smiled as he rose and walked out into the darkened living room.\n\nShe was carrying a bag of groceries in one arm, a cheap purse in the other. Her eyes widened when she saw him.\n\n\"Joe, what in the world\u2014\"\n\nHe did what he'd wanted to do for years. He swung out and knocked her flat with the back of his hand.\n\nActually, he thought about killing her. But he wanted to save that for his darling little wife. As she moaned and flailed at him weakly, he tied her with clothesline, gagged her. Once she was secured, wriggling like a fish on the floor, he dumped out her purse.\n\n\"Twenty lousy bucks,\" he complained. \"I shoulda known.\" He stuffed the bills in his pocket and picked up her keys. \"I'm going to borrow your car, need to take a little trip. A little trip with my wife. A wife's bound to go where her husband tells her to go, isn't that right?\"\n\nHe grinned as she rolled her eyes, as sick panic dulled them. \"It was real obliging of you to write all those letters to the prison. Real obliging. That's why I'm not going to mess you up too bad. I want to show you how I appreciate it.\"\n\nHe laughed when Constance moaned and babbled against the gag. \"Now, Cassie's a different thing, isn't she? She didn't stick by her husband like a proper wife, did she? But I'm going to take care of that. I'm going to teach her a real good lesson. Want to hear what I'm going to do to your daughter, old woman? Want to hear what I got planned for her?\"\n\nBecause he was enjoying the panic in her eyes, Joe hunkered down and told her.\n\nDevin squealed to a halt at the inn. His eyes scanned every bush, every tree, as he hurried around to the back and up the stairs. He didn't stop praying until he opened the door and saw Cassie at the stove.\n\nHe couldn't help it. He grabbed her, dragged her hard against him and just held on.\n\n\"Devin\u2014\"\n\n\"Sorry.\" Clamping down on every emotion, he drew back and became a cop again. \"I have to talk to you.\" He flicked a glance to the living room, where Connor and Emma sat staring at him. He started to tell Connor to take Emma to her room and stay there, then realized he was thinking like a father, not a cop. \"Joe walked off work release just over an hour ago.\"\n\nCassie's knees buckled. Devin held her up and guided her to a chair. \"Sit down, and listen. I've got people checking on his known associates, the places where he used to hang out. We'll pick him up, Cassie. Does he know you're living here?\"\n\n\"I don't know,\" she said dully. \"My mother might have\u2014 I don't know.\"\n\n\"We won't chance it. I want you to get whatever you need. I'm going to take you over to the cabin.\"\n\n\"The cabin?\"\n\n\"You'll stay with Savannah. I need Jared. I need Shane, too, or I'd have taken you over to the farm. Pull it together, Cassandra,\" he said, sharply enough to have her eyes clearing.\n\n\"I can't go to the cabin, Devin. I can't put Savannah and her children in danger.\"\n\n\"Savannah can handle it.\"\n\n\"So can I. Give me a minute.\" She needed to take a breath. \"Connor and Emma will go wherever you think they'll be safe.\"\n\n\"No, ma'am.\" Connor curled his trembling hand over Emma's. \"I'm not going anywhere without you. I'm not leaving you.\"\n\n\"Nobody's leaving anybody. You're all going where I tell you to go. Get your things,\" Devin snapped. \"Or do without them.\"\n\n\"Savannah is not responsible for me and mine,\" Cassie said slowly. \"I am.\"\n\n\"I don't have the time to be patient with you. I can't stay here and take care of you, so you're going.\"\n\nHe whirled around. Connor, his stomach queasy, saw a kind of fury he'd never seen before, not even in Joe Dolin's eyes. \"Get downstairs, into the car.\"\n\n\"I can take care of my mother.\"\n\n\"I'm counting on it, but not here. Do as I tell you, Connor.\"\n\n\"Devin, take the children, and\u2014\"\n\n\"The hell with this.\" He spun around again, picked Cassie up bodily and flung her over his shoulder. \"Out!\" he shouted at Connor, then swore when the boy's blood drained out of his face. \"Damn it, boy, don't you see I'd die before I'd hurt her? Before I'd hurt any of you?\"\n\nAnd Connor did, so clearly that the shame of it burned color back into his cheeks. \"Yes, sir. Come on, Emma.\"\n\n\"Put me down, Devin.\" Cassie didn't bother to struggle. \"Please, put me down. We'll go.\"\n\nHe set her on her feet, keeping his hands on her shoulders for a moment. \"You have to let me take care of you. You have to let me do that, at least. Trust me, Cassie.\"\n\n\"I do.\" She reached for Connor's hand. \"We do.\"\n\n\"Make it quick.\" He put a hand on the screen door, scanned quickly before stepping out. \"We've got roadblocks,\" he began. \"Helicopters are on the way. Odds are we'll have him before nightfall. How many at the inn?\"\n\n\"No one. We have a family coming in tonight, but\u2014\"\n\n\"I'll take care of it. Just don't\u2014\"\n\nWhen the shot rang out, it was so sudden, so shocking, Cassie could do nothing but gasp. Devin collapsed at her feet.\n\n\"Hi, honey.\" Joe walked forward, a grin on his face, a gun in his hand. \"I'm home.\"\n\nShe did the only thing she could do. She shoved the children behind her and faced him.\n\nShe saw the changes in him. His face was thinner, harder, as his body was. There was a scar beside and beneath his right eye, puckered and white. But the eyes themselves were the same. Brutal.\n\n\"I'll go with you, Joe.\" She knew Devin was breathing, but there was blood on his temple where the bullet had streaked. He needed help, an ambulance. The only way to save him and her children was to surrender herself. \"I'll go wherever you want. Just don't hurt my babies.\"\n\n\"I'll do whatever I want with your brats, bitch. And you'll do just what I tell you.\" He looked down at Devin, sneered. \"Not so tough now, is he? I should have aimed better.\" He squinted, laughed. \"Got a little problem with the eye, but I'll do a lot better close up.\"\n\nAs if in a dream, she saw his face, saw the gun lower. The cold came over her, the cold and the knowledge that this had happened before. Only then it had been a young, wounded soldier and a woman too weak, too frightened, to save him.\n\n\"No!\" She screamed, threw herself over Devin's body. \"He's hurt!\" She knew those words were useless, and struggled to find others. \"If you kill him, Joe, and they catch you, you'll never get out again. Do you know what happens when you kill a police officer? It isn't worth it. I said I'd go with you.\"\n\n\"You stay there, I'll just shoot through you. Then, maybe...\" He smiled again, shifted his gun toward Connor.\n\n\"Stay away from my babies!\" Like a woman possessed, she lunged, threw herself at him with a fire and fury that nearly knocked him over. Even when he hit her, she clung like a burr. Then Connor was on him, pummeling, shouting.\n\nJoe swatted him off like a fly.\n\n\"I'll teach you manners, you little brat.\" Before he could strike out with the butt of the gun, he heard the sound of sirens. \"Later,\" he said as Connor scrambled to his feet. \"I'll be back for you later.\" He had am arm around Cassie's throat, choking her, the gun to her temple.\n\nHis only escape, he saw, was the woods. \"I'll kill her!\" he shouted to anyone who could hear. \"Anyone comes after me, she's dead!\"\n\nHe dragged her away, trampling flowers.\n\nOn the ground, Emma squeezed Devin's hand. \"Please wake up. Please wake up.\"\n\nConnor crawled to him as Rafe and a deputy rushed around the house. \"He shot him, he shot him and he took Mama!\"\n\nGrim-faced, Rafe bent over his brother. \"It's not as bad as it looks.\" It helped to say it. He pulled a bandanna out of his pocket and stanched the blood. \"He's coming around,\" he murmured, and relief washed through him in a flood as Devin stirred. \"Connor, go in and call an ambulance. Hurry.\"\n\n\"No.\" Devin's eyes fluttered open. He batted his brother's hand away. \"I'm okay. Cassie\u2014\"\n\n\"You're shot, you idiot.\" But even as Rafe tried to hold him down, Devin was fighting his way up.\n\nHis vision wavered, grayed. A short stream of oaths helped steady it again. \"Where'd he take her?\"\n\n\"To the woods.\" Connor bit his lip. \"He took her into the woods. He was hurting her. I tried to stop him.\"\n\n\"Take care of your sister,\" Devin ordered. \"I want men posted around the woods. Notify Jared, tell him to get back to the cabin. He might go there. You stay with these kids,\" he ordered his deputy. \"Get them inside.\"\n\n\"I'm going in with you,\" Rafe stated.\n\n\"You can go in.\" Eyes cold, Devin drew his weapon. \"But he's mine.\"\n\nCassie did whatever she could to slow him down now that he was away from her children and Devin. She would not be a silent victim again. She scratched, she bit, she kicked.\n\n\"Forgot who's boss, didn't you? Thought you could lock me up in a cage and forget who was in charge.\" Cursing her, he shoved the gun into his waistband, so that he could use both hands to drag her. \"I'm going to have a good time reminding you.\"\n\n\"They'll find you. They'll catch you and lock you up for good this time.\"\n\n\"Maybe they'll catch me, maybe they won't.\" He stumbled along, hauling her after him and losing his direction in his fury. He hated these damn woods, the MacKade woods. \"I've had a lot of time to think about this. I know just what we're going to do. We're going to get us a car. That's what we're going to do.\"\n\nHe cursed the fact that he'd had to leave the one he'd already stolen behind.\n\n\"I've got friends,\" he muttered. \"I've got plenty of friends who'll help me out.\"\n\n\"You've got no one. You never did. Devin'll come after you, Joe, and he'll never stop.\"\n\n\"He's lying on his back and bleeding to death.\"\n\n\"He'll never stop,\" she said again. \"Nothing you do to me will come close to what he'll do to you.\"\n\n\"Got something going with him, don't you?\" Joe stopped, breathless, and dragged her head back by the hair. He thought he heard voices in his head, voices saying the words just before he did. \"You whore. I own you. Don't you forget it. I own you. Till death do us part.\"\n\n\"You're a miserable, drunken bully.\" Defiance bolted through her like lightning. \"You don't own anything, not even yourself. You're pathetic.\" She barely winced when he yanked mercilessly on her hair. \"The only thing you can beat is something weaker than you. Go ahead, Joe, hit me. It's the only thing you know how to do. But this time, damn you to hell, you're in for a fight.\"\n\nHe released her hair, using that hand to knock her sprawling on the path. The pain only energized her. Eyes hot and deadly, she got to her feet, her fists clenched.\n\nHe stepped forward, and she braced, ready, even eager, to defend herself.\n\n\"If you touch her, if you breathe on her, I'll put a hole in you.\"\n\nSlowly Joe turned. Devin was less than three yards back on the path, his weapon drawn and aimed. Rafe MacKade was behind him. As Joe's eyes darted in search of an escape, Shane stepped out of the trees. And Jared moved up the path behind Cassie.\n\n\"Drop the gun, Dolin, take it out slow and drop it, or I'll kill you.\"\n\n\"You're plenty brave, MacKade.\" Joe wet his lips as he took the gun out with two fingers, stooped to set it on the ground. \"When you've got four guns on me, and your brothers standing by.\"\n\n\"Kick it this way.\"\n\n\"Yeah, a real hero, long as it's not one-to-one.\" Joe gave the gun a shove with his foot. \"You've been helping yourself to my wife, haven't you?\"\n\n\"You don't have a wife.\" Devin turned, handed his gun to Rafe. \"Stay back,\" he demanded, then skimmed a glance over his other brothers. \"All of you.\" He looked at Cassie briefly, saw the bruises already forming. And felt hatred wash through him. \"Get to the cabin, Cassie. Savannah will take you back to the kids.\"\n\n\"You don't have to do this.\"\n\n\"Oh, yeah. I do.\" And he smiled. \"Let's go, Joe. It's been a long time coming.\"\n\n\"What's to stop one of your brothers from shooting me in the back once I beat you to a pulp, MacKade?\"\n\n\"Nothing.\" Now the smile turned feral. \"This is the last shot you're going to get at me, though, you yellow son of a bitch. So make it good.\"\n\nJoe shouted ferociously as he lunged. All Devin had to do was pivot and pump upward with a fist to send Joe reeling back.\n\n\"Tougher when it's somebody near your own size, isn't it?\" Devin taunted. \"Tougher when it's not a woman, or a little boy. Come on, you bastard. Try again.\"\n\nBlood spilling from his lip, Joe came at him like a bull. The woods cracked with the sound of bare knuckle against bone, of men grunting. Cassie forced herself not to cover her face with her hands.\n\nIt was for her. Each blow Devin threw or received was for her. So she would watch.\n\nAll the fear she'd felt of Joe ebbed as she did. He was exactly what she had called him. A pitiful bully. His size, and the wildness of his attack, helped him land a few blows. Certainly, it was that size that had him overbalancing Devin to the ground.\n\nBut even there, even outweighed, Devin dominated. His fists were fast, brutal, and the look on his face was so concentrated, she knew he felt none of the hits he took.\n\nShe didn't turn her face away from the blood, hold her hands over her ears to block out the sound. This was the end, finally the end, and she needed to bear witness.\n\nThe rage was on him so thick, so cold, that he could see nothing but Joe's face. Each time his fist hammered down, each time the power of it sang up his arms, he felt nothing but dark, deadly pleasure. His knuckles were raw, his shirt was splattered with blood, some of it his own, but he couldn't stop his fist from pumping.\n\n\"That's enough.\" Jared stepped forward to pull Devin off, and nearly got a fist in the face for his trouble. \"That's enough,\" he repeated, but it took all three of them to drag Devin to his feet.\n\n\"That's a satisfying sight,\" Rafe commented, studying Joe's battered and unconscious face. \"I guess I can't be too ticked you didn't leave a piece of him for me.\"\n\n\"Looks like he resisted arrest, right, Jare?\" Shane shouldered his rifle, scratched his chin.\n\n\"That's the way I saw it. Come on, Dev, let's haul this carcass in. You need a beer and an ice pack.\"\n\nBut the rage hadn't faded away, not completely. Devin jerked his brother's hand from his shoulder. \"Leave me alone.\" He turned, looked to where Cassie still stood, pale, bruised, eyes wide with shock. \"I'm finished.\" He took off his badge, tossed it into the dirt. \"Take him. I'm going home.\"\n\n\"Devin.\"\n\nWhen Cassie started forward, Jared put out a hand to stop her. \"Give him some time,\" he murmured, watching Devin cut through the woods, toward the farm. \"He's hurting.\"\n\nShe tried. She went to her children and comforted them. She let Regan and Savannah come to her and fuss over her bruises. She spoke to her mother, briefly, on the phone and reassured herself that, though her mother had been bruised and terrified, there was no serious damage. And, perhaps, there was some understanding between them that they'd never shared before.\n\nIn the end, she gave in and took the sedative that was pushed on her and slept like the dead through the night.\n\nBut in the morning she knew she hadn't finished facing her demons. She let Regan deal with breakfast and readied herself to go to the farm and face Devin.\n\nThe only thing she needed to take, she tucked into the pocket of her slacks.\n\n\"You're going to see Sheriff MacKade.\" Connor stepped into her bedroom doorway. His eyes were swollen and shadowed, there was a faint bruise on his cheek, and he was still so very pale. Cassie wanted badly to gather him close, but he was standing so stiff.\n\n\"Yes. I need to talk to him, Connor. I need to thank him for what he did.\"\n\n\"He'll say it was his job.\"\n\n\"Yes, I know he will. That doesn't mean I don't have to thank him. He could have been killed, Connor, for us.\"\n\n\"I thought he was dead at first.\" When his voice broke, he sucked in a breath and steadied it again. \"When he fell, and there was all the blood. I thought we were all going to be dead.\"\n\nShe shuddered, tried to keep the tears out of her voice. \"I'm sorry, Connor, for what I did, for what I didn't do. I hope one day you'll forgive me.\"\n\n\"It wasn't your fault. It wasn't ever. I shouldn't have said those things.\" He wanted to look away, but he knew that would make him a coward. He knew what cowards were like now. \"It wasn't true, and it wasn't the way I really felt. I said it to hurt you, because I felt bad.\"\n\n\"Connor.\" She held her arms out, closing her eyes tight when he raced into them. \"That part of our lives is over. I promise you it's over.\"\n\n\"I know. You were pretty brave.\"\n\nUnbearably touched, she kissed the top of his head. \"So were you.\"\n\n\"This time.\" He sucked in a deep breath. \"Sheriff MacKade stood up for us. Emma and I want to go with you. We talked about it. We want to see the sheriff.\"\n\n\"It might be better if I talked to him alone, just now. He's feeling... He's upset.\"\n\n\"I have to talk to him. Please.\"\n\nHow could she deny her child the same closure she needed for herself? \"All right. We'll go together.\"\n\nFrom his seat on the front porch of the farm, Devin saw them come out of the woods. He nearly got up and went inside, but it seemed a small and petty revenge.\n\nThey looked like a unit, he realized, and he supposed, however much it hurt him, that was what they needed to be.\n\nHis head was still aching, and his hands burned. But that was nothing compared to the pain in his gut as he watched Cassie and the children cross the wide front lawn.\n\nThere were bruises on her face, and on the boy's. Fury flashed in his blood like lightning. Then Emma broke away from Cassie's hand and raced to him.\n\n\"We came to thank you because you took the bad man away.\" She crawled right into his lap, as if she belonged there. \"You have hurts.\" Solemnly she touched her puckered lips to the cuts and bruises, to the white bandage on his temple. \"Is that better now?\"\n\nHe gave in for a moment and pressed his face into her hair. \"Yeah, thanks.\" Before Cassie could speak, he shifted Emma onto his knee. \"If they haven't contacted you, I can tell you they've already transferred him to the state prison. With the new charges\u2014the escape, the assaults, grand theft auto, the weapons possession, assault with a deadly weapon and\u2014\" he ran his fingers over his ripped knuckles \"\u2014and resisting arrest, he's not going to see the light of day again. You and your family have nothing to worry about.\"\n\n\"Are you all right?\" was all Cassie could manage.\n\n\"I'm fine. You?\"\n\n\"Just fine.\" Her fingers curled and uncurled over Connor's. \"We wanted to come and thank\u2014\"\n\n\"I was doing my job.\"\n\n\"I told her you'd say that,\" Connor said, and earned a mild glance from Devin.\n\n\"So, I'm predictable.\" He looked back at Cassie. \"You handled yourself well, Cass. You want to remember that. I've got work to do.\"\n\nAs he started to set Emma down, Cassie moved forward. \"Devin, please, don't.\"\n\n\"He hurt you.\" The words burst out of him. \"He hurt all of you, and I didn't stop him.\"\n\n\"You were shot, for God's sake. You were lying there unconscious and bleeding.\"\n\n\"The bad man was going to shoot you again,\" Emma told him. \"But Mama wouldn't let him. She lay on top of you so he couldn't.\"\n\nEvery ounce of his hot blood went cold at the thought of it. \"Damn it, Cassie, are you crazy?\"\n\n\"You needed me.\" She let out a shaky breath. \"I couldn't stand back, Devin. I did what I had to do. Now I'm going to ask you to do what you know is right.\" She took his badge out of her pocket. \"Don't give this up, Devin. Don't go.\"\n\nHe stared at the badge in her hand, then into her face again. \"You know what it's like to see something you want, you need, day after day, and know you can't have it? I'm not living like that anymore, not even for you. You won't let me be part of your life. You won't marry me, and I can't go on being your friend and nothing else.\"\n\n\"I'll marry you.\" Emma curled into him. \"I love you.\"\n\nHis heart simply shattered. He held Emma tight, then set her gently on her feet. \"I can't handle this, Cassie.\" He rose blindly. \"Go home and leave me be.\"\n\n\"Sheriff MacKade.\" Connor bolted forward, then skidded to a halt. \"I'm sorry.\"\n\n\"You've got a right to your feelings,\" Devin said steadily. \"And no need to apologize for them.\"\n\n\"Sir, I got something to say.\"\n\nDevin rubbed a hand over his face, dropped his arms. \"All right, get it out, then.\"\n\n\"I know you're mad at me. Yes, sir, you are,\" Connor said, keeping his eyes level when Devin started to correct him. \"I was mad, too, because I thought you'd come around just for me, or mostly, and then I found out it was because of Mama. And I thought if she'd let you, you'd change things, and they'd get bad again, even though you'd given your word. Bryan told me they wouldn't, but I didn't believe him. I didn't want to.\"\n\nHe had to take a deep breath. \"Yesterday, when you came to make us go to the cabin, and Mama said she wouldn't, you were mad. You were already mad, and then you were madder than anything. Weren't you?\"\n\n\"That's right.\"\n\n\"You yelled.\"\n\n\"Yeah, I did.\"\n\n\"I thought this is it, this is when he's going to hit her. You knew I was thinking it, but you weren't going to. You told me you'd never hurt her, not for anything. I knew you meant it. I knew when you went into the woods after her, you'd do anything to save her. It wasn't just because it was your job. It was because it was her. Because it was us.\"\n\nHe gathered the rest of his shaky courage and climbed the steps until he stood face-to-face with Devin. \"Even after she sent you away, even after I made her send you away, you wouldn't hurt her.\"\n\n\"I couldn't hurt her, Connor, if my life depended on it. That's how it is.\"\n\n\"Yes, sir. And she cried.\" He ignored the murmur of protest from his mother and kept his eyes on Devin's. \"After she sent you away, she cried, like she used to when she was hurt and she thought I couldn't hear. But this time I made her cry, and I want to tell you I'm sorry. I want to tell you that I don't want a father. I can't help it.\"\n\n\"All right.\" Devin knew he would fall apart in a minute. \"It's all right.\"\n\n\"I don't want a father,\" Connor hurried on. \"Except if he was you.\"\n\nThe hand Devin had laid on Connor's shoulder tightened painfully. But it was a good, solid feeling, and gave him the boost he needed to finish.\n\n\"Please, I want you to be with us all the time, like families are supposed to. I know you might not want me now, after what I did, but I swear I won't get in your way. I was stupid, and I sassed you and Mama, and you can punish me, but don't go away. You don't have to love me anymore, if you'd just\u2014\"\n\nThe boy's breath whooshed out, along with hot tears as Devin hauled him hard against his chest. \"You're too smart to say stupid things,\" Devin murmured shakily. \"I haven't stopped wanting you. I couldn't stop loving you.\"\n\n\"Don't go away.\" Connor held on for his life. \"Please, don't go away and leave us.\"\n\n\"I'm not going anywhere. I'm staying right here, okay?\"\n\n\"Yes, sir.\"\n\n\"Stop calling me sir all the damn time.\" He pressed a kiss to Connor's damp brow. Gently he used his thumb to wipe the boy's cheeks as Emma wriggled between them.\n\n\"Hold me, too,\" she demanded. \"I want you, too.\"\n\nSo he rose, the girl boosted in one arm, the boy wrapped under the other. Whatever happened now, he had no choice but to follow his heart.\n\nShe was standing there, her own eyes swimming, his badge clutched in one hand, the other pressed to her lips.\n\nIt wasn't the way he'd pictured it, asking her with two weepy children looking on. But it was going to have to do.\n\n\"No one's ever going to love you the way I do, Cassie. No one's ever going to love these children more or work harder to give them a good life. The fact is, I can't live without you, without all of you. You're my heart. For God's sake, Cassie, marry me.\"\n\nHe couldn't know what it meant to her, to hear those words, to have him say them, so simply, so plainly, while he held the children as if they were already his.\n\nOf course, they were. How foolish she'd been to ever think otherwise.\n\nHow foolish she'd been to think about doing what Abigail had done, turning away love.\n\nShe walked up the steps, took one of Connor's hands, one of Emma's. \"You are the most remarkable man I've ever known, and I love you. If you have a fault, it's that you're too patient, Devin.\"\n\n\"I'm running low right now.\"\n\n\"Then I'll make this simple. We've kept you waiting long enough.\"\n\nShe released Connor's hand only long enough to pin the badge back on Devin's shirt. Then, linked again, she lifted to her toes and kissed the man she loved in front of her children.\n\n\"We'd love to marry you, Devin. Soon.\" She laid her head on his heart. \"I think all of us have waited long enough. Very, very soon.\"\n\n##\n\nOnly Nora Roberts could create bad boys like the MacKades\u2014wild like stallions and sinfully sexy. But everyone has a weakness...and for every MacKade it's a woman.\n\nLook for more books in The MacKade Brothers series, available now wherever ebooks are sold:\n\nThe Return of Rafe MacKade\n\nThe Pride of Jared MacKade\n\nThe Heart of Devin MacKade\n\nThe Fall of Shane MacKade\n\n##\n\nDon't miss Nora Roberts's beloved Stars of Mithra trilogy, also now available as ebooks!\n\nHidden Star\n\nCaptive Star\n\nSecret Star\n\nISBN: 978-1-4592-1325-8\n\nThe Heart of Devin MacKade\n\nCopyright \u00a9 1996 by Nora Roberts\n\nAll rights reserved. Except for use in any review, the reproduction or utilization of this work in whole or in part in any form by any electronic, mechanical or other means, now known or hereafter invented, including xerography, photocopying and recording, or in any information storage or retrieval system, is forbidden without the written permission of the publisher, Harlequin Enterprises Limited, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9.\n\nAll rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of publisher.\n\nAll characters in this book have no existence outside the imagination of the author and have no relation whatsoever to anyone bearing the same name or names. They are not even distantly inspired by any individual known or unknown to the author, and all incidents are pure invention.\n\nThis edition published by arrangement with Harlequin Books S.A.\n\n\u00ae and \u2122 are trademarks of the publisher. Trademarks indicated with \u00ae are registered in the United States Patent and Trademark Office, the Canadian Trade Marks Office and in other countries.\n\nwww.Harlequin.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nContents\n\nTitle Page\n\nCopyright Notice\n\nPrologue\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nEpilogue\n\nSt. Martin's Paperbacks Titles by Sherrilyn Kenyon\n\nOutstanding Praise for Sherrilyn Kenyon and her Novels\n\nCopyright\nPROLOGUE\n\nSANTORINI, GREECE, 1990\n\nCompletely motionless, Megeara Saatsakis stood on the edge of a cliff looking out on waters so perfectly blue they were almost painful to behold. The air was fragrant with sea salt, cooking olives from the merchant carts, and bright sunlight in a homey scent that was completely unique to this region. The hot sun caressed her tanned skin while the fierce breeze whipped her plain white dress against her body. Boats glided over the gentle waves in a surreal manner that took her back to the days of her childhood when she'd walked these cliffs and shore with her father and mother while they'd done their best to instill in her what it meant to be Greek.\n\nIt was truly one of the most beautiful scenes in all the world, and any other twenty-four-year-old would love to be here.\n\nShe only wished she were one of them.\n\nInstead she hated this place with an unreasoning fervor. To her, Greece was death and sorrow.\n\nUtter misery, and she would rather have fishhooks pounded into her body than ever step foot on this land again.\n\nHer long blond hair, which she had swept back in a ponytail, slapped against her skin as she sought some peace for her troubled thoughts. But there was none to be had.\n\nOnly bottled-up rage met her.\n\nHer estranged father was dead. He'd died as he'd lived... in pursuit of a stupid, reckless dream that had taken not only his life but also that of her mother, her brother, her aunt, and her uncle.\n\n\"Atlantis is real, Geary. I can feel it radiating out to me even as I speak. It sits in the Aegean just below us, like a lost, glittering gem, waiting for us to find it and show the world what beauty it once held.\" Even now she could hear her father's hypnotic voice as he held her hand on top of the water for her to feel the softness of the waves as they whispered against her tiny palm. She could still see his handsome, enthusiastic face as he first told her why they spent so much time in Greece.\n\n\"We're going to find Atlantis and show the wonder of it to everyone else. Mark my words, babe. It's there and our family is the one that's been chosen to uncover its magic.\"\n\nThat had been his lunatic dream. One he'd spent a lifetime trying to give to her, but unlike the rest of her kookie family, she wasn't stupid enough to buy into it.\n\nAtlantis was a bogus myth made up by Plato as a metaphor for what happened when man turned against the gods. Like Lovecraft's Necronomicon, it was only a fictional invention that people wanted to believe in so badly they were willing to sacrifice everything to find it.\n\nNow her father lay in his grave on the island he'd loved so much. He'd died broken and bitter, a shell of a man who'd buried his beloved brother, his son, his wife...\n\nAnd for what? Everyone had laughed at him. Ridiculed him. He'd lost his job, along with his respectability, as a professor years ago, and the only way he'd been able to have his research published was in vanity presses.\n\nHell, even the vanity publishers had laughed at him and several had turned him down, refusing to even take his money to publish his ridiculous work. Still he'd carried on in his feverish desire to give people even more reason to laugh at him, which they'd done with relish.\n\nBut even with that, at least she'd seen him one more time before he passed and he hadn't died alone as he'd feared. Somehow, against the doctor's prognosis, her father had managed to hold on until she caught a plane from the U.S. and made it to his hospital room to see him. Though their meeting was brief, it had been enough to make peace with him so that he could die without guilt over abandoning her for his search.\n\nIf only she could have found a bit of that peace for herself. There still was no such forgiveness inside her where he was concerned. No matter how much her grandfather had tried to explain her father to her, she knew the truth. The only thing that man had ever loved had been his dream, and he had sacrificed his entire family... her entire family for it.\n\nNow at twenty-four, because of him, she had no brother and no parents.\n\nShe was utterly alone in the world.\n\nAnd her deathbed promise to her father to carry on his work burned inside her like a rampaging fire. It was one of the few times in her life that she'd been weak. But the sight of him as a frail, troubled man lying on a cold hospital bed while he desperately clung to life had torn her apart, and even though they'd barely spoken these last eight years, she hadn't had the heart to hurt him when all he wanted was to die forgiven.\n\nShe curled her lip as she watched the waves roll against the white shore. \"Find Atlantis, my ass. I won't ruin myself like you did, Dad. I'm not that stupid.\"\n\n\"Dr. Kafieri?\"\n\nShe turned at the sound of a heavily accented Greek voice to find a short, rotund man in his mid-fifties staring at her. A cousin to her father, Cosmo Tsiaris had been their family attorney here in Greece. A pseudo-partner in her father's salvage company, Cosmo had been instrumental in helping her father gain permits and investors for his antediluvian quest.\n\nAlthough she'd known Cosmo all her life, she cringed at his greeting. Kafieri had been her father's name\u2014one she'd cast off years ago after her applications to college had been rejected even though she more than met the requirements for admission. No self-respecting classics, history, or anthropology department would ever accept a Kafieri into its ranks for fear of the taint. So she'd learned to use her mother's maiden name to save her credibility and reputation.\n\nLike the rest of her immediate family, Geary Kafieri had died on these shores.\n\n\"I'm Dr. Megeara Saatsakis.\"\n\nA bright smile curved his lips. \"You married!\"\n\n\"No,\" she said simply, which made him literally deflate before her eyes. \"I legally changed my name from Kafieri eight years ago when I went back to the States and sued for emancipation from my father.\"\n\nShe could tell by Cosmo's face that he didn't understand her reasoning, and that was fine by her. With his patriarchal mind-set, he'd never comprehend it.\n\nFrowning, he didn't comment on her words as he held a small box toward her. \"I told Eneas that in the event of his death, I would make sure this was given to his daughter. That would still be you, yes?\"\n\n\"Yes,\" she said, ignoring his sarcasm. Who else would be dumb enough to claim a laughingstock as her progenitor?\n\nMegeara flinched at that thought. In all honesty, she loved her father. Even when his grief and quest had robbed him of everything, even his sanity and health, she'd still loved him. How could she not? He'd been a kind, caring father to her when she'd been a girl. It'd only been after she'd hit her teens and started questioning his research and fervor that they'd grown apart.\n\n\"Atlantis is bullshit, Dad. All this research is. I don't want to be on this stupid boat anymore. I'm young and I want friends. I want to go to school and be normal. You're wasting your time and my life!\" He'd slapped her so hard on her fifteenth birthday that she swore she could still feel the sting of it.\n\n\"Don't you dare spit on your mother's memory. On my brother's memory. They gave their lives for this.\"\n\nSix months later, so had Megeara's brother when his diving line had tangled and his tank had run out of oxygen. That had been the final straw between her and her father. She wasn't going to be Jason. She wasn't going to give up her life for someone else's dreams... ever.\n\nSo what if she'd promised her father? He was dead now. He'd never know she reneged. He'd died happy and she could finally put the past to rest and carry on with her life in America.\n\nLike her grandfather, she intended to leave this country and never step foot on it again.\n\nCosmo handed her the plain white box, then left her alone to open it.\n\nMegeara stared at it for several minutes, afraid of what she might find. Would it be some personal memento that would reduce her back to tears? She honestly didn't want to cry anymore for a man who'd broken her heart so many times that she couldn't even begin to count them all.\n\nBut in the end, her curiosity got the better of her and she opened the box. At first, there appeared to be nothing but crinkled acid-free tissue paper. She had to dig to the bottom of it all to find what it contained.\n\nAnd what she found there floored her. She stared at her palm in the bright sunlight, unable to even fathom it.\n\nThere were two items. One appeared to be a komboloi\u2014a string of worry beads similar in style to a small rosary that some Greeks used when stressed, only she'd never seen anything like this before. The age and design of it appeared to predate any form of komboloi she'd ever heard of. It had fifteen iridescent green beads made of some unknown stone that had been carved with tiny intricate family scenes of people wearing clothes unlike any she'd seen before in her research. The carvings were interspersed with five gold beads that were engraved with three lightning bolts piercing a sun. Where a komboloi might hold a small Greek piece such as a dime-sized medal, this one held a circle with writing that was similar to ancient Greek and yet very different. So much so that not even she who had been bred on ancient Greek could decipher it.\n\nLike most artifacts fresh from a dig, the komboloi had a small white tag attached to it by a red thread where her father had written finding notes:\n\n9\/1\/87\n\nsixty inches down from datum (see pg. 42)\n\nabsolute dating: 9529 B.C.\n\ngreen stone unknown\/unverified\n\nwriting unknown\/unverified\n\nThe anthropologist in her leapt to the forefront of what this could mean historically. If this date was truly absolute...\n\nIt showed a sophistication and metallurgy previously unknown. At that time, the Greeks shouldn't have had this level of skill. In fact, the precision of the carvings and engraving looked as if they were done by machine and not by hand. Eleven thousand years ago, mankind simply did not possess the tools it would take to create something this intricate.\n\nHow could this be?\n\nIntrigued, she turned her attention to the small leather pouch that lay in the bottom of the box. It, too, was tagged.\n\n7\/10\/85\n\nabsolute dating: 9581 B.C.\n\nmetal unknown\/unverified\n\nFrowning, she opened the pouch to find five coins of varying sizes. They were old... very old and heavily coated with patina. Again, there weren't coins this old. They just hadn't existed at that period in time and especially not in Greece. Like the komboloi, the coins held that same peculiar writing, but beneath said writing was something she could understand. It was the ancient Greek words for \"Atlantean Province of Kirebar.\"\n\nDear God!\n\nAgain, the coins didn't appear to be handmade, nor was their metal composite typical of anything she'd ever seen before. They were an orangish color, not silver, not gold, not bronze, copper, or iron\u2014maybe a weird combination of those metals and yet that didn't seem right, either.\n\nWhat the hell was it?\n\nEven with the patina coating them, the images and writing were as crisp, clear, and precise as those on a modern coin.\n\nHer heart pounding, she turned the largest coin over to look at the back. There was the same foreign symbol that marked the komboloi. A sun pierced by three lightning bolts. And with it were the unknown words on top of the Greek: May Apollymi protect us.\n\nMegeara stared at it in disbelief. Apollymi? Who was that?\n\nShe'd never heard that name before.\n\n\"It's a forgery.\" It had to be, and yet as she looked at it, she knew the truth. These weren't forged. Her father must have excavated them from one of his many digs in the Aegean.\n\nThis was what had kept her father going even while the rest of the world had laughed at him. He had known a truth she'd denied.\n\nAtlantis was real.\n\nAnd if it was, then her father had been doubted by everyone... even her. Grief and pain tore through her as she recalled all the arguments they'd had over the years. She'd been no better than any of the others.\n\nGod, the fights the two of them had had over this. Why had he never told her? Why would he keep a discovery of this magnitude from her?\n\nUnfortunately, she knew the answer. Because I wouldn't have believed it. Even if he'd shown it to me right in the ground where he'd found it. I would have laughed at him, too, then thrown it in his face.\n\nNo doubt he'd wanted to save himself the pain of facing her ridicule.\n\nClosing the box, Megeara held it next to her heart as she regretted every nasty word and criticism she'd ever even thought about him. How much had those words hurt? She who should have had faith in him had been as cruel as everyone else.\n\nNow it was too late to make amends.\n\n\"I'm so sorry, Daddy,\" she breathed through her fresh tears. Like everyone else, she'd assumed he was crazy. Misguided. Stupid.\n\nBut somehow he'd found these artifacts. Somehow they were real.\n\nAtlantis is real. The words chased themselves through her mind. Staring out across the blue sea, she tightened her grip on the box as she remembered her final words to him. \"Yeah, yeah, I promise. I'll look for Atlantis, too. Don't worry about it, Dad. It's in good hands.\" Those words had been rushed and passionless, and still they'd comforted him.\n\n\"It's there, Geary. I know you'll find it and you'll see. You. Will. See. You will know me for what I am, not for what you thought me to be.\" Then he'd slept for a time and he'd died only a few hours later while she'd held his hand.\n\nIn that moment of his quiet passing, she hadn't been a grown woman, she'd been a little girl all over again. One who only wanted her daddy back. One who craved someone to comfort her and tell her everything would be all right.\n\nBut there was no one in her life who could do that. And now that ludicrous, hasty promise meant something to her after all.\n\n\"I hear you, Daddy,\" she whispered to the olive-oil-laden breeze that she hoped would carry her voice to wherever he'd gone, \"and I won't let you die in vain. I'm going to prove Atlantis exists. For you. For Mom and for Uncle Theron and Aunt Athena... for Jason. If it takes me the rest of my life, I'm going to fulfill my word to you. We will find Atlantis. I swear it.\"\n\nBut even as she spoke those words that were filled with her conviction, she couldn't help wondering if she'd be able to withstand the ridicule her father had borne all of his professional life. Just six weeks ago she'd been granted her doctorate from Yale and she was supposed to begin teaching in New York this fall. She was young to have attained so much, and great things were expected of her... by her and by the institutions and professors who'd bestowed that doctorate on her.\n\nTo walk this course would be nine kinds of stupid. She would lose everything. E-v-e-r-y-t-h-i-n-g. It was a massive step she was about to take. One from which she'd never recover.\n\nMy father believed it.\n\nAnd her uncle and mother.\n\nThey had given their lives for this even while the entire world had laughed at them. Now a second generation of fools was about to follow the first down the road to ruin.\n\nMegeara only hoped that in the end she would meet a better fate than that which had greeted the first.\n\nLike father, like daughter.\n\nShe had no choice except to complete his quest because until she did, her name would be as worthless as his.\n\n\"Let the floggings begin....\"\nCHAPTER 1\n\nSANTORINI, GREECE, 1996\n\n\"My kingdom for a gun.\"\n\nShaking his head at Geary's hostile words, Brian calmly opened the car door for her as she approached their small taxi that waited in the heart of the crowded Greek thoroughfare. \"You don't have a kingdom.\"\n\nShe paused on the sidewalk to glare at him. Given the fury in her system, she couldn't believe he'd dare point out the obvious to her. She'd been known to verbally let serious blood when only half this riled. Truly, the man had no sense of self-preservation. \"And I don't have a gun\u2014looks like I'm shit out of luck all the way around, huh?\"\n\nStill, he was his ever present calm self\u2014which didn't really help her mood. For once, couldn't he get ticked off, too? \"I take it you didn't get the permits... again.\"\n\nShe could have done without that \"again\" part. Really. \"What was your first clue?\"\n\n\"Oh, I don't know. That stomping stance as you walked down the street, clenching and unclenching your fists like you're already choking someone, or maybe it's that way you're looking at me like you could claw out my eyes when I haven't done anything to piss you off.\"\n\n\"Yes, you have.\"\n\nShe could tell he was fighting a smile. Thank goodness he had the good sense to keep it hidden. \"And that is?\"\n\n\"You don't have a gun.\"\n\nHe snorted. \"Come on now, you can't shoot every Greek official who gets in your way.\"\n\n\"Wanna bet?\"\n\nBrian stepped back to let her enter the taxi first. At six three, he was a good-looking man in his mid-forties. Very distinguished and intelligent. Best of all, he was independently wealthy and more than capable of financing their latest venture in futility without complaining too much.\n\nUnfortunately, he wasn't into bribing public officials.\n\nWas it too much to ask that she find a corrupt financier? Surely Brian should have some vice, and at the moment she couldn't think of a more self-serving one than that.\n\n\"So what do we do now?\" he asked as he joined her in the car.\n\nGeary sighed, wishing she had an answer. Her team was waiting on her boat at the docks, but without the permits that allowed them to excavate the mounds she and Tory believed to be a city wall, all they could do was dive over the surface of what they'd found and do nothing more than admire it.\n\nSad comfort that. It'd been the best lead they'd had in years. \"I want another silt sample.\"\n\n\"You've already tested and retested those.\"\n\n\"I know, but maybe it will help to convince them to give us the permits.\" Yeah, right. She'd been given the run-around particularly good and the words from her latest visit still rang in her ears.\n\n\"This is Greece, Dr. Kafieri. There are ruins all around us and I will not allow you to begin tearing up the floor of the Aegean, which is a busy shipping area, when all you can give me is another this-is-Atlantis story. Really. I've enough treasure hunters trying to pilfer our national history for their own gain. I don't need any more. We here in Greece take our history most seriously and you're wasting my valuable time. Good day.\"\n\nIt was enough to make her want to bang her head on the man's desk until he either relented or had her committed. This wasn't about treasure, but trying to tell that to him had been as futile as trying to fly with wax wings.\n\n\"There has to be some way around this.\"\n\nBrian stiffened. \"I won't be a part of anything illegal.\"\n\nAnd unfortunately, neither would she. \"Don't worry, Brian. I don't want to go to jail for this, either.\"\n\nBut there had to be something else she could do....\n\nIf only the pain in her head would let up enough so that she could think. But the throbbing pain, much like the official, seemed determined to ruin her day.\n\nShe leaned back in the seat and watched the beautiful buildings and landscape of the town drift by while people went about their business on the sidewalks. How she wished she could be carefree enough to roam in and out of the stores, shopping and laughing like the majority of them. Unfortunately, she'd never once been a tourist anywhere.\n\nGeary Kafieri was always all work and no play.\n\nNeither of them spoke as the taxi wended its way through the narrow streets to the dock where their research boat was waiting. While Brian paid the fare, Geary got out and made her way up the gangway to face their team with her gloriously redundant failure.\n\nTory met her first. At fifteen and very average in height, Geary's cousin had long drab brown hair and thick glasses. She was an awkward teen who had more interest in her books than much of anything else. Even though Tory didn't remember her father, Theron, she was just like him. Finding Atlantis was her only ambition.\n\n\"Well?\" she asked, her young face expectant.\n\nGeary shook her head.\n\nTory let out an expletive that made Geary gape. \"How could they not let us excavate? What's wrong with those people?\"\n\n\"They think it's a waste of time.\"\n\nTory screwed up her face in distaste. \"That's stupid! They're stupid!\"\n\n\"Yes,\" Geary said drily. \"We're all stupid.\"\n\nTory scoffed at that. \"I'm not stupid. I'm a certified genius. But the rest... Stupid.\"\n\n\"I told you not to bother.\"\n\nGeary looked past Tory's shoulder to find her other cousin, Cynthia, joining them. Named for the Greek goddess of the hunt, Artemis, Thia hated everything to do with Greece. The only reason she was here was to get college credit and follow her latest fixation, Scott, who'd thought this would be a fun summer activity. Not to mention the small fact that had Thia stayed at home in New York, she'd have been forced to work in her mother's deli, which she hated even more than Greece.\n\nAt a cool six two, the titian-haired beauty was also one of the few women taller than Geary\u2014something that was quite a feat given the fact that Thia was barely eighteen.\n\nGeary frowned as she noted Thia's long blue skirt and white long-sleeved embroidered Grecian blouse. \"I thought you were sunbathing,\" Geary said.\n\nTory leaned forward to whisper in her ear. \"She was, and she took her top off earlier, hoping Scott would see her bare boobs and join her. He didn't, but the men on a passing boat almost fell overboard before Justina made her go belowdecks.\"\n\nThia curled her lip. \"You little nark. While you're confessing things, you should tell Geary how you almost set fire to her reports because her cat scared you and you knocked over Teddy's Bunsen burner.\"\n\nTory blushed before she pushed her glasses up on her nose. \"Genius, but not graceful. C'est moi.\"\n\nGeary smiled at the girl as Tory spoke the terrible truth. Grace had never been Tory's virtue, unlike Thia, who had more than her fair share. \"It's okay, Tor. I'd have just made you redo them.\"\n\nThia gave a heavy sigh as she cast her gaze around the deck. \"Is this not the most boring place on earth? I can't even get Scott to come up from below for more than a split second.\"\n\nObviously. If nudity didn't inspire the man to come up, nothing else would.\n\n\"He's down there with Teddy,\" Thia continued in an irritated tone, \"draped over an excavation map\u2014like that's ever going to happen. What is it about this godforsaken country that every time I bring a guy here he loses his mind?\"\n\n\"Maybe it's from being around you too long,\" Tory said, tucking a stray piece of hair behind her ear. She leaned forward to whisper to Geary in their own unique language of ancient Greek and Latin. \"I think she sucks the testosterone right out of them and then digests it for her own.\"\n\nGeary laughed.\n\nThia went instantly stiff. \"What did she say about me?\"\n\nGeary shook her head at Tory before she responded. \"Why does it always have to be about you, Thia?\"\n\n\"Because it is.\" And with that, she flounced off.\n\nTory let out a tired breath. \"One day I hope she finds someone who can put her in her place. I'm tired of watching her emasculate poor Scott. I swear she has to be part succubus.\"\n\n\"Oh, don't go there. I wouldn't wish her on anyone.\"\n\n\"Good point.\" Tory paused before she turned a probing stare on Geary. \"So tell me what happened.\"\n\nAs if she wanted to relive that misery. \"Not much to say. They refused to give us permits... again.\"\n\nTory actually stomped her foot. \"Ah, man. That's so not fair.\"\n\n\"I know,\" she said, patting Tory's arm. \"We just have to be patient.\"\n\n\"To heck with patience. At the rate they're going, I'll be in retirement and will have to dig with a cane.\" She let out a sound of supreme disgust. \"This is the closest we've ever been to finding the city. I know Atlantis is right there. I can feel it!\"\n\nA chill went down Geary's spine. Tory was just a little too close to their fathers in personality for her tastes. The same insanity that had possessed them drove Tory, too. It was like a madness in her blood that kept her working late into the night after everyone else had retired.\n\nThere were times when it truly scared Geary. All of the people in their family who'd ever shared Tory's level of dedication had met with an early death. It would destroy not only Geary but also their grandfather should anything ever happen to their youngest family member.\n\nShe was what they lived for.\n\nThen again, Geary had often suspected that Tory used it as a way to distract herself from the pain she felt at being an orphan. The poor thing had no memory of either of her parents. Their work was the only way Tory could feel close to them. It was all they'd left their daughter.\n\n\"It'll be all right, Triantafyllo.\" Geary used the nickname their grandfather had given Tory. \"Now I'm going to lie down for a bit and see if I can stop some of this headache that's brewing.\"\n\n\"Okay. I'll be below with Scott and Teddy reviewing the data that will be absolutely useless if we can't excavate. But what the heck? I'm young and have plenty of time to waste. You on the other hand...\"\n\nGeary blew her a raspberry. \"I'm not that much older than you.\"\n\nAs she sashayed off, Tory tossed back, \"Yeah, uh-huh. Get a cane, Grandma.\"\n\nGeary shook her head at Tory's play, then cringed as pain sliced through her brow and throbbed behind her eyes.\n\nBrian frowned as he joined her on deck. \"Are you okay?\"\n\n\"Another headache.\" She'd been getting a lot of those lately. Of course with her luck, it was an inoperable brain tumor and she'd probably end up at Thia's mercy so that her cousin could finally torture her without end... perish that thought. \"I'll be fine. I just need to lie down for a few minutes.\"\n\n\"If you need anything, call.\"\n\nI need a permit. Hello?\n\nIf only she could say that out loud and not lose her much-needed funding.\n\n\"I will. Thanks.\" And with that, Geary headed belowdecks to the small room she shared with Tory. There wasn't a lot of privacy on a research boat, but honestly it didn't bother Geary. Not like it had when she'd been Tory's age. The difference between them was striking. While Geary had hated the lack of personal space, Tory was ambivalent to it. All the girl cared about was their quest.\n\nBut even with their differences, Geary adored her cousin. Tory was the closest thing she'd ever had to a sister, and since Tory's parents had died before the girl reached six, their entire family had embraced her and raised her as their own.\n\nGeary smiled as she entered their room and found Tory's nightgown and matted old brown teddy bear tossed onto her bed. Tory wasn't known for her neatness.\n\n\"Okay, Mr. Cuddles, you have to stay on your side. Don't keep sneaking to my bed. I have a tendency to kick in my sleep.\" Geary set the bear on Tory's unmade bed, then folded the pink flannel gown before she placed it under Mr. Cuddles.\n\nA light smile toyed on the edges of her lips. She could hear muffled voices overhead as the boat rocked softly under her, lulling her into a bit of a stupor. She really did need some rest. Her sleep had been fitful lately. Probably a result of having too much on her mind.\n\nToeing off her shoes, she pulled back her spread and tucked herself into the narrow bed.\n\nShe fell asleep almost immediately.\n\nThe noises of the boat faded as she drifted through her dream darkness that was laced with white mist and a cooling breeze. Ever since she'd been a child, Geary had been able to rapidly fall into REM sleep... usually within five minutes, which was virtually unheard of. It was a strange sleeping disorder that no doctor had ever been able to explain.\n\nHer dreams flowing, she found herself standing on a dark beach where the snowcapped waves crashed against a foreign shore. The sound echoed in her ears as she curled her bare toes into the wet, black sand.\n\n\"Megeara.\" The deeply masculine voice was warm and erotic as it rolled with an accent so exotic and strange that it whispered through Geary like brandy-laced hot chocolate. Rich. Smooth.\n\nIntoxicating.\n\nShe groaned in her dreams as her mysterious lover appeared behind her. As always, he was breathtakingly handsome, with long black hair that tangled in the wind and with eyes so clear and blue, they seemed to glow. Every angle of his face was perfectly sculpted, and those mesmerizing eyes were set off by a pair of black brows that slashed above them. He wrapped his tanned arms around her and pulled her back against his wickedly bare chest that dipped and curved with perfect muscles.\n\nHe was divinity.\n\nAbsolute seduction.\n\nAnd for the moment, he was all hers....\n\nClosing her eyes, she allowed the scent of his raw, earthy maleness to seep into her until it made her completely drunk with pleasure. She leaned her head to the left as his hot lips brushed against her neck so that he could tenderly lick and caress her skin until her entire body burned.\n\nShe didn't know why she kept having these wildly erotic dreams. Why this incredibly sexy man haunted her. After all, Geary Kafieri wasn't known for her sensuality or femininity. Geary was as tough as nails. She'd spent her entire life fighting for her own beliefs, fighting to be her own person, and those battles hadn't left her time to cultivate the more girlie pastimes of makeup, hair, and feminine wiles.\n\nFrom the moment she'd started down this path of restoring her father's reputation, she'd been trying to prove to herself and to her colleagues and investors that she could not only compete in a male-dominated field but also rule it.\n\nAnd she'd succeeded admirably. So what if she wasn't the most ladylike of women? She had her accolades and had taken her father's failing company and turned it around in less than three years after his death. Kafieri Salvage was now one of the top companies in Greece, and while she'd rebuilt her father's company, she'd managed to carry on his private research.\n\nThat had always been enough for her.\n\nOr so she'd thought until that sultry night two months ago when Arikos had first appeared in her dreams.\n\nFrom the moment she'd laid her unconscious eyes on him, she'd been captivated.\n\nHe turned her around in his arms to face him. Biting her lip, Geary looked up into his searing blue eyes. He wore a pair of black leather pants and boots with nothing else. His wavy hair flowed around his face as the gentle breeze teased it, and strands of it became caught in the whiskers of his cheeks.\n\n\"What has you upset today, agamenapee?\" he asked in that tone that never failed to send a shiver over her.\n\nGeary leaned her head against the hollow of his muscular shoulder so that she could just breathe in his scent and let it soothe her.\n\nIf only he could be real...\n\n\"They won't give us our permits,\" she whispered, tracing the outline of his nipple and watching while it drew taut. \"And I could kill them for it. I know we've found Atlantis. I know it. I'm so close I can taste it and now... Now it's hopeless.\"\n\nShe ground her teeth in frustration, grateful that she had someone she could trust without having to put on her \"game\" face. Her staff expected her to be calm and collected at all times when what she really wanted to do was shake the official until he gave her what she needed.\n\nDamn them for it.\n\n\"I'm going to fail,\" she said, her voice catching. \"At the rate we're going, Tory's right. We'll be too old to even remember what it was we were looking for.\"\n\nArikos cupped her face in his large hands and stared at her with a frown. \"I don't understand why this is important to you.\"\n\n\"Because my father died a broken alcoholic. I want everyone who ever laughed at him to have to eat their words. I want to prove to the world that my father wasn't a fool racing against windmills. I want to keep my promise to him. I owe it to him.\"\n\nArikos tilted his head and stared into her eyes as if he could see straight into her soul. \"Finding it would make you happy?\"\n\n\"More than anything.\"\n\n\"Then it is done. I will take you to Atlantis.\"\n\nShe laughed at the absurdity. Boy, when her subconscious went off into outer space, it really went off into outer space.\n\nEven so, it meant a lot to her to have at least one person's faith. It didn't matter that he wasn't real. She needed his hypothetical support and she was grateful for it.\n\nArikos dipped his head down then and captured her lips with his. Geary moaned at the sweet taste of him. There was no one on this earth who tasted the way he did. No one who felt better in her arms, which was probably why he was relegated to her dreams.\n\nBut she was so glad to have him here. To feel the heat of his skin sliding against hers.\n\nOh, she could eat this man alive.\n\nHis hands deftly slid the white, flowing dress off her shoulders until she was naked before him while he nibbled and teased her mouth with his lips and tongue. It amazed her that she was so at ease with him, even in her dreams. In real life, Geary had never been the kind of woman to let any man whisk her off her feet. To let her passions rule her.\n\nShe was a woman of cold, hard logic and restricted emotions.\n\nIt was why she loved her dreams so much. Here she was free to have her way with Arikos without worry. There was no risk of pregnancy or disease. No worry about having to face him in the morning.\n\nNo risk of disappointment or cruel laughter. She was in control of her dreams and him. Her times with him were safe and warm and the best moments of her day.\n\nHe laid her down carefully on the sandy ground and covered her body with his. Oh, the feel of him like this was incredible. The leather of his pants caressed her legs as he separated her thighs with his knee.\n\nHe moved his mouth from hers, down toward her swollen breast that ached for his kiss.\n\nBreathless and weak, she cupped his head to her as he flicked his tongue back and forth over her taut nipple. His breath was scorching against her flushed skin.\n\n\"That's it,\" he breathed as he dipped his hand down to ease the ache that wanted him inside her. His warm fingers stroked and teased her until she was cresting an orgasm. \"Give me all of your passion, Megeara. Let me feel your pleasure. Let me taste it.\"\n\nKissing him wildly, she thrust her hips against his hand, seeking even more pleasure from him. \"I want more,\" she demanded, reaching for his zipper.\n\nHe laughed wickedly. \"And you will have it.\"\n\n\"Geary!\"\n\nThe loud shout jolted her from her dream and left her heart racing even faster than Arikos had. Geary opened her eyes to find herself lying facedown on her bed.\n\nTory burst into the room. \"You better come quick. Thia is about to drown Teddy. And I'm not kidding.\"\n\n* * *\n\nArik pulled out of the dream with a curse as he hovered in the strobilos that gave him no form or substance while he spied on the human realm. Whenever a person awoke from their dream, it left a dream god in a vast nothingness. There was no sound, no color, nothing but blackness.\n\nAll he could feel was her fleeting emotions, and he was desperate to keep those.\n\n\"Megeara...\" Arik called, wanting to return to what they'd been sharing. But he knew it was too late. His little fixation was stronger than the average human and didn't always come to him when he called for her.\n\nNot even the Lotus serum could induce her to sleep until she was ready and agreeable. All it did was give her a headache as she fought against it.\n\nDamn it. He wanted her back!\n\nHis body was aching with unsated need, but more than that he felt something strange in his chest.\n\nGrief.\n\nHe craved her and he was angry at her loss. Never once in all the history of the human world had he felt anything like this. Dream gods were supposed to be devoid of emotions... at least all except for pain. That emotion alone had been left for them so that the other gods could control and punish them.\n\nOnly he didn't feel pain in his chest. He could still feel Megeara's emotions as his own, which told just how powerful her repressed passion and anger were.\n\nShe'd started out as a passing curiosity for him. Her dreams had been vivid and colorful\u2014two things most people's were not. The average person dreamed in black-and-white with a lot of mist.\n\nMost dream gods avoided those, especially the erotic Skoti like him who were ever questing for the more daring humans. Why dance in the dreams of an unimaginative person when the point was to experience feelings and senses through the sleeper?\n\nSo his kind skipped through dreams, seeking those who could create beauty and give to the Skoti what they needed.\n\nMegeara's dreams had been awash in clever sensations. He'd first come upon her while she was bathing in a river of chocolate.\n\nRolling over on the misty ground that made up one of the dream chambers, Arik closed his eyes to summon the memory. There were still the remnants of Megeara's passion inside him even though their dream connection was severed, and that allowed him to remember the pleasure of finding her that first night.\n\nEven now he could taste the dream chocolate on his tongue as he'd licked it from her naked body. Feel the warm sensation of it sliding over his skin as they made love in it. He wondered now as he'd wondered then what that chocolate would really taste like on the mortal plane.\n\nWhy had it given Megeara so much pleasure?\n\nMost of all, he burned to know what she would really taste like. How she would smell.\n\nHis cock twitched in sweet expectation.\n\n\"Arikos?\"\n\nHe turned his head away as a bright light burst into his darkness. \"Fuck off, M'Ordant,\" he snarled, recognizing the voice of his older half brother.\n\n\"Is that anger?\"\n\nArik pushed himself up and moved to stand beside the god who was equal to his height. Like him, M'Ordant had black hair and translucent blue eyes. All of their race were marked by those colorings, along with an unearthly beauty.\n\nThis time when Arik spoke his tone was flat and level, as was befitting one of his cursed species. \"How would I know? I have no emotions.\"\n\nM'Ordant narrowed his gaze and if Arik didn't know better, he'd swear his brother was puzzled. Even though they couldn't truly feel, they'd learned to mock expressions. It made the other gods less nervous around them. \"You've been spending too much time with the human. You need to move on to another.\"\n\nThat was the way of it. A Skotos such as Arik was only tolerated to help drain humans of excess emotions. If Skoti spent too long with one person, they could, in theory, make the person go mad or even kill them.\n\nThe Skoti were normally given a single warning and if they failed to heed it, an Oneroi would be selected to either punish or eliminate them from existence. M'Ordant was one of many who monitored human sleep and who kept the Skoti in line.\n\n\"And if I don't want to leave her?\"\n\n\"Are you being argumentative?\"\n\nArik gave him an arch stare. \"How could I be?\"\n\n\"Then you are done with her.\" M'Ordant vanished.\n\nThe wisest course of action would be to heed his warning. But Arik was too drawn to his human to pay attention to M'Ordant's words. After all, that would require fear... something Arik knew nothing about.\n\nClosing his eyes, Arik could still smell the scent of Megeara's flesh. Still taste the salty sweetness of her body on his hungry tongue. Feel her touch on his skin.\n\nNo, he wasn't done with her. He was only beginning.\nCHAPTER 2\n\nLeaning against the side of the boat so that she could watch the nearby sailboats gliding over the clear blue water, Geary didn't know what was wrong with her. She was so sleepy she could barely keep her eyes open, and that wasn't like her.\n\n\"I think I have narcolepsy.\"\n\nTory paused by Geary's side before looking her up and down. \"Possibly. Did you know that seventy percent of people with narcolepsy also suffer cataleptic attacks?\" Before Geary could open her mouth to comment, Tory refuted her own theory. \"That's not you, though. I've seen you angry enough times to know that that lovely symptom doesn't affect you. Of course, narcoleptics also have vivid hallucinations either while asleep or even while awake. And, of course, sleepwalking. I know you don't sleepwalk. Have you become delusional lately?\"\n\nYes, but discussing vivid sexual fantasies with a fifteen-year-old bookworm wasn't something Geary intended to do.\n\nGeary scowled at her. \"How do you know all this? Jeez, Tory, you're a kid. Act like it.\" Before she could even blink, Tory reached out and punched her on the arm. Hard. \"Ow!\" She rubbed her biceps where Tory had hit her. \"What was that for?\"\n\n\"Unexpected and irrational emotional outbursts. Isn't that what teenagers are supposed to do? Oh, and sulk. A lot.\"\n\nGeary held her hands up in surrender. \"Fine. Have it your way, Dr. Kafieri.\"\n\nIn an expression more akin to her age, Tory gave her a gleeful grin before she went to help the boat's captain with some line he was securing.\n\nShaking her head, Geary headed back belowdecks to find Teddy and Scott, who were grumbling about Thia's presence on their team while they worked... something Geary couldn't help since she'd promised Thia's mother to watch after her this summer. Apparently, the little shrew had attacked Teddy for taking up too much of Scott's time.\n\nGeary hoped they'd get over their ire soon. She had banished her cousin to the city for a short round of shopping while they prepared to sail back to the area where Geary believed Atlantis was hiding. The last thing they needed was to have Thia gaggling about, complaining over everything.\n\nBesides, Thia lived for shopping. The shinier the object, the more she adored it. So much so, the girl had actually worn red horns last Halloween that were decorated with dangling diamond hoops. As was fitting for her, Thia had dressed up as a shopping demon.\n\nBrian had volunteered to accompany her and keep her out of trouble\u2014which, knowing Thia, was a necessary assignment. Their luck, she'd end up being either abducted into white slavery or stolen by green aliens.\n\nMeanwhile, Geary was so tired, she couldn't stand it. It was all she could do to stay awake.\n\n\"Megeara. Return to me....\"\n\nA chill went down her body as she heard the deeply erotic voice in her head again.\n\nFrom the corner of her eye she saw something move. She turned and there in the doorway, on the stairs that led to the top deck, was Arikos. Dressed completely in black, he stood to the side with wicked eyes that promised her an endless night of orgasm, and a seductive smile that froze her to the spot.\n\n\"Come, Megeara.\" His voice whispered like a phantom wind, caressing her. Lulling her.\n\nHe held his hand out...\n\nNever had she seen a more compelling pose. All she wanted was to take his hand and have him sweep her up in his arms like he did in her dreams. She wanted to strip him bare and taste the perfection of his body.\n\nTo taste those inviting lips.\n\nWithout thinking, she held her hand out toward his. So close that they were almost touching. Only a breadth of a hair more...\n\nBut he wasn't real and she knew it.\n\n\"Geary? Can you hand me my ruler?\"\n\nShe started at Teddy's voice. Dropping her hand, she glanced to her left and saw the ruler on the cluttered desktop. She barely blinked before she looked back toward the stairs.\n\nThey were empty, with no sign of Arikos waiting there for her to return, and that filled her with extreme disappointment.\n\nI am losing my mind.\n\nYeah, but what a way to go. Everyone should be stalked by such a sexy hallucination.\n\nNot wanting to think about that, she picked up the ruler and handed it to Teddy, who was watching her with a concerned frown. Even though he was only a few years older than her, he acted more like a father to her than a friend or colleague. His short brown hair was always impeccably combed, and he had jolly brown eyes and a sweet set of kind dimples. \"Are you okay?\"\n\n\"Tired.\"\n\nHe scratched his head as if baffled by her response. \"You slept fourteen hours last night.\"\n\nShe patted his arm. \"I know, but I'm still tired.\"\n\n\"Maybe you need a physical.\"\n\nMore likely my head examined. She pushed that thought aside to smile at him. \"I'll be fine. Really.\"\n\nAt least she would be if she could stop having these bizarre delusions. Even now she felt as if someone was watching her...\n\n* * *\n\nArik wanted to curse in frustration as he watched Megeara smiling at another man. Why wouldn't she succumb to his serum? To his pleas?\n\nHow could a mere mortal woman be so strong?\n\n\"Arikos?\"\n\nAs light intruded into his dark chamber once more he let out a tired sigh at the sound of his uncle Wink's voice. Arik was getting seriously tired of these interruptions when all he wanted was to be with his human target. \"What?\"\n\n\"I've been told to retrieve my sleeping serum from you. You seem to be abusing it and making your human ill.\"\n\nArik rolled over to face the older sleep god. Wink's long brown hair was braided down his back as his light gray eyes danced with mischief. Even though Wink was one of the oldest gods, he held the personality of a thirteen-year-old boy. There was nothing he loved more than to play pranks and tease\u2014two of the very things that had gotten Arik and his brethren cursed.\n\nAt one time, they'd been too easily seduced and manipulated by the other gods and had allowed themselves to be used by Wink, Hades, and the others in private jests and wars.\n\nUntil the day Zeus had put a stop to it once and for all. Funny how he'd only punished the tools and not the ones who'd wielded them.\n\nBut then, Zeus wasn't known as a god of justice.\n\n\"And if I want to keep the serum?\"\n\nWink arched a brow at that, then tsked. \"Come now, Arikos, you know the rules.\" His face sobered. \"You also know what happens to those who don't cooperate.\"\n\nOf course he did. All of his kind knew. His back bore more scars than the sky held stars. There were times when he suspected his grandfather Hypnos, who oversaw their physical punishments, was nothing more than a sadist who could only feel pleasure when he was doling out pain to others.\n\nHow cruel was it to send the Skoti in to drain humans of excess or pent-up emotions, then punish them when they didn't want to leave because they finally experienced something other than pain?\n\nBut that was the way of it.\n\nAfter his \"chat\" with M'Ordant, Arik had known it would come to this. There was no use in arguing. Wink had been sent to retrieve the Lotus serum they used on humans, and all the bribery on Olympus wouldn't sway him. Wink was only a pawn who served the sleep gods.\n\nArik pulled out the small vial and handed it to Wink, who took it with a stoic smile.\n\n\"Cheer up, old boy. There are plenty of other dreamers out there for you to play with. Mankind is generous to you that way. They live for their dreams and are possessed of them constantly.\"\n\nYes, but none of those humans held the type of uninhibited, vivid dreams of Megeara. It made Arik long to know what she'd be like outside the dream realm. What she would be like as a human...\n\nArik watched as Wink withdrew, then left him in the dream chamber to face the darkness alone.\n\nPerhaps this was just punishment after all. A son of the god Morpheus, Arik had originally been one of the Oneroi. As was customary for such, he'd been assigned humans to watch over and protect against the Skoti who sometimes preyed on them. In those days, he'd spent his life monitoring his subjects, making sure the ones under his protection had normal dreams that would either help them work through their problems or inspire them.\n\nUntil that one fateful night.\n\nHe'd gone to help one of his assignments who was ill. Because of her sickness, her dreams had become extremely vivid and emotional\u2014so much so that one of the Skoti had latched onto her. Such a thing was common and even tolerated. Skoti fed from human emotions, but so long as they kept it under control and didn't lead the dreams or interrupt the human's life, they were allowed to drain humans. It was only when the Skoti began to return repeatedly and took control of the host that they were punished.\n\nHumans held fragile psyches. A returning Skotos could easily turn human minds and either drive them insane or make them homicidal. In the worst case, a Skotos could even kill the human, which was why the Oneroi monitored them. If a Skotos spent too long with their host, then it was the Oneroi's place to step in and drive them out.\n\nIf all else failed, the Oneroi would kill the Skotos.\n\nAt one time, Arik's life had been dedicated to protecting his humans. To feel nothing and to only follow the orders of the elite Oneroi. In his day, he'd vanquished numerous Skoti without understanding or caring why they sought humans the way they did. Why they felt a burning need to risk their lives for their quest.\n\nAnd then one night... no, one encounter had changed that and brought with it a clarification that still resonated within him.\n\nBorn of a human mother and the dream god Phobetor, Solin lived on earth, but at night he ran amok in the dreams of other humans. Completely amoral, he didn't care what he did to others so long as he enjoyed himself.\n\nFor centuries, the Oneroi had been trying to stop and trap Solin. He was one of the few Skoti who'd warranted a death sentence. His voracious appetites and fighting skills were legendary among the Oneroi who'd been unfortunate enough to confront him.\n\nAnd Arik had been one of them. Still young by their years, Arik had thought to take Solin on his own.\n\nMost of the Skoti fled at the approach of an Oneroi. The Oneroi had full backing of the other gods to do whatever they had to do to control the Skoti. Since a Skotos could drain emotions from any human, they normally left without issue and didn't waste time fighting when they could simply move on to someone else.\n\nBut Solin was stronger than most. Bolder. Instead of fleeing as Arik had expected, Solin had turned the human loose on him. By their laws, Arik had been forbidden to hurt the human, and Solin had known it. Arik had tried to pry her away without harm, but the moment her lips had touched his and he'd tasted her lust, something inside him had shattered.\n\nHe'd felt pleasure and arousal for the first time in his life.\n\nAnd when the human had dropped to her knees and taken him into her mouth, he'd known his war in this was lost and his conviction shattered. In one heartbeat, he'd gone Skoti.\n\nHe'd been Skoti ever since.\n\nDrifting from one dream to the next, he'd been searching all these centuries for someone who could raise his emotions to the level of that first night. But no one had come close.\n\nNot until Megeara.\n\nOnly she was able to reach through the emptiness inside him and make him see vivid colors again. To make him feel her emotions. After all these centuries, he finally understood why certain Skoti refused to leave their partners.\n\nWhy they were willing to risk death.\n\nBecause of Megeara, he wanted to know what the world looked like through her eyes. What it tasted like. Felt like. And her ability to pull herself away from him was starting to seriously piss him off.\n\nBut what could he do? Even if he went to earth to be near her, he couldn't really experience her or her environment.\n\nHe wanted her passion. Her life force.\n\nThere might be a way to touch her...\n\nArik paused at the thought. It was true that both the Oneroi and Skoti could take human form in the mortal realm, but because of their curse, they still lacked emotions. So what was the point? They were just as cold and sterile and unable to feel in human form as they were in their own god form.\n\nThat wasn't what he wanted.\n\nNo, he wanted to be human. He wanted feelings and emotions so that he could experience her to the fullest extent possible.\n\nIt's impossible.\n\nOr was it? They were gods, with god powers. Why should such a thing be unattainable?\n\nYour powers aren't capable of such. Zeus had made sure of that when he punished them for tampering with his dreams.\n\nThen again, Arik's weren't. But there were others whose powers made a mockery of his. Gods who could make him human if they willed it.\n\nZeus would never concede such a thing\u2014he hated the dream gods too much. His children would be too afraid of him to try. But his brothers...\n\nThey were a different matter entirely.\n\nAnd Arik knew which one to barter with.\n\nHades. The god of the Underworld held no fear of anyone or anything. His powers were more than equal to any of the others', and best of all, he hated the other gods as much as they hated him. Because of that, Hades was always open to a good bargain, especially if such a bargain would irritate Zeus.\n\nIt was at least worth a shot.\n\nWith Megeara's niggling emotions retreating from him, Arik flew from the Vanishing Isle where most of the dream gods resided and descended down, straight into the heart of Hades' domain. It was dark as night here. Dismal. There were no ivory or gold halls like the ones found on Olympus. At least not until one visited the Elysian Fields, where good souls were sent to live out their eternity in paradise. Those lucky enough to attain residence there had any- and everything their hearts conceived of. They could even be reincarnated should they choose it.\n\nBut the Elysian Fields were only part of a much vaster realm. One that held nothing but misery for those who were damned to it. Especially this time of year. Three months ago the god's beloved wife, Persephone, had been sent to live with her mother in the upper realm. Until Persephone's return, Hades would be literally hell to deal with. From the moment she left until her return, he would spend his time torturing all those around him.\n\nA saner god would wait to try to deal with Hades after Persephone's return, when he was more reasonable, but Arik was desperate. The last thing he wanted was to take a chance on another Skotos finding Megeara.\n\nNo, it was now or never.\n\nBesides, Arik had never been a coward. He'd never once retreated from battle or conflict. It was what had made him one of the best of the Oneroi and what had made him one of the deadliest Skoti.\n\nHe always took what he wanted. Damned be the consequences. He had eternity to deal with those. What mattered most was the present, and that was what he focused on. Always.\n\nAs he flew past Cerberus, the three-headed dog rose up to bark at him. Ignoring it, he dove down into the catacombs made of the skulls and bones of Hades' enemies. Many of whom had been Titans and ancients who'd had the misfortune of irritating the somber god\u2014they didn't even warrant Hades torturing them for eternity. He'd relegated them to nothing more than decoration.\n\nThat alone should be a warning to Arik...\n\nBut the brave and the desperate never heeded such.\n\nArik slowed his flight as he entered the main chamber of Hades' domain. This was the only room in Hades' opulent palace that was open to outsiders.... But there was a lot more to his home than this one room.\n\nArik knew that because no one was immune to the powers of a Dream-Hunter. No one. All gods were vulnerable whenever they rested, which was why they feared the Dream-Hunters so, and it was times such as those that Arik had ventured here to see what Hades kept so secret.\n\nNow Arik faded to invisibility and rose up toward the black ceiling that glittered eerily in the dim light. Hades sat below, alone, on his throne. Made of Titan bones, his black throne had been polished until it gleamed like steel. Hard and intimidating as the god had intended, it dominated the dais where it sat. Beside it was a much smaller chair. One made of gold and cushioned with pillows the color of blood. It was where Persephone sat whenever she was home with her husband.\n\nHades stared at her throne with a look of such longing that Arik could almost feel his grief. And it wasn't until Hades moved that Arik realized the god held a small, delicate fan in his hand. One made of lace and ivory.\n\nClosing his eyes, Hades held it to his nose and gently inhaled the scent.\n\nThen he cursed and tossed the fan back to the throne by his side.\n\nA heartbeat later, he got up to retrieve it and place it more carefully in a small holder on the right arm. Obviously that was where Persephone kept it.\n\nHades froze and cocked his head as if he was listening for something. \"Who dares to enter my hall without summons?\"\n\nArik lowered himself to the floor and materialized. \"I do.\"\n\nThe god turned about slowly and narrowed his amber eyes on Arik. \"What brings you here, son of Morpheus?\"\n\nThere was no need to hide what he wanted. \"I would like to bargain with you.\"\n\n\"For what?\"\n\n\"I wish to be human.\"\n\nHades' evil laughter rang out in the hollow hall, echoing around them. \"You know how to be human, Skotos. Stop eating ambrosia and drinking nectar.\"\n\n\"That would only make me mortal and I don't want to die. I want to feel, and for that I need to be a human and not a god.\"\n\nHades approached him slowly until he stood just before Arik. \"Feel? Why would anyone in their right mind wish for that? Feelings are for fools.\"\n\nArik glanced to the fan. \"Even you?\"\n\nHades bellowed in rage as he flung out his hand and pinned Arik against the wall with his powers. The jagged bones bit into Arik's back, tearing the fabric of his clothes.\n\nArik fought the hold, but there was nothing he could do at the moment except bleed.\n\n\"For a god who doesn't wish to die, you speak of things you'd best not address.\"\n\nThe force holding him receded so fast that he barely had time to recover himself before he fell. He hovered over the floor for a heartbeat until he placed his feet on the ground.\n\nHades raised his brows in surprise. \"You're faster than most.\"\n\n\"And in my realm, I'm capable of even more feats.\"\n\n\"What are you saying?\"\n\nArik shrugged. \"Only that a god of such power should be careful. Even the great Hades has to sleep sometime.\"\n\n\"Are you threatening me?\"\n\n\"I'm only stating a fact.\" Arik looked pointedly at Persephone's throne. \"And reminding you, my lord, that there's nothing worse than allowing a Skotos to know of a weakness.\"\n\nHades narrowed his eyes before he again broke out into laughter. \"It's been a long time since anyone dared such boldness in my presence. Look around you, Skotos. Do you not see the remains of the people who have pissed me off?\"\n\n\"My name is Arik and I see everything, including the beauty and comfort of the palace you hide behind this facade of death. But in turn, I would ask you what good does it do to threaten someone who can't feel fear?\"\n\nHades inclined his head. \"Point well taken. So tell me... Arik, what bargain do you wish to propose?\"\n\n\"I want to live in the realm of the humans as one of them.\"\n\nHades tsked at his request. \"That's not so easy to attain, dear boy. No Olympus-born god can live on earth for very long.\"\n\n\"But we can live there for a time. I would go there now, but there would be no point, since I could only witness what's around me and not experience it. It's the experience I want.\"\n\n\"What good is this experience when you'll only forget it once you return?\"\n\nWhat the god didn't know was that Arik wouldn't forget. He'd remember and he wanted that memory. Unlike M'Ordant and many of the others, Arik had no knowledge of true emotions or sensations\u2014they'd been beaten out of him so long ago that he'd completely forgotten what it was like to feel. He wanted to know how much more intense feelings could be when not blocked by the curse.\n\n\"Does the why really matter?\"\n\nHades considered that for a moment. Folding his arms over his chest, he frowned at Arik. \"For what you want, there would have to be a steep price.\"\n\n\"I expected nothing less. Just tell me your fee.\"\n\n\"A soul. A human soul.\"\n\nThat was easy enough. Taking a human life wouldn't bother him. They lived finitely anyway and very few of them even bothered to appreciate the beauty that was the human existence. He, however, would savor his brief time as one of them. \"Done.\"\n\nHades clucked his tongue at Arik. \"Child, how naive of you. You agreed too soon. It's not just any soul I want.\"\n\n\"Whose then?\"\n\n\"I want the soul of the woman who has compelled you to make a deal with the devil. Surely she must have a magnificent soul for you to come here and barter with me, the most despised of all gods.\"\n\nArik hesitated. Not out of feelings for Megeara but rather because he wasn't sure he would be through with her by the time he was forced to return. \"And if I fail to complete this bargain?\"\n\n\"It will be you who suffers here in her stead. If you fail to deliver her to me, I will kill you as a man and keep your soul in Tartarus. The pain you've felt to date will be nothing when compared to what you'll suffer then. And before you reconsider, remember that you've already agreed to this. There is no going back now. Our bargain is set.\"\n\n\"How long will you give me?\"\n\n\"Two weeks and not a day more.\"\n\nArik had no time to even twitch before a strange thick blackness covered him. One moment he was standing in the middle of Hades' throne room, and in the next he was encircled by wetness.\n\nIt was water...\n\nAnd unlike in dreams, his body was heavy. Leaden. Water poured in through his mouth and nose, causing him to choke as it invaded lungs that weren't used to really breathing. He tried to swim, but the water was too thick. It seemed to be sucking him down deeper into the sea.\n\nPanic consumed him. There was nothing he could do.\n\nHe was going to drown.\nCHAPTER 3\n\n\"Geary, quick! There's a body overboard!\"\n\nOh good God, who had Thia attacked now?\n\nAggravated, Geary looked up from Tory's notes at Justina's call. Geary's second in command was pointing over the side of the boat. As Geary rushed to the side to peer over, she handed the notebook back to Tory. Sure enough, there was someone struggling in the waves. And by the looks of it, he was quickly losing his battle.\n\n\"Christof!\" Geary shouted for the boat's captain. \"We need...\" She paused as the body sank down below the hungry waves.\n\nThere wasn't time.\n\nHer heart pounding from the rush of adrenaline, Geary kicked her shoes off and dove over the side. The coldness of the water stunned her as it covered her completely. Kicking her legs, she swam upward until she broke the surface so that she could look about for him.\n\nEven though the water was clear, Geary had a hard time finding the guy below the surface. She had to keep diving down, then returning for fresh air before she dove back to search for him. Thank God she was a strong swimmer who was trained as a lifeguard and a certified diving instructor. But then, it was expected of her as an underwater recovery expert. She had to be as nimble in water as a fish.\n\nShe just wished she'd had time to get her gear before she'd come in after him. If she didn't find the guy soon, he'd be dead, especially since he hadn't resurfaced.\n\nHer lungs burned from holding her breath as she dove under the water again. Her ears were buzzing and popping from the pressure as images of him drowning consumed her.\n\nGeary had been twelve years old when Tory's father had drowned only a few miles from this very spot. Images of her father trying to save Theron's life tore through her now as she remembered her father diving for him. Her father pulling Theron out of the water and doing everything he could to resuscitate him.\n\nIt'd been awful and the last thing she wanted to do was relive it.\n\nC'mon. Don't you dare die on me. Where are you? She slowed her speed and turned about as she floated weightlessly in the sea. The light refracted and danced in the blue and green water, highlighting various fish and foliage, but there was no sight of the man she sought.\n\n\"Look down.\"\n\nShe frowned at the foreign voice in her head, not understanding the source of it, but she couldn't help obeying it. Looking down, she spotted him just below her. Even though he was trying to swim, he was sinking fast....\n\nHis long black hair danced in the water as bubbles floated around him and he waved his arms and legs to no avail.\n\nRelieved she'd found him but scared it might be too late, she headed for him as fast as she could. She came up behind him, then pulled his large body against hers and kicked them toward the surface.\n\nGood grief! The man was huge and made of solid muscle. With next to no fat on him, he was like an anchor in the water. It took a great deal of effort to get them to the surface.\n\nBy the time they broke through, both of them were sputtering and coughing.\n\n\"Hold on,\" she said to him. \"I've got you.\" Even so, she half-expected him to fight against her. Most drowning victims did.\n\nBut not him. He went limp against her as if he trusted her completely.\n\nJustina and Teddy were in the water already with a life preserver. Together, they got the man into the harness and had him hauled on board, then they followed suit.\n\nBy the time Geary was on board the Simi again, she saw the unknown man lying on the deck, covered with a blanket, while Thia was giving him mouth-to-mouth. Geary couldn't see the man's face for Thia.\n\n\"Is he dead?\" Geary asked, rushing over to them as worry tore at her.\n\nJust as she reached his side, the man coughed up a gallon of seawater. Gasping, he turned quickly to his side and started hacking and wheezing while Thia pounded him on the back to help him clear his lungs. His slick wet skin was completely bronzed and perfect, except for the deep welts that marred his back. The scars were old, but even so they were prominent enough to let Geary know how much they must have hurt when he received them. It reminded her of the way sailors were beaten for punishment back in the old days.\n\nWhy would a modern man have such scarring? Who would have beaten him like that and why?\n\nAnd he wore nothing except a thin pair of long white pants that were plastered against his perfect body... and they showed absolutely everything, right down to his religion and the fact that this man had been rather gifted in a certain department.\n\nHe might as well be naked.\n\n\"Now there's a man who doesn't believe in underwear, huh?\" Justina said in a low tone for only Geary's hearing as she wrung out her hair. \"Not that I'm not grateful for it. He has the nicest ass on the planet. No wonder Thia grabbed him for resuscitation. I wouldn't mind a little mouth-to-mouth action with that body, either.\"\n\nWhile Geary pretty much agreed with those sentiments, she didn't comment as Tory draped a blanket around her shoulders.\n\n\"Hell of a fish you found there,\" Christof said as he brought more blankets for them. He gave one to Justina and Teddy.\n\nIgnoring him, Geary knelt down beside her catch. The man held himself up with one muscular arm as he continued to breathe in short, sharp gasps. His tangled wet black hair fell over his face, completely obscuring it from her and the others. The tendons of his hands were well defined and beautiful, which made her curious as to what his face would look like.\n\nWould it be as scarred as his back or as pristine and beautiful as the rest of him?\n\n\"Are you okay?\" she asked in Greek, assuming since they were in the Aegean that he would understand her better in Greek than any other language.\n\nHe nodded as he continued to struggle to expel the water from his body. It was almost as if he wasn't used to his own lungs.\n\nHis breathing ragged, he lifted his head to look at her through the strands of his wet black hair. And as soon as their eyes met, Geary gasped and fought the urge to cross herself and spit as she came face-to-face with the intense blue eyes of her dreams.\n\nIt couldn't be....\n\nIt wasn't possible and yet there he was before her in all his almost naked glory. She knew those perfect, sardonic lips. The slash of his dark brows over eyes that were so pale a blue they radiated. She knew that strong jaw, dusted with whiskers. It was one she'd teased with her teeth and tongue for hours on end.\n\nAgainst all reason, it was him.\n\nSomething hot and needful went through her like a sharp needle as she fought the urge to reach out and touch him to make sure he was really here.\n\nArik couldn't do anything more than stare at Megeara. She was even more beautiful in reality than she'd been in her dreams. Her deep blue eyes captivated him as tendrils of her wet blond hair hung down over them. Her pale skin begged for his touch just as her partially opened mouth needed his kiss.\n\nHe started toward those lips, then coughed more as he tried to breathe through the stinging pain in his chest. His body shook uncontrollably as he was assaulted by horrifyingly intense sensations and emotions. Even the cries of the birds above him were piercing to his ears\u2014the droning of the ocean. And the heat of the sun on his skin... it was blistering. Never had he felt so out of control. Why wouldn't his body obey him?\n\nWhy the hell couldn't he stop coughing and shaking?\n\nHe half-expected Megeara to pound on his back as her accomplice had done. Instead, Megeara's touch was gentle as she lightly hit him to help dispel the water from his now human body.\n\nThen she started to gently rub his back in a circle. Chills spread over him as he felt a heat the likes of which were unimaginable. Forget the heat of the sun, this was even more scorching.\n\nNo one had ever touched him so gently and he'd never really felt a touch before, especially not against his flesh. All he wanted was to pull her into his arms and taste the taut nipples that were so apparent through her wet white shirt.\n\nIf only his body would obey him.\n\n\"I think he's going into shock,\" Megeara said to the others. \"Grab more blankets.\"\n\nAnother woman pulled Megeara away. \"Let me look at\u2014\"\n\n\"No!\" he snarled, reaching for Megeara's hand to keep her by his side. He hadn't come this far to lose sight of her now.\n\nMegeara covered his hand with hers in a soothing caress. \"It's all right. Stay calm.\" She took a blanket from a young woman with glasses, then wrapped it around him.\n\nArik closed his eyes and savored the fleeting sensation of her hands on his shoulders. The feel of her skin on his... it was electrifying. Hot.\n\nIf only he could stop shaking.\n\nGeary wasn't sure what to do. She exchanged a frown with Althea, who was on board as their physician.\n\n\"I need to check him out and make sure he's okay,\" Althea said in English.\n\nGeary agreed. \"I know.\"\n\n\"I'll be fine in a few minutes,\" the unknown man said in perfect, accented English. His voice was so deep and resonant that it literally echoed around them. Those intense, predatorial eyes pierced her. \"Just don't leave me.\"\n\nGeary found herself nodding even though the possessive command of that tone made her want to run. It wasn't in her nature to let anyone tell her what to do, but in his case, there was something unnaturally compelling about him. Alluring.\n\nHonestly, she didn't want to leave him. And that really did scare her.\n\nHer heart hammering, she used a corner of his blanket to towel dry his hair, then brushed it back from a face that was truly flawless.\n\n\"Do you prefer English or Greek?\" she asked him.\n\n\"It doesn't matter.\"\n\nWow. He was extremely bilingual. He was also extremely exposed, and the sight of him with those pants clinging to his every asset brought the most wicked images to her mind. In her dreams, she'd twisted that body of his like a pretzel and licked every inch of it.\n\nOkay, so it wasn't quite that body. In her dreams, there had been no scars. But his body was close enough to the one she was used to to evoke a fervent heat inside her.\n\nGeary brushed a drop of water off his cheek with the blanket. \"What happened to you?\"\n\nHe looked away. \"I don't know.\"\n\nThia gave her a wicked grin. \"Well, it isn't every day we fish a nearly naked god out of the sea, now is it? Glad I came back early from my shopping trip. This was definitely worth it.\"\n\nThe man snapped his head toward her and gave her a fierce scowl. It was obvious her words touched a nerve with him.\n\n\"Thia?\" Geary said in a steady tone. \"Do you mind?\"\n\nShe rolled her eyes. \"Whatever. See if I save his life next time he's drowning.\" Turning around, she headed belowdecks.\n\nChristof stepped forward. \"We should report this to the authorities.\"\n\nEven more fury snapped in those pale blue eyes. \"No!\" His tone was firm and commanding. \"No authorities.\"\n\nTeddy exchanged a frown with her. \"Why? You running from them?\"\n\n\"No. I just don't want to be interrogated when I can't remember anything.\"\n\nChristof narrowed his eyes on him. \"Do you know your name?\"\n\nHe hesitated. \"Arik.\"\n\n\"Arik what?\"\n\nHe looked up at Geary with a confusion that tugged at her heart. \"I don't remember.\"\n\nGeary tilted her head, not sure what to think. Something deep inside told her he was lying, but she wasn't sure about what. \"Did you hit your head?\"\n\nHe nodded.\n\n\"He could have amnesia,\" Tory said. \"If he fell from a boat he might have been run over by it. Or maybe he was beaten and then thrown overboard. Could be pirates.\"\n\n\"He's not bruised,\" Christof pointed out. \"And there hasn't been a lot of pirate activity here for several hundred years.\"\n\n\"Yes, but I said could. Weird and unusual things happen all the time. Did you know that there were seventy-five pirate attacks on civilian boats last year alone? Six more were against the U.S. Coast Guard. One group even tried to take over a cruise ship.\"\n\nIgnoring Tory's statistics, Geary dropped the blanket to Arik's shoulders. \"What was the last thing you remember?\"\n\n\"I... I don't know.\"\n\nA strange, warm feeling came over her as she watched him. The whole moment was so surreal. She couldn't believe she was looking at... Arikos.\n\nThat had been a dream and yet the man before her was an exact copy. A copy named Arik.\n\nCould they possibly...\n\nDon't be stupid.\n\nIt was just some strange coincidence. Maybe some sort of premonition.\n\nHer face flamed red at the thought. Well, not that kind of premonition. She wasn't about to jump naked into a pool of chocolate with this guy.\n\n\"Okay,\" she said quietly. \"Teddy, take Arik below and find some clothes for him.\"\n\nArik started to protest leaving her, then stopped himself. She was skittish of him. He could sense it. If he pressed her too much, she might bolt and push him away.\n\nThat was the last thing he wanted.\n\nNo, he must tread carefully in order to gain her trust. He was here, in her realm. And he'd have plenty of time to seduce her shortly. For now it was best to humor her.\n\nHe stood up slowly, his eyes never wavering from her gaze. As a wave crashed into the boat, he staggered slightly and almost lost his balance.\n\nMegeara reached out, her hands steadying him.\n\nArik closed his eyes as heat from her touch seared his every nerve. There was nothing to compare to the sensation of human contact\u2014to the feeling of those delicate hands touching his flesh\u2014and he couldn't wait to feel them stroking the part of him that was hard for her.\n\nHe bent his head low so that he could inhale her sweet, feminine scent of open air and woman misted with a light touch of perfume. It was even more intoxicating than it had been in her dreams, and he wanted to bask in it.\n\nEven more he wanted to smell it on his sheets and flesh. To drink her in for hours on end until he was fully sated and content.\n\nGeary tensed at the feel of Arik's hot breath against her damp skin. What was it about this stranger that set her entire body on fire?\n\nShe forced herself to step away from him even though what she really wanted to do was walk closer to that magnificently muscled body.\n\nHis eyes showed his longing as he met her gaze again and he noted her actions. \"Don't be afraid of me, Megeara.\" He all but purred in her ear. \"I would never hurt you.\"\n\nIt wasn't until he'd left that she realized he'd called her by a name no one used.\nCHAPTER 4\n\nArik cringed at the harsh sensation of denim sliding against his bare legs. The roughness of it was hard to take. How did humans stand it?\n\nThe man, Teddy, had loaned him a white shirt and jeans. But the texture of each was itchy and heavy. The clothing Arik was used to had no weight or texture. At least none that he could feel, and in dreams... well, since he was considered an Erotikos Skotos, clothing was seldom ever worn since all it did was get in the way of other, more pleasurable sensations.\n\nAfter fastening the jeans, he reached for the stiff white shirt at the same time the door burst open. He paused at the sight of Megeara standing in the low, narrow entrance-way, looking much like a puppy caught in a deluge. Her damp khaki shorts fell to her knees. She wore a baggy white shirt untucked that made her appear to be a blob of material. Or at least it would if it weren't wet. As it was, it left very little of her lush body to his imagination.\n\nIn this realm, she hid every indication of the full curves he knew she possessed. Even her thick, blond hair was pulled back severely from her face into a tight bun.\n\nBut her face was the same. Those intelligent, sharp and clear, almond-shaped eyes that took in the world around her. The light smattering of freckles across the bridge of her nose. And her lips...\n\nHe had spent entire nights kissing those luscious lips. Watching them dance over his skin as she nibbled and teased him until they were both blind with ecstasy.\n\nThe memory of it, along with the sight of her tight, puckered nipples straining against her shirt, made his entire body burn with hunger.\n\n\"How do you know my name?\" she demanded in an angry tone that was undercut by a note of apprehension.\n\nHe hesitated as he sensed her fear. He would have to play this carefully if he were to get what he wanted from her. He didn't know much about the human world, but he knew from dreams that scared hosts wouldn't let him touch them. So it only stood to reason that they'd be skittish in this realm, too. If he wanted her in his bed, he'd have to gain her trust.\n\n\"You told me.\" It wasn't a lie. On the night they'd met while they bathed in chocolate, she'd given it to him.\n\n\"No. I didn't. No one calls me Megeara. No one.\"\n\n\"What do they call you then?\"\n\n\"Geary.\"\n\n\"Then Geary you are.\"\n\n\"Yes, but that doesn't explain how you knew my name when I hadn't said it.\"\n\n\"Maybe I'm psychic.\" He'd meant that as a joke, but by the look on her face, he could tell she didn't find it amusing.\n\n\"I don't believe in psychics.\"\n\n\"Then how do you explain it?\"\n\nGeary narrowed her eyes at him. He was playing with her, and she didn't appreciate it in the least. \"Do I know you? Have we met?\"\n\nHe hesitated before he answered. \"There's no need to be afraid of me, Megeara. We did meet. Years ago when you were giving a paper at Vanderbilt.\"\n\nGeary frowned as she remembered the event clearly. That had been her first paper in public... ever. She'd been incredibly nervous. So much so that she'd stumbled on her way to the podium, dropped her pages and notes in front of everyone, and then spent ten minutes, red-faced, as she struggled to put them together again. She'd been halfway through the paper before she realized that one page had fallen underneath the heavy wooden stand and they had to stop everything again to retrieve it.\n\nThe event had left her humiliated as people laughed at her. After that fiasco, she'd been lucky anyone had ever invited her to speak on anything.\n\n\"I don't remember you.\"\n\n\"I was in the audience at the time. Dr. Chandler introduced us afterward, but we didn't really converse. You seemed a bit harried before you were pulled away by Dr. Chandler to meet her old college professor.\"\n\nShe vaguely remembered that part. The fact that he did lent credibility to his claim. It was true that she'd been preoccupied with saving some dignity at the mixer... still, a man this hot should have been branded in her memory.\n\nA teasing smile quirked one corner of his mouth up. \"You left quite an impression on me.\"\n\nShe had to bite back a laugh. Yeah, right. A guy like this would actually remember an overweight frump who'd embarrassed herself? \"I find that hard to believe.\"\n\nBut there was no laughter in his intense gaze. Only sincerity. \"You shouldn't. It's true.\"\n\nGeary frowned as she struggled to recall him from her past, but honestly, she'd been in such a fog that day that it was entirely possible they had met and she'd forgotten it. \"Why were you there?\"\n\n\"I was a student of anthropology. I asked you about Atlantis then and you were rather rude about it.\" The smile spread across his face as he teased her with his eyes.\n\nShe was still skeptical, but it did make sense. She would have taken his head off over Atlantis back then. It would also explain her blocking him out of her memory.\n\nMaybe that was why she'd been dreaming about him lately. Maybe her subconscious had remembered him and his desire to find Atlantis.\n\n\"Anyway, it's why I'm here now. Like you, I want to find Atlantis.\"\n\nShe stiffened at those words. \"Who says I'm after Atlantis?\"\n\n\"You're an American in the Aegean with a scientific team, on a boat that's outfitted for probing and excavation. What else would you be after?\"\n\n\"Any ancient artifact.\"\n\n\"Then why is it you wear an Atlantean coin around your neck?\"\n\nHer hand went straight to it. She'd had the coin mounted a month after her father's death to remind her of her promise to him. But what confused her most was the fact that the part with the writing was on the back. The part that showed to Arik was the image of the sunburst with three lightning bolts. \"How do you know that?\"\n\n\"That coin bears the symbol of Apollymi Magosa Fonia Kataastreifa.\"\n\n\"Apollymi who?\"\n\n\"The Atlantean goddess of wisdom, death, and destruction. But she was mostly referred to in Atlantean as Apollymi Akrakataastreifa. Apollymi the Great Destroyer.\"\n\nThere was no way he could know that. Not unless he'd seen the mysterious symbol somewhere else. \"Where have you seen the symbol? How do you know what it stands for?\"\n\n\"I'm from an old Greek family. There is nothing about this area that I don't know. Nothing. I also know that even if you have found Atlantis, you'll never get a permit to excavate it.\"\n\nIt was true. She'd been trying for years to gain one. But she was persona non grata here.\n\nArik's eyes narrowed on her. \"You allow me to stay on this boat as a member of your team, and I can guarantee you a permit for anything you need.\"\n\n\"You're lying.\"\n\nHe shook his head. \"I have more connections here than you can dream of. Literally.\"\n\n\"And how can I trust you?\"\n\n\"How can you not? I'm the only hope you have of obtaining what you want most.\"\n\nShe sensed a strange double entendre there. \"I don't trust you. How can you get my permits when you can't even remember your own name?\"\n\n\"I've already told you my name.\"\n\n\"Arik and nothing else.\"\n\nArik smiled at her before he took a major risk. \"Arik Catranides,\" he said, using Solin's human surname. It was a bold move given Solin's unpredictability, but his brother owed him this favor, and if he failed to cooperate, Arik would kill him.\n\nGeary stared at Arik suspiciously. For over five years she'd been bogged down in red tape as the Greek government ran her around so badly she felt like a small plastic car trapped on a racetrack in an endless loop of frustration. She'd gotten nowhere and she was quite certain she'd slammed off the track a few times and landed in a tree... face first.\n\nWas it possible for him to get her the permits she needed?\n\nNo. Hell no at that. Nothing is ever going to get them to budge and you know it. All she had to do was call his bluff and he'd retreat.\n\n\"Fine, you want to prove yourself to me, get the permits. But the only way you get to stay on this expedition is if I meet the man who signs them and I watch him put pen to paper. I don't want a forgery that lands me in jail.\"\n\n\"No forgeries. You can trust me, Geary. I promise.\"\n\nStill not sure she could, or even should, she nodded grimly, then turned to leave. Before she could make a clean exit, he gently pulled her to a stop. She expected him to say something. Instead he merely stared at her with a breathtaking expression that was part incredulous and part devouring. No man had ever looked at her like that.\n\nFace it, at six feet tall, she dwarfed most men, and though she wasn't hideous, she wasn't skinny or beautiful. She was just average and men who looked like Arik weren't interested in women who looked like her.\n\nExcept in their dreams...\n\nCould this whole day be nothing more than a delusion? Was she dreaming even now?\n\nArik wanted to tell her that he was here for her and her alone. He wanted her to know what he'd gone through to get here, but from what he knew of humans, she wouldn't react well to that knowledge. Especially not about the part where he'd bartered her soul for it.\n\nBut from the moment he'd touched her, words escaped him. He wanted to taste her, to hold her.\n\n\"I...\"\n\nShe cocked a brow in expectation.\n\nI want you with me, Megeara. Those words were on the tip of his tongue. They burned there, needing to be spoken. But to say them would cost him the very thing he was trying so hard to get.\n\n\"I need to contact my brother.\"\n\n\"Fine,\" she said softly. \"You can see him once we dock.\"\n\n\"But I don't know where he is or how to find him. I'll need your help.\"\n\nHer eyes turned suspicious once again.\n\n\"Please, Megeara.\"\n\n\"Geary,\" she said from between clenched teeth.\n\n\"Please, Geary. I have to find him.\"\n\nShe folded her arms over her chest. \"What's his name?\"\n\n\"Solin Catranides.\"\n\nHer entire demeanor was one of doubt. \"This better not be a trick, do you understand?\"\n\n\"It's not a trick.\"\n\nStill her eyes accused him. \"Fine. You stay here and I'll let you know when we're back at the marina.\"\n\n\"I shall wait most anxiously.\"\n\nShe just bet he would. Giving him a warning glare, Geary backed out of the door and shut it tight. It was only then that she could breathe again.\n\nJust what was she supposed to do now? Was there any validity to his claims? Or was he completely full of crap?\n\nUnsure of what to believe, she headed topside where Brian and Teddy were chatting.\n\n\"Are you all right?\" Brian asked as she joined them.\n\n\"I guess... oh hell, I don't know. Our newest passenger claims he can get us our permits.\"\n\nTeddy laughed in disbelief. \"What? Is he Zeus? Does he know the gods personally? 'Cause no offense, I think that's what it's going to take to get us any kind of permit.\"\n\nBrian nodded. \"I have to side with Teddy. It's beginning to look hopeless. I'm afraid I'm going to have to pull my funding.\"\n\nGeary was sick to her stomach at the news. Even though she was the co-owner of her father's salvage company, her money was so tied up that she couldn't touch the kind of cash she needed to fund these summer trips.\n\n\"C'mon, Brian\u2014\"\n\n\"I'm sorry, Geary. It's too costly and now we have no permits.\"\n\nThey'd never had permits. At least not legal ones.\n\n\"Can you give it a day? Arik swears his brother is good for it.\"\n\nTeddy snorted in contempt. \"Who is his brother? King Constantine?\"\n\n\"Some guy named Solin Catranides.\"\n\nBrian's jaw went slack.\n\nA twinge of hope went through her. \"You know him?\"\n\n\"The multi-billionaire playboy? Uh, yeah, I know of him. But I've never been able to get close enough to meet him. He's always surrounded by a harem of women trying to become his next pampered mistress.\"\n\nGeary frowned. That didn't sound like a guy who would have a brother floating in the Aegean.\n\nThen again...\n\n\"Do you know where we can find him?\"\n\n\"I can make a few calls and see if my people can locate his people.\"\n\nThat worked for her. \"Please do so. I want to know if Arik is lying.\"\n\nTeddy scratched his cheek. \"You know, it could be a different Solin Catranides.\"\n\nShe shook her head at Teddy. How many men could there possibly be with a name like that?\n\n\"Hey, you never know,\" he said defensively.\n\n\"Yeah, but what are the odds?\"\n\nTeddy laughed. \"About as good as fishing a half-naked guy of the sea.\" He looked at Brian. \"You've got to wonder about a guy like that. He wasn't drunk. What? Did he decide to take a swim twenty miles from shore? With no boat?\"\n\n\"Oh, shut up, Teddy,\" Geary said playfully.\n\nBrian left them to make his call on the satellite phone with Teddy following one step behind. But as she listened to Teddy's questions, she realized he wasn't being his usual puckish self. For once the man was making sense.\n\nWhy was Arik out here alone? How had he come to be in the sea when it was obvious the man couldn't swim?\n\n\"Are you okay?\"\n\nShe turned to find Tory behind her. \"I don't know. I'm wondering if maybe we should have left our mysterious swimmer in the water.\"\n\nTory frowned. \"How not like you. Why would you say that?\"\n\n\"There's something weird about him, don't you think?\"\n\n\"You mean aside from the fact he was almost naked in the water?\"\n\n\"Well, there is that.\"\n\nTory shrugged. \"I don't know. I'm not the one who was really talking to him. What about him bothers you so?\"\n\nGeary smiled at her. \"I don't know. Maybe I'm just tired.\"\n\n\"People only say that when they're not really willing to deal with the issue at hand. It's like when you ask a guy what he's thinking and he says 'nothing' but in reality you know he's checking out another woman and he doesn't want you to give him grief over it.\"\n\nGeary gaped at her unexpected analogy.\n\n\"It's Thia's theory.\"\n\nGeary shook her head. \"I think you need to stay away from her before she corrupts you.\"\n\n\"Nah, it's too much fun. She has the most misguided views on everything. But I think what I just said is one of the few lucid thoughts she's ever managed.\"\n\nGeary had to agree with that, too. \"All right, Doogie Howser, go back to the books.\"\n\n\"You know, that's what you always say to me when I've hit too close to home with an observation.\"\n\nShe was right, but Geary wasn't about to let her know that. \"Take your little scrawny self off and bug someone else before I make you a human sacrifice, Tor. I'm trying to think, okay?\"\n\n\"'Kay. I'll be below deck irritating Scott next if you need me.\"\n\nLaughing, Geary watched her cousin walk off. God, how she loved that little girl. There was something very infectious about her.\n\nTory passed Brian in the doorway. By the look on his face, Geary could tell he had bad news.\n\nShe met him halfway across the deck. \"What's up?\"\n\n\"Apparently Solin is an only child. He has no brothers, sisters. Hell, not even a guinea pig.\"\n\nAnger and victory whipped through her. \"I knew it! I knew he was lying.\" Geary took Brian's arm and hauled him back the way he'd come.\n\n\"What are you doing?\"\n\n\"I'm going to confront our guest with this and you're my witness.\"\nCHAPTER 5\n\nArik was fascinated by the bristled texture of the blanket on Teddy's bed. It was scratchy and irritating. Why would anyone want that next to their skin? Even the pillowcase wasn't what he'd thought it'd be. In dreams, these items were as soft as air and they slid against his skin like warm water.\n\nBut here... He shivered.\n\nThis was a strange world mankind lived in. No wonder they escaped into dreams.\n\nAnd he was tired of being here without Megeara. She was proving to be even more elusive in person than she'd been in her dreams. He didn't know where she was, but it was time he found her.\n\nHe'd just reached for the door when it swung open so fast he felt the rush of air against his skin.\n\nA warm, sweet emotion went through him as he saw Megeara. At least it started out that way until he noted the look of anger on her face.\n\n\"What?\" he asked, wondering why she was so vexed now.\n\n\"Solin Catranides is an only child.\"\n\nArik laughed at the ludicrousness of that statement. As a Dream-Hunter, Solin had thousands of siblings. Literally. \"I assure you, he is not.\"\n\nShe gestured to the man behind her. \"Tell him, Brian.\"\n\n\"I called a friend who knows him. She assured me that Solin has never mentioned a family of any sort.\"\n\nArik gave them a sardonic smile. \"I'm sure he wouldn't mention our family to a woman when it's none of her business. Get him on the phone for me.\"\n\nGeary glared at him over the commanding tone of voice he used. One thing she knew about Brian. He didn't take that tone any better than she did.\n\nBrian's eyes flickered with contempt. \"I've already made my inquiries.\"\n\n\"And they were wrong.\"\n\nBrian tossed the phone at him. \"You make your own calls, buddy.\"\n\n\"I don't know the number.\"\n\n\"Then you're screwed.\"\n\n\"Brian,\" Geary said in a gentle tone, trying to diffuse their prickliness. She took the phone from Arik and returned it to Brian. \"Can you get Solin's number for me? I want to speak to him myself.\"\n\nHe curled his lip at Arik. \"It's his brother. Shouldn't he know the number?\"\n\n\"Brian, please. He could call any number in Greece and the person who answers it could be anyone pretending to be Solin. I want to make sure that I'm speaking to the right person.\"\n\nBrian's features softened as he saw the sanity of her request. \"Fine.\" He took the phone and dialed it. After a few minutes, he pulled a pen and paper from his pocket to jot down the number. He hung up and handed it to her.\n\nGeary frowned at the number. \"You're sure this is right?\"\n\n\"It's the only Solin Catranides I know of. Whether or not it's his brother remains to be seen.\"\n\n\"Okay.\" She dialed the number and waited while Arik passed a smug look at both of them.\n\nAfter the sixth ring, a man with a deep British voice answered in Greek.\n\nGeary kept her eyes on Arik, who was watching her back with a blank expression on his handsome face. \"Is this Solin Catranides?\"\n\n\"No. Kyrios Catranides isn't available at the moment. If you would like to leave your name and number, I will add your message to the others.\"\n\nCould he have possibly said that in a snottier tone? Really, the man should teach haughty-butler school\u2014advanced studies. \"This is a bit of an emergency\u2014\"\n\n\"They always say that, thespeneice. No offense. The master has no desire to be disturbed this afternoon by anyone.\"\n\nShe narrowed her eyes on Arik, waiting for him to slip and show her if he was lying or not. \"Not even for his brother?\"\n\n\"I beg your pardon?\" The snottiness was vacant now, replaced by incredulity.\n\n\"I have a man standing in front of me who claims to be his brother.\"\n\nNow the man's tone was completely flat. \"The master has no brother, thespeneice.\"\n\nBefore she could respond, Arik pulled the phone from her hand and spoke something in a language she didn't know. It sounded as if it was Greek-based, but it was something else entirely.\n\nArik passed another smug look to Brian, then to Megeara. He was getting tired of her mistrust\u2014not that he didn't deserve it. It was merely causing him aggravation\u2014an interesting emotion that. He didn't like it. It was too... aggravating. \"He's coming to the phone now.\"\n\nTwo seconds later, Solin answered in a peeved tone of voice. \"Is this a joke?\" he asked in the language that only the gods knew.\n\nArik answered in kind. \"No, Solin, it's not. I need your help.\"\n\n\"If you are what you claim, and since you're using my native tongue, I have no doubt that you are a relative, you don't need my help.\"\n\n\"Yes, I do. I'm trapped on the human plane for two weeks without my powers and I need your assistance until I can go home again.\"\n\n\"I\u2014\"\n\n\"Don't you dare deny me,\" he said, clenching his teeth. \"Because of you I turned Skoti. You fail to help me now and I can promise you that you'll never again sleep in peace. I'll spend the rest of eternity synched only to you. Every time you close your eyes, I'll be there... beating the shit out of you.\"\n\n\"My, that's some threat you've got going.\"\n\n\"No threats. Only promises.\"\n\nSolin paused before he spoke again. \"For the record, I don't take such promises lightly.\"\n\n\"And I don't give them lightly. If you doubt me or my skills, ask M'Ordant who I am and what I'm capable of. I've come a long way from the numbed Oneroi you turned centuries ago. I want your help, Solin. I know helping someone goes against your grain, but suck it up and assist me.\"\n\nThere was a few seconds of silence as if Solin was thinking. \"If you're here as a human as you claim, I assume you have the backing of a god. Who is it?\"\n\nThere was no need to hide that from him. If he really wanted to know, it wouldn't take him long to find out. \"Hades.\"\n\nSolin snorted. \"You bargained with Hades? Are you insane?\"\n\n\"I was definitely sane and in control while I was Oneroi. Then someone changed that. What I am now is anyone's guess, even my own.\"\n\nMore silence greeted him.\n\n\"All right,\" Solin said at last. \"I don't make this a habit, but you've made me curious. What do you need from me?\"\n\n\"I need permits for an American archaeologist to excavate Atlantis.\"\n\nSolin burst out laughing. \"I know you're insane now. Have they really found the site?\"\n\n\"Does it matter?\"\n\n\"On this plane of existence, yes. You start poking there and you'll piss off people best left alone.\"\n\n\"Since the human's days are numbered, I don't think that's going to be a problem. Let her have a bit of a thrill before she dies. What's the harm?\"\n\nSolin sucked his breath in sharply between his teeth. \"No, you didn't.\"\n\n\"Didn't what?\" Arik asked.\n\n\"Promise a soul exchange with Hades. You do have nerve, I'll grant you that.\"\n\nHe wasn't sure if impressing his brother was a good thing or not, but at least Solin sounded a little more agreeable.\n\n\"Aside from the permits, what else?\"\n\n\"That's it. She wants to meet the official who grants them to make sure they're not forgeries.\"\n\n\"How soon do you need this?\"\n\nArik glanced to the other two, who were watching him expectantly. \"How soon can you do it?\"\n\nAnother brief pause. \"Give me an hour to arrange it. I have a few government friends who owe me. I just have to decide who I want to intimidate or blackmail.\"\n\nArik looked at Megeara and spoke in English. \"He needs an hour to set the permits up. Can you meet him then?\"\n\nHer jaw slackened before she nodded.\n\n\"She can be there,\" Arik told Solin.\n\n\"Good. I'll pick you up.\"\n\n\"Why?\"\n\n\"Because I have to meet the god who's this arrogant and stupid face-to-face.\"\n\nArik wasn't sure if he should be flattered or insulted. Perhaps he should be a little of both. \"Then I'll allow the good doctor to give you your directions.\" He handed the phone to Megeara, who was still gaping.\n\nGeary couldn't believe what she was hearing. Was it really that simple? Could nothing more than a phone call gain her the elusive papers she needed? \"Hello? Kyrios Catranides?\"\n\n\"Yes, and you are?\"\n\n\"Dr. Geary Kafieri.\"\n\n\"Nice meeting you, Doctor. As my brother said, I'll be needing directions on where to pick the two of you up so that we can procure your permits.\"\n\nGeary still was a bit reserved. It wasn't in her nature to trust people, and especially not after all the years she'd been trying to accomplish what the two of them seemed to be able to do in one hour. \"I thought you were an only child.\"\n\nSolin didn't hesitate with his answer. \"Yes and no. I have a number of half siblings. One of whom happens to be Arik. Now if you'll be so kind as to give me your address.\"\n\nShe did, even though she kept expecting this to be a joke.\n\n\"Very good,\" Solin said once she finished giving him the address and directions for the marina. \"I shall see the two of you in about an hour.\"\n\n\"Thank you.\"\n\nHe hung up.\n\nGeary ended the call, then handed the phone to Brian. \"He's getting us the permits. Do you think he can really do that?\"\n\nBrian shrugged. \"If anyone can, he can. Solin travels in the highest of circles. Even those I can't ascend to... which tells you exactly how much money he has.\"\n\nShe looked to Arik, whose face was completely stoic. \"And he's your brother?\"\n\n\"Yes.\"\n\nBrian cleared his throat. \"Well, if you get those permits, I'll reconsider my backing.\"\n\nThat meant a lot to her. Without his backing, they wouldn't have any choice except to pack up and go home. \"Thank you, Brian.\"\n\nHe inclined his head to her before he left them alone.\n\nArik offered her a seductive smile. \"Are you happy now?\"\n\n\"I don't know if happy is the right word. I'm still suspicious of you and your motives.\"\n\nHe clucked his tongue at her. \"After all this, how can you continue to mistrust me?\"\n\nWas he serious? \"Can you blame me? I still don't know you and here you are making grand gestures for no reason. Why would you be willing to help me?\"\n\n\"Because I find you fascinating. You were so passionate in grad school, and now you're on an impossible quest, just as I am. One has to admire that. Not to mention the small fact that you saved my life. Helping you with the permits is the least I can do.\" There was something in his eyes as he spoke that glinted and glowed. She felt like a snake with its master charmer who was luring it from its basket to the highway for it to be run over by a Mack truck.\n\n\"What do you want from me? Really?\" she asked.\n\n\"Just a simple smile. Nothing more.\"\n\n\"I find it hard to believe something so small would satisfy you.\"\n\nHis grin turned wickedly warm. \"It would at least tide me over for a bit.\"\n\nGeary wasn't sure what to make of him. On the one hand, he was helping her out in a way no one else could. He didn't owe her anything at all, and yet...\n\nCould it be as simple as he was repaying her for saving him? He was Greek and it made sense when put in that context. The Greek people had a strict code of ethics about what was right and wrong. Repayment would be something they would do without hesitation. Perhaps she was being too harsh with him.\n\n\"Okay, Arik. I'm sorry I've been so irritable to you. I just don't trust people as a rule, especially those I don't know.\"\n\n\"I understand and we did meet under extremely odd circumstances.\"\n\nA tiny smile spread across her face as she remembered hauling him on board. \"True.\"\n\nHis features softened to the most seductive look she'd ever seen from any man. \"Shall we start over?\" He held his hand out toward her. \"I'm Arik Catranides.\"\n\nShe shook his hand. \"Geary Kafieri, and I still want to know how you ended up in the sea.\"\n\nHe lifted her hand to his lips so that he could place a whisper of a kiss to her knuckles. \"And I promise you that one day you will learn the answer to that mystery.\"\n\nShe wasn't sure why, but the hairs on the back of her neck lifted as an eerie chill went down her spine. It was followed by the memory of her dreams where Arikos had bathed her in chocolate whipped cream that he'd licked off her body slowly and easily. But this wasn't that man who'd seduced her.\n\nWas it? Could it be that somehow her subconscious had held on to his memory all these years and it was only now that she needed him that she'd remembered him?\n\nIt didn't seem feasible. Yet how else could she explain his presence here on the boat and the fact that he'd been in her dreams these past weeks? She must have remembered him.\n\nAnd now that she was more relaxed, there was something about him very calm and peaceful. Something that soothed her.\n\nExcept for his eyes. They spooked her. They seemed somehow omniscient and powerful. Probing and deadly.\n\n\"So where exactly do you live?\"\n\nHe didn't answer. Instead, he moved behind her and wrapped his arms around her. It was what her dream lover had done a thousand times.\n\nShe stiffened in his embrace. \"Who are you, Arik? Why are you really here?\"\n\nHe rubbed his cheek against hers so that his whiskers sent chills over her. \"You want Atlantis, yes?\" He all but purred that into her ear as desire burned through her.\n\n\"Yes.\"\n\n\"Then what else matters?\"\n\nThe heat in her body for one thing. It was unlike any she'd ever experienced before. All her life, she'd been forced to prove herself to others. And since her father's reputation had so impugned her own, she'd gone out of her way to not allow the fact that she was a woman make the more elitist scholars disregard her. She'd focused her entire life on being a serious scientist, to the exclusion of all else.\n\nBut with Arik, it was different. He treated her as a woman and he wasn't repelled by her protective barbs. He saw her as desirable. The novelty of that alone was titillating.\n\nShe wanted to close her eyes and lean back into him. To reach up and lay her palm against his whiskers so that she could feel the muscles of his jaw work. It was what she'd do in her dreams.\n\nBut this was reality and Dr. Geary Kafieri didn't have time for such play. Even though all she wanted to do was stay where she was, she pulled away. \"I need to work.\"\n\nArik ground his teeth in frustration. But then, it was the power of her that had lured him to her in the first place. I want you to stay...\n\nThe thought had no sooner gone through his head than she turned on him with a vicious glare. \"And I told you I have things to do.\"\n\nHe frowned at the anger in her tone. \"Excuse me?\"\n\n\"You said you wanted me to stay and I can't do that.\"\n\nHe cocked his head. \"No, I didn't.\" He'd only thought it.\n\n\"I heard you loud and clear.\"\n\n\"But I didn't speak.\" How could she have heard him when he didn't have his powers?\n\nGeary didn't know what to think. Something wasn't right about this. She could sense it. She needed to get away from him until she could sort through this.\n\nWithout so much as a good-bye, she left him and headed topside so that she could get a breath of fresh air and clear her thoughts.\n\nThia met her on deck. \"Where's Mr. Cutie?\"\n\n\"Down below.\"\n\nA wicked smile curved her lips. \"Now that's where I'd like to have him... down below.\"\n\nGeary rolled her eyes at the bad entendre as a breeze brushed against her and gave her a shiver. She was still in wet clothes, and if they were truly going to meet with any Greek official she needed to change her clothes lest she offend him.\n\n\"Have at him, Thee. I'm sure he'd welcome that.\"\n\nKat laughed as she passed by them. \"I really doubt that.\"\n\nGeary frowned at the odd note in Kat's voice. At almost six and a half feet tall, she, like Thia, towered over the rest of the crew, women and men alike. Kat was also extremely Greek, with blond hair and sharp green eyes. She'd joined their team only a few weeks ago\u2014right after they'd found the wall they were trying so desperately to excavate. \"Do you know something about our guest?\"\n\nKat shrugged nonchalantly. \"Nope. Why would I?\"\n\n\"Hmmm...\" Geary wasn't so sure if she believed that or not. There was something in Kat's demeanor that said she might be withholding information.\n\n\"Kat, you're from Greece, right?\"\n\nShe laughed again. \"Born and raised. I seriously doubt anyone comes more Greek than me, why?\"\n\n\"Do you really think this guy can help us get our permits?\"\n\nKat sobered. \"I guess we'll know soon.\"\n\nBut not soon enough in Geary's book. \"All right. I need to change before we dock. I'll see you two shortly.\"\n\nKat waited until Geary had vanished belowdecks before she looked at Thia, who at eighteen bore a frightening resemblance to the goddess Artemis. \"By the way, I heard Scott was looking for you.\"\n\nHer face lit up. \"Really?\"\n\n\"Yeah. You better run before he changes his mind.\"\n\nThia couldn't have bounded off faster if she'd been wearing Hermes' enchanted sneakers. Which was definitely a good thing.\n\nThat should occupy the redhead for at least a few minutes so that Kat could go visit their latest acquisition and not be disturbed or, more to the point, overheard.\n\nAs soon as she was sure Thia was out of the way, Kat headed to Teddy's cabin, where Arik had been taken. She knocked once on the door before she pushed it open.\n\nArik stood by the window with his arms crossed over his chest as he gave her a very cool once-over. \"You're not Megeara.\"\n\n\"No, I'm not. But what I'd like to know is how a person like you made it onto this boat.\"\n\n\"A person like me?\"\n\nShe nodded as she took a step closer to him. \"Tall, dark-haired. Sexy beyond belief with eyes so blue they glow? I suppose the question of the day is who's your daddy?\"\n\n\"I beg your pardon?\"\n\n\"Who's your daddy?\" she repeated before she expanded the question. \"Morpheus or Phobetor?\"\n\nHe gave her a suspicious look. \"Who are you?\"\n\n\"Katra Agrotera. But most people in this realm call me Kat.\"\n\nShe saw the recognition in his eyes. \"Agrotera?\"\n\n\"Yes,\" she answered his unasked question. Agrotera was one of the names often attached to the Greek goddess Artemis\u2014a name often used by her servants. The very same goddess who'd sent Kat here to watch over the progress of this team. \"I'm one of her koris.\"\n\n\"What are you doing here?\"\n\n\"I used to work with Geary's father from time to time, back in the day when the two of them weren't speaking. Since she's getting a little close to the matter at hand, Artemis thought I should put a few roadblocks in front of her.\"\n\n\"And why is that?\"\n\n\"Simple. Atlantis can't be found.\"\n\nHe scoffed at her. \"You're the second person to tell me that in less than an hour. Why is it so important to Artemis that it stay hidden?\"\n\n\"The why isn't that important. Just believe me when I say you don't want to go there... in more ways than one.\"\n\nHe didn't flinch or show any emotion, which given his birth made sense to her. Still, it was time he learned to fear.\n\nWhen he spoke, his tone was level and deadly. \"Megeara wants to find it.\"\n\n\"And people in hell want ice water. The entire history of mankind is written by people wanting something they can't have. She'll get over her disappointment, trust me.\" She approached him slowly and lowered her tone to make damn sure that no one passing by in the hallway outside could overhear her next words. \"But that still doesn't explain how it is a Dream-Hunter is here, in the flesh, on this boat. I'm sure you didn't come to this plane just to help the good doctor on her quest.\"\n\nHe was more guarded than a priceless treasure. \"I wanted to know what it was like to be human. Is that a crime?\"\n\n\"On Olympus, it can be.\"\n\n\"Are you threatening me?\"\n\n\"I'm warning you to forget you ever even heard the name of Atlantis.\"\n\n\"And if I don't heed your warning?\"\n\n\"Oh, it'll get ugly. Real ugly.\"\n\nHe gave her a cocky grin. \"My flesh is used to being stripped from the bone, little girl. Is yours?\" He didn't give her time to answer before he continued. \"You can't threaten or intimidate someone who won't be able to feel any emotions in a few weeks. Pain doesn't scare me since it's all I know.\"\n\n\"You're a masochistic bastard, aren't you?\"\n\n\"Isn't that the very nature of a Skotos? After all, it's what your kind damned us to.\"\n\nKat paused at his words. It was true. What had been done to the Oneroi was deplorable and unfortunate. But it still didn't change the fact that he couldn't be allowed to uncover Atlantis. Artemis wasn't the only god who would be furious to have it exposed. This little Skotos was playing with a fire he couldn't even begin to understand.\n\n\"So you're just here to be human and experience the world? Nothing else?\"\n\n\"Nothing else.\"\n\nKat could almost accept that except for one thing. \"And how does Geary fit into this?\"\n\n\"Who says she does?\"\n\nKat laughed as she noted the speculation in his crystal eyes\u2014he was hiding something. \"Don't mistake me for an idiot. You have no god powers, I can feel it. While your kind does come to this plane from time to time to scope out victims, you don't lose your powers when you do it. You've bargained yours to be here and to be human, and you're helping Geary. Why?\"\n\n\"First you tell me why Artemis is interested in this and then I might answer you.\"\n\nHe was quick and clever. She'd give him that. \"All righty then. It appears we've come to an understanding. I stay out of your business and you stay out of mine.\"\n\n\"Fair enough.\"\n\nKat looked past him, out the porthole, to see the docks coming into sight. It wouldn't be long before he took Geary for the permits.\n\nKat stifled a shudder at that. \"Just remember one thing, Dream-Hunter. You get in my way and I will sacrifice you for my mission.\"\n\nHe laughed low in his throat. \"And to that all I can say is a resounding 'ditto.' I won't let you interfere with why I came here.\"\n\nShe arched a brow at that. \"You would dare to threaten me? Have you any idea what happened to the last man who harmed a kori?\"\n\nHe shrugged nonchalantly. \"I'm not afraid of Artemis. Even though I'm human now, I won't be for long. My powers will be returned to me in full. Both you and your mistress would do well to remember that.\"\n\nShe tsked at his arrogance. \"Oh, Arik, Artemis is the least of your problems if you succeed in finding Atlantis. There are powers so deep and dark buried with that continent that they make a mockery of Zeus. I'm only a minion in the grand scheme of things. You have much more to fear than either me or my lady-goddess. And on that pleasant note, I will leave you with only one more piece of advice.\"\n\n\"Which is?\"\n\n\"Things are very seldom what they seem. Atlantis and what happened there was a blight on the nose of many a god and many a pantheon. As you know, the gods seldom agree on anything, but they are all united when it comes to this. You would do well to leave here as soon as we dock, and find a new playmate for your dreams.\"\n\n\"So you're lying to Megeara about your presence here. Pretending to help while hindering. How noble of you.\"\n\n\"And you're here to seduce her and do what? Kill her? Is that your plan?\"\n\nHe looked away and she didn't miss the fleeting grief that darkened his eyes before he hid it. \"Does my intent matter since you've already made up your mind to kill her?\"\n\nAnger whipped through her at his words. \"I've never killed any human. I'm not that cold. Hell, I even tried to save her father and it's for him that I'm watching over Geary now, instead of allowing another kori to be here. I don't want to see her die. She's too decent a woman for that. So again, I say to you, move on.\"\n\n\"And if I can't?\"\n\n\"Then you and I are at war.\"\n\n\"Unfortunate, but I can accept that.\" He moved to stand before her and she hated the fact she had to look up at him. It was what he'd intended by the action, but it would take more than that to intimidate her. \"Stay out of my way, Katra. For your own sake.\"\n\nFine then. At least she knew the stakes. Now she just had to visit her friend and do everything she could to get Geary to throw Arik overboard as soon as possible. He was human at the moment and couldn't play with Geary's thoughts or emotions. That was a blessing.\n\n\"Oh, don't worry. I intend to be the thorn in your side until I drive you mad with fury. You may succeed in seducing Geary, but you're not going to hurt her. Not on my watch.\"\n\nArik had opened his mouth to return her comment when he heard the door behind them opening. He turned his head to see the woman Thia standing there, gawking at them.\n\n\"Am I interrupting?\" Thia asked in a snide tone.\n\nKat shook her head. \"I was just leaving.\" She gave him a stony look. \"Remember what I said.\"\n\n\"Ditto to you.\"\n\nFury blazed in her eyes before she pushed past Thia and left them alone. Arik didn't move as he considered this latest turn. So Megeara had one of Artemis's koris as a protector....\n\nThat did put a bit of a crimp in things, but it by no means deterred him. He wanted to fully experience Megeara as a man. And nothing, not even Zeus himself, was going to stop him.\n\nNow he just had to get Megeara to cooperate.\nCHAPTER 6\n\nGeary hadn't gone near Arik since she'd last left him. She had no idea whether or not she should believe him, and until she had more facts, she wanted to be as reserved as possible where he was concerned.\n\nThey'd just docked and she was in the process of putting her things away for her trip back to town.\n\nShe looked up from the table as Tory burst into the room. \"Holy Shinola, Gear, you've got to come see this!\"\n\nFrowning, she set her pad aside and followed Tory to the deck. Geary looked around but couldn't find anything that should have excited Tory. Nothing looked out of place. Christof and Althea were going over inventory while a couple of the other sailors were checking the lines. Thia was on the deck in a bikini, sunning herself.\n\n\"What's up?\"\n\nTory pointed toward the shore.\n\nGeary followed the line of Tory's finger. And as soon as she saw what Tory was pointing to, her jaw dropped.\n\nHoly Shinola nothing. Holy shit and then some.\n\nJust at the edge of the dock was a white Rolls-Royce that had a driver in full chauffeur regalia standing by the door with his gloved hands folded in front of him.\n\nBut that wasn't the impressive part. Not by a long shot.\n\nWhat made her gape was the hot piece of cheese who was on the dock, striding straight for them.\n\nWith shoulder-length black hair, the man had a gait that was just plain sexy. It was one of raw determination and extreme confidence. He wore a white linen suit with a pale blue shirt that had been left undone to show the promise of a very well-defined body. On any other man that suit might have brought his sexual preferences into question, but on this one there was no doubt. He was all male and all deadly.\n\nA dark pair of Versace sunglasses covered his eyes, which she had a sneaking suspicion were trained on her.\n\nTory cleared her throat. \"I'm going to go out on a limb here and say that he's Arik's brother. What do you think?\"\n\nYeah, that would be her guess. They both held an identical arrogant swagger\u2014as if the world were their stage and they were the only actor in town capable of playing on it.\n\nWithout a word to Tory, Geary moved forward to meet the man on the gangplank.\n\nHe paused in front of her with a hint of a grin an instant before he deftly removed the sunglasses. Her breath caught as she saw the same exact killer eyes that Arik possessed. It was followed by a dimpled smile so choice that it actually made her heart race.\n\n\"Kyrios Catranides?\"\n\nHe offered his hand to her. \"You must be Megeara. Pleased to make your acquaintance.\"\n\nShe shook his hand, but before she could let go, he brought it to his lips and laid a very romantic kiss on her knuckles. Her hand actually tingled from the sensation of his lips on her skin. \"It's a pleasure to meet you as well.\"\n\nHe released her at the same time the smile faded from his lips. His gaze slid past her.\n\nShe turned her head to find Arik standing there. He was silent and cool as he regarded his brother. So cool, in fact, she was about to get frostbite. There was definitely no love lost between them. They looked like two opposing soldiers sizing each other up before battle.\n\n\"Arik,\" Solin drawled in a velvet baritone. \"Long time no see.\"\n\nArik inclined his head to Solin. \"Yes, indeed. I hope you've been well.\"\n\nSolin laughed. \"It depends on whom you ask. Well has a variety of meanings. But I'm fit enough to cause problems. That really is all one can expect in life, no?\"\n\n\"It's all I expect out of your life anyway.\"\n\nSolin tsked at him. \"And yet here you are, asking for my help. Call me crazy, but one would expect a little less belligerency.\"\n\n\"Would they?\"\n\nSolin seemed to take his brother's challenge in stride as he turned to Geary. \"So tell me, lovely lady, where on earth did you happen to find my wayward brother?\"\n\nShe glanced at Arik over her shoulder to see him watching her before she answered. \"Floating in the sea, but he won't tell me how it is he came to be there.\"\n\n\"Knowing Arik, I'm sure he angered someone who threw him in, hoping he'd drown.\"\n\n\"Actually they threw me in hoping I'd land on someone else and drown them. Unfortunately, you swam away too fast.\"\n\nGeary had to stifle a laugh at Arik's unexpected comeback. He had a very dry sense of humor.\n\n\"Well, score one for you.\" Solin returned his sunglasses to his face. \"I have the permits waiting, but as a favor to Stefan we should not keep him late in the office or he might change his mind.\"\n\nGeary practically leapt forward. \"Most definitely not.\"\n\nAs they headed for the dock, Thia came running up behind them... still in her bikini. The top of which barely held the woman's assets in. \"May I join you?\"\n\nSolin gave her a speculative once-over that Geary was sure took in her cousin's mussed appearance that somehow managed to seen both seductive and naive.\n\n\"I think you should stay here, Thia.\"\n\nFolding her arms over her chest, which only emphasized the size of said chest, Thia pouted. But it did nothing to change Geary's mind. If anything, it only cemented her decision more. The last thing any of them needed was for Thia to hook up with a billionaire playboy.\n\nBefore Geary could wrangle the men to the car, Solin approached Thia with that deadly swagger. He gave her a proper bow before taking her hand and placing a light kiss on it. \"Don't fret, love. We'll be back.\"\n\nThia preened under his attention. At least until Arik cleared his throat. \"Isn't she a bit young for you?\"\n\nSolin answered with a deep, evil laugh. His gaze went to Geary for an instant before he released Thia and headed for his car.\n\n\"What was that?\" Geary asked Arik as they followed along after him.\n\n\"His idea of a joke. I'm afraid my brother is a bit of a head case most days. You'll have to forgive him. I'm told he has the intellect of a ten-year-old.\"\n\nSolin snorted. \"And still you aspire to my level. Wow, Arik. Does this mean you function on the intellect of an infant?\"\n\nInstead of being angry, Arik merely stared at his brother. \"Perhaps. After all, infants and I do have at least one thing in common.\"\n\n\"And that is?\"\n\nArik's gaze dropped to her breasts. \"I think you can figure it out. Then again, maybe not. You are, after all, only functioning on the level of a ten-year-old.\"\n\nGeary had never been aroused, amused, and highly offended all at the same time before. It was a strange combination. \"Could we please change this topic?\"\n\nSolin paused at the car as his chauffeur opened the door for him. \"Yes, let's.\"\n\nThey allowed her to enter first. Arik followed her in and then Solin. He sat across from them, and even though she couldn't see his eyes, she could tell his gaze was fastened on her.\n\nWhen he spoke there was no mistaking the note of humor in his tone. \"So you seek Atlantis. What an odd quest for such a beautiful lady.\"\n\nUnlike Thia, Geary wasn't buying into his act. \"You charm me, sir. I'm hardly beautiful.\"\n\n\"Not true. All women are beautiful and a woman such as you... I'm willing to bet there are some men who are willing to barter their souls just to be close to you.\"\n\nShe laughed out loud. \"You should sell snake oil. I'm told it's highly profitable.\"\n\n\"Yes, but I've already made my fortune in other things.\"\n\n\"Such as?\"\n\n\"Viagra,\" Arik said drily. \"My brother learned to take a personal problem and profit by it.\"\n\n\"It's true,\" Solin agreed with a heavy sigh. \"It pained me to see a man as young as Arik stricken with impotency. Therefore I had to do something to help the poor soul. But alas, there's nothing to be done for it. He's as flaccid as a wet noodle.\"\n\nGeary had to cover her mouth to keep from laughing out loud.\n\nArik didn't miss a beat on his comeback. \"How creative of you to project your problem onto me. But then, they say that celibacy is enough to make a man lose all reason. Guess you're living proof, huh?\"\n\n\"Are you two going to battle like this for the rest of the trip?\" Geary asked. \"Perhaps I should sit up front with the driver and give you two enough space to beat the crap out of each other and settle this like grown adolescents.\"\n\nSolin gave her a half-amused grin. \"No need for that. I think we can manage a bit of a truce... for your benefit anyway.\"\n\n\"Hmm... makes me wonder why you're being so kind to Arik and me when it's obvious you two aren't exactly friendly.\"\n\nSolin shrugged. \"We're Greek. Family is family no matter what, and we always take care of our own. Right, Arik?\"\n\n\"Yes... in more ways than one.\"\n\nAt that point, Geary gave up. There was something very odd about both men. Maybe she was crazy for even being here with them.\n\nA tremor of fear went through her at that thought. Was she crazy? She'd jumped into the car so fast...\n\nOh God.\n\nShe really didn't know anything about these guys. She'd been so excited she hadn't even paused to be her usual suspicious self.\n\n\"Are you all right?\" Solin asked.\n\n\"Fine,\" she said, trying to calm herself. But it was hard as her imagination took off with images of them raping and murdering her.\n\nSolin removed his sunglasses. \"You're looking a bit pale. You're not thinking that we've kidnapped you so that we can have our way with you, are you, Doctor?\"\n\n\"No,\" she said, hating the slight tremor in her voice. Her only consolation was that Brian knew Solin and the crew had seen his car. And all of them knew they were heading to the permits' office. \"Why would I ever think such a thing? I mean, I've known you both, what? All of fifteen minutes. Maybe Arik makes it a habit of diving into the sea to catch unsuspecting women so he can lure them into your limo.\"\n\nSolin passed an amused look to Arik. \"Is that how you work, Brother?\"\n\n\"No. Unlike some people I know, I don't like to frighten women. I find it tiresome.\" Arik turned in the seat to give her a sincere look. \"I'm not here to rape you, Megeara. I told you you were safe, and you are.\"\n\nShe didn't know why, but she believed him. \"I'm sorry. It's just been a really bad week for me. Everything has literally turned against me and I've had one disappointment too many.\"\n\nSolin arched a brow.\n\nArik glanced at his brother as he heard Solin's voice in his head. \"Disappointed her, have you? And you call yourself an Erotikos Skotos.\"\n\nHe narrowed his eyes. \"Not me, Solin. She's been harassed by your officials who won't allow her to excavate.\"\n\n\"Um-hmmm... funny, whenever I'm preoccupied with a human, she's too busy trying to get back to me in her dreams to bother with such innocuous quests.\"\n\n\"Megeara is different.\"\n\nBy Solin's face Arik could tell his brother found that hard to believe. \"So tell me, how do you find the human world? Have you ever been here before?\"\n\n\"No.\"\n\nSolin arched a surprised brow. \"Are you overwhelmed by it?\"\n\n\"Hardly. But I find parts of it confusing. It's very different from being in dreams.\"\n\nSolin grinned. \"You've no idea.\"\n\nMegeara turned to Arik. \"So why is Atlantis so important to you? I mean, if you could get permits this easily, why haven't you?\"\n\nArik hated having to lie to her, but if he didn't give her some plausible answers, as skittish as she was, she'd flee and never let him near her again. \"I didn't know where to dig for it. All of my research turned up nothing. It wasn't until I was speaking with Spiro the other day and he mentioned you that I had a clue.\"\n\n\"Spiro?\"\n\n\"Gavrilopoulos. He turned you down two weeks ago.\" And luckily she'd mentioned the event and the man's name to Arik in her dreams. \"I've been looking for you ever since to ask you about your findings. He said you were most emphatic about the site you wanted to excavate.\"\n\nShe sat back in the seat with a peeved expression. \"So you know the little weasel.\"\n\n\"Weasel?\" Solin asked curiously.\n\n\"Hmm... he laughed so hard over my request I thought he was going to choke and die from it.\"\n\nArik tried to placate her. \"He can be a little callous.\"\n\n\"Callous, nothing. He was downright rude.\"\n\n\"Well,\" Solin drawled, \"your luck is about to change.\"\n\nGeary wanted to believe that. She could use a bit of good luck in her life. And if not good, then at least mediocre.\n\nNeeding to distract herself from that line of thought, she looked at Arik. He didn't seem like the kind of guy who would be interested in anthropology. Both he and Solin seemed too self-absorbed to think about the past or the future. They appeared more the \"me, me, me, now, now, now\" kind.\n\n\"So what got you so interested in Atlantis?\" she asked Arik. \"How did you know what my necklace was?\"\n\nHis eyes gleamed with amusement. \"Do you ever ask a single question?\"\n\n\"Sorry. It's the professor in me. One question invariably leads to another, and so as not to waste time, I generally ask both and then seek the answer. And speaking of, you still haven't answered my last two.\"\n\n\"Yes, Arik,\" Solin said with a hint of laughter in his voice. \"Why are you so fascinated by Atlantis?\"\n\nArik cut a nasty glare at his brother that she couldn't even begin to fathom. Why would that question upset him?\n\n\"I'm always intrigued by the unknown,\" Arik said, glancing back at her. \"They say Atlantis is a myth, but I know better. I believe in it.\" He met Solin's gaze. \"In fact, I think the gods still walk among us, even here and now.\"\n\nSolin made a rude noise at Arik's conjecture.\n\nGeary frowned at him. After the way her father had been treated while being right, she wasn't about to laugh at anyone else's beliefs. It pained her to see Solin so cruel. \"You still haven't explained how you knew what my necklace was.\"\n\n\"I know a man who wears a similar medallion. He was the one who first told me stories of Atlantis.\"\n\nHer jaw went slack at Arik's revelation. Someone else had found one? \"Really?\"\n\nHe nodded.\n\nShe was intrigued by the possibility. \"Is he Greek? How did you meet? Could I meet him? I'd love to know where he got his necklace.\"\n\nArik shook his head. \"Again with the multiple questions.\"\n\n\"Time is fleeting and I need answers.\"\n\nHe took pity on her. \"Yes, he's Greek, and I met him back when I was very young. Sadly, he no longer mentions Atlantis. I think there's something about it that grieves him.\"\n\n\"You have no idea,\" Solin said with a laugh. \"Acheron would kill you to hear you speak of him in such a manner.\"\n\nArik kicked at his brother's foot before he turned his attention back to Geary. \"But enough about me. What changed your mind about finding it?\"\n\n\"My father. I promised him when he died that I'd find it for him.\"\n\n\"That was kind of you.\"\n\nGeary looked away as her emotions choked her. She only wished she'd been kinder to him when he'd been alive.\n\nSolin let out a long breath as if her emotions upset him, too. \"Well, let's all get maudlin, shall we?\" He reached up and pressed a button for the intercom to buzz his driver.\n\n\"Yes, sir?\"\n\n\"George, stop on the way and get us some red-hot pokers to put out our eyes. Oh, and while you're at it, I think we should see about adding salt for our wounds, too.\"\n\n\"Quite good, sir,\" the driver said in a dry tone. Then without missing a beat, he continued, \"Is there any particular place you'd care for me to stop? I've heard the market is a good place for pokers. That is, if you're agreeable to a short detour.\"\n\nSolin appeared to consider it. \"What do you two think? Run-of-the-mill pokers, or a better quality? Oh hell, why not use rusty spoons? They'd hurt more.\"\n\nGeary shook her head. \"You are a sick man.\"\n\nSolin arched a brow at her. \"So are you telling me you're going to pass on my offer?\"\n\n\"Call me crazy, but yeah. I think I'll pass.\"\n\n\"Okay. Thanks, George. It appears we'll go without the pokers after all.\"\n\n\"Very good, sir. Should I still stop for that salt?\"\n\nAgain Solin appeared to seriously consider it before he answered. \"No, I think we're fine for now.\"\n\n\"Very good, sir.\"\n\nGeary let out a nervous laugh as she glanced back and forth between Solin and Arik. The two of them were so odd. And they had the most offbeat humor she'd ever encountered. \"You two must have been a lot of fun growing up. I'll bet your poor parents are still having nightmares.\"\n\nSolin burst out laughing. \"Oh, you have no idea.\"\n\n\"You know I feel like I'm on the outside of this inside joke you two keep passing around.\"\n\n\"Ignore Solin,\" Arik said quietly. \"I told you he's demented.\"\n\n\"Yes, but I taught Arik well. Didn't I, Brother?\"\n\nGeary didn't miss the fleeting glimmer of rage in Arik's eyes. It was subtle but unmistakable.\n\nThe car slowed down and turned a corner that Geary knew better than the street where her flat was. She'd walked this way so many times over the last five years that she could do it blindfolded.\n\nThey were almost there.\n\nFrustrated doom settled like a lump in her chest as the driver parked on the street in the same exact location the taxi had used earlier that day.\n\nIt's just getting better and better.\n\nThe driver opened the door to let her out on the sidewalk. Arik came out behind her and then Solin, who left the car with a masculine flourish. Several women on the street practically swooned.\n\n\"Greetings, my lovelies,\" Solin said flirtatiously as he gave them a seductive grin.\n\nThey whispered among themselves as they continued on their way while glancing back to look at him.\n\nArik passed a droll look to Geary. \"Strange how women can't help staring at a train wreck, eh?\"\n\nSolin rolled his eyes at Arik's remark. \"Like you would know.\"\n\n\"True. I'm never a train wreck myself. I merely admire the way you skid from the tracks and burn.\"\n\nAs they approached the government building, a uniformed guardsman opened the door to admit them.\n\nGeary started for the stairs only to have Solin divert her. \"We don't go up there with the average people. Our man is this way.\"\n\nShe frowned at Arik before she followed Solin into an elegant office that was filled with Greek antiquities. The anthropologist in her was instantly fascinated by the perfectly preserved black figure vase in a glass trophy case. She'd never seen a more preserved piece. It was absolutely exquisite.\n\nShe splayed her hand against the glass as she stared in awe of the piece. \"It's from the first century.\"\n\nShe sensed Arik standing behind her. \"The battle of Troy. You can see Achilles dragging Hector around the walls.\"\n\nGeary nodded as she saw them. \"There's not a chip on it.\"\n\n\"Which is why it's in the case.\"\n\nShe turned at the perturbed voice to find a portly gentleman in his early sixties. She'd seen him here a time or two when she'd come in the past, but she had no idea of his name or job title.\n\nHe rocked back and forth on his heels as he sized her up. \"Dr. Kafieri, I presume?\"\n\n\"Yes.\"\n\nHe narrowed a look on her that said he didn't think much of her before he let out a suffering sigh. He turned toward Solin. \"I hope you won't forget this favor.\"\n\n\"Believe me, I won't.\"\n\nThe portly man gave a curt nod before he took them into a small office with a black desk that was scattered with papers.\n\nGeary's heart stopped as she saw the very things she'd been craving.\n\nThe permits.\n\nShe wanted to run to them, snatch them up, and cradle them to her chest. But without a signature and seal, they were worthless. Still, this was the closest she'd been to one. Her breath caught anxiously in her throat.\n\nWithout a word, he picked up the permits as if they didn't mean the world to her and sat behind his desk before he signed them and stamped them.\n\nShe reached for the permits without thinking, only to have him pull them away.\n\nAgain he narrowed those penetrating eyes on her. \"You do understand that any artifact you find is Greek property? I expect full records to be delivered to me on a weekly basis, along with any findings you uncover.\"\n\n\"I understand.\"\n\nHe held the papers a moment longer before he finally extended them to her.\n\nHer hand actually shook as she finally touched the permits. Honestly, she felt like crying. This was the closest she'd been to fulfilling her promise since Cosmo had given her her father's belongings. \"Thank you,\" she said, her voice cracking from the raw emotions inside her.\n\n\"Don't thank me, Dr. Kafieri. Just respect your word to me and the favor I've done today. If I regret this moment, I assure you what I feel will be a trifle to what I'll put you through.\"\n\n\"I understand, sir. Believe me, you won't regret this at all.\"\n\n\"Then see to it that I don't.\"\n\nNodding, she held the permits to her chest and turned to offer Arik a tenuous smile.\n\nArik couldn't breathe as foreign emotions seized him. Her eyes were filled with unshed tears, but it was the gratitude in them that touched him most. He'd never felt anything like this. Her pleasure was so great that he could feel it himself.\n\n\"Thank you,\" she breathed.\n\nAll he could do was nod his head at her as he struggled to understand these strange emotions in him that made no sense. His throat was tight. His heart pounding. He wanted to laugh and to cry and he didn't know why. He'd never known such confusion. No wonder Hades had profaned emotions.\n\nThey were baffling.\n\nSolin inclined his head toward the door. \"Why don't the two of you go on to the car? I'll be out momentarily.\"\n\nArik opened the door for Megeara.\n\nHe'd barely closed it when she turned on him with a giddy laugh. She threw her arms around him and kissed him on the cheek as she jumped up and down against him.\n\nHeat scorched him as her breasts pressed against his chest and her soft lips brushed his skin before she pulled away. \"I can't thank you enough for this.\" She let out a strange noise before she twirled around. \"Oh my God, I can't believe this. I can't believe I'm finally holding my permits! Legal ones, too, and I didn't have to kill anyone to get them!\" She made a strange \"yee\"-like noise before she hugged him again.\n\nUnable to stand the onslaught to his body, he pulled her to him and kissed her.\n\nGeary melted at that taste of Arik's lips. She was so excited and happy, she'd have done anything for him at the moment. Anything!\n\nOr so she thought.\n\nThe moment he started lifting the hem of her dress up, she jerked back with an indignant squeak. Her jubilation snapped straight to anger. \"What do you think you're doing?\"\n\nHe looked completely baffled by her anger. \"I thought...\"\n\n\"What? That you could lift my dress and screw me in an open hallway? Are you insane?\"\n\nSolin froze in the doorway as he heard her words. \"What did I just miss?\"\n\nShe turned on him. \"Your brother is an absolute jerk. He just lifted my dress up. Here. In public.\" And still Arik looked confused by her anger.\n\nDisgusted with them both, Geary turned and stalked back toward the car.\n\nSolin gaped at Arik. \"What did you do?\"\n\nArik held his hands up in frustration. \"She kissed me. It turned me on, so I\u2014\"\n\n\"No, you didn't,\" Solin snapped, interrupting him. \"Arik, are you an idiot? You could have exposed us all.\"\n\nRage flamed inside him at the insult. \"It's what we've done before when she's gotten the permits in her dreams. She likes the way I touch her.\"\n\n\"Yes. In dreams. This isn't a dream. You're in the human world and people don't behave like that here. Now, Brother, you understand why I venture into the dream realm. There are certain behaviors and rituals you have to practice in this world. You don't just eye a woman here and then jump her. Damn. You're lucky she didn't slap you or have you arrested.\"\n\nArik raked his hand through his hair as he understood her anger, but it didn't do anything to sate the fire in his groin. \"I came here to be with her.\"\n\n\"And you keep that up and you'll spend your time here behind bars. Damn, Arik, damn.\"\n\n\"I told you I needed your help.\"\n\nSolin ground his teeth at those words. It wasn't in his nature to help anyone. Unlike Arik, he wasn't a full god. He'd been tossed out into the world of man and left here to suffer while the rest of his kind lived on Olympus or the Vanishing Isle, far away from the prejudice and fear of humans. And if that wasn't enough, the gods themselves had come after him to punish him over a birth defect he'd never wanted. He'd barely survived their relentless attacks.\n\nNow one of them expected him to offer a help that had never been extended to him. It was almost enough to make him laugh.\n\nHe wasn't even sure why he'd come here today. Arik's threat of invading his sleep meant nothing to a man who'd had assassins after him in that realm. He'd earned his reputation for ruthlessness and he was proud of it.\n\nYet in all these centuries he'd never heard of a god trading his sanctity to be human. The only gods on this plane had been cursed or stripped. No one lived in this realm voluntarily.\n\nNo one.\n\nExcept Arik. \"Why are you here? Really?\"\n\nArik looked away without comment.\n\n\"Answer my question or I walk.\" He saw the anguish in Arik's eyes before he spoke in a low tone.\n\n\"You've always been human. You've always had feelings. You don't know what it's like to have them and then feel them leave you. The numbness is tolerable most of the time. But with Megeara...\"\n\n\"You love her?\"\n\nArik gave him a peeved glare. \"How could I ever love anyone?\"\n\nHe had a point. Self-sacrifice and such were alien concepts to Dream-Hunters.\n\nArik let out a deep breath. \"I just want to understand where her passion comes from. Why something as simple as a drink of lemonade can make her laugh. Why her eyes light up when she dances in the waves. And why the thought of her father makes her cry, even in dreams.\"\n\nSolin shook his head. Unlike his brother, Solin understood all of that. Emotions weren't a gift. They were the ultimate curse of the gods. What Arik didn't realize was that Zeus had done them a favor by demanding they be stripped of all their feelings.\n\nIt was why Solin had unleashed the human on Arik all those centuries ago. He'd been jealous of the emptiness the Oneroi lived in and he'd wanted them to suffer the way he had. He wanted them to crave things they couldn't touch.\n\nTo know what they were missing.\n\nWhat he'd done had been cruel and he knew it. But the sad thing was he felt no regret over it. How could he? Even now the Oneroi sought him in his sleep. He never had rest. No respite. They were evil bastards, all of them.\n\nAnd yet as he stood here with a brother he didn't want to claim, something foreign tickled in Solin's chest. It was pity. Compassion. Two things he'd sworn he'd never feel again for anyone.\n\nHe hated Arik for that.\n\n\"Will you help me?\" Arik asked.\n\nSolin nodded. He'd help Arik all right, but not for the reasons he thought. Solin was going to do everything in his power to let Arik be human. To know Megeara to the fullest extent that was possible, so that when she died because of him, Arik would truly understand exactly what it meant to be human.\n\nHe was going to suffer as no god had ever suffered.\nCHAPTER 7\n\nGeary sat in the car, staring at her permits as if they were the mythical Holy Grail that had miraculously dropped into her lap while George politely ignored her. She didn't know what was keeping the men. Maybe Arik had found some other woman to molest....\n\nThat bizarre thought triggered an odd jealous twinge\u2014which really made no sense. At the moment, she'd gladly pay someone to take Arik off her hands.\n\n\"George?\"\n\nHe met her gaze in the rearview mirror. \"Yes, miss?\"\n\n\"How long have you worked for Solin?\"\n\n\"Long enough, miss. Long enough.\"\n\nBoy, wasn't he the font of all information?\n\nBefore she could continue her questioning, the guys finally left the building.\n\nThey were much more subdued as they joined her in the car.\n\nSolin offered her a tight-lipped smile. \"Are you happy now, Dr. Kafieri?\"\n\n\"Thrilled beyond belief.\"\n\n\"Good.\" He cleared his throat. \"By the way, I would like to apologize for my brother's unconscionable behavior.\"\n\n\"I don't need you to apologize for me, Solin. I'm quite capable of doing it myself.\"\n\nIf George hadn't picked that exact second to pull into traffic, she'd have swung the door open and left. \"And just out of curiosity, what made you think that that behavior was appropriate?\"\n\nArik sighed. \"I wasn't thinking at all. The truth is you surprised me and I reacted poorly. For that I'm truly sorry. I would never have insulted you in any way.\"\n\nOkay, so the guy could be charming when he put forth the effort...\n\nBut she still wasn't quite ready to let him off the hook. \"Do you always make it a habit of groping a woman's private places in public?\"\n\nArik narrowed his gaze as he returned her question with one of his own. \"Do you make it a habit of throwing yourself against men and kissing them in public?\"\n\nHer face flushed as a feeling of dread consumed him. He'd offended her. Again. Damn, being human was hard.\n\n\"No, I don't,\" she snapped, her eyes blazing. \"And I can assure you that I will never do such a thing again. Especially not to you and not in public, private, or anywhere else.\"\n\n\"Good going, Arik,\" Solin said snidely in his head. \"Any more brilliant apologies and you'll be able to sell icicles on the equator.\"\n\nHe glared at Solin. \"Have you a better one?\"\n\n\"Tell her that you couldn't help yourself, you were overwhelmed by her beauty. That she is the most desirable woman you've ever kissed and because of that your emotions ran away with you.\"\n\n\"I don't think that's going to work.\"\n\n\"Trust me, it always works.\"\n\nArik wasn't so sure, but since Solin had more experience on this plane, he decided to listen to him. \"I couldn't help myself, Geary. I was overwhelmed by your desire and I've never kissed a woman more beautiful than you.\"\n\nInstead of placating her, that only seemed to anger her more.\n\n\"That's not what I said, Arik. Jeez.\"\n\n\"You were overwhelmed by my desire?\" she said, enunciating each word slowly so that they conveyed the full weight of her anger. \"What planet are you from?\"\n\n\"Moronia,\" Solin said out loud. \"Every full moon they teleport the Morons to earth and let them loose. Consider this your first encounter.\"\n\n\"Shut up, Solin,\" Arik said from between clenched teeth. Then he projected, \"I told you it wouldn't work.\"\n\n\"It would have had you said it the way I told you to.\"\n\nBefore he spoke to Megeara, Solin glared at him. \"I hope you can forgive my brother, Doctor. He was raised in the mountains where there wasn't any real civilization of any sort. He's basically a goat farmer and he isn't used to interacting with people. He lacks many social skills.\"\n\n\"Oh, thank you. Why not tell her that I wet the bed in my sleep while you're at it?\"\n\n\"If it'll work, I will.\"\n\nMegeara, gave him a gimlet stare. \"Is that the truth?\"\n\n\"Yes,\" Arik said. \"I haven't had much interaction with human people.\"\n\nShe gave a light laugh. \"Human people, huh? Are there any other kind?\"\n\nAs a matter of fact there were, but now didn't seem like the time or place to educate her on that.\n\nGrateful that her mood was lighter, he offered her a small smile. \"Can you forgive me for my behavior? Please.\"\n\nShe looked down at the permits in her lap, then smiled. \"I think I can, but only if you promise to never do something like that again.\"\n\n\"I swear it on Solin's life.\"\n\nSolin sputtered. \"Uh, excuse me?\"\n\n\"I would, but there's truly no excuse for you.\"\n\nHe folded his arms over his chest. \"Ha, ha.\"\n\nGeary rolled her eyes at them, but deep inside, their bantering play made her ache. She and Jason used to bait each other like that. It would drive her father to distraction, and the weird thing was neither she nor Jason knew why they did it.\n\nIt must have been innate sibling behavior that made them constantly pick at each other. Jason had been a smart, good-looking teenager who would have followed her father to the ends of the earth without complaint.\n\nGod, how she missed Jason.\n\n\"Are you all right, Megeara?\"\n\nShe met Arik's concerned gaze before she nodded. \"Sorry, I was just thinking about something.\"\n\n\"Something?\"\n\n\"Personal,\" she finished.\n\nHe nodded and she was grateful that he didn't pursue his line of questioning. Any time she thought about Jason, it made her cry.\n\n\"So what do we do now in preparation for the excavation?\" Arik asked, changing the subject.\n\nWhen she didn't respond quickly enough, Arik pointed toward her permits. \"Remember our bargain?\"\n\n\"I remember.\"\n\nSolin sat forward on the seat. \"What bargain is this?\"\n\n\"That I would allow Arik to participate on the team if he came through with the permits.\"\n\nSolin arched a brow. \"Really? Well, in that case I want to join you.\"\n\nGeary was aghast at his request. The boat was crowded enough with their team, they really didn't need anyone else. And especially not someone who would only get in the way. \"I don't think that's a good idea. You don't strike me as the academic type.\"\n\nA seductive smile curled his lips. \"Oh, I assure you, I'm extremely knowledgeable about this subject. So much so that some swear I actually lived in ancient Greece.\"\n\nYeah, right. Mr. Rolls and Armani in a library\u2014why did that image just not jibe with her? Oh wait, because he might get dust on his Ferragamos.\n\n\"Uh-huh. You study ancient history?\"\n\n\"All the time.\"\n\nShe narrowed her gaze on him. \"Fine. Give me the date of the Peloponnesian War.\"\n\n\"Which one?\"\n\nGeary was amazed that he knew even that much. \"The first one.\"\n\n\"It began 431 B.C. between the Peloponnesian League led by Sparta and the Delian League led by Athens. Archidamus II, who was leading the Spartans, believed that he could keep them in a land war, which Sparta was unrivaled at, that would weaken Athens. Their commander Pericles believed he could use the Athenian navy, which was the backbone of Athenian power, to weaken Sparta. Needless to say, the war lasted much longer than either side intended. And while I find the writings of Thucydides a little on the dry side, I have to admire the way Aristophanes was able to poke fun at the leaders and events of the time.\" He paused for effect. \"Of course this is all an extremely cursory overview that trivializes the entire event into an absurd CliffsNote.\"\n\nGeary had to force herself not to gape at his unexpected dissertation and commentary. \"Okay, so you're not faking. I have to say that I am impressed. It's not often I find anyone who has even the most modest understanding of what I'm talking about.\"\n\n\"Don't be. You'll find both Arik and myself quite useful when it comes to understanding antiquities.\"\n\nShe looked at Arik. \"And what was your favorite event of the war?\"\n\n\"I prefer the Peace of Nicias. Time's too precious to waste it with war and conflict.\"\n\nHis words made her smile. \"But the peace was fraught with skirmishes and ultimately broken.\"\n\n\"Yes, and doesn't it piss you off that there are always assholes who just can't let other people live in peace? Really, some people should get a life.\"\n\nHe had a point with that. In more ways than one.\n\nGeorge pulled to a stop at the marina.\n\nGeary glanced out at her boat where she could see Tory and Teddy sitting together while they compared notes. \"Well, it looks like I'm home.\"\n\nArik reached out and gently took her hand. \"What about our bargain?\"\n\nShe really hated that bargain. When they'd made it, she'd thought he was kidding, but then again, she owed him much for what he'd given her today. \"Fine. You guys can come. We're not going to do anything tonight except prepare for tomorrow's dig. Be here at dawn, sharp. I won't wait for you.\"\n\n\"We'll be here.\"\n\nSolin groaned. \"What is it with you morning people?\"\n\nGeary tsked at him. \"You don't have to join us, you know.\"\n\nArik met her gaze without humor. \"We'll be here.\"\n\n\"Then I'll see you tomorrow.\"\n\nGeorge opened her door and she left the men to themselves.\n\nArik didn't speak until after George had shut the car door. Then he turned back to Solin. \"What do you think you're doing?\" he projected the thought to his brother to keep the driver from overhearing them.\n\n\"Nothing,\" Solin said out loud.\n\nStill, Arik wasn't placated. He knew Solin was up to something. Why else would he come on Megeara's expedition when he had no reason to? \"Megeara belongs to me.\"\n\nSolin scoffed at Arik's anger. \"I have no interest in her, believe me. She's all yours.\"\n\n\"Then why are you coming?\"\n\nHis face sobered. \"To make sure you don't screw up. Have you any idea how bad it would be if they learned what you are?\"\n\nRelieved that was Solin's only reason, Arik sat back in the seat. \"They won't.\"\n\n\"No, they won't, which is why I'm going with you. I'll be there to help cover any blunders you make. Now can you at least say, 'Thank you, Solin'?\"\n\nHe said it from between clenched teeth and with no real sincerity. \"Thank you, Solin.\"\n\n\"You're welcome.\" He pressed the button for his intercom. \"George, it appears my brother is in desperate need of clothing.\"\n\n\"Very good, sir. I'll head to the store right now.\"\n\n* * *\n\nTory met Geary on the gangway. Her face was so hopeful that it brought tears to Geary's eyes. \"Well?\"\n\nGeary forced herself not to smile as she shook her head no.\n\nTory cursed until Geary handed her the permits. It took a full ten seconds before Tory realized what they were.\n\nShe literally jumped up and down. \"Oh my God!\"\n\n\"Yes.\"\n\n\"Oh my God!\"\n\n\"Yes, Tory.\"\n\nScreaming, she ran up the gangway, then stopped at the top and ran back to Geary. \"They're not forgeries, are they?\"\n\n\"No.\"\n\nShe screamed again and ran to tell Teddy.\n\nGeary laughed at Tory's exuberance. For once she was acting her age.\n\nBy the time Geary reached the deck, the entire team was there.\n\n\"You really got them?\" Teddy asked.\n\n\"I got them. We're starting at dawn.\"\n\nYou would have thought she'd brought home the winning lottery ticket\u2014but then, maybe she had. All of them had been aching for this for years and now their patience had paid off.\n\nThey were going excavating.\n\nGeary paused as she noted the reservation on Kat's face. \"Is something wrong, Kat?\"\n\n\"Oh no. I'm just stunned. I have to say that I never expected this.\"\n\n\"Yeah, I know. It's unbelievable.\"\n\n\"Yes,\" Kat said coldly, \"it is.\"\n\n\"Aren't you excited?\" Tory asked.\n\n\"Thrilled.\" But Kat's tone belied the word.\n\nGeary frowned, wondering what had Kat so upset. But the rest of the team's happiness and her own quickly had her forgetting all about it as they planned what they were going to do once they'd found Atlantis.\n\nKat stood back as the group headed below the deck and a chill went down her spine. Geary had found the location and Arik had just handed her the key to the door.\n\n\"You can't let them snoop around the remains, Katra....\" Artemis's words rang in her ears. \"If you do, they will find the seal and release the Destroyer back into this world. Should Apollymi ever be freed, I know I don't have to tell you what she would do to us. To me. You can't let her be free. Ever.\"\n\nThat was easier said than done, especially since Kat could hear Apollymi calling out to her from her prison in Kalosis. Apollymi wanted her freedom as badly as the rest of the gods wanted her trapped.\n\nAnd Kat was caught in the middle.\n\nBut at the end of the day, she knew the real reason she had to keep Atlantis hidden. If anyone ever learned the truth of what had happened to the island, the one person who would be destroyed was the one person she loved more than anyone else.\n\nAcheron Parthenopaeus.\n\nEleven thousand years ago he'd been enslaved to Artemis, and he had been mankind's champion ever since. He was the leader of the goddess's army of Dark-Hunters who protected humanity from the demons who preyed on them. And while he protected the Dark-Hunters and humans, no one had ever watched his back.\n\nExcept for Kat.\n\nFor him alone, she would do whatever she had to to keep this place sacred and buried. And if that meant sacrificing every person on this boat, including Geary, so be it.\n\nNo one would ever hurt Acheron. Not if Kat could help it.\nCHAPTER 8\n\nGeary was laughing with Teddy and Scott when she caught a shadow out of the corner of her eye. Thinking it was Kat come to join them, Geary turned her head to welcome her in. Until she saw Arik standing in the doorway\u2014just as she'd imagined him earlier that day. A chill of d\u00e9j\u00e0 vu washed over her as her humor fled. There was no denying the hunger in his light eyes as he watched her. The predatory gleam. All he had to do was extend his hand toward her and it would be her dream all over again.\n\nWas all of this a premonition?\n\nTory caught Geary's distraction and turned to look. Like Geary, she went mute, and as the rest of the crew noted the women's behavior, they followed suit. Suddenly their laughter and good cheer turned into a rigid and questioning silence.\n\nGeary cleared her throat as she approached Arik, who didn't seem to be the least bit concerned or disturbed by their silence. \"What are you doing here?\" she asked in an emotionless tone.\n\nOblivious to the tenseness of the others, he shrugged. \"I know how much this means to all of you, so I wanted to join in your celebration... that is, if you don't mind.\"\n\nThat seemed to soothe everyone in the room but her.\n\n\"Come on in,\" Teddy said, handing Arik a plastic cup of Cristal\u2014yeah, it was a cheap way to serve the best, but they'd been saving the champagne for this particular event. Geary had two more bottles stashed for when they found hard-core proof of Atlantis's location. Then the real celebration would begin.\n\nThis was, Geary hoped, just the warm-up, with everyone drinking except the divers and Tory.\n\nThe team returned to their party.\n\nMoving closer to her, Tory touched her gently on the arm. \"Are you okay?\"\n\n\"I'm fine,\" she said with a fake smile before she made her way from Tory to Arik.\n\nScott clanked his cup with Arik's. \"Man, we can't thank you enough for getting those permits. You really have no idea what this means to all of us.\"\n\nArik inclined his head before he took a drink of champagne. As soon as it touched his tongue, he gasped and sputtered, then immediately started choking.\n\nScott pounded Arik on the back as Geary took the cup from his hand.\n\n\"Are you all right?\" Teddy asked.\n\nCoughing, Arik nodded. \"I wasn't expecting it to be so...,\"\u2014he curled his lips\u2014\"so strange tasting.\"\n\n\"Strange?\" Teddy asked, gulping his. \"This is the best shit to be had.\"\n\nGeary remembered what Solin had said about Arik's sequestered upbringing. \"Have you ever had champagne before?\"\n\nHe shook his head.\n\nScott gaped. \"Get out. Where you been living, under a rock?\"\n\nArik cleared his throat. \"Not exactly. But close.\"\n\nGeary set his cup aside. \"Arik was raised in the Greek countryside, away from civilization.\"\n\nScott shuddered. \"Man, that sucks. I went up there once a couple of years ago and it was enough to convince me that I like American plumbing, if you know what I mean. And since you're from up there, I know you do.\"\n\nTeddy and Scott exchanged a horrified look before Geary pulled Arik out of the room so that she could speak to him alone. Not that they had much privacy in the hallway, but at least they weren't in direct line of sight and hearing of the others.\n\nShe crossed her arms over her chest as she narrowed an agitated glare on him. \"I thought you were going with Solin.\"\n\nArik looked less than apologetic. In fact, he was actually charming as he offered her a lopsided grin. \"I don't want to be with Solin. I want to be with you, especially while you're happy.\"\n\nOn the one hand that was flattering, but on the other it made her nervous. She didn't like the feeling of being stalked by him... granted, most of her discomfort was caused by the fact that she'd seen him in her dreams these last few months. That wasn't his fault, but still... \"Thank you, and while I can appreciate that, I don't like clingy men. I've always needed my personal space, okay? I mean, really, I barely know you.\"\n\nArik nodded as a fierce pain settled in his chest from her words. It constricted his breathing and made him literally ache. What was this sensation? He'd never felt anything like it before. Strange emotions seemed to be gathering in his throat to choke him. It was a physical pain and yet there was no physical reason for it. He didn't understand.\n\n\"Please, Megeara. Don't be angry at me. I don't have a lot of time left and I don't want\u2014\"\n\nGeary cocked her head at his offhand comment. \"What do you mean you don't have a lot of time left?\"\n\nHe tensed as if he'd let something slip he hadn't meant to. \"I meant... It's nothing. Forget I said anything.\" He started past her.\n\nGeary gently pulled at his arm to stop his retreat. \"Wait a sec. Go back to the 'lot of time' comment. What did you mean by that? Are you leaving to go back to the mountains?\"\n\nHe was sheepish now. Not the man in charge who was on comfortable ground. Something about his demeanor reminded her of a little boy. \"No.\"\n\n\"Are you going back to Solin's or your house?\"\n\nHe shook his head.\n\n\"Then what exactly did you mean?\"\n\nHe met her gaze and the anguish there actually made her heart ache for him. \"I don't have a lot of time left here in this world. I'm going to have to leave it soon... very soon.\"\n\nIn the back of her mind that was what she'd suspected he meant, but having him say it out loud cut through her a lot more than it should. She'd lost so many people who were close to her that the thought of his dying so young tore her apart. \"Are you telling me that you're dying?\"\n\nArik hesitated. He didn't want to lie to her, but in a way it wasn't a lie. He would cease to exist as a human in two weeks and he would never again be here.\n\nUltimately, he settled on being completely honest. \"My body was given an expiration date.\"\n\nGeary covered her mouth with her hand as a wave of pity washed over her. He looked so healthy and prime. How could a man like this be dying? It didn't make sense. \"Are you sure?\"\n\nHe gave a light, nervous laugh. \"Yeah. I couldn't be more positive.\"\n\n\"Oh, Arik, I'm so sorry.\"\n\n\"Don't be. I'm just glad I have any time here at all.\"\n\nThose words touched her deeply. That he could look at the positive at a time like this and not be angry or bitter over the injustice spoke volumes about his character. She couldn't imagine being told that she only had a finite time to live. How awful it would have to be.\n\n\"I don't understand why you'd help me get my permits when I'm sure you had a lot better things to do with your life.\"\n\nHis handsome features softened. \"I wanted you to have your dream before I left.\"\n\nShe couldn't understand his altruism. People just weren't that kind. \"Why?\"\n\nHe reached out and cupped her face in his warm hand. \"You live your life like it's a rare treasure to be savored. You take pleasure from the simplest of things and you never take them for granted. I saw the joy on your face and the life in your eyes when you cradled the permits to your chest. I've never seen anything lovelier. I actually thought you would cry just from the joy of touching them. I've been numb all my life, Megeara, but you... you feel on a level that I can't even imagine, and for a little while I wanted to feel that, too.\"\n\nAnd she felt like crying now at the thought of this gentle, considerate man dying. \"How long do you have?\"\n\nGrief tainted the luster of his eyes. \"Two weeks.\"\n\n\"Two weeks?\" she repeated, her chest tightening even more. \"Are you kidding?\"\n\n\"No.\"\n\nThere was no mistaking the sincerity in his gaze. The man was really dying, or at least he believed it. \"Well, maybe your doctor's wrong. Have you gotten a second opinion?\"\n\n\"I don't need one,\" he said with a bitter laugh. \"You can trust me on this. In two weeks, I won't be here, at least not with a human body.\"\n\nAnd he had come to help her in his last days....\n\n\"Oh, Arik,\" she breathed before pulling him close and hugging him. \"I'm so sorry.\"\n\nArik couldn't breathe as her breasts pressed fully against his chest. Heat coiled through him, making his entire body burn. His groin tightened and swelled as he thought of how many times they'd touched like this and yet he'd never really felt it.\n\n\"Is there anything I can do?\"\n\n\"Just be with me for a little while.\"\n\nWhy was that so important to him? \"Don't you have a girlfriend or family you'd rather be with?\"\n\n\"Only Solin, and honestly, he's not this soft. Even if he was, it'd be gross.\"\n\nStifling a laugh, she tightened her hold on him.\n\nArik leaned his cheek against her head and inhaled the sweet scent of sea and woman. There was a faint trace of peaches on her skin as her hair tickled his lips. He closed his eyes and savored the feeling of her against him. It was miraculous and wonderful, and it left him cold that he'd have to leave this behind and return to his sterile world again.\n\nAnd she would be dead because of him....\n\nHe winced with the thought. Regret had been something he'd never experienced as a Skotos, but he felt it now and it stung deep.\n\nWhat have I done?\n\nThe only consolation he had was that when he returned to the Vanishing Isle, he wouldn't have human emotions to haunt him. No regret or sorrow.\n\nNor would he have Megeara's dreams....\n\nAn aching pressure tore through him. It was raw and open and made him want to shout from the weight of it.\n\nHow did humans live with these feelings all the time? Honestly, they were enough to wear him out. To make him afraid to even move for fear of tweaking them to full capacity.\n\nBasically, emotions sucked. Hades had been right. The gods had done the dream gods a favor by making them numb. And even knowing that, he still savored his emotions.\n\nMegeara stepped away from him and took his hand in hers. The softness of her skin radiated through his body. \"C'mon. Let's go back to the party and celebrate what you've given us.\"\n\n* * *\n\nAs she stood at the prow, Kat felt a bitter chill that had nothing to do with the weather. It whispered against her skin like a light kiss, and it was the last sensation she wanted to experience tonight.\n\nUnfortunately, she'd known this was coming.\n\n\"What the hell is going on here?\"\n\nShe turned slowly toward the deep baritone voice to find an exceptionally tall and handsome man whose violet eyes were cutting even in the darkness. His dark blond hair was windswept and streaked with lighter tones that only emphasized the masculine beauty of his face. Zebulon, or ZT as he preferred to be called, was a creature of extreme power and nastiness.\n\nLike the others of his kind, no one knew when or where he'd been born. All they knew was that he held enough power to kill a god with a single act. God killers, or Chthonians as they preferred to call themselves, were a rare breed and ZT was one with an unholy attitude.\n\nHe stood before her now, dressed in jeans and a long-sleeved maroon T-shirt that had the Greek phrase \u03a3\u03b1\u03c2 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c7\u03c9, \u025b\u03af\u03bc\u03b1\u03b9 \u03d5\u03bf\u03b2\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf\u03c2\u2014\"I am watching you, be afraid\"\u2014stenciled in gray on it. How apropos, since that was what he did. Aeons ago, he and his brethren had banded together as policemen for the gods. They were the check-and-balance system for the universe.\n\nUntil they'd turned on one another for reasons known only to themselves.\n\nNow the handful who survived watched over mankind with a bitter eye and with no real leader. Rather, they were the epitome of a cold war where they were only kept in check by one another and rarely got along\u2014unless it was to go after a god who'd stepped over whatever line they'd drawn.\n\nBecause of their hostility toward one another, the earth had been divided between them for safekeeping and they were highly territorial.\n\nGreece and her surrounding areas belonged to ZT, and he tolerated very few treading on his turf, which meant that anytime Kat ventured here she got a nice visit from the crank.\n\nThe first time they'd met, she'd been a curious child who'd only wanted to see a chariot race. Her mother had sent her out with a chaperone. The sun had been bright when all of a sudden ZT had appeared out of nowhere and quickly scared her by telling her that if she ever broke Chthonian law, he would gleefully kill her.\n\nShe'd \"loved\" him ever since.\n\n\"Long time no see, ZT.\" Total sarcasm, since he kept a permanent watch on her anytime she was on earth. They'd crossed paths only two weeks ago when she'd been in the market and had sneezed, which caused her powers to shatter a nearby window. ZT had been pissed that she'd almost betrayed herself and, true to his badass form, had let her know it.\n\n\"Don't be coy with me, Katra. I know about the permits. How did that happen?\"\n\nShe shrugged. \"It was an unforeseen event, but I have it under control. There's no need for you to bother yourself.\"\n\nHis eyes flared in the darkness as he closed the distance between them. Raw esoteric power emanated from him, raising the hairs on her body. He tilted his head as if he was sensing the ether around them.\n\n\"A human god?\" he whispered.\n\n\"His time here is short and he has no powers. Again, it's nothing to worry over.\"\n\nZT curled his lip at her. \"I will decide what I worry over. Not you.\" He let out a vicious hiss. \"He's meddling with human affairs.\"\n\nEven though she knew it was foolish, she scoffed at him. \"So do I.\"\n\n\"Which is why you're on my radar. I don't like the games that Artemis plays and I like your part in them even less.\"\n\n\"Then why don't you stop her?\"\n\nHe gave a bitter laugh before he gave Kat a cutting glare. \"You're so naive.\"\n\nPerhaps she was, but it didn't change the fact that he was overreacting. \"You don't have to concern yourself with this, ZT. Really.\"\n\nA muscle worked in his taut jaw as he looked out over the dark water. When he spoke, his tone was flat and emotionless. \"I have you\u2014a god of mixed heritage\u2014on an expedition that could unleash the Destroyer from her hole. Arikos, another god, on the same team who is masquerading as a human. The demigod Solin, who I have to ride herd on constantly anyway, who gave them their permits. Megeara, a human who is sensitive and subjective to the voices of the gods. And the pissed-off goddess, Apollymi, who will do anything to be free, and once free wouldn't hesitate to destroy every one of us.\" He turned that deadly stare to Kat. \"I can't imagine why I'm concerned over this, can you?\"\n\n\"Granted it looks a little bad when you put it that way, but I can assure you that I won't let them near Apollymi's seal.\"\n\nHis doubting look was really beginning to piss Kat off. \"And does the name Pandora hold any meaning for you? Anytime you allow a human near a box they shouldn't open, what do they invariably do?\"\n\n\"It'll be different this time.\"\n\nHe made a rude noise in the back of his throat. \"Don't be arrogant, Katra. I'm tired of cleaning up the messes left by gods who thought they could do better.\" He turned toward her and there in the moonlight she saw something she'd never seen on him before. A vicious scar that ran from his hairline to his neck. It looked as if someone had once sliced open his face.\n\nBut as soon as she saw it, it faded and left him handsome and unscarred. \"Keep the seal hidden, Katra. Apollymi cannot go free.\"\n\nBefore Kat could respond, Tory came up on deck.\n\nBoth Kat and ZT froze as the girl innocently approached them.\n\nShe gave Kat a curious frown before she pushed her glasses up on her nose. \"Are you all right, Kat?\"\n\n\"Fine, Tory. I just have an old friend visiting, but he was about to leave.\"\n\n\"Oh, okay. Geary wanted me to check on you. She said you weren't feeling well.\" And before Kat could say anything, Tory held her hand out to ZT. \"Hi, friend of Kat. I'm Tory Kafieri.\"\n\nKat expected the god-killer to tear Tory's hand off or make a nasty remark. Instead he took her hand in his and shook it gently. \"ZT.\"\n\n\"ZT. What a cool name.\" She smiled up at him. \"Well, I won't bug you two anymore. It's obvious you want to be alone. I'll tell Geary that you're fine, Kat. Nice meeting you, ZT.\"\n\n\"You, too, Tory.\"\n\nKat actually gaped as Tory left them and ZT didn't blast her. She waited until Tory was gone before she spoke again. \"So you can be nice. Who knew?\"\n\n\"My niceness has a very low threshold and that little girl just sucked it dry, so don't push me, Olympian. I don't want so much as a single Atlantean stone overturned. Guard it with your life because the next time I come here, that's the price I'm going to demand for your incompetence.\"\n\nAnd before she could even flinch, he vanished.\n\n\"Nice talking to you, ZT,\" she called out after him. \"I so look forward to your visits. Next time we'll do pastries, 'kay?\"\n\nSighing, Kat rubbed her temple. This was turning into a beautiful day for her. She couldn't wait to see what was going to happen next.\nCHAPTER 9\n\nM'Ordant walked slowly through the hall that led to the Onethalamos... just in case someone was watching. It was in the Onethalamos that the three leaders of the Dream-Hunters, M'Ordant, D'Alerian, and M'Adoc, gathered to make policy, keep peace, and...\n\nIssue death warrants.\n\nProtected from the other gods and zealously guarded, this room contained all the secrets that the three of them would kill and, more important, had killed to maintain.\n\nOne of them being the fact that the three of them were no longer bound by Zeus's curse. Their feelings had come back, and with every year that passed, those emotions grew stronger, as did these Dream-Hunters' need to protect them. But outside the doors of the Onethalamos no one could ever know.\n\nInside the room, however, anything went.\n\nThe moment M'Ordant walked through the oversized gold doors, he slammed them shut with his thoughts.\n\nM'Adoc looked up from his book with an arched brow. \"Careful, adelphos. The last thing we want is for someone to know you have a temper.\"\n\n\"Yeah, and in about three seconds so will you.\"\n\nLaying the paperback aside, M'Adoc sat back in his cushioned chair to eye M'Ordant suspiciously. \"Meaning?\"\n\n\"We have a renegade.\"\n\nM'Adoc laughed. \"And this is different how?\"\n\n\"Oh, give me a second on this one,\" M'Ordant said, approaching M'Adoc's chair. \"We're not talking one of our guys went Skoti. It should be so simple. No. One of our Skoti just went human.\"\n\nIt took several seconds for that shocking bit to fully penetrate M'Adoc's mind. \"Excuse me?\"\n\nM'Ordant drew a ragged breath before he explained. \"Arikos has cut a bargain with Hades. He wanted to be human for a few weeks. The price. One human soul.\"\n\nThe color faded from M'Adoc's face an instant before rage darkened his cheeks. \"What is he doing?\"\n\n\"Fucking things up for the rest of us.\" M'Ordant slammed his fist down on the table. \"I swear, Zeus as my witness, I'm going to tear him limb from limb. How could he be so fucking stupid?\"\n\nM'Adoc shook his head. \"Enough with the vulgarity. I know you love the word, but save it.\" He growled low in his throat, letting M'Ordant know he was equally willing to break eggs, heads, and bones in this as M'Ordant was. \"Zeus and the others will question how Arik could have developed a desire so strong it would make him bargain with Hades to quell it.\"\n\n\"Yeah, and there will be hell to pay if they come knocking on our doors. If they ever find out that the curse is weakening...\" He didn't finish the sentence. He didn't have to. Unlike Arikos, he, M'Adoc, and D'Alerian were the first ones they'd rounded up at Zeus's order and punished for the Oneroi's ability to manipulate dreams for their own personal gain.\n\nTo this day, the three of them, who had been innocent in the crime but held up as examples to the others, could still feel the pain and humiliation of that torture. When it came to absolute punishment, no one could match a Greek god bent on vengeance. It was what kept the three up at night and seeking the Skoti to make sure they didn't violate the laws Zeus had set for them. The three would do anything to not relive the merciless hell they'd gone through\u2014and they were the three whom the gods would punish again if they ever learned the secrets M'Ordant, M'Adoc, and D'Alerian carried.\n\nNo one would show them mercy, and they knew it.\n\n\"Does anyone else know?\" M'Adoc asked.\n\n\"Just Hades and us.\"\n\n\"How did you find out?\"\n\nStraightening up, M'Ordant folded his arms over his chest. \"I make it my business to keep an eye on Hades and Hypnos.\" They had been the two most mischievous gods who'd gotten the rest of them cursed. \"When they sleep, I'm there every minute of it. I just don't let them know I'm spying.\"\n\n\"Good man. We have to contain this. Call out the Dolophoni. We need that bastard dead, then if Zeus finds out, we can tell him that it was an aberration unique to Arikos that we handled.\"\n\n\"You think he'll buy it?\"\n\n\"If not, we'll have to find some way to sell it to him.\" M'Adoc's phosphorescent blue eyes flickered with malice. \"I don't know about you, but I have no intention of bleeding for another one of these assholes.\"\n\nM'Ordant arched a brow at M'Adoc's word choice. He generally profaned profanity\u2014which told M'Ordant just how determined his brother was. M'Ordant held his hand out to M'Adoc. \"I hear you, adelphos, and we're in definite accord.\"\n\nM'Adoc wrapped his hand around M'Ordant's and shook it. \"Arikos dies.\"\n\n* * *\n\nGeary paused in her chat with Thia to watch Arik bite into one of the Hostess cupcakes that Tory had broken out for the celebration. His eyes actually glowed with pleasure as he tasted it.\n\nHis smile was broad and enchanting. \"This is incredible.\"\n\nTory laughed at him. \"I can't believe you've never had one before. Man, that would stink, to grow up without Hostess. This was the staple of my elementary-school lunch box.\"\n\nHe practically inhaled it. \"Do you have any more?\"\n\n\"Hang on.\" She ran from the room.\n\nExcusing herself from Thia, Geary made her way to Arik, who was frowning at the black cake that was stuck like glue on his fingers. Geary picked up a napkin as she approached him. \"It must be bare in the mountains.\"\n\nHe licked the sugar from his lips before he responded. \"I know what things are, but I haven't been able to experience them before. Like this cake. It's really very good.\"\n\n\"As the girth of my hips will attest.\"\n\nBy his face, she could tell he didn't understand she was calling herself fat. For some reason, she found that as endearing as his amazement that he couldn't easily get the cake off his fingers.\n\nSmiling, she took his hand in hers so that she could help him out. She paused at the sensation of his flesh against hers as she wiped the pads of his fingers with the napkin. He had the most beautiful hands. Large and masculine, they made her want to lick them clean. In her dreams, she would have done that in a heartbeat.\n\nHe lifted his hand so that he could place a very sweet kiss on the back of her knuckles. \"Thank you.\"\n\nGeary swallowed as white-hot desire shot through her. What was it about this man that made her literally melt? \"You're welcome.\"\n\nTory came running back in with her entire stash of junk food that she kept in a large shoe box that normally lived under her bed... protected by Mr. Cuddles of course. \"Okay, Moon Pies.\"\n\nGeary laughed in disbelief. \"You're going to share a Moon Pie? Now? You know you can't get any more of those until you go back to the States, right?\"\n\n\"It's for a good cause. We need more addicts. Besides, there's always Grandpa to bail me out with an emergency shipment if I get too desperate.\" Tory handed a chocolate-coated Moon Pie to Arik.\n\nGeary shook her head. \"Oh no, if you really want to be wicked to him, nuke it first.\"\n\nTory made a slight face of contradiction. \"Yeah, but given his reaction to the cupcake, that might overload his taste buds with pleasure and kill him.\"\n\nThis was true. The mighty Moon Pie could be orgasmically deadly when nuked... it rated right up there with the infamous Australian Tim Tam Slam and the deep-fried Twinkie. \"Good point. To be on the safe side, the first one should always be tasted at room temperature.\"\n\nArik frowned while Tory unwrapped the small, round cookielike object. And as soon as he tasted it, he was in obvious ecstasy. \"Oh my God, that's good.\"\n\nShe exchanged a mischievous grin with Tory. \"Reese's,\" they said in unison.\n\n\"Reese's?\" Arik asked, puzzled.\n\n\"Oh yeah,\" Geary laughed. \"You'll be dying the minute you bite into one.\" She started digging through Tory's shoe box until she found one. She made an evil noise in triumph as she pulled it out. \"You know, Tor, I have no idea how you stay so skinny eating all this crap. I swear I gain ten pounds just rooting through it.\"\n\n\"I'm still growing.\"\n\nGeary snorted. \"So am I, but it's out instead of up. Remind me tomorrow to start my diet again.\"\n\nArik scowled at her. \"I think you're beautiful as you are. Why would you want to change?\"\n\nSomething warm tickled her at his words. \"You're just trying to flatter me.\"\n\n\"No,\" he said seriously. \"I'm only telling you the truth.\"\n\n\"Awww,\" Tory said dreamily. \"He's such a sweetie. Can we keep him?\"\n\nGeary gave a nervous laugh. \"He's not a puppy, Tory.\"\n\n\"Yeah, but we did fish him out of trouble. In some cultures that would make us responsible for him forever.\"\n\nArik gave her a hopeful grin. \"And I wouldn't mind being kept for a while.\"\n\nGeary shook her head at them. \"You two are a terrible combination. Kind of like gasoline and fire.\" She looked past Arik's shoulder to see Kat entering the room finally. She had a very sour look on her face, as if something wasn't settling right with her. \"Hey,\" Geary called out to Kat, catching her attention. \"You okay?\"\n\nWith an extremely fake smile, Kat walked over. \"Yeah. Fine.\"\n\nTory set her shoe box to the side. \"So where's the gorgeous ZT?\"\n\nGeary frowned at the name she didn't recognize. \"Who?\"\n\nTory made a clicking noise with her teeth like a woman calling a horse. \"Kat was with this really cute guy on deck when you sent me up to check on her.\" She looked back at Kat. \"Did he not stay?\"\n\n\"No, he had to leave.\"\n\n\"ZT?\" Arik asked with a speculative gleam in his eyes. \"As in Zebulon?\"\n\nKat gave a very curt nod.\n\nTory looked back and forth between them curiously. \"You know him, too, Arik?\"\n\n\"He knows him,\" Kat said in an odd tone. Her gaze was cutting as she met and held Arik's. \"He sends you his best.\"\n\nAll emotion fled from Arik's features. \"I'll bet he does. How is old ZT doing, anyway?\"\n\n\"Charming as ever.\"\n\nThe sarcasm between them was so thick it could have been carved into an ice sculpture.\n\nArik put the unopened Reese's back inside the box as if he'd lost his appetite. \"Nice to know some things never change.\"\n\nGeary scowled even more. \"How is it that you two have a mutual friend when you didn't know each other before today?\"\n\n\"It's a small country,\" Kat said evasively. \"The old families tend to stick together and Arik's has probably known ZT's for quite some time now.\"\n\n\"Yes,\" Arik said with a wry grin. \"He's like a rash for which there's no cure. It only goes away for a bit before returning unexpectedly to ruin every pleasurable experience. He should have been named Herpes rather than ZT. Or maybe just Herpes Z, since he's a very special irritant.\"\n\nKat laughed. \"Very aptly put, hell, it's even Greek and he is creepy\u2014I'll give you that. But I wonder if he knows how you feel?\"\n\n\"I'm sure he does. He's rather astute and I'm anything but subtle.\"\n\nOkeydokey, this was getting a bit out of hand, and she wanted to avoid a herpes discussion in front of the walking\/talking medical textbook who was only fifteen years old. So, trying to stave off the animosity and veer to safer territory, Geary stepped forward. \"And on that happy note, children, I think we should all turn an eye to retiring. It's been a long day and we have a really big day tomorrow.\"\n\n\"Hear! Hear!\" Teddy concurred from the other side of the room. \"As long as we've been waiting for this excavation, I want to make sure that we all have one that's error free. We can't afford a single mistake, people.\"\n\nThere were a few grumbles, but overall everyone agreed. If they were to get cracking at dawn, then they needed their rest.\n\n\"So where's Arik bunking?\" Tory asked.\n\nGeary hesitated. There really wasn't any place to put him without inconveniencing one of the guys. Their rooms were cramped at best, and she was sure none of them wanted to bunk with a stranger.\n\nArik gave her a hopeful look that brought an unexpected smile to her face. \"I already have a roommate.\"\n\nThere was no missing Arik's disappointment. \"Who?\"\n\nTory rocked back and forth on her feet. \"Me and Mr. Cuddles.\"\n\n\"Yeah,\" Geary said, nodding, \"and Mr. Cuddles is a jealous sort. He doesn't share us well.\"\n\nArik didn't miss a beat. \"Does this mean I'll have to fight him?\"\n\n\"You'd never win,\" Tory said sweetly. \"Mr. Cuddles cheats. You think he's just a pushover teddy bear, but he's vicious, I tell you. Vicious.\"\n\nKat cast a speculative look at Arik. \"You could throw him on a hammock on deck.\"\n\nGeary considered it. It actually wasn't a bad thought. \"We will be up at dawn, so it wouldn't really wake him once we hit the deck....\"\n\nTory leaned toward Arik. \"Bet you're thinking you should have gone home, huh?\"\n\n\"No,\" he said, his tone sincere. \"I had a great time tonight.\" He looked at Tory and smiled. \"And you're right. The mighty Moon Pie is the best. Thank you for sharing your treasure with me.\"\n\n\"Anytime.\" She rose up on her tiptoes to kiss his cheek. \"Good night, Arik. I'll see you in the morning. Sweet dreams.\"\n\n\"You, too, Tory.\"\n\nKat gave him an odd look before she bid them good night and followed Tory to the hallway.\n\nThia came forward with a calculating gleam in her eyes. \"Well, if no one else wants him... I could share my bunk with him.\"\n\n\"Go to bed, Cynthia,\" Geary said sharply, \"before Justina kills you for pimping out your shared cabin.\"\n\nThia gave a weary sigh. \"I was just trying to be friendly. They always say you shouldn't sleep alone in a strange bed.\"\n\n\"Yes, and they didn't have their cousin on board to watch over them, either. One who would report any sleazy behavior to their mother. Good night, Thia.\"\n\nTossing her hair over her shoulder in a huff, she left them alone.\n\nGeary gave Arik a once-over as she realized he'd have to sleep in his borrowed jeans and shirt. \"What are you going to do for clothes?\"\n\n\"Solin said he'd return with something for me to wear in the morning.\"\n\n\"Ahh, okay. Well, I guess I'll grab a hammock and meet you topside.\"\n\nArik started to offer to go with her, but she was already feeling smothered by his presence. He would back off for a bit even though it was the last thing he wanted to do. \"All right. I'll see you on deck.\"\n\nHe headed for the stairs to the upper deck while she went the other way, deeper into the boat. He paused at the handrail, amazed by the slick feel of it. Nothing here was what he'd expected. Especially not the food. He didn't know why the gods made such a fuss over ambrosia and nectar, given how wonderful human food tasted.\n\nPerhaps the gods were in denial because they were only supposed to have the best and it bothered them to think that mankind had perfected some of their world even while they battled one another.\n\nOr maybe the gods honestly didn't know better.\n\nDismissing the thought, he climbed up to stand on deck as a light breeze whispered against his skin. The sensation was exquisite, but it was nothing compared to the sight of the city that sparkled over a velvet black landscape. The water lapped quietly against the boat as a faint jingle of music and laughter reached him. No wonder humans didn't want to die. Their world was remarkable, and their lives were made even more precious by the fact that they had so little time to spend here.\n\nHow did they do it? How did they exist knowing that the specter of death constantly haunted them? It was enough to depress anyone, and yet for the most part they were happy with their lot. They ignored their impending doom and marched on toward their death with dignity and grace while finding shards of happiness to content them.\n\nIt was truly amazing.\n\nBut then, they didn't know how long their lives would be. Decades or weeks. So they prepared for the worst and expected the best. It was actually rather noble.\n\nAnd how strange it must be for Solin and ZT and the others to live this close to walking corpses. No wonder they were closed off from everyone. Who would want to reach out and befriend someone when he or she could be torn away from you at any moment? When there was no chance of a lasting relationship. Everything here was doomed to finality.\n\nIt must be horrifying for them all.\n\nArik looked back the way he'd come and wondered what Megeara would think if she knew her life was about to end.\n\nBecause of him.\n\nHe went cold with the thought. Now there was something he couldn't dwell on. He'd been naive when he'd made his bargain with Hades. Now there was no way back from it. As M'Ordant and Wink had pointed out, there would be others to beckon Arik back to this realm once Megeara was gone.\n\nAnd yet he knew better. She was unique in this place of overwhelming emotions. In all the centuries, he'd never met anyone like her.\n\nWhere he lived, the human world seemed vague and unreal. But here it was vivid and extreme. Too extreme perhaps...\n\n\"Here you go.\"\n\nHe turned to find Megeara heading toward the prow. Her face was silhouetted by the moonlight.\n\n\"You're lucky we have these. Otherwise it'd be a pallet on the deck for you.\"\n\nArik watched as she set about stringing the hammock out. \"You like to sleep in the hammock on cool nights, don't you?\"\n\nShe looked up with a panicked expression. \"How do you know that?\"\n\nHe knew it from her dreams, but he didn't tell her that, since his goal was to soothe and seduce her, not scare her more. \"From the look on your face and the skill you're showing as you lay it out.\"\n\nShe blushed before she returned to her task. \"Yeah, I like to look up at the stars at night.\"\n\nHe knelt down to help her as she untangled some of the lines. \"And what do you find when you look up there?\"\n\nHer hands worked deftly to straighten out the canvas and lace the cord through the grommets. \"When I was a little girl, my father used to lay on deck with me and my brother and point out the constellations. Then he'd tell us stories about how the Greek gods supposedly formed them.\"\n\nHe could hear the bittersweet agony of her memory. She'd both loved and hated her father. It was a dichotomy Arik barely understood. He held no feelings whatsoever for his parents. But then, he'd never really known either one of them. Morpheus had too many children to pay attention to any one, and Arik's mother, Myst, couldn't be bothered. She was a carefree goddess who held no real affinity for anyone or anything. At least nothing he knew of.\n\nIt wasn't that he was angry over it. He was truly ambivalent. It was how things were in his world, so there was no feeling there for his parents whatsoever, not even while he was human.\n\nBut it made him wonder what it would be like to love the way Megeara loved. To feel that pain of betrayal when the person was no longer there. To have that surge of joy when the person was....\n\nHe helped her anchor a corner. \"So which story was your favorite?\"\n\nShe tugged the line to make sure it was taut. \"Orion. I always thought it was cruel and tragic that Artemis loved him, and that her own brother tricked her into killing him because Apollo was jealous and hated the fact that she was in love with a mere mortal.\"\n\n\"That's only one version of the story. The other is that Artemis killed him because he raped one of her handmaidens.\"\n\n\"I've heard that one, too, but I believe the first one.\"\n\n\"And why is that?\"\n\n\"I don't know. It just seems right.\"\n\nShe was astute and part of him wanted to confirm it for her, but he didn't dare. She'd spent her whole life studying the gods and ancient civilizations, looking for corroboration that they'd all existed, and here he was, living proof, right beside her. He wondered what she'd do if she ever learned that he was one of the gods, along with her friend \"Kat.\"\n\nIt might be a little too much to expect Geary to take it in stride.\n\nGeary was a bit nervous as she finished securing the hammock to the hooks. It hung about a foot above the deck. Not too high, but not so low it wouldn't be comfortable. Her only concern was that he might catch a chill even with blankets.\n\nWould that worsen his condition? Not that she knew what was wrong with him, but still the last thing she wanted to do was take even a moment of his life away with more illness.\n\nWhich begged the question of what was wrong with him? She wanted to know but didn't want to remind him of how short his life would be. It seemed somehow wrong.\n\nInstead, she gestured toward the hammock. \"All yours.\"\n\nHe tsked at her. \"I really wish you were talking about you and not the hammock.\"\n\n\"Yeah, I'll bet you do.\"\n\nAs she started past him, he pulled her toward him, and before she realized what he intended, he kissed her. She moaned at the sweet taste of him, at the hunger of his embrace. For the first time in her life, she wished she were more like Thia. She wouldn't have a bit of trouble taking him to her bed. But Geary wasn't like that. She'd never been a one-night-stand woman. She preferred a relationship before she got naked with someone, which was why she seldom even dated. Her quest had left her very little time for anything else in her life.\n\nBut Arik was seriously starting to bring her ethics into question.\n\nPulling back, she picked the blanket up from the hammock and held it to his chest. \"Good night, Arik.\"\n\nHe let out an exasperated breath as he took the blanket from her with a grimace. \"Good night, Megeara. May the dream gods be good to you.\"\n\nA tingle went over her at the way he purred those words. It was as if he knew she'd been dreaming of him.\n\nPutting that thought out of her mind, she left him and headed below, but as she reached the stairs she couldn't help looking back at him. He was already lying in the hammock, watching her.\n\nIn the darkness, his eyes seemed to glow.\n\nShe swore she could hear his unspoken request for her to return to him. It reminded her of the same disembodied voice that kept beckoning her to the dig site\u2014only that voice was definitely female. And it called out to her even now to find Atlantis and set her free.\n\nI am losing my mind. Maybe she should see someone about schizophrenia....\n\nBut she knew better than that. This wasn't schizophrenia. It was merely her quest calling to her to fulfill her promise. She understood that. What she didn't understand was this odd connection to Arik. Why she heard and saw him even when he wasn't around.\n\nGo to bed, Geary, and forget it.\n\nWaving good night to him, she went to her room to find Tory already in her nightshirt and bed with Mr. Cuddles tucked under her arm. Since Tory's glasses were on the nightstand by her head, she was squinting at Geary. \"I wasn't expecting you to come back.\"\n\n\"What do you mean?\"\n\n\"C'mon, Geary, I may be half-blind, but that man is the finest thing I've ever seen... blurry or not. If I were you, I wouldn't have come back here tonight.\"\n\nGeary scoffed at her. \"You're only fifteen, Tory. It's not like you've got a lot of experience under your belt to judge hot men by.\"\n\n\"Point taken, but it doesn't change the fact that he's gorgeous and he likes you. A lot. So why are you back here?\"\n\n\"Because we have a four A.M. wake-up call.\"\n\nTory sighed and shook her head. \"You're the only person I know who's more pathetic than I am. I'm an orphaned kid, Geary. You shouldn't have to be alone all the time.\"\n\n\"Oh, hush, and go to sleep before I kidnap Cuddles. And speaking of furry things, where's Kichka?\"\n\n\"I don't know. I haven't seen her. Maybe she's stuck in the hold again.\"\n\nAs if on cue, Geary's cat came running in the open door to rub against her leg. A Bengal cat, Kichka had been Tory's Christmas present a year ago, but the cat had taken such a liking to Geary that all of them had finally given up and just let Kichka own her.\n\n\"There you are.\" Geary picked Kichka up from the floor and set her on the bed while she undressed.\n\nBefore Kichka curled up on her pillow to clean one paw, she meowed loudly at Geary.\n\nTory turned over and gave Geary her back.\n\nHer thoughts on Arik, Geary turned off the lights, tucked herself into the bed, and closed her eyes while Kichka moved from the pillow to sleep on the small of Geary's back. Within seconds, the cat was purring, Tory was snoring, and the gentle roll of the boat was lulling Geary away from all her problems.\n\nAnd before Geary knew it, she was drifting to sleep.\n\n* * *\n\nArik had never slept as a human before. The weight of his body was odd, especially combined with the swinging motion of the boat and hammock. But it didn't take him long to lose himself in the realm of dreams.\n\nIt was strange to be back where he lived. His dreams were misty and cold. At least in the beginning, but after a time they began to clear and he realized something.\n\nHis powers were back.\n\nArik paused, unsure if what he felt was real. Floating above the ground, he held his hands out before him and conjured a swirling ball of flames. The heat was warm but not painful as he built the ball with his mind to a dangerous height.\n\nInvigorated, he threw it into the darkness, where it exploded even brighter than the sun. Harmless pieces of ember rained down around him as he threw his head back and laughed.\n\nOh yeah, it was good to be a sleeping dream god. He had emotions and his powers.\n\nWhich left him with one goal.\n\nMegeara.\n\nIt was time to find her. But that proved to be easier thought than done. True, he had his powers again, but he didn't have the strobilos and finding her without it proved to be rather tricky. He was also lacking Wink's serum to keep Megeara asleep. If he found her, she could wake up and leave him all over again.\n\nI'll kill her. But even as he thought it, he knew that was an empty threat. He'd never hurt the very woman he craved.\n\nFor several minutes he cruised through the subconscious realm, drifting through dreams of naked musicians writhing in money and Jell-O shots, a toy poodle attacking a Doberman, a woman who bore a strange resemblance to a lollipop who was singing with cows, and one curious incident of a hemorrhoid chasing one woman around a block of cheese until it exploded....\n\nYeah, people were very odd beings.\n\nNo wonder he left these dreamers to the other Skoti. He much preferred sexually creative women.\n\nArik paused between dreams to take a deep breath. This was wasting time, and since Megeara was planning an early morning, he needed to find her quickly.\n\nClosing his eyes, he felt the ether around him... listened to it as it breathed and whispered through his being. She was out there.\n\nAnd then he heard it. The faint sound of her laughter. Honing in on it, he willed himself to her dream.\n\nShe was on the beach again, dancing in the surf to music only she could hear.\n\nArik froze at the sight of her there with her damp hair flying around her face. Her white dress clung to her body, showing him every lush, delicious curve. Every inch of flesh that he wanted to taste.\n\nUnable to stand it, he dropped to the beach and walked toward her with a predator's intent.\n\nHis breathing ragged, he came up behind her and touched her shoulder.\n\nShe turned on him then and what happened next absolutely blew his mind.\nCHAPTER 10\n\nGeary was laughing with triumph over her day as the water splashed against her body. She was just about to go diving into the sea without her equipment to find Atlantis when she felt someone touch her arm.\n\nShe turned to find Arik there.\n\nJoy rushed through her with a heady excitement and she did what she'd been wanting to do all day\u2014she pulled him against her and kissed him for everything she was worth. The surf splashed up and over them, but instead of being cold, it was warm and soothing.\n\nShe growled at the taste of his mouth on hers, at the sensation of his hard arms enveloping her as their tongues danced together. His corded muscles rippled under her hands while he pulled her so close to him that they were virtually one person.\n\nOh, it felt so good to be held by him. To have all those muscles pressing against her body. She buried her hand in his hair, pulling at it slightly so that she could feel even closer to him.\n\n\"You are the absolute best,\" she breathed against his lips before she nipped them with her teeth. \"Thank you for the permits.\"\n\nHe nuzzled her neck, sending chills over her. \"Anytime.\"\n\nShe pulled back from him and literally ripped the white shirt from his body before she threw him on the ground and straddled him.\n\nThe beach elongated so that the waves would no longer overrun them. Instead, the sea pulled back, leaving them the damp beach, which wasn't sandy. Rather, it was like lying on a bed of rose petals.\n\nArik was completely shocked by her behavior as she groped and fondled him as if she were overwhelmingly consumed by her hunger for his body. This was the Megeara he'd expected to meet in the human realm. But she'd been so reserved all day that he'd given up hope of her receiving him like this ever again.\n\nNow she was ferocious in her lust. His head swam as she buried her lips against his neck and tongued him until he thought he'd die from the pleasure of it.\n\nHe laughed in the back of his throat. \"You're certainly rowdy tonight.\"\n\nShe slid down his body as she continued to lick and please every part of him. \"You've no idea. It's been killing me all day to be so close to you and to not touch this body. God, you're gorgeous and sexy.\"\n\nHe trembled as she worked her way from his neck to his nipples and then down his abdomen until she got to the waistband of his jeans. Normally, Arik would have melted them off, but he was curious what she'd do in this mood.\n\nShe virtually shredded his pants in an effort to get to him. She tore chunks of the denim and tossed them over her shoulder until he was completely naked. He lay there on his back, watching her as her hands explored him at length. She attacked him with fervor, and he loved every second of it. As good as it'd been before when he merely fed from her emotions, it was even better now that he had his own to draw from.\n\nHe hissed as she found his hip bone and went to work on it with her mouth while she cupped him in her hand and massaged him until he was rock hard for her. He literally shook from the ecstasy of it.\n\nHe traced the line of her back with his hand, dragging his nails gently over her flesh until chills erupted all over her body. Needing more of her, he dipped his hand down to brush through the soft tangle of curls at the juncture of her thighs. He licked his lips, already tasting her, before he sank his fingers inside her.\n\nShe was so wet and ready. He toyed with her, biding his time until he could fill her completely.\n\nGeary couldn't wait to taste him. Her eyes hooded, she watched the pleasure on his face as she toyed with his cock and he stroked her where she ached most. She loved the sensation of his hard body against her curves. The way his small hairs tickled her skin.\n\nBut it was enough of playtime.\n\nDrawing her breath in in sweet expectation, she moved to draw the tip of his cock into her mouth, where she gently tongued him. The salty-sweet taste went through her as she moved lower. She felt him shudder under her. Giddy with the sensation of it, she drew him in even deeper, wanting to taste and please him for the rest of the night.\n\nArik cupped her face as he watched her. He loved it whenever she did this. He didn't know what it was about this one act, but there was nothing more pleasing to him. With his hands on her cheeks, he felt her jaw working under his fingers while she consumed him. His body was on fire, and in the back of his mind he wondered how much better this would be while he was awake.\n\nWhat would her mouth really feel like on him?\n\nIt was weird to think about it, but he was technically a virgin. He'd never taken a woman outside of the dream realm. Until Megeara, he'd never really cared about that.\n\nBut now he did. He wanted to know what it was like to really share his body with her. To have her touch him like this while they were conscious.\n\nGeary looked up to see Arik watching her. There was a look of extreme pleasure and one of amazement in his eyes. How she wished she could see a look like that from a man in real life. She'd never really had a boyfriend for any length of time. It was why she kept men at arm's length. She didn't like being hurt. Didn't like being disappointed.\n\nIt was so much safer to be alone, and yet as she lay with Arik she wanted to know what it would be like to have someone with her. Someone who was a part of her.\n\nSomeone to trust.\n\nIn less than a day, the man Arik had given her more than anyone else ever had. He'd given her her dream. And now he was here in her dreams, loving her.\n\nWanting to feel even closer to him, she pulled away to lay her body down on his.\n\nArik closed his eyes at how good Megeara felt against him. Her flesh was warm and soft against his. He held her close, reveling in the feel of her as he cradled her with his body. She leaned back to kiss him before she slid herself onto him.\n\nHe sucked his breath in sharply as pleasure pierced every part of his being. She made love to him furiously as the surf crashed around them. The waves rolled up but didn't quite reach him.\n\nHe could focus on nothing but the slick feel of her body as she took him in all the way. He held her hips, urging her even faster.\n\nGeary rocked her hips against his, seeking peace and closeness. He was so deep inside that it made her ache. She loved having sex with this man, and she couldn't help wondering if the real Arik would be as kind and tender as her dream lover.\n\nYou could find out.\n\nYes, but that would leave her vulnerable and that was the last thing she wanted. Geary Kafieri had no desire to be hurt by anyone. It wasn't worth it. Especially since she knew Arik would be dying soon. To let him in would be to invite agony on herself, and she'd had so much pain already. Everyone she loved died. Everyone.\n\nExcept for Tory. It was why Geary was so protective of her. God help Geary if anything ever happened to her cousin. She wouldn't be able to live and she knew it.\n\nBut as she felt Arik inside her now, she wanted a future she knew she could never have, and it broke her heart.\n\nIt was her lot in life to be alone. There was no use fighting it.\n\nAt least she had him in her dreams.\n\nSmiling at him, she quickened her strokes until she finally felt that moment of absolute pleasure right before her body released and blinded her with ecstasy.\n\nArik leaned his head back as he felt her body clutching his. Lifting his hips, he buried himself deep inside her before his own orgasm claimed him and she collapsed on top of him.\n\nNow this... this was what he'd wanted. Truly, there was nothing more spectacular in this world or any other than Megeara. He held her tight against him as his heart continued to race. He'd never been more content.\n\nShe leaned up on one arm to stare down at him as she tucked a piece of hair behind her ear. Never had he seen a more beautiful woman.\n\nArik cupped her cheek in his palm, looking up at her as he laughed from his exhaustion. Megeara's eyes glowed in the moonlight as she kissed the line of his jaw.\n\n\"Now if I could only get you to do this while we're awake...\"\n\nShe laughed, which caused her breasts to brush against his chest. \"Never. I can't afford to be like this awake. No one respects a woman of easy virtue.\"\n\n\"Oh, I assure you that's not true. I would always respect you.\"\n\n\"Yeah, right.\" She pulled back to sit up.\n\nArik couldn't breathe at the sight of her resting on his hips as the moonlight cast shadows over her bare body. He reached up to trace the outline of her nipple as she glanced off into the ocean.\n\nHe watched her face as she began to frown. \"What's wrong?\"\n\n\"The sea...\"\n\nHe turned his head to look and went instantly cold. The surf was rolling against the shore, but what made him pay attention was the strange motion of it. The whitecaps began to slowly form faces, and those faces started rising out of the water in liquid forms that became solid.\n\nThe Dolophoni.\n\nChildren of the Furies, the Dolophoni were essentially the assassins of the gods. And they'd been the ones who'd rounded up the Oneroi centuries ago for Zeus to punish.\n\nNow someone had unleashed the Dolophoni against Arik. He knew it. There was no other reason for them to be here. They weren't something Megeara would have conjured, and he could feel inside that they weren't from her dreams.\n\nThey were here to kill him.\n\n\"You need to go, Megeara.\" He slid out from under her.\n\nGeary was frozen to the spot as ten people emerged from the waves, completely dry. Two women and eight men. As tall as or taller than Arik, they left the sea like a pack of rabid dogs ready to attack.\n\nTheir heads were bent low as they headed straight for Arik with a deadly swagger. Not a word or sound could be heard. Not even the surf. The air no longer moved. It was static and charged with the coming conflict.\n\nArik stood his ground as skintight black leather armor appeared over his body, and his hair magically pulled itself back into a ponytail. Spikes grew out of his forearms and one out of his left knee.\n\nOne of the men, who was at least six inches taller than Arik, had a bald head with a phoenix tattoo on one side of his face, the tail of which coiled down and around his neck. He wore a sleeveless black T-shirt that accentuated the bulging muscles of his arms, black leather pants, and a studded neck collar. Metal vambraces covered his forearms, and he carried an ax over one shoulder.\n\nAnother man was two inches shorter and much leaner and had short bright green hair that he had spiked on top of his head and over his eyes. He wore a pair of black wraparound sunglasses and carried a black staff with silver spikes poking out of both ends. His bare arms were covered with colorful tattoos, and a row of nine hoops hung from his left ear. He had two more hoops on his bottom lip.\n\nThe next man had short-cropped dark auburn hair that framed a face of perfect masculine beauty. His brown eyes flashed red as he drew an AK-47 out of the folds of his long leather coat.\n\nThe man beside him had the entire right side of his torso bare. His long black hair was pulled back in a ponytail while his left shoulder and arm were covered by black plate armor. There were scars all over his cheeks, and his black eyes were deep set into his face.\n\nTwo more appeared to be twins. They were equal in height to Arik, with short brown hair, and where one had three hoops in his right ear the other wore them in his left. Unlike the others, they were each dressed in slacks and a button-down shirt, with black leather overcoats that flowed around their booted feet. They moved slow and easy, with a fluid grace, as if synched to each other. Their faces were perfectly sculpted.\n\nTwo steps behind them was a man who had to be a minimum of seven feet tall. His blond hair was short and he was built like the Terminator, with a demeanor that would make the cyborg look weak. This man's face was rugged and harsh, and it was obvious he lived to bathe in the blood of others.\n\nThe last man was lean and wiry. Steel spikes were wrapped around his arms and over his hands. He wore tall biker boots with flames rising up from the toes to meet at the skulls at the top of them. Shirtless, he had the body of a ripped gymnast.\n\nAll of them wore expressions that said they were here for war.\n\nOne of the two women was even taller than Geary, with black hair streaked with bright green. Her hair was pulled back into a ponytail, and the green seemed to be snakes. They slithered around her shoulders, coiling about her neck as they hissed and snapped.\n\nThe other woman was much shorter but no less lethal. Trim and lean, she was corded with muscles and had bright red hair and sharp features.\n\nGeary scrambled for her clothes, not that the others even seemed to know she was there. Their attention was only on Arik.\n\n\"Who sent you?\" Arik asked defiantly.\n\nThe man with the gun answered by firing it straight at Arik. He recoiled before he backflipped to the left and threw his hand out. It functioned as a gun and returned the bullets to them. He \"fired\" more bullets with his other hand.\n\nThe group dodged them before the woman with red hair threw out a circle that exploded all over Arik. It knocked him flat on his back and sent shooting sparks into the air around them.\n\nArik hit the sand with a force so lethal it shook him to his bones. Damn them. His senses were rattled, but he'd battled enough in dreams to know that this was his domain. He might be mortal while awake, but in here he was still a god.\n\nAnd they were fucking the wrong Skotos.\n\nNo one took him out in his realm.\n\nGrowling, he flipped himself back to his feet and manifested a whip. He slung it out for the woman who'd stunned him and caught her about the waist. The cord bit into her and would have cut her in half had she been anyone but a Dolophonos.\n\nAs it was, it only cut deep and sent her to the ground.\n\nThe green-haired man paused to look at her as she writhed.\n\n\"You're strong,\" he said, betraying a set of vicious fangs. \"Not many people get a shot off on Alera.\"\n\nArik swung the whip again, causing them to dodge it. \"First mistake. I'm not a person. I'm a god. You want to fight in this realm, you need reinforcements.\"\n\nThe bald one dove at him so fast all Arik could see was the vapor trail. He caught Arik about the waist and they went down hard. Arik rolled with him, slugging him before he kicked him away. Before Arik could regain his feet, the other woman was on his back. He flipped her over his head and punched her in the chest. Without missing a beat, she sliced at him with a dagger that narrowly missed him.\n\nThe one bad thing about the weapons the Dolophoni used was that they were made by Hephaestus and that was one god who knew how to forge a weapon that hurt.\n\nMore to the point, he forged weapons that killed other gods.\n\nArik caught her around the neck with his whip, but before he could hurt her, one of the men kicked him from behind.\n\nLetting go of the whip, Arik twisted around to confront him. But first Arik had to dodge the bald man's ax. Arik caught it in both hands and kicked the giant back.\n\nHe didn't budge. All he did was laugh.\n\n\"Laugh at this, asshole,\" Arik snarled, head-butting the giant. He staggered back, releasing the ax to Arik.\n\nInstantly the other man swung his staff at Arik's feet, then his head. Arik dodged, then countered with the ax, which the man deftly ducked. He brought the staff up and planted it in Arik's ribs.\n\nArik felt the bite but didn't react to it other than to swing the ax. The man dodged it again, but one of the twins came in with an invisible block of some kind that shattered the ax into shards.\n\nCursing, Arik barely rolled out of the way of the staff swing. The man lunged at Arik's feet.\n\nArik jumped up, then caught the staff with both hands. He snatched it right, unbalancing the man before he swept his feet out from under him. Wresting the staff from the man's hands, Arik sank one end of it into the man's chest.\n\nScreaming, he disintegrated on the sand.\n\nOne down. Nine to go.\n\nArik twirled the staff around and tucked it under his arm as he turned to face the others, who treated him with more respect. They weren't as cocky now as they'd been on arrival.\n\nTheir faces were marred with disbelief as they appeared to be speaking to one another mentally. Let them. Arik didn't need to hear their thoughts to know they planned on mangling him.\n\nArik backed up as they circled around him. They were sizing him up and he knew it. Feinting at him only to back away to test his reflexes and assess his weaknesses.\n\nHe toyed with them. Giving them false impressions. False reactions. Be damned if he was that stupid. He hadn't lasted this long in dreams by allowing others to get the better of him.\n\nOne of the twins came at Arik's back. He twirled with the staff, into a crouch so that he could swipe the man's feet out from under him. Arik rose to his feet to finish the attack, but before he could the other twin knocked him back with a punch so raw it lifted him off his feet and sent him flat on his back in the sand.\n\nArik swung the staff at the same time he rolled back to his feet. He ducked the knee the bald man sent toward him and turned away from the sword the woman was trying to skewer him with.\n\nGeary could barely think as she watched the deadly dance of Arik with the others. She'd never seen anything like it.\n\nArik used the staff to lift himself up from the ground and drove his feet into the man with auburn hair. Then Arik swung around to attack the bald man and the twins at the same time.\n\nGo, Arik.\n\nBut she couldn't leave him to this on his own. Even for a dream, this was getting bloody, and honestly, this wasn't what she wanted in her subconscious.\n\nWanting control again, Geary walked over to them. \"Excuse me?\"\n\nArik paused at her call, which allowed the bald man to deliver him a staggering blow to his face. He twisted back before snarling at her. \"Run, Megeara.\"\n\n\"Run from what? They're circus freaks, and while this is mildly entertaining, I'd like to go back to what we were doing before they interrupted us.\" She waved her hands at the others. \"So you guys, go poof.\"\n\nThe twins approached her slowly. \"This isn't a game, human. Listen to the Skotos and go. We're not bound by the laws of the Oneroi. Killing humans is nothing for us.\"\n\nWas that supposed to scare her? Yeah. What had she eaten for dinner that it was manifesting like this?\n\nOh yeah. Crab cakes. Those never really agreed with her. She'd eaten two of them. Maybe that was why there were twins. Or she was just tired, which was the most likely explanation.\n\nAt any rate, she was tired of this part of the dream.\n\n\"Well, aren't you all scary in black? Oooo. What are you two masquerading as? Evil Man and his trusty sidekick Bad Boy?\" She let out a tired sigh. \"Look, this is really annoying me. I want my dream back and that means all of you have to go now.\"\n\nOne of the twins reached to grab her, but before he could, Arik was there. He pulled her by the arm, away from the others.\n\nHe paused to send a fiery blast back toward them as they ran forward. \"You have to go, Megeara.\"\n\n\"Not without you.\"\n\nArik wanted to curse the fact that she couldn't distinguish reality from her dreams. If she died on this plane, she died in her world, too. Same as him.\n\nShe stopped and grimaced. \"Why are you playing with them? Just snow them in.\"\n\nHe didn't understand what she meant until she snapped her fingers and blocks of ice encased the Dolophoni. His jaw went slack as it stopped them dead in their tracks.\n\nA human shouldn't have that ability. \"How did you do that?\"\n\n\"It's a dream, silly. I've always had control in my dreams. As a kid I used to pretend I was watching TV and if I didn't like the dream, I just changed the channel. Like this.\"\n\nSuddenly the beach was gone. They were in a summer meadow with no sign of the Dolophoni to be had.\n\nArik's jaw slackened even more as he felt the sharpness of solar heat and smelled heather and wheat. How was this possible? Humans couldn't control dreams like this. If he didn't know better, he'd swear she had Oneroi blood in her.\n\nBut she didn't. There was a scent and aura that all the gods had\u2014even those who only held a bit of god blood. Megeara had none of that. She was fully human.\n\nBefore he could ask how she'd gained control away from the Dolophoni, she captured his lips with hers. For a heartbeat all he could sense was her. With every part of him.\n\nUnfortunately, he had more to focus on than how good she tasted.\n\n\"Please, Megeara. I would love to stay with you, but I can't.\"\n\nShe frowned up at him. \"What are you talking about?\"\n\nHe kissed her on the forehead before he pulled away. She had gotten them away from the Dolophoni, but they were still out there, looking for him, and they wouldn't stop until he was dead. They wouldn't care who got in the way. All that mattered to them was completing their mission.\n\nThe last thing he wanted was to see Megeara hurt.\n\n\"I'll be with you soon.\"\n\nAnd with that, he pulled out of her dream.\n\nArik awoke in the hammock to the taste of blood in his mouth. His entire body ached to the point he could barely draw breath.\n\nWhat was going on? None of this should be happening.\n\nHe didn't know why the Dolophoni had been sent after him, but then, the why didn't matter. All that counted was the fact that they wouldn't stop until he was dead.\n\nThey'd found him in the dream realm.\n\nIt wouldn't take them long to find him in the human world, too.\n\nHolding his breath, he rolled out of the hammock and fell to the deck. He groaned as pain assailed him. He tried to stand, but his body wouldn't cooperate. With no choice, he lay on the deck, looking up at the stars that twinkled silently in the sky above.\n\nAnd as he watched them, bitter laughter filled him. How absolutely apropos.\n\nHis dream had just turned into a nightmare.\nCHAPTER 11\n\nGeary woke up at dawn feeling invigorated. She'd had a restful, dream-free sleep the night before and now she was anxious to be about the excavation. It was time to get cracking and set fire to the world.\n\nTory was already up and dressed, sitting in the corner with a flashlight as she reviewed pictures of their site. She looked like an eerie specter in the darkness.\n\n\"What are you doing?\" Geary asked.\n\nTory pushed her glasses up on her nose and gave Geary a wistful look. \"Wishing I could dive with you. It would be so cool to be down there and be the first on site, touching everything.\"\n\nGeary nodded. At two hundred feet down, it was far too deep for Tory, who was only a recreational diver. Not to mention, it was just too dangerous for her to chance. Both Jason and Tory's father had died during diving accidents. That was one family legacy Geary had no intention of bequeathing to anyone.\n\n\"Next time.\"\n\nTory sighed. \"Yeah. Just keep the live feed going so that I can see it and pretend I'm there, too.\"\n\n\"Yes, my queen. Anything else you'd like?\"\n\nTory grinned. \"A million dollars and Brad Pitt.\"\n\nAs Geary threw back the covers and left her bed, she laughed at Tory's stock answer. \"You forgot world peace.\"\n\n\"I'm feeling a bit selfish today. Teenage hormonal overdose, I think. Or just general excitement.\"\n\nGeary rolled her eyes as she went to brush her teeth. It didn't take her long to get dressed. As eager to get started as Tory, she all but sprinted to the upper deck. The sky was just starting to lighten. Pink was laced with the blue as orange broke through in ribbons that spiraled above her, promising her good weather for the dive and excavation. Closing her eyes, she inhaled the fishy scent of the sea and smiled.\n\nIt was a good day to be alive.\n\nAnd grateful to the man who'd given her this dream, she went to the hammock to wake him.\n\nOnly Arik wasn't there. He lay on the deck with his back to her. Afraid he was ill, she rushed over to him and knelt by his side.\n\n\"Arik?\"\n\nHe groaned ever so slightly as she shook him. Rolling onto his back, he opened his eyes, and she saw a slight bruise on his forehead.\n\n\"What are you doing on the deck?\"\n\nArik gestured to the hammock above him. \"I fell out of the hammock while I was sleeping.\"\n\n\"On your head?\"\n\n\"Apparently. Good thing it's hard, huh?\"\n\nShe grimaced at his misplaced humor.\n\nArik's breath caught as she gently brushed the hair back from his brow to examine his cheek and forehead. The look of concern on her face was enough to make him want to bruise himself again to see if she'd duplicate it.\n\nLuckily, he wasn't that masochistic.\n\nYet.\n\n\"You need to be more careful.\"\n\n\"I intend to,\" he said honestly. He wasn't about to let the Dolophoni get the drop on him again. While on the boat he was semiprotected, since they wouldn't cause a commotion in front of a group of humans.\n\nAt least that was the lie he was telling himself. The problem with the Dolophoni was that they didn't really have a set of rules to follow that anyone knew. You merely hoped they'd abide by the ones set down for everyone else.\n\nIn the end, they, like the Chthonians, were a law unto themselves. The only difference being that the Cthonians truly had no one holding their leash. At least with the Dolophoni, the Erinyes could call them off. Not that they did that often. The Furies had a tendency to revel in conflict, and there was nothing they loved more than a good bloodbath.\n\nMegeara sat back on her heels to stare at him. The dawn's light highlighted her hair and made it glow. Her cheeks were pink, and all he could think of was the hours he'd spent kissing those lips.\n\nAnd the hours more he'd like to spend making love to her.\n\nHe was already hard for her, wanting to taste her again. Why wouldn't she grab him in this realm the way she did in the other?\n\n\"You are so beautiful.\"\n\nShe gave him a doubting look. \"Man, you hit your head hard, didn't you?\"\n\nHe frowned. \"Why can't you take a compliment?\"\n\n\"Because I'm just not used to them. I come from a family that doesn't believe in randomly patting people on the back. The assumption is if no one's yelling at you, you're doing a great job. And no one ever complimented each other on appearances. Those are trivial. It's what's inside that matters.\"\n\nHis smile turned gentle and guileless. \"And you're even more beautiful there.\"\n\nGeary merely stared at him. What did a woman say to that? \"Thank you.\" But that was extremely inadequate for what she felt. Everything about Arik touched her deep and made her want to stay with him.\n\n\"Hey, Gear?\"\n\nShe turned at Teddy's call. \"Yeah?\"\n\n\"We got a kink in the dredge. Justina is working on it right now. I just wanted to let you know.\"\n\n\"Thanks.\"\n\nGeary pulled back and smiled at Arik. \"We've got a lot to do. You feeling up to it?\"\n\n\"Absolutely. I'm here to help.\"\n\nAnd over the next hour as they prepared the boat and equipment, he proved to be true to his word. No matter how hard or dirty the task, he lent himself to it without complaint.\n\nThey were just about to weigh anchor when Solin showed up on the docks looking perfectly groomed and highly offended.\n\nHis eyes heated, he boarded the boat and made his way straight to Geary. \"You weren't about to leave me, were you, Doctor?\"\n\nGeary didn't know what to say. Honestly, she'd forgotten about him.\n\nLuckily, Arik showed up at that moment and distracted Solin from chastising her. A stern frown creased Solin's brow as he noted the slight bruise on Arik's forehead. \"What happened to you?\"\n\n\"He fell out of his hammock last night,\" Geary explained. \"And if you two will excuse me, I want to get under way immediately.\"\n\nArik didn't speak until he was alone with Solin.\n\n\"Hammock?\" Solin said with a mocking laugh. \"It looks more like you had a run-in with something hard.\"\n\n\"I did. The Dolophoni showed up last night in my dreams.\"\n\nSolin went completely still. Anger radiated out of him with such ferocity that it actually singed Arik. One would think they'd attacked him instead. \"How many?\"\n\n\"Ten.\"\n\nSolin arched a shocked brow. \"And you lived? I have to say I'm surprised.\"\n\n\"I don't go down easily.\"\n\n\"Apparently. So how did you get away from them?\"\n\n\"They pulled back after I killed one of them, then\u2014\"\n\nSolin gaped. \"You what?\" he asked in disbelief.\n\n\"I killed one of them.\"\n\nSolin gave him a look of supreme respect. \"How did you manage that?\"\n\n\"I'm real good at what I do.\" He didn't say that arrogantly, he was only stating fact.\n\n\"Yeah, and have you any idea of the firestorm you've just unleashed for yourself? The Dolophoni don't like people to get the better of them.\"\n\n\"I know and I'm sure we'll battle again.\"\n\nSolin shook his head as he looked out over the water. One corner of his mouth lifted into an evil grin. \"So which bastard did you nail?\"\n\nArik didn't know their names, but he had a feeling Solin must have had more than his fair share of run-ins with them to be this interested. \"The one with a staff.\"\n\nSolin laughed. \"Erebos. Good man. Wish I could have seen it. Zeus knows I've been wanting to shove that staff up his ass for centuries.\" He indicated Arik's face. \"Are you sore, too?\"\n\n\"Yes.\"\n\n\"Amazing.\"\n\nAnd it was. None of this made sense. There shouldn't be a single remnant of their battle on him. With the exception of death, things that happened in the dream realm shouldn't transfer to the human plane. It just didn't happen. \"All I can figure is it has something to do with the fact that I'm Skoti and don't belong in this realm. Maybe that's why I can feel dream pain in this world.\"\n\n\"Maybe.\"\n\nSuddenly the sound of a tin whistle rent the air as the boat engines started up. Arik cocked his head as someone started playing an Irish tune. A few seconds later, he heard a beautiful voice singing the folk song \"I'm a Man You Don't Meet Every Day.\" The rest of the crew picked up the song as they pushed away from the dock and headed out of the harbor.\n\nEvery one of them was working together, and the sight of them like that warmed him.\n\nArik smiled at the camaraderie. \"They're incredible, aren't they?\"\n\n\"What? Humans?\"\n\nHe nodded.\n\n\"They can be, I suppose.\"\n\nArik watched Solin as he kept himself apart from the others, and he couldn't help wondering what it would be like to have the best of both worlds. To be able to feel and walk among humans both in this realm and in the sleep one. How could Solin be so lackadaisical about it all? Surely he had to appreciate the beauty of this world. \"So what's it like?\"\n\nSolin frowned. \"What's what like?\"\n\n\"Being human.\"\n\nHe let out an angry scoff. \"Basically sucks. I highly recommend returning to your godhood as soon as you can.\"\n\nArik didn't understand that. There was so much charm here. So much of everything. \"Listen to their song... look at the landscape. How can you not love it here?\"\n\nSolin curled his lip. \"Disease. Filth. Waste. Crime. Brutality. What's there to like?\"\n\n\"There's brutality on Olympus.\"\n\n\"True. But I hate humanity as much as I hate the gods. Both groups are selfish bastards bent on destroying everything around them. They were given a perfect world and rather than enjoy it, they'd rather destroy it and each other. Excuse me if I don't look at them with love in my eyes but rather scorn in my heart.\"\n\nArik cocked his head at the heated rancor that bled from every part of Solin. \"Yet you're helping me. Why?\"\n\nHis features blank, Solin shrugged nonchalantly. \"Nothing better to do. Eternity is boring. Really boring. I'm hoping that when you pop the seal on Atlantis, there will be a giant explosion to add some humor and interest to my life. If we're really lucky, Apollymi will come out and thoroughly entertain us with a massive fireworks display. Hell, if she does half of what she did last time, there will be belly rolls aplenty for those of us who hate the Olympians and humanity.\"\n\nArik didn't understand how anyone could get bored with the sensations of the human existence. Never mind hate it to such an extent. But then, Solin had been here for centuries. Perhaps, given time, he'd become jaded, too.\n\nAs the song ended, the crew picked up with the Beatles' \"Revolution 1.\"\n\n\"Hey, Arik?\"\n\nHe turned as Tory came running up to him with a small foil package that she handed to him. \"Frosted Pop-Tarts. Fudge. Trust me.\" Then she went bounding off again.\n\nLaughing, he realized it must be more food. She seemed bent on corrupting him.\n\nSolin wandered off while Arik broke into the package and realized that Tory had excellent taste. These things were delicious. While the Beatles melted into the Bee Gees, the boat picked up speed as it raced them to the spot where eleven thousand years ago a very pissed-off goddess had destroyed her family and sent an entire continent to the bottom of the sea.\n\nPopular legend told that it'd been Apollo who had destroyed Atlantis because their queen had ordered the death of his child and mistress. It was good propaganda for the Greek pantheon, who wanted to be thought of as the most menacing. But the truth was very different.\n\nThey were neophytes compared to the Atlanteans. Their power nothing.\n\nApollymi the Destroyer would have swept over the entire earth until nothing was left standing had she not been imprisoned in the middle of her bloodthirsty tirade by a trick of fate. Now she sat trapped in her netherworld, Kalosis, watching this one, waiting for someone to free her.\n\nEven though Arik lacked his god powers, he could hear the Atlantean goddess calling out for release. She was like a beacon, summoning people to her. It was probably why so many quested for Atlantis.\n\nThe other gods were why those quests ultimately failed. No one other than Apollymi wanted her released.\n\nHe looked up to catch Kat's gaze from where she stood on the prow. They were at odds over this, but so long as Megeara didn't disturb the seal, what was the harm in her poking around the ruins? So she'd find a few shards of pottery and maybe some jewelry. None of that would interfere with Apollymi's prison.\n\nThey were safe.\n\nAt least that was the lie he wanted to believe.\n\n* * *\n\nSolin froze as he moved along the deck and saw the exceptionally tall woman standing by the railing. Lithe and graceful, she was completely striking. But greater than her beauty was the power emanating from her. It was an aura he knew well. She was an Olympian.\n\nAnd there was nothing he hated more than Olympians.\n\nHe approached her cautiously, sizing her up and wondering how much power she carried. \"You have the presence of a god, but I don't know you.\"\n\nHer green eyes narrowed suspiciously on him, and he knew she was feeling his powers to measure him just as he was her. \"I'm a servant to Artemis.\"\n\nHe laughed at those words. \"You a servant? You have much more power than that and we both know it.\"\n\n\"And you have a lot of juice for a demigod. Makes me wonder if you haven't made a deal with someone yourself.\"\n\nSolin gave her a cocky smirk as he glanced about to make sure the humans couldn't overhear them. \"I like to keep people guessing about me.\"\n\n\"I'll bet. So what brings you here? Isn't it unusual for two Dream-Hunters to work together?\"\n\n\"Not really. There are a lot of tag-team Skoti out there who make it their habit to work together.\" He looked her up and down, taking in her delectable body. She was prime material for his kind to play with in dreams. \"I'm surprised you haven't been visited.\"\n\n\"Oh, I'm not. Artemis fed the last person who made a pass at me to a wild boar. When it comes to my dreams, she's even worse. Only the most suicidal would tread there.\"\n\n\"Oooh.\" He sucked his breath in sharply at her warning that actually made him smile with anticipation. It also made him instantly hard. \"You make it all the more tempting.\"\n\nShe returned his smile, only hers was beguiling, with a hint of malice and challenge. \"And you still haven't answered the question of the day. Why are you here, Skotos?\"\n\nHe shrugged nonchalantly. \"Originally, I was just going to screw with Arik. But I'm rethinking that now. I mean honestly, this whole situation shouldn't be the least bit interesting, but with you here that means that Artemis is extremely interested. And anything she's interested in I'm interested in, which means things around here are about to get really interesting. Wouldn't you say?\"\n\n\"Not really. Why not save yourself the headache and sod off?\"\n\n\"Oh see now, that's no way to get me to leave. You're pushing me away. Why?\"\n\n\"I find you irritating.\"\n\nHe laughed at that. \"I haven't even begun to irritate you yet. Imagine what I could do if I applied myself?\"\n\nHer eyes narrowed dangerously. \"I can imagine. I can also imagine ripping your throat out and tying my shoes with your larynx.\"\n\n\"Really, kori, you have to stop. You're seriously turning me on.\"\n\nShe screwed her face up at him. \"You're a sick bastard, aren't you?\"\n\n\"Is that not the very definition of a Skotos?\"\n\nShe stepped back from him before she looked around the boat to make sure no one was within earshot. Her gaze paused on Arik. \"As you can see, we already have one of you on board. We don't need another.\"\n\n\"That's what everyone thinks, but they were running a special. Two Skoti for one, so here I am in all my glory just to get under your skin or skirt. I'm really not particular.\"\n\n\"Yes, but there is a law that says you can send back defective merchandise. I can't think of anything more defective than you.\"\n\n\"I can. An immortal possessing god powers who passes herself off as a servant and expects the rest of us not to notice. Definitely defective, don't you think?\"\n\n\"I think it's none of your business.\"\n\n\"Hmm....\" He really was becoming intrigued by her, and that was highly unusual for him.\n\nShe cocked her head and looked at him. \"Why do you hate Arik so anyway?\"\n\nThe question was a non sequitur and surprised him. \"Excuse me?\"\n\n\"I have the powers of a god, remember? I can feel your emotions and they are ripe with malice. Why do you hate him?\"\n\nHe gave her a patronizing smirk. \"If you know that much, then you should know the answer.\"\n\n\"I can only feel emotions, I can't trace their roots. And you are eaten alive by what I feel, which also begs the question of how a Skotos has emotions that strong.\"\n\nSolin shrugged. \"I'm only a halfling, remember? We're immune to the curse.\"\n\n\"Ah,\" she said as if she finally understood.\n\nHe was intrigued by her tone. \"What?\"\n\n\"I was wrong. It's not hatred, it's envy.\"\n\nHe laughed at the very idea. Him jealous of a Skotos? Pah-lease. \"You don't know what you're talking about.\"\n\nHe could tell by her tone that she was amused by his denial. \"Yes, I do. The smell of it is all over you. You're ripe with it. Envy is just gnawing away at you like a worm inside a juicy apple.\" She tsked at him. \"Yep, there's not enough deodorant in the world to mask that odor.\"\n\nShe was being ridiculous and he was growing tired of dealing with her.\n\n\"This discussion is over.\" He started away from her.\n\n\"Wait.\"\n\nHe paused to look back. \"Yes?\"\n\n\"I've already told Arik and now I'm telling you I will not allow him, Geary, or anyone else to uncover Atlantis. Ever.\"\n\nHe sneered at her concern. \"As if I give a shit about Atlantis. I have much more self-serving interests at heart.\"\n\n\"And what would those be?\"\n\n\"As you so eloquently put it, it's none of your business. Good day, goddess. And good luck.\"\n\n* * *\n\nArik paused as they neared their destination. Apollymi's voice grew louder as the boat slowed. They were only a few feet away from where the main harbor of Atlantis had once stood. If Arik closed his eyes, he could still see it in his mind.\n\nIt'd been a bustling port, filled with merchants, pirates, and fishermen. Prostitutes, sailors, and officials had blended in seamlessly on the docks that had always been crowded to overflowing. The smell of fish, spices, and perfumes had hung heavy here as the capital city had glistened on a mountain from behind stone walls.\n\nHighly advanced, the Atlanteans had been a peaceful race who'd only wanted to help others. But Zeus and Apollo had refused to let them exist that way. The Greek gods had waged war on the Atlantean pantheon by manipulating their people.\n\nIn the end, it'd been those people who'd suffered most.\n\nPushing that thought aside, Arik glanced about the boat that was filled with people wanting to learn the truths he already knew. Humanity was better off with Atlantis at the bottom of the sea.\n\nThe crew rushed about as they set up for their doomed excavation. Arik crossed the deck to where Solin stood by the pump. \"I need a favor.\"\n\n\"Haven't I done enough for you?\"\n\nArik scoffed. \"Considering what you did to me, no. Or more to the point, hell no.\"\n\n\"I would disagree, but curiosity has me by the throat. What is it that you want now?\"\n\n\"Knowledge,\" Arik said simply.\n\n\"Of what?\"\n\n\"How to dive.\"\n\nSolin's eyes narrowed speculatively. \"Why?\"\n\nArik gave him a droll stare. \"Why do you think? I want to make sure they don't venture into the wrong area and disturb a certain goddess. I can't do that two hundred feet above her, can I?\"\n\nSolin still looked less than convinced. \"Megeara won't let you go.\"\n\n\"If I know what I'm doing, how can she stop me?\"\n\nSolin laughed. \"You've got a lot to learn about women.\" He narrowed his eyes before he placed his hands on Arik's head.\n\nArik felt a sharp stinging pain an instant before he had all the knowledge he needed to dive like a pro. Unfortunately, he also had a nosebleed. \"What the hell?\"\n\nSolin looked on him derisively. \"You're human and I just rewired your brain. It doesn't like that. As gods, we can accept these things. As a human...\" He pulled out a handkerchief and handed it to Arik.\n\nGreat. Just great. Arik wiped the blood away before he went over to Megeara, who was checking the air hoses. \"Where's my suit?\"\n\nGeary actually gaped at the unexpected question. \"Excuse me?\"\n\nHe indicated the dry suits that were beside her. \"I plan on going with you.\"\n\nHer mouth worked for several seconds before sound would actually come out. \"Uh... no. This isn't a game, Arik.\"\n\n\"And I'm not playing. I intend to go with you and help. Trust me. I know what I'm doing.\"\n\nGeary was skeptical. The last thing she needed was an amateur on board.\n\n\"He's not lying,\" Solin said as he joined them. \"I can assure you, he's part fish. Jacques Cousteau has nothing on him. Aquaman, either.\"\n\nStill, she wasn't sure as she frowned at Solin. \"You do know how dangerous this is?\"\n\n\"I wouldn't send him down there if I didn't fully believe he'd be back to piss me off.\" Solin had a delivery so dry, she could rent him out as a dehumidifier.\n\nGeary hesitated. She didn't want any of her team hurt. Or Arik, either. \"If you can swim like a fish, how is it you were drowning when we met?\"\n\nArik tensed. He'd forgotten about their meeting. Luckily he was quick with a response. \"I'd been swimming for a while when you found me and was tired. Normally, I don't have any problems. I was just lucky that day... in more ways than one.\"\n\nThere was no missing her skepticism.\n\nKat came up to her. \"What's going on?\"\n\n\"Arik wants to dive with us. I don't know.\"\n\nKat and Arik exchanged a look that was both hostile and respectful. \"Do you know what you're doing?\" Kat asked.\n\n\"Yes.\"\n\n\"Then let him go with us. What's the worst thing that could happen?\"\n\nGeary scoffed at Kat's nonchalance. \"Death.\"\n\nKat shrugged. \"You can die crossing the street, and there aren't many cars two hundred feet down.\"\n\nShe strangely had a point.\n\nKat wrinkled her nose at Geary. \"Let him go. I'll keep an eye on him. Believe me.\"\n\nKat was the only person Geary knew who was an even stronger swimmer than she was. If Kat said it was okay, then it should be. Geary looked back at Arik. \"All right. You can suit up.\"\n\nGeary watched Arik closely to make sure that he wasn't lying about his expertise as he prepared for the dive. She had to give him credit. He wore the gear as if he'd been born to it and he did know how to dress in it. There was no hesitation in his movements at all.\n\nBut that confused her. \"Tell me how someone who was raised in the mountains would have been diving.\"\n\nArik froze at the question as he tried to think up a plausible tale. \"I told you, I've been searching for Atlantis. It's hard to do that on the surface. I've spent a lot of time on research boats here in the Aegean.\"\n\n\"Hmm... you know, there's something about you that doesn't make sense. But I can't figure it out.\"\n\nHe offered her a soothing smile. \"All you need to know is that I'm here to help you.\"\n\nInstead of those words drawing her closer to him, she took a step back and her eyes flashed with suspicion. \"Yeah.\"\n\nArik wanted to curse in frustration, but he didn't have time. They were getting ready to go down. There were four of them for the dive. Geary, Kat, him, and Scott.\n\nGeary led them to the platform that would lower them, along with a dredge, to the water. None of them spoke until after they'd submerged. Arik could hear his own breathing as he followed them deeper, away from the surface light.\n\nIt was murky and dark. But it was the pressure of the water against his body that was the strangest part of it all. And the deeper he went, the worse it became. It was almost oppressive, and part of him wanted to panic. But that was ridiculous. It was only water, and he was with people who knew what they were doing.\n\nFor that matter, he knew what he was doing.\n\nGeary paused at the first dive station to let her body adjust to the pressure and depth. \"How's everyone doing?\"\n\nScott grinned. \"Great, boss lady.\"\n\nKat nodded. She looked at Arik.\n\nHe inclined his head to her. \"Fine.\"\n\nBut something belied that. It was an instinct she had, and she didn't know why. \"Are you sure?\"\n\n\"Yeah. I'm just having a vision of what would happen if someone were to jerk one of our helmets off at two hundred feet down.\"\n\nGeary screwed her face up in distaste.\n\n\"Ew,\" Kat snapped.\n\nGeary concurred. That was the kind of thing that no one wanted to think about.\n\nScott cleared his throat. \"Is it too late for me to go back? I'm not sure I want to be down here with Freddy Krueger having these kinds of visions. What's to keep him from having an experiment?\"\n\nGeary shook her head. \"Arik was just kidding. Weren't you?\"\n\n\"Absolutely. But\u2014\"\n\n\"No buts,\" the three of them said in unison.\n\nGeary patted Arik on the shoulder. \"Let's just think happy thoughts, shall we?\"\n\n\"You know,\" Tory's voice filled their heads from the link. \"Now that Arik mentioned it. At two hundred feet, given the pressure on the human body\u2014\"\n\n\"Tory!\" Geary snapped. \"Please don't give me odds or stats right now, 'kay?\"\n\n\"You're such a spoilsport.\"\n\nIgnoring the pout in Tory's tone, Geary started down to the next station. She'd factored in four stops to help them adjust. But honestly, she wanted to be able to dive straight to the site.\n\nIf only.\n\nIt took them a little bit of time to get to the area and then to stake it out. They had to be careful to anchor the grid but not accidentally damage something that might be hiding under the silt and sediment.\n\nGeary's father had drilled into her head that much of the historical Troy had been lost because of Heinrich Schliemann's fervor to find proof of it. He'd damaged as much as he'd salvaged.\n\nShe didn't want to make the same mistake.\n\nOnce the datum was set and photographed, they regrouped.\n\n\"How's everyone doing?\" she asked them.\n\nThey each gave her a thumbs-up.\n\n\"Everyone's air supply is steady?\" she double-checked.\n\nScott nodded. \"Doing great, boss.\"\n\n\"I'm good,\" Kat chimed in.\n\nArik grinned at Geary. \"Let's dig.\"\n\nSomething hot pierced her at his eagerness. He really did seem to mirror her enthusiasm. Geary headed for the first section she wanted to explore. They carefully suctioned the area until they could find what appeared to be an encrusted wall.\n\nHer hand actually shook as she touched it. She just wished she weren't wearing gloves, so that she could have the tactile sense of it. \"This isn't a natural object,\" she said, looking at Scott for verification.\n\n\"No. It's too precise.\"\n\nGeary took a picture of it while Scott scraped a sample of the sediment.\n\n\"I see you....\"\n\nShe froze at the sound of the low, seductive female voice in her head.\n\n\"You're so close, little rose. Playing with the wall. But that's not what you want, now is it?\"\n\nGeary looked at the others, but they didn't appear to hear the voice. Who are you? she asked in her head.\n\n\"I am what you seek, Megeara. I am Atlantis. Move closer to me, child. Three feet over. Dig down below the silt. There's a box waiting for you....\"\n\nIt was crazy to even entertain the idea of obeying the voice. What could it know?\n\nAnd even as she told herself to ignore it, she found herself doing what the woman had said.\n\n\"Geary?\"\n\nShe ignored Kat's voice as she dug into the silt. It swirled around her in a hazy blur. As she dug deeper, she found nothing.\n\nI am insane.\n\n\"Geary!\" Tory's voice was sharp through the intercom. \"Stop moving.\"\n\nShe froze.\n\n\"Move the camera half an inch right.\"\n\nGeary did as ordered. \"Why?\"\n\nBefore Tory could answer, Geary saw what her cousin had seen on the feed. It was the corner of something that appeared to be a box.\n\nNo...\n\nGeary held her breath as she gently pulled it free. It was encrusted with funky deposits from the sea. But that wasn't what fascinated her.\n\nThe box was ancient, with a clear design of lions pulling a chariot where a tall god holding a staff stood, directing them. It, like her necklace, held that strange indecipherable writing.\n\nHer hands shaking, she carefully pried the lid off to see what the box contained.\n\n\"What is it?\" Tory asked, her voice filled with anticipation. \"I can't see it, Geary. What's inside?\"\n\nGeary let out her breath in frustration as she realized the box was empty. \"Nothing, Tory. But the box is old.\"\n\nShe handed it off to Scott so that he could preserve it for examination topside later.\n\nHoping to find something even better, Geary had bent down to continue her search when she heard what sounded like Tory dropping something. There were muffled voices in the background like someone was arguing, but Geary couldn't tell what was going on.\n\n\"You okay, bud?\"\n\nThere was no answer.\n\n\"Tory? Christof?\"\n\nTwo seconds later, something popped loud in her ears. It was followed by nothing but empty static.\nCHAPTER 12\n\nThere was no worse feeling for Geary than to think that someone she loved was in danger and to not be able to get to them. Geary was hysterical as she forgot about her quest and swam as quickly as possible through the dark water.\n\n\"Geary!\"\n\nShe ignored Kat's call. The only thing that mattered was getting topside to see what was going on.\n\nSuddenly Arik was there, pulling Geary back. \"You've got to calm down. You're breathing too rapidly.\"\n\nShe shook her head at him. \"That sounded like an explosion. I have to get up there.\"\n\n\"All we heard was a pop.\" His tone was level and calm as those blue eyes haunted her. \"It could have been anything. All we know is the intercom is malfunctioning. You don't want to die over that, do you?\"\n\nHe was right and she hated him for it. Nodding, she pressed her hand against her helmet to listen more carefully. \"Tory? Are you there? What's happening?\"\n\nThere was still no answer.\n\n\"Tory? Christof? Justina? Thia? Dammit. Someone answer me... please.\"\n\nArik held his breath, wishing he could ease her mind. But as the silence stretched on, he knew what she did. Something had to have\u2014\n\nHis thoughts stopped as a piece of twisted metal shot past him on the right. It was followed by more pieces that rained down around them through the water.\n\nAnd it was obvious what the metal was.\n\n\"Uh, folks,\" Scott said, his voice shaking. \"I think that's our boat trying to kill us.\"\n\nYeah. Arik had a bad feeling the kid might be right. Shit.\n\nArik looked at Kat and could tell she had no more information than he did.\n\nHe closed his eyes and summoned his brother with his mind. \"Solin...\"\n\nThere was no answer from that end, either, which boded even more ominously than their exploded boat.\n\nArik let go of Megeara. \"All right, let's head up. Safely.\"\n\n\"Okay.\" But he could hear the panic and fear in Megeara's voice.\n\nHe hesitated as Megeara and Scott went first so that he could swim beside Kat. \"Any clue?\"\n\nShe shook her head glumly. \"You know if the boat's gone, so's our air supply.\"\n\nHe'd thought of that himself. So far their air was holding. Which was another reason they needed to get up immediately. They didn't have much reserve, and since he was human, he could die here, which was the last thing he wanted. Kat was the only one of them who didn't have that fear. Lucky bitch.\n\nAs they swam for the first decompression station, Arik tried his best to reach Solin repeatedly while Megeara continued to call through her intercom for the others.\n\nAnd repeatedly there was nothing from above, except for a few pieces of metal that continued to float by as they headed for the floor of the sea. Yeah, nothing like watching your life pass you by, knowing you were even more sunk than it was if you didn't get to air soon.\n\nThey reached the first tank, which held two breathing hoses. The women went first, then he and Scott. They traded the lines back and forth as they waited for their bodies to adjust to the new depth before they swam any higher.\n\nBut each of them was rattled by what could have happened to the boat and the others.\n\n\"Arikos...\"\n\nHe hesitated as he finally heard Solin's voice in his head. \"What's going on?\"\n\n\"The boat was blown up.\"\n\n\"No shit. We rather got that, as parts of it almost crushed us in the water. What happened and why didn't you stop it?\"\n\n\"Yeah, right. Stop it, my ass. I'm not messing with this one.\"\n\nArik cursed at Solin's selfishness. \"Dammit, Solin. There's not enough air in the line for us. We might not make it back.\"\n\n\"Is that supposed to mean something to me? You're all going to die anyway, right? What's the point? I still have to live here after you're all gone.\"\n\nIf Arik could lay hands on Solin, he'd kill him.\n\n\"This isn't a game, Solin.\"\n\n\"No, it isn't. And you're on your own. Good luck.\"\n\nArik ground his teeth as he took the hose from Megeara so that he could take a turn breathing. \"You better pray I don't make it up there to you.\"\n\n\"Praying's for amateurs and you got much bigger problems than me.\"\n\nArik felt Solin drifting away from his thoughts. Megeara took the hose and breathed deeply before she swam upward again. Arik and the others followed her.\n\nThey were halfway to the second station when more debris started falling through the water. What the hell? Had another boat blown up?\n\nArik swam up to Megeara as she started moving even faster.\n\n\"Megeara. Stop. We have to breathe slowly. You know that. Stay calm and focused.\"\n\nGeary wanted to shove him away from her, but she knew he was right. She couldn't afford to fight him in the water. Right now, they had reserve air at the stations. They could stay there until their bodies readjusted. Then they could hold their breath and move to the next. But this was taking too long for her tastes.\n\nShe had to know what was going on. What had happened to the others.\n\nHer heart heavy, she looked at Scott, who had tears in his eyes. \"We'll be okay, Scott.\"\n\nThe doubt he held burned through her. \"Yeah, right.\"\n\n\"No more talking,\" Kat snapped at them. \"We have to preserve what air we got.\"\n\nArik took Geary's hand and squeezed it before he urged her up through the water. Geary obeyed, but as she swam, a thousand thoughts went through her mind. Her brother had died over a malfunction in his tank.\n\nShe'd always wondered what had gone through Jason's mind in those last few minutes as he realized his life was about to end. She had to say it sucked. Memories and unfulfilled dreams poured through her with a burning intensity.\n\nShe didn't want to die. She was young and though she hadn't dated much, she still held on to the dream that one day she'd meet a great guy and have kids. That she would grow old with someone who treasured her as much as she treasured him. Was that too much to ask?\n\nThere was so much she wanted to do. And now she might never even see daylight again. It wasn't fair\u2014to come so close to her goal and die before she could finish her quest.\n\nBut the worst thought was that Tory and Thia might be dead. Justina, Teddy, Christof, and all the others...\n\nAnd it would be her fault. All her fault. God, how had her father lived with the guilt? No wonder he'd become an alcoholic. In that moment, she had an understanding of him that she'd never had before. She'd spent so much time blaming him that she hadn't even considered the blame he'd reserved for himself. The gnawing pain of knowing he'd endangered his family and the memory of them dying because of his actions.\n\nI am so sorry, Dad.\n\nIf she got out of this and if the others were okay, it was over. She would never jeopardize other people again. This was just a stupid dream that wasn't worth another drop of blood. Another life. Atlantis didn't want to be found.\n\nShe was through.\n\nSuddenly Geary realized that the water was getting lighter. Looking up, she could see the sunlight refracting in the waves over her head.\n\nNot much farther... Her joy was tempered by the fear of what she would find there waiting for her. Of what had happened to the rest of their crew. Horrific images haunted her. Thoughts of Tory and Thia lying facedown in the water. Or disfigured... of them calling out to her for help...\n\nPlease, please, please be okay.\n\nHer throat was getting tighter as she neared the top. The pain of having no air in her lungs was oppressive and painful. Her lungs burned like fire as even more panic tore through her. How tragic to die this close to their destination. Just a few feet more and she'd reach the surface.\n\nShe began unfastening her helmet as she kicked up. Her limbs felt so heavy. Her heart was pounding from the strain. She wanted to take a breath so badly but knew that she couldn't.\n\nPlease....\n\nBy the time she reached the top, she had the helmet off. Geary tossed it aside as she finally gulped air. She was shaking and cold as water rushed into her suit. But it was so good to be breathing freely again that she didn't even care.\n\nShe turned a circle in the water trying to get her bearings. The first thing she saw was the smoldering remains of their boat\u2014not that there was much of it left.\n\nHysterical, she started swimming for it, only to have someone pull at her. She turned to find Arik there.\n\n\"They're dead,\" she sobbed, pulling away from him. \"I have to go find them.\"\n\n\"They're not dead.\"\n\nAnger tore through her and as she opened her mouth to tell him not to patronize her, he pointed in the opposite direction of the boat.\n\nShe looked to find the small life raft that held Tory, Justina, Solin, Althea, Kat, Thia, Christof, and Brian. Relief tore through her with such ferocity that she sank back under the waves.\n\nArik caught her against him and helped her back to the surface. She was laughing and crying as she wrapped herself around him and kissed him. He'd never seen anything like this. It was like her emotions were completely beyond her control. She smiled at him before she left and headed for the others.\n\nBaffled, he treaded water while Kat and Scott popped up beside him.\n\n\"Thank the gods,\" Kat said after she'd tossed her helmet aside.\n\n\"What the hell happened to the boat?\" Scott asked.\n\nKat gave him a \"duh\" stare. \"It appears to have blown up.\"\n\n\"Yeah, but why?\"\n\nShe gave Arik a harsh stare. \"That's an interesting question, isn't it?\"\n\n\"Yes, it is.\"\n\nScott immediately swam after Megeara while Kat and Arik hung back.\n\n\"Do you think Zebulon had a hand in this?\" he asked her.\n\nKat shook her head. \"Not his style. He'd have simply crushed all of us and left nothing behind. No, this was a thoughtless act.\"\n\n\"Human then?\"\n\n\"I don't know, but I will find out.\"\n\nArik frowned. There was an odd note in her voice and he had a sneaking suspicion that she knew exactly who was behind this and didn't want to betray them.\n\nArik would have blamed this on the Dolophoni, but it wasn't their style, either. Their fight was with him, which meant they'd have come to fight him in the water and not bothered threatening the ones on the surface. Not to mention, the Dolophoni wouldn't have been this sloppy. No, had they been here, he would have known it. For that matter, he'd be bleeding from it.\n\nSo then who?\n\nDamn, enemies were falling out of the trees. Lucky them.\n\n* * *\n\nLaughing hysterically, Geary pulled herself onto the raft and grabbed both Tory and Thia in a giant hug.\n\n\"Hey! You're getting me wet!\" Thia snapped, shoving at her.\n\nGeary ignored Thia as she held her close. \"Thank God you guys are all right.\"\n\nTory kissed her cheek. \"And so's Kichka.\" She held up the material in her lap to show a very upset cat, who was hissing at them. \"I scooped her up on the way out.\"\n\nGeary kissed the top of her cat's furry head before she took Kichka from Tory and looked around at the others. Everyone was there and accounted for. \"What happened?\"\n\nTory indicated Solin with her thumb. \"Solin said he smelled a gas leak. If not for him, we'd all be dead now.\"\n\nGeary scowled at the explanation. \"Gas leak? How? Christof and I are always meticulous about inspecting everything.\"\n\nTory shrugged. \"I don't know.\"\n\nThey both looked at Solin, who seemed strangely imperious even though his hair was tousled and there were black stains on his immaculate suit. \"I merely smelled it and had a bad feeling that something was about to, pardon the pun, blow.\"\n\n\"Uh-huh,\" Geary said, stroking her cat. \"You have these psychic moments often?\"\n\nOne corner of his mouth lifted into a mocking grin. \"You have no idea.\"\n\nThere was such a strange note in his voice that it actually sent a chill down her spine.\n\nTeddy passed a small flask toward her. \"For you, Skipper. We're glad all of you made it back in one piece.\"\n\nGeary thanked him as Arik, Kat, and Scott joined them on the raft. She didn't miss the hostile look Arik passed to Solin before he came to sit beside her.\n\n\"You better?\" he asked.\n\nShe nodded.\n\n\"Good.\" Arik reached to pet Kichka on the head. Kichka hissed and spat at him before she reached out with one paw to pop his hand.\n\nArik jerked his hand away from her claws.\n\nGeary was stunned. In all the time she'd had her cat, Kichka had never behaved in such a manner.\n\nArik scowled before he moved out of the cat's reach.\n\n\"Kichka,\" Tory admonished. \"What's gotten into you, girl?\"\n\nIt was Solin who answered. \"She's probably just upset over what's happened. It's been a crazy, mad day.\"\n\nMaybe. But there was something very strange going on here and Geary wanted to know what it was.\n\nGeary turned her head to look at the remnants of the boat as help came. It'd been a close call. Too close. Today they had all been lucky.\n\nBut tomorrow...\n\nShe didn't want to think about that. God, what if Solin hadn't smelled the leak? What if the others hadn't listened to him? Instead of boat chunks flying by, it would have been her friends and family.\n\nThe thought sobered her.\n\n\"We have backups of all the data,\" Tory said, taking Kichka from Geary's hands. \"We can redo everything.\"\n\n\"No,\" Geary said, her tone firm. \"We're through with this.\"\n\nEveryone on the raft with the exception of Solin and Kat gaped at her.\n\n\"What are you talking about?\"\n\n\"How can we quit?\"\n\n\"Are you insane?\"\n\n\"We just got the permits! How can you even say that?\"\n\nThe questions were fired at her in rapid succession. Geary held her hands up to quell their fury. \"Look, guys, I'm not my father. I can't live knowing that I caused someone else to die. Least of all the people who are sitting here on this raft. We don't need this. I've been to one too many funerals in my life and I'm tired of it.\"\n\nTory glared at her.\n\n\"Yea!\" Thia said happily. \"Does this mean I get to shop more?\"\n\n\"Shut up, Thia,\" Scott snapped. \"Geary, think about what you're saying.\"\n\nTory held Kichka up to her chin. \"Geary's had a bad shock today. Give her time to calm down and she'll change her mind. You'll see.\"\n\nGeary started to correct her but didn't want to argue the point. Her mind was made up and there was no way she'd ever take another group this far out. Today had taught her a valuable lesson and she was going to heed it before it was too late.\n\nHer resolve set, she stayed on the raft while the others boarded the small rescue boat.\n\nArik hung back with her. \"Are you sure about this?\"\n\n\"Completely.\"\n\nShe expected him to ride her, but instead he asked a simple, flat-toned question. \"Then what are you going to do with the rest of your life?\"\n\nShe laughed. \"I don't know. It's been years since I've thought past this interminable quest. What would you do?\"\n\nA devilish light glowed in his eyes. \"Well, I personally would go back to shore. Shower and change into something nice, then go out to dinner with this guy who wants to spend a little time with me. Then I'd take him back to my place and rock his world.\"\n\nHis words warmed her and she couldn't resist teasing him. \"Daydream about a lot of guys, do you?\"\n\nHe laughed. \"No, that's just what I'd do if I were you.\"\n\nSmiling, she shook her head. \"You're relentless.\"\n\nHe gave a weary sigh. \"I'm never going to wear you down, am I?\"\n\nGeary had to admit he was gorgeous sitting there with his wet hair plastered around his sharp features, bruised though they were. And those eyes... they were the stuff of legends. What could one dinner with him really hurt? After all, he'd kept her calm today and watched after her. If not for him, she might very well have panicked and died today.\n\n\"All right. I'll take you up on it.\"\n\nHe actually gaped at her. Then his smile turned wicked. \"Shall we get naked now? I definitely want to see you rock my world.\"\n\n\"The dinner, you loon. And only the dinner.\"\n\nArik pouted playfully. \"Fine. If that's your best offer...\"\n\n\"It is.\"\n\nHe stood up and helped her to her feet, then he assisted her in climbing up after Teddy. Geary tried not to notice the strength of Arik as he easily swung her up on board the rescue boat, then climbed up behind her. He was nimble and fast.\n\nAnd he made her body melt.\n\nImages of her dreams haunted her.\n\n\"Dr. Kafieri?\"\n\nShe turned away from Arik as one of the officers came up to her. \"Yes?\"\n\n\"I need to ask you a few questions about your boat.\"\n\nNodding, she expected Arik to join the others. Instead, he stayed by her side, lending her his unspoken support while she was interrogated.\n\nKat moved to the back of the boat and remained silent as they headed for the docks. Solin joined her by the side with an intense look of displeasure etched into his features.\n\nFor a man who was completely amoral, his actions today surprised her. \"Why did you save them from the explosion?\"\n\nHe shrugged nonchalantly. \"It was a momentary lapse in judgment, I assure you.\"\n\nShe wasn't buying that argument. \"There's more human in you than you want, isn't there?\"\n\n\"I have no idea what you mean. Whatever humanity ever existed has long since died.\"\n\nUh-huh. \"Did you happen to see who was behind this?\"\n\n\"I didn't see anything, but I felt...\"\n\n\"Felt what?\"\n\n\"A presence near and dear to your heart. It appears Artemis also has an interest in ending this expedition. Perhaps you should take this matter up with her.\"\n\nKat was frozen into place as he drifted away. It was as she thought. Her anger boiling, she stopped a passing sailor. \"Where's the restroom?\"\n\nHe gave her directions to it. Thanking him, she immediately headed for it and locked herself in. Then she closed her eyes and flashed herself to Olympus to speak with said goddess.\n\nA warm breeze whispered against her skin as she opened the golden door to Artemis's temple. Dressed in a flowing white gown that set her pale features and vibrant red hair off to perfection, the goddess was lounging about on her throne while Satara, another of her handmaidens, played the harp for her entertainment.\n\nKat came to a stop before her goddess and crossed her arms over her chest. She looked to the other koris before she barked a single order, \"Leave us.\"\n\nArtemis sighed wearily. \"You don't make that whisper, Katra.\"\n\n\"It's call, matisera. The phrase is 'you don't make that call.' But today I do.\"\n\nThe other koris lifted stunned eyebrows as they waited for Artemis to blast Kat. But she knew she was safe from death. Punishment she could handle.\n\nArtemis pushed herself up. \"Fine. Koris, leave.\"\n\nThey were immediately flashed from the room. Artemis narrowed her eyes as she rose from her throne to stand beside Kat, who was a good two inches taller than the goddess she served. \"What is your breakage?\"\n\n\"Damage, matisera. The phrase is 'What is my damage?' And I want to know why you blew up the boat.\"\n\nArtemis rolled her eyes and made a disgusted sound in the back of her throat as if she couldn't believe Kat would ask such a trivial question. \"Because I felt like it.\"\n\n\"You felt like it? Good grief, matisera, have you any idea how disconcerting it is to be underwater when the boat that's supplying you your air comes floating down past you?\"\n\nShe scoffed. \"Why are you so angry? It's not like you can die. Get a grab.\"\n\n\"Grip.\"\n\n\"Whatever.\" Artemis turned on her with green eyes blazing. \"I don't care what it takes or who has to die. Preserve that seal, Katra. I heard Apollymi calling to you and to that other bitch. I know she was guiding her toward the seal. Box, my elbow. Apollymi knew what she was doing. She won't rest until she's free and I'm dead.\"\n\n\"You can relax. I'm not going to allow Apollymi to be free.\"\n\n\"No? Then act like it. Remember at the end of the day which of us has protected you. Shielded you. Nurtured you. You're nothing but a tool to Apollymi.\"\n\n\"And what am I to you, matisera? Am I not your tool?\"\n\nArtemis's face flushed with color caused by her anger. \"You know what you are to me. Now go and do as you're told. Keep that human from Atlantis.\"\n\nKat ground her teeth at the order. \"When will you learn to trust me?\"\n\n\"Trust you?\" Artemis asked in an aghast tone. \"You went behind my back to Kalosis and then bound yourself to my most mortal enemy. Why should I ever trust you again?\"\n\nThat ignited Kat's own temper. Why did Artemis always bring up something that had happened thousands, and it was thousands, of years ago? \"You know why I had to see Apollymi.\"\n\nIt didn't placate Artemis in the least. \"After all I have done for you, sacrificed for you, you slapped me in my face. If it were anyone other than you, I'd have killed you for what you've done.\"\n\n\"Then kill me.\"\n\nArtemis hissed at her, \"Don't tempt me, Katra. Ever.\"\n\n\"And don't push me, matisera. I know the source of your powers and you know the depth of mine. If we ever go to war, who do you think would win?\"\n\nArtemis curled her lip. \"You are your father's child. Impudent. Surly. Argumentative and spiteful.\"\n\nKat laughed at that. \"Strange, I would have sworn it was my mother you're describing.\"\n\nArtemis's hair flew around her as her face mottled with rage. Her incisors grew to fangs as she spat at Kat. The air around Artemis sizzled with power an instant before she raised her hands and shot a bolt at Kat.\n\nBut it didn't hit her.\n\nBefore it came near, Kat flashed herself back to the boat.\n\n\"Heed me, Katra,\" Artemis growled in Kat's mind. \"I am not one to be trifled with.\"\n\nKatra rolled her eyes at the angry voice. \"My loyalty to you is above reproach, matisera. One day, I hope you will know that.\"\n\n\"I will know it only when you help me kill Apollymi. Until that day, I will always have my doubts.\"\n\n\"I can never hurt her.\"\n\n\"Then I can never fully trust you. So long as your loyalty is split between us, you are as great a threat to me as she is.\"\n\n\"Do you honestly believe that?\"\n\n\"Believe it? I know it. And that is why your boat is lying on the seafloor. Next time you go near that seal, I will make sure that you pay for it with flesh and bone. And the humans will pay with their lives.\"\n\nThat was a beautiful thought. \"Love you, too, matisera. Thanks.\"\n\n\"Kat?\"\n\nShe jerked as she heard Tory's voice. \"Um, yeah?\"\n\n\"Are you okay in there? I heard you talking to someone.\"\n\nKat flushed the toilet before she opened the door. \"Just thinking out loud.\"\n\nBy Tory's face Kat could tell the girl didn't believe her. \"You were speaking in ancient Greek.\"\n\n\"Just practicing. You never know when it could come in handy.\"\n\n\"True. We might one day clone Aristophanes and need an interpreter.\"\n\n\"Yeah.\" Kat stepped past her and headed back to the topside. As she ascended the ladder, a whisper went through her.\n\n\"I will be free, Katra. Neither you nor Artemis can keep me bound here forever.\"\n\nKat could actually feel Apollymi's breath on her neck. Feel the touch of her hand. \"We both know why you can't leave Kalosis.\"\n\n\"And we both know why I must....\"\n\nKat had no sooner stepped on the upper deck than she met Solin, who gave her an amused stare.\n\n\"I really hate voices in my head, don't you, Kat?\"\n\nShe forced her features to remain blank. \"I have no idea what you mean.\"\n\n\"Of course not.\"\n\nAs he moved away, she stopped him. \"For the record, I'm not on their side.\"\n\nHe arched a brow at her. \"Whose side?\"\n\n\"Anyone's. My loyalty is to myself.\"\n\n\"Why are you telling me this?\"\n\nShe smiled. \"Because we are similar creatures. I have my own agenda here and I know you do, too.\"\n\n\"And what is your agenda?\"\n\n\"To survive this expedition.\"\n\nHe laughed in the back of his throat. \"That's something easier said than done, isn't it?\"\n\n\"It's beginning to appear that way.\" Kat looked over to where Arik sat with Geary. They were leaning toward each other, and even though they weren't touching, there was no missing the electricity between them. How Kat wished to feel that way toward a man, but she wasn't that type of creature. Every woman she'd ever known had been ruined by a man.\n\nShe would never be so stupid.\n\n\"What about you, Solin? What do you want out of this?\"\n\nHe gave her a harsh stare. \"Mine is simple. I only want revenge.\"\n\n\"On who?\"\n\n\"Everyone.\"\n\nBefore she could ask him to elaborate or at the very least to narrow his choice down from the few billion people on earth and in other realms, he drifted out of her reach. \"Nice talking to you, too,\" she said under her breath. She was getting really tired of gods drifting away from her.\n\nBut that was no matter.\n\nShe was stressed enough walking the tightrope between Apollymi and Artemis.\n\nGeary laughed at something Arik said. He was smiling as he looked up and caught Kat's gaze. She cocked her head at the way the two of them were acting as she felt the attraction between them. She understood the physical, but what surprised her was what she felt from Arik.\n\nHe thought he lusted for Geary, which he did. But there was more to his feelings than that. As a Skotos, he was used to taking his emotions from others. What he didn't realize was that his newfound emotions came from his feelings for Geary. They weren't Geary's feelings. They were his own. The giddiness and joy he experienced right now wouldn't be there if Geary were somewhere else.\n\nAnd in that moment, Kat understood why Solin was helping him. Solin wanted Arik to know these feelings so that when they were gone and Geary was no longer here, Arik would mourn her. It was cruel beyond belief.\n\nBe careful what you wish for, you just might get it.\n\nArik had wanted emotions and now he would experience the full array of them. May the gods have mercy on him.\n\nSympathetic pain sliced through her, but it was mitigated by the fact that none of this was her business. Arik had chosen this path.\n\nAnd he would be doomed by it.\nCHAPTER 13\n\nGeary stood in front of the full-length mirror, exhausted by her day and yet strangely thrilled at the prospect of being with Arik. She hadn't been on a date in over a year, and the last one had been particularly bad. She'd made the mistake of accepting an offer for dinner from a man she'd met at the local market. Since she'd spent a great deal of time in Europe, she was used to the differences in culture. But this guy...\n\nHe'd been commanding, controlling, and worst of all had monopolized the entire dinner conversation\u2014which had mostly been about how great he was and how he'd make the world a better place if he were emperor. Of course, in her opinion, he'd be dragged through the streets and stoned fifteen minutes after he took that office.\n\nShe should have been so lucky\u2014it was a pity the man hadn't been crowned emperor before their dinner.\n\nIt'd been the only time in her life she'd actually considered crawling out of the bathroom window to escape an obnoxious date.\n\nIf only she hadn't been in a low-cut dress and high heels....\n\nTonight she had on pants and low-heeled Clarks\u2014just in case.\n\n\"Geary, Mr. Arik is here for your date.\"\n\nShe smiled at Tory's loud voice, which was followed by a high-pitched meow from Kichka, and was again overwhelmed with gratitude that no one had been hurt today. Geary honestly wouldn't be able to survive knowing she'd killed someone in her quest.\n\nNothing was worth sacrificing a human life for.\n\nPushing that thought away before she became completely maudlin, Geary checked her makeup one more time, especially since she wasn't used to wearing it and hoped that she hadn't applied it too darkly. Or, more to the point, that she didn't look like a Kabuki actor.\n\n\"You can do this,\" she said to her reflection, trying to bolster her confidence. It was only dinner. She could survive that. There were no strings attached. Just two humans having food and good conversation...\n\nWhich she hoped wouldn't end with Arik thinking he was an all-powerful god of the known universe.\n\nShe pulled her light crocheted sweater out from under Kichka, who meowed in protest before swatting her hand with an indignant paw, then headed to the living room, where Tory was sitting with a copy of Plato's Republic in ancient Greek on her lap. Geary laughed. \"Don't you ever get bored reading that?\"\n\n\"Not really. There's always something in it that I missed the last time. The man is really, really deep.\"\n\nGeary shook her head. \"You're a sick girl, Tor. Sick, sick, sick.\"\n\n\"I know. I come by it honestly.\" She gave Geary a meaningful look over the top of her glasses.\n\n\"It's true,\" Geary agreed. \"We come from a long line of people who live to read boring texts\u2014I think it may be why we all die young. Complete boredom.\"\n\nTory stuck her tongue out at her.\n\nGeary paused as she saw Arik waiting by the door. He was positively striking in a black suit with a white silk shirt that had the top two buttons undone to show a delectably tanned neck. His black hair curled becomingly around his face and shoulders while those crystal blue eyes radiated heat and intensity. For the first time since they'd met, he was clean shaven, which made him appear somewhat more tamed and cultured. But only a tiny bit. There was still that aura of raw power that emanated from him.\n\nAs she drew near, he handed her a bouquet of white roses. Geary smiled at the gesture as she took them and cupped them to her nose so that she could inhale their sweet scent. \"Thank you.\"\n\n\"My pleasure.\" Then he crossed the room and handed a smaller bouquet off to Tory, who actually put her book down and beamed happily.\n\n\"For me, too?\"\n\nHe nodded. \"Least I could do for the woman who introduced me to fudge Pop-Tarts.\"\n\nTory squeaked as she took them and buried her face in their soft petals. \"I love roses. Thank you.\"\n\n\"Anytime.\"\n\nGeary kissed his cheek before she handed her roses to Tory to take care of. \"Are you sure you're going to be all right by yourself?\"\n\nTory scoffed. \"You're the one wigging out over today, not me. I'm fine. You two go and have fun. I have plenty of stuff here to entertain me with. Plato rocks.\"\n\nGeary glanced to the mountain of ancient Greek books on the coffee table and knew that Tory would be up all night reading. The girl really was insane. \"Okay. But if you need anything, call Teddy. He said he was staying home tonight.\"\n\n\"Will do, Captain.\"\n\nArik opened the door for Geary to walk through. She paused as she saw Solin's limousine on the street, waiting for them. \"Should I be afraid?\"\n\nHe offered her his arm. \"Not at all. Solin has already prepped me on how to behave tonight. No public gropings no matter how much you turn me on. He even showed me how to use cutlery so that I wouldn't embarrass you.\"\n\nGeary frowned, wondering if he was joking. He didn't seem to be, but surely...\n\nHer thoughts drifted as she entered the car with Arik behind her. A weird sense of d\u00e9j\u00e0 vu went through her, along with the scent of his aftershave and the strength of his body. He was a choice specimen who caused every part of her to sit up and beg for attention.\n\nHow she wished she had more of Thia in her. If she did, she and Arik would be getting naughty and naked in the back of the limo and poor George would be going blind from their raucous play. But she wasn't that type of woman. All she could do was dream....\n\nArik sucked his breath in as Megeara slid over the seat to the opposite window. The way she moved, slow and easy, reminded him of her sliding over his body. If this were a dream he'd be able to pull her to him and kiss her until they were naked and blind with pleasure.\n\nHis swollen cock burned with need. But unfortunately, this wasn't a dream and she would probably have his head if he tried... and he didn't mean the one on his shoulders.\n\n\"You look incredible,\" he said as George shut the door.\n\nHer cheeks pinkened. \"Thank you. You look pretty good yourself.\"\n\nHe smiled. \"Good. Solin can live another day.\"\n\n\"What do you mean?\"\n\n\"He's the one who told me what to wear tonight. I didn't know if I should trust him or not, though. He's not the most reliable of people.\"\n\nHer eyes softened as if she understood. \"You two have an odd relationship, don't you?\"\n\n\"You could say that. It often reminds me of a blowfish and a barracuda.\"\n\n\"Interesting analogy. So which one is you and which is Solin?\"\n\nHe winked at her. \"I'll leave that for you to decide.\"\n\nNot sure what would least insult him, Geary didn't speak while they drove to a small seaside caf\u00e9. Her heart clenched as they left the car and she realized where they were.\n\nArik paused as he noticed her hesitancy. \"Are you all right?\"\n\nShe had to force herself to respond over the lump of sadness in her throat. \"Yeah. Sorry. I was just thinking about something.\"\n\n\"What?\"\n\nShe pointed to an old brick wall across the street that was beside a set of stone stairs that had been worn over time by intensive foot traffic and the elements. \"My brother and I used to climb over that wall when we were kids. We'd pretend it was the wall of Troy.\" She gave him a sheepish look. \"Yeah, I know, we were strange children. Jason would play Hector and I was always Achilles. We'd lob dirt clods and rocks at each other until either one of us was bleeding or my father would yell at us to stop. Then we'd sneak attack each other and plot our revenge.\"\n\nShe took a deep breath to stave off the pain. \"God, how we'd play. Then when we were older, Jason used to come here to sit at this caf\u00e9 and sketch what he thought this whole area would have looked like centuries ago.\" The corner table that he would always lay claim to because it had the best view was still there just as it had always been. The table looked as if it were waiting for the young man who would never again pass this way.\n\nHer eyes misting, she looked up at Arik as all those memories ripped through her. Jason would spend hours telling her his concepts for his drawings. He'd been so precise and detailed in his descriptions that there were times when she would have sworn he had to have lived back then to know them all. How she wondered what he'd have been like now. What he'd think of her...\n\nShaking her head, she tried to dispel her bittersweet memories and the grief they caused.\n\n\"Can you imagine what the island must have looked like a thousand years ago? Two thousand?\" she asked Arik.\n\nArik wished he had his powers. If he did, he would have granted her that wish. In one heartbeat, he could have shown her exactly what this place had looked like\u2014firsthand.\n\nThen again, he did have that power in another realm. \"I'll bet when you dream tonight you'll see it.\"\n\nHe saw the doubt in her eyes before she answered. \"Sure. Why not? I dream of enough other weird stuff.\"\n\n\"Such as?\"\n\nShe blinked before she stepped away. \"Nothing. Shall we eat?\"\n\nHe hated whenever she closed herself off from him. Especially when he knew how much more she was hiding. But then, he'd known her for a while now.\n\nIn her mind, they'd just met and they were all but strangers.\n\nRegretting the necessity of that, Arik directed her toward the caf\u00e9. Solin had prepped him on how to greet the hostess and request a seat, but it was still very strange. It was odd how people left such trivial matters out of dreams. They simply cut to the chase and didn't waste valuable time with incidentals. If someone wanted to eat, they were in the restaurant, eating. There was none of this getting to it and requesting tables or waiting.\n\nDreams really were superior to reality.\n\nAfter a bit of a wait, Arik and Megeara were seated at a table that overlooked the sea. Even though it was dark, they could still hear the surf and see the whitecaps as they rolled onto shore. Lights from boats and buildings in the distance twinkled like stars that had fallen to earth, while the smell of cooking food made his stomach grumble and cramp.\n\nArik was surprised by the sensation. He'd never been really hungry before. And the sights and sounds were overwhelming as they brought a peculiar ache to his chest. He didn't understand the source of it. He felt sad and happy for no apparent reason, and when he looked at Megeara, all he wanted to do was reach out and touch her. To ask her if the sight and sounds made her feel the same way.\n\n\"I've never eaten here before,\" Megeara said as she skimmed her menu. \"What do you recommend?\"\n\nHe frowned as he looked at his menu and wondered what he should suggest. \"I don't know. I didn't think to ask that of Solin. Is that something a date normally knows?\"\n\nShe gave him an arch stare. \"Only if the date has eaten at the restaurant before.\" Then she chided him, \"Don't tell me you've never been on a date.\"\n\nArik realized he'd already made another mistake. She would never believe that he'd reached whatever age he appeared to be without having taken out a woman\u2014it would be completely illogical for a human male to have kept to himself. \"No, I have... just not like this.\"\n\nShe still wasn't buying it. \"Not like this how?\"\n\nThink, Arik, think. \"With a woman.\"\n\nHer other brow raised as she gave him an amused smile. \"So you've been on dates with men?\"\n\nSmooth move. Solin was right. He was a moron. \"No, no. What I mean is that I've never asked a woman out on a formal date. I usually spend an evening or two with them and then leave.\" There, that sounded better and it was the truth.\n\n\"Then you what?\" she asked, her voice tinged by anger. \"You leave them waiting by the phone for a call that never comes? How wonderfully kind of you.\"\n\nWhy the sarcasm? What had he said that was so upsetting?\n\n\"No, that's not what I meant.\" How could one man get himself into so much trouble with just words? But he could tell from her body language and the fury in her eyes that he wasn't helping himself in this at all. \"Why are you being so hostile to me, Megeara?\"\n\n\"I'm not hostile. I'm merely trying to understand you and the things you keep telling me. I mean, how did you manage in Nashville with such a limited understanding of people and how things work?\"\n\nNashville? What was she talking about now? He'd never heard of such a thing before. He was constantly baffled by her. \"What is Nashville?\"\n\nShe gave him a \"duh\" stare. \"Where you claim to have met me. Remember?\"\n\nHe shook his head. \"No. That was Vanderbilt where we met.\"\n\n\"Yes, and Vanderbilt is located in Nashville, Tennessee.\"\n\nArik froze as he realized what he'd just done. In her dreams she'd never made mention of what town the school was in, and since he wasn't from this plane there was no way for him to know it.\n\nHe cleared his throat as he tried, yet again, to cover his blunder. \"Oh, it's been a long time.\"\n\nInstead of being comforted, she looked even more suspicious. \"Six years isn't that long ago, especially not for a man who remembered me so well. And I don't see how a man raised in rural Greece could forget his trip to a bustling American city so easily, either. What's going on, Arik?\" She narrowed her eyes on him. \"You didn't meet me there, did you?\"\n\n\"Of course I did,\" he said defensively. He had no choice except to try to brazen this out. \"Why would I lie about that?\"\n\nGeary didn't know what to think. But something wasn't right with all of this. She could feel it in her gut and she could see it on his face. He was hiding something extremely important about their meeting. \"How should I know why you'd lie? But you're not who you say you are, are you?\"\n\n\"I am.\"\n\nYeah, right. \"Be honest with me, Arik. Who the hell are you?\"\n\n\"I've told you. I'm Contranides.\"\n\n\"Yeah, you keep saying that, but why don't I believe it?\"\n\n\"I can't imagine. It's the truth.\"\n\nStill, her gut warned her to put distance between them. Had they been alone, she would have. But they had plenty of people around them and she wanted some answers. \"Tell me the truth, Arik. Why are you here with me?\"\n\n\"I just want to spend time with you.\"\n\nWrong answer. \"You keep saying that.\"\n\n\"Because it's true. I swear it.\"\n\nShe clenched her teeth as a wave of anger went over her. Why couldn't he tell her what was going on? Honestly, she was getting tired of his cryptic ways and things about him that just didn't add up. \"I don't believe you.\"\n\n\"Then what do you believe?\"\n\nShe didn't know, but the more the thought it over, the less any of it made sense. Something wasn't right with this. With him. She knew it. And his constant denials were making her feel as if she were losing her mind.\n\nGlancing away, she caught sight of an incredibly intense man who was staring straight at them. At least six six and with an aura of \"don't look at me or I'll kick your ass,\" he was wearing a long black leather coat and dark sunglasses even though it was nighttime. He had a small goatee and short black hair. There was something ominous about him. It was as if he was looking for someone to fight and kill.\n\nShe had to drag her gaze from him, back to Arik. \"Do you know him?\"\n\nHe followed her line of vision to see the man who was now making his way toward them. There was a knowing smile on the man's sardonic lips as he paused at the table next to them. The \"killer\" whisked his coat off with a flourish, and as he moved, a tattoo of a double bow and arrow on his biceps peeked out from beneath his sleeve.\n\n\"Evening, folks,\" he said to them in Greek as he took a seat.\n\n\"Good evening,\" she answered.\n\nArik merely inclined his head. But there was no missing his tenseness. He didn't like the newcomer and it was obvious.\n\n\"Is he a friend of yours?\" she asked in a low tone.\n\nArik cursed silently at the presence of the Dark-Hunter. Immortal warriors in the service of Artemis, they protected mankind from the things that would prey on them. No doubt the Dark-Hunter could sense the essence of Arik's soul. Even though he was technically human at this time, he still had the soul of an immortal, and since Dark-Hunters were the protectors of human souls, the Dark-Hunter would know that Arik wasn't human.\n\nCould the Dark-Hunter's timing have been any worse? Megeara was already suspicious enough. The last thing Arik needed was her asking questions about immortal vampire slayers.\n\nAnd then he felt it. It was a whisper against his soul. A touch.\n\nA threat.\n\nThe Dolophoni had found him. Their presence on this plane was unmistakable. They were here and they were looking for a fight. He glanced around the restaurant and street but couldn't find anything out of place. Everyone around him, except for the Dark-Hunter, was human.\n\n\"Is something wrong?\" Megeara asked as she noticed Arik's fidgeting.\n\nHe knew the smile he offered her was extremely fake. \"No. Nothing.\"\n\n\"You don't look like nothing's wrong. You look really nervous all of a sudden. What? You owe the guy next to us money or something?\"\n\nHow he wished it were that simple. No, he owed one Greek god a human soul and a dozen more his life. Yeah... well, it was time to put a stop to at least one side of that statement. \"I just need a moment. You wait here and I'll be right back.\"\n\nGeary frowned as Arik got up and left her alone. She didn't know what concerned her more, the strange man at the table next to her who kept looking over as if he knew a secret she didn't, or Arik's peculiar behavior.\n\n\"You have an interesting friend there,\" the man said.\n\nGeary cocked her head as she picked up a slight brogue in his Greek. \"Are you Scottish?\"\n\nHe laughed before he answered her in English. \"I used to be something like that.\"\n\nGeary frowned at his words. What? Something like a Scot? Was the man a Pict? He had the bearing of one of their ancient breed.... Yeah, right. That would only make him a couple of thousand years old.\n\nShe buried her sarcasm before she spoke again. \"Do you know Arik?\"\n\nThe man nodded before he looked off into the direction where Arik had vanished. \"I met him a long time ago. He helped me out of a bad situation. He's helped a lot of people over the years.\"\n\nThere was an odd note in his voice. Dubious. And it made her wonder if Arik was a drug dealer or some other criminal. \"Helped them how?\"\n\n\"With this and that.\"\n\nThe vagueness was really starting to wear on her and it made her suspicions climb. Why wouldn't he tell her unless she was right and it was something highly illegal? Maybe they were arms dealers\u2014Solin had never answered her question about what he did for a living. Ah jeez, that would be her luck. Hooking up with wanted fugitives.\n\nGreat.\n\nShe lifted her water glass up in a mocking toast. \"Thanks for being ever so helpful.\"\n\nHe took her salute in stride. \"My pleasure really. Have a good night.\"\n\nHave a good night. Why did that not seem possible? 'Cause I'm having dinner with an arms dealer. Or some other kind of criminal. She dismissed that thought in favor of another one. Where was Arik? He should have come back by now.\n\nAs if he heard her question, the man at the other table cocked his head as if listening to the air around them. His face turned to stone before he got up and jumped over the small chain that separated the caf\u00e9 area from the street. He quickly made his way to the side of the building and vanished without even looking back at his coat.\n\nWhat was that not so stealth action?\n\nGeary knew that where he was going and what he intended was none of her business and yet she felt a deep compulsion to follow after him.\n\nDon't be stupid. He could be an undercover cop of some sort. Hell, he could even be CIA. Interpol. Scotland Yard. Even an assassin or space alien. Her imagination went wild with possibilities.\n\nBut before she could stop herself, her curiosity won out and she got up and headed in the same direction where he'd disappeared.\n\nEven as she went, she called herself every name she could think of. How stupid was this? What kind of idiot chased after a man who looked like a killer and was heading off into who knew what? I'll stick to the shadows and if it gets bad, or looks scary, I'll run right back.\n\nYou're an imbecile, Gear, a total flaming imby!\n\nBut the silent berating stopped the instant she entered the alley to find Arik in the middle of a fight with the same twins she'd seen him fighting in her forgotten dreams. In one instant, the entire fight on the beach came rushing back to her.\n\nGeary froze as she gaped at the impossibility of what she was seeing. This could not be happening.\n\nThe man she'd followed approached the twins slowly, with purpose. Arik was bleeding as he kicked one twin back and the other backed up to confront the newcomer.\n\n\"Stay out of this, Dark-Hunter,\" the twin warned the man she'd followed. \"This doesn't concern you.\"\n\nHe shook his head. \"Arikos and I go way back. You want to fight him... it involves me.\"\n\nThe twin started for him, but Arik ran at the twin and knocked him into the wall. Arik's gaze met hers and she saw his concern for her. \"Get Megeara,\" he snarled at the one called Dark-Hunter. \"Keep her safe.\"\n\nThe other twin literally ripped Arik away from his brother before he twisted open a butterfly knife and plunged it into Arik's side. Blood instantly soaked his shirt and poured over the man's hand.\n\nGeary choked on a scream as she saw the pain on Arik's face. He gasped an instant before his eyes narrowed in anger.\n\n\"We're not in the dream realm now,\" the twin snarled into his ear, \"and you're not so tough here, are you, Skotos?\"\n\nArik hissed before he head-butted the twin and knocked him away. Then Arik jerked the knife from his side and held it in a bloodied fist. \"Don't dismiss me, asshole. Here or there, I can still kick your ass.\" He moved to stab the twin, only to have the other twin lunge at him.\n\nThe Dark-Hunter caught Arik's attacker and kicked him back.\n\nGeary turned to go for the police and instead ran into a huge man who had a demeanor so lethal and a body so solid that he should rent himself out as a wrecking ball. His face bore the wrath of hell as he pushed past her and threw his hand out.\n\nAll four men hit the ground hard, as if they'd been struck by something invisible. Including Arik, who lay on his back.\n\nBut the twins shot up immediately and when she said \"shot up\" she meant \"shot\" up. They literally cleared the ground by five feet as they arced from where they'd fallen to land just in front of the newcomer.\n\nThey stood before him united in strength and power. It was as if they existed in perfect symmetry.\n\n\"Stay out of this, Zebulon,\" the one on the right warned in a ragged voice full of venom. It was so raw and primal that it sent a shiver of fear over Geary.\n\nZebulon shook his head as if he couldn't believe them. \"You guys come to my town, you don't call. You don't write. And you expect me to just let you run amok in front of the humans? Really, Deimos, don't tread here unless you want to bleed.\"\n\nThe other twin bared his teeth. \"He belongs to us.\" He turned toward Arik, then froze.\n\n\"I'm not your bitch, Phobos. You didn't drag me out of the pound to put a collar on me. Don't expect me to heel because you say so. You are on my turf now. Think about it.\"\n\nDeimos curled his lip. \"We were sent here for him. How dare you interfere with the gods?\"\n\nGods?\n\nGeary took a step back as she again heard the female voice in her head telling her to take note. Note of what? Her fleeing intelligence? The fact that she was having a massive hallucination?\n\nShe was losing her mind... she knew it. But even so, she had to check on Arik. He was bleeding profusely and lay on the ground as if he was heading into shock.\n\nZebulon scoffed at the twins. \"Did you miss my job description? Or were you just not paying attention the day I busted heads on Olympus? Fucking with you people is what I do. It's what I live for and I'm really tired of you now.\"\n\nThe twins vanished instantly.\n\nIgnoring her, Zebulon inclined his head toward the Dark-Hunter. \"You all right, Trieg?\"\n\n\"I'm not the one bleeding, ZT. That's a question best asked of Arikos.\"\n\nGeary was already at his side. Arik lay on the ground with his hand over the wound that was bleeding profusely. The blood coated his fingers and made her stomach wrench at the sight of it. The wound was so deep, she could actually see exposed bone. Sweat covered his face as he kept his jaw clenched to deal with the pain.\n\nShe brushed the hair back from his brow. \"We need to get you an ambulance.\"\n\n\"Not really,\" Zebulon snapped from behind her. \"You just need to move your butt and let me see him.\"\n\nBefore she could respond or move, Zebulon pushed her aside and ripped Arik's shirt open.\n\nGeary cringed at the ragged wound the knife had left behind. \"Don't hurt him.\"\n\nZebulon curled his lip at her. \"Do you think I came all the way over here to hurt him? If I'd wanted him hurt, I'd have left him to Tweedle Dumb and Dumber.\" Turning back to Arik, Zebulon hovered his hand about an inch over the wound, and as he moved it back and forth the wound knitted itself closed.\n\nGeary stared, dumbfounded, as shock poured through her.\n\nOf course the wound healed itself. Sure. That made perfect sense, didn't it? Arik had left her alone at the caf\u00e9 and a weird Scottishesque guy had led her to a battle with two men who'd been in her dreams, who could leap higher than the bionic kangaroo on steroids, and another scary dude who could heal gaping wounds with his hand.\n\nIt all made sense.\n\nIf you were on massive quantities of illegal drugs.\n\n\"Okay, I'm dreaming. Hallucinating. Brought on by stress. I had a hard day today and this is my mind trying to protect itself from... from stuff. Lots of stuff.\"\n\nThe three men were frowning at her, which only served to set off her temper.\n\n\"Oh, like I'm any less sane than the three of you just because I talk to myself.\"\n\nTrieg cleared his throat. \"I'm thinking you should wipe her memory, ZT. Do that Were-Hunter thing so that she goes back to normal and forgets all about us.\"\n\nZebulon scoffed. \"I'm Chthonian, Trieg. We don't do that.\"\n\nGrimacing at the response, Trieg rubbed the back of his neck. \"I'm thinking you should start.\"\n\nGeary took a step back and pointed with both index fingers over her right shoulder. \"And I'm thinking I should take myself home.\" She pointed her finger at the men, winked, and made a small clicking noise by sucking her breath between her teeth. \"You guys have a great night... with whatever it is you people do. See you later.\" She turned and took a step away, then swung back to face them. \"On second thought, no offense, I never want to see any of you again. Good night.\"\n\nWith a quick word of thanks to ZT, Arik pushed himself up from the ground and ran after Megeara. Just as she left the alley he caught up with her and pulled her to a stop.\n\n\"Megeara\u2014\"\n\n\"Geary,\" she snapped.\n\n\"Geary,\" he said, hoping to placate her as he rubbed her arm in case he might have harmed her any by stopping her. \"Please. I didn't want you to see any of that.\"\n\n\"See what?\" she asked with a bit of hysteria in her voice, \"I didn't see anything. There were no scary people there. Nothing freaky.\" She patted him on the biceps, then smiled as if nothing were wrong. \"I'm going home now and tomorrow I'm going to have the doctors check for a brain tumor. Full battery of tests. Whole nine yards. Whatever's wrong with me, we'll find it and deal with it. At this point, my vote is either tumor or space alien testing. Either one works for me.\"\n\n\"You don't have a tumor and there aren't any aliens running around here. You're not insane.\"\n\n\"No?\" Her face was aghast. \"Then what am I?\" She held her hands up before he could answer. \"No, wait. The better question is, what are you?\"\n\nArik wasn't sure how to answer. But then, there was no use keeping anything from her, since she'd already seen so much. It was time for complete honesty. \"Do you know what an Oneroi is?\"\n\nThe sarcasm in her voice was so deep it could drown a champion swimmer. \"A Greek god of sleep. I did actually have to study this stuff before they allowed me a doctorate, you know?\"\n\n\"I know,\" he said calmly. \"Oneroi are gods of sleep.\" He spoke slowly, enunciating each word carefully. \"You know me, Geary. You've known me for a long time....\"\n\nShe let out a nervous laugh and he could see the clarity in her eyes as she looked up at him. \"So what are you saying? You're an Oneroi?\"\n\nHe nodded slowly.\n\nGeary laughed. Hard. Until she realized that he wasn't joining her laughter.\n\nShe froze as a chill went over her. \"You're a god, huh? Then tell me something only a god would know.\"\n\nHe didn't even hesitate with his answer. \"The first night I met you in your dreams, you were bathing in a river of chocolate. Your entire body was coated with it and you were cupping your hands under the waterfall, then drinking the chocolate. I came up behind you and kissed your neck, then gave you a goblet that we both drank out of. You filled the cup, then poured the chocolate over me and licked\u2014\"\n\nShe placed her hand over his mouth to stop him from speaking. \"You were there.\"\n\n\"I was there.\"\n\nDisbelief poured through her. It couldn't be. It just wasn't logical. \"What about Vanderbilt?\"\n\n\"You dream about that at times. Reliving the horror of it. I snooped a bit.\"\n\nGeary dropped her hand as memory after memory of her making love to the dream Arik played through her mind. Now to find out that it was real...\n\nIt pissed her off. \"Snooped a bit? No, buddy, you've snooped a lot.\" Geary was mortified as various memories went through her mind. \"I didn't know you were real. No. You can't be real. This is crap. It's all crap. You're lying to me.\"\n\n\"It's real, Geary.\" He took her hand into his and held it against his chest so that she could feel his beating heart. \"I'm real.\"\n\nShe looked down at where he'd been stabbed. There was no blood. No tear in his clothes where she'd seen his wound with her own eyes.\n\nBut there was still blood on her hands.\n\nHis blood.\n\nHe looked just as he had when he'd picked her up at her flat. Just as he'd looked when he had left her at the table and vanished.\n\nHer gaze drifted over his shoulder to where Trieg was watching them from the shadows.\n\nShe pulled her hand away from Arik and gestured toward Trieg. \"And he's just plain odd.\" She turned away from Arik and instantly walked into Zebulon again. Okay, it bothered her that he could just appear like that out of the blue, without warning, but she'd had enough. \"And what is your problem that you keep putting yourself in my way?\"\n\nHe answered with a sadistic laugh, \"She's feisty, Skotos. I can see the appeal.\"\n\nArik snorted. \"Oh, you've no idea.\"\n\nWhen she tried to move past him, Zebulon stopped her. \"Not to be rude, but what the hell? I live for it. You can't start running your mouth about what you've seen here tonight.\"\n\nOh, that was priceless. \"Great threat you've got going there, big ZT. News flash, I didn't want to see anything. You people dragged me into this against my will, not the other way around, and who am I going to tell anyway? The last thing I want is to be dragged off and committed because I saw... something that no rational human being has ever seen before.\"\n\nZebulon gave her a cocky grin that conveyed both amusement and irritation. \"I don't think you understand what's going on here, do you?\"\n\n\"Not a clue and, no offense, I like it that way. Clueless rules.\"\n\nStill the beast wouldn't let her pass. Zebulon inclined his head toward Arik. \"The Skotos has risked his life to come here to be with you, Geary. Those two who attacked him. They're assassins and I'm sure they're going to be back. Probably with reinforcements. And now that you've seen them, they'll come for you, too, which is the only reason why I'm still talking to you. I feel morally obligated to at least warn you that they're gunning for you. Now in theory I can kill them and save you, but then that just opens up a whole can of worms and gets so messy that I really can't. I'm better off letting you die than taking them out. See my dilemma?\"\n\nShe gave a bitter laugh. \"Not really. The only dilemma I see is my imminent death that you appear ambivalent to. Hello? Did you hear any of what you just said to me?\" How could this be happening?\n\n\"I heard, but when you get to my age, you understand that some things are just best left alone. Death is only natural.\"\n\n\"Oh yeah,\" she said, sweeping his body with a derogatory glare, \"you're an old man. You're all of what? Twenty-five?\"\n\nHe was definitely amused as he responded. \"More like twenty-five thousand years old. Give or take a few hundred years. At my age we really don't count anymore.\"\n\nGeary swallowed at that deep, dry tone. \"You are joking, aren't you?\"\n\nHe shook his head.\n\nShe looked at Arik, who duplicated the gesture. Nervous and suddenly uncertain, she looked back at Zebulon. \"You're twenty-five thousand years old?\"\n\n\"Well, if you're looking for precision, twenty-seven thousand, five hundred and forty-two, but really, does it matter?\"\n\nGeary felt her jaw drop. There was no way he could be that old. \"That would put you at having been born during the Aurignacian Period.\"\n\n\"Not quite\u2014that predates me by a few hundred years. But I'm close to it.\"\n\nShe could barely comprehend what he was saying as she ran through her ancient, ancient history. \"And that would make you\u2014\"\n\n\"A Cro-Mag,\" he said with a smirk, \"so yeah, when you call me a barbaric caveman, I am. Literally. Hell, I even knew a couple of Neanderthals who once kicked my ass all over what is now Toledo, Spain. But here's the fun part. Your boyfriend over there is even older than I am and he's considered a baby by his family.\"\n\nAnd given the ludicrousness of those statements, the most screwed-up thought of all went through her head. \"You were both around during the time of Atlantis.\"\n\nThat was how Arik had known about her necklace. How he'd known about the site.\n\nOh God, it was true.\n\nThey were...\n\nShe couldn't even complete the thought. She couldn't.\n\nTrieg moved forward to touch her sympathetically on the shoulder. \"It's a bit of a stunner when you first hear about it. You should have seen my face the night I met Artemis. A bit of advice to you, love. Go with it. And on that note I need to be patrolling. Good night to you all.\"\n\nYeah, sure, let the man with flashing fangs go back to his life. Why not? She had nothing better to do than be stalked by the deadly duo who wanted her dead.\n\nAnd Mr. Freakzoid Neanderthal Cro-Mag man.\n\nSpeaking of the devil, Zebulon was watching her with an amused smirk that she dearly wanted to wipe off his face.\n\nArik was the only one who seemed to appreciate the seriousness of all this.\n\nZebulon turned his attention to Arik. \"So, bud, how long do I have to watch for the Dolophoni?\"\n\nArik let out a tired breath before he answered. \"I'll be gone from this world in two weeks... if they don't kill me first.\"\n\nZebulon nodded. \"You honestly think they're going to let you go home?\"\n\nHer anger was mirrored in Arik's eyes. \"Not really. I figure I'm basically dead one way or another.\"\n\n\"Good,\" Zebulon said drily. \"You're not as stupid as I thought you were. My only advice is for you to keep them off my streets and out of the public's eye. I don't like cleaning up these kinds of messes.\"\n\nArik looked even less amused than she felt. \"I'm not exactly the Ty-D-Bol Man myself.\"\n\n\"Then we have an understanding. Keep the riffraff off my turf or I mop the floor with all of you.\"\n\n\"I'll do my best.\"\n\nZebulon inclined his head before he literally melted into nothing.\n\nGeary was torn between outrage, hurt, and fear. Part of her wanted to slap Arik for dragging her into this, while another wanted to run as far away as she could. What won out was her sarcasm. \"Thanks so much for the date. Had a blast. Really, we must do this again sometime. I really like these near-death experiences we have whenever we're together. They're very invigorating.\"\n\nHe reached to touch her again. \"Geary\u2014\"\n\n\"Don't touch me,\" she snapped as she pulled away from him. \"Don't you dare.\"\n\nArik withdrew his hand reluctantly. He understood her anger and she was fully entitled to it. Funny how he hadn't considered how all this would affect her before he'd come here. Honestly, he hadn't cared.\n\nBut now it was different. Now he cared in ways he hadn't been able to imagine before.\n\nAnd he'd only been with her for a short time. What would it be like after they'd spent more time together?\n\nWhat had he been thinking when he made his bargain with Hades? How could he have offered her up so easily?\n\nIt was such a selfish thing to do, and now that he could feel, he understood exactly how selfish it was. And he regretted it with every part of himself. She deserved so much better than what he'd done to her.\n\nShe deserved so much better than him. What he'd done was wrong. He knew that now, but he couldn't change it.\n\nGeary shook her head. \"I just don't understand this. You lied to me about who you were. Why?\"\n\nArik swallowed as he heard the pain in her voice. It was so intense that he felt it himself. \"Why? What would you have said had I come to you and told you that I was a god from your sleep who wanted to meet you in the flesh? Would you have welcomed me in or would you have called the authorities on me?\"\n\n\"It is ludicrous,\" she admitted.\n\n\"Yes,\" he said, trying to make her understand why he needed to be near her. \"You can't imagine the world I was born into, Geary. There's no laughter there, no joy or happiness, and then one night I accidentally found you. You who laugh at the warmth of the sun when it touches your skin. You who have... what was it you called it once? A chocogasm from eating a Hershey's Kiss\u2014whatever that is. You feel things on a level most people never imagine. In all the centuries I've lived, I've never known anyone like you. And for two weeks I just wanted to be with you. To feel you, human to human, and to understand this world that is so vivid through your eyes.\"\n\nGeary didn't know what to think. No one had ever spoken so passionately to her, never mind been so passionate about her. What should she say to that?\n\n\"I just wanted to know what it was like to be human, Megeara. Just for a little while. To touch you as a man and to hear the real sound of your voice as you said my name, and not the voice that was distorted by your dreams.\" He reached for her hesitantly and took her hand into his. \"You can't imagine how good this feels when you've never known a gentle touch on your flesh.\"\n\nSomething inside her melted at the sincerity of his tone. The sincerity in those pale blue eyes. He meant every word he spoke. \"So you're not dying?\"\n\nHe shook his head. \"Not in the sense that you use the word, no. But I will have to go back to my world and most likely die there. Apparently coming here pissed off some serious people who have no intention of letting me live after this.\"\n\n\"Then why did you come here if you knew they were going to kill you for it?\"\n\n\"Honestly, I didn't know that at the time, but even if I had, I doubt it would have changed my mind. I would still have come for you.\"\n\nHow could he say that and mean it? How could seeing her be worth his life? \"You're insane, aren't you?\"\n\n\"Only when it comes to you.\"\n\nGeary closed her eyes as she let everything that had happened over the last few minutes sink in. It was awful. She felt as if something had turned her inside out. She no longer knew what to believe in. She no longer knew what was real and what wasn't.\n\nInstinctively she reached for the necklace\u2014she needed to feel something solid to help ground her. But the instant she touched it, her heart stopped as their earlier discussion went through her mind. \"You know where Atlantis is.\" It was a statement.\n\nHe nodded.\n\nDisbelief washed through her as every part of his presence in her world hit her. \"Then my father was right. It did exist. Right here. Right where he said it was.\"\n\nAgain Arik nodded to confirm her words. \"You were swimming over its harbor this afternoon when you found the box. You were right there, Geary. You really touched it.\"\n\nTears actually came to her eyes at the thought of her completing her task. Of her holding in her hand one of the keys to her promise. \"Was I really?\"\n\n\"Yes. You were right, Megeara. And so was your father.\"\n\nShe covered her mouth with her hand as she took a step back. It was one thing to suspect but an entirely different one to know.\n\n\"Then we're there,\" she said with a giddy laugh. \"We've found it.\"\n\nBut Arik didn't mirror her joy. He was tense and serious as he eyed her with warning. \"That's the good news for you. The question is, do you want to know the bad?\"\n\nNot really. She'd rather savor the good stuff. At least for a second or two. But there was no use in delaying the inevitable. As the old nursery fable went, one could never outrun trouble. There was no place far enough to avoid it. \"Oh sure, what could be worse than what happened here tonight?\"\n\nHe shrugged. \"I don't know. Does the fact that the goddess Artemis blew up your boat today top that or not?\"\n\nGeary blinked as those words sank in. Honestly, in all her wild imaginings, that one had somehow eluded her. She much preferred the thought of Arik as an arms dealer or assassin.\n\n\"Excuse me?\" she asked, hoping he might have been playing with her.\n\n\"You heard me correctly. Artemis is one of many who want you to stay away from Atlantis.\"\n\n\"And what did I do to earn this privilege?\"\n\n\"Basically the same thing you did tonight,\" a deep male voice said from behind her. \"You were meddling in a place you didn't belong.\"\n\nGeary turned at the foreign voice, then stiffened at the sight of the rest of the people who'd attacked Arik in her dream.\n\nOh. Shit.\nCHAPTER 14\n\nDread fear whipped through Geary as she saw the doom in the eyes of the ones confronting them. \"I'm really getting tired of these guys.\"\n\nArik made a rude noise in the back of his throat. \"Believe me, I share your sentiment entirely.\"\n\nThat didn't really comfort her. \"So what do you suggest we do?\"\n\nArik shrugged with a nonchalance she couldn't even begin to fathom. \"There are nine of them and two of us. They have the powers of a god and we're human.\" He gave her a very Sean Connery\/James Bond mysterious kind of smile. \"Therefore, I suggest we run. Fast.\"\n\nBefore she could even think to respond, Arik shoved her in the opposite direction of the mean army of darkness that was out to shorten their lives. Geary's heart was pounding as he took her hand and led her through the alley and down the cobbled street at a pace that would make a sprinter proud.\n\nShe had a moment when she thought it might work, but that hope died quickly as one of the women appeared out of nowhere in front of them to block their way.\n\nThe goddess tsked at Arik. \"What's the matter, Skotos? We bathed and everything. Surely you don't want to leave us without at least saying hi.\"\n\n\"Hi.\" Then without hesitating, Arik let go of Geary and kicked the goddess away.\n\nSpinning about, the goddess countered with a staggering blow to his solar plexus. Arik grimaced, then backhanded her so hard, she stumbled back.\n\nGeary grabbed at something on the woman's waist that looked like a billy club attached to her belt. She jerked it free before she struck the goddess with it across the back. A blinding light sparked on said contact and was followed by a vibrant burst of power so strong, it knocked them both apart.\n\n\"You okay?\" Arik asked.\n\nAll Geary could do was nod.\n\nHe kissed her on the cheek before he took the club and turned to face the others.\n\nRattled by the surge that continued to burn through her body, Geary stumbled away while he moved to engage the twins again.\n\n\"Megeara...\"\n\nThat deeply hypnotic female voice whispered through her mind. It was the same one that had been calling to her for weeks now. But she didn't have time for it. Geary shook her head to clear it.\n\n\"Listen to me, Megeara. Use the medallion you wear. Place it under your tongue and let me into you.\"\n\n\"What?\" she breathed.\n\n\"Just do it, child, and I will take care of them for you. Trust me. I can protect you.\"\n\nYeah, she was gone, seriously gone, and yet even as the thought went through her, what harm would there be in trying it? She and Arik were already getting their butts kicked. What harm could it do?\n\nAfter all, they were fighting a band of gods out to kill them and, given all she'd seen in the last few minutes, what were a few more leaps of faith?\n\n\"I can't believe I'm doing this.\" She pulled the small coin up and placed it in her mouth, under her tongue. She grimaced at the salty metallic taste. But that only lasted a nanosecond as something warm invaded her mouth. Whatever it was, it didn't stop there. It spread through her like lava, heating her body and making her heart race.\n\nAnd as it went through her, images filled her mind. Images of an ancient world. Of a hall filled with gold. She saw the face of a beautiful blond woman who had mercurial eyes that swirled like a silver mist.\n\nWords whispered through Geary in a language she'd never heard before.\n\nThen something snapped. Geary felt as if she'd been pushed outside of her body so that she was nothing more than a ghost, looking down on the others\u2014yet she was still in her body. Only someone else was in complete control of her. It was the strangest sensation. To be cognizant and not responsive. No matter what she tried, her body ignored her instructions.\n\nOne of the gods came up to attack her. She laughed at the warm power cascading through her before she headed toward the man to confront him. He swung at her. She ducked like a seasoned pro and elbowed him in the knee. Hissing painfully, he dipped as his knee gave way.\n\nShe rose quickly, then swung her arm so that she caught him under his chin with her fist. That blow sent him twisting, straight to the ground.\n\nDeimos approached her next. She wasn't sure how she knew that it was him, but she did.\n\nHe backed up. \"Aekyra Apollymi?\"\n\nAre you Apollymi?\n\nEven though his question had been asked in Atlantean, she understood it and, better still, answered it even against her will. \"Naiea.\"\n\nHe took another step away from her. As he did so, her hands heated up even more. She threw them out and a blast emanated from her fingertips that flooded the alley with light.\n\nTwo seconds later, the gods went flying as if she'd struck them all with lightning.\n\nArik raised his hand to shield his face as he felt the heat of an Atlantean god bolt. But what stunned him more than the fact that he felt the impact of something that hadn't existed for over eleven thousand years was that it had come from Megeara.\n\n\"Apollymi!\" Deimos snarled in Atlantean. \"This is not your fight. Back down.\"\n\n\"Naiea, Olygaia eta.\"\u2014Yes, it is, Olympian. The voice was Megeara's and at the same time she spoke the Atlantean words like a native. \"Anekico ler aracnia.\"\n\nVictory to the spider. It was an old Atlantean saying that meant \"patience wins the day.\"\n\n\"Ki mi ypomonitikosi teloson semerie.\"\n\nAnd today my patience ends. Even though her teeth were clenched, her voice was loud, clear, and angry as she snarled the words in a tone he could tell reached all the way to the halls of Olympus.\n\nMore important, it was enough to convince the Dolophoni that they didn't want even a tiny piece of the Destroyer when she was in this mood. Deimos looked to the others before he called for a retreat.\n\nThey vanished instantly.\n\nArik wiped the blood from his lips as he made his way cautiously to Megeara's side. Even though he lacked his powers, he could feel Apollymi's essence as it filled Megeara's body. Her eyes were swirling in color and dotted with silver. Rage and vengeance bled from every part of her.\n\nShe started after the others, but he pulled her to a stop before she could pursue them and possibly hurt Megeara in the process.\n\n\"Ochia, Apollymi. Anekico ler aracnia epitrepedio. Efto ler kariti u topyra.\"\n\nNo, Apollymi. Let the victory go to the spider. This is not the time or place.\n\nShe hissed at him and would have attacked had he not pulled the necklace from Megeara's lips. The abruptness of it caused her to collapse against him. He lifted her in his arms and held her there while the goddess was forced to retreat out of Megeara's body. He held her close as she trembled.\n\nGeary could barely breathe as a foreign weakness invaded every part of her. She'd been so strong before; now she was as weak as a newborn. Leaning her head against Arik's neck, she was grateful for his support, because at the moment she couldn't even lift her own arms.\n\n\"What was that?\" she asked weakly.\n\n\"The Atlantean goddess Apollymi. Even though she's trapped in Kalosis, she can reach out and at times possess people and the elements. Her powers are a pittance on this plane to what they would be if she were free, but they're still impressive.\"\n\n\"Why did she do that? Why possess me?\"\n\n\"Because she needs you to free her and if they kill you, she has no hope.\"\n\n\"I don't understand.\"\n\n\"You must have pure Atlantean blood in your ancestry. I think that's why you can hear her when most can't. It's the only thing that makes sense.... Before it was destroyed, there were two races who inhabited Atlantis. The natives who were born of the Atlantean pantheon and the Apollites who took refuge there after they were thrown out of Greece. Apollymi needs the blood of an Atlantean to break her seal and to summon her out. Because of that, she would protect you for all she's worth.\"\n\nGeary had to struggle to raise her hand to her necklace so that she could see the writing on it. \"I thought this was only a coin.\"\n\n\"No. That is the medallion that was worn by her priestesses. Whenever they were in danger, they would do as you did, place it in their mouths, and she would protect them.\"\n\nWow, hell of an insurance policy. There weren't many people who could have a goddess at their beck and call. Made Geary wonder what Apollymi would do for a false alarm.\n\nThen again, given how easily Apollymi had fought the Dolophoni off, Geary didn't want to even think about it. With that kind of power, Apollymi could easily turn on the person who wrongfully summoned her.\n\n\"Why didn't our attackers just pull it out of my mouth once I started fighting them?\"\n\n\"They're from the Greek pantheon. I doubt they knew that trick or I'm sure they would have.\" He slid her slowly down his body until she was on her feet.\n\nIt took Geary a full second before she was able to stand again. And even then, her legs were so unsteady that she held on to Arik's arm for support. The warm smell and strength of him steadied her even more, and she was grateful for his presence. \"How is it you know the trick?\"\n\nHe gave her a devilish grin. \"The benefits of being an ex-Oneroi. Since we can trip through anyone's dreams, we know a lot of tricks the other gods don't.\"\n\n\"But not how to fight those guys in human form,\" she reminded him.\n\nHe looked a bit sheepish, which she found intriguing and endearing. \"Well, not and win. But in my own body and in my realm... I'm lethal.\"\n\nGeary could feel the muscles of his biceps working under her hand as she stared up into those clear eyes. In her dreams, she'd kissed that mouth a thousand times. Had run her tongue over the stubble of his chin and had licked every inch of him. It amazed her that he'd been as captivated by her as she was by him. But more than that, she found it almost impossible to believe.\n\n\"Did you really come here for me?\"\n\nHe nodded, his eyes scorching her with their heat. \"Yes.\"\n\n\"And are you disappointed?\"\n\nOne corner of his mouth lifted into a seductive smile. \"Only that you don't rip off my clothes and have your way with me... in chocolate.\"\n\nGeary shook her head at him as she went over everything she'd discovered in the last twenty-four hours. She should be horrified, and on one level she was. But on another she was actually relieved that he was a god on the human plane. At least now she understood part of what had been happening to her.\n\nThough it didn't make it better, the explanation went a long way in saving her sanity.\n\nShe took a step back from him as she tried to comprehend everything. \"I don't really get all of this. How did you first find me in my dreams?\"\n\nHe took her hand in his and held it while he explained. \"We have rooms that are like chambers where I live. We don't have to use them, but they make connecting with humans a bit easier\u2014they can amplify our powers, and it gives us someplace to rest undisturbed while we're doing it. The only drawback is that it allows the powers-that-be an easier way to monitor us while we're at it. Whenever we're in the strobilos, we hover about and drift in and out of dreams. Whenever we find someone who is having a vivid dream we're drawn to them.\"\n\n\"And you were drawn to mine.\"\n\nHe nodded.\n\nIncredible. She couldn't imagine being able to do that. To spy on people and participate in their sleep. \"So what's it like to be in someone else's dreams?\"\n\n\"It's like bathing in Jell-O. It feels kind of thick and at times can be overwhelming. You never know what you're going to find. Many of the Skoti I know prefer nightmares, since they get such an adrenaline rush from it.\"\n\nThat didn't sound like what she'd read about the Oneroi. \"From my research, it says that you guys channel and direct the dreams, that you cause them.\"\n\n\"At one time we did. The Oneroi were extremely active in granting dreams and using them to manipulate both the gods and humanity. Then one day one of my brethren made the mistake of making Zeus crave a goat... sexually. He did it as a joke, thinking it would be funny and that the head god needed to be brought down a peg or two after insulting him. Zeus, once he came to his senses, was so outraged that he had our insistent friends, the Dolophoni, round us up and bring us to him. A small group, including the one responsible, was killed. Another group was heavily punished, and the rest of us were cursed to have no emotions of any kind.\"\n\n\"Why?\"\n\n\"Without ambition, envy, humor, and the rest of the emotional gamut, Zeus thought that it would keep us from messing with him or any of the other gods ever again.\"\n\nGeary could understand his reasoning, but it seemed a bit cruel to punish everyone over the actions of one stupid being. \"And it worked?\"\n\n\"Not exactly. Without us there to direct dreams and inspire people, certain humans and other creatures began to lose their minds. Zeus learned that we were needed to help channel pent-up feelings and to help humans and other beings determine what they desired in a contained environment. Dreams provide a necessary outlet for everyone. So the Oneroi were charged with helping others in their dreams. It worked for a time, until we realized that in a dream state we had emotions again. Fear, love, passion... they were all there, and whenever we found a special class of person they were extremely amplified. But once we leave the dreamworld, the borrowed emotions evaporate and leave us vacant and cold again.\"\n\nShe could see where this was headed. \"So some of you became addicted to emotions. Like a drug.\"\n\nHe nodded. \"Those who crave it are called Skoti.\"\n\nGeary remembered Zebulon calling Arik Skotos. \"You say that like it's a bad thing.\"\n\n\"In my world it is. Skoti are deemed uncontrollable, and if they fail to heed the warnings of the Oneroi, they're hunted down and severely punished or killed.\"\n\nShe frowned. \"Why?\"\n\n\"The gods fear us and therefore they want to control us any way they can.\"\n\n\"But if you have no feelings, how can punishment matter?\"\n\n\"Because that is the one emotion left to us.\"\n\n\"No,\" she corrected him. \"Physical pain isn't an emotion. It's a biological response to negative stimulus. No wonder you still have it.\"\n\n\"You know, the rational explanation really doesn't help. Either way, it sucks to be one of us.\"\n\n\"Sorry.\" Geary reached up to brush a lock of hair from his forehead. It was inconceivable that he was here with her and real. That he was flesh and blood and for all intents and purposes human. How strange.\n\nShe didn't know if she should be angry or flattered or both that he'd come here just to meet her.\n\nShe tightened her grip on his hand. \"And what about Solin? Is he really related to you?\"\n\n\"Yes. He's my cousin, but we dream gods consider ourselves brothers and sisters whether or not we share parents. Solin's father was Phobetor and his mother was human. He didn't know he was a demigod until he hit puberty and his powers manifested. Then his mother cast him out and the Oneroi began to hunt for him. He's hated all of us ever since.\"\n\nNow she was beginning to understand. \"Which is why he said initially that he didn't have a brother.\"\n\n\"Exactly.\"\n\nGeary fell silent as she digested that last bit. God, this was all so complicated. Unbelievable, really. How did a woman who only wanted to redeem her family name find herself in a situation like this? \"So where does this leave us?\"\n\n\"Confused?\"\n\nShe laughed. \"You have no idea.\"\n\n\"True, I didn't understand confusion until I found myself floating in deep water.\"\n\nShe let out a small laugh at his reminder of how they'd met. \"Since you're being so honest now, how did you happen to get there?\"\n\n\"Hades. After he made me human, he threw me out and put me in your path. I guess I should be grateful he didn't dump me on a busy street somewhere, under a truck.\"\n\nGeary shook her head at his humor. \"Hades.\" Bitter amusement filled her. \"And to think I used to mock my father for his belief that the ancient gods were real. I thought he was insane and I told him that on many an occasion. But he was always insistent that they had to have lived.\" She sighed as she remembered the way he'd described them and the quirks he'd invented for the pantheon. \"So what is old Hades like anyway?\"\n\n\"He's a cranky bastard who hates everyone not Persephone.\"\n\nWell, that made sense. She was his wife after all. \"So what's she like?\"\n\nArik's features softened. \"She's kind and dainty. Petite and shy. Very unassuming. She actually reminds me a lot of your cousin Tory.\"\n\n\"Really?\"\n\n\"Yes, and Thia is a dead ringer for Artemis, right down to the red hair, height, selfishness, and bitchy attitude.\"\n\nFor some reason, that didn't really surprise Geary. \"No kidding, huh? What about me? Am I like any of the gods?\"\n\nHe narrowed his eyes as if considering it before he answered. \"Athena, except she has black hair and is usually sporting her pet owl on her shoulder. But your mannerisms are very similar, and like you, she lives a life of celibacy.\"\n\n\"Oh gee, thank you for that one.\"\n\nHe lifted her hand up to kiss the back of her knuckles. In spite of her anger, the gesture warmed her. \"It's true, but that's okay. I like that about you.\"\n\n\"I'm sure you don't.\"\n\nHe cupped her face in his hands and stared at her. \"Megeara, there is absolutely nothing about you that I don't like.\"\n\n\"Can you honestly say that?\"\n\nHe stroked her cheeks with his thumbs. \"Okay, I don't like it when you walk away from me, but other than that...\"\n\nShe laughed. \"Yeah, I guess that makes sense. You transcend a dimension to come here and I brush you off. I can see where that might get annoying.\"\n\nArik smiled at her playful tone. All in all, she was taking this a lot better than he'd have thought. \"You still want to find Atlantis?\"\n\nHer face sobered. \"Not if it means anyone's life. I won't make that bargain. Believe me, there's nothing there worth my life and definitely nothing there worth someone else's.\"\n\nA twinge of guilt went through him as he realized that he'd made that bargain without hesitation.\n\n\"Is something wrong?\" she asked with a scowl.\n\n\"No,\" he lied. \"I was only amazed by the woman in front of me. By your compassion and caring.\"\n\nShe gave him a doubting stare, but for once she didn't argue. Instead she changed the subject. \"Tell me something, Arik. Was Atlantis beautiful?\"\n\n\"Like a dream.\"\n\nShe closed her eyes as if she was trying to imagine it.\n\nHe leaned forward to whisper in her ear. When he spoke, his lips brushed the soft skin of her cheek, which only whetted his appetite for her. \"Tonight while you sleep, I'll take you there and let you see it for yourself.\"\n\nJoy glowed in her eyes. \"Really?\"\n\n\"Cross my heart.\"\n\nGeary felt tears prick at the backs of her eyes at the thought of seeing it. But more than that was the fear of what else might be waiting for them in her dreams. \"Will those assassins be in our dreams, too?\"\n\nHe looked away before he answered. \"Probably. But don't worry about them. I can handle it in that realm. And if I can't you can always change the channel again. I have to say that was the neatest trick I've ever seen.\" He winked at her.\n\nShe blushed at the memory. Shaking her head, she reached up to lay her hand against the whiskers of his cheek. It was so odd to be with him, knowing all the things they'd shared. \"So how many women have you visited in their dreams?\"\n\nHe hesitated. \"Is this one of those questions that if I don't answer it correctly, you get angry at me?\"\n\nShe laughed. \"Probably.\"\n\nScowling, he hesitated before he answered. \"If it makes you feel any better, you're the only one I've ever wanted to be human for.\"\n\nIronically, that was just what she needed to hear. \"You are a fool, Arikos.\"\n\n\"Only where you're concerned.\"\n\nShe still couldn't believe that he'd come here for her. Who would have made such a bargain? \"So what do we do now?\"\n\n\"Well, if I can't have you naked in my arms, then I vote we stay alive.\"\n\nThat sounded good to her... both parts, actually. \"All right. But I want to make sure we don't endanger anyone else. Do you think Solin will help us?\"\n\n\"That's a hard question to answer. Solin is a bit self-absorbed and highly unpredictable. Although I have to say that it surprised me he rescued the others earlier, so there might be a chance he'd help us, too.\"\n\n\"Okay, then. Let's try him and see what he thinks.\"\n\nBut an hour later, after they'd gone to Solin's villa and explained it all to him, Geary learned that Arik had been right. The egotistical bastard had no intention of helping them.\n\nDressed in a pair of slacks with an open light blue shirt that showed off his tanned six-pack, he scoffed at them before he took a drink of the brandy in his hand. \"They'll come for me if I help you, and no offense, no one has ever bled for me, so I'm not about to bleed for anyone else. You can all go to hell for what I care.\" Then he slammed his empty glass down and glared at Arik. \"I've done enough helping. You've already brought Zebulon down on my ass and now the Dolophoni. It took me centuries to reach an impasse with them and I like our cold war a little too much to jeopardize it for you.\"\n\n\"I understand.\"\n\n\"Good. Now if you'll excuse me, I have some things to take care of.\" He left them alone in his study.\n\nGeary let out a sigh as she turned to face Arik, who sat beside her on the leather sofa. \"He's a fuzzy-little-bunny guy, isn't he?\"\n\nTo her surprise, Arik defended him. \"Don't judge him too harshly. You have to remember that for centuries he was persecuted by humanity and hunted by us. His resentment and anger are more than understandable.\"\n\n\"Is that compassion you feel?\"\n\nArik paused as he considered the tenderness inside him. \"Yeah, I think it is.\"\n\n\"How does it feel?\"\n\n\"Strange and comforting, but mostly disturbing.\" And it was. He wasn't sure if he liked having emotions or not. There were extreme advantages and disadvantages to both.\n\nMegeara reached over to squeeze his hand. \"So what's it like to live without them?\"\n\nHe toyed with her fragile fingers, delighting in the sensation of her hand in his. \"It's hard. Imagine a world without taste. A world where you can see the colors and all, but you can't feel it. A beautiful clear day can never choke you up. A child's laughter doesn't make you smile. You don't look at a bunny and think, How cute. You feel absolutely nothing. It's like being wrapped in thick cotton all the time.\"\n\n\"And when someone touches you?\"\n\n\"I can feel the pressure, but not the sensations. There's no blood rushing through my veins, making my heart beat faster. No excitement or chills. But the weird part is that when I'm with you, I don't even have to touch you to feel that. I get hard for you just thinking about you.\"\n\nGeary swallowed as she felt her own chills rise. No one had ever said a kinder thing to her. And as she contemplated that, she realized something else.... \"You've never slept with anyone in the flesh, have you?\"\n\n\"No.\"\n\nAmazed, she remembered the expertise he possessed. In bed, he was highly creative.\n\nHis blue eyes were haunting in their need and sincerity as he brushed his lips against her cheek, then whispered in her ear, \"There's nothing I want more than to touch you, Megeara. I want to taste your skin on my tongue. To know what it's like to slide into you while you hold me close.\"\n\nHer breasts tightened as warm heat seared her. She should be offended by his frankness. She wasn't. It oddly turned her on. No one had ever been so blatant and open with her before.\n\nAnd he had crossed worlds to be with her.\n\nHis breath scorched her neck as he nuzzled her cheek with his.\n\nHe would be gone in two weeks. There was no hope of anything with him more than a brief physical relationship. That was the last thing she wanted. \"Arik? What are the odds of your surviving this?\"\n\nHis expression turned dark as he pulled back to look at her. \"Hard to say. As a human, pretty much nil, but once Hades returns my godhead, my chances go up exponentially.\"\n\n\"So you just came here to screw me and leave?\"\n\nArik paused. Yeah, that had basically been his plan, which Hades had changed by demanding her life for it.\n\nI am a rank bastard.\n\nAnd in that moment he had a bad epiphany. \"You're right\u2014that was my intent. I'm no better than Solin. I was so fixated on the novelty of being human with you that I never thought past my blind obsession. You gave yourself so freely in your dreams that I made the assumption you lived your life that way, too. But you don't. I think that's why you're so uninhibited in dreams. You keep everything bottled up inside you.\"\n\n\"Yeah,\" she said in a low tone. \"I'm so inhibited here that it's the only place I feel free to roam without someone trying to judge me.\"\n\nHe nodded and for the first time ever, he felt guilt. Real, true, and bitter. More than that, he cared for her. He didn't want her hurt in any way, and he didn't know how to stop the course he'd already set into motion.\n\nWhen he'd seen her there in the middle of the fighting, it had made his heart ache. He'd actually been afraid for her.\n\nOh, this was getting way too complicated.\n\n\"I've made such a mistake by coming here, Megeara. I'm so sorry. I should have been content to stay in your dreams.\" If only he'd been able to. Had he stayed there, they would have driven him out of her life completely.\n\nNow he just wanted to stay here with her forever.\n\nIf only he could....\n\nGeary pulled him into her arms and held him close. She didn't know what to think about any of this, but then, maybe she shouldn't be thinking at all. She'd cherished those dreams with him. He was both naive and experienced. Trusting and suspicious. She'd never known anyone like him.\n\nNo kidding. He was a god turned human who lived an existence she couldn't even begin to understand.\n\nBut she wanted to.\n\nToday he'd given her her dream by taking her to Atlantis and letting her hold a piece of it in her hands. And if they were meant to die, then she wanted to make sure he had the one thing he'd wanted most.\n\nRising up from the sofa, she took his hand and pulled him to his feet.\n\nHe frowned at her. \"Where are we going now?\"\n\n\"Somewhere that we can be alone so that I can strip you naked and ride you into the ground.\"\nCHAPTER 15\n\nArik's breath caught in his throat as she uttered the words he'd been dying to hear since the moment he'd made his deal with Hades. A slow smile spread across Arik's face before he slung his hand out to lock the door. Nothing happened.\n\nHe cursed as he remembered his human limitations. If he wanted the door locked, he'd have to do it manually, which seriously sucked.\n\nTime's wasting. Arik raced to the door and locked it tight.\n\nMegeara was frowning at him as he returned to her side. \"What are you doing?\"\n\n\"I'm not going to take a chance on your changing your mind. By the time we find someplace else, something could happen to ruin your mood, and I don't have any chocolate here to entice you with.\"\n\nShe laughed. \"So we're just going to make out in here like horny teenagers?\"\n\n\"Works for me.\"\n\nShe looked around the room rather sheepishly. Afraid she'd chicken out, he pulled her against him and slow-danced with her. \"Come on, Megeara. Walk on the wild side with me. Let's get naked and ruin Solin's upholstery. It serves two purposes. We're happy and he's pissed.\"\n\nGeary bit her lip in indecision. Arik was adorable when he was like this. Playful and charming. How could she resist him?\n\nShe cast her gaze around the room again. \"There's no comfortable place in here.\"\n\n\"I'll be on bottom. I promise, I'll make you a good cushion.\"\n\nHe was incorrigible. \"There's nothing I can say to dissuade you from this, is there?\"\n\n\"Nope.\" He took her hand in his and pressed it against his fly so that she could feel his swollen cock in her palm. \"I'm too desperate for you. You have to take pity on me.\"\n\nGeary's heart pounded as she lifted her chin to receive his kiss. Oh, he tasted like divinity. She wrapped one arm around his neck as their tongues danced and she stroked him through his pants.\n\nArik drew a deep ragged breath as Megeara nibbled her way around his jaw, down his neck while her hand massaged him with an irritatingly slow rhythm that only added to his craving for her. The stroke of her tongue on his flesh was like being hit with electricity over and over again. Every part of him shook from the sensation. His nerve endings were alive and throbbing.\n\nGrowling, he jerked his shirt up over his head so that there would be no more barriers between her mouth and hands and his skin. He only wanted to feel her touch.\n\nShe gasped at the sight of his torso and splayed her palms against his bared chest.\n\nGeary couldn't believe the scars Arik carried. They were everywhere. Mostly faint and faded by time, she'd failed to see them when he was fished out of the water. But up close, they were extremely prominent. \"What have they done to you?\"\n\nHe ran his hand gently through her hair. \"They tried to control me.\"\n\nIt was obvious from the amount of damage that they'd failed repeatedly. Arik was truly a stubborn man.\n\nHe dipped his head to kiss her, but she pulled away. \"What will they do to you over this latest action?\"\n\n\"Don't know and right now I don't care.\"\n\n\"I care, Arik.\"\n\nHe scowled at her as if he couldn't understand her words. \"Why?\"\n\n\"Because no one should be tortured. It's wrong.\"\n\n\"It's the way of things. You break the rules and there are consequences. I'm willing to pay that price.\"\n\nShe trembled as he returned to nuzzle her neck. How could he be so nonchalant over this? But then, judging by the look of his body, it was such a common occurrence that he most likely didn't even blink over it.\n\nAs he said, to him it was normal. But to her it mattered. She didn't want him to be hurt, especially not for being with her. He was a decent man who deserved so much more than this. She couldn't stand the thought of his being punished any more\u2014it ached like a physical pain.\n\nArik was fascinated by the true softness of her skin as he toyed with her neck. By the way her body smelled so fresh and sweet. She skimmed her hands over his back while he unbuttoned her shirt. He'd expected her to be bare beneath it. Instead she wore...\n\nBondage gear. Pulling back, he scowled at the white thing that wrapped around her chest like an oversized bandage. \"What is that?\"\n\nGeary laughed as she realized that in dreams she never wore a bra. And judging by his reaction, she deduced that no one else ever had, either. \"It's a bra. Haven't you ever seen one before?\"\n\nHe ran one finger under the strap and curled his lip as if it were the most repugnant thing he'd ever seen. \"No. What's it for?\"\n\n\"It keeps me in place.\"\n\n\"Yeah, but I don't want you in place. I want you in my hands.\"\n\nOn any other man, that line might have angered her to the point of nailing him in a certain part of his anatomy and walking away. But it was said with such growling sincerity that it only made her laugh again. She reached around and unsnapped her strap. The look on Arik's face as the bra fell to the floor was priceless. No man had ever looked at her with such satisfied hunger.\n\nHe lifted one hand, then hesitated as if afraid to touch her breasts. She took his hand and led it to her. The instant he touched her, she moaned and ached at the center of her body. And all she could think of was having him hard and deep inside her.\n\nArik's breath caught in his throat as he cupped her swollen breast in his palm. It was so soft.... He'd never felt anything like it.\n\nHe brushed his thumb over the taut nipple and was delighted as she literally squirmed. Grinning, he dipped his head down to draw that peak into his mouth so that he could taste her. She let out a moan as she cupped his head to her breast.\n\nOh yeah. This was what he wanted. He reveled at the taste of her, at the way her puckered nipple rolled against his tongue. But all it did was make him burn for more of her. He wanted to see her completely naked.\n\nWith that thought in mind, he quickly unzipped her pants. She kicked her shoes off as he slid the pants down her body and then tossed them over his shoulder.\n\n\"More surprises, huh?\" he asked as he studied the small pair of panties.\n\nGeary couldn't speak as he reached up to brush his fingers between her legs. She felt on fire from his touch. And all she could think of was her dreams when he'd licked her until she'd screamed out in pleasure.\n\nHe massaged her with his thumb through this silk until she was wet and begging for him. She buried her hand in his hair, as he continued his relentless torture.\n\nArik pushed the silk material aside so that he could touch her intimately. She was so wet already and he was dying for a taste of her on his tongue. He hooked his finger in the bottom of her panties and pulled them off.\n\nHe was still in his slacks while she stood naked before him. Gods, but she was the most beautiful woman he'd ever seen. She wasn't skinny or lean but rather voluptuous and tight. He could see the outline of her bikini where her skin went from tan to a paler tawny color.\n\nBut that wasn't what held him captive....\n\nGeary licked her lips as their gazes met and locked. Arik took her hands into his and led them to the center of her body that ached for his touch. Using her thumbs, he spread the folds of her wide so that he could see the most private part of her body.\n\nHe ran one long finger down her cleft, drawing a shiver along her spine. \"You're so much softer than I thought,\" he whispered. \"Wetter.\"\n\nAnd when he slid his finger deep inside her, she almost came just from the sheer pleasure of it. Groaning low in her throat, she panted while he explored her at length. His fingers swirled and teased until she was weak from it.\n\nJust when she was at her end, he lifted himself up and replaced his fingers with his mouth. The shock of pleasure was so great that she rose to her tiptoes. She gasped his name as he tongued her with an unrivaled skill. Widening her stance, she lowered herself on his mouth as his tongue flicked inside her over and over again.\n\nUnable to stand it, she came with a small cry. But still he didn't stop. He continued to tease and torment her with his mouth and tongue until she was cresting another orgasm.\n\nWanting him inside her more than she wanted anything else, she buried her hand in his hair and pulled him away from her.\n\nArik was dazed by her taste and by the fire in his blood as she sank to the floor and captured his lips with her own. Her hands sought out his body with an unrivaled eagerness.\n\nNow this was the Megeara he'd come to know. She was relentless with her exploration. Relentless with her kissing. Shoving him back on the floor, she quickly removed his clothes so that she could run her hands over his body.\n\nHe spread his legs for her as she ran the back of her hand over his sac. He actually whimpered from the pleasure of that sensation.\n\nTaking no mercy on him, she dipped her head down and took him all the way into her mouth. Arik arched his back as unimaginable pleasure ripped through him. Everything was swirling. In all the times he'd had sex, he'd never known anything like this. Never felt any pleasure equal to it.\n\nHe tilted his head to watch her and met her hungry gaze. The look of her there tore through him. He reached down to stroke her cheek as she again took him in all the way. Her throat tickled the tip of him before she pulled back to nibble the very tip of him. He was in ultimate heaven.\n\nMegeara sucked him in hard before she pulled away again and kissed her way up his body until she was able to straddle him. Blind with ecstasy, he lifted his hips as she impaled herself on him.\n\nArik cried out from the warmth of her body surrounding his. It was all he could do not to come immediately, but he didn't want this to end so soon. He wanted to stay inside her forever.\n\nGeary smiled at the unabashed pleasure on Arik's face as he held her hips in his hands and urged her on. And she did just as she'd promised. She rode him with everything she had.\n\nFast and hard, pulling him in as deep as she could and then rising up until he was about to fall out. There was nothing better than the feel of his hardness inside her.\n\nAnd as she thrust against him, she knew they had no future together. But even so she felt closer to him than she'd ever felt to anyone.\n\nIt was like he was a vital part of her and for once she didn't feel self-conscious around him. She was completely at ease with her nudity and with her sexuality. Completely at ease with his knowing how much she craved him. There were no barriers between them now. No secrets.\n\nAnd when she came again, she actually screamed out loud.\n\nArik groaned at the sight of Megeara's ecstasy. It was so intense that it set off his own. He drove himself deep inside her as his body shook out of his control. A thousand emotions and sensations tore through him, robbing him of all reason and thought.\n\nAll he could do was feel. Feel her and the moment of pure, uninterrupted bliss as his body spasmed inside hers.\n\nShe leaned over him to kiss his lips. Arik held her there, letting his senses swirl.\n\n\"Are you all right?\" she asked, her brow creased by worry.\n\n\"I don't know,\" he answered honestly. \"I think my body just turned inside out. And I can't understand why Solin would ever venture into dreams if this is what it's like in a human body. He must be out of his mind.\"\n\nShe laughed at Arik's indignation. \"That was not what I was expecting to hear. But I'm glad you enjoyed yourself.\"\n\nHe cupped her face in his hands before he kissed her again. \"I could drink you in all day, Megeara. What is this feeling that I have inside me? The one that hurts with the thought of not being with you? The one that wants to be inside you again even though I just had you?\" He hesitated before he whispered his last question to her. \"Is this love?\"\n\n\"No,\" she answered quietly. She didn't believe in love at first sight. \"Real love takes time to build. What you feel is just infatuation.\"\n\n\"But it doesn't feel temporary.\"\n\n\"It never does at its onset. It's only in hindsight that we realize the difference between infatuation and love.\"\n\nHe didn't seem to buy her argument. \"And what if it's not?\"\n\n\"What are you saying, Arik? That you love me?\"\n\nArik fell silent as he considered it. There was no denying what he felt. But then, his feelings were very new and they would expire in only a few more days. It might be love now, but how could someone continue to love when they had no emotions?\n\nMaybe she was right. Maybe it was only infatuation.\n\nBut even as he thought that, he knew better. The very idea of returning to his old existence burned through him with so much pain that he could barely stand it. It made a complete mockery of any punishment he'd ever known. He wanted to stay with her for all eternity.\n\nAfraid of losing her, he held her close, naked body to naked body, and tried to forget how soon his departure would be.\n\nGeary lay there quietly, listening to Arik's heart beating under her cheek. How strange to be with him now knowing how finite their time together would be. There was none of the optimism of most affairs, where you hoped they'd last forever. In some ways that made her lucky. She knew to the second how finite their time was.\n\nBut it was also a curse to know when she was going to lose him\u2014because she suspected that she was already in love with him. How could she not be? He was the only man who'd ever seen the real her. In her dreams, she'd told him everything. Her hopes, her disappointments. She'd never been restrained with him. Never held back. He knew her in a way no one else did.\n\nAnd that was why she couldn't let him go.\n\n\"There has to be some way to keep you here.\"\n\nShe hadn't realized she'd spoken out loud until he answered her. \"I could stay, but I wouldn't be the man who's with you now.\"\n\n\"What do you mean?\"\n\n\"I could give up my godhood, but it would change nothing other than I'd become mortal. When my time is up, I'll revert back to what I was. I won't have any emotions and I wouldn't be able to visit you in dreams. There would be no reason to be around you then.\"\n\n\"I don't believe that. You have emotions. You feel too deeply not to.\"\n\n\"In the dreams, I was syphoning off you. Everything I felt came from you. I promise you that if I were to become mortal, that would stop. I wouldn't even have the power to feel you anymore, either physically or emotionally.\"\n\n\"How do you know that?\"\n\n\"It's the curse, Megeara. There's no cure for that. No god can alter the curse of another. I'm damned.\"\n\nShe still couldn't accept that. It wasn't in her nature to just accept things because someone said so. She was a scientist and she needed proof of his theory. \"Has any Oneroi ever gone free?\"\n\n\"No,\" he said emphatically. \"There has never been a single case of freedom for any of us. The few who tried were hunted down and killed.\"\n\n\"That's not fair. You should be able to go free if you want to.\"\n\nHe let out a deep breath as he stroked a lock of hair from her forehead. \"Who has ever said that life is fair?\"\n\n\"Maybe, but I'm going to ask Tory about this.\"\n\n\"Tory's just a child.\"\n\n\"Yes, and she's obsessed with Greek mythology. If ever there was an escape for you, she'd know it.\"\n\nArik adored the fact that Megeara was willing to try, but he knew it was hopeless. No human knew more about Greek mythology than he did. Megeara was human and he was a cursed god. All he could hope for was to find some way to keep her safe once he was gone.\n\nAs long as they were at Solin's they had a haven. Solin had told them that he had a truce with the other gods. They didn't tread into his home uninvited and he wouldn't kill them for it. But Arik and Megeara couldn't stay here every minute of the day. And she wouldn't be content to live her life here within these walls. She'd never liked cages of any sort.\n\nShe's going to be dead, so just stay here and enjoy her company until it's time to return.\n\nReturn to what? Emptiness? Coldness?\n\nThat was bullshit. He didn't want to go back to the Vanishing Isle.\n\nThen you die in her stead.\n\nArik leaned his cheek against the top of her head as she lay against him. She felt so good in his arms. So good with her naked skin resting against his. I would rather be dead than live without her.\n\nIt was true, and actually that was the only solution that made sense. He would spend his time with her and then surrender himself to Hades. Hades would torture and kill him, then everyone would be happy.\n\nYou won't be happy, dumb ass.\n\nThat was actually quite true, but even if he gave her up and went home, he'd still be tortured, not to mention the Dolophoni would kill him anyway.\n\nSo why not just let them have him and end this?\n\n\"Live your life with purpose.\"\n\nArik blinked as those words came out of the very distant past to haunt him. It'd been back in his days as an Oneroi when he'd ironically gone to help Trieg. Acheron, the leader of the Dark-Hunters, had summoned him so that they could discuss the problems Trieg was having over the death of his family and how best to help the man cope with them.\n\nTall and black-haired, the Atlantean had been even wiser than Athena. He'd been trying to make Arik understand the human psyche and essence. \"Remember, Arikos, the key to humanity is simple. Live your life with purpose. They need goals to strive for. All of Trieg's have been taken from him by his enemies, so we need to replace them with new ones that matter to him. Without goals, humanity is lost and a single man can't function.\"\n\nAcheron had been wrong about one thing. Without goals, everyone was lost. Even the gods.\n\nUntil now, Arik's goals had always been selfish in nature. As a Skotos, his goal had been to find the greatest pleasure to be had. As an Oneroi it had been to do just as he'd been told so as not to be punished. He'd never once considered anyone else's feelings or life.\n\nBut now he understood how to live with purpose. He understood sacrifice. There were things worth dying for. His was simple. Megeara. His only regret was that he wished he'd enjoyed their past more. He should have savored every second of their time together.\n\nStill, he had a few days left. Those he would make count. And when the time came, he would put his neck in the noose with no regrets.\n\nYeah, right.\n\nOkay, so there would be one single regret\u2014he'd never see or touch Megeara again.\n\nHe could die with that.\n\nAnd in the back of his mind was that same sarcastic voice laughing at him. \"Trust me, boy, you will.\"\nCHAPTER 16\n\n\"Houston, we have a problem.\"\n\nM'Adoc turned around from the bay window where he was looking out onto the waterfall behind their palace to see Deimos entering his private suite without invitation. He let out his breath slowly and silently, falling instantly into his emotionless appearance.\n\n\"How colloquial American of you, Demon.\" He raked the demigod with a practiced mocking brow that only M'Ordant or D'Alerian would know wasn't feigned, and forced his voice to remain steady and bland. \"By your unbloodied presence here I take it that you failed to kill him... again.\"\n\nDemon's eyes narrowed. \"I can do without the patronizing undertone.\"\n\nClasping his hands behind his back, M'Adoc crossed the floor to meet Deimos halfway. \"We both know I feel no such thing. But to be fair, I can deal without the incompetency. How hard is it to take out a restricted god on the mortal plane?\"\n\n\"Pretty damn impossible when he has a Chthonian and an Atlantean god standing watch over him.\"\n\nM'Adoc had to struggle to hide his confusion. \"Why would Acheron care about any of this?\"\n\n\"Not him, his mother. Remember her? Tall angry blond bitch who seriously spanked her whole family into oblivion over a hangnail?\"\n\nM'Adoc's lips itched to smile, but he was so used to catching himself that it was all too easy to keep it hidden. \"It was more than a hangnail and she's locked in Kalosis, so how can she be a problem?\"\n\n\"Not entirely, she isn't. Someone dug up one of her special little priestess medallions and it's now in the hands of the woman who has a vested interest in us not harming her boy toy... or her. Seems she has an issue with dying prematurely. Go figure.\"\n\nM'Adoc was less than amused by Demon's summation. \"Well, that sucks for you, doesn't it?\"\n\n\"It sucks for all of us, Oneroi. If you want this handled, I suggest you do it yourself.\"\n\nM'Adoc really did have to make an effort to keep his sarcastic tone at bay. \"I never thought I'd live to see the day a mere mortal could scare the Dolophoni. You guys have really grown soft over the centuries, haven't you?\"\n\nDeimos curled his lip. \"Calling me a coward is no way to goad me into suicide. As I said, we have extenuating circumstances. You're the one telling me how easy it is to kill him. Then why don't you try getting your hands' bloodied for once?\"\n\nLittle did he know, M'Adoc's hands had been coated in more blood than a seventy-five-year career surgeon's. He had no problem with executing his nuisances, he just had to be careful not to let the other gods know about it. The idea of an Oneroi taking a life without their implicit approval tended to make them jittery. \"My job is to protect.\"\n\n\"Yeah, your own back. And mine is to watch over my team\u2014one of whom is now dead.\" He took a step forward to make sure M'Adoc understood his rage. \"You know I've never shirked at killing anyone or anything. But this... this is different. I'm not going to lose another brother needlessly. This is getting out of control.\" He hesitated before he added one final comment. \"They're currently at Solin's house\u2014under his protection. I'm sure you remember him, too.\"\n\nOf course he did. He and Solin had battled it out on more than one occasion. Both of them were scarred by those fights.\n\nBut that was neither here nor there.\n\nM'Adoc gripped his hands tightly behind his back, tempted to send a bolt straight at Demon's head, but he couldn't afford to let Demon know this was anything more than a routine hit or that his failure upset him in any way. He must remain perfectly calm at all times. Demon would love to have an excuse to turn the gods against M'Adoc, and he knew that. It was a dangerous line he toed.\n\nHe inclined his head. \"Thank you for your services, Demon. I will make sure that the next time I need to consult with the Furies I summon one of the females, as they are much more vicious and competent.\"\n\nThat barb wasn't lost on Demon, who sneered at him. \"One day, M'Adoc, you're going to learn why they nick-named me Demon.\"\n\nAnd one day Demon was going to learn why M'Ordant and D'Alerian referred to M'Adoc as Fonias\u2014Slayer.\n\nIn the meantime, M'Adoc had a mess to clean up and he would make sure that this time the job was done correctly. Let Arikos have a couple of days of peace so that he'd relax a bit. Then when his guard was down, M'Adoc would take full advantage of it.\n\n* * *\n\nArik smiled as Megeara buttoned his shirt. Even though they'd slept together, this one act seemed somehow more intimate. Her hands were graceful as she wove the buttons through their holes. Her fleeting touch made his nipples harden and his body warm. The scent of her was heavy in the air, and all he wanted to do was take her away someplace private where he could be with her alone for the rest of eternity.\n\nShe glanced up at him. \"Are you all right?\"\n\n\"Fine, why?\"\n\n\"You have a strange look about you.\"\n\n\"I'm just thinking how delectably kissable your lips are.\" And before he even realized what he was doing, he was kissing her again.\n\nGeary sighed as she melted into Arik's arms. The steely feel of his body was electrifying, and it made her want to rip his clothes off and have another round with him. If only she could. But right now they had a lot of things to think about.\n\nShe pulled back as a chilling thought went through her head. \"You don't think they'll go after Tory to get to me, do you?\"\n\nArik pulled back with a scowl. \"Excuse me?\"\n\n\"The Dolophoni. They won't go after her to get me or you to come for her. Will they?\"\n\nTo her relief, he shook his head. \"Not their style. They only kill who they're sent after. They don't worry about bystanders unless the bystanders attack them. They're actually rather ethical, which for gods and assassins is an amazing feat.\"\n\n\"Then why are they coming after me if not to get to you?\"\n\n\"Someone wants you dead.\"\n\nHis emotionless tone sent a chill over her. \"Remind me later that we need to work on your tact.\" Geary shook her head as she tried to understand. \"Who could possibly want me dead? I haven't done anything.\"\n\n\"You were digging around Atlantis. It's why the boat was blown up. The gods do not, under any circumstance, want that place disturbed. And they're all willing to kill to keep its secret.\"\n\n\"What secret is that?\"\n\n\"I assume why it was destroyed. Not even we really know what happened the day it vanished. Whatever went down there went down fast, and those who know the truth have kept it hidden ever since.\"\n\nGeary cocked her head as she remembered her research. \"Plato wrote that it was human hubris that caused the gods to destroy it as punishment.\"\n\nArik scoffed. \"Plato wrote of a parable about a nation that was destroyed long before his ancestors had been born. He knew nothing of the truth. Anyone who'd ever gotten close to learning about Atlantis didn't live long enough to tell anyone else.\"\n\nShe stepped back as pain filled her. \"That's why my family's dead, isn't it? We got too close.\"\n\nHe gave her a sympathetic nod. \"I'm sorry, Megeara. But yeah. Your father was all over the real site.\"\n\nA single tear fled down her face, but she quickly wiped it away.\n\n\"Come for me, Megeara, and I will grant you vengeance on those who've wronged you\u2014those who've taken the ones you love most. Come here, child, and let us both deliver to them what they deserve. For petty vanity they took from us both the very people we loved. Help me and I will help you.\" It was the same angry woman's voice Megeara had always heard here.\n\n\"Apollymi?\" She whispered the name.\n\n\"It is I. And I will protect you, child, if you listen. I would have saved your father, but he denied my help and they killed him. I would spare you a young death.\"\n\n\"Is she talking to you?\" Arik asked in a whisper.\n\nIf it'd been anyone else asking Megeara that question, she would have denied it emphatically. \"She says the gods took from her someone she loved.\"\n\n\"Her son. At least that's what Zeus claims. Her husband, Archon, slew Apostolos, and in grief she destroyed her entire family.\"\n\nBut that didn't make sense. \"Then why does she want revenge against the Greek gods and not her own?\"\n\n\"Because Apollo has long asserted that he was the one who killed Apostolos. Back then, Greece and the Atlanteans were in a very strained truce. They'd warred against each other for centuries. The Atlanteans had tried to kill Apollo's son, but he'd managed to remove the baby from the queen's womb before she birthed him, and he substituted another child there that they killed instead. He then took his son Strykerius to Delphi, where he was raised by Apollo's priestesses.\"\n\nThat didn't make sense. \"If Apollo saved his son, why would he kill Apollymi's child?\"\n\n\"Because twenty-one years later Apollo had another son on the Greek island of Didymos. Atlantean assassins broke into the palace in the middle of the night and executed the baby and his mother, who was Apollo's sanctified mistress. To exact revenge on the Atlanteans for their crime, Apollo claims, he killed Apostolos, then cursed everyone of their bloodline to die horribly on their twenty-seventh birthday\u2014the age of his mistress at death. That is what set Apollymi off. Like Apollo she wanted revenge for the death of her son, but Apollo was the greater god, so he trapped her in Kalosis, where she now sits, plotting her revenge against him and the rest of the Greek pantheon.\"\n\nMegeara tilted her head as she caught a strange note in Arik's voice. \"But you don't believe that.\"\n\nHe looked away. \"I've met Apollymi and I know Apollo... he is not the greater god. I've never seen a god yet who would go up against the Destroyer. Even her own family was scared of her, and rightfully so. They say it took her less than ten minutes to blast all of them to oblivion while they were gathered together in their own hall. Knowing gods as I do, I'm quite sure they didn't go quietly to slaughter. But rather they put up one hell of a fight, and out of an entire pantheon there's only one of them still standing.\"\n\n\"Apollymi.\"\n\nHe nodded.\n\nA vicious thought went through her. \"Then she could deliver her promise to me? She could save you and restore my father's reputation without anyone being hurt?\"\n\nHe cupped her face in his hands and stared at her intently. \"Listen to me, Megeara. The gods don't act selflessly. None of them will help anyone without getting something out of the bargain. Ask yourself what it is Apollymi wants from you.\"\n\n\"Freedom.\"\n\nHe shook his head. \"It's never that simple, love. Apollymi wants revenge and she doesn't care who suffers for it. If she is ever released, she will destroy the entire world. All of it. No one will be able to stop her. That's why she's imprisoned and why everyone is willing to keep her there.\" His look intensified. \"She can't ever go free.\"\n\nGeary understood that. It made sense. And yet she was so close now to her goal. Her father had been right and she could prove it beyond a shadow of a doubt. She could expunge his record...\n\nBut at what cost? Was it worth it?\n\nAnd still Apollymi beckoned in her head with promises of revenge.\n\n\"Vengeance only destroys the one who seeks it.\" Geary paused as she remembered her grandfather in New York telling her that after she'd returned to the States to live. During World War II, his entire family had been slaughtered while celebrating his birthday by a raiding Nazi troop who happened upon them. Only nine years old, her grandfather had been wounded, blinded, and left for dead.\n\nWhile he was unconscious, protected by the lifeless bodies of his family, a mysterious man had come and saved him. The man had bandaged her grandfather up and brought him safely to America to start his life over.\n\nAs an angry teenager, Geary had asked her grandfather if he'd ever thought about the ones who took everything from him.\n\nHer grandfather had patted her lovingly on the hand. \"Of course I do, Megeara. I've never had a birthday since that day that I don't hear the gunfire. That I don't see them kicking in the door of our cottage to murder us all. The last thing I saw before they blinded me was my mother dying while trying to protect me. My fourteen-year-old sister being dragged off to be raped and murdered. Do you honestly think, little one, that I don't remember that day constantly and wonder why I alone survived it? If I wouldn't have been better off dead, too? Yet here I am and I'm grateful for it. Because had I died that day, there would be no you.\"\n\nRage for him had burned inside her. \"I would have gotten revenge on them. I wouldn't have been able to live until they paid for their crimes.\"\n\nHe'd nodded at her in that understanding way of his. \"I thought about that, too, and even went so far as to book passage back to Europe after the war to find them.\"\n\n\"But you didn't go.\"\n\n\"No. My Saving Angel\"\u2014his name for the one who'd brought him to America\u2014\"came to me again as if he knew what I planned and he told me that it is by our actions we are destroyed or saved. The choice is ours. He said he didn't save me that day to see me die so foolishly. And he told me that vengeance only destroys the one who seeks it. If I chose to go, he wouldn't stop me. But he asked me if the lives I sought to take would be worth the one I could make here away from the hatred and sorrow. So I chose to stay here and let go of the past. Yes, it haunts me, but it doesn't rule me. And because I stayed here, I met your grandmother and had all of you to warm my heart and to ease my sorrows. My only regret is that I've never seen the beauty of your smile with my own eyes.\"\n\nHe'd smiled tearfully at her then and patted his heart. \"But I feel its beauty here and I know there is not a lovelier, more precious child in this world than you and your cousins. I am glad that though someone did me grossly wrong, my final mark on this world is not one of countering hurt with more hurt but is one of love and friendship. We will always be known by our actions. Let them always be good ones.\"\n\nGeary had to clear her throat as the memory surged and made her eyes tear. She loved her grandfather Theo so much. He was a good man and she wouldn't hurt him ever if she could help it. He'd lost enough people in his life. She wouldn't let him bury another loved one.\n\n\"The quest has ended.\"\n\nArik's brow furrowed in disbelief. \"Has it really?\"\n\nShe nodded. \"I think the boat explosion was an omen, huh? I think we should just leave it alone before someone gets hurt.\"\n\n\"Do you think Tory will let you do that?\"\n\nHe had a point, but it didn't matter. \"I'll ship her back home if she says anything.\"\n\n\"Will she go?\"\n\n\"Kicking and screaming.\" Geary cringed at the thought of how irate the girl would be. But better alive and irate than dead and happy. \"Sometimes we don't want what's best for us\"\u2014another thing her grandfather was always saying\u2014\"but we need it anyway.\"\n\nArik never ceased to be amazed by her. He was so used to people who could only think of themselves that her altruism was baffling. That she would give up a goal that meant so much to her to keep someone safe...\n\nIt was miraculous.\n\nAnd because he knew how much it meant to her, he wanted her to have it. No one should get this close to their dream without attaining it. It seemed cruel to him.\n\nThat would be his final gift to her. Before he died he wanted to see the look of joy on her face for redeeming her father. \"How about we make a compromise?\"\n\nShe gave him a droll stare. \"How would that be possible? You said it yourself that all the gods are against this.\"\n\n\"We can try. I'll take you back to the site and we can salvage a couple of incidental artifacts that won't hurt to have known\u2014enough to prove your father wasn't insane\u2014and then we can tell Tory that the site is too unstable for excavation. Tell her part of it caved in on us and we barely escaped. We can cut the live feed and make it seem real. Then we can say that Atlantis needs to remain at the bottom of the sea where the gods intended it rest.\"\n\nGeary was stunned at the beauty of that argument. Until reality came crashing down on her. \"I'd have to give up the location in my findings report.\"\n\n\"Lie. Who would know? You can give them a location somewhere else. Tell them the site was off the banks of Mykonos.\"\n\n\"But if someone else digs\u2014\"\n\n\"They find nothing and they don't die. People have been looking for Atlantis for eleven thousand years without finding it. It's just one more chapter in this chronicle. You will still have irrefutable proof that Atlantis existed. No one will be able to argue against it.\"\n\nWould that work? It truly sounded too good to be true. \"Are you sure the gods will be appeased?\"\n\n\"I think so. I just need Kat's number from you.\"\n\n\"Why?\"\n\nArik hesitated before he answered. He wouldn't out Kat and her relationship to the gods. If she wanted Megeara to know, it was Kat's place to tell her, not his. \"We'll need another diver on the project. Just in case. She's more levelheaded than Teddy and I think she'd understand our reasons for keeping it hidden.\"\n\n\"Good point. You want me to call her?\"\n\nThat would defeat the purpose of not outing her. He needed to explain this to Kat before they went back. The last thing he needed was for her to try to kill them, too. \"I'd rather do it myself.\"\n\nStepping back, Geary eyed him warily. \"Is there something about Kat that you're not telling me?\" The suspicion in her gaze deepened. \"Is she one of you?\"\n\n\"No.\" That was the honest truth. She was in a class all by herself.\n\n\"Then I'll call her.\"\n\nHow did he keep getting himself into these messes? Kat would flip without his explanation. \"Why don't we wait until tomorrow to talk to her then? Let her rest tonight.\"\n\n\"Okay.\"\n\nHe was grateful Geary didn't push this. By tomorrow he might have a better idea.\n\nSuddenly someone was pounding on the study's door.\n\n\"Excuse me,\" Solin snapped angrily from the other side. \"Last time I checked, this was my house. Why am I locked out of my own study?\"\n\nArik moved to open the door. \"Anything to piss you off, Brother. Why else?\"\n\nSolin scoffed as he entered the room. \"Oh, that's easy enough to do. Basically the fact that you breathe does that.\"\n\nArik closed the door and turned to face him. \"Love you, too.\"\n\n\"Of course you do, like a plague on your privates.\"\n\nWell, at least Solin understood the nature of their relationship. \"So what brings you back?\"\n\n\"What part of my house did you miss?\"\n\nArik countered with his own argument. \"What about the part that we could stay here if we needed to?\"\n\nSolin opened his mouth to retort, then snapped it shut. He was silent for a few moments before he spoke again. \"I did say that, didn't I?\"\n\n\"You did.\"\n\n\"Fine,\" he said irritably, \"stay. But whatever you do, put down a blanket or something next time you two want to get frisky on my hardwood floors. That's just... disgusting.\"\n\nGeary sputtered at his indignation. \"How do you\u2014\"\n\n\"He's a demigod,\" Arik answered in an unamused tone, cutting her off. \"Never get too close to one if you want to maintain secrets.\"\n\nHer cheeks pinkened to let him know she was quite embarrassed by it. \"Well, that's not fair.\"\n\nSolin gave her an arch stare. \"You seem to have an issue with fairness, don't you?\"\n\n\"I don't like things to be disorderly, if that's what you mean. There should be a degree of fairness in the world.\"\n\nSolin snorted as he looked at Arik. \"She's priceless.\" He returned his cold stare to her. \"Sweetie, in our world, fair's got nothing to do with anything. He who has the greatest power wins. It's why we're all willing to kill each other off without flinching.\"\n\nShe cast a confused look at Arik before she responded. \"But you helped me and Arik. Why would you do that if you really feel that way?\"\n\nSolin shrugged. \"What can I say? It's so much more enjoyable to snatch victory from the hands of the gullible. You guys make the most delightful sound of agony when you're betrayed.\"\n\nThere was a part of her that wanted to think he was kidding, but another part wasn't so sure. He sounded pretty damn sincere. She glanced at Arik, who was every bit as skeptical as she was.\n\n\"Are you in with them then?\" Arik asked.\n\nSolin gave him an exasperated smirk. \"If I was, do you think I'd let you stay here?\"\n\nArik shrugged. \"I don't know. It wouldn't hurt you. It's not like letting us stay here will make them hate you any more than they already do. If anything, our presence here would piss them off, which would be a bene for you. As you said, it would be a way to snatch victory from the gullible.\"\n\nSolin turned completely stoic. His face, his demeanor, even his voice. \"I won't defend or explain my actions to you or anyone else. My motives are my own. Good, bad, indifferent.\"\n\nGeary cocked her head as she noticed something about him while he spoke. A slight tenseness on his face. \"What are you afraid of?\"\n\nSolin curled his lip at her. \"I fear nothing.\"\n\n\"You fear intimacy, don't you?\" she asked. \"With anyone. That's why you say nothing about yourself. It's why you prefer to traipse through dreams rather than sleep with women in the flesh.\"\n\n\"Thank you, Dr. Ruth.\" You would need a chainsaw to cut through the venom and sarcasm in his voice. \"But I honestly don't think you know even the most basic thing about me. So until you do, you should keep your opinions to yourself.\"\n\n\"You're right, I don't. But the question is, does anyone? Can you name me one single friend you have or have had in the past?\"\n\n\"I don't need friends. All they do is eat your food, drink your beer, then spew your secrets the first time you do something that displeases them. No offense, but when you have as many enemies as I do, you keep your secrets under lock and key. Isn't that true, Arikos?\"\n\nArik's gaze met hers and it softened in a way that made her heart speed up. \"Sometimes it pays to trust the right people.\"\n\nSolin curled his lip at them. \"Such rotten sentimentality, and gullible until the end\u2014both of which will ultimately get you killed. It is, after all, how I got you converted.\" He paused for effect before he stepped toward Geary to address her. \"You should have seen him, Megeara. He was so sure he could take me in a fight. He was getting all ready for it when I did the unexpected.\"\n\n\"And that was?\" she asked.\n\n\"I turned my human lover loose on him. She was in a dream state and had no idea what she was really doing. Arik, being the good Oneroi he was, wouldn't fight her. Protect the humans at all costs\u2014that's their credo. Unless the human is a half-breed.\" He spat the words as if they were bitter tasting on his tongue. \"Then we deserve to die for no other crime than the fact our father went slumming with a hard-on and knocked up some bitch who couldn't keep her legs crossed.\"\n\nSolin invaded her personal space, making her take a step back as his blue eyes snapped fire at her. \"So don't talk to me about fairness. I've no patience for it or you, and that, little human, is all you need to know about me.\"\n\nBacking away, Solin raked them both with a sneer. \"Stay or go. I really don't give a shit. But if you stay, I want you to continue your play upstairs in a bed, like civilized people.\" Then he turned and left them.\n\nIt took Geary a couple of minutes to recover her composure from his unwarranted rancor. \"Well, isn't he Mr. Happy Sunshine?\"\n\nArik didn't respond as he studied the floor.\n\nGeary took a moment to consider everything Solin had told them, including the piece of history that explained another mystery in their relationship. \"So he's the one who turned you. I'm surprised you would even speak to him.\"\n\nHe took a deep breath before he answered. \"Honestly, I'd rather have my brains ripped out through my nostrils, but I wanted to stay with you, and without the permits you would never have allowed me near you. I had no choice except to call on him. Besides, you can't blame him really. He has every right to hate us.\"\n\nHer chest tightened at the thought of Arik's seeking out a bitter enemy for no other reason than to be with her. It was incredibly romantic, if not somewhat stupid. \"Compassion looks good on you, Arik. You should wear it more often.\"\n\nHe took her hand into his and toyed with her fingers. \"I'm trying to, but honestly, I'd rather be wearing you.\" He offered her a smile that warmed her heart.\n\n\"Ooo, that was a good one.\"\n\nHe lifted her hand to his lips to nibble her fingers. \"It's the truth.\"\n\nGod, she was in love with this man... god... or whatever he was. They'd known each other such a short time and yet it seemed like forever. She'd confided everything to him, and here he was, trying to help her.\n\nHow could she let him go?\n\nShe already knew that answer. She couldn't. He'd come to mean too much to her. And as that thought went through her, it was followed by another. There was someone who knew more about this than Tory or even Arik.\n\n\"Apollymi?\" she let her mind shout, hoping that the Atlantean goddess hadn't abandoned her.\n\n\"Yes, child?\"\n\n\"Is there some way to free Arik from his bargain without killing him? Can he be made mortal?\"\n\n\"A god can do anything. Free me and I will grant you any wish you have.\"\n\n\"Do you swear it?\"\n\n\"On the lives of my Charontes. You free me and you will never want for anything so long as you live.\"\n\nGeary pulled Arik into her arms and held him against her. She was grateful he couldn't hear her thoughts or her conversation with the goddess.\n\nHe felt so good in her arms.... She never wanted to let him go.\n\nDon't make a pact with a god, her mind warned. In all her ancient readings she couldn't recall a single time that such a bargain had worked in the favor of the person who made it.\n\nNot once.\n\nBut that was fiction and this was real. Apollymi was real and so were Arik and Solin.\n\nGeary would allow Arik to take her back to Atlantis and then she would let Apollymi guide her. After all these centuries, the goddess would be free again.\n\nGeary's only hope was that Apollymi would keep her word. But even then, Geary's doubt was strong.\n\nWhat choice do I have?\n\nShe couldn't allow him to die if she could stop it. And she was willing to make a deal with the devil to ensure Arik's life.\nCHAPTER 17\n\nThe next four days were filled with answering official inquests about the exploding boat, dealing with the insurance company, and trying to calm down Tory, who wanted to head right back to the site and excavate even though much of their data had gone up in flames. The only ones happy about the delay were Thia, who had more time to spend with Scott and Brian, and Kichka, who was able to hunt uninterrupted for stray mice in the alley behind their flat.\n\nAnd deep inside, Geary was more than fine with the delay, too, since it meant more time with Arik. He proved himself to be a tremendous help to her. He kept her completely grounded even while her temper was snapping, and he had an unnatural ability to get the Greek officials to bend to his will. If she didn't know better, she'd swear he had his god powers back.\n\nBut he was definitely still human. He just knew how to influence people to get what he wanted.\n\nShe sighed as she lay naked in bed with him in the late afternoon. It'd been a particularly grueling day. Between dealing with her usual business with the salvage company and a couple of clients who didn't want to pay for having their cargoes retrieved or their boats towed, and the insurance company that was trying to prove she'd intentionally blown up her boat to get the money out of it, she was thoroughly exhausted.\n\nThe only good thing had been the mind-blowing sex, and now Arik was rubbing her back while she lay beside him.\n\n\"What are you thinking about?\" she asked. He'd been unusually quiet all day.\n\n\"Nothing.\"\n\nShe turned her head to look up at him. He was completely naked except for the sheet that was pooled in his lap. His hair was mussed and his lips swollen, while he had a day's worth of whiskers on his cheeks. He was a little flushed from their play, which only made his eyes paler and bluer. \"I don't believe that. You seem preoccupied. What's on your mind?\"\n\nHe squeezed her shoulder with a touch so expert it wrung a moan from her before he answered. \"You've had enough stress. I don't want to add to it.\"\n\n\"Oh, what the heck?\" she said with a smile. \"Add away. At this point, one more problem would be nothing to me.\"\n\nLaughing, he kissed the shoulder he was working on before he moved down to massage her arm. \"I was just thinking how strange it is that no one has attacked us these last few days. I keep waiting for the Dolophoni to return.\"\n\nShe propped herself up on one arm to watch him. \"Maybe Apollymi scared them off.\"\n\nHe took her free hand into his and massaged it between his fingers. The small circles went up her arm and made her literally melt. \"I don't know. They're not the type to scare all that easily.\"\n\nHe did have a point, but honestly, she preferred the thought that Apollymi had scared them off. That meant they wouldn't be back. \"So what exactly are you thinking about their absence?\"\n\n\"That they're waiting for me to get comfortable here so that they can strike while I'm not looking.\"\n\nShe liked that thought even less. \"Maybe you're just being paranoid.\"\n\n\"Do you really think so?\"\n\nNo. But she couldn't bear to say the word out loud. It was just too hard to think about. And the other thing she didn't want to think about bulldozed its way into her thoughts right behind it\u2014Arik's time here was getting shorter by the second.\n\nWhich triggered her to glance at the clock. As soon as she realized the time, she jumped up, clutching the sheet to her breasts. \"Hey, we need to get going if we're to meet Kat on time.\"\n\nArik nodded even though he was starting to dread this meeting and he didn't know why. He'd been the one to suggest they return to Atlantis, and yet he had a bad feeling that he couldn't place. Something was going to go wrong. He knew it.\n\nPerhaps he'd been human long enough to develop a degree of intuition. Or maybe he'd been attacked enough to know that the most likely place for the Dolophoni to strike next time would be underwater, where he and Megeara wouldn't be able to escape or really fight....\n\nIt was a chilling thought.\n\nBecause of that, he kept it to himself as they showered, dressed, and then headed for their rendezvous with Kat. He didn't want anything to taint Megeara's happiness after the last few days she'd had. Everyone had been chipping small pieces of her joy away, and he much preferred her smile.\n\nThis was what she'd dreamed of, and no matter what, he was going to give it to her.\n\nMegeara was beautiful in a light summer top and jeans as she drove him out to the docks, which looked like they had a gaping hole where her old boat had been. In a weird way, he missed that boat and was sad to see it gone. He could only imagine how hard it must be for Megeara, since it was the same boat her father had used on his expeditions. She didn't say it, but Arik could tell by the longing on her face as she looked to the empty moorings that she missed the boat, too.\n\nFor this excursion, they were going to use one of her smaller company boats\u2014just to make sure no one knew what their intent was. It was also small enough that they could man it with just the three of them.\n\n\"Is Tory going to be here?\" he asked.\n\nMegeara parked her car in the sandy lot that was off to the side of the marina. \"No. I told her I needed her to reconstruct the excavation maps that'd been destroyed. She has no idea that we're even heading out today. She thinks we can't move until she finishes her project.\"\n\n\"That was evil of you.\"\n\nShe gave him a shy smile. \"I think we're all a bit evil when it comes to protecting our families.\"\n\n\"Are we?\"\n\nGeary turned in the seat to look at him. \"You have no idea what I'm talking about, do you?\"\n\n\"No, not really. I mean, yes, I know the definition of family, but our families on Olympus don't work the same way yours does and we don't have the same attachment for each other.\"\n\n\"What about your mother?\" Geary asked. \"Surely she took care of you?\"\n\nHe nodded. \"True, she birthed me.\"\n\n\"And then?\"\n\n\"I was handed off to attendants who tended my needs until I was old enough to be trained.\"\n\n\"Yeah, but didn't one of the attendants love you?\"\n\nHe frowned at her. \"They were servants, Megeara, not family. There was no love, and even if there were, I was too small to remember it.\"\n\n\"How small?\"\n\nArik sat quietly, thinking, but nothing came to mind. He didn't have very many memories of childhood, and there was nothing there as he struggled to remember anyone taking care of him. \"I don't remember. That's just the way it's always been done and I'm sure I was no exception. I honestly remember nothing of being a child, except for my training.\"\n\nGeary was doing her best to understand his world, but it made no sense to her. \"And what kind of training was this?\"\n\nHe sighed as if the matter irritated him. \"Even though we're cursed, we still have residual emotions when we're born. Those have to be stripped from us and we have to be shown how to enter dreams, as well as what is allowed for us to do there and what is forbidden. Then we have to learn how to fight the Skoti who will ultimately fight us for control of the human host. It takes years to fully master our powers, and it's all very complicated.\"\n\nIt sounded like it. But one part of that stuck out in her mind. \"And how do they strip emotions from you?\"\n\n\"Usually by beating,\" he said in an empty tone. \"It's actually quite Pavlovian. You show an emotion and the punishment is such that you learn it's better to feel nothing than to suffer the consequences of having them.\"\n\n\"Does the training ever fail?\"\n\n\"Sometimes.\"\n\n\"So what do they do then?\"\n\n\"They execute us.\"\n\nShe wouldn't have been more stunned had he reached over and slapped her. \"You have got to be kidding me!\"\n\n\"No,\" he said in all earnestness. \"At least I don't think I am.\"\n\nStill, Geary was incredulous over his blas\u00e9 tone and that they would simply kill the Oneroi for no other reason than they continued to have emotions. How cruel was that? \"And all of you accept this?\"\n\nHe appeared to be equally baffled by her standpoint as she was by his. \"Have we a choice? Unless we mount a revolution against Zeus, this is what we have.\"\n\n\"Maybe it's time you did revolt.\"\n\nHe scoffed at her indignation. \"It's not that simple. The pantheons have a balance of power and you have to be extremely careful adjusting it. One wrong move and you can destroy the entire world. What good would overthrowing them be then, when we'd all be dead?\"\n\nAs much as she hated to admit it, she'd lost this argument. \"You make an awesome point.\"\n\n\"Yeah.\"\n\nGeary opened the door and got out while his words haunted her. Her poor Arik. She had to save him from this nightmare he was caught up in. She couldn't bear the thought of returning him to that life where there was no one to care for him. To hold him or love him. It just wasn't right.\n\n\"Why are the gods so callous to our suffering?\" she asked as he joined her in front of the car.\n\nHe took her hand in his before he answered. \"The world is filled with suffering. If you open yourself up to it, it will consume even a god. But not all of them are callous. ZT is one who cares.\"\n\n\"I thought you said he wasn't a god.\"\n\n\"True. He is technically human, but he does have the powers of a god and immortality, and he does care about humanity in spite of what he says. And in spite of what they've done to him, he's never lost his compassion for others. There are several more like him who feel the same way. Who protect humanity.\"\n\n\"Yes, but is there a true god who does so?\"\n\nHe thought about it a few seconds before he answered. \"Apostolos.\"\n\nGeary was surprised by his choice. \"Apollymi's son?\"\n\n\"Yes.\"\n\n\"I thought he was dead.\"\n\n\"That's what the rumor says.\"\n\n\"But you don't believe it?\"\n\nArik shrugged. \"You hang out in dreams and you hear all kinds of fascinating things. Apostolos is alive and I've heard his mother speaking with him. I know that he often tries to calm her down when she's extremely irate and threatening to destroy the world.\"\n\nGeary took a second to let that soak in. \"How ironic that the Great Destroyer's child would be the one who cares about the very people she wants to destroy.\"\n\n\"It is, but he does. He understands the grand scheme of things and consequences better than anyone else, and unlike the other gods, he won't punish people for their mistakes.\"\n\n\"Why not?\"\n\n\"Let's just say that if I had to choose between my life and his, I'd much rather live mine.\"\n\nGeary frowned. Jeez, how horrible had Apostolos's life been that Arik would make that declaration? It was a scary thought. \"Wow. You seem to know a lot about him even given the fact you hang out in dreams.\"\n\n\"Yeah, well, I've been in his a time or two, too. I just hope he never remembers it or I'm really screwed.\"\n\n\"Hi, guys.\"\n\nShe looked up as she heard Kat's voice. The tall blonde stood on the docks in a pair of shorts and a loose T-shirt.\n\n\"Hey, babe. Glad you made it.\"\n\nKat shrugged. \"Well, if you really intend to go poking around Atlantis again, I want to be there for it.\"\n\n\"I'll bet you do,\" Arik said under his breath.\n\nGeary scowled at Arik's odd tone but chose to ignore it as they neared Kat on the docks. \"Do you have all the gear ready?\"\n\n\"I do.\"\n\nGeary was grateful.\n\n\"Did you tell anyone what we're doing?\" Arik asked Kat as they joined her by the boat.\n\n\"Not at all. I know how to keep a secret.\"\n\n\"Good.\" Geary rubbed Arik's arm before she headed to the boat. \"Come on, kids, let's get this under way. We have a date with destiny.\"\n\nArik paused as Kat narrowed her gaze on him with such intensity, he could actually feel his skin burn. \"How many times do you have to be warned? I can't believe you're this stupid.\"\n\n\"I'm not. She and I have an agreement. We give her a couple of innocuous trinkets to prove that Atlantis is real and save her father's reputation, and then she will doctor her findings to help send others off on a wild-goose chase. She will help us safeguard the location of Atlantis.\"\n\nKat looked stunned by his declaration. \"Are you serious?\" she asked in a low tone that only he could hear.\n\n\"Yes. She understands why it can't be found and is in full accord.\"\n\n\"I don't believe you.\"\n\n\"Ask her.\"\n\nKat led him on board the small boat where Geary was already making preparations to sail. \"So, Gear... where's the rest of the team?\"\n\nMegeara looked a bit sheepish. \"I'm thinking we only need the three of us.\"\n\n\"Why?\"\n\nShe met Arik's gaze before she answered. \"Look, Kat, I know how much finding Atlantis means to everyone, especially you, but I've done a lot of thinking and I don't want it really found. I know it doesn't make sense to you right now, but I think this is for the best and I want you to trust me.\"\n\nKat still looked unconvinced by Geary's reasoning. \"So why are we going back?\"\n\n\"A couple of reasons. One, I want inarguable proof that it's there, to silence everyone who laughed at my dad, and two, we need to destroy the datum so that no one will see it and get curious about it. The last thing we need is someone digging down there on a site we helped set up.\"\n\nKat folded her arms over her chest as she gave Geary a doubting stare. \"Are you sure about this?\"\n\n\"Positive.\" Geary reached out to pat Kat's shoulder comfortingly. \"I'm sorry, Kat. I know you wanted to be there when we unveiled the discovery, but we can't tell anyone where Atlantis really is.\"\n\nKat shrugged. \"Don't apologize to me, Doctor. It's your excavation.\"\n\nGeary couldn't believe that Kat wasn't angry or hurt. But she was grateful for her friend's levelheadedness.\n\nOf course, there was a third reason Geary wanted to go. Apollymi. If Geary unleashed Apollymi, she would save Arik. But since Kat didn't know that and Arik would kill Geary if she mentioned it to him, she kept that to herself. No matter what, she had to save him. She couldn't stand by and let him die because he'd wanted to be with her. It was wrong and she loved him too much for that.\n\nIn less than an hour, they had the boat prepped and were under way. Kat was at the helm while Geary stayed on deck, looking out at the passing boats around them. The island was breathtaking in the background, rising up out of the water with a grand majesty unparalleled. Her father had been right. Greece was one of the most beautiful places in the world.\n\nAnd this was the first time in years she'd headed toward the site without feeling anxious and elated.\n\nThere was definitely nothing but dread in her stomach now. She glanced over to where Arik stood, checking their diving gear. For once there was no guessing about Atlantis. There was no doubt. It was there. Waiting. Just as her father had said.\n\nShe was about to show it to the world.\n\nAnd release a goddess....\n\nGeary gripped the rail as she again heard Apollymi's voice in her mind, calling out to her. She could save Arik. Keep him safe forever.\n\nThe promises sounded so good to her, especially as she watched him laying out their gear.\n\nThere's no other way. She'd done her own research and checked it with Tory. Neither of them had been able to think of or find any example of a god becoming human. At least not unless the god was cursed or there was some other extenuating circumstance that didn't apply to this case.\n\nIt was hopeless. To keep Arik, Geary would need Apollymi.\n\n\"What am I doing?\" Geary whispered. \"Don't meddle in the affairs of the gods unless you want to get eaten.\" It was a lesson that resonated through ancient literature.\n\nWho was she to tamper with fate? But as she watched Arik, she couldn't stand the thought of losing him. The thought of sending him back to die.\n\n\"It's an interesting moral dilemma, isn't it?\"\n\nGeary tensed as she heard a voice to her right. She turned her head to find a handsome man standing in the shadows, barely a silhouette. His black hair was short and the glowing blue eyes marked him as another of Arik's brethren.\n\n\"Who are you?\"\n\n\"M'Adoc,\" he said in a low tone. \"I'm one of the three leaders of the Oneroi.\"\n\nA wave of fear went through her. \"Are you here for Arik?\"\n\nHe glanced over to where Arik worked, unable to see him in the glare of the afternoon sun. \"Ultimately, yes. But I know from my dealings with humans that you're going to fight me if I try to take him, and that I don't want.\"\n\n\"You're right about that. I won't give him up. Not now. Not ever.\"\n\n\"I know. You love him. It's why I heard Apollymi laughing a few minutes ago.\" He glanced back to Arik, who was checking their small dredge. \"I give my brother kudos... to gain the love of a human is no small feat. The ability of the human heart to sacrifice for the one it loves... there is nothing on Olympus that can even begin to compete with that.\"\n\nA weird chill went over her body at his words and the way he said them. It gave her insight to this stranger. \"You've known love.\" It was a statement.\n\nHis jaw flexed as if he was gritting his teeth, and a flash of pain darkened his eyes, confirming it. \"The Oneroi know nothing of love and the Skoti know even less.\"\n\nStill, she didn't believe him. He'd known it and, by the looks of it, he'd lost it. \"Then why are you here?\"\n\n\"To warn you not to be foolish.\"\n\nWell, that was nice of him. But she didn't need his warning. She'd never been a stupid woman. \"And how am I foolish?\"\n\n\"You've given your heart to someone who has sold you out completely.\" He cast his gaze meaningfully at Arik.\n\nGeary scoffed at him. \"You're wrong. Arik loves me.\"\n\nHe shook his head. \"Arik has no conviction. If he had, he'd have never been turned by Solin.\"\n\n\"You don't know that.\"\n\n\"Oh yes, I do. Arik is weak. He's always been weak.\"\n\n\"You\u2014\"\n\n\"Sh,\" he said, cutting her off. \"Before you defend him, ask yourself this. How does a Skotos become human?\"\n\n\"He already told me that. He made a pact with Hades.\"\n\n\"Yes, and you're a woman who has spent her entire life steeped in ancient Greece. Have you learned nothing of our ways? Has a god ever given such a gift without receiving something extremely valuable in return?\" There was an ominous note in his voice.\n\n\"What are you saying?\"\n\n\"You mean so much to Arik that he bartered your life to be here. He isn't the one who will die when his time ends, sweet child.\" His blue eyes were searing with heat. \"You will.\"\n\nGeary shook her head in denial. That was total crap and she knew it. \"You're lying.\"\n\n\"I can't lie. I'm Oneroi.\"\n\n\"What the hell are you doing here?\" Arik came out of nowhere to rush M'Adoc. She expected them to fight. Instead, M'Adoc allowed Arik to grab him by his shirtfront, shove him up against the wall, and hold him there.\n\nLike a true Oneroi, there was no anger or any emotion evident as M'Adoc stared unblinkingly at him. \"I told her the truth.\"\n\n\"You what?\" Arik asked from between gritted teeth.\n\n\"I told her of your deal with Hades. That you swapped her life for your mortality.\"\n\nArik's face went pale as his eyes filled with absolute horror. He didn't deny it. Rather, he looked guilty.\n\nGeary knew then that M'Adoc wasn't lying. \"Is this true, Arik?\"\n\nHe cursed before he shoved M'Adoc again. \"I have no intention of fulfilling that bargain.\"\n\nM'Adoc looked at her. \"As I said, he has no conviction. We don't understand human emotion and we can't handle it. When he returns to his true god state, he will come back here for you and kill you. As promised.\"\n\n\"Bullshit!\" Arik roared.\n\nGeary wanted to believe in Arik's rage. She needed to. But part of her was drawn toward M'Adoc. He made a convincing argument.\n\nHe looked at his brother with those cold, unfeeling eyes and it made her wonder if that's what Arik would look like once his time had expired. \"You know it's true, Arik. When you are no longer corrupted by human emotion and Hades tells you to kill her, you will do it. You won't have a choice and you won't have any feelings left for her.\"\n\n\"Never!\"\n\n\"Not even when you're chained in Tartarus, under Hades' constant abuse?\"\n\nArik flinched. He couldn't help himself. Too many centuries of torture hit him at once, and those had been doled out by Hypnos. No one was better at making a god suffer than Hades. Everyone knew that. Arik met Megeara's worried gaze.\n\n\"He's telling the truth, isn't he, Arik?\"\n\nArik watched as she backed away from them. He released M'Adoc and turned to face her. \"Megeara, please...\"\n\nShe shook her head back and forth as she looked at him as if he were scum. That look cut him straight to the marrow of his bones. \"Why didn't you tell me about this?\"\n\n\"Because I was stupid, okay? I didn't want to hurt you.\" He reached for her, but she stepped away.\n\n\"You intended to kill me?\"\n\nHe tried to explain, but his tongue seemed to thicken in his mouth as fear seized him. How could he even begin to make her understand? \"It wasn't like that.\"\n\n\"Then explain it to me.\"\n\n\"I'd already made the deal when Hades set down the terms. I had no choice. He sent me here before I could even try to renegotiate.\"\n\n\"And so you intended to kill me,\" she repeated.\n\n\"In the beginning, yes, but\u2014\"\n\n\"But what?\" she asked, her tone ragged with pain and rage. \"There's no but here, Arik. You fully intended to kill me. How could you?\"\n\n\"He's Skoti.\"\n\n\"Shut up!\" he snarled at M'Adoc. Arik turned back toward Megeara. \"Please, baby.\" He reached for her again.\n\nShe stepped back. \"Don't touch me.\"\n\nArik couldn't breathe as he saw the tears in her eyes. The betrayal. She was hurting, he knew it. He could feel it like his own pain. It cut through him, lacerating his heart. \"I would never hurt you. You have to believe that.\"\n\n\"Great words for a man who'd planned my death from the beginning, huh?\"\n\nShe was right. How could he ever convince her that he'd changed? He was Skoti and Skoti were nothing.\n\nHe turned on his brother, hating him for telling her. \"Damn you, M'Adoc.\"\n\n\"There's nothing to damn me over, Arik. I'm not the one in the wrong here. You are. You should never have made that bargain.\"\n\nHe wanted to kill M'Adoc for this. But he was right. It'd been wrong of Arik to come here. He should have been content to stay with her in her dreams.\n\nM'Adoc spoke calmly. \"Skoti are forever selfish, Megeara. It's why we must police them. They become mad with their hunger to the point they don't care who they harm or how they harm them so long as they get what they want. Arik wanted you and so he was willing to kill you for it.\" He met Arik's gaze. \"If you truly mean what you say now, then for once, do the decent thing. Surrender yourself to me.\"\n\nEvery instinct inside Arik rebelled at the very idea of it. Every one. They would kill him and he knew it.\n\nBut then, he would die anyway. Hades would never allow him to renege on their bargain. No one backed out of a deal with the devil.\n\nAnd perhaps this was the kindest end. Megeara hated him now. She thought him the lowest of forms. If he left with M'Adoc, she wouldn't grieve for him or ask herself if maybe there was something she could have done to save him.\n\nShe would be at peace.\n\nThat would be the best gift he could give her.\n\n\"He's right, Megeara,\" Arik said, forcing all emotions out of his tone. \"I would have killed you once I returned to my natural state. I'm sorry.\"\n\nGeary couldn't breathe as he spoke those words. Part of her had still believed in him over M'Adoc. She didn't want to think that her Arik could ever harm her.\n\nBut if what they said was true...\n\nIt stung her so deep in her heart that she felt as if she'd die from the pain.\n\nArik turned to M'Adoc and whispered something to him. She couldn't hear it, but M'Adoc inclined his head before Arik sighed. \"Then I'm ready to go.\"\n\nHis crystal gaze met hers and the love she saw there singed her to the spot. \"Good-bye, Megeara.\"\n\nShe noted the satisfaction in M'Adoc's eyes.\n\nSatisfaction...\n\n\"Even in human form we feel nothing.\" M'Adoc should have no emotions at all. None.\n\nBut he was satisfied.\n\nRealizing that if M'Adoc could feel, he could lie, she opened her mouth to stop them. But before she could make a single sound, M'Adoc placed his hand on Arik's shoulder and the two of them vanished from the boat.\n\n\"No!\" Geary shouted, her heart pounding as reality crashed through her.\n\nArik was gone.\n\nHe was going to kill you, her mind tried to rationalize. But the scariest bit was that part of her didn't even care. It wanted him back no matter what.\n\nShe felt the boat slowing down.\n\nKat came out on deck and approached her slowly. \"Where's Arik?\"\n\nUnable to even begin to explain Arik's disappearance, Geary burst out laughing hysterically until she started crying and gesturing hopelessly toward the prow. She honestly felt like she was having a nervous breakdown. How could she tell Kat everything? The woman would think her crazy and who could blame her?\n\nGods in the real world? None of this would be believed. Ever.\n\nKat scowled. \"Hon, are you okay?\"\n\n\"No,\" she said, trying to gain a handle on herself. \"No, I'm not.\"\n\nKat cocked her head in a gesture that reminded Geary of Solin whenever he was \"listening\" for something preternatural.\n\n\"What are you doing?\" Geary asked.\n\nKat let out an uncharacteristic curse. \"M'Adoc was here? How could you let him take Arik?\"\n\nThat snapped Geary out of her hysteria.\n\nSurely, not...\n\n\"If you tell me that you're one of them, then I'm absolutely going to freak out.\"\n\nKat's face was deadly earnest. \"Then you better freak.\"\n\n\"Good God, is no one what they seem? Is Kichka the Egyptian goddess Bast in disguise?\"\n\n\"No, Kichka is a cat.\"\n\nUh-huh. She was supposed to believe that after the bomb Kat had just dropped on her? \"And let me guess. You're a god, too, right? Which one? Athena? Hera? Oh, what the heck? Aphrodite?\"\n\nKat gave her a droll stare. \"No. I'm not a god. I'm a servant to Artemis.\"\n\n\"Artemis?\" Yeah. That just sounded so much better... not. \"The goddess of the hunt, huh?\" Geary looked down at the wood deck. \"There must be some kind of vapors coming up off the floor of the sea\u2014like the oracle of Delphi. That's why I'm seeing and hearing all this insanity.\" She nodded, liking the sound of that argument. \"I'm hallucinating, aren't I?\"\n\n\"Oh, get a grip,\" Kat said irritably. \"If you can accept Solin, Arik, and M'Adoc, you can certainly accept me, too.\"\n\n\"One would think that, wouldn't they? But I've known you too long to think that all this time you've been hiding a secret like this from me.\"\n\n\"And now you know why I wasn't thrilled when you found Atlantis and wanted to start digging around the city.\"\n\nWell, since she put it that way, it did make sense. \"And were you in on my death plan with Arik, too? Are you the one who killed my father?\"\n\nKat's eyes blazed in anger. \"Excuse me? You don't need to be tossing around ludicrous accusations like that. I had nothing to do with your father's death. I loved that man. He was weird and strange, but I loved him and I would have done anything to keep him safe. While you were off in America, I was here with him, doing the best I could to help him and keep him alive even though he was bent on killing himself.\"\n\nTears filled Geary's eyes at the truth. \"I'm sorry, Kat. I'm just upset and I don't mean to be taking it out on you.\"\n\nKat nodded. \"No offense, but you should be sorry. How could you let M'Adoc take Arik?\"\n\n\"Arik was going to kill me.\"\n\n\"Hardly.\"\n\n\"It's true,\" Geary said past the lump in her throat. \"Arik admitted it.\"\n\nStill, Kat scoffed. \"Arik loves you, Geary. It's so obvious it's painful. No man, god or otherwise, watches over a woman the way he does you and then lets her die, never mind kills her. That's just stupid.\"\n\n\"Yeah, he loves me now, but when he loses his emotions next week, what then? M'Adoc said that he'd kill me without question. He said that Arik would have no emotions or choice but to do as they said.\" There, it sounded rational. Kind of.\n\n\"Arik?\" Kat asked incredulously. \"Do what he was told? Please. He hasn't followed the directions of any god in thousands of years. That's why he's Skoti.\"\n\nAll of Geary's fears that M'Adoc was lying came rushing back. \"What are you saying, Kat?\"\n\n\"No offense to you, kid, but what I'm saying is that you just ordered a man who loves you to his death.\"\nCHAPTER 18\n\nWith his hands cuffed behind his back, Arik didn't flinch or struggle as M'Adoc hauled him into the sanctity of the triumvirate's hall. He'd never been inside this place before, not even in dreams. It was the sacred domain of the triumvirate who zealously guarded it from all the rest of their kin.\n\nNo one knew why, but Arik had to give the guys credit. It was an opulent palace made of glass and gold that they'd constructed here. It was fit for a dream god and even Zeus would be at home... The council room where they were was decadently comfortable with padded chairs and even a laptop computer that was so out of place as to be funny... if Arik wasn't about to die.\n\nM'Ordant was seated before it, and as they came in he looked up with an unguarded expression that showed both confusion and shock\u2014two emotions he shouldn't have. \"Damn, M'Adoc, how did you manage this one?\"\n\nM'Adoc shoved Arik against the solid glass table, the corner of which bit into his hip bone and bruised him. He had to grind his teeth to keep from lashing out at M'Adoc. But Arik had given his word, and so long as the god kept his and didn't harm Megeara, Arik would submit, even though it went against every part of his genetic makeup to do so.\n\nM'Adoc shrugged as he moved to stand beside Arik. \"He surrendered willingly. In exchange for our keeping his human safe.\"\n\nThere was no missing the stunned look on M'Ordant's face.\n\n\"Was there no fight?\"\n\nArik turned his head slightly as he heard D'Alerian's deep voice from behind him. He couldn't see him, but he could feel D'Alerian's presence. Of the three of them, his had an unmistakable aura. He wasn't the most powerful, but his presence could be felt all the way to the marrow of someone's bones.\n\n\"He knew better than to fight me,\" M'Adoc said in a sinister tone.\n\n\"Get over yourself, M'Adoc,\" Arik snarled. \"You had nothing to do with this. There was no need in my staying in the human realm any longer since all you did was hurt Megeara by telling her of my bargain with Hades.\" She would never forgive him, and that hurt even more than a thousand beatings. Strange how originally the thought of her death had meant nothing to him. Now he grieved over every tear and doubly over the ones he'd caused her. \"Just take me to Hades and end it.\"\n\nM'Adoc took him by the arm. His lips were twisted as he raked Arik with a sneer. \"Oh no, Arikos. I don't think so. See, I hand you over to Hades and he's going to start questioning how it is you have so many emotions that you're willing to surrender yourself to save a simple human's soul.\"\n\n\"Because Hades made him human,\" D'Alerian answered in a dry, stoic tone. \"There won't be any questions over it. It would only stand to reason.\"\n\nM'Adoc turned on him with a hiss. \"Are you willing to take that chance?\"\n\nD'Alerian's jaw flexed. \"There's no chance involved, Adarian. He is human, by Hades' command, and he acted as a human. The god would expect no less.\"\n\nArik frowned as D'Alerian used M'Adoc's real name, Adarian. As part of their punishment, and to push them away from the idea that they were individuals of any merit, many of the original Oneroi had been stripped of their names and given new ones to designate their roles. D' meant that D'Alerian was normally assigned to watch over immortals such as the Dark-Hunters. V' designated a human helper\u2014as an Oneroi, Arik's name had been V'Arik or V'Arikos, which he now hated, since it sounded like a vein condition. And the M' was reserved for those who policed them all. There were many who called D'Alerian M'Alerian. But for reasons no one understood, D'Alerian continued to use the name they'd given him before he'd risen to the ruling ranks.\n\nM'Ordant closed his laptop as he faced them. \"He's right. We should hand him over to Hades. We don't want to cross the god of the dead. He's a nasty bugger.\"\n\nM'Adoc scoffed at them. \"And when Hades kills Arik and his immortal soul is stripped bare while Hades tortures him in Tartarus, don't you think King Badass is going to discover the fact that little Arikos can feel something other than pain without having a human host to sponsor those emotions?\"\n\nA rivulet of shock went through him. What was M'Adoc saying? Arik froze as he began to suspect that the emotions he'd thought were residuals left from Megeara's might have been his own after all. \"What's going on?\"\n\n\"Shut up, Arik,\" M'Ordant snapped angrily.\n\nM'Adoc glared at his brothers. \"We can't take a chance of them learning the truth. Ever.\" His gaze bored into D'Alerian. \"Of everyone in this room, Neco, you have the most to lose. Don't let your compassion for him stop you from doing what needs to be done.\"\n\nPain flickered across D'Alerian's face before he gave a subtle nod.\n\nThere wouldn't be any mercy given to Arik, not that he had expected any. Honestly, his welfare didn't matter. \"I don't care what happens to me,\" Arik said to M'Adoc. \"Just remember you promised to take care of Megeara.\"\n\nOne corner of M'Adoc's lips twisted up into a mocking smile. \"Oh, don't worry. I fully intend to take care of her. Immediately.\"\n\nD'Alerian scowled. \"I don't like that tone, adelphos.\"\n\nM'Adoc cast a belittling smirk at him. \"No one gives a damn what you like, Neco. She's a liability to us. She knows the location of Atlantis and she knows we exist. Would you have me leave a threat like that out there?\"\n\nHe was going back to kill her. Arik knew it with every ounce of his being.\n\n\"You swore to me, you lying bastard.\" Arik turned on M'Adoc, intending to fight, but as soon as he neared him, he felt something hot and solid pierce his stomach. Pain tore through him.\n\nArik staggered back and looked down to see a long, bloodied dagger in M'Adoc's hand. He couldn't believe it as his knees weakened from the agony of his wound.\n\nM'Adoc moved toward Arik with a merciless glint in his eyes. He buried a fist in his hair as his empty, cold gaze burned into Arik's. \"Sweet dreams, Arik,\" M'Adoc said an instant before he stabbed him again and everything went dark.\n\n* * *\n\nGeary was numb as they returned to the docks. Over and over she kept going through everything with Arik. But deep inside, she knew Kat was right. Arik had loved her. In spite of everything or maybe because of everything, they'd fallen in love with each other, and she'd just thrown him to the wolves.\n\nShe should have trusted in him. Arik wouldn't hurt her, she knew that. He might have had bad intentions in the beginning, but he wasn't like that now. Why hadn't she given him the benefit of the doubt?\n\n\"What am I going to do, Kat?\" she asked as they tied the lines.\n\nKat sighed. \"There's nothing to do. He's gone.\"\n\nGeary straightened up to stare at the taller woman. \"I can't accept that. I can't.\"\n\nBut Kat was immune to her pleading look. \"You're going to have to.\"\n\n\"Why?\" Geary asked.\n\n\"Because sometimes life just basically sucks, and this is one of those times.\"\n\n\"And if I don't want it to?\"\n\nKat shook her head. \"When has it ever listened to you?\"\n\nShe did have a point. But it didn't stop the pain inside Geary from hurting. How could she have let M'Adoc take Arik? She should have fought. Should have told him she loved him.\n\nInstead, she'd just stood there as he was taken and done absolutely nothing.\n\nDamn me, I'm such a fool. She'd waited her entire lifetime for love, and then when she'd finally found it she'd cast it aside in one moment of hurt anger. How could she have been so stupid?\n\n\"This can't be how it ends.\"\n\nKat's features softened as she neared her. \"Geary, look. Arik sacrificed himself to keep you safe. Don't ruin it by putting yourself into danger and dying. Let him go.\"\n\nShe stared at Kat. \"If someone you loved were suffering because of you, could you let him go?\"\n\nKat screwed her face up as if she was in pain. \"This isn't about me,\" she said in an anguished tone that answered Geary. \"Oh, okay, so I wouldn't stand by and let the man I love suffer when I was the one who caused it.... Damn.\"\n\n\"Yeah. Damn. We have to find some way to help him.\"\n\nKat raked her hand through her hair as if irritated beyond her tolerance. \"I don't even know how to begin to fix this.\"\n\n\"I do.\"\n\nGeary put her hand to her temple as she heard Apollymi's voice in her head. \"Not now, please.\"\n\n\"Don't dismiss her,\" Kat said out loud. \"Apollymi's probably our best hope right now.\"\n\n\"You know about...\" Of course she did. \"You hear her, too?\"\n\n\"All the time. She has a bad habit of nosing her way into pretty much everything I do. She's terribly nosy that way, but she's always a friend to me.\" She smiled before she addressed Apollymi. \"Mibreiara, have you any suggestions that don't involve one of us letting you out?\"\n\n\"That is the suggestion I prefer.\"\n\n\"Yes, but neither Geary nor I will do that. You got anything else?\"\n\n\"Yes, but it's tricky. Listen to me, my girls. You're about to have an important lesson in men and god politics.\"\n\n* * *\n\n\"Solin?\"\n\nSolin cursed as he heard Arik's voice in his head. \"I have nothing to say to you.\"\n\n\"Fine. I don't want to hear it anyway. What I need is for you to listen.\"\n\n\"Listen, my ass.\"\n\n\"I need your ears, Solin,\" he said wryly, \"not your ass.\"\n\n\"Go to hell.\"\n\n\"I'm already there.\"\n\nSolin paused as he felt something odd brush against his collar. It was the touch of the dead and he knew it, even though it'd been centuries since he'd last felt one. \"What?\"\n\nA Shade of Arik appeared before Solin. His features were ghastly white, his eyes dark and pain-filled. He wore nothing but a pair of tattered pants. \"M'Adoc killed me.\"\n\nSolin couldn't have been more shocked had he been the one who'd died. \"How?\"\n\n\"I surrendered myself to him to protect Megeara. Now he's reneged on our bargain and is heading for her. I need you to protect her from him.\"\n\nOf course he did, and Solin was through being the dupe in this. Why should he put his life on the line for anyone? Who would he go to for help once he was a Shade? No one. \"Do you think I even care?\"\n\n\"I know you do, Solin. In spite of your protestations, I can see the real man inside that you try so desperately to ignore and hide.\" He paused before he spoke again. \"Please, Brother. She's not a fighter and he won't stop until she's dead. Don't let an innocent die over nothing.\"\n\nStill Solin didn't want to get involved in this. He'd made a similar mistake before and paid dearly for it. \"Do I look like an Oneroi to you? I'm not here to protect humans. Why don't you go warn her yourself?\"\n\n\"She won't speak to me or listen. M'Adoc told her of my bargain with Hades. She hates me now.\"\n\nSolin didn't miss the tremor of pain in Arik's voice. Nor the look of absolute misery on his face. The fact that she was hurting tore Arik apart. \"You love her?\"\n\n\"Obviously more than my life,\" he said, his voice trembling from his emotions.\n\nSolin narrowed his eyes on Arik. \"It hurts, doesn't it? To have that one person you love learn the truth of what you are and hate you for it?\"\n\n\"You have no idea.\"\n\n\"Yeah, I do.\" And instead of feeling the satisfaction he'd anticipated when Arik tasted his own misery, Solin felt nothing but more pain. There was no joy in hurting someone else. At least not for him. \"Where are you?\"\n\n\"I'm on the banks of Styx. M'Adoc won't allow Charon to ferry me over for fear of Hades finding me and learning the truth. I'm sure that when Hades learns I'm dead, he'll go for Megeara to fulfill my bargain, and that I can't allow. She's innocent in this and shouldn't have to pay for my stupidity. I have nothing to give you, Solin, but please, if there's any decency left in you, don't let her die because of me. I'm begging you.\"\n\nSolin knew that kind of love. He'd tasted it once and it'd been burning on his tongue like a bitter pill for countless centuries. \"Just so you know, I've never had a drop of decency in me.\" Arik literally deflated before Solin's eyes. \"But I won't let them hurt her. Rest easy.\"\n\nEven as he spoke, he knew that would never happen. Hades wouldn't allow Arik peace once he learned where the man was, and from the looks of it, neither would M'Adoc. And for the first time in centuries, Solin honestly felt sorry for someone other than himself.\n\n\"You can trust me, Arik.\"\n\n\"Thank you.\" He inclined his head to Solin before he faded out.\n\nTaking a deep breath, Solin leaned back in his chair. His motto had always been to help no one because no one had ever helped him. He hated people.\n\nHe hated the gods most of all.\n\nAnd he had no business getting involved in this. But how could he stand by and not do something? Megeara needed protection, and unlike him, she didn't have the powers to fight against them and win. They would tear through her in no time.\n\nIf he were smart, he'd contact ZT and let the Chthonian handle it.\n\n\"Nah,\" he said with a bitter smile. \"I'd rather be vindictive than smart.\" And with that he flashed himself out of the safety of his home to search for a human.\n\nIt didn't take much to find Megeara. Her aura stood out even to the waking Dream-Hunter, especially since she was in such emotional turmoil.\n\nBut what gave him pause was the air of hopeless sadness that engulfed her. It'd been a long time since he'd seen its equal. \"You okay?\"\n\nShe jerked around at the sound of his voice to glare at him. \"What are you doing here?\"\n\n\"I have no idea, but I think it's to help.\"\n\nShe scoffed at him as she pulled down a book from her bookshelf. \"That boat already sailed. You told us we were on our own.\"\n\n\"Yeah. But the amazing thing about boats is that they occasionally turn around and come back.\"\n\n\"Or they blow up while hosting your air supply,\" Kat added meaningfully.\n\nHe turned his head to see her entering the room from his left. \"True, but not this one. Arik asked me to guard Megeara from the others.\"\n\nMegeara gave him a suspicious look. \"Why would he do that?\"\n\n\"Because he's not able to.\"\n\nStill suspicion was heavy in her eyes. She didn't trust Solin, and honestly he couldn't blame her for it. \"And why would you do that when you've already made your position clear?\"\n\nHe shrugged. \"I'm basically doing it to piss off the powers-that-be.\"\n\n\"And?\" Kat prompted.\n\n\"And what?\"\n\n\"I don't know, there just seemed to be an and attached to the way you said that.\"\n\nAnd... for some reason he didn't want to think about, he'd come to like and respect Arik. But that he'd never admit to. \"There's no and.\"\n\n\"All right then,\" Kat said, clapping her hands together. \"We're trying to save Arik. You said he came to you. Where is he?\"\n\nSolin hesitated. He'd assumed they knew already, but apparently the women had no clue what'd happened to Arik. \"He came to me as a Shade, Kat. M'Adoc killed him.\"\n\nGeary dropped the book in her hand at the unexpected news. If Kat hadn't reached out to steady her, she would have probably fallen.\n\nArik was dead.\n\nIt couldn't be\u2014and yet she could tell by the look on Solin's face that he wasn't joking.\n\n\"I can't breathe,\" she whispered as her tears gathered in her throat to choke her. \"He can't be gone.\"\n\n\"Sh,\" Kat said as she pulled Geary against her. \"It's okay, Geary.\"\n\nBut it wasn't okay. Arik was dead and it was her fault. She hadn't even fought for him. M'Adoc had come and she'd all but shoved Arik into the hands of the man who'd killed him.\n\nShe sobbed as her heart shattered. How could she have done such a thing, even in anger?\n\n\"Uh, ladies. I hate to say it, but I'm not through being a harbinger here. Arik came to me because M'Adoc is hell-bent on cleaning this situation up.\"\n\nKat stepped away from her. \"Cleaning it up how?\"\n\nHe looked meaningfully at Geary. \"Humans aren't supposed to know we exist.\"\n\nThat shocked Geary's tears away as a chill went down her spine. \"He's coming to kill me, too.\"\n\n\"Yes.\"\n\nAnger consumed her as she wiped away her tears. \"What about Tory and Thia?\"\n\n\"They don't know anything, so they're safe. But you, my dearest, are another matter entirely.\"\n\nWell, she could handle that. Her life was one thing, theirs was another. So long as they were safe, she could deal with whatever came her way.\n\nShe bent to retrieve her book of mythology from the floor. \"I can't believe this.\"\n\nSolin nodded. \"It's really quite pathetic, isn't it? Arik gives himself up because M'Adoc swears he won't hurt you, and then the lying bastard decides you need to die anyway.\"\n\nGeary froze at his words as dread consumed her. \"Arik did what?\"\n\nSolin looked ill. \"Oh, don't tell me you didn't know that, either?\"\n\n\"No,\" Kat said, stressing the word. \"She didn't.\"\n\nSolin wiped his hand over his face. \"Okay, I'm just going to stand here and be quiet.\"\n\n\"It's too late, Solin,\" Kat said from between clenched teeth. \"You've already done your damage.\"\n\n\"Wait,\" Geary said as her mind whirled with thoughts. She looked down at the leather-bound book in her hands. \"We can save Arik.\"\n\nThey exchanged a frown before Kat shook her head. \"I don't see how.\"\n\n\"Oh c'mon, you're both in the pantheon. Shades have been brought back before.\" She held the book out to them. \"Look at Orpheus and Eurydice. Hades allowed her to leave.\"\n\nSolin snorted. \"That's one example out of thousands Hades has denied, and laughed at while he did so.\"\n\nGeary glared at him. \"I thought you were going to be quiet.\"\n\n\"Sorry.\"\n\n\"As much as I hate to admit it, he's right,\" Kat said with a sigh. \"Not to mention Eurydice never made it out. Orpheus looked back before he reached the surface, and she was snatched right back into the Underworld. Hades is a selfish bastard that way. He never willingly lets go of a soul.\"\n\nApollymi cleared her throat in Geary's mind. \"Were you not listening to me earlier? Why do I even bother? Just call me Circe or Cassandra for all the attention I'm paid. Why do I have to resort to them anyway? Ferandia would be a better example, but since she's Atlantean no one knows that story, do they? No. So I have to resort to those insipid Greek tales, half of which were stolen from us. But that's another matter. Point is, no one ever listens to a trapped goddess....\"\n\nLaughing in spite of everything, Geary realized Apollymi was right, and she was going to take the goddess's earlier advice. She looked up at Solin. \"Oh, Mr. Dream Master, where's Persephone?\"\n\n\"Good girl.\"\n\nSolin narrowed his eyes. \"You're not thinking what I think you're thinking, are you?\"\n\nKat grinned. \"You want to move the mountain, you give it something it can't resist. Geary and Apollymi are right. Hades won't even look at us. But he listens to his wife. We need her.\"\n\nSolin was still shaking his head. \"And if she won't help?\"\n\n\"I'm not going to think about that,\" Geary said sternly. \"I can't afford to.\"\n\nSolin looked reluctant, but in the end, he agreed with them. \"All right then, let's go.\"\n\n\"None of you are going anywhere.\"\n\nGeary froze as M'Adoc appeared in the doorway before them.\n\nAnd he wasn't alone.\nCHAPTER 19\n\nGeary took a step back at the sight of M'Adoc and the Dolophoni... at least that's who she assumed they were. There were three of them, but she'd never seen them before. Unlike the first group, these were all women. Dressed in black leather, with black hair, nails, and lipstick, they looked mean and nasty. They were also fanged, with eyes so dark, she couldn't even see where the pupils ended and the irises began. All they needed was snakes in their hair to be even...\n\nOh wait, one of them had that. Black snakes slithered out of her ponytail to curl around her neck and hiss at them. Lovely. Just lovely.\n\nSolin moved to stand in front of Geary and Kat. \"This is over, M'Adoc.\"\n\n\"No, it isn't. Not until she's\"\u2014he indicated Geary with a jerk of his chin\u2014\"dead. Now either you and Katra can hand her over and leave or you two can bleed.\"\n\nSolin let out an exaggerated breath. \"Looks like I'm bleeding then. Unlike some people I know\"\u2014he duplicated M'Adoc's action, indicating him with his chin\u2014\"I keep my word.\"\n\nM'Adoc narrowed his eyes before he turned his head to address the women over his shoulder. \"Kill them.\"\n\nGeary tensed for the coming fight. Before she could even blink, Kat turned to her and grabbed her. Kat whispered something in Atlantean and then she kissed Geary fast on the lips.\n\nStunned beyond belief, Geary closed her eyes as she felt something hot and powerful spreading through her body, and Kat stepped away to confront the others. For a full ten seconds Geary couldn't move as indescribable power filled her. It was comparable to when she'd held Apollymi's medallion in her mouth, only this felt stronger, deadlier. And this time she was definitely in charge and not someone else. The power was incredible. It was as if her brain was alive and growing.\n\nAnd when she opened her eyes, she no longer saw in the same colors. Everything was more vibrant now. More vivid.\n\nThe snake-haired woman seized her. Without thought, Geary dodged the punch and returned the blow with one so fierce, it knocked the woman off her feet and sent her flying. Literally. She cleared the ground by a good five feet before she slammed into the wall and fell to the floor. The snakes hissed and strained in anger.\n\nKat dispelled her attacker every bit as easily. But Solin appeared to be reticent to strike his. However, when the woman backhanded him and laid open his cheek, he changed his mind. Head-butting her, he knocked her to the ground, then turned to face M'Adoc.\n\nThe three women came to their feet to renew the fight. They took a step forward in unison.\n\n\"Enough!\"\n\nGeary was expecting it to be Zebulon, but it wasn't. Instead, there was another Dream-Hunter who was leaner than M'Adoc and Solin. He appeared between them and held his hand up for the women who strangely obeyed him. His ebony hair was long and braided down his back. He was dressed all in black and held a look that said \"I'm in the mood to kill anyone who pisses me off.\" But more than that, there was an aura of power around him so strong that it actually made the hair on the back of her neck stand up.\n\n\"Whose side are you on, D'Alerian?\" Solin asked as he wiped the blood away from his face with the back of his hand.\n\n\"Ours,\" another man answered as he appeared beside D'Alerian. Equal in height to D'Alerian, he wore his black hair cropped short and had on a pair of jeans and a button-down shirt. His eyes seemed to be a blue so pale, they looked colorless. Those eyes were eerie and deadly as they settled on Geary with purpose.\n\nM'Adoc smiled in approval. \"So you finally see my way of things.\"\n\nIt was D'Alerian who answered. \"No. We can't let you kill the human. It's wrong and this isn't who and what we are. We protect, we don't assassinate.\"\n\nKat and Solin exchanged a puzzled frown.\n\n\"What's going on here, M'Ordant?\" Solin asked the newcomer.\n\n\"We're taking M'Adoc into custody.\"\n\nM'Adoc cursed. \"Are you insane? You can't do this.\"\n\nD'Alerian turned to face him. \"Yes, we can, and yes, we are.\"\n\nThe women looked confused but didn't interfere as D'Alerian covered M'Adoc with some sort of shimmery web. M'Adoc tried to fight, but it held him tight and constricted more with every movement. Finally, it was so tight that all he could do was curse them.\n\n\"It's called a diktyon,\" Kat answered Geary's unasked question. \"It's something Artemis uses to capture animals and not hurt them. Although how they\"\u2014she indicated the Dream-Hunters\u2014\"ended up with one of them, I don't know.\"\n\nD'Alerian looked at the women. \"Your services are no longer required. Furies, return.\"\n\nThey vanished instantly while M'Adoc continued to curse the others. \"Do you understand what you're doing? What the gods will do to us?\"\n\nD'Alerian's eyes were sad and dark. \"Sometimes our worst enemies are ourselves, M'Adoc. You are becoming the very thing they fear us to be, and that we can't allow.\" He met Geary's gaze. \"You understand that you can never speak a word of us to anyone?\"\n\nLike this was something she'd be bantering about? Yeah, right. \"Who would ever believe me?\" she asked seriously.\n\nD'Alerian nodded in approval. He took a small ring from his pinkie and placed it in Geary's palm. \"I know what you plan and I wish you luck with it. Give this to Persephone and tell her that Neco supports you, that you're calling in the favor she owes him.\"\n\nGeary was baffled by his actions and his words. \"What?\"\n\nHe closed her hand over the ring. \"Don't question it, Megeara. Just do it.\"\n\nGrateful and amused by his commanding tone, she couldn't help teasing him. \"You're a bossy bunch, aren't you?\"\n\nOne corner of M'Ordant's lips twisted. \"You've no idea.\" An instant later, he vanished with M'Adoc in tow.\n\nD'Alerian offered her a kind smile before he released her.\n\n\"What's really going on?\" Solin asked D'Alerian as he turned to leave.\n\nThe humor fled from his face and he returned to being stoic again as he addressed Solin over his shoulder. \"Nothing that concerns you, Skotos. Just know that we'll keep M'Adoc away from you.\"\n\nShe saw the suspicion on Solin's face. \"Given our history, why would you do that?\"\n\nThere was true regret in D'Alerian's eyes and in his stance as he turned to face Solin. The sincerity on D'Alerian's face was heart-wrenching. \"I was wrong for what I did to you, Solin. I'm sorry.\"\n\nSolin scoffed in derision. \"Rote words to you, Oneroi.\"\n\n\"No, it's heartfelt, I assure you.\" He hesitated as if considering the repercussions before he spoke again. \"Things change, Solin, and so do people. Even gods.\"\n\nSolin froze as he finally understood what D'Alerian was telling him. \"After all this time, you trust me?\"\n\nD'Alerian nodded. \"Arik did, and you proved yourself by protecting Megeara even when it could have cost you your life. You had nothing to gain and everything to lose. I think that makes you trustworthy.\" Then he did the most unexpected thing of all. He offered his hand to Solin. \"Brothers.\"\n\n\"Brothers,\" Solin said, taking D'Alerian's hand and shaking it. \"Thank you.\"\n\nHe inclined his head to them before he vanished.\n\nKat cocked her head as she frowned over their exchange. \"What did I just miss?\"\n\n\"Nothing,\" Solin said quickly. \"It's just Oneroi weirdness.\" He let out a deep breath as he faced them. He smiled at Geary. \"Last I heard, Persephone was on Olympus with her mother. I can't go there, but Kat can and she can take you with her.\"\n\nGeary didn't understand that. He was a god and should be as welcomed on Olympus as any other god. \"Why can't you go there?\"\n\n\"Solin is under a death warrant,\" Kat explained. \"There are too many gods who would kill him on sight if he were ever dumb enough to pop into their backyard.\"\n\n\"Oh,\" Geary said in understanding. How awful for him. No wonder he'd been so angry over them in the past. It sounded like he was fully entitled to it.\n\nGeary stepped forward and kissed him on his cheek. \"Thank you for your help, Solin. I really appreciate it and I'm sure Arik does, too.\"\n\nSolin nodded. \"Just make me one promise.\"\n\n\"And that is?\"\n\nHis gaze burned into hers. \"If you get Arik back, you won't be stupid again. Love is a rare thing, Megeara. Hold on to it with both hands.\"\n\nTears gathered in her eyes as she realized he was speaking from his heart and from a past he had no intention of sharing with her. But his words were too genuine to be an arbitrary recitation. \"I fully intend to.\"\n\n\"Then it's worth it.\"\n\nThat confused her. \"What is?\"\n\nSolin chucked her gently on the chin. \"Whatever they throw at me.\" He inclined his head to Kat. \"You two kids have fun, and good luck.\" Then he vanished, too.\n\nGeary frowned at Kat. \"Is it just me or was that the oddest exchange?\"\n\n\"No,\" she breathed, \"it wasn't. You're just missing part of the story. Remember, at one time the Oneroi had feelings. Some of them were in love and even had families at the time they were rounded up and punished.\"\n\nA bad feeling went through Geary as she noted the ominous tone of Kat's voice. \"What happened to their families?\"\n\n\"Let's just say Zeus was really angry.\"\n\nIt didn't take a genius to ascertain what that would have led to. \"He killed them.\"\n\nShe nodded grimly.\n\nEven though Geary had guessed it, she was still aghast that he'd be so callous to his own family members. \"All of them?\"\n\nKat nodded again.\n\nTears gathered in Geary's eyes as the magnitude of their punishment truly hit her. She couldn't imagine the horror that they must have been put through. No wonder M'Adoc was psycho. \"Did Solin\u2014\"\n\n\"No,\" Kat said quickly, cutting her off. \"He was the family they tried to destroy and he survived by the skin of his nose.\"\n\n\"Arik told me that, too. God, I feel so bad for them.\"\n\n\"All of us with any decency do, but there's nothing to be done for them as a group. Not unless you go up against Zeus, and that takes more god power than what we have.\" She offered Geary a smile. \"But right now we have one god in particular we might be able to help.\"\n\nShe was right. Before they plotted a war, they needed to win a battle, and Geary was ready to take on all of Olympus for Arik. \"Let's get on it.\"\n\n* * *\n\nGeary didn't really know what to expect from Olympus and the gods. True, she'd spent the whole of her life listening to her father and grandfather tell stories about them. But that had only been speculation on everyone's part.\n\nNow she was actually here.\n\nAnd it was scary and invigorating to know that those legends were real. That things she'd taken for granted as fiction weren't. Wow.\n\nJust like the stories told, Olympus was breathtaking. The weather was perfect. Not too hot. Not too cold. It was like a mid-spring day. The sky was so blue as to be unreal in its celestial hue, and the mountains around her were lush and green. The air was fresh and laced with a sweet scent. She'd never experienced anything like this.\n\nDreamy was the only word for it.\n\nBut what fascinated her most was that she could look down and see the world below in all its glory through the misty clouds that kept the gods isolated from the world.\n\n\"This is incredible.\"\n\nKat smiled. \"Yeah, I know.\" She looked around with pride. \"I was raised here.\"\n\nGeary couldn't imagine having a childhood like that. \"Really?\"\n\n\"Yes.\" Kat pointed to a large circular gold building at the end of the golden cobblestoned street. \"That is Artemis's temple. When I was a little girl, I used to sneak out of it and run to there\"\u2014she pointed to another temple at the opposite end\u2014\"to Athena's temple, and play with her owls.\" She laughed at the memory. \"It used to make Artemis crazy.\"\n\n\"Why?\"\n\n\"They have a long-standing feud with each other over some nonsense that happened aeons ago. And Artemis wanted me to stay as far away from Athena as I could.\"\n\n\"But you couldn't resist, huh?\"\n\nHer smile widened. \"Not really. According to Artemis, I've made a lifetime study of irritating her.\"\n\nGeary laughed as she looked around and saw three fawns sprinting across the walk ahead of them. The deer vanished into the woods, where she swore she glimpsed a centaur running. \"I can't believe this is real.\"\n\n\"Oh, believe it. Scary as it can be.\" Kat pointed to a huge hall made of gold and ivory that was on a hilltop above them. \"And that is where Zeus resides with Hera. It's the great hall where everyone gathers to mostly bicker and complain.\"\n\nIt was as magnificent as the rest of the area. Truly, it was like walking in a dream. \"Is that where we're going?\"\n\n\"No. Persephone doesn't play politics. Even though Zeus is her father, she only goes there when she's summoned, especially since Hera can't stand her because she's one of Zeus's bastards. She tries her best to stay out of their business.\" Kat pointed to another of the buildings down the street. \"She'll most likely be hanging out at her mother's temple.\"\n\nGeary followed Kat as she crossed the street and they were almost run over by a blur.\n\n\"Hermes!\" Kat shouted. \"Watch where you're going!\"\n\n\"No time...\" A faint voice drifted back to them as he vanished out of sight. It strangely reminded Geary of the Road Runner cartoon, as nothing but dust was left in his wake.\n\n\"That happen a lot?\" she asked Kat.\n\n\"Yeah, he's always in a hurry. You have to be really careful or he'll run you down. It's like being hit by a Mack truck, too. Bloody bastard.\"\n\nOkay... no hostility there.\n\nBut luckily no one else tried to mow them down as they walked the short distance to the smaller domed building. Geary paused outside to look up. It was only about half as tall as everything around it and not very large. Though it was still beautiful and bigger than any home Geary's family had ever owned, it lacked much of the awe factor that the rest of the buildings around it had. \"Why is this so small compared to the others?\"\n\nKat shrugged as she looked up at it. \"Demeter isn't one for pretense. She's very simple compared to the others.\"\n\nMoving forward, Kat opened the door to a massive foyer that was made of marble so white, it hurt her eyes to look at it. The entire room was surrounded by columns that were carved to look like people. And as Kat and Geary entered, one of the male statues on her right opened its eyes to stare at them.\n\n\"What brings you here, Katra?\" the statue asked in ancient Greek.\n\nKat was completely unfazed by the fact that a living statue was addressing her, while Geary gaped at it. \"I want a word with Persephone.\"\n\n\"She is in the garden,\" a female statue answered before she pointed to the opposite set of doors. \"But she is not in a happy mood, so be warned.\"\n\n\"Thanks, Chloe.\"\n\nBefuddled, Geary trailed along through the doors that opened of their own volition into a massive garden atrium. The wind was gentle as it stirred the scent of hyacinth and lilac. \"Ooo, nice.\"\n\nAt least that was what she thought until they heard someone cursing. Repeatedly and with relish.\n\n\"Gardening is crap, Mom,\" the light, cadent voice whined from the bushes in front of them. \"I hate it! Look at that. My manicure's going to be completely wrecked and for what? Dig the earth, plant some crap, do this, do that. Blah!\"\n\n\"Seph?\"\n\nThe bushes rustled before a small woman wiggled out from between them. Geary hid her smile as a very petite and extremely beautiful blond woman stood up. Dressed in dark green overalls and a white T-shirt, she had dirt smeared across one cheek and on the tip of her nose. Her hands were covered by large gardening gloves while green and brown leaves were stuck in her upswept hair, that still managed to be breathtaking on her. She slung her hands out and sent the gloves flying to the ground.\n\n\"Hi, Kat,\" she said as if completely unperturbed by the fact that they'd overheard her complaints. \"What's up?\"\n\n\"I wanted...\" Kat's voice trailed off as a small set of flowers started to move toward the woods.\n\nPersephone blasted them with some kind of energy from her hands. She laughed, then sent her gloves to go pull up more flowers.\n\n\"What are you doing?\" Kat asked with a frown.\n\n\"Revenge,\" Persephone said proudly. \"It's a small revenge, granted, but it's these little things in life that mean so much.\"\n\nKat cocked a brow and looked at Geary before she asked for clarification. \"Revenge on who and for what?\"\n\n\"My mother, who else?\" Persephone gestured around the lush garden. \"Sticks me in this godforsaken place nine months out of the year and thinks I ought to be grateful for it. Meanwhile all I want is to be with my husband....\" She gave them a meaningful look. \"Have you any idea how hard it is to go nine months out of the year with no sex when you're married to such a fine piece of male anatomy that he should have been the god of fertility instead of the god of death?\" She paused in her tirade as she finally saw Geary over Kat's shoulder. \"And you are?\"\n\n\"She's a friend. Megeara, meet Persephone.\"\n\nPersephone frowned sternly as she raked a gaze over Geary's body. \"You're not the Fury Megeara, are you?\"\n\n\"No, but I was named after her.\"\n\n\"Ah.\" Persephone extended her hand as her features softened. \"Nice to meet you then.\"\n\n\"You, too.\"\n\n\"So,\" Persephone said, looking back at Kat. \"What brings you two here?\"\n\n\"We\"\u2014Kat indicated the two of them with a wave of her hand\u2014\"are in need of some serious help.\"\n\nPersephone scoffed. \"I'm in need of some serious help.\" She sighed as she gave Geary a hopeless smile. \"I know we just met, Megeara, but bear with me. I'm so horny I could die and my mother's answer to my complaints... weed her prized garden. Weed! What is she, insane?\"\n\nThis was definitely more than Geary wanted to know about the goddess.\n\n\"Yeah, and on that happy note,\" Kat said with a hint of laughter in her voice, \"it's your husband who brings us here.\"\n\n\"Oh, what'd he do now?\"\n\n\"Made a pact with a Dream-Hunter who wanted to be human. Now the Dream-Hunter has been killed and we'd like to get him back from Hades.\"\n\nInteresting summation Kat had come up with. Geary would have never been so succinct.\n\nPersephone screwed her face up. \"That's a bummer. You know Hades doesn't like to let anyone leave. Ever. He's kind of attached to those souls.\"\n\n\"I love Arik,\" Geary said, her voice cracking. \"I'll do anything to get him back.\"\n\nBoth of the women cringed at Geary's words.\n\nPersephone motioned for her to lower her voice. \"Don't say that too loudly around here. There are a lot of folks who would take you up on it, and bargaining with a god is what got you guys into this mess in the first place.\"\n\n\"I'm sorry,\" Geary said quickly. \"But I do love him, with all my heart.\"\n\nKat sighed. \"The gods have really screwed her over. They've taken almost all of her family, and given that, I was thinking that we might be able to help her just this once.\"\n\nPersephone shook her head. \"You know the rules, Kat.\"\n\nGeary frowned. \"What rules?\"\n\n\"Quid pro quo,\" Kat said irritably. \"You have to give something in order to get a favor from a god.\"\n\nOh, that was just wonderful. But it still didn't stop Geary. She had to get Arik freed. \"Tell me what I have to do.\"\n\nPersephone looked surprised by Geary's response. \"She's an eager little bunny, isn't she?\"\n\n\"She's desperate, Seph. Of all people, I think you can relate to what it's like to be taken away from the one you love.\"\n\nPersephone nodded. \"Yeah, and you picked the right moment to approach me on this. I'm really missing my Hades.\"\n\nSuddenly Geary remembered the ring D'Alerian had given her. \"Wait!\" She pulled it out of her pocket. \"One of the Oneroi gave me this. He said to give it to you and to tell you that Neco wanted to call in his favor.\"\n\nShe saw the pain in Persephone's eyes as she took the ring from Geary's hand. Persephone's eyes shimmered from tears as she traced the scrollwork on it with her fingertip. \"How was he doing?\"\n\nKat gave her a sad smile before she answered. \"Well.\"\n\nPersephone placed the ring on her thumb before she nodded. \"Well, there you have it. Neco is trading in his favor for this, so it's Neco you owe.\"\n\nWho is Neco? Geary mouthed the words at Kat, wanting to understand exactly what was going on. And who she owed for this favor.\n\n\"Neco is D'Alerian.\"\n\nGeary was floored by the news. \"Why would he trade his favor in for me?\"\n\nPersephone brushed away her tears. \"Because my brother is a gentle man. He doesn't like to see anyone suffer and he hates injustice. I imagine this is his way of making amends for something he feels they've taken from you.\"\n\nStill, Geary didn't understand. \"Isn't that hard to do for someone who doesn't have any feelings?\"\n\nPersephone didn't respond.\n\nKat, on the other hand, scowled at the much shorter goddess. \"The curse on them is weakening, isn't it?\"\n\nPersephone gave a subtle nod. \"But you can't tell anyone, Kat. No one. I won't have Neco punished again. Father was more harsh on him than the others because he was his son. If he ever learns this...\"\n\n\"Don't worry,\" Kat assured her, \"I won't tell anyone. Keeping secrets is what I'm best at.\"\n\n\"Good,\" Persephone breathed. \"They've all been through enough.\"\n\nGeary shook her head. \"But I don't understand. I thought the Oneroi were the sons of Hypnos and Morpheus?\"\n\n\"Some of the Oneroi are,\" Persephone said quietly. \"Neco is my half brother. He was born of Zeus and D'Aria, one of the original Oneroi. So long as one of their parents is an Oneroi, then they inherit those powers and are given those duties.\"\n\nGeary rubbed her brow as she tried to make sense of it and fit it into her mind. \"You guys have the most convoluted family trees.\"\n\nKat laughed. \"Believe me, we know.\"\n\n\"Yeah, and you don't ever want to try and figure out Kat's. It's scary.\" Persephone craned her neck to look around the yard. \"Come on, guys, we have to hurry. If my mother returns and finds me gone, she'll launch a tornado or something.\"\n\nOne second they were in the sun-filled garden, in the next they were in a dark, stinking cavern. Geary held her hand to her nose in an effort to squelch the awful smell. \"What is that stench?\"\n\nScrewing her features up in distaste, Persephone waved her hand in front of her face. \"Cerberus's dinner. We picked a bad time to come.\"\n\nShe led them down a narrow corridor and through a door that opened into a large throne room.\n\nGeary paused in the doorway as she saw the glistening ebony walls. But what held her enthralled was the gorgeous man sitting on a black throne that was made of what appeared to be bones. With shoulder-length black wavy hair, he was absolutely stunning.\n\nAnd ripped.\n\nDressed in black leather armor, Hades had a presence that truly suited a god of death. It sent a ripple of fear over her, but even so, she understood why Persephone had been attracted to him.\n\nHe was compelling.\n\nAnd as Persephone neared him and he saw her, the look of joy on his face actually made Geary's heart ache. He rose slowly to his feet.\n\n\"Seph,\" he breathed as if he were dreaming.\n\nPersephone ran to him.\n\nLaughing, he scooped her up in his arms and twirled around with her. \"Oh, my precious Seph.\" He laughed again before he kissed her soundly.\n\nKat cleared her throat. Loudly. \"Sorry for the interruption, but before clothing starts to fly or disintegrate, I wanted to remind you two that you're not alone.\"\n\nPersephone blushed at the same time Hades growled at them. He took a step toward Kat, but his wife stopped him. \"She's right, Hades. We have to be quick before my mother finds me gone and thinks that you've kidnapped me again. The last thing we need is for her to call out my father.\"\n\nHades cursed under his breath. \"Like I fear that bastard.\"\n\n\"Hades,\" Persephone chided.\n\nHe relented, but by his face Geary could tell he did so reluctantly. \"So why are they here?\"\n\n\"They come seeking a soul.\"\n\nHe scowled at his wife. \"Whose?\"\n\n\"Arikos's,\" Kat said.\n\nHades looked even more confounded. \"The Skotos?\"\n\nGeary nodded.\n\n\"He's not here.\"\n\n\"What?\" Geary asked in disbelief, her heart sinking.\n\n\"Arikos hasn't made his way here,\" Hades repeated. \"If he had, I would know it. I have a bone to pick with that bastard, too.\"\n\nKat ignored Hades' heated tone. \"We were told he's on the other side of the river Styx and isn't able to cross. M'Adoc killed him and didn't bury him. Arik has no money to pay Charon for the crossing.\"\n\n\"Why would M'Adoc do that?\" But before they could answer, Hades shook his head. \"That little bugger. Trying to pull one on me. He doesn't want me to know Arikos is here. Bloody, freaking bastard. And you.\" He looked at Megeara. \"You're the human Arikos bartered for. You're supposed to be here in his stead. So have you come to exchange places with him?\"\n\nGeary couldn't speak as fear took root in her.\n\nBut before she could think of something to say, Persephone slapped Hades on the arm. \"Don't you even start that.\"\n\nRubbing the spot she'd hit, he scowled at her. \"Start what? Arikos and I had a bargain.\"\n\n\"So what?\" Persephone asked in an irritated tone. \"Do you mean to tell me that you are going to stand there and make her die to save him? How could you?\"\n\n\"It was a bargain,\" he said defensively.\n\n\"Yes, and my parents made a bargain, too, and look how that's turned out. I can't believe you would do this to someone else. I thought you were different.\"\n\nHis features turned placating. \"Baby, I am.\"\n\n\"No, you're not. You're just like them. You intend to break up two lovers and for what? A stupid, meaningless bargain. You who know how much it hurts to be away from the one you love, and yet you would do something so cold and mean. Oh, that's it. I'm going home to my mother's and I'm not coming back.\"\n\nHis dark eyes snapped fire. \"You have to come back. You have no choice.\"\n\nShe narrowed her gaze on him. \"You're right. I have no choice about coming here, but I have a choice about where I sleep once I'm home.\"\n\nHis face went pale as he realized he was on losing ground. \"You wouldn't dare.\"\n\nPersephone put her hands on her hips as she stared down the man who was practically twice her size. It would have been comical if Geary's future with Arik didn't hinge on the outcome. \"You break these two up and it'll be a cold day in Hephaestus's forge before you enter my bedroom. In fact, I'll get Eros to make you impotent. Yeah. Forever. That'll teach you.\"\n\nHis face completely ashen, Hades looked at Geary. \"Take him. Get his ass out of here and don't look back.\"\n\n\"Are you serious?\"\n\n\"Yes.\"\n\nPersephone winked at Geary before she pulled her husband into a tight hug. \"Now was that so hard?\"\n\nHe answered her question with one of his own. \"How long until your mother returns?\"\n\nPersephone turned to them. \"You two better hurry along and claim him. Arik will be a Shade until you return to the sunlight in the human realm. Kat, you know the way. Once he's back in the world, he'll be human and flesh. But remember, Megeara, that you have to lead him out of here yourself, and you can't look back. If you do, you'll lose him forever.\"\n\nBefore Geary could even say \"thank you,\" the two of them vanished.\n\nKat turned toward her with a laugh. \"Fun, huh?\"\n\n\"Yeah,\" Geary said lightly. \"I think I'm a bit shell-shocked. I can't believe we have no test or anything to fulfill.\"\n\n\"We don't have him out of here yet. Remember, when they say don't look back, don't.\"\n\nGeary nodded as she remembered that from her studies. They weren't out of the woods yet. And if they didn't find him soon, it would be too late.\n\n* * *\n\nArik watched as Charon passed by him on the river. The old man was a sulking bearded figure, dressed in dark brown. His pitiless gaze scanned those who were gathered on the banks of the river. Charon would only take those who had an obulos, a Greek coin, or a Persian danace to pay the ferryman's fee. Only those with the coin could go to the other side, where they were divided up\u2014those who'd done good deeds in their lives were taken to the Elysian Fields for a divine rest, and those who'd committed evil deeds were destined for Tartarus to be tortured.\n\nBut only a fool would give Charon the coin before he delivered them to the opposite bank. It was custom to show the coin to Charon, then wait and hand it over once the journey was complete.\n\nIf you couldn't show him the coin, then you were doomed to wait on the bank for a hundred years before you could cross. And if you paid Charon before he delivered you to your destination, he'd dump your soul in the river, where you'd suffer in eternal misery.\n\nPersonally, Arik knew where he was headed once he crossed the river and he could easily wait a hundred years before his torture began. Then again, he didn't have to. He was already aching from the loss of Megeara.\n\nHe felt her absence with every part of him. The despair weighed like a stack of anvils on his soul. All he wanted was to see her face one last time. To touch her cheek or to feel her hair on his flesh. Those memories seared him as he prayed for her safety.\n\n\"I hate that miserable old bastard.\"\n\nArik looked to his left as the Shade of a middle-aged man joined him.\n\nThe man was glaring at Charon, who no longer paid them any heed as he cut his way through the black water. \"I wish that boat would turn over and drown him in the river Acheron. Would serve him right if it did.\"\n\nPerhaps. Acheron was the river of woe and it was here that all the troubles of the world were gathered. It was said that if any part of your body were to touch it, those woes would seep into you and tear your body and soul asunder with grief.\n\nAll the dead must cross it to reach their final destination. It was supposed to be a symbolic journey where the dead left such concerns behind.\n\nThe man looked up at Arik. \"You didn't have a coin, either, huh?\"\n\n\"No.\"\n\nHe spat on the ground by Arik's feet. \"That's for both our families then. Leaving us stranded like this. A pox to them. May they all fall into the river Acheron and drown in their stinking misery.\"\n\nArik lifted a brow at the man's rancor. He sounded like he'd been bathing in the river Styx, where hatred flowed freely.\n\nThe man eyed him carefully. \"So what brought you here?\"\n\nArik answered without thought. \"Love.\"\n\n\"Killed yourself, did you?\"\n\n\"No. I traded my life to keep the one I love safe.\"\n\nThe man was aghast. \"Why would you do something so stupid?\"\n\n\"It's not stupid.\"\n\n\"Sure it is. Do you think she'd have done the same for you?\"\n\nAgain, Arik answered without reservation. \"Yes.\"\n\n\"You're a complete jackass if you think that.\" He made a rude noise before he wandered off.\n\n\"He's right, you know.\"\n\nArik froze as he heard the last voice he expected to hear. It was Wink, no doubt come to gloat. \"What are you doing here?\"\n\nWink shrugged. \"I hang with the dead sometimes. They can be extremely entertaining, especially the whiny ones.\" He paused and smelled the air around Arik. Wink's actions reminded him of a hound on the scent of a skunk. Finally Wink pulled back and gave Arik a gimlet stare. \"Where are your powers?\"\n\n\"Don't worry about it.\" Arik tried to move away, but Wink followed him down the bank of the river even while he was weaving between other Shades.\n\n\"What gives, Arikos?\"\n\nArik had no idea why he didn't rat out the others and tell Wink that all of them were regaining their emotions. He should. It was what they all deserved, but some misplaced sense of loyalty kept him from it. Wink would run with whatever Arik told him straight to Zeus and start trouble.\n\nArik had too much humanity in him now to do such a thing, and in the back of his mind he knew that Megeara would be disappointed in him.\n\nEven though she hated him, Arik didn't want to disappoint her.\n\nAnd still the god trailed after him. \"Arikos?\"\n\n\"Go away, Wink,\" he snapped. \"I'm dead and I just want to be left alone.\"\n\nWink took Arik's arm, then hissed and recoiled. \"You and your human?\" he said, his tone accusing. No doubt that one touch had told the god everything about how Arik had ended up here\u2014everything that had to do with Megeara anyway. \"Have you lost your mind? Why would you give up your immortality for her?\"\n\nArik couldn't explain it. It was stupid and he knew it. But it still seemed right. His life for her happiness. It strangely worked for him even though it shouldn't.\n\nHe was definitely a mental case.\n\nStill Wink wouldn't relent and leave him in peace. \"You gave up immortality for her,\" Wink repeated. \"Haven't you learned yet that humans aren't worth it? She was one of millions who are out there.\"\n\n\"No, Wink, you're wrong. She's one in a million. She's unique.\"\n\nHe snorted in response. \"So unique that she let you die for her? Believe me, there are millions of women who are that selfish.\"\n\n\"Yeah, but there's only one willing to walk through hell to get him back.\"\n\nArik came instantly to a stop as he saw Kat in the darkness. But that wasn't what stunned him most. It was the sight of Megeara moving to stand by her side.\n\nHe wanted to run to Megeara and scoop her up in his arms, but he couldn't. For one thing, he was no longer corporeal. For another, he wasn't sure she'd let him.\n\nBut she was here....\n\nGeary covered her mouth with her hand to stifle her cry as she saw what was left of Arik. His translucent skin was ashen gray. His eyes were no longer blue but rather dark and sunken. And he bore a hole in his middle that looked like someone had stabbed him.\n\n\"Arik?\" she asked hesitantly.\n\nHe appeared to be speaking, but she couldn't hear him. Terrified, she looked at Kat.\n\n\"He's a Shade now, Geary. Only the gods can hear him.\"\n\n\"What's he saying?\"\n\n\"He wants you to leave before it's too late for you.\"\n\nThat succeeded in making her tears flow down her cheeks. \"Can he hear me?\"\n\n\"Yes.\"\n\nShe turned to him. \"I'm here to get you, Arik. I won't leave unless you're with me.\"\n\nThe disbelief on his face tore through her. Even so, he held a hand out toward her. She tried to take it, but her hand went straight through his.\n\nThe Oneroi beside him snarled at her. \"You don't belong here, human. Leave.\"\n\nKat stepped between her and the man who looked as if he wanted to kill her. \"Leave her alone, Wink.\"\n\nHe turned his hostile glare to Arik. \"Don't be stupid, Arikos. She won't be able to take you out of here. No human has ever been able to resist Hades' test. And he'll make you pay double for trying to leave.\"\n\nArik hesitated. Wink was right. Until Kat and Megeara had notified him, Hades hadn't even known he was here. Now the god did. If Megeara didn't safely lead him out of here, Hades would take pleasure in torturing him forever.\n\nNo, he was already damned and tortured here without her. There was nothing Hades could do to him that was worse than the thought of her with someone else.\n\nArik needed her more than he needed anything else. He had no choice but to follow her.\n\nI love you, Megeara.\n\nGeary sobbed as she read his lips. \"I love you, too, Arik, and I'm taking you out of here. I promise.\"\n\nHe gave her a wan smile before he nodded.\n\nWink curled his lip at them. \"It's not that easy, little human. Just\u2014\"\n\nKat cut his words off by clenching his throat in her fist. \"Lay off, Wink. We're not in the mood.\"\n\n\"You can't help her,\" he choked out from his constricted windpipe. \"She's going to die, too.\"\n\n\"Then you should be happy. Now go your own way or the world will be in need of a new Sandman.\" She let go of him so quickly that he stumbled through Arik.\n\n\"I'm going. But I plan on enjoying the sight of your failure.\"\n\nKat reached for Wink again, but before she could grab him again he dissolved.\n\nGeary wiped her tears away. \"What was he going on about, Kat?\"\n\nShe let out a deep breath before she responded. \"Getting here was easy. Getting out won't be. You know the old Eagles song 'Hotel California'?\"\n\n\"Yeah.\"\n\n\"Yeah. That's our situation. And once we start out of here, if you look back or try to help Arik in any way, he's a goner and, since you're not born of a god's blood, so are you.\"\n\nA cold feeling of dread settled as a knot in the pit of her stomach. \"Nice. Could have told me this before we got here.\"\n\n\"Would that have changed your mind?\"\n\nGeary looked at Arik and felt the love inside her swell past the unsettling lump. \"No.\"\n\n\"Good, then I didn't waste my breath.\"\n\nGeary shook her head at Kat before she turned back to Arik. She wanted to touch him so badly that she ached from it. But that would be impossible until they freed him. \"Lead on, MacDuff.\"\n\n\"I'm leading, but I can't look back, either. So stick tight and remember, stay on the trail. Don't worry about anything that comes at us. Just pretend we're in a haunted house and don't get distracted.\"\n\n\"Ooooo, scary.\" But even though she was making light of this, she knew just how serious it was. One misstep and all three of them would pay dearly for it.\n\nKat led her off into the bitter darkness that was so oppressive it made her eyes ache. The only way she could tell that Kat was still in front of her was that she could hear her breathing. At least she hoped that was Kat. In the dark, Geary's imagination was playing havoc. For all she knew, it was some creepy beast bent on eating her.\n\n\"Kat?\"\n\n\"I'm right here. Keep your eyes forward.\"\n\n\"I am.\"\n\nSomething slithered beside her. Geary squeaked in dismay and had to force herself not to jump away from it. C'mon, girl, you've been on dives with sharks and eels swimming around you. You can handle this. This is nothing. Stay your course....\n\nBut in the water, she wasn't this blind.\n\n\"Don't look back,\" Kat warned again. \"They're trying to get you to glance behind you and see Arik. If you do, it's over.\"\n\nThat was easier said than done, especially since something seemed to be glowing in her right peripheral before it drifted back, away from her. And she was desperate to know if Arik was still there. There was no sound or sign of him.\n\nNothing. For all she knew, they'd grabbed him and pulled him away from her.\n\nAnd to think she'd always thought Orpheus was an idiot for checking to see if Eurydice was behind him. Now it all made sense. No wonder the demigod had been so paranoid. No doubt they'd been tormenting him every step of the way, too.\n\nWhat if Arik became distracted? What if he'd fallen and needed help?\n\nSuddenly a light flashed before them. It was followed by a shriek and a green face so horrendous that Geary actually screamed out loud. Instinctively she started to turn to Arik, but she caught herself as the loud shrieking continued. \"What was that?\"\n\n\"Gorgon,\" Kat shouted. \"Stay to the course and ignore them. They guard the barrier between the Underworld and the Outerworld. Their job is to keep us in here. Don't let them.\"\n\n\"I'm trying.\"\n\n\"Don't try, Geary, succeed.\"\n\nShe intended to.\n\nAnd as they walked, more and more gorgons appeared. They marched alongside the trio, shrieking and lashing out. Feinting toward them. But the gorgons never touched them. They only did it to make them flinch.\n\nThe gorgons were every bit as hideous as the stories had foretold. Their green skin was scaly like a snake's and they had red eyes that glowed in the darkness. There was a shuffling, slithering noise that followed them through the cavern.\n\nBut the worst was their breath that would make toxic waste proud.\n\n\"He's not there, human,\" the gorgon to Geary's right said evilly. \"You've lost him already.\"\n\n\"Shut up, Euryale,\" Kat snapped. \"Leave her alone.\"\n\nShe hissed at Kat.\n\nGeary did her best to distract herself from them. \"I thought people turned to stone if they looked at a gorgon.\"\n\n\"Only men do.\"\n\nA new fear went through her at Kat's words. \"Arik?\"\n\n\"I said men, Geary. Not gods or Shades. He's safe. Just keep moving forward and don't try to check on him.\"\n\nThat was so much easier said than done, especially when her mind was going wild with what-ifs. \"Are you sure?\"\n\n\"Well, are you stone?\"\n\nNot yet, she wasn't, but if Kat didn't take the condescension from her tone, Geary might \"stone\" her. \"I don't mean me. I meant Arik.\"\n\n\"If you turn around to see, Geary, you will lose him.\"\n\n\"I know.\" But the compulsion was so strong. It was unnatural.\n\n\"She's lying to you, human. You've lost him in the caverns. He's weeping for you, wanting you to help him.\"\n\nGeary shook her head to dispel the image in her mind of Arik doing exactly what the gorgon said. \"You're the one who's lying to me.\"\n\nThe gorgon bared her fangs before she moved to walk directly beside Geary.\n\n\"Megeara, help me.\" It was Arik's voice coming from behind her.\n\nIt's a trick, it's a trick.\n\n\"Please, Megeara. I need you. Don't let me suffer here....\"\n\n\"Stop it!\" Geary said from between clenched teeth. \"He's a Shade. I know I can't hear him and Arik would never beg like that. You're just trying to piss me off.\"\n\nOne of the gorgons behind Geary clucked her tongue. \"Poor Arik. See, she doesn't love you at all. She'd let you suffer rather than endanger herself.\"\n\nThen she heard the muffled sound of a masculine cry that sounded like it could be Arik.\n\nShe clenched her fists, struggling not to turn around and check on him. She had to know he was okay.... \"Kat,\" she whimpered. \"Help me.\"\n\n\"Don't listen, Geary. Sing a song and drown them out.\"\n\n\"Sing what?\" she asked in frustration.\n\n\"What's your favorite song?\"\n\nGeary plugged her ears with her fingers and started singing Gloria Gaynor's \"I Will Survive.\"\n\nNow it was the gorgons' turn to scream out in pain as they shrank away from Geary. Realizing that they couldn't stand her off-key harmony, Geary sang even louder.\n\n\"Stop! Stop!\" they begged.\n\nBut Geary refused. It was time someone returned the favor to them and let them exist in torment for a while.\n\nAfter she finished Gloria's song, she broke out to Wild Cherry's \"Play That Funky Music\" and then Lipps, Inc.'s \"Funky-town.\"\n\nTo her misplaced delight, the gorgons continued to writhe and moan in agony, which caused Kat to help her serenade them with more disco tunes.\n\nGeary was just finishing the Bee Gees' \"Stayin' Alive\" when she finally saw light ahead. Her heart pounded as raw excitement filled her. They were almost done.\n\nA few more steps...\n\nHer singing faltered as she struggled to hear some sign of Arik behind her. There was nothing.\n\nNothing.\n\n\"Down!\" Kat shouted an instant before a blast of fire darted over their heads.\n\nGeary squeezed her eyes shut and prayed with everything she had. She wanted desperately to touch Arik.\n\nHe's there.\n\nHe had to be. Trusting in Kat and in Arik, she opened her eyes and saw that Kat was already moving forward again.\n\nIt took some doing to make it up the sharp rocks that led to the small opening above them.\n\n\"I can't help you up, Geary,\" Kat said from in front of her. \"Like you, I can't turn around, and you can't turn to help Arik up, either, understand?\"\n\n\"Yes.\"\n\n\"Okay. Remember, we have to get clear and you have to wait right beside me, facing the east. Got it?\"\n\n\"Got it.\"\n\nAs she neared the opening, Geary's foot slipped. She slid back and cursed as the rocks cut into her hands and knees. Before she could stop herself, her head turned, but she again slammed her eyes shut.\n\nWould the gods count that?\n\nSurely not. But if she opened her eyes to double-check, it would be over.\n\nCounting to ten, she straightened her head and looked forward. \"Don't let me down, Arik. Do you hear me? You better still be there.\"\n\nWith a deep breath, she started climbing again even though her cuts stung and her body ached from the fall.\n\nIt seemed to take forever before she was clear of the cavern. Kat was outside, waiting by a small clearing that overlooked the sea.\n\nGeary joined her. \"What now?\"\n\nKat turned her head slightly to look at Geary with a frown. \"What the hell happened to you?\"\n\n\"I fell.\"\n\nKat screwed her face up in distaste and pity. \"Sorry.\"\n\nSo was Geary, especially given how ferocious the pounding pain was.\n\nBut Geary stood there, silently waiting. After a few minutes, her panic set in. \"Where's Arik?\"\n\n\"Don't look for him.\"\n\n\"I'm getting sick of that warning, Kat. He's not here....\"\n\n\"Be patient, Geary.\" Her tone was placating and calm, and it was pissing Geary off more and more.\n\n\"We're in daylight. We're clear. Why isn't he here with us?\"\n\n\"What if he's right behind you now, and you turn to look? You'll send him right back into hell.\"\n\nGeary pressed her hands to her eyes, as she wanted to weep in frustration. This was cruel and mean and it made her hate the gods for it. \"Don't die, Arik, please.\"\n\nAnd then she felt it. It was a cold touch against her cheek. Light and gentle. She'd know that touch anywhere. Lowering her hands, she saw Arik next to her, but he was still pale and gaunt.\n\nEven so, he was the best-looking thing she'd ever beheld. Before she could stop herself, she pulled him to her and kissed him senseless.\n\nArik growled as he tasted Megeara again. And the longer she kissed him, the warmer he grew. He held her tight against him, reveling in the feel of her warm body next to his. In all his life he'd never felt anything like this.\n\nHe could swear he could fly without wings right now. Never once had he even dreamed she'd come back for him, and the fact that she'd saved him...\n\nUnbelievable.\n\nMegeara pulled back to look at him, then laughed. \"You're back!\" She rained kisses all over his face.\n\nHis own joy filled him as he savored every touch of her lips on his flesh. \"I can't believe you came for me.\"\n\n\"Are you kidding? I would always come for you.\"\n\nAnd that was why he loved her so dearly.\n\n\"Uh, guys,\" Kat said, clearing her throat, \"no offense, but this is getting awkward for me. You two take care and I'll see you around.\"\n\nBefore either could speak, she vanished.\n\nArik laughed as he picked Geary up and twirled around with her. \"I can't believe you're really here and this isn't a dream.\"\n\n\"Me? Look at you....\" Geary frowned as a weird thought went through her. It was a question she hadn't even thought to answer before. \"What are you now?\"\n\n\"He's human.\"\n\nArik paused at the sound of D'Alerian's voice. He set Megeara down, expecting a fight. \"What do you want?\"\n\nD'Alerian held his hands up in surrender. \"I just wanted to make sure Megeara made it out alive. Now that you're together, I thought I'd take you both back to her home to celebrate.\"\n\n\"And we're supposed to trust you why?\"\n\nGeary put her hand on his arm to calm him down. \"Don't, Arik. We owe him everything. He's the one who called in a favor from Persephone so that I could free you.\"\n\nHe gave her a confounded look, then turned it toward D'Alerian, whose face was completely stoic. \"Why would you do that?\"\n\n\"Because I lost what I loved, Arik, and I don't want anyone to know that pain. You two have earned the right to live in peace.\"\n\nArik scoffed at his good wishes. \"M'Adoc will never allow that.\"\n\n\"Yes, he will. We'll make sure of it.\"\n\nGeary didn't miss the ominous note in D'Alerian's voice. \"What are you going to do to him?\"\n\n\"Don't worry. We won't hurt him. We're taking him someplace where he can learn compassion. It's a simple emotion, but it escapes so many. He needs to relearn it.\"\n\nThen D'Alerian held out his hand and a bright flash of light encompassed them. One moment they were outside the Underworld, and in the next they were in her flat.\n\nGeary glanced around in disbelief. It seemed like a lifetime had passed since she'd last been here.\n\nD'Alerian gave them a gentle smile. \"Treasure each other.\"\n\nGeary nodded. \"Don't worry. We will.\"\n\nHe inclined his head to them, then dissolved.\n\nAs soon as they were alone, Arik leaned his head down and nuzzled her neck with his warm lips. \"I love you, Geary.\"\n\nShe smiled as he used her nickname without her forcing him to. \"I love you, too, babe.\" She reached to take his hand into hers and pull him toward her room.\n\n\"What are you doing?\"\n\nShe glared menacingly at him. \"I'm going to make you suffer like no man has ever suffered for the lies you've told and for putting me through so much.\"\n\nHis mouth opened and closed as he looked a bit shell-shocked. Finally, he clenched his teeth and narrowed his eyes on her before his features settled down to resignation. \"And what do you plan to do to me?\"\n\nA slow grin spread across her face. \"First I'm going to strip you naked and then I'm going to bend you like a pretzel and lick your body until you beg me to stop. I'll have you begging me for mercy in no time.\"\n\n\"Hmmm,\" he breathed. \"That sounds positively awful.\"\n\n\"You've no idea.... My tongue has been known to let blood on four continents.\"\n\nHe laughed deep in his throat as she pulled him into the bedroom. \"Well, in that case, let the torture begin.\"\nCHAPTER 20\n\nD'Alerian paused inside the hall where M'Ordant was waiting for him. M'Adoc was there as well, still bound by the diktyon.\n\n\"Are they safe?\" M'Ordant asked.\n\nD'Alerian nodded before he moved toward M'Adoc, who glared his hatred at them. \"I can't believe you two have betrayed me.\"\n\nThe hostility saddened him. \"We're not betraying you, Adarian. We're going to help you.\"\n\n\"What are you planning exactly?\" M'Ordant asked.\n\n\"I'm taking him to Acheron. There's a Dark-Hunter who needs someone strong to help him with his nightmares.\" He looked at M'Adoc. \"A few months with Zarek in Alaska and I think you'll see why it's so important to let go of your hatred.\"\n\n\"Bullshit. You can't send me away.\"\n\nM'Ordant frowned. \"Why send him to Alaska? He can tend the Dark-Hunter from here.\"\n\n\"No, he can't. Here he poses a threat to us. His emotions are out of control. If any of the other gods see the way he's been behaving, we're screwed. We can deal with Wink and Hades knowing. But Zeus learning... In Alaska, no one will know.\" D'Alerian looked back at M'Adoc. \"You can stay there a short time, and once you have a better handle on yourself, I'll come get you.\"\n\n\"I won't stay there.\"\n\n\"Are you going Skoti then?\"\n\n\"Never.\"\n\n\"Then that's your assignment. Take it or leave us.\"\n\nM'Adoc's jaw twitched with his fury, but ultimately he conceded. \"Fine. I'll go. But only for a short time.\"\n\nD'Alerian nodded before he removed the diktyon. Then he flashed the two of them out of the chambers and into the human realm. They materialized in the living room of a Dark-Hunter's home in New Orleans.\n\nKyrian Hunter. A former ancient Greek general, he was now one of the Dark-Hunters who helped to guard mankind from the Daimons, or vampires as they were better known, who preyed on humanity. D'Alerian had been assigned to the general since the day Kyrian had sold his soul to Artemis for vengeance on the man who'd killed him. Nightmares had plagued him ever since.\n\nBut D'Alerian could at least mitigate them most of the time.\n\nIt took D'Alerian a moment to get his bearings in Kyrian's home as Acheron entered the room and came to a stop. At six eight, with long green hair and dressed in black leather pants and a shredded Sex Pistols T-shirt, Acheron was a hard man to miss.\n\n\"Greetings, gentlemen,\" he said, his voice thick with its Atlantean accent.\n\nBefore D'Alerian could speak, a young man on a skateboard came rolling through the room and almost collided with them. He skidded to a stop not far from Acheron, then cursed at the long black mark his wheels had left on the floor.\n\n\"I'm so friggin' dead,\" Nick Gautier whispered loudly before he kicked the skateboard up and grabbed it in a tight fist.\n\nAcheron snorted. \"Relax, Nicky, you're not as dead as I am.\"\n\n\"That's what you think. Kyrian's gonna stroke when he sees that.\" Trying to scrape the mark up with the toe of his tennis shoe, he met D'Alerian's gaze. \"So what brings you here? Kyrian's not hurt, is he?\"\n\n\"No.\"\n\nAcheron offered the eighteen-year-old a kind smile. \"They're here for me. Why don't you go see what Rosa's baking and give us a minute?\"\n\nNick frowned at him. \"You hurt?\"\n\n\"No.\"\n\n\"Then why\u2014\"\n\n\"Nick, space. Now.\"\n\nNick made a face at Acheron. \"Go, Nick, fetch. Here, boy, here,\" he groused. \"You should let me borrow one of those leather collars you wear and give me a tag with Kyrian's number on it. 'In case of loss, call my owner.'\"\n\nAcheron snorted. \"Trust me, Nicky, we're not lucky enough for you to get lost.\"\n\n\"Yeah, yeah.\"\n\nD'Alerian frowned as Nick left them alone. \"That boy has issues.\"\n\n\"You have no idea.\" Acheron closed the distance between him and M'Adoc. \"You really want to go to Alaska to help Zarek?\"\n\nM'Adoc looked askance at D'Alerian. \"I'm told I have no choice.\"\n\nAcheron nodded as if he understood. \"Well, I appreciate it anyway. The gods know he could use it. I'll take you to him tonight.\"\n\n\"Thank you, Acheron,\" D'Alerian said before he flashed himself back home.\n\n\"Ah jeez. Nick!\"\n\nAsh turned at Kyrian's irate shout to find the general standing in the doorway near the black mark Nick had left on the floor. A few inches shorter than Ash, Kyrian had short blond hair and was dressed in black. \"I'm going to kick your ass, boy! How many times have I told you no skateboards in the house?\"\n\nNick came up behind Kyrian with a face as white as chalk. Ash had seen condemned men look less panicked.\n\n\"It's not Nick's fault,\" Ash said quickly as Nick stopped behind Kyrian's back with his eyes wide. \"It's these new biker boots. Sorry. I was so stunned when M'Adoc showed up that I skidded on the floor.\"\n\nKyrian gave him a suspicious glare, but since he couldn't prove Ash was lying, he let it go. \"Well then, could you fix it?\"\n\nThe mark vanished instantly.\n\n\"Thanks.\"\n\nYou're the friggin' best ever, Nick mouthed at Ash from behind Kyrian's back. He held his hands up in a silent gesture that said Ash rocked. I love you, man.\n\nKyrian turned sharply to glare at Nick, who immediately acted as if he were just scratching his head. \"You called me, boss?\"\n\n\"No. I've called you a lot of things, but boss has never been one of them. And it never will be, either.\"\n\nNick raked his hand through his long brown hair. \"Dang, he's in a bad mood tonight. You need to get laid, boss.\"\n\n\"Shut up, Nick.\"\n\nDeciding silence on this issue was the better part of not getting his ass kicked, Nick cleared his throat. \"Well, if you guys are through ordering Fido around, he needs to go walk his mom home from work. I don't want nothing happening to her, you know?\"\n\nKyrian scoffed. \"I don't know why you bother, Nicky. You're the one who's going to be the death of her one day.\"\n\nIt was Ash's turn to scoff at that. \"Not bloody likely. I'd be the death of her before Nick would. That kid lives, breathes, and dies for that woman.\" He smiled at Nick. \"Give Cherise my best.\"\n\n\"Will do. Night, all.\"\n\nKyrian let out a heavy sigh before he grabbed his long coat from the couch and shrugged it on. \"I'm out to patrol, too. I heard there's been a lot of Daimon activity on Bourbon Street lately, so Talon and I are going to do some extra rounds. I'll see you guys later.\"\n\nAsh turned to M'Adoc, who was eyeing him strangely.\n\n\"Why don't you tell your Dark-Hunters about the Spathis? This one in particular needs to know.\"\n\nAsh hesitated. Maybe M'Adoc was right. For centuries Ash had kept silent about the group of Daimons who lived a lot longer than any of the Dark-Hunters suspected. Daimons who served his mother, Apollymi, and who came out to prey on Apollymi's enemies. But the Spathis had been virtually dormant for centuries now and he hoped they would remain so.\n\n\"We all have our secrets we don't want out, don't we, Adarian?\"\n\nM'Adoc's gaze narrowed as he caught Ash's meaning. \"You know what's happening to us, don't you?\"\n\n\"I know, but don't worry. The Greek gods aren't exactly my drinking buddies. I couldn't give two shits about them or their curses. I owe the Oneroi too much for helping me with my Dark-Hunters to ever question you.\"\n\nM'Adoc cocked his head as if he couldn't fathom Ash's reasoning. \"With this information, you could own us.\"\n\nAsh flinched as bitter, painful memories surged through him, but he banished them. \"Contrary to Nick's opinion, I don't ever want to own anyone. It's wrong to take away someone's independence.\" And on that he sought to change the subject. \"D'Alerian says that you and Zarek can help each other. I hope so. Z's too decent a man to keep suffering. If you can take any of it off him, then I'll owe you.\"\n\nM'Adoc frowned at him. \"I wouldn't say that if I were you. Being indebted to a god isn't the way to maintain independence.\"\n\n\"Yeah, believe me, I know. But it's all right, M'Adoc. I can see the future. You're going to be fine.\"\n\nM'Adoc glanced to the door where Kyrian and Nick had vanished. \"You can see my future so clearly. It's a pity you can't see your own.\"\n\n\"What's that supposed to mean?\"\n\nM'Adoc cleared his throat. \"It's not my place to say. I'm a dream god. Not one of fate. Take me to this Zarek and let me see what I can do for him.\"\n\nAsh obliged, but even as he did so he couldn't shake the feeling that something had transpired this night that he should have picked up on. As a god of fate, he knew that somehow he'd just set something into motion and, knowing his luck, it was most likely something he shouldn't have.\nEPILOGUE\n\nONE MONTH LATER\n\nGeary stood on the deck of her new boat as the water lapped gently against it. While she stared out over the crystal blue sea that was as timeless as her quest, she could hear Cynthia below playing an old Andy Gibb album. They were right above the spot where Atlantis rested. Where Geary had held the old box and had touched a tiny portion of that lost, mythical city.\n\nTwo weeks ago, she, Arik, and Kat had retrieved everything that had marked this area, and they'd destroyed every piece of evidence that Geary and her father had collected.\n\nNo one would ever know what they'd found.\n\nArik came up behind her, wrapped his arms around her waist, and kissed her on her shoulder that was left bare from her sagging tank top strap. \"Are you having second thoughts?\" he asked warmly in her ear.\n\nShe shook her head as she felt her love for him swell through her. \"How could I?\" She smiled.\n\nHe leaned his cheek against the top of her head as he rocked her gently in his arms. \"All you wanted was to save your father's reputation.\"\n\n\"And so I have. I don't care what the rest of the world thinks. I know the truth. That's enough for me.\"\n\n\"Are you sure?\"\n\nShe nodded. Even Tory had taken it better than Geary had expected. True, the girl hadn't been pleased, but she hadn't argued, either.\n\nSitting on the couch at Teddy's house, Tory had stared at Geary in disbelief. \"What do you mean we're through?\"\n\nGeary had flinched at Tory's irate voice. \"It's over, Tor. We now know our fathers weren't crazy and that they didn't die in vain. It's enough. Atlantis wasn't meant to be discovered by us. She needs to stay on the bottom of the sea forever.\"\n\nGeary had expected Tory to shout. Instead the girl merely got up calmly and gathered her books. \"I see. So are you shipping me back home to New York?\"\n\n\"Not right away. I thought we could enjoy the rest of the summer together.... Are you sure you're okay?\"\n\nTory had shrugged. \"I'm fine. The boat's gone. The research is gone, and you're giving up. How can I change any of that?\"\n\nEven so, Geary had expected more of a fight from her cousin. \"You're taking this a lot better than I thought you would.\"\n\nCradling her books to her chest, Tory had merely sighed. \"I'm a sane, rational person, Geary. I know when I can't change something. If I thought throwing a fit would sway you, I'd do so. But I know you better than that. If you say it's over, then it's done. All I can do is hope you'll change your mind one day.\"\n\nTory had then laid down her book of Plato and headed for the door. \"I'm gonna go make Thia's day by telling her the news. You two have fun.\"\n\nAnd so their zealous quest had ended with nothing more than a whisper. What had seemed so important at the time had turned out to be nothing more than a fool's errand when put in perspective. Yes, finding the island was important, but not nearly as much as enjoying the lives of the people who really mattered.\n\nAtlantis would always be there. But Tory, Thia, Scott, and the others wouldn't. At last Geary had come to understand the secret of Atlantis. It wasn't the power or the history. It was to value those around you\u2014to treasure family. To love unconditionally in spite of faults and suspicions.\n\nAnd as Tory had predicted, Thia had been ecstatic over the news that they were through seeking it.\n\nBut for Tory and Geary it was bittersweet.\n\nArik moved away from her, bringing her thoughts back to the present. \"Close your eyes.\"\n\nFrowning at him, Geary obeyed. She felt his hands at her neck an instant before something cold slid between her breasts. She opened her eyes and found a beautiful necklace there. It was a gold sun, the rays of which were outlined by diamonds. \"It's beautiful.\"\n\nHe smiled. \"Kat sent it to you. She said in her note that it was a gift from Apollymi to let you know there were no hard feelings for not releasing her.\"\n\n\"Really?\"\n\nHe nodded. \"Apollymi had said, 'Victory to the spider.' She's waited this long to be free, what's a few more centuries?\"\n\nGeary shook her head, grateful the goddess wasn't holding a grudge against them. \"I miss Kat.\" She'd left them a week ago to head off to a new assignment for Artemis. Apparently there was some woman in Greece being pursued by Daimons whom Kat was supposed to watch over.\n\n\"Yeah. She was a lot of fun. But I have a feeling we'll be seeing her again.\"\n\n\"I hope so.\" Geary turned around and took Arik's hands into hers. It was so strange to have him here with her. To know so much about him and the other world, but to not be able to share it with anyone, not even Tory or Thia.\n\nBut that was okay with Geary. She could definitely live with this secret.\n\nArik lifted her hand in his and kissed the back of her knuckles. \"So when are you going to tell me your news?\"\n\n\"And what news is that?\"\n\nHe arched a brow at her and looked down meaningfully at her stomach.\n\nGeary gaped as she caught his meaning. \"How did you know about that?\"\n\nHe gave her a wicked grin. \"I still have a lot of my powers, baby. You know that.\"\n\nShe sighed heavily. \"I wanted it to be a surprise.\" She pouted playfully until a thought occurred to her. \"Do you think the baby will inherit your powers?\"\n\n\"I don't know. Possibly.\"\n\nOh, that could be fun. She suddenly had images of Tabitha from Bewitched in her mind. Yeah. Just what she needed. A baby who held the powers of a god. But that didn't really matter. She would love her child no matter what. \"Lucy, we'll have a lot of explaining to do.\"\n\n\"Yes, we will, but first, Ricky has to make an honest woman of his Lucy.\"\n\nWarmth spread through her at his words. \"I was wondering when you were going to get around to that.\"\n\n\"It's a yes then?\"\n\nShe gave him a droll stare. \"No. I thought I'd walk through hell to reclaim you and carry your baby just for the heck of it. Who needs marriage?\"\n\n\"I do.\"\n\nShe smiled. \"Good. I can let you live another day.\"\n\nLaughing, he pulled her into his arms and held her close. \"Thank you, Geary.\"\n\n\"For what?\"\n\n\"For giving me a life that is the best dream I've ever had.\"\nAlso Available from #1 Bestselling Author Sherrilyn Kenyon\n\nThe Dark-Hunter Series\n\nFantasy Lover\n\nNight Pleasures\n\nNight Embrace\n\nDance with the Devil\n\nKiss of the Night\n\nNight Play\n\nSeize the Night\n\nSins of the Night\n\nUnleash the Night\n\nDark Side of the Moon\n\nDevil May Cry\n\nAcheron\n\nOne Silent Night\n\nBad Moon Rising\n\nNo Mercy\n\nRetribution\n\nTime Untime\n\nStyxx\n\nThe Dark-Hunter Companion\n\nThe Dream-Hunter Series\n\nThe Dream-Hunter\n\nUpon the Midnight Clear\n\nDream Chaser\n\nDream Warrior\n\nThe Guardian\n\nThe Chronicles of Nick\n\nInfinity\n\nInvincible\n\nInfamous\n\nInferno\n\nThe League\n\nBorn of Night\n\nBorn of Fire\n\nBorn of Ice\nOUTSTANDING PRAISE FOR SHERRILYN KENYON AND HER NOVELS\n\n\"Kenyon gives readers scrumptious heroes\u2014still with a dark edge and dangerous powers... she creates a world full of depth and intelligent details... and gives her stories an underpinning and a foundation in authenticity without sacrificing the breathless pace of her plots\u2014no small feat!\"\n\n\u2014Romantic Times\n\nDARK SIDE OF THE MOON\n\n\"Boundless imagination... [Kenyon] is one of the defining authors of the new wave of paranormal romance.\"\n\n\u2014Booklist\n\n\"Taut action and jaunty humor.... it contains a delicious balance of suspense and sensuality.\"\n\n\u2014Publishers Weekly\n\nUNLEASH THE NIGHT\n\n\"Kenyon defies expectations again... This depth and vulnerability of character have not been seen before in her heroes and prove a challenge for Kenyon, but the result is a thrill ride filled with magic, action, and scintillating passion.\"\n\n\u2014Booklist\n\nSEIZE THE NIGHT\n\n\"[A] taut, rollicking romance... [Kenyon] succeeds in offering a lively read containing her signature blend of brisk action, sensual thrills, and light humor.\"\n\n\u2014Publishers Weekly\n\n\"Breathtaking... Kenyon really shakes up the storyline in this latest chapter in her amazing Dark-Hunter series.\"\n\n\u2014Romantic Times (4 1\/2 starred review)\n\nKISS OF THE NIGHT\n\n\"With its frenetic, Matrix-style fight scenes and feral, leather-clad heroes, this book makes it easy to see why Kenyon's fantasy world has caught on so quickly and even inspired some readers to role-play on her Web site... an entertaining thrill ride.\"\n\n\u2014Publishers Weekly\n\nDANCE WITH THE DEVIL\n\n\"Move over, Anne Rice. Kenyon's Dark-Hunter books are changing the face of the vampire novel, making it hip, darker, and all the more appealing.\"\n\n\u2014Publishers Weekly\n\n\"Dance with the Devil cinches Sherrilyn Kenyon's place as a master of the genre! Zarek is a hero to die for, and his story will pull you in and keep you flipping pages into the wee hours; I couldn't put it down.\"\n\n\u2014Julie Kenner, author of Aphrodite's Secret\n\n\"A sensual, fast-paced read with a hero who is definitely worth the risk.\"\n\n\u2014Booklist\n\n\"Using figures from mythology, Sherrilyn Kenyon provides a deep novel that a romantic fantasy lover will cherish. A powerful story that makes believers of skeptics that ancient Gods and Goddesses, Dark-Hunters, Dayslayers, and other mythological characters walk among us... another fine myth from a superb storyteller climbing to the top.\"\n\n\u2014Baryon magazine\n\nNIGHT EMBRACE\n\n\"With her steamy, action-packed Dark-Hunter novels, Kenyon is ushering in a whole new class of night dwellers... an abundance of hot sex and snappy dialogue keep the plot both accessible and appealing. With its courageous, unconventional characters and wry humor, this fast-moving fantasy will fill the void left by the end of the Buffy the Vampire Slayer series.\"\n\n\u2014Publishers Weekly\n\n\"The second novel in Sherrilyn Kenyon's hot series is just as exciting, sexy, and thrilling as the first. Kenyon has hit on a fabulous premise that promises to be fodder for many more outstanding page-turners.\"\n\n\u2014Romantic Times\nThis is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously.\n\nTHE DREAM-HUNTER\n\nCopyright \u00a9 2007 by Sherrilyn Kenyon.\n\nAll rights reserved.\n\nFor information address St. Martin's Press, 175 Fifth Avenue, New York, NY 10010.\n\nISBN: 0-3 12-93881-0\n\nEAN: 978-0-312-93881-9\n\nSt. Martin's Paperbacks edition \/ February 2007\n\nSt. Martin's Paperbacks are published by St. Martin's Press, 175 Fifth Avenue, New York, NY 10010.\n\neISBN 9781429921527\n\nFirst eBook edition: February 2014\nDon't miss the newest DARK-HUNTER\u00a9 book\n\nSON OF NO ONE\n\nSeptember 2014\n\nRead the entire DARK-HUNTER series\n\nFantasy Lover\n\nNight Pleasures\n\nNight Embrace\n\nDance with the Devil\n\nKiss of the Night\n\nNight Play\n\nSeize the Night\n\nSins of the Night\n\nUnleash the Night\n\nDark Side of the Moon\n\nThe Dream-Hunter\n\nDevil May Cry\n\nUpon the Midnight Clear\n\nDream Chaser\n\nAcheron\n\nOne Silent Night\n\nDream Warrior\n\nBad Moon Rising\n\nNo Mercy\n\nRetribution\n\nThe Guardian\n\nThe Dark-Hunter Companion\n\nTime Untime\n\nStyxx\n\nDark Bites: A Short Story Collection\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nCollapsing Consciously Meditations\nAlso by Carolyn Baker\n\n_Reclaiming the Dark Feminine: The Price of Desire_\n\n_The Journey of Forgiveness: Fulfilling the Healing Process_\n\n_U.S. History Uncensored: What Your High School Textbook Didn't Tell You_\n\n_Coming Out of Fundamentalist Christianity: An Autobiography Affirming Sensuality, Social Justice, and the Sacred_ )\n\n_Sacred Demise: Walking the Spiritual Path of Industrial Civilization's Collapse_\n\n_Navigating the Coming Chaos: A Handbook for Inner Transition_\n\nCopyright \u00a9 2013 by Carolyn Baker. All rights reserved. No portion of this book, except for brief reviews, may be reproduced, stored in a retrieval system, or transmitted in any form or by any means\u2014electronic, mechanical, photocopying, recording, or otherwise\u2014without the written permission of the publisher. For information contact North Atlantic Books.\n\nPublished by North Atlantic Books P.O. Box 12327 Berkeley, California 94712\n\nCover art \u00a9 iStockphoto.com\/ImagineGolf Cover design by Mary Ann Casler Printed in the United States of America\n\nThe poem \"Constant,\" which appears at the end of this book, is reprinted by permission from Rebecca del Rio.\n\n_Collapsing Consciously Meditations: Further Reflections for Turbulent Times_ is sponsored by the Society for the Study of Native Arts and Sciences, a nonprofit educational corporation whose goals are to develop an educational and cross-cultural perspective linking various scientific, social, and artistic fields; to nurture a holistic view of arts, sciences, humanities, and healing; and to publish and distribute literature on the relationship of mind, body, and nature.\n\nNorth Atlantic Books' publications are available through most bookstores. For further information, visit our website at www.northatlanticbooks.com or call 800-733-3000.\n\nISBN: 978-1-58394-758-6\nContents\n\nAbout the Author\n\nEvery new beginning comes from some other beginning's end.\n\n\u2014Seneca\n\nEvery January 1st, we stand on the threshold of an extraordinary transition. On the one hand, that day is no different from the one before, except for the fact that it is marked on the calendar as the beginning of a new year. At this kind of threshold we hold newness, oldness, and sameness and in so doing, become aware of the timelessness of our existence. It is only in material reality that \"old\" and \"new\" are relevant.\n\nMany people look eagerly ahead to the coming year, as they notice the opportunity to begin anew. However, we can best utilize this opportunity by deeply evaluating the old. Today may be an ideal time to review the past year and ponder the highlights. What was useful that you wish to cherish and perhaps enhance in the coming year? What was not useful that you wish to avoid in the future? What did you learn from both?\n\nContemplating how you might want the coming year to be may be greatly enhanced by reflecting deeply on how the past year did or did not serve you. What errors do you not wish to repeat? What lessons were learned? What triumphs would you like to bring forth into the coming year and replicate? What have you learned this year about yourself? What have you learned about the sacred? What have you learned about the world?\n\nDespite the challenging times in which we live, every day can be a new one. Our spiritual journey does not vanish because we live in the midst of loss. On the contrary, living with loss _is_ our spiritual journey, as it has been for humankind for millions of years. The threshold of a new year is the end of the world as we have known it\u2014and our capacity to construct a new world, a new culture, and perhaps even a new human species.\n\nAnd in the end, the love you take is equal to the love you make.\n\n\u2014John Lennon and Paul McCartney\n\nIn tough times, giving and receiving love can be more challenging than when life feels easier and less uncertain. Yet when all is said and done, if we lose food, shelter, livelihood, our possessions, or even our lives, what may matter most is the extent to which we have loved and allowed ourselves to be loved. Many people respond to loss and social chaos with suspicion and may even ignore or reject our efforts to love them. In the midst of this turbulence, any one of us may also be inclined to reject love from others. Without love, however, whether given or received, we become something less than human.\n\nAnd what is love? Is it an emotion? Must we like someone in order to love her? Love is as much about what we do as what we feel. It is an attitude of respect, caring, patience, compassion, inclusiveness, and empathy.\n\nWho loves you when no one else will? Do you fully receive and treasure their love? How do you reciprocate? Do you allow current social and economic conditions to compel you to build walls? Instead, savor this love and open your heart to it.\n\nWhen did you last express love to a stranger\u2014someone who may be as anxious about the future as you are? Commit to reaching out today to someone you do not know well\u2014or at all\u2014and express kindness and generosity.\n\nRemember, above all, that the sacred within you loves you eternally and without exception. Take time and space to receive that love. Commune with the divine within, whose love you have done nothing to deserve. Chances are that if you can allow yourself to feel the love of the sacred, you will have more capacity for receiving love from other sentient beings\u2014and more inspiration for expressing it.\n\nThe unexamined life is not worth living.\n\n\u2014Socrates, in Plato's _Apology_\n\nThis quote, attributed to the ancient Greek philosopher Socrates, essentially sums up his purpose for being alive, namely, to examine his own life and assist others in doing the same. He repeatedly emphasized how little our human mind knows and stressed that this reality is not a liability but an asset. In fact, he also said, \"I am the wisest man alive, for I know one thing, and that is that I know nothing.\"\n\nWhen we sincerely examine our lives with utmost scrutiny, we realize the limits of the rational mind. If we are willing to continue our examination honestly, we are likely to arrive at a place of acceptance of our _un_ knowing. Like Socrates, we glimpse that within the unknowing is the Mystery that cannot be known but which incessantly sends us engraved invitations, by way of life experiences, to follow and live in awe of its presence in our lives and our world.\n\nThe Mystery presents us with countless initiations. From Socrates's perspective, his famous process of dialog was a kind of spiritual initiation. For him, the dialog was never about being right; engaging in the dialog made one a midwife at the birth of a new human being. In fact, no one who was unwilling to change could fully engage with him. The ultimate change, according to Socrates, was the bone-marrow realization that one lacks wisdom and, as a result, begins to long for it.\n\nAnd what is wisdom? There are myriad definitions, but the one to which all spiritual teachings seem to point might be described as a \"holy knowing.\" Our human experience seems to range from ignorance to uncertainty to intuition to absolute knowing. In turbulent times, it is crucial to value both our uncertainty and what we do know with certainty.\n\nWhat unknowns do you face today? As you face them, what do you absolutely know in the depths of your soul?\n\nIt is difficult to get the news from poems\n\nyet men die miserably every day\n\nfor lack of what is found there.\n\n\u2014William Carlos Williams\n\nPoetry is making an enormous comeback. I believe this is due to humankind's increasing awareness of the vacuousness of left-brain thinking. Storyteller Michael Meade comments that the world is inundated with information, but there is a dearth of real wisdom.\n\nPoems contain a depth of wisdom that cannot be found in mere information, because they are essentially eruptions of the heart and the right brain. The excerpt from the Williams poem above chillingly reminds us that people die every day for lack of what can be found in poetry. It's as if the poet is hitting us between the eyes with a pebble of truth that does not maim or injure but jolts us into full awakeness.\n\nThus I can only wonder what the world would be like if, for one day, news reporting ceased, and the airwaves, newspapers, and news websites issued one poem after another. One might argue that, if this actually happened, no one would know what was going on. Quite the contrary, I believe our species would know exactly what was going on, but we'd know it at a level far below the linear mind. We would somehow be able to take our soul's \"temperature\" and discover the ways in which it is \"normal\" or above or below \"normal.\"\n\nWhen Williams says that men die for lack of what is found in poetry, he is not talking about mere words but nutrients for the soul. Humanity is starving, and information is utterly devoid of and incapable of producing the food we need. Even as many people around us are hungry for food, shelter, and companionship\u2014even as _your_ body may long for all of these, your soul is equally hungry.\n\nToday, choose a poem to live with. Reread it, or better yet, learn all of it or just a few lines by heart. As you do, notice how it feeds you, sustains you, enlivens you, and restores your soul.\n\nAll a sane man can ever care about is giving love.\n\n\u2014Hafiz\n\nWhat did you come to this book to give? While that may strike you as a strange or even presumptuous question, the truth is that we live in a _taking,_ not a _giving,_ culture. Unless we are awake, we engage in many of our human activities as if we were on a mission\u2014a mission to get something: validation, support, a good feeling, wisdom, new friends, a romantic partner, more money, more things, or more security.\n\nIn indigenous cultures, people generally approach life with an attitude of giving. Naturally, those cultures are concerned with survival and sustaining their communities, but they believe this is best accomplished by giving. When they receive a gift, they assume that it is not intended to permanently remain with them, but that the energy must move, and the gift must be passed on in some way. This attitude results from a perspective of life and the earth community in which everything is dynamically moving and changing, and energy is flowing to and from and among all members of the community. Therefore, a gift given will return to the giver in some way, and a gift received must be passed on, so that the energy of giving and receiving never ceases. From this perspective, one can never experience scarcity, because there is always enough.\n\nPerhaps Western culture experiences so much fear of scarcity because we approach life from the perspective of the question, what's in it for me? This is particularly true in tough economic times. Aren't we all feeling needy?\n\nWhat can you give someone that you have received in this book? Practice doing this today. Notice what happens.\n\nAlso, be aware that you bring specific life experiences and gifts to the reading of this book. If that were not so, you would have discarded it by now, because you would not have been able to understand or appreciate it. So take some time to contemplate what life experiences you have that allow you to feast on what is here. What have you come to this book to give?\n\nThe journey to the underworld is needed for us to realize we are already dead, for us to encounter in full consciousness the truth about ourselves as we are.\n\n\u2014Peter Kingsley, _Reality_\n\nWhether it occurs in an actual public ceremony in one's tribe or deep within the psyche of the modern suburban dweller, sending ripples across our lives and relationships, the soul's initiatory journey to the underworld is a necessary process in our commitment to becoming conscious. While it is a rite of passage attended by deepened wisdom and ripening resilience, its overriding intention is to teach us who we truly are, not what we have come to believe we are as a result of our ego enlargement.\n\nWhat does that mean, \"who we are\"? It means that the sacred Self, the divine or the Creator within, is our authentic core, transcending the human ego and rational mind. Most sacred traditions teach that the purpose of our earthly existence is to discover that the sacred Self is our essential identity and that the ego we have been compelled to create in our human experience is \"already dead\" in the sense that we are so much more. This is the ultimate truth about us.\n\nThus, as we navigate challenging times, some profoundly healing and grounding questions to live with are: _Who_ is navigating these challenges? _Who_ is struggling to survive or even thrive? And most importantly, _who_ do I want to be as I encounter what may feel like infinite loss?\n\nIn these times, our ego designs are being individually and collectively shattered, and if we have not reclaimed or developed a relationship with the sacred Self, the losses may become untenable, perhaps even lethal. Living only from the rational mind and ego is no longer sustainable. Something more is required, because we _are_ something more. Whereas we may have spent years amassing wealth, possessions, and career successes, as well as raising children, our world and its hurting hordes are asking more from us. _Life_ is now asking more of us than we could have imagined in our youth, when we set out to accomplish our fondest hopes and dreams. The collapse of industrial civilization is a journey to the underworld for every inhabitant of planet earth\u2014an extraordinary opportunity to comprehend, perhaps for the first time, the truest truth about ourselves.\n\nI want to be with those who know secret things, or else alone.\n\n\u2014Rainer Maria Rilke\n\nIn challenging times, being alone is not wise for innumerable reasons. Not only do we need companions for our safety and survival, but we find that it's very difficult to navigate turbulence emotionally and spiritually if we are on our own. As life as we have known it unravels, we do not always have a choice about whom we will be with, and our first concern may be the physical well-being of ourselves and our neighbors. Nevertheless, it is reassuring\u2014and ultimately necessary\u2014to commune with other individuals who are able to make sense of what is happening in our world.\n\nHuman beings have myriad reactions to loss, but if we find ourselves in the company of people who cannot grasp the larger picture, we are likely to feel lonely and long for \"those who know secret things.\" For Rilke, it was so important to have wise companions that he stated clearly that he would rather be alone than in the company of those who are not attuned to the mysteries. In no way should this perspective feed our arrogance or grandiosity, but rather support us in finding our companions and in dialoging with them on a regular basis about the meaning of our mutual experiences.\n\nFor those who have been preparing emotionally and spiritually, the collapse of civilization will not be a surprise, and making sense of it as a species initiation will be easier for them than for the millions who find themselves blindsided by the demise. Part of our preparation is the cultivation of community, for without the ability to gather regularly with our companions to discuss and support one another in the chaos, we are likely to become overwhelmed.\n\nThe \"secret things\" to which Rilke refers are the mysteries of the soul, and nothing was more meaningful to him than an intimate connection with the deeper current of his life. Only our relationship with those \"secret things\" can anchor us in the tsunami of change now battering our world. Abiding in the company of others who cherish the mysteries can restore the soul in turbulent times. From the soul's perspective, if this is not possible, then it is better to be alone.\n\nThere is a Beautiful Creature\n\nliving in a hole you have dug.\n\n\u2014Hafiz\n\nAs our world becomes more challenging to live in\u2014as there is less of all things material\u2014there may, in fact, be more aspects of the sacred with which to construct our lives. Deconstruction of the outdated paradigm may well unfold in tandem with a reconstruction of our experience of what it means to be embodied humans on planet earth.\n\nCountless individuals around us\u2014and we ourselves\u2014may have begun to awaken to the beautiful creature inside us, which has settled for living in a hole of our own digging. For years, perhaps decades, we focused time, effort, and resources on achieving the American or some other kind of dream. The mortgage was a kind of certificate of our efforts to realize the dream. We may have spent those same years sitting in a cubicle or enduring brutal commutes, saving diligently for retirement, educating our kids, or upgrading to the next level of our glorious middle-class existence. Certainly, our lifestyle had its rewards, but without being fully aware of it, we were digging a hole and setting up residence in it, because we failed to meet and build a relationship with the person constructing the dream.\n\nThe dream is over, but you remain. Just who was it sitting in that cubicle, making mortgage payments, getting promotions, having children, taking vacations, and paying bills? No doubt some part of you knew there was more to life and more to you, but perhaps you didn't know how to access the \"more.\" For many of us who were married to the American dream, there was a noticeable, or even gnawing, longing for soul satisfaction\u2014more free time, more creative expression, more connection with nature, more meaning\u2014but we failed to pay attention to that longing, choosing instead to follow the dream.\n\nToday, we are left with little but the soul's calling, and the collapse of life as we have known it has created the space and time to respond to the call. We are likely to respond to the call quite differently today than we would have a decade ago, but what matters is not so much _how_ we respond but that we _do_ respond, because what the soul has wanted us to discover all along is the beautiful creature living inside the hole we have dug.\n\nI think, therefore I am.\n\n\u2014Descartes\n\nThe seventeenth-century French scientist Ren\u00e9 Descartes had a turbulent birth, which caused complications for his mother, and when he was one year old, his mother died. Descartes's father, traumatized by the loss of his wife, blamed Descartes and essentially rejected him. The young Descartes received little nurturing and may never have been held. At the age of eight, he entered a private preparatory school and later, according to his father's wishes, received a law degree. His life appears to have been largely devoid of emotion or tenderness. Small wonder that Descartes concluded, _I think, therefore I am._\n\nThe paradigm of industrial civilization is as emotionally vapid as was the life of Ren\u00e9 Descartes. It was shaped by Enlightenment philosophers such as Descartes and further embellished by the values of the Industrial Revolution. Most highly prized were the mind and intellect, particularly in the service of power and profit. Matters of the heart were delegated to women and children as men ruled the world and conquered nature.\n\nAmong the myriad aspects of the old paradigm that will not serve us in the unraveling is _I think, therefore I am._ What seems more likely is that the descendants of those who survive this enormous transition may develop maxims such as _I feel, therefore I am_ or _I dream, therefore I am._ The Cartesian model of reasoning and debate is likely to succumb to an unprecedented emphasis on inclusiveness, cooperation, and the joining of effort and resources to build a world where compassion, justice, beauty, and the sacred are robustly valued. While some may argue that in such a milieu, people will forget how to think and reason, I disagree. In fact, it may be that, for the first time since the Enlightenment and Industrial Revolution, humans will grasp the necessity of integrating the rational and the irrational\u2014the cerebral and the sacred.\n\nAt the end of the age of oil, humans continue to insist that somehow, the human mind will devise technologies that will spare us from collapse. Yet each day brings forth yet another manifestation of the limits of scientific reasoning. The unraveling compels us to shift from addiction to technology toward a soulful exploration of a post-Cartesian perspective, which may be summarized as _I am, therefore I am._\n\nIn my tradition, the fundamental idea is that the community is not formed only by the living: community extends to the realm of the dead. Therefore, the whole notion of relationship with ancestors is inspired by a desire to be able to have an ongoing relationship with this world and the other.\n\n\u2014Malidoma Som\u00e9\n\nMalidoma Som\u00e9, a shaman of the Dagara tribe of West Africa, emphasizes the presence of ancestors in our lives and our need for an ongoing connection with them. While in the West we have defined _ancestors_ as our forebears, the Dagara definition is much broader. Malidoma clarifies this, saying that our ancestors include \"those who have influenced us, assisted us as teachers, as role models, who have crossed over, but who are continuing to be there in our mind.\" Therefore, they are not necessarily our blood relatives but may include people outside our families who have served us in some way.\n\nI am aware of a host of ancestors who profoundly influenced my development throughout childhood and adulthood and into the present moment. Perhaps most memorable was my grandmother, who was my main source of nurturing in a troubled, sometimes violent, family. Other mother and father figures, teachers, mentors, and role models inspired, motivated, and guided me, and many, serving in the role of elder, blessed the gifts I brought into the world so that I might have the courage and confidence to express them.\n\nEven now, in these increasingly uncertain times, our ancestors in the other world are available to support and protect us. It is never too late to cultivate a relationship with them. What if, instead of arguing against their existence, we simply opened ourselves to having a conversation with them? What if we asked for their guidance? What if their perspective in a world beyond this one could serve us as we navigate unprecedented challenges?\n\nWho are your ancestors? What is your relationship with them? How might they be able to assist you? Are you willing to find out?\n\nAre you breathing just a little and calling it a life?\n\n\u2014Mary Oliver\n\nEleven very simple words that have the capacity to shatter one's world, if one is truly paying attention. Such is the masterful poetry of Mary Oliver.\n\nIndustrial civilization has been all about a voracious addiction to infinite, unlimited growth, and our mad obsession with \"progress\" has required us to rush, scurry, control, worry, and blaze through our days and nights with frenetic, agitated, anxious velocity, as if pursued by the furies. All of this has been defined by our civilization as \"success\"\u2014making it, winning the prize, achieving our goals.\n\nYet where in all of this is our life? Is there, in fact, a life behind or inside of the endless bustle of modernity? Now that so many things are slowing down or even coming to a standstill, we have the opportunity to find out.\n\nOne way to access the being behind the barrage of activity in which civilization has entrapped us is through the breath. In fact, wherever we are or whatever we are doing or not doing, the breath, when followed and deeply engaged, can take us to nothing less than \"the kingdom of heaven within.\" At any time, we can consciously, intentionally take time to inhale very deeply\u2014as deeply as possible\u2014hold the breath for a couple of seconds, and then let it out as slowly and as fully as we can. Doing this four or five times invariably calms and centers us. Especially helpful is to pay attention to the \"still points,\" which are the long pauses we make between one breath and the next. Deep breathing in this manner takes us out of the mental world and draws our attention down into the body and into that spacious presence that abides in our depths. In the midst of breathing deeply and with intention, it is impossible to think, and we are compelled to just simply _be._\n\nWe then discover that it is from _be_ -ing, not doing, that a real life evolves. On the one hand, whether we are employed or not, our world may be less frantic than it was ten years ago, but there may also be more stress and trauma as a result of the collapse of old structures and institutions. Now more than ever, we may need to breathe consciously. We may need to discover for the first time what an authentic life looks like.\n\nA good traveler has no fixed plans and is not intent upon arriving.\n\n\u2014Lao Tzu, Tao Te Ching, translated by Stephen Mitchell\n\nThe venerable Chinese philosopher and founder of Taoism, like all great spiritual teachers, understood that the journey is about the journey, not the destination. The rational, linear trajectory of the Western mind insists that we begin at point A and move directly to points B, C, D, and beyond. The collapse of industrial civilization and the ensuing chaos, however, often makes linear thinking and movement absurd, if not impossible.\n\nIn the past, \"reaching a goal\" may have been about earning a degree, getting a promotion, upgrading to a better house or car, saving for retirement, or living one's \"golden years\" in security and comfort. The purpose of life in a civilized world was to \"get somewhere.\" Even as we prepare for collapse, we sometimes become ensnared by the same linear thinking, such as \"I've got to have this, this, this, and this, or I won't be prepared.\"\n\nIf these turbulent times teach us anything, it is that the point of the journey is the journey. Life is now about living day to day or, more appropriately, moment to moment. It is about being present to our lives as much as humanly possible, without focusing on the end result. Do I have enough to eat for today? Do I have shelter for today? What is life asking of me today? Whom might I be able to serve today? Breathing and feeling the aliveness of the body assist us in creating space around all situations\u2014whether we name them \"good\" or \"bad\"\u2014and support our being present to every moment.\n\nTwo decades ago, few of us imagined that life would be as it is today. Perhaps we assumed that we would \"arrive\" somewhere more desirable\u2014economically, physically, socially, or spiritually. If we are not good travelers, according to Lao Tzu, we might conclude that we have done something \"wrong.\" According to the wisdom of the Tao, there is no \"somewhere\" or \"nowhere\" but only _here_ and _now._\n\nAnother way of saying this is that there is nowhere to go but within.\n\nTrust that which gives you meaning and accept it as your guide. Those who look outwards dream but those who look inwards awake.\n\n\u2014Carl Jung\n\nThe collapse of industrial civilization and the ensuing chaos has ultimately resulted from a crisis in meaning in the modern world. Although we attribute collapse to myriad factors, it appears that when all is said and done, modern humanity has failed to find meaning and has settled for all manner of things, experiences, and achievements as substitutes for meaning. In the above quote, Jung asserts that when we look to the external, we dream. His use of the word _dream_ here is not synonymous with the kind of dreaming we do when we sleep, because Jung considered that kind of dreaming monumentally important in our becoming conscious. Rather, he is speaking of fantasy; he says we are allowing ourselves to be lulled into slumber rather than waking up.\n\nMany of us have felt nurtured and empowered by that which gives us meaning, but the culture in which we live has, for nearly five thousand years, minimized or ignored the significance of meaning in our lives. It has told us not to trust that which gives us meaning but to seek satisfaction of our material needs. Today we look around us and see where that has gotten the culture, yet our hunger for meaning has not left us, and amid the chaos in which we find ourselves, we can be certain that there are millions of other human beings who share that hunger.\n\nIn this moment, how are you finding meaning in the unraveling of industrial civilization? It may be helpful to write about this, draw an image of it, or talk to a friend about it. As humanity searches through the rubble of a crumbling civilization, many people are unable to find meaning and therefore cannot abide the devastation of all things material. What can help each of us endure is the cherishing of meaning, particularly the purpose we feel for being here at this monumental time in human history.\n\nToday, seek meaning in all of your experiences and interactions. Be curious about what messages those encounters are bringing you. Nourish and fortify yourself with meaning and support someone else in finding meaning in a world that\u2014for far too many\u2014now feels utterly meaningless.\n\nEverything that is in the heavens, on the earth, and under the earth, is penetrated with connectedness, penetrated with relatedness.\n\n\u2014Hildegard of Bingen\n\nIndustrial civilization has consciously constructed its reign on the notion of separation. In the first place, it insists that humans are separate from the earth. Although it concedes that humans are the most evolved species, it argues, in concert with creationists, that humans are a species above and beyond all others\u2014that they are so superior to other animal species that they have the right to conquer and destroy them. Likewise, all other members of the earth community\u2014trees, stars, insects, lakes, birds, and rivers\u2014are considered less-than-human and therefore legitimate \"objects\" to be owned and subdued for economic profit.\n\nThe notion of separation leads invariably to subjugation, whereas a perspective or relatedness leads to stewardship. For twelfth-century Hildegard of Bingen, every atom of creation is permeated with relatedness, inseparable from every other atom. Therefore, any human \"use\" of what we call \"resources\" is contingent upon a relationship with them. For example, relationship with bodies of water compels us to keep them clean and to use water sparingly. Native Americans knew this principle well when they thanked a hunted animal for giving its life for the well-being of the tribe.\n\nAmid the chaos of a crumbling society, many people who do not understand what is happening tend to isolate and concern themselves only with \"me and mine.\" But now more than ever, we need to rediscover our relatedness, not only to each other, but to the earth. As our Native American elders did, we need to reclaim \"all my relations\" as we interact with other humans and the rest of the earth community.\n\nThe collapse of the old paradigm forces us to respect\u2014that is, to \"look again\" at\u2014the members of the earth community with whom we abide. Only by recognizing and practicing our relatedness will we have the capacity to create a new paradigm and a new culture, one in which we live in harmony with other humans and other species. Ponder today your relatedness to all living beings on this earth.\n\nThis re-enchantment with the earth as a living reality is the condition for our rescue of the earth from the impending destruction that we are imposing on it. To carry this out effectively, we must now, in a sense, reinvent the human as a species with the community of life species.\n\n\u2014Thomas Berry\n\nAs we navigate the treacherous realities of collapse, it is crucial to ask ourselves what collapse may be asking from us. This may seem like an odd question, but Thomas Berry was one of the first to pose the question, what is the _earth_ asking from us? For several years I have been utilizing this approach when confronted with any unpleasant experience. Asking this question assists me in reframing the situation from one of dread to one of possibility\u2014particularly, a possibility for transformation of consciousness.\n\nIn the above quote, Berry tells us that the only way we can rescue the earth is by becoming re-enchanted with it, and in order to do this, we must become a different kind of species. The kind of human we have become, as a result of the paradigm of industrial civilization, is a kind of human that has lost the capacity for enchantment with the earth. Only by extricating ourselves from that paradigm do we regain this capacity.\n\nI suggest that, like the earth on which we live, the collapse process is also asking something from us, and I believe that what is asking is not unlike what the earth is asking: to become a different kind of human being. At this point in human history, we stand on an evolutionary threshold. Moving to the other side of it is not as simple as stepping through a doorway from one room to another. Rather, the process of transition is protracted, challenging, and perhaps even daunting. Collapse _is_ that process, and as we move through it, we do well to continually ask, what does this initiation want from me? At the same time, it may be helpful to keep asking, what does the earth want from me?\n\nIn other words, how can I allow the ordeal of collapse to open me to becoming re-enchanted with the earth? How can I allow my humanity to be transformed and, in the process, fall madly in love with the earth community? What would re-enchantment feel like? How would I live differently?\n\nDeep play is ecstatic play. It involved the sacred and the holy. . . . Sacred places catapult people into deep play.\n\n\u2014Diane Ackerman\n\nPlay, fantasy, the imagination and free exploration of possibilities: these are the central powers of the human person.\n\n\u2014Brian Swimme\n\nIt may seem strange that in a book of meditations on the collapse of industrial civilization, I have included two quotes about play. Yet even in the darkest and most onerous of times, we need to play\u2014especially in these times. Play has always been restorative, but perhaps even more so in times when play seems most unlikely, most illusive, most inappropriate.\n\nNot only does play provide a respite from drudgery, anxiety, and uncertainty, it also energizes us and opens the mind to undiscovered creative options, to which we may not have access when we are beset with the emotional challenges of living in a collapsing world. Even individuals in Nazi death camps found humor amid the horrors of their daily lives and at times were able to secretly engage in short periods of play.\n\nMany of us realize that even in the good old days of full employment, maintaining a household, and living so-called normal lives, we did not play enough. Sometimes we felt guilty or indolent for engaging in purposeless merriment. And today, although there is more time for play, we may feel so emotionally challenged that the desire for play completely eludes us. All the more reason to find humor in our lives and surroundings. All the more reason to find ways to play and people we can play with.\n\nDiane Ackerman reminds us that play space is actually sacred space, and Brian Swimme emphasizes the \"free exploration of possibilities\" as one monumental opportunity offered by play, fantasy, and imagination. The current milieu may require some of us to be creative in order to find the time and space for play. However, the only way to discover the benefits of play in these turbulent times is to experience it firsthand. The proof of the pudding\u2014and the playing\u2014is in the tasting.\n\nI am in the hands of the unknown God. He is breaking me down to his own oblivion to send me forth on a new morning, a new man.\n\n\u2014D. H. Lawrence\n\nDuring the years following World War I, D. H. Lawrence voluntarily left England to travel abroad and spent two years in the American Southwest near Taos, New Mexico, where he socialized with such notable thinkers as Aldous Huxley, Carl Jung, and Georgia O'Keeffe. Lawrence was entranced by New Mexico and the luminaries he was privileged to associate with during those years in Taos.\n\nThroughout his adult life, Lawrence seemed to have the capacity to open to new experiences, not simply because of curiosity, but because of some sense of the divine working in his life. While, in the above quote, he refers to God as male, his openness to the divine was more characteristically feminine than masculine. Although this could be attributed to his sexual orientation, which flowed from bisexual to homosexual in his later years, Lawrence seemed to have grasped that a relationship with the sacred is about paying attention to and surrendering to something greater than oneself.\n\nWhat Lawrence is describing in the above quote is the process of being broken and humbled by life in order to be remade or transformed at the soul level. And it's important to notice that Lawrence was passionately alive and was not depressed or despairing, but he was willing to traverse the myriad emotions that life presents to us.\n\nIn this time of chaos and uncertainty, when we may not even know where the next meal is coming from, we are wise to assume that we are in the hands of an unknown energy that is breaking us down in order to transform us. The more we have practiced surrendering to something greater in less turbulent times, the less daunting our openness to the uncertainty of chaotic times will feel. We may have experienced more losses than we can count, in which case we can be certain that life is breaking us down. We must understand, however, that the breaking is not meaningless. It serves a profound and sacred purpose: the demise of who we thought we were, which makes way for a new morning and a new human being.\n\nSecurity is mostly a superstition, it does not exist in nature, nor do the children of men as a whole experience it. Avoiding danger is no safer in the long run than outright exposure. Life is either a daring adventure or nothing at all!\n\n\u2014Helen Keller\n\nWe have it on good authority\u2014a woman who lived essentially her entire life blind and deaf\u2014that security is mostly a superstition. Imagine the lack of security Helen Keller confronted as a child, unable to see or hear. Most of us are familiar with the play and movie _The Miracle Worker,_ the story of Helen's tumultuous existence with her teacher, Annie Sullivan, who taught Helen how to \"see without seeing\" and \"hear without hearing.\" Clearly, confronting life in the absence of two of the five physical senses establishes on a cellular level the knowledge that security is an illusion.\n\nAnd in this era, with civilization collapsing all around us, we need little proof of the superstition that our notions of security are. In less difficult times, we could banter about the \"adventure\" of life, but today we are living that adventure in ways that we may have never imagined we might be forced to do. We might have already concluded that _adventure_ is hardly the word we would choose for our current experience and that it should no longer be used to describe the daunting challenges we now face, because it mocks the pain of them.\n\nWhile this perspective is a normal response to upheaval, life in the midst of collapse\u2014and for as long as we remain alive on this planet\u2014is either a daring adventure, or it is meaningless and hardly worth living. In more certain times, we craved adventure, but now that each day brings more uncertainty, we may wonder what we were thinking then.\n\nNevertheless, we were all born for this day, this time. None of us arrived here by accident. Adventures usually conclude at a particular destination, and we can look back and say, \"What an adventure that was!\" Today, however, we find ourselves in the midst of an ongoing adventure with a highly uncertain conclusion. Adventure always involves some risk, and the mind- and body-numbing paradigm of industrial civilization has offered little of that. Better to be rational and scientific, do our research, and be \"secure.\" On the other hand, perhaps you are now discovering that the only security you have is in accepting the present moment and opening to whatever it holds, for indeed, _now_ is the only moment you will ever have.\n\nYour greatest protection is being conscious.\n\n\u2014Eckhart Tolle\n\nIn the tradition of ancient wisdom, Eckhart Tolle has brought phenomenal awareness to the West during the early years of the twenty-first century. The popularity of his teachings swelled at about the same time that many visionaries were beginning to speak of the collapse of industrial civilization. And upon close examination of Tolle's work, with its emphasis on the dissolution of the ego and the systems the ego has created, one cannot help but notice that his words resonate exquisitely with a planet in collapse and in search of the sacred at the core of all life in the universe.\n\nHumanity has chosen to become increasingly unconscious during the past five thousand years. Now, with each passing day, it destroys more of itself and the entire earth community with ever more cleverness, wit, and technological sophistication. It is unlikely that we can reverse even a small portion of the damage that has been done; the only recourse we have for ourselves and future generations is to commit to becoming conscious.\n\nLife in a collapsing world is often fraught with danger, and Tolle declares that our greatest protection is to be conscious. Not only must we become conscious in terms of spiritual practice, but we must be conscious of our bodies, our surroundings, our relationships, our energy level, and our intention from day to day.\n\nThe mountain climber and the martial arts champion have one thing in common: absolute, clear-eyed attention in the midst of their practice. The climber knows that if he puts his foot in the wrong place, it could mean instant death. The Aikido black belt understands that a moment of inattention in the midst of a confrontation with an adversary could cause her a fatal or wounding injury. In dangerous situations, life depends on attention, focus, and being present in the body.\n\nHumanity settled for unconsciousness; today we are reaping the consequences of that choice. Our commitment to consciousness may determine the future of those who survive us; it is the only protection any of us has.\n\nPresence is the whole atmosphere of a person or thing. . . . Presence is mainly the atmosphere of spirit that is behind them all and comes through them.\n\n\u2014John O'Donohue, _Eternal Echoes_\n\nThroughout these meditations, I refer to _presence_ or _being present._ What does that really mean? John O'Donohue defines _presence_ several times in his book _Eternal Echoes._ When he calls it \"the atmosphere of spirit,\" he is referring to life-force energy in the body. Throughout his teachings, Eckhart Tolle speaks of \"aliveness in the body.\" Presence is the life-force energy of spirit in the body, and when we feel it, claim it, and allow ourselves to be embraced by it, we can navigate the material world with much more grace, calm, and groundedness than if we are unaware of or estranged from it.\n\nConnecting with presence is the most fundamental task in our commitment to becoming conscious. Without needing to have an impeccable definition of _presence,_ we open to being embraced by it and allow ourselves to feel our life-force energy in the body, recognizing it as an indwelling spirit. We know we are experiencing presence when we are not caught in the endless feedback loop of thought. We take a deep breath, allow ourselves to feel the aliveness of the body, and then open to reverence for this aliveness that is separate from thought and ego. This allows us to retreat momentarily from the external situation and our thoughts about it. We breathe deeply, fall into presence, and surrender to what is.\n\nWhat _is_ presence? In _Eternal Echoes,_ O'Donohue assures us, \"Presence is something you sense and know, but cannot grasp. It engages us, but we can never capture its core; it remains somehow elusive.\"\n\nSo when we \"fall into presence,\" it is not as if we have a clearly defined sense of it, but an intuitive, instinctive sense of it as our true home in the body. Presence, being, the sacred, spirit, the divine within\u2014all are names for the something greater that is our core, our most authentic identity.\n\nConnecting with presence is something we must practice. For all of our lives, we have been encouraged to focus our attention outward and be defined by the external world of thought and physical and mental forms. It takes time and commitment to revere presence and return to it, as if returning to an old friend.\n\nPoetry is not luxury. It is a vital necessity of our existence.\n\n\u2014Audre Lorde\n\nThe word _prose_ means \"straightforward or direct speech.\" Prose is linear and has to do with proceeding from point A to point B. The word _poetry,_ however, is related to singing, feeling, creating. It is not linear but holistic and nonsequential. Being moved by a poem is not unlike being moved by a painting, a symphony, or a sculpture. Furthermore, writing a poem is akin to creating any monumental and meaningful work of art.\n\nSharing and writing poetry is also a profoundly social act. Poems are not meant to be locked away in silence, but shared with others in celebration of our common humanity. A powerful practice you can begin with your friends is gathering regularly, perhaps weekly, to share poems\u2014those you love that were written by others and those you write yourself. In agonizing economic times, we can discover a treasure-trove of poetry and share it with friends. Perhaps you cannot afford to give or make a material gift, but you can share or write a poem.\n\nI believe that poetry is nothing less than food for the soul, and as Audre Lorde proclaims, it is a vital necessity of our existence. Among other things, reading poetry aloud or reciting it from memory moves people out of linear thinking and into the body and the heart. I have used poetry for years in workshops and social gatherings to facilitate this shift from head to heart. Sometimes people protest that they have never \"gotten into\" poetry and don't really understand it. Of course, this is the linear mind saying, \"I can't figure this out in my head, so I feel inept, vulnerable, dumb.\"\n\nIn times of collapse and upheaval, it is imperative that we learn a different language\u2014the language of soul\u2014and poetry is indeed the language of soul. It circumvents our defenses, welcomes our longing, grief, fear, anger, despair, joy, sensuality, and confusion. Poetry is food for the soul, and now more than ever in human history, we crave the nutrients it offers.\n\nRather than thinking one perspective is preferable to the other, let's notice that both are somewhat dangerous. Either position, optimism or pessimism, keeps us from fully engaging with the complexity of this time.\n\n\u2014Margaret Wheatley, _Perseverance_\n\nOccasionally, I receive an email or blog reply from a reader who says something like, \"On the one hand, you talk about collapse and the end of life as we know it, but on the other hand, you talk about a new vision, new opportunities, and a new kind of human being. These feel like contradictions to me. It's really crazy-making. Can you explain?\"\n\nThis kind of confusion issues from the either\/or programming of industrial civilization. In that paradigm, we learn that something can only be one way or the other. In the new paradigm, however, we discover that life is complex and multifaceted, and that the collapse of the old paradigm indeed holds a new vision with many opportunities. Throughout her wonderful little book _Perseverance,_ Margaret Wheatley reassures us that this may be the most frightening, uncertain, and challenging time in human history and that it is also an era replete with new opportunities for connection, compassion, healing, and reexperiencing what it means to be human. In fact, she suggests that we are standing on the threshold of becoming an entirely new species as a result of living through these turbulent times, which are also profoundly transformative times.\n\nApparently, we are being asked to sit and live with not knowing how it will all turn out\u2014in this situation, neither pure optimism nor pure pessimism is an appropriate response. \"Much better,\" says Wheatley, \"to dwell in uncertainty, hold the paradoxes, live the complexities and contradictions without needing to resolve them.\"\n\nCan we give up the need to resolve the unknown? Can we simply abide in the unknown, holding both optimism and pessimism, without needing to know the outcome?\n\nThe outward work can never be small if the inward one is great, and the outward work can never be great if the inward is small or of little worth.\n\n\u2014Meister Eckhart\n\nWhat is your work? In present time, you may or may not be formally employed, and if employed, you may or may not be receiving payment in money. During the second decade of the twenty-first century, at the time of this writing, work is being radically redefined. Jobs, as we have known them for most of our lives, are headed toward extinction, along with a plethora of species on our planet.\n\nHumans are coming to understand that their work is not synonymous with paid employment, and millions are reinventing themselves. Still others cannot grasp the redefinition of work and have succumbed to addictions, depression, or suicide.\n\nA broader and deeper definition of _work_ is necessary in order to navigate turbulent times. In a nutshell, your truest work is what you came here to do. That may or may not be your current \"job,\" but it is your work. If you're fortunate, you are getting paid to do what your true work is, or you may be earning a living at a \"job\" so that you can more effectively do your work. A job puts food on the table; work provides meaning and a sense of purpose.\n\nMeister Eckhart tells us that we cannot have successful outer work without fulfilling inner work, and vice versa. The two \"works\" are inextricably connected. In fact, our inner work and outer work constantly inform each other\u2014complementing, supporting, and resonating or not resonating.\n\nIn a world of hurting, broken, terrified, and frequently traumatized human beings, there is no lack of healing work to be done. And in a world of profoundly downsized economies, assisting people in getting their needs met is likely to be a full-time occupation. But equally urgent is our capacity to assist others in waking up to the inner work that turbulence and trauma are calling them to do. Time to check in with self and notice any imbalance between inner and outer works in our own lives. We can _teach_ the balance of inner and outer, but do we _model_ it?\n\nGrief is the work of mature men and women. It is our responsibility to be available to this emotion and offer it back to our struggling world. The gift of grief is the affirmation of life and our intimacy with the world.\n\n\u2014Francis Weller\n\nIn his lovely little book _Entering the Healing Ground: Grief, Ritual, and the Soul of the World,_ Francis Weller confronts the notion that grief is for crybabies or that, on some level, adults don't indulge in sorrow. We live in a culture that tells us that \"just a few tears\" are okay, after which we need to \"put it behind us\" so that we can \"move on.\" From my perspective, most people in the modern world are terrified of grief and fear \"getting stuck\" in it. Michael Meade notes that the word _emotion_ contains the word _motion,_ suggesting that, like water, emotions are fluid physiological experiences that naturally flow and find their own places in the soul.\n\nAs myriad crises of industrial civilization converge and we confront the end of economic growth and therefore the end of a way of life we have known from birth, it is impossible to avoid a sense of loss and the grief that attends it. If we acknowledge the loss, the soul will quite naturally move toward grief. In fact, what is most unnatural is a denial of loss and a refusal to grieve.\n\nMany individuals argue that they don't want their lives to be only \"a bummer\" but want to also experience joy and vitality during turbulent times, even as they navigate the hardships. In _Entering the Healing Ground,_ Weller shares a story from his visit to the Dagara tribe of West Africa, the community that engages in grief rituals on a regular basis. He participated in many of these rituals with the tribe, and one day he encountered a lovely Dagara woman who was beaming with joy. He told the woman that she didn't appear to be experiencing grief in any way and asked her how, in a culture that spends so much time grieving, she could look so happy. She replied that she is so happy because she spends so much time crying.\n\nGrief restores the soul and enhances the connection with our deeper humanity. Conscious grieving allows us to feel more alive and to savor all other emotions, including joy, more exuberantly. Moreover, grief serves not only the grieving individual but the entire world. The earth community, as well as each of us, _needs_ our grief.\n\nArrested personal growth serves industrial \"growth.\" By suppressing the nature dimension of human development . . . , industrial growth society engenders an immature citizenry unable to imagine a life beyond consumerism and soul-suppressing jobs.\n\n\u2014Bill Plotkin\n\nAnyone who is paying attention to the converging crises of our planet rapidly becomes aware, if he or she isn't already, that industrial civilization has infantilized its inhabitants. As someone has said, we are a culture of two-year-olds that refuses to accept limits. Hence we are now living the consequences of our unwillingness to grow up and confront reality.\n\nIn _Nature and the Human Soul,_ Bill Plotkin outlines the developmental, emotional, and spiritual tasks of our species from infancy to death, emphasizing our need for initiatory experiences that deepen the soul and its wisdom. Speaking generally, it's easy to deduce that, as a species, we need to grow up; however, Plotkin details the maturation process of individuals who become eco-soul-centric as opposed to ego-self-centric human beings.\n\nAn eco-centric human being or culture endeavors to serve both nature and the culture of which she is a part. In an eco-centric culture, customs and traditions are rooted in interdependence. Obviously, this is in stark contrast with the ego-centric individual or culture, which perceives itself as separate and often superior to the other. Dependence on authoritarian systems or extreme independence is valued. It's all about me and mine. There is little concern for the sacred but much for the profane\u2014and the word _profane_ is related to the word _profit._\n\nAuthentic adulthood is grounded in the sacred and in one's connection with the rest of the earth community, with which the mature individual recognizes his interdependence. From that perspective, it's all about _all_ of us, including our descendants, at least seven generations forward.\n\nTurbulent times compel us to abandon our infantilized outlook and become reverent, interdependent, eco-centric adults. While we are now experiencing the end of growth on many levels worldwide, there is _no_ limit to emotional and spiritual growth. It is the one and only thing we can have as much of as we like.\n\nWhat it is we are hungering for can never be fulfilled by a mate, a job, money, a new this or that. What we hunger for is the other world, the world that sustains our lives.\n\n\u2014Clarissa Pinkola Est\u00e9s\n\nFor several years I have been writing, speaking, and teaching the urgency of preparing emotionally and spiritually for turbulent times. For some people, this is irritating, because they are more comfortable focusing on logistical preparation, such as having enough food and water, sustainable shelter, and all manner of skills necessary for navigating life after the age of oil and other resources. Frequently, a reader or listener will express his or her frustration with my insistence that no amount of logistical preparation can compensate for a lack of emotional and spiritual sustainability.\n\nIn these situations I almost always raise the issue of meaning. Why? Because without a sense of meaning and purpose, we feel lost, disoriented, and perhaps even feel that life itself is absurd. While many aspects of our lives may provide meaning\u2014family, career, creativity, hobbies\u2014all of these engender meaning as a result of something greater than their tangible, physical presence. Each brings meaning as a result of the qualities they bring _to_ the soul and which they demand _from_ the soul.\n\nI believe that each of us enters the world with a seed of the sacred within us. Either we nourish that seed and experience what flourishes as a result, or we ignore the seed for most or even all of our lives. Yet the seed remains, always waiting to be watered and tended. Industrial civilization has inculcated in us a belief that the seed is not important, that other things\u2014like houses, jobs, competition, material success, vacations, and a public persona\u2014are what matter.\n\nSome have called the seed within us a \"god-shaped hole\" that only the divine can fill. Michael Meade speaks of the \"other-inner-under world\" as the locus of our fate and destiny\u2014the wellspring of all our longing. Sustainability is about what lasts, what is eternal. In turbulent times, we increasingly discover that what we are hungering for is the seed and the other world, from which it springs, for _that_ is what most authentically sustains our lives.\n\nTraveller, your footprints\n\nAre the path and nothing more;\n\nTraveller, there is no path,\n\nThe path is made by walking.\n\n\u2014Antonio Machado\n\nIn these turbulent times, there are many roads in our lives that we will never walk on again. Or perhaps some roads are fading, and we are now able to walk on them only infrequently. Things like air travel, internet access, cell phones, foods imported from far-off lands, health care, perhaps even the daily hot shower\u2014all of these may not be accessible in the future. The collapse of industrial civilization, along with attending phenomena such as peak oil, climate change, economic meltdown, natural disasters, the dwindling of all resources, and the disintegration of all institutions, is erasing many familiar roads, both literally and symbolically.\n\nAs a result of Hurricane Irene in 2011, huge portions of the state of Vermont were obliterated by flooding. I visited Vermont shortly after this devastating event, and many people said, \"Vermont will never be the same, and some of these roads and bridges, we can never repair.\" This and many other natural disasters have radically altered landscapes, erased roads, decimated entire neighborhoods, and changed the course of rivers.\n\nHow do we find our way through such devastation, if we are fortunate enough to survive it? How do we make new \"roads\" through the wasteland of this unprecedented unraveling?\n\nMachado says that we make the road by walking. Old roads are being and will be eradicated. New roads must be constructed, but not by the engineering feats of industrial civilization, on which we have become reliant. We are allowed to look back, but we must know that there is no return and that little from the past will be useful, except for what we have cultivated as a result of our relationship with what is timeless and eternal inside us.\n\nParticipate joyfully in the sorrows of the world. We cannot cure the world of sorrows, but we can choose to live in joy.\n\n\u2014Joseph Campbell\n\nTimes of decline and dissolution are not characteristically times of joy and merriment. Long before the collapse began to unfold, clinical depression was the number one psychological disorder in the Western world. Now that collapse is well underway\u2014and all manner of loss is ubiquitous\u2014depression, rage, suicide, and addiction have become rampant in the so-called civilized world. Everywhere we turn, so much suffering and painful demise. How can we possibly experience joy in this milieu?\n\nI believe that we must work consciously to create joy somehow in our lives every day. We must \"inflict\" joy on our world in small but meaningful ways. Whether it be giving a smile to a passing stranger, playing with a child or pet, telling a joke, or just acting silly\u2014even if someone is watching\u2014moments of joy can punctuate dark times with sudden and welcome beams of light. Such pinprick holes of light in a tapestry of darkness may inspire us\u2014or someone we don't even know\u2014to keep persevering.\n\nJoy is a natural, fundamental human emotion. We cannot live long or well without feeling it. Talk about \"happiness\" abounds in industrial civilization, but little mention is made of joy, and few individuals have plumbed the depths of it. As Campbell implies, we cannot separate joy from sorrow. Sadness and loss make joy possible and palpable; joy makes both easier to bear.\n\nSo many wise women and men throughout history have held immense joy in their hearts as they navigated deep sorrow and injustice\u2014Mother Teresa, Hildegard of Bingen, Gandhi, Helen Keller, Meister Eckhart, Mary Oliver, and many more.\n\nAs old paradigms unravel, we can celebrate their demise alongside infinite opportunities for constructing new paradigms. Could there be a more exciting time in human history to be alive? We can hold grief in our hearts in proximity with immense joy and celebration, aware that our ability to feel and savor both allows us to experience the deeper layers and textures of our humanity\u2014layers and textures that we were designed to become intimately familiar with.\n\nOur systems are breaking down today\u2014all of them. And we feel it. All of our professions, all our religions, all our politics and economic and educational establishments need reinventing. They all lack feminine energy, wisdom energy. They lack cosmology and creativity. This gives hope.\n\n\u2014Matthew Fox\n\nHow can breakdown, dissolution, demise, disintegration give hope? Perhaps in order to answer this question, we need to redefine _hope._ Personally, I have not been fond of the word for some time, because it has been used incessantly by people in denial of collapse to ward off their acceptance of the inevitable. In that context, hope is something external to oneself, and it often comes from a politician or cultural icon.\n\nFor me, hope is internal, and it is synonymous with _inspiration_ \u2014inspiration that comes from the soul. In other words, it has much more to do with my internal resources than the efforts of external personalities and institutions.\n\nReflecting on my life, I see that I have lived through many dissolutions, disappointments, unfulfilled dreams, and unwanted endings. I have not only survived them but often thrived as a result. Over time and with great commitment to the deeper lessons the soul wanted to teach, I have noticed that each ending was an opportunity to open to feminine wisdom energy and thereby create something unprecedented.\n\nPicasso noted that every act of creation is first an act of destruction. Just as the losses that the earth community is currently experiencing are unprecedented, so might be whatever our species creates after the dissolution of industrial civilization. As everything breaks down, my inspiration and optimism lie in a vision of unprecedented connection between humans and with the earth\u2014a connection that embodies both the sacred feminine and the sacred masculine. This will be made possible by the demise of hubris and the end of the soul-debilitating manifestations of hubris that we've seen throughout modernity.\n\nWhat supports the life of the soul are stories and imaginations and hidden connections that can never be proven and for that reason continue to exist. Each person makes a leap of faith every time they open their eyes. Those who require absolute proofs have not yet awakened to the unique story of inner truth seeded within their own soul. Until some hint of the divine appears within, no outer proof can ever be satisfactory.\n\n\u2014Michael Meade, _Fate and Destiny_\n\nOccasionally, when I speak or write about emotional and spiritual preparation, I am met with people who argue that all this \"soul stuff\" can't be proven and is too abstract for them to wrap their minds around. With both arguments, I heartily agree. The presence and functioning of soul in our lives and in the world cannot be objectively proven. Nor can the human mind wrap itself around soul. However, if one has ever felt fear, grief, anger, or joy, if one has ever witnessed a sunset that nearly brought one to tears, if one has ever witnessed the birth of a human infant or given birth oneself, one has experienced soul in every cell of the body. Plotting our physiological responses in these situations on pie charts and graphs does not \"prove\" the existence or nonexistence of soul. As Meade emphasizes, the \"proof\" for each individual comes from the inside out and not the reverse.\n\nTo those individuals who assert that \"soul stuff\" is too abstract, I simply invite them to be courageous enough to pretend that soul is at work within their consciousness and to speak directly to it, as if it were another person. Do you have the courage, I ask, to intentionally open to soul and dialog with it about your doubts, your fears, and\u2014yes\u2014your assertion that soul is merely some phantom force made up by humans to help them get through life?\n\nI emphasize that opening to this kind of dialog requires courage, because at first blush, pun intended, the ego is going to feel a bit foolish. It may even argue that only psychotics talk to themselves. However, I challenge the soul skeptic to experiment with his or her own life, not once, but repeatedly, and notice what happens over time. Even a little bit of willingness moves us closer to soul, that mysterious force within us that is waiting for such an opening and can only be experienced, not proven.\n\nI don't think of all the misery but of the beauty that still remains.\n\n\u2014Anne Frank\n\nCertainly, in making this statement, Anne Frank was not dismissing the reality of misery but rather insisting that we balance it with beauty. Of course, balance is tricky, and neither the young author nor any of us is likely to attain perfect balance in this regard. In turbulent times, however\u2014and ours are guaranteed to become even more turbulent\u2014the capacity and willingness to create beauty may prove not only restorative but even life saving.\n\nHidden in the attic of an Amsterdam office with her family, attempting to escape apprehension by the Nazis, she had access to little that one might consider beautiful. She found comfort, however, in her cat and in her writing. In any event, she tells us to seek and savor the beauty that still remains.\n\nAs one travels throughout certain regions of the United States, such as the Rust Belt and the South, visual indications of collapse are ubiquitous. Blight, broken windows, and a distinct absence of beauty insult the eye and the soul. It is not difficult to imagine rampant ugliness engulfing cities and suburbs as collapse exacerbates. Thus, our need to find and create beauty is imperative for those who wish to sustain themselves emotionally and spiritually.\n\nPoetry, story, music, art, and dance are some of the tools available to us for creating beauty in the midst of brokenness. Each has the capacity to knit together within the soul and the senses those many opposites that seem irreconcilable in times of great unraveling: despair and joy, fear and tranquility, austerity and generosity, solitude and community, to name a few. The willingness to create beauty gives witness to our intention to persevere\u2014our commitment to living life soulfully in the midst of changes that will ultimately become soul murdering for countless individuals.\n\nCreating beauty not only is reparative for us but constitutes a generous act of healing for the community. Sharing our appreciation of beauty and resolving to cocreate more, we navigate misery with the eye, ear, and heart that refuses to concede that misery is all we have left. Thus, we rededicate ourselves to forces more meaningful and timeless than mere physical survival.\n\nI know the earth, and I am sad.\n\n\u2014Pablo Neruda\n\nIn these times of unprecedented degradation of the earth, it is not difficult to believe that the earth is a conscious being who is literally wailing in sorrow as a result of humanity's pillage and plunder of her body. Yet at the time Neruda wrote these eight words, much of the earth was still untouched by humans. Was Neruda saying that the earth is sad because of humans or that perhaps sorrow is one of her inherent characteristics, with or without humans?\n\nI write these words during autumn in Colorado. I recently took a day to ascend into the mountains and spend a few hours staring at groves of aspen trees, which radiated their heart-melting golden hues. Soon their leaves will drop to the earth, die, and be smothered by the first snowfall of the year. Whenever I contemplate autumnal dissolution, I am reminded of my mother, who was dogged by depression all of her life but mysteriously seemed to come alive in the fall of each year. I now understand that autumn was her favorite season because it is a time of endings, transitioning us from summer's blazing vitality to the dead of winter. In autumn she may have felt heard and seen by the earth.\n\nDespite its brilliant colors, clear skies, nippy nights, and chilly mornings, which revitalize in us what summer may have drained, there is a sadness in autumn that mirrors the grief held by the earth itself. Even if our planet were a pristine place unspoiled by humans, the earth would be sad. The mere changing of seasons in one year marks four endings and four beginnings in which the earth may hold both sorrow and joy. As for its human inhabitants, perhaps it feels our pain with us. In fact, some indigenous cultures, such as the Dagara tribe of West Africa, believe that the earth loves to receive our grief, because while our sorrow may feel like poison to us, it is indeed food for our \"mother.\"\n\nFor those consciously grieving anything, a restorative practice is to lie upon the earth, face down, and allow one's tears to fall into the earth. Feel yourself giving your grief to the mother; feel her holding and cherishing you in return. Then take as much time as you wish to thank her for feeling your sorrow and for all the inestimable gifts she bestows upon you from moment to moment.\n\nThe second half of life is the ultimate initiation. In it, we encounter those new, unexpected, unfamiliar, and unknowable moments that remind us that we are a sacred mystery made manifest. If we truly understand what is required of us at this stage, we are blessed with an enormous opportunity to develop and embody wisdom and character.\n\n\u2014Angeles Arrien, _The Second Half of Life: Opening the Eight Gates of Wisdom_\n\nIn previous pages we have spoken of the initiation of young people in tribal cultures and their rites of passage into adulthood and into the community. We also noted that life offers us many wanted and unwanted initiations, not the least of which is initiation into elderhood. The role of the tribal elder consists of guiding the community toward adherence to traditions and sacred practices, as well as ensuring that the youth continue to be initiated and grow into wise elders.\n\nElderhood is defined not by chronological age but in terms of wisdom. We all know people in their eighties and nineties who are not mature, wise adults, but rather they are living out a failed elderhood as a result of never finding and becoming who they really are. Likewise, we know people in their twenties and thirties who are wise beyond their years and are already living their lives as wise elders.\n\nElderhood is about discovering the sacred at one's core and thereby understanding what it is one came here to do. Authentic wisdom is about living one's life purpose in order to help guide others to theirs. This _is_ the ultimate initiation.\n\nI believe that at this point in human history, anyone who is awake to the collapse of industrial civilization and is committed to making sense of it in terms of their own and the planet's evolution is already an elder, regardless of age. Thus, those of us who are preparing emotionally and spiritually for humanity's descent are and will be elders in the planetary initiation of collapse. It is our responsibility to be stewards of collapse in our communities\u2014to step forward and assist our fellow humans in understanding the meaning and purpose of collapse and what gifts they possess that must be given in the processes of dissolution and rebirth. We are all moving in the direction of death, but what cannot die is the wisdom we contribute to the earth community as an elder\u2014of any age\u2014in humanity's rite of passage.\n\nWhat you fear will not go away: it will take you into yourself and bless you and keep you.\n\n\u2014William Stafford, \"For My Young Friends Who Are Afraid\"\n\nA woman called me for life coaching one day and said, \"I want to get rid of my fear; I want to live in peace. Fear is a negative emotion, and I want to move beyond it.\" I told the woman that I don't see any emotion as negative and that I couldn't help her get rid of fear, but that I could help her to become a student of it and to allow herself to learn what it might be attempting to teach her.\n\nFear pervades our world in times of uncertainty and loss. If we have a job, we fear losing it. If we don't have health insurance, we fear getting sick. Our culture sends us engraved invitations to be fearful of terrorists, bankruptcy, a low credit score, not having a relationship partner, and ten thousand other things that we may or may not be able to control. While these fears may be legitimate, we rarely allow them to teach us anything; instead, we merely resist or deny the fear.\n\nGavin de Becker's book _The Gift of Fear_ emphasizes that fear is an innate, instinctual emotion that has served the evolution of our species for eons. De Becker says that we need to utilize it rather than attempting to eliminate it. Naturally, this means respecting fear, not being ruled by it.\n\nLike all of the so-called negative emotions, fear serves a purpose, and that is to caution or warn us. It challenges us to become bigger people than we are and to listen to it respectfully. We can journal about it every day or we can sit quietly and dialog with it, noticing the body sensations or images that occur as we do so. In this way, we make friends with our fear and turn it into a useful ally, as opposed to a menacing enemy. In this way, fear \"takes us into ourselves and blesses and keeps us.\"\n\nOur culture tells us that we shouldn't be fearful and that if we just do the right things externally, we can eliminate our fear. Even some spiritual groups make fear an enemy. As a result, the pure emotion of fear becomes distorted into chronic anxiety, making it more difficult to deal with fear directly. If we are going to prepare for anything, especially collapse, some fear will emerge. Our work is not to erase the fear but to become wisely educated by it.\n\nThe Eye of my heart sees everything. . . . The heart is a sanctuary at the center of which there is a little space, wherein the Great Spirit dwells. . . . If the heart is not pure, the Great Spirit cannot be seen.\n\n\u2014Black Elk\n\nIn turbulent times, people respond in one of two ways: by closing their hearts, in an attempt to protect themselves, or by opening their hearts, because compassion swells within them, and they cannot do otherwise.\n\nAs I write and teach on the topic of emotional and spiritual preparation, some people insist on discounting it in favor of logistical preparation. They often do this because the logistical is familiar and because issues of soul sustenance seem foreign and abstract. Yet the collapse of industrial civilization is going to challenge our hearts as perhaps nothing else in human history has done. So many events and experiences will assault the heart, tug at the heart, break the heart, or warm the heart. Those who attempt to remain living only in their heads are going to have unimaginable difficulties with all that our daunting future will foist upon their hearts.\n\nFor this reason, I implore my readers to work on opening their hearts and to do so with safety and support. It is not with our heads that we will figure out how to navigate the turbulence, but with our hearts. As Black Elk, the great Lakota elder, reminds us, the sacred dwells at the center of the heart, but if the heart is not pure (open, vulnerable, receptive), the sacred cannot be seen. The man making this statement survived the Native American holocaust in North America and came to the end of his life not with bitterness and resentment but with an open heart.\n\nNavigating with the heart is about consciously working with the emotions\u2014especially grief, fear, anger, and despair\u2014in present time, so that when the increasing intensity of the demise reverberates through our emotional landscape, we will be able to respond more skillfully and resiliently. While calm reasoning and critical thinking in the face of crises should never be discounted, these alone will prove inadequate to weather a world of heartbreak, because the eye of your heart, not your mind, sees everything.\n\nI said to my soul, be still, and wait without hope, for hope would be hope for the wrong thing: wait without love, for love would be love of the wrong thing; there is yet faith, but the faith and the love and the hope are all in the waiting. Wait without thought, for you are not ready for thought: So the darkness shall be the light, and the stillness the dancing.\n\n\u2014T. S. Eliot, _Four Quartets_\n\nNothing is so maddening as waiting\u2014when we do not know the outcome, when we are fearful and desperate for answers or for some kind of resolution. The last thing we want to hear is, \"wait.\" When we are waiting for a diagnosis or a prognosis; when we don't know if we will get the job; when a quarrel with a lover drives both hearts into uncertainty and doubting; when we've turned in the last paper, taken the exam, but don't yet know what grade we got. Life presents us with myriad occasions to wait, and, usually, our waiting is without hope\u2014no odds or clues as to what the outcome will be.\n\nThe poet is telling us to stop relying on hope, love, or faith, and simply attend fully to the experience of waiting, because that's where we ultimately _find_ hope, love, and faith. If we attempt to use hope, love, and faith to take away the pain of waiting, we will _mis_ -use them, whereas, if we allow our bodies and minds to fully experience the sensation of waiting, we are more likely to discover the gifts disguised in the waiting.\n\nA treasure-trove of blessings abides within every human being, but the soul demands sacrifice before they can be received. It asks us to refrain from attempting to use the gifts to forego sacrifice and, instead, to commit to making the sacrifice, because within the sacrifice lie the blessings that the soul is waiting to bestow.\n\nIn times of voluntary or involuntary waiting, thought is usually not helpful. Why? Because we are not ready for thought. Thought is an activity of the rational mind, and the soul and its gifts\u2014and, yes, the waiting\u2014are all beneath the level of thought. What we _are_ ready for, invariably, is the physiological and psychological experience of waiting. By willingly and consciously attending to the waiting, opposites are transformed: darkness becomes light, and stillness becomes dancing.\n\nEverything wants to be loved. . . . You ever notice that trees do everything to get attention we do, except walk?\n\n\u2014Alice Walker, _The Color Purple_\n\nWe are wired to give and receive love. It could be argued that some other species in the animal kingdom are similarly wired. The English language, however, is extremely limited when considering the word _love._ For that notion, we have only one word, whereas ancient Greek had three: _eros,_ _philos,_ and _agape._ Some Inuit languages have nearly thirty words for _snow._ From this, we may conclude that if a concept is truly important to us, we will have many words for it. But in the language of trade and shipbuilding, from which English originated, we have only one word for love.\n\nOne of the most troublesome words for _love_ in Western culture is _eros._ Modern humans have overwhelmingly associated it with sexual love or \"the erotic,\" to the exclusion of its other meaning, which has little to do with sexuality and everything to do with connection, passion, pleasure, and delighting in life. For example, today is an extraordinary autumn day in my town. After taking myself to lunch at a local restaurant and asking to be seated on the porch, so I could enjoy eating amid the colors of autumn, caressed by a soft, crisp breeze, I walked to a park near my house and sat in the sun under a shimmering, golden aspen tree. I leaned against the tree, breathed deeply, and drank in the cobalt blue sky and the myriad colors of autumn on display in the park. Above me, the leaves rustled, and I closed my eyes and listened. I even spoke aloud, \"What are you whispering? What do you want to say to me?\"\n\nAs I returned to my writing, I felt strangely \"in love\" with life and nature. As a result of spending an hour in intimate communion with other beings in the earth community, I became deeply immersed in _eros._ The trees had whispered to me, and I had listened. The colors, smells, textures, and sounds of autumn caressed my soul and my senses.\n\nIt is especially important in turbulent times to expand our definition of _love_ and allow ourselves to give and receive it in the countless forms in which it shows up in our lives. Now, more than ever, we all want and need to be loved.\n\nPeople who have faith in life are like swimmers who entrust themselves to a rushing river. They neither abandon themselves to its current nor try to resist it. Rather, they adjust their every movement to the watercourse, use it with purpose and skill, and enjoy the adventure.\n\n\u2014David Steindl-Rast\n\nWhether we are swimming in it or riding the whitewater rapids in a boat, the river of life is a terrifying, exhilarating, breathtaking, delightful adventure. In the midst of life's anguishing moments, however, \"adventure\" is not what we normally choose to call our human experience. Rather we may conclude that life is a tragic joke or, as Shakespeare declared, \"a tale told by an idiot, full of sound and fury.\"\n\nWith age and the deepening of wisdom, we are able to gain perspective and notice that life is indeed a saga of agony and ecstasy and everything in between. If we exclude the moments of anguish, we don't really have an accurate portrait of life, and if we minimize or fail to cherish the punctuations of joy with which the human condition invariably blesses us, we end up with a dreary landscape that is both unrealistic and demoralizing.\n\nIn turbulent times, we may feel that the balance is disproportionally tilted in the direction of misery, and so it may be, which is all the more reason to \"radically scavenge\" moments of pleasure, connection, celebration, humor, and satisfaction. No matter how tough the times become, we can experience and, of course, create joy in our surroundings.\n\nAs much as humanly possible, we must be willing to swim in the river of life, because within the deeper currents are treasures waiting to be seized\u2014exquisite jewels of meaning and purpose. Most importantly, however, our willingness to swim in the river of adventure guarantees the experience of aliveness. As myriad philosophers have stated, what really matters is not so much how long we live but whether we live well. And then we must ask, what does a life well lived\u2014a life of exhilarating adventure\u2014look like? David Steindl-Rast implies that a life well lived is simply a life lived by savoring the preciousness of each moment, each event, each encounter with both the human and more than human world. In this manner, we dive into the river of life. Can you entrust yourself to the rushing river of this moment?\n\nPeople say that what we're all seeking is a meaning for life. I don't think that's what we're really seeking. I think what we're seeking is an experience of being alive, so that our life experiences on the purely physical plane will have resonance within our own innermost being and reality, so that we actually feel the rapture of being alive. That's what it's all finally about.\n\n\u2014Joseph Campbell\n\nWherever I travel in the realm of industrial civilization, I encounter human beings who are far less than alive. Body language, facial expression, and lack of animation and passion often communicate more than their predictably articulate speech ever could. Most of the time, these folks live in their heads, with little indication of sensation below the neck. Consequently, their words are not consistent with the activism with which they profess to be engaged.\n\nBut I am not surprised by the deadening effect of modernity. After all, we are socialized into numbness and encouraged to feel almost nothing except short bursts of happiness\u2014never any emotion that lasts more than five minutes. Otherwise, we risk being called \"drama queens.\"\n\nIt seems that only our indigenous friends in tribal cultures know how to feel their feelings and fully inhabit their bodies. I suspect that as collapse intensifies, we will be compelled to choose whether we will feel our emotions and work consciously with them in the context of our bodily experience, or remain numb and therefore perilously at risk of physical and emotional breakdown.\n\nIn Campbell's quote above, we hear him juxtaposing \"meaning\" and \"aliveness,\" as if to say that below our innate thirst for meaning is a deeper longing for aliveness, vitality, and wildness. Domestication may be the most damaging emotional and spiritual catastrophe of the Enlightenment, which prized intellect, reason, lack of emotion, and a ghastly disowning of the body as inferior to and perhaps even the enemy of the mind.\n\nI believe that we must commit to radical aliveness if we are to navigate turbulent times and remain emotionally and spiritually intact. Only if we settle for nothing less than full aliveness will we be able to discover the resonance of a full-spectrum outer reality of external chaos with our innermost being.\n\nThe changes we dread most may contain our salvation.\n\n\u2014Barbara Kingsolver, _Small Wonder_\n\nAs I write these words, in 2011 and 2012, it is obvious that the human race is committing suicide. For decades, many of us have tried to talk ourselves and other humans out of this massive, unprecedented annihilation. Yet if we know anything about humans, we know that, if they are hell-bent on self-destruction, they will find a way to end their lives. Likewise, if a culture is implacably committed to infinite growth, it will terminate life as we have known it and produce a planet that is virtually uninhabitable.\n\nCurrently, every human institution and most ecosystems are in an abject state of collapse. We have been committing suicide for centuries. Is this not similar to the sexual predator who molests children because he wants to get caught, knowing that only incarceration will stop his madness?\n\nAll inhabitants of planet earth are now fully engaged in the process of societal and environmental collapse. But could our societal suicide be the only way to end the ecocide? Could it be that this very apocalypse contains our salvation? Might the transformation of consciousness required to produce a new kind of human being lie in the end of life as we have known it? Is collapse saving us from ourselves?\n\nThe more we recognize this possibility, the easier it may be to navigate present and future upheavals in our lives. Surrendering regularly to what we most fear\u2014to what seems illogical, harmful, or even mad\u2014may constitute a series of \"deaths\" that will not only remake us in this lifetime but more adequately prepare us for the larger death at the end of it.\n\nI believe that in our willingness to collapse consciously lies our salvation\u2014a word I use without its usual religious connotation, but in terms of the well-being of all life on earth. We are now living through changes that we would have never consciously agreed to in an earlier time. Had we chosen to live differently, we may never have been confronted by them. In current time, we have two choices: surrender to them\u2014and discover our salvation\u2014or continue resisting, and perpetuate untold misery for ourselves and all other earthlings.\n\nI can tell you that it takes great strength to surrender. You have to know that you are not going to collapse. Instead, you are going to open to a power that you don't even know, and it is going to come to meet you. In the process of healing, this is one of the huge things that I have discovered. People recognized the energy coming to meet them. When they opened to another energy, a love, a divine love, came through to meet them. That is what is known as grace. We all sing about amazing grace. It is a gift.\n\n\u2014Marion Woodman\n\nOften when I speak of surrender, people protest that surrender is for the weak and that in these times, we need to be strong. Yet one can only know the strength required to surrender by actually surrendering and thereby discovering unspeakable empowerment.\n\nI believe that the forces that have manifested as insurmountable challenge in our lives are the same forces that imbue us with unimaginable grace, love, and power, when we surrender to them. This is not to suggest that we should roll over and capitulate to injustice or tragic loss, but that we consciously confront it, grieve it, and do whatever we can to heal and restore wholeness.\n\nWhen we develop a relationship with the sacred within, we discover that\u2014if we are willing to maintain that relationship and do the work we came here to do\u2014some energy \"comes to meet us,\" as Marion Woodman suggests. In fact, it may be that the sacred is waiting for us to surrender so that it can bestow its gifts. Is it so far-fetched to consider that something greater than the rational human mind and the ego is reaching out to us through tragic loss and through the chaos and adversity of these times for the purpose of facilitating the emotional and spiritual rebirth of humanity? Could it be that who we have thought we are is giving way to who we really have been all along?\n\nHave patience with everything unresolved in your heart and to try to love the questions themselves as if they were locked rooms or books written in a very foreign language. Don't search for the answers, which could not be given to you now, because you would not be able to live them. And the point is to live everything. Live the questions now. Perhaps then, someday far in the future, you will gradually, without even noticing it, live your way into the answer.\n\n\u2014Rainer Maria Rilke\n\nPeople frequently contact me with questions about their lives, hoping that I will have answers. I work with these individuals in many ways, but often, quite paradoxically, I help them distill their confusion into a question and invite them to stay with that question. I have found that, as Clarissa Pinkola Est\u00e9s writes, \"Asking the proper question is the central act of transformation.\"\n\nWe live in a culture that demands answers but has little patience for finding the right question or for remaining with the question\u2014as a kind of Zen _koan_ \u2014until a moment of epiphany emerges. Yet living the question, as Rilke states, is precisely what is necessary, before we are ready to receive the answers.\n\nWhatever dilemma you now face, allow a question to bubble up from inside of you. It may not be the question you were anticipating or would have asked about the situation a decade ago. Journaling may be useful, or sitting quietly and simply asking yourself what you need to ask. Dreams, walks in nature, or creating art may also be helpful. Know that the question is a gift to you from the sacred, for the purpose of deeper exploration. We might think of the question as an oar that we pull to row in the direction we need to go. Without the oar, the boat is of little value in transporting us. Therefore, the quality and condition of the oar is of utmost importance.\n\nWhat is the question that wants to be asked and lived with? Are you willing to live with it, or do you demand an answer now? If you are willing to live with it, it is very likely that you will discover how much more important the question is than the answer.\n\nThe most beautiful experience we can have is the mysterious. It is the fundamental emotion which stands at the cradle of true art and true science.\n\n\u2014Albert Einstein\n\nIn times of chaos and upheaval, one hardly thinks of the mysterious as the most beautiful experience we can have. We are often preoccupied with survival: food, shelter, and getting our basic human needs met. We feel we need to take up residence on the lowest tier of Maslow's hierarchy of needs. Who has time or energy for mystery? Isn't that somewhere near the top of the hierarchy?\n\nIf we are only creatures of survival, how pitiful we are indeed. Our souls desperately need nourishment, and without it, we will certainly perish. We may not have access to spiritual libraries, CDs, or DVDs, as we have in the past. We may even find ourselves amid people who care little about spirituality and, like us, are just trying to survive. Nevertheless, we need \"soul food\" of the interior kind.\n\nOne place to begin exploring\u2014without needing to read a book or attend a workshop\u2014is to notice the mysteries in our lives, the things we can't explain or even make sense of. We can also notice the coincidences, the synchronistic events, the events and experiences that feel illogical. We can be curious about them and linger with the curiosity. Why do we encounter a certain person at a particular time? Why do we end up at a particular place at a momentous moment?\n\nMystery is all around us at all times\u2014in our waking moments and in our dreams. The word _mystery_ is related to _mythological_ and _mystical._ In those realms, the concrete and the symbolic are joined\u2014even if only for an instant\u2014and in that joining resides the sacred. Thus, our souls can be nourished by being present with the mystery, by bowing to it, by cherishing and savoring it.\n\nWe require nothing to honor mystery. We encounter it constantly in our interactions with other humans and with animals. It permeates nature and the more-than-human world. It is meant not to be understood but to be relished as a gift beyond reason, supplying nutrients to the soul and to the soil of our creativity.\n\nI have a feeling that my boat\n\nhas struck, down there in the depths,\n\nagainst a great thing.\n\nAnd nothing\n\nhappens! . . .\n\nOr has everything happened,\n\nand are we standing now, quietly, in the new life?\n\n\u2014Juan Ram\u00f3n Jim\u00e9nez\n\nWhat could be more apt for any of us living in these times? The momentous transitions of our era are sometimes dramatic and sometimes not. We experience \"falling off the cliff\" events, but we also notice, from hindsight, that many things have changed subtly and almost imperceptibly. Chunks of great glaciers suddenly break off and fall into the ocean, and we are stunned, yet for decades, sea levels have been steadily rising, with few people in the world noticing except scientists and the residents of those areas where homes and livelihoods are threatened.\n\nSymbolically, this poem describes the slow, steady unraveling of industrial civilization. At one moment it looks as if nothing is happening, at another moment as if everything is happening, and then\u2014suddenly\u2014we discover that we are standing quietly in a new life. But let us notice that the poet says his boat has struck against \"a great thing.\"\n\nWaking up to the reality that one is actually standing in a new life is huge. While there may be similarities with the old life, nothing is really the same; everything is profoundly different.\n\nWith this realization, it is important to contemplate how this happened. How did I get from there to here? What changed\u2014not only in the world\u2014what changed in me? How has the change affected me and my life? My loved ones? My community? What is this new life I'm standing in, and who is the person standing in it? Who am I standing in it with?\n\nYou have struck against a great thing. Allow yourself to feel the full impact of that reality.\n\nTo hope means to be ready at every moment for that which is not yet born, and yet not become desperate if there is no birth in our lifetime.\n\n\u2014Emily Dickinson\n\nThe great New England poet and philosopher Emily Dickinson, who rarely left Amherst, Massachusetts, and in later years rarely left her room, was not a student of Buddhism, yet one cannot help but notice the Zen nature of the above statement. She suggests in this sentence that we are to hold the tension of opposites by being ready at any moment for a new birth and yet not despair if we never witness a birth in our lifetime.\n\nIn relation to the collapse of industrial civilization and life as we have known it, this might be called the impossible, yet infinitely crucial, perspective. At any moment, new ideas, people, and possibilities could erupt that we have never considered\u2014or that we ruled out as not feasible. Full aliveness demands that we be open to the \"what if\" alongside all that feels hopeless in our world. Yet we can remain fully alive and inspired even if we never witness a new birth.\n\nIt's all about detachment and an attitude of surrender, flexibility, and resilience. It's all about dogged commitment to creating a new world and, at the same time, having capacity to release our hopes and dreams for the future as gracefully as we embraced them.\n\nNote that Dickinson does not define hope as attachment to a particular outcome but as radical openness to possibility and magnanimous surrender to what doesn't happen, even as we dream that it may.\n\nThe first peace, which is the most important, is that which comes within the souls of people when they realize their relationship, their oneness with the universe and all its powers, and when they realize that at the center of the universe dwells the Great Spirit, and that this center is really everywhere, it is within each of us.\n\n\u2014Black Elk\n\nAs I read this statement, I am almost reluctant to write anything about it. It encapsulates virtually everything I am attempting to communicate in this book, and I feel it needs no elaboration. Yet some comments feel appropriate.\n\nBlack Elk begins by emphasizing the \"first peace,\" which he says can only be realized in the soul. He does not say it is realized with the mind, but with the soul\u2014the part of us that connects the sacred within us to the external world. We feel this intimacy with all life in our hearts and our bodies, not with the intellect. As we practice experiencing our connection with the universe, we grow increasingly peaceful, for the experience engenders deep peace. A feeling of separation is at the root of all our anxiety and despair.\n\nOur practice leads not only to peace but to the awareness that the very center of the universe is within each of us. There are seven billion \"centers of the universe,\" and yet there is only one. The Great Spirit is everywhere, and the Great Spirit is the sacred Self of every human being.\n\nBlack Elk witnessed the decimation of his culture by the white man. He and his people were no strangers to turbulent times. His culture collapsed, and life as the Oglala Sioux had known it was destroyed and subsumed by colonizers. Yet even in the face of heinous genocide, Black Elk knew in his bones that the Great Spirit inhabits every human being and that all are one with each other and the universe.\n\nWe too have been colonized\u2014by industrial civilization\u2014and must _un_ -colonize ourselves. We too are witnesses to genocide, corruption, chaos, starvation, drought, war, and disease. Yet Black Elk inspires us to practice the awareness that we are one with the universe and all its powers.\n\nWhat might happen if you were to contemplate this statement from Black Elk every day for one week? The _worst_ that might happen is that you would experience \"the first peace.\"\n\nWe would rather be ruined than changed. We would rather die in our dread than climb the cross of the present and let our illusions die.\n\n\u2014W. H. Auden\n\nFor more than a decade it has been difficult for me to accept the truth of Auden's statement. I have wanted to believe that humans would prefer to change than be made extinct, yet history has not borne out my ideal. In fact, the human species has become suicidal and seems to be willing to do whatever it takes to cling to a particular way of life, one based on power, control, and material comfort\u2014even if that means making itself extinct and the planet uninhabitable.\n\nWe have largely ignored climate change and have turned a blind eye to hydraulic fracturing, which blasts toxic chemicals into the earth in order to extract natural gas, a process that contaminates the air and drinking water for miles around it, not to mention causes earthquakes. In the United States, we have buried our heads in the sand regarding hundreds of aging, dinosaur nuclear power plants, which continue to leak and crack and which pose severe threats if exposed to extreme flooding. We seem willing to wage whatever resource wars necessary in order to maintain our way of life, even as it becomes increasingly clear that that way of life is profoundly unsustainable.\n\nI do not want to believe that human greed and attachment to material comforts are that implacable, yet it appears that they are. We live in a watershed moment in human history: our species will either allow its illusions to die and permit a radical change of consciousness, or we and most other species _will_ perish.\n\nMost writers would not end a meditation on this note. Most would hasten to offer a happy ending or the five things you can do that will make everything better. In good conscience, I cannot do that, because it would not be honest. I can only notice that time has run out; it may already be too late. And I can appeal to you, dear reader, to be an anomaly of your species; allow yourself to be \"ruined\" and your illusions to die. That will not save the earth or our species, but it will bring you home to your own soul and your deeper humanity.\n\nSooner or later, if you are on any classic \"spiritual schedule,\" some event, person, death, idea, or relationship will enter your life that you simply cannot deal with, using your present skill set, your acquired knowledge, or your strong willpower. Spiritually speaking, you will be, you must be, led to the edge of your own private resources.\n\n\u2014Richard Rohr, _Falling Upward: A Spirituality for the Two Halves of Life_\n\nIn these times, we are not likely to run out of the kinds of experiences to which Richard Rohr is referring. Despite the fact that such experiences can be wrenching, terrifying, exasperating, and overwhelming, they are ultimately gifts\u2014only disguised as tragedies\u2014painful grist for the mill of consciousness transformation.\n\nThey are asking of us two things: total and complete surrender and a willingness to persevere, no matter how much we feel like giving up. That means that, contrary to popular opinion, surrender is _not_ the same as giving up. Rather, it is the relinquishing of the human ego to something greater and a commitment to stay with the painful process until we experience internal breakthrough. This breakthrough often occurs inside us before it is manifested in the external world, and when it is manifested, it may not look as we might have envisioned it.\n\nOur greatest and most courageous response is to grasp that we are out of spiritual and emotional skills and that something beyond our ego-identity must take over. At that moment, we do not become incapacitated but rather interdependent with the sacred.\n\nComing to the end of our own private resources is a humbling and salutary experience. Humbling? We certainly understand that, but how is it salutary?\n\n_Salutary_ is related to the word _salute,_ and it literally means \"greeting good health.\" _Health_ simply means \"wholeness,\" and nothing makes us more whole than realizing the end of our ego's limits and surrendering the situation to the sacred. In this way, we cultivate an intimate relationship with the sacred Self within us. We become more of who we authentically are; we become larger human beings and perhaps even larger than the situation that is confronting us. Give thanks for those moments when you are led to the end of your own private resources.\n\nCompassion automatically brings happiness and calmness. Then, even if you receive disturbing news, it will be easier to take, as your mind is still.\n\n\u2014The Dalai Lama\n\nOne of the most challenging mental and emotional states to cultivate in our human experience is compassion. It is not something we can conjure up and make happen. As the Dalai Lama notes, it is the result of a mind that is still. Only a mind that is familiar with the inner world of stillness can begin to experience compassion. Moreover, only a human being who has suffered her own wounds and losses can begin cultivating compassion.\n\nCompassion grows when we make time to become still every day and journey within to be present with whatever is there. It also flourishes as we mindfully attend to our suffering\u2014not with the purpose of wallowing in it, but with the intention of finding meaning in it. If we are willing to engage in these practices, compassion will invariably and noticeably swell, and we will experience our suffering and that of our fellow earthlings with new eyes and ears.\n\nCompassion enhances and is enhanced by the diminishment of the human ego. Empathy increases, and we feel a deeper kinship with the earth community. With time, we discover that our first response to the suffering we witness around us, or the aggression that may be directed toward us, is not fight or flight but compassion. Compassion makes our experience of these painful realities much easier to bear, because we are not viewing ourselves as separate from our fellow creatures but as connected with them.\n\nCompassion does not remove our desire for justice; it strengthens it, because we feel in our bodies how injurious an absence of justice is to ourselves and to all of creation. The compassionate activist is like the Shambhala warrior, who has cultivated courage and a passionate defense of vulnerable beings as a result of an awakened heart. Again, he or she holds the tension of opposites: bone-marrow compassion alongside a fierce commitment to justice, both of which can evolve only through a regular practice of stillness.\n\nYou are an aperture through which the universe is looking at and exploring itself.\n\n\u2014Alan Watts\n\nWhether through the ancient wisdom of Zen Buddhism or the scientific discoveries of quantum physics, we are informed that our existence is inseparable from the universe. Mathematical cosmologist Brian Swimme uses the term _conscious self-awareness_ to describe the process of recognizing ourselves as the aperture through which the universe is observing and delighting in itself.\n\nAs we contemplate this notion\u2014no matter what our role, no matter the status of our human condition\u2014how can we not celebrate our lives as not only immensely meaningful but profoundly vast in magnitude? If Watts is correct, this radically changes how we view not only our very identity but our purpose on this planet and our relationship with it.\n\nIt means that, as an aperture, we are enormously significant, yet also merely a lens through which the cosmos knows itself. We are no more significant than any other living, breathing aperture, but no other aperture is like us. Each of us is a unique opening into the microcosm through which the macrocosm views and knows itself, and vice versa.\n\nWhen we fully contemplate these truths, we are both humbled and exalted, and we are able to authentically comprehend our humanity and our divinity. Ultimately, that means that whatever consciousness evolves in any of us has everything to do with us and, at the same time, nothing to do with us.\n\nCarry with you today this inexplicable treasure. Allow it to inform all of your activities and interactions. Whether you are aware of it in the moment or not, you are conscious self-awareness inhabiting your body and the earth. Notice what happens as you allow yourself to repeatedly contemplate this truth. You will forget this truth many times, but when you do, bring your awareness back to it, and notice\u2014this day and every day\u2014what happens when you do.\n\nTo laugh often and much; to win the respect of intelligent people and the affection of children; to earn the appreciation of honest critics and endure the betrayal of false friends; to appreciate beauty; to find the best in others; to leave the world a bit better; to know even one life has breathed easier because you have lived. This is to have succeeded.\n\n\u2014Ralph Waldo Emerson\n\nFor many years I have kept this quote in a prominent place in my home, because for me, it not only encapsulates\u2014as Emerson says\u2014the essence of success but constitutes the quintessential definition of being fully human, fully embodied.\n\nIn times of upheaval, definitions of success and of what it means to be human are invariably overturned in the light of new awarenesses dawning in collective and individual consciousness. Whenever I read this quote, I find myself asking, how might the history of Western civilization have been different, had we lived our lives according to Emerson's definition of success? Sadly, we have missed the most authentic characteristics of success in our desperation to achieve material comfort and status among our peers. This is precisely why the unraveling of the old paradigm had to occur. We now have a new opportunity; we can redefine success and discover the deeper layers of our humanity.\n\nEach time I read Emerson's words, I notice the opposites in it\u2014intelligent people juxtaposed with children, honest critics with false friends. All descriptions then culminate in the final one: to know that one life has breathed easier because you have lived.\n\nIs anyone breathing easier today because _you_ have lived? Are you breathing easier because _someone else_ has lived? What has been your sphere of influence so far in this life? What is your sphere of influence in current time?\n\nPrior to the enormous transition in which we now find ourselves, we spoke frequently\u2014and, as it turns out, somewhat irrelevantly\u2014about \"making a difference in the world.\" Indeed, many of us have made a difference, but with hindsight, we realize that it may have been too little too late. How we make a difference going forward will be unlike anything we could have imagined. Who in this hurting world needs your healing touch or just simply needs you to listen? What an exciting and terrifying time to be alive! What an opportunity to reenvision and reexperience the deepest layers of success and to taste the wine of fully embodied humanity.\n\nGod guard me from those thoughts men think\n\nIn the mind alone;\n\nHe that sings a lasting song\n\nThinks in a marrow-bone.\n\n\u2014William Butler Yeats\n\nThe Irish poet lived and wrote at the dawning of the twentieth century and the time of World War I. He grew increasingly disillusioned with industrial civilization, particularly in the wake of the \"Great War,\" which killed or wounded millions and accomplished little in the larger scheme of things.\n\nYeats was, in a sense, a mystic who grasped the necessity of holistic knowing, that is, knowing things not just in the mind but with the heart. Throughout this book, you have read and will read repeated comments about human embodiment\u2014inhabiting the body and the emotions, as well as the mind.\n\nIf the project of industrial civilization has accomplished anything, it has been the compartmentalization of the human into mind, emotions, and body\u2014a feat that is hardly praiseworthy. Indeed, it has been devastating to all of earth's inhabitants. This has been particularly injurious to men, who have been socialized by our culture to live in their heads and resist emotional and body awareness.\n\nYet Yeats and all of the purveyors of ancient wisdom have argued that we know nothing unless we know it with all of us\u2014not just our minds, but our hearts and bodies as well. The old paradigm is collapsing in part because it is not sustainable. Human beings cannot exist in their minds alone without doing horrific damage to the planet. Therefore, the collapse of the old paradigm of intellectualizing dissociation is compelling us\u2014through myriad hardships and adversities\u2014to fully inhabit our bodies and finally allow ourselves to feel and know our world with them. For how can we cherish the body of the earth or the bodies of earth's creatures if we are living only in cerebral realms?\n\nIn the words of the Harvard-trained former Green Beret officer Gary Zukav, \"You cannot find your soul with your mind; you must use your heart. You must know what you are feeling.\"\n\nTo consider persons and events and situations only in the light of their effect upon myself is to live on the doorstep of hell.\n\n\u2014Thomas Merton\n\nIn turbulent and terrifying times, it is easy to isolate, hunker down, and focus only on \"me and mine.\" When food, resources, and shelter are scarce, we are forced to make painful decisions about how much we will share our lives with others. Out of self-protection and our desire to protect our loved ones, we can become exceedingly self-focused and concern ourselves only with how events will affect us.\n\nAnother way to describe this mindset is, simply, a life without community. Merton says that living this way is another version of hell. Yet we cannot mindlessly share our resources, time, or energy with everyone we encounter. We need to cultivate discernment in our interactions. Are there situations in which we really should not involve ourselves? When do we intervene or support, and when would that be a foolish choice?\n\nPerhaps, as never before in our life experience, we need to create and maintain boundaries in order to protect ourselves and our families, but this does not mean that we become self-focused. Saying no to someone in need because not to do so would put us in danger is a wise choice\u2014and it may be heartbreaking. This dilemma evokes the oft-quoted adage \"Keep your heart open in hell.\"\n\nHow do we do this? Is it not wiser to harden ourselves, isolate, and become exceedingly self-focused? On one level, it is far easier to do this, but at what price? What does it do to us to numb ourselves to the plight of our fellow suffering humans? What happens to the human soul when it becomes myopically focused on its own needs? Is not this the hallmark of industrial civilization itself?\n\nI have no easy answers for how to navigate our heart's natural tendency to care about our fellow earthlings and the mind's awareness that we need boundaries in an unraveling world. What I can suggest\u2014again\u2014is the cultivation of a stillness practice that takes us into the inner world daily and thereby strengthens our intuition and discernment. Very often, the resolution of the dilemma lies within us rather than in the external environment.\n\nPeople usually consider walking on water or in thin air a miracle. But I think the real miracle is not to walk either on water or in thin air, but to walk on earth. Every day we are engaged in a miracle which we don't even recognize: a blue sky, white clouds, green leaves, the black, curious eyes of a child\u2014our own two eyes. All is a miracle.\n\n\u2014Thich Nhat Hanh\n\nWhat have you lost in the past five years? A job, a home, a partner, insurance benefits, a way of life, retirement savings, your career identity in the world? Have you lost a loved one to death? Have you felt stripped down to the marrow of your bones by the vicissitudes of life in a chaotic world? Any or all of these losses are painful tragedies, yet if you are reading these words, you still have your life, and that is a miracle.\n\nAn over- _load_ of sorrow and loss can over- _shadow_ the miracle of walking on this earth. Take nothing for granted, no matter how simple or \"given\" it seems. In his book _Gratefulness: The Heart of Prayer,_ David Steindl-Rast reminds us that that which is \"given\" is a gift from the universe. When we experience even the smallest and most mundane experiences of our day as gifts, we celebrate our humanity and honor the Source of every breath we take.\n\nI challenge you, dear reader, to take one day in which, during each waking moment, you set your intention to savor every experience as a gift, no matter how ordinary or mundane it may feel. Some miracles are dramatic and dazzling; others are subtle and feel so miniscule that we rarely notice them. Savoring the seemingly banal in our day-to-day existence is another way of inhabiting our bodies, walking on the earth, and practicing being present in each moment.\n\nYou may have noticed, as I have, that when you practice being radically present to life, the future is often altered\u2014at least the future that the ego thought was inevitable. In this way we discover the miracle that is life and the miracle that each of us is.\n\nThere comes a time when the pain of continuing exceeds the pain of stopping. At that moment, a threshold is crossed. What seemed unthinkable becomes thinkable. Slowly, the realization emerges that the choice to continue what you have been doing is the choice to live in discomfort, and the choice to stop what you have been doing is the choice to breathe deeply and freely again. Once that realization has emerged, you can either honor it or ignore it, but you cannot forget it. What has become known can not become unknown again.\n\n\u2014Gary Zukav\n\nMy favorite sentence in this quote from Gary Zukav is the last one. It reminds me of a scene in the movie _Thelma and Louise_ in which Thelma says, \"I've crossed a line; I can't go back.\" Nothing is ever the same after we wake up to the truth of what is so.\n\nA way of life based on infinite growth, power, control, and disconnection from the earth community is over. These turbulent times are the last gasps of a dying paradigm. We can attempt to cling to the old or embrace the new, but we can't ignore the ending of some five thousand years of humanity's grand delusion of separation from itself and the universe. We now know far too much to pretend that life as we have known it will continue.\n\nKnowledge is invariably attended by responsibility. We are always responsible for what we know and what we will do with that knowledge. There is no going back; we cannot un-know what we know.\n\nThe ruling elite of the world have known for decades that the planet could not sustain the number of humans inhabiting it, who are actively destroying it. Some of the titans of enterprise speculated that if they and their cohorts were to make the planet uninhabitable, there would always be the option of colonizing other planets and retreating there.\n\nA few other human beings knew exactly what the ruling elite knew, but they also knew that humanity had the power to reverse ecocide and live in harmony with the earth community. They chose to do something very different with their knowledge. Still others pretended to know very little outside their sphere of comfort and convenience.\n\nWhat have you done with your knowledge, and what have been the consequences?\n\nTo know is to be on the verge of being shattered; for genuine knowledge breaks the shell of what existed before.\n\n\u2014Michael Meade, _Fate and Destiny: The Two Agreements of the Soul_\n\nKnowing is closely related to awakening. In fact, unless we are wide awake, we cannot know in the manner Meade is describing\u2014in the way Gary Zukav explained in the quote in the last meditation. Sleeping minds and hearts cannot know, because genuine knowledge occurs not only mentally but emotionally and physically.\n\nThink about what you have allowed yourself to know in recent years about the world. What was it like when you allowed yourself to know about energy depletion, climate change, species extinction, the extent of corruption that engendered the economic collapse of 2008 and beyond? What was it like to know that these realities were not problems to be solved but predicaments that could only be responded to\u2014predicaments that would result in unprecedented death and suffering? Was not the shell of everything that existed before broken in a thousand pieces?\n\nYet that shattering was less devastating than it is now for millions of people who never allowed themselves to know\u2014who are currently being pulverized by realities they have refused for decades to confront. Zukav describes it beautifully when he talks about the pain of choosing to stop an activity being less severe than the pain of choosing to continue in denial and in destructive behavior.\n\nMost importantly, none of us is finished learning. The learning will continue if\u2014and only if\u2014we are willing to be shattered by what we allow ourselves to know. But the sacred Self is never shattered, even as the ego, the shell, feels eviscerated. We experience this only from hindsight, as we reflect on what we have allowed ourselves to know at various junctures of our lives and how excruciating it has been.\n\nIn this time of unraveling, we increasingly discover the dark side of humanity, but at the same time, we may be fortunate enough to know the depths of empathy, compassion, and kindness of which human beings are capable. This too can break the ego's shell, soften our hearts, and generate epiphanies of blessing and gratitude for being alive at this very painful moment in the human story.\n\nJust sit there right now\n\nDon't do a thing,\n\nJust rest.\n\nFor your separation from God,\n\nFrom love,\n\nIs the hardest work\n\nIn this\n\nWorld.\n\n\u2014Hafiz\n\nThis segment of a poem by Hafiz is one of many that I feel reluctant to comment on, because so little remains to be said about it. Yet I cannot resist singing the praises of what Hafiz is giving us in his own inimitable and magical style. He wants us to feel in our bodies how exhausting it is to perceive ourselves as separate from the sacred. Characteristic of the mystical poets is the use of sensuous exaggeration, which produces an indisputable physical resonance with the reader\u2014a poetic phrase to be reread and savored many times.\n\nAs I absorb this delicious outpouring of attentive consolation from Hafiz, my body relaxes, releases, and begins to tremble toward a tearful letting go. Can you feel it? Can you feel how weary you are from deluding yourself about your inextricable, intimate union with the divine? Are you not ready, by the end of the poem, to surrender to the cushion, the food, the drink that Hafiz offers you as you step over the threshold between worlds into ecstatic communion with the Beloved?\n\nLove the earth and sun and animals,\n\nDespise riches, give alms to everyone that asks,\n\nStand up for the stupid and crazy,\n\nDevote your income and labor to others . . .\n\nAnd your very flesh shall be a great poem.\n\n\u2014Walt Whitman\n\nIn turbulent times, few people are concerned with how to make their lives a poem. Most are preoccupied with survival. However, Whitman's advice to us may constitute the essence of the kind of simplicity that escaped industrial civilization but which we are now in a position to embrace.\n\nIn more prosperous times, we may have read Whitman's directive and called it \"romantic\" or \"quaint,\" believing that it would never actually apply to our lives. It may have felt way too simplistic, but it is important to note the difference between _simplistic_ and _simple._ In this case, _simple_ is astonishingly profound.\n\nSo it naturally follows that we must define the word _poem._ The Free Dictionary online defines a poem as \"a verbal composition the purpose of which is to convey experiences, ideas, or emotions in a vivid and imaginative way, characterized by the use of language chosen for its sound and suggestive power and by the use of literary techniques such as meter, metaphor, and rhyme.\" Poetry differs from prose in that prose is linear and goal directed, whereas poetry is nonlinear, and rather than directing the reader to an outcome, poetry is itself the end result. In other words, we might think of prose as product and poetry as process.\n\nThis is not to say that if one's life itself is a poem, it is therefore purposeless, because what Whitman describes is exquisitely purpose- _ful._ Perhaps the key to distilling his message lies in the very first word: _love._ The rest of the statement exemplifies how one lives love, rather than simply cherishing the emotion of love.\n\nWho do you want to be as you live out the demise of industrial civilization? Why not allow your flesh to become a great poem?\n\nIf God said, \"Rumi, pay homage to everything that has helped you enter my arms,\" there would not be one experience of my life, not one thought, not one feeling, not any act, I would not bow to.\n\n\u2014Rumi\n\nHow many of us can join Rumi in this statement? Notice that he does not say that he enjoyed or looks fondly on every experience, thought, feeling, or act of his life, rather, he says that he bows to them. Why? Because all have helped him enter the arms of the Beloved. Therefore, each was holy, and of course, the word _holy_ is related to _wholeness._ Each experience of our lives, no matter how profane, excruciating, vile, despicable, regrettable, or terrifying, has been part of an initiation into manhood or womanhood and, ultimately, into the wisdom of the tribal elder _if,_ and only if, we are willing to reverence it in the way Rumi demonstrates for us.\n\nI notice that Rumi does not use the language of dogma, such as \"enter the kingdom,\" nor does he use the language of commerce, such as \"enter the storehouse of wealth,\" but rather \"enter my arms.\" He longs to be held in the arms of the Beloved, and he falls on his knees before every moment of his life that has tenderly transported him there.\n\nIt is easy to join Rumi in praising the wrenching experiences of our lives from hindsight. The more daunting challenge is to value the distressing experiences of the present moment as allies escorting us into the arms of the sacred. The lover who leaves, the child who dies, the terminal illness that incapacitates, the career that vanishes, the retirement savings that evaporate\u2014how can we bow to these? Who cares, in the moment, where these experiences will ultimately carry us? We are shattered and shorn of ego identity. But it may be that the tiny flame of Rumi's words, if not extinguished, is powerful enough to sustain us through whatever hell we are traversing in real time. Perhaps all we can do from moment to moment is hold onto the words, \"even this . . . even this. . . .\" Even this is transporting me into the arms of the Beloved.\n\nIt may be that when we no longer know what to do\n\nwe have come to our real work,\n\nand that when we no longer know which way to go\n\nwe have come to our real journey.\n\nThe mind that is not baffled is not employed.\n\nThe impeded stream is the one that sings.\n\n\u2014Wendell Berry\n\nThis stunning statement from Wendell Berry depicts the ego at the end of its tether, and it is an exciting, albeit sometimes exasperating, place to be. In turbulent times, we may find ourselves puzzled about what our real work is and where our journey begins. A sense of lostness and disorientation is inherent at the end of one era and the beginning of another. But Berry assures us that the mind that is not baffled is not employed. In other words, our real work is to allow this disorientation, because in it we find the deeper work and the authentic journey that we are being called to.\n\nOf course, we need not navigate the confusion alone; we must garner the support of trusted others and know that the sacred Self knows where it is, even when the ego does not. What a bewildering but blessed time this is! In such moments we are vulnerable, tender, less defended\u2014all of which activate fertile ground for expansion and expression.\n\nLike me, you may enjoy sitting by the white water of a rushing mountain stream. Sometimes I retreat to the Rocky Mountain forests near my home in order to sit, even if only for an hour, beside a raging brook. The white noise erases from my mind the cacophony of civilization and instills, through its wild roar, the fulminating fury of Mother Earth. Paradoxically, the boisterous sound of a surging stream is calming for me, because there is music in it, as Berry notes, created by the rocks that impede the torrents of water.\n\nIn a state of being perplexed, can we also allow the music that resonates alongside the muddle, the music created by all that we perceive as a barrier?\n\nOne of the ways to heal the ancestors is to grieve them.\n\n\u2014Malidoma Som\u00e9\n\nIn the cosmology of the Dagara tribe of West Africa, ancestors are pivotal. The Dagara experience an ongoing, intimate relationship with those who have passed on to the other world. They believe that the ancestors continue to guide them and that all of the transformative inner work of the living in some way blesses the ancestors.\n\nThe grief ritual is a crucial aspect of Dagara life, and it embodies their cherished relationship with the ancestors. In it, they grieve deeply for themselves and for the world, and they literally give their grief to the ancestors because they believe that, in doing so, the ancestors are healed. In other words, from their perspective, what is poison in this world is food for the other world.\n\nWhile we do not have literal grief rituals in our culture, it makes sense that if there is an \"other\" world, our ancestors in it could be affected by our grief, particularly grief in relation to wounds that we sustained from them. Michael Meade suggests that we may be affected by ancestors as far back as five generations. Even if this is not true, it is extremely helpful to know our roots and to gather information about our ancestors in order to understand the context of our appearance on this planet.\n\nThe Dagara tradition is only one of many in Africa that reveres the presence and role of the ancestors, and not necessarily as our biological relatives. In fact, from the Dagara perspective, ancestors are all those who have left this world and departed to the other world.\n\nI have personally experienced the power and responsiveness of the ancestors by speaking to them, praying to them, and being prepared to listen to them. Their dynamic presence in the other world can be called upon by us, their descendants, for guidance and for our own healing. In our grief, we cultivate in intimate relationship with them that heals them and ourselves. Grief unacknowledged becomes toxic in our bodies, but when released and given to the ancestors, it becomes a spiritual nutrient. Grieve what is unresolved between you and your ancestors and feel how both of you are fed.\n\nEvery individual in the world, regardless of cultural background or race, has an indigenous soul struggling to survive in an increasingly hostile environment created by that individual's mind. A modern person's body has become a battleground between the rationalist mind\u2014which subscribes to the values of the machine age\u2014and the native soul. This battle is the cause of a great deal of spiritual and physical illness.\n\n\u2014Mart\u00edn Prechtel\n\nI believe that one of the principal tasks for inhabitants of industrial civilization who desire to be conscious is the process of uncolonizing themselves and redeeming the indigenous soul. Our roots are not in modernity. Our distant ancestors were not capitalists, did not define themselves by race and national citizenship, and did not follow religions that separated them from other species and from their own bodies. They were tribal people of color who lived in intimate connection with the earth, and that connection informed their cosmology and all of their relationships on this planet.\n\nThus, it is unrealistic and absurd to ignore the traumatic obscuring of the indigenous soul by the rise of industrial civilization. I believe that the fundamental emotional and spiritual disorder of modernity, with profound implications in our bodies, is our estrangement from the indigenous soul, foisted upon us violently by the dominant paradigm of infinite growth, profit, and military might.\n\nAs a result, our bodies, as Native American Shaman Prechtel asserts, have become battlegrounds between what our souls remember and what modernity insists we must think, how we must act, and who we perceive ourselves to be. Inwardly, we are soul-sick and physically debilitated by the colonization of our souls, minds, and bodies. Externally, we live in the context of the complete dissolution of industrial civilization, which will not occur overnight, but by way of a long, painful process. One result of this dissolution, however, may be the reclaiming of our original humanity. In fact, it is not unlikely that the current collapse will ultimately produce a new species of human being, one that, as a result of this agonizing initiation, will have reclaimed its indigenous soul and rediscovered wholeness.\n\nIt may be that the battle will give way to balance, in which the rational mind no longer dominates but finds its appropriate place in whole human beings who cherish the indigenous soul and body as their most precious resources, utilizing reason to sustain the embodied sacred.\n\nThe wounded healer in me knows that healing our driven selves comes from our ability to empower our bodies, hearts, minds, souls, and spirits once more, bringing them into vital unity. Spiritual healing means taking responsibility for being a whole person. We have to take responsibility for being a body, for having a heart, for possessing a mind, for awakening our soul, for opening to our spirit. We need to do right by our body, purify our relationships, use our mind for creative freedom and not enslavement, free the soul from the ego, and undertake the spiritual journey. A whole person is an inspired person, one who embodies the spirit.\n\n\u2014Gabrielle Roth\n\nCould there be a more astute authority on the integration of body, mind, and soul than this phenomenal dancer, musician, and philosopher? She begins by identifying herself as a wounded healer and then mentions \"our driven selves.\" I can think of no better word to describe the inhabitants of industrial civilization than _driven._ Since the Enlightenment and Industrial Revolution, we have been obsessed with progress, growth, achievement, profit, and status\u2014all values deeply inscribed on the human ego and dreadfully inimical to the human heart. Hence the profound wounding we all experience, healers or not.\n\nTaking responsibility for being a whole person is being foisted upon us, as the structures that created our fragmentation increasingly unravel. Have you noticed that this demise does not permit us to live in our heads and that when we try, we fall on our faces? As at no other time in human history, we are being catapulted into embodiment\u2014and often not gently or in ways we would have preferred.\n\nMost of us have never experienced full embodiment, because as children of the Enlightenment, we have regarded the mind as superior to the body. After all, it's much safer to live in the head, where bothersome emotions like grief, fear, and anger are less palpable. For many men, the mind is their only comfort zone. (Let women do the feeling; men must do the thinking.) Or perhaps we have embraced a spiritual path that initially hooked the heart but eventually became yet another avenue of escape from the body. In our effort to evolve, transcend, or \"rise above,\" we became increasingly estranged from human emotion and thus from our embodiment.\n\nDear reader, I invite you to ponder deeply the above paragraph from Roth and then move forward\u2014yes, _move_ the body and allow the emotions ( _motions_ with an _e_ at the beginning) to move you. Allow an unraveling world to embody you.\n\nWhat we call the beginning is often the end. And to make an end is to make a beginning. The end is where we start from.\n\n\u2014T. S. Eliot\n\nIn the Northern Hemisphere of our planet, this time of year\u2014spring\u2014is one of new beginnings, as we notice the vernal equinox bringing with it somewhat warmer days and shorter nights. It is sometimes called the season of new beginnings, but it is, at the same time, a time of endings.\n\nAs we observe the larger context in which spring appears in the Northern Hemisphere, it is important to notice the myriad ends\/beginnings of the current moment. Yes, the old structures of industrial civilization are collapsing, but something else is beginning. Regardless of how much chaos or upheaval we find ourselves embroiled in, this is a time of rebirth, no matter how subtle it may be.\n\nFor many years some people have challenged me to stop using the word _collapse_ to describe our predicament. I have refused to do so because in many cases, the same people who are asking me to do this are people who want to minimize the ending and maximize the new beginning. For me, however, the two are inextricably connected; I cannot speak of one without speaking of the other.\n\nAs Eliot wisely notes, the end is where we start from, knowing that it is also a beginning\u2014but the beginning cannot be fully embraced without acknowledging the ending. Endings must be consciously grieved, otherwise they keep tugging at our souls to recognize and ritualize their significance.\n\nWhat has ended or is ending for you in this unraveling? It is natural to welcome the ending of that which has been deleterious to ourselves and the earth, but even the destructive aspects of civilization may have brought good things to us, such as employment, friends, and conveniences that made our lives easier. Invariably with endings, losses cry out to be grieved.\n\nWhat is beginning for you at this time? Perhaps it feels strange, awkward, even scary. On the other hand, it may feel exciting and enlivening. How are you navigating this end\/beginning in human history?\n\nVulnerable we are, like an infant. We need each other's care or we will suffer.\n\n\u2014Catherine of Siena\n\nThroughout history, humans have needed each other in order to endure and move through crises without being broken by them. As our planet and our species confront unprecedented challenges, survival without support is virtually impossible.\n\nDizzying changes\u2014energy depletion, poverty, violence, climate chaos, homelessness, widespread illness, and the sheer harshness of day-to-day living\u2014wear us down, make us vulnerable. It is one thing to navigate these challenges with others and quite another to try to do so alone. For what we are dealing with, we need massive amounts of support. We may be spending large quantities of time being resourceful adults, but within the capable adult resides a trembling child. We desperately need companions with whom we can share our feelings and dialog about our common vulnerabilities.\n\nVulnerability is tricky. We need to have safe places where we can be vulnerable, and at the same time, we need to be able to trust the people with whom we are exposing the child within. Vulnerability is also an enormous gift that we can give and receive with each other. It softens the heart and activates compassion. Revealing our vulnerability and receiving compassion in return is a healing salve for the soul.\n\nVulnerability brings people together in ways they might not experience in so-called normal times. I see this as one of the most extraordinary blessings in the dissolution of the old paradigm. In trauma and loss, we often find each other in a capacity and with a quality of compassion not possible in earlier times.\n\nHow do you need to be cared for today? Who needs you to care for them? Take some time to reflect on the connections you have formed in the process of societal unraveling. What further connections would you like to make? What gifts do you have to contribute?\n\nPractice means to perform, over and over again in the face of all obstacles, some act of vision, of faith, of desire.\n\n\u2014Martha Graham\n\nWhat is your practice? Perhaps the first thought that comes to your mind is related to a meditation or other spiritual practice. Indeed, those are practices, but not every practice is directly spiritual in nature. You may be practicing a necessary skill, such as gardening, for the first time, or carpentry, herbal healing, homeschooling, blacksmithing, cooking, rainwater catchment, permaculture design, mechanical repair, emergency response care, and the list goes on.\n\nYou may also be practicing life skills, such as compassion, creativity, mediation, patience, communication, active listening, and more. You may be practicing living more consciously in your body rather than being lost in your mind. Perhaps you are experimenting with writing poetry or playing music.\n\nWhatever we intend to practice, life is certain to happen, creating obstacles and distractions, and we are guaranteed the unpleasant task of choosing where we will focus our attention at any given time. Martha Graham mentions three words, all of them nouns that describe actions that are often necessary as we navigate obstacles to our practice: _vision, faith, desire._ I prefer to notice them in reverse.\n\nOur practicing begins with desire\u2014a longing to actualize something\u2014but in order to do so, we must have faith in ourselves. We must believe that we are capable of achieving that for which we long. Desire and faith invariably launch a vision of what our lives may look like if we were to fulfill our desire\u2014and how we might be changed in the process.\n\nI began my commentary by distinguishing logistical and life skills from spiritual practice, but that distinction is artificial. Anything that demands vision, faith, and desire is indeed a spiritual practice, whether or not we choose to name it as such. Whatever stimulates us to exercise these qualities\u2014not only for our own enrichment, but for the healing of the earth community\u2014is unquestionably a spiritual practice. Added to vision, faith, and desire is repetition\u2014practicing over and over and over, remembering that striving for perfection annihilates the soul, whereas commitment to practice enlivens every part of us.\n\nYou gain strength, courage, and confidence by every experience in which you really stop to look fear in the face. You must do the thing which you think you cannot do.\n\n\u2014Eleanor Roosevelt\n\nWhat was your life like a decade ago? How is it different now? Are you currently living in a manner that you never believed you could manage? Did you at one time believe that you could not live without the house you lost, the career in which you had invested so much of your life, the possessions for which you had worked so hard? In the current unraveling of your world, what kinds of unthinkable events have you lived through\u2014and survived?\n\nThink about anything you have done in your life that you never believed you could do. How did that happen? Take time to reflect deeply on this, because all of us are being and will be challenged in the collapse of industrial civilization to do things that, earlier in our lives, would have felt absolutely impossible.\n\nIs there a current \"I could never do that\" belief lurking in your psyche? What fear(s) do you need to stare in the face in order to move toward doing what you believe is impossible?\n\nHer husband said that \"the only thing we have to fear is fear itself,\" and Eleanor said that when we stop and look fear in the face, we gain strength, courage, and confidence. Fear is a natural animal instinct that has enabled our evolution for millennia, and we need to respect it as such. We are more than animals, however, and we discover the depth of our humanity only as we confront, rather than flee or disown, our fear.\n\nIn order to confront fear, we need not only courage and stealth but great support. What resources do you have in your world to support you as you stare your fear in the face? Who are your allies? What gifts do they give you? What other external resources do you have? What internal resources do you have?\n\nPlease reread this page. Notice the tools offered here. What do you feel as you sit with these questions? Is there anything else going on inside you besides fear?\n\nAs a man who has devoted his whole life to the most clear headed science, to the study of matter, I can tell you as a result of my research about atoms this much: There is no matter as such. All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.\n\n\u2014Max Planck, \"Das Wesen der Materie\" [The Essence\/Nature\/Character of Matter]\n\nThese immortal words of the father of quantum physics may be far more palatable to humankind today than when they were first uttered in 1944. Planck was adamant at the end of his life that he did not believe in a personal god, but this passage, from a lecture he gave in Florence, indicates a reverence for something greater than the human mind and the material universe.\n\nAs the vast majority of humans come to the realization that \"things\" and \"stuff\" have not delivered the meaning of life, many are searching, amid the astonishing economic losses of the past decade, for a deeper purpose in their existence. Sadly, many have abandoned that quest and have taken their own lives, because all rationale for their existence vanished along with their material status. Some who remain have found meaning in family, community, nature, or service, but many more suffer from depression, despair, and feel adrift on a sea of listless despondency.\n\nThe current crisis of industrial civilization is characterized by a loss of faith in matter and the realization, however obscure, that the energy behind and vibrating through matter is that of a conscious, intelligent mind, which is indeed the matrix of all matter. Regardless of how much or how little \"stuff\" we have, we can only know with certainty our relationship with the matrix of all matter by way of our own personal experience. The good news, however, is that we _can_ experience this life-force energy, beginning with its manifestation in our own bodies. No matter how depressed or listless we are, the matrix of all matter exists within us as intelligent creativity or, if you prefer, creative intelligence.\n\nThe energy coursing through all matter holds the key to our existence\u2014why we are here and what we came here to do. This energy holds together the solar systems of the atoms in our bodies and holds the key to our aliveness. It is the matrix of all matter and the framework of our being.\n\nThe descent into the depths always seems to precede the ascent.\n\n\u2014Carl Jung\n\nI have the greatest respect for author and futurist John Michael Greer, who manages the Archdruid Report online. In _The Long Descent_ and _The Ecotechnic Future,_ Greer has brilliantly articulated humankind's predicament in the face of energy depletion, economic meltdown, and environmental catastrophe. Yet in \"Bringing It Down to Earth,\" posted on his blog in 2011, he summarizes the paramount theme of my work: _The technical dimension of our predicament is less important than the inner dimension because until we address the inner, we are doomed to worsen the severity of our situation._\n\nAfter more than a decade of awareness about peak oil and libraries of books and documentaries on preparing for it, some of the individuals who have most incisively assessed our plight are now asserting that without a transformation of our inner world, nothing constructive is going to emerge from our best attempts to address the daunting issues confronting us.\n\nGreer is pleading with us to \"bring it down to earth,\" bring our attention down to the inner world and down into our bodies. It is as if he is beseeching us to drag it to the soles of our feet and feel them nakedly ensconced on the lush, grassy earth. A writer whose style is unmistakably cerebral, Greer shifts his tone in his blog post and confronts us with what is at stake if we continue living in our heads, minimizing our inner preparation. The soul blossoms and flourishes not by going upward but by going down into the depths of emotion, body sensation, and intimate communion with nature.\n\nThe demise of industrial civilization is pulling everything downward. What would happen if, instead of resisting, we surrendered to the downward momentum and allowed ourselves to become a transfigured human species?\n\nThere have been tyrants and murderers and for a time they seem invincible but in the end, they always fall\u2014think of it, always.\n\n\u2014Mahatma Gandhi\n\nThe unraveling of industrial civilization is the inevitable consequence of humanity's lack of relatedness with the earth community. It has been a monstrosity created not only by greed but also by tyranny and murder. It could not have survived without massive corruption, exploitation, and abuse at the highest levels of society. In fact, many of its principal players have been blatant sociopaths who lacked moral conscience and declared their own egos the center of the universe.\n\nSuffering and injustice are part of the human condition, and while we must endeavor to eliminate them at every turn, the reality of our existence on this planet dictates that total elimination is impossible. Nevertheless, as all champions of social justice have declared down through the ages, the death-makers always fall, without exception.\n\nFor this reason alone, we should be inspired to persevere and to hold a vision of the next culture after this one, which we are midwifing in this moment. Although we may not see it in our lifetime, we are creating it, even as we read the words of Gandhi and revere his mission on earth. Thus, it is possible that our children's children may live in a world of unprecedented justice and compassion. It is also possible that our species, along with most others, will become extinct. There are no guarantees that either scenario will prevail.\n\nWhat we can know with certainty is that the trajectory of destruction and misery perpetrated against the earth and all species living on it will ultimately cease. This destruction is _not_ invincible, nor is it sustainable. We cannot prevent the annihilation of industrial civilization and life as we know it, but we can minimize the suffering and loss of life unleashed by tyrants and murderers by fortifying our emotional resilience and by assisting others in doing the same. Ultimately, the physical survival of the body is less important than the deepening of the human soul and our becoming who we came here to be in the midst of the dismantling of a failed paradigm.\n\nThe question is not \"Can you make a difference?\" You already do make a difference. It's just a matter of what kind of a difference you want to make, during your life on this planet.\n\n\u2014Julia Butterfly Hill\n\nIn the words of one of the great activist heroines of the late twentieth century, we are assured that it is impossible for our presence on this planet _not_ to make a difference. For 738 days in Northern California, from 1997 to 1999, Hill sat 180 feet above the ground in a tree she affectionately named Luna in order to protect the 1,500-year-old California redwood from being cut down. Not only did she protect Luna, but an agreement was reached with Pacific Lumber Company to preserve all trees within a two-hundred-foot buffer zone.\n\nHill risked her life to defend a tree. How many of us have risked or are willing to risk our lives for some part of Mother Earth's body? Yet risking our lives is not required for making a difference, because as Hill has told us, we already do. Not only have we made a difference in earlier times, whether positively or negatively, but we are making a difference today in how the collapse of industrial civilization will unfold.\n\nThe question then is, _how_ are you making a difference? What are you doing in terms of your inner world to extract as much wisdom from humanity's descent as possible? What are you doing to assist others in discovering the powerful resources within themselves that can sustain them in dark times? How are you serving your community, your neighborhood, your household?\n\nPerhaps you are making an enormous difference but underestimate your capacity to do so and therefore do not recognize your impact. Maybe you feel overwhelmed by the scope of the devastation or despair you see around you. How small one can feel in the face of so much dissolution! Yet things are not always as they seem, and it is more likely that you are making much more of a difference than you realize. Certainly, Julia Butterfly Hill did not anticipate that her 738 days of tree sitting would result in an agreement to protect many other trees from being cut down.\n\nOur work is not to control the outcome but to show up with the intention of making a difference. Any indication that we have done so is a gift to be savored.\n\nI do not allow myself to be overcome by hopelessness, no matter how tough the situation. I believe that if you just do your little bit without thinking of the bigness of what you stand against, if you turn to the enlargement of your own capacities, just that itself creates new potential. And I've learned from the Bhagavad-Gita and other teachings of our culture to detach myself from the results of what I do, because those are not in my hands. The context is not in your control, but your commitment is yours to make, and you can make the deepest commitment with a total detachment about where it will take you. You want it to lead to a better world, and you shape your actions and take full responsibility for them, but then you have detachment. And that combination of deep passion and deep detachment allows me to take on the next challenge, because I don't cripple myself, I don't tie myself in knots. I function like a free being. I think getting that freedom is a social duty because I think we owe it to each not to burden each other with prescription and demands. I think what we owe each other is a celebration of life and to replace fear and hopelessness with fearlessness and joy.\n\n\u2014Vandana Shiva\n\nIn a nutshell, we have from this brilliant, heartful activist a kind of prescription for living and working in dark times. The words that first leap out at me are _commitment, passion,_ and _detachment._ Without these, it is virtually impossible to navigate the current milieu, and with them, we have full capacity to be gloriously surprised by the impact of our efforts to make a difference. We just do what we feel called to do, and we remain free of attachment to outcome. What could be more liberating not only for ourselves, but for the world?\n\nThe reward is not necessarily in seeing the results of our efforts, because we may never see them, but in demonstrating courage and experiencing joy. Everything we have done or will do to make a difference\u2014every step we take in creating a new paradigm\u2014is all about the celebration of life. Is this not what the next culture must be about? Is not its unwillingness to celebrate life the very essence of industrial civilization and the glue that holds its unprecedented death machine together?\n\nWe cannot be passionate or committed if we are encumbered with attachment to outcome, because the attachment constricts or even kills life within us. As we set out to make a difference, what matters most, according to Vandana Shiva, is our own freedom and fullness of life.\n\nThe Beloved is all; the lover just a veil.\n\nThe Beloved is living; the lover a dead thing.\n\n\u2014Rumi\n\nThe mystical Persian poet Rumi uses the name \"the Beloved\" throughout his poetry. For Rumi, the Beloved is not a human lover but the divine within, the Atman, the sacred Self. In the above quote, _lover_ does not refer to a human being but rather the human ego. Most mystical teachings clarify that our earthly existence is a battle between the ego and the sacred Self. Our ultimate purpose in being born into a body on the earth plane is to fully experience and express the Self and diminish the ego. While the ego is necessary for our survival on earth, it invariably attempts to usurp control or, in other words, to be God. Moreover, the ego is profoundly limited in its ability to assist us in navigating tough times.\n\nThe spiritual journey takes different forms for different individuals, but the red thread that runs through the odyssey is the ego-Self (lover-Beloved) conflict. Paradoxically, a spiritual path that facilitates the expansion of the Self and minimizes the human ego is a path that is likely to deepen the quality of our human experience and our passion for life. It is possible, however, to immerse oneself in spirituality to such an extent that one loses touch with the ordinary, which may, oddly enough, nourish the ego and its designs.\n\nRumi and the mystical poets always bring us back to the congruence of the miraculous and the mundane by using language and images that are sensuous, erotic, and sometimes dripping with passion. The lover (the ego) is a dead thing, but the Beloved\u2014the divine within\u2014longs to infuse us with a delicious delight in our human embodiment. The ego deadens this holy enchantment by removing us from our bodies and catapulting us into the intellect. As if this were not offensive enough to our physicality, the ego may attempt to instill guilt, fear, shame, or other emotions to sabotage our receptivity to the Beloved.\n\nWhat opportunities will you allow today for reveling in the Beloved through your physical senses, your emotions, and your ability to create and appreciate beauty? As you open to them, notice the ego's resistance\u2014and keep allowing them.\n\nand how we are all\n\npreparing for that\n\nabrupt waking,\n\nand that calling,\n\nand that moment\n\nwe have to say yes,\n\nexcept it will\n\nnot come so grandly,\n\nso Biblically,\n\nbut more subtly\n\nand intimately in the face\n\nof the one you know\n\nyou have to love\n\n\u2014David Whyte\n\nWho is the one you know you have to love? In turbulent times, even if we are deliciously enfolded in loving community, there is yet one person we must face, knowing we have to love him or her. How well are you loving and caring for yourself these days? For many, these are agonizing, if not brutal, times. For others, who have equated self-care with material possessions and the resources that permit gym membership, trips to the spa, and exotic vacations, the capacity to love oneself may feel inaccessible. But that is so only if \"things\" are needed in order to love ourselves. The more things are stripped away, the more we are left unencumbered and alone with ourselves. For some, it is easier to take care of oneself now that life is simpler. Can we love ourselves wholeheartedly, passionately, in the midst of a downsized life? Can we admire and appreciate the face in the mirror that wears no makeup? Can we honor the healthy body that no longer ripples with Bowflex-toned muscles?\n\nThink deeply and compassionately about how these times have impacted you personally. Do you love yourself more now than ever? If not, why not?\n\nThe more you have, the more you are occupied, the less you give.\n\n\u2014Mother Teresa\n\nAt the same time that we must love ourselves ardently in this era of end\/beginning, we must also give to others as we have never given before. The downsizing, the stripping away, has lightened our load so that many of us are more available to the earth community than we have ever been.\n\nLifestyles and possessions have prevented many individuals from meeting others where they are. However, the great equalizer\u2014the collapse of industrial civilization\u2014has put the majority of humanity on a much more level playing field in ways we could have never imagined a decade ago. The irony of the human condition seems to be that the more we have, the less generosity we express, whereas the less we have, the more we want to share with those who have even less.\n\nFor many inhabitants of industrial civilization, loss of employment, foreclosure, and the diminishment of activities have created enormous space in their lives. Their stuff and the harried lifestyle they were required to live in order to keep their stuff no longer exist. Life as they have known it for most of their days on earth is over. After years of attempting to find another job, millions have given up and have begun to ask, now what?\n\nNow it is time to find our most authentic \"employment\" as human beings on this planet. Certainly, we must find a way to earn a livelihood and learn new skills that will be needed in a post-collapse world, but our livelihood need no longer be our identity. We can attend to our survival and, at the same time, fulfill our purpose in a world in chaos and dissolution. For millions, the loss of long-term employment is providing the opportunity to reevaluate what they came here to do\u2014and the chance to experience, perhaps for the first time, what it is like to generously give their time, their talents, and their hearts to others.\n\nHumans were not meant to live as we have been\u2014inordinately stressed, frenetically busy, enslaved to our possessions, and focused more on taking than giving. So much space and opportunity for sharing has been opened by our losses.\n\nShould not the mind always be in revolt so as to understand the influences that are always impinging, interfering, controlling, shaping?\n\n\u2014Jiddu Krishnamurti\n\nWhat would our world be like today if, during the past three centuries, we had practiced this maxim, posed as a question by one of the world's most extraordinary spiritual teachers?\n\nWhat if we had been conscious enough to perceive the influences of culture and popular consensus, which insisted that the earth was ours for the taking, that resources were infinite, that we could have whatever we wanted, whenever we wanted it? How might we be different? How might our planet be different?\n\nDeeply contemplating these questions is a heartbreaking exercise in futility. The pivotal question is, what have we learned from _not_ being conscious enough to question and filter overwhelming groupthink? Krishnamurti devoted his life and work to assisting humans in extricating themselves from the tyranny of the rational mind and the human ego.\n\nNotice the language he uses: \"should not the mind always be in revolt.\" He doesn't tell us to be cautious, to be wary, to be alert. He admonishes us to be _in revolt_ and, furthermore, to be _always_ in revolt. Krishnamurti's teachings reverberate with rebellion against the status quo and the cultural and familial programming that impede us from attending with laser focus to the sacred Self.\n\nThe social inculcation that deters us from intimate knowledge of the sacred Self is the same deception that has seduced our species into committing planetary plunder. Today, many humans across the globe are breaking the spells of entitlement and infinite growth with which our species has been afflicted for centuries. Wherever an honest recognition of this deception is occurring in the world, it is in our best interest to support it. It is equally important, however, that we revolt not only against centralized systems and institutions but also against the spells cast by the human ego, which will forever attempt to impinge upon us, control us, and shape us into hollowed out, less-than-human dominators who have not yet supplanted power with love.\n\nKnowing your own darkness is the best method for dealing with the darkness of other people.\n\n\u2014Carl Jung\n\nThese are unquestionably dark times. Do they also include moments of love, joy, compassion, celebration, kindness, gratitude, and inspiration? Indeed they do, but all of those lovely qualities occur in the larger context of a planet and a species in decline.\n\nIn these times, some people who were formerly known as pillars of their community, whom everyone admired for their congeniality and generosity, are becoming nearly savage in their bitterness, paranoia, and tight-fisted callousness to the needs of those around them. Others are supporting their fellow humans with unprecedented generosity. Often the responses of people to collapse are not what we would have expected from them.\n\nAs we witness violence, insensitivity, mistreatment of the innocent and vulnerable\u2014as some in our families and communities become mentally ill as a result of the trauma of collapse\u2014it is easy to judge their madness and feel superior, because we have not yet succumbed to our inner demons. What is most important, however, is to recognize that we _have_ inner demons and to discover what they are. Becoming familiar with our own darkness and even consciously dialoging with it\u2014whether by means of journaling, artwork, recording our dreams, or writing poetry\u2014can facilitate compassion. Soon we grasp that there but for grace go any of us.\n\nThe question is not, do I have inner demons? The question is, who and what are they? What triggers them? What are their impulses? How do they harm me, and how do they harm others?\n\nAt another time in his life, Jung said, \"Enlightenment is not imagining figures of light, but making the darkness conscious.\" Dark times tend to make darkness more conscious, and that is part of the purpose they serve. In fact, darkness is here to serve us. The way out is not around, above, or below the darkness, but _through_ it. If we open to it, navigate it, and work with it, darkness invariably reveals the light and accompanies us to treasures of inestimable and previously unthinkable value.\n\nThe passing life of the senses doesn't lead\n\nto knowledge of what our Self is.\n\nWhen we clearly see what our Self is,\n\nthen we shall truly know\n\nour Lord God in great joy.\n\n\u2014Julian of Norwich\n\nIf you are uncomfortable with the words _Lord_ and _God,_ you are among friends\u2014one of them being me. Personally, I prefer replacing \"our Lord God\" with \"the sacred.\" Such wording aside, in these two sentences, Julian is expressing what I have been articulating throughout these meditations: the sacred is that \"something greater\" that abides within us and within the world, within nature, within all beings. It dwells at the core of us, and\u2014like Julian\u2014I call it \"the sacred Self.\"\n\nJulian reminds us that we cannot know the sacred Self through the senses, although the senses and the emotions may very well become conduits of the sacred, if we are willing to experience them as such. Emotions link us with the heart, and the heart is intimately connected with the sacred Self.\n\nBut what difference does it really make if we understand that the sacred lives within us? Why do I repeatedly emphasize it? Because for two thousand years, institutionalized Christianity has endeavored to take the sacred out of the human and put it elsewhere. In doing so, it has convinced millions that it is impossible to connect with the sacred\u2014which, of course, is \"out there\"\u2014without relying on ecclesiastical officialdom. Who benefits from this? Certainly not the masses of humans, who long to find meaning in their life experience.\n\nA visionary far in advance of her time, Julian forsook the notion of the sacred being \"out there\" and adamantly proclaimed the presence of the sacred within us as the essential Self. Had Julian lived in another place and time, she may have had to confront more controversy. After all, from the perspective of organized religion, what could be more heretical than her notion of a sacred Self? I believe that if humans grasp the full import of this reality, it may spark an evolutionary leap. Could this be one of a plethora of purposes of the collapse of industrial civilization?\n\nThe distance between us is holy ground\n\nTo be traversed\n\nFeet bare,\n\nArms raised in\n\nJoyous dance\n\nSo that once it is crossed . . .\n\n\u2014Rafael Jes\u00fas Gonz\u00e1les\n\nSo much to notice in these words, which speak volumes about community and relationship. I first notice the opposites: distance\/crossing the distance, darkness\/light. How can the distance between people be holy? Don't we want to erase the distance and join with the other? Indeed, we all long for connection, yet the distance itself is holy.\n\nPerhaps it is holy because it provides a space in which we can feel our longing for connection. Perhaps it is holy because it allows us to consciously attend to the boundaries humans must create in their relationships with each other. Thus the distance is as sacred as the union because boundaries help us define who we are and protect us from exploitation and unhealthy relationships, which harm both us and the other.\n\nConnection always involves some risk, but with our boundaries intact, we are willing to risk all of the unknowns of relationship\u2014so much so that we remove our shoes and raise our arms in joyous dance. And thus the tracks from here to there light up the darkness of the unknown. In another passage, Gonz\u00e1les says, \"the tracks of our pilgrimage \/ shine in the darkness \/ and light our coming together \/ in a bright and steady light.\"\n\nI notice also that the poet describes the journey as a pilgrimage. First, that means it takes awhile. It does not happen instantly. Moreover, a pilgrimage is an act of devotion and exploration, motivated by an openness to being taught by the experience. What do we learn as we approach engagement?\n\nWhat lights up the connection toward which we move is the space in between ourselves and the other. Clearly, for Gonz\u00e1les, it is sacred space, a void of the unknown where so much can happen, but whatever happens there is guaranteed to illumine the ultimate experience of union, especially if we view the distance as holy, but not empty. In fact, this holy, dark space is reverberating with potential for union, but only because we value and savor the separation.\n\nFor any culture which is primarily concerned with meaning, the study of death\u2014the only certainty that life holds for us\u2014must be central, for an understanding of death is the key to liberation in life.\n\n\u2014Stanislav Grof\n\nMost individuals reading these words have not had the good fortune of growing up in a culture that was \"primarily concerned with meaning.\" For the most part, most of my readers were born into and profoundly shaped by the paradigm of industrial civilization, which cares little about meaning but a great deal about material acquisition. You are probably reading these words, however, because part of you has been attempting for some time to extricate yourself from the old paradigm. By now, you have probably discovered that it is impossible to make sense of one's life or the world in which you live without developing a relationship, however tentative, with death.\n\nWithout exception, whenever I lead groups in a \"die before you die\" exercise, many participants report that they have never felt so alive as in the moments when they were imagining their own death. Perhaps that is because death is the last and most ominous enemy. If we can confront death consciously, we soon realize that the severity of every other threat pales by comparison. When we wakefully attend to death, we discover that it is indeed as the biblical saying goes, the last enemy, not merely chronologically, but because it has the last word. If we are willing to confront it, we discover that we have nothing left to lose.\n\nAll attempts to deny the reality of the collapse of industrial civilization are attempts to deny our own death, to deny the limitations inherent in the human condition. In fact, limits of any kind are a form of death\u2014a rude intrusion into our delusionary domain of entitlement. Certainly the ego experiences limitation as remotely fatal, if not blatantly so. Yet it is precisely life's limits and our own personal limitations that, in fact, offer the key to our liberation. How many times have those very limits opened doors of unfathomable blessing to us? How often in your life have exquisite gifts evolved as a result of some limitation or loss? How often has death liberated you into unimaginable life?\n\nChoose a job you love, and you will never have to work a day in your life.\n\n\u2014Anonymous\n\nA devastating, global wave of unemployment has set many millions of individuals adrift in a sea of panic, confusion, and despair and has caused many of us to wonder whether the concept of a full-time job may go the way of the dial telephone and the typewriter. As a result, countless individuals have begun a journey of reinventing themselves. Many report that their liberation from a job they didn't particularly like\u2014or even hated\u2014allowed them to create work they had wanted to do for a long time. Others report the change opened doors to work they had no idea they would enjoy as much as they do.\n\nIs there anything more inspiring than meeting a man or woman who thoroughly delights in his or her work? Conversely, is there anything more depressing than watching millions of people depart each day for jobs they despise but keep doing because they see no other options for supporting themselves or their loved ones?\n\nClearly, losing one's job is terrifying, angering, and sometimes heartbreaking. On the other hand, when one door closes, even though there is no guarantee that another will open, the possibility of one opening\u2014or perhaps many doors opening\u2014becomes real and tangible. When attempting to reinvent oneself in a new line of work\u2014or a familiar one\u2014two qualities are crucial to consider: (1) loving the work one does, and (2) doing work that serves a practical purpose or improves the lives of individuals or the community. Doing work that we love often feels as if we aren't working at all. Add to that a sense of service in our work, and it gives new meaning to the expression \"labor of love.\"\n\nPerhaps in this moment you are abiding in limbo between past employment and your future work in the world. If you have not already done so, make the time to sit quietly and turn your attention inward. Simply ask your sacred Self to show you what needs to be done and how to do it. Open yourself to the mystery of the unknown and the possibilities that may lie within it. Ask to be shown how your talents, skills, and expertise can be used in ways that will be deeply fulfilling to you and profoundly useful to the earth community. Then listen and wait patiently for guidance and inspiration.\n\nYour soul makes sure that God's dream for you is always edging towards fulfillment even when at times the opposite seems to be the case.\n\n\u2014John O'Donohue, _Beauty: The Invisible Embrace_\n\nIf you don't like the word _God_ in this quote, do not allow that to impede the power of O'Donohue's words for you. If you are reading this book, chances are that you have some sense of your sacred Self. The name you choose to call it matters far less than your awareness that the sacred Self is your authentic Self. The point is this: something greater\u2014beyond your rational mind and ego\u2014has a dream for you. You may have a clear sense of this from time to time and, at other times, have no sense of it at all.\n\nYour soul, the psyche\u2014the part of you that serves as a bridge between the sacred Self and the external world\u2014cooperates with the sacred to bring you to greater consciousness, more awakeness and aliveness, even when the opposite seems true. Occasionally, we find ourselves in the midst of circumstances we did not plan or ask for, experiencing wrenching emotions or what feel like impossible challenges. \"How is _this_ edging me toward greater fulfillment?\" you wonder. The loss of employment, the loss of a relationship, the death of a loved one, a terminal illness, a bankruptcy, a foreclosure, a natural disaster that leaves you homeless\u2014\"if this is the path of greater fulfillment, I don't care,\" you may rail.\n\nIn such excruciating moments, it is nearly impossible to grasp that the soul is at work, bringing us to \"god's\" dream for us, because we can only feel the nightmare of our immediate experience. Yet time and distance from the horror usually reveal that the sacred was dreaming our greatness and moving mysteriously in our lives to facilitate the deepening of the soul and the expansion of awareness. Eventually we discover that, as a result of the calamity, we have become larger, more magnanimous human beings, and probably far more resilient than we could have imagined.\n\nThe tumultuous times in which we live offer us many such teaching moments. Catastrophic as they may be, something greater is dreaming our fulfillment.\n\nWhen the sacred masculine is combined with the sacred feminine inside each of us, we create the \"sacred marriage\" of compassion and passion in ourselves.\n\n\u2014Matthew Fox, _The Hidden Spirituality of Men_\n\nOne of the landmark awarenesses of our species in the twentieth century was the complexity of gender. Yet even in the nineteenth century, Carl Jung wrote of the gender opposite that abides within all of us\u2014the _anima_ within a man and the _animus_ within a woman. Jung believed that in order to become fully conscious and whole, it is necessary to integrate the _anima_ or _animus_ within the psyche. This integration he called the _sacred marriage._\n\nJung also spoke of the _sacred_ masculine and _sacred_ feminine. To the sacred masculine he attributed such qualities as strength, endurance, protectiveness, assertiveness, the willingness to take action, the capacity to make distinctions, and an awareness and defense of limits and boundaries. To the _sacred_ feminine Jung attributed relatedness, receptivity, compassion, acceptance, inclusiveness, creativity, patience, and beauty. The profane masculine or feminine, of course, consists of those qualities that exhibit the opposites of these. While all such characteristics are generalizations with myriad exceptions, their classification by Jung has been one of the most valuable contributions of his work.\n\nRegardless of how we define _masculine_ or _feminine,_ an integration or \"marriage\" of the two in the human psyche is both necessary and rare. The earliest humans revered goddesses and female deities that were associated with the earth and generative powers\u2014energies supplanted by industrial civilization, which reveres power, control, linear thinking, and material gain.\n\nIn turbulent times, we can respond to inordinate levels of stress either by digging our heels into the patterns of the old paradigm or by opening to new ones. Men can either identify more blatantly with the profane masculine, or they can open to their _anima_ and thereby develop the sacred masculine within themselves. Likewise, women can succumb to the seductions of victimhood, powerlessness, envy, and treachery, or they can cultivate a relationship with the _animus_ and empower the sacred feminine within them. The multitude of wounds resulting from the collapse of a profane masculine system have the capacity to produce a new human species in which a sacred marriage within women and men will result in a new lineage of earthlings.\n\nOn the night that the Second World War was declared, there were crowds in the street. It was a summer's night and there was a blackout. On every side you heard people crying: \"Look at the moon!\" The moon had been there every minute of their lives and they'd never seen it.\n\n\u2014Laurens van der Post\n\nAs a result of the turbulent and often traumatic times in which we live, what are you now seeing for the first time? Dark times have a way of sweeping the dross from precious gems we have failed to notice in our frantic attempt to keep a garment, already riddled with holes, from unraveling completely.\n\nPerhaps you are noticing how quickly your children are growing up, or perhaps you hear and see birds in nearby trees that, until now, you were barely aware of. Maybe it's the color of the sky at sunset or a distinct crack in a friend's voice that escaped your notice before but now lingers with you for days, amid the uncertainty that has become the hallmark of so many lives in our collapsing world.\n\nIs this part of the reason collapse is here? Is this what it means to teach us: how to notice, how to pay attention, how to be painfully present to the suffering, neglect, negation, blight, emptiness, rage, despair, and numbness\u2014but sometimes also the joy\u2014in which we seem to be immersed?\n\nIn her poem \"The Summer Day,\" the luminary bard Mary Oliver writes\n\nI do know how to pay attention, how to fall down into the grass . . . how to be idle and blessed, how to stroll through the fields,\n\nwhich is what I have been doing all day.\n\nOliver then asks\n\nTell me, what else should I have done?\n\nDoesn't everything die at last, and too soon?\n\nTell me, what is it you plan to do\n\nWith your one wild and precious life?\n\nWhile there is much work to be done in terms of preparation, survival, and service, we are being compelled to pay attention\u2014as we have never paid attention before\u2014and thereby, \"full-fill\" this one wild and precious life.\n\nRefuse to fall down.\n\nIf you cannot refuse to fall down,\n\nrefuse to stay down.\n\nIf you cannot refuse to stay down,\n\nlift your heart toward heaven,\n\nand like a hungry beggar,\n\nask that it be filled.\n\nYou may be pushed down.\n\nYou may be kept from rising.\n\nBut no one can keep you from lifting your heart\n\ntoward heaven\u2014\n\nonly you.\n\n\u2014Clarissa Pinkola Est\u00e9s\n\nWe could appropriately call this blessing a \"credo for collapse.\" It comes from a woman, a mother, an activist, a Jungian analyst, a storyteller, one who has passed through countless collapses in her own life. No one but such a pilgrim has the authority to give us this kind of poignant, powerful blessing. It deserves careful contemplation and heartfelt attending, not just for a few moments, but for a few days or weeks or years\u2014or perhaps for as long as we live.\n\nThere is nothing I can add to this blessing and nothing to say about it that would not end up detracting from the message intended by the writer. And so, as I am inclined to do with so many of the pearls of perennial wisdom in this book, I leave you alone with it. I only ask that you give it the space and reverence that a blessing such as this one, at this time in human history, is seeking to bestow upon you and the earth community.\n\nI end this missive as Clarissa often ends hers, with a simple \"Aymen,\" which in olden times meant, \"May it be so,\" or \"May it come to pass.\"\n\nWhen the Jew, the Christian, the Moslem, the Hindu and the Buddhist open themselves in prayer, in meditation, to the transcendent mystery, going beyond the word, beyond thought, simply opening themselves to the light, to the truth, to reality, then the meeting takes place. That is where humanity will be united. Only through transcendence can we find unity.\n\n\u2014Bede Griffiths, _The Other Half of My Soul_\n\nFather Bede Griffiths was a nonconventional Benedictine monk, born in India and profoundly influenced by Hindu tradition\u2014so much so that he became Swami Dayananda, a Christian yogi. A devout player in the Christian Ashram Movement, Griffiths's life was radically transformed by the union he experienced and proclaimed between all religious traditions, for he understood on a cellular level that, at their root, all traditions issue from the sacred Self. He knew this not intellectually but by practicing Christian contemplative prayer and Hindu meditation.\n\nHow few individuals can transcend\u2014which simply means _going beyond_ \u2014to such an extent that they know intuitively that all beings are one! This kind of knowing is not about amassing information, reading books, or doing academic research, but results from experiencing\u2014in one's heart and one's entire physiology---the fundamental essence of all life.\n\nWhile taught by all major religious traditions that all are one, few of us have experienced it on the level experienced by Griffiths. Whether we call it \"transcending,\" \"descending,\" or \"ascending,\" until we have been moved so palpably by universal oneness that we are permanently altered in mind, body, and soul, the human ego will argue for separation and live in alienation from the sacred in all beings.\n\nYet I believe that collapse is upon us precisely for the purpose of compelling us to experience\u2014as inexorably as Griffiths did\u2014that there is no separation between humans or between humans and the earth community. Grasping this is perhaps the missing element that could create an evolutionary leap for our species. No doubt a great deal of suffering will be required before the transcendence necessary for life-altering awareness of oneness can circumvent the human ego and carry us home to the sacred Self.\n\nWe work on ourselves in order to help others, but also we help others in order to work on ourselves.\n\n\u2014Pema Ch\u00f6dr\u00f6n\n\nPema Ch\u00f6dr\u00f6n, a Connecticut housewife turned Tibetan Buddhist nun, has delivered immeasurable wisdom to our world in her many books, courses, and workshops. She is clear, playful, and always right on target. The succinct but profound sagacity of the above statement is as instructive\u2014if not more so\u2014in turbulent times as it was when Pema wrote these words, in the relatively comfortable days of the 1990s.\n\nTimes of great unraveling are times of survival, but for those who are committed to being conscious, they must also be times of service. This book is a kind of manual for emotional and spiritual self-preservation, but without compassion and caring for others, spiritual development becomes stale and stultifying. Yet if we only serve others, we betray the evolution of the soul that is seeking fulfillment within our psyches and our lives.\n\nI have long believed and taught that we do our inner work not only for ourselves but for all the world, even for ancestors who are no longer with us. Though it may seem that sitting quietly in meditation for a half-hour is an exercise in isolation and self-absorption, in some way, it effects everyone around us. As we then do our work _in_ the world, we are more adequately resourced and we bring with us more awareness, compassion, patience, and perseverance.\n\nLikewise, as we touch the lives of others\u2014with our gifts or simply by being present with them\u2014our inner world reverberates with their presence, even if they are not grateful for our service. Even in situations where someone we serve is hostile to us, if we can see beyond appearances, we may be moved with compassion, empathy, sorrow, horror, a passion for justice, or unspeakable gratitude.\n\nI have discovered, over time, that I am both empowered and comforted when I can experience every person, animal, or plant in my path as a potential gift-giver. All come into my world bringing challenges, lessons, blessings, and epiphanies, often disguised as adversity. As a result, service and survival are no longer two, but one.\n\nWithout meditation, there is no self-knowledge. . . . Meditation is the beginning of wisdom, which is the understanding of one's own mind and heart.\n\n\u2014Jiddu Krishnamurti\n\nKrishnamurti devoted his entire life to teaching the difference between the ego, in concert with the rational mind, and the sacred Self. Much of his writing and lecturing painstakingly challenged the human race regarding what we claim to know. Krishnamurti asserted that the conditioned, enculturated mind cannot know anything except through ego and socialization.\n\nIf we want to genuinely know ourselves, we must escape the rational mind and its conditioning; we must directly access the sacred Self. Meditation, Krishnamurti and a host of other spiritual teachers proclaim, provides direct access to the sacred Self and has the capacity to transcend the many layers of the ego.\n\nIn the Western world, the last half of the twentieth century witnessed a plethora of techniques for spiritual development and creating a relationship with one's inner world. We have had these options available to us, but in less challenging times, many individuals \"didn't feel the need.\" Today we are confronted by challenges that will either drive us inward or drive us crazy. Indeed, we have a host of external dilemmas that compel us to devote as much energy to self-knowledge as we used to devote to career, establishing a family, material acquisition, and ego satisfaction.\n\nIf you are not practicing some form of meditation daily, you are limiting your self-knowledge at a time in human history when it is perhaps more urgent than it has ever been. If you _are_ practicing regular meditation, you possess an extraordinary tool for navigating a world in which all of the institutions that have conditioned us are crumbling. They will continue to disintegrate, but your sacred Self is a fixed point in a cataclysmic universe. The capacity to access it daily is nothing less than priceless.\n\nA society whose whole idea is to eliminate suffering and bring all its members the greatest amount of comfort and pleasure is doomed to be destroyed.\n\n\u2014Thomas Merton\n\nFather Thomas Merton, an incomparable mystic priest, wrote these words in the 1950s, when America was intoxicated with post\u2013World War II euphoria and the delusion that material comfort and pleasure defined the meaning of life. This was nearly six decades ago, but how incisive he was!\n\nIndustrial civilization has had at least two \"sacred\" maxims: (1) the ultimate purpose of human existence is material acquisition, and (2) humans should never have to suffer, particularly when the human in question is oneself.\n\nThese tenets are diametrically opposed to the worldview of indigenous traditions, which find meaning and abundance via an intimate relationship with the more-than-human world. Moreover, native peoples understand not only that suffering is a part of life but that a great deal of human experience is replete with suffering, punctuated by moments of joy, celebration, and exquisite consanguinity with one's community of fellow earthlings. They also recognize that when humans attempt to avoid suffering or meet their survival needs through a hostile, consumption-based relationship with the earth, they seal their fate and guarantee their own demise.\n\nIf you are reading these words, Merton's statement and the utterances of indigenous peoples must invariably resonate with you. Our deranged demand for comfort and pleasure is destroying modernity, even as this book finds its way to publication. Whatever suffering you are experiencing in this moment is not only part of the human condition but is, in some fashion, associated with the collapse of industrial civilization, a project of so-called infinite progress that\u2014as the Old Testament prophet Daniel would say\u2014\"has been weighed in the balance and found wanting.\"\n\nThe work now confronting those who survive this collapse is the construction of a new ethic and a new culture, not in search of comfort and pleasure, but fashioned on and for the sacred and the perpetual evolution of the human soul.\n\nIt is, therefore, a mistake to regard myth as an inferior mode of thought, which can be cast aside when human beings have attained the age of reason. Mythology is not an early attempt at history, and does not claim that its tales are objective fact. Like a novel, an opera or a ballet, myth is make-believe; it is a game that transfigures our fragmented, tragic world, and helps us to glimpse new possibilities by asking \"what if?\"\u2014a question which has also provoked some of our most important discoveries in philosophy, science and technology.\n\n\u2014Karen Armstrong\n\nThis book contains quotes from a number of wise women and men who, if not labeled \"mythologists,\" certainly honor and utilize the power of myth in their lives. Armstrong notes that myth is like a novel, an opera, a ballet\u2014all of which are forms of art. As we have noted, art is one manner of creating beauty in the world, and it is also intimately connected with ritual, a word that literally means \"to fit together.\"\n\nIn my years as a psychotherapist and in subsequent years as a writer and teacher, I have noticed that human beings fragmented by trauma and the vicissitudes of life are often able to \"re-collect\" themselves through art, music, theater, poetry, and dance. Perhaps even more restorative is the process of writing one's personal story or depicting it in some art form. The purpose is not for the ego to revel in \"specialness,\" but for the soul to reconnect with myth\u2014a word related to _mystery_ and _mystical._\n\nYes, myth is make-believe, a kind of game, as Armstrong notes, yet it joins us with ancient stories of \"mythic proportions\" and also inspires us toward the future with the question, \"what if?\" Always at the heart of myth is mystery\u2014questions never quite answered, symbols never clearly defined, flawed humans who often mirror parts of ourselves, both the parts that we already embrace and parts we have yet to fully recognize.\n\nMyth opens us to the mystery that is our own life: what we came here to do, what we came here to learn and to leave behind for those who follow. Perhaps the most important word in Armstrong's statement is _transfigure,_ which means \"to change the form or shape.\" As we become students of the mystery of our lives, we are transfigured as we find ourselves less interested in the \"what\" of our story and much more interested in the \"who\" that is the subject of it.\n\nIn truth everything and everyone\n\nIs a shadow of the Beloved,\n\nAnd our seeking is His seeking\n\nAnd our words are His words. . . .\n\nWe search for Him here and there,\n\nwhile looking right at Him.\n\nSitting by His side, we ask:\n\n\"O Beloved, where is the Beloved?\"\n\n\u2014Rumi\n\nIt is tragically easy to miss the beauty of these lines, if we allow the left brain to rule. Who is the Beloved? Why the pronouns \"he, his, him\"?\n\nYet more than seven hundred years ago, rivers of poetry flowed from Rumi that today continue to wash our wounds and slake seemingly unquenchable thirst in our souls. For Rumi, \"the Beloved\" was the sacred Self. For him, the Beloved was masculine, yet I have no doubt that Rumi would be thrilled if any of us replaced the masculine pronouns with feminine ones\u2014or even neutral ones, if they assist our falling in love with the Beloved.\n\nThe first phrase of this poetic fragment is the foundation all that follows. _In truth, everything and everyone is a shadow of the Beloved._ The Beloved is seeing, seeking, speaking, loving, healing through us.\n\nWhen was the last time you felt the sacred Self flowing through you into the world? When did you last feel yourself as the conduit of the Beloved's compassion, justice, mercy, fierce tenderness? Notice that this is not about what the ego experiences but a palpable sensation of something greater, something for which we may even lack words.\n\nWe all are the eyes, ears, mind, mouth, hands, and feet of the Beloved on the material plane. And at all times, we are either allowing the sacred to flow and work through us, or we are inmates in the prison of the ego.\n\nIn turbulent times, the despairing heart may ask, \"Where is the Beloved?\" We need only look in the mirror, and we will have our answer.\n\nThe world is not in any sense in danger from itself. The world is in fact not in any danger at all. It is we who are in danger.\n\n\u2014Daniel Quinn\n\nAs I read these words, I am reminded of the maxim found in all wisdom traditions\u2014our actions have consequences. Whether it be the Eastern concept of karma, which asserts that the fruits of our actions are deposited in a cosmic \"bank\" to be withdrawn or applied at a later time, or the Christian notion that \"whatsoever a man soweth, that shall he also reap,\" ancient wisdom consistently suggests that our actions have repercussions.\n\nDaniel Quinn's statement implies that the fate of the earth is less formidable than the fate of our species. After all, the earth has survived many cataclysms and renovations. This time around, humans are the ones responsible for its plunder, but not without consequences to us.\n\nWe are in danger of losing not only our lives but, far worse, our souls, as we attempt to deny, avoid, outsmart, or out-technologize the future. As Jesus declared, \"Fear not those who are able to kill the body but those who are able to kill the soul.\" Thus, our emotional and spiritual preparation for the disappearance of life as we have known it may be far more significant than our logistical preparation. If we don't come to terms with the emotional components of collapse, we will invariably be overwhelmed by them. In such a state, we are then susceptible to losing our sense of meaning and purpose and succumbing to the myriad soul-murdering onslaughts of a culture in chaos.\n\nOur species has created a dangerous world, murdering millions of members of the earth community, ravaging the ecosystems, and increasingly making the planet uninhabitable, yet the most endangered species is ultimately our own. The sum total of these realities is a situation that is nothing less than horrific. When life on this planet is nearly extinguished in an ecological holocaust, the question remains: who do we want to be\u2014whether we live or die? And more importantly, _who_ is asking that question?\n\nPeace is something that we can bring about if we can actually learn to wake up a bit more as individuals and a lot more as a species; if we can learn to be fully what we actually already are; to reside in the inherent potential of what is possible for us, being human.\n\n\u2014Jon Kabat-Zinn\n\nUpon reading this statement, I am compelled to ask, is peace possible? Clearly, the author connects world peace with individual peace and asserts that both kinds of peace result from waking up to who we really are. World peace is impossible unless we establish inner peace, and inner peace is not won without a great deal of work and conscious encounters with people and situations that may appear to be anything but peaceful.\n\nPeace is not found by avoiding turbulence, but by being willing to be present with it. We live in tumultuous times, and it may seem that peace is nothing more than a pipe dream. Yet it may also be that our openness to being taught by the struggle is the conduit to experiencing peace. This does not mean, however, that we never withdraw from the fray. A daily stillness practice of some sort is crucial fortification for staying engaged.\n\nA stillness practice facilitates coming to \"know who we already are\" and thereby \"to reside in the . . . potential of what is possible for us.\" Yet, inasmuch as we require stillness, we also require its opposite, a certain amount of clamor, in order to experience the fullness of our potential. Kabat-Zinn concludes the statement with two qualifying words, \"being human.\" At all times in our quest for peace, we must remember that we are human, and our humanity requires engagement with the tangible realities of earthly existence.\n\nOnce again, the dance of opposites invites us to participate wholly in life in order to not only experience peace but to discover the stunning miracle of who we really are. Thus we may experience a level of peace available only to those who are willing to cultivate the archetype of the spiritual warrior\u2014the woman or man grounded in stillness who is courageously willing to enter the messiness of life in order to express gentleness and compassion, as well as to take a stand for justice and mindful awareness.\n\nOur beautiful world is facing many crises. . . . It is not a time to pretend everything's good.\n\n\u2014The Dalai Lama\n\nI suspect that most readers of this book have long ago stopped pretending that everything is good. Because that is so and because you are on a path of emotional and spiritual preparation, you have no doubt experienced how challenging and even draining it is to be around people who insist that everything is good. It is nothing less than crazy-making to hear others sing the praises of the status quo or express unbridled optimism about the future.\n\nNevertheless, I think of the man who spoke the above statements, the Dalai Lama. When we see photos or videos of him, we see a man who is frequently laughing or chuckling with joy. Yet this is a man who witnessed the horrendous massacre and persecution of countless of his fellow Tibetans. How can he bubble with joy and radiate a heart-melting smile? Certainly because he has learned how to hold the tension of opposites and not be deluded that either opposite contains the sum total of reality.\n\nMost people familiar with the Dalai Lama consider him a wise man, and yet this wise, buoyant man is telling us that everything is not good. We can almost hear his booming voice reverberating in our ear. In fact, I recommend that you open to that very experience in this moment. See in your mind's eye the smiling face of the Dalai Lama and, at the same time, feel in your body the resonance of his deep, full voice saying, \"Everything is not good.\" Allow your body to feel the contrast of what you see and what you feel and just be present with those very opposite sensations. Thus, you have a palpable, physical experience of the tension of the opposites.\n\nClearly, he speaks not in anger but in love, telling us that our beautiful world is in crisis. His passion and compassion for our planet breaks his heart as he declares that everything is not good. At the same time, he never says that everything is bad, but like so many wise teachers in human history, he implores us to hold the tension of what we cannot humanly reconcile and notice what happens to us as we do so.\n\nHe who learns must suffer. And even in our sleep, pain that cannot forget falls drop by drop upon the heart, and in our own despair, against our will, comes wisdom to us by the awful grace of God.\n\n\u2014Aeschylus\n\nI will never forget the night of April 4, 1968, on which Martin Luther King Jr. was murdered in Memphis. The African American community in the United States was simmering and frequently erupting with rage in that era, at the height of the civil rights movement. At the very moment of King's assassination, Senator Robert Kennedy was in Indianapolis, campaigning to become the next president of the country. Kennedy was passionate about ending racism and bringing justice to all minorities. He instinctively realized that when the news of King's death reverberated throughout the country, an enormous retaliation from the black community would be likely. Kennedy was made aware of King's assassination before the media disclosed it, and with great skill and compassion, he announced it to the African American audience to which he had been speaking.\n\nFive years earlier, Robert Kennedy's brother, President John Kennedy, had been assassinated, and subsequently, Robert was driven to deep introspection and contemplation, which ultimately compelled him to run for president and to attempt to set the country on a steadfast course of justice and nonviolence. In the spirit of the fallen civil rights leader, on that warm spring night in Indianapolis, Kennedy pleaded for calm, restraint, and nonviolence; he ended his remarks with the above extraordinary quote from Aeschylus.\n\nOver the years I have found myself pondering these two sentences by the immortal author of timeless Greek tragedies. I invite you to do likewise, without attempting to analyze the words or the author, but simply allowing the exquisite poetry to wash over you like water and notice what happens.\n\nThe human is the being in whom the earth has become spiritually aware,\n\nhas awakened into consciousness,\n\nhas become self-aware and self-reflecting.\n\nIn the human, the earth begins to reflect on itself.\n\nIn our deepest definition and deepest subjectivity,\n\nwe humans are the earth. Conscious.\n\nYou and I are the beings in whom the earth\n\nthinks . . . knows . . . comprehends . . . analyzes . . .\n\nrationalizes . . . judges . . . remembers . . .\n\nchooses . . . acts . . . decides.\n\n\u2014Miriam MacGillis, _The Fate of the Earth_\n\nThis stunning declaration by Miriam MacGillis begs for lingering contemplation of its momentous ramifications. It asserts that human beings, in our \"deepest definition and deepest subjectivity,\" _are_ the earth. Moreover, we are the earth reflecting upon itself; we are\u2014as a result of a sacred Self within us that comingles with the sacred Self of the earth and the chemical elements that we both share\u2014the very earth itself. The linchpin of civilization's paradigm is separation, whereas the essence of earth consciousness is oneness.\n\nOn occasion, when I have shared this statement from MacGillis with readers or public audiences, I have been told that humans have sunk to the lowest rung on the species ladder and are more depraved than the lowest form of life in the universe. I suspect that, in our shame regarding our pillage and plunder of the earth, it is easy for us to succumb to such self-loathing that we may resign ourselves to having become irredeemable.\n\nYet I do not believe we are beyond redemption, because the earth is not and never will be. It has survived numerous exterminations throughout time and has managed to regenerate itself, even after millennia. Thus, if we are the earth, and it is redeemable, so are we. Any human being wishing to make amends for the behavior of our species need only humble himself or herself before the earth and acknowledge that he or she _is_ the earth. While this does not eliminate the colossal damage done by humans to this planet, and while it does not discharge human responsibility for our actions, it may alleviate our sense of separation from ourselves and from the earth\u2014the original separation, the \"original sin\" that engendered the paradigm of industrial civilization.\n\nLife is a spiritual thing, and spiritual things are silent. Those who love true life, therefore, frequently think about their death. Their life is full of silence that is an anticipated victory over death. Silence, indeed, makes death our servant and even our friend.\n\n\u2014Thomas Merton, _No Man Is an Island_\n\nTurbulent times are often times of death, and even when death appears distant, the threat of it may feel imminent. The collapse of industrial civilization is not only the collapse of a way of life that we may have known from birth, it is also the death of the collective ego. And on a more personal level, it threatens to destroy the individual human ego, and in many cases it has done or will do just that.\n\nCivilization has grown increasingly noisy over the past three hundred years\u2014so much so that silence leaves many individuals on edge, and for others it evokes abject terror. We may speculate about the causes of addiction to noise, but do they not inevitably culminate in one reality? Many human beings are terrified of being alone with themselves.\n\nSilence may remind many of their own mortality. No movement, no thinking, no breath, stillness\u2014death. Yet stillness and silence are essential conduits toward the sacred Self, and, says Merton, those who truly love life frequently think about their death. In fact, he insists, silence is not a surrender to death, but \"makes death our servant and even our friend.\"\n\nAs civilization winds down and there is less fossil fuel energy available for the myriad sources of noise in our world, we are likely to experience more silence. For some, especially those who are living in traumatic or life-threatening conditions, this may exacerbate their torment. However, if, beginning today, we make friends with silence by incorporating significant portions of it into our daily routine, including specific times of meditation and contemplation, we are not only less likely to be threatened by stillness, but we may just find ourselves falling in love with it.\n\nCelebrate all that is good and blessed about your life, realizing that gratitude is a powerful remedy.\n\n\u2014Caroline Myss\n\nIn times of loss, lost-ness, leanness, and loneliness, it may be challenging to practice gratitude. Yet, such times are optimal opportunities for doing just that. Ironically, when we swim in rivers of abundance, it's easy to take them for granted. Worse, we may begin subtly or blatantly assuming _we_ are the source of our good fortune. Indeed, we may have acquired prosperity partly as a result of our own creativity and hard work, but as millions are now coming to realize, abundance can vanish in a heartbeat, regardless of a lifetime of effort exerted to make it happen.\n\nBecause we are more than human, because the sacred Self resides in our core, all that we have originates from something greater than our own efforts. As a matter of fact, everything we have is, in some sense, a gift. Therefore, no matter how much or how little we have, gratitude is an obligation.\n\nAre you alive in this moment? Do you have food? Do you have shelter? Do you have companions? Are you reasonably healthy? Can you see, hear, taste, touch, smell? Are you able to walk, sing, smile, sleep, dream, meditate? If you have any or all of these blessings, give thanks, not once, but many times each day.\n\nSome people argue that gratitude is not only a salutary practice that helps us feel better, but that it actually opens our psychic energy to receiving more of what we need. I can only share my own experience, which is that every time I have taken a piece of paper and pen and made a list of people, events, and things for which I am grateful, another gift\u2014or more than one\u2014appears out of nowhere. Blessings may also come as opportunities to serve, heal, support, and share with others in need\u2014out of which comes more of that which we ourselves truly need.\n\nHowever mean your life is, meet it and live it; do not shun it and call it hard names.\n\n\u2014Henry David Thoreau\n\nWhile this statement by Thoreau may feel as if it is bordering on harshness, it may also be that the kind of resolve embodied in it is necessary in turbulent times. Many of our lives in present time are mean and arduous. The great early American philosopher and naturalist asks us to rise to the occasion and meet it head-on, not avoiding or complaining.\n\nThis is not to say that we never deserve a break from the rigor. Our souls and bodies need nurturance amid the distress. Yet it is useful to remind ourselves that a hundred and fifty years after Thoreau, Clarissa Pinkola Est\u00e9s wrote, in _A Letter to a Young Activist during Troubled Times,_ \"Do not lose heart; we were made for these times.\"\n\nYou did not arrive here by accident. On some level, we all chose to be here in these times, and it may require the rest of our lives on earth to understand what that means. But the challenge of this epoch is to rise to meet and live our lives. Many things within and on the perimeters of our lives may be unfair, even brutal, but it's what is so. It is in front of us and is asking something from us.\n\nRise and meet your life today with passion, with patience, with compassion, with courage, with gratitude, with resolve, and \"do not shun it or call it hard names.\"\n\nThere is nothing accidental about the fact that we in the West are starved for some real sense of meaning and crying out for something that, in spite of all our apparent sophistication and material success, we are no longer even able to name. This western civilization of ours was created for a purpose. Until we start to discover that purpose again, our lives will be meaningless. Unless we touch our roots and make contact again with the essence of our past, we can have no future.\n\n\u2014Peter Kingsley\n\nPeter Kingsley is a brilliant researcher and author, known most notably for a tome entitled Reality, which articulates the extraordinary mystical tradition lying at the heart of Western civilization. Kingsley argues that this tradition was obscured by historians who preferred to interpret the founders of the tradition in terms of their intellectual, rather than mystical, contribution to Western thought.\n\nThroughout his writings, Kingsley insists that, for Westerners, our spiritual hunger can be sated by reconnecting with our ancient mystical tradition, to be found in the teachings of the ancient Greek philosophers, who were profoundly influenced by their contact with writers and thinkers from the Far East.\n\nThe purpose for which Kingsley asserts Western civilization was created is none other than the initiation of each member of society into the sacred mysteries with which the ancient Greeks were intimately familiar. Those who obscured the mystical tradition with emphasis on the purely intellectual utterances of people like Socrates, Parmenides, and Empedocles induced Western civilization to pursue a different path. This path, lacking an awareness of a vibrant spiritual tradition, resulted in the development of industrial civilization.\n\nKingsley unmistakably declares that we can have no future if we do not return to the essence of our roots: a deep immersion in a mystical, spiritual tradition. Only there can we rediscover our purpose and encounter that which may bring meaning to our lives and the lives of countless descendants, whose physical, emotional, and spiritual well-being rests in our hands.\n\nThose who contemplate the beauty of the earth find reserves of strength that will endure as long as life lasts. There is something infinitely healing in the repeated refrains of nature\u2014the assurance that dawn comes after night, and spring after winter.\n\n\u2014Rachel Carson\n\nWhen despair for the world grows in me\n\nand I wake in the night at the least sound\n\nin fear of what my life and my children's lives may be,\n\nI go and lie down where the wood drake\n\nrests in his beauty on the water, and the great heron feeds.\n\nI come into the peace of wild things. . . .\n\nFor a time I rest in the grace of the world, and am free.\n\n\u2014Wendell Berry\n\nDawn does come after night, even if some of us may not live to see the dawn. Nevertheless, we need only immerse ourselves in nature in order to be assured and reassured that this is so.\n\nIn fact, in order to maintain our stamina, it is necessary to contemplate the beauty of the earth on a regular basis. Even if we are not feeling distressed, regular, intimate engagement with nature is a tonic for body and soul that heals, soothes, comforts, and sustains us, so that we can abide all that tests our reserves in turbulent times. Moreover, the very act of rejuvenation through nature imprints on the nervous system the reality of a receding night, a diminishing winter, and the eruption of spring\u2014the dawning of a new day, a new life, and new ways of being in the world.\n\nI invite you to deeply contemplate the above gems from Rachel Carson and Wendell Berry.\n\nBlessed be you, harsh matter, barren soil, stubborn rock: you who yield only to violence, you who force us to work if we would eat.\n\nBlessed be you, mighty matter, irresistible march of evolution, reality ever newborn; you who, by constantly shattering our mental categories, force us to go ever further and further in our pursuit of the truth.\n\n\u2014Teilhard de Chardin, _Hymn to Matter_\n\nTeilhard de Chardin was one of the first clerics in the modern world to expand his vision of theology to include the ecosystems. Strongly influenced by Teilhard, Thomas Berry began referring to himself as a geologian rather than a theologian. Teilhard's life and work became a protracted quest to reconcile spirit and matter, science and theology, the sacred and the human condition. He did not see these polarities as such, but rather as inextricably connected opposites that make up one whole.\n\nIn his _Hymn to Matter,_ the French philosopher and theologian celebrates the harshness of the physical world, which forces humanity to acknowledge its limits and respond accordingly to the material universe. Teilhard praises matter because it shatters our mental categories and forces us to go deeper into our pursuit of truth. Diverging from ecclesiastical tradition, he experienced matter as a spiritual ally rather than an enemy of the soul.\n\nAs I ponder his _Hymn to Matter,_ I ask myself if I can turn and face the turbulent times in which I live and say, \"Blessed are you, collapse of industrial civilization, because you are forcing me into the depths. Gratitude for your limits, which shatter my human ego.\" On some days I cannot speak such words with sincerity, but occasional epiphanies of the kind Teilhard experienced allow me to do just that.\n\nWe are human only in contact, and conviviality, with what is not human.\n\n\u2014David Abram, _The Spell of the Sensuous_\n\nWhat makes us human? Our DNA? The genes and chromosomes with which we were born? The fact that we walk upright? These realities define our species biologically, but psychologically, we are not fully human without contact with that which is not human.\n\nAs inhabitants of a compartmentalizing civilization, we have attempted to define ourselves exclusively in relation to our own species. Consequently, our sense of humanity is jaded by our lack of relationship with birds, bears, bushes, berries, and baby whales. The human identity of anyone who has an animal companion is profoundly informed by this relationship. An animal looks into our eyes, pants, or whines, and we notice our responses. We smile, grimace, worry, or bristle when our furry friends move or make sounds. If we are paying attention, we notice our own bodily sensations\u2014constricting blood vessels, a softening around the heart, a pervasive relaxation response throughout the body.\n\nWe generally define \"human\" as more intelligent, possessing self-consciousness, and knowing right from wrong, but these definitions emphasize our uniqueness, rather than our similarities with the more-than-human world. Ever since the Greek philosopher Protagoras stated, \"man is the measure of all things,\" we have rarely dared to dabble in research or empirical experiences that would bring our uniqueness into question. Increasingly, however, in the twentieth and twenty-first centuries, humans have become curious about their kinship with other species. One sees Jane Goodall sitting for years in the jungle with her beloved chimpanzee friends or Julia Butterfly Hill sitting for two years in a tree named Luna to protect the 1,500-year-old California redwood from being cut down. I suspect that these individuals have discovered more about what it means to be human than most of us will in a lifetime.\n\nWhat does being human mean to you? How do you know? Who have been your more-than-human teachers? What might happen to you if you were to spend even more time with them?\n\nWhen we get out of the glass bottles of our ego,\n\nand when we escape like squirrels turning in the\n\ncages of our personality\n\nand get into the forests again,\n\nwe shall shiver with cold and fright\n\nbut things will happen to us\n\nso that we don't know ourselves.\n\n\u2014D. H. Lawrence\n\nWhat a lovely meditation on what can happen to us when we completely open our hearts, our bodies, our souls to the earth! Glass bottles are fragile and easily broken\u2014and they are transparent. Thus, the ego, like a glass bottle or like a cage, entraps us, and when we go into the forest and sincerely pay attention, we will shiver with fright, but we will be changed, so much so that we will not know ourselves. There, in the wildness and raw ferocity of the wilderness, the sacred Self may be known and allowed to dance.\n\nIn the original version of the poem, Lawrence tells us more about what will happen to us and about what might happen to our relationship with civilization. Unlying, honest, authentic life will rush in, and we will be empowered, we will laugh, and the false promises of modernity will be revealed for what they are.\n\nOld things are already falling down, and every institution we can name is now crumbling like burnt paper amid our laughter, our empowerment, our celebration of who we have discovered ourselves to be\u2014not what the powers that be have told us we are.\n\nOur fears of the raw, visceral, untamed, primal elements of nature are not unfounded, because they immediately connect us with the animal we are, yet at the same time, they liberate us. They immediately reconnect us with the animal we are and, at the same time, liberate us from the robotic madness of modernity.\n\nRun forward, the way will spring open to you\n\nBe destroyed, you'll be flooded with life\n\nHumble yourself, you'll grow greater than the world\n\nYourself will be revealed to you, without you.\n\n\u2014Rumi\n\nThe beloved Persian bard forever rants and rails for the destruction of the ego so that who we think we are can give way to who we authentically are. And repeatedly he insists that the transformation will occur in precisely the opposite manner from what we would expect. \"Be destroyed, you'll be flooded with life.\" Allow the ego to be trampled, and you will become larger than the world.\n\nThe ego is often the last part of us to even perceive what is happening to us, let alone understand it. The body and emotions are usually the first to grasp what is so. The sooner we honor what they reveal, the sooner the mind can follow; yet for most individuals with origins in modernity, the reverse is usually the case. We insist not only on knowing the answers but also on appearing brilliantly in control as we grope for them.\n\nRumi begs us to give it up. Allow yourself to be humiliated, to look foolish, to appear pathetically out of control. This is not to invite abuse or abdicate personal boundaries but to recognize when the ego is impeding the emergence of the sacred Self. We will fail a thousand times in one day to surrender the ego, but what matters is not how often we succeed, rather, that we recognize the difference between the two selves and how fiercely the ego insists on total dominance of the psyche.\n\nIf we can continue to notice and to practice surrendering, we may experience moments, however brief, of the full impact of Rumi's stunning, exquisite words: \"Yourself will be revealed to you, without you.\"\n\nI slept and dreamt that life was joy. I awoke and saw that life was service. I acted and behold, service was joy.\n\n\u2014Rabindranath Tagore\n\nIn dark times, much light returns to us as we share the light of service. This is not to discount the occasions on which service breaks our hearts or taxes our bodies beyond belief, as we attempt to alleviate the suffering of a world in chaos. Yet making a difference in the lives of hurting people, animals, or ecosystems is unspeakably rewarding, even at the same time that it evokes great sorrow.\n\nBut let us not forget the connection between sorrow and compassion. If we are willing to be students of our suffering, it will teach us unspeakable compassion, out of which will grow divine love and an inability _not_ to serve the earth community.\n\nThe word _service_ is derived from the Latin _servus,_ \"slave.\" While service and slavery have very different meanings, there is a sense in which they are related. Genuine service, inspired by a divine compassion that has been cultivated by suffering, \"enslaves\" us to itself, in that we are compelled to utilize at least a portion of our energy to make a difference. \"Making a difference\" does not mean single-handedly stopping climate change, ending all wars, restoring economic prosperity, or reversing energy depletion. What it means, rather, is extending our compassion to those in our immediate environment. All around us are hungry, hurting, helpless people. Compassion borne of suffering \"enslaves\" us to give them food, listen to them, read to them, bind their wounds, play music for them, sing with them, and in myriad ways, become present with them in their suffering. How can we _not_ do so?\n\nSlaves are in bondage; those who serve are presumably free. However, when we allow divine compassion to \"enslave\" us, only then are we truly free. Only then do we discover the sublime joy of our human existence.\n\nThe most powerful agent of growth and transformation is something much more basic than any technique: a change of heart.\n\n\u2014John Welwood\n\nOur thoughts are not \"wrong\" or less important than our emotions, but so many people who are aware of the current crisis of our planet are focusing only on logistical preparation\u2014as if securing a supply of food and water, growing organic gardens, and learning new skills are the most important aspects of preparation. This emphasis on logistics results from focusing only on physical survival, to the exclusion of something greater within us, something that cannot be extinguished. Moreover, it lacks a vision of what this crisis might be asking from us for our immediate community and for generations yet unborn.\n\nThe collapse of industrial civilization will be and _is_ unequivocally traumatic, as we lose the only way of life we have ever known. Yet every person reading these words showed up on this planet at this time in history. What is your life purpose? What did you come here to _do,_ and who do you want to _be_ as the unraveling accelerates?\n\nCould it be that this collapse is the only way in which humans can experience the only thing that ultimately matters for our species and the earth: a change of heart?\n\nThe word _logistical_ originates from _logical._ When we are exclusively logical, we are living only in our heads and may be light years removed from our hearts. Welwood states that a change of heart is more basic than any technique. Might it be even more basic than all forms of logistical preparation?\n\nNo matter what we do in science or any other area, it will not help if we don't find a way to be related to each other at a deep level.\n\n\u2014David Bohm\n\nThe brilliant quantum physicist who uttered these words spent years researching and experimenting with dialog between humans. He created a particular type of dialog that later became known as the Bohm Dialogue, providing a structure and a container for authentic communication in the context of a small group. In more recent years, others, such as Marshall Rosenberg and Peter Senge, have developed similar techniques for dialog, all of which may be exceedingly useful in turbulent times.\n\nThere is no one \"right\" way to communicate, but deep listening and deep truth telling are crucial at this moment in our human experience. Never underestimate the power of simply sitting with another human being, looking into his or her eyes, and uttering, \"I'm really scared about what's happening to the earth.\" Or \"It breaks my heart to see what we have done to our planet.\" Or \"I'm so angry that humans refuse to wake up and see what's happening.\" Or \"I feel utter despair about the plight of the ecosystems.\"\n\nDeep dialog isn't about finding answers, being right, or preventing the collapse of civilization. Nevertheless, on more than one occasion, deep dialog has catalyzed options for those engaged, which have allowed them to take action on their own behalf in ways that were unimagined before dialog occurred.\n\nWe live in a culture that has never learned to communicate on a soul level. We generally speak only from our intellect, with words. In deep listening and truth telling, however, words may not even be necessary. What is necessary is that we be fully present to each other and speak from the depths of our hearts. Unless we can do this, our efforts at building a new culture for our descendants are not likely to succeed. If our intention is nothing less than to become a new species, we must learn and practice the art of deep human dialog.\n\nRitual is the original womb of art; its waters continue to nourish creativity.\n\n\u2014Starhawk, _Truth or Dare_\n\nFor some people, the word _ritual_ conjures images of incense-filled cathedrals or frenzied indigenous villagers dancing around tribal bonfires to a cacophony of drums. Or we may use _ritual_ to describe some regular routine we practice in order to implement structure and predictability in our lives.\n\nThe word _ritual_ simply means \"to fit together.\" Rituals are as old as our species and were created in the very beginning of our existence as a means of making sense of our human experience, bringing us together in community, and deeply connecting us with the sacred Self. Over the years, I have discovered the power of ritual to accomplish all of these.\n\nThe most powerful rituals are those that erupt from the earth and are not cleverly contrived or taken from a book. Perhaps you are already familiar with a variety of rituals and have experienced healing, bonding, comfort, or support as a result of practicing them. On the other hand, you may be unfamiliar with ritual altogether.\n\nIn turbulent times, rituals are exceedingly important for all of the reasons just named, particularly rituals that celebrate and incorporate the earth community. Earth-honoring rituals utilize earth elements, such as water, fire, and soil, and usually occur outdoors. They may be created in conjunction with a particular season or cosmic event, such as a solstice, an equinox, or a lunar or solar eclipse.\n\nWe may also create rituals to bless a person or a community event, but as Starhawk notes, ritual is the womb of art. Through it, our creativity is nourished and beauty expressed. In the presence of ritual, we may experience the alteration of time, a shift in consciousness, specific body sensations, or perhaps simply an overall sense of well-being. Never underestimate the power of ritual\u2014its capacity to create, transform, and connect us with each other and the earth.\n\nWhat gives light must endure burning.\n\n\u2014Viktor Frankl\n\nDuring the last two decades of the twentieth century, a variety of spiritual teachings and practices emerged, the genre of which became known as New Age. By and large, New Age teachings are fragments of ancient teachings and practices that have been reclaimed in the modern era.\n\nIt is difficult to define the New Age perspective, but in my opinion, one of its hallmarks is a fascination with light, love, hope, and positive thinking. While these qualities are noble and feel better to us than their opposites, the New Age perspective of \"positive only\" is yet another example of embracing only one side of a paradox, rather than holding the tension of opposites.\n\nThe paradox inherent in light is that it is created from fire or heat, and it is impossible to remain in it indefinitely, because eventually, light burns, sears, melts, and manifests its exact opposite. We need only recall the mythological Icarus, whose artificial wings were held together by wax and who was determined to fly toward the sun and remain in the light forever. The heat of the sun melted the wax, and Icarus fell to earth.\n\nIn dark times, \"living in the light\" may be appealing, but the appeal may prove to be seductive, especially when we consider that there is always a price to be paid for our determination to experience only the light and forsake the darkness entirely.\n\nWe most genuinely appreciate the light when we encounter it as a result of navigating the darkness. The way out is _through,_ not over, above, around, or below. If we are willing to mine the dark energies of our time, we are likely to discover gold within them. Can we hold the light in our hands and hearts as we journey through dark times\u2014without insisting that the light is all there is or should be?\n\nWhat lies behind us and what lies before us are tiny matters compared to what lies within us.\n\n\u2014Attributed to Ralph Waldo Emerson\n\nIn these eighteen words, Emerson has defined why our physical survival in this world crisis is less momentous than how we navigate it emotionally and spiritually. This assertion is not meant to diminish the preciousness of life or to contradict what I have written about the inestimable value of our human experience. Rather, my intention is to reiterate the presence and power of a sacred Self within and to invite the reader to attend to repeated shifts in perspective regarding the inner and the outer worlds.\n\nIt matters that we have enough food to eat, water to drink, shelter, and companions. It matters that we are healthy, fit, vibrant, and have the stamina to endure the physical demands of collapse. Without these, it is impossible to navigate the future with emotional stability.\n\nYet if we have all these and dismiss the significance of the inner world and all of the resources that lie therein, we are certain to discover that the external is not enough to carry us through the grueling emotional challenges we are likely to encounter.\n\nImplicit in Emerson's statement is the notion that the inner world has the capacity to provide resources that profoundly moderate our external predicament. Our relationship with and reliance on what lies within does not guarantee the survival of our bodies, but the impact of our circumstances may be tempered by an intimate awareness of our life purpose and the gifts we can offer a crumbling world. Emerson pleads for perspective and asks us not to be overwhelmed by the past or future but to access an inner domain that moderates and supersedes both.\n\nSpiritual growth is now replacing survival as the central objective of the human experience.\n\n\u2014Gary Zukav\n\nIn this sentence, the writer has captured the core issue of the collapse of industrial civilization. A paradigm based only on survival, infinite growth, unlimited progress, and perpetual material acquisition is now crumbling, along with the myriad systems and institutions engendered by it. Millions are coming to understand that these are the frail, ineffectual implements of a way of life that has fattened many wallets and bodies but left our souls famished and emaciated.\n\nSpiritual growth is not yet the central objective of the human experience, as it has been for our indigenous relatives for millennia, but the unfolding demise of unsustainable structures is likely to make it so. It may require decades, centuries, or even more time than we can imagine for humans to grasp the \"something greater\" beyond industrial civilization\u2014the \"something greater\" within themselves that alone has the capacity to bring meaning to their lives. You would not be reading this book if you were not among those entertaining the possibility that there must be far more to life than a \"secure\" job, a thirty-year mortgage, a three-car garage, a college education for your kids, and a generous nest egg for your golden years. Chances are, you've already lost all of those, and in fact, that loss may have prompted you to pick up this book.\n\nWhere do we go when the outer trappings of modernity evaporate before our eyes? We go within, of course, because there is nowhere else to go. What a gift to discover and to know with bone-marrow awareness that physical survival is not our only recourse\u2014or resource!\n\nCall it spiritual growth, discovering one's life purpose, or finding and making meaning. These must now assiduously inform our physical, day-to-day experience, and as they do, we have the capacity to grow a rich inner life and notice what difference that makes in our external experience.\n\nWherever you are is the entry point.\n\n\u2014Kabir\n\nWherever you are right now, you can begin the journey of discovering the sacred Self. It is never too late to begin exploring your inner world. It is also true, however, that the sooner we begin the journey, the more opportunity we have to prepare adequately for the external vicissitudes that will challenge us spiritually and emotionally.\n\nWhat we can also infer from Kabir's statement is that no matter how long we have been on the journey, every day is new. Therefore, we cannot assume that, because we have been exploring the inner world for a long time, we have all the answers and have arrived at the end of our journey. Our relationship with the sacred Self must begin all over again each day.\n\nIf inner work is unfamiliar to you, and you are just beginning, then you are experiencing the entry point. However, if you have spent years or even decades exploring your inner world, you are also experiencing an entry point, because in the odyssey of \"who am I and what did I come here to do?\" every moment of sitting with those questions is an entry point. In fact, if our inner work does _not_ feel as if we are standing at the entry point, perhaps we need to ask new questions. When did this become old hat? How did my journey become routine or commonplace? Where is my _re_ -entry point, and how do I get there?\n\nIf we are innocently and receptively open and attuned to life, it will constantly provide us with entry points. Each moment, each experience, if appreciated for its teaching potential, can provide a new entry point. We never know when such an opportunity will present itself. What we can know with certainty is that there is no end, no finish line, no exit point on the odyssey of evolution.\n\nWhen one of us tells the truth, he makes it easier for all of us to open our hearts to our pain and that of others.\n\n\u2014Mary Pipher\n\nMost of us pride ourselves in telling the truth, except when we are dealing with issues that are so unthinkable that we dare not speak of them. I'm talking about the extinction of species on this planet, including our own species; the possibility that natural resources, such as oil, will be depleted to such an extent that our way of life will vanish; the likelihood that the world's economies will deteriorate into a global depression, causing famine and massive homelessness. These are but a few of the truths we tell ourselves we must not know, let alone talk about.\n\nYet without exception, I have seen that when people begin speaking the truth about what is happening on this planet, they not only feel better, they also facilitate the same truth telling in others. And when individuals speak their truth and are heard by others, something profound and unequivocally healing results in both speaker and listener.\n\nSpeaking only the facts of our dying planet may result in increased knowledge, but it rarely facilitates healing. Speaking from our heads and speaking from our hearts are two vastly different forms of communication. I have repeatedly witnessed the futility of inundating people with information, yet I have also witnessed the stark emotional shifts that occur when one person can simply say, \"I want you to hear how terrified I am about what's happening in our world. I want you to hear my grief and sorrow. I am fearful for you, for myself, and for all of our loved ones. I want to tell you not from my head, but from my heart. Please hear my pain, my sorrow, and my sense of helplessness about this.\"\n\nNothing is more powerful than telling our truth from the heart. We may or may not be heard. Allowing ourselves to be vulnerable with another human being does not guarantee that he or she will respond in kind. But we cannot know the outcome until we take the risk.\n\nGratitude is a culturally subversive act.\n\n\u2014Joanna Macy\n\nWhen we give thanks in a culture focused on acquisition, progress, and unlimited growth, we invariably swim upstream. Despite the lip service we give gratitude in the United States and the one day each year that we set aside for giving thanks, how much more common are complaint, criticism, greed, envy, competition, and complacency?\n\nIndustrial civilization is about _taking_ and _taking for granted._ The exploration of the New World by Europeans in the fifteenth and sixteenth centuries was motivated by a \"taking\" mentality, and when explorers arrived on this continent, their sense of entitlement regarding the staggering abundance they found is one of the most astonishing phenomena of modern history. The impact of the legendary festival of thanksgiving celebrated by the pilgrims and their indigenous neighbors, although annually touted in the following centuries, was relatively short-lived. Almost immediately, white settlers obtusely disregarded not only the resources of the land but the treasure-trove of spiritual and eco-centric wisdom available among the native peoples, whom they very quickly exploited and eventually nearly annihilated.\n\nAlmost four centuries of conquest have produced not only an ingrained attitude of entitlement but an assumption that there is nothing to be \"grateful\" for because gratitude would imply the existence of something more momentous than one's own designs. In the act of giving thanks, one is humbled before some energy, seen or unseen, outside oneself.\n\nTherefore, gratitude implies that I am _indebted_ \u2014to the earth and to other humans who have assisted me in meeting my needs\u2014and that I cannot survive apart from relationship with these. Relationship is the antithesis of conquest and entitlement. It means that, by definition, we need each other. It is anathema to the paradigm of industrial civilization and is, thus, a culturally subversive act.\n\nBeing deeply loved by someone gives you strength, while loving someone deeply gives you courage.\n\n\u2014Attributed to Lao Tzu\n\nRelationships in a chaotic world will be challenging, whether they be friendships, a romantic partnership, or family ties. All relationships will be stressed by external conditions, and intimacy may prove to be elusive in many of them. On the other hand, the outward conditions of collapse are likely to distill much of our interaction down to the most fundamental, authentic, and transparent ways of being with each other. In current time, some individuals are creating intentional communities in preparation for collapse. What almost all of them report without exception is that an inordinate amount of time is devoted, out of necessity, to processing individual differences.\n\nIn the throes of a world unraveling, it is likely that much less processing will be required, because each person's physical survival will depend on his or her capacity to work harmoniously with others. People will get along, because if they don't, they will perish. In other words, community living in current time is a luxury; in the wake of worsening crises, it may well become a necessity.\n\nHowever we choose or are forced to live, we need each other. Having love in our lives, in whatever form, inspires and supports us to fulfill our purpose and to persevere when we sink into despair and want to give up. Loving someone else buttresses us with courage and compassion when we feel afraid. We need both forms of love in our lives in order to endure a world in crisis.\n\nThe individual who believes he can move to the hinterland by himself or with loved ones and prevail is sadly deluded. Although there are no guarantees that any of us will survive collapse, one thing is certain: it is impossible to survive in isolation. We need the experience of loving and being loved. Therefore, it is crucial that we begin creating now the kinds of relationships that we are going to require in the most chaotic hours of collapse.\n\nWho are your allies? If you don't have any, or if you need more, what do you need to do to create them?\n\nThe Universe shivers with wonder in the depths of the human.\n\n\u2014Brian Swimme\n\nI constantly encounter people who feel so much anger and bitterness toward their own species that one could almost say that they detest humanity. Their contempt is engendered by the hideous acts humans have committed against the earth and the earth community. It is completely appropriate to feel grief and remorse regarding the behavior of our species toward the ecosystems, but hating ourselves is unproductive and certainly brings no resolution to our predicament.\n\nMathematical cosmologist Brian Swimme asserts that the human species is the only one with the capacity to be consciously self-aware. This means that we alone are able to reflect on the universe and our place in it. Why is this significant? Because we have the capacity to develop such an intimate relationship with and gratitude for the earth that our character, personality, and behavior can be profoundly altered by that relationship. Moreover, whatever relationship we have with the earth community, we _are_ the earth reflecting upon itself. Our work is to fine-tune that reflection so that the reflection radically transforms who we are and how we behave in relation to the earth.\n\nWe can choose to ignore our power of conscious self-awareness\u2014and continue treating the universe as an object to be conquered and used for our designs\u2014or we can stand in awe of our capacity for conscious self-awareness and experience the result of doing so. What happens to a universe in which sufficient numbers of humans revere their conscious self-awareness? What happens to humans when they allow the universe to shiver with wonder in their depths?\n\nIt may be that the consummate meaning of our current world crisis will unfold as humans come to understand that conscious self-awareness of the universe is their ultimate reason for existence on earth.\n\nThe most profound joy has more of gravity than of gaiety in it.\n\n\u2014Michel de Montaigne\n\nThe other side of suffering is joy, and as we pass through periods of suffering, allowing ourselves to experience moments of joy assists us in tempering our sorrow with the salutary presence of humor, fun, play, celebration and gratitude. When we find ourselves on the other side of any particular ordeal, we may want to anchor our triumph with rituals that include joyful dancing, sharing food, and celebrating our delight with friends and loved ones. But we cannot plan eruptions of joy any more than we can foresee a crisis that bursts on the scene out of nowhere. It seems that both joy and sorrow have lives of their own.\n\nAs noted earlier in this book, there is a marked difference between joy and happiness. The latter tends to come and go without stirring memories of its opposite, superficially providing momentary elation or even giddiness. Joy, however, tends to resonate with its opposites in the psyche: sorrow, despair, grief, and more. Thus, the astute French philosopher Montaigne asserted that profound joy contains some gravity in it. Authentic joy is born from deep suffering and the capacity to appreciate exactly what it is relieving us from or inspiring us to be grateful for. Thus, William Blake could write, \"The deeper the sorrow, the greater the joy.\"\n\nAmerican culture has sought happiness from its inception, as if happiness were the meaning of life, and thus failed to comprehend the richness of joy. An obsession with happiness in tandem with a disownment of suffering infantilizes us and deprives us of a mature descent into the depths of our inner world, where we drink fully the cup of sorrow and therefore become capable of experiencing a quality of joy unavailable in moments of mere happiness.\n\nAs a result, in times of joy, some gravity remains in the background, which assists us in becoming more present to our joy. A particular ordeal may flash before us\u2014or a memory of all the anguish that led us to this moment of joy\u2014and we recall that the path to this singular experience of joy was watered with tears. While there is a time and place for superficial gaiety, the soul longs for moments of joy, the roots of which are always attached to its opposite.\n\nI have never let my schooling interfere with my education.\n\n\u2014Mark Twain\n\nThe late springtime of the year marks graduation for many young people. I had been taught that graduation marked the end of a particular course of study but had not learned that it also represents the launching of one's life beyond formal education. As with so many momentous occasions in our lives, graduation\/commencement is an end\/beginning.\n\nClearly not synonymous are the words _schooling_ and _education,_ as Mark Twain, the great American author and cultural critic, so wisely understood. The word _education_ has its roots in the Latin _educare,_ \"to lead out,\" whereas formal education is usually designed to pack as much information as possible into the minds of students, who casually speak of \"cramming\" for exams.\n\nOur world is choking on information but suffering from a dearth of wisdom. A person becomes wise by virtue of honoring all forms of learning and discounting none of them. She is not satisfied with simply knowing, but knowing that she knows. A crucial aspect of wisdom is the capacity to connect information and life experience with the heart and the intuition and to value the latter more than simply learning and memorizing facts. Not incidentally, Lao Tzu said, \"a wise man has no extensive knowledge; he who has extensive knowledge is not a wise man.\"\n\nIn our decaying world, formal education is becoming a glaring casualty of economic catastrophe and cultural erosion. Public schools are struggling for their very existence, and many believe that within the next decade or two, they will find themselves in the dustbin of history. So-called higher education is becoming virtually inaccessible due to its cost. Thus, it is entirely possible that within our lifetimes, public education as we know it will no longer exist. Home schools and alternative forms of education will replace the twelve years of grammar school that now occur in a state-funded brick-and-board building.\n\nWhatever form education takes in the coming decades, humans must allow the original meaning of _educare_ to prevail by de-emphasizing the amassing of information and, instead, revering the cultivation of wisdom.\n\nWhen it comes to understanding the cosmos, science and spirituality are describing two complementary aspects of reality\u2014one the nature of the material world we observe around us, the other the nature of the mind observing this world. When we consider how these understandings can be applied to the betterment of humanity, we see that science and spirituality are again complementary. To create a truly sustainable world, we need both\u2014the knowledge of science integrated with the wisdom of spirituality.\n\n\u2014Peter Russell\n\nOne of the supreme projects of the Enlightenment was to champion science and demonize spirituality of any kind. What could not be understood rationally or proven scientifically was relegated to superstition or folklore. Thus all manner of spirituality became synonymous with ignorance. In the nineteenth century, some wise women and men renounced the Enlightenment's overemphasis on reason and associated with romanticism, which touted the virtues of experiencing human emotion. Philosophically, the Enlightenment and romanticism represented polar opposites.\n\nDuring the twentieth century, great minds such as Einstein, Heisenberg, Planck, Bohm, Capra, Zukav\u2014all quantum physicists\u2014began theorizing about the relationship between energy and matter and the role of consciousness in integrating all polarities in the universe. Peter Russell's statement above is a more definitive articulation of what the quantum physicists postulated early in their research.\n\nI believe that the paradigm of industrial civilization is in decline for myriad reasons, but fundamentally because it failed to integrate science and spirituality and because the protracted polarization of science and spirituality significantly contributed to an unsustainable lifestyle. The collapse of the paradigm offers humanity not only an opportunity but an imperative: Science and spirituality must be integrated if our species is to survive on this planet. Without this integration, those who endure the collapse of industrial civilization will invariably re-create the remnants of a failed paradigm and will not have the capacity to cross the evolutionary threshold to become a new kind of species.\n\nScience and spirituality need one another, and neither a human being nor a sustainable world can evolve without their integration.\n\nWhen the first chakra is disconnected from the feminine Earth, we can feel orphaned and motherless. The masculine principle predominates, and we look for security from material things. Individuality prevails over relationship, and selfish drives triumph over family, social and global responsibility. The more separated we become from the Earth, the more hostile we become to the feminine. We disown our passion, our creativity, and our sexuality. Eventually the Earth itself becomes a baneful place. I remember being told by a medicine woman in the Amazon, \"Do you know why they are really cutting down the rain forest? Because it is wet and dark and tangled and feminine.\"\n\n\u2014Alberto Villoldo\n\nWhenever I speak of \"the feminine\" and its qualities, I am challenged by men who argue, \"But how can you say that's just 'feminine'? Men have those qualities too.\" Thus, I want to clarify that when I speak of _masculine_ and _feminine_ in relation to the current crisis, I am speaking not about literal, physical men and women, but about archetypes or universal themes found in all cultures and throughout the art, music, and literature of the world.\n\nSpeaking in terms of archetypes, the age of industrial civilization was and is an era in which masculinity is valued over the feminine. Qualities such as power, control, competition, conquest, progress, intellect, information gathering, linear thinking, and a rapid pace of living are esteemed as necessary and even instinctual. Conversely, cooperation, compassion, allowing, acceptance, wisdom, nonlinear thinking, and slower-paced organic living in harmony with natural rhythms of the ecosphere are essentially feminine qualities and are viewed as inferior or extraneous by the male-dominated global corporatocracy.\n\nI suspect that most readers of this book understand that the latter qualities of the feminine principle are now taking precedence in the hearts of those who have awakened to the unraveling of civilization. They may now constitute the ground on which you walk from day to day. Something in us hungers for a way of life rooted in the feminine, which nourishes our souls and fosters community and cooperation. Yet it will take time for us to fully embody the feminine in the way we live and relate on this planet. We say that the feminine principle is desirable, yet we may be stuck in focusing only on logistical preparation for collapse. We may feel discomfort in addressing feelings and resist forging our own spiritual path. Yet the earth, our Mother, has never left us. We need only return to her and be infused with her feminine presence and power\u2014no matter how long we have been estranged.\n\nSomeone I loved once gave me a box full of darkness. It took me years to understand that this too, was a gift.\n\n\u2014Mary Oliver\n\nThe time in which we live often feels like a box full of darkness. Sometimes it is so distressful that we cannot imagine ever perceiving it as a gift. And yet, this is the nature of evolution, the _yin_ and _yang,_ the spiral dance of our human experience. We look around and see people traumatized, terrified, starving, depressed, angry, confused, addicted, and in despair. Naturally, this is not _all_ we see, but the number of distressed individuals is growing daily. Perhaps on some days\u2014or many days\u2014we count ourselves among them.\n\nI believe that our consummate emotional and spiritual task while we remain on this planet is to discover how this box full of darkness called the collapse of industrial civilization is indeed a gift.\n\nA gift is a surprise, a blessing that we were not expecting. If it is something we wanted or asked for, we rejoice and give thanks for the gift. But if the gift is unwanted and not asked for, we immediately dismiss the possibility that it _is_ a gift and begin resisting. As a result, we may not perceive the gift in the darkness until many years later\u2014and, sadly, some people never perceive it. Yet in current time, our work is to look carefully at the box of darkness and notice what purpose it serves us and others. That task may feel absolutely absurd, yet here we are in the throes of a civilization unraveling, living through events so massive in scope that the real absurdity would be to dismiss the momentousness of what is occurring.\n\nThus, I must challenge the reader with two questions: Can you entertain the possibility that collapse is a gift for you? If so, what might be the gift(s) in it?\n\nYou may want to expand the questions beyond yourself to the planet and be curious about what gift(s) collapse is bringing to the world.\n\nKeep close to Nature's heart . . . and break clear away, once in awhile, and climb a mountain or spend a week in the woods. Wash your spirit clean.\n\nThe clearest way into the Universe is through a forest wilderness.\n\nThe gross heathenism of civilization has generally destroyed nature, and poetry, and all that is spiritual.\n\n\u2014John Muir\n\nDuring the years I lived in California, I had many opportunities to visit Yosemite National Park\u2014cared for, protected by, and named a national park through the herculean efforts of John Muir. Few human beings in modern times have experienced and fought for the intimacy Muir had with every aspect of the natural world. His spirituality, embedded in his ultimate connection with the sacred, reverberated in every mountain, waterfall, tree, and animal he encountered. For Muir, the natural world offered a plethora of conduits to something greater than any human being or any human achievement.\n\nJohn Muir's religion _was_ the more-than-human world, Yosemite Valley his temple, inhabited by the dumbfounding archangels of stately Ponderosa pines and the deafening spirits rumbling in the cascading waters gushing from icy, high-country lakes. The spires of Muir's cathedral, Half Dome and El Capitan, draw the eye toward celestial heights but relentlessly remind us of our finitude and our visceral connection with all things terrestrial.\n\nBecause he allowed himself to be transfigured by what David Abram names \"the spell of the sensuous,\" Muir realized exactly what happens to us when we surrender to nature's enchantment. Likewise, he understood what happens to a culture that forgets to be mesmerized by the earth from which its inhabitants are born.\n\nPerhaps you are fortunate enough to live or work in nature, or both. If so, you no doubt feel her indelible mark on you. But if you hunger for intimacy with her, living in a civilization starving for her breast, then take yourself to her at the earliest opportunity. Revel in what is waiting for you there, and \"wash your spirit clean.\"\n\nThe basis of optimism is sheer terror.\n\n\u2014Oscar Wilde\n\nMany of us who have been awakening, over time, to the current world crisis have been accused of fear mongering. We have been called purveyors of \"doom and gloom,\" clinically depressed, or even incontrovertibly crazy. We have been told to have a positive attitude and not revel in negativity\u2014and we have been warned that we are \"scaring people.\" Yellow happy faces pervade our culture; we're told to \"look on the bright side\" and \"keep your sunny side up.\" Most of us are regularly bombarded with maxims such as \"the power of positive thinking\" and \"we create our own reality.\" Optimism in industrial civilization has become frenetically obsessive, if not virulent, in its determination to prevail.\n\nAlways the devil's advocate, the incorrigible Oscar Wilde had his own opinion of optimism and did not hesitate to succinctly flaunt it with statements like the one above. I would add that only a culture profoundly terrified of losing what it has acquired through violation of itself, the earth, and the earth community could so rabidly pontificate about the virtues of optimism.\n\nWestern civilization's addiction to optimism is not found in parts of the globe profoundly influenced by indigenous wisdom\u2014wisdom that knows that much of the human condition consists of loss, suffering, and injustice. The ancients understood that our abundance and good fortune come from the earth and that the earth is not always kind. For them, it was crucial to live in harmony with the earth and to utilize earth-based spirituality and radical ritual to assist in navigating the precarious exigencies of their existence.\n\nIn estrangement from the earth and its extravagant wisdom, modernity has substituted \"things\" and \"happiness\" for grounded intimacy with the more-than-human world and the joy evoked through cooperation with it and with our fellow earthlings. Lives embedded in earth wisdom are lives pervaded by joy, with no need for optimism. Lives exiled from wild nature by the industrial death machine are lives engulfed in terror, where happy faces replace the unquenchable brilliance of the sun, the moon, the stars, the galaxies, and the undiscovered light that constitutes the core of the sacred Self.\n\nI say to people who care for people who are dying, if you really love that person and want to help them, be with them when their end comes close. Sit with them\u2014you don't even have to talk. You don't have to do anything but really be there with them.\n\n\u2014Elisabeth K\u00fcbler-Ross\n\nThe longer any individual survives the collapse of industrial civilization, the more likely he or she is to encounter dying people. Even in the second decade of the twenty-first century, we are witnessing the disappearance of accessible health care. In the United States, some fifty to sixty million people do not have health insurance. In many abjectly poor areas of America, bodies lie in morgues unclaimed by relatives who cannot afford to provide funerals for their loved ones.\n\nAs noted often in this book, the escalating world crisis will evoke a massive ego death in our species. The profound losses of the collapse are already stripping from us so much that we never dreamed we could live without. In that sense, we are in the midst of the pervasive, global death of an old paradigm.\n\nWhile much of the death is metaphorical, we cannot deny that the literal loss of life will be staggering. Coming to terms with this reality necessitates coming to terms with our own death, long before we experience it. This is an inherent part of emotional and spiritual preparation for the collapse of industrial civilization. Another aspect of preparation is willingness to be present with others who are dying. As K\u00fcbler-Ross notes, the most important part of this is simply to truly be with the departing person.\n\nWe may be faced with accompanying others in the dying process in the absence of their relatives or friends. Indeed, we may be the only person present for them. Two of the best preparations for this experience that I can recommend are, first, conscious preparation for our own death, and second, gaining some familiarity now with accompanying the dying.\n\nDeath is an integral part of the spiritual path, for as K\u00fcbler-Ross also notes, \"For those who seek to understand it, death is a highly creative force. The highest spiritual values of life can originate from the thought and study of death.\"\n\nEvery increase of needs tends to increase one's dependence on outside forces over which one cannot have control and therefore increases existential fear.\n\n\u2014E. F. Schumacher\n\nIn a world dramatically downsized by the collapse of industrial civilization, humanity will at last understand why _Small Is Beautiful_ is the title of Schumacher's most famous book. But what will happen to us as we come to terms with smallness? What will be the emotional repercussions of having lost so much and then attempting to build the next culture?\n\nWe will have memory of the tyranny of bigness, but will we be sufficiently fed, emotionally and spiritually, that we will have no desire to return to consumption in an attempt to fill our souls? What will it take to convert us to the simplicity of smallness of scale?\n\nPerhaps the trauma of collapse itself will bring forth such a sea change in the human psyche that we will come to deplore the grandiosity of industrial civilization. Perhaps our notion of \"needs\" will be so radically redefined by the unraveling that we will find ourselves \"needing\" less than we could ever have imagined.\n\nIn Schumacher's statement I am struck by the last three words, \"increases existential fear.\" Certainly, when we are dependent on people and systems outside of our control, we are less self-sufficient. Dependency tends to increase fear; self-sufficiency tends to diminish it. Yet even when I am reasonably self-sufficient, I still need contact with other human beings, and I may need the cooperation of the community to help me meet my needs, just as I help others in my community, in which case, I am interdependent. This is quite different from being dependent on the delivery of food and other necessities from far-away places to my local community.\n\nThus, it is safe to assert that my fear increases in direct proportion to how dependent I am on larger systems and decreases in proportion to how self-sufficient I am. The more complexity and dependence, the more fear. Smallness of scale and increased self-sufficiency are the handiwork of souls abundantly nourished in a milieu of simplicity, minimal needs, and bone-marrow magnanimity.\n\nThe beauty of the world has two edges, one of laughter, one of anguish, cutting the heart asunder.\n\n\u2014Virginia Woolf\n\nTragedy, loss, and trauma are places where we are likely to experience the two edges of the world's beauty. A society in decline is replete with both. What leaps forth to me from Virginia Woolf's statement is her definition of both as \"beautiful.\" But how can the cutting of the heart asunder be beautiful?\n\nPerhaps we need a new definition of beauty\u2014one that does not confine itself to things that make us feel good. Beauty is much larger than that, and to dismiss anguish as the opposite of beauty is to dismiss its depths in preference for the superficial.\n\nWhat is it about a Beethoven piano concerto, reverberating with sorrow, that leaves us in awe of its beauty? Or a wrenching painting by van Gogh that sears the eye with the abject misery in which he lived so much of his life? Yet we are compelled to confess the beauty of his work.\n\nAnguish, our own or that of our fellow earthlings, pulls us down into the territory of vulnerability, innocence, and imperfection that we share in our common humanity. From this well of agony, compassion is molded into the clay of our physiology. The heart is shattered and opened time and time again\u2014if we are willing to let both joy and sorrow be our teachers.\n\nHow often beauty and trauma travel together! In one moment, a passenger in a car revels in a glorious sunset, and in the next, she is the victim of a fatal crash. In one moment, a police officer proudly saves the life a child, and in the next, he is shot and killed.\n\nWe speak of black humor, the jokes or laughter we find in the midst of the most distressing situations. _Emotions_ contains the word _motion,_ which implies that they are not static, compartmentalized, or inert. Thus, we cannot know when the waters of emotion will flow from one extreme to its opposite. Laughter and anguish are the two edges of the world's beauty\u2014and of our own.\n\nWhen before the beauty of a sunset or a mountain, you pause and exclaim, \"Ah,\" you are participating in divinity.\n\n\u2014Joseph Campbell\n\n_Divinity_ may not be a welcome word to those who consider themselves atheist or agnostic. It may evoke images of \"God,\" church, religion, dogma, belief, and other notions pertaining to deity. For me, _divine_ is synonymous with \"something greater,\" something beyond the human ego and rational mind, related to that which cannot be known with the five physical senses.\n\nWhatever we choose to call this \"something greater,\" or wherever we choose to assume it resides, it is the font of beauty and awe. It is intimately connected with our emotions and therefore cannot be apprehended through reason. Campbell asserts that we participate in divinity\u2014and that we know that there is something greater\u2014when we are moved by nature or anything beautiful.\n\nI believe that the most direct conduit to the sacred, or \"something greater,\" is through nature and beauty. Moreover, when we commune with each other by way of deep listening or deep sharing, we are participating in the sacred.\n\nThe collapse of industrial civilization will redefine all things spiritual and pare them down to the most basic elements that resonate with the human soul; it will eliminate those that do not. Much that we have in the past assumed was \"spiritual\" may be found to be irrelevant or even antithetical to it, as we savor the aspects of nature and beauty that evoke joy, angst, celebration, pathos, compassion, and gratitude in our hearts.\n\nInstitutional religion, with its glaring disconnection from the earth and its obsession with personal salvation, is now being revealed as an enabler of planetary ecocide. Even worse, the prattle of designer, New Age precepts has facilitated the rape and pillage of nature by denying that it was occurring and emphasizing instead that we \"could have it all\" by simply plugging in to a chimerical \"law of attraction.\"\n\nThe spirituality of the next culture will be intensely personal and inextricably connected with nature and beauty; it will redefine all that we have previously named _divine._\n\nVoluntary simplicity involves both inner and outer condition. It means singleness of purpose, sincerity and honesty within, as well as avoidance of exterior clutter, of many possessions irrelevant to the chief purpose of life. It means an ordering and guiding of our energy and our desires, a partial restraint in some directions in order to secure greater abundance of life in other directions. It involves a deliberate organization of life for a purpose.\n\n\u2014Duane Elgin, _Voluntary Simplicity_\n\nFor a number of years we have been hearing and reading about voluntary simplicity, and many of us have been practicing it diligently. For others, depending on the extent to which they have become dependent on the trappings of industrial civilization, it may be much more difficult.\n\nVoluntary simplicity is a noble and desirable principle, but for a moment let's consider _in_ -voluntary simplicity and its impact on our lives. In the coming chaos, our lives are very likely to be pared down in ways we could never have imagined\u2014and probably would never have consented to.\n\nElgin connects voluntary simplicity directly with life purpose. Laser focus on why we came here and what we came here to do determines the role of possessions in our lives\u2014how we consume or don't consume, how we invest our resources, whom we associate with, how we care for our bodies and souls, how we use our time, and where we choose to live.\n\nThroughout history, some humans have pondered their life purpose, but most have not. For most people in the modern world, life just happens to them. If the collapse of that world does anything to humanity, I suspect that it will compel us to deeply examine our life purpose. The essential question is not \"What is the meaning of life?\" but rather \"What is the meaning of _my_ life?\"\n\nBeing able to consciously and clearly answer this question may temper the sting of sacrifice that our focus on life purpose entails. Thus Elgin speaks of some restraint in certain areas of our lives so that we may experience abundance in other areas. As he notes, voluntary simplicity is both an inward and an outward commitment. Emptiness in the inner world engenders voracious consumption. Clutter in the exterior may be a reflection of clutter in the interior. In either scenario, a \"deliberate organization of life for a purpose\" is needed.\n\nHow is one to live a moral and compassionate existence when one is fully aware of the blood, the horror inherent in life, when one finds darkness not only in one's culture but within oneself? If there is a stage at which an individual life becomes truly adult, it must be when one grasps the irony in its unfolding and accepts responsibility for a life lived in the midst of such paradox. One must live in the middle of contradiction, because if all contradiction were eliminated at once life would collapse. There are simply no answers to some of the great pressing questions. You continue to live them out, making your life a worthy expression of leaning into the light.\n\n\u2014Barry Lopez, _Arctic Dreams_\n\nThroughout this book I return repeatedly to what I believe is the fundamental underpinning of how we must navigate a world in chaos, namely, by recognizing, allowing, and opening to paradox. Lopez begins by reminding us that darkness is not only present in our world, but within ourselves\u2014a reality that is sometimes wrenching to acknowledge and explore. The word he uses to describe this reality is _horror._ Moreover, he states that one hallmark of adulthood is to recognize this horror and to grasp the irony of paradoxes. Furthermore, an adult must accept responsibility for living in the midst of paradox. Often a monumental crisis, such as the diagnosis of a terminal illness, is required for us to comprehend the paradoxes of our world: a potential death sentence delivered from the physician's lips alongside the extraordinary compassion of caregivers; the beauty of a sunset alongside an agonizing day we have just spent in pain or receiving chemotherapy. These are only some examples of the paradoxes of life that we may not see until we are confronted with crisis.\n\nA collapsing world contains a plethora of crises within the overarching crisis of the demise of industrial civilization. Each crisis within _the_ crisis will present contradiction, but Lopez implies that contradiction actually holds life together. Like _yin_ and _yang,_ positive and negative, male and female, light and dark, paradox fosters evolution and may be the glue that holds the universe, as well as our individual psyches, together.\n\nMay we cease trying to rationally explain what cannot be explained and lean into the paradoxes, for in so doing, we make our lives \"a worthy expression of leaning into the light.\"\n\nA more or less superficial layer of the unconscious is undoubtedly personal. I call it the \"personal unconscious.\" But this personal layer rests upon a deeper layer, which does not derive from personal experience and is not a personal acquisition but is inborn. This deeper layer I call the \"collective unconscious.\" I have chosen the term \"collective\" because this part of the unconscious is not individual but universal; in contrast to the personal psyche, it has contents and modes of behavior that are more or less the same everywhere and in all individuals.\n\n\u2014Carl Jung\n\nOne of the myriad contributions of Jung to the field of psychology was the concept of archetypes, which are simply universal themes or motifs present in the awareness of all people in all cultures throughout time. For example, all cultures have fascination with themes such as _mother, father, child, savior, hero, scapegoat, warrior, villain, healer, king, queen, lover,_ and many, many more. : It is perfectly natural to be fascinated with these archetypes, but often, if one minimizes or is unconscious of their universal presence, a motif can be so powerful in one's psyche that they begin to live from the archetype, losing touch with their personal identity. For example, a person who has been unjustly scapegoated as a child may find herself living out that role in adulthood, with little understanding of why she is doing so.\n\nThe collective unconscious, according to Jung, which underpins the personal unconscious and the conscious mind, is potentially connected to all other life-forms in the universe. This is why cultures on opposite sides of the world can have similar languages and traditions. For example, until the twentieth century, the Tibetan culture and the Hopi Indian culture of North America had not encountered each other, yet the two cultures have amazingly similar traditions and languages.\n\nThe decline, demise, dissolution, and collapse of a civilization is an archetype, and one that millions of humans have experienced throughout time. For this reason, a significant portion of humanity is\u2014consciously or unconsciously\u2014aware of collapse at this moment and is reacting accordingly, whether with mindful preparation or abject denial. What can we learn from other civilizations that have collapsed? What lessons are available to us from civilizations that have rebuilt or reinvented themselves? Even now, they may be speaking to us in the collective unconscious.\n\nThere have been seven disasters since humans came on the earth, very similar to the one that's just about to happen. I think these events keep separating the wheat from the chaff. And eventually we'll have a human on the planet that really does understand it and can live with it properly. That's the source of my optimism.\n\n\u2014James Lovelock\n\nIt seems to me that the operative word in Lovelock's assertion is _eventually._ How long _eventually_ takes is unknown. What feels most profound in this statement is the notion that current events are separating the wheat from the chaff.\n\nTransition, the Great Turning, the collapse of industrial civilization\u2014whatever we choose to call it\u2014is an enormous culling process, which I have repeatedly called a planetary initiation. This phenomenon may continue for decades or even centuries, but we will not \"have a human on the planet that really does understand it and can live with it properly\" until the initiation is complete.\n\nThe human species must be transformed\u2014remade from the inside out. Like the caterpillar in the cocoon, our species must be liquefied\u2014at least symbolically, if not literally\u2014in order to emerge as a metamorphosed being, intimately and inextricably connected with the earth from which it was born.\n\nThe species that evolves from this initiation may be extraordinary beyond anything we can presently imagine, but its exquisite transmutation will be the product of incomprehensible suffering. Yet in the throes of this anguish will be moments of beauty, joy, and unprecedented clarity. The juxtaposition of these realities is the only source of credible optimism.\n\nThe dire-sounding James Lovelock, who has written most poignantly about climate change and the irreversibility of global warming, nevertheless declares optimism\u2014but does so cautiously, using the word _eventually._\n\nOptimism is warranted if we are willing to hold the tension of opposites, understanding that transformation is a protracted, organic process, fraught with purifying anguish, punctuated by magnificent moments of the miraculous.\n\nSpirituality is not what we think about God or theology; it's participation. That's all it is.\n\n\u2014Anne Wilson Schaef\n\nLiving in a chaotic world, in which institutions and larger systems are unraveling, we will be compelled to form new definitions of spirituality, and perhaps most importantly, it will be crucial to separate them from religion or theology. According to psychologist and author Anne Wilson Schaef, spirituality is something far beyond doctrines, creeds, or dogma. In short, it is _participation._\n\nWhile that definition sounds exceedingly simplistic, a closer examination reveals that it is both simple and intricate. I believe that whenever we listen deeply to another person, we are performing a spiritual act. Likewise, speaking our truth, without blame or judgment, is a spiritual act.\n\nServing other human beings\u2014feeding, caring for, and healing them, being a companion to them\u2014these are ways of walking our spiritual path. Attending to the needs of animals by rescuing them; providing food, shelter, healing, and a good home; and making sure that they are protected from abuse is profound spiritual practice.\n\nCreating beauty through the arts\u2014playing music, telling stories, dancing, writing and reciting poetry\u2014all are spiritual acts. Planting, cultivating, nurturing, and harvesting a garden is spiritual practice, as is preserving food, catching rainwater, utilizing permaculture design, woodworking, beekeeping, and raising chickens or goats.\n\nIn the next culture, our definitions of spirituality will be altered as a result of an unprecedented intimacy with the earth community and other earthlings. Our experience of relationship and relatedness will be transformed by bone-marrow awareness of our inextricable connection with all beings. This awareness will be awakened by participation, and participation will be redefined by necessity and by the joy it instills in our hearts and in our communities.\n\nI accept relationship as my primary teacher about myself, other people, and the mysteries of the universe.\n\n\u2014Gay Hendricks\n\nIf the collapse of industrial civilization will challenge anything in the human psyche, it will be our relationships with each other. Every inhabitant of industrial civilization comes to this challenge with impaired or even embryonic skills. We have experienced lifelong inculcation in individualism and alienation. Simply put: we just do not live well in community.\n\nWhile in current time, we may consciously utilize our relationships as teachers, I believe that few of us are prepared for the extent to which this will be a dominant theme in a chaotic world. Like walking a spiritual path, communicating and building community will require incessant practice. We are guaranteed to make mistakes in our relationships\u2014behave obtusely; disregard the presence and needs of others; lose patience; offend others with our words or deeds; become self-absorbed, self-righteous, or self-protective. While our errors are inevitable, learning from them is not. We must choose to do so, and in order for that to happen, we must perceive every moment of every relationship as a teacher.\n\nIn community situations where our survival is at stake, we will not have the luxury of defaulting to an individualistic modus operandi. Our lives will depend on cooperation; otherwise, we _will_ perish.\n\nAuthor and spiritual teacher Jack Kornfield offers a beautiful definition of community: \"The community is created, not when people come together in the name of religion, but when they come together bringing honesty, respect, and kindness to support an awakening of the sacred.\"\n\nThe more we allow relationships of all kinds to be our teachers, the greater our potential for creating and sustaining this kind of milieu. Let us never forget that relationship work is sacred work. If we allow it to do so, it will always guide us to something greater within ourselves and within the other.\n\nEverything that slows us down and forces patience, everything that sets us back into the slow circles of nature, is a help. Gardening is an instrument of grace.\n\n\u2014May Sarton\n\nHowever quickly or slowly the current demise unfolds, it is certain that our lives will slow down dramatically in the absence of our former options for living them. Significantly fewer people will be working, economies will be in disarray, large systems will be unraveling, and much less fossil fuel energy will be available for use worldwide. Thus, we are likely to be set \"back into the slow circles of nature\" sooner rather than later.\n\nMany people who have already lost jobs are cherishing the extra time they now have to be with family and to pursue activities for which they rarely or never had time while working forty or more hours per week. Still others are fully employed but, given these economic times, find themselves doing twice as much work for the same or less compensation. For them, there is no slowing down, and the uncertainties of job security, pay cuts, loss of benefits, foreclosure, and other unknowns hover ominously over their heads.\n\nEventually, however, all of our lives will begin moving at a much slower pace. For most people, much of their time will be devoted to survival\u2014securing shelter, food, and other necessities. With less fossil fuel energy and money, our lifestyles may start to resemble those of our ancestors. Food production and preservation will be a crucial aspect of our daily lives, and acquiring what we need will happen much more slowly.\n\nOur patience will be challenged, and for millions of people, the forced slower pace of living will be nearly intolerable. Yet the gift in this phenomenon will be the way in which we will be compelled to conform to nature's cycles and seasons. Industrial civilization and its frenetic obsession with growth and progress has deprived us of these \"instruments of grace,\" with which we will be able to synchronize our souls' rhythms and the cadence of our animal nervous systems.\n\nWhen we move in the world with great flurry, we remain on the surface of life. When we are forced to slow down, we encounter the deeper currents of the life we formerly had little time to savor.\n\nThere's much more in any given moment than we usually perceive, and we ourselves are much more than we usually perceive. When you know that, part of you can stand outside the drama of your life.\n\n\u2014Ram Dass\n\nPerhaps the most fitting word to describe this statement is _detachment._ Many people assume that detachment requires a lack of caring\u2014aloofness, even obliviousness. On the contrary, _detachment_ simply means having boundaries and not engaging to the fullest extent with a particular person or situation that could be harmful to us. We care about the outcome, but we do not lose ourselves in it. Curiously, _detachment_ is also a term for a military unit that is peripheral and not central to a particular conflict. At some point a detachment could become part of the fray, but it is a separate unit. Honoring our separateness is just as important as remembering that we are all inextricably connected.\n\nLiving in a chaotic world will require both immense compassion and healthy detachment. We will need to have clear boundaries and work hard to keep them intact, even as we allow ourselves to feel the suffering of others. Again, this is an example of holding the tension of opposites, and it will require great mindfulness and support.\n\nRam Dass states one fact that is extremely important to remember when exercising detachment, namely, that things are usually not as they seem, because so much more is going on than appears to be happening. Likewise, we too are much more than we seem to be in any given situation. These awarenesses allow us to \"stand outside the drama\" of our own lives, which is yet another description of detachment.\n\nThere is more to the situation and more to ourselves than we are usually able to comprehend. But as we reflect on past experiences\u2014times when we forgot these truths\u2014and as we mindfully work with maintaining healthy boundaries, we hone the skill of detachment and more incisively perceive what is so.\n\nPeople living deeply have no fear of death.\n\n\u2014Ana\u00efs Nin\n\nWhat does it mean to live deeply? In addition to being willing to slow down and pay attention, it also means a commitment to making sense of our life experiences and finding meaning in them. _Depth_ and _meaning_ travel together and need each other.\n\nFinding meaning does not always occur as rapidly as we would like\u2014in fact, almost never. First, we must appreciate that finding meaning is the purpose of human existence. We are the only species that has the capacity to find meaning, and that fact is not incidental. As we commit to making sense of our lives and finding meaning in them, doing so becomes second nature to us. This is not to say that we will always find meaning in our experiences or that we will be able to make sense of every life experience. It simply means that doing so becomes our orientation to life, in fact, the soul's true north.\n\nWhile living life with a focus on its meaning does not eliminate the fear of death, as Ana\u00efs Nin declares, our quest for meaning decreases the senselessness and despair surrounding death. A commitment to meaning precludes superficiality in our relationships and in our activities. We understand that life is precious and, from a larger perspective, very short. What we do here matters. It matters to us and to all of the lives we touch. Moreover, it matters enormously to the earth community.\n\nLiving deeply dictates our values and how we choose to spend our time. It orients us to life as a sacred university that prepares us for advanced study beyond our mortal, human experience. In this way, the contrast between life and death is lessened, because we discern instinctively that death is a contiguous extension of life. Living deeply means living fully and richly, savoring life's complete content, not merely the aspects that please us. Perhaps _death_ is a misnomer; perhaps what lies beyond this life is a yet incomprehensible but palpable experience of consciousness, most efficiently prepared for by thoroughly inhabiting the present moment.\n\nThe intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift.\n\n\u2014Albert Einstein\n\nHow can we not be silenced and humbled when one of the greatest intellects in human history tells us that the intuitive mind is a sacred gift and that the rational mind is its servant? Yet all scientific discovery contains some aspect of intuition. Practitioners of the scientific method could not achieve success without a certain combination of the intuitive alongside the rational. What is stunning is the preference Einstein articulates for the intuitive, not only individually, but culturally.\n\nPrior to the Enlightenment or the Age of Reason, the intuitive mind was favored over the rational mind by the originators of most scientific discoveries. With the onset of the notion that the universe is nothing more than a machine, however, intuition was demeaned and reason touted as humankind's greatest achievement. Intuition was relegated to superstition, somewhat akin to religion, and therefore, in the minds of the Enlightenment thinkers, a dangerous, unreliable preoccupation.\n\nEinstein unashamedly articulated his spiritual passion alongside his monumental scientific theories because his work emerged amid a generation of physicists who were embracing quantum mechanics\u2014an orientation that no longer perceived the universe as a rational machine but rather a chaotic, unified, energetic field.\n\nThe discoveries of Einstein, Planck, Bohm, Heisenberg, and others in the field of quantum physics facilitated an unprecedented exploration of consciousness in the mid- and late twentieth century and reinforced new inquiry into the role of intuition in human behavior. Despite voluminous research on the role of intuition in our lives, Western culture has yet to recognize that the rational mind is its \"faithful servant.\"\n\nTell me, what is it you plan to do\n\nwith your one wild and precious life?\n\n\u2014Mary Oliver\n\nThese are the last two lines from Oliver's stunning poem, \"A Summer Day.\" For decades, on or about June 20, I have taken this poem deeply into my heart or physically in my pocket as I escape into nature to celebrate the summer solstice. Sometimes I have been in the company of others for the occasion, and without exception, I recite \"A Summer Day.\"\n\nWitnessing an exquisite June sunset, as the sun abides with us longer than at any other day of the year, it is easy to revel in the ecstasy of the poem, especially if we are well fed and have shelter, gainful livelihood, friends, family, our health, and our faculties. With what glorious and revivifying timbre do Oliver's words fall on our hearts!\n\nBut what about the dark days of loss, which may find us hungry, homeless, penniless, lonely, and less than robust? Many of us have already experienced such days and may experience them again. How will we meet those days?\n\nTo savor our wild and precious lives now is both salutary and farsighted, as we contemplate an uncertain future. But even in turbulent times, we can relish our wild and precious lives. In fact, we may find ourselves in situations where we have little more to relish.\n\nIndustrial civilization has tamed and diluted our wild selves. We have become domesticated, docile, subjects of its death machine. Collapse is a time for wisdom and discernment, but it is also a time for wildness\u2014for the reemergence of our wild nature and a delight in our animal bodies. For many, it will be\u2014and already is\u2014an escape from existence in a cubicle toward undomesticated abandon to nature and the radical passion that longs to burst forth from the soul.\n\nWhat _will_ you do with your one wild and precious life, and what are you doing with it even _now_ as you read these words?\n\nNo one has yet realized the wealth of sympathy, the kindness and generosity hidden in the soul of a child. The effort of every true education should be to unlock that treasure.\n\n\u2014Emma Goldman\n\nRarely do we hear anarchists speaking of kindness as beautifully as Emma Goldman does in this statement. Yet kindness lies at the core of so much for which both anarchists and social justice activists struggle. In fact, I believe that kindness must be a core value of the next culture. If it is not, our species cannot prevail, regardless of what other values we may embrace.\n\nIndustrial civilization was never about kindness, and it seems to have nearly eviscerated it from our species. Yes, we are all more than just kindness, but the paradigm of civilization, progress, limitless growth, greed, individualism, and a sense of separation have eclipsed kindness as the crowning glory of our humanity. Consumerist culture is kind only insofar as kindness promotes \"closing the deal\" or \"satisfying the customer.\" Kindness for the sake of kindness\u2014or because we are all vulnerable members of the same species\u2014is considered na\u00efve or even moronic in a cutthroat milieu.\n\nWe do not have to like another person in order to treat them kindly. And it is also true that some human beings behave heartlessly, brutally, and treacherously, and our kindness in relation to them must be tempered with stealth and no-nonsense constraints.\n\nNevertheless, a world unraveling will present us with countless opportunities to practice kindness and to teach it to our youth. It will also present us with many occasions in which we will ache for the kindness of others and feel profoundly humbled and grateful when it is demonstrated to us.\n\nAll the more reason for us to practice kindness in present time\u2014in order to liberate ourselves from the coldness of the culture of modernity and to demonstrate to our fellow earthlings how we wish to be treated. Moreover, our kindness must extend to the animals, plants, trees, soil, air, water, and all members of the more-than-human world. This is how we unlock the treasure of kindness in our children and in all beings we encounter.\n\nIf you're really listening, if you're awake to the poignant beauty of the world, your heart breaks regularly. In fact, your heart is made to break; its purpose is to burst open again and again so that it can hold evermore wonders.\n\n\u2014Andrew Harvey\n\nPerhaps you've heard it said that the only heart worth having is a broken heart. How can this be?\n\nSo much of Western culture mitigates against heartbreak, as if it were the worst thing that could befall any human being. We strategize and design our romantic encounters with, frequently, inordinate caution in order to avoid heartbreak. We are terrified of hurting or being hurt.\n\nYet Andrew Harvey says that one purpose of the heart is to break, so that it can open\u2014over and over\u2014in order to hold more wonders. In a culture that invariably hardens the heart and usurps our humanity, heartbreak is required if we are to become compassionate, caring human beings. We develop empathy not by avoiding suffering but by passing through it repeatedly.\n\nIt is no accident that, in the culture of industrial civilization, heart disease is among the top causes of death. Our hearts are \"congested\" and \"fail\" us, and we attempt to \"bypass\" them so as to avoid their being \"attacked.\" The metaphors we use in the treatment of heart disease are very telling; they could only occur in a culture of hardened hearts. Many more people die of heartbreak than our statistics reveal. Yet if heartbreak is normal, why do people die from it?\n\nI believe that heartbreak is only fatal when we resist it. When we can open to it and allow our grief to gush forth from the chest and solar plexus, heartbreak becomes a sacred rite of passage\u2014a spiritual initiation. Moreover, allowing visceral energy to flow in the body rather than constricting it can only contribute to our wholeness and well-being.\n\nIn a collapsing world, our hearts will probably be broken more times than we can imagine. In fact, the magnitude of heartbreak could become overwhelming, and therefore, it may be necessary to protect our hearts at times\u2014though not by hardening them\u2014in order to care for ourselves and prudently channel our life-force energy. The body and soul will signal their warnings. Our work is to listen.\n\nA human being has so many skins inside, covering the depths of the heart. We know so many things, but we don't know ourselves! Why, thirty or forty skins or hides, as thick and hard as an ox's or bear's, cover the soul. Go into your own ground and learn to know yourself there.\n\n\u2014Meister Eckhart\n\nI freely admit that Meister Eckhart is one of my favorite mystics. He represents a genre of clerical mystics who lived in the late Middle Ages, before the church lost its reverence for nature entirely and became the handmaiden of industrial civilization. Eckhart was a man of the heart and exudes heartful sensitivity in all of his writings. One of his assignments was to provide spiritual direction for a group of female mystics called the Beguines, but over time, it appears that they became his unintended spiritual directors, and Eckhart began championing the work of Hadewijch of Antwerp, Mechthild of Magdeburg, and Gertrude of Helfta.\n\nFor mystics, the truth is to be found in the inner world and in nature, its impeccable mirror. For mystics, true north is to be found internally and not necessarily in the teachings of the bible or the church. Eckhart wrote far more about the individual's relationship with the divine within than he did about theology in general.\n\nEven before the dawn of industrial civilization, humans armored their hearts, though in ways different from how we do today. Eckhart was well aware of the layers we use to cover the heart, and he asks us to go deep within so that we may discover the sacred Self that resides in our depths.\n\nEckhart said, \"The outward man is the swinging door; the inner man is the still hinge.\" All that we are depends on what lies inside, and our work is to discover our sacred essence and manifest it in our lives. The result, says Eckhart, will be seen in life: \"The outward work will never be puny if the inward work is great.\"\n\nIt may be that the more we know our hearts, the less need we will feel to armor them.\n\nWith our lives we make our answers all the time, to this ravenous, beautiful, mutilated, gorgeous world.\n\n\u2014Victoria Safford\n\nAs I have stated numerous times in my writing and work, our planet is not confronting us with \"problems to be solved\" but with a predicament for which there is no solution. A predicament cannot be eradicated, only responded to. Whether we call our predicament collapse, the Great Transition, or the Great Turning, our only option for dealing with it is to respond to it.\n\nWith regard to some issues, myriad options exist. With others, only a few are possible. What matters most, however, is how we live our lives.\n\nWhat strikes me in the statement by Victoria Safford is the tension of opposites in it: _ravenous, beautiful, mutilated, gorgeous._\n\nIt's all there in collapse\u2014the good, the bad, the ugly, the breathtakingly beautiful, and everything in between. But how will we live our lives in response to what is? We may have few options for that as well. Life's circumstances will compel us to live differently from how we are now living\u2014and in ways that may presently seem unimaginable. In our darkest hours, we may exclaim that we never dreamed we could live that way, _and_ in our most delightful hours, we may exclaim that we never dreamed we could live that way. The darkness may surpass our worst nightmares, and the loveliness may exceed our fondest fantasies.\n\nBut how we are forced to live is less important than how we _choose_ to live in response to the unwanted constraints of a world in chaos. Beyond doing what is necessary to survive, what meaning will we make of our circumstances? How will we reach out to and support others? What beauty will we tenaciously create amid the ugliness of ubiquitous decline? What suffering will we seek to minimize? Who do we want to be and who are we willing to become in these turbulent times and beyond? With our lives, we make our response to predicaments that have no solution.\n\nThis is not the time for illusion or evasion; it is time for transformation.\n\n\u2014David Orr\n\nWhen we can move through our denial and begin asking questions that really matter\u2014 _What is the collapse of industrial civilization really about? What is its momentous message for our species? Why am I living on this planet at the moment of this unraveling?_ \u2014when we can contemplate and struggle with the unprecedented convergence of crises in our time, then we will ultimately and inevitably understand that, whatever the final outcome of our rape, pillage, and plunder of the earth; whatever plagues, pandemics, or extinctions result; and to whatever extent humans make this planet uninhabitable, the entire saga of ascent and demise was and is yet another demonstration of the life\/death\/life cycle of embodied existence in the universe.\n\nThe beauty of any demise is that it confronts us with death, rebirth, and transformation. When we frame the current unraveling in terms of transformation, so little else matters. Blame of ourselves or the other, the transgressions of corporations and politicians, the delusional \"solutions\" proposed by proponents of \"greenwash,\" and all manner of denial and evasion are superseded by the eternal archetype of transformation. Whatever bulwarks of industrial civilization humanity has created and whatever the next culture will look like, transformation prevails.\n\nYet we must not ignore the fact that on the embodied, human level of existence, collapse is a tragedy of epic proportions. It entails death, disease, extinction, ecological degradation, and the devastation of dreams. The losses are incalculable. As a result, some may make futile attempts to \"prevent\" collapse, assuming that prevention is even possible. Others may resort to denial, despair, and nihilism. However, much of the pain of our losses may be tempered by remembering that, on another level, the life\/death\/life cycle is momentously at work and will never cease. Of all that collapse is asking of us, active, conscious transformation is its quintessential imperative.\n\nWhen asked if I am pessimistic or optimistic about the future, my answer is always the same: If you look at the science about what is happening on earth and aren't pessimistic, you don't understand data. But if you meet the people who are working to restore this earth and the lives of the poor, and you aren't optimistic, you haven't got a pulse.\n\n\u2014Paul Hawken\n\nCould holding the tension of opposites be more eloquently expressed? Likewise, Hawken's most notable book title, _Blessed Unrest,_ conveys the enormous paradox he celebrates as he witnesses both the discontent and the vision among those who are answering the call of the earth community to forge a new relationship with it.\n\nAbiding with opposites feels schizophrenic when we demand that only one or the other must be true. Holding the tension of both in our bodies produces a different feeling. The feeling may be strangely reminiscent of a pregnancy, in which as a result of holding the tension, something new begins to implant, gestate, and eventually emerge. In Eastern thought, two circles overlapping create in the center an almond-shaped area called the _mandorla_ \u2014a womb of sorts, out of which a third force or third option is born.\n\n\"Blessed unrest\" describes not only the glorious discontent among awake individuals on our planet but the tension we feel in our bodies when we refuse either to surrender to abject despair regarding our predicament or to deny it altogether and immerse ourselves in delusional optimism. Invariably, if we can hold the tension, we will experience our lives as a kind of medieval alchemical opus\u2014in which the contents are being heated to what may feel like unbearable temperatures in the process of being transformed, symbolically, into gold.\n\nAre you willing to allow yourself to be an opus in which the opposites are alchemized? Make no mistake, this is the most difficult work any human can do, but it is also the work required to create a new _kind_ of human being!\n\nThe only way to make sense out of change is to plunge into it, move with it, and join the dance.\n\n\u2014Alan Watts\n\nIf there were one simple sentence that describes the work to which I'm committed, it would be this one from Alan Watts. The fundamental intention of my work is to support people in ending their resistance to the collapse of industrial civilization and to encourage them, instead, to move with it. While this may be easier said than done, all other approaches to the plethora of losses with which we are confronted produce only more fear, frustration, and ultimately, despair.\n\nFew of us can actually _plunge_ into the mind-boggling changes of collapse, but we may be able to imagine ourselves doing so in slow motion. Most individuals begin by reading the facts about our predicament or watching a documentary about it and then sitting with that information for a while. Perhaps we are fortunate enough to have trusted others with whom we can discuss our discoveries. It is wise to take our investigation slowly, acquiring new pieces of information and sitting with them for a while before digging deeper. Moreover, it is important not to make major changes in one's life when beginning the process of awakening.\n\nAs we acquire more information and discuss it with other people who are waking up, we may never be ready to _plunge_ into what we know will be the consequences of collapse, but we can gradually develop a mindset of transition that prepares us psychologically for a daunting future. Over time\u2014and with support\u2014we become more accustomed to the notion of life without fossil fuels, a world embroiled in climate chaos, and a world where money as we have known it may not even exist.\n\nThere are two essential underpinnings for the _plunge,_ whether it be fast or slow: the support of trusted others and a spiritual path that allows us to make sense of that which might otherwise overwhelm us. As a result, we come to understand that we are far better off moving and dancing with the changes than continuing to ignore or resist them.\n\nThe story of the Western world is the story of how the peoples whose culture took shape through the religious inspiration of the Hebrew Bible, the New Testament, the humanism of the Greek world, the political-legal genius of the Romans, and a brilliant Medieval period became so entranced with a secular, scientific, industrial civilization serving limited human needs that it was willing to devastate the entire planet for the immediate benefits received.\n\nWe need to understand that the locus of the meeting of the human and the divine is in the natural world. The voice of the natural world is the resonance of the divine voice.\n\n\u2014Thomas Berry, _The Sacred Universe_\n\nIn the first paragraph of the above quote, Thomas Berry encapsulates the history of industrial civilization and conveys how the Western world lost its way and ultimately had no problem devastating the planet in order to receive the benefits of doing so. As a result, civilized cultures lost their souls, as they became obsessed with material acquisition. But it was not merely materialism that engendered soul murder. Rather, it was a loss of relationship with nature.\n\nIn _The Sacred Universe,_ Berry's final book before his death in 2009, he explains that what prevented the psyches of the Europeans who settled on the North American continent from opening to the wisdom of the indigenous populations was European culture's rejection of its own philosophical and spiritual tradition, which had its roots in a sacred connection with nature. Western culture had abandoned that connection in order to embrace the scientific and industrial revolutions. Doing so precluded the likelihood of appreciating the indigenous relationship with nature and guaranteed that conquest, not communion, would define the settlers' interaction with nature.\n\nWhat Berry proclaims must happen is a radical new relationship with the sacred _through,_ not in spite of, nature. I believe this may be the ultimate purpose of the collapse of industrial civilization. Moreover, the paramount place where the human and divine meet is in those moments when humans are entranced with forests, rivers, wildlife, sunsets, galaxies, soil, insects\u2014mesmerized by our beloved relatives, who patiently wait for our return to intimate communion with them.\n\nThe soul would much rather fail at its own life than succeed at someone else's.\n\n\u2014David Whyte\n\nThe collapse of industrial civilization will be brutal for anyone who has been trying to live someone else's life. Sadly, millions of human beings, especially inhabitants of Western societies, have never discovered their own path. This is significantly true for women, who are so often unable to extricate themselves from the role of wife or mother, even in a postfeminist world. Likewise, countless men have sat for decades in cubicles or boardrooms, living lives dictated by the bottom line, knowing that their souls were screaming to erupt and live from their depths.\n\nThe daunting challenges of an unraveling society will compel people to live their own lives, because upheaval shatters the structures of culture and families that impose roles. Life becomes a moment-to-moment quest for survival, well-being, shelter, safety, and cooperation. These are inimical to the stultifying limits with which modernity has imprisoned us. I suspect that on myriad occasions, people will rise not only to the occasion but to the kind of life they would really rather live\u2014and to the person they would really rather be.\n\nWhen the soul has nothing left to lose, it may at last discover its life purpose and rejoice that the constraints prescribed by the bulwarks of culture and consumerism no longer exist. And while the challenges of survival and well-being will be formidable, they will offer us endless opportunities to be the women and men our instincts have so long told us we are.\n\nThere will be grief, sorrow, uncertainty, disorientation, and dread, but how can there not also be spontaneous singing, dancing, and celebration of the psychological liberation from the constraints of the old paradigm? At last, we have the option of living our own lives, not someone else's, and even if we lose everything, including our physical existence, we will have tasted the succulent flavor of the roots of our being.\n\nWhether we like it or not, the depths in us are always throwing up treasure.\n\n\u2014John O'Donohue, _Beauty: The Invisible Embrace_\n\nAt this moment in the second decade of the twenty-first century, the Western world appears to be sinking more deeply into poverty. By the standards of developing nations, where some people are fortunate if they can live on two dollars a day, we hardly appear poor, but compared with the abundance we had just two decades ago, we have clearly become downwardly, not upwardly, mobile.\n\nOur economic decline, it seems, has only just begun. The collapse of industrial civilization will impose severe economic constraints on most of the Western world, with a very small sector profiting off the losses of the masses. But as we are increasingly stripped of our material comforts, we will be compelled to discover the treasures of the soul, and as O'Donohue notes, \"For the awakened imagination, there is no such thing as inner poverty.\"\n\nEverything that surfaces from the inner world is a treasure, if valued properly. Our fear, despair, anger, grief, confusion, longing\u2014all are priceless gifts from the psyche, with which we can work consciously to acquire more wisdom, stealth, resilience, humor, and creativity.\n\nThe present and coming chaos evoke a plethora of emotions and sensations that most humans are ill equipped to navigate. Few understand the treasure that lies within our so-called negative emotions. Those who are preparing for the unraveling must work with these treasures, not only to cultivate their own wisdom, but to assist others in discovering theirs.\n\nIn this way we live with less anxiety about what the depths will throw up next, and we come to value the treasure-trove of the soul that was obscured by the quest for security in civilization. Ultimately, we will come to discover that the treasure of the soul is the _only_ security there is.\n\nI have come into this world to see this:\n\nthe sword drop from men's hands\n\neven at the height of\n\ntheir arc of\n\nrage\n\nbecause we have finally realized\n\nthere is just one flesh\n\nwe can wound.\n\n\u2014Hafiz\n\nInhabitants of industrial civilization have much to learn about interdependence. Our culture teaches us to be _dependent_ on centralized systems and authority figures. Thus, we have become increasingly dependent on material security, status, and our public persona. We depend on the industrial food system to feed us; we depend on the global economy to provide clothing transported from far-off, impoverished countries; we depend on the infrastructure of our cities or suburbs to bring us clean, drinkable water; we depend on mainstream media to inform us; and we depend on technology to entertain us incessantly.\n\nMany individuals, particularly young people, decide that they are going to become fiercely _independent_ and make their own way in life. They may reject cultural norms and roles or move to isolated, rural areas in order to become totally self-sufficient. Modern survivalists prepare to become instantly independent of civilization by bugging out to their autonomous doomsteads in the event of sudden societal collapse.\n\n_Interdependence,_ however, is more difficult to practice than dependence or independence. As in the words of the poet Hafiz, interdependence assumes that \"there is only one flesh we can wound,\" and, therefore, we allow ourselves to be vulnerable yet cooperative with others\u2014and they allow themselves to risk assisting us and being cared for by us. The interdependent person essentially says, \"I don't want to depend on you, but I need you, and you need me, so let's find a way of supporting each other that works for both of us.\" In functioning interdependently, we minimize the tendency toward _codependence,_ in which one person seeks to control the behavior of the other in order to meet his or her own needs. In interdependence, we operate from an assumption of equality with the other and the reality that our own well-being _depends_ on the well-being of the other.\n\nResisting or postponing the collapse will only make it worse. Finding new ways to grow the economy will only consume what is left of our wealth. Let us stop resisting the revolution in human being-ness. If we want to outlast the multiple crises unfolding today, let us not seek to survive them. That is the mind-set of separation; that is resistance, a clinging to a dying past. Instead, let us shift our perspective toward reunion and think in terms of what we can give. What can we each contribute to a more beautiful world? That is our only responsibility and our only security.\n\n\u2014Charles Eisenstein, _Sacred Economics_\n\nHow refreshing it is for me when I encounter people who are open to and preparing for collapse! How draining it is to encounter those who attempt to convince me that I'm delusional and that collapse can be prevented or avoided or\u2014worst of all\u2014that it will never happen.\n\nEchoing Eisenstein's words above, for any of us to resist collapse is to resist \"the revolution in human being-ness.\" This demise, unraveling, Great Turning\u2014this dissolution of life as we have known it\u2014is endeavoring to pull us downward together into a planetary, collective descent for the purpose of transforming human consciousness and restoring our relationship with the earth community. At the same time that each of us is experiencing personal collapse in our individual lives (the microcosm), the macrocosm is drawing us conjointly inward and more deeply toward the earth. If we are willing to move with the descent, we have the capacity to experience a revolution in our human being-ness. If we resist, we recommit to our separation and abdicate our transformation.\n\nReunion, both on the microcosmic level of the individual psyche and in the macrocosm of human society, is waiting to happen to us. The collapse of industrial civilization is humanity's golden opportunity for reunion on every level. It beckons us to discover what can happen to us as individuals and as communities when we begin thinking in terms of what we can give to each other and what beauty we can create. Indeed, it is our only responsibility and our only security.\n\nIf we kill off the wild, then we are killing a part of our souls.\n\n\u2014Jane Goodall\n\nOne cannot watch film footage of Jane Goodall sitting in the jungle with her beloved chimps\u2014playing with them, caressing them, and communicating with them on every level\u2014without wanting to bow in honor to her passion for these phenomenally intelligent and lovable beings. She herself can probably not count the number of hours she has spent just watching and being present with them. Only a certain kind of person could invest so much time and energy with another species and consider those years the best of his or her life. But when we watch interviews with Goodall and listen to her speak of her experiences, we notice who she has become in the process.\n\nShe knows from bone-marrow, cellular-level experience what happens when we commune with the wildness of nature, which is also the wildness of our own souls. Despite the domestication of industrial civilization, a part of us yearns to experience and taste our wildness. We have discovered how inimical to our nature sitting in corporate cubicles or maintaining impeccable households can be. How antithetical we are to our hunter-gatherer ancestors, whose lives were all about survival and preserving the community, not cultivating their persona with personal trainers, tanning salons, or endless botoxing.\n\nWhen the accoutrements of industrial civilization have vanished, we are likely to have multitudinous opportunities to rediscover our wildness\u2014by plunging our hands into the dirt to grow our own food or by going into the wild to hunt or fish. No doubt beauty will be redefined when the many clothing and grooming options of modernity no longer exist. Weight loss is likely to come quite naturally as we eat more whole foods and are engaged in more arduous physical activity. Each of us is likely to discover a natural vigor and raw elegance as we become wilder men and women.\n\nIn that world, we may be appalled when we realize the extent to which killing off the wild has killed off a part of our souls, and we may well be astounded at how fully our souls are restored by reclaiming and living from the wild self.\n\nEvery happiness is the child of a separation\n\nit did not think it could survive.\n\n\u2014Rainer Maria Rilke\n\nIf we take a few moments to reflect on our lives and all the times we passed through experiences that we thought we could not and would not survive, we quickly move into sacred contemplation. At the time, all we wanted was to \"get through\" it. Whatever the situation, we may have promised ourselves or some power greater than ourselves that if we could just survive this experience, we would be wiser next time and make more discerning choices. If we could just get through this experience, we would be kinder, more compassionate, more sensitive, more helpful, more patient, more allowing\u2014more, more, more of whatever it is we feel we lack, which we fear may have brought adversity to our door. For most of us, our primary focus is on ending the pain of our situation, not on the lessons it brings us or how it brings us unexpected benefits.\n\nObviously, it is much easier to make sense of our challenges from hindsight. And as we do, it is important to notice how every one was a separation we did not think we could survive. Yet in many instances, that very separation and distress created \"children\" of happiness. Thus we notice again that adversity gives birth to happiness, and that in most cases, the deeper the sorrow, the more profound the joy.\n\nWhat is it in this turbulent world that you think you cannot survive? From whom or what do you feel separate? Indeed, many of us may not survive the collapse of industrial civilization, and sooner or later, all of us will die. We cannot say with certainty that when we die, we will experience happiness, but we can almost certainly assert that if we survive our suffering, and if we are willing to consciously make sense of it, some form of happiness, joy, or sense of well-being will reveal itself. And each moment of happiness\u2014even if it feels totally unrelated to our suffering\u2014may, in fact, have been borne from it.\n\nSteadiness is essential. Forwards, however, or backwards we will not look. Let us learn to live swaying as in a rocking boat on the sea.\n\n\u2014Friedrich H\u00f6lderlin\n\nI know of no more apt description of resilience than the last words of H\u00f6lderlin's poem \"All the Fruit.\" In these three sentences, the German poet articulates the essence of how we must navigate a chaotic world.\n\nLooking backward may cause us to long for an easier time or lose ourselves in \"if onlys\" or \"should haves.\" Regardless of how difficult life becomes or how much we'd like to go back and fix our mistakes, we may learn _from_ the past, but we must not yearn _for_ it. And no matter what our future may be\u2014no matter how uncertain, impoverished, volatile, unjust, or oppressive\u2014we must not live in terror of it. For even though we must endure it, this too will pass.\n\nWe cannot redeem the past or transform the future by dwelling on either. We _must_ learn to live swaying, as if in a boat rocking on the sea. A rocking boat sometimes moves the body backward, sometimes forward, but by being willing to sway, we become resilient and gain the capacity to create a world that is radically different from the one we have left and the one we fear lies ahead.\n\nWe do this by steadying ourselves in the present moment, for it is all we truly have. The past and future hold a certain amount of truth, but nothing is as real, tangible, and accessible to us as the \"holy now.\" Learn from the past and envision a future in which we live as we were naturally meant to live, but cherish and savor the present moment.\n\nJust by the very nature of our birth, we are on the spiritual journey.\n\n\u2014Father Thomas Keating\n\nImplicit in this statement by Thomas Keating, one of the architects of contemplative or centering prayer, is the notion that we are inherently spiritual beings. If this is so, then we existed before we became embodied human beings on earth, and our existence will not cease when we leave this planet.\n\nFrom this perspective, birth is the beginning of this part of our spiritual journey\u2014the part in which we live within the constraints of space, time, and the human body. Such a perspective raises the question of why we have become embodied human beings at this time\u2014for what purpose.\n\nIf you are reading these words and do not share this perspective, do not cease reading. Remember that this is only one perspective among many, and perspective has to do with perception. There are as many perceptions of the origin and cessation of human existence as there are humans. Keating's perception presupposes existence in spiritual form before we arrived on this planet and assumes that there is some purpose for our being here beyond simply taking up space. Moreover, he uses the word \"the\" spiritual journey, as if there is only one.\n\nWhile we could engage in philosophical or theological arguments ad infinitum about the truth of Keating's statement, he would argue that the acid test of the truth of his statement is centering prayer itself. Centering prayer is not about praying to some far-off deity, but rather contacting our center within and allowing that contact to inform us of the validity of \"the\" spiritual journey. In practicing centering prayer, many have come to believe that they have not arrived here by random accident, but that on some level, they have chosen to inhabit a body on this particular planet at this particular time. I believe that, as the Indian poet Kabir stated, if you have not experienced something, it is not true for you. If you _have_ experienced the truth of your center, it is likely to become your defining discovery.\n\nThe earth is the source and being of the people, and we are equally the being of the earth. The land is not really a place, separate from ourselves, where we act out the drama of our isolate destinies.\n\n\u2014Paula Gunn Allen, _The Sacred Hoop_\n\nAt the risk of being redundant, I repeat: we are not _part_ of the earth; we are not _children_ of the earth; we _are_ the earth. The first spiritual ecologists, such as Hildegard of Bingen, Julian of Norwich, Teilhard de Chardin, Thomas Berry, and Miriam MacGillis, argued that, because we are spiritual beings inhabiting physical bodies on earth, we are inextricably connected to it. In fact, they insisted, the earth speaks through us, because it has no other allies.\n\nOur hunter-gatherer ancestors knew this instinctively\u2014they had no other frame of reference. Their lives depended on an intimate relationship with soil, plants, animals, water, and all the elements. It did not occur to them that they could ever be other than the earth.\n\nJust as we commune intimately with other human beings who are close to us, we must now commune intimately with the earth. Some people\u2014like John Seed and Edward Abbey\u2014have lived in such day-to-day intimacy with the earth for so many years that they have had great difficulty being indoors even for short periods of time. In fact, they have no concept of being other than the earth.\n\nI'm not suggesting that we all live outdoors, but I believe it is imperative that we practice communion with the earth and continually ask the earth what it wants from us. We do this by spending intentional, contemplative time in nature and consciously communing with the earth community. By speaking to the earth as if to another human being, we develop earth intimacy and the capacity to hear and feel its communication to us. This kind of earth intimacy is guaranteed to radically alter our behavior in relation to the earth and to inform all of the decisions we make, both as individuals and with respect to our communities.\n\nThe earth is our being, and we are the being of the earth.\n\nWe're all connected, we have shared knowledge and that knowledge\u2014for better or worse\u2014is constantly evolving and changing.\n\n\u2014Thom Hartmann\n\nEverything that is in the heavens, on the earth, and under the earth, is penetrated with connectedness, penetrated with relatedness.\n\n\u2014Hildegard of Bingen\n\nIn the new paradigm that humans will create in the future\u2014and are already creating now\u2014we are inextricably, intricately, and intimately connected. We have shared knowledge and experience. If we begin practicing living in relationship with our community now, cherishing our interdependence, then when we need one another for our survival, it will not feel as awkward or vulnerable as it has in the past to share our knowledge or ask our neighbor for help.\n\nInterdependence is natural, because everything in, on, or under the earth, as Hildegard reminds us, is penetrated with relatedness. Relatedness, not isolation or fierce independence, is our natural state. If I do not know a particular skill, I may be able to ask my neighbor, or several neighbors, and if none of us knows it, we may be able to learn it together.\n\nNone of us can store food or water to last for the rest of our lives. We do and will need each other. Yes, everyone is responsible for preparing a reasonable amount of food and water for emergencies, and everyone should become relatively self-sufficient, but ultimately, we will not be able to survive without each other's help.\n\nFor the most part, our challenge, as inhabitants of industrial civilization, is not self-sufficiency as much as it is interdependence, allowing ourselves to ask for and receive help from another. And while it's true that food, water, medical supplies, tools, and other necessary items will be in short supply, and everyone is likely to fear sharing with others, we will ultimately discover that unless we share all of our resources, whether they are internal or external, our survival is in jeopardy, because we are, after all, penetrated with relatedness.\n\nAs my prayer became more attentive and inward, I had less and less to say. I finally became completely silent.\n\n\u2014S\u00f8ren Kierkegaard\n\nThere is a way of beholding which is a form of prayer.\n\n\u2014Diane Ackerman, _An Alchemy of Mind_\n\nI suspect that in the trauma and upheaval of a chaotic world, many people will be praying\u2014praying in desperation, praying in remorse, praying for mercy. As life gets increasingly daunting, I imagine that people will begin blaming everyone and everything, including the sacred, and many will renounce all manner of prayer.\n\nBut what exactly _is_ prayer? How can we forget one answer to this question, provided by Maude in _Harold and Maude:_ \"I don't pray, I communicate.\"\n\nIn the statements above, both writers speak of a manner of prayer vastly different from the wordy supplications of organized religions and their lectionaries. They speak of silence and of \"beholding,\" which is a way of paying attention, being present, honoring, and deeply witnessing. This is not to imply that verbal supplication is erroneous. The simplest and most poignant form of it is simply the cry of \"Help me!\"\n\nBut any stillness practice is a form of prayer\u2014to which both Kierkegaard and Ackerman allude\u2014because it includes silence and beholding. For centuries, religions have told us to pray, but scientific research in the twentieth and twenty-first centuries confirms the wisdom of doing so, because of the salutary effects of silence and beholding on the brain and body.\n\nIn my experience, silent beholding provides exquisite inner resilience for navigating the present and coming chaos.\n\nWhenever and wherever justice replaces injustice, healing replaces suffering, abundance and sharing replace poverty and greed, wisdom replaces ignorance, forgiveness replaces hatred, community replaces separation and isolation, vision and courage replace fear, and love replaces indifference, there the sacred world comes into being. There the soul emerges.\n\n\u2014David Spangler, _A Pilgrim in Aquarius_\n\nI am often asked what I mean by \"the sacred,\" and generally the question is asked by people who are wary of the words _soul, spiritual,_ or _sacred,_ because of their previous wounding by organized religion. Often I define _sacred_ by its most basic meanings, _set apart_ or _that for which we have reverence._ What I do actually mean, however, is that which abides in the domain of spiritual.\n\n_Spiritual_ has become such an abstract, amorphous word that it draws projections like a magnet draws iron filings. For people who identify with an atheist or agnostic orientation, it often carries a charge that relates to their negative experiences with religion.\n\nDavid Spangler's comprehensive description of the sacred, stated above, is one of the clearest and most beautiful I've ever seen. He does not define _the sacred_ but notes the conditions under which it arises or erupts. Moreover, he states that \"there, the soul emerges.\"\n\nI believe _the sacred_ is best defined when we leave the definition open and accessible to the instincts. Something within us, far beneath and beyond the wounding inflicted by organized religion, senses what it needs to sense in the word _sacred._ Whether we are atheists or religious devotees, we carry within us the ancient memory of our hunter-gatherer ancestors, who communed with the sacred in every aspect of life as a _way_ of life\u2014a way of life magnificently depicted by Spangler's succinct but explicit description of its tangible outcomes.\n\nThe white man seems to think that speech is some sort of proof of superiority over dumb creation. It is not. It is one of the many gifts the Great Spirit has given to man. To the Indian, silence is the cornerstone of character.\n\n\u2014Tom Brown, _The Search_\n\nDon't talk unless you can improve upon the silence.\n\n\u2014Vermont Proverb\n\nA chaotic world is likely to be a noisy world, but on the other hand, as much of our way of life winds down\u2014as a result of less energy, less money, and the collapse of infrastructure\u2014our lives may become much quieter. To what extent this is true may vary from place to place.\n\nOne of the things we are likely to learn in the unraveling of our world is a new appreciation for silence. Industrial civilization is all about activity, words, and filling silent spaces with sound. It is very difficult, if not excruciating, for many individuals to sit quietly with themselves. We can only imagine how torturous a quieter world will be to people who have needed the constant noise of iPods, television, video games, and radio to \"spare\" their nervous systems from silence.\n\nAs I have often emphasized, a daily stillness practice is invaluable, as is practice in deep listening, not only to the words of other people, but the voices of the earth community. The most precious gift we can give a person in distress is simply listening attentively to her. In doing so, we let her know that she is being seen and heard.\n\nA Native American saying states that the creator gave us two eyes, two ears, and only one mouth for obvious reasons. Now is the time for radical stillness and deep listening\u2014to ourselves, to the earth, and to each other.\n\nChildren say that people are hanged sometimes for speaking the truth.\n\n\u2014Joan of Arc, at her trial\n\nNot only do people of all ages say this, they say it because it's true. People are hanged in different ways for speaking the truth. Perhaps you have a personal experience of this. If so, you understand how wrenching and torturous it feels to be misunderstood or not understood at all. One speaks the truth, and the other is completely unwilling to hear it or hears something entirely different.\n\nMany readers of this book are all too familiar with being \"hanged\" for speaking the truth about the collapse of industrial civilization. We have been called \"crazy,\" \"depressing,\" and \"hopelessly pessimistic.\" Not infrequently, our spouse or another family member or a close friend decides that we have taken complete leave of our senses and chooses to take leave of _us._ Yet, like Joan of Arc, we continue to speak and live our truth.\n\nSpeaking the truth is often an abjectly lonely place, particularly if one lives in an isolated area of the world or in a location where few, if any, people are aware of the coming chaos. In this kind of situation, one must carefully consider the cost of remaining in place and living in such isolation as opposed to the alternative.\n\nIf we hold in our hearts and bodies a truth as momentous as the collapse of industrial civilization, but we cannot discuss with anyone the emotions that that knowledge elicits, a toll is taken on us, both emotionally and physically. Yet we cannot pretend not to know what we know. We need each other\u2014and not only for preparing logistically and for transforming our communities; we need each other for every kind of support as we all navigate the future.\n\nFind people among whom you will not be hanged for speaking your truth, and cherish the gift of their presence in your life. Tell them often how much you appreciate them and return kindness and support to them. In this way, you are hugged, not hanged, for speaking the truth.\n\nYou can't experience the truth of another person without feeling love. Understanding and love go together. When we nourish the soul, we automatically nourish our capacity to love another person.\n\n\u2014Jacob Needleman, \"The Sacred Impulse\"\n\nAs we share our truth with others, and they share theirs with us, the soul is deepened, and love is solidified. Soul deepening and love travel together. But what does soul deepening or soul nourishing really mean?\n\nTo answer this, we must notice what the soul loves. It thrives on creating and appreciating beauty\u2014reveling in poetry, song, story, art, and dance. Moreover, delighting in the full spectrum of our emotional color palette nourishes the soul, particularly if we are committed to consciously making sense of our feelings and learning from them.\n\nThe soul does not deepen as profoundly in so-called good times as it does in the darker days of loss and uncertainty. In fact, the soul thrives in times of downturn, depression, depletion, and even catastrophe. When humans are willing to allow the soul's descent to remake them, the outcome may vary from ecstasy to inexplicable serenity to a bone-marrow awareness that, on some level, their lives make sense despite the burden of suffering.\n\nThe end of an era, which is also the beginning of another\u2014the myriad thresholds and initiations of life\u2014nourish the soul and cultivate our capacity to love in ways that might have been impossible otherwise. This is why suffering and compassion travel together and are inextricably connected.\n\nIn the watery regions of the soul, so often moistened with tears, we are able to say with the poet Langston Hughes, \"My soul has grown deep like the rivers.\"\n\nDoubt is not a pleasant condition, but certainty is absurd.\n\n\u2014Voltaire\n\nIf any word describes the coming and present chaos, it is the word _uncertainty._ Perhaps this in itself is the most stressful of all possible exigencies of a collapsing civilization. What we have at least some certainty about, we can prepare for with a modicum of foresight. However, when we have myriad converging crises in the principal areas of modern civilization (energy, environment, and economics), certainty mercilessly eludes us.\n\nThus, we are constrained by doubt, a most unpleasant condition, and insistence on certainty in order to relieve our discomfort is often irresistible. Yet every part of us knows, consciously or unconsciously, that certainty with respect to a world unraveling does not exist.\n\nOur only recourse, then, is to nurture and fortify ourselves with the wisdom of ancient traditions that assist us in navigating life's infinite uncertainties. The ancients tell us to deepen our relationship with nature and immerse ourselves in its subtle nuances, even approaching it intentionally with questions. On countless occasions I have assigned workshop participants to wander in nature with a specific question: \"How should I deal with my grief and despair?\" \"What do I need to know in order to navigate collapse?\" \"What is my greater purpose as I confront a world in decline?\" When participants have sufficient time and support, they return from their wandering with profound shifts in awareness and with answers to questions, no matter how incipient, that definitively guide them in taking the next step.\n\nIn addition to intimate contact with nature, we can utilize meditation, journaling, working with dreams, and perhaps most importantly, staying present and attending to the most significant moment we will ever have, the one that is happening right now.\n\nThe purpose of spirituality is to fall deeply in love with being human.\n\n\u2014Matthew Fox\n\nFor years I have been a fan of the former Catholic priest Matthew Fox, who in 1993 was expelled from the Dominican order at the behest of then Cardinal Joseph Ratzinger (later Pope Benedict XVI). Fox went on to develop what he calls Creation Spirituality, consisting of the _Via Positiva, Via Negativa, Via Creativa,_ and _Via Transformativa,_ which provides a structure of earth-based spirituality that integrates indigenous wisdom, gender equity, and deep ecumenism.\n\nMost refreshing and innovative in Creation Spirituality is its honoring of the human experience as part of, rather than inimical to, the divine. One finds the sacred within the human, not separate from it. When we are most connected with the sacred, Fox asserts, we are also most present in our bodies, most honoring of our vulnerability, most compassionate toward ourselves and others, and most intimately connected with the earth community.\n\nIndustrial civilization has estranged us from our humanity and has utilized religion to split us off from our bodies, our emotions, and our wild animal nature. I wholeheartedly agree with Fox that the purpose of our spiritual path is not to make us more spiritual, however that may be defined, but more thoroughly human\u2014cherishing of our physicality, our passions, and our capacity to love and care for each other.\n\nEchoing Fox, a number of Jungian-oriented teachers and authors, such as James Hillman, Michael Meade, Thomas Moore, Marion Woodman, Andrew Harvey, Bill Plotkin, Malidoma Som\u00e9, and Clarissa Pinkola Est\u00e9s, have passionately championed the role of the human soul in spiritual practice as a bridge of sorts between the sacred within and the sacred in the world.\n\nI believe that the spirituality of the Great Turning will constitute a return to an earth-honoring, body-reverencing veneration of the sacred that reinstates the presence of Eros in our hearts on a level previously unknown in Western civilization.\n\nMy neighbor and I have the same origins;\n\nWe have the same life-experience and a common destiny;\n\nWe are the obverse and reverse sides of one entity;\n\nWe are unchanging equals;\n\nWe are the faces which see themselves in each other. . . .\n\nMy neighbor's sorrow is my sorrow;\n\nHis joy is my joy;\n\nHe and I are mutually fulfilled when we stand by each other in moments of need;\n\nHis survival is a precondition of my survival.\n\n\u2014Zulu Personal Declaration of 1825, South Africa\n\nSo characteristic of the perspective of tribal cultures, this declaration verbalizes the fundamental philosophy of the village, the tribe, the community: that your well-being is dependent on mine, and my well-being is dependent on yours. We exist side by side to honor each other, listen, speak our truth, protect each other, cry together, die together, and celebrate alongside each other. All of this, the declaration asserts, because we \"have the same origins.\"\n\nAs we create the next culture, we must model it on the principles of this declaration; otherwise, we will re-create the paradigm of industrial civilization, in which we live as atomistic individuals in a dog-eat-dog world.\n\nNotice that from the indigenous perspective, we are mutually fulfilled when we stand by each other in moments of need. In Native American folk history, elders declare that if one person in the tribe doesn't have enough, no one has enough; if one person is hurting, everyone in the tribe is hurting. These are extraordinary statements of fundamental human interdependence, grounded in a localized, village perspective that eliminates competition and virtually guarantees cooperation and abundance.\n\nThe truly wise person kneels at the feet of all creatures.\n\n\u2014Mechthild of Magdeburg\n\nA passionate thirteenth-century mystic, Mechthild was not only a Beguine nun devoted to prayer and meditation, she was also a highly educated woman and an ardent lover of nature. Mechthild experienced all of creation as exuding the sacred, a form of spirituality that was one of the inspirations of what Matthew Fox calls Creation Spirituality.\n\nAs we witness the horrific treatment of animals on our planet\u2014the slaughter of elephants, the brutal removal of rhinoceros horns, the murder of seals for their hides, and the frequent heinous torture and starvation of domestic animals by humans, we are aghast at how far from Mechthild's declaration our species is. She asks us not simply to be kind to animals, but to _kneel at their feet,_ to reverence the gifts that all species offer us.\n\nIn our despicable arrogance, we dismiss the astonishing intelligence of beings such as elephants and regard them as \"ignorant beasts\" to be exterminated for the price of their tusks, leaving their butchered bodies to rot in the Serengeti sun.\n\nIn the collapse of industrial civilization, domestic animals are not likely to fare well. The most domesticated will probably have a more difficult time than creatures who have not forgotten how to hunt, but the ability of all domestic animals to survive will be tenuous at best. Veterinary care is not likely to be widely available, if at all. Furthermore, most species will be vulnerable to vast populations of hungry humans. Only a few animals, those protected by humans (as long as they have the capacity to do so), will survive.\n\nOne of the most important gifts animals give us is their instinctual nature, and in daunting times, we will desperately need to develop and rely on our own animal instincts. We can develop these by carefully observing animals, living with them, and protecting them, and in this way, we do \"fall at their feet.\"\n\nDig within. Within is the wellspring of Good; and it is always ready to bubble up, if you just dig.\n\n\u2014Marcus Aurelius\n\nIn recent years the expression \"dig down deep\" has become popular as a motivational slogan for \"going the distance,\" discovering within ourselves the resources we need to accomplish a goal or task. Yet even in the second century, the famous Stoic philosopher and Roman emperor encouraged his listeners to excavate their inner world.\n\nI suspect that in some parts of the external world and in my own country, the fear, grief, anger, and despair of collapse will be so intense that, unless one has the capacity to dig deep into the soul and discover previously untapped resources, one will be emotionally and spiritually overwhelmed\u2014perhaps even to the point of psychosis or death.\n\nAll manner of suffering offers an opportunity to dig deep, but suffering on the scale that a world in chaos is likely to produce will compel humans to dig or, in the worst-case scenario, die. This is not to say that digging deep makes one invincible or invulnerable to death, but it does increase the likelihood of discovering the internal resources one needs\u2014resources not needed in more \"normal\" times.\n\nMy books, _Sacred Demise: Walking the Spiritual Path of Industrial Civilization's Collapse_ and _Navigating the Coming Chaos: A Handbook for Inner Transition,_ as well as this book, have been written to provide practical tools for digging. The quotes from great thinkers are a cache of perennial wisdom, upon which people have drawn throughout the ages, when their lives and communities have fallen apart. In a sense, these offer a \"shovel,\" but a shovel is only useful if it is used for digging.\n\nDigging is hard work and often leaves the body aching. In our modern world we prefer to use backhoes or to have other people to do the digging for us\u2014both literally and metaphorically. In turbulent times, however, there is no one to \"dig\" but we ourselves. Amazing, untapped resources are waiting to be discovered and appropriated, and they will be\u2014 _if_ we are willing to dig.\n\nYou don't develop courage by being happy in your relationships everyday. You develop it by surviving difficult times and challenging adversity.\n\n\u2014Epicurus\n\nThe ancient Greek philosopher Epicurus has often been accused of advocating that we drown ourselves in pleasure and avoid pain at all cost. After all, it is from his name that we get the label _Epicurean,_ denoting gluttony, hedonism, and self-indulgence.\n\nBut the philosophy of Epicurus was significantly more complicated than mere obsessive pleasure seeking, for he understood that, at times, we need discomfort\u2014and even pain\u2014in order to develop character.\n\nIn my interactions with people who have only recently discovered my work, I am often accused of fear mongering. In fact, some argue that fear is \"unspiritual\" and that, whenever we are afraid, we should immediately do something to alleviate our fear, because fear and love are opposites. If one is in fear, they say, one is not experiencing love.\n\nI beg to differ. I want to champion the perspective of Epicurus regarding the teaching potential of pain. Moreover, the late philosopher and poet John O'Donohue said that we do not become courageous by avoiding fear, but by being in it, exploring it, and moving forward anyway.\n\nLove and courage are inextricably connected with each other\u2014and with fear. We develop courage by working with, not against, our fear. The word _courage_ originates from the French word for _heart._ Fear can actually expand the heart, stimulating compassion and, therefore, greater magnanimity.\n\nMay we open to these difficult times and the adversity that they entail so that we develop courage, compassion, and a more profound quality of love than we could ever have imagined.\n\nThey intoxicate themselves with work so they don't see how they really are.\n\n\u2014Aldous Huxley\n\nWe might also say that, in the milieu of industrial civilization, people have intoxicated themselves with work so that they don't see _who_ they really are.\n\nOverwork, or what has fashionably come to be called _workaholism,_ has become the modus operandi of anyone in this culture who seeks success. In most professional positions, it is understood that one does not work an eight-hour day\u2014one must work many more hours per day, take work home, arrive at the job early, and thereby demonstrate one's competence and loyalty. Thus, most working people in our society have little time for family, hobbies, travel, or contemplation. These activities have been traditionally \"saved\" for retirement, but sadly, because some people have had so little experience savoring these pursuits earlier in life, they enter retirement and become depressed, because they just \"can't get into it.\" Work became their drug of choice, and they find it nearly impossible to withdraw from it.\n\nIn the throes of the collapse of industrial civilization, however, with increasing unemployment, many people are discovering _how_ they are and _who_ they are, as they allow themselves to engage in activities much more meaningful than earning a livelihood. Others cannot adapt resiliently to unemployment or underemployment and become mired in alcohol, drugs, and depression; some even end up taking their own lives.\n\nI believe that within the next two decades, employment, as we know it today, may no longer exist. Between increased automation of tasks and economic collapse, few people are likely to be employed. Much more time and space will be opened in peoples' lives for other pursuits, and this could be a glorious development\u2014but only if they have given themselves some permission, before losing their jobs, to discover who and how they are. Only if they have not been intoxicating themselves with work.\n\nWe can easily forgive a child who is afraid of the dark; the real tragedy of life is when men are afraid of the light.\n\n\u2014Plato\n\nOne may interpret these words from Plato in numerous ways. What does he mean by \"the light\"? Consistent with his philosophy, \"the light\" probably means \"the truth,\" or what is so.\n\nAs I reflect on Plato's statement in the context of the collapse of industrial civilization, I recognize that countless adults are terrified of what is so. Like children, adults are afraid of the dark, because they have been enculturated to believe that light, happiness, good times, prosperity, and positive thinking are not only desirable but mandatory. They immediately flee from darkness, suffering, tough times, poverty, and anything that smacks of \"negativity.\" Even in the face of all evidence to the contrary, such individuals argue that we're going through a \"downturn,\" but good times are certain to return. Their incessant mantra is \"It will come back; it always does.\"\n\nFear of what is so among adults is exactly the same as fear of the dark in children, because what is so is overwhelmingly dark. The darkness has tremendous potential for generating light in our consciousness and human relationships, but not without the suffering that darkness entails. Thus, when we disavow the dark, we also disown the light, with which darkness is pregnant and certain to ultimately deliver.\n\nIf we are going to function as adults, we must be willing to confront the darkness of what is so\u2014and extract the light that is inherent in a world in decline.\n\nThere's a lot of fear connected with the inner journey because it penetrates our illusions. Taking the inner journey will lead you into some very shadowy places. You're going to learn things about yourself that you'll wish you didn't know. There are monsters in there\u2014monsters you can't control\u2014but trying to keep them hidden will only give them greater power.\n\n\u2014Parker Palmer\n\nMy work is all about supporting people in making the inner journey. At the same time that I tout the treasures of the inner world, I must also honestly disclose that, in going inward, one will encounter \"some very shadowy places.\" But as readers of this book have discovered by now, we cannot have one without the other.\n\nNo one in the Western world was more aware of this than Carl Jung, who devoted his adult life to supporting people in integrating the sacred and the shadow within themselves\u2014as well as in the world.\n\nMy fondest hope, in terms of human consciousness in the next culture, is that we will come to understand at a bone-marrow level the machinations of the human shadow and the futility of repressing it. If we do not, then we are doomed to re-create, for the most part, the disastrous paradigm of industrial civilization.\n\nNavigating a world in chaos requires men and women who are mature enough to understand the \"both\/ands\" of life and to hold the darkness alongside the light, so that alchemical transformation can occur within themselves and within the world.\n\nIf we can face the monsters within, it is likely that we can face the monsters in a collapsing, chaotic, calamitous world. In fact, our inner demons may be profoundly more formidable.\n\nScience can point out dangers, but science cannot turn the direction of minds and hearts. That is the province of spiritual powers within and without our very beginnings\u2014powers that are the mysteries of life itself.\n\n\u2014Oren Lyons\n\nIf you do not have a moral question in your governing process, then you do not have a process that is going to survive.\n\n\u2014Oren Lyons\n\nLife will go on as long as there is someone to sing, to dance, to tell stories and to listen.\n\n\u2014Oren Lyons\n\nThese three statements from Oren Lyons, Native American faithkeeper of the Turtle Clan of the Onondaga and Seneca Nations of the Iroquois Confederacy, were spoken at different times and on different occasions, but I notice the congruence each has with the other.\n\nIn the first statement, Lyons tells us that science has limits, which I happen to believe humanity has now reached. Why is this so? The second statement answers the question. Since the Enlightenment, humanity has endeavored to polarize science and the sacred, and from the indigenous perspective, the sacred includes and manifests itself through the earth community. Science has produced penicillin, has enabled men to walk on the moon, and has resolved heretofore perplexing enigmas in human DNA, but science without spirituality has also helped produce the horrors of nuclear war and global warming, the convoluted and questionable practice of cloning, and the biotechnology of genetically modified foods. Science without a moral compass does not facilitate survival but self-destruction.\n\nYet Lyons tells us that even if we destroy ourselves, which we are clearly in the process of doing, life will go on if we can reactivate indigenous wisdom. If we can sing, dance, tell stories, and listen, if we can utilize art, music, and poetry to access our hearts by way of the right brain, we have a greater chance of not only surviving, but of creating a next culture\u2014a culture not averse to science, but integrating with it \"the mysteries of life itself.\"\n\nIn the course of history, there comes a time when humanity is called to shift to a new level of consciousness, to reach a higher moral ground. A time when we have to shed our fear and give hope to each other.\n\n\u2014Wangari Maathai\n\nI know there is pain when sawmills close and people lose jobs, but we have to make a choice. We need water and we need these forests.\n\n\u2014Wangari Maathai\n\nIn the first statement, the late Kenyan activist and Nobel Peace Prize winner tells us that a new level of consciousness is needed and then, in the second statement, gives a superb example of a dilemma that faces countless communities on this planet\u2014how does one protect the livelihoods of people whose only way of earning a livelihood comes from the destruction of the earth?\n\nSuch dilemmas cannot be solved on the level of the problem\u2014as Einstein reminded us, \"Problems cannot be solved with the same mindset that created them\"\u2014but must be addressed from a new level of consciousness. From the vantage point of that shift, new options arise that are not exclusively rational but may erupt through a kind of creativity inaccessible at lower levels of awareness.\n\nFrom lower levels of consciousness, there is only the either\/or. Earth must be sacrificed so that people can earn a living\u2014yet, ultimately, nothing and no one will survive if the planet is decimated. In this situation, over time, this \"living\" becomes a death spiral for the ecosystems.\n\nOptions discovered at new levels of consciousness may not be easy to implement; tough choices invariably need to be made, because transformation of consciousness almost always includes some form of sacrifice. Yet this is precisely when we need to \"shed our fear and give hope to each other.\" With the support of caring community, we can make sacrifices, and we can integrate opposites that were previously deemed terminally polarized. A new state of consciousness and a new state of community often create space and opportunity for solving what appears to be unsolvable.\n\nWe either make ourselves miserable, or we make ourselves strong. The amount of work is the same.\n\n\u2014Carlos Castaneda\n\nAs the collapse of industrial civilization exacerbates and societies in decline further unravel, many people will be focused only on how miserable they are. They will have no sense of the demise as a call to inner and outer work in order to become strong, vibrant, compassionate, empowered human beings.\n\nWhile there may be countless moments of joy and celebration, even in the midst of the anguish, most of us will be living drastically simplified and emotionally and physically challenging lives. With so many comforts gone, our days are likely to be preoccupied with safety and survival\u2014obtaining food, water, and shelter; mostly staying in one place; and needing to make alliances with people nearby. It is not difficult to imagine our lives resembling those of our ancestors, before the conveniences of modernity.\n\nHowever taxing a downsized life may become, let us remember that living it takes the same amount of energy as resisting and complaining about it. I suspect that as life becomes more challenging, some innocent, marginalized individuals and groups will become scapegoats. But projection and blame consume a great deal of energy.\n\nIt is important to keep ourselves informed about what is happening in the world and to have a map of the big picture regarding how we arrived at where we are and where we are being compelled by circumstances to go. However, it is far too late in the game to blame anyone or anything. Here we are\u2014in the throes of the collapse of industrial civilization. We are not here by accident; nor is it helpful to rage against a dying machine. How will we not only survive but become a more mature, wise, merciful, caring, resilient human being? Where do we want to invest our energy?\n\nIn old traditions those who acted as elders were considered to have one foot in daily life and the other foot in the other world. Elders acted as a bridge between the visible world and the unseen realm of spirit and soul. The old word for having a foot in each world is \"weird.\" Those who would truly become wise must become weird enough to be in touch with timeless things and abnormal enough to follow the guidance of the unseen.\n\n\u2014Michael Meade, _Fate and Destiny_\n\nThe word _elder_ has been so overused and misinterpreted in spiritual circles that it has become somewhat of a clich\u00e9. The worst possible distortion is the notion that _elder_ and _older_ are synonymous. In reality, one does not become an elder by simply becoming older, because the defining quality of the elder is wisdom, not age.\n\nAs we navigate the collapse of industrial civilization, it is crucial to understand the role of the elder. Other words for _elder_ might be _steward_ or _caretaker._ In the above quote, Michael Meade clarifies that the elder is a bridge between the other world and this one. She holds a solid connection with the sacred at the same time that she weathers the turbulence of a world in chaos. Surrounded by people who are deeply troubled, terrified, enraged, depressed, or suicidal, the elder steps forth and teaches those overwhelmed with angst that they are experiencing a planetary initiation, explaining what that means and why it is necessary. He assists people in connecting with the deeper, sacred Self and offers the wisdom necessary to motivate them to move _with_ the initiation, not against it, and to offer their unique gifts in ways that enhance the transformation that the initiation seeks to bring forth.\n\nHow does the elder acquire wisdom? By living through many initiations and coming to understand the importance of moving _with_ them, not against them. He or she knows that humans are only remade from the inside out, as a result of life-altering adversities that require them to reach down into the sacred Self and there discover unequivocal meaning and purpose.\n\nWhen all the trees have been cut down,\n\nWhen all the animals have been hunted,\n\nWhen all the waters are polluted,\n\nWhen the air is unsafe to breathe,\n\nOnly then will you discover that you cannot eat money.\n\n\u2014Cree Prophecy\n\nFrom the indigenous perspective, civilized humans are mad. They are so tragically disconnected from nature that they can only see its trees, animals, waters, air, and indeed, every other aspect, as an opportunity for monetary profit. For as long as we have called ourselves \"civilized,\" we have attempted to \"eat\" money by treating the ecosystems as commodities to be bought and sold.\n\nHumans now find themselves steadily approaching a situation in which the elements of nature they have sought to privatize and profitize are worthless, because those elements are either extinct or so polluted that they are useless. In this sense, the joke is on civilized humans. Ironically, we have become so voracious that we soon may have nothing to eat\u2014literally.\n\nHuman-made climate change is bombarding the planet with droughts, extreme weather, and the death of the planet's immune system to the extent that, tragically, worldwide famine is possible. Might humans actually destroy so much land and pollute so much water and air that they have nothing to eat or drink or breathe?\n\nThe mythological theme of defeat or death by way of avarice and insatiable appetites is an ancient one. Yet all of those myths lead to the reality that, regardless of how our hungers manifest themselves, that for which we are fundamentally starved is soul sustenance\u2014nutrients that can only be found in the sacred. Indigenous people have understood this for millennia, which is why they perceive our voracious consumption of the earth as madness.\n\nRecent studies have indicated that power is the ultimate addiction, followed closely by addiction to wealth. As any addict knows, the sub _stance_ is a sub _stitute_ for soul satisfaction. However, when we are sated by an intimate relationship with the earth, we are full, and there is more than enough.\n\nNinety-nine percent of who you are is invisible and untouchable.\n\n\u2014Buckminster Fuller\n\nThere is nothing in a caterpillar that tells you it's going to be a butterfly.\n\n\u2014Buckminster Fuller\n\nThis great systems theorist, architect, engineer, author, and designer was one of the towering geniuses of the twentieth century. The two sentences above reveal the visionary Fuller, who was never content with simply confronting reality but always held in his mind and heart a vision of what was possible.\n\nAs I read these two statements, I am reminded of the times in which we live and the planetary ordeal we have entered and to which I have given various names in this book: collapse, unraveling, chaos, initiation, and paradigm shift. Repeatedly I implore those who follow my work to hold in their consciousness, no matter how daunting external circumstances become, a vision of what is possible for humanity as a result of moving through our species' rite of passage.\n\nFuller was a visionary, which is evident in the first statement: most of who we are is invisible and untouchable. For that very reason, vision is not only possible but necessary. Fuller probably would not have used \"deeper Self\" to describe the \"most of who we are,\" but he was obviously aware of something within our nature that is more momentous than the rational mind. That \"something\" is the territory of vision\u2014our ability to imagine and live out what is possible, even when all external evidence appears to contradict that vision.\n\nJust as nothing in the caterpillar tells us that from its body and tireless work in the cocoon, a gorgeous butterfly will emerge, very little in the exterior landscape of today's world suggests that even one transformed human being\u2014let alone clusters of them\u2014may evolve from the misery and rubble of a collapsing civilization. In the cocoon, the caterpillar becomes a butterfly as a result of being liquefied\u2014a process that essentially remains a mystery to this day. A stunning mystery of the invisible and the untouchable.\n\nNot I, not I, but the wind that blows through me!\n\nA fine wind is blowing the new direction of Time.\n\nIf only I let it bear me, carry me, if only it carry me!\n\nIf only I am sensitive, subtle, oh, delicate, a winged gift!\n\nIf only, most lovely of all, I yield myself and am borrowed\n\nBy the fine, fine, wind that takes its course through the chaos of the world\n\nLike a fine, an exquisite chisel, a wedge-blade inserted;\n\nIf only I am keen and hard like the sheer tip of a wedge\n\nDriven by invisible blows,\n\nThe rock will split, we shall come at the wonder, we shall find the Hesperides.\n\nOh, for the wonder that bubbles into my soul,\n\nI would be a good fountain, a good well-head,\n\nWould blur no whisper, spoil no expression\n\nWhat is the knocking?\n\nWhat is the knocking at the door in the night?\n\nIt is somebody wants to do us harm.\n\nNo, no, it is the three strange angels.\n\nAdmit them, admit them.\n\n\u2014D. H. Lawrence\n\nReminiscent of Rumi's poem \"The Guest House,\" Lawrence left us, in \"The Song of the Man Who Has Come Through,\" a primer for total surrender of the ego and an adamant promise of what happens when that occurs. The poem refers to the Hesperides, the nymphs\u2014the \"three strange angels\"\u2014who tend the garden of the mythological Hera. Thus Lawrence implies that when we surrender to life's invisible, chiseling blows and consciously work with them as divinely delivered teachers, we will ultimately discover ourselves in a lush garden of unspeakably rich transformation.\n\n\"The man who has come through\" is the one who has the gifts that accrue when we allow the winds of change to blow through us and carry us to the place where we are intended to go.\n\nI offer this poem to the reader as an ongoing meditation for soul sustenance in a chaotic world.\n\nA radical inner transformation and rise to a new level of consciousness might be the only real hope we have in the current global crisis brought on by the dominance of the Western mechanistic paradigm.\n\n\u2014Stanislav Grof\n\nFor me, the most significant word in this statement is _hope._ It is not clear from this sentence how Grof interprets that word, but it must be explored more deeply. Notice that Grof does not declare that transformation of consciousness will save the world from suffering. In fact, many of his other writings would suggest that Grof understands that transformation occurs only by being willing to pass _through_ suffering, not by avoiding it.\n\nOften in this culture, when we speak of hope, implicit in the use of the word is the notion of \"salvation\" or the possibility of escape from suffering. Yet in most cases, suffering is required in order to experience transformation. And what would transformation of consciousness mean for humanity in its present state?\n\nWhen we endure suffering for extended periods of time, we invariably develop more empathy and compassion. We become more sensitive to the suffering of others, and it becomes more difficult to numb ourselves with activities and substances. We become, in a sense, bigger people; our sphere of concern is expanded. Less is taken for granted, and we experience our moments of joy, abundance, and conviviality as gifts, rather than as entitlements that we deserve simply because we exist. Suffering cultivates sensitivity to both our human and nonhuman community. Quite naturally, we notice a longing to serve and contribute our gifts to other sentient beings, and we more fully appreciate their contributions to us. In suffering, we are humbled, and the ego, rather than functioning as \"emperor of the psyche,\" begins to find its own place as a servant of the deeper Self.\n\nThis is the hope of the world, but that hope does not mitigate the demise of industrial civilization. Rather, the hope to which Grof refers is the midwife of a new culture being born from the rubble of this one.\n\nNot knowing when the dawn will come, I open every door.\n\n\u2014Emily Dickinson\n\nIn the throes of the unraveling and in and out of the shadowy crevices of so-called normal life, we never know when dawn will come. Even in the midst of abject suffering and loss, life frequently surprises us with a thread of light breaking on the distant horizon, which then swells to a radiant epiphany. We cannot predict or control an eruption of awareness or illumination, but we must know that it will come in its own time. And because we know, we must \"open every door.\"\n\nWe may or may not witness the myriad epiphanies that collapse intends to evoke on our planet. Many will not occur in our lifetime, yet we must be deeply engaged in catalyzing the transformation of consciousness. We must keep ajar the doors of our own psyche in anticipation of illuminations that we may not even have the privilege of experiencing firsthand.\n\nI am heartened that, even now, so many human beings are awakening to the reality of collapse and committing their lives to finding meaning and purpose in it. Often people tell me that I'm \"preaching to the choir,\" at which point I remind them of how important the choir actually is. The so-called choir is made up of all the potential elders or stewards of the unraveling, people who understand why it must happen and profoundly appreciate its deeper significance for our species.\n\nRegardless of how daunting the demise appears\u2014no matter how futile our preparation may feel or how small may be the odds of our physical survival\u2014the doors must remain open for a dawn that has a life of its own, outside our limited volition. What doors will you open today?\n\nI'm not afraid. . . . I was born to do this.\n\n\u2014Joan of Arc\n\nMy friends, do not lose heart. We were made for these times.\n\n\u2014Clarissa Pinkola Est\u00e9s\n\nIn workshops I often ask, \"How many of you think you are here accidentally\u2014that you just fell out of the sky one day during the twentieth century?\" To date, I have not seen one hand raised in response to the question. This does not mean, however, that every person hearing the question understands his purpose in being here at this momentous time or has a sense of the \"divine errand\" that she has been sent to this planet to complete.\n\nAn integral part of our soul deepening and training for elderhood is developing a clearer sense of our innate gifts and how we are destined to utilize them in a collapsing world. Not only did we not fall out of the sky, we did not come here empty handed. It is rare that parents actually see our gifts, and often when they do, they set out to sabotage them, though without intending to. Thus, most of us spend many years in adulthood finding and honing our gifts, often with great anguish, and others never discover them at all.\n\nDiscovering and claiming our gifts does not usually occur in \"normal\" times, because our gifts originate in the soul, and there has never been a \"normal\" soul. A person's gifts are usually discovered in troubled personal or cultural times, in unexpected ways and at unpredictable moments. The deeper and more intentionally we delve into the soul, the easier it becomes to access and contribute our gifts. The delivery of our gifts is not a left-brain project that we organize cleanly and without chaos. These troubled times compel us to do the work of soul that may have been neglected most of our lives\u2014and now cannot be avoided.\n\nMove forward and feel the fear alongside the aliveness, vitality, and purpose that these times evoke. You were born for this very moment in human history.\n\nWe're fucked, and life is really, really good.\n\n\u2014Derrick Jensen\n\nThis statement is perhaps the ultimate expression of paradox. In recent years, it has become my mantra for living in the present moment and maintaining perspective as I navigate a chaotic world. It's especially useful to know that Derrick Jensen suffers from Crohn's disease and lives with a great deal of pain on a daily basis. Yet knowing what he knows about the state of the planet and at the same time living with a painful chronic illness, he can say, \"life is really, really good.\"\n\nWhat makes life really, really good? Is it money in the bank, possessions, successful children, and an impressive retirement fund, the means with which to travel to exotic places? In a collapsing world almost no one will have these except for the very wealthy. Most individuals will become increasingly poor, will be limited in their ability to leave their immediate community, will have little access to health care, and will be forced to do much more physical work than they have ever done in their lives. Moreover, as this environmentally overtaxed planet breaks under the load of human exploitation, it may shed five or six billion of us. An early death is an almost mathematical and environmental certainty. Most of us will probably not live as long as we might have under \"normal\" circumstances, that is, with reasonably comfortable lives and the benefits of medical technology and pharmacology.\n\nIf you have been preparing emotionally and spiritually for collapse, you know that a really, really good life is one in which we treasure each moment, surrender to our vulnerability in the face of collapse, create beauty on a daily basis, share both good and bad times with loved ones, and regularly give thanks for the lives we are compelled to live. We laugh, love, play, work, serve, share, commune with nature, and savor all of the nooks and crannies of our humanity. We understand our mission on this planet, and we do everything in our power\u2014including asking for strength from the sacred\u2014in order to accomplish it. We take nothing for _granted_ and give thanks for all that we are mercifully _granted._\n\nWe are looking to brands for poetry and for spirituality, because we're not getting those things from our communities or from each other.\n\n\u2014Naomi Klein\n\nWherever I travel and whenever people communicate with me by email or in some other way, the longing for community that I hear is palpable\u2014so much so that it reverberates like an ache in the voice that catches on a lump in the throat or perhaps a sigh of anticipation similar to a young child eager to be seen and valued.\n\nConsumer capitalism has pillaged the planet and its inhabitants, leaving them emotionally empty and spiritually vacuous. A pseudo religion promising the nirvana of wealth and prosperity, consumerism demands that we not feel our feelings but, instead, obsess over our bodies, physical appearances, portfolios, and vacation homes. We are commanded to fill the desolate spaces in the soul with brands, advertising, and proper product placement.\n\nThe soul cannot be artificially assuaged. It cries out for poetry, story, song, dance, and art. Our obligation is not only to seek these for ourselves but to offer them to each other. What might happen if we sat in a circle and read or recited poems to each other? I know the power of this firsthand, having participated in a number of poetry salons when I lived in Northern California in the 1990s. People brought a cache of poems to share and fed each other's souls, providing a symbolic feast of inspiration and support, followed by a literal feast, in the form of potluck dinner.\n\nPoetry, song, story, dance, and art are the most precious gifts we can give each other, and in a collapsing world, they may be the only gifts we can afford\u2014which might be for the best, it turns out, as these are actually the ones most salutary for the soul. Consider creating a poetry salon within your community, composed of individuals who love and appreciate the language of poems. Use it as an opportunity for deep sharing and communal soul making. These are priceless gifts that indeed keep on giving.\n\nIt is not the strongest of the species that survive, nor the most intelligent, but the one most responsive to change.\n\n\u2014Atributed to Charles Darwin\n\nThis statement from Darwin can best be summarized in one word: _resilience._ Resilience is the capacity to bounce back, adapt, and regain the shape that may have been radically altered by change or trauma. In recent years I have been presenting numerous workshops on emotional resilience\u2014a quality that I believe is crucial if we are to survive and find our place in a chaotic, collapsing world.\n\nIf change comes suddenly or too quickly, our nervous systems have great difficulty adapting. However, if we make changes in anticipation of the collapse of industrial civilization, we will naturally become more resilient and, therefore, more capable of weathering the changes without being overwhelmed by them.\n\nEnter the _sacred_ \u2014a word inextricably connected with _sacrifice._ What are we willing to sacrifice in order to prepare, and\u2014as we ponder what collapse may ask of us\u2014what feels impossible to sacrifice? Considering these questions is in itself a spiritual act, because it compels us to explore the inner recesses of the soul and confront demons incorporated from consumer capitalism and its false promises of \"security\" and \"comfort.\" Someone has said that the only real \"security\" is a false sense of one. Sadly, the masses of consumers that inhabit our planet have yet to understand this. Anyone reading these words, however, has a golden opportunity to become physically, emotionally, and spiritually resilient by consciously working with the concept of sacrifice.\n\nThis is not to say that we should radically strip our lives of all comforts but rather be very present and aware in our lives of the comforts on which we depend and feel we cannot live without. We can experiment with withdrawing from them and seeing how that feels. Undoubtedly, we will feel disoriented, deprived, and resentful. We should not feel morally obligated to suddenly begin living a monastic lifestyle, because equally important as actual sacrifice is our awareness of how resilient we have yet to become.\n\nWithin tears, find hidden laughter\n\nSeek treasures amid ruins, sincere one.\n\n\u2014Rumi\n\nWhen we genuinely, deeply ponder the collapse of the way of life we have known probably since birth, we can feel unspeakably afraid. If we allow ourselves to contemplate all the losses the demise will entail, we may find ourselves consumed with sadness, our hearts broken and aching. Surges of anger may permeate the body as we consider how unjust the destruction will be, particularly as it impacts innocent lives. Subsequent to all of these feelings, we may find ourselves shutting down, overcome with numbing despair. Paralysis. Can't move. Can't speak. Total and utter hopelessness.\n\nI have just described the four of the major feelings we are likely to experience in a chaotic world of unraveling: fear, grief, anger, and despair. The one I did not mention is the one noted by Rumi. \"Within tears, find hidden laughter.\" Note that the laughter is hidden. It is our challenge to find it.\n\n\"Seek treasures amid ruins.\" Out of the rubble\u2014literal or symbolic\u2014extract treasures, because whether we realize it or not, they are there. As I have stated earlier in this book, even inmates in Nazi death camps found humor and the treasures of love, compassion, courage, and mercy in the hell they inhabited.\n\nOne useful tool for discovering laughter and treasures in a collapsing world is to pay careful attention to the ones we found at other difficult times in our lives, in those other \"little deaths\" through which we moved. What were the treasures? How did we bring them forth? Where was the humor, even in the midst of horror?\n\nRumi calls his reader \"sincere one.\" You would not be reading these words were you not sincere. Allow your sincerity to assist you in feeling all of your feelings about collapse, including laughter, joy, play, and the other treasures amid the rubble.\n\nClueless busybodies who feel that \"we must do something\" and can be spun around by any half-wit demagogue are bad enough, but the most dangerous group, and one to watch out for and run from, is a group of political activists resolved to organize and promote some program or other. Even if the program is benign, and even if it is beneficial, the politicized approach to solving it might not be. As the saying goes, revolutions eat their children. Then they turn on everyone else. The life of a refugee is a form of survival; staying and fighting an organized mob generally isn't.\n\n\u2014Dmitry Orlov\n\nMany readers of this book are familiar with the writings of Orlov, who survived the collapse of the Soviet Union in the late 1980s and early 1990s. Orlov has courageously written about collapse for at least the past ten years, and his writings grow ever more rich and poignant.\n\nIn this quote, Orlov contrasts the life of a refugee with the life of an activist, declaring that choosing to involve oneself with the powers that be is a setup; the end result is to be devoured by centralized systems.\n\nIt turns out that all of us who choose to prepare for collapse and abandon any efforts to reform the larger system will end up being refugees. We may end up without a country, a city, a neighborhood, or a home. We may be forced to wander or to live in strange venues with strange people who come into our lives for any of myriad reasons, though mostly having to do with the need for mutual survival.\n\nA refugee is a person seeking refuge from wars, revolutions, or natural disasters. She has little claim to any material possessions and feels fortunate to have escaped the melee with her life. The refugee often ends up in refugee camps with hundreds or thousands of other refugees. He lacks a sense of rootedness, place, and home. Refugee camps are intended to be transitional, leading to some kind of relocation, and ultimately resulting in finding a home.\n\nSocietal collapse invariably results in the proliferation of refugees. Protracted chaos and changes of venue are typical, and a sense of home may elude the refugee for a long time\u2014or for as long as she lives. For this reason, it is exceedingly important that we maintain a place of refuge within ourselves when a literal, external home may never be more than a memory or a great dream.\n\nThe best use of life is to spend it for something that outlasts life.\n\n\u2014William James\n\nWhat legacy are you leaving on this earth? Perhaps you don't have children, but whether or not you do, your legacy should be significant enough to be discernable by individuals living seven generations after you. I hasten to add, however, that that does not mean that your legacy must be dramatic or written in the sky. Leaving a legacy is about planting seeds, and we seldom have the opportunity to see those seeds grow and come to fruition.\n\nFor seventeen years I was a psychotherapist in private practice, and for eleven years after that, a college professor. I worked tirelessly to plant seeds in both careers. Occasionally, I hear from a former client or student who informs me of the influence my seed planting had on him or her. While this is gratifying, I am well aware that many more people with whom I planted seeds will never let me know what happened as a result.\n\nPlanting seeds, however, is an act of the deeper Self, not the ego. The ego would love to know the total outcome of seed planting so that it can evaluate how well or how poorly it planted the seeds. The deeper Self, by contrast, does not need to see tangible results; it simply trusts that its efforts will engender lush gardens of wisdom and transformation. The deeper Self is on a divine mission to express its gifts and fulfill its purpose on this planet, regardless of outcome.\n\nHumanly speaking, however, sharing appreciation with someone who has planted a seed in us is an act of great kindness. Are there people who planted seeds in you or your life whom you need to thank? It is likely that their contribution will outlast their own lives\u2014and yours as well. Therefore, thank them for how they spent their lives, even if they are no longer with us. At the same time, ponder deeply how you will spend yours.\n\nOne does not discover new continents without consenting to lose sight of the shore for a very long time.\n\n\u2014Andr\u00e9 Gide\n\nSwimming in the ocean of collapse constitutes the epitome of uncertainty. This is why so few want to hear or learn more about collapse preparation. Our humanity insists on tethering to the shore or at least on being able to see it, as we drift further into uncharted waters.\n\nCollapse, by definition, obliterates a \"shore.\" All the rules change; the patterns, definitions, and structures by which we have measured our lives and charted our course no longer exist. Furthermore, we are going to lose sight of the shore for a very long time\u2014perhaps for the rest of our lives. Indeed, as we contemplate collapse, what is so difficult to grasp is that, as we navigate it, we will discover at nearly every turn that nothing will ever be the same again.\n\nIt takes great courage to let go of the shore, and we cannot do it because someone moralistically advises us that we should. I believe that the letting go is an act of the deeper Self and not entirely within our control. Something greater within us lets go, even if we are not fully aware of what is happening. Left to its own devices, the human ego has little capacity for surrender, but when we call upon the deeper Self and our allies in nature, we may be surprised to discover how something within us has let go, even before we are consciously aware of it.\n\nIn your preparation for an unraveling world, what do you need to let go of? Do not judge yourself for clinging; only ask for assistance and notice what happens next.\n\nIt is good to have an end to journey towards; but it is the journey that matters in the end.\n\n\u2014Ursula K. Le Guin\n\nOne reason that many people prefer talking about logistical preparation instead of emotional and spiritual preparation is that logistical preparation is all about physical survival. The human mind can become obsessed with food and water storage, reskilling, taking courses, and building superbly outfitted doomsteads. I wholeheartedly support these actions, _and_ they constitute only one aspect of collapse preparation.\n\nUnless we are willing to confront our mortality and the possibility that\u2014no matter what we do to prepare\u2014we may not survive physically, we will reject or ignore emotional and spiritual preparation. Follow that course, and we have only one end in sight: staying alive.\n\nBut what happens to us when we confront our mortality and the fact that the best-laid plans of mice and men may not prevent our succumbing to any one of the plethora of perils that collapse will present? If we genuinely taste moments of mortality, the soul expands and begins charting a different and more inclusive course. You see, the soul is in love with the journey, not the outcome.\n\nHow often people have told me, \"I wouldn't miss this time in history for anything. I'm so grateful that I live now, even in this perilous time. The journey of preparation, both internally and outwardly, is more rewarding and enlivening than anything I have ever undertaken.\"\n\nIndeed, the journey _is_ what matters in the end, because it is not only an exercise in creativity, self-preservation, and the protection of loved ones. It is a journey into the deeper recesses of the soul and the treasures we brought with us in order to embark on the journey. Turbulent times ignite all manner of trouble. As African shaman Malidoma Som\u00e9 states, do not avoid trouble but rather make sure that you get into the _right_ trouble\u2014trouble that assists, rather than impedes, your transformation.\n\nIf men were ever to lose the appetite for meaning we call thinking, they would lose the capacity for asking all the unanswerable questions upon which every civilization is founded.\n\n\u2014Hannah Arendt\n\nI firmly believe that the demise of this civilization is directly related to its inability to ask the \"unanswerable questions.\" We have come to believe that if we can't answer a question, it is irrelevant to us. We pride ourselves in having the answers, and if we don't have them, we are certain that we will know where to find them.\n\nMen and women of modernity have lost interest in tough questions and reject the unanswerable ones as a waste of time. Earlier in this book the words of Rilke reminded us to \"have patience with everything unresolved in your heart and to try to love the questions themselves as if they were locked rooms or books written in a very foreign language. Don't search for the answers, which could not be given to you now, because you would not be able to live them. And the point is to live everything. Live the questions now. Perhaps then, someday far in the future, you will gradually, without even noticing it, live your way into the answer.\"\n\nTwenty-first-century humans have failed to understand that staying with a question will take them somewhere. Moreover, it is one thing to live with a tough question, for which we think there might be an answer, but what about living with one that does not have an answer? For example, what is death? What happens when we die? Have we lived before? Why do we suffer? Why did my best friend die of a terminal illness? Why did my wife leave me? Why did I have that car accident that left me in a wheelchair for life?\n\nPhysicians and many psychotherapists would encourage us not to become obsessed with the \"why\" when we are confronted with tragedy; instead, they advise us to move on. There may be value, however, in doing both. The mind says that if we could figure out the why, we could reverse our circumstances. But that should not be the motive for asking \"why.\" We can work on acceptance at the same time that we attempt to make sense of the situation. Move forward and at the same time, make meaning.\n\nAs I considered the future of all children and the juncture we are headed into, I realized that the things I developed for children in difficult circumstances were exactly what all youth needed to find their own way through the difficult times people and our planet are facing, to help shape a new and different future.\n\n\u2014Dianne Monroe\n\nDianne Monroe is an expressive-arts facilitator, writer, and parent living in Sebastopol, California. She is acutely aware of the collapse of industrial civilization and is passionately concerned with how to raise children in turbulent times. A firm believer in the Latin root of _education,_ _educare,_ \"to lead out,\" Moore is committed to honoring the sacred Self within children and supporting them in bringing it forth in the learning process, relationships, creativity, hobbies, and play.\n\nI interact regularly with parents who are deeply concerned about the future of their children in a chaotic world. Many more parents, however, are so terrified about their children's future that they are unable to face the reality of collapse at all. On the other hand, I occasionally meet parents whose children have been researching collapse for years and introduced the reality of it to them.\n\nIn my experience, whatever path parents and children take as they approach this enormous planetary transition, when they are able to honestly and lovingly discuss collapse and prepare for it together, their bonds are strengthened, and their respect for one another is enhanced.\n\nMy generation and my parents' generation have shamefully squandered the future of subsequent generations. We have literally spent their inheritance many times over. We cannot reverse the legacy we have left them, but we owe it to them to look collapse squarely in the face and speak freely with them about it. If they are not willing to discuss the future, we should not force them to do so, but we should merely let them know that we will be present with them in whatever the future holds. We cannot meet all of their emotional, financial, or physical needs, but we can love them as they and we, together, navigate the idiosyncratic unravelings of our own individual lives.\n\nThe trouble with the rat race is that even if you win, you're still a rat.\n\n\u2014Lily Tomlin\n\nMany readers of this book have only recently left the rat race; others may have been defectors for a very long time. Either way, when we leave the rat race of this harried industrial civilization, we begin to notice the toll it takes on the body, mind, and spirit. Clearly, the rat race is injurious to one's physical and mental health, one's relationships, and one's very sense of self.\n\nIf you are reading these words, it is likely that you have chosen not only to leave the rat race but to become a different species! But how is that possible?\n\nThe abundant use of rats in research experiments is no accident. Rats make good research subjects because they respond easily to stimuli and rewards. For a human to admit that he is part of the rat race is to confess that he is behaving like the rodent who scurries through mazes or is hooked up to electrodes and shocked when it does not produce the desired responses. But this is precisely what happens to human beings after years and decades of attempting to adapt to the desires of civilization's mandates.\n\nIf you have left the rat race, whether recently or long ago, you are on your way to becoming a new species\u2014not symbolically, but literally. As I have stated repeatedly, the Great Turning is fundamentally about rejecting rat races of all kinds and becoming a new variety of human\u2014one that could not permit anything like the current milieu of modernity to exist ever again on this planet.\n\nThe real measure of your wealth is how much you'd be worth if you lost all your money.\n\n\u2014Anonymous\n\nIn turbulent times, loss rules the day. In the present and future unraveling, we are all going to be a lot poorer than we are now in terms of financial wealth. Whether we have a money system in the future or we barter with buttons, the loss of material wealth is a given. The question for each of us then is, who will I be without money?\n\nTwo other questions also arise. What is wealth? What is worth?\n\nEach of us must define for himself or herself what wealth and worth are. Sometimes I feel luxuriously wealthy when I sit beside a stream in the Rocky Mountains, inhale the fragrance of pine trees, and listen to the gurgle of water splashing over rocks, creating a lullaby that puts my soul in a state of total repose. At other times, I feel inexplicably affluent when friends invite me for dinner and ask me to bring nothing but myself. When standing in front of a Monet painting in a metropolitan art museum, I can lose myself in it to such an extent that I forget that I have any wants or needs whatsoever.\n\nThe dwindling of money and the plethora of other losses that most of us are encountering and will continue to encounter in collapse will compel us to confront the meaning of _wealth_ and _worth_ in ways that we cannot presently imagine. I suspect that every activity and every relationship in our lives will be redefined and fine-tuned by a forced contemplation of those two words.\n\nI do not wish to minimize survival needs, because in Maslow's hierarchy, they are indeed the most fundamental to our existence. Because we are human, however, they do not define all of who we are. Now is the time to begin mining our inner wealth and finding the gold of the interior world that cannot be lost or stolen. What is it that you know in your bones and that you know that you know beyond the shadow of any doubt? Begin there to discover the inestimable treasures of your soul that cannot be lost.\n\nA small grove massacred to the last ash,\n\nAn oak with heart-rot, give away the show:\n\nThis great society is going to smash;\n\nThey cannot fool us with how fast they go,\n\nHow much they cost each other and the gods.\n\nA culture is no better than its woods.\n\n\u2014W. H. Auden\n\nThis stanza is the last of the poem, \"A Culture Is No Better Than Its Woods,\" and serves as the grand finale of a work saluting the beauty and necessity of wooded areas for the sanity and sanguinity of humankind. I am struck with the third line, \"This great society is going to smash,\" attesting once again to the wisdom and prescience of many poets. In ancient times, poets were considered to be prophets, not just bards. Auden is wise to the attempt of modern civilization to bypass its demise by traveling fast, but he tells us that it is precisely this that exacts a price from ourselves and from the gods.\n\nFor Auden, ultimately what matters is how much and what kind of woods a society has. Think of it: we are no better than the wooded areas we protect and maintain. Not exactly a tribute to \"man as the measure of all things.\"\n\nPoets know better than most of us that if we really want to slow down, we need to go deep into the forest, where light and shadow temper our harried rhythms and compel us to ponder trees, soil, and pervasive, inescapable wildness.\n\nIn a world where fewer and fewer wooded areas are available to us, because civilized humans perceive them as \"obstacles\" to growth and progress, we need to savor our time in the forest now\u2014before it's gone. Climate change has been destroying and will continue to destroy vast acreages of woods through drought and lethal infections of trees that leave them hollowed out and rotting. In the American West, the pine bark beetle, spurred on by climate change, has decimated millions of trees, turning miles of _timber_ into _tinder._ The toll taken on the ecosystem is immense, but the toll taken on humanity is equally precious, because we are only as good as our woods.\n\nMusic is the space between the notes.\n\n\u2014Claude Debussy\n\nThe universe functions in two ways: through rest and through activity. Nothing can continue functioning without rest, and nothing at rest can remain at rest indefinitely. Polarity is inherent in the universe and throughout our human experience.\n\nArtists, poets, and musicians\u2014and everyone who creates\u2014understand that music, words, and forms cannot exist without the spaces in between. Debussy knew this so well that he actually defines music not as the notes but as the space between them.\n\nAnd what is that space? Emptiness? Silence? Nothingness?\n\nI prefer _stillness_ or _presence._ The space between the notes is the same space that dwells within each human being, and when we listen to it attentively and with openness, the music of our lives becomes audible. It is not empty; it is not nothingness. It is replete with energy\u2014often energy that is so subtle and silent that we assume it is a vacuous void.\n\nMany people are terrified of stillness and literally find it intolerable. Yet it is in this very stillness that we discover the deeper Self and the mystery at the heart of everyone and everything in the universe. What is more, life is not the activity we generate or experience but rather the space between activity.\n\nThe collapse of industrial civilization will sometimes be noisy, but ultimately, it will result in unprecedented silence. Anyone who cannot tolerate silence is likely to suffer a painful withdrawal as the noise of our lives slowly dissipates. The truly resilient person is one who is extremely comfortable with silence and discovers the music and meaning in it.\n\nThe divine expresses itself in a dance of opposites: as present in destruction and chaos as it is in creation, order and light. What we reject as too terrifying or horrifying to imagine contains treasures of insight, compassion and wisdom.\n\n\u2014Andrew Harvey\n\nSo if we take this statement literally, then the divine is involved with the current and forthcoming destruction and chaos. For me, _divine_ is not a religious word, but a mythological and psychological one. Synonyms could be _sacred, deeper Self,_ or _something greater._\n\nThe core of my work in helping others prepare emotionally and spiritually for the Great Turning\u2014or as Michael Meade names it, the Great Churning\u2014is an emphasis on the current and coming chaos as a manifestation of the sacred. The human ego insists that talk of collapse, unraveling, or demise is too terrifying and horrifying to imagine. The mind polarizes collapse as opposite from the divine and argues for \"positive thinking\" and \"solutions.\" It cannot bear the thought or feeling of helplessness and vulnerability, not to mention acceptance of the demise as containing treasures of insight, compassion, and wisdom.\n\nA question I invite you to ask and continue asking is, where is the gold in this experience? I consider this an even more urgent question than, what lessons are inherent for me in this experience? Sometimes the ego prefers to learn lessons, while the soul looks for the golden opportunity to deepen and therefore cause us to become bigger people than we are now.\n\nThe Age of Endarkenment is here with all of its treasures, as the Age of Enlightenment fades from history, and both ages breathlessly await their integration in the human soul.\n\nWhen the American opens a . . . door in his psychology, there is a dangerous open gap, dropping hundreds of feet . . . [and there] he will be faced with an Indian or Negro shadow.\n\n\u2014Carl Jung\n\nThe unraveling of any society almost always takes a greater toll on the poor and marginalized than on the privileged and powerful. The shadow is that part of a psyche or a society that is disowned, despised, minimized, and trivialized, and of which one says, \"That's not me.\" Jung was not afraid to speak boldly about the American shadow, which is made up of African Americans, Native Americans, and many other \"shadow\" groups that Jung did not mention.\n\nAs societies come apart, the marginalized are usually blamed for the demise. Hitler's Germany is perhaps the most glaring example in modern times. Women, children, the elderly\u2014all who are vulnerable or even helpless are not only disowned but scapegoated and exploited by those who wish to flaunt their power and control and by those who feel powerless and target those with even less power.\n\nThe United States has never come to terms with its racism\u2014its deep involvement in the slave trade, its genocide of millions of Native Americans, and its blatant global imperialism, which it refuses to name as such. Thus, as collapse accelerates, we should expect to see not only intensified scapegoating of the marginalized but the marginalized fighting back or even attacking those who symbolize centuries of their oppression. For some of the scapegoated, the dissolution of society will be their opportunity to reclaim what is theirs, even if it is the crumbs that fall from the tables of the privileged.\n\nConversely, there may well be communities in which all ethnicities, ages, and genders cooperate to protect and sustain themselves\u2014but I suspect they will not constitute the majority. Part of the price some societies will pay for their collapse is hostility and violence that no one individual created but which constitute the shadow of that culture that has never been acknowledged, let alone healed.\n\nShow me a hero, and I will write you a tragedy.\n\n\u2014F. Scott Fitzgerald\n\nOne of the most prevalent archetypes in the human psyche is that of the hero. Some of the characteristics of the hero\/heroine, according to Carl Jung and Joseph Campbell are: unusual circumstances of birth\u2014sometimes born into danger or royalty; departing from family or homeland to live with others; an event that leads to a quest or journey; the hero may have a special weapon or supernatural help; the hero must prove himself or herself many times during the journey; the hero experiences atonement with the father or some authority figure; the hero experiences a homecoming in which he or she is rewarded by the community; and when the hero dies, that person is rewarded spiritually.\n\nThe hero may encounter and endure all manner of adversity, but in most cases, he or she triumphs in the end, returns from the many ordeals of the journey, and on some level, lives happily ever after. In many myths, heroes are nearly or completely invincible. Often this is connected with their unusual birth and the supernatural help they receive.\n\nThe paradigm of industrial civilization is nothing if not heroic. It teaches its inhabitants that its paradigm is essentially invincible. The very word _civilized_ connotes a special, exceptional status, and if the would-be heroes adhere to the paradigm, adopting the proscribed way of life, they are guaranteed success.\n\nBecause heroism always accrues from the ego and succeeds for the most part in inflating it, the hero invariably must fall. Perhaps this is the most valuable lesson one can acquire from studying heroes and their journeys. Heroes can certainly accomplish the journey and experience its rewards, but their downfall is sealed when they come to believe that this pattern is eternally repeatable, which they almost always do. Thus, unless awakened, the hero is fated to become a tragic victim.\n\nThere is no way to understand the attractiveness of war without understanding the unconscious seduction of the archetype of initiation.\n\n\u2014Robert Moore, _The Archetype of Initiation_\n\nThe killing of the children is the sacred secret\u2014sacred because no one will speak its name\u2014underlying all of our civilized values.\n\n\u2014Barry Spector, _Madness at the Gates of the City: The Myth of American Innocence_\n\nIn the United States, we now witness more military men and women taking their own lives after returning home than are killed in combat. How many young men and women enthusiastically join the armed forces, driven by a profound longing in their psyches to experience a spiritual initiation\u2014compelled by something larger than life itself? Months or years later, if they survive, they return to us as broken men and women, even their missing limbs less poignant than the hollowing out of their souls, now attended by a madness that seems to possess them. How seductive war is for those starving for rites of passage that will take them to the underworld\u2014supported by a cadre of wise elders, as they wrestle with life and death\u2014and then return from the jaws of hell, reborn, remade, and welcomed home by a community that recognizes their initiation for what it is and celebrates their endurance and triumph in the ordeal!\n\nBut all young men and women who look to the military to properly initiate them will only experience a failed initiation and return to us maimed and mad. As with the ancient myth of Kronos, who literally eats his children, we seduce our children to war with the promise of \"order,\" \"structure,\" \"brotherhood,\" \"manhood,\" and, of course, a college education with the GI Bill.\n\nI have no hope that wars will cease in my lifetime. Rather, I anticipate that resource wars will escalate to unprecedented levels. Nevertheless, my vision is a next culture in which sacred, fulfilled initiations of young people occur on a regular basis, and the community nourishes, rather than devours, its youth.\n\nIt is not for everyone, this path of the mysteries. Those whose interests center on getting or keeping some particular standard of living, with its privileges and comforts, usually find it unappealing at best. . . . For those who hear the call, however, the doors of the mystery schools are open today as they were in centuries past. No matter what changes shake the world, as our civilization comes belatedly face to face with the consequences of its own mistaken decisions, the patient work of the mysteries and the gradual ripening of human potential remain what they have always been.\n\n\u2014John Michael Greer, _Mystery Teachings from the Living Earth_\n\nThis rich passage from Greer's 2012 book contains several valuable nuggets of wisdom. First, as we know, emotional and spiritual preparation for the collapse of industrial civilization is widely rejected and rarely embraced. Whether a result of denial or a particular status, nothing about it appeals to the masses.\n\nYet this has always been so. Be it the Eleusinian Mysteries; Plato's mystery school; or myriad Christian, Muslim, or Jewish mysteries, exploring mystery demands sacrifice. Authentic students of mystery are always those who are willing to forego privilege, status, and monetary comforts for the sake of the treasures they believe the mysteries hold for them.\n\nGreer also suggests that following the mysteries is a response to a calling. Naturally, when we are encumbered with possessions and deafened by the cacophony of industrial civilization, it is difficult to hear the call, let alone respond to it.\n\nThe mysteries are no longer esoteric teachings \"out there\" for which we need to uproot our lives in order to respond to the call. Our lives are being uprooted daily, as larger systems around us crumble. There is no longer anywhere we need to go in order to find the mysteries. In fact, they have come to _us._ They are present in the throes of the unraveling and the multifarious manifestations of it in our lives and in the lives of everyone around us. Meanwhile, they patiently await our receptivity and our ripening.\n\nWhat does the \"Apocalypse\" mean psychologically? My essential answer is: The \"Apocalypse\" means the momentous event of the coming of the Self into conscious realization. Of course, it manifests itself and is experienced in quite different ways if occurring in the individual psyche or in the collective life of a group; but in either case, it is a momentous event\u2014literally world-shattering. This is what the content of the Apocalypse archetype presents: the shattering of the world as it has been, followed by its reconstitution.\n\n\u2014Edward Edinger, _Archetype of the Apocalypse_\n\nAs I have frequently written in recent years, _apocalypse_ is a Greek word that means \"the unveiling.\" It is also an archetype in Jungian psychology. An archetype is a universal theme or motif in individual and collective consciousness. At certain times in the life of an individual or a culture specific archetypes become activated. When an archetype begins to activate, it is the responsibility of the individual psyche to begin conscious dialog with the archetype in order to prevent domination by it. In other words, the activation of an archetype does not mean that it is destined to be played out. Its purpose, as Edinger explains, is to bring the deeper Self into conscious realization.\n\nWhen an individual or a civilization displays symptoms of collapse, the archetype of the apocalypse is revealing itself, and an internal and external transformation is being called for. When the archetype is ignored, we can expect a shattering of the inner and outer worlds, followed by a reconstitution. How long the entire process takes varies from individual to individual, from culture to culture, and if one believes in multiple lifetimes, perhaps many are required for the cycle of shattering and reconstitution to occur.\n\nSince the archetype of the apocalypse has been activated in the human psyche for at least several decades, we can safely assume that, consciously or unconsciously, every human on this planet is aware that an apocalypse is in progress. From this perspective, it makes complete sense that so few humans are willing to confront it. A spate of apocalyptic movies and television series are being created, but these are generally relegated to \"fantasy,\" \"science fiction,\" and \"horror,\" as humanity makes feeble and faltering attempts to accept the unveiling\u2014or trivialize it into unconsciousness.\n\nWe all want progress, but if you're on the wrong road, progress means doing an about-turn and walking back to the right road; in that case, the man who turns back soonest is the most progressive.\n\n\u2014C. S. Lewis\n\nIn these troubled times we all want to believe that we are \"progressive.\" Those individuals who call themselves political progressives do so on the basis of their fervent belief in human rights, civil liberties, social justice, and eco-justice. Unfortunately, most of those individuals do not understand that the system they are committed to reforming is crumbling with dizzying speed. They fail to grasp that both political parties are responsible for the demise and that neither the measures progressives espouse nor the political system of socialism, which most progressives silently or blatantly embrace, will make the collapse of industrial civilization go away.\n\nWhether one is a diehard believer in corporate capitalism or a hard-core member of Occupy Wall Street, it is time to turn around and admit that we can go no further\u2014that we must reverse course and consciously prepare for the demise of our individual lifestyles and the dissolution of all centralized systems. A new definition of _progressive_ demands our undivided attention and implementation. It is time to surrender to the demise and, at the same time, envision a next culture that we may or may not live to see come to fruition.\n\nMembers of the more-than-human world do not demand progress\u2014or even want it, as humans define it. What they seek is to be left peacefully alone in their habitats, free from toxic chemicals, from forest-massacring chainsaws, from methane burps, from human-caused natural disasters, from floating garbage dumps on the oceans the size of states, from nuclear radiation, and from the poaching of their body parts. To champion their desire for non-progress is to be declared unpatriotic or insane, or to end up on some terrorist watch list.\n\nI highly recommend constructing for yourself a new definition of _progress._ What would it look and feel like? How are you already making it happen in your home and in your community? What other \"progressive\" measures do you need to take?\n\nIt might take an all-out fatal shock treatment, close to catastrophe, to break the hold of civilized man's chronic psychosis. Even such a belated awakening would be a miracle.\n\n\u2014Lewis Mumford\n\nThe notion that civilized humanity is mad is not a recent one, as the great sociologist and philosopher of technology Lewis Mumford declared in 1970. He understood the extent to which humanity is capable of ignoring the needs and purposes of the earth community\u2014in fact, savaging them, in order to perpetuate its greed, privilege, and planetary control.\n\nForty years ago, Mumford was essentially welcoming collapse and hoping for its ability to break through civilized humanity's madness, stating that even if the awakening were late, it would be a miracle.\n\nI believe that the two sentences by Mumford quoted above provide a terse description of the demise that must occur if our species is to be prevented from killing our planet. Compare it with the \"bottoming out\" process of the alcoholic; a \"spiritual emergency,\" to use the terms of Stanislav Grof; or a descent into the underworld, as described by Carl Jung, Joseph Campbell, James Hillman, and Marion Woodman\u2014all are apt depictions of the unholy mess that humans have created, which cries out for purification and transformation.\n\nMumford does not sugarcoat his words here: \"fatal shock treatment\" and \"catastrophe.\" Frankly, I'm drawn to this quote because nearly four decades ago, before I began researching the state of the world, which I ultimately chose to name as \"collapse,\" Mumford was grasping the necessity of it and describing it as a \"belated awakening\" for humankind's \"chronic psychosis.\"\n\nI choose to call it a _blessed_ and _beloved_ awakening, the price of which will be untold death and loss\u2014yet where has transformation ever occurred without these?\n\nAny pain is bearable if it is part of a story.\n\n\u2014Isak Dinesen\n\nWhat does the author's statement mean\u2014part of a story? And why does being part of a story make pain bearable?\n\nIn all cultures, stories are an integral aspect of how people make sense of their lives. If something is part of a story, it may be unspeakably unpleasant, but it fits into a pattern, a saga, a progression from one place to another. If one understands the story, one can make sense of the suffering in it.\n\nCountless millions of human beings will be overwhelmed by the full-blown collapse of industrial civilization precisely because they do not understand the story of it and their place in the story. Other human beings, such as those who might be reading these pages, have studied the story and their role in it\u2014and have come to some awareness of their purpose in it. This does not make their pain less painful, but it may make it bearable, because they understand their place in the story and what the story calls them to be and do.\n\nDespite collapse, the story continues. What is most exciting for me is the fact that if I survive, I get to be part of writing the new story. And if I do not survive, many others who understand the story and their place in it will participate in creating a new story.\n\nNo matter how much we think we know about how collapse will continue to unfold, it is useful to remind ourselves daily that the only certainty we have about the outcome is, well, _un_ -certainty. However, if we can appreciate the story of industrial civilization, its impact on the earth community, and the inevitability of its demise, we may find our pain and loss more bearable than if, like millions of others, we were to live without a clue. My words are not intended to construct an \"us\" and \"them\" dichotomy, because many people will awaken at a late hour and may wish to help write a new story. In fact, we may be surprised at who the cocreators of the new story might be.\n\nThis is the age of loaves and fishes, people are hungry, and one good word is bread for a thousand.\n\n\u2014David Whyte\n\nRecently, I attended a conference focusing on collapse. I was the only presenter (of four) whose focus was specifically on spiritual and emotional preparation. The attendees of the conference were people who had been researching the demise of industrial civilization for years, so they were well researched and up-to-speed on the issues. Because my focus is less on external-world research and more about inner exploration\u2014or as I sometimes call it, \"Collapse as Spiritual Practice\"\u2014I was concerned that people attending my presentations would be overly skeptical of the content or, in the worst-case scenario, not show up at all.\n\nI was astonished at the number of people who _did_ attend my presentations, and even more heartened by people who spoke with me privately and said things like, \"You know, I've heard the research now for years. I've seen all the charts and graphs and watched the Power Point presentations, but that's not what I want anymore. I want what you're offering,\" or \"I'm so hungry for what you're giving us. I can chew on this for months and never feel full.\"\n\nAs I heard these kinds of comments, I reflected on the words of David Whyte's poem, \"Loaves and Fishes,\" which begins: \"This is not the age of information. . . .\" I now understand that human beings are choking on information, but they are starved for wisdom\u2014that which nourishes and fortifies the soul and assists us in making sense of a world in chaos.\n\nMy hope is that this book offers a feast of \"loaves and fishes\" to the hungry reader who knows that what we need is not more information, but more nutrients for the soul\u2014more of that which guides us to the deeper Self within and throughout the earth community.\n\nIf all mankind were to disappear, the world would regenerate back to the rich state of equilibrium that existed ten thousand years ago. If insects were to vanish, the environment would collapse into chaos.\n\n\u2014Edward O. Wilson\n\nThe pre-Socratic philosopher Protagoras is supposed to have uttered the words, \"man is the measure of all things.\" How we wish that were so, and on the level of consciousness and the capacity for introspection, it _is_ so. However, on many levels, man is certainly _not_ the measure of all things. Wilson provides one example of this in the above statement. In fact, Wilson's words evoke in me the question, on how many other more-than-human species is the environment so dependent that if they disappeared, it would collapse?\n\nI do not know the answer to this question, but I do know that humans secretly\u2014even blatantly\u2014believe that the environment depends on them. Most are incapable of imagining a planet without humans. In 2009 the History Channel produced a series entitled _Life after People,_ which theorized what would happen to the earth if we were no longer around. It essentially confirmed Wilson's assertion. The program revealed that, after a long period of chaos and savage conditions among other species, the earth would return to its ancient equilibrium. In other words, the earth is ultimately not dependent on humans at all\u2014and never was.\n\nOf course, we know this intellectually, but not in our marrow. We continue to secretly believe that we are the measure of all things and that the tree spiders, spotted towhees, aardvarks, dolphins, giraffes, sea urchins, snails, wombats, garter snakes, nightingales, newts, jellyfish, hedgehogs, and all members of the more-than-human world are incidental to our existence and rely on it for theirs\u2014or that somehow the earth would be a feckless planet without humans. In fact, this is an unequivocal inversion of reality, because it is _we_ who are dependent on these infinite life-forms, not they on us. Furthermore, we cannot know ourselves without some knowledge of and relationship with them. Together, we are all the measure of all things.\n\nEcopsychology proceeds from the assumption that at its deepest level the psyche remains sympathetically bonded to the Earth that mothered us into existence. Ecopsychology suggests that we can read our transactions with the natural environment\u2014the way we use or abuse the planet\u2014as projections of unconscious needs and desires, in much the same way we can read dreams and hallucinations to learn about our deep motivations, fears, hatreds.\n\n\u2014Theodore Roszak, _Ecopsychology_\n\nOver the past four decades, a new discipline has formed\u2014one drawing upon a variety of other disciplines such as psychology, biology, physics, chemistry, astronomy, geology, sociology, and history. Ecopsychology is the study of humanity's relationship with the environment and the environment's relationship with humanity\u2014how both are enriched or inhibited by the other.\n\nIn the 1995 ecopsychology anthology edited by Roszak, he defines the discipline as the study of connectedness. The book's subtitle summarizes its contents: \"Restoring the Earth, Healing the Mind.\"\n\nWhen we have no conscious relationship with the earth, we are profoundly limited in our relationships with others and with ourselves. After all, we are estranged from the \"Earth that mothered us into existence.\" However, when we begin to develop a conscious relationship with the earth, we are invariably changed by it, and most ecopsychologists would probably agree that the earth is also changed by it.\n\nObviously, few modern humans have such a relationship, as witnessed by the state of the ecosystems. Likewise, the state of humanity\u2014its violence, narcissism, neuroses, entitlement, terror, despair, competition, hectic and driven lifestyle\u2014all of these testify to a species tragically out of relationship with the earth.\n\nTake a moment to practice a bit of ecopsychology. Contemplate your relationship with the earth. What is it? When was the last time you checked in with the earth about your relationship with it, as you might with a beloved? When was the last time you asked it what it needs from you? Go now from the printed word and _be_ with the earth. Restore the connection.\n\nThe most beautiful experience we can have is the mysterious. It is the fundamental emotion which stands at the cradle of true art and true science. Whoever does not know it can no longer wonder, no longer marvel, is as good as dead, and his eyes are dimmed. It was the experience of mystery\u2014even if mixed with fear\u2014that engendered religion.\n\n\u2014Steven Harper, \"The Way of the Wilderness\"\n\nIndustrial civilization, direct heir of the Enlightenment, is no friend of mystery. From the moment the philosophers of the seventeenth and eighteenth centuries began penning their concepts of a scientific revolution, there was no place in them for mystery. From their perspective, mystery had led to superstition, religious dogma, subservience to theology, and the Dark Ages, from which they were ecstatically breaking free. In the so-called Age of Enlightenment, nothing can be known (or perhaps is even worth knowing) if it cannot be proven.\n\nThree centuries later, some members of our species have created a new discipline, called ecopsychology, of which Steven Harper is an enthusiastic member. He tells us that mystery is the most beautiful experience we can have and further asserts that it \"stands at the cradle of true art and true science.\" Oh, how the pendulum has swung. We can hear Descartes spinning in his grave.\n\nHarper correctly notes that mystery has engendered religion, but it has also given birth to reverence for the sacred that has nothing to do with religion. No need to believe anything, but simply _wonder._ Wonder at your own inability to solve the mysteries, but also wonder at the fact that you _can_ wonder. It is one of the few functions that makes our species unique. In fact, _wonder_ may be one of the most apt synonyms for _the sacred,_ and it certainly constitutes an intersection between what we call \"spiritual\" and the crux of our humanity. How important is wonder? Harper states unashamedly that if we can't wonder, we're as good as dead.\n\nAll human beings should function from a fundamental sense of gratitude that the cosmos has given us birth and a place within the universe . . . This feeling [of gratitude] is essential in teachers and should be instinctive in anyone entrusted with nurturing a child. Therefore, the first important thing to be worked for in spiritual knowledge is thankfulness that the universe has given a child into our keeping.\n\n\u2014Rudolf Steiner\n\nRudolf Steiner, founder of Waldorf education, lived at a time when children were to be seen and not heard; only a few visionaries were beginning to question that assumption. An avid student of the psychology of Carl Jung, Steiner was committed to understanding the changing consciousness of children and enhancing its development. He developed a system of education that provided holistic academic preparation by addressing a child's physical, emotional, and spiritual development.\n\nSteiner spoke of children as \"holy\" and devoted his life to fortifying and inspiring them to go forth into a world very different from what they encountered in their Waldorf education. The fact that Waldorf education has endured to this day\u2014and has spread throughout the world\u2014is a testimony to the soul-nourishing qualities at the core of Steiner's educational regimen.\n\nMost children are not fortunate enough to attend a Waldorf school, but with parents and caretakers who appreciate and incorporate the methodology of Waldorf, many children can receive similar training which will stand them in good stead in a chaotic world.\n\nWhether using the methods of Steiner, other educators, or simply the common sense that comes from knowing our children well and communicating with them intimately, we must prepare them for a world unraveling. That they must face such a world\u2014with all its adversity\u2014is clearly unfair, but it is the world we and they must be prepared for. If you have a child in your keeping, be thankful, and be a channel for the wisdom of your and his or her deeper Self.\n\nBeneath the veneer of civilization . . . lies not the barbarian and animal, but the human in us who knows the rightness of birth in gentile surroundings, the necessity of a rich nonhuman environment, play at being animals, the discipline of natural history, juvenile tasks with simple tools, the expressive arts of receiving food as a spiritual gift rather than as a product, the cultivation of metaphorical significance of natural phenomena of all kinds, clan membership and small-group life, and the profound claims and liberation of ritual initiation and subsequent stages of adult mentorship. There is a secret person undamaged in every individual, aware of the validity of these, sensitive to their right moments in our lives.\n\n\u2014Paul Shepard, _Encounters with Nature_\n\nIn a world where humans are killing the planet, it is ever so tempting to conclude that at their core, they are irredeemably rotten. I am often accused of being some sort of sycophantic Pollyanna because I refuse to agree with this assumption. I hasten to add, however, that I do not believe that humans are inherently virtuous, for I have long since rejected the notion that they must be one or the other: evil or holy at heart.\n\nWhat I will argue for, alongside the environmentalist Paul Shepard, is the \"secret person undamaged in every individual,\" that lies \"beneath the veneer of civilization.\" Surrounding that deeper Self is a well-defended, well-armored circumference of personal and societal inculcation that obscures the deeper Self and works mightily against it.\n\nFrom my perspective, the most important work we can do in preparation for navigating a chaotic world is to explore both the circumference and the center of who we are. What are the stories from our culture and from parents and caretakers that formed the \"full metal jacket\" around the soul? What made the outer casing so impenetrable? What is the soul\u2014or more specifically, _who_ is the soul?\n\nDo not wait for an external cataclysm to obliterate the armor, leaving the soul devastatingly exposed. Find the secret person, undamaged, and do the work necessary to nurture and protect that person. Create the bonds and boundaries that will that will replace this culture's armor and the veneer of civilization.\n\nWe're in a giant car heading towards a brick wall and everyone's arguing over where they're going to sit.\n\n\u2014David Suzuki\n\nA friend of mine recently visited her family in another state. When she arrived at their home, her father's first words were, \"Where did you park?\" Her mother's first words were, \"You can't park in the driveway; you'll have to park in the street.\" For another two or three minutes, my friend and her parents continued to discuss parking arrangements, resulting in my friend leaving the house and parking her rental car in the street. When she went back inside, her parents began discussing dinner reservations for that evening. Meanwhile, no one had actually said, \"Hello,\" or \"It's good to see you.\"\n\nSuzuki provides a perfect metaphor for industrial civilization. The car is headed for a brick wall or a gaping precipice, but the passengers are arguing about where to sit. Meanwhile, no one recognizes or speaks about the death spiral in which the car and all of the passengers are locked; everyone assumes that the car can continue its trajectory indefinitely. And since no one recognizes the death spiral, no one actually talks to the other passengers, attempting to know who they are. Nor does any one passenger seek to discover who she or he is in the midst of the death spiral, because, of course, the death spiral is not recognized or named.\n\nI'm reminded of passengers on the Titanic who knew perfectly well that the ship was sinking but continued to order drinks and carry on conversations about their stock portfolios with other passengers, equally in denial. Some ask, \"Well, what else is there to do?\" My answer: plenty. _Who_ is the person riding in the car toward the brick wall? That would be you. And who are the other passengers riding with you? Why are you fighting over a better seat? So that you can have a better view of the crash? What might happen if you began incorporating those very simple words, \"Hello, it's good to see you\"?\n\nI have suggested that at the core of the psychic alienation of Western humanity from the natural world, with its disastrous consequences of global ecological destruction is a humanist superiority complex that is a deeply rooted feature of the European psyche. For a complex variety of historical reasons, Europeans have come to experience spirituality and nature as separate or opposed.\n\n\u2014Ralph Metzner, \"The Split between Spirit and Nature in European Consciousness\"\n\nIn his brilliant 1993 article, Ralph Metzner traces the conflict between earth-based spirituality in the ancient world and the monotheistic religions that emerged alongside them. When teaching history and psychology in college, I frequently used this article to explain the phobia of the early Christian church toward nature and the feminine principle, and the consequences of this phobia in later centuries. Women and the natural world were feared by the early church fathers, and those fears led them to target earth-based spirituality.\n\nNear the end of the Middle Ages, fear of the feminine and of nature became even more pronounced, as women and myriad elements of nature, including cats, were blamed for the Black Death. However, nothing sealed the outcome of early church phobias in relation to women and nature as profoundly as the onset of the Enlightenment and its so-called Scientific Revolution. Even as the Enlightenment philosophers scoffed at the superstitious Christian church, they perpetuated the fear of women and nature by worshipping reason and disparaging any form of spirituality. The church fathers and Enlightenment thinkers laid the groundwork for the Industrial Revolution and the consummate disowning of nature and the feminine principle. Hence modernity's delusion of separation from the earth, from each other, and from ourselves.\n\nOur work in the collapse process is to reclaim our oneness with the earth community and to experience, on increasingly visceral levels, our intimate connection with all life. The calamities produced by climate change, extreme weather, and a plethora of earth changes will eviscerate our \"humanist superiority complex,\" and we will be compelled to cry out, with our Native American siblings, \"Mitakuye Oyasin!\"\u2014\"All my relations!\"\n\nAs a species we are facing our final examination; and it is a psychological examination; it is in fact an intelligence test\u2014a test of our true intelligence as a species. In essence we are being asked to let go of our self-centered thinking and egocentric behavior. We are being asked to become psychologically mature, to free ourselves from the clutches of this limited identity, and express our creativity in ways which benefit us all.\n\n\u2014Peter Russell\n\nSo many of the great minds who are speaking to the plight of our planet are framing their comments in terms of a \"test,\" an \"ordeal,\" an \"initiation.\" There was a time when humanity had a choice to end its mass murder of the planet, and at that point, it may have been possible to erase the consequences of its assaults. In the second decade of the twenty-first century, however, it seems that humanity's impact on the planet now has a life of its own. In other words, it appears that humanity has failed the examination, and now the outcome is far worse than simply getting an F.\n\nThe consequences are horrific, and it may be that a far more severe test will be given to humans than a choice to end their ecocide. I believe that the test will be far more momentous and onerous, because the next examination will be about who we are\u2014who we are without our stuff; who we are without our loved ones; who we are without the natural environment that we have destroyed; and yes, who we are beyond physical survival of the cataclysm we have created!\n\nI keep proposing that these are the consummate questions of collapse: Who am I, really? Who do I want to be as I navigate this unraveling? Who I really am and who I want to be may be very different. The first issue has to do with the deep Self, the second with the deep Self's influence on the ego. Sadly, it may require physical death for some of our species to become spiritually and psychologically mature. Others may be able to discover maturity and also survive physically.\n\nHumanity is not yet mature, nor is it intelligent. Collapse is about engendering both qualities. All prior examinations are irrelevant.\n\nThis culture devalues introspection, and many of us are trained to do whatever we can to fill (and kill) time so we never have to be alone with who and what we have become, and so we never can become who we really are and were meant to be.\n\n\u2014Derrick Jensen, _Dreams_\n\nEveryone reading these words knows of someone, or many people, who cannot tolerate being alone\u2014or cannot tolerate being alone in silence. For some people, being alone with themselves is nothing less than terrifying. Is this not one reason why we surround ourselves with electronic toys, obsessively check our smart phones from moment to moment, and need music playing while we shop, eat, get our hair cut, do our banking, and have the car repaired?\n\nNo wonder that, when I asked my college psychology students to write a paper on \"My Purpose in Life Is . . . ,\" many of them agonized for days about what I meant by \"purpose\" and barraged me with emails and questions in class about how they could possibly write such a paper. When I held a couple of in-class meditations, some of them nearly came out of their skin with anxiety as they attempted to sit quietly in silence. I honestly believe those students might have had an easier time if I'd asked them to walk on broken glass barefoot.\n\nDerrick Jensen says that in our cacophonous culture, we never have to be alone with who we have become. That is so, but likewise, we never have the opportunity to be with who we really _are_ \u2014the deeper Self beneath the ego that is participating in the murder of the planet.\n\nI can only imagine that when energy depletion and economic meltdown totally silence our electronic toys, millions will become emotionally unstable, if not mentally ill. The exceptions will be those who have already become comfortable with silence\u2014comfortable with the deeper Self, as a result of practicing being present with it regularly. May you be one of the individuals consciously working to become who you really are and were meant to be. May you learn the glorious rewards of deep, silent, undistracted introspection. The poet Mary Oliver says, \"I don't know exactly what a prayer is. I do know how to pay attention.\" That, my friend, is all you need.\n\nThis is the way the world ends\n\nThis is the way the world ends\n\nThis is the way the world ends\n\nNot with a bang but a whimper.\n\n\u2014T. S. Eliot, \"The Hollow Men\"\n\nThomas Stearns Eliot belonged to a generation of writers profoundly influenced by World War I. The \"Great War\" had a great impact on many writers, such as William Butler Yeats, Virginia Woolf, Ernest Hemingway, E. M. Forster, and D. H. Lawrence; their works often echo the pessimism of a milieu that could not quite believe that a world war had actually occurred. As I read many of the writers from this generation, I think, \"If you only knew what the world would be like a century after the war that caused you to lose all faith in humanity!\"\n\nWhat these writers could never have known is how applicable their works would be to the world a century later. In fact, if we did not know that Eliot penned the last four lines of \"Hollow Men\" soon after World War I, we might assume that it was written just yesterday to describe our current predicament.\n\nHowever, the world didn't end in Eliot's time, and it won't end in our time. What will end is life as we have known it\u2014a life where people have regular jobs, a mortgage, access to health care, an affordable college education, clean water to drink, and economic growth fueled by the energy required to make it \"limitless.\" And this world probably won't end with a bang, but with a series of whimpers resulting from the relentless deterioration of industrial civilization.\n\nIn the face of ghastly, unprecedented insanity among humans, it is easy to dismiss them as nothing but \"hollow men,\" yet something\u2014no matter how infinitesimal\u2014of humanity's goodness remains. The outcome of that goodness cannot be known, and no one reading these words will witness the traces of virtue existing in humans seven generations from now. Industrial civilization hollows us out and, at the same time, attempts to convince us that we are filled with opportunity, happiness, and meaning. Thus, the \"end of the world\" may also mean the end of our hollowness and the opportunity to be inhabited by the fullness of our own humanity for the very first time.\n\nDoing our personal work on what has caged us is crucial for humans to live productively in community. Our personal work creates the clearing needed to stand compassionately in our power, not to defend our actions of integrity, but to state boundaries and be open to staying in the conversation until an issue is resolved.\n\n\u2014Carolyn Rose Hart, _Step Up, Step Out: Share Your Gifts and Be an Agent for Change_\n\nThroughout the unraveling, myriad disagreements and miscommunications will occur. In present time, we have the luxury of choosing with whom and how we will interact. As many more people become homeless and displaced on a variety of levels, however, we may find ourselves sharing space with people that we would never choose to even have a conversation with at this moment in our lives.\n\nSome people believe that the most prudent preparation for collapse is to join an ecovillage or intentional community. Yet conflict exists in many of those communities as well. It has torn some communities apart\u2014to the point of actual dissolution. Others find it necessary to devote a significant portion of each day to processing emotional material between members. I believe that this speaks to the luxury we still have of choosing whom we will relate to and how.\n\nWhen we are compelled to spend days, weeks, or months with people we do not like and who may push every button in the psyche, and when our survival depends on it, we _will_ find a way to live together. The more we can attend to our personal wounding _now,_ rather than in the midst of conflict, the better our odds of navigating such challenges.\n\nThere are many ways to do \"personal work.\" Psychotherapy has facilitated profound healing for many, as has meditation, working with dreams, vision questing, and working directly with personal trauma. As mental health care becomes less accessible and fewer healing resources are available, however, we will be compelled to do our personal work using ever more innovative and creative methods.\n\nNo one ever \"finishes\" his or her work. It is more important that we remain in the conversation with those we need and who, at the same time, challenge us mightily, even if issues are never fully resolved to everyone's satisfaction.\n\nThe closest to being in control we will ever be is in that moment that we realize we're not.\n\n\u2014Brian Kessler\n\nThe first time I read these words I was dumfounded. Kessler could have chosen to argue in a rational, linear manner that we're not in control and could have provided all the reasons why, but instead, he made a stunning use of paradox to convey his message in a more powerful way.\n\nAs I write these words, in the summer of 2012, I'm sitting below the foothills of the Rocky Mountains in Boulder, Colorado, while nearly a dozen wildfires rage throughout the state. Temperatures have been in the triple digits or high nineties for two weeks, and winds have been gusting at very high speeds, further drying out the already-arid land, turning it into tinder-box conditions for fulminating, ferocious fires.\n\nFew argue that climate change has produced similar conditions throughout the American Southwest; ironically, some of the most groundbreaking research on the subject has been conducted here in Boulder. Another paradox.\n\nOther than packing a number of valuables and mementos and storing them in a safe venue outside my home, there is little I can do about the fires, or for that matter, climate change itself, which is inexorably out of control. Even if humanity tomorrow magically ceased conducting all activity contributing to climate change, it would persist. Climate change now has a life of its own. Move to another part of the country? Where would that be? Floods on the East Coast, tornadoes in the Midwest, raging wildfires on the Pacific Coast.\n\nParadoxically, the moment we realize we're not in control, we are. Giving up control opens up options that could not be accessed while laboring under the illusion of control. While we don't have any more control than we had before we realized we have none, what we do have is a new way of seeing the situation\u2014and that makes space for exploring new alternatives. Or as Derrick Jensen puts it, \"The moment we give up on attempting to control life, the moment we realize that we cannot control life, is the moment we enter into life.\"\n\nBecause things are the way they are, things will not stay the way they are.\n\n\u2014Bertolt Brecht\n\nSo said the maverick German playwright, director, and poet, who witnessed the rise of the Third Reich, left Germany as soon as Hitler took power, and continued to use his craft on behalf of justice and human dignity.\n\nHis words are amazingly apt in current time. Industrial civilization, constructed as it is around practices that are astonishingly unsustainable, cannot endure. Inherent in the paradigm of civilization are the seeds of its own destruction. Humans and the earth community cannot continue the civilized lifestyle, based as it is on the assumption of endless resources to which we are entitled, global economic Ponzi schemes, and the savaging of the ecosystems.\n\nCivilization _must_ collapse. It cannot be repaired, sutured, botoxed, redecorated, remodeled, or refurbished. Whether some activist humans are able to hasten the demise is debatable, but in any event, the collapse of industrial civilization is a foregone conclusion.\n\nSome well-meaning individuals who analyze world events prefer to speak of \"change\" or \"transition,\" and these are not incorrect terms, but they severely understate what we are confronting\u2014and what confronts us. I do not mean that we are facing a Hollywood blockbuster-style scenario of asteroids pulverizing the earth or rampant global pandemics\u2014although those may be part of our future. Rather, I believe we are facing a radical, unprecedented stripping away of our lifestyles and, most importantly, our human egos\u2014those things that, because they are what they are, cannot remain as they are now.\n\nWith every loss, we will be confronted with questions of meaning and purpose\u2014what our lives are actually about. We won't be able to continue identifying with our stuff, our status, or our stability. We will have to decide if we're going to live a rich (not synonymous with _wealthy_ ), meaningful life, and if our attachment to all of the accoutrements of civilization makes that impossible. Another word of wisdom from Brecht: \"Do not fear death so much, but rather the inadequate life.\"\n\nDisaster could be called a crash course in Buddhist principles of compassion for all beings, of nonattachment, of abandoning the illusion of one's sense of separateness, of being fully present, of awareness of ephemerality, and of fearlessness or at least aplomb in the face of uncertainty.\n\n\u2014Rebecca Solnit: _A Paradise Built in Hell: The Extraordinary Communities That Arise in Disaster_\n\nWhereas Naomi Klein gave us _Shock Doctrine_ in 2006, with its penetrating analysis of how disaster capitalism has successfully destroyed parts of the natural world and eviscerated human communities beyond recognition, Solnit offers a study in how communities have become astonishingly more present and solidified in the face of disaster. Both analyses are valid and represent the opposite realities of our shattered world.\n\nWhether a result of fear, empathy, vulnerability, or false heroism, when disasters occur, the majority of survivors work together to support each other and to make life as simple and stress-free as possible. Some display an \"every man for himself\" mentality, but they are vastly in the minority. This does not mean, however, that in the throes of the collapse of an entire civilization, people will remain mostly caring and compassionate. Some will; some won't.\n\nWhat Solnit says is that the quality that appears to singularly characterize most peoples' response to disaster is a sense of unity. In the midst of trauma and loss, people tend to rediscover their fundamental affinity with each other, even if they hardly know one another or don't particularly like each other. Moreover, they tend to become more present, more attentive, and, ironically, more alive. Ego encasement becomes a bit more permeable, and the deeper Self is more likely to radiate into human interactions.\n\nYears after disasters, survivors often speak fondly, even with a tone of tearful reverence in the voice, about the impact of the disaster on them emotionally and spiritually.\n\nWithin the disaster lies a stunning capacity for human connection and the creation of enduring community. Disaster both shatters and unites; it murders souls and bodies but, at the same time, makes people and communities whole in ways they could never have imagined.\n\nIsn't it bewildering . . . that everything is so beautiful, despite all the horrors that exist?\n\n\u2014Sophie Scholl\n\nI hear ancient wisdom in these words from the twenty-two-year-old Sophie Scholl, who was executed in 1943 for her participation in the resistance against the Nazi regime. Living at an extremely dark time in history, Sophie maintained her sense of humor and a passion for beauty. So few adults in the modern world can do the same.\n\nI'm struck by the word _bewildering,_ because I know this feeling in my body when I sit quietly with eyes closed and contemplate Sophie's statement. I consciously hold an image of something beautiful, and then I allow an image of something horrific to come in and be present. Then I go back to the beautiful image, then back to the horrific. I do this for a while\u2014back and forth, back and forth\u2014slowly feeling the body sensations that attend each opposite, until I gradually feel my nervous system stabilizing, and I feel the two opposites not fully integrated in my body but rather touching each other as they sit side by side in my torso.\n\nI highly recommend this exercise, not because dwelling in either opposite is a desired state or because fully integrating them is preferable to anything else. To consciously contemplate both opposites\u2014in all of their terrible wonder and horror\u2014is to consent to taste the fullness of the human condition and, in doing so, to cause it to be a bit less bewildering.\n\nWhat opposites do you need to sit with today? What polarities call you to discover them as less bewildering?\n\nI am not an atheist but an earthiest.\n\n\u2014Edward Abbey\n\nAs I write and speak of emotional and spiritual preparation for turbulent times, I frequently hear people who call themselves atheists proclaiming that they deeply appreciate my work. Whenever I write or speak, I spend a significant amount of time constructing wording that will not offend those who identify as atheist or agnostic. I do this not only because I prefer not to offend but also because, for me, spiritual preparation has little to do with religion or a deity. We have all been wounded by those concepts, and I refuse to be associated with them.\n\nMost individuals who understand the extent of our predicament are also intimately connected with the earth, and for them, a relationship with the deeper Self is most palpable when they are quietly wandering in a forest or sitting on top of a mountain or poised beside a gurgling stream. Atheists or agnostics would not describe these as spiritual experiences, but they recognize that something within them is profoundly stirred by moments of intimacy with nature.\n\nThe renowned environmental writer Edward Abbey did not use the word _God_ when he spoke of the grandeur and wonder of nature. In fact, he wrote, \"Whatever we cannot easily understand, we call God; this saves much wear and tear on the brain tissues.\" For Abbey, nature held exquisite mysteries, far beyond anything that humans could name as God.\n\nWhatever we believe or have experienced about God, religion, piety, or spirituality, if we become \"earthiests,\" we are not likely to be caught up in debates about dogma or the semantics that so often attend the use of specific words. Rather, to become an earthiest is to reverence that which we are, for we are not merely _part_ of the earth; we _are_ the earth. In the same way that we are our parents, yet not our parents, we are the earth. We carry the genetics of our ancestors, yet we are separate people. Likewise, we carry within us all of the elements of the earth from which we are born. No wonder we are so deeply moved when we revel in its mysterious beauty. No wonder it is so easy to become an earthiest.\n\nSometimes suicide isn't just an action, it's a choice in the back of the mind to save themselves from themselves.\n\n\u2014Federico Garc\u00eda Lorca\n\nWhy a page on suicide in a book of transformative truths? The Spanish surrealist poet Federico Garc\u00eda Lorca was not afraid to think and write about it, and his words feel apt for this moment in history.\n\nFirst, humanity is committing suicide and murdering the planet. If Lorca is correct, one motivation for suicide is saving oneself from oneself. In the act of suicide, an individual is exempt from having to struggle with his or her pain. And, of course, when people find a way to avoid suffering, whether through death, addiction, distraction, power, or the exploitation of others, they also forego the opportunity to venture into their core and mine the dark energies of human shadow at the center of which abides the sacred Self. So while suicide offers an escape from pain, it also guarantees that the deeper meaning of the pain will never be grasped. Humans murder themselves and the earth because they are wounded, bored, addicted, desperate, self-loathing, and alienated. Suicide and ecocide are twin energies that travel together and preclude the possibility of falling in love with the earth and one's humanity.\n\nMoreover, as collapse intensifies, this collective suicide will translate into more individual suicides than we can imagine, as millions of traumatized people choose not to endure the misery of a world unraveling. All that they have succeeded in avoiding within themselves is likely to bubble up from the unconscious and become overwhelming. Even if they could secure a lifetime supply of antidepressants, there would never be enough to construct the internal bulwark required for navigating the demise. And incidentally, I make no judgments of individuals who do not have the capacity to endure; none of us can be certain of our own endurance until we are confronted with things that feel beyond our ability to cope with.\n\nSome activists insist that aware humans must perform an intervention and stop our species from killing the earth. I salute those who commit to this daunting journey. It is certainly not for the faint of heart. Whether an intervention is feasible is uncertain. What _is_ certain is that industrial civilization is fatal for all earthlings.\n\nThe story is told of Nasrudin who took off his gold wedding ring, and when he did so, it fell through the crack in his floor into the basement. The next day, Nasrudin went into the plaza at noon, got down on his knees and began searching for something. Many people noticed this and asked him what he was looking for. Nasrudin told them that he had lost his wedding ring and was looking for it.\n\n\"Where did you lose it?\" they asked.\n\n\"It fell into the basement of my house,\" he replied.\n\n\"Then why are you looking for it here in the plaza in the middle of the day with the sun blazing down on you?\"\n\nNasrudin replied, \"Because I am scared to look for it in the darkness.\"\n\n\u2014Stories of Nasrudin\n\nThis is one of countless tales of the Sufi clown character, Nasrudin. Often within the humor of a Nasrudin story lies a profound truth. This is one of my favorite Nasrudin tales, because it depicts humanity's timeless aversion to darkness and our preference for doing everything we can to avoid it, including choosing to forfeit the golden treasures that we might find there.\n\nWe waste our time and energy looking for treasures in the light even when we know deep inside ourselves that we will never find them there. In our modern world, many people are drawn to New Age thinking, which focuses exclusively on the light. The light can be very seductive. Who would not prefer to feel blissful, hopeful, and perpetually cheerful, as opposed to so-called negative emotions, such as fear, anger, or grief? Yet we sacrifice inestimable treasures of meaning, purpose, and healing at the deepest levels in order to \"feel good.\"\n\nIndividuals who are tenaciously attached to feeling good have great difficulty confronting chaos. Rather than actually dealing with it, they tend to devise ever-new techniques for avoiding it. They buy more comforts to distract them, or they read new books or gravitate to the next spiritual teacher who assures them that suffering is optional. They will do anything to avoid the human condition, which the Buddha declared is fraught with suffering.\n\nSome people, however, experience a spiritual awakening in which they come to understand the inestimable treasures that are waiting to bless them and that can only be discovered in the dark, subterranean venues of life. Once one finds gold in the darkness, one can never be content to remain only in the light.\n\nSo live your life that the fear of death can never enter your heart. Trouble no one about their religion; respect others in their view, and demand that they respect yours. Love your life, perfect your life, beautify all things in your life. Seek to make your life long and its purpose in the service of your people. Prepare a noble death song for the day when you go over the great divide. Always give a word or a sign of salute when meeting or passing a friend, even a stranger, when in a lonely place. Show respect to all people and grovel to none. When you arise in the morning give thanks for the food and for the joy of living. If you see no reason for giving thanks, the fault lies only in yourself. Abuse no one and no thing, for abuse turns the wise ones to fools and robs the spirit of its vision. When it comes your time to die, be not like those whose hearts are filled with the fear of death, so that when their time comes they weep and pray for a little more time to live their lives over again in a different way. Sing your death song and die like a hero going home.\n\n\u2014Tecumseh\n\nAs I read these words from the great Shawnee leader, respected by so many tribes, who came so close to overthrowing the white establishment of his era, I am reminded of other nuggets of tribal wisdom from around the world. As I contemplate the first sentence and then the very last, I reflect on the African proverb that says, \"When death comes, let it find you fully alive.\" The fullness of life and the fullness of death are what Tecumseh is describing in this beautiful statement.\n\nThe more fully we revel in life and our aliveness, the less fear we are likely to encounter about our own death. This does not mean that at the end of our lives, we will have no regrets. Everyone does. What it does mean, however, is that the fullness of life is not the opposite of death, but its natural twin. Our Native ancestors could proclaim, \"It's a good day to die,\" because they were so phenomenally alive and engaged with the human and earth communities.\n\nWe all know that collapse is a kind of death\u2014the death of industrial civilization and its values. For many of us, collapse will also bring forth our physical death. But like indigenous peoples everywhere, who were slowly exterminated over the course of centuries, let us remain fully alive, even in our demise. Let us attend not only to physical survival but to our relationships with all beings. Let us create beauty as if we were destined to live forever; let us laugh, play, cry, sing, dance, make things with our hands, and speak poetry to one another.\n\nLove your life, perfect your life, beautify your life, . . . _and,_ sing your death song and die like a hero going home.\n\nYes, overthrow any government inside that makes you weep. The child blames the external and focuses his energies there; the warrior conquers the realms within and becomes gifted.\n\n\u2014Teresa of \u00c1vila\n\nBut on condition that we liberate ourselves from our own interior despots, we are the most poetic beings, the newest, the most virgin in the world.\n\n\u2014H\u00e9l\u00e8ne Cixous\n\nTeresa's words echo the original meaning of _jihad_ in the Muslim tradition\u2014a holy war on forces within oneself that obstruct one's relationship with the sacred. In the Buddhist tradition, the Shambhala warrior is the person who skillfully does battle in the world by virtue of her holy war with the ego. This quality of warriorhood never was or will be about taking lives but about \"overthrowing governments inside that make us weep.\" The natural, inevitable outgrowth of the divine inner struggle will be an enlightened passion for justice and dignity in the external world.\n\nAnything as daunting as the collapse of industrial civilization will pit us in a holy war with the \"inner colonizers\" that we have incorporated from our civilized inculcation. Unless we find a way to subdue and transform the imperialist within, we will be hard pressed to construct a next culture that is capable of living at peace with itself and the earth community.\n\nAll around us we witness individuals who childishly blame the external and focus their energies there. What else can humans do who have not been initiated into life? How else can they make sense of all that is broken in the world?\n\nThe collapse of industrial civilization is a calling to become a mature warrior, one who will overthrow the oppressive regimes within and thereby become more proficient in navigating the disintegration of a postindustrial world. Holding the outer world alongside the inner is a Herculean task, but while humanly challenging, not impossible\u2014 _if_ we begin our warrior training sooner rather than later.\n\nDo not go gentle into that good night,\n\nOld age should burn and rave at close of day;\n\nRage, rage against the dying of the light.\n\n\u2014Dylan Thomas\n\nThese lines from Thomas's poem, \"Do Not Go Gentle into That Good Night,\" about old men resisting death, were written when the poet's father was dying. Yet it is an apt poem when contemplating collapse, which, as we know, is a collective death.\n\nAs we witness so much rage in the world around us, we can become fearful, particularly of discovering any rage within ourselves. Yet as we contemplate the demise of industrial civilization, it is totally appropriate to feel our rage, as well as our grief, at the loss of so much that is sacred. As hundreds of species go extinct daily; as millions lose life and limb due to corporate greed; as hideous, disgusting dead zones form in the seas; as fresh waters become rivers of death from pollution; as CEOs accumulate billion-dollar bonuses while children starve in the streets of communities they have raped and impoverished; as veterans return to us hollowed out inside and brutally maimed outside; as natural gas companies devastate the earth with poisonous hydraulic fracturing compounds and make communities uninhabitable; as human trafficking produces millions of sex slaves annually; as the Monsanto monster designs ever more grotesque Frankenfoods; as religious fundamentalisms of all varieties oppress women and children with their appalling hypocrisy\u2014as we attempt to just live our lives in the midst of these atrocities, how can we not rage against the dying of the light?\n\nThomas suggests that elderhood includes rage, as well as other emotions. He does not instruct us, nor do I, to act out the rage but to feel it and not attempt to conceal our disgust. It is appropriate to feel our rage about the light in our world that is being snuffed out more blatantly with every passing day. How do we accept collapse, yet not go gently? How do we compassionately burn and rave at the close of day?\n\nSisters, I ain't clear what you be after. If women want any rights more than they's got, why don't they just take them, and not be talking about it?\n\n\u2014Sojourner Truth\n\nAs we move steadily into a more chaotic world, I have no doubt that women are going to be especially challenged to remain steadfast in the face of what is likely to be an attempt to erase the gains made by the feminist movement in the 1960s and thereafter. Undoubtedly, men will be challenged as well, especially if they are committed to the safety and equal status of women. Both genders will be vulnerable to unspeakable violence.\n\nWomen must prepare themselves psychologically for escalating violence against them and should learn skills to protect themselves physically. Likewise, we must learn emotional tools for living in a world where women, children, and the elderly will be targets of violence and where women may be scapegoated as the cause of society's ills.\n\nSojourner Truth admonished women not to wait for their rights to be given to them but to simply take them. We did this in the last quarter of the twentieth century, and we may need to do it again in a collapsing world, where law enforcement and judicial systems may be weakened or absent, removing barriers against rape, sexual molestation, battering, and all types of criminal acts. In the past four decades, women have learned how to construct boundaries in relationships for protecting themselves and enhancing their well-being\u2014boundaries that are likely to be sorely tested or even shattered by panicked and troubled men in a chaotic world.\n\nNow is the time not only to learn how to protect ourselves physically but to revisit and fortify our boundaries. When you say \"No,\" what are you prepared to do when you are not heard or when your \"No\" is the joke of the moment for some violent individual?\n\nWhat is more, it is important to serve and not be obsessed with one's own pain, but when you are surrounded by needy people who are terrified, angry, shattered, hungry, tired, or even suicidal, how will you take care of yourself and not become a burned-out Florence Nightingale? How will you stand strong for yourself as well as those in need?\n\nMaking reality real is art's responsibility. It is a practical assignment, then, a self-assignment: to achieve, by a cultivated sensitivity for observing life, a capacity for receiving impressions, a lonely, unremitting, unaided, unaidable vision, and transferring this vision without distortion to it onto the pages of a novel, where, if the reader is so persuaded, it will turn into the reader's illusion.\n\n\u2014Eudora Welty\n\nEven in the midst of all manner of upheaval, people still make art. They make art on the battlefield, from hospital beds and prisons, in the midst of hurricanes, in funeral parlors, and at the birth of children. I cannot imagine that even in the most chaotic world imaginable, people will cease making art. Furthermore, I suspect that some of the greatest art in which collapse is the focus has yet to be created.\n\nArt is, among other things, a way of making sense of our experience. Making it real, yes, but also a means of performing a ritual in which we and others bear witness to our experience. The ritual might be visual, such as a painting or sculpture; it may be a story that relates our experience or a song that sets our story to music; it may be a dance or a play that depicts experience through movement or theater.\n\nI believe that just as people's bodies need air, food, water, and sleep, their souls require art. Making art is an act of soul, as is receiving it and allowing it to make meaning in our lives.\n\nEudora Welty speaks of putting words on a page, without distortion, that are then turned into the reader's illusion, yet she asserts at the beginning of her statement that art is about making reality real. Reality? Illusion? Which is it? It is, of course, both. The artist selects a portion of his reality\/illusion and turns it into an art form, which is then received and interpreted by others and turned into their reality\/illusion. Art is about giving and receiving impressions, no matter how illusory, which evoke in the receiver\u2014and usually in the giver\u2014our truest truth: the reality, depth, and authenticity of the deeper Self.\n\nI laugh when I hear that the fish in the water is thirsty\n\nYou don't grasp the fact that what is most alive of all is inside your own house.\n\n\u2014Kabir\n\nIn troubled times and in all times\u2014but especially in _these_ times\u2014it is crucial to find where the soul is hidden. This is no easy task. The soul has been buried by our civilized world under the trappings of \"success\" and \"prosperity.\" It has been armored with a \"me and mine\" mentality, varnished with what the neighbors think, encrusted with the American or some other dream, and locked in the safe deposit box of \"provable scientific exploration.\"\n\nMany people tell me that they did not really discover their souls until they began preparing for collapse, and others tell me how alive they now feel because they are willing to explore their soul's treasure within the darkness that gathers around us at the end of one era and the beginning of another.\n\nOver the years I have grown especially fond of Kabir, because he so lovingly but so insistently pleads with us to discover the treasure of soul within. He begs us not to be like the fish in the water that claims it is thirsty or like someone who travels from place to place to find what is most alive.\n\nIn my work, I incessantly _im_ plore people to _ex_ plore the inner world, particularly people who argue that if they can't scientifically prove the existence of an inner world or empirically verify that it is not merely the activity of brain and neural impulses, they refuse to give it any credence. \"I'm a rational human being,\" someone asserts. \"How do I know that what you call the 'inner world' isn't just a bunch of my illusions? Neurosis? Brain farts?\"\n\nI don't know the answer to these questions, but you will\u2014if you are willing to do the long, patient, merciful excavation of the deeper Self, because until you do, \"for you, the world will never be real.\"\n\nBeauty triumphs over the suffering inherent in life.\n\n\u2014Nietzsche\n\nTreat things poetically.\n\n\u2014Ralph Waldo Emerson\n\nWhat is the difference between poetry and prose? One online source states, \"The most basic answer might be: Prose is language that has as its primary goal the _sharing of information._ Poetry has as its primary goal the _use of language itself as music._ \"\n\nAs I have stated in this book and elsewhere, our world is choking on information but starving for wisdom and beauty. To treat things poetically, as Emerson admonishes us, is to treat them with an eye toward beauty and the soul.\n\nProse is linear; the word _prose_ stems from a Latin word that means \"to move in a straight line.\" Like the soul, which cannot be accessed by moving in a straight line, poetry is cyclical, curved, indirect, and labyrinthine. When we view our world from a prose perspective\u2014in terms of \"the facts\" as we find them in news reports\u2014the soul remains hungry. When we begin to interpret these events poetically, the soul is activated and nourished, and what were simply factual occurrences begin speaking to us from a more mythic or meaningful perspective.\n\nWe need both prose and poetry, because they are two different pieces of reality and, together, provide a holistic experience of it. Prose gives us the facts regarding our suffering world, while poetry interprets the facts artistically, providing beauty, poignancy, outrage, inspiration, tragedy, ecstasy, humor, and more. Can we discover beauty in suffering without being sadistic or masochistic? I believe we can, and that may be one place where prose and poetry join to assist us in holding both opposites in the body.\n\nWhile beauty may not literally triumph over suffering, John O'Donohue writes, in _Beauty: The Invisible Embrace,_ that it is a \"climate of kindness\" that \"nurtures the sore landscape of the human heart and urges torn ground to heal and become fecund.\"\n\nIf you can look into the seeds of time,\n\nAnd say which grain will grow, and which will not, Speak.\n\n\u2014Shakespeare, _Macbeth_\n\nNone of us has a crystal ball, and even if we did, some of us\u2014like me\u2014would probably not use it. We predict, forecast, and prognosticate about the future, but none of us knows exactly how it will play out.\n\nSome people are holding out for a miracle\u2014some president or politician or movement organizer who will have such an impact on humanity that its destiny will be redirected to one of peace, environmental integrity, economic justice, and energy abundance. Others have long since submerged into despair, becoming cynical and bitter. Still others dare to lay out timelines for the collapse of industrial civilization and forecast that by a certain year or range of years, this or that will occur.\n\nForecasting is entertaining at best and foolhardy at worst. Even when a prediction is spot-on, something in us hungers for the next one, as if the correctness of one prediction will lead to the correctness of another, and another, and another. Our appetite for prediction no doubt originates in our lack of control and our sense of vulnerability. If only we could know, we tell ourselves, we would have more control and feel less susceptible to adversity.\n\nDrawing logical conclusions based on trends and diligent research is fine. No one should ignore these and just be content with unknowns. We would be foolish, for example, _not_ to pay attention to signs that portend the demise of the current paradigm. In fact, that is what the majority of humanity is doing at this moment.\n\nYet even as we gather our research and prepare for the future based on reliable information, the bottom line is that we really do not _know_ what is going to happen. Likewise, as we plant seeds in our lives that touch other people, we know not which ones will come to fruition and which ones will lie dormant.\n\nWith regard to the future, we face many more unknowns than knowns. As the English playwright and novelist W. Somerset Maugham reflected, \"It wasn't until quite late in life that I discovered how easy it is to say, 'I don't know'!\"\n\nThe civilized man has built a coach, but has lost the use of his feet.\n\n\u2014Marsilio Ficino\n\nFicino was a Renaissance humanist philosopher whose father was a physician under the patronage of the Medici family. He perceived himself as living in a golden age in which the classics of Greece and Rome were being revived; yet he also realized the perils of civilization, one of them being disconnection from the deeper Self.\n\nIn the twenty-first century, we can easily replace the word _coach_ with _car,_ and Ficino's statement becomes even more germane. Inundated with ever-new and dazzling forms of technology, we lose our basic human functioning\u2014our ability to think critically, the natural inclination toward introspection, and the slower pace of life, at which humans seem to function more comfortably. You may argue that it depends on which humans and what their definition of comfort is. Nevertheless, if we are honest, most of us will confess that we have lost our \"feet\" in the throes of modernity.\n\nTo lose our feet is to lose our grounding, particularly our connection with the earth. Technology can \"carry\" us to many places, but it cannot transport us to the soul or an intimate \"conversation\" with a tree, a deer, a snake, or a flower.\n\nMany people who choose to walk instead of ride in vehicles that burn fossil fuels discover the magic of doing so. Slow? Yes indeed, and so\u2014increasingly\u2014will be the pace of life in a collapsing world. In fact, for many people, walking will become their _only_ source of mobility. Having one's feet on the ground\u2014literally and figuratively\u2014and being able to navigate more slowly in the world solidifies our connection with it and allows us to experience it more intimately.\n\nOh, how we take for granted the use of our feet until an illness or injury compels us to stay off them. If we are fortunate enough to regain our mobility and are able to walk again, we may feel ecstatic, because we have been slowed down and forced to live life at a very different pace.\n\nAs civilization gradually disappears, we will have the opportunity to \"regain our feet,\" literally and symbolically\u2014and, thereby, the natural rhythms of body and soul.\n\nIn this world each wake-up call is an opportunity to turn back to conversation with the divine.\n\n\u2014Michael Meade, _Fate and Destiny_\n\nThroughout the course of life, the soul will create impossible dramas and persistent dilemmas in order to crack open the shell of the time-bound, little self and reveal the living seed and the original agreement it carries within it. The soul needs an outer drama that can help awaken the inner myth that it carries and the first agreement it made with the divine realm. Although limited by time and shaped by its earthly allotment, the soul instinctively longs for a reunion with the eternal. It will seek to find and feel again the divine connection in religions, philosophies, in visions and in ecstasies, and in the person of parents, teachers, in lovers and in soul companions.\n\n\u2014Michael Meade, _Fate and Destiny_\n\nCould there be a more \"impossible drama\" than the collapse of industrial civilization? Whether we fully understand it or not, the drama is an opportunity to awaken the inner myth of each of our lives. Each person's story, myth, and what some call \"original agreements\" are different, but all are connected in some way with our divine origins\u2014the deeper Self within us. In fact, according to Meade, we are always attempting to revisit and revitalize the connection.\n\nThe divine experience or experiences that collapse manifests for each person may not feel divine. They may be horrific, terrifying, deadening, or even demonic. I suspect that a common experience may be a sense of the surreal. Yet each experience is an opportunity to access the divine implicit within it.\n\nBecause humans have forgotten and forsaken the divine connection, we are already experiencing a planetary upheaval that will compel us to reunite with the sacred. And let us remember that the upheaval is of our own making. In our pathetic attempts to reconnect with the eternal, humans have decimated the earth and devoured resources, economies, and each other. Whether or not one accepts that the earth is a living organism, humans are experiencing the inevitable repercussions of our plunder.\n\nWhatever form the current wake-up call takes\u2014natural disasters, economic meltdown, wars, escalating domestic violence and suicides, addictions, pandemics, or famine\u2014it has the potential to crack open the shell of the little self and reveal the living seed within. All are opportunities to resume another conversation with the divine.\n\nWhat is this precious love and laughter budding in our hearts? It is the glorious sound of a soul waking up!\n\n\u2014Hafiz\n\nI suspect that if your soul were not awake, you would not be reading this book. But take some time to reflect on how that happened? No, it was not just one moment in time; most likely, it was many moments _over_ time.\n\nHow did your soul awaken to the reality of a world in demise? What happened within your soul when your mind understood the ramifications of collapse? What drove you to pick up this book and other resources of emotional and spiritual preparation? Take plenty of time to reflect on the process that brought you here.\n\nAs with falling in love for the very first time, the awakening of the soul for the first time is a unique, unrepeatable event. Many other events may have led up to it, but at some decisive point, one's heart is opened, and emotions flood the body. Perhaps laughter is nowhere to be found at first, but eventually, the heart does overflow with joy.\n\nAround us, we see many people, perhaps some very dear to us, whose souls have never awakened. We witness their pain, constriction, rigidity, suppressed terror, anger, and sorrow, yet we cannot make them wake up. Nor would we really want to, because awareness must dawn from within if it is to be rightly called \"awareness.\" If we are truly awake, we will feel compassion for those who are not awake, but we will not wait for them or invest energy in their awakening.\n\nA soul waking up is much like a child\u2014or any tiny being\u2014being born. Treasure and savor your own awakening. Allow the laughter and love to flow from your heart. Never lose sight or the sensation of your waking up. Savor it. Treasure its priceless value. For it is indeed why you are alive on this planet in this moment.\n\nTired of all who come with words but no language\n\nI went to the snow-covered island . . .\n\nI come across the marks of a roe-deer's hooves in the snow.\n\nLanguage but no words.\n\n\u2014Tomas Transtr\u00f6mer\n\nIn our relentlessly verbal civilization, it is easy to become weary of words. In fact, I experience moments in which I cannot hear or read another word. In those moments, the soul is overwhelmed with prose and left-brain utterances that leave the heart smothered and cold. I believe that it is important to be sensitive to such moments and to honor them by retreating from the linear and finding refuge in the poetic, as Tomas Transtr\u00f6mer does in these words.\n\nWhen he says that he's tired of all who come with words but no language, he is essentially affirming what I have just written. What the heart and soul seek is the language behind words. Language is about communication, but that is not always the case with words. In fact, it is _rarely_ the case.\n\nThe heart longs for communication, a word related to _commune\u2014_ perhaps that is why we can so easily become tired of words. Words come at us, sometimes assault us, but there is no communion or community in them\u2014and thus no authentic communication.\n\nTherefore, the poet takes himself into the wild, because the wild doesn't have words. Yet the wild is bursting with silent communication. It doesn't assault, convince, cajole, or convert. It quietly and unobtrusively blankets the land with profound messages that, when appreciated and assimilated, bring us into communion with ourselves and with the intricate, subtle, respectful language of primal wildness\u2014the language of the soul. Might this be the language of the next culture as \"words fail\" alongside industrial civilization?\n\nThe dark night is not a phrase; it is a part of experience that is always hovering close by, ready to do its job of sullying the untarnished innocence that usually arrives with spiritual awakening.\n\n\u2014Thomas Moore, _The Soul's Religion_\n\nThomas Moore, author of _Care of the Soul_ and other books, is a psychotherapist and a former Catholic monk. Greatly influenced by the Jungian psychologist James Hillman, in all of his works, Moore emphasizes the Jungian concepts of ego, shadow, and the deeper Self. When one reads Moore, one encounters consistent comments about darkness and its role in bringing us to a more meaningful and enriching spiritual path.\n\nIn the tradition of Jung, Moore begs us to be mindful of the shadow, which is always lurking and ready to sully \"our untarnished innocence.\" In other words, whenever we have a profound spiritual experience or feel that we have moved through some ordeal to a newfound level of awareness, it is wise to be aware of the relentless presence of darkness. Naturally, when we are basking in the light of some hard-won epiphany, darkness is not what first leaps to mind. We may be engrossed in gratitude, ecstasy, heart-opening intimacy with another person, or savoring a shift in perspective that creates doors where we were certain only walls existed. Yet even as we allow these moments of exquisite illumination, an awareness of darkness lurking is prudent.\n\nThe reader may protest that this promotes a paranoid way of being in the world and quashes the joy of consciousness transformation. How can one fully enjoy an epiphany if one is thinking about lurking darkness?\n\nJung once stated that if we wish to bask in the fullness of any emotion or experience, we should be prepared for an abrupt psychic shift into the other opposite extreme. If we are cherishing a glorious epiphany, we should be prepared for some darkness to rear its ugly head. Likewise, if we are writhing in the depths of sorrow, and our hearts are exceedingly heavy, we should be prepared for some unexpected ray of light to appear, which will modify, if even for a moment, our anguish.\n\nSavor the light, beware of the darkness, and know that one cannot live long without the other.\n\nPossessions make us needy.\n\n\u2014Thomas Merton, _No Man Is an Island_\n\nImagine no possessions; I wonder if you can.\n\n\u2014John Lennon\n\nOccasionally, we read news reports about someone who has found a way to live without money for months or years. I heartily admire such individuals, though I admit that, for myriad reasons, I haven't made a conscious effort to do the same.\n\nAs I have written in a number of other places, we live in a culture that infantilizes us and makes us dependent and needy. We form dependent relationships with people and with possessions. Thomas Merton explains very clearly what happens in a milieu where, fundamentally, we are not present to ourselves: \"When I am not present to myself, then I am only aware of that half of me, that mode of my being which turns outward to created things. And then it is possible for me to lose myself among them.\"\n\nMerton tells us that the remedy is _recollection._ That is, we _re-collect_ ourselves by becoming totally present to ourselves. We start with sitting quietly and exploring our feelings and their sensations in the body. We may also want to journal in order to record our _re-collection._ We consciously commit to constructing an inner life or, as I have named it in other places, an \"internal bunker.\"\n\nPeople become lost in created things because they are completely out of touch with their inner core. We can become lost not only in possessions but also in activity. How often do we hear or use the expression \"crazy-busy\"? Workaholism is the drug of choice for millions of inhabitants of industrial civilization. Addiction to work also makes us needy\u2014needy of the approval of others, needy of a paycheck or a promotion, needy of _avoiding_ the deeper current of our lives.\n\nConsider, if you are not already doing so, committing to a daily practice of _re-collection,_ through meditation, journaling, chanting, or prayer. In this way, you become fully present to yourself.\n\nSpiritual discernment is the art of critical attention that is able to recognize the Divine presence in its expected and unexpected forms. . . . Real maturity is the integrity of inhabiting that \"immortal longing\" that always calls you to new horizons. Your true longing is to belong to the eternal that echoes continually in everything that happens to you.\n\n\u2014John O'Donohue, _Eternal Echoes_\n\nIn turbulent times, we need extraordinarily astute spiritual discernment. We do not gain this by reading books but through practice and interdependence with the divine. As a result, it is possible to recognize the divine presence in its \"expected and unexpected forms.\" The collapse of industrial civilization itself may be an unexpected form of the divine\u2014an upheaval that rearranges our priorities, strips the human ego, and compels us to discover meaning and purpose on ever-new levels.\n\nTo be human is to be filled with longing; however, longing is not the same as insatiable greed. It issues not from a voracious ego, which craves more and more of what we do not really want, but rather from the soul. Ultimately, it is designed to draw us to the divine. O'Donohue tells us not only to honor the longing but to inhabit it, because it belongs to the eternal. Moreover, he says that everything that happens to us echoes the eternal.\n\nIt is important not to confuse \"echoes of the eternal\" with \"God's will.\" In other words, some may ask, how can the divine be present in the extinction of species, the death of ecosystems, the pollution of our water and air, and endless wars? I would argue that the \"echo\" of the divine is not in these evil events themselves but in how they speak to us and how we respond.\n\nEchoes of the divine are present in everything that happens to us. An echo\u2014a mirror of something spoken\u2014is, in a sense, both a new voice and the original voice, and the voice of the divine echoes in every event, every relationship, every loss, every joy. With practice, we develop the discernment necessary to recognize the voice and open our hearts and minds to simply wonder at the mysteries therein and then to ponder the message or messages hidden within the mystery.\n\nA man is alive when he is wholehearted, true to himself, true to his own inner forces, and able to act freely according to the nature of the situations he is in. To be happy, and to be alive in this sense, are almost the same. Of course, a man who is alive is not always happy in the sense of feeling pleasant; experiences of joy are balanced by experiences of sorrow. But the experiences are all deeply felt; and above all, the man is whole and conscious of being real. . . . This state cannot be reached merely by inner work. There is a myth, sometimes widespread, that a person need do only inner work, in order to be alive like this; that a man is entirely responsible for his own problems; and that to cure himself, he need only change himself. This teaching has some value, since it is so easy for a man to imagine that his problems are caused by \"others.\" But it is a one-sided and mistaken view which also maintains the arrogance of the belief that the individual is self-sufficient, and not dependent in any essential way on his surroundings. The fact is, a person is so far formed by his surroundings, that his state of harmony depends entirely on his harmony with his surroundings.\n\n\u2014Christopher Alexander\n\nThe website Group Pattern Language (http:\/\/grouppatternlanguage.org) defines Pattern Language, a term originated by Christopher Alexander, as \"an attempt to express the deeper wisdom of what brings aliveness within a particular field of human endeavor, through a set of interconnected expressions arising from that wisdom.\" Three of many notable features of design, according to Alexander are accessibility, beauty, and connection. For him, spirit and matter are not separate, and how we relate to matter\u2014how we design it, use it, and where we place it in our world\u2014says much about to how we relate to spirit.\n\nMany individuals believe that what matters most in our evolutionary journey is our commitment to inner work. As a former psychotherapist, an author, and a workshop leader on a number of spiritual topics, I adamantly encourage people to do their inner work, _and_ I am well aware that humans are inescapably formed by their surroundings. We live in an interdependent relationship with the _macrocosm,_ as well as the _microcosm._ And in the present moment, we are witnessing the collapse of the space in between them, the _mesocosm,_ or the culture of industrial civilization. Being human and willing to do inner work, by definition, compels us to be in conscious relationship with our world\u2014its chaos, crisis, suffering, dissolution, beauty, creative energies, order, and abundance. Sometimes this relationship will be harmonious and at other times, turbulent. Nevertheless, we must know that authentic inner work will always catapult us into dynamic investment in and involvement with the external world.\n\nAll shall be well, and all shall be well and all manner of thing shall be well.\n\n\u2014Julian of Norwich\n\nWhen peace, like a river, attendeth my way,\n\nWhen sorrows like sea billows roll;\n\nWhatever my lot, Thou hast taught me to say,\n\nIt is well, it is well with my soul.\n\n\u2014Horatio Spafford\n\nMany religious traditions practice some form of _equanimity,_ that is, a state of composure arising from a deep awareness of purpose in the present moment and an acceptance of it. Equanimity is in no way the same as disconnection or apathy. Rather, it is a state of being deeply involved and concerned, yet able to surrender to the wisdom of whatever is occurring. Thus, Julian of Norwich declares that all will be well.\n\nIn 1873, the four daughters of an American attorney, Horatio Spafford, were traveling to England with their mother. The ship was struck by another vessel, and 226 people were killed, including the four daughters, though their mother miraculously survived. When Spafford sailed to England on the same route as the fateful ship that carried his family, in hours of deep grief and heartbreak, he penned the Christian hymn \"It Is Well with My Soul.\"\n\nThe skeptic may protest that it is foolish to assume that all will be well or to declare in the face of the loss of one's four children that \"it is well with my soul.\" Yet we must incisively penetrate the mind's definitions of _well._ For Julian and for Horatio Spafford, _well_ did not mean \"fine,\" \"okay,\" \"acceptable,\" or \"all right.\" Rather, from the depths of their pained souls came a reverent acknowledgement that what they were experiencing\u2014which was anything but \"fine\" on the visceral, human level\u2014was somehow right and happening with a purpose that, in present time, they could not understand. Tragedies are almost never made sense of in current time. Only from hindsight can we find meaning in catastrophe, disappointment, betrayal, abandonment, disaster, abuse, or cataclysm.\n\nThe journey to equanimity is long and often wrenching. Countless losses may be encountered before we are able to even think, let alone speak, \"all shall be well,\" or \"it is well with my soul.\"\n\nThe root meaning of the word \"mystery\" is to shut one's eyes and ears. Mystery is silence, darkness. Rilke speaks in his Book of Hours of turning inward, of looking deep into himself, and he reports what he finds: \"My God is dark.\" He sees a thousand theologians plunging like divers into the light of God's name.\n\n\u2014David Steindl-Rast\n\nIn turbulent times, we are called to an extraordinary honoring of mystery. The power of mystery is no \"mystery\" whatsoever. As David Steindl-Rast notes, we need merely enter the silence and close the eyes and ears. In that silence is darkness, and\u2014as so many creative souls like Rilke have discovered\u2014within the darkness we access the divine.\n\nCertainly the darkness is not the only place where we discover the divine, but within the soul's rich, dark, fetid compost abide the nutrients that, when revered and applied to the soil of our lives, cause our lives to erupt in unfathomably lush flora\u2014burgeoning gardens of sacred fecundity.\n\nThe divine may also be experienced in the light, but light tantalizes us and causes us to open the eyes and soar away from the deeper Self to the heights of spirit. Away from the soil of soul, nothing grows. Light bursts forth; we are transported, elated, and sometimes inflated, but eventually we must return to the ground of our being. And what grows on that ground? Very little, unless we have been willing to enter the darkness and cultivate it.\n\nTurning inward is not always a comfortable process. We often dislike what we discover there: some regret, some longing, some judgment, some self-loathing. Yet this is part of the compost, which, when worked with, can nourish glorious gardens, but even this is not our final destination. Aware of the distasteful as our limited human reality, we remain in the darkness, the silence, allowing it to reveal its deeper secrets to us. We cherish the not-so-great self alongside the greater Self. And what is the outcome?\n\nMystery would not be mystery if we knew where it was taking us. Claim the distasteful, claim the divine, and notice what grows.\n\nWe have fallen into the place where everything is music.\n\n\u2014Rumi\n\nMusic is a higher revelation than all wisdom and philosophy. It is the wine of a new procreation, and I am Bacchus who presses out this glorious wine for men and makes them drunk with delight.\n\n\u2014Ludwig van Beethoven\n\nJoy, sorrow, anxiety, frustration\u2014music not only expresses it all, it echoes every human emotion in a manner distinct from the spoken word. Such is the power of music and its impact on the human nervous system. When sound is shaped into words, it has one effect, but when it reverberates, with or without words, it has quite another. Thus, Beethoven declares that it is a higher revelation than all wisdom and philosophy, and it was his joy and delight to allow stupendous sounds to pour through him like wine for humans to imbibe and savor for time immemorial.\n\nRumi declares that we have fallen into the place where everything is music. At first blush, we may assume that he specifically means \"happy\" music, but when we consider the vast range of music and its subtle tones and myriad ramifications, we understand that falling into a place of music may not necessarily be a happy place but rather a place of profound vibratory impact. Pure music has the capacity to open and remake us in ways that words cannot.\n\nMoreover, the space between the notes of music, which is anything but \"empty,\" is rich and ripe with mystery. In the void between the notes we discover the infinite, the sacred, the silence from which all sound originates. Silence, too, is transformative, because it is the core of our being. To discover anything of our essence, we must first go to the silence. This too is \"music,\" because sound and silence require each other's presence for anything even remotely worthy of the name \"music.\" We are familiar with the expression \"sounds of silence,\" the title of a Simon and Garfunkel pop hit from the 1960s, but are we familiar with the \"silence of sound\"? The silence in which all sound gestates and from which it is born. The silence we need in order to \"face the music\" of life.\n\nTrauma can be hell on earth; transformed, it is a divine gift.\n\n\u2014Peter Levine, _Waking the Tiger_\n\nSomatic Experiencing is a specific treatment for healing the wounds of trauma; it was developed by the psychologist Peter Levine. I suspect that Levine would agree with me that industrial civilization is inherently traumatizing. In fact, I have never met anyone who grew up in this culture who has not been traumatized in some fashion. Anyone living with trauma understands clearly what a hell it can be. Millions more, who do not understand the ways in which they are living _with_ trauma, are also living _out_ various versions of hell at miscellaneous times in their lives.\n\nTrauma saps our life-force energy and compels us to utilize it in ways that work against us and those we care about. It is debilitating, destructive, and often deadly. In turbulent times, we will unarguably witness an increase in levels of trauma and people acting it out on themselves and others. In some parts of the collapsing world, the severity of trauma will intensify dramatically. We cannot predict when, where, or how this will happen. What we can do is engage in trauma-healing programs, not only to learn how to deal with traumatized people around us, but also to learn how to navigate our own trauma, which will almost certainly be exacerbated in a chaotic world. Somatic Experiencing provides excellent training for trauma healing, as do a number of other techniques.\n\nWhen we consciously address our trauma\u2014or any wound sustained in the course of our human journey\u2014and when we persevere in the healing process, we invariably discover that, regardless of the final outcome, our transformed wounds become divine gifts that make us wiser, more resilient, and more whole. In fact, wholeness begins with being broken apart. Residing deep in wound is a gift\u2014or many gifts\u2014which cannot be accessed without willingly descending into it.\n\nThrough experiences of the sacred, the mind grasps the difference between what is revealed as real, potent, rich, and meaningful and that which is deficient in those qualities\u2014in other words, the chaotic and perilous flux of things, their fortuitous and meaningless appearances and disappearances.\n\n\u2014Mircea Eliade, _Ordeal by Labyrinth_\n\nWhen asked, as I often am, what I mean by \"the sacred,\" I immediately attempt to disconnect it from religion or dogma. Like Eliade, one of the words I invariably use to describe the sacred is \"meaningful.\" I believe that something within us seeks and even requires meaning in our lives. People who find life meaningless usually lack a sense of purpose and are therefore unable to experience a state of sustained contentment.\n\nA sense of meaning is unique to every individual. What is meaningful to one may not be meaningful to another at all. One person can walk away from a breathtaking sunset reflecting on a vast, glistening ocean without being moved in the slightest, yet he or she can work in a soup kitchen on a cold winter day, conversing intensely with homeless people, and leave the experience utterly transformed.\n\nAccording to Eliade, words used to delineate between sacred and profane are _real, potent, rich,_ and _meaningful,_ as opposed to _chaotic and perilous flux._ A sacred experience is replete with richness, power, and a feeling of that which is real. In it, we touch the changeless, which is both orderly and eternal. This is not to say that chaotic and turbulent experiences cannot give us a sense of meaning, but once we have savored the sacred, we have little difficulty distinguishing it from the ordinary or profane.\n\nCollapse is and will be a time of dizzying change\u2014change that will often be overwhelmingly distressing, though at other times it might feel glacial and insignificant. Nevertheless, it will prove crucial to be able to discern the difference between the sacred and that which is \"perilous flux.\" As we develop familiarity with the sacred and repeatedly taste and feel the difference in the body between the eternal and the tentative, we are likely to cultivate a kind of radar for the sacred and the profane. In that process, we may well abandon all connotations of the sacred that are even remotely related to dogma, once we see how the sacred is profoundly related to our very essence.\n\nThe call comes from the person standing in front of you who asks, \"Will you honor me? Be kind to me?\" . . . Honor the little calls. They prepare you for the big ones.\n\n\u2014David Spangler\n\nI work with many people who contact me for assistance with regard to changing careers and finding their highest purpose in life. Many have lost their jobs, weathered foreclosures, bankruptcies, and even divorce. They are acutely aware of a calling within themselves to leave the paradigm of industrial civilization behind and not only prepare themselves for its demise but discover their gifts and how to most skillfully use them.\n\nAt first, we may find ourselves being gently nudged by something in our lives that calls us to action or service. The call may resonate perfectly with us, or it may feel awkward and require us to enter new territory. A woman has the experience of being asked to spend a couple of hours each day with a dying relative. She answers the call, which, in the beginning, feels strange, even scary. But as she offers her service day after day, she discovers new emotions and awarenesses welling within her. Then, unexpectedly, she discovers that she feels compelled to be present with the dying person and really would not want to be anywhere else. Subsequent to this experience, she might feel herself drawn to work further with the dying in a hospice setting. This may lead to full-time employment doing something that she loves\u2014being present with people who have little time to live.\n\nThis is only one example of little calls coming to us in strange ways, which take us out of our comfort zone and compel us to think outside the box. They may lead to one overriding calling that we fulfill in a collapsing world\u2014or not. What matters is not that we have a clear picture of a \"big\" call but that we are willing to answer the little ones and notice where they might lead us. It's really all about just being completely present in this moment and doing what is right in front of us. We can trust life to call us to where we need to be in each moment; we need only the willingness to follow.\n\nThe spiritual gift on the inner journey is to know that creation comes out of chaos, and that even what has been created needs to be returned to chaos every now and then to get recreated in a more vital form.\n\n\u2014Parker Palmer\n\nMany indigenous stories and myths speak of unraveling and reweaving; dissolution and recreation; floods, fires, earthquakes, and pestilence that destroy places and people for the purpose of clearing away the old as part of a transformative process.\n\nI believe that the life\/death\/life cycle is not only an inherent aspect of the inner journey but of the external world as well. We are front-row observers of and dynamic participants in a culture that is being returned to chaos. For many, this is and will be a meaningless, absurd process. But hopefully, for all readers of this book, the return to chaos will be significantly illumined by what is written here.\n\nSomething\u2014call it the universe, spirit, matter, the sacred, the divine, god\u2014is attempting to recreate us in a more vital form. We humans have initiated the process of demise, but that experience does not have to be nihilistically absurd for us. Painful, frightening, frustrating, and sometimes seemingly impossible? Yes indeed, but never devoid of meaning.\n\nWhat will evolve from the chaos is anyone's guess. Myriad possibilities, many now beyond our ability to imagine, will reveal themselves with time and dissolution\u2014dissolution not only of old structures but of an old and no-longer-appropriate paradigm.\n\nWe must hold the vision of possibility alongside our experience of the chaos. The old alchemical principle assures us that if we hold these opposites together\u2014and especially if we hold them in close proximity to each other\u2014transformation is certain. We may not endure to see it come to fruition, but we will have the privilege of being participants in an unprecedented journey from creation to chaos to rebirth.\n\nWe are all connected. Metaphor knows this and therefore is religious.\n\n\u2014Natalie Goldberg, _Writing Down the Bones_\n\nWhen I present workshops in venues around the nation, I generally use storytelling with an African drum to introduce my topic. Almost without exception, people tell me that the drumming and storytelling open their bodies to hear things that they probably would not have heard had I spoken in a more linear, academic way. Why is that? Because drumming, storytelling, art, music, and poetry are similar to metaphor in that they originate in the right brain and \"carry over\"\u2014the literal meaning of _metaphor_ \u2014the linear mind to the symbolic mind.\n\nNot only does the symbolic or imaginal connect us with ourselves, it also connects us with each other. That is, it takes us to a deeper place in the psyche, where we are connected with everyone and everything. Natalie Goldberg names that \"religious,\" but I believe it is \"meta-religious,\" adding the prefix _meta-,_ \"beyond.\" So much of religion is actually literal and left-brained, whereas the imaginal is beyond religion and even beyond ourselves\u2014in other words, beyond the ego or who we have come to think we are.\n\nThus, in a chaotic world, it may become necessary, on many occasions, to communicate with people using the imaginal rather than straightforward prose. Another person may not be able to hear or understand us if we speak in linear fashion, but if we sing a song, tell a story, or recite a poem\u2014depending on the person and the circumstances\u2014we may gain more access to that person's inner world.\n\nThe imaginal, the metaphoric, stimulates the soul\u2014that part of us that serves as a bridge between the deeper Self and the external world. The Greek word for \"soul\" is _psyche_ \u2014the root word of _psychology._ In the psyche, it may be difficult to distinguish what is the deeper Self and what constitutes our emotional landscape. For this reason, in more recent years, many people have begun using the term _psycho-spiritual_ to describe work that relates to the emotions and to the sacred. Whether or not we choose to name this work \"religious,\" it is unquestionably holy in nature.\n\nListen with your feelings, see with your heart. Read the earth, listen to the wind as it speaks to you. Gather in its fragrances and touch its differences. Taste it, and see that it is good. This earth is a garden, this life a banquet. And it's time we realized that it was given to all life, animal and man, to enjoy.\n\n\u2014Tom Brown Jr., _The Search_\n\nTom Brown is a Native American naturalist and wilderness tracker. From the well of indigenous wisdom, he admonishes us to listen with our feelings and see with the heart. This manner of relating to the earth is starkly foreign to most nonnative people. Yet unless we can develop these skills, not only is our survival compromised, but our capacity for creating a new paradigm that is radically and qualitatively different from this one is significantly at risk.\n\nAround the world, those who survive the collapse of industrial civilization, which may span the lifetime of more than one generation, will at some point find themselves constructing a new culture. It is unlikely that there will be a singular \"American culture\" or a singular \"European culture.\" Rather, it is more likely that many smaller communities will emerge from the ashes of collapse. What they create going forward will depend on the stories they have heard of humanity's past\u2014how industrial civilization reached a pinnacle of so-called success, then began its demise. These individuals, in order to create the next culture, must know history, but they must also have developed an intimate relationship with the earth.\n\nWhat does that mean? Exactly what Tom Brown is describing in the above statement. The ability to listen with one's feelings and see with one's heart must be cultivated over time. Living directly off the land certainly enhances the process, but even if one has not been forced to do so, such skills can be developed.\n\nThe price of not doing so will, inexorably, be the creation of a \"next\" culture that may look and feel different from the industrial paradigm but, in fact, has many elements of the old concealed in it. Only the kind of immersion in the earth and its idiosyncrasies of which Tom Brown speaks will prevent humans from creating Civilization 2.0.\n\nCertain hurts and harms and shames can never be done being grieved. . . . Although there will be scars and plenty of them, it is good to remember that in tensile strength and ability to absorb pressure, a scar is stronger than skin. . . . Tears are a river that takes you somewhere.\n\n\u2014Clarissa Pinkola Est\u00e9s, _Women Who Run with the Wolves_\n\nWhen a natural disaster or a violent tragedy strikes a community, we see a host of memorials constructed for the dead or wounded, usually happening alongside prayer vigils, dramatic visits from politicians, and the revival of debates on issues such as climate change, gun control, the death penalty, and how to prepare for long-term disruptions of services brought about by all manner of disasters. What we almost never witness, however, is an intentional grief ritual, in which members of the community have unlimited permission to publicly mourn their losses. News coverage focuses on the tragedy for several days or a week, then moves on to other stories, with the implication that survivors of the devastation should also \"move on\" and \"put it behind them.\"\n\nAs I witness these now-predictable attempts to report such tragedies, I notice that authentic grieving is never shown. Survivors grieve in private, but grief is not merely a personal matter; it is profoundly related to the community and should be witnessed mindfully in public. An intentional grief ritual is a place where this can happen. The community witnesses the grief of the tragedy's victims, whereby compassion and deep connection with one's neighbors are dramatically increased throughout the community. This kind of public mourning is common in many indigenous communities and provides a durable fiber that weaves its members together, making them stronger as a result of weathering adversity in a common well of grief.\n\nWhen we consider tragedies such as the Oklahoma City bombing, the September 11 attacks, the Mumbai massacre of 2008, the Bhopal disaster of 1984, and countless other disasters, we encounter events for which grieving may never end. The magnitude of horror is such that the ensuing grief constitutes a bottomless pit of sorrow. In such situations, it is absurd to speak of \"moving on,\" or \"putting it behind us.\" Massive scars are inevitable, but the skin they leave is stronger than it was before the tragedy. Let us always remember that \"tears are a river that takes you somewhere.\"\n\nYour creativity needs your awareness of death for its energy, just as your muscles need long and painful workouts. Cherish your awareness of death as a gift to you from the universe.\n\n\u2014Brian Swimme, _The Universe Is a Green Dragon_\n\nYet again, the paradox\u2014the tension of opposites. We cannot live a full, rich, vital, meaningful life without profound awareness of our mortality. The two opposites absolutely need each other.\n\nWe have all had physical workouts that felt like they were \"killing\" us. In such moments we might feel as if we are dying\u2014or perhaps wish we were! Yet these very experiences enhance our energy and vitality. Not every moment of physical exercise should be excruciating, but exercise should take us out of our comfort zone and into a state of exertion that challenges our endurance in some fashion. When we are comfortable, death feels far away\u2014even unimaginable. On a regular basis, however, whether through exercise or some other challenge, we need to be reminded of its omnipresence.\n\nBrian Swimme says, \"Cherish your awareness of death as a gift to you from the universe.\" Most indigenous traditions teach that regularly confronting our death makes us more fully alive. As we encounter our mortality, the ego is humbled, and the deeper Self has the opportunity to radiate through our being and invigorate us. Gratitude for life swells, and we appreciate where we are right now, in this moment, realizing how precious it is. What greater gift could the universe give us than this consciousness-shifting illumination?\n\nLikewise, our creativity is enhanced by deep awareness of our ultimate dissolution. We create not just because we are alive but because, someday, we will die. We create in order to leave behind a fragment or a legacy of our presence on earth, something that endures long after us. We bring something to life because, although death is always lurking, we refuse to allow it the last word. Our ultimate \"triumph\" over death is a willingness to persistently engage in the dance of creativity and demise.\n\nThe magical, mythological and feminine ways of dealing with existence left behind thousands of years ago must now be reclaimed by consciousness.\n\n\u2014Edward Whitmont, _Return of the Goddess_\n\nThe good news implicit in the collapse of the paradigm of industrial civilization is the dissolution of the patriarchal imperative, which has attempted to extinguish the feminine principle from human consciousness. The late Jungian analyst Edward Whitmont tells us that it must now be reclaimed in consciousness.\n\nAnother Jungian analyst, June Singer, elaborates on this notion: \"Our feminine wisdom values truth more than information, permanence more than change. It is she in us who desires to see things whole, to see into things, to be able to anticipate and then to wait. When we are in touch with her, we take the long view and are not seduced by expediency.\"\n\nThrough the descent into collapse and chaos, humanity will be offered the opportunity to reclaim the magical, mythological, sacred feminine. The reclamation of this principle may be the watershed process that determines what kind of culture will arise from the ashes of industrial civilization. I emphasize the word _process,_ not only because collapse and whatever evolves from it will be a protracted process, but also because process is an essential aspect of the feminine principle. Collapse may entail many events, but\u2014taking the long view\u2014it will be a process fraught with adversity, horror, destruction, uncertainty, opportunity, adventure, joy, creativity, connection, and profound transformation. It will compel us to \"see things whole\" and to \"see into things\" and to \"be able to anticipate and then to wait.\"\n\nThe sacred feminine principle must be reclaimed. It will not foist itself on humanity; it requires our personal transformation and commitment to restore it to its rightful place in human consciousness. This is the fundamental task of all awake and aware women and men on this planet.\n\nThe ideal of warriorship is that the warrior should be sad and tender, and because of that, the warrior can be very brave as well.\n\n\u2014Ch\u00f6gyam Trungpa\n\nIn this culture, the word _warrior_ means little more than \"trained killer.\" Yet in ancient and indigenous cultures, it meant something entirely different. Championed by Tibetan Buddhist teacher Ch\u00f6gyam Trungpa, Shambhala warriorhood is not about taking lives but about becoming a master of one's own consciousness and bringing enlightenment to oneself and the world. Through art, politics, and virtually all disciplines, one makes war on one's own ignorance and struggles to awaken all human beings.\n\nUnlike the trained killer, the spiritual warrior is not only brave but tender. She allows herself to feel grief, anger, fear, joy, and all other emotions with equanimity, at all times keeping focus on remaining awake. A core aspect of training for the Shambhala warrior is a particular style of meditation practice on which he relies for developing and maintaining awakeness.\n\nTrungpa modeled a balanced masculinity dramatically different from the style of masculinity prevalent in industrial civilization. The Shambhala warrior, regardless of gender, is open hearted, tender, compassionate, and gentle, but also firmly committed to the path of enlightenment.\n\nIn the milieu of an increasingly chaotic world, spiritual warriors of all traditions will be desperately needed. Training in this style of warriorhood must begin now, not in the heat of unraveling. Women who are willing to access and express the sacred masculine within themselves and men who have cultivated a relationship with their sacred feminine must now prepare for warriorhood, as defined by Trungpa, as well as for myriad forms of service. At times, the tender side of the warrior will be called upon, and at other times, the courageous defender of justice must rise to the occasion, but she or he must always understand that tenderness and courage travel together and cannot be separated.\n\nIn the depth of winter, I finally learned that within me there lay an invincible summer.\n\n\u2014Albert Camus\n\nAs the earth community enters a dark night of the soul, a time of winter, it will be crucial for each of us to know that an invincible summer lies within us, and we must be able to access it. Winter is necessary in the cycle of seasons, because it brings forth the eruption of new life in springtime. Winters of the soul, however, can feel eternal and devoid of all new life. Thus, each of us needs to be intimately familiar with the internal resources that bring us courage, comfort, creativity, and perseverance.\n\nWhat is your internal, invincible summer? How do you access the warmth, caressing breezes, colors, smells, and languid leisure of a balmy summer day when you are ensnared in winter's cold, gray, blustery bleakness? Summer is not only a time of play but of gardens, extended light, sudden downpours, heightened senses, vacations, travel, and unexpected adventure.\n\nIn the \"dead\" of winter, we long for the crackling vitality of summer. Where is yours? It's important to discover not only how to access the invincible summer but how to carry it with us in the midst of adversity. In the summer, a host of infant creatures, born in spring, are nurtured to fullness of life by their mothers. Thus, a particularly life-supporting aspect of the internal summer is our ability to receive and reciprocate nurturing. In a turbulent world, humans are going to require even more nurturing than in ordinary times, yet it is quite likely that most of us will receive much less\u2014unless we construct our lives intentionally to make room for it.\n\nThe capacity to receive nurturing and to provide it for others must be one aspect of our emotional preparation for the future. Those who cannot receive it will not thrive, but those who can absorb and impart nurturing will maintain healthier bodies and spirits. A smile or touch from a trusted person, tucked lovingly in the heart and available for access, may carry us through when nothing else can.\n\nPay careful attention to the invincible summer within. Tend it well, as you would a lush, succulent summer garden.\n\nLife is difficult. . . . Once we truly know that life is difficult\u2014once we truly understand and accept it\u2014then life is no longer difficult. Because once it is accepted, the fact that life is difficult no longer matters.\n\n\u2014M. Scott Peck, _The Road Less Traveled_\n\nIn the 1980s we saw the slogan \"Life is hard, and then you die\" on millions of bumper stickers and T-shirts. This sarcastic, tongue-in-cheek expression seemed to be modern humanity's attempt to come to terms with fact that life is difficult\u2014but really it was about denial. It was as if, by mouthing the slogan, we could laugh at life's difficulties as exaggerated and somehow reassure ourselves that life isn't really _that_ difficult. After all, modern inhabitants of industrial civilization do not dare confess that life is difficult\u2014that would contradict one of the primary tenets of their profligate lifestyle. Didn't industrial civilization guarantee a life of comfort, success, and happiness?\n\nMost indigenous cultures understand that life is difficult. The Buddha said that suffering is one of the Four Noble Truths. Yet the non-indigenous world has come to believe not only that life is not difficult but that it should not be. We believe that if life somehow becomes difficult, we need to fix our condition. Perhaps we need another college degree, a vacation, a new spouse, or a promotion. If we persist long enough in our belief that life is not and should not be difficult, we become terminally entitled and reject all adversity as somehow beneath us.\n\nScott Peck asks us to accept that life is difficult, and he tells us that when we do, difficulty matters much less, because we have owned that it is an inherent part of life. This is not to say that difficulty is all there is. If we are reasonably functional human beings, we know how to play, have fun, and create joy and beauty in our lives and in our communities. Joy can be a powerful respite from difficulty and can make life worth all the trouble it entails. Indigenous people know this. They navigate suffering and celebrate between ordeals; they rarely delude themselves into believing that they will or should have a life free of difficulty. In fact, indigenous cultures utilize ritual and celebration to make sense of their suffering and to restore themselves before, during, and after passing through it. This is what mature human beings do: accept that life is difficult and create ritual, beauty, and celebration in order to temper the pain of their ordeals.\n\nRitual is the way we carry the presence of the sacred. Ritual is the spark that must not go out. . . . Rituals were designed to remind us over and over and over again of our true relationship to life: that of a grateful amazed supplicant at the feet of mystery.\n\n\u2014Christina Baldwin, _Calling the Circle_\n\nAs nonindigenous inhabitants of industrial civilization, most of us are far from our roots in ancient and traditional cultures, where rituals were practiced on a daily basis. Perhaps you, dear reader, have reclaimed your indigenous roots, however far removed you may be from them, and have reincorporated ritual into your life. Whether or not you know your specific heritage, you are human, and that means you can practice ritual anytime, anywhere.\n\nThe literal definition of _ritual_ is \"to fit together,\" which emphasizes its power to reconnect us with ourselves, one another, and the earth. However, Christina Baldwin has given us another very beautiful, simple meaning of ritual: _the way we carry the presence of the sacred._ Moreover, as the ancients knew, ritual is designed to be practiced over and over again.\n\nWest African shaman Malidoma Som\u00e9 tells us that \"a sacred life is a ritualized life, that is, one that draws constantly from the realm of the spiritual to handle even the smallest detail.\" In other words, whatever the occasion or need, we can create ritual for it. Moreover, says Malidoma, \"Ritual facilitates and provides us with a unique channel to access higher power. Certain issues don't want to be resolved mechanistically. We don't have to know how the power works; we just have to show up and let the higher forces deal with the issues. The trap we feel inside ourselves is removed once we enter into sacred space. The energies know how to push obstacles out.\"\n\nWhen I imagine a world more chaotic than the one in which we now live, I sense a profound need for ritual on a daily basis\u2014or even more frequently. Living in the stress of a collapsing world, we enter sacred space, and we feel less trapped. We access energies that help us fit together what is broken within ourselves and connect with our community to assist the healing of that which separates us. We make ritual together as \"grateful, amazed supplicants at the feet of mystery.\"\n\nEvil is separation. Theologically, it is separation from God. Psychologically, it is separation from oneself.\n\n\u2014John A. Sanford, _The Kingdom Within_\n\nI believe that the single most destructive tenet of industrial civilization's paradigm is the notion of separation. For millennia, matriarchal, earth-based religions prevailed on our planet. A number of feminine deities were revered. Ancient societies understood the intimate, inextricable connection between humans and the earth and between one another. Virtually no one lived or functioned alone; people lived in community. _Relationship_ was perhaps the fundamental value in these ancient, matriarchal cultures.\n\nBut as humans gradually began leaving the land and moving to cities, the values of industrial civilization emerged and prevailed. From the industrial perspective, humans are separate\u2014separate from the land, separate from other species, separate from each other, and ultimately separate from the myriad complexities of their own psyches. Separation invariably results in competition and conquest for survival. The Dagara people of West Africa refer to this as a state of exile. Naturally, when one is exiled, one longs to \"come home.\"\n\nI believe that we stand on the threshold of becoming a new species, not because of our vast technology, but as a result of being compelled, through the process of industrial civilization's collapse, to become related again on every level. Before that occurs, unfortunately, we may become even more estranged, as wars, competition, scapegoating, and violence escalate. Only a few individuals will be unscathed and unchanged by these atrocities\u2014those people whose wealth and power can insulate them. Most will suffer. Yet those who survive are likely to be radically altered by the price that estrangement exacts. I suspect that they will discover that without one another, they cannot survive. Nor would they want to survive, if it meant living in a world of heightened exile.\n\nThe crucial value that must ensue from collapse is connection. If our planet survives our vile abuse of it, and if any humans survive the long, ugly process of collapse\u2014even only a few\u2014may their sense of separation be so irrevocably altered that they will never again have the capacity to exile or be exiled. May they come to know in every cell of their bodies how woven into the fiber of all life they are and forever will be.\n\nThe conviction that a man who died came back to life is, of course, a paradox. But that paradox may contain the secret of its powerful appeal, . . . it speaks the language of human emotions. It addresses itself to that which may be our deepest fear, and expresses our longing to overcome death.\n\n\u2014Elaine Pagels, _The Gnostic Gospels_\n\nBe like the fox who makes more tracks than necessary, some in the wrong directions. Practice resurrection.\n\n\u2014Wendell Berry, _Manifesto_\n\nIn this time of endings, how can we speak of resurrection, that is to say, new beginnings? Honor resurrection we must, because every ending _is_ a new beginning, and every new beginning is the end of something.\n\nIn the Christian tradition, resurrection symbolizes triumph over death, and in ancient mythology, new life always issues from death. The life\/death\/life cycle is eternal. As we prepare spiritually for collapse, we must hold resurrection alongside all of the demise inherent in collapse. We will never live to see all of the myriad resurrections that occur years, decades, centuries, or even millennia after collapse. We plant seeds now, but for the most part, we will not witness the harvest.\n\nWhatever one believes about the afterlife, resurrection\u2014in some form\u2014will come. Wendell Berry tells us to practice it. Throughout this book, I have been asking us to practice becoming familiar with death and dissolution. But now I join with Berry and ask us all to also practice rising again.\n\nResurrection is a paradox, as Pagels reminds us, but so is death. Once again we find ourselves unable to escape paradox. In fact, paradox may be one of the greatest sources of resilience\u2014a secret hidden from those who place their faith in industrial civilization, but known by those who have recognized its terminal status.\n\nMake tracks erroneously. Make many mistakes. Recognize your finitude, but practice resurrection.\n\nI am the eye with which the universe beholds itself and knows it is divine.\n\n\u2014Percy Shelley\n\nThis statement by the eighteenth-century English poet contains something far more profound than we might first infer. It is same idea that Thomas Berry and Brian Swimme describe when they speak of the story of the universe and our role in it.\n\nWe must first notice that, according to Shelley, the universe is beholding itself. This clearly implies that the universe is not merely inert matter but consciousness within matter. Berry and Swimme, in what has come to be known as the New Cosmology, speak of human consciousness as one way in which the universe knows itself. Humans are actually needed by the universe. It knows itself _through_ us.\n\nWhat is more, Shelley says that through humans, the universe knows that it is divine. As we become more intimately acquainted with our divinity, and as we honor the divinity within the universe, the universe more fully appreciates its divinity.\n\nI frequently encounter human beings who are so hurt by and so angry at how humans have violated and are violating the earth that when I speak of the universe knowing itself through us, they nearly become unhinged. They rail against their own species and declare that the earth needs to shed all of humanity\u2014only then will it be able to survive. The universe, they argue, needs nothing from us.\n\nYet the New Cosmology describes in detail the intimate relationship between humans and the universe. We need the universe, not only for the air we breathe and the food we eat, but in order to know our most authentic identity. Likewise, the universe needs us so that it can know the fullness of its identity. Some humans have the capacity to understand this and reverence the universe; many more do not. Long, attentive contemplation of Shelley's statement, allowing it to sink deeply into our hearts, will invariably transform our relationship with the universe and its relationship with us.\n\nThere is no science now to construct the story of the journey we have made on this Earth, the story that connects us with all beings. Right now we need to remember that story\u2014to harvest it and taste it. For we are in a hard time, a fearful time. And it is knowledge of the bigger story that is going to carry us through.\n\n\u2014Joanna Macy\n\nThe story of the universe comes to us in part through a scientist named Brian Swimme, but science alone cannot tell the story. Humans telling and hearing the story must be open to the sacred and its role in our journey. That openness determines our capacity to \"harvest it and taste it.\"\n\nAs stated in other parts of this book, the old stories of industrial civilization and the assumptions behind those stories are rapidly disintegrating. The major difficulty we notice with those stories is that they are too small\u2014too narrow for what our souls demand. Those too-small stories speak of our separation, our estrangement from all of creation. In a time of collapse, we hunger for the larger story of our humanity\u2014and of our divinity.\n\nLook closely at the old stories and think about what new story you might write out of your heart and body. I suggest taking time to write a new story, spontaneously and creatively, from what you know of the failure of the old stories. I also recommend becoming familiar with the New Cosmology of Brian Swimme and Thomas Berry. On the one hand, this cosmology is \"new,\" but it has also been told for centuries by great mystics such as Rumi, Hafiz, Kabir, Hildegard of Bingen, Julian of Norwich, Meister Eckhart, Francis of Assisi, Teresa of \u00c1vila, and Catherine of Siena. It was not a story of creed or dogma but of our innate divinity and how our intimate relationship with the universe transforms both it and ourselves.\n\nWhether or not we physically survive the collapse of the narrow story of industrial civilization, the larger story will carry us through, because no matter what happens to the body, we will know who we always have been and always will be, despite the paltry, pathetic identity that a consumerist culture has given us. Savor the story that connects you with the earth. Allow it to carry you through all that you are encountering and will encounter in turbulent times.\n\nTo be civilized is to hold oneself in opposition to nature, which is to hold oneself in opposition to oneself, to be ashamed of the animality of the self, which to the fully civilized means the \"filth\" of self. All of this destroys any possibility of communication or entering into communion with anyone but civilized humans. If we listen to the creatures and to the elements, and even to our bodies, we are then primitive, backwards. So we learn very early to put that away. We learn to despise ourselves and to feel ashamed of our bodies, to hate the dirt and to hate everything about us, because we're human, which means we're \"hummus\": They come from the same Latin root: earth and dirt. But self-loathing is a difficult thing to acknowledge\u2014maybe the most difficult\u2014so all those characteristics we must loathe if we are to be civilized, if we are to dominate, get dumped into others who bear the shame and who end up feeling dirty.\n\n\u2014Jane Caputi\n\nPart of the old story\u2014the one that is too small for us\u2014is that earth and dirt are the same, at least symbolically, if not literally. To be of the earth is to be of filth, and onto those peoples who live close to the earth\u2014who are not obsessed with cleanliness, whose bodies are often covered with mud or dust, who have never heard of \"hand sanitizer\" or antibacterial wipes, who pray to the earth and revere the earth, who consult the crows and the stars and the trees before making decisions, who eat insects and worms and raw fish from murky streams, who sacrifice animals to the powers they believe protect and bless their communities\u2014onto these people we project our shame. All of this is, naturally, symptomatic of our contempt for nature and, therefore, our own bodies.\n\nOur opposition to the earth drives us to project all of that which we associate with the earth\u2014filth, ignorance, savagery, and primal instincts\u2014onto native peoples, whose lives are intricately intertwined with nature. All animality that we have sent away and disowned as \"not me\" then becomes attributed to tribal peoples\u2014and, for white people, that predominantly means dark-skinned people.\n\nFor this reason\u2014because we live in a world of increasingly violent shadow projection\u2014the story of the universe is desperately needed. It honors the hummus of which we are and reconnects us with our animal bodies. With this new story, our self-loathing diminishes, and we gain permission to cherish our bodily functions. We acquire some capacity for communicating with other members of the earth community besides the civilized. If we are fortunate, we increasingly un-civilize ourselves and savor and celebrate our animality.\n\nThis death urge is partly a simple desire to die to this way of living that does not serve us well, but because we in this culture have forgotten that the spiritual exists, and have devalued the metaphorical, we do not understand that this death does not have to be physical, but could be transformative. Dying to one way of being so you can be reborn transformed is the oldest metaphor in the world, one the world is built on.\n\n\u2014Derrick Jensen, _Endgame_\n\nI rarely speak of the collapse of industrial civilization without mentioning something about transformation. I do this, not only because I want people to understand that collapse does not have to be a meaningless catastrophe, but also because we simply cannot have one without the other. Jensen's statement refers, once again, to the life\/death\/life cycle.\n\nWhen people dismiss or ignore the spiritual dimension of life, collapse becomes unbearable to contemplate because it feels purposeless. Without understanding the causes of collapse and also the opportunities inherent in it, why would we want to think about it? To do so feels morbid and nihilistic, and we end up feeling more victimized by industrial civilization than we already do.\n\nEven as we speak historically of the collapse of civilizations and as we consider the sociological, political, environmental, economic, and resource implications of collapse, we must keep in mind that these constitute the _literal_ aspects of it. We must not ignore those, but within themselves, they will never provide a comprehensive picture of this demise. In addition, we must include the metaphorical and mythological aspects of collapse in order to have a larger perspective.\n\nThe process of death, transformation, and rebirth is one of the oldest metaphors known to humankind. It manifests everywhere in nature, with perhaps the most stunning representation being the metamorphosis of the caterpillar to the butterfly. Collapse is a death that becomes a rebirth that becomes another death. Although we give much thought to the next culture and what humans will create from the rubble of collapse, that too will experience, at the very least, many little deaths which will lead to an even more refined creation, and that too will result in further deaths of what has been created. The life\/death\/life cycle is both inevitable and infinite.\n\nAll places and beings of the earth are sacred. . . . The land has not been desecrated; human beings desecrate only themselves.\n\n\u2014Leslie Marmon Silko\n\nFrom the indigenous perspective of Leslie Marmon Silko, born into the Laguna Pueblo tribe of the American Southwest, we cannot desecrate or desacralize the earth. What we can do is desecrate ourselves. Indeed, this is precisely what we have done in the past three hundred years, wherever the disease of civilization has spread.\n\nBut what does it mean to desecrate ourselves? First and foremost, it means that our bodies are disconnected from the substance of the earth, as well as from its wildness. Like captive animals in a zoo, our spirits are broken, and we become domesticated, pacing in our cages, developing physical and emotional illnesses. It means that, rather than living our lives in order to give back to and replenish the earth, we have become machines of consumption and waste. It means that we have allowed ourselves to become estranged from that part of us that makes us both human and divine, the soul. Thus, we desacralize ourselves by ignoring the divine within.\n\nOnce we reconnect with the sacred within ourselves, we will be reconnected with the sacred within the land. Any so-called spirituality that keeps us estranged from our bodies and the earth hardly deserves the name. In rejoining the sacred within ourselves and the land, we will be compelled to live with the land always in our consciousness, because our survival depends on it\u2014and because its survival depends on us. We recognize that we are the caretakers of one another. Even if we lose hard-fought battles to protect the land, the land will seek to protect us, and we can rest assured that it cannot be desecrated. Like us, it will always be sacred in its essence. Purification may be necessary at some point, and in most cases, land restoration is successful. We too must purify and restore ourselves from time to time, but the sacred essence of both ourselves and the earth is never lost.\n\nIn the long history of humankind (and animal kind too) those who learned to collaborate and improvise most effectively have prevailed.\n\n\u2014Attribued to Charles Darwin\n\nMost inhabitants of industrial civilization have PhDs in individualism but failing grades in collaboration. We tend to have great difficulties living communally and are often challenged even by the level of cooperation needed to form and maintain support groups that meet only weekly. In the United States, few people have ever experienced situations in which their physical survival depended on collaboration and cooperation, yet millions of people throughout the world currently depend or have depended on others for their physical survival.\n\nDarwin is speaking not only of living interdependently but also of living resiliently. Species who learn to adapt have the best odds of surviving turbulent conditions. Because of their animal nervous systems and their capacity for being fully present in the moment, nonhumans have an easier time adapting at a moment's notice. Humans, however, usually do not adapt quickly. We have a much more difficult time with change. This is why developing resilience and preparing now for adaptation is not only wise but necessary for humans.\n\nMy friend and colleague John Michael Greer has coined the phrase \"Collapse now, avoid the rush.\" Hilarious, yet deadly serious. We can prepare now for many of the changes that will come by living close to, if not directly off, the land; living simply by downsizing and consuming only what we need; and learning skills that will prepare us to survive when services such as auto travel, mechanical repair, health care, the internet, and food delivered from the other side of our continent or other countries is no longer available.\n\nOne skill we must learn and practice with each other is clear, compassionate communication, since in the future we may have to live in close quarters with others, perhaps even strangers. Likewise, we must become emotionally literate and develop intimacy with our inner world. In some situations, it will be wise to communicate our feelings; in other situations, it will be wiser to simply cooperate and carry on the business of mutual survival. Collaboration and improvisation do not guarantee survival, but without them, survival will become impossible.\n\nThe gross heathenism of civilization has generally destroyed nature, and poetry, and all that is spiritual.\n\n\u2014John Muir\n\nThe first word in this statement to which we must attend is _heathenism._ What does it mean to be \"heathen\"? In Old English, a heathen was a person who lived on the heaths or the edges of the civilized world. He lived in a nether region between civilization and the barbarians, neither part of the culture nor completely separate from it.\n\nBut when Muir speaks of heathenism, he is not referring to this kind of nether-world existence. _Heathen_ for him means lacking in spirituality or perhaps even being anti-spiritual. Implicit in Muir's use of _spiritual_ is a connection with nature. He also includes poetry, which implies heathens follow left-brain, linear thinking.\n\nFollowing Muir's definition, even people who are deeply religious can be very unspiritual if they have no connection with nature and, as a result, seek to conquer or destroy it. Some of the most famous politicians and corporate CEOs from around the world\u2014who are upstanding members of religious groups and contribute generously to them\u2014have been pivotal figures in the destruction of the earth. In their minds, there is little connection between nature and spirituality. The sacred is \"out there\" in the celestial realms, far from the soil and \"unsanitary\" conditions of nature. Their minds are unspoiled by emotion, intuition, and the irrational. For them, logic, progress, conquest, ownership, and profit are the essential signposts on the royal road to reality. Moreover, in their self-righteous arrogance, they might name anyone who does not share their worldview as \"heathen\" or \"barbaric.\" Yet this is precisely what Muir names these individuals.\n\nThe issue here is not \"us\" and \"them\"; it is a deeper examination of what is destroying nature. Muir implies that nature is being destroyed by a worldview, an attitude that eliminates the spiritual perspective and matters of the heart. It follows, then, that if we would restore nature, we must begin with \"all that is spiritual\"\u2014poetry, beauty, and above all, a profound, bone-marrow awareness of our intimate connection with nature.\n\nWe shall continue to have a worsening ecologic crisis until we reject the Christian axiom that nature has no reason for existence save to serve man.\n\n\u2014Lynn White Jr., _The Historical Roots of Our Ecologic Crisis_\n\nThe victory of Christianity over paganism was the greatest psychic revolution in the history of our culture. By destroying pagan animism, Christianity made it possible to exploit nature in a mood of indifference to the feelings of natural objects.\n\n\u2014Lynn White Jr., _Science_\n\nAuthor of _Medieval Technology and Social Change_ and _Medieval Religion and Technology,_ the historian Lynn White traces the demise of earth-based, pagan spirituality in the wake of Christianity's triumph during the Middle Ages. This victory on the part of Christianity became a watershed in human history and in our species' relationship with the earth. With the near death of paganism came the objectification of the earth and \"the greatest psychic revolution in the history of our culture.\"\n\nWhat happened to humans as a result of this psychic revolution? We were commanded by the Christian church to put away our reverence for nature, our instinctive sense that the divine abides within nature and that, by extension\u2014because we come from earth\u2014the divine abides within us. Indoctrinated in the notion that the sacred is far from them and must be obeyed and appeased by their piety and good works, medieval men and women, over the centuries, forsook intimacy with nature in favor of ecclesiastical authority. Not all inhabitants of the medieval world abandoned pagan religions, but most did. Persecution\u2014in the form of witch burning and the hunt for heretics\u2014made the endurance of earth-based spirituality nearly impossible. Moreover, the church's myriad forms of humanitarianism made life easier for some former pagans who no longer wished to struggle against the hierarchy.\n\nThe psychic revolution\u2014or, perhaps more accurately, this open psychic wound\u2014remains with us, and our destruction of nature is a major symptom of its painful presence in the soul of our species. It is never too late to attend to this psychic wound and discover abundant healing in your reconnection with the earth.\n\nOne of the penalties of an ecological education is that one lives alone in a world of wounds. Much of the damage inflicted on land is quite invisible to laymen. An ecologist must either harden his shell and make believe that the consequences of science are none of his business, or he must be the doctor who sees the marks of death in a community that believes itself well and does not want to be told otherwise.\n\n\u2014Aldo Leopold\n\nThe words of Aldo Leopold apply not only to those who are passionate about the state of the earth but also to those who know collapse well. We witness daily the \"marks of death\" not only on the land but on people and communities. Indeed, we live in a world of wounds, but we are not destined to live in that world alone. We must not carry this weight in isolation.\n\nWe speak much about where to locate or relocate and the ideal venues for navigating collapse. One of the most important considerations, I believe, must be our proximity to other people with whom we can share preparation, resources, and, very importantly, our feelings about what we are experiencing as we prepare.\n\nIn my book _Sacred Demise,_ I used the metaphor of being a hospice worker for the world. Leopold uses one of being a doctor witnessing the marks of death. Witnessing impending death drains the soul and breaks the heart. We should not have to do it alone.\n\nOne collapse author states that ignorance is bliss\u2014until it kills you. Often we may feel that it may be easier _not_ to know the state of our world, so that we might just be able to live more carefree lives. But the genie cannot be put back in the bottle; we cannot un-know what we know. If we are committed to being compassionate human beings, we cannot simply \"harden our shell\"; nor can we pretend that the science of collapse is none of our business. We are surrounded by dying people and institutions that do not want to be told the truth about their impending doom. Holding this knowledge within ourselves is an enormous and painful responsibility. With whom can you share this burden today?\n\nWe are healed from suffering only by experiencing it to the full.\n\n\u2014Marcel Proust\n\nWhen we witness suffering in others, and when we ourselves suffer, our first instinct is to stop the pain. Most people want to find a pill, a therapy, a guru, a new partner, a new child, a new residence, a new job\u2014something, anything\u2014to stop the pain. These are our first responses to suffering; asking deeper questions about it is usually our last, and in most cases, the proper questions about suffering are _never_ asked.\n\nSome have attempted to create clever slogans about pain and suffering, such as, \"Pain is inevitable, but suffering is optional,\" meaning that our attitude toward our pain determines its effect on us. I speak of suffering in a manner similar to the Buddhist perspective on it, which is that suffering, whether we call it \"pain\" or something else, is inherent in the human condition. I do not wish to promote suffering, but even if I did, that would be unnecessary, because life gives us so much of it. What I do advocate is an understand of suffering as our teacher\u2014a terribly unwanted opportunity to descend into the deeper Self and become a wiser, more seasoned, more whole person as a result of the suffering that our humanity compels us to encounter.\n\nMarcel Proust's statement above echoes the famous utterance by Rumi: \"The cure for the pain is in the pain.\" While this may seem like an insensitive, brutal statement, many have found enormous resonance with it. If we can allow our suffering to be our teacher, we may eventually reflect upon it as a gift. If possible, our work is to hold the suffering alongside every bit of comfort, nurturing, and support we can acquire from others and from ourselves.\n\nDuring times of suffering, it is important to be very gentle with ourselves and to allow ourselves to receive as much nurturing as possible. Suffering demands that we slow down, breathe deeply, eat well, sleep peacefully, and be very merciful toward the vulnerable creature that we are. In times of suffering, the heart and body are raw, sore, aching, and hypersensitive. What is called for is openness to being taught by our pain, as well as unwavering receptivity to the comfort required to sustain us through the journey.\n\nIf we follow how emotions actually work, we begin to realize that they come to us from elsewhere, the not-me enters our premises, and enchants us into their condition. Blake also called emotions \"divine influxes,\" suggesting that they are the way the Gods flow into the soul, moving it to a more-than-human condition of excitement and fury, of sorrow and mourning, of folly and ecstatic desire.\n\n\u2014James Hillman\n\nWhen I imagine life in a chaotic, collapsing world, I sense the challenge such a world will present for men and women who have lived most of their lives in accordance with logical, linear thinking. Unless they have developed emotional literacy and have learned how to manage their emotions, such individuals will be totally unprepared to cope with the emotions that industrial civilization's demise will evoke. Hillman names a few emotions that we all experience and that are certain to arise in collapse: excitement, fury, sorrow, mourning, folly, and ecstatic desire.\n\nMalidoma Som\u00e9 notes that, for his Dagara tribe in West Africa, emotions are sacred. Echoing Hillman, he suggests that they are conduits from the divine to the human. At the same time, the Dagara believe that emotions must be ritualized\u2014held within the container of the community, blessed, and allowed to flow within ritual space.\n\nHillman suggests that emotions do not originally arise in the ego but in what I have named the deeper Self. Certainly they can become distorted or expressed in ways that harm others or oneself, and some people can become addicted to feeling certain emotions. Depending on the wounding one has experienced throughout life, specific emotions can become addictive. The \"rage-aholic\" cannot live without feeling anger. A depressed person may become addicted to sadness. Another person may find himself feeling fear and paranoia more than any other emotion, and the bipolar person struggles with swings between euphoria and depression. Trauma can distort our experience of specific emotions and their potential usefulness to us.\n\nWhat is most essential to grasp is that emotions come to us from the sacred, and our job is to work with them\u2014to utilize them in order to move from fragmentation to wholeness, from paralysis to dynamism, from being overwhelmed by emotion to being empowered by it.\n\nLife has no meaning. Each of us has meaning and we bring it to life. It is a waste to be asking the question when you are the answer.\n\n\u2014Joseph Campbell\n\nIf there is meaning in life at all, then there must be meaning in suffering.\n\n\u2014Viktor Frankl\n\nThe meaning we make from the end of life as we have known it will be different for each one of us. What is most important is that we _do_ make meaning of this unprecedented threshold on which humanity stands.\n\nJoseph Campbell would insist that we bring to this transition our own meaning\u2014that the meaning we make from it will have everything to do with who we are. In fact, it's all about bringing to this experience the meaning of our own lives from our own individual perspective.\n\nAs with so many questions in life, each one of us is the answer. This is not to say that we should ignore the constraints of society and live as if we were islands. Rather, we must not expect meaning to come from outside us in any situation, whether it is the collapse of industrial civilization or the death of a family pet.\n\nViktor Frankl, who survived a Nazi concentration camp, devoted much of his life to exploring meaning in suffering. He wrote profusely of people in concentration camps who were able to survive, emotionally and spiritually, by finding meaning in their everyday experiences, no matter how mundane or how horrific.\n\nWe are the meaning that we bring to life, and often, life will present us with overwhelming suffering. Frequently, it feels senseless and absurd. Yet it is in our hands to make meaning of what life deals us and to trust that meaning _can_ be made, because we exist and because each one of us is the answer to the question, \"What does this mean?\"\n\nAm I my thoughts, the thoughts that I am thinking? No. Thoughts come and go; I am not my thoughts. Am I my body? They tell us that millions of cells in our body are changed or are renewed every minute, so that by the end of seven years we don't have a single living cell in our body that was there seven years before. Cells come and go. Cells arise and die. But \"I\" seems to persist. So am I my body? Evidently not.\n\n\u2014Anthony de Mello\n\nIt is phenomenally difficult for the modern human to grasp that she is not her thoughts and not her body. The mind is _part_ of who we are, as is the body. Yet neither body nor mind constitutes the _essence_ of who we are. However, the body and mind are wondrous manifestations of consciousness. To study the physiology and functioning of both is to be humbled by breathtaking, uncanny miracles of intelligence and order.\n\nThe mind is greatly influenced by the human ego and almost always serves its designs, the foremost of which is the ego's desire to be in control of the individual. Thus the mind is constantly racing ahead, living in the future and creating anxiety about it, or living in the past, generating regret or resentment about what is no more.\n\nThe body is the container we inhabit that allows us to move and function in the world. It speaks volumes about who we are, but _it_ is not who we are. In this culture, we pride ourselves in \"looking good,\" but whatever appearance we present to the world is only a dim reflection of our essence. Whether we \"look good\" or not so good, the body does not reveal who we actually are.\n\nNevertheless, both the body and mind give others a great deal of information about the consciousness that inhabits them.\n\nThe collapse of industrial civilization will be traumatic for those who are tenaciously attached to their persona\u2014how the body appears and how the mind performs. The ability to impress others with one's intellect or physique is likely to become meaningless very quickly. What will become valuable is one's ability to know and live from one's essence. Consciousness, not ego achievement, will carry the day in a collapsing, chaotic, and abysmally unconscious world.\n\nThe body is a sacred garment.\n\n\u2014Martha Graham\n\nDancing: The Highest Intelligence in the Freest Body\n\n\u2014Isadora Duncan\n\nWhile it is true, as Anthony de Mello tells us, that our authentic identity is neither the mind nor the body, it is equally true that the body is an extraordinary wonder. Artists have depicted it in countless forms throughout human history. Dancers such as Martha Graham and Isadora Duncan have allowed themselves to become astonishing physical expressions of the sacred\u2014let themselves \"be danced\" by something greater than mind or body.\n\nOur bodies and nervous systems are fundamentally animal in nature. Since we live in a culture that does not understand this, our bodies have been held in captivity since birth\u2014domesticated, defanged, declawed, and profoundly disempowered. Our wild nature has been subdued and our animal instincts purged\u2014or at least dramatically distanced from us. In such a milieu, we experience the infinite, incessant wounds of civilization and become traumatized accordingly.\n\nThe good news is that the human body and nervous system have astounding capacities for healing. Our trauma can be worked with and stabilized in the body in order to reclaim much of our original wholeness. The bad news is that most modern humans are storing phenomenal quantities of wounding in the body, which impairs their well-being and human relationships, and most are blissfully ignorant of this fact.\n\nOur preparation for living in a chaotic world must include the fine-tuning of our \"sacred garment\" through healthy nutrition, regular exercise, and the incorporation of bodywork and trauma healing. All of this grounds the body to the earth from whence it came and prepares the body for the \"dance\" of demise. While we are not our bodies, our bodies contain and serve the deeper Self\u2014the \"Highest Intelligence in the Freest Body.\"\n\nIn daily life we must see that it is not happiness that makes us grateful, but gratefulness that makes us happy.\n\n\u2014David Steindl-Rast\n\nOccasionally, when I speak of gratitude, someone will retort that I'm just trying to paint a silver lining on a train wreck or that I'm selling lemons whose sourness is sugar coated. Usually I ask that person why gratitude presents such a problem. The typical response is one of militant conviction that we must never cease trying to impede industrial civilization's reign\u2014and even that we must facilitate its demise\u2014because of the dire nature of our predicament.\n\nFor me, being grateful in no way allows me to relax and abandon the work that I am called to do. If one feels compelled to facilitate the demise of industrial civilization, one should continue on that path, but one should also give thanks for what beauty, compassion, love, creativity, and simplicity still remain among fellow earthlings.\n\nMost inhabitants of civilization believe that if they are happy, _then_ they can be thankful, and their definition of happiness is all about having stuff and status. David Steindl-Rast declares that the opposite is true. Gratefulness makes us happy, and happiness is not about accumulating possessions or power. It is the state of having an open heart and appreciating all of the good things in our lives that cost nothing and that have little to do with our efforts. Gratitude also implies trust; we become vulnerable enough to say \"thank you.\" And in doing so, we express courage.\n\nGratitude also implies and encourages relationship. To give thanks to the sun for providing life-force energy for the earth is to have a relationship with it. To give thanks to another person is to honor our connection with him or her. In his essay \"A New Reason for Gratitude,\" David Steindl-Rast states, \"You can feel either grateful or alienated, but never both at the same time. Gratefulness drives out alienation; there is not room for both in the same heart. When you are grateful you know that you belong to a network of give-and-take and you say 'yes' to that belonging.\"\n\nTo live in this world\n\nYou must be able\n\nTo do three things:\n\nTo love what is mortal;\n\nTo hold it against your bones\n\nKnowing your own life depends on it;\n\nAnd, when the time comes, to let it go,\n\nTo let it go.\n\n\u2014Mary Oliver, \"In Blackwater Woods\"\n\nWhat does it mean to let go? Many use the word _detachment_ as a synonym. Others add the word _caring_ in front of _detachment_ to underscore the paradox involved in letting go. We don't let go of people or things or circumstances that we do not care about.\n\nThese lines of Mary Oliver's poem depict another kind of paradox: the seeming contradiction of holding something close, because one's life depends on it\u2014and then, at last, releasing it and letting it go. I must confess that I do not know how to do this, and I never have done it without entering what I believe is the kind of necessary process that allows us to let go. We cling and clutch until we drive ourselves mad, and all the while, we tell ourselves that we need to let go. We read books, write affirmations, pray, meditate, attend support groups, and talk with friends, but still, we hang on. Then, if we are willing to hold the tension of opposites\u2014that is, be with the clinging and watch ourselves doing just that\u2014and if we open to the paradox and allow ourselves to be taught by it, often, sometimes ever so subtly, we find ourselves letting go. It may happen in an instant, or it may happen over days, weeks, months, or years.\n\nMany reading these words are struggling with situations and relationships where letting go is necessary but seemingly impossible. Letting go of a paradigm and a way of life feels daunting, if not impossible. We cannot _make_ ourselves let go, but we can pay attention. We can watch ourselves clinging, and we can ask for grace\u2014for some part of us to surrender to what we cannot control. It takes as long as it takes. It is nothing less than a spiritual practice. And, always, it is a matter of survival\u2014we realize that what we thought our life depended on turns out to be the opposite. Then we let go\u2014in order to live.\n\nIt is not about becoming spiritual beings nearly as much as about becoming human beings.\n\n\u2014Richard Rohr\n\nIn spiritual circles, we often hear that we are not human beings having a spiritual experience, but spiritual beings having a human experience. Whereas I used to agree with that slogan, I have come to believe quite the opposite. I no longer believe that my purpose on this planet is to become a spiritual being; my purpose is to become more profoundly human, and in the process of working to that end, I have accessed unimaginable depths of the sacred. Am I a spiritual being? I don't know, but what concerns me more is whether or not I am an authentic human being.\n\nDo I feel the fullness of my joy, grief, anger, confusion, compassion, empathy, sensuality? Am I kind to all members of the earth community and honest when I feel incapable of kindness toward those who abuse or neglect it? Do I establish and maintain healthy boundaries between myself and others? Do I know where I end and someone else begins? Do I understand the difference between my ego and my deeper Self, and do I endeavor to surrender to the deeper Self when the two are in conflict? I do not have perfect answers to these questions, but they are the standard by which I measure my humanity.\n\nIf collapse, transition, unraveling, a chaotic world, or the Great Turning will accomplish anything for our species, I suspect it will be to drag us, kicking and screaming, more deeply into our humanity than we have ever been. I anticipate that it will overturn every definition we have ever held of spirituality and, most importantly, of ourselves as human beings. Our sacred moments and epiphanies are less likely to come from sitting on mountaintops and more likely to erupt as we share food with other hungry people, sit all night with someone and help them die, rescue a lost and starving animal, comfort a terrified child or elderly person, plant and tend gardens with companions, and sing and play music with complete strangers.\n\nSuch experiences can only enlighten and deepen the soul, that part of us that bridges the sometimes seemingly impassable gap between the deeper Self and a messy, brutal, unforgiving, irrational human world.\n\nWisdom is sold in the desolate market where none come to buy.\n\n\u2014William Blake\n\nThe first thing I notice about Blake's poignant statement is that wisdom is \"sold.\" In other words, there is a price for it, which is why there are so few who seek it.\n\nWisdom is dramatically different from knowledge or the amassing of information. So many are educated; so few are wise. The ancient Greek word for wisdom is _sophia,_ and the mythology of Sophia is instructive, because Sophia embodies the feminine principle. She is not the bearer of knowledge or information but of wisdom\u2014the integration of knowledge, intuition, and the heart. To know Sophia is to become embodied, that is, consciously inhabiting the body and taking one's cues from it.\n\nThe price of wisdom is exorbitant, yet beyond value. In summary, it costs us everything. Wisdom is paid for with suffering; perseverance; an openness to being taught by what life presents; and the diminishment of the human ego, alongside the expansion of the deeper Self. This culture is obsessed with Logos, the word, but cares little about wisdom. According to Jung, Logos represents the masculine principle of rationality, in contrast with the feminine principle of Eros. Sophia and Eros are eternal companions. The supreme expression of Eros is not necessarily human sexuality. Rather, Eros penetrates the body and the heart with both sensuality and suffering and prepares the psyche for the advent of Sophia. As the individual allows himself to experience the delights and also the woes of Eros, Sophia\u2014wisdom\u2014matures within him.\n\nLittle wisdom is to be found within industrial civilization except among those who reject its fundamental paradigm\u2014those who have plumbed the depths of their own being, discovering what the industrial paradigm is designed to keep us from grasping about ourselves and the earth community.\n\nExperiences such as grief, longing, vulnerability, and delighting in nature, to name only a few, open us to Eros and the presence and power of Sophia in our lives. Sophia, wisdom, desires to teach us her most treasured secrets, and the price of that instruction is a radical tempering our ego identity.\n\nDance when you're broken open\n\nDance, if you've torn the bandage off.\n\nDance in the middle of the fighting.\n\nDance in your blood.\n\nDance, when you're perfectly free.\n\n\u2014Rumi\n\nFrom Rumi's perspective, there is no good reason not to dance, and every situation calls for it. One of the most important spiritual practices we can cultivate, in my opinion, is finding occasions for joy and celebration, even in the midst of horror. Our \"dancing\" may not look like dancing, but we symbolically dance when we find reasons for gratitude, celebration, feasting, and praising what is beautiful, kind, and heartfelt.\n\nDancing in the midst of tragedy is neither irreverent nor insensitive\u2014nor does it foster denial of the wrenching emotions of loss and trauma. Rather, it affirms that tragedy is but one aspect of the human experience, overwhelming and engulfing as it may feel.\n\nThe losses resulting from collapse\u2014the terror, rage, sorrow, and despair\u2014are certain to convince many human beings that their lives consist only of these. Our work is to bring slivers of light into the dark, dreary tunnels of loss. Many humans have already stopped believing that they can dance again. So many hearts have become cynical, calcified, and atrophied in reaction to the end of life as they have known it. Can we \"dance\" by smiling, singing, and softly touching another person with kindness? Our own hearts may be breaking, but bringing joy to scenarios of death and destruction is not only for the benefit of others. We heal ourselves as we dance, as well as those we dance with.\n\nDare to dance, even when if feels like it's the last thing your body can do. Dare to dance when the others call you crazy. In the midst of incalculable loss, dare to dance the other side of the human story\u2014the side that everyone has forgotten but so desperately longs to remember.\n\nThanksgiving\n\nWe return thanks to our mother, the earth,\n\nwhich sustains us.\n\nWe return thanks to the rivers and streams,\n\nwhich supply us with water.\n\nWe return thanks to all herbs,\n\nwhich furnish medicines for the cure of our diseases.\n\nWe return thanks to the moon and stars,\n\nwhich have given to us their light when the sun was gone.\n\nWe return thanks to the sun,\n\nthat has looked upon the earth with a beneficent eye.\n\nLastly, we return thanks to the Great Spirit,\n\nin Whom is embodied all goodness,\n\nand Who directs all things for the good of Her children.\n\n\u2014Iroquois Blessing\n\nIn the United States, the national holiday of Thanksgiving is celebrated during the last week of November. Legends regarding the origin of the holiday abound in American history, but the general consensus is that the native peoples of the American Northeast included European settlers in one or more of their celebrations of gratitude for the good things in their lives. For these indigenous folk, gratitude was as much a part of their lives as breathing air. They took nothing for granted and incessantly honored their debt to the Great Spirit and the earth community for their well-being.\n\nSadly, the European settlers never fully understood the perspective of gratitude that their indigenous neighbors held. They did not recognize their dependence on or interdependence with the earth. The notion of conquest could not be expunged from their minds. I often wonder what our nation and its history would have been like had these early settlers been able to open their hearts to native wisdom and earth intelligence, instead of being obsessed with owning the land and removing the people they deemed as obstacles to doing so.\n\nI invite the reader, whether American or not, to learn the above prayer \"by heart\"\u2014and, with the heart, speak it often. It is a proven remedy for obliviousness to the abundance in which we are ensconced and a powerful medicine for deepening gratitude for all that we take for granted.\n\nI am content to follow to its source\n\nEvery event in action or in thought;\n\nMeasure the lot; forgive myself the lot!\n\nWhen such as I cast out remorse\n\nSo great a sweetness flows into the breast\n\nWe must laugh and we must sing,\n\nWe are blest by everything,\n\nEverything we look upon is blest.\n\n\u2014William Butler Yeats, from \"A Dialogue of Self and Soul\"\n\nAs a result of his exploration of spirituality, later in his life, Yeats penned \"A Dialogue of Self and Soul\" depicting a poetic conversation between the materially based human ego and the soul. The above lines are the final ones in the poem and reveal the poet's ability to stand back and view the larger perspective of suffering and the tragedies of life. To follow an event to its source simply means the capacity to view any challenging event in the context of meaning and purpose. This includes forgiving oneself and casting out remorse. Yeats implies that he is easily given to remorse, bitterness, and regret, and because of that, when he is able to let go of those emotions, he is bathed in sweetness.\n\nHis earlier poem \"The Second Coming\" is particularly appropriate when confronting collapse, but these last lines of \"Dialogue\" disclose a Yeats who has gone beyond the sentiments of \"The Second Coming.\" At the end of that poem, he declares optimistically that the horrors of the world he sees around him represent something \"slouching toward Bethlehem to be born,\" but the last lines of \"Dialogue\" reveal a deepening inspiration: the conviction that we are blessed by everything, and everything we look upon is blessed. Therefore, we must laugh, and we must sing. May we draw sustenance from these words as we confront the losses and tragedies of collapse, adopting the perspective of Yeats that all events must be followed to their source.\n\nPerhaps all the dragons in our lives are princesses who are only waiting to see us act, just once, with beauty and courage. Perhaps everything that frightens us is, in its deepest essence, something helpless that wants our love.\n\n\u2014Rainer Maria Rilke\n\nEchoing the declaration of William Butler Yeats that everything we look upon is blessed and that we are blessed by everything, Yeats's contemporary, Rilke, asks us to imagine that the travails of life are opportunities presented to us in order to take action with beauty and courage.\n\nIn turbulent times, there is no shortage of opportunities to take action in this manner. Our work is not simply to act but to act courageously and in ways that express beauty. Moreover, Rilke suggests that everything that frightens us is actually helpless and wants our love. These statements do not call us to be na\u00efve or foolish in relation to all that is malevolent. Rather, we must see through and see into the forces of evil and view them for what they are. I am reminded of Jesus's advice that we should be wise as serpents and harmless as doves.\n\nSo often the darkness in our world feels insurmountable and impenetrable. We are so easily intimidated by it. Yes, we must be fully aware of its ability to crush us, but we must also be able to see how tenuous and vulnerable it is at its core. Only the cowardly need to oppress and pillage the earth community. We must not be foolhardy in confronting evil, nor should we roll over and allow it to remain unchallenged. Bringing love to an untenable situation does not mean unquestioning acceptance of it.\n\nThe principles inherent in Rilke's admonition are golden; however, we need extraordinary guidance from the deeper Self to discover _how_ to act with courage and beauty and how to bring love to what we most fear.\n\nWhat is it that most frightens you? How might it want your love? In quiet contemplation, ask for guidance to act with beauty and courage and to bring love to the dragons of our day.\n\nEarth laughs in flowers, to see her boastful boys\n\nEarth-proud, proud of the earth which is not theirs;\n\nWho steer the plough, but cannot steer their feet\n\nClear of the grave.\n\n. . . . . . . . . . . . . .\n\nWhen I heard the Earth-song,\n\nI was no longer brave;\n\nMy avarice cooled\n\nLike lust in the chill of the grave.\n\n\u2014Ralph Waldo Emerson\n\nHow arrogant we modern humans have become. How much of the earth we believe we have under our control, and how little of anything we have the capacity to control. These two sections of the poem \"Hamatreya,\" exude Emerson's humility and reverence for the earth. As an older, wiser man, he stands back and observes young men who, because they can steer the plow, believe that they have conquered the world.\n\nHumans have not yet conquered death, but we live as if we have. Hubris has directed our species' relationship with the earth, perhaps since the moment we abandoned the values of our hunter-gatherer ancestors. Earth laughs at us, says Emerson, but she laughs in flowers. She also sings, but few hear her song. When Emerson heard it, he was no longer brave, because\u2014somehow\u2014he was reminded of death. And here is the genius of Emerson, and of anyone who listens attentively to the earth.\n\nEarth reminds us of our finitude. When we passionately listen to it, our avarice is tempered by the reality of our death. Yes, she sings and laughs in flowers, but the entire earth community attests to the life\/death\/life cycle. To be out of touch with that cycle is to be profoundly disconnected from our mortality and intoxicated with hubris. To be intimately related to the earth is to deeply feel our mortality and to be imbued with humility and gratitude for the privilege of being part of the earth's song.\n\nTo be of the Earth is to know\n\nThe restlessness of being a seed . . .\n\nThe mystery of death\n\nAnd the miracle of birth.\n\n\u2014John Soos\n\nThe poet has captured within these few lines the entire life\/death\/life cycle and the ever-shifting currents of the human condition. To be human is to constantly experience change, so much of it unwanted. He reminds us of why we cannot describe these turbulent times as an ending but must perceive them as an end\/beginning.\n\nWhen we contemplate our human journey, we may feel like a tiny leaf floating on the surface of a river, being swirled along without volition or any capacity to guide our journey. From one perspective, this is totally accurate; yet from another, it isn't\u2014we are not pawns standing passively as our destiny plays itself out. We have agency, and we have will, but we always have less choice than the ego would like us to believe. Yet let us not miss the most important words: _mystery_ and _miracle._\n\nWe are everything Soos describes, but, most notably, we begin as miracle and end in mystery. While seemingly simple and straightforward, this poem is deceivingly profound. I invite you to sit with it through several readings and plumb its depths.\n\nIf we really knew how unhappy it was making this whole planet that we try to avoid pain and seek pleasure\u2014how that is making us so miserable and cutting us off from our basic goodness\u2014then we would practice as if our hair were on fire.\n\n\u2014Pema Ch\u00f6dr\u00f6n\n\nNothing ever goes away until it has taught us what we need to know.\n\n\u2014Pema Ch\u00f6dr\u00f6n\n\nThink of it: we are not serving ourselves or the planet by avoiding pain and seeking pleasure! Pain always comes to us as a teacher, but so few humans recognize the fact. We can never swim the depths of our basic goodness until we take the hand of pain and allow it to transport us there.\n\nWe are not unhealthy or unspiritual because we want to have our basic needs met. Who does not long for three nutritious meals a day and a comfortable place to lay one's head at night? Having meaningful work and a sense of community are priceless gifts in a chaotic world. Even at this stage of collapse, we see countless individuals and families who are homeless, living in cars, relying on food banks or dumpster diving for their meals, and for whom health care is a distant dream. For many of these humans, pain is a way of life, and they may never see an end to it.\n\nBut more often, it is those who are far more comfortable who insist on avoiding pain and seeking pleasure, because, from their perspective, they are entitled to do so. For these individuals, any talk of collapse is intolerable, and those bearing news of it are immediately marginalized or discredited. What they do not realize, paradoxically, is that their frantic attempt to achieve an eternal state of pleasure is actually making them miserable. Why? Because the deepest pleasures are experienced as a result of descending into the soul's depths. Warding off the natural, inexorable gravitational force of the soul results ultimately in misery, not merriment.\n\nCollapse, whether personal or collective, is a fact of the human condition, and its anguish will not subside until we have embarked on the journey to which it calls us. While the suffering of an unraveling world may never vanish, it may lead us to our basic goodness\u2014if we are willing to stop making pleasure a necessity, as opposed to a gift.\n\nOrdinary riches can be stolen; real riches cannot. In your soul are infinitely precious things that cannot be taken from you.\n\n\u2014Oscar Wilde\n\nIf you are consciously preparing for collapse, a valuable exercise might be an assessment of what cannot be taken away from you. We live in a time of enormous loss\u2014loss of ecosystems, loss of employment, loss of home, community, friends, retirement, health care, and above all, loss of meaning. Because so few modern humans are familiar with their precious internal resources, the loss of external resources may feel insufferable, and they resort to myriad forms of self-medication or even take their own lives. In fact, the losses of our turbulent times will either drive us to discover the untold riches within or risk an implosion as a result of outward stress pushing against what feels like an inner void.\n\nOur inner riches are not uniquely ours, and having them does not mean that we are special. Actually, as we assess our inner riches, it is crucial _not_ to compare ourselves with anyone. As soon as we do, we are likely to become lost in the external world yet again. Rather, I suggest, dear reader, that you allow yourself to sit quietly, patiently, and contemplatively with your gifts, your personal qualities, and the life experiences that have helped form who you are today.\n\nNotice things like compassion, courage, empowerment, the ability to listen, empathy, tenacity, patience, generosity, inspiration, joy, creativity, loyalty, discernment, humor, and a strong sense of boundaries. Reflect on other challenging situations in your life that may have felt untenable and impassable. What lessons did you learn? No one can take those away from you\u2014ever.\n\nTry finishing this sentence: \"When I was confronted with the situation of___________, I wouldn't have survived it if I hadn't been__________.\" What aspect of your character\u2014who you fundamentally are\u2014allowed you to survive? Whatever your economic situation is or will be, answers to questions such as these reveal nothing less than pure gold\u2014riches that cannot be stolen.\n\nLife now insists that we encounter groundlessness. Systems and ideas that seemed reliable and solid dissolve at an increasing rate. People who asked for our trust betray or abandon us. Strategies that worked suddenly don't. Groundlessness is a frightening place, at least at first, but as the old culture turns to mush, we would feel stronger if we stopped searching for ground, if we sought only to locate ourselves in the present and do our work from here.\n\n\u2014Margaret Wheatley, _The Place Beyond Fear and Hope_\n\n_Turbulent times_ and _groundlessness_ may be synonymous terms for those who contemplate and prepare for collapse. In earlier times, so much seemed solid, predictable, and unshakable. Those of us who grew up in the forties and fifties remember a life that felt much simpler and more reliable than it does today. For the most part, one's biological parents both lived at home, and both were at dinner every night. Most importantly, dinner _was_ served every night, and children could count on it. Meals were \"made from scratch,\" rather than thrown together by shoving cardboard containers into a microwave. Knowing one's neighbors was a given, and mothers often baked cookies at Christmas time for the milk man, the mail man, the school bus driver, and even the family auto mechanic. In those days, it seemed that we had ground to stand on, even as the world around us grew increasingly scary.\n\nThat kind of childhood is long gone. Not only does it feel as if the sands are shifting under our feet, but we see so much crumbling around us that we never dreamed would vanish. We search for people, places, activities, beliefs, and things we can hold onto. Yet Margaret Wheatley tells us that, again paradoxically, we would feel stronger if we stopped searching for ground. To this I would add that we must understand that one of the principal purposes of collapse is to compel us to find ourselves in the midst of the chaos and, as Wheatley admonishes us, to do our work from there.\n\nAxiomatic in the old paradigm is the notion that our grounding comes from outside ourselves. The new paradigm, ushered in by a very sacred demise\u2014a host of holy endings\u2014demands that we discover _who_ we are and _where_ we are in relation to the earth community. Such _is_ our work, and like stalks proliferating from the root, all other work issues from this.\n\nSo one has to live every day dying\u2014dying because you are then in contact with life.\n\n\u2014Jiddu Krishnamurti\n\nThis morning I read a story about a family who had fallen into severe hard times and decided to take their own lives rather than attempt to endure the myriad losses that had brutally overwhelmed them. As I reflect on this tragedy, I realize that such feelings of despair can engulf any of us if our losses become unbearable. Loss can very quickly become untenable if we do not live our lives dying.\n\nAllowing the \"little deaths\" of life to occur on a regular basis, with some acceptance, is the most efficient practice for confronting the end of our days on earth. Likewise, an awareness of the collapse of industrial civilization and the ability to acknowledge that it is well underway is the beginning of preparation for releasing all that will be lost in its wake. For those who lack information about collapse or persist in denying its reality, the losses brought about by collapse will be excruciating.\n\nPreparation for collapse can be merely an exercise in survival techniques, or it can be a spiritual practice in which we understand that even our stockpiles of necessities could be taken away from us\u2014in which case, what we are left with is the relationships we have cultivated with others and with our own deeper Self, as well as the skills we have mastered for navigating collapse.\n\nAs we come into contact with death while we are living, we more fully come in contact with life. The life\/death\/life cycle continues in the minutiae of our daily existence, and letting go becomes more of a way of life than a trauma. Or as Krishnamurti notes, \"When you are no longer afraid because every minute there is an ending and therefore a renewal, then you are open to the unknown.\"\n\nTo love life, to love it even\n\nWhen you have no stomach for it\n\nAnd everything you've held dear\n\nCrumbles like burnt paper in your hands. . . .\n\nWhen grief sits with you, its tropical heat\n\nThickening the air, heavy as water\n\nMore fit for gills than lungs;\n\nWhen grief weights you like your own flesh . . .\n\nThen you hold life like a face\n\nBetween your palms, a plain face,\n\nNo charming smile, no violet eyes,\n\nAnd you say, Yes, I will take you,\n\nI will love you again.\n\n\u2014Ellen Bass\n\nThese are very difficult times to love life, and millions of people cannot or will not be able to do so. Increasingly, life feels like a black hole of loss and grief, and on many days, we really have no stomach for it.\n\nPerhaps one of the most unforgiving aspects of loss is that it always demands that we choose whether we will endure or succumb to its seemingly insufferable impositions. No one should be judged for choosing not to persevere; as inhabitants of industrial civilization, we are not well schooled in the realities of loss and grief. Whereas our indigenous ancestors understood that life is replete with sorrow and loss, punctuated by moments of joy, celebration, laughter, and loving, we have been taught\u2014whether we recognize it or not\u2014to persist in the delusion of \"happily ever after.\"\n\nThe end of life as we have known it comes to us as that plain face between our palms and asks, \"Will you take me or not? Will you persevere even when you have no stomach for it? Are you willing to forge new life on this planet and step or drag yourself across the next evolutionary threshold?\"\n\nI sit in the streets with the homeless\n\nMy clothes stained with wine\n\nFrom the vineyards the saints tend.\n\nLight has painted all acts\n\nThe same color.\n\nSo I sit around and laugh all day\n\nWith my friends.\n\nAt night if I feel a divine loneliness\n\nI tear the doors off Love's mansion\n\nAnd wrestle God onto the floor.\n\nHe becomes so pleased with Hafiz\n\nAnd says,\n\n\"Our hearts should do this more.\"\n\n\u2014Hafiz\n\nThe lure of ecstatic poetry is its hyperbole and where its passionate embellishments lead us. When we indulge in the playful exaggeration of a Hafiz or a Rumi or a Kabir, we actually may not bristle at words like \"God\" or \"the Beloved,\" because their poetry transports us to places beyond nit-picking semantics and words like _atheist_ or _agnostic_ or _spiritual._ Rumi explicitly invited us to join him in one of those places: \"Out beyond ideas of wrong-doing and right-doing, there is a field. I'll meet you there. When the soul lies down in that grass, the world is too full to talk about. Ideas, language, even the phrase 'each other' doesn't make any sense.\"\n\nIn this poem, Hafiz again brings together the opposites\u2014the homeless and vineyards tended by saints, sitting around all day laughing with his friends, and a divine loneliness at night. But, he reminds us, \"Light has painted all acts the same color.\" The ecstatic poets want us to feel, even if for only a moment, the place inside ourselves that is beyond opposites\u2014beyond labels, beyond dogma, beyond judgments. A taste of that place evokes joy and laughter and an irrepressible zaniness that leaves us with bone-marrow awareness that death, darkness, and loss do not have the last word.\n\nWhen you are joyous, look deep into your heart and you shall find it is only that which has given you sorrow that is giving you joy.\n\n\u2014Khalil Gibran\n\nThe deeper that sorrow carves into your being, the more joy you can contain.\n\n\u2014Khalil Gibran\n\nAs with so many Eastern philosophers, poets, artists, and teachers, Gibran recognizes how inseparable joy and sorrow are. Collapse author James Howard Kunstler repeatedly speaks of the lack of a sense of the tragic in American culture. By _tragic_ he does not mean horrific but that the full spectrum of life includes both joy and sorrow. In Greek tragedies, the main character often committed some thoughtless act that had horrible consequences and set in motion circumstances that he could not control\u2014an apt description of what the paradigm of industrial civilization has wrought in the past three centuries.\n\nThe more averse to sorrow humans are, the more poignant their sorrow will be when they can no longer avoid confronting it. When we lack a sense of tragedy, we fail to appreciate what is tragic and what is not. For example, the pillage of the ecosystems is tragic, but severely altering our lifestyle so that we are not increasing carbon in the atmosphere is not tragic. For the individual who has no sense of the tragic, however, of primary importance is the kind of lifestyle that annihilates the ecosystems, and being deprived of it is tragic. One reason for this inversion of perception is an absence of sorrow or a minimizing of its value in our lives.\n\nJoy and sorrow are intimately connected in the human psyche; indeed, Gibran is astonishingly accurate in his assessment that \"only that which has given you sorrow is giving you joy.\" In another passage, he states that \"the selfsame well from which your laughter rises was oftentimes filled with your tears.\"\n\nModern humans are obsessed with happiness, but few know the depths of joy, because those depths can be plumbed only as a result of living through the dregs of sorrow and discovering that both joy and sorrow issue from the same well.\n\nAlways be on the lookout for the presence of wonder.\n\n\u2014E. B. White\n\nThe job of the artist is always to deepen the mystery.\n\n\u2014Francis Bacon\n\nWhat would our lives be like if, every day, we were on the lookout for the presence of wonder? If we are watching for wonder, we may become great artists, because the artist is one who has an eye or ear for the awe inspiring.\n\nYet in reality, we _all_ are artists, because we have the capacity to create beauty, to inspire awe, and to \"deepen the mystery.\" Art is really about drawing attention to the irresolvable mystery behind everything we see, hear, taste, smell, and touch. The artist, like the spiritual teacher, communicates that something inexplicable, vast, and enigmatic dwells within the painting, the poem, the composition, or the play. The most engaging works of art are those that entice the viewer or listener into the artist's creation. What is that \"something\" that insists that we purchase a particular album of music or own a certain painting? Why do we feel compelled to memorize a specific poem or, at the very least, post it on the refrigerator door?\n\nWhy are we riveted to a painting? Is it the colors, the light and shadow, the perspective, the theme? Why do certain pieces of music become part of us? Is it the melody, the harmony, the instruments, the soloist, the chords and key changes? Our fascination with a certain creative expression is all of these and none of these. And that, precisely, is the mystery of it. We are drawn to mystery like the moth to the flame, because the soul demands wonder, and that which allures us provides nutritionally dense sustenance for the soul. Thus, keeping on the lookout for the presence of wonder is not a frivolous luxury but a crucial component of our aliveness and\u2014yes\u2014even our survival.\n\nI would rather die of passion than of boredom.\n\n\u2014Vincent van Gogh\n\nFor all of the frantic busyness and hyperactivity of industrial civilization, few of its inhabitants are truly alive. Although we hear people say, sometimes on a daily basis, \"I'm really passionate about . . .\" one is hard pressed to find passionate people in this culture. Those who display passion are often deemed \"eccentric\" or \"egotistical\" or \"hyperactive.\" In one way or other, we tend to pathologize people who display too much spunk. The mental health industry often prescribes antidepressants or relaxants for people who are deemed \"too intense\" or \"overstimulated.\"\n\nThe only acceptable affect, according to the paradigm of industrial civilization, is the appearance of numbed happiness. This is not to say that those who appear happy actually are, but within the paradigm, the appearance of being happy is of paramount importance. Any display of anger is terrifying and could result in one being labeled a terrorist. Displays of sorrow must be brief\u2014lasting no longer than a year (or less)\u2014lest one be labeled \"depressed.\" Expressing fear may result in being called \"paranoid,\" and appearing \"too\" happy could result in a diagnosis of bipolar disorder. Thus, within the parameters of industrial civilization, authentically passionate people are an anomaly and are frequently marginalized.\n\nIs it any wonder that modern humans are terminally bored and so easily addicted to consumerism, are dependent on perpetual noise in order to feel secure, and are constantly seeking new forms of stimulation through extreme sports or reality TV?\n\nIn turbulent times, which tend to stultify the soul and body, our passion can nurture both. Passion revitalizes us and augments our aliveness, and the next culture created by humans must be populated with passionate people who cherish their vitality and are unafraid to express it, as opposed to the current milieu of bored, sated, anesthetized individuals, ensconced in cocoons of stupefying numbness.\n\nWe cannot live only for ourselves. A thousand fibers connect us with our fellow men; and among those fibers, as sympathetic threads, our actions run as causes, and they come back to us as effects.\n\n\u2014Herman Melville\n\nA proper community, we should remember also, is a commonwealth: a place, a resource, an economy. It answers the needs, practical as well as social and spiritual, of its members\u2014among them the need to need one another.\n\n\u2014Wendell Berry\n\nWho are your allies in the Great Turning? Who's turning with you? Who is your community? For those focused on self-sufficiency for the Long Emergency, it may be easier to focus on the word \"self\" than on the relationships we need to cultivate in order to survive.\n\nMelville suggests that building relationships is like planting seeds or establishing electrical networks through which energy can flow between you and others. These images are true for collapse; it will be impossible to navigate turbulent times alone or by minimizing our need for community.\n\nMuch work must be done to heal the wounds this culture has inflicted on human relationships. We do not know how to forge and maintain community. We need to learn how to be close, and we also need to learn how to create and maintain boundaries. We must skillfully include, as well as exclude, others in order to create functional communities.\n\nAs Berry notes, community can meet many of our needs\u2014physical, emotional, social, and spiritual. But in order for that to occur, we must be able to need each other. Once we have entered the territory of _need,_ we are no longer living in the realm of luxury, where we can choose not to need others and therefore select the degree of our investment in one another. Now is the time to plant the seeds of community and fertilize and water them well. Now is the time to construct \"electrical networks\" of need and support, the rewards of which will come back to us over time. We need community because we have needs and because others need us as well.\n\nThe human is the being in whom the Earth has become spiritually aware, has awakened into consciousness, has become self-aware and self-reflecting. In the human, the Earth begins to reflect on itself, its meaning, who it is, where it came from, where it's going. So in our deepest definition and its deepest subjectivity, humans are the Earth\u2014conscious.\n\n\u2014Miriam MacGillis\n\nWhen we witness the madness of our species, how easy it is to feel ashamed of being human and to write off our fellow earthlings as disgustingly obtuse and hopelessly irredeemable. Yet in the Great Story, as articulated by Thomas Berry, Brian Swimme, and Miriam MacGillis, we are reminded of our primary function as human beings: to be the universe reflecting on itself. The earth\u2014and the entire universe\u2014needs us, not only to be awake, but to become students of the earth's meaning, identity, origin, and destiny, in intimate relationship with our own. In fact, we cannot fully know who we are without knowing who the earth is. Nor can the earth\u2014a vibrant, living, organic being\u2014know itself but through us.\n\nAs we look around and witness the obliviousness of most humans to any kind of camaraderie with the earth, our hearts are broken, and ire consumes us. We have fallen in love with the earth, and it pains us that so many of our fellow earthlings barely recognize that the earth exists. Environmentalists say that if we could just educate enough people about the environment, they would awaken to its plight and radically alter humanity's destruction of it. Such an undertaking addresses only the mind, however, and fails to envelope the heart.\n\nChanging our behavior toward the earth is assured when we fall in love with it. Just as we would not neglect or abuse a beloved, it is not possible to violate the earth when we are in intimate relationship with it. Without the earth, we cannot become spiritually aware, and without us, the earth cannot know itself. Together, we and the earth come to consciousness as we recognize that we are each other.\n\nYour home is regarded as a model home, your life as a model life. But all this splendor, and you along with it . . . it's just as though it were built upon a shifting quagmire. A moment may come, a word can be spoken, and both you and all this splendor will collapse.\n\n\u2014Henrik Ibsen\n\nSo said the iconoclastic, incisive nineteenth-century Norwegian playwright. Here in the twenty-first century, his words could not be more apropos. The overwhelming majority of people in the nineteenth century did not want to hear such words any more than does most of humanity today.\n\nProfoundly disturbing are the words of anyone who illumines the fragility of our lives. How dare they! We pretend that a nagging sense of our vulnerability does not lie asleep, like a furtive animal, within the deep recesses of the soul. Increasingly, inhabitants of civilization are discovering that the splendor they have constructed, whether it be modest or opulent, has been built on shifting sands. It can vanish at a moment's notice. We see around us countless individuals who have lost everything as a result of layoffs, bankruptcy, extreme weather, terminal illness, or tragic injuries that created astronomical medical bills. Such personal collapses are previews of coming attractions\u2014a culture, a country, a world inundated by devastation and loss.\n\nThe only remedy for life built on shifting sands is life built on solid rock. Such a life will have a very different appearance, because it does not consist primarily of splendor, and the rock will not be immediately visible. It is a life that prizes the rock more than the splendor\u2014a life in which the rock endures, despite the disappearance of what is built on it. Everything on it and around it can collapse, but the rock remains.\n\nWhat, dear reader, is your rock? If you lose everything built on the rock, what will carry you through? What internal resources form the rock? What have you learned, felt, experienced, or created that makes you rock-like? What allies help make you rock-solid and robust? Yes, you will feel grief, fear, rage, despair, and more\u2014and you must\u2014but these are not the rock. These come and go, while the rock remains. What have you done and what are you doing to build your life on a rock?\n\nImagine a giant asteroid on a direct collision course with Earth. That is the equivalent of what we face now [with climate change], yet we dither.\n\n\u2014James Hansen\n\nFor decades, distinguished NASA scientist James Hansen has been studying climate change. I am riveted to the first sentence in the above statement, and I must ask, what would it be like if the earth _was_ hit by a giant asteroid?\n\nFirst, many people would die. In fact, such a catastrophe might constitute the ultimate extinction event for most or all life-forms on the planet.\n\nSecond, the chemistry of the earth would be profoundly altered. The planet's axis might be radically modified and poles shifted, making it uninhabitable.\n\nThird, those who miraculously survived the collision and axis modification would have to adjust to a world without industrial civilization, which would have been destroyed.\n\nYet similar effects are in store for the earth and its inhabitants in the wake of climate change: extinction, an uninhabitable planet, and the total elimination of life as we have known it. The principal difference between the effects of an asteroid colliding with the earth and rampant climate change is simply time. The effects of the former would be rapid; the effects of the latter, protracted over years, decades, and centuries.\n\nYet, as James Hansen says, we dither. We dither because industrial civilization carries within it the seeds of its own destruction. It has the loathsome capacity to infect its inhabitants with addiction to comfort, possessions, status, image-obsession, and virulent greed. The addict loses his capacity to imagine any other manner of living and is willing to commit whatever heinous crimes he deems necessary in order to perpetuate his lifestyle.\n\nThe collapse of industrial civilization and climate change will be the ultimate rehab experience. It may prove to be the only method by which the addicts of civilization become clean and sober. Our species may not be able to quit, but\u2014as people in recovery tell us\u2014it may be forced to surrender.\n\nWhere there are humans\n\nYou'll find flies,\n\nAnd Buddhas.\n\n\u2014Kobayashi Issa\n\nOur human existence has it all\u2014flies and Buddhas. Just when we have come to believe that our species has few redeeming qualities, just when we have come to believe that humans are terminally obtuse and insensitive, if we are attentive, something or someone comes along and proves us wrong. Perhaps a child looks in our eyes and tells us she loves us; or a customer service associate unexpectedly assists us with generosity and courtesy, expressing enormous kindness and compassion; or we read about an extraordinary act of magnanimity displayed by some unlikely, sympathetic soul. It's all there in this slapdash soup called _life._\n\nMost humans prefer being surrounded by Buddhas and never having to encounter flies. And we ourselves may not always display the Buddha's qualities, but we like to be treated the way the Buddha might treat us. When we navigate life expecting to encounter only Buddhas, we are certain to be sorely disappointed. Likewise, if we expect to encounter only flies, we are guaranteed to be miserable.\n\nIn recent years I've experimented with blessing complete strangers before I actually interact with them. I quietly bless them as I approach them or before I speak with them on the phone. I cannot attest to unequivocal success with my experiment, but, more often than not, it proves useful. It anchors me in a state of calm that allows me to be present and amiable, and it encourages me to expect that the person with whom I'm interacting will rise to the glory of his or her humanity and relate to me in a reciprocal fashion. Nevertheless, this does not guarantee the complete elimination of flies.\n\nAs we interact with troubled, worn-out, overwhelmed, and battered human beings in a collapsing world, we are going to encounter more flies than we can imagine, but we should also be prepared for those unforeseen Buddha moments that melt our hearts or transport us into some delicious, unexpected epiphany.\n\nIn the deeps are the violence and terror of which psychology has warned us. But if you ride these monsters deeper down, if you drop with them farther over the world's rim, you find what our sciences cannot locate or name, the substrate, the ocean or matrix or ether which buoys the rest, which gives goodness its power for good, and evil its power for evil, the unified field: our complex and inexplicable caring for each other, and for our life together here. This is given. It is not learned.\n\n\u2014Annie Dillard, _Teaching a Stone to Talk_\n\nIf collapse is here to teach us anything, it is the value and imperative of \"riding the monsters deeper down.\" Who knows what monsters will introduce themselves to us as \"tour guides\" for the journey downward? This will be unimaginably challenging for most humans in a culture that for centuries has been fixated on soaring, ascending, up-grading, and riding high. \"Reach for the stars,\" we tell ourselves and our children. Nevertheless, the direction of the Great Turning is downward and away from the \"upscale\" existence most of us live (in comparison with the majority of our fellow earthlings).\n\nI am fascinated with how much more frequently the word _descent_ is being used by people discussing collapse. Frankly, I am thrilled that this word is finding increasing acceptance in our vocabulary, because the soul loves descent\u2014in fact, it thrives on it. It wants to take us to that \"ocean or matrix or ether which buoys the rest,\" and for which there are many names.\n\nI choose \"sacred Self\" to signify that matrix. _Matrix_ is also another word for _uterus_ or _womb._ Indeed, the soul wishes not only to take us down, but to bring us up again, reborn, transformed\u2014a new embodiment, yes, even a new kind of human being.\n\nAnnie Dillard is describing the tragic sense of life, which is actually far from tragic. Rather, it is a perspective that allows us to rise and fall, to succeed and fail, to soar to the heavens and then to suddenly find ourselves in hell, yet to be open to being made and remade by the unified field within ourselves\u2014that field that science, the rational mind, and the human ego cannot locate or name.\n\nSilence is the strength of our interior life. Silence enters into the very core of our moral being, so that if we have no silence, we have no morality. Silence enters mysteriously into the composition of all the virtues, and silence preserves them from corruption.\n\n\u2014Thomas Merton, _No Man Is an Island_\n\nMuch has been stated in this book about silence. Obviously, this is no accident, because I believe that comfort with and skillful implementation of silence is a pivotal tool for navigating turbulent times. People who do not develop an interior life in a world unraveling are perilously at risk.\n\nIn a chaotic world, we will be forced to make challenging decisions, some of them moral dilemmas. These may be easier to make if we are committed to a practice that allows us to sit deeply in silence at least once each day, if not more often. When we sit in silence, with intention to access the deeper Self, we enter into the core of our being.\n\nThe extroverted mentality of industrial civilization incessantly seduces us outward with its cacophonous noise and vapid commotion. No wonder that modern humans have become so disconnected from what we now term their \"moral compass.\" When we contemplate the sea of corruption that permeates finance, politics, and a host of institutions in this culture, we easily deduce that the perpetrators of it are abysmally unfamiliar with silence. After all, can a human being who sits in deep silence, meditation, or contemplation each day habitually commit fraud?\n\nMerton says that \"in silence we learn to make distinctions.\" In silence we stop the chatter and the protective noise of our thoughts. When we contact silence, we contact our finitude; in fact, we contact moments of death. If we wish to become more comfortable with death, we must begin by becoming comfortable with silence. In silence, the ego is invited to recede, as we allow the deeper Self to be heard and felt. As a result, according to Merton, \"Silence . . . makes death our servant and even our friend.\"\n\nThink of nothing but the source of thought.\n\n\u2014Rumi\n\nThe Enlightenment, or the Age of Reason, which unfolded in the seventeenth and eighteenth centuries, was a response to earlier centuries of preoccupation with religion, ecclesiastical supremacy, superstition, and faith. It erupted some three centuries after the horrors of the Black Death, which claimed millions of European lives and was often absurdly attributed to witches, black cats, and God's judgment on the world. A number of thinkers, such as Spinoza, Kant, Voltaire, Locke, Newton, Descartes, and Rousseau, began countering the dominance of piety with reason.\n\nOne of the principal gifts of the Enlightenment was its insistence on thinking as opposed to believing. It gave us the scientific method and provided a breath of fresh air after centuries of stultifying religious orthodoxy. The downside of this social and intellectual movement was that it began assuming that all dilemmas could be solved through thinking. If humans could rely on their own intelligence, many Enlightenment thinkers said, then they could achieve limitless progress and conquer virtually all challenges. The Industrial Revolution of the nineteenth century was built on foundations laid by the Enlightenment. The discovery of oil and its myriad applications, as well as a plethora of new technological inventions, catapulted humanity further into the delusion of infinite growth and progress. Today, in the twenty-first century, we remain enamored with the power of reason and the capacity of technology to triumph over all that impedes the advancement of industrial civilization.\n\nThinking will always be a necessary function of being human, regardless of what empires rise or fall, but humans are now being forced to move beyond thinking as their crowning achievement. We must put aside the pursuit of information in favor of cultivating wisdom. The collapse of industrial civilization is eradicating the primacy of thinking while demonstrating that, rather than attempting to think our way out of the demise, we must return to the source of thought.\n\nTwo and a half thousand years ago in the West, we were given a gift\u2014and in our childishness we threw away the instructions for how to use it. We felt we knew what we were playing with. And, as a result, western civilization may soon be nothing but an experiment that failed.\n\n\u2014Peter Kingsley, _Reality_\n\nOurs has become a culture of convenience, but learning about ourselves is rather inconvenient because it turns the world we live in upside down.\n\n\u2014Peter Kingsley, _Reality_\n\nPeter Kingsley's research confirms for him that ancient Greek thinkers such as Parmenides and Empedocles were not simply brilliant philosophers but also mystical teachers. Their lives were dedicated to imparting the mystery teachings inherent in Greek mythology and conducting sacred initiation rites into them. According to Kingsley, Western civilization has chosen to ignore the spiritual mission of the likes of Parmenides and Empedocles in favor of naming such individuals \"great thinkers.\" The cultural descendants of those who taught ancient Greek mysteries obscured the intention and message of those teachers and \"threw away the instructions\" for how to use their gifts.\n\nWhether the mysticism of Greece or a host of indigenous traditions, modern humans have all but erased ancient wisdom and embraced materialism as their guiding principle. As we observe the state of humanity and our world in the twenty-first century, it feels as if Kingsley's description of Western civilization as a \"failed experiment\" is becoming increasingly apt.\n\nIn order to reclaim this wisdom, we must \"learn about ourselves,\" as Kingsley notes. This does not mean learning intellectually but experientially\u2014by descending into the mysteries within. This is inconvenient in an extroverted culture obsessed with progress. What we learn _is_ guaranteed to turn our world upside down, and I continue to argue that initiation into the inner mysteries of who we are is the principal function of collapse\u2014the overarching task that it has come to teach us.\n\nWhat industrial civilization has created is indeed a failed experiment, but the mystery at its roots is eternal and forever accessible.\n\nThe fullness of life is in the hazards of life.\n\n\u2014Edith Hamilton\n\nWe need not look far in our world to notice hosts of humanity who live shallow, vapid lives. Of course, fullness of life is in the eye of the beholder, but why else does a culture require ghastly obsessions with consumption, entertainment, and countless forms of overstimulation, if not because its citizens feel fundamentally empty? Popular wisdom dictates that fullness of life is found in the very activities that leave most people feeling empty. If pressed about this reality, they might reply, \"But what else is there?\"\n\nIndeed, what else _is_ there?\n\nEdith Hamilton was a student of Greek mythology and appreciated, as did Jung, Joseph Campbell, Mircea Eliade, Robert Bly, James Hillman, and Michael Meade, the paradoxical in mythology and in life. Greek mythology\u2014and, indeed, all deeper wisdom\u2014reminds us that we find fullness of life in consciously working with adversity rather than evading it. I'm certain that almost everyone reading this book has his or her own story of how this principle has revealed its unequivocal truth in life. By opening to and working with the job loss, the foreclosure, the terminal illness, the divorce, the disappearance of retirement funds, or the death of a loved one, one is able to say, from hindsight, \"Through these hazards, I have experienced the fullness of life.\"\n\nEdith Hamilton's short, discomfiting statement will be yet another nugget of wisdom to carry with us into the intensifying chaos that lies in our future. Yes, disturbing on the one hand, but reassuring on the other. Of course, one must be certain that one really _wants_ fullness of life\u2014as opposed to comfort, safety, and stability. Most of us tend to vacillate between those opposing desires all the time, but it is usually not until we have tasted the vacuousness of \"certainty\" that we appreciate the richness of confronting hazards, for therein, we ultimately discover, lies the fullness of life.\n\nSo you must not be frightened if a sadness rises up before you larger than any you have ever seen; if a restlessness, like light and cloud shadows, passes over your hands and over all you do. You must think that something is happening with you, that life has not forgotten you, that it holds you in its hand; it will not let you fall. Why do you want to shut out of your life any uneasiness, any miseries, or any depressions? For after all, you do not know what work these conditions are doing inside you.\n\n\u2014Rainer Maria Rilke, _Letters to a Young Poet_\n\nRilke spent many years grappling with sorrow. Today he might be diagnosed with clinical depression, but he was more than simply intimately acquainted with grief. Prudently and tenaciously, he mined the dark energies of sorrow and savored what it had come to teach him. In fact, one might say that no sorrow was ever wasted on Rilke.\n\nPerhaps more than any other person who wrote about dark places in the psyche and in life, Rilke befriended them as if they were long-lost acquaintances who had come to redeem him. Listen to the intimacy with them suggested in this quote. Not only do we feel that intimacy, but we also feel Rilke defending these visitations of sorrow as if he were defending an old friend who had left a suspicious package at his door. We can almost feel him pleading with us not to misinterpret grief's intention, because what it so lovingly brings us is intended to transform us into beings whose depth, vibrance, and compassion cannot even be imagined.\n\nFull of sympathy, he beseeches the reader, \"Yes, I know you believe that life has abandoned you. I know that you feel like a child dangling from a precipice, holding the hand of an adult but never knowing at what moment the adult may let go of her hand. I know how engulfed by sorrow you feel right now, but please, please believe me: sorrow has come to do miraculous work inside you, and for this reason, you dare not shut misery out of your life.\"\n\nThis kind of passionate plea for dialog with sorrow could only originate with someone who had befriended it incessantly and who had come to discover the ineffable treasures concealed within the dark shadows of loss. How many of us have come to cherish sorrow at such depths? How many can even glimpse the gold within our grief?\n\nAs for me, I know nothing else but miracles,\n\nWhether I walk the streets of Manhattan,\n\nOr dart my sight over the roofs of houses toward the sky,\n\nOr wade with naked feet along the beach just in the edge of the water,\n\nOr stand under the trees in the woods,\n\nOr talk by day with any one I love,\n\nOr sleep in bed at night with any one I love,\n\nOr watch honey bees busy around the hive of a summer forenoon . . .\n\nOr the wonderfulness of the sundown,\n\nOr of stars shining so quiet and bright,\n\nOr the exquisite delicate thin curve of the new moon in spring . . .\n\nWhat stranger miracles are there?\n\n\u2014Walt Whitman, from _Leaves of Grass_\n\nThe irrepressible poet Walt Whitman was wildly in love with life. In the ecstatic utterances of \"Song of Myself\"\u2014and throughout _Leaves of_ _Grass_ \u2014we hear his passion for being alive in a human body on this planet. At the same time, Whitman had the capacity to deeply appreciate and stand in awe of the tragic. He devoted many hours to visiting the wounded during the Civil War, and his writing on Lincoln's assassination captures the depth of sorrow he felt about the tragedy, as well as the anguish reverberating through the entire nation at that moment.\n\nIn his ecstatic moments, he perceived every aspect of life as a miracle. For Whitman, the word _miracle_ did not necessarily mean something \"beautiful\" or \"positive\"; it meant that life consisted of one wonder after the next. Decades later, Einstein would write, \"There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.\"\n\nPerhaps the hallmark of our aliveness is our willingness to view every life experience as miraculous\u2014birth, death, marriage, divorce, injury, illness, success, adventure, loss, gain, failure, triumph\u2014all of it inviting us to wonder at the presence and significance of every event in our lives. Everything and everyone who comes into our lives is a miracle, just as each of us is an extraordinary, stupendous miracle.\n\nThese are the only genuine ideas; the ideas of the shipwrecked. All the rest is rhetoric, posturing, farce.\n\n\u2014Jos\u00e9 Ortega y Gasset\n\nThe Spanish philosopher declared that we are in constant dialog with the circumstances of our lives, without which we cannot understand who we really are. From time to time we feel shipwrecked by life, but the only genuine revelation of who we are emerges from this condition. Another way of saying this might be: only the shipwrecked can save themselves from drowning.\n\nEven in the throes of the unraveling, which is well underway, most refuse to say what is so\u2014refuse to admit their shipwrecked condition. The old paradigm has been defeated, and while humanity is not yet extinct, we are like sailors marooned on a desert island. We will never be able to find ourselves or be found unless we admit our shipwrecked condition. We have been defeated on many levels.\n\nEconomic catastrophe increasingly engulfs us; climate change is making our planet uninhabitable, and energy depletion and humanity's frantic search for resources is exacerbating the severity of our shipwrecked status. The more we say it is not so, the more dire our situation becomes.\n\nThe mainstream media and every institution on earth is replete with \"rhetoric, posturing, and farce.\" Most of humankind is in a state of denial or is defensively attempting to ward off the demise. But I suspect that if you are reading this book, you have already acknowledged your shipwreck. I also imagine that having done so has brought enormous relief and release.\n\nLike members of twelve-step programs, who acknowledge their defeat, we have an opportunity to experience authentic recovery from industrial civilization\u2014if we can admit that the ship of state has sunk and that the best-laid plans of the human ego could not keep our own boat afloat. Paradoxically, speaking from a place of shipwreck is where we discover our empowerment and gain the grace and courage to sail into increasingly turbulent waters.\n\nGratitude as a discipline involves a conscious choice. I can choose to be grateful even when my emotions and feelings are still steeped in hurt and resentment. It is amazing how many occasions present themselves in which I can choose to be grateful when I am criticized, even when my heart still responds in bitterness. I can choose to speak about goodness and beauty, even when my inner eye still looks for someone to accuse or something to call ugly.\n\n\u2014Henri Nouwen\n\nIf you are having difficulty giving thanks, because you see little to be grateful for, it is time for a reality check. Even if you are suffering from a terminal illness and find yourself increasingly closer to death's door, gratitude is not only appropriate but perhaps even _more_ appropriate. You can give thanks for the life you have lived up to this point, for the relationships that have graced your life, for meaningful work, for descendants, for caretakers, for five physical senses, and for breath itself.\n\nWhenever we see little or nothing for which to be grateful, we are not seeing clearly. Our inability to experience gratitude may be the result of grief, bitterness, resentment, anger, or fear, but as Nouwen states, even when we are still looking for someone to blame for our situation, we can give thanks for countless other gifts. And if one really wants to stretch oneself, one can give thanks for the situation itself as a teaching moment\u2014yet another rite of passage that has arrived to carry one farther into the soul's depths. Expressing thanks to the deeper Self for the situation in which we find ourselves and thanking those who have supported us along the way is profoundly salutary.\n\nThe sensation of gratitude is implicitly healing, but it may be expanded by putting our gratitude into words. As we savor each gift we've received, we may be reminded of the people who gave them to us, and it will be healing for those people and for us if we express our gratitude directly to them.\n\nSome individuals find direct verbalizing of gratitude challenging at best and intimidating at worst. If one cannot speak one's gratitude, then one can at least write a card, email, or letter. Saying \"thank you\" acknowledges our interdependence and facilitates healing internally and in the world. May we always remember that giving thanks, as Nouwen notes, is not merely a nicety but a discipline, a conscious choice.\n\nThere will be in the next generation or so a pharmacological method of making people love their servitude and producing dictatorship without tears, so to speak, producing a kind of painless concentration camp for entire societies so that people will in fact have their liberties taken away from them but will rather enjoy it.\n\n\u2014Aldous Huxley\n\nIn college, more than forty-five years ago, Huxley's _Brave New World_ was one of the books assigned in a psychology course I took. I recall those chilling moments and the pictures formed in my mind as I absorbed the novel. Throughout the angst evoked while absorbing Huxley's words, I told myself that I would never live to see the kind of world the author described. A pharmacological generation would never develop in my lifetime\u2014or so I told myself.\n\nYet in just four decades, a brave new world of Huxley's soma-imbibers has arrived, both literally and symbolically; all around us are people who live in \"painless concentration camps.\" Alcohol and recreational drugs are only two forms of addiction in the current milieu. Virtually anything can become an addiction\u2014television, shopping, NASCAR racing, gambling, sex, extreme sports, food, power, religion\u2014the list is endless.\n\nIndustrial civilization has provided us with whatever type of soma we choose, and millions prefer their soma to even the slightest confrontation with reality or any form of emotional intimacy with other human beings. Industrial civilization has also become dramatically authoritarian in recent years. In the United States, ghastly executive orders and congressional legislation have shredded the nation's Constitution in favor of legal, political, military, and police jurisdiction. Such eviscerations of civil liberties could only occur in a society where citizens are uninformed, dumbed down, and\u2014as a result of their soma ingestions\u2014oblivious to the encroaching tyranny around them.\n\nFor decades, members of twelve-step programs have proven that withdrawal from one's soma, developing conscious contact with one's deeper Self, and making meaningful connections with others who are liberating themselves from addiction frees us from our painless concentration camps. Mindfully discovering our inner authority redefines our relationship with external authority and its myriad attempts to establish a \"dictatorship without tears.\"\n\nHow long will you think about this painful life?\n\nHow long will you think about this harmful world?\n\nThe only thing it can take from you is your body.\n\nDon't say all this rubbish and stop thinking.\n\n\u2014Rumi\n\nThere is a life-force within your soul, seek that life.\n\nThere is a gem in the mountain of your body, seek that mine.\n\nO traveler, if you are in search of That\n\nDon't look outside, look inside yourself and seek That.\n\n\u2014Rumi\n\nThis book contains many references to death. In fact, anyone who writes about the collapse of industrial civilization is implicitly referencing death. Collapse is the death of a way of life\u2014of systems, institutions, lifestyles, and an old paradigm\u2014but it will, by definition, involve the literal, physical deaths of millions, possibly billions, of human beings. The terror of death is precisely why so few are willing to look collapse in the face. Even among those who do confront collapse there can be a level of denial, if you don't admit that, despite your most elaborate efforts to prepare, you may not survive the demise of industrial civilization.\n\nYet all of the mystics wrote from a \"die before you die\" cloister within their consciousness. All were intimately acquainted with their own finitude, as well as with the life force within their souls that had no beginning and would know no end. In the above passages by Rumi\u2014and in countless others\u2014he pleads with us to ponder and cherish and make love with the divine within, reminding us that the only thing a harmful world can take from us is the body.\n\nSeek the life force within, seek the mine inside that holds the gem that you are at your core. No matter what happens in these turbulent times, Rumi's voice reverberates and begs us to seek that which can never die\u2014that force that compelled us to wake up to collapse in the first place and that compels us to prepare, regardless of our own personal outcome.\n\nAnd so, dear reader, I would plead with you, alongside Rumi, to linger and dwell consciously and intentionally with the Beloved within. How fragile and fleeting is _this_ life; seek instead the life you will not and cannot lose.\n\nBlind guides who strain out gnats and swallow camels!\n\n\u2014Jesus\n\nThis statement is from a diatribe Jesus is said to have spoken against the Pharisees, the religious leaders of Israel who were profoundly threatened by his teachings. He accuses the Pharisees of \"swallowing camels\" and \"choking on gnats.\" In other words, he is chiding their tolerance of corruption by the wealthy and powerful, pointing out that, at the same time, they were willing to quibble for hours about minor transgressions of Jewish law, especially if those were committed by the poor and downtrodden.\n\nHypocrisy is characteristic of a society in chaos\u2014whether the first-century society into which Jesus was born, or the milieu we inhabit in the twenty-first century. Justice is neither blind nor impartial. The crimes of the rich are often ignored as a result of their well-connected status; the crimes of the poor spark outrage, and unless they are very fortunate, they are doomed to a life of incarceration and poverty.\n\nOne of the hallmarks of industrial civilization's hypocrisy has been its preferential treatment of the so-called 1 percent, while demanding a larger burden of resources and accountability from the 99 percent. Prisons in the United States are overwhelmingly populated by ethnic minorities, while Wall Street fraudsters maintain inordinate success rates in avoiding prosecution for their appalling crimes. At most, they've had their hands slapped.\n\nThe so-called religious are often the most appallingly hypocritical. Today, as in Jesus's day, they frequently claim piety as a badge of honor and exceptionalism. In a society in chaos, the religious clamor for control and austerity, fomenting endless \"us\" and \"them\" conflicts. But individuals and a society can only avoid hypocrisy through deep introspection and a willingness to confront the dark recesses of our universally fallible humanity.\n\nThe heart that breaks open can contain the whole universe. Through our deepest and innermost responses to our world\u2014to hunger and torture and the threat of annihilation\u2014we touch that boundless heart.\n\n\u2014Joanna Macy\n\nThe Great Turning, the unraveling, the collapse of industrial civilization\u2014or whatever we prefer to name it\u2014is breaking and will continue to break our hearts. At times, we will need to protect our hearts\u2014or we will be overwhelmed\u2014but as much as possible, it will serve us well to allow our hearts to be broken. Each of us has an individual physical heart, but the deeper Self of every one of us is connected with one collective heart. This is the reason that we can feel each other's pain.\n\nWe have limited options for how we respond to the pain of collapse. We can armor our hearts, live in our heads, and focus only on survival and logistical preparation. We can become overwhelmed and spiral down into emotional breakdown. Or, we can prepare, emotionally and spiritually for the far more intense scenarios of the future and intentionally practice keeping our hearts open.\n\nIf you are facing a daunting experience, practice keeping your heart open. A terminal illness or near-death experience? Organize the support you need around you and practice keeping your heart open. The death of a loved one? The loss of a career? A house burned to the ground or foreclosed upon? A divorce? Moving to a new location, where you have few friends and feel disoriented? All of these situations are immensely stressful, and many of them, profoundly heartbreaking. The great healer and spiritual teacher Stephen Levine speaks of keeping one's heart open in hell. Why allow our hearts to be broken? So that we may experience deeper levels of love and compassion than we have ever imagined, and\u2014as Joanna Macy says\u2014so that our hearts can contain the entire universe.\n\nThere is a reparative function to ritual. It sutures the tears in the soul that occur in the daily rounds of living. . . . Ritual is able to hold the long-discarded shards of our stories and make them whole again.\n\n\u2014Francis Weller, _Entering the Healing Ground: Grief, Ritual and the Soul of the World_\n\nIn a collapsing, chaotic world, ritual must become a regular part of our lives. For those not familiar with ritual, it may feel foreign, related to religion or esoteric indigenous rites. In fact, it is in our genes; our ancient ancestors were immersed in ritual for a good part of their lives. They performed rituals for greeting the day, eating meals, ending the day, hunting for food, growing crops, gathering in the harvest, celebrating the birth of a child, initiating youth, entering into marriage, becoming an elder, and of course, celebrating the life of a loved one who had died. They did not conduct these rituals because they were bored or had nothing better to do. Rather, they understood that rituals provide blessing, offer consecration, acknowledge a significant threshold in a person's life, express gratitude, and, above all, facilitate healing. Over thousands of years, our ancestors discovered the power of ritual to restore wholeness to that which is broken, both in individuals and in the community.\n\nModern humans often feel at a loss when it comes to conducting rituals or participating in them. We grope and fumble for \"instructions,\" as if we could acquire a universal manual on ritual. However, most indigenous elders would tell us that ritual must erupt out of our bodies and out of the earth. It cannot be constructed logically; it must be intuited from the soul. All elements of the earth must be involved in some way. And, of course\u2014if possible\u2014we should confer with the community to receive its input. One person can certainly intuit what ritual might be appropriate on a particular occasion, but when several others from the community are consulted, the collective input is often wondrously ingenious.\n\nAsk the earth, the land, the elements, your community, your body, and your deeper Self for guidance regarding ritual, and allow ritual to hold the long-discarded shards of everyone's story and make them whole again.\n\nTen-thousand bigger-and-better shopping malls\n\nCannot sell you your self. This is why,\n\nNo matter how many bags you fill,\n\nSooner or later you feel empty.\n\n\u2014Daniel Skach-Mills, _The Tao of Now_\n\nAmid the plethora of crises confronting humans is perhaps the most pivotal of all crises: the crisis of emptiness. Were it not for our inability to grapple with emptiness, would we have pillaged the earth to the point of creating irreversible climate change and perhaps initiating the extinction of life on this planet? Were it not for our emptiness, would we have consumed more resources and more \"stuff\" than any other society in human history? Had we the capacity to confront our emptiness, would modern, so-called civilized humans have become the most warlike and imperialistic in the history of our species?\n\nHow many individuals do you know who can sit quietly with their emptiness for even five minutes? Yet that is precisely what is needed to transform their emptiness. Rather than attempting to fill it with stuff or activities or relationships, we sit silently in it and just simply notice\u2014yes, _notice._ Nothing to do with our emptiness. Nowhere to go with it; nothing and no one to fix it.\n\nThe paradox of inner emptiness is that it isn't empty at all. It is filled with nothing and, at the same time, filled with everything. If we are willing to pay attention to our emptiness over time, we discover that part of what occupies that space is profound longing. Rumi says, \"Longing is the core of mystery. Longing itself brings the cure. The only rule is, suffer the pain. Your desires must be disciplined and what you want to happen in time, sacrificed.\"\n\nYes, there is pain in emptiness, but there is oh so much more, if we can stay with the pain and allow it to transport us to the mystery. The pain of emptiness aches to deliver us to treasures that our material consumption prevents us from ever acquiring. Stay with the emptiness, witness it, and experience yourself being filled.\n\nOn the last day of the world I would want to plant a tree.\n\n\u2014W. S. Merwin\n\nWhat would you do on the last day of the world? Where would you like to be, and with whom? Merwin says that he wants to leave the world having planted a tree, having infused more life into the ubiquitousness of death.\n\nWhat \"trees\" are you planting now? What will be your legacy? Perhaps your trees will never come to fruition\u2014or perhaps they will flourish, but you will never know how or where. Are you willing to plant them anyway?\n\nBut planting those trees isn't entirely about the consequences of doing so for the rest of the world. To plant a tree on the last day of the world is to do so for ourselves as well as the world. It is a ritual to be done for oneself, affirming our highest intentions and our purpose in being here. _Ritual,_ which means \"to fit together,\" is a way of healing what has been broken, and planting a tree in the face of death is to affirm that we are eternal and whole. No matter what the outcome, we dig the hole in the earth and lovingly set into it the small, tender plant. We caress the earth as we replace the extracted soil and water the tree, knowing that we have deposited a part of ourselves in the ground, and from that moment, our roots begin to grow toward the center of the earth.\n\nAnd so I ask again: What \"trees\" are you planting now? What will be your legacy?\n\nIf I feel life moving through me rather than always initiated by me, then I want to give more reverence to it. I can appreciate that same kind of life flowing through others and want to protect them as well. . . . If I can imagine that I am transparent and translucent, that a greater will passes through me, and that I am hollow and vacant, filled with the riches of a life I don't own or direct or understand, then death is not so strange and immortality is already familiar.\n\n\u2014Thomas Moore, _The Soul's Religion_\n\nHow familiar are you with that \"something\" flowing through you? Have you tasted it, felt it, touched it? We can physically conceptualize the flow when we look at our biological children or ponder some creative work, conceived in our minds and forged with our hands. But what of the intangible that flows through us?\n\nIt is not difficult to feel life-force energy flowing through us as we stand on this planet, but it is more difficult to imagine that it could continue flowing when we are no longer alive and in a physical body. Yet our friend Einstein assures us that energy cannot be destroyed; it can only be converted to another form.\n\nWhat happens to the energy that flows through us in lovemaking or when we become enraged or sad or terrified or when we feel as if we are dissolving in hysterical laughter?\n\nAs you approach the conclusion of this book and the total dissolution of industrial civilization, ponder Thomas Moore's statements deeply. Ponder what is being and has been created through you and the riches of a life you do not own or direct or understand. Sit with these words for several days\u2014or longer. Contemplate the familiarity of death and immortality. Savor, in the words of D. H. Lawrence, that fine wind blowing through you\u2014\"that fine wind that is blowing the new direction of time.\"\n\nWhat must I give more death to today, in order to generate more life? What do I know should die, but am hesitant to allow or to allow to do so? What must die in me in order for me to love? What not-beauty do I fear? Of what use is the power of the not-beautiful to me today? What should die today? What should live? What life am I afraid to give birth to? If not now, when?\n\n\u2014Clarissa Pinkola Est\u00e9s, _Women Who Run with the Wolves_\n\nThese poignant questions by an astonishingly wise woman trouble us, as they should. Indeed, they are personal questions for everyone reading these words, but they are also questions for the collective body of industrial civilization, because this collapse is not just about the evolution of a new kind of individual but also about the emergence of a new quality of human society.\n\nWhat are we being called as individuals to release and let die? What are we being called, as a culture, to relinquish and allow to fall into the dustbin of history?\n\nA new culture will be created by individuals who have been so stripped, so pared, so honed, so flayed by the magnitude of civilization's demise that they will become dramatically different from what they have been in the past. Were many of their acquaintances or even close friends to return in the aftermath of collapse, they would find these transformed individuals unrecognizable. The toll this transition will take on the minds and bodies of its survivors is uncertain, but it will probably be somewhere in the region of incalculable.\n\nFor decades, many of us have longed for the transformation of our species. Perhaps we fantasized, along with Steven Spielberg, about how \"close encounters\" with alien life-forms might engender a new species of humans who would be ruled by love, peace, and equality. We may be acquainted with individuals who were convinced that human consciousness would be permanently shifted and transformed on December 21, 2012, and that a golden age of enlightenment would be ushered in on that day.\n\nWhat appears far more certain than the consciousness-raising power of alien encounters or the rebirth of humanity at the conclusion of the Mayan calendar is the complete dissolution of industrial civilization and the total transformation of its survivors. They will know, with a certainty that few humans have ever had, what to let die and what to give birth to in order to generate more life.\nConstant\n\nWe live for constants,\n\nrain in winter, the cat\n\ncurled like a furry comma\n\non the edge of the bed.\n\nSometimes, many times\n\nthese don't come, instead\n\nthere is drought, the father dies,\n\nthe mother grows old.\n\nThe constant is this:\n\nthe mind insists, persists in the insane\n\ncircle of creation from chaos.\n\nMake order of mystery.\n\n\"Listen to me,\" it shouts.\n\nSo we listen.\n\nConstant chatter, constant need\n\ngrowing like a curse.\n\nThe constant is this:\n\nlife is chaos, disintegration, blooming\n\nanew into order and collapsing\n\nagain to blossom into something more perfect,\n\nthen chaos, disintegration and on.\n\nWe watch helplessly, entranced\n\nlike the magician's audience,\n\nthe hypnotist's mark.\n\nNothing to do but join hands,\n\nbow heads, say blessings\n\nto the capricious, wild\n\noriginal god.\n\n\u2014Rebecca del Rio\nAbout the Author\n\nCarolyn Baker, PhD, is a former psychotherapist and professor of psychology and history. She is nationally renowned for her writing and workshops on emotional resilience in challenging times, as well as for life coaching for individuals and groups. Her books include _Navigating the Coming Chaos: A Handbook for Inner Transition_ and _Sacred Demise: Walking the Spiritual Path of Industrial Civilization's Collapse._ She lives and writes in Boulder, Colorado. Her website is Speaking Truth to Power, at www.carolynbaker.net.\n\nSacred Activism Series Titles\n\nWhen the joy of compassionate service is combined with the pragmatic drive to transform all existing economic, social, and political institutions, a radical divine force is born: Sacred Activism. The Sacred Activism Series, published by North Atlantic Books, presents leading voices that embody the tenets of Sacred Activism\u2014 compassion, service, and sacred consciousness\u2014 while addressing the crucial issues of our time and inspiring radical action.\n\n_Occupy Spirituality_\n\nAdam Bucko and Matthew Fox\n\n_Collapsing Consciously_\n\nCarolyn Baker\n\n_Earth Calling_\n\nEllen Gunter and Ted Carter\n\nApril, 2014\n\nThe Sacred Activism Series was cocreated by Andrew Harvey, visionary, spiritual teacher, and founder of the Institute for Sacred Activism, and Douglas Reil, associate publisher and managing director of North Atlantic Books. Harvey serves as the series editor and drives outreach efforts worldwide.\n\nFor more information about the Sacred Activism Series, go to: www.nabcommunities.com\/sacredactivism\n\nNorth Atlantic Books\n\nBerkeley, California\n\nPersonal, spiritual, and planetary transformation\n\nNorth Atlantic Books, a nonprofit publisher established in 1974, is dedicated to fostering community, education, and constructive dialogue. NABCommunities.com is a meeting place for an evergrowing membership of readers and authors to engage in the discussion of books and topics from North Atlantic's core publishing categories.\n\nNAB Communities offer interactive social networks in these genres:\n\nNOURISH: Raw Foods, Healthy Eating and Nutrition, All-Natural Recipes\n\nWELLNESS: Holistic Health, Bodywork, Healing Therapies\n\nWISDOM: New Consciousness, Spirituality, Self-Improvement\n\nCULTURE: Literary Arts, Social Sciences, Lifestyle\n\nBLUE SNAKE: Martial Arts History, Fighting Philosophy, Technique\n\nYour free membership gives you access to:\n\nAdvance notice about new titles and exclusive giveaways\n\nPodcasts, webinars, and events\n\nDiscussion forums\n\nPolls, quizzes, and more!\n\nGo to www.NABCommunities.com and join today.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nRalph Ubl is professor of art history at the University of Basel in Switzerland.\n\nThe University of Chicago Press, Chicago 60637\n\nThe University of Chicago Press, Ltd., London\n\nOriginally published as _Pr\u00e4historische Zukunft: Max Ernst und die Ungleichzeitigkeit des Bildes_\n\n\u00a9 2004 Wilhelm Fink Verlag GmbH & Co. KG, Paderborn \/ Germany\n\nEnglish translation \u00a9 2013 by The University of Chicago\n\nAll rights reserved. Published 2013.\n\nPrinted in China\n\n23 22 21 20 19 18 17 16 15 14 1 2 3 4 5\n\nISBN-13: 978-0-226-82372-0 (cloth)\n\nISBN-13: 978-0-226-02931-3 (e-book)\n\nThe translator would like to thank Stephanie Ezrol for her help with the French passages.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nUbl, Ralph.\n\n[Pr\u00e4historische Zukunft. English]\n\nPrehistoric future: Max Ernst and the return of painting between the wars \/ Ralph Ubl; translated by Elizabeth Tucker.\n\npages; cm\n\nIncludes bibliographical references and index.\n\nISBN 978-0-226-82372-0 (cloth: alkaline paper)\u2014ISBN 978-0-226-02931-3 (e-book)\n\n1. Ernst, Max, 1891\u20131976\u2014Criticism and interpretation. 2. Ernst, Max, 1891\u20131976\u2014Influence. 3. Painting, Modern\u201420th century\u2014History. 4. Dadaism. 5. Surrealism. I. Title.\n\nN6888.E7U2413 2014\n\n709.2\u2014dc23\n\n2013002936\n\n  This paper meets the requirements of ANSI\/NISO Z39.48-1992 (Permanence of Paper).\n**Prehistoric Future**\n\nMAX ERNST AND THE RETURN OF PAINTING BETWEEN THE WARS\n\n**RALPH UBL** Translated by Elizabeth Tucker\n\nTHE UNIVERSITY OF CHICAGO PRESS\n\nChicago and London\nCONTENTS\n\nPreface\n\nIntroduction\n\n1. From Dada to Surrealism: The Ghost Story of Mimesis\n\n2. _Natural History_ in Service to the Surrealist Revolution\n\n3. Max Ernst and Freud\n\n4. Prehistory and Modern History: The Return of the First World War\n\nExcursus: The Earth: A Formal History of a Theme\n\n5. Prehistory and Modern History: _Europe after the Rain_ , 1933\n\nAfterword: Walter Benjamin and Max Ernst\n\nNotes\n\nIndex\n\n_Plates_\nPREFACE\n\nI wrote this book in Vienna and Berlin between 1998 and 2002. It was first published in 2004. For the present translation, I left the main body of the German original unaltered except for some minor clarifications and corrections of factual errors. The introduction has been completely rewritten and a new afterword added. These two parts include references to scholarly work published after 2004 that is of particular importance to this study.\n\n. . .\n\nLooking back at the various stages of my work on Max Ernst, I am filled with gratitude for the support I enjoyed along the way. First of all, my thanks go to Friedrich Teja Bach. Without his dedication as adviser and his scholarly example, I would have never had the courage to commit myself to this project. From my Viennese friends, I have learned more than I can say. Ever since our student days, Wolfram Pichler, Barbara Wittmann, Karin Gludovatz, Markus Klammer, and Stefan Neuner have been vital sources of encouragement and inspiration. I would not have dared to come back to my first book if it had not been the occasion of a few stimulating exchanges over the last ten years. Particularly important in this regard has been the impression of conversations with Gottfried Boehm, Michael Fried, Mark Haxthausen, Josef Helfenstein, Inka M\u00fclder-Bach, Robert Pippin, Beate S\u00f6ntgen, Nicola Suthor, Juliane Vogel, Gerhard Wolf, and Christopher Wood. I also want to thank Andrei Pop, who, at the very last moment, made two important suggestions. Last but not least, I want to express my warmest gratitude to Elizabeth Tucker for her elegant and thoughtful translation.\nINTRODUCTION\n\nMax Ernst's pictures can be read as shrewd experiments in various techniques. They test the possibilities of collage, frottage, grattage, different methods of painting and printing, but also wordplay\u2014and the mutual imitation and intersection of these devices. Louis Aragon introduced this view as early as 1923, in his text \"Max Ernst, peintre des illusions\" (\"Max Ernst, Painter of Illusions\"), in which he demonstrated the novelty and uniqueness of the Dadaist's works through a comparison with cubist collage. According to Aragon, while the cubists pasted newsprint into their pictures because they wanted to emphasize the \"reality\" of the painting\u2014in other words, its tactile support\u2014in his collages Ernst employs not materials but images: \"printed drawings, drawings from advertisements, popular images, images from dictionaries and newspapers.\" These images are subject to an array of working processes. They are cut apart, retouched, and photographically reproduced; they are reassembled, traced, and copied in paint: \"Each of these tableaux attests to the discovery of an alternative technique.\" The goal, however, is not to call attention to the material aspects of these different procedures and make their operations into the theme of art, but rather to manipulate the found images in such a way that they gain a new meaning. Aragon describes Ernst as a virtuoso of artistic procedure who plays with the operations of the human mind: Ernst has created \"a type of intellectual collage.\" That same year, 1923, Aragon's friend and fellow soldier of Dada, Andr\u00e9 Breton, wrote a text on Ernst that presents an entirely different artist. This Ernst is possessed of a \"marvelous ability\" that causes the unity of space and time, memory and personal identity, to break apart. This ability makes him a poet who does not understand his own visions anymore\u2014as Breton implies, it makes him a poet whose relationship to his visions is no longer merely \"Platonic,\" but bodily. Film provides a similar experience by placing before the eyes distant, animate phantoms and speeding them up or slowing them down, enlarging or reducing their size. The comparison with the new medium allows Breton to conceive of the artist not as the master of his own techniques, but rather as himself a medium: the same explosive power courses through him as through the inspired poet, as well as through film.\n\nWhile Breton emphasizes the artist's receptivity, Aragon is interested in the artist's procedures. This contrast between passivity and activity can also be observed in Ernst's writings. By means of their form, they make it apparent that they were meticulously fabricated. Even the title of his autobiographical notes, presented to an American readership in 1942, mixes the factual and the legendary: \"Some Data on the Youth of M.E. as Told by Himself\" anticipates a reader who will detect the ironic disparity between the gathering of data and autobiographical narration. In the first, German version, the notes are titled \"Wahrheits- und L\u00fcgengewebe\" (\"Tissue of Truth, Tissue of Lies\"). In their content, however, this and other Ernst texts are devoted to advancing the thesis that artworks are symptoms of psychic processes that evade the artist's control. As an example, there is the frequently cited passage in which Ernst relates the origin of collage:\n\nOne rainy day in 1919, finding myself in a village on the Rhine, I was struck by the obsession which held under my gaze the pages of an illustrated catalogue showing objects designed for anthropologic, microscopic, psychologic, mineralogic, and paleontologic demonstration. There I found brought together elements of figuration so remote that the sheer absurdity of that collection provoked a sudden intensification of the visionary faculties in me and brought forth an illusive succession of contradictory images, double, triple and multiple images, piling up on each other with the persistence and rapidity which are peculiar to love memories and visions of half-sleep. . . . It was enough at that time to embellish these catalogue pages, in painting or drawing, and thereby in gently reproducing only that which saw itself in me, a color, a pencil mark, a landscape foreign to the represented objects, the desert, a tempest, a geological cross-section, a floor, a single straight line signifying the horizon . . . thus I obtained a faithful fixed image of my hallucination and transformed into revealing dramas my most secret desires\u2014from what had been before only some banal pages of advertising.\n\nArt has its origin in a passive eye that is both overwhelmed and stimulated by the heterogeneity of already-existing images, and the artist's work is restricted to the performance of a secondary service (\"gently reproducing only that which saw itself in me\"). Nevertheless, at this point, one is compelled to ask: In what way does this passage describe collage? In other places, Ernst makes it clear that he does not conceive of collage as the activity that is performed with scissors and glue: \"Ce n'est pas la colle qui fait le collage.\" (It isn't the glue that makes collage.) Rather, collage is the basic principle of all the different procedures he used in his Dadaist and surrealist works, inasmuch as they all lead to the same revolutionary goal. Frottage, grattage, various printing and reproduction processes, wordplay, and also collage with glue and scissors all aim for \"the magisterial eruption of the irrational in all domains of art, of poetry, of science, in the private life of individuals, in the public life of peoples.\" Collage in this expanded sense is not a technique for the artist's manipulation; it is a form of image production that takes possession of the artist's visual experience and is capable of affecting all people. However, even when Ernst asserts that collage is not a goal-oriented fabrication but an unconscious production that subverts human intentions, the way his writing is crafted simultaneously indicates a different artist\u2014one who with meticulous care, but also with tongue in cheek, has laid a \"tissue of lies and truth\" over his past and his poetics. Ernst declined to choose between the virtuosic use of artistic devices and the surrender to an unconscious production. His writings gloss over the difference between these alternatives by showing both possibilities, depending on whether one considers their content or their rhetorical form. For Ernst, collage is always both: it is an unconscious process and simultaneously an artistic procedure with a superior capacity to emphasize the moment of fabrication.\n\nArt history, however, has not shown any interest in this vacillation. The most important works on this topic can be characterized by their taking up one of the two attitudes suggested by Aragon and Breton in 1923. On the one hand, Ernst's artful handling of diverse techniques and heterogeneous image sources presented a rich field for discoveries. It was up to art historians to determine whether a work was a collage or an overpainting and to reconstruct the stages through which it had passed: how certain images had been cut apart and reassembled, how the result had been photographed or copied in oils, how the oil painting had then been used as an underlay for a frottage. The images that Ernst had used as raw materials were traced to sources ranging from a teaching aids catalog, which figures prominently in his Cologne works, to the illustrations and popular science books of the nineteenth century, which form the basis for many of the pictures in his collage novels. On the other hand, art historians persisted in developing new, and to an extent also increasingly refined, interpretations of Ernst's pictures as symptoms of an unconscious production. Here, Freud's and later Lacan's psychoanalysis offered the theoretical framework. One also recalls that Walter Benjamin, Sigfried Giedion, and Theodor W. Adorno saw in Ernst's collages a collision of the present era with the nineteenth century, which they diagnosed as the product of modern history: modernity represses its own past and transforms it into a foreign epoch that seems as far removed as prehistoric time.\n\nIn these two approaches to Ernst's work, one can see yet another example of the well-known (and highly unproductive) division of art history in general into a concentration on material pictures and their historical genesis, on the one hand, and a theoretically inspired criticism, on the other. Rather than pursuing either of these strategies at the other's expense, I would like to propose that this division in the research is the manifestation of a conflict already virulent in Dada and surrealism between two ideas of what the art of the avant-garde was based on: It was attributed either to a new poetics that could be described through particular procedures and their implementation or to a new state of openness to powers that fundamentally evade human attempts at fabrication. To take this perspective is to position the avant-garde and art history's view of it within the larger theoretical context of the modern aesthetic of originality. Nineteenth-century romanticism through modernism can be described as the history of various attempts to reconcile originality\u2014that is, a production that even the artist cannot repeat because its sources are not readily accessed\u2014with the routine of the studio. The fabrication of a painting was conceived as an unrepeatable process dependent on singular factors that reveal within the artist a deeper origin of art. These factors include momentary bodily or emotional states, presence in a particular location, particular light conditions, certain paints, brushes, or models, and, finally, new procedures in the application of paint, in composition, and in painting in general. Under these conditions, procedures not only served in the fabrication of a painting; they were intended to relate the process of fabrication to a production that the artist himself did not control. The term \"techniques of originality\" has therefore been suitably applied. The avant-garde took up this problematics and radicalized it by giving equally urgent emphasis to both technicity and originality, thus bringing them into opposing positions. The artwork is reduced to the result of various procedures; collage or the readymade exposes an artwork's status as sheer artifact. However, in the different avant-garde movements since futurism, higher or deeper powers of production\u2014life, the unconscious, language, capital, the body, and so on\u2014are summoned as the real authorities of art. It seems to me that for the art historian, it cannot be a question of aiming for one of these two poles of artistic production. Rather, the field of tension between them must be examined. In this way, an avant-gardist such as Ernst can be understood as an artist who was committed both to the reduction of art to its procedures as well as to its channeling of unavailable powers, and who conceived of this double commitment as a productive contradiction.\n\nBefore I turn to this project, I would like to do my part to clarify what the terms \"procedure\" and \"unconscious production\" can be taken to mean. It is important to note that \"procedure\" cannot automatically be equated with a technical innovation. Ernst was right to assert that collage does not come from glue. This and other procedures can only be analyzed in their respective specific contexts of application, which themselves are difficult to isolate. To begin with, the context of application encompasses more than the single work; it also involves the ideological, social, and historical implications that were associated with a procedure at a particular point in time. Thus, I will show that the context of frottage as Ernst used it in 1925 includes (among other factors) the theory of surrealist automatism, the idea that photography was an indexical medium, the history of educational illustrations, the love triangle between Max Ernst and Gala and Paul \u00c9luard, and, finally, the relationship of frottage to other procedures such as collage or overpainting. In abstract terms, artistic procedures are not merely available or newly developed techniques; they are the result as well as the driving force of a historical process that cannot very readily be isolated.\n\nWith the same historicizing approach, I would like to approximate what I previously referred to as \"unconscious production.\" It would surely make sense to view this unconscious production in light of a given theory, such as Freud's or Lacan's psychoanalysis, Breton's automatism, or Benjamin's thoughts on the nineteenth century as the unconscious prehistory of modernism. One could then ask to what extent Ernst's works correspond to these theoretical models\u2014whether they advance Breton's optimistic conception of the unconscious or else adopt Freud's darker theses, according to which the unconscious is characterized by an essential conservatism. This course of action would be based on the assumption that theories and artworks can be compared with one another on the same plane, as if it were an essential aspect of an artwork that it support or contradict a particular theory of the unconscious. I do not wish to dispute the value of these comparisons; I also consider it unproductive to extract Ernst's art from the mesh of theory in which it was created, and which has grown increasingly dense as the number of interpretations has increased. I would only seek to loosen this mesh a little in order to put its genesis into view\u2014in order to ask how it came about that Ernst's works became intimately connected with theories of the unconscious. However, the answer I am looking for cannot be found in a history of the influence of theory upon artistic practice. The fact that Ernst himself was a careful and enthusiastic reader of Freud, that unmistakable traces of Freud can be found in his work, that he was an equally attentive reader of surrealist poetry and surrealist mission statements, that his surrealist friends in turn used his works as prompts for their own writing exercises\u2014these and other moments of exchange between pictures, readings, and texts will be brought into account, but they do not form the starting point of my thoughts. Instead, I will begin with artistic practice, in exactly those places where the different procedures that Ernst employed generate effects of the unavailable\u2014where that which has been made changes into something that evades human fabrication and that implies another authority of production.\n\nBut what fuels this unconscious production that manifests itself in the use of artistic procedures? The answer I will present in this book is painting\u2014or, more precisely, painting's remains, inasmuch as this art is dismantled by the procedures of the avant-garde and thus transformed into something foreign. Thus, I will begin my study with the thesis that avant-garde procedures developed their own unconscious by destroying painting. What I referred to earlier (somewhat vaguely) as \"effects of unavailability\" can now be organized into a group of phenomena that all relate to the dismantled components of the painting. These include perspective, which gives each object a position in the represented space and guides the viewer's gaze; the ground upon which the represented figures stand and through which they achieve weight; the picture plane, which seals the painting behind a transparent and fictive boundary; the traditional rectangular format, which makes the picture's outer boundaries inconspicuous; facture, which gives each object its specific materiality; and contour, which distinguishes the objects from one another. Once these elements had been detached from the structure of the painting and thus also distorted, they could become the basis for resistance to the very procedures that had carried out this detachment and distortion. Ernst's art becomes an art of the unconscious\u2014an art that would be closely connected with the theories of Freud, Breton, and Benjamin\u2014because in it, painting, having come under attack by the avant-garde, continues to exert an influence as a repressed power.\n\nThis avenue of approach clearly differs from the attempt to fit Ernst into the canon of modernism, for example, as a precursor to Jackson Pollock. It does no credit to the author of _Beyond Painting_ to raise Ernst to the heights of the \"great painters,\" not only because his paintings cannot sustain such a comparison (and they cannot), but above all because to place Ernst in this company is to overlook the genuine strengths of his art, which as a rule become apparent in small format and through an engagement with literature or with writing as a physical or mechanical activity. Moreover, the paintings of the late 1920s and 1930s should be understood not so much as a return _to_ painting, but as the return _of_ painting\u2014as the ghostly emergence of an art that haunts its own ruins.\n\nThe relationship of painting to collage and the readymade can be understood in quite different ways, depending on which artists are being discussed, from which art-historical perspectives. Since Clement Greenberg's classic interpretation of the cubist _papiers coll\u00e9s_ , the strongest treatments of these works\u2014by Rosalind Krauss, Christine Poggi, and Yve-Alain Bois\u2014all agree in seeing cubist collage as a reflection on painting. But in contrast to Greenberg, who understands Picasso's and Braque's collages as a detour painting followed in order to get back to itself\u2014only by means of the new technique could the problems unique to painting be solved\u2014Krauss and Bois maintain that Picasso's _papiers coll\u00e9s_ form an independent and novel system of representation. This system is no longer based on resemblance, but instead on the use of discrete signs that only acquire meaning through their difference from one another. Meanwhile, Thierry de Duve examines another instance of the transformation of painting in his groundbreaking study on Duchamp's readymades. In these works, he sees an answer to the conflicts\u2014between egalitarianism and aesthetic distinction, the use of synthetic colors and the emphasis on craft\u2014that had beset painting since industrialization and the turn to democracy.\n\nDuchamp's answer was a radical abstraction that pertains to the concept of art or, more specifically, to the negation of the particular art of painting in the general ambition to produce art _tout court_. From then on, art was no longer concerned with the challenges of a particular art form such as painting or sculpture, but instead primarily sought to answer the question of which objects can be designated as art. In an overly simplified summary of his complex deliberations, it could be said that de Duve understands Duchamp's readymade as the sublation of painting\u2014though not in Hegel's sense, since it does not result in a higher form of the conceptualization of art, but in the concept of art being reduced to an empty label. In _The Optical Unconscious_ , Rosalind Krauss introduces another perspective on the historical relationship between painting and the readymade and collage. According to Krauss, it was modernist painting and the corresponding ideology of pure seeing that produced its own unconscious by excluding from art all procedures and apparatuses that could endanger the ideal of a spontaneous, self-referential, and self-sufficient seeing. Foremost among these were collage and the readymade: the former because it confronts seeing with a cut, and the latter because it refers seeing to images that are always already given.\n\nThese different attempts to define the relationship of painting to collage and the readymade provide some indication of the larger context in which this book can be understood. Ernst, however, did not lead painting back to itself through collage (as Greenberg thought Picasso and Braque had done); nor did he create an autonomous system of representation that detached itself from its origins in painting (as Bois and Krauss argue with regard to Picasso's _papiers coll\u00e9s_ ); nor, through the use of readymades, did he arrive at a generalization of the problem of art (as Duchamp did, according to de Duve). In his works made during the Dada years, painting is destroyed, or, rather, it is mortified and preserved as a lifeless depiction of itself. Ernst's surrealism begins with this fossilized painting and revives it as undead. Unlike Duchamp, his art takes a dialectic turn; however, this ghostly dialectic restores painting not on a higher level, but in an underworld in which painting assumes the power of an unconscious authority. And in an exact inversion of the thesis of Krauss's _Optical Unconscious_ , it is not collage and the readymade that constitute the unconscious of art; rather, it is painting, which collage and the readymade have repressed.\n\nHowever, it would be inaccurate to understand Ernst exclusively in the context of the Parisian avant-garde. The mortification and haunting return of painting that is enacted in Ernst's art can only be understood in view of the specific historical conditions of his artistic career. In prewar Germany, when Ernst was attempting to become an artist\u2014not yet the artist who would take a central position in the history of avant-gardism, but the young artist adept at expressionism, which was current at the time\u2014German art of the moment was dealing with the challenge posed by French cubism. I would like briefly to introduce two of its reactions. In an art-critical text Paul Klee wrote in 1912 in conjunction with an exhibition of the Swiss artist coalition Der Moderne Bund (The Modern Alliance), he included the following reflection on cubism:\n\nIn the area of landscape, Cubism has already achieved a connoisseurship, while in the area of the figure, so it seems, it cannot avoid making itself ridiculous. I mention this because I myself have found certain inconsistencies disturbing, but above all in order to explain the justification of its final step, the omission of the object. Pure landscape can in fact tolerate more\u2014and also less: when, for example, one makes changes to the proportions of the things present within it as compared with the dimensions fixed on the retina, first feeling them out and then rethinking so as to simplify them, the result remains a landscape. More rigorous organisms cannot be as tolerant of such re-evaluations. Animals and people, which are actually there in order to live, lose more of their capacity for life with each conversion. The situation is even worse when they must be integrated into a heterogeneous pictorial organism or\u2014as with Picasso\u2014having been cut into separate motifs, must be placed wherever the pictorial idea requires. Destruction for construction's sake? Indifference to the object and, at the same time, promotion of the object through its crass mistreatment?\n\nLet us summarize Klee's argument one more time. Cubism was only able to represent that to which human beings had already given form: objects and landscapes. Living creatures, however, are not artifacts but \"more rigorous organisms\" and thus are excluded from an art that through dissection\u2014wherever \"the pictorial idea requires\"\u2014places its own being as artifact at the center. In order not to fall prey to the pure technicity of painting, Klee recommends in the next paragraph the abstract painting of Robert Delaunay, which, instead of subjecting the life of animals and people to its own artistic devices, pursues a genuine \"plastic life.\" Hans Arp, who was represented in the exhibition of Der Moderne Bund, would follow in this direction and would trace the becoming of nature in blots and lines, as I will discuss in chapter 1.\n\nA second example of this reaction formation against cubism can be found in two texts written by Walter Benjamin in 1917, \"Painting and the Graphic Arts\" (\"Malerei und Graphik\") and \"Painting, or Signs and Marks\" (\"\u00dcber die Malerei oder Zeichen und Mal\"), both of which came out of a discussion with Gershom Scholem about the nature of cubism. These two sketches were first published in the 1970s in the framework of Benjamin's collected writings and were certainly not known to Ernst. However, because Benjamin would write a fascinating commentary on Ernst nine years later, his answer to cubism is of great interest to us. These short texts, the historical importance of which was first pointed out by Yve-Alain Bois, can be reduced to two theses: that painting differs from graphic works (in this context, primarily prints and drawings) on the grounds of orientation and the form of production. For example, a drawing or print is horizontal and emphasizes its being as artifact\u2014so clearly that each line is experienced as a placed line, and each between-space as paper. As soon as the lines suggest the horizon and the between-space suggests the sky or sea\u2014thus, as soon as they take on a representational function\u2014then, according to Benjamin, the nature of a graphic work is no longer respected. A graphic work is an artifact made from lines and their underlay. By contrast, painting is oriented vertically and as a stained surface is analogous to the human body, in that an area of paint is like a mark\u2014a birthmark, a skin disease, or a blush\u2014which has risen from the depths of the body to its surface. Benjamin's theses present an answer to cubism since they were written in reaction to Scholem's conception that cubism is a mathematical art that concedes only a decorative function to color. Against this distinction between the being of law and the semblance of material, Benjamin poses another distinction: namely, the distinction between artifact and bodily symptom, between images that have been fabricated and those that appear spontaneously. Pure technicity is the province of the graphic work; however, according to Benjamin, painting after cubism could return to a mimesis of those marks that manifest spontaneously on the human body.\n\nKlee's and Benjamin's answers to cubism, while specific to the artist and the author, are also indicative of a general pattern in the reception of contemporary painting in Germany in the pre- and postwar years. Cubism was welcomed as an assault on illusionism, but it also caused distress because it was thought of as the reduction of painting to sheer fabrication, since the hope that attended the surmounting of illusionism was specifically that semblance would be replaced by an originary mimesis. This was understood as a form of imitation that consists not in representation and thus distancing, but in the re-creation of the originary authority of production, which was located partly in nature, partly in life, and partly in the human body. The difficulty, so central for the modern aesthetic, of mediating between the artwork as artifact and the artwork as the realizing of an unconscious production had already been aggravated. Before collage and the readymade led to the fundamental transformation of the visual arts in Dada and surrealism, painting itself had raised the question of whether it could be related to life, nature, and the body\u2014or, in more general terms, to an unconscious production, and thus could be considered not entirely subject to its own devices. In Germany, an influential answer to the provocation of cubism was the turn to the idea of an originary mimesis.\n\nErnst's position in this history can only be understood, however, if one takes into account an interruption\u2014the First World War\u2014that was definitive and horrifying for a whole generation. Ernst came home from four years at war and soon afterward, as a Dadaist, began insisting on the impossibility of such a homecoming. For him, there was also no going back to the originary mimesis that had preoccupied Klee and Benjamin, as well as Ernst's friend Hans Arp, before and during the war. This ideal of a painting in which nature, life, or the body expressed itself directly instead became the polemic target of his Dadaism, but it continued to have a subliminal effect. In a manner of speaking, it entered into a fossil state and ultimately, in his surrealism, achieved a ghostly afterlife. In summary, painting formed the unconscious of Ernst's surrealism, albeit under particular historical conditions: after its reduction to technique in cubism; after its re-creation as originary mimesis in the works of Klee, Delaunay, Benjamin, and Arp; and after the destruction of this type of painting by Ernst's own Dadaism. Only after these transformations did painting enter the very position, in surrealism, that previously had been occupied by nature or life: it became the source of a spontaneous, unconscious production.\n\n. . .\n\nChapter 1 is devoted to this history of painting. From there, I will show how Max Ernst's art is connected to various theories of the unconscious\u2014surrealist automatism (chapter 2), Freud's psychoanalysis (chapter 3), and the idea that modern history is a process that can be understood as analogous to psychic trauma (chapter 4). I will be interested not only in the analogies between artistic procedures and unconscious processes, but also and to the same extent in the \"disanalogies,\" for example, between Freud's mystic writing pad and Ernst's overpaintings or frottages. In the final in-depth interpretation of a painting, _Europe after the Rain_ from 1933 (chapter 5), I will pursue the idea that, under the particular political circumstances of 1933, Ernst sought an alternative to painting's ghostly return, and that he found it in a cartographic painting. A look at Benjamin's reading of Ernst (afterword) will conclude the book. The purpose of this conclusion is not to uncover in Benjamin a key to the artist's works, but to elucidate once more how art, when taken as the starting point, puts theoretical reflections on art into a new perspective.\n1\n\nFROM DADA TO SURREALISM\n\nTHE GHOST STORY OF MIMESIS\n\nWHERE PAINTING WAS: BR\u00dcHL, 1909\n\nOn a summer day in Br\u00fchl, during what he would remember two world wars later as his carefree and high-spirited youth, Max Ernst posed at his easel (fig. 1), which he had set up in a tree-lined lane that can be seen in the painting and the photograph's background. The way the young Ernst holds his neck awkwardly forward, away from his stiff collar\u2014as though he were not outside in nature, but instead were being made to stick his head through the hole in a carnival cutout\u2014brings to mind the Dadaist who would title one of his best-known collages _The Hat Makes the Man_ ( _C'est le chapeau qui fait l'homme_ ). By the spring of 1910, Ernst would parody the photo in the magazine of his graduating class, _Life at Our School_ ( _Aus dem Leben an der Penne_ ), by drawing himself in a similar pose, though as a bust, holding a gigantic palette. The caption cites Wilhelm Busch: \"A young man with a hopeful heart takes quickly to the painter's art.\" In a school magazine like this one, parody is surely an affirmative practice, since it is precisely the students' most cherished aspirations that are singled out for teasing. In the photograph from 1909, Ernst had in fact adopted the artist's pose suitable for a young painter _en plein air_. The pose emphasizes his distinctive artist's gaze and thus the mode of painting that the picture on the easel exemplifies: an impressionism already past its prime. As is well known, impressionism's fundamental tenets assert the mutuality of hand and eye. The individual act of seeing corresponds to the equally individual act of painting; artistic authenticity originates in the density, darkness, and idiosyncrasy of the body, which are expressed equally in the artist's hand and eye; and, finally, visual stimulus and tactile brushstroke\u2014the trace of the world and the trace of art\u2014become coextensive on the surface of the painting.\n\nConsequently, a painting like the one resting on the young Ernst's easel was more than a successful illusion of fleeting nature. It could also be read as an imitation of the process of perception, equivalent to the traces that nature leaves on the retina. Thus, it was a \"mimesis\" of nature that drew on perception theory to make current the ancient double meaning of the term: here \"mimesis\" means both the illusionistic simulation of phenomena as well as the active re-creation of forces operating within them\u2014it relates to both the products and the processes of nature. Modernism's art theory famously brings this double meaning into play whenever depiction or illusion is dismissed as extrinsic, while performative imitation is reclaimed for modern art as a deeper, more intrinsic relationship to nature. Within the modernist understanding of history, impressionist painting marks the last possible mediation. It depicts the phenomenal world as a fleeting apparition and simultaneously imitates it, inasmuch as \"imitating nature\" can also be understood as the subject's reflexive activity in aspiring to imitate the process by which nature is visually perceived.\n\nFIGURE 1\n\nMax Ernst as a painter, ca. 1909.\n\nSince this double meaning of mimesis plays an important role in what follows, I propose making a simple terminological distinction: I will use \"illusionism\" to refer to mimesis (or imitation in the broader sense) that takes phenomenal nature for its object; whereas I will use \"imitation\" in the narrower sense to mean a type of mimesis that aims for the active re-creation of a nature that, however it is understood, precedes phenomena. When mimesis is no longer taken seriously and is repeated with mocking unbelief, I call this parody \"mimetism (of the second order),\" following G\u00e9rard Genette. This parody targets illusionism as well as imitation, most often both.\n\nThe theorem of the living unity of illusion and imitation of nature in impressionist painting can be found in the photograph's constellation of motifs in a threefold transfer between eye and hand: first, from the seen (and photographed) section of nature to the painted one; second, from the palette to the canvas; and third, from the artist's gaze to the artist's hand. The isomorphism between the trace of light on the retina and the trace of paint on the canvas is supposed to attest to each of these three turns. But it is not necessary to relate the motifs to one another in the organic movement of a spiral. The photograph can just as well be dissected with a gaze trained in avant-garde painting critique\u2014with the gaze of Dadamax, who was responsible for the collage _The Hat Makes the Man_. Then we don't see a spiral movement anymore, but isolated elements engaged in a play of doubling\u2014photograph and oil painting, oil painting and palette, eye and hand. When the pose becomes rigid, when nature turns into a backdrop and the shirt collar stiffens, then the mediation between illusion and imitation, between visual perception and its re-creation by the artist's hand, also ceases to function.\n\nThe art-historical caesura between the impressionist unity of mimesis and its Dadaist disintegration into a play of doubling is described in what follows as a multiply delayed occurrence. It happens not as a smooth cut, but in many separate ruptures, followed by seismic shifts, which make visible how mimesis was stratified and how its return was prepared. In this phenomenon of the return by which the mimetic is brought back to the art of the avant-garde, three moments can be distinguished: First, the two aspects of mimesis\u2014illusion and imitation\u2014are exposed to various attacks, for though illusionism had already been discredited by prewar modern art, imitation of a deeper force (such as nature or life) was still a strong ideal and, therefore, a valid object of Dadaist critique. Second, the processes of anti-mimesis are combined with mimetic residue. And third, the 1920s bring the return to a mimesis that is eerily alive. While Ernst's Dadaist works preserve a mimetic residue in an essentially mortified state, his surrealism posits mortification in order to discover in it a ghostly afterlife\u2014a simulacrum of the simultaneously illusory and imitative mimesis whose living unity the young Rhineland painter had hoped to discover on his excursions into the countryside. The following attempt to describe the history of Ernst's artistic practice along the thread of its mimetic play can therefore be summarized as follows: Where painting was\u2014painting as a sensory plenitude of the visual and tactile, simultaneously imitation and illusion\u2014Dada and surrealism would come to be: Dada as painting's mortification and surrealism as its haunting return.\n\nMAX ERNST AND HANS ARP; OR, BREAKING DOWN THE BLOT\n\nLet's take the photograph further. Imagine that, sometime between 1919 and 1921, while working together on _Bulletin D_ , _Die Schammade_ , or the FATAGAGA pictures, Max Ernst and Hans Arp came across the prewar photograph, taken three years before the friends first met, and ten years before they met again under the auspices of Dada. The play of doubling that separates the oil painting from its subject and the posing artist from surrounding nature would probably have seemed to Arp like an allegory of the kind of vanity that, according to him, was the characteristic fault of illusionistic art. In those same prewar years, he himself had begun the search for a new form of nature study, which he intended to be conceived as a radical turning away from visual illusion and a regaining of an originary re-creation of nature: \"I gave myself to the study of nature, lay motionless on a table for unutterably long periods, and attempted to dream like a mountain, slowly, deeply and endlessly slowly.\" From Arp's self-experiment, it becomes quite clear that mimesis as imitation aims to produce a structural analogy to what is being imitated: the analogy between the unutterably long, motionless repose of the mountain and the sleep of the artist. Second, imitation achieves this analogy less in the product than in the process. Imitation is performative\u2014the artist plays the mountain _in actu_ , in the rise and fall of his breath, in the coming into being and passing away of his dreams. There is also a third related characteristic of imitation as Arp understands it: imitation and organicity condition each other. In the case of the mountain, this means that imitation results in bringing what is imitated to life. The structural analogy between nature and art, the performative enactment of this analogy in artwork, and the resultant bringing to life are the three aspects encompassed in Arp's dream of the ideal imitation of nature.\n\nIn Arp's ink drawings, made in the Dada years from 1916 to 1921, the illusion of nature dissolves into dark blots and light intervening spaces. Indeterminate suggestions of the figural\u2014possible proto-forms of future mountains, forests, or mythic creatures\u2014are encased like decomposing fossils (fig. 2). The amorphous facture\u2014which is not like handwriting but instead appears distinctly anonymous, accidental, and thus quasi-natural\u2014invites the viewer to an imaginative seeing of resemblances. This type of seeing does not aim to contour stable figures, so much as to suffuse the dark blots and white spaces with life, to induce in them the germination of possible images.\n\nArp dreamed of a primordial mimesis that does not yet know the play of doubling of photograph and oil painting, oil painting and palette, hand and eye. He wished for a nature that neither makes itself available nor allows for reinvention, since it is always present and effective as an originary totality. The one who imitates this nature by dreaming with the mountain and seeing with the accidental blot trusts in a maternal whole that expresses itself even in dead forms. An allegory of this desire for a universal suffusion with life can be read in the story that Arp told in which he discovered the biomorphic forms that would be the basis for his ink drawings. The setting is the shore of Lago Maggiore, at the foot of Monte Verit\u00e0. \"In Ascona, with ink and brush I drew broken branches, roots, grasses, and stones, which the lake had washed onto the shore. I simplified these forms and unified their essence in animate ovals, emblems of the eternal transformation and becoming of the body.\" Specifically, \"broken branches, roots, grasses, and stones\" become models for a drawing activity that aims to create \"emblems of eternal transformation.\" Where the waves wash inanimate or lifeless things onto shore, Arp finds elements for a primordial generation of the organic from nature's broken and discarded remains\u2014and this also means a primordial generation of animate forms from dried ink and paper.\n\nFIGURE 2\n\nHans Arp, _Untitled_ , 1917. 18 \u00d7 22 cm. Kupferstichkabinett, \u00d6ffentliche Kunstsammlung, Basel. \u00a9 2012 Artists Rights Society (ARS), New York \/ VG Bild-Kunst, Bonn.\n\nErnst admired the Dada veteran who in 1914, instead of lining up with the so-called \"August volunteers,\" fled on the last train to exile in Switzerland. However, unlike Arp, Ernst did not want to escape the play of doubling. After four long years in the field artillery, where his job was to read landscapes with diagrams in order to give coordinates to the gunner, he was no longer able to believe in a natural, originary art. A portrait of a friendship, homage to Arp's dream and at the same time the destructive analysis of it, _Arp Microgram 1:25,000_ ( _Microgramme Arp 1:25.000_ ) (fig. 3) demonstrates the difference between the two artists. The title specifies the perspective: what we see are samples of Arp's microstructure, six different samples of a thousandth of a gram of Arp, magnified by a factor of 25,000. And what do they show, these minimal samples magnified to the extreme? Crustaceans and fossil fishes, mountain ranges and crosssections of the earth's interior.\n\nThe accompanying legend suggests that the diagram could possibly be deciphered. The circle of friends comprising Paris Dada\u2014who published the magazine _Litt\u00e9rature_ , in which the diagram first appeared in 1921\u2014liked to organize social games such as questionnaires and ranked lists, and at the same time understood the secret as a power around which a society could form. Ernst used the crypto-portrait to introduce his friend, and simultaneously his own poetics, to this specific readership.\n\nFor picture 1 of _Arp Microgram_ , the legend reads: \"Arp et la sagesse de sa jeunesse\" (Arp and the wisdom of his youth). The fossil specimens\u2014ferns, fish, crustaceans\u2014presented on beveled stone slabs are like the trophies of Arp's youthful wisdom. His youth encompasses the youth of the earth, a prehuman prehistory that comes into expression in the artist. This romantic topos of the past as an impenetrable prior age to which only a certain few can give us access\u2014among them psychoanalysts, poets, neurotics, and dreamers\u2014points to Freud. In _The Interpretation of Dreams_ (a book that Ernst made heavy use of), Freud writes, \"This profound and eternal character of humanity, upon the touching of which in his listeners the poet normally calculates, is made up of the stirrings of the spirit which are rooted in childhood, in the period which later becomes prehistoric.\" The famous parallelism of ontogeny and phylogeny\u2014which dominated scientific thinking around 1900 through the enormous impact of Ernst Haeckel and which was also recognized by Freud as a precondition of psychoanalysis\u2014thus forms the code that underlies the pictures and their captions in _Arp Microgram_. Prehistory can be taken to imply childhood.\n\nPicture 2 illustrates how this primordial past continues to have an effect in the present: in sleep, which revives human prehistory again each night. The legend reads: \"Arp sismographique a) sommeil calme sans fil b) sommeil agit\u00e9\" (Seismographic Arp a) calm, wireless sleep b) agitated sleep). Depending on whether his sleep is \"wireless,\" connecting him with a peaceful dream realm, or whether he is badly shaken by nocturnal apparitions, Arp is either like a range of steep fold mountains, or like an already heavily eroded landscape of hills. Fossil specimen (picture 1) and dream (picture 2) lead back to a primordial history governed (according to Freud's _Three Essays_ ) by a polymorphously perverse sexuality, in which, according to the caption for the third picture, the \"false key of the two sexes\" is still unknown. Thus, in this picture, the combinatorics that joins various crustaceans into figures might stand for the polymorphousness of infantile desire, which likewise knows no single object.\n\nFIGURE 3\n\nMax Ernst, _Arp Microgram 1:25,000 (Microgramme Arp 1:25.000)_. From _Litt\u00e9rature_ 3 (May 1921). 22.2 \u00d7 13.8 cm, collage and ink (Spies and Metken, _Max Ernst: \u0152vre-Katalog_ , no. 408 [hereafter \"S\/M\"]).\n\nPicture 4\u2014\"Chevelure arp et les s\u00e9diments journaliers de son intelligence\" (Arp hair and the everyday deposits of his intelligence)\u2014plays on the evident formal similarity between Arp's hairdo and a crustacean: the artist's mental work is a geologic process that preserves the living nature (fern leaves, small sea creatures) of a prehuman epoch. The fossil fish in picture 5 documents Arp in a state of total nakedness, as a fish out of water. As opposed to this skeptical cameo, picture 6 can be interpreted as the natural-philosophical apotheosis of the artist whose creative medium is the primordial time of his infancy. The caption for this picture reads: \"Arp yellowstone-parc il y garde le cheveu de B\u00e9r\u00e9nice\" (Yellowstone Park Arp he keeps Berenice's hair there). Inside the earth, in geological cross-section lines, Ernst discovers the resemblance to a woman's hair and identifies this hair with the constellation Coma Berenices. The heavens are reflected in the depths of the earth; nature becomes visible as replete with correspondences. Arp, as his friend Ernst represents him, embodies this romantic nature\u2014he is its conservation area, its Yellowstone Park, where the desire for resemblances can survive unimpeded, and where nature in its whole array, from tiniest to most gigantic, is filled with images. Here, Ernst could well be citing Freud, who compared the survival of the pleasure principle after the victory of the reality principle to the establishment of national parks after industrialization: \"In the same way, a nation whose wealth rests on the exploitation of the produce of its soil will yet set aside certain areas for reservation in their original state and for protection from the changes brought about by civilization (e.g., Yellowstone Park).\"\n\nFIGURE 4\n\nFrom _Cologne Catalogue of Teaching Aids_ ( _Katalog der K\u00f6lner Lehrmittelanstalt_ ), 1914, p. 195.\n\nIn this first, purely iconographic reading, Ernst's portrait appears as an encrypted homage to Arp's ideal of the natural origin of artworks. The earth imagines, generates various types of images (from impressions to diagrams), and projects them from its own depths out among the stars. But what the iconography of this work celebrates, it simultaneously destroys\u2014with the crucial assistance of pictorial form.\n\nThe source material, taken from the _Cologne Catalogue of Teaching Aids_ (fig. 4), exemplifies specific conventions for the representation of nature, among them the diagram. Ernst accentuates what is already present in the geology illustrations, which caught his avant-gardist's eye because they make nature available in many different forms of representation and therefore make it possible to assemble nature in arbitrary relationships. Does picture 1 relate to picture 2a as picture 3 relates to picture 2b? Or do 4 and 5 add up to 6? Are 1 and 3, or 4 and 6, findings from 2 or 5? There seems to be no doubt that in _Arp Microgram_ , Ernst does what a reader of avant-garde theories would expect from a Dadaist: he summarily destroys organic imitation. The mountain that Arp wanted to bring to life through imitation is represented as having been cut into, dismantled, and measured\u2014in source images that refer to the conventions for representing nature and their mortifying consequences, and in a pictorial form that excludes the concept of art's unmediated natural origin. Because of its proximity to writing, the pictorial form of the diagram accentuates the conventionality and mediatedness of iconic signs.\n\nOne noteworthy detail remains to be mentioned. If we look at the diagrammatic connecting lines in _Arp Microgram_ more closely, we observe that all of them can still be recognized as additions drawn in by hand with a pen. The lines get thicker, are drawn over, break off abruptly, miss their target. At the picture's lower edge, they burst into a new representational form: they are no longer diagrammatic lines producing legible connections, but iconic lines representing geologic layering. It seems as if the materiality of the construction lines\u2014or their retouched quality, in which the heaviness of the hand manifests itself\u2014has solidified into a rock substratum. The diagram is now presented in a double fashion: first, before a neutral ground, which is relevant in its sheer difference from the black markings; and second, upon a naturalistic platform, as if the diagram could also be read as a pair of figures. Obviously, it does _not_ condense into a heavy and material pair of figures, but the concretization of the diagram lines into rock layers that serve as a base for the diagram can be read as a distant memory of the ground plane in an illusionistic depiction.\n\nThe difference between the small pictures and the frame of their presentation becomes permeable, both because the diagram is standing on a similar surface as the fossil specimens in picture 1, and also because several of the small pictures also show diagrams. These are natural diagrams\u2014the layers of rock inscribed inside the earth. If the natural world depicts itself diagrammatically in its geologic layers, then, conversely, diagrammatic lines on a neutral ground can take on natural characteristics and be materialized as a physical ground.\n\nWhere are these detailed observations headed? The assumption that Ernst destroys the imitation of nature in this diagram stands to be corrected, because with this destruction, the sovereignty of construction also becomes negotiable. Construction is surrounded by a self-authorizing, unavailable nature, which intrudes in the places where the diagram lines are discontinuous and thickened, in the touch-ups and flaws, that is, in the graphic parapraxes (Freudian slips). The destruction of the organic artwork, as the organic artwork was imagined by Arp, seems to bring to power an inorganic nature, which is associated with the destructive pictorial form of the diagram at the same time as it surrounds this form. There are two possible readings, depending on whether (1) illusionism or (2) imitation is selected as the target of this parodic mimetism.\n\nFirst, if it is emphasized that the diagram receives a ground plane in order to evoke an unstable memory of a pair of figures standing upright, then the diagram is related to the history of the crisis of illusionism, inasmuch as this history is particularly clearly expressed near the picture's lower edge. As has often been demonstrated (by Steinberg, Shiff, Krauss, and Pichler, among others), the picture's lower edge is a critical zone where the dissociation of the illusionistic picture begins, because in the pictorial fiction, the platform for the represented objects\u2014and thus the pictorial element that functions as a minimal condition for heaviness, material density, and location\u2014corresponds to the picture's lower edge. When the platform tips, drops away vertically, or gives way to an unfathomable void beyond the picture's lower edge (as in C\u00e9zanne or Picasso, or already in Goya), these are symptoms of the collapse of the illusionistic picture. This process is not brought to a definitive conclusion with Dada, nor does it come to rest there. Ernst plays with the illusionistic picture's isolated remains by taking the diagram\u2014a pictorial form that is oriented vertically but without implying sensory qualities such as heaviness and material density\u2014and giving it a platform, as only a heavy figure requires.\n\nSecond, inasmuch as geologic layers can be understood as a diagram, the other meaning of mimesis also comes into play\u2014not the illusionistic depiction, but the performative imitation of nature. This imitation is likewise the target of parodic mimetism. What was characteristic for mimesis as imitation, a structural analogy between nature and artistic procedure, can also be observed in _Arp Microgram 1:25,000_ \u2014the diagram lines are modeled by the layers of the earth's interior; the diagrammatic quality of nature is imitated in the diagram. However, just as the ground plane does not restore the effective qualities of an illusionistic picture (heaviness, materiality, location with a depth of field), these analogies between the diagram and the earth's layers also do not lead to a revival of a living imitation of nature. It seems that these analogies only appear as an after-effect (through retouching) in the picture's flaws\u2014the halting, inexact, material diagram lines that materialize into rock strata. It is in these remnants of production, these parapraxes of art, that the diagrammatic distancing of nature is punctuated, in order to cultivate a secondary mimetism between artistic procedure and natural process.\n\nIn summary, in the ground beneath the diagram, two separate aspects of mimesis come into contact: illusion, on the one hand, and imitation, on the other. When looking at the portrait photograph of the young late impressionist, a critical gaze can already dissect the unity in impressionism and perceive this unity as a play of doubling. Arp attempted to put an end to this play by banning illusionism and devoting art to the imitation of nature\u2014where \"imitation\" is understood as the nature-reenacting animation of an ink blot and paper. Ernst had a radically different objective. In his portrait of Arp, the anti-mimetic pictorial form of the diagram effaces both illusionism and imitation. Details added after the fact to the picture's lower edge are the only residues of the old pictorial order, fossils of illusionism and stratifications of the imitation of nature, which are preserved as anachronisms in the diagram.\n\nThese are wide-ranging thoughts on an occasional piece that Ernst used to introduce himself and his friend Arp to the French Dadaists. It is high time to justify this effort by demonstrating that the distorted preservation of mimetic remains is characteristic of Ernst's Dadaist work as a whole.\n\nPETRIFIED ILLUSIONISM\n\nIn Max Ernst's Dadaist work, many kinds of diagrams can be found: constellations, graduated scales, roulette tables, perspective schemes, and geologic cross-sections, as well as numerous machine drawings of armored vehicles, bicycles, and mechanically animated letters of the alphabet. His interest in the machine works of the avant-garde began when he discovered Francis Picabia's _Alarm Clock_ ( _R\u00e9veil matin_ ) in the _Anthologie Dada_ , published by the Zurich Dadaists (fig. 5). The print of a dismantled alarm clock reveals none of the commodity fetishism that Picabia, an auto enthusiast, had glorified in the early bachelor machines from his time in New York (e.g., _Girl Born without a Mother_ [ _Fille n\u00e9e sans m\u00e8re_ ], _Here She Is_ [ _Voil\u00e0 elle_ ], or _American Woman_ [ _Am\u00e9ricaine_ ]). Instead, _Alarm Clock_ , which Picabia presented to the European Dadas after his arrival in Zurich, is these works' ruin. Hans Arp remembers:\n\nI met Francis Picabia during his visit to Zurich. He came as the emissary of the American Dadaists to extend greetings to his colleagues in Zurich. Curious and touched, Tristan Tzara and I went to his hotel. We found him busily dissecting an alarm clock. I couldn't help but think of Rembrandt's _Anatomy_ in the art museum in Amsterdam. Truly, we had made a great leap forward into the realm of abstraction. Showing no mercy, he dismantled his alarm clock down to the mainspring, which he extracted in triumph. For a brief moment he interrupted his work in order to greet us. But without wasting too much time, he adorned a white sheet of paper with impressions of the little gear wheels, springs, hands, and other secret tiny parts of the clock. Like a dutiful postman, he zealously applied these things to the stamp pad and then to the paper, and connected the stamps to one another with lines. . . .\n\nArp's supposedly unbidden recollection of Rembrandt's _Anatomy Lesson of Dr. Joan Deyman_ suggests that Picabia's _Alarm Clock_ should be understood as a parody of illusionistic painting. The picture's object does not appear upon an illusionistic platform; instead, a material relation connects it to its underlay, the paper ground, which serves as a work surface and is stamped by Picabia. Heaviness and materiality, instead of being qualities of the represented objects, are transformed into the means of production. Picabia comments on this breaking down of illusionistic representation into literal production through his signature. In illusionism, the picture's lower edge is the correlative of the platform on which objects are placed, and simultaneously it is the site of the signature. Picabia has rotated his signature to the left, into the vertical, as if not only his alarm clock but also the pictorial field had been dismantled on a horizontal work surface. On this surface, among other remains, there lies a remnant of the destroyed illusionistic easel painting: its dismantled bottom edge, with signature.\n\nFIGURE 5\n\nFrancis Picabia, _Alarm Clock_ ( _R\u00e9veil matin_ ), from _Anthologie Dada_ 4\u20135 (1919).\n\nPicabia's parodic reflection on the image consists in that the machine diagram, by destroying itself, also recalls the destruction of the illusionistic picture and uses its remains as decoration. Ernst takes this reflexive negation as his starting point, but lends it a more complicated form, inasmuch as he employs iconic elements and generates new iconic motifs in unexpected places. Regarding iconic elements, instead of using objects (such as the components of an alarm clock), Ernst uses images from commercial intaglio plates (bicycles, roulette tables, various mechanisms). Regarding the generation of new iconic motifs, these new motifs arise precisely at the sites of error in the printing process and are predominantly geological. While Picabia dismantles the mechanism (of the alarm clock) because he does not want to represent a heavy object but wants to use the heaviness and materiality of its individual parts as a means of production for a direct imprint, Ernst turns away from this procedural literalism. Ernst also uses imprinting methods through which materiality and heaviness are literally expressed, particularly through extraneous ink traces and errors of the printing process. However, these traces and errors are immediately translated into pictorial motifs\u2014most often geological ones, the densest and most materially weighted of all.\n\nIn _Farewell My Beautiful Land of Marie Laurencin_ ( _Adieu mon beau pays de Marie Laurencin_ ; fig. 6), Ernst retouched the extraneous ink traces to form a stony cladding that surrounds the war machine and paralyzes its inner workings. In the same year, 1919, he made _Portable Handbook_ ( _Vademecum mobile, ihr seid gewarnt_ ) and _The Canalization of Refrigerated Gas_ ( _La canalisation de gaz frigorifi\u00e9_ ; figs. 7, ), also using commercial intaglio plates. In contrast to the petrified war machine, these carry off a triumph of mobility: through rotation, doubling, shifting, reflection, and optical illusion; roulettes, bicycles, constellations, and transmission belts. Mechanically reproducible production methods such as imprinting, as well as rotation, refer to Marcel Duchamp's first readymade, _Bicycle Wheel_ (fig. 9). In _The Canalization of Refrigerated Gas_ , the bicycle wheel is placed at the center of the picture as a quotation. But why does Ernst make _Bicycle Wheel_ into a ruin, when _The Canalization of Refrigerated Gas_ itself affirms the poetics of the readymade, that is, its mechanization, mobility, and repetition? The answer can again be found at the pictures' edges. In both _Portable Handbook_ and _The Canalization of Refrigerated Gas_ , the edges seem to have been trimmed away carelessly, resulting in forms that suit the aged, stained, and torn condition of the paper. A formal association can be made to the irregular ovals of Arp's ink drawings (fig. 2). At their edgeless outer limits, the traditional pictorial field seems to melt away or erode. In Ernst's two works, the traditional pictorial field is also destroyed by wear and tear: the picture's boundaries follow a line as careless and accidental as if the picture had taken shape under the influence of undirected forces. These de-organicizing natural forces, which in _Arp Microgram_ and the machine drawings surround the diagram's form, here work their effects on the diagram's support\u2014one could say they expose it to erosion and transform it into a ruin such as the _Bicycle Wheel_ at the center of the picture.\n\nFIGURE 6\n\nMax Ernst, _Farewell My Beautiful Land of Marie Laurencin_ ( _adieu mon beau pays de Marie Laurencin_ ), 1919. 40 \u00d7 28 cm, line engraving with pen and ink on paper. New York, Museum of Modern Art (S\/M no. 313). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 7\n\nMax Ernst, _Portable Handbook_ ( _Vademecum mobile, ihr seid gewarnt_ ), 1919\/20. 41.5 \u00d7 53 cm, line engraving with pen, ink, aquarelle, and gouache on newsprint. Private collection (S\/M no. 326). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn _Arp Microgram_ (fig. 3), the lines of the diagram, because of their irregularity and materiality, solidify into rock strata. In the war machine (fig. 6), retouchings made after the fact, building on the flaws in the imprint, surround the motif in a stony cladding. And likewise, in _The Canalization of Refrigerated Gas_ (fig. 8), it is through small flaws in the printing process that stony nature intrudes into the diagrammatic order. In the areas around the roulette circles, traces of ink can be seen: these deposits enclose the figures of rotation in the same way that the entire construction is contained by the eroded picture format. In all three pictures, traces of production technology can be read as parapraxes, moments at which the diagrammatic pictorial order breaks down and residues of heaviness and materiality accumulate. These residues adorn both the boundaries of the figure (armored vehicles, wheels, etc.) and the boundaries of the picture, considered either as the format (in _Portable Handbook_ and _The Canalization of Refrigerated Gas_ ) or the ground plane (in _Arp Microgram_ ). That is, these residues of illusionism appear in the critical zones where the decomposition of illusionism is most clearly manifested.\n\nFIGURE 8\n\n8 Max Ernst, _The Canalization of Refrigerated Gas_ ( _la canalisation de gaz frigorifi\u00e9_ ), 1919\/20. 54 \u00d7 38 cm, line engraving with pen, ink, aquarelle, and gouache on paper. Private collection (S\/M no. 325). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 9\n\nMarcel Duchamp, _Bicycle Wheel_ , 1951. Replica of lost 1913 original. Height 125 cm. New York, Museum of Modern Art. \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris \/ Succession Marcel Duchamp.\n\nUnder the influence of Giorgio de Chirico, whom he discovered around the same time as Picabia, Ernst intensified his engagement with illusionism. Unlike the diagrams, these pictures once again fulfill some of the minimal conditions for illusionistic representation. Their boundaries conform to the traditional rectangular format and frame a fictive world. Their objects stand upon a persectivally foreshortened platform, as is necessary in illusionism for the placement and presentation of heavy material objects. However, this platform proves to be an unstable ground, and the visibility of the fictive world is in many ways obstructed. Rectangular format, platform, and perspectival foreshortening are subject to a secondary purpose. As a decoy, they lure the gaze into the trap of a completely rigidified, hollowed-out, and broken illusion.\n\nFIGURE 10\n\nMax Ernst, _Sodalites shirkers snow-covered mountain and valley dwellers raisins and almonds beat the natives of central Europe to sea foam and following advanced denudation hurry ahead of events with the best intentions_ ( _sodaliten schneeberger dr\u00fccketh\u00e4ler_ ), also _always the best man wins_ , 1920. 15 \u00d7 22 cm, gouache, ink, and pencil (overpainting on a print) (S\/M no. 342). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn the example of _Sodalites_ ( _sodaliten schneeberger dr\u00fccketh\u00e4ler_ ) (fig. 10), the landscape serves as a platform upon which the represented plant-machines are located through perspective. However, the perspectival construction is broken and ambiguously rendered. Are the lines that divide the landscape horizontally and vertically the section lines of a geological map, or are they areas of erosion and breaks in the terrain? Are they units of measurement or features of the represented world?\n\nOr in _Stratified Rocks, Nature's Gift of Gneiss Lava Iceland Moss_ ( _schichtgestein naturgabe aus gneis_ ) (fig. 52), the extension into a depth of space, and with it the perspective, is broken down into a stratification that takes over the whole picture. The layers of rock and ice cover the illusionistic space of the landscape and, at the same time, seal the picture plane.\n\nFIGURE 11\n\nFrom _Cologne Catalogue of Teaching Aids_ ( _Katalog der K\u00f6lner Lehrmittelanstalt_ ), 1914, p. 624.\n\nOr, in the case of an untitled work also from 1920 (fig. 12), there is the question of whether the horizontal section lines depict the layered substrata of the landscape or a space receding into depth\u2014or else, as a glance at the right edge of the picture would suggest, part of a stage set, which interlocks with another part of the set behind it. The figures\u2014made from textiles, palettes, and tubes\u2014again multiply the possible interpretations. One observes the interactions that arise between their individual parts and the strata. In the middle of the picture, the horizon line merges with the joint in the central figure; the nearest and farthest points in the pictorial space collapse; the illusion is fixed to the picture plane. A firmly modeled, pyramidal tube towers directly overhead, once more giving rise to a clear spatial effect.\n\nIn all three works, a platform can be found that simultaneously displays and parodies its illusionistic function of creating the conditions necessary for location and physical concretion. Regarding location, the following should be emphasized: the horizontals and convergences of perspective construction are ambiguously rendered, so that it is impossible to decide whether what is being viewed are rock layers or cartographic symbols, elements of the pictorial world, means of their representation, or pieces of a stage set. They do not form any clearly defined outer layer or platform that would serve as a plane upon which the represented objects are located. The same fate befalls the physical concretion of the represented world. It is implied but has always already been lost. We see remarkably unspecific, essentially used-up things. The isolated parts of photographs that can be clearly recognized as textures, such as the different knitting patterns (fig. 12), remain foreign objects between textureless layers of paint. (Other photographs may have interested the artist because of their material indeterminacy, such as the model of angiosperms he painted over in _Sodalites_ [figs. 10, ]).\n\nFIGURE 12\n\nMax Ernst, _Untitled_ , 1920. 30 \u00d7 25 cm, gouache, pen, ink, and pencil (overpainting on a print), mounted on board. Private collection (S\/M no. 357). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 13\n\nMax Ernst, _Oedipus Rex_ , 1922. 93 \u00d7 102 cm, oil on canvas. Private collection (S\/M no. 496). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nMax Ernst was the \"painter of illusions,\" as Louis Aragon dubbed him in the text of the same name from 1923. This illusionism\u2014which so fascinated Aragon, Breton, and the other surrealists-to-be when Ernst's paintings were first shown in Paris in 1921\u2014can be designated as a parodic play with the products of decomposition. The famous paintings that were made between 1921 (e.g., _Celebes_ ) and 1924 (e.g., _Woman, Old Man, and Flower_  _Weib, Greis und Blume_ ]) also display the secondary, hollowed-out qualities of a fossil illusionism ([plate 3). In _Oedipus Rex_ from 1922 (fig. 13), Ernst uses the demolished perspective construction, reified into wall parts, to assemble a foreground set that obstructs the pictorial field and opens into a peculiarly abstract, empty expanse, basically inaccessible to physical beings. A firm ground is nowhere to be seen; in the foreground is a perforated plinth in which two imaginary beasts are trapped. In the dark window to the left, it is again suggested that the boundaries of the picture are inaccessible and also elude our gaze in hidden spaces; at the same time, the rectangular format of the pictorial field is repeated. While the pictorial field recedes into a purely optical, materially dulled world that is emptied of any sensual specificity, in the window a figure of destroyed tactility appears that is at once hallucinatorily powerful and unreal.\n\nAs I have attempted to demonstrate, this secondary illusionism so influential for the emergence of surrealism had already been formulated by Ernst in the overpaintings of 1920. In addition, in these paintings it becomes plainly visible that the fossil hardening and emaciation of the illusionistic image is based on procedures of modern image production. In the source picture for _Sodalites_ (figs. 10, ), besides the material dullness of the plants, Ernst was interested in their ordered arrangement, the way they can be broken down and reproduced, as well as the clear cuts through stems and blossoms. The overpainting accentuates these qualities. The fragmented perspective and the repeated, ambiguous carving up of the landscape refers to the overpainting's underlay, the mechanically reproduced image from the _Cologne Catalogue of Teaching Aids_, which Ernst drew from during this time. The result can be considered a fossil illusionism because the distorted preservation of platform, perspective, heaviness, and materiality is based on a pictorial readymade whose contours and arrangement are manifest in the surface of the picture.\n\nREPETITION OF IMITATION\n\nIt is less obvious, and in the critical literature on Max Ernst has gone unnoticed, that with this fossil illusionism, distorted remnant forms of imitation also return. These exhausted and mortified mimetisms call attention to a natural origin of artistic procedures but do not claim to present a persuasive mimetic relationship. Instead, the viewer is invited to follow the play of mimetisms through a labyrinth of multiple, competing analogies that point toward the origins of artwork.\n\nThe rock substratum in _Arp Microgram_ \u2014according to the second interpretation presented here\u2014refers to the existence of an analogy between the diagram and geologic layers; this analogy suggests a relationship of imitation between the two. In the other diagrams, it is also striking that their graphemes are connected with traces of petrifaction and rock strata. In the fossil illusionism of the over-paintings, we also encounter such exhausted natural analogies. As Louis Aragon did in 1930, one might well speak of _mise-en-sc\u00e8nes_ that present the metaphoric relationship between nature and art, and in this way raise the question of the origin of art while simultaneously parodying it.\n\nFIGURE 14\n\nFrom _La Nature_ 26 (September 1891).\n\nFor example, in _Stratified Rocks_ (figs. 52, ), the source image underneath is enclosed in layers of paint, as the fossil horse is enclosed by the Ice Age landscape. The horse's rear hoofs explicitly call attention to this analogy: they were first painted over and then traced in pen on top of the paint. There is a similar detail in _Sodalites_ (figs. 10, ): the oak blossoms on the far left and the hazelnut flower in the middle grow up from behind the stratified landscape, as if they were returning to visibility from the condition of having been painted over. Or there is the work (whereabouts currently unknown) titled _Winter Landscape: Vaporization ofthe Vulcanized Iron Bride to Produce the Necessary Bed Warmth_ ( _winterlandschaft: vergasung der vulkanisierten eisenbraut zur erzeugung der n\u00f6tigen bettw\u00e4rme_) (fig. 15): here, the \"iron bride\" is swallowed into the depths of the earth, as is the source image by the layers of overpainting. Or, most explicitly, in _Frozen Landscapes, Icicles, and Mineral Types of the Female Body_ ( _eislandschaften, eiszapfen und gesteinsarten des weiblichen K\u00f6rpers_ ; plate 1), two fields are juxtaposed along the bottom edge of the picture: the one on the right shows the wallpaper pattern of the support; the one on the left, geologic strata.\n\nFIGURE 15\n\nMax Ernst, _Winter Landscape: Vaporization of the Vulcanized Iron Bride to Produce the Necessary Bed Warmth_ ( _winterlandschaft, vergasung der vulkanisierten eisenbraut zur erzeugung der n\u00f6tigen bettw\u00e4rme_ ), 1921. 15.5 \u00d7 20.5 cm, gouache, ink, and pencil (overpainting on a print), lost (S\/M no. 410). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe list of analogies in which natural process and artistic activity play upon one another could go on, but presumably the above examples clarify how these mimetisms function. The bringing into effect of a metaphoric relationship between procedure and natural process is only as important as this relationship's hasty decomposition. To begin with, the comparison makes obvious the difference between deep-time processes of sedimentation and stratification, on the one hand, and the small-scale, small-format dilettantism of the Dadaists, on the other. What's more, the Dadaist pictures are so subdivided and so heterogeneously articulated that an imaginative, metamorphic seeing that would have the potential to instigate metaphoric exchange is never able to develop. Finally, it is not at all certain which procedures Ernst is using or, if they actually could be determined, to what they could be metaphorically compared. This last point merits further elaboration.\n\nIt is often the case that the overpaintings make themselves out to be collages. Werner Spies has dedicated a fundamental study to this hide- and- seek of procedural logic, which Aragon had already appreciated; Spies's analysis of _Katharina ondulata_ (fig. 16) should be given particular emphasis here. At first, one assumes that this work is a collage of differently patterned papers\u2014an impression that Ernst cultivated through selective retouching. He changed the pattern and outlined the pattern fragments in black, giving them the physically self-contained quality of material fragments. Only upon closer scrutiny and careful attention to the pattern does the picture reveal itself as an overpainting of a uniformly ornamented underlay, which runs continuously from the base of the figure to the individual components. Just how seriously the artist took this game of confusion is shown in the joke of procedural logic at the picture's lower edge: the section being passed off as the pictorial ground is in fact the only section that was cut from another pattern and glued on. Furthermore, in addition to Spies's observations, one notices how, to the left, a red-dotted mucilage runs down over this alleged underlay. The actually employed technique of overpainting appears as a motif within the pictorial fiction in exactly the place where Ernst collaged _on top of_ the pictorial ground a piece that is taken _for_ the pictorial ground.\n\nThe earth's interior section lines from which Ernst's geologic landscapes are constructed can be read as motifs that figure both the technique of overpainting and the cuts of collage. Thus, entirely different production-aesthetic concepts are brought into connection. Comparing earth layers with layers of paint evokes the deep-time slowness of the processes of natural history. However, seeing an analogy between earth layers and the cuttings of collage places the collagist's work in relation not to processes of nature, but to human interventions. _StratifiedRocks_ (figs. 52, ) seems to be an example of precisely this analogy: Ernst, the avant-garde artist who dissects both living and inorganic nature, takes pains to ensure that each organic connection between the parts of the horse is replaced by a mechanical one. However, _Stratified Rocks_ is not a collage\u2014it's an overpainting; and even when erroneously taken for a collage, it does not convey the impression of having been produced with freehand cuts. The rigidity of the space and the tension of the layers suggest not so much the agility of a knife or scissors, but instead directionless forces working against each other (sedimentation, glaciation, stratification).\n\nFIGURE 16\n\nMax Ernst, _Katharina Ondulata, that is Madam Hostess on the Lahn_ ( _Katharina ondulata d.i. frau wirtin a. d. lahn_ ), 1920. 31.2 \u00d7 27 cm, gouache and pencil (overpainting on a print), mounted on board. Private collection (S\/M no. 356). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nNext, let's take a look at a small-scale, small-format piece of Dada bricolage that makes a satirical masquerade of the sublime processes of _natura naturans_. It is also part of the game to play off disparate metaphors against one another. In an overpainting of knitting patterns (fig. 12), the geological analogy is confronted by a second one, between layers of paint and the stacking of objects or their insertion into one another. Let's consider the curious detail at the picture's right edge, where the underground layers unexpectedly reveal themselves as a stage set with another white layer that has been inserted into it. The figures made from knitting patterns and tubes present the same production-aesthetic concept of layering as the stacking of elements. This process can be repeated as long as the figures remain in balance or until the pictorial field has been filled up. In fact, Ernst not only stacks knitting pattern on top of tube and tube on top of knitting pattern; he also stacks stratum on top of stratum, even in the region of the sky, until the upper edge of the picture has been reached.\n\nFinally, one work takes the play of analogies further than all the others, only to lapse into an idling that allows the parody of the imitation of nature to be recognized as a mechanical repetition (plate 1). Against a winter landscape, at the lower edge of the picture stands an unfolded, three-part panel\u2014even more clearly a stage set than in figure 12\u2014that obstructs the view of the winter landscape. Evidently, this is a case of a _mise en abyme_ in the overpainting, whose layers of paint occlude the view of the overpainted wallpaper. In the unfolded panel, the procedure of overpainting is staged twice: as an object, the panel is similar to the stacked objects or interlocking parts of a stage set, but as an image, it calls attention to the analogy between layers of paint and geologic strata. On the right, we see the wallpaper pattern to be painted over, on the left, rock layers, and in between, a rendition of the wallpaper pattern in hatching. The transition from the prefabricated (readymade) wallpaper to the traditional artistic medium of hatching to the painted pictorial motif is made didactically explicit. These are three different forms of repetition: repetition as readymade in the wallpaper, repetition as artistic technique in the hatching, and repetition as natural process in the rock layers. When the work as a whole is analyzed, it becomes clear that repetition is its modus operandi.\n\nWallpaper is typical as a pictorial underlay for Ernst's overpainting, in that like the botanical models (figs. 10, ), knitting patterns (fig. 12), or diagrams of machines and animals (figs. 52, ), wallpaper exemplifies image production through serial repeatability. The comparison of wallpaper to hatching and geologic strata in the specific setting of a frozen landscape creates a metaphor that, in turn, forms the basis for a metonymy. From the serial underlay, mechanically repeated hatching lines and repetitive layers of rock emerge. This production is continued in the upper portion of the picture, following a shift from the serially produced to the serial production machines that hover on wires above the ground. The metonymy is exemplified on the level of motif and on the level of procedure:\n\nOn the level of motif, metonymy is represented through the red wheel in the center of the picture, which is accompanied by a vector pointing from the unfolded three-part panel upward toward the machine. This is the vector of metonymic shift from the serially produced wallpaper to the serially producing machines. On the level of procedure, it is shown through the incremental graphic production of the machine from elements of the wallpaper pattern. On the right, Ernst introduces a second ornament into the pattern of cone shapes, while on the left, he retouches the same cone pattern into two windmills. On the far left, there then follows a machine whose upper half consists of a monumental gear wheel. Below this, there had once been a second rotating mechanism similar to a ship's screw or propeller. Let's look more closely at this destroyed wheel: Ernst may have first drawn four propeller blades in the white spaces of the cone pattern and painted them yellow, and then doubled the number of blades by drawing four more on top of the cones. As with the windmills or the large gear wheel, in order to indicate the axis of rotation, he then added a circular mark in the middle, probably by means of a tube dipped in paint, which he used as a stamp. What happened next can best be described as \"destruction,\" especially since it results from the playful application of an idling machine. At the hub of the repetitive mechanical movement carried out by gear wheels or windmills, ship's screws or propellers, Ernst made himself into a stamping machine, and stamped until both the wallpaper pattern and the retouching could hardly be made out. As if all energy had not yet been expended, he scattered a few more imprints across the picture and then exited along a trail of stampings that lead in a playfully curving line from the site of destruction to the left edge of the picture. This line follows a movement to the dead point at which the artistic procedure of overpainting cannot generate any more mimetisms. Analogy development has reached its end, and in its place stamping appears, which playfully and compulsively repeats the serial-mechanical production of the picture's underlay (the wallpaper).\n\nImitation as the metaphoric re-creation of natural processes breaks down into various mimetisms, and these refer to a principle of production beyond imitation: that is, to repetition. Geologic layering is not the origin of the overpainting; it is only one of various possible analogies in a labyrinth of metaphors for artistic activity, such as cutting apart, stacking, interlocking, or the superimposition of stage sets. However, each of these options points back to the mechanical-serial production of the picture's underlay. The search for origins uncovers repetitions, as Michel Foucault discovered in Raymond Roussel's prose: \"At the most enigmatic moment, when all paths stop and when one is at the point of being lost, or at the absolute beginning, when one is on the threshold of something else, the labyrinth suddenly again offers the same: its last puzzle, the trap hidden in the center\u2014it is a mirror behind which the identical is located.\"\n\nIn this world of disfigured identity, surrogate creations thrive most heartily. Examples include a fossil horse (fig. 52, ), which Ernst neatly dismantled into its individual organs and then reconnected with mechanical prostheses, and tower-like plant machines (figs. 10, ), which seem to have been stuck into the lacerated ground and which inflict further cuts with the sharp edges of their shadows. In both pictures, the force of a surrogate life still pulsates, in _Stratified Rocks_ in the intestine, which inflates like a wind sock, and in _Sodalites_ in the labyrinthine coil of space between the plant towers. Vectors and ducts indicate that between mortified landscape and mortified organisms, remnant or surrogate forms of a vital exchange have been preserved. As in Roussel's language, so also here painting \"experiences a death that clings to life, and its very life is prolonged in death.\"\n\nFIGURE 17\n\nPablo Picasso, _Bowl with Fruit, Violin, and Wineglass_ December 1912\u2013January 1913). 65 \u00d7 50.5 cm, pasted paper, aquarelle, crayon, oils, and charcoal on board. Philadelphia, Philadelphia Museum of Art. \u00a9 2012 Estate of Pablo Picasso \/ Artists Rights Society (ARS), New York.\n\nCUT AND RESEMBLANCE\n\nResidues in which processual mimesis is preserved through parody can already be found in the prototypes of a collaging art, Picasso's _papiers coll\u00e9s_. The glued-in pears in his _Bowl with Fruit, Violin, and Wineglass_ (fig. 17), dilettantishly painted or cut from illustrated broadsides, are \"cut\" in two regards. By cutting the paper, the collagist accomplishes what a painter would achieve by painting a cut pear. That which is represented\u2014the motif of the cut pear\u2014is imitated through that which is produced: a pear that has been cut out of paper. In this way, Picasso's cubism gives an ironic answer to the question of the presence of an object in a picture: even if a painted fruit cannot be taken and eaten, at least it is possible to cut off a slice. A related topos of iconic criticism\u2014that painted objects above all lack one thing, their back sides\u2014was the impetus for a drawing of de Chirico's from 1917 (fig. 18), with which Ernst is known to have been familiar. The cubistically furnished interior contains two drawings. There is likely no way to relate them to one another without contradiction. However, the attempt to find a relationship between them gives rise to a reflection on representation and production, cut and fruit.\n\nFIGURE 18\n\nGiorgio de Chirico, _Autumnal Geometry_ ( _Geometria autunnale_ ), 1917. 32 \u00d7 22 cm, pencil on paper, Berlin, Neue Nationalgalerie. \u00a9 2012 Artists Rights Society (ARS), New York \/ SIAE, Rome.\n\nAre they, as one might first think, the two corresponding halves of a single pear? No, since the stem above curves to the right, and the one below, to the left. Besides, how could each half of a single pear retain an entire stem? So is it a single half of a pear, shown once from the front and once from the back? The fold in the sheet below implies that it is, and that one only has to turn the drawing over in order to see the other view of the same pear half. The play of possibilities is not yet at its end: it is surely no coincidence that de Chirico varies the placement of the cut. While the lower pear half seems to have been cut right next to the stem, the upper pear half clearly has a greater volume. Thus, it is two pear halves from two different pears. By the time one arrives at this conclusion, doubts have arisen as to the certainty with which it can be argued that the drawing below even represents a pear _half_. Couldn't it also be an entire pear, rendered by a draftsman who favors angular contours and a severe styling?\n\nOne pear cut in two halves has become one pear half-drawn twice, and this has again become two halves of two pears or one and a half pears. Surely this confusion regarding how many and which pears (or pear halves) can be seen is the humorous culmination of a reflection on the necessity of the cut in illusionistic painting: first there must be a cut through the stem, in order to get the fruit from the tree, and next there must be a cut through the fruit, in order to represent it illusionistically (that is, without a back side). Hence, Picasso's and de Chirico's critiques of illusion do not depend on the imitation of growth, as Arp's does, but instead on the imitation of the cut, which at once puts an end to growth and founds representation. In this way the meaning of imitation is transformed, in the sense that instead of bringing the imitated thing to life, imitation abuses the imitated thing with the mechanical and mortifying efficiency of the cut.\n\nErnst saw de Chirico's paintings, including the still life of pears, for the first time in Munich in 1919, and from them drew conclusions not limited to the adoption of de Chirico's imagery. It has often been demonstrated that the spatial effects analyzed here as \"fossil illusionism\" were prefigured in de Chirico's paralyzed world of stage sets, with spaces lacking materiality or atmosphere; surrogate forms such as dressmakers' mannequins, baking molds, statues, gloves, and shellfish taking the place of massive bodies; ground planes plunging away, conflicting perspectives and depths of field. Less striking, though just as important for the understanding of Ernst's Dada art, is that de Chirico's fossil illusionism also contains moments of secondary, mortified imitation\u2014for example, the halved pear as the imitation of the cut that is a prerequisite of representation. Ernst multiplies the varieties of cut, so that the cut of the collagist is compounded by the sectionings made in diagrams and overpaintings. Moreover, he expands the possibilities for perceiving the cut as imitation, so that it is made to imitate the earth's layers, geological or anatomical analysis, the action of stacking, or the shifting of scenes. These are mimetisms of the second order, which play upon the stronger production principle of repetition and parodically preserve, under repetition's hegemony, the ideal of a natural origin of artistic procedure. In view of this tightly interwoven context, which connects Ernst's procedures with Picasso's and de Chirico's, it comes as no surprise that at the point in time when Ernst fundamentally transformed his art under the sign of surrealism, he sought a renewed engagement with Picasso and de Chirico\u2014in particular, with their pears.\n\nIn 1925, in the year following the publication of the _Manifesto of Surrealism_ ( _Manifeste du surr\u00e9alisme_ ) and before his first surrealist series, _Natural History_ ( _Histoire naturelle_ ; 1925\/26), Ernst made a series of still lifes of pears, in which different procedures and forms of representation are tested. The most complex of these (fig. 19) shows three possibilities for representing a pear: in front, there is a naturalistic pear; behind that, a diagram of a pear; and in between, a pear fossil. The juxtaposition of the three forms of representation may have been preceded by a small iconic-theoretical drama enacted between naturalistic representation, diagrammatic projection, and fossilization. The naturalistic pear was projected onto the grid by means of a geometric operation. In his Dada pictures of 1919\u201321, Ernst had already employed similar procedures to transform one type of image into another. For example, in _Frozen Landscapes_ (plate 1), the landscape is unfolded in order to show both its geological cross-section and its pictorial underlay (the wallpaper), from which, in turn, the machine emerges that hovers over the landscape. The transformation progresses from pictorial landscape to geological illustration to machine diagram, and ends in stamping the surface of the picture (see above). This movement that begins with iconic painting (landscape) and ends with indexical marking (stamping) also characterizes the still life of pears from 1925. However, indexical marking is no longer a dead point, but as frottage it occupies the center of the picture and brings the picture to life. The pear frottage, which occupies the space between the naturalistic and diagrammatic pears, clearly distinguishes itself from these two types of image. The fossilized pear wins out against the grid, seeming to tear through it with its stony fault line. And the fossilized pear prevails against the naturalistic pear by provoking its own form of seeing, an imaginative seeing of resemblances. This latter is to be described at greater length in what follows.\n\nFIGURE 19\n\nMax Ernst, _Pear_ ( _Poire_ ), 1925. 29 \u00d7 18 cm, aquarelle, pencil, frottage on paper, Berlin, Neue Nationalgalerie (S\/M no. 773). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nA contemporaneous work shows another pear in an empty expanse of landscape (fig. 20). The satirical title _The Pear that Resembles Me_ ( _La poire qui me ressemble_ ) lures the viewer into a trap, since it both demands and blocks the seeing of resemblances. With its clearly determined distribution of light and shadow and its crisp contour, the naturalistic pear resists any attempt to discover resemblances within it. In this way, it is similar to the naturalistic pear in the still life discussed above, whose modeling is equally unambiguous and whose contours are equally sharply defined. One might think that these two fruits had been cut from an illustrated broadside and glued in, and that these foreign objects, starkly differentiated from the pictorial ground, were intended to block the imaginative seeing of resemblances. The frottaged pear is a different story: despite its jagged fault line, it is closely connected with the pictorial ground (the paper shimmers through the traces of graphite), and in turn forms its own pictorial ground that provokes an imaginative seeing of resemblances. The pear curves to the right with bodily insistence; it becomes possible to make out a beak and eyes\u2014virtually a whole bird family. Retouchings on the upper part of the pear, to the left, demarcate a round eye and a beak, and below there are additional marks that would stimulate additional discoveries.\n\nThe seeing of resemblances is an imaginative perception that can animate the lifeless and also give it figural form. In no way tied to art, most often practiced on objects and manifestations not produced by human hands (clouds, stones, knotholes), it accompanies the art history of the modern era as a marginal phenomenon that is nonetheless given much attention: from Leonardo da Vinci's famous stains on the wall to the discussion of \"seeing-in\" in analytic philosophy or Georges Didi-Huberman's iconic theology of marbling. It is surprising that the central pear fossil in Ernst's still life provokes this type of seeing, because it was consistently avoided in Ernst's early work (as in Picasso's cubism or de Chirico's metaphysical interiors). In his early work, a crisply cut, occasionally small-scale and materially heterogeneous, always multiply incised articulation impedes an imaginative perception that would vitalize the surface and thus \"discover\" a resemblance of the dissimilar (such as fossil pears and bird families). Another type of the seeing of resemblances was nevertheless characteristic of Ernst's Dadaism: a \"seeing apart\" ( _Zersehen_ ). This action separates a pictorial motif from its iconic context and places it into a new one in order to rediscover it as another motif. Photos of ladies' hats cut out, turned upside down, and lined up on a strip of desert now look like insects (fig. 21). Louis Aragon seized upon these manipulations and dubbed Ernst the creator of a new visual metaphorics, an art of transference of meaning different from the cubists' formal use of collage. In Picasso's cubism, the difference between a human body and the corpus of a stringed instrument can be minimal (fig. 22). Nevertheless, one object turns into another not because it inspires an imaginative and enlivening way of seeing, and not because it conceals the other within it, as in a reversible image ( _Vexierbild_ ). Instead, the transition between two motifs is always specified as an effect of breaking down and recombination: breaking down the image into sub-iconic elements (dot, straight line, curve, circle, wavy line), and their recombination through shifting, reflection, folding, or rotation\u2014in short, through operations that are carried out upon and with a surface. Meanwhile, Ernst's Dadaist method for making one motif out of another involves individual images being broken down without their iconicity as such being affected or a sub-iconic level being exposed. When Ernst cuts apart a fashion catalog, he preserves as the smallest unit images of ladies' hats. Their ambiguity is not the product of a new distribution of their sub-iconic elements; rather, ambiguity is disclosed in the original iconic element itself through its being cut out, displaced, sometimes rotated, and put into a new context.\n\nFIGURE 20\n\nMax Ernst, _The Pear that Resembles Me_ ( _La poire qui me ressemble_ ), 1924. 25.5 \u00d7 16.7 cm, aquarelle, frottage and collage on paper. Private collection (S\/M no. 771). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 21\n\nMax Ernst, _The Sandworm Who Reties Her Sandal_ ( _l'ascaride de sable_ ), 1920. 11.8 \u00d7 50.5 cm, gouache and pencil (overpainting on a print), mounted on board. Private collection (S\/M no. 360). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn 1924\u201325, when Ernst was attempting to make the Dadaist method of breaking down and \"seeing apart\" subordinate to another form of unstable visuality\u2014the seeing of resemblances as provoked by frottage\u2014he again sought out an engagement with art oriented to the procedure of the cut: that is, with Picasso's cubism and its _mise-en-sc\u00e8ne_ in de Chirico's still lifes. The still life of pears from 1925 can be understood as a working-through in which Ernst detaches himself from his Dadaist operations and their models (de Chirico, Picasso), in order to introduce a new procedure: frottage. Between the naturalistic pear, which seems to have been cut out and pasted in, and the pear diagram, he allows a third image form to emerge, which requires an imaginative seeing. While a geometric projection leads from the naturalistic to the diagrammatic pear, the pear frottage sets in motion an imaginative projection. While the transformation from the fruit to the diagram is carried out on the surface of the support, the transformation of the fossilized pear into eyes and a beak brings a depth into play\u2014the psychic depth of the imagination, which Ernst attempts to evoke with the medium of frottage.\n\nFIGURE 22\n\nPablo Picasso, _Head_ ( _T\u00eate_ ), spring 1913. 43 \u00d7 29 cm. New York, Museum of Modern Art. \u00a9 2012 Estate of Pablo Picasso \/ Artists Rights Society (ARS), New York.\n\nSURREALIST SIMULACRA\n\nIn the same year as the pear collage, Max Ernst completed more than 130 frottages. He selected 34 of these, had them reproduced in heliogravure, and published the portfolio _Natural History_ in 1926, in an edition from Galerie Jean Bucher.\n\nFirst of all, I should mention an influential change, already ascertained by Spies, that separates Ernst's surrealist pictures from his Dadaist overpaintings. These new works' underlay is no longer a mechanical-serial readymade (such as wallpaper, botanical models, or diagrams), but consists of surfaces such as wood grain, textured leather, fabric, or dried oil paint, which are indicative of natural processes. Through the technique of rubbing, these different textures transform into a homogeneous surface value. The reproduction of the frottages in heliogravure strengthens the homogenizing effect, first, because all the prints are given more or less the same format and, second, because the material accumulation of rubbing vanishes into the paper. The material heterogeneity of the Dada pictures closes them off from a unifying and enlivening mode of perception. By contrast, Ernst's surrealist frottages show a homogenous materiality, which also corresponds to a formal and iconographic uniformity.\n\nThe residual traces of rubbing provoke a seeing of resemblances that animates the picture surface. The repletion of these traces with imaginative possibilities also enables the other mode of seeing resemblances that Ernst already explored in the Dada pictures: \"seeing apart\" as the discovery of hidden figures. In _Natural History_ , numerous reversible images are concealed, of greater or lesser significance. As visual parapraxes, these beasts and goblins secreted into the contours of another pictorial motif remain secondary as compared with the vitalization of the entire surface of the frottage. For their part, however, reversible image effects underlie this seeing of resemblances, which animates the textures and infuses the picture as a whole with a ghostly vitality\u2014for example, in the sudden change from textured leather into the veins of a leaf or the leaves of a tree in _Confidences_ (fig. 23). But unlike the reversible image, which only allows seeing either one image or the other, in frottage the sudden change is not an excluding movement. The attraction lies specifically in perceiving both the leather and the leaf.\n\nFIGURE 23\n\nMax Ernst, _Confidences_ ( _Les Confidences_ ) ( _Histoire naturelle_ , print 17), 1925\/26. 42 \u00d7 25 cm (image), photoengraving from frottage (S\/M no. 806). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nA fossil tree grows out of a fossil leaf (fig. 23). Not only is the growth of petrifacts a parodic inversion, the relationship is also inverted between part and whole, leaf and tree. What's more, an inversion allows leaf and tree to grow out of textured leather (is it a ladies' handbag?). The plant kingdom grows out of the animal kingdom; the tree grows out of the leaf. This haunting growth that turns the natural order upside down arises from the mortifying technique of frottage, its fragmenting and fossilizing procedural logic.\n\nFIGURE 24\n\nMax Ernst, _He Will Fall Far from Here_ ( _Il tombera loin d'ici_ ) ( _Histoire naturelle_ , print 7), 1925\/26. 43.1 \u00d7 25.7 cm (measurement of image), photoengraving from frottage (S\/M no. 798). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nA scouring rush is placed in a landscape (fig. 24) and cut off by a line nearly as sharp as the horizon, which divides the page only slightly above it. One notices the jagged edge where the husk has been broken, through which no juices have flowed for a long time, and the charred surface more reminiscent of a fossil specimen than the skin of an organic body. But fragmentation and petrifaction are not just effects of the lunar landscape surrounding the stalk stub. The stalk may never have taken root\u2014it was a mortified growth from the start. Sure enough, exhausted movements can be discovered in the dead landscape. The title, _He Will Fall Far from Here_ ( _Il tombera loin d'ici_ ), provides a clue. There is a black mark on the horizon line: apparently the upper half is vanishing there right now\u2014or rather, the upper nine-tenths that has been broken off from the plant is vanishing right now. A sublime space opens up, as inconceivably immense as the stalk in its former state. The trajectory of the fall, at the point where the black mark cuts across the horizon, is still far from completed, as the future tense in the title indicates. At the moment when this trajectory fills the pictorial space with motion, it strikes us with greater clarity how labile the position of the stalk stump is in the landscape; indeed, its surface, despite the place where it has been broken off, is like a fleeting apparition, and even the platform of the landscape pulsates eerily.\n\nFrottaged growths never take root. Their origin is a procedure that generates still-inchoate, unstable illusion effects, located in a fossil world and isolated through the sharp edges of the picture. The cut, which in the pear collage was still opposed to frottage as a competing procedure, in _Natural History_ has been absorbed by frottage, has in a sense been naturalized\u2014either through an underlay so rich in texture that the incisions sink into the ground and become one with the frottaged world (fig. 47), or else through identification of the incision with the interior or exterior boundaries of the picture, with horizon or format. In print 17 (fig. 23), the cut on the left duplicates the boundary of the pictorial field, in order to support the seeing of resemblances and the effect they provoke of an eerie growth. The tree seems to grow out of the monstrously huge leaf; both hover as botanical phantoms above the horizon; the tree also seems to advance in the direction of the viewer by crossing the duplicated pictorial boundary.\n\nFrottage is a procedure that isolates the objects it represents from any context or presupposes this isolation. Like photography, it can be understood as indexically reproducing sections of reality and fragmenting the image of the latter. Its most suitable object, for whose scientific documentation frottage is put to use, is the archaeological or paleontological fragment\u2014or, if we recall that rubbing was the infantile precursor of photography, the single coin or the fallen leaf. The object (in _Confidences_ , a piece of leather) is laid underneath a sheet of paper, which is rubbed with soft graphite; the sheet may also be turned on the horizontal working surface. The capacity of the paper to be turned\u2014in the tree, one notices the branch at the bottom left\u2014refers to the way in which frottage, by fragmenting the world, makes it horizontal and tactile, and thus brings the fragment intimately near the viewer's body. It is thus all the more striking that tree and leaf are presented upright, as visual apparitions.\n\nThe leaf above all can be seen in a double way: as the pictorial support or as the figure of a leaf. On the one hand, it is nearly identical with the drawing paper, as if it were the tactile ground from which the figures of the frottage arise; on the other hand, it raises its unwavering central ridge into the vertical, and from there confronts the viewer as a monumental vision. The fragments, rootless tree and severed leaf, take part in a new form of continuous becoming, a type of growth that is not grounded in the soil and that can be better described as a hallucinatory blossoming. This growth expands into the depth of the pictorial field as well as into the space in front of the picture. The landscape of the background shimmers through the leaf; the tree, surrounded by a white aura, emerges from the tactile ground of the leaf and, on the left, departs from the drawing paper.\n\nTo reiterate, it is the seeing of resemblances as material vitalization that causes a penumbrous growth. This seeing of resemblances apprehends a world of second order\u2014and in this way, as in its incorporation of the cut, it essentially differs from the seeing of resemblances instigated by Hans Arp: it apprehends a world that has turned to stone and that consists only of petrified images. Arp's accidental and pre-morphous inkblots aim to be understood as an originary ground and to be animated into unstable apparitions. For Arp, the seeing of resemblances proceeds with the hope of arriving at the pre-morphic origin of images. By contrast, Ernst's approach is secondary. The rubbed textures are dead natural remains and are preserved through a mortifying technique of imprinting. The images that arise from these textures are less originary images than after-images of living nature.\n\nThis secondary, spectral animation, as well as the indeterminate location of objects near the picture plane, brings the frottages into context with a mode of representation that traditionally had been attempted with still lifes: namely, _trompe l'oeil_. Louis Marin characterizes _trompe l'oeil_ as a simulacrum and distinguishes it from classical representation by using the criterion of presence. While representation is founded on the absence of an object in order to represent that object in its ideality\u2014an ideality and recognizability produced through the transparency and distancing capacity of the medium of representation\u2014so this object made available through its absence becomes present again in _trompe l'oeil_. However, its presence is difficult to locate. If _trompe l'oeil_ materializes the transparent picture plane, does it hover in front of or come into appearance by passing through it? If the medial boundaries become porous, then the distance to the represented object as ensured in representation collapses. In this way, _trompe l'oeil_ takes on an obscure, spectral power, as if the object were present in the image of its own accord\u2014as a _doppelg\u00e4nger_ or simulacrum. D. E. Wellbery provides a synopsis of these thoughts:\n\nIn simulacrum, representation, which in mimesis is transparency to an ideal spatial depth, gains its own density; the dead (or absent) object becomes living, achieves the status of a presence, though without entirely divesting its unreal character. Mimesis separates the actuality of the observer from the virtuality of the observed through the transparent but nonetheless rigorous differentiating boundary of the medium of representation. By contrast, in simulacrum, this boundary is condensed to an imposing presence that is neither simply real nor simply unreal (ideal)\u2014it is condensed to a reality of representation, with an indissolubly ambiguous, often ghostly character.\n\nThis ghostly, neither real nor ideal effect of the presence of a phantom is termed \"surrealist\" by Marin. However, it would be too simplistic to apply Marin's considerations directly to Ernst's pear still life and subsequently to his surrealist frottages. Too simplistic, because for Marin the simulacrum opposes the classical concept of representation, which was displaced in the romantic period at latest and certainly was no longer in effect as an opposing concept for the surrealist pictures. But some opposing concept is necessary, since phantoms are known to be secondary phenomena. Ernst's surrealist simulacra are after-images of mimesis as it had functioned since the romantic period, and especially as it had served as the ideal of art in the abstraction of artists such as Hans Arp (fig. 2). They are after-images of mimesis as the reenactment of nature through the materiality of the image.\n\nThe simulacral character of frottage expresses itself in and is strengthened through a thematic analogy: _Natural History_ knows life-forms exclusively as fossils, and its mode of production generates exclusively fossil images. In the frottages, with their prehistoric image world, the artist's rubbing and imprinting technique imitates the processes of fossilization that take place inside the earth. It is critical that these mimetic analogies become visible not only as product, but also as process: Ernst not only assembles his leaves, trees, and birds from foreign textures; he allows these figures to grow out of the textures (leaf veins from nubs in leather) without causing the difference between them to vanish. In the completed image, there is a latent iconic excess that inheres in the wood, leather, and other materials or patterns. Thus, the viewer's seeing of resemblances can repeatedly give in to the fiction that it is reenacting the seeing of resemblances carried out by the artist, and that this seeing acts like fossilizing nature in generating images out of decomposing textures. The peculiar culmination of this processual analogy between the seeing of resemblances, artistic production, and fossilizing nature is that the mimetic verisimilitude is enhanced through an evident nonresemblance. I am speaking of the difference in speed between deep-time processes and the rapid production of pictures using the rubbing procedure. In the Dada works, the exact opposite was the case: mimetic verisimilitude was called into question by the non-resemblance of sublime nature and small-scale bricolage. But now, the child's technique of rubbing, which can be accomplished in an instant, releases prehistoric nature. The sudden appearance of figures on a sheet of paper under the frotteur's busy hands is a simulacrum that grows out of mortified trees and leaves.\n\nBoth the Dada pictures and the surrealist frottages follow an anachronistic movement that underlies Ernst's artwork as a whole. They fossilize the imitation of nature or let it return after it has been destroyed through diagrams, cuts, and image readymades. As parodies of mimesis, however, they clearly differ in their degree of severity. The Dada pictures present a labyrinth of mimetisms that advances toward the dead point of serial-mechanical repetition, whereas the frottages generate mortification in order to glean from it a type of surrogate organicity. In the Dada pictures, the production-aesthetic parapraxes\u2014extraneous traces of ink, retouchings, thickenings of line\u2014produce metaphoric condensations of natural process and artistic procedure. However, these never seal into a unity that organizes the picture. Instead, they are split into a double discontinuum: for one, between actual and simulated procedure (overpainting, diagram, collage); for another, between naturalistic and mechanical analogies of production (geologic layering, anatomical cross-section, superimposition of stage sets, stacking, stamping). In terms of the logic of production, this discontinuous formation of analogies refers to the mechanical-serial underlay: pictures that were made by overpainting wallpaper or diagrammatic pictures and grids reproduce the cuts of their underlay\u2014in the procedural cuts of overpainting, collage, and diagram, and in the depicted cuts of geology, anatomy, mechanics, stacking, and the construction of sets.\n\nIn the frottages, by contrast, a naturalization of the cut can be observed\u2014either through its immersion in rich textures (fig. 47) or through its function as the image boundary (fig. 23). As in the overpaintings, it is also possible to distinguish two image layers in the frottages. However, in the frottages there is an overall pulsating exchange between underlying textures and image effects (nubs in leather and leaves); and the underlay is no longer a serial-mechanical image readymade but consists of materials indicative of nature or natural processes (wood, leather, fabric, dried oil paint, etc.). The frottage transforms dead remains into a homogeneous texture and thus provokes an enlivening seeing of resemblances, which stimulates a spectral\/hallucinatory growth in the mortified residue. This growth is supported by a close production-aesthetic metaphorics (frottage\/fossilization) and a potent diaphorics (fast art\/sublime slowness of nature). As simulacra, the frottages parody modernism's mimetic ideal\u2014that what is demanded of the artist or the artist's works is an active re-creation of natural processes. The frottages bring into view a secondary growth, emaciated and mortified; the only growth that, after the distorting and cutting repetition of the Dada works, is still or is once again possible.\n\nThis surrealist afterlife of the imitation of nature, which follows upon the Dada pictures and originates in them, also transforms the illusionism of Ernst's works. In the Dada pictures of 1920\u201321, individual characteristics of traditional illusionism are in evidence, though in a prefabricated and distorted state. These include the neutral rectangular format, the perspective grid and the platform, materiality and weight\u2014fossil remains of the conditions that in illusionism provided for the effect of sensory plenitude. In the Dada pictures, sensory plenitude is instead only evoked by the hard-to-locate textures of the underlay, confusing attempts at modeling (fig. 12), cuttingly sharp shadows (fig. 10), or disgusting details such as the erect intestine in _Stratified Rocks_ (fig. 52). These attempts appear trapped in a remarkably dully painted, sensually deadened, rigid image world without tactile stimuli. By contrast, frottage, the technique of rubbing, brings the sense of touch back into play: textures and patterns appear through the interplay of pressure and resistance, soft graphite and hard ground. When the drawing paper is raised from the horizontal into the vertical, from the working surface into the frame, the tactile stimuli (wood grain, leather, fabric) transform into optical manifestations (sea swells, leaves, trees, animals, pupils; figs. 23, , 28\u201335). This metamorphosis of the seeing of resemblances releases the fossil world from its rigidity and animates the mortified spaces with the eerie afterlife of the imitation of nature. Illusion and imitation of nature, the visual and the tactile, form a unity again in the frottages. Nevertheless, this is a unity of the second degree, which comes out of a divide and glosses it over, in order to re-create what to the young Max Ernst in 1909 was still self-evident: the impressionist interplay of eye and hand.\n\nAccording to Richard Shiff, the interplay of hand and eye in impressionism is based on the precondition that in the act of painting the body of the artist expresses the singularity that guides both eye and hand. The artist's idiosyncrasies are manifest the same way in both the gaze and in the brush. In Ernst's frottages, the visual and the tactile likewise seem to form a unity: in the rubbing back and forth across a richly textural underlay, images arise that convey visually their origin in the sense of touch. The seeing of resemblances provides this mediation. It encounters a world that is not antecedent to perception but that takes form in perception's course: the wood grain begins to flow, the rough textile evokes an animate armor, or the leather pattern changes from the scaly bark of a prehistoric lepidodendron to the pulsating skin of a reptile (fig. 23). The world suddenly seems near, replete, and formable, in exchange with a tactile gaze that gives itself over to this world and makes the de-organicizing action of a fossil nature reversible. The precondition for this visual mediation of a tactile origin is the vanishing of tactile heterogeneity from the pictures. While the Dada pictures place gouache next to photography, pencil next to woodcut, stamping next to a dull area of paint, the surface of the frottages seals itself in seamless homogeneity and is additionally unified through the reproduction procedure of heliogravure.\n\nHowever, the unity of hand and eye is also divided\u2014or, more precisely, it remains palpable that the mediation of this unity plays upon a divide. In the impressionist ideal of production, the brush reacts spontaneously to the seen; and the ephemeral nature of the seen\u2014the surfaces of a body of water, trees and fields in the wind, atmospheric phenomena, and above all the play of light\u2014corresponds to the spontaneous traces of the brush. This continuous interplay of image and nature, oil paint and light, is interrupted in frottage. The hand rubs textures that lie underneath the paper and thus are withdrawn from the eye. Only after the fact does this tactility come into connection with the seen. The sensory plenitude of frottage, its connection of the visible with the graspable, the eye with the hand, plays upon a dissociation grounded in the procedure. This dissociation was often demonstrated in the Dada pictures: procedures such as stamping, reprinting, or overpainting are radical negations of the impressionist ideal according to which a body, unified on the basis of its idiosyncrasies, comes into expression. The frottages are images of a divided and deferred, parodistic rescue. What Ernst in 1909 was seeking outside in nature leads a ghostly afterlife in his surrealist natural history.\n2\n\n_NATURAL HISTORY_ IN SERVICE TO THE SURREALIST REVOLUTION\n\nFrottage came into being as the answer to automatic writing ( _\u00e9criture automatique_ ), which itself was the foundation myth of surrealism. Automatic writing is a literary practice directed toward the social totality and utopia that Lautr\u00e9amont famously demanded: \"Poetry should be made by all. Not by one.\" As an expression of the \"actual functioning of thought\" and simultaneously an \"unrestricted language,\" automatic writing claims to be a perfect communication in which the \"I\" is dissolved and individual understanding disabled. In their supposed absence, a dark power, conceived with and against Freud as the unconscious, talks to and speaks for itself. Automatic texts, which comprise only a small percentage of the bodies of work of the individual poets, are thus granted less significance in surrealism than is automatism as a practice. The origin of the surrealist revolution lies in the performance of automatic writing as the individual rehearsal of an unconscious communication that would prove its social efficacy in the surrealist group as a whole, by first constituting it as such. However, as the history of the surrealist movement demonstrates, the origin of the surrealist revolution also proves to be the surrogate of action. The various disappointments that Andr\u00e9 Breton and his followers suffered in contact with political revolutionaries repeatedly led them back to automatism, which demanded not agitation and organization, but sleep, idleness, and the absence of any goal.\n\nIn determining the relationship of frottage to automatic writing, the \"communal purpose\" of the foundation myth should not fall out of view. Before I turn to the question of how Max Ernst placed his _Natural History_ in service to the surrealist revolution, it is necessary to discuss in greater detail the problems and particularities of the production and contextualization of automatic images. The tension characteristic of the surrealists' social performance\u2014the status of automatic writing as both origin and surrogate of action\u2014is conspicuous in the surrealists' poetics, both in formulation in manifestos and in execution in literary texts and images. This tension gives form to the controversy surrounding the possibility of an automatic visual art.\n\nFIGURE 25\n\nMax Ernst, _Lesson in Automatic Writing_ ( _Le\u00e7on d'\u00e9criture automatique_ ), 1924. 17.3 \u00d7 169 cm, pen and ink on paper. Private collection (S\/M no. 564). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nORIGIN AND DOCUMENT\n\nWith the frottages, Max Ernst succeeded in achieving what his surrealist friends would recognize as his first contribution to a genuine visual automatism, and with the grattages of the following year, he was able to repeat this success in oil painting. In 1924 he had tried something completely different: with a quick and careless stroke, he had drawn various enigmatic motifs on a roll of paper, supposedly analogous to the continuity of the automatic stream of images (fig. 25). In this way, he dynamized the pictorial concept that governed his oil paintings between 1921 and 1924: the naive, artful\/artless portrayal not of objects, but of prefabricated images. _Oedipus Rex_ (figs. 13, ) is a picture of a picture because it is based on a collage of wood engravings. More importantly, it is a picture of a picture because the viewer, who generally doesn't know what it is based on, believes that he or she is witnessing the d\u00e9j\u00e0 vu of a preceding, mental image. As influential as the secondary character of Ernst's pictures was for surrealism in its emergence between 1921 and 1924, after the _Manifesto of Surrealism_ ( _Manifeste du surr\u00e9alisme_ ; 1924), his poet friends expected a new art. The paper roll was not to be granted a future as the supporting medium of automatism. Besides, the arrival onto the scene of Andr\u00e9 Masson promised a new beginning beyond the secondary painting of artists such as de Chirico or Ernst.\n\nThese expectations for visual art were articulated in a controversy that erupted following the publication of the _Manifesto of Surrealism_ in 1924. Concerning the question of whether there was such a thing as automatic painting, the dispute was set off by Breton's having nothing to say about it. After important essays in previous years on Ernst, de Chirico, Picabia, and Andr\u00e9 Derain in which Breton had stressed the centrality of painting to the avant-garde, in the _Manifesto_ he afforded the medium only two footnotes. His mistrust of painting was of an ethical nature\u2014who could be of any use as a revolutionary after having experienced success in the art market?\u2014but it also implied a theoretical problem. Painting made a question obvious that the poet had glossed over in his declarations: the question of the medium of the unconscious. How can one ever be sure that the inherent effects of a particular medium do not distort or completely occlude the messages of the unconscious? How can it be guaranteed that in sentences received as the direct precipitate of the unconscious, it is not actually just the material of language that is speaking?\n\nOne way to dispel doubts about the transparency of language is to thematize these doubts using the example of another medium\u2014that is, painting. In the _Manifesto_ , Breton advises that during half sleep, as surrealist images appear on the surface of consciousness, their outlines should simply be traced. This mechanical tracing of outlines ( _calquer_ ) is preferable to drawing because it disables the practiced conventions of the hand and the interaction with the controlling eye. The traditional culture of drawing was suspected of pre-forming the expressions of the unconscious. The uncoupling of eye and hand, censoring consciousness and executing instrument, thus became the main concern of early surrealist discussions of painting. In the first edition of _La R\u00e9volution surr\u00e9aliste_ (December 1924), Max Morise explicitly polemicizes against the traditional skill that falsifies the images of the unconscious. As an alternative, he calls for mechanical documentation and cites Man Ray's photograms as an example.\n\nOutlines like those that Breton recommends and photochemical imprints are both indexical images. Their causal relationship to the represented guarantees their documentary reliability. However, Morise goes beyond Breton, since his interest in the photogram is based implicitly on its difference from the outline. Both types of image are traces. But while Breton's outline records a mental image that already has all the qualities of a surrealist image, the photogram is a material image that possesses all surrealist qualities in itself. The outline is a mechanical and thus reliable materialization of an original mental image, which is the real object of interest. The photogram is likewise a reliable documentary trace, but as a trace it simultaneously _is_ the original surrealist image and does not refer to any pre-photographic, mental image. In other words, in the photogram it is possible to lay claim to two diverging types of authenticity: that of the documentary imprint and that of the originary, ideal image. The particular credibility of outlines, imprints, traces, and other indexical images as testaments to objects, events, or other images is commandeered by the photogram for an image that exists only as the photogram, without reference to anything preceding it. The surrealist photogram thrives on the credibility of a document produced through a mechanical procedure and is simultaneously the source where the image first appears. It testifies to the past (the past presence of the thing whose trace it documents) and simultaneously _is_ the image of a present moment as it unfolds.\n\nWhat are the consequences for painting and drawing? Morise does not discuss this point explicitly, but it can be deduced from his text that he does not grant them the paradoxical unity of being both the document and source of the surrealist image. They remain either pure document or pure source. For documentation, a \"quick and rough drawing\" suffices to fix mental images. This possibility offers only a mediated access to the treasures of the unconscious. As opposed to this split between automatic expression as mental event and the registering of this event, Morise devises another, better option, which he calls \"a real Surrealist procedure\": \"Forms and colors free themselves from objects and organize themselves according to a law that evades any premeditation, a law that is at once made and unmade when it emerges.\"\n\nThe first, worse option\u2014the \"quick and rough drawing\"\u2014was in evidence in Ernst's most recent foray, the roll of paper with its image notations that refer to a stream of mental images. Morise had had the chance to see these works in September 1924 at surrealist headquarters; plainly, he did not consider them a challenge for Masson, the painter who in 1924 embodied the future of surrealist painting. One of Masson's drawings was used as an illustration for Morise's essay. These drawings, called \"automatic,\" present what is lauded as the \"real Surrealist procedure\": the separation of forms and colors from the object, and their commitment to a principle that comes into and goes out of operation to the same extent that it is expressed. In the year of the _Manifesto_ , the surrealists, who polemicized against any kind of artistry, be it the poet's \"main \u00e0 plume\" or the painter's m\u00e9tier, put their trust in the hand of a virtuosic draftsman. Morise's statement expresses a pronouncedly modernist rejection of secondary images, combined with the requisite demand that the painter invent a medium in which the unconscious can precipitate directly. Morise thus presents painting with the alternatives of making either secondary documents of the surrealist image or original surrealist images. Apart from painting, in the photogram he recognizes a third possibility that paradoxically unifies both forms of authenticity, the originary form of the ideal image and the secondary form of the document.\n\nWith frottage (and later with grattage), Ernst attempted to avail himself of the merits of the photogram and transfer these into a technique of drawing (or painting). As an infantile mechanism through which images seem to appear of their own accord, frottage evokes an automatic originality and spontaneity of image production. As an indexical imprinting procedure, it always makes the claim of objective reproduction unfalsified by artistic convention.\n\nIt could be asked, objective reproduction of what? In archaeology and paleontology, frottage is used for the documentation of remains. In Ernst's pictures, not only remains but also the images arising from them\u2014not only the past, but also the unfolding present of an eerie growth\u2014take on the value of the documentary. But this growth, produced by the originality of the image within the entirely secondary world of frottage, is only achieved in the imaginative perception of the viewer. In the seeing of resemblances, the surrealists were able to recognize the \"actual functioning of thought\" at work. Thus, Ernst's medium made a threefold promise: to record faithful documents of automatic images, to visualize the spontaneity and originality of these images, and to experience this visualization as an effect not of the medium, but of psychic processes.\n\nIn what follows, through a discussion of other topics relevant to surrealist art, from the paradigm of the written to automatism's social aspect, it will be shown how Ernst appropriated the poetics of automatic writing in _Natural History_. In modernism, it was rather extraordinary that a visual artist should attempt to partake in the foundation myth of what at first was a purely textual activity. In fact, the most important artistic positions from impressionism to cubism made themselves out to be actively anti-literary. Ernst, who made many of his best works in other media, surely had his own reasons for preferring the literary illustrations of Max Klinger or Odilon Redon over pure painting. At the same time, however, surrealism\u2014in distinguishing itself with great firmness from \"merely literary\" movements and thus condemning any formalism, whether in poetry or painting\u2014offered the possibility of a new form of _ut pictora poesis_. It proclaimed a shared foundation myth, the activity of automatism, which was intended to be rehearsed in pictures and texts, but not to be confined to them. Performance and rehearsal require a stage. Unlike acting out, they are skilled at staging a scene, a skill that never conceals itself, even as it keeps other artifices from view. According to everything that can be observed about Ernst's working method\u2014and in light of his virtuosic use of many different procedures and the way he metaphorizes, disguises, and thematizes them\u2014it is no wonder that his appropriation of automatic writing was often carried out as the staging of a scene\u2014as the theatrification both of automatic effects and of the paradoxes that result from the tension, characteristic for automatic writing, between original and document, source and surrogate.\n\nAUTOMATISM AND THE WRITTEN\n\nMax Ernst's _Natural History_ repeatedly tempts the viewer to place its images into a rich intellectual-historical frame studded with fragments from many learned places, from _Ars combinatoria_ to Novalis to psychoanalysis. The surrealists were themselves the first to construct wide-ranging contextual fields and become intoxicated with the historical overtones of their iconography. In the following, I would like to discuss the influential topos of the writing of nature\u2014not to join in the game of surrealist erudition, but to demonstrate its relevance for the production of _Natural History_. As in the chapter on mimesis, here again it is helpful to discuss the frottages' relationship to nature in contrast to that of Ernst's Dada pictures.\n\nIn 1920 several works were made using rubbing. In _Typescript Manifesto_ ( _Typoskript-Manifest_ ), written together with Johannes Theodor Baargeld, Ernst included a frottaged coin (fig. 26). The two Dadaists assembled a catalog of different reproduction techniques (traced letters, test prints of commercial intaglio plates, steel engraving, photography, and typewriting), to which frottages were also added. However, typewriting was definitive. Language disintegrates when the keys and roller have free play. More than the repeated characters, the carriage returns, and the letters that have been struck with excessive vigor, one's attention is held by a strip without any type that cuts sharply through the text, produced by a fold in the paper when it was pressed together by the roller. In this strip, an invisible zone expands and rises to the surface of the pictorial field. This invisibility is constitutive for writing with the typewriter, since the exact place being marked remains hidden from view. The little frottage in the margin exemplifies another printing technique, which, like the typewriter, separates the eye from the hand in a similar way. Like typing, the mechanical hatching makes something visible that is hidden by the apparatus. The sign appears exactly in the shadow of the conditions of production, within the blind spot that, in typing or frottage, prevents the interplay of eye and hand.\n\nFIGURE 26\n\nMax Ernst, _Typescript Manifesto_ ( _Typoskript-Manifest_ ), 1920. 28.5 \u00d7 22.5 cm, collage and frottage on paper. Private collection (S\/M no. 335). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn 1920, during the Dada years, Ernst made an analogy between frottage and typing. In _Typescript Manifesto_ , typewriting is the paradigm of an uncoupled production mode that takes place in the shadow of the apparatus. In contrast to the automatic writing of the surrealists, this automatism does not allow any glimpse of liberation. The uncoupling of eye and hand is not, as later in surrealism, a precautionary measure for the protection of an originary spontaneity that could be restricted by the censorship of the reviewing gaze. Instead, it is the effect of an apparatus that compels mechanical repetition without any origin. If typewriting is the paradigm of Dadaist writing, then Dadaist nature must also be a user of discrete and efficiently repeatable signs. This is the real point of the overpainting _Frozen Landscapes_ (plate 1). In the first chapter, it was demonstrated how this picture stages its own production as reproduction: in the hatching, the rock strata, the machine, and finally also the stamp marks, the same principle of mechanical repetition is expressed that produced the wallpaper pattern of the pictorial underlay. But until this time, it has gone unmentioned that _Frozen Landscapes_ depicts a communication machine. This is evident in the large telegraph pole to the right, and its small offspring to the left side of the picture. The stamp marks would then be the regressive collapse of Morse code or the babbling of discrete and efficiently repeatable elementary signs. It stands to reason that a nature that produces landscapes that in turn reproduce the structure of wallpaper would also write in serial signs, or in the ruins of them. And it likewise follows that the body correlative to this nature would be broken down in strata. The complete title of the work is _Frozen Landscapes, Icicles and Mineral Types of the Female Body_ ( _eislandschaften, eiszapfen und gesteinsarten des weiblichen k\u00f6rpers_ ). This title suggests that a picture that destructively stages its own (re)production also destroys that which in the romantic tradition is understood as the origin of artistic subjectivity: nature as a female body. By making himself into a machine and stamping his picture, the artist is mocking the fact that the trace of the brush was once understood as an expression of artistic subjectivity. His facture is not _his_ at all; it is a testament neither to artistic mastery nor to a unique, natural sensibility, but instead is the destroyed code of a nature, or a woman, which it- or herself is a deconstructable body composed of discrete layers.\n\nThe break between Dada and surrealism as described in the first chapter is particularly clear in view of the written quality of Ernst's pictures. In _Typescript Manifesto_ , frottage is made analogous to typewriting. The uncoupling of eye and hand that takes place in the shadow of the apparatus is the commonality of the two printing procedures. When Ernst begins to test frottage as an artistic technique\u2014in the Dada works, frottage only provides a sample for the catalog of different mechanical procedures\u2014he becomes particularly interested in the qualities that make it different from typewriting, with a primary difference being the temporality of frottage. Whereas typewriter letters hurtle toward the page with the greatest possible speed, frottaged signs appear gradually. The typewriter is a radically spatialized medium; frottage takes time. And more importantly, in the frottages, and particularly in the prints of _Natural History_ , this temporality has not yet come to rest. It is always being punctuated by the seeing of new resemblances. In this way, the frottages achieve their genuine ambiguity as imprints and simultaneously as images that are constantly showing their ghostly emergence in a new way.\n\nErnst's surrealism emerges in the distance that separates frottage from the Dadaist paradigm of mechanical writing. Does his art thus definitively remove itself from a poetics of the written? Following the publication of the _Manifesto of Surrealism_ , the debate in the surrealist group about the position of painting particularly concerned the ideal of automatic writing. The written record was considered the ideal, because the most transparent, medium for the unconscious flow of images. With his drawings, Masson furnished evidence that the tools of the surrealist poet\u2014pen, ink, paper\u2014would also produce an automatic visual art (fig. 27). The virtuosic animation and aggression of his stroke could be interpreted as the return to writing's unconscious origin in a blind scrawl. Automatic drawing liquefies writing, transforms it from a medium of representation into a fluid substance of the unconscious. This liquid metaphorics dominates the morphology and motifs of Masson's drawings, which often present the flooding and destruction of architectural elements, as though architecture were the symbol of the established culture that opposes the unbounded stream of automatism. That the topos of the writing of nature assumes a distanced, semiotically mediated relationship to nature is glossed over in the metaphors of drifting, submersion, and deliquescence. To read this nature is to get lost in it.\n\nFIGURE 27\n\nAndr\u00e9 Masson, _Automatic Drawing_ ( _Dessin automatique_ ), from _La R\u00e9volution surr\u00e9aliste_ , 1925. \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe seductive power of the fluid continues to be felt in autobiographical retrospect. After the war, Masson would write that for the graphic expression of psychic automatism, only the following is needed: materially, paper and ink; psychically, inner emptiness, from which the automatic signs emerge \"like an unforeseeable birth.\" The white sheet of paper and the empty consciousness, the ink and the dark origin of unconscious images\u2014Masson's drawing is intended to be understood as the writing of an inner nature, whose element is a dark and restless sea contaminated with all manner of flotsam.\n\nIn the frottages, another nature is writing, with other instruments. It does not employ the typewriter, as it does in the Dada pictures, or ink, whose opaque depths it assumes in Masson's drawings. It makes do with a relatively limited number of elements in its alphabet. (The various textures can be termed an \"alphabet\" because Ernst only needed a limited number of them to produce more than 130 frottages in 1925.) Leather, straw mats, wood grain, dried oil paint, a diamond pattern, a star pattern, and twine comprise the hieroglyphs in which _Natural History_ is written. While Masson recorded the writing of a nature not yet divided into discrete signs, the frottages are constructed out of a mortifying writing. Their elements become written characters because they are the death masks of objects. Only the mortification of the imprint makes the pieces of wood or twine into the repeatable, movable, combinable letters of natural history. At the same time, as has already been emphasized, this mortification preserves a growth that constantly reappears as a simulacrum in the viewer's seeing of resemblances.\n\nErnst's ghost-writing of nature, achieved through mortification and vivid as a death mask, is characteristic of the surrealist depletion of the traditional topos. Breton, too, only knows the writing of nature as a simulacrum, as is demonstrated in an automatic text published in 1924 in _Soluble Fish_ ( _Poisson soluble_ ). At the beginning of the text, it is observed that the ground is an unfolded newspaper and that even the flowers smell like printers' ink. Two sections later, Breton discovers at the bottom edge of the newspaper's fourth page a curious fold with traces of rust from a metallic object. An imprint in a newspaper is the meeting of two surrogates that present the real world, the world of objects and events, as past and absent. The surrealist hope is precisely that reality's surrogate creations will make space for those desires that are unfulfilled in reality. Thus Breton devotes himself to interpreting the stains that bear witness to the past encounter of a newspaper and a metal object, and, by means of the imaginative seeing of resemblances, discovers in them surrealist figures. They are metonymies of the fantastic and of love: the rust stain resembles a forest (the romantic topos of the fantastic _par excellence_ ), while the object that left the stain may have been part of an Empire bed. To the Surrealists, the world of surrogates gives intimations of a hopeful future. Nature is an old, stained newspaper, but in the traces of rust, a forest of love is thriving.\n\nAt once imprint and origin, at once the ruin writing of nature and nature's afterlife\u2014in his frottages, Ernst takes up these contradictions of automatic writing and transforms them into pictorial effects. He not only assimilates the explicitly formulated directives that the poets attempted to give to the painters; more importantly, he reacts to automatic writing as it was practiced, for example, as documented in _Soluble Fish_. The problems of the new poetics that are glossed over or pushed aside as the special problems of surrealist painting in the _Manifesto_ are embraced and embellished through narrative or allegory in these automatic texts. Is automatic writing the notation or actualization of the unconscious flow of images? Is it document or origin, surrogate or inspired testimony? In _Soluble Fish_ , this aporia is negotiated similarly as in the frottages. In a world of mechanical surrogate creations, a world that is nothing more than an \"enormous unfolded newspaper\" where \"sometimes a photograph comes by\" and where the trees are headline type, the imaginative seeing of resemblances allows the blossoming of surrealist images of love and the fantastic.\n\nSURREALIST IMAGE AND AUTOMATIC SERIES\n\nSo that the signs of this new surreality can show themselves, the old reality must first be fossilized. The distinction between sign and thing, as it hardens into a surface, also begins to crack. Sign and thing cross through one another on the same mortified surface, since the signs (the printed letters of a newspaper, the imprint of an unknown object) are simultaneously the objects that comprise the world of the surrogate. There is also a penetration of these different levels in Max Ernst's frottages. The leather texture in print 17, the wood fiber in print 1, and the diamond pattern in print 7, with their characteristic instability, are both the means and the motif of representation. In the first image of _Natural History_ (fig. 29), it is a picture within a picture that inserts the world of signs into the world of objects. The slight foreshortening of its edges locates it within the stormy landscape, though it is clearly set off from this landscape as an independent image. In _Caesar's Palette_ ( _La palette de C\u00e9sar_ ; fig. 28), the leaf appears twice, once as a frottaged impression that lies on top of a rectangular field, and once as a gap through which the paper ground tears a hole in the pictorial world. Likewise, what belongs within the pictorial fiction and what should be counted among the conditions of its possibility cannot be determined. This indecision as to whether a white gap should be perceived literally as a piece of the material ground or metaphorically as an opening in a visual\/transparent world can be compared with the quality of automatic writings termed by Laurent Jenny as their \"open\" or \"undecidable figurativity.\" One consequence of the audacious metaphorics of surrealist texts is the breaking down of the distinction between literal and figurative meaning. If a metaphoric relationship can be established between anything and anything else, then it is no longer possible to decide where the literal level enters into a text. Automatic writings cancel the \"rhetorical contract\" that makes significant deviations\u2014and thus figurativity\u2014articulable. Are we perceiving (fig. 23) a plant with a snakeskin or a dissembling resemblance between leather nubs and leaves? It cannot be decided.\n\nOn an iconic level, the interpenetration of representation and the means of representation corresponds to an opening of the borders that divide nature into kingdoms. In 1927 Breton would write in _Surrealism and Painting_ ( _Le surr\u00e9alisme et la peinture_ ) that the notion of the three kingdoms of nature is \"the height of absurdity.\" As evidence, he cites the phenomenon of mimicry. Like the insect that lands on a branch in the form of an animal and flies away again in the form of a leaf, the leather nubs that grow into a leaf and a tree in print 17 (fig. 23) also gloss over the difference between the plant and animal kingdoms. Frottage is thus enriched by a contradiction: not only is it simultaneously document and presence, ruin writing and haunting growth; its medium of representation and pictorial motif also penetrate each other\u2014and in turn, this penetration of sign and signified corresponds on an iconographic level to the interpenetrability of the kingdoms of nature.\n\nFIGURE 28\n\nMax Ernst, _Caesar's Palette_ ( _La palette de C\u00e9sar_ ) ( _Histoire naturelle_ , print 20), 1925\/26. 43 \u00d7 26 cm (measurement of image), photoengraving from frottage (S\/M no. 809). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn the _Manifesto of Surrealism_ , Breton famously describes an experience in which he entered into a hypnagogic mind state. First, he was visited by the image of a man \"cut in two by a window,\" which is obviously an allegory of the surrealist penetration of representations (such as the image of a man) by their own frame (the window). However, such a representation fragmented by its own conditions can only count as automatic if it generates a series of new images. With regard to Ernst's _Natural History_ , the question arises as to why a picture series committed to the ideal of unconscious production tells the story of a prehuman nature. Automatic writing opens up the present to the prehistoric deep time of the unconscious, out of which grow the \"inadmissible flora and fauna\" that Breton hoped to encounter on his expeditions into the primordial forest of automatism. Surrealist texts in fact abound with monstrous\/seductive dragonflies, traditional or invented hybrid creatures such as the \"lyre bird\" and the \"rain bird,\" modern sphinxes stationed between prehistoric ferns. Creatures that confound traditional classifications of nature or cross the boundaries to human culture embody the audacious metaphorics of surrealism\u2014the necessary encounter of the most dissimilar things in the intersection of representation and its preconditions. For this reason, the programmatic invocation of a new nature is one of the customary incantation formulas of an avant-garde flirting with the Orphic power of the word. Looking back at Breton's first attempts at automatic writing, Louis Aragon sees his friend posing as a fantastic hunter who exhibits his prey, a \"reinvented zoology, a reinvented botany.\" The fascination with a deep time that allows new genera and species to emerge is accompanied by the desire for an unspeakable nature unseen by human eyes, which, in its prehistoric alienness, prefigures the future of the surrealist revolution.\n\nThis use of natural history to exhibit their own eloquence could be read as evidence that the surrealists only knew nature as an available material. However, this is not the case for either Breton or Aragon. Breton's digressions into natural philosophy attempted to demonstrate that the writing of nature also proceeds according to automatism and that surrealism was thus grounded in an originary unity of subject and nature, which could be regained. In _Paris Peasant_ ( _Le Paysan de Paris_ ), Aragon proposed a grounding of surrealism in transcendental philosophy: Nature is not a circumstantial given; it is a subjective construction, though its implications and consequences remain unknown to the empirical subject. Thus, nature is the subject's unconscious, which only reveals itself at \"rare thresholds.\" In his writings (which I will discuss in some detail in chapter 3), Ernst provides a psychoanalytic explanation for how it is that natural history simultaneously recounts the history of the subject. In what follows, I would like to present some of the guiding principles of surrealist prehistory. These can be particularly well studied in transitions similar to the \"rare thresholds\" that, according to Aragon, signify the unconscious: that is, in transitions between the image and its conditions.\n\nAt the beginning of surrealist natural history a storm is raging. _The Sea and Rain_ ( _La mer et la pluie_ ; fig. 29) is the title of the first print. Sea and rain belong to the repertoire of automatic writing. \"Precipitates\" (\"Les Pr\u00e9cipit\u00e9s\") was Breton's original title for the foundational text that he later decided to call _The Magnetic Fields_ ( _Les Champs magn\u00e9tiques_ ). In _Soluble Fish_ , the first line of the central text is \"La pluie est divin\" (rain is divine). The precipitate of the new language gathers in the passive hand of the poet. Likewise, the motif of rain at the beginning of the frottaged natural history announces the self-authorizing appearance of its images: where rain pours down on the sea, it strikes a picture within a picture. In turn, this picture represents a surging sea. Its waves are more clearly defined than those in the larger sea, whose congealed, large-grained mass surrounds the picture within a picture and is liquefied within it. Thus, in the place where the rain pours down, there is an image (within an image) and the hardened sea begins to move again. Probably the appearance of the circle can also be attributed to the effects of the miraculous rain. It is drawn directly beneath the rain shower in the interior picture. Unlike the foreshortened picture within a picture, the circle has been drawn frontally. It seems as if it has risen from the swells (this impression is strengthened by the fact that the circle has not been completed at the bottom).\n\nSurely, this visionary circle in the originary cosmic storm would have been allocated a deeper meaning, since Ernst's friend Paul \u00c9luard was interested in emblem books. The circle directs the gaze to the center of the interior picture. In fact, near the circle's center point, one notices the obviously retouched, many times repeated line of a wave, which traverses the entire diameter of the circle from left to right. Here, in the ground, a still formless image is stirring\u2014perhaps a descendant of the \"soluble fish\" that Breton designated as his emblem in the _Manifesto_. The mystery of the emblematic circle is analogous to the enigma of frottage, its latent iconic potential. Ernst uses frottage as a technique of the enigmatic, the concealment and simultaneously the indexing of what is concealed, the absent underlay and its obscure trace, in which the seeing of resemblances occasionally discovers hidden images.\n\nFIGURE 29\n\nMax Ernst, _The Sea and Rain_ ( _La mer et la pluie_ ) ( _Histoire naturelle_ , print 1), 1925\/26. 43 \u00d7 26.5 cm (measurement of image), photoengraving from frottage (S\/M no. 790). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nNatural history begins with an arcanum. But how does the cryptogram of the circle mark the beginning of natural history? Formally, it leads to the second print (fig. 30). Maybe the connection to the rising sun is also meteorologically motivated. The sun of the second print shines through the storm of the first and projects a rainbow that rises from the picture within a picture. Natural history begins in the second picture's reverberation within the interior picture of the first picture. Like Breton's ideal image of automatic writing (the man cut in two by the window), this beginning arises from the interpenetration of representation and the represented world: the sun's effect extends beyond the frame of the picture into the interior picture of the picture preceding it.\n\nAs this retroaction of the second print upon the first print demonstrates, from its inception, the movement of surrealist natural history is complicated by regressive moments. It does progress according to the familiar narrative schema, from originary storm to the different kingdoms of plants and animals to the first people (fig. 33). However, its procedure relates each stage of development to the relatively small textural repertoire that comprises the mortified and unchanging ground of _Natural History_. Just one example of how the procedure's tendency toward nonsimultaneity is manifest in iconography can be found in the eleventh print, _Whip Lashes or Lava Threads_ ( _Coups de fouet ou ficelles de lave_ ; fig. 31). This print occupies the caesura in natural history between the first plants (prints 6\u201310), which grow all alone in an endless expanse, and zoomorphic leaf creatures (prints 12\u201314), which form aggressive communities. Between these, print 11 shows a stone block with fossil traces. The initial tendency is to attribute them to a plant species, before one discovers, hidden in them as within a reversible image, the skeletons of two birds frozen at the climax of a wild fight. This picture between two kingdoms of nature documents a stage in natural history that is only to be reached in the far future\u2014the kingdom of birds, which begins in print 25. In order to move from the solitary and peaceful plants to the zoomorphic and aggressive ones, the surrealist natural history utilizes a picture within a picture in which fully developed animal life is already recorded. The caesura between the plant and animal kingdoms is likewise marked by nonsimultaneity, with the bread fossil in print 23, which points into the future of human culture.\n\nFIGURE 30\n\nMax Ernst, _A Glance_ ( _Un coup d'oeil_ ) ( _Histoire naturelle_ , print 2), 1925\/26. 41.1 \u00d7 26.3 cm (measurement of image), photoengraving from frottage (S\/M no. 791). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 31\n\nMax Ernst, _Whip Lashes or Lava Threads_ ( _Coups de fouet ou ficelles de lave_ ) ( _Histoire naturelle_ , print 11), 1925\/26. 27 \u00d7 42.5 cm (measurement of image), photo engraving from frottage (S\/M no. 800). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\n_Natural History_ 's dichotomous pattern of movement, which shows the progression of natural history from the originary storm to Eve as the first human being at the same time as it also tends toward regression, is connected to the series' use of procedure. First of all, it can be observed that in general, the procedure of frottage follows the schema of progress by developing from tactility into visuality. The first two prints of the series form an antithetical pair (figs. 29, ). Against three factors that remain constant in both prints (the vertical format, the form of the circle, and the cosmic theme), abrupt changes occur: from darkness to brightness, from the tactile to the optic, from seeing-in to looking back, from the near opacity of the ground to the transparency of the image filled with dazzling light. While in print 1 (fig. 29) the illusion of a storm over the sea only appears gradually against the clearly visible wood grain, in print 2 (fig. 30) frottage as a technique of tactile surface manipulation is outshone by the illusion of a heavenly body or cosmic eye. For that matter, Ernst concludes by making the only identifiable texture, a piece of leather, sink below the horizon.\n\nFIGURE 32\n\nMax Ernst, _Stallion and Bride of the Wind_ ( _L'\u00e9talon et la fianc\u00e9e du vent_ ) ( _Histoire naturelle_ , print 33), 1925\/26. 26 \u00d7 42.8 cm (measurement of image), photoengraving from frottage (S\/M no 822). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThis initial contrast is repeated in the following prints in a mitigated form and is transferred to the natural history as a whole\u2014as the development from ground to figure, from pre-morphic to iconic, from tactile to visual. The last pictures show eye creatures that break the spell of the earth\u2014and thus break free of the frottage ground\u2014and fly away (prints 29\u201331). That which rises up from the regressive\/opaque material is identified in the penultimate picture as the joy of love: the embracing horses (fig. 32) are now only connected by a few threads to the stony ground. Finally, we see _Eve, the Only One Left to Us_ ( _\u00c9ve le seule qui nous reste_ ; fig. 33). The romantic trope of the _R\u00fcckenfigur_ (figure seen from behind) visualizes the inaccessibility of whatever the figure is facing. It brings the viewer into the picture but blocks the viewer from having his or her own original experience. What the viewer gets to see is that which has already been seen. Eve casts her gaze upon a future and a landscape known only to her. We can only surmise that there, beyond the picture, the opacity of the material ground is entirely dissolved. The pictorial space is not even cut by a distant horizon line, as is present in many of the frottages. Or more precisely, the horizon line has been transformed into the contour of Eve's body. The _R\u00fcckenfigur_ embodies the boundary that cannot be crossed. Beyond this boundary, and thus beyond natural history, are pure transparency and visuality. There, the spell of the stony replacement world is lifted; the fascination of the surrogate and the seeing of resemblances are overcome.\n\nFIGURE 33\n\nMax Ernst, _Eve, the Only One Left to Us_ ( _\u00c9ve la seule qui nous reste_ ) ( _Histoire naturelle_ , print 34), 1925\/26. 43.1 \u00d7 26.1 cm (measurement of image), photo engraving from frottage (S\/M no. 823). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nHowever, the _R\u00fcckenfigur_ stands on this side of the boundary. The material\/opaque pictorial surface of frottage has been inserted into the picture's motif as the section of wall to the right. Thus, the tension between transparency and opacity, visual iconicity and tactile\/material imagination, location beyond and within the stony world of surrogates, is preserved through the final picture. The narrative resolution is the promise of an unrepresentable future; until the narrative's conclusion, the representable natural history is accompanied by a transfixing, regressive power. The seeing of resemblances always returns to the pre-morphic textures, to the traces of wood, dried paint, patterns, straw, or twine\u2014those elements that remain the same at all stages of the natural history.\n\nOne detail illustrates how artfully Ernst fostered, in his ruin writing, the regressive tendency that opposes the development toward ever-greater transparency. In the first print (fig. 29), to the left, from behind the floating picture within a picture, an elongated object protrudes, which can be identified as a comb. Since cubism, the comb has been used as a painter's tool. An instrument for the mechanization of artistic facture, it aids in the avoidance of personal handwriting. In his own paintings from the mid-1920s, Ernst used a comb to carve regular grooves into the oil ground, evoking celestial visions. In prints 4 and 5 of _Natural History_ , he used these paintings as underlays. The imprints of comb traces are transformed into frozen waterfalls and cosmic oscillations. Combing and frottage became surrealist procedures because they are relatively standardized, mechanical activities that produce audacious metaphors (leather nubs and tree bark, grooves and oscillations). In print 1 (fig. 29), the comb, which itself has now been frottaged, is brought into connection with water and waves, two stereotypical metaphors for women's hair. In fact, the same texture underlying the viscous sea appears again in Eve's hair. The comb in the first print is thus retroactively motivated: that which occupies the center of the last image as its main motif\u2014Eve's hair\u2014was already envisioned in the first image in the waves of the sea, which were formed from the same texture. Granted, this texture does not resemble either waves or hair. Where the metaphor (hair and sea waves) is lacking a common third term (gentle undulations), metonymy (hair and comb) comes to its aid and makes the exhausted figure of speech into an audacious visual figure. The comb in the first picture, which belongs as a tool in the last picture, produces the dissembling resemblance between the sea in the first print and the hair in the last, because and above all in spite of the common texture, which suggests neither hair nor sea. This is only one example of the numerous metaphoric and metonymic relationships between pictorial motif and frottage underlay that counteract the teleology of _Natural History_ \u2014but it is a highly significant example, since it shows how correspondences are made between beginning and end, densest opacity and most ethereal transparency. Whether natural history leads to an inaccessible beyond or endlessly returns to itself again remains undecided.\n\nThis undecided movement takes a new turn in a detail, Eve's shadow, which has not been addressed so far. Her shadow's being larger than her head produces the impression that, in some dark way, she is dependent upon the stained section of wall, has emerged from it and is ready at any time to return to its amorphous and inorganic world. But why, exactly, is the shadow larger than her head? Quite likely because it has been projected by a photographer's flash or spotlight. Eve thus not only presents to us a beyond about which we know no more than that we can imagine it as a landscape that has already been seen by Eve; she also presents _herself_ as already seen. Gazing at her, we see a hardened surface illuminated by a flash\u2014but this flash is nothing other than the surrealist gaze. There is a famous formulation in Breton's _Mad Love_ ( _L'Amour fou_ ): \"It is to the re-creation of this particular state of mind that Surrealism has always aspired, disdaining in the last analysis the prey and the shadow for what is already no longer the shadow and not yet the prey: the shadow and the prey mingled into a unique flash.\" This \"unique flash,\" the surrealist gaze, was there. As a mortified figure, Eve is a testament to the past surrealist moment; as a transparent figure, she marks the threshold to that moment's future.\n\nAUTOMATIC COMMENTARY\n\nDespite all these artistic devices, the automatic text\u2014and the automatic picture series\u2014cannot automatically be identified as such. The concern over its authenticity can be felt not only in text-immanent and intertextual procedures, but also in the paratextual safeguards that were put in place. For example, in _La R\u00e9volution surr\u00e9aliste_, pieces were published under the rubric \"textes surr\u00e9alistes\" (surrealist texts), with the name of the author preceding the text, followed by a colon, as if what followed was the verbatim transcription of a voice. However, the most important testament to a text's automatism comes from the past or, even better, from the future: from the afterlife of previous manifestations of the unconscious, or from their living on in later declarations or narratives, in which the prophetic words of the unconscious return in order to intervene in an event, divulge their real meaning, and give form to the narrated world. For example, toward the end of _The Magnetic Fields_ , the ominous syntagm \"wood-coal\" appears and will pursue Breton eight years later in _Nadja_.\n\nSurrealist painters use a similar method for the retroactive instigation of the automatic quality of their works when they respond to an automatic text with images, and their literary friends do likewise when they invent titles for an automatic picture series or write a foreword. The titles in _Natural History_ may have been devised in collaboration with Paul \u00c9luard, and the foreword was written by Hans Arp.\n\nMost of the titles in _Natural History_ can be interpreted as parapraxes. Instead of \"the habits of animals,\" as they would normally be described in natural science, we have _The Habits of Leaves_ ( _les moeurs des feuilles_ ), and, in fact, the monumental leaf (fig. 34) seems to have a certain bodily torque. _Iceflower Shawl_ ( _le ch\u00e2le \u00e0 fleurs givre_ ; print 4) combines two syntagms, \"flowered shawl\" and \"ice flowers,\" which, through the common appearance of \"flower,\" seem to interweave of their own accord. _The Start of the Chestnut Tree_ ( _le start du ch\u00e2taignier_ ; print 14) puts an Anglicism from the mass-cultural language of sports in place of the customary \"emergence\" or \"appearance\" from natural science, while a little bit later, the natural science term \"origin\" is used in association with a technological invention ( _The Origin of the Clock_ [ _l'origine de la pendule_ ]; print 26). _Solar Currency System_ ( _le syst\u00e8me de monnaie solaire_ ; print 31) allows economic policy discussions to be glimpsed within astronomic terminology. These _lapsus linguae_ that still allow the correct expression to be guessed are evident in automatic texts on various levels, from individual syntagms all the way to the narrative mode or genre, whose norms and topoi form the background against which the \"surrealist texts\" are set off as comparatively uncontrolled events of language.\n\nHans Arp's foreword to _Natural History_ was also published in the same year under the rubric \"surrealist texts\" and thus circulated as a document of psychic automatism. In this piece of writing, Arp makes his own division of natural history into the four kingdoms of man-made things, leaves, insects, and crystals. In the first three sections of Arp's text, man appears as a creature who arranges the universe like a furnished room. The shining sun is replaced by medals of honor; man confuses earthquakes and alarm clocks, hail and showers of candy, and teaches horses to kiss one another like presidents. This simple list of man's synthetic substitutes for nature is a variation of an artistic device that also appears prominent in _Natural History_ (at least, Arp's foreword makes it seem prominent): that is, the admixture of natural science with the symbols and products of modern civilization. Of course, this does not make prehistory seem any more familiar; instead, traces of the human take on an alien quality. Titles such as _The Start of the Chestnut Tree_, _The Origin of the Clock_ , and _Solar Currency System_ have already been mentioned; but also in the interplay of image and title, numerous slips occur that cause surrealist prehistory to stumble regularly into the quotidian world of the present. The sun casts glances ( _A Glance_ [ _un coup d'\u0153il_ ]); moons become \"little tables\" ( _Little Tables around the Earth_ [ _petites tables autour de la terre_]); ice becomes a shawl patterned with ice flowers ( _Iceflower Shawl_ ); scouring rushes assume false positions ( _False Positions_ [ _les fausses positions_ ]), a petrified leaf acts secretively ( _She Guards Her Secret_ [ _elle garde son secret_ ]). In this sense, the foreword can be read as a performative commentary that points out the device of the parapraxis and applies it to _Natural History_. Arp selects certain titles (or title elements) and motifs, and combines them into new nature\/culture montages: for example, from print 13, _Scarecrows_ ( _les \u00e9pouvantails_ ) become \"scarecrows with volcanoes and geysers in their buttonholes.\" In this manner, Arp addresses a total of fifteen frottages and ascribes them to a new kingdom of nature, a _regnum naturae_ disfigured by human surrogate creations.\n\nFIGURE 34\n\nMax Ernst, _The Habits of Leaves_ ( _Les moeurs des feuilles_ ) ( _Histoire naturelle_ , print 18), 1925\/26. 42.7 \u00d7 26 cm (measurement of image), photoengraving from frottage (S\/M no. 807). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn the next three sections, Arp's discussion of natural kingdoms\u2014the kingdoms of leaves, insects, and crystals\u2014is not a recombination of elements from _Natural History_ , but develops from the interpretive appropriation of these elements, by parodistically arranging the language of typical hermeneutic texts. However, this does not mean that nothing can be learned from his foreword for the close observation of the individual frottages. Arp displays art-critical attention to narrative detail. With a metaphorics learned from Lautr\u00e9amont (\"while the ferocious lion scents a succulent pair of newly-weds the lime-tree grows tractably on the boarded plains\"), he calls the reader's attention to the uncanny shadow cast from the space in front of print 16 onto the \"tractable\" lime tree. Let us consider the following passage more closely:\n\nleaves never grow on trees. like a mountain in a bird's-eye view they have no perspective no soap no hybrid plastron no scotch cheeks no crypt. the spectator always finds himself in a false position before a leaf.\n\nThe passage begins with a reference to the de-organicizing cut through growth that surrealist prehistory makes (\"leaves never grow on trees\"), and then in the same sentence calls attention to a specific artistic device. These leaves (figs. 23, ) ( _Bl\u00e4tter_ \u2014the printed sheets and the arboreal leaves depicted in them) have no perspective. It is in fact the case that in the frottages, lines of sight are only evident in the background and without reference to the main figure, which hovers in stark frontality directly in front of the picture plane. This uncertainty of spatial situation and proportions constantly moves the viewer into \"a false position\" vis-\u00e0-vis the leaf and produces favorable conditions for an imaginative and animating seeing-in, through which the frontal leaf takes on a bodily torque and maybe even a kind of gaze. One notices the shadow that sets off the leaf from the wooden boards and seems to provide it with a three-quarters profile. The \"false position\" is thus a mobile back- and- forth of seeing, based on whether the texture is being perceived haptically and is being animated, or whether the viewer is attempting to grasp the unstable spatial and size relationships in the picture.\n\nThe literary procedure of Arp's commentary corresponds to the poetics of automatism as it can be deduced from other surrealist texts (more so than from surrealist declarations). Arp gives particular emphasis to the following poetic attributes: evident nonsense with a logical veneer (\"with the help of his eleven and a half tails man counts ten and a half objects in the furnished room of the universe\"); obstinacy of the phonetic material (\"un mollet de ballet eucharistique\"); numerous audacious metaphors (the sound of the cuckoo's jaws is like \"a violent fall of hair\"); and an open figurativity that glosses over the difference between metaphoric and literal speech: are mirrored cupboards in fact floating in the sea during the season of the diamond harvest, or does \"armoire \u00e0 glace\" (mirrored cupboard) stand metaphorically for \"montagne de glace\" (iceberg)?\n\nThe automatic character of the text also asserts itself paratextually. The text was first enclosed as the foreword to _Natural History_ and published again in the seventh edition of _La R\u00e9volution surr\u00e9aliste_, this second time as a document of surrealist research. The second time it was published, the omission from mention of Max Ernst and the inclusion instead of illustrations by Andr\u00e9 Masson and Georges Malkine served to underscore the text's new function as an example of surrealist automatism. But the text can also be read as automatic when enclosed as a foreword in the portfolio of _Natural History_. At least from the perspective of the text, the image sequence offers a narrative normalcy (from originary storm to the appearance of human beings) against which the foreword can profile itself as a comparatively uncontrolled textual event\u2014for example, by beginning with the admixture of nature and man-made objects. On the other hand, Arp's text does not function as a commentary only\u2014or even primarily\u2014because it draws attention to particular aspects of the picture series, but instead because it retroactively asserts the series' automatic status. Audacity and richness of tropes, the cancellation of the difference between literal and figurative meaning, numerous parapraxes, and the obstinacy of the visual material are more clearly apparent in _Natural History_ after these characteristics have been adopted by and presented in the foreword.\n\nThe foreword becomes automatic writing by deforming _Natural History_ , and, likewise, _Natural History_ gains automatic characteristics from the foreword. For the surrealists, the reciprocal automatization of texts and images was the most important function of reading and viewing. These activities did not aim for understanding. They were the rehearsal of a total communication in which, instead of many individuals talking to one another, a single dark power would talk to and speak for itself. Between images, titles, and foreword, the \"communal purpose\" of automatism becomes visible as the creation of an unconscious community whose hermetic statements only become transparent in the future. In the _Manifesto_ , Breton invokes the surrealist dialogue as the model of surrealist anti-communication, which promises a future totality of communication. His prerequisite assumption is that deeper understanding is grounded in _not_ understanding. Each conversation partner simply speaks to him- or herself, and occasionally the monologues touch one another coincidentally or through a phonetic echo, and in this one way influence each other in their course. \"Words, images are only so many springboards for the mind of the listener.\" The monologue of each individual person is the foreshadowing of the one single monologue that the Surrealists seek in the depths of their psyches, in order to launch it forcibly to power. In expectation of its revolution (waiting is the surrealist condition _par excellence_ ), the group assembles and sets the scene by leading expeditions into the deep time of subjectivity. Ernst's _Natural History_ is the field guide that the adventurers have packed in their luggage\u2014and like any good field guide, it is itself an adventure.\n\nSECRETS OF PREHISTORY\n\nMax Ernst's natural history recalls an adventure that itself takes place in the depths of prehistory\u2014an adventure of childhood reading. In order to stage the automatism of his _Natural History_ and to authenticate it as an experience of \"deep time,\" Ernst refers the series to a pictorial genre originating in his and his friends' childhood. The subject may recognize his or her own deeper history in natural history, and, conversely, it may be possible to narrate the history of nature as the history of consciousness. However, the surrealists rediscover this originary unity of self and world above all in a location that is multiply determined by culture: in the child's bedroom of the late nineteenth century.\n\nGeological and paleontological illustrations began to develop their own method in the mid-nineteenth century. The most striking moments of their poetics include dramatic leaps in proportion and myopic magnifications of detail, a corresponding interest in texture, the display of the character of the fragment, and the combination of different levels of representation, as when geologic crosssections or diagrams are inscribed in a naturalistic landscape. The enormous popularity of images from deep time was also bolstered by traditional landscape panoramas, which placed their viewers in the midst of prehistoric flora and fauna. One trendsetter was Franz-Xaver Unger's book _Die Urwelt in ihren verschiedenen Bildungsperioden_ ( _The Primordial World in the Different Periods of Its Development_ ; Vienna, 1850), complete with fourteen \"scenic depictions\" that ensured the success of the work throughout Europe. The best sellers of the genre soon followed, such as Louis Figuier's _La Terre avant le d\u00e9luge_ (published in Paris in 1863; English translation, _The World before the Deluge_ , published in 1872), Camille Flammarion's _Le Monde avant la cr\u00e9ation de l'homme_ ( _The World before the Creation of Man_ ; Paris, 1886), and in Germany, Oskar Fraas's _Vor der S\u00fcndfluth!_ ( _Before the Flood!_ ; Stuttgart, 1866). All were furnished with depictions of landscape that illustrated the intact ecology of prehistoric epochs (figs. 35, ). The epistemological status of these reconstructions was precarious. Especially because of their success, they aroused suspicions as to their objectivity. The illustrators had to establish the plausibility of images that reproduce what no human being could have seen.\n\nThe pictures of the prehistoric world presented themselves as reference images for a surrealist natural history for two related reasons. For one, the methods of natural history illustration tend toward effects that the surrealists also sought, such as abrupt leaps between near and far, large and small, part and whole, texture and form, and the interweaving of diagram and illusionistic view. For another, the prehistoric picture's problematic epistemology corresponds to the precarious status of the automatic picture, which also attempts to make visible an inaccessible deep time\u2014not of outer nature, but of inner nature. Ernst, who strove to follow the program of automatism to the same degree as he understood the need to stage its aporias, found in the geological illustrations of the nineteenth century a dramaturgical source in which he was able to see the poetics of his own art being modeled.\n\nFIGURE 35\n\n\u00c9douard Riou, _Condensation and Rainfall on the Primitive Globe_ ( _Condensation et chute des eaux sur le globe primitif_ ). From Louis Figuier, _La Terre avant le d\u00e9luge_ (Paris, 1863).\n\nFIGURE 36\n\n\u00c9douard Riou, _Artist's Depiction of the Earth during the Silurian Period_ ( _Vue id\u00e9al de la terre pendant la p\u00e9riode silurienne_. From Louis Figuier, _La Terre avant le d\u00e9luge_ (Paris, 1863).\n\nThe geological illustrators developed an implicit form of visual argumentation in order to make their pictures seem less dubious. The sublimely uninhabited prehistoric views are legitimated by directly inserting, or else appending, images of the natural documents that allow such reconstructions to be made. If the shells of dead trilobites have washed ashore and lie awaiting petrifaction, then the panorama of the prehistoric sea already contains, in plain view, the fossil findings that will be available to modern research. Alternatively, these references to scientific bases can also be placed in their own illustrations, where they can be presented more explicitly as independent evidence. Hence, smaller illustrations accompany the panoramas and separately array the complete chain of fossil specimens that underlie the reconstruction of a particular plant or animal. They depict the extinct species as it can be found\u2014or, more accurately, as it is presented\u2014in a museum of natural history: as petrified and fragmented fossils. The view into prehistory is substantiated with these two artistic devices, the staging and appending of geologic documents. First, it is shown how, even on the shore of the primordial sea, nature is mindful of future research. And second, natural images that have been preserved in fragments through fossilization are depicted in their fragmented state, so that the move to the artistic illustration of the original unity of nature can be given an established starting point.\n\nThis manner of viewing prehistory also applies for the \"inadmissible flora and fauna\" (Andr\u00e9 Breton) of automatism. While recounting prehistory, _Natural History_ also tells the story of its fragmentary and mortifying documentation. Ernst's natural history likewise consists of views of a prehistoric world (prints 1, 2, 3, and others) as well as images that document the remains of this time before time (prints 4, 11, 15, and others). These two types of image constantly merge into one another, since the documentary specimens are situated in deserted prehistoric landscapes and, in general, the reconstruction views have the appearance of fossilized images. The effect and its authentication can no longer be differentiated. This situation is seen again in the use of scientific illustrations. The quality of pastness and the problematic representability of automatic images are emphasized using strategies of popular science illustration, in order to achieve a surrealist penetration of the image by the conditions of its visibility. As an example, one might consider the third print (fig. 37) of _Natural History_. The impossible view it documents is that of the earth as seen from outer space. The impossibility of this view is expressed in the doubling of the earth, which appears once as a planet circled by three moons, and again as the ground of the landscape. _The Earth Views the Earth_ ( _Terre vue de la Terre_ ) is the title of a similar frottage not included in _Natural History_. An eye that floats somewhere in the cosmos in order to observe the prehistoric metamorphoses of our planet will always be positioned on the earth. The penetration of representation by its conditions results in an imaginary situation: one is both part of the fantasy and its author, both projected into the imagined scene as well as comprising the impossible condition of its reality. From this phantasmal position, it is only consistent that the viewer perceives additional never-before-seen phenomena that likewise all prove that the viewer's imagination has always already been present in the place where he or she is about to look. In print 3 (fig. 37), the earth and its moons form a figure that takes on human features in the full, feminine mouth of the uppermost moon. Only visible in the original, a black aura surrounds all four heavenly bodies and sets them off from the slightly brighter sky. In the zone of the ground and in the central planet\u2014the two manifestations of the earth\u2014the stony crust is splitting open, in its ruins allowing a glimpse of the possibility of future figures. In the moon to the lower left, the craters produce a star pattern, and in the moon to the lower right, there appears a linear diagram, which calls attention to the possibility of thinking of the ensemble of four heavenly bodies as a constellation. In fact, to the left of the central planet, one can make out a constellation drawn into the dark night sky. What appears as a constellation from far away appears as a figure from up close.\n\nErnst's _Natural History_ restages the epistemological problems of images of deep time. It is of no little importance for the social function of the picture series that a concrete reference text is indicated: Louis Figuier's _The World before the Deluge_. This book, circulated in many editions and predominantly directed at young people, was the most successful synthesis of the traditions of illustration that are typical for the natural histories of the nineteenth century. Figuier, too, includes two different types of images\u2014objective illustrations of fossil specimens, geological sections, and other scientific documents or diagrams, as well as engravings of landscapes by \u00c9douard Riou, who would go on to become famous above all as the illustrator of Jules Verne's _Extraordinary Voyages_ ( _Voyages extraordinaires_ ). In a sublime style learned from John Martin, these panoramas provide a view of the earth's prehuman epochs, from its beginnings in an originary storm, through the Silurian, Devonian, Carboniferous, Permian, Triassic, and Cretaceous periods, to the advent of human beings.\n\nFIGURE 37\n\nMax Ernst, _Little Tables around the Earth_ ( _Petites tables autour de la terre_ ) ( _Histoire naturelle_ , print 3), 1925\/26. 44.5 \u00d7 27.5 cm (measurement of image), photoengraving from frottage (S\/M no. 792). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe first two full-page panoramas by Riou show significant similarities to the first two frottages of _Natural History_ : _Condensation et chute des eaux sur le globe primitif_ ( _Condensation and Rainfall on the Primitive Globe_ ; fig. 35) illustrates what at the time was the cutting-edge theory of the original state of the earth as an incandescent ball in outer space, whose primordial sea was formed when gradual cooling caused steam to condense in its atmosphere. In Riou's second panorama, _Vue id\u00e9ale de la Terre pendant la p\u00e9riode silurienne_ ( _Artist's Depiction of the Earth during the Silurian Period_ ) (fig. 36), the sun appears over this sea for the first time. When _The World before the Deluge_ is placed side by side with _Natural History_ (figs. 29, ), the similarity is evident in the central motifs of storm and sun, as well as in details such as, in Riou's first panorama, the atmospheric light phenomena, which return in the effects of the wood grain as well as the illumination of the center of Ernst's first print, and the sunbeams, which are materialized in a similar fashion by both Riou and Ernst.\n\nBy making these images the backdrop for his own natural history, Ernst also adopts the teleological narrative template, from original chaos to the advent of human beings. Figuier thought of his book, and especially the accompanying illustrations, as the elucidation of a natural history that culminated in the appearance of human beings. At the time Figuier's book was published, Darwin's theory of natural selection was eliciting controversial discussions but was still being met widely with rejection. Not until the sixth edition did Figuier feel the need to respond to the pressure of new intellectual trends by replacing his illustration of the first humans, a nuclear family in a paradisal natural state, with a new picture of the primal horde on the hunt. As a whole, Figuier's book is a vivid example of how, in the popular science and pedagogy of the nineteenth century, faith in science and hope for progress could be instrumentalized in the education of good citizens and God-fearing people. In a text from 1923 dedicated to Ernst, the surrealists' accusation that natural history was a modern catechism might have been applied directly to Figuier. Also, _Natural History_ 's parodic play on the biblical Genesis, in its movement from the first appearance of light to Eve, may have laid the groundwork for the surrealist anti-clericalism that was directed at church functionaries as well as the apparatus of scientific piety. Ren\u00e9 Crevel, who wrote one of the first reviews of _Natural History_ in 1927, presented the series to readers of the _Nouvelle revue fran\u00e7aise_ as a surrealist primer:\n\nWhat we so proudly call \"our education\" has to be remade from the ground up. Max Ernst is right when, under the simple title _Natural History_ , he brings together in thirty-four prints the awesome wonders of a universe whose little secrets we will never trample on again\u2014little secrets that one day will be greater than we are.\n\nCrevel's phrase \"little secrets that one day will be greater than we are\" leads one to wonder about the possible function of such veiled reproofs. More generally, it raises the question, how do \"little secrets\" come to be \"greater than we are\"? From a poetological perspective, there is another, related question. Namely, at the beginning of an automatic picture series that asserts its spontaneous originality in the motif of rain, what explains the appearance of a pedagogical best seller from the previous century and (at least superficially) the adoption of its developmental model from original chaos to human beings? Here, an observation from literary studies comes to our aid. Automatic texts often make use of narrative stereotypes and assert their automatic character in the reworking and decomposition of them. The clich\u00e9d narrative form that glimmers through an automatic text serves to constitute the text as an uncontrolled event. In our discussion, it is of particular importance that the subtext against which the automatic text is set off is itself transformed into a fragment, which is retrieved from the depths of the unconscious and inserted into the automatic text. It is no accident that these subtexts most often represent genres such as the fable or the adventure story, which refer to the reading experiences of childhood and adolescence. Breton used fixed stereotypes as foils for his own spontaneity, and at the same time received them as bottled messages from the time of his childhood. Their authenticity was all the more unimpeachable since Breton wrote his texts in school notebooks with covers depicting scenes of knights in armor and other adventures.\n\nThe genre of natural history as represented by Figuier's book was also predominantly directed at young people, in explicit competition with traditional educational literature. In the place of fantasy, with its products such as fairy tales and fables that lead impressionable young minds permanently astray, Figuier wanted to install science as the teacher. The expulsion of fables from pedagogy aims to complete a process in intellectual history that Wolf Lepenies describes as having already taken its decisive turn in the eighteenth century, with the expulsion of fables from scientific natural history. Now fables would no longer wreak their havoc in the one place where they had been permitted to survive\u2014in schoolbooks. If his advice were to go unheeded, Figuier predicted the burgeoning of such grave threats to the French nation as lies, mysticism, spiritualism, socialism, and madness. From a surrealist point of view, there was no question about it: the ones who were educated by Figuier and his ilk had to have their education \"remade,\" as Crevel states in his review. In an announcement he designed for the series, Paul \u00c9luard also emphasized that Ernst's _Natural History_ tells fables again. As a promotional commentary, he selected a passage from Condorcet's \"Eulogy of M. De Buffon,\" which characterizes the natural scientist as a maker of fables that testify to a powerful and independent imagination. \u00c9luard may have chosen this quotation not in order to draw parallels between Buffon's and Ernst's natural histories,1 but\u2014more appropriately for an advertisement\u2014in the spirit of provocation. Eyebrows were surely raised at the notion of the legacy of France's great natural scientist and stylist falling to a German painter and collagist.\n\nHowever, fables and narratives do not _return_ in surrealist natural history, since they never actually vanished from popular science teaching. As a pedagogue and best-selling author, Figuier was clever enough to adorn the narrative of the world in its becoming with entertaining digressions for his young readership. He recounts how dinosaurs were imagined as dragons in legend and literature. But his most important device consists in repeatedly evoking the amazing and sublime alienness of prehistory. Just as the illustrations alternate between documentary fossil specimens and fantastic panoramas, throughout his scientific prose, Figuier liberally scatters evocative descriptions of the prehistoric world.\n\nThe alien silence of this world was also striking to the first modern artist to bring Figuier's book, and especially Riou's illustrations, into the consciousness of the avant-garde. In his most well-known declaration, \"On Metaphysical Art,\" Giorgio de Chirico wrote that he first encountered the \"solitude of signs\" and the \"absence of human beings\"\u2014which he later would find in Poussin, Claude Lorrain, and B\u00f6cklin\u2014when he was a child, in Riou's prehuman landscapes. This \"solitude of signs,\" as de Chirico explains in the previous section, \"Madness and Art,\" is preceded by the obliteration of memory, since it is memory that produces the cohesion of the world. Without memory, there would be no connection between objects; they would break apart into disparate elements, as evinced in de Chirico's paintings. Thus, de Chirico first trained his gaze by looking at the uninhabited expanses of prehistory, because this time before any possible conscious memory prefigures the loss of memory that, according to him, forms the basis of the new painting.\n\nSurrealist automatism can be described as the radicalization of this annihilation of conscious memory by tunneling into psychic deep time. Among the \"secrets of the magical Surrealist art\" that Breton touts in the _Manifesto_ , he ultimately lists the cancellation of death:\n\nSurrealism will usher you into death, which is a secret society. It will glove your hand, burying therein the profound M with which the word Memory begins.\n\nFor surrealism, a group with certain features of a secret society, death can also be considered an association that conceals itself from, and thus undermines, conventional reality. Its initiation symbol, the glove, metaphorically represents memory's burial shroud. Drawing upon the significance of the hand in surrealist poetics (as the contemptible \"main \u00e0 plume,\" which stands for poetry as a profession, and as the greatly desired open palm, which passively receives the automatic rain), the following interpretation of Breton's allegory can be offered: The glove that conceals the \"main \u00e0 plume\" is automatism's magic instrument, burying the memory of conventions and, with them, an individual's own abilities. In this way, the doors are opened to the secret chambers of reality. In 1924, the year in which the _Manifesto_ was published, Ernst devoted a painting to this allegory. Later Breton would reformulate his poetics of deep memory using Freudian terms. In this reconception, the surrealists' wealth consists of acoustic and optical traces of memory, deposited in the unconscious, which only automatism is capable of refining into texts and images.\n\nIn his _Natural History_ , Ernst's play on Riou's illustrations\u2014his use of them as a narrative model against which the automatic natural history is set off and that the automatic natural history retrieves from out of the prehistory of childhood\u2014could only have been understood in the circle of his surrealist friends. It is all the more probable that they did understand the allusion since de Chirico had already called attention to Riou. When, around 1930, Ernst began to reveal the source material of his earlier work, he modified one of Riou's panoramas in the collage _Two Young Girls Riding across the Sky_ ( _Deux jeunes filles se prom\u00e8nent \u00e0 travers le ciel_ ; fig. 38). His surrealist friends, to whom Riou was well known as an illustrator (especially of Jules Verne's books), may have been able to identify the source exactly. If not, then they could have ascribed it to their own childhood reading material. And perhaps they received Ernst's collage as a retroactive indicator of the genre underlying _Natural History_ : this is one of Crevel's \"little secrets that one day will be greater than we are.\" At this point, these secrets still only serve to constitute a group and to deliver it bottled messages from the depths of time and the unconscious. But the \"communal purpose\" of automatism aims toward a revolutionary future that will command the entire social reality. This reality is prefigured by the avant-garde group that gives itself over to the communication conditions of the unconscious. The unconscious's signs stem from the childhood of the future surrealists, who, on both sides of the Rhine, had immersed themselves in Riou's landscape panoramas and learned how to daydream. Thus, the surrealists' unconscious is the source of their future communication and society inasmuch as this unconscious consists of out-of-date best sellers and wellworn illustrations. Only that which has at one time occupied the imagination so consumingly\u2014as only surrogates have the power to do\u2014can become an origin.\n\nFIGURE 38\n\nMax Ernst, _Two Young Girls Riding across the Sky_ ( _Deux jeunes filles se prom\u00e8nent \u00e0 travers le ciel_ ), 1929. 10 \u00d7 16 cm, collage on paper. Private collection (S\/M no. 1404). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nAnother of the \"little secrets that one day will be greater than we are\" should be presented briefly, in order to demonstrate how, in automatism, original and prefabricated images condition one another. Between _The Vaccinated Bread_ ( _Pain vaccin\u00e9_ ; print 23) and the first birds (prints 25ff.), a dragonfly nosedives over the primordial sea (fig. 39). As the first print that unequivocally represents an animal, it nevertheless remains allied with the plant kingdom. Leaves serve as wings, and a ball resembling a chestnut serves as a head. The same template that has produced the diamond-patterned bark of prehistoric plants here forms the surging sea over which the dragonfly is flying. In the constellation that appears in the firmament, Ernst repeats a device that by now has become familiar: the similarity between constellation and figure gives rise to the conjecture that constellation and dragonfly are really one and the same being, seen from up close and far away. Hence, the dragonfly repeats the leaps back and forth between the kingdoms of plants, birds, and cosmic beings. These latter will follow two stages later in the natural history [prints 29ff.]).\n\nSearching for another motivation for the appearance of the dragonfly requires setting aside the aspects of manifest devices and thematic associations, and risking a look into the semi-darkness of iconographic connotations, which the surrealists considered a medium of unconscious communication. The dragonfly appears in Figuier, too, as the most important fossil document of prehistoric insect life. In a hunting scene, Riou chooses the delicate creature as the tragic heroine who falls victim to a winged dinosaur (figs. 40, ). The central beam of light, casting the dinosaur's teeth and its prey into melodramatic silhouette, cuts through the semi-darkness of the prehistoric landscape, in which the outlines of an indeterminate number of additional reptiles can be either made out or imagined. No palm tree is empty of a dangerous winged creature; no scouring rush cannot be read as a phallic symbol. Of course, it is Ernst's dragonfly that inspires interpreting the prehistoric hunt as the primal scene of a paleo-analytic seduction theory: the print is titled _Lightning Bolts under Fourteen Years_ ( _Les \u00e9clairs au-dessous de quatorze ans_ ). In French, dragonflies, known as \"demoiselles,\" are underage girls.\n\nFIGURE 39\n\nMax Ernst, _Lightning Bolts under Fourteen Years_ ( _Les \u00e9clairs audessous de quatorze ans_ ) ( _Histoire naturelle_ , print 24), 1925\/26. 42 \u00d7 26 cm (measurement of image), photoengraving from frottage (S\/M no. 813). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nAs a book for young people in the hands of the surrealists, _The World before the Deluge_ becomes a fossil of pedagogically ambitious natural history and migrates from childhood in the late nineteenth century to just below the surface of the frottaged natural history. Among hundreds of illustrations, the image of the dragonfly remains accidentally lodged in memory\u2014as a contingent fragment that connects the surrealist picture series with a submerged past. The way in which this fragment encompasses the shared past of poets and artists who, as children, trained their imaginations by looking at Riou's landscapes clarifies the function of _Natural History_ within the surrealist group. Ernst used frottage as a technique of the secret, for the suggestion of hidden pictures, and consequently his natural history is filled with \"little secrets.\" Moreover, Ernst conceived of frottage as a technique of non-simultaneous images that document the past by giving it a ghostly afterlife, and thus the little secrets of childhood become omens of the present and future. From the perspective of revolutionaries who want to start \"our education\" over again, these images contain no lesser promise than to return to the deep time of childhood in order to create a new unconscious. Before I turn to this central fantasy of surrealism, which also characterizes Ernst's use of Freud's writings, it is necessary to examine the threshold of this new unconscious. Standing on the threshold is Gala \u00c9luard.\n\nFIGURE 40\n\n\u00c9douard Riou, _Artist's Depiction of the Earth during the Liassic Period_ ( _Vue id\u00e9ale de la terre pendant la p\u00e9riode du Lias_ ). From Louis Figuier, _La Terre avant le d\u00e9luge_ (Paris, 1863).\n\nGALA\n\nIn 1921 Gala and Paul \u00c9luard visited Cologne, and in 1922 Ernst traveled with Paul's passport to Paris, where he lived with Paul and Gala in the suburb of Eaubonne until 1924. After Paul's mysterious escape to the Far East, which ended when Max and Gala came to get him, Gala separated from Max and returned to her husband for the space of one year. In the 1950s, Ernst wrote that he had dedicated his _Natural History_ to Paul \u00c9luard, and that Paul in turn had made a gift of it to Gala.\n\nFIGURE 41\n\nFrom Louis Figuier, _The World before the Deluge_ ( _La Terre avant le d\u00e9luge_ ) (Paris, 1863).\n\nErnst may have made the selection of frottages for _Natural History_ together with Paul \u00c9luard. Several prints contain clear allusions to \u00c9luard's lyric works. The title of the third print, _Little Tables around the Earth_ ( _petites tables autour de la terre_ ), refers to his poem \"Dancing\" (\"Dans la danse\"), which evokes the fairy-tale table that covers itself on command and transforms it into an erotic wish machine. The lines \"there are women of wood dark and green\" may have led readers interested in psychoanalysis\u2014among whom Ernst was one\u2014to a passage in the _Introductory Lectures on Psychoanalysis_ where Freud describes the table and wood as \"puzzling, but certainly female symbols.\" For a surrealist, less interested in what is stable and evident than in fluctuating and enigmatic symbolism, this statement may have been particularly attractive. The \"puzzling\" femininity of table and of wood, which \u00c9luard had already treated in his interpretation of the fairy-tale command \"table, be covered,\" may also have preoccupied his friend, who made rubbings of many wooden surfaces. Frottage as a technique that makes wish images grow out of boards is the technique of \"table, be covered.\" The tables dancing around the earth are the tables of desire.\n\n\u00c9luard's poem concludes with the admission that he doesn't like the tables he is dancing on. The readily available, variable gifts of the libidinous \"table, be covered\" lead to surfeit. His poetry only grants endless allure to one single, very slender, obscurely phallic woman. She begins to haunt Ernst's art shortly after their first meeting, in the famous overpainting _Approaching Puberty . . . The Pleiades_ ( _Les Pl\u00e9iades_ ). In 1923 Ernst painted murals for the shared house in Eaubonne. In one picture, Gala, a transparent apparition, presents the interior of her body; in another, two intersecting female figures metamorphose, first into an abdomen and then (as its shadow) into a phallus. In _Eve, the Only One Left to Us_ ( _\u00c9ve, la seule qui nous reste_ ; fig. 33), the elongation of the neck and the transparency of the body are sufficient to recall the Gala myth. In addition, one detail plays on Gala's self-presentation: the hairs on Eve's neck. Gala's disinclination to use a razor or tweezers was part of the representation of herself as a passionate Russian woman. In Man Ray's famous photograph of her eyes, which was retouched by Ernst, her eyebrows almost grow together (fig. 42).\n\nThus we have arrived at the two dark caverns that were the prime objects of the courtly love and fetishization of her husband as well as her lover. _Natural History_ , which tells the story of a fantasizing and seeing nature, brings eyes from the depths of the earth into the light, subjects them to monstrous metamorphoses, and removes them to the sky as cosmic manifestations. Like the smile of his phallic mother, which Leonardo da Vinci shifts from one painting to another\u2014from the Virgin Mary, to Saint Anne, to John the Baptist\u2014the eye motif migrates through Ernst's frottages. But how might Ernst and \u00c9luard have recognized Gala's eyes in them, specifically? In the enormous fossil eye titled _The Wheel of Light_ ( _La roue de la lumi\u00e8re_ ), there are some of the most important topoi that Paul \u00c9luard employed in his poems about Gala (fig. 43). But first of all, the stony eye refers back to Gala's self-presentation (fig. 42). In contrast to her uncombed hair and ungroomed eyebrows, her eyelashes are carefully formed into bristling spikes, which crown each eye with a dangerous wreath. Ernst traced this wreath and used retouching to extend and hone each spike of a lash. In her portrait, through her lashes Gala's gaze takes on an impenetrability that is only enhanced by the glints of light positioned in each eye. The hardness of this gaze is virtually material, which explains the two dark shadows it casts over the landscape.\n\nFIGURE 42\n\nMax Ernst, _The Visible Woman_ ( _la femme visible_ ), 1925. 39 \u00d7 55 cm, photographic enlargement and pencil. Private collection (S\/M no. 787). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn _The Wheel of Light_ , Ernst repeats the armoring of the gaze with a spiked wreath and heightens its aggressiveness through a device of frottage. The leather used as an underlay transfers its animality to the stone fragment in which the eye is encased. Moreover, this snake-like quality is also semantically associated with metaphorics of petrifaction through the Medusa myth. In his retouching of Man Ray's portrait of Gala's eyes (fig. 42), Ernst already associates the threat of the gaze that casts dark shadows over the sea with a rocky mountain landscape.\n\nFIGURE 43\n\nMax Ernst, _The Wheel of Light_ ( _La roue de la lumi\u00e8re_ ) ( _Histoire naturelle_ , print 29), 1925\/26. 25 \u00d7 42 cm (measurement of image), photoengraving from frottage (S\/M no. 818). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nMany variations on the motif of the stony and petrifying gaze can be found in Paul \u00c9luard's _Capital of Pain_ ( _Capitale de la douleur_ ; 1926). Petrifaction can become dazzlement, the stony eye a gleaming sun; just as in _Natural History_ , the fossil eye bears the title _The Wheel of Light_. This title refers back to print 2, in which, conversely, the appearance of the sun is given the title _A Glance_. Just as, at the end of the natural history, the stony eye is designated a heavenly body, so, at the beginning, the sun is designated an eye\u2014what's more, this sun's surface resembles the stony waste of the moon. In contrast to this repeated castration symbolism, the eye in print 29 (fig. 43) is formed out of leaves. In \"Absences II,\" Paul \u00c9luard uses the \"window of foliage\" to present the same motif of hope that the rigid and paralyzing effect of the inaccessible gaze will yield again in fructifying love. The eye opens in a nature without bounds: \"Une fen\u00eatre de feuillage \/ S'ouvre soudain dans son visage. \/ O\u00f9 poserai-je mes l\u00e8vres, nature sans rivage?\" From the metamorphosis of the woman of stone into the microcosm of a brimming nature that welcomes the poet instead of refusing him with mineral hardness, there follows the cosmic omnipresence of her eyes in water and light. The title _Wheelof Light_, by repealing the material heaviness of the petrifaction motif and suspending it in light, suggestively anticipates what is fulfilled two prints later. _Solar Currency System_ releases the eye from the stony spell and makes an entire flock of eyes hover in space as celestial apparitions in which images of new microcosms emerge. Instead of emitting the Medusa glare, Gala's eyes become birthplaces of new worlds, which, as in \u00c9luard's \"Absences II,\" arise from a morphology of the round.\n\nThis brief iconographic examination, pursuing references to Gala's self-presentation and Paul's poetry, raises the question of whether Ernst may have seen these allusions to the _m\u00e9nage \u00e0 trois_ as more than simply the memories of an affair that was \"ancient history.\" As we have seen, the surrealists only ascribed value to memory if it tunnels into the deep time of childhood in order to penetrate to the origin of automatism, whose power lies in its involvement with surrogates. In order to count as surrealist events, recent events like the three years together in the suburbs of Paris would have to reach back into prehistory. Ernst discovered that the modern route into prehistory had been recorded in the writings of Freud. The next chapter will pose the question of what it means to be a psychoanalytic painter\u2014and at that, the first. The objective will be to show how, in Ernst's hands, psychoanalysis becomes a tool, not only for accessing the prehistory of childhood, but also for describing how the surrealist can remake his unconscious.\n\nPLATE 1\n\nMax Ernst, _Frozen Landscapes, Icicles and Mineral Types of the Female Body_ ( _eislandschaften, eiszapfen und gesteinsarten des weiblichen k\u00f6rpers_ ), 1920. 25.3 \u00d7 24.4 cm, gouache and pencil (overpainting on a print), Stockholm, Moderna Museet (Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 352 [hereafter \"S\/M\"]). Photo: Moderna Museet\/Stockholm. \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nPLATE 2\n\nMax Ernst, _Madam Hostess on the Lahn, guardian angel of the Germans, thine is the industry anatomy paleontology grant us thy jubilation_ ( _frau wirtin an der lahn, schutzengelin der deutschen, dein ist die industrie anatomie pal\u00e4ontologie schenk uns deine frohlocken_ ), 1920. 25 \u00d7 31.5 cm, collage, gouache, ink on paper, Staatsgalerie Stuttgart, Graphische Sammlung. Photo: Staatsgalerie Stuttgart (S\/M no. 405). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nPLATE 3\n\nMax Ernst, _Woman, Old Man, and Flower_ ( _weib, greis u. blume_ ), 1924. 97 \u00d7 130 cm, oil on canvas, Museum of Modern Art, New York (S\/M no. 660). \u00a9 The Museum of Modern Art\/Licensed by SCALA\/Art Resource, NY. \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nPLATE 4\n\nMax Ernst, _Europe after the Rain_ ( _l'Europe apr\u00e8s la pluie_ ), 1933. 101 \u00d7 149 cm, oil and plaster on plywood (S\/M no. 1881). Staatliche Kunsthalle Karlsruhe. \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n3\n\nMAX ERNST AND FREUD\n\nSURREALIST COMMUNICATION\n\nFrottage conceals objects so that resemblances become visible in their traces. In doing so, as was discussed in the previous chapter, it also generates a densely allusive and hermetic iconography. The secrets of this iconography refer to constellations within the surrealist group, and for their own part they set in motion a socializing effect. Thus, it would not be correct to describe _Natural History_ as a coded communication between initiates. Neither \u00c9douard Riou's woodcuts nor Paul \u00c9luard's poems contain a key to Max Ernst's pictures; nor, for that matter, does Hans Arp's foreword. A surrealist communication between images and texts is not brought about by secret knowledge, but by not knowing and by mutual misunderstandings. In the _Manifesto of Surrealism_ , with his monologic dialogue, Breton provided a model for this uncomprehending, blind, and thus automatic communication, which allows a post-revolutionary totality of unconscious communication to be glimpsed. If, in this blindness and incomprehension, secret meanings or even commentary nevertheless emerge (such as Arp's art-critical observations or Ernst's adaptation of \u00c9luard's \"table, be covered\" motif), these only attest to the fact that unconscious communication is already functioning within the surrealist group, in anticipation of the perfect transparency of the future.\n\nThis condition\u2014that every assumption of interpretative distance collapses again into the succession of parapraxes, dreams, and automatism\u2014allows Ernst's much-discussed interest in psychoanalysis to be brought into a new light. From a historical standpoint, the question of whether the artist's surrealist pictures require a psychoanalytically informed iconographic decryption or whether, as Ernst proclaimed in later years, they are banned from interpretation simply has no bearing. These alternatives were imposed after the disintegration of the surrealist group, when the artist's work became the object of art-historical research. Exponents of postwar surrealism\u2014and after them, art historians trained in iconography\u2014went in search of the psychoanalytically legible, while Werner Spies, in close contact with Ernst and under the sway of current artistic positions such as Oulipo and the _nouveau roman_ (new novel), analyzed materials and methods. But in the first years of surrealism, these alternatives had not yet been posed, since hidden meaning and iconographic innuendo had less value as knowledge (either of an esoteric or canonical nature) than as unconscious communication within the surrealist group.\n\nYet another, simpler problem has until now hampered the understanding of Ernst's interest in psychoanalysis: the rather careless use of his writing, which has predominantly been treated as a collection of supporting materials for interpretations of content or poetological commentary. It has escaped observation that in the course of decades, in accordance with changes in surrealist objectives, the artist's texts were repeatedly rewritten, printed in different types of publications, and considered as having different functions. In what follows, using the example of his first psychoanalytically inflected text, it will be demonstrated that Ernst's writings speak more powerfully when we read them in their historical context\u2014and that means above all in their material context.\n\nA HALLUCINATION IN PRINT\n\nOn October 1, 1927, _La R\u00e9volution surr\u00e9aliste_ printed Max Ernst's \"Visions of Half-Sleep\" (\"Visions de demi-sommeil\"). The title recalls the function of hypnagogic images in automatic writing as Andr\u00e9 Breton defined them in the _Manifesto of Surrealism_ : they are messages from the unconscious (like the phrase \"there is a man cut in two by the window\"), which the poet can register while barely awake, before they release a stream of automatic images. \"Visions of Half-Sleep,\" however, was not identified as a declaration. The presentation (a colon after the name \"Max Ernst\") identifies the text as a document of surrealist research, like a dream transcript or automatic text, or like the \"Journal of an Apparition\" (\"Journal d'une apparition\") by Robert Desnos, published in the same edition. The separation of the text into three parts shows that the half-asleep hallucinations have already been psychoanalytically processed. The first section is \"From 5 to 7 Years,\" the second, \"At the Age of Puberty,\" and the third, \"January 1926.\" These are the three points in time that Freud presents as critical in the etiology of a neurosis: early childhood, reawakening sexuality, and the recent occasion. Turning over a few more pages, the reader of _La R\u00e9volution surr\u00e9aliste_ comes upon an excerpt of the first French translation of _The Question of Lay Analysis_, and thus receives a brief introduction into the (pre)history of the psyche and its constitutive catastrophe, the early childhood trauma. The impulse behind Freud's text, his affirmative answer to the question of whether psychoanalysis can also be practiced without medical training, strengthened the surrealists' resolve to lay the groundwork for the new science and at the same time to compete with it.\n\nAs a document of surrealist research, \"Visions of Half-Sleep\" differs in its ambitions and function from Ernst's later texts. For example, \"How to Force Inspiration\" (\"Comment on force l'inspiration\"), published in 1933 in _Le surr\u00e9alisme au service de la r\u00e9volution_ , devotes itself explicitly to the procedures of collage, frottage, and grattage in order to clarify their position within surrealist poetics. Thus, it is not a document of surrealist research but a statement of its mission. In \"Visions of Half-Sleep,\" however, there are no guidelines for or commentary on this mission. No explicit mention is made of any of Ernst's pictures. For an illustration, there is a relief by Hans Arp, author of the foreword to _Natural History_ , which was printed in a preceding issue of _La R\u00e9volution surr\u00e9aliste_ as an automatic text, though without reference to Ernst's prints\u2014Arp's text was instead illustrated with pictures by Andr\u00e9 Masson and Georges Malkine. No text addresses an image; no image illustrates a text. The surrealist publication policy pursues the objective of presenting images and texts as the consequence of automatic expressions, between which there is no discursive distance that would allow for the establishment of a stable meta-level.\n\nWhile the layout of \"Visions of Half-Sleep\" purposely distracts from the possibility of reading it as a text about Ernst's pictures, its larger presentation context leads right back to this possibility. The same issue to publish Ernst's first psychoanalytically inflected text as a surrealist document also contains two paratextual indicators that suggest a mission statement can also be gleaned from the document. First, there is the advance publication of _The Question of Lay Analysis_ , in which the psychoanalytic developmental model that underlies Ernst's hallucinations is briefly outlined; then, on the very first page, there is a reproduction of Ernst's grattage _Paradise_ \u2014one of the pictures whose origin is staged in the text \"Visions of Half-Sleep.\"\n\nWhy go to such circuitous lengths to present \"Visions of Half-Sleep\" as a document and, inasmuch as it is a document, also as a declaration? The reader is led to receive the text in the context of the magazine, and in this way to acknowledge the authority of the avant-garde group as a whole. The danger that surrealist painters would be isolated and would attain success\u2014not least commercial\u2014on the basis of formal criteria worried Breton and shook his faith in the revolutionary reliability of his painter friends (see chapter 2). With his frottages and grattages, Ernst had for the first time achieved praise from the formalistically oriented art magazine _Cahiers d'Art_. Thus, he may have also authored his text as proof that he was committed less to art per se than to surrealist research.\n\nTHE PRIMAL SCENE AND THE DEFERRED ACTION OF THE IMAGE\n\nThe first part of \"Visions of Half-Sleep\" describes a hallucination that Max Ernst had between the ages of five and seven. It closely corresponds to the concept of the primal scene as outlined by Freud in his study on the \"Wolf Man,\" which was also iconographically influential for Ernst. But the poetological function of the primal scene only becomes clear from a second, even more famous case study, Freud's _Leonardo da Vinci and a Memory of His Childhood_. In order to understand how \"Visions of Half-Sleep\" presents a surrealist socialization fantasy, it is not sufficient to consider the first of the three hallucinations separately, as has been a common practice of scholarly research. In accordance with the underlying psychoanalytic schema, the first vision must be interpreted in view of the second and third.\n\nIn a panel of crudely painted imitation mahogany, organic forms appear: a menacing eye, a long nose, a bird's head with thick black hair. In front of the panel, a bogeyman with Ernst's father's mustache makes slow, comical movements and leaps around with spread legs. Then, with a fat, soft crayon, he draws quickly on the false mahogany, forming repulsive creatures that come to life. He gathers the creatures into a vase, and his crayon turns into a whip, which the little man, who by now is clearly Ernst's father, uses to make the vase rotate like a top. With powerful thrusts, gasping and snorting like a steam engine, he makes the spinning top whirl around Max's bed, together with \"all the horrors my father is capable of evoking in an amiable panel of false mahogany by means of his frightful soft crayon.\" Years later, in puberty, Ernst thought about his father's behavior during the night of his own conception and became aware of this early childhood hallucination, which from then on was retained by him in obsessive thoughts about his parents' coitus.\n\nAs is well known, this is also the subject of the primal scene. According to Freud, the child becomes witness to his parents' sexual intercourse and attempts to understand what is going on using the system of knowledge that has been put in place through his infantile sexual research into the origin of life and the difference between the sexes. The little observer understands the event as the aggression of the father and the castration of the mother, whom the child has previously considered a phallic being. The threat of castration, which has already been addressed to the child\u2014for example, during his first attempts at masturbation\u2014now seems to him to be particularly motivated as he witnesses its enactment by his father.\n\nThese are just a few suggestions to illustrate that Freud makes the essential currents of early childhood instinctual life (sexual research, the phallic woman, fear of castration, the Oedipus complex) converge in the primal scene and diverge from the primal scene into later instinctual life. As Freud found in his study on the Wolf Man, \"It was not only a single sexual current that started from the primal scene, but a whole set of them . . . his sexual life was positively splintered up by it.\" Even more interesting than Ernst's ingenious rendering of this theme\u2014which elicits (obvious) interpretations such as the equation of frottage with masturbation\u2014is his poetics of the primal scene. If one follows the published hallucination to the later declarations, the primal scene becomes the conceptual origin of his art. The (problematic) historicity of the primal scene, its deferred action, serves Ernst as a model for the non-simultaneity of his own pictures.\n\nThe primal scene is deferred because its effect is not solely or primarily attributable to the perception of the event but rather to the \"activation of the scene\" in later years. In the famous dream that the Wolf Man remembers from the age of four or five\u2014six or seven white wolves sitting in a walnut tree\u2014Freud recognizes the (greatly distorted) primal scene, which the patient probably witnessed at the age of one and a half. Freud defends this latter point, that the primal scene is based on the actual witnessing of the event and is not a fantasy projected back in time, with great argumentative vigor. Although the event and its witnessing come into psychic power only after the fact through the activation of the scene in later dreams, hallucinations, or obsessions, does the primal scene nevertheless mark an unconditional beginning? No: when observed from an even more distant, pre-individual past, the event and its witnessing prove to be the activation of a still more ancient scene. According to Freud, the primal scene is part of the phylogenetic inheritance of the human psyche. That is, in the depths of prehistory, it was once in fact witnessed and from then on has been preserved as a mold that reshapes the contingent experience of the child (who may have never witnessed his parents' intercourse but maybe that of two dogs, for example). \"All that we find in the prehistory of neuroses is that a child catches hold of this phylogenetic experience where his own experience fails him. He fills the gaps in individual truth with prehistoric truth; he replaces occurrences in his own life by occurrences in this life of his ancestors.\"\n\nThe construction of Ernst's childhood hallucination adheres to the deferred action exemplified by the Wolf Man's dream but displays significant differences. The artist's account, like that of Freud's patient, tells not of direct witnessing but of its distorted activation years later; in addition, both the dream and hallucination contain thematic indications that they are referring to visual perception, though to different modes of seeing.\n\nI dreamt that it was night [said the Wolf Man] and that I was lying in my bed. (My bed stood with its foot towards the window; in front of the window there was a row of old walnut trees. I know it was winter when I had the dream, and night-time.) Suddenly the window opened of its own accord, and I was terrified to see that some white wolves were sitting on the big walnut tree in front of my window.\n\nThe opening of the window concurs with the surrealists' use of the window as the metaphor for a compulsive, illusionistic image that allows a view of an externalized inner world. Also, the drawing that the Wolf Man made of his dream vision (fig. 44) corresponds to the naive, seemingly careless, and thus authentic approach to painting that Ernst attempted in his image notations rapidly applied to a roll of paper (fig. 25). As we have seen, the frottages are the product of an engagement with the critique of these types of pictures, which Max Morise granted only secondary significance, since they refer to mental images\u2014the real surrealist images\u2014that preceded them. In order to dispel this suspicion and to connect automatism directly with his procedure, Ernst began to investigate imaginative seeing. Imaginative seeing suffused rubbed depictions with resemblances, made dead traces into simulacra of nature.\n\nFIGURE 44\n\nDrawing of the Wolf Man's dream from Sigmund Freud, _Aus der Geschichte einer infantilen Neurose_ (1918).\n\nAccordingly, Ernst's childhood hallucination corrects the Wolf Man's dream, as well as his approach to drawing. The window opened as if by a ghostly hand is replaced by the seeing of resemblances in wood grain. The imitation mahogany once again invokes the positive value of the false and the secondary, which the surrealist reflections on illusion and resemblance (e.g., Breton's rust stain in an old newspaper) had always celebrated. Later, in his manifesto \"How to Force Inspiration,\" Ernst explicitly brings the false mahogany into connection with Leonardo's famous stains on the wall. In \"Visions of Half-Sleep,\" the document of an infantile trauma, this type of historical comparison would be out of place. But the allusion to Freud's freshly translated study of Leonardo, which was advertised in the same issue of _La R\u00e9volution surr\u00e9aliste_ , may not have been lost on his surrealist friends and other knowledgeable readers. This allusion underscores the surrealists' ambition to provide an independent, in many aspects divergent, contribution to the new science of psychoanalysis. Specifically, among all the psychopathologically useful evidence from Leonardo's life, paintings, and writings, Freud leaves unmentioned the famous passage in the _Treatise on Painting_ in which the master directs his students to exercise their imaginative powers by looking for images in stains on the wall. In competition with the famous screen memory of da Vinci's phallic mother as a vulture, which Freud analyzed, Ernst presents an entirely different psychoanalytic document that introduces the origins of his own art and simultaneously stages them: a hallucination, provoked by imitation mahogany (instead of a stained wall), from out of which the seeing of resemblances retrieves a more ancient image, the primal scene. In this way, Ernst presents the deferred image, which comes from the past but only gains its power through present activation, as a model for his poetics of frottage (and grattage); and at the same time, he supplements or corrects Freud's \"Leonardo da Vinci.\"\n\nOn the level of motif as well, there are many associations that connect Ernst's primal scene with frottage. As has already been observed, the many reversible images concealed in _Natural History_ refer to a psychoanalytic thesis that was particularly attractive to the surrealists: that reversible images were painting's own parapraxes in which the painter's unconscious revealed itself. Oskar Pfister saw a vulture in the folds of fabric in Leonardo's _Anna Selbdritt_ and compelled Freud\u2014who received this confirmation of his own theses with a certain skepticism\u2014to publish it in the second edition (1919) of his Leonardo study. Among many grotesque animal forms, _Natural History_ abounds with concealed images of birds. For example, fossil traces of avian creatures can easily be made out in _Whip Lashes or Lava Threads_ (fig. 31). Christa Lichtenstern has already observed that this print thematizes frottage; moreover, in the context of Ernst's reception of Freud, this print is noteworthy in its fixation of the primal scene as a prehistoric fossil.\n\nIn \"How to Force Inspiration,\" Ernst derives further poetological implications from the association of prehistory, the primal scene, and frottage. He shifts his account of the discovery of frottage to the specific situation of a rainy day in a hotel room near the beach.\n\nBeginning with a memory from childhood (related above) in the course of which a panel of false mahogany, situated in front of my bed, had played the role of optical _provocateur_ of a vision of half-sleep, and finding myself one rainy evening in a seaside inn, I was struck by the obsession that showed to my excited gaze the floor-boards upon which a thousand scrubbings had deepened the grooves.\n\nThe childhood memory encounters a present inducement through which the hallucinations of individual prehistory are activated. But the inducement, the floorboards of his hotel room, refers to another deep time that precedes individual prehistory: namely, the time it has taken for thousands of scrubbings to deepen the grooves in the wood grain and give potency to their suggestive power. Ernst implies this analogy between the \"thousand scrubbings\" and the slowness of natural processes by saying his discovery was made on a rainy day near the beach: outside, the rising and falling of waves, erosion and accretion; inside, the traces of \"a thousand scrubbings.\" Surrounded by the deep-time processes of nature and anonymous labor, the artist encounters hallucinations from his own individual deep time. Thus, Ernst also integrated into his mythopoetics of frottage and grattage Freud's theory that the primal scene as an event and a direct experience is not only determined after the fact, but is also preformed by the prehistoric past. In this way, he suggests ascribing to surrealist images a particular temporality: they come into being in the encounter of infantile hallucination and deeptime erosion, of individual and geologic prehistory.\n\nFor the reasons outlined above, Ernst refrained from making any reflective statements about his procedure in 1927. This decision accentuated a literary effect of his \"Visions of Half-Sleep\"\u2014its Ubu-esque comedy\u2014which would be lost in the declaration of 1933. While the Wolf Man's account bespeaks the terror of a nightmare, and Leonardo's screen memory approaches an erotic mystery, Ernst's father\u2014with his drawing, gasping, and whipping\u2014is a figure from Alfred Jarry's imagination. This grotesque manifestation allows the primal scene to exhibit a lesser degree of distortion in \"Visions of Half-Sleep\" than it does in the Wolf Man's dream. Ernst avails himself of what Freud describes as a characteristic of childhood fantasies: that they take revenge on the father with \"marvellous persiflage\" and \"caricatures.\" The surrealist and his father both work with pre-morphic grounds and provoke the viewer to see resemblances. The origin of surrealist painting lies in Ernst's trumping up his father's art as an origin, while his father (who dabbled as a late romantic painter) could have only understood this tribute as mockery. This parody of a homosocial Oedipal identification contrasts markedly with the artist whom Ernst, and after him many other surrealists, considered as one of their own\u2014Freud's Leonardo da Vinci.\n\nIt is obvious why Freud's biographical essay on Leonardo was seductive to the surrealists, presenting as it did an artist who held his craft in little esteem because he was looking for higher knowledge, and who generally exemplified the treatment of sexual life as a model for artistic and intellectual development. (Less attractive features, such as sexual asceticism and emotional indifference, were ignored). However, Ernst's childhood memory depicts a fantasy entirely different from Leonardo's screen memory: while the latter, according to Freud's interpretation, implies the homosexual desire of a boy neglected by his father, in the former, the father makes his appearance with the full impact, and the imposing mustache, of a Wilhelmine _pater familias_. While Leonardo's vulture, which thrusts its tail into the child's mouth, embodies the artist's phallic mother, Ernst's mother plays only a passive, supporting role in the figure of the vase that his father fills with pictures and then spins like a top. In comparison with Freud's Leonardo, the Oedipal situation is normalized. The artist-to-be develops and asserts himself against the superior competition of his father by first making his father laughable, in order then to take the fat crayon into his own hand, to conjure new monsters out of amorphic grounds.\n\nMust Ernst's \"Visions of Half-Sleep\" consequently be read as an ideal mythology of masculinity, which styles the avant-garde artist as the enemy and successor of his father, in a contracted Oedipus complex from which the mother has virtually disappeared? Does Ernst not affirm the homosocial conception that the artist must emerge victorious from a father\/son conflict beyond the mother, in order to found a new fraternal regime? Surely, the boundaries of surrealism are thus addressed, though it remains to be shown how Ernst operates within them. The Ubu-esque comedy of the first half-asleep hallucination should be brought to mind once more, since it hollows out the homosocial myth. If, as Michel de Certeau has observed, Freud's studies of artists' biographies critically invert the bourgeois cult of the individual and present its mask, regarding Ernst's use of this mask, it could be said that his Oedipus makes faces.\n\nLEONARDO'S OEDIPUS; OR, HOW MAX ERNST REWRITES HIS FAMILY ROMANCE\n\nBesides this one hallucination, \"Visions of Half-Sleep\" gives an account of two others, to which little attention has been paid in the scholarship. The subsequent hallucinations allow another Oedipus to come forward\u2014Leonardo's Oedipus. In the second section, titled \"At the Age of Puberty,\" Ernst tells of another vision he had while half asleep: a procession of normally dressed women and men appear on a distant horizon and advance toward Max's bed. Before they arrive, they divide into two groups, women going to the right and men to the left. At first, Max is struck by the youth of the women, but on closer inspection, only a few of them\u2014two or three\u2014prove to be young enough for the adolescent observer. Without looking to the other side, he knows for sure that he would commit the opposite error in observing the men. At first, all the men would seem frighteningly old, but after closer scrutiny, his father would be singled out as the only truly elderly man. Again, the primary thematic concern is seeing as scopophilia, a passive activity, as was also diagnosed in the study of the Wolf Man. The approach of the men and women from a great distance is a surrealist perspectivization, which could well be understood, with Freud, as the representation of their origin in the depths of childhood. In fact, the men and women are initially associated with the parents as models of future object choice, at whom Oedipal love (the youth of the women) and Oedipal hatred (the age of the men) are directed. However, his curious gaze and the persistent pleasure he takes in looking allow the adolescent Max to break the spell of fascination with his parents, to leave behind his father as a senile old man, and to turn his interests away from his aged mother toward younger women. After the early childhood hallucination has presented scopophilia as the source of fear of the father, in puberty this same pleasure in looking gives the adolescent the power to break out of the Oedipal family triangle.\n\nThe third vision, dated January 1926\u2014thus the time at which he had just completed the frottages of _Natural History_ \u2014shows Ernst lying stretched out on his bed, at the foot of which a tall, slender woman is standing upright. The woman and her red dress are transparent, so that it is possible to observe the surprising fineness of her skeleton. Scopophilia, which has now achieved absolute mastery in the motif of transparency, finds its supreme object in a woman standing erect, a phallic woman who, in a state of perfect surrender to the passive artist's gaze, is herself a figure of the flawless (not castrated) perfection of the visible. While his childhood was governed by a normal Oedipus complex, as an adult, Ernst has caught up with Leonardo. According to Freud's analysis, Leonardo owed his free, investigative character to a phallic woman, the single mother who had not restricted or punished his infantile sexual research with paternal authority. While in the early childhood hallucination, Ernst's mother disappears in the form of the mistreated vase, a new woman now appears, an erect and transparent figure who embodies visual perfection in accordance with surrealist scopophilia. In this way, Ernst suggests how his \"instinctual vicissitudes\" have led directly to frottage and grattage\u2014namely, two artistic procedures that place seeing above making, and thus the artist's passivity above his activity; and as the precondition of this passivity, he presents a figure of psychoanalytic mythology, the phallic woman, who, as a form, simultaneously constitutes the surrealist visual field. As the key figure of \"Visions of Half-Sleep,\" the phallic woman points to _Natural History_ , which likewise ends in a threshold figure of transparency, Eve\/Gala. Ernst thus suggests a teleological reading, which, against the intractable aspects of the natural history, accentuates the latter's progressive development, with the end result that opacity is overcome by transparency. However, the common objective of text and picture series\u2014the upright woman as the promise of a perfect visuality\u2014only comes into appearance through her being marked as the function of a mortifying and petrifying gaze. In \"Visions,\" the female apparition clearly takes on her glassy transparency for the sake, and through the effect, of the pleasure that Ernst takes in looking. In _Natural History_ , as has been discussed, it is the light of a preceding flash that will have already transfixed Eve and her shadow. It is well known that Ernst lauded frottage as a procedure that dethrones the artist as creator and relinquishes him to his passivity. In both of these optical effects\u2014transparency and the flash\u2014a mortification is revealed that always preconditions surrealism. In order to constitute the artist's passive pleasure in looking, the woman first has to be hardened into a glassy figure or transfixed in a magnesium flash. Only her becoming inorganic makes her a surrealist being. After the Dadaist overpaintings have dismantled the woman as the naturalistic origin of artistic creativity, have stratified, and finally buried her in a landscape marked by mechanical\/distorting repetition (plate 1), in the surrealist frottages she appears as a form that promises a fully transparent visuality, though under the condition that she is transfixed and mortified by a preceding gaze.\n\nTHE LIQUEFACTION OF THE UNCONSCIOUS\n\nThe progression of \"Visions of Half-Sleep\" can be summarized as follows: through scopophilia, to which he owes his first, traumatic hallucination, Max Ernst is able to gain a new, phallic mother not castrated by his father and thus arrive at the threshold of a new unconscious. Although Ernst read Freud earlier, more extensively, and in 1927 probably better than Andr\u00e9 Breton did, he evidently shared Breton's conviction that the unconscious is not so \"unyieldingly rigid\" as Freud describes it. For the surrealists, Freud's intractable unconscious harbored a conceptual danger\u2014the same danger that was assumed in the conception, established in French psychology, of automatism as a primitive function of the human psyche that tended toward mere repetition. From the perspective of the young poets, this threat was always especially tangible when the \"magic dictation\" of automatic writing and other practices devoted to the unconscious yielded unsatisfactory results\u2014that is, poetic commonplaces. In 1933 Breton made reference to the \"continual misfortune\" of automatism, and in 1928 Louis Aragon even spoke of a \"inexhaustible diarrhea.\" As early as 1924, shortly after the publication of the _Manifesto of Surrealism_ , they both made note in the logbook at surrealist headquarters that some surrealist texts were of questionable value. This banal, intractable force of repetition, which the surrealist poets had not escaped, could be attacked polemically; but also through keeping phenomena of repetition, emaciation, and mortification insistently in view, it could be held at bay and thus glossed over. In Breton's automatic texts, poetic commonplaces, narrative clich\u00e9s, and motifs of surrogates (such as a rusty imprint in an old newspaper) form a mask-like surface in which the automatic language establishes itself as a relatively uncontrolled textual event. As was extensively demonstrated in the second chapter, the material often originates in childhood and thus is intended to lead back to the time when the unconscious was still in what Freud termed a \"lava-like\" aggregate state. In the _Manifesto_, surrealism is thus heralded as a movement that discovers in childhood a space of unlived possibilities and makes these possibilities available. Thus, Freud's conviction of the \"unyieldingly rigid\" unconscious was by no means repressed or denied; rather, it was thought of as a competing conception, against which the surrealist revolution was directed, as against the petrifaction of social conditions. Surrealism's definitive fantasy was the rediscovery of the lava-like plasticity of the infantile, prehistoric unconscious. Ernst's surrealist pictures could be understood as instruments of this softening: frottage is after all a child's technique that fills the hard crusts of prehistory with simulacra of growth. Frottage provokes a seeing of resemblances that makes even rock strata, section lines, dead wood grain, or grooves in dried paint oscillate, and transforms them in a wave-like motion.\n\nFIGURE 45\n\nMax Ernst, _The Master's Bedroom_ ( _das schlafzimmer des meisters_ ), ca. 1920. 16.3 \u00d7 22 cm, collage, gouache and pencil. Private collection (S\/M no. 399). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nSince the seeing of resemblances only becomes a characteristic pictorial effect in the frottages, these works' conception of the unconscious must fundamentally differ from that of the Dadaist pictures, as well as that of the paintings that developed out of the Dadaist pictures in the years 1921\u201324. Rosalind Krauss has dedicated a fascinating analysis to the overpainting _The Master's Bedroom_ ( _das schlafzimmer des meisters_ ) of 1920 (figs. 45, ), which will be presented more extensively in what follows. Her insights, but also her blind spots, provide the impetus for a better understanding of the differences between Dada and surrealism, with regard to each one's conception of the unconscious.\n\nAs is well known, Krauss's thoughts on Ernst are formulated within the context of her critique of the modernist ideal of opticality. The correlative of this purified perception is a type of artwork distinguished by sensual presence and structural transparency. By contrast, the optical unconscious is constituted out of that which is repressed by modernism: simply stated, the optical unconscious is characterized by absences and opacity. Absences arise because perception is discontinuous or, as Krauss writes, because it pulses between on and off; opacity arises through the traces of prefabricated conceptions that have an effect on perception without being controllable by or transparent to the perceiving subject.\n\nFIGURE 46\n\nFrom _Cologne Catalogue of Teaching Aids_ ( _Katalog der K\u00f6lner Lehrmittelanstalt_ ), 1914, p. 142.\n\nErnst's overpaintings are analogous to this aspect of the optical unconscious, because in them, the pictorial ground, instead of opening up a field of pure presence, is more like a closed \"container\" that has already been filled with readymade images. But how is the pulsing temporality of the optical unconscious expressed in _The Master's Bedroom_ , where, in spite of the snake and the bat, a perfect motionlessness lies over everything? This problem compels Krauss to a rather risky construction. First, she makes a comparison with the \"mystic writing pad\" ( _Wunderblock_ ) that famously served Freud as a model for the unconscious. The stimuli registered by conscious perception leave on this outermost layer of the psychic apparatus only fleeting traces, which are immediately erased in order to ensure the capacity to receive new stimuli. While the conscious corresponds to the outer film of the \"mystic writing pad,\" the unconscious is comparable to the waxy ground, which, even after the visible drawing has been erased, remains imprinted with a mesh of traces. Further pursuing the analogy between psychic apparatus, mystic writing pad, and Ernst's overpaintings, Krauss writes that the wax tablet of the pad corresponds to the underlying page from the teaching aids catalog, whose inventory-like array of objects can be compared with the stored contents of unconscious memory. Meanwhile, the mystic writing pad's surface, which can be written on, is analogous to the overpainting, especially since the latter's skin-like thickness seems to suggest that it could be detached from the ground, as can the outermost layer of the pad. On the basis of this point of comparison, Krauss arrives at a further implication of Freud's apparatus: that the alternation between drawing and erasing stands for \"the discontinuity of the current of innervation,\" thus, for the periodicity of perception, pulsing between on and off. Now, this pulsing movement is not illustrated in _The Master's Bedroom_ ; rather, in the motionlessness of the picture, \"the sense of the gap, the detachment, the split that results from the pulse,\" comes into view. Krauss relates this interval, split, or lack\u2014which opens up in a discontinuous perception alternating between on and off, and which is registered in the frozen emptiness of Ernst's overpaintings\u2014to lack _par excellence_ ; that is, to castration. The climax of her thesis, castration's correspondence to the off function of perception, is achieved when she points out that Ernst's overpainting refers to the most extensively discussed image of castration fear, the drawing made by Freud's Wolf Man (fig. 44). Krauss is not interested in similarities of form or of motif that could likewise be demonstrated: both depict animals; the intensity of the gaze in the dream picture is transferred to the perspective in the overpainting; what's more, in the study of the Wolf Man, Freud comments, \"The whale and the polar bear, it has been said, cannot wage war on each other, for since each is confined to his own element they cannot meet.\" Instead, Krauss places the deferred action of the image at the center of her considerations. The Wolf Man's dream vision contains prefabricated motifs (such as the fairy tale of the wolf and the seven young goats, which the child's grandfather told to him) in order to activate and to distort the image of the primal scene. Ernst's technique of overpainting relates similarly to the overpainted readymade. As a stock of motifs, the page from the _Cologne Catalogue of Teaching Aids_ can be equated with the recent material (Freud's \"residue of the day\") that the dream vision processes and distorts. In addition, in its ordered arrangement, it refers to a preceding image matrix: the serial production of images. In the overpainting, this is given a deferred activation, like the repetition of the matrix of the primal scene in the Wolf Man's dream vision.\n\nDeferred action is in fact a central concept for understanding Ernst's art (indeed, it had already served as the principle of construction for his \"Visions of Half-Sleep\"). Other aspects of Krauss's interpretation warrant a more critical assessment. Her analysis of _The Master's Bedroom_ calls attention to the problem of how the deferred pictoriality of the frottages, which the artist so ingeniously stages in his \"Visions of Half-Sleep,\" can be distinguished from that of the overpaintings. This problem was of little concern to Ernst the surrealist, who endeavored to unify his works poetologically. In the 1930s, he asserted that like his surrealist pictures, his Dadaist pictures could also be attributed to visual parapraxes. They were also hallucinations, provoked by the diversity and incommensurability of the motifs assembled on a single page of the teaching aids catalog. However, this poetics in accordance with the model of frottage and grattage belies a particular characteristic of the Dada pictures: namely, the rigidity of their surfaces, which is not only displayed by the pictures with a geologic theme. _The Master's Bedroom_ is also covered by a hard, solidified film, which does not readily lend itself to interpretation via the instability of visual parapraxes. The frottages are different: they elicit a metamorphic seeing and thus lend themselves to comparison with hallucinations.\n\nIt seems to me that the hard surface of the overpaintings does not give rise to the association with a detachable skin, as Krauss proposes so that she can then attempt the analogy with Freud's mystic writing pad. The frottages are more suited to this comparison, if it actually needs to be made. In any case, in my view, the formation of analogies between pictures and artistic procedures, on the one hand, and psychic models and processes, on the other, is far more complicated. Of course, I would not deny that it has been illuminating to relate Ernst's artistic devices and the work of the unconscious to one another. But beyond this drawing of analogies, the way in which these analogies relate to the pictures must be examined. The relationship is very different according to whether the Dadaist or surrealist pictures are at issue.\n\nThe analogy between overpainting as an artistic procedure and the psychic process of repression and the distorted return of the repressed can be referred back to Freud's own geologic metaphorics, which is closely connected to his more well-known metaphors of archaeology and prehistory. In _The Psychopathology of Everyday Life_ , a work probably known to Ernst, Freud states, \"The architectonic principle of the mental apparatus lies in a stratification\u2014a building up of superimposed agencies.\" When the mimetic similarity between artistic procedures and the unconscious as the production context is too strongly emphasized, Ernst's pictures are made into transparent display cases. The association of \"overpainting\/geological layering\/layers of the psyche\" only emerges after the perspective has been narrowed to such an extent that the procedural deception (overpainting disguised as collage), the competing metaphorical condensations (not just layering, but also stage sets, stacking, anatomy), and the accentuated dissimilarity (sublime nature as opposed to intricate bricolage) have fallen out of view. Simultaneously, it is necessary to stress that in the overpaintings, psychoanalytic analogies meet the same fate as geologic, theatrical, or mechanical ones: their semantic power of integration breaks down amidst the distorting repetition that determines the relationship of image and underlay. As I attempted to show in chapter 1, this is the case for all metaphors of artistic production, and thus also psychoanalytic ones.\n\nWhile in the Dada pictures, every analogy produces difference, in the frottages, artistic technique and naturalistic process work together. Frottage and fossilization form a metaphor whose totalizing effect is additionally enhanced through non-resemblance (swift technique, slow nature). This makes the frottages more accessible to further metaphorizations, which Ernst undertakes in his writings. In \"Visions of Half-Sleep\" and with declarative directness in \"How to Force Inspiration,\" he draws the comparison between accidental, naturalistic images, the seeing of resemblances, and hallucination. In the edition of _La R\u00e9volution surr\u00e9aliste_ that printed \"Visions\" as well as the French translation of _The Question of Lay Analysis_ , the reader is also encouraged to conceive of frottage as a model of the psychic apparatus\u2014not through comparison with the mystic writing pad (about which, in 1927, the surrealists may not yet have known), but on the basis of the metaphor of the cortex ( _Rinde_ ), which Freud uses in _The Question of Lay Analysis_ to demonstrate his theorem of the ego as a surface. This idea that the conscious comes into being on the toughened surface of the psychic apparatus can easily be transferred to Ernst's frottages and, even more, to his grattages. Their pictorial surfaces are exposed to underlying layers and destabilized by them. While Freud describes the cortical toughening of the ego, Ernst's pictures show how these layers become cracked, begin to move, are carved into reversible images, are laid open to deeper layers.\n\nThe difference between the Dadaist and surrealist pictures, which Ernst glosses over in his attempts at poetological integration in the 1930s, points to a decisive turn in the artist's career, from a pictorial concept based on repetition and distortion in the Dada years, to the ghostly mimesis of the surrealist simulacra. This difference corresponds to a likewise profound breach in the conception of the unconscious. In the frottages and grattages, mortified traces transform into apparitions suffused with an eerie growth. Through the power of seeing resemblances, the fossil rigidity of the unconscious dissolves, gains metamorphic motility, sketches reversible images, and produces mysterious correspondences between textures and objects (as between the sea in the first print and the hair in the last of _Natural History_ ). To be sure, _Natural History_ repeatedly leads back to mortification as the precondition of the surrealist seeing of resemblances, as finally in _Eve_ (fig. 33), who seems transfixed by a gaze like a camera flash. This mortification and hardening is indeed the precondition of an unconscious imagined as a power that produces pictorial secrets. By contrast, the Dadaist overpaintings create zones of invisibility that do not suggest secrets. The underlay lies like a foreign body underneath the landscapes and is expressed in the landscapes predominantly in its reproducing itself as a grid, diagram, or stratification. The idea of the unconscious implied in these pictures is that of an underground that is expressed on the surface through distorting repetitions. In the next chapter, these contrasting readings of Dada and surrealism\u2014distorting repetition versus haunting growth, repetitive versus productive unconscious, foreign body versus secret\u2014will be reoriented around a new question: that of the historical traces in Ernst's pictures and the manner of their signification.\n\nIn advance, I would again like to raise the question of what makes Ernst's pictures into psychoanalytic pictures. The distinction proposed here between Dada and surrealism could be understood as the frottages' being influenced by psychoanalysis to a greater extent, since they can be brought into connection with particular psychoanalytic theses and metaphors in an unproblematic way. But is it not possible also to sketch another conception of psychoanalytic pictures, according to which pictures function not as repositories of psychoanalytic knowledge, but as its catalysts, setting in motion an unexpected and risky interpretation? The Dada pictures do this much more (and Krauss's interpretation is an outstanding testament to this catalytic effect). Like the frottages, they attest to Ernst's reading of Freud. But unlike the frottages, they contain inner resistances against analogy formation, which also includes those analogies that would bring them into connection with psychic processes or the psychic apparatus. And this resistance compels us not to use psychoanalysis as a reservoir of images and analogies that can be applied to artworks, but in fact to attempt an analysis of these works, to work with their resistances.\n4\n\nPREHISTORY AND MODERN HISTORY\n\nTHE RETURN OF THE FIRST WORLD WAR\n\nPSYCHOANALYSIS AND WAR TRAUMA\n\nWhen Andr\u00e9 Breton admitted during a radio interview in 1952 that he had already discovered \"the entirety of Surrealism's raw materials\" in the Great War, he was thinking of a certain \"war neurotic\" patient he had studied at the Saint-Dizier psychiatric clinic. The patient's daredevil behavior on the front seemed motivated by his belief that the war was a gigantic hoax being staged for the purpose of deceiving him:\n\nHis arguments . . . and the impossibility of making him give them up made a great impression on me. I've often thought, after the fact, of the extreme point he represented on a line linking the speculations of an idealist such as Fichte to certain of Pascal's radical doubts. There's no doubt that for me a certain temptation originated there, which would see the light of day several years later in my \"Introduction to the Discourse on the Paucity of Reality.\"\n\nBreton's much-criticized faith in the liberating power of the unconscious, which accounts for his fascination with the \"war hysteric's\" surmounting of reality, can be related to his position as a psychiatric assistant during the war years. In this context, his stance can be read as a symptom-like aberration within the medical discourse on the war's psychological victims. The \"flight into illness\" that Breton admired was analogous, on the part of the doctors, to a disciplinary interest in relativizing the reality of the war's horrors. The minority of military psychiatrists who posited a physical explanation for the psychological breakdown of soldiers on the front were the only ones for whom the etiological significance of violence was indisputable. The opposing thesis, that so-called \"shell shock\" was psychosomatically conditioned, had gained widespread support. As a consequence, the subject of inquiry could be expanded to include the constitutional susceptibility of the war's psychological victims, the \"disposition\" that enabled one soldier to come back \"steeled\" from the front, while another came back with the shakes. In the framework of the psychosomatic theory, the individual's prehistory fell under the doctors' scrutiny as the primary factor that enabled a violent incident to become the cause of trauma. Accordingly, military psychiatry directed its therapeutic methods against the victims' \"weakness of will\" and cited a \"pathological disposition\" in order to deny pension claims. In this way, uncertainty about the significance that could be ascribed to violence was utilized for the justification of further violent acts.\n\nAlthough psychoanalysis was aiming to prove itself within military psychiatry and ultimately to become institutionalized, the newer discipline was interested for entirely different reasons in the etiological devaluation of violent events. Following the widespread acceptance of the psychogenic explanation, psychoanalysis gained some recognition, expressed in a very selective reception of psychoanalytic terms, such as \"flight into illness.\" But this acceptance was critically relativized since one of the basic convictions of psychoanalysis was considered disproven by the war neuroses. Specifically, the sexual basis of neuroses was controverted by the obvious indication that symptoms such as trembling, deafness, paralysis, or loss of sensation referred back to the mechanized events of war and the immediate trigger of the illness (explosion, burial, barrage lasting for many hours), and not to the fixation of early childhood sexual development.\n\nIn order to assert its scientific value, psychoanalysis was obliged to devalue the violent triggering incident by demonstrating its connection with the traumatizing sexual development of the child. When lecturing at the Fifth Psychoanalytic Congress in 1918 on the problem of war neuroses, Freud's pupils found themselves confronted by a similar challenge to the one Freud seemed to have mastered with \"seduction theory\" more than twenty years before: that is, the need to transform the outer violent incident into an inner conflict. This time the problem did not involve crimes conjectured to have taken place in the seclusion of the bourgeois bedroom. Instead, it involved the effects of an industrialized war, which, in their harrowing efficiency, made plain that \"traumatic neuroses\" could be considered a modern ailment, since their appearance was closely connected with the industrialization of both civil life and warfare. A problematic that was even more topical for the development of psychoanalytic theory was the primal scene, which also involved the relationship of psychic reality to contingent events. As already mentioned, Freud decided that there was a necessary interplay between three figures: deferred activation, prehistoric preformation, and an accidental witnessing associated by metaphor (as in the copulating dogs) with parental intercourse, which was the primal scene's content.\n\nS\u00e1ndor Ferenczi's, Karl Abraham's, Ernest Jones's, and Georg Simmel's contributions to war neurosis attempted to divert attention away from the contingent event, toward sexual symptoms (impotence, homosexuality). On a theoretical level, they proposed conceiving of war trauma as a threat to narcissism. The mechanical effects of destruction and physical paralysis in modern trench warfare were not only traumatizing in themselves but moreover because of the infantile fixation of individual sexual development. Trauma as an industrially produced violent event converges with the psychic traumas of early childhood, which psychoanalysis understands as constitutive for sexuality. These traumas are based not on a single event but on the deferred relationship between two events. In his reading of Freud, Jean Laplanche summarizes this position as \"primal deceit\"\u2014deceit, because a contingent event takes on a psychic power that it only possesses through referring to a previous event, which for its own part is also not the origin of traumatizing violence, since it only becomes this after the fact, when viewed from the perspective of the contingent event: \"It may be said that, in a sense, trauma is situated entirely in the play of 'deceit' producing a kind of [rocking movement] between the two events. Neither of the two events in itself is traumatic; neither is a rush of excitation.\"\n\nBy 1916 Breton had confirmed his support of psychoanalysis, despite a knowledge of Freud that was patchy at best. He may well have hoped to use psychoanalytic techniques to follow the war neurotic on his \"flight into illness.\" The surrealists understood Freud's theory as a method by which the reality of violent events could be transferred into the \"rocking movement\" of the unconscious with its deferred activation.\n\nTrauma establishes a historical referent in events that, on account of their violence, are unavailable to any type of experience. Current attempts to utilize psychoanalytic trauma theory for a better understanding of historical and artistic processes focus on the refusal to be integrated, the deferred activation, and the compulsive return of events or their traces in theoretical and artistic works. Within art history, discussions have been framed in terms of the trauma of photography, abstraction, or the avant-garde in general, in order to emphasize the breaks and non-simultaneities of their reception. The current trend in which trauma is made to serve as a model for the event structure of all more or less violent incidents should, however, prompt one to assume a historicizing distance. It should be asked, which pictorial forms articulate (or obstruct) the conception of \"world war as psychic trauma\"? What does it mean for the art of the avant-garde that it understood and propagated its project as \"traumatic\"?\n\nOEDIPUS IN THE TRENCHES; OR, SURREALIST LISTENING POSTS\n\nFrottage is a procedure for the production of ahistorical pictures. Naturalistic remains, wooden boards, straw, string, leaves, and textured leather serve as a matrix. This difference from the Dadaist pictures, which the previous chapter attempts to illuminate from various angles, prompted Werner Spies to observe: \"In place of a reproduced material located in a cultural no-man's land, there now appears a raw material that had never been given expression in all of history or civilization. The textures that Max Ernst lays beneath the drawing paper are, in a manner of speaking, open, without history.\" This absence of history was the condition that the surrealists required in order to rewrite the prehistory of the human psyche. Iconographically, their psychic prehistory also does incorporate historical allusions. However, the interpretation of these allusions cannot be considered separately from the seeing of resemblances, to which the frottages subject all their themes.2\n\nIn _Fields of Honor, Flood, Seismic Plants_ ( _Les champs d'honneur, les inondations, les plantes sismiques_ ; fig. 47), the twelfth print of _Natural History_ , the monumental clump sitting in the foreground is tilted slightly forward. To the right, its shadow, which would have the potential to ensure a certain spatial stability, melts into a sauce. When, in this sauce, the eye and snout of an animal become visible, the brightly illuminated space is transformed into an unstable terrain where appearances answer to the viewer's gaze. The uncanny effect of the hidden shadow figure is yet surpassed by the amorphous growth that inflates the multiply domed surface of the rootless fruit and seems to push the whole thing out beyond the pictorial space.\n\nFavorable conditions for this formless, animating seeing-in are created by the homogenous surface of the frottage, as well as its grainy facture and the un stable size relationships, which generate a convulsively contracting and expanding space. In _Fields of Honor_ , this space arises from the contrast between the agglomerated fruit and the second item of vegetation, positioned far away from it at the horizon line. This latter's shadows seem not in keeping with the optical effects of the landscape; rather, they seem to fall upon the drawing paper. Likewise, the detailed rendering of the veins contributes to seeing the laurel leaves as if they were not positioned in deep space, but instead were resting on the surface of the support. One might think that someone located outside of the picture had pinned a military decoration onto the picture's outer boundary\u2014at the exact location, within the picture, of its inner boundary, the horizon. The horizon was an emblem of the surrealist movement, which aimed to go beyond perceivable reality.3\n\nFIGURE 47\n\nMax Ernst, _Fields of Honor, Flood, Seismic Plants_ ( _Les champs d'honneur, les inondations, les plantes sismiques_ ) ( _Histoire naturelle_ , print 12), 1925\/26. 26.2 \u00d7 43 cm (measurement of image), photoengraving from frottage (S\/M no. 801). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn or upon the surrealist picture\u2014applied to it from outside or, better, tacked on like the Iron Cross First Class that Ernst earned in the Great War\u2014there is a foreign body that provides the prehistoric world of _Natural History_ with a historical reference. However, the surrealist picture aims to integrate this foreign body into its mode of functioning, or to transform the historical reference, which functions through metonymy (the war), into a metaphoric element. On a barren plateau, \"seismic plants\" meet \"honors,\" clumps meet laurel leaves, while both items of vegetation press forward and compete for the viewer's gaze: one with its amorphous globules, the other with its hallucinatory sharpness. The military decoration intensifies the convulsive spatial dynamics of the surrealist picture, since it is its double position, both in deep space as well as on the surface, that overrides the natural relationships of near and far, large and small, in order to provoke the seeing of resemblances and thus to suffuse the enigmatic fossils with life. The historical reference is lost in this effect of the enigmatic that dominates the whole surrealist natural history.\n\nLike Breton, who claimed to have achieved the \"the entirety of Surrealism's raw materials\" from the study of trauma victims in the First World War, Ernst would also locate the origin of his surrealist identity in the war. In his autobiographical notes, which were first published during his exile in America\u2014and which should be read in the context of the 1940s and the search, during this time, for a new surrealist myth\u2014he writes: \"Max Ernst died the first of August 1914. He resuscitated the 11th of November 1918 as a young man aspiring to become a magician and to find the myth of his time.\" This laconic statement of death and rebirth brings into play an elementary motif of initiation rites, a familiar pattern in the fantasies of world war veterans. Veterans believed that the war was a mystery into which only they had been initiated; through the possession of its secrets, they were deeply estranged from those who had stayed at home. When one observes Ernst's autobiography as a whole, this topos\u2014that the trauma of war initiates its soldiers into secret knowledge\u2014becomes connected with a more comprehensive, universal enigma: the riddle of Oedipus. Long before the war, the mythmaking apparatus that would be brought forth from the war was already fully developed. It appears in the memories of early childhood and puberty with which Ernst's \"Tissue of Truth, Tissue of Lies\" (\"Wahrheits- und L\u00fcgengewebe\") begins. These memories\u2014his beloved bird's dying on the same day that his little sister is born, or his father as a landscape painter who cuts down trees that disrupt the subject of his pictures, or the hallucination that Ernst published in 1927 as \"Visions of Half-Sleep\"\u2014can be read as traumas in the Oedipal mold.\n\nErnst's autobiography thus performs a double movement in order to deflect the trauma of war as physical violence into the \"rocking movement\" of deferred activation: First, the war becomes a traumatic secret that grounds the artist's surrealist identity; then it becomes a psychic trauma like the infantile traumas that have already been recounted in psychoanalytically encrypted anecdotes. The First World War is thus relativized as the origin of Ernst's surrealist identity. The war does not make a singular cut but rather is related to his autobiography's central type of event: that is, the Oedipal trauma and its manifold transformations, from the terror of the father to the bliss of the phallic woman.\n\nBreton and Ernst both developed a traumatophilic conception of history in the shadow of World War II. In Ernst's autobiographical notes, the two motifs that demonstrate this conception are World War I as the secret of surrealism and the trauma of war as a belated expression of preceding psychic traumas. Both of these motifs can be found as early as 1923 in a text by Louis Aragon. Intended to serve as the introduction to a projected literary history of the avant-garde in the aftermath of World War I, the text was published under the title \"Agadir\" in the proto-surrealist magazine _Litt\u00e9rature_. The port city of Agadir was the center of the Second Moroccan Crisis, which led the Entente and Germany to the brink of war. But for his generation, Aragon writes, \"Agadir\" was a magic word that fundamentally transformed the world. It expelled young people from their world of adventure novels and cast them into the adult world of historical events. As a word that itself seemed to come from an adventure story, a hermetic name formed from auspicious sounds, a child's garbled \"Arcadia,\" it suddenly caused the opening, beyond the adolescent horizon, of the reality of military conflicts\u2014from which, one is compelled to add, Aragon's generation would never again be free. The mystery of \"Agadir,\" as Aragon explicitly writes, was the trauma that destroyed a world (that of childhood) and created a traumatomorph (that of the war and the avant-garde). \"Agadir,\" it can further be added, functions as a displacement that shifts the historical break of the war onto an incident that, through the contingent position it assumes in the biographies of Aragon's generation, prefigures the significance of the war. And through this displacement, \"Agadir\" not only anticipates the break of 1914\u201318. Conversely, it also transports the enigma that still connects the sounds \"Agadir\" with the adolescent world of adventure novels\u2014which world these sounds destroyed\u2014into the dark time of the war years. The enigmatic word that brought an end to a generation's dreamy days of youth echoes in the labyrinth of the trenches and fills them with surrealist mysteries.\n\nIn Ernst's autobiographical notes, in his _Natural History_ , as well as in Aragon's fragment \"Agadir,\" there occurs a traumatophilic back projection of the war. The war only accrues its significance in the \"rocking movement\" brought on by preceding traumas. For Aragon, his generation's entry into manhood around 1912 is the critical caesura, while Ernst views the war as one phase in a psychic prehistory that proceeds in a concatenation of traumas from early childhood to the present. As a foreign body, the war only protrudes into the surrealists' traumatophilic fantasies of history to the extent that it can be transformed into a mystery, similar to the sexual mysteries staged in psychoanalytic primal fantasies.\n\nDADA'S WORLD WAR\n\nIn the Dada pictures that Ernst made in Cologne in 1919\u201321, the years immediately following the war, there are numerous World War I motifs. Werner Spies observes that some of the machine pictures resemble artillery. In these same pictures, Hal Foster examines the iconography of the woman as a destruction machine (fig. 6) and attempts a psychoanalytic interpretation following on Klaus Theweleit's familiar theses. Ludger Derenthal is able to trace important sources for Ernst's photo collages and document several historical references (such as to the bombardment of Reims). Out of a stratified Dada landscape, insignia emerge that bring to mind political-military groups such as the Freikorps, which was on the rampage during the postwar period. In this context, the text \"The Old Vivisector\" (\"der alte vivi-sektor\") is of particular interest, since it expresses the same fascination as was felt by the young Andr\u00e9 Breton for the \"flight into illness\" as a poetic form of desertion. \"The Old Vivisector,\" published in 1921 in the last great Dada manifesto, _Dada Outdoors_ ( _Dada au grand air_ ), depicts a conversation between a general and an adjutant. In the First World War, the new destructive potential of artillery and machine-gun positions compelled a defensively oriented trench war, in which long, grueling periods of waiting could suddenly erupt into clamorous chaos and destruction. The traditional hierarchies were preserved between officers and the rank and file, despite harsh criticism of the officers' inadequate knowledge of the area, and the consequent dubious nature of their commands. Ernst amplifies the collapse of the command situation, with first the adjutant and then the general setting off on a hallucinatory desertion:\n\nThere on that hill, cried the General, I see closely spaced infantry lines. Why was I not informed of them? They are caterpillar hunters and inflorescences, the Adjutant countered. And the artillery observation booths over there? Those are the adventitious buds on their runners. To the inside left is a strong battery of seemingly large caliber, said the commander again. We don't have anything like that. Your Excellency is exactly right: they are the bellies of the ovules, the digitigrades of the future, the limbs of ones who have been buried in the snow. They surpass the spores in beauty and clarity. They are thickly covered with root hairs. Their throat canals sport fine cilia. They hide their poisonous fangs in their women's tender parts. Breathing orifice (!) and assimilation threads by the thousands. Sundew from the bottom of the cup. Forward, the General answered. The shrinking of wall cells. The germination of spores. She's an inveterate drinker.\n\nThe botanist's language employed by the adjutant also takes possession of the general. This collagist's notion itself partakes in a specific historical plausibility. Here, as in most of his work, Ernst is using something prefabricated: the employment of a mixture of military and scientific language was escalating under the influence of the war. Here is an example from medicine:\n\nThese cells essentially originate in the well-known deposits of sanguification, the different bones including the marrow being the major garrison towns in adults. How easily and how quickly these leukocytes can be mobilized can be concluded from the time it takes to rush legions from the marrow into the veins as soon as the large telegraph system of the entails is stimulated.\n\nIncorporated fragments of patriotic and militarized language also appear in several pictures' inscriptions\u2014\"rechtzeitig erkannte angriffspl\u00e4ne der assimilanzf\u00e4den auf die feste DADA 1:300000\" (plans detected in the nick of time for assimilation threads to attack the DADA fortress 1:300,000), or \"sodaliten schneeberger dr\u00fccketh\u00e4ler rosinen und mandeln schlagen die eingeborenen mitteleuropas zu meerschaum und eilen nach stattgehabter denudation den ereignissen in bester absicht voraus\" (fig. 10) (sodalites shirkers snow-covered mountain and valley dwellers raisins and almonds beat the natives of central Europe to sea foam and following advanced denudation hurry ahead of events with the best intentions). Here, military diction enters into the service of Dada. \"The natives of central Europe\" can easily be identified as the Central Powers, and their conquerors, characterized as defeatist shirkers of military duty, valley dwellers, and hedonists (\"raisins and almonds\") appear quite like the Dadaists, who are known to have first formed as a group in Zurich. Then there are titles such as _winterlandschaft: vergasung der vulkanisierten eisenbraut zur erzeugung der n\u00f6tigen bettw\u00e4rme_ ( _Winter Landscape: Vaporization of the Vulcanized Iron Bride to Produce the Necessary Bed Warmth_; fig. 15) and _Die Leimbereitung aus Knochen_ ( _The Preparation of Bone Glue_ ). These may have alluded to propaganda rumors according to which, in the German Reich, human cadavers were being industrially processed into raw materials. Simultaneously, these titles point to Ernst's artistic technique: in _Winter Landscape_ , to overpainting, and in _The Preparation of Bone Glue_ , to collage.\n\nThis abundance of references to the First World War raises some previously unexamined questions. What meaning do such motifs take on in Ernst's pictures, and in what way do these works process the time period of the war? It is tempting to make the assumption that the plentiful world war iconography of the Dada collages implies another understanding of history than the one sketched out in _Natural History_'s prehistorical frottages. That is, one might suppose that Ernst's Dada pictures show a greater permeability to historical events than his surrealist pictures, in which history is completely absorbed into a psychoanalytically conceived deep time. Brigid Doherty has interpreted Berlin Dada along these lines, emphasizing its permeability to a traumatically experienced course of history. According to her, the actions, drawings, and collages of George Grosz and John Heartfield are reproductions of shell shock. The violence in the trenches, duplicated by the methods of military psychiatrists, is carried over into the postwar period in order to denounce war as insanity and at the same time to satirize the desire for normalcy. The psychiatric establishment's warning that \"war neurotics\" could use revolution as revenge upon their doctors was taken by the Berlin Dadas as an invitation to express themselves in exactly the way that was feared. This traumatophilic fascination with the literal return of violence is connected with the simultaneity ideal of cubism. It is essentially different from surrealism, which attempts to metaphorize physical violence by catching it in a psychoanalytically woven net of meaning.\n\nIn the case of Ernst, however, every attempt at drawing a boundary between Dada and surrealism comes up against a critical difficulty, which I have not yet addressed. From a certain perspective, as Werner Spies and Rosalind Krauss have been able to demonstrate, the photo collages of the Dada period are understandable as paradigms of a genuinely surrealist visual poetics. As opposed to the Berlin Dadaists, who exhibit the constructive act, Ernst attempts to conceal his cuts through the source material (fig. 48). His photographically reproducing the photo collages gives rise to a seamless picture surface and a coloristic continuum of gray tones. By means of this technique, the incisions appear less as the interventions of an artist who cuts pictures apart, and more as cuts within the world, whose indexical depiction presents itself in the homogenous field of the photograph. Collage gains the status of a procedure that seems to dominate reality in order to excavate, within it, the hidden strata of surreality. If, in the Dada years, Ernst had already achieved the procedure that would also characterize his later surrealist works, the question as to the meaning of the numerous war motifs becomes somewhat more complicated than in the case of the Berlin Dadaists. Let us examine one of these proto-surrealist collages more closely.\n\nIn _The Chinese Nightingale_ ( _die chinesische nachtigall_ ; fig. 48), the photo of a French aircraft bomb lying in a field serves as both the pictorial ground (field) and the main motif (bomb). The ninety-degree rotation through which Ernst raises the picture into the vertical can be described as the definitive operation, in that the picture as a whole consists of arrested, potential, and actual circular movements: the rotation of the bomb corresponds to the opened fan, which in turn is answered by the folded wing of the bird's body, below. The rotation motif is set in motion by the two arms in their serpentine twining around the vertical body, in a dance that draws the viewer into a pulsating space. Two artistic devices accentuate this seductive spatial dynamic of spreading open and folding shut: one is the rotation of the grassy ground from the horizontal into the vertical format, and the other is the distribution of light and shadow.\n\nThe grassy ground\u2014which Ernst retouches with pen, especially on the right side of the picture, to transform it into a convoluted thicket\u2014intensifies the seductive gestures of the arms. It is particularly unsettling that in the left half of the picture, where the thicket is shown close up, the feminine arm sinks into it, while to the right, where the blades of grass become smaller, their mass more intricate, and the focus blurred, the arm extends beckoningly toward the viewer. To the left, the Chinese nightingale nestles into the ground; to the right, she seems to detach herself from it; above, through the fan she is firmly anchored in the visual field; below, she seems to float freely. Meanwhile, her appearance seems to be bathed in the unreal, fluctuating light of the theater stage, which illuminates each body part from a different side. In this way, so much material plasticity is divested that even the bomb seems unstable, its objecthood nearly reverting to a pre-object status.\n\nFIGURE 48\n\nMax Ernst, _The Chinese Nightingale_ ( _die chinesische Nachtigall_ ), 1920. 12.2 \u00d7 8.8 cm, collage and ink on paper, mounted on board. Private collection (S\/M no. 376). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThese qualities of space\u2014its pulsing between near and far, large and small, light and dark, stabilization and instability\u2014distinguish some of the photo collages of the Dada years and are reinvented, after a time, in the frottages. Once again, it is important to emphasize the condition of their effectiveness: none of these changes in quality is allowed to harden into discontinuity, since the picture or the chimera can only enact its seductive play when protected by the picture surface, and in the continuum of the photographic gray scale. Under this condition, which likewise is only to be found again in frottage with its homogenized facture, a foreign body can be applied to the picture. What can be said of the tacked-on laurel leaves in _Les champs d'honneur_ can also be said of the eye in _The Chinese Nightingale_ : both elements are firmly positioned in the pictorial space at the same time as they thrust forward to the picture surface.\n\nThe eye appears firmly positioned because it, more clearly than all the other pictorial elements, mutates into an organ of the aircraft bomb. In contrast to the arms, the wing, or the fan, which are set apart from the body by a spaceless distance, the eye actually sits within the surface of the bomb. But this same effective animation\u2014the dud ( _Blindg\u00e4nger_ ) is staring at us\u2014allows the eye to be perceived as an apotropaic object thrusting toward the picture surface, which, though it cannot go so far as to interrupt the seductive movement of the picture, is able to associate this movement with a threat. There are three operations that make the eye into a foreign body:\n\n1. It is the only pictorial element that remains a perfectly distinct object whose boundaries do not merge into those of a neighboring element (by contrast, one observes how arms and wing, fan and bomb fit into one another).\n\n2. The cutout eye has been rotated 180 degrees, as a result of which it has hardened into a reflective surface that opposes the rotational movement in the pictorial space. A woman's darkly shadowed gaze, implying a mysterious interior, is transformed into the impenetrable and aggressive gazelessness of a bird's eye.\n\n3. Only in the motif of the eye is the cut marked as a cut through the source material. Both the photograph and the form of the eye have been cut\u2014it is a cutout eye in a double sense, since the cut separates an existing eye motif, while this same cut produces an eye. In other words, the line of the cut imitates what is being cut out, or the indexical trace imitates the icon. For the viewer, this relationship is inverted: the photographed eye cannot be perceived independently from the cutout eye; the destructive action that has created the latter transfers itself to the former by connecting the aggressive staring of the animalistically inverted eye with the semantics of cut and blinding, Medusa and castration.\n\nEvidently, as the fairy-tale title _The Chinese Nightingale_ also suggests, this mechano-feminine monster has emerged from a regression to the infantile conflicts and fixations that psychoanalysis describes. The arsenal of industrialized war is transformed into a chimera that conceals a sexual secret and simultaneously keeps watch with a punishing glare. The foreign body that deflects the rotational movement into convulsive space does so in service to the Oedipal arcanum. Virtually automatically, an interpretation is brought into effect that refers back to the myth of the phallic woman as well as to the power of seduction and castration that emanates from her. This interpretation allows the bird, whose contours are hidden between the right forearm and bomb's right fin, to be deciphered as a symptom . . . et cetera. I will stop here, in order to relate the fact that the picture elicits this type of interpretation back to the question posed at the beginning, as to _The Chinese Nightingale_ 's reference to history. The answer must be that the picture's proto-surrealist procedural logic articulates a proto-surrealist historical fantasy. As in the frottages, which arise as an immediate reaction to Breton's first _Manifesto_ and its outlined poetics of automatism, in the photo collages, too, the world war has the character of an infantile\/sexual trauma. Instead of exploding and causing shell shock, the surrealist bomb unleashes its effect in the rocking movement of deferred action. The exceptionally absorptive power of this conception of trauma is demonstrated in that the photo collages, unlike the frottages, contain numerous world war motifs, but they are reshaped into elements of a psychoanalytic iconography.\n\nFor the onlooker not yet practiced in the mode of viewing convulsive space, what at first glance appears to be a foreign body\u2014the laurel insignia or the eye\u2014proves within the picture to be an index for a secret that lures the viewer, without its meaning ever becoming accessible. But how do these forbidden secrets come into being? What underlies them? A particular content, or even a key, is not to be found. Within Ernst's Dada works, however, a structural correspondence can be ascertained between the photo collages, which produce dangerous secrets, and another group of pictures, the geological overpaintings. The overpaintings also contain and are in themselves foreign bodies, which resist absorption by a pulsating space and the fantasizing seeing that corresponds to it.\n\n_Madam Hostess on the Lahn, guardian angel of the Germans, thine is the industry anatomy paleontology grant us thy jubilation_ ( _frau wirtin an der lahn, schutzengelin der deutschen, dein ist die industrie anatomie pal\u00e4ontologie schenk uns deine frohlocken_ ) is the title of another work made in 1920. Photos of textiles, cut out and partially overpainted, form the components of a fossil machine deposited underground (plate 2). In the title, which echoes the _Ave Maria_ , the Hostess on the Lahn\u2014a key figure from the bawdy verse commonly recited in student fraternal organizations\u2014appears as Germany's protector and playmate (\"grant us thy jubilation\"). The pillars of German nationalism (industry and economy) are resting on the dirty ground of scatophilia. This admixture of the official and the hidden corresponds, in the picture, to an inextricable confusion of outer and inner. It shows cuts through various interiors\u2014of the earth (strata), of a machine (gear wheels), of a body (blood vessels, flesh-colored masses), and of a tree (wood grain with embedded pink vermin)\u2014and at the same time is studded with motifs that indicate an outer covering: skin, hair, walls, yarn, fringe, seams, scars. Here and there, formal connections ensue between the two groups of motifs: veins merge into strands of yarn, and strands of yarn into machine parts, seams, and scars.\n\nThe de-differentiation between outer and inner is continued in Ernst's use of collage and overpainting. To summarize a few observations from chapter 1: While collage, as an operation of the cutting apart of surfaces, can be considered analogous to a surgical intervention or a geological excavation, overpainting can be understood as a procedure for the creation of new surfaces. However, as opposed to the surgeon or the geologist, the collagist who leaves his traces in Ernst's pictures does not expose any structures but loses himself in a labyrinthine interior, where he makes incisions that cannot be unequivocally located. For example, one observes the black rim below the yarn machine, which, on the left, fits snugly within the horizontal strata and, on the right, overlaps them. The layering proceeds both from the lower to the upper edge of the picture, and from pictorial support to picture surface, without these two orders being somehow mediated. Just as the collage does not reveal any inner structures, the new surface created by the overpainting cannot be conceived as a living skin. At best it is a work of provisional repair (wall parts, textiles, seams, scars), or\u2014one observes the viscous pink mass around the machine parts\u2014a rejected substance that refers to the scatophilic figure of the Hostess on the Lahn.\n\n\"Scatophilia\" is the key word that, in this instance, performs the task of psychoanalytically orienting the reference to the world war (science, industry, nationalism). German productivity is dissected by an infantile sexual researcher like the one Freud describes in his work published immediately before the war's outbreak, and it is analyzed along the lines of cloacal theory. \"Cloacal theory\" is Freud's term for the early childhood fantasy of birth as excretion. Its result is the dismantling of the mother into an anal machine, in which the mechanical elements have the function of bodily organs. Likewise, Ernst's artistic interventions lead to the mechanization and mortification of the maternal object, represented through the collaged and overpainted photos of close-up\/tactile pieces of knitting.\n\nWhile the seduction and castration threat of _The Chinese Nightingale_ attests to the plasticity of the pleasure principle\u2014which makes even a bomb into a desirable body\u2014in _Madam Hostess on the Lahn_ , this metamorphic power is brutally arrested. If one chooses to follow the psychoanalytic interpretive option as far as it can go, one arrives at the concept of the death drive. As is well known, Freud was the only psychoanalyst who drew from the war trauma a conclusion as radical as it was speculative, when, in _Beyond the Pleasure Principle_ , he introduced the death drive as an antagonistic factor underlying the libido. The compulsive, unprocessed return of traumatic events in the dreams and symptoms of those who have lived through them is an expression of this conservative force, which strives for nothing other than the return of the organism to inorganicity.\n\nArtists' use of psychoanalysis is often chastised for being naive. But suspecting the work, in its motifs of rigidification, of a death drive that the surrealists supposedly \"repressed,\" as Foster does in his main thesis in _Compulsive Beauty_ , shows a no less naively positivist approach to psychoanalysis. On Freud's theory of the death drive, Foucault remarked, \"Freud wished to explain war; but it was war that was dreamed in this shift in Freud's thinking.\" More prosaically stated, the parallel interest in de-organicizing that connects Ernst's pictures and Freud's _Beyond the Pleasure Principle_ should not be placed under the interpretive authority of psychoanalysis. It appears in the context, in the 1920s, of a widely observed metaphorics of mortification, which refers to the mineral aesthetic of prewar modernism and its distortion through the experience of world war. I will return at greater length to the example of the expressionist poetics of crystal and its debasement in Ernst's Dada pictures. But first, I would like to make a few more observations on the comparison of _The Chinese Nightingale_ and _Madam Hostess on the Lahn_.\n\nIn the preceding chapter, I tried to show that, inasmuch as Ernst's Dada overpaintings imply a psychoanalytic metaphorics of production (in which the procedure of overpainting is analogous to repression), this metaphorics breaks down as a result of the logic of disfiguring repetition. These pictures are not psychoanalytic display cases: they do not showcase models that represent the functioning of the unconscious or the psychic apparatus; nor in any way do they present an iconography that can be deciphered as a coherent text. Using the example of _Madam Hostess on the Lahn_ , I would like to describe again the negativity that characterizes the processing of psychoanalytic discourse. The picture itself directs the viewer to attempt to decipher its motifs according to the paradigm of outer\/inner, in which all the pictorial elements participate. That their differentiation is disrupted in the picture\u2014and thus it remains uncertain which layer is on the top, what is skin and what is flesh, which are hairs and which are veins\u2014repeatedly calls into question the possibility that the picture can be deciphered. Instead, the picture remains a construction that arranges numerous references to the theory of infantile sexual research: first the scatophilic hostess of the inscription, then the pieces of knitting (mother metonymies), organ fragments (the search for the origin of life), and machine parts (mechanization of the maternal body).\n\nThe photo collages are an entirely different matter. They anticipate the frottages in that their capacity to be related to the theorem of the phallic, castrating woman is not only the result of individual motifs such as the cutout eye but, moreover, is visually substantiated through corresponding spatial effects\u2014through a convulsive spatiality that simultaneously seduces and threatens the viewer. The photo collage is a picture that produces a space of hysterical seduction art under whose spell the interpretation of the picture also falls. By contrast, the overpainting is a picture that makes the act of interpretation conspicuous as always being an act of selection and recombination of different and incompatible alternatives.\n\nThis difference between the pictorial forms shows itself in the ways that the two works can be written about: _The Chinese Nightingale_ can be described in categories (seduction, dance, convulsive space, threat) denoting presence to the viewer. By contrast, _Madam Hostess on the Lahn_ only allows semantic analysis according to the paradigmatic opposition outer\/inner, and it resists any attempt to be viewed integratively. While the photo collage draws the beholder into a dynamic spatial continuum, in the overpainting her or his attention is blocked by a much worked-over, stratified, soiled, scarred picture surface; while the photo collage seduces, the overpainting, with its many barriers\u2014layers, walls, clothing, boards, rejected substances\u2014denies the possibility of immersion or even of any type of affective relation to the picture. It seems as though the foreign body (the eye or the laurel)\u2014which, in the (proto-)surrealist picture, thrusts itself to the picture surface\u2014has now itself become identical with the rigidified surface of the Dadaist picture. _Madam Hostess on the Lahn_ not only contains several variations of the foreign body motif (embedded machine parts, overpainted photos, parts of walls and textiles that have been applied and covered over, insects hidden in wood); the picture as a whole becomes a foreign body, an inextricable confusion of outside and inside with a mortified surface that repels any attempt at visual synthesis.\n\nThe fundamental difference between Ernst's surrealist and Dadaist pictures is that the former use foreign bodies as catalysts for a viewing process, while the latter themselves comprise foreign bodies of viewing. In the surrealist works, foreign bodies mark local disruptions that refer to the moment of threat in the erotic mystery, without threatening this mystery; in the Dadaist pictures, foreign bodies are the end result of mutually paralyzing procedures at work in the picture: procedures such as overpainting and collage, layering and dismantling. These form a surface covered with cuts, wounds, scars, and stains\u2014a surface that destroys the very homogeneity and closure that will be the condition of surrealist pictorial effects and mysteries. Ernst's stratified Dada overpaintings are the foreign bodies of visuality, which the (proto-)surrealist pictures strive to integrate into a world of erotic mystery.\n\nCRYSTALLIZATION\n\nThe polarity that presents itself between _The Chinese Nightingale_ and _Madam Hostess on the Lahn_ seems generally to constitute the relationship between the photo collages and the geological overpaintings. Since the overpaintings have already been analyzed at length, let us observe a second photo collage more closely. In _The Flamingos_ ( _die flamingi lassen wieder papierdotter steigen_ ; fig. 49), the harbor moles of Dunkirk dangle like \"Dover sole\" in the same water in which the flamingos are wading. The long-range clarity of the aerial view, which records the harbor with cartographic\/diagrammatic precision, suddenly reverses into a tactile close-up, confirmed by the dissected brain and especially by the unidentifiable rock formation to the right, above. This oscillation between near and far is continued in the two ropes, from which one heavenly body swings like a pendulum and the other soars like a kite. Besides flamingos, angels are the only species that can survive in this dizzying pictorial space that veers between tangibility and optical distance. In more formalistic terms, rotational operations in the homogenous pictorial field of a photographed photo collage create a seductively pulsating space that achieves positive illusion effects, as opposed to the reified illusionism of the overpaintings.\n\nA significant portion of Ernst's Dadaist works can be organized along the poles of positive photo collages and negative overpaintings. In the photo collages, there is seductive rotation, concealment of the cut, transparency of the surface, and convulsive continuity of the space. In the overpaintings, there is obstructive layering, demonstration of the cut, opacity of the surface, and rigidification of the space. In terms of content, these contrary qualities correspond to the photo collages' motifs of flight\u2014aircraft bombs, birds, airplanes, angels, flying fish, and dogs\u2014and the overpaintings' subterranean iconography. The distance that separates the two pictorial worlds from each other is traversed by projections: unintegrable, opaque details from the world of overpainting are projected into the world of photo collage (for instance, there is the curious petrifaction in the upper right of _The Flamingos_ , fig. 49), and, conversely, the convulsive space of the mutable photo collages is projected into the rigidified overpaintings (for instance, there is the pulsating, intertwining relief space of the plants in _Sodalites_ , fig. 10).\n\nFIGURE 49\n\nMax Ernst, _The Flamingos_ ( _die flamingi_ ), 1920. 29.3 \u00d7 23 cm, photographic enlargement and photomontage, mounted on board with ink. Private collection (S\/M no. 393). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nHeaven and earth, open space and labyrinth, fulfilled seeing and obstructed seeing: the experience of the front was also split along these poles, as Eric J. Leed has demonstrated in _No Man's Land: Combat and Identity in World War I_. While trench warfare was described as a destructive paralysis (contaminating, visually disempowering, and collectively proletariatizing), the image of the airman embodied lost illusions of personal valor, with all its implied qualities of visibility and the sovereign view, mobility and unprecedented freedom. The sky\u2014which was the \"cavemen's\" only view amidst a decimated, often invisible landscape\u2014revealed in the form of the heroic flyer the trench fighter's disappointed higher self. In the summer of 1914, this higher self had cheered the mobilization, in expectation that the Great War would bring the testing and confirmation, even the creation of its true identity. From the muddy maze of trenches, airplanes in the sky offered themselves as viewpoints for self-distancing and the division of consciousness. At last, from their heights, there was an unimpeded view and the hero's chance to be seen. From their heights, too, perhaps the labyrinth of trenches would appear logical and justified. This division of the experience of war between opaque earth and redeeming sky also structures statements made by several modernist artists who celebrated the industrialized war as the revelation of the hidden reality that their pictures had already been aiming for in the prewar years. Franz Marc, friend and role model to the young Ernst, describes an experience of war as could only present itself to a modernist painter influenced by Wassily Kandinsky's _Concerning the Spiritual in Art_ and Wilhelm Worringer's _Abstraction and Empathy_ :\n\nWe are waiting at the edge of a wood with our munitions wagons; the thunder of the cannons rolls like a furious storm along the whole horizon. All around are little clouds of smoke. Both thunder and clouds are already part of the landscape, like the echo that propels and doubles every shot. Suddenly, a curious whirring that passes above and away from us in an enormous arc, irregular, in constant oscillations, verging from a shrill whistle to a low rumble, like the high, wide cry of a bird of prey, always repeated, with the obstinacy of an animal that knows no other call. Then in the distance the muffled bang. These are the enemy's heavy artillery shots, which race overhead to a target unknown to us. One shot follows another; the sky is the purest autumn blue and yet we feel the high channels in which the shots storm through it. Even for the artillerists, artillery battle often has some mystical, mythic quality.\n\nMarc conjures the landscape of war as an opaque space of gun smoke and artillery fire\u2014echo, whirring, shrill whistle, low rumble, the \"high, wide cry of a bird of prey,\" and a \"muffled bang.\" Suddenly, the view that the phenomenal world dissimulates breaks through this wall of noise and opens itself to plain sight. Then projectiles become winged creatures and fill the space with vibrations and trails that crystallize into \"channels.\" A crystallized space takes form, as in the paintings of the expressionists and cubo-futurists\u2014as in Marc's animal fights or the variations of them that Ernst painted in 1917.\n\nThe crystallization of image, experience, and ego in artillery fire fulfills the desire, which already motivated Marc's prewar art, to escape the contingency of the historical and enter an eternal, transparent order. Only the one who annihilates history can hope for crystalline purity:\n\nThe world is full to suffocation. Man has placed a lien of his own cleverness on every stone. Every word is rented and mortgaged. What can be done for blessedness but give it all up and flee? But draw a line between yesterday and today?\n\nThis deed is the great task of our time, the one task for which it is worth living and dying. . . .\n\nSo let us go forth into new regions and experience the great shock that all is yet untrodden, unspoken, untilled and unexplored. The world lies before us in purity, our footsteps tremble. If we dare to go, the umbilical cord must be cut that binds us with the maternal past.\n\nThe prewar dawn that is the end of history, the venturing forth into new, untrodden regions that also severs the tie to a past perceived as impure and maternal, after 1914 becomes a quick trot in the pure morning air:\n\nI rode through a town with lovely old buildings; the women looked out the windows. The hour was russet-hued, bright and homely ( _heimlich_ ). Then I was struck by these thoughts of the danger of dead, still deadly things, and of the great purity.\n\nMight the good old devout ones have guessed what was in my mind? Something true and close, the \"danger,\" they certainly sensed.\n\nBut I trotted quickly out of the town of many memories and contagions, until I smelled the pure air of dawn.\n\nAs he rides through the village, the association of Marc's thoughts as they proceed from \"homely\" to \"deadly\" can be traced to the same literary topos that Freud will analyze only a few years later, also as a reaction to the war. The _heimlich_ (homely) in the sense of familiar, original, maternal, in becoming the _heimlich_ (hidden) in the sense of forbidden and repressed, proves to be the _unheimlich_ (uncanny): the menacing return of a past distorted through repression. Marc has fallen prey to the _heimlich\/unheimlich_ , which Freud will attempt to conceive as a problem of psychoanalytic theory. Before 1914, that from which he intended to free himself with a cut to the umbilical cord returns, \"russet-hued, bright and homely,\" as the danger of contaminating the purity of the ahistorical. The well-known thesis presented (independently) by Klaus Theweleit and Sandra Gilbert is proven once again: that the misogynistic fantasies of the prewar era escalated during the war.\n\nWhile Ernst's photo collages parody the war ideology of the untrammeled, mythic, crystalline heavens, his overpaintings derive their grotesque wit from the experiences of trench warfare. As the free-roaming imagination of the soldiers on the front is expressed in the mystery of the photo collages, their physical immobility and disempowerment is expressed in the foreign bodies of the overpaintings. In terms of motif, connections between expressionist war fantasies and Ernst's Dadaist pictures are only too clear and can be exhaustively demonstrated. Ernst's own fascination for the crystalline as a motif of redemption in the world of trench warfare can be cited as a biographical motivation. Certainly, he was familiar with Worringer's highly influential book _Abstraction and Empathy_ , through which the metaphorics of crystal gained new currency among artists. An art-critical text published during the war shows how fervently Ernst confessed to a central \"expressionist\" persuasion. As previously formulated by Worringer (who was not the only one to have written on the subject but who had the greatest impact), the age of the \"subjective\/discretionary and only individually conditioned\" was being followed by a new, at once primitive and nomothetic art. In his essay \"Vergleichung\" (\"Comparison\") published in 1917 in the _K\u00f6lner Tagblatt_ , Ernst repeated the widely held construction of history according to which the development from impressionism to the present was understood as a movement from sensuality to will. Expressionism, cubism, and futurism\u2014the recent manifestations of the \"masculine spirit\" and \"pure world formation\"\u2014emerge as the great vanquishers of hedonistic impressionism, envisioned as \"spread wide to conceive at any moment.\" \"With the advent of cubism, the world is more beautiful, more willed: the spirit of our will.\"\n\nThe essay is signed \"Max Ernst (currently in the field).\" After three years in the artillery, he certainly had no illusions about the reality of the war. His correspondence from this time bespeaks a tired irony. However, as a modern artist, he was swept away by the pathos that would prove the days and years in the field-gray uniform had served the testing and refinement of the determining, crystalline spirit. These patterns of self-perception took on the task of exalting the figure of the modern artist even under the conditions of industrialized warfare. Since the Dadaist war pictures parody the crystalline freedom of the heavens and the maternal uncanniness of the earth, they could be interpreted from a biographical perspective as instruments with which Ernst dissected and amputated his own war expressionism. Meanwhile, the obvious but until now undervalued fact that they are small, partly dilettantish, partly virtuosic pictures with lengthy inscriptions points to a media-historical context.\n\nPOSTCARDS FROM THE FIELD\n\nLike machine guns, heavy artillery, and poison gas, the large-scale implementation of military postcards constituted the modernity of the First World War. Between 1914 and 1918 in the German Reich alone, more than five billion were sent, postage free, as the most important communicative means for reassuring people at home. For artists who had been drawn into a war that promised new unity, not least the unity of modern art and general consciousness, what could be more alluring than the attempt to mix their own pictures with millions of picture postcards? In fact, several artists who served on the front were interested in this medium, which, like no other, is located at the intersection of the individual and modern history. Even if the textual field is left completely blank, the postmark proves with the assurance of an indexical sign and the authority of a state institution that the sender was an eyewitness. Nevertheless, and especially for modern artists, it can mean many different things to be an eyewitness and thus to take part in a historical event.\n\nForty-six postcards have been preserved from Otto Dix, all addressed to Helene Jakob. This is a small part of the body of work that Dix made as a soldier, including more than five hundred drawings; but in terms of the artist's objectives, it is highly significant. For the most part illustrated with his own drawings, the postcards serve to show his girlfriend in Dresden what Dix had seen with his own eyes: the company bathing, a communications trench at Angres, a machine-gun stand, a Russian, ruins, the troops' quarters in an attic, and the wreckage of Aub\u00e9rive, which Dix also describes in an ekphrasis of his drawing:\n\nShell holes in villages are full of elementary force. Everything around them seems subject to the dynamic of these violent, symmetrical craters. They are the eye sockets of the earth, and circling around them are their painfully fantastical lines. Those aren't houses anymore, no one seriously believes that. They are living creatures, a particular species with its own laws and living conditions. They are sheer holes with stones around them, or sheer skeletons. It is a strange, rare beauty that speaks here.\n\nDix's postcards from the field, his drawings and commentaries, are testaments in the emphatic sense, saying, \"I was there, I lived through the war, I experienced its unknown beauty.\" About his monumental painting _Der Sch\u00fctzengraben_ ( _The Trench_ ) (1920\/23), he would say: \"This is how it was, I saw it.\" The artists of the Jungen Rheinland (Young Rhineland), who defended him against the criticism of Julius Meier-Graefe, likewise swore to the authenticity of Dix's experience of the war: \"Dix lived through it\u2014just as Meier-Graefe demanded of artists in 1914.\" In fact, no matter whether Dix, in the enthusiasm of 1914, paints his own portrait as a bull-necked soldier and draws shell holes in the shape of flowers or, after 1918, is received as an anti-militaristic painter, his art remains constantly committed to the ideal of \"experiencing the war.\" The artist is not only an eyewitness; as a witness, he is above all a herald of \"that strange, rare beauty that speaks here\"\u2014here, in the rubble of bombed-out villages. In that Dix uses postcards as the medium of an emphatic testimony, which is certified not by the postmark so much as by the artist's capacity for experience, his art participates in the general cult of testimony that developed around military mail. As Bernd Ulrich has shown, before the beginning of the war, the reliability of eyewitness reports was rigorously tested and relativized by empirical psychology, but later the value of military mail as a testament to the war was judged according to entirely different criteria. The witnesses' inner involvement qualified them as heralds. In the prewar years, modern art, too, problematized the reality reference of its pictures. Only later did the world war create a reality that seemed to produce the referentiality of cubo-futurist paintings. Franz Marc describes artillery battle as a crystallization, similar to that which characterizes his own paintings. Otto Dix discovered the realism of fragmented pictorial spaces in the shell holes of the western front.\n\nBerlin Dada took the ideal of art as \"war experience\" and radicalized it until war experience disintegrated into disconnected moments. Shell shock was reproduced through the pictorial shock of photomontage, the literal continuation of the chain of violence into the revolutionary postwar period. If one can believe Georges Grosz, it began with military postcards:\n\nIn 1916, when Johnny Heartfield and I invented photomontage in my studio at the south end of town at five o'clock one May morning, we had no idea of the immense possibilities, or of the thorny but successful career that awaited the new invention.\n\nOn a piece of cardboard we pasted a mischmasch of advertisements for hernia belts, student song books and dog food, labels from schnapps and wine bottles, and photographs from picture papers, cut up at will in such a way as to say, in pictures, what would have been banned by the censors if we had said it in words. In this way we made postcards supposed to have been sent from the Front, or from home to the Front. This led some of our friends, Tretjakoff among them, to create the legend that photomontage was an invention of the \"anonymous masses.\" What did happen was that Heartfield was moved to develop what started as an inflammatory political joke into a conscious artistic technique.\n\nWhether or not this report is historically accurate, it contains an implicit poetics of photomontage in Berlin Dadaism. Grosz and Heartfield recognized the military postcard as a strategic pictorial form\u2014a propaganda weapon. Smuggled into normal military mail, the first Dadaist photomontages were image grenades, which, through blasted and illegible fragments, continued the war by reproducing the madness of the war through images of madness. Berlin Dada's military postcards are not testaments to an artistic view that heralds the secret beauty of barrage and shell holes. Instead, like munitions that the soldiers' councils diverted from the barracks, they are intended to function as weapons in a war that, for many, had not yet ended in November 1918.\n\nThe surviving mail that Ernst sent to his family from the field shows an entirely different person from the one who succumbs to the pathos of the crystal ideology in his art-theoretical writings of the war years. It shows the Ernst who would ask resignedly in his autobiography, \"What can he do against military life\u2014its stupidity, its ugliness, its cruelty? Screaming, swearing, vomiting with rage doesn't accomplish anything.\"\n\nThe scream of Berlin Dada was not how Ernst reacted to World War I, either in his pictures or in the four years he served\u2014during which time, unlike Grosz or Heartfield, he did not attract attention as an anti-militarist or a psychiatric case. The following postcard has been preserved from the end of 1915 (fig. 50):\n\nThis beautiful winter landscape is the farm of Mont du Crocq. Between the narrow gable to the left and the stubby tower in the middle, you see a slanting roof. Under there I wage war. The house halfway up the hill is the wireless station. To the left, the romantic grotto is the entrance to a deep underground labyrinth, the cave of our gallantry, in case of a bombardment. There no one can find us. Greetings, kisses, your Max\n\nThis card is evidently neither a weapon nor an emphatic testimony. Like most of the postcards that were sent, it serves a very obvious, familiar purpose, as an index that advertises the location and the health of a son and brother. The picture postcard performs this function so well because it relieves the sender of the impossible task of finding appropriate words for the reality of war, the cruelty of destruction, and the danger to himself. The side with the picture is a communication readymade. Merely by selecting it, the sender acquires an easily articulable topic that clears some distance from his own experience. When used within a family, as in Ernst's example, the postcard is a trace of survival that emerges from the events of war. The deciphering of this trace serves to distract from the war and dissimulate its indescribability with borrowed, prefabricated pictures.\n\nErnst's commentary redoubles this logic of distraction and dissimulation by sketching an idyll and undermining it through its sudden unmasking: \". . . a slanting roof. Under there I wage war.\" One observes the ironic na\u00efv\u00e9t\u00e9 learned from literary fairy tales of the romantic period, and the romantic sites of the wartime landscape: grotto, underground labyrinth, the cave of gallantry, Mother Earth, who promises to offer warmth and safety to the fighters. The earth's interior, which in 1917 was absorbed into the crystal mythos by painters and theorists, appears here in a metaphorics of the maternal that is approached with laconic irony. In the Dada pictures, this metaphorics will be dissected by an infantile sexual researcher.\n\nFIGURE 50\n\nMilitary postcard from Max Ernst to his family. Mont du Crocq Farm, end of 1915.\n\nNo less noteworthy are the formal correlations between wartime postcards and the Dada pictures: they are connected by the small format, the use of readymade images, as well as the irony and dissimulation characteristic of the appended commentary. Ernst borrowed the language of the fairy tale in 1915; in 1920 it was the language of the student lodging house ( _Madam Hostess on the Lahn_ ) and, again, that of the fairy tale ( _The Chinese Nightingale_ ). In fact, Ernst had two of his photo collages made into editions of postcards.\n\nThe medium of the military postcard clearly shows how artists situated their works in modern history: Dix drew emphatic testaments in order to reproduce the totality of the concept of history through the synthetic capacity of the subject. Grosz and Heartfield assembled weapons and referred metonymically to the totality of the historical situation in which they were working (though this totality was lacerated and characterized by shocks). By contrast, Ernst leaves traces of survival, the deciphering of which distracts from the witnessed event. But how can the metonymic referential function of the trace simultaneously distract from that to which it testifies? The overpaintings and the photo collages correspond to two stages in this process of detachment. First, the trace hardens into a strange and illegible image; second, this image provokes an imaginative seeing of resemblances and makes the trace available as an element of metaphoric developments. The overpaintings show the trace in its hardening, and the photo collages show it in its reworking and absorption through metaphor (as an occasion to escape, by means of the modern war machine, into a world of Oedipal compromise formations). In a divided and thus complementary distortion, on the one hand, war testimony is mortified into a foreign body with its confusion of outside and inside, and on the other hand, the foreign body is integrated into a pictorial form of the sexual secret.\nEXCURSUS\n\nTHE EARTH: A FORMAL HISTORY OF A THEME\n\nThe polarity between mortification into a foreign body and transformation of the foreign body into an Oedipal secret, between rigidification and motility, would seem to indicate that the surrealist frottages have little to do with the theme of the earth. Nevertheless, the history of the earth is the subject of Max Ernst's first surrealist series. In recounting this history, the series makes use of a pictorial form that is associated not with the geological foreign body pictures such as _Madam Hostess on the Lahn_ , but rather with the photo collages. In these and the frottages, heterogeneous textures are joined in a seamless surface that opens a pulsating space. The reproduction of the frottages through heliogravure makes this connection evident, since the material accumulations of rubbing vanish behind an optical film, as do the various cuts in the photographically reproduced photo collages. Thus, as a surrealist, Ernst tells about the earth; however, the pictorial form of his terrestrial recounting is fundamentally different from that of the geologic landscapes of the Dada years. The question is then raised as to why the theme of the earth was able to assert its primacy even after the pictorial form and process had changed. This question can only be asked if themes neither are considered as readily available to the artist's discretion, as may seem the case from a strict formalist perspective, nor are hypostasized into the origin of artwork. In the following, the theme of the earth will be analyzed in its variable relationship to the structural characteristics and problems of Ernst's poetics, and the various psychoanalytic interpretations of this theme will likewise be examined from this perspective.\n\nIn the works produced between Dada and surrealism, between the scarred surface of the wartime landscapes and the simulacra of _Natural History_ , the theme of the earth takes on a new meaning. Bodily fragments continue to be stored in the earth's interior (fig. 51), but now they are provided with an elaborated iconography of psychoanalytic, hermetic, and alchemical motifs, in order to refer the dismembered body parts to a sublimating and unifying process that designates the earth as the site of surreal image production. This drama of transformation whereby the earth, a milieu of dismemberment in the Dadaist overpaintings, becomes a milieu of synthesis is presented in _Woman, Old Man, and Flower_ ( _weib, greis u. blume_ ) from 1924 (plate 3). There are two protagonists: at the left, there is the old man with a woman on a stratified strip of landscape, and in the center, in front of the landscape stage set, there is the flower, a chimera assembled out of firm buttocks, a perforated breastplate, and an enormous fan for a head. Most interpreters take the figures as antagonistic: they consider the old man as a paternal laughingstock and the chimera as the blue flower of surrealism. Alternately, the old man is interpreted as a visionary and the blue flower as his internal image. But this alternative is too simplistic. The old man can indeed be described as an unequivocal representative of Oedipal power. His wolf face refers to traditional father fantasies made current in Freud's _Totem and Taboo_ and the Wolf Man study; what's more, he possesses the woman, and this possession corresponds to traditional illusionistic painting \u00e0 la Vel\u00e1zquez. But the amputation of his feet makes him into the riddle-solving Oedipus, and his cape is the clothing of a figure from one of the adventure stories the surrealists cherished as the treasure of their adolescence. Finally, his curious behavior is surrealist: he gazes with closed eyes into the earth's interior. In fact, he is an ambiguous figure who brings together characteristics of the despised paternal world with attributes and activities of surrealism. In its ambivalence, this figure is analogous to the grotesque father figure from Ernst's \"Visions of Half-Sleep,\" who terrifies the young Max, is presented by the adult Max to the reader's derision, and who simultaneously performs a surrealist activity, conjuring all manner of beasts from pre-morphic textures. Both the old man in the picture and Ernst's father in the surrealist text are allegories of the surrealist productivity of an Oedipal figure.\n\nFIGURE 51\n\nMax Ernst, _Men Shall Know Nothing of This_ ( _Les hommes n'en sauront rien_ ), 1920. 80.5 \u00d7 54 cm, oil on canvas, Tate Gallery, London (S\/M no. 653). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe analogy to the \"Visions of Half-Sleep\" can be pursued further: just as in the \"Visions,\" in this painting, too, the Oedipal figure is subordinated to the more powerful figure of the phallic woman, who appears in the last vision as the guarantor of a fulfilled, flawless visuality, and who also commands the scene in _Woman, Old Man, and Flower_. While the old man hunches over the eyepiece in an unnatural torsion and thus threatens to fall to the left, out of the picture, the flower stands upright and imitates with its arms the majestic rotational movement that the fan has already carried out. In doing so, its silhouette seems to emerge from the stony strip of foreground and to vault the entire space of the landscape, from the sea to the distant shore. The blue skin and the transparency of its upper body support this all-encompassing gesture, which breaks the spell of stone. But where does the blue flower of surrealism thrive? Along its right thigh, the rocks give way\u2014but this may only be a scenographic trick. Probably the flower is standing in front of the picture, in its own space, which the strangely bare potted plant, to the left, and the brass knob that is likely part of a bedstead, to the right, allow to be identified as an interior. This interior space, difficult to locate, seems to extend in the thin layer of the aesthetic boundary between the picture and the space of the viewer. The individual elements of the flower figure, especially the fan and the buttocks, occupy an indeterminate between-zone. The glints of light on each buttock do their part to lend these body parts an uncomfortable nearness\u2014as if, each time they are perceived, they emerge from the picture on their own strength and materialize exactly on the medial boundary. Here, the simulacral power that brings the represented object to life, \"though without entirely divesting its unreal character,\" is expressed satirically and is kept at a distance through satirical apotropaia. The green ribbon tied around the hips and the four yellow and blue buttons affixed to the face mask and corselet function as magic charms. The buttons send out impenetrably reflective gazes that at once fascinate and keep the viewer at a distance.\n\nImpenetrable gazes, fans, dancing arms, armor, circling movements: these elements identify the painting as a reworking of _The Chinese Nightingale_ (fig. 48). It is particularly striking that Ernst approximates this photo collage in the oil painting without aiming for the former's most important effect, the creation of a pulsating space that absorbs incisions and foreign bodies. Only through the contrasting insertion of details (such as the buttocks and the fan) into the rigid stage set space does the oil painting give rise to a simulacral effect that vitalizes the medial boundary while maintaining an unreal quality. Not until the frottages will pictures be made again that are simulacra in their entirety, and in this way pick up where photo collages such as _The Chinese Nightingale_ left off. _Woman, Old Man, and Flower_ marks a decisive transition, since, on the one hand, it seeks an engagement with simulacral pictures such as _The Chinese Nightingale_ and moreover presents an iconographic staging that anticipates the \"Visions of Half-Sleep,\" while, on the other hand, despite these motifs that clearly point ahead to the frottages to be made in the following year, the painting does not create a pulsating pictorial space that absorbs cuts and fragments, as the photo collages did, and as the frottages will go on to do.\n\nIf one attends exclusively to artistic procedures, the still life of pears from 1925 (fig. 19) is the work in which the transition from fossil illusionism to ghostly mimesis is put forward explicitly: frottage is compared with cut and diagram in order to profile frottage as a pictorial form that provokes an imaginative seeing of resemblances, which brings the picture surface to life as a pulsating space. By contrast, _Woman, Old Man, and Flower_ , made one year before the still life of pears, is the work that demonstrates the _iconographic_ passage from Dada to surrealism with particular clarity. This work shows the transformation to which the tellurian iconography had to be subject before the geological theme of the Dadaist overpaintings and diagrams could become the theme of the surrealist _Natural History_. The earth had to be transformed from a milieu of fragmentation into one that synthesizes the disparate individual parts into a figure (the figure of the phallic woman). Moreover, this iconographic stabilization is also accompanied by intimations of a formal stabilization. The most important of these is that the phallic woman is given her own platform offstage, beyond the boundaries of the picture, upon which platform she can be located as a physical figure.\n\nIn order to understand how these iconographic and formal transformations are connected to one another and related to the theme of the earth, it is first necessary to recall the function this theme had in the Dadaist pictures. In the diagrams, made in dialogue with Picabia and Duchamp (figs. 6, , ), various motifs of petrifaction or erosion occupied the outside edges of pictorial subjects and supports. These motifs functioned as thematic surrogates of the heaviness and materiality that the diagram, with its destruction of the illusionistic picture, had also negated. In the geological overpaintings of 1920\/21 (plate 1, figs. 10, , , ), Max Ernst again employed individual elements of the illusionistic picture (rectangular format, perspectival grid, plasticity, platform), in a secondary usage that preserved a negated mode of representation _as_ negated. In sum, the geological iconography became the distorted surrogate of that which the blunted and emaciated world of these pictures no longer offered: namely, heavy, material objects that can be located in space. In the critical lower third of the picture, where, in illusionism, the platform would perform its function, some works display excessive surrogate creations, replete with the psychoanalytic iconography of the infantile \"cloacal theory,\" which itself has to do with a surrogate creation: in researching the question of the origin of life (and of other children), the child is directed to mechanical surrogate parts, from which it imaginatively fabricates the maternal body as an anal machine. In _eislandschaften, eiszapfen undgesteinsarten des weiblichen k\u00f6rpers_ ( _Frozen Landscapes, Icicles and Mineral Types of the Female Body_ ) (plate 1), the mechanical dismantling of the feminine\/tellurian underground could possibly allude to this context. In several other works, softening and liquefaction of the bottom zone of the picture can be observed: the destroyed body in _Madam Hostess on the Lahn_ ( _frau wirtin an der lahn_ ) (plate 2), the mucilage in _Katharina ondulata_ (fig. 16, also titled _frau wirtin a. d. lahn_ ), the subterranean fluids in _winterlandschaft: vergasung der vulkanisierten eisenbraut zur erzeugung der n\u00f6tigen bettw\u00e4rme_ ( _Winter Landscape: Vaporization of the Vulcanized Iron Bride to Produce the Necessary Bed Warmth_ ; fig. 15), or the lowermost stratum of _schichtgestein_ ( _Stratified Rocks_ ; figs. 52, ). In this work, it is notable that none of the bones of the main motif, a fossil horse, have been preserved. The coating of the lowermost rock layer in a fluid brown color explains why the soft parts, specifically, have been retained. The geological diagram illustrates the law of metonymic displacement, which leads from product to the apparatus of production: if the earth's interior is a cloaca, then digestive organs are most well preserved. In addition, the picture's inscription identifies \"two kinds of ruptures of the perineum,\" and thus makes reference to the mythologem of the infantile sexual researcher who imagines the origin of life in the anus. In a central position, the intestine opens, waves like a windsock over the landscape, and leads through a pipe into the earth's interior. The deadened landscape of rigidified illusionism thus sustains a few distinct sensations, as strong as they are repulsive. This decline of the geological into a scatophilic iconography is an excess of content that, however, appears in a precisely determined place. That is, the place where the secondary use of illusionistic set pieces is particularly precarious: in the vicinity of the picture's lower edge, where a platform should extend into a receding space in order to receive objects with a specific heaviness and materiality. The designation of the earth as a cloaca which is the grotesque surrogate of the maternal body has thus to be understood as an iconographic symptom of the condition that a critical zone extends in the vicinity of the picture's lower edge after illusionism has broken down or has been preserved only as a surrogate.\n\nFIGURE 52\n\nMax Ernst, _Stratified Rocks, Nature's Gift of Gneiss Lava Iceland Moss_ ( _schichtgestein naturgabe aus gneis_ ), 1920. 15.2 \u00d7 20.6 cm, gouache and pencil (overpainting on a print), New York, Museum of Modern Art (S\/M no. 367). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nFIGURE 53\n\nFrom _Cologne Catalogue of Teaching Aids_ ( _Katalog der K\u00f6lner Lehrmittelan-stalt_ ), 1914, pp. 1056ff.\n\nThe earth is redeemed from the cloacal myth in the transition from Dada to surrealism. We have seen that in the oil paintings of the early 1920s, such as _Les hommes n'en sauront rien_ ( _Men Shall Know Nothing of This_ ; fig. 51), lacerated blood vessels, organs, and bowels that protrude from the terrestrial realm are no longer associated with infantile sexual research, but with a sublimating iconography composed of alchemical, hermetic, and psychoanalytic ingredients. This allows the conclusion to be drawn that, from now on, the earth is a site of production where a yet unknown entity is coming into being, which emerges in _Woman, Old Man, and Flower_ as the chimera of fan, breastplate, and lower half of the body. Its pointedly closed form extracts itself from the stone in which various organic fragments are still enclosed. Iconographically, this form is later (in \"Visions of Half-Sleep\") identified as the phallic woman, and hence appointed as the guarantor of a fulfilled, flawless and non-castrated visuality. For a history of form, it is thus crucial that the phallic woman corresponds to a new solution to the problem of the platform. The represented platform continues to be an unstable stage set upon which, despite his shadow, the old man has no foothold and cannot be located. The chimera itself, however, appears _in front of_ this stage set and may be standing firmly on a ground that evidently can also support furniture and potted plants. The shallow stage space that in most of the geological overpaintings had been formed from the ruins of illusionism (figs. 10, , ), in _Woman, Old Man, and Flower_ is split in two: on the one hand, there is the strip of landscape like a stage set along the lower edge of the picture, and, on the other hand, there is a platform on this side of the lower edge of the picture, offstage of the representation. Upon this firm ground, the simulacrum of a material and tangible body, assembled from various objects but nonetheless a unified form, makes its appearance. With particular clarity, the integration of surrogate parts and fragments into a unified form distinguishes the \"blue flower\" from the motifs of earth and femininity in Dadaism, from the _Frozen Landscapes, Icicles and Mineral Types of the Female Body_ , _The Madam Hostess on the Lahn_ , or _Stratified Rock . . . with Two Types of Perineal Ruptures_. The destructive gaze of the infantile sexual researcher is pacified; the painting stages the tellurian rebirth of form as the phallic woman, and this form stands upon firm ground (which, however, lies offstage of the picture).\n\nThe secondary use of illusionism has been transformed and, with it, the iconography of the earth. The unstable platform is duplicated and stabilized offstage; the woman as a chimera is closed into a unified form; the earth is achieved as the milieu in which this surrealist coming into form takes place. _Woman, OldMan, and Flower_ thus prepares Ernst's surrealism inasmuch as it sets the conditions for both _Natural History_ and \"Visions of Half-Sleep\": it establishes the earth as the site of surrealist image production and the phallic woman as the guarantor of a surrealist visuality. The clear formal differences between the painting and the prints made only one year later are thus all the more astounding: in 1924 there is a fastidiously worked-over, rigid, and ponderous oil painting; in 1925 there are pulsating and metamorphic surfaces. This most striking difference that separates Ernst's surrealist pictures from those of the early 1920s can also be conceived as a further stage in the formal history of the theme of the earth.\n\nFIGURE 54\n\nPablo Picasso, _Au Bon March\u00e9_ , January 1913. 23.5 \u00d7 31 cm, oil and pasted paper on board, Aachen, Ludwig Collection. \u00a9 2012 Estate of Pablo Picasso \/ Artists Rights Society (ARS), New York.\n\nAttention should be drawn to a painting from around 1923, _un tremblement de terre tr\u00e8s doux_ ( _A Very Gentle Earthquake_ ; fig. 55). It displays a surface that\u2014unlike the mortifying finish of _Woman, Old Man, and Flower_\u2014continues and thematically appropriates prewar modernism's experiments with facture. Between the erratic traces of painting and the earthquake iconography, a mimetic analogy arises, in the sense that both the surface of the painting and that of the earth become unstable. The structural problematic of the modernist painting\u2014that a zone of instability opens near to the picture's lower edge, where the platform had extended in the illusionistic picture\u2014thus undergoes a thematic motivation. The \"breaking away\" of the platform becomes a moment of analogy formation between painting and earthquake. The entering into mimetic relationship of facture and tectonics is also demonstrated by the represented scene. The main motif is a rectangular block that seems to be in the process of sinking into the earth. Its upper surface, made from strips resembling floorboards, also exhibits a black hole, which likely serves to characterize the block as a hollow box. What we see here being swallowed by the earth turns out to be one of the hollowed-out and unstable stages that Ernst discovered in the works of de Chirico, and which inspired the dismantled and unfolded landscapes of Ernst's Dadaist overpaintings. While the surrogate of a platform disappears inside the earth, the painted surface of _A Very Gentle Earthquake_ itself takes on the qualities this surrogate is no longer capable of visualizing. In the experiments with facture, heaviness and tangibility emerge as characteristics of the picture surface.\n\nFIGURE 55\n\nMax Ernst, _A Very Gentle Earthquake_ ( _Un tremblement de terre tr\u00e8s doux_ ), ca. 1923. 19 \u00d7 24 cm, oil on wood. Private collection (S\/M no. 621). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nIn the Dadaist overpaintings, the motif of the cloaca functions as the symptom of a lack, namely, the emaciation, blunting, and distortion of illusionism. In _Woman, Old Man, and Flower_ (1924), the surmounting of this lack is celebrated in the figure of the phallic woman, who arises from the mortified terrestrial realm and achieves a platform on this side of the picture. But in _A Very Gentle Earthquake_ (ca. 1923), Ernst had already tested a poetic of the earth on the basis of the mimetic analogy between the earth's crust and the crust of paint. In the picture of the earthquake, the lack is made into a fullness, the sensory plenum that presents itself, as a repeatedly painted surface, in hatches, blots, and scratches, and that moreover enters into a mimetic exchange with the tellurian theme of the painting. Ernst will draw upon this exchange in the frottages (and later in the grattages), which, in the secondary fashion examined in chapter 1, establish a mimetic analogy with geologic processes such as fossilization and erosion.\n\nThe pressure and resistance of the horizontal work surface that serves in the rubbing of textures onto paper or canvas are more than mere conditions of production; these qualities remain present in the viewing of each individual picture. For example, in _Confidences_ (fig. 23), although the leaf ( _Laubblatt_ ) addresses us in its verticality and floats away like a ghost across the medial boundary, the horizontality of the drawing paper ( _Zeichenblatt_ ) as a work surface is still present because the textures refer back to the horizontal situation in which they were made. They do so most aptly when the paper was rotated during rubbing, since the resulting textures attest with particular clarity to the tactile intimacy of production, the work with pressure and resistance that happens close to the body. It is as though the raising from the horizontal situation of working to the vertical situation of viewing did not occur as a single action, but was repeated many times in each viewing, as a pulsating movement. Again and again this movement leads from the ghostly leaf to the sheet of paper as the working surface, which the artist presses against the textured leather, works with a soft pencil, and occasionally also turns. The earth of _Natural History_ is a matrix of secondary and simulacral images released from the rigidity of secondary illusionism, because heaviness and tangibility as essential characteristics of sensory plenitude have become factors in the production process and, moreover, because they experience an aesthetic actualization in the pulsating movement of the finished picture.\n\nErnst's tellurian thematics, from his petrified diagrams to his surrealist natural history, thus take on different meanings\u2014from the site of the cloacal myth to the site of surrealist image production\u2014according to how his artistic procedures change; and, conversely, new artistic procedures are repeatedly tested on the theme of the earth, because in this theme, the crisis of the heaviness and tangibility of the image is articulated. In retrospect, from the viewpoint of surrealism, the critical transformation consisted in a (simulacral) reproduction of wholeness. Wholeness was restored in the imaginary, since the phantasm of the destroying and devouring mother was replaced by that of Leonardo's phallic mother. Simultaneously, it was restored in production, since the fossilized or otherwise destroyed qualities of tangibility and heaviness were brought back into effect in the making of frottage. And above all, it was restored in aesthetics, since the tactile procedure of frottage corresponds to a perception that is in itself tactilized: in other words, a perception that is affected by the seen and that itself affects the seen, a perception that surrenders itself to the instability of the image and also intensifies it\u2014thus, a perception that behaves actively and also passively, similar to the sense of touch. From this perspective, the surrealist seeing of resemblances can be interpreted as a secondary reproduction of imaginary, procedural, and aesthetic qualities that in Dada were taken apart from one another and mortified. This being said, in summary, it must be emphasized again that in this unity there persists a divide, which Ernst thematizes in \"Visions of Half-Sleep\" as the bodily separation of the artist and the transparent female figure, and in the last print of _Natural History_ as an unattainably pure visuality. Surrealist seeing as Ernst understood it thus only follows the model of a perception in which seeing and touching both play a role inasmuch as it simultaneously positions itself at a radical distance from palpable reality. In print 34, the figure of Eve is located on a threshold: beyond her, a realm of pure transparency opens; this side is dominated by the dazzling light of a camera flash; and between these two forms of intensified visibility, a fossil layer of tactility is spread. In view of this primacy of seeing, which has withdrawn from the physical world in order to reproduce it as secondary, it's no wonder that heaviness and density, and with them the tellurian themes, vanish again from Ernst's art. By around 1930, the time has come. At this moment, Ernst's engagement with the body of the image, which has occupied him so intensively from the petrified diagrams of Dadaism to the grattages, seems to lose significance. In what follows, I would like to outline this process, and, in doing so, I would like to turn specifically to the work that once again will confront surrealist seeing with a resistant _physis_.\n5\n\nPREHISTORY AND MODERN HISTORY\n\n_EUROPE AFTER THE RAIN_ , 1933\n\nTHE SEEING OF RESEMBLANCES TRANSFORMED\n\nBeginning in the mid-1920s, Max Ernst constantly sought new stimuli with which to test the seeing of resemblances in its different forms. In our context, it is noteworthy that in doing so, he detached himself from the thematic association with the earth as ground and matrix, as well as from the textures basic to frottage, which were indeterminate with regard to motif. Besides wood grain, textured leather, and patterns, relief postcards and other figurally predetermined underlays were now put to use. The simulacra of prehistoric nature mingle with the simulacra of obsolete visual culture. In the Loplop series, the surrealist's studio is opened to view with its various matrices that provoke the seeing of resemblances, from the linear ornament that undergoes metamorphoses and can contain reversible images all the way to the marbled paper that activates the potential and powers of the seeing of resemblances as if while in a state of idling. The latter, a seeing of resemblances that incites the animation of surfaces and lends them a pulsating depth without taking shape in distinct motifs, absorbs even materials that presumably resist this effect: specifically, the popular wood engravings that Ernst employed in his collage novels. If one compares works from 1922 with _The Hundred-Headed Woman_ ( _La femme 100 t\u00eates_ ) from 1929 (fig. 56), it is evident what power the animating seeing of resemblances had achieved after 1925. While the early collages seal their material in a hard surface, the later ones cause this surface to pulsate. Hatching gains a suggestive power similar to the dark ground that functions, in the frottages, as the pre-morphic projection screen for the seeing of resemblances. In order to illustrate the breadth of possibilities with which Ernst was working, it is also worth mentioning his trip to Maloja in 1934, where he found egg-shaped stones in the detritus of the glacier and chiseled them into metamorphic figures. From mass-produced popular illustrations to granite, every surface can become a projection screen for imaginative perception.\n\nFIGURE 56\n\nMax Ernst, _The Hundred-Headed Woman Opens Her August Sleeve_ ( _La femme 100 t\u00eates ouvre sa manche auguste_ ). From _La femme 100 t\u00eates_ , 1929, chapter 3. 30 \u00d7 14.2 cm, collage (S\/M no. 1449). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe testing of the seeing of resemblances on ever new materials and themes dissolved the relationship to the theme of the earth, which had been constitutive for the frottages of _Natural History_. This theme was now only one possibility among others, in playing with analogies for the process of making pictures. What's more, in softening the hardness of facture and formal articulation that characterized his work prior to _Natural History_ , Ernst found a new approach to the medium of painting. While his most important paintings from the early 1920s had featured a conspicuous secondariness, as if painting had been reduced to an inflexible medium for copying preexisting images, by the mid-1920s he started to become interested in the fluidity and amorphous quality of paint as a resource for picturing the very emergence of images. The transfer of the frottage method to painting was crucial. He produced his grattages by spreading a loose canvas on the floor, placing objects such as pieces of rope or boards underneath, applying paint, and then scraping it off while still wet. In a second step, the streaks, blurs, and blots that resulted from this procedure were interpreted as forests or wild creatures (fig. 57). A similar technique for creating random figurations that could be interpreted in multiple ways consisted in throwing a piece of rope onto a canvas covered with wet paint. One way or another, the aim was to transform the painting into a screen that entices and, at the same time, registers the fantasies of its beholders.\n\nIn his mission statements of the late 1920s and the 1930s, too, as has often been observed, Ernst gives central importance to such a de-differentiation of seeing and imagining. As we have seen, in \"How to Force Inspiration,\" the seeing of resemblances coalesces into the theoretical foundation of surrealist poetics and becomes the shared origin of grattage, frottage, and also collage. In his first monograph, published in 1937 by _Cahiers d'Art_ , Ernst further develops this arthistorical position. Moreover, the seeing of resemblances preoccupied the entire surrealist group\u2014the members examined its effects in joint experiments and through an array of procedures. In the most important surrealist mission statement from the early 1930s, _Communicating Vessels_ ( _Les Vases communicants_ ), Andr\u00e9 Breton devoted himself to the psychology and politics of the seeing of resemblances. \"Tout fait l'image.\" (Everything makes images.) Whatever imaginative perception absorbs brings hidden pictures to light. Leonardo's wall stains are cited as paradigmatic evidence.\n\nFIGURE 57\n\nMax Ernst, _The Court of the Dragon I_ ( _La Cour du Dragon I_ ). From _Une semaine de bont\u00e9_ , 1934, chapter 3. 15 \u00d7 12.5 cm, collage (S\/M no. 1966). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nVISIONARY POLITICS\n\nThe success of the seeing of resemblances was also the basis for the surrealist politics of the 1930s. The surrealists, specifically Breton but also Ernst, had to expend great polemic effort in order to oppose the obligation of content and party affiliation that was increasingly being demanded by Leftist writers and artists as a means of class warfare and resistance to the Fascist threat. Breton, who had already been through several conflicts with the French Communist Party pertaining to this question, insisted upon a historicity unique to art and upon art's autonomy. Ernst made it plain that he agreed with his friend entirely on this point. In his answer to the survey \"O\u00f9 va la peinture?\" (\"Where Is Painting Headed?\"), conducted in 1935 by _Commune_ , the organ of the Communist Association des \u00c9crivains et Artistes R\u00e9volutionnaires (AEAR) (Association of Revolutionary Writers and Artists), he posited that the \"manifest or latent ideological content\" of a picture did not depend on the painter's conscious decision-making. The unconscious, Breton and Ernst agree, does not grant access to a realm of individual discretion; rather, it communicates with a collective unconscious, which the individual instance represents. Through this mediation, art and poetry are brought into connection with the political questions of the present and, more importantly, the future. Expressions of the unconscious that appear obscure or nonsensical will one day attain perfect transparency. In _Political Position of Today's Art_ ( _Position politique de l'art d'aujourd'hui_ ; 1935), Breton underscores this hope by pointing out that \u00c9douard Manet's still lifes were received by his contemporaries as opaque blots of paint, while today they are remarkable for their virtually photographic precision. The same will also be the case for surrealist poetry and painting, as Breton attempts to illustrate with a geologic metaphor: \"The climate of Benjamin Peret's poetry or Max Ernst's painting will be the very climate of life.\" Surrealist art is thus a revolutionary weapon inasmuch as it anticipates a (post-)revolutionary future. The relentlessness with which Breton defended this sense of the future becomes particularly vivid in the example of the \"Aragon affair.\" Louis Aragon, who starting in 1930 had moved pronouncedly away from surrealism and toward a Soviet position, in 1932 wrote _The Red Front_ ( _Le Front rouge_ ), an incendiary poem (stylistically not at all surrealist), in which he calls for the killing of police officers and, in addition, urges opening fire on L\u00e9on Blum and the Social Democrats. The French judiciary's request that Aragon be held accountable for such acts was met, on the part of the surrealist group, by a pamphlet extolling the juridical and moral unaccountability of the poet. It was not he himself but the powers of the unconscious that were expressed in the poem; these powers were not demanding an individual act in the present but were prophesying collective action in a revolutionary future. This retreat to the unaccountability of automatic writing provoked unanimous derision from Leftist writers. The Belgian surrealists protested bitterly against their Parisian friends, and Aragon, who had authored _The Red Front_ in order to prove his revolutionary convictions to the party, declared his break with surrealism in a press release published in _L'Humanit\u00e9_. Breton went on the defensive. In _Mis\u00e8re de la po\u00e9sie_ ( _Misery of Poetry_ ), he openly admitted in what little esteem he held _The Red Front_ as a poem and once again stated the standpoint of the surrealist group, which explicitly gave him its support in _Paillasse!_ ( _Slut!_ ), another proclamation, also signed by Ernst.\n\nIn the dispute with the French Communist Party, the surrealists found themselves compelled to differentiate between artistic and political revolution, which early surrealism had specifically intended not to do. It is poetry and painting's sense of the future (and not their manifest content) that justifies why art can also become a revolutionary activity with a unique character. Artistic technique possesses its own historicity, which cannot be reduced to major political events such as the French Revolution or the Paris Commune. Technique, however, attains its own political efficacy if it is able to make seeing and imagining indistinguishable. Visual parapraxes\u2014like slips of the tongue, dreams, or chance encounters\u2014were considered symptoms, and symptoms not only bring a buried past into appearance; they also speak of a future of fulfilled desires, and in this way incite revolutionary acts. In _Communicating Vessels_ (1933), Breton defends his optimistic conception of the unconscious, on the one hand, against Freud's conviction of the implacability of the drives and, on the other hand, against the Communist disinterest in the affective. He refers to the example of his own dream of August 26, 1931, which he subjects to an extensive analysis. In this dream, there appears a cartographic reversible image:\n\nWhile going through the stock of ties again, another salesman, middle-aged, talks to me about a tie called \"Nosferatu,\" of which he used to sell a lot two years ago, but he is afraid that he has none left. I am the one to discover this tie immediately among the others. It is garnet red, and on its points there stands out in white and, at least on the visible point\u2014once it has been knotted\u2014twice, the face of Nosferatu, which is at the same time the map of France, empty, with scarcely any marks at all, on which the eastern border is very sketchily traced in green and blue, so that I think it looks like rivers, outlining in a surprising way the makeup of the vampire. I am eager to show this tie to my friends.\n\nNosferatu counts among the surrealist threshold deities, since the vampire's existence and erotic desire proves the permeability of even the most clearly defined boundary, that between life and death. The dream connects the despot of the East with Germany, and thus also with the German woman whom Breton had unsuccessfully pursued a short time before.\n\nOnce again, the dream realizes simultaneously here two sorts of desires, the first being that of speaking freely to this woman; the second, that of suppressing every cause of misunderstanding, patriotically exploitable, between France, where I live, and the marvelous country, made of thought and light, which saw Kant, Hegel, Feuerbach, and Marx born in a single century.\n\nThe garnet-colored necktie, which shows the face of Nosferatu as well as the French territory, without the rivers and with an open boundary to the East, corresponds to the desire to deterritorialize France. The visual parapraxis expresses the hope that the mythic geography of the East (as represented by Nosferatu, German philosophers, and a German woman) will annihilate the French hexagon. This trust in the Germany of \"Kant, Hegel, Feuerbach, and Marx\" would also inflect the surrealists' first official statement after Hitler's seizure of power, and for its own part is inflected by the view, dominant among Leftists and Rightists alike, that Germany was the Other _par excellence_ , more remote than India or China.\n\nBreton's theory of parapraxis elevates the seeing of resemblances to a prophetic gift that discovers political directives in geographic fantasies. When, in the same sequence, Breton arrives at the previously quoted insight, \"Tout fait l'image,\" there opens up to him an inexhaustible source for reinterpreting the world and for conceiving of this reinterpretation as a revolutionary activity. Ernst also drew from this. In his first monograph, he cites in a note the following report from Nazi Germany and moreover provides the exact source ( _Oberbadisches Volksblatt_ from February 1934):\n\nPOLITICAL OVERZEALOUSNESS\n\nPathological Imagination\n\nRecently, the authorities have received a large number of postcards, pictures, and posters, in which the scent of hidden Communist propaganda has been detected. In the hair of a head, although it is a photograph, someone imagines he has discovered the face of Lenin, together with an obscene picture in the flesh of the ear. On a poster, someone has found concealed a bashed-in skull and the head of a Communist. Granted, the viewer of this poster, which is normally pasted or hung, has to stand on his head in order to enjoy the picture-puzzle. The authorities will oppose such political overzealousness, which causes unnecessary alarm among the population and is harmful to legitimate interests. The authorities have been directed to suppress this nonsensical behavior, which can easily erupt into a dangerous psychosis.\n\nThe denunciatory frenzy shows oddly familiar, typically surrealist symptoms. It proceeds according to the motto \"Tout fait l'image.\" One might suppose that Ernst intended to point out the danger that the annihilation of the reality principle could play into the hands of the worst enemies of the surrealist revolution. Salvador Dal\u00ed's escapades involving the use of his \"paranoiac-critical method\" either to find Hitler's facial features or to conceal them in his pictures provided enough of a reason to mistrust a politics of imaginative seeing. However, if one reads the citation from the _Oberbadisches Volksblatt_ together with the main text, this interpretation proves to be false. The citation is rather intended to demonstrate the opposite: even in Nazi Germany, the unconscious is on the side of the surrealist revolution. Ernst makes use of the citation to praise his most famous procedure, collage: \"What is the most noble conquest of collage? The irrational. The magisterial eruption of the irrational in all domains of art, poetry, science, and fashion; in the private life of individuals, in the public life of nations. He who speaks of collage speaks of the irrational.\"\n\nIn the footnote that follows, the observation that collage allows the irrational to intrude into the \"life of nations\"\u2014that is, into politics\u2014Ernst cites the report from the _Oberbadisches Volksblatt_. But what does the paranoid seeing of resemblances as it consumes Germans faithful to the regime have to do with collage? It is capable of exemplifying collage's power because, in his surrealist mission statements, Ernst has attempted to trace the procedure back to visual parapraxes, and thus to relate it to the same origin as frottage and grattage (see chapter 3). Imaginative seeing is surrealism's own special weapon not least because it conducts surrealist subversion even in Nazi Germany. It finds Lenin in the hair and an obscene picture in the ear of a single head. In this way, it opened to the Germans a revolutionary future, which they in their denunciatory zeal closed off again: the future of the proletarian revolution of production conditions (Lenin) and the surrealist revolution of life (obscene pictures). The footnote with the citation from the _Oberbadisches Volksblatt_ is not included in the English version of the text, which was published as _Beyond Painting_ in 1948. I assume that, in retrospect, this specific example of the surrealist commitment to hidden images as a political weapon must have seemed rather frivolous and naive, given the political violence unleashed by Nazism. In the 1930s, however, the footnote mocked two different targets, both Nazism and the Stalinist Left, and should be seen within the intellectual-historical context of the wider discussion, led by Georges Bataille and Ernst Bloch, among others, of how the irrational, heterogeneous, romantic, and other forces counter to Enlightenment could be reclaimed from their use by Fascism.\n\nIMAGINATION BREAK\n\nIt seems as though the surrealists and Max Ernst in specific staked everything on imaginative seeing in order to create revolutionary pictures. In the following, however, I would like to assert that Ernst's most important picture of the 1930s, _l'Europe apr\u00e8s la pluie_ ( _Europe after the Rain_ ; plate 4), made in 1933, questions this faith in a seeing that draws from imaginative depths and, as an alternative, explores a pictorial form that situates imagination on the surface of the picture.\n\nThe established art-historical interpretation tends in another direction. It consists of the nearly unanimous consensus that the picture is also visionary, though in a completely different sense than Breton's and Ernst's cited statements from the 1930s would lead one to suppose: it is agreed that the painting can be understood as a premonition of the destruction that would ensue in the Second World War. The first to make this interpretation was Carola Giedion-Welcker, albeit with pronounced caution and a fine sense for surrealist defamiliarization strategies. On the occasion of the first Max Ernst retrospective, which was on view in the major museums of the Federal Republic of Germany in 1963, Giedion-Welcker, a longtime friend of the artist and at that time the painting's owner, wrote:\n\nThe subject of \"Europe after the rain,\" which appears in different variations, is first treated in 1933 in a mysterious map, like a relief map, which reflects a premonition of the bewildering territorial shifts that were yet to come. The geography, which seems familiar but here is reconstructed from fully dissociated elements (\"le d\u00e9paysement\"), carries out its deranging and deranged game with the viewer.\n\nWithout a doubt, it was Ernst's desire, and the desire of all the other surrealists, to create pictures and texts of premonition. However, admiring their visions for having held true probably says more about the moment of this admiration than about the artistic and historical situation in which the visionary pictures were created. Giedion-Welcker's interpretation is grounded in an aesthetic experience that the great art critic could only have had after 1945. Toward the end of the Second World War, Jean Dubuffet and other French artists made paintings with desiccated surfaces, as if they had been melted down and then left to harden. _Europe after the Rain_ must have seemed to its owner like a \"premonition\" of the Second World War because it anticipated the artistic methods of postwar art and may well also have inspired them.\n\nIn consideration of the year 1933, however, the question is raised as to what type of future the surrealists wanted to prophesy, and what form they wanted to give to their prophecies. With regard to the first question, which concerns the political content of surrealist visions, it is entirely improbable that prewar and postwar conceptions would agree. The surrealists longed for nothing more ardently than the demise of European civilization, but, of course, they understood this to mean something entirely different from the devastation that would in fact come to pass. Before I pursue this first question further, I will turn to the second question, that of the form of the visionary in _Europe after the Rain_.\n\nWork on the painting began with a chance occurrence. The picture support was a found object that Ernst discovered in the Billancourt studios during the filming of _The Golden Age_ ( _L'Age d'or_ ). In 1969, during a conversation with Robert Lebel, he recalled how he discovered the panel and knew right away that it was another case of \"Leonardo's famous wall, which had played such an important role in my 'Visions of Half-Sleep.'\" Like the imitation mahogany in the \"Visions,\" the studio set also serves as a surrogate for Leonardo's stained wall, as a \"visual irritant\" that provokes an imaginative perception governed by parapraxes. In one of the Loplop pictures from 1930, for which Ernst used parts of the studio set as a painting ground, the modeling of the figure of the bird in fact develops out of the plaster surface of the set. However, three years later, in _Europe after the Rain_ , an entirely different effect is produced. To be sure, Ernst's remark that the studio sets from Billancourt reminded him of Leonardo's famous wall encourages the search for comparable motifs. In several of the coastlines and also along the boundaries between smooth and rough parts of the picture surface, it might in fact be possible to detect legible forms. And in certain zones, the rough texture of the plaster suggests the feathering of birds, the motif most often concealed in Ernst's paintings. But in comparison with Ernst's other works of the late 1920s and 1930s, it is conspicuously difficult to animate the irregular relief of the picture surface and to make out forms within it. The most important organ of surrealism\u2014the imagination's eye\u2014is indeed stimulated, but it is not allowed free play, as is possible in Breton's interpretation of the dream of the Nosferatu necktie, in Ernst's memories of childhood, and in the majority of his paintings. The rubbed or scraped grounds of frottage and grattage open a suggestive ground that offers sustenance to the imagining eye. _Europe after the Rain_ may also contain hidden figures, but they remain under the spell of the desiccated texture and do not detach themselves as discernible motifs. After the catastrophe, the old continent offers to surrealist perception only an austere and thoroughly resistant surface. This becomes particularly clear in comparison with the second version of this subject, made in 1940 (fig. 60). The transfer technique (decalcomania) employed in this work is exceptionally well suited to the creation of suggestive blots and other imagination zones, which are interpreted by the artist as a half-melted, half-hardened landscape, and populated with Europa riding the bull and other mythical forms. They inspire the viewer to search for additional faces and figures in the spots and streaks. Imaginative seeing comes upon an iconographic abundance, which in part is plainly evident, and in part can be supplemented and imagined further, depending upon how responsive one is to the play of resemblances. By contrast, the painted and multiply fissured plaster in _Europe after the Rain_ deflects the gaze.\n\nOf course, coastlines and territorial boundaries have a notable aptitude for harboring reversible images, and someone very familiar with Ernst's paintings would probably be able to recognize, in the outlines of the inland body of water at the center of the picture or in the light blue zone of the sea, to the right, one of the artist's typical bird figures. (Given the title, it would be tempting to search for Europa and the bull.) But this recognizing, if it happens at all, does not function with the same ease as in many other of Ernst's surrealist paintings. It seems as if the painting forces us to look for potential or hidden figures, but we end up being unable to detach them from the devastated ground of the painting. Not even the simplest mechanism of the seeing of resemblances, which is engaged almost every time we look at a map\u2014that is, the reading of familiar territories into geographic givens\u2014seems to work when we look at _Europe after the Rain_. The distinctive figures of Europe\u2014such as the Italian boot, the French hexagon, the Iberian animal's head\u2014have vanished, and new territorial divisions have been drawn straight across the land with the arbitrariness of colonial boundaries, or else follow intricate forms whose motivation is no less opaque. The one central resemblance that certainly strikes the viewer of the painting is specifically not developed through an imaginative seeing that brings hidden elements of the picture into view, animates the formless ground, or reads the contours for their latent motifs. Rather, it is the result of an operation carried out upon the manifest surface of the picture. The object of this operation was formerly the Mediterranean Sea, the single geographic characteristic that produces the relationship to Europe's familiar form before the deluge. It is at once an evident and bewildering relationship, since it involves neither distortion (through erosion, flooding, or other external influences) nor a tectonic shift (through drifting apart or collision). The Mediterranean can really only be described as having undergone a reflection, which has shifted the Black Sea from east to west, as if Europe had been folded along the north\u2013south axis. This operation refers back to Picasso's poetics of folding and turning, for example, as in the rotation of the newspaper page in _Violin_ (fig. 58).\n\nErnst's statement from 1969 that he had once again discovered Leonardo's wall stains in the studio sets of Billancourt thus conceals how _Europe after the Rain_ differs from other influential works of surrealism. John Russell already indicated the exceptional status of this picture when he wrote: \"Nothing could be further from the rampant narcissism of conventional 'Surrealist painting' than this somber and premonitory image.\" The picture support is in fact similar to Leonardo's wall, but its mortified surface presents an impediment to the animating and figuring effect of imaginative seeing. One can also glimpse in it the reflexive negativity of the surrealist imagination, which directs itself against surrealism's own procedures and the mode of perception they provoke. _Europe after the Rain_ promotes the seeing of resemblances in the way a pre-morphic pictorial ground simply does, and it also blocks this imaginative perception. This is a reflexive form of negativity that precedes the provocation of the negated. In Europe as it has emerged again from out of the floodwaters, the earth as a matrix of images has not simply been reduced to nothing. The picture marks the zero level of surrealist perception; within it is sealed the dark, pre-morphous ground in which the seeing of resemblances follows the trace of desires, both erotic and political.\n\nFIGURE 58\n\nPablo Picasso, _Violin_ ( _Violine_ ), end of 1912. 62 \u00d7 47 cm, pasted paper and charcoal. Paris, Centre Pompidou, Mus\u00e9e Nationale d'Art moderne. \u00a9 2012 Estate of Pablo Picasso \/ Artists Rights Society (ARS), New York.\n\nREVOLUTION AND NATURAL HISTORY\n\nOperations of turning or folding, as they were previously practiced in Picasso's _papiers coll\u00e9s_ , transform the orientation of the image. It becomes a horizontal work surface and can still be experienced as such when it is viewed as a vertically oriented painting hanging on the wall. In _Europe after the Rain_ , besides the imaginary fold that has shifted the Black Sea from east to west, there are other, more concrete traces of this type of horizontal implementation of the image. I am referring to the cartographic symbols: the red lines of the ship routes are most immediately striking. While most of these connect two ports in straight lines, some follow the coasts, lead almost all the way to an island without touching it, or seem to stray without any objective. These routes indicate a population of seafarers and adventurers who have survived the flood and are now exploring the unknown shores of the destroyed continent. A ship route even crosses the inland body of water west of the \"Black Sea.\" The territorial boundaries\u2014of which some have been drawn across the relief using a straight edge and some have been drawn freehand\u2014attest to early, already cartographically documented attempts at colonization. Of actual settlements, or even cities, there is no clear trace\u2014not even at the nodal points of the sea routes. A few black circles could denote outposts. All these signs enable visualization of the horizontal condition in which maps are spread out in order to mark ways or bases; moreover, these signs indicate an openness to the future, since they enable the map's future users to orient themselves in the new Europe, and possibly also to make new markings. Perhaps these are also only projected routes and boundaries, and perhaps their purpose is to encourage investigation of a still entirely unknown area of the earth. In any case, it is reasonable to suppose that in these markings, which report of an already accomplished or else planned rediscovery of the obliterated continent, there is a political meaning: the surrealists look forward to a catastrophe in order to explore Europe after its destruction as the _terra incognita_ of revolutionary life. _Europe after the Rain_ is thus in fact a prophetic picture, but it has a different content than most art-historical interpreters think. In addition, it is differently formed than most surrealist pictures, since it does not rely upon the seeing of resemblances, as all of them do; neither does it correspond to surrealist theories about the political function of the imagination.\n\nMeanwhile, the painting's sense of the future is the product of two times: first, the pre-future of the catastrophe to which the title refers; it will have already come to pass when, second, the yet unconcluded and also unforeseeable future after the revolution begins, to which the markings on the map attest. The destruction is the precondition for registering new routes, boundaries, and way stations, which indicate the possibility of a surrealist life.\n\nThe history in surrealism of the motifs of rain, catastrophe, and geographical transformation supports this reading. In surrealism, rain has an exclusively positive meaning as the image of the passive, fluent, and fertilizing language of automatism: one need only recall the opening image of _Natural History_ or text 16 from Breton's _Soluble Fish_ , dedicated to the liberating precipitation of automatic writing.\n\nFossilization, hardening, as well as overflowing, inundation, and other geological upheavals that instigate the reversion of urban civilization to nature are read by the surrealists in the way they read rain: as allegories of unconscious powers. Ernst's earthquake pictures, paintings such as _Paris Dream_ ( _Paris-R\u00eave_ ), and many of his collages represent the surrealist territory as a post-catastrophic landscape. A world map (fig. 59) published in 1929 in the surrealism issue of the Belgian avant-garde magazine _Vari\u00e9t\u00e9s_ explicates the political thrust of these destruction fantasies. It shows the annihilation of the Western world and the growth of Germanophone, Russian, and other areas, which, from the perspective of Paris, are part of the barbaric East. From there, revolutionary hordes would invade, from whom the surrealists expected one main feat: the annihilation of the French state and its ruling class. The critique of party Communism was also formulated in geographic metaphors. Breton follows a joke about the USSR, that it is content to explore territories opened up a long time ago, with the surrealist promise to explore unknown regions and write a new geography. Another example representative of the motif world of the surrealists' historical\/political revolution fantasies is Aragon's description of geological catastrophe as sexual climax. _Europe after the Rain_ is to be understood within the context of this surrealist wish for upheavals and catastrophes, and is thus only a dark premonition inasmuch as pessimism regarding the existent cannot be separated from revolutionary hope.\n\nFIGURE 59\n\n_The World in the Time of the Surrealists_ ( _Le monde au temps des Surr\u00e9alistes_ ). From _Vari\u00e9t\u00e9s: Revue mensuelle illustr\u00e9e de l'esprit contemporain_ , 1929.\n\nAgainst the background of the political fantasies of surrealism and following a closer observation of the picture, its established interpretation as a prophecy of the Second World War is no longer self-evident. The map of Europe documents a post-catastrophic future; however, the surrealists looked forward to destructive \"d\u00e9paysement\" (disorientation) in the form of cataclysms that would clear space for new routes. With this contextual interpretation and the excavation of the \"futures past\" (Reinhart Koselleck) of the surrealists' revolutionary fantasies to which the painting attests, a more important question has already in part been answered: How did Ernst change _the form_ of the revolutionary picture? How does the reflexive negation of the seeing of resemblances determine the sense of the future in _Europe after the Rain_? The painting's sense of the future does not lie in imaginary depths, in the place where surrealism generally located it, but in the narrative lines of the ship routes.\n\nCARTOGRAPHIC PAINTING\n\nIf we again enumerate the artistic procedures that Max Ernst used in order to make the map legible as a description and design of a revolutionary future, it becomes evident that these procedures generate a painting that above all is a surface. As a surface, it closes itself off from the pulsating depth typical of Ernst's surrealism and from the simulacral\/mimetic qualities of his pictures. The painting of post-catastrophic Europe does not pilot imaginative seeing into depths, but into a space of imaginary operations carried out with and upon the surface of the picture support. While the folding has already been carried out and cannot be repeated, the picture shows cartographic operations that designate the destroyed continent as a space of narrative possibilities. This attempt to answer the political radicalization of 1933 with a picture that opens not a simulacral but an operational space is distinguished by its iconographic reserve, a quality that becomes particularly clear in comparison with Ernst's other famous political pictures of the 1930s, for example, the second version of _Europe after the Rain_ , which was made in 1940 following the Phoney War (fig. 60). The movement away from depth as a pulsating space that promises hidden pictures and iconographic significance, and toward the surface upon which all operations, past and present, remain visible, is an exact measure of the distance that Gilles Deleuze sees between \"archaeological\" and \"cartographic art.\" Surrealism may be considered archaeological art because it places its trust in the dark origins, the pre-morphic ground, or the reversible contours that elicit visual parapraxes, which bring a buried history to light and lead the way to a revolutionary future. Ernst's \"Visions of Half-Sleep\" is indebted to this model, as are his surrealist pictures, which, as deferred activations, always refer back to a dark origin and simultaneously, like Eve in _Natural History_ , promise the lifting of the spell of infantile prehistory. In _Europe after the Rain_ , this visual as well as psychic dimension of depth is broken up and hardened into a desiccated surface that allows no metamorphoses, no deferred activation, no latent images to be released. The picture's sense of the future does not emerge from the dark depths of a buried past. The new Europe offers itself as an unknown and unconscious territory of future travels and adventures.\n\nFIGURE 60\n\nMax Ernst, _Europe after the Rain II_ ( _L'Europe apr\u00e8s la pluie II_ ) (detail), 1940\u201342. 55 \u00d7 128 cm. Hartford, Wadsworth Atheneum (S\/M no. 2395). \u00a9 2012 Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nThe one past moment to which _Europe after the Rain_ refers is the catastrophe that obliterates the whole past. This consists of a series of natural cataclysms: a rain of fire that has melted the earth, a breaking up of the earth's crust that has led to the folding and reflection of the continent, and a deluge that has cooled the earth's molten surface and distributed new bodies of water like puddles across it. The traces of these cataclysms appear on the map as if it had been subject to the same forces as the continent it represents. The painting _Europe after the Rain_ seems to have originated from the same cataclysms as the new Europe that can be seen in the painting. Map and continent are not only related to one another through projection, but also and perhaps more strongly through material contact, as two kinds of traces of the same natural force. The painting is then no longer exclusively a human artifact. This double significance as natural product and instrument is correlative to two different measures of time: catastrophe and shock in the history of nature, on the one hand, and openness to the future of travel and adventure, on the other. The great upheaval seems to be an event unavailable to action; the human only appears as an agent in the post-revolutionary landscape. The routes sketch trails of a somnambulistic movement, which, after the surrealist revolution, life in general will be able to follow. Ernst abandons the position of the visionary who, seeing the future in the darkness of psychic depths, allows it to emerge in an iconography laden with significance, as in the second version of _Europe after the Rain_. As a cartographer, upon a mortified surface, he opens up a space of narrative that neither encapsulates the surrealist historical fantasies in its content, nor allows them to emerge from a mythic deep time, but rather translates them into lines of movement.\n\nFOLDING TIME\n\nIn the early 1930s, Max Ernst tested operations such as folding, shifting, and turning in the collages of the Loplop series, in which these procedures are performed by the artist's alter ego, the bird creature Loplop. While here Ernst designates his art as the product of various artistic devices and tricks, in _Europe after the Rain_ this is precisely not the case; instead, the folding originates in an event that deformed the picture support even before the cartographer's markings were registered. This in turn most directly recalls Dadaist works such as _Portable Handbook_ (fig. 7), which shows a sequence of shifts upon an eroded support. In terms of procedural logic, _Portable Handbook_ is particularly close to the first Europe picture, because _Europe after the Rain_ also does not place on view the processing of its surface as a freely available operation, but rather as a procedure that cannot be separated from the rigidified condition of the picture. Every operation carried out upon or with these surfaces is connected to the heaviness and materiality that the support takes on as the result of a geologic process.\n\nThe cut that _Europe after the Rain_ marks, as a cartographic picture by a thoroughly \"archeological\" painter in search of psychic depth, proves in the context of Ernst's oeuvre to be the fold that lays the surrealist works over the geologic Dada pictures. In 1933 Ernst again worked with a mortified surface and allowed the sovereign availability of operations carried out upon and with this surface to be wrecked by it. Nevertheless, the optimism that the shipping routes register in the postdiluvian world clearly distinguishes the painting from the works made after the failed November Revolution. In the revolutionary map of 1933, Ernst again makes use of artistic methods from his first distinctly Dadaist pictures, and at the same time corrects their mocking defeatism by forging post-revolutionary trails through the mortified surface.\nAFTERWORD\n\nWALTER BENJAMIN AND MAX ERNST\n\nIn May 1935 Theodor Adorno offered to arrange a meeting between Walter Benjamin and Max Ernst. Benjamin seemed very taken with the idea. Whether he then actually made the artist's acquaintance is unknown. However, this encounter had taken place in his thoughts many years earlier, as publicly expressed in the gloss \"Dream Kitsch\" from 1927 and the major essay on surrealism from 1929. It has therefore been seductive for art history to consult Benjamin as a theoretical authority and to relate his thoughts on the nineteenth century as the prehistory of modernism, which are partially based on an engagement with Ernst, back to Ernst again. Conversely, the question has also been raised of to what extent Ernst's pictures can be interpreted as an anticipation of Benjamin's speculations on the historical and medial constitution of modern subjectivity. A study like the one to be found in this book, which examines the problem of how Ernst's works articulate historical breaks and what interpretation of these breaks they offer, must also clarify its own relationship to Benjamin. In contrast to the stated alternatives\u2014the use of Benjamin's theses as a theoretical authority, on the one hand, or the demonstration of how Ernst's pictures anticipate these theses, on the other\u2014in what follows I would like to introduce Benjamin as a voice of dissent. That is, his writings are useful specifically because they stand in marked tension to surrealism, inasmuch as the latter came under the spell of the ghostly return of painting. At the same time, however, Benjamin's very last reflections on painting from 1938\u201340 reveal a surprising affinity to Ernst's critical turn against his own spectral art in _Europe after the Rain_ (plate 4).\n\nThe one longer passage that Benjamin devoted to Ernst occurs in \"Dream Kitsch,\" which was probably written as early as 1925 and published in 1927 in _Neue Rundschau_. In this passage, Benjamin discusses the frontispiece of the book _R\u00e9p\u00e9titions_ , jointly produced by Ernst and Paul \u00c9luard. The print was based on one of Ernst's overpaintings, with which Benjamin was certainly unfamiliar\u2014he does not address its specific procedural logic in the text. Benjamin's description and allegoresis of the frontispiece comprise only six sentences, but these are highly compressed:\n\n_R\u00e9p\u00e9titions_ is the name that Paul \u00c9luard gives to one of his collections of poetry, for whose frontispiece Max Ernst has drawn four small boys. They turn their backs to the reader, to their teacher and his desk as well, and look out over a balustrade where a balloon hangs in the air. A giant pencil rests on its point in the windowsill. The repetition of childhood experience gives us pause: when we were little, there was as yet no agonized protest against the world of our parents. As children in the midst of that world, we showed ourselves superior. When we reach for the banal, we take hold of the good along with it\u2014the good that is there (open your eyes) right before you.\n\nThe first three sentences can easily be related to Ernst's picture, in which the children stand like pupils in front of a blackboard but look off into the distance. The monumental piece of chalk to the left evidently implies another kind of writing than that practiced on the blackboard. This kind of writing leads away from the classroom and follows the hot-air balloon that is floating away to the right, or else heads in the direction of the church that towers over the horizon, or beyond, into the invisible, distant realm of modern communication suggested by the telegraph pole on the far right. According to Benjamin, one can easily see in this picture an allegory of the superiority of children with regard to the disciplinary force employed by their parents and teachers. But what is the meaning of Benjamin's last sentence, which identifies the banal with the near, the graspable, and also the good? Indeed, the picture hinges on the replacement of the near and the tactile\u2014the blackboard\u2014by the remote expanse of the sky. But one could also read from the picture, as Benjamin does, that the sky takes on the tactile qualities of what it replaces, that distant space is brought as close to the figures as the chalk and blackboard previously had been. In fact, one has the impression that the schoolboys are trying to touch the sky with their hands.\n\nAs opposed to this situation in which distant space draws close and can be touched, an obsolete ideal of art insists upon aesthetic distance. Toward the end of \"Dream Kitsch,\" Benjamin remarks, \"What we used to call art begins at a distance of two meters from the body. But now, in kitsch, the world of things advances on the human being; it yields to his uncertain grasp and ultimately fashions its figures in his interior.\" But when distant space comes closer, why does the good also become graspable? In the early notes for _The Arcades Project_ , written around 1928, Benjamin explicitly states the reason:\n\n. . . How would it look as religion or mythology? We have no tactile _< taktisch>_ relation to it. That is, we are trained to view things, in the historical sphere, from a romantic distance. To account for the directly transmitted inheritance is important. But it is still too early, for example, to form a collection. Concrete, materialistic deliberation on what is nearest is now required. Mythology, as Aragon says, drives things back into the distance. Only the presentation of what relates to us, what conditions us, is important. . . .\n\nUsing the example of Louis Aragon's _Paris Peasant_ , Benjamin accuses surrealism of reveling in the remoteness of the recent past and transfiguring its foreignness into mythology, as the romantics had done with history. Benjamin himself is concerned with restoring tactile contact to the epoch of childhood, in the future perhaps through collecting, but at the moment through his work on _The Arcades Project_ , in which the dreamlike encryption of the nineteenth century, its enraptured removal from the present, would be dispelled. The method that Benjamin prefers to use for the re-creation of this \"nearest nearness\" of the past is montage. As can further be read in his early notes for _The Arcades Project_ , he credits montage with destroying the contemplation of the past as a remote and foreign realm, but simultaneously with bringing the magic of this contemplation into full view: \"In the _Arcades Project_ , contemplation must be put on trial. But it should defend itself brilliantly and justify itself.\" As an example of this method, another early passage can be cited that leads us back to Ernst again, since in it Benjamin considers the same late nineteenth-century natural science illustrations that also formed a basis for the frottage series _Natural History_ (figs. 35, , ). (Incidentally, in the same year, 1929, that Benjamin read this and other prose pieces to Adorno and Horkheimer, Ernst exhibited the collage _Two Young Girls Riding across the Sky_ , in which he also made use of one such illustration [fig.38]).\n\nWhen, as children, we were given those great encyclopedic works _World and Mankind_ , _New Universe_ , _The Earth_ , wouldn't our gaze always fall, first of all, on the color illustration of a \"Carboniferous Landscape\" or on \"Lakes and Glaciers of the First Ice Age\"? Such an ideal panorama of a barely elapsed primeval age opens up when we look through the arcades that are found in all cities. Here resides the last dinosaur of Europe, the consumer. On the walls of these caverns, their immemorial flora, the commodity, luxuriates and enters, like cancerous tissue, into the most irregular combinations. A world of secret affinities: palm tree and feather duster, hair dryer and Venus de Milo, prosthesis and letter-writing manual come together here as after a long separation. The odalisque lies in wait next to the inkwell, priestesses raise aloft ashtrays like patens. These items on display are a rebus: and ffihow> one ought to read here the birdseed kept in the fixative-pan from a darkroom, the flower seeds beside the binoculars, the broken screws atop the musical score, and the revolver above the goldfish\u2014is right on the tip of one's tongue.\n\nThis reflection on the world of obsolete commodities is similar to the commentary on Ernst's _R\u00e9p\u00e9titions_. Old objects are worn out as if from too much handling, and by our taking hold of them again we can take part in the past from which they come, which has been detached from the present. This realization is visual: \"When we reach for the banal, we take hold of the good along with it\u2014the good that is there (open your eyes) right before you.\" Thus, from an image (Ernst's print), via the bodily appropriation of obsolete objects, Benjamin arrives at the sudden insight that is expressed in the injunction \"open your eyes.\" In the note written only shortly thereafter, he likewise begins with an image (the natural history illustration) in order to arrive at a second image (the shop window as rebus). The transition from one to the other is in turn described in a somatic metaphorics. But unlike the world of childhood, which returns as a tactile experience, the commodity world of yesterday is brought back in a manner analogous to the growth of \"cancerous tissue.\"\n\nThe description of Ernst's print in \"Dream Kitsch\" and the evocation of natural history illustrations are examples of Benjamin's interest in bringing contemplation into view in all its magic. Meanwhile, the harsh transitions between individual thoughts and the obtrusiveness of the bodily metaphorics point to his other goal of destroying contemplation and establishing in its place an intimate, bodily form of historical reference. The artifacts of the recent past appear foreign and can be assembled into seductive images, but in order for these images to become meaningful for the present, they have to come into our immediate bodily proximity, as excitations of the sense of touch or an inflammation beneath the body's surface. Only through this transformation do images of the past become \"dialectical images\" that herald an awakening from the dreaming slumber of capitalism.\n\nAs important as images were for the formulation of Benjamin's theses\u2014in addition to Ernst's collages, Albrecht D\u00fcrer's _Melancholia I_ or Paul Klee's _Angelus Novus_ could be cited\u2014he placed no hope in the modern easel painting. There survive two negative assessments Benjamin made of Ernst as a painter (both from the late 1920s), one in the context of private correspondence in which painting is polemically contrasted with writing, and the other in the essay on surrealism, in which Benjamin contrasts urban space with the space of painting.\n\nIn a letter of March 16, 1929, Benjamin thanked his school friend Alfred Cohn for the new notebook Cohn had given him:\n\nIn any case I want to send you by return post my warmest thanks for the new masterpiece, which has just arrived. It \"gave the final curve and polish\" (in the words of the _Bern Cycling News_ ) to my decision not to go to the Max Ernst exhibition at Flechtheim. The pages' front sides are years ahead of the best new painting (and their back sides, the best new painting criticism).\n\nWhy Benjamin decided not to go to the Max Ernst exhibition after receiving the new notebook and why he praised this notebook in a curious metaphor at the expense of contemporary painting and art criticism become comprehensible if we consider how important this particular underlay was for Benjamin's work. Three months earlier he had already informed Cohn, who had been supplying him with these notebooks on a regular basis since 1927, that he would soon be needing a new one\u2014he \"could not face the idea\" that his writing would be \"homeless again.\" It seems that Benjamin did not go to the Galerie Flechtheim because he wanted to make use of his new notebook right away. In fact, in the letter to Cohn, he continues to be inspired by the newly received gift. His characterization of the notebook, that its pages' front and back sides are \"years ahead of\" the best new painting and art criticism, respectively, leads one to think of the relationship between recto and verso in the notebook, and thus refers to the aspect of Cohn's \"masterpiece\" that Benjamin most prized: its particularly thin, nearly transparent paper. In a letter from February 1929, Benjamin confesses to \"a shameful weakness for this extremely thin, transparent yet excellent stationery.\" It permitted him, while writing, to perceive what had already been written, and to immerse himself not only in the individual pages, but in the notebook as a whole. What's more, his remarkably small writing attests to the importance to him of bodily nearness to the material underlay and traces of ink. In writing, Benjamin always wanted to feel the tactile space of his immediate surroundings. Against this background, it also becomes clearer why he thought modern painting was antiquated in comparison with Cohn's notebooks. He thought of painting\u2014also Ernst's painting, which was on view in 1929 at the Galerie Flectheim\u2014as an art that \"begins at a distance of two meters from the body\" and thus belongs in the past. Writing, however, is tactile in the emphatic sense. He had already discovered an allegory of this insight in Ernst's _R\u00e9p\u00e9titions_ , since in this image the conversion of distant space into the immediate space of tactility is carried out through the act and the tools of writing. While the church, the hot-air balloon, and the telegraph pole on the right as symbols of distance are contrasted with the monumental piece of chalk on the left, the schoolboys bring the distant space of opticality into the immediate space of tactility by seeming to press their writing hands to the sky. In his own writing, too, Benjamin enacted this turn away from an art that \"begins at a distance of two meters from the body,\" and he did so by explicitly dismissing painting, which he considered the quintessence of the obsolete ideal of aesthetic distance.\n\nIn the surrealism essay of 1929, Benjamin speaks pronouncedly against the painting of de Chirico and Ernst:\n\nAt the center of this world of things stands the most dreamed-about of their objects: the city of Paris itself. But only revolt completely exposes its Surrealist face (deserted streets in which whistles and shots dictate the outcome). And no face is surrealistic to the same degree as the true face of a city. No picture by de Chirico or Max Ernst can match the sharp elevations of the city's inner strongholds, which one must overrun and occupy in order to master their fate and\u2014in their fate, in the fate of their masses\u2014one's own.\n\nIn the modern city, but not in modern painting, Benjamin sees the possibility of conceiving the future of the subject and that of the collective as a common future. This thesis will be developed further in _The Work of Art in the Age of Its Technological Reproducibility_. The famous text opposes the \"contemplative\" and \"optical\" perception that easel painting requires with the \"distracted\" and \"tactile\" perception that is offered to the masses in architecture and above all in cinema. As in the early notes for _The Arcades Project_ , montage and tactility are again joined in opposition to contemplation, albeit with another goal. In _The Work of Art_ , Benjamin is concerned not with regaining contact to the past, or in laying the groundwork for an awakening out of it, but instead with the political consequences of the profound and irreversible transformation of the human sensorium through photography and film. Modern painting can only take part in this transformation inasmuch as it tends toward cinema. According to Benjamin, it has done so since cubism, and particularly radically in Dada: \"Dadaism attempted to produce with the means of painting (or literature) the effects which the public today seeks in film.\" However, because of its medial support, painting remains bound to the previous mode of contemplative seeing: \"Let us compare the screen [ _Leinwand_ ] on which film unfolds, with the canvas [ _Leinwand_ ] of painting. The image on the film screen changes, whereas that on the canvas does not. The painting invites the viewer to contemplation; before it, he can give himself up to the train of associations. Before a film image, he cannot do so. No sooner has he seen it than it has already changed.\" Painting also belongs to the past because it always preserves distance to the objects of its representation and reproduces them as wholes, while cinema intrudes into reality, as Benjamin illustrates with the famous contrast between painter and magician, on the one hand, and surgeon and filmmaker, on the other.\n\nBenjamin's diagnosis of the obsolescence of painting is very well known up to this point. To an artist such as Ernst, who since the early 1920s committed his work to the ghostly return of painting, Benjamin concedes no future. With his decision not to go to the exhibition at the Galerie Flechtheim, and later with his repudiation of Ernst's painting in the essay on surrealism, Benjamin made clear that he could no longer find in the works of this artist that which the print _R\u00e9p\u00e9titions_ had seemed to herald: pictures that were _not_ \"at a distance of two meters from the body.\"\n\nHowever, a six-page, paginated manuscript titled \"Zum Kunstwerk im Zeitalter \/ Malerei und Graphik\" (\"On the Work of Art in the Age \/ Painting and the Graphic Arts\") from 1938\u201340, in which Benjamin recorded his thoughts toward a revision of the _Work of Art_ essay, makes his relationship to painting seem no longer so very polemical. In these notes\u2014which have so far been ignored by art history and have interested Benjamin scholars in an editorial respect only\u2014Benjamin comes back to his early texts \"Painting and the Graphic Arts\" and \"Painting, or Signs and Marks,\" whose main theses I briefly summarized in the introduction: Paintings differ from graphic works because of their orientation and, moreover, paintings and graphic works are based on differing forms of production. A graphic work is horizontal and emphasizes its being as artifact, so clearly that each line is experienced as a placed line, and each between-space as paper. A painting is vertical and as a stained surface is also analogous to the human body and its marks. The mark is a sign incurred by the body, and painting is likewise a passive art according to its nature. In addition, as Benjamin's most important example, blushing, implies, this passivity can be described more specifically as becoming-seen. The paradigmatic case of blushing is, according to Benjamin, shame in the presence of mystery. He understands the flushing of the skin as the sign of a transfiguring collapse of the subject comparable to religious ecstasy.\n\nIn order to elucidate Benjamin's revision of these early speculations on painting, marks, and shame, I would first like briefly to examine the tenth footnote of the second edition of the _Work of Art_ essay, which outlines a theory of originary mimesis. Art has its source in bodily mimesis, though from the beginning in a double aspect: \"The mime presents his subject as a semblance. One could also say that he plays his subject. Thus we encounter the polarity informing mimesis. In mimesis, tightly interfolded like cotyledons, slumber the two aspects of art: semblance and play.\" Mimesis as semblance always remains related to the human body and engenders auratic artworks, while mimesis as play leads away from the human body and is strengthened by the disintegration of the auratic in modernity: \"that which is lost in the withering of semblance and the decay of aura in works of art is matched by a huge gain in [ _Spiel-Raum_ , literally 'space for play'].\" Benjamin's lifelong interest in play, in gambling as well as in children's play, is related to his attempt to conceive of subjectivity as a formation of the material world. In play, bodies are joined with things; bodies become things, things become bodies. The gambler is as closely joined to the cards or dice as to his own hand; the child imitates people and things, both the teacher and a railway train. Film, which Benjamin also calls \"Lichtspiel\" (light-play), is not considered by Benjamin to be an example of illusion; rather, he considers it the most radical form of such a penetration or \"innervation\" of body and apparatus. The cinema audience experiences this \"innervation\" and at the same time can appropriate it as a new form of sensorium. The audience is not only made into a collective by film; in the medium of film, it also creates itself as a collective. Such a relationship that is both passive and active can be termed playful. Therefore, film offers the masses a \"space for play\" in which the masses discover collective perception and simultaneously recognize it as their own.\n\nIn the notes from 1938\u201340 titled \"On the Work of Art in the Age \/ Painting and the Graphic Arts,\" the question becomes to what extent the old visual arts had already contributed, or could continue to contribute, to expanding the \"space for play.\" Beginning with the distinction between cult and exhibition value, Benjamin first establishes that the easel painting is a mobile object that has no specific location. But, he continues, the mobility of the painting proves limited when it is compared with the graphic work. (Here by \"graphic work,\" he means modern advertising and print media in addition to drawing and printing.) According to Benjamin's fundamental distinction, the painting lacks the mobility of the graphic work because it is oriented vertically and thus addresses itself to human beings as an entity that stands erect. However, the graphic work is horizontal and thus is related not to the verticality of the gaze but to the horizontal plane of locomotion. This also distinguishes its mobility from that of the painting. The easel painting is mobile because its support has no fixed location. This is also the case for the graphic work. But the graphic work is characterized by a mode of representation that takes into account the movement of the viewer, or even initiates and prepares the way for this movement. In the graphic work, the picture proves to be a field divided into various precincts that indicate possibilities of movement, as in the magic circle or the drawing on the ground in the children's game \"heaven and hell.\" In this example, the term _Spiel-Raum_ achieves its concrete basis. The graphic work opens up a \"space for play\" since it cartographizes the possibilities of future action. With this conception of graphic work as cartography and cartography as preparing the way for play and movement, Benjamin performs a crucial modification of his theory of 1917.\n\nThe thoughts that Benjamin devotes to painting in these late notes lead from the easel painting back to a primal form of painting that in turn is understood as a mark. As a mark that appears and passes away, this originary painting is likewise mobile, but in a different way from the easel painting or the graphic work. The easel painting is mobile because its support is not bound to any fixed location, and the graphic work is mobile because it marks out a \"space for play\" for its users. The originary painting is mobile because its signs appear suddenly and, like a blush of shame, also pass away again; however, the signs of originary painting do not exclusively arise from depths. More explicitly than in 1917, Benjamin states that not only blushing but also \"the writing in the palace of Nebuchadnezzar\" belong to the category of the mark. \"The mark in the most exact sense of the word\" is \"a colored configuration that appears on the wall (emerges from it or is cast upon it).\" In concluding, he returns to the diagnosis developed in the _Work of Art_ essay, that the body has lost its meaning as the ground of image production and has been replaced by a space for play in which body and apparatus are interconnected. \"From this perspective of the philosophy of history, the contemporary crisis of painting would amount to transformations that imply the atrophy of the medium of painting ( _Malerei_ ), the medium in which the mark ( _Mal_ ) is at home.\"\n\nThe manuscript \"On the Work of Art in the Age \/ Painting and the Graphic Arts\" is clear evidence that Benjamin continued until the end to meditate intensively on the nature and history of painting. In the place of clear oppositions such as that between immobile canvas and mobile film image or between optical\/contemplative painting, on the one hand, and tactile\/attacking montage, on the other, now the historical dialectic that was already outlined in the tenth footnote of the second edition is developed in the area of visual art. It begins with the antithesis of painting as mark and graphic work as map. As mark, painting belongs with bodily mimesis as semblance. It addresses its viewer with the obtrusiveness of a scar or blush. With the mobile easel painting, painting detached itself from this origin in the human body. Simultaneously, play gains significance. Its medium is the graphic work. The graphic work gives the human body access to a \"space for play,\" in that it sets the body in motion and regulates its movements. From this point of view, the magic circle or the drawing on the ground in the game \"heaven and hell\" are the first apparatuses with which the human body is joined.\n\nTo return to Max Ernst, these thoughts read like an art-critical commentary on his painting after 1925. Grattage can be understood as an attempt once more to conceive of painting as a medium of the mark (fig. 60). As opposed to the majority of post-impressionist paintings, which emphasized the application of paint in order to make the picture transparent to its own material production, the grattages were intended to be viewed as if their marks and signifying motifs had risen from the depths of the material layers of paint. During this time, Ernst also worked with pieces of rope that he first dipped in paint and then threw at a canvas strewn with sand. This procedure lends itself to comparison with Benjamin's conception of the mark as a projection on the wall. In the early text \"Visions of Half-Sleep,\" Ernst's assertion that his painting is grounded in becoming seen (see chapter 3) brings him yet closer to Benjamin's conception of painting as an art that is passive according to its nature. Also, Ernst's turn away from grattage in _Europe after the Rain_ from 1933 (see chapter 5) can be elucidated with reference to Benjamin's late thoughts on painting and graphic works. According to Benjamin in 1938, the graphic work is a mode of representation that indicates our future movements in space: \"The graphic work depicts the world in such a way that the human may traverse it. Its viewer's eye hurries ahead of his foot. No transition and no mediation leads from the easel painting to a map.\" Benjamin's opposition of paintings and maps corresponds exactly to the alternatives with which Ernst also found himself confronted in 1933 when he produced _Europe after the Rain_ , a picture in opposition to his own grattages. The painter who attempted to lead painting back to the mark took the opposing position in this one work and created a map that would guide its users in their exploration of post-revolutionary Europe.\nNOTES\n\nINTRODUCTION\n\n \"dessins imprim\u00e9s, dessins de r\u00e9clames, images de dictionnaire, images populaires, images de journaux.\"\n\n \"Chacun de ces tableaux t\u00e9moigne d'une d\u00e9couverte technique diff\u00e9rente.\"\n\n \"une sorte de collage intellectuel.\" Louis Aragon, \"Max Ernst, peintre des illusions,\" in _Les Collages_ (Paris: Hermann, 1993), 25, 28, 27.\n\n \"facult\u00e9 merveilleuse.\"\n\n \"Doesn't such an ability make the person who possesses it better than a poet, since poets do not need to be aware of their visions and, in any case, maintain strictly Platonic relations with them?\" Andr\u00e9 Breton, _The Lost Steps_ , trans. Mark Polizzotti (Lincoln: University of Nebraska Press, 1996), 61.\n\n Max Ernst, _Beyond Painting and Other Writings_ (New York: Schultz, 1948), 26\u201329. _Max Ernst_ , exhibition catalog (Zurich: Kunsthaus Zurich, 1962), I\u2013X. A persuasive analysis of Ernst's autobiographical writing, which I unfortunately overlooked while preparing the German edition of this book, is Cornelis de Boer, \"Selbstzitat Als Selbstportr\u00e4t in Max Ernsts Notes Pour Une Biographie: Die Identit\u00e4t des K\u00fcnstlers als Collage,\" in _Instrument Zitat: \u00dcber den literarhistorischen und institutionellen Nutzen von Zitaten und Zitieren_, ed. Klaus Beekmann and Ralf Gr\u00fcttemeier, _Avant-Garde Critical Studies_ 13 (2000): 81\u2013107.\n\n Ernst, _Beyond Painting_ , 14.\n\n Max Ernst, _\u00c9critures_ (Paris: Gallimard, 1970), 256.\n\n Ernst, _Beyond Painting_ , 17.\n\n Werner Spies, _Max Ernst Collagen: Inventar und Widerspruch_ (K\u00f6ln: DuMont Schauberg, 1974); Dirk Teuber, \"Max Ernsts Lehrmittel,\" in _Max Ernst in K\u00f6ln: Die rheinische Kunstszene bis 1922_, exhibition catalog, ed. Wulf Herzogenrath (Cologne: K\u00f6lnischer Kunstverein, 1980), 206\u201321; J\u00fcrgen Pech, \"Mimesis und Modifikation: Fotografische Portr\u00e4ts und ihre Verwendung im Werk von Max Ernst,\" in _Max Ernst: Das Rendezvous der Freunde_ , exhibition catalog (K\u00f6ln: Musuem Ludwig, 1991), 241\u201369; Ludger Derenthal, \"Mitteilungen \u00fcber Flugzeuge, Engel und den Weltkrieg: Zu den Photocollagen der Dadazeit von Max Ernst,\" in _Im Blickfeld: Jahrbuch der Hamburger Kunsthalle_ 2 (1994): 41\u201360.\n\n Rosalind Krauss, _The Optical Unconscious_ (Cambridge, MA: MIT Press, 1993); Hal Foster, _Compulsive Beauty_ (Cambridge, MA: MIT Press, 1993).\n\n Foster, _Compulsive Beauty_ , 157\u201392.\n\n Richard Shiff, \"Representation, Copying, and the Technique of Originality,\" _New Literary History_ 15, no. 2, \"Interrelation of Interpretation and Creation\" (Winter 1984): 333\u201363.\n\n Peter B\u00fcrger emphasizes this aspect in his theory of the avant-garde, in that he views the avant-garde as an attack on the organic artwork. B\u00fcrger, _Theory of the Avant-Garde_ , trans. Michael Shaw (Minneapolis: University of Minnesota Press, 1984).\n\n Foster, _Compulsive Beauty_.\n\n See further James Elkins, review of _Compulsive Beauty_ , by Hal Foster, _Art Bulletin_ 76 (1994): 546\u201348.\n\n Here, I am following Brigid Doherty's concise critique of the exhibition at the Met curated by Werner Spies. Doherty, \"Max Ernst: Metropolitan Museum of Art, New York,\" _Artforum_ (September 2005). Werner Spies and Sabine Rewald, eds., _Max Ernst: A Retrospective_ (Metropolitan Museum of Art) (New Haven, CT: Yale University Press, 2005). For a critical engagement with Doherty's text, one of the best on Ernst, see the afterword. Brigid Doherty, \" _Max Ernst_ : A Retrospective,\" _Artforum_ (September 2005): 295\u201397, 332, 347.\n\n Clement Greenberg, \"Collage,\" in _Art and Culture_ (Boston: Beacon Press, 1961), 70\u201383; Christine Poggi, \"Frames of Reference: Table and Tableau in Picasso's Collages and Constructions,\" _Art Journal_ 47, no. 4 (Winter 1988): 311\u201322; Yve-Alain Bois, \"Kahnweiler's Lesson,\" in _Painting as Model_ (Cambridge, MA: MIT Press, 1990), 65\u201397; Bois, \"The Semiology of Cubism,\" in _Picasso and Braque: A Symposium_ , ed. William Rubin and Lynn Zelevansky (New York: Museum of Modern Art, 1992), 169\u2013208; Rosalind Krauss, _The Picasso Papers_ (London: Thames & Hudson, 1998).\n\n Thierry de Duve, _Kant after Duchamp_ (Cambridge, MA: MIT Press, 1996).\n\n Krauss, _The Optical Unconscious_ ; another important investigation into the function of painting in the context of avant-gardism is offered by George Baker, _The Artwork Caught by the Tail: Francis Picabia and Dada in Paris_ (Cambridge, MA: MIT Press, 2008).\n\n Paul Klee, \"Die Ausstellung des Modernen Bundes im Kunsthaus Z\u00fcrich\" (1912), in _Paul Klee: Kunst-Lehre_ , ed. G\u00fcnther Regel (Leipzig: P. Reclam, 1987), 52ff.\n\n Ibid., 53.\n\n Walter Benjamin, _The Work of Art in the Age of Its Technological Reproducibility, and Other Writings on Media_ , ed. Michael W. Jennings, Brigid Doherty, and Thomas Y. Levin (Cambridge, MA: Belknap Press of Harvard University Press, 2008), 219\u201323; Benjamin, \"Letter to Gershom Scholem, 22 October 1917,\" in _The Correspondence of Walter Benjamin, 1910\u20131940_ , ed. Gershom Scholem and Theodor W. Adorno, trans. Manfred R. Jacobson and Evelyn M. Jacobson (Chicago: University of Chicago Press, 1994), 97\u2013102; Gershom Scholem, _Tageb\u00fccher nebst Aufs\u00e4tzen und Entw\u00fcrfen bis 1923_ , vol. 2, ed. Karlfried Gr\u00fcnder et al. (Frankfurt: J\u00fcdischer Verlag, 2000), 30\u201334. The most recent and most comprehensive discussion of Benjamin's early art theory is found in Annie Borneuf, \"'Radically Uncolorful Painting': Walter Benjamin and the Problem of Cubism,\" _Grey Room_ 39 (Spring 2010): 74\u201393.\n\n Yve-Alain Bois, \"Piet Mondrian, New York City,\" in _Painting as Model_ , 178\u201379, 308. For further discussion of these texts, see David E. Wellbery, \"Benjamin's Theory of the Lyric,\" in _Benjamin's Ground: New Readings of Walter Benjamin_ , ed. Rainer N\u00e4gele (Detroit: Wayne State University Press, 1988), 54\u201357; Howard Caygill, _Walter Benjamin: The Color of Experience_ (London: Routledge, 1998), 85\u201389; Charles W. Haxthausen, \"Reproduction\/Repetition: Walter Benjamin\/Carl Einstein,\" _October_ 107 (Winter 2004): 64\u201365; Heinz Br\u00fcggemann, _Walter Benjamin \u00fcber Spiel, Farbe und Phantasie_ (W\u00fcrzburg: K\u00f6nigshausen and Neumann, 2007), 143\u201353; and Brigid Doherty, \"Painting and Graphics,\" in Benjamin, _The Work of Art in the Age of Its Technological Reproducibility_ , 195\u2013217.\n\n The connection here is an etymological one. The Middle High German _Mal_ had the sense \"stain, mark, sign, sin, or shame.\" Modern usage includes _Muttermal_ (birthmark) and _Wundmal_ (scar), as well as _Schandmal_ (stigma or blemish) and _Kainsmal_ (mark of Cain). The verb _malen_ , in Modern German meaning \"to paint,\" formerly had the sense \"to apply marks or signs.\"\u2014Trans.\n\n The concept of disanalogy refers in this context to Whitney Davis's thesis that Freud chose his metaphors in part specifically to illuminate the metaphorized entity through dissimilarities. Davis, _Drawing the Dream of the Wolves: Homosexuality, Interpretation, and Freud's \"Wolf Man\"_ (Bloomington: Indiana University Press, 1995), 85.\n\nCHAPTER ONE\n\n Max Ernst, \"Souvenirs rh\u00e9nans,\" _L'\u0152il_ 16 (April 1956): 9ff.\n\n Werner Spies and G\u00fcnter Metken, _Max Ernst: \u0152uvre-Katalog_ (K\u00f6ln: M. DuMont Schauberg, 1975), no. 349, see also nos. 350, 372.\n\n \"Ein hoffnungsvoller junger Mann gew\u00f6hnt sich leicht das Malen an.\" J\u00fcrgen Pech, \"Mimesis und Modifikation: Fotografische Portr\u00e4ts und ihre Verwendung im Werk von Max Ernst,\" in _Max Ernst: Das Rendezvous der Freunde_, exhibition catalog (K\u00f6ln: Musuem Ludwig, 1991), 241ff.\n\n Here, I am following Richard Shiff's fundamental examinations, especially \"Performing an Appearance: On the Surface of Abstract Expressionism,\" in _Abstract Expressionism: The Critical Developments_ , ed. Michael Auping (New York: H. N. Abrams in association with Albright-Knox Art Gallery, 1987), 94\u2013123; and Shiff, \"C\u00e9zanne's Physicality: The Politics of Touch,\" in _The Language of Art History_ , ed. Salim Kemal and Ivan Gaskell (Cambridge: Cambridge University Press, 1991), 129\u201380. See also Rosalind Krauss, \"The Motivation of the Sign,\" in _Picasso and Braque: A Symposium_ , ed. William Rubin and Lynn Zelevansky (New York: Museum of Modern Art, 1992). On Ernst's impressionist early phase, see Martina Ewers-Schultz, _Die franz\u00f6sischen Grundlagen des \"Rheinischen Expressionismus,\" 1905\u20131914: Stellenwert und Bedeutung der franz\u00f6schen Kunst in Deutschland und ihre Rezeption der Bonner Ausstellungsgemeinschaft von 1913_ (M\u00fcnster: Lit, 1996), 118ff., 157\u201363.\n\n \"Mais l'abstraction ne joue pas simplement les moyens de l'art contre leur subordination \u00e0 l'imitation de la nature. Au contraire, en opposant l'apparence (naturelle) des objets \u00e0 l'essence de la nature, elle vise \u00e0 retrouver cette derni\u00e8re.\" (But abstraction is not simply art's means of opposing its subordination to the imitation of nature. On the contrary: by opposing the (natural) appearance of the object to the essence of nature, it aims to rediscover the latter.) Jean-Claude Lebensztejn, _L'Art de la tache: Introduction \u00e0 la Nouvelle m\u00e9thode d'Alexandre Cozens_ (Mont\u00e9limar: Editions du Limon, 1990), 397.\n\n G\u00e9rard Genette, _Mimologics_ (Lincoln: University of Nebraska Press, 1995).\n\n On the mediation of perception and the act of painting in the late nineteenth and early twentieth century, see Shiff, \"Performing an Appearance\" and \"C\u00e9zanne's Physicality\"; also see Margaret Olin, \"Validation by Touch in Kandinsky's Early Abstract Art,\" _Critical Inquiry_ 16, no. 1 (Autumn 1989): 144\u201372.\n\n Ernst, \"Souvenirs rh\u00e9nans,\" 11.\n\n \"Ich gab mich dem Naturstudium hin, lagerte w\u00e4hrend unsagbar langen Zeiten unbeweglich auf einem Tisch und versuchte, wie ein Berg, tief langsam, unendlich langsam, zu tr\u00e4umen.\" Hans Arp, _Unsern t\u00e4glichen Traum: Erinnerungen und Dichtungen aus den Jahren 1914\u20131954_ (Z\u00fcrich: Verlag der Arche, 1955), 97.\n\n The best analysis of Arp's art is given by Stefanie Poley, _Hans Arp: Die Formensprache im plastischen Werk_ (Stuttgart: G. Hatje, 1978); in addition, there is Harriet Watts, _Chance: A Perspective on Dada_ (Ann Arbor, MI: UMI Research Press, 1980); Jane Hancock, \"Die Philosophie in Arps Formensprache,\" in _Arp, 1886\u20131966_ , exhibition catalog (Stuttgart: Staatsgalery, 1986), 58\u201373; Margherita Andreotti, _Early Sculpture of Jean Arp_ (Ann Arbor, MI: UMI Research Press, 1989); Monika Schr\u00f6ter, \"Die weibliche Formkraft in Hans Arps \u00e4sthetischem Konzept,\" in _Die Weibliche und die m\u00e4nnliche Linie: Das imagin\u00e4re Geschlecht der modernen Kunst von Klimt bis Mondrian_ (Berlin: D. Reimer, 1993); and Uwe Schramm, _Raumbegriff bei Hans Arp_ (M\u00fcnster: Lit, 1995).\n\n Fundamental works on the seeing of resemblances include Ernst Gombrich, _Art and Illusion: A Study in the Psychology of Pictorial Representation_ (Oxford: Phaidon Press, 1989), 154\u201361; H. W. Janson, \"The 'Image Made by Chance' in Renaissance Thought,\" in _De Artibus Opuscula XL: Essays in Honor of Erwin Panofsky_ (New York: New York University Press, 1961), 254\u201366; Hubert Damisch, _Th\u00e9orie du nuage_ (Paris: \u00c9ditions du Seuil, 1974); Richard Wollheim, _Painting as an Art_ (Princeton, NJ: Princeton University Press, 1987); Lebensztejn, _L'Art de la tache_ ; Christa Lichtenstern, _Metamorphose vom Mythos zum Prozessdenken: Ovid-Rezeption, surrealistische \u00c4sthetik, Verwandlungsthematik der Nachkriegskunst_ (Weinheim: VCH, 1992); James Elkins, _Why Are Our Pictures Puzzles? On the Modern Origins of Pictorial Complexity_ (London: Routledge, 1999), 177\u2013230; and Dario Gamboni, \"'Fabrications of Accidents': Factura and Chance in Nineteenth-Century Art,\" _RES: Anthropology and Aesthetics_ , no. 36 (1999): 205\u201325. Unfortunately, I am no longer able to incorporate Dario Gamboni's important study on imaginative seeing in modernism ( _Potential Images: Ambiguity and Indeterminacy in Modern Art_ [London: Reaktion, 2002]).\n\n On the deficit of freedom and innovation in the history of the imitation of nature, see Hans Blumenberg, \"Nachahmung der Natur\u2014Zur Vorgeshichte des sch\u00f6pferischen Menschen,\" in _Wirklichkeiten in denen wir leben_ (Stuttgart: P. Reclam, 1981), 274.\n\n \"In Ascona zeichnete ich mit Pinsel und Tusche abgebrochene \u00c4ste, Wurzeln, Gr\u00e4ser, Steine, die der See an den Strand gesp\u00fclt hatte. Diese Formen vereinfachte ich und vereinigte ihr Wesen in bewegten Ovalen, Sinnbildern der ewigen Verwandlung und des Werdens der K\u00f6rper.\" Arp, _Unsern t\u00e4glichen Traum_ , 12.\n\n In general, on the genre of the modernist crypto-portrait, see Edith Futscher, _Diesseits der Fassade: Kryptoportraits der Moderne zwischen Bildnis und Stillleben_ (Klagenfurt: Ritter, 2001); on other crypto- and friendship portraits by Max Ernst, see Ludger Derenthal, \"Max Ernst: Trois tableaux d'amiti\u00e9,\" _Cahiers du Mus\u00e9e national d'art moderne_ 31 (1990): 73\u2013110.\n\n Sigmund Freud, _The Interpretation of Dreams_ , trans. Joyce Crick, ed. Ritchie Robertson (New York: Oxford University Press, 1999), 208\u20139.\n\n Freud, \"Formulations on the Two Principles of Mental Functioning,\" in _The Standard Edition of the Complete Psychological Works of Sigmund Freud_ , vol. 12, ed. James Strachey (Psychoanalytic Electronic Publishing, www.pep-web.org), 222. See also Freud, _Studienausgabe_ (Frankfurt am Main: Fischer, 1994), 1:363.\n\n Dirk Teuber, \"Max Ernsts Lehrmittel,\" in _Max Ernst in K\u00f6ln: Die rheinische Kunstszene bis 1922_ , exhibition catalog, ed. Wulf Herzogenrath (Cologne: K\u00f6lnischer Kunstverein, 1980), 206\u201340; Gerd Bauer, \"Max Ernst und der K\u00f6lner Lehrmittelkatalog,\" _Jahresring_ 19 (1988\/89): 205\u201324.\n\n On the diagram from a semiological perspective, see Jean-G\u00e9rard Lapacherie, \"De la grammatextualit\u00e9,\" _Poet\u00e9ique_ 59 (1985): 283\u201394.\n\n _Fehlleistung_ , literally \"faulty performance,\" commonly \"Freudian slip.\" The term \"parapraxis\" has been criticized as unwieldy, incomprehensible, and overly clinical. Nevertheless, unlike \"slip,\" \"parapraxis\" serves to retain the positive sense of _Leistung_ : something is being performed, although it runs counter to the performer's conscious intention. Because of the centrality of _The Standard Edition of the Complete Works of Sigmund Freud_ throughout this book, I prefer in general to use Strachey's terminology.\u2014Trans.\n\n See Leo Steinberg, _Other Criteria_ (Oxford: Oxford University Press, 1972), 82\u201391; Shiff, \"C\u00e9zanne's Physicality\"; Krauss, \"The Motivation of the Sign,\" 266\u201371; Christine Poggi, _In Defiance of Painting: Cubism, Futurism, and the Invention of Collage_ (New Haven, CT: Yale University Press, 1992), 58\u201389; Wolfram Pichler, \"Schminke\/Leinw and\/Caravaggio\/Goya\" (Ph.D. diss., University of Vienna, 1999), 107\u201329. Leo Steinberg's famous lecture, \"Picasso's Intelligence,\" which is fundamental in this context, has unfortunately still not been published.\n\n William Camfield, _Max Ernst: Dada and the Dawn of Surrealism_ , exhibition catalog (Museum of Modern Art, New York, 1993), 52ff.\n\n William Camfield, _Francis Picabia: His Art, Life, and Times_ (Princeton, NJ: Princeton University Press, 1979); Caroline Jones, \"The Sex of the Machine: Mechanomorphic Art, New Women, and Francis Picabia's Neurasthenic Cure,\" in _Picturing Science, Producing Art_, ed. Peter Galison et al. (New York: Routledge, 1998), 145\u201380.\n\n \"Ich lernte Francis Picabia 1917 anl\u00e4\u00dflich seines Besuches in Z\u00fcrich kennen. Er kam als Abgesandter der amerikanischen Dadaisten, um seine Kollegen in Z\u00fcrich zu begr\u00fc\u00dfen. Tristan Tzara und ich begaben uns, neugierig und bewegt, in sein Hotel. Wir fanden ihn besch\u00e4ftig beim Sezieren eines Weckers. Ich mu\u00dfte unwillk\u00fcrlich an die _Anatomie_ Rembrandts im Kunstmuseum von Amsterdam denken. Wahrlich, wir hatten einen gro\u00dfen Schritt vorw\u00e4rts in das Reich der Abstraktion getan. Erbarmungslos zerlegte er seinen Wecker bis auf die Uhrfeder, die er triumphierend extrahierte. F\u00fcr einen kurzen Augenblick unterbrach er seine Arbeit, um uns zu begr\u00fc\u00dfen. Doch ohne viel Zeit zu verlieren, versah er ein wei\u00dfes Papier mit den Abdrucken der R\u00e4dchen, Federn, Zeiger und anderen geheimen Teilchen der Uhr. Eifrig schlug er diese Dinge vom Stempelkissen auf das Papier wie ein pflichteifriger Postbeamter, verband diese Stempel miteinander durch Linien. . . .\" Arp, _Unsern t\u00e4glichen Traum_ , 63. Using the example of Picabia's Parisian works, in _The Artwork Caught by the Tail: Francis Picabia and Dada in Paris_ (Cambridge, MA: MIT Press, 2008), George Baker has recently called attention to the reflexive aspect of Dadaist negation.\n\n The German _Unterlage_ means \"base\" or \"substratum\" as well as, more generally, something that is laid underneath something else.\u2014Trans.\n\n Werner Spies, _Max Ernst Collagen: Inventar und Widerspruch_ (K\u00f6ln: DuMont Schauberg, 1974), 41ff.\n\n Since, in 1919, Duchamp's readymades were only known to a few people, predominantly American friends, the question is raised of how Max Ernst learned about this work. It was probably through Hans Arp, who had learned about it from Picabia. The exhaustive caption of fig. 8 suggests that Picabia had also told his Central European friends about Duchamp's _Large Glass_. It is well known that in the bachelor machine, illuminating gas circulates, solidifies, and dissipates in little spangles. In Ernst'sd machine, gas freezes, making little numbers grow: \"la canalisation de gaz frigorifi\u00e9 fait pousser de petits num\u00e9ros cr\u00e9pitauts \/ le c\u0153ur comprim\u00e9 s'est enfui \u00e0 temps \/ nous nous appuyons contre le laurier delphique\" (the canalization of refrigerated gas makes little sputtering numbers grow \/ the compressed heart has escaped in time \/ we lean against the delphic laurel). Incidentally, Evan Maurer has observed that a title from _Fiat Modes_ , _br\u00e4utliche mattung_ (bridal matting), alludes to _Large Glass_. Maurer, \"Images of Dream and Desire: The Prints and Collage Novels of Max Ernst,\" in _Max Ernst: Beyond Surrealism_ , exhibition catalog, ed. Robert Rainwater (New York: New York Public Library, 1986), 42; on artists' interest in illuminating gas and frozen gas, see Linda Dalrymple Henderson, _Duchamp in Context: Science and Technology in the Large Glass and Related Works_ (Princeton, NJ: Princeton University Press, 1998).\n\n Lucy Lippard, \"Dada into Surrealism: Notes on Max Ernst as Proto-Surrealist,\" _Artforum_ (September 1966): 13, Spies, _Max Ernst Collagen_ , 35ff.; Rosalind Krauss, \"Max Ernst: Speculations Provoked by an Exhibition,\" _Artforum_ 11 (May 1973): 37\u201340; Helmut R. Leppien, \"Ein Besuch bei Goltz und die Folgen,\" in _Max Ernst in K\u00f6ln: Die rheinische Kunstszene bis 1922_ , exhibition catalog, ed. Wulf Herzogenrath (Cologne: K\u00f6lnischer Kunstverein, 1980), 126\u201327; Maurer, \"Images of Dream and Desire,\" 41\u201345; Alfred M. Fischer, \"'Es werde Mode, die Kunst vergehe': 'Zu Fiat modes perat ars' von Max Ernst,\" in _Max Ernst: Druckgraphische Werke und illustrierte B\u00fccher_, exhibition catalog (K\u00f6ln: Museum Ludwig, 1990), 27\u201340; J\u00fcrgen Pech, \"Was der Taucher vor dem Sprung nicht wissen kann: Giorgio de Chirico und Max Ernst,\" in _Arnold B\u00f6cklin, Giorgio de Chirico, Max Ernst: Eine Reise ins Ungewisse_ , exhibition catalog, ed. Guido Magnaguago and Juri Steiner (Bern: Z\u00fcrich Kunsthaus, 1998), 289\u2013330.\n\n Also titled _allways the best man wins_.\n\n Louis Aragon, \"Max Ernst, peintre des illusions,\" in _Les Collages_ (Paris: Hermann, 1993).\n\n On Ernst's first Paris exhibition, see Ludger Derenthal, \"Eine surrealistische R\u00e9v\u00e9lation: Die erste Max Ernst-Ausstellung in Paris,\" in _Max Ernst: Das Rendezvous der Freunde_ , 55\u201374.\n\n Krauss, \"Max Ernst: Speculations Provoked by an Exhibition.\"\n\n On this symptomology of negated illusionism, see Ralph Ubl, \"Das Gem\u00e4lde als medialer Schwellenraum: Andr\u00e9 Breton, Giorgio de Chirico und der Gebrauch toter Bilder,\" in _Inszenierungen in Bild und Schrift_ , ed. Gerhard Neumann and Claudia \u00d6hlschl\u00e4ger (Bieleld: Aisthesis, 2004), 125\u201346; Ubl, \"Giorgio de Chirico: Exkarnation und Filiation der Malerei,\" in _Ikonologie des Zwischenraums: Der Schleier als Medium und Metapher_ , ed. Johannes Endres, Barbara Wittmann, and Gerhard Wolf (M\u00fcnchen: Fink, 2005), 385\u2013416.\n\n Louis Aragon, \"La peinture au d\u00e9fi\" (1930), in _Les Collages_ , 54.\n\n Spies, _Max Ernst Collagen_ , 57.\n\n \"Zapfenmuster.\" The German _Zapfen_ means \"cone\"; _Eiszapfen_ (literally \"ice-cone\") means \"icicle.\" So the cone shapes of the wallpaper pattern are related to the icicles of the work's title, _Eislandschaften, Eiszapfen und Gesteinsarten des weiblichen K\u00f6rpers_ ( _Frozen Landscapes, Icicles and Minerals Types of the Female Body_ ).\u2014Trans.\n\n Michel Foucault, _Death and the Labyrinth: The World of Raymond Roussel_ , trans. Charles Ruas (London: Continuum 2004), 95ff.\n\n Ibid., 56.\n\n Shiff, \"C\u00e9zanne's Physicality,\"; Shiff, \"Picasso's Touch,\" _Yale University Art Gallery Bulletin_ (1990): 39\u201347.\n\n The drawing was published in _Valori plastici_ (April\/May 1919) and became known to Ernst in September of that year. Spies, _Max Ernst Collagen_ , 35ff.; Leppien, \"Ein Besuch bei Goltz,\" 126ff.\n\n Max Ernst, who in fact _painted_ the naturalistic pear in figure 20, surely intended to deceive the viewer into taking the fruit for a cut-and-pasted readymade\u2014and he did successfully fool the authors of the catalogue raisonn\u00e9e. See Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 773. Moving from the history of artistic procedure under examination here to a microhistory, a specific context can be reconstructed for Ernst's watercolor imitation of a collaged pear. Ernst maintained a particularly close relationship with Paul and Gala \u00c9luard, who in turn knew de Chirico well enough to visit him in Rome. De Chirico even thought that the \u00c9luards supported his return to the old masters, and wrote Gala a letter (February 10, 1924) in which he shared with her his newest tempera recipe: \"Le tout melang\u00e9 ensemble forme une \u00e9mulsion parfaite. Avec cette \u00e9mulsion on broie les couleurs et on peint. On obtient une mati\u00e8re tr\u00e8s belle, pr\u00e9cieuse et lumineuse; conseillez-la \u00e0 M. Max Ernst.\" (The whole thing mixed together forms a perfect emulsion. You crush the colors and paint with this emulsion. A beautiful material is obtained, exquisite and luminous; recommend it to M. Max Ernst.) Joel de Sanna, \"Giorgio de Chirico: Letters to Andr\u00e9 and Simone Breton, to Gala and Paul \u00c9luard, Paul \u00c9luard\u2014J. T. Soby Correspondence,\" _Metafisica_ , nos. 1\/2 (2002), 62\u2013160. This request to pass the recipe along to Ernst (whom de Chirico did not know personally) is based on a misunderstanding: the surrealist painters did indeed experiment with different techniques (Ernst with collage, frottage, grattage, tracing, and many others), but the goal of these experiments was specifically to get away from the traditional artistic devices and techniques that de Chirico believed he had rediscovered. When Ernst paints a pear as naturalistically as possible in order to suggest to us that it was cut and pasted from an illustrated broadside, he is subtly leading the yearning for the old masters astray: because it stands in front of a frottage and a diagram, the painted fruit projects those features that make it most resemble a collaged fruit. It is well painted, and yet it looks as if it had been cut out and pasted in.\n\n Richard Wollheim, _Objekte der Kunst_ (Frankfurt am Main: Suhrkamp, 1982), 192\u2013210; Georges Didi-Huberman, _Devant l'image: Question pos\u00e9e aux fins d'une histoire de l'art_ (Paris: \u00c9ditions de Minuit, 1990).\n\n The prefix _zer-_ is used to denote that the action of a verb is destructive to the object. Etymologically related to _zwei_ (two), it carries the sense of breaking something into pieces. I use \"seeing apart\" as in \"breaking apart,\" \"tearing apart,\" but also because the object is seen apart from its original context.\u2014Trans.\n\n Aragon, \"Max Ernst, peintre des illusions,\" 25ff.; Aragon, \"La peinture au d\u00e9fi,\" 53ff.\n\n A _Vexierbild_ (literally \"vexing picture\") is a visual puzzle that conceals two distinct images within one set of contours, which, through a trick of perception, can be reversed.\u2014Trans.\n\n A semiologic reading for these non-metamorphic transitions and double meanings can be found in Yve-Alain Bois, \"Kahnweiler's Lesson,\" in _Painting as Model_ (Cambridge, MA: MIT Press, 1990); and Rosalind Krauss, _The Picasso Papers_ (London: Thames & Hudson, 1998).\n\n A frottage published in _Histoire naturelle_ in 1926, with the title _L'idole_ ( _The Idol_ ) (Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 808) may relate the topos of the depth of the imagination explicitly to de Chirico and thus again bring attention to Ernst's separation from de Chirico's art. Grapes can be considered the idol of painting\u2014ever since Zeuxis' masterpiece, they have stood for the deceptive power of representation. In the frottage, they are presented as a fossil imprint. With these subterranean grapes, Ernst is evidently alluding to de Chirico's text \"Zeusi l'esploratore\" (\"Zeuxis the Explorer\"), published in 1918 in _Valori plastici_. The first sentence ends with \"i nuovi Zeusi partono soli alla scoperta delle curiosit\u00e0 che s'annidano come talpe su per tutta la crosta del globo terracqueo\" (the new Zeuxi leave on their own to explore the curiosities which lodge themselves, like moles, beneath the carapace of the earth). The discoveries that the new Zeuxi make underground bring to light a world full of eyes: \"Bisogna scoprire l'occhio in ogni cosa.\" (We have to discover the eye in every object.) Eyes are one of the most frequently occurring motifs in Ernst's frottages. It is likely no accident that for an underlay, Ernst chose the rough surface of an oil painting (for the grapes and the dark outer strips) and the fabric from a canvas (for the central rectangle): in the media of traditional painting are concealed unknown, \"subterranean\" pictorial possibilities, which the subversive workings of the surrealist imagination make manifest.\n\n See Werner Spies, _Frottages_ (London: Thames and Hudson, 1968); William S. Rubin, _Dada und Surrealismus_ (Stuttgart: Hatje, 1972); Hans Holl\u00e4nder, \"Ars inveniendi et investigandi: Zur surrealistischen Methode,\" _Wallraf-Richartz-Jahrbuch_ 32 (1970): 9\u201320; Lichtenstern, _Metamorphose_ , 155\u201367; Andreas Vowinckel, _Surrealismus und Kunst: Studien zu Ideengeschichte und Bedeutungswandel des Surrealismus vor Gr\u00fcndung der surrealistischen Bewegung und zu Begriff, Methode und Ikonographie des Surrealismus in der Kunst 1919 bis 1925_ (Hildesheim: Olms, 1989), 270ff.; Elizabeth Legge, \"Zeuxis's Grapes, Novalis's Fossils, Freud's Flowers: Max Ernst's Natural History,\" _Art History_ 16 (1993): 147\u201372; Ralf Konersmann, \"Max Ernst und die Idee der Naturgeschichte,\" in _DieErfindung der Natur: Max Ernst, Paul Klee, Wols und das surreale Universum_, exhibition catalog, ed. Karin Orchard and J\u00f6rg Zimmermann (Hannover: Sprengel-Museum, 1994), 159\u201367.\n\n Spies, _Max Ernst Collagen_ , 124.\n\n See the observations of Lichtenstern, _Metamorphose_ , 158; Ludger Derenthal and J\u00fcrgen Pech, _Max Ernst_ (Paris: Nouvelles Editions Fran\u00e7aises, Casterman, 1992), 105.\n\n See Wollheim, _Objekte der Kunst_ , 192\u2013210, for his critique of Gombrich ( _Art and Illusion_ , 5).\n\n _Schachtelhalm_ , literally \"box rush.\" The scouring rush ( _Equisetum_ ) is the last surviving genus of the class _Equisetopsida_ , which was prevalent in late Paleozoic forests, where some grew as tall as large trees.\u2014Trans.\n\n The fundamental insight formulated by Werner Hofmann in his classic essay \"Ars Combinatoria,\" _Jarbuch der Hamburger Kunstsammlungen_ 21 (1976): 7\u201330\u2014that diagram and blot are not necessarily contrary, but occur in many forms of convergence and hybridization\u2014underlies these observations. Also, the integration of the cut, as Werner Spies first presented it ( _Max Ernst Collagen_ , 93), is prepared for in the photo- and woodcut collages.\n\n \"Im Simulakrum gewinnt die Darstellung, die in der Mimesis die Transparenz auf einen ideellen Tiefenraum ist, eine eigene Dichte; das Tote (bzw. Abwesende) wird lebendig, erlangt den Status einer Pr\u00e4senz, ohne jedoch seinen irrealen Charakter g\u00e4nzlich abzustreifen. Die Mimesis trennt die Aktualit\u00e4t des Betrachters von der Virtualit\u00e4t des Betrachteten durch die durchsichtige, aber nichtsdestoweniger rigoros unterscheidende Grenze des Darstellungsmediums. Im Simulakrum hingegen verdichtet sich diese Grenze zu einer aufdringlichen Pr\u00e4senz, die weder blo\u00df wirklich noch blo\u00df unwirklich (ideell) ist, zu einer Wirklichkeit der Darstellung, die einen unaufl\u00f6slich ambigen, oft gespensterhaften Charakter aufweist.\" David Wellbery, \"Verzauberung: Das Simulakrum in der romantischen Lyrik,\" in _Mimesis und Simulation_ , ed. Andreas Kablitz and Gerhard Neumann (Freiburg: Rombach, 1998), 452.\n\n \"Tromple l'oeil, by suppressing the distance between model and copy, by suspending the referential relation, traps the perceiving eye in what I shall call essence-appearance, in apparition, and it delivers the body-eye over to fascination with the double, to stupefaction; by the same token, the effect is not one of contemplation and theory, a return to the serene and peaceful truth of representation, the truth of the thing absent from the real\u2014hence the reality effect of all representation\u2014but of surreality, an impure blend of anguish and shock: the effect of presence.\" Louis Marin, \"Representation and Simulacrum,\" in _On Representation_ , trans. Catherine Porter (Stanford, CA: Stanford University Press, 2001), 316.\n\n On the anachronism of the imprint in general, see Georges Didi-Huberman, _L'Empreinte_ , exhibition catalog (Paris: Centre Georges Pompidou, 1995).\n\n \"In 1870 Duret stated that Manet brings back from the vision he casts on things an impression truly his own. This mobile eye then reflects back to record an image on its own light-sensitive and impressionable surface. Such an eye tracks and traces itself, seeing. Its vision is an image of its own individuality. Now, in the act of painting, the hand substitutes for this eye. It traces out a path of visible form as if it were an eye cast onto a visual field. According to the impressionist model, the hand responds immediately to the eye and shares all its idiosyncrasies (the body, including hand and eye, must have organic integrity). Just as the eye records its own vision, so the hand traces itself, tracing; indexically, it reveals its own character. Such an act of painting resolves the problem [. . .] of self-representation by turning outward toward nature, allowing nature to activate an impression that can only be one's own. The subject knows itself, indexes itself, only through a resisting object to which eye and hand respond.\" Shiff, \"Performing an Appearance,\" 108.\n\nCHAPTER TWO\n\n Werner Spies, _Max Ernst, Loplop: The Artist in the Third Person_ (New York: G. Braziller, 1983).\n\n Here, I am following the accounts of Laurent Jenny, _La parole singuli\u00e8re_ (Paris: Belin, 1990); and Vincent Kaufmann, _Po\u00e9tique des groupes litt\u00e9raires (Avantgardes 1920\u20131970)_ (Paris: Presses universitaires de France, 1970).\n\n Isidore Ducasse (Comte de Lautr\u00e9amont), _Po\u00e9sies and Complete Miscellanea_ , trans. Alexis Lykiard (London: Allison & Busby, 1978), 75.\n\n Andr\u00e9 Breton, _Manifesto of Surrealism_ , in _Manifestoes of Surrealism_ , trans. Richard Seaver and Helen R. Lane (Ann Arbor: University of Michigan Press, 1969), 26, 33.\n\n Besides the initial work, _The Magnetic Fields_ ( _Les Champs magn\u00e9tiques_ ) (1919), these include the texts dictated by Robert Desnos while half asleep during the fall of 1922; Breton's collection _Soluble Fish_ ( _Poisson soluble_ ), published in 1924; Michel Leiris's volume _Simulacrum_ ( _Simulacre_ ); several dozen shorter attempts by Aragon, Morise, Artaud, Leiris, and others, which were published in the columns of _La R\u00e9volution surr\u00e9aliste_ ; and finally in 1930, Breton and \u00c9luard's _Immaculate Conception_ ( _L'Immacul\u00e9e conception_ ). See Claudio Abastado, \"\u00c9criture automatique et instance du sujet,\" _Revue des Sciences Humaines_ 184 (1981): 59\u201375; Michel Murat, \"Jeux de l'automatisme,\" in _Une pelle au vent dans les sables du r\u00eave: Les \u00e9critures automatiques_ , ed. Michel Murat and Marie-Paul Berranger (Lyon: Presses Universitaires de Lyon, 1992), 15\u201325; Michel Murat, \"Les Lieux communs de l'\u00e9criture automatique,\" _Litt\u00e9rature moderne_ 1 (1988): 123\u201335; Marie-Paul Berranger, \"Poisson soluble ou Les Mains vierges dans le petite niche \u00e0 fond bleu du travail,\" in _Une Pelle au vent_ , ed. Murat and Berranger, 93\u2013111; and Jenny, _La parole singuli\u00e8re_.\n\n Margaret Cohen, _Profane Illumination: Walter Benjamin and the Paris of Surrealist Revolution_ (Berkeley: University of California Press, 1993), 77\u2013172.\n\n \"finalit\u00e9 communautaire\"; Kaufmann, _Po\u00e9tique des groupes litt\u00e9raires_.\n\n Werner Spies and G\u00fcnter Metken, _Max Ernst: \u0152uvre-Katalog_ (K\u00f6ln: M. DuMont Schauberg, 1975), nos. 564ff. The suggested date \"ca. 1923,\" in light of the following entry from the logbook at surrealist headquarters, should be adjusted to 1924. Paule Th\u00e9venin, ed., _Bureau de recherches surr\u00e9alistes: Cahier de la permanence octobre 1924\u2013avril 1925_ (Paris: Gallimard, 1988), 29: \"Mercredi 22 octobre 1924. Permanence: Simone [Breton] J. A. Boiffard [. . .] 5-3\/4 Max Ernst apporte un dessin fait selon la m\u00e9thode surr\u00e9aliste sur un rouleau \u00e9troit et tr\u00e8s long qui se d\u00e9roule. Il fait remarquer l'abrutissement progressif.\" (Wednesday, October 22, 1924. On duty: Simone [Breton] J. A. Boiffard [. . .] 5-3\/4 Max Ernst brings in a drawing, made according to the surrealist method, on a narrow, very long, unrolling scroll. He points out the increasing stupefaction.) In general, on Ernst's artistic crisis shortly after the publication of Breton's _Manifesto of Surrealism_ , see Dawn Ades, \"Between Dada and Surrealism: Painting in the Mouvement Flou,\" in _In the Mind's Eye: Dada and Surrealism_ , ed. Terry Ann R. Neff, exhibition catalog (Chicago: Museum of Contemporary Art, Chicago), 23ff.\n\n Werner Spies, _Max Ernst Collagen: Inventar und Widerspruch_ (K\u00f6ln: DuMont Schauberg, 1974), 114\u201324.\n\n Rosalind Krauss, \"Max Ernst: Speculations Provoked by an Exhibition,\" _Artforum_ 11 (May 1973): 37\u201340.\n\n There is a rich body of literature concerning this discussion, in particular Robert Desnos, \"Surr\u00e9alisme,\" _Cahiers d'Art_ 1 (1926): 210\u201313; Rosalind Krauss, \"Photographic Conditions of Surrealism,\" in _The Originality of the Avant-Garde and Other Modernist Myths_ (Cambridge, MA: MIT Press, 1985), 196\u2013209; Werner Spies, _Max Ernst Frottagen_ (Stuttgart: Hatje, 1968); Evan Maurer, \"Images of Dream and Desire: The Prints and Collage Novels of Max Ernst,\" in _Max Ernst: Beyond Surrealism_, exhibition catalog, ed. Robert Rainwater (New York: New York Public Library, 1986), 55\u201362; Werner Spies, _Loplop\u2014Die Selbstdarstellung des K\u00fcnstlers_ (M\u00fcnchen: Prestel-Verlag, 1982); Jennifer Mundy, \"Surrealism and Painting: Describing the Imaginary,\" _Art History_ 10 (1987): 123\u201335; Andreas Vowinckel, _Surrealismus und Kunst: Studien zu Ideengeschichte und Bedeutungswandel des Surrealismus vor Gr\u00fcndung der surrealistischen Bewegung und zu Begriff, Methode und Ikonographie des Surrealismus in der Kunst 1919 bis 1925_ (Hildesheim: Olms, 1989), 33ff., 310\u201316; Norbert Bandier, _Sociologie du Surr\u00e9alisme_ (Paris: La Dispute, 1999); Ralph Ubl, \"Das Gem\u00e4lde als medialer Schwellenraum: Andr\u00e9 Breton, Giorgio de Chirico und der Gebrauch toter Bilder,\" in _Inszenierungen in Bild und Schrift_ , ed. Gerhard Neumann and Claudia \u00d6hlschl\u00e4ger (Bieleld: Aisthesis, 2004).\n\n In the second issue of _La R\u00e9volution surr\u00e9aliste_ (1925, p. 2), Breton complains about the salability of painting, which makes it useless for the revolutionary ethic of surrealism: \"Est-il juste, par exemple, qu'\u00e0 talent \u00e9gal, les peintres s'enrichissent sur le sol m\u00eame o\u00f9 les po\u00e8tes pourraient mendier?\" (Is it right, for example, that with equal talent, painters thrive in the same soil where poets would have to beg?) Despite his unbridled iconophilia, this injustice would always hamper Breton's relations with painting. On May 18, 1926, Breton and Aragon launched a protest against Ernst and Mir\u00f3's collaboration with the Russian ballet. The painters' names were cleared in the next issue of _La R\u00e9volution surr\u00e9aliste_ , through the intervention of Paul \u00c9luard.\n\n \"On one hand, in automatic writing, it is not, strictly speaking, the word that becomes free; rather, the word and my freedom are now no more than one. I slide into the word, it keeps my imprint, and it is my imprinted reality; it adheres to my non-adherence. But on another side, this freedom of words means that words become free for themselves: they no longer depend exclusively on things that they express, they act on their own account, they play, and as Breton says, 'they make love.'\" Maurice Blanchot, _The Work of Fire_ , trans. Charlotte Mandell (Stanford, CA: Stanford University Press, 1995), 88.\n\n \"With a pencil and a white sheet of paper to hand, I could easily trace their outlines. Here again it is not a matter of drawing, _but simply of tracing_.\" Breton, _Manifesto of Surrealism_ , 21.\n\n _La R\u00e9volution surr\u00e9aliste_ 1 (December 1, 1924): 26\u201327. That doubts about the authentic and direct expression of the unconscious are openly raised in the discussion of visual art per se can be attributed to a poetological difficulty considered to be the unique deficiency of visual art: the insufficient plasticity and material resistance of its medium. In the third issue of _La R\u00e9volution surr\u00e9aliste_ (April 15, 1925, p. 27), Pierre Naville poses the rhetorical question, \"Comment se fait-il que ce qu'on nomme la litt\u00e9rature s'alimente presqu'uniquement de l'amour, et que les mots trouvent si facilement leur compte dans cet abandon, tandis que les arts plastiques en sont sevr\u00e9s, ou qu'il n'y transpara\u00eet que voil\u00e9 d'une fa\u00e7on tr\u00e8s ambigu?\" (How is it that what we call \"literature\" feeds almost exclusively on love, and that words so easily get their due in this abandonment, whereas the visual arts have been weaned of love, or it does not show through in them except when veiled in ambiguity?) As we have seen, the problem of transparency is no less pressing in the case of poetry. However, it seems as if this problem is explicitly articulated in programmatic texts on visual art, whereas in poetic texts, its effect is only subliminal, eliciting compensation strategies and compelling the radicalization that culminates in the assumption of a totally mortified language as the precondition for its redemption. The discussion surrounding the possibility of a surrealist painting brings to light what the surrealist program as a rule suppressed, though its metaphorics were animated by it: the importance of the artistic profession, which is positioned between psychic automatism and its artistic expression. On the sociological aspects of the relationship between surrealist poets and artists, see Bandier, _Sociologie du Surr\u00e9alisme_.\n\n \"Les formes et les couleurs se passent d'objet, s'organisent selon une loi qui \u00e9chappe \u00e0 toute pr\u00e9m\u00e9ditation, se fait et se d\u00e9fait dans le m\u00eame temps qu'elle se manifeste.\" Max Morise, \"Les yeux enchant\u00e9s,\" _La R\u00e9volution Surr\u00e9aliste_ 1 (December 1, 1924): 27.\n\n Spies, _Loplop_. The reference is to Rimbaud, \"La main \u00e0 plume vaut la main \u00e0 charrue.\" (The hand that writes is as good as the hand that plows.) Arthur Rimbaud, _A Season in Hell and Illuminations: Poems by Arthur Rimbaud_, trans. Bertrand Mathieu (Rochester: BOA Editions, 2000), 6\u20137.\u2014Trans.\n\n Ralf Konersmann describes the frottages as \"imagination machines,\" in \"Max Ernst und die Idee der Naturgeschichte,\" in _Die Erfindung der Natur: Max Ernst, Paul Klee, Wols und das surreale Universum_ , exhibition catalog, ed. Karin Orchard and J\u00f6rg Zimmermann (Hannover: Sprengel-Museum, 1994), 159.\n\n For ideological and sociological foundations, see Dario Gamboni, _The Brush and the Pen: Odilon Redon and Literature_ , trans. Mary Whittall (Chicago: University of Chicago Press, 2011).\n\n See the investigations of Holl\u00e4nder, \"Ars inveniendi et investigandi: Zur surrealistischen Methode,\" _Wallraf-Richartz-Jahrbuch_ 32 (1970); Elizabeth Legge, \"Zeuxis's Grapes, Novalis's Fossils, Freud's Flowers: Max Ernst's Natural History,\" _Art History_ 16 (1993): 147\u201372; and J\u00f6rg Zimmermann, \"Philosophische Horizonte der Histoire naturelle von Max Ernst,\" in _Die Erfindung der Natur_ , ed. Orchard and Zimmerman, 15\u201324.\n\n Spies, _Max Ernst Collagen_ , 46\u201348; Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , nos. 321\u201324.\n\n Friedrich Kittler, _Grammophone, Film, Typewriter_ , trans. Geoffrey Winthrop Young and Michael Wutz (Stanford, CA: Stanford University Press, 1999), 183\u2013266.\n\n Between the frottaged coins and the surrealist frottages, Ernst used this technique in rather incidental works. Spies and Metken, _Max Ernst: \u0152vre-Katalog_ , nos. 433ff., 593, 596. During his vacation in the Tirol in 1921, he frottaged a bull's head and its base (wood grain was utilized). It was surely no accident that the artist first tried frottage in the context of a naturalistic image while at a geographic remove from the urban centers of Dadaism. On the Dadaist turn to nature in the Tirol, see Raoul Schrott, _Dada 21\/22: Musikalische Fischsuppe mit Reiseeindr\u00fccken: Eine Dokumentation \u00fcber die beiden Dadajahre in Tirol_ (Innsbruck: Haymon, 1988).\n\n On the anti-architectural, see Denis Hollier, _La Prise de la Concorde_ (Paris: Gallimard 1974).\n\n Andr\u00e9 Masson, _Le Rebelle du surr\u00e9alisme: \u00c9crits_ (Paris: Hermann, 1976), 37: \"Mat\u00e9riellement: un peu de papier, un peu d'encre. Psychiquement: il faut faire le vide en soi; le dessin automatique prenant sa source dans l'inconscient, doit appara\u00eetre comme une impr\u00e9visible naissance. Les premi\u00e8res apparitions graphiques sur le papier sont geste pur, rythme, incantation, et comme r\u00e9sultat: pur _gribouillis_. C'est la premi\u00e8re phase. Dans la seconde phase, l'image (qui \u00e9tait latente) r\u00e9clame ses droits. Quand l'image est apparue, il faut s'arr\u00eater. Cette image n'est qu'un vestige, une trace, une \u00e9pave. [. . .] Essayons de conclure par cette d\u00e9finition: l'image prend naissance dans l'oc\u00e9an \u00e9motionnel et \u00e0 la fin s'y d\u00e9verse ou, comme il vous plaira: y retourne.\" (Materially: a bit of paper and ink; psychically, you have to empty your mind. Automatic drawing, taking its source from the unconscious, must appear like an unforeseeable birth. The first graphic manifestations on the paper are pure gesture, rhythm, incantation\u2014in effect, pure _scrawl_. That is the first phase. In the second phase, the image (which was latent) reclaims its rights. When the image has become visible, stop. This image is only a vestige, a trace, a hull. . . . Let us attempt to conclude with this definition: the image is born in an ocean of emotions and ultimately flows back into it or, if you will, returns to it.)\n\n Holl\u00e4nder, \"Ars inveniendi et investigandi.\"\n\n \"The ground beneath my feet is nothing but an enormous unfolded newspaper. Sometimes a photograph comes by; it is a nondescript curiosity, and from the flowers there uniformly rises the smell, the good smell, of printers' ink. I heard it said in my youth that the smell of hot bread is intolerable to sick people, but I repeat that the flowers smell of printers' ink. The trees themselves are only more or less interesting minor news items: a fire here, a derailment there.\" Breton, _Soluble Fish_, in _Manifestoes of Surrealism_ , 60. See Elisabeth Lenk, _Der Springender Narzi\u00df: Andr\u00e9 Breton poetischer Materialismus_ (Munich: Rogner & Bernhard, 1971), 58ff., 71ff.\n\n \"At the bottom of the fourth page the newspaper has an unusual fold that I can describe as follows: it looks as if it has been wrapped around a metallic object, judging by a rusty spot that might be a forest, and this metallic object might be a weapon of an unfamiliar shape, akin to the dawn and a large Empire bed.\" Breton, _Soluble Fish_ , 60.\n\n Even before Ernst and Breton, Louis Aragon celebrated seeing-in as a surrealist activity: _Anicet ou le panorama_ (Paris: Gallimard, 1921), 48ff.; _Le Paysan de Paris_ (Paris: Gallimard, 1926), 15, 19, and elsewhere; for background, see Jean-Claude Lebensztejn's fundamental study, _L'Art de la tache: Introduction \u00e0 la Nouvelle m\u00e9thode d'Alexandre Cozens_ (Mont\u00e9limar: Editions du Limon, 1990).\n\n The title refers to a canonical text of the surrealists, Alfred Jarry's _Caesar Antichrist_. The title character is a furious demiurge who, in the taunting creation of all possible worlds, acts like the divine model of P\u00e8re Ubu. On _Caesar's Palette_ lies a leaf, which has left a hole where it just was. Frottage becomes a tool of Jarry's annihilating world-maker, who paints not with pigments but with fragments of reality. Thus, the image as a whole not only _shows_ Caesar's palette; insofar as the title can be understood metonymically, it has been produced _with_ Caesar's palette. That it resembles canvas and that a canvas in fact forms its basis as the underlay can again be read as an appropriation of the traditional painter's paraphernalia, which have now been placed in service to the reality-annihilating art of surrealism. Jarry, _Les Minutes de sable m\u00e9morial: C\u00e9sar Antechrist_ (Paris: Fasquelle 1932); in Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 872, see the related frottage that is also named after one of Jarry's heroes: _Faustroll's Palette_ ; see Zimmermann, \"Philosophische Horizonte,\" 23.\n\n Laurent Jenny, \"L'automatisme comme mythe rh\u00e9torique,\" in _Une pelle au vent_ , ed. Murat and Berranger, 29; Jenny, _La parole singuli\u00e8re_ , 150\u201352.\n\n \"Moreover, the very idea of 'three kingdoms'\u2014animal, vegetable, and mineral\u2014is the height of absurdity. If a phyllium alights on a branch, who can be sure that it is not a leaf from the tree that flies away a little later, leaving the leaf-insect in its place?\" Andr\u00e9 Breton, _Surrealism and Painting_ , trans. Simon Watson Tyler (Boston: MFA Publications, 1972), 44\u201346.\n\n Breton, _Manifesto of Surrealism_ , 21.\n\n Ibid., 40.\n\n See Thomas M. Scheerer, _Textanalytische Studien zur \"\u00c9criture automatique\"_ (Bonn: Romanisches Seminar der Universit\u00e4t, 1974), 104ff.; and Michael Riffaterre, _Text Production_ , trans. Terese Lyons (New York: Columbia University Press, 1983), chap. 14.\n\n \"I can see Andr\u00e9 Breton calmly making a daily tableau from a fantastic quest. I wandered with him through a reinvented zoology, a reinvented botany. We stopped in front of the cages, we glanced into the greenhouses.\" Louis Aragon, _Treatise on Style_ , trans. Alyson Waters (Lincoln: University of Nebraska Press, 1991), 104\u20135.\n\n Famously argued by Karl Heinz Bohrer, \"Naturgef\u00fchl ist kein Gef\u00fchl der Natur: Die surrealistische Evokation der Natur mit R\u00fccksicht auf das romantische Erhabene,\" in _\u00c4sthetik undNaturerfahrung_, ed. J\u00f6rg Zimmerman et al. (Stuttgart\u2013Bad Cannstatt: Frommann-Holzboog, 1996), 418\u201340.\n\n Andr\u00e9 Breton, _\u0152uvres compl\u00e8tes_ , ed. Marguerite Bonnet et al. (Paris: Gallimard, 1988\u201399), 1:340; 2:341\u201392.\n\n \"Little by little, fitfully, the world dawns on me, which does not mean that it is _given_ ; on the contrary, _I gave it to myself_ , by choosing my own point of departure as the mathematician does the initial postulate. Its necessity is born of me. Thus, all nature is my contraption: what I don't know about it, assuming such ignorance to be conceivable, lies buried in my unconscious, again like the mathematician who may found his science in one stroke and yet remain unaware of its inevitable consequences. Sensual experience strikes me, then, as the mechanism of awareness, which gives you some notion of what nature becomes: nature is my unconscious. . . . But for brief instants at most, on rare thresholds, I recognize the bond uniting the data of my senses, some of them, with nature itself, with the unconscious. The exquisite awareness of a fugitive presence is the shiver of which I spoke earlier.\" Louis Aragon, _Nightwalker_ ( _Le Paysan de Paris_ ), trans. Frederick Brown (Englewood Cliffs, NJ: Prentice Hall, 1972), 101\u20132.\n\n Breton, _\u0152uvres_ , 1:1122.\n\n Ibid., 1:369\u201372. On the rich metaphorics of water in surrealism, see Michel Beaujour, \"De l'Oc\u00e9an au Ch\u00e2teau: Mythologie surr\u00e9aliste,\" _French Review_ 42 (1969): 353\u201370; Nicole Boulestreau, _La po\u00e8sie de Paul Eluard_ (Paris: Klincksieck, 1985), 77\u201381; Abastado, \"\u00c9criture automatique et instance du sujet\"; Murat, \"Jeux de l'automatisme\" and \"Les Lieux communs de l'\u00e9criture automatique\"; Berranger, \"Poisson soluble ou les Mains vierges\"; Michael Riffaterre, \"The Surrealist Libido, Andr\u00e9 Breton's _Poisson soluble_ ,\" _Dada\/Surrealism_ 17 (1988): 59\u201360; and Julia Kristeva, \"L'inqui\u00e9tante \u00e9tranget\u00e9 de l'automatisme: A propos de Poisson soluble,\" in _Une pelle au vent_ , ed. Murat and Berranger, 113\u201324.\n\n An iconographic reading of the circle motif, which does not consider its narrative function, can be found in Legge, \"Zeuxis's Grapes, Novalis's Fossils, Freud's Flowers,\" 149ff.\n\n Breton, _\u0152uvres_ 1:340; on the picture within the picture as wish fulfillment, see Spies, _Loplop_ , 35ff.\n\n Wood grain recurs most frequently as a material underlay. It connects the atmospheric phenomenon of the beginning of the world (prints 1\u20133) with Leonardo da Vinci's stained wall (print 21), forms a threatening shadow (print 12), is capable of representing the expanse of the earth's still unpopulated surface (print 3) or the Pampas (print 6) as well as the veins of leaves (prints 13, 18), and\u2014of course?\u2014a wooden board (prints 23, 26, 27). A coarse canvas netting yields a bread crust (print 23), a reptile's carapace (prints 27, 28, 30), and rock (print 29), as well as the hide of an animated horse (print 32). A diamond pattern, still unidentified, alternatively stands in for the surface of a scouring rush (print 7) and for the elongated swells of the primordial sea (prints 9, 22, 24, 30). An intricate snakeskin serves for the scales of prehistoric plants (print 17) and for a stone substratum (print 29). A radial pattern makes light phenomena (print 2), ice flowers (print 4), or eyes (prints 30, 31). Finally, the comb from print 1 finds a new, more suitable application in print 29, in which it is used to form the regular curves between the eyelashes. On the different textures, see also Werner Spies, _Frottages_ (London: Thames and Hudson, 1968).\n\n The title _The Vaccinated Bread_ ( _Le Pain vaccin\u00e9_ ) may well mean that the bread is immune from mold\u2014a preserved, rock-hard piece of bread, a bread petrifact.\n\n See Wolfgang Kemp, _Der Anteil des Betrachters: Rezaptions\u00e4sthetische Studien zur Malerei des 19. Jahrunderts_ (Munich: M\u00e4ander, 1983), 52; and Joseph Leo Koerner, _Caspar David Friedrich and the Subject of Landscape_ (London: Reaktion Books, 2009).\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 681 and elsewhere.\n\n This observation is made by J\u00fcrgen Pech, _Max Ernst, \"Histoire Naturelle\": Frottagen_ , exhibition catalog (Br\u00fchl: Max-Ernst-Kabinett, 1983), 388, though he thinks wood grain has been used as the underlay.\n\n On surrealist metaphorics, particularly the extended metaphor, see Riffaterre, \"The Extended Metaphor in Surrealist Poetry,\" in _Text Production_ , chap. 12.\n\n Andr\u00e9 Breton, _Mad Love_ , trans. Mary Ann Caws (Lincoln: University of Nebraska Press, 1988), 25.\n\n The concluding figure of Eve distinguishes Ernst's series from its most important model in the art of the nineteenth century, Odilon Redon's eight-part series _The Origins_ ( _Les Origines_ ). Ernst makes many allusions to Redon's lithographs. He certainly was familiar with the symbolist iconography of the single eye\u2014for example, one could compare the knothole eye in print 18 with Redon's illustrations of Poe's \"The Tell-Tale Heart\" or the heavenly body in Redon's _A Drowned Woman_ ( _Une noy\u00e9e_ ) with print 2 of _Natural History_. The series as a whole might also be compared to _The Origins_ , in which imaginative seeing and the Romantic conception of a history of nature as the history of the subject are likewise brought into connection. Redon's nature also exclusively produces monsters, which emerge from a prehistoric darkness coinciding with the black color characteristic of his art. From out of the suggestive indeterminacy of the first print, the viewer is met by seed-shaped eye creatures that form themselves into a dog-like figure, so slowly and laboriously that the viewer finds it difficult to traverse the space of this primordial soup world. The gaze, whose wondrous generation Redon documents in the following prints of his natural history\u2014the flower's skyward aspect and Polyphemus, the satyr's cynical expression and the female centaur's precise line of sight as she shoots a snake\u2014is in danger by the end of the series of being engulfed in darkness again. Pegasus stares helplessly at the sky and flails his front legs; his hindquarters remain stuck to the ground, where the dirty shadow of his wings keeps him firmly anchored. In the last print\u2014a hunched man stumbles into brightness\u2014it is undecided whether the light means release or blinding dazzlement. This final turn is possibly answered by that of _Natural History_ , which replaces the bedazzled Adam with the _R\u00fcckenfigur_ of Eve. Already in the previous print, the surrealist Pegasus (fig. 32) is not stuck to the earth but rises into the air above the stony ground. While Redon does not think that the organic, which in his pictures emerges from a black ooze, can possibly have a happy future, the surrealist would like to make way for one, specifically through the inorganic ruins of an extinct nature. On Redon's _The Origins_ , see Sven Sandstr\u00f6m, _Le monde imaginaire d'Odilon Redon: \u00c9tude iconologique_ (Lund: G. Wittenborn, 1955); and Douglas Druick and Peter Kort Zegers, \"In the Public Eye,\" in _Odilon Redon, 1840\u20131916_ , exhibition catalog (London: Royal Academy of Arts, 1994), 158ff. The album was originally not numbered or titled and was published in an edition of only twenty-five copies. Not until 1913 did Redon establish the title and sequence. See Ted Gott, _The Enchanted Stone: The Graphic Worlds of Odilon Redon_ , exhibition catalog (Melbourne: National Gallery of Victoria, 1990), 67.\n\n Andr\u00e9 Breton, _Nadja_ , trans. Richard Howard (New York: Grove; London: Evergreen, 1960), 27; see Marguerite Bonnet, _Andr\u00e9 Breton: Naissance de l'aventure surr\u00e9aliste_ (Paris: Corti, 1975), 189\u201391; and Georges Raillard, \"On signe ici,\" _Litt\u00e9rature_ 25 (February 1977): 12ff. Attention should also be drawn to Breton's prophetic interpretation of his own poetry in _Mad Love_.\n\n Michael Riffaterre, \"Ekphrasis lyrique,\" in _Lire le regard: Andr\u00e9 Breton et la peinture_ , ed. Jacqueline Ch\u00e9nieux-Gendron (Paris: Lachenal & Ritter, 1993).\n\n Spies, _Frottages_.\n\n English title in _Beyond Painting: Iceflower Shawl and Gulf Stream_.\n\n English title in _Beyond Painting: The Chestnut Trees Take Off_.\n\n _R\u00e9volution surr\u00e9aliste_ 7 (June 15, 1926). Arp himself translated the French version into German, in the course of which he reworked the text and made it roughly twice as long. The English translation by Ralph Manheim cited below, \"introduction to max ernst's _natural history_ ,\" was evidently made from the German version. In Max Ernst, _Beyond Painting and Other Writings_ (New York: Schultz, 1948), 124\u201325.\n\n \"la m\u00e9daille se l\u00e8ve tandis que le soleil, apr\u00e8s cinquante ans de service, se retire dans les roues calcin\u00e9es de la lumi\u00e8re. c'est l'homme qui a remplac\u00e9 les r\u00e9veille-matin par les tremblements de terre, les averses de drag\u00e9es par des averses de gr\u00eale. l'ombre de l'homme rencontrant celle d'une mouche cause une inondation. c'est l'homme aussi qui a appris aux chevaux \u00e0 s'embrasser comme des pr\u00e9sidents.\" (fifty suns and fifty medals rise while the pseudo-sun after fifty years of service retires into the calcinated wheels of light. . . . it is man who replaced alarm clocks by earthquakes showers of jordan almonds by showers of hail. the shadow of man encountering the shadow of a fly causes a flood. Thus it is man who has taught horses to embrace one another like presidents. . . .)\n\n \"les \u00e9pouvantails portant dans leurs boutonni\u00e8res des volcans et geysers.\"\n\n \"comme un lion qui flaire f\u00e9rocement un succulent couple de jeunes mari\u00e9s, le tilleul pousse docilement sur les plaines planchei\u00e9es.\" \"Tractably\" is R. Manheim's translation; Arp says the lime tree grows \"docilement\" in reference to the title of the print, _le tilleul est docile_ ( _The Lime Tree Is Docile_ ).\u2014Trans.\n\n The French version is shorter: \"les feuilles ne poussent jamais sur les arbres, comme une montagne vue \u00e0 vol d'oiseau elles n'ont pas de perspective. le spectateur se trouve toujours dans une position fausse devant une feuille.\"\n\n \"avec ces onze queues et demie l'homme compte dix objets et demi dans la chambre meubl\u00e9e de l'univers.\"\n\n \"a dancer's calf in the ecclesiastical ballet.\"\n\n \"voici le coucou l'origine de la pendule le bruit de ses m\u00e2choires ressemble \u00e0 celui d'une forte chute de cheveux.\"\n\n \"du temps de la r\u00e9colte des diamants conjugaux on rencontre sur les mers d'immenses armoires \u00e0 glace flottant sur leurs dos.\"\n\n Kaufmann, _Po\u00e9tique des groupes litt\u00e9raires_.\n\n Breton, _Manifesto of Surrealism_ , 35; Jenny, _La parole singuli\u00e8re_ , 149ff.\n\n On the history and poetics of natural science illustrations, see Martin J. S. Rudwick's fundamental study, _Scenes from Deep Time: Early Pictorial Representations of the Prehistoric World_ (Chicago: University of Chicago Press, 1992).\n\n On education and the historical turn to objectivity as a value of scientific illustrations in the nineteenth century, see Lorraine Daston and Peter Gallison, \"The Image of Objectivity,\" _Representations_ 40 (1992): 81\u2013128.\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 839.\n\n Rudwick, _Scenes from Deep Time_. Evan Maurer has already observed that _Natural History_ reflects \"Ernst's continuing interest in the illustrated books and magazines of natural science that he had begun to use as sources of illustrations for his collages in the early 1920s.\" Maurer, \"Images of Dream and Desire,\" 59. Derenthal and Pech also emphasize the \"scientific appearance.\" Ludger Derenthal and J\u00fcrgen Pech, _Max Ernst_ (Paris: Nouvelles Editions Fran\u00e7aises, Casterman, 1992), 101.\n\n Rudwick, _Scenes from Deep Time_.\n\n \"Ces cat\u00e9chismes appell\u00e9s manuels d'histoire naturelle\" (these catechisms called natural history manuals), in the text\"il fait beau\" (it's a beautiful day), written by Andr\u00e9 Breton, Benjamin P\u00e9ret, and Robert Desnos and dedicated to Max Ernst, in _Litt\u00e9rature_ 9 (1923). On the pedagogical and moralistic intention of popular natural history, see Jean-Marc Brouin and Bernadette Bensaude-Vincent, \"Nature for the People,\" in _Cultures of Natural History_ , ed. Nicolas Jardine (Cambridge: Cambridge University Press, 1996), 408\u201325.\n\n Spies, _Frottages_ ; Derenthal and Pech, _Max Ernst_ , 108. On Ernst's Catholic iconography, see David Hopkins, _Duchamp and Max Ernst: The Bride Shared_ (Oxford: Oxford University Press, 1998); J\u00fcrgen Pech, \"Studien zur religi\u00f6sen Ikonographie im Werk von Max Ernst bis 1934: der Collagenroman 'R\u00eave d'une petite fille qui voulutentrer au Carmel' (1930)\" (PhD diss., Rheinische Friedrich-Wilhelms-Universit\u00e4t, 1996).\n\n \"Ce que nous appelions bien fi\u00e8rement 'notre \u00e9ducation' est \u00e0 refaire de fond en comble et Max Ernst a raison, qui, sous le simple titre _Histoire naturelle_ , nous pr\u00e9sente r\u00e9unies en trente-quatre planches les terribles merveilles d'un univers dont notre semelle n'essaiera plus d'\u00e9craser les petits secrets, d\u00e9sormais plus grands que nous.\" Ren\u00e9 Crevel, \"Histoire naturelle,\" _Nouvelle Revue Fran\u00e7aise_ 169 (October 1927): 554ff.\n\n Laurent Jenny, \"La surr\u00e9alit\u00e9 et ses signes narratifs,\" _Po\u00e9tique_ 16 (1973): 500ff.\n\n Dominique Bozo, _Andr\u00e9 Breton_ , exhibition catalog (Paris: Centre Pompidou, 1991), 170.\n\n \"Je vais soutenir une th\u00e8se \u00e9trange. Je vais pr\u00e9tendre que le premier livre \u00e0 mettre entre les mains de l'enfance doit se rapporter \u00e0 l'histoire naturelle; et qu'au lieu d'appeler l'attention admirative des jeunes esprits sur les fables de la Fontaine, les aventures du Chat bott\u00e9, l'histoire de Peau d'\u00e2ne, ou les amours de V\u00e9nus, il faut la diriger sur les spectacles naifs et simples de la nature: la structure d'un arbre, la composition d'une fleur, les organes des animaux, la perfection des formes cristallines d'un min\u00e9ral, l'arrangement int\u00e9rieur des couches composant la terre que nous foulons sous nos pieds.\" (I am going to advance an unusual thesis. I will propose that the first book put into the hands of youth should pertain to natural history; and, instead of calling the admiring attention of young minds to the fables of La Fontaine, the adventures of Puss in Boots, the story of Donkeyskin, or the loves of Venus, attention should be directed to the pure and simple spectacles of nature: the structure of trees, the composition of flowers, the organs of animals, the perfection of a mineral's crystalline forms, the arrangement of the stratified layers that compose the very earth that we trample beneath our feet.) Louis Figuier, preface to _La Terre avant le d\u00e9luge_ (Paris: L. Hachette, 1863). On natural history as children's literature, see Harriet Ritvo, \"Learning from Animals: Natural History for Children in the Eighteenth and Nineteenth Centuries,\" _Children's Literature_ 13 (1985): 72\u201379.\n\n Wolf Lepenies, _Das Ende der Naturgeschichte: Wandel kultureller Selbstverst\u00e4ndlichkeiten in den Wissenschaften des 18. und 19. Jahrhunderts_ (M\u00fcnchen: C. Hanser, 1976).\n\n \"Ce sont aussi des esp\u00e8ces de fables, mais des fables produites par une imagination active qui a besoin de cr\u00e9er, et non par une imagination passive qui c\u00e8de \u00e0 des impression \u00e9trang\u00e8res.\" (These are also kinds of fables, but fables produced by an active imagination with the need to create, not by a passive imagination that yields to external impressions.) Condorcet, \"Eloge de M. de Buffon,\" in _\u0152uvres compl\u00e8tes de Condorcet_ (Paris: Chez Henrichs, 1804), 89.\n\n I am following the assessment of Konersmann, \"Max Ernst und die Idee der Naturgeschichte,\" 164, who calls the connections to the natural history of the eighteenth century \"associative\"; by contrast, cf. Zimmermann, \"Philosophische Horizonte.\"\n\n \"Aucun Mammif\u00e8re, aucun Oiseau n'avait encore apparu; rien n'interrompait le silence des airs, sinon le sifflement des reptiles terrestres et le vol de quelques insectes ail\u00e9s.\" (No mammal, no bird had yet appeared; nothing interrupted the silence of the air, except for the hissing of the terrestrial reptiles and the flight of some winged insects.) Figuier, _La Terre avant le d\u00e9luge_ , 175.\n\n \"The Eternal Signs: I remember the strange and profound impression made upon me as a child by a plate in an old book that bore the title _The World before the Flood_. The plate represented a landscape of the tertiary period. Man was not yet present. I have often meditated upon the strange phenomenon of this absence of human beings in its metaphysical aspect. . . . There are paintings by B\u00f6cklin, Claude Lorrain and Poussin which are inhabited by human figures, but which, in spite of this, bear a close relationship with the landscape of the Tertiary. Absence of humanity in man. Some of Ingres's portraits achieve this too. It should, however, be observed that in the works cited above (except perhaps in a few paintings by B\u00f6cklin), only the first solitude exists: plastic solitude. Only in the new Italian metaphysical painting does the second solitude appear: solitude of signs, or the metaphysical.\" Giorgio de Chirico, \"On Metaphysical Art,\" as reprinted in _Manifesto: A Century of Isms_ , ed. Mary Ann Caws (Lincoln: University of Nebraska Press, 2000), 282\u201383. On de Chirico's interest in Figuier and Riou, see Jean Clair, \"Chronos und Mnemosyne,\" in _Giorgio de Chirico_ , exhibition catalog (Munich: Haus der Kunst, 1982), 79\u201388.\n\n Breton, _Manifesto of Surrealism_ , 29, 32. Breton outlines the project of a disciplining of memory that would extend across generations, thus promising a new absolute memory\u2014a memory that encompasses and keeps present every dream and, what's more, the entire unconscious. Breton, _\u0152uvres_ , 1:317\u201320.\n\n Breton, _\u0152uvres_ , 1:369\u201372.\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_, no. 605. Pech, _Max Ernst \"Histoire naturelle,\"_ 361, already recognized the connection with Breton's _Manifesto of Surrealism_. Ernst's painting shows a man in a tuxedo, who can be identified as a waiter by the glass that replaces his head. His legs are spread wide; he is running through an interior space resembling a stage set. An enormous glove is covering his right hand; his arm, which seems to have been stuck onto his body, swings like a rubber band. His movement is obviously uncontrolled\u2014especially since his head, and thus the conscious thought process of the writer, has been replaced by a glass. This stands metonymically for its contents, in all likelihood an alcoholic fluid, which promotes the flowing of unconscious streams. In the waiter's automatism, despite the large glove, the swinging right hand, and the liquefaction of consciousness, the mysterious M returns. However, it is no longer memory as a conventionalized and available past, but inaccessible deep time, which lies beyond practiced memory and conserves the prehistoric kingdom of \"inadmissible flora and fauna.\"\n\n \"Surrealism's whole effort in particular for the last 15 years has been to obtain from the poet the instantaneous revelation of these verbal traces whose psychic charges are capable of being communicated to the perception-consciousness system (and also to obtain from the painter the most rapid projection possible of optical mnemonic traces). I shall never tire of repeating that _automatism_ alone is the dispenser of the elements on which the secondary work of emotional amalgamation and passage from the unconscious to the preconscious can operate effectively.\" Breton, \"The Political Position of Surrealism,\" in _Manifestoes of Surrealism_ , 230. In view of these crucial words, with which Ernst as a reader of Freud would have agreed, in my opinion it does not make sense to think of Ernst as a painter of \"m\u00e9moire involontaire\" (involuntary memory) (Spies, _Frottages_ ) or to cite Bergson (Maurer, \"Images of Dream and Desire,\" 68).\n\n The Loplop cycle, made during 1930, also consists largely of retroactive hints about Ernst's secrets of production. See Spies, _Loplop_.\n\n This is pointed out by Holl\u00e4nder, \"Es rauscht in den Schachtelhalmen . . . ,\" in _Ikonographia: Anleitung zum Lesen von Bildern (Festschrift Donat de Chapeaurouge)_ , ed. Bazon Brock and Achim Preiss (Munich: Klinkhardt & Biermann, 1990). At the same time, in _The Hundred-Headed Woman_ , Ernst used Riou's illustration \"Ideal Landscape of the Permian Period.\"\n\n Kaufmann, _Po\u00e9tique des groupes litt\u00e9raires_.\n\n And in English, there is the \"damselfly.\"\u2014Trans.\n\n Pierre Naville, _Le temps du surr\u00e9el_ (Paris: Galil\u00e9e, 1977); Jean-Charles Gateau, _Paul Eluard et la peinture surr\u00e9aliste (1910\u20131939)_ (Geneva: Librairie Droz, 1982); Jean-Charles Gateau, _Eluard: Le fr\u00e8re voyant, 1895\u20131952_ (Paris: R. Laffont, 1988); Paul \u00c9luard, _Lettres \u00e0 Gala, 1924\u20131948_ (Paris: Gallimard, 1984).\n\n Max Ernst, _Frottages_ , exhibition catalog (Paris: Galerie Berggruen, 1956). Also see the frottaged double portrait, Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 1063.\n\n Spies, _Frottages_.\n\n \". . . a little table that did not appear to be anything special and was made out of ordinary wood. However, it had one good quality. Whenever one put it down and said, 'table, be covered,' it would immediately be covered by a clean tablecloth, and on it would be a plate with a fork and a knife, and dishes with roasted and stewed meat, as much as there was room for on the table, and a large glass of red wine to tickle one's throat.\" \"The Magic Table, the Golden Donkey, and the Club in the Sack,\" in _The Complete Fairy Tales of the Brothers Grimm_ , trans. Jack Zipes (New York: Bantam Books), 124.\u2014Trans.\n\n \"A table for children, \/ there are women whose eyes are like sugar cubes \/ there are women as serious as love unperceived, \/ there are women whose faces are pale, \/ others like the sky the night before the wind. \/ Little table gilded for holidays, \/ there are women of wood dark and green: \/ the ones who laugh.\" In Paul \u00c9luard, _Capital of Pain_ , trans. Mary Ann Caws, Patricia Terry, and Nancy Kline (Boston: Black Widow Press, 2006). \"Petite table enfantine, \/ il y a des femmes dont les yeux sont comme des morceaux de sucre, \/ il y a des femmes graves comme les mouvements de l'amour qu'on ne surprend pas, \/ il y a des femmes au visage p\u00e2le d'autres comme le ciel \u00e0 la veille du vent. \/ Petite table dor\u00e9e des jours de f\u00eate, \/ il y a des femmes de bois vert et sombre: \/ celles qui pleurent, de bois sombre et vert: \/ celles qui rient. . . .\" Paul \u00c9luard, _\u0152uvres compl\u00e8tes_ I, ed. Lucien Scheler (Paris: Gallimard, 1968), 141. Jean-Charles Gateau refers to \u00c9luard's \"table be covered\" motif in his _Paul Eluard: Capitale de la douleur_ (Paris: Gallimard, 1994), 65ff.\n\n Sigmund Freud, _Introductory Lectures on Psychoanalysis_ , trans. James Strachey (New York: Norton, 1966), 156\u201358; Breton, _\u0152uvres_ , 2:124.\n\n Here is the other half of the poem: \"Little table too low or too high, \/ there are fat women \/ with slender shadows, \/ there are hollow dresses, \/ dry dresses, \/ house dresses that love can't get out the door \/\/ Little table, \/ I don't like the tables on which I dance, \/ I didn't realize that.\" (Petite table trop basse ou trop haute, \/ il y a des femmes grasses \/ avec des ombres l\u00e9g\u00e8res, \/ il y a des robes creuses, \/ des robes s\u00e8ches, \/ des robes que l'on porte chez soi et que l'amour ne \/ fais jamais sortir \/ Petite table, \/ je n'aime pas les tables sur lesquells je danse, \/ je ne m'en doutais pas.)\n\n Rosalind Krauss, _The Optical Unconscious_ (Cambridge, MA: MIT Press, 1993), chap. 2; in addition, on Ernst's psychoanalytic Gala iconography, see Elizabeth Legge, _Max Ernst: The Psychoanalytic Sources_ (Ann Arbor, MI: UMI Research Press, 1989), chap. 4.\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , nos. 636\u201352; Ludger Derenthal, \"Die Wandmalereine f\u00fcr Eaubonne,\" in _Max Ernst: Das Rendezvous der Freunde_ , exhibition catalog (K\u00f6ln: Musuem Ludwig, 1991), 181\u201386.\n\n \"Paul Eluard: Does Breton Enjoy Licking a Woman's Eyeball?,\" in _Investigating Sex: Surrealist Research 1928\u20131932_ , trans. Malcolm Imrie, ed. Jos\u00e9 Pierre (London: Verso, 1992).\n\n Freud, _Studienausgabe_ (Frankfurt am Main: Fischer, 1994), 10:140ff.\n\n \u00c9luard, _\u0152uvres_ 1:176ff. Caws et al. translate \"nature without banks\" (nature sans rivage) as \"unbounded space\"; see below.\u2014Trans. See also Gateau, _Paul Eluard: Capitale de la douleur_ , 124ff.\n\n \"Je sors au bras des ombres, \/ Je suis au bas des ombres, \/ Seul. \/ La piti\u00e9 est plus haut et peut bien y rester, \/ La vertu se fait l'aum\u00f4ne de ses seins \/ Et la gr\u00e2ce s'est prise dans les filets de ses paupi\u00e8res. \/ Elle est plus belle que les figures des gradins, \/ Elle est plus dure, \/ Elle est en bas avec les pierres et les ombres \/ Je l'ai rejointe \/\/ C'estici que la clart\u00e9 livre sa derni\u00e8re bataille. \/ Si je m'endors, c'est pour ne plus r\u00eaver. \/ Quelles seront alors les armes de mon triomphe? \/ Dans mes yeux grands ouverts le soleil fait les joints, \/ O jardin de mes yeux! \/ Tous les fruits sont ici pour figurer des fleurs, \/ Des fleurs dans la nuit, \/ Une fen\u00eatre de feuillage \/ S'ouvre soudain dans son visage. \/ O\u00f9 poserai-je mes l\u00e8vres, nature sans rivage? \/\/ Une femme est plus belle que le monde o\u00f9 je vis \/ Et je ferme les yeux. \/ Je sors au bras des ombres, \/ Je suis au bas des ombres \/ Et des ombres m'attendent.\" \u00c9luard, _\u0152uvres_ 1:176ff. (I go out on the arm of the shadows, \/ I rest at the foot of the shadows, \/ Alone. \/\/ Pity is a step above, and may as well stay there, \/ Virtue offers her breasts to herself \/ And grace is caught in the nets of her eyelids. \/ She is lovelier than the shapes on the tiered steps, \/ She is harder, \/ She is below, amidst the stones and shadows, \/ I have found her again. \/\/ Here clarity fights its last battle. \/ If I go to sleep, it's so as not to dream again. \/ Then what will my triumphant weapons be? \/ I open my eyes wide in the assembling sun, \/ Oh garden of my eyes! \/ Here all the fruits embody flowers, \/ Flowers of the night. \/ A window of foliage \/ Opens suddenly in her face. \/ Where will I rest my lips, unbounded space? \/\/ A woman is more beautiful than the world where I exist \/ And I close my eyes. \/ I go out on the arm of the shadows. \/ I rest at the foot of the shadows, \/ And the shadows wait for me.\") \u00c9luard, _Capital of Pain_ , 157.\n\nCHAPTER THREE\n\n On Breton's utopian future transparency of surrealist communication even outside of the group, see Andr\u00e9 Breton, _\u0152uvres compl\u00e8tes_ , ed. Marguerite Bonnet et al. (Paris: Gallimard, 1988\u201399), 2:440.\n\n \"The legibility of the individual elements juxtaposed in these paintings invites interpretation: the juxtapositions resist it. An essential component in attempting to piece together their meanings is Ernst's reading of psychoanalytic literature.\" Elizabeth Legge, _Max Ernst: The Psychoanalytic Sources_ (Ann Arbor, MI: UMI Research Press, 1989), 1. The literature on Ernst's psychoanalytic iconography is plentiful; however, the boundary between psychoanalytically informed iconography and psychoanalytic interpretation in the narrower sense is not always clearly marked. See Elizabeth Legge, \"Max Ernst's \u0152dipus Rex and the Implicit Sphinx,\" _ArtsMagazine_ 61 (September 1986): 50\u201353; Malcolm Gee, \"Max Ernst, God, and the Revolution by Night,\" _Arts Magazine_ 55 (March 1981): 85\u201391; Geoffrey Hinton, \"Max Ernst: 'Les Hommes n'en sauront rien,'\" _Burlington Magazine_ 117 (1975): 9\u201320; Jeanne Siegel, \"Max Ernst's One Night of Love,\" _Arts Magazine_ 57 (1983): 112\u201315; Laura Meixner, \"Max Ernst's Aquis Submersus as Literary Collage,\" _Arts Magazine_ 61 (November 1986): 80\u201385; Charlotte Stokes, \"Collage as Jokework: Freud's Theories of Wit as the Foundation for the Collages of Max Ernst,\" _Leonardo_ 15 (1982): 199\u2013204; and Gerd Bauer, \"Ein Meisterstreich von Dadamax,\" in _Fatagaga-Dada: Max Ernst, Hans Arp, Johannes Theodor Baargeld und der K\u00f6lner Dadaismus_ , ed. Karl Riha and J\u00f6rgen Sch\u00e4fer (Giessen: Anabas, 1995), 121\u201341.\n\n \". . . the sum total evades an exact interpretation\u2014this is the basis of the strong poetic provocation of Max Ernst's works. Their analysis supplies many separately comprehensible elements, but the final interpretation evades us despite our knowledge of the details. An attempt to explain the hermetic picture with the help of psychoanalytical methods is also doomed to failure. In the case of Max Ernst, such an attempt seems to be the most naive of all and reveals the smallest degree of comprehension.\" This is Werner Spies's basic conviction, acquired through close acquaintance with the artist, as first formulated in _The Return of la belle jardini\u00e8re: Max Ernst, 1950\u20131970_ , trans. Robert Allen (New York: Abrams, 1971), 37\u201338.\n\n This is also the case in productive and stimulating studies such as Legge, _Max Ernst: The Psychoanalytic Sources_ ; Christa Lichtenstern, _Metamorphose vom Mythos zum Prozessdenken: Ovid-Rezeption, surrealistische \u00c4sthetik, Verwandlungsthematik der Nachkriegskunst_ (Weinheim: VCH, 1992); Rosalind Krauss, _The Optical Unconscious_ (Cambridge, MA: MIT Press, 1993); and Hal Foster, _Compulsive Beauty_ (Cambridge, MA: MIT Press, 1993).\n\n On the ambiguous relationship of surrealism to psychoanalysis, see Jean Starobinski, \"Freud, Breton, Myers\" (1970), in _Psychoanalyse und Literatur_ (Frankfurt am Main: Suhrkamp, 1973), 143\u201362; J. B. Pontalis, \"Die nicht kommunizierenden R\u00f6hren (1978),\" in _Aus dem Blick verlieren: Im Horizont der Psychoanalyse_ (Munich: Kirchheim, 1991), 140\u201359; Foster, _Compulsive Beauty_ ; and David Lomas, _The Haunted Self: Surrealism, Psychoanalysis, Subjectivity_ (New Haven, CT: Yale University Press, 2000).\n\n On this text, see also Spies, _Loplop_ , 107\u20139.\n\n Ludger Derenthal and J\u00fcrgen Pech, _Max Ernst_ (Paris: Nouvelles Editions Fran\u00e7aises, Casterman, 1992), 115\u201317.\n\n See the fundamental studies by Legge, _Max Ernst: The Psychoanalytic Sources_ ; and Krauss, _The Optical Unconscious_ ; on Ernst and Leonardo in general, see Werner Hofmann, \"Max Ernst und die Tradition,\" in _Max Ernst: Das Innere Gesicht_ , exhibition catalog (Hamburg: Kunsthalle Hamburg, 1970), 14ff.; Whitney Chadwick, _Myth in Surrealist Painting, 1929\u20131939_ (Ann Arbor, MI: UMI Research Press, 1980), 87\u201396; and Spies, _Loplop_ , 101\u20136; for a summary of the effective history of Freud's Leonardo study, see Klaus Herding, \"Freuds 'Leonardo,'\" _Im Blickfeld: Jahrbuch der Hamburger Kunsthalle_ 3 (1998): 9\u201332; On Ernst and the concept of the primal scene, cf. also Foster, _Compulsive Beauty_ , 70\u201384.\n\n \". . . toutes les horreurs, que mon p\u00e8re est capable d'\u00e9veiller aimablement dans un panneau de faux acajou au moyen de son affreux crayon mou.\" The English is from Max Ernst, \"History of a Natural History\" (evidently a translation of the later version of the text, \"Comment on force l'inspiration\"), trans. Dorothea Tanning, in _Beyond Painting and Other Writings_ (New York: Schultz, 1948), 3\u20134.\n\n Freud, \"From the History of an Infantile Neurosis,\" in _The Standard Edition of the Complete Worksof Sigmund Freud_, ed. James Strachey (Psychoanalytic Electronic Publishing, www.pep-web.org), 17:43.\n\n Legge, _Max Ernst: The Psychoanalytic Sources_ , 6ff.\n\n \"Nachtr\u00e4glichkeit.\" One criticism of Strachey's \"deferred action\" or \"deferred\" for _nachtr\u00e4glich_ is its orientation toward the future, while a term such as \"retroactive\" would more suitably retain a sense of orientation toward the past. I use \"after the fact\" when possible.\u2014Trans.\n\n See Krauss, _The Optical Unconscious_ , chap. 2; and Foster, _Compulsive Beauty_ , 81\u201383.\n\n Freud, \"From the History of an Infantile Neurosis.\"\n\n Ibid., 97; see also the corresponding observations in the _Lectures_ (with which Ernst was also familiar), Freud, _Studienausgabe_ (Frankfurt am Main: Fischer, 1994), 1:362.\n\n Freud, \"From the History of an Infantile Neurosis,\" 29.\n\n Andr\u00e9 Breton, _Soluble Fish_ in _Manifestoes of Surrealism_ , trans. Richard Seaver and Helen R. Lane (Ann Arbor: University of Michigan Press, 1969), 60.\n\n _La R\u00e9volution surr\u00e9aliste_ 9\u201310 (December 1, 1927), cited in Spies, _Loplop_ , 101.\n\n On the reception history of Leonardo's wall stains in surrealism, see Lichtenstern, _Metamorphose_ , 156\u201363.\n\n Spies, _Loplop_ , 107; Siegel, \"Max Ernst's One Night of Love\"; Derenthal and Pech, _Max Ernst_ , 131.\n\n Freud, _Studienausgabe_ 10:138ff.; Oskar Pfister, \"Kryptolalie, Kryptographie und unbewu\u00dftes Vexierbild bei Normalen,\" _Jahrbuch psychoanalytischer und psychopathologischer Forschungen_ 5 (1913): 117\u201356.\n\n In print 27, _In the Stable of the Sphinx_ , there appear to be at least two hidden creatures\u2014which is not surprising from a psychoanalytic perspective, since Freud believed he detected an allegory of infantile sexual research in the riddle of the Sphinx. Freud, _Studienausgabe_ 5:100. Infantile sexual research is closely associated with the primal scene, since the little observer strives to integrate it into his fantasies about the phallic woman and castration. The \"stable of the Sphinx\" is built of suggestive boards that elicit the same imaginative activity as the imitation mahogany that produced terrifying images in the \"stable\" of the parents' bedroom. In the third board from the left, against the central crack is silhouetted an owl-like face, at which the Sphinx's gaze is possibly directed. What's more, the Sphinx itself proves to be a two-faced creature, its tail ending in a predator's head.\n\n The first bird is hidden in the center of the fossil snail that forms the body of the plant standing in the wide expanse of the pampas (print 6); others follow. On Ernst's avian mythology, see Hofmann, \"Max Ernst und die Tradition,\" 14\u201316; Spies, _Loplop_ ; Chadwick, _Myth in Surrealist Painting_ , 87ff.\n\n Lichtenstern, _Metamorphose_ , 163.\n\n \"Partant d'un souvenir d'enfance au cours duquel un panneau de faux acajou, situ\u00e9 en face de mon lit, avait jou\u00e9 le r\u00f4le de provocateur optique d'une vision de demi-sommeil, et me trouvant, par un temps de pluie, dans une auberge au bord de la mer, je fus frapp\u00e9 par l'obsession qu'exer\u00e7ait sur mon regard irrit\u00e9 le plancher, dont mille lavages avaient accentu\u00e9 les rainures.\" Max Ernst, \"Comment on force l'inspiration,\" _Le Surr\u00e9alisme au Service de la R\u00e9volution_ 5\/6 (May 15, 1933), 45; see also Ernst, _\u00c9critures_ , 53. English version from Ernst, \"History of a Natural History,\" 3\u20134.\n\n _Herrliche Persiflage_ and _Karikaturen_. Freud, _Studienausgabe_ 8:64, 77. \"The effrontery with which Hans related this fantasy and the countless extravagant lies with which he interwove it were anything but meaningless. All of this was intended as a revenge upon his father . . .\" Freud, \"Analysis of a Phobia in a Five-Year-Old Boy,\" in _The Standard Edition_ (1909); Two Case Histories (\"Little Hans\" and \"Rat Man\"), 10:129. See also in the case of the Wolf Man, Freud, _Studienausgabe_ 8:183.\n\n _Unterschieben_. The German verb\u2014literally \"to slide under,\" with the sense \"to wrongly attribute\"\u2014allows a connection to be understood here between the procedure of frottage and this act of misattribution.\u2014Trans.\n\n See the significance of the mustache for \"little Hans\" in Freud, _Studienausgabe_ 8:49, 82, and elsewhere.\n\n Cf. Susan Rubin Suleiman, _Subversive Intent: Gender, Politics, and the Avant-Garde_ (Cambridge, MA: Harvard University Press, 1990).\n\n Michel de Certeau, _Theoretische Fiktionen: Geschichte und Psychoanalyse_ (Vienna: Turia + Kant, 1997), 109, 124.\n\n Freud, _Studienausgabe_ 8:154, 208, 220.\n\n \"The fact of a dream referring to childhood may also be expressed in another way, namely by a translation of time into space. The characters and scenes are seen as though they were at a great distance, at the end of a long road, or as though they were being looked at through the wrong end of a pair of opera-glasses.\" Freud, _The Interpretation of Dreams_ (first part; 1900), in _The Standard Edition_ ,4:408.\n\n On the phallic woman's constituting a perfect, because flawless, not castrated seeing, see Mary Jacobus, _Reading Woman: Essays in Feminist Criticism_ (New York: Columbia University Press, 1986), 110\u201336.\n\n See Katharina Sykora, _Unheimliche Paarungen: Androidenfaszination und Geschlecht in der Fotografie_ (K\u00f6ln: W. K\u00f6nig, 1999); and Mary Ann Caws, \"Ladies Shot and Painted: Female Embodiment in Surrealist Art,\" in _The Expanding Discourse: Feminism and Art History_ , ed. Norma Broude and Mary D. Garrard (New York: HarperCollins, 1992), 390.\n\n For example, Breton first read _The Interpretation of Dreams_ in October 1926; Dominique Bozo, _Andr\u00e9 Breton_ , exhibition catalog (Paris: Centre Pompidou, 1991), 184.\n\n Freud, \"Leonardo da Vinci and a Memory of His Childhood,\" in _The Standard Edition_ , 9:133. On the surrealist critique of the Freudian conception of the unconscious, see Andr\u00e9 Breton, _Le Surr\u00e9alisme et la peinture_ (Paris: Gallimard, 1965), 43ff.\n\n Starobinski, \"Freud, Breton, Myers.\"\n\n \"infortune continue\"; Breton, _\u0152uvres_ 2:380.\n\n \"diarrh\u00e9e in\u00e9puisable\"; Louis Aragon, _Trait\u00e9 du style_ (Paris: Gallimard, 1928), 206ff.\n\n \"Lundi 27 octobre [1924] . . . Constat\u00e9 l'empressement un peu d\u00e9sordonn\u00e9 de chacun \u00e0 remettre des textes surr\u00e9alistes d'int\u00e9r\u00eat souvent discutable, desquels on ne voit pas bien ce qui a d\u00e9termin\u00e9 le choix.\" (Monday, October 27 [1924] . . . Remarked upon the somewhat muddled over-eagerness of whoever is notating surrealist texts, often of questionable interest. It's difficult to see how the choices were made.) Paule Th\u00e9venin, ed. _Bureau de Recherches surr\u00e9alistes: Cahier de la permanence octobre 1924\u2013avril 1925_ (Paris: Gallimard, 1988), 35.\n\n Freud, \"Instincts and Their Viscissitudes,\" in _The Standard Edition_ , 14:131; on Freud's geologic metaphorics, see Claude L\u00e9vi-Strauss, _Tristes Tropiques_ , trans. John and Doreen Weightman (London: Penguin, 1992), 55\u201357; Pierre F\u00e9dida, \"La Regression: Formes et d\u00e9formations,\" in _Les \u00c9volutions: Phylogen\u00e8se de l'individuation_ , ed. Pierre F\u00e9dida and Daniel Widl\u00f6cher (Paris: Presses universitaires de France, 1994), 48ff.; and Monique David-M\u00e9nard, \"Sympt\u00f4mes et fossiles,\" in _Les \u00c9volutions_ , ed. F\u00e9dida and Widl\u00f6cher, 253ff.\n\n \"If [man] still retains a certain lucidity, all he can do is turn back toward his childhood which, however his guides and mentors may have botched it, still strikes him as somehow charming. There, the absence of any known restriction allows him the perspective of several lives lived at once; this illusion becomes firmly rooted within him; now he is only interested in the fleeting, the extreme facility of everything. Children set off each day without a worry in the world. Everything is near at hand, the worst material conditions are fine. The woods are white or black, one will never sleep.\" Breton, _Manifesto of Surrealism_ , 3\u20134.\n\n According to Foster's thesis in _Compulsive Beauty_, chap. 1.\n\n \"Because what is projected here is a visual field that is not a latency, an ever renewed upsurge of the pure potentiality of the external, but instead a field that is already filled, already\u2014so to say the word\u2014readymade.\" Krauss, _The Optical Unconscious_ , 54.\n\n Freud, _Studienausgabe_ 3:363\u201369.\n\n Freud, \"A Note on the Mystic Writing-Pad,\" in _The Standard Edition_ , 19:231.\n\n \"In _The Master's Bedroom_ the Wunderblock's waxen slab finds its analogue in the underlying sheet of the teaching-aid page, in its inventory-like concatenation of objects, the stored-up contents of unconscious memory; while the apparatus' top sheet appears as the perspectival covering of the gouache overpainting, the skinlike thickness of which seems to be an index of the way this receptor surface is detachable from its ground. This implication of detachment and reattachment relates to a further point Freud makes about the structure of the Wunderblock and its capacity to model the very nature of sensory stimulation: This stimulation, he says, is periodic in nature. It is pulsatile. . . . In _The Master's Bedroom_ it is not that this pulsatile motion is illustrated. Indeed the scene's peculiar stillness is a striking feature of the collage. Rather, what is rendered is the sense of the gap, the detachment, the split that results from the pulse.\" Krauss, _The Optical Unconscious_ , 57.\n\n Freud, \"From the History of an Infantile Neurosis,\", 17:48.\n\n Other attempts, less ambitious than Krauss's, include Bauer, \"Ein Meisterstreich von Dadamax,\" 136, which invokes Freud's concept of the daydream from _Creative Writers and Day-Dreaming_ ; and Stokes, \"Collage as Jokework,\" which draws a parallel between collage and the mechanism of the joke as Freud explicates it in his well-known treatise; the comparison with the dream is also prevalent, according to Freud's characterization of the dream as \"something put together . . . a conglomeration of psychic images,\" \"picture-writing\" or \"rebus,\" \"a collective image . . . with contradictory features.\" Freud, _The Interpretation of Dreams_ , 87, 261, 274.\n\n In Freud, _The Standard Edition_ , 6:147.\n\n _La R\u00e9volution surr\u00e9aliste_ 9\u201310 (October 1, 1927): 31; Freud, \"The Question of Lay Analysis,\" in _The Complete Psychological Works of Sigmund Freud_ , 20:177\u2013258. _Rinde_ means \"bark,\" \"crust,\" or \"rind\" as well as \"cortex\"; in the _Complete Psychological Works_, Strachey chooses \"cortex,\" denoting the tough outer layer of a bodily organ, the outermost layer of some simple organisms, or the bark or rind of a plant.\u2014Trans.\n\nCHAPTER FOUR\n\n Andr\u00e9 Breton, _Conversations: The Autobiography of Surrealism_ , trans. Mark Polizotti (New York: Marlowe & Co., 1993), 21; Marguerite Bonnet, \"La rencontre d'Andr\u00e9 Breton avec la folie: Saint-Dizier, ao\u00fbt\u2013novembre 1916,\" in _Folie et psychanalyse dans l'exp\u00e9rience surr\u00e9aliste_ , ed. Fabienne Hulak (Nice: Z'\u00e9ditions, 1992); Hal Foster, _Compulsive Beauty_ (Cambridge, MA: MIT Press, 1993), 1ff.\n\n On this point and what follows, see the fundamendal examination by Esther Fisher-Homberger, _Die traumatische Neurose: Vom somatischen zum sozialen Leiden_ (Bern: H. Huber, 1975); Wolfgang Schivelbusch, _Geschichte der Eisenbahnreise: Zur Industrialisierung von Raum und Zeit im 19. Jahrhundert_ (Munich: Hanser, 1977); Eric J. Leed, _No Man's Land: Combat and Identity in World War I_ (Cambridge: Cambridge University Press, 1979); Bernd Ulrich, \"Nerven und Krieg: Skizzierung einer Beziehung,\" in _Geschichte und Psychologie: Ann\u00e4herungsversuche_ , ed. Bedrich Loewenstein (Pfaffenweiler: Centaurus-Verlagsgesellschaft, 1992), 163\u201392; Paul Lerner, \"'Ein Sieg deutschen Willens': Wille und Gemeinschaft in der deutschen Kriegspsychiatrie,\" in _Die Medizin und der Erste Weltkrieg_ , ed. Wolfgang Wolfgang Uwe Eckart and Christoph Gradmann (Pfaffenweiler: Centaurus-Verlagsgesellschaft, 1996); and Ruth Leys, \"Death Masks: Kardiner and Ferenczi on Psychic Trauma,\" _Representations_ 53 (1996): 44\u201373.\n\n Karl Abraham, S\u00e1ndor Ferenczi, and Ernest Jones, eds., _Zur Psychoanalyse der Kriegsneurosen_ (Wien: Internationaler Psychoanalytischer Verlag, 1919).\n\n On Freud's \"psychologizing of trauma and thus his resolution as an etiological factor,\" see Fischer-Homberger, _Die traumatische Neurose_ , 74\u201380. As early as 1964, with their reference to Freud's biological and phylogenetic speculations, Jean Laplanche and Jean-Bertrand Pontalis called attention to how, in this psychologizing, the physical trauma is always preserved, as if shifted to a deeper level. Jean Laplanche and Jean-Bertrand Pontalis, \"Fantasy and the Origins of Sexuality,\" _International Journal of Psychoanalysis_ 49 (1998): 1\u201318.\n\n Schivelbusch, _Geschichte der Eisenbahnreise_ , chap. 9.\n\n S\u00e1ndor Ferenczi: \"According to psychoanalysis, the war neuroses belong to a group of neuroses in which not only genital sexuality is implicated, as with the customary hysteria, but also a preliminary stage of it, the so-called 'narcissism,' or self-love.\" Karl Abraham: \"Trauma affects the sexuality of many persons, in the sense that it provides the impulse for a regressive transformation that takes narcissism as its objective.\" _Zur Psychoanalyse der Kriegsneurosen_, 25, 32.\n\n Jean Laplanche, _Life and Death in Psychoanalysis_ , trans. Jeffrey Mehlman (Baltimore: Johns Hopkins University Press, 1995), 41. See also Jean Laplanche, _Vie et mort en psychanalyse_ (Paris: Flammarion, 1970), 73. In Laplanche's French, \"le traumatisme se trouve enti\u00e8rement dans le jeu de 'fallace' qui produit une esp\u00e8ce de bascule entre les deux \u00e9v\u00e9nements.\" Mehlman gives \"seesaw effect\" for \"esp\u00e8ce de bascule,\" but \"bascule\" can also suggest a rocking back and forth, which perhaps is more suitable to describe this movement between an event in the present and an event in the past.\u2014Trans.\n\n Bonnet, \"La rencontre d'Andr\u00e9 Breton avec la folie,\" 125\u201327; Dominique Bozo, _Andr\u00e9 Breton_ , exhibition catalog (Paris: Centre Pompidou, 1991), 100\u2013110.\n\n Cathy Caruth, _Unclaimed Experience: Trauma, Narrative, and History_ (Baltimore: Johns Hopkins University Press, 1996).\n\n Rosalind Krauss, \"Notes on the Index Part I,\" in _The Originality of the Avant-Garde and Other Modernist Myths_ (Cambridge, MA: MIT Press, 1985), 196\u2013209; Hal Foster, _The Return of the Real: The Avant-Garde at the End of the Century_ (Cambridge, MA: MIT Press, 1996).\n\n Werner Spies, _Max Ernst Collagen: Inventar und Widerspruch_ (K\u00f6ln: DuMont Schauberg, 1974), 125, translated here; see also Spies, _Max Ernst Collages: The Invention of the Surrealist Universe_ , trans. John William Gabriel (New York: Harry N. Abrams, 1988), 123\u201324. The existing English translation could not be used here because of its extensive reworking of this difficult passage.\u2014Trans.\n\n Iconographic observations on surrealism and war can be found in Romy Golan, \"Mythes surr\u00e9alistes et imaginaires telluriques,\" in _Pens\u00e9e mythique et surrealism_ , ed. Jaqueline Ch\u00e9nieux-Gendron and Yves Vad\u00e9 (Paris: Lachenal & Ritter, 1996), 183\u2013206; Golan, _Modernity and Nostalgia: Art and Politics in France between the Wars_ (New Haven, CT: Yale University Press, 1995), 8\u201320; and Sidra Stich, _Anxious Visions: Surrealist Art_ , exhibition catalog (Berkeley, CA: Berkeley University Art Museum, 1990).\n\n As early as 1924, Louis Aragon defined the surreal as the fleeing horizon: \"We have seen then what the Surreal is about. But to really understand the concept we have to extend it; view it perhaps like the horizon which continually flees before the walker, for like the horizon this concept exists between the mind and what it knows it will never reach.\" Aragon, _A Wave of Dreams_ , trans. Susan De Muth (London: Thin Man Press, 2010).\n\n Max Ernst, who served four years in the field artillery\u2014as an \"August volunteer,\" according to a statement by his father\u2014was awarded the Iron Cross, Second and First Class, and promoted to Lieutenant of the Reserve. See J\u00fcrgen Pech, \"Da\u00df ich nicht tot bin, freut mich\" in _Fatagaga-Dada: Max Ernst, Hans Arp, Johannes Theodor Baargeld und der K\u00f6lner Dadaismus_, ed. Karl Riha and J\u00f6rgen Sch\u00e4fer (Giessen: Anabas, 1995), 17\u201347.\n\n Max Ernst, _Beyond Painting and Other Writings_ (New York: Schultz, 1948), 29.\n\n Leed, _No Man's Land_.\n\n Here, Ernst suggests reading rootless and severed plants, such as grow in _Fields of Honor_ and elsewhere, as symptoms of the paternal castration threat.\n\n Louis Aragon, _Projet d'une histoire de la litt\u00e9rature moderne_ , ed. Marc Dachy (Paris: Gallimard, 1994), 3\u20136.\n\n Jean Laplanche and Jean-Bertrand Pontalis, _Urphantasie: Phantasien \u00fcber den Ursprung, Urspr\u00fcnge der Phantasie_ (Frankfurt am Main: Fischer-Taschenbuch-Verlag, 1992).\n\n Spies, _Max Ernst Collagen_ , 42.\n\n Hal Foster, \"Armor fou,\" _October_ 56 (Spring 1991): 64\u201397.\n\n Ludger Derenthal, \"Mitteilungen \u00fcber Flugzeuge, Engel und den Weltkrieg: Zu den Photocollagen der Dadazeit von Max Ernst,\" in _Im Blickfeld. Jahrbuch der Hamburger Kunsthalle_ 2 (1994): 41\u201360.\n\n Werner Spies and G\u00fcnter Metken, _Max Ernst: \u0152uvre-Katalog_ (K\u00f6ln: M. DuMont Schauberg, 1975), no. 361.\n\n Bernd Ulrich, \"Kampfmotivationen und Mobilisierungsstrategien: Das Beispiel Erster Weltkrieg,\" in _T\u00f6ten im Krieg_ , ed. Heinrich von Stietencron and J\u00f6rg R\u00fcpke (Freiburg im Breisgau: K. Alber, 1995), 405; Ulrich Br\u00f6ckling, _Disziplin: Soziologie und Geschichte milit\u00e4rischer Gehorsams-produktion_ (M\u00fcnchen: W. Fink, 1997), 199\u2013240.\n\n \"Dort auf jenem h\u00fcgel, so rief der general, sehe ich dichte sch\u00fctzenlinien. Warum werden sie mir nicht gemeldet? Es sind puppenr\u00e4uber und bl\u00fctenst\u00e4nde, wandte der adjudant ein. Und jene artilleriebeobachtungsst\u00e4nde da dr\u00fcben? Das sind die brutknospen auf ihren leitern. Halblinks ist eine starke batterie von anscheinend gro\u00dfem kaliber, fragte der f\u00fchrer nochmals; solche f\u00fchren wir doch nicht. Ew. Exzellenz haben ganz recht: es sind die bauchteile der eizellen, die spitzeng\u00e4nger der zukunft, die gliedma\u00dfen der im schnee begrabenen. Sie \u00fcbertreffen die sporen an sch\u00f6nheit und klarheit. Sie sind mit wurzelhaaren dicht besetzt. Ihre halskan\u00e4le tragen feine wimpern. Die giftz\u00e4hne verbergen sie in den weichteilen ihrer frauen. Atem\u00f6ffnung (!) und assimilationsf\u00e4den tausendfach. Aus dem grunde des bechers der sonnentau. Vorw\u00e4rts, antwortete dieser. Die schrumpfung der wandzelle. Das auskeimen der sporen. Die unverbesserliche Trinkerin.\" _Dada Outdoors: Singing Contest in the Tirol_ ( _Dada au grand air: Der S\u00e4ngerkrieg in Tirol_) (Tarrenz bei Imst, 1921); reprinted in Herzogenrath, _Max Ernst in K\u00f6ln_ , 313.\n\n Ludwig Aschoff, \"Krankheit und Krieg: Eine akademische Rede\" (\"Disease and War: An Academic Lecture\"), quoted in Wolfgang Eckart, \"Aesculap in the Trenches: Aspects of German Medicine in the First World War,\" in _War, Violence, and the Modern Condition_ , ed. Bernd H\u00fcppauf (Berlin: Walter de Gruyter, 1997), 185.\n\n \"Schneeberger dr\u00fccketh\u00e4ler\" is a spoonerism of \"Schneeth\u00e4ler\" (dwellers in snow-covered valleys) and \"Dr\u00fcckeberger\" (shirkers [of military duty]).\n\n On propaganda rumors, see Paul Fussell, _The Great War and Modern Memory_ (New York: Oxford University Press, 1975), 116; on the point about collage, see Lucy Lippard, \"Dada into Surrealism: Notes on Max Ernst as Proto-Surrealist,\" _Artforum_ (September 1966): 13.\n\n Brigid Doherty, \"'See: We Are All Neurasthenics!' or, The Trauma of Dada Montage,\" _Critical Inquiry_ 24 (Autumn 1997): 82\u2013132.\n\n See Hanne Bergius, \"Dada, the Montage and the Press: Catchphrases and Clich\u00e9 as Basic Twentieth-Century Principles,\" in _Dada: The Coordinates of Cultural Politics_ , ed. Stephen C. Foster (New York: G. K. Hall, 1996), 129\u201332.\n\n Spies, _Max Ernst Collagen_ ; Rosalind Krauss, \"The Photographic Conditions of Surrealism,\" in _The Originality of the Avant-Garde_ , 196\u2013209.\n\n Derenthal, \"Mitteilungen \u00fcber Flugzeuge, Engel und den Weltkrieg.\"\n\n On the relationship between frottage and collage, see Spies, _Max Ernst Collagen_.\n\n For comparable artistic devices, see Rosalind Krauss, \"Corpus deliciti,\" in _L'Amour Fou: Photography and Surrealism_, ed. Jane Livingstone, exhibition catalog (Washington, DC: Corcoran Gallery of Art, 1985), 57\u2013110.\n\n On this form of representation in general, see Richard Shiff, \"Picasso's Touch,\" _Yale University Art Gallery Bulletin_ (1990): 39\u201347. A similar approach, though with an entirely different, harmlessly playful effect, can be observed with the right arm: Ernst makes a clearly visible cut through the upper arm, and then retouches it with single blades of grass, as if here the figure were lying in the field. Evidently, a semantic play is at work with _Zerschneiden_ (to cut apart) and _\u00dcberschneiden_ (to overlap). The arm's being overlapped by the grass\u2014within the illusionistic logic that Ernst implies through the retouching\u2014corresponds to its being cut by the collagist. See Shiff's analysis of similar operations in Picasso's first _papiers coll\u00e9s_ , in \"C\u00e9zanne's Physicality: The Politics of Touch,\" in _The Language of Art History_ , ed. Salim Kemal and Ivan Gaskell (Cambridge: Cambridge University Press, 1991), 160\u201366.\n\n The fairy tale by this name in German renditions of Hans Christian Andersen is known in English as \"The Nightingale.\"\u2014Trans.\n\n Also, Ernst may have been interested, in Andersen's fairy tale, in the thematizing of two related oppositions: that of organic and mechanical beauty, and that of display and the secret (which the nightingale is at the beginning and which it claims for itself at the end).\n\n _Bombenfl\u00fcgel_ , literally the bomb's \"wing.\"\n\n Just a sample: \"Frau Wirtin hat ne sechste Tante, \/ die man die 'sch\u00f6ne Rosa' nannte. \/ Der war der Damm zerrissen. \/ Und wenn man sie mal v\u00f6geln wollt,' \/ dann ward der Schwanz gerissen.\" (In a prose rendering, the Hostess's aunt named \"pretty Rosa\" suffered a perineal rupture, which made intercourse dangerous for whoever tried it.) These student drinking songs were published as \"erotische Parodien, entsprungen dem Geiste des Volkstums\" (erotic parodies borne of the folklore spirit) by Dr. Helmut and Dr. Allengo, \"Der erotische Vierzeiler h\u00f6her gebildeter St\u00e4dter,\" in _Beiwerke zum Studium der Anthropophyteia_ , ed. Dr. Friedrich S. Krauss (Leipzig: Ethnologischer Verlag, 1911), 4:210\u201337. Reference to Helmut and Allengo in Klaus Theweleit, _M\u00e4nnerphantasien_ (Frankfurt am Main: Verlag Roter Stern, 1977\u201378), 1:423. On the title, and for a precise technical analysis of the work, see also Ludger Derenthal, \"Eine surrealistische R\u00e9v\u00e9lation: Die erste Max Ernst-Ausstellung in Paris,\" in _Max Ernst: Das Rendezvous der Freunde_ , exhibition catalog (K\u00f6ln: Musuem Ludwig, 1991), 65ff.\n\n As in \"Little Hans.\" Freud, _Studienausgabe_ (Frankfurt am Main: Fischer, 1994), 8:87, 108\u201310.\n\n Foster, _Compulsive Beauty_.\n\n Michel Foucault, _Mental Illness and Psychology_ , trans. Alan Sheridan (Berkeley: University of California Press, 1987), 83.\n\n See Helmut Lethen, _Cool Conduct: The Culture of Distance in Weimar Germany_ , trans. Don Reneau (Berkeley: University of California Press, 2002).\n\n Derenthal, \"Mitteilungen \u00fcber Flugzeuge, Engel und den Weltkrieg,\" 49.\n\n The complete title reads \"die flammingi lassen wieder papierdotter steigen \/ saturn ist an den enden der seezunge festgenagelt \/ wegen der n\u00e4he des magnetischen s\u00fcdpols versagen die erzengel\" (the flamingos let paper yolks rise again \/ saturn is nailed fast to the ends of the dover sole [literally \"sea-tongue\"] \/ the archangels fail on account of proximity to the magnetic south pole).\n\n On the aerial photograph, see Andreas Haus, \"Luftbild\u2014Raumbild\u2014Neues Sehen,\" _Photogeschichte_ 12, nos. 45 \/ 46 (1992): 75\u201390; and Christoph Asendorf, _Super Constellation\u2014Flugzeug und Raumrevolution: Die Wirkung der Luftfahrt auf Kunst und Kultur der Moderne_ (Vienna: Springer, 1997).\n\n See Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , nos. 375, 385, 391\u201397, and elsewhere, e.g., nos. 352, 356\u201358, 360, 366ff., etc.\n\n \"The very ability to presume an aerial perspective over the scene of one's demise acted as a guarantee of survival and ensured that one's personal demise was only a dream. . . . The sky is charged with intense significance: it must be the residence of the observer watching himself struggle through the nightmare of war, for only then will the eye survive the dismemberment of the body.\" Leed, _No Man's Land_ , 137.\n\n Peter Dering, \"'Seine Augen trinken alles . . .'\u2014Max Ernsts Fr\u00fchwerk 1910\u20131914,\" in _Max Ernst und Bonn\u2014Student, Kritiker, Rheinischer Expressionist_, ed. Dering (Bonn: Verein August Macke Haus, 1994), 31\u201357.\n\n \"Wir liegen an einem Waldrand mit unsern Munitionswagen; gewitterartig rollt der Kanonendonner am ganzen Horizont. \u00dcberall die kleinen Sprengw\u00f6lkchen; beides geh\u00f6rt schon zur Landschaft, wie auch das Echo, das jeden Schu\u00df verdoppelt weitertr\u00e4gt. Pl\u00f6tzlich ein merkw\u00fcrdiges Surren, das in einem ungeheuren Bogen \u00fcber uns weggeht, ungleich, in steten Schwingungen, \u00fcbergehend von hellem Pfeifen in tiefes Brummen; wie der hohe weite Schrei eines Raubvogels, immer kurz hintereinander, mit dem Eigensinn des Tieres, das keinen anderen Ruf kennt. Dann in der Ferne der dumpfe Knall. Es sind schwere feindliche Artilleriegeschosse, die \u00fcber uns wegrasen, nach einem uns unbekannten Ziel. Ein Schu\u00df zieht den anderen nach; der Himmel steht im reinsten Herbstblau und doch f\u00fchlen wir die hohen Rinnen, in denen die Geschosse ihn durchst\u00fcrmen. Der Artilleriekampf hat selbst f\u00fcr den Artilleristen oft etwas Mystisches, Mythisches.\" Franz Marc, _Schriften_ , ed. Klaus Lankheit (K\u00f6ln: DuMont, 1978), 158.\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , nos. 268\u201371.\n\n Marc, _Schriften_ , 152.\n\nDie Welt ist zum Ersticken voll. Auf jeden Stein hat der Mensch ein Pfand seiner Klugheit gelegt. Jedes Wort ist gepachtet und belehnt. Was kann man thun zur Seligkeit als alles aufgeben und fliehen? als einen Strich ziehen zwischen dem Gestern und dem Heute?\n\nIn dieser That liegt die gro\u00dfe Aufgabe unserer Zeit; die eine, f\u00fcr die es sich lohnt zu leben und zu sterben. [. . .]\n\nSo wandern wir fort in neue Gebiete und erleben die gro\u00dfe Ersch\u00fctterung, da\u00df alles noch unbetreten, ungesagt ist, undurchfurcht und unerforscht. Die Welt liegt rein vor uns; unsre Schritte zittern. Wollen wir wagen zu gehen, so mu\u00df die Nabelschnur durchschnitten werden, die uns mit der m\u00fctterlichen Vergangenheit verbindet.\n\n Ibid., 208\u201310.\n\nIch ritt durch ein St\u00e4dtchen mit lieblichen alten Bauten; die Frauen sahen aus den Fenstern. Die Stunde war br\u00e4unlich, hell und heimlich. Da befielen mich diese Gedanken von der Gef\u00e4hrlichkeit der toten, wieder todbringenden Dinge und von der gro\u00dfen Reinheit.\n\nSollten die guten alten Bilderst\u00fcrmer schon meinen Sinn erraten haben? Etwas Wahres, Nahes, die \"Gefahr\" ahnten sie gewi\u00df.\n\nIch aber trabte eilend aus dem St\u00e4dtchen der vielen Erinnerungen und Ansteckungen, bis ich die reine Luft der Morgenr\u00f6te roch.\n\n Theweleit, _M\u00e4nnerphantasien_ ; Sandra Gilbert, \"Soldier's Heart: Literary Men, Literary Women, and the Great War,\" _Signs: Journal of Women in Culture and Society_ 9 (1983): 422\u201350.\n\n Ralph Ubl, \"Wilhelm Worringer, Hans Arp und Max Ernst bei den M\u00fcttern: \u00dcberlegungen zum Primitivismus der deutschen Avantgarde,\" in _Wilhelm Worringer_ , ed. Hannes B\u00f6hringer and Beate S\u00f6ntgen (Munich: Wilhelm Fink, 2002), 119\u201340.\n\n For historical details on Worringer and Ernst, see J\u00f6rgen Sch\u00e4fer, _Dada K\u00f6ln: Max Ernst, Hans Arp, Johannes Theodor Baargeld und ihre literarischen Zeitschriften_ (Wiesbaden: Deutscher-Universit\u00e4ts-Verlag, 1993), 149; for Worringer's impact in general, see Magdalena Bushart, _Der Geist der Gotik und die expressionistische Kunst: Kunstgeschichte und Kunsttheorie 1911\u20131925_ (Munich: S. Schreiber, 1990); on the function of the crystal mythos in modern art, see Regina Prange's fundamental works _Das Kristalline als Kunstsymbol Bruno Taut und Paul Klee: Zur Reflexion des Abstrakten in Kunst und Kunsttheorie der Moderne_ (Hildesheim: G. Olms, 1991); and \"Hin\u00fcberbauen in eine jenseitige Gegend: Paul Klees Lithographie 'Der Tod f\u00fcr die ldee' und die Genese der Abstraktion,\" _Wallraf-Richartz-Jahrbuch_ 54 (1993): 281\u2013314.\n\n \"Subjektiv-Willk\u00fcrlichen und nur Individuell-Bedingten\"; Wilhelm Worringer, \"Entwicklungsgeschichtliches zur modernsten Kunst,\" in _Im Kampf um die Kunst_ (Munich: Piper, 1911), 94; on the nomothetic character of primitive abstraction, see Worringer, _Abstraktion und Einf\u00fchlung_ (M\u00fcnchen: R. Piper, 1908), 52.\n\n First published in _K\u00f6lner Tagblatt_ , January 7, 1917; reprinted in Wulf Herzogenrath, ed., _Max Ernst in K\u00f6ln Die rheinische Kunstszene bis 1922_ , exhibition catalog, ed. Wulf Herzogenrath (Cologne: K\u00f6lnischer Kunstverein, 1980), 87. Surely it's no accident that \"Vergleichung,\" unlike other articles from the war years, is not included in the bibliography of the collection _Beyond Painting_ , published in 1948 when Ernst was in exile in America.\n\n \"m\u00e4nnlichen Geistes\"; \"reinen Weltformung.\"\n\n \"jederzeit zur Empf\u00e4ngnis gespreizten.\"\n\n \"Seitdem der Kubismus lebt, ist die Welt sch\u00f6ner, willenhafter: Geist von unserm Willen.\"\n\n \"Max Ernst (dzt. im Felde).\"\n\n Christine Brocks, \"Der Krieg auf der Postkarte: Feldpostkarten im Ersten Weltkrieg,\" in _Der Tod als Maschinist: Der industrialisierte Krieg 1914\u20131918_, ed. Rolf Spilker and Bernd Ulrich, exhibition catalog (Bramsche: Rasch, 1998), 154\u201363. In general, on military mail and the discourse of testimony in the First World War, see Ulrich's fundamental examination, _Die Augenzeugen: Deutsche Feldpostbriefe in Kriegs- und Nachkriegszeit 1914\u20131933_ (Essen: Klartext, 1997).\n\n \"Voll elementarer Wucht sind Granattrichter innerhalb D\u00f6rfern. Alles in der Umgebung scheint der Dynamik dieser gewaltigen symmetrischen Trichter zu unterliegen. Es sind die Augenh\u00f6hlen der Erde, was darum herum kreiselt sind ihre schmerzlich phantastischen Linien. H\u00e4user sind das nicht mehr, niemand glaubt das im Ernst. Es sind Lebewesen von besonderer Art mit eigenen Gesetzen und Lebensbedingungen. Es sind lauter L\u00f6cher mit Steinen herum, oder lauter Skelette. Es ist eine eigenartige seltene Sch\u00f6nheit, die hier redet.\" Ulrike R\u00fcdiger, _Gr\u00fc\u00dfe aus dem Krieg: Die Feldpostkarten der Otto-Dix-Sammlung in der Kunstgalerie Gera_ , exhibition catalog (Gera: Kunstgalerie, 1991).\n\n \"Dix hat erlebt\u2014so wie es 1914 Meier-Graefe von den K\u00fcnstlern gefordert hat.\" Quotation from Wolfgang Schr\u00fcck-Schmidt, \"Der Schicksalsweg des Sch\u00fctzengrabens,\" in _Otto Dix: Zume 100 Geburtstag: 1891\u20131991_, exhibition catalog (Stuttgart: Staatsgalerie, 1991), 161\u201364.\n\n Ernst Kallai, \"D\u00e4monie der Satire,\" _Kunstblatt_ 11 (1927): 97ff.: \"Das Sch\u00fctzengrabenbild k\u00f6nnte ebensogut der Gegenstand h\u00f6chster Anbetung eines fanatischen Kriegsgottverehrers, als pazifistisches Propagandamittel sein.\" (The picture of the trenches could just as well be the object of devotion of a fanatical worshipper of the god of war, as it could be a vehicle of pacificist propaganda.)\n\n Ulrich, _Die Augenzeugen_ , 14\u201334.\n\n Doherty, \"'See: We Are All Neurasthenics!'\"\n\n Hans Richter, _Dada: Art and Anti-Art_ , trans. David Britt (New York: Thames and Hudson, 1997), 117; cf. Ester Milman's analysis, \"Photomontage, the Event, and Historism,\" in _\"Event\" Arts and Art Events_ , ed. Stephen Foster (Ann Arbor, MI: UMI Research Press, 1988), 222.\n\n \"Que peut-il contre la vie militaire\u2014sa stupidit\u00e9, sa laideur, sa cruaut\u00e9? Hurler, jurer, vomir de rage ne servent \u00e0 rien.\" Max Ernst, _\u00c9critures_ (Paris: Gallimard, 1970), 25; this self-image of a resigned soldier who takes refuge in irony is also shown in the reprinted letter from the field and the photographs, p. 26ff.\n\n \"Diese sch\u00f6ne Winterlandschaft ist die Kroekbergferme. Zwischen dem schlanken Giebel links und dem stumpfen Turm in der Mitte seht Ihr ein schr\u00e4ges Dach. Unter diesem f\u00fchre ich Krieg. Das Haus auf halber H\u00f6he ist die Funkerstation. Links die romantische Felsgrotte ist der Eingang zu einem tiefen unterirdischen Labyrinth, unserer Tapferkeitsh\u00f6hle f\u00fcr den Fall einer Beschie\u00dfung. Dort kann uns keiner nich [ _sic_ ]. Herzl. Gr\u00fc\u00dfe u. K\u00fcsse Euer Max.\" For publication of this and the following postcard as well as additional important information on Ernst in the First World War, I am indebted to Pech, \"Da\u00df ich nicht tot bin, freut mich.\"\n\n See Paul Fussell's thoughts on the function of preprinted postcards, which were used by the British specifically to convey a message of survival without having to report on the cruelty of war. At the same time and above all, they made the work of the censoring authority easier, as picture postcards also did. Fussell, _The Great War and Modern Memory_ , 181\u201383.\n\n Spies, _Max Ernst Collagen_ , 67.\n\n _Geschichtsbild_.\n\nEXCURSUS\n\n See also Werner Spies, _Max Ernst Collagen: Inventar und Widerspruch_ (K\u00f6ln: DuMont Schauberg, 1974), 124ff.\n\n The latter occurs in Bachelard's writings about the earth. Gaston Bachelard, _La Terre et les R\u00eaveries de la volont\u00e9_ (Paris: Librarie J. Corti, 1948); and _La Terre et les R\u00eaveries du repos_ (Paris: Librarie J. Corti, 1948).\n\n Elizabeth Legge, _Max Ernst: The Psychoanalytic Sources_ (Ann Arbor, MI: UMI Research Press, 1989), 31\u2013103.\n\n William Camfield, _Max Ernst: Dada and the Dawn of Surrealism_ , exhibition catalog (Museum of Modern Art, New York, 1993), 150 ff. (with previous literature).\n\n The first version of _Woman, Old Man, and Flower_ , probably made in 1923, of which a photo has been preserved, was a purely polemical picture. Werner Spies and G\u00fcnter Metken, _Max Ernst: \u0152uvre-Katalog_ (K\u00f6ln: M. DuMont Schauberg, 1975), no. 659. It shows three father figures, a fat one with a mustache, which resembles Ernst's real father; behind him, the father \/ spinning top taken from the painting _Ubu Imperator_ ; and a father creature made from empty hulls and clothing. They are resting on a stage, in front of and underneath which lies a nude, also formed from an empty hull. Ernst probably chose to rework the painting because he was unsatisfied both with a satirical picture about substanceless father figures and with the simple antagonism of the composition.\n\n Legge, _Max Ernst: The Psychoanalytic Sources_ , 73ff.\n\n In David Wellbery's words regarding Louis Marin's concept of the simulacrum. Wellbery, \"Verzauberung: Das Simulakrum in der romantischen Lyrik,\" in _Mimesis und Simulation_ , ed. Andreas Kablitz and Gerhard Neumann (Freiburg: Rombach, 1998), 452.\n\n Freud, _Studienausgabe_ (Frankfurt am Main: Fischer, 1994), 8:87, 108\u201310.\n\n \"schichtgestein naturgabe aus gneis lava isl\u00e4ndisch moos 2 sorten lungenkraut 2 sorten dammriss \/ herzgew\u00e4chse b) dasselbe in fein poliertem k\u00e4stchen etwas teurer\" (stratified rocks, nature's gift of gneiss lava icelandic moss 2 kinds of lungwort 2 kinds of ruptures of the perineum \/ heart polyp b) the same thing in a finely polished box somewhat more expensive).\n\n The \"breaking away\" of the platform and the opening of an abyss between the lower edge of the picture and the picture's motif have been specifically diagnosed in the paintings of C\u00e9zanne and Picasso. See chapter 1, as well as the fundamental works by Krauss, \"The Motivation of the Sign,\" 266\u201371; and Wolfram Pichler, \"Schminke \/ Leinwand \/ Caravaggio \/ Goya\" (Ph.D. diss., University of Vienna, 1999), 107\u201329. Already with Picasso, it can be observed that this unstable zone is filled with obscene motifs in order parodically to reject the lost corporeality of the represented world. His collage _Au Bon March\u00e9_ (fig. 54) leads to the world of women but is doubly removed from the female body, since it seems as if it were made by unfolding the packaging in which ladies' underwear are sold. In the center of the picture's lower edge, white paint has been thickly applied using a stencil, and on top of this layer of paint, the following sequence of letters has been collaged: \"Lun B TROU ICI.\" Since this sequence appears on a box of ladies' underwear, it is only reasonable to deduce from it an obscene remark ( _trou ici_ , discounting the letters that have been cut away, means \"hole here\"). During the \"Picasso and Braque Symposium\" held in 1989 at the Museum of Modern Art, New York, Rosalind Krauss used the example of _Au Bon March\u00e9_ to provoke a discussion about the question of whether an iconographic reading that discovers in Picasso's paintings mainly \"schoolboy dirty jokes\" (Robert Rosenblum) is appropriate, or whether one should not primarily concentrate on the formal problem of the picture's lower edge. For me, there is no doubt that Krauss has the right approach: \"It happens in both Picasso and Braque . . . that there _is_ a hole\u2014formally speaking, now\u2014at the bottom of those paintings, just below the table ledge: a half-moon-like vacuum between the horizontal of the table and the edge of the frame. Because no pictorial event is going on there, this emptiness is often filled by adding a key and a keyhole. . . . To say, 'There's a hole here,' is thus a specific acknowledgement\u2014at least in part\u2014of one of the generic problems of the still life as a formal structure. For Picasso, there's probably also the added erotic association of the key in the lock; I wouldn't argue with that.\" In William Rubin, Kirk Varnedoe, and Lynn Zelevansky, eds., _Picasso and Braque: A Symposium_ (New York: Museum of Modern Art, 1992), 82. To this analysis, only the following can be added: the erotic association in no way contradicts the formal \/ analytical reading, nor is it simply additional information. Rather, the obscene sequence of letters is placed exactly where the heaviness, tangibility, and corporeality of painting is decided, near the picture's lower edge, where, in the illusionistic still life, a platform would have ensured the physical presence of the represented objects. In a collage that refers iconographically and formally to surrogates of the corporeal world (to underwear and its unfolded packaging), the dirty joke performs the task of addressing the absent body.\n\n Legge, _Max Ernst: The Psychoanalytic Sources_.\n\n To the left, blue and red veins, to the right, a fragmented blood vessel. The earth's interior in _Woman, Old Man, and Flower_ may also allude to the cloacal myth, by way of a joke: the eyepiece through which the old man gazes with his eyes closed could just as well be a nosepiece. The vertical cloud speaks in favor of this possibility.\n\n Regarding \"offstage\" in surrealism, see Ralph Ubl, \"Das Gem\u00e4lde als medialer Schwellenraum: Andr\u00e9 Breton, Giorgio de Chirico und der Gebrauch toter Bilder,\" in _Inszenierungen in Bild und Schrift_ , ed. Gerhard Neumann and Claudia \u00d6hlschl\u00e4ger (Bieleld: Aisthesis, 2004).\n\nCHAPTER FIVE\n\n Werner Spies, _Loplop_ , 44\u201347, 108ff.\n\n On Ernst as a sculptor, see Werner Spies, ed., _Max Ernst: Skulpturen, H\u00e4user, Landschaft_ , exhibition catalog (K\u00f6ln: DuMont, 1998); and Werner Spies and G\u00fcnter Metken, _Max Ernst: \u0152uvre Katalog_ (K\u00f6ln: M. DuMont Schauberg, 1975), nos. 2103\u201323.\n\n Max Ernst, \"Au del\u00e0 de la peinture,\" _Cahiers d'Art_ 11 (Paris, 1936).\n\n See \"Recherches exp\u00e9rimentales,\" in _Le Surr\u00e9alisme au Service de la R\u00e9volution_ 6 (May 15, 1933): 10\u201323.\n\n Andr\u00e9 Breton, _\u0152uvres compl\u00e8tes_ , ed. Marguerite Bonnet et al. (Paris: Gallimard, 1988\u201399), 1:181.\n\n Andr\u00e9 Breton, _Message automatique_ , in _\u0152uvres_ 2:377; Breton, _L'Amour fou_ , in ibid., 2:752. On the surrealist effective history of the wall stains, see Spies, _Loplop_ , 108; Christa Lichtenstern, _Metamorphose vom Mythos zum Prozessdenken: Ovid-Rezeption, surrealistische \u00c4sthetik, Verwandlungsthematik der Nachkriegskunst_ (Weinheim: VCH, 1992), 156\u201363.\n\n Robert Short, \"The Politics of Surrealism,\" _Journal of Contemporary History_ 1 (1966): 3\u201325.\n\n \"Avant sa plong\u00e9e, nul scaphandrier ne sait ce qu'il va rapporter. Ainsi, le peintre n'a pas le choix de son sujet. S'en imposer un, f\u00fbt-il le plus subversif, le plus exaltant et le tra\u00eeter d'une mani\u00e8re acad\u00e9mique, ce sera contribuer \u00e0 une oeuvre de faible port\u00e9e r\u00e9volutionnaire. [. . .] Le contenu id\u00e9ologique\u2014manifeste ou latent\u2014ne saurait d\u00e9pendre de la volont\u00e9 consciente du peintre.\" (Before his dive, no deep-sea diver knows what he will bring back. Likewise, a painter cannot choose his subject. Imposing one, be it the most subversive, the most exalted, and treating it in an academic manner, will lead to a work of little revolutionary scope. . . . Ideological content\u2014manifest or latent\u2014cannot depend on the conscious will of the painter.) Max Ernst, _\u00c9critures_ (Paris: Gallimard, 1970), 401ff.\n\n \"It is now beyond question that Surrealist works will share the same lot as all previous works that are _historically situated_. The climate of Benjamin Peret's poetry or Max Ernst's painting will be the very climate of life.\" Andr\u00e9 Breton, _Manifestoes of Surrealism_ , trans. Richard Seaver and Helen R. Lane (Ann Arbor: University of Michigan Press, 1969), 233.\n\n Jos\u00e9 Pierre, ed., _Tracts surr\u00e9alistes et d\u00e9clarations collectives I, 1922\u20131939_ (Paris: Terrain vague, 1980\u201382), 205; see Vincent Kaufmann's excellent rendition, _Po\u00e9tique des groupes litt\u00e9raires (Avantgardes 1920\u20131970)_ (Paris: Presses universitaires de France, 1970), 115\u201320.\n\n Breton, _\u0152uvres_ 2:4\u201327; Pierre, _Tracts_ , 223\u201328.\n\n Ulrich Vogt, _Le point noir: Politik und Mythos bei Andr\u00e9 Breton_ (Frankfurt am Main: P. Lang, 1982), 88\u201390; Louis Janover, _La R\u00e9volution surr\u00e9aliste_ (Paris: Plon, 1989), 162: \"Les d\u00e9bats avec le PC ont \u00e9t\u00e9 pour le surr\u00e9alisme de la plus haute importance en le r\u00e9v\u00e9lant \u00e0 lui-m\u00eame.\" (The disputes with the CP were of utmost importance to surrealism for its own self-revelation.)\n\n Andr\u00e9 Breton, _Communicating Vessels_ , trans. Mary Ann Caws (Lincoln: University of Nebraska Press, 1990), 25.\n\n See Ralph Ubl, \"Das Gem\u00e4lde als medialer Schwellenraum: Andr\u00e9 Breton, Giorgio de Chirico und der Gebrauch toter Bilder,\" in _Inszenierungen in Bild und Schrift_ , ed. Gerhard Neumann and Claudia \u00d6hlschl\u00e4ger (Bieleld: Aisthesis, 2004).\n\n Breton, _Communicating Vessels_ , 40.\n\n On the alterity of Germany, see Ladislas Mysyrowicz, \"L'Image de l'Allemagne nationalesocialiste \u00e0 travers les publications fran\u00e7aises des ann\u00e9es 1933\u20131939,\" in _Les relations francoallemandes, 1933\u20131939_ (Paris: \u00c9ditions du centre national de la recherche scientifique, 1976), 117\u201336.\n\n This article was in fact published in the _Oberbadischen Volksblattes_ on February 12, 1934 (on the first page). Ernst cites a French translation: \"DU SUPERZ\u00c8LE POLITIQUE. _Imagination maladive_. Depuis quelque temps, les autorit\u00e9s re\u00e7oivent une certaine quantit\u00e9 de cartes postales, de tableaux et d'affiches dans lesquels on flaire de la propagande communiste camoufl\u00e9. Dans la chevelure d'une t\u00eate, quoiqu'il s'agissent d'une photographie, on a voulu d\u00e9couvrir la figure de L\u00e9nine, dans l'oreille de la m\u00eame t\u00eate une image obsc\u00e8ne. On a trouv\u00e9 cach\u00e9s dans une affiche, un cr\u00e2ne fracass\u00e9 et une t\u00eate communiste. Il est \u00e0 remarquer que, pour jouir de cette affiche-devinette, coll\u00e9e et accroch\u00e9e de fa\u00e7on habituelle, le spectateur serait oblig\u00e9 de marcher sur la t\u00eate. Les lieux officiels s'opposent \u00e0 un tel superz\u00e8le qui pourrait alarmer inutilement la population et nuire \u00e0 des int\u00e9r\u00eats l\u00e9gitimes. Les lieux officiels ont re\u00e7u l'ordre d'opprimer avec toute leur \u00e9nergie ces agissements insens\u00e9s qui facilement pourraient d\u00e9g\u00e9n\u00e9rer en psychose dangereux.\" Ernst, \"Au del\u00e0 de la peinture,\" 42. Thanks to Ms. Danuta Thiel-Melerski from the City Archive of L\u00f6rrach.\n\n On the controversy surrounding Dal\u00ed's project to situate Hitler from a surrealist point of view, see Karin von Maur, \"Breton et Dali, \u00e0 la lumi\u00e8re d'une correspondance in\u00e9dite,\" in _Andr\u00e9 Breton et le surr\u00e9alisme_ , exhibition catalog (Paris: Mus\u00e9e National d'Art Moderne, Centre Georges Pompidou, 1991), 196\u2013202.\n\n \"Quelle est la plus noble conqu\u00eate du collage? C'est l'irrationnel. C'est l'irruption magistrale de l'irrationnel dans toutes les domaines de l'art, de la po\u00e9sie, de la science, dans la mode, dans la vie priv\u00e9e des individus, dans la vie publique des peuples.* Qui dit collage, dit l'irrationnel.\" Ernst, \"Au del\u00e0 de la peinture,\" 42.\n\n Ernst, _Beyond Painting and Other Writings_ (New York: Schultz, 1948), 17. I would like to thank Elizabeth Tucker for this observation.\n\n \"Bereits 1933, im Jahr der Macht\u00fcbernahme durch die Nationalsozialisten, entstand die erste Fassung von _Europa nach dem Regen_ von Max Ernst. Wie auf eine Landkarte blickt man von oben auf einen verw\u00fcsteten, deformierten Kontinent. Die Gebirge sind eingeebnet, die K\u00fcsten der L\u00e4nder nicht mehr wiederzuerkennen. Die schrundige, zerfurchte Erdoberfl\u00e4che scheint ohne jede Spur von Leben\u2014Europa nach dem n\u00e4chsten Krieg.\" (As early as 1933, the year in which National Socialism took power, Max Ernst's first version of _Europe after the Rain_ was made. As if looking at a map, one surveys from above a wasted, deformed continent. The mountains have been leveled; the coasts of countries are no longer recognizable. The earth's desiccated, rutted surface seems to be without any trace of life\u2014Europe after the next war.) Annegret J\u00fcrgens-Kirchhoff, _Schreckensbilder: Krieg und Kunst im 20. Jahrhundert_ (Berlin: Reimer, 1993), 313ff. Werner Haftmann also reaches this conclusion, _Verfemte Kunst: Bildende K\u00fcnstler der inneren und \u00e4usseren Emigration in der Zeit des Nationalsozialismus_ (K\u00f6ln: DuMont, 1986), 162.\n\n \"Das Thema L'Europe apr\u00e8s la pluie, das in verschiedenen Variationen entsteht, setzt zuerst 1933 ein, in einer reliefartigen, r\u00e4tselhaften Landkarte, die eine Vorausahnung jener verwirrenden territorialen Verschiebungen spiegelt, die noch kommen sollten. Es ist eine scheinbar altgewohnte, aber hier aus v\u00f6llig dissoziierten Elementen ('le depaysement') neu aufgebaute Geographie, die ihr verr\u00fcckendes und verr\u00fccktes Spiel mit dem Beschauer treibt.\" Carola Giedion-Welcker, \"Max Ernst,\" in _Max Ernst_ , exhibition catalog (K\u00f6ln: Wallraf Richartz Museum, 1963), 15.\n\n \"Robert Lebel: Pour _L'Europe apr\u00e8s la pluie_ , partais-tu \u00e9galement d'un 'irritateur' visuel?\u2014Max Ernst: Pendant les prises de vues de _L'Age d'or_ de Bu\u00f1uel j'ai remarqu\u00e9 au studio de Billancourt des panneaux de contre-plaqu\u00e9 irr\u00e9guli\u00e8rement couverts de peinture et de pl\u00e2tre. On les utilisait pour repr\u00e9senter sc\u00e9niquement les murs. J'\u00e9tais donc de nouveau devant le fameux mur de L\u00e9onard de Vinci qui a jou\u00e9 un si grand r\u00f4le de mes _Visions de demi-sommeil_. J'ai pu obtenir de Bu\u00f1uel le don de tous ces panneaux et ils ont servi de fond \u00e0 _L'Europe apr\u00e8s la pluie_ ainsi qu'\u00e0 _Loplop pr\u00e9sente une jeune fille_ , _Loplop pr\u00e9sente la mer en cage_, etc. sur lesquels les reliefs de pl\u00e2tres sont visibles.\" Ernst, _\u00c9critures_ , 427. (Robert Lebel: For _Europe after the Rain_ , did you also start from a visual \"irritant\"?\u2014Max Ernst: During the filming of Bu\u00f1uel's _Golden Age_ at the Billancourt studios, I noticed the plywood panels irregularly covered with paint and plaster. They are used for representing walls in scenery. I was thus once again standing before Leonardo da Vinci's famous wall, which had played such an important role in my \"Visions of Half-Sleep.\" I was able to get from Bu\u00f1uel the gift of all these panels, and they served as the foundation for _Europe after the Rain_ as well as _Loplop Presents a Young Girl_ [ _Loplop pr\u00e9sente une jeune fille_ ], _Loplop Presents the Sea in a Cage_ [ _Loplop pr\u00e9sente la mer en cage_ ] etc., in which the plaster reliefs are visible.)\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , no. 1710.\n\n See Claude Gandelman's fine analyses, _Reading Pictures: Viewing Texts_ (Bloomington: Indiana University Press, 1991), chap. 7.\n\n John Russell, _Max Ernst: Life and Work_ (New York, H. N. Abrams 1967), 113. Werner Hofmann also stressed the perfect mortification of the surface as a crucial, anti-romantic characteristic. See Hofmann, \"Max Ernst und die Tradition,\" in _Max Ernst: Das Innere Gesicht_ , exhibition catalog (Hamburg: Kunsthalle Hamburg, 1970), 18.\n\n See Leo Steinberg, _Other Criteria: Confrontations with Twentieth-Century Art_ (Chicago: University of Chicago Press, 2007), 82\u201391.\n\n Breton, _\u0152uvres_ 1:369\u201370.\n\n Spies and Metken, _Max Ernst: \u0152uvre-Katalog_ , nos. 681, 683ff., 1488\u201391, and elsewhere.\n\n Ernst's pictures of hordes can also be interpreted in this context. Ibid., nos. 1105\u201332.\n\n \"Il est bien, il est heureux que des exp\u00e9ditions sovi\u00e9tiques, apr\u00e8s tant d'autres, prennent aujourd'hui le chemin du P\u00f4le. C'est l\u00e0 encore, pour la R\u00e9volution, une mani\u00e8re de nous faire part de sa victoire. Qui oserait m'accuser de retarder le jour o\u00f9 cette victoire doit appara\u00eetre comme totale en montrant du doigt quelques autres zones, non moins anciennes et non moins belles, d'attraction?\" Breton, _\u0152uvres_ 1:190. (It is propitious and good that Soviet expeditions, after so many others, today are headed for the Pole. It's another way for the revolution to announce its victory to us. Who would dare accuse me of postponing the day when this victory must appear total, by pointing out several other compelling spots, no less ancient and no less beautiful?)\n\n \"Delivered at last, forests flowed down to the dwellings of man and ate them. Stones exploded. Plants flew unconcernedly as though they had done nothing else during their entire existence. Wakened volcanes peered at each other above the oceans, advanced toward one another and conjoined in lava loves under the kisses of craters as beneficent as rain. Waters were no longer united and coherent, but dissipated throughout the universe.\" Louis Aragon, _The Adventures of Telemachus_ , trans. R. and J. Hubert (Boston: Exact Change, 1997), 136.\n\n \"La surr\u00e9alit\u00e9 sera fonction de notre volont\u00e9 de d\u00e9paysement complet de tout.\" (Surreality will be a function of our will to complete disorientation of everything.) Breton, _Avis au lecteur pour La Femme 100 t\u00eates_, in _\u0152uvres_ 2:305. Not only in this text about Ernst, but also in his first one from the same year (1921), Breton celebrated the collagist as a master of \"d\u00e9paysement\": _\u0152uvres_ 1:246; as did Louis Aragon, \"Max Ernst, peintre des illusions,\" in _Les Collages_ (Paris: Hermann, 1993), 37.\n\n Gilles Deleuze, \"What Children Say,\" in _Essays Critical and Clinical_ , trans. Daniel W. Smith and Michael A. Greco (Minneapolis: University of Minnesota Press 1997), 61\u201367; Christine Buci-Glucksmann, _L'oeil cartographique de l'art_ (Paris: \u00c9ditions Galil\u00e9e, 1996).\n\n Spies, _Loplop_ , 11ff.\n\n Exterior circumstances support the assumption that Hitler's seizure of power and the concomitant danger of war led Ernst back to Dada and the previous war. In general, as we have seen, the political thinking of the surrealists was heavily influenced by their experiences of the First World War. French patriotism and imperialism seemed no less dangerous to them than National Socialism. A photograph published in 1934 in the Belgian surrealist magazine _Documents_ provides anecdotal but illuminating documentation that Ernst was looking back at 1914\u201318. The photograph shows the artist with a bandaged head and arm. The caption reads \"VINGT ANS AVANT: Max Ernst en 1914\" (\"TWENTY YEARS BEFORE: Max Ernst in 1914\"). J\u00fcrgen Pech has shown that the publication of this picture casts a finely woven net of surrealist allusions. Ernst with a head wound (which he did not incur in 1914) leads to Apollinaire's famous head wound, which de Chirico had prophesied in 1914 in his portrait of Apollinaire, which shows the poet as an \"homme-cible\" (shooting-gallery target). Pech, \"Was der Taucher vor dem Sprung nicht wissen kann: Giorgio de Chirico und Max Ernst,\" in _Arnold B\u00f6cklin, Giorgio de Chirico, Max Ernst: Eine Reise ins Ungewisse_ , exhibition catalog, ed. Guido Magnaguago and Juri Steiner (Bern: Z\u00fcrich Kunsthaus, 1998), 324ff.\n\nAFTERWORD\n\n Walter Benjamin, _The Correspondence of Walter Benjamin, 1910\u20131940_, ed. Gershom Scholem and Theodor W. Adorno, trans. Manfred R. Jacobson and Evelyn M. Jacobson (Chicago: University of Chicago Press, 1994), 491.\n\n Walter Benjamin, \"Dream Kitsch\" and \"Surrealism: The Last Snapshot of the European Intelligentsia,\" in _Selected Writings_ , trans. Rodney Livingstone et al., ed. Michael W. Jennings, Howard Eiland, and Gary Smith (Cambridge, MA: Belknap Press of Harvard University Press, 2005), vol. 2, part 1: 1927\u20131930, pp. 3\u20135, 202\u201321.\n\n Hal Foster, _Compulsive Beauty_ (Cambridge, MA: MIT Press, 1993), chap. 6.\n\n Brigid Doherty, \" _Max Ernst_ : A Retrospective,\" _Artforum_ (September 2005): 295\u201397, 332, 347.\n\n Brigid Doherty's interpretation (in ibid.) is based on the assumption that Benjamin wrote about _R\u00e9p\u00e9titions_ as an overpainting. However, Burckhardt Lindner rightly observes that Benjamin could not have known on the basis of the published volume that the work was an overpainting. Benjamin also does not discuss the picture's technique. Lindner, \"Versuch \u00fcber Traumkitsch: Die blaue Blume im Land der Technik,\" in _Walter Benjamin und die romantische Moderne_ , ed. Heinz Br\u00fcggemann and G\u00fcnter Oesterle (W\u00fcrzburg: K\u00f6nigshausen und Neumann, 2009), 223.\n\n Benjamin, \"Dream Kitsch,\" 4.\n\n Ibid., 5.\n\n Walter Benjamin, _The Arcades Project_ , trans. Howard Eiland and Kevin McLaughlin (Cambridge, MA: Harvard University Press, 1999), 831. On the affinity between \"Dream Kitsch\" and the first notes for _The Arcades Project_ , see Lindner, \"Versuch \u00fcber Traumkitsch,\" 229\u201346.\n\n \"Notes on montage in my journal. Perhaps, in this same context, there should be some indication of the intimate connection that ffiexists> between the intention making for nearest nearness and the intensive utilization of refuse\u2014a connection in fact exhibited in montage.\" Benjamin, _The Arcades Project_ , 861.\n\n Ibid., 866.\n\n Ibid., 874ff.\n\n On Benjamin and images, see Sigrid Weigel, _Walter Benjamin: Die Kreatur, das Heilige, die Bilder_ (Frankfurt am Main: Fischer Verlag, 2008), 265\u201396.\n\n \"Auf alle F\u00e4lle will ich Dir einen recht postwendenden, noch viel mehr herzlichen Dank f\u00fcr das neue Meisterst\u00fcck sagen, das soeben eintrifft. Es hat meinem Entschluss, mir den Besuch der Max Ernst-Ausstellung bei Flechtheim zu sparen, 'den letzten Schliff und die letzte Rundung' gegeben (um mit der Berner Radfahrer Zeitung zu sprechen). Seine Vorderseite ist der besten neuen Malerei (und seine R\u00fcckseite ihrer Kritik) um etliche Jahre voraus.\" Walter Benjamin, _Gesammelte Briefe_ ed. Christoph G\u00f6dde and Henri Lonitz (Frankfurt am Main: Suhrkamp, 1995\u20132000), 3:451, translated here.\n\n Ibid., 433. In general, on Benjamin's writing and writing materials, see the fundamental study by Davide Giuriato, _Mikrographien: Zu einer Poetologie des Schreibens in Walter Benjamins Kindheitserinnerungen (1932\u20131939)_ (Munich: W. Fink, 2006).\n\n Benjamin, _The Correspondence of Walter Benjamin_ , 345.\n\n Benjamin, \"Surrealism: The Last Snapshot of the European Intelligentsia,\" 211.\n\n Walter Benjamin, _The Work of Art in the Age of Its Technological Reproducibility, and Other Writings on Media_ , ed. Michael W. Jennings, Brigid Doherty, and Thomas Y. Levin (Cambridge, MA: Belknap Press of Harvard University Press, 2008), 38.\n\n Ibid., 53.\n\n Ibid., 35.\n\n Benjamin, _Gesammelte Schriften_ , ed. Rolf Tiedemann and Hermann Schweppenh\u00e4user (Frankfurt am Main: Suhrkamp, 1972\u201389), 7:2, 675\u201377.\n\n The connection here is an etymological one. The Middle High German _Mal_ had the sense \"stain, mark, sign, sin, or shame.\" Modern usage includes _Muttermal_ (birthmark) and _Wundmal_ (scar), as well as _Schandmal_ (stigma or blemish) and _Kainsmal_ (mark of Cain). The verb _malen_ , in Modern German meaning \"to paint,\" formerly had the sense \"to apply marks or signs.\"\u2014Trans.\n\n For more of Benjamin's thoughts on shame, blushing, and colors, see his early notes on fantasy and color ( _Gesammelte Schriften_ 6:109\u201329) and \"Der Regenbogen: Gespr\u00e4ch \u00fcber die Phantasie\" ( _Gesammelte Schriften_ 7:19\u201326). Patrick Primavesi gives a persuasive reconstruction of these thoughts in _Kommentar, \u00dcbersetzung, Theater in Walter Benjamins fr\u00fchen Schriften_ (Frankfurt: Stroemfeld\/Nexus 1998), 307\u201314, and \"Darstellung im Vergehen: Zum Begriff der Scham bei Walter Benjamin,\" in _Global Benjamin: Internationaler Walter Benjamin Kongre\u00df 1992_ , ed. Klaus Garber (Munich: Wilhelm Fink Verlag, 1999), 3:1610\u201320.\n\n Benjamin, _The Work of Art in the Age of Its Technological Reproducibility_ , 48.\n\n Ibid., 48ff. The German _Spielraum_ , literally \"space for play,\" has the sense of \"latitude,\" \"room to maneuver,\" or \"free play.\" In this passage, Jephcott et al. translate it as \"scope of play.\"\u2014Trans.\n\n Here, I am drawing entirely from Miriam Bratu Hansen, \"Room-for-Play: Benjamin's Gamble with Cinema,\" _October_ 109 (Summer 2004): 17: \"Within this anthropological-materialist framework, then, technology endows the collective with a new physis that demands to be understood and reappropriated, literally incorporated, in the interest of the collective; at the same time, technology provides the medium in which such reappropriaton can and must take place. Such a reflexive understanding of technology makes visible a different logic\u2014a logic of play in Benjamin's conception of the historic role of film.\"\n\n Similar to hopscotch.\u2014Trans.\n\n \"[Das] Mal im exaktesten Sinn des Wortes [ist] eine farbige Konfiguration die auf der Wand erscheint (aus ihr heraustritt oder auf sie geworfen wird).\" Benjamin, _Gesammelte Schriften_ , 7:2, 677.\n\n \"Die gegenw\u00e4rtige Krise der Malerei w\u00fcrde, in diese geschichtsphilosophische Perspektive versetzt also auf Ver\u00e4nderungen hinauslaufen, die auf eine Verk\u00fcmmerung des Mediums der Malerei, des Mediums, in welchem das Mal zuhause ist, schliessen lassen.\" Ibid.\n\n \"Die Graphik bildet die Welt so ab, da\u00df der Mensch sie beschreiten kann. Das Auge ihres Betrachters eilt seinem Fu\u00df voraus. Kein \u00dcbergang und keine Vermittlung f\u00fchrt vom Tafelbild zu einer Landkarte.\" Ibid., 7:2, 676.\nINDEX\n\n_Note_ : Page numbers followed by an _f_ refer to figures.\n\nAbraham, Karl, , 226n6\n\n\"Absences II\" (\u00c9luard), 109\u201310\n\nabstraction, 205nn4\u20135\n\n_Abstraction and Empathy_ (Worringer), ,\n\nAdorno, Theodor W., ,\n\n\"Agadir\" (Aragon), 137\u201338\n\n_Alarm Clock_ (Picabia), 26\u201328\n\n_Anatomy Lesson of Dr. Joan Deyman_ (Rembrandt),\n\n_Angelus Novus_ (Klee),\n\n_Anna Selbdritt_ (Leonardo da Vinci),\n\n_Anthologie Dada_ ,\n\nanti-mimesis, ,\n\nApollinaire, Guillaume, 236n36\n\n_Approaching Puberty . . . The Pleiades_ (Ernst),\n\nAragon, Louis, , ; on automatic writing, , ; automatic writing by, 211n5; on Ernst's visual metaphorics, , 209n44, 214n29; on the horizon, 227n13; on _miseen-sc\u00e8ne_ , 37\u201338; nature writing by, 78\u201379, 215n39; political ideology of, 176\u201377; on procedural logic, ; on war neurosis, 137\u201338\n\n_Arcades Project, The_ (Benjamin), 194\u201395\n\nArp, Hans, , ; on Duchamp's readymades, 207n26; foreword to _Natural History_ of, , 90\u201392, , , 217n57; irregular ink ovals of, , ; on Picabia's _Alarm Clock_ , 26\u201328; primordial mimesis of, 16\u201325\n\n_Arp Microgram 1:25,000_ (Ernst), 19\u201325, , ,\n\nArtaud, Antonin, 211n5\n\n_Artist's Depiction of the Earth during the Silurian Period_ (Riou), 94 _f_, , 1 _05f_\n\n_Au Bon March\u00e9_ (Picasso),  _68f_ , 232n10\n\n_Automatic Drawing_ (Masson), 73 _f_\n\nautomatic writing, 64\u201365, , , 211n5; Arp's commentary as, 91\u201392; Breton's views on, , , , 212n12; narrative clich\u00e9s in, ; open figurativity of, ; opening of prehistoric time in, 78\u201379; precipitates in,\n\nautomatism, 123\u201324, 224n42; artistic procedures in, 66\u201368, 212n15; in Breton's _Manifesto of Surrealism_ , ; commentary on, 87\u201392; of Ernst's frottage, 64\u201369; iconography of, 75\u201387; of Masson's drawings, 72\u201374; Morise on the photogram in, 66\u201368; opening of prehistoric time in, 78\u201379, 92\u2013105; the unconscious in, 66\u201368, , , 219n86; wide-ranging contextual fields of, 69\u201375. See also _Natural History_\n\n_Autumnal Geometry_ (de Chirico), 45\u201348\n\navant-garde, the, 100\u2013101; dismantling of painting in, 6\u20138; techniques of originality in, 4\u20137, 203n14\n\nBaargeld, Theodor,\n\nBachelard, Gaston, 232n2\n\nBataille, Georges,\n\nBauer, Gerd, 225n49\n\n_Before the Flood!_ (Fraas),\n\nBenjamin, Walter, , ; on painting and graphic art, 10\u201311, 192\u2013202, 204n25, 237n5; on unconscious production, 5\u20137\n\n_Beyond Painting_ (Ernst), , 179\u201380\n\n_Beyond the Pleasure Principle_ (Freud), 146\u201347\n\n_Bicycle Wheel_ (Duchamp), , 32 _f_\n\nBillancourt studio, 181\u201383\n\nBlanchot, Maurice, 212n13\n\nBloch, Ernst,\n\nblots and lines, , 16\u201325\n\nBlum, L\u00e9on,\n\nB\u00f6cklin, Arnold, 100\u2013101\n\nBois, Yve-Alain, , ,\n\n_Bowl with Fruit, Violin, and Wineglass_ (Picasso), 44\u201345\n\nBraque, Georges, 7\u20138, 232n10\n\nBreton, Andr\u00e9, 1\u20133; on automatic painting, ; on automatic writing, , , , 212n12; automatic writing by, 74\u201375, , , , , 211n5, 214nn27\u201328; on commercial success, , 212n12; on deep memory, , 219n84, 219n86; emblematic circle of, 79\u201381; on Ernst's illusionism, ; on the kingdoms of nature, , , ; political positions of, 176\u201379, , ; on precipitates, ; on psychoanalysis, ; on seeing resemblances, 174\u201376; on surrealist anti-communication, ; theory of parapraxis of, 177\u201379; on unconscious production, 5\u20137; on the unique flash, ; on war neurosis, 131\u201332, ; on World War I, . See also _Manifesto of Surrealism_\n\nB\u00fcrger, Peter, 203n14\n\nBusch, Wilhelm,\n\n_Caesar Antichrist_ (Jarry), 214n30\n\n_Caesar's Palette_ (Ernst), , 77 _f_, 214n30\n\n_Cahiers d'Art_ ,\n\n_Canalization of Refrigerated Gas, The_ (Ernst), , 30\u201332, 207n26\n\n_Capital of Pain_ (\u00c9luard),\n\ncartographic art, , 188\u201390. See also _Europe after the Rain_\n\ncataclysm and cleared spaces, 187\u201390, 236n36\n\n_Celebes_ (Ernst),\n\nCerteau, Michel de,\n\nC\u00e9zanne, Paul, 232n10\n\n_Chinese Nightingale, The_ (Ernst), 141\u201348, , , 228nn35\u201337\n\nChirico, Giorgio de. _See_ de Chirico, Giorgio\n\ncinema, 198\u2013200\n\ncircle, the, 79\u201381\n\nCohn, Alfred,\n\ncollage and readymades, 2\u20134; of de Chirico, 45\u201348; of Duchamp, 7\u20138, , 207n26; Ernst on the irrational in, 179\u201380; mimesis of the cut in, ; of Picasso, 7\u20138, 44\u201348, 183\u201385, 228n35; postcards of World War I as, 154\u201359; relationship to painting of, 7\u201311; techniques of originality in, 4\u20137. _See also_ frottage\n\ncollages of Ernst: collage novels of, ; foreign bodies of, 148\u201349; Loplop series of, , , 190\u201391, 220n88; mimesis of the cut in, 44\u201350, 145\u201346; overpaintings as, 40\u201342; pictorial underlay of, 40\u201344, 61\u201362, ; as post-catastrophic landscapes, 186\u201387; proto-surrealist poetics of, 141\u201349; references to wartime in, 138\u201345, 149\u201354, ; seeing resemblances in, 50\u201354; transformation of pears in, 48\u201354, , 163\u201364, 208n40, 209n46\n\n_Cologne Catalogue of Teaching Aids_ , 22 _f_, , 34 _f_, , ,  _25f_ , ,  _66f_\n\ncomb traces,\n\n_Communicating Vessels_ (Breton), 174\u201377\n\nCommunist Party of France, 176\u201380,\n\n_Compulsive Beauty_ (Foster),\n\n_Concerning the Spiritual Art_ (Kandinsky),\n\n_Condensation and Rainfall on the Primitive Globe_ (Riou), 94 _f_,\n\n_Confidences_ (Ernst), 54\u201358,\n\n_Court of the Dragon I, The_ (Ernst), 1 _75f_\n\nCrevel, Ren\u00e9, 99\u2013100,\n\ncubism, 9\u201311; Klee's and Benjamin's reactions to, 9\u201311; of Picasso, 7\u20138, 52\u201353; simultaneity ideal of,\n\nDada, 4\u20135, ; Benjamin's views on, ; disintegration of mimesis in, 15\u201325; Ernst's break towards surrealism from, 69\u201375, , 166\u201367, , 211n8, 213n23; Ernst's diagramatic illusionism in, , 26\u201337, 61\u201362, 206n19; Ernst's geologic iconography in, 37\u201344, 164\u201366; Ernst's proto-surrealist poetics in, 141\u201349; Ernst's as psychoanalytic catalysts, 127\u201330; Ernst's seeing resemblances in, 50\u201354, 208n40; foreign bodies of, 148\u201349; fossilized rigidity of surfaces in, 127\u201330; mechanical writing in, 213n23; permeability to historical events of, ; war-based photomontage in, 156\u201359; World War I references in, 138\u201349\n\n_Dada Outdoors_ ,\n\nDal\u00ed, Salvador,\n\n\"Dancing\" (\u00c9luard),\n\nDarwin, Charles,\n\nDavis, Whitney, 204n26\n\ndeath drive, 146\u201347\n\nde Certeau, Michel,\n\nde Chirico, Giorgio, , 65\u201366, 208n40; _Autumnal Geometry_ collage of, 45\u201348; Benjamin's commentary on, ; portrait of Apollinaire by, 236n36; on prehistory, 100\u2013102; still lifes of, 52\u201353, 209n46; unstable earth of,\n\nde Duve, Thierry, 7\u20138\n\nDelaunay, Robert,\n\nDeleuze, Gilles,\n\nDerain, Andr\u00e9,\n\nDerenthal, Ludger,\n\nDer Moderne Bund exhibition,\n\n_Der Sch\u00fctzengraben_ (Dix),\n\nDesnos, Robert, , 211n5\n\nDidi-Huberman, Georges,\n\ndisanalogy, , 204n26\n\nDix, Otto, 155\u201356, 158\u201359\n\nDoherty, Brigid,\n\ndragonflies, 103\u20135\n\n\"Dream Kitsch\" (Benjamin), 192\u2013202\n\n_Drowned Woman, A_ (Redon), 216n51\n\nDubuffet, Jean,\n\nDucasse, Isadore,\n\nDuchamp, Marcel, ; _Bicycle Wheel_ of, , 32 _f_; readymades of, 7\u20138, , 207n26\n\nD\u00fcrer, Albrecht,\n\nDuve, Thierry de, 7\u20138\n\nearth, the, 160\u201371, , 232n2, 232n10, 233n12\n\n_Earth Views the Earth, The_ (Ernst),\n\n\u00c9luard, Gala, , 105\u201310, 208n40\n\n\u00c9luard, Paul, , 105\u201310, 208n40, 211n5; announcement for _Natural History_ by, ; collaborations with Ernst of, 192\u201393, 195\u201397, 237n5; emblem books of, 79\u201381; iconography of,\n\nErnst, Max: autobiographical notes of, 2\u20133, 136\u201338, 203n6, 227n17; childhood hallucinations and memories of, 114\u201322; collaborations with \u00c9luard of, 106\u201310, 192\u201393, 195\u201397, 237n5; first retrospective show of, ; historical context of, 8\u201311; letters from the field of, 157\u201358; love triangle of, ; Maloja trip of 1934 of, 172\u201374; photograph of, 13\u201316; politics of the 1930s of, 176\u201380; recognition and praise of, ; World War I service of, 18\u201319, , , , 227n14\n\n\"Eulogy of M. De Buffon\" (Condorcet),\n\n_Europe after the Rain_ (Ernst), , 180\u201392, , 236n36, plate 4\n\n_Europe after the Rain II_ (Ernst), 188\u201390\n\n_Eve, the Only One Left to Us_ (Ernst), 84\u201387, , 216n51\n\nexperiments in technique, 1\u201312, 203n14; Aragon and Breton on, 1\u20133; historical framework of, 4\u20135, 177\u201378; psychoanalytic framework of, 4\u20135; raw materials of, 3\u20134; unconscious production in, 5\u201312. _See also_ collage and readymades; frottage; grattage and painting; overpaintings\n\n_Extraordinary Voyages_ (Verne),\n\n_False Positions_ (Ernst),\n\n_Farewell My Beautiful Land of Marie Laurencin_ (Ernst), 28\u201330\n\nFerenczi, S\u00e1ndor, , 226n6\n\n_Fields of Honor, Flood, Seismic Plants_ (Ernst), 134\u201336, , 227n17\n\nFiguier, Louis, , 96\u2013100, 103\u20135,  _06f_\n\nfilm, 198\u2013200\n\n_Flamingos, The_ (Ernst), , 229n45\n\nFlammarion, Camille,\n\nFoster, Hal, ,\n\nFoucault, Michel, ,\n\nFraas, Oskar,\n\nFreikorps,\n\nFrench Communist Party, 176\u201380,\n\nFreud, Sigmund, ; on the death drive, 146\u201347; dissimilar metaphors of, 204n26; on the etiology of neurosis, ; on female symbols, ; on geologic metaphorics and prehistory, , , 127\u201328; on _heimlich_ and _unheimlich_ , ; mystic writing pad of, ; on Oedipal love, 120\u201322, , 224n32; on the primal scene, 114\u201320, 126\u201327, 132\u201333, 223n12, 223n26; on the unconscious and automatism, 5\u20137, , , 122\u201330, 177\u201378, 224n42, 225n49; on war neurosis and trauma, 132\u201333, 226n4. _See also_ psychoanalytic theory\n\nfrottage, , , , 54\u201369, 213n18; analogies to typing of, 71\u201372; documentary uses of, , 213n18, 213n23; earth themes in, 160\u201371, , 232n2, 232n10, 233n12; in Ernst's pear series, 48\u201351, 163\u201364, 208n40, 209n46; Ernst's procedures for, 83\u201387; eye motifs in, 107\u201310, 216n51; as ghost-writing of nature, , 74\u201375; iconography and misunderstanding in, 75\u201387, ; mimesis of fossilization in, , 60\u201363, ; the naturalized cut in, ; passive scopophilia in, 121\u201323; poetics of the primal scene in, , 114\u201320, 223n12; seeing resemblances in, 52\u201354, 62\u201363, , 172\u201376; surrealist _trompe l'oeil_ in, 59\u201363, , 210n54; surrealist unconscious in, 123\u201330; textured underlay in, 54\u201358, 61\u201363, , , 170\u201371, 210n52, 215n44; transition to painting from, 174\u201376; war neurosis and, 134\u201338. _See also_ automatism; _Natural History_\n\n_Frozen Landscapes, Icicles, and Mineral Types of the Female Body_ (Ernst), 38\u201339, , , 71\u201372, , , plate 1\n\nFussell, Paul, 231n72\n\nGenette, G\u00e9rard,\n\ngeological and paleontological illustration, 93\u2013105; Freud on metaphorics of, , , 127\u201328; mimesis of fossilization in, , 60\u201363; overpaintings' imitation of, 37\u201344, 164\u201370\n\nGiedion, Sigfried,\n\nGiedion-Welcker, Carola,\n\nGilbert, Sandra,\n\n_Glance, A_ (Ernst), 81\u201383, ,\n\ngrattage and painting, , 6\u20138, 201\u20132; automatism of, ; Benjamin's commentary on, 10\u201311, 192\u2013202, 237n5; decalcomania technique in, 182\u201383; Ernst's transition from frottage to, 7\u201311, 174\u201376; _Paradise_ , ; passive scopophilia in, 121\u201323; premonitions of World War II in, 179\u201391, , 236n36; role of the unconscious in, 128\u201330\n\nGreenberg, Clement, 7\u20138\n\nGrosz, Georges, , 156\u201357,\n\n_Habits of Leaves, The_ (Ernst), , 89 _f_\n\nHaeckel, Ernst,\n\nhallucinations of the primal scene, , 114\u201320\n\nHansen, Miriam Bratu, 238n25\n\n_Hat Makes the Man, The_ (Ernst), , 15\u201316\n\n_Head_ (Picasso), 52\u201353\n\nHeartfield, John, , 156\u201357,\n\nHegel, Georg,\n\n_He Will Fall Far from Here_ (Ernst), 56 _f_, , 210n51\n\nHitler, Adolf, 178\u201379, 236n36\n\nHofmann, Werner, 210n52\n\n\"How to Force Inspiration\" (Ernst), , 117\u201318, ,\n\n_Hundred-Headed Woman Opens Her August Sleeve, The_ (Ernst), 172\u201374\n\n_Iceflower Shawl_ (Ernst), ,\n\niconography, 111\u201312; in Ernst's Dada works, 37\u201344, 164\u201366; of modern civilization, 88\u201390; of scatophilia, 145\u201347, 164\u201370; of the surrealist unconscious, 75\u201387,\n\nillusionism, 10\u201311, 15\u201316; Arp's response to, 24\u201325; Dadaist parodies of, 26\u201328; Ernst's engagement with, , 26\u201337, 206n19; in surrealist mimesis, 61\u201363\n\nimaginative seeing, 116\u201318\n\nimitation (as active re-creation), 15\u201316; Arp's response to, 17\u201325; Ernst's distorted forms of, 37\u201344. _See also_ mimesis\n\n_Immaculate Conception_ (Breton and \u00c9luard), 211n5\n\nimpressionism: interplay of hand and eye in, 62\u201363, 210n56; mimesis of nature in, 13\u201316\n\n_Interpretation of Dreams, The_ (Freud),\n\n_In the Stable of the Sphinx_ (Ernst), 223n22\n\n_Introductory Lectures on Psychoanalysis_ (Freud),\n\nJakob, Helene,\n\nJarry, Alfred, , 214n30\n\nJenny, Laurent,\n\nJones, Ernest,\n\n\"Journal of an Apparition\" (Desnos),\n\nKandinsky, Wassily,\n\n_Katharina ondulata, that is Madam Hostess on the Lahn_ (Ernst), ,  _1f_ ,\n\nKlee, Paul, 9\u201311,\n\nKlinger, Max,\n\nKoselleck, Reinhart,\n\nKrauss, Rosalind, ; on Ernst's _The Master's Bedroom_ , 124\u201328, 225n44, 225n46; on Ernst's surrealist poetics, ; on the modernist ideal of opticality, 124\u201325; on the optical unconscious, 125\u201330, 225n44, 225n46; on painting, readymades, and collage, ; on Picasso's _Au Bon March\u00e9_ , 232n10\n\nLacan, Jacques, , 5\u20136\n\nLaplanche, Jean, , 226n4\n\n_Large Glass_ (Duchamp), 207n26\n\nLautr\u00e9amont (Isadore Ducasse),\n\nLebel, Robert,\n\nLeed, Eric J.,\n\nLegge, Elizabeth, 221n2\n\nLeiris, Michel, 211n5\n\nLeonardo da Vinci, ; Oedipal constructs of, 107\u20138, 121\u201322, ; wall stains of, 117\u201319, 182\u201385, 215n44\n\n_Leonardo da Vinci and a Memory of His Childhood_ (Freud), , 117\u201320\n\nLepenies, Wolf,\n\n_Lesson in Automatic Writing_ (Ernst),\n\nLichtenstern, Christa,\n\n_Lightning Bolts under Fourteen Years_ (Ernst), , 1 _04f_\n\nLindner, Burckhardt, 237n5\n\n_Little Tables around the Earth_ (Ernst), , , 97 _f_,\n\nLoplop cycle (Ernst), , , 190\u201391, 220n88\n\nLorrain, Claude, 100\u2013101\n\n_Madam Hostess on the Lahn_ (Ernst), 145\u201349, , , , plate 2\n\n_Mad Love_ (Breton),\n\n\"Madness and Art\" (de Chirico),\n\n_Magnetic Fields, The_ (Breton), , , 211n5\n\nMalkine, Georges, ,\n\nManet, \u00c9douard,\n\n_Manifesto of Surrealism_ (Breton), , , ; on anti-communication, ; on deep memory, , 219n84, 219n86; emblematic circle of, 79\u201381; on hypnagogic mind states, ; on iconography of misunderstanding, ; lack of discussion of automatism in, ; on the unconscious,\n\nMarc, Franz, 151\u201353,\n\nMarin, Louis,\n\nMartin, John,\n\nMasson, Andr\u00e9, , , 72\u201374, ,\n\n_Master's Bedroom, The_ (Ernst), 124\u201328, 225n44, 225n46\n\nMaurer, Evan, 217n71\n\n\"Max Ernst, peintre des illusions\" (Aragon),\n\nMeier-Graefe, Julius,\n\n_Melancholia I_ (D\u00fcrer),\n\nMelhman, Jeffrey, 226n7\n\n_Men Shall Know Nothing of This_ (Ernst), 1 _61f_,\n\nmetonymy, 43\u201344\n\nmimesis: Benjamin's commentary on, 199\u2013200; disintegration in Dada of, 15\u201325, 61\u201362; double meaning of, 13\u201316, 205nn4\u20135; in Ernst's cut-paper collages, 44\u201354, 145\u201346; in Ernst's distorted imitation, 37\u201344; in Ernst's engagement with illusionism, 26\u201337; in Ernst's natural history frottage, 54\u201363, 209n46; in modernist abstraction, , 205nn4\u20135; surrealism's return to,\n\nmimetism of the second order,\n\nMir\u00f2, Jean, 212n12\n\n_Mis\u00e8re de la po\u00e9sie_ (Breton),\n\nmodernity\/modernism, 4\u20137; anti-literary positions in, 68\u201369; avant-garde consciousness of, 100\u2013101; Benjamin on subjectivity in, 192\u2013202; ideal of opticality in, 124\u201325; ideology of pure seeing in, ; mimesis and abstraction in, , 205nn4\u20135; nineteenth century as prehistory of, 5\u20136; psychic trauma of,\n\nmontage, 194\u201398\n\nMorise, Max, 66\u201368, , 211n5\n\nmortification, , , , ,\n\n_Nadja_ (Breton),\n\n_Natural History_ (Ernst), , 54\u201363, ; Arp's foreword to, , 90\u201392, , , 217n57; Crevel's review of, 99\u2013100; dedication of, ; deep time of prehistory in, 92\u2013105, 217n71; dragonfly in, 103\u20135; earth in, ; \u00c9luard's work on, 106\u201310; eye motifs in, 107\u201310; iconography of, 75\u201387, ; intellectual-historical frame of, 69\u201375; mimesis of fossilization in, 60\u201363; poetics of automatism in, 67\u201368; pure visuality in, ; references to World War I in, 134\u201336, ; symbols of modern civilization in, 88\u201390; teleological narrative template of, 98\u2013103; textural alphabet of, 74\u201375; threshold figure of transparency in, 121\u201322; titles in, 87\u201392; traces of underlay in, 54\u201357, ; _trompe l'oeil_ and concealed images in, 59\u201360, , 223nn22\u201323; use of parapraxis in, , , , ; use of procedure in, 83\u201387\n\nNaville, Pierre, 212n15\n\nNazism, 179\u201380, 234n17, 236n36\n\n\"Nightingale, The\" (Anderson), 228nn36\u201337\n\n_No Man's Land: Combat and Identity in World War I_ (Leed),\n\nNosferatu, ,\n\n_Oberbadischen Volksblattes_ , 179\u201380, 234n17\n\n_Oedipus Rex_ (Ernst), 36\u201337,\n\n\"Old Vivisector, The\" (Ernst), 138\u201340\n\n\"On Metaphysical Art\" (de Chirico), 100\u2013101\n\n\"On the Work of Art in the Age \/ Painting and Graphic Arts\" (Benjamin), 198\u2013201\n\n_Optical Unconscious, The_ (Krauss),\n\n_Origin of the Clock, The_ (Ernst), 88\u201390\n\n_Origins, The_ (Redon), 216n51\n\noverpaintings: cloacal functions in, ; collaborations with \u00c9luard of, 192\u201393, 195\u201397; as collage, 40\u201342; female figures in, ; as foreign bodies of visuality, 148\u201349; geologic imitation in, 37\u201344, 164\u201370; hard surfaces of, 127\u201328; Krauss on optical unconscious in, 125\u201330, 225n44, 225n46; pictorial underlay of, 40\u201344, 61\u201362; proto-surrealist poetics of, 144\u201349; on wartime disempowerment, 149\u201354, ; World War I references in, 145\u201349\n\npainting. _See_ grattage and painting; overpaintings\n\n\"Painting, or Signs and Marks\" (Benjamin), , 198\u201399\n\n\"Painting and the Graphic Arts\" (Benjamin), , 198\u201399\n\npaleontology. _See_ geological and paleontologi-cal illustration\n\n_Paradise_ (Ernst),\n\nparapraxes: historical contexts of, 177\u201380; in Ernst's Dada works, 24\u201325, , , , 206n19; in _Natural History_ , , , , ; in surrealist communication,\n\n_Paris Dream_ (Ernst), 186\u201387\n\n_Paris Peasant_ (Aragon), 78\u201379,\n\n_Pear_ (Ernst), 48\u201350, 163\u201364\n\n_Pear that Resembles Me, The_ (Ernst), 50\u201351, 208n40\n\nPech, J\u00fcrgen, 236n36\n\nP\u00e8re Ubu (character), , 214n30\n\nPfister, Oskar,\n\nphotograms, 66\u201368\n\nphotomontage, 156\u201357\n\nPicabia, Francis, 26\u201328, , , 207n26\n\nPicasso, Pablo, 7\u20138; _Au Bon March\u00e9_ of,  _68f_ , 232n10; cubism of, 52\u201353; _papiers coll\u00e9s_ of, 44\u201348, 183\u201385, 228n35\n\nPoe, Edgar Allen, 216n51\n\nPoggi, Christine,\n\n\"Political Overzealousness\" (Ernst), 179\u201380, 234n17\n\n_Political Position of Today's Art_ (Breton),\n\nPollock, Jackson,\n\nPontalis, Jean-Bertrand, 226n4\n\n_Portable Handbook_ (Ernst), , 30\u201332,\n\npostcards, 154\u201359, 231n72\n\nPoussin, Nicolas, 100\u2013101\n\n_Preparation of Bone Glue, The_ (Ernst),\n\nprimal scene, , 114\u201320, 132\u201333, 188\u201389, 223n12, 223n26\n\n_Primordial World in the Different Periods of Its Development, The_ (Unger),\n\npsychoanalytic theory, 4\u20135, 111\u201330, 221\u201322nn2\u20133; on cloacal theory and scatophilia, , 164\u201370, 232n10, 233n12; on the death drive, 146\u201347; on female symbols, ; on _heimlich_ and _unheimlich_ , ; imaginative seeing and, 116\u201318; Oedipal constructs in, 107\u20138, 120\u201322, 137\u201338, , , 224n32; on the primal scene, , 114\u201320, 132\u201333, 223n12, 223n26; role of nature in, ; on the unconscious and automatism, 5\u201312, , , 122\u201330, 224n42, 225n49; on war neurosis and trauma, 131\u201334, 226n4, 226n6\n\n_Psychopathology of Everyday Life, The_ (Freud),\n\n_Question of Lay Analysis, The_ (Freud), ,\n\nrain, , 80 _f_,\n\nRay, Man, , 107\u20139\n\nreadymades, 7\u20138, , 207n26\n\n_Red Front, The_ (Aragon), 176\u201377\n\nRedon, Odilon, , 216n51\n\nrepetition, , 71\u201372\n\n_R\u00e9p\u00e9titions_ frontispiece (Ernst), 192\u201393, 195\u201397, 237n5\n\nresemblances. _See_ seeing of resemblances\n\nreversible images, , 54\u201355, , , , , , ,\n\n_R\u00e9volution surrealiste, La_ , , , 112\u201313, . _See also_ Breton, Andr\u00e9\n\nRiou, \u00c9douard, 94 _f_, 96\u2013105,\n\nRoussel, Raymond,\n\nrubbing. _See_ frottage\n\n_R\u00fcckenfigur_ , 84\u201386, 216n51\n\nRussell, John, 183\u201385\n\n_Sandworm Who Reties Her Sandal, The_ (Ernst), 51\u201352\n\n_Scarecrows_ (Ernst),\n\nscatophilic iconography, 145\u201347, 164\u201370\n\nScholem, Gershom,\n\nscopophilia, 121\u201323\n\n_Sea and Rain, The_ (Ernst), , 80 _f_\n\nseeing apart, 51\u201352,\n\nseeing of resemblances, 50\u201354, 62\u201363, , 172\u201376, 208n40; in _Europe After the Rain_ , 183\u201391; politics of the 1930s and, 176\u201380\n\n_She Guards Her Secret_ (Ernst),\n\nshell shock, 131\u201334, , 226n4, 226n6\n\nShiff, Richard, , 205n4, 210n56\n\nSimmel, Georg,\n\n_Simulacrum_ (Leiris), 211n5\n\n_Sodalites_ (Ernst), 33\u201335, 37\u201339,\n\n_Solar Currency System_ (Ernst), 88\u201390,\n\n_Soluble Fish_ (Breton), 74\u201375, , , 211n5, 214nn27\u201328\n\n\"Some Data on the Youth of M.E. as Told by Himself\" (Ernst),\n\nSpies, Werner, , , , 222n3; on ahistoricity of frottage's textures, ; on Ernst's World War I references, ; on surrealist poetics of Ernst's collage,\n\n_Stallion and Bride of the Wind_ (Ernst), 84 _f_\n\n_Start of the Chestnut Tree, The_ (Ernst), 88\u201390\n\nstill life, , 210n54\n\nStokes, Charlotte, 225n49\n\nStrachey, James, 223n12\n\n_Stratified Rocks, Nature's Gift of Gneiss Lava Iceland Moss_ (Ernst), 33\u201335, 37\u201338, 40\u201344, ,  _65f_ , 232n10\n\nsurrealism, , , , 211n8, 212n15; annihilation of conscious memory in, , , 219n86; anti-literary position of, 68\u201369; automatic writing in, 64\u201365, , , 211n5; Benjamin's commentary on, 10\u201311, 192\u2013202, 237n5; as cartographic art, , 188\u201390; cataclysm and cleared spaces in, 187\u201390; early appearance in Ernst's work of, 141\u201349; earth themes in, 160\u201371, , 232n2, 232n10, 233n12; foreign bodies in, 148\u201349; the horizon in, , 227n13; iconography and misunderstanding in, 75\u201387, 111\u201312; opening of prehistoric time in, 78\u201379, 92\u2013105, 134\u201335; passive scopophilia of, 121\u201323; plasticity of the unconscious in, 122\u201330; political ideology of, 176\u201380; premonitions of World War II in, 179\u201391, , 234n17, 236n36; psychoanalytic theory and, 111\u201330, 221\u201322nn2\u20133; return to mimesis in, , 61\u201363; seeing of resemblances in, 50\u201354, 62\u201363, , 172\u201376, 208n40; simulacra in, 59\u201361; symbols of femininity in, 106\u201310; uses of nature in, , 78\u201379; World War I and, 131\u201334, 136\u201338; writing _vs_. visual art in, 66\u201375. _See also_ frottage; _Manifesto of Surrealism_ ; _Natural History_\n\n_Surrealism and Painting_ (Breton),\n\n\"surrealist texts\" (Arp),\n\ntechnique. _See_ experiments in technique\n\n\"Tell-Tale Heart, The\" (Poe), 216n51\n\nTheweleit, Klaus, ,\n\n_Three Essays_ (Freud),\n\n\"Tissue of Truth, Tissue of Lies\" (Ernst), 136\u201337\n\n_Totem and Taboo_ (Freud),\n\ntrauma of war, 131\u201338, 144\u201345, 226n4\n\n_Treatise on Painting_ (Leonardo da Vinci), 117\u201318\n\n_trompe l'oeil_ , 59\u201363, , 210n54\n\n_Two Young Girls Riding across the Sky_ (Ernst), 102\u20133, 194\u201395\n\n_Typescript Manifesto_ (Ernst and Baargeld), 69\u201372\n\nUbu (character), , 214n30\n\nUlrich, Bernd, 155\u201356\n\nunconscious, the, 5\u201312, 66\u201368, , 122\u201330; nature as, ; visual parapraxes of, 177\u201380; World War I's liberation of, 131\u201334. _See also_ automatism\n\nUnger, Franz-Xaver,\n\n_Untitled_ (Arp),  _8f_\n\n_Untitled_ (Ernst), 34\u201335,\n\n_Vaccinated Bread, The_ (Ernst), , 215n45\n\n\"Vergleichung\" (Ernst),\n\nVerne, Jules, ,\n\n_Very Gentle Earthquake, A_ (Ernst), 168\u201370\n\n_Violin_ (Picasso), 183\u201385\n\n_Visible Woman, The_ (Ernst), ,  _08f_\n\n\"Visions of Half-Sleep\" (Ernst), 112\u201322, , 136\u201337, ; bodily separation in, ; Oedipal constructs in, 120\u201322, ; presentation context of, 113\u201314; primal scenes in, 114\u201320, 188\u201389; on the unconscious,\n\nwallpaper, 42\u201344\n\nwar neurosis (shell shock), 131\u201338, , 226n4, 226n6\n\nWellbery, David E.,\n\n_Wheel of Light, The_ (Ernst), 108\u201310\n\n_Whip Lashes or Lava Threads_ (Ernst), 81\u201383,\n\n_Winter Landscape: Vaporization of the Vulcanized Iron Bride to Produce the Necessary Bed Warmth_ (Ernst), 37\u201339, ,\n\nWolf Man's dream, 114\u201321, 126\u201327, , 223n12, 223n26\n\n_Woman, Old Man, and Flower_ (Ernst), , 162\u201364, 167\u201370, 232n5, 232n7, 233n12, plate 3\n\nwood engravings, 172\u201376\n\n_Work of Art in the Age of Its Technological Reproducibility, The_ (Benjamin), 197\u2013201\n\n_World before the Creation of Man, The_ (Flammarion),\n\n_World before the Deluge, The_ (Figuier), , 96\u2013100, 103\u20135,  _06f_\n\n_World in the Time of the Surrealists, The_ (from _Vari\u00e9t\u00e9s_ ), 186\u201387\n\nWorld War I: crystalline skies of, 151\u201354, 229n48; Ernst's service in, 18\u201319; origins of surrealism in, 136\u201338, 236n36; postcards from the field of, 154\u201359, 231n72; references in Ernst's Dada works to, 138\u201349; war neurosis of, 131\u201334, , 226n4, 226n6\n\nWorld War II, 179\u201391, , 236n36\n\nWorringer, Wilhelm, ,\n\nwritings of Ernst: autobiographical notes among, 2\u20133, 136\u201338, 203n6, 227n17; _Beyond Painting_ , ; on collage, 2\u20133, ; on nature, ; on Nazi Germany, 179\u201380, 234n17; on psychoanalytic theory, 112\u201330; on surrealism's mission, ; on transition from sensuality to will, ; _Typescript Manifesto_ , 69\u201372; on World War I, 138\u201340. _See also_ \"How to Force Inspiration\"; \"Visions of Half-Sleep\"\n\n\"Zeuxis the Explorer\" (de Chirico), 209n46\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Symptom Sorter\n\n**FIFTH EDITION**\n\n**Keith Hopcroft**\n\n_**General Practitioner  \nLaindon Health Centre  \nBasildon, Essex**_\n\n**and**\n\n**Vincent Forte**\n\n_**General Practitioner  \nCentral Surgery  \nGorleston, Norfolk**_\n\n**Radcliffe Publishing**\n\n**London \u2022 New York**\n**Radcliffe Publishing Ltd**\n\nSt Mark's House\n\nShepherdess Walk\n\nLondon N1 7BQ\n\nUnited Kingdom\n\n**www.radcliffehealth.com**\n\n\u00a9 2014 Keith Hopcroft and Vincent Forte\n\nFirst Edition 1999\n\nSecond Edition 2003\n\nThird Edition 2007\n\nFourth Edition 2010\n\nKeith Hopcroft and Vincent Forte have asserted their right under the Copyright, Designs and Patents Act 1988 to be identified as the authors of this work.\n\nEvery effort has been made to ensure that the information in this book is accurate. This does not diminish the requirement to exercise clinical judgement, and neither the publisher nor the authors can accept any responsibility for its use in practice.\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.\n\nBritish Library Cataloguing in Publication Data\n\nA catalogue record for this book is available from the British Library.\n\nISBN-13: 978 191022 718 3\n\neISBN-13: 978 191022 748 0\n\n## CONTENTS\n\nAbbreviations\n\nIntroduction\n\n**Abdomen**\n\n**Anorectal**\n\n**Breast**\n\n**Cerebral**\n\n**Chest**\n\n**Ear**\n\n**Eye**\n\n**Face**\n\n**General physical**\n\n**Genital**\n\n**Hair and nails**\n\n**Limbs**\n\n**Neck**\n\n**Nose**\n\n**Oral**\n\n**Pelvic**\n\n**Periods**\n\n**Skin**\n\n**Urinary**\n\nIndex\nTo Ali D, my rock  \nVF\n\nTo D, H and W  \nThank you for sorting out all the symptoms I developed writing this book  \nKH\n\n## ABBREVIATIONS\n\nA&E | accident and emergency\n\n---|---\n\nABPI | ankle brachial pressure index\n\nACE | angiotensin-converting enzyme\n\nACR | albumin creatinine ratio\n\nACTH | adrenocorticotrophic hormone\n\nADH | antidiuretic hormone\n\nAF | atrial fibrillation\n\nAFP | \u03b1 fetoprotein\n\nALT | alanine-amino transferase\n\nANUG | acute necrotising ulcerative gingivitis\n\nARC | AIDS-related complex\n\nASO | antistreptolysin\n\nAST | aspartate-amino transferase\n\nBCC | basal cell carcinoma\n\nBNP | B type natriuretic peptide\n\nBP | blood pressure\n\nBPH | benign prostatic hypertrophy\n\nBTB | breakthrough bleeding\n\nBV | bacterial vaginosis\n\nBXO | balanitis xerotica obliterans\n\nCA-125 | cancer antigen 125\n\nCCF | congestive cardiac failure\n\nCKD | chronic kidney disease\n\nCNS | central nervous system\n\nCOPD | chronic obstructive pulmonary disease\n\nCPK | creatine phosphokinase\n\nCREST | calcinosis\/Raynaud's phenomenon\/oesophagealdysmotility\/sclerodactyly\/telangiectasia\n\nCRF | chronic renal failure\n\nCRP | C-reactive protein\n\nCSF | cerebrospinal fluid\n\nCT | carpal tunnel\n\nCT | computed tomography\n\nCVA | cerebrovascular accident\n\nCXR | chest X-ray\n\nD&C | dilatation and curettage\n\nDIC | disseminated intravascular coagulation\n\nDKA | diabetic ketoacidosis\n\nDM | diabetes mellitus\n\nDU | duodenal ulcer\n\nDUB | dysfunctional uterine bleeding\n\nDVT | deep vein thrombosis\n\nDXA | dual energy X-ray absorptiometry\n\nEAM | external auditory meatus\n\nEBV | Epstein\u2013Barr virus\n\nECG | electrocardiogram\n\nEEG | electroencephalogram\n\nELISA | enzyme-linked immunosorbent assay\n\nEMG | electromyography\n\nEMU | early morning urine (sample)\n\nENT | ear, nose and throat\n\nEO | epididymo-orchitis\n\nERCP | endoscopic retrograde cholangiopancreaticogram\n\nESR | erythrocyte sedimentation rate\n\nET | Eustachian tube\n\nFBC | full blood count\n\nFSH | follicle-stimulating hormone\n\n\u03b3GT | gamma glutamyl transpeptidase\n\nGAD | generalised anxiety disorder\n\nGF | glandular fever\n\nGI | gastrointestinal\n\nGI | granuloma inguinale\n\nGnRH | gonadotrophin-releasing hormone\n\nGORD | gastro-oesophageal reflux disease\n\nGUM | genito-urinary medicine\n\nHb | haemoglobin\n\nHCG | human chorionic gonadotrophin\n\n5HIAA | 5-hydroxy-indole-acetic acid\n\nHLA | human leucocyte antigen\n\nHRT | hormone-replacement therapy\n\nHSV | herpes simplex virus\n\nHVS | high vaginal swab\n\nIBD | inflammatory bowel disease\n\nIBS | irritable bowel syndrome\n\nIC | intermittent claudication\n\nIGTN | ingrowing toenail\n\nIHD | ischaemic heart disease\n\nINR | international normalised ratio\n\nITP | idiopathic thrombocytopenia purpura\n\nIUCD | intrauterine contraceptive device\n\nIVP | intravenous pyelogram\n\nIVU | intravenous urogram\n\nJCA | juvenile chronic arthritis\n\nLFT | liver function tests\n\nLGV | lymphogranuloma venereum\n\nLH | luteinising hormone\n\nLMP | last menstrual period\n\nLN | lymph node\n\nLRTI | lower respiratory tract infection\n\nLSD | lysergic acid diethylamide\n\nLUTS | lower urinary tract symptoms\n\nLVF | left ventricular failure\n\nMAOI | monoamine oxidase inhibitor\n\nMC&S | microscopy, culture and sensitivity\n\nMCV | mean cell volume\n\nMI | myocardial infarction\n\nMMR | measles, mumps, rubella\n\nMRI | magnetic resonance imaging\n\nMS | multiple sclerosis\n\nMSU | mid-stream urine (sample)\n\nNAI | non-accidental injury\n\nNICE | National Institute for Health and Care Excellence\n\nNSAID | non-steroidal anti-inflammatory drug\n\nOA | osteoarthritis\n\nOE | otitis externa\n\nO\/E | on examination\n\nOG | onychogryphosis\n\nOGD | oesophago-gastro duodenoscopy\n\nOM | otitis media\n\nOTC | over the counter\n\nPAN | polyarteritis nodosa\n\nPCOS | polycystic ovary syndrome\n\nPCR | protein creatinine ratio\n\nPCV | packed cell volume\n\nPE | pulmonary embolism\n\nPEFR | peak expiratory flow rate\n\nPF | proctalgia fugax\n\nPID | pelvic inflammatory disease\n\nPMR | polymyalgia rheumatica\n\nPMT | pre-menstrual tension\n\nPR | per rectum\n\nPRIST | paper radioimmunosorbent disc test\n\nPSA | prostate-specific antigen\n\nPTH | parathyroid hormone\n\nPU | peptic ulcer\n\nPUO | pyrexia of unknown origin\n\nPV | per vagina\n\nPVE | per vaginal examination\n\nPVD | peripheral vascular disease\n\nRA | rheumatoid arthritis\n\nRAST | radioallergosorbent test\n\nRAU | recurrent aphthous ulceration\n\nRLS | restless legs syndrome\n\nRUQ | right upper quadrant\n\nSA | septic arthritis\n\nSCC | squamous cell carcinoma\n\nSLE | systemic lupus erythematosus\n\nSOB | shortness of breath\n\nSTD | sexually transmitted disease\n\nSVT | supraventricular tachycardia\n\nTAH | total abdominal hysterectomy\n\nTATT | 'tired all the time'\n\nTB | tuberculosis\n\nTCA | tricyclic antidepressant\n\nTFT | thyroid function tests\n\nTIA | transient ischaemic attack\n\nTMJ | temporomandibular joint\n\nTN | trigeminal neuralgia\n\nTSH | thyroid-stimulating hormone\n\nTURP | transurethral resection of prostate\n\nTV | trichomonal vaginosis\n\nU&E | urea and electrolytes\n\nURTI | upper respiratory tract infection\n\nUTI | urinary tract infection\n\nUV | ultraviolet\n\nVMA | vanillyl-mandelic acid\n\nV\/Q | ventilation perfusion\n\nVT | ventricular tachycardia\n\nWCC | white cell count\n\nWRULD | work-related upper limb disorder\n\n## INTRODUCTION\n\nLife would be much simpler for GPs if patients presented with diagnoses. Unfortunately, they do not: they present with symptoms, which are frequently vague, sometimes multiple and occasionally obscure. It is up to the GP to create some order from this chaos. However, the vast majority of clinical texts adopt a diagnosis, rather than symptom-based, approach and the few which do reflect the reality of patient presentations are inevitably orientated towards hospital medicine and so are irrelevant to GPs.\n\nThis book, originally serialised, in a different form, in _Doctor_ magazine, aims to redress the balance. It analyses a multitude of symptoms commonly seen in primary care and, for each, presents differentials, distinguishing features, possible investigations and key points. The only omissions are presentations for which there are so few differentials that diagnosis is really quite simple (e.g. 'lump on elbow'); those which rarely present in isolation (e.g. nausea, anorexia); and those which are so rare that the reader would be sure to require specialist advice (our personal favourite being 'pilimiction').\n\nWritten by two full-time GP principals, its perspective is very much grass roots primary care \u2013 though informed by the latest evidence and guidance where possible or appropriate \u2013 and its appeal is therefore wide. GP registrars and young principals, relatively unfamiliar with the protean presentations possible in general practice, will be able to check their diagnostic hypotheses against the information in the book; the more experienced GP might use it as a refresher or as a pointer to a more exotic diagnosis in an unusual case; and the nurse practitioner, taking increasing responsibility as a first port of call in primary care for many patients, will find the contents unique and essential.\n\nThe popularity of the first edition and the need for second, third, fourth and fifth editions emphasises the fact that, while general practice may experience multiple reorganisations and restructurings, the bread and butter business of making sense of symptoms remains. This new edition adds a number of extra chapters and significant updates of existing chapters.\n\nEach symptom is analysed in a uniform, accessible way, as follows.\n\n#### The GP overview\n\nThis defines the symptom and its key characteristics, and gives some idea of the frequency of presentation.\n\n#### Differential diagnosis\n\nThis lists the likely diagnoses, subdivided 'Common', 'Occasional' and 'Rare'. (It should be noted that these headings are relative to the symptom in question. For example, some of the 'common' causes of delayed puberty will be rarer than some of the 'occasional' causes of acute abdominal pain \u2013 for the simple reason that abdominal pain is much commoner than delayed puberty as a presenting symptom.) Restrictions of space and imagination mean that such a differential can never claim to be exhaustive, and a lack of accurate prevalence data renders the allocation of the diagnoses to these subdivisions somewhat arbitrary, based on our experience rather than hard evidence. These are minor limitations, however; this section will invariably provide clear guidance as to the likely cause of any symptom.\n\n#### Ready reckoner\n\nThis provides a quick guide to the key distinguishing features of the five most likely diagnoses listed in the preceding section.\n\n#### Possible investigations\n\nThis section outlines those investigations likely to assist the reader in making a diagnosis. The emphasis is upon tests performed in primary care or usually arranged by the GP. Where appropriate, more complex, hospital-initiated investigations are outlined \u2013 partly because GPs may wish to let their patients know the type of tests they might anticipate after referral and also because GP access to traditionally hospital-organised investigation is increasing. All investigations discussed are categorised according to the likelihood that they will be performed, the three categories being, 'Likely', 'Possible' and 'Small print'.\n\n##### TOP TIPS\n\nThis provides a potpourri of management nuggets appropriate to each symptom, which the authors have accumulated over the years. Such hints from experience are difficult to analyse or quantify and so most are unashamedly anecdotal rather than evidence based \u2013 this should not detract from their usefulness or occasional elegance. Some might appear to stretch the scope of the book in that they cross the boundary between symptom assessment and symptom management \u2013 but the reader should bear in mind that the diagnostic process, particularly in primary care, involves hypothesis testing, and so these boundaries are, in reality, blurred.\n\nMost symptoms presented in primary care are benign, minor and self-limiting. This can occasionally lull the unwary into a false sense of security: for each presentation there exist pointers which should set alarm bells ringing. 'Red flags' highlights aspects of symptoms which suggest significant pathology and which therefore should not be missed or neglected.\n\n#### How to use this book\n\nThis _Symptom Sorter_ is designed to act as a rapid reference. It has deliberately been written in a note and list format so that, unlike weightier tomes, it is quick and easy to use. For the sake of brevity, common and well-recognised abbreviations have been used whenever possible. Its consistent style will soon breed familiarity and allow the reader to know where and how to retrieve information painlessly. To help achieve this, the symptoms are arranged in sections, each section corresponding to a system or anatomical region. In these sections, the symptoms are arranged alphabetically and, for the most part, are labelled in patient, rather than doctor, vernacular (e.g. shortness of breath rather than dyspnoea) \u2013 the exceptions being where there is no acceptable or concise 'patientspeak' version. However, as many symptoms can have a variety of descriptions (e.g. shortness of breath, dyspnoea, breathlessness, wheeziness, difficulty breathing and so on) the index is deliberately expansive and cross-referenced, and will quickly guide the reader to the appropriate pages.\n\nThe categorisation of symptoms and their arrangement in sections is a complex task which can be approached in a number of ways \u2013 for example, rashes might be divided according to distribution, size of lesion, morphology, itch and so on. Throughout, we have chosen the approach which seems most logical to us and which, whenever possible, avoids unnecessary omission or repetition; again, the index should rapidly point the reader in the right direction. Assigning symptoms to certain sections may sometimes seem arbitrary, especially when they can have such disparate causes, but this approach provides the book with a clear, understandable structure.\n\nAs GPs, we are aware that patients often present polysymptomatically. Our book, neatly dividing complaints into individual symptoms, might therefore be criticised for not accurately reflecting real primary care life. In fact, such presentations can usually be distilled down to one or two predominant symptoms; more minor symptoms often act as pointers to the actual diagnosis, a fact our 'Ready reckoners' in each chapter exploits. In the truly polysymptomatic, the book may help to define a common thread among the symptoms, thereby revealing the real diagnosis \u2013 usually, in such cases, anxiety or depression.\n\nThe book should be kept to hand for use during surgery to confirm the likelihood of a certain diagnosis or raise the possibility of others. Being comprehensive, relevant and accessible, retrieval of information will be speedy and helpful during the consultation itself (you may wish to wait until the patient is undressing behind the curtain: there should be time).\n\nThe book may be used in other ways. GP trainers could use the analysis of a certain symptom provided in the text as the basis for a tutorial. Indeed, the book could itself form part of the GP registrar's curriculum. By 'sorting' two or three symptoms a week, using the text as a guide, the registrar could, over the course of his or her practice year, cover the vast majority of presentations seen in primary care. Trainers of undergraduates, too, will find that the contents provide useful material for teaching sessions.\n\nOthers might simply like to browse, refreshing or refining their diagnostic skills and mulling over the Red flags and Top tips.\n\nFeedback received by the authors indicates that previous editions are now included in many undergraduate curricula as recommended reading, and the book is proving very popular among primary care nurses and nurse practitioners.\n\nHowever the reader uses this book, we are convinced that it will prove an essential resource. Making sense of symptoms is the essence of general practice, and any tool designed by and for GPs which contributes to this art is likely to benefit doctors and patients alike.\n\n**Keith Hopcroft  \nVincent Forte  \n _August 2014_**\n\n## **ABDOMEN**\n\nAbdominal swelling\n\nAcute abdominal pain in adults\n\nAcute abdominal pain in pregnancy\n\nChronic\/recurrent abdominal pain in adults\n\nConstipation\n\nDiarrhoea\n\nEpigastric pain\n\nRecurrent childhood abdominal pain\n\nVomiting\n\nVomiting blood\n\n### ABDOMINAL SWELLING\n\n#### The GP overview\n\nThis presentation covers both abdominal and pelvic masses, and general abdominal swelling. The patient may complain of a general increase in girth or of a discrete mass discovered accidentally; alternatively the GP might find the swelling while performing a physical examination.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  pregnancy\n\n  irritable bowel syndrome (IBS)\n\n  constipation\n\n  fibroid uterus\n\n  enlarged bladder\n\n**OCCASIONAL**\n\n  ascites (itself has many causes)\n\n  intestinal obstruction\n\n  ovarian mass (cyst or malignant tumour)\n\n  carcinoma of stomach or colon\n\n  hepatomegaly (various causes)\n\n**RARE**\n\n  splenomegaly (various causes)\n\n  pancreatic carcinoma\n\n  aortic aneurysm\n\n  massive para-aortic lymphadenopathy\n\n  hydronephrosis, renal cysts and renal malignancy\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ pregnancy test, ultrasound.\n\n_POSSIBLE :_ urinalysis, FBC, U&E, LFT, CA-125, plain abdomen X-ray.\n\n_SMALL PRINT :_ barium enema and sigmoidoscopy, paracentesis, CT scan.\n\n  Pregnancy test essential in amenorrhoeic women.\n\n  Urinalysis may reveal microscopic haematuria in renal or bladder tumours.\n\n  Abdominal ultrasound is the quickest and most efficient way to define the source of most abdominal swellings or masses. Ultrasound of pelvis\/abdomen may also be indicated according to an elevated CA-125 (see below).\n\n  Full blood count (FBC): anaemia likely in malignancy, possible in fibroids with menorrhagia; also will reveal blood dyscrasias.\n\n  Urea and electrolytes (U&E) may be deranged in gross renal disease. Liver function tests (LFTs) may give a clue to alcoholic hepatomegaly or malignancy. Low albumin in ascites.\n\n  CA-125: may be indicated in women, especially those aged 50 or more, to help exclude ovarian cancer.\n\n  Barium enema and sigmoidoscopy: useful to confirm or exclude colonic disease.\n\n  Plain abdominal X-ray: may show constipation or obstruction (in the latter case, likely to be arranged after admission).\n\n  Other tests are likely to be arranged after specialist referral, e.g. paracentesis (to investigate and relieve ascites), CT scanning (to establish nature of mass and its effects on surrounding structures).\n\n##### TOP TIPS\n\n  Take care in the history to distinguish between intermittent or variable swelling, and progressive swelling. The former will probably not be caused by serious pathology, whereas the latter may well be.\n\n  Pregnancy can catch out the unwary, particularly when dealing with perimenopausal women or teenage girls. Do not accept the claim that 'I can't be pregnant'.\n\n  Some 'swellings' turn out, on examination, to be impalpable or to represent normal anatomy. The physical examination may have a therapeutic effect. If not, explore the patient's concerns more fully and consider anxiety, depression or other psychological problems if symptoms persist.\n\n  Weight loss in conjunction with abdominal swelling should immediately suggest malignancy.\n\n  Acute onset of swelling with abdominal pain suggests obstruction, requiring urgent surgical attention.\n\n  Obesity presents difficulties in examination and can be difficult to distinguish from ascites. If in doubt, arrange an ultrasound.\n\n  Resonance on percussion does not rule out a solid mass: retroperitoneal masses will push bowel anteriorly and may be apparently tympanic.\n\n### ACUTE ABDOMINAL PAIN IN ADULTS\n\n#### The GP overview\n\nThe sudden onset of severe abdominal pain represents a genuine emergency in general practice and is a common out-of-hours call. In the true acute abdomen, the patient is obviously ill, and as the clinical condition may deteriorate rapidly, ensure that you examine the patient as soon as possible.\n\n_NOTE :_ upper and mid-abdominal pain are dealt with here. Lower abdominal pain is dealt with under 'acute pelvic pain' and specifically epigastric-type pain is covered in more detail in the epigastric pain section.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  peptic ulcer\n\n  biliary colic\n\n  appendicitis\n\n  gastroenteritis\n\n  renal colic\n\n**OCCASIONAL**\n\n  cholecystitis (may follow biliary colic, but pain is constant and fever present)\n\n  diverticulitis\n\n  acute or subacute bowel obstruction (adhesions, carcinoma, strangulated hernia, volvulus)\n\n  pyelonephritis\n\n  muscular wall pain\n\n  pancreatitis\n\n**RARE**\n\n  perforation (e.g. duodenal ulcer (DU), carcinoma) resulting in peritonitis\n\n  hepatitis\n\n  Crohn's and ulcerative colitis\n\n  ischaemic bowel\n\n  dissecting\/leaking aneurysm\n\n  diabetic ketoacidosis (DKA) and other occasional medical causes (e.g. myocardial infarction (MI), pneumonia, sickle cell crisis)\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThe only test likely to help the GP is urinalysis: this may reveal haematuria (renal colic), evidence of urinary infection, or glycosuria in DKA. In general, the following investigations will be done in hospital after acute admission.\n\n  Full blood count: WCC raised in many causes and confirms acute inflammation or infection.\n\n  U&E essential as abnormalities common with diarrhoea or vomiting. Amylase raised in ischaemic bowel and acute pancreatitis.\n\n  LFT may show raised bilirubin in biliary obstruction, and widespread derangement in hepatitis.\n\n  _Helicobacter pylori_ testing: strong association with peptic ulcer disease.\n\n  Upper GI endoscopy: to visualise upper GI tract.\n\n  Plain erect abdominal X-ray invaluable to confirm perforated viscus (air under diaphragm). Supine also necessary if obstruction suspected. Ninety per cent of renal or ureteric stones will be revealed with a plain abdominal X-ray.\n\n  Ultrasound: helpful to confirm gallstones.\n\n  IVU: for ureteric stones.\n\n##### TOP TIPS\n\n  The aim of assessment is correct disposal rather than an exact diagnosis. Colicky pain may be appropriate to manage at home; constant pain with tenderness is likely to need admission.\n\n  If treating a patient at home, arrange for review as appropriate and ensure that the patient is aware of the symptoms which should prompt urgent reassessment.\n\n  The examination is likely to contribute significantly to making the diagnosis \u2013 so take particular care and don't forget the basics such as pulse rate, temperature, bowel sounds and a rectal examination.\n\n  Beware 'gastroenteritis' masking or developing into an acute appendicitis. Make arrangements for follow-up and emphasise that constant pain needs urgent review.\n\n  Prejudice is easy if the patient has a history of functional problems or irritable bowel. Surgical pathology can happen to anyone, so be objective.\n\n  Beware the elderly patient with an irregular pulse: mesenteric infarction causes severe pain but few signs.\n\n  Don't forget to examine the hernial orifices, especially if obstruction is a possibility.\n\n### ACUTE ABDOMINAL PAIN IN PREGNANCY\n\n#### The GP overview\n\nA pregnant woman who develops this symptom is very likely to be extremely concerned that there is a threat to her pregnancy. Anxiety levels may therefore be high in the patient and her partner. Acknowledge this emotional distress by an urgent and full assessment. Listed here are causes specific to pregnancy and conditions which may be exacerbated or altered by pregnancy; 'run of the mill' causes (such as gastroenteritis, IBS and dyspepsia) may obviously occur too, but rarely create diagnostic problems and so are not considered in this section.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  symphysis pubis and ligament strain\n\n  miscarriage: 20\u201340% of pregnancies in first trimester\n\n  labour: 6% premature\n\n  placental abruption: 1\/80\u2013200 pregnancies\n\n  pyelonephritis (especially around 20 weeks)\n\n**OCCASIONAL**\n\n  constipation (common cause but only occasionally presents)\n\n  ectopic pregnancy (1\/250 pregnancies)\n\n  appendicitis (1\/1000 pregnancies)\n\n  red degeneration of fibroid\n\n  torsion\/rupture of ovarian cyst or tumour\n\n**RARE**\n\n  uterine rupture (in UK 1\/1500 pregnancies, of which 70% due to Caesarian scar dehiscence)\n\n  uterine torsion (axial rotation >90\u00b0): 90% associated with fibroids, adnexal masses and anatomical uterine anomalies\n\n  liver congestion due to pre-eclampsia\n\n  rectus sheath haematoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThese will be dictated by the clinical urgency of the situation. In severe pain, they will be done in secondary care.\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ ultrasound, FBC.\n\n_SMALL PRINT :_ laparoscopy.\n\n  Urinalysis: proteinuria in pre-eclampsia. Blood, pus cells and nitrite in urinary tract infection (UTI); the infecting organism will be confirmed on MSU.\n\n  FBC: raised WCC in UTI.\n\n  Imaging ultrasound can be diagnostic in abruption and miscarriage; the presence of an intrauterine pregnancy makes an ectopic very unlikely; ultrasound may also be helpful in detecting a rectus sheath haematoma.\n\n  Laparoscopy: to confirm ectopic pregnancy.\n\n##### TOP TIPS\n\n  Pain on standing and walking, and relieved by rest, with exquisite pubic symphysis tenderness, is 'symphyseal pain' \u2013 an often overlooked cause.\n\n  Allay understandable anxieties as appropriate \u2013 particularly regarding the well-being of the foetus or the possibility of premature labour.\n\n  Do not be too ready to diagnose UTI on the basis of an abnormal urinalysis \u2013 contamination in pregnancy is common.\n\n  Distortion of anatomy may alter symptoms and signs: appendicitis is notoriously difficult to diagnose in the second trimester. If in doubt, admit.\n\n  A woman in early pregnancy who experiences unilateral lower abdominal pain followed by light bleeding or blackish discharge has an ectopic until proved otherwise.\n\n  Don't overlook the diagnosis of premature labour. Women with no previous experience of labour pain might not consider this possibility.\n\n  Placental abruption causes severe, continuous pain with a tender, hard uterus. Vaginal bleeding may be minimal. Admit immediately.\n\n  Don't forget pre-eclampsia as a cause of epigastric pain in the third trimester: check the blood pressure (BP) and urine.\n\n### CHRONIC\/RECURRENT ABDOMINAL PAIN IN ADULTS\n\n#### The GP overview\n\nThis problem may present in any age group. The causes in children are covered in another section. In young to middle-aged adults, the cause is very likely to be benign, but this alters with age: malignancy should always be suspected in the elderly even though other causes are still commoner. A precise diagnosis sometimes remains elusive.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  IBS\n\n  recurrent UTI\n\n  chronic peptic ulcer (PU)\n\n  constipation\n\n  diverticular disease\n\n**OCCASIONAL**\n\n  gallstones\n\n  hydronephrosis\n\n  post-herpetic neuralgia\n\n  inflammatory bowel disease\n\n  ureteric colic\n\n  spinal arthritis\n\n  coeliac disease (commoner than traditionally thought: 1 in 300 adults)\n\n**RARE**\n\n  mesenteric artery ischaemia (abdominal angina)\n\n  chronic pancreatitis\n\n  subacute obstruction (adhesions, malignancy and diverticulitis)\n\n  functional (psychogenic) abdominal pain\n\n  malignancy\n\n  metabolic causes, e.g. Addison's disease, porphyria, lead poisoning\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC, ESR\/CRP, MSU, _H. pylori_ testing.\n\n_POSSIBLE :_ U&E, LFT, amylase, coeliac screen, CA-125, plain abdominal X-ray, ultrasound, IVU, barium enema, colonoscopy, gastroscopy.\n\n_SMALL PRINT :_ specialised investigations such as mesenteric angiography and further tests for rare medical causes.\n\n  Urinalysis: blood alone with stone; blood, pus cells and nitrite in UTI.\n\n  MSU: to confirm urinary infection and guide treatment.\n\n  FBC and ESR\/CRP: may suggest inflammatory bowel disease, PU or malignancy.\n\n  U&E may be deranged in hydronephrosis, renal stones or Addison's disease.\n\n  LFT and amylase: LFT may be abnormal if carcinoma suspected. Amylase may be raised in pancreatitis and bowel ischaemia.\n\n  Coeliac screen: anti-endomysial and anti-gliadin antibodies: suggest coeliac disease if positive.\n\n  CA-125: especially in women aged 50 or more, may help exclude ovarian cancer.\n\n  _H. pylori_ testing: strong association with peptic ulcer disease.\n\n  Plain abdominal X-ray: may reveal constipation, subacute obstruction or kidney stones.\n\n  IVU: for renal stones or recurrent UTI.\n\n  Ultrasound: will show hydronephrosis and gallstones. Pelvic\/abdominal ultrasound also indicated if CA-125 elevated.\n\n  Barium enema, colonoscopy: for various lower bowel disorders.\n\n  Gastroscopy: may be required to confirm PU and exclude gastric carcinoma.\n\n  Further tests such as angiography (for mesenteric ischaemia) or investigations for rare medical causes may be arranged after specialist referral.\n\n##### TOP TIPS\n\n  Simply establishing what provokes or relieves the problem can provide helpful pointers: pain occurring after eating suggests gallstones, PU, gastric carcinoma or mesenteric ischaemia; if relieved by defecation, the likely diagnoses are IBS or constipation.\n\n  In an otherwise well patient, the longer the history the less likely there is to be significant underlying disease.\n\n  Avoid repeated investigation if a patient has already been thoroughly assessed in the past \u2013 unless the individual becomes unwell or develops new symptoms. Be frank with the patient by explaining about the 'law of diminishing returns' in investigating chronic unexplained abdominal pain.\n\n  Be prepared to make a positive diagnosis of IBS in a fit young patient if the symptoms are classical and basic investigations are negative; explanation and education are the keys to effective management.\n\n  Weight loss in association with recurrent abdominal pain suggests significant pathology.\n\n  Hard enlarged left supraclavicular nodes (Troisier's sign) are pathognomic of gastric carcinoma.\n\n  Beware that constipation itself is often a symptom rather than a diagnosis. Be sure to establish and treat any underlying cause if it doesn't respond to simple treatment.\n\n  IBS is the commonest diagnosis \u2013 but consider other possibilities if the pain is always in the same site, wakes the patient at night or is associated with rectal bleeding or weight loss.\n\n### CONSTIPATION\n\n#### The GP overview\n\nConstipation is defined as the infrequent or difficult evacuation of faeces. One study of a large normal working population showed variation in frequency from three times a day to three times a week. The average GP will see about 18 presentations of constipation each year. In most cases, there is a combination of aetiological factors, and serious causes are rare.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  diet and lifestyle (inadequate fibre and ignoring the urge to defecate)\n\n  inactivity (especially in the elderly)\n\n  irritable bowel syndrome (IBS)\n\n  painful perianal conditions: fissure, haemorrhoids, abscess, florid warts\n\n  drugs, e.g. opiates, iron, aluminium hydroxide\n\n**OCCASIONAL**\n\n  poor fluid intake\n\n  acquired megacolon, e.g. chronic laxative abuse, neurological problems, scleroderma\n\n  diverticulosis (with or without stricture)\n\n  hypothyroidism\n\n  carcinoma of rectum or colon\n\n**RARE**\n\n  pressure from extracolonic pelvic masses\n\n  acute bowel obstruction (various causes)\n\n  hypercalcaemia\n\n  Crohn's disease with stricture\n\n  infants and children: behavioural ('stool holding'), Hirschsprung's disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none; if suspicion of significant underlying bowel pathology, then FBC, barium enema and sigmoidoscopy\/colonoscopy (radiology and endoscopy usually arranged in secondary care).\n\n_POSSIBLE :_ urinalysis, thyroid function tests (TFT).\n\n_SMALL PRINT :_ plain abdominal X-ray, serum calcium, ultrasound, CT scan, biopsy.\n\n  Urinalysis: specific gravity high if inadequate fluid intake.\n\n  FBC: may reveal iron deficiency anaemia if underlying carcinoma.\n\n  TFT and serum calcium: will reveal hypothyroidism or hypercalcaemia.\n\n  Plain abdominal X-ray: may reveal megacolon full of faeces; erect and supine views will show obstruction.\n\n  Barium enema, sigmoidoscopy, colonoscopy: may reveal carcinoma or diverticular disease.\n\n  Ultrasound\/CT scan: may be helpful if a pelvic mass is present.\n\n  Biopsy: of suspicious lesions or to confirm Hirschsprung's disease.\n\n##### TOP TIPS\n\n  Clarify what patients mean by constipation: they often use the term inaccurately (e.g. in reference to a perfectly 'normal' bowel habit or to describe another symptom such as tenesmus).\n\n  The longer the history, the less likely there is to be any underlying or remediable cause.\n\n  Check the medication history (including over-the-counter treatment): just about any medication can alter the bowel habit.\n\n  Look at the patient: your immediate impression may give important clues to the underlying diagnosis (e.g. hypothyroidism or weight loss in malignancy).\n\n  Constipation alone in the elderly is rarely caused by sinister pathology \u2013 but if it is accompanied by other significant symptoms, such as weight loss, rectal bleeding or mucus, or diarrhoea, carcinoma is likely.\n\n  Beware of attributing abdominal pain to constipation \u2013 the true diagnosis might be intestinal obstruction. Visible peristalsis with audible borborygmi is never due to simple constipation.\n\n  Cases of Hirschsprung's disease can present 'late' \u2013 consider the diagnosis in a child with chronic constipation, a persistently swollen abdomen and an empty rectum.\n\n  Beware of assuming that known pathology (such as diverticular disease or IBS) in an individual is the cause of constipation. If the patient has presented with constipation, then there may have been a significant change in the pattern or the nature of the symptoms.\n\n### DIARRHOEA\n\n#### The GP overview\n\nDiarrhoea is the passage of abnormally liquid and frequent stools. It is said to be chronic if it lasts more than 2 weeks. It is the fifth-commonest presenting symptom in general practice. Patients will use the term 'diarrhoea' when presenting, but they may just mean frequent stools.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute infective gastroenteritis, e.g. rotavirus, campylobacter, food poisoning\n\n  antibiotics (and other drug side effects)\n\n  irritable bowel syndrome (IBS)\n\n  diverticulitis\n\n  overflow constipation (especially in the elderly)\n\n**OCCASIONAL**\n\n  lactose intolerance\n\n  chronic infection: amoebiasis, giardiasis, hookworm\n\n  bowel neoplasia\n\n  inflammatory bowel disease (IBD): ulcerative colitis and Crohn's disease\n\n  excess alcohol\n\n  toddler diarrhoea\n\n  coeliac disease (1 in 300)\n\n**RARE**\n\n  appendicitis\n\n  laxative misuse\n\n  thyrotoxicosis\n\n  malabsorption, e.g. coeliac disease\n\n  allergy\n\n  ovarian cancer\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ if persistent, stool specimen, FBC, ESR\/CRP, TFT, anti-endomysial and anti-gliadin antibodies, faecal calprotectin.\n\n_POSSIBLE :_ urinalysis, LFT, proctosopy\/sigmoidoscopy followed by barium enema, colonoscopy, CA-125.\n\n_SMALL PRINT :_ tests for malabsorption.\n\n  One stool sample is sufficient in acute diarrhoea of more than a week to look for common infections. Follow-up needed to show clearance of salmonella.\n\n  Series of three daily stool samples necessary to look for ova, cysts and parasites in chronic diarrhoea.\n\n  FBC: Hb may be reduced and ESR\/CRP elevated in IBD and malignancy; iron deficiency anaemia in neoplasia, coeliac disease; diverticulitis \u2013 check ferritin, B12 and folate too; WCC raised in IBD and infection.\n\n  TFT: will reveal thyrotoxicosis.\n\n  Anti-endomysial and anti-gliadin antibodies: suggest coeliac disease if positive.\n\n  Faecal calprotectin: helps rule out inflammatory bowel disease.\n\n  LFT: may suggest secondaries or alcoholism.\n\n  Urinalysis: specific gravity high in dehydration.\n\n  Proctoscopy\/sigmoidoscopy followed by barium enema or colonoscopy (usually arranged by the specialist): will confirm diagnosis of malignancy, diverticulosis, carcinoma and IBD.\n\n  CA-125: to help exclude ovarian cancer \u2013 especially in women aged 50 or more.\n\n  Tests for malabsorption: such as stool fat analysis, lactose tolerance test, small intestinal biopsy (all secondary care).\n\n##### TOP TIPS\n\n  Clarify what patients mean by diarrhoea \u2013 they may be referring simply to a minor change in their normal habit or the frequent passage of normal stools.\n\n  Giardiasis is much more common than previously thought and may be difficult to isolate in stool specimens. Empirical treatment is justified if the clinical picture is suggestive (recent onset after travel of persistent fatty diarrhoea with anorexia, nausea and bloating).\n\n  IBS rarely causes nocturnal diarrhoea.\n\n  Patients with gastroenteritis should steadily improve after a few days, but may experience symptoms for up to 10 days \u2013 warn them of this.\n\n  Do not be caught out by overflow diarrhoea in the elderly. The only way to establish this diagnosis is with a PR.\n\n  Remember to ask about foreign travel and occupation, which have implications for diagnostic possibilities and management.\n\n  Weight loss in chronic diarrhoea is highly suggestive of significant pathology.\n\n  In a young and otherwise well person, it is reasonable to make a positive clinical diagnosis of irritable bowel syndrome with minimal investigation \u2013 but beware of making this diagnosis for the first time in the middle-aged and elderly. Significant pathology mimicking IBS is likely.\n\n  Carefully assess hydration in infants and the elderly with diarrhoea; if there are signs of dehydration, always admit to hospital.\n\n  Initial telephone consultation is sufficient for most cases of acute diarrhoea, but if in constant (not colicky) abdominal pain, always see and examine to exclude an acute surgical condition.\n\n  Remember that acute diarrhoea in the elderly can precipitate or aggravate renal failure \u2013 especially if they are on ACE inhibitors. Stop these drugs for the duration of the illness and ensure adequate hydration.\n\n### EPIGASTRIC PAIN\n\n#### The GP overview\n\nUp to 40% of the adult population suffer this symptom in any 1 year. Only about one in ten seek help from their GP, usually presenting with 'indigestion'. The first step involves sorting out exactly what the patient means by this term. The second is to establish whether it is acute, chronic, or acute-on-chronic. And the third revolves around management, which is often orientated towards a pragmatic, symptomatic approach rather than establishing a precise diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  non ulcer dyspepsia (NUD)\n\n  gastro-oesophageal reflux disease (GORD)\/gastritis\n\n  IBS\n\n  gallstones\n\n  duodenal ulcer\/duodenitis\n\n**OCCASIONAL**\n\n  drug related, e.g. antibiotics, NSAIDs, bisphosphonates\n\n  pancreatitis (acute or chronic)\n\n  muscular\n\n  peritonitis (perforated DU or carcinoma)\n\n  carcinoma of the stomach\n\n  gastric ulcer\n\n  oesophageal spasm\n\n  aerophagy\n\n**RARE**\n\n  myocardial infarct\n\n  pneumonia\n\n  carcinoma of the pancreas\n\n  ruptured abdominal aortic aneurysm\n\n  gastrointestinal obstruction\n\n  referred from spine\n\n  epigastric hernia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, _H. pylori_ testing.\n\n_POSSIBLE :_ LFT, upper GI endoscopy, ultrasound.\n\n_SMALL PRINT :_ serum amylase, barium swallow or meal, oesophageal manometry\/pH studies, cardiac enzymes, ECG, CXR, erect and supine abdominal X-rays.\n\n  FBC: anaemia in underlying malignancy or bleeding from peptic ulcer; WCC raised in cholecystitis and pancreatitis.\n\n  _H. pylori_ testing: strong association with peptic ulcer disease; possibly also with other gastrointestinal pathologies.\n\n  LFT: may be abnormal in gallstones or malignancy.\n\n  Upper GI endoscopy: to visualise\/biopsy the upper GI tract (in particular, to exclude carcinoma of the stomach).\n\n  Ultrasound: for gallstones; may reveal other pathology such as pancreatic disease.\n\n  Barium swallow or meal: for investigation of the oesophagus, stomach and duodenum in those unwilling to have, or unfit for, endoscopy.\n\n  Serum amylase: elevated in acute pancreatitis.\n\n  Other tests: these are likely to be initiated in secondary care after referral. They include oesophageal manometry\/pH studies (if likely oesophageal problem but a normal endoscopy), cardiac enzymes and ECG (possible acute cardiac event), CXR (pneumonia), erect and supine abdominal X-rays (obstruction), CT or MRI scan if a mass is suspected.\n\n##### TOP TIPS\n\n  The non acute case of 'dyspepsia' in primary care is likely to be managed according to the prevailing guidelines \u2013 in which case a precise diagnosis may not be necessary.\n\n  It's important to address underlying concerns. Most patients with epigastric pain do not seek medical help. Those that do usually fear significant disease such as cancer.\n\n  A diagnosis of peptic ulcer still frightens many patients, especially older age groups, as they may be unaware of recent therapeutic advances. Provide adequate explanation and reassurance.\n\n  Don't overlook the medication history, as this may provide a simple solution to the problem.\n\n  A normal endoscopy does not rule out oesophageal problems such as GORD or spasm.\n\n  Patients with the following should be referred for endoscopy\/other investigations to exclude malignant disease: new onset of symptoms \u2013 or significant change in longer term symptoms \u2013 in the over-55s with unexplained and persistent dyspepsia; weight loss; dysphagia; GI bleed; vomiting; abdominal mass; anaemia.\n\n  In an obviously unwell patient with weight loss and epigastric pain, do not be 'reassured' by a normal endoscopy. There may well still be significant pathology such as carcinoma of the pancreas.\n\n  In at least 50% of cases, carcinoma of the pancreas presents with epigastric pain rather than the classical painless, progressive jaundice.\n\n  In acute cases, do not overlook referred pain \u2013 from heart, lungs or spine.\n\n### RECURRENT CHILDHOOD ABDOMINAL PAIN\n\n#### The GP overview\n\nRecurrent abdominal pain in childhood can be a calling card for a myriad hidden agendas. More than 85 causes have been listed, and as in most areas of general practice, the trick is to sift through the morass of information to find the keys to the diagnosis and open the way to effective management. The underlying cause in the most persistent cases is usually non-organic (90% of those referred to hospital).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  recurrent viral illnesses\n\n  anxiety and depression (sometimes known as periodic syndrome or abdominal migraine)\n\n  recurrent UTI\n\n  constipation\n\n  gastritis and GORD\n\n**OCCASIONAL**\n\n  Crohn's and coeliac disease\n\n  duodenal ulcer (DU)\n\n  irritable bowel syndrome\n\n  diabetes mellitus\n\n  Henoch\u2013Sch\u00f6nlein purpura\n\n  hydronephrosis, renal stones and ureteric reflux\n\n  Meckel's diverticulum\n\n**RARE**\n\n  parasitic infestation of the gut\n\n  food allergy\n\n  sickle-cell disease\n\n  tuberculosis (TB)\n\n  Hirschsprung's disease\n\n  temporal lobe epilepsy\n\n  pica\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ FBC, blood film, ESR\/CRP, anti-endomysial and anti-gliadin antibodies.\n\n_SMALL PRINT :_ plain abdominal X-ray, abdominal ultrasound, further hospital-based investigations (after referral).\n\n  Urinalysis and MSU: urinalysis will reveal evidence of a UTI, which will be confirmed with an MSU for microscopy and culture. Urinalysis will also reveal glucose in diabetes and possible haematuria in Henoch\u2013Sch\u00f6nlein purpura.\n\n  FBC: Hb may be reduced in any chronic disorder; leucocytosis in bacterial infection; eosinophilia in parasitic infestation or genuine food allergy. Blood film may show sickling. Raised ESR\/CRP suggests organic disease.\n\n  Anti-endomysial and anti-gliadin antibodies: a positive result suggests coeliac disease.\n\n  Ultrasound: non-invasive first line investigation of renal tract. Other investigation for confirmed UTI will be arranged by the paediatrician.\n\n  Further hospital-based investigations: if there is a high suspicion of organic disease, e.g. endoscopy for DU, barium meal and follow-through for Crohn's disease.\n\n##### TOP TIPS\n\n  The majority of children with recurrent abdominal pain will not have organic pathology \u2013 take the problem seriously and assess carefully, but avoid reinforcing worries with unnecessary investigation or referral.\n\n  Explore the parents' concerns \u2013 a child's anxiety may be fed by parents unnecessarily worrying about sinister and unlikely diagnoses.\n\n  Talk to children alone \u2013 this may reveal relevant problems at home or school which they would not have been able to admit in front of parents.\n\n  If recurrent UTI is a possibility, provide the parents with the necessary bottle and lab form so that an MSU can be taken during the next episode of pain.\n\n  Organic disease is suggested by pain distant from the umbilicus which wakes the child and which is associated with loss of appetite or weight, or a change in bowel habit.\n\n  Beware the unlikely event of an acute cause for the pain supervening, e.g. appendicitis, torsion of the testis \u2013 ensure that parents know that a different, acute pain should not be dismissed as 'the same old problem', but should be presented urgently.\n\n  Children proven to have a UTI should be managed according to NICE guidelines.\n\n  Avoid colluding in parental somatisation and overlooking the existence of family dysfunction or other causes of unhappiness.\n\n  Don't forget the rare possibility of sickling in the appropriate ethnic groups.\n\n### VOMITING\n\n#### The GP overview\n\nVomiting is one of the commonest reasons for an out-of-hours call \u2013 especially for children. While most cases are self-limiting and benign, the possible causes are numerous and the symptoms can herald serious pathology. Careful assessment is required, together with a willingness to review and admit if the diagnosis remains unclear.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  gastroenteritis\n\n  acute viral labyrinthitis (and some other causes of acute vertigo)\n\n  upper respiratory tract infection (URTI) (in children, especially with marked coughing)\n\n  pregnancy\n\n  appendicitis and other causes of the acute abdomen\n\n**OCCASIONAL**\n\n  hyper- and hypoglycaemia\n\n  intestinal obstruction\n\n  pyelonephritis\n\n  ureteric calculus\n\n  migraine\n\n  medication (e.g. antibiotics and cytotoxics)\n\n**RARE**\n\n  gastroduodenal disease (e.g. pyloric stricture or stenosis, DU, carcinoma)\n\n  meningitis\n\n  cerebral haemorrhage\n\n  bulimia nervosa\n\n  severe constipation\n\n  raised intracranial pressure (e.g. tumour)\n\n  renal failure\n\n  acute glaucoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, MSU, pregnancy test, blood glucose, FBC, U&E.\n\n_SMALL PRINT :_ lumbar puncture, abdominal X-rays, IVU, OGD, CT scan.\n\n  Urinalysis: high specific gravity suggests dehydration; glucose and ketones indicate hyperglycaemia; blood, white cells and protein, with or without nitrites, suggest UTI (confirm with MSU); blood alone might indicate a renal stone.\n\n  Pregnancy test: to confirm or reveal pregnancy.\n\n  Blood glucose: will confirm hypo- or hyperglycaemia.\n\n  U&E: may be deranged by vomiting; may also reveal underlying renal failure.\n\n  FBC: raised WCC suggests underlying infection or inflammation. Haemoglobin (Hb) may be reduced in malignancy.\n\n  Lumbar puncture, IVU, abdominal X-rays, OGD and CT scans: required in a few cases depending on the clinical picture and invariably arranged by the relevant specialist.\n\n##### TOP TIPS\n\n  Vomiting in children tends to be presented early, when it may be difficult to give a definite diagnosis. Be honest about this and make sure that parents know to call you again if the symptom doesn't settle or other 'alarm' symptoms develop \u2013 or arrange a definite time for follow-up.\n\n  Remember to look for both the cause (i.e. the aetiology) and the effect (i.e. possible dehydration) \u2013 especially in the very young and the very old, when the history may be difficult to obtain and the effects of fluid loss more marked.\n\n  Check to see if the patient is on any medication. This may be causing the vomiting, or it may have serious implications for management (e.g. steroids).\n\n  Don't forget pregnancy as a cause \u2013 the patient may be 'ignoring' the possibility.\n\n  Unless the diagnosis is obviously migraine, beware the patient with vomiting and a headache \u2013 think of meningitis, subarachnoid haemorrhage or raised intracranial pressure.\n\n  Do not treat empirically with anti-emetics \u2013 these may mask the true diagnosis or cause diagnostic confusion via side effects.\n\n  Have a low threshold for admitting diabetics. Whatever the cause of the vomiting, their diabetes is liable to become uncontrolled.\n\n  Look for acid dental erosion as a clue to bulimia in recurrent vomiting.\n\n  Gastroenteritis should cause increased bowel sounds. In the patient with abdominal pain and vomiting, if bowel sounds are absent or scanty, the diagnosis is likely to be an acute abdomen.\n\n### VOMITING BLOOD\n\n#### The GP overview\n\nThis presentation may vary from a few red streaks in gastric fluid to copious quantities of bright red blood. Blood static in the stomach for a few hours will change to look dark and granular, like coffee grounds. Always do a full urgent assessment, and be prepared for the sudden need for resuscitation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  peptic ulcer (PU)\/acute gastritis\n\n  Mallory\u2013Weiss (M\u2013W) tear\n\n  oesophageal varices (cirrhosis, usually alcoholic)\n\n  malignancy: oesophagus or stomach\n\n  GORD\n\n**OCCASIONAL**\n\n  swallowed blood (nose bleeds obvious, haemoptysis less so)\n\n  foreign body or mediastinal tumour perforating oesophagus and aorta (including aneurysm)\n\n  haemobilia (blood in bile)\n\n  ingested poisons: corrosive acid and alkali, arsenic\n\n  blood dyscrasias (e.g. thrombocytopenia, leukaemia, haemophilia, aplastic anaemia)\n\n**RARE**\n\n  ruptured oesophagus (acute vomiting or trauma)\n\n  spurious: deliberate swallowing and vomiting of blood (Munchausen's syndrome)\n\n  gallstone perforation of duodenum\n\n  scurvy\n\n  polyarteritis nodosa, systemic lupus erythematosus (SLE)\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThese will be done acutely in hospital, or in general practice after an episode of haematemesis when urgent admission is not indicated.\n\n  FBC: essential for assessment of the degree of blood loss. A normal Hb does not exclude a serious bleed as haemodilution may take several hours. Will also reveal blood dyscrasias.\n\n  Upper gastrointestinal (GI) endoscopy is the gold standard for finding the cause of the bleed and biopsy of suspicious lesions.\n\n  _Helicobacter_ testing: in the presence of peptic ulceration.\n\n  LFT and \u03b3GT to assess liver function. Alcohol is a significant contributory factor in many cases.\n\n  Plain erect abdominal X-ray (in hospital) useful to look for signs of viscus perforation (air under diaphragm) and, rarely, an ectopic gallstone.\n\n##### TOP TIPS\n\n  Take a careful history \u2013 patients often confuse vomiting up and coughing up blood.\n\n  If about to visit, ask the patient not to dispose of the evidence \u2013 viewing the vomit is worth a thousand words of history.\n\n  Don't forget the relevance of the patient's drug history: non-steroidal anti-inflammatory drugs (NSAIDs), steroids and warfarin may all be associated with acute gastric erosions.\n\n  Tachycardia may be the only physical sign of a significant GI bleed.\n\n  In all acute cases, admit unless the patient is perfectly well and the cause obvious and insignificant (e.g. swallowed blood or very minor Mallory\u2013Weiss tear).\n\n  The patient may not realise the significance of coffee-ground vomit or melaena \u2013 enquire specifically about these symptoms.\n\n  Troisier's sign (enlargement of the left supraclavicular node) strongly suggests malignancy.\n\n  Oesophageal varices account for only 5% of cases, but 80% of mortality. Call for an ambulance immediately and secure intravenous (IV) access if possible.\n\n## ANORECTAL\n\nAnal itching\n\nAnal swelling\n\nAnorectal pain\n\nRectal bleeding\n\nRectal discharge\n\n### ANAL ITCHING\n\n#### The GP overview\n\nStraight out of the list of _'Embarrassing things to see your GP about',_ this is a presentation that patients love to hate. From a GP perspective, it's one that is generally straightforward to deal with, and effective treatment can usually be offered immediately, much to the patient's relief.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  fungal infection \u2013 tinea, thrush\n\n  threadworms\n\n  haemorrhoids\n\n  perianal skin tags\n\n  anal fissure\n\n**OCCASIONAL**\n\n  poor hygiene\n\n  recurrent or chronic diarrhoea\n\n  perianal warts\n\n  streptococcal perianal infection in children\n\n  trauma from sexual practices \u2013 anal intercourse and foreign body insertion\n\n  faecal incontinence, including liquid faecal seepage round impacted scybala\n\n  psoriasis\n\n  secondary to underlying diabetes\n\n  anorectal carcinoma\n\n  chemical irritation: defaecation after a very spicy meal (commonly experienced, rarely presented in practice), bubble baths, soaps, sexual lubricants\n\n**RARE**\n\n  irritation from perineal decorative body piercing (the 'Guiche')\n\n  lichen sclerosus et atrophicus (affects 1 in 100 women, 3 in 10 of these have anal symptoms)\n\n  Crohn's disease (anal\/perianal fistula)\n\n  rectovaginal fistula\n\n  rectal prolapse\n\n  any other cause of rectal discharge or anal swellings \u2013 see appropriate chapters\n\n  any serious cause of generalised pruritus \u2013 see chapter on anal itching. Rare here because pruritus ani is unlikely to be a presenting complaint\n\n  STDs, e.g. syphilis, gonorrhoea, chlamydia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ : none.\n\n_POSSIBLE_ : skin swab, FBC, ESR, fasting glucose or HbA1c, proctoscopy.\n\n_SMALL PRINT_ : none.\n\n  In general, unless there are obvious pointers to other more serious disease, investigations would usually only follow after failure of empirical treatment.\n\n  Skin swab for bacteriology may help identify local infection.\n\n  FBC, ESR: may be helpful if Crohn's disease is suspected, but only as an adjunct to referral as the appropriate management.\n\n  Fasting glucose or HbA1c is essential in recurrent or prolonged cases to exclude diabetes.\n\n  Proctoscopy is quick to do in general practice and can yield valuable information if there is an underlying rectal cause.\n\n##### TOP TIPS\n\n  Most patients will have attempted self-treatment before presenting in the surgery. This may not always have been appropriate, and could have made the problem worse.\n\n  Unless you are absolutely sure of an obvious cause, it is wise to perform a digital rectal examination to look for rectal causes.\n\n  Perianal warts imply a sexually transmitted disease contact. Refer to GUM clinic for contact tracing and treatment.\n\n  Anal itching is often associated with soreness. If it precludes a rectal examination but there is no obvious primary anal cause for itching, treat symptomatically and bring the patient back to complete the assessment when more comfortable to do so. The patient is unlikely to want to return for this without understanding a clear explanation of why it is necessary.\n\n  Four per cent of women with lichen sclerosus et atrophicus go on to develop vulval cancer. Refer if the vulva is affected, or if treatment fails.\n\n  Refer any suspicious anal lesion for biopsy.\n\n  Be confident to ask about recent sexual encounters and sexual practices if possibly relevant. Sexual history may be important.\n\n### ANAL SWELLING\n\n#### The GP overview\n\nBecause of embarrassment on the part of the patient, this may well present as a 'while I'm here' symptom. The temptation to make a diagnosis without examination should be resisted \u2013 some of the causes (such as perianal abscesses) require urgent attention and others may, rarely, provide something of a surprise (e.g. fistulae, carcinoma).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  prolapsed pile\n\n  perianal haematoma\n\n  skin tags\n\n  perianal abscess\n\n  rectal prolapse\n\n**OCCASIONAL**\n\n  warts\n\n  sebaceous cyst\n\n  sentinel pile\n\n  infected pilonidal sinus\n\n**RARE**\n\n  hidradenitis\n\n  anal fistula\n\n  carcinoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIn most cases, investigation will be unnecessary. The only exceptions are warts (in which case referral to the local GUM clinic may be required to screen for sexually transmitted disease) and possible carcinoma (in which case biopsy will be performed in secondary care). Also, any suspicion of Crohn's disease causing perianal disease would be investigated in hospital in the usual way.\n\n##### TOP TIPS\n\n  This is one of those situations in which a brief history can be taken while the patient is undressing, or during the examination. Atypically for primary care, it's the examination, rather than the history, which usually provides the definitive diagnosis.\n\n  If a discharge, as well as a lump, is mentioned by the patient, then abscesses, warts, prolapses and fistulae top the list of differentials.\n\n  The patient with an anal swelling who has obvious difficulty walking into the consulting room has either an abscess, a large perianal haematoma or strangulated prolapsed piles.\n\n  Recurrent or multiple fistulae suggest Crohn's disease.\n\n  If a prolapsed pile is very swollen and painful, it is probably strangulated, and so requires urgent surgical attention.\n\n  A persistent, ulcerating anal swelling, especially in the middle-aged or elderly, requires urgent biopsy to exclude carcinoma.\n\n### ANORECTAL PAIN\n\n#### The GP overview\n\nThis is usually severe and distressing. Because of reflex sphincteric spasm, constipation very often follows and increases the pain and suffering further. Adequate examination is also difficult for the same reason; fortunately if a PR exam is too difficult, a visual inspection can often yield the diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anal fissure\n\n  thrombosed haemorrhoids\/perianal haematoma\n\n  perianal abscess\n\n  proctalgia fugax (PF)\n\n  anorectal malignancy\n\n**OCCASIONAL**\n\n  levator ani syndrome\n\n  Crohn's disease\n\n  coccydynia\n\n  descending perineum syndrome\n\n  prostatitis\n\n  ovarian cyst or tumour\n\n  solitary rectal ulcer syndrome\n\n**RARE**\n\n  anal tuberculosis\n\n  cauda equina lesion\n\n  endometriosis\n\n  trauma\n\n  presacral tumours\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, proctoscopy, faecal calprotectin.\n\n_SMALL PRINT :_ urinalysis, ultrasound, barium enema, other imaging.\n\n  FBC\/ESR\/CRP: WCC may be raised in abscess and Crohn's disease. ESR\/CRP raised in these and carcinoma.\n\n  Proctoscopy valuable if pain allows (specialist might also take biopsy).\n\n  Faecal calprotectin: may help in diagnosing Crohn's disease.\n\n  Urinalysis: pus cells and blood may be present in prostatitis or invasive bladder tumour.\n\n  Ultrasound of pelvis if pelvic examination reveals a mass. Barium enema may be necessary to assess possible bowel involvement. In obscure cases, specialists may request other forms of imaging.\n\n##### TOP TIPS\n\n  If the patient uses dramatic language (e.g. red-hot poker) to describe fleeting pain, is otherwise well and there are no obvious abnormalities on examination, the diagnosis is likely to be proctalgia fugax.\n\n  Examine the patient \u2013 the cause is usually a thrombosed pile, anal fissure or an abscess, and these can usually be diagnosed by simple inspection.\n\n  Provide symptomatic relief but remember to deal with any underlying causes \u2013 especially constipation.\n\n  Don't forget to ask about thirst and urinary frequency: recurrent abscesses may be the first presentation of diabetes.\n\n  Preceding weight loss and\/or change in bowel habit should prompt a full urgent assessment with carcinoma and inflammatory bowel disease in mind.\n\n  Some perianal abscesses do not result in external swelling. If PR exam is prohibitively painful, consider this possibility \u2013 especially if the patient is febrile.\n\n  In florid or recurrent perianal problems, think of Crohn's disease as a possible cause.\n\n  Remember rarer causes in intractable, constant pain in a patient with no obvious signs on PR.\n\n### RECTAL BLEEDING\n\n#### The GP overview\n\nThis is a very common presenting complaint and creates a lot of anxiety in the patient. By far the likeliest causes are haemorrhoids or a fissure, but more sinister pathologies should be considered according to the clinical picture. In children, constipation causing a fissure is the most frequent cause.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  haemorrhoids\n\n  anal fissure\n\n  gastroenteritis\n\n  rectal carcinoma\n\n  diverticular disease\n\n**OCCASIONAL**\n\n  villous adenoma\n\n  trauma (especially non-accidental injury (NAI) if in children)\n\n  anticoagulant therapy\n\n  inflammatory bowel disease\n\n  colonic carcinoma\n\n**RARE**\n\n  blood clotting disorders (including anticoagulants)\n\n  bowel ischaemia\n\n  angiodysplasia\n\n  intussusception\n\n  Meckel's diverticulum (in children)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ proctoscopy.\n\n_POSSIBLE :_ FBC, ESR\/CRP, LFTs, bone biochemistry, U&E, stool for microbiology and faecal calprotectin, sigmoidoscopy, colonoscopy, barium enema.\n\n_SMALL PRINT :_ clotting screen.\n\n  FBC: check for anaemia from acute or chronic bleeding; low platelets may cause or aggravate bleeding.\n\n  ESR\/CRP raised in active inflammatory bowel disease and malignancy.\n\n  If malignancy is suspected, LFT, U&E and bone biochemistry are useful early on as a baseline.\n\n  Clotting screen: if clotting disorder a possibility; INR if on warfarin.\n\n  Stool specimen: helpful in the presence of diarrhoea. May show evidence of infective cause (especially _Campylobacter_ ) or white cells in inflammatory bowel disease. Faecal calprotectin may be useful in diagnosing IBD.\n\n  Proctoscopy: helpful in primary care in visualising haemorrhoids and proctitis to confirm a clinical diagnosis.\n\n  Sigmoidoscopy or colonoscopy are the most helpful investigations if significant pathology is suspected, and allow biopsy of suspicious lesions.\n\n  Barium enema good for revealing strictures related to the underlying pathology, but if diverticulosis present, may not rule out coexistent neoplasia.\n\n##### TOP TIPS\n\n  Eighty per cent of rectal tumours are within fingertip range. Always do a PR examination unless, in a child or young adult, the diagnosis is manifestly obvious from the history.\n\n  If blood is on the toilet paper and surface of the motions, the cause is likely to be palpable PR or visible on proctoscopy; if mixed in with the motions, referral for further investigation will be required to make a definite diagnosis.\n\n  In young adults, the diagnosis is usually clear from the history and is likely to be haemorrhoids or a fissure. In such cases, if and when you refer, to allay anxiety, emphasise that this is for treatment rather than investigation.\n\n  The presence of diarrhoea with rectal bleeding in young or middle-aged adults suggests gastroenteritis (especially _Campylobacter_ ) or colitis.\n\n  Change of bowel habit and weight loss with rectal bleeding are ominous symptoms which should prompt urgent referral.\n\n  If a child is presented with rectal bleeding without any clear cause, consider NAI.\n\n  The presence of haemorrhoids does not necessarily clinch the diagnosis \u2013 another lesion may be present, especially in the elderly.\n\n  A brisk, painless haemorrhage in an elderly patient is likely to be caused by diverticular disease. Significant amounts of blood can be lost, so assess urgently with a view to admission.\n\n### RECTAL DISCHARGE\n\n#### The GP overview\n\nThis unpleasant symptom causes considerable embarrassment and inconvenience. The presenting complaint may well be itching rather than a history of discharge or dampness. The most common causes are benign, but serious pathology is sufficiently likely to warrant thorough examination.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  haemorrhoids\n\n  anal fissure\n\n  rectal prolapse\n\n  proctitis\n\n  perianal warts\n\n**OCCASIONAL**\n\n  rectal carcinoma\n\n  IBS\n\n  anal\/perianal fistula (may be associated with Crohn's disease)\n\n  villous adenoma\n\n  poor anal hygiene\n\n  solitary rectal ulcer syndrome\n\n**RARE**\n\n  anal tuberculosis\n\n  anal carcinoma\n\n  STD, e.g. syphilis, gonorrhoea, chlamydia\n\n  AIDS\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ swab if purulent.\n\n_POSSIBLE :_ FBC, ESR\/CRP, U&E and direct visualisation techniques (see below).\n\n_SMALL PRINT :_ STD screen.\n\n  Anorectal swab for bacteriology if pus and local inflammation is present.\n\n  FBC\/ESR\/CRP: WCC and ESR\/CRP raised in inflammatory bowel disease.\n\n  U&E: profuse mucus discharge from a villous adenoma can cause hypokalaemia.\n\n  Proctoscopy easily done in practice and guides management and referral.\n\n  Sigmoidoscopy allows definitive examination and biopsy of suspicious lesions or inflammation.\n\n  STD screen \u2013 if STD suspected.\n\n##### TOP TIPS\n\n  This symptom is likely to be associated with embarrassment. A clinical examination is important \u2013 so help the patient overcome any reticence with a sympathetic approach.\n\n  Normal secretions from the anus should not be discounted \u2013 from perianal sweat glands and the anal glands themselves. This can sometimes cause a significant problem, particularly in the obese and those with poor anal hygiene.\n\n  In women, the source of the discharge may be unclear. Speculum examination and protoscopy may be required to resolve any doubt.\n\n  Sexual history may be important. Don't be coy in asking about recent sexual encounters and sexual practices if relevant.\n\n  Rectal discharge in the elderly suggests carcinoma unless proved otherwise.\n\n  A history of diarrhoea and abdominal pain suggests bowel pathology. Ulcerative colitis or Crohn's disease is most likely in this context.\n\n  Any ulcerating lesion in the perianal area other than an obvious haemorrhoid should prompt referral for biopsy.\n\n## BREAST\n\nBreast enlargement in men\n\nBreast lumps in women\n\nBreast pain\n\nNipple discharge\n\n### BREAST ENLARGEMENT IN MEN\n\n#### The GP overview\n\nSwelling of the breast tissue in a male is an embarrassing symptom, often presented behind the facade of a 'calling card'. The following differential diagnosis does not include other causes of breast swelling, which are referred to in the Top tips at the end of this section. In true breast swelling, glandular tissue is palpable behind the areola and is usually bilateral.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  puberty\n\n  drugs (spironolactone, cimetidine, digoxin, cyproterone, finasteride, marijuana)\n\n  chronic liver disease (especially alcohol)\n\n  lung carcinoma\n\n  hyperthyroidism\n\n**OCCASIONAL**\n\n  hypothyroidism\n\n  hyperprolactinaemia\n\n  haemodialysis and chronic renal failure\n\n  testicular carcinoma\n\n  adrenal carcinoma\n\n  cryptorchidism and other causes of hypogonadism\n\n**RARE**\n\n  Klinefelter's syndrome\n\n  true hermaphroditism and male pseudohermaphroditism\n\n  acromegaly\n\n  McCune\u2013Albright syndrome\n\n  hypernephroma\n\n  carcinoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ (except for obvious pubertal cause) _:_ FBC, U&E, LFT, TFT.\n\n_POSSIBLE :_ testosterone, CXR, tests of pituitary function.\n\n_SMALL PRINT :_ tumour markers, chromosome analysis, CT scan, biopsy.\n\n  FBC: many chronic systemic illnesses can cause gynaecomastia. There may be an associated normochromic, normocytic anaemia. MCV may be raised in hypothyroidism and chronic liver disease.\n\n  U&E and LFT: will reveal chronic renal and liver disease.\n\n  TFT: to diagnose thyroid abnormality.\n\n  Testosterone: reduced in hypogonadism and chronic illness including liver disease.\n\n  CXR: if lung carcinoma a possibility.\n\n  Tests of pituitary function (e.g. FSH, LH, prolactin and other more complex, hospital-based tests): to check for pituitary hormone abnormalities.\n\n  Tumour markers (usually hospital-based): AFP and HCG act as tumour markers for testicular carcinoma.\n\n  Chromosome analysis: for Klinefelter's syndrome.\n\n  CT scan (secondary care): may be necessary for testicular tumour staging and diagnosis of adrenal and renal tumours.\n\n  Biopsy: if carcinoma suspected.\n\n##### TOP TIPS\n\n  Many male breast swellings are not true breast enlargement: possibilities include simple obesity, abscess, sebaceous cyst and lipoma.\n\n  Pubertal boys will be very self-conscious about gynaecomastia. Reassure them that the problem is common and will resolve, and that they are not changing sex.\n\n  Iatrogenic causes are common \u2013 check the drug history (including over-the-counter and illicit drugs).\n\n  In a pubertal boy with a 'normal' and a 'small' testis, the 'normal' one may conceal a tumour. Check with an ultrasound if in doubt.\n\n  Apparent unilateral gynaecomastia in an adult male may be due to breast carcinoma \u2013 refer urgently if there is a hard mass, you cannot feel glandular tissue behind the areola, or you're in any doubt.\n\n  Gynaecomastia with a headache and visual disturbance may be caused by a pituitary tumour. Refer urgently.\n\n  Clubbing of the fingers in a smoker with gynaecomastia is virtually pathognomic of bronchial carcinoma. Investigate urgently.\n\n### BREAST LUMPS IN WOMEN\n\n#### The GP overview\n\nThe discovery of a lump in a woman's breast will usually create a lot of anxiety. She will probably have found it herself and with the high public awareness of breast cancer, will want reassurance or rapid action. A careful examination of both breasts and associated lymph nodes is mandatory.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  carcinoma\n\n  cyst\n\n  abscess\n\n  fibroadenoma\n\n  fibrous dysplasia\n\n**OCCASIONAL**\n\n  duct ectasia\n\n  fat necrosis\n\n  lipoma\n\n  Paget's disease of the nipple\n\n  galactocoele\n\n  multiple cysts\n\n**RARE**\n\n  tuberculosis\n\n  sarcoma\n\n  lymphoma\n\n  phyllodes tumour (benign)\n\n  Mondor's disease (thrombophlebitis)\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are few investigations worth doing in general practice other than attempted aspiration of a suspected cyst.\n\nSpecialist investigation may include aspiration, mammography, ultrasound (for example, to distinguish a solid from a cystic lump), biopsy and, when appropriate, cancer staging.\n\n##### TOP TIPS\n\n  If the lump feels cystic, consider aspiration \u2013 instant resolution of the problem will result in a very grateful patient. But warn the patient that unsuccessful aspiration, while requiring referral, does not necessarily imply a sinister diagnosis \u2013 some cysts can be very difficult to aspirate.\n\n  Check the breast again a couple of weeks after aspiration. As long as the lump has completely resolved and the aspirate was not bloodstained, no further action is required.\n\n  It is quite common for women to think they can feel a lump while the doctor has difficulty in detecting a discrete lesion. Re-examine after the patient's next period \u2013 but then make a firm management decision. If in doubt, refer rather than delay further as the woman will understandably be very anxious.\n\n  In the very elderly, breast carcinoma may run a relatively benign course, responding very well to tamoxifen alone. In certain cases it might be worth discussing the situation with your local specialist, as GP treatment will provoke far less anxiety.\n\n  Skin dimpling, local flattening of the breast and nipple alteration indicate cancer until proved otherwise.\n\n  Even if the diagnosis is likely to be a fibroadenoma \u2013 as in a young woman with a highly mobile lump \u2013 refer, as unpleasant surprises do occur.\n\n  In a post-menopausal woman, the diagnosis is almost certain to be carcinoma. Refer urgently.\n\n  A mass appearing after trauma may be fat necrosis \u2013 but recheck after a few weeks and refer if not resolved.\n\n### BREAST PAIN\n\n#### The GP overview\n\nBreast pain have a variety of innocent causes: the commonest are puberty and pregnancy. It can be a troublesome recurrent problem for women with cyclical mastalgia. Cancer is very likely to be a major concern: this is an uncommon cause and pain is an unfortunately late sign of the disease.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  pregnancy\n\n  cyclical mastalgia\n\n  cracked or inflamed nipple\n\n  breast abscess\n\n  mastitis\n\n**OCCASIONAL**\n\n  carcinoma\n\n  onset of puberty\n\n  lactation and\/or galactocoele\n\n  simple cyst\n\n  trauma\n\n**RARE**\n\n  chondritis of costal cartilage\n\n  angina\n\n  cervical spondylosis\n\n  herpes zoster\n\n  Mondor's disease (thrombophlebitis of chest wall or breast veins \u2013 rare: 0.5\u20130.9%)\n\n  tuberculosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ pregnancy test, fine needle aspiration, mammography.\n\n_SMALL PRINT :_ swab of any nipple discharge, other investigations if non-breast causes suspected.\n\n  Pregnancy test worthwhile in bilateral pain if a period has been missed.\n\n  Fine needle aspiration of a tense cyst may yield fluid for cytology and relieve the pain. If only blood is obtained, refer urgently.\n\n  Mammography may help if pain is accompanied by a mass or ill-defined nodularity but this would normally be performed in secondary care after referral.\n\n  If the aetiology is infective and the nipple is discharging, a swab may help guide treatment.\n\n  Other investigations: if a non-breast cause is suspected, other tests may be required according to the pattern of the symptoms, e.g. stress test (angina) or cervical spine X-ray (cervical spondylosis).\n\n##### TOP TIPS\n\n  Offer to examine the breasts even if you are sure from the history that there is no significant pathology \u2013 many women fear breast cancer and will find your reassurance hard to accept if they feel they haven't been taken seriously.\n\n  Don't reflexly prescribe in cyclical mastalgia; the patient's agenda is often to exclude serious disease rather than seek drug therapy.\n\n  Remember pregnancy as a cause \u2013 the patient will not always volunteer this as a possibility, even if she has just missed a period.\n\n  Unilateral breast pain with no other local signs may be an early symptom of shingles. Check the back in the T4\/5 dermatomes for a rash.\n\n  'Chest pain' may be a euphemism used by a (frequently older) woman in denial. Don't miss advanced tumour through not examining the breasts.\n\n  Cancer rarely presents with breast pain but consider this possibility in a woman complaining of constant 'pricking' breast pain.\n\n  A lactating woman with unilateral breast pain and flu-type symptoms is probably developing mastitis \u2013 treat early to avoid the development of an abscess.\n\n  If the pain is related to exertion in a late middle-aged or elderly woman, consider angina as a possibility.\n\n### NIPPLE DISCHARGE\n\n#### The GP overview\n\nNipple discharge has a number of disparate causes, from the first outward sign of a previously unrecognised pregnancy, to a late sign of an advanced carcinoma. It can cause embarrassment and concern in equally large amounts. Compared with breast pain and lumps, it is a relatively rare presenting symptom. Take it seriously and assess carefully \u2013 investigation will often be needed.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  pregnancy\n\n  duct papilloma\n\n  duct ectasia\n\n  acute mastitis\/breast abscess\n\n  areolar abscess (infected gland of Montgomery)\n\n**OCCASIONAL**\n\n  oral contraceptives\n\n  intraduct carcinoma\n\n  neonatal and peripubertal galactorrhoea (also post-lactation)\n\n  hyperprolactinaemia (drugs, prolactinoma, hypothyroidism)\n\n  duct epithelial proliferation\n\n  galactocoele\n\n**RARE**\n\n  periductal (plasma cell) mastitis\n\n  mechanical stimulation\n\n  invasive carcinoma\n\n  tuberculous abscess\n\n  mamillary duct fistula\n\n  Paget's disease of nipple\n\n  comedo mastitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none (referral for biopsy or mammogram in suspicious cases).\n\n_OCCASIONAL :_ pregnancy test, prolactin level, TFT.\n\n_SMALL PRINT :_ swab of purulent discharge.\n\n  Pregnancy test if pregnancy suspected.\n\n  Swab purulent discharge: may help guide antibiotic therapy.\n\n  Prolactin level and TFT: to check for hyperprolactinaemia or hypothyroidism in galactorrhoea.\n\n  Excision biopsy will be performed if a suspicious lump is palpable. In cases of doubt, mammography may help; surgical exploration may also be undertaken if pressure on an area of one breast consistently elicits discharge.\n\n##### TOP TIPS\n\n  Women presenting with breast discharge are likely to be afraid there may be an underlying cancer. To 'reassure' properly, make sure you address this anxiety.\n\n  If the discharge is bilateral, then serious breast disease is highly unlikely.\n\n  In a woman of child-bearing age with bilateral serous discharge, enquire specifically about pregnancy: this possibility may be deliberately concealed or genuinely overlooked by the patient.\n\n  If a pre-menopausal woman is amenorrhoeic with bilateral discharge and pregnancy has been excluded, remember the possibility of hyperprolactinaemia.\n\n  Women with new breast discharge over the age of 50 (unless there is an obvious benign cause), and all women with bloody discharge, should be referred to exclude serious pathology.\n\n  If a lump is palpable, or pressure on a certain area of the breast consistently produces the discharge, refer for probable excision biopsy.\n\n  Bright red blood from one orifice suggests duct papilloma or carcinoma.\n\n  Nipple discharge in a male is always abnormal, except occasionally in pubertal boys. Investigate or refer as appropriate.\n\n## CEREBRAL\n\nAcute confusion\n\nDizziness\n\nHallucinations\n\nHeadache\n\nInsomnia\n\nLoss of libido\n\nMemory loss\n\nVertigo\n\n### ACUTE CONFUSION\n\n#### The GP overview\n\nThere are hundreds of possible individual causes of confusion. Patients with acute confusion are usually elderly and often present out of hours via a call from an anxious relative or neighbour. The dementias constitute the chronic confusional states, which are not considered here.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  hypoxia (respiratory and cardiac)\n\n  systemic infection\n\n  cerebrovascular accident (CVA: stroke and transient ischaemic attack (TIA))\n\n  hypoglycaemia\n\n  diabetic ketoacidosis (DKA)\n\n**OCCASIONAL**\n\n  alcohol withdrawal or intoxication\n\n  cerebral infection\n\n  electrolyte imbalance and uraemia\n\n  iatrogenic (e.g. digoxin, diuretics, steroids and opiates)\n\n  myxoedema\n\n  drug abuse\n\n**RARE**\n\n  Wernicke's encephalopathy\n\n  cerebral tumour\n\n  hypo- and hyperparathyroidism\n\n  Cushing's disease\n\n  postictal state\n\n  carbon monoxide poisoning\n\n#### Ready reckoner\n\n#### Possible investigations\n\nAcute confusion has so many causes and possible presentations that it is difficult to provide a definitive guide of investigations for the GP. A number of investigations might be considered according to the clinical picture and social circumstances; in the majority of cases, though, the patient will be admitted and necessary tests therefore arranged by the hospital.\n\n_LIKELY :_ urinalysis, blood sugar (usually glucometer), pulse oximetry.\n\n_POSSIBLE :_ FBC, CXR, ECG, cardiac enzymes, TFT.\n\n_SMALL PRINT :_ calcium, digoxin levels, CT scan.\n\n  Urinalysis is very helpful if possible: look for glucose and ketones (DKA), specific gravity (dehydration), pus, blood and nitrite in UTI. Ketones alone in starvation and possibly hypoglycaemia.\n\n  A blood glucometer reading is more practical than a formal blood glucose in the acute situation to diagnose hypo- and hyperglycaemia.\n\n  Pulse oximetry: to detect hypoxia.\n\n  FBC: raised WCC in infections. Raised MCV helpful pointer to excess alcohol and myxoedema.\n\n  U&E important, especially if any signs of dehydration or on diuretics.\n\n  LFT and TFT: alcohol, disseminated malignancy and hypothyroidism should always be considered.\n\n  CXR: may reveal a cause of hypoxaemia (e.g. pneumonia, cardiac failure).\n\n  ECG, cardiac enzymes: if silent infarct suspected as cause.\n\n  Calcium: to detect possible hypo- or hyperparathyroidism.\n\n  Digoxin levels: for digoxin toxicity.\n\n  CT scan: invariably a hospital-based investigation in acute confusion: may reveal space-occupying lesion, bleed or infarct.\n\n##### TOP TIPS\n\n  The key to management is establishing that the confusion really is acute rather than a gradual deterioration of cognition. This requires a careful history from someone who knows the patient well.\n\n  Don't forget a drug history: if little information is available on a visit, check the patient's medication cupboard.\n\n  In acute confusional states, it can be difficult to obtain useful clinical pointers from the patient's history. The examination therefore assumes greater importance than usual.\n\n  It is virtually impossible to reach a firm diagnosis and treat safely in the home. Be very sure of yourself if you choose not to admit.\n\n  Central cyanosis is an ominous sign. Give oxygen, if possible, and dial 999.\n\n  In a diabetic on treatment, always check the blood sugar \u2013 remember that hypoglycaemia can produce confusion with neurological signs, mimicking a CVA.\n\n  Altered physiological responses in the elderly may result in a normal pulse and temperature even in the presence of significant infection. Don't be misled by this.\n\n  Ask if any other household members have been unwell \u2013 carbon monoxide poisoning could affect others too.\n\n### DIZZINESS\n\n#### The GP overview\n\nThis common and vague symptom can mean different things to different people. It is treated here as being a sense of light-headedness without the illusion of movement characteristic of vertigo. This is a useful distinction in practice as the causes of true vertigo are different \u2013 see p. 89. Dizziness tends to be a heartsink symptom as it is so common, has so many diagnostic possibilities, is so often linked with anxiety and other symptoms \u2013 and very often the exact cause remains obscure.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  viral illness\n\n  anxiety (and hyperventilation)\n\n  hypoglycaemia\n\n  postural hypotension (e.g. elderly and pregnancy)\n\n  vertebrobasilar insufficiency (elderly with cervical osteoarthritis)\n\n**OCCASIONAL**\n\n  acute intoxication: drugs\/alcohol\n\n  effects of chronic alcohol misuse\n\n  iatrogenic: drug therapy (antihypertensives, antidepressants)\n\n  cardiac arrhythmia\n\n  any severe systemic disease\n\n**RARE**\n\n  aortic stenosis\n\n  subclavian steal syndrome\n\n  partial seizures\n\n  Addison's disease\n\n  carbon monoxide poisoning (blocked flue)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, FBC, U&E, LFT, glucometer blood glucose.\n\n_SMALL PRINT :_ EEG, ECG\/24-hour ECG, echocardiography, CT scan.\n\n  Urinalysis for glucose: underlying diabetes may cause dizziness, either through general malaise or because of an autonomic neuropathy.\n\n  FBC: underlying anaemia will exacerbate any cause of light-headedness; raised MCV may indicate alcohol abuse.\n\n  U&E and LFT may be worth measuring if systemic disease suspected; in particular, sodium low, and potassium and urea both high in Addison's disease; LFT may be abnormal in alcohol abuse.\n\n  Glucometer blood glucose: blood glucose measurement will provide a diagnosis of hypoglycaemia only if done during an episode.\n\n  EEG: if partial epilepsy a possibility (would also then require CT scan) \u2013 both arranged by specialist.\n\n  ECG\/24 h ECG: for possible arrhythmia.\n\n  Echocardiography: for suspected aortic stenosis.\n\n##### TOP TIPS\n\n  The first step in the history is to establish what the patient means by dizziness, and, in particular, to distinguish it from true vertigo.\n\n  Dizziness is often multifactorial, especially in the elderly \u2013 so do not necessarily expect to find a single underlying pathology.\n\n  If no clear diagnosis is obvious from the history, the dizziness is long standing, and the patient presents a list of other vague symptoms yet is objectively quite well (e.g. no weight loss), the likely diagnosis is anxiety.\n\n  Don't forget that commonly prescribed drugs can cause or aggravate postural hypotension \u2013 review the patient's medication.\n\n  If the patient has episodic loss of consciousness as well as dizziness, then the chances of significant pathology are much greater: investigate or refer.\n\n  In puzzling cases, ask about other family members and type of domestic heating used. Carbon monoxide poisoning is a completely avoidable but regular killer.\n\n  If an aortic murmur is heard, refer urgently. Significant aortic stenois can cause sudden death.\n\n  Remember denial is very strong in alcoholics. If in doubt, check MCV and LFT.\n\n### HALLUCINATIONS\n\n#### The GP overview\n\nA hallucination is a sensory perception occurring without any external stimulus. This distinguishes it from an illusion, which is a distortion of a sensory perception. Hallucinations can occur in any sensory modality and may present in isolation or as part of a larger clinical problem (particularly an acute confusional state). A hallucination is often a very frightening experience for the sufferer.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  drugs (amphetamine, cocaine, LSD, ecstasy, solvents and tricyclic overdose) and drug withdrawal\n\n  extreme fatigue\n\n  alcoholic hallucinosis (delirium tremens of acute alcohol withdrawal)\n\n  febrile delirium\n\n  schizophrenia\n\n**OCCASIONAL**\n\n  severe metabolic disturbance of any cause\n\n  temporal lobe epilepsy\n\n  cerebral space-occupying lesion\n\n  psychotic depression\n\n  bereavement reaction\n\n  hypoxia\n\n**RARE**\n\n  narcolepsy\n\n  mania\n\n  post-concussional state\n\n  iatrogenic: idiosyncratic adverse drug reaction\n\n  near-death experience\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThe GP's use of investigations will depend on the clinical situation. If hallucinations are part of an acute confusional state, particularly in adults, admission is likely to be required and will result in a battery of tests to check, for example, for sources of fever, hypoxia and metabolic disturbance. The following are investigations the GP might use in patients who do not require admission or who are not presenting acutely.\n\n  Urinalysis: very useful in the acute situation, particularly in the elderly. May reveal UTI or hyperglycaemic ketotic state or severe dehydration.\n\n  Pulse oximeter: to detect hypoxia.\n\n  Glucometer blood glucose: in a known diabetic or if any glycosuria.\n\n  FBC and LFT: raised MCV and abnormal LFT suggest chronic alcohol excess.\n\n  U&E: may reveal electrolyte disturbance as underlying cause.\n\n  EEG: may suggest diagnosis of temporal lobe epilepsy or narcolepsy.\n\n  CT scan: the definitive test for a cerebral space-occupying lesion.\n\n##### TOP TIPS\n\n  Delirium in children with a fever is quite common, especially at night and is not in itself a sinister sign; assess possible causes of the fever in the usual way, and if the cause is not serious, reassure the parents as they may be quite frightened by the child's hallucinations.\n\n  Patients with anxiety, personality disorder and borderline mental illness may sometimes complain of auditory hallucinations, occasionally because experience has told them that this generates action from health professionals. Genuine auditory hallucinations are usually distressing and often in the second person (psychotic depression) or third person (schizophrenia) \u2013 and are accompanied by other hard evidence of mental illness.\n\n  Minor and transient auditory and visual hallucinations are normal in the recently bereaved \u2013 but the patient will need reassurance that he or she isn't 'going mad'.\n\n  Hallucinations caused by drugs, or by drug and alcohol withdrawal, can be terrifying and dangerous for the patient and carers, so admission is likely to be required.\n\n  Auditory hallucinations strongly suggest psychotic illness, particularly schizophrenia and depression; visual hallucinations are almost always organic in nature.\n\n  Purely olfactory hallucinations are pathognomic of temporal lobe pathology and require urgent investigation.\n\n  Tactile hallucinations are very suggestive of acute alcohol withdrawal and occasionally cocaine abuse.\n\n### HEADACHE\n\n#### The GP overview\n\nThere are almost as many causes for headache in medicine as there are disorders. This universal symptom presents a challenge to all GPs because it is common, very often non-organic, but seriously pathological just often enough to merit a thorough and usually negative examination. The chance of a sinister hidden problem is always there, but the known vast majority of benign headaches can put the clinician off guard.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  tension headache (underlying anxiety or depression)\n\n  frontal sinusitis\n\n  migraine\n\n  cervical spondylosis\n\n  eye strain\n\n**OCCASIONAL**\n\n  any acute febrile illness (common cause of headache but usually presents with other symptoms)\n\n  iatrogenic (e.g. analgesic abuse, calcium antagonists, nitrates)\n\n  chronic daily headache\n\n  reactive hypoglycaemia\n\n  fatigue\/sleep deprivation (especially in parents)\n\n  trigeminal, sphenopalatine and occipital neuralgias\n\n  temporal arteritis\n\n  post-concussional syndrome\n\n  menstrual migraines (10\u201314% of women)\n\n**RARE**\n\n  cluster headache\n\n  intracranial lesion (e.g. carcinoma, abscess, haematoma, benign intracranial hypertension)\n\n  meningitis\n\n  intracerebral haemorrhage\n\n  carbon monoxide poisoning (blocked boiler flue)\n\n  Paget's disease of skull\n\n  severe hypertension\n\n  pre-eclampsia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR.\n\n_SMALL PRINT :_ U&E, alkaline phosphatase, X-ray of sinuses, cervical spine or skull, CT scan, lumbar puncture.\n\n  FBC: WCC raised in abscess and sinusitis. ESR essential if arteritis suspected.\n\n  U&E: Na+\/K+ derangement in pituitary tumours, alkaline phosphatase raised in Paget's disease.\n\n  X-ray: may see fluid levels in sinusitis (rarely useful in diagnosis). May confirm cervical spondylosis and Paget's disease.\n\n  CT scan: to exclude intracranial lesion.\n\n  Lumbar puncture: in suspected meningitis; may also help in diagnosis of benign intracranial hypertension.\n\n##### TOP TIPS\n\n  Explore the patient's fears \u2013 the majority are worried about serious pathology, such as a brain tumour, and may leave the consultation dissatisfied unless this specific worry is addressed.\n\n  Another common concern is hypertension. Patients will expect to have their blood pressure checked, even though this is almost never the cause of the symptom.\n\n  Analgesics may paradoxically exacerbate tension headache. It is more constructive to adopt alternative approaches, such as relaxation techniques or antidepressants, as appropriate.\n\n  Headache caused by an intracranial lesion usually produces other neurological symptoms or signs.\n\n  Suspect subarachnoid haemorrhage given a history of sudden explosive headache. It is frequently described as 'like a blow to the head'.\n\n  If temporal arteritis is suspected, treat immediately. The ESR provides retrospective confirmation only.\n\n  Beware of the pregnant woman complaining of headache in the third trimester: check the blood pressure, ankles and urinalysis. Headache, particularly with visual disturbance, may be a symptom of impending eclampsia.\n\n  A new and increasing headache present on waking and increased by stooping or straining may be due to raised intracranial pressure. Check for other symptoms and signs and refer urgently if in doubt.\n\n  If a headache feels and smells like migraine, then it's a migraine regardless of age. However, beware of making this the diagnosis in the elderly without systematically ruling out more sinister causes first.\n\n### INSOMNIA\n\n#### The GP overview\n\nThis problem is commoner in women, and commonest in the elderly. Normal sleep requirement varies widely. A few people need only 3\u20134 hours per night and the average amount of sleep needed declines with age. Self-reporting of time taken to get to sleep and hours slept are said to be inaccurate, but it is the change from the individual's normal pattern that is significant in practice.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anxiety from excess psychological stress (work, relationships, finance)\n\n  clinical depression\n\n  chronic alcohol excess\n\n  poor sleep hygiene: hyperstimulation (e.g. caffeine, nicotine, drugs, exciting television films) and daytime naps\n\n  pain of chronic physical illness (e.g. osteoarthritis)\n\n**OCCASIONAL**\n\n  menopausal flushes and sweats\n\n  nocturia\n\n  external problems (e.g. snoring partner, children who disturb parental sleep)\n\n  biorhythm disruption: jet lag and shift work\n\n  respiratory problems: asthma, chronic obstructive pulmonary disease (COPD), left ventricular failure (LVF) commonest\n\n  benzodiazepine withdrawal\n\n  other medical problem, e.g. restless legs syndrome or GORD\n\n**RARE**\n\n  malnutrition and low body weight\n\n  post-traumatic stress disorder\n\n  parasomnias: nightmares, night terrors and sleepwalking\n\n  hyperthyroidism\n\n  mania\n\n  sleep apnoea (usually presents as 'tired all the time' (TATT); only 30% aware of waking)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, LFT, TFT.\n\n_SMALL PRINT :_ investigation of primary symptom leading to insomnia (see below).\n\n  FBC (MCV), LFT and \u03b3GT may show evidence of chronic alcohol misuse.\n\n  TSH will differentiate non-organic anxiety state from thyrotoxicosis.\n\n_NOTE :_ pain, nocturia, respiratory problems and sleep apnoea may require investigating in their own right.\n\n##### TOP TIPS\n\n  Uncover any underlying physical problem such as pain or nocturia and manage as appropriate \u2013 it is pointless adopting a 'sleep hygiene' approach when the problem is primarily physical.\n\n  Don't forget the role of alcohol; this is often an underlying or contributory cause, paradoxically taken by the patient to relieve the insomnia.\n\n  If the diagnosis seems likely to be tension or poor sleep hygiene, establish the patient's agenda early. Patients who simply want sleeping pills are unlikely to listen to well-intentioned advice until this issue has been discussed and resolved.\n\n  Explain to elderly patients that sleep requirements fall with increasing age and that daytime naps are to be discouraged.\n\n  Shift workers are significantly at risk of developing clinical depression. Be sure to assess carefully for this pathology in the insomniac shift worker.\n\n  Beware of young male temporary residents presenting 'urgently' with insomnia. They may well be drug addicts trying to obtain a prescription for benzodiazepines.\n\n  Bone or joint pain waking an elderly patient at night is highly significant. In the patient with known arthritis, joint replacement may be indicated; in others, it may indicate serious bony pathology such as secondaries.\n\n  Take the problem seriously even if the cause seems trivial or obvious (for example, a patient's snoring) \u2013 insomnia can be extremely debilitating, and by the time patients attend, they may be desperate for help.\n\n  Anxiety and severe weight loss with sweating and tachycardia suggests hyperthyroidism. Be sure to check TSH before deciding this is non-organic.\n\n### LOSS OF LIBIDO\n\n#### The GP overview\n\nLoss of libido can be a daunting presentation for established GP and registrar alike. This universal problem spans adulthood in both sexes. Conventional medical school teaching seems to fail to prepare the generalist; however, the didactically taught approach of systematic enquiry and examination is the key to successful management.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  depression\n\n  relationship problems\n\n  perimenopause\n\n  excess alcohol intake (and cirrhosis in men)\n\n  ageing\n\n**OCCASIONAL**\n\n  low testosterone in men\n\n  hypothyroidism\n\n  antihypertensive treatment in men\n\n  hyperprolactinaemic drugs in men (e.g. phenothiazines, haloperidol)\n\n  anti-androgenic drugs in men (e.g. cimetidine, finasteride)\n\n  anti-androgenic drugs in women (e.g. cyproterone)\n\n**RARE**\n\n  hypothalamic\/pituitary disease\n\n  renal failure\n\n  primary testicular disease or damage\n\n  adrenal disease (Cushing's and Addison's diseases)\n\n  feminising tumours in men: testis or adrenal gland\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, U&E, LFT, TFT.\n\n_SMALL PRINT :_ hormone profile.\n\n  FBC: may show evidence of general disease; MCV raised with significant excess alcohol.\n\n  U&E: check for renal failure. Na+ and K+ deranged in adrenal disease.\n\n  LFT and \u03b3GT: should reveal hard evidence of excess alcohol.\n\n  TFT: will demonstrate hypothyroidism.\n\n  Hormone profile: FSH\/LH, prolactin, oestradiol and testosterone may be useful in both sexes. Altered by primary endocrine disease, drugs and alcohol.\n\n##### TOP TIPS\n\n  This is often a 'by the way' or 'while I'm here' symptom. It may be tempting to ask the patient to return for a further appointment, but bear in mind that this may mean a lost opportunity to help the patient.\n\n  General examination is important to detect rare causes. This also demonstrates that the problem is being taken seriously.\n\n  Avoid over-medicalising the situation if it is clearly a relationship problem.\n\n  Be prepared to revise or augment your diagnosis \u2013 the problem is often multifactorial.\n\n  Don't forget iatrogenic causes and be prepared to undertake a trial without treatment.\n\n  Loss of libido may be the tip of the iceberg of significant pathology, such as depression or alcoholism \u2013 don't be distracted into a superficial approach.\n\n  Depression and relationship difficulties can cause each other and coexist. A careful history will reveal whether antidepressants and\/or psychosexual counselling is appropriate\n\n  Investigations don't often help \u2013 but lower your threshold for blood tests if the patient seems generally unwell and isn't obviously depressed.\n\n  Early hypothyroidism closely mimics depressive illness.\n\n### MEMORY LOSS\n\n#### The GP overview\n\nMemory loss is a distressing and perilous symptom for both sufferers and caring relatives. It may be due to organic or non-organic causes. Memory is classified into immediate, short-term (or recent) and long-term (or remote) memory. The type of loss varies according to the cause. Memory loss is also a feature of any cause of acute confusion; this problem is covered elsewhere (see Acute confusion, p. 68).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anxiety\/stress\n\n  depressive illness\n\n  dementia (multi-infarct, Alzheimer's disease and dementia with underlying cause, such as tumour, neurosyphilis, hypothyroidism, vitamin B12 and folate deficiency)\n\n  trauma: head injury\n\n  CVA (infarct in posterior cerebral artery territory)\n\n**OCCASIONAL**\n\n  chronic excess alcohol intake (thiamine deficiency: Korsakoff's syndrome)\n\n  subarachnoid haemorrhage\n\n  other thiamine deficiency: malabsorption, carcinoma stomach, hyperemesis gravidarum\n\n  transient global amnesia\n\n  fugue states and psychogenic amnesia\n\n  tumour of third ventricle or hypothalamus\n\n**RARE**\n\n  personality disorder\n\n  malingering\n\n  intractable epilepsy\n\n  carbon monoxide poisoning\n\n  herpes simplex encephalitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ (unless obvious depression or anxiety) FBC, TFT, LFT, calcium.\n\n_POSSIBLE :_ syphilis serology, B12 and folate levels, CT\/MRI scan.\n\n_SMALL PRINT :_ none.\n\n  FBC may show raised MCV, suggesting either alcohol abuse or B12\/folate deficiency. Check B12 and folate levels if MCV raised.\n\n  TFT: hypothyroidism is an important remediable cause of dementia.\n\n  LFT and \u03b3GT will give useful clues to alcohol intake (history likely to be unreliable).\n\n  Calcium level: may show hypo- or hypercalcaemia.\n\n  Syphilis serology: for possible neurosyphilis as underlying cause of dementia.\n\n  CT scan\/MRI: will detect space-occupying lesions, cerebrovascular disease, atrophy and subarachnoid haemorrhage.\n\n##### TOP TIPS\n\n  Patients with dementia are often unaware of, or deny, their memory loss; the problem is more often brought to the GP's attention by a concerned friend or relative.\n\n  Patients who present themselves to the GP complaining of memory loss are most likely to be suffering from anxiety or depression.\n\n  Even if a diagnosis of anxiety or depression seems obvious, patients are likely to be concerned about the possibility of dementia, which will exacerbate the situation; explaining that the problem is more to do with poor concentration than failing memory will help reassure them.\n\n  Establishing the onset gives valuable clues to the problem: a dementia pattern progressing slowly over a year or two is likely to be Alzheimer's or multi-infarct dementia; with a shorter history, an underlying cause is possible; and sudden onset of memory loss is likely to be caused by a vascular event or trauma.\n\n  It can be very difficult to distinguish between depression and dementia \u2013 and the two may coexist. Consider a trial of antidepressants.\n\n  Rapid onset of apparent dementia over 3\u20136 months or less suggests a possible underlying cause.\n\n  True memory loss after a head injury suggests significant trauma.\n\n  Depression in the elderly may mimic dementia (pseudodementia) with behavioural changes like hoarding and bad temper. Do not miss this treatable condition.\n\n### VERTIGO\n\n#### The GP overview\n\nVertigo is an illusion of movement of either the patient or his or her environment. This is both visual and positional. Associated nausea or vomiting are common and, in its acute form, it is a severe and completely disabling symptom. It must be distinguished from 'light-headedness' (see p. 71).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  benign positional vertigo\n\n  vestibular migraine\n\n  M\u00e9ni\u00e8re's disease\n\n  vestibular neuritis\n\n  Eustachian tube (ET) dysfunction (causes mild vertigo)\n\n**OCCASIONAL**\n\n  chronic otitis media\n\n  drugs: salicylates, quinine, aminoglycosides\n\n  acute alcohol intoxication (common, but unlikely to present to the GP)\n\n  neurological conditions (e.g. CVA, multiple sclerosis (MS), vertebrobasilar insufficiency, syringobulbia, cerebellar tumours)\n\n  epilepsy\n\n**RARE**\n\n  earwax (common problem but rare cause of vertigo)\n\n  syphilis\n\n  acoustic neuroma\n\n  nasopharyngeal carcinoma\n\n  post-traumatic\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are no investigations likely to be performed in primary care. Referral might lead to a number of secondary care tests, such as audiometry for cochlear function; electronystagmography, calorimetry and brainstem-evoked responses to assess vestibular function; CT or MRI scan for possible neurological conditions; EEG for suspected epilepsy; lumbar puncture in possible MS; and syphilis serology if syphilis is suspected.\n\n##### TOP TIPS\n\n  Take a careful history: the patient may use the term 'vertigo' inaccurately, or describe true 'vertigo' as light-headedness. The diagnostic possibilities for vague dizziness and true vertigo are quite different.\n\n  The vast majority of cases seen in primary care are benign positional vertigo, vestibular migraine, M\u00e9ni\u00e8re's disease or viral neuritis.\n\n  M\u00e9ni\u00e8re's disease tends to be over-diagnosed. It comprises violent paroxysms of vertigo lasting for several hours, associated with deafness and tinnitus, often necessitating urgent attention because of prostration and vomiting.\n\n  Benign positional vertigo is usually easily diagnosed by the history: the patient experiences vertigo lasting only for a few seconds, classically on turning over in bed.\n\n  Vestibular migraine is under-diagnosed. Consider it in any case of recurrent vertigo \u2013 enquire about headache before, during or after the vertigo.\n\n  The patient who has chronic otitis media and then develops vertigo probably has significant disease \u2013 especially if the fistula sign is positive (putting pressure on the external ear canal by forcibly occluding the external auditory meatus with a finger causes vertigo). Refer urgently.\n\n  Young or middle-aged patients with atypical, episodic vertigo who have other, diffuse and transient neurological symptoms may have MS.\n\n  Loss of consciousness with vertigo suggests epilepsy.\n\n  An acoustic neuroma can cause quite mild vertigo. Consider this possibility if the patient also has a unilateral sensorineural deafness and tinnitus.\n\n  A neurological cause such as a stroke is suggested by any CNS symptoms or signs; a new type of headache, especially occipital; acute deafness (otherwise unexplained); or vertical nystagmus.\n\n## CHEST\n\nAcute shortness of breath\n\nChest pain\n\nChronic shortness of breath\n\nCough in adults\n\nCough in children\n\nCoughing up blood\n\nPalpitations\n\n### ACUTE SHORTNESS OF BREATH\n\n#### The GP overview\n\nThis is a terrifying symptom for the patient, and the subjective feeling of shortness of breath is not predictably related to the type or degree of pathology. This, combined with the fact that the cause is often organic, means that a careful and urgent assessment is mandatory.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  asthma\n\n  pneumonia\n\n  acute LVF\n\n  acute exacerbation of COPD\n\n  hyperventilation\n\n**OCCASIONAL**\n\n  pneumothorax\n\n  pulmonary embolism\n\n  pleural effusion\n\n  diabetic ketoacidosis (DKA)\n\n  lobar collapse (tumour)\n\n**RARE**\n\n  aspiration pneumonitis\n\n  Guillain\u2013Barr\u00e9 syndrome\n\n  hypovolaemic shock\n\n  shock lung (adult respiratory distress syndrome)\n\n  laryngeal obstruction\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThe GP is highly unlikely to initiate any investigations at all. If the patient with acute shortness of breath is ill enough \u2013 or the diagnosis obscure enough \u2013 to warrant investigation, then the patient probably requires admission. The following therefore refers to those few cases in which the patient is reasonably well, the diagnosis unclear and the scenario not so urgent that immediate referral is required.\n\n  Urinalysis: glucose and ketones in DKA. Confirm with a glucometer reading.\n\n  Sputum culture: very occasionally helpful in infective processes not settling with first-line empirical treatment.\n\n  FBC: WCC raised in infection. Anaemia may be significant incidental finding.\n\n  Pulse oximetry \u2013 hypoxia suggests a significant problem.\n\n  CXR an essential part of assessment but usually done after admission\/referral.\n\n  Other investigations such as blood gases and ventilation\/perfusion scans might be required to clinch a diagnosis but would be arranged by the admitting team.\n\n##### TOP TIPS\n\n  If the diagnosis is likely to be hyperventilation, instruct the patient to rebreathe from a paper bag while waiting for you. This action may curtail the attack by the time you arrive.\n\n  Spacer devices can be as effective as nebulisers when managing acute exacerbations of asthma, and are more practical to use when on call.\n\n  Sudden onset of breathlessness in an elderly patient in the middle of the night is likely to be LVF. Remember that it may be have been precipitated by an infarct.\n\n  Cyanosis is an ominous sign meriting a 'blue light' ambulance and oxygen as soon as possible.\n\n  The presence of intercostal recession and use of accessory muscles of respiration indicate severe respiratory distress whatever the aetiology. Admit.\n\n  If a foreign body has been inhaled, astute telephone assessment and clear, calm advice may be lifesaving.\n\n  Acute confusion with breathlessness indicates severe hypoxaemia, metabolic disturbance or sepsis. Admit urgently.\n\n  Don't forget that pneumothorax is commoner in asthmatics \u2013 consider this diagnosis if an asthmatic suddenly becomes more short of breath.\n\n### CHEST PAIN\n\n#### The GP overview\n\nAcute chest pain is a regular visitor to general practice: it may generate more adrenaline in the physician than the patient. In spite of a constellation of causes, a good basic clinical approach will determine the diagnosis in nearly all cases, long before any necessary investigations are complete.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  angina\/MI\n\n  GORD\n\n  anxiety (Da Costa's syndrome)\n\n  pulled muscle\n\n  Tietze's syndrome (costochondritis)\n\n**OCCASIONAL**\n\n  pleurisy\n\n  peptic ulcer\n\n  biliary colic\n\n  shingles\n\n  mastitis\n\n  Bornholm disease\n\n**RARE**\n\n  pulmonary infarct\n\n  hypertrophic obstructive cardiomyopathy\n\n  pericarditis\n\n  fractured ribs\n\n  myocarditis\n\n  pneumothorax\n\n  dissecting aortic aneurysm\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ ECG.\n\n_POSSIBLE :_ FBC, CXR, pulse oximetry, secondary care cardiac investigations, OGD, ultrasound of abdomen.\n\n_SMALL PRINT : Helicobacter_ tests, ventilation\/perfusion scan (hospital-based).\n\n  ECG: may show evidence of cardiac ischaemia, pericarditis or pulmonary embolism.\n\n  FBC: WCC raised in pleurisy and may be raised in Tietze's syndrome.\n\n  CXR: may reveal chest infection, rib fracture, heart disease, cardiomyopathy or pneumothorax.\n\n  Pulse oximetry: hypoxia a sign of significant cardiac or respiratory problem in the acute setting.\n\n  Secondary care cardiac investigations: to clarify whether a cardiac cause.\n\n  Ultrasound of abdomen: to check for gallstones.\n\n  OGD: to confirm peptic ulcer or oesophagitis.\n\n  _Helicobacter_ tests useful in the presence of duodenal ulcer.\n\n  Ventilation\/perfusion scan (in hospital): to confirm pulmonary infarction.\n\n##### TOP TIPS\n\n  The history is all-important and will usually provide the diagnosis. Except in an obvious emergency, take your time getting the facts straight.\n\n  If you feel worried enough to obtain an urgent ECG then you ought to consider whether the patient really requires an urgent medical opinion or admission.\n\n  Watching the patient's hand as the symptoms are being described can provide very helpful clues. A clenched fist on the chest is worrying; a single pointing finger much less so.\n\n  Musculoskeletal pain and pleurisy both cause pain on deep inspiration \u2013 but the former usually also displays muscle or rib tenderness.\n\n  Tietze's syndrome is distinguished from costochondritis by the presence of a palpable swelling, caused by oedema, at the site of maximal tenderness. However, management is largely the same.\n\n  Always encourage the patient to contact you if the problem persists or deteriorates.\n\n  If in doubt, play safe: give aspirin (if not allergic) and admit.\n\n  Don't delay if the symptoms clearly suggest an infarct; admit the patient (via the telephone if necessary).\n\n  A normal ECG does not exclude an infarct. Treat the patient, not the test.\n\n  Symptoms of genuine and significant pathology may be clouded by various ensuing anxiety symptoms. Take time to tease them out.\n\n  Performing unnecessary tests when the diagnosis is clearly anxiety is likely to exacerbate the situation.\n\n  If the diagnosis remains unclear, examine the abdomen, especially for significant epigastric tenderness.\n\n### CHRONIC SHORTNESS OF BREATH\n\n#### The GP overview\n\nShortness of breath is defined as difficult, laboured breathing. Medical teaching tends to focus on individual pathologies; however, in practice there is often some overlap between several contributory causes and sometimes the diagnosis can only be made after therapeutic trials of treatment.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  obesity\/unfitness\n\n  COPD\n\n  anaemia\n\n  congestive cardiac failure (CCF)\n\n  asthma\n\n**OCCASIONAL**\n\n  bronchiectasis\n\n  recurrent pulmonary emboli\n\n  bronchial carcinoma with lobar collapse\n\n  pleural effusion\n\n  aortic stenosis\n\n  chronic hyperventilation\n\n**RARE**\n\n  pulmonary fibrosis\n\n  large hiatus hernia\n\n  fibrosing alveolitis\n\n  undiagnosed congenital heart disease\n\n  neurological: motor neurone disease and the muscular dystrophies\n\n  sarcoidosis\n\n  extrinsic allergic alveolitis (bird fancier's lung etc.)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ CXR, FBC.\n\n_POSSIBLE :_ peak flow, U&E, LFT, ESR\/CRP, BNP, ECG, spirometry.\n\n_SMALL PRINT :_ pulse oximetry, CT scan, V\/Q scan, pleural tap, echocardiogram, Kveim test.\n\n  CXR: the single most useful investigation. Will reveal or give clues to many of the causes listed.\n\n  FBC essential to look for anaemia; ESR\/CRP raised in carcinoma, inflammation and infection.\n\n  U&E and LFT: impaired renal function will contribute to CCF; LFT may show signs of disseminated carcinoma.\n\n  Peak expiratory flow rate variability in asthma; more comprehensive lung function tests (spirometry) are more helpful to diagnose COPD and other lung diseases.\n\n  ECG: heart failure is unlikely if the ECG is normal.\n\n  BNP: likely to be elevated in heart failure.\n\n  Pulse oximetry: helps guide assessment of severity and decisions about oxygen therapy but of little help in making the diagnosis.\n\n  Referral for more difficult cases may result in CT or V\/Q scans (e.g. for bronchiectasis or pulmonary emboli), pleural tap (diagnostic and therapeutic for pleural effusion), echocardiography (for heart valve lesions and assessment of left ventricular function) and Kveim test (for sarcoidosis).\n\n##### TOP TIPS\n\n  Cardiac failure may arise as a complication of COPD. Remember this possibility if a patient with COPD complains of gradually increasing breathlessness unrelieved by standard treatment.\n\n  Cases of breathlessness in the elderly may be multifactorial and difficult to diagnose precisely even after investigation. Do not underestimate the value of a trial of treatment, e.g. a course of steroids for possible asthma or potential reversibility in COPD.\n\n  In the young and middle-aged, sighing speech and shortness of breath worse with stress or without any clear pattern \u2013 especially if the patient does not consistently have a problem with exercise \u2013 are likely to be caused by hyperventilation.\n\n  Weight loss and clubbing with shortness of breath suggest bronchial carcinoma, though bronchiectasis is possible \u2013 arrange an urgent CXR.\n\n  Wheeze may be present in cardiac failure \u2013 crepitations may not. Look for other signs of CCF in the elderly and consider appropriate investigation and treatment.\n\n  Remember that acute causes can supervene at any time: for example, beware of pneumothorax in the asthmatic\n\n  Cardiac failure has a poor prognosis; look for an underlying cause (e.g. hypertension) and consider echocardiography with a view to starting ACE inhibitors.\n\n  Don't forget anaemia as a possible cause \u2013 contrary to popular belief this tends to cause shortness of breath rather than tiredness.\n\n### COUGH IN ADULTS\n\n#### The GP overview\n\nThis is a symptom that patients seem to fear or value \u2013 as a signifier of possible cancer or a justifier of antibiotics \u2013 far more than GPs. Most coughs are simply viral URTIs, but the GP should be aware of the various other possibilities, especially when the symptom is persistent.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  URTI\n\n  LRTI\n\n  asthma\n\n  COPD\n\n  ACE inhibitor side effect\n\n**OCCASIONAL**\n\n  smoking (including passive smoking)\n\n  lung tumour (primary or secondary)\n\n  rhinitis\n\n  GORD\n\n  LVF\n\n  bronchiectasis\n\n  aspiration (e.g. post stroke)\n\n**RARE**\n\n  TB\n\n  other medication side effect (e.g. methotrexate)\n\n  pulmonary fibrosis\n\n  fibrosing alveolitis\n\n  extrinsic allergic alveolitis\n\n  psychogenic\n\n  laryngeal carcinoma\n\n  inhaled foreign body\n\n  diaphragmatic irritation (e.g. abscess)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, spirometry, PEFR.\n\n_SMALL PRINT :_ sputum, cardiac investigations, serum precipitins, hospital-based investigations such as CT scan and bronchoscopy.\n\n  FBC: Hb may be reduced in malignancy and chronic illness; WCC raised in infections, eosinophils raised in allergic conditions.\n\n  ESR\/CRP: raised in neoplasia, infective and inflammatory conditions.\n\n  CXR: may show signs in a variety of the relevant differentials, such as LRTI, tumour and TB.\n\n  Spirometry: may show characteristic patterns particularly in asthma, COPD and pulmonary fibrosis.\n\n  Serial peak flow: may be helpful in diagnosis of asthma.\n\n  Sputum: may be useful in diagnosing TB and occasionally helps guide antibiotic treatment in LRTI or exacerbation of COPD.\n\n  Cardiac investigations: such as BNP or echocardiogram if LVF suspected.\n\n  Serum precipitins: in suspected extrinsic allergic alveolitis.\n\n  Hospital-based investigations: further investigations such as CT scan or bronchoscopy may be required to clarify CXR abnormalities or pursue clinical suspicion.\n\n##### TOP TIPS\n\n  Explain to patients that it is not unusual for the cough of a simple URTI to go on for 3 weeks \u2013 this will reduce unnecessary re-attendances.\n\n  Take a careful history of provoking factors in the case of persistent cough \u2013 this is more likely to reveal the diagnosis than is chest auscultation.\n\n  Have a low threshold for arranging a CXR in the middle-aged and elderly smoker with a cough.\n\n  ACE inhibitor-associated cough may come on many months \u2013 or even longer \u2013 after initiating treatment. It starts to improve within 1\u20134 weeks of stopping treatment but may take 3 months to settle completely.\n\n  In a persistent cough with a normal CXR and no chest signs, think asthma, GORD and rhinitis \u2013 a therapeutic trial for each may be needed to clinch the diagnosis.\n\n  Remember to ask about foreign travel. Atypical pneumonias are infrequent, and TB rare, but both can still present.\n\n  Beware of persistent cough, weight loss and voice change in a smoker \u2013 arrange an X-ray to exclude malignancy.\n\n  Night sweats with persistent cough suggest significant pathology such as TB or malignancy.\n\n  Beware the patient on immunosuppressants: these drugs may alter the clinical picture, predispose to serious complications and in some cases (e.g. methotrexate) may be the cause of the cough itself.\n\n### COUGH IN CHILDREN\n\n#### The GP overview\n\nThe symptom GPs love to hate because it can appear so trivial. Reassurance and explanation are often all that is required, and this can build a bond with parents and children. Take parents seriously and sympathetically: nocturnal cough is a destroyer of sleep and family peace.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  URTI\n\n  LRTI\n\n  post-nasal drip (e.g. post URTI, allergic rhinitis)\n\n  asthma\n\n  pertussis\n\n**OCCASIONAL**\n\n  inhaled foreign body\n\n  GORD\n\n  psychogenic\n\n**RARE**\n\n  TB\n\n  cystic fibrosis\n\n  earwax or foreign body in the ear canal\n\n  immune deficiency\n\n  interstitial lung disease\n\n  congenital, e.g. trachea-oesophageal fistula\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, CXR, serial peak flow or spirometry.\n\n_SMALL PRINT :_ pertussis serology, sweat test, secondary care investigations (e.g. for interstitial lung disease or immune deficiency).\n\n  FBC, ESR\/CRP: WCC raised in infection \u2013 marked lymphocytosis in pertussis; ESR\/CRP elevated in any inflammatory process.\n\n  CXR: may be helpful in LRTI, TB, inhaled foreign body, cystic fibrosis.\n\n  Serial peak flow or spirometry: to help confirm a diagnosis of asthma.\n\n  Pertussis serology: if a clinical suspicion of pertussis needs confirming.\n\n  Sweat test: for cystic fibrosis.\n\n  Other secondary care investigations: may be required after referral (e.g. for interstitial lung disease or immune deficiency).\n\n##### TOP TIPS\n\n  Think pertussis in any paroxysmal cough lasting more than 3 weeks \u2013 it is much more common than most people, and many doctors, realise.\n\n  Educate parents about the likely duration of URTI-related coughs and simple measures to take. Avoid prescribing, as this simply reinforces the tendency to attend the doctor for minor, self-limiting illness.\n\n  In the asthmatic child, a cough may be a sign of poor control: check treatment, compliance and inhaler technique.\n\n  Many parents panic that a cough might harm their child. An explanation that a cough is often simply a way of 'keeping the lungs clear' can defuse the situation.\n\n  An aural foreign body is an unusual but remediable cause of childhood cough.\n\n  Parents tend to focus on the cough. In the acute situation, rather more important are symptoms and signs of respiratory distress \u2013 the NICE traffic light system for febrile children is useful in the acutely coughing febrile child and will help guide the need for admission.\n\n  A dramatic and abrupt onset of coughing in a child should make you consider an inhaled foreign body.\n\n  Beware the 'poorly controlled asthmatic' who isn't thriving \u2013 this could be cystic fibrosis.\n\n### COUGHING UP BLOOD\n\n#### The GP overview\n\nPatients invariably view this relatively uncommon symptom as representing something serious \u2013 this is rarely the case in primary care. In practice, the origin of the blood may not be immediately obvious: quite often, blood from the nose or throat may be coughed out with saliva (spurious haemoptysis) and described as 'coughing up blood'.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  chest infection\n\n  pulmonary embolism (PE)\n\n  bronchogenic carcinoma\n\n  pulmonary oedema\n\n  prolonged coughing\n\n**OCCASIONAL**\n\n  bronchiectasis\n\n  mitral stenosis\n\n  polyarteritis nodosa\n\n  tuberculosis\n\n  tumour of larynx or trachea\n\n**RARE**\n\n  associated with SLE\n\n  aspergillosis\n\n  Goodpasture's syndrome\n\n  contusion due to trauma\n\n  pulmonary arteriovenous malformations (50% associated with hereditary haemorrhagic telangiectasia)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ CXR.\n\n_POSSIBLE :_ FBC, ESR\/CRP, autoantibody screen, sputum, pulse oximetry.\n\n_SMALL PRINT :_ bronchoscopy, ventilation\/perfusion scan, echocardiogram, other chest imaging (e.g. CT scan).\n\n  CXR: this is the single most valuable investigation for detecting many of the causes listed.\n\n  Sputum microbiology: may be needed to look for acid-fast bacilli of TB.\n\n  FBC and ESR\/CRP: for anaemia (LVF and malignant disease); WCC raised in infection, ESR\/CRP raised in malignancy, infection and inflammatory conditions.\n\n  Autoantibody screen: for assessing possible connective tissue disease.\n\n  Pulse oximetry: hypoxia points to significant acute lung pathology.\n\n  Hospital-based tests: various other investigations may be considered according to the likely aetiology and would usually be arranged by the hospital specialist after referral, e.g. bronchoscopy, ventilation\/perfusion scan, CT scan and echocardiography.\n\n##### TOP TIPS\n\n  In younger patients, the symptom is most commonly caused by vigorous coughing. If this is clearly the case, and the haemoptysis was minor, do not engender unnecessary anxiety by arranging a CXR.\n\n  Take a careful history. The terminology used by some patients can confuse the doctor as to whether blood was coughed or vomited up.\n\n  Don't forget that most patients \u2013 and smokers in particular \u2013 will be worried that the symptom represents cancer. Reassure firmly when appropriate, but investigate early those cases that concern you, providing an adequate explanation as to why you are arranging a CXR and making firm arrangements for follow-up.\n\n  Any smoker with significant haemoptysis should have a CXR \u2013 particularly if there are other sinister features such as shortness of breath, weight loss, persistent cough or clubbing.\n\n  PE causes sudden onset of shortness of breath with pleuritic pain. Consider this diagnosis if there is no other obvious explanation for the symptoms, especially if the patient has a tachycardia. Signs of DVT may only appear later, or sometimes never at all.\n\n  TB is on the increase in the UK. Consider this possibility in the elderly, immigrants and vagrants. It often mimics malignancy.\n\n  If haemoptysis persists, arrange referral even if the CXR is clear \u2013 some lesions may not appear on the X-ray, or may only develop after some time has elapsed. Other investigations may be required.\n\n### PALPITATIONS\n\n#### The GP overview\n\nPalpitations are presented fairly frequently to the GP, sometimes in isolation but more often immersed in other symptoms. Patients use the word 'palpitations' to describe a remarkable variety of sensations, and it is important to establish exactly what is meant. Cardiac causes are rare; anxiety about a cardiac problem, and anxiety as a cause of the symptom, are common.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anxiety (increased awareness of normal heartbeat)\n\n  sinus tachycardia (e.g. stress, fever, exercise)\n\n  atrial ectopics\n\n  ventricular ectopics\n\n  supraventricular tachycardia (SVT)\n\n**OCCASIONAL**\n\n  thyrotoxicosis (combination of sinus tachycardia and increased awareness even if ventricular ectopics are absent)\n\n  menopause (due to sudden vasodilation)\n\n  atrial fibrillation (AF \u2013 various causes, e.g. IHD, mitral valve disease, alcohol)\n\n  iatrogenic (e.g. digoxin, nifedipine)\n\n  atrial flutter\n\n**RARE**\n\n  heart block (especially with changes in block)\n\n  sick sinus syndrome\n\n  drug abuse\n\n  ventricular tachycardia (VT)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ ECG, TFT.\n\n_POSSIBLE :_ U&E, 24 h ECG or event monitor.\n\n_SMALL PRINT :_ further investigation to establish underlying cause.\n\n  ECG: may show arrhythmia itself or evidence of ischaemic heart disease or Wolff\u2013Parkinson\u2013White syndrome.\n\n  TFT: thyrotoxicosis can cause palpitations or exacerbate other causes.\n\n  U&E: electrolyte disturbance can precipitate or aggravate some arrhythmias.\n\n  24 h ECG or event monitor: to provide ECG evidence of the arrhythmia.\n\n  Further investigation of underlying cause: e.g. stress test for ischaemic heart disease, echocardiography for valve disease.\n\n##### TOP TIPS\n\n  Take time to obtain a clear history, as the patient's perception of a 'palpitation' may differ markedly from yours.\n\n  In paroxysmal cases, suggest that the patient attends the surgery or casualty urgently during an attack to obtain an ECG.\n\n  Patients can easily be taught to take their own pulse. Self-reported pulse rates can help considerably in establishing a diagnosis.\n\n  Most patients with palpitations fear heart disease, and this anxiety exacerbates the symptoms. Ensure this fear is resolved whenever possible.\n\n  Multiple, or multifocal, ventricular ectopics suggest significant ischaemic heart disease \u2013 and may herald VT or fibrillation if they follow an infarct.\n\n  Sudden onset of tachycardia in a young adult with breathlessness, dizziness, chest pain and polyuria suggests significant SVT.\n\n  Patients complaining of palpitations who are in AF are likely to have paroxysmal or recent onset AF, with significant risk of systemic embolism. Consider urgent referral for anticoagulation.\n\n  Remember that digoxin can aggravate as well as resolve some arrhythmias.\n\n## EAR\n\nDeafness\n\nEarache\n\nEar discharge\n\nTinnitus\n\n### DEAFNESS\n\n#### The GP overview\n\nDeafness is a frustrating symptom. In children it creates educational difficulties and parental worry. In adults, everyday life is fraught with difficulties, and there may be stigmatisation. Three million adults in the UK suffer some degree of persistent deafness. Congenital causes acquired in utero are not included here.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  earwax\n\n  otitis media (OM)\n\n  otitis externa (OE)\n\n  glue ear (serous otitis media)\/Eustachian dysfunction\n\n  presbyacusis (senile deafness)\n\n**OCCASIONAL**\n\n  M\u00e9ni\u00e8re's disease\n\n  otosclerosis\n\n  noise damage to cochlea\n\n  barotrauma\n\n  viral acoustic neuritis\n\n  large nasal polyps or nasopharyngeal tumour\n\n  drugs: streptomycin, gentamicin, aspirin overdose\n\n**RARE**\n\n  vascular (haemorrhage, thrombosis of cochlear vessels)\n\n  acoustic neuroma\n\n  vitamin B12 deficiency\n\n  CNS causes (e.g. multiple sclerosis, cerebral secondary carcinoma)\n\n  cholesteatoma\n\n  Paget's disease\n\n  traumatic (e.g. to tympanic membrane or ossicles)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ (in children) audiogram and tympanometry.\n\n_POSSIBLE :_ ear swab.\n\n_SMALL PRINT :_ FBC\/B12 levels, skull X-ray, further imaging.\n\n  Audiometry quantifies loss and distinguishes sensorineural from conductive hearing loss.\n\n  Tympanometry measures the compliance of the eardrum. Fluid in the middle ear flattens the compliance curve.\n\n  Swab of ear discharge: discharge can be swabbed to guide treatment in refractory otitis externa.\n\n  FBC\/B12 levels: to confirm B12 deficiency.\n\n  Skull X-ray: for Paget's disease.\n\n  Further imaging: e.g. CT and MRI scans may be arranged by specialist for suspected acoustic neuroma, multiple sclerosis or cerebral pathology.\n\n##### TOP TIPS\n\n  Take parents seriously if they suspect their child is deaf. There may be no physical signs in glue ear, and tympanometry will yield the diagnosis.\n\n  Warn patients with otitis media that hearing may take a few weeks to return completely to normal \u2013 this saves unnecessary attendances with patients complaining that 'The antibiotics haven't worked'.\n\n  In a case with no immediately alarming features and no past history of significant ear disease, it is reasonable to defer a comprehensive history and examination \u2013 instead, take a quick look at the ear canals. If the diagnosis appears to be earwax, arrange syringing. Assess in more detail only if there is no earwax or syringing doesn't solve the problem.\n\n  Remember how to perform and interpret Rinne's and Weber's tests \u2013 these are invaluable in assessing the less straightforward cases.\n\n  Remember the possibility of acoustic neuroma if there is progressive unilateral sensorineural deafness \u2013 especially if there is accompanying tinnitus, vertigo or neurological symptoms or signs.\n\n  Otherwise unexplained and persistent serous otitis media in adults may be due to nasopharyngeal carcinoma \u2013 refer for urgent examination of the nasopharyngeal space.\n\n  Sudden onset of profound sensorineural deafness is usually viral or vascular and requires same-day ENT assessment.\n\n  Otosclerosis requires early diagnosis for effective treatment. Consider the diagnosis in otherwise unexplained conductive deafness in young adults, especially if there is a family history.\n\n### EARACHE\n\n#### The GP overview\n\nThis the commonest reason for an out-of-hours call for a child. Parental distress is often as great as the child's, and appropriate advice can do much to relieve this \u2013 even over the telephone. Causes in adults are far more varied than for children and can originate in the pinna, ear canal, middle ear and from neighbouring structures (referred pain).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  infective otitis media (OM): bacterial\/viral\n\n  infective otitis externa (OE): bacterial\/fungal\/viral\n\n  boils and furuncles of the canal and pinna\n\n  trauma (especially cotton buds) and foreign bodies (including earwax)\n\n  throat problems: tonsillitis\/pharyngitis\/quinsy\n\n**OCCASIONAL**\n\n  temporomandibular joint (TMJ) dysfunction\n\n  dental abscess\n\n  impacted molar\n\n  trigeminal neuralgia\n\n  ear canal eczema\/seborrhoeic dermatitis\n\n  chondrodermatitis nodularis helicis externa\n\n**RARE**\n\n  mastoiditis\n\n  cervical spondylosis\n\n  cholesteatoma\n\n  malignant disease\n\n  barotrauma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ ear swab.\n\n_SMALL PRINT :_ X-rays of TMJ, teeth and mastoid bone, FBC, Paul\u2013Bunnell test.\n\n  Swab of ear canal useful if discharge present, after failure of empirical first-line treatment.\n\n  X-ray of mastoid bone excludes mastoiditis if mastoid clear \u2013 usually arranged by specialist. X-rays of TMJ and teeth are the remit of the dentist or oral surgeon.\n\n  FBC and Paul\u2013Bunnell test useful if glandular fever suspected. The diagnosis provides a label and guides further advice, though no specific treatment exists.\n\n  Further specialist investigations may include CT\/MRI as the only way adequately (non-invasively) to investigate the inner ear and temporal bone anatomy.\n\n##### TOP TIPS\n\n  Persistent debris in the ear canal will prevent resolution of OE and mask possible underlying causes. Aural toilet is essential.\n\n  If inserting the aural speculum causes pain, the diagnosis is likely to be otitis externa or a furuncle.\n\n  Don't forget to ask about trauma \u2013 especially the use of a cotton bud. Excavating earwax with a bud tends to produce an inflamed canal and drum, mimicking infection.\n\n  Earache can be excruciating \u2013 don't underestimate the need for adequate analgesia while you establish and treat the cause.\n\n  Consider mastoiditis if foul-smelling discharge is present for more than 10 days. Look for swelling behind the ear and downward displacement of the pinna.\n\n  Don't be too ready to diagnose otitis media in children \u2013 URTIs and crying inevitably result in some redness of the drum. Indiscriminate prescribing may lead to iatrogenic problems or the masking of the true diagnosis.\n\n  Beware the elderly patient with intractable, unexplained earache \u2013 refer to exclude a nasopharyngeal carcinoma.\n\n### EAR DISCHARGE\n\n#### The GP overview\n\nThis is often seen in swimmers and returned tropical travellers. It is frequently a sequel to water trapped behind earwax in the ear canal, which swells and encourages stasis and subsequent infection. The vast majority of cases seen settle with simple treatment, but be wary of rarer serious causes.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  boil\n\n  acute suppurative otitis media (OM)\n\n  infective otitis externa (OE): viral, bacterial and fungal\n\n  chronic suppurative otitis media\n\n  reactive otitis externa: seborrhoeic dermatitis, eczema, psoriasis\n\n**OCCASIONAL**\n\n  cholesteatoma\n\n  trauma: often a result of over-vigorous attempts to clean the ear\n\n  bullous myringitis (otitis externa haemorrhagica)\n\n  infection with foreign body (insects, beads in toddlers)\n\n  liquefying excess earwax\n\n**RARE**\n\n  mastoiditis\n\n  necrotising (or malignant) otitis externa\n\n  squamous and basal cell (rarer) carcinoma of the EAM\n\n  keratosis obturans (bolus of abnormally desquamated epithelium and earwax: associated with chronic bronchitis and bronchiectasis)\n\n  herpes zoster oticus\n\n  cerebrospinal fluid (CSF) otorrhoea\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ swab, urinalysis.\n\n_SMALL PRINT :_ skull\/mastoid X-rays, CT or MRI scan, audiometry.\n\n  Swab of ear discharge: helps guide treatment in refractory cases.\n\n  Urine for glucose: to exclude underlying diabetes if infections are recurrent (especially boils).\n\n  X-ray of the mastoid process will show a cloudy appearance in the mastoid air cells in mastoiditis.\n\n  CT or MRI scan is the best way to investigate possible invasion of temporal bone by tumour, cholesteatoma.\n\n  Audiometry may be required to assess baseline hearing loss in chronic OM, so improvement after definitive surgical treatment can be measured.\n\n  Skull X-ray: may show middle cranial fossa fracture in CSF otorrhoea (performed in hospital after significant trauma).\n\n##### TOP TIPS\n\n  Otitis externa is often recurrent: to minimise future problems, advise the patient to avoid getting water in the ear and stop using cotton buds. Also treat any underlying skin disease.\n\n  In the presence of ear discharge, pain on moving the tragus suggests otitis externa or a boil; in the case of the former, the patient tends to present with itching rather than pain.\n\n  Most cases of otitis externa and media settle with empirical treatment and so don't require a swab. Only investigate if they do not respond to first-line treatment.\n\n  If the diagnosis is not certain, be sure to follow up after initial treatment to visualise the drum; if persistent discharge makes this impossible, refer to the ENT outpatients department for aural toilet and further assessment.\n\n  Heat, tenderness and swelling over the mastoid process suggests mastoiditis: refer urgently.\n\n  If ear discharge does not clear with usual therapy, refer for microsuction of debris (aural toilet) to speed resolution and exclude significant middle-ear disease.\n\n  Very rarely, middle-ear infection causing discharge can progress centrally, causing, for example, meningitis or cerebral abscess \u2013 so refer immediately any patient with ear discharge who becomes confused or develops neurological signs.\n\n  The use of aminoglycoside or polymyxin drops in the presence of a perforated tympanic membrane carries a risk of ototoxicity (though some specialists do use such drops even if perforation is present). When using potentially ototoxic drops, be as certain as you can about what you are treating.\n\n  Beware severe otitis externa in elderly diabetics \u2013 it may be the necrotising ('malignant') form.\n\n### TINNITUS\n\n#### The GP overview\n\nThis means noises heard (nearly always subjectively) in the ears or head. They are often described as being like a whistling kettle, an engine, or in time with the heartbeat. As a short-lived phenomenon, it is very common (often with URTIs) \u2013 such cases do not usually present to the GP. More serious, persistent tinnitus occurs in up to 2% of the population. It is very distressing and can cause secondary depression and insomnia. Objective tinnitus is very rare.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  earwax\n\n  hearing loss (20% of cases: chronic noise damage and presbyacusis)\n\n  suppurative otitis media (also chronic infection and serous OM)\n\n  otosclerosis\n\n  M\u00e9ni\u00e8re's disease\n\n**OCCASIONAL**\n\n  after a sudden loud noise (e.g. gunfire)\n\n  head injury (especially basal skull fracture)\n\n  impacted wisdom teeth and TMJ dysfunction\n\n  drugs: aspirin overdose, loop diuretics, aminoglycosides, quinine\n\n  hypertension and atherosclerosis\n\n**RARE**\n\n  acoustic neuroma\n\n  palatal myoclonus (objectively detectable)\n\n  arteriovenous fistulae and arterial bruits (objectively detectable)\n\n  severe anaemia and renal failure\n\n  glomus jugulare tumours (objectively detectable)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ tympanogram, audiogram, MRI scan (all usually in secondary care).\n\n_SMALL PRINT :_ FBC, U&E, skull X-ray, angiography (the latter two in secondary care).\n\n  FBC and U&E: if anaemia or renal failure suspected.\n\n  Tympanogram for middle-ear function and stapedial reflex threshold. Audiogram to assess hearing loss objectively.\n\n  Cerebral angiography: if vascular pathology suspected.\n\n  MRI scan: the most sensitive way to examine the inner ear and skull for structural lesions.\n\n  Skull X-ray: if associated with significant head injury.\n\n##### TOP TIPS\n\n  Most patients are afraid of the diagnosis of tinnitus because of its potentially debilitating nature. If the cause is clearly self-limiting or remediable, take time to reassure the patient.\n\n  Have a low threshold for referral in persistent tinnitus. While no specific treatment may be available, this shows that you are taking the problem seriously, ensures that remediable problems won't be missed and may give the patient access to masking devices.\n\n  Be prepared to reassess ongoing tinnitus, as new symptoms may develop. For example, tinnitus may precede other symptoms in M\u00e9ni\u00e8re's disease by months or even years.\n\n  Depression in tinnitus has been severe enough to cause suicide. Make a thorough psychological assessment and consider a trial of antidepressants.\n\n  Think of otosclerosis in younger patients (15\u201330) with persistent conductive deafness \u2013 especially if there is a family history. Early diagnosis is important.\n\n  Progressive unilateral deafness with tinnitus could be caused by an acoustic neuroma. Exclude by referring for an MRI scan.\n\n## EYE\n\nAcutely red and painful eye\n\nDouble vision\n\nEyelid problems\n\nFlashes, floaters and transient visual disturbance\n\nGradual loss of vision\n\nItchy or irritating eyes or eyelids\n\nSudden loss of vision\n\n### ACUTELY RED AND PAINFUL EYE\n\n#### The GP overview\n\nThis is a common reason for an urgent surgery appointment. If a visit request is made, try to negotiate consultation in surgery, where optimal examination conditions and equipment are to hand. Carefully examine to assess acuity, state of the cornea and pupillary reflexes.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute conjunctivitis (allergic or infective)\n\n  acute iritis\n\n  acute glaucoma\n\n  keratitis\/corneal ulcer\n\n  corneal abrasion or superficial foreign body (FB)\n\n**OCCASIONAL**\n\n  episcleritis\/scleritis\n\n  keratoconjunctivitis sicca\n\n  trauma: contusion and penetrating wound, burns (arc eye and chemical)\n\n  orbital cellulitis\n\n**RARE**\n\n  carotico-cavernous fistula (rupture of carotid aneurysm)\n\n  gout (urate deposits in conjunctiva or sclera)\n\n  granulomatous disorders: TB, sarcoid, toxoplasmosis\n\n  onchocerciasis (transmitted by _Simulium_ black fly in Africa)\n\n  tumour: primary eye tumour, invasion from nasopharyngeal tumour\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIn practice, the problem is either easily treated by the GP (e.g. conjunctivitis or foreign body) or usually requires urgent referral. The GP's role in investigating the painful red eye is therefore very limited.\n\n  Swab of discharge for microbiology: very occasionally helpful in conjunctivitis not settling with usual treatment.\n\n  Blood: raised WCC and ESR\/CRP may support diagnosis of inflammatory disorders. Rheumatoid factor in suspected rheumatoid arthritis (RA); HLA-B27 usually positive in ankylosing spondylitis. The latter investigations would normally be performed at leisure rather than in the acute situation, when an underlying collagen disease is suspected (e.g. iritis).\n\n  Intraocular pressure measurement is essential if acute glaucoma is suspected. Usually done by a specialist.\n\n##### TOP TIPS\n\n  If in serious doubt about the diagnosis, refer for urgent assessment \u2013 this is one scenario where a delay in treatment can have devastating consequences.\n\n  Don't rely on the patient's subjective assessment of blurring of vision \u2013 check the visual acuity.\n\n  Remember to evert the upper lid to check for a concealed foreign body.\n\n  Review the patient 24\u201348 hours after removing a foreign body to ensure that the cornea has healed.\n\n  Never use mydriatics when examining the red eye: you may precipitate acute glaucoma.\n\n  Bilateral red eye is usually caused by conjunctivitis. If unilateral, consider other causes.\n\n  Failure to recognise herpetic corneal ulcer or acute glaucoma may lead to permanent visual loss. If in doubt, refer for urgent specialist opinion.\n\n  Never instil steroid drops unless you are absolutely sure you are managing the problem correctly and have excluded herpetic ulceration.\n\n  Take a careful history when dealing with foreign bodies. Any possibility of a high-speed impact (e.g. grinding metal) requires urgent specialist assessment to exclude intraocular foreign body.\n\n### DOUBLE VISION\n\n#### The GP overview\n\nDiplopia is nearly always binocular, with movement of one eye being limited for a number of possible reasons. Although relatively uncommon as a presenting symptom, the majority of causes are significant and therefore careful assessment is essential.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  physiological (focusing too near, or perceiving objects nearer than those focused on)\n\n  intoxication: prescribed sedation, non-prescribed drugs, especially excess alcohol, opiates, benzodiazepines\n\n  stroke\n\n  mild head injury, causing temporary diplopia\n\n  facial bone trauma \u2013 orbital and zygomatic fracture\n\n**OCCASIONAL**\n\n  mononeuropathy, e.g. diabetes, MS\n\n  orbital disease (usually associated with pain and proptosis) and after surgery (scarring limiting globe movement)\n\n  Guillain\u2013Barr\u00e9 syndrome\n\n  palsy of third, fourth or sixth cranial nerves due to intracranial space-occupying lesion (haemorrhage, tumour, aneurysm, abscess, cavernous sinus thrombosis)\n\n  myasthenia gravis\n\n  monocular diplopia: early cataract, irregularity of corneal surface, e.g. post trauma or inflammation\n\n**RARE**\n\n  ophthalmoplegic migraine\n\n  Tolosa\u2013Hunt syndrome: granulomatous or inflammatory process in anterior portion of cavernous sinus or superior orbital fissure\n\n  pseudoparalysis of ocular muscles: dysthyroid disease; Duane's syndrome (congenital fibrosis of lateral rectus)\n\n  orbital myositis\n\n  pituitary exophthalmos\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none; FBC, ESR\/CRP, lipid studies, urinalysis, fasting glucose or HbA1c.\n\n_POSSIBLE :_ TFT, X-rays, CT\/MRI scan.\n\n_SMALL PRINT :_ edrophonium test, EMG, CSF studies, angiography.\n\n  No investigation is indicated in primary care for the most common causes \u2013 referral is the likeliest course of action.\n\n  FBC, ESR\/CRP, lipid studies \u2013 if stroke suspected and admission not required. FBC and ESR\/CRP will also provide evidence of inflammatory conditions.\n\n  Urinalysis for glucose\/fasting glucose or HbA1c \u2013 to investigate possible diabetes.\n\n  TFT will reveal hyperthyroidism.\n\n  Other investigations (and possibly some of the above) are likely to be carried out in secondary care: skull and facial bone X-rays in trauma cases; CT or MRI scan (head injury, stroke, MS, space-occupying lesion); specialist neurological investigations (edrophonium test, single-fibre EMG studies, CSF examination, angiography).\n\n##### TOP TIPS\n\n  Establish if the diplopia is binocular or not. Uniocular double vision has a much narrower differential diagnosis.\n\n  Take time to clarify the symptom. Sometimes, patients complain of 'double vision' when they really mean blurring \u2013 and vice versa.\n\n  The cover test is a reliable way to find out which eye is affected.\n\n  Fourth cranial nerve palsy produces diplopia on looking downwards and inwards, often noticed when descending stairs. The patient may try to compensate by tilting the head \u2013 so-called ocular torticollis.\n\n  Intoxication in conjunction with a head injury is commonly seen in custody medicine and A&E departments. Admission for neurological observation is strongly recommended.\n\n  Diplopia of acute onset may well reflect serious pathology \u2013 refer for urgent assessment.\n\n  Intermittent diplopia should not be dismissed too readily as insignificant \u2013 remember that myasthenia gravis and multiple sclerosis are possibilities.\n\n### EYELID PROBLEMS\n\n#### The GP overview\n\nBecause of their close proximity to the eye, and their occasional cosmetic effect, eyelid problems can be disproportionately distressing to the patient. The differential diagnosis is wide \u2013 so the temptation to treat this presentation as a 'quickie' via a brief examination shouldn't distract from the need to take a careful history.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  stye\n\n  blepharitis\n\n  meibomian cyst\n\n  xanthelasma\n\n  blocked tear duct\n\n**OCCASIONAL**\n\n  periorbital oedema, e.g. orbital cellulitis, herpes zoster, angio-oedema, nephrotic syndrome, insect bite\n\n  ectropion\n\n  entropion (may be secondary trichiasis)\n\n  eczema (seborrhoeic, allergic)\n\n  ptosis (congenital, oculomotor nerve palsy, Horner's syndrome, myasthenia gravis, senile, myotonic dystrophy)\n\n  muscular problems (myokymia, blepharospasm)\n\n**RARE**\n\n  malignant growth, e.g. basal cell carcinoma\n\n  benign growths, e.g. papilloma, haemangioma\n\n  dacrocystitis\n\n  alopecia\n\n  molluscum contagiosum\n\n  lice\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ : lipid profile.\n\n_POSSIBLE_ : urinalysis, FBC, LFT.\n\n_SMALL PRINT_ : tests for ptosis, biopsy.\n\n  Lipid profile: if xanthelasma, as may indicate hypercholesterolaemia.\n\n  Urinalysis: proteinuria in nephrotic syndrome.\n\n  FBC: WCC raised in infective process, e.g. cellulitis.\n\n  LFT: hypoproteinaemia in nephrotic syndrome.\n\n  Further tests for ptosis (usually in secondary care), e.g. CXR (Horner's), edrophonium test (myasthenia), blood sugar and brain scan (oculomotor palsy).\n\n  Biopsy \u2013 if suspicion of malignancy.\n\n##### TOP TIPS\n\n  A meibomian cyst is often misdiagnosed \u2013 by patient and doctor \u2013 as a stye, particularly if it is infected.\n\n  Entropion with secondary trichiasis may be overlooked as a cause of a recurring sore, watering eye, especially in the elderly.\n\n  Myokymia \u2013 recurrent focal twitching of the orbicularis oculi \u2013 is harmless but may distress or alarm the patient.\n\n  Many patients with xanthelasma will already have been 'noted' and so have had their cholesterol measured. Enquire about this before wasting resources on further unnecessary checks.\n\n  Orbital cellulitis requires urgent inpatient treatment.\n\n  Bilateral ptosis which deteriorates through the day may indicate myasthenia gravis.\n\n  New onset of unilateral ptosis requires investigation \u2013 possible diagnoses range from diabetes to malignancy.\n\n  Loss of the eyelashes is a poor prognostic sign in alopecia.\n\n  Unilateral eyelash loss, with or without apparent blepharitis, may be a sign of an eyelid tumour.\n\n### FLASHES, FLOATERS AND TRANSIENT VISUAL DISTURBANCE\n\n#### The GP overview\n\nThis symptom can be very difficult to fathom, not least because patients often find it nigh on impossible to describe exactly what they've experienced. Patience and a painstaking approach are essential \u2013 most of the clues are likely to be in the history rather than in the examination. This section does not cover double vision, gradual loss of vision or persistent sudden loss of vision, which are dealt with elsewhere.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  'normal' floaters\n\n  migraine\n\n  posterior vitreous detachment\n\n  amaurosis fugax\n\n  retinal detachment\n\n**OCCASIONAL**\n\n  temporal arteritis\n\n  vasovagals and orthostatic hypotension (commonly cause the symptom but rarely present with it)\n\n  vitreous haemorrhage\n\n  medication (e.g. transient blurred vision with anticholinergics, blue tinge to vision with sildenafil)\n\n  poorly controlled diabetes\n\n  TIA\n\n  optic neuritis\n\n**RARE**\n\n  papilloedema\n\n  trauma\n\n  posterior uveitis\n\n  seizures\n\n  psychological\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThe GP is highly unlikely to initiate any investigations in this situation \u2013 apart, perhaps, from a blood sugar or HbA1c to check for undiagnosed diabetes or an urgent ESR in suspected temporal arteritis. Otherwise, any investigations required would be arranged by the ophthalmologist or neurologist after urgent or routine referral, depending on the clinical picture.\n\n##### TOP TIPS\n\n  It is tempting to 'bounce' some of these cases to a local optician for assessment. Resist this approach \u2013 you may delay an important diagnosis and, besides, even a thorough examination is unlikely to provide anything like as much relevant information as a careful history.\n\n  Some patients find it easier to 'draw' their visual disturbance than describe it.\n\n  Floaters are a fairly common presentation \u2013 one of the key issues to establish is the duration of the symptom. The longer they have been present, the more you can be reassured that they are 'normal'.\n\n  Remember that ocular migraine can occur without the 'usual' headache.\n\n  A sudden onset of a shower of floaters is significant \u2013 especially if accompanied by flashing lights or blurred vision. Refer urgently to exclude a retinal detachment.\n\n  Do not forget temporal arteritis as a cause of transient visual disturbance, especially in patients aged 50 or more. Treat with high-dose steroids on suspicion of this diagnosis \u2013 do not wait for the results of blood tests.\n\n  Remember to check whether a young woman with migrainous visual disturbance is on the combined contraceptive pill and advise accordingly.\n\n  Amaurosis fugax is a form of TIA and should be managed as such \u2013 apply your local 'TIA pathway'.\n\n### GRADUAL LOSS OF VISION\n\n#### The GP overview\n\nThe four major causes of gradual blindness in the world are: cataract, onchocerciasis, vitamin A deficiency and trachoma. The latter three are very rare in the UK. Cataract occurs in 75% of over-65s, but only 20% of 45- to 65-year-olds. Most cases of gradual loss of vision encountered in primary care arrive via the optician, often with a letter outlining the problem and suggesting referral to an ophthalmologist.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  cataract\n\n  chronic glaucoma\n\n  diabetic and hypertensive retinopathy\n\n  senile macular degeneration\n\n  gradual inferior retinal detachment\n\n**OCCASIONAL**\n\n  choroidoretinitis\n\n  optic neuritis (in MS)\n\n  Paget's disease of the skull\n\n  retinitis pigmentosa\n\n  intraorbital or intracranial tumours\n\n**RARE**\n\n  syphilis\n\n  cerebromacular degeneration\n\n  toxic amblyopia (tobacco, methanol, arsenic, quinine, carbon bisulphide)\n\n  choroidal melanoma\n\n  Leber's hereditary optic atrophy\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThe only investigation the GP is likely to perform is a urinalysis and\/or blood sugar or HbA1c for suspected diabetes. If glaucoma is a possibility, and the patient has not already seen the optician, then optician referral will provide information about fields and pressures. More obscure tests \u2013 such as posterior pole ultrasound and CT scan for retinal, or other, tumours; syphilis serology; skull X-ray for Paget's disease; and neurological investigations for MS \u2013 are rarely required and are inevitably arranged in secondary care.\n\n##### TOP TIPS\n\n  Opticians will tend to report cataracts in the elderly routinely. Referral for surgery is only required if the problem is significantly impairing the individual's normal activities.\n\n  The presence of a cataract in relatively young patients is unusual and should prompt referral regardless of visual impairment \u2013 there may be a rare underlying metabolic cause.\n\n  Remember that significant glaucoma or other causes of visual loss may render the individual unfit to drive.\n\n  The elderly patient with a cataract whose vision is not improved considerably with the pinhole test probably has macular degeneration too, and so is unlikely to benefit much from cataract extraction.\n\n  It can be very difficult to establish in an elderly person whether the problem really has been gradual in onset or whether the history is more sudden; if in doubt, refer urgently as the cause may be acute and remediable.\n\n  Progressive early morning headache or proptosis with gradual loss of vision suggests a tumour. Refer urgently.\n\n  Gradual or recurrent visual loss or blurring with other intermittent neurological symptoms, especially in younger patients, suggests the possibility of MS.\n\n### ITCHY OR IRRITATING EYES OR EYELIDS\n\n#### The GP overview\n\nThis is a nuisance symptom which patients present with directly, or via their optician. It can be very difficult to ascertain whether the problem is arising from the eyelid or the eye itself. Often, the symptoms affect both and the causes overlap \u2013 which is why they're considered together here.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  allergic conjunctivitis (usually hay fever)\n\n  infective conjunctivitis\n\n  dry eyes\n\n  blepharitis\n\n  blocked tear duct\n\n**OCCASIONAL**\n\n  ectropion\n\n  entropion\n\n  eczema of the eyelids\n\n  effect of contact lenses or solutions\n\n  iatrogenic (oral or local medication)\n\n  foreign body (though usually presents with pain and redness)\n\n**RARE**\n\n  pubic lice (can affect eyelashes)\n\n  floppy eyelid syndrome (chronic conjunctivitis with lax eyelids in obese, middle-aged men)\n\n  sebaceous gland carcinoma\n\n  thyroid eye disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ swab, Schirmer's test.\n\n_SMALL PRINT :_ TFT, thyroid autoantibodies, MRI orbits, biopsy.\n\n  Swab: may be necessary in persistent discharge; essential in neonates who've had sticky eyes since birth.\n\n  Schirmer's test: to assess for dry eye \u2013 may be performed by the optician.\n\n  TFT, thyroid autoantibodies, MRI orbits \u2013 may be required in suspected thyroid eye disease.\n\n  Biopsy: rarely, if sebaceous gland carcinoma suspected.\n\n##### TOP TIPS\n\n  Diagnosis can be difficult and an optician's input may be invaluable.\n\n  Enquire about the use of OTC drops and their effect \u2013 this may give clues to the underlying problem. Sometimes, the drops themselves may be the cause.\n\n  It can be easy to overlook entropion \u2013 ask the patient to squeeze the eyes shut, then suddenly open them, in which case a subtle entropion should be revealed.\n\n  Bear in mind that patients with dry eyes sometimes paradoxically complain of a stringy discharge.\n\n  Remember that thyroid eye disease can present before biochemical dysfunction \u2013 if in doubt, refer.\n\n  Do not overlook a foreign body as a possible cause, especially if the history is vague and the symptoms unilateral.\n\n  Chlamydia and gonorrhoea must be excluded in the neonate with a sticky eye or eyes from birth.\n\n  The rare sebaceous gland carcinoma causes blepharitis-type symptoms, but with localised inflammation and localised loss of lashes.\n\n### SUDDEN LOSS OF VISION\n\n#### The GP overview\n\nSudden loss of vision is a genuine GP emergency. Most causes require an urgent ophthalmological opinion as there is little that the GP can do. This particular symptom is not often encountered in general practice \u2013 a prompt appointment or visit and a careful examination are necessary to assess the situation and exclude the causes not requiring urgent specialist treatment.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute glaucoma\n\n  vitreous haemorrhage\n\n  central retinal artery occlusion\n\n  migraine\n\n  CVA or TIA\n\n**OCCASIONAL**\n\n  central retinal vein occlusion\n\n  retrobulbar (optic) neuritis\n\n  retinal detachment\n\n  temporal arteritis\n\n  posterior uveitis\n\n**RARE**\n\n  hysteria\n\n  cortical blindness (non-vascular)\n\n  optic nerve injury\n\n  quinine poisoning\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIn practice, there are none worth doing at the time, as the vast majority of cases will be referred urgently. Virtually all tests will therefore be arranged by the specialist, usually after the event, to look for underlying causes. Such investigations include the following.\n\n  Screening for diabetes: undetected retinopathy may have preceded vitreous haemorrhage.\n\n  FBC: PCV may be raised in central retinal vein occlusion.\n\n  ESR: raised in temporal arteritis.\n\n  Multiple microbiological investigations are needed for posterior uveitis.\n\n  Posterior pole ultrasound may be useful in vitreous haemorrhage to identify treatable causes.\n\n  CT scan only useful to investigate cerebral causes (CVA or cortical blindness).\n\n##### TOP TIPS\n\n  Acute visual disturbance is often difficult to diagnose accurately and very alarming for the patient. If in doubt, refer urgently, or, at the very least, review in a few hours.\n\n  The patient's assessment of visual loss, and its severity, is highly subjective \u2013 if at all possible, test it with a Snellen chart.\n\n  Always keep spare batteries handy for your ophthalmoscope!\n\n  Don't forget that the visual disturbance may be the presenting symptom of some other pathology, such as hypertension, temporal arteritis or diabetes.\n\n  Don't miss a heart murmur or carotid bruit. These may be present in retinal artery occlusion and TIA\/CVA.\n\n  A cherry red spot on the fovea is pathognomic of retinal artery occlusion.\n\n  Never use mydriatics to aid examination at the bedside: these will cloud the clinical picture and may even precipitate acute glaucoma.\n\n## FACE\n\nFacial pain\n\nFacial rash\n\nFacial swelling\n\nFacial ulcers and blisters\n\n### FACIAL PAIN\n\n#### The GP overview\n\nPain in the face may either be due to local disease of any of the major structures of the face, or conditions affecting the innervation. The latter can occur anywhere between the posterior fossa and the ends of the trigeminal nerve. A good examination is not difficult, and quicker than most as undressing is not usually required.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  maxillary\/frontal sinusitis\n\n  trigeminal neuralgia (TN)\n\n  dental abscess\n\n  temporomandibular joint (TMJ) dysfunction\n\n  shingles (herpes zoster)\n\n**OCCASIONAL**\n\n  cluster headache (periodic migrainous neuralgia)\n\n  temporal arteritis\n\n  parotid: mumps, abscess and duct obstruction (stone\/tumour)\n\n  upper cervical spondylosis\n\n  mandibular or maxillary osteitis or cyst\n\n  cellulitis\n\n**RARE**\n\n  multiple sclerosis\n\n  atypical facial pain (may be linked with depression)\n\n  nasopharyngeal and lingual carcinoma\n\n  posterior fossa tumours\n\n  gummatous meningitis and tabes\n\n  glaucoma and iritis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, sinus X-ray.\n\n_SMALL PRINT :_ X-ray of TMJ, temporal artery biopsy, sialogram, CT\/MRI scan.\n\n  FBC: WCC and ESR\/CRP raised in infection; ESR\/CRP raised in temporal arteritis and tumour (ESR more useful than CRP in suspected temporal arteritis).\n\n  X-rays: sinus X-ray of little help in acute sinusitis but may help in chronic pain to assess for possible chronic sinusitis or tumour; TMJ views and dental plain film for abscess likely to be arranged by dentist; parotid sialogram for stone\/tumour.\n\n  Temporal artery biopsy: may be necessary to clinch diagnosis of temporal arteritis.\n\n  CT\/MRI scan the only practical way to examine the posterior cranial fossa and Gasserian ganglion \u2013 a specialist investigation.\n\n##### TOP TIPS\n\n  Don't over-diagnose sinusitis \u2013 many URTIs will produce mild facial ache through a vacuum effect.\n\n  Remember that shingles can produce pain before the rash \u2013 in the acute onset of unexplained unilateral facial pain, warn the patient to report back to the doctor should a blistering rash develop.\n\n  Refer dental abscesses to a dentist without treating first, to ensure proper investigation, treatment and follow-up \u2013 and to encourage the patient to present to the correct agency in future.\n\n  If no obvious cause is found for persistent facial pain, refer to exclude sinister pathology.\n\n  Trigeminal neuralgia is usually idiopathic, but may have a serious underlying cause, especially if there is associated motor disturbance or other neurological symptoms or signs.\n\n  Temporal arteritis is a clinical diagnosis. If suspected, treat immediately with high-dose steroids to prevent blindness. ESR is for retrospective confirmation only.\n\n  If the eyeball is red and tender in frontal facial pain, consider glaucoma, iritis or orbital cellulitis. Refer urgently.\n\n  Out-of-hours dental treatment can be very hard for patients to access in the UK. You may find yourself ethically obliged to provide some form of treatment, but be sure to give the advice (and document it) that the patient should contact their own dentist as soon as the dental surgery is open. Send a copy of your clinical notes to the dentist as you would any other clinician. You should never feel pressured to work outside your areas of competence, so don't hesitate to refer to the local hospital facio-maxillary team on call if you have any doubts about diagnosis or emergency management.\n\n### FACIAL RASH\n\n#### The GP overview\n\nUnlike the other 'skin' chapters in this book, this section is presented according to the rash's distribution. This is because it is a common presentation, one with a wide differential and one which causes the patient significant concern, largely because of the cosmetic impact. Occasionally, it can be caused by, or represent, significant pathology. Individual facial spots \u2013 such as basal cell carcinoma \u2013 aren't considered here.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acne\n\n  rosacea\n\n  seborrhoeic eczema\n\n  impetigo\n\n  perioral dermatitis\n\n**OCCASIONAL**\n\n  chloasma\n\n  sycosis barbae\n\n  drug side effect \u2013 especially phototoxicity\n\n  infection, e.g. herpes zoster and simplex, cellulitis, chickenpox, slapped cheek\n\n  allergic eczema\n\n  acne excori\u00e9e\n\n  post inflammatory hypo- or hyperpigmentation\n\n  pityriasis alba\n\n  petechiae from coughing\/vomiting\/straining\n\n  other generalised skin diseases, e.g. psoriasis, vitiligo\n\n**RARE**\n\n  Stevens\u2013Johnson syndrome\n\n  SLE\n\n  mitral flush\n\n  tuberous sclerosis\n\n  lupus vulgaris\n\n  sarcoidosis\n\n  dermatomyositis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR, autoantibody screen, CPK.\n\n_SMALL PRINT :_ viral or bacterial swabs, skin biopsy, muscle biopsy.\n\n  FBC: WCC raised in any infection; may be normochromic, normocytic anaemia in SLE.\n\n  ESR, autoantibody screen: ESR likely to be raised in infection and SLE; autoantibodies may be positive in the latter.\n\n  CPK: elevated in dermatomyositis.\n\n  Viral or bacterial swabs: to help diagnosis in obscure cases or if secondary infection suspected.\n\n  Skin biopsy; muscle biopsy: the former for suspected lupus vulgaris or sarcoidosis; the latter to confirm dermatomyositis.\n\n##### TOP TIPS\n\n  Do not underestimate the possible impact of a facial rash on a patient's life. The cosmetic effect may be devastating.\n\n  A therapeutic trial of antibiotics in acne may take up to 3 months to take effect \u2013 ensure the patient is aware of this.\n\n  Remember that impetigo may simply represent superinfection of an underlying skin problem, such as eczema, which will require treatment in its own right.\n\n  Check on OTC medication usage. In particular, remember that hydrocortisone 1% cream is available over the counter \u2013 inappropriate use might aggravate rosacea and perioral dermatitis.\n\n  Parents are sensitised to non blanching rashes. They can be reassured that such a rash restricted to the face (indeed, restricted to the entire distribution of the superior vena cava) is not due to meningitis.\n\n  A complaint of dramatic facial 'sunburn' in the elderly may well represent a phototoxic reaction \u2013 check the drug history.\n\n  Beware the acute onset of unilateral unexplained facial erythema with mild oedema, especially in an elderly patient. This may well be the start of cellulitis or herpes zoster.\n\n  Warn patients with facial zoster or herpes simplex infection near the eye to report any ocular problems.\n\n  Acne excori\u00e9e may be a marker of significant psychiatric pathology.\n\n### FACIAL SWELLING\n\n#### The GP overview\n\nThis section looks at 'internal' causes of facial swelling rather than superficial skin conditions which are dealt with elsewhere (see p. 387). This problem is usually a major concern to the patient because of the disfigurement, which it is impossible to hide. The causal conditions are often very painful too.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  mumps (viral parotitis)\n\n  angioneurotic oedema (allergy)\n\n  dental abscess\n\n  trauma (especially fractured zygoma)\n\n  salivary gland stone\n\n**OCCASIONAL**\n\n  bacterial parotitis\n\n  cellulitis (including orbital)\n\n  masseteric hypertrophy (caused by bruxism)\n\n  dental cyst\n\n  myxoedema\n\n  herpes zoster (may be swelling rather than blistering initially)\n\n**RARE**\n\n  parotid tumour\n\n  maxillary or mandibular sarcoma\n\n  Cushing's syndrome\n\n  nasopharyngeal carcinoma\n\n  Burkitt's lymphoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ facial X-ray (if trauma).\n\n_POSSIBLE :_ TFT (if patient looks myxoedematous).\n\n_SMALL PRINT :_ FBC, ESR\/CRP, sialogram.\n\n  Plain facial X-ray important in trauma (view may depend on site). Also may reveal rare cases of bony tumour.\n\n  TFT, FBC, ESR\/CRP: TFT will reveal hypothyroidism; WCC raised in infective process; ESR\/CRP raised in infection and tumour.\n\n  Parotid sialogram will show obstruction of duct (stone, tumour).\n\n##### TOP TIPS\n\n  New gruff voice with diffuse facial swelling should prompt investigation for likely hypothyroidism.\n\n  Don't over-diagnose mumps in children: since the advent of MMR vaccination, this is becoming more uncommon; cervical adenitis is much more likely.\n\n  Whenever possible, direct patients with dental problems directly to the dentist, who will be able to prescribe any necessary antibiotics and analgesics.\n\n  Painless, progressive facial swelling suggests tumour or dental cyst. Urgent oral surgical referral is indicated.\n\n  Bloodstained nasal discharge in association with a unilateral facial swelling is an ominous sign suggesting malignancy.\n\n  Severe angioedema may cause respiratory tract obstruction: treat vigorously as for anaphylactic shock.\n\n  Orbital cellulitis requires urgent assessment and intravenous antibiotics.\n\n  Parotid swelling with a facial palsy suggests parotid tumour with involvement of the facial nerve.\n\n  Suspected herpes zoster in the ophthalmic division of the trigeminal nerve requires urgent antiviral treatment and ophthalmological referral.\n\n### FACIAL ULCERS AND BLISTERS\n\n#### The GP overview\n\nFacial ulcers and blisters present much earlier than similar lesions elsewhere on the body because of the cosmetic disfigurement. Smaller lesions, especially basal cell carcinomas, are often picked up coincidentally by the doctor when the patient attends for some unrelated matter. (NB: For rashes confined, or largely confined, to the face, see Facial rash, p. 153.)\n\n#### Differential diagnosis\n\n**COMMON**\n\n  impetigo\n\n  herpes simplex virus (HSV)\n\n  herpes zoster\n\n  basal cell carcinoma (BCC)\n\n  keratoacanthoma\n\n**OCCASIONAL**\n\n  squamous cell carcinoma (SCC)\n\n  ulcerating malignant melanoma and lentigo maligna (Hutchinson's freckle)\n\n  drugs (e.g. barbiturates)\n\n  acne excori\u00e9e\n\n  ulcerating dental sinus\n\n**RARE**\n\n  dermatitis artefacta\n\n  tuberculosis\n\n  pemphigus\n\n  _Actinomyces_\n\n  primary syphilitic chancre or tertiary syphilitic gumma\n\n  cutaneous leishmaniasis\n\n  cancrum oris\n\n#### Ready reckoner\n\n#### Possible investigations\n\nAcute lesions very rarely require investigation; chronic lesions pose more of a diagnostic problem. In such cases, biopsy, or excision biopsy, is the gold standard test. Cytology after scraping the lesion with a scalpel blade may be helpful in diagnosing basal cell carcinoma. Syphilis serology may very rarely be useful if primary or tertiary syphilis is suspected.\n\n##### TOP TIPS\n\n  Remember that herpes simplex can occur on the face at sites other than the lip. The appearance of the lesions and their recurrent nature should provide the diagnosis.\n\n  'Rodent ulcer' is a kinder term than basal cell carcinoma, especially for small lesions, as it is less likely to arouse unnecessary anxiety. Nonetheless, impress upon the patient the importance of attending the appointment with the specialist.\n\n  Patients with herpes zoster are at risk of a number of anxieties because of the existence of various 'old wives' tales' about shingles. Establish any fears and take time to explain the natural history of the condition, including the possibility of post-herpetic neuralgia.\n\n  In children with recurrent impetigo, consider an underlying condition \u2013 particularly eczema.\n\n  If in any doubt about the diagnosis, urgent dermatological referral for skin biopsy is indicated. Remember that chronic facial ulceration is rarely benign.\n\n  Ulceration in a previously abnormally pigmented area of skin suggests advanced local malignancy.\n\n  Beware attempting excision biopsy of facial lesions unless specially trained. Areas of cosmetic importance can be medico-legal minefields.\n\n  Ask about foreign travel: leishmaniasis develops from the bite of a Mediterranean or South American sandfly.\n\n  Beware of herpes zoster or simplex developing around the eye: significant complications may follow, so treat and follow up carefully and obtain an ophthalmological opinion if necessary.\n\n## GENERAL PHYSICAL\n\nAbnormal gait in adults\n\nAbnormal movements\n\nBack pain\n\nCrying baby\n\nDelayed puberty\n\nEpisodic loss of consciousness\n\nExcessive sweating\n\nFailure to thrive\n\nFalls with no loss of consciousness\n\nThe febrile child\n\nFeeling tense and anxious\n\nFlushing\n\nInfertility\n\nItching\n\nJaundice in adults\n\nLimp in a child\n\nNumbness and paraesthesiae\n\nProlonged fever\n\nSwollen glands\n\nThirst or dry mouth\n\nTiredness\n\nTremor\n\nWeight gain\n\nWeight loss\n\n### ABNORMAL GAIT IN ADULTS\n\n#### The GP overview\n\nVery few patients present with abnormal gait. It is more often noticed by the GP, while the patient's complaint is usually a manifestation of the gait (e.g. unsteadiness in Parkinson's disease) or of its cause (e.g. pain in arthritis). Congenital causes are not considered here as patients are most unlikely to present such problems to the GP.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  trauma (back and leg)\n\n  osteoarthritis (OA) or other painful joint problem\n\n  vestibular ataxia (vestibular neuritis, M\u00e9ni\u00e8re's disease, CVA)\n\n  Parkinson's disease\n\n  intermittent claudication (IC)\n\n**OCCASIONAL**\n\n  foot drop (peroneal nerve atrophy)\n\n  multiple sclerosis\n\n  spinal nerve root pain (especially L5 and S1)\n\n  cauda equina lesions\n\n  myasthenia gravis\n\n**RARE**\n\n  tabes dorsalis (syphilis)\n\n  dystrophia myotonica\n\n  motor neurone disease\n\n  cerebellar ataxia\n\n  hysteria\n\n#### Ready reckoner\n\n#### Possible investigations\n\nMost cases requiring tests will need referral to a specialist. The role of the GP in investigating these patients is therefore very limited.\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ X-ray, FBC, ESR\/CRP, RA factor, uric acid.\n\n_SMALL PRINT :_ scans, lumbar puncture, angiography.\n\n  FBC, ESR\/CRP, RA factor, uric acid: some forms of arthritis will result in an anaemia of chronic disorder. ESR\/CRP may also be raised. Depending on the pattern of joint pain, RA factor and uric acid may provide useful information in the diagnosis of rheumatoid arthritis and gout.\n\n  X-rays useful in bony trauma. Limited value in OA except to exclude other bony pathology.\n\n  Syphilis serology: if tabes dorsalis suspected.\n\n  If neurological signs of incoordination, consider CT\/MRI scan and lumbar puncture \u2013 usually arranged by the specialist.\n\n  Angiography: arranged by the vascular surgeon if surgery contemplated in claudication.\n\n##### TOP TIPS\n\n  Look up from the notes or computer as the patient walks in \u2013 otherwise you may miss a useful clue in the patient's gait.\n\n  If the patient actually complains of problems walking, take your time in assessing the symptom \u2013 in particular, give the patient the opportunity to demonstrate the problem by walking him or her up and down the corridor.\n\n  If the cause is not immediately apparent from the history, perform a careful neurological examination \u2013 this is a situation where there may be hard signs which contribute significantly to diagnosis.\n\n  Vestibular neuritis usually settles within a few days. If patient remains ataxic, especially with persistent nystagmus, consider a central lesion and refer urgently.\n\n  Numbness in both legs (saddle pattern) with back pain and incontinence suggests a cauda equina lesion. Admit urgently.\n\n  If the patient is ataxic and has a past history of neurological symptoms, such as paraesthesia or optic neuritis, consider multiple sclerosis.\n\n  Beware of labelling the patient as hysterical \u2013 apparently bizarre gaits may signify obscure but significant neurological pathology.\n\n### ABNORMAL MOVEMENTS\n\n#### The GP overview\n\nThis is an infrequent cause for attendance \u2013 though the public is becoming increasingly aware of conditions such as restless legs syndrome (RLS) and Tourette's, and their treatments. Obvious generalised seizures and tremor are not considered in this chapter but are covered elsewhere.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  RLS\n\n  myokymia (affecting orbicularis oculi muscles)\n\n  drug induced \u2013 including choreoathetosis, dystonias, tardive dyskinesias and akathisia (drugs include l-dopa, tricyclic antidepressants, metoclopramide and antipsychotics)\n\n  Tourette's\n\n  simple partial seizures\n\n**OCCASIONAL**\n\n  anxiety\/nervous tic (common, but rarely presented to the GP)\n\n  muscle fasciculation (e.g. benign fasciculation, motor neurone disease)\n\n  simple childhood tics (common, but infrequently presented)\n\n  dystonias (e.g. writer's cramp, blepharospasm, spasmodic torticollis)\n\n  period leg movements during sleep\n\n**RARE**\n\n  myoclonus\n\n  chorea (Sydenham's, Huntington's)\n\n  Wilson's disease\n\n  hemiballismus (e.g. post stroke)\n\n  hysterical\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, U&E, ferritin, B12, folate, TFT, fasting glucose or HbA1c, calcium.\n\n_POSSIBLE :_ CT\/MRI of brain or spinal cord, EEG, EMG, nerve conduction studies.\n\n_SMALL PRINT :_ other specialised tests (e.g. for myoclonus and Huntington's).\n\n  FBC, ferritin: to assess for iron deficiency in RLS.\n\n  U&E: renal failure is a potential cause of RLS and can be implicated in partial seizures.\n\n  B12, folate: deficiencies may cause or mimic RLS.\n\n  TFT: hypothyroidism may cause RLS.\n\n  Fasting glucose or HbA1c: diabetes may cause RLS or partial seizures.\n\n  Calcium: hypocalcaemia may be implicated in seizures.\n\n  CT\/MRI of brain or spinal cord: may be required in investigation of fasciculation and seizures (usually arranged after specialist referral).\n\n  EEG: for investigation of seizures.\n\n  Other specialised tests: usually arranged by neurologist to explore the more obscure diagnoses such as Huntington's chorea and Wilson's disease.\n\n##### TOP TIPS\n\n  When faced with odd and otherwise inexplicable movements of recent onset in a patient, remember to take a drug history.\n\n  Bear in mind that abnormal movements can be caused by a drug that the patient has been taking for some time (e.g. tardive dyskinesias).\n\n  Patients with myokymia sometimes become disproportionately anxious about the symptom, imagining all sorts of possible neurological catastrophes \u2013 they may need a lot of reassurance.\n\n  Childhood tics tend to be single; the patient with the much more significant Tourette's will probably have multiple tics.\n\n  Drug-induced dystonias may cause odd posturing and require prompt treatment. The diagnosis is easily overlooked \u2013 antipsychotics are common culprits, so it is easy to erroneously attribute the dystonia to psychiatric pathology.\n\n  Beware of the combination of personality changes and odd movements such as facial grimaces \u2013 this could be Huntington's chorea. Also, don't be misled by the lack of a positive family history \u2013 this background may have been concealed from the patient.\n\n### BACK PAIN\n\n#### The GP overview\n\nOngoing backache is a familiar presentation to all GPs, and acute back pain is one of the most common reasons for an emergency appointment in primary care. The average GP can expect about 120 consultations for this problem each year. Eighty per cent of the Western population suffer back pain at some stage in their lives: it is the largest single cause of lost working hours among both manual and sedentary workers; in the former it is an important cause of disability. Remember that many non-orthopaedic causes of back pain lie in wait, so be systematic.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  mechanical (muscular) back pain\n\n  prolapsed lumbar disc: nerve root pain\n\n  spondylosis (exacerbation)\n\n  pyelonephritis and renal stones\n\n  pelvic infection\n\n**OCCASIONAL**\n\n  the spondoarthritides (e.g. ankylosing spondylitis, Reiter's syndrome)\n\n  neoplastic disease of the spine (usually secondaries), myeloma\n\n  duodenal ulcer\/acute pancreatitis\n\n  depression and anxiety states\n\n  vertebral fracture (often compression fracture associated with osteoporosis)\n\n**RARE**\n\n  spinal stenosis\n\n  osteomalacia\n\n  aortic aneurysm\n\n  pancreatic cancer\n\n  spondylolisthesis\n\n  osteomyelitis\n\n  malingering\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, MSU, FBC, ESR\/CRP, plasma electrophoresis, blood calcium, PSA.\n\n_SMALL PRINT :_ lumbar spine X-ray, IVU, HLA-B27, CT or MRI scan, bone scan, investigations for GI cause, ultrasound, DXA scan.\n\n  Urinalysis useful if UTI suspected: look for blood, pus and nitrite as markers of infection; confirm with MSU; blood alone suggests possible stone.\n\n  ESR\/CRP elevated in malignant and inflammatory disorders.\n\n  FBC: Hb may be reduced in malignancy; a high WCC raises the possibility of osteomyelitis.\n\n  Plasma electrophoresis: paraprotein band in myeloma.\n\n  Blood calcium: elevated in myeloma and bony secondaries; reduced in osteomalacia.\n\n  PSA: if disseminated prostate cancer suspected.\n\n  Lumbar spine X-ray often not useful in mechanical pain. Consider if no resolution by 6 weeks to investigate possible underlying pathology. In younger patients, it may help diagnose sacroiliitis or spondylolisthesis; in older people, it is useful to check for vertebral collapse. Generally, if imaging is required, CT or MRI may be more helpful.\n\n  Bone scan: will detect bony secondaries and bone infection.\n\n  CT or MRI scan usually a specialist's request: good for spotting spinal stenosis, significant prolapsed disc and discrete bony lesions.\n\n  Investigations for GI cause might include endoscopy (for DU), serum amylase (for pancreatitis) and CT scan (for carcinoma of pancreas).\n\n  Ultrasound: for aortic aneurysm.\n\n  IVU: for recurrent pyelonephritis and possible renal or ureteric stones.\n\n  DEXA scan: may be required to confirm suspicion of osteoporosis.\n\n##### TOP TIPS\n\n  The vast majority are 'mechanical', and most of these improve regardless of treatment modality in 6\u20138 weeks; a positive and optimistic approach is important.\n\n  Patients often expect an X-ray. Resist requests unless appropriate \u2013 and explain why. Even if the patient doesn't make this request, consider volunteering why you're not ordering an X-ray, as this can help maintain confidence in the doctor\u2013patient relationship, especially if the symptoms take some time to settle.\n\n  If the problem is recurrent, exclude significant pathology then explore the patient's concerns. In simple recurrent mechanical back pain, it is worth discussing preventive measures and educating the patient for self-management of future episodes.\n\n  True malingering is not common, but back pain is favoured among malingerers because of its subjectivity. Beware of patients who apparently cannot straight-leg raise, yet have no problem sitting up on the couch, and patients who decline to sit down during the consultation.\n\n  The traditional 'red flags' in back pain are thought to be of very limited use because of poor specificity and sensitivity. The only ones regarded as genuinely helpful are, for spinal fracture, older age, trauma, the presence of contusions or abrasions and steroid use, and, for spinal malignancy, a past history of cancer. Current consensus is that slavish adherence to red flags should be avoided and instead the overall clinical picture and progress assessed \u2013 although an ESR may be useful in ruling out significant disease.\n\n  Bilateral sciatica, saddle anaesthesia and bowel and\/or bladder dysfunction suggests central disc protrusion: this is a neurosurgical emergency.\n\n  Consider prostatic cancer in men over 55 with atypical low back pain. Do a PR exam, together with PSA and bone assay.\n\n  Back pain without any restriction of spinal movement, or which is not exacerbated by back movement, suggests that the source of the problem lies elsewhere \u2013 consider renal, aortic or gastrointestinal disease, or pelvic pathology in women.\n\n  Tearing interscapular or lower pain in a known arteriopath suggests dissecting aortic aneurysm: admit straight away.\n\n### CRYING BABY\n\n#### The GP overview\n\nThis is a very frequent reason for an out-of-hours call. A baby's cry is almost impossible for parents to ignore. When crying continues unabated in spite of all that parents can do to settle an infant, parental distress sets in and they will turn to you for an answer and a solution.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  normal\n\n  colic\n\n  constipation\n\n  teething\n\n  viral illness\n\n**OCCASIONAL**\n\n  otitis media or externa\n\n  severe nappy rash and\/or inflamed foreskin\n\n  gastroenteritis\n\n  UTI\n\n  after immunisation\n\n  respiratory distress \u2013 severe bronchiolitis, chest infection, croupy cough\n\n**RARE**\n\n  non-accidental injury\n\n  mastoiditis\n\n  meningitis, encephalitis\n\n  septicaemia\n\n  bowel obstruction including intussusception and strangulated hernia\n\n  appendicitis\n\n  osteomyelitis\n\n  testicular torsion\n\n  undiagnosed birth injury, e.g. fractured clavicle\n\n  congenital disorders, e.g. Hirschprung's disease, pyloric stenosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none other than those performed after admission.\n\n_POSSIBLE :_ swab of any obvious discharge.\n\n_SMALL PRINT :_ see hospital investigations.\n\n  Other than a swab for obvious discharge (e.g. in otitis externa), no investigations are appropriate in general practice \u2013 if no obvious cause is found and the baby continues to be distressed, admission for observation and investigation is mandatory.\n\n  Secondary care investigations are likely to include urinalysis, MSU for bacteriology, bloods for FBC, ESR, glucose, U&E, and many others depending on the indication (e.g. CXR, AXR, lumbar puncture, blood gases).\n\n##### TOP TIPS\n\n  Babies cry on average for 1\u00bd to 2 hours per day. Some normal babies cry more than this or for long periods for no apparent reason.\n\n  Remain calm and sensitive. Parents of a crying baby are often distraught, and whatever your perception of the seriousness or not of the situation, make a thorough assessment and make sure the parents understand you are taking them seriously.\n\n  Always explain your findings and advice clearly, and make sure the parents understand you. Write things down for them if necessary. Your patient depends completely on your communication skills.\n\n  Remember the obvious \u2013 babies may cry because of tiredness, hunger, wind after feeds, boredom, and uncomfortably full nappies. Never assume that parenting skills are 100%, and do explore baby care issues that may seem to be too obvious to ask about.\n\n  Babies may be unsettled and cry more than normal for a day or two after immunisation. If a baby is crying excessively for longer than 48 hours after immunisation, it is unsafe to diagnose immunisation as the cause without clinical assessment.\n\n  Telephone advice calls may be handled with careful triage and advice alone, providing there is always a fall back plan for the parents to call back or seek further advice if things do not settle rapidly. There is no substitute for a hands-on clinical assessment, and if in the slightest doubt, always see the baby as soon as possible.\n\n  Observe in the most general way how a baby handles during examination. Irritability on handling is a very important general sign. Regardless of other examination findings, this alone can be a reason to refer for paediatric assessment.\n\n  There is no clear physiological reason why babies should develop a fever during teething, but there is no doubt this happens in some babies, in spite of traditional medical teaching to the contrary. A fever is never high if due to teething alone. The fever of teething is usually very short-lived \u2013 less than 24 hours \u2013 while a fever caused by a viral infection can go on for several days.\n\n  Babies cannot tell us what's wrong, but they can tell us _something's_ wrong. If in any doubt about the diagnosis, seek a second opinion or a paediatric assessment. Always follow your sixth-sense, intuition or personal alarm bells. Thoughtful and experienced paediatricians will respect your feeling on this, so do not worry if you can't justify your referral on textbook clinical criteria.\n\n  Be aware of the possibility of non-accidental injury as a cause for the baby crying. If you detect unusual anxiety, or unusual emotional detachment from the calling parent, make sure you see the baby and examine it thoroughly.\n\n  A baby that has been crying a lot and goes on to become lethargic (as opposed to a normal calm state) is probably very ill. Even if you are unsure of the diagnosis, follow your intuition and arrange a paediatric opinion.\n\n  A constantly bulging fontanelle is always an indication for immediate paediatric referral.\n\n  Be sure to observe and note the general muscular tone of a crying baby. Constant stiffness or floppiness are ominous signs and immediate referral is indicated.\n\n### DELAYED PUBERTY\n\n#### The GP overview\n\nDelayed puberty means delayed development of all the secondary sexual characteristics. It is a rare but serious symptom. In girls, it is usually presented as a delayed menarche (failure to menstruate by the age of 16), though it may present as failure to develop other secondary sexual characteristics from the age of 14. In boys, the defined age is 15. The following is a selection of the more important causes. (Remember that the subheadings Common, Occasional, Rare are relative \u2013 overall, this is a rare presenting symptom.)\n\n#### Differential diagnosis\n\n**COMMON**\n\n  constitutional (50% of cases in boys, 16% of cases in girls)\n\n  hyperthyroidism\n\n  Turner's syndrome\n\n  anorexia nervosa (1% of all girls in Western countries)\n\n  hypothalamic gonadotrophin-releasing hormone (GnRH) deficiency (e.g. Noonan's and Kallmann's syndrome)\n\n**OCCASIONAL**\n\n  space-occupying hypothalamo-pituitary lesion (various types)\n\n  chronic disease (e.g. diabetes, renal failure, cystic fibrosis, coeliac disease)\n\n  hyperprolactinaemia\n\n  adrenal disease: congenital adrenal hyperplasia and Cushing's disease\n\n  drugs (e.g. thyroxine, chemotherapy (both sexes); androgens, anabolic steroids (females only))\n\n  radiotherapy\n\n  growth hormone deficiency\n\n**RARE**\n\n  other ovarian problems (e.g. pure dysgenesis, autoimmune disease)\n\n  hypothyroidism if autoimmune (otherwise associated with early puberty)\n\n  pure gonadal dysgenesis\n\n  maldescent of the testes (rare nowadays: usually detected early)\n\n  trauma, infection and granulomas of hypothalamus\/pituitary\n\n#### Ready reckoner\n\n#### Possible investigations\n\nCases requiring investigation are likely to need referral to a paediatrician or endocrinologist. The role of the GP is therefore limited. A few basic tests might be arranged in primary care in probable constitutional cases, mainly to exclude underlying disease and 'reassure' patient, parents and doctor (e.g. urinalysis, FBC, U&E, TFT). More complex investigations in secondary care might include CT scanning (tumours), ultrasound of pelvis (to examine ovaries and search for nonpalpable gonads), chromosomal analysis and various tests of endocrine function.\n\n##### TOP TIPS\n\n  Delayed puberty causes worry for parents and often misery for children, who may be teased or bullied by their adolescent peers. Take their concerns seriously from the outset.\n\n  Remember to take a family history: constitutional delayed puberty often runs in families.\n\n  The majority of children brought with 'delayed puberty' will be normal, with their parents either not recognising that secondary sexual characteristics are developing or not appreciating the age range which is normal for pubertal development.\n\n  Distinguish between delayed puberty and primary amenorrhoea with otherwise normal pubertal development. The latter has different causes (e.g. vaginal atresia, cycle initiation defect and, very rarely, testicular feminisation).\n\n  Although it accounts for 50% of male cases, do not diagnose constitutional delayed puberty in boys in the presence of a very small penis or anosmia \u2013 in these situations, an underlying disease is likely.\n\n  More than 80% of cases in girls have a pathological cause, so investigation is the rule.\n\n  Short stature, malaise and symptoms or signs of hypothyroidism suggest an underlying disorder of the hypothalamus and\/or pituitary.\n\n### EPISODIC LOSS OF CONSCIOUSNESS\n\n#### The GP overview\n\nThe terminology in this area can be very confusing with words like 'syncope' and 'faints' being used imprecisely. Episodic loss of consciousness can occur in any age group, though it tends to be commoner in the elderly. It is a frightening experience for the patient, and it demands thorough examination, investigation and a low threshold for referral. For the GP, the differential widens the older the patient \u2013 and cardiac causes should not be overlooked in the elderly.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  vasovagal attacks (faints)\n\n  paroxysmal arrhythmia, e.g. Stokes\u2013Adams attacks, sinus bradycardia, SVT\n\n  epilepsy (various forms)\n\n  hypoglycaemia\n\n  orthostatic hypotension\n\n**OCCASIONAL**\n\n  cardiac structural lesion, e.g. aortic stenosis, hypertrophic obstructive cardiomyopathy,\n\npulmonary stenosis\n\n  micturition and cough syncope\n\n  sleep apnoea\n\n  Valsalva-induced syncope, e.g. weightlifting\n\n  pseudoseizures\n\n**RARE**\n\n  narcolepsy\n\n  carotid sinus syncope\n\n  hyperventilation\n\n  subclavian steal syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC; ECG (especially in elderly); if probable epilepsy, also EEG and CT scan.\n\n_POSSIBLE :_ glucometer, 24 h ECG.\n\n_SMALL PRINT :_ echocardiography, tilt-table testing.\n\n  Glucometer 'on the scene' gives diagnosis of hypoglycaemia.\n\n  FBC: anaemia will exacerbate any form of syncope and TIAs.\n\n  Standard ECG may reveal signs of ischaemia and heart block; 24 h ECG more useful for definitive diagnosis of arrhythmia.\n\n  CT scan and EEG essential if previously undiagnosed epilepsy suspected.\n\n  Echocardiography: if structural cardiac problem suspected.\n\n  Tilt-table testing: for unexplained syncope to assess susceptibility to vasovagal episodes.\n\n##### TOP TIPS\n\n  The key to diagnosis is an accurate history. This may not be available from the patient, so make a real effort to obtain an eyewitness account.\n\n  In younger patients, the diagnosis is likely to lie between a vasovagal attack and a fit; in the middle-aged and elderly, the differential is much wider and will include, for example, arrhythmias and orthostatic hypotension.\n\n  Episodic loss of consciousness is a symptom which merits diligent assessment. An accurate diagnosis has implications not only for the individual's health, but also for employment and driving.\n\n  Remember that, with a vasovagal episode, patients remaining upright (e.g. sitting or in a crowd) may develop tonic\u2013clonic movements which mimic a fit.\n\n  Unlike in syncope or seizures, the eyes are usually closed in pseudoseizures.\n\n  An eyewitness account that the patient looked as though he or she had died, together with marked facial flushing on recovery, is characteristic of Stokes\u2013Adams attacks. These can be fatal, so early diagnosis is important.\n\n  Discovery of an aortic stenotic murmur should prompt urgent referral. Severe aortic stenosis can cause sudden cardiac death.\n\n  Red flags suggesting a possible cardiac cause include a family history of sudden cardiac death, syncope during exercise and an abnormal ECG.\n\n  Syncope caused by neck pressure or head movement could be carotid sinus syncope \u2013 if recurrent, this will require a pacemaker.\n\n### EXCESSIVE SWEATING\n\n#### The GP overview\n\nUnder normal conditions, 800 mL of water is lost daily as insensible loss, mostly in sweat. Excessive sweating can at least double this figure. As a symptom, it is normally part of a package of other problems \u2013 it is unusual for the patient to present with excessive sweating in isolation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  menopause\n\n  anxiety\n\n  infections (common, acute)\n\n  hypoglycaemia: may be reactive, i.e. non-diabetic\n\n  hyperthyroidism\n\n**OCCASIONAL**\n\n  drugs: alcohol, tricyclic antidepressants, pilocarpine\n\n  alcohol and drug withdrawal\n\n  shock\/syncope\n\n  intense pain\n\n  hyperhidrosis\n\n  other infections, e.g. TB, HIV, endocarditis, brucellosis\n\n**RARE**\n\n  malignancy (e.g. lymphoma)\n\n  organic nerve lesions: brain tumours, spinal cord injury (sweating is localised to dermatome involved)\n\n  pachydermoperiostosis: localised to skin folds of forehead and extremities\n\n  hyperpituitarism\/acromegaly\n\n  rare vasoactive tumours: phaeochromocytoma, carcinoid\n\n  connective tissue disorders\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, TFT.\n\n_POSSIBLE :_ FSH\/LH, LFT, glucose.\n\n_SMALL PRINT :_ autoimmune screen, CXR, tests for uncommon infections, 24-hour urinary catecholamines, CT\/MRI scan.\n\n  FBC\/ESR\/CRP: ESR\/CRP and WCC raised in infection. Raised ESR\/CRP and anaemia possible in lymphoma and other malignancies.\n\n  TFTs: may reveal thyrotoxicosis as a cause of chronic sweating.\n\n  Glucose: in reactive hypoglycaemia only useful at the time of the sweating.\n\n  FSH\/LH: helps if diagnosis of menopause in doubt.\n\n  LFT: may reveal high alcohol intake.\n\n  CXR might reveal occult infection (especially TB) or malignancy.\n\n  Tests for uncommon infections, e.g. blood test for HIV, echocardiography for endocarditis.\n\n  Autoimmune screen: may help in confirming diagnosis of connective tissue disease.\n\n  24 h urinary catecholamines traditionally used to look for phaeochromocytoma, but low specificity makes CT\/MRI scan more useful.\n\n##### TOP TIPS\n\n  Length of history is very helpful \u2013 short-term sweating is likely to have an apparent, acute cause; if long-term, the diagnosis is more likely to be constitutional or anxiety; in the medium-term, the differential diagnosis is much wider.\n\n  Anxiety rarely causes night sweats.\n\n  Do not underestimate the potentially devastating effect of hyperhidrosis.\n\n  Lack of fever does not exclude infection. In some infections (e.g. TB, brucellosis) \u2013 and lymphoma \u2013 sweating can be out of phase with fever.\n\n  If the problem is persistent, a full examination is advisable, paying attention to the lymph nodes, liver and spleen. If no cause is apparent, have a low threshold for investigations or referral, particularly if the patient is unwell or losing weight.\n\n  Consider unusual infections in the recently returned traveller (e.g. TB, typhoid).\n\n  Episodic skin flushing (especially provoked by alcohol) with diarrhoea and breathlessness is likely to be caused by anxiety \u2013 but don't forget carcinoid syndrome as a rare possibility.\n\n### FAILURE TO THRIVE\n\n#### The GP overview\n\nFailure to thrive is defined as the failure of a child to maintain the normal rate of growth for its age and gender. A logical and systematic approach is essential to navigate through the vast differential diagnosis list.\n\n#### Differential diagnosis\n\nThere are very many causes of failure to thrive. To produce a workable list, we have given broad categories and emphasised within them the most common causes.\n\n**COMMON**\n\n  normal \u2013 the genetic components of height and weight kick in during the first 2 years, so babies of small parents may cross the centiles and appear to be 'dropping off'\n\n  neglect (emotional and physical)\n\n  feeding problems (inadequate or inappropriate feeding, physical causes, e.g. cleft palate)\n\n  vomiting from any cause (gastro-oesophageal reflux common, other causes such as pyloric stenosis rarer)\n\n  malabsorption \u2013 including cow's milk intolerance (lactose or cow's milk protein intolerance), coeliac disease\n\n**OCCASIONAL**\n\n  recurrent infections, e.g. UTI, frequent viral illnesses\n\n  metabolic\/endocrine causes, e.g. diabetes, hypo- and hyperthyroidism\n\n  common chronic infection (UTI, gastroenteritis)\n\n  syndromes, e.g. Turner's, Down's (though growth can follow centiles normally)\n\n  intrauterine growth retardation\n\n  premature delivery with complications\n\n  toxicity during pregnancy, e.g. maternal smoking, alcohol, cocaine and amphetamines\n\n  maternal medication or infection during pregnancy\n\n**RARE**\n\n  serious chronic disease, e.g. cerebral palsy, hepatic, cardiac or renal failure\n\n  severe chronic asthma\n\n  malignancy\n\n  rare (in UK) chronic infection: TB, congenital HIV, parasites\n\n  Munchausen's syndrome by proxy\n\n  cystic fibrosis\n\n  inborn errors of metabolism\n\n  rare causes of infant feeding difficulties, e.g. hypotonia, micrognathia, Prader\u2013Willi syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU, FBC, U&E, LFTs, thyroid function tests, coeliac screen, stool studies.\n\n_POSSIBLE :_ immunoglobulins.\n\n_SMALL PRINT :_ PPD skin test, radiological studies, sweat chloride test, growth hormone levels, HIV testing.\n\n  Urinalysis and MSU to seek evidence of UTI.\n\n  FBC may reveal both the effects of malnutrition (anaemia) and possible causes of failure to thrive, e.g. a raised white cell count indicating chronic infection.\n\n  U&E and LFTs can both be markers of reduced metabolic function (reduced urea, creatinine and albumin, and reduced liver enzyme activity) or the signs of primary renal or hepatic disease.\n\n  Thyroid function tests: essential and simple to detect over- or underactive thyroid.\n\n  Coeliac screen and stool studies: for malabsorption (and look too for parasites, e.g. helminths or giardiasis).\n\n  Immunoglobulins: different patterns and levels of different immunoglobulin classes are associated with a vast range of infective, autoimmune and neoplastic conditions.\n\n  Purified protein derivative (PPD) skin test (for tuberculosis) is also known as the Mantoux test. A positive reaction to the injected tuberculin PPD may indicate current TB infection, or previous exposure.\n\n  Radiological studies: bone age may be helpful to distinguish genetic short stature from constitutional delay of growth.\n\n  Sweat chloride test is a reliable diagnostic test for cystic fibrosis, which affects 1 in 2500 infants.\n\n  Growth hormone levels. Growth hormone deficiency affects about 1 in 3500 children.\n\n  HIV testing if vertical transmission of HIV is suspected.\n\n##### TOP TIPS\n\n  Remember that 3% of normal infants fall below the third centile. Ask about height of other parent and grandparents in an otherwise healthy child. A constitutionally small child's growth will still follow the centile curves albeit from a low starting point.\n\n  A third of children with psychosocial failure to thrive are developmentally delayed and have social and emotional problems.\n\n  If parents are small, and there is no other sign of an underlying problem, it is safe to wait and observe the baby with regular weight measurements. The baby should return to running parallel with the centiles after the second year.\n\n  A diagnosis of a non-organic cause of failure to thrive may be the earliest indication of a serious parent\u2013child interaction dysfunction.\n\n  Non-organically caused failure to thrive in the first year of life has an ominous prognosis. There is a high likelihood of ongoing child abuse in this group.\n\n  If child abuse is suspected, refer to paediatrics urgently \u2013 by admission to hospital.\n\n  The younger and more ill a child or baby is at presentation with failure to thrive, the more urgently assessment and action are required.\n\n  A study has shown that half of cases of organic failure to thrive were also associated with a contributory psychosocial factor. Be alert to the fact that this can be a multifactorial condition.\n\n### FALLS WITH NO LOSS OF CONSCIOUSNESS\n\n#### The GP overview\n\nThis is a common problem in the elderly and may represent an acute or chronic problem. A home visit is often necessary and can be very valuable, assisting diagnosis and management decisions.\n\n_NOTE :_ The term 'drop attacks' is inconsistently defined in the literature as 'falls with no loss of consciousness', 'falls with loss of consciousness' or may be regarded as a distinct diagnostic entity rather than a symptom. It is a term best left unused.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  orthostatic hypotension\n\n  brainstem ischaemia (vertebrobasilar insufficiency)\n\n  iatrogenic (e.g. phenothiazines, hypoglycaemics, tricyclics and hypotensives)\n\n  postural instability (osteoarthritis, quadriceps weakness)\n\n  any acute illness (e.g. sepsis, CVA)\n\n**OCCASIONAL**\n\n  lack of concentration (tripping over mats etc.)\n\n  visual disturbance\n\n  acute alcohol intoxication and chronic alcohol misuse\n\n  Parkinson's disease\n\n  cardiac arrhythmias\n\n  any cause of vertigo (e.g. labyrinthitis, M\u00e9ni\u00e8re's disease) or non-specific dizziness (e.g. anaemia)\n\n**RARE**\n\n  hypothyroidism\n\n  hydrocephalus\n\n  third ventricular tumour\n\n  diabetic autonomic neuropathy\n\n  aortic stenosis\n\n  painless ('silent') myocardial infarction\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC.\n\n_POSSIBLE :_ TFT, LFT, ECG (or 24 h ECG).\n\n_SMALL PRINT :_ CT scan, echocardiography.\n\n  Urinalysis for glucose may reveal underlying diabetes \u2013 a major cause of autonomic neuropathy \u2013 or evidence of UTI.\n\n  FBC: anaemia will exacerbate any cause of postural hypotension, or may itself cause dizziness. Sepsis is suggested by a raised WCC. A high MCV may be a useful pointer to alcohol misuse or hypothyroidism.\n\n  TFT: hypothyroidism is common in the elderly and develops insidiously.\n\n  LFT: for evidence (\u03b3GT) of alcohol misuse.\n\n  ECG or 24 h ECG is useful to identify an arrhythmia, conduction defect or MI.\n\n  CT scanning (e.g. for tumours or hydrocephalus) or echocardiography (for aortic stenosis) may be arranged by the specialist after referral.\n\n##### TOP TIPS\n\n  Failure to observe the patient's gait may mean that significant diagnoses, such as Parkinson's disease, are missed.\n\n  Recurrent falls in the elderly are often caused by a combination of factors, such as failing vision, poor lighting and trip hazards at home. A home assessment may give valuable clues.\n\n  In the acute situation, management may depend more upon the ability of the patient to remain safely at home (e.g. social support) rather than the precise diagnosis.\n\n  Don't underestimate the importance of what you prescribe in causing morbidity. Attempt to reduce polypharmacy and review therapy regularly.\n\n  In dealing with this problem, don't forget to look for cause and effect: the aetiology of the falls and any significant injuries sustained.\n\n  Sudden onset of falls in the previously well elderly patient is likely to represent acute pathology \u2013 have a low threshold for investigation or admission.\n\n  Gradual onset of recurrent falls is often multifactorial in the elderly; in younger patients, specific underlying disease is more likely, so refer for investigation.\n\n  Evidence of injury (e.g. bruises or fractures) and multiple attendance slips from A&E department indicate either a very frail, vulnerable elderly person or significant underlying illness.\n\n### THE FEBRILE CHILD\n\n#### The GP overview\n\nThis symptom probably generates more GP advice calls and parental anxiety than any other. It is nearly always caused by an infection of some kind. The list of culprits is so vast that we have concentrated on the common and occasional ones more likely to be seen in general practice in the UK.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  non-specific viral URTI, e.g. colds, flu-type illness, pharyngitis, tracheitis\n\n  gastroenteritis\n\n  otitis media\n\n  tonsillitis\n\n  chest infection\n\n**OCCASIONAL**\n\n  urinary tract infection\n\n  bronchiolitis\n\n  croup\n\n  common viral exanthems, e.g. chickenpox, roseola, hand, foot and mouth, fifth disease\n\n  appendicitis\n\n  cellulitis (especially orbital) and other significant skin infections, e.g. abscess, scalded skin\n\n  glandular fever\n\n  post-immunisation\n\n  giardiasis\n\n**RARE**\n\n  meningitis\/meningococcal septicaemia\n\n  encephalitis\n\n  hepatitis\n\n  AIDS\n\n  rare exanthems, e.g. measles, rubella\n\n  mumps\n\n  acute epiglottitis\n\n  atypical infections, e.g. brucellosis, listeriosis, Lyme disease, cat scratch fever\n\n  tuberculosis\n\n  protozoal diseases, e.g. cryptosporidium, leishmaniasis, toxoplasmosis, malaria\n\n  septic arthritis, osteomyelitis\n\n  Kawasaki's disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\nUsually none are necessary in practice. If a febrile child is ill enough to require investigation, the problem will usually be sufficiently urgent to need management by acute admission. Urinalysis, as a pointer to UTI (assuming obtaining a urine sample is feasible), is sometimes helpful in avoiding or facilitating admission. If possible, extend this to an MSU for bacteriological analysis.\n\n##### TOP TIPS\n\n  Many clinical markers (see Ready reckoner) are non-specific and are present in many different infections. It is often a matter of degree as to how likely they are a pointer to a specific pathology. For example, many infections cause mesenteric adenitis with abdominal pain, but the tenderness of appendicitis, for example, is usually far greater on examination. Respiratory rate is raised in all fevers, but a chest problem will increase it further along with the presence of chest signs. A good policy is to be as thorough as possible in an examination so as to be able to cross-reference the maximum clinical information.\n\n  Remember that parents will be worried about their child, and no matter how simple the management of this common problem appears to you, to the parent it may be the harbinger of a serious illness. Practise a calm and polite demeanour, empathy and sensitivity.\n\n  In telephone advice calls, always do three things: first, check that the parent is satisfied with your advice; second, put a robust safety net\/plan B in place with easily identifiable guidelines for the parent, e.g. 'if by X hours Y has not happened, then call back'; third, record your clinical assessment and the last two points in detail.\n\n  If in doubt, always see a child in person. Be sensitive to your intuition. If something nags you after an advice call, ring back and arrange a consultation. You will never look stupid for doing this \u2013 only careful.\n\n  Remember to follow up children in whom you've diagnosed UTI according to NICE guidelines, which recommend further investigation, varying according to the age of the child.\n\n  It is very easy to print computerised clinical notes immediately after writing them. Handing a consultation note to a parent can be invaluable to the parent (and the clinician) if the child is seen later on out of hours, when clinical records are often unavailable. The baseline findings from earlier in the day can be priceless information in the dark hours later on.\n\n  Spend time explaining the nature of fever and that the key issue is the cause of the fever rather than the fever itself \u2013 many parents are 'fever phobic'.\n\n  It's often more important to be able to distinguish between 'well' and 'ill' babies and children than it is to make a clever, precise diagnosis \u2013 the NICE 'traffic light' guidance may help.\n\n  It's easy to be tempted into complacency in the telephone or consultation management of this problem \u2013 fever is just so common. Never forget that uncommon very serious illnesses may all begin with a fever. Always be diligent and systematic in assessment, no matter how busy your winter on-call day is turning out to be.\n\n  Dehydration can kill a baby quickly. Ensure you have satisfied yourself about the state of hydration of a child. The colour and quantity of urine passed, or frequency of nappy changes are useful practical guides, together with the general 'look' of the child and the capillary refill time.\n\n  A febrile baby or child who is floppy or drowsy should be admitted immediately.\n\n  Be suspicious of the irritable and inconsolable infant. Even without other hard evidence, suspect a serious problem and arrange urgent paediatric assessment.\n\n  The petechial rash of meningococcal septicaemia is a late phenomenon. Do not be reassured by its absence. Its presence should prompt a 999 call and administration of immediate parenteral antibiotics according to local protocols.\n\n  In most cases, the height of a fever is no guide to the severity of the illness \u2013 the exception being babies, where a temperature of 38\u00b0C or more in those under 3 months is seen as a 'red' and one of 39\u00b0C or more in those between 3 and 6 months is viewed as 'amber' according to NICE.\n\n### FEELING TENSE AND ANXIOUS\n\n#### The GP overview\n\nThe patient complaining of feeling tense and anxious may induce similar feelings in the GP \u2013 because there are many possible underlying and contributory causes, the consultation may be lengthy, and the patient may well present in a crisis. A calm, methodical approach, possibly stretching over more than one consultation, will pay dividends.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  life events (may be underlying 'anxious personality')\n\n  pre-menstrual tension\n\n  generalised anxiety disorder\n\n  panic disorder\n\n  depression\n\n**OCCASIONAL**\n\n  obsessive\u2013compulsive disorder\n\n  phobias\n\n  drug side effect (for example, in the early stages of SSRI treatment)\n\n  hyperthyroidism\n\n  drug\/alcohol use or withdrawal\n\n  somatisation disorder\n\n  post-traumatic stress disorder\n\n**RARE**\n\n  psychotic illness\n\n  any cause of palpitations (may be 'misinterpreted' by the patient or others as anxiety)\n\n  organic brain disease, e.g. tumour\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIt would be very unusual for the GP to require any investigations when dealing with this symptom. Thyroid function tests would be indicated in suspected hyperthyroidism, and a blood screen, to include LFT, if alcohol was thought to be playing a significant part. Investigations in the rare event of suspected organic brain disease would usually be left to the specialist.\n\n##### TOP TIPS\n\n  It is tempting to lump many of these scenarios under a catch-all label of 'tension' or 'anxiety'. But attempts at making a more precise diagnosis are worthwhile, as this may significantly alter the management.\n\n  Do not overlook an alcohol or drug history: abuse or withdrawal may be the cause of the symptoms, or a significant contributor.\n\n  It may be worthwhile carefully reviewing the patient's old records to establish patterns of symptoms or attendance, and to check previous response to treatment.\n\n  Whenever possible, life events should not be 'medicalised' \u2013 otherwise this may, in the future, encourage re-attendance and foster dependence on treatment.\n\n  Apparent pre-menstrual tension may be a sign of some other underlying disorder \u2013 the patient may be suffering generalised anxiety disorder, for example, but may tend to focus on the pre-menstrual phase, when the symptoms are at their worst.\n\n  Do not accept a self-diagnosis of 'panic attacks' at face value \u2013 the patient may actually mean any one of a number of possible symptoms.\n\n  If the underlying diagnosis turns out to be depression, assess for any suicidal ideas or intent.\n\n  Check for any psychotic features \u2013 anxiety can occasionally be a presenting feature of serious psychotic illness.\n\n  New onset of tension or anxiety without any obvious explanation \u2013 especially in the context of personality change, neurological features or new headaches \u2013 could, rarely, reflect organic brain disease.\n\n  It's important to make diagnoses such as somatisation disorder when appropriate \u2013 otherwise the patient may suffer years of unnecessary tests and treatment.\n\n### FLUSHING\n\n#### The GP overview\n\nThis symptom presents more often in women than in men, not only because of its cosmetic importance, but also because the menopause accounts for the vast majority of presentations. It is different from emotional blushing in its context, severity, duration and extent.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  menopause\n\n  chronic alcohol misuse\n\n  rosacea\n\n  iatrogenic (e.g. calcium antagonists)\n\n  anxiety\n\n**OCCASIONAL**\n\n  polycythaemia rubra vera\n\n  hyperthyroidism\n\n  drug\/alcohol interaction: metronidazole, disulfiram\n\n  mitral valve disease (malar flush)\n\n  hyperglycaemia and hypoglycaemia\n\n  epilepsy (aura)\n\n**RARE**\n\n  carcinoid tumour\n\n  phaeochromocytoma\n\n  Zollinger\u2013Ellison syndrome\n\n  systemic mastocytosis\n\n  ACTH-secreting bronchogenic carcinoma and Cushing's syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, LFT, TFT, blood sugar.\n\n_SMALL PRINT :_ echocardiogram, EEG, urinary 5HIAA and VMA, gastrin level, further specialised endocrine tests.\n\n  FBC: raised haemoglobin and PCV in polycythaemia (may also be elevated platelets and WCC), raised MCV in chronic alcohol misuse.\n\n  Biochemistry: LFT and \u03b3GT abnormal in alcohol misuse. TFT will reveal hyperthyroidism.\n\n  Blood sugar: to reveal hypo- or hyperglycaemia.\n\n  FSH\/LH of limited use as does not correlate well with symptoms (useful only if premature menopause suspected).\n\n  Echocardiography: if mitral stenosis suspected.\n\n  EEG: for possible epilepsy.\n\n  Specialist tests might include urinary 5HIAA (carcinoid) and VMA (phaeochromocytoma), gastrin level (Zollinger\u2013Ellison syndrome) and further endocrine tests (e.g. for Cushing's syndrome).\n\n##### TOP TIPS\n\n  Most women complaining of flushing will suspect the cause is the menopause. Address this possibility in the consultation, especially in young women fearing 'an early change'.\n\n  A constantly flushed face in older men is likely to be due to alcohol, polycythaemia or rosacea.\n\n  Anxiety is likely if the circumstances fit \u2013 but bear in mind that hyperthyroidism can produce a very similar clinical picture.\n\n  It can be difficult to distinguish anxiety from menopausal symptoms in a woman of menopausal age. Flushes with sweats waking the woman at night are more likely to be caused by the menopause \u2013 but a trial of treatment is the acid test (though beware of an initial placebo response).\n\n  Diarrhoea and dyspnoea with flushing after alcohol, food and exercise suggest possible carcinoid syndrome.\n\n  Flushing followed by an episode of altered consciousness points to a significant cause, such as recurrent hypoglycaemia or epilepsy.\n\n  Do not be tempted to write this symptom off as the hot flushes of emotional blushing. While common, this problem is unlikely to present in daily practice.\n\n  Recent onset of severe flushing which is not obviously menopausal or anxiety may have a significant cause, especially if the patient has other symptoms. Have a low threshold for investigations or referral in such cases.\n\n### INFERTILITY\n\n#### The GP overview\n\nAbout 80% of couples conceive within the first year of trying, and a further 5\u201310% in the second. Infertility is defined as a failure to conceive after 2 years of regular intercourse. However, depending on the circumstances, it may be appropriate for the GP to intervene before this arbitrary 2-year cut-off. Management is likely to be dictated by prevailing national and local guidelines and protocols.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  unexplained (27%)\n\n  defective sperm (24% \u2013 may be various underlying causes)\n\n  anovulatory cycles\/defective ovulation (21%)\n\n  fallopian tube blockage (14%)\n\n  endometriosis (6%)\n\n**OCCASIONAL**\n\n  hostile cervical mucus\n\n  PCOS\n\n  other causes of amenorrhoea or hypomenorrhoea (see Absent periods, p. 374)\n\n  uterine fibroids, polyp\n\n  cervical problems: inflammation, polyps, stenosis\n\n  systemic illness, e.g. anaemia, thyroid disorders\n\n**RARE**\n\n  congenital uterine, vaginal, fallopian or ovarian malformation or absence\n\n  sexual dysfunction\n\n  adrenocortical tumours\n\n  chromosomal abnormalities: Turner's syndrome (XO), super-female (XXX)\n\n  endometrial tuberculosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_NOTE :_ amenorrhoea will need investigation in its own right \u2013 see Absent periods, p. 374. The degree of investigation undertaken for infertility will depend very much on the particular situation and local guidelines.\n\n_LIKELY :_ semen analysis, serum progesterone 7 days before anticipated onset of period.\n\n_POSSIBLE :_ if menstrual irregularity\/amenorrhoea\/unwell \u2013 FBC, FSH, LH, prolactin, TFT, U&E, testosterone. In secondary care: ovarian\/uterine ultrasound.\n\n_SMALL PRINT :_ laparoscopy and dye, hysterosalpingogram, post-coital test. In the male: further investigation of sperm problems, e.g. FSH, LH, testosterone, testicular ultrasound.\n\n  Semen analysis: may need repeating if first test sub-optimal.\n\n  Serum progesterone 7 days before anticipated onset of period: to check for ovulation.\n\n  FBC, FSH, LH, prolactin, TFT, U&E, testosterone: to check for underlying pathology if female is unwell or has menstrual problems.\n\n  Ovarian and uterine ultrasound: to exclude structural abnormalities. Can be extended to visualise ovarian follicle development and ovulation.\n\n  Laparoscopy and dye, hysterosalpingogram: to check for tubal patency.\n\n  Post-coital test: cervical mucus sampled 12 hours post-coitally is examined under high-power microscope. More than 10 motile sperm per high-power field implies satisfactory coitus, adequate sperm and antibody-free cervical mucus.\n\n  Male FSH, LH and testosterone: to check for endocrine causes of sperm production failure.\n\n  Testicular ultrasound: if anatomical abnormalities found on examination, e.g. varicocoele.\n\n##### TOP TIPS\n\n  Adopt an optimistic approach whenever possible. A surprising number of couples conceive successfully as soon as preliminary investigations are initiated!\n\n  Don't overlook health promotion opportunities \u2013 especially female rubella status and advice about taking folic acid.\n\n  Ensure you treat the couple rather than the individuals. Infertility management requires much cooperation and motivation.\n\n  In primary care, hormone tests (other than progesterone) are not necessary if the periods are normal.\n\n  Infertility will only very rarely be the presenting symptom of serious pathology. However, there are circumstances where it is important to act promptly in terms of investigation or referral, such as:\n\n  if the female\n\n\u2013 is over the age of 35\n\n\u2013 has amenorrhoea\n\n\u2013 has a history of previous pelvic surgery or PID\n\n\u2013 has any abnormality on pelvic examination\n\n  or if the male\n\n\u2013 has a history of urogenital problems or STD\n\n\u2013 has a varicocoele.\n\n### ITCHING\n\n#### The GP overview\n\nItching is the commonest presenting dermatological symptom. It is frequently distressing, and may interfere with the patient's quality of life \u2013 for example, by preventing normal sleep. Therefore, it should be taken seriously. A good history alone will reveal the diagnosis in the majority of cases. The remainder will yield to thorough examination and investigation. Dermatological referral need only be a last resort to achieve diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  contact allergy (contact dermatitis)\n\n  scabies (and other pediculoses)\n\n  atopic eczema\n\n  pityriasis rosea\n\n  psoriasis\n\n**OCCASIONAL**\n\n  urticaria (e.g. food or drug allergy)\n\n  jaundice of any cause\n\n  iron deficiency anaemia\n\n  endocrine: diabetes mellitus, hypo- and hyperthyroidism\n\n  renal failure (uraemia)\n\n  lichen planus\n\n  prickly heat\n\n**RARE**\n\n  herpes gestationis\n\n  dermatitis herpetiformis\n\n  psychogenic (includes dermatitis artefacta)\n\n  leukaemia and myeloproliferative disorders\n\n  simple pruritus: no other cause found\n\n  drug side effect (with or without rash)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, blood glucose or HbA1c, FBC, ESR, U&E, LFT, TFT.\n\n_SMALL PRINT :_ none.\n\n  Urine: dipstick for glycosuria (blood glucose or HbA1c if positive).\n\n  FBC: will reveal iron-deficiency anaemia and polycythaemia; eosinophil count may be raised in allergic conditions; WCC may be very high in leukaemia; ESR may be elevated in lymphoma.\n\n  U&E: will reveal uraemia.\n\n  LFT: deranged liver enzymes and raised bilirubin in liver disease.\n\n  TFT: both hypo- and hyperthyroidism can lead to skin changes which cause itching.\n\n##### TOP TIPS\n\n  An itchy, unidentifiable rash which is worse at night is likely to be scabies, particularly if any contacts are affected.\n\n  Warn the patient that scabies treatment may take a week or two fully to relieve symptoms \u2013 otherwise, the patient may apply the treatment repeatedly, causing a chemical irritation and diagnostic confusion.\n\n  The books usually state that psoriasis doesn't itch \u2013 but it certainly can, so don't let this symptom put you off the diagnosis.\n\n  It is usually very difficult to identify the allergen in a single episode of urticaria. Tell the patient to keep a note of foods or medicines just ingested so that, in the event of recurrence, the culprit can be identified.\n\n  If no obvious cause, always examine the abdomen and lymph nodes: do not miss lymphadenopathy, or enlarged liver, spleen or kidneys.\n\n  Don't be tempted not to examine the itchy, malodorous self-neglected patient: poor personal hygiene may deceptively mask some other identifiable underlying cause.\n\n  Beware of apparently florid eczema appearing for the first time in an elderly patient \u2013 this may be a manifestation of serious underlying pathology.\n\n  Don't forget iatrogenic causes \u2013 enquire about any drugs recently prescribed.\n\n### JAUNDICE IN ADULTS\n\n#### The GP overview\n\nPatients rarely present with the complaint of 'turning yellow'; more often \u2013 though still infrequently \u2013 the GP notices jaundice during an examination of the patient. A systematic clinical assessment together with relevant laboratory investigations will help pinpoint the cause.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  gallstones in common bile duct\n\n  viral hepatitis (e.g. glandular fever, hepatitis A, B, C)\n\n  carcinoma of head of pancreas\n\n  hepatic carcinoma (usually metastases)\n\n  alcoholic cirrhosis\n\n**OCCASIONAL**\n\n  alcoholic hepatitis\n\n  primary biliary cirrhosis\n\n  drugs: chlorpromazine, isoniazid, anabolic steroids, methyldopa, paracetamol overdose\n\n  haemolytic anaemia (many causes, such as autoimmune, malaria, drugs)\n\n  venous congestion: cardiac failure, constrictive pericarditis\n\n  cholangitis (and stricture in common bile duct afterwards)\n\n  pancreatitis\n\n**RARE**\n\n  carcinoma of bile duct\n\n  leptospirosis\n\n  Rotor, Dubin\u2013Johnson and Mirizzi syndromes\n\n  cholestasis or fatty liver of pregnancy\n\n  genetic: Gilbert's syndrome, Wilson's disease, \u03b11-antitrypsin deficiency, galactosaemia, glycogen storage diseases, haemochromatosis\n\n  amyloidosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC, LFT, hepatitis serology.\n\n_POSSIBLE :_ ultrasound, antimitochondrial antibody.\n\n_SMALL PRINT :_ serum amylase, secondary care tests (e.g. ERCP, liver biopsy).\n\n  Urinalysis: if bilirubin is present in the urine, the jaundice is cholestatic. If present with urobilinogen, it is hepatocellular. If not, it is obstructive.\n\n  LFT: bilirubin very high in biliary obstruction. AST and ALT raised in hepatic causes. Alkaline phosphatase rises moderately in hepatic causes and markedly in biliary obstruction and primary biliary cirrhosis.\n\n  FBC: anaemia in chronic illness. Raised WCC in hepatitis. May be macrocytosis, reticulocytosis and other red cell abnormalities in haemolytic anaemia. MCV raised by alcohol.\n\n  Hepatitis serology: may reveal cause of viral hepatitis.\n\n  Serum amylase: raised in pancreatitis.\n\n  Antimitochondrial antibody test: positive in over 95% of patients with primary biliary cirrhosis.\n\n  Ultrasound useful to assess liver, pancreas and gall bladder: may reveal stones, primary tumours and metastases.\n\n  Referral may result in various other specialised tests including ERCP and liver biopsy, to establish the underlying cause.\n\n##### TOP TIPS\n\n  Remember to look at the patient: if significant jaundice is present it will probably strike you at first glance.\n\n  In younger patients, the diagnosis is likely to be viral hepatitis. In older age groups, the differential is much wider.\n\n  Don't forget iatrogenic causes. Remember too that the presence of jaundice implies liver dysfunction, so take great care if prescribing any medication.\n\n  If the patient is well, with no pain and fever, it is reasonable to arrange initial investigations \u2013 especially LFT \u2013 and arrange for review in a day or two. Most other cases will require admission.\n\n  Remember to ask about foreign travel, contact with travellers, drug misuse and sexual history if necessary in the suddenly jaundiced febrile patient.\n\n  Painless progressive jaundice suggests carcinoma of pancreas. Refer urgently.\n\n  An enlarged, knobbly, hard liver is nearly always caused by metastases.\n\n  Beware of restlessness, poor concentration and drowsiness. These suggest fulminant hepatic failure.\n\n### LIMP IN A CHILD\n\n#### The GP overview\n\nThis is an infrequent but alarming presentation, as it may herald significant pathology and may be difficult to manage properly in an uncooperative toddler. Assessment should be methodical and patient, and follow-up arranged unless the diagnosis is obvious at the outset.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  trauma, including foreign body in foot (especially toddlers)\n\n  irritable hip (transient synovitis)\n\n  acute viral infection with arthralgia\n\n  pauciarticular juvenile chronic arthritis (JCA: 1 in 1000)\n\n  slipped femoral epiphysis (usually over 10 years old)\n\n**OCCASIONAL**\n\n  Perthes's disease (1 in 2000 between 4 and 10 years old)\n\n  septic arthritis\n\n  idiopathic scoliosis\n\n  congenital dislocation of the hip (5\u201310 per 1000)\n\n  acute lower abdominal pain \u2013 especially appendicitis\n\n  unequal leg length\n\n  neurological, e.g. cerebral palsy\n\n**RARE**\n\n  acute osteomyelitis\n\n  rheumatic fever\n\n  autoimmune disorders (e.g. SLE, dermatomyositis)\n\n  rickets\n\n  genuine juvenile rheumatoid arthritis\n\n  malignancy affecting bone\n\n  Duchenne's muscular dystrophy\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, X-ray.\n\n_POSSIBLE :_ autoimmune screen.\n\n_SMALL PRINT :_ calcium, phosphate, alkaline phosphatase, creatine kinase, ASO titre, blood culture.\n\n  FBC and ESR\/CRP: WCC and ESR\/CRP elevated in an underlying inflammatory or infective cause.\n\n  Hip X-ray: may reveal fracture, slipped femoral epiphysis, congenital dislocation, Perthes's and other significant disorders \u2013 but may be normal in the presence of serious pathology.\n\n  Rheumatoid factor and autoimmune screen may be helpful if a connective tissue disorder is suspected.\n\n  Serum calcium, phosphate and alkaline phosphatase: calcium and phosphate low, alkaline phosphatase high in rickets.\n\n  Creatine kinase: markedly elevated in muscular dystrophy.\n\n  ASO titre is raised in 80% of cases of rheumatic fever.\n\n  In hospital, blood culture may identify the infecting organism in osteomyelitis and septic arthritis.\n\n##### TOP TIPS\n\n  Never forget to examine the soles of the feet and between the toes for obvious and potentially simple to treat, non-serious causes of limp.\n\n  It's worth investing some time gaining the child's confidence \u2013 this will enable you to make a proper assessment and feel positive about your management.\n\n  Parents may try to rationalise the symptom by recalling a recent minor episode of trauma which is likely to be purely coincidental.\n\n  Don't forget referred pain. Hip pathology can cause pain in the knee.\n\n  Marked restriction of movement and\/or dramatic bony tenderness suggests a significant problem \u2013 especially fracture, septic arthritis and osteomyelitis.\n\n  Fever with a limp requires an urgent specialist opinion. Admit to exclude osteomyelitis or septic arthritis.\n\n  Beware the fat pubertal boy with groin pain and a limp: slipped femoral epiphysis is likely.\n\n  Do not confine your assessment to the hip: for example, abdominal pain, especially appendicitis, can make a child limp.\n\n### NUMBNESS AND PARAESTHESIAE\n\n#### The GP overview\n\nParaesthesiae and numbness are taken here to mean sensations of tingling, pins-and-needles, subjective numbness, and feelings of cold and heat. They may appear spontaneously or be a result of touching the area of skin concerned. Patients are often alarmed and may make an immediate association with serious disease. The differential is huge but most cases in primary care involve anxiety, nerve entrapment or cerebrovascular disease.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anxiety with hyperventilation\n\n  carpal tunnel (CT) syndrome\n\n  sciatica\n\n  diabetic neuropathy\n\n  cervical spondylosis\n\n**OCCASIONAL**\n\n  multiple sclerosis and dorsal myelitis\n\n  peripheral polyneuropathy (especially alcohol; also vitamin B12 and folate deficiency, iatrogenic, metabolic, connective tissue disorder, malignancy and rare causes such as Guillain\u2013Barr\u00e9, leprosy)\n\n  stroke and TIA\n\n  trauma\/compression involving a peripheral nerve or spinal cord\n\n  migraine with focal neurological signs\n\n**RARE**\n\n  intramedullary spinal cord tumour\n\n  syringomyelia\n\n  trauma, brain tumour and epilepsy affecting sensory cortex\n\n  hysteria\n\n  vascular: ischaemic heart disease, peripheral vascular disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, blood sugar or HbA1c.\n\n_POSSIBLE :_ FBC, LFT, \u03b3GT, U&E, serum calcium, B12 and folate, TFT, nerve conduction studies.\n\n_SMALL PRINT :_ autoimmune screen, cervical spine X-ray, secondary care investigations (MRI\/CT scan, lumbar puncture, carotid imaging, angiography, myelography).\n\n  Urinalysis: to pick up glycosuria in undiagnosed diabetes.\n\n  Blood sugar or HbA1c: to confirm diabetes.\n\n  FBC: to look for macrocytosis (sign of alcohol excess or B12\/folate deficiency). May be anaemia of chronic illness or malignancy.\n\n  LFT and \u03b3GT if alcoholic neuropathy suspected.\n\n  Metabolic screen (including U&E, calcium, LFT, B12 and folate).\n\n  TFT: hypothyroidism can cause a polyneuropathy or precipitate carpal tunnel syndrome.\n\n  Autoimmune screen: to help diagnose a connective tissue disorder.\n\n  Nerve conduction studies: to confirm a diagnosis of nerve compression prior to surgical treatment.\n\n  X-ray cervical spine: confirms clinical diagnosis of cervical spondylosis, but not really helpful as positive findings common and don't correlate well with symptoms, and the investigation is unlikely to alter the management.\n\n  Secondary care investigations might include: lumbar puncture (MS, Guillain\u2013Barr\u00e9 syndrome), carotid imaging (TIA), CT or MRI scan (spinal pathology or compression, MS, cerebral tumour, syringomyelia), angiography (vascular causes), myelography (cord compression).\n\n##### TOP TIPS\n\n  Intermittent perioral paraesthesiae are pathognomic of hyperventilation.\n\n  Use a logical approach: a careful history will often reveal the likely underlying problem. For example: well-demarcated area in anatomically explicable distribution \u2013 peripheral nerve entrapment; larger area, one limb \u2013 root compression; whole side of body \u2013 cerebral lesion; hands and feet \u2013 peripheral neuropathy; legs alone \u2013 cord lesion.\n\n  Wasting of the thenar eminence suggests significant CT syndrome which will require decompression.\n\n  Remember to tell women taking the combined oral contraceptive who develop migraine with focal symptoms to use an alternative method of contraception.\n\n  Sudden and progressive bilateral leg symptoms with sphincter disturbance suggest cord compression: admit immediately.\n\n  Intermittent paraesthesiae in varying distributions \u2013 especially with other features, such as vertigo or optic neuritis \u2013 suggest MS.\n\n  Patients with TIAs with an 'ABCD2' score of 4 or more should be seen by a specialist within 24 hours; those with lower scores should be seen within 1 week.\n\n  The borders of sensory loss in hysteria are often sharply demarcated and do not correspond to an anatomical pattern.\n\n  Constant, progressive paraesthesiae, especially with other neurological symptoms or signs, suggest significant pathology. Refer urgently.\n\n### PROLONGED FEVER\n\n#### The GP overview\n\nGPs deal with fevers on a daily basis \u2013 the vast majority are caused by viruses and are self-limiting. This section deals with a different and much less common scenario: a temperature above normal for 3 weeks or more. It may be continuous or swinging. Many more causes exist than can be named here, but those likeliest in general practice are listed.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  glandular fever (GF)\n\n  abscess (anywhere)\n\n  chronic pyelonephritis (recurrent UTI)\n\n  carcinoma (especially bronchial)\n\n  rheumatoid arthritis (RA)\n\n**OCCASIONAL**\n\n  lymphoma and leukaemia\n\n  systemic lupus erythematosus, polyarteritis nodosa, polymyositis\n\n  Crohn's disease and ulcerative colitis\n\n  drug idiosyncrasies\n\n**RARE**\n\n  malaria and other tropical diseases\n\n  Lyme disease\n\n  tuberculosis, syphilis\n\n  actinomycosis\n\n  HIV infection: AIDS\n\n  infective endocarditis\n\n  factitious\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, LFT, U&E, urinalysis, MSU.\n\n_POSSIBLE :_ Paul\u2013Bunnell test, CXR, autoimmune screen.\n\n_SMALL PRINT :_ faecal calprotectin, blood cultures, malaria films, syphilis serology, HIV test and a variety of other secondary care-based tests.\n\n  FBC, ESR\/CRP, U&E, LFT: anaemia will be revealed in a variety of disorders (e.g. malignancy, connective tissue disorders); WCC raised in many inflammatory or infective processes and also some blood dyscrasias. Elevated ESR\/CRP a non-specific finding in many of the illnesses listed. Abnormal U&E or LFT may point to an underlying renal or hepatic problem.\n\n  Urinalysis, MSU: may be proteinuria, haematuria and evidence of infection in chronic pyelonephritis.\n\n  Paul\u2013Bunnell test: may be positive in glandular fever.\n\n  CXR: will show signs of malignancy (e.g. lung, lymphoma), occult infection and TB.\n\n  Autoimmune screen: may suggest a connective tissue disorder.\n\n  Faecal calprotectin: if symptoms suggest possible IBD.\n\n  Secondary care-based tests: a number of tests may be performed after referral to the specialist in cases which remain obscure after initial assessment and investigation. These include further microbiological tests (e.g. stool, blood cultures), blood tests (e.g. for malaria, syphilis and HIV), isotope scans, ultrasound and CT scans (for occult infection or malignancy), tuberculin testing (for possible TB) and esoteric tests for tropical diseases.\n\n##### TOP TIPS\n\n  Prolonged fever is usually an uncommon presentation of a common disorder (unless there has been recent travel), so review the situation regularly and encourage the patient to report new symptoms, which may help reveal the diagnosis.\n\n  Refer early if the patient is unwell or has lost weight; if not, arrange initial investigations yourself.\n\n  Don't always accept self-reporting of this symptom at face value. Flushing or sweating may be misreported as 'fever'. If in doubt, get the patient to record the temperature over a period of time.\n\n  Always take a travel history, and specifically enquire about insect bites and compliance with antimalarial therapy. Remember, too, occupation and recent contact with infectious illness.\n\n  Tuberculosis is rare but on the increase in the UK. Consider this diagnosis, particularly in\n\nAsian immigrants and vagrants.\n\n  Itching with prolonged pyrexia suggests leukaemia or lymphoma.\n\n  Refer to a tropical medicine centre a patient with PUO who has recently been abroad somewhere exotic \u2013 in such a case, the differential is much wider and the possibility of an obscure pathology therefore much greater.\n\n  Factitious prolonged fever is rare, but possibly more common among health staff; beware the health worker with apparent fever but who never feels hot and who never appears unwell, especially if basic investigations are all normal.\n\n  Don't forget the possibility of infective endocarditis in a patient with a cardiac murmur.\n\n### SWOLLEN GLANDS\n\n#### The GP overview\n\nThere are very many causes of swollen glands (lymphadenopathy), but in general it is possible to narrow the list of possible causes down to a manageable few by careful history and examination. Age, geography (or travel history) and distribution of enlarged glands have a considerable influence on the differential diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  local infection (e.g. URTI, tonsillitis)\n\n  generalised viral infection (e.g. glandular fever, rubella)\n\n  malignancy: secondary metastasis\n\n  white cell malignancy: lymphoma, leukaemia, myeloma\n\n  septicaemia\n\n**OCCASIONAL**\n\n  sarcoid\n\n  cat scratch fever (especially in children)\n\n  rubella\/measles\n\n  SLE\n\n  rheumatoid arthritis\n\n  tropical\/subtropical sexually transmitted infection: lymphogranuloma venereum (LGV), granuloma inguinale (GI)\n\n**RARE**\n\n  syphilis (primary or secondary)\n\n  HIV: AIDS and AIDS-related complex (ARC)\n\n  tuberculosis\n\n  tropical infections: leprosy, filariasis, trypanosomiasis, tularaemia\n\n  drug reactions (e.g. phenytoin, penicillins, sulphonamides)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none if localised; FBC and ESR\/CRP if generalised.\n\n_POSSIBLE :_ Paul\u2013Bunnell test, CXR, acute and convalescent sera, HIV testing, lymph node biopsy.\n\n_SMALL PRINT :_ autoimmune blood tests, syphilis serology, cultures and scrapings for LGV and GI, Kveim test, CT scan, indirect fluorescent antibody blood test for _Bartonella henselae._\n\n  FBC: atypical lymphocytes reflect acute viral infection; many of the causes listed will result in a raised WCC and ESR\/CRP. Hb may be low in malignancy and connective tissue disease; WCC and film may show evidence of lymphoma or leukaemia.\n\n  Paul\u2013Bunnell test: to confirm glandular fever.\n\n  CXR: may reveal carcinoma, TB, lymphoma, sarcoid and the source of septicaemia.\n\n  Serology: acute and convalescent sera may confirm specific viral infection.\n\n  Kveim test for sarcoid.\n\n  Abdominal and chest CT scan is a sensitive test to detect pelvic, para-aortic, mesenteric, hilar or paratracheal node enlargement (e.g. in lymphoma).\n\n  Autoimmune blood tests: may help in diagnosis of connective tissue disorder.\n\n  Culture\/scrapings (GUM clinic): for LGV and GI.\n\n  Syphilis serology, HIV testing: for syphilis and AIDS.\n\n  Indirect fluorescent antibody test for _B. henselae:_ this is the causative organism in cat scratch disease.\n\n  Lymph node biopsy may be necessary to reach a definitive diagnosis.\n\n##### TOP TIPS\n\n  Normal cervical lymph nodes are often palpable in children; they swell with URTIs and may be presented by anxious parents fearing significant disease.\n\n  Remember geography: a young adult from the UK with persistent cervical nodes is likely to have Epstein\u2013Barr virus (EBV) infection while, in Africa, the likeliest diagnosis would be tuberculosis. Swollen groin glands in the latter group might be caused by LGV or GI.\n\n  Patients often attach great significance to swollen glands. It is worth explaining that lymphadenopathy usually represents a normal part of the immune system's defence against infection and does not in itself require attention from the doctor unless there are unusual features.\n\n  Unexplained and persistent cervical lymphadenopathy in the middle-aged and elderly should prompt urgent ENT assessment to exclude nasopharyngeal carcinoma.\n\n  An enlarged left supraclavicular node (Troisier's) in a patient with weight loss suggests gastrointestinal carcinoma.\n\n  Generalised, persistent lymph nodes with weight loss and sweats in a young adult suggest glandular fever, lymphoma or AIDS.\n\n  A slowly enlarging, non-tender cervical node in an unusual site is likely to be malignant.\n\n### THIRST OR DRY MOUTH\n\n#### The GP overview\n\nThe complaint of thirst rings alarm bells in doctor and patient alike. Diabetes clearly needs to be excluded but the differential may need to be extended beyond this in the light of negative initial tests. Dry mouth tends to create less concern but can sometimes herald significant pathology and may be a serious nuisance to the patient.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  diabetes mellitus\n\n  dehydration\n\n  medication (e.g. tricyclic antidepressants, antihistamines)\n\n  mouth breathing (usually through nasal blockage)\n\n  anxiety\n\n**OCCASIONAL**\n\n  normality (children sometimes presented because 'they are always thirsty')\n\n  smoking\n\n  excess alcohol\n\n  Sj\u00f6gren's syndrome\n\n  hypercalcaemia\n\n  chronic kidney disease (CKD)\n\n**RARE**\n\n  pregnancy (common condition but rarely causes significant thirst)\n\n  compulsive water drinking\n\n  sickle-cell disease\n\n  previous head\/neck irradiation\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, fasting blood glucose or HbA1c.\n\n_POSSIBLE :_ FBC, ESR\/CRP, U&E, calcium, rheumatoid factor and other autoantibody screen.\n\n_SMALL PRINT :_ serum and urine osmolality, sickle-cell screen.\n\n  Urinalysis: glycosuria in diabetes, specific gravity raised in dehydration and reduced in diabetes insipidus and compulsive water drinking, may be proteinuria and\/or microscopic haematuria in CKD.\n\n  Fasting blood glucose or HbA1c: to definitively diagnose diabetes.\n\n  FBC\/ESR: Hb may be reduced and ESR elevated in Sj\u00f6gren's linked to connective tissue disorder; Hb may also be reduced in CKD.\n\n  U&E: may suggest dehydration or CKD.\n\n  Calcium: elevated in hypercalcaemia.\n\n  Rheumatoid factor and other autoantibodies: Sj\u00f6gren's may be linked to rheumatoid arthritis, SLE or other connective tissue disease.\n\n  Serum and urine osmolality: serum osmolality raised and urine osmolality low in diabetes insipidus; in compulsive water drinking, serum osmolality low.\n\n  Sickle-cell screen: to detect sickle-cell anaemia.\n\n##### TOP TIPS\n\n  The assessment of thirst does not stop at the exclusion of diabetes mellitus \u2013 consider other causes.\n\n  Intermittent dry mouth in an anxious individual also reporting episodic perioral paraesthesiae is likely to be caused by anxiety \u2013 perhaps aggravated by certain medications the patient might be taking for the problem.\n\n  Do not underestimate the complaint of dry mouth, especially in the elderly \u2013 it can cause significant distress.\n\n  Remember hypercalcaemia, particularly in palliative care patients \u2013 this is a potentially remediable cause of troublesome thirst.\n\n  Children who are 'always thirsty', have been like that for as long as the parents can remember and are otherwise well will not have diabetes \u2013 though the parents may feel short-changed if this isn't tested for.\n\n  In the acute presentation of thirst, it is essential to exclude diabetes immediately \u2013 a very high glucose level with ketonuria will require admission.\n\n  Beware the thirsty elderly patient with an acute illness, particularly if the patient is on ACE inhibitors \u2013 he or she may be significantly dehydrated and developing renal failure.\n\n  Dry eyes and joint swellings in association with a dry mouth may indicate Sj\u00f6gren's.\n\n  Beware that the elderly with diabetes may complain of dry mouth rather than thirst.\n\n### TIREDNESS\n\n#### The GP overview\n\nFeeling tired all the time is such a common presenting symptom that 'TATT' has become the universal GP acronym. In around 85% of first consultations the cause is apparent by the end. Although the vast majority turn out to have a non-organic cause, it is clearly important not to get lulled into ignoring important physical illness. A structured approach can turn this heartsink calling card into a rewarding and successful consultation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  true depressive illness\n\n  stress (overwork, young children, boredom, etc.)\n\n  anaemia\n\n  acute post-viral fatigue\n\n  hypothyroidism\n\n**OCCASIONAL**\n\n  diabetes mellitus\n\n  iron deficiency (in the absence of anaemia)\n\n  coeliac disease\n\n  any cause of insomnia (see p. 80)\n\n  chronic post-viral fatigue syndrome ('ME')\n\n  major organ failure (heart, liver, kidney)\n\n  hyperthyroidism\n\n  substance misuse\n\n  drug therapy (\u03b2-blockers, diuretics)\n\n**RARE**\n\n  malignant disease\n\n  chronic infection (e.g. TB, hepatitis, HIV)\n\n  chronic neurological disorders (Parkinson's disease, myasthenia gravis, MS, motor neurone disease)\n\n  other endocrine disorders and deficiency states (e.g. Addison's disease, hypo- or hypercalcaemia, hyponatraemia, hypokalaemia)\n\n  connective tissue diseases (RA, polymyalgia rheumatica (PMR), SLE)\n\n  carbon monoxide poisoning\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC, blood glucose or HbA1c, TFT.\n\n_POSSIBLE :_ ESR\/CRP, U&E, LFT, ferritin, calcium, anti-endomysial and anti-gliadin antibodies.\n\n_SMALL PRINT :_ CXR, autoantibody screen, further blood tests as indicated such as HIV, glandular fever test, hepatitis serology.\n\n  Urinalysis: simple screen for diabetes and renal disease.\n\n  Blood glucose or HbA1c: for diabetes.\n\n  TFT for hypo- or hyperthyroidism.\n\n  FBC: the anaemias, infection and alcohol abuse.\n\n  ESR\/CRP: raised in a host of causes; not diagnostic but suggests a possible underlying physical cause.\n\n  U&E: deranged in renal failure, hyponatraemia, hypokalaemia and Addison's disease.\n\n  LFT: for liver disease (malignancy, alcohol abuse and hepatitis).\n\n  Ferritin: iron deficiency may cause tiredness in the absence of anaemia.\n\n  Calcium: hyper- or hypocalcaemia may cause tiredness.\n\n  Anti-endomysial and anti-gliadin antibodies: will suggest a diagnosis of coeliac disease.\n\n  Autoantibody screen: for connective tissue disease.\n\n  Further blood tests: these will be dictated by the clinical picture and might include HIV, glandular fever or hepatitis testing.\n\n  CXR: may reveal malignancy, cardiac failure or TB.\n\n##### TOP TIPS\n\n  Tiredness as a presenting symptom, in the absence of other significant volunteered symptoms \u2013 particularly weight loss or gain \u2013 is unlikely to have a physical cause.\n\n  The longer tiredness has been a problem, the less likely you will find _any_ remediable cause.\n\n  Make eye contact with the patient and shake hands \u2013 your first impressions as to whether or not the patient is actually 'ill' are likely to prove correct, and may give early clues to easily overlooked causes such as Parkinson's disease.\n\n  If you suspect depression, enquire directly about relevant symptoms \u2013 you do not have to 'exclude' physical illness before making a presumptive diagnosis of this sort.\n\n  Investigations ordered are often more therapeutic than diagnostic.\n\n  Ask about other household members' health and well-being. Carbon monoxide poisoning would be likely to affect them too.\n\n  Do not be too trigger-happy with the usual 'blood screen'. Many patients 'promised' blood tests if their symptoms persist another month do not return.\n\n  Take tiredness associated with weight loss seriously \u2013 this combination suggests malignant disease or hyperthyroidism.\n\n  Don't miss depressive illness by being coy in enquiry or colluding with patient denial.\n\n  Don't miss easy-to-find signs when physical illness seems a real possibility \u2013 check pulse rate, mucous membranes for pallor, lymph nodes, chest and abdomen.\n\n  Consider a rare cause if the symptoms progress and the patient starts to look unwell.\n\n### TREMOR\n\n#### The GP overview\n\nThis is rhythmic movement of parts of the body. There are three clinical types: rest tremor (worst at rest), postural tremor (worst in a fixed posture, e.g. outstretched arms) and intention tremor (worst during voluntary movement). Tremor may be noticed by the GP during an assessment for some other problem, or it may be the presenting symptom. In the latter case, the patient may be embarrassed by the lack of 'self-control', so a sympathetic approach is important.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  anxiety\n\n  thyrotoxicosis\n\n  drug withdrawal (e.g. opiates, benzodiazepines, alcohol)\n\n  benign essential tremor (familial)\n\n  Parkinson's disease\n\n**OCCASIONAL**\n\n  adverse drug reaction (e.g. phenothiazines, \u03b2-agonists)\n\n  carbon dioxide retention (e.g. COPD)\n\n  multiple sclerosis (MS)\n\n  cerebellar ataxia: many causes, including tumour, acoustic neuroma, Friedreich's ataxia, CVA, abscess\n\n**RARE**\n\n  fulminant hepatic failure\n\n  Wilson's disease\n\n  tertiary syphilis\n\n  hysterical: usually restricted to one limb and is very gross\n\n  meningoencephalitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ TFT.\n\n_POSSIBLE :_ FBC, LFT.\n\n_SMALL PRINT :_ syphilis serology and, in secondary care, MRI scan, lumbar puncture, visual evoked response, serum caeruloplasmin\/blood copper, blood gases.\n\n  FBC: macrocytosis in chronic alcohol excess.\n\n  LFT: for evidence of alcohol abuse or liver failure.\n\n  TFT: if hyperthyroidism suspected.\n\n  MRI most sensitive test for picking up CNS demyelination and tumours (e.g. cerebellar).\n\n  Lumbar puncture: CSF electrophoresis may show oligoclonal bands in MS, or evidence of meningoencephalitis.\n\n  Visual evoked response: prolonged in MS.\n\n  Syphilis serology: in suspected syphilis.\n\n  Blood gases: will reveal carbon dioxide retention.\n\n  Serum caeruloplasmin\/blood copper: to diagnose Wilson's disease.\n\n##### TOP TIPS\n\n  Patients who present with their tremor are invariably worried about significant disease, especially Parkinson's disease. Ensure that these anxieties are resolved during the consultation.\n\n  Essential tremor is characteristically suppressed by a small dose of alcohol. This can be a useful pointer from the history.\n\n  A tremor can have more than one cause and may not necessarily follow the neat patterns described, especially in the elderly.\n\n  The tremor of early Parkinson's disease usually causes the patient few problems. It may therefore be noticed by the doctor when the patient presents with other symptoms, or be presented by concerned relatives or friends.\n\n  Have a low threshold for arranging TFT: anxiety can closely mimic thyrotoxicosis and vice versa.\n\n  Parkinson's disease may well present with a consultation for frequent falls. Look for other signs such as mask face, small handwriting, slow movement, festinant gait and difficulty rising from chair.\n\n  Think of alcohol problems in isolated middle-aged and elderly men developing postural tremor.\n\n  Intention tremor with nystagmus or dysarthria suggests significant cerebellar pathology.\n\n### WEIGHT GAIN\n\n#### The GP overview\n\nBy far the commonest cause of this problem is simple obesity. This is 50% commoner in women, who also present more often than men. It may be the primary problem presented or it may be a 'while I'm here' symptom \u2013 either way, it tends to be viewed as a 'heartsink' symptom, as such patients may have unrealistic expectations of what the doctor can achieve.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  simple obesity (usually hereditary component with poor diet and lack of exercise)\n\n  hypothyroidism\n\n  pregnancy\n\n  oedema of any cause (e.g. cardiac failure, renal failure, hepatic failure)\n\n  alcohol excess\n\n**OCCASIONAL**\n\n  menopause\n\n  iatrogenic (e.g. steroids, insulin, sulphonylureas, oestrogen, pizotifen)\n\n  polycystic ovary syndrome\n\n  large single ovarian cyst\n\n  physical or mental disability restricting activity (e.g. CVA, Down's syndrome)\n\n**RARE**\n\n  anabolic steroid abuse\n\n  Cushing's syndrome\n\n  hypothalamic lesion or hypopituitarism\n\n  insulinoma\n\n  rare genetic syndromes such as Prader\u2013Willi syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ TFT.\n\n_POSSIBLE :_ urinalysis, FBC, LFT, U&E, pregnancy test, cardiac investigations if the cause is possible cardiac failure, pelvic ultrasound.\n\n_SMALL PRINT :_ secondary care investigations for endocrine dysfunction.\n\n  Urinalysis: proteinuria may be present in oedema caused by underlying renal disease.\n\n  FBC: anaemia may precipitate cardiac failure; MCV raised in hypothyroidism and alcohol excess.\n\n  TFT: will reveal hypothyroidism.\n\n  U&E: deranged in renal failure and Cushing's syndrome.\n\n  LFT: to check for liver failure, evidence of alcohol abuse and hypoproteinaemic states.\n\n  Pregnancy test: to exclude pregnancy if this is a possibility.\n\n  Pelvic ultrasound: will confirm ascites, ovarian cysts and pregnancy.\n\n  Cardiac investigations: CXR, ECG, BNP and echocardiography if the underlying cause is possible cardiac failure.\n\n  Secondary care investigations for endocrine dysfunction: for insulinoma, hypopituitarism, Cushing's syndrome.\n\n##### TOP TIPS\n\n  Patients presenting with weight gain, especially if it is a long-term or fluctuating problem, usually have a clear agenda of their own \u2013 such as wanting a blood test or drug treatment. Much time can be wasted giving unwanted advice about diet and exercise which the patient will have heard before. Establish the patient's agenda early in the consultation.\n\n  Parents anxious about weight gain in their children often want to be reassured that the child's 'glands' are normal. If the child's height is in proportion to its weight, or the parents have a similar physiognomy, and the child is otherwise well, it is highly unlikely that there is an underlying cause.\n\n  Patients with Down's syndrome have a relatively high prevalence of hypothyroidism \u2013 consider this possibility if a patient with Down's syndrome experiences inexplicable weight gain.\n\n  Investigation is usually clinically unnecessary, but may be useful to clear the way to dealing with the real cause.\n\n  Correction of hypothyroidism may not solve a weight gain problem, merely unmask an underlying simple obesity. Don't falsely raise expectations.\n\n  Establish whether the weight gain is diffuse or whether there is predominantly abdominal swelling. The latter may indicate pregnancy, ascites or a large ovarian cyst.\n\n  Recent onset of weight gain in an elderly patient suggests probable congestive cardiac failure or possible hypothyroidism.\n\n  Young women may not consider, or accept, a diagnosis of pregnancy as a cause of their weight gain. Arrange a pregnancy test if in any doubt.\n\n  The yield from routine blood tests is low \u2013 but consider investigation if the patient is unwell or the history atypical.\n\n### WEIGHT LOSS\n\n#### The GP overview\n\nThe significance of weight loss should not be underestimated: in about one-third of patients, there is no specific cause, but in the rest, a serious underlying pathology is found. The minority of these are psychiatric; 90% have organic illness. Thorough assessment from the start is the rule.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  'normal' stressful life events without psychiatric illness (e.g. changing job, divorce, redundancy, bereavement, exam pressure)\n\n  clinical depressive illness\n\n  eating disorders: anorexia nervosa and bulimia nervosa\n\n  hyperthyroidism: thyrotoxicosis and iatrogenic (excess thyroid replacement)\n\n  malignancy anywhere\n\n**OCCASIONAL**\n\n  anxiety (of whatever cause)\n\n  any terminal illness (e.g. malignancy, motor neurone disease)\n\n  substance misuse: alcohol, opiates, amphetamines, laxatives\n\n  organ failure: cardiac, renal and hepatic\n\n  undiagnosed diabetes mellitus\n\n  chronic inflammatory conditions (e.g. RA, SLE)\n\n  gastrointestinal disease (e.g. peptic ulcer, inflammatory bowel disease, coeliac disease, parasites)\n\n**RARE**\n\n  any chronic infection (especially tuberculosis)\n\n  endocrine: Addison's disease, phaeochromocytoma\n\n  AIDS\n\n  malnutrition (rare in developed countries, common worldwide)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC, ESR\/CRP, TFT, U&E, LFT, CXR, blood glucose or HbA1c.\n\n_POSSIBLE :_ autoimmune screen, HIV test, bowel investigations such as faecal calprotectin or anti-endomysial and anti-gliadin antibodies, other hospital-based investigations.\n\n_SMALL PRINT :_ none.\n\n  Urinalysis: for possible undiagnosed diabetes; proteinuria in renal failure.\n\n  FBC and ESR\/CRP: Hb may be reduced and ESR\/CRP elevated in malignancy and any chronic disorder.\n\n  U&E: abnormal in renal failure and sometimes in eating disorders; sodium reduced, potassium and urea elevated in Addison's disease.\n\n  TFT: will confirm hyperthyroidism.\n\n  LFT: deranged in alcoholism and liver disease.\n\n  Blood glucose or HbA1c: to confirm diabetes.\n\n  Autoimmune screen: may be helpful in suspected connective tissue disorder.\n\n  HIV test: if AIDS suspected.\n\n  Bowel investigations such as faecal calprotectin or anti-endomysial and anti-gliadin antibodies: if IBD or coeliac disease suspected.\n\n  CXR: may reveal carcinoma, TB, lymphadenopathy or cardiac failure.\n\n  Other investigations (usually hospital-based) may be required according to the symptoms accompanying the weight loss and the results of preliminary investigations.\n\n##### TOP TIPS\n\n  Weight loss needs to be taken seriously but can be complex and time-consuming to assess. If presented as a 'by the way' at the end of a consultation for some other matter, it is reasonable to reverse the normal approach by arranging basic blood tests and urinalysis first, and booking a follow-up appointment, with the results, for a more complete assessment.\n\n  Establish whether episodes of weight loss have happened before. Patients, or their records, may indicate, for example, that they always lose weight when stressed.\n\n  Check that the patient really has lost weight. The history may not be clear, and there is often a record available (e.g. new patient check or health promotion data) of previous weight.\n\n  Look at the patient. The obviously cachectic will have significant disease and require urgent and thorough investigation.\n\n  Rapid weight loss with malaise and respiratory or gastrointestinal symptoms strongly suggest a physical cause.\n\n  Think of eating disorders in young females: look for acid dental erosion on palatal surfaces of upper teeth as a giveaway sign of recurrent vomiting.\n\n  Weight loss in a child is caused either by significant organic pathology or emotional abuse. Look out for signs of non-accidental injury (NAI) during physical examination.\n\n  Depression with weight loss is a difficult problem; it may be the primary cause or the change in mood may be secondary to some physical illness. Either way, don't overlook significant depression while you arrange investigations; there is nothing to be lost in starting antidepressants while you continue to exclude a physical cause, so long as you explain your strategy to the patient.\n\n## GENITAL\n\nErectile dysfunction\n\nHaemospermia\n\nPainful intercourse\n\nPenile pain\n\nPenile ulceration\/sores\n\nScrotal swelling\n\nTesticular pain\n\nVaginal discharge\n\nVulval irritation\n\nVulval swelling\n\nVulval ulceration\/sores\n\n### ERECTILE DYSFUNCTION\n\n#### The GP overview\n\nThis is the partial or complete failure to achieve a satisfactory erection. The inability to ejaculate (ejaculatory erectile dysfunction) is not dealt with here. It presents fairly often to GPs and will probably do so increasingly frequently as new treatments are developed and publicised.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  depression\/anxiety\n\n  excessive alcohol intake\n\n  relationship dysfunction\n\n  vascular: arterial insufficiency (arteriopathy) or excessive venous drainage\n\n  iatrogenic (e.g. prostatic cancer treatments, hypotensives, some antidepressants)\n\n**OCCASIONAL**\n\n  testosterone deficiency (may be primary or secondary)\n\n  diabetic autonomic neuropathy\n\n  trauma: pelvic or spinal fracture, trauma to penis, post-TURP\n\n  anatomical: phimosis, tight frenulum\n\n  excessive cigarette smoking\n\n  Peyronie's disease\n\n  drug abuse (e.g. heroin, amphetamines)\n\n**RARE**\n\n  fetishism (erection only possible with unusual stimuli)\n\n  spinal cord compression: tumour\n\n  thrombosis of a corpus cavernosum\n\n  neurological: tabes dorsalis, multiple sclerosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis.\n\n_POSSIBLE :_ blood sugar or HbA1c, FBC, LFT, endocrine assays (testosterone, prolactin, FSH\/LH, TSH), cholesterol.\n\n_SMALL PRINT :_ Doppler flow studies, angiography, intracorporeal prostaglandin injection test, MRI scanning \u2013 all likely to be specialist initiated.\n\n  Urinalysis: an essential easy screen for undiagnosed diabetes.\n\n  Blood sugar or HbA1c: to confirm diabetes.\n\n  FBC and LFT possibly helpful in alcohol excess (raised MCV and possible LFT abnormalities).\n\n  Testosterone levels reduced in primary or secondary hypogonadism. Prolactin, FSH\/LH and TSH check pituitary function. Erectile dysfunction may be a sign of cardiovascular disease \u2013 so in certain patients a cholesterol level would be warranted.\n\n  Doppler flow studies of superficial and deep penile arterial flow assess arterial sufficiency. Angiography may be necessary if symptoms suggest significant lower limb arterial insufficiency associated with impotence.\n\n  Intracorporeal prostaglandin injection test: immediate and prolonged response indicates neurological problems. Good initial response with rapid failure indicates excessive venous drainage.\n\n  Possible neurological causes will occasionally require further investigation (e.g. MRI scanning for cord lesions or MS).\n\n##### TOP TIPS\n\n  Establish whether the patient can get an erection at any time (e.g. early morning). If he can, then the cause is unlikely to be organic. Take a positive approach \u2013 many psychological causes are transient.\n\n  Don't be too quick to diagnose anxiety as the underlying problem. This may be the effect, rather than the cause, of the impotence.\n\n  Erectile dysfunction is often presented as a 'by the way' or 'while I'm here' symptom. The temptation is to invite the patient to book a further appointment, but bear in mind that this may represent a lost opportunity to help, as he may not return.\n\n  Demonstrate that you are taking the problem seriously \u2013 for example, by performing an appropriate examination or by inviting the patient's partner to attend a subsequent appointment.\n\n  Sudden onset of erectile dysfunction with saddle anaesthesia and sphincter disturbance indicates a cauda equina lesion. Refer urgently.\n\n  An erection which is consistently lost after a predictable period is likely to be organic \u2013 probably vascular \u2013 in origin.\n\n  Early morning headache and visual disturbance suggests a pituitary fossa tumour.\n\n  Do not forget that alcohol and drug abuse are possible causes. Look in the notes for clues and make specific enquiry, as these problems are unlikely to be volunteered.\n\n  Erectile dysfunction may well be a marker for vascular disease elsewhere. Extend your assessment as appropriate.\n\n### HAEMOSPERMIA\n\n#### The GP overview\n\nThis is an uncommon presentation \u2013 but one we may see increasingly frequently as men become less reticent about coming to the doctor. As with any symptom involving leakage of blood, anxiety levels tend to run high. This shouldn't transfer to the GP, though \u2013 it is unusual for the symptom to have a sinister cause.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  unknown (at least 50%; the majority are probably secondary to forgotten or unnoticed trauma)\n\n  prostatitis\n\n  post-operative (prostate surgery, biopsy or extracorporeal shock wave lithotripsy)\n\n  genito-urinary infection (epididymo-orchitis, urethritis, UTI)\n\n  trauma (to testicles or perineum)\n\n**OCCASIONAL**\n\n  blood clotting disorder or anticoagulation\n\n  calculi in the prostate\n\n  carcinoma: prostate, testicles, bladder or seminal vesicles\n\n**RARE**\n\n  tuberculosis\n\n  schistosomiasis\n\n  malignant hypertension\n\n  structural problems (such as urethral strictures or polyps)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ FBC, ESR\/CRP, PSA, urethral swab.\n\n_SMALL PRINT :_ INR, clotting screen, seminal fluid culture, transrectal ultrasound, prostate biopsy, urethroscopy.\n\n  Urinalysis: protein, nitrites, leucocytes and possible haematuria in any genito-urinary infection or prostatitis. Haematuria possible in malignancy and schistosomiasis.\n\n  MSU: to confirm infection and identify pathogen.\n\n  FBC and ESR\/CRP: WCC may be elevated in infection; Hb may be reduced and ESR\/CRP raised in malignancy; ESR\/CRP also raised in infection.\n\n  PSA: the pros and cons of this test might be discussed as a pointer to prostatic carcinoma.\n\n  Urethral swab: if urethritis suspected (best taken at GUM clinic).\n\n  INR, clotting screen: if patient on warfarin or a bleeding disorder suspected.\n\n  Other investigations (usually hospital-based): these might include seminal fluid culture to investigate deep-seated infection; transrectal ultrasound and prostatic biopsy for detailed investigation of prostate; urethroscopy\/cystoscopy if felt to be a structural urethral or bladder problem.\n\n##### TOP TIPS\n\n  A frank and open approach, using plain language, is important for the patient to feel comfortable and capable of describing an accurate history.\n\n  Do not underestimate the patient's level of anxiety \u2013 and ensure it's properly addressed. Most men with this symptom are convinced they have serious pathology.\n\n  The approach with this symptom has more to do with deciding on further action than establishing a precise diagnosis. This is because assessment in primary care rarely reveals any underlying pathology \u2013 management is more likely to be influenced by the patient's age and the history than the clinical findings (see below).\n\n  Men under the age of 40 with short-lived symptoms do not require referral, as the chance of significant pathology is miniscule. Older men \u2013 and those with persistent or recurrent haemospermia, or abnormalities on initial assessment \u2013 require referral for further assessment.\n\n  A serious underlying cause is rare but should be considered in men over the age of 40 who have more than one episode.\n\n  The chances of significant pathology are increased by the finding of microscopic haematuria \u2013 refer these cases.\n\n### PAINFUL INTERCOURSE\n\n#### The GP overview\n\nThis term is taken to apply to women. It causes much misery and may be embarrassing for a woman to discuss with her doctor. As a result, it may be the 'hidden agenda', presenting as a nonexistent 'discharge' or vague 'soreness down below'. Alternatively, it may be the underlying cause of a presentation of infertility or stress. Tact and sensitivity are the most valuable diagnostic and therapeutic tools in these situations.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  pure vaginismus: psychogenic spasm and dryness\n\n  vulvovaginitis (especially infection, e.g. bacterial or fungal vaginosis, ulceration, bartholinitis)\n\n  menopausal vaginal dryness (atrophic vaginitis)\n\n  endometriosis\n\n  pelvic inflammatory disease (PID) and cervicitis\n\n**OCCASIONAL**\n\n  post-partum perineal repair\n\n  pelvic congestion (pelvic pain syndrome)\n\n  fibroid and retroverted uterus, ovaries in pouch of Douglas\n\n  pelvic adhesions (post-surgical or PID)\n\n  cystitis, urethritis\n\n**RARE**\n\n  congenital abnormality\n\n  large ovarian cyst or tumour\n\n  vulval dysplasia\n\n  urethral caruncle\n\n  unruptured hymen\n\n  anal fissure, thrombosed piles, perianal abscess\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ high vaginal\/cervical swabs.\n\n_POSSIBLE :_ urinalysis, MSU, urethral swab, ultrasound, laparoscopy (in secondary care).\n\n_SMALL PRINT :_ FBC, ESR\/CRP, vulval biopsy (secondary care).\n\n  Urinalysis for nitrite, pus cells and blood useful to rule out UTI.\n\n  MSU will help guide treatment in UTI.\n\n  If abnormal discharge, take high vaginal and cervical swabs to establish nature of pathogen. Urethral swab useful if possible urethritis (usually at genito-urinary medicine (GUM) clinic).\n\n  FBC may show raised WCC in chronic PID.\n\n  ESR\/CRP elevated in PID.\n\n  Pelvic ultrasound can define lie of the uterus and ovaries, presence of cysts and gross endometriosis.\n\n  Investigations after referral may include laparoscopy (e.g. for endometriosis and PID) and vulval biopsy (for suspected dysplasia).\n\n##### TOP TIPS\n\n  Superficial dyspareunia (pain at the introitus) is usually caused by infection, vaginismus or atrophy; deep dyspareunia (deep pain) may be caused by pelvic pathology.\n\n  If a sexually transmitted infection could be the cause, refer to a GUM clinic: these are best equipped for thorough screening, counselling and contact tracing.\n\n  The patient complaining that her 'vagina feels too small' to accommodate her partner's penis probably has vaginismus.\n\n  Deep dyspareunia which is long-standing and positional is 'collision' dyspareunia and is very unlikely to be due to significant pathology.\n\n  Deep dyspareunia usually resolves immediately on withdrawal; if it lasts a day or two after intercourse, it may well have a psychological basis.\n\n  Relationship problems may cause dislike of intercourse which presents as pain. Disharmony may be the cause rather than result of the problem.\n\n  Cyclical dyspareunia with generalised pelvic pain and heavy, painful periods suggest endometriosis or PID: refer for gynaecological opinion.\n\n  Pelvic tumour is rare in this context, but consider this possibility in the older woman presenting with deep dyspareunia of recent onset.\n\n  Examine the menopausal or perimenopausal woman complaining of persistent superficial dyspareunia \u2013 vulval dysplasia, rather than atrophic vaginitis, may be the cause.\n\n### PENILE PAIN\n\n#### The GP overview\n\nPain in the penis occurs not just as a result of local causes, but also by referral from remote lesions. It frequently generates embarrassment for the patient, who may also be frightened that he has a sexually transmitted disease. The diagnosis will often be clear after a carefully taken history and appropriate examination.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  balanitis (fungal, bacterial or allergic)\n\n  acute urethritis\n\n  phimosis (e.g. balanitis xerotica obliterans)\n\n  urinary calculus (at any point in ureter or urethra)\n\n  prostatitis\/prostatic abscess\n\n**OCCASIONAL**\n\n  herpes simplex (and rarely zoster)\n\n  carcinoma of bladder or prostate\n\n  trauma: torn frenulum, zipper injury, urethral injury or foreign body\n\n  acute cystitis\n\n  Peyronie's disease (pain usually on erection)\n\n  paraphimosis\n\n  tight frenulum\n\n**RARE**\n\n  anal fissure\/inflamed haemorrhoid\n\n  carcinoma of penis\n\n  carcinoma of rectum\/anus\n\n  tuberculosis of urinary tract\n\n  schistosomiasis ( _Schistosoma haematobium_ ): common in Africa and the Middle East\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU, swabs.\n\n_POSSIBLE :_ FBC, ESR\/CRP, PSA.\n\n_SMALL PRINT :_ IVU, cystoscopy, terminal stream urine.\n\n  Urinalysis: may reveal proteinuria, haematuria, pus cells and nitrites in the presence of infection; haematuria alone with a stone. Will also reveal glycosuria in the previously undiagnosed diabetic (may present with candidal balanitis).\n\n  MSU (for MC&S): to establish pathogen in UTI (may also reveal infective agent in prostatitis).\n\n  Swabs for microbiology: urethral swab if urethritis likely (best performed at GUM clinic). In balanitis with discharge, a swab may help guide treatment.\n\n  FBC and ESR\/CRP: WCC and ESR\/CRP raised in significant infection and inflammation (e.g. prostatitis or prostatic abscess). ESR\/CRP may be raised in malignancy.\n\n  PSA: consider this test if carcinoma of the prostate a possibility.\n\n  IVU more useful than ultrasound to investigate the urinary tract if stone or carcinoma suspected, or if chronic UTI suspected.\n\n  Terminal stream urine: for schistosomiasis.\n\n  Cystoscopy: may be required in secondary care to confirm and treat stone or tumour.\n\n##### TOP TIPS\n\n  The man who has symptoms suggesting cystitis but who has sterile pyuria on MSU probably has urethritis.\n\n  GUM clinics are organised to undertake full investigation, counselling and contact tracing. Referral is essential if STD is likely.\n\n  Prostatitis is often forgotten as a diagnosis \u2013 but is very difficult to diagnose with certainty, especially when chronic. A trial of a prolonged course of antibiotics may be justified.\n\n  Painful intercourse \u2013 usually a sudden pain \u2013 accompanied by bleeding suggests a torn frenulum. This often occurs in a younger man who is frequently very alarmed by the event. Reassure him explaining that this is not sinister and is easily treatable.\n\n  Pain after micturition suggests cystitis. This is unusual in men, and further investigation is indicated if recurrent.\n\n  Intermittent pain with passage of blood clots interspersed with painless haematuria suggests a carcinoma (bladder, ureter (rare) or kidney).\n\n  Remember that candidal balanitis may be the first sign of diabetes.\n\n  Refer the elderly man with an adherent foreskin and balanitis. There could be an underlying carcinoma.\n\n### PENILE ULCERATION\/SORES\n\n#### The GP overview\n\nPresentation of this symptom is nearly always accompanied by fear of sexually transmitted disease, even in elderly or no longer sexually active men. There are a number of other causes, many of which are significant and require investigation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  herpes simplex virus (HSV)\n\n  boil\/infected sebaceous cyst\n\n  balanitis: bacterial or fungal\n\n  trauma: zipper injury commonest; also torn frenulum, bites, self-mutilation\n\n  balanitis xerotica obliterans (BXO)\n\n**OCCASIONAL**\n\n  herpes zoster\n\n  Reiter's syndrome: circinate balanitis\n\n  allergic contact eczema\n\n  chancroid (soft sore: _Haemophilus ducreyi_ )\n\n  granuloma inguinale ( _Klebsiella granulomatis:_ tropical infection)\n\n  lymphogranuloma venereum (tropical infection)\n\n**RARE**\n\n  syphilis (chancre)\n\n  carcinoma of the penis\n\n  tuberculosis\n\n  dermatological conditions, e.g. Beh\u00e7et's syndrome, lichen planus\n\n  fixed drug eruption\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ swab, syphilis serology.\n\n_POSSIBLE :_ urinalysis, FBC, ESR\/CRP.\n\n_SMALL PRINT :_ patch testing, biopsy.\n\n  Urinalysis: in balanitis, may detect undiagnosed diabetes.\n\n  Swab: may reveal infectious cause, e.g. herpes simplex, _Candida_ , chancroid, lymphogranuloma venereum and granuloma inguinale (if STD suspected, then other appropriate swabs and blood tests for coexistent disease will be performed at GUM clinic).\n\n  FBC and ESR\/CRP: raised WCC and ESR\/CRP in significant infection or inflammation (e.g. Reiter's syndrome).\n\n  Syphilis serology: if syphilis suspected ( _Note:_ may take up to 3 months to become positive after initial infection).\n\n  Patch testing: if allergic contact eczema a possibility.\n\n  Biopsy (in secondary care): to confirm suspected malignancy or reveal underlying skin condition (e.g. lichen planus).\n\n##### TOP TIPS\n\n  Take a full sexual history, even in the older patient. If STD is suspected, refer to a GUM clinic for investigation, counselling and contact tracing.\n\n  A diagnosis of HSV may induce a number of worries in the patient, some of them well founded, others less so. Give the patient plenty of time to talk through the diagnosis and its implications properly.\n\n  Whatever the cause, the patient is very likely to fear a STD. Ensure that inappropriate anxieties are resolved.\n\n  Enquire after coexistent or previous dermatological problems in obscure cases \u2013 this may provide the diagnosis (e.g. lichen planus).\n\n  A history of travel or sexual contact with travellers is important: a number of the more obscure causes are 'tropical'.\n\n  Take a sexual history: syphilis is rare generally but is more common in homosexuals.\n\n  Balanitis and urethritis, arthritis and conjunctivitis form the triad of Reiter's syndrome. Always make a thorough general systemic enquiry.\n\n  A single, unexplained, persistent ulcer needs thorough investigation as significant disease (infection or malignancy) is likely.\n\n  Remember the possibility of underlying diabetes in severe or recurrent candidal balanitis.\n\n### SCROTAL SWELLING\n\n#### The GP overview\n\nScrotal swellings can occur at any age. They arise most commonly from the testicle and its coverings, the spermatic cord and the skin. Greater awareness of testicular cancer has resulted in increasing numbers of young men attending the GP, usually with benign lumps.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  inguinal hernia\n\n  sebaceous cyst\n\n  hydrocoele\n\n  epididymal cyst\n\n  epididymo-orchitis (EO)\n\n**OCCASIONAL**\n\n  torsion of the testis\n\n  iatrogenic sepsis: surgery and catheterisation\n\n  haematocoele\n\n  varicocoele\n\n  congestive heart failure\n\n  post-vasectomy swelling (aseptic), including haematoma, inflammatory reaction to spilt sperm\n\n  trauma \u2013 haematoma\n\n**RARE**\n\n  testicular tumour (seminoma, teratoma)\n\n  ascites\n\n  inferior vena caval thrombosis\n\n  tuberculosis and syphilis (not rare abroad)\n\n  elephantiasis (filariasis)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, MSU, ultrasound.\n\n_SMALL PRINT :_ FBC, U&E, LFT, CXR, urethral swab, AFP and \u03b2-HCG.\n\n  Urinalysis may show signs of UTI, but may be negative in epididymo-orchitis, as may MSU.\n\n  If urethra discharging take urethral swab for gonococcus and _Chlamydia._\n\n  Ultrasound useful to distinguish solid from cystic swelling.\n\n  FBC, AFP and \u03b2-HCG essential baseline investigations if solid tumour suspected \u2013 would be arranged by the specialist after referral.\n\n  CXR may show cannonball metastases if carcinoma has spread.\n\n  May require further investigations such as U&E, LFT, syphilis serology if underlying pathology (e.g. ascites, cardiac failure, syphilis) suspected.\n\n##### TOP TIPS\n\n  Don't forget that the patient's main fear is likely to be cancer. Broach this even if the swelling is obviously benign.\n\n  Examine the patient standing. Many lumps are easier to feel this way and some \u2013 especially varicocoeles \u2013 may disappear on lying down.\n\n  In the older patient, with bilateral swelling, there is likely to be some underlying disease process such as cardiac failure.\n\n  Consider arranging an ultrasound if a patient remains very anxious about an obviously cystic swelling, or if you have any doubt yourself \u2013 a normal result will relieve both parties.\n\n  Seminoma may often feel smooth and mimic a large normal testis. Do not rely on the absence of clinical features of malignancy \u2013 if the patient feels there has been a change in the testis, act on it.\n\n  It can be difficult to distinguish between hernias and hydrocoeles in babies. Hernias require surgical attention; hydrocoeles may resolve within the first year of life. If in doubt, refer.\n\n  Remember that a hydrocoele may be caused by an underlying malignancy; in younger patients, always refer, while in the elderly, examine the testis carefully after aspiration.\n\n  Left supraclavicular nodes may be involved after tumour spread to para-aortic nodes (can be massive). Examine the abdomen and chest if any suspicion of malignancy.\n\n  If any suspicion of torsion \u2013 admit urgently.\n\n### TESTICULAR PAIN\n\n#### The GP overview\n\nThis is an uncommon symptom in everyday general practice. Though commonest in the young adult, it can affect all age groups. In its acute form, it is excruciating and disabling. In the chronic form it is usually described as a dull ache or dragging sensation. It is the former which causes the GP the most diagnostic difficulty and anxiety.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute orchitis (mumps, and less commonly scarlet fever and flu)\n\n  acute epididymo-orchitis (EO) (UTI and sexually transmitted infection)\n\n  torsion of the testis\n\n  epididymal cyst\n\n  referred from ureteric stone\n\n**OCCASIONAL**\n\n  varicocoele\n\n  haematocoele\n\n  hydrocoele\n\n  trauma (fractured testis)\n\n  undescended or misplaced testis\n\n  torsion of the appendix testis\n\n  post-vasectomy pain\n\n  idiopathic chronic testicular pain (accounts for 25% of chronic cases)\n\n**RARE**\n\n  testicular carcinoma (teratoma and seminoma)\n\n  incarcerated or strangulated inguinoscrotal hernia\n\n  syphilis\n\n  referred from spinal tumours\n\n  neuralgia testis\n\n  tuberculosis of the testis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ urethral swab, ultrasound.\n\n_SMALL PRINT :_ lumbosacral spine and abdominal X-rays, syphilis serology.\n\n  Urinalysis: protein, blood and pus cells in EO. Blood alone with stone.\n\n  MSU: will identify UTI.\n\n  Urethral swab for gonococcus and _Chlamydia_ necessary if STD suspected.\n\n  Plain lumbosacral spine and abdominal X-rays are valuable to investigate referred testicular pain (stones and spinal pathology).\n\n  Ultrasound is good at 'seeing' if a testicular mass arises from the body of the testis or its coverings, and whether solid or not.\n\n  Syphilis serology: if syphilis suspected.\n\n##### TOP TIPS\n\n  In an adult, relief of pain by elevating the testicle suggests epididymitis.\n\n  A negative urinalysis does not exclude epididymitis.\n\n  In mild, chronic testicular ache, examine the patient standing up, otherwise you may miss a varicocoele.\n\n  A sudden onset of excruciating pain associated with nausea suggests torsion of testis \u2013 especially in children and adolescents. Admit immediately.\n\n  Repeated episodes of spontaneously resolving pain may represent recurrent, self-correcting torsion. Refer for possible orchidopexy and warn the patient to report urgently if there is severe and persisting pain.\n\n  If non-gonococcal\/chlamydial epididymitis is clinically suspected, treat immediately with a broad-spectrum antibiotic.\n\n  If epididymitis does not settle with antibiotics, consider abscess formation \u2013 admit for IV antibiotics or surgical drainage.\n\n### VAGINAL DISCHARGE\n\n#### The GP overview\n\nVaginal discharge is usually a symptom of the reproductive years, but can occur at any age. It can be influenced by the menstrual cycle, use of 'the pill', age, pregnancy and sexual activity. Treatment is often simple, but if it fails, or if there are risk factors for STDs, it is sensible to refer to a GUM clinic.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  excessive normal secretions\n\n  thrush\n\n  bacterial vaginosis (BV)\n\n  trichomonal vaginosis (TV)\n\n  cervicitis (gonococcus, _Chlamydia_ , herpes)\n\n**OCCASIONAL**\n\n  cervical ectropion\n\n  cervical polyp\n\n  lost tampon, ring pessary or other foreign body\n\n  IUCD\n\n  bartholinitis\n\n  salpingitis\n\n**RARE**\n\n  vulvovaginal neoplasia\n\n  cervical or uterine neoplasia\n\n  sloughing intrauterine fibroid\n\n  pyometra\n\n  pelvic fistula\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ high vaginal swab (HVS).\n\n_POSSIBLE :_ endocervical swab, urethral swab, urine testing for _Chlamydia_ , blood sugar or HbA1c.\n\n_SMALL PRINT :_ other specialist investigations.\n\nMost GPs would confine themselves to the HVS, endocervical swab and urine test. Those with a special interest might undertake the microscopy themselves.\n\n  HVS is simple and readily detects _Candida,_ BV and TV.\n\n  Wet saline microscopy shows clue cells in BV, motile trichomonads in TV.\n\n  Gram stain of cervical or urethral exudate shows negatively staining diplococci in up to 85% of gonococcal infections.\n\n  Endocervical swab for ELISA is the gold standard for detecting _Chlamydia._\n\n  DNA amplification testing of first-catch urine (not MSU) specimens for _Chlamydia_ is noninvasive and relatively acceptable to patients.\n\n  Blood sugar or HbA1c: to detect diabetes in severe or recurrent thrush.\n\n  Specialist investigations might include D&C or hysteroscopy (for possible malignancy) and barium enema (for pelvic fistula).\n\n##### TOP TIPS\n\n  It is reasonable to diagnose and treat thrush empirically in a woman with classical symptoms who has had the problem before \u2013 many women successfully self-medicate and only attend to obtain their treatment free, via a prescription. But if in any doubt about the diagnosis, examine and investigate as appropriate.\n\n  It is worth investing time with the patient suffering confirmed recurrent thrush \u2013 advice supplemented by written patient information may help minimise future problems.\n\n  Make sure you have all the appropriate swabs (HVS, endocervical, urethral) to hand \u2013 you never know when you might need them.\n\n  Excessive concern about normal secretions might mask a sexual problem or worry \u2013 enquire discreetly about this.\n\n  Recurrent or florid thrush may be a presentation of undiagnosed diabetes mellitus. Ask about thirst, polyuria and tiredness and check a fasting glucose if any suspicion of underlying diabetes, or there is a positive family history of diabetes.\n\n  Vaginal discharge is an uncommon symptom before puberty. Don't forget the possibilities of abuse or a foreign body.\n\n  Always conduct a full pelvic examination in the post-menopausal woman with vaginal discharge. Malignancy is one of the likeliest causes.\n\n  A florid erosion is likely to be caused by chlamydial cervicitis \u2013 take a swab and treat appropriately.\n\n  If you suspect a sexually transmitted disease, refer to the GUM clinic for full assessment and contact tracing. Refer urgently to the GUM clinic or duty gynaecologist if there are systemic flu-like symptoms and fever with pelvic pain and vaginal discharge.\n\n### VULVAL IRRITATION\n\n#### The GP overview\n\nVulval irritation encompasses soreness and itch and is a very common presentation in primary care. Embarrassment may mean a 'calling card' symptom has first to be presented. Sometimes it is a 'calling card' itself, being easier to talk about than a psychosexual problem. With sensitivity, the real issues should emerge during the consultation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  thrush: _Candida_ infection\n\n  _Trichomonas vaginalis_\n\n  chemical: bubble baths, detergents, 'feminine hygiene' douches\n\n  trauma: insufficient lubrication during intercourse\n\n  atrophic vaginitis\n\n**OCCASIONAL**\n\n  ammoniacal vulvitis from incontinence\n\n  skin disorders (e.g. eczema, psoriasis, lichen planus)\n\n  infestations: threadworms, pubic lice\n\n  psychosexual problems\n\n  other infections (e.g. genital warts or herpes)\n\n**RARE**\n\n  diabetes (without _Candida_ infection)\n\n  vulval dysplasia (various other terms for this include lichen sclerosis et atrophicus, leukoplakia)\n\n  vulval carcinoma\n\n  general disorder causing pruritus (e.g. jaundice, leukaemia, chronic renal failure, lymphoma)\n\n  psychogenic (no organic or psychosexual problem)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ HVS (if discharge present).\n\n_POSSIBLE :_ urinalysis.\n\n_SMALL PRINT :_ FBC, LFT, U&E, fasting sugar or HbA1c, vulval biopsy.\n\n  Urinalysis for sugar: diabetes predisposes to thrush and glycosuria in itself can cause vulvitis.\n\n  FBC, LFT, U&E: if vulvitis is part of a generalised pruritus, or if the patient is generally unwell, these blood tests may reveal blood dyscrasias or renal or hepatic dysfunction.\n\n  Fasting sugar or HbA1c to diagnose or rule out underlying diabetes.\n\n  HVS: identifying the pathogen if discharge is present will help management in puzzling or recurrent cases.\n\n  Vulval biopsy (secondary care): multiple biopsies are required if vulval dysplasia or carcinoma are suspected.\n\n##### TOP TIPS\n\n  It is easy to make an erroneous diagnosis of UTI in a patient with vulvitis: external dysuria and contamination of urine with pus cells and blood (especially if there is an associated discharge) may mislead the unwary. Helpful pointers are the presence of external vulval irritation and the absence of frequency or urgency.\n\n  In obscure cases, check the skin elsewhere. Vulval irritation may be a manifestation of a primary skin disorder, such as eczema or psoriasis.\n\n  The aetiology may be multifactorial with, for example, some primary cause leading to secondary chemical irritation from over-washing or the use of douches. A careful history is needed to unravel the underlying cause and exacerbating factors.\n\n  Recurrent candidal infection is a particular problem. Various therapeutic strategies are available, but it is important to take time to explore the woman's perception of the cause, explain the diagnosis and resolve any exacerbating factors.\n\n  Post-menopausal atrophic vaginitis causes soreness rather than itch. Dysplasias and some carcinomas produce intense irritation. Examine these patients and, if in doubt, refer for biopsy.\n\n  Consider diabetes in florid or refractory cases of _Candida_ infection.\n\n  Significant psychosexual problems may present with vulval irritation. Adopt a sympathetic, open approach. Take particular note of any comments made during the physical examination as this sometimes prompts the patient to reveal the true problem.\n\n  Persistent vulval irritation may rarely be a symptom of significant systemic disease. Consider this if the patient has generalised pruritus elsewhere and seems unwell in herself.\n\n  If the cause is sexually transmitted (e.g. genital herpes or warts), exclude other infections by referring to a GUM clinic.\n\n### VULVAL SWELLING\n\n#### The GP overview\n\nVulval swellings may originate in the vulva, or appear there after displacement from their origin. They often present as 'a lump down below' \u2013 an expression which belies the variety of possible causes. They generate a lot of anxiety but are rarely sinister.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  boils\n\n  sebaceous cysts\n\n  viral warts (condylomata acuminata)\n\n  Bartholin's cyst\n\n  inguinal hernia (may extend down to labium major)\n\n**OCCASIONAL**\n\n  varicose vein, varicocoele of vulva\n\n  Bartholin's abscess (infected Bartholin's cyst)\n\n  fibroma, lipoma, hidradenoma\n\n  uterine prolapse, cystocoele, rectocoele, enterocoele (hernia of the pouch of Douglas)\n\n  urethral caruncle (meatal prolapse)\n\n**RARE**\n\n  cervical polyp\n\n  carcinoma (95% are squamous)\n\n  endometrioma\n\n  hydrocoele of the canal of Nuck\n\n  traumatic haematoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are no investigations likely to be performed in primary care: the diagnosis is almost always established by history and examination. If it isn't, then referral is usually required.\n\n##### TOP TIPS\n\n  Remember that, to many patients, a lump means cancer until proven otherwise. You may only require a cursory glance to reassure yourself that the problem is insignificant \u2013 but the patient may interpret your approach as dismissive or inadequate. Ensure that the patient's anxieties are resolved by adequate examination and explanation.\n\n  If the lump is not obviously apparent, or is poorly defined, examine the patient standing: this may reveal a hernia, varicocoele or prolapse.\n\n  A varicocoele of the vulva has a characteristic 'bag of worms' feel. It often appears and gets worse during pregnancy.\n\n  A persistent, ulcerating lump in the vulva must always be referred for biopsy to exclude carcinoma, even though some benign lumps can ulcerate (e.g. hidradenoma).\n\n  Check for lymphadenopathy: hard inguinal nodes with a painless lump are highly suggestive of malignancy. The lump can occasionally be a metastasis itself.\n\n  Women with genital warts may have coexisting sexually transmitted infection: refer to the local GUM clinic for appropriate investigation and, if necessary, contact tracing.\n\n### VULVAL ULCERATION\/SORES\n\n#### The GP overview\n\nThis often unpleasant symptom is uncommon, but very important, as many causes are significant and require specialist investigation, treatment and follow-up. Patients often have difficulty visualising or describing these types of lesions, so adequate examination is vital in establishing the diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  herpes simplex virus (HSV)\n\n  thrush (particularly if very excoriated)\n\n  vulval dysplasia\n\n  squamous cell carcinoma (SCC): 95% of vulval malignancies\n\n  excoriated scabies\n\n**OCCASIONAL**\n\n  allergic contact eczema\n\n  chancroid: _H. ducreyi_ (tropical)\n\n  granuloma inguinale: _K. granulomatis_ (tropical)\n\n  lymphogranuloma venereum: _Chlamydia trachomatis_ (tropical)\n\n  other malignancies (e.g. BCC, melanoma, adenocarcinoma, sarcoma)\n\n  herpes zoster\n\n**RARE**\n\n  syphilis and yaws\n\n  Beh\u00e7et's syndrome\n\n  tuberculosis\n\n  fixed drug eruption\n\n  dermatological disorders (e.g. pemphigus and pemphigoid)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, swab.\n\n_POSSIBLE :_ fasting sugar or HbA1c, and (in secondary care) biopsy.\n\n_SMALL PRINT :_ syphilis serology, patch testing.\n\n  Urinalysis: glycosuria may be present in undiagnosed diabetes presenting with severe or recurrent _Candida_ infection. Fasting sugar or HbA1c to rule out or diagnose undiagnosed diabetes.\n\n  Swab for microscopy and culture: may help in the diagnosis of various infections such as HSV, _Candida_ , chancroid, granuloma inguinale and lymphogranuloma venereum (if STD suspected, then other relevant swabs and blood tests for coexistent infection will be performed at GUM clinic).\n\n  Syphilis serology: if syphilis a possibility ( _Note:_ serology may not become positive for up to 3 months after infection).\n\n  Patch testing: may help in the diagnosis of allergic contact eczema.\n\n  Biopsy (in secondary care): for any persistent ulcer to confirm diagnosis \u2013 may reveal carcinoma, vulval dysplasia or underlying skin disorder.\n\n##### TOP TIPS\n\n  A diagnosis of HSV can be traumatic for a woman. Spend time discussing the nature of the problem and its recurrent nature, including implications for sexual partners and future pregnancies.\n\n  If the patient suffers recurrent vulval ulceration, offer to see her as an 'urgent' during the next attack to visualise the lesions and arrange microbiological testing.\n\n  In obscure cases do not confine the history and examination to the vulva. Lesions elsewhere (e.g. with pemphigus or Beh\u00e7et's syndrome) may give the clue needed to make the diagnosis.\n\n  History of travel and sexual contact with travellers is very important as there are a number of 'tropical' causes.\n\n  The more straightforward causes (HSV and severe excoriated thrush) usually result in multiple ulcers, with the diagnosis being obvious from the history and examination. Take very seriously any single persistent vulval sore: significant disease is likely.\n\n  If you suspect a sexually transmitted disease, refer urgently to the local GUM clinic for appropriate investigations and contact tracing.\n\n  Remember the possibility of undiagnosed diabetes in severe _Candida_ infection.\n\n  The pregnant woman near term with primary HSV is in danger of transmitting the virus to her newborn \u2013 a situation with a significant mortality and morbidity. Contact the obstetrician urgently to arrange probable elective Caesarian section.\n\n## HAIR AND NAILS\n\nAbnormal nails\n\nExcess body hair\n\nHair loss\n\nItchy scalp\n\n### ABNORMAL NAILS\n\n#### The GP overview\n\nWomen are far more likely to present with this symptom than men, often for cosmetic reasons. The likeliest queries relate to normal nails that break easily or are not smooth. Other abnormalities may be detected during physical examination rather than being volunteered by the patient, and may indicate significant pathology.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  psoriasis\n\n  fungal infection: onychomycosis\n\n  trauma to nail bed\n\n  trauma due to biting (also hang nail)\n\n  onychogryphosis (OG)\n\n**OCCASIONAL**\n\n  trophic changes (Beau's lines) \u2013 appear 2\u20133 months after severe illness\n\n  hand eczema\n\n  longitudinal ridging (onychorrhexis)\n\n  chronic paronychia\n\n  clubbing (various causes)\n\n  autoimmune disease (e.g. alopecia areata, lichen planus)\n\n  koilonychia (spoon nails): iron deficiency anaemia and Plummer\u2013Vinson syndrome\n\n**RARE**\n\n  subungual melanoma\n\n  leuconychia (e.g. liver disease, diabetes)\n\n  lamellar nail dystrophy\n\n  yellow nail syndrome (may be accompanied by chronic bronchitis, bronchiectasis or pleural effusion)\n\n  exfoliative dermatitis: nail shedding\n\n  nail dystrophy due to poor circulation (e.g. Raynaud's disease)\n\n  epidermolysis bullosa\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ nail clippings for mycology.\n\n_POSSIBLE :_ urinalysis, FBC.\n\n_SMALL PRINT :_ LFT, CXR.\n\n  Nail clippings for mycology may be the only way to differentiate psoriatic nail dystrophy and onychomycosis.\n\n  Urinalysis worthwhile if nails are unusually white: this can occur in diabetes.\n\n  FBC may confirm iron deficiency anaemia in koilonychia.\n\n  LFT: to assess liver function in leuconychia.\n\n  CXR worthwhile if chest symptoms with clubbing or yellow nails.\n\n##### TOP TIPS\n\nDon't confine your examination to the nails \u2013 useful clues may be found elsewhere, e.g. patches of psoriasis or coexisting tinea corporis.\n\n  The commonest differentials are psoriasis and fungal infections. The latter are usually asymmetrical.\n\n  Patients usually worry about vitamin or calcium deficiencies \u2013 these are never the real cause.\n\n  By the time Beau's lines are obvious to the patient, 3 months or so will have passed from the precipitating event \u2013 look back in the records for aetiological clues.\n\n  Subungual melanoma is rare and is easily confused with the much more common subungual haematoma. Possible pointers include nail destruction, extension of pigment onto the nail fold and longitudinal bands of pigment. If in doubt, refer.\n\n  Clubbing is really an abnormality of the fingertips; if noted, be alert to signs of major pulmonary or cardiac disease. Carcinoma of the lung is the commonest cause.\n\n  Don't assume crumbly white nails are caused by fungus. Before embarking on lengthy antifungal treatment, try to confirm the diagnosis with nail clippings.\n\n  Severely bitten nails may be a minor symptom of a major anxiety disorder. Be aware of the possible need to explore psychological issues.\n\n### EXCESS BODY HAIR\n\n#### The GP overview\n\nThis is defined as excess growth of terminal hair in women in male distribution sites (i.e. chin, cheeks, upper lip, lower abdomen and thighs). It presents as a cosmetic problem. Ethnic origin must be taken into account: Mediterraneans and Indians grow more than Nordics. Japanese, Chinese and American Indians grow the least. In the UK, according to surveys, up to 15% of women believe they have excess body hair, although only a minority present to the GP.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  constitutional (physiological)\n\n  polycystic ovary syndrome (PCOS): 50% of cases\n\n  anorexia nervosa\n\n  menopause\n\n  iatrogenic, e.g. phenytoin, minoxidil, danazol, glucocorticoids\n\n**OCCASIONAL**\n\n  congenital adrenal hyperplasia (1 in 5000)\n\n  anabolic steroid abuse\n\n  ovarian tumours: arrhenoblastoma, hilus cell tumour, luteoma\n\n  adrenal tumours: carcinoma and adenoma\n\n  congenital (1 in 5000 live births) and juvenile hypothyroidism\n\n**RARE**\n\n  acromegaly (incidence 3 per million)\n\n  porphyria cutanea tarda\n\n  Cushing's syndrome (incidence 1\u20132 per million)\n\n  hypertrichosis lanuginosa\n\n  Cornelia de Lange syndrome (Amsterdam dwarfism)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ serum testosterone, pelvic ultrasound, FBC, U&E, TFT.\n\n_SMALL PRINT :_ FSH\/LH, other tests of endocrine function and specialised imaging techniques (for adrenal\/pituitary disorders), urinary porphyrins.\n\n  Serum testosterone: probably the most useful investigation. Mild elevation (up to three times the normal value) suggests PCOS; levels above this indicate a possible tumour.\n\n  FBC, U&E: possible iron deficiency anaemia and electrolyte disturbance in anorexia; U&E may be deranged in adrenal disorders.\n\n  FSH\/LH and TFT: the former may help to confirm menopause and may point towards PCOS (elevated LH, normal FSH); the latter reveals hypothyroidism.\n\n  Other tests of endocrine function and imaging techniques: to investigate possible adrenal and pituitary disorders (usually undertaken in secondary care).\n\n  Pelvic ultrasound: multiple ovarian cysts characteristic of PCOS; may also reveal ovarian tumour.\n\n  Urinary porphyrins: for porphyria.\n\n##### TOP TIPS\n\n  Mild, long-standing hirsutism does not require investigation.\n\n  Enquire about self-medication, especially in athletes \u2013 anabolic steroids may occasionally be the cause.\n\n  Take the problem seriously and be prepared for questions about cosmetic treatments such as bleaching, depilatory creams and electrolysis.\n\n  Sudden and severe hirsutism is the most important marker for serious underlying pathology.\n\n  Other clues suggesting a possible hormone-secreting tumour include amenorrhoea, onset of baldness at the same time as hirsutism and a patient who seems generally unwell.\n\n  Consider psychological factors: hirsutism can cause \u2013 or be the presenting complaint in \u2013 significant depression.\n\n  Recent onset of significant headache and visual field defect raise the possibility of a pituitary adenoma.\n\n### HAIR LOSS\n\n#### The GP overview\n\nA distressing symptom for both genders: men fear loss of potency, and women are horrified at the cosmetic disaster unfolding. The psychological significance of increased hairfall is all too easy to overlook in a typical busy surgery. Take care to acknowledge that the problem is being taken seriously.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  androgenic alopecia (male pattern baldness)\n\n  seborrhoeic dermatitis\n\n  alopecia areata\n\n  contact allergic dermatitis\n\n  tinea capitis\n\n**OCCASIONAL**\n\n  bacterial folliculitis\n\n  telogen effluvium\n\n  endocrine: myxoedema, hypopituitarism and hypoparathyroidism\n\n  traction alopecia\n\n  lupus erythematosus\n\n**RARE**\n\n  secondary syphilis\n\n  trichotillomania\n\n  morphoea\n\n  iatrogenic, e.g. chemotherapy, anticoagulants\n\n  malnutrition\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ Wood's light test, hair and scales for mycology.\n\n_SMALL PRINT :_ FBC, ESR\/CRP, U&E, TFT, FSH\/LH, prolactin, autoimmune tests, syphilis serology.\n\n  _Microsporum_ infections will fluoresce green under a Wood's (UV) light.\n\n  Send scrapings and hair for mycology if the scalp looks abnormal.\n\n  FBC, ESR\/CRP and autoimmune tests may help identify autoimmune causes, e.g. SLE.\n\n  Syphilis serology: old-fashioned, but syphilis is on the increase.\n\n  U&E, TFT, FSH\/LH\/prolactin will effectively screen for endocrinopathy.\n\n##### TOP TIPS\n\n  Alopecia areata is occasionally associated with other autoimmune diseases. Further assessment is sensible, even at a later consultation.\n\n  Remember that in telogen effluvium, the traumatic event \u2013 such as a significant illness or childbirth \u2013 will have taken place about 4 months before the onset of hair loss, so the connection is unlikely to be made by the patient.\n\n  The patient invariably fears total hair loss \u2013 ensure that this is broached and that a realistic prognosis is given.\n\n  Reassure men with male-pattern baldness who are anxious about their potency. In this case it signifies the opposite. Its onset is related to adequate circulating testosterone. Eunuchs tend not to go bald.\n\n  Lymphadenopathy in association with alopecia may suggest an infective process: consider bacterial folliculitis.\n\n  Alopecia areata has a particularly poor prognosis if there are several patches, there is loss of eyebrows or eyelashes, or if it begins in childhood.\n\n  Scarring alopecia should prompt the clinician to look for general signs of lupus erythematosus.\n\n  Trichotillomania in children is usually simply due to habit; in adults, though, it is more often a sign of significant psychological disturbance.\n\n### ITCHY SCALP\n\n#### The GP overview\n\nThis might appear a mundane symptom to the GP, but it can cause the patient significant distress. It's usually a welcome presentation, as it's perceived as a quickie \u2013 one of those problems in which a brief examination can be performed at the same time as an equally brief history. This approach generally pays dividends, but the problem can sometimes be more complicated than it might at first appear.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  head lice\n\n  seborrhoeic eczema\n\n  psoriasis\n\n  lichen simplex\n\n  impetigo (may be underlying head lice or eczema)\n\n**OCCASIONAL**\n\n  allergic\/contact eczema\n\n  other fungal infections\n\n**RARE**\n\n  stress\/depression\n\n  viral infection, e.g. chickenpox \u2013 a common problem but only rarely presents with scalp itch\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIt is highly unlikely that any investigations will be required at all. Confirmation of a fungal infection may be obtained via fluorescence under a Wood's light or by microscopy and culture of scalp and hair samples.\n\n##### TOP TIPS\n\n  Whatever the actual diagnosis, scratching will perpetuate or complicate the problem and so should be discouraged.\n\n  Consider examining the skin and nails, as these may provide useful additional clues.\n\n  The diagnosis in a child is very likely to be head lice; the list of differentials increases the older the patient.\n\n  Scalp impetigo in a child \u2013 particularly if it relapses rapidly \u2013 suggests an underlying problem such as head lice or eczema. This needs treating too, or the symptom will persist.\n\n  In an otherwise puzzling case, consider psychological causes \u2013 stress and depression can sometimes present with scalp itching.\n\n## LIMBS\n\nAcute single joint pain\n\nArm and shoulder pain\n\nCalf pain\n\nFoot pain\n\nHand and wrist pain\n\nLeg ulcers\n\nMultiple joint pain\n\nPainful muscles\n\nRecurrent hip area pain in an adult\n\nRecurrent knee pain\n\nSwollen ankles\n\nSwollen calf\n\n### ACUTE SINGLE JOINT PAIN\n\n#### The GP overview\n\nThis is a very common problem in primary care. Usually, there are few physical signs, although occasionally a genuine monoarthritis with all the classical signs of inflammation will present. Overall, the most likely aetiological factor is trauma, though other conditions may already affect a joint. In the elderly, an exacerbation of osteoarthritis is common; this condition may also cause multiple joint pain. The knee is probably the single most frequently affected joint.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute exacerbation of osteoarthritis (OA)\n\n  traumatic synovitis\n\n  gout\/pseudogout\n\n  chondromalacia patellae (CP) and other anterior knee pain syndromes\n\n  traumatic haemarthrosis (e.g. after cruciate ligament injury)\n\n**OCCASIONAL**\n\n  fracture\n\n  Reiter's disease\n\n  psoriatic arthritis\n\n  rheumatoid arthritis (RA)\n\n  patellar tendinitis, Osgood\u2013Schlatter's disease\n\n**RARE**\n\n  septic arthritis (SA)\n\n  haemophilia\n\n  local tropical infections (e.g. Madura foot (mycetoma pedis), filariasis)\n\n  malignancy (usually secondary)\n\n  avascular necrosis\n\n  recurrent joint subluxation\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, uric acid, X-ray, joint aspiration (in monoarthritis of large joint).\n\n_SMALL PRINT :_ rheumatoid factor, clotting studies\/factor VIII assay, arthroscopy.\n\n  FBC\/ESR\/CRP: WCC and ESR\/CRP raised in infection, systemic inflammatory conditions; Hb may be reduced in the latter.\n\n  Uric acid: once attack has subsided, useful to add weight to clinical diagnosis of gout (especially if considering treatment with allopurinol).\n\n  Rheumatoid factor may be useful if symptoms suggest possible RA.\n\n  X-ray: essential if fracture suspected. May also reveal OA, avascular necrosis, malignancy and help to distinguish between RA and psoriatic arthritis.\n\n  Sterile aspiration of joint fluid: to look for pus (septic arthritis), blood (haemarthrosis) and crystals (gout\/pseudogout).\n\n  Clotting studies\/factor VIII assay: if haemophilia a possibility.\n\n  Arthroscopy: may be required urgently in secondary care if trauma has resulted in a haemarthrosis.\n\n##### TOP TIPS\n\n  Autoimmune blood tests can be misleading in possible arthritis. The diagnosis should be clinical; blood testing simply adds weight and prognostic information to the clinical assessment. Positive tests can be found in normal patients \u2013 beware of inappropriately labelling an insignificant problem as a significant arthritis on the basis of a blood test.\n\n  Gout is very painful, will limit movement and may cause a slight fever. Septic arthritis gives a similar picture but with marked restriction of movement and, usually, a high fever. If in doubt, arrange urgent assessment.\n\n  In obscure cases, question and examine the patient carefully. For example, in Reiter's disease, symptoms of urethritis or conjunctivitis may have been minimal or forgotten; in psoriatic arthritis, there may only be insignificant skin lesions.\n\n  If one joint is red, very hot, intensely painful with marked limitation of movement and systemic illness, septic arthritis must be excluded: admit.\n\n  Haemarthrosis usually develops rapidly after trauma and indicates significant damage requiring immediate referral; effusion due to synovitis usually takes a day or longer to accumulate and is less urgent.\n\n  Septic arthritis is notoriously easy to miss in a patient with coexisting RA. The systemic signs may be absent and the diagnosis may mistakenly be viewed as a flare-up of rheumatoid arthritis.\n\n  A young adult male with a monoarthritis of the knee not caused by trauma is likely to have Reiter's disease.\n\n### ARM AND SHOULDER PAIN\n\n#### The GP overview\n\nArm pain is a common presentation with a wide differential. Many generalised disorders, such as arthritis, neuropathy and polymyalgia, cause widespread symptoms, which may involve the arm \u2013 these are not considered here. Instead, this section concentrates on pain specific to the arm, or pain characteristically referred to the arm.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  simple muscular strain\n\n  epicondylitis (tennis or golfer's elbow)\n\n  subacromial bursitis\/capsulitis\n\n  cervical spondylosis\n\n  angina\n\n**OCCASIONAL**\n\n  bicipital tendonitis\n\n  acromioclavicular and sternoclavicular joint pain\n\n  acute calcific tendonitis\n\n  de Quervain's tenosynovitis\n\n  carpal tunnel (CT) syndrome\n\n  cervical rib\n\n  brachial and ulnar neuritis (including post-herpetic pain)\n\n  cervical and thoracic disc prolapse\n\n**RARE**\n\n  GORD\n\n  malignancy: local bone cancer, spinal cord, spine and lung\n\n  subclavian aneurysm\n\n  multiple sclerosis\n\n  syphilitic aortitis\n\n  thoracic outlet syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, TFT, ECG\/secondary care cardiac investigations, nerve conduction studies, chest and neck X-ray.\n\n_SMALL PRINT :_ other X-rays\/bone scan, MRI scan, lumbar puncture, syphilis serology.\n\n  FBC, ESR\/CRP: may be anaemia and raised ESR\/CRP in inflammatory or malignant conditions.\n\n  TFT: myxoedema and carpal tunnel syndrome significantly associated.\n\n  Neck X-ray sometimes useful to confirm diagnosis of cervical spondylosis and assess its severity \u2013 but cervical spondylosis on X-ray does not correlate well with symptoms and may just be an incidental finding.\n\n  ECG\/secondary care cardiac investigations: to pursue possible diagnosis of angina.\n\n  Nerve conduction studies: will help confirm nerve entrapment (e.g. carpal tunnel).\n\n  CXR: for apical tumour.\n\n  Other X-rays\/bone scans: if bony tumour (especially secondaries) suspected. Calcium deposits may be seen in acute calcific tendonitis.\n\n  MRI scan, lumbar puncture: if MS suspected; scanning may also be helpful to visualise possible cord lesion (all likely to be arranged after specialist referral).\n\n  Syphilis serology: in the rare case of possible syphilis.\n\n##### TOP TIPS\n\n  It is tempting to view arm pain as a welcome 'quickie'; in fact, a careful history is important to exclude the more unusual serious pathologies and the examination should usually serve only to confirm an already formulated diagnosis.\n\n  Patients with arm pain \u2013 especially if it is accompanied by intermittent paraesthesiae \u2013 are often inappropriately concerned that the diagnosis may be angina or a stroke. Make sure these fears are properly explored.\n\n  The natural history of many of the more common problems (e.g. subacromial bursitis, epicondylitis) can be quite prolonged. Making this clear from the outset helps maintain the patient's trust in you if the symptoms do take some time to settle.\n\n  Getting the patient to indicate the site of pain is a useful ploy in shoulder discomfort. Diffuse pain is typical of capsulitis and subacromial bursitis, whereas, with sternoclavicular or acromioclavicular joint problems, or bicipital tendonitis, the area is more likely to be very localised.\n\n  Beware of persistent paraesthesiae with arm pain, especially if the patient also complains of arm or hand weakness; either there is serious nerve compression or some other significant neurological pathology.\n\n  Angina may present only with arm pain. Enquire carefully to establish the pattern of the pain.\n\n  Apical lung tumour (Pancoast tumour) may cause severe arm pain long before any signs are evident. Investigate smokers with unexplained arm pain.\n\n  Consider the other less common diagnoses if the pain is severe and persistent, the diagnosis is not obvious from the history and the patient displays unrestricted arm movements.\n\n### CALF PAIN\n\n#### The GP overview\n\nCalf pain is usually innocent, except when accompanied by swelling. It is often caused by cramp, which is especially common in the elderly. In this group it can cause significant distress, through the havoc wreaked on sleep. Some of the less likely diagnoses, such as peripheral vascular disease, have important implications, so careful assessment is necessary.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  idiopathic (simple) cramp (including night cramps)\n\n  muscle stiffness (unaccustomed exercise)\n\n  cellulitis\n\n  peripheral vascular disease (PVD; intermittent claudication)\n\n  muscle injury (e.g. strain)\n\n**OCCASIONAL**\n\n  referred back pain (L4 and 5)\n\n  referred knee pain (arthropathy, infection)\n\n  alcoholic or diabetic neuropathy\n\n  cramps caused by underlying hypocalcaemia or electrolyte imbalance\n\n  ruptured Baker's cyst\n\n  deep vein thrombosis (DVT)\n\n  thrombophlebitis\n\n**RARE**\n\n  motor neurone disease\n\n  multiple sclerosis\n\n  muscle enzyme deficiency\n\n  psychological: muscle tension\n\n  lead and strychnine poisoning\n\n  ruptured Achilles tendon\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, WCC, ESR\/CRP, U&E, LFT, blood glucose or HbA1c, D-dimer.\n\n_SMALL PRINT :_ ultrasound, venogram, angiography.\n\n  Urinalysis: check specific gravity, glucose and protein (over and under-hydration, diabetes, renal failure as occasional causes of 'simple' cramp).\n\n  WCC and ESR\/CRP: both raised in infection. ESR\/CRP raised in arthropathy.\n\n  U&E and calcium: check renal function and electrolyte imbalance (e.g. from diuretics; hypocalcaemia).\n\n  LFT and blood glucose or HbA1c: if suspect alcoholism or diabetes resulting in a neuropathy.\n\n  D-dimer (usually in hospital): a raised level suggests a DVT, but is not conclusive.\n\n  Consider referring urgently for ultrasound of lower limb veins with or without venography if DVT suspected.\n\n  Angiography will be arranged by the specialist if peripheral vascular disease is suspected.\n\n##### TOP TIPS\n\n  Save the patient unnecessary investigation and possible anticoagulation by taking a careful history. A muscle tear and a DVT can both produce calf swelling and warmth. The former, though, is preceded by a dramatic and sudden pain in the calf, sometimes described as being like a kick or a gunshot.\n\n  It can be difficult to distinguish a simple muscle strain from claudication. Muscular pains tend to produce discomfort as soon as the patient stands; claudication usually starts after the patient has walked a predictable distance.\n\n  Patients with superficial phlebitis will fear the more serious DVT. Explain the difference to them.\n\n  Do not overlook Achilles rupture. The presentation may sometimes be less dramatic than you would expect.\n\n  Children rarely present with cramp (though it is common): avoid labelling as 'growing pains' unless serious causes are excluded.\n\n  Consider investigating the adult patient with recent onset of apparently simple cramps if associated with general malaise (these will be in the minority).\n\n  Claudication accompanied by nocturnal pain in the ball of the foot suggests critical ischaemia \u2013 refer urgently.\n\n  Homans's sign is positive in virtually all painful conditions of the calf. It is not diagnostic of DVT.\n\n### FOOT PAIN\n\n#### The GP overview\n\nPain in the foot is difficult for patients to ignore and so will often present with a relatively short history. Local causes predominate, but remember to think further afield: referral through S1 (lateral border of the foot) and L5 (dorsum of the foot to the big toe) nerve roots may occur. Ankle pain is not considered here.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  gout\n\n  verruca\n\n  bunion\/hallux valgus\n\n  infected ingrowing toenail (IGTN)\n\n  plantar fasciitis\n\n**OCCASIONAL**\n\n  Morton's neuroma\n\n  metatarsalgia\n\n  arthritis (osteo and rheumatoid)\n\n  Achilles tendonitis\/bursitis\n\n  oedema\n\n  foreign body\n\n**RARE**\n\n  march fracture\n\n  Sever's disease (apophysitis of the calcaneus), usually a problem of adolescence\n\n  osteochondritis: navicular = K\u00f6hler's disease; head of second or third metatarsal = Freiberg's disease\n\n  osteomyelitis and septic arthritis\n\n  erythromelalgia and painful polyneuropathy\n\n  ischaemia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ urinalysis, FBC, ESR\/CRP, rheumatoid factor, uric acid, X-ray.\n\n_SMALL PRINT :_ bone scan, angiography.\n\n_NOTE :_ if the cause is oedema, this will need investigating in its own right (see Swollen ankles, p. 312).\n\n  Urinalysis may reveal glycosuria in previously undiagnosed diabetic with neuropathy (suspicion of neuropathy may in itself require further investigations).\n\n  FBC\/ESR\/CRP: raised WCC and ESR\/CRP in infection and severe inflammation.\n\n  Rheumatoid factor: of prognostic help if foot pain is part of clinical diagnosis of rheumatoid arthritis.\n\n  Uric acid: if gout suspected, especially with recurrent attacks and if considering prophylaxis.\n\n  X-ray useful if suspect possible arthritis, osteomyelitis, march fracture, osteochondritis, radio-opaque foreign body. If clinical suspicion high and X-ray unhelpful, bone scan may be more useful.\n\n  Angiography: if ischaemic foot with rest pain.\n\n##### TOP TIPS\n\n  The vast majority of causes are obvious from the history or from a cursory examination. The harder you have to think, the more likely that there may be an obscure cause requiring investigation.\n\n  It can be difficult to distinguish gout from a severely inflamed bunion. With gout, the patient may have had previous episodes, the onset tends to be sudden, the joint is extremely tender and joint movements are very limited.\n\n  Important pointers can be picked up in the history, especially for some of the less common causes. Thus, Morton's neuroma causes a sharp pain often radiating to the third and fourth toes, relieved by removing the shoe; plantar fasciitis is described as 'walking on a pebble', especially after resting; and a march fracture results in a pain which initially comes on predictably with exercise and which then becomes continuous, with local bony tenderness and possibly a lump.\n\n  If a known arteriopath complains of pain in the ball of the foot disturbing sleep then the diagnosis is probably critical ischaemia. Refer urgently.\n\n  Fever and systemic illness with localised extreme bone pain and signs of local infection is acute osteomyelitis or septic arthritis until proved otherwise. Admit.\n\n  Pain with no obvious signs \u2013 particularly tenderness \u2013 in the foot suggests ischaemia, neuropathy or an L5\/S1 nerve root lesion.\n\n  If no cause is evident but the patient has very localised sole tenderness away from the heel, consider a foreign body.\n\n### HAND AND WRIST PAIN\n\n#### The GP overview\n\nThis may be the presenting problem but just as often it is a 'while I'm here' symptom. The differential diagnosis is quite wide but 'arthritis' is often uppermost in the patient's mind. A brief history and focused examination should provide the correct diagnosis quite rapidly in most cases.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  osteoarthritis (especially the carpometacarpal joint of the thumb and the distal interphalangeal joints of the fingers)\n\n  carpal tunnel syndrome\n\n  trauma (e.g. sprain, scaphoid fracture)\n\n  rheumatoid (or other inflammatory) arthritis\n\n  tenosynovitis\n\n**OCCASIONAL**\n\n  ganglion\n\n  gout\n\n  Raynaud's disease or syndrome\n\n  infection (e.g. paronychia, pulp space)\n\n  work-related upper limb disorder (WRULD)\n\n  trigger thumb or finger\n\n  other nerve entrapment, e.g. ulnar nerve, cervical root pain\n\n  complex regional pain syndrome\n\n**RARE**\n\n  infected eczema (common, but rarely presents with pain)\n\n  writer's cramp\n\n  peripheral neuropathy\n\n  Dupuytren's contracture (usually painless)\n\n  diabetic arthropathy\n\n  osteomyelitis\n\n  Kienb\u00f6ck's disease (avascular necrosis of the lunate)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ X-ray, FBC, ESR\/CRP, rheumatoid factor, uric acid.\n\n_SMALL PRINT :_ blood screen for underlying causes in peripheral neuropathy or Raynaud's syndrome, if clinically indicated.\n\n  X-ray: may show a fracture in trauma, joint erosions in RA, the typical features of OA, and sclerosis or collapse of the lunate in Kienb\u00f6ck's disease.\n\n  FBC: Hb may be reduced in inflammatory arthritis; WCC raised in infection.\n\n  ESR\/CRP: raised in infective and inflammatory conditions.\n\n  Rheumatoid factor: may support a clinical diagnosis of RA.\n\n  Uric acid: an elevated level (post episode) supports a diagnosis of gout.\n\n##### TOP TIPS\n\n  OA of the fingers can be relatively abrupt in onset and inflammatory in appearance compared with OA at other sites.\n\n  Explore the patient's occupation \u2013 this will provide valuable information regarding the possible cause and effect of the problem.\n\n  Simply asking the patient to point to the site of the pain can help distinguish two of the most commonly confused differentials: OA of the carpometacarpal joint of the thumb and de Quervain's tenosynovitis. In the former the pain is relatively localised to the base of the thumb; in the latter the discomfort \u2013 and certainly the tenderness \u2013 is more diffuse.\n\n  Pain from a ganglion can precede the appearance of the ganglion itself \u2013 or the ganglion may be fairly subtle, only appearing on wrist flexion.\n\n  Remember that RA is a clinical diagnosis \u2013 don't rely on blood tests. Early referral minimises the risk of long-term joint damage.\n\n  If in doubt over tenderness in the anatomical snuffbox after a fall on the outstretched hand, refer for A&E assessment \u2013 a missed scaphoid fracture can cause long-term problems.\n\n  Do not underestimate pulp space infection \u2013 this can cause serious complications such as osteomyelitis or bacterial tenosynovitis. It may need IV antibiotics or incision and drainage.\n\n  Thenar wasting suggests significant compression in carpal tunnel syndrome \u2013 refer.\n\n### LEG ULCERS\n\n#### The GP overview\n\nChronic leg ulcer is a major problem in the UK, costing the National Health Service up to \u00a3600m per annum. It is reckoned that nearly 1% of the population may be affected by leg ulceration at some time during their lives. Recurrence is common. The vast majority have a vascular underlying cause.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  venous disease: 70\u201380% of leg ulcers\n\n  peripheral arterial disease: about 15% of leg ulcers\n\n  associated with systemic disease \u2013 diabetes (5% of ulcer patients), rheumatoid arthritis (8%), vasculitis\n\n  gross oedema due to systemic causes, e.g. CCF, renal disease, osteoarthritis, severe obesity, prolonged immobility from any cause\n\n  chronic infection, e.g. after trauma, insect bite\n\n**OCCASIONAL**\n\n  drug misuse\n\n  after primary herpes zoster\n\n  primary malignancy \u2013 squamous cell carcinoma, melanoma, malignant change in an existing ulcer\n\n  secondary malignancy \u2013 metastases\n\n**RARE**\n\n  tropical infections\n\n  AIDS\n\n  tuberculosis\n\n  systemic drug reaction\n\n  factitious \u2013 self-inflicted (Munchausen's, personality disorder)\n\n#### Ready reckoner\n\n_NOTE :_ In practice it is very difficult indeed to separate the causes of leg ulcers through clinical examination alone. This is a guide to likely causes only.\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, TSH, LFTs, U&E, fasting glucose or HbA1c and RA factor, ABPI.\n\n_POSSIBLE :_ swabs for bacteriology, cardiovascular assessment if appropriate.\n\n_SMALL PRINT :_ Duplex ultrasound.\n\n  FBC, ESR\/CRP, CRP, TSH, LFTs, U&E, fasting glucose or HbA1c and RA factor as a basic screen for systemic causes and background disease.\n\n  Swabs for bacteriology are only useful if there is clinical evidence of viable tissue infection, e.g. cellulitis.\n\n  Full cardiovascular assessment if any suspicion of arterial insufficiency.\n\n  Ankle brachial pressure index (ABPI) in both legs by handheld Doppler. Sensitivity of up to 95%; if less than 0.8 assume arterial disease is present. Limited usefulness in patients with microvascular disease, e.g. RA, DM, systemic vasculitis; may cause spuriously high ABPI.\n\n  Specialist: Duplex ultrasound is the investigation of choice to assess arterial and venous insufficiency.\n\n##### TOP TIPS\n\n  Be systematic in the clinical notes: describe the edge (e.g. rolled, punched-out), base (e.g. sloughy, necrotic, granulating), location, morphology (may help detect rarer causes), and surface area (serial measurements of surface area of an ulcer are a good index of healing).\n\n  Palpation of peripheral pulses is not a reliable guide to arterial sufficiency. Use the ABPI.\n\n  Lipodermatosclerosis is a red or brown patch of skin on the lower leg, usually on the medial side, just above the ankle. This and venous eczema are indications of superficial venous valve failure, even in the absence of varicose veins. They may represent disease amenable to surgery, so refer for a vascular opinion.\n\n  Deep ulcers involving deep fascia, tendon, periosteum or bone may well have an arterial component.\n\n  Mixed ulcer aetiology may confuse the clinical picture and make treatment choices harder. Refer for a specialist opinion if in any doubt.\n\n  More than 50% of leg ulcer patients are sensitive to one or more allergens, including lanolin, topical antibiotics, cetyl stearyl alcohols, balsam of Peru and parabens. These may contribute to non-healing and cause discomfort to the patient. Refer the patient for patch testing if dermatitis is associated with a leg ulcer.\n\n  Topical antibiotics do not contribute to healing and are frequent sensitisers \u2013 avoid using them.\n\n  Pain from an ulcer is most frequently associated with an arterial aetiology.\n\n  Refer for biopsy if the ulcer has an atypical appearance or distribution, or fails to heal within 12 weeks of treatment. Beware neoplastic change in an existing ulcer. This is rare, but not to be missed.\n\n  Compression bandaging is dangerous in diabetes and arterial insufficiency. Do not prescribe it until they are ruled out. If in doubt about ABPI, refer for a vascular opinion.\n\n### MULTIPLE JOINT PAIN\n\n#### The GP overview\n\nThe range of causes of multiple joint pain spans acute, chronic and chronic relapsing conditions. In the elderly, the commonest problem is multiple osteoarthritis; in middle age, inflammatory conditions predominate; and in the young, systemic conditions are more likely.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  rheumatoid arthritis (RA)\n\n  psoriatic arthropathy\n\n  viral polyarthritis, e.g. hepatitis, rubella\n\n  connective tissue diseases, e.g. SLE, systemic sclerosis, polymyositis, polyarteritis nodosa, giant cell arteritis\n\n  multiple osteoarthritis (OA)\n\n**OCCASIONAL**\n\n  the spondoarthritides: ankylosing spondylitis, Reiter's disease, enteropathic arthritis, Beh\u00e7et's syndrome, juvenile chronic arthritis\n\n  Henoch\u2013Sch\u00f6nlein syndrome\n\n  malignancy (usually secondary)\n\n  iatrogenic: corticosteroid therapy, isoniazid, hydralazine\n\n  hypertrophic pulmonary osteoarthropathy (due to lung cancer)\n\n  sarcoidosis\n\n**RARE**\n\n  sickle-cell crisis\n\n  amyloidosis\n\n  rheumatic fever\n\n  atypical systemic infections, e.g. Lyme disease, Weil's disease, brucellosis, syphilis (secondary)\n\n  decompression sickness (the bends)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, autoantibodies.\n\n_POSSIBLE :_ urinalysis, U&E, HLA-B27, joint X-rays, synovial fluid aspiration.\n\n_SMALL PRINT :_ blood film, serology, creatine phosphokinase, CXR, bone scan, Kveim test.\n\n  FBC, ESR\/CRP, blood film: WCC and ESR\/CRP raised in acute inflammation and infection. Anaemia of chronic disease may be seen, and blood film will reveal sickle cell.\n\n  Autoantibodies: rheumatoid factor is positive in most cases of RA, but is also positive in many autoimmune diseases and chronic infections; antinuclear factor is positive in 90% of cases of SLE but a similar result is obtained in 30% of cases of RA and also in many other diseases.\n\n  Urinalysis: may reveal proteinuria or haematuria if there is renal involvement in connective tissue disease.\n\n  U&E: to check for renal failure via renal involvement in multisystem connective tissue disease.\n\n  HLA-B27: a high prevalence in spondoarthritides.\n\n  Serology: may be useful to diagnose viral, or atypical systemic, infections. ASO titres, if rising, suggest recent streptococcal infection (e.g. in rheumatic fever).\n\n  Creatine phosphokinase: elevated in polymyositis.\n\n  Joint X-rays: hand X-rays may show characteristic features helping to distinguish between RA and psoriatic arthritis; pelvic and lumbar spine X-rays may show the typical changes of ankylosing spondylitis (if negative, and clinical suspicion high, a bone scan may be helpful); X-rays of affected joints may confirm clinical diagnosis of OA.\n\n  CXR: may reveal lung malignancy.\n\n  Synovial fluid analysis: helps distinguish inflammatory from infective and crystal arthropathies.\n\n  Sarcoidosis: Kveim test may help confirm diagnosis.\n\n##### TOP TIPS\n\n  The connective tissue diseases can all affect almost every organ system. Take a full history so as not to miss a clue or complication.\n\n  Check the skin as this may contribute to the diagnosis (e.g. scaly rash in psoriasis, butterfly rash in SLE, thickening of skin in sclerosis and heliotrope rash in dermatomyositis).\n\n  Don't overvalue autoimmune blood tests. Most diagnoses of arthritis are clinical, blood tests simply providing confirmatory or prognostic information.\n\n  Suspect Reiter's syndrome in a young male with an inflammatory oligoarthritis of the lower limbs.\n\n  An insidious onset of symmetrical polyarthritis in the 30\u201350 age range, with early morning stiffness, pain and swelling of hands and feet, suggests RA.\n\n  Pain in the wrists and ankles of a middle-aged or elderly smoker with clubbing and chest symptoms strongly suggests hypertrophic pulmonary osteoarthropathy caused by underlying lung cancer.\n\n  Don't overlook the patient's occupation as this may be relevant in certain cases: for example, in vets and farm workers, brucellosis and Weil's disease are possible infective causes.\n\n### PAINFUL MUSCLES\n\n#### The GP overview\n\nThis symptom has a multitude of causes. A careful history is required to distinguish between muscle and joint pain, and between muscle pain and weakness. In some of the underlying pathologies, these symptoms may coexist. Cramp, causing very transient muscle pain, is covered elsewhere (see Calf pain, p. 288).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  overuse (including strain injury)\n\n  acute viral illness\n\n  depression\n\n  polymyalgia rheumatica (PMR)\n\n  side effects of statins (myalgia much more common than myositis)\n\n**OCCASIONAL**\n\n  vitamin D deficiency\n\n  referred joint pain (e.g. from hip to thigh, neck to shoulder, shoulder to arm)\n\n  fibromyalgia\n\n  chronic fatigue syndrome\n\n  connective tissue disease, e.g. RA, SLE, polyarteritis nodosa (PAN), scleroderma\n\n  peripheral vascular disease: intermittent claudication\n\n  neuropathy: diabetic, alcoholic\n\n  Bornholm disease (epidemic myalgia, devil's grip)\n\n  hypothyroidism\n\n  drugs: statins, clofibrate, street drug withdrawal, chemotherapy, lithium, cimetidine\n\n**RARE**\n\n  polymyositis (usually more weakness than pain)\n\n  adult and childhood dermatomyositis\n\n  underlying malignancy\n\n  porphyria\n\n  Guillain\u2013Barr\u00e9 syndrome and poliomyelitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP.\n\n_POSSIBLE :_ urinalysis, autoimmune blood tests, TFT, LFT, blood sugar or HbA1c, creatine phosphokinase (CPK), vitamin D levels.\n\n_SMALL PRINT :_ joint and chest X-rays; in secondary care, angiography, electromyography, muscle biopsy, lumbar puncture, urinary porphyrins.\n\n  Urinalysis: glycosuria in undiagnosed diabetes.\n\n  FBC and ESR\/CRP: Hb may be depressed in connective tissue disease and PMR. WCC and ESR\/CRP raised in any inflammatory disorder (ESR more useful than CRP in PMR); MCV elevated in hypothyroidism and alcohol abuse.\n\n  Autoimmune blood tests: may be helpful if connective tissue disorder suspected.\n\n  TFT: will confirm hypothyroidism.\n\n  Blood sugar or HbA1c, LFT: the former to confirm diabetes; the latter may help in confirming an alcohol problem. Both may cause a neuropathy resulting in muscle pain.\n\n  CPK: raised in acute inflammatory and viral myopathies.\n\n  Vitamin D levels: vitamin D deficiency is increasingly being recognised and may present with muscle pain and\/or weakness.\n\n  Joint X-rays: if referred pain from primary joint pathology suspected.\n\n  Angiography: for peripheral vascular disease.\n\n  Electromyography and muscle biopsy (both in secondary care): to confirm diagnosis of polymyositis or dermatomyositis.\n\n  Lumbar puncture: to examine CSF in hospital in suspected Guillain\u2013Barr\u00e9 syndrome or poliomyelitis.\n\n  Urinary porphyrins: to exclude porphyria.\n\n  Other investigations for suspected underlying malignancy, e.g. CXR.\n\n##### TOP TIPS\n\n  In polysymptomatic patients with muscle pain but no objective signs and normal blood tests, consider fibromyalgia, depression and chronic fatigue ( _Note:_ these problems may coexist).\n\n  The diagnosis of PMR is clinched by a trial of prednisolone (20 mg\/day). In PMR, this treatment should lead to total resolution of symptoms within a few days.\n\n  Muscle pain is more likely to be associated with significant pathology in the very young and old than the middle-aged, when psychological causes and overuse are the most likely.\n\n  Always remember PMR in the older patient complaining of aching pain and stiffness in the hip and shoulder girdle muscles which is worse in the mornings.\n\n  If considering PMR, or initiating treatment in this condition, enquire after symptoms of temporal arteritis. About 30% of patients develop this complication, and are at risk of blindness.\n\n  Muscle pain with significant and progressive weakness (e.g. difficulty climbing stairs or getting out of a chair) suggests polymyositis, hypothyroidism, vitamin D deficiency or malignancy.\n\n  Significant underlying disease (e.g. PMR, polymyositis, dermatomyositis or connective tissue disease) is likely if there is an arthritis associated with the muscle pain.\n\n### RECURRENT HIP AREA PAIN IN AN ADULT\n\n#### The GP overview\n\nHip area pain is a common presentation in the middle-aged and elderly, and the patient will often attribute it to osteoarthritis. This diagnosis may well be correct, although the differential is wide \u2013 besides, the patient's view of what actually constitutes the 'hip' may be at odds with the anatomical truth. The differential for the child with hip pain is very different \u2013 see Limp in a child, p. 209.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  muscular\/ligamentous strain\n\n  osteoarthritis\n\n  trochanteric bursitis\n\n  referred from back\n\n  meralgia paraesthetica\n\n**OCCASIONAL**\n\n  inflammatory arthritis\n\n  avascular necrosis\n\n  hernia\n\n  complications of a total hip replacement, e.g. loosening, infection\n\n  spinal stenosis\n\n  iliotibial band syndrome\n\n  acetabular labral tear\n\n**RARE**\n\n  impacted fracture\n\n  dislocation\n\n  bony pathology, e.g. secondaries, Paget's\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ : X-ray.\n\n_POSSIBLE_ : FBC, CRP, autoantibodies, HLA-B27, alkaline phosphatase, urinalysis.\n\n_SMALL PRINT_ : arthroscopy, bone scan, lumbar spine MRI (all in hospital).\n\n  X-ray: may show evidence of osteoarthritis, avascular necrosis, fracture, dislocation, hip replacement loosening and bony pathology. Spinal X-ray may reveal spinal pathology as a cause.\n\n  FBC, CRP: CRP may be elevated and Hb reduced in inflammatory arthritis. CRP and WCC raised in infection of joint prosthesis.\n\n  Autoantibodies: for clues about the aetiology of inflammatory arthritis.\n\n  HLA-B27: a high prevalence in spondoarthritides.\n\n  Alkaline phosphatase: raised in Paget's disease.\n\n  Urinalysis: may reveal proteinuria or haematuria if there is renal involvement in inflammatory arthritis.\n\n  Arthroscopy: diagnostic and potentially therapeutic in labral tear.\n\n  Bone scan: may reveal bony secondaries.\n\n  Lumbar spine MRI: for evidence of spinal stenosis; might reveal other causes of pain referred from spine.\n\n##### TOP TIPS\n\n  Check what the patient means by 'hip'. Most don't realise that the hip joint is actually in the groin.\n\n  An X-ray may not be necessary, even if the clinical picture suggests hip arthritis \u2013 but the patient may well expect one, so ensure it is at least discussed.\n\n  Examine the patient standing up \u2013 this may reveal a hernia as the cause.\n\n  Localised lateral pain aggravated by lying on the affected side is likely to be caused by trochanteric bursitis.\n\n  Remember the possibility of loosening or infection in joint replacements.\n\n  Consider avascular necrosis if a patient on long-term steroids develops severe hip pain.\n\n  Beware that the elderly can sometimes remain weight bearing \u2013 albeit with pain and a limp \u2013 after an impacted hip fracture.\n\n  Significant depression may aggravate or result from hip arthritis pain \u2013 consider a trial of antidepressants.\n\n### RECURRENT KNEE PAIN\n\n#### The GP overview\n\nRecurrent knee pain is a very common presentation with a wide differential. Classification of causes isn't helped by changing and confusing nomenclature. As ever in general practice, a careful history and examination will provide useful clues \u2013 but management will often be dictated more by degree of disability and the patient's wishes than by making a precise diagnosis.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  ligament sprain\/minor soft tissue injury\n\n  osteoarthritis\n\n  cartilage injury\n\n  chondromalacia patellae\/patellofemoral pain\n\n  Osgood\u2013Schlatter's disease\n\n**OCCASIONAL**\n\n  recurrent monoarthritis, e.g. gout, pseudogout, Reiter's\n\n  as part of polyarthritis, e.g. rheumatoid, ankylosing spondylitis, psoriatic arthritis\n\n  iliotibial tract syndrome\n\n  bursitis\n\n  referred from hip or back\n\n  ligament rupture\n\n  patellar tendonitis\n\n  Baker's cyst\n\n  loose body\n\n  bone disease, e.g. Paget's\n\n  recurrent dislocation of the patella\n\n  medial shelf syndrome\n\n  osteochondritis dissecans\n\n**RARE**\n\n  haemochromatosis\n\n  recurrent haemarthroses, e.g. coagulation disorder\n\n  osteosarcoma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ : X-ray.\n\n_POSSIBLE_ : FBC, CRP, uric acid, MRI, autoantibodies.\n\n_SMALL PRINT_ : HLA-B27, joint aspiration, hip or back investigations if referred pain suspected, alkaline phosphatase, serum ferritin, coagulation screen.\n\n  X-ray: may give clues to many of the possible causes, or confirmatory evidence when clinical suspicion is high \u2013 for example, with osteoarthritis, bony loose body, Paget's, osteochondritis dissecans.\n\n  FBC, CRP: CRP elevated and Hb may be reduced in inflammatory polyarthritis.\n\n  Uric acid: typically elevated in gout.\n\n  MRI: useful to assess soft tissue such as cartilage, especially if surgery is being considered.\n\n  Autoantibodies: if inflammatory polyarthritis suspected.\n\n  HLA-B27: a high prevalence in spondoarthritides.\n\n  Joint aspiration: may be diagnostically useful if an effusion is present \u2013 for example, revealing positively birefringent crystals in pseudogout. In practice, usually performed after specialist referral.\n\n  Hip or back investigations: appropriate radiology may be necessary if it is thought the knee pain is referred from these areas.\n\n  Alkaline phosphatase: elevated in Paget's.\n\n  Serum ferritin: elevated in haemochromatosis.\n\n  Coagulation screen: if coagulopathy suspected.\n\n##### TOP TIPS\n\n  Patients place great value on X-rays whereas, in practice, they may contribute little to management of straightforward recurrent knee pain. To prevent an unsatisfactory outcome, consider proactively broaching the fact that an X-ray may be unnecessary.\n\n  Insisting that the patient accurately localises the pain \u2013 if possible \u2013 may usefully limit the diagnostic possibilities.\n\n  'Straightforward' osteoarthritis can become quite suddenly more painful, often for no obvious reason \u2013 exacerbations and remissions are part of the natural history of the disease.\n\n  Keen sports people often present with recurrent knee pain and are unlikely to indulge in the GP's time-honoured 'wait and see' approach. Earlier investigation or intervention may prove necessary.\n\n  Do not forget that knee pain may be referred \u2013 if the cause isn't immediately apparent, examine the hip, especially in children.\n\n  Anterior cruciate ligament injury is quite easily missed in casualty in the acute stage. It may only present later with an unstable knee.\n\n  Osteosarcomas are rare \u2013 but most commonly occur near the knee. Beware of unexplained constant, increasing pain waking the patient at night. Swelling and inflammation will only appear later.\n\n### SWOLLEN ANKLES\n\n#### The GP overview\n\nThis is one of the commonest presenting complaints in the elderly and, in this age group, may be linked to recurrent falls. As a result, it is frequently the reason for a home visit request. In younger age groups, it is much rarer, but much more likely to signify serious pathology.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  congestive cardiac failure (CCF)\n\n  drug reaction \u2013 especially calcium antagonists\n\n  gravitational (venous insufficiency, often with poor mobility)\n\n  obesity\n\n  pelvic mass (including pregnancy)\n\n**OCCASIONAL**\n\n  cirrhosis\n\n  premenstrual syndrome\n\n  anaemia\n\n  renal: acute or chronic nephritis, nephrotic syndrome\n\n  protein-losing enteropathy, e.g. coeliac disease, inflammatory bowel disease\n\n**RARE**\n\n  malnutrition\n\n  inferior vena cava thrombosis\n\n  filariasis\n\n  Milroy's disease (hereditary lymphoedema)\n\n  ancylostomiasis (hookworm)\n\n  angioneurotic oedema\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, FBC, U&E, LFT.\n\n_POSSIBLE :_ TFT.\n\n_SMALL PRINT :_ CXR, pelvic ultrasound, further investigation of underlying cause.\n\n  Urinalysis: for proteinuria.\n\n  FBC: look for anaemia of chronic disorder, raised MCV (alcohol abuse).\n\n  U&E: will reveal underlying renal failure; sodium low in CCF and cirrhosis.\n\n  LFT: may reveal hypoproteinaemia (e.g. in cirrhosis, protein-losing enteropathy and nephrotic syndrome).\n\n  CXR: pulmonary oedema and pleural effusion in CCF.\n\n  Pelvic ultrasound: for pelvic mass.\n\n  Further investigation of underlying cause: this might involve ECG, BNP and echocardiography (CCF), CT scan (pelvic mass), renal biopsy (nephritis) and bowel investigations (enteropathy).\n\n##### TOP TIPS\n\n  In the elderly, the cause is often multifactorial, with immobility playing a major role.\n\n  Proper assessment can take time: consider spreading the work over a couple of consultations, using the intervening time to arrange and assess investigations.\n\n  Ankle swelling is usually symmetrical, though venous insufficiency in particular can affect one side much more than the other. But if only one ankle is swollen, consider deep vein thrombosis, a ruptured Baker's cyst or cellulitis.\n\n  Don't forget that many drugs (such as calcium antagonists and NSAIDs) can cause marked ankle swelling.\n\n  If no cause is obvious in an elderly person, examine the abdomen and also consider a rectal examination.\n\n  The younger the patient, the greater the chance of significant pathology \u2013 especially renal.\n\n  Marked swelling of recent and sudden onset is likely to be significant regardless of age.\n\n### SWOLLEN CALF\n\n#### The GP overview\n\nSuch has been the publicity about 'economy class syndrome' that this presentation \u2013 and the closely related symptom, 'calf pain' (see p. 288) \u2013 has become quite common. The worry the patient has about a possible DVT can prove quite 'infectious', with the GP anxious not to miss this significant problem. In most cases, a careful history, backed up by appropriate examination, should reveal the true cause.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  cellulitis\n\n  most causes of swollen ankles (see p. 312)\n\n  muscle strain\/rupture (especially rupture of plantaris tendon)\n\n  ruptured Baker's cyst\n\n  DVT\n\n**OCCASIONAL**\n\n  ruptured Achilles tendon\n\n  varicose eczema\n\n  phlebitis\n\n**RARE**\n\n  muscle herniation through fascia (especially tibialis anterior)\n\n  muscular neoplasm\n\n  pseudohypertrophy (as in muscular dystrophy)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY_ : none (unless sent to hospital).\n\n_POSSIBLE_ : FBC, ESR\/CRP and other investigations for swollen ankles (see p. 312); usually in hospital \u2013 D-dimer, ultrasound, venography.\n\n_SMALL PRINT_ : (in hospital) radio-iodine labelled fibrinogen uptake test.\n\n  FBC, ESR\/CRP: elevated white cell count and ESR\/CRP in cellulitis.\n\n  D-dimer: raised level suggests DVT but is not conclusive.\n\n  Ultrasound: may help diagnose DVT and useful in confirming ruptured Baker's cyst as the cause.\n\n  Venography, radio-iodine labelled fibrinogen uptake test: hospital test which may be used to confirm DVT.\n\n##### TOP TIPS\n\n  The swelling resulting from a muscle rupture can be impressive \u2013 but a typical history with pain (described as 'Like being shot in the calf') preceding the swelling should clinch the correct diagnosis.\n\n  Varicose eczema is often misdiagnosed as cellulitis. Clues are that it is commonly bilateral, itches more than hurts and is not accompanied by fever.\n\n  Anxiety about possible DVT may cloud the presentation: careful questioning may reveal that swelling is, in fact, long-standing and\/or bilateral, making DVT very unlikely.\n\n  Patients with unexplained DVT are three to four times more likely than controls to have an underlying malignancy \u2013 so, once the DVT has been dealt with, consider appropriate investigation.\n\n  In high-risk patients \u2013 such as those who have just returned from a long haul flight \u2013 your index of suspicion for DVT should be raised.\n\n  When the history suggests muscular rupture, ensure that the Achilles tendon is intact.\n\n## NECK\n\nDifficulty swallowing\n\nHoarseness\n\nNeck lumps\n\nSore throat\n\nStiff neck\n\nStridor in children\n\n### DIFFICULTY SWALLOWING\n\n#### The GP overview\n\nThis symptom can mean several things, and a careful history is necessary to tease out the precise problem: difficulty in initiating swallowing; a sensation of food sticking somewhere; painful swallowing; also included here is the sensation of 'something in the throat' even when not trying to swallow anything.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  globus hystericus\n\n  any painful pharyngeal condition, e.g. pharyngitis\n\n  GORD\n\n  benign stricture\n\n  oesophageal carcinoma\n\n**OCCASIONAL**\n\n  pharyngeal pouch\n\n  pharyngeal carcinoma\n\n  compression by mediastinal tumours (e.g. lymphoma, bronchial carcinoma)\n\n  oesophageal achalasia\n\n  gastric carcinoma\n\n  xerostomia (the elderly, post-parotidectomy and Sj\u00f6gren's syndrome)\n\n  foreign body\n\n  drugs: NSAID-associated oesophagitis, failure to take bisphosphonate tablets correctly\n\n**RARE**\n\n  Plummer\u2013Vinson syndrome\n\n  Chagas's disease (South American trypanosomal infection)\n\n  scleroderma (CREST syndrome), polymyositis and dermatomyositis\n\n  neurological disorders, e.g. myasthenia gravis, bulbar palsy\n\n  motor neurone disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ (unless obvious globus or local pharyngeal cause) FBC, ESR\/CRP, barium swallow or endoscopy.\n\n_POSSIBLE :_ CXR, LFT.\n\n_SMALL PRINT :_ pharyngeal swab, CT scan thorax.\n\n  FBC and ESR\/CRP: may reveal evidence of malignancy or iron-deficiency anaemia.\n\n  LFT if malignancy suspected: abnormality suggests hepatic spread.\n\n  Barium swallow useful in the frail and to safely demonstrate stricture or motility problems if no absolute dysphagia for liquids (risk of aspiration).\n\n  Flexible upper GI endoscopy allows visualisation and biopsy of suspicious lesions.\n\n  Throat swab occasionally helpful in painful pharyngeal lesions.\n\n  CXR if suspicion of mediastinal tumour of any cause.\n\n  CT scan or further imaging may be arranged by the specialist to further define mediastinal tumours.\n\n##### TOP TIPS\n\n  A young patient under stress who can swallow food and drink without problems but who feels there is 'something stuck' almost certainly has globus. Reassurance usually resolves the situation.\n\n  Remember to ask about medication \u2013 recent onset of painful dysphagia may be caused by severe oesophagitis secondary to drugs such as alendronate, NSAIDs and slow-release potassium supplements.\n\n  Take time with the history: difficulty in swallowing can mean a number of different things, and the diagnosis is much more likely to be revealed by careful questioning than by examination.\n\n  Recent onset of progressive dysphagia with weight loss in an elderly patient is caused by oesophageal carcinoma until proved otherwise.\n\n  A palpable hard lymph node in the left supraclavicular fossa (Troisier's sign) is strongly associated with gastric carcinoma.\n\n  Beware of patients who have a long history of oesophagitis but who complain of increasing or unusual dysphagia: they may have developed a stricture, or even a carcinoma.\n\n  If endoscopy does not reveal a cause but the symptom continues, remember rarer causes, such as extrinsic compression on the oesophagus or a neurological problem. Consider a barium swallow, or referral to a neurologist if there are other neurological symptoms or signs.\n\n### HOARSENESS\n\n#### The GP overview\n\nHoarseness may start suddenly and last a few days (acute), or arise gradually and continue for weeks or months (chronic). The history will clarify this and point the way forward in management. Acute hoarseness rarely causes any diagnostic problem or concern; the less common chronic case raises more worrying possibilities and usually requires referral.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute viral laryngitis\n\n  voice overuse (shouting, screaming)\n\n  hypothyroidism\n\n  smoking\n\n  sinusitis\n\n**OCCASIONAL**\n\n  GORD\n\n  benign tumours: singer's nodes, polyps\n\n  crico-arytenoid rheumatoid arthritis\n\n  functional (hysterical) aphonia\n\n**RARE**\n\n  acute epiglottitis\n\n  laryngeal carcinoma\n\n  recurrent laryngeal nerve palsy\n\n  physical trauma (e.g. after intubation)\n\n  chemical inhalation trauma\n\n  rare inflammatory lesions (e.g. TB, syphilis)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ TFT, CXR, direct or indirect laryngoscopy.\n\n_SMALL PRINT :_ throat swab.\n\n  TFT: in chronic hoarseness to exclude hypothyroidism.\n\n  CXR: to check for thoracic lesions causing recurrent laryngeal nerve palsy.\n\n  Indirect laryngoscopy: useful for a GP with the necessary skills; most will refer to an ENT specialist.\n\n  Direct laryngoscopy: using a flexible fibre-optic endoscope. This is a specialist investigation allowing close-up views and biopsy of suspicious lesions.\n\n  Throat swab: useful, very rarely, if hoarseness is associated with a persisting pharyngitis.\n\n##### TOP TIPS\n\n  In acute laryngitis, don't forget to tell the patient to rest the voice, and remember that occupational factors are important: use of voice (e.g. by telephonists) or working in smoky environment (e.g. a pub) will aggravate and prolong symptoms, causing diagnostic confusion.\n\n  If you suspect a malignancy, arrange an urgent CXR immediately prior to referral. The referral can then be made to the correct specialist (chest rather than ENT) if a lung lesion is present, thus expediting appropriate management.\n\n  Don't forget transient hoarseness caused by intubation \u2013 GPs are seeing this increasingly often as patients spend less post-operative time in hospital.\n\n  Every adult patient with hoarseness for 3 weeks or more showing no signs of improvement has carcinoma of the larynx until proved otherwise.\n\n  GORD is a common cause in the elderly, but beware of making this diagnosis without specialist investigation first.\n\n  Epiglottitis is rare but if you suspect it in any patient, admit immediately \u2013 and don't examine the throat.\n\n  Hypothyroidism is easily overlooked \u2013 prompt diagnosis can save unnecessary anxiety and investigation.\n\n### NECK LUMPS\n\n#### The GP overview\n\nA lump in the neck usually means just one thing to a patient: cancer. This is rarely the cause in practice, and a careful examination and explanation can be more anxiolytic than a bucket-load of benzodiazepines. Occasionally further investigation is needed.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  reactive lymphadenitis due to a local infection\n\n  prominent normal lymph nodes\n\n  goitre\n\n  sebaceous cyst\n\n  thyroglossal cyst\n\n**OCCASIONAL**\n\n  branchial cyst\n\n  pharyngeal pouch\n\n  cervical rib\n\n  actinomycosis\n\n  primary lymphoma or secondary neoplastic metastasis\n\n**RARE**\n\n  tuberculosis of cervical lymph nodes (king's evil; scrofula)\n\n  thyroid carcinoma\n\n  carotid body tumour or aneurysm\n\n  sarcoidosis\n\n  cystic hygroma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ TFTs if thyroid swelling.\n\n_POSSIBLE :_ FBC, ESR, CXR.\n\n_SMALL PRINT :_ thyroid ultrasound, radioisotope studies, barium swallow, biopsy.\n\n  TFT in all cases of thyroid enlargement: may reveal hypo- or hyperthyroidism.\n\n  FBC and ESR in persistent enlarged nodes: check WCC and investigate further if abnormal or if ESR high.\n\n  CXR: may reveal primary lung carcinoma, lymphoma, or other more obscure pathologies.\n\n  Thyroid ultrasound and\/or radioiodine studies if lump felt within the thyroid \u2013 usually arranged by endocrinologist after referral.\n\n  Barium swallow: to confirm and outline a pharyngeal pouch.\n\n  Biopsy: specialist procedure to establish nature of a persistent, suspicious neck lump.\n\n##### TOP TIPS\n\n  Establish the patient's concerns: cancer fear is common with this symptom.\n\n  Unless the lump is obviously suspicious, employ the 'diagnostic use of time' \u2013 a judicious delay often resolves the problem, or it may reveal the true diagnosis.\n\n  Suspicious lymphadenopathy usually involves a single, gradually enlarging and non-tender lymph node.\n\n  Children with normal or reactive neck glands are often presented by anxious parents. Take time to explain the nature of the problem to properly allay fears and prevent inappropriate repeat attendances.\n\n  A neoplastic-type lymph node enlargement without any obvious cause should be referred urgently for detailed ENT assessment.\n\n  A neck lump fixed to skin and without a punctum should arouse suspicion: urgent biopsy should be done once a primary ENT tumour is ruled out.\n\n  Dysphagia with a neck lump is a serious symptom unless associated with a transient sore throat. Further investigation by endoscopy is necessary.\n\n  Beware of a hard swelling developing rapidly in the thyroid \u2013 carcinoma must be excluded.\n\n### SORE THROAT\n\n#### The GP overview\n\nThis presenting symptom is the king of superlatives in general practice. It is the commonest \u2013 the average GP will see about 120 cases each year \u2013 the most over-treated, the most controversial and usually the most mundane. It is also probably the most welcome, as consultations are often short, even when self-management is explained rather than a prescription given.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  mild viral pharyngitis (with URTI)\n\n  tonsillitis\/streptococcal pharyngitis ('strep throat')\n\n  glandular fever\n\n  quinsy (peritonsillar abscess)\n\n  oropharyngeal candidiasis\n\n**OCCASIONAL**\n\n  GORD\n\n  glossopharyngeal neuralgia and cervicogenic nerve root pain\n\n  trauma: foreign body or scratch from badly chewed crispy food\n\n  other viral or bacterial infections, e.g. Vincent's angina, herpangina, herpes simplex, gonorrhoea\n\n  aphthous ulceration\n\n  acute or subacute thyroiditis\n\n**RARE**\n\n  cardiac angina\n\n  carotidynia\n\n  blood dyscrasia (including iatrogenic)\n\n  epiglottitis\n\n  diphtheria\n\n  oropharyngeal carcinoma\n\n  retropharyngeal abscess\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ throat swab, FBC, Paul\u2013Bunnell test.\n\n_SMALL PRINT :_ upper GI endoscopy, biopsy, cardiac investigation (all secondary care).\n\n  Throat swab: use is controversial, mainly because of low specificity and sensitivity. Practical use only in persistent sore throat or treatment failure.\n\n  FBC: may show atypical lymphocytes in glandular fever; also will reveal any underlying blood dyscrasia.\n\n  Paul\u2013Bunnell test for glandular fever if malaise and fatigue persist.\n\n  Upper GI endoscopy may be necessary to diagnose GORD.\n\n  X-ray\/laryngoscopy: if suspicion of foreign body.\n\n  Cardiac investigation: in rare case when referred symptoms cause pain in throat.\n\n  Biopsy of suspicious lesions important to investigate possible malignancy.\n\n##### TOP TIPS\n\n  Consultations for severe sore throats usually boil down to a decision whether or not to prescribe antibiotics. There is no easy or reliable way to distinguish clinically between bacterial and viral causes, so the situation becomes an exercise in pragmatism, though the Centor criteria may help. Even in 'true' streptococcal throats, antibiotic treatment probably only reduces the duration of symptoms by about 24 hours and is unlikely to influence the likelihood of complications.\n\n  Mild sore throat with an URTI is usually just one of a 'package' of symptoms presented, along with rhinorrhoea, cough, headache and so on. The cause is invariably viral and antibiotics have no role to play.\n\n  Throat swabs only help management in obscure or persistent cases (and even then usually contribute little).\n\n  In adolescents and young adults whom you decide to treat with antibiotics for 'strep throat', explain that the symptoms can also be caused by other infections such as glandular fever. This will help maintain the patients' confidence in you if they return with the sore throat persisting after the course of antibiotics.\n\n  Remember that this apparently trivial symptom can occasionally herald a serious problem. In particular, enquire about medication (the first sign of drug-induced agranulocytosis may be a sore throat).\n\n  A true foreign body stuck in the throat will lodge in the supraglottic area and may not be seen orally. Refer if in doubt.\n\n  Admit if any suspicion of epiglottitis \u2013 and do not examine the throat.\n\n  Consider a possible underlying problem (such as diabetes or immunosuppression) in the younger patient with oropharyngeal candidiasis which has no obvious cause.\n\n  Quinsy can cause a respiratory obstruction. Never attempt conservative management, but admit for surgical drainage.\n\n  Florid 'tonsillitis' is unusual in the middle-aged or elderly \u2013 consider investigations for a blood disorder or oropharyngeal carcinoma.\n\n### STIFF NECK\n\n#### The GP overview\n\nThe most common causes of acute neck stiffness are benign and easily managed in general practice. However, this symptom causes disproportionate panic in parents of febrile children thanks to extensive media coverage of meningitis. This anxiety can spill over into adult illness behaviour, with the result that a troublesome but harmless symptom may be misinterpreted as the harbinger of serious pathology.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute torticollis (positional, draughts)\n\n  cervical spondylosis\n\n  viral URTI with cervical lymphadenitis\n\n  whiplash injury\n\n  meningism due to systemic infection (e.g. pneumonia)\n\n**OCCASIONAL**\n\n  other forms of arthritis, e.g. rheumatoid (RA) and ankylosing spondylitis\n\n  abscess in the neck\n\n  hysteria\n\n  intracerebral haemorrhage\n\n  cerebral tumour (primary or secondary)\n\n**RARE**\n\n  meningitis\n\n  vertebral fracture\n\n  bone tumour (primary or secondary)\n\n  atypical infections: tetanus, leptospirosis, sandfly fever, psittacosis\n\n  brain abscess\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, Paul\u2013Bunnell test, ESR\/CRP, rheumatoid factor, HLA-B27.\n\n_SMALL PRINT :_ bone biochemistry, X-ray cervical spine, bone scan, other hospital-based tests.\n\n  FBC and Paul\u2013Bunnell: in unresolved or resistant URTI, check these parameters if glandular fever suspected.\n\n  ESR\/CRP, rheumatoid factor and HLA-B27: will help in the diagnosis of possible RA and ankylosing spondylitis in the young and middle-aged with unresolving neck stiffness.\n\n  Neck X-ray: for possible fracture (at hospital); of limited value in cervical spondylosis \u2013 symptoms do not correlate well with X-ray findings. May reveal serious bone pathology, but bone scan more useful for this.\n\n  Bone biochemistry: consider this if bony secondaries or myeloma are possible diagnoses.\n\n  Hospital-based tests: these might include lumbar puncture (for meningitis) and scans for cerebral lesions.\n\n##### TOP TIPS\n\n  Neck tenderness due to cervical lymphadenopathy in an URTI is infinitely more common than meningitis, but is often misreported as 'neck stiffness'.\n\n  Only advise soft collars in the majority of stiff necks for a maximum of 48 hours. Though comfortable, they tend to delay resolution. Instead, suggest adequate analgesia, heat and mobilisation.\n\n  Warn patients with whiplash injury that symptoms may take many months to settle completely \u2013 this saves repeated futile and frustrating consultations.\n\n  Meningococcal petechiae are usually a late sign and can be missed unless the febrile child with a stiff neck is undressed and examined.\n\n  Pain and stiffness may be the only symptoms of vertebral fracture or subluxation, which can occur without cord involvement \u2013 significant trauma merits A&E referral.\n\n  Thunderclap headache preceding neck stiffness suggests subarachnoid haemorrhage: admit straight away.\n\n  Consider serious bony pathology if pain and stiffness are relentless and wake the patient at night \u2013 especially if there are other worrying symptoms, or the patient has a past history of carcinoma.\n\n### STRIDOR IN CHILDREN\n\n#### The GP overview\n\nAcute stridor is a very frightening experience for both child and parents. The respiratory effort can lead to hyperventilation, making things worse. 'Difficult' or 'noisy' breathing in a child quite commonly leads to a request for an out-of-hours visit in winter. The usual cause is viral croup, producing mild, harmless stridor \u2013 but serious cases do occur. A calm exterior and a methodical approach are the keys to effective management.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  viral croup (laryngotracheobronchitis)\n\n  acute epiglottitis\n\n  acute laryngitis\n\n  acute airways obstruction: foreign body (small toy, peanut)\n\n  laryngeal paralysis (congenital: accounts for 25% of infants with stridor)\n\n**OCCASIONAL**\n\n  laryngomalacia\n\n  laryngeal trauma\n\n  bacterial tracheitis\n\n  pseudomembranous croup (staphylococcal)\n\n  upper airway burn\n\n**RARE**\n\n  laryngeal stenosis\n\n  laryngeal tumours (papilloma, haemangioma) and mediastinal tumours\n\n  laryngeal oedema (angioneurotic: oedema also present in other tissues)\n\n  anomalous blood vessels, e.g. double aortic arch\n\n  diphtheria\n\n  retropharyngeal abscess\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are no investigations likely to be performed in primary care. The following might be performed in hospital: FBC (WCC raised in infection), lateral X-ray of pharynx (enlarged epiglottis in epiglottitis), CXR (may show foreign body, distal collapse or external compression of larynx or trachea) and laryngoscopy (for direct visualisation of the larynx).\n\n##### TOP TIPS\n\n  In practice, the first step is to exclude those conditions requiring immediate admission (epiglottitis or inhaled foreign body), leaving a probable diagnosis of viral croup. Management then depends on the child's general condition \u2013 in particular, the level of respiratory distress.\n\n  Children with viral croup may have marked stridor and some recession when crying. It is reasonable to observe such children at home provided these signs disappear when the child is settled.\n\n  When managing a child at home, make absolutely sure that the parents understand the signs of deterioration. If in doubt, arrange review.\n\n  The toxic child with low-pitched stridor (often not marked), severe sore throat or difficulty in swallowing, and respiratory distress has epiglottitis until proved otherwise. Admit immediately and do not examine the throat (this can provoke respiratory obstruction).\n\n  Restlessness, rising pulse and respiratory rate, increasing intercostal recession, fatigue and drowsiness are ominous signs: admit urgently regardless of precise diagnosis.\n\n  Consider an inhaled foreign body if the onset is very sudden and there are no other symptoms or signs of respiratory infection.\n\n## NOSE\n\nBlocked nose\n\nNosebleed\n\nRunny\/discharging nose\n\n### BLOCKED NOSE\n\n#### The GP overview\n\nThis extremely common symptom is very familiar to all GPs. The most frequent cause, the common cold, is not included here, since nasal obstruction in itself is not usually the presenting symptom. The majority of causes of nasal obstruction are benign, but care should be taken to consider referral in those few cases that do not respond promptly to simple treatment.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  vasomotor rhinitis\n\n  allergic rhinitis (seasonal and perennial)\n\n  nasal polyps\n\n  adenoidal hypertrophy (in children)\n\n  nasal septal deviation (may affect 20% of adults)\n\n**OCCASIONAL**\n\n  overuse of over-the-counter nasal decongestants (rhinitis medicamentosa)\n\n  chronic sinusitis\n\n  papilloma\n\n  trauma (including septal haematoma)\n\n  foreign body (especially toddlers)\n\n**RARE**\n\n  iatrogenic, e.g. doxazosin\n\n  carcinoma of nose and sinuses: squamous cell, adenocarcinoma\n\n  other rare nasal tumours, e.g. melanoma, teratoma\n\n  unilateral choanal atresia\n\n  fibroangioma of puberty\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ sinus X-rays, allergy tests.\n\n_SMALL PRINT :_ FBC, adenoid X-ray, CT scan.\n\n  Sinus X-rays may be helpful in confirming chronic sinusitis.\n\n  Allergy tests may identify sensitivities to particular allergens.\n\n  FBC: eosinophilia in allergic rhinitis (though rarely required for diagnosis).\n\n  Adenoid X-ray: will confirm adenoidal hypertrophy.\n\n  CT scan defines size and extent of nasopharyngeal carcinoma and sinus pathology.\n\n##### TOP TIPS\n\n  Nasal obstruction is usually the cause rather than the result of recurrent sinusitis. Dealing with the underlying problem may well relieve the sinusitis.\n\n  Nasal allergy and polyps often coexist. It is worth trying medical treatment to relieve the symptoms before referring for polypectomy.\n\n  Very sudden nasal blockage with profuse watery rhinorrhoea is likely to be caused by vasomotor rhinitis.\n\n  If adenoidal nasal obstruction is present with possible hearing or speech difficulties, or possible sleep apnoea, refer with a view to adenoidectomy.\n\n  Unilateral bloodstained discharge in an older patient with nasal blockage is sinister and suggests malignancy.\n\n  Soft swelling either side of the septum following trauma suggests septal haematoma. This should be evacuated to reduce the risk of cartilage necrosis and infection.\n\n  Check what over-the-counter nasal sprays the patient has been using, remembering the possibility of rhinitis medicamentosa.\n\n  A toddler with a unilateral nasal blockage and foul-smelling discharge is likely to have a nasal foreign body.\n\n### NOSEBLEED\n\n#### The GP overview\n\nThis is commonest in the very young and the very old. It may present routinely as a recurrent problem, or in the acute situation when the patient cannot control the bleeding. The latter cases usually result in trivial haemorrhage by clinical standards, but may create a disproportionate amount of alarm. Occasionally, a prolonged nosebleed can cause significant hypovolaemia, especially in the elderly.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  spontaneous (from Little's area; may be aggravated by nose-picking and sneezing)\n\n  nasal infection and ulceration\n\n  drugs, e.g. anticoagulants\n\n  allergic rhinitis (and atrophic rhinitis)\n\n  hypertension (often with atherosclerosis)\n\n**OCCASIONAL**\n\n  nasal sprays, e.g. corticosteroids\n\n  septal granulomas and perforations\n\n  severe liver disease\n\n  tumours of nose and\/or sinuses\n\n  abnormal anatomy: septal deviation\n\n  trauma: nasal fracture\n\n**RARE**\n\n  leukaemia\n\n  thrombocytopenia\n\n  coagulopathy: haemophilia, Christmas and von Willebrand's diseases\n\n  vitamin deficiencies: C and K\n\n  hereditary haemorrhagic telangiectasia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none (INR if on warfarin).\n\n_POSSIBLE :_ FBC, clotting studies.\n\n_SMALL PRINT :_ LFT, sinus X-ray, CT scan.\n\n  FBC: to check for thrombocytopenia or other signs of blood dyscrasia.\n\n  LFT: severe (e.g. alcoholic) liver disease causes clotting problems.\n\n  Raised INR may reflect severe liver disease or warfarin overdose.\n\n  Further clotting studies: if disorders such as haemophilia or von Willebrand's disease are suspected.\n\n  Sinus X-ray\/CT scanning (usually secondary care): if tumour a possibility.\n\n##### TOP TIPS\n\n  Emergency calls for children with acute epistaxis can usually be dealt with by clear, calm and authoritative advice on the telephone. First-aid measures should also be advised in the elderly, but hospital referral may prove necessary, as the bleeding can be considerable and more difficult to stop.\n\n  When children are presented in the surgery, establish any parental concerns. The main problem is often a fear of a blood disorder such as leukaemia rather than the inconvenience of the symptom itself.\n\n  In young or middle-aged adults with recurrent bleeding and ulceration, consider cocaine abuse.\n\n  Adults with recurrent epistaxis may well expect their blood pressure to be taken; either go ahead and take it, or, if the cause is obviously not hypertension, explain why there is no need.\n\n  Severe nosebleed unresponsive to standard first-aid measures is best dealt with in hospital \u2013 especially in the elderly. Refer urgently to ENT or A&E.\n\n  If recurrent nosebleeds with purpuric bruising, check FBC and coagulation screen urgently.\n\n  Beware of recent onset of persistent unilateral bloodstained discharge and obstruction in the middle-aged and elderly. Carcinoma of the nose, nasopharynx or sinus is possible.\n\n  Patients on warfarin should have an urgent INR and review of their dosage requirements.\n\n### RUNNY\/DISCHARGING NOSE\n\n#### The GP overview\n\nThough not usually viewed as a significant symptom, this is very bothersome to those that suffer it. Many of the pathologies overlap with the causes of the 'Blocked nose' \u2013 please refer to this chapter too, as appropriate. The cause is rarely sinister, but referral for further assessment may be necessary if it persists in spite of treatment.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  URTI including common cold\n\n  allergic rhinitis (seasonal or perennial)\n\n  vasomotor rhinitis\n\n  sinusitis\n\n  infected nasal mucosal lesion (herpes simplex, impetigo)\n\n**OCCASIONAL**\n\n  polyps\n\n  rhinitis medicamentosa\n\n  cocaine abuse\n\n  cluster headaches (though these will usually present with the pain)\n\n  chemical irritation (smoke, fumes)\n\n  nasal foreign body (usually seen in toddlers or children)\n\n  drug side effects, e.g. a-blockers, calcium channel blockers\n\n**RARE**\n\n  malignancy\n\n  CSF rhinorrhoea following head injury\n\n  barotrauma ('sinus squeeze')\n\n  _Corynebacterium diphtheriae_ infection\n\n#### Ready reckoner\n\n#### Possible investigations\n\nUsually none. Sinus X-rays, CT scan and allergy tests likely to be ordered by specialist after referral rather than in primary care.\n\n##### TOP TIPS\n\n  Use an auriscope with the largest available speculum to look into the nasal cavity \u2013 tell patients to hold their breath or the lens will steam up.\n\n  Intermittently runny nose associated with a nasal obstruction that is dependent on position, e.g. disappears when lying down, suggests a single nasal polyp.\n\n  Patients who describe their runny nose as 'just like turning on a tap' probably have vasomotor rhinitis.\n\n  Remember to enquire about non-prescribed medication. Cocaine abuse, or the use of OTC sympathomimetic drops may be very relevant in making a diagnosis; and the response to any OTC treatments such as intranasal steroids might help guide diagnosis and further treatment.\n\n  Intranasal foreign bodies are relatively common in toddlers \u2013 beware the unilateral foul-smelling nasal discharge in a child.\n\n  A persistent bloodstained discharge requires investigation, especially if associated with unilateral nasal obstruction.\n\n  Clear unilateral nasal discharge after direct trauma to the face may represent CSF leakage from an ethmoidal skull fracture. Occasionally this can present some time after the injury so beware the late presentation.\n\n## ORAL\n\nBad breath\n\nBleeding or painful gums\n\nMouth lumps and marks\n\nMouth ulcers\n\nPainful tongue\n\n### BAD BREATH\n\n#### The GP overview\n\nThis common symptom is usually caused by poor dental hygiene. As a presenting complaint it is seen far more often by dentists than GPs. It may be detected by a doctor examining a patient for an unrelated complaint, and rarely but significantly can herald serious pathology.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  poor dental hygiene\n\n  heavy smoking\n\n  gingivitis (including acute necrotising ulcerative gingivitis (ANUG), acute and chronic gingivitis)\n\n  excess alcohol intake (acute and chronic)\n\n  discharging dental abscess\n\n**OCCASIONAL**\n\n  ketohalitosis of starvation \u2013 especially in people using high-protein, high-fat, low-carbohydrate weight reduction diet regimens (and pre-operatively starved patients)\n\n  drugs, e.g. disulfiram\n\n  acute or chronic sinusitis\n\n  subjectively perceived (non-existent) halitosis (sometimes a form of somatisation)\n\n  GORD or acute gastroenteritis with reflux of gas\n\n**RARE**\n\n  bronchiectasis\n\n  liver failure \u2013 hepatic foetor is said to smell like a freshly opened corpse; this is due to mercaptans in expired air\n\n  true delusional subjective halitosis as part of psychiatric condition, e.g. severe depression with nihilism, psychotic illnesses\n\n  rare oral or nasal conditions, e.g. pyogenic granuloma, discharging sinus from abscess\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIt is unlikely that the GP will initiate any investigations other than LFT if alcohol is a likely aetiology, or liver failure is suspected. CXR may be helpful if bronchiectasis is a possibility. Dentists may carry out the following investigations.\n\n  Plaque and bleeding indexes \u2013 for oral hygiene.\n\n  Basic periodontal examination \u2013 measurement of pocket depth between gums and teeth carried out with a specially marked probe.\n\n  Oral pantomograph X-ray to investigate general state of teeth.\n\n##### TOP TIPS\n\n  If a dental cause is suspected, rather than telling the patient to make their own appointment with a dentist, give a short referral letter for them to hand over. This will ensure and record that you have carried out proper management and may speed things considerably for the patient.\n\n  The vast majority of cases will involve oral hygiene. Be certain to exclude this, and other physical causes, before deciding that the halitosis is entirely subjective. As with other forms of somatisation, active management \u2013 with a psychiatric referral if necessary \u2013 is preferable to sending the patient away, only to return to make more appointments.\n\n  If no obvious oral cause is found initially, extend the history and examination to include the respiratory and gastrointestinal systems.\n\n  ANUG is a 'spot diagnosis' \u2013 the stench usually precedes the patient by some distance.\n\n  In the absence of an obvious cause, do not neglect to examine the head and neck including the buccal cavity and nasal airway. Failure to do so could mean that a serious local cause is missed.\n\n  Do not overlook alcohol abuse as a possible underlying cause.\n\n  If the impact of the symptom seems out of all proportion to any objective sign of a problem, consider depression.\n\n### BLEEDING OR PAINFUL GUMS\n\n#### The GP overview\n\nThe primary cause of this symptom is nearly always infection, usually because of poor dental hygiene: an endemic problem worldwide. Systemic problems may also cause gum pain or bleeding. While a dental referral is likely to be the end result, it is worth checking for general causes or easily remediable problems before directing the patient to the dentist.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  gingivitis\/periodontal (gum) disease\n\n  pregnancy gingivitis\n\n  acute necrotising ulcerative gingivitis (ANUG): Vincent's stomatitis\n\n  trauma: poorly fitting dentures\n\n  drugs: warfarin overdosage, long-term phenytoin\n\n**OCCASIONAL**\n\n  aphthous ulceration\n\n  acute herpetic gingivostomatitis (occasionally EBV)\n\n  autoimmune disease: lichen planus, SLE and others\n\n  oral neoplasia (commonest is SCC) ( _Note:_ may bleed but usually painless)\n\n  blood dyscrasias (especially acute myeloid leukaemia)\n\n**RARE**\n\n  malabsorption (including scurvy)\n\n  chemical poisoning: mercury, phosphorus, arsenic and lead\n\n  hereditary haemorrhagic telangiectasia\n\n  lymphangioma\n\n  cavernous haemangioma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC.\n\n_SMALL PRINT :_ swab, INR, autoimmune screen, Paul\u2013Bunnell test.\n\n  FBC: to check for blood dyscrasias and malabsorption.\n\n  Swab may help if obscure infective cause.\n\n  Urgent INR if patient on warfarin.\n\n  Paul\u2013Bunnell test: EBV infection may cause gingivostomatitis.\n\n  Autoimmune screen if autoimmune disease suspected.\n\n##### TOP TIPS\n\n  Patients with manifestly 'dental' problems may attend the GP because they view the doctor's service as cheaper or more accessible. Direct them firmly to the dentist to discourage inappropriate attendance in the future.\n\n  Review the patient's medication \u2013 it is easy to overlook iatrogenic causes of gum soreness or bleeding.\n\n  Patients with aphthous ulcers are likely to have read that their problem is associated with vitamin deficiencies or systemic illness. In primary care, it almost never is.\n\n  Ulcerative gingivitis can often be diagnosed as soon as the patient walks into the consulting room, because of the characteristic odour.\n\n  Children with primary attacks of herpetic gingivostomatitis can become quite ill and dehydrated. Consider early review or admission.\n\n  Petechiae on the soft palate in conjunction with gingivostomatitis raise the possibility of EBV infection, acute leukaemia or scurvy.\n\n  Enquire about skin problems elsewhere, or you may miss a significant diagnosis: SLE, pemphigus, pemphigoid, bullous erythema multiforme, epidermolysis bullosa and lichen planus can all affect the mouth.\n\n### MOUTH LUMPS AND MARKS\n\n#### The GP overview\n\nMouth lumps and marks can be unfamiliar territory \u2013 partly because it is rarely an area of expertise for many GPs, and partly because many mouth problems are picked up by, or presented to, dentists in the first place. A proportion of patients will choose a GP as the first port of call, so a working knowledge of the area is useful.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  apical tooth abscess (gumboil)\n\n  aphthous ulceration\n\n  Fordyce spots (tiny white or yellow spots, on mucosa opposite molars and vermilion border of lips; they are sebaceous glands)\n\n  oral candida infection\n\n  mucocoele (solitary cystic nodule inside lip)\n\n**OCCASIONAL**\n\n  lichen planus\n\n  trauma \u2013 bitten cheek\n\n  ranula\n\n  torus \u2013 benign maxillary or mandibular outgrowth of bone (very common but usually asymptomatic so not commonly seen)\n\n  premalignant coloured areas: erythroplakia (red), leukoplakia (white), speckled leukoplakia\n\n(red and white), or verrucous leukoplakia\n\n  geographical and hairy tongue\n\n  tonsillar concretions\n\n  other forms of oral ulceration (see Mouth ulcers, p. 356)\n\n**RARE**\n\n  malignancy \u2013 SCC or melanoma\n\n  pachyderma oralis (from irritants)\n\n  heavy metal poisoning (lead, bismuth, iron) \u2013 a dark line below the gingival margin\n\n  cancrum oris\n\n  sublingual dermoid cyst\n\n  sublingual gland tumour\n\n  pigmentation due to oral contraceptive pill \u2013 black or brown areas anywhere in the mouth\n\n  Addison's disease \u2013 bluish hue opposite molars\n\n  Peutz\u2013Jeghers spots \u2013 brown spots on the lips\n\n  telangiectasia \u2013 may be a sign of Osler\u2013Weber\u2013Rendu syndrome\n\n  Stevens\u2013Johnson syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_OCCASIONAL :_ FBC, ESR, CRP, ferritin, B12 and folate, fasting glucose or HbA1c, swab of lesion, HIV test.\n\n_SMALL PRINT :_ biopsy (performed at hospital).\n\n  FBC, ESR, CRP and HIV are useful if immune deficiency (e.g. as a background to _Candida_ infection) is suspected; FBC and ESR or CRP may be helpful in suspected malignancy too.\n\n  Ferritin, B12 and folate deficiency is sometimes associated with oral aphthous ulceration \u2013 worth checking these in cases of recurrent or chronic ulceration (and see other possible investigations in Mouth ulcers, p. 356).\n\n  Fasting glucose or HbA1c to investigate possible diabetes if candidal infection otherwise unexplained.\n\n  Mouth swab to confirm candidal infection, though a diagnostic trial of treatment is often the practical first step.\n\n  Biopsy: of suspicious lesions \u2013 this is inevitably performed in secondary care.\n\n##### TOP TIPS\n\n  Recurrent oral aphthous ulceration is a feature of a few systemic diseases (e.g. coeliac disease, Crohn's disease, Beh\u00e7et's disease and AIDS) so be prepared to re-evaluate the history and widen the net of information gathering in repeat presentations.\n\n  It is tempting to give antibiotics for a dental abscess, but the old surgical maxim 'if there's pus about, let it out' still holds true. Antibiotics may help reduce pain and surrounding infection but are only a temporary measure and may delay definitive treatment in those trying to avoid seeing a dentist, and increase the risk of complications. Encourage patients to see a dentist in the first place: offering a referral letter is helpful and may help to overcome any possible barrier to urgent access to a dentist at the dental reception desk.\n\n  Always examine lumps by palpation from inside as well as outside the mouth. Wash latex gloves before the examination. Glove powder tastes foul!\n\n  Always refer the patient with permanent red or white buccal mucosal patches. Biopsy is indicated.\n\n  If an ulcer fails to heal within a few weeks, especially if it is painless, refer for a specialist opinion as a suspected malignancy.\n\n  Do not fail to examine regional lymph nodes. Enlarged nodes would be a significant finding, especially if they are non-tender and persistent.\n\n### MOUTH ULCERS\n\n#### The GP overview\n\nThis symptom may often make a GP feel baffled \u2013 largely because it tends to be overlooked, or dealt with only briefly, during medical training. In fact common causes are simple to detect and treat, and it is clearly important to detect the occasional serious problem at an early stage. Examination could hardly be simpler, and a dentist may well have a clearer idea if referral is necessary in more obscure cases.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  trauma\n\n  recurrent aphthous ulceration (RAU)\n\n  acute necrotising ulcerative gingivitis (ANUG)\n\n  thrush\n\n  iron-deficiency anaemia (also vitamin B12 and folate deficiency)\n\n**OCCASIONAL**\n\n  Coxsackie virus: herpangina, hand, foot and mouth\n\n  inflammatory bowel disease: ulcerative colitis and Crohn's disease\n\n  coeliac disease\n\n  herpes simplex and zoster\n\n  glandular fever (EBV): infectious mononucleosis\n\n  erosive lichen planus\n\n**RARE**\n\n  carcinoma: squamous cell, salivary gland\n\n  autoimmune: Beh\u00e7et's syndrome, pemphigoid, pemphigus, bullous erythema multiforme\n\n  syphilitic chancre or gumma\n\n  leukaemia, agranulocytosis (may be iatrogenic)\n\n  tuberculosis\n\n  HIV infection\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC.\n\n_POSSIBLE :_ urinalysis, vitamin B12 and folate, coeliac screen.\n\n_SMALL PRINT :_ swab, autoantibody screen, syphilis and HIV serology, biopsy.\n\n  FBC: essential basic investigation for anaemia and rarer blood dyscrasias.\n\n  Urinalysis: check for glycosuria. Underlying diabetes may predispose to infective causes (especially _Candida_ ).\n\n  Vitamin B12 and folate: to establish if underlying vitamin deficiency (especially if MCV raised).\n\n  Coeliac screen: anti-endomysial and anti-gliadin antibodies suggest coeliac disease if positive.\n\n  Swab: may help confirm doubtful diagnosis of ANUG \u2013 confirms presence of Vincent's organisms.\n\n  Autoantibody screens and HLA tests may be useful if autoimmune causes are suspected.\n\n  Syphilis or HIV serology: if syphilis or HIV are suspected.\n\n  Biopsy: required in persistent ulcer of uncertain aetiology (secondary care investigation).\n\n##### TOP TIPS\n\n  Consider vitamin or iron deficiency, especially if the patient has glossitis and angular cheilitis as well as oral ulceration.\n\n  The patient with sore, ulcerated gums and foul halitosis has ANUG; the smell is sometimes apparent as soon as the patient walks in.\n\n  Patients with RAU often believe they are suffering from a vitamin deficiency; in fact, this is rarely the case, but be sure to broach this with them and consider a blood test as this may reinforce your reassurance.\n\n  Enquire about skin problems elsewhere in an obscure case \u2013 this may give a clue to the precise diagnosis.\n\n  Faucial ulceration and petechial haemorrhages of the soft palate and pharynx are likely to be caused by glandular fever.\n\n  A solitary, persistent and often painless ulcer could be malignant \u2013 especially in smokers. Refer urgently to the oral surgeon for biopsy.\n\n  Ask about bowel function: diarrhoea, abdominal pain and bloodstained stools with mucus suggest associated inflammatory bowel disease.\n\n  Don't forget to enquire about medication: blood dyscrasias are a rare but significant side effect of some treatments (e.g. gold, carbimazole), and oral ulceration may be the first sign.\n\n  Oral candidiasis is common in the debilitated and those with dentures, but much less so in the otherwise apparently fit. In the latter cases consider underlying problems such as immunosuppression or diabetes.\n\n### PAINFUL TONGUE\n\n#### The GP overview\n\nPain in the tongue is usually caused by something immediately apparent on examination, but there are a few less obvious causes. This is something much more likely to be seen by a dentist, but is not strictly dental and therefore a working knowledge of the symptom is firmly within the GP remit.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  geographic tongue (benign migratory glossitis) \u2013 painful in some cases\n\n  candidal infection (e.g. post-antibiotic, steroids and uncontrolled diabetes)\n\n  trauma (bitten, burnt from hot food or drink)\n\n  anaemia: iron, vitamins B6 and B12, and folate deficiency\n\n  aphthous ulceration\n\n**OCCASIONAL**\n\n  viral infection, e.g. herpes simplex, hand, foot and mouth\n\n  median rhomboid glossitis (superficial midline glossitis)\n\n  burning mouth syndrome (also known as glossodynia)\n\n  fissured tongue (doesn't commonly cause pain)\n\n  glossopharyngeal neuralgia\n\n  lichen planus\n\n**RARE**\n\n  carcinoma of the tongue\n\n  Beh\u00e7et's disease\n\n  pemphigus vulgaris\n\n  drugs, e.g. reserpine, mouthwashes, aspirin burns\n\n  Moeller's glossitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC.\n\n_POSSIBLE :_ vitamin B12, folate and ferritin assays, swab.\n\n_SMALL PRINT :_ biopsy.\n\n  FBC initially to screen for anaemia.\n\n  Vitamin B12, folate and ferritin assays: if indicated by FBC.\n\n  Swab of tongue may be helpful if appearance not obviously candidal.\n\n  Biopsy of suspicious lesion to determine diagnosis (especially if possible carcinoma or pemphigus).\n\n##### TOP TIPS\n\n  Take note of self-medication. Aspirin sucked for toothache can cause a mucosal burn.\n\n  A long history of soreness with spicy or bitter foods suggests geographic tongue or median rhomboid glossitis.\n\n  A miserable, mildly febrile child with a painful tongue caused by numerous ulcers is likely to have a viral infection such as herpes simplex or hand, foot and mouth disease.\n\n  Check the skin for other lesions in obscure cases \u2013 this may reveal the diagnosis (e.g. pemphigus, lichen planus).\n\n  Patients with recurrent aphthous ulcers often erroneously believe they are deficient in vitamins \u2013 broach this concern with them.\n\n  If an ulcer in an adult fails to heal within a few weeks of presentation, refer urgently (though most oral neoplastic lesions are initially painless).\n\n  The border of geographic tongue changes shape within weeks. This is not the case with more serious pathology.\n\n  In candidal infections without an obvious cause, consider underlying diabetes or immunosuppression.\n\n  Glossodynia characteristically produces burning pain on the tip of the tongue: a 'burner' is a dentist's heartsink and the symptom may signify underlying depression.\n\n## PELVIC\n\nAcute pelvic pain\n\nChronic pelvic pain\n\nGroin swellings\n\n### ACUTE PELVIC PAIN\n\n#### The GP overview\n\nThis is nearly always seen in women rather than men. In its mildest form it is experienced universally at some time or other associated with periods, ovulation or sexual intercourse. In its severest form it is the commonest reason for urgent laparoscopic examination in the UK.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acute pelvic inflammatory disease (PID)\n\n  urinary tract infection (UTI)\n\n  miscarriage\n\n  ectopic pregnancy\n\n  ovarian cysts: torsion, rupture\n\n**OCCASIONAL**\n\n  pelvic abscess (appendix, PID)\n\n  endometriosis\n\n  pelvic congestion (exacerbation of pelvic pain syndrome)\n\n  prostatitis (men)\n\n  functional (psychosexual origin)\n\n**RARE**\n\n  misplaced IUCD (perforated uterus)\n\n  referred (e.g. spinal tumour, bowel spasm)\n\n  proctitis\n\n  invasive carcinoma of ovaries or cervix\n\n  fibroid degeneration\n\n  strangulated femoral or inguinal hernia\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ HVS, cervical swab, urinalysis, MSU.\n\n_POSSIBLE :_ FBC, ESR\/CRP, pregnancy test, ultrasound, laparoscopy (all usually arranged by hospital admitting team).\n\n_SMALL PRINT :_ none.\n\n  Urinalysis: look for nitrites and pus cells to make diagnosis of UTI.\n\n  MSU will confirm UTI and guide antibiotic treatment.\n\n  HVS for bacteria including gonococcus and endocervical swab for _Chlamydia_ if purulent discharge present.\n\n  ESR\/CRP: elevated in PID.\n\n  Pregnancy test: positive in ectopic and miscarriage.\n\n  FBC: raised WCC helps confirm PID and UTI if not being admitted. Also elevated in pelvic abscess.\n\n  Urgent ultrasound helpful if miscarriage or ectopic pregnancy suspected.\n\n  Cases referred to hospital are likely to undergo laparoscopy.\n\n##### TOP TIPS\n\n  In miscarriage, pain follows bleeding. In ectopic pregnancy, the sequence is usually reversed.\n\n  Remember that there may be no bleeding with an ectopic pregnancy \u2013 or that the vaginal loss may be a light, blackish discharge.\n\n  PV bleeding will cause haematuria on urinalysis. Only diagnose UTI if the symptoms are suggestive and urinalysis also shows nitrites and pus cells.\n\n  Severe unilateral pain and tenderness PV around 6 weeks after last menstrual period (LMP) suggests ectopic pregnancy, even with no bleeding. Admit urgently.\n\n  If PID does not settle within 48 hours of appropriate antibiotic treatment, consider abscess formation.\n\n  Don't forget to check femoral and inguinal canals for a possible strangulated hernia.\n\n### CHRONIC PELVIC PAIN\n\n#### The GP overview\n\nPelvic pain is defined as chronic if it has been present for three cycles or more. The difference between this and 'normal' period pain is one of intensity and duration. It is one of the commonest reasons for referral to a gynaecology clinic and for a woman to see her GP in the first place.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  endometriosis\n\n  chronic pelvic inflammatory disease\n\n  pelvic congestion\n\n  irritable bowel syndrome\n\n  physiological (mittelschmerz, primary dysmenorrhoea)\n\n**OCCASIONAL**\n\n  recurrent UTI\n\n  musculoskeletal pain (back pain, pubic symphysis pain)\n\n  uterovaginal prolapse\n\n  benign tumours: ovarian cyst, fibroids\n\n  chronic interstitial cystitis\n\n  IUCD\n\n  adhesions (from previous surgery)\n\n**RARE**\n\n  malignant tumours (ovary, cervix, bowel)\n\n  diverticulitis\n\n  lower colonic cancer\n\n  inflammatory bowel disease\n\n  subacute bowel obstruction\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ MSU, CA-125.\n\n_POSSIBLE :_ laparoscopy, ultrasound, HVS and cervical swab.\n\n_SMALL PRINT :_ FBC, ESR\/CRP, bowel and back imaging.\n\n  MSU detects UTI. Red cells alone may be present in interstitial cystitis.\n\n  CA-125 \u2013 especially in women aged 50 or more to help rule out carcinoma of the ovary.\n\n  FBC, ESR\/CRP: WCC, ESR\/CRP may be raised during exacerbation of chronic PID.\n\n  HVS and cervical swab for _Chlamydia_ may help in determining the infective agent in PID.\n\n  Ultrasound is helpful if there is a palpable mass or if CA-125 is elevated.\n\n  Laparoscopy is the investigation of choice for diagnosing PID, endometriosis and pelvic congestion.\n\n  Further investigations, such as bowel and back imaging, might be undertaken by the specialist after referral.\n\n##### TOP TIPS\n\n  A 'forgotten' coil can cause cyclical pelvic pain.\n\n  If the pain links with periods, establish whether it is primary or secondary dysmenorrhoea \u2013 the latter is far more likely to have a pathological cause.\n\n  In some cases the diagnosis will remain obscure. Avoid colluding with obviously erroneous diagnoses and try to adopt a constructive approach without over-investigating the patient.\n\n  Don't overlook non-gynaecological causes.\n\n  Bloating is a very common gynaecological symptom, but is characteristic of IBS. A trial of antispasmodics may aid diagnosis.\n\n  Women over 35 at first presentation and those with a mass should be referred for a gynaecological opinion.\n\n  Misdiagnosis of PID without reliable evidence will delay the real diagnosis and lead to repeated courses of unnecessary antibiotics.\n\n  Ovarian cancer nearly always presents late. Have a low threshold for investigation.\n\n  Beware the diagnosis of endometriosis. Even if confirmed at laparoscopy, remember that many women with similar findings are asymptomatic. Discuss this openly with the patient \u2013 this will help prevent dysfunction if she does not improve with anti-endometriotic treatment.\n\n### GROIN SWELLINGS\n\n#### The GP overview\n\nMost causes of lumps in the groin are non-urgent. Many patients do not realise this, however \u2013 the development of a groin swelling often heralds an urgent appointment, either because the patient fears sinister pathology, or because the patient knows the diagnosis but erroneously perceives it as an emergency. GPs generally welcome the problem, as diagnosis and disposal are usually straightforward.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  sebaceous cyst\n\n  palpable lymph nodes (LNs) \u2013 'normal' or secondary to an infection\n\n  inguinal hernia\n\n  femoral hernia\n\n  saphena varix\n\n**OCCASIONAL**\n\n  retractile testicle\n\n  abscess (local)\n\n  metastatic tumour (usually as skin-fixed lymphadenopathy)\n\n  hydrocele of spermatic cord\n\n  low appendix mass, pelvic\/inguinal tumour\n\n  lipoma\n\n**RARE**\n\n  abscess (psoas)\n\n  lymphoma\n\n  femoral artery aneurysm\n\n  neurofibroma\n\n  undescended or ectopic testis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, GUM screen.\n\n_SMALL PRINT :_ pelvic ultrasound.\n\n  FBC and ESR\/CRP useful if diffuse lymphadenopathy found, especially if no evidence of local cause or other significantly enlarged nodes found. Hb may be reduced and ESR\/CRP elevated in malignancy; WCC and ESR\/CRP elevated in abscess, infection and blood dyscrasias.\n\n  Urethral, vaginal or endocervical swabs indicated if any associated discharge and\/or suspicion of STD.\n\n  Pelvic ultrasound useful if pelvic mass suspected.\n\n##### TOP TIPS\n\n  A large saphena varix can look very much like a small hernia. Try the Valsalva test (see Ready reckoner) and look for evidence of varicose veins.\n\n  If the cause is local lymphadenopathy, look for local infective causes and don't forget to consider STDs.\n\n  Don't be surprised to find no abnormality \u2013 normal groin nodes in a slim person, and a normally retractile testis can cause great anxiety in patients and parents.\n\n  If the history suggests a hernia, but nothing is obvious on examination, get the patient to raise the intra-abdominal pressure with a vigorous cough or by raising the legs straight up while lying on the couch \u2013 and remember to examine the patient standing up, too.\n\n  Femoral herniae (commoner in women) are at high risk of strangulation, so always refer.\n\n  Undescended testis in the adult carries a high risk of malignancy. If the testis is not descended by the age of 1 year, then operative intervention is indicated.\n\n  If lymphadenopathy is the cause, look elsewhere for abnormal lymph nodes and investigate or refer if any are found. Hard, skin-fixed nodes suggest metastatic malignancy \u2013 refer urgently.\n\n  An acutely painful and irreducible groin lump suggests a strangulated or incarcerated hernia. If in any doubt, refer for urgent surgical assessment.\n\n## PERIODS\n\nAbsent periods\n\nHeavy periods\n\nIrregular vaginal bleeding\n\nPainful periods\n\n### ABSENT PERIODS\n\n#### The GP overview\n\nThis symptom causes substantial anxiety in the sexually active woman: the first unexpectedly missed period suggests pregnancy; prolonged absence raises the concern that something is seriously amiss. In contrast, management is usually straightforward, and helped by acknowledging the anxiety.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  pregnancy\n\n  physiological: rapid weight (10\u201315%) loss, and severe emotional stress\n\n  menopause (including premature ovarian failure)\n\n  polycystic ovary syndrome (PCOS)\n\n  drugs: phenothiazines, metoclopramide, valproate, cytotoxics\n\n**OCCASIONAL**\n\n  hypo- and hyperthyroidism\n\n  anorexia nervosa\n\n  excessive exercise\/training\n\n  severe systemic illness of any kind\n\n  contraception (progestogen-only pill, long-acting reversible contraception \u2013 common cause but rarely presented as a symptom)\n\n**RARE**\n\n  adrenal disorders: Addison's disease, Cushing's disease, congenital adrenal hyperplasia\n\n  Sheehan's syndrome\n\n  arrhenoblastoma, bilateral ovarian tumours\n\n  prolactinoma, other pituitary tumours\n\n  rare structural or chromosomal abnormalities (primary amenorrhoea)\n\n  anterior pituitary failure (Simmonds's disease)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ pregnancy test.\n\n_POSSIBLE :_ FBC, U&E, TFT, FSH\/LH, testosterone, prolactin, ultrasound.\n\n_SMALL PRINT :_ CT with or without other imaging.\n\n  Pregnancy test whatever contraception is used: urinary HCG. Remember small false negative rate.\n\n  FBC, U&E, TFT: to assess for general severe systemic illness, adrenal disorders and hypo- or hyperthyroidism.\n\n  FSH, LH and testosterone: LH and testosterone high in PCOS, FSH very high in menopause.\n\n  Prolactin levels high in prolactinoma and with some drugs (e.g. phenothiazines).\n\n  Ultrasound useful to show multiple ovarian cyst formation and is a reliable check of pregnancy.\n\n  Specialist will arrange CT or similar imaging if prolactinoma suspected.\n\n##### TOP TIPS\n\n  Amenorrhoea is common in young women, especially at times of stress; once pregnancy has been excluded and in the absence of any worrying symptoms or signs, only investigate if the problem persists beyond 6 months.\n\n  It is important to confirm a possible diagnosis of premature menopause, as the patient will require hormone-replacement therapy (HRT).\n\n  The same pathologies can cause both amenorrhoea and oligomenorrhoea, therefore take the same clinical approach to both.\n\n  Do not accept too readily the claim that there is 'no chance of pregnancy'; if in any doubt, arrange a pregnancy test.\n\n  Consider anorexia \u2013 an emaciated body may be well hidden under baggy clothing, and the disease often presents with the absence of periods.\n\n  Early morning headache and visual disturbance associated with amenorrhoea suggest possible intracranial pathology \u2013 refer urgently.\n\n  Before attributing amenorrhoea to weight loss, make sure that the weight loss itself hasn't been caused by thyrotoxicosis.\n\n### HEAVY PERIODS\n\n#### The GP overview\n\nThis is a common presenting complaint. The average GP can expect about 100 women to consult each year for menstrual problems (female GPs rather more) and many of these will be for menorrhagia. Normal menstrual blood loss is 20\u201380 mL. In practice, measurement is not feasible, so the definition rests on what the woman reports, although some efforts can be made to establish whether or not the bleeding is 'excessive'.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  dysfunctional uterine bleeding (DUB)\n\n  cervical or endometrial polyps\n\n  endometriosis\n\n  fibroids\n\n  puberty and perimenopause\n\n**OCCASIONAL**\n\n  hypothyroidism (and hyperthryoidism)\n\n  IUCD\n\n  iatrogenic (contraceptive pill, HRT)\n\n  cystic glandular hyperplasia (metropathia haemorrhagica)\n\n  chronic PID\n\n**RARE**\n\n  adrenal disorders and hyperprolactinaemia\n\n  liver disease, especially alcoholic\n\n  clotting disorder\n\n  endometrial carcinoma\n\n  tuberculous endometritis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC.\n\n_POSSIBLE :_ TFT, ESR\/CRP, transvaginal ultrasound and, after referral, endometrial sampling and hysteroscopy.\n\n_SMALL PRINT :_ LFT, HVS, clotting studies, endocrine assays.\n\n  FBC to check for anaemia and thrombocytopenia.\n\n  Check possible thyroid dysfunction with TFT.\n\n  ESR\/CRP: elevated in PID.\n\n  LFT: if clinical suspicion of liver disease.\n\n  Clotting studies: if other history of abnormal bleeding or bruising.\n\n  Transvaginal ultrasound useful for confirming fibroids and suggesting endometrial pathology.\n\n  HVS very occasionally useful in chronic PID with discharge.\n\n  Endocrine assays: for hyperprolactinaemia or adrenal disorders.\n\n  Investigation after referral is likely to include endometrial sampling and\/or hysteroscopy.\n\n##### TOP TIPS\n\n  Self-reporting of the heaviness of the menstrual flow is notoriously unreliable. Attempt an objective assessment by enquiring about the number of pads or tampons used, flooding, the presence of clots, and by checking an FBC for iron-deficiency anaemia.\n\n  Establish the woman's agenda. This presentation may be the passport to a prescription (e.g. the contraceptive pill in a young woman) or to discussion of a specific anxiety (e.g. fears about possible cancer or a need for hysterectomy).\n\n  Don't forget to enquire about a 'long-forgotten' coil.\n\n  In a young woman who has painless heavy periods, is otherwise well and has a normal vaginal examination, it is reasonable to make a presumptive diagnosis of DUB and treat empirically.\n\n  Establish whether the problem really is simply 'heavy periods'; if the bleeding is chaotic, or there is also intermenstrual or post-coital bleeding, the chances of a structural lesion are much higher \u2013 ensure the patient is appropriately investigated.\n\n  Blood clots suggest significant bleeding; do not forget to arrange an FBC.\n\n  Menorrhagia with secondary dysmenorrhoea, dyspareunia and pelvic tenderness on examination suggest endometriosis or chronic PID.\n\n### IRREGULAR VAGINAL BLEEDING\n\n#### The GP overview\n\nIrregular vaginal bleeding presents commonly in primary care \u2013 particularly to female GPs, as the patient will often anticipate a pelvic examination. This chapter covers all causes of this symptom throughout life including prepubertal, causes in early pregnancy and postmenopausal (causes in late pregnancy are not covered as they constitute a quite different clinical scenario). The key to appropriate management usually lies in a careful history.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  dysfunctional uterine bleeding (DUB)\n\n  breakthrough bleeding (BTB) on contraceptive pills and long acting reversible contraception; also HRT\n\n  cervical polyp or erosion\n\n  cervicitis and PID\n\n  post-menopausal atrophic vaginitis\n\n**OCCASIONAL**\n\n  endometrial polyps\n\n  ovulatory bleeding (associated with mittelschmerz)\n\n  hypothyroidism (and less commonly hyperthyroidism)\n\n  perimenopause\n\n  bleeding during early pregnancy (20% of all pregnancies in first trimester); also miscarriage, ectopic pregnancy\n\n**RARE**\n\n  uterine bleeding in the newborn\n\n  carcinoma (ovary, fallopian tube, uterus, cervix and vagina)\n\n  cystic glandular hyperplasia (metropathia haemorrhagica)\n\n  pyometra\n\n  hydatidiform mole (5% go on to chorionic carcinoma)\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, specialised gynaecological investigation (see below) for post-menopausal or intermenstrual bleeding.\n\n_POSSIBLE :_ HVS and endocervical swab, colposcopy, transvaginal ultrasound.\n\n_SMALL PRINT :_ TFT.\n\n  FBC to check for anaemia in heavy bleeding. Raised WCC in PID.\n\n  TFT: to check for possible thyroid dysfunction.\n\n  HVS and endocervical swab: to attempt to establish pathogen in cervicitis and PID.\n\n  Transvaginal ultrasound: to detect uterine and ovarian pathology, hydatidiform mole and to establish nature of problem in early pregnancy.\n\n  Colposcopy: if significant cervical pathology suspected.\n\n  Specialised gynaecological investigation (e.g. hysteroscopy and endometrial sampling): performed in secondary care, particularly for intermenstrual and postmenopausal bleeding.\n\n##### TOP TIPS\n\n  Try to distinguish between intermenstrual bleeding and irregular periods, as the likely causes are different (the former suggests a structural lesion, the latter is likely to be DUB). Simply asking the patient if the bleed feels like a period, with associated period-type symptoms, may help.\n\n  Consider asking the patient to keep a menstrual diary. Very slight bleeding occurring consistently in mid-cycle with a slight pain and with no other worrying features suggests an ovulatory bleed.\n\n  Remember that missed doses, diarrhoea and vomiting, and the first few months of treatment can cause breakthrough bleeding when using oral hormonal preparations. It is worth waiting a few more cycles before changing treatment.\n\n  Post-menopausal bleeding is always abnormal. Even if atrophic vaginitis is present, do not assume this is the cause: this symptom requires a full assessment.\n\n  A very inflamed-looking cervix with a purulent discharge is likely to be caused by _Chlamydia_ infection: consider referral to the local GUM clinic for further investigation and contact tracing.\n\n  A 'recent' normal cervical smear can provide false reassurance; if the cervix looks suspicious, do not take another smear \u2013 remember, this is a screening, rather than diagnostic, test. Instead, refer urgently for colposcopy.\n\n  Beware of 'breakthrough bleeding' in patients on hormonal contraception or HRT. If it persists, consider other causes and investigate or refer.\n\n  Unilateral pelvic pain with vaginal bleeding within a fortnight of a missed period suggests an ectopic pregnancy. Admit urgently.\n\n### PAINFUL PERIODS\n\n#### The GP overview\n\nPainful periods are extremely common: 50% of women in the UK complain of moderate pain, and 12% suffer severe, disabling pain. Primary dysmenorrhoea is pain with no organic pathology, usually starting when ovulatory cycles begin. Secondary dysmenorrhoea is associated with pelvic pathology, and appears later in life.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  primary dysmenorrhoea\n\n  endometriosis\n\n  chronic PID\n\n  IUCD\n\n  pelvic pain syndrome ('venous congestion')\n\n**OCCASIONAL**\n\n  retroverted uterus\n\n  cervicitis\n\n  chocolate cyst of ovary\n\n  endometrial polyp\n\n**RARE**\n\n  uterine malformation\n\n  imperforate hymen\n\n  uterine hypoplasia\n\n  cervical stenosis\n\n  psychogenic\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ FBC, ESR\/CRP, HVS and endocervical swab, ultrasound, laparoscopy.\n\n_SMALL PRINT :_ none.\n\n  FBC for anaemia if periods also heavy. WCC and ESR\/CRP raised in PID.\n\n  HVS and endocervical swab for _Chlamydia_ if vaginal discharge present \u2013 may help establish pathogen in chronic PID.\n\n  Ultrasound helpful to define uterine enlargement or other abnormalities and to detect ovarian cysts.\n\n  Laparoscopy is the usual investigation after referral to secondary care: will make diagnosis of PID and endometriosis.\n\n##### TOP TIPS\n\n  Explore the patient's agenda: young adolescents may use the symptom of painful periods as a passport symptom to obtain a prescription for the contraceptive pill.\n\n  Beware of the diagnosis of endometriosis. Even if detected laparoscopically, this may not be the actual cause of the patient's pain (endometriosis is often asymptomatic). Unless it is explained that the treatment offered is not necessarily a panacea, the patient is likely to get frustrated at the apparent lack of progress.\n\n  A long-forgotten IUCD can be a cause of dysmenorrhoea. Enquire specifically about this possibility \u2013 and check the notes too.\n\n  Explain to patients early on in your management that a precise organic diagnosis isn't always possible; this will help maintain a good doctor\u2013patient relationship, which will facilitate the subsequent unravelling of any significant underlying psychological problems.\n\n  The chances of organic pathology are greater if the patient has secondary dysmenorrhoea which is severe enough to disturb sleep.\n\n  Half of women who undergo laparoscopy for secondary dysmenorrhoea have no obvious organic pathology. Consider psychological problems and avoid over-investigating and over-referring \u2013 a number of these women end up having surgery (e.g. TAH) but even then continue to have pain.\n\n  Consider other pathologies if the patient presents acutely with a self-diagnosis of 'severe period pain': non-gynaecological causes of pelvic pain such as appendicitis, renal colic or UTI may occur at the expected time of the period.\n\n  While painful periods are depressing, true clinical depression may lower the pain threshold of an otherwise normal woman and should not be missed.\n\n## SKIN\n\nBlisters (vesicles and bullae)\n\nErythema\n\nMacules\n\nNodules\n\nPapules\n\nPurpura and petechiae\n\nPustules\n\nScales and plaques\n\n### BLISTERS (VESICLES AND BULLAE)\n\n#### The GP overview\n\nBlisters are skin swellings containing free fluid. Up to 5 mm they are called vesicles, larger than 5 mm they are called bullae. The fluid can be lymph, serum, extracellular fluid or blood. Some conditions cause both kinds of blister, but others mainly one or other type. Pustules are dealt with elsewhere (see p. 403).\n\n#### Differential diagnosis\n\n**COMMON**\n\n  trauma: skin friction, burns (thermal and chemical), insect bites\n\n  herpes simplex\n\n  herpes zoster\n\n  childhood viruses: hand, foot and mouth disease, chickenpox\n\n  eczema (pompholyx and other acute eczemas)\n\n**OCCASIONAL**\n\n  pemphigus and pemphigoid\n\n  dermatitis herpetiformis\n\n  secondary to leg oedema (various causes)\n\n  bullous impetigo\n\n  drug reactions, e.g. ACE inhibitors, penicillamine, barbiturates\n\n  erythema multiforme\n\n**RARE**\n\n  pemphigoid (herpes) gestationis\n\n  porphyria\n\n  toxic epidermal necrolysis (Lyell's syndrome)\n\n  epidermolysis bullosa\n\n  allergic vasculitis\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are unlikely to be any investigations that will prove useful in primary care. Usually, the problem is either self-limiting or the cause obvious; in obscure cases, referral may result in skin biopsy to establish the diagnosis. Patch testing may also be useful to identify possible allergens in contact dermatitis, especially if occupational.\n\n##### TOP TIPS\n\n  Herpes zoster involving the ophthalmic division of the trigeminal nerve will affect the eye in about 50% of cases. The likelihood is increased if there are blisters on the side of the nose. Ensure that the patient knows to seek help urgently if the eye becomes red or painful, or there is blurring of vision.\n\n  In uncomplicated herpes zoster, explain the natural history of the condition to the patient, resolving any worries (old wives' tales abound) and warning about the possibility of post-herpetic neuralgia.\n\n  Follow up unexplained rashes. The bullae of pemphigoid, for example, may be preceded by itching, erythema and urticaria by several weeks.\n\n  Herpes simplex and varicella zoster infections can become severe and disseminated in the immunosuppressed: admit. Similarly, herpes simplex can result in a serious reaction (Kaposi's varicelliform eruption) in patients with atopic eczema.\n\n  Pregnant women with chickenpox are at significant risk of severe varicella pneumonia; there are also risks to the foetus. Follow the detailed guidance in the 'Green Book' ( _Immunisation against Infectious Disease_ , Her Majesty's Stationery Office) when dealing with pregnant women who have been in contact with chickenpox.\n\n  Toxic epidermal necrolysis (scalded skin syndrome) can develop rapidly in infants and children, causing serious illness. Admit urgently if you suspect this diagnosis.\n\n  Pemphigus is a serious condition affecting a younger age group (usually middle-aged) than pemphigoid. Inpatient care is usually required.\n\n### ERYTHEMA\n\n#### The GP overview\n\nErythema is a reddening of the skin due to persistent dilation of superficial blood vessels, and can be local or generalised. It is distinguished from flushing (see p. 197) by its permanence: flushing is transient.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  cellulitis\n\n  gout\n\n  burns: thermal, chemical, sunburn\n\n  toxic erythema: drugs (e.g. antibiotics, NSAIDs), bacteria (e.g. scarlet fever), viruses (e.g. measles, slapped cheek syndrome)\n\n  rosacea\n\n**OCCASIONAL**\n\n  palmar erythema, e.g. pregnancy, liver disease, thyrotoxicosis\n\n  phototoxic reaction to drugs, e.g. phenothiazines, tetracyclines, diuretics\n\n  'deck-chair legs' (prolonged immobility)\n\n  erythema multiforme (various causes)\n\n  systemic lupus erythematosus (erythematous, photosensitive butterfly rash)\n\n  erythema ab igne (reticulate pattern)\n\n**RARE**\n\n  fixed drug eruption\n\n  livedo reticularis: connective tissue disease\n\n  seroconversion rash of HIV\n\n  erythema nodosum: sarcoidosis, streptococci, tuberculosis, drugs\n\n  erythema induratum (Bazin's disease: tuberculosis)\n\n  erythema chronicum migrans: Lyme disease\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ uric acid (if possible gout).\n\n_POSSIBLE :_ FBC, ESR\/CRP, LFT, TFT.\n\n_SMALL PRINT :_ autoimmune studies, serology, CXR, ASO titre.\n\n  FBC\/ESR\/CRP: WCC and ESR\/CRP raised in significant infection; Hb may be reduced (normochromic normocytic) in connective tissue disorder.\n\n  Autoimmune studies: if connective tissue disorder a possibility.\n\n  Serology: may help if suspect infective cause for erythema multiforme; also useful in assessing immune status in a pregnant woman exposed to slapped cheek syndrome, and in diagnosis of HIV infection and Lyme disease.\n\n  Uric acid: to confirm clinical suspicion of gout (when attack has subsided) especially if considering allopurinol.\n\n  LFT, TFT: if palmar erythema present in non-pregnant patient \u2013 to detect alcohol excess or hyperthyroidism.\n\n  Other investigations for erythema nodosum: if a non-drug cause is possible, investigations likely to include CXR (for TB, sarcoidosis) and ASO titre (for streptococcal infection).\n\n##### TOP TIPS\n\n  Toxic erythema caused by drugs tends to be itchy; if due to infection, it does not irritate but is accompanied by fever.\n\n  Remember that there is often a delay before a drug causes toxic erythema \u2013 therefore, symptoms may only appear after a course of treatment (especially antibiotics) has been completed.\n\n  'Deck-chair legs' is erythema of the lower legs, sometimes with oedema and blistering, in the immobile. It tends to be mistakenly diagnosed as persistent or recurrent cellulitis.\n\n  A violent local erythema, rapidly darkening and blistering and recurring at the same site, suggests a fixed drug eruption.\n\n  Erythema nodosum and multiforme may be caused by significant disease, including, very occasionally, malignancy. If the patient is generally unwell or has other significant symptoms, investigate urgently or refer.\n\n  Take a travel history: Lyme disease is endemic in forested areas. If not diagnosed and treated early, it can have significant complications.\n\n  Erythema multiforme with blistering and ulceration of the mucous membranes is Stevens\u2013Johnson syndrome. Though rare, it is a very serious illness requiring urgent hospital treatment.\n\n  Enquire about joint symptoms: many causes of erythema (e.g. erythema multiforme, butterfly rash, livedo reticularis) are linked to a connective tissue isorder.\n\n  Remember to take a drug history, including over-the-counter medications. This may reveal the underlying cause in toxic erythema, erythema nodosum and multiforme, and phototoxicity.\n\n  Remember that parvovirus can cause serious problems in pregnancy \u2013 check serology in women with suggestive symptoms, or exposure to a case.\n\n### MACULES\n\n#### The GP overview\n\nA macule is a flat, demarcated, abnormally coloured area of skin of any size. It may be red (e.g. drug eruption), dark red (e.g. purpura), brown (e.g. a flat mole) or white (e.g. pityriasis versicolor). Purpura is described elsewhere (see p. 400). There is some crossover between erythema (see p. 390) and red macules.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  drug reaction\/allergy\n\n  flat mole (junctional naevus)\n\n  non-specific viral exanthem\n\n  sun-induced freckles (including solar lentigines)\n\n  chloasma\n\n**OCCASIONAL**\n\n  measles and rubella\n\n  post-inflammatory hypo- or hyperpigmentation\n\n  _caf\u00e9 au lait_ spot (creamy brown) and Mongolian spot (brown or slate-grey)\n\n  Berloque dermatitis (brown: chemical photosensitisation, e.g. bergamot oil)\n\n  depigmentation: vitiligo, pityriasis versicolor, pityriasis alba\n\n**RARE**\n\n  infections: macular syphilide, tuberculoid leprosy, typhoid (rose spots in 40%)\n\n  Albright's syndrome\n\n  neurofibromatosis (associated with more than six _caf\u00e9 au lait_ spots)\n\n  pathological freckles: Hutchinson's freckle, Peutz\u2013Jeghers syndrome\n\n  naevus anaemicus (permanent vasoconstriction due to neurovascular abnormality)\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are very few relevant investigations to consider and they would be required only exceptionally, as the diagnosis is usually clinical: skin scrapings for mycology or fluorescence under Wood's light may help in the diagnosis of pityriasis versicolor; acute and convalescent serum samples may confirm rubella; serology for syphilis may be appropriate with an unusual macular rash; and very occasionally, a skin biopsy may be required to clinch an obscure diagnosis.\n\n##### TOP TIPS\n\n  A drug eruption can take 2 weeks to appear from the time of the first dose \u2013 so don't be misled by the fact that a course of antibiotics may have been completed some days before the related drug rash develops.\n\n  Pityriasis versicolor may be misdiagnosed as vitiligo. If in doubt, take scrapings for mycology or examine under Wood's light.\n\n  Odd lines of hyperpigmentation on the sides of the neck are likely to be Berloque dermatitis \u2013 a photosensitive rash caused by oil of bergamot, present in perfumes.\n\n  Hutchinson's freckle is a giant, variegated freckle, seen in elderly sun-exposed skin. There is a high risk of malignant change, so refer.\n\n  Rubella is rare, but may become commoner as a result of media coverage of 'immunisation scares'. Establish whether or not a young woman presenting with a rubella-type rash is pregnant \u2013 if she is, confirm her rubella status.\n\n  A child with very many freckles on and around the lips may have Peutz\u2013Jeghers syndrome. This is associated with small bowel polyposis.\n\n  Vitiligo tends to have a poor prognosis in Caucasians, especially if it is widespread and affecting lips and extremities.\n\n### NODULES\n\n#### The GP overview\n\nSkin nodules are larger than papules \u2013 more than 5 mm diameter. However, their depth is more significant clinically than their width. Some are free within the dermis; others are fixed to skin above or subcutaneous tissue below. The causes are various; the patient is usually concerned about the cosmetic appearance or malignant potential.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  sebaceous cyst\n\n  lipoma\n\n  basal cell carcinoma (BCC)\n\n  warts\n\n  xanthoma\n\n**OCCASIONAL**\n\n  dermatofibroma (histiocytoma)\n\n  squamous cell carcinoma\n\n  nodulocystic acne\n\n  kerato-acanthoma\n\n  gouty tophi\n\n  chondrodermatitis nodularis chronica helicis\n\n  rheumatoid nodules and Heberden's nodes\n\n  pyogenic granuloma\n\n**RARE**\n\n  malignant melanoma (becoming more common in the UK)\n\n  vasculitic: erythema nodosum, nodular vasculitis, polyarteritis nodosa\n\n  atypical infections (e.g. leprosy, treponema, lupus vulgaris, fish tank and swimming pool granuloma, actinomycosis)\n\n  lymphoma and metastatic secondary carcinoma\n\n  sarcoidosis\n\n  pretibial myxoedema\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none (skin biopsy or cytology if doubt about the lesion or clinical diagnosis of possible carcinoma).\n\n_POSSIBLE :_ lipid profile, FBC, ESR\/CRP, urate, rheumatoid factor, urinalysis.\n\n_SMALL PRINT :_ TFT, Kveim test, further investigations guided by clinical picture (see below).\n\n  Excision biopsy is the definitive investigation for achieving a diagnosis; cytology from skin scrapings can be used to diagnose BCC.\n\n  Lipid profile: xanthomata require a full lipid profile to define the underlying hyperlipidaemia.\n\n  Urinalysis: if suspect inflammatory or vasculitic skin lumps, as may reveal proteinuria if associated with systemic and renal disorders.\n\n  FBC and ESR\/CRP: ESR\/CRP raised in inflammatory disorders and malignancy; may also reveal anaemia of chronic disease or malignancy (including lymphoma).\n\n  Check urate if gouty tophi are clinically likely.\n\n  Rheumatoid factor: nodules are usually associated with positive rheumatoid factor.\n\n  TFT: to diagnose Graves's disease with pretibial myxoedema.\n\n  Kveim test: may contribute to a diagnosis of sarcoidosis.\n\n  Further investigations according to clinical picture: some lesions, such as erythema nodosum, may require further investigation to establish the underlying cause; histological confirmation of skin secondaries may similarly require further assessment, although the overall condition of the patient may mean this is a futile exercise.\n\n##### TOP TIPS\n\n  Look at the lesion under the magnifying glass \u2013 this may reveal suspicious signs such as ulceration or a rolled, pearly edge.\n\n  In uncertain cases which do not require urgent attention, record your findings carefully (including precise dimensions) and review in a month or two.\n\n  Stoical patients may underestimate the significance of a suspicious lesion, particularly if you discover it during a routine examination \u2013 if you are referring them for biopsy, impress upon them the need to attend their appointment.\n\n  Establish the patient's concern, which will usually centre on worries about cosmetic appearance or cancer. This will result in a more functional consultation and a more satisfied patient.\n\n  Night sweats and itching with skin nodules raises the suspicion of lymphoma. Examine lymph nodes, liver and spleen carefully.\n\n  The elderly patient complaining of a lesion in a sun-exposed area which 'just won't heal' may well have a squamous or basal cell carcinoma.\n\n  The appearance of a nodule in a mole is highly significant and requires referral.\n\n  A patient with nodulocystic acne requires referral to a dermatologist for possible treatment with 13- _cis_ -retinoic acid.\n\n  The unwell middle-aged or elderly patient who develops bizarre and widespread skin nodules over a period of a few weeks probably has an underlying carcinoma.\n\n### PAPULES\n\n#### The GP overview\n\nPapules are solid, circumscribed skin elevations up to 5 mm in diameter. If they are larger, they are called nodules \u2013 these are dealt with elsewhere (see p. 395). (Clearly, many nodules start life as a papule; to avoid confusion, if they are generally 'nodular' by the time they present to the GP, then they are dealt with in that section, and not repeated here.) They are usually round but the shape, and the colour, may vary. They may be transitional lesions, e.g. becoming vesicular, or about to ulcerate.\n\n_NOTE:_ there are more causes of papules than can be listed here. This is a sensible selection.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  acne\n\n  scabies\n\n  viral wart and molluscum contagiosum\n\n  Campbell de Morgan spot\n\n  skin tag\n\n**OCCASIONAL**\n\n  viral illness\n\n  milia\n\n  insect bites\n\n  early seborrhoeic wart\n\n  xanthomata\n\n  guttate psoriasis\n\n  pityriasis lichenoides chronica, lichen planus\n\n  prickly heat\n\n  keratosis pilaris\n\n**RARE**\n\n  malignant melanoma, early basal cell carcinoma, Kaposi's sarcoma\n\n  Darier's disease\n\n  acanthosis nigricans\n\n  pseudoxanthoma elasticum\n\n  tuberous sclerosis\n\n#### Ready reckoner\n\n#### Possible investigations\n\nIn practice, very few investigations are needed with this presentation: a lipid screen is required in the presence of xanthomata; genital warts require referral for screening for other STDs; thorough investigation may be needed in the very rare case where underlying malignancy is possible (e.g. acanthosis nigricans); and obscure rashes or solitary papules may occasionally require excision biopsy for a definitive diagnosis.\n\n##### TOP TIPS\n\n  Bear in mind that skin cancer is usually uppermost in the patient's mind, especially in subacute or chronic cases \u2013 so provide appropriate reassurance.\n\n  In obscure solitary lesions, record clinical findings carefully and arrange to review in due course.\n\n  Itchy, asymmetrical grouped papules are likely to be insect bites, although the patient may take some convincing!\n\n  An enlarging dark blue or blue-black papule may be a malignant melanoma, blue naevus or Kaposi's sarcoma. Refer for urgent opinion.\n\n  Brown, skin-coloured papules crowded around the nose of a child may be tuberous sclerosis. This can be associated with serious systemic pathology so refer for expert opinion.\n\n  An intensely itchy papular rash which is worse at night and has no other obvious cause is likely to be scabies, even if scabetic burrows are not evident \u2013 treat on suspicion.\n\n### PURPURA AND PETECHIAE\n\n#### The GP overview\n\nPurpura are reddish-purple lesions which do not blanch with pressure. When less than 1 cm in diameter, they are called petechiae; if larger, they are known as ecchymoses. The problem often presents as 'bruising easily' \u2013 a common 'while I'm here' complaint in primary care. Most cases are normal, with causative minor trauma simply being forgotten or unnoticed.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  trauma\n\n  senile purpura\n\n  liver disease (especially alcoholic cirrhosis)\n\n  increased intravascular pressure, e.g. coughing, vomiting, gravitational\n\n  drugs, e.g. steroids, warfarin, aspirin\n\n**OCCASIONAL**\n\n  vasculitis, e.g. Henoch\u2013Sch\u00f6nlein purpura, connective tissue disorders\n\n  thrombocytopenia, e.g. idiopathic thrombocytopenia purpura (ITP), bone marrow damage (e.g. lymphoma, leukaemia, cytotoxics), aplastic anaemia\n\n  renal failure\n\n  infective endocarditis\n\n**RARE**\n\n  paraproteinaemias, e.g. cryoglobulinaemia\n\n  inherited clotting disorders, e.g. haemophilia, Christmas disease, von Willebrand's disease\n\n  infections, e.g. meningococcal septicaemia, Rocky Mountain spotted fever\n\n  vitamin C and K deficiency\n\n  disseminated intravascular coagulation (DIC)\n\n  congenital vessel wall abnormalities, e.g. Ehlers\u2013Danlos syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ FBC, ESR\/CRP, INR (if on warfarin).\n\n_POSSIBLE :_ LFT, U&E, coagulation screen, plasma electrophoresis.\n\n_SMALL PRINT :_ autoimmune testing, further hospital investigations (see below).\n\n  FBC, ESR\/CRP: FBC may reveal thrombocytopenia or evidence of blood dyscrasia. ESR\/CRP and WCC may be raised in blood dyscrasia, connective tissue disorder and infection.\n\n  LFT, U&E: for underlying liver or renal disease.\n\n  INR: if on warfarin.\n\n  Autoimmune testing: if possible connective tissue disease causing vasculitis.\n\n  Coagulation screen: to test haemostatic function, e.g. bleeding time, prothrombin time, activated partial thromboplastin time.\n\n  Plasma electrophoresis: for hypergammaglobulinaemia, paraproteinaemia and cryoglobulinaemia.\n\n  Further investigations (usually secondary care): to investigate underlying cause, e.g. skin biopsy to confirm vasculitis, bone marrow biopsy if possible marrow infiltrate.\n\n##### TOP TIPS\n\n  Multiple bruises of varying ages on the legs of young children who are otherwise well and have no other stigmata of clotting disorder or abuse are likely to be non-pathological.\n\n  Senile purpura should be easy to diagnose from the history and examination. Reassurance, rather than investigation, is required.\n\n  A few petechiae on or around the eyelids of a well child can be caused by vigorous coughing or vomiting. If the history is clear, explain the cause to the parents \u2013 but emphasise that the appearance of an identical rash elsewhere requires urgent attention.\n\n  The distribution of purpura can give useful clues to the diagnosis. On the legs, platelet disorders, paraproteinaemias, Henoch\u2013Sch\u00f6nlein purpura or meningococcal septicaemia are likely; lesions on the fingers and toes indicate vasculitis; and senile and steroid purpura tend to affect the back of the hands and arms.\n\n  Purpura caused by vasculitis tend to be raised palpably above the skin.\n\n  Remember that the rash of meningococcal septicaemia can appear before a child is obviously systemically unwell. If there is any suspicion that this might be the diagnosis, give parenteral penicillin and admit immediately.\n\n  The absence of a family history does not exclude a significant inherited bleeding disorder: these disorders can arise spontaneously.\n\n  Take very seriously the pale patient with purpura: a bone marrow problem is likely. Arrange an urgent FBC or admit.\n\n  Always do a full surface examination of a child with bruising, not forgetting the anogenital area. Keep non-accidental injury in mind.\n\n  Never give an IM injection if a serious bleeding disorder is suspected.\n\n### PUSTULES\n\n#### The GP overview\n\nPustules are raised lesions less than 0.5 cm in diameter containing a yellow fluid. They signify infection to most people, and will often present in an urgent appointment, as they are likely to have appeared suddenly. Patients will often expect antibiotic treatment. This will not always be necessary, so be prepared to offer a clear explanation and an appropriate alternative.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  impetigo\n\n  other staphylococcal infections, e.g. early boils, folliculitis, sycosis barbae\n\n  herpes simplex and zoster\n\n  acne vulgaris\n\n  rosacea\n\n**OCCASIONAL**\n\n  perioral dermatitis\n\n  hidradenitis suppurativa (axillae and groins)\n\n  candidiasis (satellite vesicopustules around moist eroded patch)\n\n  pustular psoriasis (palmar and plantar commoner than generalised pustular psoriasis of von Zumbusch)\n\n**RARE**\n\n  dermatitis herpetiformis\n\n  Beh\u00e7et's syndrome\n\n  viral: cowpox and orf ( _Note:_ chickenpox is vesicular, not pustular)\n\n  hot tub folliculitis (superficial _Pseudomonas_ infection)\n\n  drug induced\n\n#### Ready reckoner\n\n#### Possible investigations\n\nThere are very few investigations likely to prove useful or necessary in primary care. The presence of widespread or recurrent candidal or staphylococcal lesions might necessitate a urinalysis or blood sugar to exclude diabetes; a swab of pus may help confirm a clinically suspected infective agent; and in very obscure cases, a skin biopsy might prove helpful.\n\n##### TOP TIPS\n\n  Take time to explain to the patient the nature of the problem in recurrent staphylococcal infections. Exclude diabetes, check carrier sites and reassure that the patient's 'hygiene' is not in question. A prolonged course of antibiotics may be helpful.\n\n  Check self-treatment in rosacea and perioral dermatitis. Treatment with OTC topical steroids will exacerbate the problem. Warn the patient that the condition may worsen before it improves on withdrawal of this inappropriate treatment.\n\n  Papules and pustules around the mouth and eyes, often with a halo of pallor around the lip margin, are caused by perioral dermatitis. Treat with antibiotics, not topical steroids.\n\n  Widespread, severe and recurrent staphylococcal lesions suggest diabetes or possible immunosuppression.\n\n  Localised pustular psoriasis can be very resistant to standard treatments, so have a low threshold for referral. The very rare generalised form can make the patient dangerously ill: admit urgently.\n\n  Remember that herpes simplex or zoster infections in the immunocompromised can become disseminated and severe.\n\n  Ocular problems in rosacea can be complicated and troublesome: refer to an ophthalmologist.\n\n### SCALES AND PLAQUES\n\n#### The GP overview\n\nSkin scales represent an abnormally fast piling up of keratinised epithelium. Scales and plaques are common at all ages and have a variety of causes. The presentation will centre on cosmetic appearance, itching, fears about serious disease or a combination of these.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  psoriasis\n\n  eczema (in all its various forms)\n\n  fungal infections (e.g. scalp, body, feet)\n\n  seborrhoeic dermatitis\n\n  seborrhoeic keratosis\n\n**OCCASIONAL**\n\n  lichen simplex\n\n  lichen planus (usually scaly only on legs)\n\n  solar keratosis\n\n  pityriasis versicolor and rosea\n\n  juvenile plantar dermatosis\n\n  guttate psoriasis (scaly papules)\n\n**RARE**\n\n  malignancy: Bowen's disease and mycosis fungoides (cutaneous T-cell lymphoma)\n\n  drug induced (e.g. \u03b2-blockers and carbamazepine)\n\n  ichthyosis (various forms)\n\n  keratoderma blenorrhagica (part of Reiter's syndrome)\n\n  pityriasis lichenoides chronica\n\n  secondary syphilis\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ none.\n\n_POSSIBLE :_ Wood's light, skin scrapings\/hair samples, patch testing.\n\n_SMALL PRINT :_ skin biopsy, syphilis serology, FBC, ESR\/CRP, fasting glucose or HbA1c.\n\n  Green fluorescence under UV (Wood's) light is diagnostic of microsporum fungal infection.\n\n  Skin scrapings and hair samples for mycology: will help differentiate fungal infections from similar rashes.\n\n  Skin biopsy: may be the only way to achieve a firm diagnosis in obscure rashes, and is essential if malignancy suspected.\n\n  Patch testing: to establish the likely allergen in allergic contact eczema.\n\n  Syphilis serology: if justified by clinical features or obscure pattern.\n\n  FBC and ESR\/CRP: may suggest significant underlying disease (e.g. T-cell lymphoma); ESR\/CRP also elevated in Reiter's disease.\n\n  Fasting glucose or HbA1c: check for diabetes in recurrent fungal infections.\n\n##### TOP TIPS\n\n  To help distinguish between fungal and eczematous rashes, look at the symmetry and edges of the lesions. Fungal rashes are usually asymmetrical with a scaly, raised edge.\n\n  In the presence of a fungal rash, look for infection elsewhere (e.g. groins and feet) and treat both areas, otherwise the problem is likely to recur.\n\n  In uncertain cases, explain to the patient that the real diagnosis may only become apparent as the rash develops (the typical example being the herald patch of pityriasis rosea looking like initially tinea corporis) \u2013 invite the patient to return for reassessment if your initial treatment proves unsuccessful.\n\n  A symmetrical, glazed, scaly and fissured rash on the soles of a trainer-loving child or adolescent is juvenile plantar dermatosis.\n\n  Eight per cent of people with psoriasis will have arthropathy, which is usually also associated with nail changes. Check the nails and ask about joint symptoms in patients with psoriasis.\n\n  Erythroderma \u2013 universal redness and scaling caused, rarely, by psoriasis, eczema, mycosis fungoides and drug eruptions \u2013 renders the patient systemically unwell. Urgent inpatient treatment is required.\n\n  A solitary, well-defined, slowly growing scaly plaque on the face, hands or legs of the middle-aged or elderly patient is probably Bowen's disease \u2013 but it can easily be mistaken for an isolated patch of eczema or psoriasis.\n\n  If a pityriasis rosea-like rash extends to the palms and soles, with fever, malaise, sore throat and lymphadenopathy, consider secondary syphilis.\n\n## URINARY\n\nBlood in urine\n\nExcessive urination\n\nFrequency\n\nIncontinence\n\nNocturia\n\nRetention\n\n### BLOOD IN URINE\n\n#### The GP overview\n\nBright red blood in the urine causes instant alarm in a patient, and usually generates an emergency appointment or an out-of-hours call. Blood may also be picked up by dipstick testing or MSU during the assessment of some other problem or in a routine medical. This is often less frightening even when disclosed to the patient, but should prompt full investigation.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  UTI\n\n  bladder tumour\n\n  renal\/ureteric stones\n\n  urethritis\n\n  prostatic hypertrophy\/carcinoma of prostate\n\n**OCCASIONAL**\n\n  jogging and hard exercise\n\n  renal carcinoma\n\n  chronic interstitial cystitis\n\n  anticoagulant therapy\n\n  nephritis\/glomerulonephritis\n\n**RARE**\n\n  renal tuberculosis\n\n  polycystic kidney disease\n\n  blood dyscrasias: thrombocytopenia, haemophilia, sickle-cell disease\n\n  infective endocarditis\n\n  schistosomiasis (common abroad)\n\n  trauma\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU, FBC, U&E, ACR\/PCR.\n\n_POSSIBLE :_ PSA, ultrasound, plain abdominal X-ray, IVU, cystoscopy.\n\n_SMALL PRINT :_ urethral swab, CT scan, urine cytology, renal biopsy, angiography.\n\n  Urinalysis: pus cells and nitrite in UTI. Pus cells alone in urethritis, TB and bladder tumour. Presence of protein suggests renal disease.\n\n  Urine microscopy and culture to establish pathogen in infection. May show casts in renal disease.\n\n  FBC and U&E help establish basic renal function and any associated anaemia or leucocytosis; consider PSA \u2013 usually elevated in prostatic carcinoma.\n\n  ACR\/PCR: to quantify any proteinuria.\n\n  Urethral swabs if urethritis (best done at GUM clinic).\n\n  If painless haematuria, ultrasound may show renal tumour or polycystic kidneys; CT may be more useful.\n\n  IVU is investigation of choice if renal\/ureteric stones are suspected (when pain is present); plain abdominal X-ray useful when attack has settled (reveals 90% of stones). IVU also required if ultrasound, abdominal X-ray and cystoscopy are all negative.\n\n  Specialist investigations include cystoscopy, urinary cytology, renal biopsy and angiography.\n\n##### TOP TIPS\n\n  Microscopic haematuria in an asymptomatic menstruating woman can be ignored temporarily; repeat the urinalysis at mid-cycle.\n\n  Remember that there are other less common causes of spurious haematuria \u2013 sometimes the blood may be coming from the rectum or vagina. Assess each case carefully and be prepared to rethink if symptoms persist but urological investigations prove negative.\n\n  Some food pigments, beetroot and certain drugs (e.g. nitrofurantoin) can colour the urine red \u2013 confirm haematuria with urinalysis to save the patient unnecessary tests.\n\n  Painless frank haematuria is an ominous sign indicating possible malignancy.\n\n  Beware of recent onset of recurrent cystitis with haematuria in the elderly. The underlying cause may be a bladder tumour, especially if the haematuria (micro- or macroscopic) does not settle with treatment of the infection.\n\n  Renal tumours can sometimes present with renal colic, as blood clots in the ureters mimic the effects of stones. A useful clue is that the bleeding may precede the pain.\n\n  Haematuria requires emergency admission if there is significant blood loss or clot retention.\n\n### EXCESSIVE URINATION\n\n#### The GP overview\n\nPolyuria is a highly subjective symptom and one which presents rather less often than urinary frequency (which is dealt with separately, see p. 416). Most of the causes of polyuria listed here are also, by implication, causes of polydipsia \u2013 the only causes of true polydipsia not included are those due to dehydration.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  diabetes mellitus (DM)\n\n  diuretic therapy\n\n  chronic renal failure (CRF)\n\n  hypercalcaemia (e.g. osteoporosis treatment, multiple bony metastases, hyperparathyroidism)\n\n  alcohol\n\n**OCCASIONAL**\n\n  potassium depletion: chronic diarrhoea, diuretics, primary hyperaldosteronism\n\n  relief of chronic urinary obstruction\n\n  drugs: lithium carbonate, demeclocycline, amphotericin B, glibenclamide, gentamicin\n\n  cranial diabetes insipidus (hypothalamo-pituitary tumour, skull trauma, sarcoidosis or histiocytosis X)\n\n  Cushing's disease from excessive corticosteroid doses and ACTH-secreting bronchial carcinoma\n\n  sickle-cell anaemia\n\n  early chronic pyelonephritis\n\n**RARE**\n\n  psychogenic polydipsia (compulsive water drinking)\n\n  supraventricular tachycardia\n\n  DIDMOAD syndrome (diabetes insipidus, diabetes mellitus, optic atrophy, deafness: autosomal recessive)\n\n  familial cranial diabetes insipidus (autosomal dominant inheritance)\n\n  familial nephrogenic diabetes insipidus (males only: X-linked recessive)\n\n  Fanconi syndrome\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, fasting glucose or HbA1c.\n\n_POSSIBLE :_ FBC, U&E, serum calcium.\n\n_SMALL PRINT :_ blood film, further specialist investigations (see below).\n\n  Urinalysis: glucose and possible ketones in diabetes; possible haematuria and proteinuria with renal problems; specific gravity very low in diabetes insipidus and psychogenic polydipsia.\n\n  Fasting glucose or HbA1c: to confirm diabetes mellitus.\n\n  FBC: normochromic anaemia in CRF; film for sickle-cell anaemia.\n\n  U&E: to detect potassium deficiency and abnormalities suggesting CRF.\n\n  Serum calcium: elevated in hypercalcaemia.\n\n  Further specialist investigations: many of the aforementioned 'causes' will need further investigation in secondary care to establish underlying aetiology (e.g. ultrasound and renal biopsy in CRF, water deprivation test for diabetes insipidus, CT scan if possible pituitary lesion, and so on).\n\n##### TOP TIPS\n\n  Take time to clarify the symptoms. It is essential to differentiate polyuria from frequency, as the causes are very different.\n\n  Remember alcohol as a possible cause, especially in young males. Patients can be surprisingly slow to make quite obvious connections.\n\n  Refer for more detailed investigation if the symptoms are clear-cut and baseline tests draw a blank.\n\n  Diabetes mellitus is not the only cause of polyuria with thirst. If urinalysis is negative for sugar, consider diabetes insipidus or hypercalcaemia.\n\n  Weight loss and cough in a smoker with polyuria suggests a possible ACTH-secreting tumour. Arrange an urgent CXR.\n\n  If urinalysis reveals glucose and ketones in a known or new diabetic, arrange for urgent assessment with a view to admission for stabilisation.\n\n  Renal disease is likely in patients with polydipsia who have blood and protein on urinalysis.\n\n### FREQUENCY\n\n#### The GP overview\n\nThis means an increased frequency of micturition, and is usually associated with the passage of small amounts of urine. It is not the same as increased production of urine (see Excessive urination, p. 413). It is a commonly presented problem, affecting women far more often than men: the average GP will deal with around 60 cases of cystitis (the main cause) each year. Terminology can be confusing \u2013 we now use 'LUTS' for what we used to describe as 'prostatism'.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  infective cystitis\n\n  anxiety\n\n  overactive bladder syndrome\n\n  bladder calculus\n\n  lower urinary tract symptoms (LUTS) in men\n\n**OCCASIONAL**\n\n  interstitial cystitis (non-infective)\n\n  prostatitis\n\n  pregnancy\n\n  ureteric calculus (in lower third of ureter precipitates reflex frequency)\n\n  urethritis, pyelonephritis\n\n  iatrogenic (e.g. diuretics)\n\n  bladder neck hypertrophy\n\n  'habit frequency'\n\n**RARE**\n\n  pelvic space-occupying lesion, e.g. fibroid, ovarian cyst, carcinoma\n\n  secondary to pelvic inflammation: PID, appendicitis, diverticulitis, adjacent tumour\n\n  bladder tumour (benign or malignant)\n\n  post-radiotherapy fibrosis (testicular, ovarian and prostatic cancer)\n\n  tuberculous cystitis\/renal TB\n\n  fibrosis secondary to chronic sepsis from long-term catheter drainage\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU, urinary frequency volume chart.\n\n_POSSIBLE :_ urethral swab, PSA, uroflowmetry, urodynamic studies, plain abdominal X-ray, IVU, cystoscopy, CA-125.\n\n_SMALL PRINT :_ pelvic ultrasound, U&E, pregnancy test and three EMUs for TB.\n\n  Urinalysis: protein, nitrites, leucocytes and possible haematuria in infection; possible stone or tumour if blood alone.\n\n  MSU: microscopy may show abnormal epithelial cells, blood, pus and help identify pathogen in infection.\n\n  Urinary frequency volume chart \u2013 may be useful in men with LUTS.\n\n  Swab any urethral discharge present for _Chlamydia_ and gonorrhoea.\n\n  CA-125 \u2013 may be useful if ovarian cancer suspected.\n\n  EMU: for pregnancy test; also three EMUs to check for TB if suspected (e.g. sterile pyuria).\n\n  U&E: check if assessment suggests chronic sepsis or outflow obstruction.\n\n  PSA: consider this if LUTS in male.\n\n  Specialist tests include: uroflowmetry (for LUTS, urodynamic studies (for unstable bladder), IVU and cystoscopy (for stones and tumours) and ultrasound (for pelvic masses or if CA-125 elevated).\n\n##### TOP TIPS\n\n  Frequency due to anxiety is typically long term, worse with stress and cold weather, and is associated with a normal urinalysis.\n\n  It is reasonable to make an empirical diagnosis of overactive bladder syndrome in a non-pregnant female with frequency in whom CA-125, pelvic examination and urinalysis are entirely normal.\n\n  An unrecognised pregnancy may present with frequency: ask about periods, and do a pregnancy test if a period has been missed.\n\n  In the elderly, a bladder tumour may present as cystitis. If a new, recurring problem, or haematuria attributed to the cystitis does not settle with antibiotics, consider referral.\n\n  Do not ignore sterile pyuria on the MSU: possible causes include urethritis and TB.\n\n  The adult patient with frequency who has persistent microscopic haematuria but no other abnormalities on urinalysis may have a stone or tumour. Refer.\n\n  Appendicitis can cause mild frequency and pyuria. Do not be misled by the urinalysis into an inappropriate diagnosis of UTI: act according to the clinical findings.\n\n  UTI in infancy is a major cause of renal failure. Manage according to NICE guidance.\n\n### INCONTINENCE\n\n#### The GP overview\n\nIncontinence is involuntary micturition. It is not a common presenting symptom, embarrassment tending to inhibit patients, but it is often mentioned as a 'while I'm here' or noted by the doctor, typically because of the characteristic odour when visiting an elderly patient. It may present more frequently in the future as the problem receives more publicity and patients realise that help is available. The population prevalence in women is around 10%, but is probably much higher in older age groups.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  stress incontinence (with or without prolapse)\n\n  infective cystitis\n\n  overactive bladder syndrome: idiopathic or secondary to other problems, e.g. CVA, dementia, Parkinson's disease\n\n  chronic outflow obstruction, e.g. prostatic enlargement, bladder neck stenosis, urethral stenosis\n\n  after prostatectomy (usually temporary)\n\n**OCCASIONAL**\n\n  chronic UTI\n\n  interstitial cystitis\n\n  bladder stone or tumour\n\n  after abdomino-pelvic surgery and radiotherapy\n\n  fistula: vesicovaginal\/uterine, ureterovaginal (surgery and malignancy)\n\n  polyuria (any cause, e.g. diabetes, diuretics \u2013 particularly if compounded by immobility in the elderly)\n\n**RARE**\n\n  after pelvic fracture (direct sphincter damage with or without neurological damage)\n\n  congenital abnormalities: short urethra, wide urethra, epispadias, ectopic ureter\n\n  sensory neuropathy, e.g. diabetes and syphilis\n\n  multiple sclerosis, syringomyelia\n\n  paraplegia, cauda equina lesion\n\n  psychogenic\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ PSA, U&E, ultrasound, IVU, urodynamic studies, uroflowmetry.\n\n_SMALL PRINT :_ fasting sugar or HbA1c blood sugar, syphilis serology, cystoscopy, neurological investigations.\n\n  Urinalysis: to test for infection and diabetes.\n\n  MSU: to confirm infection and guide antibiotic treatment.\n\n  Fasting sugar or HbA1c and syphilis serology: if diabetes or syphilis a possible cause of neuropathy.\n\n  PSA: consider this if LUTS or prostatic enlargement on examination.\n\n  U&E: to assess renal function in chronic outflow obstruction.\n\n  Ultrasound good for assessing renal size non-invasively: may suggest outflow obstruction or chronic infection.\n\n  IVU best for looking for renal scarring of chronic UTI, structural anomalies and demonstrating residual urine; may also reveal site of outflow obstruction and fistulae.\n\n  Specialist investigations may include: urodynamic studies (helpful to distinguish between urge and stress incontinence), uroflowmetry (prostatism), cystoscopy (may reveal cause of outflow obstruction, stone or tumour) and neurological investigations (e.g. imaging of spinal cord).\n\n##### TOP TIPS\n\n  Incontinence has many causes, but can often be broadly categorised into one of three groups: stress incontinence (e.g. with coughing), urge incontinence ('when I've got to go, I've got to go') and continuous, like water over the edge of a dam (e.g. through a vesicovaginal fistula, or in overflow from a chronically distended bladder).\n\n  The aetiology may be multifactorial, particularly in the elderly. Mobility, vision, distance to the toilet and ongoing medication may all be relevant.\n\n  Overactive bladder syndrome and stress incontinence can be difficult to distinguish. The latter rarely causes nocturnal incontinence, while it may be a feature of overactive bladder syndrome. If in doubt, refer for urodynamic studies.\n\n  Adopt a sympathetic approach. Incontinence can have a devastating impact on self-esteem and seriously affect a patient's social and sexual functioning.\n\n  Incontinence with saddle anaesthesia and leg weakness suggests a cauda equina lesion. This is a neurological emergency: refer urgently.\n\n  Continuous incontinence suggests significant pathology, such as a fistula, chronic outflow obstruction or a neurological problem.\n\n  Never empty the huge bladder of chronic retention in one go. This can cause bleeding and renal complications. Admit for catheterisation and controlled release.\n\n  Adult-onset nocturnal enuresis suggests chronic retention.\n\n### NOCTURIA\n\n#### The GP overview\n\nNocturia may present in isolation or it may be a manifestation of other urinary disturbances such as polyuria or frequency. Surprisingly, in older age groups, it is as common in women as men. Occasional nocturia is, of course, quite normal \u2013 the symptom should only be viewed as pathological when it causes disruption or distress.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  age related (in part caused by a reduction in bladder capacity)\n\n  excess fluid at bed time (especially alcohol)\n\n  any cause of swollen ankles (the recumbent posture redistributes the fluid load at night) \u2013 see Swollen ankles, p. 312\n\n  cystitis\n\n  LUTS\n\n**OCCASIONAL**\n\n  overactive bladder syndrome\n\n  lower urinary tract obstruction (other than prostate problems)\n\n  any other cause of urinary frequency (see Frequency, p. 416)\n\n  diabetes mellitus\n\n  any other cause of polyuria (see Excessive urination, p. 413)\n\n  sleep apnoea (causes overproduction of urine)\n\n**RARE**\n\n  anxiety\n\n  drug side effect (rare because drugs likely to cause a diuresis are usually taken in the morning)\n\n  diabetes insipidus\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_NOTE :_ urinary frequency, polyuria, or swollen ankles as 'causes' of nocturia will need investigating in their own right \u2013 see the relevant sections for more details on each of these topics.\n\n_LIKELY :_ urinalysis, MSU, urinary frequency volume chart.\n\n_POSSIBLE :_ blood sugar, PSA.\n\n_SMALL PRINT :_ cystoscopy, urodynamic studies, IVP\/ultrasound, water deprivation test.\n\n  Urinalysis: protein, nitrites, leucocytes and possible haematuria in infection; glucose in diabetes; specific gravity very low in diabetes insipidus.\n\n  MSU: to confirm infection and identify pathogen.\n\n  Urinary frequency volume chart \u2013 to help distinguish nocturnal polyuria (increased urine production at night) from reduced bladder storage capacity.\n\n  Blood sugar: to confirm diabetes mellitus.\n\n  PSA: pros and cons of this test may be discussed if assessment raises the possibility of prostate cancer.\n\n  Specialist tests include: cystoscopy and IVU\/ultrasound (for lower urinary tract obstruction), urodynamic studies (for unstable bladder) and water deprivation test (for diabetes insipidus).\n\n##### TOP TIPS\n\n  In the elderly, the cause is often multifactorial.\n\n  The effects \u2013 such as disturbed sleep, a disrupted household, exhaustion and occasional incontinence \u2013 may be more important to the patient than the specific diagnosis.\n\n  Swollen ankles \u2013 of any aetiology \u2013 are frequently overlooked as an underlying cause.\n\n  Nocturia may just be a manifestation (albeit the most distressing) of polyuria or urinary frequency. Focus your approach on the underlying problem.\n\n  Exclude diabetes \u2013 but remember that it is not the only cause of polyuria, nocturia and thirst.\n\n  A habitual 'nightcap' may be the cause of nocturia \u2013 and may be a pointer to an underlying alcohol problem, especially in solitary elderly males.\n\n### RETENTION\n\n#### The GP overview\n\nRetention is failure to empty the bladder completely. The acute form characteristically affects men, presents urgently and requires immediate catheterisation or hospitalisation. Chronic retention may produce few symptoms and may only be discovered during palpation of the abdomen.\n\n#### Differential diagnosis\n\n**COMMON**\n\n  prostatic hypertrophy: benign, rarely carcinoma\n\n  anticholinergic drugs: bladder stabilisers and tricyclic antidepressants\n\n  constipation\n\n  bladder neck obstruction\/urethral stricture\n\n  UTI (including prostatitis and prostatic abscess)\n\n**OCCASIONAL**\n\n  urethral calculus\n\n  'holding on' (leads to prostatic congestion)\n\n  pelvic mass: retroverted, gravid uterus or fibroid uterus\n\n  acute genital herpes (via local inflammation and interference with neurological control of detrusor reflex arc)\n\n  clot retention (e.g. after bleed from tumour or post-TURP bleed)\n\n  balanoposthitis in children (if very painful)\n\n**RARE**\n\n  neurological: MS, syphilis, spinal cord compression\n\n  pedunculated bladder tumour\n\n  traumatic rupture of urethra\n\n  foreign body inserted into anterior urethra\n\n  phimosis\n\n  psychological\n\n#### Ready reckoner\n\n#### Possible investigations\n\n_LIKELY :_ urinalysis, MSU.\n\n_POSSIBLE :_ U&E, PSA, ultrasound, IVU, cystoscopy.\n\n_SMALL PRINT :_ neurological investigations, prostatic biopsy, urethrography (all hospital-based investigations).\n\n  Urinalysis of any urine available may confirm a UTI as the cause; may also reveal microscopic haematuria if a stone or bladder tumour.\n\n  MSU: will confirm infective agent in UTI.\n\n  U&E: renal failure may follow chronic retention.\n\n  PSA may be worth considering if preceding symptoms of prostatism or abnormal prostate on examination.\n\n  Specialist tests may include: renal ultrasound (reveals obstruction and pelvic masses), IVU (may reveal site of obstruction and will provide information about renal function), cystoscopy (may be diagnostic and therapeutic for stones, stricture, bladder outflow obstruction and bladder tumour), neurological investigations (e.g. spinal cord imaging if cord lesion suspected), prostatic biopsy (if suspicious area of prostate palpable) and urethrography (for stricture).\n\n##### TOP TIPS\n\n  Do not overlook faecal impaction in the elderly patient as a cause of urinary retention.\n\n  'First-aid' relief of retention when the cause is a painful perineal condition (e.g. balanoposthitis, herpes simplex or UTI) may be achieved by encouraging the patient to urinate while immersed in a warm bath.\n\n  Anuria can be mistaken for retention. A straightforward clinical assessment should differentiate the two conditions.\n\n  A history suggesting a disc prolapse with urinary retention indicates possible cord compression \u2013 admit immediately.\n\n  Sudden stoppage of urine with a pain like a blow to the bladder and passage of a few drops of blood is pathognomic of urethral calculus.\n\n  Beware of any drugs with anticholinergic side effects in patients with a history of outflow obstruction \u2013 they may precipitate acute retention.\n\n  Avoid catheterisation when sepsis is likely (e.g. possible UTI) \u2013 instrumentation may result in septicaemia. Instead, admit to hospital for catheterisation under appropriate antibiotic cover.\n\n  Do not catheterise the patient with chronic retention; admit for controlled drainage. Sudden decompression can result in haematuria and renal complications.\n\n## INDEX\n\nabbreviations v\u2013vi\n\nabdominal carcinoma abdominal pain ,\n\nabdominal swelling\n\nconstipation 18\u201320\n\ndiarrhoea\n\nepigastric pain\n\nabdominal masses, epigastric pain\n\nabdominal pain\n\nacute 9\u201311\n\nin pregnancy 12\u201314\n\nchildhood, recurrent 27\u20139\n\nchronic 15\u201317\n\nconstipation\n\nlimp in a child ,\n\nrectal discharge\n\nabdominal swelling 6\u20138\n\nweight gain\n\nabdomino-pelvic surgery, urinary incontinence\n\nabnormal gait in adults 165\u20137\n\nabnormal movements 168\u201370\n\nabscess\n\nfacial pain\n\nfebrile child\n\ngroin swelling 370\u20131\n\nprolonged fever 215\u201316\n\ntremor\n\nacanthosis nigricans, papules 398\u20139\n\nACE inhibitors\n\nblisters\n\ncough 103\u20135\n\ndiarrhoea\n\ndry mouth\n\nachalasia, swallowing difficulties\n\nAchilles tendon, ruptured\n\ncalf pain 288\u20139\n\nswollen calf\n\nAchilles tendonitis, foot pain\n\nacne\n\nfacial rash 153\u20134\n\nnodulocystic ,\n\npapules 398\u20139\n\npustules 403\u20134\n\nacne excori\u00e9e\n\nfacial rash ,\n\nfacial ulcers and blisters\n\nacoustic neuroma\n\ndeafness 116\u201318\n\ntinnitus ,\n\ntremor\n\nvertigo ,\n\nacromegaly\n\nbreast enlargement in men\n\nexcess body hair\n\nexcessive sweating\n\nACTH-secreting tumours\n\nexcessive urination\n\nflushing\n\nactinomycosis\n\nfacial ulcers and blisters\n\nneck lumps\n\nnodules\n\nprolonged fever\n\nAddison's disease\n\nabdominal pain 15\u201316\n\nabsent periods\n\ndizziness 71\u20132\n\nloss of libido\n\nmouth lumps\n\ntiredness 224\u20135\n\nweight loss 233\u20134\n\nadenocarcinoma\n\nblocked nose\n\nvulval ulceration and sores\n\nadenoidal hypertrophy, blocked nose 338\u20139\n\nadenoma\n\npituitary\n\nexcess body hair ,\n\nvillous\n\nrectal bleeding\n\nrectal discharge 49\u201350\n\nadrenal disorders\n\nabsent periods 374\u20135\n\ndelayed puberty\n\nexcess body hair\n\nheavy periods 377\u20138\n\nloss of libido 83\u20134\n\nadrenal tumours\n\nbreast enlargement in men\n\nexcess body hair 272\u20133\n\nadrenocortical tumours, infertility\n\nadverse drug reactions, _see_ drug side effects\n\naerophagy, epigastric pain\n\nageing\n\nloss of libido 83\u20134\n\nnocturia 422\u20133\n\nagranulocytosis, sore throat\n\nairway obstruction, stridor 335\u20136\n\nakathisia, abnormal movements\n\nAlbright's syndrome, macules\n\nalcohol misuse\n\nbad breath 348\u201350\n\nbreast enlargement in men\n\nconfusion 68\u20139\n\ndiarrhoea 21\u20132\n\ndizziness 71\u20133\n\ndouble vision\n\nerectile dysfunction 238\u201340\n\nexcessive sweating 182\u20134\n\nexcessive urination 413\u201314\n\nfalls 188\u20139\n\nflushing 197\u20138\n\ninsomnia 80\u20131\n\njaundice 206\u20137\n\nloss of libido 83\u20135\n\nmaternal\n\nmemory loss\n\nnocturia\n\nnumbness and paraesthesiae 212\u201313\n\npalpitations\n\ntension and anxiety 194\u20135\n\nthirst and dry mouth\n\ntiredness\n\ntremor 228\u20139\n\nvertigo\n\nvomiting blood\n\nweight gain 230\u20131\n\nweight loss 233\u20134\n\nalcohol withdrawal, hallucinations 74\u20136\n\nalcoholic neuropathy\n\ncalf pain 288\u20139\n\nmuscle pain\n\nnumbness\n\nallergic rhinitis\n\nblocked nose 338\u20139\n\ncough\n\nnosebleed\n\nrunny nose 344\u20135\n\nallergies\n\nabdominal pain 27\u20138\n\ncough\n\ndiarrhoea\n\nfacial swelling\n\nitching 203\u20134\n\nleg ulcers\n\nmacules\n\nalopecia\n\neyelid problems ,\n\ntraction\n\nhair loss 275\u20136\n\nalopecia areata\n\nabnormal nails\n\nhair loss 275\u20137\n\n\u03b1-blockers, runny nose\n\n\u03b11-antitrypsin deficiency, jaundice\n\naluminium hydroxide, constipation\n\nalveolitis\n\ncough 103\u20134\n\nshortness of breath ,\n\nAlzheimer's disease, memory loss 86\u20137\n\namaurosis fugax, transient visual disturbance 139\u201341\n\namblyopia, toxic, vision loss\n\namenorrhoea\n\nexcess body hair\n\ninfertility 200\u20132\n\nprimary ; _see also_ periods, absent delayed puberty\n\naminoglycosides\n\near discharge\n\ntinnitus\n\nvertigo\n\namoebiasis, diarrhoea\n\namphetamines\n\nerectile dysfunction\n\nhallucinations\n\nmaternal, failure to thrive\n\nweight loss\n\namphotericin B, excessive urination\n\namyloidosis\n\njaundice\n\nmultiple joint pain\n\nanaemia\n\nabdominal swelling\n\naplastic\n\npurpura and petechiae\n\nvomiting blood\n\narm and shoulder pain\n\nblood in urine\n\ncoughing up blood\n\ndizziness\n\nepigastric pain 25\u20136\n\nexcessive sweating\n\nfailure to thrive\n\nfalls 188\u20139\n\ninfertility\n\nirregular vaginal bleeding\n\npainful periods\n\nrectal bleeding\n\nshortness of breath , 100\u20132\n\nswollen ankles\n\ntinnitus 125\u20136\n\ntiredness\n\ntongue pain 359\u201360\n\nweight gain\n\nanaemia of chronic disease\n\nabnormal gait\n\njaundice\n\njoint pain\n\nnodules\n\nnumbness and paraesthesiae\n\nswollen ankles\n\nanal fissure\n\nanal itching 38\u20139\n\nanorectal pain 43\u20134\n\npainful intercourse\n\npenile pain\n\nrectal bleeding 46\u20137\n\nrectal discharge 49\u201350\n\nanal fistula, anal swelling\n\nanal itching 38\u201340\n\nanal\/perianal fistula\n\nanal itching\n\nrectal discharge\n\nanal swelling 41\u20132\n\nanalgesic abuse, headache 77\u20138\n\nancylostomiasis, swollen ankles\n\nandrogenic alopecia, hair loss 275\u20136\n\nandrogens, delayed puberty\n\naneurysm, dissecting\/leaking\n\nangina\n\narm and shoulder pain 285\u20137\n\nbreast pain 60\u20132\n\nchest pain 97\u20138\n\nsore throat\n\nangiodysplasia, rectal bleeding\n\nangioedema\n\neyelid problems\n\nfacial swelling\n\nangioneurotic oedema\n\nfacial swelling 156\u20137\n\nswollen ankles\n\nankles, swollen 312\u201314\n\nnocturia 422\u20133\n\nswollen calves 315\u201316\n\nankylosing spondylitis\n\nback pain\n\njoint pain 300\u20131\n\nknee pain\n\nneck stiffness 332\u20133\n\nred eye\n\nanorectal carcinoma\n\nanal itching\n\nanal swellng 41\u20132\n\nanorectal pain 43\u20135\n\npenile pain\n\nrectal discharge ,\n\nanorectal pain 44\u20135\n\nanorexia nervosa\n\nabsent periods ,\n\ndelayed puberty 177\u20138\n\ndiarrhoea\n\nexcess body hair 272\u20133\n\nweight loss\n\nanterior knee pain syndromes, single joint pain 282\u20133\n\nanti-androgenics, loss of libido\n\nantibiotics\n\ndiarrhoea\n\nepigastric pain\n\nerythema\n\nvomiting\n\nanticholinergics\n\ntransient visual disturbance\n\nurinary retention ,\n\nanticoagulants\n\nblood in urine\n\nhaemospermia\n\nhair loss\n\nnosebleed\n\nrectal bleeding\n\nantidepressants\n\ndizziness\n\nerectile dysfunction\n\nheadache\n\nloss of libido\n\nweight loss\n\nantihistamines, thirst and dry mouth\n\nantihypertensives\n\ndizziness\n\nloss of libido\n\nantipsychotics, abnormal movements ,\n\nANUG (acute necrotising ulcerative gingivitis)\n\nbad breath 348\u20139\n\nbleeding gums 351\u20132\n\nmouth ulcers 356\u20137\n\nanuria, urinary retention\n\nanxiety\n\nabdominal pain 27\u20138\n\nabdominal swelling\n\nabnormal movements\n\nabnormal nails\n\nback pain\n\nchest pain 97\u20139\n\ndizziness 71\u20132\n\nerectile dysfunction 238\u20139\n\nexcessive sweating 182\u20133\n\nflushing 197\u20138\n\nfrequent urination 416\u201317\n\nhallucinations\n\nheadache\n\ninsomnia\n\nmemory loss 86\u20137\n\nnocturia\n\nnumbness and paraesthesiae 212\u201313\n\npalpitations 112\u201313\n\nthirst and dry mouth 221\u20132\n\ntremor 227\u20139\n\nweight loss 233\u20134\n\naortic aneurysm\n\nabdominal swelling\n\nback pain 171\u20132\n\ndissecting ,\n\nepigastric pain\n\naortic arch, double, stridor\n\naortic stenosis\n\ndizziness 71\u20133\n\nfalls 188\u20139\n\nloss of consciousness ,\n\nshortness of breath\n\naphonia, functional, hoarseness\n\naphthous ulceration; _see also_ mouth ulcers\n\nbleeding gums 351\u20132\n\nmouth lumps and marks 353\u20134\n\nsore throat\n\ntongue pain 359\u201360\n\napical tumour, arm and shoulder pain 286\u20137\n\nappendicitis\n\nabdominal pain , 11\u201312, ,\n\ncrying baby\n\ndiarrhoea\n\nfebrile child 191\u20132\n\nlimp in a child ,\n\npainful periods\n\nurinary frequency ,\n\nvomiting\n\nappendix, low mass, groin swelling\n\nareolar abscess, nipple discharge 63\u20134\n\narm and shoulder pain 285\u20137\n\narrhenoblastoma\n\nabsent periods\n\nexcess body hair\n\narsenic\n\nbleeding gums\n\nloss of vision\n\nvomiting blood\n\narterial bruits, tinnitus\n\narterial disease, leg ulcers 297\u20138\n\narterial insufficiency\n\nerectile dysfunction 238\u20139\n\nleg ulcers 298\u20139\n\narteriovenous fistula, tinnitus\n\narthritis\n\nabnormal gait 165\u20136\n\nfoot pain\n\npsoriatic\n\nknee pain\n\nmultiple joint pain\n\nsingle joint pain 282\u20133\n\nspinal\n\nabdominal pain\n\nascites\n\nabdominal swelling 6\u20138\n\nscrotal swelling 253\u20134\n\nweight gain 231\u20132\n\naspergillosis, coughing up blood\n\naspiration, cough\n\naspiration pneumonitis, shortness of breath\n\naspirin burns, tongue pain 359\u201360\n\naspirin overdose\n\ndeafness\n\ntinnitus\n\nasthma\n\nchronic, failure to thrive\n\ncough 103\u20137\n\ninsomnia\n\nshortness of breath 94\u20135, 100\u20131\n\natherosclerosis\n\nnosebleed\n\ntinnitus\n\natrial ectopics, palpitations 112\u201313\n\natrial fibrillation, palpitations ,\n\natrial flutter, palpitations\n\nautoimmune diseases\n\nabnormal nails\n\nbleeding gums 351\u20132\n\ndelayed puberty\n\nhair loss\n\nmouth ulcers 356\u20137\n\nmultiple joint pain\n\navascular necrosis\n\nhand and wrist pain\n\nhip pain 306\u20138\n\nsingle joint pain 282\u20133\n\nback pain 171\u20133\n\npelvic pain\n\nreferred\n\ncalf pain\n\nhip pain 306\u20137\n\nknee pain\n\nBaker's cyst\n\nknee pain\n\nruptured\n\ncalf pain\n\nswollen ankles\n\nswollen calf 315\u201316\n\nbalanitis\n\npenile pain 247\u20139\n\npenile ulcerations and sores 250\u20132\n\nbalanitis xerotica obliterans (BXO)\n\npenile pain\n\npenile ulcerations and sores 250\u20131\n\nbalanoposthitis, urinary retention 425\u20136\n\nbarbiturates\n\nblisters\n\nfacial ulcers and blisters\n\nbarotrauma\n\ndeafness\n\nearache\n\nrunny nose\n\nbartholinitis\n\npainful intercourse\n\nvaginal discharge\n\nBartholin's abscess, vulval swelling\n\nBartholin's cyst, vulval swelling 264\u20135\n\nbasal cell carcinoma (BCC)\n\neyelid problems\n\nfacial ulcers and blisters 159\u201360\n\nnodules 395\u20137\n\npapules\n\nvulval ulceration and sores\n\nBeau's lines, abnormal nails 270\u20131\n\nBeh\u00e7et's syndrome\n\nmouth lumps and marks\n\nmouth ulcers\n\nmultiple joint pain\n\npenile ulcerations and sores\n\npustules\n\ntongue pain\n\nvulval sores 266\u20137\n\nbenzodiazepine withdrawal, insomnia ,\n\nbenzodiazepines\n\ndouble vision\n\ntremor\n\nbereavement reactions, hallucinations 74\u20135\n\nBerloque dermatitis, macules 393\u20134\n\n\u03b2-blockers\n\nscales and plaques\n\ntiredness\n\ntremor\n\nbile duct carcinoma, jaundice\n\nbiliary cirrhosis, jaundice 206\u20137\n\nbiliary colic\n\nabdominal pain\n\nchest pain\n\nbiliary obstruction\n\nabdominal pain\n\njaundice\n\nbiorhythm disruption, insomnia\n\nbirth injury, crying baby\n\nbisphosphonates\n\nepigastric pain\n\nswallowing difficulties\n\nbladder, enlarged, abdominal swelling\n\nbladder neck hypertrophy, frequent urination\n\nbladder neck obstruction, urinary retention 425\u20136\n\nbladder neck stenosis, urinary incontinence\n\nbladder of chronic retention\n\nbladder stone\n\nfrequent urination 416\u201317\n\nincontinence\n\nbladder tumour\n\nabdominal swelling\n\nanorectal pain\n\nblood in urine 410\u201312\n\nhaemospermia\n\npenile pain\n\nurinary frequency ,\n\nurinary retention 425\u20136\n\nblepharitis\n\neyelid problems 136\u20138\n\nitchy eyes 144\u20135\n\nblepharospasm\n\nabnormal movements\n\neyelid problems\n\nblindness, _see_ vision loss\n\nblisters 388\u20139\n\nbloating\n\ndiarrhoea\n\npelvic pain\n\nblood, coughing up, _see_ coughing up blood\n\nblood clotting disorders\n\nhaemospermia\n\nheavy periods\n\nknee pain\n\nnosebleeds\n\npurpura and petechiae 400\u20132\n\nrectal bleeding 46\u20137\n\nblood dyscrasias\n\nabdominal swelling\n\nbleeding gums 351\u20132\n\nblood in urine\n\ngroin swellings\n\nmouth ulcers 357\u20138\n\nnosebleeds\n\nprolonged fever\n\npurpura\n\nsore throat 329\u201330\n\nvomiting blood 33\u20134\n\nvulval irritation\n\nblood in urine 410\u201312\n\nabdominal pain ,\n\nhaemospermia\n\nhip pain\n\nmultiple joint pain\n\npelvic pain\n\npenile pain 248\u20139\n\nprolonged fever\n\nurinary frequency\n\nurinary retention\n\nblood vessels, anomalous, stridor\n\nbody hair, excess 272\u20134\n\nbody piercing, anal itching\n\nboils\n\near discharge 122\u20133\n\nearache 119\u201320\n\npenile ulcerations and sores\n\npustules\n\nvulval swelling 264\u20135\n\nbone marrow damage, purpura and petechiae ,\n\nbone pain\n\nfoot pain\n\ninsomnia\n\nbone tumour\n\narm and shoulder pain 285\u20136\n\nfacial swelling\n\nlimp in a child\n\nneck stiffness\n\nbony metastases, excessive urination\n\nbony pathologies\n\nabnormal gait\n\nhip pain 306\u20137\n\ninsomnia\n\nneck stiffness 333\u20134\n\nboredom\n\ncrying baby\n\ntiredness\n\nBornholm disease\n\nchest pain\n\nmuscle pain\n\nbowel ischaemia, _see_ ischaemic bowel\n\nbowel malignancy\n\ndiarrhoea\n\npelvic pain\n\nbowel obstruction\n\nabdominal pain , 15\u201316\n\nabdominal swelling 7\u20138\n\nconstipation\n\ncrying baby\n\nepigastric pain\n\npelvic pain\n\nbowel spasm, pelvic pain\n\nBowen's disease, scales and plaques ,\n\nbrain abscess, neck stiffness\n\nbrain disease, organic, tension and anxiety 194\u20136\n\nbrain tumours\n\nexcessive sweating\n\nnumbness\n\nbrainstem ischaemia, falls\n\nbranchial cysts, neck lumps\n\nbreast abscess\n\nbreast enlargement in men\n\nbreast lumps in women 57\u20138\n\nbreast pain 60\u20132\n\nnipple discharge\n\nbreast carcinoma\n\nbreast enlargement in men 54\u20136\n\nbreast lumps in women 57\u20139\n\nbreast pain 60\u20132\n\nnipple discharge 63\u20134\n\nbreast enlargement in men 54\u20136\n\nbreast lumps in women 57\u20139\n\nbreast pain 60\u20132\n\nbreath, bad 348\u201350\n\nbronchial carcinoma\n\nbreast enlargement in men\n\ndifficulty swallowing\n\nexcessive urination\n\nprolonged fever 215\u201316\n\nshortness of breath ,\n\nbronchiectasis\n\nabnormal nails\n\nbad breath 348\u20139\n\ncough\n\nshortness of breath 100\u20132\n\nbronchiolitis\n\ncrying baby\n\nfebrile child\n\nbronchitis, chronic\n\nabnormal nails\n\near discharge\n\nbronchogenic carcinoma\n\ncoughing up blood 109\u201310\n\nflushing\n\nbrucellosis\n\nexcessive sweating ,\n\nfebrile child\n\nmultiple joint pain ,\n\nBTB (breakthrough bleeding), irregular vaginal bleeding 380\u20132\n\nbubble baths\n\nanal itching\n\nvulval irritation\n\nbulbar palsy, swallowing difficulties\n\nbulimia nervosa\n\nvomiting ,\n\nweight loss\n\nbunions, foot pain 291\u20132\n\nburns\n\nblisters\n\nerythema 390\u20131\n\nred eye\n\nbursitis\n\narm and shoulder pain 285\u20137\n\nfoot pain\n\nhip pain 306\u20137\n\nknee pain\n\nBV (bacterial vaginosis), vaginal discharge 258\u20139\n\ncaf\u00e9 au lait spot\n\ncalcium antagonists\n\nflushing\n\nheadache\n\nswollen ankles 312\u201313\n\ncalcium channel blockers, runny nose\n\ncalf pain 288\u201390\n\ncalves, swollen 315\u201317\n\nCampbell de Morgan spot, papules 398\u20139\n\ncampylobacter\n\ndiarrhoea\n\nrectal bleeding 47\u20138\n\ncancrum oris\n\nfacial ulcers and blisters\n\nmouth lumps\n\n_Candida_ ; _see also_ thrush\n\nmouth lumps \u20134\n\nmouth ulcers 357\u20138\n\noropharyngeal\n\nsore throat 329\u201331\n\npenile ulcerations and sores 251\u20132\n\npustules\n\ntongue pain 359\u201361\n\nvaginal discharge\n\nvulval irritation ,\n\nvulval sores 267\u20138\n\ncarbamazepine, scales and plaques\n\ncarbimazole, mouth ulcers\n\ncarbon bisulphide, vision loss\n\ncarbon dioxide retention, tremor 227\u20138\n\ncarbon monoxide poisoning\n\nconfusion ,\n\ndizziness ,\n\nheadache\n\nmemory loss\n\ntiredness ,\n\ncarcinoid syndrome\n\nexcessive sweating ,\n\nflushing 197\u20139\n\ncardiac arrhythmia\n\ndizziness 71\u20132\n\nfalls\n\ncardiac failure\n\nconfusion\n\nfailure to thrive\n\njaundice\n\nscrotal swelling\n\ntiredness\n\nweight gain 230\u20131\n\nweight loss\n\ncardiac ischaemia, chest pain\n\ncardiac structural lesion, loss of consciousness\n\ncardiomyopathy, hypertrophic obstructive\n\nchest pain\n\nloss of consciousness\n\ncardiovascular disease, erectile dysfunction\n\ncarotico-cavernous fistula, red eye\n\ncarotid body tumour, neck lumps\n\ncarotid sinus syncope, loss of consciousness ,\n\ncarotidynia, sore throat\n\ncartilage injury, knee pain 309\u201310\n\ncat scratch fever\n\nfebrile child\n\nswollen glands 218\u201319\n\ncataracts\n\ndouble vision\n\nvision loss 142\u20133\n\ncauda equina lesions\n\nabnormal gait ,\n\nanorectal pain\n\nerectile dysfunction\n\nincontinence ,\n\nCCF (congestive cardiac failure)\n\nleg ulcers\n\nscrotal swelling\n\nshortness of breath 100\u20132\n\nswollen ankles 312\u201313\n\nweight gain\n\ncellulitis\n\ncalf pain 288\u20139\n\nerythema 390\u20131\n\neyelid problems\n\nfacial rash ,\n\nfacial swelling\n\nfebrile child\n\nleg ulcers\n\norbital\n\neyelid problems ,\n\nfacial pain ,\n\nfacial swelling\n\nred eye\n\nswollen ankles\n\nswollen calves 315\u201316\n\ncentral disc protrusions, back pain\n\ncerebellar ataxia\n\nabnormal gait\n\ntremor\n\ncerebellar tumours, vertigo\n\ncerebral abscess, earache\n\ncerebral carcinoma, secondary, deafness\n\ncerebral haemorrhage, vomiting\n\ncerebral infection, confusion\n\ncerebral palsy\n\nfailure to thrive\n\nlimp in a child\n\ncerebral space-occupying lesions\n\ndouble vision\n\nhallucinations 74\u20135\n\nmemory loss\n\ncerebral tumour\n\nconfusion\n\nnumbness\n\nstiff neck\n\ncerebromacular degeneration, vision loss\n\ncerebrovascular accident (CVA)\n\nabnormal gait\n\nconfusion 68\u201370\n\nfalls\n\nmemory loss 86\u20137\n\ntremor\n\nurinary incontinence\n\nvertigo\n\nvision loss 147\u20138\n\nweight gain\n\ncerebrovascular disease\n\nmemory loss\n\nnumbness\n\ncervical adenitis, facial swelling\n\ncervical carcinoma\n\nirregular vaginal bleeding\n\npelvic pain ,\n\nvaginal discharge\n\ncervical ectropion, vaginal discharge\n\ncervical lymphadenitis, neck stiffness\n\ncervical mucus, hostile, infertility\n\ncervical osteoarthritis, dizziness\n\ncervical polyp\n\nheavy periods\n\ninfertility\n\nirregular vaginal bleeding\n\nvaginal discharge\n\nvulval swelling\n\ncervical rib\n\narm and shoulder pain\n\nneck lumps\n\ncervical spondylosis\n\narm and shoulder pain 285\u20136\n\nbreast pain 60\u20131\n\nearache\n\nheadache 77\u20138\n\nneck stiffness 332\u20133\n\nnumbness and paraesthesiae 212\u201313\n\nupper\n\nfacial pain\n\ncervical stenosis, painful periods\n\ncervicitis\n\nirregular vaginal bleeding 380\u20131\n\npainful intercourse\n\npainful periods\n\nvaginal discharge 258\u201360\n\nChagas's disease, swallowing difficulties\n\nchancroid\n\npenile ulcerations and sores 250\u20131\n\nvulval sores 266\u20137\n\nchemical burns\n\nblisters\n\nerythema\n\nchemical inhalation, hoarseness\n\nchemical irritation\n\nanal itching\n\nrunny nose\n\nvulval irritation 261\u20132\n\nchemotherapy\n\ndelayed puberty\n\nhair loss\n\nmuscle pain\n\nchest infection\n\nchest pain\n\ncoughing up blood 109\u201310\n\ncrying baby\n\nfebrile child 191\u20132\n\nchest pain 97\u20139\n\nbreast pain\n\nchickenpox\n\nblisters 388\u20139\n\nfacial rash\n\nfebrile child\n\nitchy scalp\n\nchild abuse, failure to thrive\n\nchildhood tics, abnormal movements ,\n\nchlamydia\n\nanal itching\n\nirregular vaginal bleeding\n\nitchy eyes\n\npainful periods\n\npelvic pain ,\n\nrectal discharge\n\nscrotal swelling\n\ntesticular pain\n\nurinary frequency\n\nvaginal discharge 258\u20139\n\nchloasma\n\nfacial rash\n\nmacules 393\u20134\n\nchlorpromazine, jaundice\n\nchoanal atresia, unilateral, blocked nose\n\ncholangitis, jaundice\n\ncholecystitis\n\nabdominal pain\n\nepigastric pain\n\ncholestasis, jaundice\n\ncholesteatoma\n\ndeafness\n\near discharge 122\u20133\n\nearache\n\nchondritis of costal cartilage, breast pain\n\nchondrodermatitis nodularis chronica helicis, nodules\n\nchondrodermatitis nodularis helicis externa, earache\n\nchondromalacia patellae (CP)\n\nknee pain 309\u201310\n\nsingle joint pain 282\u20133\n\nchoreoathetosis, abnormal movements\n\nchorionic carcinoma, irregular vaginal bleeding\n\nchoroidoretinitis, vision loss\n\nChristmas disease\n\nnosebleeds\n\npurpura and petechiae\n\nchronic infection\n\ndiarrhoea\n\nfailure to thrive 185\u20136\n\njoint pain\n\nleg ulcers\n\ntinnitus\n\ntiredness\n\nurinary incontinence\n\nweight loss\n\nchronic pain, insomnia 80\u20131\n\ncimetidine\n\nbreast enlargement in men\n\nloss of libido\n\nmuscle pain\n\ncirrhosis\n\nalcoholic\n\njaundice 206\u20137\n\npurpura\n\nswollen ankles 312\u201313\n\nCKD (chronic kidney disease), thirst and dry mouth 221\u20132\n\nclaudication, intermittent\n\nabnormal gait 165\u20136\n\ncalf pain 288\u201390\n\nmuscle pain\n\nclavicular fracture, crying baby\n\ncleft palate, failure to thrive\n\nclofibrate, muscle pain\n\nclot retention\n\nblood in urine\n\nurinary retention\n\nclubbing\n\nabnormal nails 270\u20131\n\nbreast enlargement in men\n\nmultiple joint pain\n\nshortness of breath\n\ncoagulopathy\n\nknee pain\n\nnosebleed\n\ncocaine\n\nhallucinations ,\n\nmaternal\n\nfailure to thrive\n\nnosebleeds\n\nrunny nose 344\u20135\n\ncoccydynia, anorectal pain\n\ncochlear vessels, thrombosis of, deafness\n\ncoeliac disease\n\nabdominal pain 15\u201316, 27\u20138\n\ndelayed puberty\n\ndiarrhoea 21\u20132\n\nfailure to thrive\n\nmouth lumps\n\nmouth ulcers 356\u20137\n\nswollen ankles\n\ntiredness 224\u20135\n\nweight loss 233\u20134\n\ncoffee-ground vomit ,\n\ncolic, crying baby 174\u20135\n\ncolitis, rectal bleeding\n\ncolonic cancer\n\npelvic pain\n\nrectal bleeding\n\ncolonic disease, abdominal swelling\n\nconfusion, acute 68\u201370\n\nmemory loss\n\nshortness of breath\n\ncongenital adrenal hyperplasia\n\nabsent periods\n\ndelayed puberty\n\nexcess body hair\n\nconjunctivitis\n\nitchy eyes 144\u20135\n\nred eye 130\u20132\n\nsingle joint pain\n\nconnective tissue disorder\n\nerythema 390\u20132\n\nexcessive sweating 182\u20133\n\njoint pain 300\u20132\n\nlimp in a child\n\nmuscle pain 303\u20135\n\nnumbness and paraesthesiae 212\u201313\n\nprolonged fever\n\npurpura and petechiae 400\u20131\n\nswollen glands\n\nthirst and dry mouth\n\ntiredness 224\u20135\n\nweight loss\n\nconstipation 18\u201320\n\nabdominal pain , 15\u201317, 27\u20138\n\nabdominal swelling 6\u20137\n\nanorectal pain\n\ncrying baby 174\u20135\n\noverflow diarrhoea 21\u20133\n\nurinary retention 425\u20136\n\nvomiting\n\ncontact dermatitis\n\nblisters\n\nhair loss 275\u20136\n\nitching 203\u20134\n\nitchy scalp\n\nscales and plaques\n\nvulval ulceration and sores 266\u20137\n\ncontact lenses, itchy eyes\n\ncontraceptives, oral\n\nabsent periods 374\u20135\n\nheavy periods 377\u20138\n\nirregular vaginal bleeding 380\u20132\n\nmouth lumps\n\nnipple discharge\n\npainful periods\n\nvisual disturbance\n\nCOPD (chronic obstructive pulmonary disease)\n\ncough 103\u20134\n\nexacerbation of\n\nshortness of breath 94\u20135\n\ninsomnia\n\nshortness of breath 100\u20131\n\ntremor\n\ncorneal abrasion, red eye 130\u20131\n\ncorneal ulcer, red eye 130\u20132\n\nCornelia de Lange syndrome, excess body hair\n\ncorpus cavernosum thrombosis, erectile dysfunction\n\ncortical blindness, vision loss 147\u20138\n\ncorticosteroids\n\njoint pain\n\nnosebleeds\n\n_Corynebacterium diphtheriae_ , runny nose\n\ncostochondritis, chest pain 97\u20138\n\ncough\n\nin adults 103\u20135\n\nin children 106\u20138\n\ncroupy\n\ncrying baby\n\nexcessive urination\n\npurpura and petechiae 400\u20131\n\ncough syncope, loss of consciousness\n\ncoughing up blood 109\u201311\n\ncowpox, pustules\n\ncow's milk intolerance, failure to thrive\n\nCoxsackie virus, mouth ulcers\n\ncramps\n\ncalf pain 288\u201390\n\nmuscle pain\n\ncranial nerve palsies, double vision 133\u20134\n\nCREST syndrome, swallowing difficulties\n\nCRF (chronic renal failure)\n\nbreast enlargement in men\n\nexcessive urination 413\u201314\n\nvulval irritation\n\nCrohn's disease\n\nabdominal pain , 27\u20138\n\nanal itching 38\u20139\n\nanal swelling\n\nanorectal pain 43\u20135\n\nconstipation\n\ndiarrhoea\n\nmouth lumps\n\nprolonged fever\n\nrectal discharge\n\ncroup\n\nfebrile child\n\nstridor 335\u20136\n\ncrying baby 174\u20136\n\ncryoglobulinaemia, purpura and petechiae 400\u20131\n\ncryptorchidism, breast enlargement in men\n\ncryptosporidium, febrile child\n\nCSF (cerebrospinal fluid) leakage ear discharge 122\u20133\n\nrunny nose 344\u20135\n\nCT (carpal tunnel) syndrome\n\narm and shoulder pain 285\u20136\n\nhand and wrist pain 294\u20136\n\nnumbness and paraesthesiae 212\u201314\n\nCushing's disease\n\nabsent periods\n\nconfusion\n\ndelayed puberty\n\nexcessive urination\n\nloss of libido\n\nCushing's syndrome\n\nexcess body hair\n\nfacial swelling\n\nflushing 197\u20138\n\nweight gain 230\u20131\n\ncyanosis\n\nconfusion\n\nshortness of breath\n\ncyproterone\n\nbreast enlargement in men\n\nloss of libido\n\ncystic fibrosis\n\ncough 106\u20138\n\ndelayed puberty\n\nfailure to thrive 186\u20137\n\ncystic glandular hyperplasia\n\nheavy periods\n\nirregular vaginal bleeding\n\ncystic hygroma, neck lumps\n\ncystitis\n\ninfective\n\nfrequent urination 416\u201318\n\nincontinence 419\u201320\n\ninterstitial\n\nblood in urine ,\n\nfrequent urination\n\nincontinence\n\npelvic pain 367\u20138\n\nnocturia\n\npainful intercourse\n\npenile pain 247\u20139\n\ncystocoele, vulval swelling\n\ncysts\n\nbreast lumps in women 57\u20138\n\nbreast pain 60\u20131\n\ncytotoxics\n\nabsent periods\n\npurpura\n\nvomiting\n\ndacrocystitis, eyelid problems\n\ndanazol, excess body hair\n\nDarier's disease, papules\n\ndeafness 116\u201318\n\ndeck-chair legs 390\u20131\n\ndecompression sickness, joint pain\n\ndehydration\n\nconfusion\n\ndiarrhoea 22\u20133\n\nfebrile child\n\nhallucinations\n\nthirst and dry mouth 221\u20133\n\nvomiting\n\ndelirium, febrile, hallucinations 74\u20135\n\ndemeclocycline, excessive urination\n\ndementia\n\nconfusion\n\nmemory loss 86\u20138\n\nurinary incontinence\n\ndental abscess\n\napical, mouth lumps and marks 353\u20135\n\nbad breath 348\u20139\n\nearache\n\nfacial pain 150\u20131\n\nfacial swelling 156\u20137\n\ndental cysts, facial swelling ,\n\ndental hygiene\n\nbad breath 348\u20139\n\nbleeding gums 351\u20132\n\ndental sinus, ulcerating, facial ulcers and blisters\n\ndepigmentation\n\ndepression\n\nabdominal pain 27\u20138\n\nabdominal swelling\n\nback pain\n\nbad breath ,\n\nerectile dysfunction 238\u20139\n\nexcess body hair\n\nfacial pain\n\nheadache\n\nhip pain\n\ninsomnia 80\u20132\n\nitchy scalp 278\u20139\n\nloss of libido 83\u20135\n\nmemory loss 86\u20138\n\nmuscle pain 303\u20135\n\npainful periods\n\npsychotic\n\nhallucinations 74\u20136\n\ntension and anxiety 194\u20136\n\ntinnitus\n\ntiredness 224\u20136\n\ntongue pain\n\nweight loss 233\u20135\n\ndermatitis; _see also_ contact dermatitis; seborrhoeic dermatitis\n\nexfoliative, abnormal nails\n\nperioral\n\nfacial rash 153\u20134\n\npustules 403\u20134\n\ndermatitis artefacta\n\nfacial ulcers and blisters\n\nitching\n\ndermatitis herpetiformis\n\nblisters\n\nitching\n\npustules\n\ndermatofibroma, nodules\n\ndermatomyositis\n\nfacial rash 153\u20134\n\njoint pain\n\nlimp in a child\n\nmuscle pain 303\u20135\n\nswallowing difficulties\n\ndermatosis, juvenile plantar, scales and plaques 405\u20136\n\ndermoid cyst, sublingual, mouth lumps and marks\n\ndescending perineum syndrome, anorectal pain\n\ndevil's grip, muscle pain\n\ndiabetes\n\nabnormal nails 270\u20131\n\nanal itching 38\u20139\n\nanorectal pain\n\ncalf pain\n\ndelayed puberty\n\ndizziness\n\ndouble vision 133\u20134\n\nearache 123\u20134\n\nexcessive urination\n\neyelid problems\n\nfailure to thrive\n\nincontinence 419\u201320\n\nleg ulcers ,\n\nmouth lumps and marks\n\nmouth ulcers\n\nnocturia 423\u20134\n\npenile pain\n\npustules\n\nscales and plaques\n\nsore throat\n\ntiredness\n\ntongue pain\n\ntransient visual disturbance 139\u201340\n\nundiagnosed\n\nerectile dysfunction\n\nmuscle pain\n\nnumbness\n\npenile ulceration and sores\n\ntransient visual disturbances\n\nvulval ulceration and sores 267\u20138\n\nvomiting\n\nvulval irritation 261\u20132\n\ndiabetes insipidus\n\nexcessive urination 413\u201315\n\nnocturia 422\u20133\n\nthirst and dry mouth\n\ndiabetes mellitus (DM)\n\nabdominal pain 27\u20138\n\nexcessive urination 413\u201314\n\nitching\n\nleg ulcers\n\nnocturia 422\u20133\n\nthirst and dry mouth 221\u20132\n\ntiredness\n\nundiagnosed\n\nvaginal discharge\n\nweight loss 233\u20134\n\ndiabetic arthropathy, hand and wrist pain\n\ndiabetic autonomic neuropathy\n\nerectile dysfunction\n\nfalls 188\u20139\n\ndiabetic neuropathy\n\ncalf pain\n\nfoot pain\n\nmuscle pain 303\u20134\n\nnumbness and paraesthesiae 212\u201313\n\nurinary incontinence 419\u201320\n\ndiabetic retinopathy, vision loss 142\u20133,\n\ndiaphragmatic irritation, cough\n\ndiarrhoea 21\u20133\n\nabdominal pain\n\nchronic\n\nanal itching\n\nexcessive urination\n\nconstipation\n\nrectal bleeding\n\nrectal discharge\n\nDIC (disseminated intravascular coagulation), purpura and petechiae\n\nDIDMOAD syndrome, excessive urination\n\ndigoxin\n\nbreast enlargement in men\n\nconfusion\n\npalpitations ,\n\ndiphtheria\n\nsore throat\n\nstridor\n\ndisc prolapse\n\ncervical and thoracic, arm and shoulder pain\n\nlumbar, back pain 171\u20132\n\nurinary retention\n\ndislocation, hip pain 306\u20137\n\ndisseminated malignancy\n\nconfusion\n\nshortness of breath\n\ndisulfiram, flushing ,\n\ndiuretics\n\ncalf pain\n\nconfusion 68\u20139\n\nerythema\n\nexcessive urination 413\u201314\n\nfrequent urination\n\nincontinence\n\ntinnitus\n\ntiredness\n\ndiverticular disease\n\nabdominal pain ,\n\nconstipation 18\u201320\n\ndiarrhoea 21\u20132\n\npelvic pain\n\nrectal bleeding 46\u20138\n\nurinary frequency\n\ndizziness 71\u20133\n\nfalls 188\u20139\n\nDKA (diabetic ketoacidosis)\n\nabdominal pain 9\u201310\n\nconfusion 68\u20139\n\nshortness of breath 94\u20135\n\ndouble vision 133\u20135\n\ndouches, vulval irritation 261\u20132\n\nDown's syndrome\n\nfailure to thrive\n\nweight gain 230\u20132\n\ndoxazosin, blocked nose\n\ndrug abuse, _see_ substance abuse\n\ndrug side effects\n\nabsent periods 374\u20135\n\nbleeding gums 351\u20132\n\nblisters\n\nerythema\n\nexcessive urination\n\nleg ulcers\n\nmacules 393\u20134\n\nnocturia\n\nnosebleeds 341\u20132\n\npurpura and petechiae 400\u20131\n\nrunny nose\n\nscales and plaques\n\nswallowing difficulties 320\u20131\n\nswollen ankles 312\u201313\n\ndrug side effects\n\nabnormal movements 168\u20139\n\nhallucinations\n\nitching ,\n\nswollen glands\n\ntension and anxiety\n\ntongue pain\n\ntremor\n\nurinary retention\n\ndrug withdrawal\n\nexcessive sweating\n\nhallucinations\n\nmuscle pain\n\ntremor 227\u20138\n\ndry eyes\n\ndry mouth\n\nitchy eyes 144\u20135\n\ndry mouth 221\u20133\n\nDuane's syndrome, double vision\n\nDUB (dysfunctional uterine bleeding)\n\nheavy periods 377\u20138\n\nirregular vaginal bleeding 380\u20131\n\nDubin\u2013Johnson syndrome, jaundice\n\nDuchenne's muscular dystrophy, limp in a child\n\nduct ectasia, breast lumps in women , 63\u20134\n\nduct epithelial proliferation, nipple discharge\n\nduct obstruction\n\nfacial pain\n\nfacial swelling\n\nduct papilloma, nipple discharge 63\u20135\n\nduodenal ulcers (DU)\n\nabdominal pain ,\n\nback pain\n\nchest pain\n\nepigastric pain 24\u20135\n\nDupuytren's contracture, hand and wrist pain\n\nDVT (deep vein thrombosis)\n\ncalf pain 288\u201390\n\ncoughing up blood\n\nswollen ankles\n\nswollen calf 315\u201317\n\ndysmenorrhoea\n\nprimary\n\npainful periods 383\u20134\n\npelvic pain\n\nsecondary\n\nheavy periods\n\npainful periods\n\npelvic pain\n\ndyspareunia, painful intercourse 245\u20136\n\ndysphagia, epigastric pain\n\ndysthyroid disease, double vision\n\ndystonias, abnormal movements ,\n\ndystrophia myotonica, abnormal gait\n\near discharge 122\u20134\n\nearache 119\u201321\n\nearwax\n\ncough\n\ndeafness 116\u201317\n\nearache\n\ntinnitus 125\u20136\n\nvertigo\n\neating disorders, weight loss 233\u20135\n\nEBV (Epstein\u2013Barr virus)\n\nbleeding gums 351\u20132\n\nmouth ulcers\n\nswollen glands\n\necstasy, hallucinations\n\nectopic pregnancy\n\nabdominal pain ,\n\nirregular vaginal bleeding ,\n\npelvic pain 364\u20136\n\nectropion\n\neyelid problems\n\nitchy eyes\n\neczema; _see also_ contact dermatitis\n\nallergic\n\nfacial rash\n\npenile ulcerations and sores 250\u20131\n\natopic\n\nblisters 388\u20139\n\nitching 203\u20134\n\nblisters 388\u20139\n\near discharge\n\nearache\n\neyelid problems\n\nhand\n\nabnormal nails\n\ninfected\n\nhand and wrist pain\n\nitchy eyes\n\nscales and plaques 405\u20137\n\nvaricose\n\nswollen calves 315\u201316\n\nvenous\n\nleg ulcers\n\nvulval irritation 261\u20132\n\nEhlers\u2013Danlos syndrome, purpura and petechiae\n\nelectrolyte disturbance\n\nhallucinations\n\npalpitations\n\nelectrolyte imbalance\n\ncalf pain 288\u20139\n\nconfusion\n\nelephantiasis, scrotal swelling\n\nencephalitis\n\ncrying baby\n\nfebrile child\n\nendocarditis\n\nblood in urine\n\nexcessive sweating 182\u20133\n\nprolonged fever ,\n\npurpura\n\nendocrine disease, primary, loss of libido\n\nendocrine dysfunction, weight gain\n\nendometrial carcinoma, heavy periods\n\nendometrial polyps\n\nheavy periods\n\nirregular vaginal bleeding\n\npainful periods\n\nendometrioma, vulval swelling\n\nendometriosis\n\nanorectal pain\n\nheavy periods 377\u20139\n\ninfertility 200\u20131\n\npainful intercourse 244\u20136\n\npainful periods 383\u20134\n\npelvic pain , 367\u20139\n\nenterocoele, vulval swelling\n\nenteropathic arthritis, joint pain\n\nenteropathy, protein-losing, swollen ankles 312\u201313\n\nentropion\n\neyelid problems 136\u20137\n\nitchy eyes 144\u20135\n\nEO (epididymo-orchitis)\n\nhaemospermia\n\nscrotal swelling 253\u20134\n\ntesticular pain 256\u20137\n\nepicondylitis, arm and shoulder pain 285\u20136\n\nepidermal necrolysis, toxic, _see_ scalded skin syndrome\n\nepidermolysis bullosa\n\nabnormal nails\n\nbleeding gums\n\nblisters\n\nepididymal cysts\n\nscrotal swelling\n\ntesticular pain 256\u20137\n\nepigastric pain 24\u20136\n\nepiglottitis\n\nacute\n\nfebrile child\n\nhoarseness ,\n\nstridor 335\u20136\n\nsore throat ,\n\nepilepsy\n\nflushing 197\u20139\n\nintractable\n\nmemory loss\n\nloss of consciousness 179\u201380\n\nnumbness and paraesthesiae\n\npartial\n\ndizziness\n\ntemporal lobe\n\nabdominal pain\n\nhallucinations 74\u20135\n\nvertigo 89\u201391\n\nepiscleritis, red eye\n\nepispadias, urinary incontinence\n\nEpstein\u2013Barr virus (EBV)\n\nbleeding gums\n\nmouth ulcers\n\nswollen glands\n\nerectile dysfunction 238\u201340\n\nerythema 390\u20132\n\nerythema ab igne\n\nerythema chronicum migrans\n\nerythema induratum\n\nerythema multiforme , 390\u20132\n\nblisters\n\nbullous\n\nbleeding gums\n\nmouth ulcers\n\nerythema nodosum 390\u20132\n\nerythroderma ,\n\nerythromelalgia, foot pain\n\nerythroplakia, mouth lumps and marks\n\nEustachian tube dysfunction\n\ndeafness 116\u201317\n\nvertigo 89\u201390\n\nexercise, excessive\n\nabsent periods\n\nblood in urine\n\neye strain, headache 77\u20138\n\neye tumour, red eye\n\neyelid problems 136\u20138\n\neyes, acutely red and painful 130\u20132\n\nfacial bone trauma, double vision 133\u20134\n\nfacial pain 150\u20132\n\nfacial rash 153\u20135\n\nfacial swelling 156\u20138\n\nfacial ulcers and blisters 159\u201361\n\nfactitious illness\n\nleg ulcers\n\nprolonged fever ,\n\nfaecal impaction, urinary retention\n\nfailure to thrive 185\u20137\n\nfallopian tube blockages, infertility 200\u20131\n\nfallopian tube carcinoma, irregular vaginal bleeding\n\nfalls with no loss of consciousness 188\u201390\n\nFanconi syndrome, excessive urination\n\nfat necrosis, breast lumps in women ,\n\nfatigue; _see also_ ME\n\nextreme\n\nhallucinations 74\u20135\n\nheadache\n\nfebrile child 191\u20133\n\ncough\n\nneck stiffness\n\ntongue pain\n\nfebrile illness, acute, headache\n\nfeeding problems, failure to thrive 185\u20136\n\nfeminising tumours, loss of libido\n\nfemoral artery aneurysm, groin swelling\n\nfemoral epiphysis, slipped, limp in a child 209\u201311\n\nferritin deficiency, mouth lumps and marks\n\nfetishism, erectile dysfunction\n\nfever, prolonged 215\u201317\n\nfibroadenoma, breast lumps in women 58\u20139\n\nfibroid, red degeneration of, abdominal pain\n\nfibroid degeneration, pelvic pain\n\nfibroma, vulval swelling\n\nfibromyalgia, muscle pain ,\n\nfibrous dysplasia, breast lumps in women 57\u20138\n\nfifth disease, febrile child\n\nfilariasis\n\nscrotal swelling\n\nsingle joint pain\n\nswollen ankles\n\nswollen glands\n\nfinasteride\n\nbreast enlargement in men\n\nloss of libido\n\nfixed drug eruption\n\nerythema 390\u20131\n\npenile ulcerations and sores\n\nvulval ulceration and sores\n\nflashes, _see_ visual disturbance, transient\n\nfloaters, _see_ visual disturbance, transient\n\nfloppy eyelid syndrome, itchy eyes\n\nfluid intake, inadequate, constipation\n\nflushing 197\u20139\n\nfolate deficiency\n\nmemory loss 86\u20137\n\nmouth lumps\n\nmouth ulcers 356\u20137\n\nnumbness 212\u201313\n\ntongue pain\n\nfolliculitis\n\nbacterial\n\nhair loss 275\u20136\n\npustules\n\nfontanelle, bulging, crying baby\n\nfood allergy, _see_ allergies\n\nfood poisoning, diarrhoea\n\nfoot drop, abnormal gait\n\nfoot pain 291\u20133\n\nFordyce spots, mouth lumps and marks 353\u20134\n\nforeign body (FB)\n\nblocked nose ,\n\nin ear\n\ncough 106\u20137\n\nearache\n\nin foot\n\nfoot pain 291\u20133\n\nlimp in a child\n\ninhaled\n\ncough , 106\u20138\n\nshortness of breath\n\nstridor 335\u20136\n\nitchy eyes ,\n\npenile pain\n\nred eye 130\u20132\n\nrunny nose 344\u20135\n\nsore throat 329\u201331\n\nswallowing difficulties\n\nurinary retention\n\nvaginal discharge ,\n\nvomiting blood\n\nforeskin\n\nadherent\n\npenile pain\n\ninflamed\n\ncrying baby\n\nfractures\n\nimpacted, hip pain ,\n\nsingle joint pain\n\nfreckles 393\u20134\n\nFreiberg's disease, foot pain\n\nfrenulum\n\ntight\n\nerectile dysfunction\n\npenile pain\n\ntorn\n\npenile pain 247\u20138\n\npenile sores\n\nFriedreich's ataxia, tremor\n\nfugue states, memory loss\n\nfungal infections\n\nabnormal nails 270\u20131\n\nanal itching 38\u20139\n\nitchy scalp 278\u20139\n\nscales and plaques 405\u20136\n\nfuruncles, earache 119\u201320\n\ngalactocoele\n\nbreast lumps in women\n\nbreast pain\n\nnipple discharge\n\ngalactorrhoea, nipple discharge\n\ngalactosaemia, jaundice\n\ngallstones\n\nabdominal pain , 15\u201316\n\nchest pain\n\nectopic\n\nepigastric pain 24\u20135\n\njaundice 206\u20137\n\nperforating duodenum vomiting blood\n\nganglion, hand and wrist pain 294\u20135\n\ngastric carcinoma\n\nabdominal pain 16\u201317\n\nmemory loss\n\nswallowing difficulties ,\n\ngastric ulcer, epigastric pain\n\ngastritis\n\nabdominal pain 27\u20138\n\nepigastric pain\n\nvomiting blood\n\ngastro-oesophageal reflux; _see also_ GORD\n\nfailure to thrive\n\ngastroduodenal disease, vomiting\n\ngastroenteritis\n\nand abdominal pain\n\ncrying baby\n\ndiarrhoea ,\n\nfailure to thrive\n\nfebrile child 191\u20132\n\nrectal bleeding 46\u20138\n\nvomiting 30\u20132\n\ngastrointestinal bleeding, epigastric pain\n\ngastrointestinal carcinoma, swollen glands\n\ngeneralised anxiety disorder 194\u20135\n\ngenito-urinary infection, haemospermia 241\u20132\n\ngentamicin\n\ndeafness\n\nexcessive urination\n\ngeographic tongue 359\u201361\n\nGI (granuloma inguinale) penile ulcerations and sores 250\u20131\n\nswollen glands 218\u201319\n\nvulval ulceration and sores 266\u20137\n\ngiant cell arteritis\n\njoint pain\n\nmultiple joint pain\n\ngiardiasis\n\ndiarrhoea 21\u20132\n\nfailure to thrive\n\nfebrile child\n\nGilbert's syndrome, jaundice\n\ngingivitis\n\nbad breath 348\u20139\n\nbleeding gums 351\u20132\n\ngland tumour, sublingual, mouth lumps and marks\n\nglands, swollen 218\u201320\n\nabdominal swelling\n\ngroin swelling\n\nitching\n\nweight loss\n\nglandular fever (GF)\n\nearache\n\nfebrile child\n\njaundice\n\nmouth ulcers 356\u20137\n\nneck stiffness\n\nprolonged fever 215\u201316\n\nsore throat 329\u201330\n\nswollen glands 218\u201320\n\ntiredness\n\nglaucoma\n\nacute\n\nred eye 130\u20132\n\nvision loss 147\u20138\n\nvomiting\n\nchronic\n\nvision loss 142\u20133\n\nfacial pain ,\n\nglibenclamide, excessive urination\n\nglobus hystericus, swallowing difficulties 320\u20131\n\nglomerulonephritis, blood in urine\n\nglomus jugulare tumours, tinnitus\n\nglossitis, tongue pain 359\u201360\n\nglossodynia, tongue pain ,\n\nglossopharyngeal neuralgia\n\nsore throat\n\ntongue pain\n\nglue ear, deafness 116\u201317\n\nglycogen storage diseases, jaundice\n\nglycosuria, abdominal pain\n\nGnRH deficiency, delayed puberty 177\u20138\n\ngoitre, neck lumps \u20137\n\ngold therapy, mouth ulcers\n\ngonococcus\n\npelvic pain\n\nscrotal swelling\n\ntesticular pain\n\nvaginal discharge\n\ngonorrhoea\n\nanal itching\n\nitchy eyes\n\nrectal discharge\n\nsore throat\n\nurinary frequency\n\nGoodpasture's syndrome, coughing up blood\n\nGORD (gastro-oesophageal reflux disease)\n\nabdominal pain 27\u20138\n\narm and shoulder pain\n\nbad breath\n\nchest pain 97\u20138\n\ncough , 105\u20136\n\nepigastric pain 24\u20136\n\nhoarseness ,\n\nsore throat 329\u201330\n\nswallowing difficulties 320\u20131\n\nvomiting blood\n\ngout\n\nabnormal gait\n\nerythema 390\u20131\n\nfoot pain 291\u20132\n\nhand and wrist pain 294\u20135\n\nknee pain 309\u201310\n\nred eye\n\nsingle joint pain 282\u20133\n\ngouty tophi, nodules 395\u20136\n\ngranulomas\n\ndelayed puberty\n\nnodules\n\npyogenic\n\nbad breath\n\nnodules\n\nseptal\n\nnosebleeds\n\ngranulomatous disorders, red eye\n\nGraves's disease, nodules\n\ngroin swelling 370\u20132\n\ngrowth hormone deficiency, failure to thrive\n\nGuillain\u2013Barr\u00e9 syndrome\n\ndouble vision\n\nmuscle pain\n\nnumbness and paraesthesiae 212\u201313\n\npainful muscles\n\nshortness of breath\n\ngumma, syphilitic\n\nfacial ulcers and blisters\n\nmouth ulcers\n\ngums, bleeding or painful 351\u20132\n\nhaemangioma\n\ncavernous\n\nbleeding gums\n\neyelid problems ,\n\nhaemarthroses, recurrent, knee pain\n\nhaemarthrosis, traumatic, single joint pain 282\u20134\n\nhaematocoele\n\nscrotal swelling\n\ntesticular pain\n\nhaematoma\n\nintracranial, headache\n\nperianal\n\nanal swelling 41\u20132\n\nanorectal pain 43\u20134\n\nrectus sheath, abdominal pain 12\u201313\n\nseptal\n\nblocked nose ,\n\nhaematuria, microscopic ; _see also_ blood in\n\nurine\n\nabdominal swelling\n\ndry mouth\n\nfrequent urination\n\nhaemospermia\n\nurinary retention\n\nhaemobilia, vomiting blood\n\nhaemochromatosis\n\njaundice\n\nknee pain 309\u201310\n\nhaemodialysis, breast enlargement in men\n\nhaemolytic anaemia, jaundice 206\u20137\n\nhaemophilia\n\nblood in urine\n\nnosebleed 341\u20132\n\npurpura\n\nsingle joint pain 282\u20133\n\nvomiting blood\n\nhaemorrhage\n\nintracerebral\n\nheadache\n\nstiff neck\n\nsubarachnoid\n\nheadache\n\nmemory loss 86\u20137\n\nstiff neck\n\nvomiting\n\nhaemorrhagic telangiectasia\n\nbleeding gums\n\ncoughing up blood\n\nnosebleeds\n\nhaemorrhoids\n\nanal itching 38\u20139\n\nconstipation\n\ninflamed\n\npenile pain\n\nprolapsed\n\nanal swelling 41\u20132\n\nrectal bleeding 46\u20138\n\nrectal discharge 49\u201350\n\nthrombosed\n\nanorectal pain\n\npainful intercourse\n\nhaemospermia 241\u20133\n\nhair loss 275\u20137\n\nhallucinations 74\u20136\n\nhaloperidol, loss of libido\n\nhand, foot and mouth disease\n\nblisters\n\nfebrile child\n\nmouth ulcers\n\ntongue pain 359\u201360\n\nhand and wrist pain 294\u20136\n\nhead injury\n\ndouble vision 133\u20135\n\nmemory loss ,\n\nrunny nose\n\ntinnitus\n\nheadache 77\u20139\n\ncluster\n\nfacial pain\n\nrunny nose\n\nearly morning\n\nabsent periods\n\nerectile dysfunction\n\nvision loss\n\nexcess body hair\n\nthunderclap\n\nneck stiffness\n\nhearing loss, tinnitus 125\u20136\n\nheart block\n\nloss of consciousness\n\npalpitations\n\nheart disease, undiagnosed congenital, shortness of breath\n\nheart failure, _see_ cardiac failure\n\nheavy metal poisoning, mouth lumps and marks\n\nHeberden's nodules ,\n\nhelminths, failure to thrive\n\nhemiballismus, abnormal movements\n\nHenoch\u2013Sch\u00f6nlein purpura 400\u20131\n\nabdominal pain 27\u20138\n\nHenoch\u2013Sch\u00f6nlein syndrome, joint pain\n\nhepatitis\n\nabdominal pain 9\u201310\n\nfebrile child\n\njaundice 206\u20137\n\nmultiple joint pain\n\ntiredness 224\u20135\n\nhepatomegaly 6\u20137\n\nhermaphroditism, breast enlargement in men\n\nhernia\n\nepigastric pain\n\nfemoral\n\ngroin swelling 370\u20132\n\npelvic pain ,\n\nhip pain 306\u20137\n\ninguinal\n\ngroin swelling 370\u20131\n\npelvic pain ,\n\nscrotal swelling 253\u20134\n\nvulval swelling 264\u20135\n\ninguinoscrotal\n\ntesticular pain\n\nheroin, erectile dysfunction\n\nherpangina\n\nmouth ulcers\n\nsore throat\n\nherpes\n\nblisters\n\nurinary retention\n\nvaginal discharge\n\nvulval irritation ,\n\nherpes gestationis, itching\n\nherpes simplex\n\nblisters 388\u20139\n\nfacial rash ,\n\nfacial ulcers and blisters 159\u201361\n\nmouth ulcers\n\npenile pain\n\npenile ulcerations and sores 250\u20131\n\npustules 403\u20134\n\nrunny nose\n\nsore throat\n\ntongue pain 359\u201360\n\nurinary retention\n\nvulval ulceration and sores 266\u20138\n\nherpes simplex encephalitis, memory loss\n\nherpes zoster\n\nblisters 388\u20139\n\nbreast pain\n\neyelid problems\n\nfacial pain\n\nfacial rash ,\n\nfacial swelling ,\n\nfacial ulcers and blisters 159\u201361\n\nleg ulcers\n\npenile sores\n\npustules 403\u20134\n\nvulval sores\n\nherpes zoster oticus, ear discharge\n\nherpetic gingivostomatitis, bleeding gums 351\u20132\n\nhiatus hernia, shortness of breath\n\nhidradenitis, anal swelling\n\nhidradenitis suppurativa, pustules\n\nhidradenoma, vulval swelling 264\u20135\n\nhilus cell tumour, excess body hair\n\nhip\n\ncongenital dislocation of, limp in a child 209\u201310\n\nirritable, limp in a child 209\u201310\n\nhip pain\n\nrecurrent 306\u20138\n\nreferred, knee pain\n\nHirschsprung's disease\n\nabdominal pain\n\nconstipation 18\u201320\n\ncrying baby\n\nHIV\/AIDS\n\nerythema 390\u20131\n\nexcessive sweating 182\u20133\n\nfailure to thrive 185\u20137\n\nfebrile child\n\nleg ulcers\n\nmouth lumps\n\nmouth ulcers 356\u20137\n\nprolonged fever 215\u201316\n\nrectal discharge\n\nswollen glands 218\u201320\n\ntiredness 224\u20135\n\nweight loss 233\u20134\n\nhoarseness 323\u20135\n\nhookworm\n\ndiarrhoea\n\nswollen ankles\n\nHorner's syndrome, eyelid problems 136\u20137\n\nHRT (hormone-replacement therapy), irregular\n\nvaginal bleeding ,\n\nHuntington's chorea, abnormal movements 168\u201370\n\nHutchinson's freckle\n\nfacial ulcers and blisters\n\nmacules 393\u20134\n\nhydatidiform mole, irregular vaginal bleeding 380\u20131\n\nhydralazine, joint pain\n\nhydrocephalus, falls 188\u20139\n\nhydrocoeles\n\ngroin swelling\n\nscrotal swelling 253\u20135\n\ntesticular pain\n\nvulval swelling\n\nhydronephrosis\n\nabdominal pain 15\u201316,\n\nabdominal swelling\n\nhymen, unruptured\n\npainful intercourse\n\npainful periods\n\nhyperaldosteronism, excessive urination\n\nhypercalcaemia\n\nconstipation 18\u201319\n\nexcessive urination 413\u201314\n\nthirst and dry mouth 221\u20132\n\ntiredness\n\nhypercholesterolaemia, eyelid problems\n\nhyperemesis gravidarum, memory loss\n\nhypergammaglobulinaemia, purpura and petechiae\n\nhyperglycaemia\n\nconfusion\n\nflushing 197\u20138\n\nvomiting 30\u20131\n\nhyperglycaemic ketotic state, hallucinations\n\nhyperhidrosis, excessive sweating 182\u20133\n\nhypernephroma, breast enlargement in men\n\nhyperparathyroidism\n\nconfusion 68\u20139\n\nexcessive urination\n\nhyperpigmentation, post-inflammatory\n\nfacial rash\n\nmacules 393\u20134\n\nhyperpituitarism, excessive sweating\n\nhyperprolactinaemia\n\nbreast enlargement in men\n\ndelayed puberty\n\nheavy periods 377\u20138\n\nnipple discharge 63\u20134\n\nhyperprolactinaemic drugs, loss of libido\n\nhyperstimulation, insomnia\n\nhypertension\n\nheadache 77\u20138\n\nloss of vision\n\nmalignant, haemospermia\n\nnosebleeds 341\u20132\n\ntinnitus\n\nhypertensive retinopathy, vision loss\n\nhyperthyroidism\n\nabsent periods\n\nbreast enlargement in men 54\u20135\n\ndelayed puberty 177\u20138\n\ndouble vision\n\nerythema\n\nexcessive sweating 182\u20133\n\nfailure to thrive\n\nflushing 197\u20138\n\nheavy periods\n\ninsomnia ,\n\nirregular vaginal bleeding\n\nitching 203\u20134\n\nneck lumps\n\ntension and anxiety 194\u20135\n\ntiredness 224\u20136\n\ntremor\n\nweight loss 233\u20134\n\nhypertrichosis lanuginosa, excess body hair\n\nhypertrophic pulmonary osteoarthropathy, joint\n\npain ,\n\nhyperventilation\n\ndizziness\n\nloss of consciousness\n\nnumbness 212\u201313\n\nshortness of breath 94\u20135, ,\n\nstridor\n\nhypocalcaemia\n\nabnormal movements\n\ncalf pain 288\u20139\n\ntiredness\n\nhypoglycaemia\n\nconfusion 68\u201370\n\ndizziness 71\u20132\n\nexcessive sweating 182\u20133\n\nflushing 197\u20139\n\nloss of consciousness 179\u201380\n\nreactive\n\nexcessive sweating\n\nheadache\n\ntiredness\n\nvomiting 30\u20131\n\nhypoglycaemic drugs, falls\n\nhypogonadism\n\nbreast enlargement in men 54\u20135\n\nerectile dysfunction\n\nhypokalaemia\n\nrectal discharge\n\ntiredness 224\u20135\n\nhyponatraemia, tiredness 224\u20135\n\nhypoparathyroidism, confusion 68\u20139\n\nhypopigmentation, post-inflammatory\n\nfacial rash\n\nmacules\n\nhypopituitarism\n\nhair loss\n\nweight gain 230\u20131\n\nhypotensives\n\nerectile dysfunction\n\nfalls\n\nhypothalamic lesion, weight gain\n\nhypothalamic\/pituitary disease, loss of libido\n\nhypothalamo-pituitary space-occupying lesion, delayed puberty\n\nhypothalamo-pituitary tumour, excessive urination\n\nhypothalamus\/pituitary trauma, delayed puberty\n\nhypothyroidism\n\nabsent periods\n\nbreast enlargement in men 54\u20135\n\nconfusion\n\nconstipation 18\u201319\n\ndelayed puberty\n\nexcess body hair 272\u20133\n\nfacial swelling\n\nfailure to thrive\n\nfalls 188\u20139\n\nheavy periods\n\nhoarseness 323\u20135\n\nirregular vaginal bleeding\n\nitching 203\u20134\n\nloss of libido ,\n\nmemory loss\n\nmuscle pain 303\u20135\n\nneck lumps\n\nnipple discharge 63\u20134\n\nnumbness and paraesthesiae\n\ntiredness 224\u20135\n\nweight gain 230\u20132\n\nhypotonia, failure to thrive\n\nhypovolaemic shock, shortness of breath\n\nhypoxaemia\n\nconfusion\n\nshortness of breath\n\nhypoxia\n\nconfusion 68\u20139\n\nhallucinations 74\u20135\n\nhysteria\n\nabnormal gait\n\nabnormal movements\n\nnumbness ,\n\nstiff neck\n\ntremor\n\nvision loss\n\nIBS (irritable bowel syndrome)\n\nabdominal pain , ,\n\nabdominal swelling 6\u20137\n\nconstipation 18\u201319\n\ndiarrhoea ,\n\nepigastric pain 24\u20135\n\npelvic pain 367\u20138\n\nrectal discharge\n\nichthyosis, scales and plaques\n\nIGTN (infected ingrowing toenail), foot pain 291\u20132\n\niliotibial band syndrome\n\nhip pain\n\nknee pain\n\nimmobility, prolonged\n\nerythema\n\nleg ulcers\n\nimmune deficiency\n\ncough 106\u20137\n\nmouth lumps\n\nimmunisation\n\ncrying baby 174\u20135\n\nfebrile child\n\nimmunosuppression\n\ncough\n\nmouth ulcers\n\npustules\n\nsore throat\n\ntongue pain\n\nimpacted molar, earache\n\nimpetigo\n\nfacial rash 153\u20134\n\nfacial ulcers and blisters 159\u201360\n\nitchy scalp 278\u20139\n\npustules 403\u20134\n\nrunny nose\n\ninborn errors of metabolism, failure to thrive\n\nincontinence\n\nfaecal, anal itching\n\nurinary 419\u201321\n\nvulval irritation\n\ninferior vena cava thrombosis\n\nscrotal swelling\n\nswollen ankles\n\ninfertility 200\u20132\n\ninflammatory arthritis\n\nhand and wrist pain\n\nhip pain 306\u20137\n\ninflammatory bowel disease (IBD)\n\nabdominal pain 15\u201316\n\nanorectal pain\n\ndiarrhoea 21\u20132\n\nmouth ulcers ,\n\npelvic pain\n\nprolonged fever\n\nrectal bleeding 46\u20137\n\nrectal discharge\n\nswollen ankles\n\nweight loss 233\u20134\n\ninguinal tumour, groin swelling\n\ninsect bites\n\nblisters\n\neyelid problems\n\nleg ulcers\n\npapules 398\u20139\n\nprolonged fever\n\ninsomnia 80\u20132\n\ntiredness\n\ninsulin, weight gain\n\ninsulinoma, weight gain 230\u20131\n\nintestinal obstruction\n\nabdominal swelling\n\nconstipation\n\nvomiting\n\nintoxication\n\nconfusion\n\ndouble vision 133\u20135\n\nintracranial abscess, headache\n\nintracranial lesions\n\ndouble vision\n\nheadache 77\u20138\n\nintracranial pressure, raised\n\nheadache\n\nvomiting ,\n\nintracranial tumour, vision loss\n\nintraduct carcinoma, nipple discharge\n\nintraorbital tumour, vision loss\n\nintrauterine growth retardation, failure to thrive\n\nintravascular pressure, increased, purpura and\n\npetechiae 400\u20131\n\nintubation, hoarseness 323\u20134\n\nintussusception\n\ncrying baby\n\nrectal bleeding\n\niritis\n\nfacial pain ,\n\nred eye 130\u20131\n\niron deficiency, tiredness 224\u20135\n\niron deficiency anaemia\n\nabnormal nails 270\u20131\n\nconstipation 18\u201319\n\ndiarrhoea\n\nexcess body hair\n\nheavy periods\n\nitching 203\u20134\n\nmouth ulcers 356\u20137\n\nswallowing difficulties\n\ntongue pain\n\nischaemia, foot pain 291\u20133\n\nischaemic bowel\n\nabdominal pain 9\u201310,\n\nrectal bleeding\n\nischaemic heart disease\n\nloss of consciousness\n\nnumbness\n\npalpitations 113\u201314\n\nisoniazid\n\njaundice\n\nmultiple joint pain\n\nitching 203\u20135\n\nprolonged fever\n\nitchy eyes 144\u20136\n\nITP (idiopathic thrombocytopenia purpura)\n\nIUCD (intrauterine contraceptive device)\n\nheavy periods 377\u20138\n\npainful periods 383\u20134\n\npelvic pain , 367\u20138\n\nvaginal discharge\n\njaundice 206\u20138\n\nitching\n\nvulval irritation\n\nJCA (juvenile chronic arthritis)\n\njoint pain\n\nlimp in a child 209\u201310\n\njoint pain\n\nacromioclavicular and sternoclavicular arm and shoulder pain ,\n\ninsomnia\n\nmultiple 300\u20132\n\nreferred\n\nmuscle pain\n\nsingle 282\u20134\n\njoint replacement, hip pain 306\u20138\n\njoint subluxation, recurrent, single joint pain\n\njoint swelling, symmetrical, hand and wrist pain\n\njunctional naevus, macules 393\u20134\n\nKallmann's syndrome, delayed puberty\n\nKaposi's sarcoma, papules 398\u20139\n\nKaposi's varicelliform eruption\n\nKawasaki's disease, febrile child\n\nkeratitis, red eye 130\u20131\n\nkerato-acanthoma\n\nfacial ulcers and blisters 159\u201360\n\nnodules\n\nkeratoconjunctivitis sicca, red eye\n\nkeratoderma blenorrhagica, scales and plaques\n\nkeratosis, solar, scales and plaques\n\nkeratosis obturans, ear discharge\n\nkeratosis pilaris, papules\n\nketohalitosis of starvation, bad breath\n\nKienb\u00f6ck's disease, hand and wrist pain 294\u20135\n\nking's evil, neck lumps\n\nKlinefelter's syndrome, breast enlargement in men 54\u20135\n\nknee pain 309\u201311\n\nreferred\n\ncalf pain\n\nK\u00f6hler's disease, foot pain\n\nkoilonychia, abnormal nails 270\u20131\n\nKorsakoff's syndrome, memory loss\n\nl-dopa, abnormal movements\n\nlabour, premature, abdominal pain 13\u201314\n\nlabral tear, hip pain 306\u20137\n\nlabyrinthitis\n\nfalls\n\nvomiting 30\u20131\n\nlactose intolerance, diarrhoea\n\nlamellar nail dystrophy, abnormal nails\n\nlaryngeal nerve palsy, recurrent, hoarseness 323\u20134\n\nlaryngeal obstruction, shortness of breath\n\nlaryngeal oedema, stridor\n\nlaryngeal paralysis, stridor 335\u20136\n\nlaryngeal stenosis, stridor\n\nlaryngeal trauma, stridor\n\nlaryngeal tumour\n\ncough\n\nhoarseness ,\n\nstridor\n\nlaryngitis\n\nhoarseness 323\u20134\n\nstridor 335\u20136\n\nlaryngomalacia, stridor\n\nlaxative abuse\n\nconstipation\n\ndiarrhoea\n\nlead poisoning\n\nabdominal pain\n\ncalf pain\n\nLeber's hereditary optic atrophy, vision loss\n\nleg oedema, blisters\n\nleg ulcers 297\u20139\n\nlegs, unequal length, limp in a child\n\nleishmaniasis\n\nfacial ulcers and blisters ,\n\nfebrile child\n\nlentigo maligna, facial ulcers and blisters\n\nleprosy\n\nnodules\n\nnumbness and paraesthesiae\n\nswollen glands\n\ntuberculoid, macules\n\nleptospirosis\n\njaundice\n\nneck stiffness\n\nleucocytosis\n\nabdominal pain\n\nblood in urine\n\nleuconychia, abnormal nails 270\u20131\n\nleukaemia\n\nbleeding gums\n\nitching 203\u20134\n\nmouth ulcers\n\nnosebleeds 341\u20132\n\nprolonged fever ,\n\npurpura\n\nswollen glands 218\u201319\n\nvomiting blood\n\nvulval irritation\n\nleukoplakia\n\nmouth lumps and marks\n\nvulval irritation\n\nlevator ani syndrome, anorectal pain\n\nlibido, loss of 83\u20135\n\nlice\n\neyelid problems\n\nitchy eyes\n\nitchy scalp 278\u20139\n\nvulval irritation\n\nlichen planus\n\nabnormal nails\n\nbleeding gums 351\u20132\n\nerosive\n\nmouth ulcers\n\nitching\n\nmouth lumps\n\npainful tongue 359\u201360\n\npapules\n\npenile ulcerations and sores 250\u20131\n\nscales and plaques\n\nvulval irritation\n\nlichen sclerosus et atrophicus, anal itching ,\n\nlichen simplex\n\nitchy scalp 278\u20139\n\nscales and plaques\n\nligament rupture, knee pain\n\nligament sprain, knee pain 309\u201311\n\nlimp in a child 209\u201311\n\nlingual carcinoma, facial pain\n\nlipodermatosclerosis, leg ulcers\n\nlipoma\n\nbreast enlargement in men\n\nbreast lumps in women\n\ngroin swellings\n\nnodules 395\u20136\n\nvulval swelling\n\nlisteriosis, febrile child\n\nlithium, muscle pain\n\nlithium carbonate, excessive urination\n\nlithotripsy, haemospermia\n\nlivedo reticularis, erythema ,\n\nliver cancer, jaundice 206\u20137\n\nliver congestion, abdominal pain\n\nliver disease\n\nabnormal nails\n\nchronic\n\nbreast enlargement in men 54\u20135\n\nerythema\n\nheavy periods 377\u20138\n\nitching\n\nnosebleeds 341\u20132\n\npurpura and petechiae 400\u20131\n\ntiredness\n\nweight loss\n\nliver failure\n\nbad breath 348\u20139\n\nfailure to thrive\n\nfulminant\n\njaundice\n\ntremor 227\u20138\n\nweight gain 230\u20131\n\nlobar collapse, shortness of breath ,\n\nloss of consciousness\n\nepisodic\n\ndizziness\n\nvertigo\n\nlow body weight, insomnia\n\nLRTI (lower respiratory tract infection), cough 103\u20134, 106\u20137\n\nLSD, hallucinations\n\nlung carcinoma\n\nabnormal nails\n\narm and shoulder pain\n\njoint pain 300\u20132\n\nneck lumps\n\nprolonged fever\n\nlung disease, interstitial, cough 106\u20137\n\nlung tumour\n\nbreast enlargement in men 54\u20135\n\ncough\n\nlupus erythematosus, hair loss , ; _see also_ SLE\n\nlupus vulgaris\n\nfacial rash 153\u20134\n\nnodules\n\nluteoma, excess body hair\n\nLUTS (lower urinary tract symptoms)\n\nfrequent urination 416\u201317\n\nincontinence\n\nnocturia\n\nurinary retention\n\nLVF (left ventricular failure)\n\ncough\n\ninsomnia\n\nshortness of breath 94\u20135\n\nLyell's syndrome, blisters\n\nLyme disease\n\nerythema 390\u20132\n\nfebrile child\n\nmultiple joint pain\n\nprolonged fever\n\nlymph nodes\n\npalpable\n\ngroin swelling 370\u20131\n\nprominent normal\n\nneck lumps 326\u20137\n\nlymphadenitis, reactive, neck lumps 326\u20137\n\nlymphadenopathy, _see_ glands, swollen\n\nlymphangioma, bleeding gums\n\nlymphogranuloma venereum (LGV)\n\npenile sores 250\u20131\n\nswollen glands 218\u201319\n\nvulval sores 266\u20137\n\nlymphoma\n\nbreast lumps in women\n\nexcessive sweating 182\u20134\n\ngroin swellings\n\nitching\n\nneck lumps 326\u20137\n\nnodules 395\u20137\n\nprolonged fever 215\u201317\n\npurpura\n\nscales and plaques 405\u20136\n\nswallowing difficulties\n\nswollen glands 218\u201320\n\nvulval irritation \nmacular degeneration, vision loss 142\u20133\n\nmacules 393\u20134\n\nMadura foot, single joint pain\n\nmalabsorption\n\nbleeding gums 351\u20132\n\ndiarrhoea 21\u20132\n\nfailure to thrive 185\u20136\n\nmemory loss\n\nmalaria\n\nfebrile child\n\njaundice\n\nprolonged fever 215\u201316\n\nmalignant melanoma\n\nnodules\n\npapules 398\u20139\n\nulcerating\n\nfacial ulcers and blisters\n\nmalingering\n\nback pain ,\n\nmemory loss\n\nMallory\u2013Weiss tear, vomiting blood 33\u20135\n\nmalnutrition\n\nfailure to thrive\n\nhair loss\n\ninsomnia\n\nswollen ankles\n\nweight loss\n\nmamillary duct fistula, nipple discharge\n\nmania\n\nhallucinations\n\ninsomnia\n\nmarch fracture, foot pain 291\u20132\n\nmarijuana, breast enlargement in men\n\nmasseteric hypertrophy, facial swelling\n\nmastalgia, cyclical, breast pain 60\u20131\n\nmastitis\n\nbreast pain 60\u20132\n\nchest pain\n\nnipple discharge 63\u20134\n\nmastocytosis, systemic, flushing\n\nmastoiditis\n\ncrying baby\n\near discharge 122\u20134\n\nearache 119\u201321\n\nMcCune\u2013Albright syndrome, breast enlargement in men\n\nME (chronic fatigue syndrome)\n\nmuscle pain ,\n\ntiredness\n\nmeasles\n\nerythema\n\nfebrile child\n\nmacules\n\nswollen glands\n\nMeckel's diverticulum\n\nabdominal pain\n\nrectal bleeding\n\nmedial shelf syndrome, knee pain\n\nmediastinal tumours\n\nstridor\n\nswallowing difficulties 320\u20131\n\nvomiting blood\n\nmegacolon, acquired, constipation 18\u201319\n\nmeibomian cysts, eyelid problems 136\u20137\n\nmelaena, vomiting blood 34\u20135\n\nmelanoma ; _see also_ malignant melanoma\n\nblocked nose\n\nchoroidal\n\nvision loss\n\nleg ulcers\n\nmouth lumps\n\nsubungual\n\nabnormal nails 270\u20131\n\nvulval ulceration and sores\n\nmemory loss 86\u20138\n\nM\u00e9ni\u00e8re's disease\n\nabnormal gait\n\ndeafness\n\nfalls\n\ntinnitus 125\u20136\n\nvertigo 89\u201390\n\nmeningism, neck stiffness 332\u20133\n\nmeningitis\n\ncrying baby\n\nearache\n\nfebrile child\n\ngummatous\n\nfacial pain\n\nheadache 77\u20138\n\nneck stiffness 332\u20133\n\nvomiting ,\n\nmeningoencephalitis, tremor 227\u20138\n\nmenopausal flushes, insomnia\n\nmenopausal vaginal dryness, painful intercourse\n\nmenopause\n\nabsent periods 374\u20135\n\nexcess body hair 272\u20133\n\nexcessive sweating 182\u20133\n\nflushing 197\u20138\n\npalpitations\n\nweight gain\n\nmenorrhagia 377\u20139\n\nabdominal swelling\n\nmeralgia paraesthetica, hip pain 306\u20137\n\nmercury poisoning, bleeding gums\n\nmesenteric artery ischaemia, abdominal pain\n\nmesenteric infarction, abdominal pain\n\nmetabolic disturbance\n\nhallucinations 74\u20135\n\nshortness of breath\n\nmetastases\n\ngroin swelling ,\n\njaundice 206\u20138\n\nleg ulcers\n\nneck lumps\n\nnodules\n\nvulval swelling\n\nmetatarsalgia, foot pain\n\nmethanol, vision loss\n\nmethotrexate, cough ,\n\nmethyldopa, jaundice\n\nmetoclopramide\n\nabnormal movements\n\nabsent periods\n\nmetronidazole, flushing\n\nMI (myocardial infarction)\n\nabdominal pain\n\nchest pain\n\nepigastric pain\n\nfalls 188\u20139\n\nmicrognathia, failure to thrive\n\n_Microsporum_ infections, hair loss\n\nmicturition, _see also_ urination\n\npain after\n\npenile pain\n\nsyncope\n\nloss of consciousness\n\nmigraine\n\nheadache 77\u20139\n\nnumbness and paraesthesiae ,\n\nophthalmoplegic\n\ndouble vision\n\ntransient visual disturbance 139\u201341\n\nvestibular, vertigo 89\u201390\n\nvision loss 147\u20138\n\nvomiting\n\nmilia, papules\n\nMilroy's disease, swollen ankles\n\nminoxidil, excess body hair\n\nMirizzi syndrome, jaundice\n\nmiscarriage\n\nabdominal pain 12\u201313\n\nirregular vaginal bleeding\n\npelvic pain 364\u20135\n\nmitral flush, facial rash\n\nmitral stenosis\n\ncoughing up blood\n\nflushing\n\nmitral valve disease\n\nflushing\n\npalpitations\n\nmittelschmerz\n\nirregular vaginal bleeding\n\npelvic pain\n\nMoeller's glossitis, tongue pain\n\nmolluscum contagiosum\n\neyelid problems\n\npapules\n\nMondor's disease\n\nbreast lumps in women ,\n\nbreast pain\n\nMongolian spot\n\nmonoarthritis, recurrent, knee pain\n\nmorphoea, hair loss\n\nMorton's neuroma, foot pain 291\u20132\n\nmotor neurone disease\n\nabnormal gait\n\nabnormal movements\n\ncalf pain\n\nshortness of breath\n\nswallowing difficulties\n\ntiredness\n\nweight loss\n\nmouth breathing, thirst and dry mouth 221\u20132\n\nmouth lumps and marks 353\u20135\n\nmouth ulcers 356\u20137\n\nmouthwashes, tongue pain\n\nmucocoele, mouth lumps and marks 353\u20134\n\nmulti-infarct dementia, memory loss 86\u20137\n\nmultiple sclerosis (MS)\n\nabnormal gait ,\n\narm and shoulder pain\n\ncalf pain\n\ndeafness 116\u201317\n\ndouble vision 133\u20134\n\nerectile dysfunction 238\u20139\n\nfacial pain\n\nnumbness and paraesthesiae 212\u201314\n\ntiredness\n\ntremor 227\u20138\n\nurinary incontinence\n\nurinary retention\n\nvertigo 89\u201391\n\nvision loss 142\u20133\n\nmumps\n\nfacial pain\n\nfacial swelling 156\u20137\n\nfebrile child\n\ntesticular pain\n\nMunchausen's syndrome\n\nfactitious illness\n\nvomiting blood\n\nMunchausen's syndrome by proxy, failure to thrive\n\nmuscle enzyme deficiency, calf pain\n\nmuscle fasciculation, abnormal movements 168\u20139\n\nmuscle herniation, swollen calves\n\nmuscle rupture, swollen calves 316\u201317\n\nmuscle stiffness, calf pain 288\u20139\n\nmuscle strain\n\narm and shoulder pain 285\u20136\n\ncalf pain 288\u20139\n\nhip pain 306\u20137\n\nmuscle pain\n\nswollen calves 315\u201316\n\nmuscles\n\npainful 303\u20135\n\npulled\n\nchest pain 97\u20138\n\nmuscular dystrophy\n\nlimp\n\nshortness of breath\n\nswollen calf\n\nmuscular neoplasm, swollen calves\n\nmusculoskeletal pain\n\nchest pain\n\npelvic pain\n\nmyalgia, epidemic, muscle pain\n\nmyasthenia gravis\n\nabnormal gait\n\ndouble vision ,\n\neyelid problems ,\n\nswallowing difficulties\n\ntiredness\n\nmycosis fungoides, scales and plaques ,\n\nmydriatics\n\nloss of vision\n\nred eye\n\nmyelitis, dorsal, numbness and paraesthesiae\n\nmyeloma\n\nback pain 171\u20132\n\nneck stiffness\n\nswollen glands\n\nmyocarditis, chest pain\n\nmyoclonus\n\nabnormal movements 168\u20139\n\npalatal\n\ntinnitus\n\nmyokymia\n\nabnormal movements 168\u20139\n\neyelid problems 136\u20137\n\nmyositis, orbital, double vision\n\nmyotonic dystrophy, eyelid problems\n\nmyxoedema\n\narm and shoulder pain\n\nconfusion 68\u20139\n\nfacial swelling\n\nhair loss\n\npretibial, nodules 395\u20136\n\nnaevus anaemicus, macules\n\nNAI (non-accidental injury)\n\ncrying baby ,\n\npurpura\n\nrectal bleeding ,\n\nweight loss\n\nnail dystrophy, abnormal nails 270\u20131\n\nnails, abnormal 270\u20131\n\nnappy rash, crying baby\n\nnarcolepsy\n\nhallucinations 74\u20135\n\nloss of consciousness\n\nnasal fracture, nosebleeds\n\nnasal infection, nosebleeds 341\u20132\n\nnasal mucosal lesion, infected, runny nose 344\u20135\n\nnasal polyps\n\nblocked nose\n\ndeafness\n\nrunny nose 344\u20135\n\nnasal sprays, nosebleeds 340\u20131\n\nnasal tumours\n\nblocked nose\n\nnosebleeds ,\n\nnasopharyngeal carcinoma\n\nblocked nose\n\ndeafness ,\n\nearache\n\nfacial pain\n\nfacial swelling\n\nswollen glands\n\nvertigo\n\nnasopharyngeal tumour, red eye\n\nnausea\n\ndiarrhoea\n\ntesticular pain\n\nnear-death experience, hallucinations\n\nneck, stiff 332\u20134\n\nneck abscess, neck stiffness\n\nneck lumps 326\u20138\n\nneglect, failure to thrive 185\u20136\n\nnephritis\n\nblood in urine\n\nswollen ankles 312\u201313\n\nnephrotic syndrome\n\neyelid problems 136\u20137\n\nswollen ankles 312\u201313\n\nnerve compression\n\narm and shoulder pain\n\nnumbness and paraesthesiae 212\u201313\n\nnerve entrapment\n\nhand and wrist pain\n\nnumbness 212\u201313\n\nshoulder pain\n\nnerve root pain\n\ncervicogenic\n\nsore throat\n\nspinal, _see_ spinal nerve root pain\n\nneuralgia\n\nheadache\n\npost-herpetic, abdominal pain\n\nneuritis\n\nbrachial and ulnar, arm and shoulder pain\n\noptic\n\nabnormal gait\n\nloss of vision\n\nnumbness\n\ntransient visual disturbance\n\nvision loss\n\nvestibular\n\nabnormal gait ,\n\nvertigo 89\u201390\n\nviral acoustic\n\ndeafness\n\nneurofibroma, groin swelling\n\nneurofibromatosis, macules\n\nneurosyphilis, memory loss\n\nnifedipine, palpitations\n\nnight sweats, cough\n\nnipple discharge 63\u20135\n\nnipples, cracked or inflamed, breast pain 60\u20131\n\nnitrates, headache\n\nnocturia 422\u20134\n\ninsomnia 80\u20131\n\nnocturnal enuresis\n\nnodules 395\u20137\n\nnoise damage\n\nchronic, tinnitus\n\ndeafness\n\nnon ulcer dyspepsia (NUD), epigastric pain 24\u20135\n\nNoonan's syndrome, delayed puberty\n\nnose\n\nblocked 338\u20139\n\nrunny\/discharging 344\u20135\n\nnosebleeds 341\u20133\n\nNSAIDs\n\nepigastric pain\n\nerythema\n\nswallowing difficulties 320\u20131\n\nswollen ankles\n\nvomiting blood\n\nnumbness and paraesthesiae 212\u201314\n\nobesity\n\nabdominal swelling\n\nbreast enlargement in men\n\nleg ulcers\n\nshortness of breath 100\u20131\n\nswollen ankles 312\u201313\n\nweight gain 230\u20132\n\nobsessive-compulsive disorder, tension and anxiety\n\nocular muscles, pseudoparalysis of, double vision\n\noculomotor nerve palsy, eyelid problems 136\u20137\n\noedema\n\nfoot pain 291\u20132\n\nleg ulcers\n\nweight gain 230\u20131\n\noesophageal carcinoma, swallowing difficulties 320\u20132\n\noesophageal spasm, epigastric pain\n\noesophageal varices, vomiting blood ,\n\noesophagitis\n\nchest pain\n\nswallowing difficulties 321\u20132\n\noesophagus, ruptured, vomiting blood\n\noestrogen, weight gain\n\nonchocerciasis\n\nloss of vision\n\nred eye\n\nonychogryphosis, abnormal nails 270\u20131\n\nonychomycosis, abnormal nails 270\u20131\n\nonychorrhexis, abnormal nails\n\nopiates\n\nconfusion\n\nconstipation\n\ndouble vision\n\ntremor\n\nweight loss\n\noptic nerve injury, vision loss\n\noral neoplasia, bleeding gums\n\norbital disease, double vision\n\norbital fracture, double vision\n\norchitis, testicular pain 256\u20137\n\norf, pustules\n\norgan failure\n\ntiredness\n\nweight loss\n\norganic nerve lesions, excessive sweating\n\noropharyngeal carcinoma, sore throat ,\n\northostatic hypotension\n\nfalls 188\u20139\n\nloss of consciousness 179\u201380\n\ntransient visual disturbance\n\nOsgood\u2013Schlatter's disease\n\nknee pain 309\u201310\n\nsingle joint pain\n\nOsler\u2013Weber\u2013Rendu syndrome, mouth lumps and marks\n\nosteitis, facial pain\n\nosteoarthritis (OA)\n\nabnormal gait 165\u20136\n\nfalls\n\nhand and wrist pain 294\u20135\n\nhip pain 306\u20137\n\ninsomnia\n\njoint pain 282\u20133, 300\u20131\n\nknee pain 309\u201311\n\nleg ulcers\n\nosteochondritis, foot pain 291\u20132\n\nosteochondritis dissecans, knee pain 309\u201310\n\nosteomalacia, back pain 171\u20132\n\nosteomyelitis\n\nback pain 171\u20132\n\ncrying baby\n\nfebrile child\n\nfoot pain 291\u20133\n\nhand and wrist pain ,\n\nlimp in a child 209\u201311\n\nosteoporosis\n\nback pain 171\u20132\n\nexcessive urination\n\nosteosarcoma, knee pain ,\n\notitis externa (OE)\n\ncrying baby\n\ndeafness 116\u201317\n\near discharge 122\u20133\n\nearache 119\u201320\n\notitis media\n\nchronic, vertigo ,\n\ncrying baby\n\ndeafness 116\u201318\n\near discharge 122\u20133\n\nearache 119\u201321\n\nfebrile child 191\u20132\n\ntinnitus 125\u20136\n\notosclerosis\n\ndeafness ,\n\ntinnitus 125\u20137\n\noutflow obstruction\n\nfrequent urination\n\nurinary incontinence 419\u201321\n\nurinary retention\n\novarian cancer\n\nabdominal pain\n\nabdominal swelling\n\ndiarrhoea\n\nirregular vaginal bleeding\n\npelvic pain , 367\u20139\n\nurinary frequency\n\novarian cysts\n\nabdominal pain\n\nabsent periods\n\nanorectal pain\n\nchocolate\n\npainful periods\n\npainful intercourse\n\npainful periods\n\npelvic pain 364\u20135,\n\nurinary frequency\n\nweight gain 230\u20132\n\novarian failure, premature, absent periods 374\u20135\n\novarian masses, abdominal swelling\n\novarian problems, delayed puberty\n\novarian tumours\n\nbilateral, absent periods\n\nexcess body hair 272\u20133\n\novaries, in pouch of Douglas, painful intercourse\n\noveractive bladder syndrome\n\nfrequent urination 416\u201317\n\nnocturia\n\nurinary incontinence 419\u201321\n\novulation, defective, infertility\n\novulatory bleeding, irregular vaginal bleeding 380\u20131\n\npachyderma oralis, mouth lumps and marks\n\npachydermoperiostosis, excessive sweating\n\nPaget's disease\n\nhip pain 306\u20137\n\nknee pain 309\u201310\n\nof the nipple\n\nbreast lumps in women\n\nnipple discharge\n\nof the skull\n\ndeafness 116\u201317\n\nheadache 77\u20138\n\nloss of vision 142\u20133\n\npain, intense, excessive sweating\n\npainful intercourse 244\u20136\n\npenile pain\n\npalmar erythema 390\u20131\n\npalpitations 112\u201314\n\ntension and anxiety\n\nPancoast tumour, arm and shoulder pain\n\npancreatic carcinoma\n\nabdominal swelling\n\nback pain\n\nepigastric pain\n\njaundice 206\u20138\n\npancreatitis\n\nabdominal pain 9\u201310,\n\nacute\n\nback pain 171\u20132\n\nchronic\n\nabdominal pain\n\nepigastric pain 24\u20135\n\njaundice 206\u20137\n\npanic disorder, tension and anxiety\n\npapilloedema, transient visual disturbance\n\npapilloma\n\nblocked nose\n\neyelid problems\n\nstridor\n\npapules 398\u20139\n\nparacetamol overdose, jaundice\n\nparaphimosis, penile pain\n\nparaplegia, urinary incontinence\n\nparaproteinaemias, purpura and petechiae 400\u20131\n\nparasites\n\ndiarrhoea\n\nfailure to thrive 185\u20136\n\nweight loss\n\nParkinson's disease\n\nabnormal gait 165\u20136\n\nfalls 188\u20139\n\ntiredness 224\u20135\n\ntremor 227\u20139\n\nurinary incontinence\n\nparonychia\n\nchronic, abnormal nails\n\nhand and wrist pain\n\nparotid tumour, facial swelling ,\n\nparotitis, bacterial, facial swelling\n\nparoxysmal arrhythmia, loss of consciousness 179\u201380\n\nparvovirus, erythema\n\npatella, recurrent dislocation of, knee pain\n\npatellar tendinitis, single joint pain\n\nPCOS (polycystic ovary syndrome)\n\nabsent periods 374\u20135\n\nexcess body hair 272\u20133\n\ninfertility\n\nweight gain\n\npelvic abscess, pelvic pain 364\u20135\n\npelvic adhesions, painful intercourse\n\npelvic congestion\n\npainful intercourse\n\npainful periods\n\npelvic pain , 367\u20138\n\npelvic fistula, vaginal discharge 258\u20139\n\npelvic fracture\n\nerectile dysfunction\n\nurinary incontinence\n\npelvic infection, back pain 171\u20132\n\npelvic inflammation, frequent urination\n\npelvic masses\n\nabdominal swelling\n\nextracolonic\n\nconstipation 18\u201319\n\ngroin swelling\n\nswollen ankles 312\u201313\n\nurinary frequency\n\nurinary retention 425\u20136\n\npelvic pain\n\nacute 363\u20135\n\nchronic ,\n\npainful periods\n\npelvic pain syndrome\n\nacute pelvic pain\n\npainful intercourse\n\npainful periods\n\npelvic space-occupying lesion, frequent urination\n\npelvic tumour\n\ngroin swelling\n\npainful intercourse\n\npemphigoid\n\nbleeding gums\n\nblisters 388\u20139\n\nmouth ulcers\n\nvulval ulceration\n\npemphigus\n\nbleeding gums\n\nblisters 388\u20139\n\nfacial ulcers and blisters\n\nmouth ulcers\n\ntongue pain 359\u201360\n\nvulval sores 266\u20137\n\npenicillamine, blisters\n\npenicillins, swollen glands\n\npenile pain 247\u20139\n\npenile ulcerations and sores 250\u20132\n\npenis, carcinoma of\n\npenile pain\n\npenile ulcerations and sores\n\nperianal abscess\n\nanal swelling 41\u20132\n\nanorectal pain 43\u20135\n\npainful intercourse\n\nperianal conditions, constipation 18\u201319\n\npericarditis\n\nchest pain 97\u20138\n\nconstrictive, jaundice\n\nperimenopause\n\nheavy periods 377\u20138\n\nirregular vaginal bleeding\n\nloss of libido 83\u20134\n\nperiods\n\nabsent 374\u20136\n\nheavy 378\u20139\n\npainful 383\u20135\n\nperiorbital oedema, eyelid problems\n\nperipheral neuropathy\n\nhand and wrist pain 294\u20135\n\nnumbness and paraesthesiae\n\nperipheral vascular disease (PVD)\n\ncalf pain 288\u20139\n\nmuscle pain 303\u20134\n\nnumbness and paraesthesiae\n\nperitonitis\n\nabdominal pain\n\nepigastric pain\n\npersonality disorders\n\nfactitious illness\n\nhallucinations\n\nmemory loss\n\nPerthes's disease, limp in a child 209\u201310\n\npertussis, cough 106\u20137\n\npetechiae 400\u20132\n\nbleeding gums\n\nfacial rash\n\nneck stiffness\n\nPeutz\u2013Jeghers spots\n\nmacules 393\u20134\n\nmouth lumps and marks\n\nPeyronie's disease\n\nerectile dysfunction\n\npenile pain\n\nphaeochromocytoma\n\nexcessive sweating 182\u20133\n\nflushing 197\u20138\n\nweight loss\n\npharyngeal carcinoma, swallowing difficulties\n\npharyngeal pouch\n\nneck lumps 326\u20137\n\nswallowing difficulties\n\npharyngitis\n\nearache\n\nfebrile child\n\nhoarseness\n\nsore throat 329\u201330\n\nswallowing difficulty\n\nphenothiazines\n\nabsent periods 374\u20135\n\nerythema\n\nfalls\n\nloss of libido\n\ntremor\n\nphenytoin\n\nbleeding gums\n\nexcess body hair\n\nswollen glands\n\nphimosis\n\nerectile dysfunction\n\npenile pain 247\u20138\n\nurinary retention\n\nphlebitis, swollen calves\n\nphobias, tension and anxiety\n\nphosphorus poisoning, bleeding gums\n\nphototoxicity\n\nerythema\n\nfacial rash ,\n\nphyllodes tumour, breast lumps in women\n\npica, abdominal pain\n\nPID (pelvic inflammatory disease)\n\nchronic\n\nheavy periods 377\u20139\n\npainful intercourse\n\npainful periods 383\u20134\n\npelvic pain\n\ninfertility\n\nirregular vaginal bleeding 380\u20131\n\npainful intercourse 244\u20136\n\npelvic pain 364\u20139\n\nurinary frequency\n\npilocarpine, excessive sweating\n\npilonidal sinus, infected, anal swelling\n\npituitary exophthalmos, double vision\n\npituitary tumour\n\nabsent periods\n\nbreast enlargement in men\n\nerectile dysfunction\n\nheadache\n\npityriasis alba\n\nfacial rash\n\nmacules\n\npityriasis lichenoides chronica\n\npapules\n\nscales and plaques\n\npityriasis rosea\n\nitching 203\u20134\n\nscales and plaques\n\npityriasis versicolor\n\nmacules 393\u20134\n\nscales and plaques\n\npizotifen, weight gain\n\nplacental abruption, abdominal pain 12\u201314\n\nplantar fasciitis, foot pain 291\u20132\n\nplantaris tendon, ruptured, swollen calves\n\npleural effusion\n\nabnormal nails\n\nshortness of breath , 100\u20131\n\nswollen ankles\n\npleurisy, chest pain 97\u20138\n\nPlummer\u2013Vinson syndrome\n\nabnormal nails\n\nswallowing difficulties\n\nPMR (polymyalgia rheumatica)\n\nmuscle pain 303\u20135\n\ntiredness\n\nPMS (pre-menstrual syndrome)\n\nanxiety 194\u20135\n\nswollen ankles\n\npneumonia\n\nabdominal pain\n\natypical\n\ncough\n\nconfusion\n\nepigastric pain 24\u20135\n\nshortness of breath 94\u20135\n\nstiff neck\n\npneumothorax\n\nchest pain 97\u20138\n\nshortness of breath , ,\n\npoisoning\n\nbleeding gums\n\nvomiting blood\n\npoliomyelitis, muscle pain 303\u20134\n\npolyarteritis nodosa\n\ncoughing up blood\n\nmultiple joint pain\n\nmuscle pain\n\nnodules\n\nprolonged fever\n\nvomiting blood\n\npolyarthritis\n\nknee pain 309\u201310\n\nsymmetrical\n\njoint pain ,\n\npolycystic kidney disease, blood in urine\n\npolycythaemia, itching\n\npolycythaemia rubra vera, flushing 197\u20138\n\npolymyositis\n\nmultiple joint pain 300\u20131\n\nmuscle pain 303\u20135\n\nprolonged fever\n\nswallowing difficulties\n\npolymyxin, earache\n\npolyneuropathy\n\nfoot pain\n\nperipheral, numbness and paraesthesiae 212\u201313\n\npolyuria\n\nnocturia 422\u20134\n\nurinary incontinence\n\nporphyria\n\nabdominal pain\n\nblisters\n\nmuscle pain 303\u20134\n\nporphyria cutanea tarda, excess body hair 272\u20133\n\npost-concussional state\n\nhallucinations\n\nheadache\n\npost-nasal drip, cough 106\u20137\n\npost-partum perineal repair, painful intercourse\n\npost-radiotherapy fibrosis, frequent urination\n\npost-traumatic stress disorder\n\ninsomnia\n\ntension and anxiety\n\npost-TURP, erectile dysfunction\n\npost-vasectomy\n\nscrotal swelling\n\ntesticular pain\n\npost-viral fatigue, tiredness\n\nposterior fossa tumours, facial pain\n\nposterior vitreous detachment, transient visual disturbance\n\npostictal state, confusion\n\npostural hypotension\n\ndizziness 71\u20132\n\nfalls\n\npostural instability, falls 188\u20139\n\npotassium depletion, excessive urination 413\u201314\n\npotassium supplements, swallowing difficulties\n\nPrader\u2013Willi syndrome\n\nfailure to thrive\n\nweight gain\n\npre-eclampsia\n\nabdominal pain 13\u201314\n\nheadache ,\n\npregnancy\n\nabdominal swelling 6\u20137\n\nabsent periods 374\u20136\n\nacute abdominal pain 12\u201314\n\nbreast pain 60\u20131\n\nearly\n\nirregular vaginal bleeding\n\nerythema ,\n\nfatty liver of\n\njaundice\n\nfrequent urination 416\u201317\n\nnipple discharge 63\u20134\n\nthirst and dry mouth\n\ntoxicity during\n\nfailure to thrive\n\nvomiting 30\u20131\n\nweight gain 230\u20132\n\npregnancy gingivitis, bleeding gums 351\u20132\n\npremature delivery, failure to thrive\n\npresacral tumours, anorectal pain\n\npresbyacusis\n\ndeafness 116\u201317\n\ntinnitus\n\nprickly heat\n\nitching\n\npapules\n\nproctalgia fugax (PF), anorectal pain 43\u20134\n\nproctitis\n\npelvic pain\n\nrectal bleeding\n\nrectal discharge 49\u201350\n\nprolactinoma\n\nabsent periods 374\u20135\n\nnipple discharge\n\nprolonged coughing, coughing up blood 109\u201310\n\nproptosis\n\ndouble vision\n\nloss of vision\n\nprostate calculi, haemospermia\n\nprostate cancer\n\nback pain 172\u20133\n\nblood in urine 410\u201311\n\nhaemospermia 241\u20132\n\nnocturia\n\npenile pain\n\nprostatectomy, urinary incontinence 419\u201320\n\nprostatic abscess\n\npenile pain 247\u20138\n\nurinary retention\n\nprostatic hypertrophy\n\nblood in urine 410\u201311\n\nurinary incontinence 419\u201320\n\nurinary retention 425\u20136\n\nprostatism, _see_ LUTS\n\nprostatitis\n\nanorectal pain 43\u20134\n\nhaemospermia 241\u20132\n\npelvic pain\n\npenile pain 247\u20138\n\nurinary frequency\n\nurinary retention\n\npruritus, anal itching\n\npseudodementia, and memory loss\n\npseudogout\n\nknee pain 309\u201310\n\nsingle joint pain 282\u20133\n\npseudohermaphroditism, male, breast enlargement in men\n\npseudohypertrophy, swollen calves\n\npseudoseizures, loss of consciousness 179\u201380\n\npseudoxanthoma elasticum, papules\n\npsittacosis, neck stiffness\n\npsoriasis\n\nabnormal nails 270\u20131\n\nanal itching\n\near discharge\n\nfacial rash\n\nguttate\n\npapules\n\nscales and plaques\n\nitching 203\u20134\n\nitchy scalp 278\u20139\n\njoint pain 301\u20132\n\npustular 403\u20134\n\nscales and plaques 405\u20137\n\nvulval irritation 261\u20132\n\npsoriatic arthropathy, multiple joint pain\n\npsychogenic amnesia, memory loss\n\npsychotic illness\n\nbad breath\n\nhallucinations\n\ntension and anxiety ,\n\nptosis, eyelid problems 136\u20138\n\nPU (peptic ulcer)\n\nabdominal pain 9\u201310, 15\u201316\n\nchest pain 97\u20138\n\nepigastric pain\n\nvomiting blood 33\u20134\n\nweight loss\n\npuberty\n\nbreast enlargement in men 54\u20135\n\nbreast pain\n\ndelayed 177\u20138\n\nfibroangioma of\n\nblocked nose\n\nheavy periods 377\u20138\n\npubic symphysis pain\n\nabdominal pain 12\u201313\n\npelvic pain\n\npulmonary arteriovenous malformations, coughing up blood\n\npulmonary embolism (PE)\n\nchest pain\n\ncoughing up blood 109\u201311\n\nshortness of breath ,\n\npulmonary fibrosis\n\ncough 103\u20134\n\nshortness of breath\n\npulmonary infarction, chest pain 97\u20138\n\npulmonary oedema\n\ncoughing up blood 109\u201310\n\nswollen ankles\n\npulmonary stenosis, loss of consciousness\n\npulp space infection, hand and wrist pain ,\n\npurpura 400\u20132\n\npustules 403\u20134\n\npyelonephritis\n\nabdominal pain ,\n\nback pain 171\u20132\n\nchronic\n\nexcessive urination\n\nprolonged fever 215\u201316\n\nurinary frequency\n\nvomiting\n\npyloric stenosis\n\ncrying baby\n\nfailure to thrive\n\npyometra\n\nirregular vaginal bleeding\n\nvaginal discharge\n\nquadriceps weakness, falls\n\nquinine\n\ntinnitus\n\nvertigo\n\nvision loss ,\n\nquinsy\n\nearache\n\nsore throat 329\u201331\n\nRA (rheumatoid arthritis)\n\nabnormal gait\n\ncrico-arytenoid\n\nhoarseness\n\nfoot pain\n\nhand and wrist pain 294\u20136\n\njoint pain 282\u20134, 300\u20132\n\njuvenile\n\nlimp in a child\n\nleg ulcers 297\u20138\n\nmuscle pain\n\nneck stiffness 332\u20133\n\nprolonged fever 215\u201316\n\nred eye\n\nswollen glands\n\ntiredness\n\nweight loss\n\nradiotherapy\n\ndelayed puberty\n\nurinary incontinence\n\nranula, mouth lumps and marks\n\nRAU (recurrent aphthous ulceration), mouth ulcers 356\u20137\n\nRaynaud's disease\n\nabnormal nails\n\nhand and wrist pain\n\nrectal bleeding 46\u20138\n\nabdominal pain\n\nconstipation\n\nrectal carcinoma, rectal bleeding 46\u20137,\n\nrectal discharge 49\u201351\n\nrectal prolapse\n\nanal itching\n\nanal swellng 41\u20132\n\ndischarge 49\u201350\n\nrectocoele, vulval swelling\n\nrectovaginal fistula, anal itching\n\nred eyes, _see_ eyes, acutely red and painful\n\nreferred pain\n\nearache\n\nepigastric\n\nknee 310\u201311\n\nlimping\n\nmuscular\n\npelvic\n\nReiter's syndrome\n\nback pain\n\njoint pain 282\u20134, ,\n\nknee pain\n\npenile sores 250\u20132\n\nscales and plaques 405\u20136\n\nrelationship problems\n\nerectile dysfunction 238\u20139\n\nloss of libido 83\u20134\n\npainful intercourse\n\nrenal colic\n\nabdominal pain 9\u201310\n\nhaematuria\n\npainful periods\n\nrenal cysts, abdominal swelling\n\nrenal disease\n\nblood in urine\n\nexcessive urination\n\ngross\n\nabdominal swelling\n\nleg ulcers\n\npurpura\n\ntiredness\n\nweight gain\n\nrenal failure; _see also_ CRF\n\ncalf pain\n\ndelayed puberty\n\ndiarrhoea\n\nfailure to thrive\n\nfrequent urination\n\nitching\n\nloss of libido 83\u20134\n\npurpura and petechiae\n\nthirst and dry mouth\n\ntinnitus 125\u20136\n\ntiredness\n\nurinary retention\n\nvomiting 30\u20131\n\nweight gain\n\nweight loss\n\nrenal stones\n\nabdominal pain ,\n\nback pain 171\u20132\n\nblood in urine 410\u201311\n\nvomiting\n\nrenal tumours\n\nabdominal swelling\n\nblood in urine 410\u201312\n\nbreast enlargement in men\n\nreserpine, tongue pain\n\nrespiratory distress, crying baby\n\nrespiratory problems\n\nchest pain\n\ninsomnia 80\u20131\n\nretinal artery occlusion, vision loss 147\u20138\n\nretinal detachment\n\nloss of vision\n\ntransient visual disturbance 139\u201341\n\nvision loss ,\n\nretinal vein occlusion, vision loss 147\u20138\n\nretinitis pigmentosa, vision loss\n\nretinopathy, vision loss 142\u20133\n\nretropharyngeal abscess\n\nsore throat\n\nstridor\n\nrheumatic fever\n\njoint pain 300\u20131\n\nlimp in a child 209\u201310\n\nrheumatoid nodules\n\nrhinitis; _see also_ allergic rhinitis\n\natrophic, nosebleeds\n\ncough , 105\u20136\n\nvasomotor\n\nblocked nose 338\u20139\n\nrunny nose 344\u20135\n\nrhinitis medicamentosa\n\nblocked nose ,\n\nrunny nose\n\nrib fractures, chest pain 97\u20138\n\nrickets, limp in a child 209\u201310\n\nRLS (restless legs syndrome)\n\nabnormal movements 168\u20139\n\ninsomnia\n\nRocky Mountain spotted fever, purpura and\n\npetechiae\n\nrosacea\n\nerythema 390\u20131\n\nfacial rash 153\u20134\n\nflushing 197\u20138\n\npustules 403\u20134\n\nroseola, febrile child\n\nrotavirus, diarrhoea\n\nRotor syndrome, jaundice\n\nrubella\n\nfebrile child\n\nmacules 393\u20134\n\nmultiple joint pain\n\nswollen glands\n\nsacroiliitis, back pain\n\nsalicylates, vertigo\n\nsalivary gland carcinoma, mouth ulcers\n\nsalivary gland stones, facial swelling 156\u20137\n\nsalmonella, diarrhoea\n\nsalpingitis, vaginal discharge\n\nsandfly fever, neck stiffness\n\nsaphena varix, groin swelling 370\u20131\n\nsarcoid\n\nred eye\n\nswollen glands 218\u201319\n\nsarcoidosis\n\nerythema 390\u20131\n\nexcessive urination\n\nfacial rash 153\u20134\n\njoint pain 300\u20131\n\nneck lumps\n\nnodules 395\u20136\n\nshortness of breath 100\u20131\n\nsarcoma\n\nbreast lumps in women\n\nfacial swelling\n\nvulval ulceration\n\nscabies\n\nexcoriated\n\nvulval ulceration and sores 266\u20137\n\nitching 203\u20134\n\npapules 398\u20139\n\nscalded skin syndrome\n\nblisters 388\u20139\n\nfebrile child\n\nscales and plaques 405\u20137\n\nscalp, itchy 278\u20139\n\nscaphoid fracture, hand and wrist pain ,\n\nscarlet fever\n\nerythema\n\ntesticular pain\n\nSCC (squamous cell carcinoma)\n\nbleeding gums\n\nblocked nose\n\nfacial ulcers and blisters\n\nleg ulcers\n\nmouth lumps\n\nmouth ulcers\n\nnodules ,\n\nvulval ulceration and sores 266\u20137\n\nschistosomiasis\n\nblood in urine\n\nhaemospermia 241\u20132\n\npenile pain 247\u20138\n\nschizophrenia, hallucinations 74\u20136\n\nsciatica, numbness and paraesthesiae 212\u201313\n\nscleritis, red eye\n\nscleroderma\n\nconstipation\n\nmuscle pain\n\nswallowing difficulties\n\nscoliosis, idiopathic, limp in a child\n\nscrofula, neck lumps\n\nscrotal swelling\n\nscurvy\n\nbleeding gums 351\u20132\n\nvomiting blood\n\nsebaceous cyst\n\nanal swelling\n\nbreast enlargement in men\n\ngroin swellings 370\u20131\n\ninfected\n\npenile ulcerations and sores\n\nneck lumps 326\u20137\n\nnodules 395\u20136\n\nscrotal swelling 253\u20134\n\nvulval swelling 264\u20135\n\nsebaceous gland carcinoma, itchy eyes 144\u20136\n\nseborrhoeic dermatitis\n\near discharge\n\nearache\n\nhair loss 275\u20136\n\nscales and plaques 405\u20136\n\nseborrhoeic eczema\n\nfacial rash 153\u20134\n\nitchy scalp 278\u20139\n\nseborrhoeic keratosis, scales and plaques 405\u20136\n\nseborrhoeic warts, papules\n\nsedatives, double vision\n\nseizures\n\npartial\n\nabnormal movements 168\u20139\n\ndizziness\n\ntransient visual disturbance\n\nseminoma\n\nscrotal swelling ,\n\ntesticular pain\n\nsenile purpura 400\u20131\n\nsensory neuropathy, urinary incontinence\n\nsentinel pile, anal swelling\n\nsepsis\n\nchronic\n\nfrequent urination 416\u201317\n\nfalls 188\u20139\n\niatrogenic\n\nscrotal swelling\n\nshortness of breath\n\nseptal deviation\n\nblocked nose\n\nnosebleed\n\nseptal perforation, nosebleeds\n\nseptic arthritis\n\nfebrile child\n\nfoot pain ,\n\nlimp in a child 209\u201311\n\nsingle joint pain 282\u20134\n\nsepticaemia\n\ncrying baby\n\nmeningococcal\n\nfebrile child ,\n\npurpura and petechiae 400\u20132\n\nswollen glands 218\u201319\n\nSever's disease, foot pain\n\nsexual dysfunction, infertility\n\nsexual practices\n\nanal itching ,\n\nrectal discharge\n\nshingles\n\nbreast pain\n\nchest pain\n\nfacial pain 150\u20131\n\nfacial ulcers and blisters\n\nshock, excessive sweating\n\nshock lung, shortness of breath\n\nshortness of breath\n\nacute 94\u20136\n\nchronic 100\u20132\n\nsick sinus syndrome, palpitations\n\nsickle-cell anaemia\n\nexcessive urination 413\u201314\n\nthirst and dry mouth\n\nsickle-cell crisis\n\nabdominal pain\n\njoint pain\n\nsickle-cell disease\n\nabdominal pain 27\u20139\n\ndry mouth\n\nhaematuria\n\nsildenafil, transient visual disturbance\n\nSimmonds's disease, absent periods\n\nsinger's nodes, hoarseness\n\nsingle joint pain, acute 282\u20133\n\nsinus bradycardia, loss of consciousness\n\nsinus tachycardia, palpitations 112\u201313\n\nsinusitis\n\nblocked nose\n\nchronic\n\nbad breath\n\nblocked nose 338\u20139\n\nfrontal\n\nheadache 77\u20138\n\nhoarseness 323\u20134\n\nmaxillary\/frontal\n\nfacial pain 150\u20131\n\nrunny nose 344\u20135\n\nSj\u00f6gren's syndrome\n\nswallowing difficulties\n\nthirst and dry mouth 221\u20133\n\nskin disorders\n\nvulval irritation\n\nvulval sores\n\nskin friction, blisters\n\nskin tags\n\nanal itching 38\u20139\n\nanal swelling 41\u20132\n\npapules 398\u20139\n\nskull fracture, basal, tinnitus\n\nskull injury, excessive urination\n\nslapped cheek syndrome\n\nerythema 390\u20131\n\nfacial rash\n\nSLE (systemic lupus erythematosus)\n\nbleeding gums 351\u20132\n\ncoughing up blood\n\nerythema\n\nfacial rash 153\u20134\n\njoint pain 300\u20132\n\nlimp in a child\n\nprolonged fever\n\nswollen glands\n\nthirst and dry mouth\n\ntiredness\n\nvomiting blood\n\nweight loss\n\nsleep apnoea\n\ninsomnia 80\u20131\n\nloss of consciousness\n\nnocturia\n\nsleep deprivation, headache\n\nsleep hygiene, insomnia 80\u20131\n\nsmoking\n\nbad breath 348\u20139\n\ncough 103\u20134\n\ndry mouth\n\nerectile dysfunction\n\nexcessive urination\n\nhoarseness 323\u20134\n\nmaternal\n\nfailure to thrive\n\nmouth ulcers\n\nsoft tissue injury, knee pain 309\u201310\n\nsolitary rectal ulcer syndrome\n\nanorectal pain\n\nrectal discharge\n\nsolvents, hallucinations\n\nsomatisation disorder, tension and anxiety\n\nsperm problems, infertility 200\u20131\n\nspinal cord carcinoma, arm and shoulder pain\n\nspinal cord compression\n\nerectile dysfunction\n\nnumbness and paraesthesiae 212\u201314\n\nurinary retention ,\n\nspinal cord injury, excessive sweating\n\nspinal cord lesions, arm and shoulder pain\n\nspinal cord tumour, intramedullary, numbness and paraesthesiae\n\nspinal fracture\n\nback pain\n\nerectile dysfunction\n\nspinal nerve root pain\n\nabnormal gait\n\nback pain\n\nspinal stenosis\n\nback pain 171\u20132\n\nhip pain 306\u20137\n\nspinal tumour\n\npelvic pain\n\ntesticular pain\n\nspine, neoplastic disease of, back pain\n\nspine cancer, arm and shoulder pain\n\nspironolactone, breast enlargement in men\n\nsplenomegaly, abdominal swelling\n\nspondoarthritides\n\nback pain\n\nhip pain\n\nknee pain\n\nmultiple joint pain 300\u20131\n\nspondylolisthesis, back pain 171\u20132\n\nspondylosis, back pain 171\u20132\n\nstaphylococcal infection, pustules 403\u20134\n\nstarvation\n\nbad breath\n\nconfusion\n\nstatins, side effects of, muscle pain 303\u20134\n\nSTDs (sexually transmitted diseases)\n\nanal itching 38\u20139\n\ngroin swelling\n\npapules\n\npenile pain\n\npenile ulcerations and sores\n\nrectal discharge 49\u201350\n\ntesticular pain 256\u20137\n\nvaginal discharge\n\nvulval swelling\n\nvulval ulceration and sores\n\nsteroid eye drops, red eye\n\nsteroid purpura\n\nsteroids\n\nanabolic\n\ndelayed puberty\n\nexcess body hair 272\u20133\n\njaundice\n\nweight gain\n\nconfusion\n\nhip pain\n\nintranasal\n\nrunny nose\n\npainful tongue\n\npurpura\n\nshortness of breath\n\nvomiting\n\nvomiting blood\n\nweight gain\n\nStevens\u2013Johnson syndrome\n\nerythema\n\nfacial rash\n\nmouth lumps\n\nsticky eyes, itchy eyes 145\u20136\n\nStokes\u2013Adams attacks, loss of consciousness ,\n\nstreptococcal infection\n\nanal itching\n\nerythema 390\u20131\n\nsore throat 329\u201330\n\nstreptomycin, deafness\n\nstress\n\nabsent periods\n\ninsomnia 80\u20131\n\nitchy scalp 278\u20139\n\nmemory loss\n\npalpitations\n\nshortness of breath\n\ntiredness\n\nweight loss 233\u20134\n\nstress incontinence 419\u201321\n\nstridor in children 335\u20136\n\nstroke\n\nconfusion\n\ndouble vision 133\u20134\n\nnumbness\n\nvertigo\n\nstrychnine poisioning, calf pain\n\nstye, eyelid problems 136\u20137,\n\nsubclavian aneurysm, arm and shoulder pain\n\nsubclavian steal syndrome\n\ndizziness\n\nloss of consciousness\n\nsubstance abuse\n\nconfusion\n\ndizziness\n\nerectile dysfunction ,\n\nhallucinations 74\u20136\n\nleg ulcers\n\npalpitations\n\ntension and anxiety 194\u20135\n\ntiredness\n\nweight loss\n\nsulphonamides, swollen glands\n\nsulphonylureas, weight gain\n\nsunburn, erythema\n\nsuper-female syndrome, infertility\n\nSVT (supraventricular tachycardia)\n\nexcessive urination\n\nloss of consciousness\n\npalpitations 112\u201314\n\nswallowed blood, vomiting ,\n\nswallowing difficulties 320\u20132\n\nsweating, excessive 182\u20134\n\nsycosis barbae\n\nfacial rash\n\npustules\n\nSydenham's chorea, abnormal movements\n\nsynovitis, traumatic, single joint pain 282\u20133\n\nsyphilis\n\nabnormal gait 165\u20136\n\narm and shoulder pain\n\nfacial ulcers and blisters\n\nhoarseness\n\nincontinence 419\u201320\n\njoint pain\n\nmacules\n\nmouth ulcers\n\npenile ulcerations and sores 250\u20132\n\nprolonged fever 215\u201316\n\nscales and plaques\n\nscrotal swelling 253\u20134\n\nsecondary\n\nhair loss 275\u20136\n\nscales and plaques ,\n\nswollen glands 218\u201319\n\ntesticular pain\n\ntremor 227\u20138\n\nurinary retention\n\nvertigo 89\u201390\n\nvision loss 142\u20133\n\nvulval ulceration and sores 266\u20137\n\nsyphilitic aortitis, arm and shoulder pain\n\nsyphilitic chancre\n\nfacial ulcers and blisters\n\nmouth ulcers\n\nsyringobulbia, vertigo\n\nsyringomyelia\n\nincontinence\n\nnumbness and paraesthesiae 212\u201313\n\nsystemic infection\n\nconfusion 68\u20139\n\nstiff neck\n\nsystemic sclerosis\n\njoint pain\n\nmultiple joint pain\n\ntabes dorsalis\n\nabnormal gait 165\u20136\n\nerectile dysfunction\n\ntachycardia, vomiting blood\n\ntardive dyskinesias, abnormal movements 168\u20139\n\nTB (tuberculosis)\n\nabdominal pain\n\nanal\n\nanorectal pain\n\nrectal discharge\n\nbreast lumps in women\n\ncough\n\nendometrial\n\ninfertility\n\nerythema 390\u20131\n\nhaemospermia\n\nhoarseness\n\nleg ulcers\n\nmouth ulcers\n\nneck lumps\n\npenile pain\n\npenile ulcerations and sores\n\nprolonged fever 216\u201317\n\nrenal\n\nblood in urine 410\u201311\n\nfrequent urination 416\u201318\n\nscrotal swelling\n\nswollen glands 218\u201319\n\ntesticular pain\n\ntiredness 224\u20135\n\nvulval ulceration and sores\n\nweight loss 233\u20134\n\ntear ducts, blocked\n\neyelid problems 136\u20137\n\nitchy eyes 144\u20135\n\nteething, crying baby 174\u20136\n\ntelangiectasia, mouth lumps and marks\n\ntelogen effluvium, hair loss 275\u20136\n\ntemporal arteritis\n\nfacial pain 150\u20132\n\nheadache ,\n\nmuscle pain\n\ntransient visual disturbance 139\u201341\n\nvision loss 147\u20138\n\ntendonitis\n\nbicipital, arm and shoulder pain 285\u20137\n\npatellar, knee pain\n\ntenosynovitis\n\narm and shoulder pain\n\nhand and wrist pain 294\u20136\n\ntension and anxiety 194\u20136\n\ncalf pain\n\nteratoma\n\nblocked nose\n\nscrotal swelling\n\ntesticular pain\n\ntestes\n\nmaldescent of\n\ndelayed puberty\n\ngroin swelling ,\n\nretractile, groin swelling 370\u20131\n\ntesticular carcinoma\n\nbreast enlargement in men 54\u20136\n\nhaemospermia\n\ntesticular pain\n\ntesticular disease, loss of libido\n\ntesticular torsion\n\nabdominal pain\n\ncrying baby\n\nscrotal swelling ,\n\ntesticular pain 256\u20137\n\ntesticular tumour, scrotal swelling\n\ntestosterone deficiency\n\nerectile dysfunction\n\nloss of libido\n\ntetanus, neck stiffness\n\ntetracyclines, erythema\n\nthiamine deficiency, memory loss\n\nthird ventricular tumour, falls\n\nthirst 221\u20133\n\nthoracic outlet syndrome, arm and shoulder pain\n\nthreadworms\n\nanal itching 38\u20139\n\nvulval irritation\n\nthroat, sore 329\u201331\n\nthroat problems, earache 119\u201320\n\nthrombocytopenia\n\nblood in urine\n\nheavy periods\n\nnosebleeds 341\u20132\n\npurpura 400\u20131\n\nthrombophlebitis\n\nbreast lumps in women\n\nbreast pain\n\ncalf pain\n\nthrush\n\nanal itching\n\nmouth ulcers 356\u20137\n\nvaginal discharge 258\u201360\n\nvulval irritation 261\u20132\n\nvulval sores\n\nvulval ulceration and sores 267\u20138\n\nthyroglossal cyst, neck lumps 326\u20137\n\nthyroid carcinoma, neck lumps ,\n\nthyroid disorders, infertility\n\nthyroid dysfunction\n\nheavy periods\n\nirregular vaginal bleeding\n\nthyroid eye disease, itchy eyes 144\u20136\n\nthyroiditis, sore throat\n\nthyrotoxicosis\n\nabsent periods\n\ndiarrhoea 21\u20132\n\nerythema\n\nexcessive sweating\n\npalpitations 112\u201313\n\ntremor 227\u20139\n\nweight loss\n\nthyroxine, delayed puberty\n\nTIA (transient ischaemic attack)\n\nconfusion\n\nloss of consciousness\n\nnumbness and paraesthesiae 212\u201314\n\ntransient visual disturbance ,\n\nvision loss 147\u20138\n\nTietze's syndrome, chest pain 97\u20138\n\ntinea, anal itching\n\ntinea capitis, hair loss 275\u20136\n\ntinnitus 125\u20137\n\nvertigo 90\u20131\n\ntiredness 224\u20136; _see also_ fatigue\n\ncrying baby\n\nTMJ (temporomandibular joint) dysfunction\n\nearache 119\u201320\n\nfacial pain 150\u20131\n\ntinnitus\n\nTN (trigeminal neuralgia)\n\nearache\n\nfacial pain 150\u20132\n\ntobacco, vision loss\n\nTolosa\u2013Hunt syndrome, double vision\n\ntongue\n\nfissured, tongue pain\n\nhairy, mouth lumps and marks\n\npainful 359\u201361\n\ntongue carcinoma, tongue pain 359\u201360\n\ntonsillar concretions, mouth lumps and marks\n\ntonsillitis\n\nearache\n\nfebrile child 191\u20132\n\nsore throat ,\n\nswollen glands\n\ntorticollis\n\nacute\n\nneck stiffness 332\u20133\n\nocular\n\ndouble vision\n\nspasmodic\n\nabnormal movements\n\ntorus, mouth lumps and marks\n\nTourette's syndrome, abnormal movements 168\u201370\n\ntoxic erythema 390\u20132\n\ntoxoplasmosis\n\nfebrile child\n\nred eye\n\ntrachea-oesophageal fistula, cough\n\ntracheitis\n\nfebrile child\n\nstridor\n\ntransient global amnesia, memory loss\n\ntrauma\n\nabnormal gait 165\u20136\n\nabnormal nails 270\u20131\n\nanorectal pain\n\nbleeding gums 351\u20132\n\nblisters 388\u20139\n\nblocked nose\n\nblood in urine\n\nbreast pain\n\ncoughing up blood\n\nearache 119\u201320\n\nerectile dysfunction\n\nfacial swelling 156\u20137\n\nhaemospermia 241\u20132\n\nhand and wrist pain 294\u20135\n\nhoarseness\n\nlimp in a child 209\u201310\n\nmemory loss 86\u20137\n\nmouth lumps and marks\n\nmouth ulcers 356\u20137\n\npenile ulcerations and sores\n\npurpura and petechiae 400\u20131\n\nrectal bleeding\n\nred eye\n\nscrotal swelling\n\nsore throat\n\ntesticular pain\n\ntongue pain 359\u201360\n\ntransient visual disturbance\n\nvertigo\n\nvulval irritation 261\u20132\n\ntraumatic haematoma, vulval swelling\n\ntremor 227\u20139\n\ntrichiasis, secondary, eyelid problems 136\u20137\n\n_Trichomonas vaginalis_ , vulval irritation 261\u20132\n\ntrichotillomania, hair loss ,\n\ntricyclic antidepressants\n\nabnormal movements\n\ndry mouth\n\nexcessive sweating\n\nfalls\n\noverdose, hallucinations\n\nurinary retention\n\ntrigger thumb or finger, hand and wrist pain\n\nTroisier's sign\n\nabdominal pain\n\ndifficulty swallowing\n\nswollen glands\n\nvomiting blood\n\ntropical diseases\n\nleg ulcers\n\npenile sores\n\nprolonged fever 215\u201316\n\nsingle joint pain\n\nswollen glands\n\ntrypanosomiasis, swollen glands\n\ntuberculous abscess, nipple discharge\n\ntuberculous cystitis, frequent urination\n\ntuberculous endometritis, heavy periods\n\ntuberous sclerosis\n\nfacial rash\n\npapules 398\u20139\n\ntularaemia, swollen glands\n\nTurner's syndrome\n\ndelayed puberty 177\u20138\n\nfailure to thrive\n\ninfertility\n\nTV (trichomonal vaginosis), vaginal discharge 258\u20139\n\ntyphoid\n\nexcessive sweating\n\nmacules\n\nulcerative colitis\n\nabdominal pain\n\ndiarrhoea\n\nmouth ulcers\n\nprolonged fever\n\nrectal discharge\n\nupper airway burn, stridor\n\nuraemia\n\nconfusion\n\nitching 203\u20134\n\nureter, ectopic, urinary incontinence\n\nureteric colic, abdominal pain\n\nureteric reflux, abdominal pain\n\nureteric stones\n\nabdominal pain\n\nback pain\n\nblood in urine 410\u201311\n\ntesticular pain 256\u20137\n\nurinary frequency\n\nvomiting\n\nureterovaginal fistula, urinary incontinence\n\nurethra\n\nrupture of, urinary retention\n\nshort or wide, urinary incontinence\n\nurethral caruncle\n\npainful intercourse\n\nvulval swelling\n\nurethral injury, penile pain\n\nurethral stenosis, urinary incontinence\n\nurethral stones, urinary retention ,\n\nurethral strictures\n\nhaemospermia\n\nurinary retention\n\nurethritis\n\nblood in urine 410\u201311\n\nfrequent urination ,\n\nhaemospermia 241\u20132\n\npainful intercourse 244\u20135\n\npenile pain 247\u20138\n\npenile sores\n\nsingle joint pain\n\nurinary frequency\n\nurinary calculus, penile pain 247\u20138\n\nurinary frequency 416\u201318\n\nnocturia 422\u20133\n\nurinary infection, abdominal pain\n\nurinary retention 425\u20137\n\nurinary tract obstruction, nocturia 422\u20133\n\nurination, excessive 413\u201315\n\nurticaria\n\nblisters\n\nitching 203\u20134\n\nURTIs (upper respiratory tract infection)\n\ncough 103\u20134, 106\u20137\n\nearache\n\nfacial pain\n\nfebrile child 191\u20132\n\nneck stiffness 332\u20133\n\nrunny nose 344\u20135\n\nsore throat 329\u201330\n\nswollen glands 218\u201319\n\nvomiting 30\u20131\n\nuterine bleeding in the newborn, irregular vaginal\n\nbleeding\n\nuterine carcinoma\n\nirregular vaginal bleeding\n\nvaginal discharge\n\nuterine fibroids\n\nabdominal swelling 6\u20137\n\nheavy periods 377\u20138\n\ninfertility\n\npainful intercourse\n\npelvic pain\n\nsloughing\n\nvaginal discharge\n\nurinary retention\n\nuterine fistula, urinary incontinence\n\nuterine hypoplasia, painful periods\n\nuterine malformation, painful periods 383\u20134\n\nuterine prolapse, vulval swelling\n\nuterine rupture, abdominal pain\n\nuterine torsion, abdominal pain\n\nuterovaginal prolapse, pelvic pain\n\nuterus\n\nperforated, pelvic pain\n\nretroverted\n\npainful intercourse\n\npainful periods\n\nUTI (urinary tract infection)\n\nabdominal pain , 15\u201316, 27\u20139\n\nback pain\n\nblood in urine 410\u201311\n\nchronic\n\nincontinence 419\u201320\n\npenile pain\n\ncrying baby\n\nfailure to thrive 185\u20136\n\nfebrile child 191\u20133\n\nhaemospermia\n\nhallucinations\n\npainful intercourse\n\npainful periods\n\npelvic pain 364\u20135,\n\nrecurrent\n\npelvic pain\n\nprolonged fever 215\u201316\n\nscrotal swelling\n\ntesticular pain\n\nurinary frequency\n\nurinary retention 425\u20137\n\nvomiting\n\nvulval irritation\n\nuveitis\n\nloss of vision 147\u20138\n\ntransient visual disturbance\n\nvaginal bleeding, irregular 380\u20132\n\nvaginal carcinoma, irregular vaginal bleeding\n\nvaginal discharge 258\u201360\n\nvaginismus, painful intercourse 244\u20135\n\nvaginitis, atrophic\n\nirregular vaginal bleeding 380\u20132\n\npainful intercourse 244\u20136\n\nvulval irritation 261\u20133\n\nvalproate, absent periods\n\nValsalva-induced syncope, loss of consciousness\n\nvaricocoele\n\ninfertility 201\u20132\n\nscrotal swelling 253\u20134\n\ntesticular pain 256\u20137\n\nvulval swelling 264\u20135\n\nvaricose veins\n\ngroin swellings\n\nleg ulcers\n\nvulval swelling\n\nvascular drugs, erectile dysfunction\n\nvasculitis\n\nallergic\n\nblisters\n\nleg ulcers 297\u20138\n\nnodular\n\npurpura and petechiae 400\u20131\n\nvasoactive tumours, excessive sweating\n\nvasovagal attacks, loss of consciousness 179\u201380\n\nvenous congestion\n\njaundice\n\npainful periods\n\nvenous disease, leg ulcers 297\u20138\n\nvenous drainage, excessive, erectile dysfunction 238\u20139\n\nvenous insufficiency, swollen ankles 312\u201313\n\nventricular ectopics, palpitations 112\u201314\n\nventricular tachycardia (VT), palpitations ,\n\nverruca, foot pain 291\u20132\n\nvertebral fracture\n\nback pain\n\nneck stiffness ,\n\nvertebrobasilar insufficiency\n\ndizziness 71\u20132\n\nfalls 188\u20139\n\nvertigo\n\nvertigo 89\u201391\n\nfalls\n\nvomiting\n\nvesicovaginal fistula, urinary incontinence\n\nvestibular ataxia, abnormal gait 165\u20136\n\nVincent's angina, sore throat\n\nVincent's stomatitis, bleeding gums\n\nviral exanthem, macules 393\u20134\n\nviral illness\n\nabdominal pain 27\u20138\n\ncrying baby 174\u20135\n\ndizziness 71\u20132\n\nfrequent\n\nfailure to thrive\n\nmuscle pain 303\u20134\n\npapules\n\nviral infection\n\nacute\n\nlimp in a child\n\nswollen glands 218\u201319\n\nitchy scalp\n\npainful tongue 359\u201360\n\nviscus perforation\n\nabdominal pain\n\nvomiting blood\n\nvision loss\n\ngradual 142\u20133\n\nsudden 147\u20138\n\nvisual disturbance\n\nfalls\n\ntransient 139\u201341\n\nvisual field defect, excess body hair\n\nvitamin B6 deficiency, tongue pain\n\nvitamin B12 deficiency\n\ndeafness 116\u201317\n\nmemory loss\n\nmouth lumps and marks\n\nmouth ulcers 356\u20137\n\nnumbness and paraesthesiae 212\u201313\n\ntongue pain\n\nvitamin C deficiency\n\nnosebleeds\n\npurpura and petechiae\n\nvitamin D deficiency, muscle pain 303\u20135\n\nvitamin K deficiency\n\nnosebleeds\n\npurpura and petechiae\n\nvitiligo\n\nfacial rash\n\nmacules 393\u20134\n\nvitreous haemorrhage\n\nloss of vision 147\u20138\n\ntransient visual disturbance\n\nvoice overuse, hoarseness 323\u20134\n\nvomiting 30\u20132\n\nabdominal pain\n\nepigastric pain\n\nfailure to thrive 185\u20136\n\npurpura and petechiae 400\u20131\n\nrecurrent, eating disorders ,\n\nvomiting blood 33\u20135\n\nvon Willebrand's disease\n\nnosebleeds 341\u20132\n\npurpura\n\nvulval carcinoma, vulval irritation\n\nvulval dysplasia\n\npainful intercourse 244\u20136\n\nvulval irritation 261\u20132\n\nvulval ulceration and sores 266\u20137\n\nvulval irritation 261\u20133\n\nvulval swelling 264\u20135\n\nvulval ulceration and sores 266\u20138\n\nvulvitis, ammoniacal, vulval irritation\n\nvulvovaginal neoplasia, vaginal discharge\n\nvulvovaginitis, painful intercourse 244\u20135\n\nwarfarin\n\nbleeding gums 351\u20132\n\nhaemospermia\n\nnosebleeds 342\u20133\n\npurpura and petechiae 400\u20131\n\nrectal bleeding\n\nvomiting blood\n\nwarts\n\nanal itching 38\u20139\n\nanal swelling 41\u20132\n\nnodules 395\u20136\n\npapules 398\u20139\n\nrectal discharge 49\u201350\n\nvulval irritation ,\n\nvulval swelling 264\u20135\n\nwater drinking, compulsive\n\nexcessive urination\n\nthirst and dry mouth 221\u20132\n\nweight gain 230\u20132\n\nweight loss 233\u20135\n\nabdominal pain\n\nabsent periods 374\u20136\n\nanorectal pain\n\nconstipation 19\u201320\n\ncough\n\ndiarrhoea\n\nepigastric pain\n\nexcessive urination\n\nrectal bleeding\n\nshortness of breath\n\ntiredness 225\u20136\n\nWeil's disease, joint pain ,\n\nWernicke's encephalopathy, confusion\n\nwhiplash injury, neck stiffness 332\u20133\n\nWilson's disease\n\nabnormal movements 168\u20139\n\njaundice\n\ntremor 227\u20138\n\nWolff\u2013Parkinson\u2013White syndrome, palpitations\n\nwriter's cramp\n\nabnormal movements\n\nhand and wrist pain\n\nWRULD (work-related upper limb disorder), hand and wrist pain\n\nxanthelasma, eyelid problems 136\u20137\n\nxanthoma\n\nnodules 395\u20136\n\npapules 398\u20139\n\nxerostomia, swallowing difficulties\n\nyaws, vulval ulceration and sores\n\nyellow nail syndrome, abnormal nails 270\u20131\n\nzipper injury\n\npenile pain\n\npenile ulcerations and sores\n\nZollinger\u2013Ellison syndrome 197\u20138\n**CPD with Radcliffe**\n\nYou can now use a selection of our books to achieve CPD (Continuing Professional Development) points through directed reading.\n\nWe provide a free online form and downloadable certificate for your appraisal portfolio. Look for the CPD logo and register with us at: www.radcliffehealth.com\/cpd\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"# Table of Contents\n\n  1. Title Page\n  2. Table of Contents\n  3. Copyright\n  4. Dedication\n  5. Prologue\n  6. Chapter One\n  7. Chapter Two\n  8. Chapter Three\n  9. Chapter Four\n  10. Chapter Five\n  11. Chapter Six\n  12. Chapter Seven\n  13. Chapter Eight\n  14. Chapter Nine\n  15. Chapter Ten\n  16. Chapter Eleven\n  17. Chapter Twelve\n  18. Chapter Thirteen\n  19. Chapter Fourteen\n  20. Chapter Fifteen\n  21. Chapter Sixteen\n  22. Chapter Seventeen\n  23. Chapter Eighteen\n  24. Chapter Nineteen\n  25. Chapter Twenty\n  26. Chapter Twenty-One\n  27. Chapter Twenty-Two\n  28. Acknowledgments\n  29. About the Author\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nACCLAIM FOR WENDY LESSER'S\n\n_The Genius of Language_\n\n**One of the 25 Best Books of 2004**  \n ** _The Seattle Times\/Post-Intelligencer_**\n\n\"When we talk about a mother tongue we are talking about more than language, we are talking about our sense of our selves at the deepest layers of our being... [A] fine collection... The essays are informative, well written and several are moving.\"\n\n\u2014 _The Times Literary Supplement_\n\n\"Both intimate and global, this entirely unique guided tour through some of the regions beyond English will be seized upon and treasured by anyone who wants to know more about how the languages we speak make and unmake us.\"\n\n\u2013Esther Allen, chair of the PEN American Centers Translation Committee\n\n\"Excellent... The writers peel back the layers of their relationship to the language of their birth as well as to... their adopted English.\"\n\n\u2014 _Santa Cruz Sentinel_\n\n\"As a translator, and a long-time student of languages, I find the topic of mother tongues and learned languages fascinating, and the treatment it receives in these compelling essays is revelatory.\"\n\n\u2014Edith Grossman, translator of _Living to Tell the Tale_ by Gabriel Garc\u00eda M\u00e1rquez\n\n\"A rich and surprising book brimming with love of culture and respect for language.\"\n\n\u2014 _Tucson Citizen_\n\n\"This delightful collection... vividly recounts the process that anyone who loves words goes through: the process of falling under the spell of languages seemingly infinite potential.\"\n\n\u2014 _Publishers Weekly_\n\n_\"A_ wonderful book that is both intellectually stimulating and a great pleasure to read.\"\n\n\u2014Lara Vapnyar, author of _There are Jews in My House_\n\n\"[A] collection that should heighten anyone's awareness of the potential and the limitations of the English language.\"\n\n\u2014 _San Jose Mercury News_\n\n_\"A_ brilliant collection of writers thinking brilliantly about one of the most intimate aspects of their lives: language.\"\n\n\u2014Andr\u00e9 Aciman, author of _Out of Egypt_\n\n#\n\nWENDY LESSER\n\n_The Genius of Language_\n\nWendy Lesser is the founding editor of _The Threepenny Review_ and the author of six books of nonfiction. Her reviews and essays appear in periodicals and newspapers around the country. She lives in Berkeley California.\nALSO BY WENDY LESSER\n\n_The Life Below the Ground_\n\n_His Other Half_\n\n_Pictures at an Execution_\n\n_A Director Calls_\n\n_The Amateur_\n\n_Nothing Remains the Same_\n\n_Hiding in Plain Sight:  \nEssays in Criticism and Autobiography_ (EDITOR)\n\nIN MEMORY OF LEONARD MICHAELS  \n1933-2003\n\n# _Contents_\n\n_Introduction_ Wendy Lesser\n\n[BANGLA  \n _The Way Back_ Bharati Mukherjee](Less_9780307485397_epub_c01_r1.htm)\n\n[CHINESE  \n _Yes and No_ Amy Tan](Less_9780307485397_epub_c02_r1.htm)\n\n[CZECH  \n _Trouble with Language_ Josef Skvoreck y](Less_9780307485397_epub_c03_r1.htm)\n\n[DUTCH  \n _Circus Biped_ Bert Keizer](Less_9780307485397_epub_c04_r1.htm)\n\n[FRENCH  \n _French Without Tears_ Luc Sante](Less_9780307485397_epub_c05_r1.htm)\n\n[GERMAN  \n _Prelude_ Thomas Laqueur](Less_9780307485397_epub_c06_r1.htm)\n\n[GIKUYU  \n _Recovering the Original_ Ngugi wa Thiong'o](Less_9780307485397_epub_c07_r1.htm)\n\n[GREEK  \n _Split Self_ Nicholas Papandreou](Less_9780307485397_epub_c08_r1.htm)\n\n[ITALIAN  \n _Limpid, Blue, Poppy_ M. J. Fitzgerald](Less_9780307485397_epub_c09_r1.htm)\n\n[KOREAN  \n _Personal and Singular_ Ha-yun Jung](Less_9780307485397_epub_c10_r1.htm)\n\n[POLISH  \n _On Being an Orphaned Writer_ Louis Begley](Less_9780307485397_epub_c11_r1.htm)\n\n[RUSSIAN  \n _The Mother Tongue Between Two Slices of Rye_ Gary Shteyngart](Less_9780307485397_epub_c12_r1.htm)\n\n[SCOTS  \n _Boswell and Mrs. Miller_ James Campbell](Less_9780307485397_epub_c13_r1.htm)\n\n[SPANISH  \n _Footnotes to a Double Life_ Ariel Dorfman](Less_9780307485397_epub_c14_r1.htm)\n\n[YIDDISH  \n _My Yiddish_ Leonard Michaels](Less_9780307485397_epub_c15_r1.htm)\n\nAbout the Contributors\n\n# _Introduction_\n\n# Wendy Lesser\n\nThe original idea for _The Genius of Language_ was given to me by an editor, Alice van Straalen. Why not find a dozen or more writers who now write in English but who originally spoke another language, she suggested, and get them to write essays on the differences between their two languages? Normally I am against accepting ideas from editors, but this one struck me as such a good one I couldn't resist. It is an appealing notion\u2014that there is some hidden ur-language seeping into or shaping or otherwise influencing the language in which the writer now writes. Perhaps even monolingual writers have this feeling; that may explain why one friend of mine, a poet whose sole language has always been English, heard the idea for the book and exclaimed, \"Oh, I want to be in it!\"\n\nWhen I invited the fifteen writers included here to participate in the project, I urged them to be as autobiographical as they wished. The story of original languages, I suspected, would make itself felt not just on the linguistic or literary level, but also in the way people felt about their lives. And because those lives involved a move\u2014often a forced move\u2014from one country or family culture to another, these stories might well tell us something about the larger historical or political issues of our time.\n\nBut that was secondary. What mattered most to me, at the beginning, was to uncover the sources of writing in writers I admired, to burrow in behind the acquired layers and get at the inherent nature, the native quality, the \"genius\" of the work. Of course, what I expected and what I eventually got were not identical. Writers are like cats: you can't herd them. Life (and editing) would be far more boring if you could.\n\nThe minute you consider the category\u2014writers who came to English after first speaking another language\u2014the name Joseph Conrad invariably springs to mind. He is the great ancestor, the supervisory ghost, in a book like this. (Some people would add Vladimir Nabokov's name as well, but to my mind Conrad is the far greater writer: I am convinced that _Lord Jim_ and _Chance_ and _The Secret Agent_ will be read long after _Lolita_ and _Pale Fire_ have bit the dust.) Since the essays in the present collection were commissioned especially for the book, I was, for obvious reasons, unable to secure a contribution from Conrad. So I am sneaking him in the back door by quoting him up front.\n\nAt the beginning of the 1919 edition of his autobiographical _A Personal Record,_ Conrad takes great pains to dispel the impression that he _chose_ to write in English. His first language, of course, was Polish, and his father, a well-known Polish patriot who died when Conrad was twelve, was apparently an acknowledged master of the language. Growing up in Poland, Conrad knew French, as he says, \"fairly well and was familiar with it from infancy.\" Yet when he came to write fiction, it was the English language that seized his imagination:\n\nThe truth of the matter is that my faculty to write in English is as natural as any other aptitude with which I might have been born. I have a strange and overpowering feeling that it had always been an inherent part of myself. English for me was neither a matter of choice nor adoption. The merest idea of choice had never entered my head. And as to adoption\u2014well, yes, there was adoption; but it was I who was adopted by the genius of the language, which directly I came out of the stammering stage made me its own so completely that its very idioms I truly believe had a direct action on my temperament and fashioned my still plastic character.\n\nConrad acknowledges that his relationship to the language is necessarily different from that of a native speaker. But the degree of intimacy he feels is, if anything, greater:\n\nA matter of discovery and not of inheritance, that very inferiority of the title makes the faculty still more precious, lays the possessor under a lifelong obligation to remain worthy of his great fortune... All I can claim after all those years of devoted practice, with the accumulated anguish of its doubts, imperfections and falterings in my heart, is the right to be believed when I say that if I had not written in English I would not have written at all.\n\nSeveral of the writers collected here would and even do say the same thing about themselves. It is the crossing of a boundary, the alienation from the original tongue, which made writers of them. So a book in which various bilingual authors are asked to consider the differences between their original language and English will inevitably be as much about the adopted language as it is about anything else. In using the phrase \"the genius of the language,\" Conrad was referring to English, and it is that experience\u2014the process of being embraced or enveloped by English, whether through books or movies or other people\u2014 that you will witness over and over in these pages.\n\nStill, the intention of the book was also to stimulate some reflections on the particularities of the language of origin: that is, to get these writers to express in English the singular characteristics of their mother tongue. It is never an easy task to explain the mechanism of such literary and linguistic border-crossing (Conrad himself calls it \"a task which I have just pronounced to be impossible\"), but the authors I've collected here all give it a good try. Even as you read about each writers discovery of English, you will also be learning something about the peculiar nature, the indwelling spirit, of French, or Greek, or Korean, or Russian. Sometimes the revelation will not be explicit: you will need to listen closely to pick up the tonal subtleties that inflect the adopted tongue. Sense of humor, sentence rhythm, use of adjectives, attention to time and place, penchant for anecdotal-ism or analysis, construction of the first-person narrative voice\u2014 these are all qualities that the writer may carry over from his or her original language.\n\nI should say a few words about what this anthology is not. It is not a scholarly work in the field of linguistics or anthropology or comparative rhetoric. Nobody in this volume is a trained linguist; hardly anyone here even holds an academic post. The contributors are all writers, and they have been chosen almost entirely on the basis of their quality as writers in English, since that is the only way I could know their work. It was my assumption that we would learn something from fiction-writers and playwrights and critics and literary journalists that we would _not_ learn from scholars of language.\n\nNor is this a representative collection, whatever that might be in a world occupied by approximately six thousand different languages. I have made some effort to get a bit of geographical distribution, but the effort is relatively weak. Huge gaps are evident: Arabic, Japanese, Portuguese (to name but a few of the major missing languages). There is only one writer from the entire African continent. There are too many former Europeans. And there are too many Jews. Part of this can be blamed on geopolitics: the vicissitudes of twentieth-century history drove large numbers of Jews into the English language, and many of them became writers. But part is no doubt the result of my limited perspective, in that my own family background is European-Jewish. You might also keep in mind that I _did_ have to persuade each of the contributors to do the onerous work of writing an essay. In this context, I am reminded of the passage from Shakespeare's _Henry IV, Part One_ in which Glendower claims, \"I can call spirits from the vasty deep,\" to which Hotspur sensibly responds: \"Why, so can I, or so can any man; \/ But will they come when you do call for them?\" What you find here are the spirits that came.\n\nThe contributions do not all follow the rules\u2014or rather, I allowed myself to break the rules in acquiring the essayists, and the individual authors went on to break more rules in writing their essays. A rigid adherence to the from-foreign-language-to-English rubric would have eliminated Ariel Dorf-man (who forgot his early Spanish and had to relearn it after English) and Amy Tan (who acquired English and Chinese simultaneously) and Nicholas Papandreou (who was born in Berkeley and spoke English before Greek) and James Campbell (who never did speak Scots, though he understood the infusions of it in his mothers and grandmothers speech). An insistence on the now-they-write-only-in-English rule would have left out Josef Skvorecky (who has written books in English but continues to depend more heavily on Czech) and Ngugi wa Thiong'o (who wrote in English for many years but reverted to Gikuyu for political and personal reasons) and Bert Keizer (who writes and speaks primarily in Dutch). In place of a strictly one-way journey I have substituted a looser bilingualism which allows all my authors to comment on the relationship between English and their other languages. My notion of a \"language\" has been loose, too, in that I have included some that might be viewed as variants or subsets of others (Yiddish a version of German, Scots a version of English). All this latitude has produced a certain variety of approach, as has the age span of the writers, who range from their twenties to their seventies; each generation has a different story about coming to English, just as each country or culture does.\n\nStill, there are common threads in these essays. Perhaps the most common is a tendency to equate the language of childhood with childhood itself. The tangibility of childhood experience\u2014 the tendency to join the word with the thing, so that, in Randall Jarrells terms, \"a word has the reality of a thing: a thing that can be held wrong side up, played with like a toy, thrown at someone like a toy\"\u2014is repeatedly invoked here. We have all been exiled from childhood, but because most of these writers have actually been exiled from their childhood _-place,_ they seem to feel that something of that lost experience still exists somewhere, accessible (if at all) only through language. You can see this most strongly, perhaps, in M. J. Fitzgerald's chapter on Italian and Gary Shteyngart's on Russian, but it is also prominently there in Bharati Mukherjee's piece on Bangla, Thomas Laqueur's essay about German, Nicholas Papandreous discussion of Greek, and Ha-yun Jung's reflections on Korean. When Louis Begley conflates the Polish interiors of his childhood with a Polish story he once read, or when Luc Sante describes French as \"my secret identity, inaccessible to my friends,\" we are being offered a glimpse of the writer's imaginary homeland, compounded of fiction, memory, and loss.\n\nThere is an irony to my editing a book about exile\u2014for that is what a book about the genius of language is bound to be, just as it is also bound to be about geography, and culture, and politics, and history. Not only am I embarrassingly monolingual, possessed of the merest remnant of my little high-school Spanish and even tinier high-school Russian, but I am also the very opposite of an exile. I now live about forty miles from where I grew up, and out of my fifty years I have spent no more than six or seven away from California, the state where I was born. What all these exiled authors long for\u2014a return to a lost place of origin, generally one from which they or their families were violently expelled\u2014I have automatically. And yet the possession of this birthright is not so automatic that I am unconscious of it. Driving over the bridge to San Francisco, especially at dawn or twilight, I frequently look out at that familiar landscape and think of the lines from Brechts _Caucasian Chalk Circle_ about why one loves one's country: \"Because the bread tastes better there, the sky is higher, the air smells better, voices sound stronger, the ground is easier to walk on. Isn't that so?\" Sometimes one understands the truth of things best through absences and lacks (as Conrad grasped English more tightly because it was discovered, not inherited). It is in this way that I understand what Brecht, exiled in California when he wrote those lines, was feeling; and it is in this way that I understand what exiled writers mean when they write about their lost languages.\n\nAnother absence, too, haunts this book. Shortly before I turned in the final manuscript, in May of 2003, my dear friend Leonard Michaels died quite suddenly of complications from lymphoma. I cannot adequately convey how essential a writer he was: to me as a reader, to the magazine he helped me with for more than twenty years, and to the landscape of American literature, which he partially shaped. Perhaps his beautiful essay on Yiddish\u2014his last major piece of prose\u2014will begin to suggest something of what we have lost. He was a master of the precisely descriptive sentence, and his humorously apt phrases often came back to me in the course of daily life; I remember thinking, just before I learned he was ill, that I would not like to exist in a world that did not contain Lenny Michaels. Now I will have to, as will we all. It seems fitting, though hardly consoling, that a collection about the genius of language should be dedicated to his memory.\n\n# BANGLA\n\n#\n\n# _The Way Back_  \nBharati Mukherjee\n\nThere is a reason why the language we inherit at birth is called our mother tongue. It is our mother, forgiving, embracing, naming the world and all its emotions. Though I have lived for the last forty years in cities where English or French is the language of the majority, it's Bangla that exercises motherly restraint over my provisional, immigrant identity. Mother-Bangla is fixed; I haven't learned a new word nor had new thought or feeling in Bangla for nearly half a century. _I don't need to._ According to group-norms, as a native-born speaker, I have automatic membership in the world's most articulate, most imaginative and most intelligent club. With its brazen appeals to love and vanity, enforced with coercive guilt, the language sabotages irony towards the community's billowing self-esteem. Like a child whose mother might tipple or stray, I look for excuses, cannot condemn. To my inner Bengali I remain constant, as it does for me.\n\nHow exclusive can a club of nearly a quarter-billion members be? Bangla is the language of Bangladesh, the eighth most populous nation in the world, and of the Indian state of West Bengal, the second-largest linguistic group in India. Millions more, documented or not, have settled abroad. Impressive numbers aside, every Bengali, to her at least, is a majority of one. We harbor the faith, implanted by myth and history, of our exalted place in the hierarchy of breeding and culture.\n\nTo international relief agency workers, Bangla is the mother tongue of esurient poverty, but to the heirs of _shonar bangla,_ golden Bengal of harvest-ready paddyfields and fish-filled rivers, it is the mother tongue of poetry, passion and abundance. It is also the language of nostalgia and of tentative hope: nostalgia for the Hindu-Muslim harmony that existed in undivided Bengal before its vindictive partition by the fleeing British; and hope for the shared mother tongue, devotion to the possible tomorrow that will transcend the religious furies exploited by today s politicians. I think a shared language is stronger than divisive religions. (Based on my travels in Bangladesh, I think Hindu and Muslim Bengalis could cross the abyss between them. It's the national politics of India and corrupt fiefdoms in Bangladesh that get in the way.)\n\nUp to age eight, I lived exclusively in Bangla. My father was the sole support of forty to fifty relatives, who lived with us crowded together in the ground-floor apartment of a two-storied house in a homogenously Hindu, Bangla-speaking, middle-class neighborhood of Kolkata (until recently mispronounced and misspelled as Calcutta by colonialists). All the adults in our large household had been born in villages or towns in the Dhaka (then \"Dacca\") district of East Bengal (now Bangladesh); all their children, my sisters and cousins, in the thriving capital, Kolkata, in West Bengal. Among themselves, the adults spoke the dialect of Dhaka, the children the Bangla of Kolkata. I had no idea as a child that linguists considered the Dhaka dialect \"deviant,\" and Kolkata the standard. In our home the Dhaka dialect, ban-gal, was the language of authenticity. You are what dialect your forefathers spoke even if you yourself have lost fluency in it because of successive migrations. We were East Bengalis or _Bangal_ first, then Bengali. We distanced ourselves from West Bengalis or _Ghoti_ who surrounded us and considered us interlopers. We conducted ourselves as _Bangal,_ exiled permanently from our ancestral homeland.\n\nTo be born a displaced _Bangal_ was to inherit loss of, and longing for, one's true home. Identity had to do with mother tongue, but home was the piece of land that our forefathers had owned, the soil that they had slept and walked on. To be cast out of your _janma bhumi_ or ancestral birth-soil is to be forever doomed. Unlike dialects, which can be transported by migrants, the loss of _janma bhumi_ is permanent. The diasporic Bengali may own real estate in the country of her adoption, but that real estate can only be her residence, as provisional as her immigrant identity, her home. I think now that this intimate braiding of inherited language, place and identity is why Bengalis never took to the British system of primogeniture. Generation by generation, the extent of ancestral land owned by an individual male was whittled down to specks and strips. It didn't matter that the shrunken land was unprofitable for cultivation and support of large families. Second, third, fourth... eighth, ninth. Sons didn't emigrate; they just stayed and got poorer. Evaluation was symbolic, not economic.\n\nEven as a child I picked up on our linguistic nuancing of house _(basha, bari),_ room _(ghar),_ land _(jomi),_ soil _(bhumi)_ and homeland _(desh)._ My cousins and I were alert to the moral of the countless children's stories about villagers willing to starve rather than sell off inherited strips of _bhumi._ The most menacing refrain from a popular poem we learned by rote was a rich, greedy landowners threatening a desperately poor farmer, \"Do you get it, Upen \/ I'm going to buy up your land.\" Even now in comfortable San Francisco, with every mortgage and property tax payment, I can be thrown into panic by that simple refrain. (Refinance? Never! Lock it in and forget about it, like immigrants or Depression-era survivors who distrust banks.) The inherited culture insists that accidents of impulse and geography have made homelessness my permanent condition; the adopted culture tries to persuade me that home is where I choose to invest love and loyalty.\n\nBetter colleges, better job opportunities, natural disasters like floods and famines, and periodic flarings of Hindu-Muslim antagonism had induced my parents' immediate families to migrate to Kolkata in the 1920s and 1930s, though they still owned land in their hometowns. My father had been sent to Kolkata to stay with a childless aunt and uncle-in-law so that he could get a sound English-language undergraduate education at St. Xavier's College, run by Jesuits from Belgium. By then a scholarship student, he stayed for graduate degrees in applied chemistry. After that, as the most educated though not the eldest of nine sons, he dutifully looked for a job in Kolkata, the most prosperous of Bengal's cities. Dependents and family friends also in need of free food and lodging started arriving from the east as soon as they'd heard that he was job-hunting. For a while, the house-guests commuted from Kolkata to their homes in provincial towns of eastern Bengal, where they still jointly owned an ancestral strip.\n\nThe Hindu-Muslim riots of 1946, the fiercest in communal memory, drove the last of the Mukherjee clan out of their hometown, where Muslims constituted an overwhelming majority. The refugees brought with them tales of arson, rape and looting. From their tears and nightmare-hour screams, I learned the special resonance that _bhoi,_ the Bangla word for fear, carries. There is no English equivalence for the scale of terror that _bhoi_ implies. In words such as _bhoi,_ the individual experience of fear is shot through with the memory of unspeakable communal suffering. The partitioning of Bengal in 1947 transformed my vast, extended family from commuters shuttling between residence of convenience and homeland to political refugees stranded in a city where they could never belong. Kolkata was their begrudged place of asylum.\n\nThat exilic melancholy was passed on to me in infancy. We refugees were different from, and superior to, _Ghoti_ Bengalis. Which end of the soft-boiled egg do you crack, and should we go to war over it? We rejected matchmaking between _Ghoti_ families and ours. We made fun of the _Ghoti_ inability to pronounce the \"l\" sound in _lebu_ (lemon) and _loochi_ (deep-fried bread). Whenever the East Bengal soccer team played Mohun Bagan, the West Bengal team, we conducted ourselves not as well-brought-up young women but as rowdy soccer fans. The _Ghoti,_ in turn, stereotyped us as provincial bumpkins. We directed our Us-versus-Them pugnacity to people who spoke dialects other than the Dhaka one within the _Bangal_ community. Long before I had heard of Freud, I had enacted \"the narcissism of small differences.\"\n\nIn our house, bangal was the language of passion and of discipline. Unhappy wives threatened death by fasting in bangal; virtuous virgins gossiped about neighborhood sluts in it; headstrong young uncles swore at each other in it. Whenever my father had to assume the unwelcome role of patriarch and punish unseemly behavior by a relative, he first, in bangal, consulted his widowed mother, an autocratic upholder of conservative traditions. I remember bangal, however, mainly for the ancient doggerels that my paternal grandmothers coterie of tobacco-chewing, osteoporosis-bent widows recited for their deft delivery of sexist cruelty. Day in and day out, these widows tormented my mother for having borne three daughters and no sons. I may be walking on Haight Street, but I still hear them repeat their malicious couplets: _Puter mutey kori \/ Meyer galai dori_ (There's money to be made off a son's piss \/ There's rope to hang a daughter with).\n\nMy mother tongue transmitted unambiguously the society's values and taboos. Literacy turned women rebellious, unsubmissive, and unmarriageable. My mother, who had been married off the moment she had finished high school, was abused regularly by her in-laws, first for daring to express the hope that she might enroll in a women's college, and later for wanting to send her three daughters to the best elementary school for girls, which happened not to be in our neighborhood. There were harangues and beatings. I didn't realize at the time that I was not just a child-spectator of a scene of authorized sado-masochism, but that I was witness to the last chapters of a long, cultural mega-upheaval. Only very recently, while researching the battle between the Hinducentric traditionalists and the Europhilic reformists (the Brahmo-Samaj) for a novel, I came across a doggerel that has relocated those childhood scenes of pain inflicted and pain accepted in the context of nationalist struggles. _Likhey porey ranhr_ **_I_** _'buley jano udome shanhr._\n\nThe doggerel is in a colloquial Bangla long out of usage.\n\n_An educated woman is a woman without a husband \/ Like an untethered ox, she roams around._ On the surface, its moral is familiar and unambiguous: educated women threaten tradition. But the literal translation does not convey the anonymous versifiers hidden linguistic ambiguities and cultural codes. _Ranhr_ describes a husbandless woman, who might be a widow (an unfortunate fate in a society that advocates _sati),_ or an unmarried woman (a worse fate in a society that expects a woman to worship God through worshipping her husband), or a prostitute. The adjective _udome_ that characterizes the ox of the doggerels analogy can be correctly translated as both \"untied\" and \"naked.\" _Buley_ is the third-person conjugation of the verb _bula_ (to roam, loiter, barge in). These ambiguities evoke malicious connections between a woman allowed to learn the three Rs and a loitering whore, who preys on the very society that has cast her out. But, for the Bangla-speaker, this pejorative connotation is undermined or even negated by the great medieval Bengali poet Govinda Das s application of the verb _bula_ to the wanderings of poets. In our linguistic community, the poet or _kavi_ is not only an artist and entertainer, as she or he might be in the West; the poet is the visionary revealer of ultimate cosmic mysteries. I inherited a mother tongue charged with contradictions and nuances. The capacity of diction to imply opposite meanings has fed me even as I write fiction in English.\n\n(Bangla is not unique in that regard. An Israeli poet friend of my husbands, Ronny Someck, once described modern Hebrew as an amalgam of two sources: the ancient Bible and the modern army. According to him, many words and most poems in that language can be read in at least two ironic, sometimes comic and often-pornographic ways.)\n\nSo it was that through _chhara_ (lullabies), proverbs, and domestic squabbles, I intuited the dues and privileges of membership in the Bangla-speakers' club. My formal training in ladylike behavior was undertaken by my paternal grandmother (the same full-throated widow who harangued my mother for wanting to educate us sisters). She herself had been taught to read as a child by her older sister, a childless woman whose husband\u2014 picked out by my paternal great-grandfather for his potential to earn a good living as a doctor in Kolkata\u2014turned out to have progressive ideas. Every night, though we children would have preferred to listen to ghost stories told with dramatic sound effect by a retired servant who had stayed on with us, this grandmother recited episodes from the Bangla-version of the _Ramayana_ and the _Mahabharata._\n\nOf the two epics, the _Ramayana_ was my favorite. What young girl could not thrill to the sheer romance? The handsome and virtuous prince, Ram, legal heir to the throne of his polygamous father, is banished to the forest. His aged father is in thrall to a young wife with ambitions to get her own issue appointed as crown prince. (Didn't we know such stories from the neighborhood, or, indeed, from our own family?) The subsequent adventures of the banished Ram include wife kidnapping and Homeric-scale war for wife-rescuing, martial victory and the final installation of Ram the rightful king. How pale in comparison are Helen and Paris, Greece and Troy. It's _such_ a good story, with its incorporation of shape-changing demons and deities into everyday reality. I thrilled to the narrative clutter, the strong plot and the balancing of violence and tenderness. And under all the convolutions and endless adumbration, it is true to the psychological tensions within the extended family. I was already in training to be a novelist.\n\nAs a writer, I'm sometimes taken to task for linking too many narratives, turning family stories into murder mysteries, indulging myself in violence, ramming ancient history into contemporary reality, dipping into voices and situations (the discredited charge of \"appropriation\") far beyond my personal experience. Well, blame the _Rarnayana._\n\nMy habit of scavenging the worlds legacy for ideas and for vocabulary, then reshaping them until they comfortably _belong_ to me, is something I must have picked up unconsciously, in infancy. For all our proud insistence on exclusivity, Bengalis (particularly those from Kolkata) have absorbed words from peoples we have come into contact with through trade, wars, migration and marriage, without feeling it a violation of our core identity. Persian was the pre-British court language (and still in circulation during the British Raj); therefore, from Persian we have _golap,_ the word for rose, and _ostad,_ the term for a master musician. Even the famous Anglo-Indian, Victorian word _durbar,_ the elaborate imperial tent and ceremonial display of power, and the refreshing drink, _sharbat,_ come from Persian. From a long exposure to Portuguese traders came words like _toali_ for towel, _janla_ for window, _almira_ for almirah or large wooden cabinet. From the British, who made \"Calcutta\" the traditional seat of power, we got the courts, the schools, parks, the museums and luxurious residences. We also got the first brush with the authority of the English language\u2014and two hundred years' indoctrination in our backwardness. Indianized English words, such as _tebil_ for table, _pulish_ for police, _kabard_ for cupboard, and (my favorite) _tikoji_ for tea-cozy, trail a certain colonial shame. Other aspects of the British linguistic presence, which I will return to later, are far more interesting, or damaging, according to one's interpretation.\n\nIn our home, we operated in two distinct forms of Bangla. We spoke chalit or colloquial Bangla, which was peppered with Indi-anized foreign words that over centuries had lost their foreign-ness, but when we wrote letters or school essays, we switched to sudha or pure Bangla, which was weighted with Sanskritized words and literary formality. The choice of colloquial or literary depended on the when, why and with whom we needed to communicate. As a child I accepted the uncrossable immensity of the gulf between the oral and the written. It wasn't until I returned from three years of elementary school in England, where I'd had to learn English from scratch in a hurry and to turn in weekly essays on walks taken, birds sighted, holidays in Margate (which I had written in the simple English I spoke and for which I had received top grades), that I rebelled against the Bangla tradition of treating the colloquial as an inferior form unfit for the page.\n\nI was eight when I was initiated into bilingualism. At the school in Sloane Square, I spoke the English I heard around me. At home I spoke the Bangla I had in our Kolkata house. My _Bangal_ identity was not at all threatened by my growing fluency in the former colonizer's language. The shock came from suddenly seeing myself as a minority, a brown girl in a white school. I was still a member of the world's most elite linguistic club, but nobody in school knew that.\n\nWithin the first weeks of my immersion in English I realized that the Bangla alphabet was far larger. English letters were arranged higgledy-piggledy, without logical sequence, conforming to no order but the arbitrary \"alphabetical.\" Bangla orders its vowels separately from its consonants. And when it comes to consonants, we owned sounds that no English schoolgirl, or teacher, could copy. The \"Bh\" in my name, Bharati, was unpronounceable. The nasalizations could not be copied. Consonants were arranged in rows according to where inside the mouth the tongue positioned itself. We had three distinct \"n\" sounds, three distinct \"r\" sounds, and though this precision made dictation exercises very stressful, I began to take pride in the lingual dexterity that this precision demanded.\n\nIn contrast, English had too thrifty an alphabet. It tolerated imprecise noises uttered by the lazy-tongued. English had no use for the nuanced nailing of extended-family relationships. Their \"aunts\" and \"uncles\" seemed disrespectful and generic compared to our \"fathers-side older brother\" or \"mothers second sister.\" Next to the drawn-out vowels that made spoken Bangla a euphonic language, English sounded harshly energetic. I missed the onomatopoeic phrases in Bangla that mimicked the blowing of wind, the drizzle of rain and gurgle of waterfalls. I missed, too, the echo words (sometimes as ingenious and coded as cockney rhyming slang), the repetitions\u2014such as _garam garam,_ hot hot (and its infinite expansion: good good, rich rich, fat fat)\u2014that emphasize caution, respect or possible ridicule. In English, such repetition comes across as simple-mindedness, proto\u2013Peter Sellers. Other forms of repetition, like _bosta phosta,_ bags and baggage, vary the initial consonant for a dismissive implication. My father was a master of the colloquial Bangla put-down, even while introducing us to distinguished visitors. By the simple act of artful doubling, he could slip across the counter-intelligence that Mr. So-and-So had earned his money dishonestly, had wastrel sons and a spendthrift wife.\n\nI envied the English their recording of time passing. They used \"yesterday\" \"today\" and \"tomorrow\" to mark the past, the present and the future while in Bangla we made use of \"kal\" (time) for all three tenses. My British school chums and I had fundamentally opposed views of Time (also \"kal,\" movement). And as I began to become more fluent in my second language, I became conscious of their syntactical differences. Where my English friends said, \"I'm going now\" or \"I'm reading,\" in Bangla I would have to say, \"I now am going\" or \"I a book am reading.\" My brain felt hardwired for constructing sentences in the order my mother tongue dictated.\n\nI began to invest in mother-tongue nostalgia for the storytelling hour with my grandmother and the rowdiness of the relatives who, in Kolkata, I had hidden under beds to avoid. That synthetic nostalgia somehow quickened into a deep longing for the dramatic tales that my mother used to tell about daring young freedom-fighting martyrs from ancestral towns like ours. She told these stories as I ate dinner, sitting on a braided-bamboo mat on the floor along with my many live-in cousins and two sisters. Because of the size of our extended family, we ate in batches: children first, then adult male relatives, then women relatives, then servants. As we ate, my mother, a kavi-seer in her way, launched into stories of teenage boys and girls risking torture in prison, banishment to the penal Andaman Islands and death by hanging in order to take over a police station or an armory. My mother extracted long, pearly bones by macerating chunks of curried carp and then hand-fed me rice and fish balls. She had to make herself heard above squabblers' cries of \"I want the fish head!\" and \"No, it's my turn. You had the head last night. It's not fair, just because my father doesn't...,\" and the shrill calls of birds settling for the night in the foliage of deodar trees lining the sidewalk.\n\nIn the Kolkata remembered oceans away, twilight was a hinge-moment between professing trust in the values of imported Enlightenment and submitting to invisible, cosmic forces. My mother was a skillful deliverer of tales, an improviser (like jazz musicians, I think now) who could keep me tense, entranced, through each re-telling and make the tale's familiar ending come off as unpredictable. Her voice melted my physical surroundings.\n\nAs a novelist, I now melt down the cultural borders of my legacies. The fluid concept of time inherited through Bangla's use of _kal_ and the \"magic realism\" inherited from the Hindu epics inform my writing about immigrants in North American cities. Now I write in my third language, American, another \"deviant dialect\" of the E. M. Forsterian British I learned as authoritative, and in which, in fact, I wrote my first novel and earliest stories.\n\nMy mother tongue was a linguistic primer, a thin whitewash over all that is pre-conscious and pre-rational. It was in English that I began to analogize. Successive coats of French and English have faceted Bangla, but it still shines through. It is the odd fate of so many of us in the global community, not just those of us from India but from other homelands at ease in family-time and epic storytelling, that a second language, a school language, was necessary to liberate their minds from their bodies, their self from their community.\n\nTwo selves exist within the language-adoptee, as with any adoptee\u2014what might have been, what was lost, and the good fortune, the delivery from want and frustration. For a writer, the melting of a mother tongue is the madeleine, the way back, and the way in, an early loss with the deepest memory, the mother of all plots.\n\n# CHINESE\n\n#\n\n# _Yes and No_  \nAmy Tan\n\nOnce, at a family dinner in San Francisco, my mother whispered to me: \"Sau-sau [Brothers Wife] pretends too hard to be polite! Why bother? In the end, she always takes everything.\"\n\nMy mother thinks like a _waixiao,_ an expatriate, temporarily away from China since 1949, no longer patient with ritual courtesies. As if to prove her point, she reached across the table to offer my elderly aunt from Beijing the last scallop from the Happy Family seafood dish.\n\nSau-sau scowled. _\"B'yao, zhen b'yao!\"_ (I don't want it, really I don't!) she cried, patting her plump stomach.\n\n\"Take it! Take it!\" scolded my mother in Chinese.\n\n\"Full, I'm already full,\" Sau-sau protested weakly, eyeing the beloved scallop.\n\n\"Ai!\" exclaimed my mother, completely exasperated. \"Nobody else wants it. If you don't take it, it will only rot!\"\n\nAt this point, Sau-sau sighed, acting as if she were doing my mother a big favor by taking the wretched scrap off her hands.\n\nMy mother turned to her brother, a high-ranking communist official who was visiting her in California for the first time: \"In America a Chinese person could starve to death. If you say you don't want it, they won't ask you again forever.\"\n\nMy uncle nodded and said he understood fully: Americans take things quickly because they have no time to be polite.\n\nI thought about this misunderstanding again\u2014of social contexts failing in translation\u2014when a friend sent me an article from the _New York Times Magazine._ The article, on changes in New York's Chinatown, made passing reference to the inherent ambivalence of the Chinese language.\n\nChinese people are so \"discreet and modest,\" the article stated, there aren't even words for \"yes\" and \"no.\"\n\nThat's not true, I thought, although I can see why an outsider might think that. I continued reading.\n\nIf one is Chinese, the article went on to say, \"One compromises, one doesn't hazard a loss of face by an overemphatic response.\"\n\nMy throat seized. Why do people keep saying these things? As if we truly were those little dolls sold in Chinatown tourist shops, heads bobbing up and down in complacent agreement to anything said!\n\nI worry about the effect of one-dimensional statements on the unwary and guileless. When they read about this so-called vocabulary deficit, do they also conclude that Chinese people evolved into a mild-mannered lot because the language only allowed them to hobble forth with minced words?\n\nSomething enormous is always lost in translation. Something insidious seeps into the gaps, especially when amateur linguists continue to compare, one-for-one, language differences and then put forth notions wide open to misinterpretation: that Chinese people have no direct linguistic means to make decisions, assert or deny, affirm or negate, just say no to drug dealers, or behave properly on the witness stand when told, \"Please answer yes or no.\"\n\nYet one can argue, with the help of renowned linguists, that the Chinese are indeed up a creek without \"yes\" and \"no.\" Take any number of variations on the old language-and-reality theory stated years ago by Edward Sapir: \"Human beings... are very much at the mercy of the particular language which has become the medium for their society... The fact of the matter is that the 'real world' is to a large extent built up on the language habits of the group.\"\n\nThis notion was further bolstered by the famous Sapir-Whorf hypothesis, which roughly states that one's perception of the world and how one functions in it depends a great deal on the language used. As Sapir, Whorf, and new carriers of the banner would have us believe, language shapes our thinking, channels us along certain patterns embedded in words, syntactic structures, and intonation patterns. Language has become the peg and the shelf that enables us to sort out and categorize the world. In English, we see \"cats\" and \"dogs;\" what if the language had also specified _glatz,_ meaning \"animals that leave fur on the sofa,\" and _glotz,_ meaning \"animals that leave fur and drool on the sofa\"? How would language, the enabler, have changed our perceptions with slight vocabulary variations?\n\nAnd if this were the case\u2014of language being the master of destined thought\u2014think of the opportunities lost from failure to evolve two little words, _yes_ and _no,_ the simplest of opposites! Ghenghis Khan could have been sent back to Mongolia. Opium wars might have been averted. The Cultural Revolution could have been sidestepped.\n\nThere are still many, from serious linguists to pop psychology cultists, who view language and reality as inextricably tied, one being the consequence of the other. We have traversed the range from the Sapir-Whorf hypothesis to est and neurolinguis-tic programming, which tell us \"you are what you say.\"\n\nI too have been intrigued by the theories. I can summarize, albeit badly ages-old empirical evidence: of Eskimos and their infinite ways to say \"snow,\" their ability to _see_ differences in snowflake configurations, thanks to the richness of their vocabulary, while non-Eskimo speakers like myself founder in \"snow,\" \"more snow,\" and \"lots more where that came from.\"\n\nI too have experienced dramatic cognitive awakenings via the word. Once I added \"mauve\" to my vocabulary I began to see it everywhere. When I learned how to pronounce _prix fixe,_ I ate French food at prices better than the easier-to-say _\u00e0 la carte_ choices.\n\nBut just how seriously are we supposed to take this?\n\nSapir said something else about language and reality. It is the part that often gets left behind in the dot-dot-dots of quotes: \"... No two languages are ever sufficiently similar to be considered as representing the same social reality. The worlds in which different societies live are distinct worlds, not merely the same world with different labels attached.\"\n\nWhen I first read this, I thought, Here at last is validity for the dilemmas I felt growing up in a bicultural, bilingual family! As any child of immigrant parents knows, there's a special kind of double bind attached to knowing two languages. My parents, for example, spoke to me in both Chinese and English; I spoke back to them in English.\n\n\"Amy-ah!\" they'd call to me.\n\n\"What?\" I'd mumble back.\n\n\"Do not question us when we call,\" they scolded me in Chinese. \"It is not respectful.\"\n\n\"What do you mean?\"\n\n\"Ai! Didn't we just tell you not to question?\"\n\nTo this day, I wonder which parts of my behavior were shaped by Chinese, which by English. I am tempted to think, for example, that if I am of two minds on some matter it is due to the richness of my linguistic experiences, not to any personal tendencies toward wishy-washiness. But which mind says what?\n\nWas it perhaps patience\u2014developed through years of deciphering my mothers fractured English\u2014that had me listening politely while a woman announced over the phone that I had won one of five valuable prizes? Was it respect\u2014pounded in by the Chinese imperative to accept convoluted explanations\u2014that had me agreeing that I might find it worthwhile to drive seventy-five miles to view a time-share resort? Could I have been at a loss for words when asked, \"Wouldn't you like to win a Hawaiian cruise or perhaps a fabulous Star of India designed exclusively by Carter and Van Arpels?\"\n\nAnd when this same woman called back a week later, this time complaining that I had missed my appointment, obviously it was my type A language that kicked into gear and interrupted her. Certainly, my blunt denial\u2014\"Frankly I'm not interested\"\u2014was as American as apple pie. And when she said, \"But it's in Morgan Hill,\" and I shouted, \"Read my lips. I don't care if it's Timbuktu,\" you can be sure I said it with the precise intonation expressing both cynicism and disgust.\n\nIt's dangerous business, this sorting out of language and behavior. Which one is English? Which is Chinese? The categories manifest themselves: passive and aggressive, tentative and assertive, indirect and direct. And I realize they are just variations of the same theme: that Chinese people are discreet and modest.\n\nReject them all!\n\nIf my reaction is overly strident, it is because I cannot come across as too emphatic. I grew up listening to the same lines over and over again, like so many rote expressions repeated in an English phrase-book. And I too almost came to believe them.\n\nYet if I consider my upbringing more carefully, I find there was nothing discreet about the Chinese language I grew up with. My parents made everything abundantly clear. Nothing wishy-washy in their demands, no compromises accepted: \"Of course you will become a famous neurosurgeon,\" they told me. \"And yes, a concert pianist on the side.\"\n\nIn fact, now that I remember, it seems that the more emphatic outbursts always spilled over into Chinese: \"Not that way! You must wash rice so not a single grain spills out.\"\n\nI do not believe that my parents\u2014both immigrants from Mainland China\u2014are an exception to the modest-and-discreet rule. I have only to look at the number of Chinese engineering students skewing minority ratios at Berkeley, MIT, and Yale. Certainly they were not raised by passive mothers and fathers who said, \"It's up to you, my daughter. Writer, welfare recipient, masseuse, or molecular engineer\u2014you decide.\"\n\nAnd my American mind says, See, those engineering students weren't able to say no to their parents' demands. But then my Chinese mind remembers: Ah, but those parents all wanted their sons and daughters to be _-pre-med._\n\nHaving listened to both Chinese and English, I also tend to be suspicious of any comparisons between the two languages. Typically, one language\u2014that of the person doing the comparing\u2014 is often used as the standard, the benchmark for a logical form of expression. And so the language being compared is always in danger of being judged deficient or superfluous, simplistic or unnecessarily complex, melodious or cacophonous. English speakers point out that Chinese is extremely difficult because it relies on variations in tone barely discernible to the human ear. By the same token, Chinese speakers tell me English is extremely difficult because it is inconsistent, a language of too many broken rules, of Mickey Mice and Donald Ducks.\n\nEven more dangerous to my mind is the temptation to compare both language and behavior _in translation._ To listen to my mother speak English, one might think she has no concept of past or future tense, that she doesn't see the difference between singular and plural, that she is gender blind because she calls my husband \"she.\" If one were not careful, one might also generalize that, based on the way my mother talks, all Chinese people take a circumlocutory route to get to the point. It is, in fact, my mothers idiosyncratic behavior to ramble a bit.\n\nI worry that the dominant society may see Chinese people from a limited\u2014and limiting\u2014perspective. I worry that seemingly benign stereotypes may be part of the reason there are few Chinese in top management positions, in mainstream political roles. I worry about the power of language: that if one says anything enough times\u2014in _any_ language\u2014it might become true.\n\nCould this be why Chinese friends of my parents' generation are willing to accept the generalization?\n\n\"Why are you complaining?\" one of them said to me. \"If people think we are modest and polite, let them think that. Wouldn't Americans be pleased to admit they are thought of as polite?\"\n\nAnd I do believe anyone would take the description as a compliment\u2014at first. But after a while, it annoys, as if the only things that people heard one say were phatic remarks: \"I'm so pleased to meet you. I've heard many wonderful things about you. For me? You shouldn't have!\"\n\nThese remarks are not representative of new ideas, honest emotions, or considered thought. They are what is said from the polite distance of social contexts: of greetings, farewells, wedding thank-you notes, convenient excuses, and the like.\n\nIt makes me wonder, though. How many anthropologists, how many sociologists, how many travel journalists have documented so-called \"natural interactions\" in foreign lands, all observed with spiral notebook in hand? How many other cases are there of the long-lost primitive tribe, people who turned out to be sophisticated enough to put on the stone-age show that ethnologists had come to see?\n\nAnd how many tourists fresh off the bus have wandered into Chinatown expecting the self-effacing shopkeeper to admit under duress that the goods are not worth the price asked? I have witnessed it.\n\n\"I don't know,\" the tourist said to the shopkeeper, a Cantonese woman in her fifties. \"It doesn't look genuine to me. I'll give you three dollars.\"\n\n\"You don't like my price, go somewhere else,\" said the shopkeeper.\n\n\"You are not a nice person,\" cried the shocked tourist, \"not a nice person at all!\"\n\n\"Who say I have to be nice,\" snapped the shopkeeper.\n\n\"So how does one say 'yes' and 'no' in Chinese?\" my friends ask a bit warily.\n\nAnd here I do agree in part with the _New York Times Magazine_ article. There is no one word for \"yes\" or \"no\" \u2014but not out of necessity to be discreet. If anything, I would say the Chinese equivalent of answering \"yes\" or \"no\" is _discrete,_ that is, specific to what is asked.\n\nAsk a Chinese person if he or she has eaten, and he or she might _say chrle_ (eaten already) or perhaps _meiyou_ (have not).\n\nAsk, \"So you had insurance at the time of the accident?\" and the response would be _dwei_ (correct) or _meiyou_ (did not have).\n\nAsk, \"Have you stopped beating your wife?\" and the answer refers directly to the proposition being asserted or denied: stopped already, still have not, never beat, have no wife.\n\nWhat could be clearer?\n\nAs for those who are still wondering how to translate the language of discretion, I offer this personal example.\n\nMy aunt and uncle were about to return to Beijing after a three-month visit to the United States. On their last night I announced I wanted to take them out to dinner.\n\n\"Are you hungry?\" I asked in Chinese.\n\n\"Not hungry,\" said my uncle promptly, the same response he once gave me ten minutes before he suffered a low-blood-sugar attack.\n\n\"Not too hungry,\" said my aunt. \"Perhaps you're hungry?\"\n\n\"A little,\" I admitted.\n\n\"We can eat, we can eat then,\" they both consented.\n\n\"What kind of food?\" I asked.\n\n\"Oh, doesn't matter. Anything will do. Nothing fancy, just some simple food is fine.\"\n\n\"Do you like Japanese food? We haven't had that yet,\" I suggested.\n\nThey looked at each other.\n\n\"We can eat it,\" said my uncle bravely, this survivor of the Long March.\n\n\"We have eaten it before,\" added my aunt. \"Raw fish.\"\n\n\"Oh, you don't like it?\" I said. \"Don't be polite. We can go somewhere else.\"\n\n\"We are not being polite. We can eat it,\" my aunt insisted.\n\nSo I drove them to Japantown and we walked past several restaurants featuring colorful plastic displays of sushi.\n\n\"Not this one, not this one either,\" I continued to say, as if searching for a Japanese restaurant similar to the last. \"Here it is,\" I finally said, turning into a restaurant famous for its Chinese fish dishes from Shandong Province.\n\n\"Oh, Chinese food!\" cried my aunt, obviously relieved.\n\nMy uncle patted my arm. \"You think like a Chinese.\"\n\n\"It's your last night here in America,\" I said. \"So don't be polite. Act like an American.\"\n\nAnd that night we ate a banquet.\n\n# CZECH\n\n#\n\n# _Trouble with Language_  \nJosef \u0160kvoreck\u00fd\n\nI'm told that the first decade of life decides life for as many decades as it takes a man to return to his Maker. Judging by the evidence of my life, I believe it's true.\n\nI was born in a small town built on an ancient route through a pass between two mountain ranges by which caravans of merchants entered the vast valley of Bohemia, reaching eventually its heart. In the times of the Celts and the Germans it was just a fortress, Marobudum, but after the influx of Slavic tribes it grew into a town, then city, then metropolis called Prague. On medieval maps the mountain pass was called _Porta regni,_ the Gate to the Kingdom, i.e., the kingdom of the mightiest tribe in the region, the Czechs.\n\nWhen I was a child, America was far, very far away: only an echo of some unreachable reality. And yet, my earliest cultural memory came from that misty midregion of Weir. Sitting on my mother's lap in the local cinema, a mere pre-schooler, I saw Fatty Arbuckle's two-reeler _Saved by Fido_ and I remembered it, in the twisted memory of a child, till the day some sixty years later, in Canada, when I read a book on Hollywood silent comedies.\n\nMy second cultural inspiration came also from North America; this time, however, it was literary. The author of the book was an American from Michigan, James Oliver Curwood, and the novel was _Men of Brave Hearts_ (I hope that was the title\u2014 I'm translating from my Czech memory). It was the first part of a trilogy set in the Canadian North, with Mounties as heroes, chasing (and saving) beautiful Indian girls\u2014the Rose Marie stuff. My father gave it to me for Christmas\u2014in those days principal Christmas presents were not toy cars but books. I was to wait but didn't till next Christmas for further adventures of Mounties and their paramours. I bought the second part myself, out of savings from my weekly allowance, but to my dismay at the end of the book I found the publisher's note that, regrettably, Mr. Curwood died suddenly without finishing his trilogy.\n\nHere fate intervened. I sat down and finished the saga for the late Mr. Curwood: _The Mysterious Cave,_ my opus number one. Father was so impressed by the eighteen-page novel that he typed it up for me and drew a frontispiece. He copied it, on translucent paper, from an illustration in a Karl May novel. In those days novels came out illustrated.\n\nOf course I thought that Mr. Curwood was a Canadian. Who else would write so convincingly about the Canadian wilderness? (Convincingly, that is, for me, who knew a big zero about the wilderness in the north of America.)\n\nNevertheless, my aroused literary passion didn't end with this mistake. About two years later, the pseudo-Canadian Curwood was replaced in my affection by an American who, unfortunately, _was_ Canadian. I don't know why I was convinced that Ernest Thompson Seton was a Yankee, and that _Two Little Savages_ was set in Chicago and on a farm nearby. Probably I overheard my mother's visitors discussing the then fashionable translation of an American novel called _The Jungle,_ with its gloomy descriptions of life and animal deaths in the Windy City. Ladies in those distant pre-Oprah days sometimes actually discussed books.\n\nAgain, it wasn't until some sixty years later, in Toronto, which on a winter day can be pretty gloomy, that I found (in a city guide) that the Don Valley, some ten minutes' walk from our house, was the scene of Seton's young hero's depressed roamings in the deep forests around the gloomy city; that Ian lived in my adoptive home of Toronto, not in Illinois; and that Seton was not an American.\n\nLater, in another chapter of my early life, I almost died of pneumonia\u2014as with nearly everything in my childhood, it was in some sort of \"pre\" days; in this case, pre-antibiotics\u2014and consequently, when I miraculously survived, doctor's orders excluded me from participation in boys' sports, of which, before my illness, I had been an avid enthusiast. I even dreamed of introducing rugby football to my school. Arbuckle, Curwood and Seton had already made me an Anglophile. Rugby had been played in Prague since the mid-nineteenth century but it never gained popularity. After World War Two, when I once went to see a championship game in Prague, there were more people in the field than in the stands. I was sitting there, the lonely man in the company of five or six suffering wives of the diehards who were giving each other bloody noses on the grass.\n\nThe rugby dream ended anyway, even without my pneumonia: I was unable to buy a rugby ball. They were not on the market.\n\nExcluded from the companionship of ball-kicking boys, I read. My father's home library provided me with quite a lot of translations because my father followed tradition. The Czech nation, or rather its language, was reborn on translations, mostly of English and American classics. The Czechs had lost the ancient sovereignty of their kingdom (in the same year that the Pilgrim Fathers landed in America), and with it\u2014almost\u2014its language, too. Reading those precious and very bad translations, I shook with horror with A. G. Pym on the hull of the capsized _Grampus_ ; I did things I never dared to do otherwise with Penrod; I sailed down the mystery of the Mississippi, through incomprehensible bloody feuds between the Grangerfords and the Shepherd-sons; I flew in a balloon over the Sahara desert with Tom Sawyer, and, yes, I also felt the unoppressive humidity of the tropical forest where Tarzan lived with his family of apes. My _education sentimentale._\n\nThanks to this abundance of translated novels I had no need of English, which was not taught at the gymnasium. There we had to master Latin, German and French, this last the \"language of diplomats.\" Before WWII this may have been true; English certainly wasn't the _lingua franca_ of the world, as it is nowadays.\n\nThen biology interfered. My glands began to do their disturbing work and I fell in love with an American boy. Actually he was British, which I again didn't know, for he lived in Hollywood, California. His name was Freddie Bartholomew and, at a Sunday afternoon matinee in the same cinema where I had seen Arbuckle from my mother's lap, I saw Freddie in _Little Lord Fauntleroy._ Many if not all young boys, I'm told, go through an early homosexual phase. Mine was strictly an affair of the soul, and anyway, I was saved from the then very sad fate of a gay person by an unquestionable female, also from Hollywood, named Judy Garland. I jilted Freddie for her after I'd heard her singing in the race-track drama _Thoroughbreds Don't Cry._\n\nThe platonic affair had a side effect; I decided to learn English so as to be able to write Judy a love-letter. Which at long last brings me to my theme.\n\nI acquired a little pocketbook entitled _Teach Yourself English_ and became deeply immersed in Judy's tongue. I was an only child in a relatively affluent family and my parents believed in easing the burden of school mass instruction by hiring private tutors for me. One day my French tutor, Mrs. Hlavackova, called on my mother with sad news: she, asserted the French lady, had never had such a stupid pupil as I was. (Only she said \"untalented.\") It would be a sorry waste of money to continue paying for my French lessons.\n\nMother was not pleased; however, she found my little English textbook, and instead of punishing me, she provided me with another private teacher, Miss Pokorna, to instruct me in English. She was a wise lady, my poor mother\u2014dead at fifty from high blood pressure, which I was diagnosed with at fifty and, at seventy-eight, I'm still here. She died in another one of those \"pre\" times.\n\nI can't resist a digression. The most beloved of my private language tutors was Mr. Neu, the cantor of the local synagogue, who taught me to speak an almost perfect German. His first name dated also from the \"pre\" days, for his parents innocently named him Adolf. He perished in Theresienstadt. Before he died, in the twilight days when Jews in my native town were still allowed the luxury of a \"Jewish Caf\u00e9,\" our German lessons changed into nostalgic meditations of the past. _\"Was wir Juden schon alles mitgemacht haben!\"_ sighed Mr. Neu, and then he would tell me about the days of World War One, which were also bad; there was very little food, but he would give private German lessons and refused to be paid in money, though he accepted _Zucker, Mehl, sogar Fleisch._ The German he taught was radically different from the one I used to hear later, in the Messerschmitt factory, or watching the weekly Ufa newsreels in the local cinema. The Big Boss of Czech schools in those days was one Inspektor Werner, whose method of inspection was to burst unexpectedly in on a teacher unprepared for the horror, listen for half an hour to his stuttering instruction, then attack him with language best characterized by the tubercular Mr. Pro-pilek, a teacher of Russian and Baltic languages (which, naturally, were not taught in the Protektorate). He once underwent the trial of the scowling brows of Inspektor Werner, then the explosion of his gutter diatribe. But when Inspektor Werner briefly stopped to catch his breath, Mr. Propilek made the memorable pronouncement of: _Ich lehre Goethes Deutsch, Herr Inspektor. Ich lehre nicht Schweindeutsch._ This, miraculously, made the bloodthirsty Nazi shut up. Inspektor Werner survived Mr. Neu by a mere couple of years. After the war's end they hanged him.\n\nMr. Propilek was later banned from teaching Russian by the Commies.\n\nNext to Mr. Neu in my affection was my gymnasium German instructor, Dr. Eva Althammer, a pretty young blonde, an _echt Deutsch_ with a quintessentially Aryan name. She, however, never joined the Nazis, loved Rilke, and married a man by the name of Svorcik. Thus she committed minor _Rassenschande,_ and because she refused to join the party and her husband wouldn't change his nationality to German, with an appropriate change of the orthography of his name, Inspektor Werner threatened her with Ravensbr\u00fcck. Werners threats were never to be taken lightly; some people paid with their life for such mistakes. So Mrs. Althammer-Svorcik turned to a friend of my fathers, a Czech doctor, who advised her to get pregnant. The Nazis, he correctly maintained, in their twisted race theory, wouldn't send an innocent unborn German baby to a concentration camp. Before Werners threats, she and her husband intended to wait for peace to have babies, but now she followed the doctors advice and survived. Many years later, when she became my first German translator, I gave her the only poem I wrote in German, when I was her admiring pupil in the Quinta\u2014a gross imitation of her beloved Rainer Maria Rilke:\n\n_Bald kommen Winterst\u00fcrme mit dem weissen Schneen_\n\n_Und langsam wird zum Koder alte liebe Pfad._\n\n_In meinem Herze kalte Winde wehen..._\n\nBack to my English beginnings. About a year after she'd hired Miss Pokorna, my mother enquired about my progress. Miss Pokorna with great enthusiasm assured her that I was her best student ever.\n\nMy English was a labor of love.\n\nBy the way, I managed to send the love-letter to Judy on December 2nd, 1941. By that time we no longer lived in Czechoslovakia but were second-rate citizens of the Protektorate B\u00f6hmen und M\u00e4hren in the German Reich. Mail to America, however, was still functioning; the States were not to join the war until a few days later. Chances are that Judy got my letter. In any case, she never answered, but soon I didn't mind. Local beauties were replacing Judy in my heart.\n\nAnd with Miss Pokorna I read my first English and American writers in the original tongue. My tutor was a graduate of a British boarding school attended when her father was stationed in London as a business representative of some sort, and so her English was real. Why her family had returned to Czechoslovakia a few days before it was annexed by the Reich I don't know. In those days very few people had an idea of the bottomless evil that was Hitler.\n\nShe had a good private library of English and American authors, and after I quickly mastered basic grammar, she made me read Shaw, O'Neill, Oscar Wilde, Kipling, Mark Twain and O. Henry.\n\nI was sixteen, seventeen, the author already of several unfinished early novels about the glorious career of a Czech saxophonist in Hollywood nightclubs, and the author also of a more mature and finished (though unpublished) novel called _An Inferiority Complex._ It featured the heroines of my later novels Irena and Marie, and also the Kostelec jazz band.\n\nLike every young person, I avidly read poems: the poets of the Czech \"poetism\" movement, Karel Capek's unique translations of modern French poetry. Miss Pokorna lent me a volume of T. S. Eliot. The war was going badly for the Reich; that meant it was going well for us involuntary citizens, and I was drafted into the _Totaleinsatz_ in a Messerschmitt subsidiary which manufactured fighter planes and Stukas. There, against the background noises of drills and pneumatic hammers, I recited to myself:\n\n_Because I do not hope to turn again_\n\n_Because I do not_\n\n_hope Because I do not hope to turn_\n\nI do not pretend that I understood the meaning of Eliot's verses. But they had a quality of magic incantation\u2014probably my first touch of the magic of language.\n\nThe war ended, and in a bookstore in Prague I bought a copy of Hemingways _A Farewell to Arms,_ published in Sweden but in English. I read it and I understood what Josef Hora, a Czech poet, had meant when, long before M\u00e1rquez, he wrote about magic realism in prose. Unlike the Spanish-American leftist, what Hora meant was not introducing supernatural elements in prose, or falsifying historical data to suit one's ideological purpose, but the novelist's duty to take infinitesimal care of each and every word, the way a lyrical poet does.\n\nReading Hemingway's story of Catherine and Fred, I saw that this was precisely the way Hemingway wrote: the way novels should be written. Henceforth I tried to follow the advice of the poet and the example of the novelist. Many, many years later a Czech literary critic, Premysl Blazicek, very kindly said about my novel _The Bride of Texas_ that each sentence in that book was perfect.\n\nWhether he was right or not, I don't dare to judge. What is certain, however, is that no American critic would say that, because they don't know my writings. I mean: they don't know how these books are written since they don't read the obscure language of the westernmost Slavs.\n\nMy inner language, since the murky days of sickly adolescence, was English. I even said my evening prayers in that foreign tongue, and I voraciously read books from Miss Pokorna's private library. Years later, as a student of philosophy, I belonged to the privileged few in Stalin's Reich who could get almost all important modern works in the original language from the seminar library of the English department. At the same time, the more I immersed myself in these books, the more I ignored Czech literature. In that respect I probably could be a Guinness laureate. When, at long last, my novel about the Czech Communist Army appeared in America (it was banned in Czechoslovakia, not published there until after the fall of the Evil Empire) and American reviewers made the obvious comparisons to _The Good Soldier Schweik,_ I still hadn't read the notorious classic.\n\nAmericans are linguistically very tolerant, very nice. I was often congratulated on my very good or even excellent English. On each such occasion I grinned politely because I knew only too well that it was just American politeness. English was still my very limited inner language, grammatically more or less correct, but idiomatic? \"Do you now write in English?\" was the usual follow-up question. Only articles, I would answer. For writing articles you don't have to be at home in a language. I never dared to say \"essays,\" always \"articles,\" because I painfully felt that for that noble genre my English inner self was lamentably inadequate.\n\nI often remembered the kind Miss Pokorna, long after she got married to a theatrical producer in Prague, long after I started a new and different life in Canada. In her assessment of my linguistic talent, either she was wrong or the English proficiency standards in my native town were comfortably low. True, I learned English rather quickly\u2014to a degree. A degree that enabled me to read books (with and soon without a pocket dictionary) and to discuss them intelligently. Yes, to discuss literature, and that came in handy when I found myself teaching literary courses at the University of Toronto. My abilities in non-literary conversations, such as talking to people in eateries and bars, were, and remained, restricted. In short, I reached some kind of limit, a barrier, a boundary, and I was never able to surmount the obstacle.\n\nI read Conrad because he must have faced a similar problem when, as an adult, he had to learn a foreign language in which he later so incomparably excelled. Graham Greene always maintained that Conrad was the best English stylist of the twentieth century. I read and re-read _Heart of Darkness,_ covered with sweat while stumbling through the richness of his vocabulary, awed by the music of the dark sentences. Then, from Ford Madox Ford's memoir, I learned that Conrad's _spoken_ English was far from perfect, perhaps not even very good. But what help was it to me? Conrad _did_ write like a demiurge. I didn't. Certainly not in my acquired language. In my native language?\n\nA strange thing and Henry Miller got it right. Surrounded by the sounds of the foreign language\u2014speaking, on a daily basis, my very good English, as friends kindly assured me\u2014my eyes, my ears, my inner receptive organs became attuned to Czech to a much higher degree of precision than back in Bohemia. I awakened to aspects of my mother tongue of which formerly I was unaware, having used them subconsciously, mechanically. The sex appeal of feminine endings, the lure of verbal aspects, the capricious scherzos of prefixes, such things.\n\nNevertheless, I continued to read in English. The magic of Faulkner, the biting manner of Waugh, Chandler's vistas of nature and street, the beauty of the lingo, Hemingway's early hypnotic brevity. Such things. And I wrote.\n\nDid I imitate? Perhaps. If so, only in the sense of _Imitatio Christi,_ with language as my religion, as my home. That's why I was never nostalgic. The old country, with its fifty years of nonsensical but cruel dictatorships, was not my home anymore. The language was. Czech. The language of my mother, of my writing.\n\nAnd yet, prodded by those well-meaning friends, I once tried to cross the barrier and wrote a novel in English. It was a joy to feel the moulding of sentences, of dialogues, of descriptions in an acquired language, with here and there some peach of an idiom, overheard in a pub or maybe suggested by Marlowe\u2014the detective, not the murdered playwright. The joy, as it were, of being an additional human being, because writing in an additional tongue. That joy.\n\nHowever, it didn't last long. At my publishers, editor girls went to work and when the book was out, they didn't earn much praise. Something is lost, the reviewers decided. When he writes in English, something is missing. The Genius of Language, perhaps?\n\nWere they right? Don't ask me. What those kind editor girls earned was my admiration, mixed, however, with the bitter feeling of defeat. The girls didn't correct many grammatical errors; they just made my English English. In their hands my vocabulary blossomed to an almost Conradian opulence. Yet these were blossoms created by those girls' hands, not mine.\n\nThe reviewers never read my novels as I had written them in my \"small\"\u2014for most American critics, even \"obscure\"\u2014language. They read only translations. And I thought of my early days, of the Sinclairs and Dickenses and Dreisers, not to speak of Cur-woods and Setons and Edgar Rice Burroughses, all of them enjoyed in dubious\u2014no: bad\u2014no: horrible translations, translations really insulting to sensitive speakers of the obscure language of the westernmost Slavs. And I wondered. What made me enjoy Mr. Babbitt, who constantly used the second person plural in addressing his children, his wife, his closest friends? What made me ignore the shocking impoliteness of characters who addressed their physicians with the disrespectful \"Doctor,\" not \"Mr. Doctor\"? What made me so imperceptive of the twisted sentences that slavishly followed the word order of the originals? Did they sound alluringly exotic? Sweetly foreign? What made me not wonder about a military band in Thackeray whose bulky musician played very loudly on the dulcimer? All that?\n\nSurely, there was nothing resembling genius in the language of those translations. Yet the novels spoke to me, with great intensity. So strongly that they decided my future. True, there was Eliot, whom I first read in English, then years later in a supposedly good Czech translation, who, after the true magic of \"Because I do not hope...,\" was almost torture. Was it Josef Hora's labor devoted to each word which the translator, paid by that word, obviously neglected? Something else?\n\nFor a decade of my life, when my own efforts were banned, I turned translator myself. The experience taught me to appreciate my excellent translators in Canada. I bitterly learned what it was to cleanse your text of the abundance of auxiliary verbs so foreign to Czech, of the prevalence of the passive voice, of possessive pronouns used with parts of the human body, all these and other translators' errors which so uglified the American magic of Faulkner, the British acrimony of Waugh, the translu-cency of Hemingways diction. Would any American monolingual or even\u2014in the major languages\u2014bilingual or trilingual reviewer dare to say what the late Czech critic said about my sentences?\n\nNo, and it wouldn't be their fault. Although they were unaware of my originals, their reviews were rarely scathing, often favorable. What about language, then? What is it that makes even books that present only a ragged shadow of their model enjoyable, even enthralling? What makes a teenager in a landlocked little country ruled and butchered by foreign invaders and mighty Big Daddies enter the skin of an illiterate boy from Missouri, of a nigger slave\u2014enter a world as far away as the stars?\n\nYes, language can be of supreme beauty. But there is more to works of fiction than just language. Style in Chandlers sense, the experience of Dickens but also that of Henry James, of life's martyrdom or of life's sweet mellowness, and many other things.\n\nLet's leave it to the horses, they have bigger heads.\n\nOr perhaps to the elephants.\n\n# DUTCH\n\n#\n\n# _Circus Biped_  \nBert Keizer\n\n_Die Grenze meiner Sprache sind die Grenze meiner Welt,_ said Wittgenstein. The limits of my language are the limits of my world. You would almost leap to: my language is my world. An aphasiac's nightmare is the waking-up after a stroke to discover that everybody speaks Chinese. This is the worst possible way of being catapulted into a foreign language, because you find yourself out of a world in a flash.\n\nMy journey into English was luckily a much more gradual affair, which started with the trying-on of funny little hats while in the back of my mind there already hovered a mirage of me striding in full regalia down the path toward\u2014oh, I don't know, a beach in California, an Oxford quadrangle, a glen in Scotland, or a pub in Dublin\u2014obvious places, anyway, which in fact I never got to.\n\nThe first word was \"YES,\" encountered in a boys' book about cowboys and Indians. White Feather was the stony Indian and Eagle Eye the impetuous cowboy. We pronounced it \"coyboy\" for some reason, though there was nothing coy in sight, I can assure you. Neither did we connect \"cow\" to anything bovine, because it is impossible to imagine a daredevil in full gallop swinging a lasso in swirls of dust, amidst excited cries, next to the proverbial sluggishness of a Dutch cow in a dreamy meadow. The poor dears would be very upset at such an unnecessary display of sound and fury, so deeply foreign to their drowsy domain.\n\nBut White Feathers YES was emblematic as my first English utterance, charged with sturdy manliness, an inborn determination to remain unruffled (though I am not sure about how far the waves of English commingle here, for it may well be the case that I confuse several avenues down which this language came washing over me). White Feather and Eagle Eye\u2014in Dutch: Witte Veder en Arendsoog\u2014were probably assembled by their author, J. P. Nowee (who from his own experience couldn't tell a cowboy from a parking space), out of the motley crowd that fought its way onto the Normandy beaches in 1944, bringing cigarettes, gum, and a way with the girls, from which assets he picked the boy-ingredients and mixed them up with what he had gathered from the movies about prairies.\n\nYES!\n\nI didn't know how to pronounce it, because I had never seen the letter Y, and inwardly mumbled something like \"aye.\"\n\nIn 1952, English to me meant soldiers marching merrily down the road, not your biting Nazi maniacs, but free spirits in good order, always willing to break the ranks for a laugh or a drink, and carrying hardly any bullets in their rifles. \"The Yellow Rose of Texas,\" with the militaristic undertow of the snare drum, was the song for me. Or the melancholy sound of \"Tom Dooley.\" \"Hang on your hat, Tom Dooley\" is what I heard, because that is closest to the Dutch idiom of placing your hat on a hook. The next phrase sounded to me like \"Good boy you're boy goodbye,\" and on these words the picture arose of a good boy who took off his hat and was sung to by an admiring circle of old wise men. Then Tennessee Ernie Ford's \"I owe my soul to the company store,\" from which I gathered that if you lived long enough in an American wood, you would turn into something like this poor brute who could talk to the world no more, only hammer blows on it.\n\nAll this hale-and-heartiness was a far cry from the surly male who next made his entry, and this summing-up inadvertently takes on the obvious hue of my own emerging maleness, which must have traveled down that road from fairy-tale cowboy through adolescent sulk to youthful doubt, ending up in an instant of unquivering certainty (which we'll talk about some other time), and from there on down the relentless slope, struggling fiercely\u2014if only to instill the idea in the audience that I am fighting my way down, not just sliding.\n\nI forgot to mention an all-pervasive and therefore hardly noticeable aspect of all this: the fact that a Dutch boy in the second half of the twentieth century should build up a considerable part of his bulwark against the world using English bricks. Allied bricks.\n\nI even imagined, in that stage when I was feeling myself into the English language, that I could dream up the English equivalent of Dutch words by mentally staring at their essence and transmuting that in the furnace of my inner appreciation into the English expression I was looking for. Somewhere in my mental attic there still subsist the remains of these verbal phantoms, which at the time felt to me like proper English words. This effort at linguistic alchemy, throwing in my Dutch iron in order to extract a nugget of English gold, was not entirely silly, because many Dutch words are entangled with English words\u2014and I am not talking deep linguistics or sophisticated etymology here. What I mean are simple look-alikes or sound-alikes, which were as enticing as they were confusing to a boy not even vaguely in command of words. Consider \"girl\" and the Dutch equivalent, \"meisje.\" Now, these two words would never run into each other in my mind, but \"maid\" and \"meid\" are almost twins in appearance and sound, though wholly different in meaning: \"meid\" denoting (outside the household) a gal rather than a girl, and gals being more fun, so much fun, even, that a certain boundary may be crossed beyond which a \"meid\" becomes a hussy.\n\nAll this fumbling with English words and sounds is pre-literate and was mainly brought home to me through songs, for I rarely heard English spoken. I cannot remember one word from the Roy Rogers movies we saw, and I only recently discovered that his horse was named Trigger, when I saw a documentary about the famous cowboy in which he proudly sat next to his beloved horse, now in stuffed condition\u2014a bizarre sight that made me uncomfortable for Triggers sake.\n\nI was raised in a Dutch Roman Catholic family, my mother a farmers daughter, my father the son of a village painter, the two of them setting up house in a small provincial town in the 1930s. We grew up under the awning of the last vestiges of medieval Christendom. But that's hindsight. Well into the mid-fifties, we were comfortably stuck in a homely nineteenth-century version of the Middle Ages. Dutch Roman Catholicism had retreated into an alley, far away from the busy traffic rushing past on the intellectual highways of Europe. Wittgenstein had died already, Beckett was approaching Nobel stature, but in our parish the Trinity remained an inscrutable Mystery.\n\nWe had no idea we lay under siege. What would have made us think that, anyway? Perry Como didn't sound like much of a threat. Every little statue in our parish church stayed exactly where it was when he sang. I never caught on to pre-Army Elvis, and only realized he was around in his chubby phase. But then, in 1963, the Beatles came along, and under the spell of that sound we forgot all about statues and strolled out of the church with a laugh, being dealt a smart kick in the ass by the Rolling Stones on the threshold, just to make sure. We left our parents sitting there, and for all I know that is where they still linger.\n\nDutchmen of my generation like to think that they fought themselves free of the clutches of the Church in heroic fashion. As if they entered into a fierce exchange with their elders\u2014nay, with God Himself, preferably\u2014from which, surprisingly, they emerged triumphant. Now, this is more or less how things would have gone if they had woken up in the thirties, where ostracism would have been the reward of anyone dropping out of the sacred community; but in 1963 it was all grassy meadows, \"an endless breaking of the bank,\" that was beckoning us.\n\nThe Beatles pulled the rug from under that particular brand of unsmiling masculinity, the D-Day heroes who kept their cigarettes going under enemy fire and never stopped chewing their gum, even while burning thousands of innocent people with their carpet-bombings in Germany. \"It was a terrible job but it had to be done.\" _You can stuff your terrible jobs_ is what the Beatles said.\n\nI may have made a hash of Tom Dooley and the Company Store, and only sensed a sunny Sundays contentment in Magic Moments, but the Beatles' lyrics I actually understood. \"I want to hold your hand\" is not exactly unfathomable, though in \"Love Me Do\" I initially mistook \"Do\" for a girl's name. It was the Beatles' English that ushered me into the language proper. Before they sang it to me, English was atmosphere, not language as a charged verbal message. From then on, the possibility lurked for me that one day I might say \"I\" in English.\n\nWhen I moved to England in 1968, the last strains of Vera Lynn were still in the air\u2014something you wouldn't think when listening to the Stones' \"Jumpin' Jack Flash,\" which came out that summer (\"I was drowned with a spike right through my head\"). But that was the inexplicable charm of the place in those days. The Second World War was really over, a mere mirage now over the white cliffs of Dover; the last GIs were finally out of the way; and an entirely new way of being young was invented on the spot (I know, but that's how it felt), seemingly out of nothing, and outrageously whipped into a more daring frenzy by the kids of those same GIs, who were about to be slaughtered in Vietnam, which lacked all Normandy glamor.\n\nThe first thing that struck me when I moved to England was the fact that youngsters there didn't understand much of Bob Dylan's lyrics either. I reluctantly abandoned my theory that \"Like a Rolling Stone\" was a tribute to Mick Jagger's way of life, but didn't get much in exchange. I suppose Dylan's texts are still a bit of a problem: shallow nonsense or sheer genius stuff.\n\nThough my brother told me not to (it would spoil everything, especially my \"freedom\"), I fell, after or while asking many questions, into the arms of the first girl who talked to me for longer than five minutes. She came from London, she lived in Cornwall, she was doing her A-levels. I was a waiter in the local hotel and spouted English by the mile, sounding awfully American\u2014a remnant of the Continents debt to our liberators, an accent which was not exactly an endearing asset in those days, because of Vietnam. My surname, which was always pronounced in the \"Kaiser Bill\" manner, was kindly pardoned as betraying my \"Austrian\" descent, because nobody wanted to besmear me with any German connections, and to the blissfully ignorant people I moved amongst, \"Austrian\" meant \"German, but all right really.\" To certain English people, it did seem unlikely that there was such a thing as a Dutch language at all. Once, while having tea in the house of a vicar, father of a friend, I was asked by the wife in the sweetest tones imaginable, \"I wonder, do you Dutch have a language?\"\u2014by which she sought my confirmation of her opinion that probably we merely spoke a ragbag of dialects, assembled over the years from passing marauders such as Romans, Celts, Frisians, Vikings, Franks, and Saxons, out of whose verbal droppings we, the local monkeys, somehow fabricated what we took to be a language. This made me so angry that I practically hissed at her: \"No, ma'am, we do not have a language, we just bark at each other from the trees!\" My friend hastened me into the garden to cool down while his father attended to Mummy, who was quite taken aback by my \"vicious and entirely uncalled for snarl.\"\n\nMy girlfriend led me firmly out of sundry other confusions (don't ever try to work out the difference between the _ough_ in _plough, rough,_ and _thorough,_ but do pronounce them differently), reminding me of the old joke that in English words are not written as they are spoken\u2014unlike the case in everyone's native language. I never quite mastered the \"th\" and sounded unbearably silly while singing along with Mary Hopkins song, \"Dose were de days, my friend.\" Though I didn't shrink into a mere figure of fun, I certainly became ridiculous in unsought ways by having to plod along in this borrowed garb. Anyone who is not fluent in a strange language sheds about 30 to 40 IQ points; that's quite a dive, which few intellects will be able to sustain without some damage to their underlying ego. Being surrounded by benevolent English speakers, I was quickly helped back on my feet, only to fall into the next trap: idolatry.\n\nIt takes a few years to get well acquainted with the clich\u00e9s in a language, and the list can never be exhaustive, but there is a stage where you are innocent of the difference between a worn-out phrase and a gem of personal expression. So at one stage I found sayings such as _looking like death warmed over, green around the gills, get stuffed, kicked the bucket, deaf as a post,_ and _blind as a bat_ a delight, until I found them out as the dead doornails they really are.\n\nOne thing I picked up very quickly was the possibility of positioning people socially by listening to their language. Gutsy, arrogant, boaster; working-class-and-doomed, working-class-and-angry working-class-and-on-the-way-up; hardly, reasonably, highly, and over-educated; arty type, peasant\u2014the millions of possibilities according to which we stack people socially, using our judgment or wallowing in prejudice\u2014all that is soon learned in any European language. Mainly, I suspect, because our own tribe consists of similar strata. I mention this because, during my stay as a doctor in Africa, I had to go without all these pointers and could only tell a peasant from a genius by using clothes, body care, and face, which often pointed in the wrong direction.\n\nI cannot remember any half-reading in English, the way I now do in Chateaubriand's _M\u00e9moires d'outre-tombe,_ looking up words, pondering phrases, and sometimes not getting anywhere. But I still have my copy of _Ulysses,_ read in 1970, my second year in England, and there I have rather annoyingly marked the words I didn't understand. This is what I was stuck with after the first fourteen pages: _threadbare, fretted, breeks, skivvy's, whinge, lunged, rashers, gaud, mosey, flagged, barbicans, prepuces, kine, crone, upbraid_ \u2014and, if you want to know, I am still, or again, stuck with \"mosey\" and \"barbicans.\" Then I reached \"Agenbite of inwit,\" which I understood so well that I grew suspicious: surely it cannot mean THAT!\n\nBut it did.\n\nThere are a number of \"barbicans\" which I keep looking up and noting down, only to forget them again. Whenever I try to unmask them, I first note down what I think their true face is, before tracing them in the dictionary, and by this method I have arrived at a number of \"solutions\" which, in intention, surely must stem from those earlier alchemistic efforts in which I would divine the English equivalent to certain Dutch words.\n\nHere's a short list of intuitive lexicography:\n\n_propinquish_ \u2014propose in halting manner.\n\n_barbican_ \u2014wooden appendix protruding from ceiling.\n\n_obstreperous_ \u2014striped in obnoxious fashion.\n\n_inchoate_ \u2014darkly unsuited.\n\n_anodyne_ \u2014biting liquid used for facial cleansing after shaving incident. _maverick_ \u2014unusual in old Balkan way.\n\n_uxorious_ \u2014forcefully nagging.\n\n_enjoined_ \u2014merged unknowingly.\n\n_sedulous_ \u2014sable covered.\n\n_spinnaker_ \u2014dizzy sailor.\n\n_emunctory_ \u2014oozing orifice.\n\n_arcane_ (as in \"arcane knowledge\")\u2014ancient lore, whispered beneath arches or under arc-lights.\n\nLooking at this list, I am struck by a difference between Dutch and English which is never very apparent to English-speakers: the extent to which their language is _not_ German\u2014I mean Teutonic\u2014I mean stemming from those dense forests between the North Sea coast in the west and, say, the outer reaches of Poland in the east, in the time before the Romans brought the tribes there to heel. In the above list, I think only \"spinnaker\" can be traced to the forests; the rest crossed the Channel with William the Conqueror, hailing ultimately from Rome and Greece. There is a funny difference between the way an Englishman uses words like _anthropology, psychology, democracy, orifice, noxious,_ etc.\u2014as if they are his own\u2014and the way a Dutchman deals with his Graecisms and Latinisms, such as _democratie, extreem, theocratie, pieteit, psychologie, notaris, reparatie, bibliothecaris, secretaris, amfibie, mobiel, ornithologie,_ etc. In Dutch these are clearly funny birds, wholly unlike \"schuur\" (barn); but in English they have, for historical reasons, sunk deeper into common parlance, so that these Mediterranean borrowings are now being paraded as the only possible and at the same time wholly English word. For instance, \"History\" (\"Historie\" in Dutch). We have, in addition to the fob from down south, the real thing (\"Geschiedenis\") at our disposal, but a false sense of cosmopolitanism adheres to the English \"History\" and is supposedly lacking in the Dutch \"Geschiedenis.\" I think that this kind of misconception may be an additional factor accounting for the silly veneration with which the English language is often approached in my country.\n\nMy first proper read in English was Russell's _History of Western Philosophy._ For much of 1969 I was doing the washing-up in the kitchen of an old castle in Devon\u2014Dartington Hall\u2014 which once belonged to Henry the Eighth, and in which, amongst other things, a College of Art was established. It was an unlikely stop on the way to university, but there I found myself doing the dishes while my girlfriend was attending the dance & drama course. In my vain efforts to scale the rocky heights of philosophy, I had brought Karl Jaspers' book on Kant with me in Dutch translation. And it got me nowhere. I couldn't follow it and felt repulsed. I started to nurse rather grim feelings towards that impenetrable fortress of philosophy, within the walls of which I hoped people were studying man's ultimate questions, possibly even resolving them, but so far without letting me in.\n\nI couldn't find an entrance anywhere and was about to go into a sulk when one day, in the family library at Dartington, I stumbled onto Russell's _History of Western Philosophy._ From the sombre citadel I had dreamt up, this sprightly man came trotting out to me, and with one wave of his hand dispelled all my sulks and misgivings. His boyish charm, his malicious humor, and above all his keen sharpness swept me along on an unforgettable journey through twenty-five centuries of Western thinking. He was such a delight to read because he personally went out into the field with, or against, the philosophers he described. I had never read anyone who was so impressively knowledgeable, authoritative, and funny at the same time. Some quotes: \"Erasmus was incurably and unashamedly literary.\" On Machiavelli: \"It is the custom to be shocked by him.\" On Spinoza: \"Intellectually, some others have surpassed him, but ethically he is supreme. As a natural consequence, he was considered during his lifetime, and for a century after his death, a man of appalling wickedness.\" On Hume and Rousseau: \"Rousseau was mad but influential, Hume was sane but had no followers.\"\n\nMuch later I learned that Russell's _History_ sadly lapses when he gets to the nineteenth century, but that is not relevant here. For me, as I was at that time, lingering in half-sulk in front of that Venerable Temple of Western Wisdom, my greatest luck was to be greeted by Bertrand Russell at the entrance. He is a witty and brilliant host, and even if not all those at the party get the attention they deserve, his company was an overwhelming compensation. I still cherish my copy with the pencil portrait by Robin Guthrie on the jacket.\n\nUp till then I had never encountered in Dutch a man who was so clever, learned, and amusing. I hate to say this, because I don't want to run my country down in any way, but I have to mention it here as stealthily as I can: there's an awful lot of intensely dull writing going on in Dutch academia, and it has been going on there for decades on end, and will go on for decades to come, this time in English, I'm afraid\u2014almost a guarantee of drabness, in view of the local command of the venerated tongue.\n\nIn Holland you would, for instance, find it hard to encounter the likes of Gilbert Murray, J. B. Bury, Francis Cornford, Maurice Bowra, H. D. F. Kitto, Moses Finley, Benjamin Jowett, W. K. C. Guthrie\u2014to mention only the names that readily come to my mind, with their brilliant commentaries and wonderful translations from classical history and philosophy. Reading Plato in Dutch in the sixties, I would have had to hack my way through a dense undergrowth of grammatically correct but hopelessly lifeless prose, emanating inevitably from the boring provincialism of the translators, and leaving the reader with the wrappings of a mummy. But reading Plato in English, I was immediately swept along. The insight, the clothes, the flirting, the markets, the courtyards, the personalities, the jokes\u2014in English, the whole thing sprang to life inescapably. Plato's _Republic,_ which I read in the summer of 1969, has forever remained one of my happiest encounters on paper.\n\nI will not bore you with the ins and outs of the Dutch educational system. Be it said, however, that it lacks the possibility which was so gloriously rampant in the English public school system, of maintaining a broad and deep connection with the ancients. In Holland, and even more so, of course, in the intellectually humble milieu I stem from (I may be confusing the two), Plato was sadly entombed in the dusty glass cupboard of the Past.\n\nBy 1970 my familiarity with English had far outgrown anything I had ever achieved in German or French. In some areas, I even think that my English was better than my Dutch\u2014mainly in matters intellectual. There was a stage where the intrusion went so far that whenever I dreamt about my family back in Holland, they spoke English to me and each other. I never thought the parrot could travel so deeply into a man's soul. My spoken English was virtually accentless then, though people with a keen ear did suspect I might have something to do with South Africa. (An accent I did not know at the time. When I did get to know it, I thought someone was having me on, because to a Dutchman a South African sounds like a Dutchman who is deliberately refusing to pick up the right accent\u2014somewhat in the way the French and English often diligently abuse each others parlance by wilfully contorting the other nations speech.)\n\nSo I waded\u2014or swam, rather\u2014deeply into English, and read Kant, Schopenhauer, Nietzsche and, most important to me, Wittgensteins _Tractatus_ and _Philosophical Investigations_ in English translation, though I could easily have handled these works (language-wise) in the original German. As a consequence, I made the common mistake of thinking about Wittgenstein as a British philosopher with a vague Viennese past, which accounted for his annual trip there around Christmas. This is, of course, a silly view of Wittgenstein, whom I later came to know as primarily a Viennese thinker, one who only halfheartedly took up his sojourn in Britain, as so many other Viennese were forced to do in the thirties, if they were Jewish.\n\nReading those German philosophers in English was a bit daft, as it misled me into a wrongheaded assessment of some aspects of their thinking. Luckily, quite the reverse happened when I read Proust in Scott Moncrieff s translation. I may have missed out on bits of the Germans in English, but I would have missed out on lots of Proust in Dutch, had there been a translation around at the time. There is quite an enormous societal overlap between late-nineteenth-century French and English ways of being artistic, a snob, a litterateur, a dandy, a Jockey Club member, a society hostess, a scion of a noble house, a maid-servant, or a hopelessly middle-class riser. All these are ways of being which certainly had their vague equivalents in our parts, but not with the added dash of being relevant in the international political and cultural arena of the time. I mean, you could take a coach from Wildes Bedford Square to a soir\u00e9e at the Duchesse de Guer-mantes'. No such clear connection existed between Amsterdam, on the one hand, and Paris and London, on the other. It is unimaginable for any Dutch author to be met in Amiens the way Ruskin was encountered there by Proust. This is not a reflection on Dutch authors of the period, but rather an expression of the international situation at the time. How many Lithuanian, Latvian, Finnish or Basque novelists are internationally coming to the fore at present?\n\nIn 1972, after having graduated in philosophy, I returned to Amsterdam to take up medicine. I had been away in England for almost five years, and soon after my arrival, to my utter dismay, I sat through the spectacle of a medical professor solemnly writing out in English, on the blackboard in the lecture theater, the reasons for which general practitioners refer patients to consultants in hospitals:\n\n_because we do not know._\n\n_because we cannot do._\n\n_because we need support._\n\n\u2014giving the final touch of imbecility by the gnarled way in which he pronounced these three simple phrases.\n\nIn Holland, and in many other parts of the world, the type of idolatry I was talking about earlier is one of the most repulsive effects of the fact that English is now lording it globally. In my own profession, there is a lamentable inclination to use English phrases when talking about matters that can be described perfectly well in our own language. What to think of a designation such as \"the blue toe syndrome,\" used to refer to a patient with a circulatory problem? Doctors have dropped Latin but now seize on English phrases in which to wrap up the rather humdrum contents of many of their concepts. \"Irritable bowel syndrome\" is a standard diagnosis which can easily be phrased in Dutch, but then it loses that clinical polish which keeps the patient at bay. The doctor needs these fancy phrases to protect his domain.\n\nComputer idiom is inevitably adapted\u2014 _download, boot start, surf, chat, online, mail, hard disk, update, delete, crash, e-mailing,_ etc.\u2014but the English is subsequently maltreated when the verbs are being conjugated. For \"downloaded\" you get \"ge-download\" (don't bother to pronounce it) or \"downgeload.\" For \"surfed,\" past tense, you obtain \"surfte.\" \"Deleted\" becomes \"ge-deleted,\" I guess. And \"crashed\" turns into \"crashte.\" How would you like \"downloadized\" or \"surfetted\" or \"croshed\" or \"be-mailed\" or \"chatten\" as conjugations?\n\nThese are ugly results. Uglier still are the many academic Ph.D. theses written in Holland in English. I don't think it really matters when you are dealing with atoms, bridges, teeth, arteries, or gamma rays, but when you are writing about people and ought to throw in a little of your own personality in order to infuse some life into the thing, the handicap of having to do this in English is severely debilitating. People rarely realize this and therefore tend to use English as if it were a dead language, like Esperanto, with an equally lifeless outcome.\n\nBut ugliest of all are the scenes in which Dutchmen think they can speak English fluently, often lured on by repeated assurances of native speakers (who will say, \"But your English is marvelous!\" \u2014a thing they wouldn't dream of saying to one of their countrymen). Thus I once witnessed, in agony, a colleague of mine being slaughtered on a BBC program. He had walked into the studio, in all innocence, for a frank but fair exchange on the (to him) crystal-clear subject of euthanasia. Ludovic Kennedy was there to help him out, Michael Ignatieff was in the chair, and the opposition consisted of Dame Ciceley Saunders\u2014the Holy Virgin of the Hospice Movement\u2014and two supercilious British neurologists who, talking down from incredible heights of arrogance, explained to my colleague that he was killing off his patients in his ignorance of proper medical treatment. Now, the poor man's English was not so bad that he couldn't say, \"The coronary arteries are on the surface of the heart\" or \"My uncle Dick was a butcher,\" but to fight off these two, he needed really to speak the language, to live an emotion in words\u2014words which now utterly failed him. Overwhelmed by righteous indignation but blocked by his lack of English, he was reduced to a spluttering heap. It was a horrible lesson about the emptiness of knowing a few words and phrases and about the fullness of a spoken language. There is a vast difference between showing someone the way to the railway station in English and showing him the way to Plato. This is often overlooked by city-map speakers.\n\nLiving in Holland, I encounter many people who speak a little English, mostly of too dull a variety to sharpen my own English on. So I only run into proper English on paper or on television or in the cinema. As a consequence, my passive English is still all right, but my active English barely keeps in shape, and I fear it is not getting any more vigorous.\n\nWriting in English at first felt to me like trying to plough a stretch of marble: an ungainly procedure, ruining some pretty nice material, and the result was nil. I feel reasonably comfortable now writing in English\u2014though please note that is something I would never say about writing in Dutch. Why not? Well, it's the difference between a natural biped (man) and a circus biped (dog). You wouldn't ever say to a human that you admire the way he manages so well on two legs, while a dog is applauded for just this feat. The dream of a foreign writer using English is that the natives will forget about his dogginess and say to each other: I just love the way he moves.\n\nBut, comfortable or not, I still have to shrug off a slight resentment at having to put on these funny clothes in order to be let in. I suppose that I could counter this by pitying you for missing out on certain Dutch authors whose virtues I couldn't begin to try to expound to you\u2014no more than I could give someone an idea of Jimi Hendrix's guitar-playing by whistling a few notes. Though I wouldn't argue absolutely against this possibility, the fact is that I cannot do it right here.\n\nThere remain, however, many areas where I do not know my way around in English, and I am not only thinking of barbicans and obstreperous arcaneness, but terms of carpentry, for instance, or automobile parts, or shipping terms, or bird names. I will always confuse grebes, sparrows, thrushes, curlews, and snipes. I am, in English, strictly a pigeon\u2013blackbird\u2013duck man (but only ornithologically, he added hastily).\n\nI am, to put it briefly, not a native speaker, and I don't mind.\n\n# FRENCH\n\n#\n\n# _French Without Tears_  \nLuc Sante\n\nMy parents and I emigrated to the United States from the French-speaking part of Belgium when I was a child. The move was made for pressing economic reasons and was lamented by my parents from the start; only intermittently did they have the leisure and lightheartedness to plunge into the adventure of their new surroundings. My mother spoke no English at first; my father relied on a weak memory of the language from his secondary studies, and he tended to mix it up with the more vigorous strain of German in him, acquired from growing up in a town scant miles from the linguistic frontier. In America my parents had few French-speaking acquaintances. The isolation was hardest on my mother, who was uneducated (both my parents left school in their middle teens), came from a particularly restricted and provincial background, and stayed at home, while my father, more cosmopolitan by nature, at least had the opportunity to mix with Americans and immigrants from other countries during his working hours. My mother therefore seized upon any and all instances of French in American life. A French-derived surname spotted on a signboard could cheer her up for an hour; a drive with my parents would be punctuated by my mother happily reading aloud from the roadside: _Chez Pierre!_ _Maison de Beaut\u00e9!_ When I watched cartoons on Saturday mornings, the whole family would gather for Pepe le Pew, the Gallic skunk forever making romantic advances to horrified black and white cats: _L'arnour, toujours_ **_I_** _'amour..._\n\nDuring my first year of school in America, my mother drilled me in French for an hour every day when I got home. That was shrewd of her; at first I was so discombobulated by the shift in languages between home and school that for an hour or so on either side I was effectively unlanguaged, nearly aphasic. The drills were as effective at getting me back to French as schoolyard peer pressure was in forcing English upon me in the morning. After a while I could slide between languages with relative ease, and when my mother and I returned to Belgium for sojourns lengthy enough that I was sent to school there, I engaged the curriculum as if I had never been away at all. Those trips, made when I was not quite eight and not quite nine, respectively, marked a significant shift in our lives. Previously, my parents had maintained the hope that our stay in America was to be temporary, but when my maternal grandparents sickened and died, which made those trips necessary, an important link was severed. My fathers parents were already long dead, and there was not much immediate family left. We were on our own, and might as well stay where opportunities grew densest. This decision did not improve the morale of the household. Thereafter my parents would try to maintain a semblance of Belgium in our home, but the enthusiasm was gone, and the simulacrum shifted, steadily if invisibly, away from its model. In the same way, the family language was progressively mongrelized. While keeping the pronunciation and syntax of French it became _franglais._\n\nFor me the French language very nearly became detached from its base, like so many of our household customs, which had lost their connection to any wider world and hovered in a vacuum, fetishes that might as well have been invented by my parents to keep me alienated from my peers. But I had a fortuitous link to the world of francophone children: my fathers sister and her husband, small-town newsagents, subscribed me to my favorite Belgian comic magazine. I read _Spirou_ every week for ten years, and through it subcutaneously absorbed not just the living language but also a sense of daily life in a Belgium that was then changing much more rapidly than my parents realized. The comic weeklies (the others were _Tintin_ and _Pilote,_ the latter published in France) had no American equivalent; they combined about a dozen serial comic strips, on double-page spreads, with a handful of single-page gags, along with games, contests, educational tidbits, and some prose fiction I never so much as glanced at. I didn't care much about stories; I cared passionately about graphic style, and this affected my reading\u2014I disdained the ostensibly serious yarns, with their conventionally realist draftsmanship, in favor of the wildest and funniest drawings. The funny strips also happened to be the most unbridled in their use of language, reveling in the singular ability of French to generate wordplay, puns in particular.\n\nFrench-speaking children are schooled in puns from the start. Of course, this could be said of speakers of English and maybe every other language as well\u2014that's what riddles are for. For example, I date my true immersion in English from the moment I understood the humor of Q: When is a boy not a boy? A: When he turns into a store. But puns lie much thicker on the ground in French, in large part because the language is so much more rigorous and willfully delimited than the sprawling mass of English, an elegantly efficient two-stroke engine to the latter's uncontainable Rube Goldberg mechanism. French does not necessarily have fewer sounds than English, but the protocols governing their order and frequency make their appearances predictable\u2014hence the profusion of sound-alike phrases and sentences, which fueled Surrealism and ensure the ongoing appeal of Freudian and post-Freudian ideas in the French-speaking world: _Les dents, la bouche. Laid dans la bouche. Les dents la bouchent. L'aidant la bouche._ Etc. These phrases, which sound exactly alike, respectively mean \"the teeth, the mouth;\" \"ugly in the mouth;\" \"the teeth choke her;\" \"helping her chokes her.\" You don't need to have been psychoanalyzed by Jacques Lacan to see from these examples how language can assist thought in swiftly tunneling from the mundane to the taboo. Children are instinctively aware of this, even and perhaps especially if they are being raised Catholic and are thus trained in the finer points of repression.\n\nThe most internationally famous characters in _Spirou_ were Les Schtroumpfs, known in the English-speaking world as the Smurfs, small blue elfin creatures who lived in a toadstool village. In their English-language animated appearances they could be cloyingly cute, but in French they were spared this fate by their language, marked by an incessant use of the (invented) word _schtroumpf,_ employed as noun, verb, adverb, adjective, and interjection. Every reader, no matter how young, understood this usage without a gloss, because it parodied the French conversational trope of substituting catchalls such as _true, chose,_ and _machin_ for words that cannot immediately be called to mind, in any grammatical position. What _schtroumpf_ highlighted was the ability of such dummy words to suggest words prohibited from writing or speech, regardless of the fact that the actual words _schtroumpf was_ substituting for were always clear from context. _True_ or _chose_ became neutral from exposure, but _schtroumpf_ subliminally spoke to the unconscious; its surface strangeness could make it mean things that the child's mind does not yet know but can imagine with tantalizing vagueness.\n\nNot all the wordplay was so freighted, of course. In the Ast\u00e9rix series (tales of a Mutt-and-Jeff pair of winged-helmeted first-century Gauls, serialized in _Pilote),_ the characters' names were always elaborate puns that turned on their suffixes, -ix for the Gauls and -us for the enemy Romans (to pick two that don't require lengthy glosses, one of the former was Madamboevarix, one of the latter Volfgangamad\u00e9us). Deciphering such names\u2014 and puns of that sort were rife in all the funny strips\u2014provided an agreeable gymnastic exercise, especially if it took a week or two of rolling the name around before it clicked open like a combination lock. Meanwhile, the adventures of Tintin, the boy reporter, a Belgian (and eventually international) institution since the 1920s, featured as a recurring character Captain Haddock, an alcoholic and irascible but good-hearted old sea dog. He was noted for his pratfalls, and even more for the streams of insults he would launch at villains, thieving wildlife, cars that splashed puddle water at him on the street, or small boys who had hit him in the head with a ball: _Accapareurs! Coloquintes! Ophi-cl\u00e9ides! Patapoufs! Cloportes! Anthropophages! Catachr\u00e8ses! Moujiks! Rhizomes! Ectoplasmes! Anthropopith\u00e8ques! Analpha-b\u00e8tes! Cornichons! Va-nu-pieds! Saltimbanques! Monies \u00e0 gaufres! Protozoaires!_ (Monopolists, bitter apples, serpents [the musical instrument], fatsos, woodlice, cannibals, catachreses, muzhiks, rhizomes, ectoplasms, Anthropopitheci Erecti, illiterates, gherkins, ragamuffins, mountebanks, waffle irons, protozoa.) It was an explosion in the dictionary, _Finnegans Wake_ on a matchbook cover, a fantastically liberating surge of pure unshackled language. The comics provided an important lesson: Language could be a medium of fun, and not just safe, approved fun, either, but wild, anarchic, disruptive fun. There was nothing lazy or slapdash about the comics' employment of words, though; that much was clear even to an eight-year-old. Therefore, the appendix to the lesson was that fun could best be achieved through a thorough grounding in ballistics and a heightened sense of precision.\n\nThe value of precision was something I had been learning all my life, perhaps subliminally from my father. He had quit school at fourteen to go to work; his father had done likewise; his grandfather had been illiterate. Nevertheless, both my father and his father were great readers. There was always at least one crowded bookcase in our home, much of its contents having been brought with us across the ocean\u2014he was not only a great reader, but a great rereader. The books were diverse, to my eye, ranging from somber hardbound volumes in slipcases to lurid paperbacks I imagined as containing all the secret lore of the tribe of adults. Looking at them today (I made a point of saving the library's core after my parents sold their house and, almost immediately thereafter, died), I realize that the great majority of the books were bestsellers, items prominently displayed in Belgian bookstores between the late 1940s and the late 1950s. The lurid paperbacks, in fact, were nearly all published by the pioneering firm Marabout, a French-language phenomenon equivalent to a downmarket Penguin, that happened to be headquartered in our otherwise not very literary home town.\n\nMy fathers books, then, could have been found in many other middle-class Belgian households of the period, and today they profusely line the shelves of secondhand bookshops. Not only are there no rarities among them, but scarcely any would have been seen among the effects of Parisian tastemakers. Few of them would be considered literary; not many date from before the period in which they were acquired. They were, nevertheless, the result of discriminating selection, and what they all had in common was style. There was not just one style among them, since they included popular novels, popular history, travel narratives, war memoirs, and humorous vignettes, but all of them answered my fathers requirements. He was a stickler for _le mot juste,_ that very French, very positivistic idea that there is one, and only one, exact word capable of expressing a particular idea in a particular circumstance. Style for him was a matter of both precision and elegance, which were entwined in any case. His classics included La Fontaines fables, Moli\u00e8re's comedies, Victor Hugo's poems, and the late-nineteenth-century plays of Edmond Rostand, especially _Cyrano de Bergerac._ All these he cited continually, sometimes because they fit the occasion, sometimes because he merely wanted to savor their music.\n\nAt some early point in my life he inculcated in me the very model of elegance, the end of _Cyrano._ The dying hero tells his friends that _quelque chose que sans un pit, sans une tache \/ J'emp\u00f6rte malgr\u00e9 vous_ (something spotless and unwrinkled, that despite you I'm taking with me). He lifts high his sword, proclaims _et c'est_ (and it is); the sword drops from his hand and he falls into the arms of his companions. Roxane kisses his forehead, asks _C'est?_ Cyrano opens his eyes, recognizes her and says, smiling, _Mon panache._ Curtain. _Panache_ literally means the plume of a hat, as worn by a seventeenth-century gentleman, but it also means what it does in English, only more so. Thus we have the pun in the last breath of life, the expression of wit as an exemplary act of heroism, the manifestation of a principle in the very utterance of its name. I was reminded of this years later when I learned that apprentice _toreros_ call themselves \"students of elegance,\" but if the Spaniards are equally capable of deeming elegance to reach its summit when it brushes against death, only the French could conceive of the matter as intrinsically verbal.\n\nElegance and precision are necessary allies; together they indicate the presence of truth. Nowhere is this axiom more clearly illustrated than in the fables of Jean de la Fontaine (1621\u20131695). Nearly every francophone can recite, at least, _tout flatteur vit aux d\u00e9pens de celui qui Vecoute_ (every flatterer lives at the expense of whoever listens to him), from the fable of the Crow and the Fox. I knew the tune before I knew the words, as it were\u2014the phrase was burned into my mind before I could define the word _de-pens_ (expense), and although I had a fair idea of what the phrase meant, it was as much a mantra as a moral. By the time I was of age to understand all the implications of the phrase, I knew its music to be a further guarantee of its wisdom. So it was with a sense of deep familiarity that, when I was in my twenties and by then a working writer, I first read Flaubert's famous letter to George Sand:\n\nWhen I come upon a bad assonance or a repetition in my sentences, I'm sure I'm floundering in the false. By searching I find the proper expression, which was always the only one, and which is also harmonious. The word is never lacking when one possesses the idea. Is there not, in this precise fitting of parts, something eternal, like a principle? If not, why should there be a relation between the right word and the musical word? Or why should the greatest compression of thought always result in a line of poetry?\n\nMy father never read Flaubert, and yet he had transmitted to me something of his essence\u2014in part because some of Flaubert's ideas had existed in French literature long before he articulated them, and in part because some had been broadly disseminated since his time. By the time I read the letter, its message was already for me an article of faith.\n\nNevertheless I avoided my fathers books, and to this day _I've_ read very few. The obvious Freudian interpretation is probably not irrelevant, although more pedestrian reasons seem just as valid now as they did then. I was bored by the very idea of most of them: the mountaineering sagas of Frison-Roche, the broad peasant comedies of Arthur Masson, the orotund Catholic and patrician moral tales of Jean de la Varende. The only books I plucked from his shelves were the crime novels, by Simenon and others, which he hated and never read, but which his sister and her husband, who were wonderful people as well as newsagents but who regarded all books as indiscriminate product, sometimes threw into the parcels they sent our way. These and the comic magazines constituted the bulk of what was available for me to read in French in my youth. In English, though, I was trying as well as I could to cultivate precociously advanced tastes\u2014 I wanted to find literature as hip as the music I enjoyed. Another sort of gap between the languages was forming.\n\nThen, when I was just the right age, we traveled to Montreal to take in Expo 67. It was the first time any of us had been in a French-speaking country in more than four years, and I was at least as excited by the prospect of visiting bookstores as by the fair itself. In my recollection, possibly telescoped by time, a center-city _librairie_ was our very first stop. I don't know whether I had anything particular in mind before going in, but I came away with two books. One of them was Andr\u00e9 Bretons _Anthologie de l'humour noir,_ an excellent choice if one made by happenstance\u2014\"black humor,\" a literary genre spawned by Lenny Bruce as much as anybody, was all the rage in the U.S. then, and that's what I thought I was getting. The other was a fat paperback anthology of French poetry, published by Marabout. I wasn't very much interested in poetry, except maybe stray bits of Beat stuff I'd seen here and there, but in flipping through the volume I noticed that many of the poems looked different from what I'd generally been exposed to: some had very long lines, some were studded with proper nouns, some were even in prose, if such a thing was possible. That night I lay on my bed in the motel room in Longueil and opened the book to\n\n_A la fin tu est las de ce monde ancien_\n\n\"In the end you are tired of this old world.\" Thus began \"Zone,\" by Guillaume Apollinaire.\n\n_Berg\u00e8re \u00f4 tour Eiffel le troupeau des ponts b\u00eale ce matin._ \"Shepherdess o Eiffel Tower the flock of bridges is bleating this morning.\" The poem was speaking directly to me, to me alone, as proven on the second page: _Voil\u00e0 la jeune rue et tu n'es encore qu'un petit enfant \/ Ta m\u00e8re ne t'habille que de bleu et de blanc._ \"Here is the young street and you are but a little child \/ Your mother only dresses you in blue and white,\" which was exactly true of my early childhood; that _tu_ clinched it. _Tu regardes les yeux pleins de larmes ces pauvres \u00e9migrants I Us croient en Dieu Us prient les fernrnes allaitent des enfants I Us ernplissent de leur odeure le hall de la gare Saint-Lazare._ \"You look with your eyes filled with tears at the poor immigrants \/ They believe in God they pray the women suckle infants \/ They fill with their odor the hall of the Saint-Lazare station\"\u2014I had been there and seen that! Furthermore, the poem seemed to be about a yearning for modernity in the face of confusion as to the truth of religion, a clairvoyant depiction of my own central inner drama of the time. But there was more: the poem was fluid, rhyming but in an elastic meter like an improvised song, with phrases strung together without punctuation but always clear in their meaning, with an unlabored syntax close to conversational, with capitalized names like cherries in a box of chocolates, with sudden movements in time and space executed with a casual legerdemain, with a flash and whirl and continual surprise that was just what I wanted from the modern world but with a palpable kindness that reassured me as the poem flung me about.\n\nAt that moment I became a French modernist, and I suppose I've never stopped being one, despite appearances. French was capable of astounding feats unavailable to most languages, it seemed to me. In his poem _\"L'union libre,\"_ one of the most erotic works in all literature, Andr\u00e9 Breton wrote: _Ma femme \u00e0 la bouche de cocarde et de bouquet d'etoiles de derni\u00e8re grandeur \/ Aux dents d'empreintes de souris blanche sur la terre blanche I\u00c0 la langue d'arnbre et de verre frott\u00e9s I Ma fernrne \u00e0 la langue d'hostie poignard\u00e9e._ What does it give in English? \"My woman with her cockade mouth, the mouth of a bouquet of stars of the greatest magnitude \/ With teeth of the footprints of a white mouse on the white earth \/ With her tongue of polished amber and glass \/ My woman with her tongue of a stabbed host.\" It's not terrible, maybe, but it has none of the music or the magic, in part because of the tendency of English to condensation and bluntness, away from the silken chains of prepositional phrases that give French its incantatory power. Of course, languages are never equivalent, can never be measured on the same scale, but when French lyricism is translated into English, the English version always sounds lead-footed, boorish, resolutely unsexy. Take the phrase _hostie poignard\u00e9e_ \u2014the profanation of the transubstantiated body of Christ in the form of a white disk of bread. The French phrase enacts the violence ono-matopoeically following the serene _hostie_ with the triple puncture of _poignard\u00e9e,_ and you even see the dagger, the _poignard._ In English, \"stabbed host,\" pretty much the only way of expressing the thought in less than a sentence, suggests a murder-robbery in a highway diner as reported over a police radio, while musically it is a coarse cluster of dentals, and it is over in a second and leaves no echo.\n\nThe French language opened poetry to me, and I wrote as well as read it throughout my teens, albeit in English since I did not trust my command of the nuances of French. Eventually I came to love English-language poetry as well, but never quite in the same way. Had I been stopped on the street and ordered to recite a line of poetry, I would automatically have said:\/' _ai tendu des cordes de clocher \u00e0 clocher; des guirlandes de fen\u00eatre \u00e0 fen\u00eatre; des chatnes d'or d'etoile \u00e0 \u00e9toile, et je danse_ (\"I stretched ropes from spire to spire, garlands from window to window, gold chains from star to star, and I dance\"\u2014Rimbaud).\n\nMidway through college, I stopped writing poetry altogether. I doubted my talent, but I also had found what I thought was the authentic music of the American language, in the prose of Dashiell Hammett, Raymond Chandler, and James M. Cain. \"They threw me off the hay truck about noon,\" the opening sentence of _The Postman Always Rings Twice,_ seemed to exemplify in nine words all the highest virtues of American prose. It was plain, unadorned demotic speech, resolutely laconic and flat, containing a whole landscape of gas stations and bus depots and bars, of dollar bills and cigarette butts and spit, stuff I had encountered in daily life that seemed to stare down literature and dare it to cross the line in the dirt. It defied all the verities and aesthetics of the university in which I was a half-reluctant conscript, of course, but no less significant was the fact that it embodied the inverse of everything I thought I knew about French.\n\nThe importance of both causes was emphasized by the fact that my epiphany occurred in Paris, where I was attending a summer program sponsored by my college that was devoted to the very latest manifestations of French critical thought. What was I doing there? The previous year I had signed up for a course given in the French department on Surrealism, a subject of enduring interest to me. Not ten minutes into the first class I was at sea. The instructor, a recent Parisian transplant, drew cryptic diagrams on the blackboard while issuing a rapid-fire stream of references, quotes, unfamiliar Greek-derived words, and puns. The latter, at least, I could appreciate, although they were unfunny and often ponderous (an English-language one, \"French Freud,\" was to reappear continually as a catch-phrase), and were redolent of forced play; it was like watching academics dance at a disco in order to make a point. As a chronic shirker of math requirements, I was dismayed by the diagrams and the scientific or pseudo-scientific tone of the propositions; I had never heard of Lacan or Derrida; and what did any of it have to do with Surrealism? It seemed to me the equivalent of getting to know someone by administering chloroform and then dissecting her or him on a slab. Somehow I completed the year, and achieved a grade that did not disgrace me. I can only imagine that I signed up for the summer program because it would get me to Paris with an educational alibi. Somehow I even managed to obtain financial aid for the adventure.\n\nThe courses were a mixed lot. The art historian was genuinely riveting, although I remember more vividly the specialist in modern fiction, who seemed to devote the entirety of his analysis of _Madame Bovary_ to rolling names and phrases around in his mouth until they became puns by force of will, for instance mangling \"Charles Bovary\" until it yielded up _charivari._ Finally I was sick of puns, sick of the alleged _jouissance_ of language, very nearly sick of French itself, and I hiked up to Galignani, the venerable English bookshop on the Rue de Rivoli, and picked up American books primarily composed of words of one syllable. But when I look at my notebooks from that time I am forced to acknowledge that every choice I made was saturated by the French spirit, the version prevailing at the time in particular. My approach to American crime fiction was that of an outsider, was informed by the _S\u00e9rie Noire_ collection, by the ideas on American movies held by the critics at _Cahiers du Cin\u00e9ma,_ by Sartre's enthusiasm for the work of Faulkner, which he said resembled the view out the back window of a moving car. Despite myself I was in love with the chic that imbued all manifestations of the French intellect. I was seduced by the French tendency to wrench words and phrases and even entire narratives from all context the better to prize them as artifacts, the way oily rags become art when framed with a broad white mat. I was enthralled and frustrated in equal measure by the French literature of the time that seemed intended for admiration rather than actual reading, dependent for its effect on a title, or an allusion to something classical and recondite, or a typographical decision, such as a thin scatter of fragments around the page like so many notes on a refrigerator door, or a block of unpara-graphed and perhaps unpunctuated prose running on for the entirety of a slim volume.\n\nNot long after my return I lost contact with the French-speaking world once again. This was to prove a pattern, with French waxing and waning in my life at long intervals, like the moon of a large planet. I did not set foot in my native land for another fifteen years, and then for a decade I went there annually, ostensibly to do research for a book. I made friends; I acquired a neighborhood and a set of site-specific habits; I got so that I could regain my fluency in the language within 24 hours of deplaning. It was then that I discovered several other kinds of French. The language that appeared in the media, in advertisements, and in the mouths of the more urban and well-connected people I met was quite different both from the tongue I had learned as a child and that which appeared in the books I read. It was bright and cold and hard-edged, implied technology and market research and modern accounting practices. I knew that it had its American parallel, which I generally avoided and often mocked, but I took this kind of French personally, like a slap. When some of its words leaked into my conversation because I had no friendly synonyms at hand in which to express a particular thought, it felt like an unhealthy imposture, as if I had caught myself putting on a gold tie\u2014I suppose I felt like a class traitor, antiquated notion though that may be. Even unarguable statements made using those words felt like lies, since the language so clearly had been produced in a laboratory.\n\nBut I also immersed myself in argot, _la langue verte_ (the green tongue). I had encountered it before, notably in my mid-twenties, when I hung out on both continents with a group of radical offspring of French academic families who affected _ver-lan_ (backslang, then just on the cusp of becoming chic) and conducted entire conversations in prison slang without glossing anything for my benefit, making me feel excluded and desperately unhip. But by a decade or two later the lingo had penetrated more deeply into the everyday speech of ordinary folk, and I absorbed a good deal of it from reading, in particular from crime novels of the 1950s and '60s. American slang, whatever its origin, tends to fill particular lexical slots, usually pertaining to highly charged categories of meaning\u2014sex and drugs and crime in particular. French slang is even more rooted in crime, but it is defiant rather than furtive. It is an entire language, a parallel verbal world that mocks the formal protocols of the master language. Unlike the American variety, it contains words for every sort of thing, for \"door\" and \"table\" and \"cup.\" Some of it is ancient, dating back to the time of Fran\u00e7ois Villon and beyond; some of it actually derives from Romany and it continues to loot other languages, in pointed contrast to official French, which proscribes loan-words. It is a highly metaphorical language, as slang tends to be, with an insolent, blaring music and a staccato beat: _Quand le bruit se refand que la poule tape aux fafs dans un coin, vous voyez les tapis se vider de tons les tri-cards._ Literally, this would more or less mean: \"When the noise spreads that the hen is tapping for papers in a corner, you see the carpets emptying themselves of all the tricksters.\" What it signifies is: \"When word gets around that the cops are checking IDs in the neighborhood, all the parolees instantly vanish from the bars.\" I derive deep satisfaction and sensual pleasure from argot, as little as I use it in the course of things. It is almost as if French and American had mated in the night and produced another tongue with all of the advantages of both, and none of the pomposities.\n\nFrench was once the international language of diplomacy but is no more. It barely hangs on to its association with the courtly arts. It has been forced into retreat, in one domain after another, before the Anglo-Saxon juggernaut. This diminished status has occasioned both a resentful provincialism and the unfortunate tongue of technocrats and _biznessmen._ Its literature is rarely and haphazardly translated into English these days. The romance of French among poetically inclined American youths has waned considerably, in part because of the very success of French theory and its particular brand of double dutch, the nuances of which sometimes require a profound knowledge of classic literature, although this is not always apparent to English speakers. Where I live, in rural America, it is an obscure joke. It is my mother tongue, although I will probably seldom encounter again the specific variety of it I heard while growing up, since it was the instrument of a class that has changed drastically and to which I have lost most of my connections. I don't even employ it every week, let alone every day, and yet one way or another it informs every decision I make in the screen language I employ in order to pass unmolested in the land where I have lived for most of my life without ever shedding my internal foreignness. French is my secret identity inaccessible to my friends. Sometimes I feel as though I have it all to myself.\n\n# GERMAN\n\n#\n\n# _Prelude_  \nThomas Laqueur\n\nI seem to have had a peculiar loyalty to the German language from about as early as a child can articulate views. I was told by my parents that when they urged me as a three-year-old to learn Turkish, so that I might communicate more effectively with my playmates in Istanbul, where they had come in their flight from Hitler, I would have nothing of it. Let them learn German, I supposedly said; Turkish \"ist eine h\u00e4ssliche Sprache.\" My feelings about speaking German, and more generally about being European, have become stronger as what few real connections I had ever had to the language, to Germany, or to Europe have all but disappeared.\n\nGerman was my mother tongue. I mean this partly in the usual sense\u2014my first language was German. But it is also true that I spoke it almost entirely with my mother, my grandmother and their women friends. Only certain words and phrases are spoken by men or to men in my linguistic fantasy life. German is almost entirely a self-contained family language for me, but it is also the language of a world\u2014real, remembered, and misremembered\u2014 that my parents lost, a world that now exists almost entirely in my imagination, but which I maintain as a way of mourning them and theirs.\n\nI spoke only German until we left Turkey in November 1949. A stop in London with relatives was still all German, as were a brief few weeks in New York. My mothers brother\u2014my Onkel Otto\u2014and his wife lived in Manhattan near Fort Tryon Park, in the middle of a German-Jewish ghetto. Later, when we had settled in West Virginia, my mother visited them periodically and came back complaining about how insular their world was. I think I understand what she meant: one could not forget that one was living in exile there, amidst one's countrymen on the cliffs above the Hudson. In contrast, my family's relationship to its native language could not have been more cut off from its roots than ours was in the coal villages and towns where I grew up. I do not think my parents thought of themselves as living in a diaspora because they had no one with whom to share their loss.\n\nAfter New York, we lived for a few months in my father's sister's boarding house near the University of Texas. She specialized in housing foreign students. My Tante Eli and her husband had gone to Yugoslavia when Hitler came to power. When Hitler attacked Belgrade, they made their way south from Dubrovnik and Mosta to Albania, in the hope of being captured by the possibly benign Italians instead of by the certainly murderous Germans. They succeeded, and spent the war until 1942 in a Ca-labrian internment camp; then they were liberated by the British Eighth Army and headed north with it as translators. By the time they had to earn a living in Texas, they had Italian and Yugoslav and colloquial English in addition to very good school French\u2014and Latin, in case an ancient Roman turned up. This was my first sustained exposure to English.\n\nI remember being grumpy about learning a new language while in Austin. I do not remember saying what my parents claimed were my first words in my new language: \"me no eat fruit.\" I find this unlikely given that I have no memory of ever not liking any fruit, but still, this is family lore.\n\nAfter three months of crowded living, my mother, paternal grandmother, younger brother and I joined my father in a hollow near Montgomery, West Virginia, where he had secured a job as a pathologist in a private coal-field hospital. A friend from Istanbul, also a pathologist, had found a job near there the year before, through a Jewish relief agency. I have no memory of speaking English during our months in that hollow, just up from a railroad track. I think my mothers English was not very good, so we didn't see much of the neighbors. Tante Biba and Onkel Peter, the friends from Istanbul, lived twenty or thirty miles away, and with them I of course spoke German. Then on to Bluefield, the \"air-conditioned city,\" where coal poured in from the southern West Virginia bituminous coal-fields to one of the Norfolk and Western Railroads biggest train yards. It was here that I started to learn English seriously. I remember no hostility this time, although I do remember being teased about my German accent for many years to come. Unlike my brother, who is three years younger than I am, I never acquired the mountain accent, and I still sound foreign in those parts.\n\nIt was in Bluefield that I discovered German was a language that people other than my parents and a few friends actually spoke. It was not, as I had unselfconsciously assumed, a family code. This revelation came as follows: I was having a screaming fight in German with my brother, in front of the Pen Mar Grocery, a half-block from our house on North Street; he was three and I six. The issue was how much of a two-barrel popsicle I was going to share with him. A lady came up to us and said, in German, that she would give us a nickel so that each of us could have a treat of our own. I do not remember buying a second popsicle, but I do remember being very excited at finding someone else of our linguistic species. I rushed home with the big news.\n\nFrau Bressler, as she was called, had asked where we lived; I had told her. She visited. Frau Bressler had married Herr Bressler, who was many years her senior, after a long courtship. He had some sort of a disease that had caused his hands to shrivel into reddish, claw-like appendages, and he worked repairing small electrical appliances and meters. The Bresslers were poor; she was a southern German Catholic. (This I deduced on a visit last summer, from books about a papal visit to Bavaria I saw on the coffee table of her house.) Frau Bressler became one of my mothers close friends despite their very different circumstances. She also became our regular\u2014indeed only\u2014babysitter when my parents were away for more than an evening.\n\nThere was a third German in Bluefield, Frau Snelling, who had married\u2014after the war, I assume\u2014an alcoholic West Virginia forester. I associate her, however, not with making German a more public language for me, but rather with my first noteworthy failure in my efforts to be a good little German boy. The traumatic moment came when her mother, Frau W\u00f6ppekind, visited from Germany. I did, on meeting her, remember to address her with the formal _Sie,_ as I had been told to do. I did not, however, remember to bow. \"Mach eine Verbeugung,\" said my mother, not pleased with my lapse. I do not recall what Frau W\u00f6ppekind said, but I do remember that she seemed manifestly taken aback by \"der Bube's\" ill manners.\n\nSo now there were three strangers who spoke German in my world. I knew they were strangers because I addressed them as Frau or Herr instead of Onkel or Tante, which is what I called almost all other German-speaking adults. The fact that, following local custom, I called American adults who were close friends by their first names made our linguistic isolation palpable. Eddie and Janie and CO. and Hazel were simply from another universe, where other laws pertained. I was in my late twenties before I could comfortably address grown-up Europeans by their first names, and even then it was not easy. The crisis came when I got a job at Berkeley and was placed on a committee with two older colleagues: Paul Alexander, a saintly, extravagantly learned Byzantinist who was on the fringe of my family circle (the best friend of a cousin by marriage), and Nicholas Riasanovsky, a famous Russian historian. We were to give out money for graduate research projects. I could not call Alexander \"Onkel Paul,\" as I might otherwise have done; \"Onkel Nick\" was of course out of the question. And I could not address colleagues as \"Professor.\" So \"Paul\" and \"Nick\" it was, but not without a mental gulp. I still find this blurring of boundaries difficult.\n\nThere were two exceptions to this first-name rule: my mothers closest friends from Istanbul. Both were known by their nicknames. One, still alive, is \"Dicke\" or \"die Dicke\" (\"the fat one\"), who was supposedly once fat; the other was \"Schweinchen\" (\"piglet\"), whose nickname is a corruption of her maiden name, Schwerine. Schweinchen was sometimes Tante Paula; Dicke was always Dicke.\n\nGerman, in other words, constituted a world that I knew intimately but also not at all. I had, growing up, only the vaguest sense that people outside our family circle actually lived and functioned in our private language. Although I spoke it fluently, I got things having to do with the public\/private distinction seriously wrong. The _du\/Sie_ question was never easy. In our family of course, I used the familiar; likewise with family friends. I could use _Sie,_ but it did not come naturally. I had to be coached and reminded, a formula for screwing up, a sign not so much of bad character but cultural cluelessness. Dickes husband, Wie-gand, was said to be _vornehm_ (\"refined,\" \"high class\"). I do not know on what this view was based, but when he visited it was said to be important that I, age seven, not _dutzt_ him. I think I succeeded. But there were embarrassing lapses. When I was eleven or twelve, we visited Boston and made a pilgrimage to the butcher shop of Herr Thyssen, who was my parents' longdistance purveyor of German food. It came every few weeks to Beckley and Bluefield, packed on dry ice, via Greyhound bus: _Kaiserjadgwurst, Leberwurst, Blutwurst,_ and other wursts I can only say and not spell; _stinkerk\u00e4se_ (my name for Limburger cheese); every kind of dark bread. At Herr Thyssens shop, introductions were made and I lapsed into _du;_ he was clearly taken aback; my mother was appalled. There was nothing to do but try to disappear.\n\nThe same problem came up in regard to tone of voice and distance from one's interlocutor. I seemed to have always been off. \"Mami\" Putschar, the German-speaking wife of a Hungarian pathologist in Charleston whom we visited occasionally, always said to me that I sounded like a _Feldwebel._ Frankly I did not know what this was (it is a sergeant), because it is not the sort of word that comes up in family life; I did not play soldiers with anyone who knew the language. But it was clear that this was not a good way to sound. In college, where\u2014in the persons of \u00e9migr\u00e9 professors\u2014I met my first \"stranger Germans,\" I knew that I was somehow standing too close to them when I spoke. It took time to get the right range.\n\nMy family's and my German was entirely cut off from Germany and from everything that had happened to the language since the 1930s. (The one exception was a pilgrimage to New York when I was in high school, to see the Brucke theater do Schiller's _Don Carlos._ This was the first and I think only German play my parents saw after the mid-1930s.) There were lots of German speakers in my life, but none had had any connections with the real sources of the language for decades. They were an odd assortment of \u00e9migr\u00e9s, some native speakers, others part of the German cultural penumbra. In Beckley where we had moved in 1956, there was only a Ukrainian orthodox couple who spoke German. He had studied medicine in Germany after escaping from the east; she was a self-consciously romantic sort who spoke a hyperbolic, soulful, Russian-accented version of my mother tongue. During the summers there were also my uncles and aunts, who came to visit our cottage by a lake in southwestern Virginia; there were my mother's buddies from Istanbul, and some even from her late twenties and early thirties in Germany; there was, early on, my grandmother's sister-in-law, who spoke a Polish-accented German; there were several Hungarians, including a voice teacher from Juilliard who had the deepest voice of anyone I knew; there was an Austrian nurse who had somehow linked up with a West Virginia dermatologist named Locks-ley who spouted Shakespeare at the slightest provocation. And there were Max and his wife, who owned a bakery in the small town of Pulaski, Virginia, near our lake; both had tattoos on their arms from Auschwitz. Why they wanted to speak German with my mother is unclear. I did not wonder about it at the time. They also spoke Yiddish with my Onkel Otto when he visited. In any case, this was an eccentric linguistic universe.\n\nI dwell on all of these childhood memories because German is for me the language of memory and loss, a linguistic _Prelude._ My German is, first of all, a connection with a pre-Oedipal me. I have never made love in German; I know no slang words for matters sexual, and few slang words of any sort. I would not know what it would mean to feel sexual in German. The gigantic impact of linguistic adolescence\u2014when one comes to own one's language as a separate person, when it becomes something belonging to one's generation\u2014is lost on me. My German is frozen, amber-like, not only in pre-war history but in childhood; with some few exceptions, it is emotionally fixed. The word for carraway seed, _k\u00fcmmel,_ is an adjective for a kind of bread on which one eats corned beef or chopped liver, i.e. rye bread; for me it describes a man who terrified me as a small child, _der k\u00fcmmel Mann,_ a beggar with a pox-marked face who stood outside our Istanbul apartment. Too little has happened to me in German to make the regular public uses of words mean what they should.\n\nPowerful German words generally feel like they come from my mother; phrases, dicta, from my dad. _Sanft_ \u2014\"soft\" or \"gentle\"\u2014 I associate with her, although the phrase in which it comes back to me in the first instance is not hers. In my mind's ear it is from Schiller's \"Ode to Joy\"\u2014... _Wo dein sanfter Fl\u00fcgel weilt_ (\"where your gentle wing may come to rest\"). I think of the word in connection with my birthdays. On the evening of September 6th, from as early as I can remember until I was ten or so, my father and I would lie on a couch, I enfolded in his arms, and listen to a recording of Beethoven's Ninth. For the first years in West Virginia, it was the old 78s of the Furtw\u00e4ngler recording that would clack-clack-clack down until the whole stack had to be turned. Sometime around second grade, we switched to the Toscanini \"long play\" version that, miraculously, played for twenty-five minutes without a clack and went by, in its wild tempos, considerably faster than Furtw\u00e4ngler's more Germanic version. The ritual, however, did not change with conductors: lights were dimmed; during _die Neunte_ there would be no talking or interruptions by other family members; we were alone. I wonder how, before record-players, Germans of my parents' class and generation learned their reverence for _die Neunte_ (\"the Ninth\")\u2014 which, without further modification, can only mean Beethoven's. In German, or at least my parents' German, one puts just a little bit more emphasis on the article _die_ and lingers just an instant on the noun _Neunte_ than one would in a phrase like \"the ninth symphony of Schubert\" or \"the ninth symphony of Mahler.\" I know that this work still has considerable cultural clout in Germany, or at least did until recently; the great national work of the nineteenth century, it was what Bernstein conducted at the fallen Berlin Wall. But I have no sense whether men and women of my generation would say the words like my parents did and feel what I learned to feel. Like so much German, I know these two words of the language almost entirely in isolation from all but friends and family.\n\n_Geboren_ \u2014the adjective \"born\"\u2014is a mother word. She, and only she, and no one since she died, would address me on my birthday with the redundant silliness of _mein einziger Erstgeborener_ (\"my only firstborn\"). The suffix _-lein_ that produces the diminutive in German is also my mother's: my father might occasionally have addressed me as Thomaslein\u2014I do not remember\u2014 but my mother always did. Tommy, which is what they called me in West Virginia, has always sounded silly to me; Tom is just a name; Thomaslein is very sweet. _Traurig_ (\"sorrowful\") is a mother word, although I think my mother was in fact far happier than my father. She could not keep a tune for more than two measures but loved to sing a song called \"Die Lorelei,\" the lyrics of which were by her favorite poet, Heinrich Heine. I have her copy of his complete poems that she kept on her night table and read most days of her life.\n\n_Ich wei\u00df nicht, was soll es bedeuten,_\n\n_Da\u00df ich so traurig bin,_\n\n_Ein M\u00e4rchen aus uralten Zeiten,_\n\n_Das kommt mir nicht aus dem Sinn._\n\n\"I do not know what it means, that I am so sorrowful; I cannot get out of my head a tale of the most ancient of times.\" This is roughly how I feel about things German in general: a \"M\u00e4rchen\" fairy-tale built of projections and fantasies and memories that I cannot erase and that leave me melancholy.\n\nIn my family, we spoke German at the dinner table until I left for college because my grandmother claimed that she neither spoke nor understood English. This was clearly false\u2014she read English papers and watched English TV\u2014but feigning ignorance allowed her to maintain the fiction of otherworldly incompetence that she seems to have cultivated all her life and that kept her entirely out of public view. She did not venture outside family circles during her twenty-three years in America. My grandmother was born in 1873, in the waning of the Biederman era, the youngest daughter of six children. She went to school long enough to learn French; she played piano well; she and my grandfather lived for music, which they played four hands. They had heard Brahms conduct, early in their lives together, as well as many of the other great German conductors of the nineteenth century. (I know all this from their concert diary, which I inherited when my father died.) My grandmother could do all sorts of needlework. But she could not\u2014or at least did not, in anyone's memory\u2014so much as boil an egg. She stayed in Germany until December 1939 on the grounds that she did not want to leave her Bechstein grand piano. In America, she dressed and acted like a lady of a distant century, seemingly unaware that the world around her had changed. (She did read about the deaths of her contemporaries in the Aufbau, and remained alert until the ninety-fifth of her one hundred years.) The first of my fantasy Germanies is hers. The words I associate with her are _Es geht rapide bergab_ \u2014\"things are going rapidly downhill\"\u2014something she said about herself from when she was in her late seventies to when she went gaga in her late nineties.\n\nWith my mother I spoke German exclusively until she died; I have not spoken it regularly since 1992. With my father I spoke only English, the grown-up language, the language in which I talked of science and medicine and politics. He did, in fact, speak English much better than my mother, but it was only much later, when I heard him on a Dictaphone machine summarizing an autopsy, that I realized how heavily accented his English was, almost parodically so.\n\nThere were, as I said, exceptions to this linguistic segregation. The few bits of really grown-up German I know, and the minimal sense I have of the rhythms of the language, are from sayings or maxims, _Sprichw\u00f6rter,_ that come from my father. (I wish I could rattle off those wonderful torrents of dependent clauses and finish up with the verb, as grown-up German speakers do.) Likewise curses come through him. _Mit der Dummheit k\u00e4mfen G\u00f6tter selbst vergebens_ (\"With stupidity the gods themselves fight in vain\") was a big one, as was Kant's categorical imperative, which was recited with a special tone of reverence. I loved its sounds and the fact that there was only one such rule, even if it took a while to understand what it meant: _handle nur nach derjenigen Maxime, durch die du zugleich wollen kannst, da\u00df sie ein allgemeines Gesetz werde_ (\"act only on that maxim by which you can at the same time will that it should become a universal law,\" which I understood in the still grander form, \"act as if the maxim of your action were to become through your will a universal law of nature\").\n\n_Donnerwetter_ (\"thunder weather\") was the prelude to an explosion of my fathers anger and was often followed by _noch ein mal_ (\"once again\"). This malediction was frequently associated with the threat that if we continued to misbehave my mother would call my father, who would then say _ein machtwort_ \u2014 literally, a \"word of power,\" but really more like the definitive warning of the super-ego. Since one of the other big sayings in my family was _Quod licet Jovi, non licet bovi_ (Latin sayings had the authority of German ones), which meant \"What is allowed to Jupiter is not allowed to the ox,\" the \"thunder weather \/ words of power\" combo carried a certain mythological terror. The Latin saying itself was used mostly to explain why my reading of the categorical imperative was mistaken in holding that the maxim for some action of my fathers included him. So, if it was okay for him to be late when we all knew that lateness was indefensible as a universal principle, the old _Jovi_ exception was adduced. I thought that this was fudging on the universality principle but got nowhere with this line of argument.\n\nThe only curses I know are my fathers, and they are ridiculously quaint. He would reproach my mother with _Was glaubst du das ich bin, ein Dukatenscheisser_ (\"What do you think I am? Someone who shits ducats?\") every month as he was paying the bills. He was terribly anxious about money having no one to back him up if he failed, but he must have known full well that his wife was frugal and extremely efficient at household management. _Lech mich am arse_ (kiss my ass) was another, always attributed to Schiller's G\u00f6tz von Berlichen. I have never used any of these phrases in public because I have no idea whether they mean anything in the outside world.\n\nTwo words belong to both parents and have universal resonance for me: _Unsinn_ and _vern\u00fcnftig._ Again, I do not know whether other German-speakers my age feel this way about them; for all I know, they resonate as they do for me only in my private language. _Unsinn_ (\"nonsense,\" \"absurdity\") had many applications and was often used as an expletive. But it is one of the few words from childhood that carried over into adolescence. _Mache keinen Unsinn_ (\"Don't do anything stupid\") was the standard caution before my going out on a date. It did not apply to my driving, which was impeccable, but to \"parking\" on one of the hundreds of miles of strip mine roads around where we lived and necking the evening away. (There was nothing else to do in Beckley but this nonsense had other things to recommend it.) _Unsinn_ and _sei vern\u00fcnftig_ (\"act reasonably\") are the only German words that have any personal association with sex for me. They have other meanings, of course. Being _vern\u00fcnftig_ meant being governed by reason in all matters and applied to life generally, but in the absence of any other post-pubescent words, they still have a peculiar ring of sexual danger.\n\nAlthough, as I said, I spoke English with my father, my sense of German as a language of loss comes through him. I felt strongly as I was growing up that he simply did not get what it meant to live in another culture. My mother, whose English was wildly ungrammatical and full of Germanic neologisms, got on well with the locals. She made a joke of misunderstanding, as when on her citizenship examination she answered that _ja, ja,_ she \"had been and was still a member of the Communist Party.\" She had been told by friends that if she did not understand a question\u2014she often missed what people said if they spoke quickly or with especially pronounced mountain accents\u2014 she should simply answer \"yes.\" Beckley and Bluefield abounded with Toni Laqueur malaproprisms. But she fit in. My father was clueless. He somehow translated my high school graduation as _Abitur,_ an occasion for much ceremony and for a punch bowl of Champagne and liquor-soaked fruit. This did not go over well with my high school friends. He tried at my parents' New Year's Eve parties to have everyone wear tuxedos and listen to Beethoven's Ninth. This also did not find wide acceptance.\n\nAnd he seemed to have no sense of what his own past meant after Hitler. We had a recording of German university student drinking songs that we played often. He knew all of them; I even knew them. He had a picture of himself and his university fraternity brothers wearing their uniforms and displaying sabers. He had a small dueling scar above his hair line. None of this struck him as odd or ironic. Perhaps this is just an instance of the strategy my parents shared, attempting to mitigate the pain of having lost their homeland by neither assimilating nor living in a diaspora community, among others who had been displaced. They lived as much as they could in a bubble, eating food and speaking a language and listening to music that no one around them appreciated or understood. My German has inherited something of their cultural autarchy.\n\nI do not want to suggest that I speak a childish German, or that I cannot get by doing adult things in the language. But whenever I do something grown-up in German, I am self-conscious about doing it; I am aware of the temporal chasm between now and then (\"then\" being the lives of my parents and my own childhood). My father never went back to Germany; my mother went back once, in 1955, to visit an old friend who had returned. She lucked upon Central Castings nightmare of a taxi driver, who went on about all the good things the Nazis had done and how Americans misunderstood _die Hitlerzeit._ Never again. So both of them remained passionately German, but without any real contact with Germany. They drank only German wines. They staged an elaborate German Christmas complete with candles on the tree (until neighbors told them that American trees, cut a month in advance, would go up in flames). They listened almost exclusively to German music\u2014 _Parsifal_ was on for Easter. They thought that the French were wrong to occupy the Rhineland in 1920, and wrongheaded about much else besides. So I lived a childhood produced by the children of nineteenth-century Jews, who imagined the land of Goethe and Schiller with little of its reality or recent history.\n\nI went to Germany for the first time in 1992, when I was forty-seven. I was there as a tourist and spoke of little but rooms, food, and schedules. The first time I actually said anything in German that was neither about travel nor about the sorts of things one talks about in families\u2014that is, the first time I felt that German was for me a public language\u2014was in the summer of 1995 at a conference in Frankfurt. I asked a question in German of a journalist; he understood and answered; I asked a follow-up. I translated in whispers the lectures of colleagues for my wife, and found that I was good at it. On subsequent trips I have given my own lectures in German, sometimes at the request of my hosts but sometimes just because I wanted very much to reclaim the language for my parents.\n\nI love being in Germany among my friends; it is a return to a place and a language and a cultural tradition that my parents never ceased to mourn. The people I know there are to a person cultivated, intelligent, liberal and welcoming. But I have no illusions about the phantasmic\u2014arguably delusional\u2014attachment I have to place and language. In 1995, my wife and I visited my mothers hometown, Holzminden an der Weser, a small city of about 30,000 not far from Hanover. It is in what was the heartland of Nazi electoral strength. My grandparents' house looked exactly as it did in pictures, almost entirely unchanged. The river Weser ran swiftly less than a hundred meters from the little meadow where my grandfather, a grain merchant, kept a few cows and chickens.\n\nI knocked on the door of the house, and an old woman appeared at the window. I asked her if she had lived here for a long time. Yes, she had always lived there. Well, I said, my grandparents had once lived there. No, not possible, she said before she relented: who were they? Their name was Weinberg. _\"Ach ja, die Juden. Feine leute.\"_ (\"Ah yes, the Jews. Fine people.\") Her father, a carpenter, had bought the place from my grandfather in the Hitlerzeit. This must have been in the early 1940s, just before he was deported to Theresienstadt and on to Auschwitz. She shared with me what she knew about where one swam in the Wesser (swimming was my mothers great love, and I had heard a lot about the rivers quick currents and what one had to do to navigate it). She told me the location of the Catholic girls' _gymnasium_ that my mother had attended. I then asked her whether my wife might take a picture of me in the window in which she was sitting. (I have a picture of the house with my maternal great-grandparents in the top window, my grandparents in the middle window, and my mother and her siblings in the window where my interlocutor was sitting.) Suddenly she ceased to understand my German. The conversation was over; I could reclaim only so much.\n\n# GIKUYU\n\n#\n\n# _Recovering the Original_  \nNgugi wa Thiong'o\n\nHe lay on his tummy on a high table in the assembly hall with all the students and staff present. Two teachers held his head and legs and pinned him to the table and called him monkey as the third whip lashed his buttocks. No matter how horribly he screamed and wriggled with pain, they would not let him go. Scream Monkey. Eventually the shorts split and blood spluttered out, some of it on the shirts of those who held him down, and only then did they let him go. He stood up barely able to walk, barely able to cry, and he left, never to be seen in the precincts of that government school or any other again; I have never known what happened to him. His fault? He had been caught in the act of speaking Gikuyu in the environs of the school, not once, not twice, but several times. How did the teachers come to discover his sins?\n\nSpeaking African languages in the school compound was a crime. If a student caught another speaking an African language, he would pass a token called a monitor to the culprit, who would carry it around his neck till he caught another speaking the forbidden tongues; he would pass the dreaded thing to the new culprit, and so on\u2014children spying on one another, all day, or even tricking each other into speaking the leprous language. The one with the monitor at the end of the day was the sinner and would be punished. The above recipient of whiplashes had been a sinner for so many weeks that it looked as if he was deliberately defying the ban on Gikuyu. The teachers were determined to use him as an example to teach others a lesson.\n\nThis was the Kenya of the fifties in the last century. The country was then a British settler colony with a sizeable white settlement in the arable heartland, which they then called White Highlands. But from its colonization in 1895, Kenya was always contested, the forces of colonial occupation being met by those of national resistance, with the clash between the two sides climaxing in the armed conflict of the fifties, when Kenyans grouped around Mau Mau (or, more appropriately, Kenya Land and the Freedom Army) took to the forests and mountains to wage a guerrilla struggle against the colonial state. The outbreak of the war was preceded by a heightened nationalist cultural awareness, with songs, poetry, and newspapers in African languages abounding. The outbreak of the war was followed by a ban on performances and publications in African languages. A similar ban applied to African-run schools\u2014they were abolished.\n\nI first went to Kamandura primary, a missionary set-up, in 1947. But we must have been caught up by the new nationalist awareness, because there were rumors that missionary schools were deliberately denying us children real education _(G\u00fcth-imira ciana \u00fcgi)._ Such schools were alleged not to be teaching Africans enough English, and some of us were pulled out of the missionary school and relocated to Manguu, a nationalist school where the emphasis was on the history and culture of Africans. In religion, some of the nationalist schools, which called themselves independent, aligned themselves with the orthodox church, thus linking themselves to the unbroken Christian tradition of Egypt and Ethiopia, way back in the first and fourth centuries of the Christian era.\n\nI was too young to know about this linkage; all I knew was that I was going to a school where we would be taught \"deep\" English alongside other subjects and languages, in our case, Gikuyu. I can't remember if the English in the nationalist school was \"deeper\" than that taught in the previous school\u2014I doubt if there was any difference in approach to the teaching of English\u2014but I do recall that a composition in Gikuyu was good enough to have me paraded in front of the class, in praise. That is how to write good Gikuyu, the teacher said after reading it aloud to the class. So in the nationalist school of my early primary schooling, mastering Gikuyu and knowing English were not in conflict. One got recognition for mastering one or both.\n\nThis peaceful co-existence of Gikuyu and English in the classroom changed suddenly a few years later, when the African independent schools were shut down, with some of them resurrected as colonial state-run institutions. Manguu was one of these and the emphasis on humiliating the Gikuyu language-users, as the pre-condition for acquiring English, was the most immediate outcome of the changes. It was under the new dispensation that terror was unleashed on Gikuyu. The screaming student was being thrashed to take him out of the darkness of his language to the light of English knowledge.\n\nI enjoyed English under all dispensations, but the image of the screaming student haunted me and even puzzled me for a long time. The student was hounded out of the school for speaking Gikuyu, the language I had once been praised for writing well. Maybe there was something wrong with the teachers who had so praised me; the evidence of this was that they had all lost their jobs under the new colonial position on the importance of English. The new teachers, all African, all black, all Gikuyu, devised all sorts of methods for associating African languages with negative images, including making linguistic sinners carry placards that asserted that they were asses. It was a war of attrition that gradually eroded pride and confidence in my language. There was nothing this language could teach me, at least nothing that could make me become educated and modern. Gore to the students who spoke Gikuyu; glory to those who showed a mastery of English. I grew up distancing myself from the gore in my own language to attain the glory in English mastery.\n\nThere were rewards. A good performance in English meant success up the ladder of education. And it was this that took me from Manguu, under its colonial tutelage, to Alliance High School, the most prestigious institution for Africans at the time, and eventually to Makerere University College in Kampala, Uganda, where I studied English Literature.\n\nIt was there, at Makerere, in the sixties\u2014the heyday of decolonization, with country after country in Africa becoming independent\u2014that I wrote what eventually became _The River Between, Weep Not Child,_ and several short stories and plays, all of them in English. When later I went to Leeds University in England, I wrote the novel _A Grain of Wheat._ I truly felt joy in trying to make English words sing and capture the color and contours of my life. During the composition of _A Grain of Wheat,_ much of it done in my room at Bodington, a student residence hall near the Yorkshire Moors (the setting of Bront\u00eb's _Wuthering Heights),_ I often played Beethoven's Fifth Symphony in the background, and I aspired to similarly weave several movements into a seamless whole. I wanted to climb on English words to the highest peak of the mountain of human experience. But why choose English as the vehicle of my ambitions? It was not a question of choice. By this time in my education, and with everything that surrounded me at schools in Africa and abroad, writing in English seemed the most natural thing to do. I had been socialized into regarding writing in English as normal and desirable, even when the subject matter was the drama of decolonization and independence, a major theme in my work.\n\nIn all my writings I drew on the life and culture of Gikuyu and the African peoples. Their history, particularly that of anti-colonial resistance, was at the center of my writing. But this history and culture were negotiated through Gikuyu and other African languages. Mau Mau fighters against the British colonial state, in their hideouts in the forests and mountains, did not strategize and plan in English; they talked Gikuyu, Kiswahili, and other Kenyan languages. Yet I wrote as if they were doing so in English. I heard their voices in Gikuyu but wrote them down in English sounds. What I was doing, of course, was a mental translation. This means that for every novel that I wrote in English, there was an original text. What happens to this original text, since in fact it exists only in the mind and is not written down? It is lost, and we can only access it through English. In my educated hands, Gikuyu language, culture, and history came out wearing an English-language mask.\n\nI believe there is genius in every language. It does not matter how many people speak it: the genius of a language is not dependent on the quantity of its speakers. I was taking away from the genius of Gikuyu to add to the genius of English. I was taking away from the product of one genius to enrich the form of another.\n\nBut language is not simply an arrangement of sounds. Language is the people who speak it. There was more to my act of writing in English than simply enriching it at the expense of Gikuyu. I was taking away from the people who created Gikuyu and its genius, making sure that they could only access the rendering of their history through another tongue. In my early writings, I did not think about this; I was thrilled to see myself in print and reviewed in the English-language press, in Africa and abroad. But the situation was beginning to annoy me. For whom was I writing?\n\n_A Grain of Wheat_ came out in 1967. I was visiting Beirut, Lebanon, when I got a copy of the _London Observer,_ which carried a tribute to the novel. The warm sentiment was the same in the rest of the press reviews. But despite the very good reception, I felt uneasy about the implications of the linguistic form. Back in Leeds, in an interview with a student newspaper, I said that after _A Grain of Wheat_ I did not think I would write novels anymore. Why? Because I knew about whom I was writing but I did not know for whom I was doing it. The people about whom I wrote so eloquently would never be in a position to read the drama of their lives in their own language. On looking back, what I now find striking was that I thought of _not writing anymore_ instead of switching languages to write in the one accessible to the subjects of the narratives. I still accepted English as the only possible means of my literary deliverance. What a choice is implied in my response! Write in English or not at all. And indeed my next novel, _Petals of Blood,_ which came out in 1977, was written in English although littered with Gikuyu and Swahili words\u2014almost as if, in the text, I was announcing the contradiction in my position and practice.\n\nTwo events in my life changed my relationship to English and Gikuyu. In 1976, while a Professor of Literature at Nairobi University, I was invited to work at Kamirithu Community Education and Cultural Center, a village near Limuru town, thirty or so kilometers from Nairobi. It was an education and cultural project charged with providing literacy skills and cultural material. Dramatic performance became a natural means of achieving both. The Kamirithu community spoke Gikuyu, and there was no way that we could work in the village without working in the language of the community. For the first time in my life, I was being forced by the practical needs to face the Gikuyu language. _Ngaahika Ndeenda_ (in English, _I Will Marry When I Want),_ co-authored with Ngugi wa Mirii, was the immediate outcome, and it was received warmly by the community. But it was received with hostility by the post-colonial state. The performance took place on November 11, 1977; on December 31, I was arrested by the Kenyan government and detained at Kamiti Maximum Security Prison. I have written about this in many of my books, principally _Detained: A Writer's Prison Diary._\n\nWhat I have not mentioned in those narratives is the parallel between those events at Kamirithu in 1977 and those of my primary-school experience almost thirty years earlier. This was the first time I had been seriously engaged in writing in Gikuyu since that early effort at Manguu in its nationalist phase. The teachers and the students of the nationalist school had praised me for writing in Gikuyu. In the same way, the reception of the play by the community years later was fabulous. The applause in the primary school was followed by the takeover of the school by the colonial state, with terror unleashed on the speakers of the language in the school compound. The screaming student was forced out of the school. In my case, the community's applause was followed by my imprisonment; I was forced out of all classrooms and later into exile.\n\nIt could be argued that it was my contact with Kamirithu that reconnected me with the genius of Gikuyu. To a certain extent, this is true. The peasantry had retained their faith in the language. They kept it alive by using it. I learnt a great deal from them. But it was in Cell 16, at a Maximum Security prison, that I really came into contact with the genius of Gikuyu. I had been imprisoned by an African government for writing in an African language. Why? The question made me revisit the language, colonialism and my relationship to both. I had to find a way of connecting with the language for which I had been incarcerated. It was not a matter of nostalgia. I was not being sentimental. I needed to make that contact in order to survive. It was an act of resistance. So I wrote the very first novel ever written in Gikuyu, on toilet paper, in a room provided \"free\" by the post-colonial state. The novel, _Caitaani Mutharabaim,_ was published in 1980 to oral critical review. It was subsequently translated into English under the title _Devil on the Cross_ and came out in 1982 to literary acclaim in the English-language press\u2014but this time only _after_ it had been reviewed by the community. By the time I came out of prison in 1978, the decision had already been made. I would no longer write fiction in English (except through the translation of an existing Gikuyu text); from then onwards, Gikuyu would be the primary language of my creative acts. I have not looked back since.\n\n_Devil on the Cross_ was followed by _Matigari,_ and I have just finished a manuscript that I started in May 1997. When _M\u00fcrogi wa Kagogo_ comes out, it will be the longest novel ever written in Gikuyu. (The English translation will be published under the title _Wizard of the Crow._ )\n\nMore important has been the rise of other novelists, poets, and playwrights in Gikuyu. A new literature has been born. If this tradition has a discernible beginning and a location, it is in Cell 16, in Kenya's Kamiti Maximum Security Prison in 1977\u201378. Or maybe it was earlier, on the day I witnessed the plight of the screaming student. In trying to run away from his plight, perhaps I was running towards his fate. Only the genius of the language kept me alive to tell the tale.\n\n# GREEK\n\n#\n\n# _Split Self_  \nNicholas Papandreou\n\nGrowing up bilingual meant growing up with two cultures, two opposing identities. The Greek language was, in the first case, the language of politics, meaning the speeches of my father and grandfather. \"Greece to the Greeks,\" my father cried out in the mid-1960s; or, in my grandfathers more _apophthegmatic_ (in todays parlance, sound-bite) Greek: \"The King reigns but the people rule.\"\n\nGreek, then, was their language, and they had a famously firm hold on it. Theirs was the language of the humble men who gathered inside our kitchen during campaigns, of modern Athenians with razor-thin ties and dark suits, of women in black with absurdly thick fingers, much thicker and stronger than my mothers or my half-Polish grandmothers. These women believed it was their god-given birthright to stretch what little of my flesh they could grab hold of.\n\nYet it was the language of my mothers Anglo-Saxon Chicago side that ultimately won my heart. When we moved to Greece from Berkeley in the early sixties, so my father could enter politics, English automatically became my refuge, a way to prevent the complete loss of my embryonic identity.\n\nIn _Richard the Second,_ Thomas Mowbray reacts to his banishment from England: \"Within my mouth you have enjailed my tongue,\" he says, which is \"so deep a maim.\" Of course, my tongue was not fully imprisoned, since along with an ample supply of books, I had English as our in-house language.\n\nI stuck solidly to my English, meaning that I read British books (and learned to say _Bloody 'ell_ and _Blimey!_ \u2014which I was sure derived from a British rendition of _Blame me!)_ but also introduced comic-book expressions when beating up my younger brother, Andy. My less-than-Homeric blows to his small chest were accompanied by rapturous cries of \"zap!\" \"pow!\" and, for the execution, \"kablooey!\" I was always delighted to discover new words, especially slang. When an American teenager asked me where the toilet was so he could \"take a leak,\" I was bowled over. I imagined our bodies to be like badly built ships from which water leaked out. When an American family moved in next door\u2014I learned later the father helped put mine in jail\u2014I learned that \"man\" could be thrown into a sentence just about anywhere, and that \"cool\" meant, well, cool, man.\n\nYet Greek was all around, with classes in geometry, algebra, and geography, in a version of the language known as \"katha-revousa\" (meaning clean or pure). The language brought with it all the attendant cultural sidebars: priests grilled alive by Turks, women who jumped off cliffs rather than be taken by the enemy, and the Bridge of Arta, which reminded me of the story of Sisyphus\u2014the bridge would be fixed in the day but would collapse at night, and so a virgin was built into the bridge and this successfully reversed the trend. There was also the story of the World War Two collaborator who chopped off and then sold partisans' heads to the Germans like cabbage. When the war was over the man was caught, sliced lightly all over his skin with razor blades, then buried in a sand dune in Thessaly.\n\nI couldn't wait to tell my friends \"back home\" about the lamb we had for a pet, about the sheer steepness of the Isthmus of Corinth, about the shark I saw hanging by a hook on the island of Hydra, about the taste of souvlaki with pita and the caterpillars that hung in white sacks from the branches of pine trees. There were soon no friends on the other side, yet I was still possessed by the need to tell them everything that was different from America. It took me years to realize that I carried the other side with me at all times.\n\nBut I was most impressed by the enormous crowds that came to listen to my father and my grandfather, through whom I learned and imitated a rhetorical speech-making Greek. \"Greece of Christian Greeks catholically protestant,\" my grandfather hurled at the dictators when he was under house arrest. Even then, at the age of eleven, I marveled at how he squeezed three religions into one active phrase. Other sayings of his joined the pantheon of national tradition: _Many a people has deposed a king, never has a king deposed the people,_ or _All regimes boast political parties, only democracies have opposition parties._ The rhetorical expertise of both men put pressure on me to speak a Greek that was better than the average\u2014a pressure so daunting that, I now realize, I soon abandoned the effort and threw myself squarely into the camp of the possible.\n\nI remember selecting from my parents' library the thickest book I could find, presumably because the thickest book would provide me the greatest protection, which is how I ended up reading, at the age of nine, the sorry life of an architect written by someone with an unpronounceable first name (Ayn Rand's _Fountainhead)._ But I quickly strayed into the adventures of Biggles and Blyton's Secret Five, the Hardy Boys and every single Drew Sisters book I could secure from my sister, Gayle-Sophia. I refused to call her Sophia and persisted in her nicely American Gayle (after the actress Gail Storm, whom my parents had apparently taken a liking to in the fifties, when she was born). The rest of us had solidly Greek names: Nick, Andy, George.\n\nIt was my godfather, also a George, who got me thinking more about language. \"Why is a spoon called a spoon?\" he asked.\n\n\"That's silly,\" I recall answering. \"Because it's a spoon! And that's a fork, so it's called a fork!\"\n\nI hadn't yet realized that he was a fan of Magritte's. I liked my godfather because he looked precisely the way a godfather should look: three-piece suits, a smart tie, a hat, a cane, a well-trimmed mustache, with a distinct air of aristocracy.\n\n\"Do you know what your name means?\" he asked me when we sat in the dining room in our home in Paleo Psycheko.\n\n\"My name means... well, it means Nick!\"\n\n\"But in its full version,\" he offered, \"what does it really mean?\"\n\n\"You mean Nicholas?\"\n\n\"Two words in there. Can you see them?\"\n\n\"No.\"\n\n\"Nike and Laos, victor of the people.\"\n\n\"Cow!\" (I didn't know the whole expression yet.) So Greek words really did have secrets.\n\nI was off. I easily pried apart brother George's first name: _geo_ for earth and _orge_ for the verb plough\u2014though I had to look _orge_ up. George was no more nor less than a farmer. Little Andy with his blond hair and the black tuft sprouting out from the crown, who actually spoke only Greek, had a name that meant simply Man, like Oriana Fallaci's book _Un Uomo,_ about her Greek lover. Sophia, however, didn't have a synthetic name and hers meant simply Wisdom. Names like hers were less fun because there was no puzzle, no secret.\n\nThe bakers wife, Euphony, was fair game. When my sister once came home with a loaf of bread, I shouted: \"You phony! I bet you didn't buy it from Mrs. Good-Sound!\" Alexander, a friend, meant Man-Repellent. Thinking I was ahead of the game, I challenged my mother (who was having a harder time with Greek than I was) by demanding she tell me a word I didn't know, in any language. She threw out an easy one at first\u2014 \"sludge,\" I think it was\u2014which I proceeded to answer. Then came a far more difficult one, which I still remember to this day, amazed she knew such a long word. It was the word \"eleemosynary.\" I admitted defeat. \"Look it up,\" she advised. I discovered, to my delight, it had a Greek root\u2014 _eleimosini,_ meaning the quality of being charitable or charitableness.\n\nI began to look for English words which were in fact Greek\u2014 except that you would never think they were. I made a list of such words: For example, the word \"cemetery\" _(kimitirio)_ simply meant a sleeping place. The word \"police,\" familiar the world over, derived from the word _polis._ The word \"zone\" was the Greek word for what we wore around our waist\u2014a belt. My all-time favorite is a word you'd never think was Greek: \"disaster,\" meaning a bad alignment of the stars.\n\nI started to drive the family nuts by finding words that either sounded awful or made a lot of noise when you said them loudly, since I had now become the most word-infected family member:\n\n\"Dad, stop making all that _cacophony!\"_\n\n\"Mom, that souvlaki's really gonna hurt my _esophagus!\"_\n\n\"I can't concentrate with all the _susurrus_ from your newspaper!\"\n\n\"Dad, sometimes you are _a.pompous_ pop!\"\n\nFor a brash statement like that I could get popped myself, since my father, especially on his return from America, was growing less and less beholden to American child psychologists and had reverted more and more to the traditional forms of control\u2014 Ottoman law, as we called it, applied sporadically but effectively with the help of a _zoni_ (belt) to our behinds.\n\nOnce I had worked out first names _(Cleanthes_ \u2014Bouquet, _Calliope_ \u2014Beautiful-faced), there appeared a whole new treasure where I least expected it: Greek surnames. My sister and I would translate surnames to see how dumb they sounded in English: Mister Kalovelonis was Mister Goodneedle, while Mister Kalambokis was his Royal Highness Mister Corn. Our all-time favorites were the derogatory surnames like Mrs. Low-Butt and Mrs. Fat-butt, the famous Buttley sisters, like my mothers high-school heroines, the Andrews Sisters.\n\nThe last name of one of my fathers deputies made no sense but was certainly fun to say, if you could spit it out without stuttering: _Papapanayotou._ Three pa's in a row\u2014try them apples on for size. Our surname, with its double papa (our greatgrandfather was a priest, hence the Papa) was nothing compared to Mister Papapanayotou. My gleeful rendition of his name each day caused his name to be repeated by nearly all the household. \"Oh dear Mister Papapanayotou,\" my mother might exclaim for no reason.\n\nWhen he showed up one night, my father made a big thing of introducing him to me, then did me the awesome favor of actually adding yet another \"pah\" to the train. For days I savored the delightful extra\u2014Papa-pa!-panayotou. I don't think the owner of the surname thought twice about this delicious distortion, but I treasured it for weeks and kept seeing my father's slight grin as he machine-gunned the whole thing into the hallway, specifically for my pleasure. In a way I was being acknowledged as the family's linguist.\n\nDuring the dictatorship of 1967\u20131974, with my father in jail, we called on the American side of the family to visit us. One such member was a medal-studded Lieutenant Colonel who had just returned from service in Vietnam. Walking around Athens with all six-foot-five of him, in full military decoration, ignoring curfew, we were able finally to stand outside Averoff prison on Alexandras Street, where my father was being held. This was not just a thrill, a small act of revenge; it also reinforced the sense that English offered much more protection than did Greek.\n\nWe moved to Sweden in 1968, after my father, with the help of President Johnson\u2014who was quoted as saying, in full Texan drawl, \"Let that damn sunuvabitch out\"\u2014was amnestied by the dictators. A twelve-year-old loosed on Stockholm, I dipped briefly but excitedly into the Englishness of Swedish. Besides the blatant and unheard of pix of full-breasted vix which hung on just about every newsstand in the city, I was transfixed by the word for entrance and exit\u2014the blatancy of the _infart_ and _utfart_ strewn all over the place. Adolescence is nothing if not delight in the scatological (Greek for \"study of excrement,\" as opposed to eschatological, \"the theology of death or endings\"). My favorite: the word for constipation in Swedish was _ferstoppning,_ meaning exactly what it says, thank you.\n\nOn the run, we finally ended up in Canada in the last year of the decade, under the good graces of the then Prime Minister Pierre Trudeau, who offered political asylum to my father if he wouldn't overdo his criticism of America (a restraint my father was unable to obey). Strangely enough\u2014or perhaps not so strangely\u2014my real home, the one I thought of as my real home was for many years the country of Canada. And Canadians\u2014 well, they spoke pretty much like we did but, to my great delight, not exactly. When I played basketball, the referee might shout \"Eeyoot of Bounds!\" Objects were \"yea high,\" highways had \"soft shoulders,\" and a decent-sized snowplow weighed \"two ton\" without the pluralizing ess. You could talk like you were a hardware employee showing a customer the goods and get away with it: \"Well, there you've got _your_ Phillips Screw and _your_ five-inch dead bolt... \"The wonderful possessive _your_ gave you instant ownership over all such male objects. There was also a machine called a \"snowblower\" which, besides snow, would churn out pebbles, animals and, in at least one James Bond film, a couple of bad humans. Ski-Doos raced across the snow at night in the vast white space: an upgraded version of Dr. Zhivago.\n\nGoing to school in rural Ontario\u2014where, for some reason that I never discovered, my father decided we should live\u2014I learned that the business end of a scythe was called a _snath,_ that Viceroy butterflies look like Monarchs but don't have the same flight pattern, and that Lord Strathcona drove the last spike of the Canadian Pacific Railway on November 7, 1885. Swamps were called _muskegs,_ a frozen pond thundered when you walked on it, trapped air bubbles looked like crystal balls, a hockey puck traveled up to a hundred miles an hour, a solid slapshot was as satisfying as any slam dunk, and, contrary to popular wisdom, when it got really cold it didn't snow. Driving along Route 13 in King City one cold afternoon, we passed the small _kimitirio_ with its snow-laden crosses sticking up like frozen spinning jacks. I turned to my mother. \"Mom, when I die, this is where I want to be buried.\" Not in Berkeley, not in Greece, not in Sweden, but here, in King City, Ontario. I had never seen her cry before because of something I said.\n\nIt was in Canada I first heard a third but instantly recognizable language, one which I sort of knew without ever having learned it. It was the English spoken by first-generation Greeks, what the community of bi-culturals like me now informally calls Gringlish. Gringlish usually takes English verbs or even nouns and pops them directly into the sentence. Will you park the car becomes, in Gringlish, \"Tha kanis _park_ to _caro?\"_ How many blocks away do you live becomes \"Posa blockya makria?\"\n\nI dislike the word _Gringlish_ because it sounds like a combination of two evil heroes, Grendel and the Grinch. I prefer a word of my own invention, which is perhaps derogatory but more to the point: Dinerese. In the Greek diners spread across Route One, in Chicago or in Florida, in perhaps the most famous Greek eatery of all, Astoria's Neptune Diner (nested neatly beneath the Triborough Bridge), you can still hear this language. \"The Greek people,\" a phrase much liked and much used by my father, in Dinerese becomes \"the Greek peep.\" Greeks love the peep. Peeps of the world unite. Long live the peep. Fast-speaking Greeks dismiss the distance between words. Like a hut kupukuf-fee? No, you sumunabeets?\n\nMy favorite interchange occurred while I was in college, when a Greek Greek who had learned English only from his law books and who worked part-time at a Greek pizza place in New Haven encountered an unhyphenated America. The conversation went something like this, as best as I can recall:\n\n\"Ordered a double cheese 'zah, half-pep, half anch.\"\n\n\"I am sorry. What was that?\"\n\n\"Half-pep half-anch, man. The full spread.\"\n\n\"I'm sorry. I don't speak colloquial.\"\n\n\"You don't speak what?\"\n\n\"Slang. That's it. I don't speak slang.\"\n\n\"Who's speaking slang? I'm speaking English.\"\n\n\"Do you mock me, sir? Do you deride me?\"\n\n\"C'mon, man! I just want my 'zah!\"\n\n\"You think you're in your home, that you can talk like this?\"\n\n\"Man, this ain't no home, this is Naples effing Pizza last time I looked. Which planet you from?\"\n\n\"Planet is the ancient Greek word for wanderer, sir. I know precisely my origins, sir, from Arta, in Western Greece, sir, where they once built the bridge.\"\n\n\"Sheesh! Get back on that ship and return to wherever!\"\n\nStrangely enough, the Greek I had learned as a kid in the \"home\" country was a passport into restaurants, brought instant connections in college with others whose surnames began with _Papa_ or ended with _opoulos,_ and afforded me instant, no-questions-asked entry into a distinctly raunchy world of nightclubs owned or run by Greeks: Mykonos, Zorba's or, in Baltimore, Towson Bouzouki.\n\nThe burden of the Greek language weighed heavily on me because, more and more, I was being called on to represent my father, who in the late seventies now led the chief opposition party, bent on bringing socialism to Greece. I attended \"caucuses\" on Cyprus, helped organize the Greeks At Yale (lovely acronym for that), and gave numerous fund-raising speeches in hardcore Greek-American communities located in Astoria, Baltimore or Chicago. I was a stand-in for my father, who was climbing the steps to the palace of power. One place to which I was obliged to return time and again was the Crystal Palace in Astoria, Queens. Here I could find the whole enchilada of Greek-American linguistic abortions. The Crystal Palace was the prime location for thousands of Greek-American events over two to three decades: political rallies, wedding receptions, dances, baptisms\u2014a Coppolian ethnic-American setting of sheer kitsch. Much later I realized that there once existed a real Crystal Palace, built over a hundred and fifty years ago in England, \"the crystal edifice that can never be destroyed,\" as Dostoyevsky puts it in _The Underground Man._ Though I am no longer enmeshed in that particular strain of ethnic America called Astoria, and though Astoria has now lost much of its Greekness, I drew some meaning from this indirect link to the Russian writer, even if the meaning is no greater than a micro nano bit.\n\nBy the age of twenty-nine, I had acquired yet another language. A Ph.D. in economics taught me everything there was to know about _transcendental logarithmic cost functions, variance covar-iance matrices_ and _three stage least squares estimators._ Never having returned to Greece except for summers until my studies were completed, I returned for my military service in the late eighties, exactly a week after defending my doctorate on a comfortable Ivy League campus.\n\nThat's how I found myself on the island of Lemnos, in Northern Greece, inducted into the Greek Air Force. I could have relied on my American citizenship to avoid military service altogether, but such an act would have been highly unpatriotic; besides, I actually liked the idea of wearing a uniform and carrying a gun and not reading another economics article. There was also this: I imagined bumping into an officer who had arrested my father the night of the coup, the same one who had pointed a machine gun at my face. The thought excited me. I am sorry to report that such a meeting never occurred and that the extreme right-wing officers saluted me as I did them.\n\nThe island was honed of volcanic rock, the home of Poseidon. The barracks were full of raw eighteen-year-olds most of whom spoke with distinct regional accents. I immediately felt like an intruder, a jokester, a false twin who would soon be discovered for pretending to be the Prime Ministers son. I lived, thought and wrote in English, yet my father was by now not only Prime Minister but also Minister of Defense. This made him head honcho of all soldiers. As the son of the nationalist leader, I was supposed to be the automatic expert on all matters Greek, to know the Heroes of the Revolution, to know which minister served what post and what year, and, worst of all, to make no spelling mistakes on all documents for which I was responsible.\n\nYet for all the pressure, there was one tremendous benefit for a word-infected person like me: army slang. Greek army slang.\n\n\"With someone else's ass it's easy to pretend you're gay,\" I heard one soldier say after returning from a particularly cold night shift. Another soldier who'd stubbed his toe at night on the bunkbeds shouted: \"Screw the donkey that ate Christ's palm fronds on the road to Nazareth!\" If you dropped your rifle you would most likely think of God and shout, \"Screw the Virgin Mary's ear!\" (This was a reference to Immaculate Conception; certain Fathers of the Church once held that such a conception had occurred via the good Mary's auricular orifice.)\n\nNot that the soldiers had no sense of their ancient heritage:\n\n\"Halt! Who goes!\"\n\n\"Friend!\"\n\n\"What did Ulysses say when he got to Ithaca?\"\n\n\"Screw me!\" Or:\n\n\"How did Ulysses spy on the Trojans?\"\n\n\"Through the wooden horse's wooden butthole.\"\n\nWe also used more standard passwords: Hercules-Lion, Achilles-Patroclus, Sophocles-Oedipus\u2014things the Turks, the enemy about fifty kilometers away, were supposed to have no idea about and could never answer properly, even if they had wanted to.\n\nMy greatest fear at the time was to be called up in front of the thousand or so soldiers to say the Lord's Prayer. Because of all the back and forth between countries, I had missed the teaching of it in either language and for the life of me couldn't remember it. Each night, as we were all standing in line, the commander would call out a name at random and ask the specified \"grunt\" to come recite the prayer. While waiting for the name to be called, I would try to remember the prayer, filling in the empty Greek parts with what I remembered in English, then translating it back into Greek. But this was a puzzle best not done under pressure, next to a thousand breathing bodies. Fortunately, the stars were not once in disorder. My name was never called.\n\nAll this was a rich linguistic pillow in which to sleep at night. When I finally decided to write, I saw that the friction of the two languages had great value and could convert the trivial into the metaphorically rich. A clever person is an \"eagles talon,\" a tall man is a \"Cypress-lad,\" a piano is a \"tooth-mattress,\" the earth is an \"ant-sphere,\" a boy's erect penis is a \"fakirs flute.\" \"Never scowl at the lowest steps,\" a saying goes, \"since you need them to get to the palace.\"\n\nI discovered rhyming couplets from the island of Crete which I tried to translate:\n\n_Others shrivel up from the times, the wars and years_\n\n_but me, I shrivel up with the pains and the fears._\n\n_The wind beats my clothes and the sun eats my knives_\n\n_and a small little love eats up my insides._\n\nThere was gold then in them thar hills. \"I can hear the salty smell,\" a village woman once told me when the wind brought with it bits of the sea. I was shocked by the confounding of the senses, and later learned that she had just expressed what philosophers call \"synesthesia,\" where one sense \"leaks into\" the other. (Ah, there's that unexpected four-letter word coming back at me.) A more literary example of a synesthete is Nabokov, who in his autobiography _Speak, Memory,_ tells of seeing colors when he hears the alphabet pronounced\u2014a trait he refers to as \"colored hearing,\" or _audition color\u00e9_ in French, which, I guess, sounds more sophisticated.\n\nMy Greek grandmother, Sophia Mineiko Papandreou, half-Greek and half-Polish, offered me a name of someone she once knew, a little girl named Eulalia. She told me it came from the prefix _ef,_ meaning good, and _lalia,_ meaning speech. It was a name I found much later in my readings of Americans from the South, when I decided to become a writer myself. I did actually meet a Eulalia, on the island of Syros, with its Catholic and Orthodox churches jutting up into the sky competing for space. I was presenting my first book, _A Crowded Heart,_ written in English but available in Greek translation. The crowd was decent-sized. After all, my surname guaranteed me instant recognition, and one effect was a small but constant crowd at my book readings.\n\nAn aging actor, Constantine, had been chosen to read a section from my book. He read it as if it were ancient Greek tragedy, shouting at the top of his voice, and once the applause subsided\u2014an applause which was obliged to rival his efforts\u2014 he sat down next to me. While others continued to speak about my book (the mayor, a deputy from my fathers party, a high-school teacher with two books of poetry under his belt, and god knows how many others), he struck up a conversation with me as if we weren't on the podium. I kept hunching lower in the hope this would induce him to lower his voice. Suddenly he squeezed my thigh excitedly. \"See that girl there, over there, with the dark hair and those eyes? You see her? She once had a speech impediment but I corrected it with four years of lessons in orthophony Take one guess what her name is.\" That's how I met the only Eulalia I have ever known. I even got the chance to sign her name in my book, which I did with a calligraphic flourish. She was indeed a tall, dark-haired beauty and she did indeed speak with perfect diction, the way a Eulalia should, but nope, I never saw her again.\n\n_Lalia_ in Greek means voice or language or tongue. In the Swedish the word for Speak is _tala. Tala svensk?_ In Danish _lalle_ is a drunken persons babble. When I started to learn some Spanish I thought I heard an echo _of lalia_ in _habla,_ with that _la_ at the end of it. From a Brazilian acquaintance I heard _fala_ for talk. _Thinkparier._ Or even parlance.\n\nOne summer, back in Greece between college years, I visited the pine-filled island of Skiathos. A fisherman took me to his favorite beach, Lalaria. \"Why is it called Lalaria?\" I asked.\n\nHe had an answer (when doesn't a Greek?). \"You see those rocks there?\" He pointed to the large round stones like ostrich eggs that formed the beach. \"When the sea hits those stones they talk. La la. Close your eyes and listen.\"\n\nI think all members of my family are wounded by language in one way or another. Brother George, in the words of his detractors, \"is our first Minister of Foreign Affairs who actually speaks a _second_ language\u2014Greek.\" My mother doesn't do television interviews because she is worried she will place a feminine pronoun to a masculine noun, and this after leading the Greek women's movement for decades. My sister has escaped to Canada and her little son now speaks fluent Canadian. For a long time, my younger brother Andreas prepared his economics classes at the university down to the last word, so that he wouldn't make any grammatical mistakes. My father, always burdened with the suspicion that he was too American, commanded both languages fluently. Ironically enough, he was perhaps the only member who never suffered from the mistake bird of language.\n\nI now treasure the split. English acts as a passport into unexplored territory, into the terrain of my fictional Greece, into the Greece of my memory, the Greece of my childhood.\n\n# ITALIAN\n\n#\n\n# _Limpid, Blue, Poppy_  \nM. J. Fitzgerald\n\nIn early 1959, while we were living in a small seaside town along the Ligurian coast, Italian television made a documentary of us. The documentary was called _Una famiglia Americana in Italia_ \u2014 An American family in Italy.\n\nThe decision by the fledgling RAI to make such a documentary may have had a lot to do with the reversal: an American family in Italy rather than the thousands of Italian families forced to emigrate to America. But I imagine that one of the reasons they chose our family from what was not an insignificant population of Americans in Italy was that we were not in fact living an expatriate life in Milan, Florence, or Rome, among larger or smaller clusters of other Americans, but were living integrated in the small-town life of Levanto, the only Americans there year round.\n\nA second irresistible reason in that children-loving country must surely have been the presence of a clutch of six duckling children, three boys and three girls, all more or less the same age (there is a mere seven years between the oldest of us and the youngest), all pale and freckly all fair-haired and, like all children, all absolutely deserving of the bruising pinch on the cheek and the exclamation \"bella!\" Of the four of us who were in school, three went to the local elementary school: the two boys in the boys' section, I across the green-and-white-tiled corridor in the girls' section. My older sister was at boarding school at Trinit\u00e0 de' Monti in Rome, an exclusive enough place run by French nuns, where the student population consisted of fewer than a hundred daughters of the higher middle classes, mostly from the poorer south\u2014Calabria, Puglia, Basilicata, Campania, Sicilia\u2014but with a sprinkling from the more northern regions and from Lazio itself. By the time I joined her briefly in 1963, there were a couple of Americans with whom I walked in irrepressible sobs to the American Consulate to sign the condolence book following the 22nd of November. But there were none when she went in the fall of 1958.\n\nA third consideration may have been my father's status as _Il Professore_ who seemed not to have a job\u2014he made a poem of my brother's famous boast among his peers that the reason why _il babbo_ was at home all day was that at night he was a burglar\u2014 even though the study where he spent his days translating the allocated lines of Homer's _Odyssey_ was an inviolate place we were rarely privileged to visit. The running of the household was squarely on my mother's shoulders, with the help of two live-in maids, Maria and Alice. Each went home once a month to villages high in the Ligurian mountains, with ten dollars for their families.\n\nFor months, it seemed, a television crew came and went, stayed days or perhaps it was weeks, then left and then returned. I remember having to walk again and again down the brick and stone stairway that led up to the olive and vine grove behind the house. I remember being incapable of sitting still in class with hands behind my back, before they were to film me playing _F\u00fcr Elise_ on the piano with my father. I remember dashing in and out of my father's study: it took what seemed like hours to set up the piano just for the filming.\n\nI stopped taking piano lessons shortly after, many months before we gathered in the priests house to see the documentary. As my father predicted when he sighed at my stubbornness\u2014I don't want to, I hate it, I don't care\u2014I regret it today with as much intensity as I then detested the dull _solfeggi,_ the thump of the right hand on the left to achieve a metronome beat, and the endless repetition of the scales. I am pretty sure now that I pouted and stomped and insisted in Italian, and that my father expressed his regret and disappointment in English. But I have no memory of it. The only memory I have of this use of two languages is that when my parents did not want us to understand something, they would spell it out in English. We understood their speech in English completely, but spelling, reading, and writing defeated us all until we were in our teens and had begun the migration to the English-speaking world.\n\nIn the early fall of 1959, before school started in October, we moved to Florence, and for three years we rented the _piano nobile_ of the Brewster villa: the priest's house was attached to the small chapel that was itself a wing of the villa. Sometime in late 1959 or 1960\u2014I have a dim memory of rain and chill, so it may have been the winter\u2014we crowded into the parlor with a number of other folk who did not have television sets, and sat on the sagging sofa or the hard straw chairs or the jute-covered brick floor to watch the documentary. My parents took small sips of the artichoke cordial Cynar, the _digestive_ that was the preferred after-dinner drink to offer guests in modest households, and we were offered _caramelle_ under the disapproving eye of our mother, helpless to fight the rules of hospitality which required candy to be distributed to children on great occasions.\n\nOnly the setting has stayed with me from that very first viewing\u2014that, and the inch-long hard fruit candy with the soft center, wrapped in crinkly cellophane. It was suddenly in every _Bar_ and _Alimentari,_ but I had never tasted it before, and it immediately replaced my favorite _mentine_ \u2014the tiny sugar drops in various flavors that are miraculously still for sale in Levanto. The word _candy_ does nothing to my taste buds, whereas the word _caramella_ brings instantly back the sweet crunch of the teeth through the shell, softened by sucking for as long as possible, to the soft center.\n\nI did not see the documentary again for over twenty-five years, not until my father was diagnosed with cancer and undertook the six-month-long battle to contain the spread from the lungs to the liver. I spent long days with him in the summer of 1984, while he fought the losing campaign with chemotherapy and radiation. My brother succeeded in spiriting a copy of the sixteen-millimeter documentary from the archives of RAI, had it formatted to tape, and had copies made. He came by with the copy for my father on a beautiful summer afternoon: the sun shone on the canvas shades without dazzling its way in, merely giving a fullness of white light to the quiet sick room. My brother and I flanked our father as he lay on the hospital bed that could be raised and lowered. He sat up a little, his legs over the folded end of the mattress, his back and head against the raised top of the mattress, and we watched the official, thirty-minute-long edition of our life twenty-five years before.\n\nThe most startling thing at that second viewing was how very halting my parents' Italian was. But whereas my mother valiantly improvised her way through the hesitation and the mistakes into comprehensibility my father was utterly dependent on the note pad he kept glancing at as he spoke to the interviewer, and his pronunciation was atrocious. As I sat by him in the muted summer light, I realized with shock that as an adult, after I had learnt English, after our family migrated back to the United States and then broke up, I had not heard him speak Italian, and save for the occasional, absolutely untranslatable word, I had not spoken to him in Italian. With my mother it was a constant slide between the two languages, used easily and interchangeably. Her Italian was strongly accented, but she would speak it whenever she could. My father seemed to me at that moment to have abandoned Italian as completely as he had our family life.\n\nI glanced at him beside me on the sickbed he would not rise from, his beautiful nose ever more aquiline in the gaunt features, the green of his eyes darker in the deepening sockets, the long expressive thumb at rest from its characteristic gesture\u2014 forming a circle with the middle finger while he clicked his tongue against the roof of his mouth to make a point. After all, it was twenty years since the migration started in the fall of 1964, when he took up his position at Harvard; twenty years since the four eldest of his children had been dispersed to boarding schools in four different countries. Childhood in Italy, the momentous source of experience and impression that had determined so much of my life, the eleven years between October 1953 and October 1964, were no more than a brief interlude in the life of my parents, an interlude that my mother clung to, and that my father had shaken off. Perhaps he never did speak Italian with us? Perhaps he spoke in English, and it was sufficient for him to understand the Italian we spoke to communicate with us?\n\nThe tape came to an end and my father grinned at us, sighed deeply, but made no comment. Embarrassed at his lack of fluency in the language I spoke as fluently as English, I did not think then to ask him whether he had ever spoken to us in Italian at all. Although he watched the documentary a number of times with the rotation of siblings who were in and out as constantly as I was during those last months, he and I never watched it again together, neither during the rest of that last summer when I was there for days at a time, nor in the blaze of the red and yellow fall, when I would fly back from England regularly to be with him. I have seen it only once since then. I will perhaps look at it again before I finish the composition of this piece. I feel a strong reluctance to do so, partly from a measure of shame at our privileged life\u2014a shame engendered by a political correctness I simultaneously view with great suspicion. And partly I am reluctant for reasons I cannot quite articulate, as nebulous as my fear of the ache of nostalgia and as clear as the possibility of the stab of regret at the loss and then the death of my father. And now there is the rawer bereavement at the recent death of my mother: in my memory of the documentary, she still wears the long Native American braids she had in the photos of us taken on the beach in Sestri Levante and Levanto.\n\nWhen an assortment of us coincided by my fathers bedside, we talked about the documentary and remembered those years, which were by then really the only link between all of us. But the talking, I see now, came from us, not from him. As I write, I realize that he had supplied a context for our whole lives that was not only not his, but that was as distant from his background as it could possibly be. The blue of the Mediterranean, the dazzling light of heat refracting, the arid steep slopes of the olive groves, the ground that brought to the surface more stone with each rainfall\u2014gathered daily _-per fare una casa,_ to make a home, by the sweetly retarded son of the _contadino_ who looked after those trees and picked the olives and the grapes\u2014 the acrid smell of the _mosto_ in the barn where we too trod and gleefully squashed the grapes with bare and dancing feet, the delicate scent of mimosa, the medlar we climbed into, the hydrangea whose leaves made crowns, and the ubiquitous pines and magnolias\u2014what had they in common with the landlocked Midwest of the United States, of which I knew nothing in 1984 except that corn grew as high as an elephants eye? What did our childhood days filled with sibling squabbles and friendships have in common with the small household in which my father grew up against the backcloth of a loss too huge to fathom, the death of his mother and brother?\n\nIt is such a different context from mine that it has only been possible for me even to begin to comprehend it since my own motherhood has shown me how the loss of me would affect my son, and since life's turn has dragged me kicking and screaming to the same landlocked Midwest, mere hours away from my fathers birthplace, Springfield, Illinois, where he lived those all-important years to adolescence in the singular quiet companionship of an ailing father, with winters of ice-skating in a dazzle of deep snow rather than summers of snorkeling through a dazzle of sea. A sea whose intensity of blue can only be contained in the Italian word _azzurro._ The word _blue,_ an anodyne descriptive term, does not convey that childhood sea, but _azzurro_ brings it all back with a violence in memory that invariably hits me like a punch in the stomach.\n\nI think now, as I write and look back at us watching _Una famiglia Americana in Italia_ on that sunny afternoon of his last summer, that perhaps my father never left the Midwest\u2014unlike my mother, who truly did leave Texas and actively chose the life she led. I think that perhaps my father merely found himself migrating from Springfield to Choate, to Harvard, to New York, to New England, to Italy; just as I feel I have never truly left Italy and have simply found myself migrating from country to country, borne by necessity rather than driven by choice.\n\nBut what about the choice I made _not_ to read or speak Italian? It was a very deliberate decision, taken in the wake of the earnest aspiration to \"be a writer.\" I had to choose between the two languages, but why did I not choose Italian? At sixteen, when the decision was taken, Italian and English were in equipoise\u2014if anything, Italian was the language in which I was more fluent. After two years at boarding school in England, my spoken English still suffered from Italian intonations, and my written English from errors consequent on an Italianate and Latinate sentence construction.\n\nI had always been a ravenous reader, although never a precocious one: at seven I lived on a regular diet of _Piccole Donne, Piccole Donne Crescono, I ragazzi di Jo_ (Little Women) and _Heidi._ At eight I read _Cuore_ perhaps a hundred times. At ten I read in Italian the pre-teen bright blue hardbacks about English boarding schools that I could buy for five hundred lire (stolen, I'm afraid, from my mothers handbag) at the newspaper kiosk at the bottom of the tree-lined avenue that led to the square where we lived in Florence. And after we moved to Perugia in the fall of 1961, I read every single one of the hundred or so books the previous owner of our house left behind: 1930s paperback editions of Emilio Salgari as well as romances by forgotten authors, in one of which a turbaned and bearded man converted to Mahomet to be able to reach the woman he loved in a harem... I did not at twelve read _I Promessi Sposi,_ or _Mastro Don Gesu-aldo,_ or the whole _Iliad,_ or even the whole of the _Zabaglione,_ extracts of all of which we were reading in our first year of secondary school. I did not read Pavese, Sciascia, Bassani, Buzzati, or even Calvino, whose _Fiabe Italiane_ would provide the final impetus fifteen years later for \"Creases,\" the first story I wrote that defined my ambition as a writer. The determining book at thirteen, during my eighteen months at boarding school in Rome, was a three-volume novel translated into Italian from German, of a decidedly satisfying symmetry: A gifted Austrian violinist of seventeen marries a man exactly twice her age. They have twin girls who are seventeen in the second volume, where they meet and marry twins. In the third volume their children in turn are seventeen, and the saga ends with the girl of the first volume, now an old woman of roughly my present age, playing the violin after her husbands death at sixty-eight, overheard by her thirty-four-year-old children and seventeen-year-old grandchildren.\n\nDuring that year, 1963, I wrote my first novel. In Italian, of course. A gifted teenage pianist dying of leukemia meets a girl whose boarding school abuts his hospice, and a doomed romance blossoms between them... or maybe _she_ is dying of leukemia and is consoled by his wonderful playing of Chopin from the conservatory which abuts the hospice. It was a short novel, the hard-won length of a notebook, written in collaboration with two schoolmates during the endless evening homework hours, and laboriously typed out on my fathers Olivetti 22 during school holidays. I wrote poems then, too, in Italian: all critiqued by my classmates with comments ranging from _mi -place_ to _bella_ to _carina._\n\nBut I remember also the misery of those days. The school did not have a library, and the random selection of books on random shelves scattered here and there around the school was so quickly devoured that I had nothing new to read. And whereas in my childhood I had been content to read the same book over and over again, by then I wanted a new book every day. Even when I got home for the holidays, the situation was no better. There were shelves and shelves of unreadable English books, but the supply of Italian ones had run out, and there was no kiosk at the end of the avenue where I could replenish my supply with stolen money: in Perugia, the only stores within walking distance in the poor village above the house were a meager _Ali-mentari_ always swarming with flies, and a general goods store that only stocked the photo-romances I was explicitly forbidden to get. I remember trying in vain to slow my reading down so that I would not come to the end of a beloved book ( _he cinque Sorelle d'America,_ for example, Five American Sisters) and have to scour to find something I had not read.\n\n_Al centro,_ where we would saunter sometimes up and down the length of the cobbled main street in the early evening, from the Duomo to Piazza Italia and back to the Duomo\u2014the _passeg-giata_ I was just beginning to dimly know as an occasion for flirting and courtship\u2014there were two bookstores. But even there, no book appealed to me; they were either the same blue-backed hardbacks I had bought with stolen money in Florence, or uninspiring grown-up novels. New _Gialli Mondadori,_ the detective and murder mysteries that had superseded stories of stowaway children in English country houses as reading that could be bought from newspaper kiosks, only appeared once a month or so. I would have devoured one of those in the car on the twenty-minute ride home, if the road had not been so winding it would have made me car-sick. I certainly devoured them before falling asleep, incapable of putting them down until the mystery had been solved. And then time stretched out in an eternity of not reading. The crossword puzzles in _La Settirnana Enigmistica_ and the random accumulation of odd facts that made up much of the rest of the weekly publication, with its impossible rebus and the silly cartoons, did not substitute for the deep immersion in lives not my own that took place when I was reading a novel. Nor did reading _Grazia_ or _La Domenica del Corriere_ whenever I could lay my hands on them, nor did the comic books, although _Topolino, Nembo Kid,_ and _Tex Wilier_ were always around\u2014 demanded by my siblings as earnestly as I demanded the _Gialli_ from the newspaper kiosk\u2014and I did read them. I had even been driven to reading the Jack London novels that my friend Luciana thrust on me, and which I did not enjoy at all: adventure was not my dream as much as romance.\n\nJust at this moment of cusp, in my second teenage year, as the world of childhood receded so rapidly I did not even have occasion to regret it, I was sent to the longed-for boarding school in England. We did not have the grey uniforms with coveted striped ties, but we did have to wear berets and blazers. Eventually I even indulged in midnight feasts of crumpets and marmite washed down with the cider I did not like but pretended to, and the beer I could not even pretend to like. Eventually I did get into trouble for smoking in my room. But the great revelation of English boarding school was not social\u2014I was too fundamentally introverted for that. It was literary. We read _A Midsummer Night's Dream_ and especially _Romeo and Juliet,_ which satisfied every particle of my dreamy romantic, and sentimental heart. We read _Jane Eyre._ There had been nothing like _Jane Eyre_ in my reading so far. None of the romantic heroes of the 1930s novels in the attic of our house, or the stories in the hybrid magazines of photo-romances that I hid under the mattress whenever I had managed to sneak one home, had ever been blinded or handicapped in any way: the passion that I had felt at twelve for Leopardi, because he was a great _hunchback_ poet, was finally being justified. And then I read _Wuthering Heights._ Bad brooding Heathcliff, how I loved you: as much as Catherine did, both of us with all of our misguided romantic hearts and souls.\n\nBoarding school was the perfect setting for me. As long as I did as I was told when I was told to do it, I was left alone. I basked in that friendly indifference. The library had shelf after shelf of books that appealed to me\u2014Georgette Heyer, Agatha Christie, Charlotte Bront\u00eb, Kathleen Winsor, Emily Bront\u00eb. To access them, all I had to do was learn to read English as quickly as I was learning, after the first mute months, to speak it. Patricia High-smith, Dorothy Sayers; by the end of my second year, _Pride and Prejudice_ and the wonderful revelation of Jane Austens astringent romanticism; George Eliot's ocean-sized scope in _Middle-march;_ Shakespeare's _Antony and Cleopatra; Twelfth Night,_ with Viola and her doppelg\u00e4nger brother both in love, and an end as satisfying as the ending of the three-volume saga of my twelfth year, but in a language that exactly mirrored the excessive drama in my heart.\n\nReturning to Italian writing after gorging on this diet of English\u2014Leopardi's _L'Infinito_ was restrained in comparison to Dylan Thomas's _Fern Hill_ \u2014was like returning to milk toast after scones with clotted cream. I loved the accumulation of adjectives that a language so rich in words could indulge in, instead of the nuances in the repetition of the same adjective that gives Italian its power. I loved the exaggeration of English, the curlicues of language, its baroque quality. Many of the churches, much of the painting, and peoples gesturing in Italy are baroque. But the language itself is severe: its beauty lies in elegant simplicity and the hypnotic power of its sound. And when it is distorted by the wrong rhetoric in an attempt to \"enrich it,\" it becomes impenetrable without gaining in power. English has to work to be elegant and simple, because its sounds are rarely if ever as spellbinding as Italian, and so much of its nature is tortuous. But how fabulous the honed expression of that tortuousness can be for a girl who sees so clearly reflected in this language her fervent and histrionic self: not one word to describe her feelings, but half a dozen variations. At sixteen, there was no contest. English was the language of my sensibility, the language with which I would write poems as full of words as Dylan Thomas's, novels as rich with emotion as _Villette,_ dramas as powerful as Christopher Fry's _The Boy with a Cart._\n\nI ran away from Italian, carrying the parcel of intensity and aspiration, nourishing myself with the rich language of my parents, although I never consciously acknowledged then that it was my parents' language. I traveled with it for ten years, and ended up jettisoning the original aspiration: I no longer wanted to emulate Charlotte Bront\u00eb, Dylan Thomas or Christopher Fry. I did not even want to emulate Virginia Woolf or James Joyce or Nabokov, nor even William Golding or Hopkins or T. S. Eliot. I ended up hitching rides of ambition here and there, but without a destination, homeless. Until I read Dino Buzzati's _Sessanta Racconti_ and then Italo Calvino\u2014 _Il Barone Rampante, Il cavaliere dimezzato, Se una notte d'inverno,_ and most especially _Fiabe Italiane._ Then my ambition became to write English with an Italian accent. In fact I wanted to write English with more than just an accent: my ambition became to throw out all that had first drawn me to English, all the baroque quality, and write English as if it were Italian. Now, while living in London, I could write my first volume of stories, _Rope-Dancer,_ where most of the stories were inspired by Buzzati and Calvino. Now, while commuting to Southampton, I could write my first spare novel, _Concertina._ The two books were published concurrently a few months after my fathers death in January 1985\u2014less than a year after I sat beside him to watch _Una famiglia Americana in Italia,_ when I found myself so shocked by the poverty of his spoken Italian. Had I come full circle?\n\nNo. The end of my fathers life was the beginning of my life as a writer, and I was not in a circle or even a spiral, unless it be a twister. The role of Italian in all this turns out to be more complicated, even if I did physically move back to Perugia in 1986. I set myself up as a freelance writer, believed I could survive on an income patched together from writing and translations, supplemented by giving English lessons. But I had not come home as I thought when I made the move, and I had not found my style as I thought when I was composing my first works. English was playing its unfathomable role. Looking out at the brown-green Umbrian hills towards the ancient blue ridge of mountains, I dropped the Italian accent, abandoned the ambition to write English as if it were Italian. My next novel, _The Placing of Kings,_ was inspired by _Pride and Prejudice_ \u2014by the most English of English writers.\n\nBy the time I finished it, I knew that living in England was not living at home. But I also knew that if Italy was home, it was not providing me with a living, just as it had not provided generations of Italians with a living (albeit my kind of labor was not backbreaking physical labor), and I found myself taking on what would be perhaps the greatest challenge of all, by emigrating to America, to my fathers Midwest. I had only been on the East Coast before, and no visit had lasted longer than the summer I spent with my father in 1984.\n\nNow I watch myself trying to adapt to a world where the sense of history and of time are dizzyingly relative, a world where a European background and education are held in awe and despised at the same time, a world where Italian\u2014my childhood language, the language with which I still add and subtract, the language of the only nursery rhymes I remember, the language of the only tongue twisters I can twist my tongue around\u2014is _exotic._\n\nBut is it in fact any more exotic than we must have seemed to Italians in 1957, when the documentary crew recorded our daily life in a small town that had rarely seen an American family living its daily life\u2014a small town that had never seen a bunch of American kids speaking Italian as if it were their native tongue? Is this finally the circle I've been in all my life, this being perceived as \"exotic,\" which pertains to anyone in any place who is not native of that place\u2014whether it be an Italian or a Somali here in the States, an American or an Asian in Italy, an Asian or an Italian in Somalia? But unlike the Asian-American, the Somali-American, the Mexican-American, the Italo-American, I cannot enjoy the solidarity of belonging to a group, because I cannot say I am Italian here and I cannot say I am Italian in Italy. In fact, I cannot even say I am English here, or English in England. Nor can I say or think of myself as American, because I am not. I have to live straddling the three cultures I have absorbed. Yet in only one do I really feel at home.\n\nThis, I discover, is what I meant when I wrote that I feel as if I have never left Italy, in a way that parallels my father never having left Springfield, Illinois. The experience of childhood, which always determines _some_ adult experience, for some determines _all_ adult experience. However far my father went physically from that quiet, solitary, and thoughtful childhood steeped in American nineteenth-century culture, he was never willing or able to be free from it, and it informed all his adulthood, through every twist and change of twentieth-century intellectual life. I think that, however far I find myself from Italy, the country and the language will always determine all that I do.\n\nHome remains the language I can no longer write as fluently as I write English, the Italian that now feels as clogged at the end of the pen as flour and water between the fingers\u2014 _impastata_ is the only word, but useless because it communicates nothing to English readers who have not watched _contadine_ prepare for a wedding feast by making fresh pasta and _gnocchi,_ and cannot see the laboring and chapped hands placing some of the mixture in front of me, and cannot feel the exact stickiness the word conveys.\n\nHome is the words: the word _limpido,_ for example, that does not limp as \"limpid\" does when denied the strong accent of the final _o,_ but somehow contains in full the transparency of air on autumn days after the first violent sea storms of the year had exhausted themselves against the poor jagged cliffs, just in time for the picking of the last grapes. The sky would turn _azzurro_ once more, in time for the final nets to be put down under the olive trees; at the next storm, all the olives would be shaken from the trees clinging to the steep terraced hill and roll down the slopes. Perhaps a landslide\u2014how much more frightening the words _frana_ and _valanga_ are\u2014would cut a deep slice off the outer edge of the already narrow road winding down into Levanto from the house, and up and around to Bonassola.\n\nHome is the word _incendio,_ containing so much more than \"a fire\" ever could: the sight of the forest on the other side of the bay, on the mountain between Levanto and Monterosso, ablaze, fire rising high in the ink-black sky, a stippling of stars with orange sparks. The sounds, not of sirens, but of church bells ringing, and human voices calling and echoing, and behind those sounds the crackle of wood in flames; the smell of smoke drifting for days, for weeks across the bay to our house. Every time we looked across, the mountain lay bald and exposed. Only long after we had left did the low scrub appear, and only now, forty-five years later, are the small portions that have not been built up green with young oak-sapling and walnut trees once more.\n\nHome is the word _papavero,_ poppy, that conjures up with such clarity the song that was sung in those years, in fields where the flower gloried among the ticklish hay, **_h_** _o sai ehe i papaveri son alti alti alti e tu sei piccolina, lo sai ehe i papaveri son alti alti, sei nata paperina ehe cosa ci vuoifar,_ and trails in its wake the other song, _Aveva una casetta piccolina in Canad\u00e0, con vasche_ _pesciolini e tanti fiori dt Ulla, e tutte le ragazze ehe passavano di la dicevano ehe bella la casetta in Canad\u00e0._ These songs were never heard on the radio, but always in those fields, always to the accompaniment of the accordion and sometimes the harmonica, instruments only the men played. Is it one long summer picnic forever imprinted on my memory, or was it a series of outings with Maria and Alice and their village boyfriends? I cannot remember the details, but the songs and the words unlock the smell of the hay, the taunts and the teases of endless hot afternoons, and the playing, singing, and dancing which started as soon as the heat abated and continued by the light of paraffin-filled rags wrapped around thick sticks and made into torches. No words in English have this power, to take me back home to childhood.\n\n# KOREAN\n\n#\n\n# _Personal and Singular_  \nHa-yun Jung\n\nIn Korean, the first-person singular is an elusive voice. The simple English sentence \"I want an apple\" sounds awkward when translated, word for word, into Korean. A Korean person is much more likely to say something that could be translated as \"It would be nice to have an apple.\" Omitting \"I\" is never a grammatical defect; on the contrary, the sentence sounds more polished without it. Rarely will you hear a Korean speak\u2014or write\u2014consecutive sentences that start with I-this, I-that. \"I\" seems content to crawl behind the curtain at the first given moment.\n\nAnd when it comes to possessive forms, \"our\" is often used in the place of \"my.\" \"My country,\" \"my people,\" \"my neighborhood\" are all very unusual expressions in Korean, even when one is speaking as an individual. This is all the more evident when referring to family members; even when the speaker is an only child, one will say \"our\" mother or father, never \"my.\"\n\nI (there's that \"I\"!) am a single parent, but when I speak Korean I say \"our son.\" And in English I am prone to saying things like \"So then we had to go fill out a prescription for our kid\" to a casual acquaintance I might see while in line at Starbucks. Then I quickly realize, from the slightly confused look on the persons face, that to the American ear what I said sounds like a slip. \"Poor thing. After four years, she still hasn't gotten over her divorce,\" my listener probably concludes.\n\nI used to have a sibling, but I no longer do; yet I refer to my mother as \"our mom\" in Korean, just as my deceased father is still \"our dad.\" And my brother, dead or alive, will always be _uri dongsaeng,_ \"our little brother.\"\n\nPerhaps this is why I write in English, and not in the language that I was born into.\n\nOur family arrived in Bangkok, Thailand, in 1975. A bitter winter chill had seen us off in Seoul, the city I had lived in all nine-and-a-half years of my life, but as soon as we stepped out of the airport terminal the humid heat of this new land filled my nostrils. My brother, two years younger than I, frowned at the glare of the sun. His eyes were too sensitive. But I lifted my face and greeted the light, felt the warmth spread all over. It really was summer here in the middle of November, just as my parents had explained, with fascination.\n\nAnd it would be summer, sweltering summer, all year long, all through the three years of our stay. These years would remain in my memory as a single season made up of nothing but sun and rain, which made everything around me grow; the grass, the insects, the opulent orchids, the sky-high coconut trees, the fiery red salamanders that lurked under bushes like fat lumps of lust. And this was where I would grow, in surprising spurts, into an unfamiliar female body.\n\nWe moved into a gated housing complex designed for expatriate families, with landscaped lawns and tennis courts and a maid quarter\u2014a narrow block of bare cement walls and floors and a wash room with only a tap\u2014attached to every unit. Despite the new and exotic setting, at first this protected world did not appear so different from our old neighborhood in Seoul. We had lived in a Japanese colonial-style house nestled in a winding web of hilly roads, only doors away from our grandparents and uncles and aunts, surrounded by servants and drivers, locked inside a set of reliable routines and rituals. Our lives, it seemed, would go on the same way, whether we were back home or on the other side of the South China Sea.\n\nMy brother and I had grown up listening to accounts of our parents' wedding. It had taken place at the height of both our grandfathers' careers, attended by the Prime Minister and other dignitaries, allegedly causing a traffic jam downtown. My mother's father was the founder and principal of the prestigious Seoul High School, which he envisioned as the Korean version of Eton. My father had been one of his students, one of the many boys\u2014\"All of them,\" as my mother still claims\u2014who had competed for the attention of the five beautiful daughters at the principal's on-campus residence. By the time they got married, my father and mother were graduates of top universities, and my paternal grandfather, a revered lawyer, had entered a political career as the face of integrity in the Park Chung Hee military regime's ruling party. Ever since I can remember, my brother and I were constantly being referred to as so-and-so's grandchildren, and we believed this was a blessing that would see us through in life, that our place in the world had already been shaped for us.\n\nIn Bangkok, my father, in his mid-thirties, was the commercial attach\u00e9 for the South Korean embassy, headed for a future in public office. My mother was his pretty, petite wife, an English Literature major who cooked and sewed and read Dr. Spock, who, upon arriving in this new country, eagerly ventured into attending antique auctions and bridge games and shopping at Jim Thompson.\n\nEnglish was the language spoken at Ruamrudee International School, an institution founded by Redemptorist fathers, whose name meant \"union of hearts\" in Thai. The Bangkok expatriate community was huge, and the Americans had their own school, as did the British, the French, and the Japanese. As a result, the student population at Ruamrudee was truly international, with a Third World focus: a majority of rich Thai and Chinese kids, plus Indian, Filipino, Scandinavian, and even a scattered few from the Eastern bloc. Coming from the fanatical anti-Communist society of the Park Chung Hee era, I remember being at a complete loss as to how I was going to look at or talk to Nikola, a classmate with wide brown eyes and soft curls who was the son of a Romanian diplomat.\n\nThe school, a concrete cube of a building with a bare cement courtyard, was actually a mere annex to the Holy Redeemer Church that dominated the grounds. The huge white cathedral had a grand, slanted roof, tiled in blue and red and lots of gold, identical to those of the royal palaces and Buddhist temples all around Bangkok. Yet it was a church, with crosses and stained glass and figures of Jesus Christ. Its facade was lined with a long row of doors that led straight into the pews. These doors were always open, with cool winds blowing through them, in and out, no matter how scorching the weather. It was an alien mystery to me, just like the children at school who looked so different from the people back in Korea. These kids made me feel unremarkable and bland, with their pierced ears, hairy arms, gold chains, oily smells and all sorts of head wraps, from turbans to fist-sized knots that sat on the heads of long-lashed, round-eyed Sikh boys.\n\nThere were no American students at Ruamrudee, but our curriculum was American, as were our principal, Mr. Maxwell (who had a buttery roar of a voice and a balloon-like belly that hung over his belt), and the fathers and sisters who led the Lord's Prayer during assembly, following the raising of the Thai flag. We learned about the fifty states and their capitals, about Lewis and Clark and Paul Revere, though most of us had never been to the United States. In music class, our Filipino teacher taught us to sing \"John Henry,\" \"My Favorite Things\" and \"America the Beautiful.\"\n\nMy lessons in English began with picture books showing a word for every letter of the alphabet. At age nine, I could absorb without complication what words meant and how sentences were structured, but learning how to pronounce these unfamiliar vowels and consonants took much conscious effort. Korean is phonetically a far less flamboyant language than English; it almost sounds monotonous in comparison, each syllable of a word uttered in equal length and stress. The _f, r, v, z_ and _th_ sounds are all non-existent in Korean and I would have to tighten my tummy, strain my throat, and purse my lips, trying to figure out exactly how to roll and twist and keep afloat my tongue. It felt like an extremely secretive, personal effort, all taking place deep inside of my mouth. \"U is for um-brel-la\u2014\" I would try to read out loud, reminding myself of the difference between the _r_ and the _l_ sounds; then I would try to figure out the stress and completely go blank\u2014 _How did that go again? How do you make your mouth spit out all those sounds at the rush of the second syllable? I sound like I am about to throw up!_ At assembly every morning I stood in the sun-soaked courtyard, my feet already sweaty inside thick white socks, and listened for the smooth, effortless twang in Mr. Maxwells every word\u2014 \"A-a-n an-nou-ncement from the high school da-a-nce committee... \"\u2014then tried to join everyone in prayer, bringing my hands together, closing my eyes, mouthing the words to practice my pronunciation, though I had no idea what they meant or who and where \"our father\" was.\n\nThe rest of this country, beyond our home, school and department stores, my brother and I saw only from inside of our air-conditioned, chauffeur-driven Volvo with the diplomatic plates. On our way to and back from school, we passed several marketplaces filled with fume and clutter, streets overflowing with three-wheeled _tuk-tuks,_ trucks decorated with paintings of women in bikinis, packed buses rattling along with passengers dangling in the doorway, tilting the vehicle. Local kids, skinny and shrill-voiced, blocked the cars at every red light, peddling orchid garlands, offering to clean the windshield, begging for coins. When we rolled down the car window for only a short moment, the steamy odor of the outside world rushed in\u2014a pungent mix of _phakchi_ and jasmine rice, sweat and smoke, sweet mango and slow-burning incense. The whole and skinned poultry that hung upside down from hooks outside the storefronts looked gruesome, but their meaty aroma made my mouth water. My mother never let us buy food from the street. She claimed it was unreliable, especially for my brother, who got rashes from shellfish and fell sick from MSG at restaurants. There seemed to be danger and death on these streets, but at the same time the bustle was bursting with life, like luscious blossoms sprouting after the monsoon.\n\nNot long after arriving in Bangkok, I began having repeating nightmares. In the middle of the night, I would wake up crying, waving my arms and legs, struggling to pull away the swarm of worms and caterpillars and snakes crawling all over my body. My bed was overflowing with them, and the more I tried to get away, the deeper I would sink. Even when I opened my eyes, the sweat on my skin felt slimy, like those slithery, faceless creatures. These nights, dark and dank like a deep hole, had to be battled by myself. No one else noticed. And in the morning, as my brother and I headed out into the city, I saw dead frogs on the driveway, squashed flat on the asphalt by late returning cars, already dry and shapeless in the scalding sun, thin as paper cutouts.\n\nTowards the end of his life, my father would cringe and curse when he talked about Thailand. \"The heat, that is what made me ill,\" he said.\n\nThe liver does not reveal its ailments through pain; instead, it takes away pleasures\u2014of the senses, of physical energy, of simple optimism. Hepatitis robbed my father of the ability to enjoy taste and he spent decades talking about food with desperate nostalgia. He could never appreciate Thai cuisine; he had believed at first\u2014before he was diagnosed\u2014that his health was deteriorating because of the overwhelming unfamiliarity of this new food. Just the thought of fish sauce, lemongrass, coconut milk and long grain rice would later make him nauseous, bringing back early physical memories of a disease that would eventually kill him, gradually hardening his liver into an ugly yellow block, the color of a stinky, overripe durian.\n\nOver our first summer break in Bangkok, my father began coming home for lunch and a nap, then reluctantly returning to the embassy when the mid-day sizzle had subsided, if only slightly. When this didn't help, he figured the lethargy was due to a lack of exercise and shot hoops under the sun before his nap and played more golf on weekends. \"It was practically suicide,\" he would say years later, \"clueless that I was helping sick cells multiply themselves, inviting them to take over my body.\"\n\nWhen a new Korean ambassador arrived the following year, my father had been through several leaves and a month-long stay in the hospital. He did not want to attend the ambassadors first family luncheon\u2014he rarely got out of bed on Sundays\u2014but he had no choice.\n\nThe residence was a modern house with wide, white rooms and long panels of glass that looked out into a garden, a pool, and an atrium displaying strangely twisted rocks as if they were art. I was now one of the older kids in the group. Two other girls and I were exploring the rooms, crossing the vast emptiness of each, wondering what all these rooms were for. We came across a long, high-ceilinged marble hallway, where we found a group of boys pointing and laughing at something, at someone. It was my father, seated in an armchair placed against a tall wall, his elbows on the arm rests, his hands clasped in front of him, his head dipped as if in a low bow, his expression so solemn and somber that you would think he was praying. But he was asleep. He could not make it through the day without a nap and I knew it, but I wanted to slap him awake. Behind him hung a huge painting of nothing, made up entirely of different shades of blue. He looked utterly isolated, unable to keep up with the others, unaware of the ridicule, barely holding himself together in that small corner. My brother followed other boys into the hallway and joined in the laughter; I could not tell if he knew what was going on. I turned away, as if this man had nothing to do with me.\n\nMy brother and I did not fly back to Korea for our grandfathers funeral. We still had a month to go until vacation\u2014I was in fifth grade and my brother in third\u2014so we dutifully attended school, doing our homework, watching \"Six Million Dollar Man\" and \"Charlie's Angels\" while our maid Wong served us grilled cheese sandwiches or bowls of Ichiban ramen.\n\n\"There would be too much going on,\" our mother told us, when we demanded their reason for leaving us behind. My brother and I could not see, at least not then, what our grandfather's death meant and might do to the family. My fathers father was 84 years old when he died, a paterfamilias who had under his wing a family of almost thirty, including his two wives, two brothers, thirteen children, and the widows, widowers, and offspring left behind by the four sons and two daughters that he had lost during the Korean War. Five years before his death, he had taken an unexpected turn towards political dissent, when he openly declared, as chairman of the ruling party, that he would vote against President Park Chung Hee's constitution reforms, a move that would ensure Park's third\u2014then fourth and fifth\u2014term in office. The law was passed, of course, and our grandfather won quiet respect but also faced persecution. When he died, he did not have much to leave behind but the overbearing burden of his legend.\n\nMuch later, after our family returned to Korea that year, my brother and I were shown a silent home movie of the funeral. We sat in the study that our grandfather had occupied for decades, once stacked with dusty old books and decorated with stoic calligraphy. Now its walls were covered with cheap wallpaper too thin to mask the old stains and the bookshelves were scattered with mahjong pieces and Johnnie Walker bottles. The room had been taken over by my fathers older brother, who would in a few years go bankrupt and lose everything, including the old house. In the movie, I saw everyone in our extended family\u2014the women in traditional white cotton _hanbok,_ the men in black suits and conical mourners' hats\u2014weeping, as if they were in fear, as if the funeral would not be the end but the beginning of something terrifying. My fathers face looked crumbled, as if his eyes and nose and lips had been rearranged. I stared at him on the screen, astounded at the unfamiliarity and though there was no sound, I slowly realized that he was wailing, like an animal in pain. It was the first time I had seen him cry.\n\nIt was while we were by ourselves in Bangkok, our parents still away in Seoul, that Mrs. Shanti died. She was from India and taught high school, wore beautiful saris of bold colors and a thick pile of thin, shiny bracelets that chimed when she walked by. I remember seeing her at the cafeteria only several days before we heard the news, the dark make-up on her eyes glamorous as always, her strides sure but light, as if she were gliding an inch above the ground. It was a stroke that took her.\n\nThe entire school was sent to pay respects to Mrs. Shanti at the cathedral. All the classes lined up along the aisle, proceeding toward the altar. I had no idea what to expect once I got there. The morning sun glazed the doorways along both sides of the nave with white, foamy light. I had never been in here. It was not as dark as I had imagined, but the air was dense and damp, as if I could feel the silent chill dripping in long, heavy drops, landing on my forehead, on my bare arms, down the back of my neck. When the wind blew in from the outside, it carried with it faint remnants of the greasy heat that was starting to boil up.\n\nThere I stood, in front of the altar, next to a casket placed at an angle to its left. Mrs. Shanti lay in white satin, dressed in a stiff new sari, her face as purple as her blouse, white, puffy cotton balls pushed deep into her nostrils. I imagined the dead Mrs. Shanti dressing herself for one last time, putting on her jewelry and her lipstick, pulling up her hair in a smooth chignon, then lying back inside her coffin, closing her eyes, giving in with a stale sigh. She looked alive, yet the decay was so palpable it made me nauseous. I wondered if my grandfather looked like this, buried under the ground, the traditional grave mound\u2014 still only a fresh, bald pile of earth\u2014pressing down on him. I drew a cross on my chest, not knowing what else to do, and walked back into the daylight that blinded my eyes.\n\nThat year, I got my period and grew so much that by the time I got to sixth grade I was the tallest in my class. I let my hair grow, and the tomboyish bob cut that my mom had always insisted on quickly dissolved into thick, untamable abundance.\n\nI was now allowed to go to parties, though only during the day, often taking a cab on my own. The girls in my class talked endlessly about our first school dance, coming up around Christmas, but by then I would have left the country. The last party I went to was a birthday bash for Marissa, a Filipino friend, at her mansion-like home with an overgrown garden. She had many older siblings and their friends had been invited as well. To my dismay Chi-wen, an eighth-grader from Taiwan, with his wobbly Mandarin accent and uncool no-brand tennis shoes, kept making eye contact. I felt lithe and light\u2014which was rare, after all that growing\u2014in my faded jeans and gauzy white peasant blouse.\n\nAs the late afternoon heat simmered down, we moved outside, bored with not having much to do but sip Fanta and listen to Boney M and play Monopoly. The garden was shady under the looming trees and I lingered at the edge of the patio, hesitating to step onto the thick grass.\n\n\"A snake!\" someone called out, pointing to the shade of a banana tree, draped with huge paddle-shaped leaves, bearing bubbly clusters of new fruit.\n\nI could not see the snake, but it seemed as if the entire garden was moving, overflowing with batches of fleshy leaves, fresh green coils stretching from vines, carnivorous-looking flowers with petals like perked-up lips. I could not move. Just as in my dreams, I felt as if I were drowning in a pool of snakes, wrapping around my limbs and pulling me down.\n\nOther kids gathered in front of the tree and joined in as the high school boys began to throw rocks, some striking at the target with a stick. Everyone was evidently having fun, the girls' squeals breaking into laughter, the boys cheering, as if this had been planned as part of the festivity all along. Then, finally someone called out, \"I got it!\" It was Chi-wen, holding up the black serpent, shiny and sinuous. He looked over at me with a happy, harmless grin. I kicked at my feet, stumbling to turn around, and pushed myself out the gate, panting all the way down the alley to the busy street. I stopped the first cab I saw. As I slammed the creaky backseat door, Chi-wen tapped the roof of the car, out of breath.\n\n\"You okay?\" he asked.\n\n\"That was so gross,\" I said, exaggerating my _o_ 's.\n\n\"Sorry, sorry.\" He held up his arms, to show he no longer had the dead snake.\n\n\"Soi Srinakorn, fifteen _baht?\"_ I said to the cab driver, negotiating the fare with the few Thai words I could handle, lengthening the vowels and rolling the _r_ 's, declaring, _I am an English-speaking foreigner._ \"Bye,\" I said to Chi-wen and as the taxi pulled away, I felt a hot rush of anticipation, leaving him standing there, flushed and forlorn.\n\nWhen we returned to Korea in November of 1978, we were greeted by the first cold we experienced in three years\u2014biting, harsh and oppressive. It was the same month that we left, but this time the winter air felt so strange I was scared to breathe. My toes would go numb in my sneakers when I was outside and I could not smell anything, as if there were no room in this frozen air for smells to move around.\n\nAlmost everyone we knew had now moved to high-rise apartments that had started to go up all around southern Seoul, the new, modern half of the 500-year-old capital. Our parents moved us into a three-bedroom unit in Banpo, an entire neighborhood built around two hundred identical gray buildings. The four of us lived on the first floor of Bldg. #56 for thirteen years, trapped inside that rectangular cell, divided into smaller squares sealed from the outside world like air-tight containers. In my memory, that apartment is always brown, with the drapes drawn; always tidy, everything in its place, because things were seldom moved, like the silver wine goblets with elephant engravings that my mother had brought back from Thailand. They sat on the kitchen cabinet shelf for years and years, turning black like cavities.\n\nMy first day at Sehwa Girls' Junior High was in March. We had been back in Seoul for three months, but it was still cold and I was uncomfortable under the layers of bulky thermal underwear and ill-fitting uniform, a spartan black jacket and skirt. My eyes were puffy from crying all night, after getting my hair cut to the required one-centimeter-below-the-ears length. I sat in the classroom with seventy other girls, all wearing the exact same clothes and shoes, with the exact same hair, parted and pinned to one side, the backs of our necks shaved blue with electric hair clippers. A stern portrait of President Park Chung Hee hung above the blackboard next to Taegeuk-ki, the national flag.\n\nEvery morning at the school gate, the senior student inspectors checked our hair and uniforms as we bowed. And every morning I would feel like someone was going to see through me, despite my perfect get-up, and pull me out of the orderly procession to interrogate: _Why don't you look like all the others? Why don't you fit in?_\n\nThe Korean that I reencountered in Seoul was a language of hierarchy, of honorifics and humble submission. Because I was no longer the content, innocent nine-year-old, the intimate way that I used to talk was not of much use. I had to constantly remind myself of my relation with the person I was speaking to, for it dictated every single pronoun and verb form I was to utter. And I began to learn, instinctively, what was speakable and unspeakable. When a teacher scolded you, the only way to respond was, \"It was my fault.\" Trying to explain was considered talking back. In Korean, I did not know how to speak of everything I missed about Bangkok\u2014the heat, the colors, the dance I never got to attend, the way Chi-wen had looked at me in that lush garden.\n\nAt the same time, my English lessons at school retreated to the alphabet, see-saw-seen, I-we\/you-you\/he-she-they _What is your name?_\n\nOur family had come back to the place where we used to live but it seemed that we had accidentally stepped into another life. Everything changed. To follow in his late fathers footsteps, my father quit the ministry and ran for a seat in the National Assembly, but failed. He never would regain the old glory; instead he lost his money, his health and his bond with the rest of his family. My mother began to go through long periods of depression, which would slowly progress into bouts of frenzied hope, only to crash back down. She was diagnosed with bipolar disorder and there were many afternoons when I came home from school to find out that she had again been hospitalized. It would take weeks, sometimes months, until she came home again, with shaking fingers and slurred speech and unkempt hair.\n\nWe were no longer \"us.\" My brother and I went about doing what we had to in order to grow up, but separately, alone behind the doors of our rooms.\n\nI started to keep a diary in English. I no longer used English with other people; it had become a language that was completely personal and singularly mine. Recently I dug out the old, yellowed notebook from a box that I have carried with me across states and countries and continents. What amazed me was that my writing was entirely devoid of descriptions or details, with hardly a mention of friends or family filled with sentences that began with \"I,\" relentlessly pouring out how _I_ felt, over and over, as if the words were coming from solely within myself, a place disconnected with the outside world, a place where no one or nothing else could find a way in.\n\nOn Sept. 25th, 1981, the day I woke to find my mother unconscious, with a bottle of rat poison by her bedside, I came home from the hospital, closed the door, sat at my desk and wrote, in English: \"I don't know where to begin. I can't believe it.\"\n\nMy mother would live on, through more suicide attempts and manic sprees, through the deaths of others.\n\nThree weeks before the second-year memorial service for our father, my brother, who had been missing for over a month, was found in his car, frozen dead stiff at 28, in a parking lot by the Han River, a short ten-minute drive on the riverside expressway from the Banpo apartment, where our mother had been waiting and waiting. That night in Boston, I got on a plane that would take me back to Seoul, flying over the white glaciers of Alaska, where there was no summer. I was four months pregnant.\n\nAnd I live on, not feeling whole in Korean or in English. For me, one language is complementary to the other, one always lacking a capacity that the other has. And I have a fear, constantly, of not quite being understood in just one language: _Do you know what I am trying to say? Do you know who I am?_\n\n# POLISH\n\n#\n\n# _On Being an Orphaned Writer_  \nLouis Begley\n\nIn the late summer of 2002, I completed the draft of a new novel, _Shipwreck._ Publishing house gears grind slowly when they process fiction that is not expected to bring down a shower of gold, so that I received my editors comments only a short while before the Christmas vacation. I made the final revisions during that two-week period and countless hours I later stole from my work as a lawyer. Only one of my editors suggestions concerned structure. It was easy to handle. The rest were the sort of take it or leave it pencil notations\u2014some of them extraordinarily helpful\u2014that a gifted reader cannot resist making on the margins of a manuscript. Nevertheless, dealing with these pencil squiggles took an extraordinary amount of time, and left me even more despondent and apprehensive than is usual in that desolate time of the year. My spirits did lift when I finally read the retyped novel. Once again, I began to think that what I had written was comely. But I knew that it would take very little to shatter my calm, perhaps nothing more than the need to reread in order to repair foolish inconsistencies in the plot or the time line that the copy editor, who has worked on all my novels, will track down with her unfailing instincts of a bounty hunter. I know the reasons for my slowness and uncertainty, and I don't confuse them with the doubts that assail every novelist when he delivers a finished manuscript to his publisher\u2014an act in its way as grave as giving up a child for adoption. Until that moment you had in your possession a sheaf of pages with which you communed daily devoting to them non-stop thought and care. Suddenly it's adieu forever: the book will pass into the hands of strangers, and who is to say how they will treat it? Will it meet indifference or hostility or favor, will it be understood? All you can do is wish it luck. Sometimes you think you would rather stand naked among the noontime crowd in Times Square than witness your book's ordeal.\n\nThe soil in which my special anxieties are rooted is the precarious relationship between me and the English language. The circumstances are as follows: Polish is my mother tongue. Since by reason of events over which I had no control I write instead in English, I am an orphaned writer. English is not even my first or second language; I did not begin to learn it until I was eleven or twelve, in preparation for leaving Poland. That is where I was born almost seventy years ago, and where I lived until early October of 1946, around the time of my thirteenth birthday. The next stop was France. But my parents and I remained in Paris only for a few months; we left for New York at the end of February 1947. Despite many absences, some of which were relatively long\u2014when I was away at college and law school, during my military service in the U.S. Army in Germany, and in the years when I worked at my law firm's office in Paris\u2014New York has been my home ever since.\n\nI did not return to Poland until 1994, when I went to Warsaw on legal business. Several visits to Warsaw followed, all quite short, and only one of them, in 1995, I made as a writer. The reason was the publication, in Polish translation, of my first novel, _Wartime Lies,_ which had appeared in the original English in 1991. In it I recount the life in Poland, during World War II, of a Polish Jewish boy, whom I call Maciek, and his aunt; in a number of respects, although my novel is in no sense an autobiography, Macieks adventures resemble my own. This is especially true of the boy's chaotic and inadequate education. Before I arrived in New York, my schooling had consisted of nothing more than sporadic lessons in Warsaw during the War from a memorable tutor who risked her life to give them, whatever my mother managed to teach me, and one year, directly after the War ended, at a _gimnazjum_ in Cracow. Naturally, all of this was in Polish, which my parents and I, just like Maciek and his aunt, spoke to each other. The only foreign language I learned during the War was German. Now my German is mostly dormant; it stirs when I urgently need it. Sometime in the interval between the Soviet occupation in September 1939 of Eastern Poland, where my parents and I then lived, and Germany's breach of the Ribbentrop-Molotov Pact in 1941, whereupon the Wehrmacht drove away the Soviet troops and commenced its thrust into Russian territory, I also studied Russian. I have forgotten it completely. Learning French came later. Paradoxically, that is a language in which I am completely at ease. I always speak it with my wife, it being her native tongue and the language of our courtship.\n\nAnd Polish? My connection with it has remained unbroken. I still use it in conversations with my mother and her Polish housekeeper, and the occasional Poles I encounter in New York. When I was in Warsaw, it never occurred to me to speak English or French to the many Poles who are fluent in those languages. It was natural to use Polish. It is a rare night when I do not have nightmares about World War II in Poland; they are invariably in Polish. When I do sums in my head, or count wine glasses set out on the sideboard for a cocktail party we are about to give, I begin in Polish, and switch to English only after I have become conscious of the absurdity of what I am doing. The songs I sing off key in the car to fight against drowsiness are as often as not Polish folksongs and military airs I learned as a child; I still remember hundreds of lines of Polish poems my mother required me to memorize. Only the other week, I received a remarkable letter in Polish from an eleven-year-old schoolgirl living in Warsaw, provoked by her having read _Wartime Lies_ with her grandfather and having been to see Roman Polanski's film _The Pianist._ She put some hard questions about the cruelty portrayed in both works and expressed the hope that I would answer in Polish, if I still remembered it. I composed the reply immediately and asked a Polish friend to correct gross mistakes. He told me there weren't any. Omissions of letters and words abounded, just as they do in my English texts, in part because I think faster than I write in longhand or type. Of course, I have never stopped reading Polish novels and Polish poetry. Thus, among the most moving literary discoveries I have made in the last two decades have been the Polish poet Zbigniew Herbert and the Polish novelist and playwright Witold Gombrowicz, who stand at the pinnacle of twentieth-century literature. Close after Gombrowicz, and of the same generation as Herbert, comes the short story writer and essayist Gustaw Herling-Grudzinski. I came upon his work reviewing a collection of his short stories. He too has his place in my pantheon.\n\nSometimes cultural change turns out to be far slower than one would expect. After a reading I gave in Warsaw in 1995, from _Wartime Lies,_ someone in the audience asked me to say any poem I had learned as a child. Without a moments thought, laughing with pleasure, I launched into lines about a railroad car full of grotesquely fat men eating fat (and delicious) sausages, Polish _kielbasy._ This charming poem, by Julian Tuwim, the greatest of Polish poets active between the two World Wars, was one that every Polish child of my generation with more or less educated parents knew by heart and loved. As I recited it, though, my exhilaration gave way to sadness and fear that this fragment of pre-War Poland so vivid in my memory would seem alien, and perhaps ludicrous, to a roomful of post\u2013Cold War Poles. My astonishment and joy were extreme when I heard the audience speaking the lines in unison with me. My past was alive in their present.\n\nA curious reader might ask about my English. Of course, it is very good. All my education was in the United States, after I left Poland. By the time I was a high school senior, in 1949, I wrote English well enough to win a city-wide short story contest administered by New York University. The prize was a leather-bound volume of the Oxford Book of English Poetry, which in due course I gave to my older son, a lover of literature who drowned it in the bathtub in which he was soaking. I was an English major in college, and composed short stories for the sort of creative writing courses one took until, in the spring of my junior year, I decided I had better stop because I had nothing to say. It would be more precise and less coy to admit that there was nothing that I was ready to write and stand by as my truth. Afterward, I became engaged in the kind of law practice that requires one to produce, as the major part of one's work, letters of advice, memoranda of law, contracts and briefs, advancement in the profession depending largely on the view one's elders took of the quality of the writing. It was therefore not wholly accurate to think of me as a novice when, in 1989, at the age of fifty-six, I completed my first novel. I had in reality been honing my skills relentlessly, although not in the conventional manner. Even more important for the development of English as my best language, and the only language in which I want to write works of imagination, was serving in the U.S. Army as an enlisted man and having three American children, two of them born before I was thirty and the third only two months after my thirtieth birthday. G.I. talk and the language of the nursery and playground games: had I not experienced them fully, my written English, however literary and elegant I might make it, would be like a dressed-up corpse.\n\nEven though I am told that my writing does not show signs of _rigor mortis,_ it is a fact that I write slowly and laboriously, pausing after every word I set down. I change it countless times and repeat the process with each sentence and paragraph before I can move forward. The vision of Trollope composing the Bar-chester novels in a railroad car, traveling desk balanced on his knees, with hardly an erasure or addition needed before the manuscript went off to the publisher, fills me with admiration, envy, and dull despair. I too can perform on the high wire when I write a legal text or an essay; writing fiction I need to keep my feet on the ground. Perfectionism and perennial dissatisfaction with everything I do are not alone to blame: it seems to me that when I write in English I lack normal spontaneity, let alone the unbeatable self-assurance a writer needs to soar or to be outrageous. I know that I do manage from time to time to be outrageous in my fiction, but the stress falls on the verb \"manage.\"\n\nNothing about those effects is instant. The truth is that even today, after an immersion of more than fifty-five years in the English language, I am never completely confident that I have gotten right whatever it is that I write down, certainly not on the first try. Knowing objectively that often\u2014perhaps most often\u2014 in fact I do, is not a consolation. In that respect only, I am not unlike my great countryman, Joseph Conrad. But Conrad had more of an excuse: he began to learn English only at the age of twenty-one; he was thirty-eight when his first novel, _Almayer's Folly,_ was published, and he had spent most of the intervening years on the high seas. Vladimir Nabokov's command of the English language is a different case altogether. English was in effect Nabokov's first language: he learned to read it before he could read Russian. Becoming thoroughly proficient in a \"civilized\" tongue, usually the French, and leaving the vernacular for later, to be absorbed as a part of growing up, was usual in the nineteenth century among Slavic upper-class families. There was a series of English and French governesses who took care of the Nabokov children, and it wasn't until Vladimir was seven that his father, alarmed by his sons' backwardness in their native language, engaged the schoolmaster from the village adjoining the family estate to teach them to read and write in Russian.\n\nTo go back to the torment I experienced revising the manuscript of my most recent novel, its immediate cause was the number of times my editor was in essence questioning my diction, the correctness of the way I expressed myself in English. It didn't matter that he wasn't always right. What hurt was the contrast between his instinctive grasp of how one would normally say whatever it was that I wanted to express and my doubts: my need to grope to find the way, to test each sentence by reading it aloud. He had kept his birthright\u2014the ability to use his mother tongue in his calling\u2014and I had lost mine. Whether I would have had the spontaneity and freedom that I feel I lack as a novelist writing in English, a language I didn't master until I was fourteen, if I had become a novelist writing in my native Polish, is an intriguing question to which obviously there is no reliable answer. I think, though, that my love for the Polish language, and the way in which it has remained present in my conscious and subconscious memory, are favorable indications.\n\nAn even more difficult question is whether I would have become a writer of fiction at all if my parents had not, driven by the need to escape from the ghosts of so many of our unburied dead and the oppressive weight of Stalinism, left Poland with me in 1946, as soon as they had acquired the means to do so\u2014if they had not chosen the adventure of penniless emigration over the comfort, however precarious, of remaining in Cracow. I do not believe that a novelists talent is one of those that bear fruit regardless of circumstances, being probably different in this regard from the gift for music or the sciences, and, for that matter, certain kinds of physical prowess. I am convinced that my having become able, rather late in life, to write fiction was the result of the peace I finally made with the past and its effect on my identity. But important aspects of that past\u2014the extermination of Polish Jews by Germans during World War II, and the attitude of the vast majority of non-Jewish Poles toward the abject humiliation and then slaughter of some three million of their Jewish fellow citizens\u2014were taboo as subjects for open discussion in post-War Poland under the Communist regime. They remained such even after the advent of Solidarity and well into the 1990s. Indeed, the taboo may not have been fully lifted until the revelations in 2001 by the historian Jan T. Gross in his book _Neighbors: The Destruction of the Jewish Community in Jedwabne, Poland,_ concerning the pogrom in 1941 in the very small town of Jedwabne, in the course of which half of its population, some 1,600 Jews, were savagely murdered by the other (Catholic) half. The publication of Gross's book was followed by a fierce debate in Poland that brought many issues into the open and disposed of many myths. In the conditions that prevailed in Poland after the War ended, a not inconsiderable number of Jews who survived in Poland and did not leave, and of Jews who returned to Poland or were born there after the War, including some who gained prominence in public life, were careful not to let their being of Jewish birth or faith be evident, in any event not to be evident enough to expose them to more or less attenuated forms of harassment or to put their careers at risk. The analogy to how, under German occupation, my mother and I, and many of those same Polish Jews and their parents, avoided the ghetto and extermination by passing for Catholic Poles, is painful. I cannot believe that in those conditions such literary talent as I have would have led me to write novels; it might have developed in other directions or not have progressed beyond sustaining my love for books and language. In fact, I am not aware of any Polish Jews of my generation or younger writing novels or poetry in Polish; perhaps there are some who are not well known, or whose being Jewish simply has not come to my attention. The contrast is striking when one looks at the generations of Polish Jews that came to maturity in Poland between the two World Wars, out of which came many major figures in Polish literature, among them Julian Tuwim and Herling-Grudzinski, whom I have already mentioned, and Josef Wittlin.\n\nThe literary silence of Polish Jews living in Poland since World War II may be of no significance: there is no orderly progress based on nationality in the production of writers, and whether writers are Jews or not is not of significance except when they use in their work specifically Jewish material to which the accident of upbringing gave them special access. In my own case, I wrote my first novel on the quintessentially Jewish subject of survival during the War, because it was an experience that had seared me. I must have felt that to write about other subjects and to remain silent about the War in Poland was impossible; perhaps it was even dishonorable. That task completed, I have had no further interest in Jewish or Polish material. To the contrary, I have dealt in novel after novel with great human themes that are not limited by national, religious or, for that matter, social boundaries: the effect on us of losing those we love the most; our profound and total loneliness, from which only the power of Eros liberates us; the randomness of the catastrophes that befall us; and the hash we make of relationships that count for us the most. My settings have been those that I thought suited best the story I was going to tell, and that I know sufficiently Thus my characters have found themselves on the east coast of the United States, in France and Italy, on a Greek island, in Brazil, and in Tokyo and Beijing.\n\nNevertheless, I have never been free of the pull of Poland. Or is it in reality the pull of the Polish language and Polish literature? This is a question I have put to myself more than once, reflecting on how I constructed the setting of the story I told in _Wartime Lies._ My first novel was a work of fiction based in part on recollections of what happened to me and, in at least equal part, on stories I heard during the War and soon afterward about the misadventures of others. Writing it, I soon realized that I had forgotten the topography of every place I had known in Poland, except the town where I was born, and that even that memory had been stripped down by time to a few startling essentials. Therefore, when I constructed the itineraries of the little boy and his aunt, to find my way I had to pore over the street maps of Lw\u00f3w and Warsaw. I remembered best the interiors, for the simple reason that avoiding denunciation and capture made it preferable to go out into the city as little as might be thought appropriate for a young woman and a little boy leading an ostensibly normal life. There were as well, alas, certain outdoor scenes in my novel the visual memory of which is etched in my skin like the message that the great harrow wrote with acid on the back of the prisoner in Kafka's \"In the Penal Colony.\"\n\nOne of the very happy memories I had been able to retain\u2014 I did not use it in _Wartime Lies_ because for a number of reasons it did not fit\u2014was of a visit to my grandparents' small property in a remote Polish countryside. I discovered, however, thinking intensively about the image in my mind of the low, weatherbeaten wood house, a connection between it and my reading, a year or two after World War II ended, _Pan Tadeusz,_ Adam Mickiewicz's great verse epic, published in 1834, about the life of provincial Lithuanian gentry, as the poet remembered it from the time of Napoleon's Russian campaign. Before World War I, Lithuania was a part of Poland, so far as Poles and the Lithuanian upper class were concerned; it had been so ever since the Lithuanian Grand Prince Jagiello was baptized in 1386, married Jadwiga, the daughter of the King of Hungary, and thus, by a dynastic sleight of hand, ascended with her to the Polish throne. The point of this historical digression is to explain why a Polish school-boy, reading _Pan Tadeusz,_ naturally did not make much of a distinction between Lithuania and Poland. In any event, this particular schoolboy being me, with my special past, made a leap that was even greater. Something in Mickiewicz's elegy, which haunted me like a forgotten melody, was nothing more or less, I realized, than my recollection of the summer and early autumn at my grandparents' property. The feat of self-aggrandizement or empathy or imagination was prodigious: what possible resemblance could there be between the modest manor house of a well-to-do Polish Jew who bought and sold agricultural produce, and the life in that fictional house, the grand domain of Mic-kiewicz's aristocratic and superbly old-fashioned Judge Soplica? But how much is imagination and how much is specific recollection in _Pan Tadeusz?_ The poet was born in 1798. The events he described took place in 1811 and 1812, when he was a boy. In 1824 Mickiewicz was exiled by the Russians for revolutionary activities and never returned. How much could he have really remembered during the composition of a work completed more than twenty years after the fact? What were the shards from which he fashioned his gloriously detailed and textured description of the countryside and of a society that perished with its hunts, balls and quarrels? The answer is that for a poet or a novelist the distinction between what was remembered and what was imagined and made up is shadowy and unimportant. What matters is the irresistible, magnetic force exerted by a place, by a language, and, I will add, by a literature.\n\nMickiewicz wrote, of course, in Polish, which was his mother tongue. The gentry did not speak Lithuanian. It may be that the operation that I performed so many years ago on _Pan Tadeusz,_ when I was eleven or twelve, and reconstituted when at the age of fifty-six I wrote _Wartime Lies,_ exemplifies my abiding connection with the country of my birth. It is made of words, and what they evoke. Thus, in Gombrowicz's masterpiece, _Ferdydurke,_ I was able to find, in the description of the school into which the adult narrator is suddenly thrust as though in a nightmare, the buried memory of the brutality of the students and certain teachers in the _gimnazjum_ I attended in Cracow, and in the scene in which the narrator and his cousin discuss with relish the gentleman's sport of slapping servants, hall porters, barbers\u2014anyone who can't hit back\u2014my memory of being slapped on the face as a child and seeing that done to others. I did not read _Ferdydurke_ until the late 1980s. In consequence, it may be even more surprising that some time ago I found a description of such blows to the face in a schoolroom\u2014obviously in Poland, although I did not say so\u2014in a short story I wrote as English homework, in 1948, in the high school I attended after my parents and I came to live in New York.\n\nThere should be a banner on the wall each writer faces as he sits behind his desk with the injunction AVOID CONCLUSIONS! embroidered in the appropriate language. And yet, I cannot escape asking myself about my innermost feelings. Would I have preferred remaining in Poland after the War to an emigration which, from the point of view of a very young Polish patriot\u2014or to use a more modest formulation, a child who was brought up to love his country\u2014could be called exile? I take into account here the fact that my initial reluctance to return to Poland as a visitor, in large part responsible for my first visit not having taken place until 1994, was related to the illegality of my parents' and my departure, and to fears (which may well have been baseless) of official chicanery, or worse, should we come back. Would I have preferred, if such a thing had been possible, to become a writer using the Polish language? As to the first question, the answer is easy. Had I remained in Poland I would not be myself today. I am not sure that I would like the person I might have become. Fortune is an ever-watchful goddess; she does not rest, and revels in striking those who seem happy. Nevertheless, I will dare to admit that I consider myself lucky to have become American, and to have the life I have had. To answer the second question involves heartbreak. I have old scores to settle with Poland. They are being liquidated by time, which has swept away most Poles who were old enough at the time of World War II to bear responsibility for what was done then or not done. As for Poles living in Poland today, I reserve judgment. The Polish language has been a source of undiluted joy for me, and it pains me to make an admission that may make me seem unfaithful to my first love. But the plain truth is that I consider myself also supremely lucky to be an American novelist, using a language of incomparable beauty and access to readers, a language that for all the difficulties I have described is totally my own. In this respect as well, my case is very different from Nabokov's, as he described it in an afterword to _Lolita:_\n\nMy private tragedy, which cannot, and indeed should not, be anybody's concern, is that I had to abandon my natural idiom, my untrammeled, rich and infinitely docile Russian tongue for a second-rate brand of English, devoid of any of those apparatuses\u2014the baffling mirror, the black velvet backdrop, the implied associations and traditions\u2014which the native illusionist, frac-tails flying, can magically use to transcend the heritage in his own way.\n\nTragedy, surely, but one that was embedded in triumph. I know Nabokov's Russian novels only in English translation; since the translations were heavily reworked by him, it is difficult for me to distinguish the English in which they appear from the English Nabokov used in the works he wrote during his second incarnation, beginning with _The Real Life of Sebastian Knight,_ which appeared in 1941. Both groups of work are masterpieces of illusionism and self-consciousness that borders on mannerism in the use of language. Were the characteristics of his Russian style different? I am incapable of providing an answer. Therefore, I follow Nabokov's advice, and take the tragedy seriously only because Nabokov's intimate wound was surely very real, as is the wound inflicted by every exile, whatever its circumstances and aftermath. The wound is one that never heals, even if one can say with Nabokov, as I do, quite heartlessly: \"The break in my own destiny affords me in retrospect a syncopal kick that I would not have missed for worlds.\"\n\n# RUSSIAN\n\n#\n\n# _The Mother Tongue Between Two Slices of Rye_  \nGary Shteyngart\n\nWhen I return to Russia, my birthplace, I cannot sleep for days. The Russian language swaddles me. The trilling _r_ 's tickle the underside of my feet. Every old woman cooing to her grandson is my dead grandmother. Every glum and purposeful man picking up his wife from work in a dusty Volga sedan is my father. Every young man cursing the West with his friends over a late morning beer in the Summer Garden is me. I have fallen off the edges of the known universe, with its Palm Pilots, obnoxious vintage shops, and sleek French-Caribbean Brooklyn bistros, and have returned into a kind of elemental Shteyngart-land, a nightmare where every consonant resonates like a punch against the liver, every rare vowel makes my flanks quiver as if I'm in love.\n\nLying in bed in my hotel room I am hurt to the quick by the words from an idiotic pop song: \"Please don't bother me,\" a cheerful young girl is singing on a Russian music channel, \"I'm going back to my mama's house.\"\n\nIf I'm in some cheap Soviet hell-hole of a hotel, I can hear the housekeepers screaming at each other. \"Lera, bitch, give me back my twenty.\"\n\n\"You, Vera, are the bitch,\" says her colleague. These words _Ti, Vera, suka_ replay themselves as an endless mantra as I sink my face into a skimpy, dandruff-smelling pillow from Brezhnev times. For the time being, Lera and Vera are my relatives, my loved ones, my everything. I want to walk out of my room and say, in my native tongue, \"Lera, Vera, here is twenty rubles for each of you. Ladies, dear ones, let's have some tea and cognac in the bar downstairs.\"\n\nIf I'm in a Western hotel, one of Moscow's Marriotts, say, I try to tune into the airplane-like hum of the central AC and banish Russian from my mind. I am surrounded by burnished mahogany, heated towel racks, and all sorts of business class accoutrements (\"Dear Guest,\" little cards address me in English, \"your overall satisfaction is our ultimate goal\"), but when I open the window I face a stark Soviet-era building, where the Veras and Leras carry on at full pitch, grandmas coo to children, young men while away the morning hours in the courtyard with beer and invective.\n\nIn order to fall asleep, I try speaking to myself in English. \"Hi there! Was' up? What are you doing Thursday? I have to see my analyst from 4:00 to 4:45. I can be downtown by 5:30. When do you get off work?\" I repeat the last words to my phantom New York friend over and over, trying to regain my American balance, the sense that rationalism, psychiatry and a few sour apple martinis can take care of the past, because, as the Marriott people say, overall satisfaction is our ultimate goal. \"When do you get off work? When do you get off work? When do you get off work? Hi there!\" But it's no use.\n\n_Please don't bother me, I'm going back to my mama's house._\n\n_Lera, you bitch, give me my twenty rubles._\n\nAnd in a final insult, an old Soviet anthem from my youth, hummed through the back channels of memory, the little chutes and trap doors that connect the right brain and the left ventricle through which pieces of primordial identity keep falling out.\n\n_The seagull is flapping its wings_\n\n_Calling us to our duty_\n\n_Pioneers and friends and all our comrades_\n\n_Let us set out for the journey ahead_\n\nSliced down the middle, splayed like a red snapper in a Chinatown restaurant, stuffed with _kh_ and _sh_ sounds instead of garlic and ginger, I lie in a Moscow or St. Petersburg hotel bed, tearful and jet-lagged, whispering to the ceiling in a brisk, staccato tone, maniacally naming all the things for which the Russian language is useful\u2014ordering mushroom and barley soup, directing the cab driver to some forgotten grave, planning the putsch that will for once install an enlightened government. _Khh... Shh... Rrrrr._\n\nHome at last.\n\n_Veliky moguchi russki yazik._ The Great and Mighty Russian Tongue is how my first language bills itself. Throughout its seventy-year tenure, bureaucratic Soviet-speak had inadvertently stripped it of much of its greatness and might (try casually saying the acronym OSOAVIAKHIM, which denotes the Association for Assistance of Defense, Aircraft and Chemical Development). But in 1977 the beleaguered Russian tongue can still put on quite a show for a five-year-old boy in a Leningrad metro station. The trick is to use giant copper block letters nailed to a granite wall, signifying both pomp and posterity, an upper-case paean to an increasingly lower-case Soviet state. The words, gracing the walls of the Technological Institute station, read as follows:\n\n1959\u2014SOVIET SPACE ROCKET\n\nREACHES THE SURFACE OF THE MOON\n\nTake that, Neil Armstrong.\n\n1934\u2014SOVIET SCIENTISTS CREATE\n\nTHE FIRST CHAIN REACTION THEORY\n\nSo that's where it all began.\n\n1974\u2014THE BUILDING OF THE BAIKAL-AMUR\n\nMAIN RAILROAD TRUNK HAS BEEN INITIATED\n\nNow what the hell does that mean? Ah, but Baikal-Amur sounds so beautiful\u2014Baikal the famous (and now famously polluted) Siberian lake, a centerpiece of Russian myth; Amur (amour?) could almost be another word Russian has gleefully appropriated from the French (it is, in fact, the name of a region in the Russian Far East).\n\nI'm five years old, felt boots tight around my feet and ankles, what might be half of a bear or several Soviet beavers draped around my shoulders, my mouth open so wide that, as my father keeps warning me, \"a crow will fly in there.\" I am in awe. The metro, with its wall-length murals of the broad-chested revolutionary working class that never was, with its hectares of granite and marble vestibules, is a mouth-opener to be sure. And the words! Those words whose power seems not only persuasive, but, to this five-year-old kid already obsessed with science fiction, extra-terrestrial. The wise aliens have landed and WE ARE THEM. And this is the language we use. The great and mighty Russian tongue.\n\nMeanwhile, a metro train full of sweaty comrades pulls into the station, ready to take us north to the Hermitage or the Dos-toyevsky museum. But what use is there for the glum truth of Rembrandt's returning prodigal son or a display of the great novelist's piss pots, when the future of the human race, denuded of its mystery, is right here for all to see. SOVIET SCIENTISTS CREATE THE FIRST CHAIN REACTION THEORY. Forget the shabby polyester-clad human element around you, the unique Soviet metro smell of a million barely washed proletarians being sucked through an enormous marble tube. There it is, kid, in copper capital letters. What more do you want?\n\nSome two years later, in Queens, New York, I am being inducted into a different kind of truth. I am standing amidst a gaggle of boys in white shirts and skullcaps, and girls in long dresses wailing a prayer in an ancient language. Adults are on hand to make sure we are all singing in unison; that is to say, refusing to wail is not an option. _\"Sh'ima Yisrael,\"_ I wail, obediently, _\"Adonai Elo-heinu, Adonai echad.\"_\n\nHear, O Israel, the Lord is our God, the Lord is One.\n\nI'm not sure what the Hebrew words mean (there is an English translation in the prayer book, only I don't know any English either), but I know the tone. There is something plaintive in the way we boys and girls are beseeching the Almighty. What we're doing, I think, is supplicating. And the members of my family are no strangers to supplication. We are Soviet Jewish refugees in America (\"refu-Jews,\" the joke would go). We are poor. We are at the mercy of others: Food Stamps from the American government, financial aid from refugee organizations, second-hand Batman and Green Lantern t-shirts and scuffed furniture gathered by kind American Jews. I am sitting in the cafeteria of the Hebrew school, surrounded first by the walls of this frightening institution\u2014a gray piece of modern architecture liberally inlaid with panes of tinted glass\u2014with its large, sweaty rabbi, its young, underpaid teachers, and its noisy, undisciplined American Jewish kids, and, in a larger sense, surrounded by America: a complex, media-driven, gadget-happy society, whose images and language are the lingua franca of the world and whose flowery odors and easy smiles are completely beyond me. I'm sitting there, alone at a lunch table, a small boy in over-sized glasses and a tight checkered Russian shirt, perhaps the product of some Checkered Shirt Factory #12 in Sverdlovsk, and what I'm doing is, I'm talking to myself.\n\nI'm talking to myself in Russian.\n\nAm I saying \"1959\u2014soviet space rocket reaches the surface of the moon\"? Its very possible. Am I recounting the contents of the Vorontsovski Palace in Yalta, where, just a year ago, I proved myself smarter than the rest of the tour group (and won my mothers undying love) by pointing out that the palace resembled the contours of a neighboring mountain? It could be. Am I nervously whispering an old Russian childhood ditty (one that would later find its way into one of my stories written as an adult): \"Let it always be sunny, let there always be Mommy, let there always be blue skies, let there always be me\"? Very possible. Because what I need now, in this unhappy, alien place, is Mommy, the woman who sews my mittens to my great overcoat, for otherwise I will lose them, as I have already lost the bottle of glue, lined notebook and crayons that accompany me to first grade.\n\nOne thing is certain\u2014along with Mommy, and Papa, and one sweet kid, the son of liberal American parents who have induced him to play with me\u2014the Russian language is my friend. It's comfortable around me. It knows things the noisy brats around me, who laugh and point as I intone my Slavic sibilants, will never understand. The way the Vorontsovski Palace resembles the mountain next to it. The way you get frisked at the Leningrad Airport, the customs guard taking off your hat and feeling it up for contraband diamonds. The way SOVIET SCIENTISTS CREATE[D] THE FIRST CHAIN REACTION THEORY in 1934. All this the great and mighty Russian language knows. All this it whispers to me at night, as I lie haunted by childhood insomnia.\n\nTeachers try to intervene. They tell me to get rid of some of my Russian furs. Trim my bushy hair a little. Stop talking to myself in Russian. Be more, you know, _normal._ I am invited to play with the liberals' son, a gentle, well-fed fellow who seems lost in the wilderness of Eastern Queens. We go to a pizza parlor, and, as I inhale a slice, a large string of gooey Parmesan cheese gets stuck in my throat. Using most of my fingers, I try to pull the cheese out. I choke. I gesture about. I panic. I moo at our chap-erone, a graceful American mama. _Pomogite!_ I mouth. Help! I am caught in a world of cheap endless cheese. I can see a new placard for the Leningrad metro. 1979\u2014first soviet child chokes on capitalist pizza. When it's all over, I sit there shuddering, my hands covered with spittle and spent Parmesan. This is no way to live.\n\nAnd then one day I fall in love with cereal. We are too poor to afford toys at this point, but we do have to eat. Cereal is food. It tastes grainy easy and light, with a hint of false fruitiness. It tastes the way America feels. I'm obsessed with the fact that many cereal boxes come with prizes inside, which seems to me an unprecedented miracle. Something for nothing. My favorite comes in a box of a cereal called Honey Combs, a box featuring a healthy white kid\u2014as a sufferer of asthma, I begin to accept him as an important role-model\u2014on a bike flying through the sky (many years later I learn he's probably \"popping a wheelie\"). What you get inside each box of Honey Combs are small license plates to be tied to the rear of your bicycle. The license plates are much smaller than the real thing but they have a nice metallic heft to them. I keep getting MICHIGAN, a very simple plate, white letters on a black base. I trace the word with my finger. I speak it aloud, getting most of the sounds wrong. MEESHUGAN.\n\nWhen I have a thick stack of plates, I hold them in my hand and spread them out like playing cards. I casually throw them on my dingy mattress, then scoop them up and press them to my chest for no reason. I hide them under my pillow, then ferret them out like a demented post-Soviet dog. Each plate is terribly unique. Some states present themselves as \"America's Dairy-land,\" others wish to \"Live Free or Die.\" What I need now, in a very serious way, is to get an actual bike.\n\nIn America the distance between wanting something and having it delivered to your living room is not terribly great. I want a bike, so some rich American (they're all rich) gives me a bike. A rusted red monstrosity with the spokes coming dangerously undone, but what do you want? I tie the license plate to the bicycle, and I spend most of my day wondering which plate to use, citrus-sunny FLORIDA or snowy VERMONT. This is what America is about: choice.\n\nI don't have much choice in pals, but there's a one-eyed girl in our building complex whom I have sort of befriended. She's tiny and scrappy, and poor just like us. We're suspicious of each other at first, but I'm an immigrant and she has one eye, so we're even. The girl rides around on a half-broken bike just like mine, and she keeps falling and scraping herself (rumor is that's how she lost her eye) and bawling whenever her palms get bloodied, her head raised up to the sky. One day she sees me riding my banged-up bicycle with the Honey Combs license plate clanging behind me and she screams \"MICHIGAN! MICHIGAN!\" And I ride ahead, smiling and tooting my bike horn, proud of the English letters that are attached somewhere below my ass. Michigan! Michigan! with its bluish-black license plate the color of my friend's remaining eye. Michigan, with its delicious American name. How lucky one must be to live there.\n\nVladimir Girshkin, the struggling young immigrant hero of my first novel, _The Russian Debutante's Handbook,_ shares a few characteristics with me, notably his penchant for counting money in Russian, which, according to the book, is \"the language of longing, of homeland and Mother, his money-counting language.\" And also, I might add, the language of _fear._ When the ATM coughs out a bushel of cash or I am trying to perform a magic trick with my checkbook, trying to glean something from nothing, I leave English behind. American dollars, the lack of which constitutes an immigrants most elemental fright, are denominated entirely in the Russian language. And so with shaking hands, the fictional Vladimir Girshkin and the all-too-real Gary Shteyngart count a short stack of greenbacks, a record of our worth and accomplishment in our adopted land: _'Yosem-desyat dollarov... Sto dollarov... Sto-dvadtsaf dollarov... \"_\n\nMany of my dreams are also dreamt in Russian, especially those infused with terror. There's one, for instance, where I emerge into a sepia-toned Manhattan, its skyscrapers covered by the chitinous shells of massive insects with water-bug antennae waving menacingly from their roofs. \"What has happened?\" I ask an unmistakably American passer-by, a pretty young woman in a middle-class pullover.\n\n_\"Nichevo,\"_ she answers in Russian (\"it's nothing\"), with a bored Slavic shrug of the shoulders, just as I notice a pair of insect-like mandibles protruding from the base of her jaw. And I wake up whispering _bozhe moi, bozhe moi._ My God, my God.\n\nAnd when terror informs my waking world, when an airplane's engines for some reason quit their humming mid-flight, when a big man with murder in his nostrils turns the corner and walks right into me, I think _Za shto?_ What for? Why me? Why now? Why am I to die like this? Is it fair? It's a question addressed not to the Heavens, which I'm guessing are fairly empty of God, but to the Russian language itself, the repository of my sense of unfairness, a language in which awful things happen inexplicably and irrevocably.\n\nAfter we come to the States, many of my more adaptable fellow immigrants quickly part ways with their birth languages and begin singing Michael Jackson's \"Billie Jean\" with remarkable accuracy and hip-swinging panache. The reason I still speak, think, dream, quake in fear, and count money in Russian has to do with a series of decisions my parents make when we're still greenhorns. They insist that only Russian be spoken in the home. It's a trade-off. While I will retain my Russian, my parents will struggle with the new language, nothing being more instructive than having a child prattle on in English at the dinner table.\n\nOur house is Russian down to the last buckwheat kernel of kasha. When English does make its appearance, it is scribbled on a series of used IBM punch cards from my father's computer classes. I handle the punch cards with the same awe as I do the Honey Combs license plates, intrigued as much by their crisp, beige, American feel as by the words and phrases my father has written upon them, English on one side, Russian on the other. I remember, for some reason, the following words\u2014\"industry _(promishlenost),_ \"teapot\" _(chainik),_ \"heart attack\" _(infarkt),_ \"symbolism\" _(symvolizm),_ \"mortgage\" _(zaklad),_ and \"ranch\" _(rancho)._\n\nThe second decision is mostly economic. We cannot afford a television, so instead of the Dukes of Hazzard, I turn to the collected works of Anton Chekhov, eight battered volumes of which still sit on my bookshelves. And when we find a little black-and-white Zenith in the trash can outside our building, I am only allowed to watch it for half an hour a week, not enough time to understand why Buck Rogers is trapped in the 25th century or why the Incredible Hulk is sometimes green and sometimes not. Without television there is absolutely nothing to talk about with any of the children at school. It turns out these loud little porkers have very little interest in \"Gooseberries\" or \"Lady with Lapdog,\" and it is impossible in the early 1980s to hear a sentence spoken by a child without an allusion to something shown on TV.\n\nSo I find myself doubly handicapped, living in a world where I speak neither the actual language, English, nor the second and almost just as important language\u2014television. For most of my American childhood I have the wretched sensation that fin-de-si\u00e8cle Yalta with its idle, beautiful women and conflicted, lecherous men lies somewhere between the Toys \"R\" Us superstore and the multiplex.\n\nAround this time, I start writing in English with gusto. I write for the same reasons other curious children write: loneliness, boredom, the transgressive excitement of building your own world out of letters, a world not sanctioned by family and school. The latter becomes my target. While I patiently wail my _\"Sh'ma Yisrael,\"_ praying that God will indeed take mercy on me, that he will make the young Hebrew School Judeans stop teasing me for my cardboard sweater and my anxious, sweaty brow, for being a bankrupt Russian in a silver-tinseled American world, I also decide to act.\n\nI write my own Torah. It's called the Gnorah, an allusion to my nickname Gary Gnu, the name of an obscure television antelope which I have never seen. The Gnorah is a very libertine version of the Old Testament, with lots of musical numbers, singing prophets, and horny eleven-year-old takes on biblical themes. Exodus becomes Sexodus, for instance. Henry Miller would have been proud.\n\nThe Gnorah is written on an actual scroll, which I somehow manage to type up sideways so that it looks like an actual Torah. I hit the IBM Selectric keyboard with a giddy, nerdish excitement. Thousands of sacrilegious English words pour out in a matter of days, words that aren't inflected with my still-heavy Russian accent. Impatiently I blow on passages deleted with white-out, knowing somehow that my life is about to change. And it does. The Gnorah receives wide critical acclaim from the students of the Solomon Schechter Hebrew Day School of Queens\u2014a relief from the rote memorization of the Talmud, from the aggressive shouting of blessings and counter-blessings before and after lunch, from the ornery rabbi who claims the Jews brought on the Holocaust by their over-consumption of delicious pork products. The Gnorah gets passed around and quoted. It doesn't quite make me acceptable or beloved. Only owning a twenty-seven-inch Sony Trinitron and a wardrobe from Stern's department store can do that. But it helps me cross the line from unclubbable fruitcake to tolerated eccentric. Tell me, is there anything writing can't do?\n\nThe Gnorah marks the end of Russian as my primary tongue and the beginning of my true assimilation into American English. I want to be loved so badly, it verges on mild insanity. I devote most of my school hours, time that should be spent analyzing Talmudic interpretations of how a cow becomes a steak, writing stories for my classmates, stories that poke fun at our measly lives, stories filled with references to television shows I barely know, stories shorn of any allusion to the Russia I've left behind or to the pages of Chekhov patiently yellowing on my bookshelves. A progressive young teacher sets aside time at the end of the English class for me to read these stories, and, as I read, my classmates yelp and giggle appreciatively, a great victory for the written word in this part of Queens.\n\nBut soon my pre-adolescent writing career is cut short. My family is not so poor anymore and can afford to shell out one thousand dollars for a salmon-colored twenty-seven-inch Sony Trinitron. The delivery of this Sony Trinitron is possibly the happiest moment of my life. Finally in a real sense, I become a naturalized citizen of this country. I turn it on, and I never turn it off. For the next ten years, I will write almost nothing.\n\nI have begun this essay with a sleepless trip to contemporary Russia, a trip bathed in the anxious sounds of the mother tongue, and I have come to the end with a child's farewell to the language that once choreographed his entire world. But memory, which in the Russian sense is often just a flimsy cover for nostalgia, begs for a different ending.\n\nSo I will conclude elsewhere, at a place called the Ann Mason Bungalow Colony in the Catskill Mountains. Even the poorest Russian cannot live without a summer _dacha,_ and so every June we, along with other Russian families, rent one of a dozen of little barrack-style bungalows (white plaster exterior with a hint of cheap wood around the windows) not far from the old Jewish Borscht Belt hotels. My mother and I sneak into the nearby Tamarack Lodge, where Eddie Fisher and Buddy Hackett once shared a stage, to witness giant, tanned American Jews lying belly-up next to an Olympic-size outdoor pool or sleep-walking to the auditorium in bedroom slippers to watch Neil Diamond in _The Jazz Singer._ This is probably the grandest sight I have come across in the ten or eleven years of my existence. I immediately vow to work hard so that one day I can afford this kind of lifestyle and pass it on to my children (the Tamarack Lodge has since closed; I have no children).\n\nBack at the Ann Mason Bungalow Colony, we survive without daily screenings of _The Jazz Singer_ and the pool can fit maybe a half-dozen small Russian children at a time. Ann Mason, the proprietor, is an old Yiddish-spouting behemoth with three muu-muus to her wardrobe. Her summer population during weekdays consists almost entirely of Soviet children and the grandmothers entrusted with them\u2014the parents are back in New York working to keep us all in buckwheat. The children (there are about ten of us from Leningrad, Kiev, Kishinev and Vilnius) adore Ann Masons husband, a ridiculous, pot-bellied, red-bearded runt named Marvin, an avid reader of the Sunday funny papers whose fly is always open and whose favorite phrase is \"Everybody in the pool!\" When Ann Mason cuts enough coupons, she and Marvin take some of us to the Ponderosa Steakhouse for T-bones and mashed potatoes. The all-you-can-eat salad bar is the nexus of capitalism and gluttony we've all been waiting for.\n\nAnn Masons Bungalow Colony sits on the slope of a hill, beneath which lies a small but very prodigious brook, from which my father and I extract enormous catfish and an even larger fish whose English name I have never learned (in Russian it's called a _sig;_ the Oxford-Russian dictionary tells me, rather obliquely, that it is a \"freshwater fish of the salmon family\"). On the other side of the brook there is a circular hay field which belongs to a rabidly anti-Semitic Polish man who will hunt us down with his German shepherd if we go near, or so our grandmothers tell us.\n\nOur summers are spent being chased by these grandmothers, each intent on feeding us fruits and farmer's cheese, which, along with kasha in the morning, form the cornerstones of our mad diets. Shouts of _\"frukti!\"_ (fruits) and _\"tvorog!\"_ (farmer's cheese) echo above the anti-Semite's mysterious hay field. By sundown a new word is added to the grandmothers' vocabulary, _\"sviter!\"_ \u2014a desperate appeal for us to put on sweaters against the mountain cold.\n\nThese children are as close as I have come to compatriots. I look forward to being with them all year. There is no doubt that several of the girls are maturing into incomparable beauties, their tiny faces acquiring a round Eurasian cast, slim-hipped tomboyish bodies growing soft here and there. But what I love most are the sounds of our hoarse, excited voices. The Russian nouns lacing the barrage of English verbs, or vice versa _(\"babushka, oni poshli_ shopping _vmeste v ellenvilli\"_ \u2014\"grandma, they went shopping together in Ellenville\").\n\nFresh from my success with the Gnorah, I decide to write the lyrics for a music album, popular American songs with a Russian inflection. Madonnas \"Like a Virgin\" becomes \"Like a Sturgeon.\" There are paeans to babushkas, to farmers cheese, to budding sexuality rendered with a trilled _r_ that sounds sexier than we think. We record these songs on a tape recorder I buy at a drugstore. For the album cover photograph I pose as Bruce Springsteen on his _Born in the USA_ album, dressed in jeans and a t-shirt, a red baseball cap sticking out of my back pocket. Several of the girls pose around my \"Bruce.\" They are dressed in their finest skirts and blouses, along with hopeful application of mascara and lipstick. \"Born in the USSR\" is what we call the album. _(\"I was bo-ho-rn down in-uh Le-nin-grad... wore a big fur_ shapka _on my head, yeah... \")_\n\nWe await the weekends when our parents will come, exhausted from their American jobs, the men eager to take off their shirts and point their hairy chests at the sky, the women to talk in low tones about their husbands. We cram into a tiny stationwagon and head for one of the nearest towns where, along with a growing Hasidic population, there is a theater that shows last summers movies for two dollars (giant bag of popcorn with fake butter\u2014fifty cents). On the return trip to the Ann Mason Bungalow Colony, sitting on each others laps, we discuss the finer points of _E.T. the Extra-Terrestrial_ I wonder aloud why the film never ventured into outer space, never revealed to us the wrinkled fellows planet, his birthplace and true home.\n\nWe continue our Russo-American discussion into the night, the stars lighting up the bull's-eye of the anti-Semitic hay field, our grandmothers mumbling the next day's rations of kasha and sweaters in their sleep. Tomorrow, a long stretch of non-competitive badminton. The day after that, Marvin will bring out the funny papers and we will laugh at Beetle Bailey and Garfield, not always knowing why we're laughing. It's something like happiness, the not knowing why.\n\n# SCOTS\n\n#\n\n# _Boswell and Mrs. Miller_  \nJames Campbell\n\nWhen James Boswell took the low road from Scotland to London in 1762, to seek his fortune and eventually to write the _Life of Johnson,_ he required no passport to cross the border; but as he went, he imagined his whole being receiving the stamp of improvement. Boswell's overwhelming purpose in life was to better himself; in order to do so, he was ready to slough off the rough Scots \"Jamie,\" and admit the politer, anglicized James. In London, however, Boswell encountered an unexpected and unwanted reminder of home on the southern air. \"Mrs. Millers Glasgow tongue excruciated me,\" he wrote in his journal for March 17, 1763. \"I resolved never again to dine where a Scotchwoman from the West was allowed to feed with us.\"\n\nThe Scotchwoman from the West must have made an awful din. Boswell suggests a genteel table upset by a barking ruffian. We notice that, while he \"dines,\" Mrs. Miller \"feeds.\" It comes as a surprise to learn that Mrs. Miller was the wife of Thomas Miller, Lord Advocate of Scotland, the country's highest ranking legal figure. She would have been considered, and would have thought herself, a member of the gentry. Boswell's annoyance and embarrassment tell us that it was common for respectable society figures in mid-eighteenth-century Scotland to speak a form of what is called Older Scots, a generic designation for the dialect tongue that wags across time, from the middle ages to the present day. Boswell himself could only have avoided sounding like Mrs. Miller by making a positive effort not to.\n\nHow, exactly, does Mrs. Miller speak? She says \"aff\" for off and \"oot\" for out; \"ben the hoose\" to mean indoors, and \"greetan\" for weeping. Mrs. Miller gets wired intae her dener, while Boswell and the others are carefully keeping their elbows off the table. She uses idioms and peculiar grammatical constructions which he has been taking pains to expunge for years: \"Ah doot Jamie canny tell a rich wumman bi a puir,\" she thinks, sensing Boswell's snobbish contempt. \"He haes a face on him aye that wad soor milk.\" Catching his angry eye, she cries out, \"Dinna fash yersel',\" before turning back to her \"parridge,\" the common name she gives to the tastiest of dishes. To Boswell and the assembled company, Mrs. Miller seems incapable of grasping the difference between \"those\" and \"they\" (or thae), and equally incapable of pronouncing the flat \"a,\" so that references to those apples in the dish over there inevitably tumble out of her mouth as \"thae aipples,\" no matter how she tries to prevent it happening. She speaks of the dish as \"thon ashet yonder.\" She havers on about her \"faither\" and \"mirra\" and the \"wee wean,\" her child, and \"hoo i wiz glaiket but bonny forby.\" When she does use the flat \"a,\" it's in the wrong place: water, for example, drips off Mrs. Millers tongue as \"waa'er.\" Imagine these deviations spread across the entire field of English speech, and you have some idea of the sound that \"excruciated\" Boswell.\n\nMrs. Miller cannot be allowed back to the table; if she should be, Boswell will refuse to join in. A few weeks earlier, he had reflected that it would be wiser in future\u2014more socially advantageous, in other words\u2014to avoid contact altogether with the compatriots who arrived in London and came knocking on his door. Particularly those who embarrassed him by speaking in \"the abominable Glasgow tongue.\"\n\nI know Mrs. Miller well. I can hear her clearly. With a few shifts in flats and sharps, a slight increase or reduction in the incidence of glottal stops (try removing any hint of a \"t\" from \"waa'er\" and replacing it with an emphasis on the second syllable), her descendants in Glasgow speak today as she did two and a half centuries ago. Mrs. Millers speech reflected the natural West of Scotland way of talking; it was Boswell, powdering his palate from a compact of airs and affectations, who was trying to groom himself to \"talk suddron\" (southern). It was fashionable among some, though not all, Scottish ladies and gentlemen of the day to do so. Boswell came from a well-to-do family of landowning lawyers from the rural Southwest, and attended Glasgow University. He had taken lessons in elocution in Edinburgh from Thomas Sheridan, father of the playwright Richard Brins-ley and he would have been familiar with the little books of \"Scotticisms,\" published in the 1750s and 60s, containing alphabetical lists of words and phrases that Scots in public life were advised to avoid, especially when indulging in commercial or social intercourse with their English or foreign counterparts. Mrs. Millers offence, on being \"allowed to feed\" at a polite table in London, was to disregard the presumed linguistic etiquette.\n\nBoswell's objections make him sound like a boor and even a traitor. But he was less of a snob than he might seem. His attachment to the great natural democrat Dr. Johnson was genuine and philosophically grounded, and his feeling for Scotland was deep. Eventually, he married a Scotchwoman from the West, and lived with her in Edinburgh, in the East. However, his severe attitude to the Glasgow tongue is just as familiar to someone who was born and brought up in the city\u2014as I was\u2014as the tongue itself. The two ways of speaking may be separated into \"Glasgow\" and \"Glesca,\" after the different pronunciations of the city's name. The tongue has divided families, neighbors and neighborhoods; it has drawn a notional segregation marker through the city. A refined Glasgow speaker might go out of his way to avoid contact with a Glesca speaker. Each would recognize the social standing of the other as soon as they opened their mouths. Mr. Glasgow might treat Mr. Glesca and his \"patter\" as a topic of couthie humour, a kind of Caledonian minstrelsy, which is calculated to amuse; similarly, the Glesca man can only bring himself to pronounce the official name of his city, \"Glasgow,\" in a pointed, comical way. To attempt it in ordinary conversation would be to invite ridicule from his friends. The Glasgow man probably believes (without having given it much thought) that the other who says \"Glesca\" does so out of an inability to pronounce \"Glasgow.\" It is possible that neither is aware that the \"lower\" pronunciation reflects the medieval spelling of the city's name; in the fourteenth and fifteenth centuries, Glasgow was \"Glescu,\" and must have been pronounced that way by the Boswells and Mrs. Millers alike.\n\nThe linguistic division, which developed fully round about 1600 with the Union of the Crowns of England and Scotland, has also split individuals. There is no better illustration of the double-sidedness of the Scottish tongue than the national poet himself. For both his daily life and his poetry, Robert Burns had two dialects: Older Scots and Standard English. Sometimes he employed them both in a single sentence, or poetic couplet, as in the well-known lines,\n\n_The best-laid schemes o' mice an men_\n\n_Gang aft agley._\n\n\"To a Mouse,\" like many Burns poems, is written in a combination of Scots and English, but the dominant flavor is Scots, even when dialect vocabulary is used scarcely\u2014\"Wee, sleekit, cow'rin, tim'rous beastie.\" The same was true of Burns's everyday speech. To the neighboring farmers in Ayrshire (Boswell's county, as it happens, some fifty miles to the south-west of Glasgow), Burns spoke like this:\n\nI'm sitten down here, after seven and forty miles ridin, e'en as forjesket and forniaw'd as forfoughten cock, to gie you some notion o' my landlowper-like stravaguin sin the sorrowfu' hour that I sheuk hands and parted wi' auld Reekie....\n\nI hae daunder'd owre a' the kintra frae Dumbar to Selcraig, and hae forgather'd wi' monie a guid fallow and monie a weelfar'd hizzie\u2014I met wi' twa dink quines in particular, ane o' them a sonsie, fine fodgel lass, baith braw and bonnie.\n\nYet he was capable of adopting a cultivated manner when circumstances required it. For the men and women with whom he socialized in the drawing-rooms of Edinburgh (\"auld Reekie\"), on whom he sometimes was forced to depend financially, he adopted a different voice altogether:\n\nI cannot bear the idea of leaving Edinburgh without seeing you\u2014I know not how to account for it\u2014I am strangely taken with some people; nor am I often mistaken. You are a stranger to me; but I am an odd being: some yet unnamed feelings; things not principles, but better than whims, carry me farther than boasted reason ever did a Philosopher.\n\nThe letter in Scots, dealing with one of Burns's favorite subjects (the \"twa dink quines\" might elsewhere be described as two comely wenches), is to the poet's friend William Nicol, a schoolmaster; the other, written in the same year, 1787, is to Agnes McLehose, a more genteel-sounding Scotchwoman from the West than Mrs. Miller, with whom Burns conducted a brief courtship.\n\nScots has numerous regional variations, of which Burns's Ayrshire and Mrs. Miller's Glesca are only two. All are related, and all forms of Scots are likewise linked to standard English. The use of Scots for day-to-day purposes was still common in the time of Burns and Boswell. Two hundred and fifty years before that, it was universal. For the past century and a half, however, Scots has been declared dead, or regarded as petering out (or else it is in the throes of a revival). Robert Louis Stevenson, born into a middle-class Edinburgh family in 1850, picked up a fair sprinkling of Scots from servants and gardeners, which he put to spirited use in poems and letters, and occasionally in short stories such as \"Thrawn Janet;\" but he was aware as he did so that he was indulging a linguistic form of nostalgia. Stevenson recalled his grandfather, born a year or two after Burns's effusions, as \"one of the last, I suppose, to speak broad Scots and be a gentleman.\" The country folk of present-day Ayrshire, tuning into _EastEnders_ and _Friends,_ and conversing via the transatlantic line with their emigrant cousins in North America, no longer talk to one another as Burns did to Willie Nicol, nor do they use much of the vocabulary that gives poems such as \"To a Mouse\" and \"Tam o' Shanter\" their distinct fibre. There is, in a sense, less space for the dialect; the distances that separate groups of people have shrunk, and we are apt to address our neighbors in a language we trust they will understand.\n\nYet Scots is still alive. It is current in ways that may be barely noticed. Boys and girls in the streets of Glasgow today for example, would find the idiom of this sixteenth-century ballad quite familiar; should you be passing by you might hear them speak in a way that is close to it:\n\n_As I was a-walkin all alane_\n\n_I spied twa corbies makkin a mane._\n\n_The tane untae the ither did say-o:_\n\n_Whaur sall we gang tae dine the day-o._\n\n_In ahent yon auld fell dyke,_\n\n_I wot there lies a new slain knight._\n\n_And naebody kens that he lies there-o_\n\n_But his hawk an his hound an his lady fair-o._\n\n\"Doubles\" or \"doubling\" are often evoked in discussions of Scottish literature, with reference made to Stevenson's _Strange Case of Dr. Jekyll and Mr. Hyde_ and James Hogg's _Confessions of a Justified Sinner,_ and even to poems like \"Tam o' Shanter,\" which inhabit a split-level reality\u2014this world, and the world of ghaisties, witches and warlocks into which Tam stumbles. Indeed, \"doubling\" is an actual feature of the language in which a large portion of Scottish literature is composed. There is a formulation, originally made by the poet Edwin Muir but so often cited as to have become a commonplace, that modern Scottish writers who make use of the dialect feel in one language (Scots) while they think in another (English).\n\nThe conflict between the two elements continues to occur in Scotland today. It was played out in my family living room on the southside of Glasgow in the late 1950s and early 60s. We were a typical working-class family with typical aspirations to be more middle-class. I was carefully brought up to speak properly (or, as the people who don't speak properly say \"speak polite\"). But in my mid-teens, as part of a private revolution, I began a linguistic migration back to the Older Scots. I didn't know then that that was the name of the dialect which I heard all around me (\"As I was a-walkin all alane\"); there must have been something in the rougher way of talking that suited my adolescent storm. My parents took the position, let's say, of Boswell, whereas I found myself cast as Mrs. Miller. The process coincided with my falling in with a new crowd of friends, who came from a poorer, indeed notorious, area of Glasgow, the Gorbals. There was nothing in my friends' behavior to deserve the stamp of notoriety, but geography is itself the marker of repute in most big cities, and Glasgow is no exception. Each evening, to the alarm of those who cared for my welfare and my future, I walked past the neatly trimmed hedges of our street and strolled into the world of tenement closes, pens, yards, and dunnies. And, like Boswell but in reverse, I exchanged my tongue on the way. Leaving my jacket at home and pittan oan ma jaiket; leaving Glasgow and daun'erin owre tae Glesca.\n\nIt is hardly unusual for teenagers to have one language for the playground and another for the classroom or, as in this case, one for the street and another for the living room. Here the scene was dramatized into a choice between dark and light, like the choice Tam o' Shanter faced as he rode home on his grey mare Meg after an evening sat \"bousin at the nappy.\" My mother, like most of the mothers round about, had barely heard of Boswell; but she knew all about \"the abominable Glasgow tongue.\" By gentling their vowels, and those of their children, my parents were doing what generations of lowly folk had done before: they were trying to \"get on,\" or, in Boswellian terms, to make themselves welcome to feed at the table. The streets of Glasgow were crowded with people who had not got on. They were poor, they were out of work, they drank too much and had troublesome dealings with the law, and frequently with everyone else who came near them. It seemed they could not even negotiate the vowels and consonants of the language, the Queens English, with proper competence. _Who'll give you a job when you speak like that?_ It was held up before us as a character failure.\n\nThere were respectable people who said \"grun\" when they meant ground, who couldn't shape the \"ou\" in house, or the \"ea\" in dead and bread, but said \"hoose\" and \"deid\" and \"breid;\" who said \"hame\" for home; who had difficulty in completing simple words, such as of, all, Dad (o', a', Da'), and could not master the pronunciation of blind, but had to settle for \"blinn\" instead. But mostly they lived in the country, like my mother's adored Uncle Willie, a shepherd on a Highland farm, where a Scots tongue was regarded as a \"hertsome\" thing. In the country, the broad Scots accent sounded healthier, just as the milk and the eggs that came straight from the farm on to our breakfast table tasted better.\n\nIf these acceptable Scots speakers were not country folk, they were elderly, rooted in old ways and the nineteenth century, like Grandma and Grandpa. For it is an oddity of the linguistic politics I am discussing that when my sisters and I went to visit our grandparents, born within the lifetime of Robert Louis Stevenson, we were greeted by the auld tongue. \"C'wa ben the hoose,\" our grandmother would say on our arrival. A light fall of rain she'd call a \"smirr,\" or a \"smirrie rain;\" wet children were \"fair drookit.\" If she should \"jalouse,\" or suspect, a cold, she gave us loathesome brandy. Like many of her class and generation, her speech, though principally English, dawdled naturally and frequently amid the Older Scots. She said \"gang\" for going, \"havers\" for nonsense, called boys and girls \"chiels and quines,\" called a drain a \"stank.\" If it was \"dreich\" outside, it was \"a scunner\" to her. Many of the words from Burns's vocabulary would have come naturally, and still do to large numbers of Scottish people: \"bide\" for live, for example; \"thole,\" to endure. Grandma would never have heard of Hugh MacDiarmid, the greatest twentieth-century practitioner of Scots, but she would have understood his verse:\n\n_Mars is braw in crammasy,_\n\n_Venus in a green silk goun,_\n\n_The auld mune shak's her gowden feathers,_\n\n_Their starry talk's a wheen o' blethers,_\n\n_Nane for thee a thochtie sparin,_\n\n_Earth, thou bonnie broukit bairn!_\n\nMeanwhile, my elder sisters were being sent to an elocution teacher to comb out as many tholes, bides, dreichs, and drookits from their speech as possible. They pranced around the house saying, \"How\u2014now\u2014brown\u2014cow,\" in theatrical fashion. Boswell, recalling his own youthful instruction from Thomas Sheridan, would have smiled on them.\n\nMy mother and Boswell had a formidable range of good intentions in their armory. These included education, respectability, worldly acceptance. On our side, Mrs. Miller and I (though I did not yet realize it) had literature. Only much later did I understand that the language spoken by my friends in the Gor-bals, by Grandma and Grandpa and Uncle Willie with his shepherd's crook, and by the chiels and quines in Ayrshire and throughout Scotland, was not corrupt at all. Each regional variation, including the Glesca dialect, was derived from the Older Scots, the language used by the great fifteenth-century \"makaris,\" William Dunbar and Robert Henryson. The boys and girls in the neighboring streets, who said \"thae aipples,\" did not do so out of an inability to pronounce \"these apples,\" or because they found it an embarrassment to \"speak polite,\" as when prodded by teachers to say \"Glasgow\" instead of \"Glesca;\" they were simply, unwittingly, carrying on the Older Scots idiom which centuries of elocutional refinement had failed to smooth out. \"Doon,\" \"gaun,\" \"grun,\" \"dinnae\" and a thousand other features of present-day Glasgow speech are retentions from a way of talking that was once common to all the people of Lowland Scotland (as \"gotten\" and \"the fall,\" no longer used in English English, are retained in the American). \"Thae aipples yonder, lyan oan the brae ahent the dyke, are sweit and bonie\" is a sentence which my Gorbals comrades and the poet William Dunbar (1460\u20131513) would understand, as one. \"These apples on the hill, over there behind the wall, are sweet and delicious\" is not.\n\nDo I \"feel\" in Scots, despite thinking in English? On occasion, yes, especially to accommodate certain rushes of skepticism or joy that I take to be native. Or to express anger, or engage with children and animals. Scots words are apt to make a particular appeal to me, and Scots poetry, in the higher range, pleasures me like no other. Dunbar was quick to insist that he was a lesser poet than Chaucer, but Dunbar's poetry speaks to me in ways that Chaucer's never does and never could. It finds the familiar in me. The language of Dunbar's poetry, and that of certain colleagues writing five hundred years later, comes across as something half-remembered, like a first language since superseded. When I arrive at Glasgow Central Station these days, a wave of recognition breaks over me as I step off the train. The speech in the air around me carries experiences which, though I may not have realized it till then, were obscured by the invisible wall that separates Scotland from England.\n\nWhen I went south to live in London, at the age of thirty, I admit it, I did a Boswell. I straightened out my tongue. It had never been \"abominable\"\u2014at least not since those adolescent days\u2014but it had what others were pleased to refer to as a \"lilt.\" When I heard myself speak on a tape recorder or on the radio, I would be surprised at how strong my accent was. But gradually it faded. It happened without my noticing. I didn't shoo it away, or plot my advancement among the London literati by honeying the knobbled surfaces and thistled joints of my syllables. I excuse this fact, when people remark on it (invariably to my annoyance), by telling myself that my voice is mimetic by nature, that my tendency is to sound like those by whom I wish to be understood, that my Scots voice hasn't gone away, it's just concealed beneath these southern clothes. Or thae suddron claes.\n\nThe paradox\u2014our own family paradox\u2014is that while my accent traveled southwards, that of my parents went back in the opposite direction. In recent years, my father, in particular, would announce \"ah cannae thole it,\" usually in reference to a politician or something else that he found \"a right scunner.\" He never talked like that in the days when I was being persecuted for the company I kept. He would have said of the politician \"I can't stand him,\" that he found him annoying. In his last years, living at the lower fringe of the Highlands among people who speak a mild modern form of Scots, he found his vowel sounds drawn back to the streets where he had grown up\u2014not that far, it so happens, from the stamping ground of my notorious Gor-bals cronies. I noticed a certain self-consciousness as he modulated into this voice, often for my benefit, but also a pleasure, a relaxation, at being reacquainted with his older tongue.\n\nMy mother would have no qualms about sticking with the \"Glasgow\" way. But one day, during my fathers last illness, when he responded to doctors orders by failing to take his medicines when he should have, threatening to go out when he shouldn't, and generally behaving obstinately, she sat down in her usual chair with an air of great weariness and turned to me.\n\n_\"Thrawn,_ I think is the word,\" she said. Her precision took me aback. Thrawn means, literally, twisted or crooked, but it has a more common figurative sense, which is not hard to see: difficult, stubborn. Thrawn was indeed the word. I believe I had never heard her use it before. She must have been saving it up.\n\nAs for Boswell, several years after his abomination of Mrs. Miller, he was back in London, having in the meantime returned to Scotland to marry and set himself up as a lawyer. An entry in the Journal, March 30, 1772, finds him in a Covent Garden coffee house with Johnson, contemplating the idea of moving his practice south for good (he never did). \"Mr. Johnson is not against it; and says my having any Scotch accent would be but for a little while.\" Here he gives himself away. Almost a decade after having seen off Mrs. Miller, despite his lessons in \"pronunciation\" from Thomas Sheridan, Jamie is still talking native.\n\n# SPANISH\n\n#\n\n# _Footnotes to a Double Life_  \nAriel Dorfman\n\nI should not be here to tell this story.1\n\nIt's that simple: there is a day in my past, a day many years ago in Santiago de Chile, when I should have died and did not.2\n\nThat was the place, the house of death.3 That's where I caught pneumonia one Saturday night in February of 1945, when my parents had gone out by themselves for the first time since we had arrived in the States\u2014and I carefully use that verb, to catch, aware of its wild ambiguity, still unsure, even now, if that sickness invaded me or if I was the one who invited it in. But more of that later. To save his life, that boy was interned in a hospital, isolated in a ward where nobody spoke a word of Spanish. For three weeks, he saw his parents only on visiting days and then only from behind a glass partition.4\n\nMy parents have told me the story so often that sometimes I have the illusion that I am the one remembering, but that hope quickly fades, as when you arrive at a movie theater late and never discover what really happened, are forever at the mercy of those who have witnessed the beginning: _te internaron en ese hospital,_ my mother says slowly, picking out the words as if for the first time, _no nos acordarnos del nombre, 5_ there is a large glass wall, it is a cold bare white hospital ward, my parents have told me that every time they came to see me, tears streamed down my face, that I tried to touch them, I watch myself watching my parents so near and so far away behind the glass, mouthing words in Spanish I can't hear. Then my mother and my father are gone and I turn and I am alone and my lungs hurt and I realize then, as I realize now, that I am very fragile, that life can snap like a twig. I realize this in Spanish and I look up and the only adults I see are nurses and doctors. They speak to me in a language I don't know. A language that I will later learn is called English. In what language do I respond? In what language can I respond?\n\nThree weeks later, when my parents came to collect their son,6 now sound in body but in all probability slightly insane in mind, I disconcerted them by refusing to answer their Spanish questions, by speaking only English. \"I don't understand,\" my mother says that I said\u2014and from that moment onward I stubbornly, steadfastly, adamantly refused to speak a word in the tongue I had been born into.\n\nI did not speak another word of Spanish for ten years.7\n\nOut there,8 at the edge of my tongue, within reach of the Spanish words I hardly knew how to formulate, a real challenge was lying in wait for me: the people who spoke that language, the guardians of a plenitude of things and experiences that were to sensually surround my body and demand a name. That Spanish out there contained my future. It contained the words of Garc\u00eda Lorca I would say to Ang\u00e9lica one day _Verde que te quiero Oerde,_ the lover-like green of desire, and the words of Quevedo I would say to my country, _Mir\u00e9 los rnuros de la patria rnia,_ watching the walls of my fatherland crumble, and the words of Neruda I would say to the revolution, _Sube a nacer conmigo, hermano,_ rise and be born with me, my brother, and the words of Borges I would whisper to time, _los tigres de la memoria,_ the tigers of memory with which I would try to fool death once again. I would realize one day that the word for hope in Spanish, the word _esperanza,_ hides within its syllables the sound and meaning of _esperar,_ to wait, that there was in the language itself a foretelling of frustration, a warning to be cautious, to hope but not to hope too much because the experience of those who forged those syllables tells them that we end up, more often than not, being violated by history.9\n\nNot only wonders, in Spanish: also learning with it how to avoid responsibility. A day comes back to me\u2014I must have been sixteen\u2014the first time I realized that Spanish was beginning to speak me, had infiltrated my habits. It was in carpentry class and I had given a final clumsy bang with a hammer to a monstrous misshapen contraption I had built and it broke, fell apart right there, so I turned to the carpentry teacher and _\"Se rompi\u00f3,\"_ I said, shrugging my shoulders.\n\nHis mouth had twisted in anger. _\"Se, se, se,\"_ he hissed. \"Everything in this country is _se,_ it broke, it just happened, why in the hell don't you say I broke it, I screwed up. Say it, say, _Yo lo romp\u00ed, yo, yo, yo,_ take responsibility, boy.\" And all of a sudden I was a Spanish speaker, I was being berated for having used that form of the language to hide behind, I had automatically used that ubiquitous, impersonal _se,_ I had escaped into the language, _escap\u00e9 lenguaje adentro,_ merged with it.10\n\nI became conscious then of the other elusive ways in which the language allowed its devoutest followers to pass the buck on to others, the proliferation of passive forms and the overemployment of the _hay que, hab\u00eda que, habr\u00eda que_ (approximately, \"it should be necessary to...\") which, in years to come, would drive me crazy, people all around me endlessly discussing in smoke-filled rooms what should be done and very few of them effectively doing anything. But by then I had gone deeper into the language and learned that this multiplication of possibilities and parallel paths could also be a virtue, could also enrich the language. I had come to explore the verb system in Spanish, perhaps the richest in the Indo-European family of languages. I had come to adore the fluid use of time that Spanish plays with, I had internalized the subjunctive, to mentally live a plurality of forms of time that had not yet occurred, a time that was suspended and waiting to occur, a time that existed in the mind even if it had no chance of materializing in history, the construction of alternative imaginary universes always haunting the hard reality of our hearts trapped in the prison house of today and now and right here.11\n\nI was not aware of what was happening to my mind: it was a subtle, cunning, camouflaged process, the vocabulary and the grammatical code seeping into my consciousness slowly, turning me into a person who, without acknowledging it, began to function in either language. Although from the very beginning I did not allow my new language to enter into a dialogue with the older one. I stubbornly avoided comparing their relative merits, what one could offer me that the other could not. It was as if they inhabited two strictly different, segregated zones in my mind, or perhaps as if there were two Edwards,12 one for each language, each incommunicado like a split personality, each trying to ignore the other, afraid of contamination. I did not attempt\u2014 or even contemplate the possibility of\u2014cross-fertilization: to weigh the caliber and performance of one against the other would have meant creating a territory from which to think the phenomenon, a common space they both shared within me. It would have meant admitting that I was irrevocably bilingual, opening the door to questions of identity that I was much too vulnerable and immature to face: Who is it that speaks Spanish? Is it the same youngster who speaks English? Is there a core that is unchanged no matter what dictionary you reach for? And which is better equipped to tell a particular story? And how is it that your body language changes when you switch from one to the other? Is it a different body? Questions that only many years later, only now that I have agreed to their coexistence, can I begin to register.13\n\n1 That is how my memoir _Heading South, Looking North: A Bilingual Journey_ began.\n\nBegin, began, beginning: in a manner of speaking. Because it took me almost nine months to come up with that first line and the five or six lines that followed. It was, in fact, exactly nine, but I hesitate to say so, as this evokes an abusively glib parallel between gestation\/childbirth and creation\/writing. But nothing glib about the nine months of hard work, scribble, clack, _tecleo continuo,_ day after day, _d\u00eda y noche._ Writing and rewriting the first ten pages of the book over and over again until I wondered if my wife had been right to tell me that I would go insane before I had completed the first chapter.\n\nForget the many other reasons why self-scrutiny can be so agonizing and zero in on what was really bugging me: I couldn't for the life or death of me decide in which of my two languages to write the story of my life. They had been _disput\u00e1n-dome_ for most of my existence, each of them dominating my life monolingually, for long stretches of years freezing the other out of power and articulation. Until I got tired of being a child pulled this and pushed that way by two distraught parents insulting each other in a language the other pretended not to know but that the disputed offspring understood all too well. Tired of being a husband with two squabbling wives or a mistress with two lovers or maybe I was the bed where the two vocabularies coupled or... choose your metaphor, _tu met\u00e1fora._ What matters is that by the time I had decided to write the memoir, these two sides of my brain, these two tongues lodged in the cavity called my _cabeza_ \u2014also known as a head\u2014 had declared a truce, had decided to stop waging war because I needed them both to survive exile, to make a living (you get paid once in Spanish and once in English and between both payments, _sabes,_ you manage to get one whole meal for one family of four). I needed them because of the dictatorship in Chile: how to deny the possibility of transmitting twice over to an increasingly deaf and indifferent world the story of my ravaged land\u2014which would, presumably, lead to my being able to convince twice as many people. And that armistice led me to believe that I could now tackle the story of my life, I could at least give it an ending that did not conclude in strife and dividing walls.\n\nBut no sooner did I start to write the first sentences of that autobiography in one of the languages, say English, than the Spanish misbehaved abominably, blocked those words as if they were alien, an in flagrante case of linguistic adultery. And the same menace of divorce\u2014\"you do this, boy, and I am outta here\"\u2014if I tried to spin the tale in Spanish, my English telling me it would not tolerate such treachery. I am using a metaphor, of course. _Claro que s\u00ed._ Languages do not exist as characters in a play. They may talk about sex but they don't\u2014you know\u2014 _do it;_ talk about battles, but don't fire real bullets. By conferring an independent life on them, I am merely trying to express their extraordinary power over me, how I felt that their double boycott of my writing functioned in the most concrete of ways. _Muy simple._\n\nSo how did this boycott work?\n\nWhenever I wrote anything about my life, in either language, it simply sounded... false, _falso,_ fraudulent, _fraudulento._ And the Spanish, by the way, has that _lento_ adhered to its tail, that sense of a fraud that is slow, that persists, that prolongs itself inside your mind. And the English is, therefore, at least for me, peremptory and cutting, something not to be forgiven, that \"t\" at the end terminating all altercations. Nothing _lento_ about my English language self. In a word (and there's the rub, it can't be expressed in a word at all, _ni en una palabra):_ jealousy\/ _celos,_ they paralyzed me by making me feel that anything I stated on paper in one language about the other would not pass the test.\n\nSo what was it that they actually objected to, my two _amantes?_\n\nEach had agreed to allow the other right of passage, rites of passage, as long, I came to realize, _as long as the story being told was not theirs._ Spanish said: _yo voy a contar lo que pas\u00f3, porque me pas\u00f3 a m\u00ed._ And English repeated (or anticipated first) the same words: I am the one who will tell what happened, because it happened to me. I could argue with each that it had happened to me, and therefore to both of them, to the two zones of poor me\u2014but they argued back that it could be expressed first only by one rather than the other, because languages can incorporate any number of loans from another tongue, but at their moment of enunciation demand exclusivity. And so nine months went by. So desperate those months, so crippled by the certainty that I could not venture one word in either language without feeling that I was betraying one or the other. Because this was not a quick fling, it wasn't an article I wrote for the _New York Times_ that had to be in English or one that I wrote for _El Pa\u00eds_ in Madrid that had to be in Spanish, a novel like _Konfi-denz_ that I penned in Spanish and then translated into English and then went back and corrected the Spanish version with what I had learnt while transposing it to my other tongue. Oh no, this was the far more serious matter of moving in, settling down, with one or with the other, choosing one over the other, giving it primacy, bragging dibs, establishing a hierarchy and _una primogenitura._ Favoring one over the other to tell the story of their troubled relationship. Imagine two countries at war who have fought each other to a standstill and find, at the moment of the armistice, that the final treaty will only be in one of the tongues. The next day they would be at war again, right? This time over whose language would be the one used to define the terms, the frontiers, the reparations, the repatriation of enemy soldiers. Otherwise, who knows what contumacious and abstruse clauses and particulars the rival entity has smuggled into the final draft. I'll tell you how raving and frenzied I got: one dawn, after an almost sleepless night, I grabbed a pad of paper and scratched out some words in... French. C'est vrai! A language I can stutter, barely bring myself to write. You want neutral?, some worm in my brain asked. _Quieres neutralidad?_ Well, you got it... Yes, that's what I got: an impartial arbiter to speak for me, but she (is French a she?, voluptuously so, I think) had no capacity for writing anything eloquent or intimate: so again, I had no memoir\n\nThat madness at dawn may, however, have been what saved me. I had hit the bottom of the pit. I think it was the next morning (it certainly makes this a more interesting story if there is an immediate consequence) that I decided that enough was enough, _basta,_ it was time to let my languages know who was in charge. If you do not let me decide, I said to them, I will end up in a mental institution and my words will be neither in English or Spanish but a combination of the two of you closer to sheer jarring gibberish.\n\nFor reasons that I prefer to keep under strict lock and key here inside, I then proceeded to choose English as the vehicle for my life, give English first rights\u2014but temporarily, I promised, just to get this damn thing into the world; and then, I turned and murmured to my Spanish, _te voy a dejar que re-escribas por entero el libra,_ I'll let you write your own version of my life.\n\nIt was a trap and maybe my Spanish knew it\u2014or maybe at that time she, he, it, didn't care. Maybe she knew that once the story had been established in a certain way, once I had told it in English, it would be basically invariable.\n\nWhich turned out to be the case: my rewriting of the memoir in Spanish after I completed it in English followed the structure, story, explorations of history and of the mind which its rival language had set out. Spanish had to overflow its words inside the house that English built.\n\nAnd yet, how changed was that house as it filled with Spanish.\n\nIt was not the same book.\n\n2 Look at the first lines in the Spanish text, read them: _\"Si estoy contando esta histo- ria, si la puedo contar, es porque alguien, muchos aftos atr\u00e1s en Santiago de Chile, muri\u00f3 en mi lugar.\"_ Meaning: \"If I am writing this story, if I can tell it, it is because someone, many years ago in Santiago de Chile, died instead of me.\"\n\nIt took me a good few weeks to figure that one out, that I could not merely transfer and smooth the English words into Spanish, that Spanish was going to demand that I keep at least part of my promise and allow a slightly different version of my existence to circulate in the world.\n\nSee how the Spanish elongates and complicates the brief and unaffected early formulation (I should not be here to tell this story) of the English? Beginning with an If\/ _Si,_ making existence more conditional, adding a second if\/ _si_ to vaguely suggest how halting this process has been. But more crucially: the Spanish required that from the very start I include the fact that someone had died in my place, instead of me. I could not remember my survival in Spanish without remembering immediately the Spanish-speaking person who was dead while I was alive. I had never spoken in English to the man, Claudio Jimeno, who died in my stead. He makes an appearance very soon in the English version of the memoir, of course, but not in the first lines. My Spanish, therefore, was not willing to leave aside or behind the community it carried inside its vowels and grammar, needed to thrust that reciprocal dimension of my life straight into the reader's mind. Without delay Because this was my _historia:_ not just my story, but my simultaneous history, the history I had both made and suffered. Not that I had blocked myself from spelling all this out in my English. My memoir is, among other things, an exploration of how we shape history as it shapes us, how a language speaks us as much as we speak it. But in the sensuous Spanish pounding out of the specific words, the emphasis was altered, the landscape found itself widened, it was established from the get-go that this story would not be mine alone. Someone else died. And the first words that someone heard in his life, the last words he heard in his life, the first and last words I said to him, to my friend Claudio, were in Spanish. The revolution he died for, that I did not give my life for, was lived by both of us, in the Spanish of Chile.\n\nBut this other version is also determined by what sounds better. \"When I should have died and did not,\" well, that resonates in me dramatically, perhaps even elegantly, the way in which the succession of eis re-enforce one another, the way in which that _not_ at the end closes any door to doubt. The Spanish translation _cuando deb\u00ed haber muerto y no lo hice_ is weak, awkwardly constructed, repeating the ugly _bs (deb\u00ed haber)._ Worse still, the use of the verb _hacer (hice)_ misconstrues the original meaning in English, by positing the survival as more active than it really was. To give an approximate equivalent of the first sentence in the English language memoir I would have had to write something like _cuando la muerte vino por m\u00ed y no me encontr\u00f3_ (\"when death came for me and did not find me\"), but that is a thought and a structure of feeling I wanted to reserve for later in the text.\n\n3 We're already in the second chapter (I've been born into Spanish and Argentina and at a very young age am already being subjected to displacement in time and space), and I'm referring to my arrival in New York at the age of two and a half\u2014 speaking not a word of the language in which I write this footnote.\n\n4 Note the change to the third person. Distance. My demon and my savior and my instrument: _distancia._ I am apparently already trying to compensate for the fact that this is my English language persona speaking about what happened to _\u00e9l, that one, that kid, ese ni\u00f1o._\n\nI was not there\u2014this Ariel who writes this in English now, who wrote that in English when I began the second chapter of the memoir. Is it the Spanish that, growling gently inside, demanded some signal in its direction, that nod to its past dominion? Maybe. But it turns out that Spanish itself did not know, does not remember, what happened next, has kept not one _pedacito_ of _rememoranza,_ is as orphaned as English\u2014as the child himself felt himself to be... So the next line had to be: My parents have told me the story so often.\n\n5 And must have told it in Spanish, because that is how I recall their own memory, narrating me, my mother, into the future with the tongue that she herself had used to hijack a different identity from the Yiddish she had first heard floating over her infantile head as she headed from Europe to Argentina on that steamer from Hamburg four years before the First World War was to ravage her continent of birth.\n\nBut this is also a device\u2014not merely stylistic\u2014that I use throughout this memoir... and in some of my journalism as well as my later novels. Introduce Spanish directly into the text (or English if the text is in Spanish), often without explaining or translating, no help to the reader, you're on your own, as I was, shipwrecked in a sea of words we don't understand. A tiny taste of what it means to be adrift in someone else's language. Or maybe, in this case, it is a way of assuaging the Spanish in which, after all, this event first happened and was first told, by giving it a token presence\u2014 letting it sit down at a remote corner of the table and not be submitted to translation's perfidious traffic.\n\nA technique, by the way, that I first noticed in a number of Spanish American novels\u2014for instance by the Peruvian Jos\u00e9 Mar\u00eda Arguedas or the Paraguayan Augusto Roa Bastos. These authors, who came from bilingual societies (Quechua\/ Spanish and Guaran\u00ed\/Spanish, respectively), interjected many native words into the Spanish text and eliminated footnotes or glossaries, forcing their readers to infer the meaning from the context, refusing to let us off the hook. You don't understand? _No comprendes, carajo?_ Too bad! Learn Guaran\u00ed!\n\n6 Here my devious double-crossing mind makes a transition, tries to have it both ways, find a middle ground between the Spanish in which this occurred and the English in which it is expressed: _my parents_ and _their son,_ an attempt to be object and subject.\n\n7 Perhaps I can now elucidate the deepest reason why I chose to write that memoir in English. _Heading South, Looking North: A Bilingual Journey_ is organized in two alternating series of chapters. One sequence starts by stating that I was almost killed in Santiago during the military coup of 1973 and goes on, in the ensuing chapters, to follow me as I survive death on several occasions until the day when I finally am able to go into exile, a situation that will force me to live away from Chile and therefore be bilingual. The other sequence starts with my birth in Argentina (my birth into Spanish) and, after a first chapter which ends with my switching from Spanish into English (the one I am commenting on here), proceeds to show how history finally sent me back to Latin America, to my native tongue, how the day when I could have died and did not was awaiting me in Chile. The central events that determine the two sequences are both traumatic, moments when death circled me, whether in the New York hospital as a child, or on the streets of Santiago as a young adult. And both these shattering events were lived by me in Spanish.\n\nSo, why English?\n\nI think it may have been because it was the best way of dealing with the ordeal, using the measured framework of the English words to contain the pain, to look at those circumstances in a sort of roundabout, indirect fashion. English as a sort of oblique mirror that allowed me to see the events in a different (or at least tolerable) light, work through this confession, show myself, perhaps reveal myself, use the distance, treat myself as an almost fictional object. So much so that very often, when, later on, I was reworking the text in Spanish, I would find myself sick and trembling, faint with anxiety, asking my book how I had dared to write this, what naked madness must I have gone through and tamed in order to finally bring out into the open such secret thoughts. And those events, after all, those two close encounters with death, were at the origin of my conversion to the English in which I was writing this; they were, in some bizarre manner, the mothers, _las madres_ of this very language. Or maybe the _padres,_ maybe the Spanish words had inseminated an English child in my brain. Whatever the gender, this much seems to be true: to unveil one's origins, to journey to where it all started, we may need to use a different tongue, create an alter ego and trust him with the furtive truth we have told no one. You can't journey to your origin without a translator of some sort by your side. And a consolation: the ultimate reconciliation of my languages in this memoir, perhaps in this commentary as I write it. The very fact that I can write it may be proof that they are finally beginning to trust one another\n\n8 By now, I have been forced to leave the States and journey to Chile, forced to speak and write the detested Spanish language.\n\n9 If I do not remember the first transition of my life (from non-language to Spanish) or the second one (from Spanish into English)\u2014and have had to basically invent the way in which I experienced them\u2014this return to Spanish is one that needs to be expressed as a real process. But that does not mean that I know exactly how that happened, that I was conscious, while it was happening, of what was going on. It was a long, drawn-out seduction, a back and forth operation, a crossing and recrossing of the borders of my own mind. And so I do not focus on one event as much as offer an approximation, anticipate a future when Spanish will be present in all its glory. As if it had been calling to me, _Ilam\u00e1ndome todos esos a\u00f1os de exilio,_ during all those years of banishment.\n\n10 And here comes English to the rescue! The presence of that other language inside me, back then, and also now, does not allow me to hide as much as I would have wanted, as many of my compatriots can and still do. Not that English-language speakers are any less adept than those who practice Spanish at squirming out of accountability. Kissinger uses his (foreign-learned) English as skillfully as Pinochet uses his (barely learned) Spanish to avoid, both of them, facing the crimes they are alleged to have committed against humanity. Being bilingual does not exempt you from any of the horrors of the human. No one language condemns you to laziness or efficiency, mendacity or truth. If you dispose of two languages, therefore, you can lie twice as much\u2014but also have a good extra whack at the truth, if you are so inclined.\n\n11 The language I wrote my memoir in may be English, but the aesthetic seems to be resolutely Spanish American, the creation of that parallel conjuring up of what might have been, what still might be, language as the one irrevocable site of free dom, my life as not only what happened, but what almost happened. Life as a series of footnotes to a text written by someone else, more powerful, apparently in command. Latinos: embracing our margins as if we had chosen them instead of history imposing that marginality upon us.\n\n12 In my insanity to become American and leave my Spanish past behind, I had rebaptized myself Edward. The few friends I have left over from adolescence still call me Ed.\n\n13 Most of this paragraph was not in the original manuscript. John Glusman, my friend and editor at Farrar, Straus, believed that I needed, at some point in my memoir, to examine these problems. How did I do it? What does it mean to be, feel, live bilingually? At what point does Spanish take over, when does English concede? How in the hell did it, does it, actually _happen?_\n\nAnd what I wrote then, in response to his legitimate questions, was as far as I could go, at least back then. Back then? Even now I do not dare to venture any deeper into that territory.\n\nAs if the reader had not already realized that this collaboration between my two languages, my two loves, is a precarious and fragile one, that can be all too easily upset. Dangerous, certain questions. Like a sweetheart asking if she makes love better than the other one, the wife, the legitimate spouse.\n\nI am wary of opening up anything that could disturb the balance I have somehow struck between my two recently reconciled but still potentially antagonistic vocabularies. There is someone inside me that makes the decision about when to speak Spanish and when to speak English. Frequently, most of the time, this is decided for me. I answer, generally, in the language in which I am addressed: I sit at the computer and set the language for Spanish if I am writing for _Revista Proceso_ in Mexico or reach for the English book of synonyms if I am planning to publish the piece in the _Washington Post;_ I use Spanish with my graduate students and English with my undergrads\u2014and so on and so forth, _y as\u00ed es la cosa._ But there are many solitary moments in my day, inevitably, for any author, when I am left alone with _mis dos idiomas,_ and I have to decide which of them will receive my full attention. And I do not intend, for the moment, to ask myself how I reach that conclusion, why one at a certain moment sprints out of my fingers onto the keyboard or simmers to the surface when I am looking at a tree during a walk through the woods. I don't want to know, I don't want to legislate, I don't want it to be anything other than spontaneous, automatic, surfacing from some depth that I prefer not to gaze into.\n\nQuestions which, if I had asked them when I was first starting this journey toward duality, would have made me clamp down, suffocate Spanish again, deny its right to a voice. And my Spanish knew this, and cooperated, was glad to be once again inside my head, did not call attention to its gains, was not stupidly going to let itself crow victory when suddenly, in the middle of a sentence in English, a word in Spanish would make its upstart appearance as if nothing was more natural in the world, given that there was no English equivalent for that untranslatable turn of phrase. My Spanish did not demand that I examine why I needed that precise word when I had an infinitude of English at my tongues end, why it was irreplaceable. Having smuggled itself in, my Spanish was wise enough not to corner me. Instead, quite simply, it grew. And grew. And grew.14\n\n14 This much I do know. It is still growing. As is the English. Even when I do not use one of the two languages, when one of them is relegated to the attic of my life, that language, be it English or Spanish, continues to grow\u2014and reign. At this very moment, as I compose this, my Spanish is whispering instructions, suggestions, blowing rhythms my way, shaping the rival's choices. Creating between the two of them, something that is not quite one hundred percent English or Spanish, but something quite other, _creciendo ambos._ I swear it is true, I hope it is true.\n\n_Juro que es cierto._\n\n_M\u00e1s bien: espero que sea cierto._\n\n# YIDDISH\n\n#\n\n# _My Yiddish_  \nLeonard Michaels\n\nIn Paris one morning in the seventies, walking along rue Mahler, I saw a group of old men in an argument, shouting and gesticulating. I wanted to know what it was about, but my graduate school French was good enough only to read great writers, not good enough for an impassioned argument or even conversation with the local grocer. But then, as I walked by the old men, I felt a shock and a surge of exhilaration. I did understand them. My god, I possessed the thing\u2014spoken French! Just as suddenly, I crashed. The old men, I realized, were shouting in Yiddish.\n\nLike a half-remembered dream, the incident lingered. It seemed intensely personal, yet impersonal. Meaning had come alive in me. I hadn't translated what the old men said. I hadn't done anything. A light turned on. Where nothing had been, there was something.\n\nPhilosophers used to talk about The Understanding as if it were a distinct mental function. Today they talk about episte-mology or cognitive science. As for The Understanding, it's acknowledged in IQ tests, the value of which is subject to debate. It's also acknowledged in daily life in countless informal ways. You're on the same wave length with others or you are not. The Paris incident, where I rediscovered The Understanding, made me wonder if Descartes's remark, \"I think, therefore I am,\" might be true in his case, but not mine. I prefer to say, \"I am, therefore I think.\" And also, therefore, I speak.\n\nUntil I was five, I spoke only Yiddish. It did much to permanently qualify my thinking. Eventually I learned to speak English, then to imitate thinking as it transpires among English speakers. To some extent, my intuitions and my expression of thoughts remain basically Yiddish. I can say only approximately how this is true. For example this joke:\n\nThe rabbi says, \"What's green, hangs on the wall, and whistles?\"\n\nThe student says, \"I don't know.\"\n\nThe rabbi says, \"A herring.\"\n\nThe student says, \"Maybe a herring could be green and hang on the wall, but it absolutely doesn't whistle.\"\n\nThe rabbi says, \"So it doesn't whistle.\"\n\nThe joke is inherent in Yiddish, not any other language. It's funny, and, like a story by Kafka, it isn't funny. I confess that I don't know every other language. Maybe there are such jokes in Russian or Chinese, but no other language has a history like Yiddish, which, for ten centuries, has survived the dispersion and murder of its speakers.\n\nAs the excellent scholar and critic Benjamin Harshav points out, in _The Meaning of Yiddish,_ the language contains many words that don't mean anything\u2014 _nu, epes, tockeh, shoyn._ These are fleeting interjections, rather like sighs. They suggest, without meaning anything, \"so,\" \"really,\" \"well,\" \"already.\" Other Yiddish words and phrases, noticed by Harshav, are meaningful but defeat translation. Transparent and easy to understand, however, is the way Yiddish serves speech\u2014between you and me\u2014 rather than the requirements of consecutive logical discourse; that is, between the being who goes by your name and who speaks to others objectively and impersonally. For example, five times five is twenty-five, and it doesn't whistle.\n\nYiddish is probably at work in my written English. This moment, writing in English, I wonder about the Yiddish undercurrent. If I listen, I can almost hear it: \"This moment\"\u2014a stress followed by two neutral syllables\u2014introduces a thought which hangs like a herring in the weary droop of \"writing in English,\" and then comes the announcement, \"I wonder about the Yiddish undercurrent.\" The sentence ends in a shrug. Maybe I hear the Yiddish undercurrent, maybe I don't. The sentence could have been written by anyone who knows English, but it probably would not have been written by a well-bred Gentile. It has too much drama, and might even be disturbing, like music in a restaurant or an elevator. The sentence obliges you to abide in its staggered flow, as if what I meant were inextricable from my feelings and required a lyrical note. There is a kind of enforced intimacy with the reader. A Jewish kind, I suppose. In Sean O'Casey's lovelier prose you hear an Irish kind.\n\nWittgenstein says in his _Philosophical Investigations,_ \"Aren't there games we play in which we make up the rules as we go along, including this one.\" _Nu._ Any Yiddish speaker knows that. A good example of playing with the rules might be Montaigne's essays, the form that people say he invented. _Shoyn,_ a big inventor. Jews have always spoken essays. The scandal of Montaigne's essays is that they have only an incidental relation to a consecutive logical argument but they are cogent nonetheless. Their shape is their sense. It is determined by motions of his mind and feelings, not by a pretension to rigorously logical procedure. Montaigne literally claims his essays are himself. Between you and him nothing intervenes. A Gentile friend used to say in regard to writing she didn't like, \"There's nobody home.\" You don't have to have Jewish ancestors, like those of Montaigne and Wittgenstein, to understand what she means.\n\nI didn't speak English until I was five because my mother didn't speak English. My father had gone back to Poland to find a wife. He returned with an attractive seventeen-year-old who wore her hair in a long black braid. Men would hit on her, so my father wouldn't let her go take English classes. She learned English by doing my elementary school homework with me. As for me, before and after the age of five, I was susceptible to lung diseases and spent a lot of time in a feverish bed, in a small apartment on the Lower East Side of Manhattan, where nobody spoke anything but Yiddish. Years passed before I could ride a bike or catch a ball. In a playground fight, a girl could have wiped me out. I was badly coordinated and had no strength or speed, only a Yiddish mouth.\n\nFor a long time, Yiddish was my whole world. In this world family didn't gather before dinner for cocktails and conversation. There were no cocktails, but conversation was daylong and it included criticism, teasing, opinionating, gossiping, joking. It could also be very gloomy. To gather before dinner for conversation would have seemed unnatural. I experienced the pleasure of such conversation for the first time at the University of Michigan, around 1956. It was my habit to join a friend at his apartment after classes. He made old fashioneds and put music on the phonograph, usually chamber music. By the time we left for dinner, I felt uplifted by conversation and splendid music. Mainly, I was drunk, also a new experience. Among my Jews, conversation had no ritual character, no aesthetic qualities. I never learned to cultivate the sort of detachment that allows for the always potentially offensive personal note. Where I came from, everything was personal.\n\nFrom family conversation I gathered that, outside of my Yiddish child-world, there were savages who didn't have much to say but could fix the plumbing. They were fond of animals, liked to go swimming, loved to drink and fight. All their problems were solved when they _hut geharget yiddin._ Killed Jews. Only the last has been impossible for me to dismiss. Like many other people I have fixed my own plumbing, owned a dog and a cat, gotten drunk, etc., but everything in my life, beginning with English, has been an uncertain movement away from my _hut geharget_ Yiddish childhood. When a BBC poet said he wanted to shoot Jews on the West Bank, I thought, _\"Epes._ What else is new?\" His righteousness, his freedom to say it, suggests that he believes he is merely speaking English, and antisemitism is a kind of syntax, or what Wittgenstein calls \"a form of life.\" But in fact there is something new, or anyhow more evident lately. The _geharget yiddin_ disposition now operates at a remove. You see it in people who become hysterical when they feel that their ancient right to hate Jews is brought into question. To give an example would open a boxcar of worms.\n\nIt's possible to talk about French without schlepping the historical, cultural, or national character of a people into consideration. You cannot talk that way about Yiddish unless you adopt a narrow scholarly focus, or restrict yourself to minutiae of usage. The language has flourished in a number of countries. Theoretically, it has no territorial boundary. The meaning of Yiddish, in one respect, is No Boundaries. In another respect, for \"a people without a land,\" the invisible boundaries couldn't be more clear. There is mutual contempt between what are called \"universalist Jews\" and Jewish Jews. It's an old situation. During the centuries of the Spanish Inquisition, Jews turned on Jews. In Shakespeare's _The Merchant of Venice_ \u2014assuming the merchant Antonio is a gay converso, or new Christian, and Shylock is an Old Testament moralistic Jewish Jew\u2014the pound of flesh, a grotesquely exaggerated circumcision, is to remind Antonio (who says, \"I know not why I am so sad\") of his origins.\n\nThe first time I went to a baseball game, the great slugger Hank Greenberg, during warm-up, casually tossed a ball into the stands, a gift to the crowd of pre-adolescent kids among whom I sat. My hand, thrusting up in a blossom of hands, closed on that baseball. I carried it home, the only palpable treasure I'd ever owned. I never had toys. On Christmas nights I sometimes dreamed of waking and finding toys in the living room. _Tokeh?_ Yes, really. If there is a support group for Christmas depressives, I will be your leader. The baseball made me feel like a real American. It happened to me long before I had a romance with the mythical blond who grants citizenship to Jews. By then I was already fifteen. I had tasted _traif_ and long ago stopped speaking Yiddish except when I worked as a waiter in Catskills hotels. What Yiddish remained was enough to understand jokes, complaints, insults, and questions. As guests entered the dining room, a waiter might say, \"Here come the _vildeh chayes,\"_ or wild animals. One evening in the Catskills I went to hear a political talk, given in Yiddish. I understood little except that Yiddish could be a language of analysis, spoken by intellectuals. I felt alienated and rather ashamed of myself for not being like them.\n\nFamily members could speak Polish as well as Yiddish, and some Hebrew and Russian. My father worked for a short while in Paris and could manage French. My mother had gone to high school in Poland and was fluent in Polish, but refused to speak the language even when I asked her to. Her memory of pogroms made it unspeakable. In Yiddish and English I heard about her father, my grandfather, a tailor who made uniforms for Polish army officers. Once, after he'd worked all night to finish a uniform, the officer wouldn't pay. My grandfather, waving a pair of scissors, threatened to cut the uniform to pieces. The officer paid. The Germans later murdered my grandfather, his wife, and one daughter. Polish officers imprisoned in Katyn forest and elsewhere were massacred by Stalin. This paragraph, beginning with the first sentence and concluding with a moral, is in the form of a _geshichte,_ or Yiddish story, except that it's in English and merely true.\n\nAt the center of my Yiddish, lest I have yet failed to make myself clear, remains _hut geharget yiddin,_ from which, like the disgorged contents of a black hole in the universe, come the jokes, the thinking, the meanings, and the meaninglessness. In 1979, American writers were sent to Europe by the State Department. I went to Poland and gave talks in Warsaw, Poznan, and Cracow. I was surprised by how much seemed familiar, and exceedingly surprised by the intelligence and decency of the Poles, a few of whom became friends and visited me later in America. One of the Poles whom I didn't see again was a woman in Cracow with beautiful blue eyes and other features very like my mother's. I was certain that she was a Jew though she wore a cross. I didn't ask her questions. I didn't want to know her story. I could barely look at her. I detest the word \"shiksa,\" which I've heard used more often by friendly antisemites than Jews, but in my personal depths it applies to her.\n\nAs suggested earlier, in Yiddish there is respect for meaning-lessness. If the woman in Cracow was passing as a Catholic, was she therefore a specter of meaninglessness who haunted me, the child of Polish Jews, passing as an American writer? A familiar saying comes to mind, \"If you forget you are a Jew, a Gentile will remind you,\" but, in the way of forgetting, things have gone much further. Lately, it might take a Jew to remind a Jew that he or she is a Jew. Then there is a risk of ruining the friendship. For an extreme example, I have had depressing arguments with Jewish Stalinists who, despite evidence from numerous and unimpeachable sources that Stalin murdered Jews because they were Jews, remain Stalinists. It's as if they would rather die than let personal identity spoil their illusions. Thus, the Jewish face of insanity says to me, \"Stalin was a good guy. He just got a bad rap.\" A demonic parallel to this mentality is in the way Nazis used material resources, critical to their military effort, to murder Jews even as the Russian army was at the gates. They would rather die etc. In the second century, Tertullian, a father of the Christian church, insisted that absurdity is critical to belief. His political sophistication seems to me breathtaking, and also frightening in its implications. As the believers multiply everywhere, it becomes harder to believe\u2014rationally\u2014in almost anything.\n\nParadox as a cognitive mode is everywhere in Yiddish. It's probably in the genes and may explain the Jewish love of jokes. The flight from sense to brilliance effects an instant connection with listeners. Hobbes calls laughter \"sudden glory,\" which is a superb phrase, but I've seen the Jewish comics, Lenny Bruce and Myron Cohen, reduce a nightclub audience to convulsive and inglorious agonies of laughter. When I worked in the Cats-kills hotels I noticed that it was often the _tumler,_ or the hotel comic and hell raiser, to whom women abandoned themselves. Jerry Lewis, formerly a _tumler,_ said in a televised interview that at the height of his fame he \"had four broads a day.\" As opposed to Jerry Lewis, Hannah Arendt preferred disconnection. She used the snobbish word \"banal\" to describe the murderer of millions of Jews, and later said in a letter that despite the abuse she had received for using that word, she remained \"light hearted.\"\n\nFamily was uncles and aunts who escaped from Poland and immigrated to the United States. They stayed with us until they found their own apartments. I'd wake in the morning and see small Jews sleeping on the living room floor. My Aunt Molly, long after she had a place of her own, often stayed overnight and slept on the floor. She was very lonely. Her husband was dead, her children had families of their own. A couch with a sheet, blanket, and pillow was available, but she refused such comforts. She wanted to be less than no trouble. She wore two or three dresses at once, almost her entire wardrobe. She slept on the floor in her winter coat and dresses. To see Molly first thing in the morning, curled against a wall, didn't make us feel good. She was the same height as my mother, around five feet, and had a beautiful intelligent melancholy face. I never saw her laugh, though she might chuckle softly, and she smiled when she teased me. She used to _krotz_ (scratch) my back as I went to sleep, and she liked to speak to me in rhymes. First they were entirely Yiddish. Then English entered the rhymes.\n\n_Label, gay fressen._\n\n_A fish shtayt on de tish._\n\n_Lenny, go eat._\n\n_A fish is on the table._\n\n_Shtayt_ doesn't exactly mean \"is.\" \"Stands on the table\" or \"stays on the table\" or \"exists on the table\" would be somewhat imprecise, though I think \"A fish exists on the table\" is wonderful. I once brought a girlfriend home, and Aunt Molly said, very politely, \"You are looking very fit.\" Her \"fit\" sounded like \"fet,\" which suggested \"fat.\" My girlfriend squealed in protest. It took several minutes to calm her down. The pronunciation of \"fet\" for \"fit\" is typical of Yiddishified-English, which is almost a third language. I speak it like a native when telling jokes. The audience for such jokes has diminished over the years because most Jews now are politically liberal and have college degrees and consider such jokes undignified or racist. A joke that touches on this development tells of Jewish parents who worry about a son who studies English literature at Harvard. They go to see Kit-tredge, the great Shakespeare scholar, and ask if he thinks their son's Yiddish accent is a disadvantage. Kittredge booms, \"Vot ekcent?\"\n\nAs a child I knew only one Jew who was concerned to make a _bella figura._ He was a highly respected doctor, very handsome, always dressed in a fine suit and, despite his appearance, fluent in Yiddish. His office was in the neighborhood. He came every morning to my father's barber shop for a shave. A comparable miracle was the chicken-flicker down the block, a boisterous man who yelled at customers in vulgar funny Yiddish. This man's son was a star at MIT. In regard to such miracles, an expression I often heard was \"He is up from pushcarts.\" It means he went from the Yiddish immigrant poverty to money or, say, a classy professorship. The day of such expressions is past. In the sixties there were Jewish kids who, as opposed to the spirit of Irving Howe's _The World of Our Fathers,_ yelled, \"Kill the parents.\" The suicidal implication is consistent with the paradoxical Yiddish they no longer spoke.\n\nIf I dressed nicely to go out, my mother would ask why I was _fapitzed,_ which suggests \"tarted up.\" Yiddish is critical of pretensions to being better than a Jew, and also critical of everything else. A man wants to have sex or wants to pee\u2014what a scream. A woman appears naked before her husband and says, \"I haven't got a thing to wear.\" He says, \"Take a shave. You look like a bum.\" Henry Adams speaks of \"derisive Jew laughter.\" It is easy to find derision produced by Jews, but Adams's word, aside from its stupid viciousness, betrays the self-hate and fear that inspires antisemitism among the educated, not excluding Jews. Ezra Pound called his own antisemitic ravings \"stupid.\" The relation of stupidity and evil has long been noted.\n\nJewish laughter has a liberal purview and its numerous forms, some very silly, seem to me built into Yiddish. Sometime around puberty, I decided to use shampoo rather than handsoap to wash my hair. I bought a bottle of Breck. My father noticed and said in Yiddish, \"Nothing but the best.\" I still carry his lesson in my heart, though I have never resumed using handsoap instead of shampoo. What has this to do with Yiddish? In my case, plenty, since it raises the question, albeit faintly, \"Who do you think you are?\"\n\nWhat I have retained of Yiddish, I'm sorry to say, isn't much above the level of my Aunt Molly's poems. But what good to me is Yiddish? Recently in Rome, during the High Holidays, a cordon was established around the synagogue in the ghetto, guarded by the police and local Jews. As I tried to pass I was stopped by a Jew. I was amazed. Couldn't he tell? I said, \"Ich bin a yid. Los mir gayen arein.\" He said, \"Let me see your passport.\" _La mia madrelingua_ wasn't his. This happened to me before with Moroccan Jews in France. I've wondered about Spinoza. His Latin teacher was German, and the first Yiddish newspaper was published in Amsterdam around the time of his death. Did he know Yiddish?\n\nI'm sure of very little about what I know except that the Yiddish I can't speak is more natural to my being than English, and partly for that reason I've studied English poets. There is a line in T. S. Eliot where he says words slip, slide, crack or something. \"Come off it, Tom,\" I think. \"With words you never had no problem.\" Who would suspect from his hateful remark about a Jew in furs that Eliot's family, like my mother's ancestors in Vienna, was up from the fur business? Eliot liked Groucho Marx, a Jew, but did he wonder when writing _Four Quartets,_ with its striking allusions to Saint John of the Cross, that the small dark brilliant mystical monk might have been a Jew?\n\n\"Let there be light\" are the first spoken words in the Old Testament. This light is understanding, not merely seeing. The Yiddish saying, \"To kill a person is to kill a world,\" means the person is no longer the embodiment, or a mode of the glorious nothing that is the light, or illuminated world. This idea, I believe, is elaborated in Spinoza's _Ethics._ Existence\u2014or being\u2014entails ethics. Maybe the idea is also in Wittgenstein, who opens the _Tractatus_ this way: \"The world is everything that is the case.\" So what is the case? If it's the case that facts are bound up with values, it seems Yiddish or Spinozist. Possibly for this reason Jewish writers in English don't write about murder as well as Christians. Even Primo Levi, whose great subject is murder, doesn't offer the lacerating specificity one might expect.\n\nIn regard to my own writing, its subterranean Yiddish keeps me from being good at killing characters. The closest I've come is a story called \"Trotsky's Garden,\" where I adopt a sort of Yiddish intonation to talk about his life. I'd read a psychological study that claimed Trotsky was responsible for murders only to please Lenin, his father figure. If so, his behavior was even worse than I'd thought. I wrote my story out of disappointment. I had wanted to admire Trotsky for his brilliant mind, courage, and extraordinary literary gifts. His description of mowing wheat in his diaries, for example, almost compares with Tolstoy's description of the same thing in _Anna Karenina._ Yiddish can be brutal, as, for example, _Gay koken aff yam,_ which means \"Go shit in the ocean,\" but in regard to murder what Jew compares with Shakespeare, Webster, Mark Twain, Flannery O'Connor, Cormac McCarthy, or Elmore Leonard? The Old Testament story of Abraham and Isaac, which is of profound importance to three faiths, stops short of murder, but it is relevant to the children in contemporary religious terrorism.\n\nA story by Bernard Malamud begins with the death of a father whose name is Ganz. In Yiddish, \"ganz\" means \"all\" or \"the whole thing\" or \"everything.\" Metaphorically, with the death of Ganz, the whole world dies. Everything is killed. Malamud couldn't have named the father Ganz if he had written the story in Yiddish. It would be too funny and undermine all seriousness. The death of a father, or a world-killed-in-a-person, is the reason for Hamlet's excessive grief, a condition feared among Jews for a reason given in the play: \"All the uses of this world seem to me weary, stale, flat, and unprofitable.\" Because Hamlet Senior is dead, Hamlet Junior is as good as dead. Early in the play he jokes about walking into his grave, and the fifth act opens, for no reason, with Hamlet in a graveyard, and then he actually jumps into a grave. On the subject of grief, in \"Mourning and Melancholia,\" Freud follows Shakespeare. Like Hamlet, who demands that his mother look at the picture of his father, Freud makes a great deal of the residual, or cathectic, force of an image. Again, regarding my Yiddish, when I once wrote about my fathers death, I restricted my grief to a few images and a simple lamentation: \"He gave. I took.\" My short sentences are self-critical, and have no relation to the work of writers known for short sentences. They are only Yiddish terseness seizing an English equivalent.\n\nShakespeare's short sentences\u2014like \"Let it come down,\" \"Ripeness is all,\" \"Can Fulvia die?\"\u2014seem to me amazing. I couldn't write one of those. This confession brings a joke instantly to mind. The synagogue's janitor is beating his breast and saying, \"Oh, Lord, I am nothing.\" He is overheard by the rabbi who says, \"Look who is nothing.\" Both men are ridiculed. A Jewish writer has to be careful. Between schmaltz and irony there is just an itty bitty step.\n\nMy mother sometimes switches in midsentence, when talking to me, from English to Yiddish. If meaning can leave English and reappear in Yiddish, does it have an absolutely necessary relation to either language? Linguists say, \"No. Anything you can say in German you can say in Swahili which is increasingly Arabic.\" But no poet could accept the idea of linguistic equivalence, and a religious fanatic might want to kill you for proposing it. Ultimately, I believe, meaning has less to do with language than with music, a sensuous flow that becomes language only by default, so to speak, and by degrees. In great fiction and poetry, meaning is obviously close to music. Writing about a story by Gogol, Nabokov says it goes la, la, do, la la la etc. The story's meaning is radically musical. I've often had to rewrite a paragraph because the sound was wrong. When at last it seemed right, I discovered\u2014incredibly\u2014the sense was right. Sense follows sound. Otherwise we couldn't speak so easily or quickly. If someone speaks slowly and sense unnaturally precedes sound, the person can seem too deliberative; emotionally false, boring. I can tell stories all day, but to write one that sounds right entails labors of indefinable innerness until I hear the thing I must hear before it is heard by anyone else. A standard of rightness probably exists for me in my residual subliminal Yiddish. Its effect is to inhibit as well as to liberate. An expression popular not long ago, \"I hear you,\" was intended to assure you of being understood personally, as if there were a difference in comprehension between hearing and really hearing. In regard to being _really_ heard, there are things in Yiddish that can't be heard in English. _Hazar fisl kosher._ \"A pig has clean feet.\" It is an expression of contempt for hypocrisy. The force is in Yiddish concision. A pig is not clean. With clean feet it is even less clean. Another example: I was talking to a friend about a famous, recently deceased writer. The friend said, \"He's _ausgespielt.\"_ Beyond dead. He's played out. So forget it. Too much has been said about him.\n\nCultural intuitions, or forms or qualities of meaning, dancing about in language, derive from the unique historical experience of peoples. The intuitions are not in dictionaries but carried by tones, gestures, nuances effected by word order, etc. When I understood the old men in Paris I didn't do or intend anything. It wasn't a moment of romantic introspection. I didn't know what language I heard. I didn't understand that I understood. What comes to mind is the assertion that begins the Book of John: \"In the beginning was the word.\" A sound, a physical thing, the word is also mental. So this monism can be understood as the nature of everything. Like music that is the meaning of stories, physical and mental are aspects of each other. Yiddish, with its elements of German, Hebrew, Aramaic, Latin, Spanish, Polish, Russian, Rumanian, is metaphorically everything. A people driven hither and yon, and obliged to assimilate so much, returned immensely more to the world. How they can become necessary to murder is the hideous paradox of evil.\n\nWhen I was five years old, I started school in a huge gloomy Victorian building where nobody spoke Yiddish. It was across the street from Knickerbocker Village, the project in which I lived. To cross that street meant going from love to hell. I said nothing in the classroom and sat apart and alone, and tried to avoid the teachers evil eye. Eventually, she decided that I was a moron, and wrote a letter to my parents saying I would be transferred to the \"ungraded class\" where I would be happier and could play ping-pong all day. My mother couldn't read the letter so she showed it to our neighbor, a woman from Texas named Lynn Nations. A real American, she boasted of Indian blood, though she was blond and had the cheekbones, figure, and fragility of a fashion model. She would ask us to look at the in-sides of her teeth, and see how they were cupped. To Lynn this proved descent from original Americans. She was very fond of me, though we had no conversation, and I spent hours in her apartment looking at her art books and eating forbidden foods. I could speak to her husband, Arthur Kleinman, yet another furrier, and a lefty union activist, who knew Yiddish.\n\nLynn believed I was brighter than a moron and went to the school principal, which my mother would never have dared to do, and demanded an intelligence test for me. Impressed by her Katharine Hepburn looks, the principal arranged for a school psychologist to test me. Afterwards, I was advanced to a grade beyond my age with several other kids, among them a boy named Bonfiglio and a girl named Estervez. I remember their names because we were seated according to our IQ scores. Behind Bonfiglio and Estervez was me, a kid who couldn't even ask permission to go to the bathroom. In the higher grade I had to read and write and speak English. It happened virtually overnight so I must have known more than I knew. When I asked my mother about this she said, \"Sure you knew English. You learned from trucks.\" She meant: while lying in my sickbed I would look out the window at trucks passing in the street; studying the words written on their sides, I taught myself English. Unfortunately, high fevers burned away most of my brain, so I now find it impossible to learn a language from trucks. A child learns any language at incredible speed. Again, in a metaphorical sense, Yiddish is the language of children wandering for a thousand years in a nightmare, assimilating languages to no avail.\n\nI remember the black shining print of my first textbook, and my fearful uncertainty as the meanings came with all their exotic Englishness and devoured what had previously inhered in my Yiddish. Something remained indigestible. What it is can be suggested, in a Yiddish style, by contrast with English. A line from a poem by Wallace Stevens, which I have discussed elsewhere, seems to me quintessentially goyish, or antithetical to Yiddish:\n\n_It is the word_ pejorative _that hurts._\n\nStevens affects detachment from his subject, which is the poet's romantic heart, by playing on a French construction: \"word _pejo-_ _rative,\"_ like _mot juste,_ makes the adjective follow the noun. Detachment is further evidenced in the rhyme of \"word\" and \"hurts.\" The delicate resonance gives the faint touch of hurtful impact without obliging the reader to suffer the experience. The line is ironically detached even from detachment. In Yiddish there is plenty of irony but not so nicely mannered or sensitive to a readers experience of words. Stevens s line would seem too self-regarding; and the luxurious subtlety of his sensibility would seem unintelligible, if not ridiculous. He flaunts sublimities here, but it must be said that elsewhere he is as visceral and concrete as any Yiddish speaker.\n\nI've lost too much of my Yiddish to know exactly how much remains. Something remains. A little of its genius might be at work in my sentences, but this has nothing to do with me personally. The pleasures of complexity and the hilarity of idiocy, as well as an idea of what's good or isn't good, are in Yiddish. If it speaks in my sentences, it isn't I, let alone me, who speaks.\n\nWhen asked what he would have liked to be if he hadn't been born an Englishman, Lord Palmerston said, \"An Englishman.\" The answer reminds me of a joke. A Jew sees himself in a mirror after being draped in a suit by a high-class London tailor. The tailor asks what's wrong. The Jew says, crying, \"Vee lost de empire.\" The joke assimilates the insane fury that influenced the nature of Yiddish and makes it apparent that identity for a Jew is not, as for Palmerston, a witty preference.\n\n# _About the Contributors_\n\nLOUIS BEGLEY lives in New York City. His seventh novel, _Shipwreck,_ was published in September 2003 by Alfred A. Knopf.\n\nJAMES CAMPBELL worked as a printer in Glasgow, before going on to university in Edinburgh. Between 1978 and 1982, he was the editor of the literary quarterly _New Edinburgh Review._ His books include _Invisible Country: A Journey through Scotland_ (1984), _Talking at the Gates: A Life of James Baldwin_ (1991), and _This Is the Beat Generation_ (2001). He lives in London, where he works for the _Times Literary Supplement._\n\nARIEL DORFMAN, the Chilean expatriate writer and human rights activist, holds the Walter Hines Page Chair at Duke University. His books, written both in Spanish and English, have been translated into over thirty languages, his plays staged in more than one hundred countries. He has received numerous international awards, including the Danish ALOA prize for best foreign book _(Heading South, Looking North)_ and the Laurence Olivier Award (for the play _Death and the Maiden)._ His novels include _Widows, Konfidenz, The Nanny and the Iceberg,_ and _Blake's Therapy._ His latest works include poetry _(In Case of Fire in a Foreign Land),_ a new play _(Purgatory),_ and a travel book _(Desert Memories)._\n\nM. J. FITZGERALD is Director of the Creative Writing Program at the University of Minnesota and lives in Minneapolis with her partner, Brian, and son, Robert. Her most recent story, \"The Invention of Greek Statues,\" was published in _Literary Imagination,_ the review of the Association of Literary Scholars and Critics.\n\nHA-YUN JUNG'S writing has appeared in various publications including _Prairie Schooner, The Threepenny Review,_ the _New York Times,_ and _Best New American Voices 2001._ She has received fiction fellowships from the University of Wisconsin-Madison and the Radcliffe Institute for Advanced Study and also a translation grant from the Korean Literature Translation Institute. After living in the United States for seven years, she recently returned to Korea, where she is at work on a novel.\n\nBERT KEIZER works as a geriatrician and writer in Amsterdam. In 1994 he published his first book, _Het Refrein is Hein,_ which he translated, or rather rewrote in English, under the title _Dancing with Mister D._ He has also written a novel about his work as a doctor in Africa and an appraisal of Wittgenstein. In the Netherlands he writes a weekly column in a national daily; in the United States, he appears regularly in _The Threepenny Review._ He is presently engaged in translating a selection of Emily Dickinsons letters into Dutch, an effort which by its very nature will go entirely unnoticed in the English-speaking world.\n\nTHOMAS LAQUEUR was born in Istanbul, Turkey, on September 6, 1945, and grew up in Beckley, West Virginia. He graduated from Swarthmore College in 1967 and received a Ph.D. in history from Princeton in 1973. Since then he has taught at Berkeley. He writes regularly on history and culture for the _London Review of Books,_ the _Times Literary Supplement,_ and other publications. His books include _Making Sex: Body from the Greeks to Freud_ (which has been translated into over a dozen languages) and _Solitary Sex: A Cultural History of Masturbation._ He is now at work on a new project called _The Dead Among the Living._\n\nLEONARD MICHAELS was born in New York in 1933. He taught for many years in the English Department at the University of California, Berkeley and after his retirement he divided his time between California and Italy. His books of fiction and nonfiction include _Going Places, I Would Have Saved Them If I Could, The Men's Club, Shuffle, Sylvia, To Feel These Things,_ and _A Girl with a Monkey._ A final collection of stories about the character Nachman is to be published by Penguin Putnam. He died on May 10, 2003.\n\nBHARATI MUKHERJEE is the author of six novels _(Desirable Daughters, Leave It to Me, The Holder of the World, Jasmine, Wife, The Tiger's Daughter)_ and two collections of short stories _(The Middleman and Other Stories, Darkness);_ she is also the co-author, with Clark Blaise, of two nonfiction books _(The Sorrow and the Terror, Days and Nights in Calcutta)._ A member of the American Academy of Arts and Sciences, she has received a Guggenheim Fellowship, a Senior Canada Council Fellowship, and an NEA grant. She teaches at the University of California, Berkeley.\n\nNICHOLAS PAPANDREOU'S most recent novel, _Kleptomnemon,_ imagines a world where people s memories are stolen, then traded on the marketplace. His forthcoming book, _Politics in the First Person,_ analyzes the role of narrative and story-telling in the lives of famous Greek politicians. He works part-time at the Ministry of the Aegean with a team of dreamers who want to save the islands from the wrath of over-construction. Most of his published short stories, essays, magazine articles and book excerpts are available on his website: www.nikos-papandreou.gr.\n\nLUC SANTE was born in Verviers, Belgium. He is the author of _Low Life, Evidence,_ and _The Factory of Facts_ and co-editor, with Melissa Holbrook Pierson, of _O.K. You Mugs: Writers on Movie Actors._ He is the recipient of a Whiting Writer's Award, a Guggenheim Fellowship, an Award in Literature from the American Academy of Arts and Letters, and a Grammy, for album notes, and is a Fellow of the American Academy of Arts and Sciences. He is Visiting Professor of Writing and the History of Photography at Bard College, and he lives with his wife and son in Ulster County, New York.\n\nGARY SHTEYNGART was born in Leningrad, USSR, in 1972, and came to the United States seven years later. His novel, _The Russian Debutante's Handbook,_ won the Stephen Crane Award for First Fiction, was named a New York Times Notable Book, and was chosen as a best book of the year by the _Washington Post Book World_ and _Entertainment Weekly._ His work has appeared in _The New Yorker, Granta, GQ,_ the _New York Times,_ and many other publications. He lives in New York City.\n\nJOSEF SKVORECKY, who was born in 1924, is the author most recently of _When Eve Was Naked._ Among his earlier books are _The Cowards, The Bass Saxophone, The Engineer of Human Souls, The Bride of Texas,_ and many others. He lives with his wife, Zdena, in Toronto, Ontario.\n\nAMY TAN is the author of _The Joy Luck Club,_ a beloved and internationally best-selling novel that explores the relationships of Chinese women and their Chinese-American daughters. She is also author of _The Kitchen God's Wife, The Hundred Secret Senses,_ and two children's books. Her latest novel, _The Bone-setter's Daughter,_ was published in 2001, and in the fall of 2003 she published a collection of nonfiction work entitled _The Opposite of Fate._\n\nNGUGI WA THIONG'O was born in Limuru, Kenya, in 1938, and was educated at Makerere University College in Kampala, Uganda. With eight novels, a book of short stories, a memoir, and five plays to his credit, Ngugi is one of the leading African writers and scholars at work today. His novels have been translated into more than thirty languages and have earned him a number of prizes, including the East African Novel Prize, the Paul Robeson Award for Artistic Excellence, Political Conscience and Integrity, and the Zora Neale Hurston-Paul Robeson Award for artistic and scholarly achievement. He was recently named University of California Irvine's Distinguished Professor in the School of Humanities and director of the International Center for Writing and Translation, and in 2003 he was elected an honorary member in the American Academy of Arts and Letters.\nCopyright \u00a9 2004 by Wendy Lesser  \n\"The Way Back\" copyright \u00a9 2004 by Bharati Mukherjee  \n\"Yes and No\" copyright \u00a9 2004 by Amy Tan  \n\"Trouble with Language\" copyright \u00a9 2004 by Josef Skvoreck y  \n\"Circus Biped\" copyright \u00a9 2004 by Bert Keizer  \n\"French Without Tears\" copyright \u00a9 2004 by Luc Sante  \n\"Prelude\" copyright \u00a9 2004 by Thomas Laqueur  \n\"Recovering the Original\" copyright \u00a9 2004 by Ngugi wa Thiong'o  \n\"Split Self\" copyright \u00a9 2004 by Nicholas Papandreou  \n\"Limpid, Blue, Poppy\" copyright \u00a9 2004 by M. J. Fitzgerald  \n\"Personal and Singular\" copyright \u00a9 2004 by Ha-yun Jung  \n\"On Being an Orphaned Writer\" copyright \u00a9 2004 by Louis Begley  \n\"The Mother Tongue Between Two Slices of Rye\" copyright \u00a9 2004 by Gary Shteyngart  \n\"Boswell and Mrs. Miller\" copyright \u00a9 2004 by James Campbell  \n\"Footnotes to a Double Life\" copyright \u00a9 2004 by Ariel Dorfman  \n\"My Yiddish\" copyright \u00a9 2004 by The Estate of Leonard Michaels\n\nAnchor Books and colophon are registered trademarks of Random House, Inc.\n\np. cm.  \n1. English language\u2014Style. 2. English language\u2014Study and teaching\u2014Foreign  \nspeakers. 3. Authors, American\u201420th century\u2014Biography. 4. Immigrants\u2014United  \nStates\u2014Language. 5. Bilingualism\u2014United States. 6. Second language acquisition.  \n7. Authors, Exiled\u2014Language. 8. Language and languages. 9. Authorship.  \nI. Lesser, Wendy.  \nPE1421.G456 2004 820\u2014dc22 2003060894\n\neISBN: 978-0-307-48539-7\n\n_Author photograph \u00a9 Mike Minehan_\n\nwww.anchorbooks.com\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## Emeril 20\u201340\u201360\n\nFresh Food Fast\n\n## Emeril Lagasse\n\nWith Photography by Steven Freeman\n\n## This book is for all my wonderful fans,\n\nyou terrific home cooks out there who keep me inspired and keep me going. Thanks for sticking by me. I hope the recipes within these pages help make your lives a little less hectic and a whole lot tastier. (Remember, it's all about food of love.)\n\n## Contents\n\nAcknowledgments\n\nIntroduction: Fresh Food Fast: It's As Easy As 1\u20132\u20133\n\n20 Minutes or Less\n\nSoups\n\nStarters\n\nSalads and Dressings\n\nSandwiches\n\nPasta\n\nRice and Beans\n\nVegetables\n\nSeafood\n\nPoultry\n\nMeat\n\nDesserts\n\n40 Minutes or Less\n\nSoups\n\nStarters\n\nSalads\n\nSandwiches\n\nPasta\n\nRice and Beans\n\nVegetables\n\nSeafood\n\nPoultry\n\nMeat\n\nDesserts\n\n60 Minutes or Less\n\nSoups\n\nStarters\n\nPasta\n\nRice and Beans\n\nSeafood\n\nPoultry\n\nMeat\n\nDesserts\n\n60 Minutes +\n\nSearchable Terms\n\nAbout the Author\n\nOther Books by Emeril Lagasse\n\nCredits\n\nCopyright\n\nAbout the Publisher\n\n## Acknowledgments\n\nEmeril 20\u201340\u201360 would never have made it to the table on time without the help of everyone below\u2014\n\nMy amazing family, who is always, always there for me\u2014Alden, EJ, Meril, Jessie, Jillian, Mom, Dad, Mark, Wendi, Katti Lynn, Dolores, Jason, and baby Jude. I love you all.\n\nMy incredible Culinary team who always make it happen against all odds\u2014Charlotte Martory, Alain Joseph, Stacey Meyer, Angela Sagabaen, and Kamili Hemp-hill. Two down, eight to go!\n\nMy supportive Homebase team\u2014Eric Linquest, Tony Cruz, Dave McCelvey, Marti Dalton, Chef Chris Wilson, Chef Bernard Carmouche, Chef Dana D'Anzi, Tony Lott, Scott Farber, and George Ditta.\n\nPhotographer Steven Freeman and his on-the-ready photography associates, Kevin Guiler and Josh Maready.\n\nMy associates at Martha Stewart Living Omnimedia\u2014Martha, Charles, Robin, Lucinda, and the hospitable test kitchen staff.\n\nMy super M's\u2014Mara Warner Jones, Michelle Terrebonne, and Maggie McCabe. Mimi Rice Henken and TJ Pitre, for their assistance with the photo shoot and photo editing.\n\nThe very talented design team who made the photos come alive\u2014Jed and Elias Holtz and Charissa Melnik.\n\nMy pal, Sal Passalacqua. Shelley Van Gage, for helping me look my best.\n\nOur partners at HarperStudio, for their vision\u2014Bob Miller, Debbie Stier, Sarah Burningham, Julia Cheiffetz, Katie Salisbury, Sally McCartin, Jacqui Daniels, Mary Schuck, Leah Carlson-Stanisic, Kim Lewis, Lorie Young, Nikki Cutler, Doug Jones, Kathie Ness, and Ann Cahn.\n\nOur partners at All-Clad, T-Fal, and Wusthof.\n\nMy friends at Leonard Simchick Prime Meats and Fresh Poultry and at Pisacane Fish Market\u2014you guys are the best.\n\nAll the terrific employees at my restaurants and Homebase who make it happen every day.\n\nMy dear friends Frank and Richard Santorsola.\n\nSherif, for getting me where I need to go each and every day.\n\nJim Griffin, my terrific agent and friend.\n\nMy friend and trusted counsel, attorney Mark Stein.\n\nThank you all from the bottom of my heart.\n\n## Introduction\n\nFRESH FOOD FAST: IT'S AS EASY AS 1\u20132\u20133\n\nIs this world getting faster? More and more folks are asking me the same question these days: \"Emeril, how can I find the time to prepare exciting meals at home without sacrificing quality? How do you do it?\" They tell me that with the passing of each year, there's less time available. Even with all the new technology that's supposed to simplify our lives, we're pulled in a thousand directions. And when children are involved in the equation\u2014forget it! There's always a soccer game to be played, homework to be done, questions that need to be answered, and visits to the gym (we have to keep our \"machines\" running smoothly!). I'm sure you get the picture\u2014because, trust me, these folks aren't telling me anything that I don't live myself 365 days a year. But I honestly do believe that with a few basic principles under the belt and a fresh mind-set, anyone is capable of putting delicious, well-balanced meals on the table in the time they have available. And that, my friend, is the philosophy behind this book:\n\nMake the meals you want in the time you have.\n\nTo help you kick things up and simplify your life all at the same time, I've collected close to 160 recipes here, from startlingly simple ones that can be ready in next to no time, to some more lavish recipes that might take a bit of time to simmer but that are still a cinch to put together. The recipes are divided into three basic categories: The 20-minute chapter is comprised of dishes that can be on the table in 20 minutes or less; the 40-minute chapter is for dishes that fall between 20 and 40 minutes (with many of them hovering around the 30-minute mark); and the 60-minute chapter is peppered with a few special recipes that can take anywhere from 40+ to 60+ minutes to get to the table. While these longer \"splurge\" recipes may seem \"time-luxurious,\" they can still be perfect for a not-so-hectic weekday, a lazy Sunday afternoon, or an end-of-the-week meal. You know what time you have. Be honest with yourself and you're on your way to making this collection of recipes work for you.\n\nIn cooking quick, fresh meals at home, there are many steps we can take to make the process easier than we had imagined:\n\nRead the recipe(s) from beginning to end once you've decided what you want to cook. This can be done anytime\u2014on a lunch break during the day or even during your daily commute (riding, not driving!). The point is that at some time before you actually start cooking, you need to have an idea of what you'll be doing. Take a few minutes and run things through in your mind, visualizing the ingredients called for and the steps you will take to prepare each dish. This way, when you actually begin to cook, there are no surprises and you're ready to go. Practicing this kind of methodical thinking will become second nature and will transform you into a speed demon in the kitchen!\n\nReady to begin? Read through the recipe (again) and gather all of your ingredients and equipment. You'll spare yourself from going back and forth, making prep time a breeze. Also, make a note to yourself if you notice a task in the recipe that can be tackled while doing something else. For example, zero in on inactive time. Inactive time in the kitchen basically refers to time that does not require your full attention, such as waiting for something to marinate, bringing a liquid to a boil, or finishing something in the oven. Be smart and use that time wisely to do other things, such as finishing prep (like chopping herbs or grating cheese for a garnish), beginning another recipe, or catching up on tasks around the house (what's that, dirty dishes in the sink?). This way once dinner is on the table, you're free to sit and enjoy with friends and family.\n\nKeep a well-stocked pantry at all times, so you're ready when the hunger hits. While fresh is my preference when available and when it's in season, there are certain packaged items that I always have on hand, such as pasta, rice, canned beans, and tomatoes, just to name a few. You know what you like. I also think of my freezer as a friend in creating fresh food fast, since many things do just fine in the freezer for short periods of time. These things come in handy when going to the store isn't an option. Frozen vegetables and fruits are often examples of produce that were picked and quickly processed at the peak of ripeness, making these a better option than off-season, pricey supermarket finds that traveled way too far. A small package of nuts kept in the freezer, defrosted and quickly toasted, can be just the added note a salad needs. A well-wrapped loaf of day-old French bread freezes well, too, and can quickly be transformed into tasty croutons to really make that soup or salad sing. And hey, don't forget about your biggest ally\u2014your fridge. Keep yours well stocked with prewashed greens and other produce that will help get you through your week, along with the dairy basics and your personal favorite condiments. Hey, what are those door shelves for, anyway? With a little prior proper planning (the three Ps, as I like to call it), you're on your way to an impromptu meal in no time.\n\nFresh herbs rule! Though we use many dried herbs and seasonings in the recipes in this book, since that is what many cooks typically have on hand, if you have a little green space out back or a roomy kitchen window, I would suggest keeping a few pots of herbs going year-round. Fresh herbs are quick to snip when needed and add an unmistakable touch to any dish. My kids love watering the herbs at the end of the day\u2014it's one of the fun things that we do together, making them feel more connected to the cooking process, too.\n\nMultitasking is important in the home kitchen, too, just like we cooks do in the restaurant kitchens. For example, if you need to mince some garlic for tonight's dinner, make a little extra (or lagniappe, as we say here in New Orleans) for tomorrow's feast. You can also do the same for chopping onions and scallions. Just reserve them in airtight containers and store them in the refrigerator for up to 4 days. You can apply the same idea when cooking pasta or rice: make a little extra, save yourself some time, and have it ready for another night's dinner. You might decide to turn these items into a delicious salad or add them to a bro-thy soup. It's just good kitchen economics. I also make stocks in big batches when I have the time, often utilizing the carcasses of roasted poultry from previous meals, and then freeze the stock in small portions for later. And of course, with a little creative thinking, leftover roast poultry, meats, and roasted or grilled veggies can come together to create fabulous sandwiches, salads, pastas, and risottos.\n\nBut, you know, whether speaking of ingredients available or of time on hand, in the end, it's all about doing the best with what you have and enjoying yourself along the way. Everyone's lives are different, so I've given you many options here. Whether you have 20 minutes, 40 minutes, or 60 minutes to put a meal together, and whether you're prepared because you've stocked your pantry well, kept some tasty leftovers to use in a salad, or shopped efficiently and only have to walk to your fridge, remember that the most important things about cooking at home are as follows: have fun, cook well, and eat great food!\n\n## 20 Minutes OR LESS\n\nSOUPS\n\nSimple Italian Wedding Soup\n\nSweet Pea Soup\n\nSTARTERS\n\nShrimp and Chorizo Tapas\n\nHerbed Olives\n\nRoasted Red Pepper Hummus\n\nBalsamic-Marinated Cremini Mushrooms\n\nBruschetta\n\nMozzarella and Tomato Bites with Kalamata Olive Drizzle\n\nSALADS AND DRESSINGS\n\nEmeril's Salad\n\nReal Caesar Salad\n\nSpinach Salad with Bacon and Fried Eggs\n\nOrange, Walnut, and Goat Cheese Salad\n\nCucumber Ribbon Salad\n\nAntipasto Pasta Salad\n\nCantaloupe, Prosciutto, and Arugula Salad\n\nOrange, Fennel, and Black Olive Salad\n\nIceberg Wedges with Cherry Tomato Vinaigrette\n\nSalad Tropicale\n\nButtermilk Dressing\n\nBalsamic Vinaigrette\n\nRed Wine Vinaigrette\n\nHerb Vinaigrette\n\nSANDWICHES\n\nSausage and Pepper Po-Boy\n\nBacon, Lettuce, Avocado, and Tomato Sandwich with Basil Mayo\n\nSteak and Cheese Sandwiches\n\nPressed Roast Turkey, Pesto, and Provolone Sandwiches\n\nProsciutto and Mozzarella Panini\n\nOpen-Face Turkey and Cheese Sandwich\n\nKicked-Up Tuna Melts\n\nFish Tacos with Black Bean Salsa\n\nPASTA\n\nFettuccine with Peas and Ham\n\nOrange, Currant, and Pine Nut Couscous\n\nLinguine alla Carbonara\n\nEmeril's Shrimp and Pasta with Garlic, Lemon, Crushed Red Pepper, and Green Onions\n\nRICE AND BEANS\n\nKicked-Up Shrimp Fried Rice\n\nTurkey and Wild Rice Salad\n\nAromatic Jasmine Rice\n\nSpicy Sausage, Bean, and Cheese Nachos\n\nVEGETABLES\n\nPan-Roasted Asparagus with Shiitake Mushrooms and Cherry Tomatoes\n\nGarlicky Bok Choy\n\nBroiled Zucchini\n\nSaut\u00e9ed Yellow Squash with Carrots and Tarragon\n\nRoasted Carrots with Fresh Thyme\n\nEmeril's Saut\u00e9ed Cucumber with Basil and Mint\n\nGlazed Radishes\n\nSEAFOOD\n\nBroiled Catfish with Fresh Thyme, Garlic, and Lemon\n\nBroiled Salmon with a Warm Tomato-Lemon Vinaigrette\n\nGaaahlicky Sizzling Shrimp\n\nSouthern-Style Pan-Fried Catfish\n\nBlue Corn\u2013Crusted Rainbow Trout with Cilantro-Lime Sour Cream\n\nTrout \u00e0 la Meuni\u00e8re\n\nClassic Moules Marini\u00e8re\n\nPOULTRY\n\nStir-Fried Chicken with Cashews\n\nSaut\u00e9ed Chicken Breasts with Dijon Herb Sauce\n\nChicken Salad with Fresh Herbs and Celery\n\nMEAT\n\nLamb T-Bones with Rosemary-Balsamic Butter Sauce\n\nSteak au Poivre\n\nNew York Strip with Beurre Ma\u00eetre d'H\u00f4tel\n\nLamb Chops with Mustard Herb Crust\n\nMinute Steaks Teriyaki-Style\n\nBoneless Pork Chops Parmigiana\n\nSpicy Pork Stir-Fry with Green Beans\n\nMushroom-Smothered Steaks\n\nDESSERTS\n\nEmeril's Late-Night Parfaits\n\nBrown Sugar\u2013Baked Bananas\n\nCandied Hot Fudge Sundaes\n\nPeanut Butter\u2013Chocolate Chip Cookies\n\nMelon with Amaretti Cookie Crumbles\n\nFresh Berries with Balsamic Drizzle and Almond Cream\n\nFlamb\u00e9ed Strawberry Sauce for Angel Food Cake or Ice Cream\n\nSoups\n\nSIMPLE ITALIAN WEDDING SOUP\n\nPrep time: 10 minutes Cook time: 1 minute Total: 11 minutes\n\nThis simple classic soup is a no-brainer for this quick and easy cookbook. Feel free to use either chicken or beef stock or a combination of the two\u2014whatever you prefer will work here. Kind of an Italian version of egg drop soup, if you will.\n\n6 cups chicken stock, or canned, low-sodium chicken broth\n\n2 cups beef stock or canned, low-sodium beef broth\n\n4 large eggs, beaten\n\n\u00bc cup plus 2 tablespoons finely grated Parmigiano-Reggiano cheese\n\n\u00bc cup plus 2 tablespoons finely chopped mixed fresh herbs (such as parsley, marjoram, and basil)\n\nPinch of ground nutmeg\n\nSalt and freshly ground black pepper\n\n1. Pour the stock into a 3-quart saucepan and bring to a simmer.\n\n2. While you are waiting for the stock to simmer, prep the rest of the ingredients. Combine the eggs, Parmesan, herbs, and nutmeg in a medium mixing bowl and whisk to combine.\n\n3. When the stock is simmering, use a large fork to drizzle in the egg mixture while continuously stirring the stock. Continue to simmer until the egg is just set, about 45 seconds. Remove from the heat and season with salt and pepper to taste. Serve immediately.\n\n4 to 6 servings\n\nSWEET PEA SOUP\n\nPrep time: 7 minutes Cook time: 9 minutes Total: 16 minutes\n\nThis soup is sublime\u2014in method, in timing, in appearance, and in taste. Sweet peas are simmered in a flavorful broth and pureed with fresh spinach. Dollop with the lemony sour cream. Enjoy sir, enjoy madam.\n\n\u00bd cup sour cream\n\n1 teaspoon freshly squeezed lemon juice\n\n\u00bd teaspoon lemon zest\n\n2 tablespoons butter\n\n1\u00bc cups thinly sliced onion (about 1 medium onion)\n\n1 medium clove garlic, thinly sliced\n\n3 sprigs fresh mint\n\n3 sprigs fresh parsley\n\n4 cups vegetable stock or canned, low-sodium vegetable broth\n\n1 pound frozen green peas\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground white pepper\n\n1\u00bd cups packed prewashed spinach (about 2 ounces)\n\n1. Combine the sour cream, lemon juice, and lemon zest. Cover and refrigerate until ready to use.\n\n2. Melt the butter in a 4-quart or larger pot over medium-high heat. Add the onion and garlic and cook until soft and translucent, 4 minutes.\n\n3. Gather the herbs into a bunch and secure with kitchen twine. Add the vegetable stock and the herb bundle to the onion, cover, and bring to a boil. Uncover, reduce the heat to a simmer, and cook for 3 minutes. Increase the heat to high, return the soup to a boil, and add the peas. Bring the soup back to a boil, reduce the heat, and simmer for 2 minutes longer. Remove the soup from the heat. Remove the herb bundle. Stir in the salt, white pepper, and spinach.\n\n4. Puree the soup using an immersion blender or in two batches using a blender (see Note). Serve immediately with a dollop of the lemony sour cream.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.\n\nAbout 1\u00bd quarts, 4 to 6 servings\n\nStarters\n\nSHRIMP AND CHORIZO TAPAS\n\nPrep time: 7 minutes Cook time: 13 minutes Total: 20 minutes\n\nImpress your guests with this unique combination of spicy chorizo and sweet shrimp...so simple and tasty! The trick is to use the best chorizo sausage you can find, the freshest shrimp available, and a good-quality olive oil.\n\n1 tablespoon plus \u00bc cup Spanish olive oil\n\n1 pound firm (smoked) chorizo, cut on the diagonal into \u00bd-inch-thick slices\n\n1\u00bd cups thinly sliced onions\n\n1 tablespoon minced garlic\n\n\u00bd cup dry (fino) sherry\n\n1\u00bd pounds medium shrimp, peeled and deveined\n\n1 tablespoon Spanish paprika\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n3 tablespoons freshly squeezed lemon juice\n\n2 tablespoons minced fresh parsley Crusty bread, for serving\n\n1. Place a large skillet over medium-high heat, and add the 1 tablespoon olive oil. When it is hot, add the sliced chorizo and saut\u00e9, turning as necessary, until it begins to brown around the edges on both sides, 4 to 6 minutes. Add the onions and cook, stirring occasionally, until caramelized around the edges and softened, 4 to 6 minutes. Add the garlic and cook, stirring, for 30 seconds. Add \u00bc cup of the sherry and cook for 1 minute.\n\n2. Add the shrimp, paprika, salt, and pepper and cook, stirring occasionally, until the shrimp are pink and just cooked through, about 4 minutes. Add the remaining \u00bc cup sherry, the remaining \u00bc cup olive oil, and the lemon juice and parsley; stir to combine. When heated through, remove from the heat.\n\n3. Serve immediately on small plates, with any accumulated cooking juices spooned over the top. Pass the bread at the table.\n\n6 to 8 servings\n\nHERBED OLIVES\n\nPrep time: 8 minutes Total: 8 minutes\n\nIf you buy olives that are already marinated, make them special by adding the fennel, thyme, rosemary, and orange peel. Or marinate plain olives as outlined below. A perfect starter or snack any time of the day!\n\n1 pound assorted olives, such as\n\nKalamata, Cerignola, Greek, Gaeta, and\/or Ni\u00e7oise, drained\n\n1\u00bd cups olive oil\n\n1 onion, julienned (see Note)\n\n4 cloves garlic, thinly sliced\n\n4 bay leaves, torn into pieces\n\n1 teaspoon fennel seeds\n\n2 sprigs fresh thyme\n\n1 sprig fresh rosemary\n\n3-inch-long strip of orange peel, julienned\n\n12 coriander seeds\n\n\u00bc teaspoon crushed red pepper, or to taste\n\n1. Combine all the ingredients in a large bowl, and stir well. Serve immediately, or marinate overnight before serving.\n\n2. Transfer any leftovers to glass jars with lids, and refrigerate for up to 2 weeks. (Allow to return to room temperature before serving.)\n\nNotes: Don't discard the remaining olive oil after the olives are gone\u2014it is delicious served as a dipping oil for crusty French bread or drizzled over pasta or grilled vegetables.\n\n\"Julienned\" means cut into thin strips about the size of matchsticks.\n\nAbout 1 quart\n\nROASTED RED PEPPER HUMMUS\n\nPrep time: 18 minutes Total: 18 minutes\n\nI absolutely love hummus, and what a spin on the classic\u2014adding roasted red peppers, crushed red pepper, and cumin. We suggest serving this with bread or chips, but it also works well as a dip with crudit\u00e9s or even as a spread for sandwiches!\n\nTwo 15-ounce cans chickpeas, drained\n\n\u00be cup (about 6 ounces) jarred roasted red peppers, drained and coarsely chopped\n\n3 large cloves garlic\n\n1\u00bd teaspoons salt, plus more to taste\n\n\u00bd teaspoon crushed red pepper\n\n\u00bd cup tahini\n\n\u00bd cup water\n\n4 to 6 tablespoons freshly squeezed lemon juice, plus more for drizzling\n\n4 to 6 tablespoons olive oil, plus more for drizzling\n\n1\/8 teaspoon cayenne pepper\n\n1\/8 teaspoon ground cumin\n\n1 teaspoon chopped fresh parsley\n\n\u00bc teaspoon sweet paprika\n\nPita bread or pita chips, for serving\n\n1. Place the chickpeas and chopped roasted red pepper in the bowl of a food processor.\n\n2. On a cutting board, use the side of a knife to mash the garlic cloves with \u00bc teaspoon of the salt, forming a smooth paste. Add this to the processor along with the crushed red pepper, tahini, water, 4 tablespoons of the lemon juice, 4 tablespoons of the olive oil, the remaining 1\u00bc teaspoons salt, cayenne, and cumin. Process until smooth, stopping to scrape down the sides of the bowl as needed. Taste, and adjust the seasoning by adding more salt, lemon juice, and \/ or olive oil as needed.\n\n3. Transfer the hummus to a wide, shallow bowl for serving. Drizzle with lemon juice and olive oil, and sprinkle with the parsley and paprika. Serve with pita bread wedges or chips.\n\nNote: If you prefer a traditional plain hummus, simply omit the roasted red peppers and decrease the garlic to 2 cloves.\n\n3 cups\n\nBALSAMIC-MARINATED CREMINI MUSHROOMS\n\nPrep time: 18 minutes Total: 18 minutes\n\nDid you know that cremini mushrooms are the little baby siblings of the portobello mushroom? The addition of Italian herbs and grated pecorino makes this really yummy.\n\n\u00bc cup balsamic vinegar\n\n2 teaspoons minced garlic\n\n1\u00bd teaspoons dried Italian herbs, crushed with your fingers\n\n1 teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\n\u00be cup extra-virgin olive oil\n\n2 pounds cremini mushrooms, wiped clean, stemmed, and quartered\n\n\u00bc cup finely grated Pecorino Romano cheese\n\n1. In a small bowl, whisk together the balsamic vinegar, garlic, Italian herbs, salt, and pepper. Gradually add the olive oil, whisking constantly.\n\n2. In a large bowl, combine the mushrooms and marinade, and toss to coat evenly. Cover and refrigerate until ready to serve, up to 24 hours.\n\n3. Return to room temperature before serving, and garnish with the cheese. Serve with toothpicks or on small plates with forks.\n\nAbout 1 quart\n\nBRUSCHETTA\n\nPrep time: 12 minutes Cook time: 4 minutes Total: 16 minutes\n\nA simple Italian delight: slices of crusty bread topped with diced fresh tomatoes that have been tossed with olive oil, herbs, garlic, and salt. Wait until the tomatoes are in their peak season for the best flavor. Try this as a starter to any quick meal. Be sure to use a timer when toasting the bread slices\u2014these toast really fast!\n\n1 pound medium-ripe tomatoes, cored and cut into \u00bd-inch dice\n\n\u00bd cup thinly sliced fresh basil\n\n\u00be teaspoon fine sea salt\n\n\u00bd teaspoon freshly ground black pepper\n\n6 tablespoons extra-virgin olive oil\n\n1 teaspoon minced garlic\n\nThirty \u00bd-inch-thick slices Italian or French bread (as crusty as you can find)\n\n1. Position a rack as close as possible to the broiler element and preheat the broiler.\n\n2. In a small serving bowl, combine the tomatoes, basil, sea salt, pepper, and 3 tablespoons of the olive oil. Set aside.\n\n3. In another small bowl, combine the garlic and the remaining 3 tablespoons olive oil.\n\n4. Lay the slices of bread on a baking sheet. Broil on both sides until golden and crispy, about 1\u00bd minutes per side. Remove from the oven and brush the tops of the slices with the garlic oil. Return the baking sheet to the broiler for 30 seconds. Transfer the toast slices to a large platter and serve immediately, with the tomato mixture alongside for guests to spoon over the toasts.\n\n6 to 8 servings\n\nMOZZARELLA AND TOMATO BITES WITH KALAMATA OLIVE DRIZZLE\n\nPrep time: 20 minutes Total: 20 minutes\n\nThis dish makes a gorgeous presentation: the contrasting colors of the bright red cherry tomatoes, green basil, and white mozzarella skewered together will make your guests feel that you put in a lot of effort to impress them (but it's pretty easy to put together, and I won't tell if you don't). Drizzle the olive mixture on top and, oh, baby, your guests will be screaming for the next course!\n\n20 to 25 small grape or cherry tomatoes, halved\n\n40 to 50 small fresh basil leaves\n\n8 ounces ciliegine (see Note), drained and patted dry, halved\n\n\u00bc cup chopped pitted Kalamata olives\n\n\u00bd cup extra-virgin olive oil\n\n\u00bc teaspoon minced garlic\n\n\u00bc teaspoon crushed red pepper\n\n2 tablespoons minced fresh oregano or marjoram\n\n\u00bc teaspoon salt\n\nKosher salt, for garnish\n\n1. Using small decorative cocktail skewers or bamboo toothpicks, skewer 1 tomato half, 1 basil leaf, and 1 ciliegine half onto each skewer, kebab-fashion, with the basil leaf sandwiched between the mozzarella and tomato halves and the cut sides of the tomato and mozzarella facing each other. Repeat until you have used all of the ciliegine halves. You should have about 42 filled skewers. Arrange on a small serving platter and set aside.\n\n2. In the bowl of a blender or food processor, combine the olives, olive oil, garlic, crushed red pepper, oregano, and salt, and process until smooth. Drizzle the olive mixture over the tomato-mozzarella skewers. Sprinkle the entire platter with kosher salt, and serve immediately.\n\nNote: Ciliegine are cherry-size balls of fresh mozzarella, available at upscale Italian markets and some gourmet grocery stores. If these are unavailable in your area, simply substitute 1-inch cubes of fresh mozzarella.\n\nAbout 42 hors d'oeuvres\n\nSalads and Dressings\n\nEMERIL'S SALAD\n\nPrep time: 8 minutes Total: 8 minutes\n\nThis is a simplified version of the signature salad served at Emeril's Restaurant in New Orleans. It's been on the menu since day one. Try it and you'll see why folks keep asking for it after all these years.\n\n8 cups (about 4 ounces) loosely packed assorted baby salad greens\n\n\u00bd cup (about 3 ounces) grated pepper Jack cheese\n\n\u00bc teaspoon chopped fresh rosemary\n\n\u00bc teaspoon chopped fresh thyme\n\n\u00bc cup (about 2 ounces) whole oil-packed sundried tomatoes, drained and julienned 2 tablespoons balsamic vinegar\n\n2 tablespoons extra-virgin olive oil\n\nSalt and freshly ground black pepper\n\nSimple Croutons (40 Minutes or Less) or store-bought croutons, for serving (optional)\n\n4 tablespoons alfalfa sprouts or broccoli sprouts\n\n1. To assemble the salad, place the greens, cheese, and chopped herbs in a large mixing bowl. Add the julienned sundried tomatoes. Drizzle the balsamic vinegar and extra-virgin olive oil over the salad, and season with salt and pepper to taste. Toss to coat well.\n\n2. Divide the salad equally among four salad plates. Place 4 or 5 croutons on each plate, and top each salad with 1 tablespoon of the sprouts. Serve immediately.\n\n4 servings\n\nREAL CAESAR SALAD\n\nPrep time: 15 minutes Total: 15 minutes\n\nI bet you never dreamed that this classic salad could be on your table in less than 20 minutes!\n\n1 egg\n\n3 large canned anchovy fillets\n\n2 cloves garlic\n\n2 tablespoons freshly squeezed lemon juice\n\n1\u00bd teaspoons whole-grain mustard\n\n\u00be cup olive oil\n\n\u00bc cup extra-virgin olive oil\n\n\u00be cup (about 2 ounces) grated Parmigiano-Reggiano cheese\n\n1\/8 teaspoon Tabasco sauce\n\n\u00bc teaspoon Worcestershire sauce 1 teaspoon salt\n\n1 teaspoon freshly ground black pepper\n\n3 romaine hearts (one 12-ounce bag), cut into 1-inch pieces, rinsed and spun dry\n\n2 cups Simple Croutons (40 Minutes or Less) or store-bought croutons\n\n1. Place the egg, anchovies, garlic, lemon juice, and mustard in the bowl of a food processor and process until smooth, about 1 minute. While the machine is running, slowly drizzle in the olive oil and the extra-virgin olive oil until completely incorporated, smooth, and thick.\n\n2. Stop processing and add \u00bc cup of the cheese, the Tabasco, the Worcestershire, \u00be teaspoon of the salt, and \u00be teaspoon of the pepper. Pulse to combine.\n\n3. Transfer the Caesar dressing to a small container. (Covered, it will keep in the refrigerator for up to 1 week.)\n\n4. In a large mixing bowl, combine the lettuce, the remaining \u00bd cup cheese, remaining \u00bc teaspoon salt, remaining \u00bc teaspoon black pepper, and the croutons. Toss to combine. Add \u00bd cup plus 2 tablespoons of the dressing, and toss again. Serve immediately.\n\nNote: We advise using caution when consuming raw egg products; children or other individuals with compromised immune systems should take care due to the slight risk of salmonella or other food-borne illness. To reduce this risk, we recommend using only fresh, properly refrigerated, clean grade A or AA eggs with intact shells.\n\n4 servings (1\u00be cups dressing)\n\nSPINACH SALAD WITH BACON AND FRIED EGGS\n\nPrep time: 10 minutes Cook time: 9 minutes Total: 19 minutes\n\nBacon, eggs, and spinach. Oh, baby. What a classic combination, what a simple way to get dinner on the table in no time.\n\n10 ounces thick-sliced bacon, cut into \u00bd-inch-wide pieces\n\n1 tablespoon olive oil\n\n4 large eggs\n\n\u00bc cup minced shallots\n\n4 tablespoons red wine vinegar\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bc teaspoon freshly ground black pepper, plus more to taste\n\nOne 10-ounce bag prewashed spinach, any thick stems removed\n\n1. Set a 10-inch saut\u00e9 pan over medium heat and add the bacon. Cook, stirring often, until most of the fat has been rendered and the bacon is crisp, 7 to 8 minutes.\n\n2. During the last few minutes the bacon is cooking, heat the oil over medium heat in another large nonstick saut\u00e9 pan. Add the eggs and cook until the white is firm and the yolk is cooked to your liking, about 1 minute for a runny yolk and longer for a firmer yolk. Set the pan aside and keep warm.\n\n3. Use a slotted spoon to transfer the bacon pieces to a paper towel\u2013lined plate. Add the shallots to the pan and saut\u00e9 until fragrant and soft, about 1 minute. Add the red wine vinegar, salt, and pepper. Stir to combine and then remove from the heat.\n\n4. Place the spinach in a large bowl, and working quickly, add about half of the hot vinaigrette from the pan. Toss carefully.\n\n5. Divide the spinach equally among four serving plates. Arrange an egg on top of each mound of spinach, season with salt and black pepper, and sprinkle with the crisp bacon. Drizzle with additional vinaigrette if desired, and serve immediately.\n\n4 servings\n\nORANGE, WALNUT, AND GOAT CHEESE SALAD\n\nPrep time: 19 minutes Total: 19 minutes\n\nThere is something immensely satisfying about the combination of simple greens, toasted nuts, fresh oranges, and goat cheese. If you like, add thin slices of grilled or broiled chicken breast to this salad for a complete meal.\n\n\u00bd cup roughly chopped walnuts 2 medium oranges\n\n8 ounces mesclun salad greens or spring greens mix\n\n2 shallots, cut into thin rings (about 1\/3 cup)\n\n5 tablespoons olive oil 2 tablespoons balsamic vinegar\n\n\u00bc teaspoon salt, plus more to taste\n\n1\/8 teaspoon freshly ground black pepper, plus more to taste\n\n1\/8 teaspoon sugar\n\n3 ounces soft, mild goat cheese, crumbled\n\n1. Preheat the oven to 350\u00b0F.\n\n2. Spread the walnuts on a small baking sheet and toast until fragrant and lightly colored, 5 to 6 minutes. Set aside until cooled slightly.\n\n3. Cut away the peels from the oranges, leaving no white pith. Holding them over a small bowl, segment the oranges, catching any juices (see 20 Minutes or Less).\n\n4. In a large bowl, combine the greens, orange segments, and shallot rings.\n\n5. In a small bowl, combine the oil, vinegar, salt, pepper, sugar, and 2 tablespoons of the reserved orange juice. Whisk to combine. Taste, and adjust the seasoning if necessary. Drizzle the dressing over the salad, tossing to coat it evenly. Add the goat cheese and chopped walnuts, and toss gently to combine. Season lightly with salt and pepper if desired, and serve immediately.\n\n4 to 6 servings\n\n1 Using a sharp knife, cut away the peel from the orange on all sides, leaving the orange completely free of any white pith. Working over a small bowl, segment the orange by cutting in between the membranes on both sides of each segment.\n\n2 Use the edge of the knife to help release the segments into the bowl.\n\n3 Repeat until the orange is completely segmented. Squeeze the membranes over the bowl to release any remaining juices.\n\nCUCUMBER RIBBON SALAD\n\nPrep time: 19 minutes Total: 19 minutes\n\nThis simple cucumber salad is equally good with or without smoked salmon.\n\n3 tablespoons olive oil\n\n3 tablespoons minced shallot\n\n2\u00bd tablespoons freshly squeezed lemon juice\n\n2 tablespoons chopped fresh mint\n\n2 tablespoons chopped fresh dill\n\n\u00bd teaspoon grated lemon zest\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground white pepper\n\n1 teaspoon superfine sugar\n\n3 English (seedless) cucumbers, peeled\n\n4 ounces smoked salmon (optional)\n\n1. Whisk together the olive oil, shallot, lemon juice, mint, dill, and lemon zest in a small bowl; season with the salt, white pepper, and sugar.\n\n2. Cut the cucumbers in half crosswise. Using a vegetable peeler, peel long strips to form ribbons, and place the ribbons in a salad bowl. Add the dressing and toss to combine.\n\n3. Arrange the cucumber ribbons on four salad plates. If using, tuck slices of salmon in among the ribbons, or drape them loosely over all. Serve immediately.\n\n4 first-course servings\n\nANTIPASTO PASTA SALAD\n\nPrep time: 11 minutes Cook time: 8 minutes Total: 19 minutes\n\nEveryone who knows me knows that I'm a big fan of charcuterie of all sorts\u2014salami, pepperoni, sausages, you name it. Here the most basic salami elevates a simple pasta salad to something more substantial. Save time by buying pre-packaged thinly sliced salami.\n\n2 tablespoons plus \u00bd teaspoon salt\n\n1 pound fusilli pasta 1 tablespoon minced garlic\n\n\u00bc cup balsamic vinegar\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bd teaspoon crushed red pepper\n\n\u00bd cup extra-virgin olive oil\n\n\u00bd cup plus 2 tablespoons finely grated Pecorino Romano cheese\n\n\u00bd cup sliced pitted Kalamata olives\n\n\u00bd cup finely sliced pepperoncini\n\n\u00bd cup julienned salami slices\n\n2 tablespoons chopped mixed fresh herbs (such as parsley and basil)\n\n1. Combine the 2 tablespoons salt and 4 quarts of water in a large pot and bring to a boil over high heat. Add the pasta and cook, stirring occasionally to keep the pasta from sticking together, until just tender, 6 to 8 minutes.\n\n2. While the pasta is cooking, assemble the remaining ingredients and make the vinaigrette: Using the back of a wooden spoon, make a paste of the garlic and remaining \u00bd teaspoon salt in a large bowl. Whisk in the balsamic vinegar, black pepper, and crushed red pepper. Gradually whisk in the olive oil.\n\n3. Drain the pasta and rinse it under cold running water. Drain again. Transfer the pasta to a large mixing bowl and add the vinaigrette, along with the cheese, olives, pepperoncini, salami, and herbs. Toss to mix. Serve immediately, or cover and refrigerate until ready to serve (see Note) (let the salad return to room temperature before serving).\n\nNote: If the salad sits for any length of time before you serve it, you may need to drizzle a bit of extra vinegar and olive oil over it and toss to mix.\n\n4 to 6 servings\n\nCANTALOUPE, PROSCIUTTO, AND ARUGULA SALAD\n\nPrep time: 20 minutes Total: 20 minutes\n\nThis salad should be attempted only when melons are in season\u2014it's a great addition to an al fresco dinner in the heat of summer. So simple, so refreshing, so satisfying. Don't skip the prosciutto\u2014it's what makes this dish. Try using the best-quality prosciutto that you can find, preferably a variety from Italy or Spain. It can be pricey, but a little goes a long way here.\n\n\u00bc cup champagne vinegar or white wine vinegar\n\n1 tablespoon minced shallot\n\n\u00bd teaspoon minced garlic\n\n\u00bd teaspoon Dijon mustard\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bc teaspoon freshly ground black pepper, plus more to taste\n\n\u00bd cup vegetable oil or vegetable\u2013olive oil blend\n\n1 tablespoon minced mixed fresh herbs (such as basil, chives, and parsley)\n\n8 ounces fresh arugula, rinsed and spun dry\n\n\u00bd cup thinly sliced red onion\n\n1 cantaloupe, halved, seeded, peeled, and cut into thin wedges\n\n6 to 8 thin slices prosciutto, torn into bite-size pieces\n\n1. In a mixing bowl, combine the vinegar, shallot, garlic, mustard, salt, and pepper and whisk to combine. While continuously whisking, add the oil in a slow, steady stream until completely incorporated. Whisk in the herbs, and set aside while you prepare the salad.\n\n2. In a large bowl, combine the arugula and red onion. Drizzle in \u00bc cup of the vinaigrette and toss to combine. Add more vinaigrette to taste, if desired, and season lightly with salt and pepper. Toss gently to combine.\n\n3. Arrange the cantaloupe wedges on a large serving plate, top with the arugula salad and the prosciutto. Serve immediately.\n\nNote: Any unused vinaigrette can be stored in a nonreactive, airtight container in the refrigerator for up to 2 days.\n\n4 to 6 servings\n\nORANGE, FENNEL, AND BLACK OLIVE SALAD\n\nPrep time: 13 minutes Cook time: 17 minutes Total: 20 minutes (prep and cook times overlap)\n\nThis classic Mediterranean flavor combination is hard to beat.\n\n2 cups freshly squeezed orange juice\n\n2 medium fennel bulbs (7 ounces each)\n\n1 large red onion (10 ounces)\n\n1\u00bd cups pitted Kalamata olives, drained\n\n6 large California navel oranges (12 ounces each)\n\n\u00bd teaspoon grated orange zest\n\n2 tablespoons red wine vinegar\n\n1 tablespoon minced shallot\n\n1 teaspoon honey\n\n\u00bc cup extra-virgin olive oil, plus more for drizzling (optional)\n\nSalt and freshly ground black pepper\n\n1. Place the orange juice in a small saucepan and bring to a boil. Continue to cook at a simmer until reduced in volume to about \u00bc cup, about 17 minutes (be careful near the end of cooking, as this can easily burn if left unattended). Allow the reduced orange juice to cool to room temperature.\n\n2. While the orange juice is reducing, trim off and discard the ends of the fennel bulbs, reserving a few fronds for garnish. Cut the fennel bulbs into thin crosswise slices, and place them in a salad bowl. Cut the onion into thin julienne (see 20 Minutes or Less). Pit the olives and cut them in half. Add the onion and olives to the salad bowl.\n\n3. Using a sharp knife, cut away the peels from the oranges, leaving no white pith. Holding them over a small bowl, segment the oranges, cutting in between the membranes to release each segment and letting them fall into the bowl (see 20 Minutes or Less). Refrigerate until you are ready to serve the salad.\n\n4. Transfer the cooled reduced orange juice to a mixing bowl, and add the orange zest, red wine vinegar, shallot, and honey. Whisk thoroughly to combine. Add the oil in a thin, steady stream, whisking all the time, until completely incorporated and emulsified. Season with salt and pepper to taste.\n\n5. Arrange the orange segments over the fennel, red onion, and olives, and drizzle with the vinaigrette. Toss gently to combine. Serve the salad garnished with some of the reserved fennel fronds, and drizzled with additional olive oil if desired.\n\n4 servings\n\nICEBERG WEDGES WITH CHERRY TOMATO VINAIGRETTE\n\nPrep time: 20 minutes Total: 20 minutes\n\nIceberg sometimes gets a bad rap, especially as of late, with the advent of fancy specialty lettuces and microgreens, but sometimes there is nothing like a cool, crisp wedge of iceberg with a tasty dressing. Here I've made it special by serving it with a simple cherry tomato vinaigrette and two of my favorite cheeses.\n\n1\/3 cup sliced shallots\n\n3 tablespoons balsamic vinegar\n\n3 tablespoons red wine vinegar\n\n\u00be cup extra-virgin olive oil\n\n1 pint cherry tomatoes, halved 1 tablespoon chopped fresh basil\n\n1 tablespoon chopped fresh parsley\n\n1\u00bc teaspoons salt 1\u00bc teaspoons freshly ground black pepper\n\n1 large head (about 1 pound) iceberg lettuce, cored and cut into 8 wedges\n\n1 cup (about 4 ounces) grated fresh mozzarella cheese\n\n1 cup (about 4 ounces) grated or crumbled blue cheese\n\n1. In a medium bowl, combine the shallots, balsamic vinegar, and red wine vinegar. While stirring with a spoon, drizzle in the extra-virgin olive oil. Add the tomatoes, basil, parsley, 1 teaspoon of the salt, and 1 teaspoon of the pepper. Mix well to combine, and set aside.\n\n2. Arrange 2 wedges of iceberg against each other on each of four serving plates. Season the iceberg with the remaining \u00bc teaspoon salt and \u00bc teaspoon black pepper. Sprinkle \u00bc cup mozzarella and \u00bc cup blue cheese over each plate. Then generously spoon the tomato vinaigrette over all, dividing it evenly among the salads.\n\n4 servings\n\nSALAD TROPICALE\n\nPrep time: 20 minutes Total: 20 minutes\n\nHere we have a composed salad. Instead of tossing everything together, the ingredients are arranged over the lettuce in neat, separate piles for an interesting display. This salad takes its name from the combination of hearts of palm, tomatoes, and avocados. These three tropical ingredients, combined with just about any dressing you like, make for a refreshing salad that just screams, \"Take me to the islands!\"\n\n1 large head Bibb lettuce\n\n\u00bc teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\nOne 14-ounce jar hearts of palm, drained and cut diagonally into quarters\n\n2 medium-size vine-ripened tomatoes, each cut into 8 wedges\n\n2 firm-ripe Hass avocados, halved, seeded, and sliced\n\n1 cucumber, halved lengthwise and cut into \u00bc-inch-thick half-moons\n\n\u00be cup Herb Vinaigrette or Buttermilk Dressing (20 Minutes or Less)\n\n1. Halve and core the Bibb lettuce. Gently separate the leaves; rinse and spin dry. Divide the lettuce among four plates, arranging the leaves so as to create a bowl on each plate. Sprinkle the lettuce bowls with the salt and pepper.\n\n2. Divide the heart of palm pieces, tomatoes, avocados, and cucumbers evenly among the four plates, forming neat individual piles of each within the lettuce bowls. Spoon 3 generous tablespoons of vinaigrette or dressing over each salad. Serve immediately.\n\n4 servings\n\nBUTTERMILK DRESSING\n\nPrep time: 10 minutes Total: 10 minutes\n\nThis creamy dressing just needs to be stirred together and that's it. Kick it up by adding minced herbs to your liking, or add up to \u00be cup crumbled blue cheese to the basic recipe for a rich and creamy blue cheese dressing. Perfect for a crisp wedge of iceberg lettuce!\n\n5 tablespoons buttermilk\n\n\u00bc cup mayonnaise\n\n\u00bc cup sour cream\n\n1 tablespoon freshly squeezed lemon juice\n\n2 tablespoons minced green onion tops\n\n1 teaspoon minced garlic\n\n\u00bc teaspoon finely grated lemon zest\n\n\u00bc teaspoon salt\n\n\u00bc teaspoon freshly ground white pepper\n\n\u00bc teaspoon Emeril's Original Essence or Creole Seasoning\n\nWhisk all the ingredients together in a nonreactive mixing bowl. Use immediately, or store in a covered nonreactive container in the refrigerator for up to 3 days.\n\n1 cup\n\nCreole Seasoning\n\n2\u00bd tablespoons paprika\n\n2 tablespoons salt\n\n2 tablespoons garlic powder\n\n1 tablespoon black pepper\n\n1 tablespoon onion powder\n\n1 tablespoon cayenne pepper\n\n1 tablespoon dried oregano\n\n1 tablespoon dried thyme\n\nCombine all the ingredients thoroughly.\n\n2\/3 cup\n\nBALSAMIC VINAIGRETTE\n\nPrep time: 8 minutes Total: 8 minutes\n\nA simple vinaigrette that is equally at home on mixed greens, grilled veggies or chicken, or pasta salads.\n\n5 tablespoons balsamic vinegar\n\n1 teaspoon Dijon mustard\n\n1 teaspoon sugar\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\n1 clove garlic, smashed\n\n5 tablespoons extra-virgin olive oil\n\n5 tablespoons olive oil\n\nIn a nonreactive mixing bowl, whisk together the vinegar, mustard, sugar, salt, pepper, and garlic. While whisking, add the oils in a thin, steady stream until they are completely incorporated and the dressing is emulsified. Use immediately, or refrigerate in a covered nonreactive container for up to 1 week.\n\nThis is best if allowed to sit for at least 10 minutes to allow the garlic to infuse. (Discard the garlic before serving.)\n\nAbout 1 cup\n\nRED WINE VINAIGRETTE\n\nPrep time: 10 minutes Total: 10 minutes\n\nThis is a classic French-style vinaigrette. Serve it over mixed greens, poached leeks or asparagus, grilled chicken, or even poached fish.\n\n\u00bc cup red wine vinegar\n\n2 tablespoons minced shallot\n\n1\u00bd teaspoons Dijon mustard\n\n\u00be teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\n\u00be cup canola or vegetable oil\n\nIn a medium bowl, whisk together the vinegar, shallot, mustard, salt, and pepper. While constantly whisking, add the oil in a thin, steady stream until it is completely incorporated and the vinaigrette is emulsified. Serve immediately, or refrigerate in a covered nonreactive container for up to 2 days.\n\n1 generous cup\n\nHERB VINAIGRETTE\n\nPrep time: 18 minutes Total: 18 minutes\n\nUse whichever herbs you like, or whichever you have on hand, to make this simple vinaigrette.\n\n\u00bc cup white wine vinegar\n\n1 tablespoon minced shallot\n\n1 tablespoon Dijon mustard\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bc teaspoon freshly ground black pepper, plus more to taste\n\n\u00bd cup vegetable oil (such as canola)\n\n2 tablespoons extra-virgin olive oil\n\n4 tablespoons chopped mixed fresh soft herbs (such as parsley, chives, basil, and tarragon)\n\nIn a medium bowl, whisk together the vinegar, shallot, mustard, salt, and pepper. While constantly whisking, add the oils in a thin, steady stream until they are completely incorporated and the dressing is emulsified. Stir in the herbs, and adjust the seasoning to taste if necessary. Serve immediately or refrigerate in a covered nonreactive container for up to 2 days.\n\n1 generous cup\n\nSandwiches\n\nSAUSAGE AND PEPPER PO-BOY\n\nPrep time: 8 minutes Cook time: 10 to 12 minutes Total: 18 to 20 minutes\n\nThis is my rendition of the traditional Italian sandwich, which is also popular at New York City's street festivals\u2014it is a favorite at the San Gennaro festival in Little Italy. I have added a New Orleans spin here by kicking the mayo up with a little Louisiana hot sauce.\n\n2 tablespoons olive oil\n\n1 red bell pepper, thinly sliced\n\n1 green bell pepper, thinly sliced\n\n2 cups sliced yellow onions\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 teaspoon dried Italian herbs\n\n1 pound mild Italian sausage, removed from casings and crumbled\n\n\u00bd cup mayonnaise\n\n1 tablespoon Louisiana hot sauce or other red hot sauce\n\n1 loaf soft French bread, cut into four 6-inch sections, each section sliced in half horizontally\n\n4 ounces sliced provolone cheese\n\n1. In a large saut\u00e9 pan, heat the olive oil over medium-high heat. When the oil is hot, add the bell peppers, onions, Essence, and Italian herbs. Cook until the vegetables have softened, 4 to 5 minutes. Transfer the peppers and onions to a paper towel\u2013lined plate.\n\n2. Add the crumbled sausage to the hot pan and cook until the meat is cooked through and lightly browned, about 5 minutes. Set aside.\n\n3. Preheat the broiler.\n\n4. In a small bowl, combine the mayonnaise and the hot sauce.\n\n5. To assemble the po-boys, spread the cut sides of each piece of bread with the spicy mayonnaise. Divide the sausage evenly among the 4 bottom pieces of bread, then top evenly with the peppers and onions. Arrange the cheese slices over the filled bottom halves and over the cut side of each top portion (cut the cheese to fit as necessary).\n\n6. Place the sandwich halves on a baking sheet, cheese sides up, and heat under the broiler just until the cheese melts, 1 to 2 minutes. Press the sandwich halves together, and serve immediately.\n\n4 servings\n\nBACON, LETTUCE, AVOCADO, AND TOMATO SANDWICH WITH BASIL MAYO\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nThe addition of creamy avocado slices takes this simple classic up a notch and makes it even more delicious and satisfying. And the basil mayo, well, now, what's not to love?\n\n16 slices thick-cut bacon\n\n8 slices (about \u00bd inch thick) brioche, challah, or other soft white bread\n\n\u00bd cup mayonnaise\n\n2 packed tablespoons finely chopped fresh basil\n\n8 to 12 thin slices ripe tomato 8 red-leaf lettuce leaves, rinsed and patted dry\n\n1 ripe avocado, halved, seeded, and sliced\n\n1. Place the bacon in a large skillet or saut\u00e9 pan over medium heat, and cook until crisp, 6 to 8 minutes. Drain on paper towels while you prepare the remaining ingredients.\n\n2. Toast the bread slices in a toaster to the desired color.\n\n3. Combine the mayonnaise and basil in a small bowl, and stir to mix well.\n\n4. Spread the mayo evenly over one side of each bread slice. Cut or break the bacon slices in half, and arrange on one half of each sandwich. Place 2 or 3 slices of tomato, 2 lettuce leaves, and about one-quarter of the avocado slices on top of the bacon on each sandwich. Top with the other half of the bread, cut the sandwiches in half, and serve immediately.\n\n4 servings\n\nSTEAK AND CHEESE SANDWICHES\n\nPrep time: 7 minutes Cook time: 13 minutes Total: 20 minutes\n\nSteak and tangy blue cheese\u2014ooohhh, how could you go wrong? So fast and yet so good\u2014we could make Philly jealous.\n\n1 baguette or other crusty country bread, about 24 inches long\n\n6 tablespoons butter: 4 tablespoons melted, 2 tablespoons at room temperature\n\n2 tablespoons olive oil\n\n4 cups thinly sliced onions\n\n\u00be teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n1 pound top sirloin steak\n\n\u00bc cup Worcestershire sauce\n\n\u00bd cup (about 4 ounces) grated or crumbled blue cheese\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Cut the baguette into four 6-inch lengths. Split each piece in half horizontally, and brush the insides with the melted butter. Lay the bread, cut side up, on a baking sheet and toast in the oven until crispy, about 10 minutes. Set aside.\n\n3. While the bread is toasting, heat the olive oil in a 14-inch skillet over medium-high heat. Add the onions, \u00bd teaspoon of the salt, and \u00bc teaspoon of the pepper. Cook, stirring as needed, until the onions are nicely browned, about 10 minutes.\n\n4. While the onions are cooking, slice the steak with a very sharp knife into 1\/8-inch-thick slices (see Note). Season the steak with the remaining \u00bc teaspoon salt and \u00bc teaspoon pepper.\n\n5. Increase the heat under the skillet to high, move the onions to one side of the pan, and add the softened butter. When it has melted, add the steak and cook for 2 minutes without stirring. Add the Worcestershire and \u00bc cup of the blue cheese. Using tongs or a metal spatula, mix the onions, meat, and cheese together and cook for 1 minute longer. Remove from the heat.\n\n6. Divide the steak mixture, and any accumulated pan juices, evenly among the 4 bottom portions of the toasted bread. Sprinkle with the remaining \u00bc cup cheese, and cover with the toasted bread tops. Serve immediately.\n\nNote: You may find the steak easier to slice thinly if you place it in the freezer for 10 to 15 minutes before slicing.\n\n4 servings\n\nPRESSED ROAST TURKEY, PESTO, AND PROVOLONE SANDWICHES\n\nPrep time: 5 minutes Cook time: 12 to 15 minutes Total: 17 to 20 minutes\n\nTalk about a delicious quick lunch: slices of turkey and provolone cheese sandwiched with an herbaceous pesto\u2014yum! Try serving these with the Cream of Tomato Soup on 40 Minutes or Less for a true power meal.\n\n8 slices ciabatta or other crusty Italian or hearty white sandwich bread\n\n4 tablespoons prepared basil pesto\n\n4 slices (about 4 ounces) provolone cheese\n\n8 slices (about 8 ounces) roast turkey\n\n1\u00bd tablespoons extra-virgin olive oil\n\n1. Lay the bread slices on a clean work surface, and spread 1\u00bd teaspoons of the pesto over one side of each slice of bread. Divide the cheese evenly among the slices (depending on the size of your cheese slices, you may need to cut them in half so that you can have cheese on both sides). Divide the roast turkey slices evenly among 4 of the bread slices. Place the remaining cheese-topped slices on top of the turkey-topped slices to form 4 sandwiches. Brush the outside of each sandwich with some of the olive oil.\n\n2. Preheat a grill pan over medium heat. When it is hot, add the sandwiches, in batches if necessary, and weight them with a sandwich press or another skillet (or other heavy object). Cook until the sandwiches are golden brown and crisp and the cheese has melted, 4 to 6 minutes per side. Remove the sandwiches, cut in half on the diagonal, and serve immediately.\n\nNote: Though we prefer the crisp, ridged exterior you get when these sandwiches are cooked in a grill pan, they can also be cooked in a saut\u00e9 pan or panini press; the cook time will vary slightly.\n\n4 servings\n\nPROSCIUTTO AND MOZZARELLA PANINI\n\nPrep time: 8 minutes Cook time: 8 minutes Total: 16 minutes\n\nThis is a classic Italian panini. The texture of true Italian ciabatta really makes this sandwich special because it grills up so nice and crisp on the outside.\n\n\u00bc cup extra-virgin olive oil\n\n1 tablespoon balsamic vinegar\n\n2 teaspoons minced fresh oregano, or 1 teaspoon dried\n\n1 teaspoon minced garlic\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\nEight \u00bd-inch-thick slices ciabatta or other rustic Italian white bread\n\n4 ounces thinly sliced mozzarella cheese\n\n4 ounces thinly sliced prosciutto\n\n6 ounces jarred roasted red peppers, drained, and torn into 1-inch-wide pieces\n\n1. Whisk the olive oil, vinegar, oregano, garlic, salt, and pepper together in a small bowl to blend.\n\n2. Arrange the slices of bread on a flat work surface, and using a brush, spread the vinaigrette evenly over one side of each slice. Divide the mozzarella equally among the bread slices. Top 4 of the bread slices with the prosciutto and red peppers, and then place the remaining 4 slices of bread on the top, vinaigrette side down, to form 4 sandwiches.\n\n3. Heat a large skillet or grill pan over medium heat. Add the sandwiches and cook, pressing them occasionally with a large spatula or the bottom of a small heavy saucepan, until the bread is golden brown and the cheese has melted, about 4 minutes per side. Serve immediately.\n\n4 servings\n\nOPEN-FACE TURKEY AND CHEESE SANDWICH\n\nPrep time: 12 minutes Cook time: 7 minutes Total: 19 minutes\n\nThis unique combination of flavors is a true showstopper. Trust me on this one\u2014and enjoy!\n\n6 ounces soft, mild goat cheese\n\n1 tablespoon finely chopped fresh parsley\n\n1\u00bd teaspoons minced garlic\n\n\u00bc teaspoon grated lemon zest 1 tablespoon olive oil\n\n8 ounces button mushrooms, wiped clean and sliced\n\n\u00bd teaspoon salt\n\n5 ounces prewashed spinach, any thick stems removed\n\n4\u00bd-inch-thick slices hearty white bread (such as sourdough), toasted\n\n\u00bc cup chopped walnuts, toasted\n\n8 thin slices tomato\n\n8 to 10 ounces sliced roasted turkey breast\n\n4 ounces sliced Emmenthaler or provolone cheese\n\n1. Combine the goat cheese, parsley, \u00bd teaspoon of the garlic, and the lemon zest in a small bowl and mix well. Set aside.\n\n2. Heat the olive oil in a large nonstick saut\u00e9 pan over medium-high heat. When it is hot, add the mushrooms and \u00bc teaspoon of the salt, and cook, stirring occasionally, until they are soft and caramelized, about 4 minutes. Move the mushrooms aside and add the spinach and the remaining \u00bc teaspoon salt to the pan. Stir with the mushrooms and cook until the spinach has almost completely wilted, about 1 minute. Stir in the remaining 1 teaspoon garlic and cook for another 30 seconds. Remove the mushrooms and spinach from the pan and drain off any excess liquid. Set aside until ready to use.\n\n3. Position a rack close to the broiler element and preheat the broiler.\n\n4. Spread the goat cheese mixture evenly over the slices of toasted bread. Sprinkle the walnuts on top. Add 2 slices of tomato on each sandwich, and divide the turkey evenly among the sandwiches, arranging it over the tomatoes. Divide the mushroom-spinach mixture among the sandwiches, then top with the cheese.\n\n5. Place the sandwiches on a baking sheet and broil until the cheese is melted and bubbly, about 1 minute. Serve hot.\n\n4 servings\n\nKICKED-UP TUNA MELTS\n\nPrep time: 15 minutes Cook time: 4 to 5 minutes Total: 19 to 20 minutes\n\nAt our house, tuna melts were always on the menu when I was growing up. Add the cheese and just the right amount of heat (whether you griddle it or do a quick broil), and you've got comfort served on a plate. This is a classic!\n\nFour 5-ounce cans solid white tuna packed in water, drained\n\n\u00bc cup plus 1 tablespoon mayonnaise, plus more for spreading\n\n\u00bc cup finely chopped red onion\n\n1 tablespoon plus 1 teaspoon nonpareil capers, drained\n\n1 tablespoon freshly squeezed lemon juice\n\n1 teaspoon freshly ground black pepper\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon dried oregano, crumbled between your fingers\n\n4 slices rustic white bread or other dense white bread\n\n8 thin slices tomato\n\n4 ounces sliced provolone cheese\n\n1. Position a rack about 6 inches from the broiler element and preheat the broiler.\n\n2. Combine the tuna, mayonnaise, red onion, capers, lemon juice, pepper, salt, and oregano in a medium bowl and stir until thoroughly combined.\n\n3. Arrange the bread slices on a baking sheet and spread additional mayonnaise over each slice. Divide the tuna salad evenly among the bread slices, then top with the tomato slices. Arrange the sliced provolone evenly over the sandwiches. Place the baking sheet under the broiler and cook until the cheese is golden and bubbly, 3 to 4 minutes. Serve hot.\n\n4 open-face sandwiches, 2 to 4 servings\n\nFISH TACOS WITH BLACK BEAN SALSA\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nUse any kind of fresh fish you like for these tacos. (Keep in mind when buying fish that a truly fresh fish should smell like the sea.) Here chunks of the lightly crusted fish combine with the flavors of lime juice, garlic, and black beans, making for a very delicious taco. Kid-friendly, for sure, but also sure to win over any adult.\n\nOne 15-ounce can black beans, rinsed and drained\n\n\u00be cup olive oil\n\n1 jalape\u00f1o, minced (and seeded, if you prefer, for a milder salsa)\n\n1 tablespoon freshly squeezed lime juice\n\n1 teaspoon minced garlic\n\n1\u00bd teaspoons salt\n\n2 pounds skinless firm white fish fillets (such as snapper), trimmed and cut into 3-inch pieces\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bd cup cornmeal\n\nEight 6-inch flour tortillas\n\n3 cups thinly sliced or shredded romaine lettuce\n\n4 lime wedges\n\n\u00bd cup sour cream\n\n1. In a medium bowl, combine the black beans, \u00bc cup of the olive oil, and the jalape\u00f1o, lime juice, garlic, and \u00bd teaspoon of the salt. Set aside.\n\n2. Season the fish fillets evenly with the remaining 1 teaspoon salt and the black pepper. Dredge quickly in the cornmeal, shaking to remove any excess, and set aside.\n\n3. Heat a 12-inch saut\u00e9 pan over high heat. Toast each of the tortillas for 30 seconds on one side in the hot saut\u00e9 pan. Transfer to a plate and cover loosely with a clean kitchen towel to keep warm.\n\n4. In the same saut\u00e9 pan, heat the remaining \u00bd cup olive oil over medium-high heat. Add half of the fish fillets, and saut\u00e9 until just cooked through, about 2 minutes per side. Transfer them to paper towels to drain. Repeat with the remaining fillets.\n\n5. To assemble the tacos, place 2 tortillas on each plate, and divide the shredded lettuce among them. Spoon the black bean mixture onto the lettuce, and divide the fish fillets among the tortillas. Squeeze the lime wedges over the fish, dollop with the sour cream, and fold the tortillas to close. Serve immediately.\n\n4 to 6 servings\n\nPasta\n\nFETTUCCINE WITH PEAS AND HAM\n\nPrep time: 8 minutes Cook time: 12 minutes Total: 20 minutes\n\nVegetable group? Check. Meat group? Check. Grain group? Check. Dairy group? Check. This dish has it all, making it a great one-dish meal. Serve it with a nice green salad and some hot crusty bread. One of my all-time favorites.\n\n\u00bd teaspoon salt, plus more for the pasta water\n\n1 pound fettuccine\n\n2 teaspoons olive oil\n\n1 cup diced onion\n\n\u00bc cup thinly sliced shallot\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 teaspoon minced garlic\n\n8 ounces ham steak, diced (about 2 cups)\n\n1 cup frozen green peas\n\n2 cups heavy cream\n\n\u00bc cup chopped fresh parsley\n\n\u00bd cup grated Parmigiano-Reggiano cheese, plus more for serving (optional)\n\nFreshly ground black pepper, for serving (optional)\n\n1. Bring a large pot of salted water to a boil and cook the fettuccine until al dente, about 12 minutes.\n\n2. While the pasta is cooking, make the sauce: In a 12-inch (or larger) saut\u00e9 pan, heat the olive oil over medium-high heat. Add the onion, shallot, Essence, and the \u00bd teaspoon salt, and cook until the onions are translucent, about 2 minutes. Add the garlic, ham, and peas and cook for 2 minutes longer. Increase the heat to high, add the cream, and bring it to a boil. Then reduce the heat to medium and simmer until the cream thickens, about 5 minutes.\n\n3. Drain the pasta in a colander, reserving 1 cup of the cooking water.\n\n4. Add the drained pasta, \u00bd cup of the reserved cooking water, and the parsley and cheese to the sauce and cook, tossing constantly, until heated through and well combined, about 1 minute. If the mixture seems dry, add the remaining cooking water, a little at a time, as needed.\n\n5. Remove the pan from the heat and transfer the pasta mixture to a large serving bowl or to individual bowls. Sprinkle with additional cheese and black pepper if desired.\n\n4 to 6 servings\n\n1 Beginning the sauce.\n\n2 Adding the pasta.\n\n3 Putting it all together.\n\nORANGE, CURRANT, AND PINE NUT COUSCOUS\n\nPrep time: 12 minutes Cook time: 3 minutes Inactive time: 5 minutes Total: 20 minutes\n\nThis simple couscous side dish is a nice complement to many entr\u00e9es\u2014and it's pretty, too. The bright flavors of the orange segments and currants create bites of contrasting tastes, and the toasted pine nuts add a bit of texture.\n\n2 cups water\n\n2 tablespoons extra-virgin olive oil\n\n\u00bd cup diced red onion or shallots (small dice)\n\n\u00bd cup diced carrots (small dice)\n\n\u00bd cup dried currants 1 teaspoon grated orange zest\n\n\u00be teaspoon salt\n\n\u00bd teaspoon freshly ground white pepper\n\nOne 10-ounce package couscous\n\n1 orange, peeled and segmented (see 20 Minutes or Less)\n\n\u00bc cup pine nuts or chopped walnuts, lightly toasted\n\n1 tablespoon chopped fresh mint or parsley\n\n1. Combine the water, olive oil, red onion, carrots, currants, orange zest, salt, and white pepper in a small saucepan, and bring to a boil. Cook at a slow simmer for 2 minutes to make a flavorful broth.\n\n2. Meanwhile, place the couscous in a medium heatproof bowl.\n\n3. Pour the hot broth over the couscous and cover with plastic wrap. Allow to steam for 5 minutes.\n\n4. Remove the plastic wrap. Add the orange segments, pine nuts, and mint, and fluff the couscous with a fork. Serve hot.\n\nAbout 4 cups, 4 servings\n\nLINGUINE ALLA CARBONARA\n\nPrep time: 6 minutes Cook time: 12 to 14 minutes Total: 18 to 20 minutes\n\nThis dish takes its inspiration from many wonderful meals at my friend Mario Batali's restaurant, OTTO, in New York City. Try my quick and easy take on it\u2014my kids go crazy for this dish.\n\nSalt\n\n1 pound linguine\n\n\u00bc cup extra-virgin olive oil 1 tablespoon unsalted butter\n\n6 ounces bacon, cut into \u00bd-inch-wide pieces\n\n1 yellow onion (6 ounces), minced 2 teaspoons minced garlic\n\n3 large egg yolks\n\n\u00be cup finely grated Parmigiano-Reggiano cheese, at room temperature\n\nFreshly ground black pepper, to taste\n\n1. Bring a large pot of salted water to a boil, and cook the linguine until al dente, about 9 minutes.\n\n2. While the pasta is cooking, assemble the remaining ingredients and make the sauce: Heat the oil and butter in a large saut\u00e9 pan over medium heat. Add the bacon and cook until it is beginning to crisp, about 5 minutes. Add the onion and garlic and cook until soft, about 2 minutes. Remove the pan from the heat and set aside.\n\n3. Drain the pasta in a colander, reserving a small amount of the cooking water, and return the pasta to the pot. Place the pot over high heat, add the bacon-onion mixture, and stir until the pasta is coated with the mixture and heated through, 1 minute.\n\n4. Whisk the egg yolks in a small bowl and add them to the pasta, along with the grated cheese. Remove the pot from the heat and toss the pasta until it is well coated. (If needed, add a bit of the reserved cooking water to help toss the pasta.) Season with salt and pepper to taste, and serve immediately.\n\n4 servings\n\nEMERIL'S SHRIMP AND PASTA WITH GARLIC, LEMON, CRUSHED RED PEPPER, AND GREEN ONIONS\n\nPrep time: 8 minutes Cook time: 12 minutes Total: 20 minutes\n\nThe flavors of this dish are all in the title\u2014it's spicy, savory, and bright, and it makes me happy, happy, happy any time of the day. So easy and fast, too!\n\n1 pound linguine\n\n1\u00bc pounds large shrimp, peeled and deveined\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n8 tablespoons (1 stick) unsalted butter\n\n1 tablespoon minced garlic\n\n\u00bd cup dry white wine\n\n\u00bc cup freshly squeezed lemon juice\n\n\u00bd teaspoon salt, plus more for the pasta water\n\n1\/8 teaspoon freshly ground black pepper\n\n1 teaspoon crushed red pepper\n\n\u00bc cup chopped green onion tops\n\n2 tablespoons chopped fresh parsley\n\n1. Bring a large pot of salted water to a boil and cook the linguine until al dente, about 9 minutes.\n\n2. While the pasta is cooking, toss the shrimp with the Essence in a medium bowl. Place 4 tablespoons of the butter in a 14-inch skillet set over high heat. Add the shrimp, spreading them evenly in one layer. Cook for 2 minutes, and then turn them to the other side. Add the garlic and cook for 30 seconds. Add the wine, lemon juice, and 2 tablespoons of the remaining butter, and cook for 1\u00bd minutes. Season the shrimp with the \u00bd teaspoon salt and the black pepper.\n\n3. Drain the pasta in a colander, reserving \u00bd cup of the cooking water. Add the pasta, reserved cooking water, crushed red pepper, and green onion tops to the sauce in the skillet. Toss until everything is heated through and the pasta is well coated, about 1 minute.\n\n4. Remove the skillet from the heat, add the remaining 2 tablespoons butter and the parsley, and toss to combine. Serve hot.\n\n4 to 6 servings\n\nRice and Beans\n\nKICKED-UP SHRIMP FRIED RICE\n\nPrep time: 12 minutes Cook time: 6 minutes Total: 18 minutes\n\nI love this so much that I usually make extra to eat the day after. Like all stir-fry dishes, it's important that you have all your ingredients prepared and ready to go once you start cooking. Since you must cook this over high heat, take care to constantly toss or stir your ingredients so that nothing gets overcooked. If you don't own a big wok or a skillet that is large enough to do this in one batch, do it in two; just be sure to wipe the pan clean with paper towels in between batches.\n\n4 tablespoons peanut or vegetable oil\n\n3 eggs, lightly beaten\n\nSalt and freshly ground black pepper\n\n8 ounces medium shrimp, peeled and deveined\n\n\u00bd teaspoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n3 green onions, white and green parts separately minced\n\n2 teaspoons minced ginger\n\n2 teaspoons minced garlic\n\n3 cups cooked white rice\n\n1 tablespoon dark Asian sesame oil\n\n2 cups (about 12 ounces) frozen stir-fry vegetables, defrosted\n\n2 tablespoons plus 1 teaspoon soy sauce\n\n1. Heat 1 tablespoon of the peanut oil in a large skillet or wok over high heat. When it is hot, add the eggs and a pinch of salt and pepper, and quickly stir until the eggs are fully cooked, moving the skillet off and on the heat as necessary, about 40 seconds. Transfer the eggs to a paper towel\u2013lined plate and set aside. Chop the eggs into small pieces when cool enough to handle.\n\n2. Add 1 tablespoon of the remaining peanut oil to the skillet.\n\n3. In a small bowl, season the shrimp with the Essence and a pinch of salt and pepper. When the oil is hot, add the shrimp to the skillet, in batches if necessary, and cook until pink and lightly caramelized, about 2 minutes per side. Remove from the skillet and set aside.\n\n4. Add the remaining 2 tablespoons peanut oil to the skillet. Add the green onion bottoms (white portion), the ginger, and the garlic, and cook until fragrant, about 15 seconds. Add the rice and cook, tossing, until it is hot and golden, about 2 minutes. Add the sesame oil and the stir-fry vegetables, and cook until heated through, 1 minute. Add the soy sauce, the reserved cooked eggs, and the shrimp, and cook until everything is warmed through, about 1 minute.\n\n5. Season with salt and pepper to taste, garnish with green onion tops, and serve.\n\n4 servings\n\nTURKEY AND WILD RICE SALAD\n\nPrep time: 15 minutes Total: 15 minutes\n\nObviously this is a great choice for the fall season, when leftover holiday turkey is often found lurking in the fridge, but I'm such a fan of roast turkey that I make it year-round.\n\n4 cups cooked wild rice mix (see Note)\n\n2 cups diced roast turkey breast\n\n1 cup assorted dried fruits (such as cranberry, cherry, and apricot), coarsely chopped\n\n\u00bd cup coarsely chopped almonds, toasted\n\nJuice and grated zest of 1 orange\n\n\u00bc cup olive oil\n\n2 tablespoons red wine vinegar\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\nCombine all of the ingredients in a large bowl, and toss well to combine. Let stand for 5 minutes, and then serve.\n\nNote: For testing purposes we used a blend of basmati and wild rice, but any cooked rice blend or mix of cooked rices will work here. If the prepared rice has been refrigerated, however, you will need to microwave or steam it briefly to soften it slightly before combining it with the other ingredients.\n\n8 cups, about 8 servings\n\nAROMATIC JASMINE RICE\n\nPrep time: 5 minutes Cook time: 15 minutes Total: 20 minutes\n\nBring a taste of Asia into your home with this deliciously nuanced jasmine rice. This dish would be great served alongside Stir-Fried Chicken with Cashews (20 Minutes or Less), Stir-Fried Beef and Broccoli (40 Minutes or Less), or Spicy Pork Stir-Fry with Green Beans (20 Minutes or Less).\n\n2 cups jasmine rice\n\nOne 1-inch piece fresh ginger, peeled and halved\n\nGrated zest of 1 lime\n\n1\u00bd cups unsweetened coconut milk\n\n1\u00bd cups chicken broth or canned, low-sodium chicken broth or water\n\n\u00bd teaspoon salt\n\n1 tablespoon chopped fresh cilantro\n\n1 tablespoon crushed peanuts, for garnish\n\n1. Combine all the ingredients except the cilantro and peanuts in a 2-quart saucepan, and stir well to combine. Make sure the aromatics are fully submerged in the rice.\n\n2. Place the pan over high heat and bring the liquid to a boil, stirring occasionally to prevent the rice from sticking to the bottom of the pan. Once it reaches a boil, immediately reduce the heat to low and cover the pan. Cook for 10 minutes. Remove the pan from the heat and allow the rice to steam, covered and undisturbed, for 5 minutes.\n\n3. Discard the ginger. Add the cilantro and gently fluff the rice with a fork. Transfer the rice to a deep serving bowl and garnish with the peanuts. Serve hot or warm.\n\n4 to 6 servings\n\nSPICY SAUSAGE, BEAN, AND CHEESE NACHOS\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nThis filling snack is perfect for football season, but don't feel that you have to wait until game time\u2014it's great anytime and can even form the basis of a simple supper when paired with a nice green salad.\n\n1 pound fresh hot sausage (such as chorizo), removed from the casings and crumbled\n\n\u00bc cup finely chopped green onion bottoms (reserve tops separately)\n\n1 tablespoon chopped garlic\n\nTwo 15-ounce cans pinto or black beans, drained\n\n\u00be cup chicken stock or canned, low-sodium chicken broth or water\n\n\u00bd teaspoon chili powder\n\n\u00bc teaspoon ground cumin\n\n\u00bc teaspoon salt\n\n12 ounces large (restaurant-style) tortilla chips\n\n3 cups grated pepper Jack or sharp cheddar cheese, or a combination\n\n\u00bc cup pickled jalape\u00f1o slices, or to taste\n\nYour favorite salsa, for serving\n\nSour cream, for serving\n\n1. Preheat the oven to 450\u00b0F.\n\n2. Place a large skillet over medium-high heat, add the sausage, and cook until it is nicely browned and the fat is rendered, about 5 minutes. Use a slotted spoon to transfer the sausage to a paper towel\u2013lined plate to drain.\n\n3. Add the green onion bottoms and the garlic to the fat remaining in the skillet, and cook until fragrant and soft, about 20 seconds. Add the beans, mix well, and cook until heated through, about 1 minute. Add the chicken stock, chili powder, cumin, and salt. Mash the beans with the back of a heavy wooden spoon or a potato masher until chunky-smooth.\n\n4. Reduce the heat to medium and continue cooking until completely warmed through, 1 to 2 minutes. Remove from the heat.\n\n5. Spread half of the tortilla chips in one even layer on a large oval ovenproof platter or in a large baking dish. Top with half of the beans, half of the sausage, half of the cheese, and half of the jalape\u00f1os. Repeat with another layer of chips, beans, sausage, cheese, and jalape\u00f1os. Bake until the cheese is melted and the mixture is hot, 2 to 3 minutes.\n\n6. Remove from the oven, and serve garnished with chopped green onion tops, salsa, and sour cream.\n\n4 to 6 servings\n\nVegetables\n\nPAN-ROASTED ASPARAGUS WITH SHIITAKE MUSHROOMS AND CHERRY TOMATOES\n\nPrep time: 8 minutes Cook time: 11 to 12 minutes Total: 19 to 20 minutes\n\nAsparagus comes in all sizes and can be cooked any number of ways\u2014and to me they are all great\u2014but I especially love asparagus that has been quickly saut\u00e9ed and then blasted in a very hot oven. In this recipe I have added shiitake mushrooms to give the dish an earthy flavor that strikes a nice balance with the sweetness of the cherry tomatoes.\n\n3 tablespoons olive oil\n\n1 large shallot, sliced crosswise into rings\n\n1 pound asparagus, woody portion of stems removed\n\n4 ounces shiitake mushrooms, wiped clean, stemmed, and quartered\n\n2 ounces cherry or grape tomatoes, quartered\n\n1 teaspoon fresh thyme leaves\n\n1 tablespoon freshly grated Parmigiano-Reggiano cheese\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Heat 2 tablespoons of the olive oil in a large ovenproof saut\u00e9 pan over medium-high heat. When the oil is hot, add the shallot and cook for 30 seconds. Add the asparagus and cook for 3 minutes. Push the asparagus to one side of the pan, and add the remaining 1 tablespoon olive oil and the shiitake mushrooms. Cook for 3 to 4 minutes, allowing the mushrooms to brown. Add the tomatoes and the thyme, and cook for another 2 minutes, tossing the ingredients together.\n\n3. Transfer the pan to the oven and cook for 3 to 4 minutes, or until the asparagus is crisp-tender.\n\n4. Transfer the asparagus mixture to a serving platter, garnish with the cheese, and serve immediately.\n\n4 side-dish servings or 2 appetizer servings\n\nGARLICKY BOK CHOY\n\nPrep time: 2 minutes Cook time: 5 to 6 minutes Total: 7 to 8 minutes\n\nIf you cannot find baby bok choy in your area, feel free to substitute the same amount of regular bok choy. Regular bok choy will need to be cut into 1\u00bd-inch lengths on the diagonal and stirred occasionally while cooking.\n\n2 tablespoons canola oil\n\n1 teaspoon crushed red pepper\n\n1 pound baby bok choy, split in half lengthwise\n\n\u00bd teaspoon salt\n\n2 tablespoons roughly chopped garlic\n\n\u00bc cup chicken or vegetable stock, or canned, low-sodium chicken or vegetable broth, or water\n\n2 tablespoons butter\n\nPlace a 14-inch saut\u00e9 pan over medium-high heat, and add the oil. When it is hot, add the crushed red pepper and cook until fragrant, about 30 seconds. Add the bok choy, cut sides down, and cook for 1 to 2 minutes. Add the salt, garlic, and chicken stock and cook for 3 minutes, until the stock is mostly reduced. Add the butter to the pan, and when it has melted, turn the bok choy so that it is evenly coated. Remove from the heat and serve immediately.\n\n4 servings\n\nBROILED ZUCCHINI\n\nPrep time: 5 minutes Cook time: 15 minutes Total: 20 minutes\n\nIn this preparation, zucchini is caramelized and roasted with garlic under the broiler\u2014delicious!\n\n2 pounds zucchini\n\n4 cloves garlic\n\n\u00bc cup olive oil\n\n1\u00bd teaspoons kosher salt\n\n1 teaspoon freshly ground white pepper\n\n4 sprigs fresh thyme\n\n1. Position a rack 5 or 6 inches from the broiler element and preheat the broiler.\n\n2. Rinse the zucchini and pat it dry. Cut the zucchini in half crosswise; then cut each half lengthwise into 6 to 8 wedges. Smash the garlic cloves, and cut each clove in half.\n\n3. Place the zucchini, garlic, and all the remaining ingredients in a large bowl, and toss well to coat.\n\n4. Place the zucchini mixture in a 12-inch ovenproof skillet, and broil, tossing the pieces occasionally, until it is well caramelized, about 15 minutes.\n\n5. Remove from the broiler, discard the thyme sprigs, and serve immediately.\n\n4 to 6 servings\n\nSAUT\u00c9ED YELLOW SQUASH WITH CARROTS AND TARRAGON\n\nPrep time: 12 minutes Cook time: 8 minutes Total: 20 minutes\n\nThis is a beautiful, versatile side dish that can go with just about any entr\u00e9e. While testing recipes, we enjoyed it with Crispy Pan-Roasted Chicken with Garlic-Thyme Butter (40 Minutes or Less)\u2014in fact, the combination was a knockout.\n\n2 tablespoons unsalted butter\n\n1 tablespoon olive oil\n\n8 ounces carrots, cut into small dice\n\n1\u00bd pounds large yellow squash, halved lengthwise and then sliced into \u00bc-inch-thick half-moons\n\n2 tablespoons thinly sliced shallot\n\n1\u00bd teaspoons kosher salt\n\n\u00be teaspoon freshly ground white pepper\n\n1 tablespoon chopped fresh tarragon\n\n1. Set a 12-inch saut\u00e9 pan over medium-high heat, and once it is hot, add the butter and olive oil. When the butter has melted, add the carrots and cook, stirring often, for 2 minutes.\n\n2. Add the yellow squash, shallot, salt, and white pepper to the pan and continue to cook, stirring often, until the squash has wilted and released most of its moisture, about 6 minutes. Sprinkle with the tarragon and toss to blend. Serve hot.\n\n4 to 6 servings\n\nROASTED CARROTS WITH FRESH THYME\n\nPrep time: 5 minutes Cook time: 10 to 12 minutes Total: 15 to 17 minutes\n\nCarrots are an inexpensive, all-purpose root vegetable that are often overlooked. This delicious dish is a cinch to prepare and is easily jazzed up with any number of additions. Feel free to experiment by using different herbs, adding a handful of raisins, or drizzling with a splash of balsamic vinegar.\n\n2 tablespoons olive oil\n\n1\u00bd pounds carrots, cut diagonally into 1\u00bd-inch lengths\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n4 sprigs fresh thyme\n\n2 tablespoons unsalted butter\n\n1 tablespoon honey\n\n1. Preheat the oven to 450\u00b0F.\n\n2. Place a medium ovenproof saut\u00e9 pan over high heat and add the olive oil. When the oil is hot, add the carrots, salt, and pepper and cook for 2 minutes on each side. Add the thyme sprigs and butter, and drizzle with the honey.\n\n3. Transfer the pan to the oven and roast for 6 to 8 minutes, or until the carrots are golden and crisp-tender. Serve immediately.\n\n4 servings\n\nEMERIL'S SAUT\u00c9ED CUCUMBER WITH BASIL AND MINT\n\nPrep time: 5 minutes Cook time: 12 minutes Total: 17 minutes\n\nThis is a great way to enjoy cucumbers\u2014and one preparation that most folks don't think about when thinking cucumber. Delicate and buttery, these are especially wonderful alongside saut\u00e9ed fish.\n\n2 tablespoons butter\n\n2 tablespoons olive oil\n\n4 cups sliced peeled, seeded cucumber\n\n1 tablespoon chopped fresh basil\n\n1 tablespoon chopped fresh mint\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n1 tablespoon freshly squeezed lime juice\n\n1. Place a large saut\u00e9 pan over medium-high heat, and add 1 tablespoon of the butter and 1 tablespoon of the olive oil. When the mixture is hot and bubbly, add 2 cups of the cucumbers. Cook until the cucumbers are lightly browned on one side, 2 to 3 minutes. Toss the cucumbers and brown on the other side, another 2 minutes. Add half of the basil, half of the mint, \u00bd teaspoon of the salt, and \u00bc teaspoon of the pepper. Transfer the cucumbers to a serving dish, and repeat the process with the remaining ingredients.\n\n2. Drizzle the lime juice over the cucumbers, toss, and serve warm.\n\n4 servings\n\nGLAZED RADISHES\n\nPrep time: 5 minutes Cook time: 15 minutes Total: 20 minutes\n\nThe end result of this unique preparation reminds me of turnips...we forget that radishes and turnips are close cousins. The pearly pink color really adds a wow factor to the dining table, too. A perfect side next to roast chicken or a simple saut\u00e9ed fish fillet.\n\n1\u00bd pounds radishes, ends trimmed\n\n2\u00bd cups water\n\n\u00bc cup sugar\n\n2 tablespoons unsalted butter\n\n1 teaspoon salt\n\n\u00bd teaspoon finely ground white pepper\n\n4 teaspoons chopped fresh mint or tarragon\n\n1. Cut the radishes into lengthwise quarters. Meanwhile, have a 12-inch saut\u00e9 pan heating over medium heat.\n\n2. Add the radishes to the hot pan and raise the heat to high. Carefully add the water, sugar, butter, salt, and white pepper (the liquids may splatter). Bring to a boil and cook, stirring occasionally, until the radishes are easily pierced with a fork and most of the liquid has evaporated, about 15 minutes.\n\n3. Remove the radishes from the heat and sprinkle with the mint; toss to blend, and serve hot or warm.\n\n4 servings\n\nSeafood\n\nBROILED CATFISH WITH FRESH THYME, GARLIC, AND LEMON\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nThere are two types of catfish I prefer: one is from the Mississippi Delta, where my wife's family is from, and the other is from Des Allemands, Louisiana, the self-proclaimed \"catfish capital of the world.\" This is a very quick dish, and is always best when made with fresh catfish. However, if quality frozen catfish is available where you live, that'll work too.\n\nFour 6-to 8-ounce skinless catfish fillets\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground white pepper\n\n1 tablespoon minced garlic\n\n3 tablespoons extra-virgin olive oil\n\n1 teaspoon fresh thyme leaves\n\nJuice of 1 lemon\n\n1 tablespoon chopped fresh parsley (optional)\n\nLemon wedges, for serving (optional)\n\n1. Position a rack about 4 inches from the broiler element and preheat the broiler.\n\n2. Line a rimmed baking sheet with aluminum foil. Season the catfish fillets on both sides with the Essence, salt, and white pepper, and place the fillets on the prepared baking sheet.\n\n3. In a small bowl, combine the garlic, olive oil, and thyme. Using a small spoon, spread the garlic mixture evenly over the catfish. Then drizzle the fillets with the lemon juice.\n\n4. Transfer the baking sheet to the broiler. Cook the catfish for 6 minutes. Then rotate the baking sheet front to back, and continue cooking until the fish flakes easily when pressed with your fingers at the thickest part, about 4 minutes longer.\n\n5. Remove from the oven and serve immediately, sprinkled with the parsley and garnished with lemon wedges if desired.\n\n4 servings\n\nBROILED SALMON WITH A WARM TOMATO-LEMON VINAIGRETTE\n\nPrep time: 6 minutes Cook time: 9 minutes Total: 15 minutes\n\nMost people really enjoy salmon because of its \"steak-like\" texture and wonderful flavor. Wait until you try it with a tomato-lemon vinaigrette! Oh, baby, this is not only tasty but healthy and beautiful, too.\n\n\u00bc cup plus 1 tablespoon extra-virgin olive oil\n\n1 tablespoon fresh marjoram leaves\n\n\u00bd teaspoon finely grated lemon zest 2 cups cherry tomatoes, halved\n\n1\u00bd teaspoons salt\n\n\u00be teaspoon freshly ground black pepper\n\n\u00bc cup freshly squeezed lemon juice\n\nFour 6-ounce salmon fillets\n\n1. Position a rack 6 to 8 inches from the broiler element and preheat the broiler. Line a rimmed baking sheet with parchment paper.\n\n2. Heat the \u00bc cup olive oil in a 10-inch saut\u00e9 pan over high heat. When it is hot, add the marjoram and lemon zest and cook until fragrant, about 15 seconds. Add the tomatoes, \u00bd teaspoon of the salt, and \u00bc teaspoon of the pepper. Cook until the tomatoes begin to wilt, about 2 minutes. Stir in the lemon juice and mix well. Remove from the heat and set aside.\n\n3. Arrange the salmon fillets on the prepared baking sheet. Brush the fillets with the remaining 1 tablespoon olive oil and season with the remaining 1 teaspoon salt and \u00bd teaspoon pepper. Broil until the salmon is browned and cooked through, 5 to 6 minutes.\n\n4. Remove from the oven, and serve with the tomato-lemon vinaigrette spooned over the top.\n\n4 servings\n\nGAAAHLICKY SIZZLING SHRIMP\n\nPrep time: 10 minutes Cook time: 4 to 5 minutes Total: 14 to 15 minutes\n\nGarlic meets shrimp. This is really simple and equally delicious eaten over pasta, over steamed white rice or creamy grits, or over nothing at all, with crusty French bread for sopping up all the garlicky goodness.\n\n2 pounds medium shrimp, peeled and deveined\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd teaspoon salt\n\n2 tablespoons olive oil\n\n2 tablespoons unsalted butter\n\n2 tablespoons roughly chopped garlic\n\n6 tablespoons shrimp or chicken stock or canned, low-sodium shrimp or chicken broth\n\n2 tablespoons freshly squeezed lemon juice\n\n1 tablespoon chopped fresh parsley\n\n1. In a medium bowl, toss the shrimp with the Essence and the salt. Set aside.\n\n2. Heat a large cast-iron pan over medium-high heat, and add the olive oil. When the oil is hot, add the butter and garlic. Once the butter is nearly melted (about 20 seconds), add the shrimp and cook, stirring occasionally, until they are cooked through, about 3 minutes.\n\n3. Add the stock, lemon juice, and parsley, and cook for 30 seconds. Remove from the heat. Serve immediately.\n\n2 to 4 servings\n\nSOUTHERN-STYLE PAN-FRIED CATFISH\n\nPrep time: 12 minutes Cook time: 8 minutes Total: 20 minutes\n\nA quintessential Southern dish. If you're not from the South, here's your chance to transport your guests to a place they may never have been to. If you are a Southerner, you will arrive in a place that's nostalgic and new all at once. Delicious.\n\n\u00bd cup buttermilk\n\n2 tablespoons whole-grain mustard\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 tablespoon minced garlic\n\n2 teaspoons Louisiana hot sauce or other red hot sauce\n\nFour 6-to 8-ounce skinless catfish fillets\n\n1 cup all-purpose flour\n\n2\/3 cup cornmeal\n\n1 tablespoon salt\n\n2\/3 cup vegetable oil, for frying\n\nKicked-Up Tartar Sauce (recipe follows), for serving, optional\n\nLemon wedges, for serving\n\n1. Whisk together the buttermilk, mustard, Essence, garlic, and hot sauce in a small bowl. Place the fillets in a tray or baking dish that is just big enough to hold them. Pour the buttermilk mixture over the fish, making sure they are evenly coated. Set aside to marinate while you assemble the remaining ingredients.\n\n2. In a second tray or shallow baking dish, whisk together the flour, cornmeal, and salt.\n\n3. One at a time, remove the fillets from the buttermilk mixture, allowing any excess to drip off. Transfer the fish to the flour-cornmeal mixture and dredge to evenly coat. Place the breaded fish on a plate, and set aside.\n\n4. Heat the vegetable oil in a large saut\u00e9 pan over medium heat. When the oil is hot, carefully add 2 fillets to the pan, presentation side down, and cook until they are golden brown and crisp, about 4 minutes. Turn and cook until golden on the second side, 3 to 4 minutes. Using a fish spatula, transfer the fillets to a paper towel\u2013lined plate. Repeat with the remaining 2 fillets.\n\n5. Serve hot, with the Kicked-Up Tartar Sauce and lemon wedges, if desired, alongside.\n\n4 servings\n\nKicked-Up Tartar Sauce\n\nPrep time: 10 minutes Total: 10 minutes\n\nWe wouldn't just give you any ole tartar sauce recipe. Creole mustard, hot sauce, cayenne, and tarragon.... Quick to whip up and definitely worth the effort. Heinz who?\n\n1 cup mayonnaise\n\n\u00bc cup finely chopped cornichons, dill pickles, or dill pickle relish, drained\n\n2 tablespoons minced shallots\n\n2 tablespoons minced green onion tops\n\n1 tablespoon finely chopped drained nonpareil capers\n\n1 tablespoon chopped fresh parsley\n\n2 teaspoons Creole mustard or other spicy whole-grain mustard\n\n\u00bd teaspoon Louisiana hot sauce\n\n\u00bc teaspoon cayenne pepper\n\n\u00bc teaspoon dried tarragon, crushed between your fingers\n\nCombine all the ingredients in a small bowl and stir well to blend. Refrigerate until ready to serve. This will keep for 1 week.\n\nAbout 1\u00bd cups\n\nBLUE CORN\u2013CRUSTED RAINBOW TROUT WITH CILANTRO-LIME SOUR CREAM\n\nPrep time: 8 minutes Cook time: 6 minutes Total: 14 minutes\n\nYou'd better make extra batches of this dish\u2014my Culinary Team couldn't keep their hands off it when we tested it in our kitchen! Before I knew it, the fish was long gone and not even a crumb was left on the plate.\n\n\u00bd cup sour cream\n\n1 tablespoon chopped fresh cilantro\n\n1 tablespoon freshly squeezed lime juice\n\n\u00bc teaspoon cayenne pepper\n\n1\u00bc teaspoons salt\n\nGenerous \u00bd teaspoon plus 1\/8 teaspoon freshly ground black pepper\n\n1 cup blue cornmeal\n\n1 teaspoon ground coriander\n\n1 teaspoon ground cumin\n\nFour 6-ounce skinless rainbow trout fillets\n\n4 tablespoons vegetable oil\n\n2 tablespoons butter\n\nLime wedges, for serving\n\n1. Combine the sour cream, cilantro, lime juice, cayenne pepper, \u00bc teaspoon of the salt, and the 1\/8 teaspoon black pepper in a small bowl and stir to mix well. Set aside.\n\n2. Combine the blue cornmeal with the coriander and cumin in a shallow dish, and whisk to blend.\n\n3. Season the fish fillets on both sides with the remaining 1 teaspoon salt and generous \u00bd teaspoon black pepper. Then dredge them in the blue cornmeal mixture, shaking to remove any excess.\n\n4. Heat 2 tablespoons of the vegetable oil in a large nonstick saut\u00e9 pan over medium-high heat. When it is hot, add 1 tablespoon of the butter. When the butter has melted, add 2 fillets, underside down, and cook until the skin side is golden brown and crisp, 2 to 3 minutes. Flip the fillets over and cook briefly on the presentation side until the fish is just cooked through, 1 to 2 minutes. Remove from the pan and keep warm. Wipe the pan clean, add the remaining 2 tablespoons oil and 1 tablespoon butter, and repeat with the remaining fillets.\n\n5. Serve each fillet with a dollop of the cilantro-lime sour cream, and garnish with lime wedges.\n\n4 servings\n\nTROUT \u00c0 LA MEUNI\u00c8RE\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nThis is a classic New Orleans dish in which thin trout fillets are dredged in flour, then quickly saut\u00e9ed and finished with a simple lemony butter sauce. Any fresh trout from the gulf, the brook, or da bayou will do.\n\nFour 6-ounce skinless trout fillets\n\n1 teaspoon salt, plus more to taste\n\n1 teaspoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd cup Wondra flour (see Note)\n\n4 tablespoons olive oil\n\n8 tablespoons (1 stick) unsalted butter, cubed, at room temperature\n\n1 tablespoon minced shallot\n\n2 tablespoons dry white wine\n\n2 tablespoons freshly squeezed lemon juice\n\n\u00bd cup thinly sliced almonds\n\n2 tablespoons chopped fresh parsley\n\nFreshly ground white pepper\n\n1. Preheat the oven to 200\u00b0F.\n\n2. Season the trout fillets with the salt and the Essence. Lightly dredge the seasoned trout in the Wondra, shaking to remove any excess.\n\n3. Set a 10-inch saut\u00e9 pan over medium-high heat and add 2 tablespoons of the olive oil. Once the oil is hot, place 2 fish fillets in the pan and cook until golden, about 2 minutes per side. Place the cooked fillets on an ovenproof serving platter and keep warm in the oven while you cook the remaining fillets in the same manner with the remaining 2 tablespoons of olive oil.\n\n4. Once all the fillets are cooked, return the empty saut\u00e9 pan to the stovetop and reduce the heat to medium. Add the butter to the pan, and when it has melted, add the shallot. Cook for 30 seconds. Then add the white wine, lemon juice, almonds, and parsley. Continue to cook for 30 to 40 seconds, swirling the pan occasionally. Season the sauce with salt and white pepper to taste, and remove from the heat.\n\n5. Remove the platter from the oven, pour the sauce over the fish, and serve immediately.\n\nNote: Wondra is an instant flour most typically used for sauce because it dissolves quickly. We have found that its fine texture is perfect for a thin crisp coating on seared fish.\n\n4 servings\n\nCLASSIC MOULES MARINI\u00c8RE\n\nPrep time: 10 minutes Cook time: 8 minutes Total: 18 minutes\n\nA classic white wine\u2013based dish, with a touch of cream and lots of shallots and garlic. In the Proven\u00e7al region of France, they add tomatoes to this dish; feel free to do so if you like. You can enjoy this with either crusty bread or toasted slices of French bread, as in the Bruschetta recipe on 20 Minutes or Less. In French bistros, these mussels are traditionally served with pommes frites.\n\n3 tablespoons unsalted butter\n\n6 tablespoons chopped shallots\n\n1 tablespoon minced garlic\n\n2 sprigs fresh parsley, plus 2 tablespoons chopped fresh parsley for garnish\n\n2 sprigs fresh thyme 1 cup dry white wine\n\n\u00bc cup heavy cream\n\n\u00bd teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n4 pounds (about 4 dozen) live mussels, well scrubbed, rinsed, and debearded\n\nCrusty French bread, for serving\n\n1. In a large deep saut\u00e9 pan or a large wide saucepan, melt the butter over medium-high heat. Add the shallots, garlic, and herb sprigs and cook, stirring, until the shallots are soft and fragrant, about 1 minute. Add the wine, heavy cream, salt, and pepper, and bring to a boil.\n\n2. Add the mussels, cover the pan, and cook, shaking the pan occasionally, until the mussels have opened, 5 to 6 minutes.\n\n3. Remove the pan from the heat and discard any mussels that have not opened. Transfer the mussels and their liquid to a large, deep serving bowl and garnish with the chopped parsley. Serve immediately, with French bread for dipping.\n\n4 servings\n\nPoultry\n\nSTIR-FRIED CHICKEN WITH CASHEWS\n\nPrep time: 12 minutes Cook time: 6 minutes Total: 18 minutes\n\nHoisin sauce is a Chinese condiment that is easy to find in most supermarkets; its distinct flavor really makes this dish come alive. If you'd like, feel free to substitute an equal amount of shrimp or turkey for the chicken. This dish would be right at home served with the Aromatic Jasmine Rice on 20 Minutes or Less.\n\n\u00bd cup plus 2 teaspoons chicken stock or canned, low-sodium chicken broth\n\n\u00bc cup hoisin sauce\n\n2 tablespoons soy sauce\n\n3 tablespoons vegetable oil\n\n1\u00bc pounds boneless, skinless chicken breasts, cut crosswise into \u00bd-inch-thick even slices\n\n\u00bc cup finely chopped green onion bottoms, plus 2 tablespoons thinly sliced green onion tops\n\n2 teaspoons minced garlic\n\n\u00bd teaspoon crushed red pepper, or to taste\n\n1 large red bell pepper, stemmed, seeded, and julienned (see 20 Minutes or Less)\n\n1 teaspoon cornstarch\n\n\u00bd cup roasted cashews\n\nSalt and freshly ground black pepper\n\nCooked white rice, for serving\n\n1. In a small bowl, combine the \u00bd cup chicken broth, the hoisin sauce, and the soy sauce. Set this sauce aside.\n\n2. Heat the oil in a wok or a large saut\u00e9 pan over high heat. When the oil is hot, add the chicken and cook until it just turns opaque, 2 to 3 minutes. Add the green onion bottoms, garlic, and crushed red pepper and cook until fragrant, about 30 seconds. Add the red bell pepper and cook until just tender, about 1 minute.\n\n3. Add the sauce to the pan and mix well. In a small bowl, combine the cornstarch with the remaining 2 teaspoons chicken broth and stir well. Add the cornstarch mixture to the stir-fry and bring to a boil. Continue to simmer until the sauce begins to thicken, about 45 seconds.\n\n4. Remove from the heat and stir in the green onion tops and cashews. Season with salt and pepper to taste. Serve over cooked white rice.\n\nAbout 4 servings\n\nSAUT\u00c9ED CHICKEN BREASTS WITH DIJON HERB SAUCE\n\nPrep time: 6 minutes Cook time: 14 minutes Total: 20 minutes\n\nSimply described, this dish is a classic! Serve it with a green salad and hot, crusty, buttered bread. Don't make me say it: \"plate-lickin' good\"!\n\nFour 6-to 8-ounce boneless, skinless chicken breasts\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground white pepper\n\n2 tablespoons olive oil\n\n2 tablespoons chopped shallot\n\n1 cup dry white wine\n\n1 tablespoon Dijon mustard\n\n\u00bd cup heavy cream\n\n1 teaspoon chopped fresh tarragon\n\n1 teaspoon chopped fresh parsley\n\n1. Season the chicken on both sides with the salt and white pepper. Heat the olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Place the chicken in the pan, and cook until golden, about 4 minutes per side.\n\n2. Increase the heat to high, add the shallot and white wine, and cook for 4 minutes. Remove the chicken and set it aside on a serving plate.\n\n3. Whisk the mustard and heavy cream into the pan, and bring the sauce to a boil. Then reduce the heat to medium and simmer for 2 minutes, until thickened and bubbly.\n\n4. Add the tarragon and parsley, and remove from the heat. Spoon the sauce over the chicken, and serve immediately.\n\n4 servings\n\nCHICKEN SALAD WITH FRESH HERBS AND CELERY\n\nPrep time: 8 minutes Cook time: 12 minutes Total: 20 minutes\n\nThis delicious chicken salad is a breeze to put together, especially if you make the most of your time by prepping the dressing and chopped ingredients while the chicken is roasting in the oven. Feel free to make this up to two days in advance. Serve it in sandwiches or with greens for a light salad entr\u00e9e.\n\n2 pounds boneless, skinless chicken breasts\n\n2 teaspoons salt\n\n\u00bd teaspoon freshly ground black pepper\n\n2 tablespoons olive oil\n\n1 cup mayonnaise\n\n1 teaspoon minced garlic\n\n\u00bd teaspoon Dijon mustard\n\n\u00bd cup finely diced celery, plus \u00bc cup chopped celery leaves\n\n1\/3 cup finely chopped red onion\n\n3 tablespoons chopped mixed fresh herbs (such as parsley and tarragon)\n\n\u00bd teaspoon celery seeds\n\n\u00bc teaspoon cayenne pepper\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Rinse the chicken briefly under cool running water, then pat dry with paper towels. Season on both sides with the salt and pepper.\n\n3. Heat the olive oil in a 12-inch ovenproof saut\u00e9 pan over high heat. Add the chicken to the pan and cook for 2 minutes. Turn the chicken over and immediately place the pan in the oven. Roast for 10 minutes, or until the chicken reaches an internal temperature of 165\u00b0F when tested with an instant-read thermometer. Remove from the oven and set aside to cool.\n\n4. In a large mixing bowl, combine the mayonnaise, garlic, mustard, celery, celery leaves, red onion, herbs, celery seeds, and cayenne pepper. Mix well.\n\n5. When the chicken is cool enough to handle, cut it into \u00bd-inch dice, add it to the mayonnaise mixture, and mix well. Serve immediately, or transfer to the refrigerator to chill.\n\n4 to 6 servings\n\nMeat\n\nLAMB T-BONES WITH ROSEMARY-BALSAMIC BUTTER SAUCE\n\nPrep time: 6 minutes Cook time: 14 minutes Total: 20 minutes\n\nYou don't have to go to a restaurant to have great lamb chops. People assume that lamb is difficult to cook, or too fancy to cook at home, but wait until you see how easy these little T-bones are.\n\nEight 2-inch-thick lamb T-bone chops (about 2 pounds)\n\n\u00bc cup plus 2 teaspoons olive oil\n\n\u00bd cup balsamic vinegar\n\n2 teaspoons minced garlic\n\n1\u00bd tablespoons chopped fresh rosemary\n\n\u00bd teaspoon salt\n\n2 shallots\n\n\u00bd cup dry red wine\n\n6 tablespoons (\u00be stick) butter, cut into medium dice\n\n\u00bd teaspoon coarse sea salt\n\n\u00bc teaspoon freshly ground black pepper\n\n1. Place the lamb chops in a gallon-size resealable plastic bag. In a small bowl, combine the \u00bc cup olive oil, \u00bc cup of the balsamic vinegar, the garlic, \u00bd tablespoon of the rosemary, and the salt; stir together with a fork. Pour the marinade into the bag with the lamb chops, seal, and set aside at room temperature while you prepare the remaining ingredients.\n\n2. Mince the shallots (about 2 tablespoons), and place them in an 8-inch skillet. Add the red wine and remaining \u00bc cup balsamic vinegar, and bring to a boil over medium-high heat. Cook until the mixture has reduced to a syrupy consistency and the entire surface of the sauce is bubbly, 7 to 8 minutes. Reduce the heat to low and whisk in the butter in three separate additions, fully incorporating each addition before adding more; do not allow the sauce to boil. Add the remaining 1 tablespoon rosemary, remove from the heat, and set aside. (Keep the sauce warm, covered, until ready to serve but do not allow it to boil.)\n\n3. Remove the lamb chops from the marinade, lightly pat them dry with a paper towel, and set them on a plate. Season both sides of the chops with the sea salt and black pepper. Heat the remaining 2 teaspoons olive oil in a 10-inch saut\u00e9 pan over medium-high heat. When the oil is hot, add the chops and cook for 3 minutes. Reduce the heat to medium, turn the chops over, and cook for 3 minutes for medium-rare. Set the chops aside to rest briefly before serving.\n\n4. Serve the chops drizzled with the rosemary-balsamic butter sauce.\n\n4 servings\n\nSTEAK AU POIVRE\n\nPrep time: 5 minutes Cook time: 15 minutes Total: 20 minutes\n\nIn a classic steak au poivre, peppercorns are coarsely crushed by hand using the bottom of a skillet...but in this quick and easy version, the pepper is simply coarsely ground in a peppermill. The pepper coating on the outside of the steaks roasts in a hot, dry cast-iron skillet, drawing out the natural oils from the peppercorns and imparting a deep flavor. The cream and brandy finish the dish with just the right richness.\n\n\u00bc cup coarsely ground black pepper\n\nTwo 2-inch-thick rib-eye steaks (about 1 pound each)\n\n1\u00bd teaspoons kosher salt\n\n\u00bc cup brandy or cognac\n\n2 tablespoons Worcestershire sauce\n\n1 tablespoon freshly squeezed lemon juice\n\n\u00bd cup heavy cream\n\n2 tablespoons butter\n\n2 teaspoons chopped fresh parsley\n\n1. Sprinkle the pepper over both sides of the steaks, and gently press it into the meat with the heel of your hand.\n\n2. Sprinkle the salt over the bottom of a large cast-iron skillet. Heat the skillet over high heat. When the salt begins to brown, add the steaks. Lower the heat to medium-high and cook, without disturbing, for 5 minutes. Turn the steaks over and cook for 5 minutes more.\n\n3. Carefully add the brandy or cognac to the pan, taking care as it may ignite (allow the flames to burn off). Transfer the steaks to a platter to set aside to rest.\n\n4. Add the Worcestershire, lemon juice, and cream to the skillet and cook for 1 minute. Then whisk in the butter. Remove the pan from the heat, and stir in the parsley. Spoon the sauce over the steaks, and serve immediately.\n\nNotes: If you prefer a higher degree of doneness, preheat the oven to 400\u00b0F. After browning the steaks on both sides, transfer then to a baking sheet and place it in the oven to cook further while you finish the sauce in the skillet.\n\nFor thinner steaks: Change the cook time to 3 minutes per side for 1-inch rib-eyes.\n\nFor a lighter version of this sauce: substitute \u00bc cup beef broth for \u00bc cup of the cream.\n\n2 to 4 servings\n\nNEW YORK STRIP WITH BEURRE MA\u00ceTRE D'H\u00d4TEL\n\nPrep time: 5 minutes Inactive time: 5 minutes Cook time: 12 to 14 minutes Total: 22 to 24 minutes\n\nWhoa! You're pulling out all the stops here! Gorgeously seared New York strips with a slab of flavorful butter, on the table in fifteen minutes! Pour the wine and pass the salad.\n\n8 ounces (2 sticks) unsalted butter, at room temperature\n\n\u00bc cup minced fresh parsley\n\n3 teaspoons freshly squeezed lemon juice\n\n4\u00bd teaspoons salt\n\n2\u00bc teaspoons freshly ground black pepper\n\nFour 12-to 14-ounce boneless New York strip steaks, fat trimmed\n\n4 teaspoons olive oil\n\n1. Preheat the oven to 450\u00b0F.\n\n2. Place the butter in a medium bowl. Add the parsley, lemon juice, \u00bd teaspoon of the salt, and \u00bc teaspoon of the pepper, and stir until combined. Refrigerate while you cook the steaks.\n\n3. Heat a large cast-iron skillet over medium-high heat.\n\n4. Rub both sides of the steaks with the olive oil, and season evenly with the remaining 4 teaspoons salt and 2 teaspoons pepper. Place the steaks in the hot skillet and cook for 4 minutes on each side. Then transfer the skillet to the oven and roast to the desired degree of doneness, 4 to 6 minutes for medium-rare. An instant-read thermometer inserted into the thickest part of the meat should register 130\u00b0F for medium-rare, 140\u00b0F for medium.\n\n5. Remove the skillet from the oven; let the steaks stand for 5 minutes before serving.\n\n6. When ready to serve, top the steaks with spoonfuls of the flavored butter, to taste, or slice the steaks crosswise into 1\/3-inch-thick slices and serve with the butter. (Any unused butter can be stored in an airtight container in the refrigerator for up to 1 week.)\n\n4 to 6 servings\n\nLAMB CHOPS WITH MUSTARD HERB CRUST\n\nPrep time: 9 minutes Cook time: 11 minutes Total: 20 minutes\n\nSear. Slather. Coat. Roast. Eat. Impress your friends with this one.\n\n\u00bd cup Dijon mustard\n\n2 tablespoons minced garlic\n\n1 cup unseasoned dry breadcrumbs\n\n\u00bc cup finely grated Parmigiano-Reggiano cheese\n\n\u00bc cup chopped fresh rosemary\n\n2 teaspoons dried Italian herbs\n\n2 tablespoons vegetable oil\n\nSixteen 2-to 3-ounce baby lamb chops\n\n2 teaspoons salt\n\n1 teaspoon freshly ground black pepper\n\n1. Preheat the broiler, and line a large baking sheet with aluminum foil.\n\n2. Combine the mustard and garlic in a small mixing bowl, and set aside. Combine the breadcrumbs, cheese, rosemary, and dried herbs in a shallow dish; whisk to mix well, and set aside.\n\n3. Heat the vegetable oil in a large saut\u00e9 pan over medium-high heat. Using a paper towel, pat the lamb chops dry. Season them with the salt and pepper. Add the lamb chops to the pan and sear on both sides until nicely browned and caramelized, about 2 minutes per side.\n\n4. Transfer the chops to a plate and using a basting brush, lightly coat them with the mustard mixture. Then dredge them in the breadcrumb mixture. As the chops are coated, transfer them to the prepared baking sheet. Place the baking sheet under the broiler, and cook until the chops develop a nice golden crust and reach an internal temperature of 145\u00b0F when tested with an instant-read thermometer, about 5 minutes.\n\n5. Remove the baking sheet from the oven and let the lamb chops rest for 5 minutes before serving.\n\n4 main-course servings\n\nMINUTE STEAKS TERIYAKI-STYLE\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nYou will not believe this stir-fry! Portion your meat, slice your onions, peppers, and carrots, cook it all over high heat, and finish with the sauce. Done.\n\n2 pounds top sirloin steak, \u00bc inch thick, cut into 6 or 8 portions\n\n\u00be cup soy sauce\n\n6 tablespoons rice wine vinegar\n\n\u00bc cup chopped green onions, white and green parts\n\n2 tablespoons sugar\n\n2 teaspoons chopped garlic\n\n1 teaspoon cornstarch\n\n\u00bd teaspoon crushed red pepper\n\n3 tablespoons vegetable oil\n\n1\u00bd cups thinly sliced onions\n\n1 medium bell pepper, thinly sliced (about 1 cup)\n\n1 medium carrot, halved lengthwise and thinly sliced (about 1 cup)\n\nSteamed white rice, for serving (optional)\n\n1. Place the sirloin in a resealable plastic bag, add \u00bd cup of the soy sauce, seal, and set aside for 10 minutes.\n\n2. While the steaks are marinating, make your sauce: In a small bowl, combine the remaining \u00bc cup soy sauce, vinegar, green onions, sugar, garlic, cornstarch, and crushed red pepper. Set aside.\n\n3. Remove the steaks from the marinade, lightly pat them dry, and set them aside on a plate.\n\n4. Heat 2 tablespoons of the vegetable oil in a 12-inch saut\u00e9 pan over medium-high heat. Add half of the steaks and cook until nicely browned, 1\u00bd minutes per side. Transfer them to a serving platter, and repeat with the remaining steaks.\n\n5. Add the remaining 1 tablespoon vegetable oil to the pan, and when it is hot, add the onions, bell pepper, and carrot. Cook, stirring occasionally, until crisp-tender, 3 minutes. Stir the sauce, add it to the pan, and cook for 1 minute longer. Spoon the vegetables and sauce over the steaks. Serve immediately, over steamed white rice if desired.\n\n4 to 6 servings\n\nBONELESS PORK CHOPS PARMIGIANA\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nEveryone knows how delicious Chicken Parmesan can be, but I decided to do a spin-off with pork. Your friends and family will be astonished at how quickly this dish comes together.\n\n2 pounds boneless thin-cut pork chops (about 8 small chops)\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bc cup all-purpose flour\n\n1 egg\n\n2 tablespoons milk\n\n1 cup fine dry unseasoned bread crumbs\n\n\u00bd cup finely grated Parmigiano-Reggiano cheese\n\n4 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd cup olive oil\n\n1 cup jarred marinara sauce, plus more (heated) for serving with pasta if desired\n\n2 cups grated mozzarella cheese\n\nCooked pasta, for serving (optional)\n\n1. Preheat the broiler, and line a large baking sheet with aluminum foil.\n\n2. Season the pork chops on both sides with the salt and pepper. Set three shallow pans side by side. Place the flour in one, the egg and milk in another, and the breadcrumbs and cheese in the third pan. Season the flour with 1\u00bd teaspoons of the Essence, the egg-milk mixture with 1\u00bd teaspoons of the Essence, and the breadcrumbs with 1 teaspoon of the Essence. Stir the flour to incorporate the Essence; beat the eggs, milk, and Essence to blend; and toss the breadcrumbs with the cheese and Essence to combine.\n\n3. Dredge the pork chops in the flour and shake to remove any excess. Working with one at a time, dip the pork chops in the egg wash to coat, then transfer them to the breadcrumb mixture and coat evenly, shaking to remove any excess.\n\n4. Set a 12-inch saut\u00e9 pan over medium-high heat and add the olive oil. When the oil is hot, place half of the breaded pork chops in the pan and cook until golden brown, 1\u00bd to 2 minutes per side. Transfer the browned pork chops to the prepared baking sheet. Repeat with the remaining pork chops.\n\n5. Spread 2 tablespoons of the marinara sauce over each of the pork chops, and top each with \u00bc cup of the grated mozzarella. Place the baking sheet under the broiler and cook for 2 to 2\u00bd minutes, until the cheese is bubbly and lightly browned in spots and the chops are just cooked through. Remove from the oven and, if desired, serve over cooked pasta with additional marinara sauce.\n\n8 cutlets, 4 to 6 servings\n\nSPICY PORK STIR-FRY WITH GREEN BEANS\n\nPrep time: 12 minutes Cook time: 8 minutes Total: 20 minutes\n\nNowadays you can easily find fresh, beautiful, washed packaged green beans. Grab 'em and go. Here is a quick and delicious way to serve them.\n\n3 tablespoons soy sauce\n\n\u00bc teaspoon freshly ground white pepper\n\n1 pound ground pork\n\n\u00bc cup chicken stock or canned, low-sodium chicken broth\n\n3\u00bd tablespoons hoisin sauce\n\n\u00bd teaspoon crushed red pepper\n\n1 tablespoon rice wine vinegar\n\n\u00bd teaspoon cornstarch\n\n\u00bc cup peanut oil\n\n12 ounces green beans, rinsed, ends trimmed, cut into 4-inch lengths\n\n3 tablespoons thinly sliced garlic\n\n1\u00bd teaspoons dark Asian sesame oil\n\nCooked white rice, for serving (optional)\n\n1. In a mixing bowl, combine 2 tablespoons of the soy sauce, the white pepper, and the ground pork. Mix well to combine, and then set aside.\n\n2. Make the sauce by combining the chicken stock, hoisin sauce, crushed red pepper, rice vinegar, cornstarch, and the remaining 1 tablespoon soy sauce in a bowl. Set aside.\n\n3. Heat a wok or saut\u00e9 pan over high heat until hot. Add the peanut oil, and when the oil is smoking, add the green beans and cook, stirring frequently, until they are slightly wrinkled, 3 to 5 minutes. Using a slotted spoon, transfer the beans to a paper towel\u2013lined plate, and set aside.\n\n4. Add the garlic to the wok and cook briefly until fragrant, about 10 seconds. Add the ground pork and stir-fry until it is no longer pink, about 1\u00bd minutes. Stir the sauce mixture, add it to the wok, and stir to combine. Bring the liquid to a boil and cook until it begins to thicken, about 45 seconds.\n\n5. Return the green beans to the wok and drizzle with the sesame oil. Cook briefly until warmed through. Then serve immediately, over hot rice if desired.\n\n4 servings\n\nMUSHROOM-SMOTHERED STEAKS\n\nPrep time: 10 minutes Cook time: 10 minutes Total: 20 minutes\n\nCube steaks, also referred to as minute steaks, are not cubes at all. They are called cube steaks because of the cubelike pattern that has been pounded into them with a meat tenderizer. These steaks can be cooked very quickly, will remain tender and juicy, and are very affordable...need I say more?\n\n4 cube steaks (about 1\u00bd pounds total)\n\n\u00be teaspoon salt\n\n\u00be teaspoon freshly ground black pepper\n\n4 tablespoons olive oil\n\n1 pound mushrooms, wiped clean, stemmed, and sliced (about 4 cups)\n\n\u00bd cup chopped green onions, white and green parts, plus more for garnish\n\n1\u00bd tablespoons minced garlic\n\n\u00bd cup dry red wine\n\n3 tablespoons butter, cut into 4 pieces\n\n1. Season the steaks on both sides with \u00bd teaspoon of the salt and \u00bd teaspoon of the black pepper.\n\n2. Heat 2 tablespoons of the olive oil in a 14-inch saut\u00e9 pan over medium-high heat. Add the steaks and cook until nicely browned on one side, 2 minutes. Remove from the pan and set aside.\n\n3. Add the remaining 2 tablespoons olive oil to the pan. Then add the mushrooms, green onions, and garlic and saut\u00e9 until browned, about 2 minutes. Move the mushrooms to the edge of the pan and return the steaks, browned sides up, along with any accumulated juices, to the pan. Add the wine and cook for 3 minutes. Add the remaining \u00bc teaspoon salt and \u00bc teaspoon black pepper. Dot each steak with a pat of butter and cook for 1 minute longer.\n\n4. Transfer the steaks to a serving platter or individual plates, and spoon the mushrooms and their juices over them. Sprinkle with chopped green onions and serve immediately.\n\n4 servings\n\nDesserts\n\nEMERIL'S LATE-NIGHT PARFAITS\n\nPrep time: 7 minutes (including 5 minutes inactive) Total: 7 minutes\n\nThe great thing about these parfaits is that they can be assembled and ready in no time. Don't worry if you don't have the exact ingredients called for; any type of cookie and ice cream you have on hand can work wonderfully together.\n\n1 pint vanilla ice cream\n\n\u00bd cup crumbled biscotti cookies or other cookie crumbs of choice\n\n\u00bc cup Frangelico or other nut-flavored liqueur\n\n2 tablespoons roughly chopped hazelnuts or walnuts, lightly toasted\n\n1. Remove the ice cream from the freezer and let it soften slightly, about 5 minutes at room temperature.\n\n2. Into the bottom of four parfait or ice cream dishes, scoop about 2 tablespoons of the vanilla ice cream. Top with about \u00bd tablespoon of the cookie crumbs, and drizzle with 1\u00bd teaspoons of the liqueur. Continue layering the ingredients, ending with liqueur on top.\n\n3. Serve immediately, garnished with the chopped nuts.\n\n4 servings\n\nBROWN SUGAR\u2013BAKED BANANAS\n\nPrep time: 10 minutes Cook time: 8 minutes Total: 18 minutes\n\nIn New Orleans we have a classic dessert known as Bananas Foster. This simplified oven-baked version is every bit as good, but oh, what a walk in the park to prepare.\n\n2 tablespoons unsalted butter\n\n3 bananas, peeled, sliced in half lengthwise and crosswise\n\n1\/3 cup packed light brown sugar\n\n\u00bd teaspoon ground cinnamon\n\n\u00bc teaspoon freshly grated nutmeg\n\nJuice of 1 orange\n\n\u00bc teaspoon grated orange zest\n\nLight rum or banana liqueur, for drizzling (optional)\n\nVanilla ice cream, for serving\n\n1. Position a rack about 8 inches from the broiler element and preheat the broiler to high.\n\n2. Butter the bottom of a flameproof 11-by 7-inch baking dish with 1 tablespoon of the butter, and arrange the banana slices in it in one even layer, cut sides down. Set aside.\n\n3. In a small saucepan over medium-high heat, combine the remaining 1 tablespoon butter with the brown sugar, cinnamon, nutmeg, and orange juice. Cook, stirring, until the sugar begins to melt, 1 to 2 minutes. Remove from the heat and stir in the orange zest. Pour the sauce evenly over the bananas.\n\n4. Place the baking dish under the broiler and cook, turning the bananas over midway through, until the sauce is slightly thickened and bubbly and the bananas are tender, about 6 minutes total.\n\n5. Remove the baking dish from the broiler, and if desired, drizzle with rum. Set aside to cool briefly before serving.\n\n6. Serve the bananas over vanilla ice cream, with some of the sauce drizzled over the top.\n\n4 servings\n\nCANDIED HOT FUDGE SUNDAES\n\nPrep time: 15 minutes Cook time: 5 minutes Total: 20 minutes\n\nDecadent and delightful...\n\n\u00be cup heavy cream\n\n\u00bc cup light corn syrup\n\n\u00bd cup semisweet chocolate chips\n\n2 tablespoons unsalted butter\n\n\u00bd teaspoon vanilla extract\n\n3 ounces (about \u00be cup) of your favorite chocolate\u2013peanut butter candy bar (such as Reese's Peanut Butter Cups or Snickers bars), coarsely chopped\n\n2 large ripe bananas, peeled and sliced Vanilla ice cream, for serving\n\n1\u00bd cups lightly sweetened whipped cream, for serving (optional)\n\nCrushed roasted salted peanuts, for garnish\n\n6 maraschino cherries, for garnish\n\n1. Carefully heat the heavy cream and corn syrup in a small saucepan. When the mixture is hot, remove the pan from the heat, add the chocolate chips, and let sit undisturbed for about 3 minutes.\n\n2. Whisk the cream mixture until smooth, and return the saucepan to low heat. Warm, stirring the sauce frequently, until it is heated through, 1 to 2 minutes. Add the butter in pieces, stirring until incorporated and smooth. Remove from the heat and let sit for 1 minute to cool. Then stir in the vanilla extract and chocolate candy bar pieces.\n\n3. Divide the sliced bananas equally among six sundae glasses or ice cream dishes. Spoon the chocolate sauce over the bananas, then top with scoops of ice cream. Place a dollop of whipped cream over the top, if desired, and then sprinkle with roasted peanuts. Top each sundae with a cherry and serve immediately.\n\n6 servings\n\nPEANUT BUTTER\u2013CHOCOLATE CHIP COOKIES\n\nPrep time: 5 minutes Cook time: 10 minutes Total: 15 minutes\n\nThese are by far the easiest and best-tasting peanut butter cookies you will ever make. A perfect recipe for kids\u2014no fuss, no muss.\n\n1 cup creamy peanut butter\n\n\u00bd cup granulated sugar\n\n\u00bd cup packed light brown sugar\n\n\u00bd cup semisweet chocolate chips\n\n1 large egg, beaten\n\n1 teaspoon vanilla extract\n\n1. Position two oven racks in the center of the oven and preheat the oven to 350\u00b0F.\n\n2. Combine all the ingredients in a bowl, and stir with a wooden spoon until smooth.\n\n3. Divide the dough into 24 portions, about 1 heaping tablespoon each. Roll each portion between your hands to form a smooth ball. Place the balls of dough on ungreased cookie sheets, spacing them 1 inch apart. You should get about 12 cookies per sheet. Using a fork, press on the dough in two directions to form a crosshatch pattern.\n\n4. Bake the cookies, rotating the sheets between oven racks and turning them back to front midway, until the cookies are puffed and lightly golden, about 10 minutes. Remove the baking sheets from the oven and let the cookies cool on the sheets. Then remove them with a metal spatula.\n\nAbout 24 cookies\n\nMELON WITH AMARETTI COOKIE CRUMBLES\n\nPrep time: 10 minutes Inactive time: 5 to 10 minutes Total: 15 to 20 minutes\n\nThis recipe relies on the ripeness of your fruit and is best prepared when melons are in season. Tip: If the melons you purchase are not ripe when you buy them, place them in a brown paper bag, add a banana, seal the bag, and set aside for a day or two.\n\n1 ripe honeydew melon, halved, seeded, fruit scooped into 1-inch balls with a melon baller\n\n1 ripe cantaloupe, halved, seeded, fruit scooped into 1-inch balls with a melon baller\n\n\u00bd pint fresh strawberries, hulled and quartered\n\n\u00bc cup sugar, or more to taste (this will depend on the sweetness of the fruit)\n\n3 tablespoons amaretto liqueur\n\n\u00bd cup crumbled amaretti cookies, shortbread, or vanilla wafer cookies\n\n1. Combine the honeydew, cantaloupe, strawberries, sugar, and amaretto in a large mixing bowl, and toss gently but thoroughly to combine. Cover with plastic wrap and let sit for 5 to 10 minutes before serving.\n\n2. Divide the fruit among six to eight small dessert bowls, and sprinkle the crumbled cookies evenly over the top of each dessert. Serve immediately.\n\n6 to 8 servings\n\nFRESH BERRIES WITH BALSAMIC DRIZZLE AND ALMOND CREAM\n\nPrep time: 10 minutes Cook time: 5 minutes Total: 15 minutes\n\nBalsamic vinegar and fresh fruit is classic Italian fare. Here, we reduce the balsamic vinegar to bring out its sweetness, then pair it with fresh berries and a simple cream cheese blend for one knockout combination.\n\n\u00bd cup balsamic vinegar\n\n1\u00bd cups sour cream\n\n\u00bd cup plus 2 tablespoons cream cheese, at room temperature\n\n\u00bc cup confectioners' sugar\n\n\u00bd teaspoon almond extract\n\n1 pound fresh strawberries, hulled and sliced or quartered\n\n1 cup fresh raspberries\n\n1 cup fresh blackberries or blueberries\n\n3 tablespoons chopped almonds, lightly toasted\n\n1. Pour the balsamic vinegar into a small saucepan and bring to a boil. Cook until reduced to \u00bc cup (about half the original volume) and syrupy, about 3 minutes. Transfer to a small bowl and set aside to cool while you assemble the desserts.\n\n2. In a mixing bowl, combine the sour cream, cream cheese, confectioners' sugar, and almond extract. Whisk until the sugar has dissolved and the mixture is very smooth.\n\n3. When ready to serve the dessert, divide the berries evenly among six dessert bowls. Place a generous dollop of the sour cream mixture over the berries, and then drizzle with some of the balsamic drizzle. Garnish with the almonds, and serve immediately.\n\n6 servings\n\nFLAMB\u00c9ED STRAWBERRY SAUCE FOR ANGEL FOOD CAKE OR ICE CREAM\n\nPrep time: 10 minutes Cook time: 3 minutes Total: 13 minutes\n\nAiry angel food cake is a perfect vehicle for this orange-scented strawberry sauce. There is a simple elegance to this dessert, and it is practically guilt-free. Serve the sauce over ice cream if you're feeling more indulgent.\n\n2 tablespoons unsalted butter\n\n1 pound fresh strawberries, hulled and quartered\n\n\u00bc cup sugar\n\n2 tablespoons brandy\n\n2 tablespoons orange-flavored liqueur (such as Triple Sec, Cointreau, or Grand Marnier)\n\n4 cups store-bought angel food cake, cut into 1-inch cubes, or 2 pints vanilla ice cream\n\n1 cup lightly sweetened whipped cream, for serving (optional)\n\nFresh mint leaves, for garnish\n\n1. Set a 12-inch saut\u00e9 pan over medium-high heat and add the butter. Once the butter has melted, add the strawberries and sugar and saut\u00e9, stirring often, for 1\u00bd to 2 minutes. Remove the pan from the heat and add the brandy and liqueur. Carefully tilt the pan toward the open flame to ignite the liquor. Once lit, swirl the pan until the flames die down, 30 to 45 seconds. Alternatively, if using an electric stove, simply reduce the sauce over high heat for 45 seconds instead of flaming it. (If serving over ice cream, you may wish to allow the sauce to cool slightly before serving.)\n\n2. Divide the cake or ice cream evenly among four small dessert bowls. Spoon the warm strawberries over the top, and garnish with the whipped cream, if desired, and mint leaves.\n\n4 servings\n\n## 40 Minutes OR LESS\n\nSOUPS\n\nBroccoli and Cheese Soup\n\nQuick Red Bean Soup\n\nChicken and Rice Soup\n\nCarrot Ginger Soup\n\nSpicy Smoked Sausage, Tomato, and Mushroom Soup\n\nHot and Sour Soup\n\nPotato and Leek Soup\n\nGarden Vegetable Soup\n\nPotato and Turkey Hot Dog Soup with Herbs\n\nCream of Tomato Soup\n\nSTARTERS\n\nShrimp and Zucchini Fritters with Roasted Red Pepper Mayo\n\nCreamy Shrimp and Green Onion Dip\n\nSALADS\n\nSeared Shrimp Salad\n\nEmeril's Noodle Salad\n\nGarden Vegetable Salad\n\nSimple Croutons\n\nChickpea Salad with Tabbouleh\n\nSANDWICHES\n\nChicken Queso Burgers\n\nChili-Rubbed Shrimp Wraps\n\nSpicy Pork Wraps with Creamy Coleslaw\n\nChicken Patty Pockets with Minted Yogurt Sauce\n\nOven-Crispy French Fries with Paprika-Parmesan Salt\n\nPASTA\n\nOrzo \"Risotto\" with Tomato, Mozzarella, and Basil\n\nShiitakes and Bacon with Penne\n\nPenne with Sausage and Escarole\n\nSpaghetti with Caramelized Onions and Anchovies\n\nThree-Cheese Baked Macaroni\n\nShrimp and Linguine Fra Diavolo\n\nPenne alla Puttanesca\n\nBeef Stroganoff with Egg Noodles\n\nPasta Primavera\n\nRICE AND BEANS\n\nGreen Onion Rice Pilaf\n\nBasic Risotto\n\nBlack Bean Cakes\n\nTurkey and Pinto Bean Tostadas\n\nCreamy White Beans with Sausage\n\nVEGETABLES\n\nBacon Braised Green Beans\n\nCreamed Mustard Greens\n\nSaut\u00e9ed Mushrooms with Fresh Thyme\n\nSesame Eggplant\n\nSpicy Braised Greens\n\nButtermilk Mashed Potatoes\n\nSEAFOOD\n\nIndian-Inspired Shrimp with Coconut, Chiles, and Tomatoes\n\nFish en Papillote\n\nSwordfish with Puttanesca Relish\n\nRoasted Scrod with Herbed Breadcrumbs\n\nShrimp and Feta, Greek-Style\n\nSalmon with Orange Butter Sauce\n\nBaked Flounder with Carrots, Spinach, and an Asian Vinaigrette\n\nPOULTRY\n\nBoursin Cheese, Spinach, and Pecan-Stuffed Chicken Breasts\n\nHoney-Lemon-Thyme Cornish Game Hens\n\nOven-Roasted Chicken Wings\n\nCrispy Pan-Roasted Chicken with Garlic-Thyme Butter\n\nChicken Cordon Bleu\n\nTurkey Saltimbocca\n\nPanko-Crusted Chicken Tenders\n\nMEAT\n\nSloppy Joes\n\nCountry-Fried Steak with White Gravy\n\nSausages and Sauerkraut\n\nStir-Fried Beef and Broccoli\n\nQuick and Easy Lamb Kebabs\n\nThin-Cut Pork Chops with Rosemary-Balsamic Glazed Shallots\n\nDESSERTS\n\nKicked-Up Snickerdoodles\n\nSkillet Corn Cake with Stewed Cherries\n\nSoups\n\nBROCCOLI AND CHEESE SOUP\n\nPrep time: 20 minutes Cook time: 15 minutes Total: 35 minutes\n\nThis classic combination of flavors lends itself well to this simple, creamy soup. By cooking the broccoli just right, the soup retains a vibrant bright green color. Serve this as a starter to any meal or with a sandwich or salad for a complete meal. So good for you, too.\n\n2 tablespoons olive oil\n\n1\u00bd cups thinly sliced yellow onions\n\n1 tablespoon sliced garlic\n\n1 teaspoon salt\n\n\u00bc teaspoon cayenne pepper\n\n5 cups chicken stock or canned, low-sodium chicken broth\n\n4 cups broccoli florets\n\n1\u00bd cups (6 ounces) shredded medium sharp cheddar cheese\n\nSimple Croutons (40 Minutes or Less), for garnish (optional)\n\n1. Heat the olive oil in a 6-quart stockpot over medium heat. When it is hot, add the onions, garlic, salt, and cayenne pepper. Saut\u00e9 until the onions are soft and translucent, 4 to 5 minutes.\n\n2. Add the chicken stock and bring to a boil. Once the stock is boiling, add the broccoli and cook until fork-tender, about 5 minutes.\n\n3. Remove the soup from the heat and let it cool slightly. Then puree the soup, in batches, in a blender, adding the cheese in three additions while blending (see Note). Adjust the seasoning if necessary, garnish with croutons if desired, and serve hot.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.\n\n1\u00bd quarts, 4 to 6 servings\n\nQUICK RED BEAN SOUP\n\nPrep time: 10 minutes Cook time: 26 minutes Total: 36 minutes\n\nThis soup is inspired by one of my favorite New Orleans classics: red beans and rice. You can easily serve a bowl of this soup garnished with some cooked white rice for a heartier meal. Oh, yeah, baby, let the New Orleans in me come out!\n\n1 tablespoon vegetable oil\n\n6 ounces (1 cup) smoked ham or smoked sausage, finely chopped\n\n1\u00bd cups chopped yellow onions\n\n\u00bd cup finely chopped celery\n\n\u00bd cup finely chopped green bell pepper\n\n2 bay leaves\n\n\u00bd teaspoon cayenne pepper\n\n2 tablespoons minced garlic\n\nFour 15.5-ounce cans red beans, drained\n\n6 cups chicken stock or canned, low-sodium chicken broth\n\n2 tablespoons chopped green onions, green and white parts, for garnish\n\n1. Heat the vegetable oil in a Dutch oven or large pot over high heat. Add the ham, onions, celery, bell pepper, bay leaves, and cayenne, and cook until the vegetables are lightly caramelized and very tender, about 6 minutes. Add the garlic and cook, stirring, for about 30 seconds. Add the red beans and chicken stock, and bring to a boil. Reduce the heat and simmer uncovered, stirring occasionally, for 10 minutes.\n\n2. Using a potato masher, mash some of the beans slightly to thicken the broth. Continue cooking for another 10 minutes, or until the soup has thickened and the flavors have come together.\n\n3. Remove from the heat and discard the bay leaves. Stir in the green onions, and adjust the seasoning if necessary. Serve hot.\n\n2 generous quarts, about 8 servings\n\nCHICKEN AND RICE SOUP\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nI think most of us are reminded of our childhoods when we think of chicken and rice soup. It really can hit the spot sometimes\u2014so simple and yet so delicious. Keep in mind that if the soup is made in advance, the rice will continue to soak up the broth as it sits. Feel free to add more broth as you like.\n\n2 pounds boneless, skinless chicken breasts, cut into \u00be-inch dice\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 tablespoon olive oil\n\n2 cups diced onions (small dice)\n\n1\u00bd cups diced carrots (small dice)\n\n1\u00bd cups diced celery (small dice)\n\n1 tablespoon minced garlic\n\n1 teaspoon dried basil\n\n1 teaspoon salt\n\n\u00bc teaspoon crushed red pepper\n\n2 quarts chicken stock or canned, low-sodium chicken broth, plus more if needed\n\n\u00bd cup uncooked long-grain white rice (see Note)\n\nOne 5-ounce bag prewashed spinach\n\n1. Place the chicken in a medium bowl, season with the Essence, and set aside.\n\n2. Heat the olive oil in a 6-quart (or larger) soup pot over medium-high heat. Add the onions, carrots, and celery and cook, stirring occasionally, until the onions are translucent, about 5 minutes.\n\n3. Add the garlic, basil, salt, and crushed red pepper, and continue to cook for 1 minute. Add the chicken and cook for 3 minutes. Add the broth and the rice, cover the pot, and bring to a boil over high heat. Remove the cover, reduce the heat to a simmer, and cook until the rice is just tender, about 12 minutes.\n\n4. Stir in the spinach, and serve immediately.\n\nNote: If you have cooked white rice on hand, omit the uncooked rice and simply stir in about 1\u00bd cups cooked rice just before you add the spinach.\n\n3\u00bd quarts, 6 to 8 servings\n\nCARROT GINGER SOUP\n\nPrep time: 8 minutes Cook time: 22 minutes Total: 30 minutes\n\nThe ginger in this soup gives it a nice little kick. Though we suggest serving it hot, it can also be nice ice-cold on a hot summer day.\n\n4 tablespoons (\u00bd stick) butter\n\n2 pounds carrots, cut into large dice (about 4 cups)\n\n2 cups diced onions (medium dice)\n\n\u00bc cup (about 2 ounces) peeled and sliced fresh ginger\n\n6 sprigs fresh thyme, tied in a bundle with kitchen twine\n\n2 teaspoons salt\n\n\u00be teaspoon freshly ground white pepper\n\n6 cups water\n\nSour cream, for garnish (optional)\n\n1. Melt the butter in a 6-quart (or larger) soup pot over high heat. Add the carrots, onions, ginger, thyme bundle, salt, and white pepper, and cook for 2 minutes. Then add the water, cover the pot, and bring to a boil. Remove the cover, reduce the heat to medium-low, and simmer for 15 minutes.\n\n2. Remove the pot from the heat, and remove the thyme bundle. Blend the soup until it is completely smooth, using an immersion blender or in three batches in a blender (see Note).\n\n3. Transfer the pureed soup to a 4-quart pot or other serving dish. Stir to combine, and adjust the seasoning to taste. Serve hot, garnished with a dollop of sour cream if desired.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.\n\n2\u00bd quarts, about 6 servings\n\nSPICY SMOKED SAUSAGE, TOMATO, AND MUSHROOM SOUP\n\nPrep time: 10 minutes Cook time: 30 minutes Total: 40 minutes\n\nMake sure to use a good-quality sausage here. My favorite would be chorizo, but it is also wonderful with andouille or other spicy smoked pork sausage. This soup is thick and hearty\u2014a \"manly-man\" soup, if you will.\n\n1 tablespoon olive oil\n\n1 pound firm (smoked) chorizo or other spicy smoked sausage, diced or crumbled into \u00bd-inch pieces\n\n8 ounces button mushrooms, wiped clean and quartered, or diced if very large\n\n1\u00bd cups diced onions\n\n\u00bd cup diced red bell pepper\n\n\u00bd cup diced green bell pepper\n\n2 tablespoons thinly sliced garlic\n\nTwo 28-ounce cans whole tomatoes, roughly chopped, with juices\n\n4 cups chicken stock or canned, low-sodium chicken broth\n\n\u00bc cup coarsely chopped fresh soft herbs (such as marjoram and\/or basil)\n\n1\u00bd teaspoons salt\n\n\u00be teaspoon crushed red pepper\n\n1. Heat the oil in a large nonreactive saucepan or Dutch oven over medium-high heat. When it is hot, add the sausage and cook, stirring occasionally, until it is browned around the edges, about 4 minutes.\n\n2. Add the mushrooms, onions, bell peppers, and garlic to the pan and cook, stirring occasionally, until the vegetables are soft and lightly browned, 7 to 8 minutes.\n\n3. Add the tomatoes and their juices, chicken stock, herbs, salt, and crushed red pepper and bring to a boil. Reduce the heat to a simmer, and cook until the flavors have married, about 15 minutes. Serve hot.\n\nAbout 3 quarts, 8 to 10 servings\n\nHOT AND SOUR SOUP\n\nPrep time: 15 minutes Cook time: 9 minutes Total: 24 minutes\n\nOh, yeah, babe. Cure your cold with this one! This is a perfect balance between spicy and sour. It's an Asian classic\u2014and it doesn't get much simpler than this. If you're a vegetarian, feel free to substitute veggie stock and cubes of tofu for the chicken broth and chicken strips. For less heat, reduce the red pepper.\n\n8 cups chicken stock or canned, low-sodium chicken broth\n\n6 ounces thinly sliced mushrooms (such as shiitake or button)\n\n\u00bc cup soy sauce\n\n\u00bc cup minced fresh ginger\n\n2 tablespoons minced garlic\n\n\u00be teaspoon crushed red pepper\n\n3 tablespoons cornstarch\n\n\u00bc cup plus 3 tablespoons freshly squeezed lime juice\n\n8 ounces skinless, boneless chicken breast, cut into thin strips\n\n1 teaspoon dark Asian sesame oil\n\n2 tablespoons thinly sliced green onion tops\n\n1. Combine the stock, mushrooms, soy sauce, ginger, garlic, and crushed red pepper in a 4-quart pot. Cover and bring to a boil. Remove the cover, reduce the heat to a simmer, and cook until the mushrooms are tender, 7 to 8 minutes.\n\n2. Whisk the cornstarch and the lime juice together in a small bowl. Add the cornstarch mixture and the chicken to the soup. Bring to a boil, and cook until the soup thickens, about 1 minute.\n\n3. Stir in the sesame oil and sliced green onions, and serve hot.\n\nAbout 2\u00bd quarts, 4 to 6 servings\n\nPOTATO AND LEEK SOUP\n\nPrep time: 10 minutes Cook time: 30 minutes Total: 40 minutes\n\nThe trick to this soup lies in not overcooking the potatoes. Cook them until they are just tender, then quickly puree them to make sure that they don't become overly starchy. Though I just love this soup served hot on a cool, crisp day, you could also serve it chilled\u2014an especially nice option for make-ahead meals.\n\n1 large or 2 small leeks (about 1 pound)\n\n2 bay leaves\n\n20 black peppercorns\n\n4 sprigs fresh thyme\n\n2 tablespoons butter\n\n2 slices bacon, diced\n\n\u00bd cup dry white wine\n\n5 cups chicken stock or canned, low-sodium chicken broth\n\n1 to 1\u00bc pounds russet potatoes, peeled and diced into 1-inch cubes\n\n1\u00bd teaspoons salt\n\n\u00be teaspoon freshly ground white pepper\n\n\u00bd to \u00be cup cr\u00e8me fra\u00eeche or heavy cream\n\n2 tablespoons snipped fresh chives\n\n1. Trim the green portions of the leek, and using 2 of the largest and longest leaves, make a bouquet garni by folding the 2 leaves around the bay leaves, peppercorns, and thyme. Tie into a package-shaped bundle with kitchen twine, and set aside. (Alternatively, tie the 2 leek leaves, bay leaves, peppercorns, and thyme together in a piece of cheesecloth.)\n\n2. Using a sharp knife, halve the white part of the leek lengthwise. Rinse the leek well under cold running water to rid it of any sand. Slice thinly crosswise and set aside.\n\n3. Melt the butter in a large soup pot over medium heat, and add the bacon. Cook, stirring occasionally, until the bacon is soft and has rendered most of its fat, about 5 minutes. Add the chopped leeks and cook until wilted, about 5 minutes. Add the wine and bring to a boil. Add the reserved bouquet garni and the chicken stock, potatoes, salt, and white pepper. Bring to a boil. Then reduce the heat to a simmer and cook for about 20 minutes, or until the potatoes are tender and the soup is very flavorful.\n\n4. Remove the bouquet garni and puree the soup using an immersion blender or in batches in a blender (see Note). Stir in the cr\u00e8me fra\u00eeche, and adjust the seasoning if necessary. Serve immediately, with some of the snipped chives sprinkled over the top of each bowl of soup.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.\n\nAbout 1\u00bd quarts, 4 to 6 servings\n\nGARDEN VEGETABLE SOUP\n\nPrep time: 12 minutes Cook time: 23 minutes Total: 35 minutes\n\nDon't be afraid to make this straightforward, veggie-packed soup your own by using the vegetables you especially like or you have on hand. For instance, replace the zucchini and yellow squash with frozen green peas and frozen corn. If you're a tomato lover, two cups of chopped canned tomatoes can be substituted for two cups of the broth. Also, you could gild the lily by adding three cups of cooked macaroni or other small pasta, or a bit of cooked rice, right before serving. The sky's the limit here.\n\n6 sprigs parsley\n\n2 bay leaves\n\n2 tablespoons olive oil or butter\n\n2 cups diced onions\n\n1\u00bd cups diced carrots\n\n1\u00bd cups diced celery (small dice, with or without leaves)\n\n2 tablespoons minced garlic\n\n8 ounces button mushrooms, wiped clean, stemmed, and quartered (about 2 cups)\n\n4 quarts beef, chicken, or vegetable stock or canned, low-sodium beef, chicken, or vegetable broth\n\n1\u00bd teaspoons salt\n\n\u00be teaspoon freshly ground black pepper\n\n2 cups broccoli or cauliflower florets, cut into bite-size pieces\n\n1 cup diced zucchini (large dice)\n\n1 cup diced yellow squash (large dice)\n\nOne 10-ounce bag prewashed spinach\n\n\u00bd cup grated Parmigiano-Reggiano cheese (optional)\n\n1. Tie the parsley sprigs and bay leaves together with a piece of kitchen twine. Set aside.\n\n2. Heat the olive oil in a 6-quart (or larger) soup pot over high heat. Add the onions, carrots, celery, and the parsley bundle, and cook for 2 minutes, stirring frequently. Add the garlic and mushrooms, and cook for 3 minutes. Add the broth, salt, and pepper. Cover the pot and bring to a boil. Remove the cover, reduce the heat to medium-low, and simmer for 12 minutes.\n\n3. Add the broccoli, zucchini, yellow squash, and spinach to the soup. Simmer for 5 minutes.\n\n4. Remove the parsley bundle and serve the soup hot, garnished with the grated cheese if desired.\n\n4 quarts, 6 to 8 servings\n\nPOTATO AND TURKEY HOT DOG SOUP WITH HERBS\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nMy mom, Hilda, used to make this soup for me, my brother Mark, and my sister Dolores when we were growing up, and we ate it up like no one's business. Here I've subbed turkey hot dogs for the regular variety, but really any good-quality hot dog will do. Your kids are gonna love this one!\n\n2 tablespoons olive oil\n\n8 ounces turkey hot dogs, cut into\n\n\u00bd-inch-thick rounds\n\n1 cup thinly sliced onion\n\n1 tablespoon minced garlic\n\n1 tablespoon chopped fresh thyme\n\n1\u00bd teaspoons chopped fresh sage\n\n1\u00bc to 1\u00bd pounds russet potatoes, peeled and diced into 1-inch cubes\n\n1 quart chicken stock or canned, low-sodium chicken broth\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n1 bay leaf\n\n1 cup finely diced tomatoes\n\n\u00bd cup heavy cream\n\n1 tablespoon chopped fresh parsley\n\n1. Heat a 2-quart saucepan over medium-high heat and add the olive oil. When it is hot, add the hot dogs and cook, stirring occasionally, until they are caramelized on both sides, about 3 minutes. Remove the hot dogs from the pan and set them aside.\n\n2. Add the onion and garlic to the pan and cook, stirring occasionally, until the onion is wilted and the garlic is fragrant, about 4 minutes. Add the thyme and sage and saut\u00e9 for 1 minute. Add the potatoes, chicken stock, salt, pepper, and bay leaf, and bring to a boil. Reduce the heat to a simmer and cook until the potatoes are tender, about 20 minutes.\n\n3. Remove the bay leaf, and using an immersion blender (or in batches in a blender), quickly puree the soup until smooth (see Note).\n\n4. Return the hot dogs to the soup and add the tomatoes, heavy cream, and chopped parsley. Rewarm gently and serve hot.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.\n\nAbout 1\u00bd quarts, 4 to 6 servings\n\nCREAM OF TOMATO SOUP\n\nPrep time: 10 minutes Cook time: 24 minutes Total: 34 minutes\n\nIf you've never made homemade cream of tomato soup, you don't know what you're missing! This soup is the perfect accompaniment to any grilled sandwich, but I especially love it next to the Prosciutto and Mozzarella Panini on 20 Minutes or Less. Use the balsamic vinegar to finish at the end if you like. And if you're fresh out of cream, the soup is still delicious without it. Just tell everyone, \"It's simply tomato soup.\"\n\n2 tablespoons olive oil\n\n1 cup chopped onion\n\n\u00bd cup chopped carrot\n\n\u00bd cup chopped celery\n\n1 tablespoon minced garlic\n\nTwo 28-ounce cans whole tomatoes, with juices\n\n2 cups chicken stock or canned, low-sodium chicken broth\n\n1 teaspoon salt 1 teaspoon freshly ground black pepper\n\n\u00bd cup heavy cream\n\n3 tablespoons chopped mixed fresh herbs (such as marjoram and basil)\n\n1 tablespoon balsamic vinegar (optional)\n\n1. Set a 3-quart pot over medium-high heat. Add the olive oil, and when it is hot, add the onion, carrot, and celery. Saut\u00e9 the vegetables, stirring occasionally, until the onion is translucent, 3 minutes. Add the garlic and saut\u00e9 until fragrant, about 30 seconds. Add the tomatoes with their juices and the chicken stock, and bring the liquid to a boil. Reduce the heat to a simmer, season with the salt and pepper, and cook for 5 minutes.\n\n2. Break up the tomatoes a bit with the back of a wooden spoon or spatula, and continue to cook until the soup is slightly thickened, about 15 minutes.\n\n3. Remove the soup from the heat and puree it using an immersion blender (or in batches in a blender) until smooth (see Note). Stir in the heavy cream and the herbs, and rewarm until hot. Finish with the balsamic vinegar, if desired, and serve.\n\nNote: Please use caution when blending hot liquids; blend only small amounts at a time, with the blender carefully covered and a kitchen towel held over the top.\n\nAbout 2 quarts, 4 to 6 servings\n\nStarters\n\nSHRIMP AND ZUCCHINI FRITTERS WITH ROASTED RED PEPPER MAYO\n\nPrep time: 25 minutes (including Roasted Red Pepper Mayo) Cook time: 15 minutes Total: 40 minutes\n\nWhat a great hors d'oeuvre or party food...let's serve it to the family! It's fun, easy, and delicious.\n\nRoasted Red Pepper Mayo (recipe follows)\n\n2 cups all-purpose flour\n\n1\u00bd teaspoons baking powder\n\n2 teaspoons salt\n\n1\u00bd teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less), plus more if desired\n\n\u00bc teaspoon cayenne pepper\n\n1\u00bc cups milk\n\n2 eggs\n\nVegetable oil, for frying\n\n1\u00bd zucchinis (about \u00bd pound)\n\n1 pound medium to large shrimp, peeled and deveined\n\n1 tablespoon freshly squeezed lemon juice\n\n1 tablespoon chopped fresh parsley, plus more for garnish\n\n1. Make the Roasted Red Pepper Mayo and set aside.\n\n2. In a medium bowl, combine the flour, baking powder, 1\u00bd teaspoons of the salt, \u00be teaspoon of the Essence, and the cayenne. In a small bowl, whisk together the milk and eggs. Add the liquid ingredients to the dry ingredients, and whisk until smooth. Set aside.\n\n3. Heat 4 to 6 inches of oil to 350\u00b0 F in a 6-quart pot or deep-fryer.\n\n4. While the oil is heating, cut the zucchinis into small dice (about 2 cups), and place in a small bowl. Cut the shrimp into \u00bc-inch pieces and add to the bowl, along with the remaining \u00be teaspoon Essence, remaining \u00bd teaspoon salt, and the lemon juice and parsley. Mix to combine. Add this mixture to the batter and stir to incorporate.\n\n5. Using a 2-tablespoon scoop, carefully drop 10 portions of the batter into the hot oil. Cook, turning the fritters as necessary, until nicely browned on all sides, 4 to 5 minutes total. Drain on paper towels and repeat with the remaining batter.\n\n6. Sprinkle chopped parsley over the fritters, season with additional Essence if desired, and serve immediately, with the Roasted Red Pepper Mayo alongside.\n\nAbout 30 fritters\n\nRoasted Red Pepper Mayo\n\n\u00bd cup diced roasted red pepper (from about \u00be cup packed jarred roasted red pepper)\n\n\u00be cup mayonnaise\n\n1 tablespoon minced garlic\n\n\u00bc teaspoon salt\n\n1\/8 teaspoon cayenne pepper\n\nCombine all the ingredients in the bowl of a food processor, and process for 30 seconds. Scrape down the sides of the bowl with a rubber spatula, and process for 30 seconds longer. Store in an airtight container in the refrigerator for up to 1 week.\n\n1\u00bd cups\n\nCREAMY SHRIMP AND GREEN ONION DIP\n\nPrep time: 20 minutes Cook time: 10 minutes Total: 30 minutes\n\nThis is an exceptional dip. You probably won't even make it to the table with this one. If you plan to share, have a Plan B.\n\n1 pound medium shrimp, peeled and deveined\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 tablespoon olive oil\n\n8 ounces cream cheese, at room temperature\n\n3 tablespoons mayonnaise\n\n\u00bd cup chopped green onions, white and green parts\n\n1\/3 cup minced celery\n\n2 teaspoons freshly squeezed lemon juice\n\n1 teaspoon finely grated lemon zest\n\n\u00bd teaspoon salt\n\n\u00bd teaspoon freshly ground white pepper\n\n\u00bc teaspoon cayenne pepper\n\n\u00bc teaspoon Worcestershire sauce\n\nCrackers or French bread toasts, for serving\n\n1. Combine the shrimp, Essence, and olive oil in a mixing bowl.\n\n2. Heat a nonstick skillet over medium-high heat. When the skillet is hot, add the seasoned shrimp and cook until lightly golden and just cooked through, 3 to 4 minutes. Transfer the shrimp to a plate or shallow bowl and place in the freezer until chilled, 5 to 10 minutes.\n\n3. While the shrimp are chilling, combine all the remaining ingredients (except the crackers) in a mixing bowl. Stir until smooth and creamy.\n\n4. When the shrimp have chilled, remove them from the freezer and coarsely chop them. Add them to the cream cheese mixture and stir well to combine. Taste, and adjust the seasoning if necessary. Serve as is or chilled, with your favorite crackers.\n\nAlmost 3 cups, 6 to 8 servings\n\nSalads\n\nSEARED SHRIMP SALAD\n\nPrep time: 17 minutes Cook time: 8 minutes Total: 25 minutes\n\nThis light and refreshing salad makes for a perfect warm-weather meal when avocados are in season.\n\n\u00bc cup freshly squeezed orange juice\n\n2 tablespoons freshly squeezed lime juice\n\n1 teaspoon honey\n\n\u00be teaspoon crushed red pepper\n\n\u00bd teaspoon soy sauce\n\n\u00bd teaspoon plus a pinch of salt\n\n\u00bc cup plus 2 tablespoons olive oil\n\n1 pound large shrimp, peeled and deveined\n\nOne 5-ounce bag prewashed mixed greens (about 8 cups)\n\n2 oranges, peeled and segmented (see 20 Minutes or Less)\n\n\u00bd cup thinly sliced red onion\n\n1 ripe avocado, halved, seeded, and thinly sliced\n\n1. Combine the orange juice, lime juice, honey, \u00bd teaspoon of the crushed red pepper, the soy sauce, and \u00bc teaspoon of the salt in a small nonreactive bowl. Whisk to blend. In a slow, steady stream, whisk in the \u00bc cup olive oil. Set the vinaigrette aside.\n\n2. Heat 1 tablespoon of the remaining olive oil in a 12-inch saut\u00e9 pan over medium-high heat. In a bowl, toss the shrimp with the remaining 1 tablespoon olive oil, and season with \u00bc teaspoon salt and \u00bc teaspoon crushed red pepper. Add the shrimp to the pan, in two batches if necessary, and cook until they have curled and are just cooked through, 2 minutes on each side. Transfer the shrimp to a paper towel\u2013lined plate and reserve.\n\n3. In a large mixing bowl, combine the greens, oranges, red onion, and pinch of salt. Whisk the vinaigrette, and add 3 tablespoons to the salad. Toss lightly to combine, and then divide the salad among four serving plates. Take 3 to 4 slices of the avocado and fan them out on top of each salad. Divide the shrimp evenly among the salads. Drizzle a little more vinaigrette over the shrimp and avocado, and serve immediately.\n\n4 servings\n\nEMERIL'S NOODLE SALAD\n\nPrep time: 15 minutes Cook time: 10 minutes Total: 25 minutes\n\nThough we call for egg noodles here, feel free to use other noodles, such as soba, rice noodles, linguine, fettuccine, you name it!\n\nSalt for the pasta water\n\n1 pound egg noodles\n\n\u00bd cup salted peanuts\n\n6 tablespoons soy sauce\n\n\u00bc cup rice wine vinegar\n\n2 tablespoons dark Asian sesame oil\n\n2 tablespoons honey\n\n1 teaspoon minced fresh ginger\n\n1 teaspoon minced garlic\n\n\u00bd teaspoon crushed red pepper\n\n1 large seeded cucumber, cut into \u00bc-inch-thick slices (about 4 cups)\n\n1\u00bd cups grated carrots\n\n\u00bd cup thinly sliced green onions, cut on the diagonal\n\n1. Bring a large pot of salted water to a boil. Add the noodles and cook according to the package directions until just tender. Drain the noodles in a colander, rinse them under cold running water until cool, and set aside.\n\n2. Set an 8-inch (or smaller) skillet over low heat. Add the peanuts and toast, tossing as needed and being careful not to let them burn, until the oils begin to release and the nuts are fragrant, 2 minutes. Set aside. When they are cool enough to handle, chop the nuts.\n\n3. In a medium bowl, whisk together the soy sauce, rice wine vinegar, sesame oil, honey, ginger, garlic, and crushed red pepper. Add the cucumber, carrots, and green onions. Add the noodles and peanuts, and mix thoroughly. Serve immediately, or refrigerate until ready to serve. (Can be stored, covered, in the refrigerator for up to 2 days.)\n\n4 servings\n\nGARDEN VEGETABLE SALAD\n\nPrep time: 20 minutes Inactive time: 20 minutes Total: 40 minutes\n\nThis is a marinated salad: the vegetables are simply mixed together and tossed with the dressing. The flavors mingle in just twenty minutes! Delicious.\n\n4 cups sliced cabbage (\u00bd-inch-wide slices)\n\n3 cups broccoli florets, cut into bite-size pieces (about 2 small heads)\n\n2 cups halved and sliced yellow or zucchini squash (\u00bc-inch-thick slices)\n\n1\u00bd cups sliced carrots (\u00bc-inch-thick slices)\n\n\u00bd cup thinly sliced red onion\n\n\u00bd cup thinly sliced radishes\n\n\u00bc cup diced red bell pepper\n\n1 teaspoon salt\n\n\u00be teaspoon freshly ground black pepper\n\n\u00bc cup cider vinegar\n\n1 teaspoon Dijon mustard\n\n1 teaspoon sugar\n\n\u00bd cup plus 2 tablespoons vegetable oil (such as canola)\n\n\u00bc cup finely grated Parmigiano-Reggiano cheese\n\n1. In a medium bowl, combine the cabbage, broccoli, squash, carrots, red onion, radishes, and red bell pepper. Season with \u00be teaspoon of the salt and \u00bd teaspoon of the black pepper, and mix well. Set aside.\n\n2. In a small bowl, whisk together the vinegar, mustard, sugar, remaining \u00bc teaspoon salt, and remaining \u00bc teaspoon pepper. While constantly whisking, add the oil in a thin, steady stream until completely incorporated. Stir in the cheese.\n\n3. Pour the dressing over the vegetables and mix thoroughly. Cover with plastic wrap and refrigerate for at least 20 minutes, or as long as overnight, before serving.\n\n4 to 6 servings\n\nSIMPLE CROUTONS\n\nPrep time: 5 minutes Cook time: 30 minutes Total: 35 minutes\n\nI like to make a big batch of croutons when I find myself with a day-old baguette and a little time, and then store them in airtight containers so that I have them on hand for whenever I need them. They will keep this way for a couple of weeks\u2014as long as you make sure to cook them until they're totally crisp and golden. My kids love snacking on them, and they really add such a nice crunch and texture to so many dishes.\n\n1 French baguette (about 12 ounces), preferably day-old, cut into \u00bd-inch dice\n\n\u00bd cup extra-virgin olive oil\n\n\u00be teaspoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1\/8 teaspoon salt\n\n1\/8 teaspoon freshly ground black pepper\n\n1. Preheat the oven to 300\u00b0F.\n\n2. Combine all of the ingredients in a large mixing bowl, and toss quickly to coat the croutons well. Transfer the croutons to a baking sheet and spread them out in a single layer. Bake, rotating the baking sheet front to back midway through, until crisp all the way through and golden, 25 to 30 minutes. Remove from the oven and set aside to cool completely. Store in an airtight container at room temperature.\n\nAbout 6 cups\n\nCHICKPEA SALAD WITH TABBOULEH\n\nPrep time: 25 minutes Total: 25 minutes\n\nThe addition of bulgur wheat to this simple chickpea salad makes for an unexpectedly delicious combination. Who would've thought? Oh, yeah, babe\u2014good and good for you, too!\n\n\u00bd cup bulgur wheat\n\n1 cup hot water\n\n\u00bd cup chopped green onions, white and green parts\n\n\u00bc cup chopped fresh parsley\n\n\u00bc cup chopped fresh mint\n\n\u00bc cup diced oil-packed sun-dried tomatoes\n\n\u00bc cup plus 2 tablespoons freshly squeezed lemon juice\n\n2 teaspoons minced garlic\n\n1 teaspoon fine sea salt\n\n\u00be teaspoon crushed red pepper\n\n\u00be cup extra-virgin olive oil\n\n\u00bd cup (about 4 ounces) crumbled feta cheese\n\nTwo 14-ounce cans chickpeas, rinsed and drained\n\n1. Place the bulgur in a mixing bowl, add the hot water, and set aside to soak for 25 minutes.\n\n2. While the bulgur is soaking, prep the remaining ingredients. Combine the green onions, parsley, mint, sun-dried tomatoes, lemon juice, garlic, sea salt, and crushed red pepper in a medium bowl. Gradually whisk in the olive oil.\n\n3. Drain the bulgur, squeezing it to remove any excess liquid, and add it to the herb-tomato mixture. Fold in the feta cheese and chickpeas. Taste, and adjust the seasoning if necessary. Serve at room temperature or chilled.\n\n4 servings\n\nSandwiches\n\nCHICKEN QUESO BURGERS\n\nPrep time: 12 minutes Cook time: 28 minutes Total: 40 minutes\n\nI know you love burgers and cheese, and this chicken version is super-cheesy. Because of the generous amount of cheese, be sure to use a nonstick skillet. The griddled onions make the burger extra-special. After this, you're sure to be making them for all your sandwiches.\n\n2 eggs, lightly beaten\n\n8 ounces cheddar cheese, grated (about 2 cups)\n\nOne 4-ounce can (about 1\/3 cup) minced green chiles, drained\n\n2 tablespoons chopped fresh oregano\n\n3 teaspoons chili powder\n\n1 teaspoon ground cumin\n\n2\u00bd teaspoons salt\n\n2 pounds ground chicken thigh meat\n\n\u00be cup fine unseasoned dry breadcrumbs\n\n1\u00bd cups minced onions\n\n1 cup olive oil\n\n8 hamburger buns\n\nLettuce, tomato, and mayonnaise, for garnishing burgers (optional)\n\n1. In a large bowl, combine the eggs, cheese, chiles, oregano, 1\u00bd teaspoons of the chili powder, the cumin, and 2 teaspoons of the salt. Add the ground chicken and the breadcrumbs, and mix until well blended. Divide the mixture into 8 portions, and shape each portion into a 1-inch-thick patty. Lay the patties on a tray, cover with plastic wrap, and refrigerate until ready to cook.\n\n2. Combine the onions, \u00be cup of the olive oil, remaining 1\u00bd teaspoons chili powder, and remaining \u00bd teaspoon salt in a small bowl. Divide the onion mixture evenly among the cut sides of the hamburger buns, and spread it out with a brush to coat.\n\n3. Heat a 12-inch nonstick skillet over medium-high heat. In batches, toast the buns, coated sides down, in the skillet until the onions cook slightly and stick to the buns and the bread is lightly toasted, 2 to 3 minutes. Set the buns aside and wipe the skillet clean.\n\n4. In the same nonstick skillet, heat 2 tablespoons of the remaining olive oil over medium heat. Add 4 patties and cook until an instant-read thermometer registers 165\u00b0F when inserted into the center of a patty, about 4 minutes per side. Set the patties aside; keep warm. Repeat with the remaining 2 tablespoons olive oil and 4 patties. Place the patties between the toasted buns and serve immediately, with lettuce, tomato, and mayonnaise, if desired.\n\n8 servings\n\nCHILI-RUBBED SHRIMP WRAPS\n\nPrep time: 20 minutes Cook time: 8 minutes Total: 28 minutes\n\nThese shrimp marinate for just ten minutes. For a twist, skewer and grill them instead of saut\u00e9ing. They can be served as part of an hors d'oeuvre tray, in a salad, with pasta tossed in olive oil, salt, and pepper, or as we've outlined here, in a tortilla.\n\n8 tablespoons olive oil\n\n\u00bc cup plus 1 tablespoon freshly squeezed lime juice\n\n3 tablespoons chopped fresh cilantro\n\n4 teaspoons minced garlic\n\n1 teaspoon Mexican chili powder\n\n1\u00bd pounds large shrimp, peeled and deveined\n\n\u00bd cup sour cream\n\n\u00be teaspoon salt\n\nFour 14-inch flour tortillas\n\n8 ounces red-leaf lettuce, rinsed, spun dry, and cut into 1-inch pieces\n\n1 medium tomato, diced\n\n1 Hass avocado, halved, seeded, and thinly sliced\n\n\u00bd cup grated Monterey Jack or pepper Jack cheese\n\n1. In a medium bowl, combine 6 tablespoons of the olive oil, the \u00bc cup lime juice, 2 tablespoons of the cilantro, the garlic, and \u00bd teaspoon of the chili powder. Add the shrimp and set aside to marinate at room temperature for 10 minutes, turning them every few minutes.\n\n2. In a small bowl, combine the sour cream, the remaining 1 tablespoon lime juice, the remaining 1 tablespoon cilantro, \u00bc teaspoon of the remaining chili powder, and \u00bc teaspoon of the salt. Stir well and set aside.\n\n3. Heat the remaining 2 tablespoons olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Remove the shrimp from the marinade and season with the remaining \u00bd teaspoon salt and \u00bc teaspoon chili powder. In two batches, cook the shrimp for 2 minutes per side. Remove from the pan and allow to cool slightly before assembling the sandwiches.\n\n4. Lay the tortillas on a clean work surface. Arrange one-fourth of the shrimp across the lower third of each tortilla, leaving about 2 inches of space on either side. Divide the lettuce, tomato, avocado slices, and grated cheese evenly among the tortillas, scattering them over the shrimp. Then drizzle some of the dressing over each. Fold both sides of each tortilla in toward the center, then roll the lower edge of the tortilla up, burrito-style, forming a wrap. Position the wraps on a serving plate, seam side down, and slice each in half on the diagonal. Serve immediately, or refrigerate and serve within 1 hour.\n\n4 servings\n\nSPICY PORK WRAPS WITH CREAMY COLESLAW\n\nPrep time: 10 to 14 minutes Cook time: 16 minutes Total: 26 to 30 minutes\n\nNo marinade needed. The tenderloins are coated in an intensely spiced rub, seared, sliced, and cloaked in a cool, creamy slaw. These babies pack a punch! Who's makin' this sandwich?\n\n2 tablespoons light brown sugar\n\n1 tablespoon chili powder\n\n2 teaspoons dry mustard\n\n2 teaspoons salt\n\n\u00be teaspoon dried oregano\n\n\u00be teaspoon cayenne pepper\n\n2 pork tenderloins (about 1 pound each), tail ends tucked under and tied\n\n2 tablespoons olive oil\n\nSix 10-inch flour tortillas\n\nOne 14-ounce bag coleslaw mix, or 14 ounces shredded mixed cabbage and carrots\n\n1\/3 cup buttermilk\n\n\u00bc cup mayonnaise\n\n\u00bc cup minced green onions, white and green parts\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Combine the brown sugar, chili powder, dry mustard, 1\u00bd teaspoons of the salt, the oregano, and \u00bd teaspoon of the cayenne in a small bowl. Generously rub the mixture all over the pork tenderloins.\n\n3. Heat the olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Add the tenderloins and sear, turning frequently, until evenly crusted on all sides (they will be dark in color because of the sugar), 2 to 4 minutes. Transfer the pork to a baking sheet and roast in the oven for 12 minutes, or until the internal temperature registers 145\u00b0F on an instant-read thermometer inserted into the center.\n\n4. Remove the tenderloins from the oven, transfer them to a cutting board, and tent with foil. Let rest for at least 10 minutes before slicing. While the pork is resting, wrap the tortillas in aluminum foil and place them in the oven until warmed through, 6 to 8 minutes.\n\n5. Make the coleslaw by combining the coleslaw mix with the buttermilk, mayonnaise, green onions, remaining \u00bd teaspoon salt, and remaining \u00bc teaspoon cayenne pepper. Toss well to combine, and set aside until ready to serve.\n\n6. To assemble the pork wraps, slice the pork tenderloin in thin slices at an angle. Place \u00bc cup of the coleslaw on the lower half of each tortilla, leaving about 2 inches on either side, and then divide the pork slices evenly among the tortillas. Fold in both sides of the tortillas, and then roll the tortillas up, burrito-style. Slice in half and serve immediately.\n\n4 to 6 servings\n\nCHICKEN PATTY POCKETS WITH MINTED YOGURT SAUCE\n\nPrep time: 19 minutes Cook time: 7 minutes Total: 26 minutes\n\nYou don't need a giant roasting spit for mouthwatering meat. You're on the Emeril Express here. Fill pita pockets with lettuce, tomatoes, and alfalfa sprouts for good measure, add your spicy baked chicken patties, and dollop with the Minted Yogurt Sauce.\n\n1 pound ground chicken thigh meat\n\n2 large egg whites, lightly beaten\n\n\u00bd cup fine unseasoned dry breadcrumbs\n\n\u00bc cup finely chopped onion\n\n\u00bc cup chopped fresh parsley\n\n1 tablespoon minced garlic\n\n1 teaspoon salt\n\n\u00bd teaspoon cayenne pepper\n\n\u00bd teaspoon ground coriander\n\n\u00bc teaspoon ground nutmeg\n\n\u00bc teaspoon ground cumin\n\n2 tablespoons olive oil\n\nPita bread, for serving\n\nLettuce, sliced tomatoes, and alfalfa sprouts, for garnishing sandwiches (optional)\n\nMinted Yogurt Sauce, for serving (optional; recipe follows)\n\n1. Position a rack as close as possible to the broiler element and preheat the broiler.\n\n2. In a large bowl, combine the chicken, egg whites, breadcrumbs, onion, parsley, garlic, salt, and spices. Mix until well blended, using a large spoon. Line a rimmed baking sheet with aluminum foil, and drizzle with 1 tablespoon of the olive oil. Divide the seasoned meat into 8 portions, 2\u00bd to 3 ounces each, and place them on the prepared baking sheet. Shape each into an oval patty and flatten it slightly. Brush or drizzle the remaining 1 tablespoon olive oil over the patties.\n\n3. Broil until lightly browned, about 7 minutes, or until an instant-read thermometer registers 165\u00b0F when inserted into the center of a patty. Serve on pita bread, with lettuce, tomatoes, sprouts, and Minted Yogurt Sauce, if desired.\n\n4 servings\n\nMinted Yogurt Sauce\n\nPrep time: 10 minutes Total: 10 minutes\n\n1 cup plain yogurt\n\n\u00bd cucumber, peeled, seeded, and cut into small dice (about 1 cup)\n\n2 tablespoons chopped fresh mint\n\n1 tablespoon freshly squeezed lime juice\n\n\u00be teaspoon salt\n\n\u00bc teaspoon ground cumin\n\n\u00bc teaspoon sweet paprika\n\n1\/8 teaspoon cayenne pepper\n\nPinch of sugar\n\n1. Place the yogurt in a fine-mesh strainer set over a bowl, and let it drain for 3 minutes while you assemble the other ingredients. Discard any liquid that drains from the yogurt.\n\n2. Combine the cucumber, mint, lime juice, salt, cumin, paprika, cayenne, and sugar in a small bowl. Stir to blend.\n\n3. Add the drained yogurt and stir to combine. Serve immediately, or refrigerate, covered, for up to 1 hour to allow the flavors to come together before serving.\n\nAbout 2 cups\n\nOVEN-CRISPY FRENCH FRIES WITH PAPRIKA-PARMESAN SALT\n\nPrep time: 10 minutes Cook time: 30 minutes Total: 40 minutes\n\nThese fries are not just for burgers. While they're in the oven, make the Crispy Pan-Roasted Chicken with Garlic-Thyme Butter (40 Minutes or Less), the Steak au Poivre (20 Minutes or Less), the Classic Moules Marini\u00e8re (20 Minutes or Less), or the Broiled Salmon with a Warm Tomato-Lemon Vinaigrette (20 Minutes or Less)! Do I need to keep going? Ahhhh, you get it!\n\n2 large baking potatoes (about 1\u00bd pounds), scrubbed well\n\n\u00bc cup olive oil\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00be teaspoon salt\n\n\u00bc cup finely grated Parmigiano-Reggiano cheese\n\n1 tablespoon sweet paprika\n\n\u00bd teaspoon garlic powder\n\n\u00bc teaspoon onion powder\n\n1. Preheat the oven to 425\u00b0F.\n\n2. Pat the potatoes dry and cut them lengthwise into \u00bd-inch-thick slices. Turn each side flat and slice again lengthwise into even \u00bd-inch-thick fries. Place the potatoes in a mixing bowl and add the olive oil, Essence, and \u00bc teaspoon of the salt. Toss well to combine. Then transfer the fries to a large baking sheet and arrange them in one even layer so that they are not touching. Roast, scraping the potatoes from the baking sheet with a metal spatula and turning them over halfway through, until golden brown and crispy, 30 minutes.\n\n3. While the potatoes are cooking, combine the cheese, paprika, garlic powder, onion powder, and the remaining \u00bd teaspoon salt in a small bowl and stir to blend.\n\n4. When the potatoes are crisp and brown, remove them from the oven and transfer to a serving platter. Sprinkle the potatoes with the Paprika-Parmesan Salt, and serve hot.\n\n4 to 6 servings\n\nPasta\n\nORZO \"RISOTTO\" WITH TOMATO, MOZZARELLA, AND BASIL\n\nPrep time: 10 minutes Cook time: 20 minutes Total: 30 minutes\n\nHere you'll find orzo cooked in the style of risotto. This works equally well as a quick and easy side dish or as a vegetarian entr\u00e9e when served with a nice salad or grilled veggies.\n\n3 tablespoons olive oil\n\n1\/3 cup minced red onion\n\n1 tablespoon minced garlic\n\n2 cups orzo pasta\n\n4 cups chicken stock, or canned, low-sodium chicken broth, heated\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n2 cups diced fresh tomatoes\n\n\u00bd cup diced fresh mozzarella cheese\n\n1\u00bd tablespoons thinly sliced basil\n\n1. Heat the olive oil in a medium saucepan over medium heat. When it is hot, add the red onion and garlic and cook until fragrant, about 30 seconds. Add the orzo and stir well to coat.\n\n2. Gradually add the hot chicken stock in \u00bd-cup increments, stirring until all of the stock has been absorbed before adding more, until the stock is completely incorporated and the pasta is just tender, 14 to 16 minutes. Season with the salt and pepper.\n\n3. Add the tomatoes to the orzo and cook until they are just heated through, 1 to 2 minutes. Add the mozzarella and cook for another 2 to 3 minutes, or just until it is incorporated.\n\n4. To serve, spoon the risotto into bowls and garnish with the basil.\n\n4 servings\n\nSHIITAKES AND BACON WITH PENNE\n\nPrep time: 15 minutes Cook time: 17 minutes Total: 32 minutes\n\nTalk about tasty\u2014this dish is a real winner. The combination of shiitake mushrooms and bacon really works here, trust me. If you're a pancetta-lover like me, feel free to substitute about 4 ounces of diced pancetta for the bacon. Mama mia!\n\n1 pound penne pasta\n\n4 slices thick-cut bacon, cut into \u00bd-inch pieces\n\n7 ounces shiitake mushrooms, wiped clean, stemmed, and thinly sliced\n\n1\u00bd cups diced onions (small dice)\n\n2 teaspoons minced garlic\n\nTwo 14.5-ounce cans diced tomatoes, with juices\n\n\u00bc cup chopped fresh basil\n\n1\u00bd teaspoons salt, plus more for the pasta water\n\n1 teaspoon freshly ground black pepper\n\n\u00bc cup extra-virgin olive oil\n\nGrated Parmigiano-Reggiano cheese, for garnish (optional)\n\n1. Bring a large pot of salted water to a boil. Add the penne and cook until just tender, about 11 minutes. Drain the pasta in a colander.\n\n2. While the pasta is cooking, heat a 12-inch (or larger) saut\u00e9 pan over medium-high heat. Add the bacon and cook, stirring occasionally, until crisp, 5 to 6 minutes. Add the mushrooms and onions and cook until the mushrooms are browned and the onions are soft, 5 to 6 minutes. Add the garlic and cook until fragrant, about 30 seconds.\n\n3. Add the drained pasta to the pan, along with the tomatoes, basil, salt, pepper, and olive oil. Cook, tossing often, until the pasta is thoroughly combined and heated through, about 5 minutes.\n\n4. Serve immediately, garnished with grated cheese if desired.\n\n4 to 6 servings\n\nPENNE WITH SAUSAGE AND ESCAROLE\n\nPrep time: 12 minutes Cook time: 20 minutes Total: 32 minutes\n\nThose of you who don't love escarole had better try it again...the sweetness of the Italian sausage here really complements its flavor beautifully. A simple pasta dish for any day of the week.\n\n1 pound penne pasta\n\n1 teaspoon olive oil\n\n2 medium onions, cut into small dice (about 2 cups)\n\n1 red bell pepper, cut into medium dice (about 1 cup)\n\n\u00bd teaspoon salt, plus more for the pasta water\n\n\u00bd teaspoon freshly ground black pepper\n\n1\u00bd pounds sweet Italian sausage, casings removed (or bulk sausage)\n\n2 teaspoons minced garlic\n\n1 bunch escarole or mustard greens, rinsed, stemmed, and torn into bite-size pieces (about 8 cups)\n\n\u00bd cup grated Parmigiano-Reggiano cheese\n\n\u00bc teaspoon crushed red pepper\n\n2 tablespoons extra-virgin olive oil\n\n1. Bring a large pot of salted water to a boil. Add the penne and cook until just tender, about 11 minutes. Drain the pasta in a colander, reserving 1 cup of the cooking water, and set aside.\n\n2. While the water is heating and the pasta is cooking, heat the olive oil in a 14-inch saut\u00e9 pan over medium heat. Add the onions, bell pepper, \u00bd teaspoon salt, and black pepper, and cook until the vegetables are soft, about 4 minutes. Add the sausage and cook, breaking the pieces up with the back of a wooden spoon, until browned, 6 to 8 minutes. Add the garlic and escarole, and cook for 5 minutes longer.\n\n3. Add the cooked pasta and the reserved cooking water, and stir gently to combine. Simmer just until everything is heated through, about 2 minutes.\n\n4. Transfer the mixture to a large serving bowl. Add the cheese and crushed red pepper, and toss to combine. Drizzle with the extra-virgin olive oil, and serve immediately.\n\n6 to 8 servings\n\nSPAGHETTI WITH CARAMELIZED ONIONS AND ANCHOVIES\n\nPrep time: 10 minutes Cook time: 25 minutes Total: 35 minutes\n\nThe amount of anchovies in this recipe may seem alarming, but, yes, two whole tins is correct here, and makes for a deliciously sweet and salty pasta sauce. Oh, this dish brings me back to the south of Italy in a heartbeat.\n\n1 pound spaghetti\n\n1\/3 cup olive oil\n\n2 tablespoons unsalted butter\n\n8 cups thinly sliced onions\n\n2 tablespoons thinly sliced garlic\n\nTwo 2-ounce cans flat anchovy fillets (packed in olive oil, not salt), well drained\n\n1\u00bd teaspoons kosher salt, plus more for the pasta water\n\n1 teaspoon freshly ground black pepper\n\n2 tablespoons chopped fresh parsley\n\n\u00bd cup extra-virgin olive oil\n\n1. Bring a large pot of salted water to a boil. Add the spaghetti and cook until just tender, about 9 minutes. Drain the pasta in a colander, reserving 1 cup of the cooking water, and set aside.\n\n2. While the water is heating and the pasta is cooking, set a 14-inch saut\u00e9 pan over medium-high heat and add the olive oil and butter. Once the butter melts, add the onions and cook, stirring occasionally with a heat-resistant rubber spatula, until they have softened and caramelized, about 20 minutes. (Should the onions get too dry and begin sticking in spots before they are all caramelized, add a bit of water, stir, and continue cooking.)\n\n3. Add the garlic and anchovies to the pan and cook, stirring, until fragrant, 2 to 3 minutes.\n\n4. Add the drained pasta and the reserved pasta water to the pan and season with the 1\u00bd teaspoons salt and the pepper. Cook, tossing to combine, until the pasta is heated through and the water has nearly evaporated, 3 to 4 minutes. Remove from the heat, add the parsley, and drizzle with the extra-virgin olive oil. Toss to combine, and serve immediately.\n\nNote: It is not necessary to chop the anchovies; they will break into small pieces while cooking.\n\n4 to 6 servings\n\nTHREE-CHEESE BAKED MACARONI\n\nPrep time: 8 minutes Cook time: 20 minutes Inactive time: 10 minutes Total: 38 minutes\n\nThis super-easy custard-style macaroni has the perfect blend of cheeses and bacony goodness for any mac-n-cheese lover.\n\n8 ounces elbow macaroni\n\n3 ounces bacon (about 3 strips), sliced crosswise into \u00bd-inch pieces\n\n1\u00bd teaspoons minced garlic\n\n3 eggs\n\n1\u00bd cups evaporated milk\n\n\u00bd teaspoon salt, plus more for the pasta water\n\n\u00bc teaspoon cayenne pepper\n\n1\/8 teaspoon freshly grated nutmeg\n\n6 ounces sharp cheddar cheese, grated (about 1\u00bd cups)\n\n2 ounces Monterey Jack cheese, grated (about \u00bd cup)\n\n1 ounce Parmigiano-Reggiano cheese, finely grated (about \u00bd cup)\n\n1. Preheat the oven to 425\u00b0F.\n\n2. Bring a large pot of salted water to a boil. Add the macaroni and cook until just tender, about 6 minutes. Drain, and set aside.\n\n3. While the pasta is cooking, heat a small saut\u00e9 pan over medium heat and add the bacon. Cook until the fat is rendered and the bacon is crisp, about 6 minutes. Add the garlic and cook until fragrant, 30 seconds to 1 minute. Drain the fat from the bacon-garlic mixture, and transfer the mixture to a medium bowl.\n\n4. Add the drained macaroni to the bacon mixture, and stir to combine.\n\n5. In a large bowl, whisk the eggs and evaporated milk together. Add the \u00bd teaspoon salt, cayenne, nutmeg, and grated cheeses, and mix well. Add the macaroni-bacon mixture, and stir well to blend.\n\n6. Transfer the macaroni to an 8-or 9-inch square baking dish or gratin dish of similar size. Using a spoon, gently spread the mixture to form an even layer. Place in the oven and bake for 12 minutes. Remove the macaroni and cheese from the oven and let it rest for at least 10 minutes before serving.\n\n4 to 6 servings\n\nSHRIMP AND LINGUINE FRA DIAVOLO\n\nPrep time: 15 minutes Cook time: 12 minutes Total: 27 minutes\n\nThis is hot as the devil! The classic Fra Diavolo preparation uses lobster, but here I've simplified it a bit for the home cook with sweet shrimp, preferably from the Gulf of Mexico. Talk about making me smile!\n\n1 pound linguine\n\n6 tablespoons extra-virgin olive oil\n\n1 cup chopped onion\n\n3 tablespoons minced garlic\n\n2 to 3 teaspoons crushed red pepper, to taste\n\n1\u00bd cups tomato sauce\n\n2 tablespoons tomato paste\n\n1\u00bd pounds shrimp, peeled and deveined\n\n1 teaspoon salt, plus more for the pasta water\n\n2 tablespoons chopped fresh parsley\n\n\u00bd cup grated Parmigiano-Reggiano cheese (optional)\n\n1. Bring a large pot of salted water to a boil. Add the linguine and cook until barely tender, about 8 minutes. Drain, reserving 1 cup of the cooking water, and set aside.\n\n2. While the pasta is cooking, set a 14-inch saut\u00e9 pan over medium-high heat and add the olive oil. Once the oil is hot, add the onion and cook until lightly caramelized and wilted, 3 to 4 minutes. Add the garlic and saut\u00e9 until fragrant, about 30 seconds. Add the crushed red pepper and saut\u00e9 briefly; then add the tomato sauce and tomato paste. Cook until the sauce has reduced by about half, about 3 minutes. Add the shrimp to the sauce and cook for 2 minutes.\n\n3. Add the pasta and the reserved cooking water to the pan, and cook until the pasta is heated through and coated with the sauce, 3 minutes. Season the pasta with the salt, and garnish with the parsley. Toss to combine, and serve with the grated cheese if desired.\n\n6 servings\n\nPENNE ALLA PUTTANESCA\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nPuttanesca sauce is nothing more than the combination of tomatoes, onion, garlic, basil, anchovies, and capers\u2014ingredients typically kept on hand in most Italian households and one of my favorite sauces of all time. Make a big batch of this and freeze it in small containers so you can pull some out for a quick and easy meal anytime.\n\nSalt\n\n\u00bc cup extra-virgin olive oil, plus more for serving\n\n1 large onion, chopped\n\n\u00bd teaspoon crushed red pepper\n\n8 cloves garlic, minced\n\nOne 28-ounce can Italian plum tomatoes, roughly chopped or broken into pieces, with juices\n\n1 cup halved pitted Kalamata olives, drained\n\n\u00bc cup nonpareil capers, drained, liquid reserved separately\n\n5 canned anchovy fillets, or to taste, finely chopped\n\n\u00bd teaspoon dried basil, crushed between your fingers\n\nFreshly ground black pepper\n\n1 pound penne rigate pasta\n\nFinely grated Parmigiano-Reggiano cheese, for serving (optional)\n\n1. Bring a large pot of salted water to a boil.\n\n2. While the water is heating, heat the oil in a large nonreactive saucepan or skillet over medium-high heat. Add the onion and crushed red pepper, and saut\u00e9 until the onion is tender and beginning to caramelize, about 4 minutes. Add the garlic and cook for 1 minute, stirring. Add the tomatoes, olives, capers, anchovies, and basil, and bring the sauce to a boil. Reduce the heat to low and simmer, uncovered and stirring occasionally, until the sauce has thickened, 15 to 20 minutes.\n\n3. Meanwhile, cook the pasta in the boiling water until just tender, about 11 minutes. Drain the pasta, reserving the pot it was cooked in. Set the pasta aside.\n\n4. Remove the sauce from the heat, and season it with salt and pepper to taste. Add 1 to 2 teaspoons of the reserved caper juice, to taste.\n\n5. Return the penne to the pasta pot over medium-high heat. Add half of the pasta sauce and cook, stirring, until heated through, about 2 minutes. Serve hot, with more sauce ladled on top of each serving if desired, drizzled with additional extra-virgin olive oil. Garnish with grated cheese, if desired.\n\n4 to 6 servings\n\nBEEF STROGANOFF WITH EGG NOODLES\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nNow this is what real beef and mushrooms taste like! Splurge on 1\u00bd pounds of good rib-eye steak and feed six. The beef is quickly saut\u00e9ed. The sauce is full of browned mushroom goodness. Put them together and add your noodles. Whoa!\n\n8 ounces extra-wide egg noodles\n\n1\u00bd pounds rib-eye steak, sliced into \u00bd-inch-thick strips\n\n2 teaspoons salt, plus more for the pasta water\n\n1 teaspoon freshly ground black pepper\n\n4 tablespoons olive oil\n\n3 tablespoons butter\n\n1\u00bd cups thinly sliced onions\n\n12 ounces button mushrooms, wiped clean, stemmed, and sliced (about 4 cups)\n\n1 tablespoon chopped garlic\n\n1 tablespoon all-purpose flour\n\n2 cups beef stock or canned, low-sodium beef broth\n\n\u00bd cup sour cream\n\n1 tablespoon chopped fresh parsley\n\n1. Bring a large pot of salted water to a boil. Add the noodles and cook until just tender, about 8 minutes.\n\n2. While the noodles are cooking, season the beef with 1 teaspoon of the salt and \u00bd teaspoon of the pepper.\n\n3. Heat 2 tablespoons of the olive oil in a 12-inch saut\u00e9 pan over high heat. In two batches, brown the beef strips for 1 to 2 minutes per side. Transfer the beef to a plate and set aside.\n\n4. Add the butter to the saut\u00e9 pan, and when it has melted, add the onions. Reduce the heat to medium and cook until the onions are soft, about 4 minutes. Add the mushrooms and continue to cook, stirring as needed, until nicely browned, about 7 minutes.\n\n5. When the noodles are cooked, drain them, transfer them to a bowl, and toss with the remaining 2 tablespoons olive oil. Cover to keep warm until ready to add to the sauce.\n\n6. Add the garlic and the remaining 1 teaspoon salt and \u00bd teaspoon pepper to the mushroom mixture and cook, stirring, for 1 minute. Sprinkle with the flour, and stir. Increase the heat to high and whisk in the broth. When the liquid comes to a boil, reduce the heat to a simmer and cook for 5 minutes.\n\n7. Return the beef, and any juices that have accumulated on the plate, to the saut\u00e9 pan. Whisk in the sour cream and parsley, and remove the pan from the heat. Fold in the warm noodles, and serve immediately.\n\n6 servings\n\nPASTA PRIMAVERA\n\nPrep time: 15 minutes Cook time: 15 minutes Total: 30 minutes\n\nThis colorful sauce is a simpler, lighter version of the primavera sauce most often encountered. Feel free to substitute other veggies, as desired. As you will see, this makes a very large batch of pasta, enough for six to eight healthy appetites.\n\n1 pound rotini or penne pasta\n\n5 tablespoons butter\n\n2 cups chopped red onions\n\n1 cup chopped red bell pepper\n\n2 teaspoons salt, plus more for the pasta water\n\n1 teaspoon freshly ground black pepper\n\n1\u00bd tablespoons minced garlic\n\n1 pound zucchini or yellow squash (or a mixture of both), halved lengthwise and cut crosswise into \u00bd-inch half-moon pieces (about 4 cups)\n\n2 cups frozen mixed peas and carrots\n\n1 cup diced canned tomatoes, with juices\n\n\u00bd cup finely grated Parmigiano-Reggiano cheese, plus more for serving\n\n\u00bd cup thinly sliced basil (optional)\n\n1\/3 cup extra-virgin olive oil\n\n1. Bring a large pot of salted water to a boil. Add the rotini and cook until just tender, about 11 minutes. Drain, reserving \u00be cup of the cooking water, and set aside.\n\n2. While the pasta is cooking, melt the butter in a 14-inch saut\u00e9 pan over medium-high heat. Add the onions and bell pepper and cook until soft, 4 minutes. Add the 2 teaspoons salt, black pepper, and garlic, and cook for 1 minute longer. Add the squash and continue to cook, stirring as needed, for 2 minutes. Add the peas and carrots and cook for 2 minutes. Add the tomatoes and continue to cook, stirring, for 2 minutes.\n\n3. Add the pasta and the reserved cooking water to the sauce and cook until the pasta is heated through and the ingredients are well combined, 1 to 2 minutes. Remove from the heat and fold in the Parmigiano-Reggiano and basil, if desired. Then drizzle with the extra-virgin olive oil. Garnish with more cheese, if desired, and serve hot.\n\n6 to 8 servings\n\nRice and Beans\n\nGREEN ONION RICE PILAF\n\nPrep time: 10 minutes Cook time: 20 minutes Total: 30 minutes\n\nThis simple rice dish goes well with so many things, it's not even funny. The method is foolproof. All you need is a timer to keep you from worrying about the rice while it's cooking, leaving you free to tend to other things...\n\n\u00bc cup olive oil\n\n\u00bd cup chopped onion\n\n1 teaspoon minced garlic\n\n2 cups long-grain white rice\n\n2 teaspoons salt\n\n1 teaspoon freshly ground white pepper\n\n3 cups chicken stock or canned, low-sodium chicken broth or water\n\n\u00bc cup thinly sliced green onion tops\n\n1. Place a 2-quart ovenproof saucepan over medium-high heat and add the olive oil. Once the oil is hot, add the onion and cook, stirring often, until translucent and beginning to soften, about 4 minutes. Add the garlic and saut\u00e9 for 30 seconds. Add the rice and saut\u00e9, stirring, until fragrant, 3 to 4 minutes. Season the rice with the salt and white pepper.\n\n2. Add the chicken stock to the pan and cook, stirring occasionally, until the water comes to a boil. Then cover the pan, reduce the heat to low, and cook for 20 minutes.\n\n3. Remove the pan from the heat and let the rice stand, covered, for 5 minutes. Remove the lid, add the green onion, and toss with a fork to combine.\n\n6 cups, 6 to 8 servings\n\nBASIC RISOTTO\n\nPrep time: 5 minutes Cook time: 25 minutes Total: 30 minutes\n\nThink while stirring, \"Buttery, cheesy, and creamy...,\" \"Buttery, cheesy, and creamy...,\" \"Buttery, cheesy, and creamy...\" Aah, finito.\n\n\u00bc cup olive oil\n\n\u00bc cup finely chopped shallots\n\n2 cups Arborio rice\n\n1 teaspoon salt\n\n\u00bc teaspoon freshly ground white pepper\n\n\u00bd cup dry white wine\n\n6 cups chicken stock, or canned, low-sodium chicken broth, heated\n\n1 tablespoon butter\n\n\u00bd cup finely grated Parmigiano-Reggiano cheese\n\n1 tablespoon fresh thyme leaves\n\n1. Heat the olive oil in a 14-inch saut\u00e9 pan over medium-high heat. Add the shallots and cook, stirring with a heat-resistant rubber spatula, until fragrant and soft, about 1 minute. Add the rice and cook until the grains are opaque, about 2 minutes. Then add the salt, white pepper, and white wine. Continue cooking, stirring the rice as needed, until nearly all the liquid has been absorbed.\n\n2. Reduce the heat to medium, stir in \u00be cup of the hot broth, simmer, and stir until nearly all the liquid has been absorbed. Continue in this manner, adding the broth in \u00be-cup increments and only adding more once the previous addition has been absorbed, until all the broth has been used and the risotto is tender and creamy, about 20 minutes.\n\n3. Fold in the butter, cheese, and thyme. Remove from the heat, adjust the seasoning if necessary, and serve immediately.\n\n4 servings\n\nBLACK BEAN CAKES\n\nPrep time: 16 minutes Cook time: 8 minutes Total: 24 minutes\n\nTalk about \"knock your socks off\"! These bean cakes end up crispy and crusty on the outside, but oh so tender and creamy on the inside. A true study in contrasts, this dish is elevated to notches unknown when served garnished with your favorite guacamole, salsa, and sour cream.\n\n7 tablespoons olive oil\n\n1 small onion (5 to 6 ounces), cut into small dice\n\n2 teaspoons minced garlic\n\n\u00bd cup all-purpose flour\n\n2 tablespoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\nTwo 15.5-ounce cans black beans, drained and quickly rinsed\n\n2 tablespoons chopped fresh cilantro, plus more for garnish\n\n1 egg, lightly beaten\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bd teaspoon ground cumin\n\n\u00bd teaspoon ground coriander\n\n2 teaspoons hot sauce\n\n1. Heat 1 tablespoon of the olive oil in a medium saut\u00e9 pan over medium heat. When it is hot, add the onion and cook until soft and lightly caramelized, about 3 minutes. Add the garlic and cook until fragrant, about 30 seconds. Remove from the heat and set aside to cool.\n\n2. Place the flour in a shallow bowl or plate, and stir in the Essence. Set aside.\n\n3. In a medium mixing bowl, mash the black beans well with the back of a fork\u2014the mixture should be relatively smooth, with no whole beans remaining. Stir in the cooled onion mixture, cilantro, egg, salt, pepper, cumin, coriander, and hot sauce and mix well. Divide the mixture into 8 evenly sized patties (about 1\/3 cup each).\n\n4. Heat the remaining 6 tablespoons olive oil in a medium nonstick skillet over medium heat. When the oil is hot, dust the patties in the seasoned flour mixture and carefully transfer them to the hot skillet (the cakes will be delicate). Cook the cakes until golden brown on both sides and heated through, about 2 minutes per side.\n\n5. If necessary, season with more salt. Garnish with chopped cilantro, and serve immediately.\n\n4 servings\n\nTURKEY AND PINTO BEAN TOSTADAS\n\nPrep time: 10 minutes Cook time: 25 minutes Total: 35 minutes\n\nFor this open-face taco we crisp the tortillas in the oven. Hold on to your sombreros\u2014you're in for a real treat.\n\n4 tablespoons olive oil\n\n1 medium onion, chopped\n\n1 pound ground turkey (preferably 85\/15 blend)\n\n2 tablespoons Mexican chili powder\n\n1\u00bd tablespoons chopped garlic\n\n1\u00bd teaspoons ground cumin\n\n1 teaspoon salt\n\nTwo 15-ounce cans pinto beans, drained and briefly rinsed\n\n1\u00bd cups chicken stock or canned, low-sodium chicken broth\n\n15 fresh cilantro sprigs, stems and leaves chopped separately\n\n8 to 10 corn tortillas\n\nCondiments as desired for serving (such as grated cheese, chopped tomato, chopped red onion, shaved lettuce, salsa, pickled jalape\u00f1os, sour cream)\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Set a skillet over medium-high heat, add 2 tablespoons of the olive oil and the onion, and cook, stirring, for 2 minutes. Add the turkey, chili powder, garlic, cumin, and salt, and cook, stirring and breaking up the meat as it browns, for 5 minutes. Add the beans, chicken stock, and chopped cilantro stems, and bring to a boil. Cook, mashing the beans frequently against the bottom and sides of the skillet, until the mixture is thickened to a refried bean consistency and the flavors have come together, about 15 minutes.\n\n3. While the turkey-bean mixture is simmering, brush the tortillas on both sides with the remaining 2 tablespoons olive oil. Place them on baking sheets and bake in the oven until crisp, 12 to 15 minutes. Remove from the oven, and transfer to paper towels to drain and crisp briefly before serving.\n\n4. Stir the cilantro leaves into the turkey-bean mixture. Spread the mixture over the crisped tortillas, and garnish with condiments as desired.\n\n8 to 10 tostadas, 4 to 6 servings\n\nCREAMY WHITE BEANS WITH SAUSAGE\n\nPrep time: 5 minutes Cook time: 25 minutes Total: 30 minutes\n\nThis creamy bean dish can work well as either an appetizer or a main course, and is delicious when eaten with pieces of crusty French bread. Don't forget to drizzle it with the best extra-virgin olive oil you have on hand, Italian-style, as this small gesture really makes a big difference.\n\n3 tablespoons olive oil\n\n1 tablespoon minced garlic\n\n12 ounces hot smoked sausage, sliced into \u00bd-inch-thick rounds\n\nOne 14.5-ounce can diced tomatoes, with juices\n\nFive 15-ounce cans cannellini beans, rinsed and drained\n\n1 tablespoon chopped fresh rosemary\n\n1 teaspoon chopped fresh thyme\n\n3 cups fresh spinach or arugula, rinsed and spun dry\n\nSalt and freshly ground black pepper\n\nExtra-virgin olive oil, for serving\n\nCrusty French or peasant bread, warmed, for serving (optional)\n\n1. Heat the oil in a large heavy Dutch oven over medium heat. Add the garlic and saut\u00e9 until fragrant, about 30 seconds. Add the sausage and tomatoes. Increase the heat to medium-high and simmer for 2 minutes. Add the beans and bring to a boil. Then reduce the heat to a simmer and cook until the beans are tender and flavorful, about 20 minutes.\n\n2. Remove the pot from the heat and stir in the rosemary, thyme, and spinach. Season with salt and pepper to taste, and serve hot in wide shallow bowls, drizzled with extra-virgin olive oil. Pass the French bread, if desired.\n\n2 quarts, 4 to 6 servings\n\nVegetables\n\nBACON BRAISED GREEN BEANS\n\nPrep time: 8 minutes Cook time: 17 minutes Total: 25 minutes\n\nThis modern take on the Southern classic will leave you wanting more! Since this version is quick-cooked, the beans stay crisp-tender\u2014but oh, they're coated with saut\u00e9ed onions and little pieces of bacon. Don't make me talk about it!\n\n1 tablespoon olive oil\n\n6 slices bacon, diced\n\n1 cup thinly sliced onion\n\n2 tablespoons sliced garlic\n\n2 pounds green beans, rinsed, ends trimmed\n\n1 cup chicken stock or canned, low-sodium chicken broth or water\n\n1 teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n1. Set a Dutch oven over medium heat, and add the olive oil. Once the oil is hot, add the bacon and cook, stirring occasionally, until it is well browned, about 5 minutes. Add the onion and garlic and cook, stirring occasionally, until the onion is translucent, 3 to 4 minutes. Add the green beans and toss to combine with the bacon and onion.\n\n2. Increase the heat to medium-high and add the chicken stock. As soon as the stock begins to boil, place the lid on the pan and cook the beans for about 6 minutes. Remove the lid, season the beans with the salt and pepper, and toss well. Replace the lid and cook until the beans are tender, 1 or 2 minutes longer.\n\n3. Remove from the heat and transfer the beans to a serving dish or small platter to serve.\n\n4 to 6 servings\n\nCREAMED MUSTARD GREENS\n\nPrep time: 16 minutes Cook time: 23 minutes Total: 39 minutes\n\nEveryone loves creamed spinach, but check out this version of the dish made with mustard greens and you'll be an instant convert. We have also made this with turnip greens and kale with good success\u2014truthfully, any type of green will do, and each type adds its own unique flavor. To stay within the forty-minute time frame, make sure you start the sauce while the greens are cooking.\n\nSalt\n\n6 pounds mustard greens, rinsed well, tough stems and ribs removed\n\n3 tablespoons canola oil\n\n1 cup finely chopped onion\n\n2 large cloves garlic, minced\n\n1 cup half-and-half\n\n8 ounces Neufch\u00e2tel cheese, cut into pieces\n\nPinch of freshly grated nutmeg\n\nKosher salt and freshly ground black pepper\n\n1. Bring a large stockpot of salted water to a boil. Prepare a large bowl of ice water. Add half of the mustard greens to the boiling water, a little at a time, pushing them down into the water. Let the water return to a boil and then cook until tender, about 10 minutes. Using tongs, transfer the greens to the bowl of ice water. Repeat with the remaining greens. Drain the greens very well and squeeze them dry in a kitchen towel. Finely chop, and set aside. (If your stockpot is large enough, you may be able to do this in one batch and save yourself some time.)\n\n2. While the greens are cooking, heat the canola oil in a 12-inch skillet over medium heat. Add the onion and garlic, and cook until softened, about 6 minutes. Stir in the half-and-half and the Neufch\u00e2tel, and simmer, stirring occasionally, until thickened and creamy, about 3 minutes.\n\n3. Add the greens to the cream mixture and cook, stirring, until warmed through, about 3 minutes. Season with the nutmeg, add salt and pepper to taste, and serve.\n\n6 cups, 4 to 6 servings\n\nSAUT\u00c9ED MUSHROOMS WITH FRESH THYME\n\nPrep time: 15 minutes Cook time: 20 minutes Total: 35 minutes\n\nThis mushroom dish is the perfect accompaniment to the New York Strip with Beurre Ma\u00eetre d'H\u00f4tel on 20 Minutes or Less!\n\n2 tablespoons olive oil\n\n2 tablespoons unsalted butter\n\n2 pounds button mushrooms, wiped clean, stemmed, quartered\n\n1 cup chopped yellow onion\n\n1 tablespoon thinly sliced garlic\n\n1\u00bd teaspoons fresh thyme leaves 1\u00bd teaspoons salt\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00be cup chicken stock or canned, low-sodium chicken broth\n\n2 tablespoons Worcestershire sauce\n\n1. Heat a 12-inch cast-iron skillet (or other heavy skillet) over medium-high heat, and add the olive oil. When the oil is hot, add the butter, and once it has melted, add the mushrooms, onion, garlic, thyme, salt, and pepper. Cook, stirring occasionally, until most of the liquid has been released from the mushrooms and has evaporated, about 15 minutes.\n\n2. Add the chicken stock and cook until nearly evaporated, about 4 minutes. Add the Worcestershire sauce and cook, stirring, for 1 minute. Remove from the heat and serve immediately.\n\n4 to 6 servings\n\nSESAME EGGPLANT\n\nPrep time: 11 minutes Cook time: 12 minutes Total: 23 minutes\n\nI'm a huge eggplant fan, and love it just about any way you can imagine. Here is a quick and easy preparation that I make for the family on weeknights. Note: If you have a large enough pan, you can cook this dish in one batch and in half the time.\n\n\u00bd cup peanut oil\n\n2 pounds eggplant, cut into \u00be-inch dice\n\n1\u00bd teaspoons salt\n\n4 green onions, bottoms minced and tops sliced, reserved separately\n\n2 tablespoons minced garlic\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bd teaspoon crushed red pepper\n\n1 tablespoon sesame seeds\n\n1 tablespoon dark Asian sesame oil\n\n1. Heat \u00bc cup of the oil in a large nonstick saut\u00e9 pan. When it is hot, add half of the eggplant and \u00be teaspoon of the salt. Cook until the eggplant is nicely browned and softened, stirring as necessary to promote even browning, about 5 minutes. Add half of the green onion bottoms, 1 tablespoon of the garlic, \u00bc teaspoon of the black pepper, \u00bc teaspoon of the crushed red pepper, \u00bd tablespoon of the sesame seeds, and \u00bd tablespoon of the sesame oil. Toss to combine. Transfer to a plate and set aside.\n\n2. Repeat with the remaining ingredients.\n\n3. Combine both batches in the same pan, add the green onion tops, and heat briefly until warmed through, 1 minute. Serve immediately.\n\n4 servings\n\nSPICY BRAISED GREENS\n\nPrep time: 10 minutes Cook time: 20 minutes Total: 30 minutes\n\nWould you like some amazing-tasting greens? This is the recipe! Make this once, and the next time you'll do it twice! To make this even faster, use the prewashed greens from your favorite produce section.\n\n3 tablespoons olive oil\n\n4 ounces smoked bacon slices, cut into 1-inch pieces (use turkey bacon if you prefer)\n\n1 cup sliced yellow onion\n\n1 tablespoon minced garlic\n\n2 pounds collard greens, mustard greens, beet greens, Swiss chard, or a combination, ribs removed, chopped into bite-size pieces, and rinsed\n\n\u00bd teaspoon crushed red pepper, or more to taste\n\n1 cup chicken stock or canned, low-sodium chicken broth\n\n2 tablespoons butter\n\n\u00bc teaspoon salt, or more to taste\n\n1. Heat a 12-inch saut\u00e9 pan over medium-high heat and add the olive oil. When it is hot, add the bacon and cook, stirring often, until it is well browned, about 4 minutes. Add the onion and cook, stirring often, until softened, about 3 minutes. Add the garlic and cook for 30 seconds. Add the greens and cook, stirring frequently, until softened, 3 minutes.\n\n2. Add the crushed red pepper and the chicken stock, and bring the liquid to a boil. Cover the pan and reduce the heat to medium-low. Cook until the greens are tender, about 10 minutes.\n\n3. Remove the lid, increase the heat, and bring the liquid to a high simmer. Cook for 5 minutes or until the liquid has reduced by half. Stir in the butter, and season with the salt. Serve hot.\n\n4 servings\n\nBUTTERMILK MASHED POTATOES\n\nPrep time: 10 minutes Cook time: 30 minutes Total: 40 minutes\n\nThis basic recipe is perfect as is, but feel free to embellish it with other ingredients such as goat cheese, grated cheddar cheese, roasted garlic, or sliced green onions. You just can't make the Simple Turkey Meatloaf on 60 Minutes or Less without making a batch of these to serve alongside!\n\n2 pounds Yukon gold potatoes, peeled and diced into 1-inch cubes\n\nSalt\n\n8 tablespoons (1 stick) butter, cubed\n\n1\u00bd cups buttermilk\n\nFreshly ground white pepper\n\n1. Place the potatoes in a pot of salted water and bring to a boil. Reduce the heat to a simmer and cook until the potatoes are fork-tender, 12 to 15 minutes.\n\n2. Remove the pot from the heat and drain the potatoes. Return the potatoes to the pot and set it over medium heat. Stir the potatoes constantly for 2 to 3 minutes to remove any excess liquid.\n\n3. Add the butter to the potatoes and using a handheld masher, mash the butter into the potatoes. Gradually add the buttermilk, mashing until the desired texture is achieved. Season the potatoes with salt and white pepper to taste, and serve.\n\n6 to 8 servings\n\nSeafood\n\nINDIAN-INSPIRED SHRIMP WITH COCONUT, CHILES, AND TOMATOES\n\nPrep time: 23 minutes Cook time: 15 minutes Total: 38 minutes\n\nThis dish is chock-full of complex flavors that'll make your guests think that you worked for hours in the kitchen. Who needs to know otherwise? Serve it with basmati rice for an authentic taste of India.\n\n2 tablespoons butter\n\n2 teaspoons mustard seeds\n\n2 teaspoons cumin seeds\n\n1 cup finely chopped red onion\n\n4 green serrano or jalape\u00f1o chiles, seeded and finely chopped\n\n2 tablespoons minced fresh ginger\n\n1 tablespoon minced garlic\n\nOne 14.5-ounce can diced tomatoes, with juices\n\nOne 13.5-ounce can unsweetened coconut milk\n\n1\u00bd pounds medium shrimp, peeled and deveined\n\n2 teaspoons salt\n\n2 tablespoons chopped fresh cilantro\n\n2 tablespoons chopped fresh chives (optional)\n\nSteamed basmati rice, for serving\n\n1. Melt the butter in a 14-inch saut\u00e9 pan over medium-high heat. Add the mustard seeds and cumin seeds and toast until fragrant, about 1 minute. Add the onion, chiles, ginger, and garlic and saut\u00e9, stirring occasionally, until the onion is translucent, about 3 minutes. Add the diced tomatoes and their juices, and cook until the liquid is reduced by half, 3 to 4 minutes. Raise the heat to high and add the coconut milk. Simmer until the milk is reduced by half, 3 to 4 minutes.\n\n2. Season the shrimp with 1\u00bd teaspoons of the salt. Add the shrimp to the pan and cook, stirring as needed, until they are curled, pink, and just cooked through, about 3 minutes.\n\n3. Remove the pan from the heat. Season with the remaining \u00bd teaspoon salt, and garnish with the cilantro and chives, if desired. Serve immediately over steamed basmati rice.\n\n4 to 6 servings\n\nFISH EN PAPILLOTE\n\nPrep time: 25 minutes Cook time: 12 minutes Total: 37 minutes\n\nA meal in a pouch! It doesn't get much simpler than this!\n\n1\u00bd cups thinly sliced red cabbage\n\n1 large onion (about 12 ounces), thinly sliced\n\n4 parsnips (about 12 ounces), cut into 1\/8-inch-thick rounds\n\n\u00bc cup chopped mixed soft herbs (such as parsley and tarragon)\n\n1\u00bd teaspoons salt\n\n\u00be teaspoon freshly ground white pepper\n\n3 tablespoons freshly squeezed lemon juice\n\n1 teaspoon honey\n\n6 tablespoons olive oil\n\nFour 6-ounce skinless fish fillets (such as red snapper or striped bass)\n\n1. Preheat the oven to 375\u00b0F.\n\n2. Place the cabbage, onion, parsnips, herbs, \u00bc teaspoon of the salt, and 1\/8 teaspoon of the white pepper in a mixing bowl and toss to combine; set aside.\n\n3. In a small mixing bowl, whisk together the lemon juice and honey. While whisking, add the olive oil in a slow, steady stream. Season with \u00bc teaspoon of the salt and 1\/8 teaspoon of the white pepper. Set aside.\n\n4. Using a paper towel, pat the fish fillets to dry them. Season each fillet on both sides with the remaining 1 teaspoon salt and \u00bd teaspoon white pepper.\n\n5. Assemble the packets: Fold four 14-inch squares of aluminum foil in half, forming four rectangles. Open the rectangles, and divide the cabbage mixture evenly among them, placing it just to one side of the fold. Stir the lemon-oil mixture well, and drizzle 1 tablespoon over each portion of cabbage. Place a fish fillet on top of each, and drizzle another tablespoon over each fillet. Fold the other side of the foil over the ingredients, and then fold the edges inward two or three times to form an airtight packet that is sealed on all sides.\n\n6. Place the packets on a baking sheet, and bake until the packets are puffed and the fish is cooked through and flakes easily when touched with a fork, about 12 minutes (the cook time may vary slightly, depending on the thickness of the fillets).\n\n7. Remove the baking sheet from the oven, and open the packets carefully (there will be a lot of steam escaping from the packet). Serve immediately.\n\n4 servings\n\nSWORDFISH WITH PUTTANESCA RELISH\n\nPrep time: 15 minutes Cook time: 10 minutes Total: 25 minutes\n\nUnlike puttanesca pasta sauce, this is more of a relish that is basically uncooked, save for a quick warming when you're finishing the swordfish steaks in the saut\u00e9 pan. The recipe uses only half of the relish that is made here; any unused puttanesca relish can be enjoyed over cooked chicken breasts, grilled fish, with pasta, or even as a topping for crostini or bruschetta. The leftover relish will keep in an airtight nonreactive container in the fridge for up to 3 days.\n\nOne and a half 14-ounce cans petite diced tomatoes, with juices\n\n\u00be cup halved pitted Kalamata olives\n\n1\u00bd tablespoons extra-virgin olive oil\n\n1 tablespoon red wine vinegar\n\n2 teaspoons minced garlic\n\n2 teaspoons anchovy paste or finely chopped canned anchovy fillets\n\n2 tablespoons nonpareil capers, drained\n\n4 tablespoons chopped fresh basil\n\n1 teaspoon freshly ground black pepper\n\nFour 6-ounce swordfish steaks, about 1\u00bd inches thick, patted dry\n\n\u00bd teaspoon salt\n\n2 tablespoons olive oil\n\n1. Combine the tomatoes and their juices with the olives, extra-virgin olive oil, vinegar, garlic, anchovy paste, capers, 2 tablespoons of the basil, and \u00bd teaspoon of the black pepper in a medium bowl and set aside.\n\n2. Season the swordfish steaks with the salt and the remaining \u00bd teaspoon black pepper. Heat the olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Add the swordfish and cook for 4 minutes. Turn the steaks over and cook for an additional 4 minutes.\n\n3. Add 2 cups of the puttanesca relish to the fish and cook for 2 minutes. Transfer the fish to a serving platter or individual plates, and spoon the warm puttanesca relish over it. Garnish with the remaining 2 tablespoons basil, and more relish if desired.\n\n4 servings, 4 cups puttanesca relish\n\nROASTED SCROD WITH HERBED BREADCRUMBS\n\nPrep time: 18 minutes Cook time: 12 minutes Total: 30 minutes\n\nScrod, cod, baby cod, haddock\u2014any of these fish would work beautifully in this simple preparation, which reminds me of my New England upbringing. If you're feeling indulgent, try drizzling the fillets with extra olive oil or melted butter just before serving.\n\nNonstick cooking spray\n\n3 tablespoons unsalted butter\n\n3 tablespoons minced shallot\n\n\u00bd cup fine unseasoned dry breadcrumbs\n\n1\u00bd teaspoons salt\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bc teaspoon grated lemon zest\n\n2 teaspoons chopped fresh parsley\n\n2 teaspoons chopped fresh chives\n\n2 teaspoons chopped fresh thyme\n\n2 tablespoons olive oil\n\nFour 6-to 8-ounce skinless young cod, scrod, or haddock fillets\n\n1. Preheat the oven to 400\u00b0F. Lightly grease a baking sheet with nonstick cooking spray.\n\n2. Melt the butter in a small saucepan over medium-high heat. Add the shallot and saut\u00e9 until tender, 1 to 2 minutes. Transfer the shallot and butter to a medium bowl, and set aside to cool briefly.\n\n3. Once the shallot-butter mixture has cooled slightly, add the breadcrumbs, \u00bd teaspoon of the salt, \u00bc teaspoon of the pepper, and the lemon zest, parsley, chives, and thyme. Stir to blend. Drizzle the olive oil into the breadcrumb mixture, tossing until the mixture is moistened.\n\n4. Pat the fillets dry with a paper towel, and season with the remaining 1 teaspoon salt and \u00bc teaspoon pepper. Place the fish on the prepared baking sheet. Top each fillet evenly with the breadcrumb mixture. Place the baking sheet in the oven and bake until the crust is golden brown and the fillets are just cooked through, 10 to 12 minutes, depending on their thickness.\n\n4 servings\n\nSHRIMP AND FETA, GREEK-STYLE\n\nPrep time: 5 minutes Cook time: 25 to 30 minutes Total: 30 to 35 minutes\n\nThe test kitchen made a big splash in our office the day this dish was prepared. Folks were coming out of the woodwork like crazy! Now, I must warn you: this is a bold dish that makes no apologies. If you are sensitive to spicy foods, I would suggest using less crushed red pepper than is called for here. And do not forget the crusty bread, as it's an absolutely essential part of this dish. It's just what's needed for sopping up the delicious pan juices.\n\n\u00bd cup plus 3 tablespoons extra-virgin olive oil\n\n2 tablespoons minced garlic\n\nTwo 14.5-ounce cans petite diced tomatoes, with juices\n\n\u00bd cup clam juice\n\n2\u00bd teaspoons chopped fresh oregano\n\n1 teaspoon crushed red pepper\n\n\u00bc cup nonpareil capers, drained\n\nSalt and freshly ground black pepper\n\n2 pounds large shrimp, peeled and deveined\n\n\u00bd cup Pernod\n\n8 ounces Greek feta, crumbled\n\n1 loaf peasant bread, for serving\n\n1. Preheat the oven to 450\u00b0F.\n\n2. Heat the \u00bd cup olive oil in a large saucepan over medium-high heat. When it is hot, add the garlic and cook until fragrant, about 1 minute. Add the tomatoes and their juices, clam juice, oregano, crushed red pepper, and capers, and cook until the sauce has thickened slightly, 4 to 6 minutes. Season to taste with salt and pepper.\n\n3. While the tomato sauce is cooking, heat the remaining 3 tablespoons olive oil in a large saut\u00e9 pan. When the oil is hot, add the shrimp and cook until they are just pink on both sides, about 2 minutes. Remove the pan from the heat, and add the Pernod. Return the pan to the heat and shake it carefully to ignite the alcohol. When the flames have died down, season the shrimp lightly with salt and pepper. Do not overcook; the shrimp should not be cooked through at this point.\n\n4. Spoon the tomato sauce into a large casserole or individual gratin dishes. Nestle the shrimp down in the sauce, and crumble the feta evenly over the top. Bake for 10 to 12 minutes, or until the shrimp are just cooked through and the sauce is bubbly.\n\n5. Remove from the oven and serve immediately, with pieces of crusty bread for dipping.\n\nNote: If using an electric burner, simmer the Pernod in the pan over medium-high heat for 30 seconds.\n\n6 servings\n\nSALMON WITH ORANGE BUTTER SAUCE\n\nPrep time: 8 minutes Cook time: 22 minutes Total: 30 minutes\n\nThe sauce for this simply saut\u00e9ed salmon is reminiscent of some of the serious sauces found in French cuisine but don't be fooled\u2014the dish can be on the table in no time flat.\n\n\u00be cup freshly squeezed orange juice\n\nOne \u00bd-inch strip of orange zest\n\n\u00bc cup julienned shallots\n\n\u00bd bay leaf\n\n1 teaspoon whole black peppercorns\n\n1 clove garlic, smashed\n\n1 sprig fresh thyme\n\n1\/3 cup dry white wine\n\n\u00bd cup heavy cream\n\n8 ounces (2 sticks) cold unsalted butter, cut into small pieces\n\nSalt and freshly ground black pepper\n\nFour 6-ounce salmon fillets, skin on\n\n2 tablespoons canola or vegetable oil\n\n1. Combine the orange juice, strip of orange zest, shallots, bay leaf, peppercorns, garlic, thyme, and wine in a saucepan, and bring to a boil. Reduce the heat to a simmer and cook until the liquid has reduced by three-fourths, about 8 minutes. Add the heavy cream and cook until the liquid has reduced by half, about 4 minutes longer. Whisk in the cold butter little by little, whisking until the sauce is smooth and thick and all the butter is incorporated, 3 to 4 minutes; do not allow the sauce to boil. Strain the sauce through a fine-mesh sieve into another saucepan, pressing on the solids to extract all the liquid; discard the solids. Season the sauce with salt and pepper to taste, and keep warm until ready to serve. (Do not allow the sauce to boil or it will separate.)\n\n2. Season the salmon on both sides with \u00bd teaspoon salt and \u00bc teaspoon black pepper. Heat the canola oil in a large skillet over medium-high heat. Add the salmon, skin side down, and cook until lightly browned and crisp, 3 to 4 minutes. Flip the salmon over and cook until the fish is just cooked through, about 2 minutes.\n\n3. Serve the salmon, skin side up, immediately, with the orange butter sauce spooned around the fillets.\n\n4 servings\n\nBAKED FLOUNDER WITH CARROTS, SPINACH, AND AN ASIAN VINAIGRETTE\n\nPrep time: 18 minutes Cook time: 12 minutes Total: 30 minutes\n\nMy wife, Alden, loves fish and this is one of the ways we enjoy fresh fillets at home with the family. So easy, it's just about foolproof\u2014as long as you take care not to overcook the fish. It's also important to use the freshest fish you can find. If flounder is unavailable or too pricey, feel free to substitute any mild, white-fleshed fish fillets, but note that if you use thick fillets, the cook time will vary accordingly.\n\n6 tablespoons extra-virgin olive oil\n\n2 tablespoons freshly squeezed lemon juice\n\n2 tablespoons soy sauce\n\n2 teaspoons dark Asian sesame oil\n\n2 teaspoons honey\n\n1 teaspoon minced fresh ginger\n\n1 teaspoon minced garlic\n\n2\u00bc teaspoons salt\n\n1\u00bc teaspoons freshly ground white pepper\n\n8 ounces prewashed baby spinach, stemmed, and roughly chopped\n\n2 carrots, sliced into ribbons with a vegetable peeler\n\n1\/3 cup mixed fresh parsley, cilantro, and tarragon leaves\n\nFour 6-ounce skinless flounder fillets\n\n1. Preheat the oven to 375\u00b0F.\n\n2. In a medium bowl, combine the extra-virgin olive oil, lemon juice, soy sauce, sesame oil, honey, ginger, and garlic, and whisk well to combine. Season with \u00bc teaspoon of the salt and \u00bc teaspoon of the white pepper.\n\n3. In a separate bowl, combine the spinach, carrots, and herbs, and drizzle with 6 tablespoons of the vinaigrette; toss well to combine. Arrange the salad in a glass or ceramic casserole or baking dish. Season the flounder on both sides with the remaining 2 teaspoons salt and 1 teaspoon pepper. Lay the fish over the salad, and drizzle 1 tablespoon of the vinaigrette over each of the fillets. Place in the oven and bake until the vegetables are wilted and the fish flakes easily when pierced with a fork, about 12 minutes.\n\n4. Remove from the oven and serve the fish with some of the wilted vegetables.\n\n4 servings\n\nPoultry\n\nBOURSIN CHEESE, SPINACH, AND PECAN\u2013STUFFED CHICKEN BREASTS\n\nPrep time: 26 minutes Cook time: 14 minutes Total: 40 minutes\n\nEverybody loves stuffed chicken breasts, so make 'em happy. If you don't have a piping bag, fill a heavy resealable plastic storage bag with the filling, snip a corner to create a diagonal opening, and squeeze.\n\n3 tablespoons butter\n\n4 ounces prewashed fresh spinach\n\n8 ounces pecans, toasted and chopped (about 1 cup)\n\nOne 5.2-ounce Black Pepper Boursin cheese (or other flavor Boursin)\n\nSix 6-to 8-ounce boneless, skinless chicken breasts\n\n1\u00bd teaspoons salt\n\n\u00bd teaspoon freshly ground black pepper\n\n\u00bd cup all-purpose flour\n\n3 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd cup milk\n\n1 egg\n\n1 cup unseasoned dry breadcrumbs\n\n2 tablespoons olive oil\n\n1. Preheat the oven to 400\u00b0F. Line a baking sheet with parchment paper, and set it aside.\n\n2. Melt 1 tablespoon of the butter in a small saut\u00e9 pan over medium heat. Add the spinach and cook until wilted, about 2 minutes. Remove the spinach from the pan, drain all the liquid, and chop the spinach. Transfer it to a small bowl. Add the chopped pecans and the Boursin to the spinach, and stir to combine. Using a rubber spatula, transfer the mixture to a piping bag fitted with a medium tip. Set aside.\n\n3. Lay the chicken breasts flat on a clean work surface. With a paring knife, cut a long slit in the thick side of each breast, slicing about three-fourths of the way down and 2 inches deep to form a deep pocket (be careful not to go through the other side). Stuff each breast with the filling by piping it in tightly. Secure the opening with a toothpick. Season the breasts on both sides with \u00bd teaspoon of the salt and the black pepper.\n\n4. Combine the flour and 1 teaspoon of the Essence in a shallow baking dish or bowl. In a second shallow pan, combine the milk, egg, and 1 teaspoon of the Essence. In a third shallow pan, combine the breadcrumbs, remaining 1 teaspoon Essence, and remaining 1 teaspoon salt. Dredge each chicken breast first in the flour, then in the egg wash, and lastly in the breadcrumbs; set aside.\n\n5. Heat the olive oil and the remaining 2 tablespoons butter in a large skillet over medium-high heat. Add the chicken and saut\u00e9 until browned, 2 minutes on each side. Place the seared chicken breasts on the prepared baking sheet, place it in the oven, and bake until cooked through and the temperature registers 165\u00b0F on an instant-read thermometer, about 10 minutes.\n\n6. Remove the toothpicks before serving.\n\n6 servings\n\nHONEY-LEMON-THYME CORNISH GAME HENS\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nWhat would this book be without a Cornish hen recipe? Gotta have one! Cornish hens are tiny enough for everybody to have their own, easily served. Having guests? Perfect.\n\nFour 1-pound Cornish game hens\n\n8 tablespoons (1 stick) butter\n\n1 tablespoon plus 1 teaspoon chopped fresh thyme\n\nGrated zest of 2 lemons\n\n4 teaspoons salt\n\n1 teaspoon freshly ground black pepper\n\n2 tablespoons freshly squeezed lemon juice\n\n1\/3 cup honey\n\n1 tablespoon plus 1 teaspoon soy sauce\n\n1. Position a rack in the center of the oven and preheat the oven to 500\u00b0F. Line a large baking dish with aluminum foil, and set it aside.\n\n2. Rinse the hens well, inside and out, under cool running water. Pat them dry with paper towels.\n\n3. Combine 4 tablespoons of the butter, the thyme, and the lemon zest in a small bowl and use a fork to blend well. Divide the mixture into 4 portions, and spread one portion under the skin covering the breast of each hen. Combine 1 teaspoon of the salt with \u00bd teaspoon of the black pepper, and season the cavities of the hens. Truss the hens, fold the wing tips back and tuck them under, and arrange the hens breast side up in the prepared baking dish.\n\n4. Combine the lemon juice, honey, soy sauce, and the remaining 4 tablespoons butter in a small saucepan, and warm over low heat until heated through. Divide the honey mixture in half, and set aside one portion. Use some of the remaining portion to baste the tops of the hens well. Season the hens with the remaining 3 teaspoons salt and \u00bd teaspoon pepper.\n\n5. Transfer the baking dish to the oven and cook, basting the hens with the honey mixture about every 5 minutes, until they are nicely browned, 20 to 25 minutes. If necessary, tent the hens with foil during the last few minutes of cooking to prevent over-browning.\n\n6. When the hens reach an internal temperature of 165\u00b0F, remove the dish from the oven and let them rest for about 5 minutes. Discard the honey mixture used for basting. Use the reserved honey mixture to drizzle over the hens before serving.\n\nNote: To check the temperature, insert an instant-read thermometer in the thickest part of the breast, avoiding any bones.\n\n4 servings\n\nOVEN-ROASTED CHICKEN WINGS\n\nPrep time: 15 minutes Cook time: 20 minutes Total: 35 minutes\n\nIf you don't want to bother with cutting the chicken wings yourself (though it only takes 5 minutes), buy little drumettes from your grocer instead. Add a little lemon juice, some herbs and spices, and roast in a smokin' hot oven. You won't have to clean the fryer or hide from your doctor. Need I say more?\n\n4 pounds chicken wings, wing tips removed and discarded, separated at the joint\n\n\u00bc cup freshly squeezed lemon juice\n\n1 tablespoon freshly ground black pepper\n\n1 tablespoon garlic powder\n\n1 tablespoon onion powder\n\n2 teaspoons salt\n\n2 teaspoons dried thyme\n\n\u00bd teaspoon cayenne pepper\n\n4 tablespoons (\u00bd stick) butter, melted\n\nYour favorite sauce (such as barbecue sauce, ranch dressing, or blue cheese dressing), for serving (optional)\n\n1. Preheat the oven to 500\u00b0F. Line a large rimmed baking sheet with aluminum foil.\n\n2. In a large mixing bowl, combine the wings and the lemon juice and mix thoroughly. Add the pepper, garlic powder, onion powder, and salt. While crushing it between your fingers, add the thyme. Mix again, and add the cayenne and melted butter. Mix thoroughly a final time. Then transfer the wings to the prepared baking sheet, and arrange them in one layer.\n\n3. Roast in the oven for 10 minutes. Rotate the pan and roast for 10 minutes longer, until the wings are nicely browned and cooked through. Serve as is or with your favorite dipping sauce.\n\n4 to 6 servings\n\nCRISPY PAN-ROASTED CHICKEN WITH GARLIC-THYME BUTTER\n\nPrep time: 5 minutes Cook time: 23 to 25 minutes Total: 28 to 30 minutes\n\nOne whole chicken, cut in half, cooks in half the time! Go figure. You will not believe how something so simple can be so good. I mean, isn't it just chicken? Prepare to swoon.\n\nOne 3\u00bd-pound chicken, halved, with the breastbone, backbone, and first two digits (tips) of the wings removed\n\n1 tablespoon kosher salt\n\n1\u00bd teaspoons freshly ground white pepper\n\n4 teaspoons olive oil\n\n2 tablespoons unsalted butter, at room temperature\n\n1 teaspoon minced garlic\n\n1 teaspoon fresh thyme leaves\n\n1. Preheat the oven to 400\u00b0F.\n\n2. Season the chicken halves on both sides with the salt and white pepper. Set a 12-inch cast-iron skillet over high heat, and when it is hot, add the olive oil. Swirl the skillet to coat it evenly, and then lay the seasoned chicken halves, skin side down, in the skillet. Sear until golden, about 3 minutes.\n\n3. Transfer the skillet to the oven and roast until the chicken is nearly cooked through and the skin is crispy, about 17 minutes. Turn the chicken over and continue to roast, skin side up, until it is cooked through, 3 to 5 minutes.\n\n4. While the chicken is roasting, combine the butter with the garlic and thyme in a small bowl, and stir well to blend.\n\n5. As soon as the chicken is removed from the oven, spread the garlic butter over the skin and serve immediately.\n\n2 to 4 servings\n\nCHICKEN CORDON BLEU\n\nPrep time: 25 minutes Cook time: 10 minutes Total: 35 minutes\n\nTo make things even easier, these babies can be filled and breaded up to a day ahead of time and cooked later\u2014just six minutes in the pan and four minutes in the oven! You just wait for them to get hot and melty in the center.\n\nFour 6-ounce boneless, skinless chicken breasts (about 1\u00bd pounds)\n\n\u00bd teaspoon salt\n\n\u00bc teaspoon freshly ground black pepper\n\n6 ounces sliced Swiss cheese (4 to 6 slices)\n\n4 ounces thinly sliced prosciutto or Black Forest ham\n\n1 cup all-purpose flour\n\n2 eggs\n\n2 tablespoons milk\n\n1 cup fine unseasoned dry breadcrumbs\n\n1 tablespoon plus 2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n3 tablespoons olive oil\n\n1. Butterfly each chicken breast; then cut down the middle to separate the halves so that you have a total of 8 pieces of chicken. Place the chicken between two pieces of plastic wrap, and pound each piece, using a mallet or the bottom of a heavy skillet, to a thickness of about \u00bc inch. Lay the 8 chicken pieces on a baking sheet, and sprinkle each side with the salt and pepper. Divide the cheese evenly among 4 of the chicken pieces. Arrange the prosciutto slices evenly over the cheese. Top each \"filled\" piece of chicken with one of the \"unfilled\" pieces, trying to tuck in any of the cheese or prosciutto that extends over the edges. Secure the chicken pieces together using toothpicks along both long edges.\n\n2. Preheat the oven to 400\u00b0F, and line a baking sheet with parchment paper.\n\n3. Place the flour in a shallow bowl. In another shallow pan, combine the eggs and milk with a fork. Place the breadcrumbs in a third shallow pan. Season the flour with 1 tablespoon of the Essence. Season the breadcrumbs with the remaining 2 teaspoons Essence. Dip each chicken scallop \"sandwich\" in the flour, and shake to remove any excess. Then dip it in the egg-milk mixture, and finally dip it in the seasoned breadcrumbs. Set the breaded chicken pieces on a plate.\n\n4. Heat the olive oil in a 14-inch saut\u00e9 pan over medium-high heat. Arrange all 4 chicken \"sandwiches\" in the pan and cook until nicely browned on one side, about 4 minutes. Turn the \"sandwiches\" over and cook for 2 minutes longer. Then transfer them to the prepared baking sheet and place in the oven. Cook for 4 minutes, or until the cheese is melted and bubbly and the chicken is just cooked through. Remove the toothpicks and serve immediately.\n\n4 servings\n\nTURKEY SALTIMBOCCA\n\nPrep time: 12 minutes Cook time: 13 minutes Total: 25 minutes\n\nTurkey cutlets are available at most groceries today\u2014no pounding needed. That makes it easy! Once browned, the turkey is finished in the oven while you complete the sauce.\n\n5 tablespoons olive oil\n\n16 large fresh sage leaves\n\n1 teaspoon salt plus more for seasoning\n\nEight 4-to 6-ounce turkey breast cutlets\n\n\u00be teaspoon freshly ground black pepper plus more for seasoning\n\n8 thin slices (about 4 ounces) prosciutto\n\n\u00bd cup all-purpose flour\n\n1 cup chicken stock or canned, low-sodium chicken broth\n\n1 cup dry white wine\n\n6 tablespoons (\u00be stick) butter, cut into 3 pieces\n\n1. Preheat the oven to 400\u00b0F. Line a baking sheet with parchment paper. Place a slotted spoon and a paper towel\u2013lined plate near the stove.\n\n2. Heat 1 tablespoon of the olive oil in an 8-inch saut\u00e9 pan over medium-high heat. Fry 8 of the sage leaves, in batches if necessary, for about 10 seconds, and then quickly remove them from the pan with the slotted spoon and transfer them to the lined plate to drain. Sprinkle the leaves with a little salt, and set aside. (The leaves will be nicely green and crisp when they have cooled. This can be done a day in advance; keep the fried sage leaves in a closed container at room temperature.)\n\n3. Lay the turkey cutlets on a clean, flat work surface. Season the cutlets on both sides with 1 teaspoon salt and \u00be teaspoon pepper. Lay one of the remaining (uncooked) sage leaves down the middle of each cutlet, and then roll each cutlet into a tight cylinder. Wrap a slice of prosciutto around each roll.\n\n4. Place the flour in a small shallow bowl, and carefully dredge the turkey rolls in the flour. Set aside.\n\n5. Heat the remaining 4 tablespoons olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Add the turkey rolls, seam side down, to the pan and brown for 2 minutes. Turn them over and continue to brown on all sides for 2 to 3 minutes longer. Transfer the turkey rolls to the prepared baking sheet (reserve the saut\u00e9 pan), place it in the oven, and bake for 8 minutes or until the internal temperature registers 165\u00b0F on an instant-read thermometer.\n\n6. While the turkey is baking, make the sauce: Add the chicken broth and white wine to the hot saut\u00e9 pan over medium-high heat. Cook, scraping up any browned bits from the pan, for about 7 minutes or until the liquid is reduced to 1\/3 cup. Whisk in the butter in three separate additions, and remove the sauce from the heat. Season to taste with salt and pepper.\n\n7. Spoon 1 tablespoon of the sauce over each turkey roll, and top each with a fried sage leaf. Serve immediately.\n\n6 to 8 servings\n\nPANKO-CRUSTED CHICKEN TENDERS\n\nPrep time: 15 minutes Cook time: 15 minutes Total: 30 minutes\n\nA chicken comes with two tenders\u2014one under each breast. In this recipe, you separate the tenders and then cut the rest of the breasts into the tender shape. If you buy chicken breasts that do not include the tender, don't worry; just cut the breast into strips and call them all \"tenders.\"\n\n2 pounds boneless, skinless chicken breasts\n\n1\u00bc teaspoons salt\n\n\u00bd teaspoon freshly ground black pepper\n\n4 cloves garlic\n\n1 egg\n\n\u00bd cup buttermilk\n\n\u00bc cup Crystal hot sauce or other Louisiana red hot sauce\n\n2\u00bd cups panko breadcrumbs\n\n\u00bd teaspoon sweet paprika\n\n1\u00bc cups canola oil or other vegetable oil\n\n1. Remove the tenders from the chicken breasts and place them in a medium-size bowl. Cut the chicken breasts into strips that are similar in size and shape to the tenders, about 1 inch wide and 4 to 5 inches long. Add the chicken strips to the bowl, season with 1 teaspoon of the salt and the black pepper, and mix well to combine.\n\n2. Smash the cloves of garlic with the flat side of your knife. Sprinkle the remaining \u00bc teaspoon salt over the garlic, and chop and mash the garlic to form a paste. Transfer the garlic to a medium bowl and whisk in the egg, buttermilk, and hot sauce. Pour this mixture over the chicken pieces and mix well.\n\n3. In another medium bowl or in a gallon-size resealable plastic food storage bag, combine the panko crumbs and paprika. Place half of the chicken pieces in the crumbs and toss to coat evenly. Remove the pieces from the crumbs, shake them lightly, and then transfer them to a small baking sheet or platter. Repeat with the remaining chicken. (The breaded pieces may be stacked on top of one another.)\n\n4. Heat \u00be cup of the canola oil to 350\u00b0F in a 12-inch (or larger) saut\u00e9 pan over high heat. Once the oil is hot, lightly shake loose any excess crumbs from the chicken. Add half of the chicken pieces to the oil, reduce the heat to medium, and cook until the chicken is golden on both sides and just cooked through, 2 to 3 minutes per side. Transfer the cooked chicken to a paper towel\u2013lined plate to drain. Increase the heat under the pan to high, add the remaining \u00bd cup canola oil, and when it is hot, add the remaining chicken pieces. Reduce the heat to medium and cook the chicken in the same manner. Serve hot or at room temperature.\n\nNote: Panko breadcrumbs are most widely used in Japanese cooking. They are made from crustless bread and create a crispier coating.\n\n4 servings\n\nMeat\n\nSLOPPY JOES\n\nPrep time: 10 minutes Cook time: 20 minutes Total: 30 minutes\n\nA school cafeteria classic that takes me back...definitely an oldie but goodie! The Sloppy Joe mixture is even better if left to sit overnight, or for up to three days, in the refrigerator before serving. Or, hey, make a big batch on the weekend and freeze it in airtight containers for busy weeknight meals.\n\n2 tablespoons plus 2 teaspoons olive oil\n\n2 cups diced onions\n\n\u00bd cup diced celery (small dice)\n\n\u00bd cup diced green bell pepper (small dice)\n\n1 teaspoon freshly ground black pepper\n\n\u00be teaspoon salt\n\n2 teaspoons minced garlic\n\n1\u00bd pounds lean ground beef (about 91% lean)\n\n3 tablespoons dark brown sugar\n\n3 tablespoons Worcestershire sauce\n\n2 cups tomato sauce\n\n1 cup beef or chicken stock or canned, low-sodium chicken broth\n\n2 teaspoons hot sauce (optional)\n\n4 to 6 hamburger buns\n\n1. Heat the 2 teaspoons olive oil in a 12-inch saut\u00e9 pan over medium-high heat. Add the onions, celery, bell pepper, black pepper, and salt, and cook until the vegetables are soft, about 2 minutes. Add the garlic and cook for 1 minute longer. Add the beef, breaking it up with the back of a spoon, and brown for 2 minutes. Then add the brown sugar, Worcestershire, tomato sauce, and beef broth, and bring to a boil. Reduce the heat to a simmer and cook for 15 minutes, stirring as needed. Add the hot sauce if desired, and remove from the heat.\n\n2. While the mixture is cooking, preheat the broiler.\n\n3. Arrange the buns, open-faced, on a baking sheet and lightly brush the cut sides with the remaining 2 tablespoons olive oil. Broil until the buns are golden, about 3 minutes. (Alternatively, grill them in a grill pan or toast them in a skillet, oiled side down, over medium-high heat.)\n\n4. Generously spoon the Sloppy Joe mixture over the toasted bun bottoms. Top with the bun tops, and serve immediately (with forks).\n\n4 to 6 servings\n\nCOUNTRY-FRIED STEAK WITH WHITE GRAVY\n\nPrep time: 10 minutes Cook time: 30 minutes Total: 40 minutes\n\nWe've added bacon to this classic home-style dish, and boy, talk about kickin' it up a notch. Serve this alongside the Buttermilk Mashed Potatoes on 40 Minutes or Less\u2014they're the perfect vehicle for this rich, creamy gravy and crispy fried steaks. I can taste it now!\n\n8 ounces sliced bacon, cut crosswise into \u00bd-inch pieces\n\n1 cup plus 1\u00bd tablespoons all-purpose flour\n\n3 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 large egg\n\n2 to 3 cups milk, as needed\n\nFour 6-to 8-ounce cube steaks\n\n1\u00bd teaspoons salt\n\n1 teaspoon freshly ground black pepper\n\nVegetable oil, for frying, as needed\n\n\u00bd cup minced yellow onion\n\n1. Cook the bacon in a 12-inch saut\u00e9 pan over medium-high heat until just crisp, about 5 minutes. Remove the pan from the heat. Using a slotted spoon, transfer the bacon to paper towels to drain, leaving the fat to cool in the pan.\n\n2. Combine the 1 cup flour with 1 teaspoon of the Essence in a shallow bowl or pan. Whisk the egg, \u00bd cup of the milk, and 1 teaspoon of the Essence in another shallow bowl or pan.\n\n3. Season the steaks all over with 1\u00bc teaspoons of the salt and \u00bd teaspoon of the black pepper. Dredge the meat in the seasoned flour, then dip in the egg wash, letting the excess drip off, and then dredge in the flour a final time. Set the steaks aside on a tray.\n\n4. Pour the cooled bacon fat into a liquid measuring cup, and add enough vegetable oil to measure \u00bd cup total. Return the mixture to the saut\u00e9 pan and heat over medium-high heat until it is hot but not smoking. Carefully add 2 of the steaks and fry until golden, 3 minutes per side. Transfer them to paper towels to drain. Add the remaining 2 steaks to the pan, dipping them in the flour one more time before frying if necessary, and cook in the same manner. Transfer the steaks to paper towels.\n\n5. Add the 1\u00bd tablespoons flour to the pan and cook, whisking constantly, for 1 minute. Add the onion and cook, stirring often, until softened, about 3 minutes. Whisk in 1\u00bd cups of the milk, the remaining 1 teaspoon Essence, the remaining \u00bc teaspoon salt, and the remaining \u00bd teaspoon black pepper. Return the bacon to the pan and bring the gravy to a boil. Reduce the heat to medium-low and simmer for 5 to 6 minutes, until the sauce has thickened and there is no raw flour taste. Add additional milk as necessary to achieve the desired consistency.\n\n6. Serve the steaks with the hot gravy.\n\n4 servings\n\nSAUSAGES AND SAUERKRAUT\n\nPrep time: 10 minutes Cook time: 21 minutes Total: 31 minutes\n\nThe type of beer you use here will vary the flavor. If you prefer a slightly bitter flavor to complement your sausage, use a dark or amber beer. A lighter American-style beer will lend a sweeter flavor. Make it with what you like. Make it with what you have! And if you're out of beer, hey, use either more stock or apple cider.\n\nOne 28-ounce jar sauerkraut, drained\n\n4 ounces sliced apple-cured bacon, cut crosswise into \u00bd-inch pieces\n\n1\u00bd medium yellow onions, sliced (about 2 cups)\n\n1 tablespoon unsalted butter\n\n1\u00bd pounds kielbasa or other smoked sausage, cut into 3-inch lengths, casing scored lightly on two sides\n\n2 tablespoons minced garlic\n\n4 fresh thyme sprigs\n\n2 bay leaves\n\n\u00bd teaspoon coarsely ground black pepper\n\n1 cup chicken stock or canned, low-sodium chicken broth\n\n1 cup beer\n\n1. Place the sauerkraut in a colander and rinse it briefly to remove some of the salt from the brine (don't rinse it too much, or you will lose a lot of the flavor). (Alternatively, if the sauerkraut is not excessively salty, use as is.) Press to release most of the excess liquid, and set aside.\n\n2. Cook the bacon in a large nonreactive skillet or Dutch oven over medium heat for 5 minutes until the fat is rendered. Add the onions and cook for 2 minutes longer. Move the bacon and onions to the side of the skillet and increase the heat to medium-high. Add the butter, and when it has melted, add the sausage. Cook, turning occasionally, until it is browned on both sides and the scores are beginning to open up a bit, 1 to 2 minutes per side.\n\n3. Add the garlic, thyme sprigs, bay leaves, and pepper to the skillet and cook, stirring, for 30 seconds. Add the drained sauerkraut. Toss to combine, scraping up the brown bits that have accumulated in the pan. Add the chicken stock and beer, and bring to a boil. Cover, reduce the heat to a simmer, and cook for 10 minutes.\n\n4. Remove the bay leaves and serve hot.\n\n4 servings\n\nSTIR-FRIED BEEF AND BROCCOLI\n\nPrep time: 15 minutes Cook time: 12 minutes Total: 27 minutes\n\nOnce you've assembled the ingredients, this dish goes together like one-two-three. You'll find it easiest to slice the beef in thin strips if it has been partially frozen. And, hey, if you have one very large wok or skillet, this can be done in one batch, cutting the cook time in half. How do you like that!\n\n\u00bc cup soy sauce\n\n2 tablespoons minced garlic\n\n4 teaspoons grated fresh ginger\n\n4 teaspoons Chinese black vinegar or balsamic vinegar\n\n3 teaspoons hoisin sauce\n\n3 teaspoons honey\n\n1\u00bd teaspoons crushed red pepper\n\n2 tablespoons cornstarch\n\n4 tablespoons peanut oil\n\n1 large head broccoli, cut into florets\n\n2 pounds beef sirloin steak, thinly sliced across the grain\n\nSteamed white rice, for serving\n\n1. Combine the soy sauce, garlic, ginger, vinegar, hoisin sauce, honey, crushed red pepper, and cornstarch in a small bowl. Mix well, and set aside.\n\n2. Heat a wok or a large skillet over high heat. When it is hot, add 2 tablespoons of the peanut oil and half of the broccoli and cook, shaking the work and stirring frequently, until the broccoli begins to soften, 2 to 3 minutes. Add half of the beef and cook, stirring, for 1 to 2 minutes. Add half of the sauce and stir to evenly coat. Cook until the sauce thickens and the broccoli is crisp-tender, about 1 minute.\n\n3. Transfer to a serving platter and keep warm while you repeat the process with the other half of the ingredients. Serve immediately with steamed white rice.\n\n4 to 6 servings\n\nQUICK AND EASY LAMB KEBABS\n\nPrep time: 15 minutes Cook time: 6 to 8 minutes Total: 21 to 23 minutes\n\nIf you have the time, these simple kebabs can benefit from longer marinating. Why not prep the meat and marinade, set it aside to marinate overnight in the refrigerator, and then pull it out of the fridge the next day and make the kebabs? If you're like me and love a good sauce, these kebabs would be great served with Minted Yogurt Sauce (40 Minutes or Less).\n\n\u00bc cup plus 1 tablespoon olive oil\n\n2 tablespoons chopped fresh rosemary\n\n2 teaspoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1\u00bd pounds boneless lamb (such as leg), cut into 1-inch cubes\n\n2 green bell peppers, cut into 1-inch squares\n\n2 onions, cut into 1-inch pieces\n\n8 large white mushrooms, wiped clean, stems trimmed\n\n8 large cherry tomatoes\n\nSalt and freshly ground black pepper\n\n1. Combine the \u00bc cup olive oil, rosemary, and Essence in a small bowl. Place the lamb in the bowl and let it marinate while you assemble the remaining ingredients, at least 10 minutes.\n\n2. Arrange the lamb, alternating with the vegetables, on eight 8-inch metal skewers and place them on a large baking sheet. Brush the vegetables with the remaining 1 tablespoon olive oil, and season the skewers on all sides with salt and pepper to taste.\n\n3. Position a rack about 6 inches from the broiler element, and preheat the broiler.\n\n4. Broil the kebabs, turning them once midway, until the meat is well browned, 6 to 8 minutes for medium-rare. Serve hot.\n\n4 servings\n\nTHIN-CUT PORK CHOPS WITH ROSEMARY-BALSAMIC GLAZED SHALLOTS\n\nPrep time: 10 minutes Cook time: 15 to 17 minutes Total: 25 to 27 minutes\n\nThese quickly seared pork chops are out-of-the-box good when paired with the simple pan sauce. Serve with hot white rice or buttered noodles and a simple vegetable for a complete meal.\n\nFour 6-ounce center-cut pork chops\n\n2 teaspoons kosher salt\n\n1 teaspoon freshly ground white pepper\n\n2 tablespoons Wondra flour (see Note, 20 Minutes or Less)\n\n2 tablespoons olive oil\n\n1 tablespoon unsalted butter\n\n1 cup thinly sliced shallots\n\n1 teaspoon minced garlic\n\n\u00bd teaspoon chopped fresh rosemary\n\n\u00bd cup balsamic vinegar\n\n1\u00bd cups chicken stock or canned, low-sodium chicken broth\n\n1. Season the pork chops on both sides with the salt and white pepper. Dust each pork chop lightly with the Wondra, and set aside.\n\n2. Set a 12-inch cast-iron skillet over medium-high heat, and add the olive oil and butter. When it is hot, place the pork chops in the skillet and sear for 2 minutes per side. Remove the chops from the skillet and set them aside. Add the shallots to the skillet and cook, stirring often, until lightly caramelized, about 2 minutes. Add the garlic and rosemary and cook until fragrant, about 30 seconds.\n\n3. Add the balsamic vinegar and deglaze the skillet. When the vinegar has nearly evaporated (about 1\u00bd minutes), add the chicken stock. Increase the heat to high, and return the pork chops to the skillet. Baste the pork chops with the stock and cook until the liquid has reduced to a sauce consistency, 6 to 8 minutes.\n\n4. Remove from the heat, and serve hot.\n\n4 servings\n\nDesserts\n\nKICKED-UP SNICKERDOODLES\n\nPrep time: 25 minutes Cook time: 14 minutes Total: 39 minutes\n\nEveryone loves snickerdoodles! We've sneaked in a bit of cayenne pepper for a surprising little kick.\n\n2\u00be cups all-purpose flour\n\n1 teaspoon baking soda\n\n\u00bd teaspoon salt\n\n\u00bd cup solid vegetable shortening, at room temperature\n\n8 tablespoons (1 stick) unsalted butter, at room temperature\n\n1\u00bd cups plus 5 tablespoons sugar\n\n2 large eggs\n\n1 tablespoon ground cinnamon\n\n\u00bc teaspoon cayenne pepper\n\n1. Preheat the oven to 350\u00b0F.\n\n2. Sift the flour, baking soda, and salt together into a bowl and set aside.\n\n3. In a separate medium mixing bowl, using a handheld or standing electric mixer, combine the shortening and butter and beat until smooth. Add 1\u00bd cups plus 2 tablespoons of the sugar, and continue beating until light and fluffy, about 3 minutes. Add the eggs, one at a time, beating well after each addition. Add the sifted flour mixture and mix until the dough just comes together.\n\n4. Combine the remaining 3 tablespoons sugar with the cinnamon and cayenne in a small bowl.\n\n5. Divide the dough into 1\u00bd-tablespoon portions and roll them between your hands to form 1\u00bd-inch balls. Roll the balls in the spiced sugar. Then divide them evenly among two unlined cookie sheets, spacing them about 1 inch apart. Flatten the balls into \u00bd-inch-thick disks.\n\n6. Bake, rotating the sheets back to front midway, until the edges of the cookies are golden brown, 12 to 14 minutes. Let the cookies cool on the baking sheets on wire racks.\n\nAbout 30 cookies\n\nSKILLET CORN CAKE WITH STEWED CHERRIES\n\nPrep time: 15 minutes Cook time: 25 minutes Total: 40 minutes\n\nTo make the best use of your time, prepare the deceptively delicious cherry sauce while the corn cake is baking. The cake is best served either warm or at room temperature, with the stewed cherries spooned over the top. Since frozen cherries are available year-round, you'll never need to wait for cherry season to enjoy this dessert.\n\n1 cup all-purpose flour\n\n1 teaspoon baking powder\n\n\u00be teaspoon salt\n\n6 tablespoons yellow cornmeal\n\n2 large eggs\n\n1 large egg yolk\n\n2\/3 cup milk\n\n\u00bd cup olive oil\n\n2 teaspoons grated lemon zest\n\n1\u00bd cups sugar\n\n2 tablespoons unsalted butter\n\nTwo 10-ounce bags frozen pitted cherries\n\n\u00bc cup freshly squeezed lemon juice\n\n1. Place a 10-inch cast-iron skillet in the oven and preheat the oven to 350\u00b0F.\n\n2. In a medium bowl, sift together the flour, baking powder, and salt. Stir in the cornmeal.\n\n3. In a separate medium bowl, whisk together the eggs, egg yolk, milk, olive oil, and lemon zest until frothy. Add \u00be cup of the sugar, and whisk to combine. Pour the wet ingredients over the dry ingredients and mix just until the batter is smooth.\n\n4. Swirl the butter in the hot cast-iron skillet until it has melted, and then pour the batter into the skillet. Return the skillet to the oven and cook until the center is set, about 25 minutes.\n\n5. While the cake is baking, set a 10-inch skillet over high heat, and add the cherries, lemon juice, and remaining \u00be cup sugar. Cook until the cherries have released most of their juice, 10 to 12 minutes. Remove from the heat and set aside until ready to use.\n\n6. Allow the cake to cool in the skillet for 5 minutes. Then slice it into wedges and serve with the stewed cherries spooned over the top.\n\n8 servings\n\n## 60 Minutes OR LESS\n\nSOUPS\n\nEmeril's New-Style Caldo Verde\n\nShrimp and Corn Chowder\n\nSTARTERS\n\nCaramelized Onion and Goat Cheese Tart\n\nPASTA\n\nBetter Than Mama's Chili-Mac\n\nRICE AND BEANS\n\nChicken and Mushroom Risotto\n\nPortuguese Rice with Tuna\n\nChili-Beans\n\nSEAFOOD\n\nSeared Salmon with Lentils\n\nRoasted Whole Red Snapper with Orange, Rosemary, and Kalamata Olives\n\nBrazilian Fish Stew\n\nShrimp \u00c9touff\u00e9e\n\nPOULTRY\n\nRoast Turkey Breast with Bacon and Sage\n\nRoast Chicken with Shallot-Garlic Butter\n\nMEAT\n\nPork Loin with Apples and Prunes\n\nGrilled Marinated Pork Tenderloins with an Orange-Apricot Glaze\n\nDESSERTS\n\nPeach-Blueberry Crisp\n\n60 MINUTES +\n\nBraised Chicken Thighs\n\nSpinach and Mushroom Lasagna\n\nSimple Turkey Meatloaf\n\nSoups\n\nEMERIL'S NEW-STYLE CALDO VERDE\n\nPrep time: 15 minutes Cook time: 45 minutes Total: 60 minutes\n\nIf I had to choose one dish to represent my childhood, it would be this. I call this version \"new-style\" because the kale is cut into thin strips and is cooked only until crisp-tender, which differs from the more traditional version. Ines, my Portuguese friend back home, would be proud. Serve this with crusty bread alongside.\n\n2 tablespoons olive oil\n\n1\u00bd cups finely chopped yellow onions\n\n1 tablespoon minced garlic\n\n2 pounds Idaho potatoes, peeled and cut into \u00bd-inch cubes\n\n7 cups chicken stock or canned, low-sodium chicken broth\n\nSalt and freshly ground black pepper, to taste\n\n\u00bd teaspoon crushed red pepper\n\n8 ounces kale, large stems and ribs removed\n\n8 ounces firm (smoked) chorizo or other hot smoked sausage, diced or crumbled\n\n\u00bd cup chopped fresh cilantro\n\n\u00bc cup chopped fresh parsley\n\n2 tablespoons chopped fresh mint\n\n1. Heat the olive oil over medium-high heat in a large soup pot, and add the onions and garlic. Cook until the onions are wilted, 4 minutes. Add the potatoes and chicken stock, cover, and bring to a boil. Season with salt and pepper, and add the crushed red pepper. Reduce the heat to a simmer and cook, uncovered, until the potatoes are tender, 20 minutes.\n\n2. While the potatoes are cooking, thinly slice the kale. Set aside.\n\n3. When the soup is thick and the potatoes have begun to break down, add the sausage and cook for 5 minutes. Stir in the kale and simmer until the leaves have softened but are still slightly crunchy and the flavors have melded, 15 minutes. Stir in the cilantro, parsley, and mint, and season to taste with salt and pepper. Serve hot.\n\n4 to 6 servings\n\nSHRIMP AND CORN CHOWDER\n\nPrep time: 15 minutes Cook time: 35 minutes Total: 50 minutes\n\nThis version of a classic Louisiana country soup is simplified by the use of frozen sweet corn, making it easy to cook up year-round. But if fresh local corn is in season when you decide to give this soup a spin, by all means use that instead.\n\n4 tablespoons (\u00bd stick) butter\n\n6 ounces smoked sausage, cut into \u00bc-inch dice (about 1 cup)\n\n2 cups diced onions (small dice)\n\n1 tablespoon minced garlic\n\n1\u00bd teaspoons salt\n\n1 teaspoon dried thyme\n\n\u00bd teaspoon cayenne pepper\n\n\u00bc cup all-purpose flour\n\n7 cups shrimp or chicken stock or canned, low-sodium shrimp or chicken broth\n\n1 pound Idaho potatoes, peeled and cut into \u00bd-inch dice (about 2 cups)\n\n3 cups frozen sweet corn\n\n1 pound shrimp, peeled and deveined\n\n\u00bd cup heavy cream\n\n\u00bd cup chopped green onions, white and green parts (optional)\n\n1. Melt the butter over medium-high heat in a 6-quart (or larger) stockpot. Add the sausage and cook until it is browned and the fat is rendered, about 2 minutes. Add the onions and cook until translucent, about 3 minutes. Add the garlic, salt, thyme, and cayenne, and cook for 1 minute.\n\n2. Sprinkle the flour into the pot and cook, stirring often, for 2 minutes.\n\n3. Whisk in the stock, and add the potatoes. Increase the heat to high, cover the pot, and bring to a boil. Uncover, reduce the heat to a simmer, and cook for 15 minutes.\n\n4. Add the corn and cook for 5 minutes. Add the shrimp and heavy cream, and cook until the shrimp is just cooked through, 2 to 3 minutes.\n\n5. Remove the pot from the heat and serve the soup hot, garnished with the chopped green onions if desired.\n\n2\u00bd quarts, 6 servings\n\nStarters\n\nCARAMELIZED ONION AND GOAT CHEESE TART\n\nPrep time: 15 minutes Cook time: 40 minutes Total: 55 minutes\n\nI present to you a tart reminiscent of the French pissaladi\u00e8re. (To make this one more authentic, simply replace the cheese with anchovies and top the baked tart with Ni\u00e7oise olives.) For a twist, you could garnish the tart with toasted pine nuts and fresh figs for an elegant appetizer. Another idea would be to add saut\u00e9ed spinach. As is or fancied up, this delectable tart is irresistible!\n\n1 egg\n\n\u00bc cup plus 1 tablespoon water\n\nAll-purpose flour, for rolling out puff pastry\n\n1 sheet frozen puff pastry, thawed in the refrigerator\n\n4 tablespoons (\u00bd stick) butter\n\n4 large onions, thinly sliced (about 8 cups)\n\n1\u00be teaspoons salt\n\n1\u00bc teaspoons freshly ground black pepper\n\n4 ounces goat cheese\n\n2 tablespoons chopped fresh basil\n\n1\u00bd teaspoons minced garlic\n\n1\u00bd teaspoons extra-virgin olive oil\n\n1. Line an 11 \u00d7 17-inch rimmed baking sheet with parchment paper.\n\n2. In a small bowl, mix the egg and the 1 tablespoon water with a fork.\n\n3. Lightly flour your work surface and roll out the puff pastry to an 18 \u00d7 12-inch rectangle (this is more easily done by rolling from corner to corner, on a diagonal, adding more flour as necessary). Transfer the dough to the lined baking sheet. (Another tip: Fold the pastry in half, then, like closing a book, in half again. Unfold it on the baking sheet.) Fold a 1-inch edge toward the center and press it lightly with your fingers to seal it, to form a border along all sides. Brush the edges with the egg mixture. Prick the pastry inside the border thoroughly with your fork. Refrigerate for at least 20 minutes or as long as overnight before baking.\n\n4. Preheat the oven to 400\u00b0F.\n\n5. While the pastry is chilling, cook the onions: Melt the butter in a 12-inch skillet over high heat. Add the onions, 1\u00bd teaspoons of the salt, and 1 teaspoon of the pepper, and cook, stirring intermittently, until the onions begin to caramelize, about 10 minutes. Reduce the heat to medium-low and continue to cook, stirring frequently, for 7 minutes. Increase the heat to high, add the \u00bc cup water, and stir, scraping up any caramelized bits, and cook for 3 minutes longer. Remove the skillet from the heat, transfer the onions to a small baking sheet, spread them out, and chill in the freezer for 5 minutes.\n\n6. While the onions are chilling, make the filling by combining the goat cheese, basil, garlic, extra-virgin olive oil, remaining \u00bc teaspoon salt, and remaining \u00bc teaspoon black pepper in a small bowl with a rubber spatula. Set aside until ready to use.\n\n7. Bake the puff pastry in the oven for 5 minutes.\n\n8. Remove the onions from the freezer and stir to help cool them (it's okay if the onions are still somewhat warm). Spread the onions evenly over the pastry. Drop heaping teaspoonfuls of the goat cheese mixture all over the pastry. Return the tart to the oven and bake, rotating it halfway through, until the edges are golden and the cheese has begun to brown, 15 minutes. Serve immediately.\n\n6 to 8 servings\n\nPasta\n\nBETTER THAN MAMA'S CHILI-MAC\n\nPrep time: 10 minutes Cook time: 40 minutes Inactive time: 5 minutes Total: 55 minutes\n\nNow this is a scrumptious casserole. It is super-simple, and best of all, it's a huge pan of bowl-you-over goodness that will feed the entire family...and then some!\n\n2 teaspoons olive oil\n\n2 medium onions, cut into small dice (about 3 cups)\n\n2 jalape\u00f1os, stemmed and minced (optional)\n\n2 to 3 teaspoons salt, plus more for the pasta water\n\n2 pounds extra-lean ground beef\n\n5 tablespoons Mexican chili powder\n\n1 tablespoon dried Mexican oregano\n\n2 tablespoons minced garlic\n\nOne 28-ounce can whole plum tomatoes, broken with your hands, with juices\n\nTwo 15-ounce cans kidney beans, drained\n\n\u00bd cup water\n\n1 pound elbow macaroni 1 pound medium sharp cheddar cheese\n\nSour cream, for serving (optional)\n\n1. Heat the olive oil in a 6-quart soup pot over medium-high heat. Add the onions, jalape\u00f1os (if desired), and 2 teaspoons of the salt. Cook until the onions are soft, 2 minutes. Add the ground beef, chili powder, oregano, and garlic, and cook, breaking up any clumps of meat with a spoon, for 5 minutes. Add the tomatoes and their juices, beans, and water. Stir, and bring the chili to a boil. Then reduce the heat to a simmer and cook until thickened to chili consistency, about 20 minutes. Taste, and add 1 more teaspoon salt if needed.\n\n2. While the chili is simmering, preheat the oven to 400\u00b0F.\n\n3. Place a 9 \u00d7 13\u00bd-inch or other 3-quart baking dish on a baking sheet. Bring a pot of salted water to a boil, add the macaroni, and cook until just tender, 6 minutes. Drain in a colander, rinse under cool running water, and set aside. Grate the cheddar cheese and set it aside.\n\n4. Once the chili has finished cooking, fold in the cooked macaroni and one-third of the cheddar. Transfer the chili-mac mixture to the baking dish and top with the remaining cheese. Bake until the chili is heated through and the cheese has melted, about 10 minutes.\n\n5. Remove from the oven and let cool for 5 minutes before serving. Garnish each portion with a dollop of sour cream, if desired.\n\n6 to 8 servings\n\nRice and Beans\n\nCHICKEN AND MUSHROOM RISOTTO\n\nPrep time: 15 minutes Cook time: 31 to 36 minutes Total: 46 to 51 minutes\n\nEveryone loves risotto, right? Well, brown some chicken, saut\u00e9 some mushrooms, and turn a simple risotto into a complete meal.\n\n1 pound boneless, skinless chicken thighs, cut into bite-size pieces\n\n1\u00bd teaspoons salt\n\n1 teaspoon freshly ground black pepper\n\n6 tablespoons olive oil\n\n1 shallot, thinly sliced\n\n1 pound shiitake mushrooms, wiped clean, stemmed, and sliced\n\n2 cups Arborio rice\n\n\u00bc cup balsamic vinegar\n\n\u00be cup red wine\n\n6 cups chicken stock, or canned, low-sodium chicken broth, heated\n\n2 tablespoons butter\n\n\u00bd cup finely grated Parmigiano-Reggiano cheese\n\n1 tablespoon fresh thyme leaves\n\n1. Season the chicken with the salt and pepper, and set aside.\n\n2. Heat 4 tablespoons of the olive oil in a 14-inch saut\u00e9 pan over medium-high heat. Add the shallot and cook, stirring with a heat-resistant rubber spatula or a wooden spoon, until fragrant and soft, about 1 minute. Add the mushrooms and cook until nicely browned, 3 minutes.\n\n3. Move the mushrooms and shallots to the edge of the pan, and add the remaining 2 tablespoons olive oil. Add the chicken and cook without stirring until browned on one side, 2 minutes. Add the rice and cook, stirring, until the grains are opaque, about 2 minutes.\n\n4. Add the balsamic vinegar and the wine, and continue cooking, stirring frequently, until nearly all of the liquid has been absorbed, 1 minute. Reduce the heat to medium, stir in \u00be cup of the hot broth, simmer, and stir until nearly all the liquid has been absorbed, about 3 minutes. Continue in this manner, adding more broth, \u00be cup at a time, and not adding more until the previous addition is absorbed, until all the broth has been added and the risotto is tender and creamy, 20 to 25 minutes.\n\n5. Fold in the butter, cheese, and thyme. Remove from the heat, adjust the seasoning as necessary, and serve immediately.\n\n6 to 8 servings\n\nPORTUGUESE RICE WITH TUNA\n\nPrep time: 10 minutes Cook time: 30 minutes Inactive time: 5 minutes Total: 45 minutes\n\nThis moist rice dish is classic Portuguese\u2014it'll knock your socks off. The tuna melts into the flavorful rice along with the onions and peppers and all but disappears. This is a perfect side dish or light entr\u00e9e when served with a green salad and crusty bread.\n\n1\u00bd tablespoons olive oil\n\n1 medium onion, cut into small dice (about 1\u00bd cups)\n\n1 green bell pepper, cut into small dice (about 1 cup)\n\n3 large cloves garlic, minced (about 1 tablespoon)\n\n1 \u00bc teaspoons salt\n\n\u00bc teaspoon freshly ground black pepper\n\n\u00bc teaspoon crushed red pepper\n\nOne 14.5-ounce can petite diced tomatoes, drained\n\nOne 4.5-ounce can good-quality solid tuna packed in olive oil\n\n1\u00bd cups long-grain white rice\n\n1\u00bd cups chicken stock or canned, low-sodium chicken or vegetable broth\n\n2 tablespoons chopped fresh parsley\n\n1. Heat the oil over medium-high heat in a 4-quart pot. Add the onion and bell pepper and cook until soft, about 3 minutes. Add the garlic, salt, pepper, and crushed red pepper and cook until fragrant, about 1 minute. Add the tomatoes and cook for 4 minutes. Stir in the tuna and its oil, breaking it up with a spoon and mixing well, and cook for 1 minute. Add the rice, stir well to coat, and cook for 1 minute. Stir in the stock and bring to a boil. Cover the pot and reduce the heat to a gentle simmer, and cook until the rice is tender and most of the liquid has been absorbed, 20 minutes.\n\n2. Remove the pot from the heat and let the rice sit, covered, for 5 minutes.\n\n3. Add the parsley and using a fork, gently mix it into the rice, fluffing the rice at the same time. Serve immediately.\n\n6 generous cups, 4 to 6 servings\n\nCHILI-BEANS\n\nPrep time: 15 minutes Cook time: 31 to 36 minutes Total: 46 to 51 minutes\n\nWhen they taste it, people will think that this quick-cook chili simmered on the stove all day. Garnish it with the typical suspects if you like\u2014hey, a little cheddar cheese, sour cream, green onions, and jalape\u00f1o peppers never hurt anyone!\n\n4 slices thick-cut apple-smoked bacon, diced\n\n2 medium onions, diced (about 3 cups)\n\n\u00bd red bell pepper, diced (about 1 cup)\n\n1\u00bd pounds lean ground chuck\n\n3 tablespoons Mexican chili powder\n\n2 teaspoons ground cumin\n\n1 tablespoon minced garlic\n\n1 teaspoon dried Mexican oregano\n\n1\u00bd teaspoons salt\n\nOne 14-ounce can petite diced tomatoes, with juices\n\nTwo 14.5-ounce cans pinto beans, drained and rinsed\n\nOne 14.5-ounce can black beans, drained and rinsed\n\n2 cups water\n\n1. Cook the bacon in a heavy Dutch oven over medium-high heat until the fat is rendered and the bacon is crisp, about 5 minutes.\n\n2. Add the onions and bell pepper and cook, stirring occasionally, until the vegetables have softened, about 3 minutes.\n\n3. Add the ground chuck, chili powder, cumin, garlic, oregano, and salt. Cook, stirring occasionally and breaking the meat into small pieces, until the meat is well browned, about 8 minutes.\n\n4. Add the tomatoes with their juices, beans, and water, and bring to a boil. Reduce the heat to a simmer, partially cover the pot, and cook, stirring occasionally, until the meat and beans are tender and the sauce is thick and flavorful, 15 to 20 minutes. Serve hot.\n\n2 quarts, 4 to 6 servings\n\nSeafood\n\nSEARED SALMON WITH LENTILS\n\nPrep time: 10 minutes Cook time: 50 minutes Total: 60 minutes\n\nThere is something about the combination of salmon and lentils that always makes me come back for more, so I prepare this dish often at home for the family. If you can find French green lentils, get them\u2014they're worth it because they hold their shape well after cooking. Beluga lentils would also work well here. Whatever type of lentil you use, don't overcook them\u2014they should be just tender when you drain them.\n\n9 tablespoons olive oil, plus more for drizzling (optional)\n\n1\u00bc cups diced red onion (small dice)\n\n\u00bd cup diced celery (small dice)\n\n2 cups French green lentils\n\n8 cups chicken stock or canned, low-sodium chicken broth\n\n1 bay leaf\n\nSalt\n\nFour 6-ounce salmon fillets, skin on\n\nFreshly ground black pepper\n\n1 cup diced red bell pepper (small dice)\n\n\u00bc cup chopped fresh parsley\n\n3 tablespoons balsamic vinegar\n\n1 teaspoon grated lemon zest\n\n1. Heat 3 tablespoons of the olive oil in a medium saucepan over medium-high heat. When it is hot, add 1 cup of the onion and the celery and cook, stirring, until soft, about 4 minutes. Add the lentils, chicken stock, and bay leaf, and bring to a boil. Reduce the heat to a simmer and cook, uncovered and stirring occasionally, until the lentils are just tender, 35 to 45 minutes. Season to taste with salt, and then drain the lentils in a colander. Set aside while you prepare the salmon.\n\n2. Season the salmon fillets on both sides with salt and pepper to taste. Heat 3 tablespoons of the remaining olive oil in a medium saut\u00e9 pan over medium-high heat. Place the salmon in the pan, skin side down, and cook until golden brown and crisp, 3 to 4 minutes. Turn the fillets over and cook to the desired degree of doneness, about 2 minutes for medium-rare. Transfer the salmon to paper towels to drain briefly.\n\n3. Transfer the drained lentils to a mixing bowl and add the remaining \u00bc cup red onion, the bell pepper, parsley, balsamic vinegar, lemon zest, and the remaining 3 tablespoons olive oil. Toss well to combine, and adjust the seasoning if necessary. Divide the lentils among four wide, shallow bowls, and top each with a salmon fillet. Serve immediately, drizzled with olive oil if desired.\n\n4 servings\n\nROASTED WHOLE RED SNAPPER WITH ORANGE, ROSEMARY, AND KALAMATA OLIVES\n\nPrep time: 20 minutes Cook time: 35 to 40 minutes Total: 55 to 60 minutes\n\nI know that there are a lot of you out there who shy away from cooking whole fish at home. But, hey, it's a great technique that leaves you with super-moist, tender fish and delicious pan juices. It's a knockout presentation. Take a walk on the wild side and give this one a try.\n\nTwo 3-pound whole red snappers, scaled and gutted\n\n2 tablespoons kosher salt\n\n2 teaspoons freshly ground white pepper\n\n2 large oranges, each sliced into 4 thick rounds\n\n6 sprigs fresh rosemary, plus 2 tablespoons fresh rosemary leaves\n\n\u00bc cup julienned orange zest\n\n\u00bd cup sliced garlic cloves (\u00bc-inch-thick slices)\n\n1 cup sliced pitted Kalamata olives\n\n1\u00bd cups dry white wine\n\n1 cup freshly squeezed orange juice\n\n\u00be cup extra-virgin olive oil\n\n1. Preheat the oven to 425\u00b0F.\n\n2. Cut 2 parallel diagonal slices, about 3 inches apart and \u00bc inch deep, into each side of the fish. This will make the fish easier to serve. Season the snappers on both sides with the salt and white pepper. Place the orange slices in a large roasting pan, arranging them in two columns of 4 slices each. Place 3 rosemary sprigs across each of the rows of oranges. Lay the fish on top of the orange slices, and scatter the rosemary leaves, orange zest, garlic, and olives evenly over the 2 fish. Pour the wine and orange juice around the fish, and drizzle each fish with 2 tablespoons of the olive oil.\n\n3. Place the pan in the oven and roast, uncovered, for 20 minutes.\n\n4. Remove the pan from the oven and drizzle 2 more tablespoons of the olive oil over each fish. Return it to the oven and continue to roast until the fish is well caramelized and the flesh flakes easily when tested with a fork, 15 to 20 minutes.\n\n5. Remove the pan from the oven and drizzle the fish with the remaining 4 tablespoons olive oil. Serve immediately, with the pan drippings drizzled over the fish.\n\nNote: To serve the fish, use a large spoon to lift the fish section by section between the diagonal cuts. When the fish from the first side has been served, remove the bone by lifting the tail. Serve the remaining fillets.\n\n4 to 6 servings\n\nBRAZILIAN FISH STEW\n\nPrep time: 25 minutes Cook time: 20 minutes Total: 45 minutes\n\nThe flavors and colors of this dish will wow any seafood lover, trust me. Now, I've gotta tell you, it is a bit on the spicy side, so feel free to reduce the cayenne peppers if you're not a spice lover. With white rice served alongside, this is a complete meal.\n\n2\u00bd pounds red grouper, skin on, scaled and cut into 2-inch pieces (or substitute redfish, flounder, striped bass, escolar, or any other white-fleshed fish)\n\n3 tablespoons freshly squeezed lime juice\n\n\u00bc cup olive oil\n\n1\u00bd cups thinly sliced onions\n\n2 to 3 fresh cayenne chiles, stemmed, seeded, and roughly chopped\n\n1 tablespoon minced garlic\n\n2 tablespoons tomato paste\n\nOne 14.5-ounce can diced tomatoes, with juices\n\n\u00bd cup fish or chicken stock, or canned, low-sodium chicken broth, or water\n\n2 teaspoons salt\n\nOne 14.5-ounce can unsweetened coconut milk\n\n2 tablespoons chopped fresh cilantro\n\nSteamed white rice, for serving\n\n1. Place the fish in a large nonreactive mixing bowl, add the lime juice, and set aside to marinate while you proceed with the recipe.\n\n2. Heat a large saut\u00e9 pan over medium-high heat, and add the olive oil. Once it is hot, add the onions and cayenne peppers and saut\u00e9, stirring often, until the onions are translucent, 3 to 4 minutes. Add the garlic and saut\u00e9 for 30 seconds. Add the tomato paste, diced tomatoes, stock, and 1 teaspoon of the salt, and stir well to incorporate. Bring the mixture to a boil. Season the fish with the remaining teaspoon of salt. Then add the fish (with the lime juice) and the coconut milk. Stir to combine, and bring the liquid to a boil. Cover the pan, reduce the heat to medium-low, and cook until the flesh of the fish starts to flake, about 10 minutes.\n\n3. Remove the cover, sprinkle the cilantro over the fish, and serve accompanied by steamed white rice.\n\n6 to 8 servings\n\nSHRIMP \u00c9TOUFF\u00c9E\n\nPrep time: 16 minutes Cook time: 44 minutes Total: 60 minutes\n\nMost of the active time here is chopping the vegetables; once the sauce is together, the simmering time leaves you free to do other things. If you have more time available, the sauce can be simmered over low heat for a longer time. This is even better if made a day in advance; simply reheat it gently over low heat.\n\n6 tablespoons (\u00be stick) unsalted butter\n\n4 cups chopped onions\n\n2 cups chopped green bell peppers\n\n2 cups chopped celery\n\n2 tablespoons minced garlic\n\n\u00bd cup all-purpose flour\n\nOne 14.5-ounce can diced tomatoes, with juices\n\n2 tablespoons Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n3 cups shrimp or chicken stock or canned, low-sodium shrimp or chicken broth\n\n3 pounds medium shrimp, peeled and deveined\n\nSalt and cayenne pepper, to taste\n\nSteamed white rice, for serving\n\n\u00bc cup chopped fresh parsley\n\n1. Melt the butter in a large Dutch oven over medium-high heat. Add the onions, bell peppers, celery, and garlic. Cover the pot and cook, stirring occasionally, until the vegetables are tender, 8 to 10 minutes.\n\n2. Add the flour and cook, stirring constantly to pick up the browned bits from the bottom of the pot, until the roux is a golden brown color, 3 to 4 minutes.\n\n3. Add the tomatoes with juices and Essence and continue to cook, stirring, for 2 minutes. Add the stock and bring to a boil. Reduce the heat to a steady simmer and cook until the sauce is thick and flavorful, about 25 minutes.\n\n4. Add the shrimp and cook until they are just cooked through, 2 to 3 minutes. Season with salt and cayenne pepper to taste, and spoon over cooked white rice in wide, shallow bowls. Garnish with the parsley, and serve.\n\n3 quarts, 6 to 8 servings\n\nPoultry\n\nROAST TURKEY BREAST WITH BACON AND SAGE\n\nPrep time: 15 minutes Cook time: 40 to 42 minutes Total: 55 to 57 minutes\n\nWhen folks tell me that they don't want to fool with roasting an entire turkey, I understand. It can be an undertaking, for sure. But this herbed turkey breast is another thing altogether. It goes together in no time, and the meat ends up incredibly moist and flavorful. I would have no problem whatsoever serving this as the centerpiece of a small Thanksgiving get-together. Just don't skip the bacon!\n\n3 slices thick-cut bacon\n\n1\u00bd tablespoons minced garlic\n\n1 tablespoon chopped fresh sage\n\n1\u00bd teaspoons chopped fresh rosemary\n\n1 teaspoon chopped fresh oregano\n\n1 teaspoon chopped fresh thyme\n\n3 tablespoons unsalted butter, at room temperature\n\n2 teaspoons kosher salt\n\n1 teaspoon freshly ground black pepper\n\nOne 2-pound boneless turkey breast, rinsed and patted dry\n\n1. Preheat the oven to 500\u00b0F. Line a shallow roasting pan or baking dish with aluminum foil.\n\n2. Cook the bacon in a medium skillet over medium-high heat until crisp, about 6 minutes. Transfer the bacon to paper towels to drain, reserving 1 tablespoon of the rendered bacon fat separately.\n\n3. When the bacon has cooled, chop it fine and transfer it to a small bowl. Add the garlic, sage, rosemary, oregano, thyme, butter, 1 teaspoon of the salt, and \u00bd teaspoon of the pepper. Using a small spoon, mix thoroughly to combine and form a paste.\n\n4. Using your fingers, gently loosen the skin on both sides of the turkey breast so that it is separated from the flesh. Divide the herb paste in half, and gently spread half between the skin and the flesh on each side of the breast. Season the outside of the turkey breast with the remaining 1 teaspoon salt and \u00bd teaspoon pepper. Brush the turkey all over with the reserved bacon fat.\n\n5. Place the turkey breast in the prepared pan and roast, uncovered, for 10 minutes. Reduce the heat to 400\u00b0F and cook for another 20 to 25 minutes, or until an instant-read thermometer inserted into the deep portion of the breast registers 165\u00b0F.\n\n6. Remove the pan from the oven and allow the turkey to rest for 10 minutes before carving it into thin slices. Serve with the pan drippings.\n\n6 to 8 servings\n\nROAST CHICKEN WITH SHALLOT-GARLIC BUTTER\n\nPrep time: 10 minutes Cook time: 45 minutes Total: 55 minutes\n\nEveryone needs a good roast chicken in their lives every now and again, and in my family, Sunday dinners are often all about roast chicken. Let's face it, once the chicken is prepped and in the oven, well, as they say, it's all gravy, baby. Take a look at the photos if you've never carved a chicken\u2014it just takes practice, and once you get the hang of it, you can practically do it with your eyes closed.\n\n4 tablespoons unsalted butter, at room temperature\n\n2 tablespoons minced shallot\n\n1 tablespoon chopped fresh parsley\n\n1 teaspoon minced garlic\n\n1 teaspoon salt\n\n\u00be teaspoon freshly ground black pepper\n\n1 whole chicken (about 3 pounds), excess fat and giblets removed, rinsed and patted dry\n\n1. Preheat the oven to 450\u00b0F.\n\n2. In a small bowl, combine the butter with the shallot, parsley, garlic, \u00bc teaspoon of the salt, and \u00bc teaspoon of the pepper. Rub 1 tablespoon of the flavored butter on the inside of the chicken, and sprinkle the cavity with \u00bc teaspoon of the salt and \u00bc teaspoon of the pepper. Use your fingers to gently loosen the skin covering the breast, and place \u00bd tablespoon of the flavored butter under the skin on each side of the breast (working from the top end) and under the skin of each thigh (working from the bottom end of the breast). Reserve the extra flavored butter for brushing on the chicken when finished.\n\n3. Tuck the wings behind the bird and tie the leg ends together with kitchen twine. Sprinkle the remaining \u00bd teaspoon salt and \u00bc teaspoon pepper all over the chicken. Set the chicken in a shallow baking dish (just large enough to hold the chicken) and place it in the oven with the legs to the back. Roast for 20 minutes. Baste the chicken with any accumulating juices and roast for 15 minutes longer. Baste the chicken again and roast until the thigh registers 165\u00b0F when tested with an instant-read thermometer, 10 to 15 minutes.\n\n4. Transfer the chicken to a cutting board, and let it rest for 10 minutes. Brush the warm chicken with the remaining flavored butter, if desired, then carve and serve with the pan juices spooned over the top.\n\n4 to 6 servings\n\n1 After the chicken has rested for at least 10 minutes and has been set on a carving board, remove the kitchen twine.\n\n2 Using a sharp knife, begin to separate the leg and thigh from the breast by slicing between the leg and breast, through the skin, to the joint.\n\n3 To further detach and expose the ball joint, gently pry the leg and thigh away from the breast by pressing it toward your cutting surface. With the tip of your knife, slice through the joint to completely separate the leg quarter. This will involve cutting through to the backbone. Set the leg quarter aside. To separate the leg from the thigh, cut through the joint. If you meet resistance, reposition the knife blade.\n\n4 To remove the breast, slice against the breastbone along the rib. You will have to angle the knife so that the blade runs flat against the ribs (you are not cutting through the ribs; you want to remove the breast from the ribs). Use the tip of your knife to separate the meat from the wishbone.\n\n5 Make another slice along the bottom length of the breast (running parallel to the wing), remove the meat, and set aside. Detach the wing by cutting through the joint. Carve the other side in the same manner.\n\nMeat\n\nPORK LOIN WITH APPLES AND PRUNES\n\nPrep time: 10 minutes Cook time: 50 minutes Total: 60 minutes\n\nThis impressive pork roast is a great dish for the fall, when apples are in season at your local farmer's market and you can purchase fresh apple cider. Cozy up to a nice bottle of wine and you're there.\n\n\u00bd cup dried prunes\n\n1\u00bc cups fresh apple cider\n\nOne 2\u00bd-pound boneless pork loin\n\n2 teaspoons salt\n\n1 teaspoon freshly ground black pepper\n\n3 tablespoons olive oil\n\n2 small onions, cut into \u00bd-inch-wide wedges\n\n2 Pink Lady or Honeycrisp apples, peeled, cored, and cut into \u00bd-inch-wide wedges\n\n2 cloves garlic, sliced\n\n\u00bc cup chicken stock or canned, low-sodium chicken broth\n\n4 fresh thyme sprigs\n\n2 tablespoons cider vinegar\n\n4 tablespoons (\u00bd stick) cold unsalted butter, cut into small pieces\n\n1. Preheat the oven to 325\u00b0F.\n\n2. Place the prunes in a small bowl, add \u00bc cup of the apple cider, and set aside to soak while you proceed with the recipe.\n\n3. Season the pork loin with 1\u00bd teaspoons of the salt and \u00be teaspoon of the black pepper.\n\n4. Heat the oil in a large Dutch oven over high heat. When it is hot, add the pork loin and cook until browned on all sides, 8 minutes. Remove the pork from the pan and transfer it to a baking sheet or platter.\n\n5. Reduce the heat to medium and add the onions, apples, and garlic to the Dutch oven. Cook, stirring occasionally, until caramelized, about 3 minutes. Add the soaked prunes (with any remaining juices) and cook, stirring occasionally, for 3 minutes. Add the remaining 1 cup apple cider and the chicken stock, and bring to a boil.\n\n6. Return the pork loin to the pot, add the thyme sprigs, and bring to a simmer. Cover, and transfer the pot to the oven. Cook, undisturbed, until the pork registers 145\u00b0F on an instant-read thermometer, about 30 minutes.\n\n7. Remove the pot from the oven and transfer the pork to a platter. Tent it with aluminum foil to keep warm.\n\n8. Place the Dutch oven over medium-high heat and bring the onion-apple mixture to a boil. Cook until slightly reduced, about 3 minutes. Remove and discard the thyme sprigs. Add the vinegar, remaining \u00bd teaspoon salt, and remaining \u00bc teaspoon black pepper, and stir to combine. While stirring, add the butter, little by little, until it is completely incorporated. Do not allow the sauce to boil or it will separate. Remove from the heat.\n\n9. To serve, spoon some of the sauce onto a serving platter. Slice the pork into thin slices, and arrange them over the sauce. Spoon more sauce over the pork slices, and serve immediately.\n\n4 to 6 servings\n\nGRILLED MARINATED PORK TENDERLOINS WITH AN ORANGE-APRICOT GLAZE\n\nPrep time: 5 minutes Marinating time: 20 minutes Cook time: 22 to 25 minutes Total: 47 to 50 minutes\n\nWe tested this recipe in a grill pan in our indoor kitchen, but if you're in the mood to fire up the outdoor grill at your house, please, by all means. Your cook time will likely be less, so have your thermometer handy and make sure not to cook the pork beyond 145\u00b0F.\n\n\u00bd cup cider vinegar\n\n6 tablespoons olive oil\n\n2 tablespoons minced garlic\n\n2 pork tenderloins (about 1 pound each)\n\n\u00bd cup apricot preserves\n\n\u00bd cup chicken stock or canned, low-sodium chicken broth\n\n1 teaspoon grated orange zest\n\n2 teaspoons salt\n\n1 teaspoon freshly ground white pepper\n\n1. In a 1-gallon resealable plastic food storage bag, combine the vinegar, olive oil, garlic, and pork tenderloins. Seal, and set aside to marinate at room temperature for 20 minutes.\n\n2. While the pork is marinating, prepare the glaze: Combine the preserves, chicken stock, and orange zest in a small saucepan and bring to a boil. Cook until reduced by half, about 5 minutes. Set aside.\n\n3. Set a grill pan over medium-high heat.\n\n4. Remove the pork from the marinade and season it with the salt and white pepper. Place the pork in the hot grill pan and cook for 5 minutes. Turn it over and cook for another 5 minutes. Continue to cook the pork, turning it every few minutes, until it is cooked through, 12 to 15 minutes longer.\n\n5. Remove the pork from the pan and set it aside to rest briefly. Brush with the orange-apricot glaze, slice, and serve hot.\n\n4 servings\n\nDesserts\n\nPEACH-BLUEBERRY CRISP\n\nPrep time (including topping): 20 minutes Cook time: 40 minutes Total: 60 minutes\n\nThis sinfully delicious crisp can be made any time of the year since it uses frozen peaches and blueberries. Prep it and let it bake while you're enjoying dinner, and it'll be hot and bubbly when you're ready to dive in!\n\n2 teaspoons unsalted butter\n\nOne 1-pound bag (4 cups) frozen peaches\n\nOne 1-pound bag (3 cups) frozen blueberries\n\n\u00bd cup sugar\n\n3 tablespoons all-purpose flour\n\n1 teaspoon vanilla extract\n\nCrisp Topping (recipe follows)\n\nWhipped cream, cr\u00e8me fra\u00eeche, or vanilla ice cream, for serving (optional)\n\n1. Preheat the oven to 375\u00b0F. Grease a 2\u00bd-quart baking dish with the butter.\n\n2. Combine the peaches, blueberries, sugar, flour, and vanilla extract in a large bowl. Toss well to mix. Transfer the fruit to the prepared baking dish, and cover with the topping. Place the baking dish on a parchment-or foil-lined rimmed baking sheet to catch any juices that may bubble over.\n\n3. Bake until the crisp is browned on top and the juices have thickened around the edges, about 40 minutes. Serve with whipped cream, cr\u00e8me fra\u00eeche, or vanilla ice cream, if desired.\n\n4 to 6 servings\n\nCrisp Topping\n\n6 tablespoons (\u00be stick) cold unsalted butter, cut into small pieces\n\n2\/3 cup all-purpose flour\n\n2\/3 cup old-fashioned rolled oats\n\n\u00bd cup packed light brown sugar\n\n\u00bc cup packed dark brown sugar\n\n1 teaspoon ground cinnamon\n\n\u00bd teaspoon finely grated nutmeg\n\n\u00bc teaspoon salt\n\nCombine all the ingredients in the bowl of a standing electric mixer fitted with the paddle attachment, and process on low speed until the mixture is crumbly and coarse. (Alternatively, combine the ingredients in a bowl, and using a pastry blender, two knives, or your fingers, cut the butter into the dry ingredients until the mixture resembles coarse crumbs.)\n\nEnough topping for 1 crisp\n\n60 Minutes +\n\nBRAISED CHICKEN THIGHS\n\nPrep time: 5 minutes Cook time: 65 minutes Total: 70 minutes\n\nWhen I'm preparing this dish, it's all I can do to wait until it's finished simmering to get a taste of the awesome gravy. You've simply gotta cook some rice to eat with this, no doubt about it.\n\n6 chicken thighs (about 2 pounds), trimmed of any excess skin or fat\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n1 teaspoon salt\n\n\u00bd cup plus 1 tablespoon all-purpose flour\n\n2 teaspoons olive oil\n\n3 tablespoons unsalted butter\n\n2 cups thinly sliced yellow onions\n\n1 tablespoon minced garlic\n\n6 sprigs fresh thyme, tied in a bundle, or 2 sprigs fresh rosemary\n\n\u00bc teaspoon freshly ground black pepper\n\n3 cups chicken stock or canned, low-sodium chicken broth\n\n\u00bc cup chopped fresh parsley\n\nSteamed white rice, for serving\n\n1. Season the chicken all over with the Essence and \u00bd teaspoon of the salt. Place the \u00bd cup flour in a small bowl, and quickly dredge both sides of each thigh in the flour, shaking to remove any excess. Set aside.\n\n2. Heat 1 teaspoon of the olive oil in a 10-to 12-inch flameproof casserole or saut\u00e9 pan over medium-high heat. Add 2 tablespoons of the butter, and when it has melted, place the chicken, skin side down, in the pan. Brown for 2 minutes on each side. Remove the chicken from the pan and set aside.\n\n3. Add the remaining 1 tablespoon butter to the pan, and when it has melted, add the onions, garlic, thyme bundle or rosemary sprigs, remaining \u00bd teaspoon salt, and the black pepper. Cook, stirring as needed, until the onions are translucent, about 4 minutes. Sprinkle the 1 tablespoon flour over the onions and cook for 2 minutes longer. Then whisk in the chicken stock and increase the heat to high. Return the chicken, skin side down, to the pan, and bring the stock to a boil. Reduce the heat to medium-low, cover the pan with a heavy, tight-fitting lid, and simmer for 15 minutes.\n\n4. Uncover the pan, stir the bottom of the pan to prevent scorching, and turn the chicken skin side up. Cover the pan, and simmer for 20 minutes longer.\n\n5. Stir the bottom of the pan a final time, re-cover, and simmer for 20 more minutes.\n\n6. Remove the pan from the heat and discard the herb bundle or rosemary sprigs. Transfer the chicken to a serving platter. Add the parsley to the sauce, stir to combine, and then spoon the sauce over the chicken. Serve with steamed white rice.\n\n4 to 6 servings\n\nSPINACH AND MUSHROOM LASAGNA\n\nPrep time: 20 minutes Cook time: 74 to 78 minutes Inactive time: 20 minutes Total: 114 to 118 minutes\n\nThough this dish is a bit of a splurge timewise, it is definitely worthwhile. Go on, indulge. As far as lasagna goes, this recipe is actually very simple, and boy, is it delicious. Keep in mind that most of the time is inactive, when the lasagna is either baking in the oven or resting after baking. A great Sunday take-it-to-work-Monday kinda meal.\n\n2 tablespoons olive oil, plus more for the pan\n\n1\u00bd cups diced onions\n\n1 tablespoon plus 1 teaspoon minced garlic\n\n1 package (about 8 ounces) sliced cremini mushrooms\n\n1 teaspoon salt\n\n10 cups (about 10 ounces) prewashed fresh spinach, chopped\n\n4 cups your favorite jarred marinara sauce or other tomato sauce for pasta\n\n\u00bc teaspoon freshly ground black pepper\n\n1\u00bd teaspoons dried Italian herbs\n\n3 cups low-fat cottage cheese, drained of excess liquid in a strainer\n\nOne 8-ounce package no-cook lasagna noodles\n\n12 ounces part-skim mozzarella cheese, shredded (3 cups)\n\n2 cups finely grated Parmigiano-Reggiano cheese\n\n1. Preheat the oven to 375\u00b0F.\n\n2. Heat a large nonstick skillet over medium heat, and add 1 tablespoon of the olive oil. Add the onions and saut\u00e9 until soft and translucent, 3 to 4 minutes. Add the 1 tablespoon garlic and cook for 30 seconds. Then add another \u00bd tablespoon of the olive oil, the mushrooms, and \u00bc teaspoon of the salt. Continue to cook until the mushrooms are soft and wilted, 5 to 6 minutes. Remove them from the skillet and set aside.\n\n3. Add the remaining \u00bd tablespoon olive oil and 1 teaspoon garlic to the same skillet. Add the spinach and saut\u00e9, stirring, until wilted, 3 to 4 minutes. Drain the spinach and return it to the skillet. Add the marinara sauce, \u00bd teaspoon of the salt, the black pepper, and the Italian herbs. Simmer for 3 to 4 minutes. Remove the skillet from the heat and set aside.\n\n4. In a bowl, combine the remaining \u00bc teaspoon salt and the cottage cheese. Lightly grease a 9\u00bd \u00d7 13-inch baking dish with olive oil.\n\n5. Arrange one even layer of lasagna noodles (3 to 4 noodles) in the baking dish so that most of the bottom is covered, taking care that the noodles are not overlapping. Spread a layer of the cottage cheese mixture over the noodles, then top with a third of the mushrooms. Spoon a third of the tomato sauce over all, then top with a third of the mozzarella and a third of the Parmigiano-Reggiano. Repeat the layers two more times, ending with the Parmigiano-Reggiano. Cover the dish with aluminum foil and bake for 45 minutes.\n\n6. Uncover the dish and continue baking until the cheese bubbles and is lightly browned, about 15 minutes. Let the lasagna cool for at least 20 minutes before cutting. Serve hot.\n\n6 to 8 servings\n\nSIMPLE TURKEY MEATLOAF\n\nPrep time: 15 minutes Cook time: 45 to 50 minutes Total: 60 to 65 minutes\n\nMy second daughter, Jillian, hasn't eaten red meat since I can remember. She got me started using ground turkey in many things that are typically made with ground beef. Here is an example of one dish that I have come to especially enjoy. And talk about good sandwiches the next day! The perfect accompaniment is, of course, Buttermilk Mashed Potatoes (40 Minutes or Less).\n\n1\u00bd pounds ground turkey, preferably 85\/15 blend, or a mix of ground breast and thigh meat\n\n2\/3 cup chopped yellow onion\n\n\u00bd cup chopped red or green bell pepper\n\n\u00bd cup unseasoned dry breadcrumbs\n\n1\/3 cup chopped celery\n\n1 large egg, lightly beaten\n\n\u00bd cup ketchup\n\n1 tablespoon minced garlic\n\n1 tablespoon Emeril's Original Essence or Creole Seasoning (20 Minutes or Less)\n\n\u00bd teaspoon salt\n\n\u00bd teaspoon freshly ground black pepper\n\n1 tablespoon hot sauce\n\n1. Position a rack in the center of the oven and preheat the oven to 375\u00b0F.\n\n2. Place the turkey in a large mixing bowl. Add the onion, bell pepper, breadcrumbs, celery, egg, 1 tablespoon of the ketchup, the garlic, Essence, salt, and pepper. Mix gently but thoroughly until the ingredients are well combined. Transfer the turkey mixture to a 1-pound loaf pan, and form it into a domed loaf shape.\n\n3. Place the remaining ketchup in a small bowl, and stir in the hot sauce. Spoon the ketchup mixture evenly over the meatloaf, spreading it with the back of a spoon.\n\n4. Bake until the meatloaf is browned on top, cooked through, and an instant-read thermometer inserted into the center registers 165\u00b0F, 45 to 50 minutes.\n\n5. Remove the pan from the oven and let the meatloaf rest for 5 minutes before serving.\n\n4 to 6 servings\n\n## Searchable Terms\n\nNote: Entries in this index, carried over verbatim from the print edition of this title, are unlikely to correspond to the pagination of any given e-book reader. However, entries in this index, and other terms, may be easily located by using the search feature of your e-book reader.\n\nNote: Page references in italics refer to recipe photographs.\n\nA\n\nAnchovies\n\nand Caramelized Onions, Spaghetti with, 152\n\nPenne alla Puttanesca, 155\n\nReal Caesar Salad, 16\n\nSwordfish with Puttanesca Relish, 178, 179\n\nApples and Prunes, Pork Loin with, 240\u201342, 241\n\nArugula, Cantaloupe, and\n\nProsciutto Salad, 22, 23\n\nAsparagus, Pan-Roasted, with\n\nShiitake Mushrooms and\n\nCherry Tomatoes, 60\n\nAvocado(s)\n\nBacon, Lettuce, and Tomato Sandwich with Basil Mayo, 35\n\nChili-Rubbed Shrimp Wraps, 140\u201341, 141\n\nSalad Tropicale, 27\n\nSeared Shrimp Salad, 132, 133\n\nB\n\nBacon\n\nBraised Green Beans, 166, 167\n\nCountry-Fried Steak with White Gravy, 202\u20133\n\nand Fried Eggs, Spinach Salad with, 17\n\nLettuce, Avocado, and Tomato Sandwich with Basil Mayo, 35\n\nLinguine alla Carbonara, 50\u201351\n\nand Shiitakes with Penne, 149\n\nBalsamic Vinaigrette, 30\n\nBananas, Brown Sugar\u2013Baked, 100\n\nBasil\n\nBruschetta, 11\n\nMayo, Bacon, Lettuce, Avocado, and Tomato Sandwich with, 35\n\nMozzarella and Tomato Bites with Kalamata Olive Drizzle, 13\n\nTomato, and Mozzarella, Orzo \"Risotto\" with, 148\n\nBean(s)\n\nBetter Than Mama's Chili-Mac, 222\u201323, 223\n\nBlack, Cakes, 162, 163\n\nBlack, Salsa, Fish Tacos with, 44\n\nChickpea Salad with Tabbouleh, 137\n\nChili-Beans, 226, 227\n\nGreen, Bacon Braised, 166, 167\n\nGreen, Spicy Pork Stir-Fry with, 95\n\nPinto, and Turkey Tostadas, 164\n\nRed, Soup, Quick, 114\n\nRoasted Red Pepper Hummus, 8, 9\n\nSausage, and Cheese Nachos, Spicy, 58, 59\n\nSeared Salmon with Lentils, 228\u201329, 229\n\nWhite, Creamy, with Sausage, 165\n\nBeef\n\nBetter Than Mama's Chili-Mac, 222\u201323, 223\n\nand Broccoli, Stir-Fried, 205\n\nChili-Beans, 226, 227\n\nCountry-Fried Steak with White Gravy, 202\u20133\n\nMinute Steaks Teriyaki-Style, 91\n\nMushroom-Smothered Steaks, 96, 97\n\nNew York Strip with Beurre Ma\u00eetre d'H\u00f4tel, 88, 89\n\nSloppy Joes, 200, 201\n\nSteak and Cheese Sandwiches, 36\u201337\n\nSteak au Poivre, 86\u201387\n\nStroganoff with Egg Noodles, 156\u201357, 157\n\nBerries\n\nFlamb\u00e9ed Strawberry Sauce for Angel Food Cake or Ice Cream, 107\n\nFresh, with Balsamic Drizzle and Almond Cream, 104, 105\n\nMelon with Amaretti Cookie Crumbles, 103\n\nPeach-Blueberry Crisp, 244\u201345, 245\n\nBlueberry(ies)\n\nFresh Berries with Balsamic Drizzle and Almond Cream, 104, 105\n\n-Peach Crisp, 244\u201345, 245\n\nBok Choy, Garlicky, 62, 62\n\nBrazilian Fish Stew, 232, 233 Breads\n\nBruschetta, 11\n\nSimple Croutons, 136, 136\n\nBroccoli\n\nand Beef, Stir-Fried, 205\n\nand Cheese Soup, 112, 113\n\nGarden Vegetable Salad, 135\n\nGarden Vegetable Soup, 122, 123 Bruschetta, 11\n\nBurgers\n\nBlack Bean Cakes, 162, 163\n\nChicken Queso, 138\u201339\n\nButtermilk\n\nDressing, 28, 29\n\nMashed Potatoes, 173, 173\n\nC\n\nCabbage\n\nFish en Papillote, 176\u201377\n\nGarden Vegetable Salad, 135\n\nGarlicky Bok Choy, 62, 62\n\nSausages and Sauerkraut, 204\n\nSpicy Pork Wraps with Creamy Coleslaw, 142\u201343\n\nCaesar Salad, Real, 16\n\nCake, Skillet Corn, with Stewed Cherries, 212, 213\n\nCantaloupe,\n\nMelon with Amaretti Cookie Crumbles, 103\n\nProsciutto, and Arugula Salad, 22, 23\n\nCapers\n\nPenne alla Puttanesca, 155\n\nShrimp and Feta, Greek-Style, 182\u201383\n\nSwordfish with Puttanesca Relish, 178, 179\n\nCarrot(s)\n\nEmeril's Noodle Salad, 134\n\nGarden Vegetable Salad, 135\n\nGinger Soup, 116, 117\n\nMinute Steaks Teriyaki-Style, 91\n\nPasta Primavera, 158, 159\n\nRoasted, with Fresh Thyme, 66\n\nSpinach, and an Asian Vinaigrette, Baked Flounder with, 186\n\nand Tarragon, Saut\u00e9ed Yellow Squash with, 64, 65\n\nCashews, Stir-Fried Chicken with, 80\n\nCatfish\n\nBroiled, with Fresh Thyme, Garlic, and Lemon, 70, 71\n\nSouthern-Style Pan-Fried, 74\n\nCheese\n\nBetter Than Mama's Chili-Mac, 222\u201323, 223\n\nBoneless Pork Chops Parmigiana, 92\u201394, 93\n\nBoursin, Spinach, and Pecan\u2013Stuffed Chicken Breasts, 188\u201389\n\nand Broccoli Soup, 112, 113\n\nChicken Cordon Bleu, 194\u201395, 195\n\nChicken Queso Burgers, 138\u201339\n\nGoat, and Caramelized Onion Tart, 220\u201321\n\nGoat, Orange, and Walnut Salad, 18, 19\n\nKicked-Up Tuna Melts, 42, 43\n\nLinguine alla Carbonara, 50\u201351\n\nMozzarella and Tomato Bites with Kalamata Olive Drizzle, 13\n\nOrzo \"Risotto\" with Tomato, Mozzarella, and Basil, 148\n\nOven-Crispy French Fries with Paprika-Parmesan Salt, 146,, 147\n\nPressed Roast Turkey, Pesto, and Provolone Sandwiches, 38, 39\n\nProsciutto and Mozzarella Panini, 40\n\nReal Caesar Salad, 16\n\nSausage, and Bean Nachos, Spicy, 58, 59\n\nSausage and Pepper Po-Boy, 34\n\nShrimp and Feta, Greek-Style, 182\u201383\n\nSpinach and Mushroom Lasagna, 248\u201349, 249\n\nand Steak Sandwiches, 36\u201337\n\nThree-, Baked Macaroni, 153\n\nand Turkey Sandwich, Open-Face, 41\n\nCherries, Stewed, Skillet Corn Cake with, 212, 213\n\nChicken\n\nBreasts, Boursin Cheese, Spinach, and Pecan\u2013Stuffed, 188\u201389\n\nBreasts, Saut\u00e9ed, with Dijon Herb Sauce, 81\n\ncarving, 239\n\nCordon Bleu, 194\u201395, 195\n\nCrispy Pan-Roasted, with Garlic-Thyme Butter, 193\n\nHot and Sour Soup, 119\n\nand Mushroom Risotto, 224\n\nPatty Pockets with Minted Yogurt Sauce, 144\u201345, 145\n\nQueso Burgers, 138\u201339 and Rice Soup, 115\n\nRoast, with Shallot-Garlic Butter, 238\u201339\n\nSalad with Fresh Herbs and Celery, 82\n\nStir-Fried, with Cashews, 80\n\nTenders, Panko-Crusted, 198\u201399\n\nThighs, Braised, 246\u201347, 247\n\nWings, Oven-Roasted, 192\n\nChiles\n\nBrazilian Fish Stew, 232, 233\n\nChicken Queso Burgers, 138\u201339\n\nCoconut, and Tomatoes, Indian-Inspired Shrimp with, 174, 175\n\nChili-Beans, 226, 227\n\nChili-Mac, Better Than Mama's, 222\u201323, 223\n\nChili-Rubbed Shrimp Wraps, 140\u201341, 141\n\nChocolate\n\nCandied Hot Fudge Sundaes, 101\n\nChip\u2013Peanut Butter Cookies, 102, 102\n\nChowder, Shrimp and Corn, 218\n\nCoconut milk\n\nAromatic Jasmine Rice, 57\n\nBrazilian Fish Stew, 232, 233\n\nChiles, and Tomatoes, Indian-Inspired Shrimp with, 174, 175\n\nCookie(s)\n\nAmaretti, Crumbles, Melon with, 103\n\nEmeril's Late-Night Parfaits, 98, 99\n\nKicked-Up Snickerdoodles, 210\u201311\n\nPeanut Butter\u2013Chocolate Chip, 102, 102\n\nCorn and Shrimp Chowder, 218\n\nCorn Cake, Skillet, with Stewed Cherries, 212, 213\n\nCornish Game Hens, Honey-Lemon-Thyme, 190\u201391, 191\n\nCouscous, Orange, Currant, and Pine Nut, 48, 49\n\nCreole Seasoning, 29\n\nCroutons, Simple, 136, 136\n\nCucumber(s)\n\nEmeril's Noodle Salad, 134\n\nRibbon Salad, 20, 20\n\nSalad Tropicale, 27\n\nSaut\u00e9ed, with Basil and Mint, Emeril's, 67\n\nD\n\nDesserts\n\nBrown Sugar\u2013Baked Bananas, 100\n\nCandied Hot Fudge Sundaes, 101\n\nEmeril's Late-Night Parfaits, 98, 99\n\nFlamb\u00e9ed Strawberry Sauce for Angel Food Cake or Ice Cream, 107\n\nFresh Berries with Balsamic Drizzle and Almond Cream, 104, 105\n\nKicked-Up Snickerdoodles, 210\u201311\n\nMelon with Amaretti Cookie Crumbles, 103\n\nPeach-Blueberry Crisp, 244\u201345, 245\n\nPeanut Butter\u2013Chocolate Chip Cookies, 102, 102\n\nSkillet Corn Cake with Stewed Cherries, 212, 213\n\nDips and spreads\n\nCreamy Shrimp and Green Onion Dip, 131\n\nKicked-Up Tartar Sauce, 75\n\nRoasted Red Pepper Hummus, 8, 9\n\nRoasted Red Pepper Mayo, 130\n\nE\n\nEggplant, Sesame, 170, 171\n\nEggs\n\nFried, and Bacon, Spinach Salad with, 17\n\nLinguine alla Carbonara, 50\u201351\n\nSimple Italian Wedding Soup, 4\n\nEscarole and Sausage, Penne with, 150, 151\n\nF\n\nFennel, Orange, and Black Olive Salad, 24\u201325\n\nFish. See also Anchovies; Shellfish\n\nBaked Flounder with Carrots, Spinach, and an Asian Vinaigrette, 186\n\nBlue Corn\u2013Crusted Rainbow Trout with Cilantro-Lime Sour Cream, 76\n\nBroiled Catfish with Fresh Thyme, Garlic, and Lemon, 70, 71\n\nBroiled Salmon with a Warm Tomato-Lemon Vinaigrette, 72\n\nen Papillote, 176\u201377\n\nKicked-Up Tuna Melts, 42, 43\n\nPortuguese Rice with Tuna, 225\n\nRoasted Scrod with Herbed Breadcrumbs, 180, 181\n\nRoasted Whole Red Snapper with Orange, Rosemary, and Kalamata Olives, 230\u201331\n\nSalmon with Orange Butter Sauce, 184, 185\n\nSeared Salmon with Lentils, 228\u201329, 229\n\nSouthern-Style Pan-Fried Catfish, 74\n\nStew, Brazilian, 232, 233\n\nSwordfish with Puttanesca Relish, 178, 179\n\nTacos with Black Bean Salsa, 44\n\nTrout \u00e0 la Meuni\u00e8re, 77\n\nFlounder, Baked, with Carrots, Spinach, and an Asian Vinaigrette, 186\n\nFritters, Shrimp and Zucchini, with Roasted Red Pepper Mayo, 128\u201330, 129\n\nFruit. See also specific fruits Turkey and Wild Rice Salad, 56\n\nG\n\nGaaahlicky Sizzling Shrimp, 73, 73\n\nGarlicky Bok Choy, 62, 62\n\nGinger Carrot Soup, 116, 117\n\nGrains. See also Rice\n\nChickpea Salad with Tabbouleh, 137\n\nGreen Beans\n\nBacon Braised, 166, 167\n\nSpicy Pork Stir-Fry with, 95\n\nGreens. See also Spinach\n\nCantaloupe, Prosciutto, and Arugula Salad, 22, 23\n\nEmeril's New-Style Caldo Verde, 216, 217\n\nEmeril's Salad, 14, 15\n\nIceberg Wedges with Cherry Tomato Vinaigrette, 26\n\nMustard, Creamed, 168\n\nOrange, Walnut, and Goat Cheese Salad, 18, 19\n\nPenne with Sausage and Escarole, 150, 151\n\nReal Caesar Salad, 16\n\nSalad Tropicale, 27\n\nSeared Shrimp Salad, 132, 133\n\nSpicy Braised, 172\n\nH\n\nHam. See also Prosciutto\n\nChicken Cordon Bleu, 194\u201395, 195\n\nand Peas, Fettuccine with, 46, 47\n\nQuick Red Bean Soup, 114\n\nHerb(ed). See also specific herbs\n\nOlives, 7\n\nVinaigrette, 32\n\nHot Dog, Turkey, and Potato Soup\n\nwith Herbs, 124, 125\n\nI\n\nIceberg Wedges with Cherry\n\nTomato Vinaigrette, 26\n\nL\n\nLamb\n\nChops with Mustard Herb Crust, 90\n\nKebabs, Quick and Easy, 206, 207\n\nT-Bones with Rosemary-Balsamic Butter Sauce, 84\u201385, 85\n\nLasagna, Spinach and Mushroom, 248\u201349, 249\n\nLeek and Potato Soup, 120\u201321, 121\n\nLentils, Seared Salmon with, 228\u201329, 229\n\nM\n\nMeat. See Beef; Lamb; Pork\n\nMeatloaf, Simple Turkey, 250, 251\n\nMelon\n\nwith Amaretti Cookie Crumbles, 103\n\nCantaloupe, Prosciutto, and Arugula Salad, 22, 23\n\nMinted Yogurt Sauce, 144\u201345, 145\n\nMushroom(s)\n\nBeef Stroganoff with Egg Noodles, 156\u201357, 157\n\nand Chicken Risotto, 224\n\nCremini, Balsamic-Marinated, 10\n\nHot and Sour Soup, 119\n\nOpen-Face Turkey and Cheese Sandwich, 41\n\nQuick and Easy Lamb Kebabs, 206, 207\n\nSaut\u00e9ed, with Fresh Thyme, 169\n\nShiitake, and Cherry Tomatoes, Pan-Roasted Asparagus with, 60\n\nShiitakes and Bacon with Penne, 149\n\n-Smothered Steaks, 96, 97\n\nSpicy Smoked Sausage, and Tomato Soup, 118\n\nand Spinach Lasagna, 248\u201349, 249\n\nMussels\n\nClassic Moules Marini\u00e8re, 78, 79\n\nN\n\nNachos, Spicy Sausage, Bean, and Cheese, 58, 59\n\nNoodles, Egg, Beef Stroganoff with, 156\u201357, 157\n\nNoodle Salad, Emeril's, 134\n\nNuts\n\nBoursin Cheese, Spinach, and Pecan\u2013Stuffed Chicken Breasts, 188\u201389\n\nEmeril's Noodle Salad, 134\n\nStir-Fried Chicken with Cashews, 80\n\nO\n\nOlive(s)\n\nAntipasto Pasta Salad, 21\n\nBlack, Orange, and Fennel Salad, 24\u201325\n\nHerbed, 7\n\nKalamata, Drizzle, Mozzarella and Tomato Bites with, 13\n\nKalamata, Oranges, and Rosemary, Roasted Whole Red Snapper with, 230\u201331\n\nPenne alla Puttanesca, 155\n\nSwordfish with Puttanesca Relish, 178, 179\n\nOnion(s)\n\nCaramelized, and Anchovies, Spaghetti with, 152\n\nCaramelized, and Goat Cheese Tart, 220\u201321\n\nSausage and Pepper Po-Boy, 34\n\nSteak and Cheese Sandwiches, 36\u201337\n\nOrange(s)\n\nButter Sauce, Salmon with, 184, 185\n\nCurrant, and Pine Nut Couscous, 48, 49\n\nFennel, and Black Olive Salad, 24\u201325\n\nRosemary, and Kalamata Olives, Roasted Whole Red Snapper with, 230\u201331\n\nSeared Shrimp Salad, 132, 133\n\nsegmenting, 19\n\nWalnut, and Goat Cheese Salad, 18, 19\n\nP\n\nPasta\n\nBeef Stroganoff with Egg Noodles, 156\u201357, 157\n\nBetter Than Mama's Chili-Mac, 222\u201323, 223\n\nEmeril's Noodle Salad, 134\n\nFettuccine with Peas and Ham, 46, 47\n\nLinguine alla Carbonara, 50\u201351\n\nOrange, Currant, and Pine Nut Couscous, 48, 49\n\nOrzo \"Risotto\" with Tomato, Mozzarella, and Basil, 148\n\nPenne alla Puttanesca, 155\n\nPenne with Sausage and Escarole, 150, 151\n\nPrimavera, 158, 159\n\nSalad, Antipasto, 21\n\nShiitakes and Bacon with Penne, 149\n\nand Shrimp, Emeril's, with Garlic, Lemon, Crushed Red Pepper, and Green Onions, 52, 53\n\nShrimp and Linguine Fra Diavolo, 154\n\nSpaghetti with Caramelized Onions and Anchovies, 152\n\nSpinach and Mushroom Lasagna, 248\u201349, 249\n\nThree-Cheese Baked Macaroni, 153\n\nPeach-Blueberry Crisp, 244\u201345, 245\n\nPeanut Butter\u2013Chocolate Chip Cookies, 102, 102\n\nPea(s)\n\nand Ham, Fettuccine with, 46, 47\n\nPasta Primavera, 158, 159\n\nSweet, Soup, 5\n\nPecan, Boursin Cheese, and Spinach\u2013Stuffed Chicken Breasts, 188\u201389\n\nPepper(s). See also Chiles\n\nMinute Steaks Teriyaki-Style, 91\n\nPasta Primavera, 158, 159\n\nProsciutto and Mozzarella Panini, 40\n\nQuick and Easy Lamb Kebabs, 206, 207\n\nRoasted Red, Hummus, 8, 9\n\nRoasted Red, Mayo, 130\n\nand Sausage Po-Boy, 34\n\nPork. See also Bacon; Ham; Sausage(s)\n\nChops, Boneless, Parmigiana, 92\u201394, 93\n\nChops, Thin-Cut, with Rosemary-Balsamic Glazed Shallots, 208\n\nLoin with Apples and Prunes, 240\u201342, 241\n\nStir-Fry, Spicy, with Green Beans, 95\n\nTenderloins, Grilled Marinated, with an Orange-Apricot Glaze, 243\n\nWraps, Spicy with Creamy Coleslaw, 142\u201343\n\nPotato(es)\n\nButtermilk Mashed, 173, 173\n\nEmeril's New-Style Caldo Verde, 216, 217\n\nand Leek Soup, 120\u201321, 121\n\nOven-Crispy French Fries with Paprika-Parmesan Salt, 146,, 147\n\nShrimp and Corn Chowder, 218\n\nand Turkey Hot Dog Soup with Herbs, 124, 125\n\nPoultry. See also Chicken; Turkey\n\nHoney-Lemon-Thyme Cornish Game Hens, 190\u201391, 191\n\nProsciutto\n\nCantaloupe, and Arugula Salad, 22, 23\n\nChicken Cordon Bleu, 194\u201395, 195\n\nand Mozzarella Panini, 40\n\nTurkey Saltimbocca, 196\u201397\n\nPrunes and Apples, Pork Loin with, 240\u201342, 241\n\nR\n\nRadishes\n\nGarden Vegetable Salad, 135\n\nGlazed, 68, 69\n\nRed Snapper, Roasted Whole,\n\nwith Orange, Rosemary, and\n\nKalamata Olives, 230\u201331\n\nRice\n\nBasic Risotto, 161\n\nChicken and Mushroom Risotto, 224\n\nand Chicken Soup, 115\n\nJasmine, Aromatic, 57\n\nPilaf, Green Onion, 160\n\nPortuguese, with Tuna, 225\n\nShrimp Fried, Kicked-Up, 54\u201355, 55\n\nWild, and Turkey Salad, 56\n\nRisotto\n\nBasic, 161\n\nChicken and Mushroom, 224\n\n\"Risotto,\" Orzo, with Tomato,\n\nMozzarella, and Basil, 148\n\nSalad dressings. See also specific salad recipes\n\nBalsamic Vinaigrette, 30\n\nButtermilk Dressing, 28, 29\n\nHerb Vinaigrette, 32\n\nRed Wine Vinaigrette, 31, 31\n\nSalads\n\nAntipasto Pasta, 21\n\nCaesar, Real, 16\n\nCantaloupe, Prosciutto, and Arugula, 22, 23\n\nChicken, with Fresh Herbs and Celery, 82\n\nChickpea, with Tabbouleh, 137\n\nCucumber Ribbon, 20, 20\n\nEmeril's, 14, 15\n\nGarden Vegetable, 135\n\nIceberg Wedges with Cherry Tomato Vinaigrette, 26\n\nNoodle, Emeril's, 134\n\nOrange, Fennel, and Black Olive, 24\u201325\n\nOrange, Walnut, and Goat Cheese, 18, 19\n\nSeared Shrimp, 132, 133\n\nSpinach, with Bacon and Fried Eggs, 17\n\nTropicale, 27\n\nTurkey and Wild Rice, 56\n\nSalmon\n\nBroiled, with a Warm Tomato-Lemon Vinaigrette, 72\n\nwith Orange Butter Sauce, 184, 185\n\nSeared, with Lentils, 228\u201329, 229\n\nSandwiches\n\nBacon, Lettuce, Avocado, and Tomato, with Basil Mayo, 35\n\nChicken Patty Pockets with Minted Yogurt Sauce, 144\u201345, 145\n\nChicken Queso Burgers, 138\u201339\n\nChili-Rubbed Shrimp Wraps, 140\u201341, 141\n\nFish Tacos with Black Bean Salsa, 44\n\nKicked-Up Tuna Melts, 42, 43\n\nOpen-Face Turkey and Cheese, 41\n\nPressed Roast Turkey, Pesto, and Provolone, 38, 39\n\nProsciutto and Mozzarella Panini, 40\n\nSausage and Pepper Po-Boy, 34\n\nSpicy Pork Wraps with Creamy Coleslaw, 142\u201343\n\nSteak and Cheese, 36\u201337\n\nSauces\n\nMinted Yogurt, 144\u201345, 145\n\nStrawberry, Flamb\u00e9ed, for Angel Food Cake or Ice Cream, 107\n\nTartar, Kicked-Up, 75\n\nSauerkraut, Sausages and, 204\n\nSausage(s)\n\nAntipasto Pasta Salad, 21\n\nBean, and Cheese Nachos, Spicy, 58, 59\n\nCreamy White Beans with, 165\n\nEmeril's New-Style Caldo Verde, 216, 217\n\nand Escarole, Penne with, 150, 151\n\nand Pepper Po-Boy, 34\n\nQuick Red Bean Soup, 114\n\nand Sauerkraut, 204\n\nShrimp and Chorizo Tapas, 6\n\nSpicy Smoked, Tomato, and Mushroom Soup, 118\n\nScrod, Roasted, with Herbed Breadcrumbs, 180, 181\n\nSeafood. See Fish; Shellfish\n\nSesame Eggplant, 170, 171\n\nShellfish. See also Shrimp\n\nClassic Moules Marini\u00e8re, 78, 79\n\nShrimp\n\nChili-Rubbed, Wraps, 140\u201341, 141\n\nand Chorizo Tapas, 6\n\nand Corn Chowder, 218\n\n\u00c9touff\u00e9e, 234, 235\n\nand Feta, Greek-Style, 182\u201383\n\nFried Rice, Kicked-Up, 54\u201355, 55\n\nGaaahlicky Sizzling, 73, 73\n\nand Green Onion Dip, Creamy, 131\n\nIndian-Inspired, with Coconut, Chiles, and Tomatoes, 174, 175\n\nand Linguine Fra Diavolo, 154\n\nand Pasta, Emeril's, with Garlic, Lemon, Crushed Red Pepper, and Green Onions, 52, 53\n\nSeared, Salad, 132, 133\n\nand Zucchini Fritters with Roasted Red Pepper Mayo, 128\u201330, 129\n\nSloppy Joes, 200, 201\\\n\nSnickerdoodles, Kicked-Up, 210\u201311\n\nSoups\n\nBroccoli and Cheese, 112, 113\n\nCarrot Ginger, 116, 117\n\nChicken and Rice, 115\n\nEmeril's New-Style Caldo Verde, 216, 217\n\nGarden Vegetable, 122, 123\n\nHot and Sour, 119\n\nItalian Wedding, Simple, 4\n\nPotato and Leek, 120\u201321, 121\n\nPotato and Turkey Hot Dog, with Herbs, 124, 125\n\nRed Bean, Quick, 114\n\nShrimp and Corn Chowder, 218\n\nSpicy Smoked Sausage, Tomato, and Mushroom, 118\n\nSweet Pea, 5\n\nTomato, Cream of, 127\n\nSpice mix. See Creole Seasoning\n\nSpinach\n\nBoursin Cheese, and Pecan\u2013Stuffed Chicken Breasts, 188\u201389\n\nCarrots, and an Asian Vinaigrette, Baked Flounder with, 186\n\nand Mushroom Lasagna, 248\u201349, 249\n\nOpen-Face Turkey and Cheese Sandwich, 41\n\nSalad with Bacon and Fried Eggs, 17\n\nSweet Pea Soup, 5\n\nSquash. See also Zucchini\n\nGarden Vegetable Soup, 122, 123\n\nPasta Primavera, 158, 159\n\nYellow, Saut\u00e9ed, with Carrots and Tarragon, 64, 65\n\nStarters\n\nBalsamic-Marinated Cremini Mushrooms, 10\n\nBruschetta, 11\n\nCaramelized Onion and Goat Cheese Tart, 220\u201321\n\nCreamy Shrimp and Green Onion Dip, 131\n\nCreamy White Beans with Sausage, 165\n\nHerbed Olives, 7\n\nMozzarella and Tomato Bites with Kalamata Olive Drizzle, 13\n\nPan-Roasted Asparagus with Shiitake Mushrooms and Cherry Tomatoes, 60\n\nRoasted Red Pepper Hummus, 8, 9\n\nShrimp and Chorizo Tapas, 6\n\nShrimp and Zucchini Fritters with Roasted Red Pepper Mayo, 128\u201330, 129\n\nSpicy Sausage, Bean, and Cheese Nachos, 58, 59\n\nStew, Brazilian Fish, 232, 233\n\nStrawberry(ies)\n\nFresh Berries with Balsamic Drizzle and Almond Cream, 104, 105\n\nMelon with Amaretti Cookie Crumbles, 103\n\nSauce, Flamb\u00e9ed, for Angel Food Cake or Ice Cream, 107\n\nSwordfish with Puttanesca Relish, 178, 179\n\nT\n\nTabbouleh, Chickpea Salad with, 137\n\nTacos, Fish, with Black Bean Salsa, 44\n\nTart, Caramelized Onion and Goat Cheese, 220\u201321\n\nTartar Sauce, Kicked-Up, 75\n\nTomato(es)\n\nBacon, Lettuce, and Avocado Sandwich with Basil Mayo, 35\n\nBetter Than Mama's Chili-Mac, 222\u201323, 223\n\nBruschetta, 11\n\nCherry, and Shiitake Mushrooms, Pan-Roasted Asparagus with, 60\n\nCherry, Vinaigrette, Iceberg Wedges with, 26\n\nCoconut, and Chiles, Indian-Inspired Shrimp with, 174, 175\n\nKicked-Up Tuna Melts, 42, 43\n\n-Lemon Vinaigrette, Warm, Broiled Salmon with a, 72\n\nMozzarella, and Basil, Orzo \"Risotto\" with, 148\n\nand Mozzarella Bites with Kalamata Olive Drizzle, 13\n\nPenne alla Puttanesca, 155\n\nSalad Tropicale, 27\n\nShrimp and Feta, Greek-Style, 182\u201383\n\nSoup, Cream of, 127\n\nSpicy Smoked Sausage, and Mushroom Soup, 118\n\nSwordfish with Puttanesca Relish, 178, 179\n\nTortillas\n\nChili-Rubbed Shrimp Wraps, 140\u201341, 141\n\nFish Tacos with Black Bean Salsa, 44\n\nSpicy Pork Wraps with Creamy Coleslaw, 142\u201343\n\nSpicy Sausage, Bean, and Cheese Nachos, 58, 59\n\nTurkey and Pinto Bean Tostadas, 164\n\nTostadas, Turkey and Pinto Bean, 164\n\nTrout\n\n\u00e0 la Meuni\u00e8re, 77\n\nRainbow, Blue Corn\u2013Crusted, with Cilantro-Lime Sour Cream, 76\n\nTuna\n\nMelts, Kicked-Up, 42, 43\n\nPortuguese Rice with, 225\n\nTurkey\n\nBreast, Roast, with Bacon and Sage, 236\u201337\n\nand Cheese Sandwich, Open-Face, 41\n\nHot Dog and Potato Soup with Herbs, 124, 125\n\nMeatloaf, Simple, 250, 251\n\nand Pinto Bean Tostadas, 164\n\nRoast, Pesto, and Provolone Sandwiches, Pressed, 38, 39\n\nSaltimbocca, 196\u201397\n\nand Wild Rice Salad, 56\n\nV\n\nVegetable(s). See also specific vegetables\n\nGarden, Salad, 135\n\nGarden, Soup, 122, 123\n\nKicked-Up Shrimp Fried Rice, 54\u201355, 55\n\nW\n\nWild Rice and Turkey Salad, 56\n\nY\n\nYogurt Sauce, Minted, 144\u201345, 145\n\nZ\n\nZucchini\n\nBroiled, 63\n\nGarden Vegetable Salad, 135\n\nGarden Vegetable Soup, 122, 123\n\nPasta Primavera, 158, 159\n\nand Shrimp Fritters with Roasted Red Pepper Mayo, 128\u201330, 129\n\n## About the Author\n\nEMERIL LAGASSE is a chef, restaurateur, and bestselling author of thirteen popular cookbooks, including Emeril's New New Orleans Cooking, which introduced his creative take on Creole cuisine, and Emeril at the Grill: A Cookbook for All Seasons. He is the proprietor of eleven restaurants in New Orleans, Las Vegas, Orlando, Miami, Gulfport, Mississippi, and Bethlehem, Pennsylvania.\n\nAs a national TV personality, Emeril hosts Emeril Green, an original series exploring fresh and seasonal ingredients on the eco-lifestyle network Planet Green, and is the food correspondent for ABC's Good Morning America. His show Essence of Emeril can be seen on the Food Network, and Emeril Live appears on both Fine Living and the Food Network.\n\nIn 2002, Emeril established the Emeril Lagasse Foundation to support children's educational programs, life skills development, culinary training, and cultural enrichment. As of May 2009, the foundation has contributed $3 million to organizations in New Orleans and on the Gulf Coast.\n\nwww.emerils.com\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.\n\n## Also by Emeril Lagasse\n\nEMERIL AT THE GRILL\n\nEMERIL'S CREOLE CHRISTMAS\n\nEMERIL'S DELMONICO: A RESTAURANT WITH A PAST\n\nEMERIL'S NEW NEW ORLEANS COOKING\n\nEMERIL'S POTLUCK: COMFORT FOOD WITH A KICKED-UP ATTITUDE\n\nEMERIL'S THERE'S A CHEF IN MY SOUP!: RECIPES FOR THE KID IN EVERYONE\n\nEMERIL'S THERE'S A CHEF IN MY FAMILY!: RECIPES TO GET EVERYBODY COOKING\n\nEMERIL'S THERE'S A CHEF IN MY WORLD!: RECIPES THAT TAKE YOU PLACES\n\nEMERIL'S TV DINNERS: KICKIN' IT UP A NOTCH WITH RECIPES FROM EMERIL LIVE AND ESSENCE OF EMERIL\n\nEVERY DAY'S A PARTY: LOUISIANA RECIPES FOR CELEBRATING WITH FAMILY AND FRIENDS\n\nFROM EMERIL'S KITCHENS: FAVORITE RECIPES FROM EMERIL'S RESTAURANTS\n\nLOUISIANA REAL AND RUSTIC\n\nPRIME TIME EMERIL: MORE TV DINNERS FROM AMERICA'S FAVORITE CHEF\n\n## Credits\n\nPhotography by Steven Freeman\n\nCover design by Mary Schuck\n\n## Copyright\n\nEMERIL 20\u201340\u201360. Copyright \u00a9 2009 by Emeril\/MSLO Acquisition Sub, LLC. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nAdobe Digital Edition September 2009 ISBN 978-0-06-196592-0\n\n10 9 8 7 6 5 4 3 2 1\n\n## About the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\n25 Ryde Road (PO Box 321)\n\nPymble, NSW 2073, Australia\n\nhttp:\/\/www.harpercollinsebooks.com.au\n\nCanada\n\nHarperCollins Publishers Ltd.\n\n55 Avenue Road, Suite 2900\n\nToronto, ON, M5R, 3L2, Canada\n\nhttp:\/\/www.harpercollinsebooks.ca\n\nNew Zealand\n\nHarperCollinsPublishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\nhttp:\/\/www.harpercollins.co.nz\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\nhttp:\/\/www.harpercollinsebooks.co.uk\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\nhttp:\/\/www.harpercollinsebooks.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2014 by Nicky Epstein  \nPhotographs copyright \u00a9 2014 by Potter Craft\n\nAll rights reserved.  \nPublished in the United States by Potter Craft, an imprint of Crown Publishing Group, a division of Random House LLC, a Penguin Random House Company, New York.  \nwww.pottercraft.com  \nwww.crownpublishing.com\n\nPOTTER CRAFT and colophon is a registered trademark of Random House LLC.\n\nLibrary of Congress Cataloging-in-Publication Data  \nEpstein, Nicky.  \nKnitting reimagined \/ Nicky Epstein.\u2014First edition.  \npages cm  \nIncludes index.  \n1. Knitting\u2014Patterns. I. Title.  \nTT825.E64226 2014  \n746.43'2\u2014dc23 2013028642\n\nISBN 978-0-385-34625-2  \nEbook ISBN 978-0-385-34626-9\n\nDesign by Jan Derevjanik  \nPhotographs by Rose Callahan  \nCover design by Jan Derevjanik  \nCover photographs by Rose Callahan\n\nThe author and publisher would like to thank the Craft Yarn Council of America for providing the yarn weight standards and accompanying icons used in this book. For more information, please visit www.YarnStandards.com.\n\nv3.1\n\nTo my husband, Howard, who has encouraged me to \"take the road less traveled\"... and that has made all the difference!\n\n# CONTENTS\n\nINTRODUCTION\n\nHOW TO USE THIS BOOK\n\n[Chapter 1  \nDIRECTIONALS](Epst_9780385346269_epub_c01_r1.htm)\n\nReckoning Rectangles Shawl\n\nRenaissance Castle Tunic\n\nOn the Block Topper\n\nJe Ne Sais Quoi Cape\n\nThe Deep End Shawl\n\nRoyal Lace Coat with Hood\n\nWelted Button Tuck Cardi\n\n[Chapter 2  \nCOOL CONSTRUCTION](Epst_9780385346269_epub_c02_r1.htm)\n\nWeekend Warrior Wraparound\n\nNouveau Wrap Cardigan\n\nOn the Edge Dress\n\nGlory Rising Circle Cardigan\n\nDirectional Vest\n\nShape-Shifter Vest\n\n[Chapter 3  \nWOVEN WEAVES](Epst_9780385346269_epub_c03_r1.htm)\n\nCrisscross Weave Tank\n\nBraided Vitality Pullover\n\nEtiquette Unchained Pullover\n\nPixilated Weave Drapelette\n\nChaos Couture Pullover\n\n[Chapter 4  \nSTITCH IMPACT](Epst_9780385346269_epub_c04_r1.htm)\n\nSpring Forward Dress or Tunic\n\nButtons and Bows Manteau\n\nDressage Pony Poncholette\n\nSpirits Fly Pullover\n\nVictory Fair Isle Pullover\n\nQuintessential Cable Pullover\n\nEdging Epilogue Dress\n\nABBREVIATIONS\n\nKNITTING TECHNIQUES\n\nRESOURCES\n\nACKNOWLEDGMENTS\n\nINDEX\n\n_About the Author_\n\n# INTRODUCTION\n\nTwo roads diverged in a wood, and I\u2014\n\nI took the lesser traveled by,\n\nAnd that has made all the difference.\n\n\u2014Robert Frost\n\nI consider Knitting Reimagined the destination I've arrived at after a thirty-year journey on a less-traveled road: designing hundreds of published pieces, authoring twenty-five knit and crochet books, and developing and teaching unconventional techniques of knitting. One of my priorities\u2014and passions\u2014over the years has been creating unique designs. I have never adhered to the adage that \"everything has been done in knitting.\" I respect and love traditional knitting techniques, stitches, and patterns, but there comes a time to break new ground, a time for experimentation and improvisation, and a time to rethink and reimagine typical structures and shapes in hand-knitting.\n\nMy goal was to fill this book with chic, wearable, but uniquely atypical garments that will appeal to knitters of all skill levels. The stitches are easy, as are the techniques to make the designs, but the resulting structures and shapes are unconventional, unexpected, and, if I do say so myself, showstopping. Hopefully these pieces break interesting new ground in hand-knitting, without being radically over-the-top avant-garde. Knitting Reimagined has twenty-five original designs using a variety of forward-looking techniques that will transform your yarn into sophisticated, adaptable knitted garments.\n\nThe designs run the gamut from tailored to bohemian, structured to unstructured. They are made with a variety of yarns, including handspun, hand-dyed, novelty, and cashmere. Everything is detailed for you in the instructions and diagrams: stitches, shapes, angles, openings, lengths, button closures, tucks, twists, layering, and more. Also noted are skill levels and approximate time frames to complete each project. You'll find Reimagine It sidebars that offer a few more ideas you can try when knitting the design. Perhaps there will be a suggestion to spark a new idea of your own. What you knit is an expression of yourself, so reimagine what will make each piece uniquely you.\n\nJoin me on this road less traveled. I think you'll find it a very surprising and inspiring one.\n\nHappy Knitting,  \nNicky\n\n# HOW TO USE THIS BOOK\n\nThere are three key questions that knitters ask when they choose a project to knit.\n\n1. Is it hard to make?\n\n2. How long does it take to make it?\n\n3. How much does the yarn cost?\n\nOf course, we all vary in our mastery of knitting techniques. We knit with our different experiences, at different speeds, and have different amounts of money that we want to invest. But I've included a general skill level and a ballpark of the amount of time each pattern requires to offer some guidance and give you a frame of reference.\n\n## Skill Levels\n\nBeginner friendly: Basic stitches, minimal shaping, simple finishing\n\nIntermediate: More intricate stitches, shaping, and finishing\n\nAdvanced: For experienced knitters able to tackle more complicated stitches, shaping, and finishing\n\n## Time\n\nQuick: These projects are quick and can be done in a weekend or a few days.\n\nWeeks: These projects take at least a week or more to complete.\n\nMonths: Depending on how much you are knitting, these projects can take a month or more to complete.\n\nchapter 1\n\nDIRECTIONALS\n\nReaders of my book Knitting Block by Block already know how creative, easy, and fun it is to combine the simplest shapes and transform them into gorgeous, wearable designs. These directional knits show just a few ways you can construct garments using only rectangles, angles, and squares; think of it as working easy block puzzles. As you can see in the projects in this chapter, the 90-degree corners and angles lend themselves to beautiful draping over the body, but these pieces require little to no shaping. The structure and fit are formed simply by how the shapes are positioned and sewn together!\n\nReckoning Rectangles Shawl\n\n## RECKONING  \nRECTANGLES  \nshawl\n\nThis sheer beauty is a breeze to make with two lush, easy-to-knit rectangles and no additional shaping. Using sequined mohair yarn in a soft lavender adds sparkle and glamour. Knit the edging at the same time, then simply overlap the edgings and sew the rectangles together at the shoulders, leaving a neck opening. The nature-inspired leaf and floral embellishments are appliqu\u00e9d onto the front lace panel. The rectangles will drape gracefully over the shoulders and flatter all body types.\n\nreimagine it\n\nThis wrap can easily be turned into a shrug. Just sew the two rectangles together down the shoulder seam to the wrist, then sew two sleeve seams, leaving a center opening of approximately 22\" (56cm). Use any color you like, or try the same idea with a different stitch pattern and embellishment.\n\nSKILL LEVEL\n\nTIME\n\nSIZE\n\nS\/M (L\/XL), shown in S\/M\n\nFINISHED MEASUREMENTS\n\nEach piece measures 12\" \u00d7 50\" (16\" \u00d7 54\") [30.5 \u00d7 127 (40.5 \u00d7 137)cm]\n\nGAUGE\n\n18 stitches and 24 rows = 4\" (10cm) in stitch pattern on smaller needles\n\nTake time to check gauge.\n\nMATERIALS\n\nSkacel Schulana Kid-Paillettes (42% kid mohair, 40% polyester, 18% silk), 0.87oz (25g), 137 yd (125m); 6 (8) balls of #380 Pale Mauve\n\nSize U.S. 7 (4.5mm) straight needles and double-pointed needles, or size needed to obtain gauge\n\nSize U.S. 8 (5mm) straight needles, or one size larger than gauge needles, for casting on and binding off\n\nTapestry needle\n\nRemovable stitch markers\n\nThree 8mm crystal beads for the flower\n\n### Open Honeycomb Stitch\n\n(over an odd number of stitches)\n\nRow 1 (RS): Purl.\n\nRow 2: Purl.\n\nRow 3: K1, *yo, ssk; repeat from * to end.\n\nRow 4: Purl.\n\nRepeat rows 1\u20134 for pattern.\n\n### FRONT\/BACK (MAKE 2)\n\nWith larger needles, cast on 55 (71) stitches.\n\nChange to smaller needles and knit 8 rows.\n\nNext row (RS): K3, work in Open Honeycomb stitch to the last 3 stitches, k3.\n\nRepeat this row, working Open Honeycomb stitch between Garter stitch edges, until piece measures 49 (53)\" [124.5 (134.5)cm] from the cast-on edge, ending with row 1.\n\nKnit 8 rows.\n\nBind off with larger needle.\n\nNote: The stitch pattern will bias; block lightly to shape.\n\n### FINISHING\n\n#### Flower\n\nCast on 10 stitches with larger needle, leaving a 12\" (30.5cm) tail.\n\nRow 1: *Kfb; repeat from * to end\u201420 stitches.\n\nRow 2: Knit.\n\nRow 3: *Kfb; repeat from * to end\u201440 stitches.\n\nRow 4: Knit.\n\nRow 5: *K4, kfb; repeat from * to the last 5 stitches, k5\u201447 stitches.\n\nRows 6\u201317: Work in Open Honeycomb stitch.\n\nRow 18: *K2tog; repeat from * to the last stitch, k1\u201424 stitches.\n\nRow 19: *K2tog; repeat from * to end\u201412 stitches.\n\nRow 20: Repeat row 19\u20146 stitches.\n\nPass the 2nd, 3rd, 4th, 5th, and 6th stitches, one at a time, over the first stitch and off the needle. Fasten off the last stitch.\n\nWeave the tail through the cast-on edge, gather tightly, and secure. Sew the cast-on edge to the bound-off edge to form the Flower and secure. Sew 3 crystal beads to the Flower center.\n\n#### Leaves (MAKE 9)\n\nCast on 5 stitches with larger needle.\n\nRow 1 (RS): K2, yo, k1, yo, k2\u20147 stitches.\n\nRow 2 and all WS rows: Purl.\n\nRow 3: K3, yo, k1, yo, k 3\u20149 stitches.\n\nRow 5: K4, yo, k1, yo, k4\u201411 stitches.\n\nRow 7: Knit.\n\nRow 9: Ssk, k7, k2tog\u20149 stitches.\n\nRow 11: Ssk, k5, k2tog\u20147 stitches.\n\nRow 13: Ssk, k3, k2tog\u20145 stitches.\n\nRow 15: Ssk, k1, k2tog\u20143 stitches.\n\nRow 17: Slip 1, k2tog, psso\u20141 stitch.\n\nFasten off.\n\n#### I-cord\n\nWith double-pointed needles, cast on 5 stitches.\n\n*Do not turn. Slide the stitches to the other end of the needle, k5; repeat from * for 17\" (43cm).\n\nNext rnd: Do not turn. Slide the stitches to the other end of the needle, k2tog, k1, k2tog\u20143 stitches.\n\nLast rnd: Do not turn. Slide the stitches to the other end of the needle, k3tog\u20141 stitch.\n\nFasten off.\n\n### ASSEMBLY\n\nLay both pieces side by side lengthwise, wrong sides facing up. Mark each side of the center 9 (10)\" [23 (25.5)cm] with removable stitch markers for the neck opening. Mark 6\" (15cm) out from each neck marker. Overlap the Back garter stitch edge over the Front garter stitch edge from the neck marker to the outer marker for each side, pin in place, and sew securely. The piece drapes over the shoulders and opens at the arms. Sew the Flower, Leaves, and I-cord to the Front using the photo and schematic as a guide. Sew one leaf to the Back.\n\nRenaissance Castle Tunic\n\n## RENAISSANCE CASTLE  \ntunic\n\nFeaturing an enchanting castle motif, this versatile tunic moves from medieval to modern. Like the Reckoning Rectangles Shawl this piece is also created with two rectangles, but with a totally different style. The castle motif is worked using stockinette stitch, reverse stockinette stitch, and seed stitch to create a texture resembling brocade, or relief needle work. Adorned with my castle buttons, the sides are closed at the waist, but of course can be sewn together or left open. This design drapes down the front and back and is crowned with a seed stitch cowl neck and matching buttons.\n\nreimagine it\n\nYou can omit the castle, add more colors, and do some bold striping, or, perhaps, sew the side seams, add sleeves, and attach two 10\" \u00d7 12\" (25.5cm \u00d7 30.5cm) blocks that can be sewn together to make a hood.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in S\/M\n\nFINISHED MEASUREMENTS\n\nWidth: 22 (26)\" [56 (66)cm]\n\nLength: 29 (30)\" [74 (76)cm]\n\nNote: This tunic is worked from the bottom up.\n\nGAUGE\n\n20 stitches and 24 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nCascade Lana D'Oro (50% alpaca, 50% wool), 3\u00bd oz (100g), 219 yd (200m); 4 (5) skeins of #1086 Hare (A), 2 (2) skeins of #1049 Charcoal (B)\n\nSize U.S. 7 (4.5mm) straight needles, or size needed to obtain gauge\n\nRemovable stitch markers\n\nTapestry needle\n\nThree 1\u215c\" (3.5cm) buttons (JHB's Nicky Epstein Carcassone #92725)\n\n### BACK\n\nWith B, cast on 110 (130) stitches.\n\n#### Seed Stitch Edging\n\nRow 1 (RS): *K1, p1; repeat from * to end.\n\nRow 2: *P1, k1; repeat from * to end.\n\nRepeat rows 1 and 2 (seed stitch) for 1\u00bd\" (3.8cm), ending with a WS row.\n\nNext row (RS): With B, work 7 stitches in seed stitch, join A and k96 (116), join a second ball of B and work the last 7 stitches in seed stitch.\n\nNote: When changing colors, twist yarns on the wrong side to avoid gaps.\n\nContinue working 7 stitches at each end in seed stitch with B and the center 96 (116) stitches in stockinette stitch with A until piece measures 13\u00bd\" (34.5cm) from the cast-on edge, ending with a WS row.\n\nMark each end of the next row with removable stitch markers to indicate the button placement.\n\nNext row (RS): Work 7 stitches in seed stitch, k5 (15), work Back chart over the next 86 stitches, k5 (15), work 7 stitches in seed stitch.\n\nWork as established until 58 rows of the Back chart are complete.\n\nContinue in seed stitch and stockinette stitch until piece measures 13 (14)\" [33 (35.5)cm] from the button markers.\n\n#### Shape Shoulders\n\nBind off 1 stitch at the beginning of the next 10 (14) rows, bind off 2 stitches at the beginning of the next 12 (16) rows, then bind off 4 stitches at the beginning of the next 6 (8) rows\u201452 stitches.\n\nBind off.\n\n### FRONT\n\nCast on and work Seed Stitch Edging same as for the Back.\n\nNext row (RS): With B, work 7 stitches in seed stitch, join A and k5 (15), work Front chart over the next 86 stitches, k5 (15), join a second ball of B, and work the last 7 stitches in seed stitch.\n\nNote: When changing colors, twist yarns on the wrong side to avoid gaps.\n\nMark each end for button when it is the same length as back markers.\n\nWork as established until 150 rows of the Front chart are complete.\n\nPiece should measure 26\u00bd\" (67.5cm) from the cast-on edge.\n\n#### Shape Shoulders and Neck\n\nBind off 1 stitch at the beginning of the next 10 (14) rows, bind off 2 stitches at the beginning of the next 12 (16) rows, then bind off 4 stitches at the beginning of the next 6 (8) rows.\n\nAt the same time, when 96 stitches remain, shape neck.\n\nNext row (RS): Bind off 2 stitches, k31, join a second ball of A and bind off 30 stitches, knit to the end.\n\nWorking on both sides at the same time, continue shaping shoulders as established and work neck shaping as follows:\n\nAt each neck edge, bind off 4 stitches twice, then 3 stitches once.\n\nComplete shoulder shaping.\n\n### FINISHING\n\nSew shoulder seams.\n\n#### Collar\n\nWith B, cast on 42 stitches.\n\nWork in seed stitch for 25\" (63.5cm). Bind off.\n\nPlace a marker 4\" (10cm) down the right front neck from the shoulder seam. Pin the long edge of the collar around the neck opening, working clockwise from the marker and extending the last 4\" (10cm) inside the collar to the shoulder seam. Sew in place. Sew a button to the collar overlap as pictured.\n\nMatch the Front and Back, overlapping the front seed stitch band over the back band at the button markers. Sew a button at each side through both bands.\n\nRenaissance Castle Tunic  \n~return to the beginning of the project~\n\nOn the Block Topper\n\n## ON THE BLOCK  \ntopper\n\nThe topper is quickly becoming a new classic design and is also a favorite of mine for gift giving. By sewing just four blocks together you can create an easy cowl or topper. I call this the \"magic\" topper because it can be worn multiple ways, has easy sizing and fit, and offers endless design possibilities. Use any combination of yarns, stitch patterns, and knitted embellishments. Of course, there are thousands of edgings to choose from to frame your masterpiece! This design is so satisfying that you will want to make it over and over for yourself and for others.\n\nreimagine it\n\nThe possibilities here are truly endless. You just need four blocks. They can be any stitch pattern, of one or many colors, and made with your favorite embellishments or yarns. For a wealth of block ideas, refer to my book Knitting Block by Block. It is sure to inspire you. Take a look at a few examples of how to reimage this design for your own unique style!\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L, XL), shown in size S\n\nFINISHED MEASUREMENTS\n\nLength (including waist and neck ribbing): 19\u00bd (20\u00bd, 21\u00bd, 22\u00bd)\" [49.5 (52, 54.5, 57)cm]\n\nBlock size: 12 (13, 14, 15)\" [30.5 (33, 35.5, 38)cm] square\n\nGAUGE\n\n18 stitches and 24 rows = 4\" (10cm) in stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nCascade Eco Cloud (70% undyed merino wool, 30% undyed alpaca), 3\u00bd oz (100g), 164 yd (150m); 3 (3, 4, 4) balls of #1802 Ecru (A), 3 (3, 4, 4) balls of #1803 Tan (B)\n\nSize U.S. 9 (5.5mm) straight needles, or size needed to obtain gauge\n\nSize U.S. 8 (5mm) 16\" (40cm) circular needle\n\nSize U.S. 8 (5mm) 32\" (80cm) circular needle\n\nTapestry needle\n\nStitch markers\n\nCable needle\n\nStitch holders\n\nPiece of cardboard, approximately 11\" \u00d7 11\" (28cm \u00d7 28cm)\n\n### BLOCKS (MAKE 4, TWO EACH WITH A AND B)\n\nWith larger needle, cast on 54 (58, 63, 67) stitches. Work in stockinette stitch until piece measures 12 (13, 14, 15)\" [30.5 (33, 35.5, 38)cm]. Bind off.\n\nPlace 3 blocks side by side, B-A-B, with RS facing up and sew together. (See diagram.) Flip over so the WS faces up. Place the second A block, RS up, on top of the first A block. Fold the top edges of the B blocks down to meet the side edges of the second A block, and sew them together.\n\n#### Bottom Edging\n\nNote: Two markers will be next to each other until rnd 1 is worked.\n\nWith RS facing, longer circular needle, and color A, and starting at a corner, *pick up and k1 corner stitch, place marker, pick up and k147 (159, 171, 183) stitches evenly to the next corner; place marker; repeat from * once more\u2014296 (320, 344, 368) stitches. Place marker for the beginning of the round, and place marker before corner stitch.\n\nRnd 1: Slip beginning-of-round marker, [m1p, slip marker, k1, slip marker, m1p, *k3, p3; repeat from * to 3 stitches before the next marker, k3] twice\u2014300 (324, 348, 372) stitches.\n\nRnd 2: Slip beginning-of-round marker, [p1, m1p, slip marker, k1, slip marker, m1p, p1, *k3, p3; repeat from * to 3 stitches before the next marker, k3] twice.\n\nContinue to work edging as established, increasing 1 stitch at each side of the corner stitch and working the increased stitches in purl, until the edging measures 1\u00bc\" (3cm). Bind off in rib.\n\n#### Cable Appliqu\u00e9s (MAKE 2, ONE EACH WITH A AND B)\n\nWith larger needle, cast on 12 stitches.\n\nRows 1 and 3 (WS): K2, p8, k2\n\nRow 2: P2, k8, p2.\n\nRow 4: P2, slip the next 4 stitches to a cn and hold in front, k4, k4 from cn, p2.\n\nRows 5, 7, and 9: Repeat row 1.\n\nRows 6, 8, and 10: Repeat row 2.\n\nRepeat rows 1\u201310 for pattern until piece measures 38 (41, 44, 47)\" [96.5 (104, 112, 119.5)cm]. Do not cut yarn. Place stitches on a holder.\n\n### COWL\n\nWith the RS facing, shorter circular needle, and B, start at a shoulder seam and pick up and k102 (108, 114, 126) stitches evenly around the neck opening. Place marker and join for working in the round.\n\nWork in k3, p3 rib for 2\" (5cm).\n\nChange to A and continue in rib for 2\u00bd\" (6.5cm).\n\nRemove marker and work back and forth in rib as established for 1\u00be\" (4.5cm), creating a slit. Bind off in rib.\n\n### FINISHING\n\nPlace a piece of cardboard inside the garment to keep from sewing the front to the back.\n\nGather the cast-on edges of the 2 cables and secure.\n\nPin the cast-on end of cable A to the bottom left corner of one Block A. Lay it diagonally up just past the midpoint of the block and pin in place. Then bring it back to the upper left corner, over the shoulder, diagonally past the midpoint on the second block A, ending back at the lower right corner. Adjust the cable length as needed, and bind off. Repeat for the other cable, working in the opposite direction and looping it under the first cable so that it forms crosses on both front and back. Adjust the cable length as needed, and bind off. Gather the bound-off edges and sew them into a corner seam. Sew the cables in place (see diagram).\n\nOn the Block Topper~ return to the beginning of the project ~\n\nThis is one of my favorite projects to teach because my students learn to think \"outside the block\" and discover that they can easily make unique designs with just a little imagination. I hope these topper examples will encourage you to reimagine any design in this book by simply changing the color, selecting a different stitch pattern, or adding an embellishment.\n\nGarter striping makes lovely angles when the blocks are sewn together.\n\nA combination of two multi-cable and two plaid blocks topped with a turtleneck and 2\u00d72 mitre edging.\n\nLEFT: Bubble stitch blocks worked in a lovely multicolored yarn, topped with a flower. RIGHT: Counterpane lace trimmed with I-cord.\n\nMulti-cabled blocks with a buttoned edge on one of the seams.\n\nJe Ne Sais Quoi Cape\n\n## JE NE SAIS QUOI  \ncape\n\nYou will be amazed at how easy and quick this stunning design is to make. Truly one of my designs that looks complex but is not, it is constructed with cleverly placed rectangles and simple stitch pattern repeats that when put together give the design a curved appearance. Notice how the longest rectangles fold in half to create a unique shoulder curve when gathered. The cape is made with a bulky alpaca and large needles for a quick knit that looks like haute couture and is luxurious to wear.\n\nreimagine it\n\nThis piece is made in rectangles so you can easily change the stitch patterns, but make sure the yarn is the same weight and gauge to keep the fit. Cuffs can also be added to the arm opening.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nShoulder width: approximately 15 (18)\" [38 (45.5)cm]\n\nLower edge: 76\u00bd (92)\" [194.5 (233.5)cm]\n\nLength: 23 (25)\" [58.5 (63.5)cm]\n\nGAUGE\n\n12 stitches and 14 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nThe BagSmith Blissa Botanicals (85% baby alpaca, 10% extra-fine merino wool, 5% polyamide), 8 oz (226g), 70 yd (64m); 6 (8) balls of Dusty Plum\n\nSize U.S. 15 (10mm) straight needles, or size needed to obtain gauge\n\nTapestry needle\n\nOne 1\u215c\" (3.5cm) button (JHB's Nicky Epstein Carcassone #92725)\n\n### SIDE PANEL (MAKE 2)\n\n#### Open Rib Stitch\n\n(multiple of 7 stitches + 3)\n\nCast on 38 (45) stitches.\n\nRow 1 (RS): P3, *k4, p3; repeat from * to end.\n\nRow 2: K1, yo, k2tog, *p4, k1, yo, k2tog; repeat from * to end.\n\nRepeat rows 1 and 2 for pattern. Work until piece measures 46 (50)\" [117 (127)cm].\n\nBind off.\n\n### LEFT FRONT AND COLLAR (MAKE 1)\n\n#### Elongated Basket Weave\n\n(multiple of 18 stitches + 10)\n\nCast on 28 stitches.\n\nRow 1 (RS): K11, p2, k2, p2, k11.\n\nRow 2: P1, k8, [p2, k2] twice, p2, k8, p1.\n\nRow 3: K1, p8, [k2, p2] twice, k2, p8, k1.\n\nRow 4: P11, k2, p2, k2, p11.\n\nRows 5\u20138: Repeat rows 1\u20134.\n\nRow 9: Knit.\n\nRow 10: [P2, k2] twice, p12, [k2, p2] twice.\n\nRow 11: [K2, p2] twice, k2, p8, [k2, p2] twice, k2.\n\nRow 12: [P2, k2] twice, p2, k8, [p2, k2] twice, p2.\n\nRow 13: [K2, p2] twice, k12, [p2, k2] twice.\n\nRows 14\u201317: Repeat rows 10\u201313.\n\nRow 18: Purl.\n\nRepeat rows 1\u201318 for pattern 7 (8) more times. Piece should measure 41\u00bd (46\u00bd)\" [105.5 (118)cm].\n\nBind off.\n\n### RIGHT FRONT (MAKE 1)\n\n#### Garter Dash Stitch\n\n(multiple of 10 stitches + 12)\n\nCast on 32 (42) stitches.\n\nRow 1 (RS): K5, p7, *k4, p6; repeat from * to end.\n\nRows 2, 4, and 6: Purl to the last 6 stitches, k1, p5.\n\nRow 3: K5, p1, knit to the end.\n\nRow 5: K5, p2, *k4, p6; repeat from * to the last 5 stitches, k5.\n\nRow 7: Repeat row 3.\n\nRow 8: Repeat row 2.\n\nRepeat rows 1\u20138 for pattern. Work until the piece measures 22 (24)\" [56 (61)cm], ending with a WS row.\n\nButtonhole row 1 (RS): K3, bind off 2 stitches, work in pattern to end.\n\nButtonhole row 2: Work in pattern to the bound-off stitches, cast on 2 stitches, p3.\n\nContinue in pattern until piece measures 23 (25)\" [58.5 (63.5)cm].\n\nNote: The stockinette stitches at the front edge will curl, forming a cord-like edge.\n\n### BACK\n\n#### Seeded Zigzag Stitch\n\n(multiple of 9 stitches)\n\nCast on 45 (54) stitches.\n\nRow 1 (RS): *[K1, p1] twice, k4, p1; repeat from * to end.\n\nRow 2: *P4, [k1, p1] twice, k1; repeat from * to end.\n\nRow 3: [K1, p1] 3 times, *k4, [p1, k1] twice, p1; repeat from * to the last 3 stitches, k3.\n\nRow 4: P2, *[k1, p1] twice, k1, p4; repeat from * to the last 7 stitches, [k1, p1] twice, k1, p2.\n\nRow 5: K3, *[p1, k1] twice, p1, k4; repeat from * to the last 6 stitches, [p1, k1] 3 times.\n\nRow 6: *[K1, p1] twice, k1, p4; repeat from * to end.\n\nRow 7: Repeat row 5.\n\nRow 8: Repeat row 4.\n\nRow 9: Repeat row 3.\n\nRow 10: Repeat row 2.\n\nRepeat rows 1\u201310 for pattern. Work until piece measures 19\u00bc (20\u00be)\" [49 (53)cm].\n\nContinuing in pattern as set, decrease 1 stitch each side every row 13 (15) times\u201419 (24) stitches.\n\nBind off.\n\n### FINISHING\n\nFold one Side Panel in half with the right sides together. Sew one side of the panel closed, leaving a 6\" (15cm) opening approximately 9 (10)\" [23 (25.5)cm] from the bottom edge for the armhole. With a doubled length of yarn, tightly gather the open rib stitches across the shoulder line of the panel to 4\u00bd (5)\" [11.5 (12.5)cm] and secure to form a nice rounded shoulder. Repeat for the other panel.\n\nPlace the Back between the right and left Side Panels, lining it up from the bottom edges to the back neck, and sew in place.\n\nThe Right Front is sewn to the right Side Panel.\n\nThe Left Front is sewn to the left Side Panel, over the left shoulder and across the back neck, over the right shoulder and across the top edge of the Right Front, ending at the p1 stitch of the edging.\n\n#### Closure\n\nSew the button to the Left Front, 1\" (2.5cm) from the shoulder and the Side Panel, to correspond with the buttonhole.\n\nJe Ne Sais Quoi Cape  \n~return to the beginning of the project~\n\nThe Deep End Shawl\n\n## THE DEEP END  \nshawl\n\nThis design has a mix of curves, rectangles, and a sharp elongated angle that come together perfectly to create a very unique shawl. There is a bit more knitted construction in this piece, but only two seams connect the three pieces. A cable loop closure is knit into the piece leaving an opening and the deep end point goes through the cable slit to close the shawl. A cable cord edging is knit in on the edges or can be knit separately and sewn around the edges instead. Wear this shawl for a casual elegant look.\n\nreimagine it\n\nI would love to see this stitch pattern made by selecting a solid color for A and a bold variegated yarn for B. Another way to go would be making the two-color stitch pattern with a solid color and then using a contrasting color for the cable slit and cord.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nBack width: 25\u00bc (28)\" [64.5 (71)cm]\n\nGAUGE\n\n18 stitches and 36 rows = 4\" (10cm) in box pattern on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nPlymouth Yarn Baby Alpaca Aire (100% baby alpaca), 3\u00bd oz (100g), 218 yd (199m); 2 (3) balls of #5403 Charcoal (A); 5 (6) balls of #5002 Olive (B)\n\n2 pairs of size U.S. 9 (5.5mm) straight needles and a set of 2 double-pointed needles, or size needed to obtain gauge\n\nSize U.S. 2 (2.5mm) 16\" (40.5cm) circular needle (or smaller) to hold stitches\n\nStitch holder\n\nCable needle\n\nTapestry needle\n\n1 covered coat hook and eye (optional)\n\n#### Box Pattern\n\n(multiple of 3 stitches)\n\nRow 1 (RS): With B, knit.\n\nRow 2: With B, purl.\n\nRow 3: With A, k1, slip 1 wyib, *k2, slip 1 wyib; repeat from * to the last stitch, k1.\n\nRow 4: With A, k1, slip 1 wyif, *k2, slip 1 wyif; repeat from * to the last stitch, k1.\n\nRepeat rows 1\u20134 for pattern.\n\n#### Cables Pattern\n\n(multiple of 8 stitches)\n\n4\/4 RC: Slip 4 stitches to cn and hold in back, k4, k4 from cn.\n\n4\/4 LC: Slip 4 stitches to cn and hold in front, k4, k4 from cn.\n\nRow 1 (RS): Knit.\n\nRow 2 and all WS rows: K2, purl to the last 2 stitches, k2.\n\nRow 3: Knit.\n\nRow 5: K2, *4\/4 RC; repeat from * to the last 2 stitches, k2.\n\nRows 7 and 9: Knit.\n\nRow 11: K6, *4\/4 LC; repeat from * to the last 6 stitches, k6.\n\nRow 12: Repeat row 2.\n\nRepeat rows 1\u201312 for pattern.\n\n#### Cable I-cord\n\n(over 6 stitches)\n\n2\/2 LC: Slip 2 stitches to cn and hold in front, k2, k2 from cn.\n\nRows 1\u20135: Knit. Do not turn, slide stitches to the other end of the needle.\n\nRow 6: K1, 2\/2 LC, k1.\n\nRepeat rows 1\u20136 for pattern.\n\n### BACK\n\nWith larger straight needles and B, cast on 114 (126) stitches.\n\nWork in Box pattern until piece measures 6\u00bc (6\u00bd)\" [16 (16.5)cm], ending with a WS row 4.\n\nDecrease row (RS): K1, ssk, work in Box pattern as established to the last 3 stitches, k2tog, k1.\n\nRepeat Decrease row every 4th row 16 (18) more times\u201480 (88) stitches.\n\nWork 3 rows even in Box pattern.\n\n#### Shoulder Shaping\n\nContinuing in Box pattern as established, bind off 6 (7) stitches at the beginning of the next 6 rows\u201444 (46) stitches.\n\nBind off.\n\n### LEFT FRONT\n\nWith larger straight needles and B, cast on 63 (69) stitches. Work in Box pattern until piece measures 14 (15\u00bc)\" [35.5 (38.5)cm], ending with a WS row 4.\n\nNext row (RS): Work 46 (48) stitches in Box pattern, insert the smaller circular needle into the front of the last 28 stitches worked and leave it on the RS of the work to be used later for the Cable Tab, work in Box pattern to end.\n\nContinue in Box pattern for 5\" (12.5cm), ending with a WS row 4.\n\nLeave stitches on the needle.\n\n### TAB\n\nPlace the Tab stitches from the small circular needle onto the second set of larger needles, ready to work a RS row. Join B to the first stitch and work rows 1\u201312 of Cables pattern for 5\" (12.5cm), ending with a WS row.\n\n#### Join Tab\n\nReturn to the Left Front stitches.\n\nJoining row (RS): With B, k18 (20), *knit 1 stitch from the Tab together with 1 stitch from the Left Front; repeat from * until all the Tab stitches have been worked, knit to end\u201463 (69) stitches.\n\nContinue in Box pattern for 6\u00bd\" (16.5cm), ending with a WS row 4.\n\n#### Side Shaping\n\nNext row: Bind off 28 (29) stitches, ssk using the 1 stitch on the right-hand needle as the first stitch, knit to end\u201434 (39) stitches.\n\nWork 3 rows in Box pattern as established.\n\nDecrease row (RS): K1, ssk, knit to end\u201433 (38) stitches.\n\nContinuing in Box pattern as established, repeat Decrease row every 4th row 15 (17) more times\u201418 (21) stitches.\n\nWork 3 rows even in Box pattern as established.\n\n#### Shoulder Shaping\n\nBind off 6 (7) stitches at the beginning of the next 3 RS rows.\n\nFasten off the last stitch.\n\n### RIGHT FRONT\n\nWith B, cast on 3 stitches.\n\nRow 1 (RS): Knit.\n\nRow 2: Purl.\n\nRow 3: With A, k1, slip 1 wyib, k1.\n\nRow 4: K1, slip 1 wyif, k1.\n\nRow 5: With B, k2, kfb\u20144 stitches.\n\nRow 6: Purl.\n\nRow 7: With A, k1, slip 1 wyib, k2.\n\nRow 8: K2, slip 1 wyif, k1.\n\nRow 9: With B, k3, kfb\u20145 stitches.\n\nRow 10: Purl.\n\nRow 11: With A, k1, slip 1 wyib, k2, slip 1 wyib.\n\nRow 12: Slip 1 wyif, k2, slip 1 wyif, k1.\n\nRow 13: With B, k4, kfb\u20146 stitches.\n\nRow 14: Purl.\n\nRow 15: With B, k1, slip 1 wyib, k2, slip 1 wyib, k1.\n\nRow 16: K1, slip 1 wyif, k2, slip 1 wyif, k1.\n\nContinue to increase 1 stitch at the end of every Box pattern row 1 as established until a total of 60 (66) increases have been completed and there are 63 (69) stitches on the needle.\n\nWork 7 rows even in Box pattern, ending with a WS row 4.\n\nNext Row (RS): With B, k33 (38), k2tog, k28 (29).\n\n#### Side Shaping\n\nNext row (WS): Bind off 28 (29) stitches, purl to end\u201434 (39) stitches.\n\nWork Box pattern rows 3 and 4 as established.\n\nDecrease row (RS): With B, knit to the last 3 stitches, k2tog, k1\u201433 (38) stitches.\n\nContinuing in Box pattern as established, repeat Decrease row every 4th row 15 (17) more times\u201418 (21) stitches.\n\nWork 4 rows even in Box pattern as established.\n\n#### Shoulder Shaping\n\nBind off 6 (7) stitches at the beginning of the next 3 WS rows.\n\nFasten off.\n\n### FINISHING\n\nWith the RS together, sew the 28 (29) bound-off stitches of the fronts to the side edges of the Back, the decrease edges and shoulder bind-offs, one after the other, as one seam.\n\n#### Cable I-cord Trim\n\nWith dpns and a doubled strand of B, cast on 6 stitches. With the RS facing, start across the cast-on edge of the Left Front.\n\nJoining row: K5, slip 1, pick up and k1 stitch, psso.\n\nRepeat the Joining row, working rows 1\u20136 of Cable I-cord, picking up and attaching 1 stitch on the body of the shawl for each I-cord row.\n\nWhen all cast-on stitches have been worked, work 5 rows of Cable I-cord without attaching them, for the corner.\n\nContinue to work Cable I-cord, attaching it evenly (approximately every 3rd row) up the side edge of the Left Front, across the Back, down the side edge of the Right Front, and around the front edges back to the start, working 5 unattached rows at each corner. Use the Kitchener stitch to graft the 2 ends together. Sew hook and eye (optional) at V-point on each side, under the cable edge where the fronts cross.\n\nThe Deep End Shawl[  \n~return to the beginning of the project~](Epst_9780385346269_epub_c01_r1.htm#c01-s5)\n\nRoyal Lace Coat with Hood\n\n## ROYAL LACE  \ncoat with hood\n\nLike something out of a fairy tale, this design is created by sewing seven rectangles together and adding a detachable hood. Each rectangle is made with graduated arrow lace that knits up like a dream and converges into stockinette stitch forming lovely lace points on all the rectangles. The sleeves add structure for a more tailored fit, and the shape is perfectly flattering for any body size. Always in style, it's a piece that should be in everyone's closet. The sample pictured uses my Knight's Amour buttons and Fleur de Lis clasp from JHB.\n\nreimagine it\n\nKeep the lace pattern and construction, but try using a fabulous subtly colored, hand-dyed yarn. Add a different button and closure choice for a completely new look.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size L\/XL\n\nFINISHED MEASUREMENTS\n\nBust: 39\u00bd (47\u00bd)\" [100 (120.5)cm]\n\nLength: 23\u00bc (25)\" [59 (63.5)cm]\n\nGAUGE\n\n16 stitches and 24 rows = 4\" (10cm) in stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nCascade Cloud (70% merino wool, 30% baby alpaca), 3\u00bd oz (100g), 164 yd (150m); 8 (9) skeins of #2109 Red\n\nSize U.S. 10 (6mm) needles, or size needed to obtain gauge\n\nSize U.S. 9 (5.5mm) needles\n\nStitch markers\n\nStitch holders\n\nTapestry needle\n\nSix 1\u215b\" (2.8cm) buttons (JHB's Nicky Epstein Knight's Armour #92723)\n\n1 clasp (JHB's Nicky Epstein Fleur de Lis #4042)\n\n#### Pattern 1\n\n(multiple of 8 stitches + 1)\n\nRow 1 (RS): K1, *ssk, [k1, yo] twice, k1, k2tog, k1; repeat from * to end.\n\nRow 2: P1, *p2tog, [p1, yo] twice, p1, p2tog tbl, p1; repeat from * to end.\n\nRow 3: K1, *yo, ssk, k3, k2tog, yo, k1; repeat from * to end.\n\nRow 4: P2, *yo, p2tog, p1, p2tog tbl, yo, p3; repeat from * to the last 7 stitches, yo, p2tog, p1, p2tog tbl, yo, p2.\n\nRow 5: K3, *yo, s2kp, yo, k5; repeat from * to the last 6 stitches, yo, s2kp, yo, k3.\n\nRow 6: Repeat row 2.\n\nRow 7: Repeat row 1.\n\nRow 8: P1, *yo, p2tog, p3, p2tog tbl, yo, p1; repeat from * to end.\n\nRow 9: K2, *yo, ssk, k1, k2tog, yo, k3; repeat from * to the last 7 stitches, yo, ssk, k1, k2tog, yo, k2.\n\nRow 10: P3, *yo, s2pp, yo, p5; repeat from * to the last 6 stitches, yo, s2pp, yo, p3.\n\nRepeat rows 1\u201310 for pattern.\n\n#### Pattern 2\n\n(multiple of 8 stitches +1)\n\nRows 1 and 3: K1, *ssk, [k1, yo] twice, k1, k2tog, k1; repeat from * to end.\n\nRow 2 and all WS rows: Purl.\n\nRow 5: K1, *yo, ssk, k3, k2tog, yo, k1; repeat from * to end.\n\nRow 7: K2, *yo, ssk, k1, k2tog, yo, k3; repeat from * to the last 7 stitches, yo, ssk, k1, k2tog, yo, k2.\n\nRow 9: K3, *yo, s2kp, yo, k5; repeat from * to the last 6 stitches, yo, s2kp, yo, k3.\n\nRow 10: Purl.\n\nRepeat rows 1\u201310 for pattern.\n\n### BACK\n\nWith larger needles, cast on 85 (101) stitches.\n\nKnit 2 rows.\n\nRow 1 (RS): [K1, p1] 3 times, place marker, work 73 (89) stitches in Pattern 1, place marker, [p1, k1] 3 times.\n\nRows 2\u201370: Keeping 6 stitches at each end in rib as set, work Pattern 1 rows 2\u201310 between markers once, then work rows 1\u201310 another 6 times.\n\nRow 71: [K1, p1] 3 times, slip marker, work 73 (89) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 72\u201380: Keeping 6 stitches at each end in rib as established, work Pattern 2 rows 2\u201310 between markers.\n\nRow 81: [K1, p1] 3 times, remove marker, k8, place marker, work 57 (73) stitches in Pattern 2 as established, place marker, k8, remove marker, [p1, k1] 3 times.\n\nRows 82\u201390: Work 6 stitches in rib, 8 stitches in stockinette stitch, Pattern 2 rows 2\u201310 between markers, 8 stitches in stockinette stitch, and the last 6 stitches in rib.\n\nRow 91: [K1, p1] 3 times, k8, remove marker, k8, place marker, work 41 (57) stitches in Pattern 2 as established, place marker, k8, remove marker, k8, [p1, k1] 3 times.\n\nRows 92\u2013100: Work 6 stitches in rib, 16 stitches in stockinette stitch, Pattern 2 rows 2\u201310 between markers, 16 stitches in stockinette stitch, and the last 6 stitches in rib.\n\nRow 101: [K1, p1] 3 times, k16, remove marker, k8, place marker, work 25 (41) stitches in Pattern 2 as established, place marker, k8, remove marker, k16, [p1, k1] 3 times.\n\nRows 102\u2013110: Work 6 stitches in rib, 24 stitches in stockinette stitch, Pattern 2 rows 2\u201310 between markers, 24 stitches in stockinette stitch, and the last 6 stitches in rib.\n\nRow 111: [K1, p1] 3 times, k24, remove marker, k8, place marker, work 9 (25) stitches in Pattern 2 as established, place marker, k8, remove marker, k24, [p1, k1] 3 times.\n\nRows 112\u2013120: Work 6 stitches in rib, 32 stitches in stockinette stitch, Pattern 2 rows 2\u201310 between markers, 32 stitches in stockinette stitch, and the last 6 stitches in rib.\n\nSize S\/M only, skip to row 131.\n\nRow 121: [K1, p1] 3 times, k32, remove marker, k8, place marker, work 9 stitches in Pattern 2 as established, place marker, k8, remove marker, k32, [p1, k1] 3 times.\n\nRows 122\u2013130: Work 6 stitches in rib, 40 stitches in stockinette stitch, Pattern 2 rows 2\u201310 between markers, 40 stitches in stockinette stitch, and the last 6 stitches in rib.\n\nRows 131\u2013150: Work 6 stitches in rib, 73 (89) stitches in stockinette stitch (remove markers on row 131), and the last 6 stitches in rib.\n\nPlace stitches on a holder.\n\n### LEFT FRONT\n\nWith larger needles, cast on 45 (53) stitches.\n\nKnit 2 rows.\n\nRow 1: [K1, p1] 3 times, place marker, work 33 (41) stitches in Pattern 1, place marker, [p1, k1] 3 times.\n\nRows 2\u201310: Work 6 stitches in rib, slip marker, work Pattern 1 rows 2\u201310 between markers, slip marker, work the last 6 stitches in rib.\n\nRows 11\u201370: Repeat rows 1\u201310 six more times.\n\nRow 71: [K1, p1] 3 times, slip marker, work 33 (41) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 72\u201380: Work 6 stitches in rib, Pattern 2 rows 2\u201310 between markers, and the last 6 stitches in rib.\n\nRow 81: [K1, p1] 3 times, remove marker, k8, place marker, work 25 (33) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 82\u201390: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 91: [K1, p1] 3 times, k8, remove marker, k8, place marker, work 17 (25) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 92\u2013100: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 101: [K1, p1] 3 times, k16, remove marker, k8, place marker, work 9 (17) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 102\u2013110: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nSize S\/M only, skip to row 121.\n\nRow 111: [K1, p1] 3 times, k24, remove marker, k8, place marker, work 9 stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 112\u2013120: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRows 121\u2013139: Work in rib stitches as set and 33 (41) stitches between markers in stockinette stitch.\n\n#### Neck\n\nBind off 8 (10) stitches at beginning of next WS row. Continue to bind off at the beginning of WS rows, 3 (4) stitches twice, and 2 stitches 3 times. Place the remaining 25 (29) stitches on a holder.\n\n### RIGHT FRONT\n\nWith larger needles, cast on 45 (53) stitches.\n\nKnit 2 rows.\n\nRow 1: [K1, p1] 3 times, place marker, work 33 (41) stitches in Pattern 1, place marker, [p1, k1] 3 times.\n\nRows 2\u201310: Work 6 stitches in rib, work Pattern 1 rows 2\u201310 between markers, work the last 6 stitches in rib.\n\nRows 11\u201370: Repeat rows 1\u201310 6 more times.\n\nRow 71: [K1, p1] 3 times, slip marker, work 33 (41) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 72\u201380: Work 6 stitches in rib, Pattern 2 rows 2\u201310 between markers, and the last 6 stitches in rib.\n\nRow 81: [K1, p1] 3 times, slip marker, work 25 (33) stitches in Pattern 2, place marker, k8, remove marker, [p1, k1] 3 times.\n\nRows 82\u201390: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 91: [K1, p1] 3 times, slip marker, work 17 (25) stitches in Pattern 2, place marker, k8, remove marker, k8, [p1, k1] 3 times.\n\nRows 92\u2013100: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 101: [K1, p1] 3 times, slip marker, work 9 (17) stitches in Pattern 2, place marker, k8, remove marker, k16, [p1, k1] 3 times.\n\nRows 102\u2013110: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nSize S\/M only, skip to row 121.\n\nRow 111: [K1, p1] 3 times, slip marker, work 9 stitches in Pattern 2, place marker, k8, remove marker, k24, [p1, k1] 3 times.\n\nRows 112\u2013120: Working rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRows 121\u2013138: Work in rib stitches as set and 33 (41) stitches between ribs in stockinette stitch.\n\n#### Neck\n\nBind off 8 (10) stitches at beginning of next RS row. Continue to bind off at the beginning of RS rows, 3 (4) stitches twice, and 2 stitches 3 times. Place the remaining 25 (29) stitches on a holder.\n\n### SLEEVES (MAKE 2)\n\nWith larger needles, cast on 43 stitches.\n\nKnit 2 rows.\n\nRow 1: K1, place marker, work 41 stitches in Pattern 1, place marker, k1.\n\nRows 2\u201314: Keeping the first and last stitches in stockinette stitch, work Pattern 2 rows 2\u201310 then rows 1\u20134 between markers.\n\nRow 15 (increase row): K1, m1, slip marker, work Pattern 2 row 5 between markers, slip marker, m1, k1.\n\nRepeat increase row every 6th row 8 (10) more times, working m1 one stitch in from each edge, working increases in stockinette stitch as follows:\n\nRows 16\u201320: Keeping the first 2 and last 2 stitches in stockinette stitch, work Pattern 2 rows 6\u201310 between markers.\n\nRow 21 (increase row): K1, m1, k1, remove marker, k8, place marker, work 25 stitches in Pattern 2, place marker, k8, remove marker, k1, m1, k1.\n\nRows 22\u201330: Keeping stitches at each end in stockinette stitch, work Pattern 2 rows 2\u201310 between markers\u201449 stitches. (Row 27 is an increase row.)\n\nRow 31: K12, remove marker, k8, place marker, work 9 stitches in Pattern 2, place marker, k8 remove marker, k12.\n\nRows 32\u201340: Working stitches at each end in stockinette stitch, work Pattern 2 rows 2\u201310 between markers\u201453 stitches. (Rows 33 and 39 are increase rows.)\n\nRow 41: Knit across, removing markers.\n\nContinue in stockinette stitch, increasing as established, until you have 61 (65) stitches. If necessary, work even in stockinette stitch until Sleeve measures 16\" (40.5cm) from the cast-on edge, or to desired length.\n\nBind off.\n\n### HOOD  \nRIGHT SIDE\n\n#### Bottom Edge\n\nCast on 40 (48) stitches with larger needles.\n\nWork in knit 1, p1 for a total of 9 rows.\n\n#### Body\n\nRow 1 (RS): [K1, p1] 3 times, place marker, work 33 (41) stitches in Pattern 2, place marker, k1.\n\nRows 2\u201310: Work rib stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 11: [K1, p1] 3 times, slip marker, work 25 (33) stitches in Pattern 2, place marker, k8, remove marker, k1.\n\nRows 12\u201320: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 21: [K1, p1] 3 times, slip marker, work 17 (25) stitches in Pattern 2, place marker, k8, remove marker, k9.\n\nRows 22\u201330: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 31: [K1, p1] 3 times, slip marker, work 9 (17) stitches in Pattern 2, place marker, k8, remove marker, k17.\n\nRows 32\u201340: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 41: [K1, p1] 3 times, slip marker, work 0 (9) stitches in Pattern 2, place marker, k8, remove marker, k25.\n\nRows 42\u201350: Work rib and stockinette stitches as set; for size L\/XL only, work Pattern 2 rows 2\u201310 between the markers.\n\nRow 51: [K1, p1] 3 times, knit to end, removing all markers.\n\nRow 52: Purl to the last 6 stitches, [k1, p1] 3 times.\n\nRepeat rows 51 and 52 until Hood measures 12\" (30.5cm) from the cast-on edge, ending with a WS row.\n\nBind off.\n\n### LEFT SIDE\n\n#### Bottom Edge\n\nWith larger needles, cast on 40 (48).\n\nStarting on the WS, work k1, p1 for a total of 9 rows.\n\n#### Body\n\nRow 1 (RS): K1, place marker, work 33 (41) stitches in Pattern 2, place marker, [p1, k1] 3 times.\n\nRows 2\u201310: Work rib stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 11: K1, remove marker, k8, place marker, work 25 (33) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 12\u201320: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 21: K9, remove marker, k8, place marker, work 17 (25) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 22\u201330: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 31: K17, remove marker, k8, place marker, work 9 (17) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 32\u201340: Work rib and stockinette stitches as set, work Pattern 2 rows 2\u201310 between markers.\n\nRow 41: K25, remove marker, work 0 (9) stitches in Pattern 2, slip marker, [p1, k1] 3 times.\n\nRows 42\u201350: Work rib and stockinette stitches as set; for size L\/XL only, work Pattern 2 rows 2\u201310 between markers.\n\nRow 51: Knit to the last 6 stitches, [p1, k1] 3 times, removing all markers.\n\nRow 52: [P1, k1] 3 times, purl to end.\n\nRepeat rows 51 and 52 until Hood measures 12\" (30.5cm) from the cast-on edge, ending with a WS row.\n\nBind off.\n\n### FINISHING\n\nWith the right sides together, use the 3-needle bind-off to join the shoulders, leaving the 35 (43) Back neck stitches on a holder.\n\n#### Neck Band\n\nWith RS facing and smaller needles, pick up and k21 (25) stitches from Right Front neck edge to the right shoulder seam; k35 (43) Back neck stitches, increasing 2 stitches evenly across [37 (45) stitches]; pick up and k21 (25) stitches from the left shoulder seam down to the Left Front edge\u201479 (95) stitches.\n\nRow 1 (WS): P1, *k1, p1; repeat from * to end.\n\nRow 2: K1, *p1, k1; repeat from * to end.\n\n### MAKE BUTTONHOLES\n\nRow 3: Rib 3, [bind off 3 stitches, rib 11 (14)] twice, bind off 3 stitches, rib 11 (15), [bind off 3 stitches, rib 11 (14)] twice, bind off 3 stitches, rib 3.\n\nRow 4: Work in rib, casting on 3 stitches over the bound-off stitches.\n\nContinue to work in rib until band measures 1\u00bd\" (3.8cm). Bind off in rib.\n\nReinforce buttonholes with whipstitch.\n\nMeasure and mark 7\u00bd (8)\" [19 (20.5)cm] down from the shoulder on both the back and the front. Sew Sleeves between the markers under the rib edging (see photograph).\n\nSew sleeve seams.\n\nWith the right sides together, sew the top and back seams of the Hood. Sew buttons onto the Hood to correspond with the neck-band buttonholes.\n\nSew the clasp to the front at the neck (see photograph).\n\nRoyal Lace Coat with Hood  \n~return to the beginning of the project~\n\nWelted Button Tuck Cardi\n\n## WELTED BUTTON  \ntuck cardi\n\nThis asymmetrical style has an unusual drape and shape, and traditional armhole shaping for fit. The multiple buttons and buttonholes do much of the work by creating graceful folds and flowing tucks. The two front shapes are completely different, but when tucked at the sides and sewn into the back it creates a perfect fit. The sleeve cuffs have a little ruching and each is topped with a button. You need to enjoy the stockinette stitch because there is a lot of it! It is the perfect piece for therapeutic, stress-free, or on-the-go knitting.\n\nreimagine it\n\nMulticolored dyed yarns often create a one-of-a-kind look, such as the diamond patterning on the back. Yours may end up on the front! Try keeping the welted button side in one color, with the other side in two-color striping and the sleeves in another color. Consider adding a pocket to the left side.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nBust: 39 (50\u00bd)\" [99 (128)cm]\n\nBack length: 32\u00bd (34\u00bd)\" [82.5 (87.5)cm] at back point edge\n\nSleeve length 20\" (51cm) before gather\n\nNote: The front and back of this sweater are made from the top down. The sleeves are worked from the cuff up. The sleeves are intentionally long.\n\nGAUGE\n\n20 stitches and 24 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nBlue Heron Yarns Rayon Metallic (88% rayon, 12% metallic), 8 oz (226g), 550 yd (503m); 3 (4) skeins of Water Hyacinth\n\nSize U.S. 7 (4.5mm) straight needles, or size needed to obtain gauge\n\nTapestry needle\n\nEleven \u00be\" (2cm) buttons (JHB's Feng Shui #96992)\n\n### BACK\n\nCast on 83 (90) stitches.\n\nRows 1\u201345 (1\u201335): Starting with a purl row, work 45 (35) rows in stockinette stitch.\n\n#### Armhole Shaping\n\nRow 46 (36) (RS): K1, m1, knit to the last stitch, m1, k1\u201485 (92) stitches.\n\nRow 47 (37): Purl.\n\nRows 48\u201351 (38\u201355): Repeat rows 46 and 47 twice (rows 36 and 37 nine times)\u201489 (110) stitches.\n\nRow 52 (56): Cast on 4 (8) stitches, knit to the end\u201493 (118) stitches.\n\nRow 53 (57): Cast on 4 (8) stitches, purl to the end\u201497 (126) stitches.\n\nRows 54\u201371 (58\u201375): Work 18 rows in stockinette stitch.\n\nRow 72 (76): K1, m1, knit to the last stitch, m1, k1\u201499 (128) stitches.\n\nRows 73\u201389 (77\u201393): Work 17 rows in stockinette stitch.\n\nRows 90\u2013143 (94\u2013147): Repeat rows 72\u201389 (76\u201393) 3 times\u2014105 (134) stitches.\n\nWork even in stockinette stitch until piece measures 16\" (40.5cm) from the underarm, ending with a RS row.\n\n#### Shape Lower Edge\n\nRows 1, 3, and 5 (WS): Purl.\n\nRow 2: Bind off 5 stitches, knit to the end\u2014100 (129) stitches.\n\nRows 4 and 6: Bind off 4 stitches, knit to the end\u201492 (121) stitches.\n\nRows 7\u201348 (7\u201354): Repeat rows 1\u20136 seven more times (rows 1 and 2 twenty-four times).\n\nFasten off the last stitch.\n\n### RIGHT FRONT\n\nCast on 14 (18) stitches.\n\nRow 1 (WS): Purl.\n\n#### Neck Shaping\n\nRow 2: Knit to the last stitch, m1, k1\u201415 (19) stitches.\n\nRows 3\u201324: Repeat rows 1 and 2 eleven more times\u201426 (30) stitches.\n\nRows 25 and 27: Purl.\n\nRow 26: Knit.\n\nRow 28: Knit to the last stitch, m1, k1\u201427 (31) stitches.\n\nRows 29\u201344 (29\u201334): Repeat rows 27 and 28 eight (three) more times\u201435 (34) stitches.\n\nRow 45 (35): Purl.\n\n#### Armhole Shaping\n\nRow 46 (36): K1, m1, knit to the last stitch, m1, k1\u201437 (36) stitches.\n\nRow 47 (37): Purl.\n\nRows 48\u201351 (38\u201355): Repeat rows 46 and 47 twice (36 and 37 nine times)\u201441 (54) stitches.\n\nRow 52 (56): Cast on 4 (8) stitches, knit to the last stitch, m1, k1\u201446 (63) stitches.\n\nRows 53 and 55 (57 and 59): Purl.\n\nRow 54 (58): Knit.\n\nRow 56 (60): Knit to the last stitch, m1, k1\u201447 (64) stitches.\n\nRows 57\u2013244 (61\u2013256): Repeat rows 1\u201328 six (seven) more times, then rows 1\u201320 once (zero times) more\u2014135 (155) stitches.\n\nRow 245 (257): Purl.\n\nRow 246 (258): Bind off 5 stitches, knit to the last stitch, m1, k1\u2014131 (151) stitches.\n\nRows 247\u2013268 (259\u2013280): Repeat rows 245 and 246 (257\u2013258) eleven times\u201487 (107) stitches.\n\nRow 269 (281): Purl.\n\nRow 270 (282): Bind off 5 stitches, knit to end\u201482 (102) stitches.\n\nRows 271 (283): Purl.\n\nRow 272 (284): Bind off 5 stitches, knit to the last stitch, m1, k1\u201478 (98) stitches.\n\nRows 273\u2013300 (285\u2013312): Repeat rows 245\u2013272 (257\u2013284) once\u201421 (41) stitches.\n\nRows 301\u2013310 (313\u2013332): Repeat rows 271 and 272 five times (271\u2013272 ten times).\n\nFasten off the last stitch.\n\n### LEFT FRONT\n\nCast on 14 (18) stitches.\n\nRow 1 (WS): Purl.\n\n#### Neck Shaping\n\nRow 2: K1, m1, knit to the end\u201415 (19) stitches.\n\nRows 3\u201344 (3\u201334): Repeat rows 1 and 2 twenty-one (sixteen) more times\u201436 (35) stitches.\n\nRow 45 (35): Purl.\n\n#### Armhole Shaping\n\nRow 46 (36): K1, m1, knit to the last stitch, m1, k1\u201438 (37) stitches.\n\nRows 47\u201350 (37\u201354): Repeat rows 45 and 46 twice (nine times)\u201442 (55) stitches.\n\nRow 51 (55): Cast on 4 (8) stitches, purl to the end\u201446 (63) stitches.\n\nRow 52 (56): K1, m1, knit to the end\u201447 (64) stitches.\n\nRow 53 (57): Purl.\n\nRows 54\u201371 (58\u201379): Repeat rows 52 and 53 nine more times (56 and 57 eleven more times)\u201456 (75) stitches.\n\nWork even in stockinette stitch until piece measures 16\" (40.5cm) from the underarm, ending with a WS row.\n\n#### Shape Lower Edge\n\nRow 1 (RS): Knit.\n\nRow 2: Bind off 6 stitches, purl to the end\u201450 (69) stitches.\n\nRow 3: Knit.\n\nRow 4: Bind off 5 stitches, purl to the end\u201445 (64) stitches.\n\nRows 5\u201319 (21): Repeat rows 1\u20134 four (five) more times; for size L\/XL only, work rows 1 and 2 once more.\n\nFor size S\/M, fasten off the last stitch; for size L\/XL, bind off the remaining 3 stitches.\n\n### SLEEVES\n\nCast on 45 (51) stitches.\n\nKnit 4 rows.\n\nStarting with a purl row, work in stockinette stitch for 4\" (10cm), ending with a WS row.\n\nIncrease row (RS): K1, m1, knit to the last stitch, m1, k1.\n\nContinuing in stockinette stitch, repeat Increase row every 4th row 2 (19) more times, then every 6th row 11 (0) times\u201473 (91) stitches.\n\nWork even until piece measures 20\" (51cm) from cast-on or to desired length, ending with a WS row.\n\n#### Cap Shaping\n\nBind off 4 (8) stitches at the beginning of the next 2 rows\u201465 (75) stitches.\n\nDecrease 1 stitch at the beginning of the next 6 rows\u201459 (69) stitches.\n\nWork 2 rows even.\n\nDecrease 1 stitch at the beginning of the next 38 rows\u201421 (31) stitches.\n\nBind off 2 stitches at the beginning of the next 4 rows, then 3 stitches at the beginning of the next 2 (4) rows\u20147 (11) stitches.\n\nBind off the remaining stitches.\n\n### FINISHING  \nBOTTOM EDGINGS\n\n#### Back\n\nWith the RS facing, pick up and k105 (134) stitches. Knit 3 rows. Bind off knitwise on the RS.\n\n#### Right Front\n\nWith the RS facing, pick up and k144 (174) stitches. Knit 3 rows. Bind off knitwise on the RS.\n\n#### Left Front\n\nWith the RS facing, pick up and k56 (75) stitches. Knit 3 rows. Bind off knitwise on the RS.\n\nSew both shoulder seams.\n\n#### Buttonhole Band\n\nWith the RS facing, pick up and k200 (214) stitches, starting at the bottom edge of the Right Front and ending at the shoulder seam.\n\nRow 1 (WS): Knit.\n\nRow 2: K7, *bind off 2 stitches, k16 (17); repeat from * 8 more times, knit to the end.\n\nRow 3: Knit, casting on 2 stitches over each bind off on the previous row.\n\nBind off.\n\n#### Button Band\n\nWith the RS facing, pick up and k111 (119) stitches, starting at the shoulder seam of the Left Front and ending at the bottom edge.\n\nKnit 3 rows. Bind off.\n\n#### Neck Band\n\nWith the RS facing, pick up and k55 (54) stitches, starting at the right shoulder seam and ending at the left shoulder seam.\n\nKnit 3 rows. Bind off. Sew side edges to Right and Left Front bands.\n\nSet in the Sleeves. Sew left side and sleeve seams.\n\nFollowing the schematic, measure and make two tucks on the Right Front side edge.\n\nSew right side and sleeve seams. Measure 7\" (18cm) from the bottom and top center of the sleeve, and mark with a pin. Sew 9 buttons to the Left Front following the schematic, evenly spaced (approximately 2\u00bc\" [5.5cm] apart). With a threaded tapestry needle, work a running stitch to the pin, gather, and secure the yarn. Sew a button to the bottom to gather. Repeat for second sleeve.\n\nWelted Button Tuck Cardi  \n~return to beginning of project~\n\nchapter 2\n\nCOOL CONSTRUCTION\n\nThis chapter showcases a variety of nontraditional knit garments with creatively constructed shapes and interesting details. They are all inspiring and exciting to make, whether you are combining a cleverly placed scarf and a snap to create shaping, constructing a jacket from two knitted circles with sleeves extending from the center of the circle, or working short rows and cabled edgings at the same time for an avant garde vest. This chapter is sure to spark some new ideas and perhaps inspire you to explore the possibilities of knitting without using the traditional front, back, and two sleeves\u2014just for the thrill of it!\n\nWeekend Warrior Wraparound\n\n## WEEKEND WARRIOR  \nwraparound\n\nA bounty of colors is worked in short rows to create a large spellbinding circle. Instead of binding off the last row, the live stitches are threaded with a corresponding scarf, shaping the neck. A snap or two at the waist does the rest of the shaping for the tunic. To wear it as a cape just throw it around your shoulders and go. You can also bind off the last row and weave a scarf through the neckband instead of leaving the front stitches live. And yes, Weekend Warrior can easily be made in one weekend.\n\nreimagine it\n\nLive stitches and a scarf are used here to shape the neckline, but if you bind off those stitches, the cast-on and bound-off edges can become the front opening of a fun poncholette. Color combinations and multicolor yarn offer endless possibilities. An I-cord or even a thin belt could replace the scarf or ribbon.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size S\n\nFINISHED MEASUREMENTS\n\nInner edge: 18\u00bd (27\u00be, 37)\" [47 (70.5, 94)cm]\n\nOuter edge: 112 (130\u00bd, 149\u00bc)\" [284.5 (331.5, 379)cm]\n\nWidth: 16 (17\u00bd, 18\u00be)\" [40.5 (44.5, 47.5)cm]\n\nNote: The Body is made in one circular piece using short rows.\n\nGAUGE\n\n9 stitches and 18 rows = 4\" (10cm) in garter stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nHPKY Flame (100% merino wool), 3\u00bd oz (100g), 100 yd (91m); 1 ball of Grape (A), 1 ball of Lilac (B), 1 (2, 2) balls of Orchid (C), 1 ball of Blue (D), 1 ball of Teal (E), 1 ball of Green (F), 1 ball of Seafoam (G), 1 ball of Purple (H)\n\nSize U.S. 15 (10mm) needles, or size needed to obtain gauge\n\n2 size 10 snaps\n\nTapestry needle\n\nMatching silk scarf (sample is shown with a hand-dyed scarf from HPKY) or ribbon\n\n### Circle Short Row Pattern\n\nRow 1 (RS): K3, turn.\n\nRow 2: Knit.\n\nRow 3: Knit to the gap, k3, turn.\n\nRow 4: Knit.\n\nRows 5\u201322 (5\u201324, 5\u201326): Repeat rows 3 and 4 until there are 3 stitches after the gap, end after a WS row.\n\nRow 23 (25, 27): Knit.\n\nSIZE S ONLY\n\nRow 24: Purl.\n\nRepeat rows 1\u201324 for pattern.\n\nSIZE M ONLY\n\nRow 26: Slip 1, p2, knit to the end.\n\nRow 27: Knit.\n\nRow 28: Purl.\n\nRepeat rows 1\u201328 for pattern.\n\nSIZE L ONLY\n\nRows 28 and 30: Slip 1, p2, knit to the end.\n\nRows 29 and 31: Knit.\n\nRow 32: Purl.\n\nRepeat rows 1\u201332 for pattern.\n\n### BODY\n\nWith A, cast on 36 (39, 42) stitches. Work 3 repeats of Circle Short Row pattern.\n\nWork 2 repeats each of Circle Short Row pattern with B, C, D, and E.\n\nWith F, work 3 repeats of Circle Short Row pattern.\n\nWork 2 repeats each of Circle Short Row pattern with G, H, and C.\n\nWith E, work 1 repeat of Circle Short Row pattern.\n\nDo not bind off; cut yarn and secure.\n\n### FINISHING\n\nFold one end of the matching silk scarf or ribbon, thread it through the live stitches on the needle, and gather it for the neckline. Sew half the snap at the center of the cast-on edge on the RS and the other half at the center of the G-to-H color-change row on the WS. Sew 2nd snap where the piece wraps at the waist line.\n\nTie scarf\/ribbon at either the neck, back, or the side.\n\nWeekend Warrior Wraparound  \n~return to the beginning of the project~\n\nNouveau Wrap Cardigan\n\n## NOUVEAU  \nwrap cardigan\n\nMake a grand entrance at the office, the theatre, a ball game, or any other event wearing this cardigan. Nouveau Wrap Cardigan is made with stockinette stitch and trimmed with an easy cable rib stitch. For drape and texture I chose a sensuous heathered alpaca yarn. Study the schematic to understand how the rectangles and curves come together for fit and at the same time include the scarves. One scarf end goes through a slit over the right shoulder and the other wraps over the left side and falls to the back.\n\nreimagine it\n\nI can see this design reworked in many ways using different colors and\/or stitch patterns. Try using a two-color small repeat pattern, a check or a houndstooth pattern repeat for the stockinette stitch area, and then continue with a single color for the cable scarf and edging.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size S\n\nFINISHED MEASUREMENTS\n\nBack width: 18 (20, 22)\" [45.5 (51, 56)cm]\n\nBack length: 19\u00bd (20\u00bd, 21\u00bd)\" [49.5 (52, 54.5)cm]\n\nGAUGE\n\n21 stitches and 28 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nBlue Sky Alpacas Melange (100% baby alpaca), 1\u00be oz (50g), 110 yd (100m); 12 (14, 15) balls of #808 Olive\n\n3 Size U.S. 6 (4mm) straight needles, or size needed to obtain gauge\n\nStitch holders\n\nRemovable stitch markers\n\nTapestry needle\n\n#### Baby Cable Rib\n\nRT: K2tog keeping both stitches on the left-hand needle, then knit the first stitch again, removing both stitches from the needle.\n\nRow 1 (RS): P2, *k2, p2; repeat from * to the end.\n\nRow 2: K2, *p2, k2; repeat from * to the end.\n\nRow 3: P2, *RT, p2; repeat from * to the end.\n\nRow 4: Repeat row 2.\n\nRepeat rows 1\u20134 for pattern.\n\n### BACK\n\nCast on 94 (106, 116) stitches. Work in stockinette stitch for 18 (19, 20)\" [45.5 (48.5, 51)cm], ending with a RS row.\n\nNext row: P30 (34, 37) and place stitches on a holder for the left shoulder, bind off 34 (38, 42) stitches for the back neck, p30 (34, 37), and place stitches on a holder for the right shoulder.\n\n#### Bottom Edging\n\nWith the RS facing, pick up and k94 (106, 116) stitches across the cast-on edge.\n\nSetup row (WS): K2,*p2, k2; repeat from * to end.\n\nWork 8 rows of Baby Cable Rib.\n\nBind off in rib.\n\n### LEFT FRONT\n\nCast on 23 (26, 28) stitches.\n\nRow 1 (WS): Purl.\n\nRow 2: Knit to the end, cast on 2 stitches\u201425 (28, 30) stitches.\n\nRepeat rows 1 and 2 two (three, four) more times\u201429 (34, 38) stitches.\n\nRow 1: Purl.\n\nRow 2 (RS): Knit to the last stitch, m1, k1\u201430 (35, 39) stitches. Repeat Rows 1 and 2 35 (35, 37) more times\u201465 (70, 76) stitches.\n\nStarting with a purl row, continue in stockinette stitch until piece measures 18 (19, 20)\" [45.5 (48.5, 51)cm] from the cast-on edge, ending with a WS row.\n\nNext Row (RS): K30 (34, 37) and place these stitches on a holder for the shoulder, knit to the end.\n\n#### Scarf\n\nContinue in stockinette stitch on the remaining 35 (36, 39) stitches for 26\" (66cm) more, ending with a WS row. Work 8 rows in Baby Cable Rib. Bind off in rib.\n\n### RIGHT FRONT\n\nCast on 23 (26, 28) stitches.\n\nRow 1 (WS): Purl.\n\nRow 2: Cast on 2 stitches, knit to end\u201425 (28, 30) stitches.\n\nRepeat rows 1 and 2 two (three, four) more times\u201429 (34, 38) stitches.\n\nRow 1: Purl.\n\nRow 2 (RS): K1, m1, knit to the end\u201430 (35, 39) stitches.\n\nRepeat Rows 1 and 2 35 (35, 37) more times\u201465 (70, 76) stitches.\n\nStarting with a purl row, continue working in stockinette stitch until piece measures 14 (15, 16)\" [35.5 (38, 40.5)cm] from the cast-on edge, ending with a WS row.\n\n#### Scarf Slit\n\nNext RS row: K54 (56, 59), place the last 22 stitches worked on a holder, knit to the end.\n\nNext WS row: P11 (14, 17), cast on 22 stitches using the cable cast-on method, p32 (34, 37).\n\nContinue in stockinette stitch until piece measures 18 (19, 20)\" [45.5 (48.5, 51)cm] from the bottom cast-on edge, ending with a RS row.\n\nNext row (WS): P30 (34, 37) stitches and place them on a holder for the shoulder, knit to end.\n\n#### Scarf\n\nContinue in stockinette stitch on the remaining 35 (36, 39) stitches for 26\" (66cm) more, ending with a WS row. Work 8 rows in Baby Cable Rib. Bind off in rib.\n\n#### Slit Facing\n\nWith the RS facing, place the 22 stitches from the holder onto a needle. Work 8 rows in Baby Cable Rib. Bind off in rib.\n\nUsing the 3-needle bind-off method, join shoulders. Mark 9 (9\u00bd, 10)\" [23 (24, 25.5)cm] down from the shoulder seams on both Fronts and Back.\n\n### SLEEVES\n\nWith the RS facing, pick up and k94 (100, 106) stitches evenly spaced between markers on the Fronts and Back.\n\nStarting with a purl row work 7 rows in stockinette stitch.\n\nDecrease row (RS): K1, ssk, knit to the last 3 stitches, k2tog, k1.\n\nRepeat Decrease row every 4th row 25 (24, 25) more times\u201442 (50, 54) stitches.\n\nWork even in stockinette stitch until sleeve measures 16 (16\u00bd, 17)\" [40.5 (42, 43)cm].\n\nWork 12 rows in Baby Cable Rib. Bind off in rib.\n\n### FINISHING\n\n#### Fronts and Neck Facing\n\nWith the RS facing, starting at the top shoulder side of the Right Front scarf, pick up and k116 stitches to the shoulder seam, k34 (38, 42) Back neck stitches, pick up and k116 stitches to the top of the shoulder side of the Left Front scarf\u2014266 (270, 274) stitches.\n\nSetup row (WS): K2, *p2, k2; repeat from * to end.\n\nWork 10 rows of Baby Cable Rib. Bind off in rib.\n\n#### Right Front and Scarf Side Edging\n\nWith the RS facing and starting at bottom side seam, pick up and k262 (266, 270) stitches evenly spaced to the top edge of the scarf.\n\nSetup row (WS): K2, *p2, k2; repeat from * to end.\n\nWork 10 rows of Baby Cable Rib. Bind off in rib.\n\n#### Left Front and Scarf Side Edging\n\nStarting at top edge of the scarf and ending at the bottom side seam, work the same as for the Right Front and Scarf Side Edging.\n\nSew side and sleeve seams.\n\nNouveau Wrap Cardigan  \n~return to the beginning of the project~\n\nOn the Edge Dress\n\n## ON THE EDGE  \ndress\n\nEverything about this design is soft, sensual, and feminine. It is made in stockinette stitch using a lightweight wool blend metallic yarn. The main feature of this dress is the side and bottom edging. Simple rectangles are edged with a corresponding color in faux fur yarn and then spot sewn to the dress creating a very stylish and unique detail. The front and back are knit with increases and decreases designed to hug and flatter the body. The teardrop openings at the neck add another sexy detail but could be sewn closed instead.\n\nreimagine it\n\nEnvision the edging done with a colorful floral duplicate stitch, intarsia pattern, or a Fair Isle repeat. An easier idea might be colorful bold stripes. For a more textural look, use seed stitch in place of stockinette stitch.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L, XL), shown in size S\n\nFINISHED MEASUREMENTS\n\nBust: 31\u00bc (35\u00bc, 39, 43\u00be)\" [79.5 (89.5, 99, 111)cm]\n\nLength: 31 (32, 32\u00be, 33\u00bd)\" [79 (81, 83, 85)cm]\n\nGAUGE\n\n19 stitches and 30 rows = 4\" (10cm) on smaller needles in stockinette stitch with A\n\nTake time to check gauge.\n\nMATERIALS\n\nBerroco Flicker (87% baby alpaca, 8% acrylic, 5% other fibers), 1\u00be oz (50g), 189 yd (173m); 10 (11, 12, 13) balls of #3317 Dark Taupe (A)\n\nBerroco Marmot (100% nylon), 1\u00be oz (50g), 93 yd (85m); 2 (3, 4, 5) balls of #3703 Moonstone (B)\n\nSize U.S. 8 (5mm) straight needles and 16\" (40.5cm) circular needle\n\nSize U.S. 7 (4.5mm) straight needles, or size needed to obtain gauge\n\nStitch holders\n\nTapestry needle\n\n### FRONT\n\nWith B and larger needles, cast on 110 (124, 138, 152) stitches. Knit 3 rows.\n\nChange to A and smaller needles. Work in stockinette stitch for 8\" (20.5cm), ending with a WS row.\n\nBind off 15 stitches at the beginning of the next 2 rows.\n\nBind off 2 (2, 3, 3) stitches at the beginning of the next 6 rows\u201468 (82, 90, 104) stitches.\n\n#### Waist Shaping\n\nDecrease 1 stitch at the beginning of the next 8 (12, 12, 16) rows\u201460 (70, 78, 88) stitches.\n\nWork even in stockinette stitch until piece measures 15\" (38cm) from the cast-on edge, ending with a WS row.\n\nIncrease row (RS): K1, m1, knit to the last 3 stitches, m1, k1.\n\nRepeat Increase row every RS row 4 more times\u201470 (80, 88, 98) stitches.\n\nContinue even in stockinette stitch until piece measures 18\" (45.5cm), ending with a WS row.\n\n#### Raglan Shaping\n\nBind off 4 (5, 6, 7) stitches at the beginning of the next 2 rows\u201462 (70, 76, 84) stitches.\n\nDecrease row (RS): K2, ssk, knit to the last 4 stitches, k2tog, k2.\n\nContinuing in stockinette stitch, repeat Decrease row every RS row 3 (9, 11, 17) more times, then every 4th row 12 (10, 10, 8) times. Place the remaining 30 (30, 32, 32) stitches on a holder.\n\n#### Lower Front Side Edging\n\nWith the RS facing, B, and larger needles, pick up and k40 stitches along one 8\" (20.5cm) side edge of the lower front. Knit 3 rows.\n\nBind off. Repeat on the other side.\n\n#### Front Bottom Panel\n\nWith B and larger needles, cast on 110 (124, 138, 152) stitches. Knit 3 rows.\n\nChange to A and smaller needles. Work in stockinette stitch for 5\" (12.5cm), ending with a WS row.\n\nChange to B and larger needles. Knit 3 rows. Bind off.\n\nWith the RS facing, B, and larger needles, pick up and k28 stitches along one side of the panel. Knit 3 rows. Bind off. Repeat on the other side.\n\n### BACK\n\nWith B and larger needles, cast on 120 (134, 150, 168) stitches. Knit 3 rows.\n\nChange to A and smaller needles. Work in stockinette stitch for 8\" (20.5cm), ending with a WS row.\n\nBind off 15 stitches at the beginning of the next 2 rows, bind off 2 (2, 3, 3) stitches at the beginning of the next 6 rows, then decrease 1 stitch at the beginning of the next 0 (4, 2, 10) rows\u201478 (88, 100, 110) stitches.\n\nWork even in stockinette stitch until piece measures the same as the front to the armhole, ending with a WS row.\n\n#### Raglan Shaping\n\nBind off 4 (5, 6, 7) stitches at the beginning of the next 2 rows\u201470 (78, 88, 96) stitches.\n\nDecrease row (RS): K2, ssk, knit to the last 4 stitches, k2tog, k2.\n\nContinuing in stockinette stitch, repeat Decrease row every RS row 3 (9, 11, 17) more times, then every 4th row 12 (10, 10, 8). Place the remaining 38 (38, 44, 44) stitches on a holder.\n\n#### Lower Back Side Edging\n\nWith the RS facing, B, and larger needles, pick up and k40 stitches along one 8\" (20.5cm) side edge of the lower back. Knit 3 rows. Bind off. Repeat on the other side.\n\n#### Back Bottom Panel\n\nWith B and larger needles, cast on 120 (134, 150, 168) stitches. Knit 3 rows.\n\nChange to A and smaller needles. Work in stockinette stitch for 5\" (12.5cm), ending with a WS row.\n\nChange to B and larger needles. Knit 3 rows. Bind off.\n\nWith the RS facing, B, and larger needles, pick up and k28 stitches along the side of the panel. Knit 3 rows. Bind off. Repeat on the other side.\n\n### SLEEVES  \n(MAKE 2)\n\nWith B and larger needles, cast on 52 (56, 60, 62) stitches. Knit 3 rows.\n\nChange to A and smaller needles. Work 6 (6, 2, 6) rows in stockinette stitch.\n\nIncrease row (RS): K1, m1, knit to the last stitch, m1, k1.\n\nRepeat Increase row every 8th (6th, 6th, 4th) row 6 (9, 10, 14) more times\u201366 (76, 82, 92) stitches.\n\n#### Raglan Shaping\n\nWork raglan shaping same as the Front. Place the remaining 26 stitches on a holder.\n\n### SIDE PANELS (MAKE 2)\n\nWith B and larger needles, cast on 128 stitches. Knit 3 rows.\n\nChange to A and smaller needles.\n\nWork in stockinette stitch for 5\" (12.5cm), ending with a WS row.\n\nChange to B and larger needles. Knit 3 rows. Bind off.\n\nWith the RS facing, B, and larger needles, pick up and k28 stitches along one side of the panel. Knit 3 rows. Bind off. Repeat on the other side.\n\n### FINISHING\n\nThe edgings may roll. Lightly steam flat.\n\nSew the Back raglan seams. Sew the Front raglan seams, leaving a 3\u00bd\" (9cm) opening 1\" (2.5cm) down from the top edge. Sew sleeve and side seams, leaving the lower front and back side edges open.\n\n#### Neck Band\n\nStarting at the Back right neck seam, place all stitches from the holders onto the circular needle\u2014120 (120, 128, 128) stitches. With B, knit 3 rows. Bind off.\n\n#### Assembly\n\nWith B, sew the Front and Back Bottom Panels for 1\" (2.5cm) at 6 points evenly spaced across the lower edges of the dress, starting and ending at the side edges.\n\nWith B, sew the Side Panels to the lower Front and Back side edges of the body for 1\" (2.5cm) at 5 points evenly spaced across, starting and ending at the cast-on edges of the body and leaving the ends over the bottom panels free.\n\nOn the Edge Dress  \n~return to the beginning of the project~\n\nGlory Rising Circle Cardigan\n\n## GLORY RISING CIRCLE  \ncardigan\n\nThis design consists of two large circles knitted from the outer edge to the center stitches, which are then used to continue knitting the ribbed sleeve. You heard me correctly: The circle extends into the sleeve! A variety of intriguing stitches and some easy color work are used to create the circle itself, which is shaped by strategic placement of decreases in between the stitches and row counts. The two circles are overlapped and sewn at the back, while the front collar is shaped by folding the outer edge back.\n\nreimagine it\n\nThis could be created using two strong contrasting colors like black and white. And why not try making three-quarter sleeves instead? You could even reimagine this piece using different circles or experimenting with other closures. It might be tricky, but it could be worth the effort.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L\/XL), shown in size M\n\nFINISHED MEASUREMENTS\n\nBust: 36 (40, 50)\" [91.5 (101.5, 127)cm]\n\nLength: 22 (26, 29\u00bc)\" [56 (66, 74.5)cm]\n\nSleeve length: 15 (16, 17)\" [38 (40.5, 43)cm]\n\nGAUGE\n\n16 stitches and 22 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nSkacel Schulana Accordion (80% merino wool, 20% super-kid mohair), 1\u00be oz (50g), 93 yd (85m): 10 (13, 15) balls of #13 Blue (A); 1 ball of #00 Natural (B); 1 ball of #09 Navy (C); 1 ball of #02 Orchid (D)\n\nSize U.S. 9 (5.5mm) 24\" (60cm), 32\" (80cm), 40\" (100cm), and 47\" (120cm) circular needles as needed, and a set of 5 double-pointed needles, or size needed to obtain gauge\n\nStitch markers\n\nCable needle\n\nTapestry needle\n\n3 toggle buckles\n\n### Pattern Stitches\n\n1\/1 RC: Slip 1 stitch to cn and hold in back, k1, k1 from cn.\n\nMB (make bobble): [K1, p1] twice, k1 all in the same stitch. Turn, k5; turn, pass the 2nd, 3rd, 4th, and 5th stitches over the first stitch.\n\n### CIRCLE (MAKE 2)\n\nWith longest circular needle and A, cast on 276 (328, 368) stitches loosely. Place marker and join for working in the round, being careful not to twist the cast-on stitches.\n\nWork 16 (24, 26) rnds of ribbing as follows:\n\n*K2, p2; repeat from * around.\n\nDecrease rnd 1: K1 (2, 2), k2tog, *[k1, k2tog] 3 (4, 3) times, k2, k2tog; repeat from * around\u2014191 (226, 255) stitches.\n\nWork 12 (16, 22) rnds of garter stitch as follows:\n\nPurl 1 rnd, knit 1 rnd.\n\nDecrease rnd 2:\n\nSize S only: *P2tog, p3; repeat from * to the last 6 stitches, p2tog, p4\u2014153 stitches.\n\nSize M only: [P2tog, p3] twice, *p2tog, p2, [p2tog, p3] 10 times; repeat from * around\u2014180 stitches.\n\nSize L only: *P2tog, p3; repeat from * around\u2014204 stitches.\n\nWork 7 (9, 15) rnds in Cable Rib as follows, ending with rnd 1:\n\nRnd 1: *K2, p1; repeat from * around.\n\nRnd 2: *1\/1 RC, p1; repeat from * around.\n\nDecrease rnd 3:\n\nSize S only: P2tog, p3, [p2tog, p2] twice, *[p2tog, p3] twice, p2tog, p2; repeat from * around\u2014120 stitches.\n\nSize M only: *P2tog, p3, p2tog, p2; repeat from * around\u2014140 stitches.\n\nSize L only: *P2tog, p2, ([p2tog, p3] twice, p2tog, p2) 7 times; repeat from * around\u2014160 stitches.\n\nKnit 2 rnds.\n\nNext rnd: *K2, MB, k2; repeat from * around.\n\nKnit 2 rnds.\n\nDecrease rnd 4: *P2tog, p3; repeat from * around\u201496 (112, 128) stitches.\n\nChange to B.\n\nNext rnd: With B, knit.\n\nWork 9 rnds of Chart 1.\n\nDecrease rnd 5: With B, *[k2tog, k1] 4 times, k2tog, k2; repeat from * around\u201466 (77, 88) stitches.\n\nWork 6 rnds of Chart 2.\n\n### SLEEVE\n\nChange to A.\n\nRnd 1: With A, knit, decreasing 0 (3, 2) stitches evenly around\u201466 (74, 86) stitches.\n\nRnds 2\u20134: K2, *p2, k2; repeat from * around.\n\nDecrease rnd: K2tog, work in rib to the last 2 stitches, ssk\u20142 stitches decreased.\n\nContinuing in pattern as established, repeat Decrease rnd every 4th rnd 14 (16, 18) more times\u201436 (40, 48) stitches.\n\nWork even in rib until sleeve measures 15 (16, 17)\" [38 (40.5, 43)cm] or to desired length.\n\nBind off all stitches loosely in rib.\n\n### FINISHING\n\nPlace the circles side by side with the beginning of rounds together. Overlap one circle's ribbing on top of the other by 10 (12, 14)\" 25.5 (30.5, 35.5)cm], measured along the beginning of the garter stitch section with the beginning of the rounds at the center. Pin in place, allowing the ribbing overlap to curve outward. Sew both right side and wrong side in place using a [whipstitch.\n\nFold the ribbed sections to the outside along the front edges and the top of the neck to form a shawl collar. Sew 3 toggle closures to the Front (see photograph for placement).\n\nGlory Rising Circle Cardigan  \n~return to the beginning of the project~\n\nDirectional Vest\n\n## DIRECTIONAL  \nvest\n\nAlthough easy to knit, the avant-garde construction will interest any knitter who likes the challenge of unusual shaping (take a quick look at the diagram). This design is shaped by using short rows and at the same time knitting the cabled edging around the front, back, and armholes. A cabled back insert pulls it all together, and cord closures embellish the front. Once you have made one of these vests, you will want to make it again and again.\n\nreimagine it\n\nI can see this vest reworked using a hand-painted yarn, but be careful the yarn isn't so busy that you lose the short-row and cabled edging patterns. Also try this with a clasp instead of the I-cord closure. Adding sleeves is another option.\n\nSKILL LEVEL\n\nTIME\n\nSIZE\n\nOne size\n\nFINISHED MEASUREMENTS\n\nBack width (at underarm): 18\" (45.5cm)\n\nBack length: 21\" (53.5cm)\n\nGAUGE\n\n14 stitches and 22 rows = 4\" (10cm) in stockinette stitch\n\nMATERIALS\n\nMadelinetosh Tosh Chunky (100% superwash merino wool), 3\u00bd oz (100g), 165 yd (151m): 4 skeins in Oceana\n\nSize U.S. 10 (6mm) straight needles\n\nSize U.S. 9 (5.5mm) set of 2 double-pointed needles\n\nCable needle\n\nStitch markers\n\nTapestry needle\n\n#### Pattern Stitches\n\nM2kp: Insert the left-hand needle, from front to back, under the horizontal bar between the last stitch worked and next stitch, k1 in the back loop, p1 in the back loop\u20142 stitches increased.\n\nM2: Insert the left-hand needle from front to back under the horizontal bar between the last stitch worked and next stitch, k1 in the back loop, k1 in the front loop\u20142 stitches increased.\n\nPkp: [P1, k1, p1] into the same stitch\u20142 stitches increased.\n\n2\/2 RC: Slip 2 to cn and hold in back, k2, k2 from cn.\n\n#### Cable Rib\n\n(multiple of 9 stitches + 5)\n\nRow 1 (RS): *[P1, k1] twice, p1, k4; repeat from * to the last 5 stitches, p1, [k1, p1] twice.\n\nRow 2: *[K1, p1] twice, k1, p4; repeat from * to the last 5 stitches, k1, [p1, k1] twice.\n\nRow 3: *[P1, k1] twice, p1, 2\/2 RC; repeat from * to the last 5 stitches, p1, [k1, p1] twice.\n\nRow 4: Repeat row 2.\n\nRepeat rows 1\u20134 for pattern.\n\n### CENTER BACK INSERT\n\nCast on 32 stitches.\n\nRows 1\u201328: Work 7 repeats of Cable Rib pattern.\n\nRow 29 (RS): P1, ssk, work in pattern to the last 3 stitches, k2tog, p1\u201430 stitches.\n\nRow 30: K1, p2tog, work in pattern to the last 3 stitches, p2tog tbl, k1\u201428 stitches.\n\nRows 31\u201342: Repeat rows 29 and 30\u20144 stitches after row 42.\n\nRow 43: P1, k2tog, p1\u20143 stitches.\n\nRow 44: S2kp\u20141 stitch.\n\nFasten off.\n\n#### Right Armband\n\nCast on 14 stitches.\n\nWork 60 rows in Cable Rib (15 cables), ending with row 4.\n\nNote: From this point on, \"Cable Rib\" refers to working these 14 stitches as established.\n\nIncrease row (RS): P1, [k1, p1] twice, k4, [p1, k1] twice, pkp\u201416 stitches.\n\nNext row: K1, p1, Cable Rib.\n\nContinue in pattern as established until 28 rows (7 more cables) have been completed, ending with row 4.\n\n### RIGHT FRONT\n\nRow 1 (RS): Work 13 stitches of Cable Rib, m2kp, p1, k1, p1\u201418 stitches.\n\nRow 2: [K1, p1] twice, place marker, Cable Rib.\n\nRow 3: Cable Rib, slip marker, m2kp, [k1, p1] twice\u201420 stitches.\n\nRow 4: [K1, p1] 3 times, slip marker, Cable Rib.\n\nRow 5: Cable Rib, slip marker, m2kp, [k1, p1] 3 times\u201422 stitches.\n\nRow 6: K1, [p1, k1] twice, p3, slip marker, Cable Rib.\n\nRow 7: Cable Rib, slip marker, m2kp, k3, p1, [k1, p1] twice\u201424 stitches.\n\nRow 8: K1, [p1, k1] twice, p4, k1, slip marker, Cable Rib.\n\nRow 9: Cable Rib, slip marker, m2kp, p1, k4, p1, [k1, p1] twice\u201426 stitches.\n\nRow 10: K1, [p1, k1] twice, p4, k1, p1, k1, slip marker, Cable Rib.\n\nRow 11: Cable Rib, slip marker, m2kp, k1, p1, 2\/2 RC, p1, [k1, p1] twice\u201428 stitches.\n\nRow 12: K1, [p1, kl] twice, p4, k1, [p1, k1] twice, slip marker, Cable Rib.\n\nRow 13: Cable Rib, slip marker, m1, place marker, Cable Rib\u201429 stitches.\n\nRow 14: Cable Rib, slip marker, p1, slip marker, Cable Rib.\n\nRow 15: Cable Rib, slip marker, m2, knit to marker, slip marker, Cable Rib\u201431 stitches.\n\nRows 16 and 18: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 17: Cable Rib, slip marker, m1, knit to marker, slip marker, Cable Rib\u201432 stitches.\n\nRows 19\u201325: Repeat rows 15\u201318 twice, then rows 15\u201317 once more\u201438 stitches; 10 stitches between markers.\n\nNote: From this point until row 291, stitch counts will not include Cable Ribs.\n\nRow 26: Cable Rib, slip marker, p3, w&t.\n\nRow 27: K3, slip marker, Cable Rib.\n\nRow 28: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRows 29\u201334: Repeat rows 15\u201317 once, then rows 26\u201328\u201413 stitches.\n\nRows 35\u201346: Repeat rows 29\u201334 twice\u201419 stitches.\n\nRow 47: Cable Rib, slip marker, m2, k3, p16, slip marker, Cable Rib\u201421 stitches.\n\nRow 48: Cable Rib, slip marker, k16, p5, slip marker, Cable Rib.\n\nRows 49 and 50: Repeat rows 17 and 18\u201422 stitches.\n\nRows 51\u201356: Repeat rows 29\u201334\u201425 stitches.\n\nRow 57: Cable Rib, pbf, slip marker, knit to marker, slip marker, Cable Rib\u201426 stitches.\n\nRow 58: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 59: Bind off 14 stitches, place marker, m2, knit to marker, slip marker, Cable Rib\u201428 stitches.\n\nRow 60: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRow 61: P1, slip marker, m1, knit to marker, slip marker, Cable Rib\u201429 stitches.\n\nRow 62: Cable Rib, slip marker, p3, w&t.\n\nRow 63: K3, slip marker, Cable Rib.\n\nRow 64: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRow 65: P1, slip marker, m2, knit to marker, slip marker, Cable Rib\u201431 stitches.\n\nRows 66\u201370: Repeat rows 60\u201364\u201432 stitches.\n\nRow 71: P1, slip marker, m2, k3, purl to marker, slip marker, Cable Rib\u201434 stitches.\n\nRow 72: Cable Rib, slip marker, knit to 5 stitches before marker, p5, slip marker, k1.\n\nRow 73: P1, slip marker, m1, knit to marker, slip marker, Cable Rib\u201435 stitches.\n\nRow 74: Cable Rib, slip marker, p3, w&t.\n\nRow 75: K3, slip marker, Cable Rib.\n\nRow 76: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRow 77: P1, slip marker, m2, knit to marker, slip marker, Cable Rib\u201437 stitches.\n\nRow 78: Cable Rib, slip marker, p17, w&t.\n\nRow 79: Knit to marker, slip marker, Cable Rib.\n\nRow 80: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRows 81\u201396: Repeat rows 73\u201380 twice\u201443 stitches.\n\nRows 97\u2013122: Repeat rows 71\u201396\u201454 stitches.\n\nRow 123: Purl to the 2nd marker, slip marker, Cable Rib.\n\nRow 124: Cable Rib, slip marker, knit to end.\n\nRow 125: P1, slip marker, k1, ssk, knit to marker, slip marker, Cable Rib\u201453 stitches.\n\nRow 126: Cable Rib, slip marker, p3, w&t.\n\nRow 127: K3, slip marker, Cable Rib.\n\nRow 128: Cable Rib, slip marker, purl to 3 stitches before the next marker, p2tog tbl, p1, slip marker, k1\u201452 stitches.\n\nRow 129: Repeat row 125\u201451 stitches.\n\nRow 130: Cable Rib, slip marker, p17, w&t.\n\nRow 131: K17, slip marker, Cable Rib.\n\nRow 132: Repeat row 128\u201450 stitches.\n\nRows 133\u2013148: Repeat rows 125\u2013132 twice\u201442 stitches.\n\nRows 149\u2013152: Repeat rows 125\u2013128\u201440 stitches.\n\nRows 153 and 154: Repeat rows 123 and 124.\n\nRow 155: P1, slip marker, k2, m1, knit to marker, slip marker, Cable Rib\u201441 stitches.\n\nRow 156: Cable Rib, slip marker, p3, w&t.\n\nRow 157: K3, slip marker, Cable Rib.\n\nRow 158: Cable Rib, slip marker, purl to 2 stitches before the next marker, m1p, p2 slip marker, k1\u201442 stitches.\n\nRow 159: Repeat row 155\u201443 stitches.\n\nRow 160: Cable Rib, slip marker, p17, w&t.\n\nRow 161: K17, slip marker, Cable Rib.\n\nRow 162: Repeat row 158\u201444 stitches.\n\nRows 163\u2013182: Repeat rows 155\u2013162 twice, then rows 155\u2013158 once\u201454 stitches.\n\nRows 183 and 184: Repeat rows 123 and 124.\n\nRow 185: P1, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u201452 stitches.\n\nRow 186: Cable Rib, slip marker, p17, w&t.\n\nRow 187: K17, slip marker, Cable Rib.\n\nRow 188: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRow 189: P1, slip marker, k2tog, knit to marker, slip marker, Cable Rib\u201451 stitches.\n\nRow 190: Cable Rib, slip marker, p3, w&t.\n\nRow 191: K3, slip marker, Cable Rib.\n\nRow 192: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRows 193\u2013208: Repeat rows 185\u2013192 twice\u201445 stitches.\n\nRow 209: P1, slip marker, k3tog, k2, purl to marker, slip marker, Cable Rib\u201443 stitches.\n\nRow 210: Cable Rib, slip marker, knit to 3 stitches before the next marker, p3, slip marker, k1.\n\nRows 211\u2013236: Repeat rows 185\u2013210\u201432 stitches.\n\nRow 237: P1, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u201430 stitches.\n\nRow 238: Cable Rib, slip marker, p3, w&t.\n\nRow 239: K3, slip marker, Cable Rib.\n\nRow 240: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRow 241: P1, slip marker, k2tog, knit to marker, slip marker, Cable Rib\u201429 stitches.\n\nRow 242: Cable Rib, slip marker, purl to marker, slip marker, k1.\n\nRows 243\u2013246: Repeat rows 237\u2013240\u201427 stitches.\n\nRow 247: Cast on 14 stitches, Cable Rib 13 stitches, p2tog, place marker, knit to marker, slip marker, Cable Rib\u201454 stitches total, 26 stitches between markers.\n\nRow 248: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 249: Cable Rib, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u201424 stitches.\n\nRow 250: Cable Rib, slip marker, p3, w&t.\n\nRow 251: K3, slip marker, Cable Rib.\n\nRow 252: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 253: Cable Rib, slip marker, k2tog, knit to marker, slip marker, Cable Rib\u201423 stitches.\n\nRow 254: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 255: Cable Rib, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u201421 stitches.\n\nRow 256: Cable Rib, slip marker, p3, w&t.\n\nRow 257: K3, slip marker, Cable Rib.\n\nRow 258: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 259: Cable Rib, slip marker, k3tog, k2, p16, slip marker, Cable Rib\u201419 stitches.\n\nRow 260: Cable Rib, slip marker, k16, p3, slip marker, Cable Rib.\n\nRow 261: Cable Rib, k2tog, knit to marker, slip marker, Cable Rib\u201418 stitches.\n\nRow 262: Cable Rib, slip marker, p3, w&t.\n\nRow 263: K3, slip marker, Cable Rib.\n\nRow 264: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 265: Cable Rib, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u201416 stitches.\n\nRow 266: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRows 267\u2013280: Repeat rows 261\u2013266 twice\u201410 stitches.\n\nRow 281\u2013284: Repeat rows 263\u2013266\u20148 stitches.\n\nRow 285: Cable Rib, slip marker, k2tog, knit to marker, slip marker, Cable Rib\u20147 stitches.\n\nRow 286: Cable Rib, slip marker, purl to marker, slip marker, Cable Rib.\n\nRow 287: Cable Rib, slip marker, k3tog, knit to marker, slip marker, Cable Rib\u20145 stitches.\n\nRow 288: Repeat row 286.\n\nRows 289\u2013290: Repeat rows 285\u2013288\u20142 stitches.\n\nRow 291: Cable Rib, slip marker, k2tog, slip marker, Cable Rib\u201429 stitches.\n\nRow 292: Cable Rib, remove marker, p1, slip marker, Cable Rib.\n\n### LEFT FRONT\n\nRow 293: Cable Rib, slip marker, k3tog, continue in pattern to end\u201427 stitches.\n\nRow 294: Work in pattern to marker, slip marker, Cable Rib.\n\nRows 295\u2013302: Repeat rows 293 and 294\u201419 stitches.\n\nRow 303: Cable Rib, slip marker, k2tog, p1, k1, p1\u201418 stitches.\n\nRow 304: K1, p3tog, Cable Rib\u201416 stitches.\n\nNext row: Cable Rib, k1, p1.\n\nContinue in Cable Rib as established until 27 rows (7 more cables) have been completed, ending with row 3 of pattern.\n\nDecrease row (WS): K3tog, work in pattern to end\u201414 stitches.\n\n#### Left Armband\n\nWork 60 rows in Cable Rib (15 cables).\n\nBind off in pattern.\n\n### FINISHING\n\nSew the cast-on ends to the bound-off ends of the armbands. Beginning at the center back point, sew the Center Back Insert to the center of the V. Sew the armbands to the back armholes and the sides of the insert, matching the small increases in each band to the top edge of the insert.\n\n#### I-cord Closure (MAKE 2)\n\nCast on 3 stitches onto a double-pointed needle.\n\nRow 1: Do not turn. Slide stitches to right end of needle, k3.\n\nRepeat Row 1 for 24\" (61cm). Bind off.\n\nStarting at the center, form a spiral approximately 1\u00bd\" (3.8cm) wide, leaving remaining 14\" (35.5cm) for the tie, and sew to secure. Knot the ends of each cord.\n\nSew each closure to the Front (see photograph for placement).\n\nDirectional Vest  \n~return to the beginning of the project~\n\nShape-Shifter Vest\n\n## SHAPE-SHIFTER  \nvest\n\nThe fusion of two slightly shaped rectangular pieces becomes a forward-looking, nontraditional vest. I will let you in on a designer secret: This vest started out as sleeves for a coat, but when I put the pieces on my mannequin they overlapped, and with a little twisting and stitching, became the Shape-Shifter Vest. The contrasting cord trim is sewn on and twisted to resemble the organic shape of a tree branch. Knitting accidents like this one can be wonderfully inventive.\n\nreimagine it\n\nTry this design using a different rib stitch for the overall pattern; just be careful of your gauge. Or try using a lovely multicolored hand-dyed yarn for a stylish look. It might also be interesting to use a textured bobble or cabled I-cord for the edging. See my book Knitting over the Edge for a variety of I-cord patterns.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nBust: 40 (48)\" [101.5 (122)cm]\n\nLength: 27\u00bd (28)\" [70 (72.5)cm]\n\nGAUGE\n\n12 stitches and 16 rows = 4\" (10cm) in stockinette stitch on larger needles\n\n18 stitches and 10 rows = 4\" (10cm) in Eyelet Rib pattern on larger needles\n\nMATERIALS\n\nCascade 128 Superwash (100% superwash merino wool), 3\u00bd oz (100g), 128 yd (117m); 7 (9) skeins of #1926 Tan (MC), 1 skein of #1981 Brown (CC)\n\nSize U.S. 10\u00bd (6.5mm) straight needles, or size needed to obtain gauge\n\nSize U.S. 10 (6mm) double-pointed needles\n\nStitch holders\n\nTapestry needle\n\n#### Eyelet Rib Pattern\n\n(multiple of 7 stitches + 3)\n\nRow 1 (RS): P3, *k4, p3; repeat from * to end.\n\nRow 2: K1, yo, k2tog, *p4, k1, yo, k2tog; repeat from * to end.\n\nRepeat rows 1 and 2 for pattern.\n\n### LEFT\/RIGHT SIDE  \n(MAKE 2)\n\nWith larger needles and MC, cast on 52 (59) stitches.\n\nWork in Eyelet Rib pattern for 18 (19)\" [45.5 (48.5)cm], ending with a WS row.\n\nNote: Lengthen or shorten here as desired.\n\n#### Armhole Shaping\n\nBind off 7 (9) stitches at the beginning of the next RS row, then bind off 2 (3) stitches at the beginning of the following RS row\u201443 (47) stitches.\n\nContinue in pattern as established for 16\" (40.5cm), ending with a WS row.\n\nCast on 2 (3) stitches at the beginning of the next RS row, then cast on 7 (9) stitches at the beginning of the following RS row\u201452 (59) stitches.\n\nContinue in pattern as established for 18 (19)\" [45.5 (48.5)cm]. Bind off.\n\n### FINISHING\n\nFold each piece in half widthwise at the center of the armhole and sew the side seam. Join the two sides following the diagram, adjusting for fit as necessary.\n\n#### I-cord Trim\n\nWith dpns and B, cast on 5 stitches and work I-cords to the lengths listed below. When each cord is the desired length, do not bind off; place stitches on a holder.\n\n\u2022 2 Armhole Cords, 18 (19)\" [45.5 (48.5)cm] each\n\n\u2022 1 Body Cord, 96 (100)\" [244 (254)cm]\n\n\u2022 1 Lower Left Front Cord, 6 (7)\" [15 (18)cm]\n\nSew the Armhole Cords to armholes, beginning with the cast-on end at the underarm seam, lengthening or shortening the cord as needed.\n\nStarting at the lower right front, sew the Body Cord to 13 (14)\" [33 (35.5)cm] of the edge, leave the next 7\" (18cm) of the cord and front edge unattached, continue sewing up the right front neck and down the right center back, leaving the last \u00bd\" (13mm) of cord and back edge unattached; sew the corner.\n\nFollowing the diagram above, loop around the unattached sections and continue twisting around the 7\" (18cm) unattached section. Sew the twist section and continue to sew up the left front and left back neck. Adjust the length if necessary, bind off, and sew to the front.\n\nAdjust the length of the Lower Left Front Cord if necessary, bind off, and sew to the lower left front edge.\n\nchapter 3\n\nWOVEN WEAVES\n\nSome of my previous designs that have included weaving have proven so popular with knitters and seemed to have such an impact in our knitting world that I decided to devote an entire chapter to woven pieces. Although they may look complicated, you'll quickly discover that they are as simple to make as an easy braid or a child's loomed potholder. Color choices here also have a big impact; with hand-dyed variegated yarn, each piece is one of a kind, because you never know where the colors will end up in the finished garment.\n\nCrisscross Weave Tank\n\n## CRISSCROSS WEAVE  \ntank\n\nThis peaceful design is so simple to knit even though it uses two weaving techniques, braiding and woven weaves. The shoulder straps are three braided cords, woven together and sewn in place at the back for a more exciting detail and fit. Squares sewn into the side seams add an unusual drape to the design, complementing the thick and thin cotton ribbon yarn used for a zen look. A perfect piece for hot weather or winter wear over another garment.\n\nreimagine it\n\nTo make a more fitted piece, leave out the squares sewn into the sides and sew the whole side seam. I can see this tank made longer and becoming a sexy dress or even a gown, especially if you use a yarn with some bling in it.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size S\n\nFINISHED MEASUREMENTS\n\nBust: 33 (38\u00bd, 44)\" [84 (98, 112)cm]\n\nLength: 25 (27, 29)\" [63.5 (68.5, 74)cm]\n\nGAUGE\n\n20 stitches and 24 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nTahki Yarns Ripple (100% mercerized cotton), 1\u00be oz (50g), 142 yd (130m); 5 (6, 8) balls of color #10 Pewter\n\nSize U.S. 8 (5mm) straight needles, or size needed to obtain gauge\n\nSpare needle for holding stitches\n\nStitch holders\n\n6 large safety pins\n\nTapestry needle\n\n### POCKET LININGS\n\n(MAKE 2)\n\nCast on 19 stitches. Work in stockinette stitch for 3\u00bd\" (9cm). Leave on a spare needle.\n\n### FRONT\n\nCast on 100 (116, 132) stitches. Work 10 (8, 6) rows in stockinette stitch.\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every 10th row 8, (9, 10) more times\u201482 (96, 110) stitches.\n\nAt the same time, when piece measures 5\u00bd\" (14cm) from the cast-on edge, make the first pocket as follows:\n\nNext Row (RS): Knit to the last 30 stitches and place the next 19 stitches onto a holder; knit the 19 held stitches of Pocket Lining 1, knit to the end. Continue in stockinette stitch and decrease as established until piece measures 9\" (23cm) from the cast-on edge for the second pocket.\n\nNext Row (RS): K10, place the next 19 stitches onto a holder, knit the 19 held stitches of Pocket Lining 2, knit to the end. Continue as established until piece measures 16 (17, 18)\" [40.5 (43, 45.5)cm], ending with a WS row.\n\n### Shape Armholes\n\nBind off 5 (7, 8) stitches at the beginning of the next 2 rows\u201472 (82, 94) stitches.\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every RS row 3 (4, 6) more times\u201464 (72, 80) stitches.\n\nWork even in stockinette stitch until armhole measures 5 (5\u00bd, 6)\" [12.5 (14, 15)cm], ending with a WS row.\n\n### Neck Shaping and Crisscross Strips\n\nNext Row (RS): K24 and place these stitches on a holder, bind off the next 16 (24, 32) stitches, [k8 and place these stitches on a safety pin] twice, k8.\n\nWork in stockinette stitch on these 8 stitches for 19 (20, 21)\" [48.5 (51, 53.5)cm] and place stitches on a safety pin.\n\nSlip the next set of 8 stitches onto a needle and work in stockinette stitch for 19 (20, 21)\" [48.5 (51, 53.5)cm]. Repeat for the remaining set of 8 stitches.\n\nRepeat for the opposite set of 24 stitches.\n\n### BACK\n\nWork same as the Front to the neck shaping, omitting the pockets\u201464 (72, 80) stitches. Decrease 6 (9, 17) stitches evenly across the next row, leaving the remaining 58 (63, 63) stitches on a spare needle.\n\n### SIDE SQUARE GUSSETS (MAKE 2)\n\nCast on 40 stitches and work in stockinette stitch for 8\" (20.5cm). Bind off.\n\n### FINISHING\n\nMake 6\" (15cm) braids at the front ends of the strips and stitch to secure them, then crisscross the strips following the photo (right), adjusting length if necessary. Place the 6 strips onto a needle and join, evenly spaced, to the 58 (63, 63) Back stitches, using the 3-needle bind-off as follows:\n\n[Bind off 8 strip stitches together with 8 Back stitches, bind off 2 (3, 3) Back stitches only] 5 times, bind off 8 strip stitches together with 8 Back stitches.\n\nStarting at the underarm, sew the side seam, leaving 8\" (20.5cm) up from the bottom open. Insert the Side Square Gusset and sew two consecutive sides, as a triangle, into the opening. Repeat for the other side seam.\n\nWith the RS facing, place the front pocket stitches onto a needle. Bind off the stitches and using a long tail, sew the edges of the pocket linings to the inside of the garment.\n\nPress seams lightly.\n\nCrisscross Weave Tank  \n~return to the beginning of the project~\n\nBraided Vitality Pullover\n\n## BRAIDED VITALITY  \npullover\n\nFeaturing a bold all-over rib pattern, this pullover is worked in three pieces: a front, a back, and a combined bodice and sleeves made in one piece. The dramatic design is all about the center braid on the front of the sweater. The sleeves and shoulders have been knitted cuff to cuff and are sewn to the front and back pieces. The rectangular neck band is knit separately and sewn on, bringing this revolutionary pullover together in a medley of lovely color striping.\n\nreimagine it\n\nThis was such a fun, quick piece to make. I used Lion Brand's Da Vinci yarn, which comes in many great colors you can choose from for this piece. Imagine this pullover made longer into a tunic or dress. You could wear it with fun leggings or tights and a cool pair of boots!\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nBust: 43 (48)\" [109 (122)cm]\n\nLength: 23\u00bc (25\u00bc)\" [59 (64)cm]\n\nSleeve length to underarm: 16 (17)\" [40.5 (43)cm]\n\nGAUGE\n\n13 stitches and 17 rows = 4\" (10cm) in k2, p2 rib\n\nTake time to check gauge.\n\nMATERIALS\n\nLion Brand Da Vinci (53% wool, 47% acrylic), 1\u00be oz (50g), 55 yd (50m); 14 (16) skeins of #207 Quartz\n\nSize U.S. 10\u00bd (6.5mm) straight needles, or size needed to obtain gauge\n\nStitch markers\n\nWaste yarn or stitch holders\n\nMatching worsted-weight yarn for seaming\n\nTapestry needle\n\n### K2, P2 Rib\n\n(multiple of 4 stitches + 2)\n\nRow 1 (RS): *P2, k2; rep from * to last 2 stitches, p2.\n\nRow 2: *K2, p2; repeat from * to last 2 stitches, k2.\n\nRepeat rows 1 and 2 for pattern.\n\n### FRONT\n\nCast on 70 (78) stitches.\n\nWork 6 rows in k2, p2 Rib pattern.\n\nSetup row (RS): Work 17 (21) stitches in rib, place marker, work [12 stitches in rib, place marker] 3 times, work 17 (21) stitches in rib.\n\nWorking on the last set of 17 (21) stitches only, work in rib as established for 17\" (43cm), ending with WS row. Place stitches on a holder.\n\n*Join new yarn to the next set of 12 stitches ready to work a WS row, and work in rib as established for 20\" (51cm), ending with WS row. Place these stitches on a holder; repeat from * for the next 2 sets of 12 stitches.\n\nJoin new yarn to the last set of 17 (21) stitches ready to work a WS row, and work in rib as established for 17\" (43cm), ending with a WS row.\n\nFollowing the photograph, braid the three center strips tightly. Place all 70 (78) stitches back onto the same needle ready to work a RS row.\n\nWork 4 rows in rib as established.\n\nBind off in pattern.\n\n### BACK\n\nCast on 70 (78) stitches. Work in rib pattern until piece measures 17\" (43cm). Bind off.\n\n### RIGHT SLEEVE\n\nCast on 34 (38) stitches. Work in rib pattern for 8\" (20.5cm).\n\nIncrease row (RS): Work 1 stitch in rib, m1p, work in rib to the last stitch, m1p, work 1 stitch in rib.\n\nRepeat Increase row every 4th row 7 more times, working increased stitches into pattern and using either m1 or m1p as needed\u201450 (54) stitches.\n\nWork even in rib as established until piece measures 16 (17)\" [40.5 (43)cm] or desired length to underarm, ending with a WS row.\n\nBind off 5 stitches at the beginning of the next 2 rows\u201440 (44) stitches.\n\n#### Yoke\n\nWork even in rib for 7 (8)\" [18 (20.5)cm], ending with a WS row.\n\n#### Neck Shaping\n\nNext row (RS): Work 18 (20) stitches in rib, join new yarn and bind off the next 4 stitches, work 18 (20) stitches in rib. Work both sides at the same time in rib as established for 8\u00bd\" (21.5cm), ending with a WS row.\n\nNext row (RS): Using one yarn across all stitches, work 18 (20) stitches in rib, cast on 4 stitches, work 18 (20) stitches in rib\u201440 (44) stitches.\n\nContinue in rib as established for 7 (8)\" [18 (20.5)cm], ending with a WS row.\n\n### LEFT SLEEVE\n\nContinuing in rib, cast on 5 stitches at the beginning of the next 2 rows\u201450 (54) stitches.\n\nWork 4 rows even in rib.\n\nDecrease row (RS): Work 1 stitch in rib, p2tog, work in rib to the last 3 stitches, p2tog, work 1 stitch in rib.\n\nRepeat Decrease row every 4th row 7 more times, using either k2tog or p2tog as needed\u201434 (38) stitches.\n\nWork even in rib until left sleeve measures 16 (17)\" [40.5 (43)cm] from underarm.\n\nBind off.\n\n### FINISHING\n\nNote: Use a plain worsted-weight yarn for sewing.\n\nSew Front and Back side seams. Sew sleeve seams. Pin yoke to Front and Back, matching side and underarm stitches, and sew in place, easing yoke stitches across the Front and Back.\n\n#### Neck Band\n\nCast on 17 stitches.\n\nRow 1 (RS): *K2, p2; repeat from * to the last stitch, k1. Work in pattern as established for 21\" (53.5cm). Bind off. Sew cast-on edge to bound-off edge. Using the k1 edge as the seam stitch, sew Neck Band to the neck opening, centering the seam at center back.\n\nEtiquette Unchained Pullover\n\n## ETIQUETTE UNCHAINED  \npullover\n\nYou are sure to be in the spotlight when you wear this colorful woven pullover. This piece is reminiscent of other designs I've created over the years but with several twists. It is knit from the top down using stockinette stitch strips in a multicolor yarn, which assures each of these will look like a true original. Whether you are a novice or an advanced knitter, I think you will enjoy the excitement of making this piece. The woven strips require some extra fussing and sewing but it is worth it.\n\nreimagine it\n\nEnvision this pullover as a vest: Simply leave off the sleeves. Using the Mochi Plus multicolor skeins also ensures your piece will look different than mine. Every project becomes one of a kind. This technique is also lovely in solid colors. Try belting the finished garment.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L, XL), shown in size M\n\nFINISHED MEASUREMENTS\n\nBust: 35 (37\u00bd, 41\u00bd, 48)\" [89 (95, 105.5, 122)cm]\n\nLength: 27 (28, 29, 30)\" [68.5 (71, 74, 76)cm]\n\nNote: Length is approximate. Garment is draped on the bias and is highly elastic. It is worked from the top down.\n\nSleeve length to underarm: 17\u00bd (17\u00bd, 18, 18)\" [44.5 (44.5, 45.5, 45.5)cm]\n\nGAUGE\n\n18 stitches and 22 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nCrystal Palace Mochi Plus (80% wool, 20% nylon), 1\u00be oz (50g), 95 yd (87m); 13 (14, 16, 18) balls of #608 Rainbow Trout\n\nSize U.S. 10 (6mm) needles, or size needed to obtain gauge\n\nTapestry needle\n\n### FRONT (MAKE 2)\n\nNote: Neck and strips have some natural roll.\n\nStarting at the shoulder, cast on 28 (31, 34, 37) stitches. Purl 1 row.\n\nIncrease row (RS): *K1, m1; repeat from * to end\u201456 (62, 68, 74) stitches.\n\nWork even in stockinette stitch until piece measures 7 (7, 7\u00bd, 8)\" [18 (18, 19, 20.5)cm], ending with a WS row.\n\n### Strip 1\n\nK19 (21, 23, 25), leaving the remaining stitches on the needle.\n\nIncrease row (WS): P1, m1, purl to the last stitch, m1, p1\u201421 (23, 25, 27) stitches.\n\nWork even in stockinette stitch for 20 (21, 21\u00bd, 22)\" [51 (53.5, 54.5, 56)cm], ending with a RS row.\n\nKnit 4 rows. Bind off.\n\n#### Strip 2\n\nWith the RS facing, join yarn to the next stitch and k18 (20, 22, 24).\n\nIncrease row (WS): P1, m1, purl to the last stitch, m1, p1\u201420 (22, 24, 26) stitches.\n\nContinue as for Strip 1.\n\n#### Strip 3\n\nWork the same as Strip 1.\n\n#### BACK\n\nMake 2 pieces the same as the Front.\n\n#### SLEEVES\n\nNote: The sleeves are knit from the top down.\n\nCast on 64 (64, 68, 72) stitches.\n\nWork in stockinette stitch for 2\" (5cm), ending with a WS row.\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every 6th row 12 (12, 11, 9) times, then every 4th row 2 (2, 4, 7) times\u201434 (34, 36, 38) stitches.\n\nWork even in stockinette stitch until sleeve measures 17\u00bd (17\u00bd, 18, 18)\" [44.5 (44.5, 45.5, 45.5)cm], ending with a RS row.\n\nKnit 4 rows. Bind off.\n\n#### FINISHING\n\nGently press strips, allowing the sides of each strip to roll slightly to the wrong side.\n\nWeave the Fronts together following the diagram above.\n\nFlip the piece over so the wrong side faces up. Tack the strips in place at each corner as pictured in the diagram.\n\nWeave the Backs together the same as the Fronts.\n\nSew the shoulder seams. Sew in the Sleeves. Sew the side and sleeve seams.\n\nEtiquette Unchained Pullover  \n~return to the beginning of the project~\n\nPixilated Weave Drapelette\n\n## PIXILATED WEAVE  \ndrapelette\n\nAn interesting woven I-cord wakes up the casual drape of this go-anywhere design. It is knit from side to side using rib-stitch cuffs and a stockinette-stitch body with a series of decreases to shape the deep flattering angle. The knitted cord looks as though it is knitted into the background but it is actually woven, pinned, and sewn on after the piece is knit. Try a contrasting color for the cord if you like, substitute one of the many cord applications and motifs in my other books, or make up your own!\n\nreimagine it\n\nEven if you leave the background piece the same, there are hundreds of different ways you can embellish this sweater using knitted cords, flowers, leaves, bobbles, colorwork, or embroidery! Or try a multicolored yarn with complementary colors for the embellishments.\n\nSKILL LEVEL\n\nTIME\n\nSIZE\n\nOne size\n\nFINISHED MEASUREMENTS\n\nBust: 73\u00bd\" (186.5cm)\n\nCuff to cuff: 36\u00be\" (93.5cm)\n\nLength: 13\u00be\u201325\u00bd\" (35\u201365cm)\n\nGAUGE\n\n16 stitches and 22 rows = 4\" (10cm) in stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nFyberspates Scrumptious Aran (45% silk, 55% merino), 3\u00bd oz (100g), 180 yd (165m); 7 balls of #409 Teal Green\n\nSize U.S. 9 (5.5mm) straight needles, or size needed to obtain gauge\n\nSize U.S. 7 (4.5mm) straight needles and double-pointed needles\n\nStitch holders or large safety pins\n\nTapestry needle\n\n### FRONT\n\nWith larger needles, cast on 55 stitches.\n\nWork 2 rows in stockinette stitch.\n\nIncrease row (RS): K1, m1, knit to the end.\n\nContinuing in stockinette stitch, repeat Increase row every 4th row until there are 102 stitches. Work even for 2\u00bd\" (6.5cm), until piece measures 34\u00bd\" (87.5cm). Bind off.\n\n### BACK\n\nWork same as Front, working Increase row as follows:\n\nIncrease row (RS): Knit to the last stitch, m1, k1.\n\n### FINISHING\n\nMatch the Front and Back pieces WS together and sew the shoulder seams, leaving a 9\" (23cm) neck opening in the center.\n\n#### Cuffs\n\nMark 25 rows down on each side (Front and Back) of the shoulder seam. With the RS facing and smaller needles, pick up and k50 stitches between the markers.\n\nWork in k2, p2 rib for 3\" (7.5cm) or desired length. Bind off in rib. Repeat on the other side.\n\nSew cuff and side seams.\n\n#### Weave\n\nFollow the diagram opposite for strip placement. Pick up and k6 stitches for each strip. Work in stockinette stitch for 10\" (25.5cm) for each straight strip and 15\" (38cm) for each spiraled strip. End each strip by threading the end through the 6 stitches, pulling it tight, and securing. Weave the strips, spiral the ends, and tack them in place, allowing the sides of the strips to curl and form tubes.\n\nPixilated Weave Drapelette  \n~return to the beginning of the project~\n\nChaos Couture Pullover\n\n## CHAOS COUTURE  \npullover\n\nSplendid textural yarn and braiding create this dazzling pullover that is one of a kind, thanks to a variety of different yarns that are hand-tied together. The combination of knitted details\u2014the braided sleeve, diagonal ribbed treatment, and cable edgings\u2014takes this design in a new direction. If you are substituting different yarns, be aware that combining a cohesive mix of yarn textures will add to the overall richness of this look.\n\nreimagine it\n\nIt is hard to consider using a different yarn since the beautiful textural variety of the Prism yarns adds to the bold depth of the braided sleeve. (Any piece knit using Prism yarns will always be a one-of-a-kind sweater!) The slanted rib at the bottom edge could be lengthened and belted, or worked in a mix of fun colors.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size M\n\nFINISHED MEASUREMENTS\n\nBust: 39\u00bd (46\u00be, 55\u00bd)\" [100 (118.5, 141)cm]\n\nLength: 18\u00bc (19, 19\u00bd)\" [46.5 (48.5, 49.5)cm]\n\nGAUGE\n\n20 stitches and 28 rows = 4\" (10cm) in stockinette stitch using size U.S. 6 (4mm) needles with yarn A\n\nApproximately 18 stitches and 24 rows = 4\" (10cm) in stockinette stitch using size U.S. 8 (5mm) needles with yarn B (yarn is a mix of textured yarn so the gauge varies throughout the skein)\n\n12 stitches and 16 rows = 4\" (10cm) in stockinette stitch using size U.S. 10 (6mm) needles with yarn C\n\nTake time to check gauge.\n\nMATERIALS\n\nPrism Symphony (80% merino, 10% cashmere, 10% nylon), 2 oz (57g), 118 yd (108m); 7 (8, 9) balls of Mushroom (A)\n\nPrism Layers Stuff (rayon, cotton, nylon, kid mohair, bamboo, Tencel, wool, cashmere), 6\u20138 oz (170\u2013226g), 300 yd (274m); 1 (1, 1) ball of Mushroom (B)\n\nPrism Plume (100% nylon), 2.8 oz (79g), 45 yd (41m); 2 (2, 3) balls of Mushroom (C)\n\nSize U.S. 6 (4mm) straight needles, or size needed to obtain gauge\n\nSize U.S. 8 (5mm) needles, or size needed to obtain gauge\n\nSize U.S. 10 (6mm) needles, or size needed to obtain gauge\n\nCable needle\n\nStitch markers\n\n4 stitch holders\n\nWaste yarn for provisional cast on\n\nTapestry needle\n\nNotes\n\n\u2022 Yoke depth and sleeve width are approximate since the braids are elastic. The bust is based on the stockinette stitch gauge. The cables draw the fabric in to create gathers.\n\n\u2022 Sleeves and yoke are made in strips and then braided to shape one piece for the sleeves, shoulders, and neck opening.\n\n### Diagonal Wave Rib\n\n(multiple of 6 stitches + 3)\n\n3\/3 RPC: Slip 3 to cn and hold in back, k3, p3 from cn.\n\nRows 1, 3, and 5 (RS): K3, *p3, k3; repeat from * to the end.\n\nRows 2 and 4: P3, *k3, p3; repeat from * to the end.\n\nRow 6 *3\/3 RPC; repeat from * to the last 3 stitches, k3.\n\nRows 7, 9, and 11: P3, *k3, p3; repeat from * to the end.\n\nRows 8 and 10: k3, *p3, k3; repeat from * to the end.\n\nRow 12: P3, *3\/3 RPC; repeat from * to the end.\n\nRepeat rows 1\u201312 for pattern.\n\n### FRONT\n\nWith smallest needles and A, cast on 99 (117, 129) stitches. Work 18 rows in Diagonal Wave Rib.\n\nWork in stockinette stitch until the piece measures 5\u00bd (6, 6\u00bd)\" [14 (15, 16.5)cm] from the cast-on edge, ending with a WS row.\n\nRow 1 (RS): K14 (20, 26), place marker, work Diagonal Wave Rib over the next 21 stitches, place marker, k29 (35, 35), place marker, work Diagonal Wave Rib over the next 21 stitches, k14 (20, 26).\n\nRows 2\u201324: Work in patterns as established.\n\nWork 12 rows in stockinette stitch.\n\n#### Armhole Shaping\n\nMark the center 27 stitches.\n\nContinuing in stockinette stitch, work 12 rows of Diagonal Wave Rib over the center 27 stitches.\n\nAt the same time, after 6 rows, bind off 3 (5, 7) stitches at the beginning of the next 2 rows, then 2 (2, 3) stitches at the beginning of the next 4 rows\u201485 (99, 103) stitches.\n\nBind off.\n\n### BACK\n\nMake same as the Front.\n\n### FRONT\/BACK SLEEVES AND YOKE\n\nNote: This piece starts with the sleeves on the left side, divides into two side strips to accommodate the braid, is worked across the yoke, down the right side, and joined to end at the opposite cuff.\n\n#### Left Side\n\nWith medium needles and A, cast on 51 (57, 63) stitches. Work 18 rows in Diagonal Wave Rib.\n\nWork 6 (6, 4) rows in stockinette stitch.\n\nIncrease row (RS): K1, m1, knit to the last stitch, m1, k1.\n\nContinuing in stockinette stitch, repeat Increase row every 6th (6th, 4th) row 3 (3, 5) more times\u201459 (65, 75) stitches.\n\nPurl 1 WS row.\n\nDividing row (RS): K7 (10, 15), attach a second ball of A and k45, place these 45 stitches on a holder, k7 (10, 15).\n\nWorking both sides of the sleeve at the same time, continue to work Increase row every 6th row at the outside edges 10 more times\u201417 (20, 25) stitches each side.\n\nWork even on both sides until the sleeve measures 17\" (43cm) or the desired length to the underarm. Make note of the number of rows worked so it can be repeated on the other side.\n\n#### Armhole Shaping\n\nBind off 3 (5, 7) stitches at each armhole edge, then 1 stitch every other row 2 (2, 3) times\u201412 (13, 15) stitches each side.\n\n#### Yoke\n\nWork even in stockinette stitch on both sides for the front and back yokes for 18\u00bd (22, 24\u00bc)\" [47 (56, 61.5)cm].\n\n#### Right Side\n\nAt each armhole edge, increase 1 stitch every RS row 2 (2, 3) times, then cast on 3 (5, 7) stitches\u201417 (20, 25) stitches each side.\n\nWork the same number of rows noted from the first sleeve.\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches of the other side, ssk, k1.\n\nRepeat Decrease row every 6th row 9 more times. Work 3 rows in stockinette stitch.\n\nJoining row (RS): K7 (10, 15), cast on (preferably with a provisional cast on) 45 stitches, k7 (10, 15)\u201459 (65, 75) stitches.\n\nContinue decreasing 1 stitch at each outside edge every 6th row 4 (4, 1) more time(s), then every 4th row 0 (0, 5) times\u201451 (57, 63) stitches.\n\nWork 6 (6, 4) rows even in stockinette stitch.\n\nWork 18 rows in Diagonal Wave Rib.\n\nBind off in rib.\n\n### CENTER SLEEVE AND YOKE STRIPS\n\nNotes: The number of rows worked in each strip differ because of yarn variations. Knit each strip to the indicated length, ending all strips with a WS row.\n\n#### LEFT SIDE\n\n#### Strip 1\n\nWith the RS facing, place the rightmost 11 stitches onto medium-size needles. With B, k1, m1, k9, m1, k1\u201413 stitches.\n\nWork in seed stitch for 23 (26, 28)\" [58.5 (66, 71)cm] as follows:\n\nRow 1: K1, *p1, k1; repeat from * to end.\n\nRepeat row 1 for pattern.\n\nPlace stitches on a holder.\n\n#### Strip 2\n\nWith the RS facing, place the next 10 stitches onto the largest needles.\n\nWith C, k1, m1, k8, m1, k1\u201412 stitches.\n\nWork in reverse stockinette stitch (purl on RS, knit on WS) for 23 (26, 28)\" [58.5 (66, 71)cm].\n\nPlace stitches on a holder.\n\n#### Strip 3\n\nWith the RS facing, place the next 13 stitches onto the smallest needles.\n\nWith A, work in stockinette stitch for 23 (26, 28)\" [58.5 (66, 71)cm].\n\nPlace on a holder.\n\n#### Strip 4\n\nWith the RS facing, place the last 11 stitches onto the medium-size needles.\n\nWith B, k1, m1, k9, m1, k1\u201413 stitches.\n\nWork in stockinette stitch for 23 (26, 28)\" [58.5 (66, 71)cm].\n\nPlace on a holder.\n\n#### Braiding\n\nFollowing the diagram (right), braid the 4 strips together. Pin and sew them to the inner sleeve edges. Twisting 2 strips each for the front and back yokes, pin in place and set aside.\n\n#### RIGHT SIDE\n\nRemove the waste yarn, and work the strips the same as the left side. When dividing strips for the yoke, make sure to match the ones from the left side, adding or removing rows as necessary to meet at the center neck edge. Pin in place.\n\n#### FINISHING\n\nGraft the strips using Kitchener stitch and sew them to the sleeve and yoke side strips.\n\nMark the center 9\" (23cm) for the neck opening and tack the strips together at each side.\n\nSew the sleeve seams. Sew the side seams.\n\nSew the front and back yoke edges to the Front and Back, matching underarm shaping.\n\nChaos Couture Pullover  \n~return to beginning of project~\n\nchapter 4\n\nSTITCH IMPACT\n\nThis chapter includes interesting and unique stitches, color work, and edgings that all go beyond \"business as usual\" stitches. By that I mean they jump out and shout, \"Look how special I am!\" Within these patterns you'll find a lovely repeat tuck stitch, a bulky knit bold leaf stitch, and two exciting motifs\u2014a dressage horse and a scary skull spirit. Edgier versions of traditional stitches include an unusual use of Fair Isle knitting and a cool slant using a scalloped edging. Adding such elements to a design transforms a garment into something far from ordinary. I hope you enjoy knitting and wearing these bold fashion statements as much as I do.\n\nSpring Forward Dress or Tunic\n\n## SPRING FORWARD  \ndress or tunic\n\nBig needles, big yarn, big bold stitches, and a fabulous dress (or tunic) that takes little time and effort. What could be more fun and creatively rewarding? Spring Forward features flattering, bold lines that update a classic knitted leaf. The shape is knit in two pieces from the bottom up to the bodice, and the sleeves are made up to the armholes. Then all four pieces are placed on one circular needle and knit in a rib pattern with graduated decreases to shape the yoke. Be prepared to hear, \"Wow! Where did you get it?\"\n\nreimagine it\n\nConsider making a shorter version and continuing the neckline to create a cowl neck. Cascade Yarns also offers Magnum Print (a multicolor version of Magnum), which can be used for a completely different look. Or combine that print with a solid yoke and a multicolor bottom.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS\/M (L\/XL), shown in size S\/M\n\nFINISHED MEASUREMENTS\n\nBust: 36 (45)\" [91 (114)cm]\n\nOverall length: 30\u00bc (32\u00bc)\" [77 (82)cm]\n\nGAUGE\n\n8 stitches and 12 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nMATERIALS\n\nCascade Magnum (100% Peruvian highland wool), 8.8 oz (250g), 123 yd (112m); 6 (7) skeins of #2422 Orchid\n\nSize U.S. 15 (10mm) straight needles and 24\" (61cm) circular needle, or size needed to obtain gauge\n\nStitch markers (optional)\n\nStitch holders\n\nMatching worsted-weight wool for seaming\n\nTapestry needle\n\n### Leaf Pattern\n\n(begins with 3 stitches)\n\nRow 1: K1, yo, k1, yo, k1\u20145 stitches.\n\nRow 2 and all WS rows: Purl.\n\nRow 3: K2, yo, k1, yo, k2\u20147 stitches.\n\nRow 5: K3, yo, k1, yo, k 3\u20149 stitches.\n\nRow 7: K4, yo, k1, yo, k4\u201411 stitches.\n\nRows 9 and 11: Knit.\n\nRow 13: Ssk, k7, k2tog\u20139 stitches.\n\nRow 15: Ssk, k5, k2tog\u20147 stitches.\n\nRow 17: Ssk, k3, k2tog\u20145 stitches.\n\nRow 19: Ssk, k1, k2tog\u20143 stitches.\n\nRow 21: Sk2p\u20141 stitch.\n\nThere are 2 stitches decreased after each leaf is complete.\n\n### FRONT\n\nCast on 54 (62) stitches.\n\nRow 1 (RS): K2, *p2, k2; repeat from * to end.\n\nRow 2: P2, *k2, p2; repeat from * to end.\n\nRow 3: Purl.\n\nRow 4: K26 (30), k2tog, k26 (30)\u201453 (61) stitches.\n\nRow 5: P10 (14), k1, [p7, k1] 4 times, p10 (14).\n\nRow 6: K10 (14), p1, [k7, p1] 4 times, k10 (14).\n\nRows 7\u201324 (7\u201330): Repeat rows 5 and 6.\n\nAt the same time, decrease every 16th row 4 times as follows:\n\nDecrease row (RS): P1, p2tog tbl, work to the last 3 stitches, p2tog, p1.\n\nNote: See photographs for leaf placement on the dress. The leaves are numbered 1 through 5 from right to left. The single columns of RS knits are the stems on which leaves will be knit.\n\nWork leaves on stems 1 and 5 as follows:\n\nLeaves 1 and 5 setup rows (RS): Work to 1 stitch before stem 1, work row 1 of Leaf pattern over the next 3 stitches, work to 1 stitch before stem 5, work row 1 of Leaf pattern over the next 3 stitches, work to the end. (If desired, place markers before and after Leaf pattern stitches.)\n\nContinue in patterns and decreases as established, ending with row 4 of Leaf pattern.\n\nLeaf 3 setup row: Work in patterns as established to 1 stitch before stem 3, work row 1 of Leaf pattern, work in patterns as established to the end.\n\nContinue in patterns and decreases as established, ending with row 16 of Leaf 3.\n\nLeaf 2 setup row: Work in patterns as established to 1 stitch before stem 2, work row 1 of Leaf pattern over the next 3 stitches, work in patterns as established to the end.\n\nContinue in patterns and decreases as established, ending with row 6 of Leaf 2.\n\nLeaf 4 setup row: Work in patterns as established to 1 stitch before stem 4, work row 1 of Leaf pattern over the next 3 stitches, work in patterns as established to the end.\n\nWhen the last Leaf is complete, ending with a WS row, [purl 1 row, knit 1 row] twice, decreasing 0 (1) stitch on the last WS row\u201436 (42) stitches.\n\nPlace 36 (42) stitches on a holder.\n\n### BACK\n\nWork same as the Front.\n\n### RIGHT SLEEVE\n\nCast on 18 (22) stitches.\n\nRow 1 (RS): K2, *p2, k2; repeat from * to end.\n\nRow 2: P2, *k2, p2; repeat from * to end.\n\nRow 3: P9 (11), m1p, p9 (11)\u201419 (23) stitches.\n\nRow 4: Knit.\n\nRow 5: P9 (11), k1, p9 (11).\n\nRow 6: K9 (11), p1, k9 (11).\n\nRows 7 and 8 (7\u201310): Repeat rows 5 and 6.\n\nAt the same time, increase every 6th row 4 times as follows:\n\nIncrease row (RS): P1, m1p, work to the last stitch, m1p, p1\u201427 (31) stitches.\n\nBegin Leaf pattern on RS center k1.\n\nLeaf setup row 9 (11): Work to 1 stitch before the stem, work row 1 of Leaf pattern, work to the end.\n\nContinue in patterns and increases as established until sleeve measures 16 (17)\" [40.5 (43)cm], ending with a WS row\u201425 (29) stitches. Place stitches on a holder.\n\n### LEFT SLEEVE\n\nWork same as the Right Sleeve, starting the Leaf pattern 9\" (23cm) from the cast-on edge.\n\n### YOKE\n\nPlace stitches onto the circular needle, ready to begin a WS row.\n\nStarting at the center back, place 18 (21) Back stitches, 25 (29) Left Sleeve stitches, 36 (42) Front stitches, 25 (29) Right Sleeve stitches, 18 (21) Back stitches onto the circular needle\u2014122 (142) stitches.\n\nBeginning with a WS row, work back and forth as follows:\n\nRow 1 (WS): P2, *k2, p2; repeat from * to end.\n\nRow 2: K2, *p2, k2; repeat from * to end.\n\nRow 3: Repeat row 1.\n\nRow 4: K2, *p2, k2tog; repeat from * to the last 4 stitches, p2, k2\u201493 (108) stitches.\n\nRow 5: P2, *k2, p1; repeat from * to the last 4 stitches, k2, p2.\n\nRows 6\u20139: Work in rib as established.\n\nRow 10: K2, *p2tog, k1; repeat from * to the last stitch, k1\u201463 (73) stitches.\n\nRow 11: P2, *k1, p1; repeat from * to the last stitch, p1.\n\nRows 12\u201315: Work in rib as established.\n\nBind off in pattern.\n\n### FINISHING\n\nWith matching worsted-weight yarn, sew the top 9\u00bd (10)\" [24 (25.5)cm] at each side of the Sleeves to the Front and Back. Sew the side seams. Sew the sleeve seams.\n\nSpring Forward Dress or Tunic  \n~return to the beginning of the project~\n\nButtons and Bows Manteau\n\n## BUTTONS AND BOWS  \nmanteau\n\nThe colors, buttons, and tailored shape give this delightful-to-wear design a slightly vintage appeal. Although it has classic knit shaping, using the tuck stitch texture makes it new and interesting. This versatile design can be worn as either a dress or a jacket. If you'd like, the bows can be omitted and the waist belted. As you can see, this piece has a lot of stockinette stitch along with the entertaining tucks, and though not difficult, it is a time-consuming piece.\n\nreimagine it\n\nThink about choosing a color for the tucked bottom section and switching to a contrasting color at the beginning of the waist darts. Then choose a third color for the collar and bows. Remove bows or add even more to create the look you want. The piece could also be easily converted into a shorter cardigan style.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M\/L, XL), shown in size M\/L\n\nFINISHED MEASUREMENTS\n\nBust: 39\u00bc (45\u00bd, 58)\" [99.5 (115.5, 147.5)cm]\n\nOverall length: 31 (32, 33)\" [79 (81, 84)cm]\n\nSleeve length to underarm: 10 (10\u00bd, 11)\" [25.5 (26.5, 28)cm]\n\nGAUGE\n\n20 stitches and 28 rows = 4\" (10cm) in stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nKollage Milky Whey (50% milk, 50% soy), 1\u00be oz (50g), 137 yd (123m); 9 (12, 16) balls of #7610 Milky Green (A); 2 (2, 3) balls of #7604 Cameo (B)\n\nSize U.S. 6 (4mm) straight needles, or size needed to obtain gauge\n\nSize U.S. 4 (3.5mm) set of 5 double-pointed needles\n\nStitch markers\n\nRemovable stitch markers\n\nStitch holders\n\nTapestry needle\n\nEight \u00be\" (2cm) buttons (JHB's #12594)\n\n5 sew-on snaps (optional)\n\n### LEFT FRONT\n\nWith larger needles and B, cast on 55 (75, 95) stitches.\n\n### Hem\n\nWork in stockinette stitch for 1\" (2.5cm), ending with a WS row.\n\nChange to A and knit 3 rows.\n\nWork in Tuck Stitch pattern as follows:\n\nRows 1\u201316: Work in stockinette stitch.\n\nRow 17 (RS): K15, *k5, insert the dpn through these 5 stitches on the WS, k15; repeat from * to end. Place a marker in the last stitch on the WS.\n\nRows 18\u201326: Work 9 rows in stockinette stitch.\n\nRow 27: K15, *knit the next 5 stitches together with the 5 stitches from the dpn below, k15; repeat from * to end, knitting the last stitch together with the marked stitch below.\n\nRows 28\u201344: Work 17 rows in stockinette stitch.\n\nRow 45: K5, *k5, insert a dpn through the back of these 5 stitches, k15; repeat from * to the last 5 stitches, k5. Place a marker in the last stitch on the WS.\n\nRows 46\u201354: Work 9 rows in stockinette stitch.\n\nRow 55: K5, *knit the next 5 stitches together with the 5 stitches from the dpn below, k15; repeat from * to the last 10 stitches, knit the next 5 stitches together with the 5 stitches from the dpn below, k4, knit the last stitch together with the marked stitch below.\n\nRow 56: Purl.\n\nRepeat rows 1\u201356 once, then work rows 1\u201344 once more.\n\n### Shape Waist\n\nRow 1: K15, *ssk, k1, k2tog, k15; repeat from * to end\u201451 (69, 87) stitches.\n\nRows 2 and 4: Purl.\n\nRow 3: K14, *ssk, k1, k2tog, k13; repeat from * to the last 14 stitches, k14\u201447 (63, 79) stitches.\n\nRow 5: K13, *ssk, k1, k2tog, k11; repeat from * to the last 13 stitches, k13\u201443 (57, 71) stitches.\n\nWork even in stockinette stitch for 4 (4\u00bd, 5)\" [10 (11.5, 12.5)cm], ending with a WS row.\n\n### Shape Armhole\n\nNext row (RS): Bind off 3 (5, 8) stitches, knit to the end\u201440 (52, 63) stitches.\n\nNext row: Purl 1 row.\n\nDecrease row (RS): K1, k2tog, knit to the end\u201439 (51, 62) stitches.\n\nRepeat Decrease row every RS row 2 (2, 8) more times\u201437 (49, 54) stitches.\n\nWork even in stockinette stitch until armhole measures 5 (5\u00bd, 6)\" [12.5 (14, 15)cm], ending with a RS row.\n\n### Neck Shaping\n\nAt the beginning of WS rows at the neck edge, bind off 10 (12, 14) stitches once, 2 stitches 3 (4, 5) times, then 1 stitch 2 (2, 3) times\u201419 (27, 27) stitches.\n\nWork even in stockinette stitch until armhole measures 8 (8\u00bd, 9)\" [20.5 (21.5, 23)cm]. Place shoulder stitches on a holder.\n\n### RIGHT FRONT\n\nWith larger needles and B, cast on 55 (75, 95) stitches.\n\n### Hem\n\nWork in stockinette stitch for 1\" (2.5cm), ending with a WS row.\n\nChange to A and knit 3 rows.\n\nWork in Tuck Stitch pattern as follows:\n\nRows 1\u201316: Work in stockinette stitch.\n\nRow 17 (RS): Place a marker in the first stitch on the WS. K15, *k5, insert a dpn through the back of these 5 stitches, k15; repeat from * to end.\n\nRows 18\u201326: Work 9 rows in stockinette stitch.\n\nRow 27: Knit the first stitch together with the marked stitch below, k14, *knit the next 5 stitches together with the 5 stitches from the dpn below, k15; repeat from * to end.\n\nRows 28\u201344: Work 17 rows in stockinette stitch.\n\nRow 45: Place a marker in the first stitch on the WS. K5, *k5, insert a dpn through the back of these 5 stitches, k15; repeat from * to the last 5 stitches, k5.\n\nRows 46\u201354: Work 9 rows in stockinette stitch.\n\nRow 55: Knit the first stitch together with the marked stitch below, k4, *knit the next 5 stitches together with the 5 stitches from the dpn below, k15; repeat from * to the last 10 stitches, knit the next 5 stitches together with the 5 stitches from the dpn below, k4, knit the last stitch together with the marked stitch below.\n\nRow 56: Purl.\n\nRepeat rows 1\u201356 once, then work rows 1\u201344 once more.\n\nWork same as the Left Front until the armhole shaping, ending with a RS row.\n\n### Shape Armhole\n\nNext row (WS): Bind off 3 (5, 8) stitches, purl to end.\n\nDecrease row (RS): Knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every RS row 2 (2, 8) times more\u201437 (49, 54) stitches.\n\nWork even in stockinette stitch until armhole measures 5 (5\u00bd, 6)\" [12.5 (14, 15)cm], ending with a WS row.\n\nWork neck shaping and remainder of Right Front same as Left Front, working neck bind-offs at the beginning of RS rows.\n\n### LEFT AND RIGHT BACKS\n\nMake Left and Right Backs same as Left and Right Fronts, omitting the neck shaping. Place 37 (49, 54) stitches on a holder.\n\n### SLEEVES\n\nWith larger needles and B, cast on 57 (61, 67) stitches.\n\n### Hem\n\nWork in stockinette stitch for 1\" (2.5cm), ending with a WS row.\n\nChange to A and knit 3 rows. Mark the center 5 stitches.\n\nWork 16 rows in stockinette stitch.\n\nIncrease row (RS): K1, m1, knit to the last stitch, m1, k1.\n\nRepeat Increase row every 4th row 7 (8, 9) more times\u201473 (79, 87) stitches.\n\nAt the same time, starting on the same row as the first increase, work 3 Tuck Stitches on the center 5 stitches as follows:\n\nRow 1 (RS): Knit to the center 5 stitches, insert a dpn through the back of these 5 stitches, knit to the end.\n\nRows 2\u201310: Work 9 rows in stockinette stitch.\n\nRow 11: Knit to the center 5 stitches, knit the next 5 stitches together with the 5 stitches from the dpn below, knit to the end.\n\nRows 12\u201328: Work 17 rows in stockinette stitch.\n\nRepeat rows 1\u201328 once, then work rows 1\u201311 once more.\n\nWork even in stockinette stitch until sleeve measures 10 (10\u00bd, 11)\" [25.5 (26.5, 28)cm].\n\n### Shape Sleeve Cap\n\nBind off 3 (5, 8) stitches at the beginning of the next 2 rows, then decrease 1 stitch each side every RS row 16 (17, 18) times. Bind off 3 stitches at the beginning of the next 8 rows. Bind off the remaining 11 stitches.\n\n### FINISHING\n\nSew the back seam.\n\nJoin shoulder seams using the 3-needle bind-off method. Set in the Sleeves. Sew the side and sleeve seams. Sew hems to the wrong side.\n\n### Left Front Band\n\nWith the RS facing and A, pick up and k141 (147, 153) stitches from neck edge to the bottom edge, working through 3 layers over the tucks. Work in k3, p3 rib for 1\" (2.5cm). Bind off in rib.\n\n### Right Front Band\n\nWith the RS facing and A, pick up and k141 (147, 153) stitches from the bottom edge to the neck edge, working through 3 layers over the tucks. Work 3 rows in k3, p3 rib.\n\nButtonhole row (RS): *Work 14 (15, 16) stitches, bind off 3 stitches; repeat from * 7 more times, work to the last 5 (3, 1) stitches as set.\n\nNext row: Work in rib as established, casting on 3 stitches over the bound-off stitches of the previous row.\n\nContinue in k3, p3 rib until rib measures 1\" (2.5cm). Bind off in rib.\n\nSew buttons to Left Front band to correspond with buttonholes.\n\n### Neck Band\n\nWith the RS facing and smaller needles, pick up and k95 (107, 131) stitches evenly around the neck opening, including the front bands. Work in k1, p1 rib for 3\u00bd\" (9cm).\n\nWork Picot Bind-Off as follows:\n\nBind off 2 stitches, *slip the last stitch back onto the left-hand needle, cast on 3 stitches using the cable cast-on, bind off 5 stitches; repeat from * to the last stitch, bind off the last stitch.\n\n### Lace Ruffle Collar (optional)\n\nWith larger needles and B, cast on 295 (323, 379) stitches (a multiple of 14 stitches plus 1 more).\n\nRow 1 and all WS rows: Purl.\n\nRows 2, 4, 6, 8, 10, and 12: K1, *yo, k3, ssk, yo, sk2p, yo, k2tog, k3, yo, k1; repeat from * to end.\n\nRow 14: K1, *k2tog; repeat from * to end\u2014148 (162, 190) stitches.\n\nRow 16: P1, [k1, p1] twice, *k2 (2, 2) ssk; repeat from * to the last 7 (5, 5) stitches, k2 (0, 0), [p1, k1] twice, p1\u2014114 (124, 145) stitches.\n\nRow 17: Purl, decreasing 11 (7, 10) stitches evenly between the two sets of rib stitches\u2014103 (117, 135) stitches.\n\nWork in k1, p1 rib across all stitches for 1\" (2.5cm). Bind off in rib.\n\nSew 5 snaps evenly across the edge of the collar and around the neck band to attach it.\n\n### Bow (MAKE 3)\n\nWith larger needles and B, cast on 15 stitches.\n\nWork in k1, p1 rib for 4\" (10cm). Bind off in rib.\n\n### Tie (MAKE 3)\n\nWith larger needles and B, cast on 5 stitches.\n\nWork in k1, p1 rib for 1\u00bd\" (3.8cm). Bind off in rib.\n\nWrap the tie around the center of the bow and sew the bound-off edge to the cast-on edge.\n\nNotes: Adjust the fit of this jacket by connecting 2 adjacent darts (waist decrease points) and sewing a bow over each join. Make a pleat between the front decrease darts and attach a bow on each side. Make one pleat at center back and attach bow.\n\nButtons and Bows Manteau  \n~return to the beginning of the project~\n\nDressage Pony Poncholette\n\n## DRESSAGE PONY  \nponcholette\n\nAny horse lover would be happy wearing this piece, sporting a classic jumping horse and other sophisticated details. The main feature is the horse motif that can either be knitted in or worked in duplicate stitch after the piece is knit. The triple cuff detail includes a rolled edge, a rib, and a two-color houndstooth pattern. An I-cord drawstring calls attention to one side, and an elegant braided cord collar at the neck adds extra interest to this universally flattering style.\n\nreimagine it\n\nSans horse the piece is instantly reimagined. Omitting the horse will give you a subtler look, and you can still keep the comfy fit and casual style of the design. Instead of the horse, you can also add any motif of your choice, for example, a flower or a cat. The shape provides the perfect canvas for you to reimagine this stylish design.\n\nSKILL LEVEL\n\nTIME\n\nSIZE\n\nOne size\n\nFINISHED MEASUREMENTS\n\nBust: Oversized 58\u00bd\" (148.6cm)\n\nOverall length (including waist ribbing): 24\u00bd\" (62.2cm)\n\nSleeve length to underarm: 12\u00bd\" (31.7cm)\n\nGAUGE\n\n20 stitches and 28 rows = 4\" (10cm) in stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nAslan Trends Royal Alpaca (100% alpaca), 3\u00bd oz (100g), 220 yd (200m); 5 balls of #6365 Sable (A); 2 balls of #19 Black (B); 1 ball of #1570 Pale Blue (C)\n\nSize U.S. 7 (4.5mm) 24\" (61cm) circular needle, or size needed to obtain gauge\n\nSize U.S. 5 (3.75mm) double-pointed needles\n\nStitch markers\n\nTapestry needle\n\nWaste yarn\n\nSafety pin\n\n### Houndstooth Check Pattern\n\nRow 1 (RS): K1 B, *k1 A, k3 B; repeat from * to the last 3 stitches, k1 A, k2 B.\n\nRow 2: *P3 A, p1 B; repeat from * to end.\n\nRow 3: *K3 A, k1 B; repeat from * to end.\n\nRow 4: P1 B, *p1 A, p3 B; repeat from * to the last 3 stitches, p1 A, p2 B.\n\nRepeat rows 1\u20134 for pattern.\n\nNote: This garment is knit from side to side. Read ahead as multiple shapings occur at the same time. The silhouette of the horse is worked in duplicate stitch, or can be knitted in intarsia instead.\n\n### BACK\n\nStarting with B and larger needle, at the right back sleeve, cast on 30 stitches. Work 9 rows in stockinette stitch.\n\nShoulder Increase row (RS): K1, m1, knit to end.\n\nContinuing in stockinette stitch, repeat Shoulder increase row every 10th row 10 more times.\n\nAt the same time, when sleeve measures 4\" (10cm), ending with a WS row, increase on next and every RS row 6 times as follows:\n\nUnderarm Increase row (RS): Work to the last stitch, m1, k1\u201440 stitches for body.\n\nAfter all underarm increases have been completed, cast on 70 stitches for body using the cable cast-on method at the beginning of the next WS row.\n\nWork 43 rows in stockinette stitch, continuing shoulder increases as established, ending with a WS row\u2014117 stitches.\n\nChange to C and work 20 rows even in stockinette stitch.\n\nChange to A and work 10 rows even in stockinette stitch.\n\n#### Back Neck\n\nPlace a marker at the shoulder edge, work 59 rows even in stockinette stitch, place marker at the next shoulder edge.\n\nShoulder decrease row (RS): K1, k2tog, knit to end.\n\nContinuing in stockinette stitch, repeat Shoulder decrease row every 10th row 10 more times.\n\nAt the same time, when piece measures 24\u00bd\" (62.2cm) from the body cast on, work bottom curve on WS rows as follows:\n\nBind off 1 stitch every WS row 6 times, 1 stitch every 4th WS row 5 times, then bind off 59 stitches for body\u201440 stitches.\n\nUnderarm decrease row (RS): Work to the last 3 stitches, ssk, k1.\n\nRepeat Underarm decrease row every RS row 5 more times\u201434 stitches.\n\nWhen all shoulder decreases have been worked, work 9 rows even in stockinette stitch on 30 stitches.\n\nBind off.\n\n### FRONT\n\nStarting at the right front sleeve, with B and larger needle, cast on 30 stitches. Work 9 rows in stockinette stitch.\n\nShoulder increase row (RS): Knit to the last stitch, m1, k1.\n\nContinuing in stockinette stitch, repeat Shoulder increase row every 10th row 10 more times.\n\nAt the same time, when sleeve measures 4\" (10cm), ending with a WS row, increase on next and every RS row 6 times as follows:\n\nUnderarm increase row (RS): K1, m1, work to the last stitch.\n\nAfter all underarm increases have been completed, cast on 70 stitches for body using the cable method at the beginning of the next RS row.\n\nWork 43 rows in stockinette stitch, continuing shoulder increases as established, ending with a WS row\u2014117 stitches.\n\nChange to C and work 20 rows even in stockinette stitch.\n\nChange to A and work 10 rows even in stockinette stitch.\n\n#### Front Neck Shaping\n\nDecrease row (RS): Knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every RS row twice more, then every 4th row 3 more times\u2014111 stitches.\n\nWork 26 rows even in stockinette stitch.\n\nIncrease row (RS): Knit to the last stitch, m1, k1.\n\nRepeat Increase row every 4th row twice more, then every RS row 3 more times\u2014117 stitches.\n\nPurl 1 WS row.\n\nShoulder decrease row (RS): Knit to the last 3 stitches, ssk, k1.\n\nContinuing in stockinette stitch, repeat Shoulder decrease row every 10th row 10 more times.\n\nAt the same time, when piece measures 24\u00bd\" (62.2cm) from the body cast-on, shape the bottom curve, then the underarm, on RS rows as follows:\n\nBind off 1 stitch every RS row 6 times, 1 stitch every 4th row 5 times, then bind off 59 body stitches.\n\nUnderarm decrease row (RS): K1, k2tog, work to end.\n\nRepeat Underarm decrease row every RS row 5 more times.\n\nWhen all shoulder decreases have been worked, work 9 rows even in stockinette stitch on 30 stitches.\n\nBind off.\n\nWith B, work Dressage Pony chart on the Front in duplicate stitch, starting the first stitch of the chart in the 3rd stitch from the ribbing and the first row in the C section.\n\n### SLEEVES\n\nSew Front to Back from cast on to neck edge.\n\nWith the RS facing, larger needle, and B, pick up 60 stitches evenly along the sleeve edge. Work 12 rows in Houndstooth Check pattern. Keeping the needle in the stitches, thread waste yarn through the stitches for later use.\n\nNext row (RS): With B, knit.\n\nWork in k2, p2 rib for 1\u00bd\" (3.8cm). Bind off.\n\nPlace the stitches on the waste yarn onto larger needle, ready to work a WS row.\n\nWith C, purl 1 WS row.\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1.\n\nContinuing in stockinette stitch, repeat Decrease row every 4th row 8 more times\u201442 stitches. Bind off. Repeat for second sleeve.\n\n### FINISHING\n\n#### LOWER EDGE RIB\n\n#### Front\n\nWith the RS facing and matching colors across the bottom edge, with B and larger needles pick up 39 stitches, with C pick up 18 stitches, and with A pick up 117 stitches\u2014174 stitches. Work in k2, p2 rib in established colors for 1\" (2.5cm). Bind off.\n\n#### Back\n\nWith the RS facing and matching colors across the bottom edge, with A and larger needles pick up 117 stitches, with C pick up 18 stitches, and with B pick up 39 stitches\u2014174 stitches. Work in k2, p2 rib in established colors for 1\" (2.5cm). Bind off.\n\n#### SIDE CASING\n\n#### Front\n\nWith the RS facing, larger needles, and A, starting at the rib edge, pick up 6 stitches along the side edge of the rib and 33 stitches across the bound-off stitches\u201439 stitches.\n\nKnit 1 WS row.\n\nKnit 1 RS row. Continue in stockinette stitch until casing measures \u00be\" (2cm) from the pickup. Bind off.\n\n#### Back\n\nWith the RS facing, larger needle, and A, starting at the 33rd bound-off stitch from the left, pick up 33 stitches across the bound-off stitches and 6 stitches along the side edge of the rib\u201439 stitches.\n\nKnit 1 WS row.\n\nKnit 1 RS row. Continue in stockinette stitch until casing measures \u00be\" (2cm) from the pickup. Bind off.\n\nSew underarm and side seams. Fold Front and Back casings to the WS and sew in place.\n\n#### I-cord (MAKE 2)\n\nWith dpns and A, cast on 3 stitches. *Do not turn work. Slide the stitches to the other end of the needle, k3; repeat from * until cord measures 30\" (76cm).\n\nK3tog and fasten off.\n\nAttach a safety pin to one end and insert an I-cord through each casing. Secure the tops of the I-cords to the casing and knot the ends of the cords.\n\n#### NECK BAND\n\nWith A, cast on 21 stitches. Work in k1, p1 rib for 6\u00bd\" (16.5cm), ending with a WS row.\n\nNext row (RS): *Work 7 stitches in rib, join another strand of A; repeat from * once, work 7 stitches in rib.\n\nWork each strip separately in stockinette stitch for 6\" (15cm), ending with a WS row.\n\nBraid these three separate strips together, ending with the first strand ready to work a RS row.\n\nJoining row (RS): With the first strand of yarn, k21, dropping the other strands of yarn.\n\nContinue across all stitches in k1, p1 rib for 6\u00bd\" (16.5cm). Bind off.\n\nSew cast-on edge to bound-off edge. Center the cable at the front neck and sew in place.\n\nDressage Pony Poncholette  \n~return to the end of the project~\n\nSpirits Fly Pullover\n\n## SPIRITS FLY  \npullover\n\nAn admittedly faux Goth look, the shape and fit of this piece is one of my favorites. The extra-deep raglan sleeve is quite flattering. The skull, crosses, and edging motifs are created with duplicate stitch, so you can also create this piece with or without the motifs. The yarn used for the background creates a wonderful uneven striping that is beautiful on its own, but also makes the perfect background for the skull spirit. Although made with easy stockinette stitch, this piece requires a lot of knitting and some advanced color work. Let your needles fly!\n\nreimagine it\n\nAnything goes here. Reimagine this basic shape without the motifs, give it a different background color, and add stitch patterns or your own motif or two if you wish.\n\nSKILL LEVEL\n\nTIME\n\nSIZE\n\nOne size\n\nFINISHED MEASUREMENTS\n\nBust: 54\" (137cm)\n\nOverall length: 30\" (76cm)\n\nSleeve length: 12\u00bd\" (32cm)\n\nGAUGE\n\n16 stitches and 24 rows = 4\" (10cm) in stockinette stitch\n\nTake time to check gauge.\n\nNote: Skull motif is worked in duplicate stitch after the piece is knit.\n\nMATERIALS\n\nCascade Casablanca (60% wool, 25% silk, 15% mohair), 3\u00bd oz (100g), 220 yd (201m); 5 skeins of #12 Jemstones (A)\n\nCascade Lana D'Oro, (50% alpaca, 50% wool) 3\u00bd oz (100g), 219 yd (200m); 1 ball each of #1055 Black (B), #1086 Hare (C), and #1089 Dark Grey and Medium Grey Twist (D)\n\nCascade 220 Heathers (100% Peruvian highland wool), 3\u00bd oz (100g), 220 yd (201m); 1 skein each of #2425 Provence (E) and #2450 Mystic Purple (F)\n\nPlymouth 24K (82% nylon, 18% lam\u00e9) 1\u00be oz (50g), 187 yd (171m), 1 ball of #0002 Silver (G)\n\nSize U.S. 9 (5.5mm) straight needles, or size needed to obtain gauge\n\nTapestry needle\n\nSize U.S. G-6 (4mm) crochet hook\n\n### BACK\n\nWith B, cast on 117 stitches. Starting with a WS row, work 6 rows in stockinette stitch.\n\nKnit 1 WS row for turning ridge.\n\nWork 6 rows in stockinette stitch.\n\nContinue in stockinette stitch in the following color sequence: 10 rows F, 2 rows E, 10 rows F, 12 rows B.\n\nChange to A and work in stockinette stitch until piece measures 14\" (35.5cm) from the turning ridge, ending with a WS row.\n\n### Raglan Shaping\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1.\n\nRepeat Decrease row every RS row 5 times, every 4th row 14 times, then every RS row 20 times\u201437 stitches.\n\nBind off.\n\n### FRONT\n\nWork same as Back, ending with 12 rows of B.\n\nChange to D and work 8 rows in stockinette stitch.\n\nChange to A and continue same as Back until 53 stitches remain, ending with a WS row.\n\n### Neck Shaping\n\nContinue raglan shaping at armholes and at the same time, work as follows:\n\nNext row (RS): Work 15 stitches, join a second ball of A, bind off the next 23 stitches, work the remaining 15 stitches to the end.\n\nWorking on both sides at the same time, at each neck edge, bind off 2 stitches twice and 1 stitch once.\n\nWhen all raglan shaping is complete, bind off the remaining 2 stitches on each side.\n\n### SLEEVES (MAKE 2)\n\nWith B, cast on 44 stitches. Work in K1, p1 rib for 1\u00bd\" (3.8cm).\n\nIncrease row (RS): K1, *kfb; repeat from * to the last stitch, k1\u201486 stitches.\n\nWork in stockinette stitch for 1\" (2.5cm), ending with a WS row.\n\nChange to A.\n\nIncrease row (RS): K1, m1, knit to the last stitch, m1, k1\u201488 stitches.\n\nRepeat this Increase row every RS row 13 more times\u2014114 stitches.\n\nContinue even in stockinette stitch until piece measures 12\" (30.5cm) from the cast-on edge, ending with a WS row.\n\n### Raglan Shaping\n\nDecrease row (RS): K1, k2tog, knit to the last 3 stitches, ssk, k1\u2014112 stitches.\n\nRepeat Decrease row every 4th row 9 more times, then every RS row 35 more times\u201424 stitches.\n\nBind off.\n\n### COWL COLLAR\n\nWith B, cast on 115 stitches.\n\nWork in stockinette stitch in the following color sequence: 6 rows B, 10 rows F, 4 rows B, 8 rows E, 4 rows B, 22 rows A, 8 rows B, 6 rows D.\n\nWith D, knit 5 rows.\n\nBind off loosely.\n\n### SKULL, ARROW, AND DIAMOND MOTIFS (OPTIONAL)\n\nWork motifs in duplicate stitch following the charts and referring to photographs for placement.\n\n### FINISHING\n\nSew raglan seams. Sew side and sleeve seams, using a full stitch at each selvedge. Fold and sew hem to WS of the garment. Sew the Collar seam. With the WS of the Collar facing the RS of the garment, pin the cast-on edge of the Collar to the neck opening. With B and the crochet hook, join them using a slip stitch.\n\nSpirits Fly Pullover  \n~return to the beginning of the project~\n\nVictory Fair Isle Pullover\n\n## VICTORY FAIR ISLE  \npullover\n\nWith its beautiful fit and easy but intricate stitch work, this sweater is a new classic for your wardrobe. I am a big fan of traditional Fair Isle knitting, but I always try to kick it up a notch by mixing the treasured technique with other stitch patterns and textures, or by giving it nontraditional placement, as you can see here. Be warned, this piece is a major time investment and not for a novice knitter. There is Fair Isle color work (in this case, with lots of ends to work in) and the combination of the colors and stitch pattern is seriously exciting knitting.\n\nreimagine it\n\nImagine omitting the fur on this piece and using a cabled or bobbled I-cord in its place. Instead of the Fair Isle pattern, make the V and sleeves in an easy stripe or a solid color.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size M\n\nFINISHED MEASUREMENTS\n\nBust\/lower edge circumference: 35 (40, 45)\" [89 (101.5, 114)cm]\n\nLength: 24\u00bd (26\u00bd, 27\u00be)\" [62 (67.5, 70.5)cm]\n\nGAUGE\n\n24 stitches and 32 rows = 4\" (10cm) in stockinette stitch using size U.S. 6 (4mm) needles.\n\n26 stitches and 30 rows = 4\" (10cm) over Lattice pattern using size U.S. 6 (4mm) needles.\n\n27 stitches and 27 rows = 4\" (10cm) over Fair Isle pattern using size U.S. 6 (4mm) needles.\n\n16 stitches = 4\" (10cm) over stockinette stitch using size U.S. 10\u00bd needles with fur yarn (row gauge not necessary)\n\nTake time to check gauge.\n\nMATERIALS\n\nRowan Felted Tweed (50% wool, 25% alpaca, 25% viscose), 1\u00be oz (50g), 191 yd (175m); 6 (6, 7) balls of # 175 Cinnamon (A), 1 each of the following: #167 Maritime (B), #158 Pine (C), #153 Phantom (D), #178 Seasalter (E), #184 Celadon (F), #150 Rage (G), #161 Avocado (H), #154 Ginger (J), #177 Clay (L), #181 Mineral (M)\n\nPrism Plume (100% nylon) 2.8 oz (79g), 45 yd (41m), 2 skeins of Denali\n\nSize U.S. 4 (3.5mm) 16\" (40.5cm) and 32\" (81cm) circular needles\n\nSize U.S. 6 (4mm) 16\" (40.5cm) circular needles, or size needed to obtain gauge\n\nSize U.S. 6 (4mm) 32\" (81cm) circular needles, or size needed to obtain gauge\n\nSize U.S. 10\u00bd (6.5mm) 16\" (40.5cm) circular needles\n\nStitch holders\n\n25 stitch markers\n\nTapestry needle\n\n### Pattern Stitches\n\nLT (left twist): With the right-hand needle behind the left-hand needle, skip 1 stitch and knit the second stitch through the back loop; then insert the right-hand needle into the back of both stitches (the skipped stitch and the second stitch) and k2tog.\n\nRT (right twist): K2tog, leaving both stitches on the left-hand needle; then insert the right-hand needle through the front of the first stitch just knitted together, and knit the first stitch again; then slip both stitches from the needle together.\n\n### Lattice Pattern\n\n(multiple of 8 stitches + 2)\n\nRow 1 (RS): K1, *LT, k4, RT; repeat from * to the last stitch, k1.\n\nRow 2 and all WS rows: Purl.\n\nRow 3: K2, *LT, k2, RT, k2; repeat from * to end.\n\nRow 5: K3, *LT, RT, k4; repeat from * to the last 7 stitches, LT, RT, k3.\n\nRow 7: K4, *RT, k6; repeat from * to the last 4 stitches, k4.\n\nRow 9: K3, *RT, LT, k4; repeat from * to the last 7 stitches, RT, LT, k3.\n\nRow 11: K2, *RT, k2, LT, k2; repeat from *to end.\n\nRow 13: K1, *RT, k4, LT; repeat from * to the last stitch, k1.\n\nRow 15: K8, *LT, k6; repeat from * to the last 2 stitches, k2.\n\nRow 16: Purl.\n\nRepeat Rows 1\u201316 for pattern.\n\n### BACK\n\nWith A and 32\" (81cm) smallest-size circular needle, cast on 114 (130, 146) stitches.\n\nWork back and forth on the circular needle to make the hem. Beginning with a knit row, work 8 rows in stockinette stitch, ending with a WS row.\n\nPurl 2 rows.\n\nChange to the 32\" (81cm) medium-size circular needle.\n\nWork 134 (150, 150) rows in Lattice pattern.\n\n#### Shape Underarms\n\nNext row (RS): Bind off 4 (5, 7) stitches, work in pattern to end.\n\nNext row: Bind off 4 (5, 7) stitches, purl to end\u2014106 (120, 132) stitches.\n\nPlace stitches on a holder.\n\n### FRONT\n\nNote: Treat the Fair Isle V-neck panel as an intarsia shape, twisting yarns on the WS when changing from Lattice pattern to Fair Isle panel and back to Lattice pattern.\n\nWork same as Back until 48 (64, 64) rows of Lattice pattern have been completed.\n\n#### Fair Isle V Panel\n\nRow 1 (RS): Work 57 (65, 73) stitches in Lattice pattern, drop yarn. With color of Row 1 of Fair Isle V chart, m1, drop yarn. Attach a new ball of A and work 57 (65, 73) stitches in Lattice pattern\u2014115 (131, 147).\n\nRow 2: Purl, keeping in patterns and colors as established.\n\nRow 3: With A, work in Lattice pattern to 2 stitches from V, k2tog, drop yarn. With color(s) of next row of Fair Isle V chart, m1, k1, m1 in colors of chart, drop yarn(s). With A, ssk, work in Lattice pattern to end.\n\nRow 4: Purl, keeping in colors as established.\n\nRepeat rows 3 and 4, keeping patterns as established, until row 86 of Fair Isle V chart has been completed and piece measures same as the Back to the underarm shaping.\n\n#### Shape Underarms\n\nRow 87: Bind off 4 (5, 7) stitches, work in patterns as established to the end.\n\nRow 88: Bind off 4 (5, 7) stitches, purl to end, keeping in patterns and colors as established\u2014107 (121, 133) stitches. Place stitches on a holder.\n\n### SLEEVES (MAKE 2)\n\nWith A and 16\" (40.5cm) smaller sized circular needle, cast on 49 (49, 65) stitches. Work back and forth on the circular needle to make the hem: Beginning with a knit row, work 8 rows in stockinette stitch, ending with a WS row. Purl 2 rows.\n\nChange to 16\" (40.5cm) medium sized circular needle.\n\nRow 1 (RS): With A, k1, work 18 (18, 26) stitches of Lattice pattern, drop yarn. With color(s) of Sleeve charts, work 11 stitches of Sleeve chart. Attach a new ball of A, work 18 (18, 26) stitches of the Lattice pattern, k1.\n\nRow 2: Purl, keeping in patterns and colors as established.\n\nRepeat rows 1 and 2, working through 128 rows of Sleeve chart and keeping in patterns as established.\n\nAt the same time, working increases into Lattice Pattern, increase every 6th row 18 (20, 16) times as follows:\n\nIncrease row (RS): K1, m1, work in patterns as established to the last stitch, m1, k1\u201485 (89, 97) stitches.\n\nWork even in patterns until row 126 has been completed.\n\nRow 127: Bind off 4 (5, 7) stitches, work in patterns as established to the end.\n\nRow 128: Bind off 4 (5, 7) stitches, purl to the end, keeping in patterns and colors as established\u201477 (79, 83) stitches. Place stitches on a holder.\n\n### YOKE\n\nWith RS facing, 32\" (81cm) medium sized circular needle, and D, k77 (79, 83) Left Sleeve stitches, k107 (121, 133) Front stitches, k77 (79, 83) Right Sleeve stitches, and k106 (120, 132) Back stitches\u2014367 (399, 431) stitches. Place marker and join for working in the round.\n\nPreparation round (RS)\n\nSize S: With B, k2, k2tog, k61, k2tog, k127, k2tog, k61, k2tog, [k26, k2tog] 3 times, k24\u2014360 stitches.\n\nSize M: With B, [k10, k2tog] twice, k45, m1, k15, m1, k111, m1, k24, m1, [k28, k2tog] 5 times, k30\u2014396 stitches.\n\nSize L: With B, k378 (to center back), m1, knit to end\u2014432 stitches.\n\nAll sizes: Place a marker every 18 stitches, using 20 (22, 24) markers around.\n\n#### Work Yoke Chart\n\nSize L only: Work rounds 1\u20134 and 8\u201311 of Collar chart, then continue as for all sizes.\n\nAll sizes: Work 44 rounds of Yoke chart, working all decreases as k2tog, and changing to shorter circular needle as necessary\u2014100 (110, 120) stitches.\n\n### COLLAR\n\nRemove all markers and shift stitches around the needle to begin the new round at the center back neck.\n\nSizes M and L only: With C, *k9 (10), k2tog; repeat from * around\u2014100 (110) stitches.\n\nAll sizes: Place a marker every 10 stitches, using 10 (10, 11) markers around.\n\nWork 43 rounds of Collar chart.\n\n#### Make Hem\n\nRemove markers and change to smaller circular needle.\n\nWith A, knit 1 round, purl 1 round, knit 8 rounds.\n\nBind off very loosely.\n\n### FINISHING\n\nWeave in all the ends. Sew side, sleeve, and underarm seams. Sew collar, sleeve, and body hems to the WS. Block lightly.\n\n#### Add Fur Trim to Body\n\nWith the largest size circular needle and fur yarn, cast on 280 (300, 320) stitches. Knit 1 row. Bind off. With A, whipstitch the fur trim to the body of the sweater, outlining the entire Fair Isle section, following photographs.\n\n#### Add Fur Trim to Sleeve Cuffs\n\nWith largest size circular needle and fur yarn, cast on 30 (30, 35) stitches. Knit 1 row. Bind off. With A, whipstitch the fur trim to the sleeve edge. Repeat for the second sleeve.\n\nVictory Fair Isle Pullover  \n~return to the beginning of the project~\n\nQuintessential Cable Pullover\n\n## QUINTESSENTIAL CABLE  \npullover\n\nPeplums are always a favorite fashion silhouette. This design uses an easy cable rib pattern for a sleek fit, but the real interest here is threefold: the very unusual way the sleeves are made, the flap edging made in graduated short to long lengths on the bottom and sleeve edges, and the bold oversize rib collar that enhances the neck shaping and shoulder detail. The sleeve shaping is entirely innovative, so study the diagram for a better understanding of this new sleeve style technique. Also note the stunning side seaming.\n\nreimagine it\n\nYou'll want to keep the unique sleeve construction and the flaps, so instead you may want to reimagine this pullover into a dress using a yarn with sequins or metallics. You can also leave off the collar for a sleek V-neck.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size S\n\nFINISHED MEASUREMENTS\n\nBust: 32\u00bd (37\u00bd, 41\u00bc)\" [82.5 (95, 105)cm]\n\nLength above flap joining to shoulder: 20\u00bd (23\u00be, 24\u00be)\" [52 (60, 63)cm]\n\nSleeve length from wrist to underarm: 17\u00bc (18, 18\u00be)\" [44 (45.5, 47.5)cm]\n\nNote: This pattern is written for size S; the M and L sizes are achieved by using larger-sized needles. Both stitch and row gauges are important.\n\nGAUGES\n\nFor S: 26 stitches and 29 rows = 4\" (10cm) on size U.S. 7 (4.5mm) needles in Cable and Rib pattern, slightly stretched\n\nFor M: 24 stitches and 26 rows = 4\" (10cm) on size U.S. 8 (5mm) needles in Cable and Rib pattern, slightly stretched\n\nFor L: 22 stitches and 24 rows = 4\" (10cm) on size U.S. 9 (5.5mm) needles in Cable and Rib pattern, slightly stretched\n\nTake time to check gauge.\n\nNotes: All decreases are made 1 stitch in from the edges as k2tog or p2tog as needed for continuity of pattern stitch. All increases are made 1 stitch in from the edges as m1k or m1p as needed for continuity of pattern stitch.\n\nMATERIALS\n\nPlymouth Select Merino Superwash (100% superwash fine merino wool), 3\u00bd oz (100g), 218 yd (196m); 9 (10, 11) skeins of #0027 Caraway\n\nSize U.S. 7 (8, 9) [4.5 (5, 5.5)mm] needles, or size needed to obtain gauge\n\nCable needle\n\nRemovable stitch markers\n\nStitch holders\n\nTapestry needle\n\n### Pattern Stitch\n\n2\/2 RC: Slip 2 stitches to the cn and hold in back, k2, k2 from cn.\n\n### Cable and Rib Pattern\n\nRow 1 (RS): P1, k1, p1, *k4, [p1, k1] twice, p1; repeat from * to the last 7 stitches, k4, p1, k1, p1.\n\nRow 2: K1, p1, k1, * p4, [k1, p1] twice, k1; repeat from * to the last 7 stitches, p4, k1, p1, k1.\n\nRow 3: P1, k1, p1, 2\/2 RC, [p1, k1] twice, p1; repeat from * to the last 7 stitches, 2\/2 RC, p1, k1, p1.\n\nRow 4: Repeat row 2.\n\nRepeat rows 1\u20134 for Cable and Rib pattern.\n\n### BACK\n\n#### First Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 6 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Second Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 7 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Third Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 8 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Fourth Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 9 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Fifth Flap\n\nWork same as Third Flap.\n\n#### Sixth Flap\n\nWork same as Second Flap.\n\n#### Seventh Flap\n\nWork same as First Flap, but do not cut yarn, turn.\n\n**Joining row (WS): K1,*p1, k1, p4, [k1, p1] twice, k1, p4, k1, p1, k2tog, repeat from * to the last 18 stitches, p1, k1, p4, [k1, p1] twice, k1, p4, k1, p1 k1. Place a marker at each end of the row\u2014127 stitches.\n\nRows 1\u201332: Working in Cable and Rib pattern, decrease 1 stitch each end of every RS row 15 times\u201497 stitches. Place a marker at each end of row 31.\n\nRows 33\u201336: Work in pattern as set. Place a marker at each end of row 36.\n\nRows 37\u201350: Work in pattern, increasing 1 stitch at each end of every RS row 7 times\u2014111 stitches.\n\nRows 51\u201382: Work in pattern, increasing 1 stitch each end of every 4th row 8 times\u2014127 stitches.\n\nRows 83\u201386: Work even in pattern.\n\n#### Armhole Shaping\n\nRows 87 and 88: Bind off 8 stitches, work in pattern to end\u2014111 stitches.\n\nRows 89\u201394: Work in pattern, decreasing 1 stitch at each end of every RS row\u2014105 stitches.**\n\nRows 95\u2013148: Work even in pattern.\n\nMark the center 41 stitches for the back neck and place all stitches on a holder.\n\n### FRONT\n\n#### First Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 6 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Second Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 5 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Third Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 4 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Fourth Flap\n\nCast on 19 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 3 times, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\n#### Fifth Flap\n\nWork same as Third Flap.\n\n#### Sixth Flap\n\nWork same as Second Flap.\n\n#### Seventh Flap\n\nWork same as First Flap, but do not cut yarn, turn.\n\nWork from ** to ** of Back.\n\nRow 95 (RS): Work in pattern over 52 stitches, join a second ball of yarn and bind off 1 stitch, continue in pattern over last 52 stitches.\n\nRow 96: Working both sides at the same time, work even in pattern.\n\nRows 97\u2013128: Work in pattern, decreasing 1 stitch at each neck edge every RS row 16 times\u201436 stitches each side.\n\nRows 129\u2013144: Work in pattern, decreasing 1 stitch at each neck edge every 4th row 4 times\u201432 stitches each side.\n\nRows 145\u2013148: Work even in pattern.\n\nPlace stitches on a holder.\n\n### SLEEVES (MAKE 2)\n\n#### Flaps\n\nCast on 10 stitches.\n\nRow 1 (RS): P1, k1, p1, k4, p1, k1, p1.\n\nRow 2: K1, p1, k1, p4, k1, p1, k1.\n\nRow 3: P1, k1, p1, 2\/2 RC, p1, k1, p1.\n\nRow 4: Repeat row 2.\n\nRepeat rows 1\u20134 twice more, then work rows 1\u20133. Cut yarn and leave stitches on needle.\n\nMake 2 more flaps the same as the first, but do not cut yarn after the third flap, turn.\n\nJoining row (WS): K1,* p1, k1, p4, k1, p1, k2tog; repeat from * to the last 9 stitches, p1, k1, p4, k1, p1, k1\u201428 stitches. Place a marker at each end of this row to mark wrist.\n\n#### Sleeve Shaping\n\nRow 1 (increase row): Work in pattern to the last stitch, m1, p1.\n\nRows 2\u2013119 (2\u2013109, 2\u2013105): Continue in pattern, repeating the increase row every 4th row 0 (5, 7) more times, then every 6th row 19 (14, 12) times\u201448 stitches.\n\nWork 6 (8, 8) more rows in pattern, ending with a RS row.\n\n#### Cap\n\nBind off 8 stitches, work in pattern to end\u201440 stitches.\n\nRows 1\u20136: Decrease 1 at the end of the next 3 RS rows\u201437 stitches.\n\nRows 7\u201349: Work 43 rows in pattern as established. Place a marker at each end of row 49 for the top of the sleeve.\n\nRows 50\u201392: Work 43 rows in pattern as established.\n\nRows 93\u201397: Increase 1 at the end of the next 3 RS rows\u201440 stitches.\n\nNext row (WS): Cast on 8 stitches, work in pattern to end\u201448 stitches.\n\n#### Sleeve Shaping\n\nWork 6 (8, 8) rows even in pattern.\n\nRow 1 (RS decrease row): Work in pattern to the last 3 stitches, decrease 1, p1.\n\nRows 2\u2013115 (2\u2013105, 2\u2013101): Repeat RS decrease row every 6th row 19 (14, 12) more times, then every 4th row 0 (5, 7) times\u201428 stitches.\n\nPlace a marker at each end of the last row for the wrist.\n\n#### Flaps\n\nNext Row (WS): K1,* P1, k1, p4, k1, p1, kfb; repeat from * to the last 9 stitches, p1, k1, p4, k1, p1, k1\u201430 stitches.\n\nRow 1 (RS): P1, k1, p1, 2\/2 RC, p1, k1, p1. Leave the remaining stitches on the needle or place on a holder.\n\nRow 2: K1, p1, k1, p4, k1, p1, k1.\n\nRow 3: P1, k1, p1, k4, p1, k1, p1.\n\nRow 4: Repeat row 2.\n\nRepeat rows 1\u20134 twice, then rows 1\u20133.\n\nBind off.\n\n*Join yarn to next stitch on a holder and work the next flap same as the first; repeat from * for the third flap.\n\n### COLLAR\n\nCast on 145 stitches.\n\nRepeat rows 1\u20134 of Cable and Rib pattern 9 times.\n\nMark the center 41 stitches for the back neck. Bind off.\n\n### WRIST BAND (MAKE 2, OPTIONAL)\n\nCast on 10 stitches.\n\nRow 1 (RS): P1, k1, p1, k4, p1, k1, p1.\n\nRow 2: K1, p1, k1, p4, k1, p1, k1.\n\nRow 3: P1, k1, p1, 2\/2 RC, p1, k1, p1.\n\nRow 4: K1, p1, k1, p4, k1, p1, k1.\n\nRepeat rows 1\u20134, 14 more times.\n\nBind off.\n\n### FINISHING\n\nWith the RS facing, join the 32 shoulder stitches of the Front and the Back using the 3-needle bind-off, bind off the 41 Back neck stitches, use the 3-needle bind-off to join the remaining 32 shoulder stitches.\n\nStarting at right edge marker at the top of the Sleeve, weave a length of yarn over the ribs and under the cables to the left marker, then back again to the right marker. With RS together, sew the center sleeve seam. Set the sleeve cap into the armhole and adjust the gathers at the top using the photo as a guide. Sew optional wrist bands between wrist markers. Sew the sleeve seam from the Joining\/dividing row markers to the underarm. Sew the side seams from the Joining row markers to the underarm, matching waist markers.\n\nPin the WS of the Collar to the RS of the neck opening, matching the center 41 stitches of the Collar to the back neck and the ends to the point of the V. Sew in place and tuck the seam to the WS.\n\nQuintessential Cable Pullover  \n~return to the beginning of the project~\n\nEdging Epilogue Dress\n\n## EDGING EPILOGUE  \ndress\n\nBeautiful silk yarn with sparkling sequins and an asymmetrical design create a romantic silhouette perfect for an evening out. The neckline is sexy but wearable. The bold scalloped bottom edge flows into a quiet eyelet stitch to make the front and back. A double overlay of the scalloped edges along the top creates off-the-shoulder glamour. Three Swarovzki crystals are sewn at each scallop for a bit more glitz. The double overlay is used again for a wide shoulder strap across the left shoulder.\n\nreimagine it\n\nIf you stay with the stunning yarn, consider lengthening the pattern to make it a dramatic dress or gown. Another way to go would be to use a cotton yarn (watch the gauge) for a more casual summer look.\n\nSKILL LEVEL\n\nTIME\n\nSIZES\n\nS (M, L), shown in size M\n\nFINISHED MEASUREMENTS\n\nWidth: 15\u00be (20\u00be, 26)\" [40 (53, 66)cm]\n\nLength from shoulder: 25\u00bc (27, 28\u00bd)\" [64 (68.5, 72)cm]\n\nLength from underarm: 18 (19, 20)\" [45.5 (48.5, 51)cm]\n\nGAUGE\n\n18 stitches and 22 rows = 4\" (10 cm) stockinette stitch on larger needles\n\nTake time to check gauge.\n\nMATERIALS\n\nTilli Tomas Disco Lights (100% silk), 3\u00bd oz (100g), 225 yd (206m); 4 (5, 7) skeins in Ocean Spray\n\n2 size U.S. 8 (5mm) 29\" (73cm) circular needles, or size needed to obtain gauge\n\nSize U.S. 6 (4.25mm) 29\" (73cm) circular needle\n\nWaste yarn or stitch holders\n\nTapestry needle\n\n### Scallop Pattern\n\n(multiple of 23 + 2)\n\nRow 1 (RS): K1, *k8, k2tog, yo, k1, p1, k1, yo, ssk, k8; repeat from * to the last stitch, k1.\n\nRow 2: K1, *p7, p2tog tbl, p2, yo, k1, yo, p2, p2tog, p7; repeat from * to the last stitch, k1.\n\nRow 3: K1, *k6, k2tog, k1, yo, k2, p1, k2, yo, k1, ssk, k6; repeat from * to the last stitch, k1.\n\nRow 4: K1, *p5, p2tog tbl, p3, yo, p1, k1, pl, yo, p3, p2tog, p5; repeat from * to the last stitch, k1.\n\nRow 5: K1, *k4, k2tog, p2, yo, k3, p1, k3, yo, k2, ssk, k4; repeat from * to the last stitch, k1.\n\nRow 6: K1, *p3, p2tog tbl, p4, yo, p2, k1, p2, yo, p4, p2tog, p3; repeat from * to the last stitch, k1.\n\nRow 7: K1, *k2, k2tog, k3, yo, k4, p1, k4, yo, k3, ssk, k2; repeat from * to the last stitch, k1.\n\nRow 8: K1, *p1, p2tog tbl, p5, yo, p3, k1, p3, yo, p5, p2tog, p1; repeat from * to the last stitch, k1.\n\nRow 9: K1, *k2tog, k4, yo, k5, p1, k5, yo, k4, ssk; repeat from * to the last stitch, k1.\n\nRow 10: K1, *p11, k1, p11; repeat from * to the last stitch, k1.\n\nRow 11: K1, *k11, p1, k11; repeat from * to the last stitch, k1.\n\nRow 12: Repeat row 10.\n\nRepeat rows 1\u201312 for pattern.\n\n### Eyelet Pattern\n\n(multiple of 23 + 2)\n\nRow 1 (RS): K1, *k4, k2tog, yo, k1, yo, ssk, k2, pl, k2, k2tog, yo, k1, yo, ssk, k4; repeat from * to the last stitch, k1.\n\nRow 2: K1, *p11, k1, p11; repeat from * to the last stitch, k1.\n\nRepeat rows 1 and 2 for pattern.\n\n### FRONT\n\nWith larger needles, cast on 71 (94, 117) stitches.\n\nKnit 2 rows.\n\nWork rows 1\u201312 of Scallop pattern twice, then work rows 1\u201310 once more.\n\nWork rows 1\u20132 of Eyelet pattern until piece measures 18 (19, 20)\" [45.5 (48.5, 51)cm] from the beginning, ending with a RS row.\n\n#### Armhole and Short Row Shaping\n\nRow 1 (WS): Bind off 2 (2, 4) stitches; work in pattern as established to the last 5 stitches, w&t.\n\nRow 2 and all RS rows: Work in pattern as established to end.\n\nRows 3 and 5: Bind off 1 stitch, work in pattern to 4 (5, 5) stitches before the last wrapped stitch, w&t.\n\nRow 7: Work in pattern to 4 (5, 5) stitches before the last wrapped stitch, w&t.\n\nRepeat row 7 every RS row 0 (1, 15) more times.\n\nNext row (WS): Work in pattern to 3 (4, 4) stitches before the last wrapped stitch, w&t.\n\nRepeat the last row every RS row 13 (14, 1) more times\u20148 (5, 8) stitches remain before the last wrapped stitch\n\nNext row (RS): Knit to the last 10 stitches, lifting the wraps and working them together with the wrapped stitch, work 10 stitches in pattern\u201467 (90, 111).\n\nNext row: Work 10 stitches in pattern, purl to end.\n\nNext row: Knit to the last 10 stitches, work 10 stitches in pattern.\n\nNext row: Work 10 stitches in pattern; purl to end.\n\nPlace all stitches on a holder.\n\n### BACK\n\nWork the same as the Front to Armhole and Short Row Shaping, ending with a WS row.\n\nRow 1 (RS): Bind off 2 (2, 4) stitches, work in pattern as established to the last 5 stitches, w&t.\n\nRow 2 and all WS rows: Work in pattern as established to end.\n\nRows 3 and 5: Bind off 1 stitch, work in pattern to 4 (5, 5) stitches before the last wrapped stitch, w&t.\n\nRow 7: Work in pattern to 4 (5, 5) stitches before the last wrapped stitch, w&t.\n\nRepeat row 7 every RS row 0 (1, 15) more times.\n\nNext row (WS): Work in pattern to 3 (4, 4) stitches before the last wrapped stitch, w&t.\n\nRepeat the last row every RS row 13 (14, 1) more times\u20148 (5, 8) stitches remain before the last wrapped stitch.\n\nNext row (WS): Purl to the last 10 stitches, lifting the wraps and working them together with the wrapped stitch, work 10 stitches in pattern\u201467 (90, 111).\n\nNext row: Work 10 stitches in pattern, knit to end.\n\nNext row: Purl to the last 10 stitches, work 10 stitches in pattern.\n\nPlace all stitches on a holder.\n\n### LARGE SCALLOP BAND (MAKE 2)\n\nWith larger needle, cast on 140 (186, 232) stitches.\n\nWork Rows 1\u201311 of Scallop pattern.\n\nRow 12: Purl, decreasing 26 stitches evenly across\u2014114 (160, 206) stitches.\n\nPlace all stitches on a holder.\n\n### SHOULDER SCALLOP BAND\n\nCast on 71 (71, 94) stitches.\n\nWork Rows 1\u201312 of Scallop pattern.\n\nWork 8 rows in stockinette stitch.\n\nPlace all stitches on a spare needle.\n\nMake a second Shoulder Scallop Band, working the same as the first to row 12.\n\nWith the RS facing up on both bands, place the first band behind the second band.\n\nNext row (RS): *Using the smaller needle, knit 1 stitch from the second band together with 1 stitch from the right band; repeat from * to end\u201471 (71, 94) stitches.\n\nKnit 5 rows. Bind off.\n\n### FINISHING\n\nPlace the shoulder stitches from the Front and Back (the 10 stitches worked in pattern) onto separate needles. Join the shoulder stitches using the 3-needle bind-off.\n\nPlace the remaining 114 (160, 206) stitches onto the larger needle, and the 114 (160, 206) stitches of one Large Scallop Band onto the smaller needle. With the RS facing up on both pieces, place the body needle inside the Large Scallop Band needle. Starting at the underam with the second larger needle, *knit 1 stitch from the band together with 1 stitch from the body; repeat from * around.\n\nWork 8 rows in stockinette stitch.\n\nPlace the stitches of the second Large Scallop Band onto the larger needle. Using the smaller needle, join the second band the same as the first band.\n\nWork 5 rows in stockinette stitch.\n\nBind off.\n\n### Armband\n\nWith the RS facing and smaller needle, starting at the underarm, pick up and k64 (70, 78) stitches evenly around the armhole. Knit 4 rows. Bind off knitwise on the WS.\n\nSew the side seams, starting at the uppermost row of the Scallop pattern, leaving slits at the lower edge. Leave the edges of the Large Scallop Bands unsewn.\n\nPin the ends of the Shoulder Scallop Band underneath the upper edges of the Front and Back, adjusting as necessary to fit. Sew in place. Weave in ends.\n\nEdging Epilogue Dress  \n~return to the beginning of the project~\n\n# ABBREVIATIONS\n\nInstructions for cables, bobbles and special increases\/decreases are included with the pattern.\n\ncn \u2022 cable needle\n\ndpn \u2022 double-pointed needle\n\nk \u2022 knit\n\nkfb \u2022 knit in front and back of the same stitch\n\nm1 \u2022 make 1 by lifting the bar between the last stitch and the next stitch, k1 tbl\n\nm1p \u2022 make 1 by lifting the bar between the last stitch and the next stitch and p1 tbl\n\np \u2022 purl\n\np2tog tbl \u2022 purl 2 stitches together through the back loops\n\npbf \u2022 purl in back and front of the same stitch\n\npm \u2022 place marker\n\nrnd \u2022 round\n\nRS \u2022 right side\n\ns2kp \u2022 slip 2 stitches as if to k2tog, k1, pass the 2 slipped stitches over the k1\n\ns2pp \u2022 slip 2 stitches as if to p2tog, p1, pass the 2 slipped stitches over the p1\n\nsk2p \u2022 slip 1 stitch knitwise, k2tog, pass the slipped stitch over the k2tog\n\nsm \u2022 slip marker\n\nssk \u2022 [slip 1 knitwise] twice, insert LH needle through the front of the 2 slipped stitches and k2tog tbl\n\nSt st \u2022 stockinette stitch: knit on the RS, purl on the WS\n\ntbl \u2022 through the back loop\n\ntog \u2022 together\n\nw&t \u2022 wrap and turn (see here for detailed instructions)\n\nWS \u2022 wrong side\n\nwyib \u2022 with yarn in back\n\nwyif \u2022 with yarn in front\n\nyo yarn over\n\n# KNITTING TECHNIQUES\n\n3-NEEDLE BIND-OFF\n\nWith the right sides of the two pieces facing each other and the needles parallel, insert a third needle knitwise into the first stitch of each needle, wrap the yarn around the needle as if to knit, as shown. Knit these two stitches together and slip them off the needles. Repeat to end, binding off at the same time.\n\nCABLE CAST-ON\n\nWith a slip knot on the left needle, insert the point of the right needle knitwise into the stitch on the left needle. Wrap the yarn around the right needle as if to knit. Draw the yarn through the first stitch to make a new stitch, but do not drop the stitch from the left needle. Transfer the new stitch to the left needle.\n\nFor each successive stitch to be cast on, insert the point of the right needle between the two stitches on the left needle. Wrap the yarn around the right needle as if to knit and pull the yarn through to make a new stitch, but again do not drop the stitch. Transfer the new stitch to the left needle. Repeat for the required number of stitches.\n\nDUPLICATE STITCH\n\nThread a tapestry needle with the desired color. *Draw the needle from the back to the front at the base of the V of the knitted stitch to be covered. Bring the thread over the right-hand leg of the stitch, down into the stitch in the row above it, over the left-hand leg and back down into the base of the same stitch. Repeat from * working from right to left and bottom to top until the desired area is covered.\n\nI-CORD\n\nWith two double-pointed needles, cast on 3 (4, 5) stitches. *Do not turn work. Slide the stitches to the other end of the needle, k3 (4, 5); repeat from * until cord measures desired length and fasten off.\n\nINTARSIA\n\nStep 1: On the knit side, drop the old color. Pick up the new color from under the old color and knit to the next color change.\n\nStep 2: On the purl side, drop the old color. Pick up the new color from under the old color and purl to the next color change.\n\nRepeat these two steps.\n\nKITCHENER STITCH\n\nTo set up: Thread a tapestry needle with a strand of yarn three to four times the length of the edge to be grafted together. Hold the working needles parallel to each other with the tips pointing in the same direction and the right sides facing up. Insert the tapestry needle purlwise into the first stitch of the front knitting needle, draw the yarn through, without dropping the stitch from the needle. Insert the tapestry needle knitwise into the first stitch on the back knitting needle, and draw the yarn through without dropping the stitch.\n\nTo graft: *Insert the tapestry needle knitwise into the first stitch on the front needle, and drop the stitch from the needle. Insert the tapestry needle purlwise into the next stitch on the front needle. Draw the yarn through and leave the stitch on the knitting needle. Insert the needle purlwise into the first stitch on the back needle. Drop the stitch from the knitting needle. Insert the tapestry needle knitwise into the next stitch on the back needle, and draw the yarn through without dropping the stitch.* Repeat from * to * until all live stitches have been grafted.\n\nCOLOR STRANDING \/ FAIR ISLE\n\nHolding colors left\n\nHolding colors right\n\nStranding one-handed: On the knit side, drop the working yarn. Bring the new color (now the working yarn) over the top of the dropped yarn and work to the next color change. Repeat these two steps when changing colors.\n\nOn the purl side, drop the working yarn. Bring the new color under the dropped yarn and work to the next color change. Repeat these two steps when changing colors.\n\nStranding two-handed: On the knit side, hold the working yarn in your right hand and the nonworking yarn in your left hand. Bring the working yarn over the top of the yarn in your left hand and knit with the right hand to the next color change.\n\nThe yarn in your right hand is now the nonworking yarn, the yarn in your left hand is the working yarn. Bring the working yarn under the nonworking yarn and knit with the left-hand needle to the next color change. Repeat these two steps when changing colors.\n\nOn the purl side, hold the working yarn in your right hand and the nonworking yarn in your left hand. Bring the working yarn over the top of the yarn in your left hand and purl with the right hand to the next color change.\n\nThe yarn in your right hand is now the nonworking yarn, the yarn in your left hand is the working yarn. Bring the working yarn under the nonworking yarn and purl with the left-hand needle to the next color change. Repeat these two steps when changing colors.\n\nSLIP STITCH (CROCHET)\n\nInsert the hook into a chain or other stitch, wrap the yarn around the hook, and then draw the yarn through both the stitch and the loop already on the hook.\n\nWHIPSTITCH\n\nThread a strand of yarn into a tapestry needle. Holding two knitted pieces, stitch over the edge where the seams align to create small, evenly spaced diagonal stitches.\n\nWRAP AND TURN (FOR SHORT ROWS)\n\nKnit to the specified stitch, and with yarn in back, slip this stitch onto the right needle. Bring the yarn from back to front, wrapping the stitch.\n\nSlip the same stitch back to the left needle.\n\nTurn the work. Bring the yarn to the front or back (depending on whether you are knitting or purling) to complete the wrap. Finish working the row.\n\n# RESOURCES\n\nAslan Trends\n\n8 Maple Street\n\nPort Washington, NY 11050\n\n(800) 314-8202\n\nwww.aslantrends.com\n\nThe BagSmith\n\n24000 Mercantile Road, Suite 7\n\nBeachwood, OH 44122\n\n(888) 879-7224\n\nwww.bagsmith.com\n\nBerroco, Inc.\n\n1 Tupperware Drive, Suite 4\n\nN. Smithfield, RI 02896-6815\n\n(401) 769-1212\n\nwww.berroco.com\n\nBlue Heron Yarns\n\n8737 Brooks Drive, #108\n\nEaston, MD 21601\n\n(410) 819-0401\n\nwww.blueheronyarns.com\n\nBlue Sky Alpacas, Inc.\n\nP.O. Box 88\n\nCedar, MN 55011\n\n(888) 460-8862  \nwww.blueskyalpacas.com\n\nCascade Yarns\n\n1224 Andover Park East\n\nTukwila, WA 98188\n\nwww.cascadeyarns.com\n\nCrystal Palace\n\n160 23rd Street\n\nRichmond, CA 94804\n\nwww.straw.com\n\nFyberspates\n\nUnit 1 + 6 Oxleaze Farm Workshops\n\nBroughton Poggs\n\nFilkins Lechlade\n\nGloucestershire GL7 3RB\n\nUnited Kingdom\n\n+44 1367 850880\n\nwww.fyberspates.co.uk\n\nHPKY (Hand Painted Knitting Yarns)\n\n7 Tiffany Lynn Court\n\nWentzville, MO 63385\n\n(636) 332-9931\n\nwww.hpkyllc.com\n\nJHB Buttons\n\n1955 South Quince Street\n\nDenver, CO 80231\n\n(800) 525-9007\n\nwww.buttons.com\n\nKollage Yarns\n\n3591 Cahaba Beach Road\n\nBirmingham, AL 35242\n\n(888) 829-7758\n\nwww.kollageyarns.com\n\nLion Brand Yarn\n\n34 West 15th Street\n\nNew York, NY 10011\n\nwww.lionbrand.com\n\nMadelinetosh\n\n7515 Benbrook Parkway\n\nBenbrook, TX 76126\n\n(817) 249-3066\n\nwww.madelinetosh.com\n\nPlymouth Yarn Company, Inc.\n\n500 Lafayette Street\n\nBristol, PA 19007\n\n(215) 788-0459\n\nwww.plymouthyarn.com\n\nPrism Yarns\n\n3140 39th Avenue North\n\nSt. Petersburg, FL 33714\n\nwww.prismyarn.com\n\nRowan Yarns\n\nGreen Lane Mill\n\nHolmfirth\n\nWest Yorkshire, England\n\nHD9 2DX\n\n+44 1484 681881\n\nwww.knitrowan.com\n\nSkacel Schulana\n\n(800) 255-1278\n\nwww.skacelknitting.com\n\nTahki Stacy Charles, Inc.\n\n70\u201360 83rd Street, Building #12\n\nGlendale, NY 11385\n\n(718) 326-4433\n\nwww.tahkistacycharles.com\n\nTilli Tomas\n\n(617) 524-3330\n\nwww.tillitomas.com\n\n# ACKNOWLEDGMENTS\n\nThis unconventional book required a publisher with courage, imagination, and foresight. That publisher is Potter Craft, an imprint of Penguin Random House, and I want to thank the entire team for their ongoing support and great work.\n\nMany thanks to:\n\nMy editors Betty Wong, who made my vision a reality, and Caitlin Harpin, who skillfully and beautifully made it all come together.\n\nStephanie Huntwork, art director, for the beautiful look she created that was simply perfect for my designs.\n\nRose Callahan, photographer, whose talent and enthusiasm is elegantly expressed in these gorgeous photos, and her assistants Demetrius Fordham and Bowen Rodkey.\n\nMeg Goldman, stylist, whose great sense of style led to a harmonious wardrobe that beautifully blended with my knitted garments, and assistant Tara Ferri for all her hard work.\n\nYuko, hair and makeup artist, who made our beautiful models, Sanita and Nadine, look even more gorgeous.\n\nMy intrepid team of knitters, who in this book truly outdid themselves, especially dealing with the reimagined and off-the-beaten-track designs that looked challenging, but eventually, when they understood the premise, admitted were not much more difficult to execute than traditional knitting.\n\nMy talented and \"always there\" knitters Nancy Henderson, Jo Brandon, Eileen Curry, Mary Taylor, and Dianne Weitzul, along with Sammi Sherwin, Eva Wilkins, Marla Fialkow, Dana Vessa, and Megan Hand. I cannot imagine a better group of talented people. And to Eve Eng for easy-to-follow instructions.\n\nMy supportive friends Debbie Rufrano, Ashley Panaro, Emily Brenner, Christine Farrow, and Ryan Brandon.\n\nThank you as well to Cascade Yarns and to all the yarn manufacturers for their support and to Faryl Robin at Seychelles for the shoes, Karen Ko at K2o and Joan Goodman at Pono for their amazing jewelry, and to Jennifer Ouellette for the hats.\n\nAnd last, but certainly not least, thank you to Melody Remillard and the entire staff of Grey Towers National Historic Site for their hospitality and for allowing us to photograph the pieces at this beautiful estate. If you're ever in the Milford, Pennsylvania, area you must visit this magnificent and historic wonder.\n\n# INDEX\n\nNote: Page numbers in italics indicate projects.\n\nAbbreviations, See also Icon definitions\n\nAdvanced projects\n\nabout: icon for\n\nButtons and Bows Manteau, 4.1\n\nChaos Couture Pullover, 3.1\n\nDressage Pony Poncholette, 4.1\n\nGlory Rising Circle Cardigan, 2.1\n\nVictory Fair Isle Pullover, 4.1\n\nBeginner projects\n\nabout: icon for\n\nOn the Block Topper, 1.1\n\nReckoning Rectangles Shawl, 1.1\n\nWeekend Warrior Wraparound, 2.1\n\nBind-off, 3-needle, bm1.1\n\nBlock topper, 1.1\n\nBook overview\n\nBraided Vitality Pullover, 3.1\n\nButtons and Bows Manteau, 4.1\n\nCable cast-on\n\nCable pullover, 4.1\n\nCape and topper. See also Coats and such\n\nJe Ne Sais Quoi Cape, 1.1\n\nOn the Block Topper, 1.1\n\nWeekend Warrior Wraparound, 2.1\n\nCardigans\n\nGlory Rising Circle Cardigan, 2.1\n\nNouveau Wrap Cardigan, 2.1\n\nWelted Button Tuck Cardi, 1.1\n\nCastle tunic, 1.1\n\nCast-on, cable\n\nChaos Couture Pullover, 3.1\n\nCoats and such. See also Cape and topper\n\nButtons and Bows Manteau, 4.1\n\nDressage Pony Poncholette, 4.1\n\nRoyal Lace Coat with Hood, 1.1\n\nColor stranding \/ Fair Isle\n\nCool construction projects\n\nabout: overview of\n\nDirectional Vest, 2.1\n\nOn the Edge Dress, 2.1\n\nGlory Rising Circle Cardigan, 2.1\n\nNouveau Wrap Cardigan, 2.1\n\nShape-Shifter Vest, 2.1\n\nWeekend Warrior Wraparound, 2.1\n\nCrisscross Weave Tank, 3.1\n\nThe Deep End Shawl, 1.1\n\nDirectional projects\n\nabout: overview of\n\nThe Deep End Shawl, 1.1\n\nJe Ne Sais Quoi Cape, 1.1\n\nOn the Block Topper, 1.1\n\nReckoning Rectangles Shawl, 1.1\n\nRenaissance Castle Tunic, 1.1\n\nRoyal Lace Coat with Hood, 1.1\n\nWelted Button Tuck Cardi, 1.1\n\nDirectional Vest, 2.1\n\nDrapelette, pixilated weave, 3.1\n\nDressage Pony Poncholette, 4.1\n\nDresses\n\nButtons and Bows Manteau, 4.1\n\nOn the Edge Dress, 2.1\n\nEdging Epilogue Dress, 4.1\n\nSpring Forward Dress or Tunic, 4.1\n\nEdging Epilogue Dress, 4.1\n\nEtiquette Unchained Pullover, 3.1\n\nFair Isle \/ color stranding\n\nFair Isle pullover, 4.1\n\nGlory Rising Circle Cardigan, 2.1\n\nIcon definitions\n\nI-cord, 1.1, bm1.1\n\nIntarsia\n\nIntermediate projects\n\nabout: icon for\n\nBraided Vitality Pullover, 3.1\n\nCrisscross Weave Tank, 3.1\n\nThe Deep End Shawl, 1.1\n\nDirectional Vest, 2.1\n\nOn the Edge Dress, 2.1\n\nEdging Epilogue Dress, 4.1\n\nEtiquette Unchained Pullover, 3.1\n\nJe Ne Sais Quoi Cape, 1.1\n\nNouveau Wrap Cardigan, 2.1\n\nPixilated Weave Drapelette, 3.1\n\nQuintessential Cable Pullover, 4.1\n\nRenaissance Castle Tunic, 1.1\n\nRoyal Lace Coat with Hood, 1.1\n\nShape-Shifter Vest, 2.1\n\nSpirits Fly Pullover, 4.1\n\nSpring Forward Dress or Tunic, 4.1\n\nWelted Button Tuck Cardi, 1.1\n\nJe Ne Sais Quoi Cape, 1.1\n\nKitchener stitch\n\nManteau, buttons and bows, 4.1\n\nNouveau Wrap Cardigan, 2.1\n\nOn the Block Topper, 1.1\n\nOn the Edge Dress, 2.1\n\nPixilated Weave Drapelette, 3.1\n\nPoncholette, dressage pony, 4.1\n\nPullovers\n\nBraided Vitality Pullover, 3.1\n\nChaos Couture Pullover, 3.1\n\nDressage Pony Poncholette, 4.1\n\nEtiquette Unchained Pullover, 3.1\n\nPixilated Weave Drapelette, 3.1\n\nQuintessential Cable Pullover, 4.1\n\nSpirits Fly Pullover, 4.1\n\nVictory Fair Isle Pullover, 4.1\n\nQuintessential Cable Pullover, 4.1\n\nReckoning Rectangles Shawl, 1.1\n\nRenaissance Castle Tunic, 1.1\n\nResources\n\nRoyal Lace Coat with Hood, 1.1\n\nShape-Shifter Vest, 2.1\n\nShawls\n\nThe Deep End Shawl, 1.1\n\nReckoning Rectangles Shawl, 1.1\n\nShort rows, wrap and turn for\n\nSkill levels\n\nSlip stitch (crochet)\n\nSpirits Fly Pullover, 4.1\n\nSpring Forward Dress or Tunic, 4.1\n\nStitches. See Techniques\n\nStitch impact projects\n\nabout: overview of\n\nButtons and Bows Manteau, 4.1\n\nDressage Pony Poncholette, 4.1\n\nEdging Epilogue Dress, 4.1\n\nQuintessential Cable Pullover, 4.1\n\nSpirits Fly Pullover, 4.1\n\nSpring Forward Dress or Tunic, 4.1\n\nVictory Fair Isle Pullover, 4.1\n\nStranding, color\n\nTank, crisscross weave, 3.1\n\nTechniques\n\ncable cast-on\n\ncolor stranding \/ Fair Isle\n\nduplicate stitch\n\nI-cord, 1.1, bm1.1\n\nintarsia\n\nKitchener stitch\n\nslip stitch (crochet)\n\n3-needle bind-off\n\nwhipstitch\n\nwrap and turn (for short rows)\n\n3-needle bind-off\n\nTime (clock) icons defined\n\nTopper, 1.1\n\nTunics\n\nRenaissance Castle Tunic, 1.1\n\nSpring Forward Dress or Tunic, 4.1\n\nVests\n\nDirectional Vest, 2.1\n\nShape-Shifter Vest, 2.1\n\nVictory Fair Isle Pullover, 4.1\n\nWeekend Warrior Wraparound, 2.1\n\nWelted Button Tuck Cardi, 1.1\n\nWhipstitch\n\nWoven weave projects\n\nabout: overview of\n\nBraided Vitality Pullover, 3.1\n\nChaos Couture Pullover, 3.1\n\nCrisscross Weave Tank, 3.1\n\nEtiquette Unchained Pullover, 3.1\n\nPixilated Weave Drapelette, 3.1\n\nWrap and turn (for short rows)\n\nWraparound, 2.1\n\n# ABOUT THE AUTHOR\n\nNicky Epstein is the beloved knitwear designer and best-selling author of numerous books, including Knitting Block by Block, Knitting in Circles, the Knitting on the Edge series, Knitting on Top of the World, and Nicky Epstein's Knitted Flowers. Must-haves in the libraries of designers and knitters alike, her award-winning knitting and crochet books range from highly original resource books to knitting\/travel books to collections for Barbie and 18\" dolls. She is a three-time winner of the National Independent Book Publisher's Award for Best Craft Book of the Year.\n\nHer innovative and fashion-forward designs have appeared in every major knitwear magazine, in museum exhibits, and on television. She loves to share her expertise and enthusiasm for knitting with countless fans around the world and has traveled and taught throughout the United States and in England, Australia, New Zealand, Italy, Canada, Argentina, Uruguay, and France, gaining a loyal following. She hosts popular knitting tours and recently launched the Nicky Epstein Knitting Club. A line of her designs and videos can be found on her website: www.nickyepstein.com.\n\nNicky lives in New York City but is constantly on the go, sharing her love of knitting with a fun-loving army of fans.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nThis is a work of fiction. Names, characters, places, and incidents are either products of the author's imagination or, if real, are used fictitiously.\n\nCopyright \u00a9 2010 by PJ Haarsma\n\nAll rights reserved. No part of this book may be reproduced, transmitted, or stored in an information retrieval system in any form or by any means, graphic, electronic, or mechanical, including photocopying, taping, and recording, without prior written permission from the publisher.\n\nFirst electronic edition 2010\n\nThe Library of Congress has cataloged the hardcover edition as follows:\n\nHaarsma, PJ.  \nThe softwire : awakening on Orbis 4 \/ PJ Haarsma. \u20141st ed.  \np. cm. \u2014 (The softwire)  \nSummary: As the Scion's guardian, Johnny Turnbull is expected to begin training as a Space Jumper, a role he promised his girlfriend, Max, he would never take on, and which might not be enough to save his sister and friends when Orbis is threatened.  \nISBN 978-0-7636-2712-6 (hardcover)  \n[1. Computers \u2014Fiction. 2. Space and time \u2014Fiction. 3. Mercenary troops \u2014Fiction. 4. Science fiction.] I. Title. II. Title: Awakening on Orbis 4. III. Title: Awakening on Orbis four. IV. Series.  \nPZ7.H111325Soi 2010  \n[Fic] \u2014dc22 2009032482\n\nISBN 978-0-7636-5238-8 (electronic)\n\nCandlewick Press  \n99 Dover Street  \nSomerville, Massachusetts 02144\n\nvisit us at www.candlewick.com\n\n#\n\n\"Stop it!\" I begged.\n\n\"I can't,\" Theylor whispered. \"Ketheria _must_ suffer this.\"\n\n\"When will it be over?\"\n\nMy sister's body convulsed while suspended over a thick block of chrome deep within the Keepers' lair on Orbis 1. I turned away.\n\n\"This is one of the fourteen steps of the awakening,\" Theylor reminded me. \"We have discussed this.\"\n\n\"But look at her. Her eyes are going to pop out of her head!\"\n\n\"The _glow_ is doing that. Her eyes will remain firmly inside their sockets,\" he assured me. \"That is nothing more than an illusion.\"\n\n_Glow_ was a clumsy description of what was happening to Ketheria. Her skin was shining with a lustrous golden light that pulsed brighter, not with each heartbeat but with some otherworldly measure that I was not privy to. I stood by, helpless, and watched as the metal block she floated above refused to absorb the glow, tossing it back while her body deflated with each throb of light. During one convulsion, Ketheria's head lobbed sideways and her eyes seemed to focus on mine.\n\n\"Ketheria!\" I called out, but her vacant stare just bore straight through me. I don't think she had a clue that I was even in the room with her.\n\nI felt Theylor place his slender hand on my shoulder. I turned and looked into the eyes of his left head. I had grown to trust Theylor over the last three rotations on the Rings of Orbis, and I searched his bluish face now to find any justification for my sister's suffering.\n\n\"The Nagools have been waiting their whole lives for this moment,\" he said. \"They will do everything to make the Scion's \u2014\"\n\n\" _My sister,_ Theylor. She's my _sister,_ nothing more,\" I corrected him. \"We traveled from Earth to work on these rings just like the zillions of other aliens who come here every rotation to do the exact same thing. Once our debt is paid, we get to start a new life of our own \u2014 as _Citizens._ That was the deal. Not this! The _Scion_? The _Tonat_? None of it makes sense to me, Theylor. I don't want any of it, and I'm certain Ketheria doesn't, either. You guys are the ones calling her the Scion. That word doesn't mean anything to me.\"\n\nTheylor bowed his head before he continued. \"Even if you do not want this to happen, you cannot deny that it _is_ happening. This is self-evident.\" Theylor motioned toward Ketheria. \"Yet for some reason, you resist believing what you see right before your very eyes. Your sister _is_ the Scion, Johnny Turnbull. I assure you that the Nagools will do everything possible to make her awakening a painless experience. I do not understand your anger.\"\n\n\"I'm not angry,\" I whispered. \"I'm confused.\"\n\n\"Some things are easier to accept if you simply trust the Universe.\"\n\nThat was easy for him to say. He wasn't a knudnik.\n\n\"Come now,\" he added. \"I must get you to the spaceport. There is not much time before the Orbis 4 shuttle launches. Your new work rule has already started.\"\n\nDespite the fact that everyone believed Ketheria was the new Scion, the Trading Council insisted that we finish our fourth rotation of indentured service, and the Keepers and the Nagools did not argue. They believed that the Scion had to awaken along the path that the Universe had predicted for him or her, no matter how dangerous that path may be. It was a small miracle that they even let me follow Ketheria to Orbis 1 once she got sick. I had hoped she might come out of her awakening and we could travel to Orbis 4 together, but that did not seem likely now.\n\n\"Please,\" I begged. \"I was hoping for a little more time.\"\n\n\"I am sorry. I have done everything I could just to let you stay this phase.\"\n\n\"I've been here a whole phase?\"\n\nTheylor nodded, and I remembered the moment when I had found Ketheria in the middle of one of her spells. It was right before we were all about to leave to meet our new Guarantor on Orbis 4. Ketheria had started doodling those little spirals on the walls while everyone else was packing. Soon afterward she slipped into the catatonic state that now consumed her. Usually Ketheria came out of one of her spells by the end of a cycle, but this time she hadn't. Instead, the glow had started. I told Vairocina, my friend inside the central computer, the moment it began, and she contacted Theylor immediately. Soon Theylor and an army of Nagool masters converged upon Ketheria. As they shuttled both her and me to Magna on Orbis 1, Theylor informed me that the glow was the Source working through her, making connections with the rest of the universe. Max, Theodore, and everyone else were shipped off to Orbis 4.\n\n\"Johnny,\" Theylor called. \"I'm sorry, but it is time.\"\n\nReluctantly, I turned from my frozen vigil and followed Theylor out of the room, entrusting my sister to the Nagools. When I thought about waking up the next cycle not knowing a thing about my sister's condition, I felt as if a Neewalker had clamped his hands around my throat, trapping the air inside me. It scared me and I hated it.\n\nTheylor paused next to one of the many treelike pillars that supported the largest cavern of the Keepers' lair located beneath the city of Magna and waited for me. I looked up to where the deep blue stone pillars made contact with the roof. Bands of yellowish light oscillated from the top of the support with a beat that was oddly reminiscent of the glow. I then followed Theylor as he navigated around huge pools of black water that glittered from deep below the surface.\n\nA small domed craft waited on a rail of shorter pillars next to a platform. I followed Theylor aboard, and we sat in silence during the short trip to the surface. It gave me time to think. What had happened? Why Ketheria? No one had any answers for me, but that didn't mean no one knew. I had heard stories of other Scions and the horrible fates they met. No matter what part of the universe they came from, new Scions were always persecuted by one group or another, tested until they broke. I had also heard tales of the Tonat, the guardian entrusted to protect the Scion once the awakening was complete. That's who they wanted me to be. But if the Tonat was so important, why was I leaving my sister behind? It didn't make sense, but then when had my life on the rings ever made sense? No one ever explained anything here. Three rotations had taught me that most people on the Rings of Orbis protected their knowledge more than they did an Orodi Orb.\n\nOnce we disembarked from the small craft, I followed Theylor up a wide staircase that led to two metal doors, scuffed and marred by eons of use. At the top of the stairs, Theylor paused and turned to me. \"Others may have come to see the Scion,\" he said, as if it were a warning.\n\n\"I thought no one knew where Magna was located.\"\n\n\"Idolatry has a unique way of bringing light to the blind,\" he replied.\n\nWhen Theylor pushed back the thick doors, the glassy glow from a distant star burnished my eyes, and faster than my pupils could contract, a throng of aliens burst upon us.\n\n\"Who are these people, Theylor?\"\n\n\"Worshippers,\" he replied, holding up his hand to the crowd. The effect seemed to push the people back, which allowed us to move forward. \"I did not expect to see so many. I am afraid news of the Scion has spread quickly. This is not good.\"\n\n\"Why is the Scion so important to them?\"\n\n\"It has been a very long time since a Scion has been discovered. In fact, most people thought it was no longer possible. Your sister is their last hope.\"\n\nThere must have been thousands of people gathered there. Every one of them seemed to be whispering something at me. Hushed pleas called to me from every side as we pushed through the crowds.\n\n\"They worship my sister?\"\n\n\"They worship the Scion,\" he said, as if Ketheria was a separate entity entirely. \"And some even worship the Tonat.\"\n\n\"I'm not the Tonat, Theylor. The Trust said _I_ have to make that choice, and I don't want to be a Space Jumper. In order to be the Tonat, I have to be a Space Jumper.\"\n\n\"I believe the Trust merely presented you with that choice as a gesture.\"\n\n\"A gesture of what?\"\n\n\"To appease your fierce need to control your own existence. I wonder how much choice you actually have in this matter.\"\n\n\"What does that mean, Theylor? It _is_ my choice.\"\n\nBut Theylor did not respond. It frustrated me to be fed these cryptic answers all the time.\n\n\"If this is so important, why won't you tell me anything else?\" I shouted as more people pushed in on us, but Theylor did not answer. His attention was now on the crowd. More and more people rushed toward us, and the crush was beginning to smother me. One alien tugged at my vest, another simply rubbed her hands over me, while another squawked in my face. Theylor tried to force them back, but that only created an opening for more to pour into.\n\n\"Theylor!\"\n\n\"I'm trying,\" he grunted.\n\nThe crowd now engulfed me. I could no longer see the sky and had lost sight of Theylor in a sea of wanting hands.\n\n\"I have nothing to give you!\" I shouted. \"I can't help you.\"\n\nThen someone struck me. It was a quick blow to my forehead, but still painful.\n\n\"Death to the Tonat!\" the alien screamed, but the crowd turned on the assailant. At least a dozen worshippers descended on my attacker and delivered blows much worse than the one he had given me.\n\n\"Stop!\" I screamed at them, but the crowd swallowed the brawling aliens. Then I saw the flash of a Zinovian Claw, a nasty little weapon that was often equipped with a poison cartridge. \"Look out!\" I screamed, pointing at the weapon. The effect was instant. The same punishment dealt to the first detractor was unleashed on the claw-toting alien. Fights were now breaking out between different groups, and my body flowed helplessly with the energy of the crowd.\n\n\"Theylor! Help me!\"\n\nSuddenly the crowd blew apart. Bodies wrenched away from me like metal shavings pulled helplessly toward a huge magnet. Theylor stood in the opening, flanked by two Space Jumpers, who immediately descended upon me. Theylor moved calmly, but I can't say the same for the mass of worshippers.\n\n\"Do you wish for war, Keeper?\" a shocked Citizen shouted.\n\n\"We will crush you!\" another added as the effect of seeing the heavily armed mercenaries rippled through the crowd.\n\n\"This is a taste of your life now, whether you accept your fate or not,\" Theylor whispered to me. \"Just imagine what this is going to be like for Ketheria. She is going to need you.\"\n\n\"I won't do it, Theylor. This is not our battle. I don't know how any of this happened. I'm just a kid from Earth.\"\n\nOne of the Space Jumpers grunted.\n\n\"But she is your _sister,_ \" Theylor pleaded.\n\n\"And I will protect her, but it has to be in my own way.\"\n\nTheylor breathed deeply. I knew he didn't like my answer. \"You are naive. Your actions risk your life, they risk your sister's life, and they might even risk every life in this universe. We will talk of this again,\" he said, motioning to the Space Jumpers at my side. And then I was gone.\n\nA moment later, I was standing in a field, rubbing the smell of sweat-soaked socks out of my nose. The Space Jumper to my left had released me. I looked up and saw the city of Nacreo gleaming in the distance.\n\n\"Hello, Johnny Turnbull. Or do you prefer JT?\" a voice called out.\n\nI turned; a tall, strong-looking humanoid was standing next to one of the Keepers' fliers. He was wearing a heavy-looking overcoat flung back to expose his tall black boots. When I noticed the small stalactites of flesh that hung from his jawline, I suddenly realized that I had seen this alien before. At Odran's! The dinner party! This alien was a Trading Council member.\n\n\"You're the \u2014 the \u2014\" I stammered.\n\n\"Hach. I believe we've met once before. I am your new Guarantor.\"\n\nMy first Guarantor was a weaselley little rat named Weegin. The Keepers replaced him with Odran, a vile creature whose scruples were worse than his appearance (and trust me, his appearance was disgusting). My third Guarantor was my friend, a human named Charlie Norton. A cycle never passed when I did not think about him. But Hach was nothing like my previous Guarantors, even Charlie. I'd never forget the way he had confronted his fellow Trading Council member for insulting us at Odran's party. Hach stood confidently, with his hands cupped, waiting patiently for my reply.\n\n\"Hello. Yeah \u2014 JT. That's what my friends call me,\" I told him.\n\n\"I look forward to being your friend, then, JT. If you'll follow me, the Keepers have arranged transportation to the spaceport.\" Hach motioned toward the flier. It was nothing more than a wheel with a cockpit near the center. \"Normally I have a driver, but I couldn't resist flying one of these things.\"\n\nThe Space Jumper to my right pushed me toward my new Guarantor. The unexpected force tripped me up, and I fell to my knees. Hach spun around and unleashed a small staff from beneath his cloak. I had the keen sense to duck as he thrust his right arm in front of him, unfolding the device like a double-sided whip. In one complete motion, Hach caught the Space Jumper around the ankles and pulled. The Space Jumper toppled to the ground.\n\n\"What's it like from down there?\" Hach hissed at the fallen Space Jumper.\n\n\"Wow,\" I mumbled. I had never seen anyone take out a Space Jumper before. I didn't think it was even possible.\n\n\"Come, JT,\" Hach ordered, and turned toward the flier. Both Space Jumpers glared at me as I walked past. The air around them seemed to ripple as the light folded in on them, and then they were gone, jumping back to wherever they had come from.\n\nI had seen one of the Keepers' fliers on Orbis 3. It was basically a large wheel piloted from a cockpit positioned off center and lower to the ground. This cockpit remained in its position as the wheel spun around. Hach took his seat at the controls while I climbed in and sat up behind him. I snuggled in, and the seat conformed to my body.\n\nHach uplinked to the control panel using the neural port embedded behind his left ear, and the cockpit closed in around us. The clouded glass slowly became transparent. I was seated slightly higher than Hach, so I had an unobstructed view of what was in front of me. The craft rolled forward and then lifted into the air, picking up speed as the huge wheel spun faster and faster. Soon it was spinning so fast that I could barely see it except for the slight distortion it made in my vision.\n\n\"Comfortable?\" Hach asked, his voice soft in my ears, amplified through the smart material behind my head.\n\n\"Um, yes,\" I said. \"Thank you.\" I was not used to having a Guarantor care about my well-being. (Except for Charlie, of course, but that was different.) Most of the Citizens on the Rings of Orbis treated knudniks with the same respect they gave the dirt between their toes, if they had toes. I wasn't going to let Hach's seemingly open manners go to waste.\n\n\"Hach, may I ask you something?\" I said.\n\n\"You may.\"\n\n\"How did you become our Guarantor?\"\n\nThere was a pause before Hach answered. History had taught me that this sort of pause was usually followed by a lie.\n\n\"You were given to me.\"\n\n\"By whom?\"\n\n\"Your last Guarantor.\"\n\n\"Charlie?\"\n\n\"Yes.\"\n\nI didn't know what to say. When could Charlie have entrusted us to anyone? The attack had left him unconscious and he'd never come out of it. During our first rotation back on Orbis 1, Madame Lee had murdered Max's first Guarantor, Boohral, and since he had not willed his knudniks to anyone in time, the Keepers redistributed Max and the others (much to the anger of Boohral's brood). I could only assume the same would have happened after Charlie died.\n\n\"When?\" I said. \"How?\"\n\n\"I am not at liberty to say,\" he replied. \"Policy of the Trading Council.\"\n\nAnd there it was again, the same nonanswer to my most important questions. Why couldn't anyone on these stupid rings just tell the truth? Every response was a diversion.\n\n\"Can I ask another one?\"\n\n\"How many do you plan on asking?\"\n\n\"I \u2014 I . . . don't know.\"\n\n\"You may ask me three more questions. I'm sure that's all we'll have time for before we reach the city of Nacreo.\"\n\nThree questions? I had a million, and Hach was a Trading Council member. Didn't they know everything? I watched the city growing in front of me as the flier sped toward the spaceport. Three questions. Better start now.\n\n\"Do you know what's happening to my sister?\"\n\n\"Of course. Everyone does. That's an odd waste of a question since I'm sure you, too, are aware of that answer.\"\n\n_But that's not what I meant!_ I knew about the awakening. I knew it was some sort of transformation. That's how the Keepers explained it, but I didn't believe it. There was no way my sister or I had anything to do with the salvation of the universe.\n\n\"No, I meant what does it _mean_?\"\n\n\"It means your sister is the Scion and you are the Tonat.\"\n\n\"I know that, too.\" I resisted the urge to call him a split-screen. \"You don't understand. No one knew I was coming to Orbis. I can't possibly have anything to do with all of this. Our parents smuggled us onto the _Renaissance_ in hopes of a better life, away from Earth. That's all. Did _you_ know that?\"\n\n\"I did.\"\n\n\"So then I don't get it. I don't understand how my sister is the Scion or how I'm supposed to be the Tonat. What's going to happen to us if what they say is true?\"\n\n\"Oh, it _is_ true, but that, I'm afraid, is question number four, and we've arrived at the spaceport. Get ready for a jolt. I can't land these things as well as the Keepers can.\"\n\n#\n\nOnce inside the spaceport, I gazed at the starships nestled in their docking bays as I waited for Hach on a crystal bench. _Is this their first time to the Rings of Orbis?_ I wondered. _Are they here to do business with the Trading Council, or are their bellies filled with aliens looking for a better life?_\n\n\"Keep going if _that's_ what you're looking for,\" I whispered to them.\n\nI tried to remember what the _Renaissance_ had looked like sitting in the same spot after its one-way journey from Earth, but I could not recall ever seeing our ship docked in the spaceport. What would have happened if we had never arrived? I suddenly wondered. Where would I be right now if I had listened to Switzer and helped him escape with the _Renaissance_? Would I have been able to pull it off? I didn't even know I was a softwire back then.\n\nI watched the largest ship unhook and push back, moving like a Samiran in the crystal-cooling tank. The nose of the starcraft pushed away from me as if sniffing out the open space behind it. Suddenly I sprang from the bench and rushed to the window. I wanted to be on that ship! At that very moment, as I pressed against the glass, I wanted nothing more than to feel the thrust of the engines against my chest. I couldn't explain why, but my stomach surged with a huge gulp of regret as the starship disappeared into deep space. Then Hach called for me, and I followed him to our ship.\n\nInside the shuttle to Orbis 4, the drill was familiar. I was placed in a lower cabin while Hach sat in an area reserved for Citizens. _Where would the Scion sit?_ I wondered as I glanced at the empty plastic benches. _If I had chosen to be the Tonat, would I still be sitting here?_\n\n_\"Your actions risk your life, they risk your sister's life . . .\"_\n\nI shook Theylor's dire words from my head and tried to focus on the activity visible through the portal that lined the shuttle's cabin, but the huge passenger shuttles tethered to the spaceport only reminded me of my friend Toll. Would he have approved of my choice? Of course not. He would have insisted I follow in my father's footsteps \u2014 a Space Jumper whom I had never met and whose origin was still a mystery to me. Argh! So many things pointed to a conclusion that I was simply not willing to accept. I was _not_ just some kid from Earth. Neither was Ketheria, _but why_? How did this happen?\n\nEven if it was true, I still refused to become a Space Jumper. I had my reasons, and the most important one was waiting for me on Orbis 4. I had made a promise to Max that I would _never_ become a Space Jumper. I knew full well how much she detested Space Jumpers, and it would kill me to have her think of _me_ like that. I _was_ going to watch over my sister, but I would protect Ketheria in my own way, no matter how difficult that proved to be.\n\nAfter the shuttle pulled away, I must have fallen asleep with my face pressed against the glass. I woke to find Hach standing over me.\n\n\"Get up,\" he ordered. \"We're here.\"\n\nI wiped the drool off the window and looked out at the spaceport of Orbis 4. The ring was in shadow, and the inky darkness was pierced by a multitude of lights that glittered throughout the port. Red beacons flashed across the sky, alerting incoming ships to the tallest buildings inside the port, while glaring white spotlights interrogated the docked spaceships, exposing the fatigue of deep-space travel. Gold, orange, and green lights advertised the locations of the different trading chambers within the port, while frosted blue lights wove their way through the different levels, one after another, like a stream of frozen water. The spaceport on Orbis 4 was a busy place.\n\nMy temples throbbed and I felt nauseous as I jogged to keep up with Hach. Then it dawned on me that this was the farthest I had ever been from Ketheria. Ever since we were young, I had never liked being apart from her for long periods of time, and whenever we were separated, I became distracted by a weird empty feeling that I had always assumed was simply anxiety. I never really paid much attention to it, but after the awakening started, it had become more noticeable. This was the worst I had felt since the start of her awakening, and it was also the farthest I had been from her \u2014 a fact that was not lost on me.\n\nI followed Hach as he marched across the spaceport and through one of the many exits. Once outside, we descended a broad set of stone steps that led into a city. When I caught sight of the city, I gasped. Hach turned and saw me gawking.\n\n\"Welcome to the dumping ring,\" he said.\n\n\"Who calls it that?\"\n\n\"I do. They call it Murat. I believe it was named after some Nagool. Now, come, we have people waiting for us.\"\n\nI followed my Guarantor into the city. Murat looked like a way station, a shantytown constructed from used materials fastened to anything that was standing. Metal and glass structures like you might find on the other rings were buried under an erratic framework of multilevel trading chambers that sold what looked like the worthless trinkets Switzer and Dalton had scavenged on Orbis 2. The ones that sold useful items, like food or tools, were the busiest, but most of the action seemed to come from customers haggling over prices.\n\nAs I was walked deeper into Murat, I noticed that a lot of the signs were painted on boards or scratched right into the concrete and that the central computer did not translate half of them. Along the cramped streets, I also witnessed small fires burning in the open, where aliens roasted small creatures \u2014 skin, fur, and all. The smell was disgusting, as if you had burned the hair on your arm.\n\n\"This is where you work?\" I said.\n\n\"Of course not,\" Hach replied. \"My industries are off-ring, mostly on Ki and Ta. This is where _you_ will work.\"\n\n_What could I possibly do here?_ my mind cried.\n\nI stepped to the side as an alien with a narrow chin and a hunched back scurried toward Hach and offered him what I thought was a chemical analyzer, though the tool was too mangled to be certain. Hach took the used item from him and tossed the alien a tiny crystal in return. Two other aliens emerged from the shadows and pawed at the alien's new bounty. Farther down the path, Hach gave the broken device to another alien, sitting alone in the street.\n\n\"Sad, isn't it?\" he remarked. \"This is what they risked their lives for. They've traded rotation after rotation of indentured service for this meager existence.\"\n\n\"I don't get it,\" I said. \"How do they end up like this?\"\n\n\"It's simple, really. The First Families like what they have, and they work diligently to keep it for themselves. When knudniks complete their work rules, most of them don't have the finances or the skill set required to live as proper Citizens. Do you know what it costs to live on Orbis 3? Even if half the people here pooled their resources, they couldn't afford a dwelling on the ring. Here lies the Rings of Orbis's dirty little secret. And it could be even worse.\"\n\n\"Worse?\"\n\n\"Orbis 4 would be overflowing with refugees if these aliens weren't offered passage through the wormhole after their work rule ended. If I were them, I wouldn't stay here, either.\"\n\n\"But don't the Keepers know about this? I can't imagine them allowing it.\"\n\n\"But that's the genius of the Citizens, especially the Trading Council. You see, the Trading Council controls the wealth. The Keepers do not have to do anything; the Citizens pay for everything, but that leaves the Keepers without any hard currency. Don't get me wrong: the Keepers are wealthier than you can imagine, but they waste their money here. They try to do what they can, but it's a futile effort. There are too many forces working against them.\"\n\n\"But you're on the Council. You sound like you hate this. Why can't you do anything?\"\n\n\"I _was_ on the Trading Council, but I'm not anymore. That is why _you_ are here,\" he said.\n\nHach was no longer a council member! When had that happened? And more important, why?\n\nI must have been gawking. \"Don't worry,\" he said. \"I left the Trading Council of my own accord.\"\n\nThen Hach stopped in front of a private flier. I looked back down the street toward the spaceport. I knew it was rude and dangerous to question my Guarantor, but I had to ask.\n\n\"Why didn't we just \u2014\" I pointed back toward the spaceport.\n\nHach cut me off. \"I wanted you to see Murat. I wanted you to experience the city for yourself. I felt it was important for you to know. You'll understand my motives later.\"\n\nAn alien with thick legs and muscular arms emerged from the ruby-red craft.\n\n\"I trust your trip was satisfactory,\" the alien said.\n\n\"Yes, I believe it was,\" Hach replied, motioning toward me.\n\nThe alien looked at me and smiled. _Another knudnik,_ I thought. It was undeniable. The hopeless look in his eyes gave him away. Did I look like that?\n\nThen the alien reached for the door. I thought he was opening it for Hach, but another alien with taut, glowing-white skin, stepped out of the flier and strode toward me. As he stood up, the skin at the edges of his collar and sleeves seemed to ripple before settling.\n\n\"It's good to see you again, Queykay,\" Hach said, but the candescent alien stepped around my Guarantor without even a glance. Queykay was the same height as Hach but walked with his chin raised, forcing him to look down upon anyone he spoke to. Hach ignored the snub and continued. \"Queykay ba Torel, meet JT. JT, meet your new liaison with the Trading Council. It was not possible for me to be your Guarantor and sit on the Council, but the Trading Coun \u2014\"\n\n\"The Trading Council feels the pulse of its Citizens,\" Queykay interrupted him. \"And we feel it is necessary that a member of our elite supervise the arrival of a Scion, independent of your Guarantor's responsibilities. Besides the honor her presence bestows on the multitude of Citizens on Orbis 4, your sibling's existence creates many security risks. The Tonat cannot be in all places at once.\"\n\n\"I'm not the Tonat,\" I informed him.\n\n\"I thought this was taken care of,\" he said as he turned to Hach.\n\n\"It will be,\" Hach assured him. \"I am aware of the arrangement, and I will deliver as promised.\"\n\n_What arrangement?_\n\nQueykay stared at me, sizing me up before speaking. \"So if you are not the Tonat, then who are you?\" he asked. I could hear the sarcasm in his voice.\n\n\"I'm . . . I \u2014\" I stumbled for an answer. \"I'm the Softwire.\"\n\n_What a stupid answer,_ I thought. I didn't want to do this. I saw the other knudnik get into the flier and took it as my cue to get away from this guy. I tried to step around Queykay, but his hand darted out from his burgundy cape and pressed against my face. His skin felt damp and sickly except where a large crystal ring encircled his finger. Repulsed, I pulled away from him, glancing up his sleeve.\n\n\"But you are still a knudnik,\" he hissed. \"It would be healthy if you remembered that.\"\n\nI didn't like Queykay. It was one of those instant feelings you get. His entire demeanor seemed polished to make me feel inferior. And he was creepy. When I had pulled away from him just then, something rattled my senses. It wasn't even possible, my mind reasoned. For the tiniest moment of time, I swear that I saw a hundred tiny red eyes blink at me from the depth of his black silk shirt. I shook it off. Weird. I hoped I didn't have to be around this guy too much. I did not see a long-lasting friendship in our future.\n\nHach leaned in and mumbled something to Queykay. Then he motioned for me to get in the flier. I slipped into a seat at the back, glad to be out from under Queykay's glare, as the flier lifted above the crowds and then turned toward the ring's edge on my right. After a short trip across the tattered city, the flier settled down atop the only building I could see that appeared to be constructed from a set of plans rather than the wire and guesswork that seemed to hold the rest of the city together.\n\nWhen the flier settled, I waited for my Guarantor to exit first, but he turned to Queykay. He was about to speak when Queykay raised his finger and focused on me. They were both staring when Hach motioned for me to get out.\n\n_My pleasure._\n\nI scrambled out of the flier and moved as far away as possible. While I waited for their conversation to finish, I surveyed what I assumed was my new home. The ship was resting on the lower roof of a multilevel structure, some sort of landing pad, I figured. Behind the flier, about a hundred meters away, I spotted an entrance to the second level \u2014 a curved structure comprised of nothing but tall black windows. I looked for some sign of life but saw none. _Is Max inside?_ I wondered. I hoped so.\n\nI turned away from the black windows and moved to the edge of the lower roof. I glanced below and spotted a long, walled walkway that led away from the front of the building. The concrete path ended at a huge open square. _What does Hach do here?_ I wondered. Or rather, what was _I_ going to do here?\n\n\"It's impressive, isn't it?\" Hach called out.\n\nI turned and saw him walking toward me. Queykay and the flier were nowhere to be seen.\n\n\"Don't worry \u2014 he's gone.\"\n\n\"Would I be punished if I said that I hoped he was gone for good?\"\n\n\"I won't punish you, but don't let _him_ hear that. Just stay out of his way, all right? The Trading Council needs to appear to be in control despite the fact that they were caught completely off guard. They need the Citizens to believe that the Scion is in _their_ pocket. They don't want the Keepers taking all of the credit.\"\n\n\"She's not in their pocket,\" I told him.\n\n\"That doesn't matter. Appearances can have just as much influence as fact.\"\n\n\"It sounds like more politics to me,\" I said. \"What is this place?\"\n\n\"For you? This is home. For me, this is a unique partnership. I hope it does us both some good.\" Then he let out a deep breath and turned away. \"Come, there are some people eager to see you, I'm sure.\"\n\n\"Excuse me, Hach?\"\n\nHe turned back toward me. No Citizen I ever met liked to be addressed by a knudnik. Inside I cringed, waiting for his punishment, but none came. \"Yes?\" was all that he said.\n\n\"I was hoping that the work I have to perform, you know, what you want me to do, could keep me close to my sister. I've always looked out for her and \u2014\"\n\nHach interrupted me. \"Don't worry, Softwire. I'm counting on that.\"\n\nI wasn't used to getting my way with a Citizen, but I knew when to keep my mouth shut. If Hach was expecting me to stay close to my sister, then that could only mean that he was counting on me to play the Tonat. Even Queykay seemed to expect it. The way those two had confided in each other made me think they were planning something. _But what?_\n\nI followed Hach toward the wall of black glass. As he neared the middle, he waved something in the air and the glass plates parted in response. When I stepped through the door, I was greeted by a digi three times my height, hanging in a hallway that ran parallel to the curve of the glass. I could see at least twenty digis, lit with pink lights that appeared to float above the polished floor. Some digis showed images of aliens I had never seen while others displayed places I had never visited.\n\nHach had turned right and was marching down the hall. \"This way,\" he called over the echo of his boots striking the glasslike floor.\n\nPast the last of the enormous digis, Hach stopped under the center of an arched doorway, where cooler light spilled out from the room beyond, along with a familiar chatter. I knew instantly who was in there. I rushed past Hach.\n\n\"JT!\" Theodore shouted as I entered the room. He sprang from the floor. Theodore was as tall as I was now, and he had let his hair grow into a shaggy mop. It reminded me of the mane on a Garin, the knudniks that served the Trading Council. Everyone rushed toward me \u2014 all of the eighteen kids who had lived together with Ketheria and me as knudniks on the Rings of Orbis, though we were only a small fraction of the total number of kids from the _Renaissance._\n\nI scanned the room for Max. When I saw her, my stomach tightened and sent a jolt to my heart. She looked up, tucked her hair behind her ear, and smiled. She was so pretty \u2014 I couldn't take my eyes off her. Some of the other kids gathered around me and created a barrier between me and Max before she slipped out of sight entirely.\n\n\"How's Ketheria?\" Grace asked.\n\n\"Where is she?\" asked someone else.\n\nMy replies were quick but friendly. I didn't have the nerve to break through the crowd and go over to Max. Something inside me still hesitated when it came to showing my affection toward her in front of the others.\n\n\"Is the awakening finished?\" Theodore asked.\n\n\"No,\" I answered. \"Ketheria is still with the Nagools.\"\n\nThat's when I noticed Theodore's clothes. He should have been wearing his vest and the tattered clothes he had owned since the _Renaissance,_ but he wasn't.\n\n\"What are you wearing?\" I asked him.\n\n\"This?\" he said, pulling at the burlaplike robe wrapped around his body. \"It's actually quite comfortable. Don't laugh. You have to wear one, too.\"\n\nI looked around. Everyone was wearing these dull robes; some had different-colored cords tied around the middle or scarves draped around their necks.\n\n\"I'm not wearing that,\" I told him.\n\n\"Actually, you won't,\" Hach remarked. \"The Tonat requires something a little different.\"\n\nI looked around and found Max standing behind Grace. I saw her frown when Hach mentioned the word _Tonat._\n\n\"I'm not the Tonat,\" I insisted. \"I have no intention of becoming a Space Jumper.\" I looked directly at Max when I said it.\n\n\"But the others on the ring don't need to know that,\" Hach argued. \"Remember what I said about the power of appearance? I simply need you to _pretend_ to be the Tonat. Will you at least agree to that?\"\n\n\"Pretend? Why would I would I pretend to be something I don't want to be?\"\n\n\"Because that's our new job,\" Max said, now standing to my left and looking at Hach. \"Well, yours, anyway. This whole building has been designed for it. Once Ketheria is finished with the awakening, this building will become a shrine and fees will be charged so others may visit her.\"\n\n\"Correct,\" Hach exclaimed. \"Humans are so much more observant than they give you credit for. Personally, I have no need for these fables. OIO and its teachings have no room in my life, but that doesn't mean there aren't plenty of individuals on the rings who feel differently.\"\n\n\"And will pay a lot for the privilege to see the Scion,\" Max added.\n\n\"As well as the Tonat,\" Hach reminded her. \"The Trading Council sees a great profit in this little charade.\"\n\nI didn't know what to say. I just stared at Hach. Could it be true? Were we going to be put on display like animals in a zoo?\n\n\"What about everyone else?\" I asked.\n\n\"Every deity needs her disciples,\" Hach replied.\n\n\"Ketheria won't go for it.\"\n\n\"I would not count on that. When your sister is delivered to me, she will be well on her path to becoming the Scion.\"\n\nWith that, Hach turned and headed out of the room. \"Queykay will return later with instructions on preparing for the Scion's arrival.\"\n\n\"Queykay?\" I cried.\n\n\"I take it you've met him,\" Max commented.\n\n\"Although I am your Guarantor, the Trading Council's needs supersede any authority I may have over you. They, along with the Keepers, are an integral part of this arrangement. I know you will not like to hear this, but when I am not around, Queykay is in charge. He is to be treated with the same respect you would afford me, if not more.\"\n\n\"You're leaving?\" I called after him. \"But I don't understand. I thought we were working for you.\"\n\nHach stopped at the doorway. \"My new contracts require my presence on Ta. I have simply provided the building to house the Scion. You are in capable hands with Queykay. This is what's best for everyone, you included. Please trust me when I tell you that this arrangement is far too complicated to explain.\"\n\nHach looked at me, waiting for a response, but I said nothing. Anything that I wanted to say would only make matters worse for everyone else.\n\n\"Remember,\" he added, \"you could be sleeping in Murat this cycle, and eating one of those things you saw grilling in the street.\" Then he left. I turned away, grinding the palm of my hand into my forehead.\n\n\"I don't understand what's wrong, JT,\" Grace said. \"I think it's wonderful. One more rotation, and then we're free, with Ketheria as the Scion. I can't think of a better scenario.\"\n\n\"I can think of a few,\" I muttered, glancing at everyone in the room, dressed in those stupid robes. Max caught me looking.\n\n\"What? You don't like them?\" she said, smiling and smoothing out the material on her stomach. \"They're really quite comfortable.\" She came over to me and took my hand. I instantly felt better. \"C'mon, you have to admit this place is better than the trash belts at Weegin's. Remember that radiation gel?\"\n\nI followed Max away from everyone else and out of the back of the room into another glassed hallway. She led me past small pools of water cut into the stone floor and stopped at a crystal bench under a yellow tree that grew right inside the building. The black windows, despite their color, provided an ocean of light, as if a beaming sun were hanging right outside.\n\n\"Are the windows lights?\" I asked, pointing toward them.\n\nShe didn't answer. Instead she grabbed my arm, pulled me close, and kissed me.\n\n\"I've been waiting a long time to do that,\" she said.\n\n\"You can do it again if you like.\"\n\nMax leaned toward me, and this time I kissed her.\n\n\"How long is this going to go on for?\" Theodore interrupted as he walked toward us. Max pulled away.\n\n\"Not long _now,_ \" I complained.\n\n\"Good, because there are some things we need to discuss,\" he said, and thrust something in my face. It looked like the taps we used to get at the Illuminate on Orbis 3.\n\n\"What is it?\" I asked him.\n\n\"Queykay has us handing them out all over Murat,\" Max said.\n\n\"No one refuses them,\" Theodore added. \"Some people even try to resell them.\"\n\nI poked into the tap with my softwire. Accessing something like a tap was almost as easy as breathing for me now. I no longer thought about the mechanics; I simply concentrated on the outcome, and the contents of the device filled my thoughts. The tap contained moving images of Ketheria with the glow all about her. She was smiling and touching people softly, people who were kneeling in front of her. Some were crying, some rejoicing. It was all strangely eerie, as if Ketheria were some sort of god. This was followed by more images of crowds streaming up the walkway that led to this building. The fictitious events played out inside my head stronger than my most vivid memories.\n\n\"I don't get it. Ketheria has never been here before,\" I said.\n\n\"It's an advertisement,\" Max informed me.\n\n\"The Trading Council is going to have Ketheria hold sermons or something,\" Theodore said.\n\n\"Hach knows about this?\"\n\nMax nodded.\n\n\"What does it mean? They're using _us_ to start a religion?\"\n\n\"OIO is not a religion,\" she argued.\n\n\"But some aliens distort it for their own gain,\" Theodore said. \"They prey on those who worship the Ancients.\"\n\n\"He's right,\" Max said. \"I think that's what the Trading Council is attempting to do here, but OIO is actually a philosophy. It's the art and science of cosmic energy. It helps you to interpret the events in your life so you might gain control. They believe that everything, even your thoughts, goes out into the cosmic soup and has the potential to affect everyone else. Through this energy, we are connected with everything in the universe, no matter how close or how far. Nagools try to master this energy, releasing only constructive energy while avoiding deconstructive energy. But even they look at the arrival of Ketheria as a messiah, as if she's going to help them tip the scales of the deconstructive energy they claim plagues our universe.\"\n\n\"They think it's that bad?\" I asked.\n\nTheodore scoffed. \"Have you seen it out there? This place is a hellhole.\"\n\n\"And Hach is trying to capitalize on this?\" I said.\n\n\"The Trading Council is. I'm certain of it,\" Theodore whispered.\n\n\"Queykay has been the one making us hand out the taps,\" Max reminded him. \"I think most of Hach's work is with mining or something.\"\n\n\"You just wander around the city handing out taps? Isn't Queykay afraid we're going to try to escape?\"\n\n\"Where would we go? This is by far the best place on the ring. Besides, our staining would make it easy for them to find us.\"\n\nI shook my head. \"It doesn't make sense.\"\n\n\"Yes, it does. Hach is a Citizen,\" Theodore reminded me.\n\n\"I know, but you remember him, don't you? Remember how he acted toward that other Citizen at Odran's party?\"\n\nMax was nodding. \"The one who didn't like knudniks much? I think her name was Pheitt.\"\n\n\"Hach was golden.\" Theodore smiled.\n\n\"I can't picture him doing this. Why would he want to start a religion? You said he was into mining? This doesn't make sense.\"\n\n\"Well, he's doing it,\" Theodore insisted.\n\n\"Vairocina?\" I called out. \"You can show yourself. It's just us.\"\n\nParticles of light pooled in front of us, and a figure began to form. Vairocina had started changing her appearance ever since she had begun projecting her holograph for us. Each time it seemed as if she was trying to look a little older. Her new look was not lost on Theodore, who always straightened up whenever Vairocina appeared.\n\n\"Hi, V!\" Theodore gushed.\n\n\"Hello, Theodore. Hello, everyone.\"\n\nI looked at Theodore and mouthed, _\"V?\"_ He only shrugged.\n\n\"Hi, Vairocina,\" I said. \"I know I'm always asking for your help, but . . . well, I need your help again.\"\n\n\"You know I will always help you in any way I can.\"\n\n\"Thank you. This time I was wondering if you could help us with our new Guarantor. Hach told me earlier that he gained possession of us from Charlie.\"\n\n\"He did?\" Max interrupted, and I nodded to her.\n\n\"Can you check and see how that happened? I can't figure out when Charlie had any interactions with Hach. Charlie never mentioned him. It just seems strange that he would leave us to Hach. Will you see if you can dig anything up?\"\n\n\"Certainly,\" she replied.\n\n\"Did you ever find out how Charlie got all that wealth when he became a Citizen?\" Max asked her.\n\n\"After his demise, I was certain you wouldn't be interested anymore. I did find a trail, but it ended at the Keepers.\"\n\n\"The Keepers?\" I said.\n\n\"Do you wish me to look further?\"\n\n\"No. You're right. It's not important anymore, but I would appreciate any information about Hach and how we came into his possession.\"\n\n\"I'm on it already,\" she said, smiling and blinking.\n\n\"Bye, V!\" Theodore said.\n\n\"Good-bye, Theodore.\"\n\nVairocina's image mixed with the light in the room, and then she was gone.\n\n\"You _like_ her,\" Max squealed.\n\n\"I do not,\" Theodore said.\n\n\"Yes, you do!\"\n\nTheodore shot me a look. \"Don't look at me,\" I argued. \"I mean, you do act a little . . . weird whenever she's around.\"\n\n\"Me? _I'm_ weird? Maybe I should leave so you guys can get back to sucking on each other's faces.\"\n\n\"Theodore!\" Max exclaimed, but I was nodding, hoping my friend would leave quickly.\n\n#\n\nAnother entire phase passed and still there was no sign of Ketheria. I was nervous, and to make matters worse, Theylor had not sent me a single screen scroll updating me on my sister's condition. _How long does this step of the awakening take?_ I worried.\n\nI was lying in my sleeper, squeezing my temples with the palms of my hands, when Theodore barged into my room. The pain in my head had been coming in waves ever since I woke.\n\n\"Come on, malf!\" Theodore cried, but then stopped short. \"What's wrong with you?\" he said.\n\n\"Don't get too close,\" I groaned. \"I just might throw up on you.\"\n\n\"We have a lot of these taps to deliver,\" he complained while hoisting up a large blue sack. \"How are you going to help if you look like that?\"\n\n\"I can't. You go. Queykay's not here, is he?\"\n\n\"I don't know. He might have left already. He did ask for you, though. Max covered. She'll be here in a sec.\"\n\nI sat up, making the pain worse, as if my brain were trying to squeeze out around my eyes. I couldn't help but think of Ketheria and the glow.\n\n\"No,\" I said. \"I can't. Tell Max I'm not feeling well or something. Tell her I'll catch up with you guys later.\"\n\n\"She thinks you're avoiding her, you know.\"\n\n\"Did she say that?\"\n\n\"Not exactly.\"\n\n\"I'm not doing this on purpose. I can't control it. I go from zero to puke in a nanosecond. Can you imagine if I tossed in front of Max? Not to mention while I was kissing her!\"\n\nTheodore cringed. \"No. I don't want to imagine that.\" He squeezed his eyes shut and shook his head. \"Argh! Too late. All right, I'll cover for you again. Get some sleep, but you'll have to do them _all_ next cycle.\"\n\nI nodded as Theodore left. Then I lay down, careful to place my head on the sleeper as gently as possible. I was aching to see Max, but not like this. I tried to avoid her whenever I felt this way. I knew it wasn't right, but I couldn't face the alternative. After a few moments on my back, another wave of nausea crested in my throat and I jumped off my sleeper despite the cries from my splitting head. I made it to the bathroom just before I threw up.\n\nSatisfied that my stomach was empty and hoping that the others had already left to flood Murat with more propaganda, I set off to find the chow synth. I still hadn't adjusted to the layout of the building, so every stroll was an adventure. The place was enormous. I swear it was as big as a space station, but I needed some water. Outside my room I saw a knudnik pulling a train of double-shelved carts filled with bowls and flowers. The alien deftly maneuvered the six or seven linked carts around the corner; none of them even came close to scraping the wall. It had been like this for the whole phase. Knudniks and construction-bots were everywhere. Queykay was building some sort of shrine in preparation for Ketheria's arrival, but I didn't care. It just meant that Ketheria would be here soon.\n\nI was following the train-pulling knudnik down an enormous hallway when I heard Max and Theodore coming toward me.\n\n\"This is ridiculous. If he's this sick, then he needs to see a doctor, Theodore,\" I heard Max say.\n\nI was trapped. If I headed back in the direction of my room, I was certain she would see me. What would I say to her?\n\n\"Trust me, Max. I saw him. He's in no condition to see anyone,\" Theodore said.\n\nThe knudnik with the carts stopped, as if he, too, were reacting to their voices. For a moment I thought about crawling into the cart, but I didn't see anything that could hide me.\n\n\"This is stupid,\" I muttered. _Just deal with it,_ I thought.\n\nThen the wall to my right seemed to split apart as two seamless stone doors swung open. A couple of Argandians, squat knudniks with yellow, scaled bellies, waddled through the opening. The alien pulling the carts greeted them, and they continued down the hall in the direction of Max and Theodore.\n\nI dove inside the room before the doors swung closed.\n\nI waited for my eyes to adjust to the light, or the lack of it. A soft blue glow arched around the perimeter, but it was not enough for me to see anything. I groped the wall, looking for some sort of control panel that I might push into, but I found only smooth stone, cold and indifferent under my fingertips. _What is this place?_ I wondered. I walked slowly toward the blue glow. _That light has to be controlled by something,_ I thought. I reached out in front of me, swiping at the air for any obstacles lurking in the dark, when _BAM!_ I hit my shin on something hard and sharp. The pain bolted up my leg, and when I reached down, expecting to find blood, I struck my forehead on another stupid barrier.\n\n\"Of all the \u2014!\" I yelled, and then I was gone.\n\nAt first I didn't know what had happened, but the rancid smell of feet gave it away. I had _jumped._ I hadn't tried to jump; I just did. _But how?_\n\nThe smell of feet was too much for my weak stomach, and I unloaded the meager contents of my stomach onto my boots. Embarrassed, I wiped my mouth and looked around. Thankfully, there was no one watching. I was alone in an alley, except for a bunch of garbage and busted shipping crates that gave no clue to my whereabouts. As I moved away from the smell of my own vomit, the space rippled while the light closed in, forming a single point. It meant only one thing to me.\n\nSpace Jumpers.\n\nTwo of them surfaced on either side of me \u2014 tall, imposing figures clad in silvery chest plates and thick leather boots. One wore a helmet that covered half of his face. Him I recognized, from my encounter last phase on Orbis 1, but the other I had never seen before.\n\n\"You again,\" the familiar one grunted.\n\n\"For a group of individuals who are supposed to be banished from the Rings of Orbis, you sure do show up a lot,\" I remarked.\n\n\"Here, take this,\" the masked Jumper ordered. \"Keep it with you at all times.\"\n\n\"What is it?\" I asked, holding up the smooth disc he'd handed me.\n\n\"That device informs us that it's just you trouncing through space,\" he replied.\n\n\"So we don't have to babysit you anymore,\" the other spat.\n\nThe new Space Jumper glared at him before telling me, \"What you're doing is illegal, and we have no way of telling who you are without a belt. This is for your own safety.\"\n\n\"We should arrest the _popper,_ \" the surly one complained. \"Let him spend a few rotations in slow-time.\"\n\n\"Take it up with the Trust,\" the other Space Jumper snapped at him. \"You have your orders.\"\n\n\"How do I get back?\" I asked him.\n\n\"You're smart; figure it out,\" one of them replied, and then they were gone.\n\n\"Wait!\" But there was no one left to answer. The Space Jumpers had left me there, and I had no idea where _there_ was. I tried to jump back home, but nothing happened. I closed my eyes and concentrated on my new home, but still \u2014 nothing. I simply remained there as motionless as the metal crates that surrounded me, only feeling a little more stupid.\n\n\"Great. Now what?\"\n\nI stepped away from the crates and found myself behind a small group of trading chambers. _Is this Murat?_ I wondered. I had no idea. I could be anywhere, couldn't I? I mean, I had no clue how I was jumping, let alone where. _Spontaneous space jumping?_ _This is great,_ I thought. I knew everything about my softwire abilities but nothing about this new, uncontrollable ability to move through space. I knew that Space Jumpers were softwires, but they all used belts to jump. I also knew that the Trust trained softwires to jump, but I had no training. So how was I jumping without a belt?\n\nSuddenly I wished Charlie were there. He never answered much, either, but at least he had a way of making me feel better about not knowing anything.\n\nThe streets were clogged with aliens as scruffy-looking as the tiny trading chambers wedged together, some even built right on top of one another. I saw stuff for sale that wasn't much better than the scraps we used to throw out at Weegin's World. Max was right. If this _was_ Murat, Murat was a dump. _How do these aliens even survive?_ I wondered. I was waiting for an opening in the traffic of well-worn trams and smoking cargo hovers when I heard my name.\n\n\"Johnny Turnbull!\"\n\nI spun around and saw a Keeper disembarking from one of those circular fliers.\n\n\"Drapling?\"\n\n\"How are you, child?\" Drapling said, rushing toward me.\n\nDrapling was my least favorite Keeper. From the moment I met him on Orbis 1, he had treated us humans with such contempt that I had always tried to avoid him. But then he had changed. It started after the staining on Orbis 2. Now Drapling was . . . well, nice. Almost too nice. I wondered if he'd known about Ketheria and me back then. Did he _have_ to be nice to us now for some reason? I didn't know, and I knew he would never tell me. Drapling was just like everyone else. Despite his new friendly attitude, he still had that look on his face and that slight pause before he spoke, as if he were going over some list of rules in his head, deciding what information he could divulge without really telling me anything.\n\n\"Hi, Drapling,\" I said. \"What are you doing here?\"\n\n\"Someone informed me of your misfortune,\" he said as his left head smiled.\n\n\"Someone? How did you find me?\"\n\nDrapling held up a small device, cradling it in his long bluish fingers. \"I told you the staining would help us protect you.\"\n\n\"From whom?\" I mumbled.\n\n\"Excuse me?\"\n\n\"Drapling, I'm sure you must have better things to do than chase an errant knudnik around the rings. Besides, what are you doing on Orbis 4, anyway?\"\n\nDrapling was next to me now. He was trying to shake his left head, but it just sort of wobbled. He wasn't that good at Earth gestures. \"Keepers spend most of their time on this ring,\" he said. \"You have no idea about our work on this ring, do you?\"\n\n\"So we're on Orbis 4?\"\n\n\"Yes. Let me show you what we do.\"\n\nI felt awkward walking through the streets with Drapling, almost as if we were friends. He informed me that I was somewhere on the far side of Murat. He asked if I was hungry and how I'd gotten here. They were simple questions, nothing too probing, since I didn't know myself. I asked him when Ketheria was coming.\n\n\"Soon,\" he replied.\n\n\"That's all you know?\"\n\n\"No one can say for sure. Remember, this is only the beginning of her awakening. It will take your sister a while to adjust to her new responsibilities. The Nagools will help her through each step. Ketheria will know when she is ready.\"\n\nHow my sister even had a clue about what was happening to her was beyond me.\n\n\"Why here?\" I said.\n\n\"What do you mean?\" Drapling replied.\n\n\"Why this place? I mean, if Ketheria is going to be so important, why are they bringing her to a dump like Murat? Is it because of our work rule? Somehow I think Ketheria would be better served on a different ring.\"\n\n\"That is where you are wrong,\" Drapling said. \"Murat needs her the most, and it is here she will find her strongest advocates.\"\n\n\"You mean, this is the place with the most knudniks who are willing to give up what little they own to have an audience with her?\"\n\nDrapling stopped and turned toward me. Both his heads were focused on me, and they were both frowning. \"You know so little,\" he whispered. \"You have much to learn.\"\n\n\"I couldn't agree more, but the funny thing is that no ones ever tells me the truth.\"\n\n\"Let us try to change that. Shall we start here?\" Drapling said, gesturing toward a slanted glass building flanked by several impressive spires.\n\n\"What happens here?\"\n\n\"I'll show you,\" he said, and stepped toward the building.\n\nThe doors disappeared, and I followed Drapling inside. I noticed that the air in the building was much cooler, like in the Keepers' home below Magna. Even the glass walls changed the light to a bluish color similar to the tone down in the caverns. I saw dozens of Keepers strolling across the atrium and even more knudniks waiting on narrow benches that lined the long room.\n\n\"When their work rules are completed, many former knudniks are unable to adapt to the Citizens' way of life. We help arrange passage for them through the wormhole. This allows them to find another planet more suited to their needs.\"\n\n\"So you just get rid of them?\"\n\n\"The Descendants of Light struggle every cycle to improve the living conditions of new Citizens. Some don't want to leave, and we help them as well.\"\n\nI looked at the aliens sitting, waiting. Would that be me at the end of the rotation? Assuming the Citizens declare my work rule finished, that is. Each alien looked either worried or just plain exhausted. Had they failed? Was their dream to live as Citizens on the Rings of Orbis simply a bust?\n\n\"So if I understand this correctly, the Trading Council makes promises of a better life to lure knudniks here, but it's a promise they never plan to keep. When these individuals learn they've been lied to, you ship them off to another world. Problem solved.\" I turned to Drapling. \"Do you like doing the Council's dirty work?\"\n\n\"Your judgment is too harsh. It is quite expensive to live on the Rings of Orbis, despite our best efforts. Many Citizens are descendants of the First Families, and their wealth is, well, unimaginable. It makes it impossible for the newcomers to compete.\"\n\n\"Or maybe the Citizens just like it that way. Isn't it true that when you become a Citizen, you receive a percentage of the proceeds from the crystals harvested from the moon?\" I said.\n\n\"That is correct, but before you can receive your share, you must first establish a place of residence and prove to the Trading Council that you are a stable and contributing member of this society.\"\n\n\"So then the other Citizens aren't too big on sharing what they have with these new Citizens, especially the ones who used to slave for them.\"\n\nDrapling looked at me but said nothing. Then he leaned in and whispered to me, \"Some observations are better kept to oneself. Especially since there is a new Scion on the horizon.\" He nudged me toward the door and then in a louder tone said, \"You know, many of those who want to stay on the rings reapply for work rule. I admit the circumstances are slightly different, but it does give the new Citizens more time to establish their residency.\"\n\n\"You mean they volunteer to be knudniks again?\" I was astonished.\n\n\"Queykay is one such alien,\" Drapling informed me. \"The rules become a little blurred at times, but look at Queykay. He is a very respected Citizen on the rings now. He even sits on the Trading Council, and he was once a knudnik, just like you.\"\n\nI still didn't like him any better.\n\n\"But despite the success of a few individuals like Queykay, many do struggle. I cannot deny that, but Ketheria will change everything.\" Drapling was whispering again. \"She will give us hope. She can restore the dream that was the Rings of Orbis. This is where she belongs.\"\n\n\"You mean _them,_ \" I corrected him.\n\n\"Excuse me?\"\n\n\"You said she would give _us_ hope. Don't you mean _them_?\" I said, pointing to the aliens on the benches. \"I think the Keepers already have it pretty good around here. How can the Scion help the Keepers?\"\n\n\"I believe I was referring to the work class here on the rings. You must be mistaken.\"\n\nHe said _us._ I was sure of it.\n\nHe lifted his hand toward the exit. I took this as my cue to leave.\n\n\"If you ever find yourself lost, I want you to know that you can come here,\" he said. \"And if you are ever in trouble, please tell your friend in the computer to find me. I will come immediately.\"\n\nI looked at Drapling as he walked ahead of me back onto the streets of Murat. He was up to something.\n\n\"Drapling?\"\n\nHe stopped and turned. \"Yes?\"\n\n\"What are you up to? What do you want?\"\n\n\"What do you mean?\"\n\n\"Let's face it. When we first met, you were not this nice to me, or to any of us, for that matter. Does this have anything to do with Ketheria?\"\n\n\"How can you say that? I remember several instances where my actions displayed nothing but kindness toward you.\"\n\n\"Yeah, ever since the staining. Is that when you realized Ketheria was the Scion? What are you trying to get from us?\"\n\n\"You are forgetting Odran's. Do you not remember my concern for your well-being when Odran forced you into the crystal-cooling tank? That was well before your staining.\"\n\n\"But you still let me get inside,\" I reminded him.\n\n\"The sanctity of the rings was at stake. Everyone was at risk, not just you.\"\n\nBoy, he was good at lying.\n\n\"JT!\"\n\nI turned and saw Max waving at me from across the street. I was glad that I was feeling better. Theodore and the others were marching down a narrow path that separated the stacks of trading chambers. I must admit that they were an imposing group dressed in their matching robes.\n\n\"Your friends can show you the way home, JT,\" Drapling said.\n\n\"Thanks for your help, Drapling.\"\n\n\"Remember, do not hesitate to contact me if you need anything,\" he said, and then he slipped away.\n\n\"What are you doing here?\" Theodore asked. \"And was that Drapling?\"\n\n\"It's a long story. What are you guys doing?\"\n\n\"We're all done,\" Max exclaimed, holding up an empty sack. \"Let's get something to eat.\"\n\n\"We don't have any money,\" I said.\n\n\"Wait until you see what these robes can get us.\"\n\nMax led us to a narrow eating-house a few streets away. Tables were constructed from scraps of metal that looked like they were torn right off one of the shuttles.\n\n\"Far cry from the Earth News Caf\u00e9, huh?\" Theodore whispered.\n\n\"Yeah, but the food's good,\" one of the other kids remarked.\n\n\"I still don't know how we're going to pay,\" I said.\n\n\"Watch and learn,\" Max boasted.\n\nAfter we entered the caf\u00e9, an alien leaped out from behind the tiled counter. \"You're back! I'm honored,\" he shouted, throwing his hands up in the air. \"Do you see who my caf\u00e9 attracts?\" he added, addressing the other patrons. \"Sit, sit! I will bring you my finest dish. No charge. Nothing but the best for those who share their existence with the Scion.\"\n\nI was attempting to decipher the caf\u00e9's odd odor \u2014 pungent spices mixed with a strong detergent \u2014 when the round little alien spotted me. His mouth hung open and he pushed past the other kids. The alien dropped to his knees and began rubbing my feet with his thick hands. He was mumbling something, but I couldn't understand a word.\n\n\"Please don't do that,\" I whispered.\n\nThe alien bolted upright and said, \"Whatever you ask me, I will do. Please sit. I, Kasha, will serve you personally.\"\n\nHe led me to a seat near the front and shoved a patron off one of the short stools. He cleared the table with the sweep of his hand. Everyone in the caf\u00e9 turned and looked.\n\n\"Dodu! Clean this mess!\" he shouted, and an alien appeared at his side, snatching the broken dishes off the floor.\n\n\"Is this comfortable for you?\" Kasha asked me.\n\n\"Please don't do this,\" I told him. \"Please don't make a fuss.\" He saw me looking at the other patrons.\n\n\"I understand. The Tonat wishes to have privacy. I will respect this.\" Kasha stood up and clapped his hands. Four aliens rushed from the back, dragging metal stands, each draped with purple cloth. Kasha directed them to place the dividers around my table. I only groaned and shook my head.\n\n\"You're coming out with us more often,\" Theodore gushed, stepping around the cloth dividers with the others.\n\nI looked at Max. Her pained expression rattled me more than the caf\u00e9's odor.\n\n\"I'm sorry,\" I said to her.\n\n\"It's not your fault,\" she mumbled.\n\n\"Are you crazy?\" one of the kids said, eavesdropping on our conversation. \"You've got it made here, JT.\"\n\n\"Shut up,\" I told him.\n\nMax sat to my right while Theodore sat across from me. Grace dragged a stool across the floor and squeezed next to Max. She and Max had grown closer, ever since Grace started hanging out with the tall boy with black hair. I couldn't remember his name because he was always changing it. Even on the _Renaissance_ he was always trying different names, the same way Ketheria would rummage through other people's clothes and try on different shirts.\n\nThe other kids were also trying to squeeze through the dividers, and the table was getting a little crowded.\n\n\"This is ridiculous,\" I said, standing up and pushing back the dividers.\n\nKasha flew to my side, wringing his hands. \"Is something wrong?\" he gasped.\n\n\"We don't need these, Kasha. Thank you, but we can eat without them,\" I told him.\n\n\"As you wish.\" Kasha clapped his hands once more. The aliens returned and dragged the barriers away, scraping the metal against the concrete floor. Some of the other patrons were staring now. Not my favorite feeling in the world, that's for sure.\n\nI sat back down. \"A lot of people on Murat act like him,\" Theodore pointed out.\n\n\"It's weird. I don't like it,\" I told him.\n\n\"Better get used to it,\" Grace said. \"Especially when Ketheria gets here.\"\n\nThat worried me even more.\n\n\"Forget Kasha. I'm just glad you were feeling well enough to join us,\" Max said, slipping her hand onto my knee. My mind became focused on her warm touch. \"How did you find us?\"\n\n\"I jumped here. Can you believe it?\"\n\n\"What?\" she said, pulling her hand off my knee. The empty space now felt like a hole in my leg.\n\n\"Wait, it's not like that. It was an accident.\"\n\n\"Well, you're acting like it's golden to be one of those vile mercenaries.\"\n\n\"No, I'm not,\" I pleaded. \"You know I don't want to be a Space Jumper. I told them, no way. You have to believe me. I told the Keepers. I told Hach. I told everyone. I didn't _try_ to jump. I couldn't control it.\"\n\n\"I believe you, but do you really think the Citizens on the rings will? Do you think they're just going to let you pop around the rings as you please? You know how they feel about Space Jumpers around here. If someone sees you and complains, you'll be banished for sure, whether you _say_ you're a Space Jumper or not.\"\n\nI never thought about that. \"I guess you're right,\" I mumbled. I was staring at my feet. How could I have been so stupid? If I were banished, I would never see Max again.\n\n\"You have to start thinking like that, JT. We're still knudniks.\"\n\n\"I'm the last person you have to remind.\"\n\n\"Really?\"\n\nI looked up at Max, but she was already talking to Grace as if the two of them had been in conversation the entire spoke. Max hated anything to do with space jumping, especially when it involved me.\n\n\"What do you mean by 'an accident'?\" Theodore asked.\n\nI turned to Theodore and sighed. \"It was weird,\" I said. \"I was roaming around in the dark, trying to find some sort of control panel, when I bumped into something and the next thing I knew I was at the back of some trading chamber in Murat.\"\n\n\"How did you do that?\"\n\n\"I don't know. I didn't _try_ to jump, it just happened,\" I whispered.\n\n\"You better get that looked at.\"\n\n\"Get what looked at? It's not like I have some sort of switch that I turn on and off. I don't even have a belt.\"\n\nI sat back as Kasha and the others arrived with bowls of steaming liquid and placed them in front of us. I could see chunks of meat and vegetables bobbing in the brown stew. It smelled like cinnamon and apples.\n\n\"This is the dish my father served to me as a boy. An ancient recipe my people share during the Hudshuka. Now I make it for you. Please, enjoy!\" Kasha exclaimed.\n\nKasha passed out an assortment of mismatched spoons, one to each of us, and then crowded in next to me. He was grinning wildly, waiting for me to try his dish. I only asked the Universe that it would taste as good as it smelled.\n\nI dipped the spoon into the bowl and scooped up the broth, trying to avoid the chunks of meat until I knew what they were. I lifted the spoon to my lips, blowing on it gently. Kasha leaned in even closer as I tasted it. It was sweet with a meaty thickness. I liked it. I smiled and nodded to Kasha. \"It's good,\" I told him. \"Really good. Thank you.\"\n\nKasha bolted upright and clapped his hands. \"He likes it! He _really_ likes it! The Tonat likes my _hudspa._ The Tonat is eating at my caf\u00e9. I will rename this dish after the Tonat,\" he cried, turning back toward me.\n\n\"I am not the Tonat,\" I seethed through clenched teeth.\n\n\"I don't understand. You _are_ the Tonat. You will be the greatest Space Jumper ever to step on these rings, and _you_ ate my food.\"\n\n\"I am not the Tonat!\" I yelled. I felt my face flush with rage. Kasha was staring at me. \"Stop looking at me!\" I slammed my fists on the table.\n\nAnd then I was gone \u2014 again.\n\nI tumbled backward onto a soft patch of grass and leaves, my hands still ringing from the contact with Kasha's table.\n\n\"This is ridiculous!\" I screamed, and fell back onto the ground. It was darker here, wherever _here_ was. I looked up at the lights sparkling on the far side of the ring. Was that Murat? There were more patches of twinkling lights spread across the ring but none in the vicinity of where I landed. Where was I now?\n\n\"Vairocina!\"\n\nDrapling sent a small shuttle to pick me up. By the time I arrived back home, everyone was already sleeping.\n\n\"That's a nasty tic you've picked up. I hope you don't make it a habit,\" said Queykay, stepping out from the shadows.\n\n\"Tell me about it,\" I replied. I tried to slip past him, but he reached out and caught me by the shoulder. I turned and looked at him. _What now?_ I thought, but Queykay didn't say anything. He just stared at me, studying my face. It was creeping me out.\n\n\"It's not my fault, Queykay.\"\n\nHe pulled his lips back, sucking in air as if charging his lungs to speak. His teeth were narrow and pointed and just as white as his skin.\n\n\"What!\" I asked.\n\n\"The Scion arrives next cycle. You may not be the Tonat, but I'm certain she will ask for you. Do not mess this up.\"\n\n\"Ketheria? Ketheria is coming?\"\n\nThe alien let go of my shoulder and turned away without a word.\n\n\"Queykay!\" I called after him. \"Mess what up?\" But he was already gone.\n\n#\n\nI woke the next cycle to the hum of cart-bots. I stepped outside my room and saw a dozen motorized baskets writhing through the corridors like the jointed tail of a sea dragon. The only other time I had seen this much commotion was during the preparations for the Harvest of Life back on Orbis 2, and that only happened once every seventy rotations. I reached out and snatched a peachlike item from a bowl of fruit. It was one of the few items I could actually recognize on the carts, besides the enormous bouquets of flowers.\n\nI stopped a Honine carrying three bolts of silk and asked him, \"What's all this for?\"\n\n\"The Scion is coming,\" he gushed, and hurried past me.\n\n\"When?\" I yelled after him in vain.\n\nI turned to go back into my room and found Max standing in the doorway.\n\n\"Hi,\" I said softly.\n\n\"Can we talk?\" she said.\n\n\"Sure.\"\n\nOnce we were inside my room, the door closed behind us. The building was big enough that each of us had our own room, but they weren't really more than large closets with a sleeper. Max found a spot on the floor and leaned back on a pillow against the wall.\n\n\"I'm sorry about last cycle,\" she said. \"I didn't realize how bad it was.\"\n\n\"Me neither. That's what I tried to explain to you.\"\n\n\"You don't have to, JT. As much as I don't want to accept it, you _are_ a Space Jumper.\"\n\n\"I'm _not_ a Space Jumper. Being a Space Jumper requires training. It's something I have to choose, and I haven't chosen it.\"\n\n\"Why not?\"\n\n\"You know why not.\"\n\n\"Say it again anyway.\"\n\n\"Max!\"\n\nMax stood up and wrapped her arms around me. \"Say it,\" she whispered.\n\nWhen Max was this close to me, I usually wasted most of my time worrying about when it was going to end. This time I forced myself to focus on Max.\n\n\"It's because of you. I'm doing it for you. You know that.\"\n\nMax pushed me away.\n\n\"For _us_!\" she said.\n\n\"That's what I meant. I made you a promise.\"\n\n\"But you understand _why,_ don't you? I don't want you to do this just for me.\"\n\n\"But Max, I don't really understand.\"\n\n\"Space Jumpers are horrible, JT. They're trained for one purpose \u2014 to destroy things. They're hired killers, just like Neewalkers. They have no life outside of the Trust, and they certainly don't get to make any choices for themselves. I know that's not you, JT. If you choose that path, there will never be an _us_!\"\n\nI slumped on her pillow.\n\n\"I don't know who I am anymore, Max. When I came to the Rings of Orbis, I just wanted a home. A place to belong, to grow up. A place to have some _fun._ But I don't get to have any of that.\"\n\n\"Me neither, but we still _can._ We simply need to stay out of their way. The Keepers, the Space Jumpers, the Citizens, the Neewalkers, they've been fighting since the beginning of time. They're not going to stop whether you're a Space Jumper or not.\"\n\n\"Then what _am_ I, Max? I'm certainly not normal.\"\n\n\"You're mine,\" she said, and climbed into my lap. \"That's all I care about.\"\n\n\"I know _that._ \"\n\nMax frowned.\n\n\"I mean, c'mon, Max. They're not telling us everything. First the softwire thing, then Ketheria becomes the Scion, now I'm bouncing through space if I sneeze. I don't get it. It has to mean _something._ I'm really beginning to believe we're here for some other purpose.\"\n\nMax didn't say anything. She just stared at me for an uncomfortably long time.\n\n\"What?\" I said.\n\n\"No, we're not. We're a bunch of kids from Earth. You've said that a million times. It's just a freak coincidence what happened to you and Ketheria. How come no one else from the _Renaissance_ has any of these abilities or powers? Don't you find _that_ strange? If someone had planned this, then why aren't we all softwires, or Scions, or Space Jumpers? Why just you guys? Look, we have one rotation left, and then we can go do what we want. They'll forget about us, pin their hopes on some new knudnik, and we can live life the way you always said we would.\"\n\n\"That's another thing. What's _that_ going to be like? I don't think I can live like those other knudniks in Murat. It's horrible.\"\n\nMax sat back. \"I know,\" she mumbled. \"And it's so sad. Do you know some of them become knudniks again?\"\n\n\"Drapling told me. I'm certainly not going to make that choice.\"\n\nMax kissed me. \"We'll figure it out together, all right? Nothing's changed. Everything is still the same. Nothing will come between us.\"\n\nThere was the sound of a chime in the room, and then the door opened. Queykay was standing there. I thought I noticed him grimace slightly when he saw Max in my lap.\n\n\"Your sister is here,\" he said. \"She's asking for you.\"\n\nBefore I realized what I was doing, I had pushed Max away from me and rushed toward the door.\n\n\"JT!\" Max complained.\n\nI turned and found her on the ground. Had I pushed her that hard?\n\n\"I'm sorry,\" I said, and helped her off the floor. I made a mental note to make it up to her later.\n\n\"I want to see her, too,\" Max said, pushing me out of the room.\n\nIt was easy to tell where Ketheria was. All the knudniks who had been working in the building were now crowded around the open door of the room where I'd made my first unwanted jump. Max and I tried to nudge our way through the crowd, but they were having none of that. I looked at Queykay, and he rolled his eyes.\n\n\"Move back!\" he demanded, and the crowd parted. Then he turned to me and said, \"The Citizens should be thankful that you're not the Tonat. I doubt you could make the Scion's enemies very nervous.\"\n\nI ignored Queykay and examined the circular room. It was filled with Citizens and bathed in a soft blue light from huge crystals lodged in the ceiling. Some aliens were draped in clothes that sparkled under the godly light, while others wore fitted suits with their Citizen's emblem displayed prominently, as if their arrogance wasn't enough to announce their status. I found Ketheria engulfed in some sort of ceremonial chair at the center of the room under the brightest crystal. I could swear the air around her seemed to sparkle. She was dressed in a pure white version of the robes we all wore.\n\n\"Wow,\" Max whispered.\n\nI hung back, watching the crowd watch Ketheria. With each delicate move she made, an anxious shiver rippled through the crowd. Ketheria's hair seemed fuller and longer, covering most of the silver band that was still wrapped around her head. The amber crystal at the center glowed brighter than I remembered.\n\n\"She looks different,\" Max said.\n\n\"It's scary,\" I said.\n\n\"These people worship her.\"\n\n\"That's even scarier.\"\n\nAliens dressed in navy jumpsuits and masked in ashen helmets that extended beyond their chins to form narrow chest plates kept the perimeter of the room clear. Queykay strode along the open passage, and Max and I followed. When we stepped onto the riser supporting Ketheria and her chair, she saw us.\n\n\"JT!\" she exclaimed. She stood up and glided over to hug me. I swear her feet never touched the ground.\n\nOver her shoulder, I watched the crowd react to her affection. Their admiration now seemed to consume me as well.\n\n\"Are you all right?\" I whispered. \"The last time I saw you, you didn't look too good. I was worried.\"\n\n\"I'm wonderful,\" she said, squeezing me tighter. \"I feel fantastic.\"\n\nShe pulled away and then hugged Max. \"I've missed you so much,\" she gushed.\n\n\"Me, too,\" Max replied.\n\nThen Ketheria took Max's hand and placed it in mine. She smiled at Max, and I felt my skin flush. Then I saw them. Her eyes. Ketheria's pupils were gone. Well, they were still there, but they were strange-looking. As if the glow was still circulating through her eyes, weaving through her irises, surfacing occasionally, and then settling back where her pupils once were. I caught Max staring.\n\n\"There's so much to do,\" Ketheria exclaimed. \"So much.\"\n\n\"What?\" I asked. \"What do you have to do?\"\n\n\"We! _We_ get to do it.\"\n\nJust then Queykay began to address the crowd. \"The Trading Council is pleased with your presence. The honorable Citizen Hach Ba Fay and myself welcome you, our closest friends, to this privileged viewing, but as you are aware, the Scion will not receive guests until the next phase,\" Queykay announced. \"The Scion must begin her work. Believe me when I tell you how excited I am that the Universe has chosen a Scion from among us. It echoes our belief that the Rings of Orbis are truly an important place. Again, the Trading Council thanks you for your patronage, and we are looking forward to a long and special relationship with the Scion. I am personally handling her security for the Trading Council, so everyone can leave here knowing that she is safe. You are all welcome to return to hear the Scion speak at the Cycle of Witnessing. Admission to this rare and distinguished event will be available for a nominal fee.\"\n\nMany in the crowd moaned, but most turned for the door. No complaining, no pushing or shoving, just an orderly reaction to Queykay's announcement. As the visitors streamed out, six Nagool masters squeezed in and waited patiently in the wings.\n\nKetheria acknowledged them and then whispered to us, \"We'll talk more later. I have so much to tell you.\"\n\nShe turned away and moved toward the Nagools.\n\n\"Ketheria?\" I said, but the Nagools had swallowed her up. I felt cheated. Where was my little sister? Who was this diplomat they had replaced her with? Queykay nudged me toward the door.\n\n\"Don't touch me,\" I spat, not caring whether he'd punish me for my rudeness.\n\nQueykay squeezed my shoulder, digging his fingers into my skin.\n\n\"Do as you're told, knudnik,\" he sneered.\n\n\"But why do I have to leave? She's my sister.\"\n\n\"That is irrelevant. She is far more important than that label. Now, do as you're told and run along, knudnik.\"\n\nI reached across with my robotic arm and grabbed Queykay's wrist. I applied just enough pressure to make him grimace and his fingers retract from my skin.\n\n\"Do not underestimate me,\" I warned him.\n\n\"JT, what's gotten into you?\" Max cried. \"Stop it!\"\n\nThen I felt a stinging sensation run up my arm as if something bit me. I pulled away, and Max grabbed my wrist, dragging me toward the door. I looked back and caught Queykay smirking.\n\nOutside the room, Max continued to drag me away from everyone else. When we were alone, she pushed me up against the wall.\n\n\"What are you doing?\" she hissed.\n\n\"What am _I_ doing?\n\n\"Yes! That's a Trading Council member you just assaulted. Do you want to get thrown into one those blue cells again, or worse? What's wrong with you?\"\n\n\"With _me_? There's nothing wrong with me. I want to see Ketheria! If that even is Ketheria. Did you see her eyes?\"\n\n\"Yes, her eyes seemed a little strange, but she said it herself: she feels fantastic. You should be happy right now.\"\n\n\"Well, I'm not. That's not my sister in there. She's changed.\"\n\n\"You're the one who's changed, JT. You avoid me for a whole phase \u2014 I have no idea why. You freak out at the slightest thing and then disappear into thin air.\"\n\n\"It has to do with her. I'm sure of it.\"\n\n\"Then what happened at Kasha's? Ketheria wasn't there. It was just you getting angry again. Soon you'll be jumping who-knows-where around the ring.\"\n\n\"If I could just talk with \u2014\"\n\n\"I don't have time for this. Figure it out and let me know.\" With that, she turned and stormed away.\n\n\"Max! Wait!\"\n\nI stood frozen, my brain unhinged, waiting for any part of my consciousness to take control. My body ached as the sound of Max's boots against the hard floor faded. _Go after her!_ But I turned to find Ketheria instead.\n\nI returned to the hall where I had left my sister. I didn't feel completely in control. I was relieved to not feel the nausea and headaches I had been experiencing for phases, but still I felt different. I had an overwhelming urge to be near my sister. I only hoped nothing was wrong. I found the room still lit, but empty. I spotted another door at the back of the room behind Ketheria's chair. I slipped through it and moved quietly, hoping Ketheria was still inside with the Nagools.\n\n\"Ketheria?\" I whispered. The room was so still that I could hear my own heartbeat.\n\nI moved into another room, hoping to find Ketheria, but instead I found two Space Jumpers guarding an archway. I knew my sister was through there.\n\nWhen they saw me, the Jumpers stiffened and closed the space between them. I hesitated but tried to act as if I was supposed to be there. As I moved toward them, the Space Jumper on my left said, \"She's not seeing anyone right now.\"\n\n\"I'm . . . I'm the Tonat,\" I lied, hoping it would work.\n\n\"The Tonat? Really?\" the other one said. \"As far as I understand, you've chosen _not_ to be one of us. Instead, you let the insects who run the Council do what you were born to do. Where is the courage in that? Now, get out of here before you get hurt.\" His words hissed through a face mask that covered the lower part of his face.\n\n\"Well, then, tell my _sister_ that her _brother_ wants to see her.\"\n\n\"Go home, _popper,_ \" scoffed the first one.\n\n\"What does that mean?\"\n\nThe other one gave a knowing grunt.\n\nI moved forward, trying to squeeze between them. They both reached for their weapons and pressed their shoulders together. The clacking of metal and the hum of their plasma rifles changed my mind.\n\n\"Ketheria!\" I shouted. \"Ketheria!\"\n\nIt felt childish, but what else could I do? It was my _sister_ back there. Why couldn't I see her? _They would let you pass if accepted your destiny. You are the Tonat,_ someone whispered inside my head. It wasn't Vairocina \u2014 that's for sure. The voice rattled me. I stumbled back, waiting to hear it again.\n\n\"Go home,\" one of them growled.\n\n\"I don't have a home!\"\n\n\"JT?\" I heard Ketheria's voice from inside the room.\n\n\"Ketheria! These space monkeys won't let me through!\"\n\nMy sister squeezed between the Space Jumpers. \"It's all right,\" she told them. \"We're finished.\"\n\nThree Nagool masters slid out from behind her and slipped away. My skin prickled as their robes brushed against me. One glanced back at Ketheria, and she nodded, smiling. All I could do was stand there and stare. I was not part of Ketheria's world anymore.\n\n\"I'm glad you came,\" she whispered to me. \"Come. There's lots I want to tell you.\"\n\n\"Finally,\" I said under my breath.\n\n\"Popper,\" one of the Space Jumpers muttered as I elbowed my way past them.\n\nInside Ketheria's room, I stepped around bowls of fruit and flowers. The same ones I had seen delivered earlier that cycle. Long silks clung to the pale stone walls, and lights seemed to sparkle through pinholes in the rock. The floor was padded with a thick carpet that swallowed up the soft tones that resonated off metallic bowls of water placed about the room like sculptures.\n\nI spotted Nugget sitting in the corner. He saw me as well and jumped to his feet.\n\n\"Nugget! What are you doing here?\"\n\n\"I take care of Ketheria,\" he proclaimed, standing in front of me with his fists on his waist and his feet planted shoulder-width apart. The bald little beast had hardly changed. His pink skin seemed a little more burgundy, but he still sported the same thick, protruding lower jaw, and he had it raised proudly (as usual).\n\n\"Good for you,\" I said.\n\n\"Someone has to do it,\" he grunted.\n\nI chuckled. \"You, too, huh?\"\n\n\"Let him be,\" Ketheria said. \"He makes a good bodyguard.\"\n\n\"Do you need a bodyguard now?\" I asked.\n\n\"So they tell me.\"\n\nI didn't want to get into an argument with my sister, so I turned away to admire the room.\n\n\"What is this place?\" I asked.\n\n\"It's my place,\" she replied.\n\n\"You mean you're not staying with us?\" I knew it was a dumb question.\n\nKetheria shook her head, smiling and resting herself on a long cushion on the floor near a shallow metal pan. A blue flame flickered from is center. Ketheria tapped the cushion, inviting me to sit, which I did. I couldn't help but think how much older she seemed.\n\n\"Isn't it amazing?\" she gushed.\n\n\"Isn't _what_ amazing?\"\n\n\"All of this,\" she replied, stretching her arms out.\n\n\"Is it? I wouldn't know.\"\n\nKetheria frowned. \"Why do you have to be like that? Aren't you the one who always talked about having some sort of purpose on Orbis, something to do? That's all you ever talked about. I loved listening to your stories about the rings when we were on the _Renaissance._ Now we have everything you ever wanted. Why is this so hard for you to accept?\"\n\n\"Because I don't see it that way at all.\"\n\n\"What do you mean?\"\n\nI stood up, feeling an argument coming on again. I inspected the room once more and noticed several antechambers that led from this main room, each with the same sparkling walls. The place was so still, I could hear myself breathe.\n\n\"We didn't get to pick this,\" I said, turning back to her. \"You didn't choose to be the Scion. I don't even know what you are, really.\"\n\n\"Some things choose us, JT.\"\n\nI shook my head. \"It doesn't make sense. Why us? I think they know more and they're not telling us. That scares me, Ketheria, and it should scare you as well. You know their history, their greed. This doesn't feel right.\"\n\n\"Who? Who are you talking about?\"\n\n\" _Them,_ Ketheria. All of _them._ The Trading Council, the Citizens, the Trust, even the Keepers. This is their world, not ours.\"\n\n\"I'm not the one who has changed, JT. Listen to yourself. This is not the brother I know. You couldn't wait to get to the Rings of Orbis and start a new life, and now you're going to be the Tonat.\"\n\n\"No, I'm not!\"\n\nKetheria cocked her head at me. The swirling in her eyes intensified, and I looked away. It was freaking me out. When I turned back, Ketheria had a large book opened on her lap and she was reading something. The book's pages were yellowed, and its edges looked tattered.\n\n\"What's that?\" I asked.\n\n\"It's a book,\" she replied.\n\n\"I know that, but why do you have it?\"\n\n\"The Nagools gave it to me.\" Ketheria closed the book and slipped it partly under the cushion. It was too thick to go willingly.\n\n\"Don't do that, Ketheria.\"\n\n\"Do what?\"\n\n\"Be like them,\" I said. \"Toy with your answers when you know the truth. I'm your brother. I deserve the truth.\"\n\n\"Sit down,\" she said. It was an order, and I obeyed.\n\n\"You must stop this. It's only a book. It helps me understand the reactions of those around me. The awakening is not finished yet. There are fourteen stages in all, and there is much I have to learn. That's why I have this book. That is all, nothing more.\"\n\n\"What do you have to learn?\"\n\nThen Ketheria reached up to the metal band that wrapped around her head and removed it with a click.\n\n\"I didn't know you could do that,\" I said.\n\n\"The Nagools showed me how.\"\n\nKetheria turned it over and pointed at the underside. \"See that?\" she said.\n\nI leaned in and saw an OIO symbol carved into the metal behind the amber crystal. \"Yeah. What does it mean?\"\n\n\"The person who made this knew he was making it for me.\"\n\n\"That's impossible.\"\n\n\"That's the first thing you have to change, JT. Anything is possible. Learn that now. You are so bound to a false vision of how you _think_ your life should be that your eyes are closed to everything around you. Don't feel bad, though. Most of the universe is like that.\"\n\nKetheria pushed the metal crown toward me. \"Take it,\" she urged.\n\nThe metal was warm in my hands. I rolled it around and ran my fingers over the OIO symbol. \"So what does it mean, then?\" I asked, trying to sound open to her ideas.\n\n\"The person who made that \u2014\"\n\n\"I thought that guy Tinker made it.\"\n\n\"He did. He knew the Scion would be forced to wear it some cycle. Tinker is a believer, and he worked with the Nagools to create something that would enhance the Scion's abilities while allowing the Citizens to believe that he or she was under its control. That's why Tinker was so freaked out when he met us. He recognized his work and knew who I was. Who _we_ were.\"\n\n\"But what does that mean?\"\n\n\"There are many, many forces at work here, JT. This is bigger than me; this is bigger than _you._ \"\n\nI stood up again. \"You make me sound like some self-centered fool.\"\n\nKetheria didn't reply.\n\n\"I'm not just thinking about myself, Ketheria,\" I argued. \"Did you ever think that we might just be tokens, meaningless gambits for them to use?\"\n\n\"Pawns in their game?\"\n\n\"Yes!\"\n\n\"Now _you_ are a clich\u00e9. This _is_ our game, JT. That's what I'm trying to tell you. You _are_ the Tonat. There is no choice. This is your life now whether you accept it or not.\"\n\n\"Well, I won't.\"\n\n\"Max will understand,\" she said. \"Would you like me to talk to her?\"\n\n\"Max! Are you reading my mind right now, Ketheria? That's not fair. Get out of my head.\"\n\n\"Max is worried about losing you.\"\n\n\"Stop it, Ketheria. I don't want to talk about Max.\"\n\n\"She's smart. More than you give her credit for. She knows what the life of a Space Jumper will be like, and there is no way that life can include her. It's Max that's being selfish here, JT.\"\n\n\"Don't say that!\"\n\n\"I'm sorry, but it is true.\"\n\n\"I'm leaving,\" I announced.\n\n\"Will you come and visit again?\" Ketheria asked me as if my being upset had little relevance. She opened her book again, and this time she made a mark in it.\n\n\"What's that for? Why are you doing that now?\"\n\n\"I explained that to you already,\" she said. \"Are you going to come back?\"\n\n\"I doubt it,\" I snapped. \"Your goons here won't let me in, and why do you have Space Jumpers guarding your room, anyway? They're supposed to be banished. If Queykay finds out, he'll surely notify the Council. He's a member, you know. They could cause a lot of trouble, Ketheria.\"\n\n\"Many things are changing, JT.\"\n\n\"Yeah, well, I'm not one of them.\"\n\nI stormed out of Ketheria's room and past her Jumpers.\n\n\"See ya, popper,\" one of them joked.\n\nI spun around. \"What does that mean?\" I yelled, and shoved him. The other Space Jumper slammed the butt of his rifle into my stomach. I heard them both laughing as my body and mind were torn through space and time yet again.\n\n#\n\nThe stink and decay were familiar to me now, so I knew I had jumped to Murat. The nausea returned, too; I felt my stomach tighten and push toward my throat. I was once again in an alley. I leaned against an abandoned transport \u2014 one wheel was missing, and the engine had been ripped out. I figured it wasn't going anywhere, so I just lay back and closed my eyes. The cold metal felt soothing against my neck, and I took this private moment to catalog the recent events in order to establish some direction in my life.\n\nIt was obvious to me that the awakening was changing Ketheria on a deeper level than I had even imagined. She seemed so different to me now. I felt like the Rings of Orbis had taken my sister from me. It was one more reason to hate it here. It was almost as if she was on _their_ side now, but I couldn't figure out when that line had been drawn. And where did that leave me? Ketheria had always been on _my_ side. I felt more isolated than I had on the _Renaissance._\n\nMy problems with Max were not making any of this easier. How could I keep my promise to her now? Could I really take care of Ketheria if I was no longer part of her inner circle? The fact that I was being shut out of Ketheria's life simply dumbfounded me. What if being the Tonat was the only way I could protect my sister? But if I became the Tonat, I was certain that I would lose Max, and I was not prepared to do that.\n\nSitting against the cold machine and feeling sorry for myself certainly wasn't going to help me. I knew that much. I needed a plan. What could I use to my advantage? Well, I could move freely about the ring, while Space Jumpers had to stay in the shadows. Maybe there was some way to control this spontaneous jumping. I also had my arm. Its robotics had come in handy more than once. And there was my alliance with Vairocina. That relationship was very important to me. But best of all, I could get inside their precious central computer whenever I pleased. _My side_ was looking pretty good. It was time to learn what they knew.\n\n\"Vairocina?\" I said.\n\n\"Yes, JT?\"\n\n\"Can you determine my location?\"\n\n\"Yes. You're in Murat. What are you doing there?\"\n\n\"Long story. Do you know about the place where the Keepers do their charity work on Orbis 4?\"\n\n\"I believe it's called the Center for Relief and Assistance. It's 3.7 kilometers from where you are now.\"\n\n\"Great. Which way?\" I asked, getting up.\n\n\"Up ring. It will be on your right near the center of the city but, JT, I would like to talk about the information you asked me to find.\"\n\n\"Oh, sure, but can it wait, Vairocina? I need to talk with a Keeper right now. We'll chat when I'm done.\"\n\n\"Certainly.\"\n\nI may not have decided to be the Tonat yet, but I was certain I could garner valuable information while they tried to convince me. Drapling's defenses were down. He wanted me to become the Tonat so bad, I could almost smell it on him. I was going to leverage his desire to get something from _me_ to get what I wanted from _him:_ information.\n\nA large, scarred metallic orb drifted over me. Six bluish lights crawled along the orb's surface and scanned the area around me. _Security?_ The searchlights converged on the ship's belly and focused on me for only a moment. The orb then rotated and drifted away. I figured Vairocina must have sent it, so I followed my makeshift escort.\n\nMurat's buildings grew taller as I marched toward its center. Instead of building out, Murat had built upward. I began to notice a larger, more modern city beneath the refuse. Skyscrapers fashioned in the images of those you might find on Orbis 1 poked through the city's poorer framework. I followed a narrow canal of green silt, which seemed to flow in and out of the city, and stopped just inside the densest part of Murat. Next to me, a metal and glass pod cracked open and a gangly alien unfurled himself. I looked up and saw more pods mounted above that one, each attached to the same narrow beam that arched up and over my head. I could see more aliens lumbering inside the dull, well-worn pods. They were roughly the same size and shape as the nurture pods we had used on the _Renaissance._ In one capsule, I saw a female with two small children. I could not even imagine having to _live_ inside one of those things.\n\nI moved away from the pods and squinted through the mountains of oxidized metal framework that formed a forest of trading chambers and makeshift shelters. There was no sign of the Center for Relief and Assistance, so I kept moving.\n\n\"Vairocina? Can you tell if I'm close to the Center yet?\"\n\nThere was a pause before she replied. \"You need to walk about six hundred meters and you will find it on your left.\"\n\n\"Thank you.\"\n\nAs Vairocina predicted, the slanted glass structure appeared like a distress beacon amid the chaos of Murat. A steady stream of aliens flowed through the Center's pristine doors, and I stepped into the flow.\n\nInside I found more aliens, sitting at O-dats, while three Keepers emerged from light chutes located behind a large oval counter. I watched as each Keeper retrieved a new Citizen and then left with him or her through the same chute. I searched for an empty O-dat, but they were all occupied, with at least two or three aliens waiting their turn. How many aliens did the Keepers help? I decided to find out for myself.\n\n\"Excuse me,\" I said to the Saliman standing in front of me. I could always spot a Saliman because they had big hornlike ears that pointed backward. \"Are you a Citizen?\"\n\n\"Lot of good it does me,\" the alien grunted, waving one of her thick pink forearms in the air for emphasis. \"The First Families have the system so rigged, I was better off as a knudnik.\"\n\n\"What will the Keepers do for you?\"\n\n\"Keepers? Don't confuse the Descendants of Light with the Keepers. They may look the same, but they are definitely two different breeds. Watch out for the DOL,\" she said in a gossipy tone, and then glanced at the Keeper standing a few meters to her right.\n\nAnother alien had been eavesdropping on our conversation. \"Get your pass and get out of here,\" he said.\n\n\"Pass?\" I said.\n\n\"Through the wormhole,\" replied the Saliman. \"It's the one good thing about the DOL. They love to see you go.\"\n\n\"And I'm going to keep them happy,\" added the other one. I turned toward that alien. His face was almost as wide as his shoulders. It was a Roshilon. His eyes blinked at me as they struggled to peer around his big bony face. \"There's no way I can afford to live on these rings, and there is absolutely no way I can afford the tax to travel through that wormhole. The DOL arranges your passage for free. It's the best assistance this place can offer.\"\n\n\"Is everyone here leaving?\" I whispered.\n\n\"If they're smart, they will.\"\n\nThe Saliman squatted in front of the terminal and reached up with his short arms. I stood back, digesting what the aliens had told me, when I heard my name.\n\n\"Johnny Turnbull!\" Drapling called out to me. \"What a pleasure. Are you lost again?\"\n\nI saw the Saliman glance at me over his shoulder.\n\n\"No,\" I said. \"I was hoping we could talk.\"\n\n\"Absolutely,\" Drapling cried. \"Follow me.\"\n\n\"Get your pass!\" hissed the Saliman.\n\nI followed Drapling through the light chute, then emerged to find him waiting next to a tall green crystal anchored in the wall. It was one of many that lined the polished stone hallway.\n\n\"This way,\" he said, motioning.\n\nI followed Drapling down the hallway and across the glossy floor. Below my feet, buried about ten centimeters under the clear floor, I could see rows and rows of loosely arranged hand-fashioned copper slabs. These plates were separated by globs of rust-colored grout, just the sort of sloppy brick-and-mortar job one might find in Murat.\n\n\"What are these?\" I asked, pointing at the floor.\n\n\"Keepers who have served their purpose beyond all expectations,\" he replied.\n\n\"They're graves?\"\n\n\"This is much more honorable,\" he replied.\n\n\"Walking on them is honorable?\"\n\n\"They're still serving the greater good. I find that honorable, don't you?\"\n\n_No! How is spending eternity as a paving stone honorable?_ I stepped away from the tombstones, tiptoed along the grout, and followed Drapling into a room near the end of the hallway. He sat in a sloped chair made from some sort of greenish, silky material. There was no place for me to sit.\n\n\"How is Ketheria?\" he asked.\n\n\"She says she's fine,\" I replied, walking past amber lights embedded in the floor.\n\n\"I take it you do not agree.\"\n\n\"Tell me about the Tonat, Drapling. I may have been hasty in my decision.\"\n\nDrapling stood up. There was that pause again. He was thinking about the proper response, but at least I had his attention.\n\n\"What would you like to know?\" he asked carefully.\n\nHe was stalling. I could feel it.\n\n\"Have there ever been other Tonats?\"\n\n\"Of course.\"\n\n\"Is the Tonat always a sibling of the Scion?\"\n\nDrapling got up and slid along a narrow table near the back of the room. I watched as he lifted a thin carafe and poured an opaque liquid into a fragile fluted glass. He placed the glass on the table before answering me. I knew this entire charade was just a way to give him time to think about his answer.\n\n\"That is hard to say,\" he finally responded.\n\n\"Well, is it a condition of their relationship?\"\n\n\"We are all connected in more than one way, Johnny. Even you and I are connected within this universe. OIO tells us \u2014\"\n\n\"Just answer the question, please,\" I interrupted.\n\n\"I am confused as to the point of your question. I should inform you that I have not met any other Scions. Your sister is my first. A Scion is a rare and extraordinary individual, but unfortunately most do not live long. Your species has destroyed many.\"\n\n\"You mean humans?\"\n\nDrapling nodded but said no more. I think he was waiting for my response to his doomsday claim, but I didn't take the bait. \"Not every Scion has found his or her Tonat in time. I believe Ketheria is unique in having a Tonat who is also her brother, although you seem indifferent to her safety.\"\n\n\"I care for my sister more than you know. Tell me more about this _connection._ \"\n\n\"You must be feeling it now. You became sick as the awakening started, did you not?\"\n\n\"I did.\"\n\n\"See? And your movements through space and time, despite being so unorthodox, are, I am convinced, also connected to your relationship to the Scion. As her awakening continues and as you move closer to your destiny, I am convinced this little anomaly will disappear.\"\n\n\"How is that possible?\"\n\n\"You are connected to her like no other creature in this universe. You feel her pain and sense her danger. This will grow stronger as the awakening continues. You cannot escape this. A Tonat is burdened with all the pain the Scion experiences. You are like a valve that releases this pressure so that she may live in the light and bring harmony to the universe. You, in turn, must use this pain to protect her. You must fashion this energy as a soldier fashions a weapon. It is your greatest strength. That is why you are feared. As the Scion grows and takes on the pain of all those suffering in the universe, so, too, does your power grow. The longer the Scion lives, the more feared you become. This pain and suffering will strengthen you and allow you to do what others cannot. Feel her, sense her, think like her. There is not a Space Jumper in the universe that will protect your sister as you will. You and the Scion are connected like no other individuals in this universe.\"\n\n\"But _how_ did this happen, Drapling? Who made this connection? I'm certain not every brother and sister has this kind of connection.\"\n\nDrapling sat the glass down, but did not respond \u2014 again.\n\n\"Drapling, tell me how this happened. Please!\"\n\n\"You may want to sit down,\" said a voice from the door. I spun around to find Theylor entering the room. He extended his arm toward the chair. \"Please, sit,\" he said. \"I believe the answer to your question might not please you.\"\n\n#\n\nI slept straight through the next two cycles. My head was so full, I couldn't hold it up, anyway. I spent the first diam of the third cycle simply staring at the ceiling. Ketheria came to visit me while I was awake, but I pretended to be asleep. I figured her telepathy would give me away, but she let me be, all the same. Even Queykay left me to myself, but there was nothing unusual about that.\n\nI finally got up and washed. Moments later I heard a soft tapping at my door.\n\n_Go away,_ I thought\n\nWhen I didn't answer, they did go away. I had already made my decision about what I was going to do, and I didn't need others trying to change my mind. Despite the odds, I was even more resolved to have it my way now. _How dare they do this to me? How_ dare _they?_\n\nWhen I felt ready, I ventured out into the building. The first person I wanted to find was Max. I found her in her room with Grace and that other kid, whatever his name was.\n\n\"Hi.\" I waved from the doorway.\n\n\"JT!\" Max shrieked, and bolted to her feet. Grace got up, too, and kicked the other kid to do the same.\n\n\"We were just on our way out,\" Grace declared. \"Good to see you up, JT. Max, we'll talk about it more later.\"\n\n\"Sure,\" she replied, looking anxious for them to leave.\n\nI nodded at the other kid as he and Grace slipped out of the room.\n\n\"What's his name now?\" I whispered to Max.\n\n\"Dante,\" she replied.\n\n\"What were you guys talking about?\"\n\n\"Nothing. Sit. Are you all right? Theylor said you got sick in Murat. He told us not to disturb you.\"\n\n\"Was that you who knocked earlier?\"\n\nShe smiled. \"Yeah. I'm sorry. I just needed to apologize about the other cycle. Knowing you were here and I couldn't talk to you \u2014 well, it was driving me crazy. Ketheria is your sister and that _is_ precious. I had no right to say what I did. It was horrible. Can you forgive me?\"\n\n\"You don't need to be forgiven, Max. I was the one who was acting like a malf \u2014 to everyone. I know that now, but that's going to change \u2014 I promise. It's just like you said: one more rotation and then we can do whatever we want. We can even leave the rings if you want. I'll lead the way.\"\n\n\"Really?\"\n\n\"Really,\" I assured her.\n\nMax just stood there, smiling. \"So, now what?\"\n\n\"Let's have some fun.\"\n\nA large furry knudnik with thick arms appeared at the door. It was a Garin, and they were only assigned to Trading Council members. \"Queykay sent me to retrieve you. Your sister has requested you.\"\n\nI looked at Max and then back to the messenger. \"Tell Queykay I'm sure the Council can deal with the Scion. Tell Ketheria I'll come by later. Much later.\"\n\nI grabbed Max's hand and pulled her out of the room as the Garin stepped aside. I really don't think he knew what to do, but I didn't care.\n\n\"You do not have permission to leave,\" he challenged me.\n\nI turned and faced him, Max's hand firmly in mine. \"This is not your fight. Are you going to stop me?\" Max stepped next to me.\n\nThe Garin sucked the air through his teeth, and I adjusted the controls in my arm just in case.\n\n\"Well?\" Max said.\n\nHe glared at us for another moment. \"I must report this,\" he spat, and then stormed off.\n\n\"You should get sick more often,\" Max teased.\n\nI could only smile. The fact was that my stomach had been doing backflips ever since I refused to go to Ketheria, and I was afraid to open my mouth in fear of what would come out.\n\n_Live with it,_ I told myself. _That's your new motto._\n\nTypical of Max, she had already found a secret route into Murat. I followed her through one shortcut that was nothing more than a crack in a concrete barrier. The maze of trading chambers and living quarters was like second nature to her.\n\n\"Where are we going?\" I asked.\n\n\"It's a surprise.\"\n\nI followed Max down a series of steps that ended in a small amphitheater carved into the foot of one of Murat's superstructures. The building's green glass bathed the entire courtyard in its reflection of a distant, dying sun. Max found a spot on the stone seats and settled into the eerie afterglow. I must admit, the effect did a pretty good job of masking the city's decay.\n\n\"What are we doing here?\" I whispered as we sat among other aliens, some of whom seemed to be sleeping.\n\n\"This is a special cycle on the Rings of Orbis. Not one that everyone celebrates, but quite a few do. Look up in the sky.\"\n\nMax pointed down ring and up about sixty degrees. I followed her finger to see what she was pointing at. \"See it?\" she said. \"The rings. They spell OIO.\"\n\nIn the sky, Orbis 1 and Orbis 3 were positioned next to each other, and Orbis 4, the ring we were on, ran up between them. It did spell OIO \u2014 well, kind of, anyway.\n\n\"The golden thing is that OIO works in any language. It's really a symbol.\"\n\n\"I always thought the central computer translated it for us.\"\n\n\"Everyone gets the same translation. That's one of the things that makes it so special. The alignment happens once every rotation.\"\n\n\"What's going to happen now?\" I asked.\n\n\"It's a celebration. Remember that place you took me to on Orbis 3? The place with the musician?\"\n\n\"He was amazing.\"\n\n\"Then I think you might like this. Watch,\" she said, holding her fingers to her lips.\n\nI looked at the stage near the bottom of the amphitheater. A few aliens were setting up musical instruments among the rubbish. Single notes washed over me as they tuned their stringed devices. I watched more musicians join the group, and the air soon resonated with a cacophony of notes and sounds as they set up their instruments. I fidgeted in my seat, anxious to hear them play. Max looked at me and smiled.\n\n\"Thanks,\" I whispered.\n\nThen she leaned toward me and rested her head against my shoulder at the precise moment the musicians came together. A wall of sound fell upon us, and anyone who had been sleeping now sat up. It was amazing that amid all this atrophy, a sound so pure and so promising could lift me up and turn my dingy surroundings into the most exquisite concert hall in the universe. I sat with Max and listened without saying a word. We let the music fill in the spaces around us, and for that moment, I had everything I had ever dreamed about when I was on the _Renaissance._ It did not matter what they had planned for me. It didn't even matter what Theylor said they had done to me. I could resist it. I knew I could.\n\nThen I threw up. The feeling came so fast, I barely had time to react. As my mouth filled with vomit, I tore away from Max, horrified that I might puke on her. I unloaded the contents of my stomach on the unfortunate alien to my right.\n\n\"JT! Are you all right? What's wrong? Are you still sick?\" she cried.\n\nI couldn't face Max. I was so embarrassed and I didn't want her asking why I was still sick because I don't think could have lied to her just then.\n\n\"I'm sorry,\" I said to the alien next to me, but he didn't seem to mind. Instead he picked through the remains on his shirt as if I had passed him the leftovers of my meal (which I kind of did, in a way).\n\nI wiped my mouth and turned back to Max. \"I guess I'm not a hundred percent yet,\" I said.\n\n\"Let's go back,\" she insisted.\n\n\"No, I said. I love this. I'm sorry.\"\n\n\"Don't be sorry. You're sick. What if it's something serious?\"\n\nIt was serious, _very_ serious. But I couldn't tell her I was going to be like this for the rest of my life and that it was only going to get worse. My head was splitting now, and my underarms were soaked as well. If Max knew the truth, if she knew what Theylor had told me, I was certain that she would never accept me and I would lose her.\n\n_Live with it!_ I reminded myself.\n\n\"Let's get some water,\" she said.\n\n\"Good idea.\"\n\nReluctantly, I left the amphitheater, following Max back up and into the street, mumbling to her the entire time that I was sorry.\n\nAfter I assured Max that I was fine, I told her, \"That's such a golden place. How did you ever find it?\"\n\n\"That's the thing, JT. There is so much like that here in Murat, but the Trading Council won't fund any of it. In fact, they made it illegal for certain groups, like those musicians, to even perform concerts anymore.\"\n\n\"That's stupid.\"\n\n\"But it's happening. The city is jam-packed with these little pockets of creativity. It's really inspiring. I mean, despite the conditions these people are forced to live under, they are still able to connect to the Source.\"\n\n\"The Source?\"\n\n\"Creativity is the best way to connect to the Universe,\" she replied matter-of-factly as she dashed into the surface street, pausing for a makeshift tram to pass.\n\n\"Do you really believe all that stuff, Max?\"\n\n\"You mean OIO?\"\n\n\"Yeah. I don't get it. It just smells like another system of rules.\"\n\nMax stopped in front of a fountain where water bubbled out of a plastic pipe. \"Here,\" she said. \"Drink this.\"\n\n\"Is it clean?\" I asked her.\n\n\"Crystal,\" she replied, and I drank. \"OIO's not like that at all. It's really an investigation of truths and principles that guide our Universe. It helps a lot of people remain calm in the presence of all the trouble and chaos around them. You ever see a Nagool get upset?\"\n\n\"None that I can think of. So that's it? It just makes you calm?\"\n\nMax turned to me and chuckled. \"No! You really don't get this stuff, do you? I'm so surprised. Look. It's very simple. The Universe is energy. Our thoughts and actions contribute to this energy and have influence over every creature within it. Negative or deconstructive energy created by individuals, and even societies as a whole, contribute to behaviors that are self-destructive, like a hidden virus undetected in our psyche. That deconstructive energy feeds certain forces in our Universe and has the power to corrupt entire cultures \u2014 look at the Trading Council. Even when they know their actions are destructive, they continue because they are addicted to this energy. Remember Theodore and those tetrascopes?\"\n\n\"What does this have to do with Ketheria?\"\n\n\"Some say the Universe chooses a Scion. The balance of constructive and deconstructive energy flowing from the Source is very delicate. The universe can self-destruct under the sheer mass of unopposed deconstructive energy. When Ketheria has completely awoken these negative forces will no longer influence her. Her nodes will be in perfect sync with the brightest part of the Source. Nagools consider Scions to be the only enlightened individuals in the universe. A Scion's presence alone can raise the consciousness of another individual by absorbing all their deconstructive energy. To become conscious is the greatest gift a Scion can give you, but it scares the crap out of the Trading Council.\"\n\n\"Why?\"\n\n\"There is a direct link between consciousness and a sense of self. When your nodes are clogged by deconstructive energy, you feel worthless and incapable of achieving anything, which makes you very easy to control. The Trading Council likes their knudniks that way. On the other hand, a higher consciousness can make you feel like you can do anything. Even run these rings.\"\n\n\"No wonder so many Scions have been killed,\" I said.\n\n\"It reminds me a lot of the way you acted around Switzer on the _Renaissance._ You really kept your head when most people wouldn't.\"\n\n\"I don't consider those the finer moments in my life. I'd rather forget them,\" I said.\n\nMax smiled, moved toward me, and put her arms around me. \"Did you like me back then?\"\n\n\"Did you like me?\" I asked, resting my nose against hers.\n\nWhen I breathed, Max winced and pinched my lips together with her fingers. \"You shouldn't talk. C'mon, let's get something to eat. Something that will settle your stomach.\"\n\n\"What? Does my breath stink? Great!\"\n\n\"C'mon. I know another place,\" she yelled as she ran ahead.\n\nI looked at Murat a little differently after what Max had told me. Instead of seeing trading chambers simply filled with junk, I began to notice exquisite little dolls fashioned from scraps of plastic and thread as well as detailed paintings on discarded scraps of metal or wood, all hung neatly in the chambers and ready for sale. Windows were no longer stacked with discarded electronics but rather parts used by skilled technicians repairing anything their customers could bring them. Despite the obstacles created by the First Families, these new Citizens had carved out an existence for themselves.\n\n_Just like you,_ I whispered to myself. I sure was going to try.\n\nI saw Max stop under a huge splash of red light outside a tiny chamber. The doorway was so small, I was forced to turn sideways to enter, and once I was inside, the smell of cooking grease violated my senses. To my right, I saw three Bachaks stuffed behind a tall counter lined with mismatched metallic stools. I watched as these brawny-looking aliens with thick forearms jammed pouches of fried foods under tiny light chutes that delivered the food to smaller tables along the wall. Max and I sat at the farthest table from the counter. She was giggling as we sat.\n\n\"Golden place, huh?\" she said.\n\n\"Small,\" I remarked.\n\n\"This is only part of it.\" Then Max knocked on the wall behind her bench. A few moments later, part of the wall slid back.\n\n\"Max!\" cried the alien who opened the door. I stepped back. It was a Belaran. Her inky black skin and sharp features immediately brought back memories of Madame Lee, who had tried to kill me on Orbis 1.\n\n\"Hi, Tic. I brought my friend, the one I told you about. I hope you don't mind,\" Max said.\n\nTic looked me over and smiled. \"Of course not. Come in!\" Max squeezed past, and I followed. \"Who would refuse the Tonat?\" Tic whispered as I passed. I spun around to look at her, but the alien's back was to me while she locked the little door.\n\nI turned back and followed Max down the narrow hallway and into a much larger, circular room. We stepped over cushions scattered on the floor, and I ducked under one of two metal pots that hung from the center of the ceiling. The pots leaked blue smoke that wove its way through the silks also hanging from the ceiling. Max plopped onto one of the cushions, and I did the same as Tic gathered some drinking glasses. The Belaran appeared much older than Madame Lee and walked with a slight stoop. To me, Tic seemed like a bland version of the warrior I once knew, but I was still nervous. Belarans had a fierce reputation.\n\n\"JT wasn't feeling well, and I didn't want to go home,\" Max said.\n\n\"You are always welcome here,\" Tic exclaimed. \"I have just the thing that will help your friend as well.\"\n\nWhen Tic left the room, I whispered to Max, \"What's a Belaran doing here?\"\n\n\"We are not all as fortunate as some of our race,\" Tic answered for herself, returning to the room with three glasses.\n\n\"I'm sorry,\" I said. \"I didn't mean to be rude. I was just surprised to see someone from Zinovia in Murat.\"\n\n\"Zinovia is an amazing planet, but far too ruthless for my tastes. I like the simplicity of Murat. Don't you?\" she said as she passed me a glass and then another to Max. I noticed that mine was filled with a black liquid while Max's was yellow. \"Drink it. It will help your stomach.\"\n\n\"How did you know it was my stomach?\" I asked.\n\nTic did not reply. Instead, she took a sip from her own glass and glanced at Max. \"I trust you're feeling fine,\" she said.\n\n\"Yes, but I do need to use your bathroom.\"\n\n\"Of course. You know where it is.\"\n\nAfter Max left the room, I sat in awkward silence while Tic just stared at me. I tried to drink the liquid, but the smell only twisted my stomach more.\n\n\"Trust me: it will help you. I have more if you need it.\"\n\n\"Thanks, but this will pass.\"\n\n\"Will it?\" she asked.\n\nI glanced up at Tic. What did she know?\n\n\"The Belaran believed that they possessed the Scion at one point, as well you know,\" she whispered. \"In fact, I believe you met her once.\"\n\n\"Who?\"\n\n\"Madame Lee believed that she was the Scion, or at least she wanted to be. Such a taste for power, that one.\"\n\n\"Really?\"\n\n\"She was livid after the Keepers had proven her unworthy. She even had a Tonat.\"\n\n\"Where is he?\"\n\n\"Why do you assume it was a male?\"\n\n\"I'm sorry. Where is _she_?\"\n\n\"Dead. The genetic alterations killed her, as often happens when individuals try to force what should be a natural process.\"\n\n\"You know about that?\"\n\n\"It's written all over you.\"\n\n\"Don't tell Max. Please!\"\n\n\"Don't tell her what? That the boy she loves has been genetically altered by the Trust to protect his sibling? That her partner will forever feel the tug of the Scion even to the point of physical ailment?\"\n\n\"Max loves me?\"\n\n\"You miss the point. As the Tonat, your cell structure has been coded to respond to the needs of the Scion. Even if you do not want to be the Tonat, you cannot escape its effects.\"\n\n\"I know. The Keepers told me already. Theylor explained to me that even if I choose _not_ to be the Tonat, my genetic structure will fight me every step of the way.\" I stood up. \"I can't believe they did this to me.\"\n\n\"Understand that the Trust, those five patriarchic Space Jumpers, are wired to do one thing: create the Space Jumper that protects the Scion. That's all they care about. The Trust knows what they are doing. They've been getting ready for this just as long as the Nagools. Maybe longer. The Scion needs a Tonat.\"\n\n\"But this is supposed to be _my_ life. How can I have my own life and choose what I want to do when a bunch of aliens have already rewired me to protect another?\"\n\n\"Do you think it's fair not to tell Max? Do you think that's fair to either of you?\"\n\n\"Nothing seems fair on the Rings of Orbis.\"\n\n\"Then you should feel welcome here.\"\n\n\"What are you guys talking about?\" Max asked as she returned.\n\nI looked at Tic. I wasn't ready to tell Max. _Please don't,_ I thought.\n\n\"JT was telling me how my little concoction was making him feel much better.\"\n\n\"Golden!\" Max exclaimed. \"I knew it was a good idea to come here.\"\n\n\"You must take some home with you,\" Tic insisted. \"In case you feel a relapse.\"\n\n\"Thanks,\" I replied.\n\nWe sat and talked with Tic for some time. She was so much different from Madame Lee, although I sometimes caught glimpses of a ferocious warrior hidden in her chiseled bone structure and jet-black skin. Tic told us she had lived on Orbis since before the Citizens' uprising on Orbis 3. That's when she moved to Murat and began living like a knudnik. I was interested, but my thoughts began to drift away from the conversation. I was thinking about what to do with Max. Would she find out on her own? She wouldn't let me live like this, even if I explained to her that it wasn't that bad. Theylor told me that the symptoms would get worse the farther I traveled from Ketheria, but I figured I could live with throwing up every once in a while. Besides, I could always get more of Tic's magic drink.\n\nThe truth was, all _I_ wanted was to be with Max. The fact that my genetic structure had been altered to help protect Ketheria was not going to get in the way. Whoever did this to me had no right to do so. I would protect my sister, but I would have my own life as well. I had always protected her in the past, and I didn't see why I couldn't continue to do so. Ketheria had enough Space Jumpers around her, anyway. One more wasn't going to help. I would keep this a secret from Max and have my own life. At least that was my plan.\n\nBy the time Hach had returned from his business dealings, my plan seemed to be working. I had programmed the chow synth to create more of Tic's drink, and I drank it in private at the start of every cycle. I also tested Theylor's distance theory and began going out with the other kids to distribute taps around Murat. The only new symptom was a sharp headache stabbing at my temples as I ventured farther from home. It was still plenty of distance to lead my own life, despite what the Keepers had warned.\n\nThe guilt, however, was something I couldn't escape. Ketheria must have known that I had no intention of becoming the Tonat. She _was_ a telepath. But if she was disappointed in me, she never let it show. In fact, it seemed to me that Ketheria did her best to accept my decision; she was always assuring me that she was well protected. I made her promise to inform me if anything seemed out of the ordinary or dangerous. The thing was, Ketheria seemed so loved by everyone around her that I didn't understand why there was so much worry about her safety, anyway. Still, to demonstrate my ability to protect her, I made alliances with the knudniks who served her, cleaned her room, and worked in the building. I asked them to report anything suspicious. They eagerly agreed to help, but only reported that Ketheria spent every cycle with the Nagools. When I questioned her, she said that she was learning about the Universe and preparing for the Cycle of Witnessing. She never left the building, and the Space Jumpers always guarded her door. It seemed like a boring existence to me, but Ketheria looked happy and she was safe. That's all that mattered.\n\n\"There are some in the universe who are appalled by your sister's very existence,\" Hach told me when I asked about her security a few cycles later. He was dining in his room and had asked me to join him. \"Her existence is an affront to their own beliefs, and they refuse to see the truth.\"\n\n\"So you believe in it too?\" I asked.\n\n\"There's nothing to believe. OIO does not ask you to have faith in anything or follow anyone; it simply is. They say OIO is a seamless part of your own existence.\"\n\n\"I still don't get it.\"\n\n\"You don't have to. Listen to me. It is believed that the Ancients _made_ the first softwires.\"\n\n\"Made?\"\n\nHach stuffed a piece of meat in his mouth and shook his head. \"I don't know folklore. Don't drill me on it, but it's common knowledge that the Ancients gave them the technology for their belts, and we know the Ancients picked the first Trust \u2014 that council of Jumpers who now govern and train all Space Jumpers. You see, the Ancients knew that there would be many individuals in this universe, even believers, who would see any Scion as a threat. They have been right in the past. I'm afraid you can't fight this.\"\n\n\"I've heard that,\" I mumbled. \"So you're not upset with all these Space Jumpers hanging around here?\"\n\n\"I'm not, but I do worry for the Keepers. They are running a huge risking by parading them in the open like this. I can't help but feel they are taunting the Trading Council. The Keepers cannot afford a war. We've made certain of that.\"\n\n\"Then why do you allow the Space Jumpers to remain here?\"\n\n\"I suppose some things are worth the risk. At least the Keepers feel that way,\" he said, stabbing another piece of meat with his fork and winking at me. It was nice to have Hach back.\n\nA knudnik entered the room. \"Your guests have arrived,\" he announced.\n\nHach swallowed and said, \"Good. Have them wait. I'm not finished with the Tonat just yet.\"\n\nThe thin alien nodded before leaving. Hach put down his fork and knife and said, \"I know you're not happy with this arrangement, but as I'm sure you are well aware, you are in no position to deny me.\"\n\n\"I've been in Murat. I've passed out the taps. I've done everything that has been asked of me.\"\n\nHach nodded. \"The Scion's first public appearance is a few cycles away. The Cycle of Witnessing. I certainly can't have a bunch of Space Jumpers lining the stage, now, can I? Your presence will be required.\"\n\n\"Why? These people worship her. No one is going to hurt her,\" I complained.\n\nHach leaned on his elbows. \"On the planet of Sorlinda, maybe ninety million light-years from here, a very advanced society discovered that a Scion was among their ranks. They rejoiced. They celebrated. As far as they were concerned, _they_ were the chosen ones, but as they waited for the Scion to fully awaken, others on the planet decided that they, too, were worthy of this title. If the Universe had chosen one Sorlindian, why shouldn't it choose them all?\"\n\nHach grabbed another hunk of meat from the tray in front of him and plopped it onto his plate. He sliced it as he spoke. \"Come the Cycle of Witnessing, a powerful arm of its government seized the Scion and ceremonially sliced her up, serving her flesh for consumption to anyone in attendance at the Witnessing. They passed around pieces of her on plates, just as you might do at a banquet.\" Hach shoved a piece of meat into his mouth for emphasis. My stomach rolled over once, but it had nothing to do with my illness.\n\n\"Where was the Tonat?\" I asked.\n\n\"They had tricked the Tonat and drugged him. He was unconscious during the entire event. When he awoke, he was so enraged that he slaughtered every single person who had attended the Witnessing, and there were many. He piled the dead bodies in a pyramid on the exact spot where the Scion had died. Then he stole their precious metals and entombed the bodies in a silver shrine so no one would ever forget what they had done to the Scion. It was quite ghastly. In fact, the Tonat is now considered a monster in Sorlindian folklore.\"\n\n\"The Keepers never spoke of this,\" I said.\n\n\"I can see why. Even your own people have destroyed their share of Scions. One fable talks of a Scion who was nailed to a piece of wood while he was still alive; those who worshipped him stood around and watched him suffer. He died eventually, of course.\" Hach sipped from a goblet. \"Shall I go on?\"\n\nI shook my head. \"You're just trying to scare me,\" I said.\n\n\"I know you want to live your own life, but believe me when I tell you that your sister _needs_ you. I do not trust the Council. Do it for your sister, Ketheria, not for the Scion.\"\n\nThat was a dirty trick. \"Fine,\" I said. \"But since I'm not trained as a Space Jumper, I don't know what I'll do if anything happens at the Witnessing.\"\n\n\"Presentation will be our best defense.\" Hach said, and stood up, pushing his chair back. \"People fear the Tonat,\" he called out as he left the room. \"Use that.\"\n\n#\n\nThe Cycle of Witnessing was sort of like Ketheria's coming-out party. Anyone who had heard rumors about the Scion was now allowed to see her firsthand (if they paid the fee, of course). I stood back and watched as the corridors of Hach's newly created temple buzzed with gossip and the workers scrambled about under Queykay's precise, military-like instruction. He oversaw every detail, including the new robes he ordered us to wear. It was clear that the Trading Council wanted everyone to know that they were in charge of the Scion.\n\nThe new cream-colored robes were detailed with a broad, deep red collar marked with the OIO symbol. My outfit was different, though. Instead of a robe, I was given bloodred pants that matched the collars on everyone's robe. My pants flared behind my legs, leaving a short train of fabric as I walked. I wore a belt marked with the OIO symbol and a long, double-breasted jacket lined with gold buttons, each one sporting the Orbis emblem.\n\n\"I think you look . . . great!\" Max exclaimed, tugging at the jacket as I got dressed in my room.\n\n\"I look ridiculous,\" I complained.\n\n\"No, you don't. In fact, you look impressive.\"\n\nMax put her arms around me and kissed me.\n\n\"What's that for?\" I asked.\n\n\"I don't know,\" she said, blushing. \"Just because.\"\n\n\"Well, your robe looks very nice, too. You wear it . . . well,\" I said, searching for the proper word.\n\n\"You think?\" Max said, rubbing her hands over the material. It was so hard for me to compliment her without sounding like a malf. Did I tell her how much I liked the way the material stretched over her legs as she walked? Or the way it clung to her waist? It sounded stupid in my head. I couldn't imagine saying it out loud. I was staring at her and she saw me. Max kissed me again. \"Thanks,\" she whispered in my ear.\n\nTheodore charged into my room. \"Have you seen all the people out there?\" he exclaimed. \"The place is already full, and it doesn't start until the next spoke. No wonder the Trading Council wants to charge for this.\" He turned to Max. \"Queykay also wants us to hand these out among the crowds.\"\n\nTheodore held up four huge sacks filled with more taps. The bags were three times as big as anything we had ever handed out in Murat.\n\n\"Wow!\" Max remarked. \"Where is he, anyway?\"\n\n\"Not a clue. These were left outside my door with a note.\"\n\n\"We better get going. It will take all the time we have to hand them out,\" Theodore said.\n\n\"Queykay left me a note with my clothes. He asked me to stay with Ketheria,\" I told them.\n\nMax frowned. \"Aw, come with us. Please. I was getting used to you in your new clothes.\"\n\n\"Yeah, that's way more golden than what we have to wear,\" Theodore added.\n\nI looked at Max and Theodore, with those huge bags of taps. \"What can it hurt?\" I said. I'd get to the stage in time. No one was going to carve up Ketheria and feed her to the Citizens. I would much rather be with Max than stand around like some stupid trophy. My stomach reacted to my thoughts, turning over once before settling. \"Let me grab something, and then we'll drop by Ketheria's room before I help you with the taps.\"\n\nI made sure not to let Max see me chug Tic's drink from the chow synth. The liquid's effect on my stomach was instant, and I was thankful for my encounter with the Belaran even though her warning about Max crept back into my head. _I_ will _resist their genetic tampering,_ I said to myself. _I don't care what they're trying to do. I will live my own life, and right now I want to be with Max._\n\nTheodore handed off two of the sacks, one to Grace and one to Dante (I think that's what he was calling himself at the moment). Then we stepped outside with the remaining bags. I stopped when I saw the crowd outside waiting to see Ketheria. I think every alien from all four rings was standing in the open courtyard. I looked out and saw more aliens pouring over the concrete walls and stuffing themselves into the walkway leading to the building. Over the farthest points of the audience, huge O-dats floated in the sky as everyone was straining to see the platform extending from the roof and over the crowd.\n\n\"How many people do you think are here?\" Max wondered aloud.\n\n\"More than I ever imagined,\" I mumbled.\n\nI walked next to Theodore while he handed out the taps to aliens thrusting their hands toward him. No one pushed or grabbed at Theodore, but you could tell they were anxious to get any information they could. As I walked through the crowd, people whispered and moved out of my way. Max stayed at my side, scooping taps from her sack and handing them to anyone who wanted them. Theodore then fell in behind.\n\n\"We're going to need more than this,\" Theodore declared.\n\n\"Can't they share?\" I asked.\n\n\"I don't think they want to. They keep these taps like souvenirs.\"\n\nI tried to read the faces. Some immediately averted their eyes when I looked at them, and I even caught the odd daring sneer. What did they think I was going to do? Chop _them_ all up and eat them? I imagined parents threatening their unruly offspring with horrible stories about the Tonat. Would I ever live up to those fears? _Impossible,_ I thought. I mean, look at the outfit I was wearing. Who would be afraid of me?\n\nA Choi stepped in front of me and grabbed my hand with her scaly paw. The familiar knobby stumps poking out from her shoulder blades reminded me of Weegin, our first Guarantor. The alien rubbed my hand along her face, mumbling something I couldn't understand, an odd display of affection by a Citizen over a knudnik. Did they realize I was still a knudnik? I stopped and looked at Max. _What do I do now?_ my eyes pleaded, but Max only shrugged. With my other hand, I stroked the top of the Choi's bumpy head. Others, seeing my reaction, poured in around us as if by invitation. They pawed at my shoulders, my hair, and even my legs. A sea of hands engulfed Max and me, reaching out to touch any part they could reach. There was no way to move.\n\n\"Stop!\" I shouted.\n\nThe aliens pulled back as if I had used some sort of invisible battering ram. The Choi in front of me started crying. Dirty yellow tears puddled at my feet, and I moved around her to get away. I had no idea what to do.\n\n\"This is creepy,\" Max whispered.\n\n\"Let's get out of here,\" I said.\n\n\"What about Theodore?\"\n\n\"He'll understand.\" I turned to him and called out, \"Theodore, we'll be back in a bit. We'll meet near the right side of the platform when Ketheria comes out.\"\n\n\"Hey!\" Theodore tried to protest, but the aliens reaching for their taps closed in and smothered him. Max and I slipped into the crowd.\n\n\"I feel bad leaving him like that,\" Max said.\n\n\"He'll be fine,\" I assured her, straining to locate a private spot among the throng of aliens. I grabbed Max's hand and pulled her along as the crowd parted for us.\n\n\"Where are we going to go?\" she asked.\n\n\"Anywhere,\" I replied.\n\nBeneath one of the huge O-dats floating over the crowd, I found a tower holding speakers and smaller O-dats; it was draped in some sort of black material. I poked my head under the cloth. Nothing but the metal frame. _Perfect,_ I thought. I lifted the cloth and motioned Max to get under. She laughed and looked over her shoulder. It felt good to be doing something I wanted to do.\n\nInside, the light barely penetrated the thick material. \"I can hardly see you,\" I whispered, but Max only chuckled in response.\n\n\"Use your hands then,\" she said, and placed mine on her face. I ran my fingers around the edges of her face, across her cheeks, and along her lips. We were silent as I explored her face, but I swear I could have started a fire in the space between my fingers and her soft skin. Her warm breath seemed to quicken against the palm of my hand and then I cupped the back of her neck.\n\n\"Kiss me,\" she said, and I obliged.\n\nWe remained tangled in each other's arms, invisible to the pageantry that surrounded us. I felt happy, truly happy. This was all I wanted \u2014 to be alone with Max. We could have been hidden under those drapes for a parsec or an entire light-year. I had no idea. Time was not relevant at that moment; I just wanted more of it.\n\nWithout warning I felt like a Neewalker had dug his claws into me and ripped out my stomach. I pushed away from Max and doubled over in the dark.\n\n\"What's wrong?\" Max cried.\n\n\"I don't know,\" I croaked. I could hardly breathe.\n\n\"JT! Are you sick again?\"\n\nClawing at the fabric, I dug my way into fresh air. The light slammed into my eyes with a searing bolt of pain to my brain, and I screamed. Or maybe Max did \u2014 I couldn't tell. I stared into the crowd and a found a Neewalker, directly in front of me, setting up a long-range plasma rifle. I cried out, lunging toward the creature, but found myself with my hands wrapped around the throat of a terrified knudnik, scrambling to get away from me. I spun around, but the Neewalker was gone. _What's happening to me?_\n\n\"JT! What are you doing?\" Max cried.\n\nThe pain seized my brain again and squeezed mercilessly. Another flash, as if someone pointed the sun directly into my eyes, but this time I saw Queykay running along the rooftop.\n\n\"Queykay!\" I screamed, and pushed through the crowd, stumbling toward him.\n\nMax was at my side. \"JT, Queykay's not here. Talk to me. What's happening?\"\n\nI looked at the roof. She was right. Queykay was gone. Then another flash. More pain.\n\n\"JT! Talk to me. Please!\"\n\nI turned to Max and saw a wormhole pirate standing behind her. Max saw me staring and spun around. Then I saw another and another. The pirates revealed weapons cloaked at their hip.\n\n\"Max, something's wrong!\" I yelled.\n\n\"I see that.\" She grabbed me by the shoulders. \"Look at me, JT. Tell me what's happening to you.\"\n\n\"I'm seeing things. It has something to do with Ketheria \u2014 I'm certain of it.\"\n\nAnd as I spoke her name, my sister stepped out onto the rooftop platform.\n\n\"I have to get to her, Max!\"\n\nI lunged forward as the crowd around me broke free like a solar flare. The sound was deafening. A wave of people pushed toward the platform, and Max was swallowed up in the rush.\n\n\"JT!\"\n\nBut I couldn't respond. Another bolt of pain stabbed at my head, and this time I saw another alien remove a strange-looking device from under his emerald-colored cape. The device was electric of some sort, its long barrel crackling from some unknown power source. I blinked and he disappeared.\n\nI spun toward the platform. I could see my sister waving to the crowd.\n\n\"Welcome,\" her voice echoed through the loudspeakers. \"I know some of you have traveled far, and I thank you.\"\n\n\"Ketheria!\" I screamed.\n\nAnd then I heard it. I knew instantly.\n\nA sting ripped over the crowd, like an electrical cable had been cut free. There was a short pause followed by a crack. I looked up and saw Queykay moving away from Ketheria, just as I had envisioned earlier. Then I saw Ketheria fall. The crowd gasped and fell silent. I heard nothing but the sound of Ketheria's body hitting the platform before the air was gobbled up with mayhem. As some screamed, others began to run. Instantly, more than a dozen Space Jumpers appeared on the platform.\n\nThat's when I felt it.\n\nI didn't try to jump. It was as if someone was _making_ me jump. My mind and my body struggled to gain control while some psychic tether yanked me through space and time.\n\nFor a nanosecond, the world about me appeared frozen before I was torn from my current moment in time and placed next to my sister on the platform without lapse. The chaos and pandemonium roared back in as the universe caught up with me. Four Nagool masters moved toward my sister, and I spotted Drapling running across the platform. I turned to see where Queykay had gone, but I could not find him. Then four Jumpers closed in around me. Each one grabbed onto Ketheria, and then they were gone, my sister with them. All of this happened within a single breath.\n\n\"Wait! Where are they taking her?\" I screamed, staring at the spot where they had just been. I ran to the next Jumper, but he disappeared before I even got close. Each Jumper followed, one by one. \"Where are you taking her?\" I screamed at them.\n\nWith Ketheria gone, my head felt like it was going to crack open and its contents spill out on the platform. I was drowning in a wave of nausea as I tried to focus, to unlock some hidden link to the Space Jumpers or to Ketheria herself. Surely the Trust had wired me with something like that. I fixated on being with Ketheria, being off this platform, being next to her. Precious seconds slipped away. _Where are they?_ Then my mind unhinged and I pushed myself forward, not from where I was standing but rather through space and time. Instead of darkness, my mind exploded with pure light, and I felt relief as I slipped away from the here and now. I let the Universe guide me and prepared myself for what was to come. I even welcomed the sickly scent of smelly feet.\n\n#\n\nI had an idea where I was. The pain in my head had subsided the moment I jumped, and even my stomach felt better.\n\nKetheria was close.\n\nI recognized the steel beams and walls rooted in the black rock from the last time I had been held here a rotation ago. The Space Jumpers had taken Ketheria to the Trust \u2014 I was certain of it. Standing in a wide corridor, I saw blue light glowing along the ceiling's edge and sparkling on the textured floor, just as I remembered. The fact that I had left the Rings of Orbis and jumped to the Trust was not an easy concept to understand, but, as with my softwire abilities, I chose to simply accept it. Right now I only wanted to find Ketheria.\n\nThe Trust was powerful, and my last meeting with them had not been a favorable one. They could inflict pain without laying a finger on me, and they could track my movements whenever they pleased. Besides that, whenever the Keepers spoke of the Trust, I always sensed both fear and respect in their tone. When Theylor had informed me that I was genetically altered to be the Tonat, he told me that it was under their instruction. Theylor also told me that the Trust had been instrumental in arranging our trip from Earth. The Trust's main mission, he told me, was to search for a Scion and then, once he or she was found, to help the Scion awaken. But who gave them this role? I assumed it had been the Ancients.\n\nI heard footsteps around the corner to my right. I bolted across the corridor and slipped down another hallway on my left. I had to move quickly, before I ended up in some hole waiting to find out what had happened to Ketheria. I knew she wasn't dead; I sensed it. _Is that from their programming?_ I wondered. Or was it simply because Ketheria was my sister? I couldn't even imagine how my body would react if she died.\n\nI slipped along the corridor, searching for some connection to the computer that ran the place. What would I say to Ketheria once I found her? Certainly this incident proved that the security around my sister was inadequate, despite their manpower. I knew Ketheria would say that none of this was my fault, but it was. It had been my decision to handle security in this manner, yet the first time Ketheria appeared in public, she was attacked. What was I thinking? I should have been next to her. It _was_ my fault.\n\n_Max!_ I had simply left her in that mob of aliens. _Have I failed everyone? Would she even understand?_ I hoped she would.\n\nI stopped in front of a glass panel embedded in the wall. It was the best chance I had. I pushed into the device.\n\n\"Ouch!\"\n\nI pulled out immediately. My teeth were ringing from the shock, and I could taste a peculiar metal tinge on the roof of my mouth. I tried to shake it off, but it stuck to me like radiation gel. The Trust was using some sort of security device to keep people out of their computer. I moved on, looking for something else, but all I found were more and more corridors with more useless little panels.\n\nAgain the sound of footsteps resonated down the hall. There were lots of them this time, and they were marching. I searched for a place to hide and found a small impression in the bedrock. I pushed against the wall, trying to make myself as small as possible, when the wall opened up and I fell inside.\n\n\"It doesn't take a softwire to use one of those door panels,\" said a voice from somewhere within the room. Actually, it was more like the voice was everywhere in the room.\n\n\"Who said that?\" I called out.\n\n\"I did,\" the deep voice replied.\n\nThe room wasn't much. In fact, all I saw were two sloped chairs similar to the ones in Drapling's rooms. There was also a small table. That was it.\n\n\"Please sit if you feel more comfortable,\" the voice offered.\n\n\"Thanks,\" I mumbled, with no intention of sitting.\n\nThe walls were the most active things in the place. Lights seemed to flash through the room as if circulating through the rock. Tubes and pipes filled in most of the blank spaces, and then I saw it. To my left, in one corner of the room, I saw a hand, as if it was stuck in the rock. A small control panel was placed within reach of the hand's fingers. I began to notice more body parts spread out through the rock and around the entire room.\n\n\"Who are you?\" I asked.\n\n\"That's rather rude,\" the voice replied.\n\n\"I'm sorry. This is a little strange for me. I'm worried about my sister. She's hurt, and I want to get to her.\"\n\n\"Ketheria is fine. They are attending to her now.\"\n\n\"You know! Can I see her? Do they know who did it?\"\n\n\"They know who wasn't protecting her.\"\n\n\"Oh.\"\n\nI knew it. And so did they. I wasn't going to live this one down, that's for sure. I failed Ketheria on my initial test. The anger flushed through my skin. Anger at myself.\n\n\"I do not know why you resist it,\" the voice said. \"It is your destiny,\"\n\n\"Says who?\"\n\n\"Says me.\"\n\n\"Who are you?\"\n\n\"Every cell in your body has been programmed to protect that girl. It does not matter how hard you try to avoid your responsibility; you cannot escape it. That sickness you feel? Can you imagine what that would be like if she left this galaxy?\"\n\n\"I can handle it. I handled it when she came here,\" I said.\n\n\"This asteroid is simply orbiting the rings, you fool.\"\n\nWell, that explained all this rock.\n\n\"What you feel now is nothing compared to what you will feel if you don't stay close to the Scion. You'll need more than those foolish little potions you drink.\"\n\nThe lights in the rock flared with the sound of the voice, as if agitated.\n\n\"How do you know this? Tell me who are you,\" I said.\n\n\"I'm surprised you haven't figured that out. You haven't read those files I gave you, have you?\"\n\n\"Files? No one gave me any files,\" I complained.\n\n\"Were you not given a device by a Space Jumper and told to keep it with you at all times?\"\n\nI reached into my pocket and touched the cold metal disc given to me by one of the Space Jumpers.\n\n\"Yes,\" I muttered.\n\n\"Were you not curious to examine the device, to see what you were asked to carry around with you? Obviously not. This is why you need training. You must join us, son. Protecting the Scion is the one thing I cannot program into you.\"\n\n_Program into me?_ I removed the disc and pushed inside it. Nestled behind the intricate tracking device was a single nugget of data. I willed it open, and my parents' missing files flashed in front of me, all 321 of them.\n\n\"You're . . .\"\n\n\"Quirin,\" the voice replied.\n\n\"My father?\"\n\n\"By definition.\"\n\nI didn't look for the chair; I simply sat on the floor and stared at the body parts spread around the room.\n\n\"How can it be?\" I whispered. \"What happened to you?\"\n\n\"When the _Renaissance_ was attacked, I attempted a jump that I knew was not possible. I tried to take everything with me and move backward through time. I failed, and now I will be like this forever.\"\n\n\"But my mother. You left her there.\"\n\n\"Your mother never left Earth.\"\n\n\"What?\"\n\n\"You will find everything in those files. I'm afraid most of it will upset you, and I caution you against telling the others any of what you find, but that is your choice. I would rather use this time to help you accept your fate. The Scion needs you.\"\n\nThe weight of this revelation tightened around my chest. I tried sitting up to let some much-needed oxygen into my lungs, but could breathe in only a little. It seemed every time some inexplicable part of me was explained, I was left feeling emptier and even more hollow. A million new questions crept in to fill the void. Why had I not seen my father sooner? What did he mean that my mother never left Earth? Would he be a part of my life now? If he was my father, was I even human?\n\nI rubbed the smooth disc across the palm of my hands, afraid to look inside it again. How could my life have strayed so far from my own desires? Where was the promise of happiness on the Rings of Orbis? I didn't want my father to be some freak pieced together inside some asteroid. I wanted the man in the photo \u2014 the human who had left Earth to create a better life for his family.\n\n\"I don't want any of this,\" I told him.\n\n\"You're willing to sacrifice the future of the universe for your own selfish desires. This obsession with free will is ridiculous. There are no options after this. That is why everything has been sacrificed to save the last possible Scion. There isn't a species left in the universe that can fill this role. We've tried.\n\n\"Only a Scion can raise the consciousness of the universe. Without a Scion, the universe will implode under the mass of deconstructive energy that its inhabitants will produce. But before that happens, the universe will be fed upon by beings I can only begin to describe. Ketheria is the last possible candidate. If we fail, the universe will fall.\"\n\n\"Why? Why her? Why me?\"\n\n\"Only your ego, that sense of yourself, would ever ask such useless questions. There is no _you_ or _her_ in this equation. The fact that you are involved is nothing more than a random outcome. You were simply the embryo I reached for. You were nothing more than a group of cells that took the genetic information I gave it. The others did not survive. . . .\"\n\n\"What do you mean, 'the others'?\"\n\n\"It is far too risky to prepare only one Tonat. Once the Scion was stable, I needed to make absolutely certain to link a Tonat. We could not wait for it to happen naturally. Had the ship not been attacked, I would have tried several combinations. I must admit, though, your abilities have exceeded my expectations. I am grateful you survived. I believe the remaining candidate would have proven even more difficult.\"\n\n\"There's another Tonat?\"\n\n\"You are the only Tonat.\"\n\n\"You are not making any sense. This is only confusing me more.\"\n\n\"Before the _Renaissance_ was attacked, I had time to prepare two cell specimens to receive the required genetic coding. You and one other survivor. The one you call Switzer.\"\n\nNow I really couldn't breathe. I swallowed hard, hoping that some oxygen would sneak in as well.\n\n\"I need to know more,\" I croaked.\n\n\"The wormhole pirate you helped capture had received similar genetic coding to yourself. He was prepared as a backup, if you will \u2014 a replacement should something have gone wrong with you. I fear his sociopathic behaviors and narcissistic tendencies are the direct results of my procedures. If I had had more time to give him the attention required, I would have adjusted those anomalies. I would have made him more like you.\"\n\n\"So there would be two Tonats running around the rings?\"\n\n\"No, I would have destroyed one of you before the _Renaissance_ ever arrived on the Rings of Orbis. In fact, I would have destroyed all the embryos except for you and the Scion. That was the plan.\"\n\n\"You would have killed everyone?\"\n\n\"Yes. Why does this matter?\"\n\n\"It matters.\"\n\nA world without Max? Without Theodore? Without Switzer? I felt dizzy even though I was already sitting. I wanted to lie down.\n\n\"I don't feel good,\" I said.\n\n\"This is why you need more training. Your emotions are too strong. Your sense of self blinds you. Stay with me and become who you are destined to be. I offer you one of the greatest roles in the history of the universe. Your name will be emblazoned in the hearts of every creature in every galaxy. With Ketheria as the Scion, every human in the universe will be given a unique path to enlightenment. She will guide humans to fulfill their new role in this universe. The Ancients are gone, my son. When the Scion completes the fourteenth step of the awakening, then humans can succeed the Ancients and bring harmony to this universe. It is our last defense against the Knull. An entire universe will stand behind you. You are poised for greatness, John Turnbull. You only need to accept it.\"\n\nI could only think about Max and Theodore and Grace and even the guy whose name I had trouble remembering, and Switzer. They existed only because the Trust tried to grow their own Scion and Tonat \u2014 Ketheria and me. Switzer is sitting in a cell, probably for the rest of his life, because of me. Now, more than ever, I wished I had waited until I was inside his spaceship on Orbis 3, so he could have escaped. At least he would have some sort of life right now.\n\n\"It still matters,\" I mumbled. \"It matters a lot.\"\n\nEvery one of them had suffered life as a knudnik on the Rings of Orbis. Suffered because of me and Ketheria. Of course I didn't think Quirin should have actually followed through with his mission and destroyed everyone, but I wanted desperately to be back on the _Renaissance,_ trying to help Switzer steal the ship.\n\nI was staring at my hands. It would be Switzer standing here right now if a single one of my cells had mutated in some errant fashion. It would be Switzer poised for all this \"glory,\" not me. I was feeling worse by the second. I really didn't want to be me right now.\n\nI looked up at the wall. \"Wait,\" I said. \"Did you do this on purpose? Did you bring Ketheria here so I would hear this?\"\n\n\"You had to come on your own.\"\n\n\"So does this mean that Switzer is my brother or something?\"\n\n\"You both share my genetics.\"\n\nWow! Too much. I needed to get out of here.\n\n\"You, however, have proven to be a much better candidate. I abandoned him for you.\"\n\n\"I want to see my sister,\" I whispered.\n\n\"But we are not finished.\"\n\n\"I was finished a long time ago,\" I replied. \"In fact, I want to go back to the rings. You said Ketheria is all right, didn't you? I want to go home.\" The word struck me as odd. I had never referred to the rings as my home before. \"I need time to swallow all of this.\"\n\nQuirin did not reply.\n\n\"Look, I'll come back so we can talk again,\" I lied, purely to let him believe that I was considering the situation.\n\n\"Certainly. If you remained, however, and if you studied, you could jump back to the rings yourself.\"\n\n\"I said we would talk some more. That is all I can offer you right now.\"\n\n\"There is no way you can fight this,\" he warned me.\n\n\"I can find a way.\"\n\n\"Then I suggest you act quickly. It is dangerous on the rings for your sister.\"\n\n\"Then keep her here.\"\n\n\"The awakening must be allowed to continue. She must do so among those whom she will enlighten.\"\n\n\"Or those who will kill her.\"\n\n\"We must not interfere.\"\n\n\"You are _way_ beyond that,\" I reminded him.\n\nThe door to Quirin's room opened, and a Space Jumper stepped in, someone I had never seen before. This Jumper was not as militarized as the others and was dressed in a sleek grayish material. His belt, however, was still prominent.\n\n\"He will return you to the rings,\" Quirin said. \"Your sister will be fine. She will follow you there shortly.\"\n\n\"Good-bye . . . Quirin,\" I said. There was no way I could call him Father, not now.\n\n\"Good-bye, son.\"\n\nThe Space Jumper returned me to Ketheria's room on Orbis 4. We did not speak even once, and that suited me fine. I was eager to get back to my own room, lock the door, and open my parents' files. For so long I had wanted to see what was on those restricted documents, but now I had almost forgotten about them. When Madame Lee destroyed the copy from the _Renaissance,_ I had thought that was the end of it.\n\nWhat would I find now? I was almost too afraid to look.\n\n\"JT!\" Theodore stopped me as I stepped into the corridor outside Ketheria's chamber. \"Where's Ketheria? Is she dead? Where did she go?\"\n\n\"She's fine. The Trust has her.\"\n\n\"Is that where you went?\"\n\nI nodded.\n\n\"All those Space Jumpers!\" he exclaimed. \"Everyone is talking about war. So many people saw how easily they moved about the rings. The Citizens are really going crazy. Queykay arrived with the entire Trading Council. They put all these new laws into place. You can't even move from ring to ring now. They said it was to protect the Scion, but Hach was furious. They took him with them when they left.\"\n\n\"Where's Max?\"\n\n\"She's with Grace and that guy \u2014 what's his name?\"\n\nI shook my head.\n\n\"Come on, I know Max will be glad to see you. They're having a meeting or something. Max has Grace getting everyone together.\"\n\nTheodore tried to guide me in the direction he was going, but I pulled away.\n\n\"I can't,\" I told him.\n\n\"Why not?\"\n\n\"There's something I've got to do first.\"\n\n\"Fine,\" he said, turning back in my direction. \"We'll go after that.\"\n\n\"Alone,\" I said.\n\n\"Oh.\"\n\n\"I'm sorry.\"\n\n\"Don't be. I understand. We'll come by later. Max will want to see you when she finds out that you're back.\"\n\n\"I don't want you to tell her you saw me,\" I said.\n\nTheodore just looked at me. \"What's wrong, JT? Let me help you.\"\n\nI shook my head. \"Not this time. It's not like that.\"\n\n\"But \u2014\"\n\n\"I'm sorry. Really.\" I turned and headed toward my room. \"Don't tell her, all right?\" I yelled back.\n\nBut Theodore did not respond.\n\nInside my room, I sat at the edge of my sleeper and slipped the disc from my pocket. I ran my finger along the impression that circled the face of the disc. I pushed in and grabbed a random file.\n\n**NEXUS ACCESS 12B-532-AFG**\n\n**TIME POINT: 12:45.2: 227**\n\n**CONTACT: QUIRIN NE YARNOS**\n\n**The genetic enhancements on specimen 1325b appear to manifest the desired traits far more rapidly than the original candidate (specimen 334). My fears that I may not be able to procure an alternate now seem unfounded. Cleavage of the Scion zygote coupled with the experimental regermination may prove to have created an exceptional candidate for the Tonat.**\n\n**If success continues, I will destroy specimen 334 prior to the standard schedule.**\n\n**_Transmission successful._**\n\n_It's true!_\n\nI pulled out of the storage device. The residual burn of Quirin's entry still sparkled against the inside of my forehead. It would fade, I knew, but the knowledge would stay with me forever.\n\nWas _I_ specimen 1325b? I had to be. When Theylor admitted that I had been genetically altered, I never once thought it happened on the _Renaissance._ I assumed it was when they replaced my arm, or when I almost drowned in the cooling tank on Orbis 2.\n\nIf this was true, then Switzer must have been specimen 334. What did they do to him? Did Quirin start to turn Switzer into the Tonat and then stop? If Quirin had never touched Switzer, would he be like the other kids from the _Renaissance_? Out of the two hundred children born on that ship, Switzer was the only one we feared. He was a monster, and they had made him that way. It was not fair.\n\n_If success continues, I will destroy specimen 334 prior to the standard schedule._\n\nThey had planned to flush everyone on that seed-ship. The thought horrified me. Where were all of our parents? Were they already dead? This didn't make sense. Quirin said my mother was still on Earth.\n\n\"JT?\" Vairocina's voice whispered inside my head.\n\n\"Yes,\" I replied, welcoming the distraction.\n\n\"I was hoping we could talk now.\"\n\n\"It's not a good time.\"\n\n\"But I think it is too important for you to wait any longer. I've located the information you requested.\"\n\n\"What information?\" I stood up and paced the room.\n\nVairocina formed to my left. \"You asked me to gain a better understanding of your transfer to the Citizen Hach,\" she replied.\n\n\"I did, didn't I? I'm sorry. I forgot. Did you find anything?\"\n\n\"Yes. It seems that Charlie had made arrangements for his possessions to be distributed through an advocate.\"\n\n\"Is that normal?\"\n\n\"Absolutely. Advocates are usually appointed by the Trading Council to ensure a fair and honest allocation of assets in the event there are no relatives.\"\n\n\"Like that would ever happen. No wonder I ended up with Hach.\"\n\n\"No,\" Vairocina interrupted. \"The unusual aspect is that Charlie had _already_ chosen an advocate well before he died. In fact, the same cycle he became a Citizen and only moments after an enormous sum of chits was transferred to his personal holdings.\"\n\n\"Who? Who did he assign?\"\n\n\"Well, it wasn't a single person really. . . .\"\n\n\"Who, Vairocina?\"\n\n\"The Descendants of Light. Drapling signed the transfer. Drapling is also the one who then bequeathed you to Hach.\"\n\n\"What?\"\n\n\"The Descendants of Light. The trans \u2014\"\n\n\"I heard you. I just can't believe it. It doesn't make sense. Charlie was no fan of the DOL, especially Drapling. I even heard them fighting once when I was half-conscious. I'm sure it was Charlie and Drapling. This just doesn't make sense.\"\n\nI sat down on my sleeper and hoped the room would stop spinning.\n\n\"Is there anything else I can find for you, JT?\"\n\n\"No,\" I said, \"but there is something you can do.\"\n\n\"Certainly.\"\n\n\"Can you tell Drapling I need to meet him? It's an emergency.\"\n\n\"I'll do it now.\"\n\nI pushed back into the storage device. This time I discovered a small interface along with the hidden data. _Only a softwire would find this interface,_ I thought. The interface displayed a projection mode, which I initiated, then pulled out of the device and sat the metal disc on the lid of my sleeper. Using my softwire, I interacted with the display projected on the wall in the same manner I would with an ordinary O-dat. I opened another file at random.\n\n**NEXUS ACCESS 6F-448-MGH**\n\n**TIME POINT: 14:40.9: 814**\n\n**CONTACT: QUIRIN NE YARNOS**\n\n**Long-range subspace tracers have detected a distress signal in the Dorvum system. Ion Signatures identify craft as a Zinovian class 4 cruiser: nonmilitary issue.**\n\n**Threat: none.**\n\n**Current systems are stable. Recommend jump interaction. Please advise.**\n\n**_Transmission successful._**\n\nZinovian class? Was that Madame Lee? Did she trick him? I opened another file.\n\n**NEXUS ACCESS 11C-102-MKL**\n\n**TIME POINT: 03:03.1: 019**\n\n**CONTACT: QUIRIN NE YARNOS**\n\n**AI programming was initiated as per instructions. Earthlike histories for all specimens now loaded. Scion and Tonat entered as siblings. The AI has been time-stamped to display death of parents, children, and crew upon arrival. Birth sequence initiated for Scion and Tonat and scheduled to arrive on the Rings of Orbis near their thirteenth Earth year.**\n\n**_Transmission successful._**\n\nKetheria and I were supposed to arrive on the rings as the same age. I tried to think of Ketheria as seventeen, just like me. It was too weird. What would she look like? I retrieved the first file I had opened. What did the words _zygote_ and _cleavage_ mean? Were Ketheria and I twins? Impossible!\n\n\"JT?\" Vairocina whispered inside my head.\n\n\"Did you find Drapling?\"\n\n\"He will meet you at the Center for Relief and Assistance.\"\n\n\"When?\"\n\n\"Whenever you are ready.\"\n\n\"I'm ready now.\"\n\n#\n\n\"I want to see Switzer,\" I told Drapling. We were seated in the same room underneath the Center for Relief and Assistance.\n\nBoth of Drapling's heads converged on me. \"I'm sorry, but that is impossible,\" he stated.\n\n\"I'm learning quickly that nothing is impossible. I want to see him, Drapling. He shouldn't be there. What happened to Switzer is not his fault. In fact, it is more the Rings of Orbis's fault than his.\"\n\n\"Your friend is a criminal, a wormhole pirate. He will never see the surface of the ring for as long as he lives. Your request is denied.\"\n\nI stood up, marched over to Drapling, and leaned in close. Any other person would have shrunk back into their seat, but Drapling remained indifferent. \"Listen carefully,\" I whispered. \"This is not a request. I don't need your answers anymore. I _know._ Do you understand me? I met with Quirin. He gave me the missing files. As far as I can figure, you guys sent him to snatch a bunch of embryos from Earth and grew your own Scion. Except your plan didn't work out the way you wanted it to, and now you have all of _us._ Switzer isn't just a friend; he's my brother.\"\n\n\"Technically, I believe you would call him a half brother. You share only the genetic coding of Quirin. You and the Scion share a common human egg.\"\n\n\"Shut up! No matter how you spin it, the life Switzer now has is directly the result of _your_ actions.\"\n\n\"This means nothing. You feel this way only because you have been adjusted to exhibit a greater care for your sibling. For Ketheria, _not_ for Randall Switzer.\"\n\nI pulled back and stepped around Drapling. It took eight strides across the shiny floor to reach the door. I _pushed_ into the control panel and locked it. My old feelings for Switzer were gone. _Now_ I wished that I had helped him steal the _Renaissance_ when we first arrived. _Now_ I wished I had let him escape on Orbis 3. I had to make it up to him before I did anything else. It was not right for him to be forgotten and left to rot away in some cell. I wanted to apologize to him. I wanted to apologize for everything.\n\nI turned toward Drapling. He hadn't moved a muscle. \"I know about the Descendants of Light,\" I told him. \"I know that you helped Charlie become a Citizen. Yet I was wondering if Theylor and the other Keepers know everything. If I remember correctly, when Theylor came to greet us on the _Renaissance,_ he was expecting to find a ship filled with adults. Or was that a ruse as well?\"\n\nDrapling did not respond. Instead he looked toward the glasses at the far side of the room.\n\n\"Is something wrong, Drapling? Was that information restricted? I don't know what it means yet, but I think I can use it.\"\n\n\"You cannot comprehend the magnitude of the situation,\" Drapling declared.\n\n\"Let me see him.\"\n\nDrapling spun toward me.\n\n\"Why can't you just accept it? You _are_ the Tonat. This is your destiny!\"\n\n\"Let me see him!\"\n\nDrapling just stared at me from across the room. Then, without speaking, he stood up and strode toward the door.\n\n\"Please unlock the door,\" he said. \"And follow me.\"\n\nDrapling led me from the room, across the floor of tombs, and past the light chute we used to descend from the Center for Relief and Assistance. He stopped at the end of the corridor in front of another chute.\n\n\"Can this take us off the ring?\" I asked. \"Switzer's at the Center for Science and Research on Orbis 1, isn't he?\"\n\n\"No,\" Drapling replied. \"I am afraid you underestimate your adversary.\"\n\n\" _I_ could never escape from the Center,\" I reminded him.\n\n\"You never tried.\"\n\nHad Switzer tried to escape? I doubted he was the easiest prisoner for the Keepers to deal with.\n\n\"Please understand that if Quirin had failed with your enhancements, Switzer would have been the Tonat. All that remained for Quirin to do was to initiate the proper gene activation sequence and Switzer, too, would have been a softwire.\"\n\n\"I thought softwires were rare. If it's so easy to cook one up in a test tube, then why don't you just make an army of them?\"\n\nDrapling was about to punch the code into the panel next to the purplish light chute when his right head turned and said, \"Believe me when I tell you that we have tried, but it is not as simple as you suggest. You must appreciate how fortunate Quirin was to have two workable specimens at his disposal. I only wonder if Quirin may have made the wrong choice, as Switzer certainly exhibits the ruthlessness required for the job.\"\n\n\"It sounds to me like you built a monster that you cannot control and now you're going to imprison him for the rest of his life to avoid dealing with your own mistake.\"\n\n\"What do you suggest? That we kill him?\"\n\n\"No! Let him go. You made him this way.\"\n\n\"We also made you.\"\n\nThe edge in Drapling's voice and the sneer rubbed across his face reminded me of the Keeper I first encountered when I had arrived on the Rings of Orbis. Any niceties he had recently shown were gone. I watched him over my shoulder as I stepped into the chute.\n\nWhen I exited the light chute, the first thing I felt was cold. In fact, it was freezing. I moved away from the chute and into a gray, lifeless corridor. Drapling was behind me in the next instant.\n\n\"Where are we?\" I asked. I could see my breath in front of me.\n\n\"Deep within Orbis 4,\" Drapling replied, stepping past me.\n\nHe walked beneath a single bluish light source embedded in the ceiling of the concrete corridor, and I followed, avoiding his purplish robe, which dragged behind him. I glanced at my surroundings and noticed frost gathering in places where the walls met the ceiling. I could only assume Switzer was somewhere behind one of these walls.\n\nA green electrical field blocked us from continuing down the corridor. I watched as Drapling turned and placed his hand inside a device mounted on the right wall. The green force field appeared to drop away and run along Drapling's arm before scanning his entire body. He removed his hand and ordered me to follow him.\n\nOnce we passed through this entry point, I saw a series of thick chrome doors along the wall every three meters or so. Each door flashed an LED symbol embedded right at the Keeper's eye level.\n\n\"What is this place?\" I asked Drapling.\n\n\"It does not have a name.\"\n\n\"What do _you_ call it?\"\n\n\"Terminus,\" he mumbled.\n\n\"Why is it so cold?\"\n\n\"I hadn't noticed.\"\n\nDrapling stopped in front of the second door. I waited as he reached inside his left sleeve and a small bench slid out from the wall on the opposite side of the hallway. He sat but motioned me toward the metal door.\n\n\"What am I supposed to do?\"\n\nDrapling reached under his sleeve again, and this time the door disappeared. A paler version of the electrical field I saw earlier remained in its a place.\n\nOn the other side, I saw Switzer lying on the floor.\n\n\"Switzer!\" I cried.\n\nHis body only jerked in response. I turned to Drapling. \"Open it!\"\n\nDrapling reached under his sleeve, and the field fell away. I raced in and knelt next to Switzer. He was balled up in a fetal position, clutching his stomach and moaning. Someone had removed the piece of screen over his eye but had left the wires sticking out of his face.\n\n\"Switzer!\"\n\nHe cracked open his eyes and stared at me through the pus that had crusted around his lids.\n\n\"Switzer, it's me, JT!\"\n\nSwitzer didn't respond. Instead, his body convulsed as if he was trying to throw up, but nothing came out. I looked around the room. By the looks of the mess near the toilet I didn't think he had much left to throw up, anyway. I turned to Drapling. He had replaced the barrier to the room.\n\n\"Does he have the same thing as me, the sickness when Ketheria is too far away?\" Drapling nodded. \"How could you leave him like this?\" I screamed at him. \"Do you have any idea what that feels like?\"\n\n\"You appear to be doing fine,\" he reminded me.\n\n\"That's because \u2014\"\n\nI thrust my hand into my pocket. I had the medicine that Tic had given me. I pulled it out and chewed off the lid as I propped Switzer's head up.\n\n\"Drink this. It will help. I know what you're going through. I would be just like you if it wasn't for this stuff.\"\n\nI expected Switzer to resist, but he raised his chin a little and parted his lips. They were dried and cracked. Scabs had grown over the smallest crevices, but thick, bleeding sores were visible at the corners of his mouth. I poured the liquid over his lips. He pawed at my hand, forcing me to pour more liquid into his mouth.\n\nHe pulled his head away and fell back onto the floor, his arms flung out. Then he bellowed with laughter and cried, \"Sweet golden universe! Where was that stuff when I needed it?\"\n\nI sat back, relieved to see the liquid working so quickly. \"I'll bring you more. You won't have to feel that way again.\"\n\n\"You can't even imagine what I felt,\" he mumbled, and tried to sit up.\n\n\"Yes, I can,\" I replied.\n\nSwitzer looked at me and then looked out at Drapling. \"What's he doing here?\"\n\n\"Forget him. We need to talk.\"\n\n\"About what? You want to rub it in my face a little? You want to tell me how you were right all along?\"\n\n\"No, Switzer. _You_ were right. You were right about everything. They did this to you. They made you this way. They messed with your genetic code, trying to create a security force for Ketheria. I'm sorry.\"\n\n\"Sorry? For what?\"\n\n\"For everything. I should have listened to you. I should have helped you get out of here.\" I leaned toward him. \"I should have left with you. It's my fault you are in here.\"\n\nSwitzer pushed himself up to his sleeper and then struggled to stand. He walked over to the door and stared at Drapling. \"Don't take the blame for everything, split-screen. These two-headed space monkeys aren't telling you the whole story. I'm sure of it.\"\n\n\"I don't need them now. I know all the answers now. I found the files from the _Renaissance._ \"\n\nSwitzer turned to me. \"And?\"\n\n\"On the _Renaissance,_ they messed with your genetics to create a Space Jumper who would be the Tonat. Then they made me. You were supposed to be . . . well, let's say that when the ship was attacked, they didn't have time to finish with their experiments and you were born. They made you who you are and they should be responsible for you. You should be out there learning to be a Space Jumper. Enjoying your life. Preparing for great things.\"\n\n\"But somehow I'm stuck in here, regretting my every waking moment.\" Switzer turned and looked at me. \"And don't think that an apology is enough, Dumbwire.\"\n\nHe was right. Words weren't going to change anything. I stood up and marched toward the door.\n\n\"Drapling. You have to let him out. Let Switzer study with the Space Jumpers. Let him go with the Trust.\n\n\"I will not let Switzer roam free,\" he insisted.\n\n\"The Trust will take him away from the Rings of Orbis. He won't stay here. He'll leave Orbis. I know he will.\" I turned to Switzer. \"Won't you?\"\n\n\"Forget it. He's not going to do it,\" Switzer mumbled.\n\n\"Drapling, Switzer's wanted to leave here ever since he knew about the Rings of Orbis. He has the same genetic enhancements I have, so let him fulfill his role as a Space Jumper. You owe that to him, Drapling. It's the right thing to do. He deserves at least _that,_ not a life like this.\"\n\nDrapling stood up and walked toward the force field.\n\n\"Fine,\" he said. \"But with one condition.\"\n\n\"Anything,\" I replied.\n\n\"You go with him.\"\n\n#\n\nI returned to my room and found Max kneeling in front of my sleeper with half a dozen tools scattered about the room. She had the front panel of my sleeper on her lap and was picking through a fistful of knotted wires.\n\n\"I don't know why that makes me nervous,\" I said.\n\n\"Should you be?\" she asked without looking up, and then placed the ball of wire between her teeth. With her hands free, she snapped the plastic panel back into place and snatched a laser drill off the floor to secure her handiwork. She looked satisfied and stood up, tossed the wires onto the sleeper, and turned to me.\n\n\"Don't I need those?\" I asked.\n\n\"You could have let me know you were back,\" she said.\n\nI walked toward her and took her hand.\n\n\"I'm sorry, Max. I really am.\"\n\n\"What happened?\"\n\nWhat was I going to say? I knew _too_ much now. Should I tell Max that her entire existence was a mistake? Should I tell her that the Trust had messed up and she was never meant to be alive? The life that she and the others had suffered on Orbis was all because of Ketheria and me. What about the others, who I hadn't seen since we arrived? What were they suffering?\n\n\"I went to see Switzer,\" I told her.\n\n\"What?\"\n\n\"He shouldn't be there, Max.\"\n\nMax pulled away, but it felt like someone had chopped my hand off. \"What caused this turn of events, JT? You've hated him ever since you were born. He tried to kill you. He was directly involved in Charlie's death and who knows what else? Trust me: he's supposed to be wherever he is.\"\n\n\"It's not his fault, Max.\"\n\n\"What do you mean? I can't believe you can even say that. It's certainly not _your_ fault. I don't get this, JT. You went to see Switzer instead of coming to see me? Ketheria I could understand, but him?\"\n\n\"Max, you _don't_ understand.\"\n\n\"Apparently I don't. I had no idea what happened to you. First you start going crazy, then someone tries to assassinate Ketheria, and then, _bang,_ you're gone, too! I was going crazy wondering what happened to you. If it wasn't for Theodore telling me you were all right, I don't know what I would have done.\"\n\n\"Theodore told you I was here?\"\n\n\"Don't get mad at him. He was just being a good friend.\"\n\n\"Some friend,\" I mumbled.\n\n\"You could take a few lessons from him, Johnny Turnbull.\"\n\nMax turned and walked out of my room. I didn't try to stop her. Maybe it was the best thing, anyway. I didn't have a clue how to tell her that I was leaving.\n\nKetheria returned the next cycle along with an army of Space Jumpers and a half dozen Nagool masters.\n\n\"This isn't good,\" Theodore said as we watched Ketheria's entourage pile into the antechamber of her room. I caught Queykay watching from down the corridor. He did not look happy.\n\n\"What do you mean?\" I asked Theodore.\n\n\"All those Space Jumpers, JT. The Council is having a fit. They're saying that the Keepers have broken the treaty.\" Theodore glanced toward Queykay. \"Look at him. I would stay out of his way if I were you.\"\n\n\"But I thought most of the people on the rings loved this OIO stuff. Isn't Ketheria their leader now or something?\"\n\n\"Not really. The OIO philosophy is basically a set of tools to aid in enlightenment. The Scion acts as a seed. Her purpose is to awaken the Universe and help it protect itself against the Knull. I can understand why the Council is nervous. Who's going to listen to them now?\"\n\nI turned toward Theodore. \"Where did you learn this stuff?\" I asked him.\n\nTheodore grabbed me by the arm and pulled me away from the crowd. He stopped when it appeared no one could hear him. \"From Grace and Diablo.\"\n\n\"Diablo?\"\n\n\"That's what he's calling himself now.\"\n\n\"And you know this because?\"\n\n\"I have joined their group. Max started it, actually.\"\n\n\"What group?\"\n\n\"Shhh!\" Theodore pulled me farther down the corridor, but I didn't think anyone could hear us anyway.\n\n\"We call ourselves Knudnik Nation. We're convinced that if the Citizens go to war against the Keepers again and we, the knudniks, work behind the scenes to undermine the Citizens' efforts, then we can sway the outcome of the war. Do you know how much business on these rings is dependent upon knudniks? Just by collectively refusing to work, we could bring the Citizens' cycle-to-cycle activity to a halt. We have so much power! We simply need to unite. Our biggest hurdle is to get the word out. The taps have helped. Max has hacked into them and we've begun leaving little messages after the original propaganda. They're only viewed once and then the tap is destroyed. There's no way it can come back to us.\"\n\nI didn't know what to say. It was as if someone had shone a light on Theodore. He had been in the room all along, but no one had ever noticed him. Theodore was empowered by this mission in a manner I had never seen.\n\n\"Well, what do you think? Join us. We could use your softwire abilities. You could use it to spread the message and connect to the other rings inside the central computer. Even Vairocina could help.\"\n\nI shook my head. \"I don't think I can,\" I told him.\n\n\"Why not? Max is there. She's practically our leader. I think Ketheria would actually promote it. She knows how evil the Citizens are; that's why the Council is so afraid of her.\"\n\n\"It's not that.\"\n\n\"Then what is it?\"\n\nHow was I going to explain to Theodore that I was leaving to become a Space Jumper in order to free the one person we had both despised? Our hate for Switzer was a common bond that Theodore and I had shared our entire lives. If I told him now, I knew I would be hurting our friendship, maybe permanently.\n\n\"It's nothing,\" I said. \"Of course I'll join. Are you crazy? When do they meet? Is it soon? I'll come with you.\"\n\n\"That's golden, JT. I knew I could count on you. This is going to strengthen our effort like you can't believe. We meet in Murat next cycle. I don't have the location yet, but I'll let you know the moment I do.\"\n\n\"Sure,\" I said, and motioned back toward Ketheria. \"I want to check on my sister. I still haven't seen her yet.\"\n\n\"Oh, of course. Go. Do you want me to say anything to Max for you? We're going to deliver some more taps.\"\n\n\"You and Max?\"\n\n\"And a bunch of us,\" he said.\n\n\"No,\" I replied. \"I have to deal with Max myself. It's only fair.\"\n\n\"I'll tell her you're with Ketheria; she'll understand,\" he said as he turned away.\n\n\"No!\" I yelled after him. \"Don't say anything else. Please. Let me deal with Max.\"\n\n\"All right, but remember: say nothing. To anyone.\"\n\nI nodded as I watched Theodore trot down the corridor in the opposite direction of Queykay. I was jealous. _I_ wanted to be going to see Max. I wanted to make plans with them, but all I was going to do was disappoint Max and Theodore. I couldn't tell them. Not yet. I needed a better reason, one that everyone would understand and one that would not expose the fact the each and every kid from the _Renaissance_ was never meant to be alive.\n\nInside the first chamber of Ketheria's room, I found two Nagools discussing something quietly. They both looked up when I entered and smiled. I returned the gesture.\n\n\"We welcome your participation,\" the one Nagool said, his voice like still water.\n\n\"And thank you for your decision,\" the other added.\n\n\"You're welcome,\" I muttered. Truth was, Nagools made me nervous. I didn't understand them, and I didn't want to. In fact, I didn't really know what they did. I made a mental note to ask Ketheria.\n\nInside Ketheria's main room, I found my sister sitting up in her bed. Seated on the far side were Theylor and another Keeper I had never seen before. This Keeper had only one head. I had never seen that before, either.\n\n\"Hello, Johnny Turnbull,\" Theylor said when I entered.\n\n\"Hi, Theylor. I didn't know you were here.\"\n\n\"I wanted to make sure your sister was comfortable. She has been through a great deal.\"\n\n\"Hi, Ketheria,\" I whispered.\n\nShe smiled weakly, her eyelids looking heavy upon her eyes. I saw a yellow bandage wrapped around her left shoulder and arm. Tiny sensors protruded from the bandage, and one of those blue med-lights glowed in a semicircle over the headboard of her sleeper. Seeing my sister wounded by some unknown attacker only reinforced my decision to become a Space Jumper. Suddenly, my own desires seemed selfish and childish to me. Was I feeling this way because of their genetic tampering? _No,_ I thought. Ketheria needed me. No one had to alter any part of me to understand that.\n\n\"Does anyone know what happened?\" I asked Theylor. I moved closer to my sister and let my fingertips caress her hand. She felt warm.\n\n\"She was attacked by a long-range plasma rifle,\" he informed me. \"The most disturbing aspect is that our security sensors never picked it up.\"\n\n\"What does that mean?\"\n\n\"Someone had to program the sensors around her platform to ignore the signature of that weapon.\"\n\n\"Who?\"\n\n\"Our first suspicion was that someone close to her had done it. Someone with access to the platform, but we have questioned everyone. I am afraid that we have found nothing.\" The Keeper stood up and walked toward me. \"Your acceptance will go a long way to eliminate these holes in their security. I am confident you will rise to be the greatest Tonat ever. An achievement only possible as Ketheria's brother.\"\n\n\"So you know about my plans?\" I said.\n\n\"This is a great cycle for everyone. I am proud of you.\"\n\n\"Let's hope I can live up to the hype,\" I said.\n\n\"You will,\" he said, and then Theylor and the single-headed alien slipped away, leaving me with Ketheria.\n\n\"Thanks,\" she whispered.\n\n\"Don't thank me yet. I haven't told you about the details.\"\n\nKetheria closed her eyes and attempted to smile.\n\n\"You know?\"\n\nShe nodded.\n\n\"Aren't you upset?\"\n\n\"I'm proud of you as well. To put your hatred aside and seek a path to help undo the misfortune set upon another \u2014 a person with whom you have such a tattered history. I couldn't be more happy, JT.\"\n\n\"I'm afraid,\" I told her.\n\n\"I know that as well.\"\n\n\"Not just for me, but about what Max will think.\"\n\nKetheria winced as the light around her bed pulsed red.\n\n\"Don't talk anymore,\" I whispered, and helped her to lie on her pillow. \"Get your rest.\" But I don't think she heard me. Whoever was monitoring Ketheria had put her to sleep as soon as the pain registered. I stood over the bed and watched my sister rest. _Such a little girl with such an enormous responsibility,_ I thought. How could I _not_ protect her? Whatever the Universe had planned for Ketheria, I knew in my heart that she needed me. I had made the right decision. Now I had three cycles to convince Max of my decision before I left the Rings of Orbis.\n\n_It simply wasn't enough time._\n\nWhen I left Ketheria's room, I found the two Nagools still waiting outside. One of them drifted my way when Ketheria's door closed behind me.\n\n\"She's sleeping,\" I whispered.\n\nThe Nagool simply reached into his robe and removed something with the OIO symbol marked on one side. He handed it to me.\n\n\"What's this?\" I asked, turning the card over in my hand.\n\n\"It's an OIO key,\" he replied.\n\n\"What do you do with it?\"\n\n\"It's simply a reflection of the energy that is moving through you right now. Use this gift as you see fit.\"\n\n\"Um, thanks, I guess.\"\n\nWhen he turned back to the other Nagool, I slipped past them and returned to my room. I flopped on my sleeper and looked at the OIO key. I turned it over and brushed my fingers over the raised letters. I tried to push into the thing, thinking it was some sort of computer device, but there was nothing. It was simply a piece of plastic. It read:\n\n_Many entities in this universe feed on fear. They seek out fear, and when they find it, they encourage it. Their efforts are often subtle but effective, and you are completely unaware of their presence._\n\n_Understand that your fears are learned and compounded by others around you. Simply let this energy pass through your nodes and do not give it attention, as this fear is not yours._\n\n_Fear Nothing._\n\nI read the words again. There was something in their meaning that struck a chord deep within me, like the music I enjoyed so much. It felt like the OIO key was speaking directly to me, as if the author had followed me my entire life, experienced everything I had, and eavesdropped on that inner voice that only I heard. Is this what OIO was all about?\n\nI read the card one more time. The words empowered me. They allowed me to release the ownership of my fears and look at my needs with intense clarity.\n\nIt was time to talk to Max.\n\nBefore I even placed my feet on the floor, there was a knock at my door.\n\n\"Come in,\" I said, hoping it was Max. The door disappeared, and Hach entered my room.\n\n\"I've been informed about your decision,\" he said. \"I am pleased by this, especially after the incident with your sister.\"\n\n\"It was a little more than an incident,\" I pointed out. \"Someone tried to kill her. Someone who might even be involved with this place.\"\n\nHach checked the door. He seemed nervous. \"So you know? This is the reason for my visit. The Trading Council believes that one of you \u2014 one of the humans \u2014 is responsible for the attempt on your sister's life.\"\n\n\"One of us!\" I jumped off my sleeper. \"Are you crazy?\"\n\n\"May I remind you that I am still your Guarantor? And please keep your voice down. I understand the mood of many of the knudniks and the new Citizens on Orbis 4. I hear the whispers of war. Many feel cheated and rightly so, but it is no reason to upset the balance that we have worked so hard to maintain on these rings.\"\n\n\"At the expense of others,\" I reminded him.\n\nHach could only nod.\n\n\"Has anyone looked at the Council?\" I asked.\n\n\"Don't be ridiculous. The Council needs the Scion.\"\n\n\"So what are you getting at?\"\n\nHach checked the door again. \"At first I, too, thought it might be the Council that staged the attempt on the Scion. It certainly helps their position. But Queykay informed me that someone has been tampering with the taps. A concerned Citizen returned one to us, but the additional information had been wiped. I have authorities attempting to retrieve the missing data and trace the source of the tampering, but I must assume that one of the renegade groups of Citizens on 4 has something to do with this. My concern is that a few of _you_ my have been persuaded to join their ranks.\"\n\n\"Why are you telling me this?\"\n\n\"If this is true, I can only warn you that such an action is a threat to their lives.\"\n\nWhat had Theodore gotten himself into? Then I remembered: it wasn't just him. Max was involved, too, and so were Grace and that other kid.\n\n\"You look as is if you have just remembered something,\" Hach said.\n\nI stared at Hach and tried to put on my best liar's face. \"I don't know anything. Look, I'll be gone in a few cycles. My new training should help me track this person. It's not one of us, I assure you. I'll find whoever it is \u2014 I promise.\"\n\n\"That's good, because that _is_ your job now. You are the Tonat. Even if you discover that your friends are the culprits, Queykay will make you punish them. He may not own you, but he will use you.\" Hach turned to leave. \"Get some rest,\" he said. \"You look tired.\"\n\n\"Thanks,\" I muttered as the door closed. I fell back onto my sleeper. _What had I done?_ I couldn't leave my friends with Queykay. I needed to know what he knew. I jumped off my sleeper and peeked out the door to make sure Hach was gone. I slipped into the corridor but realized I had no idea where Queykay stayed when he was here, if he even was here.\n\n\"Vairocina?\" I whispered.\n\n\"Yes, JT.\"\n\n\"Do you know where Queykay rests when he visits us?\"\n\n\"Not exactly, but I do know there is a section of your building accessible only by council members.\"\n\n\"Can you help me locate it?\"\n\n\"It's only for council members,\" she reminded me.\n\n\"That's never stopped us before.\"\n\nUsing my staining, Vairocina located where I was in the building and directed me to the far side of the complex. I figured a good conversation with Queykay might reveal a few of his suspects, or at least get him thinking in another direction.\n\n\"I believe these are his quarters,\" Vairocina stated, and I stopped in front of a set of double brass doors stamped with the Orbis emblem.\n\nI knocked. No one answered.\n\n\"Is he here?\" I asked.\n\n\"It is impossible for me to know,\" she replied.\n\nLooking for an access point into the central computer, I spotted the entry pad to the left of the doors. When I pushed inside, I found Vairocina waiting for me.\n\n\"He is a Trading Council member,\" she reminded me. \"I don't want to see you get in any trouble.\"\n\n\"Something's not right about this guy. If I'm going to go away, I need to be certain he won't hurt my friends. I just want to talk to him.\"\n\n\"What if this is not his room? What if he is not even here?\"\n\n\"Then it won't hurt to look around a little.\"\n\nVairocina paused before she stepped aside. \"Thanks,\" I whispered, and then I unlocked the door and slipped out of the central computer and into Queykay's room.\n\nI was glad that he was gone. It would be easier to snoop around to find some answers than actually trying to get it out of him myself. I crept down the entry to his quarters, staying close to the cold walls. The only light seemed to emanate from plants spilling out of tall vases that were set back in the walls. It reminded me of an underwater cave.\n\nAt the end of the main hall, I spotted an open door. Warm light spilled into the hallway and mixed with the cool green light from the hall plants. I stopped just outside the door and peeked around the corner.\n\nWhat I saw caught my breath. Queykay was naked, lying on a stone slab, his robe on the floor beside him. His porcelain skin was covered with hundreds of little wriggling wormlike creatures that seemed to swell in unison as they nurtured themselves off Queykay's body. I stared in horror when one of the parasites, no more than six centimeters long, pulled away from Queykay. The moist sucking sound made me choke as the creature turned in my direction. Its beady red eyes lit on fire when they caught mine. Then it opened its bloody mouth and screamed.\n\nI ran.\n\nI did not leave my room the next cycle. I had no intention of bumping into Queykay and his brood. I had wanted to see Max, but everything I needed to say was now bottlenecked by the enormous amount of information that had been dumped on me. The missing files I had read before falling asleep only thickened the logjam with more of Quirin's reports.\n\nThe history of our parents, whom we thought had died on the _Renaissance,_ had been manufactured and placed within Mother purely for Ketheria's and my benefit. The depth of their elaborate ruse was actually inspiring, as Quirin and the Trust had created backup contingencies for every possible scenario. What they never anticipated, however, was two hundred of us arriving on the Rings of Orbis. Their intentions had been to destroy all the embryos and blame it, along with the death of the adults, on the failed cryogenics. Madame Lee's attack now looked like a convenient coincidence, but her actions must have been the catalyst for Quirin to abort his mission.\n\nComing to grips with the fact that there never were any human adults aboard the _Renaissance_ was like cutting the tether that secured me to the ship. I grew up thinking my parents had chosen to come to the Rings of Orbis to begin a new life. I had openly adopted that dream for Ketheria and myself, but now . . . all of that was a mirage. The dreams, the hopes, and even the girl I loved weren't ever meant to be. She wasn't meant to exist, according to the plan set forth by the DOL and the Trust. But why? So they could rig the outcome of their fate with the Knull? That's what it seemed like to me. And what was the Knull, anyway?\n\nThe craziest part of all was that I was going to accept this new reality and become a Space Jumper, the Tonat, or whatever they wanted me to be, so I could protect my sister. I was also forcing them to release my lifelong enemy \u2014 who apparently was my half brother. Argh! How in the universe could I explain any of this to Max? It didn't even make sense to me.\n\nI let another cycle slip past. I stopped in on Ketheria while they were changing the sensor bandages. Whoever had attacked her had done a nasty job on her shoulder. The medical AI was attempting to fabricate new bone tissue before tackling her muscles. I couldn't stomach anymore. I left Nugget with Ketheria and returned to my room to find Theodore waiting.\n\n\"You're coming next cycle, right?\" he asked.\n\n\"Sure,\" I told him. \"When?\"\n\n\"Third spoke. I don't know the location yet. It's kept secret until the very last moment.\"\n\n\"Are you sure this is safe, Theodore?\"\n\nHe looked at me and chuckled. \"Since when did you get concerned about _safe_?\"\n\n\"How does this group feel about Ketheria? You know, about OIO and all of the stuff that's going on.\"\n\n\"There are a lot of believers. Most agree that Ketheria's awakening is a sign \u2014 a sign that it's time to act. I think most of them would follow Ketheria to the corners of the universe.\"\n\n\"And you?\"\n\n\"What do you mean? I know Ketheria's your sister, but she's every bit a sister to me as well. We are a family.\"\n\n\"In more ways than one,\" I muttered.\n\n\"What?\"\n\n\"Nothing,\" I said. \"Do you ever wonder why Ketheria and I are the only siblings from the _Renaissance_?\n\n\"What are you getting at, JT?\"\n\n\"Think about it \u2014 all those kids and only one pair of siblings. Seems odd to me, don't you think?\"\n\n\"More odd than you being a softwire and Ketheria being the Scion?\"\n\n\"Good point.\"\n\n\"What's wrong, JT?\"\n\nI wanted to tell Theodore everything I had found out. I wanted to share it with someone. I needed to. I couldn't keep all of this inside me. But how could I tell Theodore he was a mistake? Just one of a thousand embryos brought along for the ride \u2014 just in case?\"\n\n\"My dad is alive,\" I told him.\n\n\"What?\" Theodore jumped off my sleeper.\n\n\"I met him after Ketheria was attacked. He's the reason why Ketheria and I are the way we are. He's not human. Well, not completely. He was a Space Jumper, just like they said. He messed around with our genetics. The Keepers wanted us this way.\"\n\n\"That explains a lot.\"\n\n\"Does it?\" I asked him.\n\n\"Well, first off, it explains your softwire abilities and Ketheria's awakening. No one else on the ship has the powers you guys have. _That_ I always found strange. It was only natural you were siblings. If your evolution had been affected somehow by space travel, why wouldn't it have happened to some others?\"\n\n\"You sound envious.\"\n\n\"Why wouldn't I be? I would kill to have some of what you have. Why was it that only you two got to race down the evolutionary highway at the speed of light?\"\n\n\"We aren't the only ones.\"\n\n\"What do you mean? Who else? There's another softwire?\"\n\n\"Switzer.\"\n\n\"No!\"\n\n\"And we share the same genetic code from Quirin, my father.\"\n\n\"You and Switzer are brothers?\"\n\n\"Kind of, but not really. We both received genetic anomalies from Quirin, so in some sense we _are_ connected genetically. Ketheria and I are actually from the same human female egg.\"\n\n\"So you and Ketheria are twins?\" Theodore slumped onto my sleeper, the weight of the information visible in his posture.\n\n\"Just imagine how I feel,\" I said. Theodore's blank stare told me he was trying to digest everything.\n\n\"And now Switzer is locked up.\"\n\n\"Because of this, the way I see it.\"\n\n\"Doesn't seem fair, does it?\" he remarked.\n\n\"I feel the same way,\" I whispered.\n\n\"What are you going to do?\"\n\n\"I've already done it. I've agreed to become a Space Jumper if they let me take Switzer. He doesn't deserve to be where he is. It's not his fault.\"\n\n\"JT!\"\n\n\"I know.\"\n\n\"Max is not going to like this.\"\n\n\"You can't tell her. I want to tell her.\"\n\n\"When? You've been avoiding her like a case of space scratch. Everyone sees it. She's crazy about you, you know.\"\n\n\"Does she say that?\"\n\n\"Everyone knows.\"\n\n\"I'll tell her at your meeting tomorrow. It's right before I leave. I think it will be easier that way.\"\n\n\"For who?\"\n\nI glared at Theodore, but he was right. It wouldn't be easier for Max. I'd been sitting with it for cycles, and now I was just going to dump it on her at the last minute. Now I felt twice as bad.\n\n\"You're not helping any,\" I said.\n\n\"She deserves to know,\" he said as he headed for the door.\n\n\"I can't.\"\n\n\"But you should.\"\n\nWhen Theodore opened the door, a messenger-bot entered with a screen scroll.\n\n\"I didn't think knudniks could get those,\" Theodore said.\n\n\"I think my cycles as a knudnik are numbered,\" I mumbled.\n\n\"Fortunately, mine are, too,\" he said, and left me alone.\n\nI pushed into the screen scroll thinking it was from Theylor. The scroll read:\n\n**Human: Turnbull, J.**\n\nAn escort will arrive on the next cycle during the third spoke to retrieve you. Your training will begin the moment your escort arrives. You are prohibited from bringing any possessions.\n\n**THE TRUST**\n\nThe third spoke! It figured. Well, they would have to find me, because I was going to that meeting with Theodore.\n\nAt the start of the next spoke, I went to say good-bye to Ketheria. I didn't know how long I would be gone, but I assumed it would be a short trip. How else was I supposed to protect her? I figured the Trust would make me uplink a few codecs, give me a belt, and then ship me back here before anyone knew I was gone.\n\nThe two Space Jumpers standing outside Ketheria's room only glanced at me as I slipped past them. I found my sister sitting up. She was alone and sipping from a small ceramic bowl.\n\n\"You look better,\" I told her.\n\n\"I don't feel it yet,\" she groaned. \"Who would want to do this to me?\"\n\n\"I'm afraid more people than you realize.\"\n\nKetheria gestured for me to sit on her sleeper, and I did.\n\n\"We need to talk before you go,\" she said.\n\n\"About?\"\n\n\"The Nagools have a ritual they call awakening a self. It is a fourteen-step spiritual journey that allows the individual to discover some part of him or herself, usually an important part on their path to enlightenment. Even I'm experiencing these steps on my path to truly becoming the Scion.\"\n\n\"Theylor told me about that, but why fourteen?\"\n\n\"Fourteen nodes in and around your body . . . fourteen levels to the labyrinth . . . fourteen keys to enlightenment. Fourteen is an important number. But that's not why I bring it up. Sometimes it can take an entire lifetime to awaken a single self. Sometimes it never happens. Space Jumpers use the same technique to awaken the selves that exist within them. Above all else, they believe in courage, self-discipline, and integrity.\"\n\n\"So? Those are good traits, aren't they?\"\n\n\"Yes, but they don't let you wait a lifetime to awaken that self. This is one aspect of their training I do not agree with. They use force to awaken those parts within you whether you're ready or not.\"\n\n\"How do they do that?\"\n\n\"I don't know exactly. I can't get anyone to tell me, but I wanted to warn you. I know you can be a little . . . stubborn sometimes, but that won't work with the Trust. You need to open your mind and allow them to do their work. Otherwise, I'm worried that the training will be a horrible experience for you.\"\n\nMy sister's eyes ballooned with tears ready to pop. I loved her for that. \"I can take care of myself,\" I whispered. \"They're just going to zap me with a couple of codecs and then I'll be back. Don't worry. Everything will be all right.\"\n\n\"I know,\" she said, and tried to smile, \"but this is different. Please listen to me and try to understand. Don't fight them. Let it happen. It will be good for you, for me, for us.\"\n\nI held Ketheria's hand in mine. \"I promise,\" I told her. \"I know it seemed I was resisting before, but I will make this work. Besides, the training will help me to protect you so that nothing like this will ever happen to you again.\"\n\n\"Good. Now, I also want you to think of something else to say to Max. What you're going to say won't work with her. In fact, it will probably make matters worse. Try something different. Tell her how much you feel for her, and blame this on me. She can't stay mad at me.\"\n\n\"Out of my head, little sister!\"\n\nKetheria laughed and swiped at a wayward tear that trickled onto her cheek. \"All right. But really, think of something else to say to Max. I'll be fine. They'll let you come back and see me, I'm sure. You're not in prison.\"\n\nI got up and walked to the door. Before I left, I turned to Ketheria and asked, \"Hey, with all your power to read minds and whatever other miraculous things you can do now, did you see who tried to kill you?\"\n\nKetheria shook her head. \"I was too busy soaking up all the love coming toward me. That's why I need a Space Jumper, big brother. Finding the bad guys is your job.\"\n\nKetheria's eyes were welling up again. She looked past me, as if trying to fight back the tears. She stood up, and I moved toward her. \"Ketheria . . .\"\n\nShe shook her head and pointed toward the door. I turned to find Nugget standing there.\n\n\"You're so handsome!\" Ketheria gushed.\n\nNugget stood at attention while holding his chin up like some dignitary. His big clumsy feet poked out from an absurd-looking military uniform with mismatched epaulettes and a crude set of medals pinned to his chest. I leaned in and noticed that the medals were made from pieces of plastic and crystal rocks stitched or pinned to his jacket.\n\nI tried not to laugh. \"What's going on?\"\n\n\"Nugget is on a mission,\" Ketheria said, and moved next to him. Nugget stood a little taller. \"I have made him a mediator on his home planet. It's one of his fourteen keys.\" Ketheria rubbed his chin, and I remembered the first time she did that back on Orbis 1, when she was the only one who could control him. \"Nugget is going to return to Krig and help reunite the Choi and the Choival. Nugget will do so with the blessing of the Scion. He will be an extremely important person on his planet. They will write songs about Nugget some cycle.\"\n\nKetheria couldn't stop the tears now. Nugget practically fell on her and threw his arms around her waist.\n\n\"It's all right,\" she cooed. \"I will miss you so much, you can't even imagine.\"\n\n_What was she doing?_ I wanted to step in. I wanted to say something. The race wars on Krig were legendary. Nugget didn't stand a chance. This was a death sentence for him.\n\n\"No, it's not,\" Ketheria said, reading my mind. \"Nugget is going to do great things. If he has any trouble, he knows that I will be in his heart. All he has to do is reach out to the Source, and I will be there for him.\"\n\n\"I don't know about this, Ketheria,\" I said.\n\n\"Of course, it will be difficult, but I know Nugget can do it.\"\n\n\"I am not afraid,\" he declared.\n\n\"But \u2014\"\n\n\"Walk us out, will you, JT?\"\n\nNugget spun on his heels and marched through the door.\n\n\"Why him, Ketheria?\"\n\n\"Why not him? You're judging him on his size.\"\n\n\"I'm judging him on more than that. You've seen what the Choival did to the Choi. Look at Weegin's wings.\"\n\n\"Trust me, JT. I have a knack for these things now. Nugget will perform magnificently in his new role, and he will bring peace to his planet like they have never experienced.\"\n\n\"You can see the future now?\" I asked, almost mocking her.\n\n\"No, but I can see into Nugget's Source, and he is the only one capable of performing this task, just as you are the only one capable of being the Tonat. It's your judgment of his ability that clogs his nodes and weakens him. So many creatures in this universe are repressed by the thoughts and discriminations of others. You need to stop that, JT. You must stop that now.\"\n\nKetheria reached for my hand and then smiled, taking the edge off her demand. I obliged, and we followed Nugget past the Space Jumpers and out into the open hall. Ketheria stopped short and glanced down the hall to her right. I could hear some sort of commotion coming toward us, as if someone was being dragged down the corridor. The sound was quickly succeeded by the spectacle of four faceless guards dragging the kid whose name I could never remember over the stone floor. Queykay marched behind them as Max and Grace hurried to keep next to him.\n\n\"Max!\" I cried, and rushed toward her.\n\nThe four guards stepped between us and took an aggressive position between me and the kid. Their long chrome chest plates extended and then locked together to form a barrier.\n\n\"What are you doing?\" I yelled.\n\n\"Leave it alone, JT,\" Max called out.\n\n\"He is my prisoner,\" Queykay responded as he stopped in front of the wall of guards.\n\n\"What did he do?\" I asked\n\n\"Darja didn't do anything!\" Grace said, and kicked one of the guards with her boot. The guard only glanced over his shoulder at her.\n\nQueykay threw back his crimson robe and reached behind the guards as if digging into a shipping crate. He surfaced with his rigid white fingers ensnared in Max's hair and dragged her to the front.\n\n\"Maybe I should arrest everyone to prove my point,\" he said with a snarl.\n\n\"Don't you touch her!\" I yelled.\n\n\"I can do whatever I want. I am a Trading Council member. _You_ are nothing.\"\n\nQueykay's brood scrambled down his arm and reached for Max's hair as well. They were getting bigger now, more than ten centimeters long. Queykay barked something at them, and they scurried back into his robe. Max's toes clicked on the floor as he lifted her higher off the ground.\n\n\"Stop it!\" Ketheria yelled at him. \"Put her down.\"\n\n\"If you hurt her, Queykay, I swear, I'll \u2014\"\n\n\"You'll do nothing unless I tell you to do it,\" he hissed. I watched in vain as Max clawed at the alien's hand.\n\n\"We'll see about that,\" I said, and moved toward the guards.\n\n\"Don't!\" Max screamed, but I would not stop.\n\nAs I moved, I adjusted the strength and torque settings in my right arm and used it as a battering ram on the four guards. I hit the middle one, hoping to buckle its armor and send them scattering. As my arm made impact, I heard the metal crunch and then snap apart as the guards scattered. I spun around, ready to attack Queykay, but the guards re-formed and surrounded me, their plates locking together again without a scratch.\n\n\"How pathetic,\" Queykay sneered. \"You? As the Tonat?\"\n\n\"Let them go!\" I yelled from within my makeshift cell.\n\n\"He is my prisoner. He is charged with treason. I have proof that he has been tampering with the taps and spreading lies around the rings. His punishment is death.\"\n\n\"No!\" Grace cried.\n\nNugget ran toward Queykay, his big snout open. He clamped onto Queykay's leg as the guards broke rank and turned on him. As Nugget bit down, Queykay screamed and released Max from his grip. She fell to the floor and rolled away from Queykay. I moved toward her.\n\n\"I'm fine,\" she said.\n\nI looked up and saw Queykay remove a long, silvery talon from under his cloak.\n\n\"Don't!\" Ketheria screamed.\n\nHe raised it over Nugget's head. The four guards that circled me broke rank and lashed at Nugget with black metal prods. In an instant, they had each secured one of Nugget's limbs with a lasso of sorts that was attached to the end the prod. They pulled, lifting Nugget off the ground as I leaped toward them.\n\n\"You're hurting him!\" Max yelled.\n\nJust as Queykay was about to bring the talon down upon Nugget, the hallway went silent. Queykay slid into a state of motionlessness, his eyes locked on Ketheria. The air around us was so still, I could hear my heartbeat. I could even hear Max breathing behind me.\n\nI looked down the corridor and then back the other way. It seemed as if everything was a little less colorful, a little less in focus. Frozen in their attack, Queykay and his goons did nothing as Ketheria moved toward Nugget and untangled him from the metal prods.\n\n\"Are you doing this, Ketheria?\" I asked.\n\nBut she didn't say anything. She worked quickly to get Nugget loose, and I moved in to help, snapping the prods in half with my robotic arm.\n\n\"What sort of powers _do_ you have?\" I whispered.\n\nOnce Nugget was free, she looked at me and said in a hushed voice, \"I don't really know. I find new ones every cycle.\"\n\nWith everyone still locked in some sort of alternate reality, Ketheria turned to me and said, \"I suggest you leave.\"\n\n\"Wait!\" I protested. \"He's not safe.\" I pointed at Darja. \"Queykay will take revenge on him \u2014 I guarantee you that.\"\n\n\"Queykay will not remember a thing. Nor will the guards. At least I don't think so,\" she replied, and turned to Darja. \"Where were you when this all started?\"\n\n\"I was in my room,\" he said.\n\n\"Go back there. I will meet you there shortly. Queykay will attempt to arrest you as he did before, but I will be there with a Keeper and a Nagool. They will not let Queykay take you. Max, Grace, take him there now, please.\"\n\nThen Ketheria put her arm around Nugget's shoulder and started to lead him away.\n\n\"But wait. I don't know when I will see you again,\" I said.\n\n\"You will. That's all that matters.\"\n\nKetheria disappeared down the corridor and I turned to Max, but she had already disappeared with Grace and Darja.\n\n\"Max?\"\n\nI looked up at Queykay, who was still staring at the spot where Nugget once was. I stepped toward him and reached up with my hand, waving it in front of his icy face.\n\n\"I wish you could stay like this forever,\" I whispered.\n\nWith my forefinger and thumb, I picked at the edge of Queykay's sleeve and lifted it up, hoping for a glimpse of the tiny creatures nurturing themselves off Queykay's body. Before my eyes were able to crack the darkness of his sleeve, two bloodred eyes launched themselves at me. A zipper's worth of pointed white teeth sparkled as they broke from the shadows, and I fell back, horrified. The little worm landed on my leg screeching as it tore at my pants, trying to burrow itself into my leg. With a quick sharp blow, I struck the abomination with my right hand and watched it skitter across the stone. I was on my feet and down the corridor before it even turned around.\n\n#\n\nI sat up in my sleeper. _Theodore will be here soon,_ I thought. I had tried to sleep, but instead I'd lain awake berating myself the entire spoke. I was ashamed at how useless I had been against Queykay and his goons. What good was I going to be to Ketheria? What could the Trust teach me that wasn't better than these powers she kept developing? I hoped I had made the right decision.\n\nSince I couldn't sleep, I decided to stop at the chow synth before I went to the meeting with Theodore. I needed to get that food dispenser to manufacture Tic's lifesaving potion into some sort of solid form, like the food tablets we ate at Weegin's World. There was no one at the chow synth when I entered, and I was glad for that. I poked into the synth's chip, and despite the infinite array of choices, I could not figure out how to change a liquid to a solid. The synth would let me freeze and dehydrate, which I thought might work, but it would not let me make a tablet. I pulled out and called Vairocina for help.\n\n\"I was wondering when you were going to call,\" she said.\n\n\"Can you help?\"\n\n\"Let me link through the chow synth,\" she replied as she materialized before me.\n\n\"I guess you know I'm leaving.\"\n\n\"I do,\" she said.\n\n\"I should have called you to say good-bye.\"\n\n\"Good-bye? I assumed I was coming with you.\"\n\n\"Oh.\" I hadn't thought about bringing Vairocina.\n\n\"I was told I could not bring anything \u2014 I mean anyone!\" I said, correcting myself and trying to sound as if I had already thought about this. \"Besides, I would feel much better if I knew you were still here keeping an eye on everyone for me. I have no idea what my training is going to be like or even how long I will be gone. I really need you here. I'm sure if anything goes wrong, you can get a message to me through Theylor or Drapling.\"\n\n\"So, this is good-bye then?\"\n\n\"It's more like see you in a while.\"\n\nVairocina dropped her head as if she were looking at the ground.\n\n\"It's getting boring in here,\" she whispered. \"It's so much more exciting when I'm inside your arm, going places with you.\"\n\n\"I'll be back, and it will be like I never left. I need you here, V,\" I said, hoping Theodore's nickname would help.\n\nShe smiled. It was a small smile, but it was better than nothing. \"I'll try to find the link the Keepers use with the Trust, and maybe I can communicate with you through that,\" she said.\n\n\"That would be perfect,\" I told her. \"Now, any chance you can help make this thing spit out some magic tablets for me? I'm going to need a lot.\" I was also thinking about taking some for Switzer.\n\n\"My pleasure,\" she replied.\n\nBy the time I returned to my room, Theodore was waiting for me to go to the secret meeting. I had half expected to find a couple of Space Jumpers waiting for me instead.\n\n\"I thought Max was going to be with you,\" I said.\n\n\"She went with Grace.\"\n\n\"Does she know I'm coming?\"\n\n\"Yes, she does, but Darja doesn't think this is a good idea.\"\n\n\"Darja?\"\n\n\"You know, that kid. He changed his name again.\"\n\nI shook my head. \"How long do these meetings take?\" I asked him.\n\n\"Usually most of the spoke,\" he replied.\n\nWith the tablets spread between two pockets in the legs of my pants, I slipped Quirin's disc into my back pocket. I had become used to traveling light, and I saw nothing else worth taking. I tossed my skin onto my sleeper.\n\n\"You're not coming back, are you?\" Theodore asked. There was a little worry in his voice.\n\n\"Not for a while, I'm beginning to think.\"\n\nTheodore checked and rechecked each corridor leading to the far end of our building and away from Queykay's quarters, counting his steps as we moved. When he was sure no one was watching, he overrode the security panel at the back exit of the building using a crude-looking uplink attached to his neural port. A small blinking box dangled in the middle of the hardwire. When it turned blue, the door opened.\n\n\"How did you learn to do that?\" I whispered.\n\n\"Max showed me.\"\n\nAs we sprinted across the open compound and into Murat, Hach's suspicions that one of us was involved in the attack on Ketheria trickled into my thoughts. _That's just ridiculous,_ I told myself.\n\nThe city was rotating into shadow, and the building lights blinked on in sequence as the city was slowly swallowed up by the encroaching darkness. I stayed close to Theodore as he picked up the pace and raced toward the darkness. This was my favorite time on the rings. Sparkling reflections of red and gold replaced the dirt and grime, which seemed to wash away with the receding light. Things that once looked decrepit and uninviting now shimmered in the golden dusk.\n\nThis was also the same time of cycle that Max had taken me to the concert in Murat and the time we'd held hands while watching the musician play those strange glass bowls on Orbis 3. This was also the time Max and I would leave the Labyrinth and head home after a Quest-Nest match last rotation. It was in this same dusky glow that Max and I would walk in the garden behind Charlie's, and it was the same time of the cycle that Max and I often strolled through Murat. I loved this time of cycle.\n\nTheodore and I stopped in front of the podlike living quarters that stacked up and over the street. Using a control pad, he punched in a long access code, one I would never have been able to remember.\n\n\"We can't possibly be meeting in one of these things, can we?\" I said.\n\n\"Keep it down,\" he whispered. \"We're not.\"\n\nOne pod broke rank, rotated silently to the street level, and then cracked open. Theodore reached inside and pulled out a tap that he attached to his neural port.\n\n\"What's it say?\" I asked.\n\nHe pulled the tap out and dropped it on the ground. Then he crushed it under the heel of his boot. \"I got it,\" he announced, and turned up the street. All I could do was stare at the shards of plastic on the ground. What had happened to the Theodore I knew? The one who avoided trouble like a Trefaldoor avoided a lie?\n\n\"Wait up!\" I called after him.\n\n\"It's not far,\" he said, pointing up ahead.\n\n\"Does everyone find out about the meeting this way?\"\n\n\"Pretty much. It changes, though, when we think someone might be watching us.\"\n\nI turned and looked behind me to see if that was the case, but I saw no one. \"Who sets this up? I mean who's going to these great lengths to keep this hidden?\"\n\n\"We are,\" he replied.\n\n\"I know that, but who's the leader, who's the head guy?\"\n\n\"Oh, we've never met them. We're just one cell. There are hundreds, just like us, waiting all over the ring.\"\n\n\"Waiting for what?\"\n\n\"For orders.\"\n\n\"Orders for _what_?\"\n\n\"Well, obviously we're waiting for the big order, the one that tells us we're going to fight for our freedom. When the Keepers and the Trading Council go at it, we'll be ready to spring into action. But mostly it's little things. Like that tap back there. I'm sure someone in some cell was given an order to place that there. They had no idea why; they just did it.\"\n\n\"Have you ever had one of these orders?\"\n\n\"Lots.\"\n\n\"Wait,\" I said, stopping in the street. \"What's going on here, Theodore?\"\n\n\"What do you mean?\"\n\n\"Everyone running around following these orders, with no connections to the big picture. Think about it. How easy would it be to set up an assassin to take out Ketheria? No one would really know what had happened. All these little things could be put in place without anyone ever knowing they were helping to kill the Scion. Think about that.\"\n\nTheodore stopped and cocked his head, his eyes wide. \"We wouldn't kill Ketheria. The Scion is one of us. The Scion is a knudnik, JT. She came here just like we did, to work for the Citizens, to labor in their system, to dream _their_ dream. We have more in common with the Scion than we do with anyone here.\"\n\nWas this where I told my friend that he had nothing in common with the knudniks on the Rings of Orbis? Did I tell him that he never had parents who chose to come to Orbis to work \u2014 that his life was an accident? Did I tell him that if Madame Lee had never attacked the _Renaissance,_ he would have been flushed with the rest of the embryos on the seed-ship?\n\nOf course I didn't.\n\n\"I'm sorry,\" I mumbled. \"I didn't mean to imply anything. It's just that Hach told me that someone on the inside was involved.\"\n\n\"And you thought it was _me_? You thought I would try to kill Ketheria? Are you crazy? I can't believe you would think that.\" Theodore glanced over his shoulder toward the building to my right.\n\n\"What's wrong?\" I asked.\n\n\"I don't know if you should come, JT. People in there aren't going to take that kind of thinking lightly. We are the last people who would try to kill the Scion.\"\n\n\"Theodore, I'm sorry. I won't say anything. Please, I need to talk to Max. I'll be _gone_ by next cycle.\"\n\n\"I don't even know if she'll talk to you. Now is not the right time. You should have done this earlier, JT.\"\n\n\"But I didn't. This is my only chance. Please. I won't say a word. I'm sorry for even thinking it.\"\n\nTheodore waited. It was a long pause. He was actually considering not bringing me. The thought freaked me out a little.\n\n\"Not a word, then?\" he whispered.\n\n\"I promise.\"\n\nAs we slipped off the main street and down an unlit alley, I was reminded that this was the second time I had followed Theodore to some unfamiliar place under his direction. The first one was the Shed, where he'd sneaked off to use a tetrascope. I only hoped this was a better place than that. When Theodore stopped, there was no light chute this time, no industrial cavern, just an unmarked metal door. He opened the door without knocking, and I followed him down a narrow hallway lit with golden glass balls that were embedded in the mottled walls.\n\n\"Whose place is this?\" I whispered.\n\n\"I don't really know. No one ever tells me,\" he replied.\n\n\"Has the meeting started?\"\n\n\"I don't think so.\"\n\nWe passed several unmarked doors before stopping in front of a double metal door at the end of the hall on the right-hand side. I waited as Theodore ran his hand over some sort of scanner before reaching down to grab a cable that hoisted the door up. I don't know why the scanner bothered me. Maybe because it meant that there was some record of the people who were allowed entry into whatever club was now meeting on the other side. I had always thought anonymity was the best defense when doing something you weren't supposed to.\n\nThere were several people on the other side of the door. They looked up at Theodore and me when we entered. I saw Grace and that kid immediately.\n\n\"What's he doing here?\" the kid said. I didn't like his accusatory tone. If any knudniks were involved in the attack on Ketheria, I would check him out first. Maybe Queykay had something when he picked this kid up.\n\n\"What's wrong with JT coming?\" Theodore confronted him in my defense.\n\n\"He's a softwire. He's not on our side.\"\n\nThere was that side issue again. This little group was quickly getting on _my_ bad side.\n\n\"He's still a knudnik,\" Theodore argued.\n\n\"Hardly,\" the kid spat.\n\n\"What is your name now, anyway?\" I interrupted.\n\n\"Why do you want to know, so you can run and tell them?\"\n\n\"Tell who? Queykay? It seems he knows you're up to something already. Need I remind you that you're stained, just like me? They don't need your name to find you.\"\n\nGrace jumped in. \"His name is Ganook now. He won't be changing it again,\" she said, smiling and placing her hand on his shoulder. Grace's smiled seemed to put the kid at ease. _Ganook?_ His choice, not mine.\n\n\"Besides, I think you should be thanking me for rescuing you last cycle.\"\n\n\"You? Rescue _me_? The Scion did everything. In fact, as I recall, you were rather useless.\"\n\nI stepped toward the kid. Even Theodore moved next to me.\n\n\"That is enough,\" a voice said behind me.\n\nI turned and saw a slender alien approaching me. Large green eyes eclipsed his narrow forehead, and two small bones protruded from his slanted shoulders. The bones supported a deep burgundy cloth that wrapped around him, almost like a loose cocoon.\n\n\"Remember: we are against no one,\" he said, his voice deep and soothing. Instantly, I felt relaxed. \"We are for freedom and the sanctity of the moons, just as the Ancients were so long ago. We only wish to awaken from the dream, the dream of the Trading Council.\"\n\n\"You must agree with the Keepers then. They want the same thing,\" I said.\n\n\"Some, maybe. But greed has corrupted many of them as well.\"\n\n\"Then what _do_ you guys want?\" I asked him.\n\n\"Freedom from the way of life that has destroyed the hearts and minds of so many here on the rings. Freedom from the tyranny of the Citizens and freedom from the deconstructive energy that plagues these rings and anyone who walks among us. It was not always like this, you know. The Rings of Orbis were once the glowing epicenter of true Source energy.\n\n\"So I've heard,\" I said.\n\n\"Please join us. My name is Horgan.\"\n\nHorgan extended his arm and motioned toward a large green and gold curtain suspended from the ceiling. I could see more light through the sheer material and more people. The room was larger than I had realized.\n\nHorgan walked next to me as the group moved toward the curtain.\n\n\"We could benefit from the abilities of a softwire. I'm sure you appreciate this. Your powers could single-handedly disrupt the lives of the Citizens.\"\n\n\"I'm afraid I've heard that one before as well.\"\n\n\"You are far more powerful than you realize, and so many of your friends are already with us.\"\n\nOn the other side of the curtain, I saw Max sitting at a huge round table in the center of the room. Her shadow, created by open fires placed around the table like gigantic candles, flickered on the wall. Everything was afire in an orange glow.\n\n\"I can see that,\" I replied, \"but I'm afraid I can't stay long.\"\n\n\"Maybe after you sit with us, you will decide to stay longer.\"\n\n\"Maybe,\" I said.\n\nWhen Max saw me, she looked at Theodore and then back at me. Max turned away and struck up a conversation with Grace, who had slipped in next to her.\n\nMy ears were burning. I almost used the sweat from my palms to cool them off. Max was going to freak when I told her I was leaving to train as a Space Jumper. I knew it. I hesitated to sit next to her, but I knew there was no more time to waste. In fact, my escort could show up at any second to take me away. I was moving toward the empty seat next her when a small Honine, her forehead spotted with small spikes, sat down before I could.\n\n\"Excuse me. Do you mind if I sit here? I would like to talk to my friend,\" I said. I made sure Max could hear the last part so at least she would be aware of my intention.\n\nThe Honine smiled and moved away without incident. I sat next to Max, but she continued to talk with Grace. I hadn't been this close to Max in a while. Just her scent made my eyes close, and I secretly breathed her in. When I opened my eyes, she was looking at me.\n\n\"What are you doing?\" she hissed.\n\n\"I was just \u2014 I mean, I . . .\" It was hard to talk over the lump that was now formed in my throat. Why was this so difficult?\n\n\"I mean, what are you doing here?\" Max said.\n\n\"I need to talk to you about something.\"\n\n\"We've had plenty of time to talk since your return. I don't think this is the appropriate place for you to be apologizing to me.\"\n\n\"Who said I was \u2014? I mean, I _am_ sorry.\"\n\nMax kind of grunted. \"Do you even know what you're sorry for?\"\n\n\"I know what I'm going to be sorry for,\" I mumbled.\n\n\"What?\" she asked, but then Horgan stood up and raised his arms to quiet everyone down.\n\n\"Welcome, all. There are some new faces around the table. It makes me happy to see our ranks swelling.\"\n\nMax's arm was about ten centimeters away from mine \u2014 so close. I wanted to touch it. I wanted to tell her what I was about to do while I was holding her, not like this. Theodore was right. It was the wrong place and the wrong time. I was such a malf.\n\n\"Max, there's something I need to tell you.\"\n\n\"Shhh,\" she shushed me as Horgan continued.\n\n\"As we discussed last time, signs of war are increasing. It was just this cycle that the Trading Council suspended all new petitions for work rule from inquiring races.\"\n\n\"What?\" I said out loud.\n\nHorgan paused and nodded. \"Oh, yes,\" he said. \"The Council has even suspended education for knudniks and Citizens nationalized in the last five rotations. And there is talk to suspend all work-rule expirations until the Council feels it appropriate to continue normal contracts. They want to make sure they have control of everyone and everything if they choose to go to war against the Keepers.\"\n\n\"But why would they want war?\" I asked.\n\n\"The Keepers have broken the treaty set in place almost two thousand rotations ago. The Council is using the Keepers as scapegoats and rallying the Citizens into a unified frenzy over the need to eliminate all Space Jumpers, and the Keepers with them. The Keepers' arrogance has given the Trading Council the perfect enemy to go to war against.\"\n\nThe crowd seated at the table fidgeted in their seats each time Horgan spoke. Some nodded anxiously while others called out in agreement.\n\n\"But why would they want war? They will only hurt themselves,\" I yelled over the others. They went quiet when I spoke.\n\n\"War will give them more power. War will give them more control over the crystal moons with less interference from the Keepers and the Trust. The Citizens could rule the rings the way they want to, without restriction, and _that_ would be devastating for the likes of you and me on the Rings of Orbis.\"\n\nHorgan began speaking to the entire crowd now. \"Can you imagine what it would be like on the Rings of Orbis if the Trading Council controlled the laws? Can you imagine how many races would suffer under the hand of slavery? Enticed by wealth and greed, many more people would flock to the rings unaware of the consequences that await them.\n\n\"The Trading Council also wants to set their own prices for the crystals harvested from Ki and Ta. Many civilizations that rely on these energy sources would be shackled under the exorbitant prices set by the Council. With no intervention from the Keepers, economies would grind to a halt, societies would crumble, and, all the while, the Council would grow fatter from their obscene profits.\"\n\n\"And how do you plan to stop them?\" I asked.\n\n\"Do we have to explain ourselves every time someone new comes along?\" Ganook complained. \"This is a waste of time.\"\n\nI stared at the kid I had grown up with on the _Renaissance._ \"What's your problem?\" I asked him. \"What do you have against me?\"\n\n\"He has nothing against you,\" Grace interrupted, putting her hand on the kid's shoulder.\n\n\"Let _him_ answer,\" I said.\n\nGanook stood up, shaking off Grace. He kicked his chair back and walked away from the table. Then he turned toward me. \"Don't you see it? You are their instrument,\" he growled. \"Just your presence gives the Trading Council another excuse to act.\"\n\n\"But if you want the Keepers to stay in power, aren't you going to need the Space Jumpers on your side? When I become a Space Jumper, I \u2014\"\n\nI stopped mid-sentence and looked at Max. She spun around, her mouth agape.\n\n\"I'm sorry. I can explain,\" I whispered.\n\nMax's eyes filled with tears. She shoved her chair back and ran from the room. Grace got up to go after her, but I stopped her. \"Let me, please,\" I said.\n\nI chased Max through the curtains and found her against the wall, her face in her hands. Her shoulders shook in unison with her sobs. I placed my hand on her back.\n\n\"Don't!\" she cried, and pulled away.\n\n\"Max, please, it's not what you think. I don't have a choice.\"\n\n\"We always have a choice!\"\n\nWithout warning, an explosion ripped through the building. Chunks of concrete tore through the curtain, and I threw myself over Max as the debris rained down upon us. I heard screams and felt my skin turn warm as the open fires leaped from the their containers and crawled over anything that would burn.\n\n\"Theodore!\" Max cried from underneath me. She pushed me off, and we both jumped up. Max sprinted through the dust and smoke that now choked the room.\n\n\"Theodore!\" I yelled, but I heard only moans and crying as the last pieces of rubble trickled to the ground. \"Be careful, Max!\"\n\nI bumped into Grace, clawing her way through the smoke and debris. She was bleeding from her forehead, and her hands were covered in blood.\n\n\"Ganook!\" she screamed, looking around. \"Ganook!\"\n\nI grabbed her by the shoulders. Tears and blood raked through the dust on her face. \"Grace! Grace, we'll find him,\" I yelled, shaking her back to reality.\n\n\"He turned blue!\"\n\n\"Blue? Who turned blue? What do you mean?\"\n\n\"Ganook. He turned blue just before the explosion! His whole body.\"\n\n\"Blue? Grace, you're hurt. Sit down.\"\n\n\"No! He was a target. They wanted him. They knew!\"\n\nIt was Queykay. I was certain of it. If Ketheria's little stunt had not completely erased his memory, then a trace on Ganook could have exposed his whereabouts. Could they simply kill us from an O-dat whenever they wanted? I certainly hoped not.\n\n\"JT, find Theodore,\" Max cried.\n\n\"I'm here. I'm all right,\" he shouted from somewhere in the smoke. \"You have to find Grace. Make her stay where she is. Don't bring her over here.\"\n\n\"Why!\" Grace screamed. \"GANOOK!\" She ran into the dust before I could grab her.\n\nI was wading through the debris, looking for Max, when I saw streams of light materialize in the dust. They rippled before merging into a single point. A moment later, I was staring at two Space Jumpers standing in front of me.\n\n\"Not now!\" I cried, but the Space Jumpers looked at the destruction around me and worked quickly. One armed his plasma rifle, while the other moved toward me and gripped my forearm.\n\n\"What happened here?\" he said.\n\n\"I don't know. Let me find out. Give me a minute. Please! Max!\"\n\nMax stepped through the dust and saw me standing between the two Space Jumpers. \"Please, Max, not like this,\" I pleaded. \"Try to understand. I never wanted to hurt you.\"\n\nMax didn't say anything. She just stood there amid the debris. Tears streaked the soot that had settled on her face. She lifted her right hand as if to wave good-bye. I didn't know for sure because I was ripped away before I could respond.\n\n#\n\nMy arrival at the Trust was as uneventful as a solar flare on the surface of the sun. The two Space Jumpers dumped me into my room and left without even a welcome. The stone floor and metal fixtures were an exact copy of the room they had put me in before, the first time I was taken to their comet. I looked up and realized, however, that something was different. This time I had a roommate.\n\n\"Of all the rocks in the universe, they have to put me with you?\" Switzer grumbled.\n\nI sat on the floor, Max's image still emblazoned on my mind. \"Give me a minute and I'll call someone to see if they have anything else available, something more suiting a wormhole pirate.\"\n\n\"At least your sarcasm has gotten better,\" Switzer said, and leaned back on his sleeper. He hoisted his huge boots over the edge and let them clunk on the metal as he clamped his thick hands behind his even thicker neck. It was difficult for me to get used to the older Switzer. I had to look carefully, past the scars and muscles, to see the kid I had grown up with. If it wasn't for his cocky attitude, I don't know if I would have recognized him at all. \"But really,\" he went on in his deeper, Switzer-the-man voice. \"Thanks. I thought I was going to rot in that awful cell. I owe you one.\"\n\nHis grateful remark caught me off guard. So much had happened between the two us, and none of it was pleasant. Yet I couldn't help but glance at him and blame myself for his very existence. \"It was the least I could do,\" I said without sarcasm.\n\n\"Yeah, I've been thinking about that. Why _did_ you do it, anyway? From what I've seen, you had a pretty good thing going on down there. What possessed you to drag me out of prison?\"\n\n\"I told you already. It's not fair what they did to us. They made you the way you are. They are responsible for you. They can't just lock you away. Your actions are just as much their fault as yours.\"\n\n\"Not sure I see it your way, split-screen. I've done a lot of bad things that had nothing to do with them, but thanks all the same.\"\n\n\"Please stop calling me that.\"\n\n\"What?\"\n\n\"Split-screen, Dumbwire, or whatever witty little tag you can come up with. Call me JT or don't call me anything.\"\n\nSwitzer paused before muttering, \"Sure. Whatever you want. Hey, how did your girlfriend take it?\"\n\nI stood up and moved in front of his sleeper. \"Do not talk about Max,\" I told him. \"Ever! Understand that I do not want to be here. I did it to get you out. My goal is to get through this stupid training and then get back to the Rings of Orbis. I need to protect Ketheria. You're not the only one I feel responsible for. When you're done with your training, I want you to accept some post in another galaxy, all right? But after you become a Space Jumper, I never want to see you again. Understood?\"\n\n\"Wow, one minute you're feeling sorry for me and the next minute you never want to see me again. What's with that?\"\n\n\"Nothing that I'm going to tell you,\" I said as I turned away.\n\n\"Suit yourself,\" he replied, and leaned back only to sit right back up again. \"Hey, do you have any more of that stuff you gave me back on the rings? I could really use some right now. Just a little. I don't want to get used to it.\"\n\nI reached into my pocket and pulled out the bag of tablets I had made for Switzer. I took a couple from the bag and popped them into my mouth. My head had been killing me ever since I'd arrived. \"Here,\" I said, and tossed the bag to him. \"They'll do the trick.\"\n\nSwitzer caught the bag and held it up, admiring it. Then he looked at me and said, \"You'll make a good wife some cycle, split-screen.\"\n\nSwitzer's teasing was the least of my concerns. I went to sleep worrying about how they'd gotten to Ganook and woke up wondering what Max was doing. How long was I going to be here? _It couldn't be that long,_ I told myself. Then I remembered slow-time. The Keepers had used it in the Center for Science and Research, and the Trust even mentioned it the first time I was here. Surely they must be using it now. If they wanted me to protect Ketheria, how could I do it from here? All I had to do was finish their little course and I would be done. _Keep an open mind._ That's what Ketheria had said. How hard could that be?\n\nSwitzer's sleeper was closed, but I could still hear him snoring. I pushed back the lid of my own sleeper and stared at the metal door. A soft blue light lined the perimeter. I figured some sort of computer chip controlled all the doors on this ship, or rock, or whatever it was that they called this thing. I glanced over at Switzer. He was out cold. Since no one said we had started training yet, I figured now was a good a time as any to do a little exploring.\n\nI slid off my sleeper, stuffed my feet into my boots, and went to the door. I pushed into the blue light and was surprised to find a rudimentary locking device, which I merely nudged open. The last time I was on this thing, the Trust had kept me locked up using a far more elaborate security system. _It's not like I could leave here, anyway,_ I thought as the door disappeared. Outside, I dragged my fingers along the stony wall and slipped down the corridor.\n\nI had no idea where I was going, so I followed the polished support girders that reflected the frigid glow from caged lights mounted above my head every meter or so. It was cold, and I could smell a slight medicinal scent lingering in the air.\n\nI turned right down another corridor. _The place is bigger than I thought._ I found a short set of stairs at the end and climbed them into a small atrium. I stepped toward a large door at the far end, and it disappeared. Once inside, I found myself looking out at the stars through an enormous glass dome. It was some sort of observation deck, like the one we had on the _Renaissanc_ _e._ I loved that place. I spent so many cycles staring out at the stars, wondering what my new home would be like.\n\nI went up to the glass to look out over the ship, but what I saw really wasn't a ship at all. It was just a big rock \u2014 a huge comet falling through space. Behind me, a brilliant white tail of dust and ice lit up the empty blackness, and it was _empty._ There were no planets on the horizon, no nearby stars to light up the ship. But worse than that, the most glaring absence of all was that of the Rings of Orbis. I searched everywhere, running from side to side of the observation deck, but I could find no sign of it. _Where was I?_ Everything I knew was gone. The home I had struggled to accept was nowhere to be found. _What had I done?_ The enormity of my decision settled on my shoulders and forced me to the ground. I had felt alone in my life before, but never like this. Sitting there, on top of the comet, with nothing in sight, I felt _more_ than alone. I felt dead.\n\n\"Do I really snore that bad?\" Switzer asked.\n\nI opened my eyes. I was still in the observation deck. I must have fallen asleep on the floor. \"What time is it?\" I mumbled, looking up at Switzer. There was a strange device hovering near his head, a golden light suspended over a metal spike like a torch.\n\n\"Um . . . you got something here,\" Switzer whispered, pointing at the corner of his mouth. I reached up and wiped away the drool that must have escaped while I was sleeping.\n\n\"Thanks. Who's that?\"\n\n\"Him?\" Switzer said, thumbing at the thing floating in the air. \"That cheery little fellow is our escort. I think we're going to meet the rest of our playgroup.\"\n\n\"What are you doing here?\"\n\n\"I didn't want to start my first cycle of school without my new buddy.\"\n\n\"Enough with the sarcasm. Do you have any of those tablets? My head is killing me.\"\n\nSwitzer tossed me a couple tablets and said, \"Try to suck it up a little and use those only when it hurts. I don't know how long we're going to be on this icicle.\"\n\n\"So you know where we are?\"\n\n\"I do now,\" he replied, walking over to the edge of the observation deck.\n\n\"Orientation is now assembling. Your presence is required. Please follow me,\" the floating thingy announced as its light flashed red.\n\n\"Ah, the universal color of danger. How long do we plan on letting them wait?\" Switzer said, turning to me.\n\n\"I don't,\" I told him, and sprang to my feet. \"Time to be a Space Jumper.\"\n\nWe followed our escort down several corridors before the light led us to a small lift suspended over a huge open area. The place was so big that I think I could have flown a ring shuttle through it. Directly across from our lift were a couple of enormous cylinders like two giant spacescopes standing next to each other, balanced on their lenses. I squinted to see what they were made of, but the black metal was punctured with an assortment of bright yellow, green, and white lights that made it hard to tell. I could see that each pillar was constructed from odd-size sections stacked one upon the other and that the sections were simply too numerous to count. _Theodore would have tried to count them, though,_ I thought.\n\nThen I heard some unseen motor clunk into action, and the two massive pillars began to rotate as we descended to the floor. I watched the enormous structures peel back and the room we were in flood with a brilliant blue light that hung at the center of the next room.\n\n\"This is what I call an entrance,\" Switzer remarked as we both stepped off the lift. There was another alien already on the floor, watching the pillars part. I elbowed Switzer and he glanced at our new companion.\n\n\"We aren't the only new kids at school,\" he said.\n\nWe followed the red light into the next room. It was circular, lined with balconies. It was difficult to see all the way to the top because the light source was blinding near the room's apex. I shielded my eyes and discovered rows and rows of Space Jumpers awash in the harsh light and staring down at us.\n\nSwitzer was gawking as well. \"Guess being a softwire ain't as special as you thought,\" he whispered.\n\nI guess it wasn't. I knew that all Space Jumpers were softwires and from what I'd understood, it was an especially rare ability. But when I walked past the cylinders and looked up, I almost felt common.\n\nSwitzer, the new guy, and I walked under a huge egglike structure suspended by a thick metal cable at the center of everything. The thing must have been the size of a small spaceship. In fact, I actually wondered if it _was_ a spaceship of some sort.\n\n\"Well, I'm impressed,\" Switzer said.\n\n\"Me too,\" I replied.\n\nThe other guy, an alien with thin tentacles that sprouted from his head, his arms, and even his back, did not respond. Instead, his many tentacles cautiously flicked about as if they were licking the air.\n\n\"What are we supposed to do?\" I whispered to Switzer.\n\n\"Nothing. At least that's what I normally do when the odds are like this. Let them make the first move,\" he replied.\n\nThe egg thing started to descend, and Switzer and I moved back. Circling the center of the egg was a metal support mounted with half a dozen spotlights. Three of them fired up and focused on us. The two giant cylinders slowly swung closed as the egg began to speak.\n\n\"The Source has bestowed its most important gift upon you,\" a voice boomed from the egg. I recognized that voice. It was one of the Trust. \"Do you accept this gift?\"\n\n\"Yes!\" cried the alien next to us.\n\n\"Absolutely,\" Switzer exclaimed.\n\nI looked at Switzer. I guess this was it. I was going to do it. I was doing it for Ketheria, but I was also doing this for me in some weird way. I was finally able to make that \"choice\" I was always grumbling about.\n\n\"Yes,\" I added, with much less enthusiasm than the others.\n\n\"Your softwire ability is a prerequisite, but your Source is the admission into our family,\" the Trust continued. \"Do you accept these terms?\"\n\n\"Yes!\" cried the other alien.\n\n\"You bet,\" Switzer added.\n\n\"What does that mean?\" I whispered to Switzer. \"Source? What source? I don't get it. What terms?\"\n\n\"Just say yes, split-screen. There is no pamphlet to read, and I don't think they pay for the ride home if you want out now.\"\n\n\"Yes,\" I replied, looking up at the egg. I couldn't believe I was taking advice from Switzer.\n\n\"A Space Jumper is nothing without his family. His family understands his gift, and his family shares his unquestionable belief in the trinity,\" the egg said.\n\n\"Courage! Integrity! Self-discipline!\" These three words stormed down upon us from the hoards of Jumpers who filled the balconies all around us.\n\n\"Your new family will guide you in your acceptance of these tenets. Only when these selves are awakened within you can you claim to be a Space Jumper.\"\n\nThe Space Jumpers responded by cheering and thumping on whatever was close to them, even if that included each other.\n\n\"Piece of cake,\" Switzer boasted.\n\n\"Please step forward as your name is called,\" the Trust ordered. \"Gora Bloom!\"\n\nThe alien with the tentacles practically jumped.\n\n\"Your connector will be Sul em Pah; your monitor will be Kebin Tam.\"\n\nThe crowd applauded, and Gora looked like he or she (I couldn't tell) had won some sort of award.\n\n\"Randall Switzer!\"\n\n\"Here goes nothing. Wish me luck,\" he whispered as he stepped toward the egg.\n\n\"Your connector will be Che Tort; your monitor will be Temasos.\"\n\nThe crowd cheered again, and Switzer held out his arms. This made them cheer a little louder. Switzer always loved attention.\n\n\"Johnny Turnbull!\"\n\nDutifully, I stepped toward the egg as Switzer returned to his original spot. The crowd had stopped cheering, and they were all staring at me. _Did they know about me?_ I hated feeling this way, like I already knew things weren't going to be fair for me. I knew it was just in my head, but that's the way I always felt.\n\n\"Your connector will be Brine Amar.\" The Space Jumpers cheered just like they had for Switzer and Gora. _See?_ I told myself. _It's all in your head._ \"Your monitor will be Quirin Ne Yarnos.\"\n\nThat shut them up. The cheering had stopped immediately at the mention of Quirin's name, and I shook my head. I knew it. It was always like this. I moved back toward Switzer.\n\n\"You always have to be different, don't ya?\" Switzer muttered.\n\nGora turned and spoke to us for the first time. His lipless mouth seemed to cut his face in half. \"No one gets a member of the Trust for a monitor, especially Quirin,\" he said.\n\n\"Is that good or bad?\" I asked.\n\nGora grunted. \"I suppose that depends on you. I envy you, but I would not change places with you for a million yornaling crystals.\"\n\n\"There's a lot I would do for a million yornaling crystals,\" Switzer interrupted.\n\n\"Not this,\" Gora spat, and all his tentacles lay down at once.\n\nI looked up and saw the Space Jumpers start to shuffle away as the huge egg rose silently toward the blinding light above my head. \"Doesn't anyone explain things properly?\" I grumbled.\n\n\"Why?\" Switzer replied. \"Why do you always have to _know_? You live in an alien world. Maybe they don't explain it because it's natural to them. Maybe it's like breathing for them. I don't see you going around explaining breathing to everyone you meet.\"\n\n\"It's not the same thing,\" I argued.\n\n\"Isn't it? Why do you feel so entitled to get an answer for everything? It looks like things have been going along just dandy here before you showed up. You think they're going to change how things are done just to suit you? Maybe it's you who needs to do a little changing. Keep your mouth shut and your eyes and ears open. Something tells me that this is a place you need to figure out on your own.\"\n\n\"Strong advice,\" Gora added.\n\nSwitzer made it sound so easy, but I still couldn't let it go. \"Then what do we do now?\" I asked, but it felt like more of a complaint.\n\n\"You follow me,\" said a voice walking toward us. The light was strong across the open floor, and three figures emerged out of the glare. They were Nagools.\n\n\"See?\" Switzer whispered. \"Let it come to you.\"\n\n\"I am Brine Amar,\" the Nagool greeted us. \"This is Sul em and Che. Please follow me.\"\n\nBrine Amar appeared a little more colorful than most of the Nagools I had encountered. While his face was still ashen, he sported colors of shimmering sapphire around his eyes and brow. His forehead formed a ridge just above this blue marking that crested up and over his head. As the Nagools turned back in the direction they had come from, I could see that this bone ran all the way to the back of his skull. Thin brown hair grew on the sides of his head \u2014 maybe to even it out, I thought, but it still left him with a long, oval-shaped head.\n\nWe followed slowly, as each of the Nagools shuffled their feet so delicately that it looked as if they were floating. I was careful not to catch their robes as I walked, staring at their high collars. Brine Amar's was decorated with a thick bouquet of colored animal feathers, while the others were a plain golden material.\n\nAs we were leaving, I glanced up at the balconies. They were all empty.\n\n\"Where did they go?\" I said.\n\n\"Oh, would you shut up!\" Switzer cried.\n\nI glared at Switzer as we slipped through a tall, crowned doorway and into a coolly lit corridor. Wherever the Nagools had taken us, it was much different from the rock and metal section of the comet we were staying on. The corridor was molded from some sort of plastic, and a soft bluish light reflected off threads in the Nagool's robes. I noticed that crystal markers identified the different hallways and doors that we passed.\n\n\"I guess these guys live a little better than we do,\" I whispered, and then asked, \"Why are you called connectors?\"\n\nSwitzer jabbed me in the ribs.\n\n\"Ow. Can't I ask?\"\n\n\"Certainly you may,\" Brine Amar said, turning to look at us. \"That is part of our role: to help you understand the magnitude of your responsibility. Sometimes the answer to a seemingly insignificant question can mean the difference between life and death in your travels. As your connector, I am here to help your nodes establish a permanent link with the Universe. A Space Jumper's greatest accomplice is his connection with the Source, but understand that this is not a simple task. In order to make this link, you must be willing to abandon the dream that you are currently in. Through self-discipline and trust in yourself, you will be able to manifest a new dream, a dream in which you are in perfect harmony with the energy that flows through our universe. The Source is abundant. It can provide everything you need. It can connect you with the infinitesimal components of the atom and put you in harmony with the orbit of the largest planets. The Source is your point of singularity. The enlightened state is a Oneness in which there is no division of parts. The Source will present everything you need to reach your true potential as a Space Jumper. You simply need to accept it.\"\n\nI only understood about a quarter of what Brine Amar said, but I liked the fact that he was answering questions. \"All those Space Jumpers I just saw, have they all reached this enlightened state?\"\n\n\"Again with all the questions,\" Switzer scoffed.\n\n\"Oh, no,\" replied Brine Amar.\n\n\"It is a difficult task,\" Sul em Pah remarked.\n\n\"Some spend a lifetime reaching and never move from the spot you occupy right now,\" Che Tort added.\n\n\"Most of the universe is unable to resist the negative distractions,\" Brine Amar said.\n\n\"Do I need this connection to be a Space Jumper?\" Switzer asked.\n\n\"Be careful \u2014 that sounds like a question,\" I said.\n\n\"The path is just as important as the destination,\" Brine Amar replied.\n\n\"I'll take that as a no,\" Switzer said.\n\n\"But it will make your task much easier,\" Sul em Pah added.\n\nThe corridor opened into a small glass atrium similar to the Spaceway stations on Orbis, only much smaller.\n\n\"This is where we depart,\" Brine Amar announced.\n\n\"How do we get back?\" I asked.\n\n\"I'm sorry. I meant, this is where we will separate and begin your individual orientation. The goal of a connector is to gain a deeper understanding of the candidate so that we may lay out the best path for your awakening. One-on-one encounters are a necessary part of this process. We will meet four times every cycle, between your training sessions. I thought we might begin with a relaxing tour of your environment. It may be our last chance before you begin.\"\n\nThe glass doors opened up and allowed us to access three small fliers, each large enough to hold three or four people. I followed Brine Amar into the middle one. I sat next to him in the back. Some sort of bot piloted each flier \u2014 it was hard to tell, as the pilot was encased in a dark green shield near the nose of the craft, but I was able make out someone moving inside. _Only a robot could see through that cockpit,_ I thought.\n\nThe Nagools did not say good-bye to each other, but I nodded at Switzer and he did the same. We pushed away before he did, and as the flier wobbled over the expanse that opened up below me, I really got a sense of how big this place was.\n\n\"Are we on a ship or a comet?\" I asked.\n\n\"Both,\" Brine Amar replied. \"The Trust built a base within the comet and then eventually harnessed the mass to move at their will. The comet provides perfect cover when their work requires proximity to a young or primitive civilization. Are you aware of the nickname the other Space Jumpers have for the comet?\"\n\n\"No,\" I said.\n\n\"They refer to it as the Hollow.\"\n\n\"Why the Hollow?\"\n\n\"They've called it that for so long now that I don't believe anyone really remembers, but it certainly makes this place seem more friendly. Don't you agree?\"\n\nBrine Amar did not seem like a typical Nagool to me. My history had afforded me many encounters with Nagools, and they always seemed extremely isolated. In fact, I had hardly ever heard them speak.\n\n\"Friendlier? Maybe,\" I replied. \"Brine Amar, will I be able to go back to the Rings of Orbis and visit my friends and my sister? I was hoping I could see them soon.\"\n\n\"I'm afraid that is not possible. You belong to the Trust now. It is up to them to decide when you are fit to leave.\"\n\n\"Well, how long will my training take?\"\n\n\"That is up to you.\"\n\n\"That's not an answer.\"\n\n\"What would you like the answer to be, then?\"\n\n\"I would like you to say _soon._ I would like you to tell me that I can see my friends very soon.\"\n\n\"If that is what you want, then you must believe that you will see them as soon as you want. As I said, it is up to you.\"\n\n\"That doesn't make any sense at all.\"\n\n\"It will. That is why you are here. You will understand in time.\"\n\nTime was the one thing I didn't feel like wasting on this rock.\n\nI glanced out the window as our flier cleared the bay where we had boarded. Below us, I saw half a dozen spacecraft parked in some sort of landing bay that was carved into the side of the comet. I guess Space Jumpers need spacecraft on occasion as well. I didn't bother to ask. Near the top and on the far side of the bay, we entered an oval shaft that looked like some sort of connector tube to another part of the comet. When we emerged from the other end, Brine Amar gestured below us and said, \"I'm sure you are familiar with this.\"\n\nI was. Below me I could see a Quest-Nest arena, more like the one on the _Renaissance_ than the one on Orbis 3.\n\n\"You will be spending a lot of time here. I understand you are quite accomplished.\"\n\n\"Switzer's better,\" I remarked, watching the Space Jumpers watching the labyrinth. They were running some sort of drill, and the different participants were waiting for their turn.\n\n\"Are they training as well?\" I asked.\n\n\"I believe they are refreshing their skills.\"\n\n\"I don't mean to be rude. . . .\"\n\n\"It is impossible for you to offend me,\" he asserted.\n\n\"That's good to know, but I was just wondering about _your_ involvement, you know, as a Nagool, I mean. I thought Nagools were against violence and this sort of thing,\" I said, pointing at the Quest-Nest arena. \"I remember witnessing a group of Nagools protesting outside the Labyrinth before the Chancellor's Challenge.\"\n\n\"The Citizens on the Rings of Orbis use the Space Jumpers' training field in a perverse manner. It is true that we oppose all violence, but Space Jumpers are not here to initiate violence; they are merely here to protect and facilitate this universe's path to enlightenment. You will never find a Space Jumper who strikes first or one who uses his powers for personal gain. A Space Jumper serves the Ancients, prepares for enlightenment at the hands of the Scion, and eventually, if needed, will stand against the Knull.\"\n\n\"I've heard the Knull mentioned before, but I don't know anything about it,\" I said.\n\n\"You will know everything in time.\"\n\nBrine Amar and I moved through more tubes and even smaller sections of the Hollow. My escort pointed out one of the eating commons, a relaxation area, a holographic recreation area, which looked interesting, a medical center, and even a commerce area with little Trading Chambers.\n\n\"This is like a city,\" I exclaimed.\n\n\"Much bigger, really. I've only shown you the highlights.\"\n\n\"Is that some sort of robot?\" I asked, pointing to an individual attending one of the Trading Chambers. \"That person there. The one with that metal thing around the back of his neck. He doesn't look like a Space Jumper.\"\n\n\"That is a Honock. You'll find many here at the Hollow.\"\n\n\"Honock? I'm sorry, but that didn't seem to translate. Is that a race?\"\n\n\"No, they are workers for the Trust.\"\n\n\"Knudniks?\"\n\n\"No, not at all. How do I explain? The Space Jumpers . . . make them.\"\n\n_\"They make them?\"_\n\n\"Yes, in a way. During their travels, Space Jumpers come across individuals who have been killed, often wrongly so. When possible, the Space Jumpers can save certain parts of the unfortunate individual. It can be some tissue or some memories, and if they're lucky, a little bit of their Source energy, and then they instill that essence into these Honocks. They really are machines by most definitions, but depending on how much of the individual is salvaged, they are every bit as functioning an individual as you or I.\"\n\nI thought about Vairocina. I wondered if she would be able to use a Honock for a new body. She had been searching for so long. I felt a tinge of regret for not bringing her.\n\n\"Do they ever leave the Hollow?\"\n\n\"Oh, no, I don't believe they could survive outside this environment.\"\n\nHonocks certainly sounded like knudniks to me. I couldn't help but wonder if a Honock ever wished that Space Jumpers hadn't saved them. I'm sure Max would spend a few cycles trying to find one or two who agreed.\n\nOur flier returned to our original docking station while I was thinking about Max. I saw only one other flier parked next to us, but I couldn't remember if it was Switzer's craft or Gora's. I followed Brine Amar out of the small station and down the posh corridor.\n\n\"Your room is down here, two decks below. It will not take you long to learn the layout. It's one big circle, with most facilities located in or near the center. There is a schematic for you to upload if you like,\" he informed me.\n\nBrine Amar paused in front of his door.\n\n\"When do I see you again?\" I asked him.\n\n\"I will find you,\" he replied.\n\n\"All right. What do I do now?\"\n\n\"Return to your room. There is someone waiting there for you.\"\n\n\"For me?\" Who was it? Someone I knew? Suddenly I was anxious to find out. I spun around to go, but Brine Amar stopped me.\n\n\"JT, excuse me, but could I ask you a favor?\"\n\nI turned back around. \"Yeah, sure. What do you need?\"\n\n\"It seems the security access code on my door is not working. Do you think you could use your softwire to allow me entrance?\"\n\n\"Me?\"\n\n\"You are a softwire, are you not? I think that task would be quite easy for you.\"\n\n\"You're right, it's just that when most people ask me to help them with my softwire, it's usually for something illegal.\"\n\n\"I assure you my intentions are to merely gain access to my room so I may rest until we meet again.\"\n\n\"Of course. I didn't mean to imply \u2014\"\n\n\"You didn't.\"\n\nI approached the door and pushed into the control pad. Once inside the chip, I expected to find something blocking the access sequence, but there was nothing. I simply switched the \"access granted\" algorithm to _true_ and the door disappeared.\n\n\"There you go,\" I said.\n\n\"Your ability must come in handy,\" Brine Amar remarked.\n\n\"It just gets me in trouble mostly.\"\n\n\"Well, you helped me, and I thank you.\"\n\n\"It's just a door,\" I said.\n\n\"To you, maybe.\"\n\nNagools certainly were strange creatures. I turned and hurried down the hall, anxious to greet my guests.\n\n\"JT!\" Brine Amar called out.\n\n_What did he want now?_\n\n\"Yes?\" I said, stopping and turning.\n\n\"Max is not waiting for you in your room, nor is Theodore, or your sister.\"\n\n\"How do you know about them?\" I asked.\n\n\"There is a Space Jumper waiting to begin your training.\"\n\n\"Oh,\" I moaned.\n\n\"You're a Space Jumper now,\" he said solemnly as I turned and walked away.\n\n_Not yet,_ I thought, and took my time going back to my room. Who was I fooling, thinking that my friends where waiting for me in my room? _Split-screen._ The thought of Max and Theodore made me wonder what they were doing. I hoped they were all right. I hoped Max was avoiding Queykay. If only I had some way to know. It might make living on this comet a little easier.\n\nJust as Brine Amar had promised, there was a Space Jumper waiting when I returned.\n\n\"Take your time, why don't ya!\" Switzer cried when I walked in. \"You kept our new friend waiting. Benas, isn't it?\" Switzer asked as he slapped the Space Jumper on the shoulder. The jumper nodded in reply.\n\n\"He's not much of a talker,\" Switzer remarked.\n\nBenas then stepped away from Switzer. I think he must have found Switzer's familiarity uncomfortable or at least odd. I took a moment to marvel at what Benas was wearing. It was a suit fashioned from a shimmering gray material with the texture of rough concrete. I couldn't tell if the pieces of metal and leather layered over the suit were decoration or padding. His Space Jumper's belt hung casually around his waist, but I could see no weapons attached to his body. I liked the leanness of his suit. I wondered if it was his choice.\n\n\"You are requested to dine with the others,\" Benas announced.\n\n\"Food?\" Switzer said. \"But I thought we would jump right into training. I'm interested to find out what kind of firepower this place holds.\"\n\nBenas looked at him. It wasn't the kind of look friends shared, at least not my friends. \"You are many phases away from weaponry training,\" Benas scolded him. \"If you even make it that far. Pseudos usually don't last long at the Hollow.\"\n\nI wouldn't say Benas pushed passed me to get to the door, but I honestly believe that if I hadn't moved, he would have walked right through me.\n\nSwitzer stared after him and then said, \"Now, what's that all about?\"\n\n\"Welcome to my world,\" I replied, and followed Benas.\n\nThe eating area was the same one Brine Amar had showed me during the orientation. I wasn't really hungry. Those pills I took to combat the effect of being away from Ketheria always curbed my appetite. They also made it hard to sleep. I figured I should eat something, though. I didn't know how hard they worked new recruits, so I felt it best to assume I would need my energy.\n\nThe food commons was a large open area with individual tables that rose out of the floor, similar to the ones Odran used in his private quarters on Orbis 2. These seating stations, however, were able to move and connect into larger groups. I could see that several of them were now joined and about four dozen Jumpers were seated around the room in various-size groups.\n\nAlong one wall, behind the tables, I spotted rows and rows of little latch doors that protected a zillion little compartments of food. The food wall had an eerie similarity to the chow synth on the _Renaissance._ It was another similarity with my past that proved that my whole life had been orchestrated by the Trust.\n\nSwitzer had seemed unusually quiet ever since the comments made by Benas. I caught him checking out the other Space Jumpers in the room, and they were checking us out as well. Benas had slipped away the moment we arrived and had linked up with another group. It felt like he didn't want to be seen with us.\n\nAs I walked up to the food wall, a Space Jumper to my left whispered, \"Hey, popper.\"\n\n\"What did you say?\" Switzer retorted, but the Space Jumper ignored him and went on talking with his friends.\n\n\"This isn't good,\" Switzer grumbled behind me.\n\n\"Better get used it,\" I said. \"This was my life on the _Renaissance_ \u2014 my _whole_ life, for that matter.\"\n\n\"What I don't understand is why you put up with it.\"\n\nAnother Space Jumper whispered, \"Pseudos,\" as we walked past. Switzer spun around and scanned the crowd for the culprit, but everyone appeared to be minding their own business.\n\n\"What was I supposed to do?\" I asked him.\n\n\"I'll show you what you were supposed to do.\"\n\nSwitzer turned and headed toward the largest group of Space Jumpers.\n\n\"Switzer, where are you going?\"\n\n\"Just watch and learn,\" he called out.\n\nI followed him but kept my distance. He strolled over to the table and stopped in front of the largest Space Jumper. This was one of those militarized Jumpers, the ones who used every spare inch of his suit to hold some sort of weapon. There were more weapons on the table than there were items to eat. I watched in horror as Switzer reached forward and shoved his fingers into the guy's food. Then he scooped out a handful of something I would never attempt to eat and shoved it into his mouth.\n\n\"Mmmm, that's good,\" he cried with his mouth full. It was loud enough that everyone turned to watch. \"Where'd you get that? Mind if I have some more?\"\n\nThe Space Jumper stared at Switzer as if he had just popped a circuit. When Switzer reached for another scoop, the Space Jumper clamped onto Switzer's forearm. The guy's paw was almost as big as Switzer's arm, and that's saying something, because Switzer is a big guy.\n\n\"Switzer!\" I cried, but he didn't listen. After the Space Jumper clamped onto him, Switzer swung around, his fist clenched, and caught the Jumper right under the chin. The whole room heard the crack. The big Space Jumper fell backward, unconscious. Switzer moved around the table and pushed the guy's legs away. Then he took his seat. The other Jumpers at the table all stood up, but Switzer ignored them. I think it was the complete lack of fear on Switzer's face that scared the other Jumpers and thwarted any retaliation on behalf of their friend. They remained motionless, leaning toward Switzer and staring at him.\n\n\"What?\" Switzer snapped. \"He told me he was done. Didn't you hear him?\"\n\nSwitzer dove into the guy's food.\n\n\"What's your name?\" one of the Space Jumpers asked.\n\n\"Switzer,\" he replied. \"Do you need me to write that down? Because you should remember it.\"\n\nThe Jumper did not respond, but he did sit down. Switzer scooped the last little bit of the slop from the plastic bowl and licked his fingers. Then he stood up and said, \"Oh, that was good. I have to get some more. You guys want anything while I'm up?\"\n\nThe rest of the Space Jumpers at the table either shook their heads or grunted, concentrating on their food again. Switzer strolled over to where I was standing and used his fingers to push my chin up, closing my mouth. His fingers were still wet from the guy's food.\n\n\"I think that's been taken care of,\" he whispered.\n\nI followed Switzer to the food wall and asked, \"Why that guy? Is it because he was the biggest?\"\n\n\"No. Have you seen these guys fight? He was the only one I saw whose belt was on the table. I didn't want him slipping out and popping back up behind me. That's not a fair fight.\" Switzer looked over his shoulder and saw the Space Jumper getting up and returning to his seat. Switzer nodded at the guy, and he returned the nod. \"The fact that he was the biggest is just a bonus,\" Switzer added.\n\nI grabbed some food behind Switzer \u2014 a few things I could recognize \u2014 and we sat at a table with some of the other Jumpers. Switzer was already making alliances and starting friendships. As I ate and watched him, I realized that I would never be like him. If we were somehow picked from the same gene pool, then why were we so different? How come he could adapt so quickly to this new world when all I did was resist?\n\nAs a couple of Space Jumpers got up to leave, I overheard Switzer making plans with them to meet at Quest-Nest. I just shook my head.\n\n\"So I was just supposed to walk up to you on the _Renaissance_ and knock your stars out?\" I asked. \"When we were eight years old?\"\n\n\"Better than trying it now,\" he replied.\n\nAfter we ate, I was informed by tap that we were to meet with our advisors. I uplinked directions to Quirin's quarters and then mumbled good-byes to Switzer. For the first time in my life, I found myself craving Switzer's company. His altercation at mealtime garnered us immediate respect with the other Jumpers, but I knew that the respect afforded me was due to my association with him, not from my own actions. Walking alone left me feeling exposed and I knew there was no way I was going to knock someone out, no matter how effective Switzer made it look.\n\nFortunately, I made it to Quirin's without the need to knock someone's head off and slipped inside his quarters.\n\n\"I was pleased to hear of your decision to participate, although I was concerned about your request to include the other human,\" he said. His voice seemed to come from every corner of the room.\n\n\"That other human is partly my brother, according to you. It was not fair what you did to us. It was the least I could do for him.\"\n\n\"I had hoped these sentiments would have been erased with your gene therapy.\"\n\n\"What do you mean?\"\n\n\"You were not programmed to be so empathetic toward other life-forms. It was a human trait I clearly underestimated. Yet this Randall Switzer, from what I have heard, does not seem to share your feelings for your own kind.\"\n\n\"What do you mean by _programmed_? You make it sound like I'm some sort machine.\"\n\n\"In a way you are. Aren't we all?\"\n\n\"No.\"\n\n\"Shall I say _designed,_ then? As the Tonat, you will be placed in many situations where your only concern should, and will, be the survival of the Scion. You cannot fight that even if you try. All other life-forms will be of no consequence to you. They simply cannot be. That is why you have difficulty understanding the OIO philosophy. It was part of your design. As much as I believe in the value system, I could not have the questions it raises clouding your decisions.\"\n\n\"Well, I think there may be some flaws in your design,\" I told him.\n\n\"Then that is where we will start,\" he replied.\n\nThe light in the room focused on a point near the center of the floor. A sort of workstation with a seat and a panel of O-dats placed in a semicircle around the metal stool emerged.\n\n\"Sit,\" Quirin ordered.\n\nAs I obeyed, several wires snaked out from points in the wall, or rather from Quirin, that is, and connected with my workstation.\n\n\"We must upload traditional Space Jumper protocols into your cortex before physical training can begin. These rules, procedures, and themes are at the core of your studies. I will be uploading large chunks of data, so be prepared. The sooner this is done, the sooner your physical training can begin.\"\n\nThe O-dats lit up, and I began to feel a little nauseous. Hoping Quirin would not notice, I reached into my pocket and took a tablet, pretending to rub my nose as I slipped it into my mouth.\n\nBut Quirin saw it just the same. \"You must learn to control that discomfort,\" he said. \"It is a tool that allows you to physically estimate your distance from the Scion. Masking its effect is not the proper way to master the tool's important function.\"\n\n\"Some tool,\" I muttered.\n\n\"I am placing the files I need you to upload on the terminal in front of you. Please access the computer and install each one in sequence.\"\n\n\"Wait,\" I said. \"How is Switzer doing this? He's not a softwire. I thought you had to be a softwire to be a Space Jumper. Are they doing it with his hardware?\"\n\n\"The genetic structure for this gift was always within Switzer. The sequence was never initiated, as I found you a much better candidate. Once the genetic coding is manipulated, Switzer will share the same abilities as any other softwire. That is why he survived the jump from Orbis 2. The belt recognized the dormant genes and made a connection with him. Otherwise he would have died.\"\n\n\"He'll be just like me,\" I said.\n\n\"In some ways; in others not. Your ability to jump without a belt is a unique skill that no other Space Jumper has, although you are unable to control it. The mishaps you experienced early on, jumping without warning, will disappear. I'm convinced they were the result of Ketheria's awakening.\n\n\"Your softwire is merely used to connect with the belt. A Space Jumper's belt does most of the work under normal situations. It stores entry points throughout the universe and allows a Space Jumper to return to wherever he has been. You, on the other hand, simply need to remember a place in order to jump there. Unfortunately you cannot jump to a place you have never been without the use of a belt.\"\n\n\"But I've jumped to places I'd never been before. I've showed up in back alleys in Murat and other places on the rings,\" I told him.\n\n\"But you did not control those jumps. They were simply sparked by your emotions and sent you adrift through the Source. You are lucky you did not jump to the center of a black hole.\"\n\n\"Oh,\" I replied.\n\n\"May I continue?\"\n\n\"Sorry.\"\n\n\"Your unique ability to jump without a belt parallels your ability to _push_ into a computer, as you call it. Softwires merely connect to a computer without hardware and interface with the data. You can actually enter a computer with your mind, manipulating its contents in ways we are unable to do. Although I did manipulate your genetic structures, I believe you were the only candidate capable of evolving in this manner. But I don't know why. You are unique. You are the future of our kind.\"\n\n\"And you did all this for me?\"\n\n\"Yes.\"\n\n\"Thanks,\" I said. \"I guess.\"\n\nThe O-dats lit up, and I began downloading file after file. Most of it I didn't even acknowledge outside of the title. I knew that the information would come to me when I needed it, as long as I accessed it often enough to store it inside my long-term memory. That was the purpose of my physical training, Quirin said, to ingrain the information into my unconscious.\n\nBy the time we were finished, I figured I was going to have to do a lot of training to use all of this information. I uploaded data files big enough to knock out a whole class back at the Center for Wisdom, Culture, and Comprehension. I was exhausted and had to drag myself back to my room.\n\nThe moment I was settled in my sleeper, Switzer burst into the room.\n\n\"You've been hiding this your whole life!\" he cried.\n\n\"What?\"\n\n\"This!\" he yelled, tapping his head. \"I can't believe you've been ashamed of this.\"\n\n\"What are you talking about?\" I asked him.\n\nSwitzer was prancing around the room as if he had just discovered a new planet.\n\n\"Your softwire,\" he cried. \"They just turned mine on, or whatever it is they do. It's incredible!\"\n\n\"Oh, that,\" I scoffed. \"If I recall, you gave me a lot of crap over it when they discovered mine.\"\n\nSwitzer walked over to me and placed his hand on my shoulder. \"Look, I'm sorry about that, all right? How was I supposed to know? I was just a kid, remember? And besides, how could someone else fathom this power?\" Switzer took up prancing around the room again.\n\n\"It's done more harm than good,\" I reminded him.\n\n\"That's where you're wrong. Do you even realize the power you have inside your head? I think half the training here is going to be about not abusing this.\"\n\n\"Abusing it? What are you talking about?\" Suddenly, I felt extremely nervous about the fact that I had brought Switzer here. \"Look,\" I said, standing up. \"Don't do anything stupid, all right? I convinced them to bring you here because of what they did to us. It wasn't fair. Your situation is the exact result of their actions. Don't go proving me wrong, please.\"\n\n\"Don't get your uplink in a tangle. I'm not going to do anything. But that's why we haven't gotten the warmest reception around here. All these other Space Jumpers evolved this ability. It made them exceptional on whatever planet they came from. You and I, on the other hand, were tinkered with. They needed the process speeded up so baby-malf could be the Scion. Humans were their last hope.\"\n\n\"Don't call her that.\"\n\n\"Sorry. Old habits.\"\n\n\"So that's what they meant by calling us psuedos?\"\n\n\"Yep.\"\n\n\"What about this popper thing I keep hearing?\"\n\n\"Even _I_ can figure that one out. It's because you were popping in out of space and time when you were getting angry. And you were doing it without a belt. No one else can do that, by the way.\"\n\n\"I've heard.\"\n\n\"I think it makes them a little jealous. We can't jump without a belt.\"\n\n\"We?\" I said.\n\n\"Us _softwires._ I'm one of you now, my friend!\"\n\nSwitzer punched me in the shoulder and laughed out loud. I couldn't believe how much he was enjoying this. I had spent my entire life hiding my softwire ability, and he was wearing it like some sort of medal. I couldn't even imagine acting like that. I wondered if the other Space Jumpers were stuffed with this much pride about their condition.\n\nSwitzer flopped onto his sleeper, still smiling.\n\n\"What are you thinking about?\" I asked.\n\n\"The future,\" he replied.\n\n\"What about it?\"\n\n\"How great it's going to be!\"\n\nAs I lay in my sleeper, waiting to sleep, I tried to see Switzer's point of view, but it was impossible. Being a softwire was not something I looked at with such optimism. As a kid, my abilities had only garnered me ridicule and shame, but when I thought about it, Max was excited when my abilities were first discovered. It was just the Space Jumper part she didn't like. And I had promised her I wouldn't become one. _So much for keeping my promise._\n\nInsomnia. A side effect from the tablets that I kept popping despite Quirin's instructions. While Switzer snored in his sleeper, I lay in mine, staring at the darkened lid. It was no use. I wasn't going to fall asleep. I pushed the lid back and sat up. I was really missing Max. I wanted to know what she was doing, and the same with Theodore and Ketheria. I pulled on my clothes and headed for the observation deck.\n\nSomeone else was sitting there when I arrived. It was a Honock. He turned and looked at me when I entered.\n\n\"You bad!\" he hissed.\n\nI remembered that voice.\n\n\"Hey, I know you.\" It was the same voice I heard the first time I was taken to the Hollow, when I had popped during the Chancellor's Challenge on Orbis 3. This Honock was the one outside my room. \"What's your name?\"\n\nThe Honock stood up and moved away from me. \"You bad,\" he repeated.\n\n\"No, I'm not. My name is JT. What's yours?\"\n\nHe didn't reply. His back was against the glass, and he was sidling along it back toward the entrance. I didn't push him. I kept my distance.\n\n\"I'm not bad. Why do you keep saying that?\"\n\nHe pointed at my waist. \"You bad.\"\n\n\"What?\" I said, patting my waist. \"I'm bad because I don't have a Space Jumper's belt? Why would that make me bad?\"\n\n\"You bad!\" he yelled, and bolted for the door.\n\n\"Wait! Tell me why!\" But the Honock was gone. At first I thought about following him. It made me wonder if Honocks even slept. How much of them were machine, anyway? Instead of following him, I sat near the glass and looked out at the stars. \"Where are you, Max?\" I whispered. \"I miss you.\"\n\n#\n\nThe next cycle, I was forced to endure Switzer's whistling as he strutted around the room, getting ready for the cycle's training. I allowed myself to take pride in the fact that he wouldn't be feeling this way if I had not gotten him out of that hole and brought him to the Hollow. Despite the rotations of abuse I'd taken from Switzer when we were kids, and even later when he was a wormhole pirate, I could see that he was a completely different person now. Secretly, I took a little credit for his change.\n\nI started my cycle with a visit to Brine Amar, who asked me if I could help him fix his O-dat. I had to use my softwire ability again, and I started to wonder if the Nagool only thought of me as his own little handyman. Where was the connection to the Source? Where was the guidance? At least he was extremely thankful, and I liked using my softwire to _help_ people, for a change.\n\nOnce at Quirin's, I uploaded more files \u2014 simulated experience memories, or SEMs, as he called them. This cycle, I learned how to pilot a shuttle that I had never even been on. Now, that was definitely something I would like to try. I could only imagine how much fun Switzer was having.\n\nDuring mealtime, I caught myself thinking about Max again, as well as Ketheria. Switzer was knee-deep in friends now, and even I began to feel a little camaraderie with everyone sitting at our table. But enjoying myself made me feel guilty for not knowing what was happening down on the rings. I wished there was some way to contact my friends, but when I had mentioned this to Quirin, he'd quickly shut me down, saying it was out of the question.\n\nDuring the sleep spokes, I often found myself back in the observation deck. This routine continued for many cycles, but the Honock never showed again. I did spot him once, working behind the food wall, but he acted as if he didn't know me. I don't know why the Honocks interested me so much. Maybe I saw them as knudniks and felt some sort of connection to them.\n\n\"You look like crap,\" Switzer said to me one cycle as he was headed out our door.\n\n\"Thanks,\" I replied.\n\n\"No, I mean it. When was the last time you slept?\"\n\n\"I don't remember.\"\n\n\"Are you still popping those pills?\"\n\n\"Yeah, aren't you?\"\n\n\"No. I'm fighting it,\" he replied, but I found that hard to believe. The headaches, the nausea, it was too much to endure this far away from Ketheria.\n\n\"How?\" I asked.\n\n\"I just _am._ It's not as hard as you think. You have to put the pain to the back of your mind and focus on something else.\"\n\nSwitzer would have made a better Tonat, I told myself. \"I can't,\" I said. \"I've tried.\"\n\n\"Well, try harder. You think you're going to be able find a lifetime supply of those things when we're done? What if you get stationed in another galaxy?\"\n\n\"What do you mean _stationed_ in another galaxy? Who told you this? I'm going back to the Rings of Orbis. I'm supposed to be protecting Ketheria. I'm not going anywhere.\"\n\nSwitzer had stopped at the door, but now he walked back to my sleeper. \"Give it up,\" he growled. \" _This_ is your life now. You are an instrument of the Trust, a protector for the Ancients. You have a far greater purpose than all the split-screens on Orbis combined.\" Then he left.\n\nSwitzer _definitely_ would have made a better Tonat.\n\nThe next cycle was the first phase of our physical training. Using the Quest-Nest arena at the most physically demanding settings I had ever seen, a team of seasoned Space Jumpers ran us through coordination, endurance, flexibility, and strength drills. And then we did it again. The playing field had been replaced with what was mostly an obstacle course, which re-formed on me when I was too slow. With each run at the course, the computer would slip in new elements that required the use of another SEM, usually one that I had uplinked in a previous session with Quirin. The Space Jumpers had us take single turns, as well as switching out partners, using Gora, Switzer, and one of the trainers. We were often in pairs.\n\n\"Do Space Jumpers always work as pairs when they are on missions?\" I asked the instructor, a big militarized Space Jumper.\n\n\"Concentrate on the now, popper. You're in no condition to be thinking about a mission,\" he barked.\n\n_So much for camaraderie,_ I thought.\n\nWhenever Switzer was asked to \"run the Nest,\" as he began to call it, he didn't just walk up to it; he attacked it. Each obstacle was something else for him to conquer. I had to admire how good he was at it. He even completed one run ahead of the trainer. Not something he let slip by, either.\n\n\"You should have used that immobility cube on those spheres near the end. I find it acts as an adhesive on inanimate objects,\" he boasted.\n\nThe trainer did not snap at him. In fact, he seemed to be absorbing what Switzer was telling him. I often saw them discussing a move Switzer had tried or an unorthodox manner in which he employed his weapon.\n\n\"You were made for this,\" I said to him once at mealtime.\n\n\"Technically, I was,\" he said, his mouth full of something green. \"But my experiences as Captain Ceesar taught me to be resourceful. I have to think it's going to get a lot harder than this if we are to live up to the Space Jumpers' reputation.\"\n\n\"What do you mean?\" I asked. Switzer stuffed something with a tentacle into his mouth. I guess being a wormhole pirate had also broadened his appetite.\n\n\"Out there, in the real universe, these guys are gods,\" Switzer whispered. \"On some planets, the mere mention of a Space Jumper can send an enemy scurrying for cover. I just thought it would be a little tougher.\"\n\n\"Or maybe you're just that good,\" I told him.\n\n\"Hey, don't worry \u2014 you're going to get better.\"\n\n\"I don't need your pity, thanks.\"\n\n\"I didn't mean it that way. Look, what are you now, seventeen? I have ten, maybe fifteen years' experience on you. That's all. How can anyone expect to be good at this right off the launch? That's what the training's for. I'm sure you're much better than me when you're using that thing in your head and jumping around computers and stuff. I'm still grasping working the interfaces on O-dats.\"\n\n\"You really like this life, don't you?\"\n\n\"It's better than what I used to do.\"\n\nWe finished our meal in silence. We were both exhausted and sore. It even hurt to stand up. As I limped back to our room, I hoped I might get some sleep, but it did not come. I lay in my sleeper, thinking about what Switzer had said, about the reputation of Space Jumpers elsewhere in the universe. I thought about how they were feared. Max had heard those stories as well. I'm sure a lot of people on the rings shared the same sentiment. I wondered if Max would ever take me back now.\n\nOver the next few cycles, I watched Switzer begin to dominate the trainers during the Quest-Nest drills. I then decided that instead of sitting there and grumbling about it, I would watch him and learn. Switzer was good, often combining two movements at once. But when I tried to repeat the task, it was simply impossible for me.\n\n\"Don't give up!\" he encouraged me.\n\nAnd I didn't. Whenever I trained with him, I stayed close, trying to reenact his movements even if it resulted in a painful drop to the floor or being blindsided by a moving obstacle.\n\nDespite my exhaustion, I still wasn't sleeping.\n\nOne spoke, I caught another Honock in the observation deck. This one did not resort to calling me bad, but he was still afraid of me. I wondered if they had always been afraid. Had this individual almost been killed by a Space Jumper, only to be reincarnated as a machine and forced to live among us? How horrible would that be? No wonder he thought I was bad.\n\n\"Do the Honocks ever ask to go home?\" I asked Brine Amar during one session.\n\n\"No. Why do you ask?\"\n\n\"I've seen them in the observation deck, looking out at the stars, almost as if they were reminiscing about something. I thought they might be thinking about their former lives.\"\n\n\"Honocks are not designed that way. Yes, some of their personality is maintained, but you really must think of them as machines, just as you would a cart-bot or an android.\"\n\nBefore I left, though, he made the oddest request.\n\n\"Could you do me a favor?\" he asked. \"Would you play in a match of Quest-Nest with Randall Switzer? I would enjoy that very much.\"\n\n\"Me? I guess. Sure, why not? Whom will we play against?\"\n\n\"No, I want you to play against Randall Switzer. You will use one of your instructors as your partner.\"\n\n\" _Against_ him?\"\n\n\"Yes. Do you mind?\"\n\n\"Um, no. I guess not.\"\n\n\"When?\"\n\n\"Now.\"\n\nInstead of training this spoke, it appeared I was going to enter the labyrinth and play against Switzer. When I arrived, Switzer was already waiting, as were most of the other Space Jumpers I had seen in the Hollow.\n\n\"Just like old times,\" he said.\n\n\"You know?\"\n\n\"Yeah, my guy asked me to play you.\"\n\n\"Mine, too. Don't you think that's weird? I mean everyone in the Hollow is here.\"\n\n\"Why should I? I'm looking forward to watching you lose.\"\n\nSwitzer yanked the helmet over his head and launched into the labyrinth. One of my trainers walked up and asked, \"You want to be the bait or the tracker?\"\n\n\"Tracker,\" I said under my breath, and pulled a helmet off the wall.\n\nThe labyrinth on the Hollow was different from the one I had been using on the Rings of Orbis, in that it did not have a sort. I was glad that I did not have to think of a sort strategy to use against Switzer. I was comforted by the fact that I would not be floating in a vacuum while trying to navigate multidimensional mazes after the door opened. No, this match would be familiar to me. This would be just like on the _Renaissance._\n\nWaiting for the computer to set, I felt the sudden urge to run. What if I was wrong? What if it was different here as well? _But you've already been practicing here,_ I reminded myself. _Concentrate!_ When the door peeled away, it was like stepping back onto the _Renaissance._ It was exactly the same! I sprinted along the curved purple walls and jumped over the blue lights embedded in channels on the floor. I knew the first obstacle of metal crates was just ahead to my right and the immobility cube would be waiting on the other side.\n\nI dragged the metal crate next to the other two and used it to hop up and over a half wall. I grabbed the immobility cube and then used the ladder I knew would be there to sidestep two more obstacles, just as I had done so many times before on the _Renaissance. This is almost too easy,_ I thought. It was obvious to me now that Quirin had designed our Quest-Nest on the _Renaissance._ Didn't they know I had played this version many, many times before? Was this some sort of trick? It certainly must have been boring for Brine Amar to watch.\n\nI was ready for the four frontier pilots hiding in the deep trenches past the next doorway. An additional immobility cube and a plasma rifle took them down before they even started to scream. I ran across the darkened room, past a sparking electrical circuit that provided the only light. I was about to run through the doorway when I stopped.\n\nThe door was on the wrong side of the room. It should have been on the left, but it was on the right, on the other side of a snaking electrical wire that was torn loose from the wall. I wouldn't even have even thought about it if everything hadn't been so exact up until this point, right down to the color of the lights and scars on the walls. It was a complete reenactment of the _Renaissance._\n\nExcept for this door.\n\nI approached the door and saw that it was slightly open, as if someone had tried to open it but it had jammed. Had Switzer already been here and messed with the door to slow me down? I searched for some sort of control panel near the door, but there was none. I peered through the crack and saw a pink light flickering on the other side, so I figured this was the way to go. I even tried to pry it open with the butt of my fedaado blade, but the door was stuck.\n\nI flexed my arm, used my softwire to adjust the torque and pain levels and then clobbered at the door with my right arm. The metal buckled, but I knew I was going to have to destroy the door to get through. I had to admire Switzer for jamming it like this. The delay was definitely going to set me back.\n\nI pounded on the door several times; each effort widened the crack a little more. When I peered through again, it was mostly black beyond except for the beams of pink light that crisscrossed the darkness. I figured it was coming from some light source on the wall. With one final lunge, I hammered at the door, and whatever had jammed it came loose. The door jerked to the right with such a jolt that I lost my footing and fell forward, through the opening.\n\nIt was a hole.\n\nI was falling through the beams of pink lights that flicked on as I passed them. _This is going to hurt,_ I thought, but I didn't scream. Who would hear me, anyway?\n\nAnd then I hit water. Water isn't as hard as concrete, but it still hurts. I felt my right leg jam up into my hip socket and the water crash in on my face.\n\nIt was over. I had lost.\n\nI looked up through the tiny beams of pink light, waiting for the labyrinth to turn off. The water would drain away, and I would be left with my sore leg and my loser self. I had wanted to beat Switzer. I knew I didn't have a chance, but I wanted to beat him.\n\n\"Hey!\" I called out when no one turned the maze off. \"I'm down here. You win!\"\n\nBut the water was not draining. In fact, I noticed that the water level was creeping upward, swallowing the little beams of pink light as it rose. Soon there were just as many pink lights in the water as there were above me.\n\n_Is this part of the match?_\n\nWhen I floated up to the door that I had fallen through, some sort of force field swallowed the opening before the water could spill out. The well, or tunnel or whatever I had fallen into, was filling up.\n\nAs I passed the broken door, I searched frantically for some sort of computer device. I knew I wasn't supposed to use my softwire to aid myself in the game, but that was on Orbis. Everyone here was a softwire. Everyone was on equal ground. I searched the walls for something to push into, but I found nothing. _What put up that force field?_ I wondered. Where was the computer that ran this thing, for that matter? _Surely there must be something around here to manipulate,_ I thought.\n\nI could see the top of the tunnel, only a few meters above my head. There wasn't enough time to figure out why, nor was there anyone around who could answer my questions. There had to be a way out of here. If they weren't turning the labyrinth off, then that meant the match wasn't over. There must be a way out.\n\nI swam to the edge and groped the walls for some sort of plate or panel, anything really, but the walls were smooth like glass. I couldn't even find a seam, nothing except for the little holes where the pink beams of light shone. Was that it?\n\nMy hand was now against the ceiling. There was nothing else to try. I held my breath and dove under, close to one of the lights. There was nothing to interface with, so I tried to push in, but I found nothing. I moved to another light, hoping it might be different, but again, nothing. I surfaced, gasping for air. There were only about fifteen centimeters left for me to breathe in. Surely they wouldn't let me die in here. This wasn't the Chancellor's Challenge. _Do I simply give up?_ I wondered. _What if Ketheria was with me? Would I give up then?_\n\nI dove back into the water, searching for any light that was different in any way. Near the bottom, I spotted one that appeared slightly brighter than the others. I dove deeper, knowing that this was my last chance, since the water would surely be at the ceiling by now. In front of the light, I groped for a computer device but still found nothing! This time, I pushed in and discovered a single computer chip attached to a sensory timer. It was the smallest thing I had ever pushed into before, but at least it was something. I knew it had to connect to something larger. I forced myself inside the circuit and pushed against the resistance. It felt as if something was scratching at my skin, pawing for something to latch on to, something to hold me back with. The narrow corridor I found inside the chip then opened into a larger chamber of data cells stacked one on top of the other. I noticed that the one at the center was highlighted with a bright yellow glow as if it were a beacon to let me know I had arrived.\n\nA simple manipulation of the data sequence, clearly marked for anyone who found it, set the tank on drain, and I felt the rush of water pull me down to the floor. The water drained through a grate that ran the circumference of the tank. I waited as the floor extended downward, enlarging the grate until I could simply walk out. My bait was waiting for me.\n\n\"Well done!\" he cried. \"You're the first.\"\n\n\"First what?\"\n\n\"The first one to ever escape that trap. Nice job!\"\n\nThe trek back was simple. The water obstacle was gone, and I exited the maze in front of the bait. Those watching erupted in cheers as we stepped out. A few Space Jumpers who had made friends with Switzer were coming up to me and clapping me on the back, hanging close to me and discussing the match with whoever walked up. I listened as some, in the excitement, even answered questions that were directed to me. It felt good.\n\nSwitzer was nowhere to be found.\n\nBrine Amar walked up to me and said, \"I must say, I am impressed. I'm sure your name will be carved into a rock around here somewhere. You should be proud of your accomplishment.\"\n\nAnd I was. A little voice tried to creep up and remind me how scared I'd been, how I thought about quitting. _But I didn't,_ I argued. I had figured it out and now I was here. The winner. If I had given up, I would have been the loser. If I had given up in real life, I would have been dead.\n\nSwitzer finally came out, shaking his head. \"Nice job!\" he cried. There were no jabs at my abilities, no taunts about next time. Switzer seemed genuinely proud of what I had done. \"I couldn't get out of that tank. I tried everything. The softwire thing didn't help me at all.\"\n\n\"Ah, you just need a little practice. I'll show you,\" I called out to him as he walked over to the group of Space Jumpers standing around me. I expected the other Space Jumpers to turn to Switzer and leave me to myself, but they didn't. He joined us as if it was the most natural thing in the universe.\n\n\"Go on, smile,\" he whispered. \"You deserve it. You won.\"\n\nThings were different after our Quest-Nest match. I don't just mean with Switzer and me, but with everyone at the Hollow. I enjoyed my friendships with Max and Theodore, and even my sister \u2014 in fact, I cherished them \u2014 but this was different. We were all joined by an ability we shared, a talent that had at one time made me an outsider, but not now, not here. Here you wore that ability like a badge. Your softwire was your admission to the Hollow, and it didn't matter anymore if some alien had tampered with your genetic structure or you had come to it through your own natural evolutionary process.\n\nDespite our past, I felt that a bond had now formed between Switzer and me. We were in this together. I had finally accepted that. The fact that we _had_ actually been in this together from the beginning did not escape me, either, but I was glad we had found a way to overcome all that. I looked forward to his company, and he had even begun asking me for help with his softwire ability. Switzer and I had become friends.\n\nThen I found Charlie.\n\n#\n\nI tried to stop using the tablets like Switzer had suggested. It wasn't that I couldn't handle the nausea or the headaches \u2014 I really did want to live without the pills. The real problem was that whenever I began to feel sick, I thought about home. I thought about Ketheria and Max and Theodore. I ached to know what they were doing and to be near Max again. The Hollow provided an excellent distraction, but the sickness was a strong reminder of the people I had left behind.\n\nAs I did on most sleepless spokes, I snuck off to the observation deck instead of staring at the lid of my sleeper. I found that the stars relaxed me, and if I was lucky, I might even drift off for a diam or two. When I entered the deck this time, however, I wasn't alone. A Honock was there, staring at the stars. At first, I thought it was the one that was afraid of me, but when I got near and it didn't run off, I assumed it had to be a different one.\n\nLeaving him to himself, I sprawled out in front of the glass, a good five meters away just to be polite. As I sat down, I glanced over to see if my presence had disturbed him.\n\nBut there was something about him that looked familiar to me.\n\nI looked over my shoulder, trying not to stare. He was definitely a Honock \u2014 I could tell from the way his facial skin was pulled over his metal skull \u2014 so there was no way I could have known him. The skin stopped right at the jawline exposing a multitude of wires and metal bones. But despite the hardware, his profile was undeniable. It was Charlie Norton.\n\n\"Charlie?\" I whispered.\n\nThe Honock turned and looked at me. It was Charlie, all right. The thick nose. The rugged chin. \"Hello,\" he replied.\n\nI jumped up. \"Charlie, it's me. JT!\"\n\nI scrambled over to my old Guarantor, my friend \u2014 the person they told me was dead.\n\nHe scooted back a little when I rushed forward. His movements were precise and machinelike \u2014 very \"un-Charlie,\" but there was no denying the resemblance.\n\n\"Don't you remember me, Charlie?\"\n\nI sat still and let him examine me. I waited as his eyes searched my face, and I stared back at his waxen skin. I had never been this close to a Honock before, but I could see why Brine Amar treated them like machines. The facial skin was an excellent plasticlike imitation, right down to the fake pores. I heard a humming sound coming from him, sort of like the whir of many tiny motors. Each gesture he made had a slight pause and the exacting execution of a machine.\n\nWhat had they him done to my friend?\n\n\"Ketheria?\" Charlie said.\n\n\"No, JT. But do you remember Ketheria? She was my sister. She loved you.\"\n\n\"Peanut Butter.\"\n\n\"Yes! Yes! Ketheria loves peanut butter.\"\n\n\"I don't eat peanut butter.\"\n\n\"You did,\" I told him.\n\nCharlie turned and looked back out at the stars, casually sitting next to me as if the two of us had done so a dozen times before. It hurt to think that he didn't remember me, but what did I expect? I didn't even know what Charlie _was_ anymore. I looked at his profile again, following the metal and wires down in to the collar of his green jumpsuit.\n\n\"What have they done to you, Charlie?\" I whispered softly. I didn't think he heard me, but he turned and smiled anyway.\n\nIt hit me that I was not as surprised by his presence as one might think. They had brought Switzer back from the dead, didn't they? Why not Charlie?\n\n\"What are you looking for?\" I asked him.\n\nCharlie turned to me and said, \"Chicago.\"\n\n\"I'm afraid we are a long way from home, Charlie. A very long way.\"\n\nI wanted to blame the Rings of Orbis for what had happened to Charlie, but Orbis had nothing to do with it. Charlie was they way he was because of one person. Randall Switzer. It was Switzer's malf of a plan to steal the Ancients' Treasure that had gotten Charlie killed. Charlie's presence was a glaring reminder of every selfish act Switzer had ever committed in his life \u2014 from his tyranny on the _Renaissance_ to his utter lack of concern when he entered us in the Chancellor's Challenge. Switzer was an animal, pure and simple. He was not my friend. Who was I kidding? I hated Switzer at that moment.\n\n\"Charlie, do you know who I am?\"\n\n\"You're JT.\"\n\n\"I am, but do you remember me? Do you remember being my Guarantor on the Rings of Orbis? Do you remember Max or Theodore?\"\n\n\"Your friends.\"\n\n\"Yes!\"\n\n\"My friends.\"\n\n\"Yes!\"\n\n\"They're sleeping.\"\n\n\"Who's sleeping?\"\n\n\"Our friends.\"\n\nWas Charlie talking about the other Space Jumpers at the Hollow? I'm sure he had seen me training with them. \"No, Charlie. I don't know what our friends are doing right now. They're far away. Too far, I'm afraid.\"\n\n\"I watch them. I watch you. I watch your friends now.\"\n\nI looked through the glass and out into the stars. Could he see the Rings of Orbis? \"Don't you mean my friends here, Charlie? On the Hollow? Do you watch me and the other Space Jumpers?\"\n\n\"I watch them.\"\n\n\"I thought so,\" I mumbled.\n\n\"But I watch Theodore and Max and Ketheria and you.\"\n\n\"What?\" I cried.\n\n\"Shh!\" Charlie whispered. \"I am not allowed. I watch when they aren't watching me.\"\n\n\"Charlie, you can see Max? Where? Show me.\"\n\n\"It is forbidden. I am not allowed.\"\n\n\"You have to show me, Charlie. Where is it? Is it here on the ship? I have to see them, Charlie. Show me, please!\"\n\nCharlie covered my mouth with his hand. To my surprise, it was warm. Charlie looked over his shoulder toward the door and then stood up. I followed him, something I had done so often that the back of his head had been emblazoned into my memory, but this was different; everything was different. The confidence in his walk was gone; there was a Honock-like stiffness to his movements, and the hardware around the base of his brain looked unnatural. Plus, he was bald.\n\nOutside the observation deck, Charlie moved quickly, even with his awkward gait. I followed him down four decks and through the Honocks' living quarters. He never once paused to see if I was keeping pace. When he did stop, the only exit I could see in the dimly lit corridor was a ventilation grate that covered most of the wall.\n\n\"What is this place, Charlie?\" I asked him.\n\n\"Shh!\" he replied, and glanced back down the corridor. He lifted the thick metal grate out of the wall and set it down with a clank. The thing must have weighed sixty kilos, yet he moved it as if it was a scrap of plastic.\n\n\"Go,\" he whispered, and nodded for me to enter the tunnel.\n\nI heard the grate grind back into place as Charlie followed, covering our tracks. A greenish electrical glow at the end of the tunnel was my only guide as I slid my foot forward to make sure I was walking on solid ground. When I reached the light, I discovered a patchwork of O-dat terminals, wires, and computer parts, all linked to a control panel that looked as if it had been hacked open.\n\n\"Charlie, did you make all of this?\" I asked, staring at his makeshift control center. I think the hideaway was fashioned from some sort of a utility shaft off the main ventilation system.\n\nI moved aside so Charlie could sit at a small stool he must have taken from the meal room. Then he pulled a scrap of polymer out of his pocket and attached the crude device to a hardwire that he coaxed from his arm. The makeshift O-dat sparkled to life, and Charlie began copying numbers into the larger O-dats in front him. After a few moments, I saw the screens light up with unrecognizable coordinates, each with small blinking dots near the right of the screen. I could see that one of the dots was moving slightly.\n\nI looked at Charlie as he rested his head on his hands and stared longingly at the screens. What was he looking at? The markings made no sense to me whatsoever. They looked like some sort of radar coordinates, or maybe he had hooked into a deep-space probe. I couldn't break it to him that he was just staring at nothing, especially if he thought these were Max and Ketheria. Or _should_ I tell him? I didn't know how much a Honock understood. What sort of feelings he had, if any at all. The creature in front of me looked like Charlie, but he _wasn't_ Charlie. My friend never had a nest of computer circuitry mounted to the base of his skull. Suddenly, I felt sad. What had I done to him? What had Switzer done?\n\n\"That's nice, Charlie. They look good,\" I croaked, pushing down the lump trying to free itself from my throat.\n\nThen, as if as an afterthought, Charlie flipped a yellowed toggle switch mounted to the frame of his contraption. The screens exploded with images of Max, Theodore, and Ketheria. They were all asleep except Max.\n\nI pushed myself in front of Charlie, practically jumping into his lap.\n\n\"How are you doing this?\" I grabbed the piece of polymer. It was filled with numbers and symbols. \"What are these? Where did you get them?\"\n\nCharlie typed in some more numbers, and an image of my room on Orbis 4 came up on one of the previously blank O-dats.\n\n\"Charlie, how do you know how to do this? Who gave you these? Did someone put surveillance equipment in our rooms? Charlie, tell me!\"\n\n\"I don't remember,\" was all he said, and then he went back to staring at the screens.\n\nMax was not in her sleeper. She was sitting at her chair, her back to the camera. There was something in her hands, something she was looking at, but I couldn't make it out. How long had it been since I'd seen her? She looked just as she did in the image that was burned into my memory. Maybe her hair was a little longer, but that was it. The sight of her flooded my senses with her smell, her touch, the smoothness of her skin, the sound of her voice \u2014 all of it was inside me right now, igniting an ache I had been trying to bury. I wanted to be back on the Rings of Orbis. I wanted to be back on the rings _right now._\n\nI watched as Max placed whatever she'd been holding on her table and then walked toward her sleeper. She slipped out of her robe and climbed in. As the lid closed, she looked up \u2014 toward whatever was looking at her, I thought. For a nanosecond I was looking straight into her eyes again. It felt as if I had been impaled with a rod of hot metal. After the lid closed, I stood there staring at her sleeper, and then I slumped onto the floor. Charlie flipped the toggle switch, and the green dots came back up. He just sat there and watched them.\n\nWhen I returned to my room, I found Switzer already up.\n\n\"Up early or just getting in?\" he asked.\n\nI couldn't talk to Switzer. Not right now. I crawled into my sleeper and closed the lid. I didn't even look at him. Knowing what he did to Charlie made me sick to my stomach.\n\n\"I'll take that as just getting in. You got to lay off those tablets, JT.\"\n\n\"Shut up,\" I muttered, and closed my eyes. How could we ever be friends? How could I ever forget what he did?\n\nSwitzer thumped his big hand on the lid of my sleeper.\n\n\"C'mon. Big cycle. We start looking like Space Jumpers. New uniforms. I want something mean-looking. What do you want to get?\"\n\nI didn't reply. I couldn't. Not a single word. I wanted to throw back the lid and rip Switzer's throat out. I wanted to hit him, punch him once for every one of the millions of stupid things he had done in his life. I wanted to hit him until he was dead.\n\n\"Suit yourself,\" he grumbled, and then I heard the door disappear.\n\nFinally, I breathed. Then, for the first time in a long time, I slept. I didn't dream; my body simply wanted to check out for a while, to turn my brain off. When I woke, Switzer was back in his sleeper. I had spent the entire cycle asleep. No one had bothered me. No one had come looking for me when I missed my scheduled appointments with my connector or my meeting with Quirin. I found that odd, but I was relieved to find some freedom left in my life.\n\nI got up and searched for Charlie. He was not in the observation deck, so I traced my steps back to the ventilation shaft. The huge grate was in its proper place, and when I went to remove it, I was forced to adjust the settings to my arm. How strong was Charlie now? I wondered. The clunk of the grate hitting the floor resonated down the hallway, but I didn't wait to see if I had alarmed anyone. With the grate back in place I headed toward the green glow.\n\nCharlie was not at his screens, and all the O-dats were turned off. I searched the counter under the screens for the polymer, but that was nowhere to be found. _Where are you, Charlie?_ I sat waiting for him to return, but he did not come. Instead, I stared at the blank O-dats. I wanted so badly to turn them on. I could simply push in and take a little peek into his handiwork. He wouldn't mind, I convinced myself, and so I sat at his stool and fired up the portal that was my only link to my old life.\n\nCharlie's computer creation seemed designed for only one purpose: to spy on us. Once inside his array, I found hundreds and hundred of unlabeled files containing digis of Max, Theodore, Grace, Ketheria, me, and most of the other kids who had lived with Charlie on Orbis 3. Somehow Charlie had managed to locate each of us on Orbis 4. It was easy to figure out how he had done it \u2014 the staining. Everyone had been stained on Orbis 2 with a genetic mark that allowed Citizens to track us. After Switzer had jumped (and taken my arm with him), the Trading Council demanded that we get stained or be put to death. I remember Charlie railing against the staining at the time. I wondered if they gave him the tools to find us after he became our Guarantor. It all seemed plausible, but who had installed the surveillance equipment in our rooms and why? And how did a Honock find out about it?\n\nInstead of watching the files inside Charlie's computer, I pulled out and displayed them on the O-dats. File after file, each of us sleeping or sitting in our rooms. Sometimes there was nothing but long passages of our empty rooms. I could find no pattern to what Charlie had saved. Every clip had a time stamp, the earliest dated right after we arrived on Orbis 4. Charlie, and whoever else, had been watching us since the moment we had arrived.\n\nI pulled up a recent clip of Ketheria and began to watch. She was sitting with two Nagools, nodding as they spoke. What were they saying? All this equipment and no sound? It didn't make sense. I watched as one of the Nagools got up and left. Ketheria began to fidget, pulling at her robe and looking past the remaining Nagool. Then the first Nagool returned with another alien. He had his arm around the alien's waist, supporting him as he stumbled toward my sister. I think the creature was another Nagool. He had the same ashen complexion and OIO symbol marked on his face as the others, but he wasn't wearing the traditional Nagool garb.\n\nI stared at the O-dat in fascination as the Nagool helped the alien kneel in front of Ketheria, and then the alien dropped his head onto her lap. The two other Nagools moved back and out of the frame. She looked up as the others left and then seemed to focus on the alien in her lap. Ketheria's shoulders slumped, and then her whole body flexed. The room became brighter, and I realized that Ketheria was glowing. The light seemed to radiate from her skin, getting brighter and brighter as she stroked the Nagool's forehead with her right hand, all the while clutching his left hand in hers.\n\nThe light continued to build around her, and she began to tremble. She was shaking now, and then she threw her head back as if to scream. The O-dat flashed white as the glow around Ketheria exploded, making it too bright for the surveillance equipment to register an image. When the flash subsided, so had the glow around Ketheria. She was no longer trembling, but her hair was soaked with sweat. The two other Nagools rushed back in, and one caught Ketheria as she collapsed onto the makeshift sleeper. The third Nagool, the one that had needed to be helped in, stood up without the aid of the others. His skin looked tighter, and he stood straight, smiling at the other Nagools.\n\nDid Ketheria do that to him? What had I just watched? Was Ketheria healing people now?\n\nI opened another file that had been recorded after my arrival at the Hollow. An image of Max jumped to the screen, making my stomach lurch. She was seated on the floor in her room, and I could tell she was crying. Theodore was seated next to her. When he put his arm around her, I immediately felt angry. _I_ should be there comforting her, not Theodore. What was he doing, anyway? I pulled up another file of Max. This time Queykay was in her room, leaning over her. It looked as if he was yelling at her. About what, though? Max was holding her ground, at least she was trying to, but Queykay seemed very angry, very threatening. He waved his arms about, forcing himself into her personal space. At one point, I thought he might even hit her, but then he stormed out. Grace entered. She kept glancing over her shoulder as she spoke closely with Max. What was going on? I wondered. I flipped over to a file with the same time stamp. A file for Theodore, but he was not in his room. I couldn't help but feel that something was wrong. I searched Charlie's computer to find some sort of live feed like the one he had shown me before.\n\n\"What are you doing?\" Charlie said from behind me.\n\nI spun around on the stool to find Charlie looming over me, a plastic pipe in his hand. He swung and struck me in the head before I had time to react.\n\nI awoke to a pounding headache and a nice lump near the top of my head. Someone had put me back in my room. I pushed the lid aside and sat up. The blood rushed from my head, leaving nothing but the pain, which exploded and filled my entire skull. I pried open my eyelids to see if Switzer was sleeping. He wasn't there; I was glad about that. Had Charlie brought me back here? Had Switzer seen him? I couldn't tell what spoke it was, but despite the rocket that kept launching inside my head, I was hungry.\n\nI found Switzer eating with our usual group of Space Jumpers, but he didn't see me enter the food commons. Switzer was holding up something attached to his uniform and showing it to the others while they ate. It looked like some sort of fabricated metal pocket. Our new uniforms! I had forgotten. _Who cares?_ I muttered to myself while I searched the food wall for something to eat.\n\n\"Where have you been?\" Switzer asked, coming up behind.\n\n\"Nowhere,\" I replied, and kept pulling food out and stuffing it into my mouth before even putting it on my plate.\n\n\"How can you be nowhere? You weren't in training. You weren't around to get your uniform, and you certainly weren't in the room.\"\n\n\"What are you now, my Guarantor? I was doing stuff.\" I slammed the last door and pushed past him.\n\n\"Stop taking that crap. Those tablets are making you unbearable,\" he called after me.\n\n\"Yeah? What's your excuse?\" I said, and left the food area.\n\nWhy had Charlie hit me? Had he not remembered me? I wanted to go back and find out. I also wanted to get something to protect myself with, or at least be ready to use my arm if Charlie attacked me again. He was strong, though. I needed to be prepared. Maybe there was something in my room to use, I thought.\n\nWhen I returned to my room, Brine Amar was waiting for me.\n\n\"You've missed our scheduled appointments,\" he said.\n\n\"Sorry, I haven't been feeling well,\" I told him. It wasn't really a lie. My head was killing me.\n\n\"Did you go to the infirmary?\"\n\n\"No.\"\n\n\"What is ailing you?\"\n\n\"Um . . . my head. I'm having bad headaches.\"\n\n\"They can fix that immediately. Come. I'll show you the way if you don't know,\" he said, standing up.\n\n\"Now?\"\n\n\"Yes, now, unless there is somewhere else you are off to, though I don't know where that could be since you're supposed to be with me right now.\"\n\nDespite the fact that he had told me that I could not offend him, I detected a tinge of anger in Brine Amar's voice. Should I tell him about Charlie? What if that led to Charlie's little surveillance room? I couldn't risk that. I wanted to see Max again. I wanted to know how Ketheria was healing people.\n\n\"You know, I just had something to eat and I'm starting to feel a little better. Why don't we just have our session? I apologize if I messed up our schedule.\"\n\n\"That is fine as well. We have much to do.\"\n\nI tried to concentrate on what Brine Amar asked of me, but I could not stop thinking about Charlie. How long had he been here? Why couldn't he remember anything? How much of him was really Charlie? My connector was instructing me to upload file after file of Space Jumper history, most of it dry and boring, while I thought of Charlie. The process reminded me of my first rotation at the Center for Wisdom, Culture, and Comprehension.\n\n\"What do I need this for?\" I complained.\n\n\"Do you not wish to understand the rich heritage created by the many Space Jumpers that have come before you?\"\n\n_Not really,_ I said to myself, but I had to admit, Brine Amar was right. There were a lot of Space Jumpers who had done some amazing things in the universe. Softwires didn't seem so rare when you looked at their entire history. I did notice, however, that there was little mention of the Tonat. Instead, I found stories that involved other categories of Space Jumpers. I read about units of militarized Space Jumpers, as well solitary Jumpers called Cenots, who worked undercover within unknowing civilizations. I read of one Cenot who lived among a warring race known as Forlians for more than forty rotations. He single-handedly created a new form of government that, once adopted by its people, allowed the Forlians to reach the stars and assist the Ancients in constructing the Rings of Orbis. I wondered why I had never heard of the Forlians before and why I had never encountered any on the rings. Orbis certainly held a lot of mysteries. Did anyone truly know everything?\n\nI went slowly with the Cenot files. I found myself admiring these aliens and their dedication to the Ancients, but I could not understand why they sacrificed their entire lives yet had lapsed into obscurity. It just seemed strange.\n\n\"These Cenots,\" I said to Brine Amar. \"Are there any here? You know, at the Hollow?\"\n\n\"I don't believe so. It takes a special kind of Space Jumper to spend his or her life in isolation.\"\n\n\"Her? I wondered about that. So there are female Space Jumpers as well?\"\n\n\"Certainly. Some of the best.\"\n\n\"I don't know how those Cenots do it,\" I said, shaking my head.\n\n\"Acceptance,\" Brine Amar replied. \"Something you have much trouble with, I'm afraid. I hoped we would be much further along than this before you commenced the next stage of training.\"\n\n\"Next stage!\" I cried. \"I thought we were done. I thought we were getting our uniforms and that was it. Switzer has his now. I thought I could go back to the rings.\"\n\n\"But you don't even have your Burak yet.\"\n\n\"Burak?\"\n\n\"Your Space Jumper's belt. Your most important possession.\"\n\n\"But I don't need a belt.\"\n\n\"You will if you ever need to jump _with_ someone. _You_ may be able to jump without your burak, but you would surely kill anyone you ever tried to jump with.\"\n\n\"Cenots don't jump with anyone,\" I reminded him.\n\n\"I'm referring to a rescue attempt. What if you were forced to jump and take the Scion with you? Without a belt, you would be unable to make the proper temporal allowance for your passenger. I shudder to think of the result.\"\n\nI remembered the numerous times I had fantasized about jumping off the ring and taking Max with me. What if I had killed her? What else didn't I know? There was so much to learn, but I didn't want to stay here any longer. I wanted to be with my friends.\n\n\"When does the new training start?\" I asked.\n\n\"When you are ready,\" he replied.\n\n\"I'm ready.\"\n\n\"Are you?\"\n\nBrine Amar's tone frightened me. \"You don't sound very confident about my progress. What about Switzer? How is he doing? Perfect, I'm sure.\"\n\n\"Why don't we start with your uniform? It is a big step in accepting who you are.\"\n\n\"I know who I am,\" I told him.\n\nBrine Amar didn't say anything, but he didn't need to; his silence said enough. He raised his hand toward the door, and I shrugged. I could use some new clothes, anyway.\n\nBrine Amar took me to the small cluster of trading chambers located near the recreation area, just past the food commons. I had yet to visit this area. I had heard Switzer talking about going there with a few other Jumpers, but I had not been invited.\n\n\"Who works here?\" I asked. \"Do they live on the Hollow?\"\n\n\"Most are Honocks. I believe there are also a few people who failed to complete the program but chose to stay.\"\n\n\"Or maybe the Trust didn't want them to leave and reveal the Hollow's whereabouts.\"\n\nBrine Amar looked at me. \"This is not an evil place, JT,\" he said. \"You really must open your mind before it's too late.\"\n\n\"Too late for what? You don't have to be so cryptic. Just tell me.\"\n\n\"Here we are,\" Brine Amar said, stopping in front of an open trading chamber. It was nothing more than a few tables arranged under a concrete shelter.\n\n\"But where are the uniforms?\" I said, staring at the bolts of material and piles of rubber, metal, and plastic.\n\n\"You don't think they stock uniforms for every species, do you? Your uniform is made specifically for you. It is unique, as unique as your softwire.\"\n\n\"That ability seems pretty common around here,\" I told him.\n\n\"You look only at what you see.\"\n\nA Honock entered from a door in the back wall. I could tell he was a Honock by the hunk of hardware around the back of his neck.\n\n\"Welcome,\" he said, beaming. \"One last straggler for a fitting?\"\n\nThis Honock seemed a little more \"awake\" than Charlie or the Honock that was afraid of me. This one acted like a normal person, despite the fact he was mostly metal, wires, and circuits.\n\n\"Just me. It must not be that busy around here,\" I said, pushing the Honock for information.\n\n\"Oh, you would be surprised. I also do repairs and tailoring. The Trust keeps me very busy, but it's always an honor to create a suit for a new Space Jumper. I've made every Space Jumper suit for . . .\" The Honock looked up as if trying to remember. Then he gave up and shook his head. \"Well, for a very long time. That's for sure, isn't it, Brine Amar?\"\n\n\"Indeed, Potu. Your work is worn by almost everyone at the Hollow.\"\n\nPotu looked at me, smiling. I was surprised by how alert he seemed to be. Why wasn't Charlie like this?\n\n\"Have you been assigned yet? I don't think so. Let me see. Something simple to start. A uniform you can build on. Do you see yourself in a militarized unit?\" Potu asked me.\n\n\"Actually, what about the suits you make for the Cenots? I think that's more what I'm looking for.\"\n\nPotu's smile faded, and he looked at Brine Amar, as if for guidance. From the corner of my eye, I saw Brine Amar nod gently.\n\n\"Then follow me,\" Potu instructed.\n\nPotu led me to another section of his trading chamber, where there was some type of scanning machine with four pillars. Potu ask me to stand at the center of it while he adjusted each pod, raising them to my height.\n\n\"This gives me a multidimensional rendition of your exact body shape. I can make you a suit that fits better than your own skin,\" he boasted.\n\n\"Sounds good,\" I said. \"Just make sure to leave my old skin where it is.\"\n\nPotu chuckled as he continued to set his fitting machine. _So, Honocks understand humor,_ I thought. Then Potu moved in and allowed me a closer look at the hardware attached to the base of his neck. Without asking, I pushed inside his little computer. Immediately, I found a simple interface that would allow other Space Jumpers access to Potu's mind. Not every variable was accessible. Some were password-protected. My softwire allowed me to push past that kind of simple security programming and look at the hardware in its entirety. It was fascinating \u2014 so many hardwire connections to his soft tissue, synaptic chips with chemical impulses, and electric pulse generators, all in that little construction at the back of his head.\n\nI pulled out, and Potu was staring at me. \"Don't do that,\" he scolded me.\n\nHow did he know? \"I'm sorry,\" I whispered. \"I just \u2014\"\n\n\"It's polite to ask first,\" he interrupted. \"I'm more than just a machine.\"\n\nPotu continued with his work and then punched at an O-dat on the wall. A beam of green light sprouted from each of the four posts and scanned my entire body. It was over before I blinked.\n\n\"Done,\" he said. \"Come back next cycle.\" The coldness in his voice was easy to detect.\n\n\"Don't I get to pick a style or anything?\"\n\n\"You already mentioned you were interested in a Cenot standard. That's what you'll get.\"\n\nI was embarrassed by what I had done, but I was also intrigued by how much emotion Potu displayed.\n\nI stepped out of the fitting machine and walked toward Brine Amar. \"I guess that's it,\" I said.\n\n\"It's just the start,\" he replied. \"With your new suit, you can begin the next stage of your training.\"\n\nI skipped my next meal and went looking for Charlie. I waited outside the ventilation grate and slipped into the shadows whenever I heard someone coming. If it was Charlie, I planned to warn him, so as not to scare him again. I didn't think I could take another blow to the head. I was convinced that Charlie's hidden O-dat setup was illegal, and I didn't want anyone tearing it down before I got a chance at it again. While I sat thinking about seeing Max again, Charlie lumbered up to the grate. I stepped out of the shadow and out of the range of his swing.\n\n\"Charlie, it's me, JT.\"\n\nCharlie stood there with his head cocked to one side, as if he was trying to remember who I was. Maybe he hadn't recognized me the last time and had taken me for an intruder. Charlie's awareness of the people around him seemed so much different from Potu's. Why? Was Charlie's brain damaged before they turned him into a Honock? I think I knew another possible reason. I was eager to find out.\n\n\"Charlie, don't you remember me?\"\n\n\"Ketheria?\"\n\n\"Yes, my sister. Peanut butter.\"\n\n\"Peanut butter!\"\n\nThere was something wrong with his memory. His short-term memory was either gone or disabled.\n\n\"We're friends, remember? You showed me your screens.\" I pointed at the grate. \"Max and Theodore. I would like to see them again.\"\n\n\"JT?\"\n\n\"Yes!\"\n\n\"My friends.\"\n\n\"Can we see them?\"\n\nCharlie turned and pulled the grate from the hole. I slipped inside, and Charlie followed. This time when Charlie sat at the controls, I looked at the computer circuitry behind his head. Then I remembered Potu's warning.\n\n\"Charlie, do you mind if I take a look?\" I asked, pointing at the metal.\n\nCharlie reached behind his neck and brushed his fingers across the device. Then he shrugged. That looked like a yes to me.\n\nThe interface inside Charlie's brain was just as complex as Potu's. After a little searching, I discovered controls for memory, emotion, coordination, and bodily functions, such as metabolism and heart rate. The Space Jumpers had completely mapped Charlie's brain and applied controls to every function. Could this be done to someone who was still alive? I wondered. The thought of Citizens controlling knudniks this way sent a shiver down my spine.\n\nWhen I pulled out, Charlie was leaning on his hands and staring at the green dots on the screen.\n\n\"Can you make them so I can see them? You know, see the images?\"\n\nCharlie flipped the switch, and there was Max. She was in her room with Grace and Theodore. Why was Theodore there every time I saw Max? I shook it off and looked at what they were doing. Grace and Theodore seemed to be watching the door. Max opened a panel in the wall and reached in to pull something out. It was slightly out of view, so I couldn't really see as it was. I stared at Max's face. It was so close to the secret camera that was spying on her that she was almost life-size.\n\nWhen she pulled back, Theodore turned to help her. What did she have? They placed whatever it was on the floor in front of the sleeper, and Max stood up. When Max reached for her tools, I caught a full view of what they had hidden. It was a plasma rifle, and a big one, too. What was Max doing with a weapon?\n\nCharlie reached up and flipped to another screen.\n\n\"Wait!\"\n\nBut Charlie was looking at an image of Ketheria now.\n\n\"No,\" he replied. \"Don't like guns.\"\n\n\"Charlie, I'm going to make a couple of adjustments, if you don't mind. You know, with that thing there on your neck.\"\n\nCharlie looked back at the screens and then at me. \"Sure,\" he said.\n\nI pushed back into Charlie's brain and went over the controls once more. What would make Charlie a little more coherent? How much would I need to tinker in order to give him back his memory, or his old attitude? These controls weren't like dials that you set from one to ten. It was more like restricting blood flow through a vein with your fingertips. I would have to go slowly.\n\nThe first thing I did was to adjust the memory variable. Was it possible to awaken memories of his childbirth if I went too far? And which way was more? What if I cut off still more of his short-term memory? I hesitated. I bet Max would know what to do.\n\nI made a tiny adjustment to his memory and pulled out.\n\n\"Charlie?\" I said.\n\nHe turned and looked at me.\n\n\"Yes, JT?\"\n\nThat was good. At least that was in the right direction.\n\n\"Charlie, where are you?\"\n\nHe cocked his head and looked at me. \"Here,\" he said, his tone slightly mocking.\n\n\"How did you get here?\"\n\nHe looked around as if he was trying to remember. Maybe I needed to adjust it more if he couldn't remember how he got here.\n\n\"Forget it,\" I told him. \"Watch your friends. I need to do a little more.\"\n\nCharlie turned back to the screens, and I slipped back inside. Maybe the variables were tied together in some way. Memories can elicit strong emotions, so maybe I needed to work the two controls together. I pushed through the memories and the emotion controls, working the two of them together. I increased both a little more than before. Was that too much?\n\nCharlie answered that for me. From inside his head, I felt his fingers clamp around my neck. I heard screaming. Was that me or was it Charlie? I pulled out and found Charlie on top of me, his hands clamped around my throat.\n\n\"Charlie!\" I croaked.\n\n\"What did you do to me? Where am I?\" he screamed. \"Where did you go this time? I do everything for you kids and this is how you treat me?\"\n\nSpit flew from Charlie's mouth, and the blood vessels in his eyeballs beat red with fire.\n\n\"Can't . . . breathe . . .\"\n\n\"Do you know what I sacrificed to take care of you? Can you even imagine what I've gone through? For what? You're nothing but a worthless, whiny punk!\"\n\nCharlie was going to kill me. At least he was trying to. I felt the familiar blackness creeping in around the edges of my vision. I pushed back into Charlie's brain controls and sloppily grabbed whatever I could find. Charlie's grip weakened, and he slumped backward. I sat up, coughing and rubbing my neck. My head throbbed from the lack of oxygen.\n\n\"Charlie?\"\n\nHe said nothing. He just sat there, drooling, staring at a spot on the floor. Then he slumped over.\n\n\"Charlie!\"\n\nNothing. He was practically catatonic. What had I done? How did I fix this? I stood up and moved him to the stool. He didn't resist. It was like pushing around a huge weather balloon. I slipped back inside his brain, looking for traces of what I had touched. Everything looked normal, the same as the first time I entered Charlie's brain. _Do I go get help?_ I wondered. _But where?_\n\nI began tweaking the variables one by one, pulling out each time to check Charlie's reaction. After a few tries, I got the drooling to stop, but he still wasn't responding to his name. _What am I going to do? I can't leave him like this. He'll never find his way out of here._ After a few more adjustments, Charlie seemed a little more alert, but he still wasn't what he had been.\n\n\"Charlie, c'mon, get up. Can you follow me?\"\n\nHe just stared at me like the little ones used to do on the _Renaissance._ Once I got him up, I pushed him toward the ventilation grate. If anyone could help fix him, Quirin could.\n\n#\n\n\"You knew he was here the whole time and you didn't tell me?\" I cried. Charlie was standing next to me in Quirin's quarters.\n\n\"You are not here to make friends,\" Quirin said. His voice was sharp. \"You are here to learn. Besides, I gathered that your emotional needs were met by the other human you so adamantly requested to participate.\"\n\n\"Switzer? I only wanted him here because I thought it was the right thing to do after what _you_ did to him.\"\n\n\"I thought it was the right thing for the Honock after what _you_ did to _him._ \"\n\n\"Me? I didn't do this! If anyone is to be blamed, it's Switzer. He had one of his wormhole pirates kill Charlie.\"\n\nI cringed at the awful sound of crunching bones as Quirin shifted in his rock bed.\n\n\"This is enough,\" his voice boomed. \"I will not tolerate insubordination. You are an instrument, controlled by us, in service to protect the Scion. Your wishes are irrelevant. I have only appeased you so far because humans are the last chance.\"\n\n\"Against what? Some invisible force that's eating up other universes? Sounds pretty far-fetched to me. It's just another story to oppress those stupid enough to believe it.\"\n\n\"Enough!\"\n\nI felt the stone walls of Quirin's room shudder as if his anger had lifted the rock. Even I knew when to shut up. I just needed to get my training over with and get back to the Rings of Orbis.\n\n\"I'm sorry,\" I said.\n\n\"I don't want your apologies. I want your commitment. I want you to stand up and be the man I created. Your life is filled with pathetic self-awareness that only hampers your ability to act. It is time to awaken what is in you, to awaken the Space Jumper.\"\n\nThe door to the room opened, and two Space Jumpers walked in and seized Charlie.\n\n\"Wait!\" I cried, but I could only watch as they led Charlie, unresisting, from the room. \"Quirin, what are they going to do to him?\"\n\n\"Fix what you did,\" he spat.\n\n\"What are _you_ going to do _me_?\"\n\n\"Train you. I am going to turn you into the softwire you are destined to be.\"\n\nAfter two more Space Jumpers entered the room, the gravity of the moment settled upon me.\n\nI just had my first argument with my father.\n\nI did not look back as the Space Jumpers escorted me out.\n\nI had expected to be led back to my room, but they turned in the other direction, securing me by the arms before I could resist. It wasn't necessary, though. I wasn't going to resist. As they marched me down the corridor, I thought of my sister's warning. The one where she told me that the Trust used force to awaken parts within you whether you were ready or not. Would Quirin use that force on me? I was not scared. I couldn't have been more ready.\n\nI was taken to an area behind the labyrinth, where I was loaded into a light chute and then dumped into a part of the Hollow that I had never seen before. I looked around the huge cavernous space and couldn't see where the room ended. In the distance, I spotted a tiny light blinking through the mist on the horizon, but I could not figure out where I was. I followed the Jumpers a few meters to my left, but instead of finding solid flooring, I discovered individual platforms that seemed to float over a murky abyss. The center platform was the only one rooted to something, but I could not see what. When I looked down, the cylinder supporting the platform merely disappeared through a bluish fog. Each platform was rimmed with a cool, electric light; and a narrow railing, only wide enough to let one person pass, surrounded the entire area.\n\n\"Is this where I train?\" I asked.\n\n\"This is where everyone trains,\" the Space Jumper to my left replied.\n\n\"Why not in the labyrinth?\"\n\n\"This is where you learn to use your burak.\"\n\nI wanted to shout, \"I don't need a belt!\" But what was the use? No one would listen, anyway.\n\nI looked up toward the ceiling and spotted an alien descending swiftly to the center pod. The creature sprouted thick pointed tentacles from the back of his head that made him look as if he were caught in a wind tunnel. His broad shoulders were pulled back, and he carried a long staff in his right hand. The smug look on his face alone told me that I was in trouble.\n\nWithin an instant, the alien was next to me \u2014 he was a Space Jumper. He began sniffing me like an animal.\n\n\"So you're the one they speak so much of. A Tonat. How privileged are we?\"\n\nThe alien circled me. Each step was a cautious gesture with a threatening glance, and I should have been scared, even terrified, but I suddenly found myself fighting the urge to laugh. I clenched my teeth and stared past the vile creature. I had an intense moment of clarity in which I saw this guy as a caricature of every alien I had ever scuffled with. Suddenly, I was outside of my body looking down at this whole absurd ceremony. All I saw was a kid \u2014 a kid from a planet called Earth. And now I was living in a comet with this animal towering over me, strutting about, as if _I_ were a threat that needed to be dealt with. Maybe it was a nervous reaction, I don't know, but finally I couldn't help myself anymore, and I snickered.\n\n\"I'm sorry,\" I mumbled, but the alien wasn't accepting apologies.\n\nHe swung around and struck me in the stomach with his muscular left arm. The blow pushed my stomach up into my lungs, leaving no room for air. I buckled over, gasping, and the alien brought the staff up into my face. The pain exploded across my nose and I tumbled backward, slipping under the railing. At the very last moment, I reached out and caught the vertical support, but the rest of me dangled over the empty void.\n\n\"Is _that_ funny?\" he asked.\n\nI couldn't breathe, let alone answer him. Blood poured down the back of my throat, and reality came crashing down upon me. Now I was scared, _really_ scared.\n\n\"Is that funny?\" he screamed. His voice echoed across the void.\n\nI shook my head. That was all I could do.\n\n\"For the rest of your pitiful life, you will remember your time with me as your easiest cycles. I am your best friend now. Once you are placed in the universe, everyone will be your enemy. You will be hunted like a common cochark, and even your own family will loathe you.\" The alien knelt in front of me, his tone growing softer. \"But _I_ will love you, and so will your brethren. A Space Jumper's plea can be heard across the galaxy, and they are the only ones who will ever answer you. You have one purpose in your life now, and that is to serve the Ancients, to serve the Scion.\"\n\nThen he kicked my hand and sent me falling into the abyss.\n\nI awoke in my sleeper with all my appendages intact. Had I dreamed it? The pain in my fingers and my nose told me that everything was real, but how had I gotten here? Switzer was sitting up, putting his suit on. There seemed to be a new piece attached to it, a metal plate over his heart.\n\n\"So you met Chausau, huh? I wouldn't look in any mirrors for a few cycles.\"\n\nI sat up but didn't reply. Every time I looked at Switzer now, I saw the wormhole pirate and thought about what he did to Charlie.\n\n\"Still not talking to me?\" He stood up and stomped his thick boots on the floor as if they were new. \"Suit yourself, but I'm gonna need your help here. I've done so much in my life, I don't know what part you want me to apologize for.\"\n\n\"All of it,\" I muttered as Switzer marched from the room, but he poked his head back in.\n\n\"You know you're not being fair,\" Switzer said.\n\n\"Fair? What do you call fair? Your whole life has only been about yourself. If there's not something in it for you, then forget about it. Why don't you just get out of here? Go back to whatever wormhole you were hiding in and leave me alone, _Captain Ceesar._ \"\n\n\"You have no idea what it was like for me. You think I roamed around the universe pillaging whatever I wanted, like it was some perpetual Birth Day celebration?\"\n\n\"You certainly seemed proud of your actions.\"\n\n\"I wasn't even sixteen years old when I popped up onto that pirate ship! They were brutal. Simply brutal. I begged the Universe to let me be a knudnik again, but no one answered. I _fought_ for my position in their world, literally.\n\n\"Life as a wormhole pirate is nothing like they whisper in the back rooms of your cushy little school. If you've got the stomach for it, I'll tell you about the time I almost froze to death, abandoned on a mining moon with my best friend. I held him in my arms like a little one, wishing he would hurry up and die so I could gut him and then crawl inside his dead carcass to keep warm. You don't want to know what I did with his insides.\"\n\n\"You murdered _my_ best friend.\"\n\n\"That was not my intention, JT. Charlie was an accident. I only wanted him out of the way for a while. How was I supposed to know he would have an allergy to the stuff? I'm sorry. I really am. Besides, he's not dead. Look at him! I think he looks pretty good.\"\n\n\"Get out of here.\"\n\n\"It was an accident.\"\n\n\"Get out of here!\"\n\nI fell back into my sleeper and heard the door close. Then I swung my feet around and over the edge. I didn't want to go back to Chausau. Who was I kidding? I wasn't cut out for this. Switzer should be the Tonat. It was like he relished this stuff. I couldn't stand him for what he did to Charlie, but I had to admit that the Hollow had changed him. He acted with purpose now and a sense of belonging. Why couldn't I find that?\n\nDespite my best efforts to resist, I put my feet on the floor and stood up. I didn't have to be a softwire to imagine the consequences for missing Chausau's training. That's when I realized I hadn't taken one of those tablets for quite some time. When I went looking for it, I found the nausea and the headache lurking inside me, but now it was more of a gauge, a tool to tell me how far away Ketheria was. The more I concentrated on the feeling, the more sick I felt. Ketheria was far away. I pushed the sickness back down, but it didn't go easily. I reached into my pocket, fished out a tablet, and popped it into my mouth. I didn't need anything getting in the way of my training. But I knew that was just an excuse.\n\nIn the food commons, I grabbed an olack, a sweet fleshy fruit Switzer had shown me, from the food wall. I also grabbed a bowl of protein grains. I ate some of it while walking, but I tossed most of it. Out of the corner of my eye, I caught Switzer with some of the other Space Jumpers. I tried to picture the old Switzer I knew popping up on that pirate ship as a kid. He had to have been scared. I couldn't even imagine trying to crawl inside another person to keep warm, let alone a friend.\n\nWhen I reached Chausau's training facility, I found several other Space Jumpers, including Gora Bloom, already waiting on the outside platform. I also noticed several Honocks, but Charlie was not among them. I wondered where he was. I wanted to see him. Not just so I could look at his surveillance monitors again, but because I wanted my old friend back.\n\n\"I heard we're getting our belts,\" Switzer remarked as he stepped out from the light chute.\n\nThe best I could do was a grunt. When I offered Switzer no more, he walked over to Gora. Chausau entered from above, just as he had last cycle, but this time with a Honock in tow. The Honock, floating behind Chausau, concentrated on the platform below as if to make sure he wouldn't miss it. Clutched in his arms was a collection of Space Jumper belts.\n\n\"I have here your most prized possession!\" Chausau shouted to us from the middle platform. \"This single item has as much value as the Source used to ignite your existence. Lose this and you may as well lose your life!\"\n\nChausau took the belts from the Honock and held them up.\n\n\"Well, what are you waiting for? Come and get them.\"\n\nI looked at Gora and Switzer. They were trying to find a way across to Chausau, but there was none. His platform was too far to physically jump across the void to, and there was no craft to take them. Was I supposed to jump? I could do it without a belt, although I had only done it around Ketheria, or when I got upset. Even that little glitch had seemed to fade, however. Was this a test?\n\nI concentrated on the center platform, trying to will myself there, but it was no use. I might as well have been trying to move the platform to me. If I was a Space Jumper who could jump without a belt, someone was going to have to show me how. I was relieved to think the burak would take that pressure off me. I was anxious to get that belt.\n\n\"Well?\" Chausau called out. Then he was next to Switzer. \"That's a joke. Of course you can't do it without the belt. That was the whole point.\"\n\nI watched him hand a belt to Gora and then to Switzer. I could see Gora's eyes light up as he cradled the belt in his hands. Then Chausau turned to me. There were no more belts left.\n\n\"But I _was_ hoping to be surprised by you,\" he cried. \"Are the rumors false?\"\n\n\"What rumors?\"\n\n\"What rumors?\" he said, loud enough for everyone to hear. \"The rumor that you can move through space and time without a belt. That you possess an ability no other Space Jumper has ever exhibited . . . until now.\"\n\nChausau's face was centimeters from mine, and the intense air from his nostrils pushed against my skin. I didn't dare move.\n\n\"Care to demonstrate this extraordinary ability?\"\n\n\"I can't,\" I mumbled.\n\n\"Speak up!\"\n\n\"I can't!\"\n\nChausau circled me. \"You can't or you won't? I've been told that you never require a belt. That you can jump whenever you like.\"\n\n\"You're wrong. It's not like that,\" I told him. \"It's tied somehow to my sister. I can't just jump when I feel like it.\"\n\n\"I'm _wrong_?\" The other Space Jumpers standing along the railing chuckled. \"I'm never wrong. Just ask any one of them. The fact that you can't jump whenever you want has to do with this,\" he said, and stabbed my head with his finger. \"It has nothing to do with this.\" Chausau then stabbed at my heart. \"But that's what you're here for. I will awaken that part of you that can control your gift, to make it part of you, a function as automatic as breathing. That is, if it's even true.\"\n\nChausau walked away. \"Put your new belts on!\" he cried. \"Look at them! Are they not beautiful? They are yours now. Take care of them as if your life depends on them \u2014 because it does.\"\n\nI stood there as the other Space Jumpers cheered. I watched Switzer and Gora slip their belts around their waists and admire them. I couldn't look at the others, though. I only stared at the belts. I hated Chausau for singling me out like this. I didn't want the other Space Jumpers to look at me without a belt, naked and waiting for ridicule. Switzer looked up and ours eyes met. I expected to see that stupid smirk on his face. I assumed he would be the first to ignite my long and torturous humiliation, but he didn't. No, it was much worse than that. When I looked at Switzer, I saw pity. Pity for me. I wanted to die.\n\nChausau assigned a Space Jumper and a Honock to Switzer and then to Gora and instructed them to walk the new owners through their belt's interface.\n\n\"What am I supposed to do?\" I asked as Chausau walked past me.\n\n\"Come with me,\" he said, and my nostrils immediately filled with the stench of feet.\n\nChausau released his grip, and I found myself standing with him on the center platform.\n\n\"Warn me next time you're going to do that, will ya?\" I said.\n\n\"Hopefully there won't be a next time. Quirin has informed me of the details of your ability, and it seems quite simple. Unlike those who use a belt, you cannot jump with anyone in tow. Only a belt can create the proper time distortion for that. If you try, you will kill them. And you do not need to put coordinates in a belt as we do; you simply need to know _where_ you're jumping.\"\n\n\"That seems pretty limiting, don't you think? Why don't you just give me a belt, like them, so I can learn this properly? Why burden me with those limitations?\"\n\n\"It's not the limitations that count. It's the freedom to jump whenever you want. Imagine yourself torn apart in battle protecting the Scion. If you had a belt and it was lost or damaged you would be useless. But not you. You can still jump. It is a gift beyond comprehension. They will fill moons with stories about you.\"\n\n\"I don't want anyone to write anything,\" I spat.\n\n\"That is not up to you. Your burden is to master your gift. That is your only concern. Do you see the spot where you were just standing?\" he asked, pointing back to the railing near the light chute.\n\n\"Yes,\" I grumbled.\n\n\"That is your goal. That is where you are going.\"\n\n\"How? Aren't you supposed to teach me?\"\n\n\"Did someone teach you to breathe?\"\n\n\"That's a stupid question.\"\n\n\"Really? When you concentrate on your breathing, you can block out most of the deconstructive energy that flows through you. When you concentrate on your ability, you can block out most of the Universe, as well as its physical form. You may then choose what part of the Universe will be manifested in your presence. _You_ choose your own reality, Space Jumper. Whether you are here or there is up to you. Matter does not exist until it is observed; until then, it's simply waiting for you.\"\n\nThen Chausau jumped, and a moment later I spotted him near the light chute where I had started.\n\n\"That doesn't make any sense!\" I yelled after him, but he only turned and walked over to Switzer.\n\nHow was I going to get over there? What kind of training was this? I looked around the platform, but for what? I had no clue. Maybe I could find something to tell me what to do, but the circular markings I saw on the floor were nothing more than decorations.\n\n_You have to concentrate._ But I just stood there with my arms crossed, watching the others. Gora and Switzer looked like they were swapping notes as the other Jumpers explained the workings of their new belts. Why wasn't that me? Why did I always seem to be the one like _this_? Abandoned in the middle of nowhere and left to figure things out for myself. I hated it. It wasn't fair.\n\n\"Hey, I can't do it!\" I yelled. Switzer looked up as my cry echoed back to me. I cringed as my whining replayed across the void.\n\nI sat down near the center of the platform in an attempt to concentrate, but concentrate on what? My breathing? That seemed like a bunch of nonsense to me. I closed my eyes and took in a deep breath.\n\nNothing.\n\nThe sound of laughter from the others across the void crept into my mind. _Push that out. Concentrate on the spot. Picture it in your head._\n\nStill nothing.\n\n\"This is stupid!\" I cried, standing up. No one looked across at me this time, not even Switzer. \"Hey!\"\n\nI stared at the spot near the light chute. I pictured myself standing there. I clenched down on my back teeth, trying to focus harder on the location. I even bulged my eyes for effect.\n\n\"Oh, this is useless,\" I said, and flopped back down. This time, I sat at the edge of the platform and let my feet dangle over. I wanted to throw something down the void, to hear how long it took before it hit bottom. At that moment, I almost pushed myself off the platform. I almost threw myself down the void just to see what would happen. My arms tensed, and I even felt the rush inside my stomach just before someone does such a stupid thing. Of course it was a stupid thing. Chausau wouldn't have kicked me over if there were any chance I could die, but he wasn't here right now was he? Why even take the chance?\n\nI looked up to see where Chausau was, but I couldn't find him. He had left. So, too, had Switzer and Gora. I watched the other Space Jumpers gather their things and leave as well.\n\n\"Hey, what about me?\" I yelled.\n\nA moment later, I was alone. Now all I _could_ hear was my own breathing. Immediately I began to wonder how long Chausau would leave me here. What if I got hungry? What if I had to go to bathroom? _I could always void over the void,_ I thought.\n\nHow pathetic. Look at me. What a malf.\n\nAll I had to do was jump. I had done it before. _But that was different,_ I tried to reason. Ketheria was in danger. That's what triggered my jump, and the time in the food chamber I was simply angry. _Then get angry._\n\nI clenched my fist and snorted. Then I narrowed my eyes. I recalled every clich\u00e9 I could think of to express my anger. Then I thought about Switzer and Queykay, even Odran and Weegin. I thought about the time Weegin tried to sell us, and I thought about my life on the _Renaissance._ The anger smoldered, and soon a small flame ignited all those things I wanted to say but never did. I was getting very angry now. I could feel it. I stood up and concentrated on the spot near the light chute. Oh, I was mad, really mad. . . .\n\nNothing.\n\nI screamed. I guess I wasn't mad enough. I lay back on the platform with my hands behind my head. My headache, the one always lurking just behind my eyes, raged against my forehead now. I reached into my pocket for one of the tablets, hoping for relief, but I paused before popping it into my mouth. These things weren't stopping me, were they? I didn't eat much or even sleep much when I took them, and it made me wonder if they were part of the problem. It seemed unlikely, but I still tossed the tablet over the edge. In fact, I turned my pocket inside out and let all of the tablets tumble into the void. I waited to hear if the tablets ever hit bottom, but I never heard a thing. _Must be too small to make a sound, anyway,_ I told myself.\n\nThen the lights went out.\n\nI think I fell asleep shortly after the place went dark. I remember staring up at the soft blue haze in the ceiling and wondering about Max. I didn't dream about her. In fact, I don't remember dreaming about anything at all. I woke to the sound of Space Jumpers filing into the training area. I saw Switzer and Gora and then immediately looked up. Chausau should be descending upon my platform like he always did, but I saw nothing. Instead I heard him across the way. He had used the light chute, bypassing me completely. They were ignoring me, I realized.\n\nI sat and watched Switzer make his first jump with the belt. After several tandem jumps with another Space Jumper, he moved from one platform to another. The others cheered and gathered around him, thumping him on his back. I could see him smiling even from where I sat. I hope he enjoyed the smell of stinky feet.\n\nGora was next. I watched him fidget with his belt as if he was checking and re-checking something, or everything, while the others waited patiently. When he did manage to jump, he found himself teetering close to the edge of the other platform. So close that he almost missed his mark entirely. Everyone remained breathless, even me, until he steadied himself and found solid footing. He turned and thrust his arms into the air, his tentacles wiping widely about, and a cheer erupted.\n\n\"Goodie for you,\" I said.\n\nThey made it look so easy with the belt. _Why not just give me one?_ I could learn the harder way later. I wanted to be next so badly. I wanted the others to cheer as I jumped from platform to platform. I knew I could skip across every one of them if they just gave me one of those stupid belts. I sat back, concentrating on the platform they were all standing on. I would show them. I'd jump over there and demand a belt. I focused hard, willing every cell of my body away from the space I was currently occupying. I dove deep inside, looking for any exit out of my current reality. Something tingled. I was close. A little more.\n\nNothing.\n\nI looked up, and Chausau was making them jump again. I couldn't hear what he said, but everyone stood still, listening carefully. I turned my back to them and started thinking about my stomach. Despite the nausea, I was starving. Right now I was so hungry, I'd even eat that slop Odran used to feed us. I tried to concentrate on the food hall. Maybe the hunger would help me. But I found nothing to grab on to. Why wouldn't they give me some sort of instruction? This was ridiculous.\n\nI sat firmly rooted in my self-pity and continued to watch the triumphs of the Hollow's newest recruits. Switzer seemed more confident than Gora, jumping much faster and even able to pick up an object from one platform while jumping to another. I saw him for only a split second before he appeared on the platform far to my right. Gora was a little slower. I supposed he was having trouble setting the coordinates in the belt fast enough. Maybe he wasn't that attuned to working with his softwire. I wanted to jump so bad that the pain was worse than the hole now growing in my stomach.\n\nAfter nearly a cycle, they all left, not a single one of them acknowledging me or even glancing in my direction. I curled up on the floor, hoping sleep would come quickly. I did try to jump once more, but it was useless. I didn't know what to do.\n\n\"Wake up,\" something shouted as I felt the weight of a boot push against my ribs.\n\nI opened my eyes to find Chausau standing above me.\n\n\"I can't do it,\" I mumbled, and curled back up.\n\nThis time Chausau kicked me. \"That's because you're weak.\"\n\n\"No, it's because no one has shown me how to do it. You show Switzer and Gora how to work their belts. Everyone is there helping them learn, but I'm supposed to figure it out myself. Why?\"\n\n\"You were given all of the skills to perform your task. In fact, you have already done it. No one can show you because you are the only one who can do it. Yet you sit here as if someone owes you something. As if it is our responsibility to take your hand and do it for you. You are pathetic, and you are lazy.\"\n\n\"Yeah, well, you're ugly.\"\n\nFive other Space Jumpers appeared on my platform. Each of them was carrying some sort of teal-colored piece of metal. The strain in their muscles told me that the pieces were heavy.\n\n\"On your knees, Jumper,\" Chausau ordered.\n\n\"What are you doing?\" I asked.\n\n\"On your knees!\" Chausau kicked me again, lifting me off the floor. I got on my knees. Two of the Space Jumpers moved in and placed their pieces of metal on either side of me. They pushed them together, slamming against my knees. I lifted my knees, and the pieces slid together perfectly, stopping at my waist, with barely enough room left for me. I could feel the rough iron against my legs, my hips, even my feet.\n\n\"Chausau, what are you doing?\"\n\n\"Put your arms out in front of you.\"\n\n\"Why?\"\n\n\"Do it!\" Chausau screamed in my face, and my arms shot forward, a survival reflex.\n\nThe other two Jumpers placed two more pieces of the puzzle around my chest, shoulders, and arms, locking me in the metal cocoon. The only part of me that was exposed was my head. I couldn't hold the weight of the thing, and I toppled forward, slamming against the platform. The sound of metal striking metal rang across the void.\n\n\"What are you doing, Chausau?\" I pleaded.\n\nI tried to look up, but the metal was too close to the base of my skull. All I could do was stare at their feet.\n\n\"You wanted help. This is how I will help you.\"\n\n\"By locking me up? I already can't get off this platform.\"\n\n\"You're not staying here,\" he whispered.\n\nI watched the feet move away from me. \"Where are you going?\" I screamed.\n\nThen I heard a clink, and then another. I felt myself lift off the platform. Something was pulling me into the air. When I cleared the platform, I could see the other Jumpers watching as I pulled away. Chausau was not there.\n\n\"Chausau, where are you?\"\n\n\"I'm right here,\" I heard him say from somewhere behind me. He must have been standing on me as I was pulled into the air.\n\n\"Please tell me what's going on.\"\n\n\"Your mind is your worst enemy, Softwire. The Nagools tried to bring you through your awakening their way, but now we will do it my way.\"\n\nThis is what Ketheria was talking about. What were they going to do me? I could hardly see the platform below me as the mist around me grew thicker. I could feel the cold moisture against my face, but then it was gone, as if we had slipped through some atmospheric cloud. I pushed against the metal cocoon, but there was no wiggle room. The metal fit as close as my own skin. But the more I pushed, the less I began to feel the metal. I couldn't move, yet I could no longer feel anything pushing against my skin. It was maddening.\n\nIn front of me I could see the plate metal wall, scarred with hooks and bare bolts. The space closed in, tapering as I rose higher and higher. Then I passed another metal cocoon bolted to the wall. It looked identical to the pieces that were placed around me. Was he going to mount me to the wall like some trophy? My metal mold swung toward the wall. Then I heard the sound of a small motor as I rotated up, my hands and knees now pointing toward the wall in front of me. To my left was another one of the metal molds. It was empty.\n\nMy cocoon hit the wall with a clank, and Chausau was now balancing himself on my arm. I heard more whirring and clicking as I was fastened in place on the wall.\n\n\"Please, Chausau, don't do this,\" I begged.\n\n\"A disciplined mind is a Space Jumper's best ally.\"\n\n\"I don't want to be a Space Jumper, Chausau, really I don't.\"\n\n\"Hush now. You will thank me when it is over.\"\n\nThen I felt a vibration at the back of my neck, and something began to crawl across my skull, then over my eyes, and then over my mouth. My lips brushed against the crimpled surface. It was more metal, but the sensation of something pressing against my mouth soon disappeared. Finally the thing sealed itself around my neck. I screamed out, \"No!\" and felt my own hot breath fall back against my skin. Chausau did not answer. All I could hear was the sound of my own breathing, and soon I was no longer able to feel the warmth of it pushing back against my face.\n\n\"Chausau!\" I screamed once more, and the material covering my face spilled into my open mouth. I clenched down, pushing whatever it was out of my mouth, but then the sensation of that slipped away as well. I didn't know if the stuff had worked its way down my throat or if I had been successful in pushing all of it out.\n\nMy mind scrambled for some solution, rattling in my skull, bouncing around in my head as it searched for some way out of this. I pushed outward with my mind, looking for any sort of computer device to interface with, but there was nothing. Even the interface for my arm was no longer available to me.\n\nI needed to move, but I felt paralyzed. I thought I was moving, but I couldn't feel it. I was panicking. _Calm down,_ I told myself. Think about what Ketheria said. This is their method. Chausau mentioned that the Nagools had tried with me, but they had failed. Failed at what? My ability to jump without a belt? What was it?\n\n_Relax. I can't!_\n\nMy mind was screaming again. I hated it. I wanted out, but I couldn't think of a single thing to do. I felt like I was going crazy. I wanted out of this contraption so bad, I would rather die than spend another parsec in this thing.\n\n_Then die._\n\n_What do you mean?_\n\n_Give in._\n\n_Give in to what?_\n\n_Just give in._\n\n_I can't!_ my mind screamed.\n\nI think I fell asleep. I'm not quite sure. Maybe my brain had simply shut down. When I awoke, the futility of my situation poured back into my metal mold, but I did not panic. I was surprisingly calm. I would not let it control me this time. I knew the only way out of here was to jump, and the only way to do that was to discipline my mind.\n\nI noticed that even the sound of my breathing was gone. I could see nothing, hear nothing, feel nothing, and smell nothing. Every part of me was completely deprived of stimuli, and I was left with only myself to talk to. _This is how Vairocina must have felt,_ I thought. _Now I understand why she searched so desperately to find a real body._ But _I_ was not my body. I could still think. In fact, I was talking. Talking to myself. What was this sense of self that was left to wander about my head? As if I was talking to someone else entirely. It had no physical form, yet it demanded attention. It argued with me, influenced me, praised me, and even berated me, more often than not. But where did it come from? Who controlled it? Me?\n\n\"Hello?\" I said, as if to call out that part of me, to identify it as a separate entity.\n\nIt did not respond. Or did I not respond?\n\nDespite my lack of contact with the physical world, \"I\" still existed. \"I\" existed independently of my race, my job, my clothes, even what others thought about me. For so long, I had defined myself by those things, but here, now, they meant nothing. They served no purpose whatsoever. Suddenly, I craved to see something, feel something, or hear something. I needed to know that I still existed.\n\n_But why?_\n\nMaybe _I_ didn't even exist. For all I knew, I could be dead. Was this what death felt like? I hoped not. But then it struck me. I realized something about the \"I\" alone with me in this mold. I realized that this was the person who loved Max, who loved Ketheria and Theodore and even the Keepers. Suddenly, despite the lack of stimuli, I felt a flood of love and warmth rush through my body. I did have a body! If I could have felt it, I know I would have felt tears against my skin. I imagined my tears rolling down my cheeks and dropping off my face. In my mind, I reached down to the puddle of tears with my fingertips and raised them to my mouth. The salty teardrops touched my tongue, and I slept again.\n\nI awoke to a blinding white light. I closed my eyes, but the light persisted, boring through to the back of my skull. Was someone doing this, or was I hallucinating the light? I tried to will the light away, but it remained. I even imagined myself lying at the bottom of a lightless tunnel, but still the white light pierced its depths. Who was doing this? _Please stop,_ I begged them.\n\n\"JT? Wake up. JT?\"\n\n\"Hello?\" I croaked.\n\nThe light was still there, but softer somehow, as if someone had managed to gray the edges.\n\n\"Are you all right?\"\n\n\"Who is that?\"\n\nIt sounded like Ketheria, or was it Brine Amar? The voice seemed to change pitch every time I recognized an inflection or the speech pattern.\n\n\"You must help yourself if you are going to leave here,\" the voice said. It was Brine Amar.\n\n\"Why are you doing this to me? I don't know how to jump without a belt. I can only do it around Ketheria. I think Quirin made it that way.\"\n\n\"Yes, you can, but you refuse to release your physical body.\" The voice did not belong to Brine Amar. It was Ketheria.\n\n\"Who is this?\" I asked again.\n\n\"Your mind is sick, and the disease is fear.\"\n\n\"I'm not afraid of anything.\"\n\n\"You are afraid of yourself.\"\n\nMy mind tuned out, and I turned off.\n\nWhen I woke next, I was standing on the observation deck of the _Renaissance._ The ship was orbiting a dying star, and the golden light burned away the edges of everything I saw. I shielded my eyes to look at the star. _We're too close,_ I thought. _The ship will burn up._\n\n\"Mother!\" I called out, but the computer did not answer.\n\nOf course not. The _Renaissance_ was gone. Dismantled and sold for parts the moment after we arrived on the Rings of Orbis, almost four rotations ago.\n\nWas this a dream?\n\n\"Hello, JT.\"\n\nI spun around. It was Max!\n\n\"Max!\"\n\nThe soft cream-colored robe she wore on Orbis 4 sparkled in the intense sunlight. The edge of her hair burned blond. She was beautiful. I rushed toward her and took her hands in mine.\n\n\"Max, I'm so sorry. I should never have left you. Can you forgive me? Please. I love you so much. I want it to be just like you said with Ketheria and Theodore. Together. We can have that!\"\n\nMax shook her head gently and held her finger to my lips. I breathed in her scent, she was so close. My mind filled with flowers and sunlight and fresh running water. I wanted to hold her.\n\n\"You've done the right thing,\" she whispered. \"I see the importance of the task that has been placed upon you. I was so selfish to want something else. You have greatness in your future. I want you to have that. You deserve it.\"\n\n\"But \u2014\"\n\n\"You need to get out of here, JT,\" Max insisted. \"You have to jump. We need you. It is time.\"\n\n\"Time for what?\"\n\n\"Time for you.\"\n\n\"We can stay here, Max. Just like you wanted it. We can live here on the _Renaissance_ again. We'll travel the stars, just you and me. We don't have to go.\"\n\nI pulled her close to me. I could feel her warmth under her robe as she pressed against me. Was this real? I pushed my lips against hers. I could taste her.\n\n\"This _is_ real, Max. I know it. Can't you feel it? I don't know how I did it, but we're together again on the _Renaissance._ \"\n\nMax pulled away, gently, as if not to make a statement by her actions. She reached for my hand and led me away from the observation deck. Below deck, I took the lead, remembering each corridor, stairwell, and quarters. I had forgotten how much I missed the _Renaissance._ I pulled Max into the chow synth and saw that Mother had created all of our favorite foods, but why hadn't the computer answered me?\n\n\"Max, Mother is not responding.\"\n\nBut Max was gone. My hand was still outstretched, but it was empty.\n\n\"Max?\"\n\n\"Hey, JT.\"\n\nI turned toward the voice, a new voice. I thought I recognized it but wasn't quite sure. Out of the light stepped a young man, maybe thirty Earth-years old. I recognized something in his face, and his hair was still the same unmanageable brown nest I knew.\n\n\"Theodore?\" I asked the man.\n\n\"You look well, JT.\"\n\n\"You look . . . older. Why are you so much older than me? We're supposed to be the exact same age.\"\n\n\"We still are.\"\n\nI looked at my hands. They looked like they always did, and they also looked liked Theodore's \u2014 older. \"I don't understand.\"\n\n\"That's part of the problem, JT. Everything is not meant to be understood. It is impossible for you to comprehend and control every aspect of the universe. I know you always try, but sometimes you just have to accept what is. I don't know if you noticed, but I gave up counting things. That had been my way of trying to control situations. You have to trust that no matter what comes at you, you will manage. You have to trust yourself, JT.\"\n\nI felt a tinge of jealousy. \"How do you know?\" I said. \"If anyone was always afraid, it was you. Do you know how many times I wished you had spoken up to Switzer? To defend yourself? You were the one who was afraid, Theodore.\"\n\n\"But I'm not anymore. This is not a competition. You can see that, can't you? Now it's your turn. I know you can do it, JT. Just let go.\"\n\n\"Let go of what?\"\n\n\"Your mind,\" said a voice from behind me.\n\nI spun around and saw Ketheria sitting on my sleeper. It was just like the first time we tried to leave the _Renaissance._ A huge sack was slumped at her feet, and in her hands she held the crude locket containing the image of our parents.\n\n\"Your entire image of yourself is based on a lie,\" she said, holding up the makeshift locket. \"That was their only mistake. They created a false dream for you \u2014 a dream of a reality that never existed. And you are holding on to that dream in the same manner the knudniks do on the Rings of Orbis. You believe that if you allow your own domestication at the hands of the Citizens, you, too, will share in all their wealth and glory. You dream of parents who left Earth for a new life, parents who never existed. You dream of a life on the Rings of Orbis with me, Max, and Theodore living free, but no one on the rings is ever free. It has all been a lie, JT.\"\n\nKetheria squeezed the locket in her hand. When she opened it again, the locket was gone and in its place was a handful of space dust. Ketheria turned her palm over, and the shimmering particles drifted across the cabin.\n\n\"Now it is time to wake up from _that_ dream and create your own dream. Humans are destined to play a marvelous role in this universe. A role that awakens oppressed cultures from their own dreams, not by war but through love and kindness. We are all that is left. You and I are standing at the event horizon, JT. You and I will lead them into the light. But you must act now. I can't wait any longer for you, JT. It is time. You must trust me. You must trust yourself.\"\n\nKetheria's eyes burned with the glow. She was standing now, reaching out to me.\n\n\"Now, JT! There is no more time!\"\n\nHer mouth hung open, ripping at the edges and consuming all the light in the room as she screamed. The sound tore me away from the _Renaissance,_ and I reached out, trying to hold on, but I was swallowed by the darkness bearing down on me.\n\nI stood up. I was alone.\n\nI gazed out from the center platform of the training area. An alarm ripped through the silence. _Was I out of the cocoon? Was this real?_ I looked up to see if I could spot the metal mold, but it was too far. If I had jumped, then I could do it again. Ketheria needed me. Whatever had happened inside that thing, I knew in my heart that Ketheria needed me.\n\nWithout thinking about my body and with complete trust that I could do it, I focused on the platform across the void. It was sort of like the pushing I did when I entered a computer. I accepted the fact that I would succeed. I did not drag my body with me, or anything else, for that matter. I simply placed the \"I\" inside of me on the platform, as light from the room began to pull into my eyes. I breathed deeply, welcoming the smell of stinky feet.\n\nI stood on the platform across the void and looked back at where I had just been. I expected a rush of pride to swell up inside me, but none came. There was no need. I smiled. The alarm, which was now sounding at regular intervals, told me that something was wrong. I pictured my room in the Hollow and willed myself there.\n\n\"Where did you come from?\" Switzer said as I appeared in front of him.\n\n\"I jumped here,\" I told him.\n\nSwitzer's eyes ignited, and a smile stretched across his face. He leaped toward me, arms outstretched, ready to thump me on my back, but he stopped.\n\n\"You still pissed at me?\" he asked, his old mask slipping over his face.\n\n\"No,\" I said. \"That was then. There's nothing I can do about it, and you've worked hard to gain my trust. Just like a friend would do.\"\n\nSwitzer smiled.\n\n\"What's the alarm about?\" I asked.\n\n\"I was about to go look. It just started.\"\n\n\"I'll go with you,\" I said.\n\nBut before we could leave, the door to the room disappeared and one of Switzer's friends rushed in. \"The Scion is gone!\"\n\n\"What do you mean?\" I cried.\n\n\"She's gone! Someone took her.\"\n\nI pushed in front of him. \"She can't be gone. She's stained. Track the staining.\"\n\n\"We have. There's nothing,\" he replied. \"They're sending a team to look for her.\"\n\nThe Space Jumper turned to leave, and I was close behind.\n\n\"Where are you going?\" Switzer cried.\n\n\"I know how to find her,\" I said.\n\n\"I'm coming with you.\"\n\nI ran all the way to Charlie's little surveillance room. While I thumped on the grille to his hideout, I cursed at myself, thinking I could simply have jumped.\n\n\"Charlie! It's me!\"\n\n\"What's a Honock going to do?\" Switzer asked.\n\n\"Wait.\"\n\nCharlie came to the grill and spotted Switzer. One look and he turned away.\n\n\"Charlie, it's all right. He's with me. He won't hurt you.\"\n\n\"I didn't hurt you the first time,\" Switzer added.\n\n\"Charlie, Ketheria's in trouble, We have to find her.\"\n\nCharlie peeked through the grate. After an excruciatingly long moment, he hoisted the gate off it moorings. I slipped past Charlie and dashed to the monitors. They were already on. I scoured the screens for any sign of Ketheria, but there was nothing. In fact, there was no sign of Max or Theodore, either.\n\n\"They're all gone,\" Charlie informed me.\n\n\"What's this?\" Switzer asked, looking around the room.\n\n\"There's no time to explain. We have to get assigned to that search party. Meet me back at our room.\" I turned to Charlie. \"Keep looking for them, will you? Search everywhere you can. There must be a sign of one of them somewhere.\"\n\nThen I jumped to Quirin's room. I didn't even shoot for outside; I went straight in.\n\n\"Just because you've accepted your skill does not mean you should abandon your manners,\" he said.\n\n\"I want to be on that search party you are sending for my sister.\"\n\n\"Impossible,\" he replied. \"Tensions are too delicate on the rings. I cannot have a Space Jumper moving about freely. It will tip the scales in the Council's favor. We cannot have a war.\"\n\n\"But I have to go! I am the Tonat, aren't I?\"\n\n\"I have sent four experienced Jumpers who will find the Scion without anyone ever seeing them. You will finish with your training.\"\n\n\"How will they find her? The staining doesn't work anymore,\" I told him.\n\n\"That is impossible. The staining is foolproof. I invented it. This is merely a glitch in the system. The staining will locate her much more easily than you can.\"\n\n\"You don't know about the Scion. She can do things you don't know about.\"\n\n\"There is no argument that will change my mind. Go back to your quarters. This will be over shortly.\"\n\n\"Quirin, please \u2014\"\n\n\"Go!\"\n\nI left his room, certain he was wrong. Ketheria was trying to tell me something. I knew it. When I returned to my room, I found Switzer lying on his bunk.\n\n\"Get up,\" I ordered him.\n\n\"Where are we going now?\" he asked, sitting up.\n\n\"We're going home,\" I informed him.\n\n\"Have fun.\" Switzer flopped back onto his sleeper.\n\nI slipped through space, compressing the distance between us. The room blurred, and in the exact same moment, I was standing next to Switzer. Using my right arm, I grabbed him by his uniform and dragged him off his sleeper. With one clean jerk, I hoisted him high in the air, his shocked expression staring down at his feet, which were nowhere near the ground now.\n\n\"You're coming with me, and you're bringing Charlie. I don't have a belt, so I can't jump with him. It could kill him. You're bringing him back for me. You owe me that, and you owe him at least that. This is not the life he was supposed to live. Charlie doesn't deserve to live here any more than you deserved to live in that hole I rescued you from.\"\n\n\"Fine, enough with the melodrama. Put me down.\"\n\nI let Switzer fall to the floor but he jumped in midair, reappearing soundly on his feet.\n\n\"Nice,\" I said, nodding at his acrobats. \"Now let's get Charlie. I need to get to Max and Theodore.\"\n\n#\n\n\"I don't want to go,\" Charlie snapped after I told him what we were doing.\n\n\"Suits me,\" Switzer said, turning back down the hall.\n\n\"Wait,\" I cried. \"Charlie, look at the screens. Ketheria's gone. Max and Theodore will know where she is. We have to find them. I want you to come with me.\"\n\nCharlie looked at the screens but shook his head. While he stared, I slipped into the computer device at the back of his head. With the precision of a laser drill, I adjusted the controls for memory, then pulled out.\n\n\"What did you do?\" Switzer grumbled. \"He's crying now.\"\n\nCharlie was staring at the screens and weeping. Maybe I should have adjusted the controls for emotions when I was in there, but I didn't have time.\n\n\"Charlie? You all right?\" I asked him.\n\nHe turned and said, \"Let's go find them.\"\n\nAs Charlie stood up, Switzer said, \"Wait. I can't jump without the coordinates. I'm not like you. All you have to do is bring up a memory. I'm sorry, but I can't take him to Orbis 4.\"\n\n\"Charlie, that surveillance system we watch \u2014 there must be some sort of coordinates link with it, right? I mean, if they're spying on people, they must know where they're looking. Can you pull up the coordinates off one of those cameras and give them to Switzer?\"\n\nCharlie parked himself in front of the O-dats without speaking. After a few moments, a stream of digits rolled across the screens.\n\n\"I only need one,\" Switzer scoffed.\n\nCharlie tapped on one screen, and a single string of symbols flashed on the O-dat.\n\n\"That's JT's room,\" he said.\n\n\"Grab it,\" I told Switzer.\n\nSwitzer interfaced with the O-dat and uplinked the info Charlie had provided. \"All set,\" he replied.\n\n\"You go first,\" I told him.\n\n\"Why, you don't trust me?\"\n\n\"Just go first. I'll see you in my room.\"\n\n\"Come on, big guy,\" he said to Charlie. \"Let's hope you stay in one piece.\"\n\n\"Don't say that!\" I said.\n\nBut they were gone. The air rippled the moment Switzer latched on to Charlie while any light reflecting off them broke apart and scattered across the room. I envisioned my room on Orbis 4 and jumped right behind them.\n\nWithin the same breath, I was standing next to my old sleeper with Switzer and Charlie, trying to rub the smell of feet out of my nose.\n\n\"I know you were messy on the _Renaissance,_ \" Switzer said, looking around my room, \"but this is ridiculous.\"\n\nSwitzer nudged an errant pot with his toe, a pot I had never seen before. It rolled toward my sleeper, the lid of which hung at a reckless angle, as if someone had tried to rip the sleeper from its moorings. Someone or something had destroyed my room.\n\n\"I didn't leave my room like this,\" I said. \"Something is wrong. Hurry, we need to find them.\"\n\nOutside my room, the walls were marred with charcoal streaks, results of a plasma rifle. I maneuvered around an overturned bench and noticed that a chunk of the wall was completely missing. Switzer stepped over a discarded wall panel and crushed the remains of an uprooted plant.\n\n\"What happened here?\" I asked. \"And where is everyone?\"\n\n\"Someone was either looking for something or just felt like trashing the place,\" Switzer answered.\n\n\"I saw it,\" Charlie murmured. \"It looked like a bit of both.\"\n\n\"Max's room is right up here,\" I told them.\n\nMy stomach flipped once when I said her name and then settled at an uncomfortable angle when I paused outside her room. How would she react when she saw me? How would I react? I was so confident about finding them, but the thought of speaking to Max made me uncertain. Switzer reached past me and thumped on the door. No one answered. I pushed into the door's control panel and opened it.\n\n\"Max! Max, you here?\" I called toward the bathroom without entering her room, but there was no sign of her.\n\nSwitzer whistled. \"Wow, and I thought you were the messy one.\"\n\n\"Be quiet. It's obvious someone has trashed the place. Let's check Theodore's room.\"\n\n\"He's not there, either,\" someone said behind me.\n\nI turned to find Queykay, flanked by four armed guards, standing at the top of the corridor. The faceless creatures, protected by long, chrome chest plates, took cautious positions around Queykay, as if ready for a fight. I had no intention of giving them one.\n\n\"They're all gone, including your sister,\" he said.\n\n\"Where are they?\" I demanded to know.\n\n\"I was hoping that maybe you knew where they were.\"\n\n\"Why? Because it doesn't look good that the Trading Council doesn't know where the Scion is? Are you afraid of her power?\"\n\n\"The Trading Council has far more power than you will ever have.\"\n\nThe four armed guards took a step toward me, their plasma rifles readied in their grips.\n\n\"Actually, I do know where one of them is,\" he added as an afterthought. \"The one you call Theodore has been arrested for treason. I personally stopped his plot to overthrow the Council.\"\n\n\"That's not true!\" I cried, and the guards raised their weapons at me.\n\n\"Then what _is_ true?\" he asked.\n\nWhat had Theodore gotten himself into? Where were the others?\n\n\"Do you have something to tell us?\" Queykay said.\n\n\"I don't have to tell you anything. You have no authority over me.\"\n\n\"That's where you are wrong. Things have changed since you left, Softwire. The Trading Council has invoked certain privileges, and I am here to enforce them. For one, strolling about the ring with a convicted wormhole pirate is definitely grounds for treason charges.\" Queykay pointed a long bony finger at Switzer. \"Seize them!\" he ordered.\n\n\"Stop!\" Hach cried as he rushed into the corridor just behind the security guards. \"What are you doing, Queykay? This is still _my_ house,\" he insisted.\n\n\"The Trading Council has jurisdiction over it now.\"\n\n\"That's ridiculous!\"\n\n\"The Chancellor has suspended all civil rights, but how could you know that, with the time you spend on Ki and Ta tending to your privileged mining rights?\" Queykay turned back to us. \"Arrest the enemy,\" Queykay growled, pointing at me. \"He is a Space Jumper. The Council has clearly \u2014\"\n\n\"The Council does nothing _clearly,_ \" Hach interrupted. He pointed at the armed guards. \"Who are these men?\"\n\n\"These are the Trading Council's Preservation Forces.\"\n\n\"Preservation Forces? I know nothing of this.\"\n\n\"The Council has enlisted their services, and I have been given authority to use them. I signed the order myself. If you continue to interfere, then I will have them arrest you as well.\"\n\n\"I will not abide by this foolish rule!\" Hach cried, and reached inside his long jacket.\n\nAs Hach stepped back to remove his weapon, the four guards turned their focus on him. \"Now!\" Switzer hissed, and we both jumped at the same time.\n\nI moved across the room and refocused just behind the guard on the far right of Queykay. I stepped into him, and knocked his rifle away with my strong arm. Switzer struggled with the two guards closest to Queykay, while Charlie lunged for the remaining one. I watched Charlie bend the guard's gun in half as if it were made of rubber and then fling the guard thirty meters over his head and down the hall. The guard slid along the floor and hit the wall with a thud. He did not move again.\n\nFrom the corner of my eye, I watched Queykay fling back his thick cloak as he squared off against Hach. Queykay's brood, the disgusting creatures that fed off his body, launched at Hach the moment the light hit them. The squirrely creatures were at least a foot long now, and more than two dozen of them landed on Hach. He dropped his weapon, clawing at the creatures as they bit into his neck and face.\n\n\"No!\" I cried.\n\nWhile I watched Hach writhing under the sickening creatures, Queykay thrust a Zinovian Talon into Hach's ribs. I grabbed Queykay's arm, but when he pulled back, I saw the empty cartridge from the talon dangling from Queykay's hand. His brood must have tasted the poison now running through Hach's veins; they scurried back to their father like maggots scrambling over rotten meat. I ran to Hach's side.\n\n\"Don't fight it,\" Queykay whispered to Hach. \"It will be over quickly. There is no room on the rings for sympathizers. The Keepers have broken their agreement, and they must be punished. Their time on the Rings of Orbis is finished, with or without the Scion.\"\n\nI held Hach's head as Queykay casually reloaded his talon.\n\n\"I'm sorry,\" Hach breathed. \"I thought I was doing good. I thought you and the Scion would be safe here. I failed.\"\n\n\"Do you know where they are?\" I whispered.\n\n\"They disappeared after Theodore was taken. I think Ketheria knew. They are nowhere to be found. I should have been here.\" Hach's voice was almost a whisper.\n\n\"What did Ketheria know? Hach!\" His eyes began to roll back into his head. I turned to Queykay. \"Help him!\" I cried, but he only smiled.\n\n\"Your Guarantor was given lucrative mining rights to new fields on the crystal moons in exchange to protect you. He made this deal with the Keepers. I have no intention of helping him.\"\n\nI looked at Hach, his face and neck a bloody mess. \"Is this true?\"\n\nHe nodded.\n\n\"Why?\"\n\nHach swallowed hard but did not reply.\n\n\"That's a question for the Keepers,\" Queykay hissed. \"They have broken the treaty, and now we will take what is ours.\"\n\nSwitzer stepped toward Queykay as he removed a small communication device from his robe and raised it to his mouth. \"I have the Tonat,\" he said into the device.\n\n\"Wow, for a Trading Council member, you are one dumb alien!\" Switzer cried. Then he jumped across the room and refocused next to Charlie. \"We're Space Jumpers!\"\n\nAs if on cue, I jumped at the same time Switzer did. I couldn't look at Hach. I knew he was dying. I didn't want to think about the deal he had made with the Keepers. I liked Hach and wanted to leave it that way, whether he was dead or alive.\n\nI knew where Switzer was going. He only had the coordinates for one place on Orbis 4, and I jumped there. I wished it was a little farther away from Queykay, but it would have to do. I refocused back in my room, with Switzer and Charlie at my side.\n\n\"They made some pretty good improvements to you,\" Switzer said to Charlie, lifting his right arm and inspecting it like part of a ship's engine.\n\nCharlie pulled his arm away. \"I wish I could remember you,\" he said.\n\n\"No, you don't,\" Switzer replied.\n\n\"What do we do now?\" I asked.\n\n\"I don't know; this is your plan.\"\n\n\"We need to get out of here, but you can't jump without coordinates. We have to walk out.\"\n\n\"That's not going to happen without some firepower.\"\n\n\"Well, whatever we do, we have to do it quickly. Vairocina?\"\n\n\"JT? You're back!\" her voice rang inside my head. \"I'm so pleased to hear your voice.\"\n\n\"Well, it's not a scheduled visit, that's for sure, but listen, I'm in trouble. I need some help with burak coordinates for Orbis 4. Is there any way to uplink them from my room?\"\n\n\"Why don't you \u2014?\"\n\n\"It's not for me. It's for Switzer.\"\n\n\"Oh.\"\n\n\"Vairocina?\"\n\n\"Hurry up,\" Switzer urged.\n\n\"The only O-dat I see working near you is located inside Ketheria's chambers,\" she replied.\n\n\"It's not far,\" I told Switzer.\n\n\"JT, someone has placed a query on your whereabouts within the central computer. They are commencing a trace at this very moment,\" she informed me.\n\nI remembered what they did to Ganook before I left. I assumed they had placed a trace on him and then sent some sort of explosive to finish him off. Would Queykay try to kill me?\n\n\"Can you do anything to slow them down?\" I asked her.\n\n\"It will not be permanent.\"\n\n\"Do what you can,\" I told her. \"And get ready with the coordinates for Switzer.\" I turned to Switzer and Charlie. \"We have to move now. If I start turning blue, move away from me quickly.\"\n\n\"What does that mean?\" Switzer asked.\n\n\"Just do it.\"\n\nKetheria's room was not far away, but I had no idea where Queykay and his goons had gone. I was certain that he had initiated the trace, but I was also sure he didn't think I would still be in the building. I knew that once he discovered I was still there, I would have very little time to get the coordinates into Switzer's belt. I took the lead and slipped through the debris, careful not to attract attention. Inside the main chamber, I headed straight to Ketheria's room.\n\n\"I don't remember her being the tidy one,\" Switzer said as all three of us filled her room. Charlie went to a table near Ketheria's bed, where some of her personal things were still arranged neatly, as if she would be right back. Charlie picked up a hairbrush and cradled it in his clumsy hands.\n\n\"You miss her, don't you?\" I whispered, and he only smiled.\n\n\"Can we get on with this?\" Switzer grunted.\n\n\"Vairocina, you ready?\"\n\n\"Almost,\" she replied. \"There is an O-dat in her room. Have Switzer interface with the device.\"\n\n\"Over there,\" I told Switzer, pointing to the O-dat screen on the table.\n\n\"Tell him that in the system's local memory, there is a file marked _charts._ It's big, but I just grabbed everything I had.\"\n\n\"Look for something called charts,\" I called out to him.\n\n\"Got it!\" he replied.\n\nCharlie returned the brush to its original place and looked around the room. \"No one would hurt Ketheria,\" he said.\n\n\"That's where you are wrong,\" Queykay said, stepping into the room, the Talon already drawn.\n\n\"Switzer?\"\n\n\"Not yet,\" he croaked, and I nudged Charlie to move toward him.\n\n\"It was simple. You did not want to be the Tonat. With you out of the way, the Council could control the Scion. We were in charge. Everyone feared she was not safe; we proved that. The Council would protect her, _for their Citizens._ Everything would have run smoothly, but you just couldn't play along, could you?\"\n\n\"You staged her assassination attempt, didn't you? She read your mind. That's why she's not here. You can't be trusted.\"\n\n\"I'll make you a deal,\" Queykay continued, slinking into Ketheria's room. I could see the Preservation Forces moving in the shadows of the other room and Queykay's army of assassins rippling at the collar of his cloak. \"You help me find her, and I'll spare your life.\"\n\n\"What about _her_ life?\" I said.\n\n\"The Scion will be very useful to the new Council. Of course we would protect her.\"\n\n\"Don't believe him,\" Switzer groaned. His voice was strange. I looked over and saw that he was swaying slightly.\n\n\"Switzer?\"\n\nHis knees buckled, and Queykay raised his talon. I jumped across the room and refocused right under the weapon. I sprang straight up and knocked the weapon from Queykay's grip as it discharged, the poison spear now safely lodged in the ceiling.\n\n\"You stupid \u2014\"\n\nBut before he could finish the insult, I landed on Queykay and delivered a thunderclap to his neck.\n\n\"You really are dumb,\" I said as Queykay crumpled underneath me. I saw the Preservation Forces scrambling in the other room, so I pushed into Ketheria's door and jammed the energy field, preventing the others from entering. Two of Queykay's brood had escaped and sunk their teeth into my robotic arm. I already had the pain turned off. I felt nothing, but more were crawling toward my feet.\n\n\"Wrong arm, kids,\" I said, and flicked them off. They screamed as they splattered against the wall. The others hesitated between their father and me before choosing to retreat. They were not very brave now that he was unconscious.\n\n\"Switzer!\" I turned and saw him slipping to his knees as Charlie pulled on his right arm.\n\n\"Vairocina, what did you give him?\"\n\n\"I told you it was a lot of information. I didn't know what exactly to leave out or if you would have another chance to do this again. He will be fine in a moment.\"\n\n\"We don't have a moment!\"\n\n\"Switzer! Switzer!\" I yelled. I turned and saw the guards hacking their way through the wall, bypassing the energy field all together. A chunk of the wall landed on the floor. They were almost through.\n\n\"Uh . . . too much . . .\" he groaned, his eyes rolling back in his head. I remembered when this happened to Theodore; he was out for a whole cycle.\n\n\"Switzer. Do you have it all? Come on. We have to go!\"\n\nCharlie helped me get Switzer to his feet. He was barely conscious. There was no way he could jump, let alone take Charlie with him.\n\nThe forces were almost through the wall.\n\n\"Take his belt,\" Charlie said.\n\n\"What?\"\n\n\"Take his belt and jump with all three of us.\"\n\n\"But I've never used a belt,\" I told him. \"I don't know \u2014\"\n\n\"The belt is only there to create the pathway for me and him. You do not even need it.\"\n\n\"But what if it doesn't work. What if I \u2014\"\n\n\"You'll do fine. You're a Space Jumper.\"\n\nI unclipped the belt from Switzer's waist. It was warm. I slipped it around my own and let it hang down over my right thigh. There was no time to adjust it. I held up the other side with my left arm.\n\n\"Grab him and get close to me,\" I told Charlie. \"And if this doesn't work, I'm sorry.\"\n\nI pictured the city of Murat in my head and interfaced with the belt. I activated the only button on the belt that I had ever seen before \u2014 the button that I thought had once killed Switzer, back on Orbis 2. It felt like forever before the smell of stinky feet invaded my senses.\n\n#\n\nI half expected to find Max waiting for me. I half expected to find Charlie in little pieces at my feet and Switzer lost in some corner of the universe, but none of that came true, especially the Max part. Before I jumped, I had pictured the concert area she had shown me on Murat, recalling my amazement that she had remembered my interest in music and thinking it was one of the finest moments of my life. Standing on the stone steps with only Charlie and Switzer at my side, I looked up as the ring unrolled its shadow across the city.\n\n\"Where are you, Max?\" I whispered.\n\n\"We'll find her,\" Charlie replied.\n\n\"Vairocina?\" I called out.\n\n\"You're safe!\" she cried.\n\n\"You have to tell me if anyone puts another trace on me, all right? And don't worry about Switzer. According to the central computer, he's dead, but you need to give me as much time as you can. I can't risk what they did to Ganook.\"\n\n\"Of course,\" she said.\n\n\"Can you find any sign of Theodore?\"\n\n\"No, I've been looking. I cannot trace his staining unless a Citizen initiates it, and can find no record of this happening. The Keepers may have done it, but I have not been able to breach their security.\"\n\n\"Then you have to get ahold of Theylor for me. I need to meet with him, but be very careful. I don't know who's on what side anymore. After you establish a meeting place, check to see if a trace is placed on me. If there is, then we'll know what side Theylor is on.\"\n\n\"Where do you want to meet him?\"\n\n\"Across from the Center for Relief and Assistance. I think I might know where they are holding Theodore.\"\n\nWhile we waited for Theylor, Vairocina discovered two different traces placed on me.\n\n\"Can you trust him?\" Switzer said.\n\n\"I want to,\" I replied, refusing to believe that it was Theylor who had initiated the trace. \"We need to keep moving.\"\n\nThe more I jumped around Murat, the more obvious it seemed that war was coming to the Rings of Orbis. Vairocina had given Switzer plenty of coordinates for Murat, and this enabled us to move freely. Each time we jumped, we found barricaded streets, closed trading chambers, and barely a person to be seen. The three of us stuck out everywhere we jumped, so we glued ourselves to the shadows.\n\nWhen Vairocina informed me that Theylor was waiting for us, I jumped once more to shake off anyone who might be tracing me. Switzer was walking on his own now, but not saying too much except to warn me.\n\n\"I don't trust those two-headed space monkeys,\" he hissed.\n\n\"I trust Theylor,\" I reassured him.\n\nAcross from the Keepers' aid center, I saw Theylor standing near one of those sleeper arches that Theodore had used. I watched him open a capsule and place something inside it. He closed it and slipped away.\n\n\"I'll get it,\" Charlie said. \"You stay here.\"\n\nTheylor was being careful. That worried me. How bad were things now? Charlie returned with a tap and handed it to me. I pushed in and grabbed Theylor's message.\n\n\"Wait here,\" I told them, and stepped out into the open courtyard. I felt naked.\n\nOn the tap were simple instructions, yet they were odd just the same. Theylor instructed me to walk across the stone plaza and jump when I reached the center. I was supposed to jump to the darkened alley directly across from the center, less than thirty meters away.\n\n\"Good,\" Theylor said when I had refocused in the shadow of the empty alley. \"If anyone was following you, they would assume you jumped far from here. Follow me.\"\n\nThe Keeper turned away, his purple robe brushing against my leg. Another fifty meters down the alley and the Keeper pushed opened the dull-looking door of a lifeless building. Inside, the air felt bitter, as if trapped in a long-sealed metal container.\n\n\"How are you?\" he asked, and offered me a metal crate for a seat.\n\n\"I'm good, Theylor, but I can't say the same for the Rings of Orbis.\"\n\n\"No, you are right. It seems the Council wants a war and the Descendants of Light are willing to oblige.\"\n\n\"Drapling?\"\n\nTheylor removed a small light source from the depths of his robe and placed it on the floor. The blue light exposed the veins glowing under his skin like circuits in a computer. \"They feel empowered by the presence of the Scion. They believe the Ancients will return now,\" he confessed.\n\n\"Will they?\"\n\n\"It is foolish to believe that the Ancients are still alive, Johnny. This is the reason we have worked so hard to bring you and your sister to the Rings of Orbis. Humans were the last chance for this universe. The Ancients sacrificed everything when they found your Earth, so isolated from everything else in the galaxy. Yet they feared that humans were too far along in their evolution to ever seed a Scion. To solve this, they moved backward through time, to find the precise moment to best alter the human race and seed your fate. It was almost ninety thousand rotations ago that this journey began. You were the very last component of the intricate project. But what the Ancients did is something that breaks all rules of physics. You cannot go backward in time without destroying what you leave; it is a one-way journey. However, the Ancients knew that unless they did so, someone like your sister could never be born.\n\n\"This was their sacrifice, and they have succeeded. The Scion has almost completed her enlightenment. We are so close to fulfilling the dream of the Ancients. Ketheria will have the power to enlighten every one of us and connect us all to the Source. It is our only defense against the Knull.\"\n\n\"How can she do this from the rings?\"\n\nTheylor did not answer.\n\n\"But she's gone now. What's going to happen?\" I pressed.\n\nTheylor shook his head. I was suddenly aware of how familiar he was now with Earth gestures.\n\n\"Why can't you find her?\" I said. \"That's why you agreed to the staining in the first place, isn't it? How else could you keep track of her? You knew back then.\"\n\n\"We do not understand how she's doing it. Drapling is livid. Somehow she has cloaked her staining. She is nowhere to be found. In our defense, we had never stained a Scion before.\"\n\n\"You never stained my sister before,\" I reminded him.\n\n\"You must find her and bring her to us.\"\n\n\"I don't think she'll come.\"\n\n\"Why do you say that?\"\n\n\"Because if she trusted you, she would have gone to you already.\"\n\n\"It is the Descendants of Light who cannot be trusted. They want to _use_ the Scion just as the Council would use a weapon. They believe she can unite the knudniks and the new Citizens against the Council and the First Families. The DOL have never forgotten the humiliation and loss caused by the War of Ten Thousand Rotations. They are convinced that the Scion can bring them revenge.\"\n\n\"Can she?\"\n\n\"She is the Scion. Of course she can.\"\n\n\"I have no idea where she is,\" I told him.\n\nBoth of Theylor's heads stared at me, each one creased with anxiety.\n\n\"But \u2014\"\n\nI interrupted him. \"Do you know where Theodore is? Is he where they held Switzer?\" If anyone knew Ketheria's whereabouts, it was Theodore. I was certain of it.\n\nTheylor shook his head and said, \"We have been trying, all of us.\"\n\n\"Trying what?\" I asked.\n\n\"To keep him alive. The Trading Council has charged him with treason. The penalty is death, even for a Citizen.\"\n\n\"You have to stop them!\"\n\n\"The Council has convinced the Citizens that unless the Scion is controlled, she is a threat to their well-being. The Citizens have granted the Council control over every aspect of life on the rings, and they have banned the Keepers from participating in any decisions. This in itself is grounds for war, but the Trust does not want to attack until the Scion is located. I am afraid that Theodore, as he has been so close to you and Ketheria, is merely being used as a pawn in all of this. They are using him as an example of the Council's ability to deal with the growing rebellion of knudniks and to convince the Citizens that they can use force over the Scion's power.\"\n\n\"How can I get to him?\"\n\n\"I am afraid it's impossible. He is guarded more carefully than the Ancients' Treasure.\"\n\n\"I already got through that defense,\" I reminded him.\n\nAs I feared, Theodore was being held in the same facility that Switzer had called home for so many phases. Theylor could not provide entry, as the Keepers had been banned from entering the holding area, and he was convinced that the Trading Council would carry out Theodore's sentence, with or without a war.\n\nI first thought about simply jumping into Theodore's holding cell, but I didn't know which one it was, nor did Theylor. In fact, he believed they were using Theodore as bait to lure me into a trap. But if they were, why wouldn't they simply declare his location to me?\n\n\"Oh, this is ridiculous,\" Switzer said, bolting to his feet, as Charlie and I were discussing an assortment of strategies. \"Why can't we just walk into the Keepers' building, use the light chute, and jump our way through? That's what we do. We're Space Jumpers! How many times do I have to remind you of that important fact?\"\n\n\"He might be right,\" Charlie agreed.\n\n\"And remember: we have the Hulking Honock here. He's a regular superhero, if you ask me. I'm sure he could crack a few heads if they overwhelm us.\"\n\n\"You can't do that,\" I told him. \"We don't have any weapons, and we don't have a clue what their defenses are. For all we know, they could have changed the path of the chute and directed it right into a holding cell that we can't jump out of. We may be Space Jumpers, but you seem to have forgotten most of what we learned at the Hollow.\"\n\n\"Well, my way has worked a million times before,\" he said, almost pouting.\n\n\"You're not a wormhole pirate anymore, Switzer. Look, I'm going to do this myself. I'm going to jump in there and grab Theodore and then jump out. You don't have any coordinates, and at least I know how to get inside.\"\n\n\"How are you going to bring him out? You don't even have a belt!\"\n\n\"I'll use yours.\"\n\n\"And leave me here with nothing? Did you pop a chip? What if you get caught? No way. You're not taking my belt.\"\n\n\"Switzer. There's no other way!\"\n\n\"JT, another trace has been placed on you,\" Vairocina whispered in my ear.\n\n\"How long do we have?\" I asked her.\n\n\"They're close, maybe a fraction of a diam. If you jump now, you might be able to shake the trace.\"\n\n\"Thanks,\" I told her, and turned to Switzer. \"Come on. We can't stay here.\"\n\nWe jumped to the far side of Murat, near the restaurant I went to with Max and Theodore. My whole life seemed marked by moments with them. I had to find them.\n\n\"Look, Switzer, do you have a better solution?\" I asked, pushing Charlie into the shadows of a trading chamber.\n\n\"Of course. We can all go. You jump with me. My belt will store the coordinates. We jump back out and then grab circuit-man here.\"\n\n\"You really don't trust me, do you?\"\n\n\"It's not that I don't trust you; it's that I don't trust them. I've got a good thing here. I lose this belt and I have nothing. There's no way I could get back to the Hollow, and I don't even want to think about what the Council will do to me. I really thank you for getting me out of that hole, but I cannot risk going back in there. If you want your little friend, then we do it my way. I'm not letting this belt out of my sight.\"\n\n\"What if we jump right into _them_ and give our plan away? Then we lose the element of surprise.\"\n\n\"I'm afraid there ain't much to give away,\" he said.\n\n\"Let's do it, JT,\" Charlie urged. \"I don't think we have much time.\"\n\nA clatter was seeping through the stillness. I peeked out of the shadows and glanced up the empty alley. On the horizon, a jagged line of darkness was gobbling up the sky.\n\n\"Who are they?\" I whispered.\n\n\"You want to stick around here and find out?\"\n\n\"Fine! Give me the belt. Charlie, we'll be right back. And stay hidden!\"\n\nCharlie nodded as Switzer unlocked his belt and moved next to me. I slipped the belt around my waist and grabbed Switzer. I was so angry with him, I wanted to throw him through the jump.\n\nI remembered the corridor Drapling had taken me to in order to see Switzer. That's where we jumped. As we refocused, Switzer and I crouched with our backs to each other. It was a standard position we learned using SEMs at the Hollow when jumping tandem into a hostile environment. I had never tried it before, but I needed it.\n\nFour armed guards from the Council's Preservation Forces were marching straight toward the spot where we had materialized.\n\n\"Again!\" Switzer shouted, and I jumped behind the four guards as they readied their plasma rifles.\n\nBefore they could spin around, Switzer and I dropped to the floor and swept their feet out from under them. The four guards collided into one another as Switzer and I each secured a plasma rifle. Since I had the belt, I knew Switzer was unable to jump again, so I refocused back to my original position. My hope was to keep the guards disoriented.\n\n\"Hey!\" I screamed. I greeted the first guard with my right foot. I spun around and planted my new plasma rifle in the belly of the second guard. Then I turned to help Switzer.\n\n\"Could you take longer,\" he said. Switzer stood triumphantly over the remaining guards, who were unconscious, with both of their rifles slung over his shoulder.\n\n\"Show off,\" I muttered under my breath. Switzer was a natural.\n\nI stared down the corridor. No one else was coming.\n\n\"What's wrong?\" Switzer asked.\n\n\"I don't know. It seems too easy.\"\n\n\"I tell ya, we're Space Jumpers!\"\n\n\"Still.\"\n\n\"Think they contacted anyone?\"\n\n\"I don't think they had time,\" I told him.\n\n\"Your guys might have.\"\n\n\"Funny.\"\n\n\"So where's split-screen?\"\n\n\"Don't call him that.\"\n\n\"Fine, any idea where we might locate _Theodore_?\"\n\n\"He has to be in one of these rooms. You never had any guards watching your cell. I guess they weren't as afraid of you.\"\n\n\"Now, _that's_ funny,\" Switzer said.\n\nI found Theodore in the third cell down from Switzer's. He sprang from the floor when he saw me.\n\n\"JT! How \u2014?\"\n\n\"Don't worry about that. We have to get out of here.\"\n\n\"You want to go get circuit-man first?\" Switzer asked.\n\n\"What's he doing here?\" Theodore asked.\n\n\"Don't worry \u2014 he's on our side,\" I said.\n\nTheodore only snorted.\n\n\"We could leave you in there, you know,\" Switzer said with a sneer.\n\n\"No, we won't,\" I argued.\n\nI jumped to the other side of the energy field and refocused next to Theodore.\n\n\"Golden!\" he cried.\n\n\"Grab on to me,\" I instructed.\n\nOutside of the cell, both of us refocused next to Switzer. \"Now you,\" I told him.\n\n\"Give me my belt. I can take both of you,\" Switzer protested.\n\n\"You're ridiculous,\" I said, unlatching the belt and handing it to Switzer. He slipped it lovingly around his waist and then held his arms out to both of us.\n\n\"Come to Papa!\"\n\n\"Oh, shut up,\" I snapped.\n\nBack on the surface, Charlie was waiting patiently, shrouded by the shadows. The angry mob I had seen in the distance now spilled through the streets of Murat. I had to shout in order for Charlie to hear me.\n\n\"Who are they?\"\n\n\"Knudniks, Citizens, all of them angry,\" Charlie replied. \"Very angry.\"\n\n\"Charlie?\" Theodore said, his voice almost cracking.\n\n\"Hi, Theodore,\" he said. \"I missed you.\"\n\n\"But . . . I thought . . . he . . .\" Theodore was gawking at Charlie, then Switzer, then me.\n\n\"It's hard to explain, but, yes, that's Charlie.\"\n\n\"Part of me,\" he corrected me.\n\n\"The best part,\" I pointed out, and Charlie smiled.\n\n\"Enough with the family reunion,\" Switzer butted in. \"I don't like hanging out here in the middle of their party.\"\n\nI looked down the alley and saw that the mob was blocking the entrance. I watched hundreds of aliens file past, some with metal pipes, some with sticks, a few even with real weapons. Zinovian claws were popular, but I also saw a Fedaado blade and even a Choi cril.\n\n\"They mean business,\" I said.\n\n\"They're everywhere,\" Theodore remarked. \"Ever since the Council began adding more and more restrictions on our way of life, people have been rallying. They want a war.\"\n\n\"So does the Council,\" I replied.\n\nSuddenly I felt a horrible rumble that dampened the sound of the crowd. The shock stretched down the alley and called up the stone beneath my feet.\n\n\"What was that?\" Theodore cried.\n\nThe crowd was turning. I watched a Honine backtrack and then fall. Another retreating alien stomped on his chest, and the Honine screamed in vain. I jumped to the end of the alley.\n\n\"JT!\"\n\nPanic. The crowd was rushing from something, but I could not see what that was. Another explosion. I jumped to the top of the building across the street to get a better look.\n\n\"What do you see?\" Theodore called out.\n\nI didn't speak at first. Not because I couldn't see what was coming, but because I couldn't believe it. An army of Neewalkers was marching, rolling, and flying through Murat. One machine, or monster (there wasn't much difference), rolled over anything in its path while firing at anything above it. A Neewalker, strapped to the controls, artfully maneuvered the rolling tank against the outmatched aliens. I counted more than a dozen of the machines before I jumped back into the alley.\n\n\"It's bad,\" I told them.\n\n\"What do we do?\" Switzer asked.\n\n\"We should help them,\" I said.\n\n\"Let's do it.\"\n\nNeewalker defense strategies were a vital part of a Space Jumper's training. These nefarious creatures were often at the heart of conflicts in this star sector. At least this is what we were taught at the Hollow, and it happened to be the norm on the Rings of Orbis.\n\n\"Are you thinking what I'm thinking?\" Switzer asked. His faced brightened with anticipation.\n\n\"Remember: I can push into those stilts and disable them once you knock them out. It will take you too long to decipher the interface.\"\n\n\"Always trying to show off,\" he muttered.\n\n\"And I'll handle those rollers as well.\"\n\n\"Why do you get all the fun? _Those_ I can handle.\"\n\n\"Fine,\" I agreed. \"But let's get the guys on the ground first. Those machines look like they'll take forever to turn around.\"\n\n\"Got it.\"\n\n\"I'll be right back,\" I told Theodore, and turned to Charlie. \"Make sure \u2014\"\n\n\"I will.\"\n\nSwitzer and I jumped behind the center roller. We refocused next to a Neewalker. We dropped fast, swiping out the stilts with our legs. My good arm was far more effective, and I grabbed the first Neewalker and snapped its stilt. As it fell, I pushed inside the stilt chips and trashed anything I could find. Switzer and I took out more than a dozen Neewalkers before they even knew what had hit them, and even then they couldn't find us. When one spotted us, we jumped to the other side of the battalion, working in unison. I jumped a nanosecond behind Switzer, waiting for the Neewalker to fall before taking out its computer. I began to see glimpses of Switzer as he broke through time and space and refocused next to the unprepared Neewalker, as if a ghost image of him revealed his whereabouts between dimensions. I found the effect extremely useful in trying to stay close to him.\n\nSoon Neewalkers began to abandon their broken stilts, but their fins were useless on the streets of Murat. The rollers crushed many of them as they frantically searched the skies for Switzer and me.\n\n\"JT!\" Switzer called.\n\nI turned to Switzer, who was strapped into one of the rollers, firing on our enemy. I watched the Neewalkers turn and run while the crowd of angry aliens moved in on them.\n\n\"Get out of there!\" I shouted at him.\n\n\"I couldn't resist!\"\n\nOne of the other rollers saw Switzer and returned fire with a direct hit.\n\n\"Switzer!\"\n\nI jumped next to the attacking machine and pushed into the controls. The machine was useless by the time it tried to fire again. I jumped to Switzer's machine and found it pitched wildly on its side. Switzer was coughing and swiping at the smoke as a small fire licked at the cockpit, but he was still alive.\n\n\"Get out of there, Switzer!\" I grabbed him by the collar with my good arm and hoisted him out, and we both jumped to the ground.\n\n\"The Tonat!\" I heard someone cry from the crowd, and a group of aliens near the front line rushed in and smothered me. \"The Tonat! The Tonat is helping us!\"\n\nThe words echoed through the crowd.\n\n\"It's time to go, Switzer!\" I cried out.\n\nThe aliens were trying to lift me up. Hands grabbed at me from all sides, like kids reaching for a pouch of toonbas.\n\n\"Tonat! Tonat! Tonat!\"\n\n\"Now, Switzer!\" I shouted.\n\nWe both refocused in the alley.\n\n\"That was amazing,\" Theodore cried.\n\n\"Dazzling would be a better word,\" Switzer argued. \"Maybe even stunning, but we can't stay here. There'll be more.\"\n\n\"Come on, this way,\" Theodore instructed.\n\nWe followed Theodore through the streets. My blood was pumping; I was filled with pride in our victory.\n\n\"That was golden,\" I told Switzer.\n\n\"I gotta tell ya, you're fast,\" he complimented me.\n\n\"Did you see the looks on their faces?\"\n\n\"They didn't have a clue what was happening to them.\"\n\nSwitzer followed Theodore up another street, and I fell behind.\n\n\"You were amazing, JT. The way you and Switzer worked in unison, those Neewalkers didn't have a chance,\" Charlie said.\n\n\"Thanks, Charlie.\"\n\nIt seemed obvious to me why Space Jumpers worked in tandem. Switzer and I had performed like a single machine connected by some kind of cosmic cable as we sliced through space, refocusing in the exact position required to chop down our enemy. I wondered if I could work with Switzer, as in permanently. When Space Jumpers were teamed together by the Trust, the only thing that separated them was death. What would that be like? What would Max say?\n\nTheodore stopped outside a building draped in permanent shadow. The plastic structure was the color of despair, and if you didn't have a reason to be here, you would never even see it.\n\n\"Where are we?\" I asked.\n\n\"The hideout,\" he whispered as he pushed the door open and stepped inside. \"This is where Max and Ketheria are. Hey, everyone,\" he called out into the darkness, \"look who I found!\"\n\n\"Theodore! No!\" I cried.\n\nBut it was too late. Switzer and Charlie had followed Theodore inside. My cry was muffled by Ketheria's scream as she caught sight of Charlie. I stayed back as she jumped up from her metal crate and charged at her old friend. Gone was the Scion, the person who had the weight of the Universe placed on her shoulders. Instead, I saw the little girl I knew as my sister, the little girl who loved the man once called Charlie.\n\nThe Honock scooped her up in his clumsy arms and hoisted her into the air like a piece of solar paper. She dripped tears of joy on his face.\n\n\"Charlie! How! Oh, Charlie,\" she cried.\n\nMax walked up behind her. She was looking at Charlie, but she saw me as well. She moved slowly, and then, as if everything that had happened between us recently melted away, as if the Rings of Orbis had melted away, she rushed toward me. That was my signal, and I flung myself at her. We collided in the middle of the room, and I gulped her in. How long had I dreamed about this moment? With my face in her hair, I felt her tears on my neck. I squeezed her tighter.\n\n\"I'm sorry,\" she said.\n\n\"I love you,\" I whispered back, and she held me tighter.\n\nWe were all together again. Despite what had happened and who we had become, it was still us \u2014 just the kids from the _Renaissance._ This is what Max had always wanted. This is what _I_ had wanted. It should have been a wonderful reunion, but it lasted only a nanosecond.\n\n\"JT, a trace has been placed on Theodore,\" Vairocina whispered in my mind. \"It was triggered the moment you left his cell. A mobile force has picked up the signal. They have you as well, I'm afraid.\"\n\n\"I know,\" I replied silently with Max in my arms. It crushed me to let the outside in during that moment. I knew we were in trouble as soon as Theodore found this place. That was why they just let me take Theodore. They knew we would come here. \"How long do we have?\"\n\n\"Not long. Not long at all.\"\n\nIt was the most difficult thing I ever had to do, but I unlocked from Max's embrace.\n\n\"Listen, everyone!\" I called out. There must have been thirty people or more spread out across the dusky rooms. When I spoke, more kids stepped out from their hiding places. \"Do you have any weapons?\"\n\n\"What's wrong, JT?\" Max asked.\n\nI looked over at Theodore. He knew. \"They're coming,\" I whispered.\n\nThose sitting on crates jumped up. Someone cried out, but Max took control.\n\n\"We planned for this! Everyone get ready!\" she ordered. \"Grace! The windows. Theodore, raise the barricade.\" Then she turned to me. \"We have weapons.\"\n\n\"Good,\" Switzer cried. \"We're going to need them.\"\n\nHe tossed a plasma rifle to Theodore, who caught it in midair and came closer to me.\n\n\"JT, I'm sorry. I wasn't thinking \u2014\"\n\n\"Stop,\" I interrupted him. \"It's all right. This moment was inevitable. We had to take a stand eventually.\" I turned to my sister. \"Ketheria!\"\n\nShe was still in Charlie's grip. Seated on his crossed forearms, she was gently examining the metal and wires exposed in his neck. She was smiling, but her eyes were close to unloading their payload of regret. I let her have another moment. It was an expensive gesture, but I let them have it.\n\nThen I spoke again. \"Ketheria!\"\n\nShe turned to me slowly, as if forcing herself to come back to this moment.\n\n\"How are you blocking the staining?\"\n\n\"It's easy,\" she replied. \"I sort of let my mind drift around it, and then it's not there anymore.\"\n\n\"Can you do it for everyone?\"\n\nShe nodded.\n\n\"Then do it now, please.\" I turned inside. \"Vairocina, how long \u2014?\"\n\nThe wall behind Charlie blew apart. Shards of plastic and stone rained down on him and Ketheria.\n\n\"They're here!\" Vairocina said.\n\n\"Move!\"\n\nI jumped outside and refocused for a nanosecond. I was gone again before I could swallow my surprise. \"I think they sent an entire battalion,\" I whispered to Switzer.\n\n\"That just means more fun for us,\" he gloated. \"Theodore, you ready?\"\n\n\"Ready!\"\n\n\"JT! Can you create a distraction, give us more time?\" Max called out. She was tossing weapons to anyone within range. Four other people piled crates and other pieces of metal into a makeshift barricade. It wouldn't be enough. Not even close. We were dead.\n\n\"Yes!\" I told her, and turned to Charlie. I pointed at Ketheria, still in his arms. \"Protect her!\"\n\nHe nodded.\n\n\"Switzer, this time we have to take the big machines out first. There are four of them. I don't know if we can confuse them like the Neewalkers. These guys are going to be ready for us.\"\n\n\"No one is ever ready for _me,_ \" Switzer said, and looked back at Max and the others. \"You think these guys can hold while we go out there?\"\n\n\"They have no choice,\" I whispered.\n\n\"Down the street, then on the backside. Go,\" he ordered, and we refocused behind the battalion of Preservation Forces.\n\nIn front of us, hovering on the flanks of the battalion, I could see four of the metal monsters \u2014 two on each side.\n\n\"Take the one on the far left and fire across the battalion. You get one shot,\" Switzer said.\n\nI refocused inside the weapon's cockpit.\n\nThey were waiting for me.\n\nTwo guards from the Preservation Forces tossed a net at me the moment I refocused. The mesh burned my skin as it touched me. The net was weighted with some sort of electrical spheres that were moving together, trying to close the loop. Something told me that if they touched, I would never get out of this net. I jumped back.\n\nSwitzer was already waiting.\n\n\"What was that?\" he cried, rubbing at his skin.\n\n\"They know we're here.\"\n\n\"So much for that plan. Time to show you what I learned during my missing years. Wait here.\"\n\nBefore I could protest, Switzer jumped. A few moments later, he was standing next to me again holding two plasma cannons. There was a trickle of blood running down his forehead.\n\n\"Where did you get those?\" I exclaimed.\n\n\"Up my ass, where do you think? From out there! Here.\"\n\nSwitzer tossed me a fist-size object. It was spiked like a space mine.\n\n\"Explosive?\" I asked.\n\nHe nodded as one of the huge metal tanks blasted another hole in the hideout. \"Get out of the way after you toss that thing,\" he ordered. \"And toss it hard. It blows on contact. They'll know what's happening after the first one. I figure we jump as we toss. We have to do this together. You ready?\"\n\n\"Absolutely.\"\n\n\"You take the two on the left; I've got these two. On three.\"\n\n\"One . . .\"\n\n\"Switzer?\"\n\n\"What?\"\n\n\"Thanks.\"\n\n\"Shut up and go blow stuff up.\"\n\nI nodded and smiled.\n\n\"Three!\"\n\nI jumped within a meter of the first hover tank and felt the burn from its turbine reaching for my skin. No one saw me as I wound up and drilled the explosive into the rear-mounted engine. I jumped before it hit the tank. I refocused behind the next tank and felt the pressure from the first explosion race me to the next tank. Even the air was trying to get away, knocking over everything in its path. As I felt myself fall, the explosive slipped from my fingers.\n\n_It blows on contact._\n\nI watched the spiked device roll over in the air as debris from my first strike pelted my skin. I refused to close my eyes in fear the explosive would speed up. I pushed myself through the empty space, reaching for the explosive before I even refocused. I stretched out and grabbed the prickly metal from some other dimension, refocusing on my back. I hurled the hunk of metal at the belly of my target. When I looked up, I saw the butt of a plasma rifle coming down from the sky, guided by a Preservation guard. The rifle and the guard both disappeared with the exploding hover tank. Obviously things standing straight up were ripped away first. I jumped before I was forced to follow them.\n\nBack behind the action, I watched the last hover tank rip apart and the bulk of the beast land lopsided on the ground. Several troops were crushed after being knocked down by the explosion and unable to get out of the way. Switzer was next to me before they even cried out.\n\n\"Now for the messy part,\" he said, and hoisted the cannon onto his shoulder. He looked at me before firing into the crowd. \"Why don't you go see if they need help inside?\"\n\n\"You all right here?\"\n\n\"Perfect,\" he said, grinning.\n\nAs I refocused inside, I heard the first cannon blast from outside. Grace cried out as the walls of the building echoed the explosion.\n\n\"Don't worry,\" I told her. \"That's us. We got the tanks, too. It's almost over.\"\n\nEvery one of the kids in the room was armed and ready, hidden by a crate or a wall or some sort of makeshift barricade, most of the items just hunks of garbage. One blast from a cannon out there would blow all of this apart. Maybe they should know the Scion was in here. It would probably save a few lives.\n\n\"I'll be back,\" I whispered to Max, and she nodded.\n\nI jumped outside and refocused away from the fight. I searched the mayhem for Switzer and found him to my left, jumping through the troops. I watched one guard turn where Switzer had refocused and fire. The errant round sailed past Switzer as he jumped again. After the guard watched one of his comrades fall from his own gun, he dropped his weapon and ran. Others followed his lead, but as the guards began to scatter, I heard a roar rushing up behind me. I turned to witness thousands and thousands of knudniks and Citizens marching to join the battle. The Preservation Forces were about to be unmatched.\n\nI jumped next to Switzer. \"Having fun?\"\n\n\"More than you can imagine,\" he replied.\n\n\"It's time to go,\" I told him. \"That mob is even larger than before, and they're headed this way. We need to leave before more troops arrive.\"\n\n\"I'm right behind ya.\"\n\nWe refocused inside.\n\n\"Theodore, come here!\" I called to him, and then found my sister. \"Ketheria, we are going to leave, but we'll be back. When we return, can you do that thing you do and cloak us in your staining? Both me and Theodore. You have to be fast. Can you do that?\"\n\n\"Sure,\" she said. \"Is Charlie staying here?\"\n\n\"For now,\" I told her, and turned to Switzer.\n\n\"Let's jump back to Hach's and wait for the trace to be picked up there. The moment they have us, we'll jump back here and slip under Ketheria's cloak.\"\n\n\"Then what?\" he said.\n\n\"I haven't thought that far ahead, but I know we can't stay here.\"\n\nSuddenly, the door behind Switzer swung open. Everyone turned and readied their weapons in the direction of the door as Drapling strolled into our hideout with three other Keepers following.\n\n\"The Scion can handle all of this,\" Drapling announced. \"She can take care of this fighting. She is coming with us. The Descendants of Light will show her how to use her powers. The Scion will restore order. Our order.\"\n\n\"That's not going to happen,\" I told him.\n\nMax stood next to me. \"How did you find us?\"\n\n\"I imagine everyone on the rings is converging upon this point,\" he said, and then raised his hand toward Ketheria. \"Come, my child. Now is the time for you to fulfill your destiny.\"\n\nA surge of kids jumped up and surrounded Ketheria. We made an imposing posse of plasma-toting teenagers.\n\nDrapling stepped back, his arms still reaching out to Ketheria. \"This is ridiculous. She is the Scion! You have to let the prophecy fulfill itself. You must not intervene in these matters.\"\n\n\"Like you, Drapling?\" I said.\n\nDrapling would not look at me. He wouldn't take his eyes off Ketheria, and she wouldn't leave Charlie's side. I could see the yearning in Drapling's eyes. His prize was right in front of him! I looked at Charlie. \"Don't let him touch her,\" I whispered, and he nodded. I might as well have locked Ketheria in a safe.\n\n\"C'mon, guys,\" I said to Switzer and Theodore. \"Be ready, Ketheria. We'll come right back here. Max, please make sure everyone is ready.\"\n\n\"For what?\"\n\n\"To leave,\" I told her.\n\n\"To leave where?\"\n\n\"To leave the Rings of Orbis.\"\n\nWe jumped back to Hach's and hid in an empty room down the corridor from Theodore's room. The stillness of the air made the building feel empty and lifeless.\n\n\"I don't think there's anyone here,\" Theodore whispered.\n\n\"What do we do now?\" Switzer asked.\n\n\"We wait. Vairocina, let me know when a trace has been placed on one of us, will you, please?\"\n\n\"Already waiting for it,\" she replied.\n\n\"I'm sorry about back there, JT. I wasn't thinking,\" Theodore mumbled.\n\n\"It was my fault. I should have told you.\"\n\n\"That's enough, girls,\" Switzer scoffed. \"Look, we found them, can we get back to the Hollow now? Pick your favorites and let's jump back. I'm hungry.\"\n\n\"I'm not leaving any of them,\" I argued. \"In fact, I'm not going back to the Hollow.\"\n\n\"What \u2014?\"\n\n\"JT.\" Vairocina materialized in front of us. \"The trace has been placed. The Trading Council has mobilized an even larger force, and they're heading in your direction.\"\n\n\"Perfect. Now \u2014\"\n\n\"JT, they have no intentions of capturing you. The Trading Council has given orders for you to be killed on sight.\"\n\n\"JT!\" Theodore cried.\n\n\"It's all right,\" I assured him. \"If they wanted me dead, they could have done it already. They could have killed me like they did Ganook.\" I turned to Vairocina. \"Thanks. One last thing.\"\n\n\"Don't say it like that,\" she replied.\n\n\"Don't worry. I have no intention of dying this cycle,\" I told her. \"Listen, can you jam the signal from the staining? I don't know how, but is there some way of thinking I'm still here after I leave?\"\n\n\"I don't know if it's possible, but I could try some sort of echo. I might need a little time, and it certainly won't last. I'm sure they'll figure it out.\"\n\n\"That's all I can ask.\" I turned to Switzer and Theodore. \"Switzer, straight back to Ketheria. Let her put Theodore under the cloak.\"\n\n\"What about you? You're not staying here by yourself.\"\n\n\"Just until Vairocina's ready. It's me they want. Not you two.\"\n\nHe nodded. \"Then what?\"\n\n\"Then it will be time to go. We're not wanted here anymore,\" I said.\n\n#\n\nI sat alone in my old room and waited for Vairocina's cue. It wasn't much of a room now, but that did not matter anymore. I wasn't scared, either. In fact, I was quite excited by what was coming next. We would all leave the rings together, I thought, including the Scion. And better yet, I would have Max with me. With her and Switzer at my side, we were an invincible force.\n\n\"It's ready,\" Vairocina said.\n\n\"How much time do I have?\"\n\n\"Fraction of a diam, not much more.\"\n\n\"I'll take it. Thank you.\"\n\n_What am I going to do without Vairocina?_ I suddenly wondered. I didn't even know how I was going to say good-bye.\n\nThe moment I returned, Ketheria confirmed that we were now protected from any attempts to trace our genetic stain. I did not question her methods. I simply trusted them.\n\nOutside the hideout, I could hear the war cries from knudniks charging the Preservation Forces.\n\n\"They're pushing them back,\" I said to Switzer.\n\nHe nodded. \"I figure the rebels are getting squashed or being forced back into the city. Either way, I don't think we have much time.\"\n\n\"I know you have a plan,\" Max whispered, slipping next to me and wrapping her arm in mine. I took a moment to enjoy her touch.\n\n\"I always knew you two had a thing,\" Switzer added, and Max smiled.\n\n\"I do,\" I told her. \"But it wasn't my idea. This idea was presented to me a long time ago, but I refused to listen.\"\n\n\"Tell me! Don't be so cryptic,\" she begged.\n\nTheodore, Grace, and a few other kids from the _Renaissance_ had gathered around me. I looked over at Switzer. \"It was really his idea,\" I told them, thumbing in his direction. \"Before we ever arrived on the Rings of Orbis, all I ever thought about was coming here and starting a new life with my sister. Remember our observation deck?\"\n\n\"Of course,\" Grace answered.\n\n\"Well, I would lie there dreaming about what my life on the Rings of Orbis would be like.\"\n\n\"We all did that.\"\n\n\"Yeah, but I had imagined a utopia. A place where they handed out chits and no one went sick or hungry. It was childish. In my imagination, this was a perfect place. I gobbled up every story they planted in Mother and wished away every moment so I could get here sooner. Even when Theylor told us about our fate, I refused to let go of my dream. I would not even consider that the Rings of Orbis might be a cruel place, motivated by greed, a place where success was achieved only by sacrificing others.\"\n\n\"It wasn't always like that,\" Drapling cried out. \"The Rings of Orbis were different. The Trading Council changed everything. This is why the Scion is here.\"\n\n\"Shut up!\" Switzer growled. \"Or I will come over there and do it for you, you two-headed space freak.\"\n\n\"I should have listened when Switzer convinced you guys to take the _Renaissance._ \"\n\n\"No!\" Max said.\n\n\"Let him talk,\" Switzer argued.\n\n\"If I had known back then that I was a softwire, I could have pushed into the ship's computer, or at least I should have tried. You don't know how many times I have thought about that moment, over and over and over again. I know now that I should have listened. We _should_ have taken the _Renaissance_ and never looked backed.\"\n\n\"What are you saying?\" Theodore asked.\n\n\"I say we do that now.\"\n\n\"The _Renaissance_ is gone, split-screen,\" Switzer reminded me.\n\n\"No, he wants to steal a new starship and leave the rings,\" Max said, smiling.\n\n\"Have you flipped a chip?\" Theodore cried. \"We can't steal a starship. Who's going to fly it?\"\n\nSwitzer's face sparkled with surprise, quite a feat for such a scarred mug. He put his hand up and grinned sheepishly. \"Captain Ceesar, at your service,\" he gloated.\n\n\"And Switzer and I have been versed on a zillion different spacecraft in our training,\" I added.\n\n\"It's still crazy,\" someone else complained.\n\nI turned to Max. \"Isn't this what you always wanted? We can be together, all of us, away from here. No Scion, no Tonat, no Space Jumpers. Just _us._ We can find a new world to live on, and we'll never look at the Rings of Orbis again.\"\n\n\"Where did the Keepers go?\" someone asked.\n\nI turned and they were gone. \"Drapling?\" No answer.\n\n\"I don't think Twin-Top ran off to book us a seat on the shuttle,\" Switzer said.\n\n\"If we're going to go, then we go now,\" I told the group.\n\n\"Wait!\" Grace cried. \"Can we think about this?\"\n\n\"What for?\" Switzer said, throwing his arms up and stomping to the back of the room.\n\n\"We don't have time,\" I argued.\n\n\"Just wait!\" Grace said.\n\nGrace and two other kids broke into their own group, then four other kids did the same. Theodore glanced at them.\n\n\"It will be all right,\" I told him.\n\n\"I know it will.\" He stepped toward me. \"Of course I'm with you.\"\n\n\"I think it's perfect,\" Max said. \"Especially the part about us being together. Not just you and me. I mean, of course I love that, but I want all of us to be together, even Switzer.\"\n\nI looked over at Switzer, who was now perched on a metal shipping crate. Ketheria was next to him. They were whispering about something, and I could only assume she was forgiving him for everything he had done in their past. Inside, I smiled (I wouldn't dare let Switzer see me). Their reconciliation was necessary if we were going to live together on a starship again. I could not even guess how long it would take to find a new home.\n\n\"They have a lot to discuss,\" Max whispered.\n\n\"He's different now,\" I told her. \"He's not the Switzer we used to know.\"\n\nMax took my hands and turned me to face her. She was intoxicatingly close to me. It took everything to keep my eyes open.\n\n\"I'm sorry,\" she whispered.\n\n\"Don't,\" I said.\n\n\"No, I want to. I wasn't fair to you. I ignored the pressures they placed on you. I did not want to admit what they had done to you, _to us._ I'm so sorry. I love you, and I don't ever want us to be apart again. It just hurts too much.\"\n\n\"We won't. I promise.\"\n\n\"All right!\" Grace said. \"We'll go. But Switzer cannot be in charge. It has to be Ketheria.\"\n\nI looked over at Switzer, knowing he would protest, but Switzer was staring over Ketheria's shoulder. His faced showed no sign that he had heard the group's objection.\n\n\"Switzer?\" I called out to him.\n\n\"What's wrong with him?\" Max whispered.\n\n\"Switzer!\"\n\n\"What?\" he grunted, shaking off his trance.\n\n\"They won't let you be captain. They want Ketheria,\" I told him.\n\n\"What?\" Switzer protested. \"I'm the captain.\"\n\n\"Then no deal,\" Grace said.\n\nI glared at Switzer.\n\n\"Fine,\" he grumbled, although his protest was unusually weak. \"But I'm not calling her Captain Ketheria.\"\n\nKetheria glanced at Switzer. \"I'm ready,\" she said.\n\n\"There's still a war going on out there,\" I warned them, \"and Switzer's right: Drapling didn't leave to reserve a seat for us. It's a long way to the spaceport, maybe four kilometers.\"\n\nJust then I heard a _WHUMP._ In fact, I felt it. Even the air pushed against me.\n\n\"What was that?\" Grace cried.\n\n\"We better hurry,\" Max whispered.\n\n\"Can't you jump there and take us?\" Theodore asked.\n\n\"First, there are too many of you to jump at once with Switzer, and I don't have a belt to help.\"\n\n\"I don't know how far my cloaking works, either,\" Ketheria said.\n\n\"We can't risk it. We need to stay together and move as a group.\"\n\nThe building shivered from another blast, coaxing the dust and debris from the ceiling.\n\n\"Can we go?\" Grace demanded.\n\n\"I will take the lead and Switzer will follow last. Everyone else pair up and keep Ketheria protected. Charlie, you stay with her, in the middle. Don't talk to anyone, and keep your head down!\" I yelled.\n\nPeople began pairing off, moving Ketheria to the middle. Max came up behind me. \"I'm with you,\" she whispered.\n\n\"Stay close,\" I said, and kissed her on the cheek. I thought of Vairocina's warning. \"But not too close. Give yourself some running room.\"\n\n_I'm not dying this cycle._\n\nOnce outside the building, I could see intense fighting still raging to my left. The knudniks appeared to be holding their own as the Preservation Forces hunkered down into a building at the edge of Murat.\n\n\"Don't look,\" I whispered to Max as I stepped around the aftermath of Switzer's cannon.\n\n\"Oh, that's disgusting!\" Grace cried.\n\nMy plan was to race around the far side of Murat in order to reach the spaceport. The military aircraft (and there were a lot of them) were converging over the center of the city, so most of the conflict was happening there. The detour added a kilometer to our run, but there were too many of us to risk getting caught in the skirmish. I was certain that once we reached the spaceport, operations would be so chaotic on the landing pads that Switzer and I could jump inside a ship and leave orbit before anyone even knew we were there.\n\n\"Ketheria, what's the range on that ability of yours?\"\n\n\"I don't know,\" she called out.\n\n\"Then stay close, everyone!\"\n\nI treated the city as nothing more than an obstacle course. To me, it was just another map in a game of Quest-Nest, and my bait was the spaceport. Actually, my bait was a shiny new spaceship ready to take me to my new home, far away from here.\n\nAs we raced past the busted buildings and abandoned trading chambers, I concentrated on the prospects of a new life and it sparked an excitement in me. The energy moved my legs forward unconsciously, and I occasionally glanced behind to make sure everyone was keeping close.\n\nWe moved as one group over barricaded alleys and crumbled buildings, slowing only when a quick climb seemed faster than finding a new way around a fallen structure or mountain of garbage.\n\n\"I didn't know things had gotten so bad here,\" I called out to Max.\n\n\"The Trading Council really wants the rings.\"\n\n\"They won't go without a fight.\"\n\n\"You weren't the only knudnik who thought the Rings of Orbis should have been their utopia.\"\n\nWe had run about a kilometer when I was forced to pull up.\n\n\"Stop!\" I cried out.\n\nIn front of us was an enormous hole in the ring. Some sort of bomb or missile had destroyed an entire city block, preventing us from going any farther. I couldn't tell what had caused the damage, but whatever it was, it was big. Scary big.\n\n\"I hope they ran out of whatever did that,\" Theodore remarked.\n\nThe guts of Orbis 4 lay open at our feet, like some kind of busted space shuttle abandoned by its mechanic.\n\n\"This just happened,\" I said. \"The dust has hardly settled and parts are still burning. That must have been the sound we heard back at the hideout.\"\n\n\"We can't cross this,\" Max said.\n\nShe was right. To my right I could see rows of factories turned into mountains of rubble by the explosion.\n\n\"Just go around it,\" Switzer ordered.\n\n\"We can't,\" Max said. \"I've been here before, handing out taps in the city. Those factories go on forever. That would be a very long detour.\"\n\n\"It looks like someone knows what they're doing,\" I said.\n\n\"They're cutting off access, keeping everyone in the center,\" Switzer pointed out.\n\n\"Or they're making it very difficult for anyone to leave,\" I added.\n\n\"There's only one person who knows what we're trying to do,\" Max whispered.\n\n\"Think Double-Dome would risk baby-malf's life like that?\" Switzer asked.\n\nI frowned at him.\n\n\"Sorry,\" he mumbled.\n\nI looked to my left. The long street still sparkled, a reminder of the city that once was. Only now it led directly into the conflict, a route I wasn't prepared to take, but I saw no other choice.\n\n\"We don't have to go all the way in,\" I told Switzer. \"We could work our way in just a little, to cut back over and up.\"\n\n\"I know this street, too,\" Theodore said. \"Every alley dumps into the center of Murat.\"\n\n\"Maybe we should send a reconnaissance group out and map a route to avoid that,\" I offered.\n\n\"No, JT,\" Max said. \"If one goes, we all go. We stay together now. You said so yourself. Think of the staining.\"\n\n\"I agree,\" someone else called out.\n\nI looked back up toward the factories.\n\n\"We could jump \u2014\" I started to say, but Max cut in.\n\n\"No,\" she snapped. \"Look!\"\n\nFar down the street, I spotted two enormous hover tanks as they rounded the corner. Behind them was a wave of Preservation Forces. I couldn't tell if they were retreating or moving toward us. Circular fliers spun overhead, firing into the crowds.\n\n\"Look! Knudniks!\" Max cried. They were fighting the Preservation Forces hand to hand. \"We're trapped.\"\n\n\"Let's move!\" I cried. \"Everyone into the alley!\" I pointed to an opening in the building between the hole and the fighting moving toward us.\n\n\"Vairocina,\" I called out. \"I'm stuck in the city, trying to make my way around Murat's industrial core. Can you see where I am and find me the shortest route around it?\"\n\n\"I am unable to locate your whereabouts, JT. The manner in which you are blocking the staining is very effective. More than three dozen attempts to trace your location have been attempted \u2014 unsuccessfully, I might add. If you give me a bearing, I can pull up a schematic to help you navigate,\" she offered.\n\nA small speck tumbled toward us like an extinguished star giving up its spot in the night sky. The speck grew larger, and for a moment, I didn't comprehend what it was. In fact, I was mesmerized by the curve of its trajectory as it sailed past my head. Only when the thing disappeared inside the factories next to us, did I realize.\n\n\"No!\" I cried, but a deafening _WHUMP_ rolled over me, flinging me backward as the air seemed to disappear, as if it was being sucked into outer space.\n\nThe ground vomited as I hit it, tossing me back up and mixing me into the debris. The blast from the explosion refused to subside, as if it were taunting me. I couldn't get a bearing on anything, or anyone, and it felt as if someone had set the ring spinning out of control. Suddenly, I slammed to a stop. A large hunk of factory followed me to the ground and crushed my robotic arm. I pulled my arm out and watched my fingers curl back, almost touching my wrist. The pain shot up my arm before my fingers snapped back into place, lifeless.\n\n\"Switzer!\" I cried out. Even I was surprised that this was my first word.\n\nI turned into the swirling debris. \"Max!\"\n\nI tried to focus on something \u2014 anything to make my world settle, but everything was in motion.\n\nCharlie slumped, lifeless, on the ground.\n\nTheodore crawled on his knees.\n\nGrace wandered, bleeding.\n\nSwitzer was nowhere to be seen.\n\n\"Max!\"\n\nI jumped out of the chaos, to a place just beyond the explosion. I refocused and saw another gaping hole in the center of the street as debris swirled about the opening as if some vortex had been ripped open by the blast. I could see some people standing; some were lying on the ground. Some I could hardly see at all.\n\n\"Ketheria!\"\n\nThe Preservation Forces were now at my heels, but they were too busy fighting the knudniks. I jumped back to the highest point of the rubble and refocused atop the aftermath and inside the growing tornado. I tried to flex my crushed arm again, but it would not respond. It flopped at my side, useless.\n\n\"Max!\"\n\nStill no answer.\n\n\"Max!\"\n\nIn the hole, about twenty meters below me, I could see the purple stream of a light chute, untethered and flailing about like a broken gas line. The stream crackled and hissed across the black void. Another uprooted chute intersected with the first one, igniting a light storm whenever they touched.\n\n\"Max!\"\n\n\"JT,\" my sister called out.\n\n\"Ketheria! Where are you?\"\n\n\"Down here,\" she cried. \"I have Max.\"\n\nI fell on my belly and peered over the edge. Ketheria's crimson hands clung to a utility pipe sprouting from the rubble and over the hole in the ring. Max, her hair matted with blood, was lying unconscious on a chunk of concrete just above Ketheria's head and slightly to my right.\n\n\"Ketheria! Are you all right? Is Max alive?\"\n\n\"I don't know. Be careful. I think everything is really loose.\"\n\n\"Hold on!\"\n\nI jumped to the far side of the hole so I could get a better look. Max was barely on the rock, and there was no way to jump to Ketheria. _Where was Switzer?_\n\nI refocused on a small metal girder just above the girls. I struggled to keep my footing as the girder tilted severely toward my sister. My right arm was now switching between functioning and useless as I looked for a way to secure myself. I jammed my legs between the girder and a slab of concrete. As I reached over the edge, another explosion set the world in motion yet again.\n\n\"Ketheria!\"\n\nThe blast heaved Max into the air while the rock underneath her tumbled into the void. The busted light chute gobbled it up. Ketheria reached out and caught hold of Max's shirt while my left hand clamped onto Ketheria's right wrist. My other hand, the bad one, snagged Max's shirt. It wasn't much, but it was holding.\n\n\"JT, help me!\" Ketheria begged.\n\nKetheria's plea for help ignited some part of me that found strength I never knew I had. My mind focused on my contact with Ketheria while I shifted my weight to help Max. She was heavy.\n\n\"Max!\" I pleaded.\n\n\"I have to let go of her, JT!\" Ketheria said. A red trail grew on Max's shirt as she slipped through Ketheria's bloody fingers.\n\n\"Wait! I don't have her!\"\n\nI concentrated hard to maintain what little hold I had. I interfaced with my arm, but there was only a patchwork of controls at my disposal now, and most of those were unresponsive. I only managed to squeeze a little more strength out from it.\n\n\"JT!\"\n\n\"All right! I have her.\"\n\nKetheria grabbed the pole again as Max's shirt ripped.\n\n\"You don't have her, JT. Use both hands!\"\n\nKetheria was holding on to the metal pipe sticking out of the concrete, but my hand would not release her wrist.\n\n\"I can't!\"\n\n\"Yes, you can. I'm fine!\"\n\n\"No. I can't.\" I stared at my left hand clamped around Ketheria's wrist. \"My mind won't let me.\"\n\nAs much as I wanted to let go of Ketheria and use both hands to pull Max to safety, something inside of me refused to let go of my sister, to let go of the Scion.\n\n\"That's not you, JT! That's what they did to you! That's the coding working. The coding the Trust put inside of you. You love her, JT. Fight it! Let go of me!\"\n\nMax's shirt ripped again.\n\n\"JT!\" Ketheria screamed.\n\nI tried. I tried so hard to let go of Ketheria, but my mind refused the logic.\n\n\"Max, wake up, please,\" I whispered.\n\nEverything in my vision now began to swim together in the purple light. Tears fell from my face and sparked against the chute.\n\n\"JT, it's not you! Let go of me and grab Max, please!\"\n\nI thought of every moment I'd ever had with Max. The first time she helped me with the hidden files on the _Renaissance,_ the first time she held my hand, even our first kiss. My hand wriggled on Ketheria's wrist, but it was not enough. I could not let go.\n\nMax's shirt ripped again \u2014 a final time.\n\nKetheria grabbed at Max as she fell, and I like to think I tried as well. My left hand stayed on Ketheria while my right hand scratched at the air.\n\nI didn't scream. I didn't cry out. Instead, I told Max I loved her as her body plunged into the purple light chute.\n\n\"I'm so sorry, JT,\" Ketheria whispered through her sobs. \"I'm so sorry for this.\"\n\nI stared at the purple chute for a while. _This couldn't be happening. Max? Max! This isn't real,_ I tried to tell myself, but I knew Max was gone. I could hear Ketheria sobbing, and I could hear the war raging over my head, but I could also hear my breathing over it all, for some weird reason.\n\nAs I stared at the purple light chute, waiting for time to reverse itself, I felt my hate for the Rings of Orbis burn my insides. I hated everything they had done to me. I hated them for everything I had lost and everything I'd never had.\n\n\"It's not your fault, JT.\"\n\n\"Yes, it is,\" I whispered.\n\n\"It's not. It's this place, these people.\"\n\n\"I know that, but it won't bring Max back.\"\n\n\"I'm so sorry,\" Ketheria whispered again as she stood up. I was still staring down the hole where Max fell. \"They did this to you, JT, and they'll do it again. I have to stop them.\"\n\nWithout looking, I said, \"What do you mean?\"\n\n\"It's my destiny.\"\n\n\"What is?\"\n\n\"To save them.\"\n\n\"Save who?\"\n\n\"Save everyone.\"\n\nI finally turned toward my sister and away from where Max had fallen. In the back of my mind, I was aware that my life was still moving forward. \"Ketheria, what are you talking about? Here, grab on to me. Can you pull yourself up at all?\"\n\n\"JT?\" It was Switzer.\n\n\"Where were you?\" I screamed. \"I needed you! You could have helped me.\"\n\nSwitzer was kneeling on the far side of the hole. I could see blood gushing from a nasty cut over his right eye, and his left arm was clearly busted.\n\n\"Switzer!\" my sister cried, trying to look over her shoulder. \"Is that you? I'm ready.\"\n\n\"Ready for what?\" I said.\n\n_Max is gone._\n\n\"JT, I must suffer this,\" she said. \"It will not happen if I do not do this. It is the last thing I must do before I can truly awaken.\"\n\n\"Ketheria, tell me what you are going to do,\" I demanded.\n\n\"I have to do this, and I have to do it without you. I see that now. I'm sorry.\"\n\nThen Ketheria bit down on my hand. \"Ow!\" My fingers loosened just enough for her to slip from my grip, and she let go of the bar.\n\nShe tumbled into the hole.\n\n\"Ketheria!\"\n\nSwitzer was next to her in an instant and plucked her out of the purple air. He refocused on the far side of the hole, just as he had practiced at the Hollow. I did the same, surfacing in the center of the battle. Preservation Forces were fighting hand to hand with knudniks and Citizens alike. I couldn't help but think that Switzer, with Ketheria in his arms, was a far better Space Jumper than me.\n\n_Max is gone._\n\n\"What are you doing, Switzer?\" I said.\n\n\"Getting a little payback. Something you should have done a long time ago.\"\n\n\"Put her down!\"\n\n\"No. I'm not like you, buddy. Things are black or white for me. You spend too much time in here,\" he said, pointing at his head. \" _This_ is a good deal, and I'm going to take it.\"\n\n\"Deal? What deal?\"\n\n\"It was my idea,\" Ketheria said. \"Don't blame him.\"\n\n\"What are you doing?\"\n\n\"Put me down, Switzer, but don't let go until I say.\"\n\n\"Ketheria. I don't understand. Tell me, please,\" I pleaded with her.\n\n_Max is gone._\n\nKetheria did not reply. She stood perfectly still with her feet together and lifted her arms to the side. Then the glow within her eyes seemed to expand. The golden luminescence flowed from her eyes and formed a radiant coronet around her head before dropping to her feet. When the light hit the ground, it exploded outward like the birth of a new galaxy. The circle of light engulfed everyone in its path. Citizen, knudnik, and soldier alike dropped their weapons and bathed in the stream of light now flooding Murat. I could not tell how far the light was going, but soon it flowed as far as I could see.\n\nEveryone just stood there and stared at the people next to them with this perplexed look on their faces, as if they were trying to understand how they had gotten here. Soon some people were tending to the fallen and no one was fighting anymore. I gawked as Preservation Forces stepped down from their tanks and pulled knudniks from the rubble. Was Ketheria doing this? I turned toward my sister as the stream of light faded and eventually stopped flowing. Then she turned toward me slowly. There was something different about her. I didn't know if it was her eyes or her smile. She looked as if she was capable of understanding anything.\n\nThen she smiled at me and said, \"Good-bye, JT.\"\n\n#\n\n\"Are you sure you want to do this?\" I asked Charlie.\n\n\"I have a few debts to repay,\" he replied.\n\nCharlie and I were seated in the spaceport on Orbis 4. His shuttle was about to leave, and he had asked me to visit him before he left.\n\n\"Your real name isn't Charlie, is it?\"\n\nHe shook his head and said, \"Harlan. Harlan Admunsen.\"\n\n\"I like _Charlie_ better.\"\n\n\"Then let's leave it like that.\"\n\n\"I'm going to mi \u2014\" I started to say, but he interrupted me.\n\n\"Don't get me crying. Something might start to rust. You never did turn down those emotion levels, either,\" he complained. \"But thanks for making me feel like myself again. You know, with the . . .\" Charlie pointed at the metal around the back of his skull.\n\n\"No problem,\" I said.\n\nI stared at the floor, swallowing the lump in my throat.\n\nThen he said, \"They still might find her.\"\n\nI shook my head, unable to talk.\n\n_\"The shuttle for Orbis 2 is now boarding.\"_\n\nFinally, I croaked out, \"No.\" I looked up at him. \"I wish I could feel what everyone has been feeling since the awakening. Theylor said I was designed to not experience it. The enlightenment had no effect on me. Just another gift from the Trust to ensure that their fighting machine stays true to its mission. I don't mind, though. . . .\"\n\n\"I'm sorry.\"\n\n\"No, I think it's helped a little, with Max, you know, to fill that hole up a little bit.\"\n\n\"You're going to be all right,\" he said, and clamped his hand on my shoulder. It was a painful blow.\n\n\"Ouch! You may act like the old Charlie, but you have superhuman strength now.\"\n\n\"Sorry.\"\n\n_\"The shuttle for Orbis 2 is now boarding.\"_\n\n\"I gotta go,\" he said, standing up. \"Give me a hug and then go and get on with your life.\"\n\nI laughed.\n\n\"Hey! None of that. You know what you have to do. You are a Space Jumper. They're going to write stories about you one cycle, JT.\"\n\nI stood and smiled. Then I gave Charlie a hug. \"Come find me,\" I whispered.\n\n\"I will,\" he croaked.\n\n_\"The shuttle for Orbis 2 is now boarding.\"_\n\nCharlie broke away. \"Go on, go say good-bye to Theodore,\" he ordered.\n\n\"Good-bye. Charlie.\"\n\n\"Good luck, kid.\"\n\nCharlie picked up his bag and joined the line for the shuttle. He didn't turn back to look at me again, but I waited until he disappeared through the loading door. Even after everything that had happened, Charlie still managed to avoid most of my questions. I wondered where he was off to. I wondered if I would ever know.\n\nI set out to find the New Arrival Processing Center, to say good-bye to Theodore. Theodore was now helping the new knudniks arriving on the Rings of Orbis. He had wanted to take this cycle off work, but I told him not to. I figured it would be easier to say good-bye that way.\n\nNo one referred to the new arrivals on the rings as knudniks anymore, and they were no longer indentured to the Citizens, either. Ketheria's enlightenment had spread fast through the rings, even reaching the Trading Council, who structured a new power deal with the Keepers.\n\nWatching everyone file through the spaceport, I couldn't help but feel that they looked a little happier in their home. But the Rings of Orbis were no longer home for me. For so long I had thought that this is where my life would end up, but now I realized that it was only the starting point. My home was always where my friends were. I had had a home on the _Renaissance,_ and I had had a home on the rings when we were all together, despite the conditions. I just didn't see it.\n\nI missed Max terribly. I hoped for so long that the light chute had transported her to another place on the ring, but after a long and fruitless search, the Keepers were unable to locate her. Phase after phase, I blamed myself for that moment. I was unable to turn off the creature inside of me long enough to help the girl I loved. I could still get angry thinking about it, but at least the awakening had created enough space for me to move on with my life.\n\nI stopped outside the entrance to the New Arrival Processing Center. I watched Theodore talking with each alien, directing them to the R5s and then helping them uplink information for their adjustment period on the rings. He moved from alien to alien, more confident than I had ever seen him. Theylor said it was from the enlightenment, but I couldn't get a sense of that. In fact, I think Theodore's change had nothing to do with Ketheria. I think he'd simply found his passion.\n\n\"JT!\" Theodore called out when he finally saw me.\n\nI waved at him and walked into the room. \"I wouldn't have guessed in a billion rotations that we would have ended up like this,\" I said.\n\nTheodore looked back at the aliens huddled near the R5s.\n\n\"I know! It's crazy, isn't it? But you know, I feel so empowered. It's hard to explain. I just want to help. I want to make the rings a better place, and it's not just me. The sentiment is spreading through every ring.\"\n\n\"It's almost as if the Ancients have come home,\" I replied.\n\n\"That's what Theylor said!\"\n\n\"I guess we did what they needed.\"\n\n\"But what does it mean? You know, for us? What will happen when Ketheria goes around the universe waking everyone up?\"\n\nI looked at Theodore and shrugged. \"I don't know. Like this, I suppose.\"\n\n\"Then it's going to be great.\"\n\n\"You can still change your mind.\"\n\nTheodore shook his head. \"No. My place is here now. You know, _you_ could stay. Switzer can take care of Ketheria. Even you said he's changed.\"\n\n\"I don't think so. Switzer saw a good deal and he took it. He will always be looking for what's best for him. Ketheria will have no effect on him. One of those deals is going to hurt Ketheria some cycle, and I can't have that. Besides, I think she expects me to find her. It was something she said. You know . . . after . . .\"\n\nTheodore looked at his feet, and I did the same. It was a routine we went through whenever there was nothing left to discuss but Max. Theodore had helped me the most after the accident, and I was grateful for that. I hoped he knew.\n\n\"Thanks,\" I whispered just in case he didn't.\n\nHe nodded and smiled. \"Will I ever see you again?\"\n\n\"I'm going to make it a promise,\" I said. \"One that I will keep.\"\n\n\"Then I'll see you soon, my friend. Stay safe,\" he said, and hugged me.\n\n\"You too.\"\n\nTheodore broke away and waved as he returned to the new arrivals. I watched him slip back into his work, and then I turned for the door. I spotted two Keepers walking through the spaceport and wondered if Theylor had arrived yet. He had personally disbanded the Descendants of Light after it was discovered that they were the ones who bombed the exit points from Murat. I wondered if Drapling even knew what he had done. Theylor never mentioned what happened to him.\n\nMy instructions stated that I meet Theylor at the wormhole launch located at Gate 5 on the far side of the Spaceport 1. Whenever a knudnik's work rule had expired, the knudnik was offered the choice to stay and petition for citizenship or take a free trip through the wormhole. Not once had I ever thought about taking that trip. I always saw the rings as my final destination, yet here I was, ready to leave the place I had so wanted to call home.\n\nThe cycle's traffic was sparse near the gate. I figured few knudniks opted for the wormhole option anymore. I spotted Theylor waiting near the gate. Both his heads were smiling.\n\n\"I have a present for you,\" he said.\n\n\"That sounds intriguing. Is it something to eat?\"\n\n\"You must be hungry, but you can eat on the other side. It's best to travel through the wormhole on an empty stomach.\"\n\n\"So what is it?\"\n\n\"Come, I'll show you.\"\n\nI followed Theylor through the gate. He was the first person I had ever met from the Rings of Orbis. It was appropriate that he be the last I see. Throughout everything, Theylor had never changed. He was the same alien as he was the first time I met him.\n\nI stepped through the gate and onto a curved platform. Theylor moved toward the huge windows that arched up and over our heads. On the other side of the glass was a small spacecraft docked at the portal. The slick flier glistened under the warm floods that spilled down on the ship.\n\n\"That's mine?\" I asked.\n\n\"Well, you certainly cannot walk through the wormhole. Did you ever wonder why the trip was so expensive? You need a vessel.\"\n\n\"And you're giving this one to me?\"\n\n\"It is our gift. A token of our gratitude for everything you have done for the Rings of Orbis. I took it upon myself to make a few upgrades and enhancements,\" he said. \"I see you've done the same.\" He glanced at my right arm.\n\n\"You know?\" I said.\n\n\"Of course. She was worried that we might still need her, but I assured her that we could cope in her absence. Besides, I believe she would have missed you more than she will miss us.\"\n\nI held up my right arm and fiddled with a thick piece of jewelry made of silver metal and black bands of rubber that now clung to my wrist. It looked like a bracelet, but it was one that I could never take off, for it was actually attached to my arm.\n\n\"Vairocina made the addition,\" I whispered.\n\n\"A girl needs a little room,\" she teased inside my head, \"especially if I'm going to traipse around the universe locked inside your arm.\"\n\n\"It looks nice,\" Theylor remarked. \"No one will ever know. Would you like to see your new ship?\"\n\n\"You know I have to do this, right, Theylor?\" I said.\n\n\"Of course I do.\"\n\n\"And I have to do it alone.\"\n\n\"As you have always stated.\"\n\n\"Vairocina gave Switzer the coordinates of every place she had ever visited in the galaxy when she uploaded those coordinates in Ketheria's room. Switzer can only jump to those star systems. I figure I'll simply do the same thing. I'm certain I can pick up his trail somewhere along the line.\"\n\nTheylor reached into his robe and removed a Space Jumper's belt.\n\n\"Then you'll need this when you find her,\" he said.\n\nI took the belt in my hands. All my thoughts and emotions for Max ignited inside my chest. If only I hadn't been so stubborn, if only I had accepted my fate sooner, then I would have had one of these stupid things. I could have jumped to Max after she fell and then jumped to safety. It was the most costly mistake of my life, but one I would never make again.\n\n\"Thanks,\" I mumbled, and slipped the belt around my waist. \"I still don't understand why Ketheria had to leave without me. Switzer is not the most trustworthy person.\"\n\n\"She had to. It was part of her awakening. The fourteenth and final step required her to let go of the thing she cherished most in this universe. That, I am afraid, was you. Only when she released you from your duties could she truly be the Scion. The results were immediate, as you remember.\"\n\nI nodded.\n\n\"When you emerged from the cocoon on the Hollow, you, too, completed the final step of your awakening. Without it, you would never have become a Space Jumper and Ketheria would be on her own forever.\"\n\n\"I'll find her soon,\" I told him.\n\n\"I know you will.\"\n\n\"Ready, Vairocina?\"\n\n\"Absolutely,\" she said.\n\n\"Good-bye, Theylor. Thank you for everything.\"\n\nThe Keeper smiled, and both heads nodded. \"Drink deep from the Source, my friend,\" he said.\n\nI looked back at the spaceport before stepping onto the ship. The tallest spire was reaching for the eclipse as the ring laid its shadow across the city. _I will never miss this place,_ I told myself. As I hesitated outside the bay, I felt a deep pain in my stomach and a wave of nausea rose up in my throat. I smiled, not because I was leaving but because I knew that Ketheria was still within my reach.\n\n# ACKNOWLEDGMENTS\n\nYou always hear about writers locked away, toiling over their manuscripts for years, before unleashing them on the world. It sounds like a solitary process, but it's far from that. I would personally like to acknowledge those who have namelessly helped me bring the Softwire series to life. Thank you so much. I really mean it.\n\nTo Eddie, for getting the ball rolling in the first place.\n\nTo Liz, Lynne, and Michael for finding the Softwire books a home at Candlewick.\n\nTo Sarah for putting up with me and making me a better writer.\n\nTo Laura, for your patience and always taking my phone calls. :)\n\nTo Lisa, for your big bookstore support when others were silent.\n\nTo Denise and the girls at KNTR \u2014 your support has never wavered. Thank you. Ninjaritas for everyone!\n\nTo Jim, thank you for your insight and encouragement.\n\nTo Faith for living up to your name and helping me carry the torch.\n\nTo Nard \u2014 the keeper of the case. Thank you for always being there.\n\nTo Michelle, for your amazing novel studies.\n\nTo Alan, for your encouragement and friendship. And your voice.\n\nTo Nathan, for your relentless support and unwavering friendship. Thank you.\n\nTo my fans for your great letters and e-mails. They really kept me writing.\n\nTo the Citizens of Orbis for helping me create an unbelievable place to hang out online at the ringsoforbis.com\n\nTo teachers and librarians who invited me into your schools and placed books in the hands of kids who might never have heard of them.\n\nTo all of the independent bookstores that love the Softwire series and hand-sell my books. I can't thank you enough and tell you how important that is.\n\nTo Frank, the best friend I could ever have in my corner. Thank you so much.\n\nTo Sky, for being patient with Daddy and loving me no matter how often I had to go away to promote the Softwire books.\n\nAnd to Marisa, for your love, your understanding, and for sticking it out with me. You're amazing. I love you.\n\n#\n\nPJ HAARSMA has always been transfixed by what lies beyond our solar system. He says, \"When the mother ship finally arrives, and they ask if there are any humans who want to go for a spin, I'll be the first to sign up.\" When he's not gazing at the stars waiting for his ride, you can find him on the Rings of Orbis, the online universe that he created for the Softwire series, which has spawned a legion of loyal fans. He has a degree in science and lives in southern California with his wife and daughter. To learn more about PJ Haarsma, visit his website at www.pjhaarsma.com.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nCharlotte Markham and the House of Darkling\n\nMichael Boccacino\n\nDedication\n\nFor my mother\nContents\n\nCover\n\nTitle Page\n\nDedication\n\nPart 1\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nPart 2\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nPart 3\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nAcknowledgments\n\nP.S.: Insights, Interviews & More . . .\n\nAbout the author\n\nAbout the book\n\nRead on\n\nAdvance Praise for Charlotte Markham and the House of Darkling\n\nCredits\n\nCopyright\n\nAbout the Publisher\nPart 1\n\nThe Other Side\nCHAPTER 1\n\nThe Unraveling of Nanny Prum\n\nEvery night I dreamt of the dead. In dreams those who have been lost can be found, gliding on fragments of memory through the dark veil of sleep to ensnare themselves within the remains of the day, to pretend for a moment like a lifetime that they might still be alive and well, waiting by the bedside when the dream is done. They never were, but I could not stop myself from wishing for the possibility that everything I remembered was a mistake, a nightmare taken too literally by the imagination. But morning always came, and with it the startling realization that the dead continued to be so, and that I remained alone.\n\nThat night the pleasant rest of black, unthinking oblivion gave way to a dimly lit ballroom without any ceiling or walls, a place lost in the bleak abyss of time. Crystal chandeliers hung above the marble flooring untethered to any surface, threatening to crash down upon the guests, who were dressed in moldering finery that would have been out of fashion decades before. The dance began with a slow, melodious waltz that felt akin to a waking sleep, and I let it wash over me, swaying with the rhythm until someone from behind took me into his arms. I did not need to see his face; I knew who it was. My late husband, Jonathan, turned with me across the ballroom, faster and faster, never reaching any wall or barrier, never colliding with another couple, until he dipped me deeply. My mother and father were next to us, warm and whole, younger than I ever remembered them being. This was the dance of the dead.\n\nThe music stopped. My husband let go of me and bowed before retreating into the dark place beyond the ballroom. The room began to fill with people I did not recognize\u2014leering strangers with faces that were really masks, ready to slip at any moment. My parents disappeared into the crowd. I tried to find them, but the crowd was too large and the music began again, this time an eerie, cruel sound, a broken music box filled with regret. A man appeared before me dressed all in black, his features cloaked in shadow. As he took my hand I knew with a certainty that only dreams can provide that he was not a stranger; we had met before. His hands were cold and his lips, though I could not see them, were smiling. The other dancers spun around us until they blurred together. He pulled me close against his body, into the darkness that surrounded him until I was falling, the chandeliers trailing away as I spun through the void, screaming into nothingness.\n\nI woke upon the realization that the screams were not my own. A woman was shrieking in the night. At first I was deeply annoyed, for anyone blessed with the company of another could at least have the decency to keep their nocturnal enjoyments to themselves. But then I wondered at the length of the cry, and the tone. Whatever was happening didn't sound very pleasurable, and if it was meant to be, then both parties involved had failed. There was something primal and finite in it, and when it stopped it did not begin again. The sound had come from outside my window, and for a moment I thought to tell my father, but then I remembered that he was dead and my heart fell as I lost him all over again. The feeling passed quickly, as it was something I was accustomed to; the same thing happened at the end of every dream.\n\nI shook my head, refusing to dwell on it. A woman was in trouble, and there were not many who lived within the confines of the estate that I would not count as my friends. I threw off the blankets and ran to the wardrobe, pulling out my warm dressing gown. Winter was coming, and the house was growing colder every evening. I pulled my hair over one shoulder, like my mother used to, thinking how much it was like hers\u2014soft and pale gold in the moonlight, lacking only her distinctive scent of lilac and jasmine. I observed myself quickly in the mirror. Every photograph of my mother had been lost in a fire years before, and when in need of comfort or strength I could sometimes find traces of her in my own features. Though I was taller than she had been, I had the same short, pointed nose and lips that were always slightly parted, as if I had something to say (which I often did), and hazel eyes like my father's. I slid the robe over my white cotton nightgown, the one Jonathan had loved so much, and left my room.\n\nEverton was a large country house, and while it had once been very fine, it had fallen into a comfortable state of disrepair well before my arrival nine months earlier. The burgundy carpets in the hallway were worn and fraying at the edges; the gaslights, turned down to candle flames with just enough light to cast rich black shadows along the walls, were tarnished; the floral pattern of the wallpaper cracked and withered on the vines as it peeled away from the walls. This condition was not for lack of trying. Mrs. Norman, the housekeeper, seemed to hire new maids daily in her futile efforts to bring the house back to its former glory, but it was no use. The manor continued to crumble away. Just the week before, the cook claimed to have seen mice scurrying about her kitchen. The other servants had started to whisper that the spirit of the house, if there ever were such a thing, had died with its mistress the year before.\n\nFor my part I did not mind the imperfections of the place. There was a warmth to it, a kind of intimacy that only comes with age, like the creases around the mouth that appear after years of excessive smiling, or a favorite blanket worn down from friendly use. It was certainly less intimidating than the cold, austere manors found in the larger towns and cities. Everton was happily flawed, like any person of true merit. It was a house of character, and I sustained that thought as I padded down the dark hallway.\n\nThe children had their nanny in a room connected to the nursery, but all the same I felt responsible as their governess to look in on them. Nanny Prum was known to drink after putting the children to bed. She was a very silly drunk, tripping over carpets and talking to birdcages as if they were party guests in a very high-pitched voice that was not at all like the deep baritone she usually employed while sober. Because of her predilections she slept very deeply, and a random sound in the night was unlikely to disturb her whereas it could very well tip the younger of the two boys into a web of nightmares that the both of us would then have to spend the remainder of the evening cooing and coddling away.\n\nThe door opened as I approached it, and a small head with wild blond hair emerged from the gloom, peering in my direction with round green eyes.\n\n\"Charlotte?\"\n\n\"Go back to sleep, James.\" I took him gently by the hand and led him back into the room, but not before he stuck out his bottom lip with indignation.\n\n\"But I heard a noise and Nanny isn't in her room and I'm scared,\" he said in a single breath. I sat him down on his bed and smoothed out his hair, brushing it away from his face as his older brother, Paul, growled dangerously from beneath a mound of covers at the other end of the room, apparently as resolute in not being disturbed by the nocturnal rustlings of the house as his five-year-old brother was in taking part in them. James had left Nanny Prum's door half open.\n\n\"Are you sure she's not there?\" I asked him in a voice just above a whisper. The little boy nodded carefully, wide-eyed and eager to be of help in the strange business of adults that only takes place when children are asleep in bed. I lifted him so that he straddled my waist and entered the nanny's room.\n\nThe bed was indeed empty, and I began to worry. Nanny Prum was not the sort of person to leave the children unattended, and she was certainly not the type to wander the grounds of the estate at night, even while intoxicated. She was a woman of some physical substance, and there were few people in the village who were not intimidated by her girth.\n\nI tucked James back into bed and stroked his forehead until he fell asleep again. Paul continued undeterred from his slumber, and I sat in Nanny's rocking chair curled into a blanket like an old maid, which was how I felt\u2014full of maternal feeling for the children and anxiety at the absence of my friend and confidant. Only a year before I would have been lying next to my husband in bed, the mistress of my own estate. How odd are the places one finds oneself as time passes. It's best not to look back, but how can one resist? I slept very briefly, the specters of the past only just uncoiling from my subconscious like a blot of ink unspooling itself in a pool of water, before the door to the nursery was opened by one of the maids.\n\n\"Mrs. Markham?\" she whispered in surprise. I put a finger to my lips and met her by the door, careful not to wake the children. She appeared very frightened, and I placed my hand over hers. She was shaking.\n\n\"What is it, Ellen?\"\n\nThe maid closed her eyes and grasped the silver cross that hung around her neck with callused fingers. She was a stout, rotund woman, never one to talk out of turn and hardly ever intimidated by anything, but all decorum seemed to have left her as she took my hand and kissed it. Her lips were as rough as her hands looked.\n\n\"Oh, thank the Lord, Charlotte! When I went to your room and found it empty, I was certain that . . .\" She stopped herself and sighed. \"You're needed in the kitchen.\"\n\n\"At this hour?\"\n\n\"It's a dreadful thing, too dreadful to mention so close to the ears of the children, be they sleeping or awake. I'll keep watch over them while you're gone.\"\n\nShe patted my hand but would tell me no more than she already had, so I left the boys in her care. The house was still dark, but now there were footsteps in addition to my own, and voices. In another room, a woman who was not Nanny Prum spoke quickly in a trembling voice. I crept along the hallway, down the grand staircase, through the dining room, and into the kitchen, where a small group of people had gathered over a pale figure collapsed on the cool stone floor. It was Susannah Larken, the apprentice seamstress from the village, wife of the local barkeep, and my friend.\n\nHer head was in the large lap of Mrs. Mulbus, the cook, who knelt on the ground and stroked the side of the poor girl's face, which was now nearly as red as her hair. Mrs. Norman, the housekeeper, and Fredricks, the butler, stood anxiously beside them.\n\nI bent down and took her hand. The wild look in Susannah's eyes abated slightly, and her breathing returned to normal.\n\n\"Oh, Charlotte, it was dreadful!\" She blinked away tears and began to sob.\n\nMrs. Norman, a severe, controlling woman with a hook nose and an anxious, birdlike disposition, continued speaking where my friend could not. \"There's been a murder,\" she said with a hungry, ghoulish enthusiasm.\n\nI wanted to slap the housekeeper's face for her repulsive insensitivity, but I restrained myself as Susannah sat up and continued her story.\n\n\"I was taking Mr. Wallace home from the pub. He'd had a bit too much to drink, and Lionel was busy behind the bar. Mrs. Wallace couldn't be bothered to collect him. You know how that woman is.\"\n\nI nodded in agreement. Mildred Wallace was the village busybody, eager to know everyone else's business so that she might forget her own. For years her husband had been the most loyal customer of the Larken brothers' pub, the Crooked Stool, but she continued to deny it, telling anyone who would listen how much her dear Edgar loved his nighttime strolls about the village.\n\nSusannah curled her lip into a sneer. \"Wouldn't lift a finger to help a soul, not even her own husband. I took him home to his cottage and went back by the path along the lake. That's when I heard the scream, that terrible sound, and I saw them at the edge of the forest behind Everton. There was a man standing over a woman on the ground, a man dressed all in black.\" Suddenly I remembered the man from my dream. My mouth went dry and a chill prickled across the surface of my skin. I brushed the thought aside as mere coincidence and begged her to go on.\n\n\"Lionel had given me the club, just in case I had any trouble.\" She fingered the wooden bat at her side, a small, heavy thing with just enough force to knock some sense into a drunken attacker, but perhaps not enough to ward off someone with murder on his mind.\n\n\"I ran over to help her, but there was nothing to be done . . .\" Her voice gave out, and she closed her eyes as if to stop herself from seeing it all again. I squeezed her hand, and brought it to my cheek.\n\n\"Who was it, Susannah?\"\n\nShe took a deep breath and opened her eyes.\n\n\"It was Nanny Prum . . . all in pieces. Like she'd come apart from the inside.\"\n\nI looked up at the others, but none of them could meet my gaze. They were all lost in shock. Even Mrs. Norman's unpleasant interest in the matter had soured. For myself I could not believe that something so horrific could possibly have happened in a village as quiet as Blackfield, at a house as great and noble as Everton. I believed Susannah and everything she said, but just as I did when I woke from my nightmares, wishing them real and everyone I ever loved still alive and well, I hoped that there had been a misunderstanding, some mistake, perhaps a play of shadows and moonlight over the ground that had made the situation more grotesque than it actually was.\n\n\"The constable . . .\" I spoke up weakly but felt as if I might be sick, for when I said it aloud I knew that there could have been no mistake. Susannah, having worked for many years in a dress shop and in a pub, had an eye for detail no matter how small. Something unspeakable had happened to Nanny Prum in the woods. Who would tell the children?\n\nFredricks spoke up in a wavering, nervous voice that was not much different from the one he normally used. \"Mr. Darrow and Roland have already gone to fetch him.\"\n\n\"He saved my life . . .\" Susannah's eyes began to glaze over again with a look lost in terror and madness. Her nails dug into the flesh of my hand. \"When I ran to help her, the man in black tried to come toward me. He smelled dreadful, like the very depths of Hell. It was so strong it burned my throat. I nearly fainted, but then Roland was there and the man fled into the woods. He saved my life.\" She started to sob again, but then caught herself. \"Someone must tell Lionel.\"\n\n\"Of course.\" I looked to Fredricks, and he left to fetch Susannah's husband, who was probably still closing down the pub. Mrs. Mulbus made a pot of tea while we waited for the constable to arrive. He was not much help when he did.\n\n\"Looks like wild animals to me\" was the first thing he said after he swept into the kitchen with Roland, the groundskeeper of the estate, whose burly appearance belied a gentle, soft-spoken disposition. He nodded to me as he leaned against the wall, recovering from what I could only imagine to be an extended state of exhaustion and shock. Constable Brickner, a portly, balding man with a weak chin and a mustache too large for his face, was not a popular man. Whatever the crime, he didn't inspect it so much as pass judgment on it, disregarding facts and eyewitness accounts in favor of his own infallible opinion. Luckily, his opinion was easily swayed by whomever he last spoke with, and all anyone had to do to win an argument was to be the last one to speak to him before the case was closed.\n\nBehind him, the door stood open, the dark of the forest beyond fractured in the moonlight until Mr. Darrow, the master of the house, stepped forward to follow the constable inside, his skin pale and radiant against the shadows, his dark blond hair tangled and windswept, his cheeks mottled from the cold. He looked at me directly as he crossed the threshold into the kitchen, and in his eyes I saw that it was as bad as anything I could imagine. We were the closest to Nanny Prum, and for a moment it was just the two of us in the kitchen, framed in that moment of time by the beginnings of grief and an almost conversational familiarity with death.\n\n\"But there was a man, I saw him!\" Susannah was feeling much better now and sitting at the cutting table, eating from a plate of biscuits that Mrs. Mulbus was nearly forcing into her mouth with meaty fingers.\n\nBrickner stroked his mustache and squinted. \"Surely no man could do that.\" He did not elaborate, but the emphasis on the word was enough for me to envision what was left of my friend on the floor of the forest. Nanny Prum had been a force of nature in her own right, and whatever happened to her, it would have taken an enormous amount of strength in addition to the obvious brutality.\n\n\"Perhaps he found the body before you did and ran off for fear of being mistaken for a killer.\" Constable Brickner picked up two biscuits, and then another, eating each in a single bite. Eventually Susannah pushed the plate toward him, but Mrs. Mulbus took it away with a look of disgust before he could finish them all off.\n\n\"He came toward me when I tried to go to her. He was going to attack me.\" Susannah's voice began to rise, but Brickner shook his head with blustering confidence.\n\n\"No man in town could do such a thing. Must have been an animal. I'm sure of it.\"\n\nSusannah stood from her chair, but Lionel came in at that moment with Fredricks and instead of launching herself at the constable, as she appeared ready to do, she collapsed into the arms of her husband. He took her home, a sobbing mess of nerves, while Mr. Darrow joined Roland and Brickner in collecting the remains of Nanny Prum. Mrs. Mulbus cleaned the kitchen after everyone had left, while Mrs. Norman and I went back to our rooms.\n\n\"Someone will have to tell the children,\" said the housekeeper.\n\nI wondered if she was assigning me the task or asking to do it herself, so all I gave her was a weary nod of the head. I did not look forward to the days ahead. The children had already seen too much death in recent years with the passing of their mother, the late Mrs. Darrow, the year before, and the loss of another woman from their lives was bound to do untold damage to the boys' already broken hearts.\n\n\"Tomorrow,\" I said quietly. When I relieved Ellen of her watch over the children, they were, fortunately, fast asleep. I returned to my room and slipped out of my winter robe, back beneath the covers of the bed, already chilled from my absence. But I could not sleep. I rarely dreamt of the same thing twice, but that night I could not repel the fear that if I closed my eyes I would find myself once more in the infinite and mysterious ballroom, my lost loved ones now accompanied by the large, imposing figure of Nanny Prum, dancing amid a room full of strangers led by the man in black, further and further into the darkness.\n\nI rose from my bed and paused at the door to my room, fingertips grazing the doorknob with trepidation, knowing full well what would happen if I left my chamber and wandered the dim corridors of Everton until I found refuge in the place I went when the nightmares became too much to bear.\n\nI opened the door and stepped into the hallway. The air was cool inside the house, but the carpet was soft and warm against the soles of my feet. I ventured up the stairwell of the east wing and came to a set of double doors I had discovered on a similar evening months earlier, just after my arrival at Everton.\n\nThe loss of Jonathan had felt especially fresh that night, but I refused to cry alone in my room. I needed to separate myself from the sorrow, to put it somewhere and lock it away for safekeeping, where it would be waiting for me to take it out again on other lonely nights, so that I might explore it in the darkness that lay beyond midnight.\n\nThe space behind the double doors might have once been a music room, but the instruments had been covered in sheets and stacked against the walls for storage. The only piece of furniture that remained was a simple divan. I had found Mr. Darrow standing before the window gazing pensively into the night. I tried to turn back, but he had already noticed my presence and beckoned for me to join him.\n\n\"This was her favorite room. She played music, you know . . . the harp, piano, violin . . . They said she was a prodigy when she was young.\"\n\n\"Jonathan liked the accordion. It was utterly ridiculous, but he always made me laugh with the dance he did as he played.\" I smiled at the memory and noticed that Mr. Darrow was staring at me with a curious look, as if he were searching for something in my eyes. I turned away.\n\n\"There are times I'm in town, or in the city, when I see a woman from behind. I know it can't be her, but her hair may be just right, and her dress so familiar that I want to take her into my arms before she can turn around and break the illusion. Am I insane?\"\n\n\"Grief makes us all mad. I often imagine that I'm able to speak with him one last time.\"\n\n\"What do you say?\"\n\nMy throat tightened, but still I smiled, a portrait of Victorian composure despite the maelstrom of pain and regret that spun so quickly in my chest I felt as if it would tear my flesh away in strips from the inside out, until nothing would be left and everything I felt erupted out of me to devour the world.\n\n\"So many things. What would you say to your wife?\"\n\n\"I would tell her about the boys, as best I could. She loved them so. I'm afraid I've been rather distant with them. I would ask for her forgiveness on that point as well.\"\n\n\"She would forgive you.\"\n\n\"You're a kind soul, Mrs. Markham.\"\n\n\"We're only as kind as people perceive us to be.\" I almost finished by calling him Jonathan, but caught myself before the word could form on my lips. Instead it stayed with me, calming the whirl of emotion that had built inside, and even though I kept the name in the silence that settled over us as we sat down beside one another on the divan, I knew that I had given it to him already. From that night on, we became nocturnal confidants, meeting in the sanctuary of the music room whenever fate and mourning compelled us to happen upon one another and remember aloud our lost loved ones as the sky beyond the windowpane turned with the stars into morning. At times our sessions together would only end when the sun threatened to appear over the horizon; at others they would continue on until a lull in the conversation became punctuated by a prolonged stare, or an accidental touch of one hand against the other charged the space between us with something unspoken and unacknowledged. We filled the music room with many things, but always left them there when we were done.\n\nIt was with great relief and little surprise that I found Mr. Darrow on the divan the night of Nanny Prum's murder, and together we sat in the darkness to wordlessly become reacquainted with the third member of our party: death.\nCHAPTER 2\n\nAn Inconvenient Holiday\n\nThe funeral was held at St. Michael's Church, a little toy parish on top of a hill overlooking the quaint village graveyard overrun with wildflowers and ivy. The vicar, Mr. Scott, a middle-aged bachelor with hair so fine and delicate it seemed to float over his head like a halo, gave an unusually somber sermon, only sporadically interrupted by Mr. Wallace's drunken outbursts. The poor man hadn't stopped drinking since he'd heard what almost happened to Susannah after she took him home from the pub the week before. His wife, Mildred, stood stiffly beside him clutching his arm, trying not to grunt as she struggled to keep him standing through the service. The two of them swayed back and forth, nearly making the rest of the mourners seasick. In a way, I felt that Nanny Prum would have approved of the spectacle.\n\nI stood with the Darrow family at the front of the church, before the colossal casket that Mr. Darrow had purchased. Nanny Prum had no family, or at least none that she ever spoke of, and so Mr. Darrow had spared no expense with the costs of the funeral. She had been a large woman in life, and the size of her coffin only highlighted the strangeness of her demise. Stranger still were the boys themselves. James held my hand and fidgeted in his seat, unable to stay interested in even something as extreme as the murder of his nanny. He did not cry, whereas his older brother, Paul, wept so profusely that the entire front of his shirt grew damp with tears. I tried to comfort him\u2014to take his hand, to kiss his forehead, as I had seen Nanny Prum do so many times before\u2014but he refused to be touched, preferring instead to be alone in his grief.\n\nMr. Darrow sat on the other side of the children with a glazed expression, his bright blue eyes lost in some distant memory. I had come into his employment only three months after the death of his wife, but I imagined her funeral must have looked very similar to Nanny Prum's: the same people, the same graveyard, even the same time of year. Death, it seemed, was another season, an inconvenient holiday put aside like any unpleasant responsibility, only to reappear once it's been forgotten, a reminder that time has passed, life has changed, and that nothing ever stays the same.\n\nThe next day Mrs. Norman and I put away Nanny Prum's things. Constable Brickner and his men had already been through her room looking for clues, despite his insistence against the evidence that the attack had been perpetrated by some wild animal. They were careless and cavalier in their search efforts, leaving the dresser drawers overturned on the bed and her belongings scattered across the floor.\n\nThere was very little clothing, only a couple of severe, high-collared black cotton dresses and a soft velvet maroon gown that she wore for special occasions. There were also books: a King James Bible, a volume of fairy tales, and a novel of some romantic melodrama told in three parts. I found a small wooden chest next to her bed filled with scraps of paper, pieces of jewelry, and faded pictures. I supposed it was a memory box, and I imagined her going through it each night before bed, thinking of all the children she had ever raised, gone off into the world, grown and with families and memories of their own. Perhaps she consoled herself by thinking that they thought of her every now and then when they remembered their youth, and that they might have smiled when they did.\n\n\"I warned her this would happen.\" Mrs. Norman stood in the shadow of the wardrobe, sliding dresses from their hangers and folding them in a sharp, mechanical fashion with her thin, birdlike fingers. She did not turn in my direction as she spoke. The wardrobe was now empty, and the contents of the room were slowly being siphoned away.\n\n\"What did you tell her?\"\n\nThe housekeeper went still for a moment, and then craned her neck toward the entrance of the room, listening to the ambient sounds that filled the house: women gossiping and snickering beneath their breath, heavy footsteps on creaking floorboards, a distant cough, metal scraping against wood . . . She added to them as she crept to the door and closed it with a soft click. Mrs. Norman took my arm and sat us on the bed, where she brought her face close to mine and began speaking with quiet urgency.\n\n\"That she was in terrible danger.\" A wave of dread went through me, and as she continued I could only think of the man in black standing over Nanny Prum's body, poised to lash out at Susannah. \"Someone must watch over this family, now that dear Mrs. Darrow has left us, God rest her soul, and I do what I can in my way. I clean up afternoon tea, every afternoon tea, and one can't help it if one sees something in the leaves.\" She pursed her lips, and appeared for a moment wearier than I had ever seen her before. \"Someone must watch over them, and warn them when necessary of the things that are coming. There is evil here. I did my part, I warned Nanny Prum, but she failed to heed my advice.\"\n\nThis was the longest conversation I'd had with the housekeeper in my nine months at Everton, and while her words were jarring, she spoke with a conviction that I could not ignore. \"What did you tell her, Mrs. Norman?\"\n\n\"There was a man in her life. Who, I'm not sure. But he meant her harm, and from what Susannah Larken saw, he meant her a great deal more than that.\"\n\n\"Have you told anyone else?\"\n\n\"What would be the point? Most people no longer believe.\" Mrs. Norman suddenly took my hand and looked into my eyes. \"Do you believe, Charlotte?\"\n\nI thought of my childhood in India, of the holy men and mystics, and of my mother gasping for air in her sickbed. I had been alone with her when she finally died, my father shouting at the doctor just outside the door. I never told him about the man in black who suddenly appeared next to her bedside. The room was dimly lit, so I could not make out his features, but when he moved in to touch my mother's body, I launched myself at him, kicking and biting with all my might. In the instant I reached him he was gone. My father reentered the room with the doctor a moment later and lost himself in his grief. There had been no time for the man to escape unnoticed and so I said nothing, thinking it all a dream until years later when my father and I were dining in the conservatory of our estate.\n\nOne moment he was smoking his pipe, gesticulating wildly at the azaleas as he explained his feelings about a certain political party, a wreath of smoke around his head, and the next he was grabbing his chest and slumping to the tiled floor. I cradled him in my lap and refused to cry before the doctor could arrive. When the bell rang, my father's manservant left us to answer the door, but I could sense we were not truly alone. The man in black stood by my side, wiping a bead of sweat from my father's brow. I knew then that he had died in my arms.\n\n\"Who are you?\" I shouted at the stranger. He placed a gloved hand beneath my face and tilted it to meet his own. Even at such a close proximity his features were occluded by a perpetual shroud of gloom. I shrank back in horror and clung tightly to my father, but the man stepped away from us, the plant life in his immediate vicinity shriveling to brown decay and dust.\n\nWhen I was married I told Jonathan about what I had experienced with my mother and father. At first I wasn't sure he had believed me, but then he wrapped his arms around my waist and whispered in my ear: \"I believe that the world is far more complicated than we could ever possibly understand. Perhaps you saw a hint of something that most people aren't meant to see. Death comes to us all, my love.\"\n\nAnd so it did. The fire came for my husband only a few months later, and as he lay dying on the charred remains of our estate, I was met by the man in black for a third time. I was too weak to attack him or to even shout after him as he closed Jonathan's eyes with his gloved fingers. But I did ask him a question: \"Do you cause this, or are you simply a vulture come to pick at the bones of my life?\"\n\nHe tilted his head to the side, but whether or not it was some kind of response, I never knew. In the morning they found me still clutching my husband's body, the trees and grass around us suddenly withered and dead, though the fire had never made it to the forest.\n\nWith each passing year I became more convinced that it was as Jonathan had said; for whatever reason, Death made himself known to me as he took the souls of my loved ones to the Other Side. I said none of this to Mrs. Norman. Instead I met her gaze and replied: \"I believe that the world is far more complicated than we could ever possibly understand.\"\n\n\"So you'll believe me if I tell you that you're in danger?\"\n\n\"What sort of danger?\"\n\n\"The same as Nanny Prum. A man waits for you. He watches you.\"\n\nMy face suddenly grew very hot, though I could not decide if it was due to panic or anger. Susannah had seen a man dressed all in black. If it was the same one that I had encountered, then was I being stalked by Death? And if so, then who might he take next? I was nearly shaking.\n\n\"How do I stop him?\"\n\n\"Be careful. Be watchful.\" She lifted the trunk from the bed and took it out of the room, providing a knowing look in my direction, and then said nothing more on the matter. We put everything else away, into boxes and cloth sacks, and left them in the hallway. Roland would load Nanny Prum's belongings into the wagon and take them to the church. There would be a bazaar at the start of winter, and the people of Blackfield would pick through Nanny Prum's things, dispersing her memories like seeds on the wind.\n\nIn the days that followed the funeral, I moved into the room connected to the nursery and filled the empty spaces with pieces of myself. I couldn't help but wonder, in the morbid way of all people who have lost more than once, what would be made of my things if I were to die before my time. There was the wedding ring I placed in a drawer of the table next to my bed, unable to wear it any longer, the weight of the thing too great a burden to bear; a lock of my mother's hair, bound in a thin blue ribbon, her scent intact, that I used for a place holder in the book upon my nightstand; my father's pipe, with a crack in its bowl, a dried husk of Sunday afternoons in his study, on his lap, reciting poetry, now with my mother's jewelry in a small box in the wardrobe. This was where I kept my memories, ensconced in little tokens that would be meaningless to anyone else. I wondered how people would remember me, what might cause them to stop, many years later, and pause for a moment to recall a woman named Charlotte.\n\nTo remember Nanny Prum I kept an ivory brooch that she used to wear about her throat, engraved with the image of a woman. Perhaps it was her mother or her grandmother? I never asked. Perhaps she had bought it secondhand at a bazaar, or took it to remember a lost friend as I did. It was elegant in a simple way, and it reminded me of the time we first met, during my first day at Everton.\n\nJonathan had not been the only one to perish in the fire. Six members of the household staff had died as well, leaving families with no means to support themselves. Against the wishes of our lawyer, Mr. Croydon, I used what wealth I had at my disposal to provide them with an element of comfort, though it could never have replaced the loved ones they had lost. I could not have lived with myself otherwise, and the thought of making a new home for myself, orphaned, widowed, and alone, was too much to bear. I still had my father's military pension. It was not enough to continue the kind of life I had been accustomed to, and so Mr. Croydon begrudgingly agreed to find me some kind of employment as a governess, where I could insert myself into someone else's life and family, if only for a little while.\n\nWithin a matter of weeks I found myself wandering the hallways of Everton, admiring a painting of a bleak, gray landscape, the signature of the work an enigmatic \"L. Darrow,\" when a hefty woman in heavily starched black skirts and the aforementioned brooch waddled down the corridor to meet me.\n\n\"Mrs. Markham!\" Her voice barreled off the walls in a loud, rolling crescendo. She approached me with open arms, and her thin, wide lips stretched into an effortless smile rounded off by apple-colored cheeks. The woman would have been plain if her sense of cheerfulness hadn't been so infectious. I found myself laughing as the stranger embraced me with thick, fleshy arms.\n\n\"Wonderful to meet you, my dear! I'm Nanny Prum.\" She released me from her grip and I quickly tried to catch my breath.\n\n\"The pleasure's all mine.\"\n\nThe woman cackled and slapped me on the back with a hand like a round steak. She took my arm and led me down the hallway.\n\n\"I expect we'll be like sisters, or at the very least a pair of silly aunties to little James and Paul! Such good boys, sweeter children I've never known. Very different temperaments, mind you, but sweet just the same. I wouldn't go so far as to call them angels, because they are children, after all, but their hearts are in the right place. Although I suppose you're more concerned with their minds, hmm?\"\n\n\"Their well-being is my primary concern.\"\n\nNanny Prum nodded in approval and led us up the grand staircase, careful to avoid the holes in the red fabric that covered the stairs, and the cracks that had started to appear in the wood.\n\n\"So you've met Mr. Darrow?\"\n\n\"Yes, he seems to be a fine sort of gentleman.\"\n\n\"To be sure, very fine and rather strapping if you ask me. But then his late wife, the boys' mother, was also very beautiful. Such a pair they made! So sad that she was taken from this world so young. But it is not for us to dwell on such things. We must help the children forget.\"\n\n\"I'm not sure I would say forget, exactly . . .\"\n\nIn hindsight, it was the first and only disagreement we ever had, and one, I'm sorry to say, that I would win. I would not let the children forget their late mother, or their nanny.\n\nWe turned the corner at a painting of a nocturnal landscape with a castle looming in the distance. Nanny Prum pointed to it as we passed.\n\n\"Mrs. Darrow enjoyed the fine arts: painting, singing, sculpting, that sort of thing. Although I daresay she had a rather morbid aesthetic.\" She stopped at the end of the hallway and entered the nursery.\n\nThe children were waiting for us. Paul was nearly thirteen at the time, thin and pale with dark hair like his mother, whose portrait hung in Mr. Darrow's study, and intense blue eyes. His brother, James, was four; a sandy blond\u2013haired little boy who wore a light, playful expression on his round, dimpled face. He held a small bouquet of wildflowers, and bowed politely while his older brother leaned against the wall.\n\n\"We are very pleased to meet you.\" said James.\n\n\"And I must say I'm very pleased to meet the both of you! I knew there was already one gentleman at Everton, but I had no idea I'd have the pleasure of acquainting myself with two more. And what lovely flowers!\" I clasped my hands together in approval and smiled as I was expected to do, even though I had no idea what I would do with them. Flowers always made me nervous, especially when given as gifts. One is expected to keep them alive, to help them flourish for a short while, and if they do not, well, then what does that say about a person? Too much can be inferred from such a failure: Is she simply incapable of keeping anything alive? Dear me, I do hope she does a better job with the Darrow children! And so on and so forth. \"I haven't smelled anything so wonderful since I was a little girl in India.\"\n\nJames immediately perked up. \"You lived in India?\"\n\n\"Yes, for many years. My father was stationed there when I was about your age.\"\n\n\"Did you get to see a cobra?\"\n\n\"Within striking distance! But thankfully it was under the spell of a snake charmer at the time, so it was probably much less of an adventure than you'd like to think. But I did see many wonderful things, and if you're at all interested I'm sure we could devote a small portion of our class time to the discussion of the Far East.\"\n\n\"Please, ma'am!\"\n\nPaul appeared to be uncomfortable during this entire exchange. He looked around as if he couldn't wait to leave.\n\n\"I do apologize, Paul. Sometimes I forget that not everyone is interested in hearing my stories.\"\n\n\"Oh, it's not that, ma'am. It's just that we're late, you see.\"\n\n\"Late?\"\n\nNanny Prum pushed me aside and took the boy by the arm. \"Not now, Paul dear. We can do that some other time.\"\n\nBut I continued to press the issue. \"On the contrary, I have no intention of disrupting anyone's daily routine, especially if they have an appointment elsewhere.\"\n\n\"It can wait.\"\n\n\"Really, I would feel just dreadful if I imposed on anyone my first day here.\"\n\nNanny Prum sighed, shrugged her large shoulders, and released the boy. \"If you insist. I'll help the boys with their coats.\"\n\nI assisted Nanny Prum with this endeavor and followed them out of the house into the sunlight. She and I walked beside one another as the boys trailed ahead of us, Paul with his hands in his pockets, quiet and distant, and James, skipping along singing some nonsensical song at the top of his lungs. We followed the path down to the entry gate and turned onto the road that led to the village of Blackfield below, situated comfortably at the base of the hill that provided Everton with a justified air of importance.\n\nThe village was full of thatched buildings and cobblestone streets. It was a small, wholesome sort of place despite having two pubs. Nearly all the people of Blackfield found this to be a sign of progress and perhaps evidence that the village was slowly becoming a town, except for Mildred Wallace, who complained bitterly to anyone who would listen that one pub was sinful, but two was simply decadent. After a while it was left to poor Mr. Wallace to listen to her ravings, as the rest of the villagers would turn around corners or move indoors as soon as they saw Mildred coming. It went without saying that this had something to do with Mr. Wallace's frequent patronage of both establishments.\n\nWe continued down the road until the buildings were overtaken by farmland and rolling hills. St. Michael's stood on its little hill at the edge of the town, a small country church with stone walls and a quaint, well-kept vicarage. A graveyard sat between them, and as we neared it Paul hastened his pace. He was through the gates and winding around the tombstones before I realized what we had come to do. James caught up to his brother and they knelt before a sizable headstone that bore their mother's name. The earth was still settling where she had been buried. Nanny Prum held me back as the boys chattered with enthusiasm at the plot of dirt that held their late mother; she spoke to me in a soft whisper as different from her normal voice as was possible while the children updated their mother on everything that had transpired since their previous visit.\n\n\"They come every day, the poor dears. I'm not sure that James even understands what's happened to her. He didn't cry at the funeral. But Paul . . . it cut him deeply.\" Even as she spoke it was apparent that Paul was taking the most time talking to his mother, and James became distracted by a pair of butterflies. To Nanny Prum's horror, he chased after them, hopping from one grave plot to another. She shouted after him. \"James Michael Darrow, stop that at once!\"\n\nPaul ignored his brother and continued his conversation with his mother, but he was surprised when I joined him before the tombstone.\n\n\"Please don't stop on my account,\" I said to him. The sun was heavy in the sky.\n\nThe boy squinted at me, and then looked on as his nanny chased after his brother, the hem of her skirts in her hands as she tried to avoid offending the dead. \"She thinks it's strange.\"\n\n\"It's difficult for people who have never lost someone close to them to understand what it's like.\"\n\nPaul looked back at the tombstone and traced the inscription of his mother's name with his fingers. \"I was the only one who wasn't there when she died.\" He looked up at me, a question in his expression.\n\n\"Why was that?\" I asked.\n\n\"I couldn't bear to see her like that. I tried, I truly did. I held her hand and I kissed her cheek, but then she would begin to wail. It wasn't even her anymore. It was as if something had taken her place, this thing that lived inside her skin, not even human, only just alive. I didn't want her to see me cry, so I kept away. I'm a coward.\"\n\n\"You are no such thing,\" I said, placing my hand tentatively on his shoulder. \"She knew how you felt, I'm sure of it.\"\n\n\"I dream about her almost every night,\" he sighed.\n\nI thought of my own dreams. I looked forward to them more than most things in life, and even when they turned into nightmares, I still found a sense of relief in seeing my mother dance or hearing Jonathan laugh. They became real in a way that memory could not make them.\n\n\"That must be wonderful.\"\n\n\"Sometimes. But when I wake up I have to remember that she's gone.\"\n\nJames shrieked in the distance. Nanny Prum had lifted him by the waist and held him under her arm. The little boy was screaming so ferociously that the vicar stepped out of his cottage to see if someone was being murdered in the graveyard.\n\n\"Is everything all right?\" Mr. Scott was a few years younger than my father would have been had he survived, and his hair billowed over his head, threatening to blow away completely as he looked frantically for the source of the screaming.\n\nNanny Prum waddled over to him, full of cheerful exuberance. \"Yes, Vicar, quite all right! Little James must learn to respect the dead.\"\n\nPaul and I left the grave and joined the others before the cottage. Nanny Prum set James back onto the ground and straightened her dress, even though it was so heavily starched that it was impossible for it to wrinkle. She introduced me to the vicar, and he inquired about my family before becoming embarrassed by the question when he learned that I had none.\n\nWe continued our daily trips to the graveyard, despite Nanny Prum's apprehensions about appearing to be too morbid, and every day Mr. Scott made a point of it to greet us before we left. It was something the boys and I would continue to do even after she died.\n\nNanny Prum's tombstone was not so very far from Mrs. Darrow's grave, and while the boys had considerably less to say to their late nanny than they did to their late mother, they did keep her updated on the happenings at Everton. Paul even began to bring her bits of gossip he had overheard from Ellen and the other servants, and while I scolded him for eavesdropping, I did not dissuade him from continuing to bring her news she might have enjoyed while she was alive.\n\nJames frequently grew tired of the game before it was time to turn back, and so he would go through the graveyard familiarizing himself with its other residents, having grown out of the habit of skipping from grave to grave thanks to the persistent conviction of Nanny Prum's large, heavy hand against his bottom.\n\nPaul always seemed less morose on the walks back to Everton. We had a specific route that we followed through Blackfield, starting with Mr. Ingrams at the blacksmith shop, who pulled and twisted burning metal in a shower of sparks like it was taffy, then to Mr. Wallace, who was the local clockmaker in addition to his duties as the village drunk. None of the clocks kept the same time, and so every moment in the shop was punctuated with the clanging of chimes or the cooing of a cuckoo, but the boys didn't mind. James relished the noise, and Paul watched with genuine interest when Mr. Wallace opened the clocks to show him the interlocking cogs and springs. He even smiled.\n\nAfter the clock shop it was impossible to avoid Mrs. Totter's bakery. She always kept the door open, even on the coldest days of the year, and the scents of freshly baked cakes and chocolate croissants and mincemeat pies all tangled together, snaking through the streets of Blackfield, a crisp, golden brown siren's call promising a warm, full stomach. I only let the boys purchase one item a week, usually a cookie the size of a dinner plate, but for the most part our trips to Mrs. Totter's were largely exercises in excruciation.\n\nMrs. Willoughby's dress shop was next, since that was where Susannah worked as an apprentice seamstress, and the week after Nanny Prum's funeral we found her alone in the front of the shop, pushing over a dressing mannequin out of frustration. She jumped when she saw us in the doorway, and then blushed furiously at her own display, her emotions laid bare with the mannequin rolling about on the floor.\n\n\"Hello,\" she said in a surprised, falsely tranquil voice, brushing away a strand of red hair that had fallen into her eyes. She bent down to pick up the dummy. \"How was the church?\"\n\nI frowned at her while her back was turned, but quickly removed the expression from my face. This was how the people of Blackfield referred to our daily visits to the graveyard. Most of them couldn't bear to name it for what it was, except for Susannah, who normally had no qualms about describing things exactly as they were.\n\n\"It was fine.\" I smiled, not judging her for attacking the lowly mannequin, who, for all I knew, might have had it coming after a day of uncooperative participation in the fine art of dressmaking. \"And how are you?\"\n\nThe question uncorked the remaining emotions that Susannah had thus far been successful in containing. She gripped the mannequin so tightly that I could have sworn she bent its metal frame. She sat down with a beleaguered sigh. \"Dreadful, Charlotte, just dreadful!\"\n\nI sat down beside her, amid the reams of fabric, which Paul examined thoughtfully with his hands, listening but not really listening, lost in his own morose thoughts, while James pulled needles of all shapes and sizes from their pincushions and stuck them into chairs and tabletops with a violence unique to little boys, absolutely determined to not be bored by the dull conversation of adults.\n\nSusannah went on. \"Brickner came by the shop, to tell me, to tell me, that I was mistaken in what I saw\u2014the hysterical ravings of a frightened woman. Can you imagine?\" She was waving her hands as she spoke, before catching herself, and stopping. \"Well, I suppose you can. But I wasn't hysterical, not then anyway, not that night. I know what I saw. Wild animal indeed!\" Susannah sat back, deflated, and looked at me with searching eyes, waiting for me to say something.\n\n\"It goes without saying that I believe you, and so does half the village for that matter.\"\n\nShe seemed genuinely relieved by this, and whatever was taken from her by Constable Brickner's lazy certainty was replaced with newfound confidence, stronger than before, like mended bone.\n\n\"Then why doesn't Brickner?\" she asked.\n\n\"Simple people like simple explanations,\" I said before I could stop myself. Paul stifled a small laugh.\n\nSusannah smiled briefly, and then became dark and serious once more. \"It's not for myself that I want to be believed, you know. I don't need justice served, but she most certainly does.\" She nodded curtly and looked past the boys at some distant memory of Nanny Prum, who some months before perhaps sat in that very same spot, having a very similar conversation about something moderately less sensational than murder.\n\n\"I would think,\" I said carefully, trying not to get her hopes too high, \"that Mr. Darrow would be of a similar mind.\"\n\n\"Could he be convinced to speak with Constable Brickner?\"\n\n\"I don't know about convincing, really, but it could certainly be suggested, yes.\"\n\nSusannah leapt up from her chair and threw her arms around my neck. \"Oh thank you, Charlotte!\"\n\n\"No need to thank me yet. Wait until I've at least done something mildly useful!\" I hugged her back and extricated myself from her embrace, rising to leave with the children before she found it necessary to inflict upon me the same force of emotion that she showed to the poor dressing mannequin, who now appeared fatally dented along one side. Paul helped me pull James away from the cutting shears before he could do any damage to them. The three of us continued on our way through Blackfield, rising with the incline of the path to the steep hill that held Everton, hidden behind a line of trees dusted with autumn.\n\nWhen we arrived back at the manor, I sent the boys inside and wandered through the wood at the back of the estate until I found the flattened patch of earth where Nanny Prum had met her end. Even with the onset of fall, it was impossible to deny the peculiar ring of dead grass and brittle, shriveled plant life that encircled the place where she had died, the center of it still stained with flakes of dried blood as dark as the shadows that cloaked the man in black's smiling, hidden lips.\nCHAPTER 3\n\nThe Mistress of Everton\n\nMr. Darrow was not the sort of tragic figure to stay forever locked away in his study, which was itself one of the last bastions of Everton's former glory, with waxed oak-paneled walls, a tidy, ash-free hearth, and antiquities that had been collected from around the world by the master of the house and his late wife, whose adventures had been chronicled on the marble globe that stood at the center of the room, their travels marked by glittering jeweled pinpoints of color. No one ever saw Mr. Darrow tend to the cleaning of his study, but as he did not allow anyone else to do it for him, everyone supposed that that was what he did when he was alone, perhaps beating the curtains with sullen determination as he stared longingly at the portrait of his late wife that hung above his desk.\n\nBut he was not always in his study. There were a number of occasions where I went to the door, in need of additional pocket money for school supplies, or to seek his permission for a day trip to a particular castle or battlefield, and I would find the room empty. No one knew where he went, and no one could say that they saw him depart. He did not like to leave Everton, and so I imagined him haunting the shadows of the hallways or drifting into unused rooms, resting on a chair that had long ago been covered over in cloth sheets, or pausing before a clock to wonder if it had stopped with the death of his wife.\n\nHe did not usually dine with us, and so the children and I were visibly surprised when two weeks after Nanny Prum's funeral he sat down at the head of the table in the dining room, looking rather dour until Fredricks, the butler, who either was very drunk or had completely forgotten how to use his own hands, lost hold of the soup kettle and tipped it down the front of his uniform. After establishing that the liquid was not so hot as to be scalding, everyone laughed together, and even Mr. Darrow seemed to enjoy himself for a brief moment, but then he looked at the other end of the table, where the late Mrs. Darrow would have sat, and he retreated into his comfortable melancholy. I do not know why he chose that night to join us. Perhaps the reality of Nanny Prum's death had sunk in and he needed the company, or perhaps he was reminded of his children's existence and sought to comfort them with his distant, sad presence.\n\nSoon after dessert, the grandfather clock began to chime. I took the boys upstairs to change for bed, saying good night to Mr. Darrow and leaving him alone at the table to finish off the contents of a wine bottle. I worried about his state of mind, but I was too busy adjusting to my new responsibilities with the children to think too intently on it.\n\nIn the days that followed Nanny Prum's death, it had quickly been decided that the most practical thing would be for me to take on her duties. I already spent a great deal of time with the boys as their tutor, and if I were able to control their lives beyond the schoolroom it would make life that much easier for everyone else. I would have been remiss to turn down such an offer, especially since it was accompanied by a nanny's salary in addition to my own. I understood the value of such an opportunity and was eager to please the apparent generosity of Mr. Darrow.\n\nIt was not so awkward a transition as might be imagined. As governess I had still been expected to deal with the well-being of the children at every occurrence of illness or injury, and I easily transitioned from in-class paper cuts and bleeding nostrils to bed-wetting and bad dreams.\n\nThat night after Mr. Darrow's surprising appearance at dinner, I tucked James into bed and read him a story while Paul perused some obscure volume of poetry, his back toward me. He was very much like his father, especially in the eyes\u2014very blue, large, and thoughtful. I wondered what Mrs. Darrow had been like, if she had been as erratic and adventurous as James, or something else altogether. When James fell asleep, or at least pretended to in order to get me out of the room, I left them alone. I did not have to go far.\n\nThe nanny's room had two windows that overlooked the forest behind the house, and while the grounds of Everton seemed to respond to Roland's care better than the house itself did to Mrs. Norman's, the wood still had a wild look about it, especially at night, when the depths of the forest were hidden even from the light of the moon.\n\nI changed into my nightgown and slid into bed, trying not to think of anything. Not of Nanny Prum and the dead circle of undergrowth that marked her demise behind the house, or of poor Mr. Darrow, probably still drinking himself into oblivion. But then the very act of not thinking about something is rather self-defeating, since one must think about the things one wants to avoid before nothing can be thought of. With that last tangled thought, my attention drifted inadvertently to Mr. Darrow.\n\nIt did not escape me that my new position had effectively made me into a mother of two. I was, for all intents and purposes, their guardian. I saw to it that the boys were fed, bathed, schooled, and looked after. I was the closest thing to a mistress of Everton since Mrs. Darrow's passing. I cannot deny that this pleased me, or that I did not spend those last moments before sleep thinking about my time with Mr. Darrow in the music room, hands beside one another, almost touching, and what would happen should one of us decide to cross that fraction of space . . .\n\nI drifted off and dreamt of my childhood in India, of temples and jungles and crumbling, many-armed statues, tigers and cobras and monkeys, and of our home in the colonial residency of Lucknow. It was a familiar scene, the same one that ended every dream about my youth: my mother's room, with the lights dimmed as she lay in bed with cholera, her body slowly drying out and withering into the shape of a child, gasping desperately for air, each moment a struggle until death. My father was nowhere to be found, and the servants avoided coming into the room. This part of the dream always ended the same way, would only end if I crawled into the bed to be with her, over sheets and pillows and shawls, unable to find her until at last I reached the center of the bed and realized that I must enact her death, live through it in order to wake. I would feel someone standing over the bed and observing me writhing in the damp sheets, a man dressed all in black. I struggled to see his face, but it was dark, and despite the fact that it was a dream, I could not will the face into existence. It was as if it were separate from my mind and from the dream of India. I tried to sit up from the bed, but instead sank down deeper into the mattress. However, unlike in the other occurrences of the dream, this time the figure whispered to me:\n\n\"Children need their mothers.\"\n\nIt was a woman's voice, and as she retreated to the other side of the room I died on the bed, choking for air with long, unbearable pauses between each breath until I was back at Everton, gasping as sweat trickled down my chest and face. I threw the blankets off the bed so as not to get them damp, and was about to change when there was a knock on the door to the nursery.\n\n\"Charlotte?\"\n\nJames opened it without waiting for me to respond. He had been crying, his face as wet with tears as mine was from perspiration. Both of the boys seemed to have unique problems with nightmares. Being the more courageous of the two, James often dreamt himself into outlandish adventures featuring ghouls, mummies, and spider women, just to name a few from his well-populated menagerie of monsters. Occasionally, the capacity of his imagination to invent horrors would outpace his actual ability to tolerate them, and he would wake up in the middle of the night absolutely convinced that the spider on the windowsill was an agent of the malevolent Spider Queen, intent upon making him pay for the theft of her enchanted silver webbing.\n\nPaul was another matter altogether. Quite often he would already be awake when I entered their room to take care of his brother, angry that he had been aroused from a most wonderful nightmare, as he was that night when I took James back to his bed.\n\n\"I was at a ball,\" he said as James nestled his head against my chest. \"And Mother was there. She was young and beautiful. I tried to move across the dance floor to speak to her, but it was too crowded. Every person in the ballroom was dancing with something inhuman, and Mother took the hand of a creature who only pretended to look like a man. She waved to me from across the room, and I was about to go save her, but then James started screaming about the stupid Spider Queen.\" He shot his brother a spiteful look.\n\n\"She's not stupid!\" James lunged out of the bed, his tear-streaked face furrowed in rage, but because he was in my arms and only five years old, I was able to hold him back without much effort.\n\n\"It's nothing to me if you want to kill one another,\" I told them. \"I imagine that it would be much easier to care for one child as opposed to two. But I daresay your father would be furious with whichever one of you murders the other. If violence and murder are the methods you choose to use when dealing with family, then we can only surmise the tactics you might use when dealing with your peers would be that much worse. We would be forced to lock you away in the attic for the good of the village. I don't believe that such an existence would be a very pleasant one, but then it's not up to me to make your decisions for you.\"\n\nThe boys had no idea which one of them I was talking to, and as they tried to sort out what I had said, their anger abated. I tucked them into bed so tightly that it was difficult for them to move about, even Paul, who was completely horrified that I had the gall to treat him like his younger brother. Rather than struggle against it, they both gave in to the hour of the night and fell asleep. I watched them for a while to make sure there was no relapse of flared tempers, and when I was satisfied that they were truly asleep, I retired to my room. By then the notion of sleep had left me completely, so I changed into a fresh nightgown, combed my hair, read for a bit, and finally decided to make myself a cup of tea.\n\nI always preferred Everton at night. It was not a noisy house, the sort of place that creaked or groaned a whole symphony of innocuous sounds that, when taken together, could twist a shadow into something tangible and dangerous. It was simply dark and quiet without any pretense, with the kind of rich, musty smell that only comes with age.\n\nMr. Darrow was still in the dining room. He was startled when he saw me, but not drunk. The wine bottle was gone and had been replaced by a full afternoon tea spread, despite the fact that it was well past two in the morning. There was a large pot of black tea, still steaming, the usual cream and sugar, as well as tea sandwiches, scones with clotted cream, and a chocolate tea cake that sat on his plate, conspicuously untouched. He invited me to join him, and though there were many chairs to choose from, I sat down beside him rather impulsively and felt my face flush. He was a very handsome man, and the dining room was not the private refuge of the music room.\n\n\"Good evening, Mr. Darrow.\"\n\n\"Trouble sleeping?\"\n\n\"James had a nightmare.\" He looked concerned, and half rose from his chair, but I touched his arm and he sat once more, his eyes lingering where I had touched him. He gaped for a moment until I reassured him. \"It's all right. He's asleep again.\"\n\nMr. Darrow regained his composure. \"And now you're not.\"\n\n\"An occupational hazard, I'm afraid. May I?\" I reached for a teacup, but he grabbed it before I could and filled it with the black, aromatic contents of the teapot.\n\n\"Darjeeling.\"\n\n\"Wonderful.\"\n\n\"Scone?\"\n\n\"Please. Mrs. Mulbus has truly outdone herself, considering the hour.\"\n\n\"You underestimate my cowardice, Mrs. Markham. I would never approach dear Mrs. Mulbus's door at this time of night demanding food and drink. There is a specter of death that seems to hover over Everton, and I do not wish to tempt it.\" There was a weariness in his gaze, his blue eyes peering over the lip of his cup, framed by strands of his dark blond hair, fixated once more on the other end of the table, as empty and silent as the rest of the house. But then he smiled and reached for a scone. \"Besides, I'm not completely incapable.\"\n\n\"Of course not.\"\n\n\"I mean, why must one wait until the afternoon to have afternoon tea? That part of the day is so busy that one can rarely enjoy it.\"\n\n\"I quite agree.\"\n\n\"But I must say that it is nice to have your company outside of the music room.\"\n\nI tipped my cup toward him, and he returned the gesture. We sipped our tea in silence.\n\n\"You know, Mr. Darrow\"\u2014I finished my cup and set it in the saucer, carefully choosing my words\u2014\"if you require company during the daylight hours, I'm sure I could make the children available.\"\n\nHe nodded thoughtfully and refilled my cup. \"Yes, what must you think of me? Locked away in my study or wandering the halls of the house. I only seem to reclaim some semblance of my old life after everyone's gone to sleep. I'm afraid I've become something of a ghoul. It's rather pathetic.\"\n\n\"That's not how I meant it.\"\n\n\"But it is. Lily's been dead for . . . God, has it nearly been a year? And now Nanny Prum. How did you recover from the loss of your husband?\"\n\n\"I'm not sure one ever does. It still hurts for me to think of him. I miss him so.\" I relaxed as I said aloud what I always felt: his absence. I could tell that I was beaming as I thought of him. I dabbed my lips with a napkin. \"But there is something fortifying about the pain. It reminds me of how much I loved him, and my love is equal to the pain. It protects me, and it grows stronger the more I think of him. I'm sure that someday it will be the same for you and Mrs. Darrow.\"\n\n\"Perhaps, but then again . . .\" His brow contracted into an expression of anxiety as he finished eating a cucumber sandwich. \"Mrs. Norman will be up soon, and I like to avoid her when possible.\" He smiled weakly at me and stood from the table.\n\n\"Mr. Darrow, before you go. About Nanny Prum.\"\n\n\"Yes?\"\n\n\"Well, it's Susannah Larken. She's one of the most honest, reliable people that I know, and\u2014\"\n\n\"I believe her.\"\n\n\"You do?\"\n\n\"Every word that she said. I've been meaning to speak to Brickner about the way he's handling the investigation, or lack thereof, but I've been preoccupied.\"\n\n\"I'd be very grateful.\"\n\n\"In that case, I'll be sure to pay him a visit tomorrow. You can count on it.\"\n\nHe nodded to me, and together we moved the plates and scraps of food back into the kitchen, leaving them in the sink for Jenny, the scullery maid, to take care of the next day. With that done, we awkwardly parted ways, and I found that I had to restrain myself from looking back at him as he left for his chambers. I hated myself for my slavish devotion to propriety, but what else did I have to be devoted to?\n\nI went to my room next to the nursery. I felt relieved at the thought of allaying Susannah's concerns, and at the possibility of finding whoever had killed Nanny Prum. Sleep came, but not without a struggle to quiet my mind.\n\nI dreamt of Heatherdale, my family's estate, where Jonathan and I lived for three glorious years. The dream was a recurring one, but as with the others, it was always different. I found it strange that this time I was detached from the scene, watching myself sleep in bed with my husband, his strong arms around my waist. I felt larger than myself, my body not a body at all, wrapped between the bones of the house, shifting them dangerously out of place. There was also a great heat radiating from my skin, curling the wallpaper to black char, eating away at the wooden beams that supported the house, and it caused me to expand with reckless abandon. I cackled, and sparks erupted from my throat in a plume of black smoke.\n\nJonathan woke gasping for air. He shook the other Charlotte awake, and together they ran through the house. But it was too late. I was all around them, singeing their skin and hair, choking them quickly back to sleep. Jonathan noticed a curtain that I had not yet touched, one of the only things that had not been burned. He wrapped it around his wife's body and picked her up in his arms, even as she struggled and screamed for him to stop, and plunged them both into the flames.\n\nI tried to stop him. I tore at his skin until it blistered and cracked, at his hair until it burned down to the scalp, but still he ran through the remains of the manor, not stopping until he was outside, collapsing into a ruined heap while his wife cried over him, begging him to wake up as the man in black observed the scene in silence, the light from the flames extending his shadow over the dying underbrush.\nCHAPTER 4\n\nA Lesson in Dreaming\n\nThe next morning, after taking breakfast downstairs with the children\u2014Mr. Darrow was good as his word and had already left to meet with Constable Brickner\u2014I marched the boys up to the schoolroom to begin their daily lessons.\n\nDuring the first few weeks after my arrival, Mrs. Norman and I scoured the empty rooms of the manor, lifting the covers off of antiquated pieces of furniture in search of practical desks, and kicking up small clouds of dust as we traveled through parlors, bedrooms, and servants' quarters that hadn't been used in generations. We eventually discovered a small attic at the top of the east wing of the house.\n\nIt was large enough that it had a proper staircase rather than one that had to be pulled down from the ceiling, and very little was actually stored up there. The ceiling was low, and both sides of the room slanted at an angle, reaching up to a point like a church steeple. It felt very much like a small country schoolhouse, with windows on each end of the wide room, and I knew when we found it that it was just what we needed.\n\nHaving discovered a desk in an unused study (presumably belonging to Mr. Darrow's father or grandfather), I had Fredricks and Roland carry it up into the attic. It was not an easy task, but then, as I reminded them, neither was the education of children. The desk was placed at the front of the room for my own use, before a blackboard Nanny Prum had kept in the nursery. Two low tables were found for the boys, and they were placed far enough apart as to avoid easy physical confrontation. The back of the schoolroom contained many of the items that had been found there upon its discovery. I arranged some old end tables, rusted gaslights, and empty picture frames into a sort of artists' corner, stocked with supplies I had brought to the house myself. An intellectual education was of course deeply important, but I felt that an aesthetic curriculum was equally worthwhile, especially in light of the late Mrs. Darrow's rather prolific creative accomplishments.\n\nEach day I began the boys' lessons with arithmetic. Mornings were best suited to intense study, as it loosened the mind for the interpretation of literature later on in the day. Whenever I felt that I was losing their attention or interest, I would quickly end whatever it was they were doing and challenge them to tackle some artistic accomplishment.\n\nThat afternoon, in the difficult time before lunch when children begin to think with their stomachs even though the next mealtime is at least an hour away, I was still fixated on the problem of dreams. The boys looked tired, and I myself hadn't been able to sleep very well after the nightmares of the past few evenings. It was silly that we should all suffer so much from self-inflicted trauma. I once read that dreams were the products of unacknowledged emotion and feeling, and that setting them down with either words or images often lessened their power. To understand fear was to control it.\n\nPaul yawned. James followed suit, and I was obliged to set down the book of poems they were reading through and send them to the back of the schoolroom.\n\n\"That's quite enough of that. I have a new task for you.\"\n\nJames yawned again. \"But it's nearly time for lunch!\" He clutched his stomach as if he would waste away to nothing before the end of the hour.\n\nI ignored him. \"You are to describe your dreams from last night with either drawings or prose.\"\n\nBoth of the boys grabbed the colored pencils on the table and ignored the mention of prose, despite the fact that this was my favorite medium. I frowned but said nothing, remembering that their late mother had created the majority of the artwork that decorated the walls of Everton.\n\nPaul began scribbling furiously without hesitation. He paused every few minutes to look out the window, and continued drawing, in fine detail, a meticulous landscape of Everton from a bird's-eye view. James had more trouble deciding what to draw. He had many dreams every night, and so choosing the most exciting and violent one to illustrate was no small task. Eventually he settled on what he knew best and began sketching the hulking black thorax of the Spider Queen.\n\nWhen the boys had finished, I led them back to their desks and asked them to present their artwork. James, who always demanded to go first and threatened to throw a magnificent tantrum if he didn't get his way, had found no purchase with me by using this tactic, despite a glorious performance involving impressive physicality with thrown chairs, toppled tables, and broken vases, during which I clapped and cheered him on as if I had paid for the privilege of his outburst, and eventually he relented to alternating turns with his brother. Nevertheless, it was his turn to present first. He stood from his chair and moved to the front of the room beside the desk.\n\n\"I drew the Spider Queen.\" The paper contained a black blob of a body with eight spindly appendages, but the face of the creature was very much like that of a young woman, with curly silver hair and pretty features. \"She lives in a cave beneath my bed and eats up the goblins whenever they try to steal my breath. Sometimes she has me over for tea, and sometimes we're friends, but other times she sends her children after me because I've stolen some of her treasures.\" He stopped and clutched his stomach again in an effort to remind me of his delicate and very hungry condition, but I did not let him return to his seat.\n\n\"But why would you steal from her? It sounds as if she's doing you a favor by gobbling up the goblins.\"\n\nJames looked at me as if I were quite slow. \"To buy back Mother's soul from the Goblin King.\"\n\nMy heart sank, and for a moment I did not know how to respond. What was there to say? It was a beautiful, sad sentiment, but I quickly recovered. \"Why would he have your mother's soul? She went to Heaven.\"\n\nThe boy thought about this and shrugged his shoulders. \"I dunno. It was just a dream I had.\" I nodded to him that he could return to his chair.\n\nPaul stood up and took his place before the blackboard. He held up his re-creation of Everton, which now resembled something like a treasure map.\n\n\"Last night I dreamt that I went to Mother's house.\"\n\nI took a deep breath and wrung my fingers together. This was not going at all the way I had expected. But then, what did I expect? The boys had lost their mother. Of course they were dreaming of her. I knew that they were dreaming of her. I had lost my mother nearly fifteen years before and still dreamt of her. It was not something that ever truly went away. The three of us would perhaps always be bound by our grief, never truly finished with the long nightmare of loss. But if that were true then we were also bound together searching in our dreaming for new memories of the mothers we lost. Children need their parents in whatever form they're available, and I shivered for a moment as I thought back to one of my dreams the night before.\n\nChildren need their mothers.\n\nPaul continued to explain his drawing.\n\n\"She came for me in the night and led me through a wood.\" He pointed to his illustration of the old-growth forest behind the house. \"The wood turned into an orchard, and there was a great house. Mother said we couldn't go inside just yet, that we had to do it in person. She's waiting for us.\"\n\nThe specificity of his dream was unnerving. I folded my hands on the desk and peered carefully at the young man. \"Why would she do that?\"\n\nHe looked down at the ground, his large blue eyes fixated on nothing particular but still lost in thought. He spoke without looking up. \"She misses us.\"\n\nWith those three words he nearly reduced me to tears. I felt the years of built-up sorrow at the loss of my own mother materialize as a tightening in my chest and the prickling sensation behind the eyes that heralded the impending arrival of tears.\n\n\"Paul\"\u2014my voice almost broke\u2014\"your mother is gone.\"\n\nHe finally looked up, his face creased in a pleasant, knowing expression that should have been impossible for someone who had just turned thirteen. \"I know. But every now and then, when we're in the village, and I see the back of a lady with long black hair, I always hope that it's her; that everything I remember is wrong. That she didn't die. It was all just a misunderstanding.\"\n\nIt was almost identical to what his father had told me that first night in the music room. We stared at each other in silence for what felt like a long while, until James grew tired of not being the center of attention and spoke up.\n\n\"Can we go?\"\n\n\"Lunch isn't for twenty more minutes,\" I reminded him.\n\n\"Not to eat; into the woods.\" He pointed at his brother's map.\n\n\"Whatever for?\"\n\nThe boy shrugged, and his older brother spoke up. \"Aren't some dreams true?\"\n\nWhile I wanted the boys to find solace in the idea that their dreams were not real, I hadn't anticipated them finding so much relief in the notion that they were. If I took them into the woods, they would find nothing there and be forced to face the fact that their mother was truly gone. If I did not, then it was likely they would find some way to use the map when I wasn't looking, and the last thing I wanted was for the boys to go off into the wilderness on their own, especially in light of the fact that whoever had killed Nanny Prum was still at large. There was only one option: I would take them, show them the reality of death, and deal with the consequences as they came.\n\n\"I suppose we could take our lunch outside this afternoon; I do enjoy picnics.\"\n\nAt the mention of a meal James clutched his stomach again and did his best to look pathetic. Paul looked over his handmade map with a frown, but said nothing.\n\n\"Does that sound like something you'd both enjoy?\"\n\nThe older boy folded the map carefully and placed it in his pocket. \"Yes, thank you.\" He smiled placidly. I was beginning to notice that he was unreadable when he wanted to be. It was an unnatural quality in one his age, and I made a note to watch him more carefully.\n\n\"Can we go now?\" James whined, taking my hand before I could respond and leading me down to the parlor, where I had them wait as I negotiated our picnic with Mrs. Mulbus. Luckily, Jenny was in the village running an errand, and so the usual shouting and arguing that accompanied a visit to the kitchen was replaced with complaints from Mrs. Mulbus about the scullery maid's tardiness and general laziness. As I provided a sympathetic ear to the cook's woes, she had no trouble procuring a basket full of finger sandwiches, slices of roast chicken, bread, cheese, pork pies, and fruit for our afternoon adventure. She even supplied a sturdy blanket for the occasion.\n\nAfter I'd collected the boys, we set out from the back of the house and found a patch of grass at the edge of the forest still awash in sunlight. It was unseasonably warm despite the impending arrival of winter. I spread the picnic blanket over the ground and laid out the contents of the basket. As we filled our stomachs, the shadows of the tree branches marked the length of the meal like a sundial, and when we finished, I fell back into the tall grass. The children danced around me in circles like giants among the dying wildflowers, happy and full, finally collapsing into a breathless, red-faced heap of tousled hair and grass-stained shins.\n\n\"We all fall down!\" James giggled into the hem of my dress as Paul tugged at his leg in an effort to twist it off. The little boy squealed, and I sat up with a dramatic sigh.\n\n\"Paul, must you do that to your brother's leg?\"\n\n\"It wouldn't come off when I pulled.\"\n\n\"I imagine it might be difficult to continue the day's activities if you have to carry your brother's leg around.\"\n\n\"Maybe, but he won't give me back my map.\"\n\n\"James?\"\n\n\"But I want to look at it!\"\n\n\"It seems to me that I've done a very poor job of teaching you the importance of sharing, and it may be time for a precocious little song.\"\n\nJames scrambled up and serenaded me with a series of high-pitched screams, a sound that, despite the ringing it left in my ears, spoke of the intimacy and affection that had quickly formed between us in the weeks since the loss of Nanny Prum. Paul put his hands over his ears and attempted to trip his brother.\n\n\"Yes, you should run! You've heard me sing! But by now you must also be aware that I happen to find threats and subterfuge a much more effective means of communicating with inscrutably dense children.\"\n\nJames stopped running and turned to squint back at me. \"What's dense?\"\n\nI leapt from the ground and snatched the paper from the young boy's hands. I was so quick he barely had time to register what had happened before I handed the homemade map over to his brother.\n\n\"Paul, what does dense mean?\"\n\n\"That we still have a lot to learn about the world.\"\n\n\"That will do for now, I suppose.\" I kissed James on the head and lifted him into the air, placing his legs so they straddled my waist. He scowled but put his arm around my neck anyway.\n\n\"Now, where does it say to go next?\"\n\nPaul held the map close to his face. It was eerily accurate as he compared it to the landscape, looking across the field toward the overgrown forest up ahead.\n\n\"Over there, into the woods.\"\n\n\"Off we go then.\" I set James back on the ground next to his brother and gathered up the remains of the picnic into the basket. As we marched away from the field, the sun slid behind the twisting, knotted tree branches and the ground swelled with half-buried roots and rocks, both big enough to trip over and small enough to get trapped at the bottom of a shoe.\n\n\"Paul, how much farther?\" I asked, becoming a bit nervous as the shadows grew longer.\n\n\"It was just ahead in my dream.\"\n\nI said nothing for a moment, prepared to let reality speak for itself as it tore away the curtain of hope to reveal the cruel actuality of death, which had been unable to grab hold of the children, even though James, at least, had been at their mother's side when she had passed on\u2014a prime example of the power of the heart to overwhelm the mind. \"And what do you expect to do if there isn't anything there?\" I said after a while.\n\n\"I'll keep dreaming.\" Paul said this matter-of-factly, without turning away from the task at hand, stepping over underbrush, moss-covered boulders, and rotted logs with complete determination. I held James's hand and continued my minor lecture as we walked.\n\n\"Dreams are my favorite things in the world. Sometimes they even come true, but sometimes we must learn when to wake up.\"\n\nPaul ignored me and pointed excitedly at something up ahead. \"There!\"\n\nThe path ended at a small fallow creek, but began again on the other side to disappear around a dark, massive cage of roots at the base of an ancient oak tree. Whatever lay beyond the magnificent tree was obscured in a thick, roiling patch of fog. James wrenched himself free from my hand and leapt over the creek, bounding into the mist before I was able to stop him.\n\n\"James!\"\n\nI quickly hoisted my dress up to my waist and jumped over the brook, glancing back at Paul to wave him on. Together we chased his brother into the mist.\n\nThe air around us grew heavy with a dampness that remained even as the fog subsided, and we found ourselves in the middle of a vast orchard. While it had been daylight mere moments before, the moon now hung low in the sky, larger than I had ever seen. It was so vast and oppressive I felt that if I were to reach toward the sky I might be able to push the orb back where it belonged, high above on the black velvet mantle of the night.\n\n\"It's nighttime here.\" Paul was behind me, hugging himself against the cool air.\n\n\"Perhaps I misjudged the time . . .\" I said with uncertainty as I took his hand very tightly into my own and began to march between the rows of squat orchard trees. \"We must find your brother.\"\n\nPaul was silent as he walked, his knuckles white as he peered between the trees at the shadows that stretched out to us when we passed, sensing us with hungry anticipation.\n\n\"Is this the place you dreamt of?\"\n\nPaul shivered against the chill in the air, observed the heavy moon in the sky, and shook his head slowly. \"No. There was an orchard, but it was different.\"\n\nNormally, I would have been very interested in such a sudden change of landscape\u2014and apparently, time\u2014but I was anxious to find James. My heart began to pulse in my ears, throbbing so intensely that my body seemed to reverberate with each beat. I refused to panic. Instead, I felt a heightened awareness of the atmosphere around me, of the curling fog behind a distant tree, of the rustle of branches around us, the movement of the shadows in our direction as we passed, of the very alien nature of the place that Paul's map had led us to.\n\n\"James!\" My voice did not echo through the air, nor could we hear the sounds of our footsteps on the hard, cold earth. Still, I continued to call out until I was hoarse. Paul dragged behind me, gasping every time he looked back to where we had come from, seeing nothing but a gloom as opaque and tangible as the fog that had heralded our arrival to this strange, dark land. It was building around and behind us, pushing us toward a destination neither of us wanted to think about.\n\n\"Charlotte . . .\"\n\n\"We'll leave as soon as we find your brother.\"\n\nI stopped at what appeared to be the main thoroughfare and peered in both directions, trying to decide which way James might have gone. Behind me, Paul pressed himself against the nearest tree, as if to block the creeping darkness from his line of vision. Thin branches and twigs cracked and broke around his body, and his head grazed the bottom of a low-hanging piece of fruit with enough force to knock it free. It dropped down into his hands.\n\nIt was about the same size and shape as a grapefruit, but before he could get a good look at it, he glanced up at me, clearly frightened, sensing that something was wrong. The fruit quivered, and with a wet, tearing sound it began to unroll from the inside out, the air laced with the scent of peaches as the thing in his hands untwisted its arms and legs from the pulpy interior of its body and wrenched its head free from its shell. A baby's face blinked at us with pale blue eyes as Paul dropped it onto the ground with a look of utter terror, backing away, his gaze transfixed on the thing as it fell onto its back, protected by what was formerly the leathery skin of the fruit.\n\nIt smiled at him with thin, sharp teeth.\n\nPaul let out a manic, consuming scream that frittered away the last remaining edges of his youthful courage and curiosity and exploded into his legs. He ran past me, past the trees, never looking directly at them, at the fruit, perhaps afraid that it might look back; his voice never breaking through the air as he shrieked, never echoing, but rather circling back in on him like a vulture, an eater of dead things, pecking away his every last hope, every rational thought, every instinct but the one that told him to run for as long as it took to escape.\n\nI trailed behind him, struggling to follow his voice, which was quickly muffled by the rows of trees, but as he was running in a straight line I was able to catch up to him when he stopped, panting and heaving at the edge of the orchard before a great house as grand as anything either of us had ever seen before.\n\nThe doors of the house stood open, and silhouetted against the light that streamed over the threshold as vibrantly as the darkness churned in the orchard was a woman, tall and regal, even at a distance. James was at her side, clinging to her waist as she descended the steps leading up to the house with slow deliberation, almost gliding to the ground, a beautiful phantom with a small, worried smile as she approached Paul and gently touched his face. He collapsed against her and sobbed so loudly into her shoulder that I was unable to dispute the name that he immediately and distinctly gave her:\n\n\"Mother!\"\nCHAPTER 5\n\nBargains with the Dead\n\nFor a moment, I could only stand at the foot of the great house in an attempt to catch my breath, my mind reeling, searching for some way to escape with the children. Lily Darrow was dead. There had been witnesses and a funeral and a painting commissioned to hang over the desk in her husband's study, a portrait of a raven-haired creature with glittering eyes like cracked jade and a playful expression of mock superiority, which he stared at for hours on end when he didn't think the servants were watching. And yet . . . the likeness was so startling that I had to shake myself of the very notion that the woman before us could possibly be the late Mrs. Darrow.\n\nThey had mourned her. What kind of wife and mother would allow her family to feel the things that her death had inflicted upon them if she were not well and truly dead? It was unfathomable. This was some act of trickery, a cruel impostor toying with the emotions of children. I would not stand for it.\n\nPaul sobbed into the woman's shoulder, crying and apologizing\u2014\"I'm sorry I wasn't there, I'm so sorry\"\u2014as she stroked his head and cooed away his sorrows. I stepped forward, stopping as I happened upon the crinkled hand-drawn map of the forest left behind on the ground of the orchard, a thing crafted from the scraps of dreams. How could anyone have influenced the boy to lead us into the woods? Was such a thing even possible? There were so many questions, all of them overshadowed by the one thought I could not ignore:\n\n\"No one ever comes back,\" I said.\n\nJames pulled his face away from the skirts of the mystery woman, and looked her over carefully before returning my pleading gaze with a confused expression. In his eyes I could see that there was no doubt the woman he clung to was his mother.\n\nPaul didn't bother to remove his head from the other woman's shoulder. He had awoken from his nightmare and it had all been some terrible misunderstanding. Everything he hoped for had come true.\n\n\"But she has. She's alive again.\"\n\nThe woman ran her fingers through Paul's hair and raised his chin so she could look into his eyes. \"No, my love, I am not.\"\n\nHis face fell, and he slowly backed away from her, dragging his brother with him. I quickly grabbed them both by the shoulders a little more roughly than I meant to and held them tightly before they could run off.\n\nThe great house before us was more appealing than the dark, oppressive gloom of the orchard, with the shadows that twitched and snaked about the ground, but I would not hesitate to escape the way we had come. I considered the woman who purported to be Mrs. Darrow. To anyone who might ask, I would deny that I had any belief in ghosts, but then what of the man in black? A mysterious shade prone to the company of corpses was just as unlikely as the resurrection of a young mother taken before her time. Was she a liar, a ghost, or something else altogether?\n\nMy heart continued on its perilous drop deep into the nether reaches of my chest, and I realized with no small amount of revulsion that I might soon begin to panic in the way I had witnessed other women do, as I was perhaps expected to do. But I refused to faint or swoon; the fearful emptiness I felt inside instead began to swell, blood raging in my ears, until it changed into something solid and substantial. Nothing would happen to the children\u2014I would not allow it. It was a very odd sensation, unlike anything I had ever experienced. We were in danger, true danger, and it thrilled me to know that I was equal to the challenge presented.\n\nPerhaps the woman saw this fearlessness in my gaze, which I imagine had become rather hard and fiery, for it was then that she sighed, her comfortable, dignified composure falling away. She folded her hands before her like the woman from the portrait in Mr. Darrow's study and began to look a bit desperate.\n\n\"Please, I've come back to you.\" She took a step forward, and the boys huddled against me. The woman stopped again and smiled faintly. \"I suppose I should have expected as much. I've been gone from you both for so long.\"\n\n\"And Father.\" Paul had let go of me, but did not move toward the other woman. He spoke with a slight edge in his voice.\n\nAt the mention of Mr. Darrow, the woman quickly looked back at the house and then returned her gaze to Paul. \"Please don't be angry with me. I never wanted to leave you, which is why I've come back. I would have returned to Everton, but there were rules I had to agree to.\"\n\n\"Come home with us.\" James stepped away from me to join his brother's side.\n\nThe woman shook her head. \"This is my home now, and you're welcome here anytime you'd like.\" She motioned to the massive house.\n\nThe boys looked at one another, and then to me, but I remained unconvinced.\n\n\"Do forgive me for being skeptical,\" I said, trying to contain the smoking flesh and boiling blood surging beneath the confines of my skin. \"But how can we be sure that you are really Mrs. Darrow, and not some impostor intent on doing us harm?\"\n\n\"I can see that my husband chose well.\" She paused at this, making an implication that was not lost on me. \"If that were my intention, why would I be reasoning with you to believe my story? Wouldn't I have done something by now to prove your point?\"\n\n\"I do not pretend to understand the whims of the dead.\"\n\n\"A wise decision. So you believe me then?\"\n\nI shot her a steely glare and changed the subject. \"This place is hardly fit for children.\"\n\n\"How can you be sure? You have not yet been inside. The House of Darkling can be whatever you choose to make of it.\" She had begun to regain her confidence, much to my dismay, and her lips formed into a tight smile at her own cleverness. I would have none of it. I turned with the children pressed against me and began to trot briskly back into the orchard. The woman called after us, desperate once more, which was the only way I would deal with her. Desperate people are more likely to make mistakes.\n\n\"Please! You must give me the chance to prove myself! Ask me anything. Something that only Lily Darrow would know.\"\n\nI stopped at the edge of the orchard and turned around slowly, searching my mind for every scrap of information I had ever learned about my employer's late wife.\n\nPaul spoke up before I could. \"What was the name of the lullaby you used to sing to us?\"\n\nThe woman smiled and closed her eyes for a moment, as if listening for the music on the soft wind that began to blow through the orchard trees, the fruit on their branches swaying in the breeze. \" 'Every Night at Everton.' We made it up together, and it was different every night depending on what had happened during the day.\"\n\nThis was enough proof for both of the children. They returned to their mother's side and hugged her tightly, immediately sorry that they had ever doubted her intentions. I, on the other hand, was dubious even as the thrill of danger dissipated into caution. When a person died, they did not come back to their children, but if this had somehow been reversed for Mrs. Darrow\u2014and I was not convinced that it had, nor that this was not some elaborate ruse to take advantage of the children of a wealthy widower\u2014then why had none of my loved ones been able to do the same? It was for this reason and this reason alone that I followed them into the great house with unease, remembering those that I had lost and hoping against hope that if what the woman said were true, then perhaps the place contained more than one departed soul.\n\nThe parlor was small and intimate, the walls lined with square wooden panels and elaborate tapestries depicting the room itself filled with a strange pantheon of creatures, perhaps from some obscure mythology or religion with which I was unfamiliar. As I stared at the fabric, examining the intricate patterns and threading, I realized with some bewilderment that the shapes were changing, reknitting themselves from left to right in an impossible act of defiance of the rules of scientific propriety, recasting the occupants of the room until I recognized myself and the children as carefully constructed embroideries. I reached out to pull aside the tapestry, but I stopped myself before I could uncover the mechanism enabling its manipulation. Against my better judgment I preferred to believe, if only for a little while longer, that the house and the alleged Mrs. Darrow were part of something extraordinary.\n\nA squat, muted chandelier hung low from the ceiling, casting the room in dim amber light. I sat on the edge of a thick leather armchair, determined not to sink back so far as to be rendered incapacitated should the strange situation spiral any further out of my control, even as I promised myself that it would not. To my bewilderment the cushions expanded as if the chair were fighting against me so that I might be more comfortable. Was it possible for furniture to become offended? I firmly kicked the leg behind my right foot, and the chair regained its former shape.\n\nBefore sitting down, Mrs. Darrow gently touched three of the wooden panels along the wall, each of them clicking open to reveal the different components of a full afternoon tea spread. She removed cups and a steaming pot from the first, a pedestal of finger sandwiches and scones from the second, and a chocolate tea cake from the last. I fought to ignore a pang of sympathy as I realized that the cake was the same as Mr. Darrow had provided during our midnight tea. She left it on a plate to the side of the table, as conspicuously untouched as her husband's had been. I imagined the two of them sitting alone in two different houses, Mr. Darrow at Everton and Mrs. Darrow at the place called Darkling, staring at the empty chairs that surrounded them, silence shrieking, with tea cakes perched on lone plates like ceremonial offerings to memories not quite dead.\n\nI eyed the boys carefully as they sat beside the woman who claimed to be their mother, sprawled on a plush divan before the large stone fireplace at the front of the room. The flames contorted into various shapes, casting shadows of flickering animals and their masters along the back wall. The children marveled at the trick for a long while, slowly drifting off to sleep as the alleged Mrs. Darrow watched me back, her eyes gleaming in the firelight, as dangerous and silver-green as a cat's.\n\n\"Your tea will cool,\" she said. Both boys were nearly asleep in her lap; even Paul, who was too old for that sort of thing.\n\nI looked at the saucer and brought the cup to my lips, careful to seal them to the rim so as not to allow any of the liquid into my mouth. I was already at a disadvantage if the woman meant us any harm, a fact I firmly kept at the forefront of my mind, and sitting in the parlor of a woman who claimed to be dead, in a strange land with shadows that crawled and pieces of fruit that walked, the least I could do was avoid a potentially poisoned cup of tea.\n\nI brought the cup away from my mouth and placed it back onto the saucer sitting primly in my lap. The other woman turned away from me and gazed into the fire.\n\n\"Is my husband well?\" The tone of her voice was emotionless and elusive. It reminded me of Paul's.\n\nI considered the question before I answered. A vague, bland answer might lead to an informal inquisition, but a detailed one might hint at a relationship that was more involved than was true.\n\n\"I've been with the Darrow family for nine months, and in that time I've come to know Mr. Darrow as two very different people. The first man smiles when someone says something clever and eats as voraciously as the groundskeeper. But then there are times when he sees something either in the house or in the boys that makes him grow very distant. He is my employer, and so I don't pretend to know him as one might know a friend, but whenever he is overcome by such an episode, I've begun to suspect that he's thinking of what he has lost, and there is a sadness in his distance that leads me to believe that he may forever remain two people: one struggling to enjoy life, and the other trapped in sorrow.\"\n\nThe woman did not look away from the fireplace. Her chest rose and fell in an uncomfortable quiet broken only by the ambient sounds of the room\u2014the embers in the fire crackling; the grandfather clock chiming; something shuffling across the floor in one of the upper chambers of the great house. I brought the cup to my lips again, pretended to drink, and set it back onto the saucer.\n\n\"You are very thorough, Mrs. Markham.\"\n\nI placed the saucer on the table between us and stood to circle the room. I spotted a bookcase filled with obscure texts whose titles seemed to be in a language I had never seen before. I fingered the spines lovingly and turned back to Mrs. Darrow. \"The one detail I find myself most curious about at present is the fact that I'm having a conversation with the alleged late wife of my current employer.\"\n\nThe woman smiled, and the stoic decorum that had framed her every action since we entered the house partially melted away. She lifted herself gingerly out from beneath the children and stood before the fireplace.\n\n\"You are right to be suspicious of me.\"\n\n\"The children believe you. Who am I to disagree with them? But by all accounts Mrs. Darrow died.\"\n\n\"So I did.\"\n\n\"I have lost many people from my life\u2014my mother to cholera, my father to a heart attack, and my husband to a fire\u2014and when they died, they did not come back.\" The end of the last word sharpened in the air for a pregnant moment until I began again. \"While I do not doubt the power of a mother's love for her children, I will not believe that the love of my family was somehow inferior to yours.\" I said this evenly, without hope of masking the jealous curiosity that had replaced my confidence, but I was determined nevertheless to have a cordial, honest discussion. \"In order for this conversation to continue, I feel that I must ask\u2014why you?\"\n\nThe woman did not appear to be surprised at the directness of my question; in fact she seemed relieved. The alleged Mrs. Darrow spoke facing the fireplace, a silhouette before the flames.\n\n\"When I first took ill, I told anyone who would listen that I would conquer my sickness, that I would not accept anything less than a full recovery; that God was testing me. I followed the doctor's instructions: I continued with my social engagements, I ate healthily, exercised regularly, and yet each day I grew more weary.\n\n\"Food began to make me queasy, and I lost the ability to stand under my own volition. I became confined to my bed and slowly wasted away until the skin hung from my bones, loose and pallid. People came to my bedside and whispered words of solace and comfort, but that was of little consolation after I went blind, and even less when I lost my hearing.\n\n\"You would think that a person in such a state would be adrift in the darkness, but I could still feel; I could still smell. I knew when my family was nearby; when Henry kissed my forehead or pushed a strand of hair from my face, when James took my hand into his to keep me company, just as I knew that Paul could not bring himself to see me bedridden. I knew that I would die and that they would miss me, though death was a thing I craved more and more each day.\n\n\"The very act of breathing slowly drove me to the brink of insanity. Even in my delirium I found it ironic how a thing that gives one life could become the most unbearable part of living. The moments between each breath grew further and further apart, a series of contractions as I delivered my own end, until finally, I stopped.\n\n\"I realized I had died when I opened my eyes and could see again. A man stood before me, as unremarkable and ordinary a person as I have ever encountered. He wore a black suit and a bowler hat, and he held out his hand. He said nothing, but he did not have to. I knew who he was, and what he expected me to do. Free from illness, I felt revitalized, elated even, and yet something whispered to me: a voice in the place between life and death. It spoke to me, whereas Death did not; it told me I was special, and that exceptions could be made for any rule. It told me the story of my life, one that did not end with a woman in her sickbed.\n\n\"The forgettable man in the ordinary black suit grew fainter and fainter, retreating down a corridor made of light until he was gone altogether. The voice grew more substantial, until there was a hand, and it took me someplace else . . . to a place for the Things That Do Not Die.\"\n\nI felt a chill run through my body. I was near a window, and the darkness outside seemed to press against it, flexing the glass with an ominous groan.\n\n\"And here you stand,\" I whispered.\n\n\"I do not know why it was different for me. Perhaps I was in the right place at the right time. Regardless of why the opportunity presented itself, I took it. Children need their mothers, little boys most of all.\"\n\nI paused at that turn of phrase. The old nightmare of my mother's death returned, as did the voice of the mysterious woman from my dream, who, it was now so obvious, sounded very much like Mrs. Darrow. My heart fluttered with a mixture of anger and fear. I approached the divan with the sleeping children, clutching the lip of the seat.\n\n\"The children can't stay here. It isn't safe.\"\n\n\"Nothing on the estate would dare hurt the children.\"\n\n\"And their governess?\"\n\nMrs. Darrow, for by that point I could no longer pretend to think of her as anything else, stepped closer and put her hand over my own. She was warm to the touch, more so than any living person I had ever encountered. With the children between us, I relaxed for a moment.\n\n\"I mean no one any harm,\" she said.\n\nI looked her carefully in the eyes, their catlike quality replaced by something more somber and quiet. Suddenly her intrusion into my dream seemed more sad than threatening.\n\n\"I've dreamt of you. You tricked us into coming here.\"\n\n\"I did what I had to do in order to see my children.\"\n\n\"What is it that you want from them?\"\n\n\"More time.\"\n\n\"To what end? You have passed on, and it can't be healthy for them to meet you somewhere in between.\"\n\n\"Is it any worse than allowing them to grow up without me? You must have seen what happens to some children who lose their parents.\"\n\nA barrage of heartless, foulmouthed little boys passed before my mind's eye, hitting and shouting, stealing and spitting, raping scullery maids in the middle of the night.\n\n\"That can be avoided.\"\n\n\"Yes, it can. That's why I came here. They don't have to be without me. I don't have to be gone.\"\n\nThe woman moved her hand up, grasping my wrist. There was a desperation in her grip.\n\n\"You never were.\"\n\nMrs. Darrow dropped her hand away and turned back to the fireplace. The flames licked at the embers, which had stacked themselves into something like a house.\n\n\"I'd like them to visit Darkling, when they can. Time passes differently here, and it would be as if they'd never left Everton. My husband would never know.\"\n\n\"You don't wish to see him?\"\n\n\"He can never know.\"\n\n\"He's lost without you.\" My throat tightened as I said the words.\n\n\"You must not tell him!\" The woman's voice raised in pitch, waking the boys with a start.\n\n\"Mother?\"\n\nMrs. Darrow was back at their sides before they could lift their heads, kissing their faces gently as she lifted them off the divan. I felt that I had touched upon something important, perhaps even powerful. She was afraid, and her feelings for her husband were clearly complicated. Suddenly the situation became very manageable. She was no different from any other person and could be manipulated if necessary. I was surprised at the callousness I had discovered within myself. It was not in my nature to have thoughts so overtly cunning, but then I had never been faced with such a dangerous situation. I wondered if the person one becomes when faced with such things is the person one truly is, or only a temporary mask worn to survive. Again, a shiver ran through me.\n\nThe woman looked from the boys to me, and then smiled sweetly at her children. \"I'm afraid I'm very tired. Our visit must come to an end.\"\n\n\"But, Mother, we just got here!\"\n\n\"Please don't leave us!\"\n\nMrs. Darrow and I stared at each other during this exchange, and I searched myself deeply for a response I would not live to regret. There would be consequences from dabbling with the dead, this I felt most certain of all. But was there also not a wealth of things to learn? The veil of death had been lifted for this regal, beautiful woman, so stoic and frail, broken by the end of her own life in a way that perhaps even the children would be unable to fix. Were there others like her? I attempted to stifle the thought before it became fully formed, but it was too late. It blossomed in my mind, accompanied by images of Jonathan and my parents.\n\nI could not ignore the strength of Mrs. Darrow to overturn the rules of existence, to find a way out of death, and to fight for her children. There was something powerful in such love, such conviction, such devotion, and at the same time such desperation, a weary stubbornness to deny what must occur. I could not refuse her or the children or my own curiosity, despite the danger I sensed in the agreement I was about to make.\n\n\"There will be other visits, and I imagine that we'll be seeing your mother again soon enough,\" I told the boys.\n\n\"She can't come home with us?\" James stuck out his bottom lip. His eyes welled up with tears.\n\n\"Oh, my darling James, I wish that I could. But this is my home now.\"\n\n\"Can we bring Father next time?\"\n\n\"I'm afraid you mustn't, Paul.\"\n\n\"Why not?\"\n\n\"It's almost like a spell that's keeping me from leaving you forever, and if you tell your father, it will be broken. Do you understand?\"\n\nBoth boys nodded and hugged their mother tightly around the neck. She kissed each of them roughly on the lips and turned them back over to my care.\n\n\"Mrs. Markham, it was truly a pleasure meeting you.\"\n\nI took the hands of the children into my own and squeezed them, searching for further conviction.\n\n\"And you, Mrs. Darrow.\"\n\n\"Please, call me Lily.\"\n\nThe woman escorted us out of the parlor and into the main foyer of the House of Darkling. Many stories above, I saw someone leaning against the railing of the grand staircase, watching us and smoking casually in the dark. But before I could look more closely, the doors swung open and the damp, cool air of the orchard swept over us. Lily kissed me affectionately on the cheek, and ushered us out the door.\n\n\"Please come back as soon as you can,\" she said.\n\nThe boys waved at their mother as we set out for the main path between the trees. Something howled in the distance, the sound of it languishing in the chilled air. We continued into the gloom until we came to the familiar wall of mist, and passed through to return to sunshine and the world of the living.\n\nThat evening, I struggled to put the children to bed. James sang and squealed slowly toward exhaustion, jumping up and down on his bed and playing so loudly that I was finally forced to threaten him with an ancient form of Indian torture I had learned as a young girl in Asia. This prompted an inevitable inquiry into my life abroad, and soon the excitement the boys felt over their rediscovered mother was eclipsed by curiosity, and they were able to listen to my tales of the Far East as they drifted off to sleep.\n\nI left the nursery dabbing the perspiration from my forehead with a handkerchief. I was about to return to the schoolroom to organize my lesson plan for the following day when I realized I was not alone in the hallway.\n\n\"Mrs. Markham.\"\n\nI jumped and quickly laughed at myself. I had been so absorbed in the strange events of the day that I had not noticed Mr. Darrow standing behind me, the pale illumination from the gaslights suffused in his golden hair. He was a tall man, with a lithe frame and a distinguished, sharp face that was more beautiful than handsome, and the way his hair glowed in the darkness gave his appearance a quality that bordered on angelic.\n\n\"Mr. Darrow! I apologize, I didn't notice you.\"\n\n\"From the sounds of it, the boys had you on the run this evening.\" He smiled at me and pointed down the corridor. \"Would you mind joining me in the study before turning in for the night?\"\n\nMy stomach twisted into knots. Did he suspect that something was amiss? Or\u2014I could not help but let my mind wander to the strange, romantic thoughts one cannot avoid while traveling through a dark, old house at night\u2014perhaps he had something else in mind? Even in light of meeting his late wife, I could not deny that I found the second possibility rather exciting. Is that not the way most stories go? With the young governess falling in love with her handsome widowed employer and living happily ever after? We both deserved some happiness. Even the dead, it seemed, were entitled to get the things they wanted most.\n\n\"Not at all.\"\n\nThe gaslights flickered above us in their cracked glass husks, fighting against the darkness that attempted to envelop the house. I was unpleasantly reminded of the agreement I had made earlier in the day. I wrung the handkerchief in my hands.\n\n\"Are you all right?\" Mr. Darrow paused at the door to his study and nearly touched my shoulder, but he seemed to catch himself and placed his hand against the door instead, pushing it open. I stood stupidly in the hallway, rousing myself from the day's events and wondering what his hand would have felt like against my skin.\n\n\"Yes, perfectly fine.\" I followed him inside.\n\nHe sat behind his spotlessly clean desk and folded his hands, framed beneath the portrait of his late wife. At the other side of the room, the door to the office stood open.\n\n\"The children seem very happy,\" he said.\n\n\"They have a wonderful home and a loving father. How could they not be?\"\n\n\"You are doing wonders for them, and I want you to know that I intend to compensate you accordingly.\"\n\n\"Mr. Darrow, I can assure you that my current salary is more than generous.\"\n\n\"A fact of which I'm quite aware, but even so, I think you deserve a raise. It's been a difficult year for our family . . .\" His voice cracked, and for a moment I was unsure if he would be able to continue. \"You were right the other evening, I'm afraid I have been growing distant with the children.\"\n\nIt was true that since Nanny Prum's death, Mr. Darrow had been keeping even stranger hours than usual, having afternoon tea in the middle of the night, taking meals in his study, and on the rare occasions that he did join us he drank too much. He was in mourning again, not just for his late employee, but, I felt sure, for his wife. It was a pain I knew too well.\n\n\"With all due respect, you must not ignore the impact of your loss.\"\n\nMr. Darrow attempted to smile, but instead he looked faraway and sad. \"Quite right. We used to spend every weekend together as a family, but ever since Lily . . . They remind me so much of her.\"\n\n\"Perhaps you could reclaim your time with them on the weekends? We could plan an outing by the lake.\"\n\n\"Perhaps.\" He sat back in his chair and sniffed the air. \"Mrs. Markham, have you chosen a new perfume?\"\n\nI felt my chest tighten. I must have gotten too close to Mrs. Darrow during our visit to the House of Darkling.\n\n\"No, why do you ask?\"\n\n\"It's nothing; I suppose I'm just tired. I don't mean to keep you.\"\n\nHe stood from his chair and leaned against the mantel behind his desk, averting his eyes from the portrait of his wife that hung above it, keeping them trained on the fireplace.\n\n\"The lake. Yes, that would be splendid.\"\n\n\"Good night, Mr. Darrow.\"\n\nI crept out of the room and left him to his thoughts.\nCHAPTER 6\n\nA Question of Spirits\n\nThe next morning I woke before sunrise. I had decided in the middle of the night to run a very specific sort of errand to help alleviate some of my concerns about the promise I had made to the children the day before, and so I chose a long black dress from my wardrobe\u2014the kind of severe uniform one would expect a typical governess to wear. I hardly thought of myself as ordinary, and while I much preferred to wear something lighter and more colorful, I had set my mind on this particular mission and knew that it would be best to dress for the occasion. I even wore the brooch I had found in Nanny Prum's room.\n\nThe house was dark and quiet except in the kitchen, where Mrs. Mulbus berated Jenny for failing to adequately clean the soup kettle.\n\n\"There's a ring of filth around the rim!\" As she was a big woman, she lifted the heavy pot without any effort, flailing it in the air above Jenny's head.\n\n\"Yes, Mrs. Mulbus. Of course, Mrs. Mulbus.\"\n\n\"Don't get sharp with me, Jenny Saxon!\"\n\n\"I wouldn't think of it, mum!\" Without turning from the sink, Jenny performed a small curtsy.\n\nMrs. Mulbus slammed the pot down on the counter and clutched her chest. \"You'll be the death of me, I swear it!\"\n\n\"I shall make it my personal duty to put wildflowers on your grave every Sunday morning.\"\n\nThere was a sudden murderous look in Mrs. Mulbus's eyes, and as she motioned to grab for the heavy pan once more, I spoke up to make my presence known.\n\n\"Good morning, Mrs. Mulbus.\"\n\nThe cook turned away from Jenny, who had never turned away from the sink, halfheartedly scrubbing the dishes that always seemed to be there.\n\n\"Good morning to you, Mrs. Markham. I hope we didn't wake you?\"\n\n\"Nonsense. I have an errand to run this morning.\"\n\n\"This early?\"\n\n\"I'm afraid it must be done before the children wake.\"\n\n\"Of course. Might you want something to eat before you leave?\"\n\nThe kitchen was small for such a large house, but it was filled with food. Baskets of fruit and freshly baked breads, smoked meats hanging above the butcher block, stacks of pungent cheeses, rows of spices from India and the Far East (bought by catalog in a very sweet effort to appease my palate), jars of jellies and preserves, and large glass containers filled with caramel toffees. I picked up an apple and tucked it away into the small basket I carried at my side. \"A piece of fruit will do nicely.\"\n\nThe cook was visibly disappointed, and as I left the house through the back of the kitchen, I heard Mrs. Mulbus launching a new tirade at the scullery maid: \"Spots! On the silver!\"\n\nI passed Roland as I crossed the grounds, and we smiled at one another as the shouting continued in the kitchen.\n\n\"Early morning for you, mum?\"\n\n\"I should hope that I'm a little young to be called 'mum'!\"\n\n\"Sorry, Mrs. Markham, just trying to be respectful is all.\"\n\n\"I can't imagine you being anything less.\" The young man had started at Everton a few weeks before my arrival. There wasn't much need for a gardener as the grounds were rather small, but Fredricks was getting older and someday soon would be unable to tend to his duties. When Roland wasn't outside, he followed the butler around attempting to understand the old man's mumbling and increasingly senile instructions. The week before last, Fredricks had asked him to bring Mr. Darrow a box of his favorite cigars, which was alarming since the only member of the Darrow family who had ever enjoyed smoking was Mr. Darrow's father, and he had been dead for over fifteen years.\n\n\"Roland, have you noticed anything strange around the grounds since the night Nanny Prum was attacked?\"\n\n\"How do you mean?\"\n\n\"I'm not sure exactly. Everything about what happened was so unusual . . . and with the murderer still at large, I worry for our safety.\"\n\n\"I walk the grounds with the rifle for a bit every night before I turn in, but no trouble yet. Not even that strange smell that was all over the place it happened. No, I think the bastard that did it is long gone. Pardon my language.\"\n\n\"If you see anything suspicious, do let me know. Even the smallest thing could be important.\"\n\n\"Of course. Miss.\"\n\n\"Now that's more like it.\"\n\nRoland winked at me and tugged on his hat, about to set off toward the caretaker's shed, but he stopped and nervously wrung his hands together like a schoolboy.\n\n\"And how is Mrs. Larken?\"\n\n\"Susannah is coping as best she can. She's very grateful to you.\"\n\nHe blushed. \"Good woman, there.\" He looked dazed for a moment, and then continued. \"If you see her, tell her that I'm keeping an eye out, so she has no cause to be scared.\"\n\n\"I'm sure she'll be very pleased to hear that, Roland.\"\n\nHe nodded once more, and I descended the hill toward the village below. Blackfield was just beginning to come to life. Mr. Wallace was in his shop with key in hand, hurriedly winding each clock face to match up with its siblings, but he was too hungover and slow, and the clocks chimed at him sporadically out of spite. Mr. Rookway, the butcher, stood on a ladder behind his shop window hanging the day's offerings of plucked geese, sausage links, and cured beef. I stopped in the dress shop to see Susannah, but Mrs. Willoughby had her busily sorting through containers of buttons for a set with a mother-of-pearl finish she desperately wanted to use for Mrs. Reese's dinner gown but had apparently misplaced. I promised Susannah that I would come back after my errand and continued on my way to the church.\n\nSt. Michael's was a small country church with stone walls and a family of sparrows nesting within the steeple. The vicarage sat behind it, a modest little cottage with a half dozen birdhouses carefully placed in the surrounding trees. Mr. Scott had been trying for the last three weeks to convince the sparrows to move out of his church, but with little success. His sermons had taken on an air of hysteria as he struggled to shout over the annoying but otherwise lovely birdsongs while the faithful attempted to dodge being sanctified by any unwanted sacrament from above. Despite all this, attendance at the Sunday morning service had never been higher.\n\nThe sun was just coming up. I shifted my basket to the other hand and rapped sharply on the door. There was a stumbling, a muffled curse, and the door wrenched open to reveal Mr. Scott.\n\n\"Mrs. Markham?\" His hazel eyes were watery in the young sunlight.\n\n\"Good morning, Vicar. I hope I haven't disturbed you?\"\n\nHis hair was mussed and his collar crooked. He fumbled with them as he replied. \"Of course not! I always have time for the devoted.\" He motioned for me to enter the cottage.\n\nIt was as small on the inside as it appeared from the exterior, sparsely decorated with every surface covered in what appeared to be birdhouses in varying degrees of construction. Mr. Scott moved a birdhouse from one of the chairs and motioned for me to sit beside him. \"Now, what seems to be the trouble?\"\n\n\"What do you make of spirits?\"\n\nHe looked disappointed. \"I wouldn't know. I don't touch the stuff. Man of the cloth, you know.\"\n\n\"Not spirits, spirits. As in apparitions of the formerly living.\"\n\nHe paused and rubbed his chin. \"Well, I can't say that I've ever seen one.\" He looked at me strangely, as if I'd suddenly grown a pair of horns.\n\nI quickly elaborated. \"Neither have I, of course. But I've been reading the children ghost stories, and James asked me if all spirits were evil. I didn't want to answer him until I'd consulted an expert.\"\n\n\"Expert?\" Mr. Scott squinted for a moment, and then blushed. \"Oh, you mean me? Mrs. Markham, you flatter me. I'm afraid I don't know any more about ghosts and spirits than you do.\"\n\n\"But surely the Bible says something on the subject?\"\n\n\"Well\"\u2014he stood from his chair and paced the room\u2014\"I do recall a line from the Gospel of Luke. I believe it says that spirits are not permitted to return to the earth without a valid purpose, such as offering up a warning.\"\n\n\"So not all spirits have malicious intent?\"\n\n\"It's hard to say. The Book of Job mentions that demons have no power that God himself does not allow. So I suppose that spirits must work along the same lines. They might wish a person harm, but only as a test of faith.\"\n\n\"I see. And how would one defend themselves from spirits or demons, if God had decided to test them?\"\n\nMr. Scott opened his mouth to speak, but closed it again. He narrowed his eyes in amusement. \"James was very specific with his questions. He sounds like a clever young man.\"\n\n\"He does have an excellent governess.\"\n\n\"So I see. Well, if one is being tested by God, then there really is no protection available, but none should be needed. God is a force for good in the world, and he loves each of his children. On the other hand, the demons and spirits you mentioned may be given their power by God, but are more frequently the tools of Lucifer in his quest to lure mankind into damnation. Such creatures use temptation and trickery over physical harm or injury. The best defense against such tactics is good judgment and honorable choices.\"\n\n\"I see.\" I looked around the room and noticed a small crucifix hanging above a doorway. \"And what about holy relics? Crosses, holy water, that sort of thing?\"\n\n\"It couldn't hurt, I suppose, but again, these are complex questions. What good would holy articles do against forces that have their power sanctioned by God, regardless of whether or not they're in the employ of the Devil?\"\n\n\"Such are the mysteries of life, I suppose.\" I said this with sullen annoyance at the complexity of my situation.\n\n\"Indeed.\" The vicar nodded with heavy importance.\n\nI stood and turned for the door.\n\n\"I do apologize, Vicar. I didn't mean to interrupt your morning routine. You've been most helpful.\"\n\n\"No trouble at all, always happy to assist. I suppose I'll see you on Sunday?\"\n\n\"I certainly hope so,\" I said darkly under my breath, thinking of the agreement I had made with the late Mrs. Darrow. I left the vicarage and headed for the path back to the village.\n\nIn the dress shop, Susannah was stacking small paper button boxes, Mrs. Willoughby apparently having found the buttons with the pearl finish for Mrs. Reese's gown. Susannah flinched when I entered the shop, the little bell at the top of the door ringing shrilly to mark my arrival. She dropped the boxes she was holding, scattering lacquered buttons of various sizes across the floor with a small yelp. She closed her eyes and bent down to pick them up.\n\n\"Is everything all right?\" I knelt down to help her, and soon we had the buttons all gathered up and stowed in boxes on the back shelf. I sat Susannah down in a chair behind the counter.\n\n\"No, everything is not all right. Mrs. Willoughby knew that I didn't want to be left alone, and she still went to prepare for tea with Cornelia Reese. I've been a bundle of nerves ever since.\"\n\n\"Whatever for? It's just after sunrise.\"\n\nSusannah narrowed her eyes and lowered her voice in a conspiratorial tone. \"There's a strangeness in Blackfield.\"\n\n\"That's no surprise, what with all the strange people.\" I smiled at her playfully, but she maintained her serious demeanor, so I quickly changed my expression to match her sense of gravity.\n\n\"Ever since that night in the woods, I've felt that there's someone watching me.\"\n\n\"You've seen someone?\"\n\n\"That's just it, I'm not sure. I swear I've seen the same figure out of the corner of my eye when I'm walking through town, but it's always gone when I look at it directly.\n\n\"It's only natural to be nervous after what you've been through. Perhaps it's Roland? He seems a bit smitten with you.\"\n\n\"The groundskeeper? No, there are other things. When I'm alone, I can't seem to keep the lights lit, whether it be candle or gas. And I've been noticing a smell, the same one I noticed near Nanny Prum's body. I fear that someone may be following me.\"\n\n\"Have you told Lionel?\"\n\n\"Of course. He's worried sick, the poor thing. He walks me to the shop every morning, and walks me home in the evenings. Won't let me work in the pub at all anymore.\"\n\n\"Perhaps we ought to tell someone?\"\n\n\"Like who, Brickner? He barely believes me as it is.\" Even so, the constable had been proceeding with the investigation in the wake of his conversation with Mr. Darrow, though there was still very little evidence to go on and minimal progress had been made.\n\nI sat with Susannah until Mrs. Willoughby returned from her tea with Cornelia Reese. Susannah gave her employer an icy reception, which melted into a heated discussion on who, exactly, was the owner of the dress shop. This then escalated into which of them was the better seamstress, until eventually each woman became so frustrated with the other that they broke down and began to sob in one another's arms, promising everlasting friendship and, on Mrs. Willoughby's part, a more careful consideration of her apprentice's nervous state of mind. I left them to their reconciliation and returned to Everton, where I spent most of my morning deflecting the boys' requests to visit their mother.\n\n\"But you promised!\" James kicked at the leg of his desk.\n\nI stood in front of the chalkboard, my skin powdered with flecks of white dust. I pointed at two sets of arithmetic problems written neatly on the board, one for each of the boys. \"And you promised that you would finish your lessons for the day. I hardly think your mother would want to be visited by such lazy children.\" I again pointed forcefully at the board. \"Now, would either of you care to solve your equation?\"\n\nJames kicked the leg of his desk again, but raised his hand to attempt a solution. He was wrong, but it was a step in the right direction. By midafternoon the boys had calmed down and I noticed them looking out the window with wistful, silent agony. I steeled myself against their longing, and I would continue to do so until I was confident that I could protect them from whatever lay beyond the veil of mist in the old-growth forest.\n\nWhen it was lunchtime, I took the boys to the dining room. Everton buzzed with the flurry of daily life. Fredricks left Mr. Darrow's study with an air of distinction, carrying the tray of an empty tea set with hands so tremulous that Roland walked carefully beside him, catching stray cups and saucers before they crashed to the floor. Meanwhile, Mrs. Norman circled around the nervous maids like a vulture, the young women frantically dusting and sweeping to avoid her wrath.\n\nI was suddenly reminded of her warning about the mysterious man who was waiting for me and told the boys that I would meet them in the dining room. Paul and James were more than happy to be rid of me, as I had made it perfectly clear that for the time being I would not change my mind. I returned to the landing on the stairs where Mrs. Norman ran her finger along the railing, testing it for dust in the presence of a young maid named Catherine. It was clean, much to Mrs. Norman's dissatisfaction, and she dismissed the maid as I approached.\n\n\"Good afternoon, Mrs. Markham.\"\n\n\"Hello, Mrs. Norman. I was hoping you might be able to help me with something.\"\n\nThe older woman was immediately suspicious. \"How might I be of service?\" she asked with not a small hint of sarcasm.\n\n\"I was wondering if you might be able to tell me about spirits.\"\n\nI could see immediately that it warmed Mrs. Norman's stony heart to know that when it came to the occult, people thought of her, but she maintained her haughty composure and raised a single eyebrow.\n\n\"There is much to say on the subject. What is it that you wish to know?\"\n\n\"Are they predominantly good or evil?\"\n\nMrs. Norman waved her hands impatiently. \"You ask a question that has no answer. Are people predominantly good or evil? I should say that every man, woman, and child has a great capacity for both, but the discipline for only one or the other. I suppose it depends on whose spirit you are seeing.\"\n\n\"I'm asking more for the sake of curiosity than from experience. I can't say that I've been lucky enough to receive any visits from the Other Side.\"\n\nShe observed me carefully. \"I see. In that case, it depends upon the relationship between the spirit and the person it's sought out. Spirits only return when they have unfinished business. I suppose the only way a person would be in danger is if they either caused the death of the person who became the spirit or meant to impede it on the completion of its otherworldly task.\"\n\nI waited for Mrs. Norman to say more, but it seemed that she had finished doling out advice on the supernatural.\n\nShe raised her eyebrow again. \"Is there anything else?\"\n\n\"No, you've been most helpful. Thank you.\"\n\nShe seemed to falter for a moment. \"The other afternoon in the nursery . . . the man who watches. Have you found him yet?\"\n\nI had been trying not to think about the housekeeper's warning. \"No, I have not.\"\n\n\"Find him before he finds you.\" Mrs. Norman went up the stairs and left me alone on the landing. I felt very weary. I had been going round and round with my dilemma, trying to decide if such extraordinary events warranted disregard of the suspicion I felt deep in my heart. There was something wrong with Mrs. Darrow's proposal. There was something wrong with the mysterious house in which she lived. And yet, I could not shake the feeling that there would be something terribly wrong with me if I did not assist the boys in sharing a little more time with their late mother. Had I had such an opportunity with my mother, even at present, I would have risked life and limb for a moment's conversation. There is always so much left unsaid, even when one has the opportunity to say good-bye, and despite my apprehension, I continued to return to the same conclusion: I would take the boys to visit their mother. I would be guarded, and I would bring the appropriate articles of protection, even if I doubted their usefulness, and I would remain alert for the slightest sign of mischief. Heaven and Earth had been moved to accommodate Mrs. Darrow, and I could not find it in my heart to deny what was so obviously a mother's right to demand. As for the woman herself, I believed that she could be managed. She loved her family, and if she wished to have access to them, then she would do so on my terms.\n\nI found the children in the dining room devouring plates of roasted pheasant and slabs of cheese on toast. For all my inner turmoil, I had lost my appetite and was anxious to continue with my plan before I lost my resolve.\n\n\"Would you care to take the remainder of your lessons outside?\"\n\nThe boys looked up at me, their mouths full of food. Before they even swallowed they were dashing away from the table to grab their coats out of their room. I took a heavy shawl and placed it in my favorite basket.\n\nAs I took the children out the back of the house, we could hear Mrs. Mulbus still shrieking at Jenny. \"How could you scrub the fine china?!\"\n\nOnce outside, the boys took my hands and pulled me along toward the forest. I closed my eyes for a moment and let them lead the way as if I were caught on the sharp autumn wind.\n\nThe light dwindled as we entered the woods, and suddenly everything was silent except for the sound of our shoes crunching through the dried leaves and twigs that littered the ground. We came to the tall cage of roots at the base of the large oak tree, and were overcome by the thick, swirling mist that separated the living from the dead.\nPart 2\n\nThe Human Fashion \nCHAPTER 7\n\nThe House of Darkling\n\nThe light from the moon cast pale, sharp silhouettes that danced between the orchard trees. I wondered if there was such a thing as daytime there. The boys tried to race ahead, but I kept hold of their hands and struggled successfully against them. Despite the safety that had been assured to us by Lily Darrow, I did not trust anything about the place. If the former mistress of Everton was strong enough to turn back death, then I was obviously in no position to deny her something she had worked so hard to earn. It's not every day that the natural order of the universe becomes subverted, and if death could be turned back once, then perhaps it could be done again.\n\nIf the woman proved to be malicious in her intent, then I hoped that I would be able to deal with her when the time came. I had put on a silver cross necklace before we left Everton, but even as it pressed against my skin, it did little to soothe my apprehension as the gloom moved around us.\n\nWe walked along the path between the trees. Paul was careful not to stray far from us, and he avoided looking too closely at the bulbous pieces of fruit that twitched on the ends of the branches. Something moved up ahead, more solid than the ominous darkness that swirled languidly around us as we headed for the House of Darkling in the distance. It stepped into the center of the path and knelt down on one knee.\n\nIt was a child, a boy of about nine or ten, smartly dressed in a black waistcoat, the chain of a gold pocket watch hanging from his side. His face was long and thin, fixed with an expression of sly amusement, and yet there was something soft about his features. His face lacked any lines or creases; there was an indistinctness about him that was unsettling. I assumed it was just the light, but his skin had an orange pallor similar to the color of a peach. The boy rose from his bow and placed a finger before his lips, wordlessly silencing the children and me before leading us the rest of the way through the orchard.\n\nThe doors to the great house stood open much as before. Without the pressing anxiety that had accompanied my first visit I was better able to get a sense of the place. The entrance from the orchard was actually at the back of the house. The entryway sprawled out toward the massive oak doors at the front of the foyer, which were taller than half a dozen men standing head to toe, but the entrance was empty save for the grand staircase that spiraled away into the distance many floors above us, and the strange glittering tiles that covered the floor.\n\nThe tiles nearest to the walls were made of stone, and the band after that of rough marble, and then glazed ceramic. The tiles themselves were rather plain, but at the center of the display was a mosaic made from shards of metal and glass, the color of which shifted depending upon where one stood. As we walked over the various strata on the floor, the room changed. In the ring of stone the place was empty just as it had been before, but once we moved over the line of marble, the walls, which were tastefully paneled in wood, began to glow with an inner warmth revealing intricate etchings that seemed to tell a single story, the light burning through the designs, each panel a stained-glass window made from wood. The elegant crystal chandeliers that hung in empty space above the room began to bloom with liquid flame, light erupting out of them like stars to illuminate the corners of the space where gilded curios and antique end tables held glittering, unknowable things: strange pools of water that rippled in place but did not drip or cascade onto the floor; an iridescent apple with skin so glossy and sleek that the light it invited made it appear translucent; a portrait of a crying old woman whose tears smeared the paint; a pair of shears so sharp they seemed to cut the very light that touched their edges. But these baubles were nothing compared to the transformation that occurred at the center of the room in the mosaic. The floor was blazing with a radiant fire, pulsing in time to the silent song of the universe, throbbing with life and energy, searing not the eyes but something secret in the soul.\n\nI gasped and the children paused in wonder, but our young guide kept moving us forward, and in the next band of tiles, the one made of glazed ceramic, the room dimmed. The liquid flame of the chandelier became drawn out and stretched away from where it hung in space, translated through the crystals so that the entryway became a living world of color that shimmered and danced like the northern lights. The inner warmth of the walls gave way to delicate, incandescent fractures in the wood, smoldering and cracking like dying embers. The curios and end tables held different objects in this version of the room: a small silver harp with lines of thin shadow instead of strings, theater masks that shuddered under the weight of the emotion in their expressions, a leathery flower that grew from a pot of diamonds, a trail of black ink in a water-filled glass vase that twisted itself into the shape of a face, staring pensively at us with a gaping mouth as we passed. The mosaic on the floor withdrew into itself, the vibrancy muted in a faint red sunset flecked with blue and gold that inspired a feeling of melancholy I was not unfamiliar with.\n\nAt last we crossed over the center of the room, and as we did the pieces of glass and metal in the mosaic flared with a pale, cool light that cast all else into shadow and reflected off every surface like a million distant stars. We were lost in our own private universe, a singular nocturne that would not end so long as we stayed in the circle at the center of the entryway. For a moment I felt at peace and marveled at the power of true silence, for all sound had been extinguished. But the boy pulled us onward, and we passed across the other half of the room, into dusk, into dawn, and back into the true emptiness of the place.\n\nIt was a strange house. As feeble a thought as this was, I could not find any others to adequately describe my opinion of the House of Darkling. The little silver cross that hung below my throat was even less comforting than before.\n\nThe boy in the black waistcoat continued to guide us through the manor without slowing down. I would have worried about getting lost in a house with so many twisting corridors, but as I was being strung along by two boys who currently had more in common with a pair of excited bloodhounds straining against their leashes than with the polite, refined young men I was striving to create, I remained unconcerned.\n\nWe were led down a tall, mirrored hallway with flickering gaslights, past oval windows covered in silver latticework and a collection of heavy oak doors. The door at the end stood open, a curved wall of coarse stone panels visible beyond the threshold. The boy took us inside.\n\n\"Children?\" Lily Darrow rose from a plush green leather chair at the center of a magnificent library. The room was entirely round and four stories tall, with each subsequent ring of bookshelves smaller than the one beneath it, leading up to a domed glass ceiling, beyond which the moon hung ominously between the clouds. An ornamented footbridge led from the fourth level to a closed door. Lily opened her arms, and the boys were quick to enter her embrace. She squeezed James and kissed Paul on the forehead.\n\n\"You've come back to me . . . it's been so long.\" Her eyes trailed away and stared into space, until Paul put his hand on her shoulder.\n\n\"But, Mother, we were here just yesterday.\"\n\n\"Of course, yes. Time passes differently here. Days and years can become so confusing.\" She shook the thought away. \"I see you've met Duncan?\" She gestured to our young guide. Now that we were inside, I could clearly see that the discoloration of his skin was not simply a trick of light in the orchard. His complexion was indeed a soft shade of orange. The boy bowed out of the room and winked at us as he left. \"He serves the master of the house. Good help is difficult to find, and so Mr. Whatley grows his own.\" I connected Duncan's appearance to the fruit in the orchard, and how it had wandered off during our previous visit.\n\n\"This Mr. Whatley grows people?\"\n\n\"Duncan is not a person. Not yet, at least. Perhaps someday.\" She took a breath and smiled, seeming to become more like herself again. \"We have much to do. I expect you'll be staying the night?\" She glanced in my direction expectantly, and I pressed my lips together in a flat, expressionless smile.\n\n\"I'm afraid we didn't bring any other clothing,\" I replied.\n\n\"Of course you didn't. People would become suspicious.\"\n\n\"Just as they would if we didn't return home in a timely fashion.\"\n\n\"There's no need to worry. An entire day may pass for you here while a minute passes for everyone in Blackfield.\"\n\nI suppose this was meant to alleviate my concern, but it only made me feel very sorry for Lily. If what she said were true, then our last visit must have taken place years ago. She seemed much the same as before\u2014but then the dead could not be expected to age\u2014regal, beautiful, but with a solemn undercurrent of fragility, as if the weight of her own virtues might cause her to collapse.\n\nPaul and James looked in my direction and then back to their mother, sensitive to the subtle power struggle embedded in our exchange. I was again conscious of the silver cross hanging from my neck, completely useless against the decidedly un-supernatural force of Lily Darrow's verbal persuasion.\n\n\"Then I suppose it's not a problem.\"\n\n\"Splendid. Would you care for a tour?\" It wasn't a question. She twirled on the spot and waved her arms at the shelves of books. \"This is the library, naturally. Charlotte, you're more than welcome to use it whenever you'd like, but do be careful. The books here have a reputation for their cunning. Some readers enter for an evening's diversion and are never heard from again.\"\n\n\"Mmm.\" I could not stop myself from delivering a rather patronizing smile, but the other woman failed to notice.\n\n\"At the top of the library is Mr. Whatley's study. As I said, he is the master of the house, and always very busy. Do not disturb him unless you've been invited to do so. I expect you'll be meeting him soon enough.\" Lily looked away from us for a moment and seemed to flinch, but it was a quick movement, and I could not be sure that it wasn't caused by something mundane like a speck of dust caught in her eye. I nearly asked about Mr. Whatley and his connection to our hostess, but I held my tongue. That was a conversation that did not need to take place in front of the children, who looked up at their mother with rapt attention and an almost luminous affection. There was not a moment when one of them wasn't holding her hand or placing his forehead against her. I did not even toy with the idea of discouraging that sort of behavior. If my mother had suddenly returned from the dead, I would most likely do the same, so long as my mother still looked as graceful and alive as the former mistress of Everton.\n\nLily swept out of the library, the boys trailing behind her. I followed suit, and as I struggled to keep up with them, I was overcome by the suspicion that the day was going to feel much longer than it actually was. As we passed the large oval windows in the hallway, I took the opportunity to survey the estate. There were hills in the distance, speckled with thin, barren trees. A light mist roiled close to the ground, and far off a stark, short metal gate marked the edge of the estate.\n\nWe continued down the hallway and turned a corner at a marble sculpture of some amorphous, many-headed creature with knots of tentacles twisting out from both ends of a sleek, tubular body. I was glad when the boys passed it without really seeing it, as it was the sort of thing that would give James nightmares, if I didn't suffer from them first. I suddenly wished that I had brought some holy water from St. Michael's Church. Even if it were useless, it might have improved my appraisal of the situation I had allowed myself and the children to fall into.\n\nLily opened a set of large doors trimmed in gold leaf and took us inside a cavernous ballroom lined with rough stone pillars that could have been plucked from the bowels of the earth. The floor was a smooth black and white marble chessboard. The walls were gilded in silver and set with exotic glittering jewels of every imaginable color. Red velvet curtains clung to the sides of the windows.\n\n\"We don't entertain nearly as much as we'd like.\" Her voice echoed through the massive chamber. I estimated that Everton would fit comfortably within the ballroom twice over. \"But we expect to hold a ball sometime in the near future. Have you learned to dance yet?\" She lifted James into her arms and swung him through the air. He threw back his head and giggled with abandon.\n\nPaul looked at her strangely. \"Father hasn't held any parties.\"\n\nHis mother set James back down on the floor and seemed to notice that both boys were dressed all in black, still mourning the death that hadn't taken.\n\n\"Yes, of course. How callous of me.\"\n\n\"Can't we bring Father with us?\"\n\nShe was quick to respond. \"It's quite out of the question, and any mention of this place will close it off forever.\"\n\nPaul stepped toward me, perhaps taken aback by the unpleasant reminder that mothers could be bossy.\n\n\"Don't worry, Mother,\" James said. \"We can keep a secret. Paul brought a hedgehog into the house and kept it in the wardrobe for a whole week before Mrs. Norman found it and screamed like a girl, but I didn't tell a soul.\"\n\nLily patted her younger son on the head, visibly aware of the emotional divide that had appeared between her two children. \"Thank you, James. Shall we continue?\" She led us out of the ballroom and into a labyrinth of tight, narrow corridors, twisting and turning through the house, past the dining hall and the kitchens, the parlor, the greenhouse, the craft room, the baths, until the children were lagging as far behind as I was, perspiring and out of breath. When she realized she was twenty feet ahead of everyone else, Lily stopped and folded her hands with the graciousness of every great hostess. \"As you can see, the house is rather large. Perhaps we should survey the grounds?\"\n\nI began to sigh, feeling the burn of exhaustion in my legs, but masked it with a carefully timed cough. Lily placed an embroidered silk handkerchief into my hand. \"Are you all right?\" she asked.\n\n\"Yes, thanks. It must be dusty,\" I answered, and when she looked personally affronted, \"in my room at Everton. With my schedule I'm afraid I've let it become a bit untidy.\"\n\nLily pursed her lips and turned to face the wall behind us. Much like every other room in the house, it was paneled with wooden rectangles of various sizes. She pressed against one the size of a small door, and it swung away to reveal a stable with a gray open carriage. A silver horse was attached to the reins, its sleek body covered not in hair but in flower petals.\n\nThe boys both rushed over to the animal while I fingered the other wall panels in the hallway.\n\nI pressed against a smaller one, and it clicked open to display a stuffed miniature satyr perched in a birdcage. \"You're living in a cabinet of curiosities.\"\n\n\"But in a cabinet of curiosities, things stay in one place. The House of Darkling is always shifting to provide you with the things you want most.\" She said this without any inflection in her voice, her eyes drifting from me to her children. James patted the horse on its flank while Paul ran his fingers gently over the delicate petals that sprouted out of its skin.\n\n\"His name is Specter,\" Lily said softly.\n\nThe name was very appropriate. The animal was like something out of a dream, tall and ethereal, glowing like the perpetual moon that hung low in the sky above. Specter snorted and nodded at the carriage.\n\nThe seats were thickly padded and covered in a soft, smooth material that looked like leather but felt like velvet. James sat snugly against his mother, and across from them Paul, back to his sullen, gloomy self, sat as far away from me as he could manage. The carriage set off through the dark interior of the house, exiting beside the orchard.\n\nAs we rounded the side of Darkling, the fruit trees fell away and a large pond became visible in the distance. A lonely tree jutted out over the water. A rowboat drifted listlessly from a small, battered dock on the length of rope that tied it in place. The water at the center of the pond suddenly began to bulge with the release of air bubbles, but no one emerged from beneath the surface. I warmed myself against the chill night air, and despite himself Paul scooted closer to my end of the carriage.\n\nThe front of the great house was naturally more elaborate than the back. A circular driveway framed a fountain unlike anything I had ever seen before. Metal rods protruded out of a dark hole in the earth, all of them different heights, while pale blue swaths of liquid light bounced from one pole to another, cascading back into the black pit in a shower of electric sparks.\n\n\"That's the Star Fountain,\" Lily explained.\n\n\"It's beautiful. What is it made out of?\" I asked.\n\n\"Stars, of course.\"\n\n\"Of course.\" I could not take my eyes off the fountain. It was like having the creation of the universe in the middle of one's yard. Just who was this Mr. Whatley?\n\nSpecter trotted diligently away from the house, leading us over dark green hills flecked with drops of dew. We passed a squat iron fence that ran the length of the property, barely tall enough to keep out small animals. Lily observed it with narrowed eyes as she addressed her sons. \"Stay back from the fence, and under no circumstances are you to leave the property.\"\n\n\"Why not, Mother?\"\n\n\"The neighbors don't care for humans.\"\n\nA half mile away, the fence became a formidable, ornate gate twined with ivy. It was closed. A thick fog roiled ominously just beyond the property line. James pointed it out to his mother. \"It's just like in the orchard. Does it go all the way back to Everton?\"\n\n\"No.\" She did not elaborate any further, despite a lingering look of curiosity on her younger son's face. But Specter whinnied, and James forgot all about it. Paul did not. He looked toward me with a dark expression.\n\n\"Do you ever leave this place, Lily?\" I asked as carefully as one might speak to a dangerous animal that was at an advantage by having very large, sharp teeth.\n\n\"No. The estate is safe because that is what Mr. Whatley wishes it to be. The place beyond the gates is not quite as predictable.\"\n\n\"And what place might that be?\" I asked.\n\nBefore she could answer, Specter returned us to the front of the house. She looked as if she were about to reply, but then thought better of it and stepped down out of the carriage.\n\n\"I'm sure you must all be exhausted. Let me show you to your rooms.\" Lily entered the house and led us up the grand staircase, crossing over the rings of the floor tile, from sunrise to sunset, into the eastern wing of the mansion.\n\nShe took us into a room adorned in red and gold, with two beds set against separate alcove windows, two wardrobes, and a coffin-size toy chest. \"Boys, you'll be sleeping here.\"\n\nJames jumped onto one of the beds, claiming it as his own.\n\nPaul walked over to one of the windows and peered outside. \"Does the sun ever come up here?\"\n\nLily shut the drapes and turned Paul to the bed. \"There's nothing more beautiful than the night sky.\"\n\nThe boys undressed, and I hung their clothes in the wardrobes for when we would return home. Whenever that might be, I thought cynically.\n\nLily handed the boys fresh sets of pajamas. \"I picked these out myself.\"\n\nJames pulled his over his head immediately, but Paul stared at the clothing with suspicion. \"From where?\" he asked.\n\nLily wore a bewildered expression. \"A catalog. You've grown very curious in the past year.\"\n\n\"I've grown up a lot in the past year.\"\n\nHis mother smiled weakly and kissed him on the forehead. \"Would you like a bedtime story?\"\n\nHowever much Paul had grown up, it was not enough for him to reject being read a bedtime story curled next to his mother. He jumped into bed next to his younger brother and waited for Lily to join them. I immediately felt out of place, but was not sure why. It could not have been that I was jealous. Why should I have been? I wanted the boys to be closer to their mother. That was why I had gone against my better judgment and taken them back to the House of Darkling. That was the point. But suddenly I felt as if I were intruding upon something very private and intimate. I rose to leave the room, but Lily asked me to stay. She ran her fingers through James's hair and did not look up. I sat on the bed opposite the Darrows and waited for her to begin. She reached across the nightstand and grabbed hold of a knob set into the wall. An alcove swung open, and from it she extracted a book entitled Laurel Parker Wolfe's Tales of The Ending. She began to read:\n\nThe Sleeping King\n\nOnce upon a time, there was a castle in the sky and nothing else. From every turret and tower, only darkness could be seen. This suited the king quite well, for while his kingdom consisted of only the castle and the void, he had very few responsibilities. He was very old and tired, being immortal and having been king since before the beginning of Everything, and he had little to do but sleep in his chambers. But this was often easier said than done, for the king had five young sons, who took great pleasure in having noisy brawls in the stairwells.\n\nOne day, after the king had been stirred from an especially good dream, he banished the five princes from the castle and forced them out into the void. The princes found this to be a most unfortunate fate, for there was nothing but blackness and oblivion, and it was certainly very dull. For a long time they amused themselves with more fights and brawls, but after an aeon or two they grew tired of even this diversion and stood around in the dark looking for something to do.\n\nThe oldest of the princes was eager to return to the castle, for there were comfortable beds and large banquets every evening. He tried to enter, but the doors were firmly shut against him, and the servants had been given strict instructions from the king not to allow the princes back inside.\n\n\"We must find a way to appease Father,\" he said, and after he and his brothers had thought long and hard, they decided that the best way to win favor with the king was to flatter him with achievements and gifts. The oldest prince, having come up with the idea, was the first to attempt to win his father's forgiveness.\n\n\"I shall light the void for him, so that he might see his kingdom.\" With that, he began to cry, and every place his tears fell a star was born. He plucked out his eyes and threw them as far as he could, leaving behind a trail of glittering galaxies. When he was finished, he had his brothers place him beneath his father's tower, for he could no longer see, and he called out to the king. \"Father, see what I have done for you!\"\n\nThe old king, who had enjoyed an unbroken sleep much longer than he was used to, came to the balcony and was blinded by the unusual brightness.\n\n\"Bah! Now there is work to be done! Someone must keep the stars in the sky, and take them away when they burn out! Go, my son, and tend to your creation.\"\n\nThe oldest prince was most dismayed, but did as he was told and left to lord over the stars.\n\nThe second oldest prince, being much less intelligent than his older brother, learned nothing from this exchange. As soon as his father went back to sleep, he declared his own ambition to win his father's approval. \"I shall make a ground, so that he may travel his kingdom and have something to do besides sleep!\"\n\nWith that, he tore the bones from his body and ground them to dust, scattering them across the void and creating land. When he was finished, he had his brothers place him beneath his father's tower, for he could no longer walk, and he called out to the king. \"Father, see what I have done for you!\"\n\nThe old king, who had only just managed to get back to sleep, was unpleasantly reminded of how life had been when his sons were allowed to live in the castle, and was hardly in the mood to be impressed. He came to the balcony and was shocked by the vastness of his kingdom.\n\n\"Bah! Now there is work to be done! Someone must scout the land and find out where it ends! Go, my son, and tend to your creation.\"\n\nThe second oldest prince was most dismayed, but did as he was told and crawled across the land.\n\nThe middle brother was a very arrogant creature and, having lived in the shadow of his older brothers for so many years, was eager to succeed where they had failed. As soon as his father went back to sleep, he began preparations for his own creation.\n\n\"I shall make a sea, so that the crashing of the waves might soothe him as he sleeps.\" With that, he took a knife and cut deep into his breast. The blood from his body pooled over a portion of the land, until the castle stood on the shore of an immense ocean. When he was finished, he had his brothers place him beneath his father's tower, for he was very weak, and he called out to the king. \"Father, see what I have done for you!\"\n\nThe old king, who was now growing rather irritable at having been awoken so many times, stomped onto the balcony and was shocked by the appearance of the sea.\n\n\"Bah! Now there is work to be done, for the closeness of the water will only bring storms and flooding! Someone must sail the seas and warn us of bad weather! Go, my son, and tend to your creation.\"\n\nThe middle prince was most dismayed, but did as he was told and traveled across the ocean.\n\nThe second youngest brother was more cunning than all the rest, and while his brothers had failed, he had devised a scheme that would not only please his father but also improve his station in life. As soon as the king went back to sleep, he began to set his plan in motion.\n\n\"I shall give him subjects, so that they might worship him.\" The prince said this with a sly smile, and created the first subjects from his own flesh, until there was nothing left of him but tendon and bone. When he was finished, he had his brother place him beneath his father's tower, for he was but a skeleton, and he called out to the king. \"Father, see what I have done for you!\"\n\nThe old king was growing angry. He threw himself onto the balcony and was shocked to find that he now had actual subjects to govern.\n\n\"Bah! Now there is work to be done!\" he exclaimed. But before he could continue, the second youngest son interrupted his father.\n\n\"Yes, but I am happy to do it, Father! I will govern your subjects while you sleep!\"\n\nThe king said nothing for a moment, and then his face twisted into an expression of bemused spite. \"Ah, but as I am the king, there is no need for you to concern yourself with governance! Although someone will need to look after my subjects. They are ephemeral things, and even now they are dying. Go, my son, and tend to your creation.\"\n\nThe second youngest prince gaped for a moment, and then smiled. \"A Lord of the Dead is a king of all things, and the realm has room for but one ruler. I shall return when the dying is done, and on that day my father may find himself unable to ever sleep again.\" With that, he left to walk among the subjects of the kingdom as Death.\n\nThis left the youngest prince outside the castle, but no longer in emptiness. He was surrounded by land, sea, stars, and the specter of Death, each of which represented the failure of one of his brothers. Being the cleverest of the king's sons, he had learned from their mistakes. He had watched his brothers try and fail to win their father's approval even though none of them had attempted to consider the old king himself. The youngest did not wait for his father to return to his room. As soon as the king had sent away the second youngest prince, he called out to the king from beneath the balcony.\n\n\"Father, I will not offer you the baubles and trifles of my brothers, but the thing you want most.\"\n\nThis caught the king's attention, and his temper abated. \"And what would that be, my son?\"\n\n\"Sleep.\"\n\nThe youngest prince ripped his heart from his chest and opened it. The spark within became the moon in the sky, and the land darkened with moonlight and shadows. Instead of the dusty lands and the sprawling sea that had formerly surrounded the castle, there was now a black-green hill overlooking an empty moor. The subjects of the kingdom were nowhere to be found. The young prince went back to his place beneath the balcony.\n\nThe king was much impressed, but did not fully understand what had happened. \"What have you done?\"\n\n\"I have given you a place to end, my father. Sleep here, for while we are immortal and may never die, you will now know peace until Everything has run its course.\"\n\nThe king said nothing, and the youngest prince was unsure whether he had done something very wise or very foolish, but then his father began to cry. He left his balcony and threw open the doors to the castle. The king embraced his son for a long while, and when he was finished he set the young prince upon his throne. There were banquets and celebrations. The newly created subjects of the kingdom were invited, and the other princes even returned home for a brief while to begrudgingly join in the festivities. But when it was over, the old king returned to his chamber and went to sleep for the rest of eternity. The young prince governed in his stead and remains in his castle at The Ending of All Things, beloved by his people as the wisest and most generous creature in the land.\n\nThe boys were already fast asleep by the time Lily finished the story and set the book back on the nightstand. She kissed them both on the cheek and disentangled herself from between the children.\n\n\"Sleep well, my darlings.\"\n\nShe turned and left the room. I observed the boys one last time and followed her into my own quarters for the evening, which were decorated in muted blues and deep purples. A single four-poster bed sat at the center of the room, adrift in a sea of azure floor coverings.\n\n\"That was quite a story,\" I said.\n\n\"And somewhat strange, even for creation myths. But I thought it prudent to share as you'll be spending more time here. The people of this place have a different view of life and death.\"\n\n\"I'm afraid I'm unclear as to which they prefer.\"\n\n\"A perceptive observation. It would most certainly depend upon whom you ask.\" Lily lingered silently for a moment before going back to the doorway. \"Breakfast is at nine.\"\n\n\"How can you tell when it's morning?\"\n\n\"You can't. I\u2014\" She stopped herself. \"I'll fetch you and the children when it's time.\"\n\nLily Darrow left and closed the door behind her, leaving me alone to wonder why someone who purported to be the lady of the house was showing her guests to their rooms.\nCHAPTER 8\n\nInterrupted Moonlight\n\nI dressed for bed, but couldn't sleep. The House of Darkling was flush with ambient noises: creaking floorboards, raspy breathing from down the hall, a scuttling sound just beyond the door to my room. These, combined with the occasional smell of ammonia, were enough to keep anyone awake. I wondered if the children were having any trouble sleeping. At least they had each other.\n\nThe wall opposite my bed was made up of wardrobes from different eras, all of them carpentered together into an oversized curiosity chest with half-moon handles on each of the little doors. I opened them one by one and explored the many novelties of Mr. Whatley's collection. There was a hand mirror that sapped what light there was from my room and used it to illuminate the world beheld in the reflection, a glass eye that rolled in one continuous figure eight, something mysterious in a velvet purse that pulsed like a human heart, a pedestal of crystal phials glittering with liquid light, and a wax dollhouse in a constant state of melting as its tiny wax occupants were crowned in flames instead of hair. I paused at this last compartment and they seemed to notice that I was staring at them, for the largest one, whom I assumed to be the patriarch of the house, leapt down from the opened alcove and onto the floor. Four other candle people followed, and they marched in a single-file line to the door of my room, waiting patiently for me to open it.\n\nI wondered at Lily's comment about the house knowing what its inhabitants wanted most, for I found that the natural urges of the body required me to turn away from the dollhouse and leave my room in search of a lavatory. If there was one thing I had not wanted to do, it was to traipse through the house at night (whenever that might be) without Lily Darrow at my side. She had assured us that it was safe, but the strange noises emanating from every nook and shadow told a different tale altogether. There was an elaborate beauty about the place, but it was so bombastic that I could not shake the suspicion that it hid something more sinister. Lily was not being entirely forthcoming about how she had found her way here, to say nothing of the enigmatic Mr. Whatley, who had yet to introduce himself. I made up my mind to take the children back to Everton as soon as they'd had breakfast, but for the immediate future, I had to find the lavatory. I pulled on a housecoat from the wardrobe and opened the door.\n\nThe wax men stuck their flaming heads around the lip of the door and looked in both directions before jumping out into the corridor. They waved me onward. The hallways were empty but filled with the same noises that had disturbed my sleep. Trailing behind in the shadows of my new acquaintances, I went from one door to another, attempting to recall which of them led into the lavatory. The first one I tried opened into what appeared to be a small earthen burrow, the kind dug out by rabbits and voles. But this room was nearly as large as my own, and I crossed myself, thankful that it had been unoccupied. The next door was the correct one, and when I was finished I found myself to be quite awake.\n\nNormally in such a situation I would read until I fell asleep, but as it was I had not thought to bring along a book. I remembered the large, impressive library at the other end of the mansion. Surely I could find my way there without any trouble? While it was true that the house was eerie and strange, it had so far only proved itself to be odd, rather than dangerous. Besides, what better way to uncover the true nature of the place than to explore it alone? I would simply collect a volume or two from the library and return to my room.\n\n\"Could you please take me to the library?\" I whispered to the lead candle man. He nodded briskly and took off down the grand staircase.\n\nI could not ignore the number of small sounds that seemed to come from all corners of the house. Dripping water, something heavy dragging along uneven floorboards, the clanking of dishware; none of them were very loud, but taken altogether they sounded as if there were a great deal of activity occurring just beyond whatever closed door I happened to stand next to. The sounds followed me down the hallway with the large oval windows looking out over the estate. The metal gate at the entrance was barely visible in the mist, but still closed. I pushed open the doors at the end of the corridor and entered the library. The wax men stayed behind in the hallway, obviously sensitive to the realities of vast quantities of paper.\n\nI wasn't exactly sure what I was looking for. The book titles I had examined in the parlor weren't even in English, but then I supposed that if Lily could read them, then so could I. The first circle of the library was by far the largest, and I started there. Each of the shelves was labeled with a small silver plaque, some of them with familiar subjects, like agriculture, astrology, and astronomy, and others with more abstract areas of interest, such as death, demagogy, and demonology. I stopped at one of the larger sections in the Es, labeled \"Every Place There Is,\" which I interpreted as Travel, and took away a book entitled Balthazar.\n\nI opened the front cover, looking for some sort of description, but found only lines of elaborate calligraphy in a language I could not understand. But that did not seem to matter. The library disappeared completely, and I found myself on a low sea cliff overlooking a smooth, sandy beach, book still in hand. I nearly stumbled over the precipice in shock and wonderment, but quickly caught myself. Lily had warned me about the cunning nature of Darkling's literature. As I regained my composure, I turned around and saw a magnificent scarlet-colored castle, or perhaps a fortress, clinging to the edge of the rock. Women paraded along the walkways with pastel parasols, while the men wore expensive-looking suits and black top hats. There was a breeze coming in from the ocean, and the calls of seagulls overhead. I closed the book, and the scene at the beach disappeared with it. I turned around to observe the library, but it was the same as it had been the moment before. I placed the book under my arm, careful not to open it, and removed another book from the shelf, this one entitled India. The candle men were still waiting for me when I returned to the hallway.\n\n\"Back to my room, please,\" I said. They took me through the House of Darkling a different way than we had come, into an indoor forest with branches made of bone, past a bar whose walls were built out of packing crates, and into a room so dark I began to feel claustrophobic, staying close enough to my guides that I nearly stepped on them when they came to a sudden halt. They huddled together and extinguished themselves, leaving me alone and anxious in the chamber until another light appeared before us.\n\nA candelabra lit with quivering flames hovered in the emptiness, revealing nothing until it crept closer and a small hand appeared wrapped around its brass base, followed by the mischievous face of the boy, Duncan. For a moment I was sure that he had seen me, but he continued forward without a word. A stranger followed behind him. Even in the gloom I could tell that it was a large man, and he held a hat in front of his chest.\n\nI followed them. The candle men grabbed hold of my robe in an effort to stop me, but I shook them off. If I was going to protect the children, then there were secrets here that I had to learn.\n\nDuncan strode slowly through the house and paused before a marble bas-relief sculpture set into the wall, which depicted a litany of faces (human and otherwise) agape in agony.\n\n\"Oh yes, please. Please . . .\" said the man with a hint of desperation. His eyes were small and watery, set too deeply in a fat, chinless face, with a wattle that trembled as his body shook with excitement. Duncan's expression of bemusement never changed as he pressed a finger into the eye socket of one of the smaller characters from the sculpture, pushing it back until it clicked into place. The wall swung open slowly, heavy with the weight of the marble, and the boy stepped aside. He looked back in my direction and brought a finger to his lips before joining the stranger in the secret room, leaving the door open for me to follow. I accepted this rather blatant invitation and trailed behind them.\n\nI entered a circular chamber enveloped in concentric rings of billowing silk veils. They turned slowly in place, so that instead of walls there were only spinning layers of gauzy partitions, with gaps at random intervals in the fabric. To move from one ring to the next I had to step quickly through the openings until I stood just beyond the center of the room, where Duncan was strapping the large man into a metal chair. A wheeled table stood beside them, and on it a silver tray that held a smoky-colored phial with a white label I could not make out, a syringe, a set of forceps, and what looked to be a single lump of sugar.\n\n\"Yes, yes . . . I've been waiting so long.\" The man closed his eyes. Tears streamed down his corpulent cheeks as he relaxed into the chair, and the boy removed the stopper from the phial to extract the contents with the syringe. He injected a black liquid into the center of the sugar cube and set the needle aside, using the forceps to place the modified confection into the open, eager mouth of the stranger.\n\nThe man bit down with a crunch and immediately strained against the chair's straps as his entire body began to convulse. Duncan ignored this and tidied up the tray, sealing the phial and capping the syringe before glancing once again in my direction with a sly, knowing look. The man in the chair had stopped moving. The boy initiated the process of releasing him as I backed out of the room to find the candle men still waiting for me, their flames reignited and beckoning me from the darkness.\n\nI could not be sure of what I had just seen, but it was most certainly sordid and secretive. Why Duncan had allowed me to witness such a thing was even more vexing than the fact that such a room existed within the confines of Darkling. I was being toyed with, and I was not pleased.\n\nWe soon reached the wing where the children and I were to spend the night. Moonlight streamed through the window at the end of the corridor. I had opened the door to my room when a shadow passed over the wall. The flames of the wax men sputtered out, and they ran back to their alcove.\n\nI spun around, but there was nothing behind me. It happened again, and this time I noticed something moving beyond the window, interrupting the moonlight. Curiosity got the better of my fear, and I padded quietly to the end of the hallway to investigate.\n\nBelow the window was the pond, and an elderly man whom I could only assume to be the groundskeeper was shoveling wet, viscous slabs of something that looked like meat from a wheelbarrow into the pond. For a moment nothing happened. The meat plunked into the water and sank immediately. But then something bubbled and bulged beneath the water as it had during our tour of the estate, and a tentacle emerged like a headless snake, and then another, and another, until a half dozen of them swayed in the water, twisting and curling, finally snapping at the hunks of meat already in the pond. The old man wiped his brow with his shirtsleeve and continued emptying the contents of his wheelbarrow. The limbs of the creature submerged in the depths of the water shuddered hungrily.\n\nI backed away from the window, my heart beating against the silver cross I still wore. I did not scream. I did not swoon. But I felt with complete certainty that I had to get the children out of the house as soon as possible.\n\n\"Don't be frightened.\" A voice came from behind me. I whipped around.\n\nLily Darrow stood at the other end of the hallway, staring at me with calm reserve.\n\n\"How could you invite your children to such a terrible place?\"\n\n\"I'll admit that it is strange, but terrible . . . no.\"\n\n\"The thing in the pond\u2014\"\n\n\"Was eating.\" She approached the window and looked out over the estate. \"Just because it does not look or behave as we do is not enough to make it evil. It would never harm you or the children.\" The shadows of the tentacles in the pond passed over us, and she observed the books under my arm. \"I see that you've been to the library.\"\n\n\"Among other places,\" I said with a small measure of disdain, but I would not reveal to her what I had seen. Perhaps Lily herself was unaware of the kind of business that this Mr. Whatley trafficked in. \"I couldn't sleep.\"\n\n\"Neither could I. I've been worried about the children.\"\n\n\"As have I.\"\n\n\"I would never let anything happen to them, you must realize that.\"\n\n\"And how would you protect them from things I can scarcely find the words to describe?\" I gestured to the window.\n\n\"I cannot, but the master of this house is perfectly suited to do so.\"\n\n\"I find myself growing exceedingly suspicious over the arrangement you've made with this Mr. Whatley, who, I might point out, has yet to introduce himself.\"\n\n\"He's a very busy man. But he'll be at breakfast tomorrow. When you meet him, you'll understand why I agreed to stay. Please, if you still don't trust me after tomorrow morning, you can take the children and never bring them back.\" She gently touched my arm and looked into my eyes. She was a cipher of a woman, an odd amalgamation of strength and fragility, and I hoped that would be enough to keep whatever lived in the House of Darkling at bay.\n\n\"Then why do you worry?\"\n\n\"They've grown.\" She softened a little, and I could not keep myself from doing the same.\n\n\"Children have a tendency to do that when you're not looking.\"\n\n\"So much time has passed. Perhaps I should have let them go. I've been terribly selfish.\"\n\n\"Nonsense.\" I put my hand over hers and smiled in sadness. \"Any chance for them to know you is too important to ignore. I wish I had known my mother a little better, before she died.\"\n\nWe stood facing one another in silence, the other woman's head tilted to one side, as if she were deciding something. Then she embraced me softly.\n\n\"Until tomorrow.\"\n\nShe turned and left the hallway. I realized that I had no idea where she slept, and I wondered again what she had been doing up while the rest of us were in bed.\n\nThe noises in the house never subsided, but I was able to escape them with the help of the books from the library. The lines of strange calligraphy in the volumes translated themselves into the sights and smells of India. I wandered through the streets of Lucknow and Bombay, traveled through the courtyards of the Taj Mahal dressed in nothing more than my nightgown, a barefooted ghost unobserved by all. Each page took me to another part of the country, and I explored each of them until my legs were ready to give out from under me. I closed the book and returned to my room at the House of Darkling, where I immediately dropped onto the bed and fell into a deep sleep.\n\nI dreamt that my mother walked along a windswept moor in nothing but a nightgown identical to the one that I wore, still dead but walking all the same. I tried to convince her to come back inside, to a house in the distance, but she told me to let her be. The wind continued tearing at her, stripping off her clothes, her hair, and then her flesh as I looked on in horror, helplessly grasping at her against the storm, crying out as she literally ran through my fingers.\n\nI was already dressed when there was a sharp knock at the door to my room.\n\n\"Yes?\"\n\nA maid entered. She was young, with the same sallow, peach-colored complexion as Duncan. She motioned for me to join her in the hallway. I checked on the boys, but their room was empty. My chest tightened. I swept down the grand staircase, across the segmented foyer of the entryway, lights flashing and dimming with each step, and found the dining hall at the other end of the house.\n\nWhen I entered I felt as if I had stumbled upon some medieval banquet. The massive table was laid out with slabs of meat still on the bone, platters of sliced fruit, tureens filled with egg and cheese, urns of coffee and tea, great heaps of fish, and a number of other delicacies that I could scarcely identify.\n\nJames and Paul were seated next to one another, across from their mother. A pale teenage girl with sleek blond hair sat next to Lily. The heads of the table were occupied by two gentlemen. The first I recognized as the large man from the night before, seemingly recovered from the previous evening's indulgences as he dined on a tray of sausage and ham. The second I could only assume to be the oft-mentioned Mr. Whatley. The two men stood from their chairs.\n\n\"Charlotte, so good of you to join us,\" said Lily. \"May I introduce you to Mr. Samson\"\u2014the heavyset gentleman bowed his head, still chewing\u2014\"and of course, Mr. Whatley.\"\n\nI nodded congenially. Mr. Whatley was imposing, not in terms of weight or girth, but in proportion, not quite a giant, but oversized compared to what one would think of as normal. His substantial hands picked up a napkin from the table and dabbed it at the corners of his mouth, his thin lips framed by a rugged face, the kind that is never completely clean-shaven. He had a windswept look about him, his hair wild and disheveled, his clothing very fine but rumpled, his shirt not entirely tucked in, his collar askew, yet the most interesting thing about him was his eyes\u2014so dark that no light escaped them, giving off no reflection. They were unreadable, and as we looked at one another from across the table I felt a swell of apprehension. He was not a man to trifle with.\n\n\"Welcome, Mrs. Markham.\" His voice was deep and commanded the same power as his eyes, yet there was a swagger to it, as if he couldn't be bothered to take anything seriously. \"Please. Sit.\"\n\nIt sounded less like an invitation and more like a command, and so I hovered for a moment just to see what he would do. Whatley had gone back to his breakfast, and when he noticed that I had not complied with his request, he leaned forward and spoke to me again.\n\n\"So good of you to bring the children.\" He gestured with his hand at the chair Lily had offered me.\n\n\"Children need their mothers, little boys most of all,\" I said. I nodded imperceptibly to Mrs. Darrow and took my seat next to James and Paul. I did not wish to come off as impertinent or as a troublemaker, at least not just then. I wanted to test him, and so I had. He seemed patient, the sort of predator that prefers to lie in wait.\n\n\"Alas, there is not a strong maternal feeling among the people of The Ending, Mrs. Markham,\" said Mr. Samson between bites.\n\n\"Then I am sorry for you.\" I placed my napkin on my lap and began to serve myself.\n\nMr. Whatley smirked at me, his thin lips pulled to the left corner of his mouth. \"Are you now?\"\n\n\"Yes. There is nothing like a mother's love. One must value it while one can.\" I nodded to the boys and to Lily.\n\n\"So long as it benefits the child?\" said Mr. Whatley in a more serious tone.\n\n\"Naturally.\"\n\n\"Then it's for the best that most of ours depart. Some mothers have a hunger for their children that can't be satisfied by love.\" He said this with relish, his face an expression of mock sympathy in everything but his eyes, which observed me without emotion like a reptile's. \"It's not without precedent, even in the world of the living.\"\n\nSamson cleared his throat and brought a napkin to his lips. \"Come now, my friend. Must you use that phrase? 'World of the living,' indeed. We are just as alive, are we not?\"\n\n\"Perhaps even more so, as we do not die. But then again, most of us do not exactly live, either.\"\n\nThe large man chortled at this. \"Conserve some of that wit, Whatley, for when I finally convince you to join us.\"\n\n\"Please, sir. No politics at the table.\" The pale, blond girl seated next to Mr. Whatley spoke up with refined enthusiasm. She was severely beautiful in a cold sort of way, with arched eyebrows and an upturned nose. Her eyes were so pale that they were colorless, and they shone with enough light to compensate for the dark opacity of the man who sat to her right.\n\n\"You have my apologies, my dear.\"\n\n\"Quite all right, Mr. Samson. My father hardly needs the encouragement; he can be ever so boorish as it is. I've tried to train him the best I can, but he's simply hopeless.\"\n\n\"I prefer impertinent,\" said Mr. Whatley.\n\n\"It's all the same to me. It will never do at all if you hope to marry me off.\"\n\n\"Perhaps I don't want you to marry, Olivia.\"\n\n\"Of course you do, Father. Don't patronize me. I simply detest being patronized. You want me married as much as I do or you would never have procured a governess, let alone a human governess.\" She looked at Mrs. Darrow and smiled gratefully before patting her hand. \"Father demands the best for me.\"\n\n\"The best?\" I asked. It felt a strange sort of thing to say, using mankind as an indicator of quality, like the mink in a fur coat, or a particular kind of tea.\n\n\"Humans are the height of fashion.\"\n\n\"For now,\" said Mr. Whatley dismissively.\n\n\"Whatever for?\" I asked. I was most confused.\n\nMr. Samson folded his hands together and placed them beneath his lack of chin. \"We are immortal things, Mrs. Markham, and there is nothing duller than eternity. To pretend for a time that we know what it is like to be mortal is a mercy, however short-lived it might be.\"\n\n\"One must stay current with the trends of society.\" Olivia sipped on her tea, pinkie extended.\n\nMr. Whatley leaned back in his chair with lazy pompousness. \"On the contrary, I could do without society altogether. It's such a tiresome thing.\"\n\n\"All I require of you is that you're mildly agreeable until after my coming-out ball,\" said the girl to her father.\n\n\"I have already agreed to your terms, my love. Am I not wearing this silly thing?\" He plucked at the skin of his face so that it stretched and snapped back into place like a rubber mask. Paul jumped in his chair, while James cackled maniacally at the display. For myself I pushed my plate away, having thoroughly lost my appetite.\n\n\"You wear it, but not well,\" Olivia went on.\n\n\"Then you must be more specific the next time you make a bargain. One must be careful with what one agrees to.\"\n\n\"A valuable lesson,\" said Mr. Samson as he stood from the table.\n\n\"Leaving us so soon?\" asked Mr. Whatley.\n\n\"I'm afraid I must be on my way. Thank you again for your hospitality. You shall not forget what we discussed?\"\n\n\"I shall consider it, nothing more.\"\n\n\"That will be acceptable. Good day to you all.\" The large man departed the dining room, leaving a lull in the conversation that I was eager to break.\n\n\"I'm afraid the children and I must shortly return to Everton. Their father will soon begin to worry.\"\n\nMr. Whatley leaned over the table and sipped at his tea. \"Mr. Darrow, you say? What is he like? Lily has been rather cryptic.\"\n\nMrs. Darrow jumped in immediately with a flash of an innocuous, sociable expression. \"Mr. Whatley, why don't you show them the main collection after breakfast? It is ever so interesting.\"\n\n\"Splendid idea!\" If he noticed Lily's obvious intrusion into the conversation at the mention of her husband, he pretended not to, although his eyes remained darkly mysterious.\n\n\"What do you collect, exactly?\" I asked out of politeness more than curiosity, and in an effort to divert attention from the apparently uncomfortable topic of Mr. Darrow.\n\n\"It's better to show than to tell.\" Mr. Whatley winked at me without any regard to propriety, and I looked down at my plate as I blushed, completely unprepared to deal with such a person. I had expected some cold, hard miser, the sort who would flourish in an eternal night like the one present at the House of Darkling, stifling all beauty and withering all life. But Mr. Whatley was strangely playful and in control of a quiet power that, when taken with his disheveled appearance, was quite striking and a contrast to the beautiful melancholy of the master of Everton.\n\nWhen everyone else was finished eating, Mr. Whatley stood from the table. \"Well then, by now you must have seen some of the eccentricities of the house?\" He didn't wait for a response. \"A cabinet of curiosities is enough for some. A small wardrobe with antiquities and keepsakes from the places one has been. But a true collector lives and breathes his collection.\" He began to leave the dining hall, continuing to talk as he went and expecting us to follow close behind, which we did. At the end of one hallway, beneath a portrait of an austere-looking woman with tentacles instead of hands, he pointed out a stone dais that held a statue of a man, the shadow of which passed over a circle of markings with various labels, like CHILDHOOD, YOUTH, and MIDDLE AGE. In another room there was a spinning wheel that spun water, and a vase that, when held and turned clockwise, changed its pattern so that it was always different.\n\n\"The House of Darkling is my life's work, full of oddities and marvels, some of them trinkets and some of them more . . . useful.\" He led us into the library and up the spiral staircase. When we reached the top level of the library, he went across the bridge and, after turning back to us with a dramatic pause, opened the door to his collection.\n\n\"This way, if you please.\"\n\nThe room was like a mausoleum, starkly furnished in ivory, opal, and alabaster, with a tall ceiling that opened up into a glass dome identical to the one found in the library. The natural light of the evening sky was enough to illuminate the gargantuan room, which extended along the entire length of the house, branching off every now and then into corridors with similar items on display, galleries within galleries, like chapels in a cathedral.\n\n\"A collector is only as good as his collection, they say. This place holds all of my more valuable items. These are my Emotions.\" Mr. Whatley pointed to the first section. Both sides of the hallway were lined with alabaster statues lit from behind in small pools of light. They were ancient things, reminiscent of the golden ages of Greece or Rome, naked, handsome figures, over four dozen of them, each one in a different pose. I approached one labeled Envy. The statue was of a man, his arms folded and his eyes looking sideways at something with an expression of distaste. As I looked at it I began to feel very insignificant. Mr. Whatley had certainly accomplished quite a lot if he had the luxury to work on such an expansive personal compilation of relics and antiquities, whereas what had I done but lose every person I had ever loved? I folded my arms and began to watch Mr. Whatley from the corner of my eye, before Lily pulled me away from the statue. The sensation left me as quickly as it came, and I was myself again.\n\n\"It's best not to get too close to some of them,\" she said.\n\nWe continued down the corridor, and I kept an eye on the boys so they did not linger too long before any of the statues, especially the ones like Lust that were too obscene to warrant a description. Beyond the display of emotions were landscapes painted on panes of glass. They seemed to provide their own illumination, pulsing faintly in the gloom.\n\n\"These are perhaps some of my favorite pieces,\" said Mr. Whatley.\n\n\"What are they?\" Paul spoke up as he peered at a painting of a sprawling metropolis.\n\n\"Places. Or doorways to places.\"\n\nPaul reached out to touch the glass picture of the landscape, but Mr. Whatley grabbed his hand away with gentle control. \"They're not to be touched, unless you wish to become stranded there without any hope of returning. Besides that, they're incredibly fragile and easily ruined. If you find one that piques your interest, simply ask and I can create a doorway in the orchard, like the one to Everton.\"\n\n\"Are you familiar with Everton, Mr. Whatley?\" I asked.\n\n\"Only from Lily's stories. I have not yet had the pleasure. Perhaps someday soon you would be good enough to give me a guided tour?\"\n\nI was so taken aback by this that all I could do was nod tersely in agreement.\n\nOlivia gave a dramatic sigh. \"Father, must you show them everything? My lesson with Mrs. Darrow was supposed to begin twenty minutes ago.\"\n\n\"Whatever love wants. There will be plenty of time to see the rest, and there is still much more to see.\" He led everyone out of his collection and locked the door behind us. As we went down the staircase to the bottom of the library, Lily pulled me aside and spoke to me as we walked.\n\n\"Duncan can escort you back through the orchard, but I hope you'll be able to visit us again soon? Perhaps the day after next?\" There it was, laid bare before us\u2014the moment of truth. Would we return? Darkling was certainly a very interesting place, and the Whatleys were odd but not obviously threatening. This did not release them from my suspicions, but I softened toward them. Mr. Whatley appeared to be a man of knowledge, and there was much that could be learned at the House of Darkling, things that would prove impossible at Everton. This alone was worth exploring, and considering it in conjunction with Lily's desire to continue her relationship with her children, I saw little reason to decline the invitation. I could not ignore what I had seen of the creature in the pond, or Mr. Samson's interlude with Duncan, but if there was one thing I had learned as a girl in India it was not to presume to understand a thing before one had all the requisite facts.\n\n\"I suppose we could.\"\n\n\"Excellent. I'm planning a surprise for the boys.\"\n\n\"There's no need to go to any trouble. Having their mother back from the dead is more than enough excitement for one week.\"\n\nLily lowered her voice and slowed our pace to put some distance between our conversation and the others. \"As you can see, my position here is a professional one, just as yours is at Everton.\"\n\nI gulped down a knot of guilt that rose in my throat, remembering the way I felt when seated next to Mr. Darrow in the music room, alone in the middle of the night . . .\n\n\"Can I count on you to return?\"\n\n\"I said I would return with the children and I meant it. I don't entirely trust this place, but I understand that you do. For now, that's enough.\"\n\n\"Good.\" She squeezed my wrist. At the bottom of the library we bade the Whatleys good-bye. Olivia went to prepare for her lesson, and Mr. Whatley, after taking my hand into his own large fingers and kissing it, nodded to Lily and began wandering aimlessly through the house, stopping every so often to admire the pieces of his collection that decorated each interior. Lily escorted us across the foyer of the entryway, a kaleidoscope of rooms within rooms, to the back entrance into the orchard, where Duncan was waiting for us. She kissed the children good-bye and watched from the steps of the great house as we disappeared through the trees.\nCHAPTER 9\n\nBazaar and Bizarre\n\nI had never been to a village bazaar before the one in Blackfield. Mrs. Mulbus spent the whole week leading up to it baking mincemeat pies, spice cakes, and chocolate biscuits, too busy to even bother shouting at Jenny, who sulked from the lack of attention and loudly broke a number of dishes with more than a little dramatic flair, all the while looking over her shoulder at her tormentor with something like desperation in her eyes. When she could be bothered, Mrs. Mulbus would tut quietly to herself, and Jenny would happily scowl back at her with affectionate venom.\n\nWhen I passed by Mrs. Mulbus's table with the boys and Mr. Darrow, she snuck some biscuits into the hands of the children, thinking I hadn't noticed. I was too struck by the normality of it to say anything. Following the discovery of Darkling, it seemed a long while since I had been plagued by something as simple as the children spoiling their appetites. The boys ran ahead of us and crammed the biscuits into their mouths, gulping down crumbs and wiping the bits of chocolate from their faces onto their gloves, as pleased with their own cunning as they were with their secret snack.\n\nAutumn was ending. There was very little green left in all of Blackfield. The surrounding forests shook in the breeze like dying embers, brilliant patches of gold and red erupting from the trees as showers of sparks into the ashen sky. James kicked through the piles of parchment-colored leaves that littered the grounds of St. Michael's Church. Mr. Scott walked arm in arm with Cornelia Reese, who was not only the richest woman in the village but also the catalyst for the bazaar itself. Having come from the city, she made no effort to hide her displeasure with the quaint nature of our little church, and she told anyone who would listen\u2014Mr. Scott most of all\u2014how she intended to see to it that St. Michael's be cultivated into a proper place of worship befitting the level of patronage she could offer. And so, every Sunday for the past few months, poor Mr. Scott had reminded everyone who could hear him over the din of birdsong wafting down from the rafters to do their part. The turnout at the bazaar spoke very highly of his place in the esteem of the villagers, for although many people could not stand to see Cornelia Reese succeed, it was apparent that this sentiment had been overcome by those who wished to see the vicar prosper.\n\nThere were other tables from Everton in addition to the cook's. Ellen and some of the other maids had a display of handmade dolls with simple button eyes but exquisitely detailed dresses. Mrs. Norman sat enclosed in a small tent, looking haughty and mysterious in a cloth turban that was woefully inaccurate if she intended to conjure the image of an Indian swami. Some of the villagers, Cornelia Reese in particular, seemed mortified by the idea of having a fortune-teller on the grounds of the church. They glared at Mrs. Norman as they passed by her booth, some of them more than once to make their opinions known, and all of them crossed themselves with exaggerated devotion. If Mrs. Norman noticed, she neglected to give them the benefit of a reaction, for hers was the busiest of all the tables, and by the end of the day no one had contributed more to the proliferation of St. Michael's Church than the housekeeper from Everton.\n\nThe bazaar was a boisterous affair, despite the occasional social posturing. Mr. Watersalt, the carpenter, had built a small puppet theater next to Ellen's table and was demonstrating the usefulness of her handmade dolls with a bit of theatrical flair and a vast assortment of tiny, high-pitched dolls' voices. Mildred Wallace, who was usually too concerned with the lives of others to enjoy her own, tried to show anyone who would listen the ornate clock her husband had constructed that, on the hour, displayed a whole array of carved, lifelike figures that very much resembled herself. Even Mr. Darrow seemed to forget his melancholy. He greeted everyone he passed with a dashing smile and carried James on his shoulders until the boy was persuaded to join a group of children in a complicated game of tag that mostly involved running in circles around the church and screaming as loudly as possible.\n\n\"It occasionally astounds me that he can be so happy,\" Mr. Darrow observed.\n\n\"Children are more resilient than we are, but they do still need us to set an example,\" I said, obliquely referencing Mr. Darrow's unfulfilled promise to spend more time with his sons.\n\n\"You are a wise woman, Mrs. Markham.\"\n\n\"You flatter me, Mr. Darrow.\"\n\n\"Perhaps I should do so more often.\"\n\nWe gazed at one another, lost in the moment until Constable Brickner cut between us. Mr. Darrow shook his hand with enthusiasm, much to the other man's surprise, for he was immediately suspicious of such goodwill. Together they went to the Larken brothers' table, which was copiously populated by both many kinds of ale and the majority of the men of the village. I asked after Susannah, since I hadn't yet seen her at the bazaar, but Lionel had lost track of her the hour before last. Mr. Darrow found Fredricks, who was well ahead of everyone else in his enjoyment of the festivities. It was then that Mr. Darrow took his leave of us to purchase a pint for his old friend and confidant.\n\nThat left Paul and me to wander the grounds of the church alone. We drifted away from the noise and the laughter and the scent of food, into the graveyard. The tombstone of Lily Darrow was unchanged from our last visit, and yet it meant so many other things than it had before. Paul touched the chiseled numbers that marked the date of her death.\n\n\"It's still here.\"\n\n\"What did you expect to find?\"\n\n\"I don't know . . . maybe a crack running through it? Just something different.\"\n\nWe stood beside one another in silence. I didn't know what to say. I placed my hand on his shoulder, and he continued.\n\n\"I wanted to see her again so badly. I dreamt of her every night, and every morning I would wake up and remember that she was gone. It made me sad, but it was worth it just to pretend for a while that everything was all right. But somehow this is worse, because it's real, and I still have to leave her. I can hug her, but she's still dead and Father is alone. We can't take her back with us, and everything is still broken.\"\n\n\"Would you rather that she never came back?\"\n\n\"No. I don't know. I wish nothing ever had to change.\"\n\n\"That's all life is. It must change, or else we never would.\"\n\nPaul looked so sullen, his bright blue eyes dull with a sadness that resembled his father's more and more every day. I ran my fingers through his soft black hair. \"We don't have to go back, you know, not if you don't want to.\"\n\n\"Yes, we do. I'm not ready to say good-bye, and neither is she.\" Paul stood from his mother's grave and returned to the bazaar without any pretense of enjoyment. I followed behind him, until Roland caught my gaze with a friendly, nervous wave. He was dressed in his best Sunday clothes, and he had attempted to slick down his dark hair with copious amounts of pomade, but instead of refining his appearance the waxy substance sharpened the strands into asymmetrical spikes that lent him a feral, yet also innocent look.\n\n\"Good bazaar, eh?\"\n\n\"I can see that you've dressed for the occasion.\"\n\n\"A fellow's got to look nice once in a while, or he's not much of a fellow at all. Is Mrs. Larken all right?\"\n\n\"I suppose so. I haven't seen her in a day or two\u2014\" He ran past me, toward a disheveled young woman with wild red hair. He slid an arm beneath her and sat her on the ground. Her hands were bleeding.\n\n\"Susannah?\"\n\n\"Charlotte!\" She smiled at me with relief, and patted Roland's arm. \"I've so much to tell you!\"\n\n\"What on earth has happened to you?\"\n\n\"You're going to think I'm insane.\" She put her head in her hands, smearing blood onto her forehead. It was difficult to get the groundskeeper to leave her side, but eventually he relented and agreed to fetch Lionel, occasionally glancing back at us with a dark expression. I took Susannah into the church and sat her down in a pew before the altar.\n\n\"I could never think you mad,\" I told her.\n\n\"At least that makes one of us.\"\n\n\"Let's start at the beginning. What happened to you?\"\n\nShe sat back in the pew and smoothed down her hair before taking a deep breath. She told me her story.\n\n\"I'd brought along a special cask of ale for the reverend. Lionel forgets himself sometimes when he starts drinking with the boys, and I didn't trust him to keep it set aside, so I hid it in the cellar of the church for safekeeping. But when I went down to retrieve it, the room had changed, Charlotte. There was a door where there had never been one before. For a moment I thought I had gotten turned around and discovered some new chamber beneath the parish, but no . . . it was the same old stone walls, and the cask of ale was on the table right where I had left it.\n\n\"There was nothing special about the door aside from the fact that it hadn't existed just a few hours before. It was made of cherrywood, with no special markings and a plain brass doorknob to match. I was about to leave, but then it opened inward by itself. I didn't want to know what was inside, you must believe me. I tried to go back up the stairs, but there was a darkness on the other side of that door that spilled into the cellar. The entire room went black, and soon I couldn't tell which way was up. I felt along the walls trying to find my way out, and then I saw a light.\n\n\"I went toward it, desperate to get out of that accursed place, but was disappointed to find myself staring into a mirror. I spun around in an attempt to locate the source of the light, but the rest of the room was still awash in gloom. I pressed my forehead against the glass, starting to feel exasperated, when a pair of black hands slid around my throat. Gloved hands. His hands. I tried to scream, but he was already choking the life out of me. I tried to thrash against him, but I could find no one behind me, just the hands closing tighter around my throat, and yet, I wasn't dying. In fact the light before my eyes multiplied, and I was surrounded by a half dozen similar mirrors. My reflection was different in each of them. In the closest one, I was drowning underwater. In the next, I was burning alive. There were scenes of me with my throat slit, being mauled by a wolf, shot in the head\u2014every terrible way that I've ever been afraid to die, forced upon me. I felt myself growing faint. The hands were tightening their grip around my throat, and the mirror images multiplied again.\n\n\"I saw a vision of myself the same way I had seen Nanny Prum . . . coming apart from the inside. In that moment, even as I began to lose consciousness, I felt something rise up out of me, from some deep place I didn't know I had. I stopped trying to pry the fingers from around my throat, and with all my might, I punched my fist through the looking glass.\n\n\"Every mirror shattered at once. I grabbed ahold of a glass shard and cut at the hands still clutching my neck. They shuddered and fumbled against me, trying to regain their grip, but then stopped altogether. We were no longer alone in the darkness. There were other women with us, visions of myself flayed, burned, bleeding . . . all of them stepping through the broken glass to lunge at the man in black with a fury I could never have dreamt I was capable of. I turned away and ran into the gloom that surrounded us, until the world felt solid beneath my feet once more and I could feel the cool stone walls of the church cellar. I turned around to close the door, but in its place was nothing but a pile of soot and cinders.\"\n\nShe stared at me when she was done and waited for me to say something. I didn't know what to believe until she unclenched her bloodied hands to reveal a small shard of mirror glass.\n\n\"I didn't dream it, Charlotte. It happened. What do I do?\"\n\nMy mouth tasted like ash. My mother, my father, Jonathan, Nanny Prum, and now Susannah . . . all of them set upon by a mysterious man in black.\n\n\"Be careful. Be watchful.\" Mrs. Norman's warning became my own. \"It's time for you to find your husband and to tell him what's happened.\"\n\nA man waits for you. He watches you.\n\nBut why? What did he want from me? The specter of Death had hung over my life since I was a girl, taking everyone I had ever loved. But then, with a wave of triumph, I remembered: Death is not an absolute. I knew someone who had fought against it and won, and I realized that with her help, I would be able to put an end to this horror once and for all.\nCHAPTER 10\n\nA Dangerous Game\n\nThe next afternoon, I took the children back to the House of Darkling. A young man of sixteen or seventeen years was already waiting for our party on the other side of the swirling mist. He bowed before us in greeting, and I nearly introduced myself to him. But Paul touched my arm.\n\n\"Is that Duncan?\" he asked.\n\nI observed the young man's face as he rose back to full height. There were certainly traces of the impish, mute little boy in the appearance of the stranger, as they both wore the same frozen, knowing smiles, but we had only been gone a matter of days and the pigmentation of this young man's skin was more like that of a human, whereas Duncan had retained a distinctive shade of orange. I could not believe that the two were the same until the young man brought a finger to his lips.\n\n\"Have we really been gone that long?\" I said softly, my mind wandering to thoughts of Lily Darrow, alone in Darkling for what must have felt like years, though it was probable that the Whatleys kept her busy with Olivia's education, and there was certainly enough mischief to keep her occupied were she not otherwise indisposed. I wondered how she passed the time, and then I remembered the room with the gauzy silk veils, and Mr. Samson strapped to the chair. I shivered against the coolness of the air and warmed myself by keeping pace with Duncan.\n\n\"How do you know when we're coming through?\" I asked.\n\nHe gestured to the trees with long, spindle-like fingers, and as he did, the branches twitched and swayed. The hanging pieces of fruit turned to face us, drawn to Duncan's presence. He escorted us the rest of the way out of the orchard and into the house, leading us past a room filled with the softest, most beautiful music I had ever heard, though there were no obvious musical instruments visible in the space. We found another room whose windows looked out onto a sunlit mountaintop that was, as far as I could tell, nowhere near Mr. Whatley's estate. Another room was shaped like the inside of a gazebo and made entirely of glass, and as we passed through it, I was certain that I could see the town of Blackfield in its reflection. I had no time to dwell on this, for Duncan guided us briskly through the house, the slow, languid pacing of his youth replaced by the urgent certainty of adulthood.\n\nWe found Lily and Olivia in a small parlor, both of them seated before easels with panes of painted glass identical to the ones in Mr. Whatley's collection. Lily was helping the girl mix a particular shade of green for a rolling hillside when we entered, and as she saw us, she dropped her palette in a splatter of paint. She could say nothing for a moment as she extricated herself from behind her canvas, kneeling down to hug both of the children with an audible sigh of relief. When she rose she greeted me with a polite peck on the cheek.\n\n\"You've returned,\" she said, slowly recovering from her daze.\n\n\"Of course, Mother. We missed you!\" James buried his head in her skirts.\n\nShe smiled weakly and stroked his cheek before turning to her pupil. \"Olivia, will you excuse me for a moment? There's something I want to show the children.\"\n\nThe girl nodded with her typical cool indifference, too involved in the creation of her landscape, which seemed to move even as she refined the details.\n\nLily led us from the parlor, across a drawbridge set between two cascading, lavender-scented waterfalls, through a room where it was snowing and I had to pull the boys apart as they pelted one another with balls of ice, and finally into an empty banquet hall that could have been lifted from some medieval castle, the ceiling supported by roughly hewn wood beams, and the walls made of crumbling, porous stone. There was a door at one end, a ghastly thing forged from black wrought iron that snaked around the frame like ivy, with a silver knocker set in the center.\n\nLily stood before it. \"Now, tell me what you see.\"\n\n\"It's a door,\" said James.\n\n\"Yes, but what kind of door?\"\n\n\"A heavy one, made from oak,\" answered Paul. \"With metal rivets set in the wood.\" His little brother shot him a look unique to siblings, a combination of disbelief and pity that he could possibly be related to someone so dim.\n\n\"That's not it at all. What about the gargoyles?\" James pointed to the top of the door, where I could see nothing but black metal loops like vines.\n\nLily stepped between them. \"The door is different for everyone. To some it might show the thing you need most, to others, a version of your life that you did not live. Some say it can even tell the future. Shall we find out what it has in store for us?\" I was about to object, for there are some things that children are not prepared to know, but she had already opened the door. There was darkness on the other side of the threshold, and it descended upon the stone hall to surround us in a singular void. I could still make out the Darrows as a dozen points of light circled around them, taking the shapes of framed paintings.\n\nThe first depicted Lily in a sickbed at Everton, one hand to her forehead, the satin sheets rumpled and positioned like something out of a romanticist's studio as a doctor took her pulse. Suddenly the picture came to life, startling the four of us as the doctor's voice echoed through the abyss, hollow and distant.\n\n\"Madam, I do believe you shall recover!\"\n\nThe scene ended, and the trio proceeded to the next moving frame, where it was Christmastime at Everton. The house was decorated with an attention to detail that I could never hope to match. Lily sat by the fireplace observing her family. An older version of Paul carried a little boy in his arms, and the woman who might have been his wife held a little girl by the hand as they helped the children choose their toys from the magnificent Christmas tree. A teenage James was on the other side of the room, trapping a giggling young woman beneath the mistletoe and kissing her scandalously on the cheek. Mr. Darrow joined his wife by the fire and took her hand in his. I blushed. These were private moments, and yet they would never, could never happen.\n\nMy discomfort was readily visible and threatened to change into something else altogether. I could not put into words the anger I felt in that moment. I had been betrayed. The children were supposed to say good-bye to their mother. That was what I had brought them for, but instead Lily allowed them to wallow in their loss, to obsess over the things that could never be, the lives that could not be lived. And yet, was I so very different? Did I not dream of Jonathan or my mother or father every night? My anger shifted to myself. There was a danger in what we were doing.\n\nI backed away from them and sought my way out of the room. There was another point of light in the distance, and I moved toward it, hoping for some sort of exit, but unfortunately it was another of the floating frames. Then I realized that was wrong. I could see myself in its surface, fractured a million different ways. These were not paintings; they were mirrors, and a piece of this one was missing. Even in the splintered looking glass I could see the look of understanding as it crossed my face, curdling into revulsion, and then anger.\n\nI remembered the blood on Susannah's hands, and the sound of Nanny Prum's scream as it cut through the night all those weeks ago. But most of all I remembered the man in black; the phantom from my youth who had followed me into adulthood, striking down everyone I had ever loved.\n\nA man waits for you. He watches you.\n\nWere the specters from my past and present one and the same? How had a relic from the House of Darkling found its way into the basement of St. Michael's Church, and why had it been used against Susannah?\n\nI willed myself out of the darkness, groping about for solid walls until I felt the edges of the door and slid back into the empty stone room. Duncan was waiting with a small piece of parchment that declared Mr. Whatley's desire to speak to me in private.\n\nThe young man led me deep into the great house, down many flights of stairs to a room that resembled a Turkish bath. Despite the copious veils of steam wafting through the air, I could see Mr. Whatley at the other end of the chamber, half-submerged in a murky mineral bath. He tilted his head back until the ends of his hair trailed through the water. The pool was large, and ripples formed where it was impossible for him to make them. I noticed something gliding beneath the water very much like an eel or a snake, and then I realized that it was a tentacle. There were at least a half dozen of them traveling away from Mr. Whatley's body, dipping in and out of the water in a languid, thoughtful sort of way. Yet his face was still human, as rough and wild as it had been upon our first meeting.\n\n\"Ah, Mrs. Markham.\" He smirked at my discomfort in seeing him in such casual repose.\n\n\"Mr. Whatley,\" I said sharply. I took a breath in an effort to calm myself, and he pointed for me to sit on a marble bench at the edge of the water with a hand that still resembled a man's. I knew I should have been horrified to see such a creature, especially on so intimate a level, but I felt nothing like terror, as I was too angry to have any fear. It had burned away the moment I found the shattered looking glass and connected the specter of the man in black to the House of Darkling. Who else but a collector such as Mr. Whatley would have possession of an oddity like the mysterious, ever-changing door?\n\n\"I know I should apologize for calling on you in what I'm told are improper conditions in your culture, but I won't.\"\n\n\"Should I be impressed by your rudeness?\"\n\n\"Perhaps. I only share my daughter's interest in the human fashion where it suits me. Otherwise I am only ever myself.\"\n\n\"How lucky for you. May I ask the purpose of this meeting?\"\n\n\"It's rare that both Lily and the children are preoccupied. What did you think of their little game?\"\n\n\"I found it somewhat less interesting than the one you're playing.\"\n\n\"Is that so?\" He splashed at the water, playfully distracted as I narrowed my eyes at him.\n\n\"I think it unlikely a collector such as yourself would let any of his antiquities be put to work without some notion of how they were to be utilized.\"\n\n\"It sounds as though you believe part of my collection has been put to ill use,\" he said with his sideways smirk.\n\n\"There are things happening in Blackfield that defy explanation, unless the answer lies in our recent excursions to the House of Darkling. The timing is rather suspicious.\"\n\n\"Perhaps the two are merely a coincidence?\"\n\n\"Or it is as I've said, and a game is being played.\"\n\nThe master of Darkling tilted his head to one side in a brief moment of contemplation. \"On that point I must disagree with you, for a game cannot be played alone. There must be two players.\" He stared at me from across the water, something hungry in his gaze that weighted the statement, twisting it into a kind of invitation that hung in the air with the currents of steam, chilling the heat of my anger and confidence until I began to shake. I folded my arms in an effort to mask my nerves. I had not been entirely prepared for his boldness, but I refused to be intimidated by him. I thought of Susannah, and straightened myself as I replied.\n\n\"I would imagine there to be stakes involved?\"\n\n\"Naturally. If you are able to prove a connection between Darkling and Blackfield, then I can promise you that whatever is happening will come to an end.\"\n\n\"And if I fail?\" I kept my voice even.\n\n\"I do not enjoy being accused of treachery in my own home.\" His expression suddenly grew dark, and the water became still as if in response to the change in his temperament. \"If you fail, I will take something from you of my own choosing to add to my collection. Are you sure enough of yourself to take such a risk?\"\n\nI stood from the marble bench and knelt down to the lip of the pool, lowering myself over it to look into the black pits of his eyes. \"That would depend on the rules.\"\n\n\"The only rule is to win.\"\n\n\"Very well.\" I rose and flicked the condensation from my hands. \"Then how do we begin?\"\n\n\"With a question: Do you think you'll continue to bring the children here?\"\n\n\"After today I'm not inclined to.\"\n\n\"They'll come to hate you for it.\"\n\n\"That is a sacrifice I'm willing to make.\"\n\n\"How brave of you. I doubt that would please Mr. Darrow.\"\n\n\"I was under the impression that you've never had the pleasure of his acquaintance?\"\n\n\"I feel that I know him already. It is a very familiar story, is it not? The widower who hires a beautiful young governess to tend to his children. The secret romance, social barriers broken, a spectacular wedding at the end. They all live happily ever after.\"\n\nI tried to read his face to gauge the intent of his words, but the steam was too thick and his eyes retained their vacant, unreadable blackness. I folded my hands and walked along the edge of the shallow pool.\n\n\"It may be a familiar story, but it is not one that I've had the privilege to live out. I know very little of happiness.\"\n\n\"And you won't if you do not let Lily and the children end on their own terms. They must be the ones to finish it.\"\n\n\"You insult my integrity. My only interest is in the well-being of the children.\"\n\nMr. Whatley dipped beneath the surface of the water and swam to the other end of the pool. Duncan stood near the stairs with a robe. Whatley stepped out of the pool, his entire body unabashedly visible to me, completely human, completely male, muscular and imposing. I felt myself blush and was glad for the darkness. He stepped into his robe, and Duncan handed him a cigar.\n\n\"Do not insult my intelligence. You would be a fool not to hope for such a union. Besides, what the children need is a mother. Preferably one who is of the living.\"\n\nI could not argue against that. I thought of Mr. Darrow and our conversations in the music room, and our midnight tea parties. With a shock of revulsion I wondered if our relationship was genuine, or if it was something I had, on some subconscious level, planned from the start as a game of my own. The gentleman bit off the end of the cigar but did not seem to spit it out. Duncan lit the tip of it, and Whatley deeply inhaled the smoke.\n\n\"And what of Lily?\" I asked. \"Will she live happily ever after as well?\"\n\nMr. Whatley pulled the cigar out of his mouth and smiled again. \"Perhaps. But ever after is a very long time. Good luck to you, Mrs. Markham. It's your move.\"\n\nDuncan was suddenly at my side. Mr. Whatley disappeared into a tunnel that continued on deeper into the baths, the glowing end of his cigar sliding away with him into the gloom.\nCHAPTER 11\n\nThe Stolen Sun\n\nDuncan led me to the entrance of the baths and watched as I walked up the stairs to the rest of the house. I passed by the library, rising four floors into the air, perpetual moonlight bathing the books in a soft blue glow. I could not hope to match Mr. Whatley unless I knew more about what I was involving myself with. I ran my fingers along the leather spines, and noticed a small stack of books next to the plush leather chair that Lily had been sitting in the first time we found her in the room. One of them was entitled Dreams of Blackfield. I closed the door to the library, took to her chair, and opened the book.\n\nMy eyes trailed over the lines of unintelligible calligraphy, and suddenly I was in Mr. Darrow's study. The man himself was slumped in his chair, quietly taking an impromptu afternoon nap. I found myself pushed toward him, nearly against my will, gliding across the room until I was at his side. He opened his eyes.\n\n\"Charlotte?\"\n\n\"You can see me?\"\n\n\"Yes, of course.\" He rose from his chair and stood very close to me. I could feel his breath on my face. \"I always see you here.\" He touched my cheek with trembling fingers, and I sighed with relief.\n\n\"Mr. Darrow\u2014\"\n\n\"Henry. My name is Henry.\"\n\n\"Henry.\"\n\nHe pulled me against his body and kissed me deeply on the lips. I returned the gesture and ran my fingers through his golden hair. He pushed me against the wall and jolted me out of the reverie. I was back in the library.\n\n\"Oh dear.\" I set the book on top of the pile, thought better of it, and set it back in my lap. Flushed with excitement, I had no idea how real my experience inside the book had been. What would happen the next time that we saw one another? It was difficult not to be attracted to Henry Darrow. He was very handsome, sensitive, and financially secure. Yet the attraction felt wrong, and it could not be blamed on the fact that Mr. Whatley had pointed it out. The dream of becoming the next Mrs. Darrow had begun the moment I met him, as had the loathsome idea of coming off like some sort of temptress, some fortune-seeking harpy who was using her position with the children to secure the good favor of her employer. I was not that woman; I refused to be, and so long as I could not be sure of my own intentions, I would refuse myself any happiness just to ensure that my actions were entirely pure and unquestionable.\n\nI went through the other titles in Lily's collection: Ode to the Balthazar, Eternal Death, Human Fashions, and Mysteries of The Ending. I took this last one with the other I had just read and carried them upstairs to add to my growing collection. As I passed the boys' room, the door was open and I saw the children settling in beside their mother for another bedtime story. I was about to return to my own room, but Lily saw me standing in the hallway and gestured for me to sit down with them. She began to read:\n\nThe Stolen Sun\n\nOnce upon a time, there was a caravan of gypsies traveling through the countryside. The youngest member of the clan was a girl named Spada. She was as inquisitive as she was beautiful, and each time the caravan stopped to set up camp, she would start out into the surrounding woodlands to see what sorts of interesting things she could find. As the forests could be dangerous, her mother and father would have to go after her before she got lost and left behind, for winter was coming and the caravan had to make it over the mountains before the first frost.\n\nOne day, the gypsies set up camp after an especially long trip and Spada went into the forest in search of something to eat. Her parents were busy tending the horses, as they had briars caught in their hooves, and in no time at all she was as lost as she could be, wandering through the woods as the sun began to set. There was a chill in the air, and Spada, who was usually fearless in the face of everything, grew worried that she would be unable to find shelter for the night. No sooner had she almost given up than she found a magnificent house in a clearing.\n\nIt was ancient, made of rough, large stones and timber, but the windows were full of light and the smoke coming from the chimney was sweet with the smell of baking. She approached the house with little hesitation and pulled the rope next to the entryway. A short, squat man with curling whiskers answered the door and was more than happy to take in the lost gypsy girl.\n\n\"You may stay the night,\" he said. \"But you must stay the entire night, for the forest is dangerous and I'm certain your family would rather have you lost than dead.\"\n\nSpada found the sense in this, and agreed to spend the entire night in the strange little man's home. He led her through the house to a large dining room, where they dined on many succulent dishes, and to a room with high ceilings where she was given a comfortable bed. She quickly fell asleep beneath a pile of soft blankets.\n\nThe girl slept for quite some time, so long in fact that she was surprised upon waking to see that the sun had still not risen. Spada found this to be very strange, and she left her room to learn how much time had passed. She located the little man in a parlor with a great black fireplace and told him of her concern.\n\n\"But my dear,\" he said, \"you've been here but an hour. I suppose you've enjoyed it so much that it must have seemed longer.\"\n\nSpada found the sense in this and was about to return to her room when the little man invited her to a game of cards. No longer tired, the girl played with him for quite some time until they were both feeling hungry again, and the little man called for his servants to prepare the dining room once more. Spada and her new friend ate many succulent dishes, and when she asked to retire for the remainder of the evening, she was led to another bedroom altogether, with an even larger bed and pillows so delicate she felt as if her head were resting on air.\n\nWhen she awoke Spada was certain she must have slept at least half the day away, but when she looked out the window she was dismayed to find that the sun had still failed to rise. She rushed through the great house and found the little man seated in a study filled with books and paintings. She told him of her concern.\n\n\"I agree that the night seems very long indeed,\" he said. \"But that is only because we have done so much in such a short time.\"\n\nSpada found the sense in this and was about to return to her room when the little man suggested that she join him in playing music. Coming from a family of musicians, the girl found this to be a very practical way to pass the time, and together they played and sang until their fingers hurt and their voices were raw. The little man called to his servants to prepare the dining room, and for the third time that evening Spada feasted on many delicious dishes. When they were finished the little man excused himself for a moment and left the girl alone in the company of his butler.\n\nThe servant, who always observed his master with a small measure of disdain, began speaking to the girl in a hushed whisper as soon as the little man had left. He warned her that she had been tricked, and that the little man had stolen the sun from the sky and hidden it somewhere in the house to keep her with him for one long, eternal night. Spada thanked the butler for telling her, but found that this information neither frightened nor upset her. In fact, all she could feel for the master of the house was sympathy and a little pity.\n\n\"He must be very lonely if he is willing to go to such lengths to keep me here,\" she said. \"If the sun is in the house, then I shall find it and prove to him that he does not need to use such tricks to make us friends.\"\n\nThe little man returned to the dining room and escorted Spada to another fantastic bedroom, this one with a bed lined in lullabies. She slept very soundly, but when she awoke she did not go in search of the little man. Instead, she went through the house and examined every reflective surface in search of the sun. She peered into mirrors and silver goblets, golden doorknobs, and gilded cages, looking carefully for anything that contained the sparkle of daylight. When she had satisfied herself that every reflection in the house was natural, she found the little man waiting for her in the kitchen wearing a ridiculous chef's hat. Neither of them brought up the unending night. Instead they baked all of Spada's favorite pies and cakes and ate everything that they made until their stomachs were ready to burst.\n\nIn due time, the little man escorted her to a new bedroom with a plush, delicate bed lined with dreams. This time, before he left her he paused at the door and wished her a pleasant evening. She drifted off to sleep.\n\nWhen she awoke, Spada set off into the house in search of every candle flame and burning fireplace that might contain the stolen sun. She peered into every gaslight that lined the hallways, and into every room with a blazing hearth, and when she was satisfied that she had examined every available source of firelight, she found the little man in an empty ballroom. He wore his best dancing shoes and seemed eager to teach her his favorite steps, but the look of defeat on Spada's face was enough for him to ask her what was wrong.\n\n\"I know that you have stolen the sun,\" she said without any anger or accusation, much to the little man's surprise, \"and I've been looking for it in every place I could think of. I was certain that if I could find and return it, you would see how you did not need to trick me to win over my friendship. I would have given it freely, and do still.\"\n\nThe little man was very clever and usually good at anticipating every possible outcome of a situation, but Spada's declaration caught him off guard. His eyes glistened with tears more brightly than was possible, and Spada discovered that he had hidden the sun not in his house, but in his heart. So great was his affection for the gypsy girl that he could no longer keep it to himself. His chest welled with emotion, and as it did the ballroom filled with sunlight, which streamed out through the windows of the house and into the sky above the forest, calling to the gypsies still searching for the lost Spada.\n\nWhen the girl's family arrived at the great house, they were invited into the ballroom, where everyone played music and danced and sang, and Spada never left the little man's side, not even when the sun had set and the first frost of winter licked across the earth. The mountains would wait until spring, for a true friendship was as rare as the sun in the sky.\n\n\"Was that a true story?\" James yawned and lifted his head from his mother's shoulder as she finished. Lily closed the book and set it on the nightstand. Paul, who was wide awake, kept his head in her lap and stared quietly into space.\n\n\"Every story starts with a bit of truth, no matter how small,\" she said.\n\n\"Then they probably changed the ending to make it happy,\" said the boy as he stretched his arms over his head. \"Her family wouldn't have been glad at all. The little man tried to steal her away.\"\n\nTheir mother was becoming visibly uncomfortable. Children were supposed to fall asleep with a bedtime story.\n\n\"I suppose he must have been very lonely,\" said Lily.\n\n\"Lots of people are lonely,\" said Paul without moving. He blinked and sighed. \"That doesn't make it all right to do something wrong.\"\n\nJames sat up and reached over his mother to pick up the book of fairy tales. \"Do you think they were all dead? Did the little man kill them?\"\n\n\"Why would you say such a thing?\" Lily asked with surprise that bordered on horror. \"Winter was coming. They might have died if they went into the mountains. Perhaps he saved them.\" She kissed him on the forehead and began to extricate herself from her sons. \"Although that's a very interesting interpretation.\"\n\nShe brushed the side of Paul's face, but stopped when he said: \"Perhaps he was collecting the gypsies.\" Her eyes flashed and revealed something secret, but it happened so fast that it was difficult to read. She kissed him on the cheek, and together we closed the door to the bedroom. We stood facing one another in the hallway, and Lily observed the books clutched under my arm.\n\n\"You returned to the library.\"\n\n\"Yes, the books here are quite fascinating, as was the fairy tale.\"\n\n\"Some stories are truer than we realize, and others less than we might wish,\" she said.\n\nA silence passed between us, unbroken until I lifted the books to keep them from slipping, minutely aware that I had tactfully placed my arm to hide the title of Dreams of Blackfield.\n\n\"I hope you don't mind that I've taken them?\"\n\n\"Not at all. In fact I encourage your curiosity. The more you know about this place, the more comfortable you'll feel bringing the children.\"\n\nMoonlight came in through the window at the end of the hallway, broken up only by the shadows of the languid tentacles rising from the pond outside.\n\n\"Mr. Whatley asked to speak with me today.\"\n\n\"Did he?\" Lily's voice was unreadable. \"Whatever for?\"\n\n\"He wanted to know if I would continue to bring the children,\" I said, leaving out the bargain I had struck with the master of Darkling. Lily waited for me to go on as I struggled to find the right words. \"I said that I would, for now.\"\n\n\"Splendid!\"\n\n\"I'm not so sure that it is.\" I tightened my grip on the books. \"How much do you know about Mr. Whatley?\"\n\n\"I believe that he's some kind of politician.\"\n\n\"But what sort of man is he?\"\n\n\"I should think you'd be aware by now that he's no man at all.\"\n\n\"Yes, of course, but has he ever given you any reason to fear him?\"\n\n\"We made a bargain, and whatever he might be, he keeps his promises.\"\n\nThat did not answer my question. I attempted a different tactic.\n\n\"I worry that Darkling could be harmful to the children.\"\n\nLily narrowed her eyes and folded her arms. \"I'm not sure I understand.\"\n\n\"Mourning is difficult enough by itself, but to extend it indefinitely . . . they might never recover.\"\n\n\"I see.\" Her voice was a piercing whisper. \"Then why bring them at all?\"\n\n\"Because I understand what it means to lose a loved one. You can all heal together, but at some point it will have to end.\"\n\nShe held my gaze for a long while, until I began to feel uncomfortable and looked away. Lily's ire deflated, and suddenly she looked small and weak. \"What do you propose?\"\n\n\"An end to it. Three additional visits. No more than that.\"\n\nEven in the dark I could see that she went pale. \"I'm tired, Mrs. Markham.\"\n\n\"I did not mean to overstep myself\u2014\"\n\n\"You've made your point. I shall consider it carefully.\"\n\nLily left the corridor for her room, a place I realized I had still not seen. I hoped she would think on my proposal. To protect the Darrows, I had to sever the ties between Darkling and the world of the living, but I could not do so until I more fully understood the connection.\n\nI continued down the hallway and found my own room, too tired to explore the mysteries of the books from the library.\n\nI dreamt of Sundays with my father. In the years after we returned from India with my mother's body, we had started to spend time together on the Sabbath day locked away in the conservatory, reading books, playing chess, and telling each other stories both fictional and true, no small number of which pertained to my late mother.\n\n\"The end is nigh, my peppercorn.\" That was his my father's pet name for me, his peppercorn, for he felt that I was very beautiful, but rather fiery if not handled with care.\n\nHe lit his pipe and began to smoke, which meant he was growing tired and that we would soon be off to bed. But the smoke did not disperse into the air to attach itself to his hair and clothing. It circled around his head before gathering beside him, a cloud of noxious fumes taking the shape of a man. My father grew weary the more he smoked, and the cloud became more substantial. Black. It watched him sit back in his chair as he dropped his pipe and stopped breathing, eyes open, mouth slack.\n\nIt lingered for a moment beside his lifeless body before scattering through the air to attach itself to everything in the room, a memory extinguished and remade.\nCHAPTER 12\n\nMysteries of The Ending\n\nThe next morning Lily did not come down for breakfast. Instead, she gave Duncan a note for the children, which mentioned something about an upset stomach and resting for their next visit (which she hoped would be soon). I held my tongue, while Mr. Whatley tipped his glass in my direction and stared at me from across the table with the devilish smirk that seemed a permanent fixture on his rugged, impish face. We did not linger after our meal.\n\nAs we approached Everton, I was so lost in thought that I did not notice Mr. Darrow standing at the door of the house waiting for us, and I had no time to prepare myself for our first encounter since I had inadvertently invaded his dream.\n\n\"Hello, boys!\" Mr. Darrow lifted James into the air and tussled Paul's hair with his free hand. \"Having fun with Charlotte?\"\n\n\"It's a lovely day, so I thought we'd go for a walk.\" I spoke up before the boys could respond to their father's question. Adults are much better liars than children.\n\n\"Splendid idea! I wish you'd thought to invite me along. I ended up falling asleep at my desk.\" He looked at me and blushed when he noticed that I was looking back at him. He turned his attention to the boys. \"Are you finished with your lessons yet?\"\n\nMy stomach tightened. Our lessons seemed such distant things. What kind of governess had I become? I chastised myself, but not too severely. The boys were young and would recover from whatever lapses these interludes with their mother had caused in their education. Some things were more important than arithmetic, such as being available to their reclusive father.\n\n\"Yes, I believe we are for the day.\" The boys turned to one another, beaming at their good fortune.\n\nHenry clapped his hands together. \"Excellent! I believe we have an appointment by the lake?\"\n\nThe boys rode their bicycles ahead, flicking the bells on their handlebars with gusto at every passing pedestrian as I rode next to Mr. Darrow. We traveled past the bakeshop and the butcher, the blacksmith, and the sweets shop that sold the caramel toffees the boys loved so much, past St. Michael's Church and the vicarage. We continued on until the barren autumn spokes of the forest obscured the village of Blackfield, and we found a grassy clearing overlooking the lake. Everton was visible across the water.\n\nIt was unseasonably warm, as if summer had decided to make one last appearance before the onset of winter. I removed a picnic basket from its perch on the back of my bicycle and began to set out our lunch, but while my back was turned the boys stripped down to their underwear and jumped into the lake.\n\n\"You'll catch your death!\" I said anxiously, but Mr. Darrow laughed and sat down on the blanket.\n\n\"More for us, then,\" he said.\n\nI looked up and smiled uncomfortably.\n\n\"Is everything all right?\" he asked.\n\n\"Yes, I'm just afraid I haven't gotten much sleep lately.\"\n\n\"You're working too hard, and that's not good for anyone. Especially the children.\"\n\n\"They are very energetic.\"\n\n\"Which is why I want to be more involved, just like this. Spending time with them and getting to know them better. Being a father instead of a distant figure locked away in his office, snoring.\" He smiled again, and I tried to stop myself from thinking him rather dashing, but failed.\n\n\"That's why I so enjoy working for you, Mr. Darrow. You have a true interest in the well-being of your children.\"\n\n\"My name is Henry.\"\n\n\"Sir?\"\n\n\"You can call me Henry, if you like. We're outside. There aren't as many rules out here.\"\n\n\"Mr. Darrow, I am still your employee.\"\n\n\"Nonsense. You're part of the family. Since I can call you Charlotte, it's only fair.\"\n\n\"All right then, Henry.\"\n\n\"Charlotte.\"\n\nWe stared at one another comfortably, wordlessly, at complete ease in each other's company. Then the boys plopped down on the blanket, wet as dogs.\n\n\"We're hungry!\"\n\n\"What have I told you about toweling yourselves off?\" I said with a bemused expression.\n\n\"But I'm wet like a fish!\" James sucked his cheeks between his teeth and puckered his lips. \"Phfee?\"\n\nI grabbed the nearest towel and locked the boys' heads in the crooks of my arms. Henry watched in amazement as I wrestled them dry.\n\n\"There! Almost passable. At least it'll do for now.\" I released them and turned around to serve tea from a sealed flask when I heard a pair of giggles followed by the appropriate splashes.\n\nI turned to Henry. \"You couldn't have stopped them?\"\n\n\"I wanted to see you dry them off again. It was quite impressive.\" He smiled playfully and dashed off to the edge of the lake, calling, \"Lunchtime!\"\n\nWhen he finally succeeded in getting them out of the water and dried off on his own (I refused to help him), he walked back with the boys, his gaze carefully fixed in my direction. I suddenly felt my heart very acutely, beating against the confines of my chest, and I was glad that he happened to be out of reach lest I might have re-created the scene from his dream and made a fool of myself in front of the children.\n\nIt was a relief to bask in the sunlight again after so much time at the House of Darkling. I spread my dress over the blanket on the ground and took in the mild air of the lakeshore. Henry removed his hat and sat down with a pleasant sigh and a half smile, seeming to lose his perpetually tortured expression.\n\nHe spoke without opening his eyes. \"Would you care to take the boat across the lake?\"\n\n\"What a lovely idea. You prepare the vessel, and I shall untangle the children.\"\n\nThey had started fighting after they left the water, and it was no small task to separate them. Paul had already directed a few choice blows into his brother's shoulder and thigh, and James was insistent that there be retribution. But when I threatened to drown them both like unwanted kittens, they believed me and sat quietly on opposite sides of the boat. Henry pushed us off, and we began to glide through the water.\n\nThe lake was surrounded by great hills and a smattering of trees. The steeple of St. Michael's Church was visible in the distance, and trails of smoke from the cottages in the village laced the air with the smell of freshly baked pies and roasted nuts. The boat rocked gently against the tepid current.\n\n\"So tell me, boys, what have you been learning from Charlotte?\"\n\n\"If we don't do our lessons, she's going to skin us like a pair of Indian tigers!\" James said excitedly, thrilled to be compared to anything as vicious as a tiger.\n\nHenry smiled broadly and began to chuckle.\n\n\"One must be firm with children, Mr. Darrow.\"\n\n\"Oh absolutely!\" He had to stop to catch his breath. \"Boys, I was unaware that I had unleashed such a force of nature upon you. Your mother would be very pleased.\"\n\nJames spoke up before Paul or I could think to stop him. \"But she is, Father!\"\n\nPaul immediately stood up and lunged at his brother. The boat leaned precipitously to one side, and then turned over completely. I took Henry's arm instinctively as we fell into the water, a plume of spray ascending into the air when we splashed into the cold lake. We were still near the shore, but I grabbed James anyway and lifted him to my side, scrambling out of the water like a drowned rat, our clothes sticking to us uncomfortably and releasing pockets of water with each sloshing step. The four of us collapsed at the edge of the lake. Henry took out his handkerchief, but saw that it too had been completely soaked and threw it on the ground.\n\n\"Paul, what on earth has gotten into you?\"\n\nThe boy looked at me nervously, and then to his father.\n\n\"I'm afraid it's my fault, Mr. Darrow. James has been telling fibs, recently\u2014\" James opened his mouth to speak, but I gave him a forceful look, willing him to be quiet. He was. \"\u2014and I've been trying to break him of the habit. Paul was simply a little overzealous with my instructions.\"\n\nHenry scratched his wet head, and pushed a strand of sopping dark blond hair away from his face. I thought he suddenly looked very boyish.\n\n\"I see. Well then, I suppose we had better get back to the house and change before any of us catch cold.\" He extended his arm and pulled me up from the ground, but when I stood, I neglected to release it.\n\nI felt an unpleasant combination of euphoria and dread. I could not deny that it pleased me to be walking arm in arm with Henry Darrow. I would never be able to forget my late husband, but Henry made the loss of Jonathan bearable somehow. When I was around him, the pain I always felt transformed itself into something else. He gave me hope that I could be happy again.\n\nBut then there was the other part of it, the echo of the conversation I had had with Mr. Whatley. What kind of person was I that I presumed to take an interest in a man whose wife had not been\u2014and was still not quite\u2014dead for much more than a year? I never thought of myself as a schemer or a seductress, but it was difficult to avoid the comparison. A match with Mr. Darrow would be advantageous to say the least. The only thing I could rely upon was the strong arm that looped through my own, and I held on to it more tightly than perhaps I should have. I wondered if this was not lost on the boys, who trailed behind us in silence.\n\nLionel Larken was waiting for me in the kitchen when we arrived back at Everton, trying not to come between the cook and the scullery maid, for Mrs. Mulbus held a large paring knife in her hand and was pointing it severely in Jenny's direction while muttering something about a nick in the stew kettle. He looked haggard and tired. After I had dried off and changed, I sat with him in the parlor.\n\n\"It's Susannah.\"\n\n\"What's happened? Not another attack?\" I felt the color draining from my face.\n\n\"That's just it, I don't quite know. I'm not sure that she's been sleeping. I woke up the night before last and found her wide awake, pacing around the cottage. She was staring out the windows with this funny look in her eyes, but she wouldn't tell me what it was all about. I finally got her back into the bedroom, but she insisted upon touching every shadow in the room to make sure there wasn't anything hiding in them.\n\n\"But that's not the worst of it. Mrs. Willoughby called on me last night. She said that Susannah had an episode in the shop yesterday. She left her alone again in the afternoon to make some deliveries, and when she came back, the store was in ruins. Susannah said that she thought she saw something like a rat in a pile of scrap fabric. She went to see what it was to get it out of the store, but there was nothing there. Instead, and these are her words, the pieces of fabric started knitting themselves together around her throat, strangling her, covering her nose and mouth, binding her hands so she couldn't struggle. She was able to free one of her hands and took a hot iron to her throat. Whatever was happening to her stopped, and she burned all of the fabric scraps behind the store. My God, Charlotte, you should see her throat. Dr. Barberry said she'll be fine and all healed in a few weeks, but I'm worried for her.\"\n\n\"Do you believe her story?\"\n\n\"I believe that she believes it, and my wife is no fool. Never was very superstitious. If she said she saw something unnatural, then I believe her. But what do you do for someone chasing ghosts in the night?\"\n\n\"These things are only happening when she is alone. You must watch her. Never let her out of your sight.\"\n\n\"There must be something else we can do.\"\n\n\"Leave it to me.\"\n\nThat night as I was preparing for bed, I removed Mysteries of The Ending from my basket. I remembered Lily's warning, and how the books had affected me back in the library, but my anger and suspicion overpowered any sense of caution. Nanny Prum had died an unnatural death, and so would Susannah unless I could better comprehend Whatley's intentions. To work against him, I first had to understand him. I sat on the bed and opened the book.\n\nAs I read over the strange, otherworldly characters, an icy wind plucked at my nightdress. I looked up to close the window, but instead of being in my bedroom at Everton I stood before a crumbling castle with toppled-over towers and a drawbridge that looked as if a very large bite had been taken out of its side. Across it stood a ruinous doorway with a knocker made of chains as thick as my neck.\n\nThe book was still open in my hands, and I closed it around my index finger to keep my connection to the place open but tenuous in case it proved to be dangerous. The door was thrown backward before I could even knock, answered by a vile, mad-looking thing shaped like a little girl. Her hair was falling out in clumps, and where her eyes should have been there were two tiny black keyholes set in the gray flesh of her face, both of them oozing a dark, foul liquid I did not care to identify. I gasped with a sharp intake of breath, and she snapped at me with black, broken teeth, almost catching my arm until she was yanked backward by a loop of chain around her throat.\n\nThe little girl lay sprawled. The chain that encircled her neck extended across the coarse stone floor and halfway up a staircase, where it was wrapped around the gloved wrist of a woman dressed in an aging ball gown, ratty and frayed around the edges, with holes worn down to the petticoats beneath, and a ring of skeleton keys that clattered at her side. But even so the woman held herself with a composure that bordered on the majestic, and when she spoke it was with a voice as rich as velvet, deeply commanding and aware of its power to instruct and control.\n\n\"You do not belong here,\" she said softly. In her other hand she held a battered candelabra whose candlesticks were lit with tiny blue flames. I stood upon the threshold, still shaking from my encounter with the unpleasant little girl, my insides knotted together with fear.\n\nThis was a mistake. I contemplated closing the book and turning back to Everton, to slide back beneath my covers and allow Whatley to do as he would, but what of Susannah and the Darrows? I inhaled deeply, the air stale with decay, and entered the castle.\n\nThe blue light from the candles was reflected in the chain held in the woman's hands. It not only bound the strange girl who had greeted me so poorly but trailed away behind her to every corner of the room, lashed to an infinite line of creatures hiding in the shadows, all of them thin and emaciated, some crawling along the floor while others were strung up the walls by their shackles. The lady of the castle noticed my wandering gaze and spoke again.\n\n\"You do not belong, and yet still you enter.\"\n\n\"There are things I must know.\"\n\n\"Your questions will have a price.\"\n\n\"I have no money.\"\n\nShe smiled coldly. \"I trade in a different sort of currency.\"\n\n\"What sort?\"\n\n\"Answers for questions. Mr. Whatley is not the only collector in The Ending.\"\n\n\"How do you\u2014?\"\n\n\"The stench of Darkling clings to you. This way.\" She descended the staircase and beckoned me into a room hidden behind a tattered curtain, the loop of keys at her waist chiming as she moved. I kept myself away from the walls as I followed, afraid to brush against the rest of her children, who slumped from their chains in a living death, staring into space with their empty keyhole eyes.\n\nI found the woman in a small salon, seated at a table before a window that looked over a nocturnal landscape of mountains pressing into the pale flesh of the moon. She pulled the chain that encircled her wrist, and a serving boy with dirty fingernails set two crystal goblets in front of us, filling them with the scarlet contents of a dusty cask. The scent of blackberry and rye rose from the liquid, though I did not dare to taste it.\n\n\"I will give you three answers and three answers only.\"\n\nI opened my mouth to argue, but glanced again at the chain around the woman's wrist and thought the better of it. She lifted the circle of skeleton keys from her side, unhooked the catch, and placed three of them on the table. The boy with the cask still stood before us, lock-shaped eye sockets staring out the window.\n\n\"Your first answer.\" The woman picked up the nearest key, all rusted brass and crooked teeth, and stuck it into the small, dark keyhole where the child's right eye should have been. I cringed and looked away as she turned it with a click of bone and metal. The boy began to speak with a thin, androgynous voice.\n\n\"You cannot hope to stop the master of Darkling. The games he invents are centuries in the making and will end only when they are played out. The most you can hope to do is twist the outcome to your own devices.\"\n\nWhen the boy was done speaking, I turned to the woman. \"I never asked my question.\"\n\n\"These are the answers you need, not the ones you want. The next answer.\" She lifted the second key and inserted it into the child's left eye socket. This time he spoke with the voice of an old man, high and reedy.\n\n\"We are the things that do not die, born to end with the world, but not before. We are your gods and your monsters, indifferent and unsated, waiting for a close that might never come. 'The Ending, The Ending, full of nighttime portending, the place for the Things Above Death. In great houses they wait for the Season to abate, and for time to give up its last breath.' \"\n\nThe woman touched the boy's neck and pushed down the stained collar that crowded against his chains, revealing a third keyhole at the base of his throat. She used the last key. The boy spoke with a voice more appropriate for his appearance, small and quiet, with a singsong melancholy.\n\n\"There are only the dead and the damned. Remember that when the man in black comes for you. He is coming soon, Charlotte.\" With that, the woman collected her keys and placed them back upon the metal circle at her waist.\n\nI felt as if I were going to be sick. He is coming soon, but the boy had not said when, and would not. Three answers only. I stared ahead as dumbly as the dirty children, not seeing the lady of the castle until she placed a length of cold metal chain into my hands, the end of it looped around the neck of the boy with dirty fingernails.\n\n\"My payment,\" she said.\n\n\"I don't understand.\"\n\nShe ignored me and returned to the entry hall, her children screaming as she dragged them off the floor and down from the rafters, up the length of the staircase into the hidden depths of the castle, illuminated only by the blue glow of the candelabra.\n\n\"Answers for questions. You will take him back with you, to watch, to listen. To remember.\"\n\nThat was simply too much. I threw down the fetters she had given me. \"I will do no such thing!\"\n\nShe never heard me, for the world turned over on itself and I was back in my room at Everton. Head spinning, I regained my balance in time to see the end of the chain slip around the corner of my open bedroom door as the boy with the keyhole eyes escaped into the hallway, the rattle of metal muffled by the carpet.\n\nThe children.\n\nI dropped the book and bolted out of my chambers. The creature was moving in the opposite direction of the nursery, dashing through the corridor on all fours with such speed I could never hope to keep up. He disappeared down the stairwell, chains slapping against the banister in a dull echo I was sure had awoken the entire manor.\n\nI cornered the creature in the kitchen and grabbed hold of his shackles. \"Stop this at once!\" I hissed at him, and he responded by pressing his body against the wall, sinking into it as if he were stepping into a bath. He leered at me with a mouthful of rotten teeth as his face disappeared into the skin of the house. I wrenched and twisted the chains around my arms, but still they followed the boy into the walls of Everton. Link by link they slipped away from me until at last the final loop remained, a thin brass key dangling from its end. The teeth caught on the skin of my hands, friction burning as I lost hold of it. Then I was alone in the kitchen, sobbing despite myself.\n\nNumb with exhaustion, I did the only thing I could think of: I made a cup of tea. I sipped it quietly in the dark as I glared at the wall, at yet another problem I had not the faintest idea how to solve. The woman had said that he would only observe us, but that could hardly be trusted. What have I done? I was as bad as Lily, stubborn beyond reason and steadfast in a belief that my own abilities were unmatched; a belief that was apparently misguided as I had released something wicked into Everton.\n\nI wondered if Lily felt the same.\n\nThrough the window, the grounds of the estate were bathed in moonlight, just like the House of Darkling. With the thought of the place, I set my cup onto its saucer so severely that it fell off the table and cracked into a dozen pieces. I cursed myself under my breath and knelt down to collect the ceramic fragments, so absorbed in my own melancholy and frustration that I didn't notice Henry standing over me.\n\n\"Are you all right?\"\n\nI stood, holding the sharp pieces too tightly in my hand.\n\n\"Yes, I hope I didn't wake you?\" I glanced again at the place in the wall where the dirty little boy had escaped, and then into Henry's soft sapphire eyes. Every anxiety, doubt, and fear that had been building within me melted into the ether.\n\n\"No, I've been awake. I had tea prepared, but no one else came down.\"\n\n\"How thoughtful.\"\n\n\"I had hoped you would join me.\"\n\n\"I'm sorry, I didn't realize.\"\n\n\"How could you have known? I should have made my intentions clear.\"\n\nHe gently took the pieces of the broken cup out of my hands, his fingers lingering against my own. Even after he moved away, I could feel the echo of his touch singing through my body. I gasped softly and backed away, but he held my arm, pulling me against his chest and pressing his lips to my own. The song I felt at his touch erupted into a chorus of emotion, the darkened space around us blazing with invisible light. We parted, and I rested my head against his shoulder to catch my breath.\n\n\"Henry.\"\n\nHe stiffened at the sound of his own name, releasing me and beginning to stammer. \"I'm so sorry\u2014don't know what I'm doing\u2014if I've offended you\u2014\"\n\n\"You haven't.\" I tried to take his hand, but he pulled away.\n\n\"I've taken advantage. You're in my employ. The children\u2014\" He backed away, shaking his head. \"Forgive me, but I've made a terrible mistake.\" Mr. Darrow spun around and nearly ran from the room, leaving me alone with the broken pieces of the cup in the sink, which were not nearly as sharp or painful as the ones I felt inside.\nCHAPTER 13\n\nDeath Revisited\n\nAfter a fitful night of straining to listen for the rattle of chains or the scraping of dirty fingernails against the walls, I rose for breakfast. Mr. Darrow was nowhere to be found, nor was he at lunch, yet his presence weighed heavily on me. The dread I felt at the prospect of our next meeting was so suffocating that when the boys asked to go on their late afternoon walk, I was outside waiting for them before they could even find their coats. We left under the pretense of visiting the cemetery, and found the cloud of mist just beyond the cage of roots in the forest.\n\nI did not know what I would say to Lily. How could I look her in the face?\n\nWhy should you be ashamed to take what is yours?\n\nI tried unsuccessfully to silence the voice in my mind that had grown steadily louder since my conversation with Mr. Whatley. The tone was selfish and callous, but also powerful and assured, threatening to overrule the well-mannered woman I had always prided myself (perhaps foolishly) on being. I remained undecided about whether or not this was something to resist.\n\nDuncan was waiting for us in the orchard, even taller than before and ever closer to my own age. His skin had lost all of its discoloration, leaving him nearly human in every way but for the impish smile that was imprinted upon his face like a mask.\n\nThere were two crisp suits waiting for the boys in their room.\n\n\"It would appear that we are expected to dress for dinner,\" I observed. I attempted to help James change, but he became jealous over his older brother's silver cuff links, and Paul did not take kindly to having his little brother attempt to strangle him from behind. I split them apart as best I could, each brother taking something from the other\u2014like hair and skin\u2014during the course of the separation, and I threatened them both with a form of Indian torture so terrible that I couldn't describe it to them for fear that it would scar their delicate, youthful psyches.\n\nI left them alone and returned to my own quarters. A dark green gown had been placed over the blank fabric body of the dressing mannequin next to the wardrobe. I slid it off the dressmaker's dummy and held it against my body. It was exquisite and sleek, much more extravagant a thing than I could ever have afforded, let alone purchased from Mrs. Willoughby's dress shop. I thought about Susannah and the promise I had made to her husband. Leave it to me. Yet what progress had I made? I still only half understood the game I was playing. I refused to dwell on what would happen to my friend if I failed.\n\nI began to undress, and when I was down to my underthings, Lily entered the room dressed in a slender silver gown encrusted with glittering jewels. She closed the door. I was startled and tried to hide my nakedness by pressing the dress against my body, and I snapped at her. \"Lily!\"\n\nShe did not avert her eyes or apologize for intruding. She wordlessly walked toward me and took hold of the dress. She helped me into it without a word, her hands moving along my body, finally lacing up the back. When she was finished we turned to one another, our faces mere inches apart. I wondered if she could see in me what had happened the night before with her husband, what I felt for him.\n\n\"I wanted to apologize for the way things were left,\" she said. \"You were right to question me.\"\n\nShe tightened the corset of the dress, and I could only muster a reply half above a whisper.\n\n\"I only want what's best for the children.\"\n\nLily brushed a strand of hair away from my face. \"Let me fix your hair,\" she said. She sat me down in front of the vanity mirror and undid the tightly wound bun at the back of my head, releasing blond hair over my shoulders and combing through it gently, our faces in stark contrast to one another: Lily's, ethereal with her sharp green eyes and midnight-black tresses that took on a blue sheen beneath the gaslights; the pointed features of my own softened by the light flush of my complexion, now even richer in contrast with her paleness. I looked beneath my reflection for any trace of my mother, gathering strength from it. I wondered what Lily saw when she looked at herself.\n\n\"We're not so different, you and I,\" she said. \"Either in our current occupations or in the things that we want.\" Our eyes met in the mirror. She knows. I smiled weakly and closed my eyes as the comb ran through my hair, massaging my scalp.\n\n\"If you know what I want, then I hope you'll enlighten me, because I'm not so sure that I do.\"\n\nLily placed her hands on my shoulders and set down the comb. \"We have lost too much. Life has been cruel. But we must not lose our capacity to love. We must soldier on, for life is short and death is long. We must try to love, to move on, to embrace the opportunities presented to us.\"\n\nI opened my mouth but then closed it again. I couldn't respond, at least not right away. She knew what had transpired between Mr. Darrow and me. I didn't know how, but she was aware and she approved. I doubted whether I would have been as lenient with Jonathan had our places been reversed, but then I could not begin to fathom the things that Lily Darrow had been through.\n\nThere was a charged silence between us, long enough for me to wonder if I was mistaken. Perhaps she was speaking more about her own situation than mine. Lily ran her fingers through my hair, gathering it up in her hands and pinning it into place.\n\n\"One must make sacrifices to get the things one wants most,\" she said.\n\n\"Do you think so?\"\n\n\"I know so.\"\n\n\"And what's your sacrifice?\"\n\n\"Don't you remember? I died.\" She held a hand mirror to the back of my head, showing how she had styled my hair high onto the top of my head. I didn't fully believe her. There was something else in the mirror's reflection, an expression of anguish highlighted by the glistening of her eyes, as if she were on the verge of tears. \"I'm hosting a dinner party this evening. It will give me a chance to introduce the children to the society here. Most of the guests have never seen a human child before. In fact they fear them.\"\n\n\"It seems a strange thing to fear two young boys.\"\n\n\"Humans are not permitted in The Ending, and Mr. Whatley has put himself at great risk to fulfill our bargain. Allowing mortals into a place without Death has caused something of a sensation among the people here.\"\n\n\"And what sort of people are they, exactly?\"\n\n\"Surely you must know by now. You've read some of the books. 'The remains of the day forged from shadows and clay, endless and moribund in twain. As the worlds flicker out and Death flits about, the Old Ones sip tea and champagne.' \"\n\n\"How dreadful.\"\n\n\"But beautiful in its way.\"\n\n\"Where do they come from?\"\n\n\"I don't think they're from anywhere. They always were and always will be. 'The Ending, The Ending, full of nighttime portending, the place for the Things Above Death. In great houses they wait for the Season to abate, and for time to give up its last breath.' \"\n\nI stood from the chair in front of the vanity mirror, and once again turned to face Lily.\n\n\"Such a long wait, I would imagine. It sounds very lonely,\" I said.\n\n\"I suppose that's why they've become so interested in the ways of mankind. To them, we're an amusing distraction, like animals in a cage. A pleasant diversion from their own complicated society.\"\n\n\"What will you do when they grow tired of this 'pleasant diversion'?\"\n\n\"I expect humanity will be entirely extinguished before that could ever happen.\"\n\n\"And what will happen when the boys grow into men and stop visiting?\"\n\nWe stood up and moved toward the door of the bedroom.\n\n\"I believe that we agreed upon two more visits after this one, did we not?\" She opened the door and held it for me. \"What will you do when they no longer need a governess?\"\n\n\"Move on.\" With that, I turned in to the hallway.\n\nWe found that the boys had mostly dressed, despite the fact that James's head seemed to have been intentionally wedged into the sleeve of his jacket, and Paul was missing a rather sizable patch of hair. Together and wordlessly, Lily and I smoothed out their rumpled clothing, combed over the places where Paul's scalp was visible, and did our best to make the boys look presentable. When we were finished, we left for the drawing room. James skipped ahead of us while Paul shuffled morosely behind him, leaving Lily and me to walk side by side in silence.\n\nThe other dinner guests had already started to arrive and stood around the drawing room with drinks in hand, making small talk and gossiping whenever they were out of earshot to do so comfortably.\n\nThey were an eclectic group to say the least. There was a kindly-looking older couple doing their best to resemble humans, but their skin bulged uncomfortably in the wrong places, as if it had been put on very hastily or at least without much understanding of how it was supposed to fit. The couple excitedly greeted Lily, who introduced them to the children and me as Mr. and Mrs. Arthur Puddle.\n\n\"SO PLEASED TO MAKE YOUR ACQUAINTANCE!\" said Mrs. Puddle as if she were speaking to a very slow child.\n\nLily cut in before I could reply. \"There's no need to shout, Mrs. Puddle. Most humans have ears.\"\n\n\"Is that what you call them? How odd.\" She felt the sides of her head and found her own ears. They shifted uneasily, as if they had been pinned to her skull. James stifled a giggle just as Mr. Whatley joined us with a younger couple who seemed to more closely follow his philosophy of individuality.\n\nThe woman, if she could be called a woman, wore a sheath of netting over the entire surface of her body instead of skin. Her innards pushed uncomfortably through the gaps in the fabric, red and glistening. Still, she was roughly shaped like a person, and she had features resembling eyes and lips that were fixed in a perpetual expression of haughty disdain. Mr. Whatley introduced her as Miss Yarborough, and she nodded to me without a word of greeting. The children gazed upon her with rapt fascination rather than disgust, for she was very much a living, breathing variation on some of the anatomical diagrams we had studied in class. I had to swat James's hand away before he could poke a finger into the wet flesh beneath her netting.\n\nThe gentleman, on the other hand, was most talkative. He had no body at all, as he was made of some thick, gaseous substance that coiled itself into a humanoid figure, but this did nothing to prohibit him from being the liveliest person in the room. He had no facial features, and as Mr. Whatley introduced him, he changed from the color of silver mist to a deep blue.\n\n\"Mrs. Markham, this is Mr. Snit,\" said Mr. Whatley.\n\nThe gentleman bowed deeply, so deeply in fact that it seemed he was mocking the very custom of bowing. He took my hand into his cool, misty tendrils and kissed it. \"The pleasure is mine a thousand times over, my dear lady.\"\n\nI blushed politely, and Miss Yarborough rolled her eyes. \"Do sober up, Snitty, or you shan't make it through dinner.\"\n\nMr. Snit turned an indignant shade of red. \"One must have a certain level of intoxication, my dear Miss Yarborough, to put up with you for an entire evening.\"\n\nLily pulled me away to the other side of the room for further introductions. There was a blond boy a few years younger than Olivia who was more beautiful than Miss Whatley could ever hope to be; his mother, a formidable but pleasant-looking woman; and a pair of tall, gangly creatures that resembled oversized centipedes, both of them with twelve different limbs along dappled undersides that curved into the air as they held themselves to conversation level, bulbous eyes blinking sideways.\n\n\"Mrs. Markham, I would like to introduce you to Mrs. Aldrich and her son, Dabney,\" said Lily, referring to the woman and her son.\n\nMrs. Aldrich nodded. \"It's a pleasure to meet you, Mrs. Markham.\"\n\n\"Good evening,\" said the boy. When he spoke, everyone in the room seemed to stop to stare at him, watching his perfect lips shape themselves around the words. If he noticed the attention, he pretended not to.\n\n\"And this is the Professor and Mrs. Baxter.\"\n\n\"How do you do?\" I greeted both of the centipede creatures, but as I blinked I noticed something odd about them. In the moment just before my eyes closed, and again just after they opened, the Baxters seemed to disappear. It was an odd sensation, and so I tried not to blink as I faced them.\n\n\"Hello,\" the Baxters said in unison, smiling together.\n\nFortunately I was rescued by the sudden incursion of Mr. Samson. He took my arm and steered me to the other side of the drawing room.\n\n\"Mrs. Markham! Pleasure to see you again, my dear.\" I could smell the bourbon on his breath.\n\n\"And you, Mr. Samson,\" I said primly, releasing myself from his grip.\n\n\"What do you think of the party?\"\n\n\"An interesting collection of guests to say the least.\"\n\n\"I don't much care for them myself. Present company excluded, of course.\" He glared at the only guest I had not yet met, a gentleman with a squat, flat face, tendrils of graying hair obscuring the place where one might typically find a mouth and chin. His body was sheathed in plates of calcified, translucent skin, and in place of arms or legs he had boneless, trunk-like appendages that protruded through cracks in the dried-out husk of his flesh.\n\n\"It's impolite to stare,\" said Mr. Whatley, approaching us from behind.\n\nMr. Samson turned and poked his host in the chest. \"Do you insult me, sir? By inviting that . . . creature?\"\n\n\"Mr. Cornelius is a valued member of society, just as you are. As I have told you in the past, I play no favorites. Your disagreements are your own.\"\n\n\"You're playing a dangerous game, Whatley.\"\n\n\"They are the only kind worth playing. Wouldn't you agree, Mrs. Markham?\" He gave me a knowing look and smiled with his lopsided smirk.\n\nMr. Samson huffed away, leaving me alone with the master of Darkling.\n\n\"A fair game is more interesting than a dangerous one,\" I replied, the edge in my voice unmasked.\n\nTo my surprise, Mr. Whatley nodded. \"It can be difficult to distinguish between the two, especially when an opponent does not realize the advantages she possesses.\" To this I had no retort, and I glared at him more from confusion than from dislike. He nodded at me and announced to the group that dinner was to begin. We slowly wandered into the dining hall.\n\nThe room looked nothing like it had during our previous meals, as it was now exquisitely decorated for the occasion. Bolts of lightning had been captured and mounted on black metallic pedestals along the sides of the room. The walls were adorned with the glass paintings from Mr. Whatley's private collection, each depicting a different landscape: a bleak, cratered wasteland in one; a rocky seashore with a familiar-looking scarlet fortress in the distance of another; a massive, decaying metropolis in one of the grander pieces; and even a representation of the woods just beyond Everton. Upon closer inspection, I became aware that these were not merely pictures but actual windows into the places they showed. In the portrait of Everton, I could see the trees swaying and coils of fog drifting in the breeze.\n\nMr. Whatley sat at one end of the table and his daughter at the other. I found my name written on a dainty placard and took my seat between Paul and Mr. Puddle, who was next to his wife, while Mr. Cornelius sat beside the Baxters. Miss Yarborough was between Mr. Whatley and Mr. Samson, while Lily and James were between Mr. Snit and Mrs. Aldrich, whose son, Dabney, was next to Olivia and across from Paul. Once everyone was seated, the usual dinner conversation began as we waited for the first course.\n\n\"How do you find The Ending?\" Mrs. Aldrich addressed me from across the table.\n\n\"I'm not sure that I have found it, as of yet,\" I responded coyly. \"It is rather mysterious.\"\n\nMrs. Aldrich nodded in agreement. \"We are a mysterious people, Mrs. Markham. All of us in our own little fiefdoms, our own great houses, separated by worlds upon worlds. But that is as it should be. Eternity is a very long time, and any society, no matter how enlightened it may be, is bound to tear itself apart just to prove that it can.\"\n\n\"Like the Romans.\" Paul jumped in and, to my surprise, smiled across the table at Dabney Aldrich.\n\nThe other boy spoke up, and once again, everyone stopped what they were doing to listen to whatever it was he had to say.\n\n\"I'm afraid I must disagree with my mother, for if we truly lived in an enlightened society, a difference of opinion would not be enough to set the worlds onto a path of madness.\"\n\nMiss Yarborough was about to respond when a servant wheeled a cart into the dining hall. On top of it sat a saut\u00e9 pan and a small burner with a blue flame. Mr. Whatley rose from his chair and addressed his guests.\n\n\"The universe is a very large place, but not as vast as one might imagine. There are some commonalities that bind us all together, and here, even in The Ending, it is a custom to formally welcome newcomers with a gathering of friends. Thank you, everyone, for joining us this evening in welcoming our guests.\" He gestured to the Darrows, and there was polite applause as Mr. Whatley picked up a knife from the table. He held it out in front of him.\n\n\"It is our tradition that the host of any gathering make an offer of friendship, and the best thing that anyone can hope to give is a piece of themselves. With that being said\u2014\" Mr. Whatley's human hand unraveled into a conjoined grouping of tentacles. He sliced off one of the smaller limbs with the knife, and the hand re-formed no worse for wear. The foot-long piece of flesh fell into the saut\u00e9 pan, and the servant quickly divided it into sixteen equal portions, tossing them in the air to brown them on all sides. When he was finished, he rolled the cart around the table and served each of the guests a cooked piece of Mr. Whatley.\n\nPaul stared at his plate in horror, and then gave me a pleading look. Across the table Lily nodded at us to consume what we had been given, then scolded James for eating his before everyone else had been served. Paul and I looked at one another, picked up our forks, and placed our slices of tentacle into our mouths. It tasted as if a piece of squid had been stuffed with venison. The thought of where it came from was more revolting than the actual flavor, but it was certainly nothing that I was eager to try again. Paul choked it down and quickly guzzled copious amounts of ice water.\n\nEager to pick up their conversation where it had been left off, Miss Yarborough spoke up in a crisp, intelligent voice. \"There is a difference between simple disagreement and outright rebellion. There are rules that make the universe what it is. Some of those can be broken, but to ignore others is to forget ourselves; we can only be the things that we are, and to pretend otherwise is an act of ignorance.\"\n\nMr. Samson jumped in. \"A culture that never changes grows stagnant. If we cannot change, then we have nothing to lose and even less to learn, which is why humankind has overtaken our own.\"\n\n\"The essence of our culture is not something to be slouched off to appease some insufferable fad,\" answered the bearded creature Mr. Whatley had referred to as Mr. Cornelius. \"And a dangerous one at that. With all due respect to our distinguished guests, humans do not belong in The Ending.\"\n\n\"A fad would imply a lack of social relevance,\" said Dabney. \"Look around this table, and you will see the beginnings of a movement. We wear the skins of men to remind our brethren of how far we have fallen. If we refuse to evolve, then we might as well be human.\"\n\nMiss Yarborough scoffed. \"That is only a half-truth. You wear them to play at mortality, to pretend for a moment that your time is at an end. You wear them as a sign of solidarity, but it is only a badge of weakness.\"\n\n\"Hear, hear,\" said Mr. and Mrs. Baxter in complete synchronization.\n\nOlivia sighed and shook her head. \"Why must the way one dresses mean anything beyond simple aesthetic appeal? I for one am perfectly happy when I look as I truly am, but even more so when I am dressed as a human girl because it is something I choose to be. It makes me even more like myself.\"\n\n\"My darling daughter, if that is true then you have had more selves than all the guests seated at this table.\"\n\n\"Father is rather old-fashioned, I'm afraid.\"\n\nWhatley nodded and continued. \"A person may be fashionable, but one can only bend so far before one snaps and forgets oneself altogether.\"\n\nMr. Samson glared at him, but said nothing as servants returned carrying tureens of steaming soup.\n\nI lifted the lid off of mine and found the bowl filled with a light blue sky. Dollops of cloud drifted across the surface of the broth, and as it cooled they turned to steam and wafted into the air. I dipped my spoon into the soup and brought it to my lips. It tasted of the cool wind that descends upon the land between winter and spring. It was very refreshing.\n\n\"At least you were good enough to invite Mrs. Darrow to stay on as my governess,\" said Olivia to her father. \"I've been learning ever so much.\"\n\n\"A young woman must be prepared in this day and age to understand both sides of an argument. We live in dangerous times,\" said Mr. Whatley in his ironic way, his eyes flickering from me to Lily, and then down to his daughter. I was unsure if he had been talking about himself or about the brewing discontent within The Ending.\n\n\"Indeed.\" The Professor and Mrs. Baxter finished their bowls of soup at the exact same time and pushed them away. \"We are quite afraid that civil war will soon be upon us.\"\n\n\"It will never come to that,\" said Miss Yarborough.\n\nDabney nodded in agreement. \"Cooler heads will surely prevail.\"\n\nMr. Puddle leaned back in his chair. \"Speck and Ashby are capable politicians. Surely some agreement can be reached.\"\n\nI was transfixed by this entire exchange, but Mrs. Puddle shook her head with an exasperated sigh. \"I do apologize, but I simply can't listen to any more talk of politics.\" With that, she pulled her ears off of her head and put them into her purse.\n\n\"Quite right,\" said a rose-colored Mr. Snit. \"Why dwell on such depressing matters after a fourth glass of wine? Someone say something cheerful or I shall be forced to sing a filthy drinking song.\"\n\nMiss Yarborough passed her hand through his body to grab hold of something solid at the center of his misty form. He squeaked and remained seated in his chair.\n\nMr. Samson ignored this and seemed to rise out of his state of agitation. \"Young Mr. Aldrich is up for a very important apprenticeship,\" he said.\n\nDabney blushed, and Mrs. Aldrich smiled modestly. \"One must not become overconfident, Mr. Samson,\" she said.\n\nThe chef entered the dining hall. He was an older, rotund gentleman with pasty skin and the curling mustache traditional for someone in his profession. The dinner guests applauded politely. The other servants raced around the room and gave each of the diners a small hollow pastry. The chef extracted a knife from his belt and held it in the air like it was part of some magician's trick.\n\nNot again, I thought. But he took the knife and cut into his wrist. For any other occurrence of bodily mutilation at the dinner table I would have covered my eyes, but this was such a strange sight that it transcended revulsion. At first, nothing happened. There was no blood. He put the knife back into his belt, took out a medium-size mallet, and stood behind Mr. Whatley. He held his wrist over the plate, and after a moment, something bulged down his sleeve. There was something literally under his skin, crawling down his arm between layers of flesh and muscle, searching for the slit in his wrist. The incision bulged unpleasantly, and a small lizard slithered wetly out of his wound and into the hollow pastry. With his other hand the chef smacked the pastry smartly with the mallet, crushing both it and the lizard in a single blow. He moved to Miss Yarborough and did the same for her. When he got to Olivia, he had to put the mallet under his arm and hold the incision in his wrist closed, for there were more lizard creatures trying to escape than there were guests in the dining hall. James clapped excitedly and couldn't wait to taste his very own lizard, but Paul gagged and I stomped on his foot with the heel of my shoe.\n\n\"We must be polite, Paul,\" I said under my breath. The chef bowed out of the room when he was finished, and I nibbled at the edge of the pastry shell but was hesitant to dig into the meaty center of the thing. Paul drank more water and refused to touch his lizard pastry. He looked up eagerly to Dabney and asked, \"What sort of apprenticeship?\"\n\nThe other boy looked down and smiled humbly. \"Mr. Samson is too kind. I will soon be competing for a position with Mr. Speck.\"\n\n\"It's one of the most prestigious apprenticeships in all the worlds,\" said Mrs. Aldrich with more than a little hint of pride.\n\n\"What does this Mr. Speck do, exactly?\" I asked, for I could no longer suppress my natural curiosity.\n\nMr. Whatley, who had been sipping a glass of wine while the other guests finished their lizard pastries, set down his goblet. \"Whatever needs to be done; Speck and Ashby are the leaders of our political parties.\"\n\nThe servants returned to clear away the plates and set out a small dish of green salad in front of each guest. After the last course I was relieved to eat something that had not previously been sentient. In no time at all the servants took away the salad plates and gave everyone an eggcup filled with what appeared to be a multicolored egg. I picked up my spoon and was about to dig into it, but then I saw the clouds hovering around the surface of the thing. Upon closer inspection I could see tiny continents and mountains, rivers and valleys. There was a whole world sitting in my eggcup.\n\nOlivia whispered to me, \"Don't worry, it's uninhabited.\"\n\n\"Thank heavens,\" I said. I scooped off the top of the tiny world with my spoon, revealing a warm red-orange center within, and ate it. It was crispy and smooth at the same time, with a hint of peppermint. At first it was cool on my tongue, but when I tasted the core of it I burned my mouth. Even before it slid down the entire length of my throat, I began to feel very full and could barely think of taking another bite. The other diners finished with their eggcups, and the servants cleared the table and brought out the final course. It was a small bowl filled with something dark but insubstantial. Olivia informed me that it was Sweetened Shadows, and after tasting it I had to agree that the name was appropriate. It had a hint of sweetness to it, but no distinct flavor. It was a pleasant way to end a very strange meal.\n\nMr. Whatley stood from the table, smiling congenially at his guests. \"Shall we retire to the drawing room?\"\n\nThe evening ended the way it began, with everyone engaged in polite conversation until Duncan entered carrying a tray of cordials. Mr. Whatley made another toast to good health, and slowly the other guests began to take their leave of the affair.\n\nMr. and Mrs. Puddle, who each shook my hand before their departure, were transfixed by the fact that my fingers could only bend in one direction and would not release me until Lily intervened and saw them out the door, leaving me to be approached by the bearded creature with the flat face.\n\n\"Mrs. Markham. We have yet to be formally introduced. I am called Cornelius.\" He bowed before me, which was for the best as I did not relish the thought of shaking one of his trunk-like limbs. \"I hope we did not offend you with all the talk of politics.\" As he spoke, I could see appendages moving themselves to form words behind the curtain of his beard. The effect was deeply off-putting, but I kept my gaze focused on his dark onyx eyes.\n\n\"On the contrary, I found it most intriguing, though I confess I still do not understand the reason for the disagreement.\"\n\n\"As with most arguments, it is an old one hardly worth discussing.\" He ushered me to a far corner of the room where we could sit beside one another on a chaise longue beneath the window. \"I do hope you'll forgive me for saying so, but I could not help overhearing your exchange with our host earlier this evening.\"\n\nI stiffened. \"That was a private conversation.\"\n\n\"Which is why I beg for your forgiveness. But you and I are of a similar mind. A game should only be played when both sides are evenly matched. Our host would do well to remember that.\"\n\n\"Then we are in agreement.\"\n\n\"I observed you listening intently to our dinner conversation. What did you make of our host?\"\n\n\"His words appease one faction of The Ending, while his appearance pacifies another.\"\n\n\"Precisely. And yet, Mr. Whatley has a reputation for decisiveness. It is perplexing that he should be uncertain of where his loyalties lie, and a situation I am eager to end.\"\n\nI peered at him carefully, absorbing his words and searching his black eyes for some hint of his character. \"That is a most vexing problem, but why should you bring it to my attention?\"\n\n\"There have never before been living humans in The Ending. For some, that alone could be viewed as an act of treason, but others are not so sure. I only wish to find some token of Mr. Whatley's allegiance to Ashby's cause, so that we might put away our suspicions.\"\n\n\"And what cause might that be?\"\n\n\"To preserve our traditions, and, if I may be frank, some sense of stability. Speck and his compatriots are dabbling in things that could upset the balance of your world in addition to our own. You are here more than I; there are many rooms in the House of Darkling, and I am unable to explore them all. You can see my dilemma. It is for his own good, wouldn't you agree?\"\n\n\"Why should I wish to do anything for the good of Mr. Whatley?\"\n\n\"Because I can offer a token of protection against whatever game he has you playing.\" Mr. Cornelius reached into his beard and extracted a small iron key. He handed it to me before any of the other guests noticed. \"Provide me with evidence of Whatley's allegiance, and I will protect you from him.\" He gestured to the key. \"One turn in any lock will send for me.\"\n\nI took the key and slid it into the folds of my dress. \"How can I trust you?\"\n\n\"I'm afraid, my dear, that you do not have much of a choice.\"\n\nMr. Cornelius and the Baxters left without saying good-bye to anyone, but none of the other guests seemed very offended. Mrs. Aldrich went to collect her son, who was seated next to Paul in a far corner of the room. They were deep in conversation and very cross when they had to part. More strange than anything during the evening was the abrupt disappearance of Paul's perpetual gloom. He began to smile and continued to do so even after Dabney left the house.\n\nMiss Yarborough and Mr. Snit were the last to depart. The gentleman was so intoxicated by the end of the evening that he turned a nasty shade of green and had to be gathered into a milk jar to be carried out by his proud, haughty companion.\n\nLily and I said good night to the Whatleys and took the children to their room. As before, Lily took Laurel Parker Wolfe's book of fairy tales from the compartment beside the bed. The children wrapped themselves against her as she began to read.\n\nThe Seamless Children\n\nOnce upon a time, there was a village in the forest. The people were happy and prosperous for a long while until a sickness spread throughout the land and took the lives of many small children. The villagers were distraught, for the local doctor could not cure them, and with the coming of winter they would certainly not survive. It was quickly decided that the only thing to do would be to seek the help of the mysterious old wizard who lived in a cave deep in the wood.\n\nThe wizard, who was lonely and did not receive many visitors, was glad to help the villagers, but he cautioned them on the dangers of what they asked him to do. \"The young ones will not survive the winter as they are. Bring them to me if you will, but I warn you that the task you present me with is a difficult one. I may not be able to undo it.\"\n\nThe villagers thanked the wizard for his warning but were so desperate to save their children that they bundled them in heavy fur blankets and carried them to the cave in the wood without a second thought. The wizard spread a large cloth sheet over the ground and asked that the children be laid in a circle on top of it. When this was done, he moved his hands through the air in an intricate pattern of unseen glyphs, all the while reciting some secret spell under his breath. The children grew smaller, down to the size of babes, and the wizard gathered the ends of the cloth sheet, shaping it into a sack. He twisted the ends together until they were closed, and kept turning them against the bulk of the shrunken children until their little faces pressed roughly against the fabric. The wizard continued the motion until the fabric came apart in many places, but the children were no longer inside. Instead, the pieces of cloth had been split and sewn together to form a family of rag dolls, one in the place of each child.\n\nThe villagers, who were not sure what to make of this change, had very little time to dwell on it, for as soon as the wizard finished he fell to his knees and died. The strain of the spell had been too great, and even as the doll children rose to their little cloth feet and stretched their little cloth arms, their parents backed out of the cave, unable to deal with what had been done to them and choosing instead to imagine that they had died along with the wizard.\n\nIt was a long, lonely winter for the dolls who had once been children. They buried the wizard as best they could, some of them ruining the threading of their delicate hands as they dug into the cold, hard earth. Most of them chose to live in the wizard's cave, but others tried to return home to the families that had abandoned them. These reunions never ended well, for even at their best, the parents still looked upon the dolls as little ghosts in cloth skin. The lucky ones were able to return to the cave, but more often than not they were burned and buried or torn apart in hopes of releasing whatever remnant of the child remained hidden within the body of the rag doll.\n\nThe single winter they had been made to survive came and went. Years passed, and the doll children slowly began to fall apart; threading unraveled, cloth skin split, and the stuffing that kept them whole fell out in clumps. It was decided among them that they could not go on for much longer. Some of them had heard during their unhappy trips to the village that there was a fairy in a nearby forest, and if the stories were true, she would be able to undo what the wizard had done.\n\nTogether the dolls traveled across the land, avoiding hawks and wolves by sleeping in hollowed-out logs, crossing large brooks so that their bodies became waterlogged, until finally they came to a small house made of colored glass at the edge of the forest. The company of dolls collapsed before the door, summoning just enough strength to rap their cloth fists against the smooth entrance. The fairy found them in a sad state of disrepair and gathered them into her arms, setting them before a comfortable fireplace to rest and dry. When they were well enough to speak, she sat with them on the ground to listen to their plea.\n\n\"Good fairy,\" they said, \"we who were once children wish to go back to what we were.\"\n\nThe fairy petted their soft cloth heads and nodded in understanding.\n\n\"Are you sure that is what you truly wish?\" she asked them. \"Instead, I could fix up your holes and strengthen your threading. You could go on forever, if you wish. The life of a child is hard and sometimes short.\"\n\nBut the doll children had come for one reason only and could not be dissuaded from their wish. The fairy placed them all into a large cauldron and cooked them for seven days and seven nights in boiling water until their cloth became skin and their stuffing turned to flesh, until their little bodies swelled and grew to their original sizes. They slept for the entire day when she was done. She found clothing for them, and a wagon with a horse, so that when they awoke they were ready to return to their families.\n\nIt was a bittersweet reunion. The families were very guilty about what they had done to their young ones, but the children who had once been dolls could not be bothered with anger or grief, for when the good fairy had returned them to their original forms, she had also brought back the incurable sickness that had driven their parents to seek the old wizard's help. They were put to bed, and surrounded by the families that had left them alone for so many years, they died as they should have done years before.\n\nLily closed the book and set it down in her lap. During the story James had squirmed away from her and moved to his own bed. Paul stayed by her side, his eyes closed and his arm around his mother.\n\n\"I don't think I like bedtime stories anymore,\" said James.\n\n\"I must admit that this one is a bit sad, but it has a very good moral.\"\n\n\"Accept your fate or suffer the consequences?\" Paul yawned and stretched, releasing her from his embrace and settling into his own bed.\n\n\"No, not at all.\" Lily gaped for a moment, put off by his comment. \"Enjoy the time you have with the ones you love. No matter what the situation.\"\n\nJames pulled his covers up to his chin as his mother tucked him in. \"Do you think they were happy to die?\" he asked her.\n\n\"Not to die, but to be with their families again, yes.\" She kissed him on the forehead and turned to do the same for Paul, but he was already under the blankets with his head on his pillow and his eyes closed.\n\n\"How could they be?\" said the older boy. \"They were abandoned for years and years. You can't forget something like that.\"\n\n\"But they did. When you only have so much time together, you must move on.\" Lily knelt beside him and kissed him on the cheek.\n\nHe squinted when she did so and turned away from her. \"I'm tired.\"\n\nLily looked as if she'd been struck across the face. She left the room before I could and leaned against the wall outside. I closed the door and joined her in the hallway.\n\n\"Are you all right?\"\n\n\"Yes, I'm fine. That story . . . I was trying to prepare them. You were right that these visits cannot go on for much longer. We have to move on. I have to let them go. Good night, Charlotte.\" She kissed me on the forehead the same as she had done to the boys and walked alone down the dark hallway to wherever it was that she went during our slumber. I did not have time to dwell on this.\n\nI entered my room and found the curio set in the wall, opening the panel that held the tiny candle men. They were playing a game of keep-away with the smallest of their group, tossing his wax head back and forth between them when I cleared my throat in disapproval.\n\n\"Could you please take me to the dark room? You know the one I mean.\" They ended their game and hopped down to the floor, and together we paraded through the House of Darkling to the bas-relief sculpture of marble faces twisted in suffering. I pressed my finger into the eye socket of one of the characters, just as Duncan had done, and the wall behind the sculpture clicked open.\n\nIt was empty save for the frightening metal chair with its restraints and the wheeled table that stood beside it. I passed through the gauzy veiled partitions to the center of the chamber as the candle people waited for me outside, refusing to enter.\n\nI knelt down and opened a compartment beneath the table, revealing a dozen rows of the smoky-colored phials, each of them featuring a white label with a concise description written in a refined, elegant hand. There were phials labeled STRANGLED, INFIRMED, IMPALED, DROWNED, BURNED, SHOT, EATEN, FROZEN, and so on and so forth, each classified by some form of misfortune. I placed one marked DISMEMBERED into my pocket and paused at BURNED. I pulled out the stopper and sniffed at the contents. It smelled of charcoal. Not exactly appetizing, but to deliver Cornelius something useful, I had to know what I had found. I dipped a finger into the black fluid so that a small trace of it was left clinging to my skin, and placed it into my mouth, ingesting it as I had seen Mr. Samson do.\n\nSuddenly I was not myself. The air was burning around me, enveloped by wreaths of flame and smoke. I was carrying something important bundled in cloth, running from room to room, my skin blistering and cracking, hair crisping against a blackened scalp. When I burst through the door into the cool night air, it pressed against the searing of my flesh, only intensifying the pain until I collapsed to the ground.\n\nThe parcel I clutched unraveled to reveal the scorched face of a woman with blond hair and a short, pointed nose, a face I beheld in the mirror whenever attempting to conjure the image of my late mother, a habit that always lent me strength, though at that moment the sight of it stripped away every shred of energy and composure I had left. I screamed, and as I screamed, the world slipped away into the blackness of oblivion, where I remained for what felt like an eternity, frozen in terror at the sight of my own body sprawled on the floor as I realized that the person running through the burning house had been Jonathan, and that, for a moment, I had felt what he had as he died.\n\nSomeone grabbed my shoulder. I spun around, nearly dropping the phial, only to find Duncan standing behind me, a finger pressed to the wry smile on his lips.\n\n\"Now you know.\"\n\nFor a moment I thought the mute had spoken, but then Mr. Samson appeared behind him, his eyes bloodshot, the lines in his face more visible than even at dinner.\n\n\"Death. All of them.\" I motioned to the smoke-colored phials with an exhausted wave of my hand.\n\n\"Many different kinds of death.\" Mr. Samson sat his large body on the metal chair and allowed Duncan to strap him in. \"We dress as humans for many reasons, but for most of us, we simply wish for peace. Endless peace. This is as close as we are likely to ever get. Ashby, Cornelius, and the others . . . they fear us. Death has never been allowed into The Ending. If that were to change, then death for some would mean death for all.\" He pointed to one of the phials, and Duncan began to make his preparations.\n\n\"The death I saw. It was my husband's.\" I could scarcely get the words out.\n\n\"Is that so? A most peculiar coincidence. But then again, few things are coincidental with Mr. Whatley. He does enjoy his games. Despite what you think, or what he might wish you to think, he is a great man. He gives us peace, if only for a few moments.\"\n\nI gestured to the phials, hands shaking. \"Where does he find them?\"\n\n\"He collects them. Observes them as they are happening, or so I'm told. Though I wouldn't put it past him to cause a few deaths in order to expand his collection.\"\n\nMy stomach lurched, and I had to concentrate very intently in order to stop myself from being sick. The man in black from my past had never been the specter of Death. It was Whatley who had plagued my life since I was a girl. But was he simply an observer of my misfortune, or had he taken my family from me to set in motion the series of events that would slowly steer me toward Everton, to Darkling, and into the diabolical game I now played with him? I had learned something important, for while I still did not fully comprehend the scope of his intentions, I now at least knew how to solve the puzzle of our game. The strongest connection between Darkling and Blackfield was not Susannah or Nanny Prum or even Lily Darrow. It was me.\n\n\"He must be stopped,\" I said through gritted teeth.\n\n\"On the contrary, Mrs. Markham. You have your own death. Leave us with ours.\"\n\nDuncan held a darkened sugar cube before Mr. Samson's lips, and the gentleman bit down on it with a satisfied crunch. As his body began to convulse, I backed out of the room and retreated to what comfort could be provided by a strange bed in the strange house of the man in black.\nCHAPTER 14\n\nLock and Key\n\nThe children and I ate breakfast alone, and when we were finished Duncan escorted us back through the orchard. We arrived at Everton, and I took the children up to the schoolroom to continue their lessons. It would be difficult to say which of us was more subdued. The boys answered questions without complaint and barely looked up when Ellen entered to inform me that Mr. Darrow required my presence in his study.\n\nI meandered through the halls of Everton, and though I failed to find any trace of the boy with the keyhole eyes, I was unable to shake the feeling that I was being watched. The sensation only intensified when I entered Mr. Darrow's study. The master of Everton stood from his desk when I found him.\n\n\"Mrs. Markham.\"\n\n\"Mr. Darrow. I didn't see you at breakfast.\" I took my seat across from him, and something cool brushed against my leg. I looked down in time to catch sight of a length of chain disappearing around the side of the desk. I felt the color drain from my face, but Mr. Darrow appeared oblivious and remained where he was, leaning forward on the palms of his hands, apprehensive and clearly anxious. I resisted the urge to cry out or leap after the shackles, focusing myself entirely on his gaze.\n\n\"I've had a lot to think about,\" he said.\n\n\"As have I.\"\n\nMr. Darrow opened his mouth to speak, but no words came out as he looked into my eyes, searching for something to say. Behind him, the boy with the keyhole eyes crept noiselessly up the wall beside Lily's portrait, sniffing at it, his head twisted in an unnatural position so that his abominable face never left our direction, broken teeth poking out from between pale gray lips.\n\n\"What happened last night . . . I'm afraid I put you in a very bad position.\"\n\n\"You mustn't think that,\" I replied, nervous that my voice might betray the growing horror I felt at the sight of the creature perching itself in the corner of the ceiling. I tentatively placed my hand over Mr. Darrow's. He pulled away.\n\n\"But I do, and there's only one thing to be done about it.\" He sat down roughly in his chair and looked away from me. \"I must send you away.\"\n\nThe anxiety I felt over Darkling and the terrible childlike creature faded beneath the weight of his words as they hung in the air between us, stinging. \"You can't possibly mean that.\"\n\n\"I took advantage of your kindness and friendship, I see that now. It's for your own good.\"\n\n\"I don't wish to leave. Doesn't that amount to anything?\" I felt it all slipping away\u2014the storybook ending, my future with the children, Henry's happiness, Lily's redemption, my victory over Whatley\u2014the pieces were sliding out of place.\n\n\"It can't. We have to think of the children. Your reputation.\"\n\n\"My reputation is perfectly intact.\"\n\n\"And I'd like to keep it that way. The servants will start to talk.\"\n\nThen I began to understand. The anger I felt won out, and the mask made of rules and restraint that had been threatening to slip from my grasp finally did.\n\n\"I see. Of course, how silly of me, it's not my reputation that concerns you.\"\n\n\"That is not what I meant.\"\n\n\"Did you ever stop to think that perhaps it was I who took advantage of you?\"\n\n\"No, never!\"\n\n\"Well, perhaps you should have. Perhaps you are simply stronger than me, because I'll never be strong enough to deny the way I feel about you.\" He stared at me dumbly, mouth agape.\n\n\"Will you tell the children or should I?\" I carried on. \"I can't imagine what it will do to them to have another woman taken from their lives.\"\n\n\"I should be the one to tell them.\"\n\n\"Then I'll go and collect my things.\"\n\n\"Please don't be angry,\" he pleaded.\n\n\"I'm not angry, you fool. I'm heartbroken.\"\n\nHe moved his mouth in a mechanical sort of way, but no sound came out. I turned away from him and stormed out of his study, throwing the door shut with a satisfying crack that echoed down the hallway. I leaned against the wall to steady myself, fighting the urge to break down and cry.\n\nContrary to what I'd told Henry, I was angry. Not just with him, but at myself for becoming vulnerable to an illusion. I had overreached, placing my hopes into a fairy tale that was not real, that hardly ever happened. Men like Henry Darrow did not fall in love with their servants. They raped them and sent them away to raise their bastard children in poverty. I stopped myself. I was becoming dramatic. Mr. Darrow had taken no liberties with me save for our kiss in the kitchen, and again, I felt my loss. Men like Henry Darrow did not come along very often, and that made it all the worse.\n\nI contemplated the different ways I could attempt to keep him until my mind drifted into dark places. There is always blackmail. I pushed the voice out of my head, but it returned again as quickly as it came. Lily could help you. She wants you together. You have not lost. At this I shuddered, for I did not wish to win or lose Henry Darrow. I wanted him to love me.\n\nTears welled in the corners of my eyes, blurring my vision to the point that I almost dismissed the shape of a head materializing through the closed oak door to Mr. Darrow's study. I lunged for it without thinking, grasping the boy with the keyhole eyes by the scruff of his neck and pulling him all the way out into the hall.\n\nI looped his chains around my wrist and lifted him beneath the crook of my arm, trapping him as I ran down the corridor to my room. I passed by a single maid dusting the draperies, but I moved so quickly that she did not give me a second glance as the creature was roughly the same size as James. He squirmed beneath my grip while I struggled with the door, but I won out and slammed it closed, dropping onto my bed with the childlike thing in my lap.\n\n\"Now, what to do with you?\" Mysteries of The Ending was locked away in a trunk at the foot of my bed, and even if I were able to get it open while keeping hold of the boy, I had no idea what his mistress might do if I showed up with him in hand. He writhed against me, and the chain jangled on the floor where the brass skeleton key connected to the last link caught on the fabric of my dress. I reached for it as he slipped away, grabbing hold of the key as the creature bolted toward the door. I yanked on the chain with all my might, and the boy jerked backward to the floor. I stepped over him to straddle his chest as I inserted the key into his right eye socket with the familiar click of metal against bone. He immediately went still and began to speak in a hushed, androgynous whisper.\n\n\"They meet in the night.\" I almost asked him to elaborate, but he continued without pausing for breath in a second, more familiar voice.\n\n\"For what purpose?\"\n\n\"A sharing of affections. And it would appear that they drink tea.\"\n\n\"Most curious. Either way it shouldn't be much longer. You should prepare yourself.\"\n\n\"Am I still permitted to take the seamstress?\"\n\n\"To what end?\"\n\n\"I wear many skins for you, and I've always wanted to have someone to repair them.\"\n\n\"As you will, so long as you do as we discussed.\"\n\n\"Of course, Mr. Whatley.\"\n\nWith that, the boy with the keyhole eyes stopped speaking and began to melt into the floor. Refusing to lose him again, I looped the length of chain around my bedpost and watched as it was stretched taut, the other end of the shackles trailing away into the solid wood flooring as the boy struggled to free himself.\n\nDespite all of my other failures, I took some solace in the fact that I had managed to detain him, and could make my way back to the dilapidated castle in order to return him to his master. Still, there was much left undone. Who would help the boys say good-bye to their mother? Who would protect them from the House of Darkling? The answer was simple, really. If I was to be dismissed, then I would tell Mr. Darrow of his wife's bargain. If what Lily said were true, then the doorway between Everton and The Ending would be closed, and while the boys might hate me for it, I would at least get the satisfaction of knowing that they were safe from things I could neither fully explain nor understand.\n\nThe conversation I'd just overheard with the help of the boy with the keyhole eyes confirmed that Whatley's intentions were at least malicious toward Susannah. Considering how transparent he had been in his plotting, I found this hardly surprising. In fact, it was mildly comforting to know that I had at least discovered a part of his methodology. With a servant of Whatley's living at Everton, there was little choice but to send Susannah away to escape whatever designs had been fixed on her. If I was indeed the link between Darkling and Blackfield, then I hoped my departure and the removal of the door between the two worlds would be enough to divert Whatley's attention away from the remaining members of the Darrow family.\n\nI felt sorry for Lily. She would never know what had happened, and I appeared unable to save her from whatever Mr. Whatley had in store. But I could not allow her relationship to continue without my supervision. It was too unpredictable.\n\nI dug my valise out of the wardrobe and began to assemble my belongings. I couldn't bring myself to pack them away, so I laid everything into piles on the bed, carefully arranging and rearranging them until well after nightfall, when there was a knock at my door. I draped a blanket over the chain that disappeared into the floor and opened the door.\n\nEllen looked tired. \"It's Mr. Darrow. He didn't come down for dinner, and when Roland went to bring him his plate, he was gone. We've searched the whole house, but he's nowhere to be found, and no one knows where he's run off to. What are we to do?\"\n\nI turned back to the spread of clothes I had assembled and braced myself from feeling too acutely the pain of my impending departure.\n\n\"Perhaps he's gone to the village?\"\n\n\"Mr. Darrow never leaves the grounds of Everton if he doesn't have to, and never without telling anyone. It's not like him at all.\"\n\n\"Are the horses accounted for?\"\n\n\"Yes. And the carriages and the bicycles. He couldn't have gone far, wherever he went, but with the weather getting on the way it is, he might catch his death.\" It was not so cold as to be freezing, but it was cool enough to signal the start of winter. Soon the lakes would turn to ice, and snow would begin to fall, and the house would prepare for the holidays. I had so wanted to spend Christmas at Everton. I snapped myself back to the matter at hand before I trailed off into self-pity.\n\nEllen and I went down to the kitchen and were joined by Roland, Fredricks, Mrs. Norman, and Mrs. Mulbus, the group of us deciding what to do and whether or not a search party should be sent out. Before we had made much progress in our pursuit of the master of the house, the doorbell rang and we found Mr. Scott on our doorstep with Mr. Darrow clinging to his side.\n\n\"Found him in the graveyard. Must have been there for hours.\"\n\nIndeed, he looked very pale. I took him from the vicar and put his arm over my shoulder. He could walk, but only just. I helped carry him up to his bedroom, while Ellen kindled a fire in the hearth.\n\n\"What have you done to yourself?\" I whispered as I tucked him into bed.\n\n\"She's gone\" was all he said before slipping off to sleep. His forehead began to burn, and I summoned Dr. Barberry, who prescribed lots of fluids and plenty of rest.\n\nHenry remained bedridden for a number of days, and I attended to his every need without regard to propriety. The children flitted in and out at will, monitored by Ellen and content with the fact that their father continued to be alive and well. The servants could talk all that they wanted, but I would tend to Henry until he was recovered and then I would leave.\n\nA week after he became ill, I found his bed empty when I went to bring him his breakfast. I went downstairs, where he was in the dining room with the children. I tried to leave before he saw me, but I was too late. He invited me to eat with them. Color had returned to his cheeks, and even some of the dourness that had lingered in him for so long seemed to have abated.\n\n\"I take it you're feeling better, Mr. Darrow?\"\n\n\"Yes, thanks to your diligence.\"\n\n\"In the future I hope that you will do everyone a courtesy and refrain from sitting out in the cold until you become ill.\"\n\n\"I was careless. About many things.\" He smiled at me weakly, not out of exhaustion or illness, but as a matter of apology.\n\n\"Do you think so?\"\n\n\"More than that. I was wrong. I hope you'll ignore what I said.\"\n\n\"I could never do that. But we can speak later. The boys and I have a schedule to maintain.\"\n\n\"Of course. Some other time.\"\n\n\"Perhaps.\"\n\nAnd with that, everything was back to the way it had always been. I maintained my position and Henry kept his detachment, staring into the bottom of his teacup as I ushered the children up to the schoolroom. It was difficult to begin our lessons again after a weeklong absence, and progress was slow, but after four hours of reviewing everything from the fall of ancient Rome to the Pythagorean theorem, the boys were well primed and in need of a break. I didn't have to ask them what they wished to do. I could see the longing in their eyes radiating out of them.\n\nIt was the longest gap we'd had between our visits to Darkling, and no sooner had Duncan brought us to the steps of the house than Lily Darrow appeared at the door, red-faced and breathless as if she had run the length of the manor to meet us.\n\n\"You've come back!\" Her eyes were wide and glistening, though whether it was from fear or joy I could not be certain. She hugged the children for a long while in a simple, moving gesture until James pulled away.\n\n\"Father was sick so Charlotte took care of him.\"\n\n\"I'm sorry to hear that. Thank you for looking after my husband.\" She did not look at me as she said the words. \"Come. I'm afraid I'm in the middle of my lessons with Olivia, but you may join us.\" Lily took us to the ballroom, where Olivia was dancing languidly before Mrs. Aldrich and her son, Dabney. The woman stopped Olivia and placed her manicured hands beneath the girl's arms, moving them into a more rigid position above her head.\n\n\"The Dance of Infinite Sorrow requires the arms to be kept over the head at all times, as if you're holding the moon against the sky. Do you understand?\"\n\nOlivia bit the inside of her cheek and nodded politely but broke form when she saw that she had an audience. \"Lily, you brought the children! How delightful!\"\n\nPaul became indignant at being referred to as a child. \"I'm not a child! I'm thirteen.\" He said this a little more loudly than he needed to and quickly greeted his friend Dabney with a hearty handshake. The eerily handsome young man laughed and embraced Paul with a hug instead.\n\n\"Olivia is to have a coming-out ball,\" explained Lily. \"And now I must insist that we continue with our lessons. Dabney?\"\n\nThe boy left Paul's side and took Miss Whatley's hands into his own. They swept across the ballroom as Lily counted time, a graceful blur of golden hair and pale-skinned beauty. When Lily was satisfied with their progress, and only after Mrs. Aldrich approved, we sat for dinner.\n\nDabney's mother chatted with Miss Whatley at one end of the table while her son conversed quietly with Paul at the other, leaving Lily and me to catch up. James sat silently eating his meal, an uncharacteristic gloominess seemingly borrowed from his older brother, but other than that it was a perfectly delightful evening. I looked over to Paul halfway through the meal. He was smiling and laughing, as children should. He seemed completely transformed in the company of the older boy, and it dawned on me that perhaps there was some good in the House of Darkling after all.\n\nThe Aldriches left after the meal, and as always, Lily escorted us up to our rooms. But this time she kissed the boys good night without tucking them in, and the book of fairy tales was not taken out. No one mentioned this change, and so neither did I, but it was obvious that something had shifted, or was shifting, even in a place where such a thing was never supposed to happen.\n\nI waited in the hallway for Lily as I always did, not expecting her to look so tired and worn down when she turned away from the door to face me.\n\n\"Is everything all right?\" I asked. She seemed on the verge of tears.\n\n\"Life is never what one expects, and death is only worse.\"\n\n\"What's happened?\"\n\n\"Nothing I can't manage.\" Her eyes darted around the hallway, and she led me into my room, closing the door behind us.\n\n\"Please . . . when you leave tomorrow, don't come back.\"\n\n\"I don't understand. I promised you one last visit after this one. What's happened?\"\n\n\"I wanted to say good-bye. Now it's done.\" She clutched the knob of the door, uncertain about something.\n\n\"But is it done for the children?\" I asked.\n\n\"It must be. For their own good.\"\n\n\"What shall I tell them?\"\n\n\"Something that will make it better.\"\n\n\"There's no such thing.\"\n\nLily nodded glumly and then left me alone to consider how I might tell the boys that their mother had left them again, only this time of her own free will. It consumed me, most of all because I did not understand why someone with such love and determination would give up everything she had fought so hard to earn. More important, I was not done with Darkling myself.\n\nWhen I dressed for bed I removed the smoke-colored phial labeled DISMEMBERED and placed it beside the iron key given to me by Mr. Cornelius. I had still not sent him proof of Whatley's loyalty to Ashby's adversaries. Despite the murder of Nanny Prum and the continued harassment of Susannah, he was helping his people. What kind of murderer did such a thing? I wondered if I had something wrong, or if things would only get worse if I involved myself in a larger game than the one I was playing with Mr. Whatley.\n\nI could not sleep, and so I returned to the library to collect my thoughts. I did not know what I was looking for, exactly. Perhaps I sought something to take my mind off the plight of the Darrows, or maybe it was something more substantial, to fix what could not be mended. I read through the various titles on the shelves until my eyes drifted up to the door on the fourth floor that led to Mr. Whatley's study. I walked up the spiral staircase with slow uncertainty, not sure what it was I intended to do, until I was in front of the door, about to try the handle, when it opened.\n\n\"Mrs. Markham.\" He wore a robe and nothing else, his muscled chest visible between the conveniently placed folds of the fabric.\n\n\"Mr. Whatley.\"\n\n\"Trouble sleeping?\" He smirked at me with his usual indolent, boyish grin, so self-satisfied that it made me question my sanity for ever thinking that there could be something good and decent about him.\n\n\"Something like that.\"\n\n\"I wouldn't know. I don't sleep.\"\n\n\"How sad for you.\"\n\n\"Come in, if you'd like.\"\n\nI considered his offer. I pictured him all in black, standing over the corpses of my mother, my father, Jonathan, and Nanny Prum. I had no evidence that he had actually killed them; just that he had somehow taken their memories of death, and that a piece from his collection had been used against Susannah and presumably Nanny Prum. But I still lacked the most important part of the puzzle: why me? I pushed past him without a word.\n\nThe room was much the same as it had been before. Mr. Whatley's chest glowed nearly as luminously as the alabaster statues that lined the walls of the first part of his collection. He continued on, past the glass portraits that led to real places, a room of trees that changed seasons every few seconds, fountains built out of water that flowed with streams of liquid marble, until we reached the end and turned in to a high alcove that resembled a cathedral, with vaulted ceilings and a bed at the far end that was like an altar. It was quite grandiose and ridiculous, but at the same time perfectly appropriate for someone of his character.\n\n\"Would you care to see my true collection?\"\n\n\"That wasn't it?\"\n\n\"Of course not. A true collector keeps the rarest items private.\" He walked to the wall behind the bed and found a hidden panel that opened a secret door. He slid inside, his features disappearing into shadow. I followed him, for I knew that he would not harm me. He would not end our game without a grander confrontation; of that much I was certain.\n\nThe secret room was a miniature version of the library, but instead of books it held a wide variety of people, each of them standing motionless on a labeled pedestal, their eyes closed as if they were sleeping, waiting to be stirred from a long, sad dream. There were men and women, old and young, beautiful and plain, of various colors and sizes, something different for every occasion. It did not take me long to find Lily, positioned at the center of the display like a living doll, her chin resting on her chest. At the sight of her my face grew flushed with anger.\n\nThis is where she sleeps. I had imagined her in another wing of the house, perhaps in a room adjacent to Olivia's or, as I already had a sense of Mr. Whatley's true character, one that was convenient to his own.\n\n\"What do you think?\" Mr. Whatley's eyes, normally black and soulless, shimmered an inhuman silver-green in the dark of the room.\n\n\"It's disgusting.\" I spoke between clenched teeth.\n\n\"Perhaps a little. But all completely voluntary.\"\n\n\"That's even worse.\"\n\n\"They're all people of exceptional character. They did what they had to do in order to get what they wanted most.\"\n\n\"Which was what?\"\n\n\"That depends. They all had different desires. What is it that you want most, Mrs. Markham?\"\n\n\"To free her.\" I gestured to Lily. \"And to beat you.\"\n\nMr. Whatley chuckled, the sound of it reverberating through the room. \"You are hardly alone with the latter sentiment. Living humans in The Ending have begun to make Mr. Ashby's friends uncomfortable.\"\n\n\"I do not intend to stay after I've stopped you.\"\n\nHe raised an eyebrow, and the perpetual smirk stuck on his face twisted into one of greed. \"Are you so certain of yourself?\"\n\n\"You don't frighten me.\"\n\n\"Perhaps I should.\" He came closer to me, close enough to brush a strand of hair away from my face, which he did with his large hand. His touch was different from Henry's\u2014rougher and empty of any emotion, but possessed with a power, like his voice and his eyes, that made me want to fall into him and pull away at the same time. I did not dare to move.\n\n\"Everything I've ever loved has been taken from me, piece by piece, year by year, to place me where I now stand. I have nothing left to fear.\"\n\n\"You think yourself very clever, yet I am older than you could imagine and that much more powerful. The only thing you have that I do not is a death, which some might consider a disadvantage. Do you think this will end well?\"\n\n\"Not for you.\"\n\n\"I'm not afraid to lose so long as everyone else does. You'd do well to remember that, Mrs. Markham\"\n\n\"But that's not how the story goes. Someone has to win.\"\n\n\"Indeed. But whose story is it? Yours or mine?\"\n\n\"I suppose we'll have to wait and see.\" I turned away from him, confident that he would not do me any physical harm while my back was turned, and moved toward the door.\n\n\"Best of luck to you, Mrs. Markham. I can't wait to see what will happen next.\"\n\nI walked the entire length of the long room feeling his gaze at my back, unwilling to turn around to see if he was following me. I reached the door to the library and closed it behind me. The place had lost all the comfort that it once offered. I returned to my room, but not before checking once on the boys to make sure they were still comfortably asleep in their beds, unthinking and unworried about what their mother might have done in order to see them again.\n\nI lifted the smoke-colored phial and the iron key from where I had left them on my bed and stood before my bedroom door. I recalled the words of Mr. Cornelius.\n\nOne turn in any lock will send for me.\n\nI inserted the key into the lock and turned. The door opened into a dank, chilly room with walls covered in a hard, emerald film. A shape appeared on the ceiling, and Mr. Cornelius scuttled down the wall to greet me, the feelers behind his beard formed into a smile.\n\n\"Mrs. Markham.\"\n\n\"There is a room in the House of Darkling where those who so desire might taste a human death. Is that the proof you require?\" I handed him the phial labeled DISMEMBERED.\n\n\"It should do nicely.\" He tucked it into his beard and turned, pressing his flat face against the wall, pincers emerging from the graying tendrils of his facial hair to work into the glossy green surface, cutting and slicing at it until a chunk broke away into the stout trunk-like appendages he used as hands. He held it out to me. \"A token of protection.\"\n\nIt was a clear disk of petrified green amber, with a single glyph scratched into it.\n\n\"What do I do with it?\"\n\n\"Keep it close. Do be careful, Mrs. Markham. He won't like this, not one bit.\" He escorted me back to the door.\n\n\"Let it never be said that I'm an unworthy opponent.\"\n\nWe parted company, and when I attempted to retrieve the iron key from the lock of my door, I was not surprised to find it missing. Our bargain had been completed, and I was once again on my own. I crawled into bed clutching the green amber disk to my breast, and for the first time in all our visits to Darkling, I slept peacefully.\nCHAPTER 15\n\nThe Christmas Guest\n\nWinter finally descended upon Blackfield a few days later. The barren branches of the forest became gloved in white snow, stretched out beneath the pale gray sky and arranged around the glass shores of the lake in a frozen ballet bereft of movement or song, dormant until spring, when the ice would melt and stream off of them, their forms glistening from the extended concentration of holding a single pose all season long.\n\nBut the villagers themselves would not be forced into hibernation. Indeed, the winter months were some of the busiest of the year. After the bazaar there were dinner parties and special church services, winter markets and quilting circles, not to mention the grandest event of all: the Blackfield Christmas Ball.\n\nIt was not really a ball so much as it was a local dance festival, but since it was held in the home of Cornelia Reese it could not be called anything else, nor discussed without the highest reverence, at least not within the presence of Mrs. Reese. The Reeses lived in the largest house in the village, and although it was only slightly bigger than Everton, that was enough for Mrs. Reese to declare it the only proper place to hold such an event, for it could accommodate the entire population of Blackfield. There was nothing the woman liked more than to take pity upon the poor dregs of society, the common folk who were not so well off as she, so that they might know, for at least one evening, some happiness in their sad and dreary lives. Despite this fact, or perhaps because of it, the people of Blackfield had no problem converging upon the Reese estate, which was called Arkham Hall, and conversing very loudly among themselves about how dreary the interiors had become since the previous year, all the while piling copious amounts of food into their mouths and purses, completely willing to play along with Cornelia so long as they could take advantage of her generosity. After all, a party was a party.\n\nThat afternoon I cornered James after our lessons and threw him over my shoulder. He giggled and kicked his legs, always a willing participant in any sort of violence, but he did not pass up the opportunity to make a scene.\n\n\"Help! I'm being murdered!\"\n\nThis was in very bad taste, considering what had happened to Nanny Prum. Fortunately Paul was old enough to be aware of such sensitivities and swatted his brother on the back of the head as we left the schoolroom. James, like many little boys of similar temperament, did not care for the idea of a bath. There were too many mud puddles to splash through, too many frogs to capture, too many trees to climb to bother with such provincial chores as bathing. However, once submerged in water, after the thrashing of arms and legs subsided, he was quite at home being naked and wet, imagining himself to be a fish and slipping through my hands as I struggled to soap him up.\n\nWhen we finished I carried him back to his room, careful not to let him out of my sight before we left the house. Paul was nearly finished getting dressed, meticulously combing his dark hair. I was glad that Lily could not see him then. It wouldn't be long before he would be a young man, no longer requiring the services of a governess or a nanny. I placed James into his brother's care as I began my own preparations for the evening, and threatened him with Indian curses if he did anything to undo my work with his brother's appearance.\n\nFor myself I sat before the mirror and released my hair from the top of my head, smoothing it out with a brush and pinning it back into place with small jeweled pins that were once my mother's. That evening she did not appear in my reflection. I put aside my governess's uniform and stepped into an evening gown the color of deep midnight, the corset flecked with silver beads that gave the impression of stars in the night sky.\n\nJust before dusk the children and I met Mr. Darrow in the foyer of Everton. His mood was unreadable, as he did not meet my gaze, but he had done a reasonable job of dressing himself for the occasion. His blond hair had been slicked back so that it did not hide his handsome blue eyes, and he wore a dark suit with a cream-colored vest and a deep blue tie that coincidentally matched the color of my gown. I doubted that he noticed, but I would not be the one to make mention of it. I was merely the governess.\n\nMrs. Mulbus and Jenny had already departed to prepare for the evening's festivities, and the other servants left the house in packs, wearing heavy coats over their finest dresses and suits. Even Mrs. Norman looked mildly less dreary than normal, wearing a feathered hat that made her resemble something like a peacock. She was escorted out of the house by old Fredricks, who secreted away a small silver flask in the folds of his jacket with the hand that was not holding Mrs. Norman's.\n\nRoland brought a covered carriage around to the door and helped us inside. The boys immediately sat on the same side of the vehicle, forcing Mr. Darrow and me to sit uncomfortably beside one another. Fortunately the trip was a short one, and soon we were pulling into the modest driveway of Arkham Hall.\n\nWhile the house might have been only slightly larger than Everton, it was better kept and much more ornate. Cornelia Reese still traveled to the city with some regularity to maintain her social calendar, and she always returned with a small caravan of antique dealers and artists to fortify the appearance of whatever room had lost her good favor.\n\nOur carriage pulled around a marble fountain meant to emulate the antiquities of ancient Rome. It was a ghastly thing, with streams of water pouring out of eye sockets and battle wounds, but very appropriate for the home of Cornelia Reese. One of the footmen helped me out of the carriage, and with James's hand secured into my own, I entered the foyer of the house.\n\nThe festivities had already begun to pour out of the ballroom, with red-faced guests standing in the hallways, glasses of wine in hand, all of them talking much too loudly over one another. Backs were patted, hands were placed in front of peals of laughter as one party made a wry observation about another, and certain gentlemen made untoward advances that their wives would not soon forget. There were also plenty of children threading through the throngs of adults, and it took James very little time to find a playmate suitable to warrant his escape from my supervision. He ran off into the crowd, but not before turning back to deliver a devilish sort of grin. I knew then that it was going to be a trying evening.\n\nThe ballroom was tall and narrow, with a promenade along the second floor that allowed those not inclined to dance to observe and enjoy those who were. I looked out over the crowd, and a feeling of hopelessness descended upon me. The villagers seemed so happy to be together, and I dreaded the thought of something happening to disrupt that. If I could not stop Mr. Whatley, which of them might be next?\n\nI found Mr. Scott; he had done his best to tame his hair for the occasion, but it still floated over his head with wispy abandon.\n\n\"I hope you've sorted out your problem with spirits?\" he said with some difficulty over the volume of the music, for there was a small orchestra playing beneath us.\n\n\"I am sad to say that I have not.\"\n\n\"Ah, so James remains curious?\"\n\n\"Belligerently so.\"\n\nThe vicar nodded knowingly.\n\n\"The boy reminds me of myself. I too have always been curious about such affairs, which prompted my entrance into the servitude of the Lord.\" He smiled, impressed by his own piety, but then faltered at his pride. He went on. \"I have continued to think on the matter, and I've come to the conclusion that spirits must not have malicious intent.\"\n\n\"Do you think so?\"\n\n\"They mustn't. They may act with some cruelty, but only to bring about some change in the world in the name of God.\"\n\n\"How can you be sure?\"\n\n\"My dear, I cannot be sure of anything. I am a man of faith. But then this is all hypothetical, is it not?\"\n\nI thanked him for his advice and carefully maneuvered down to the first floor, where I spotted Susannah and Lionel dancing together. They were radiant, staring fixedly into each other's eyes despite the speed of the song, turning with one another around and around, all the while laughing with abandon until the music stopped and Susannah met my gaze. She appeared much healthier than the last time I had seen her. The wild look in her eyes was gone, and there was a comforting peace about her. Lionel went to fetch his wife a drink, and Susannah kissed my cheek in greeting. She could not seem to stop fingering the clear green disk of amber that hung at her throat.\n\n\"It suits your coloring,\" I said.\n\n\"It does, doesn't it?\" She ran a hand through her wild red hair. \"Wherever did you find it?\"\n\n\"Ancient Indian talisman to ward off evil spirits,\" I lied.\n\n\"Whatever it is, it's working. Nothing has happened since you gave it to me.\" I began to wonder what was happening back in The Ending between Cornelius and Whatley, but then I shook myself. The evening was supposed to be a celebration, and Darkling would have no part in it. For the first time in weeks I felt a shred of relief and vindication. While I still had not solved the puzzle of Mr. Whatley's intentions, I had been able to circumvent at least one of them, though with my friend safe the rest of the village remained vulnerable. It was for this reason that I had kept the boy with the keyhole eyes chained to the bedpost in my room. The promise of further snippets of secret conversations was worth the rattling and scraping sounds that persisted throughout the later hours of the evening as he attempted and failed to free himself from my control. Luckily, he seemed disinterested in the boys, preferring instead to collect the secrets of adults.\n\nRoland tapped Susannah on the shoulder and cleared his throat. \"Excuse me, ma'am, but might I have this dance?\" He had exchanged his dusty workman's clothes for a brown tweed suit, though his hair was still an unkempt tangle, ironic given his position as groundskeeper of Everton.\n\nSusannah took his hand. \"Roland, I would dance with you all night if you asked,\" she said to the man who had saved her life. They spun off onto the dance floor as her husband returned with their drinks, watching them with sullen jealousy. I tried to join him to keep him company but was cut off by the sudden appearance of Mr. Darrow.\n\n\"Mrs. Markham.\"\n\n\"Mr. Darrow.\"\n\nThat seemed to be the end of our conversation until the music began to play again and Mr. Darrow nervously offered me his hand. \"Would you care to dance?\"\n\n\"Are you sure that's wise? People will begin to talk.\"\n\n\"Let them.\"\n\n\"That is quite a change of heart.\"\n\n\"Life is too brief a thing to dwell on the opinions of others, especially when there is dancing to be done. Shall we?\"\n\nI took his hand and we joined the other villagers on the ballroom floor. The song was very fast, and we spent more time being thrown from one partner to the next than actually dancing with one another, and so it was a relief when the music slowed and each dancer was allowed to return to his or her original partner. Henry took my hands into his and looked at me, perhaps really seeing me for the first time since that evening in the kitchen, looking into me as we swirled and spun to the music, the hem of my dress floating as we turned, grazing his legs. We were not so very close together, but the interlacing of our hands channeled a friction through the empty space between us that dimmed the rest of the room, changing the music into something that could only be for us. I did not want it to end, and for a long while it seemed that it never would. We danced and danced until I could no longer feel my legs, just his touch against my own and the deep, primal thumping in my chest.\n\nBut it did end, and when it was over our hands stayed connected. We wove through the house, past a happily scandalized Cornelia Reese, who would have something dreadful to talk about with her friends and confidants for at least a whole month; past Mrs. Norman in her peacock hat, whose tight, severe lips broke into a faint smile of approval, until we were outside in the frosted gardens of Arkham Hall. Beneath the black and empty sky, behind a hedge dusted with ice, Henry pulled me against his chest and kissed me, passionately, deeply, with nothing like reserve or anxiety, our lips moving in tandem with one another, all fear falling away, unraveling as something new knit itself together, something good and pure and strong, full of promise and hope. I was losing myself in the moment until a stark and frenzied shriek broke through the chilled air.\n\nIt was a woman's scream, and I already knew who it must have been. I pulled away from Henry and ran through the gardens, following the echo of it, bounding off stone statues and empty bird fountains, a shadow of a sound frozen in the ice. I sprinted as fast as I could, the ends of my dress in my hands, the wind ripping against my skin as I darted around a lattice strung with withered vines, and found Susannah sprawled on the cold ground, her hands before her face as something unfurled above her, a shadow untwisting itself from the dark, separating itself, becoming tangible and textured, wet and glistening, the surface of it freezing from the cold and cracking as it slid a tendril around Susannah's neck and lifted her into the air, her feet scraping at the ground for some purchase but finding none.\n\nThe moment the creature touched her, the small, clear disk of amber that hung about her throat began to glow. A dull green light pulsed through the air, folding itself around Susannah, searing the flesh of the creature until it dropped her out of shock and anguish.\n\nImmediately all the people I had ever lost swam before me, and the pain congealed into anger, into hate, and into action as I launched myself at the thing while it was caught off guard, kicking, biting, tearing at it with my fingernails, until it lashed me across the face with a dark, unknowable appendage. I fell to the ground, and the horror hovered over me, blocking out what little light there was from behind the luminous clouds, the smell of it overpowering, equal parts ammonia and brimstone. I did not fear death, for I knew that my loved ones were waiting for me, but as the creature crept toward me, a shot rang out and I felt a damp spray across my face. The thing winced and quivered, seeming to retreat into itself and pausing at the inevitable opening of doors and stirring of voices from the house, contracting into something man-shaped before fleeing ever more deeply into the frozen gardens.\n\nI touched the side of my face, unsure if the wetness I felt was my own blood or the property of the thing that had attacked me. Susannah was sitting up, holding her knees in a fetal position, rocking back and forth. Henry was next to her, a smoking pistol in hand.\n\nHe helped me to my feet. \"Are you all right?\"\n\n\"Yes, I\u2014I'm fine.\" That was not the man in black, I wanted to say, that was something else altogether. There were parts of Whatley's game that I still did not understand, and that worried me.\n\nI did not let go of Henry's hand when I was standing again. I went over to my friend. Her eyes were unfocused and she was muttering to herself.\n\n\"You must keep them closed. Never open. When they're open it all comes out, all apart in the darkness . . .\"\n\nI knelt down beside her and kissed her on the forehead. \"Who was it, Susannah? Who attacked you?\"\n\n\"Wolf in sheep's clothing. Monster under the bed.\"\n\nHenry shook his head and put his hand on my shoulder. \"We'll get Dr. Barberry.\"\n\n\"Someone will have to find Lionel first.\"\n\n\"Of course.\"\n\nThe voices from the house were growing more distinct. I pointed to the pistol still tightly clutched in Henry's hand. \"You had a gun.\"\n\n\"I still do.\"\n\n\"Why?\"\n\n\"There's a madman on the loose.\"\n\n\"Do you really think it was a man?\"\n\nHe looked at me strangely but had no time to dwell on it, for we were quickly surrounded by the other party guests, clasping at the throats of gowns and jackets, gasping at the macabre scene in the garden\u2014the three of us standing over a frozen pool of black blood.\n\nLionel was summoned to take his wife back to their cottage, but not before the doctor looked her over with a dour expression. He mentioned to the barkeep that there were places for people who suffered extreme trauma, comfortable places where they could be made well again, but Lionel would have none of it.\n\nHenry and I collected the children and rode home in silent reflection. When we arrived at Everton I ushered the children to their rooms, feeling Henry's gaze on me as we traveled up the stairs, unable to stop myself from looking back to meet his eyes.\n\nThe boy with the keyhole eyes was cowering in the corner when I returned to my room, holding his knees against his chest as he pointed with a broken, blackened fingernail in the direction of my bed. Waiting on my pillow was a parchment envelope with a blue wax seal. I slit it open with a penknife and read:\n\nYou are cordially invited\n\nTo the coming-out ball of\n\nMiss Olivia Whatley\n\nTomorrow evening, dusk\n\nI reread the invitation a dozen times over and set it on my nightstand while I dressed for bed. Someone had placed the letter in my room, someone from the House of Darkling. This wasn't a simple admission of guilt; this was an outright declaration of war, though perhaps not enough to prove a connection between Whatley and Everton, effectively ending our game.\n\nI wrestled with the implications of this well into the night, and when I finally found sleep I dreamt that I danced with Mr. Whatley on the brink of an abyss, turning and turning until neither of us could be certain which was about to slip off the edge.\nCHAPTER 16\n\nMrs. Whatley\n\nThere was no way I could avoid bringing the children back to Darkling. If I went alone Mr. Whatley might realize that something was amiss. And what if the children came in search of me, alone in The Ending, vulnerable to whatever machinations the master of Darkling had set in motion? I could not allow it, as I believed Whatley to be responsible for everything that had happened in Blackfield in addition to the sad, doll-like imprisonment of Lily Darrow.\n\nWhen we crossed the threshold from the forest into the House of Darkling, it was immediately apparent that something was different, for the fruit children in the trees had been unraveled from their leathery rinds, each of them holding tiny candles in their pygmy hands, the orchard transformed into a flickering sea of teardrop stars. Even Duncan was different, dressed in black coattails and bowing low in greeting before he escorted us to the great house, which glowed with a preternatural sheen from the inside out. It was actually welcoming.\n\nWe could hear the distant sounds of the party\u2014laughter, shouting, the clinking of glasses, an echo of music. Thanks to the arrival of the invitation I'd had enough forethought to dress the children for the evening's festivities, though neither one was happy to wear formal attire again so soon after the Christmas Ball. For myself, I had chosen a high-collared gown with an opal brooch at the throat.\n\nWhen we arrived, the ballroom seemed to be in the middle of some sort of decorative metamorphosis, for the skin of the stone pillars that lined the massive room cracked and fell away, revealing tree trunks of nearly the same girth. The jewels set in the brushed metal of the walls dropped off, exploding on the ground with pops of colored light to the delight and annoyance of some of the party guests attempting to converse and enjoy their cocktails. Out of these fresh alcoves grew sinewy vines, creeping up the walls, which themselves became less opaque and more like mirrors, until the ballroom began to resemble a never-ending forest with flowers growing up through the grout between the black and white marble tiles of the floor.\n\n\"You shouldn't have come back.\" Lily appeared at my side and whispered into my ear, her voice earnest and desperate, but then the boys saw her and she was transformed into the mother they knew and loved, never sad or upset, always pristine and composed. She would have kissed them both if the lights hadn't gone out. A spotlight appeared at the front of the ballroom as Olivia Whatley entered wearing an ice blue evening gown, holding the arm of her father. There was a polite smattering of applause as they circled the room, and then he left her with Dabney in a wide, empty circle as the guests stepped backward to give her space. The handsome young man wore an expensively tailored plum-colored suit that contrasted strangely with the mysterious way he held himself: his arms out, head tilted skyward, almost in a trancelike state. Olivia stood before him, and he placed his hands around her neck.\n\nIt was then that I felt growing warmth in the pit of my stomach, traveling up my chest, into my throat, and I thought I was going to be sick until my lips parted and erupted into a sound that was like a song, or at least a part of one. I thought that I had gone insane, or that Mr. Whatley had done something to me in retaliation for my impertinent threat the last time we spoke, but then I looked around and realized that every other guest sang a different part of the same song, a five-hundred-part harmony blending together in the echoing cavern of the ballroom.\n\nDabney had disappeared from the room, and Olivia moved as we sang, swaying gently to the Dance of Infinite Sorrow. The five ice sculptures that stood over the tables of food at the back of the room creaked to life and stepped down from their perches to join her on the dance floor. She moved from one to another, slowly at first in a languid, dreamlike fashion, until one of them struck her across the face.\n\nI felt my own cheek in horror, unable to forget the night before, but Olivia immediately fought back, pushing the dance partner who had struck her hard enough to tip him over, and he shattered into a million shards of ice over the ground. This seemed to upset the others, for they surrounded her, tearing at her dress so that it fell away from her body, leaving her naked and vulnerable. The sculptures clung to her, her nudity never completely visible, and as they did our chanting grew faster and the temperature of the room started to rise. The wooden pillars scattered throughout the room burst into flames; the ice dancers melted as Olivia's skin blistered over. It was an awful, terrible sight, and while I despised her father I wanted to help the poor girl, but Lily kept me firmly in place.\n\nThe pillars stopped burning and turned back to stone. Olivia, although severely burned, did not appear to be in any pain. She reached to the back of her head and peeled away large strips of her charred flesh, revealing the beautiful, healthy girl who had entered the room with her father mere moments before. The liquid remains of the ice sculptures collected themselves together of their own accord and slid toward her, up her legs and torso, freezing so that they took the shape of the same ice-colored ball gown that had been torn asunder.\n\nWe stopped singing as abruptly as we had started, and Olivia bowed deeply. Dabney appeared out of nowhere behind her, taking her hand. All I could think to do was applaud. I had no idea what had happened, or what it signified, but it had been most extraordinary. The other guests joined in while the chef from the dinner party many nights before pushed through the throngs of people with a wheeled cart and collected the scraps of viscera and skin left over from the finale of Miss Whatley's performance. He placed them into a crystal bowl, and Dabney performed a quick blessing over the mess while Olivia began to greet her guests. I decided then that I would not be tasting the hors d'oeuvres.\n\nMore traditional music began to play and the mingling turned into dancing. Olivia moved from one partner to the next, men and women alike, sometimes dancing with creatures that had no easily identifiable sex. Lily danced with James, free for a moment from any anxiety or fear, and I saw Paul accept an invitation from Dabney to join him in a rather slow waltz.\n\nI was content to be by myself for a moment, and I spent the time observing the other guests. There were the Baxters, flickering in and out of sight as I stared at them; Mrs. Aldrich, at the center of a group of well-dressed women, doubtless bragging about her son; the Puddles, standing beside the large frame of Mr. Samson, already red-faced and laughing too loudly at one of Mr. Puddle's jokes; Mr. Snit, drifting from one cluster of people to the next, changing color as he did so and covertly absorbing the cocktails of whoever happened to be standing closest to him. Miss Yarborough stood at an opposite corner of the room in the same sheath of netting as the last time we crossed paths, her disdainful expression visible and well honed despite the skinless nature of her face. I spotted Mr. Cornelius standing off to the side of the room with a circle of strange-looking creatures, hunched over things with amphibious faces and spiny backs covered in quills of bone, all of them speaking to one another in strained whispers. He saw me as I approached and excused himself from his friends with some discomfort. The others peered at me with suspicion.\n\n\"Mr. Cornelius.\"\n\n\"Mrs. Markham. I had not expected to find you here this evening.\" His onyx eyes shifted from side to side.\n\n\"We were invited.\"\n\n\"I see.\"\n\n\"I must again thank you for your help. Your token of appreciation was most effective.\"\n\n\"I am glad to hear it.\"\n\nI could see that my presence was making him visibly uncomfortable. His trunk-like appendages folded over one another, as if he were wringing them together.\n\nI pressed on, undeterred. \"I do wish that we could continue our arrangement. I fear that the game is not yet over, and I am nearly out of moves.\"\n\n\"On the contrary, Mrs. Markham. You have more control in this game than you know\u2014\" He pulled me close, the pincers behind his beard clicking together. \"You should not stay long this evening.\" My lips parted to form the start of a question, but then Mr. Cornelius looked behind me and smiled. \"Ah, Mr. Whatley.\"\n\nThe master of Darkling observed us through the crowd with his sideways smirk. Whatley greeted the other gentleman and took my hands into his large ones, leading me into throngs of dancers without asking for my permission.\n\n\"You look quite ravishing this evening, Mrs. Markham.\"\n\n\"And you appear as if you've just come in from a storm.\" His dark hair was a wild, windswept tangle, and his suit, fine as always, was unkempt and disheveled.\n\n\"I try to be consistent.\"\n\n\"Ah, the success of lowered expectations.\"\n\n\"The only expectations that matter are my own, and I always seem to meet them.\"\n\n\"How lucky for you.\"\n\n\"Luck has nothing to do with it. I play to win.\"\n\n\"And when you fail?\"\n\n\"I'll let you know when that happens.\"\n\n\"It might be quite soon.\"\n\n\"Do you think so?\"\n\nThe music reached a crescendo, and Whatley pulled me against his body. I blushed furiously. I tried to break away from him, but he held me firmly in place, refusing to let me go until he was good and ready. Finally he winked at me and retreated into the crowd. I looked around for Lily, but Whatley was already moving toward her. He said something in her ear, and she nodded unhappily while he waved at the musicians to stop playing. Mr. Whatley addressed the party guests.\n\n\"My friends, thank you for joining us on this most special occasion, as my daughter, Olivia, reaches maturity and sets off to make her mark upon the worlds. It is often difficult for a parent to let go of his children, but I am happy to say that I have found some solace, for soon I will be remarried. May I introduce you all to the future Mrs. Whatley.\" He took Lily's hand into his own as a smattering of applause moved through the crowd, accompanied by some uncomfortable murmuring, not the least of which was a short, angry squeak from James, who stood beside his brother and me, looking terribly confused. \"But what about Father?\" he stammered.\n\nMr. Whatley did his best to look sympathetic, but ended up only appearing to be condescending. \"My dear boy, he is living and your mother is dead. There can be no hope for such a pairing.\"\n\nBut James would hear none of it. He ran out of the ballroom in tears.\n\n\"James!\" Lily went after him, and suddenly I heard the familiar clicking of Mr. Cornelius's pincers close to my ear.\n\n\"Take them and do not return. Now. Run.\"\n\nA chair crashed against the wall. A glass shattered to the tiled floor. Silence filled the room as the partygoers looked around in confusion.\n\nAnd then someone screamed.\n\nA woman pointed to the body of Dabney Aldrich, slumped on the floor, holding a gash that had appeared in his throat as he pinched the bloodless folds of skin together. The slimy interior of his true body, the one within the angelic human slip he wore, began to spill out down the front of his chest. One of the amphibian-faced creatures stood over him, spat out a hunk of Dabney's flesh, and roared.\n\nThe ballroom erupted into madness. Paul stared agape at his friend, unable to speak in the wake of such shock. He reached out to help pick him up off the floor, but I grabbed Paul's hand and ran, sprinting away from Whatley's guests as they ripped each other apart, bodies falling, never dead or dying, simply in pieces, the crowd pushing for the door all at once, blocking it until Mr. Cornelius tore them aside to make room for us.\n\n\"Remember what I've said.\" He nodded to me, and then launched himself into the brawl, his beard parting to reveal a hideous cluster of sharp, dangerous-looking appendages that sank themselves into the corpulent neck of Mr. Samson.\n\nPaul and I spun through the house, from one room to the next, corridor after corridor, until we were in the entryway and outside, down the steps to the orchard, where we found Lily and James. The boy would have nothing to do with his mother, but we had no time to sort out any recent emotional baggage.\n\n\"I want to go home,\" he said. I grabbed his hand as I ran past.\n\nPaul tightened his grip, breathless from our escape. \"What about Mother?\"\n\n\"Lily, you are coming with us,\" I called out behind me, but Mrs. Darrow shook her head in earnest.\n\n\"I can't. You know I can't.\" But then the chaos of the ballroom erupted into the orchard. A body was thrown from a second-story window, and the attacker leapt out after its victim, continuing to claw at it in midair before they had even hit the ground.\n\n\"Oh, I think you can.\" The four of us dashed between the trees.\n\nLily was on the verge of tears. \"But I'm not ready,\" she said desperately.\n\nJames seemed to soften, seeing her in such a pathetic state, unable to defend himself against his mother's misery. We found the wall of coiling fog.\n\n\"Mr. Whatley is doomed, you cannot stay. But we can say good-bye, here and now,\" I said to her.\n\n\"I'm so frightened.\"\n\n\"The boys will be with you to the end. Isn't that what you always wanted?\"\n\n\"What I wanted was more time. There are so many things I wanted to teach you,\" she said, taking the hands of her sons into her own. \"Marry for love. See the world.\" We approached the veil of mist that separated the living from the dead. \"Cherish every day with your children. Don't let your father become lonely. Treasure every single moment.\" Lily kissed both of her sons, the three of them in tears.\n\n\"We will, Mother,\" Paul replied, red-faced from the dual exhaustion of running and crying.\n\n\"I don't know what will become of me, but know that I love you both so very much,\" said Lily, her words breaking apart before they even left her lips. Still, she nodded to me, and together the four of us crossed over the threshold from Darkling into Everton.\n\nBut this time, there was a figure in the mist.\n\n\"I'm afraid I cannot allow any of you to leave.\" Mr. Samson was not himself. His human body had been completely torn apart save for his face, revealing a puckered mass of red flesh and many-jointed tendrils.\n\n\"We are returning home, sir,\" I said to him. \"We are done with The Ending.\"\n\n\"But we are not done with you. We must have humans in The Ending. There must be retribution for what has transpired this evening.\" His body shuddered, and four boneless limbs slithered along the ground to grab hold of us. I kicked them away, and Lily launched herself at Mr. Samson. He flung her to the ground like a rag doll. I leapt behind him and jumped onto his back, pushing my fingers into what I hoped were his eyes, digging at them with all my might, but I could not keep my grip. He tossed me aside, my head bouncing against something solid. My vision filled with stars and I drifted out of consciousness, the names of the Darrows on my lips as the mist swirled above me until I heard someone calling my name, softly at first, then gathering substance like an echo in reverse.\n\n\"Charlotte?\"\n\nI opened my eyes. The sky blazed with the pink and purple bruising of twilight.\n\n\"Charlotte? What on earth are you wearing?\" Henry leaned over me, and I realized that we must have looked very foolish wearing evening clothes in the middle of a winter afternoon. I was glad to have such a high-collared dress, and the opal brooch to hold it tightly closed against my throat. But then I remembered . . .\n\nI sat up and whipped around. Paul and James were nowhere to be found, and behind the cage of roots, a path led into the heart of the forest. The mist was gone. I spun in place three times, my heart sinking.\n\n\"We came looking for you. You've been gone for hours,\" said Henry with growing concern. \"We started to worry that\u2014\" He grabbed me by the arms. \"Where are the children?\"\n\n\"They were just with me!\" I said in a voice hoarse with horror. I collapsed against him, unable to process what had happened, let alone speak it aloud. But still he asked me the question I dreaded:\n\n\"Charlotte, where are my children? What's happened?\"\n\nWords came out of my mouth, but I did not hear them. My voice cracked, and with it my entire world fell to pieces.\n\n\"They're gone.\"\nPart 3\n\nThe Ending\nCHAPTER 17\n\nAn Interrupted S\u00e9ance\n\n\"I believe you.\"\n\nI had spent the better part of an hour seated on the frozen floor of the forest trying to explain to Mr. Darrow all that had happened during the past few weeks, and nothing surprised me more than those three simple words he uttered when I had reached the end of my tale.\n\n\"Completely? Without any question?\"\n\n\"After what was done to Nanny Prum, and after seeing the thing that attacked Susannah Larken . . . yes, I believe you.\"\n\nI threw my arms around him and rested my chin on his shoulder.\n\n\"We will find them.\"\n\nI didn't know what to say. In a way I was grateful that the game of secrets had ended and that Mr. Darrow believed me, but in that moment I hated myself. In my vanity and arrogance I had used the children as pawns in a larger game between Mr. Whatley and myself, and that I had lost them was no one's fault but my own. It did not matter that my intentions were pure, I had put them in danger and now they were gone, to be used as collateral in the civil war of The Ending. Or just as likely, to be stored away next to their mother in Mr. Whatley's secret chamber.\n\nIf Whatley survived. But I knew that was foolish. Creatures like Whatley always survived. It was the innocent who suffered.\n\nOnce I realized that we were all in danger, I should have severed our connection to Darkling and accepted the consequences. But I knew that wouldn't have been enough. There was still Susannah to consider, and Nanny Prum.\n\n\"We should get you back to the house,\" Mr. Darrow said to me. We stood together in the forest as the other men from the search party joined us. Roland looked on me with sympathy as I shivered in my evening gown against the cold midwinter's day. Mr. Darrow took off his jacket and placed it over my shoulders. I was too tired to argue. I wanted to throw off his kindness and race to find the children, but I had no idea where to begin. We walked out of the forest and into the sunlight. The warmth against my skin did nothing to alleviate the dread I felt growing in my chest, which swelled as my mind wandered to the night before, when Susannah was attacked by the mysterious, otherworldly assailant.\n\n\"Susannah, is she well?\"\n\nMr. Darrow's eyes swept over me with pity, as if I could be foolish enough to believe that she had indeed recovered from nearly being killed by some nameless, shapeless monstrosity only days before Christmas.\n\n\"The last time I saw her I was with you. But it is difficult to forget her condition, since it is all the servants have been talking about. I understand that she continues to rave about impossible things, and that she screams whenever her husband leaves her side. Dr. Barberry attempted to take her away this morning, but Lionel would not hear of it. He is certain that she will recover on her own.\"\n\n\"And she will. I've never met anyone as strong as Susannah Larken.\"\n\nMr. Darrow said nothing more until we were halfway back to Everton. \"We must talk to Lily.\"\n\n\"I have no idea how to contact her.\"\n\n\"She's dead,\" he replied as if he had lost her a second time. His voice cracked, and he cleared his throat. \"There are ways, are there not?\"\n\n\"What do you mean?\"\n\n\"Mrs. Markham, after everything that's happened, I don't think it's out of the question for us to contact my wife through a medium.\" He said this in what I hoped was a tone a little more harsh than he intended, even though I knew I deserved it.\n\n\"You're perfectly right, Mr. Darrow.\" I was aware that we had not addressed each other by our first names in quite some time. \"But where do we find one?\"\n\nEven as I said it, it sounded like the most ridiculous question in the world. We turned to one another at the same time and sprinted the rest of the way to Everton.\n\nWe found Mrs. Norman on the second floor of the house, busy reducing Jessica the chambermaid to tears.\n\n\"You call this dusted?\"\n\nThe girl winced at the infliction of each word. \"Sorry, Mrs. Norman, I thought\u2014\"\n\n\"No. You did not think. Not even to a degree that would be halfway acceptable.\"\n\nMr. Darrow intervened. \"Erm, Mrs. Norman?\"\n\nThe housekeeper turned to her employer, and her icy demeanor melted into something that was nearly pleasant.\n\n\"Yes, Mr. Darrow?\" She spoke in a clipped, mannered rhythm that only highlighted the woman's predilection for structure and rules.\n\n\"Could I have a word for a moment?\"\n\n\"Certainly.\" She backed away from the girl, who scampered off like a wounded animal.\n\n\"Mrs. Norman, it's no secret that you have a great interest in the supernatural.\"\n\n\"It is true, the Other World does hold a great deal of fascination for me, especially ever since dear Mr. Norman passed on.\" She crossed herself and kissed the cross that hung at her throat.\n\n\"Have you tried to contact him in spirit form?\"\n\n\"A number of times, yes. And I was successful, once.\" She began to talk in an excited, confidential tone. She lowered her voice and looked around to make sure that we weren't being overheard. \"He helped me find a shawl I had misplaced.\"\n\n\"Ah. Well. We would like to hold a s\u00e9ance. To contact Mrs. Darrow.\"\n\n\"A s\u00e9ance?\" The housekeeper looked at me as if she had not noticed me standing there in such close proximity to Mr. Darrow. She rubbed her chin. \"I've never performed one before; I typically use the cards, you see, but yes, I suppose it could be done. I would need time to consult my books\u2014\"\n\nI interrupted her. \"I'm afraid it's rather urgent.\"\n\n\"The dead can usually wait,\" said Mrs. Norman in a sharp tone that was much closer to her usual voice. But Mr. Darrow began to lose patience.\n\n\"I'm afraid that this time they can't.\" He spoke to her in the same tone he had used toward me, that of a master instructing a servant.\n\nMrs. Norman and I flinched at the same time, but she nodded in acquiescence.\n\n\"Let me collect my things upstairs, and I'll meet you in your study, Mr. Darrow.\" She lowered her voice to a whisper. \"The servants are skittish when it comes to the supernatural.\" She went down to the servants' quarters while Mr. Darrow went to his study and I to my room, where the chains looped around the bedpost disappeared into the floorboards, the creature held by them off in some lower part of the house scrounging for private conversations.\n\nI quickly changed out of the dress I'd worn to Darkling. I almost put on the black, somber governess's uniform that made me look and feel much older than I was, like an elderly spinster, the sort of woman I might still have become, alone and bitter, raising other people's children in a vicarious, unfulfilling life. But it seemed too much like a death shroud. I could not bear to imagine what was happening to the children. I found a casual blue cotton dress with white pinstripes and made my way to Mr. Darrow's study. Mrs. Norman had still not arrived, and the master of Everton stood gazing at his wife's portrait with not a small hint of dejection. He turned away from it when he noticed that I was standing in the doorway.\n\n\"We will find them, Henry,\" I said, regressing to our intimacy with more confidence than I felt. He smiled at me weakly and was about to reply when Mrs. Norman barreled into the room carrying a heavy-looking carpetbag in her arms. She dropped it onto Henry's desk with some relief and began unloading various pieces of occult paraphernalia.\n\n\"Mrs. Markham, set the candles in a circle around the desk. Mr. Darrow, pull those chairs here.\" She pointed to the sides of the desk and removed an old, heavy book with leather binding from the bottom of the bag, sending decks of tarot cards and phials of powders and liquids to the floor. She did not seem to notice the mess and opened the book.\n\n\"Sit and join hands.\" We formed an uncomfortable circle in the middle of Henry's study. \"Relax, breathe in and out.\"\n\nWe did this for some time, until the room was heavy with silence and the smell of incense burning with the candles.\n\nShe continued. \"Now, our beloved Lily Darrow, we ask that you commune with us and move among us.\" We sat in silence, waiting for something to happen. Mrs. Norman repeated the phrase a dozen times. Nothing happened. The room grew stuffy, and my fingers felt clammy in the hands of the others. I became very conscious of Henry's touch. I tightened my grip on his hand without thinking and was surprised when he returned the gesture. Then, the air in the room suddenly grew colder.\n\n\"If you are with us, please rap once,\" said Mrs. Norman in the dreamy sort of way one would expect a medium to sound, as if she had been practicing for this moment for some time. A knock sounded everywhere and nowhere, echoing and distant, perhaps upon some tabletop in another plane of existence.\n\n\"If we are communing with Lily Darrow, please rap once more.\" The ethereal rap sounded again. \"I would like to invite the spirit of Lily Darrow to use my body as vessel to speak with us directly.\" The temperature continued to drop. The candles went out, and Mrs. Norman leaned forward in her chair until her head was hanging over the top of the desk. She snapped back up as the candles reignited themselves, but kept her eyes firmly closed. Henry and I looked at one another.\n\n\"Lily?\"\n\nMrs. Norman spoke with her own voice, but it was higher and more melodic, with none of the cold authority that so characterized her.\n\n\"Henry?\"\n\n\"Yes, love! I'm here with Charlotte.\"\n\n\"Oh, Charlotte!\" Mrs. Norman let out a small sob. \"I'm so sorry, I didn't want any of this to happen, you must believe me!\"\n\nI did not answer her directly. \"Are the children all right?\"\n\n\"Yes, but Mr. Samson refuses to let us leave, and he has forced Mr. Whatley to close the portal. The war has begun.\"\n\n\"But why are the children important?\"\n\n\"He means to use them to destroy The Ending.\"\n\nHenry looked at me in confusion, but I pressed on.\n\n\"There must be some way to bring them back?\" I asked.\n\n\"I still know so little of The Ending, but there are different doorways, different methods of entry. The books in the library\u2014\"\n\n\"Oh!\" I gasped, and my lips formed into a perfect circle of surprise. Lily\/Mrs. Norman looked around the room blindly.\n\n\"Did something happen?\" they asked.\n\n\"No, I just remembered, there are books from Darkling in my room!\" I thought it best not to mention the eyeless, childlike thing chained to my bed.\n\n\"I'm afraid I'm terribly confused,\" said Henry. We both ignored him.\n\n\"Which ones did you take?\" asked Lily through Mrs. Norman.\n\n\"I believe there's one called Mysteries of The Ending.\"\n\nThe medium and the spirit within her became visibly troubled. \"You can use the book, but you must be careful. You will have to travel a great distance through The Ending to reach Darkling, but even then, what would you do when you got here?\"\n\n\"Humans are a threat to The Ending, and I think I know why.\"\n\nBefore I could explain any further, there was a knock at the door and Roland entered the room carrying a clattering tray of tea and biscuits. Fredricks had taken ill when he returned from the ball, and the young gardener had been fulfilling his duties as he had been training to do. Lily stopped speaking through Mrs. Norman, the connection severed by the intrusion as the housekeeper recovered and became herself again.\n\nRoland closed the door and set the tray onto the middle of the desk. \"Tea, sir?\"\n\nHenry became very cross. \"I am indisposed, Roland. Please take this away and make sure that we are not interrupted again.\"\n\nBut the young man ignored him. He placed a saucer and a cup before each of us and began to pour the tea, his hands shaking as he did so that the liquid spilled over the table.\n\n\"Roland!\" Mrs. Norman lunged for her book, knocking over a candlestick onto a pile of Henry's papers, which ignited immediately into a small fire. The gardener suddenly grabbed her arm and lifted her into the air until her feet scraped at the carpeting. He threw her across the room so that she collapsed against a shelf of books.\n\nHenry stood from his chair indignantly. \"What is the meaning of this?\"\n\nRoland struck him across the face, sending him roughly to the ground. The room was burning now. I backed away, not from the flames but from the young man as he began to shake uncontrollably, his face contorting in pain, wrinkling in strange ways as if it were not a face at all. I realized that it wasn't when his throat bulged oddly, as if something were pushing its way out of his chest. I pulled Henry away from him and then ran to lift Mrs. Norman to her feet just as the boy hunched over the table and made a terrible retching sound. A tentacle slid out of his mouth, protruding from between his lips until the girth of the thing was so great it tested the very limits of his mouth. His throat continued to expand, and his cheeks split in a spatter of blood as the tentacle continued rolling out of his throat until it was two feet in length. It coiled into the air above the remains of his head and made a slashing motion at his throat. A nest of tendrils and feelers spilled out of the gash in his esophagus, over his chest, each one tearing at his flesh and undoing him, releasing him so that he expanded, a writhing mass of movement and gore, to his full stature, many feet taller than any human being could ever be. Roland opened like a flower, a blossom of black, wet, slithering appendages curling from the pulp of human mess, and was no more.\n\nWe did not stand around to wait for the end of his metamorphosis. The room was nearly engulfed in flames, and the three of us crept toward the door until Henry wrapped his fingers around the doorknob and pulled Mrs. Norman and me over the threshold into the hallway, slamming the door closed behind him. The creature shrieked from within the room, a fluttering, high-pitched buzzing sound that left me nauseated and oddly half asleep, as if it were all a dream that I could wake up from if only I concentrated a little harder. But then the door shook as the thing that had been Roland threw itself against it, tearing me from my reverie.\n\n\"We must get to my room,\" I said to Henry.\n\n\"Are you certain?\"\n\n\"Trust me.\" I took his hand and we escaped down the hallway, but not before the door gave way and the creature slid out of the study, contracting to fit through the doorframe, a nightmare born into Everton, silhouetted by the blazing of the fire that had started to lick at the ceiling. The monstrosity filled the corridor, became a pulsating wall of tendon and viscera, churning and twisting toward us with a shapeless appendage that reached for my ankle but failed, moving on instead to Mrs. Norman, who was not as fast, and finding purchase around her leg. The housekeeper fell to the ground but grabbed hold of a cabinet against the wall, pulling it over and knocking a marble bust of some dead and forgotten member of the Darrow family to the floor, where it cracked into two sharp pieces. She seized a piece even as the creature slithered over her, hungry and unstoppable, and brought it sharply into the body of the thing.\n\nThe beast quivered, perhaps more in shock than in pain, for it did not release her from the struggle. Instead it parted Mrs. Norman's flesh as easily as it might have dipped into a pool of water. She briefly cried out in pain, but then the thing entered her mouth through the back of her head, severing the top of her skull and sending it to the floor.\n\nHenry and I did not wait to see what would happen next. We took off through the house, warning away every servant we saw, telling them to run for their lives even as we went ever deeper, the creature still behind us and the fire ripping through the innards of the mansion, nearly stopping us at the stairwell as a wall curled away in a sheet of flames. But with the deadly alternative behind us, we pushed on, mostly unharmed save for a singed strand of my hair, the scent of which cut me more deeply than Roland ever could.\n\nJonathan.\n\nHistory would not repeat itself. I refused to allow it, to allow anyone else to die. Everton might burn, but we would not be inside when it did.\n\nWe found my room at the end of the hallway and barricaded ourselves inside, the fire already smoking around us as I searched desperately for the books from Darkling. I had packed them away after Mr. Darrow sacked me, and I hadn't found the time to unpack since he'd changed his mind. I frantically dug through my valise, came up empty-handed, and then went through my steamer trunk, finally locating the volumes beneath a heavy thesaurus. They were tied shut with ribbon as a precaution against any further unwanted visits from the strange eyeless children, and try as I might, I could not get the small knot undone.\n\n\"I can't get it open.\" I handed the books to Henry, and as I went back to my trunk in search of a pair of scissors, the door to my room cracked completely in half and was thrown inward against him. The thing that had been Roland stood in the doorway. It hissed at us again, but before it could enter the room a length of blue chain looped around one of the ceiling beams exposed by the fire and was stretched taut. The ceiling collapsed in a hail of sparks and heavy timbers, piercing through the creature and pinning it to the floor, where it withered and blistered in the flames. The dirty little boy with the keyhole eyes appeared beside Henry and helped him to his feet.\n\n\"What is that?\" Henry gestured to the strange little child.\n\n\"A friend, it would seem.\" I picked up the largest wooden splinter that I could find and turned to the creature trapped beneath the debris.\n\n\"You killed them both\u2014Roland and Nanny Prum,\" I said to it. Something bubbled to the surface of the creature's viscous skin and broke open, revealing a small mouth with needle-sharp teeth.\n\n\"And Mrs. Norman,\" it said with sick pleasure. \"But there never was a Roland. Only me.\"\n\n\"But why?\"\n\n\"To set the game into motion.\" It squirmed in an effort to escape, but failed and resigned itself to its inevitable fate. \"Nanny Prum had to die to place the children under your complete care. And you delivered them, just as he knew you would.\"\n\nHenry turned to me for a moment as if to say something, but hesitated and began again with his attempt to untie the knot of ribbon around the books.\n\nI went on with my interrogation. \"Why did Whatley take the children?\"\n\n\"Whatley?\" The creature seemed confused by the question, but then its mouth, even without the benefit of any other facial expression, spread wide into a condescending smile. \"How little you understand what you are meddling with. It was Samson who kidnapped the children. Whatley has only ever tried to protect them.\" The monster began to laugh at us, sounding somewhere between its human and inhuman vocal register.\n\nI felt the swatch of wood in my hands and stabbed it into the beast's throat until I felt the wet crush of its flesh between the makeshift spear and the floor. The laughter went on uninterrupted.\n\nSmoke was pouring into the room, and the heat was unbearable. I could scarcely catch my breath. I took the stack of books from Henry, procured the shears from the bottom of my trunk, and cut the ribbon that held them all closed.\n\n\"Are you ready?\" I asked.\n\nHenry looked at the creature still cackling to itself on the floor. \"We have no choice, do we?\"\n\nI had nearly opened the book when I realized that all of my belongings would burn in the fire even if I did not. I went to my nightstand and quickly collected my old wedding ring, my mother's lock of hair, and my father's pipe, which I kept in my memory box. They were the last relics of my family, the only pieces of my past. I would not leave them behind.\n\nThe boy with the keyhole eyes had fastened his chain around Roland's neck like a leash, and he held out his hand to me. I took it and went back to Henry, now choking, and together the three of us opened the front cover to Mysteries of The Ending.\nCHAPTER 18\n\nCharlotte Underground\n\nAt the entrance to the castle the book grew hot in my hands and turned to ash without igniting, the connection to Everton dying in the flames of the house. There was nothing to do but move forward. Henry watched as the ashes scattered to the wind. \"Our home . . .\"\n\nI put my hand on his shoulder. \"Houses can be rebuilt, better than they ever were before.\" He nodded glumly, and the grimy little boy showed us inside the crumbling stone fortress, dragging with him the bloodied, muttering remains of the creature who had been Roland the groundskeeper. The lady of the castle, still resplendent in her decaying elegance, greeted us from halfway up the collapsed staircase. The boy handed her the length of chain, and she patted him on the head as she brought her face close to the creature bound in shackles.\n\n\"Do you know who I am?\" she asked.\n\n\"Yes, my lady.\" The thing could not meet her hard, steely gaze.\n\n\"And you agree to willingly serve me?\"\n\n\"Yes, my lady,\" it said with a whimper.\n\n\"Then it is done.\" She handed his chains to her other children, and they took Roland away, deeper into the unseen parts of the castle. The little boy with the dirty fingernails stayed behind with his mistress. The woman beckoned for us to follow her into the room at the top of the stairs. Henry put his arm around me, shaking, though it was unclear which of us he meant to comfort.\n\nThe chamber was made of glass and windows, like a solarium, if such a place could exist in The Ending. The oppressive moon hung low in the sky, and we were so high up in the castle that the horizon was lost in a dark sea of stars. The woman seated herself in a high-backed silver throne and observed us without expression.\n\n\"You wish for safe passage into The Ending,\" she said.\n\nThis seemed to rouse Henry from his dejection. \"My children are being held against their will.\"\n\n\"Of this I am aware. Hostages. Their continued presence in our world has set us on the brink of war. Would it not be careless of me to escalate the situation with two more humans?\"\n\n\"On the contrary,\" I replied, \"their rescue would only serve to mollify the tensions.\"\n\nThe woman was silent for a moment. She stroked the chains that circled her wrists. \"You assume that you could get to Darkling untouched. Not even I could promise such a thing.\"\n\n\"That is a risk we are willing to take.\"\n\n\"There is also the matter of payment.\" She smiled, her teeth glittering like the metal of her fetters.\n\nI stepped forward. \"You trade in answers to unasked questions. Yet you haven't asked us what we will do once we get to Darkling. If you agree to send us on our way, I'll tell you.\"\n\nThe lady of the castle leaned back in her silver throne and motioned me forward with a slender index finger. I whispered my plan to her. At first she said nothing, and then a clicking sound escaped her throat, growing louder until it echoed off the glass walls of the solarium. She threw her head back and laughed: a dry, broken, thousand-year shriek of mirth that sent the dirty, eyeless boy scampering off into the dark corridors of the castle. I backed away from her, perplexed.\n\n\"You think I'm foolish?\" I asked when she was done.\n\n\"I will allow you to pass. Is that not enough?\" She lifted herself from her throne and took us onto an adjoining terrace that looked over a forest of black, jagged trees, a dirt path winding between them. A solitary crow stood on the ledge. The lady of the castle whispered softly to it, words that we could not hear, and the bird flew off into the night sky. The woman herself led us down stone steps etched into the side of a cliff, into the mouth of the wood.\n\n\"Follow the trail to the temple on the other side of the forest. Tell the cleric that the Blue Lady has sent you. You will travel along Mr. Samson's own rebel underground. Ironic, is it not?\"\n\n\"Why would Samson's people help us?\"\n\n\"He has kidnapped two children, a crime even in our land. His actions do not help the cause.\" The echo of the wind sounded around us without the benefit of a breeze, and the limbs of the trees creaked overhead. The only other sound was the clinking of the chains that trailed behind the Blue Lady, up the stairs, and around the throats of her grimy, blind children.\n\n\"Thank you for your help,\" I said.\n\n\"Do not thank me yet.\" She started to cackle again as she walked back up along the cliff, to the terrace of her castle. Henry stared anxiously into the woods and stepped in front of me before I could start along the path.\n\n\"What did you tell her?\" he asked, looking into my eyes.\n\n\"The answer she wanted to hear.\" I moved around him and would say nothing more on the subject.\n\nThe trees were tall and lean, and as we went on the forest dwindled, plant life shifting in color from dark green to ash gray. The bark appeared to be chipped and dusty, ready to crumble at the slightest touch. Many trees had been pushed over and broken, the remaining stumps jutting sharply into the air. Then the forest stopped completely, and I stared out at a vast gray wasteland.\n\nIt was a dreary, oppressive place, an infinite desert pocked with craters, without stones on the ground or stars in the sky. There was only ash and the glow of the moon, rendered sallow and pale on the desolate landscape. We walked for what felt like miles until we stood on a ledge above an even greater expanse of gray nothingness. There was a building in the distance below.\n\nWe navigated down the slope of the rock with relative ease, and when we arrived at the bottom I finally heard it: a collective, raspy breathing coming from everywhere and nowhere. I felt surrounded, but then the little temple was still a mile off and there was no one around for as far as I could see. The path wove through the craters and the sound continued, rising and falling, a thousand different mouthfuls of air, as we walked to the lone building in the distance. I finally found the source of the sound just before we entered the temple.\n\nIt was coming from one of the pits, from a sad-eyed creature clawing at the sides of the cavity with a dozen bloodied appendages worn away from unsuccessful attempts at escape. I heard the gasping sound again, and the cavity in the earth closed in on the emaciated thing that still clung to the wall. Its eyes widened as it saw me, and remained open even as the pit closed around it.\n\nHenry pulled me away, and we approached the entrance to the temple. It was not a Christian building. The image of a serpent devouring itself was etched above the door. I knocked, and a small hunchbacked man answered.\n\n\"Are you the cleric?\" I asked.\n\nThe man nodded.\n\n\"The Blue Lady sent us,\" said Henry.\n\nThe cleric observed us with watery eyes and moved away from the door. A narrow stairwell curled deep into the earth. We followed him down to the bottom, into an amphitheater with a glowing pool of water at the center instead of a stage. There were many tunnels leading off the room into other parts of the earth, more dark places with shadowy things that I had had quite enough of.\n\nOur host stopped at the edge of the pool. \"If you seek sanctuary, you must bathe in the pool.\"\n\n\"Must we?\" I said under my breath. I considered myself a very open-minded person, but I was growing quite tired of the endless customs and traditions of the people of The Ending. Still, as there wasn't anyplace else for us to go, we had little choice but to comply. I unbuttoned my blue pin-striped dress as Henry took off the pieces of his suit. Truth be told, our clothes were becoming increasingly tattered, and it felt good to step out of them and even better to enter the gleaming pool of water.\n\nWe turned to one another after we were submerged, our nudity hidden by the opacity of the water. We kept our distance to maintain what little propriety remained to us, and we peered at each other across the pool.\n\n\"Are you all right?\" I asked. I had never seen Henry look so haggard.\n\n\"You were very brave.\"\n\n\"I'm afraid bravery had very little to do with it. I'm rather fond of my life and quite willing to run in order to preserve it. If I had been brave I would have done something to save Mrs. Norman.\"\n\nOnce I said this, I realized it sounded like an attack on his masculinity and courage, which I didn't mean for it to be. I could not get the image of Mrs. Norman out of my head. She was the courageous one, with her final, rebellious act of violence. She had been a deeply unpleasant woman but did not deserve the fate she had been handed, which we had inadvertently dealt her. It had not been her fight or her mistake. It was mine, and I had failed her. I said as much out loud, but Henry shook his head.\n\n\"You couldn't have known.\"\n\n\"I could have. I should have. But\"\u2014I scooped a handful of clean water over my face to soothe my singed skin and lips\u2014\"there will be plenty of time for self-loathing after we've found the children.\"\n\nWhen the hunchbacked cleric was satisfied with our informal baptism in the pool, he handed us robes and took us down a corridor lit by phosphorescent motes of dust that floated through the air, occasionally catching at my hair and skin, and causing me to glow the way that people do when light shines through their fingertips, pink and translucent. Henry brushed away one that had stuck to my forehead. The touch of his skin energized me despite my exhaustion.\n\n\"Thank you,\" I said.\n\n\"No trouble at all.\" Light cracked through his fatigue, perhaps the beginning of a smile before it faded again into weariness.\n\nWe were brought to a low, small room with no furniture save for a recess in the center that had been filled with furs and blankets. The luminous specks of dust had settled into the fibers of the sheets, and as we stepped into the alcove we quickly became covered in the stuff, our skin brought to life, our flesh an incandescent hearth at the center of the room. Our host left us, and we attempted to sleep, but I could not leave my hands alone, hypnotized by the trails of light they left behind. Henry was equally fascinated, and together we made as many shapes as we could think of, writing our names in the air, brushing ourselves off so that the flecks of dust became floating stars above our berth, until we settled in beside one another.\n\n\"Charlotte?\"\n\n\"Yes, Henry?\"\n\n\"What would you say to Jonathan, if you saw him again after all these years?\"\n\nI closed my eyes and summoned my husband's face, even as I could feel the warmth of Henry's body beside me. It surprised me to find that my feelings for both men were not mutually exclusive.\n\n\"I would tell him I loved him, that I will always love him. No matter what the future might hold.\"\n\nHenry inhaled deeply, lost in thought for a long while. And then: \"When Lily became ill, I stayed by her side day and night, wasting away as she did. I thought that if I could be there for her in every way possible, she might draw some small shred of strength from me. But she didn't. There was nothing to be done but watch her slip away. She died in my arms. I felt it happen, the breath leaving her body one last time. I kissed her then. Part of me hoped that I could draw her back from the Other Side, while she was still warm.\"\n\nI took his hand into mine.\n\n\"I don't know if I can face her,\" he said.\n\n\"You can. You will.\"\n\n\"But what do I say?\"\n\nI did not know how to respond. We remained in the dark, our fingers entwined, and soon I drifted off into a dreamless, peaceful sleep.\n\nIn the morning, the hunchbacked cleric collected us and provided a change of clothing from the singed rags we had been wearing upon our arrival. We bathed quickly one last time in the pool, washing the glowing motes of dust from our skin and hair, before we dressed in what appeared to be servants' uniforms and were escorted back up the stairwell to the entrance of the underground temple.\n\nOutside, I could still hear the gasping sounds from the pits in the ground. I turned to the cleric. \"Those pits around the temple . . .\"\n\n\"A political prison.\"\n\nI stopped, disgusted.\n\n\"Is that what the rebellion is working against?\" asked Henry.\n\n\"That is the rebellion at work. In times such as these the question of right and wrong becomes a complicated one.\" The cleric led us down the side of a hill behind the building, to an empty seashore of tepid waves, with a rickety boathouse perched at the end of a dock. He went inside and came out pulling a small rowboat through the water by a dirty cord.\n\n\"In you go,\" he said. Henry got into the boat first and then helped me off the dock. The hunchback cast us off and lumbered through the vessel, nearly capsizing it, to sit at the front. He looked back at Henry. \"Oars,\" he said.\n\nThe former master of Everton took the oars and cut across the dark water, through a sea of green-black islands with rolling hills and barren trees that scratched into the air with clawed branches. I sat back and looked up at the stars in the velvet sky. Nothing moved among the trees and hills of the empty coastline, whose barrenness could not match the unease I felt watching Henry push us across the surface of the water, our destinies entwined as we moved blindly beyond the confines of our story, away from master of the house and governess, simply two people searching in the night for two lost children, and perhaps for themselves.\n\nThere was land on the horizon, black and cold in the moonlight with a thin spire of smoke climbing above its charcoal shores. The lamp of a crumbling lighthouse turned atop a precarious heap of rubble and cracked walls.\n\nAs we neared, the rocky coast gave way to a deserted shoreline of squalid cottages, a sad little town of molding walls and broken-toothed windows, huddled together against the lip of a demolished harbor. Nonetheless, it was not unoccupied. The smoke we had seen from the sea was coming from the chimney of a hut at the end of the lane, its windows glowing with the promise of fire.\n\nThe hull of the boat scraped against the shoreline, and Henry hopped into the water to drag the dinghy onto the beach. He extended his hand and helped me over the side. The hunchbacked cleric followed, leading us from the beach onto a path of ruined, uneven cobblestones. We walked silently down the narrow road to the house and knocked on the door.\n\nA woman answered. She was shapely and round, with buttermilk skin and red ringlets of hair. The hunchback whispered something to her, and she opened her arms in greeting.\n\n\"Welcome. Please, come in.\" She led us to sit in front of the hearth.\n\nThe hut was small and decorated for Christmas. The frail tree in the corner of the room was gray, even by firelight, and clung to the sparse ornamentation perched on its branches. There were other guests seated next to us, old, broken, and decayed, staring deeply into the flames.\n\nA cauldron hung above the embers, its bubbling contents hidden by brown, sticky foam. The woman brought us three coarse wooden bowls. She took a ladle from the wall and dipped it into the kettle. The head of the brew dissolved, and we could see something moving beneath the surface.\n\n\"No thank you, we've already eaten,\" I lied, starving, though the cleric heartily devoured his bowl of brown. The eyes of the other guests never left the flames. The woman sat at a table by the door to peel carrots. She placed one into a small cage that swung above her head, the animal inside gnawing at the stick as white, gelatinous foam dripped from its maw. It might have been a ferret, but the slaver had smeared its fur so that it was an indistinguishable mass of pelt and teeth. I slipped my hand through the crook of Henry's arm.\n\n\"There is trouble tonight,\" said the woman without looking up from her task.\n\n\"What do you mean?\" asked Henry.\n\n\"The one who will take you is late. He is never late.\" She smiled as she fed the caged creature another sliver of carrot. We moved closer to the hearth, joining the others as they lost themselves in the fire. I imagined that the flames formed the walls of a house, and inside a small family of embers burned away their bright little lives to keep it intact.\n\nHenry broke into my reverie. \"I still haven't the faintest idea what we plan to do.\"\n\n\"You escape with the children, and I will sort out Mr. Whatley.\"\n\n\"Alone?\"\n\n\"Hardly.\" I turned to him, widening my eyes in an effort to end the conversation. I was unsuccessful.\n\n\"Why must you be so cryptic?\"\n\nAnnoyed, I put my lips to his ear and whispered sharply. \"I have little experience traveling through rebel undergrounds, but I would imagine that they are not safe for private conversation.\" I motioned to the others seated beside us, all of them nearly catatonic save for one man who stood suddenly, kicking over his chair.\n\nThe woman with the red hair shouted at him. \"Pipe down there.\"\n\nBut instead of reseating himself, the stranger leaned his head back, the surface of his skin gathering like beads of melted wax traveling up his face, a strand of it pulling away from his body, a thread of flesh rising into the air to attach itself to the ceiling.\n\nOur hostess gasped a single word: \"No . . .\"\n\nHis body blasted apart with a wet tearing sound, sinewy tendrils erupting out of a husk of red meat to embed themselves in the walls and ceiling, scrambling around the room in search of prey. Where it touched the other guests, their flesh became its flesh, merged together and absorbed into an ever-expanding mass, none of the victims dead or dying, simply devoured whole.\n\nAs one of the tendrils made to slide around my leg, the hunchbacked cleric threw himself in its path, the thing entering his back and swelling around him. I did not have time to cry out, for Henry pulled me through the door, and together we ran into the night, looking back just long enough to see the hut crack and collapse, the plaster and rock consumed into the growing girth of the beast.\n\nThe town came to life, screaming. Doors opened all around us, voices calling out for their loved ones as they ran into the streets; creatures and creatures in human skins, sobbing and shouting, pushed into each other. A young man ran past us carrying a glass bottle of jet-black liquid, a swatch of cloth sticking out of the opening. He lit it and chucked it at the monster in the demolished hut, but it fell short and landed at the base of the structure. At first I thought he had failed, but then the ground split open with a bone-chilling crack and began to fall away, creating an abyss where there had not been one mere seconds before. The creature scuttled for purchase at the edges of the chasm, but it had already grown too heavy for its own good and descended into the darkness below. The crowd of onlookers cheered momentarily, but then the rift continued to expand. Houses and whole streets succumbed to the schism, the edges of the earth flaking away into the void below.\n\nHenry and I followed the crowd into a forest on the outskirts of town, winding between the trees until the chaos was behind us. When we were far enough from the townspeople that we would not draw unwanted attention, we collapsed against one another.\n\n\"Are you all right?\" he asked.\n\n\"He saved me,\" I said incredulously.\n\n\"Who?\"\n\n\"The cleric. Why would he do such a thing? He barely knew me.\"\n\nHenry put my hand into his. \"We should keep going.\"\n\n\"Where?\"\n\nHe pointed behind me to a wide road. It had once been paved over, for the edges of the thoroughfare were crusted with pieces of crushed red brick, but the center of the highway had been worn to dirt from good use. It was the largest road that I had ever seen. It went on for as far as my eyes could see, and it was wide enough to accommodate twenty carriages riding beside one another.\n\nHenry helped me to my feet, and we scampered down the side of a hill to reach the large expanse of road, but before we could leave the forest a person stepped into our path. He had a flat, open gash of a face, and a body that would have been snakelike had it not been for the heavy, muscular appendages that trailed along both sides of his torso. The person called out to us. \"Hello, my friends!\"\n\nWe stared at the man wordlessly.\n\n\"My companion and I are faced with a bit of an imposition. We're travelers like yourselves, you see.\" Another person appeared behind him. He was taller than any man, with arms and legs like very strong sticks, and half a mouth, with no bottom jaw but many long, sharp teeth. \"We've lost the third member of our party.\"\n\n\"I wish we could help you, but we must be on our way,\" I replied.\n\n\"But you can help us,\" said the man who was almost, but not quite, like a snake. \"Our friend was with you, the large gentleman in the hut. You were ever so quick to part with him.\" The two highwaymen leered at one another. I tried to run, but the stick man grabbed ahold of me and threw me to the ground, my head throbbing in pain though I refused to let him see it by meeting his gaze.\n\n\"Eager to leave us as well?\" said the snake. \"We should be insulted. It's a good thing Mr. Ashby is asking for you alive.\"\n\n\"A good thing indeed.\" The stick man's breath was dry and sour.\n\n\"But still . . . he would be ever so cross with us if we grabbed the wrong person. Best make sure you're human, after all. Blood will tell, as they say.\" He removed a long, thin knife from his jacket and cut away my cloak. He was about to do the same to the rest of my clothes when Henry leapt onto his back and began to strangle him with his arms. The snake man shrieked and dropped his knife to the ground, while the other swatted Henry through the air and into the forest.\n\nThe highwaymen congratulated themselves and hovered over me, unaware of a shadow moving behind them, black and dangerous, shifting along the ground, alive. As it passed over the stick man, his limbs shattered into thousands of bloodless little pieces and the back of his head was knocked through his mouth.\n\nThe snake man shrieked at the sight of his fallen companion, and he took off down the road. But the moon was high, and the shadow stretched after him as well. It embraced him so tightly that he stopped moving and fell to the earth, his skin peeled away in one swift motion. His flesh followed with a moist rip. But as this was The Ending, neither of the highwaymen could die even if he so desired. The stick man quivered in a fetal position on the ground and attempted to slurp the contents of his head back into his mouth. His partner, all bloody pulp and bone, pulled himself up the road, gathering his flesh and skin.\n\nThe shadow subsided as it lifted Henry from the floor of the forest and set him into a wagon near the side of the great road. As it approached me, it took the shape of a middle-aged man, impish and dark-featured, with a slender finger pressed against his sly lips.\n\n\"Duncan!\"\n\nWhatley's manservant brought me to my feet and climbed into the driver's seat. He did not wait for me to join him before whipping the side of the animal that pulled the wagon. The squat, headless thing with hundreds of fleshy flaps of skin squirmed forward like so many caterpillars, but it proved very fast as it took off. I quickly retrieved the knife that had been dropped by one of the highwaymen and barely had enough time to jump into the back of the vehicle as it started forward. I tucked the weapon into the folds of my dress.\n\n\"Why did you come for us?\" I hardly expected an answer, but Duncan reached into his coat pocket and extracted a parchment envelope with a blue wax seal. I tore it open and read the two words written in Lily Darrow's handwriting: Trust him. I handed it to Henry.\n\n\"He did save our lives,\" he said. That was hardly sufficient, but given what few options we had, I reluctantly turned to face the back end of the wagon. We twisted down through the dark hills of The Ending, the wagon lumbering along the hard, beaten earth between patches of broken brick.\n\nWe passed by great houses and manors with candlelit windows and oddly shaped figures scuttling about behind them. I wondered if the boys would remain unchanged after having lived among such strangeness. There were other orchards, and a lone clock tower on a small island at the center of a lake. Henry and I soon settled into the quiet rhythms of the wagon. We took turns sleeping while the other remained alert, but no one passed us by. We were completely alone.\n\nThe hills grew taller as we continued, and the road sloped upward along the side of a mountain. There was movement in the valley below. Shapes in the darkness bounded after one another, their ridged backs glistening with sweat in the moonlight. Barbed tentacles lashed cruelly at tender undersides, and talons tore through flesh and fat and bone. Blood was thick on the ground, black as midnight.\n\nDuncan did not bother looking down at the carnage below, and the creatures seemed to be paying us little if any attention, for we went down the other side of the mountain without trouble.\n\nI realized Henry and I had started holding hands, but I could not recall when it happened, or why. I felt so many things all at once\u2014fear, anger, exhilaration, doubt\u2014each of them vying for my attention, surging through my body in alternating waves of anxiety and relief, to the point that I shut them all out and focused instead on the fact that I was glad to have Henry by my side at that moment. With my other hand I felt for the last relics of my former life, which I had rescued from the burning cinder of Everton.\n\nJonathan. I fell asleep despite my anxieties. My body demanded rest, and I dreamt that I went to a traveling carnival with my family.\n\nWe drifted from tent to tent, from the jugglers to the fortune-teller and finally to the magic man. He stood on his collapsible stage, a man in black making doves appear from his throat and fire dance at his fingertips. He called my mother from the audience and placed a sheet over her body. With a clap of his hands she disappeared. Then he took Jonathan, who vanished in a flash of light. My father was the last person chosen from the audience. The magic man placed him in a chair and levitated him into the air and out of sight. I clapped and clapped when the show was over. I waited at the entrance for my family to return, but then the gypsies packed up their tents and drove away in their caravans, leaving me alone on an empty hill.\n\n\"Charlotte.\" My eyes fluttered open. Henry's face hovered above me. I sat up and realized that I recognized the landscape. There was the orchard and beyond it, the House of Darkling alight with activity. There were other carriages on the road now, all of them in a long caravan to the front entrance by the Star Fountain. Duncan avoided this, steering us through the tall black iron gates, past human-shaped guards who waved us through without a second glance, and around the back to a servants' entrance, where arm in arm, Henry and I entered into the House of Darkling.\nCHAPTER 19\n\nThe Man in Black\n\nThe lower floors of the manor were characterized by crooked hallways and room upon room of sweating, anxious servants, all of them frantic to respond to the panels of angry, chiming bells that heralded the needs of the hundreds of guests upstairs. Duncan did not allow us to linger, urging us onward until we were expelled from the inner workings of the house into the wing where the children and I had slept during our visits. To my surprise, he opened the door to my own quarters and pushed us inside.\n\nA woman stood in the center of the room, her back to the entrance. She was dressed in a white gown that flowed from an ivory bodice of lace down the curves of her body into a pool of silk on the floor. A veil hid her face, but Henry knew her all the same.\n\n\"Lily . . .\" he said breathlessly.\n\n\"Henry?\" She sounded weary and sad, but her voice left her entirely when her eyes turned to meet her husband's. He took a single step forward, then another, and another, as if approaching a dream, careful to hold it for as long as possible before it slipped away. When he reached her he pushed back the veil to stroke the side of her face. She trembled, closing her fingers around his wrist. They stayed in the same pose, a silent conversation playing out in their mutual gaze, which remained uninterrupted by any further physicality. They simply looked into one another.\n\nI could not help but feel a slight twinge of jealousy at the sight of them together, even as I reminded myself that they were still husband and wife. A shadow passed over Lily's face, and she became very melancholy.\n\n\"I'm afraid you might be too late.\" She motioned to her wedding gown.\n\nHenry seemed to see it for the first time and backed away in confusion. \"You can't be. You mustn't.\"\n\nShe ignored him. \"I sent Duncan to fetch you after we last spoke, so that you could see the children safely home. They do not belong here.\"\n\n\"Neither do you. Come with us,\" I said.\n\n\"We tried that once before, don't you remember? They'll never release me. I'm the only one in The Ending who has ever died. They worship me, and Whatley will marry me.\"\n\n\"You deserve peace, my love,\" Henry interjected.\n\n\"This is exactly what I deserve,\" she said bitterly. \"I am glad you came, Henry. I had so wanted to see you one last time.\"\n\n\"I don't understand. Why didn't you summon me here with the boys? Your death broke something inside of me, Lily. I would have been here in an instant.\"\n\n\"That's exactly why I tried to keep you away. I went so far as to tell the children there was a spell keeping this place connected to Blackfield, and if they spoke to you of their time here, it would be forever broken. I was afraid, Henry. To say good-bye to the children was my duty as their mother. Their hearts will mend. I couldn't bear to see yours break all over again.\"\n\nI felt a pang of foolishness at my own gullibility, but it was washed away by all the other emotions that swept over me during this exchange.\n\nLily must have seen this, for she closed her eyes and summoned the strength to turn her husband away. \"They'll be coming for me soon. I think you had better leave.\"\n\n\"We can help, Lily,\" I said.\n\n\"How? What could you possibly do?\" When I did not respond, she turned away from us once more. \"Take the children back.\" Her voice cracked as she spoke, and she nodded to Duncan. He went to the wardrobe and extracted two black cloaks, draping them over us. The hoods hung low before our faces, like shrouds.\n\nLily handed me a silver skeleton key without looking in my direction. \"You may find passage back to Everton in Mr. Whatley's study. But for myself there is no other choice.\"\n\n\"That is where you're wrong.\" I tucked the key away into the folds of my dress. \"There is an alternative to The Ending.\"\n\n\"Death offers his gift but once, if at all. Now, please leave me. The ceremony will begin soon and you had better find the children.\"\n\nWe pleaded with her, but she ignored us and continued to prepare for her wedding. I pulled Henry grudgingly out of the room, but as we left he spoke to her one last time.\n\n\"I love you.\"\n\nShe observed us in the reflection of her vanity mirror but remained silent as she watched us leave. I thought I noticed tears in her eyes, but Duncan was already weaving us through the house, carefully out of sight of the other guests, who lingered on the periphery of Darkling, voices raised, cutlery scraping together, heels clacking against tile; ghosts who lived just beyond the edge of sight.\n\nWhen Duncan entered the dark room, I stopped and touched his arm. \"We need to find the children,\" I whispered. He observed me drily, the smile on his face slipping for a moment as he pushed his finger into the eye socket of one of the marble faces, opening the door to the circular chamber enveloped in concentric rings of silk veils. A boy sat on the metal chair, his feet dangling just above the floor.\n\n\"There you are,\" he said. \"I was worried you weren't\"\u2014his face fell as we pushed back our hoods, the words dying on his lips\u2014\"coming.\"\n\n\"What on earth are you doing here?\" I asked him. James hopped down from the chair and approached us with caution and very adult suspicion. He was dressed in a black suit with a gray vest, a red cummerbund circling his waist. Yet even disregarding the finely tailored clothing, he held himself differently than the last time I'd seen him. He did not look any older despite the years that had doubtlessly passed for him since our separation, but nevertheless there was something changed in him.\n\n\"You came back,\" he said to me.\n\n\"I never meant to leave.\"\n\n\"But you did.\" He hugged his father in a mechanical gesture without any emotion.\n\nHenry did not seem to notice. \"My boy,\" he said. He smoothed out the curls of his son's blond hair with unguarded sentimentality, but James pulled away, his face contorted in confusion.\n\n\"I'm not a child.\" He shuffled back to the metal chair in the center of the room, where he removed a smoke-colored phial from his pocket. It was labeled INFIRMED.\n\n\"James, put that down!\"\n\n\"Do you even know what it is?\" Though he appeared to be only five, he spoke with all the stoic assurance of an adolescent.\n\n\"Someone's death.\"\n\n\"Not just anyone's.\" He held it to the dim light, picking at the stopper in a distracted way. \"I remember the night she died.\" He looked to his father. \"You don't think that I do, but you're wrong. You left me alone with her to talk to the doctor, and she started to make a sound. There were noises coming out of her; she was gasping, and her eyes were wet, like she was drowning from the inside out. I think she was crying.\n\n\"I tried to give her a hug, but she jerked away from me, like I had hurt her. So I just stood by her side. I heard the doctor say that she was blind by then, but I felt like she could see me because she grabbed my hand. She pulled me close and tried to whisper something, but she couldn't speak right. The words were all broken. But then she said it again and again, and I realized that what she said was 'I want to die.'\n\n\"One night I asked her about it, if she remembered me there and if I helped make it easier. I wanted her to know that I cried when she was gone, but it only upset her and she ran from the room before she could answer. I haven't asked her since.\"\n\n\"And that's her death?\" I gestured to the phial still in his hands.\n\n\"I think so. I found it hidden in her room. I was waiting to open it, and tonight seemed appropriate. Duncan was going to help me.\" He held it out to his father. \"Would you like to try it instead?\"\n\nHenry went very pale, and a bead of sweat dripped from his brow. Yet he did not reach out to accept the phial. \"No, thank you, James. I think we both experienced enough of your mother's death firsthand.\"\n\nThe boy nodded and handed the glass container to Duncan, who secreted it away in the folds of his jacket.\n\n\"Are you taking us home?\" James's green eyes found my own, and I could barely hold back tears of guilt.\n\n\"Yes, of course we are. I'm so sorry, James. It's my fault you were trapped here alone.\"\n\nFrom the expression on his face I was certain that he felt pity for me. \"We weren't alone. Mother was here, and Mr. Whatley.\" I noticed he hadn't corrected me, but then he had no reason to. It was my fault that the door between Everton and Darkling had been closed, but that he had confirmed it made the changes wrought in him since the last time we had been together all the more clear. He had grown up.\n\n\"Did Mr. Whatley hurt you?\" I asked, looking over his face for any signs of abuse.\n\n\"No, not at all. He protects us.\"\n\n\"From what?\"\n\n\"His friends.\"\n\nI found myself in the strange position of feeling gratitude to the master of Darkling. Fortunately Duncan chose that moment to usher us all out of the chamber and back into the claustrophobic darkness of the other room. We pulled our hoods over our faces and followed him in silence, James taking his father's hand as we began to encounter the other wedding guests, mysterious figures garbed in cloaks identical to ours, human-shaped creatures much like Whatley, Samson, and all the rest of their circle.\n\nDuncan escorted us into the medieval banquet hall that Lily had shown us on one of our prior visits, but instead of containing the mysterious ever-changing door that had tormented Susannah, it was now filled with row upon row of hospital beds, all of them occupied by poor creatures in varying states of decline.\n\nThe patient nearest to us might have once resembled an oversized earthworm, but it had been torn into pieces, its stumps bound in white gauze and placed along the length of the bed, struggling to squirm together in sequential rhythm despite the fact that they were no longer part of the same whole. Another victim was riddled with perforations in its head and torso, and thick metal spokes had been placed into the gaps of its flesh to brace the body against complete collapse. There were no doctors or nurses to tend to the wounded, only a dark-haired boy who stood at the other end of the room, struggling to help place what was left of Dabney Aldrich into a human-shaped suit.\n\nPaul wiped a streak of sweat from his brow and scowled at his little brother. \"I told you not to visit me here, James.\"\n\n\"Come along, Paul,\" I said.\n\nHe gaped at us, not understanding until we came close enough for him to see beneath the shrouds of our cloaks. He gently laid Dabney back onto the bed. The other boy's face was just as angelic and beautiful as it had been before, but beneath his neck his human body was matted with the pieces of his actual one, bound together in bandages in a hopeless effort to give him something of a human shape.\n\nPaul ushered us to a far corner of the room, away from his patients. \"You've been gone for so long. I didn't think we'd ever leave,\" he stammered, rubbing the back of his head with his hands as if he were trying to decide something.\n\n\"What on earth are you doing here?\" asked his father.\n\n\"The night of the engagement marked the start of the war. Someone had to tend to the victims, the ones damaged beyond repair. Their families either are in worse shape or have disowned them.\" He gestured to Dabney, and I recalled the haughty dignity of Mrs. Aldrich. I could not imagine her having the compassion or the patience for long-term care. \"I've stayed by their sides and helped as best I could.\"\n\n\"That's very brave of you,\" I said to him.\n\n\"No, they're the brave ones. They endure without the hope and mercy of death. I only manage to find strength in their resolve.\" His eyes shifted to me for a moment, wordlessly referencing our conversation by his mother's gravestone all those months before.\n\nDabney stirred on his bed. \"Paul.\" His voice was a weak shadow of what it used to be. Still, the elder Darrow boy lifted him upright and helped dress the remains of his body for the wedding as we looked on in discomfort. When he was done, Paul placed his friend in a wheelchair. The other boy reached out and took his hand. \"Are you leaving us?\" he asked.\n\n\"I'm taking you to the wedding, just as you wanted, my friend.\"\n\nDabney smiled and stared off into space as Paul wheeled him out of the makeshift infirmary.\n\n\"Is Mother coming with us?\" Paul whispered. Henry and I exchanged glances. \"We can't leave her here.\"\n\n\"And we won't. Leave everything to me,\" I said with a note of finality. I replaced the hood of my cloak as Duncan led us to the dining hall, where the guests had been informed that the wedding was about to begin.\n\nA thousand wedding guests in hooded shrouds began to file into the ballroom, lavishly decorated for the event. Silver cages filled with firebirds hung from the ceiling. Sad, languid music was being orchestrated on a twenty-foot-long harp that took a dozen people to play it, some of them standing on ladders.\n\nThe Darrows and I sat down with Dabney while Duncan bowed to us and retreated to the far end of the hall. Across the aisle Olivia chattered flirtatiously with some of the younger guests, throwing her head back gently in a demure scoff, pleasantly scandalized by some rude observation. Her eyes flittered over us but did not stop. She gazed at her father with an expression carefully guarded by a well-practiced blank smile that did not show in her eyes as he took his place at the front of the aisle, smirking victoriously.\n\nThe music died out and the room became hushed in silence. The harp players began to strum their instrument until the notes resembled a wedding march. Lily Darrow stood at the entrance to the ballroom, dressed in her elaborate white wedding gown. She strode down the aisle. When she reached the section we were sitting in, she turned to us with a weak smile.\n\n\"Don't do it, Mother,\" James whispered to her loudly enough for everyone to hear. Lily looked from the children to Henry. Mr. Whatley grew increasingly impatient at the end of the aisle. She turned back to the task at hand, and Mr. Whatley glowered at the boys triumphantly. I removed the small, thin knife that I had taken from the highwayman and hidden in the folds of my dress.\n\n\"Do you trust me?\" I asked Henry.\n\n\"Of course.\"\n\n\"Then stay by my side and help me with the children.\"\n\n\"What are you\u2014?\"\n\nI took a deep breath and, hands shaking with conviction, plunged the knife into my chest. It did not feel as I thought it would. I had imagined more pain, more terror, but it was all very numb. The world slowed down, and I collapsed into Henry's arms in slow motion. Lily turned to see what had happened, her veil fluttering before her eyes, one step behind the progress of time. She ran back down the aisle to my side. I pushed the hood of the cloak away from my face, and the crowd stood up to observe the chaos, blurry figures at the edge of my vision that I felt I recognized, but could not quite make out.\n\nThey were elated.\n\nI heard Mr. Whatley above the crowd, demanding to know what had happened, for he could not see over the throngs of wedding guests despite his unnatural height. Henry cradled me in his arms, unsure what to do and trying to calm the children, who were beside themselves in shock and horror.\n\n\"It's all right,\" I choked, trying to comfort them even as my teeth became slick with blood. \"Look!\"\n\nA storm had gathered outside, the moon obscured by a writhing tempest of black clouds that spilled down from the sky and into the horizon, churning over the bleak pine-colored hills of The Ending in a frigid, swirling vortex that pressed against the windows until they shattered inward. The firebirds extinguished themselves, and the lights in the room went out. A doorway made of night opened to meet us where we stood, and from within it there appeared the shape of a man clad all in black.\n\nI had summoned Death to The Ending.\n\nSome of the wedding guests began to cry with tears of joy, while others knelt in reverence.\n\n\"Really, there's no need for that,\" said the man as he stepped forward. Mr. Samson appeared beside him.\n\n\"We bid you welcome to The Ending, my lord.\"\n\n\"While that's very kind of you, I am no one's lord, and I'm afraid I have more pressing matters to attend to.\" He observed me on the floor with Lily, a pool of blood spreading before us, staining the hem of her white dress. The man bent over me and looked at my injury. He gestured to the knife. \"Shall we remove this?\"\n\nI nodded to Lily, and she pulled the knife from my torso. I winced, gasping for air, the pain of it nearly causing me to black out, but I gritted my teeth together and bore through it.\n\n\"There, all better,\" he said drily. If I hadn't been in so much pain I would have laughed aloud. \"Now, to the matter at hand. One of you has been dead for some time, and the other is dying in a place where death has never before occurred. What am I to do with the two of you?\"\n\nI sat up, and a gout of blood spilled down the front of my chest. \"If I may, sir, there is only one reasonable course of action.\"\n\n\"And what is that, Mrs. Markham?\" asked the man. At the utterance of my name, the crowd fell away, and Mr. Whatley finally saw me with his fianc\u00e9e. His eyes went very wide, and for what I assumed was the first time in a very long time, he was speechless.\n\n\"A soul must be taken,\" I continued. \"Lily passed on, but not completely, and I cannot die in a place where death does not exist. Reason would follow that you should take her into the light, and let me keep my life.\"\n\n\"A reasonable point, but wrong all the same. Death did not exist here, until now. Hello.\" He turned and waved to the crowd of onlookers with good cheer. \"But this is still The Ending. I am, if anything, a man of the people, and the people of The Ending are different. New rules are needed.\"\n\n\"Please, sir, take us with you,\" Samson blubbered at his side.\n\n\"Yes, some of you would like that very much, but others would prefer to persist, even though they might say otherwise. I can sense it throughout this room. Normally it doesn't matter, I would take each and every one of you all the same, but you don't die. If I left, you would simply keep ticking away until the end of time. That is where you're different, and that's why I will give you the chance to decide. Come or stay, live or die.\" He spun around again to face Lily and me. \"The same goes for each of you. Which will it be?\"\n\nLily Darrow looked at her children, and at her husband. Tears began to stream down her cheeks. \"I think I've kept this gentleman waiting long enough.\"\n\n\"This is outrageous!\" bellowed Whatley, taking several steps forward until the man in black raised his hand with unveiled antipathy toward the master of Darkling.\n\n\"Do not interrupt us again, sir, or I shall be encouraged to take you instead. Do I make myself clear?\"\n\nMr. Whatley fumed and glared, but remained silent.\n\nJames clung to his mother's side, the adolescent confidence he had earned temporarily forgotten in the wake of his mother's decision. \"No, Mother, you can't!\" he cried.\n\n\"I must accept my own death if any of you is to ever live your own lives. I'm sorry if I've been selfish, but I love you so much I couldn't bear leaving you behind.\" She hugged the boys. Henry caressed the side of his wife's face.\n\n\"I'm sorry, Henry.\"\n\n\"Never be sorry.\"\n\n\"Do you still love me?\"\n\n\"Until the end of time.\"\n\nHe kissed her softly on the cheek. I felt my heart pounding in my chest, but then it could have been due to the loss of blood. As they parted, she wiped her eyes. \"Charlotte, you will see to it that they get home to Everton?\"\n\n\"Yes, of course.\" The pain was settling into my body now, not softened but endured.\n\n\"Thank you . . . for everything,\" she said as she took her place beside the gentleman who was Death.\n\n\"I'll return in a moment,\" said the man. \"I imagine that you wish to be next?\" He gestured to Mr. Samson, who nodded excitedly, nearly beside himself with joy.\n\nSuddenly Paul stepped forward, pushing Dabney's wheelchair in front of him. \"If you please, sir. The injured should be taken first.\" Paul placed a hand on Dabney's ruined shoulders, and the other boy nestled his head against Paul's arm with unspoken intimacy.\n\nThe man in black nodded in agreement. \"An admirable observation.\" He spun around and gestured to the broader crowd. \"Would all interested parties please line up? I do love a good queue.\"\n\nPaul wheeled Dabney toward the door made of night and knelt beside him. I could not hear what they said, but by the end of it they were both crying, and Dabney watched Paul return to us as his mother accepted the hand of Death.\n\n\"Are you ready?\" he asked Lily.\n\n\"No, but I imagine few people ever are,\" she replied.\n\nTogether they passed through the door made of night and were engulfed by it, their forms obscured and faded in a dim burst of light even as the door persisted.\n\nMr. Whatley shrieked and collapsed to the floor in visible pain. Olivia ran to his side and took his arm to help him to his feet as the crowd began to murmur with excitement, some of the guests lining up beside Dabney and Mr. Samson, hand in hand, to follow the man in black into the afterlife.\n\nI could see Mr. Whatley staggering up, somehow diminished in Lily's absence. He met my gaze and cackled with manic abandon, his body shaking with the timbre of his voice. \"You warned me, but I didn't believe you. You threatened, and I ignored you. You've stolen my wife from me, Mrs. Markham!\"\n\n\"Father, please!\" Olivia had not released his arm. She held on to him very tightly, her fingers digging sharply into the fabric of his suit jacket.\n\n\"Boys, it's time to go. Help me up.\" I put my arms around Henry's shoulders and fastened a piece of cloth around my side to lessen the bleeding.\n\n\"You're simply leaving, just like that?\" spat Mr. Whatley at his guests as they stood waiting for Death to return. \"What do you think Ashby and Cornelius will do when they find out what's happened?\"\n\n\"I don't suppose I'll be inclined to care by that point,\" said Mr. Samson. \"There needn't be a war at all. We could simply die.\"\n\nWhatley tried to escape from Olivia's grasp, but she held herself tight against him until he reached down with both hands and released her fingers from his arm, then pushed through the crowd as Henry, the children, and I left the ballroom, a thin trail of blood marking our path.\n\n\"Markham!\" he bellowed after us.\n\nJames looked up in fear. \"Where are we going?\"\n\n\"Quickly, to the library!\" I said. We turned down a corridor, but I stopped in shock. Before us was one of the blurry figures that had been standing over me in the ballroom, made all the more clear as I continued to hemorrhage. It was my mother, wrapped in bedclothes, with dried, bloodied mucus crusted beneath her chin.\n\n\"Mother?\"\n\n\"It's time to rest, my darling.\" She smiled and held her arms apart to embrace me. But I don't believe that anyone else saw her, for Henry half carried me away as the boys led us into the library.\n\n\"Stay with us, Charlotte!\" he cried. We threw ourselves into the room and bolted the door shut. I wanted to go back to my mother, but I was becoming more and more confused. I tried to focus on the task at hand\u2014I had to save the Darrows. I had to save myself.\n\n\"The end is nigh, my peppercorn.\" I saw my father in the green leather armchair that had been Lily's favorite. He had his pipe in his hand and a halo of smoke encircling his head. I wanted to run to him, to drop into his lap, to cry into his shoulder, to have him kiss away the pain in my chest, but instead Henry drove us onward.\n\n\"Up the stairs, into the study!\" I could barely speak, focusing all my energy on each step as I leaned against Paul and Henry. The pain in my side throbbed with each beat of my heart. I wondered briefly what would happen when I had no more blood to spill.\n\nAs we rounded the third floor of the library, Mr. Whatley banged roughly on the door, then ripped it off its hinges.\n\n\"More games? How delightful! Shall I come after you then?\" His body shuddered and strained against his suit, shredding it and the fa\u00e7ade of human skin beneath it as his voluminous tendrils and appendages released themselves from the confines of human clothing. He stretched and threw himself against the wall, using his many limbs to climb each bookshelf as if it were a step.\n\nI urged the Darrows to quicken their pace. \"Hurry, we're almost there!\"\n\nAs we reached the door to Whatley's study, I extracted the silver skeleton key Lily had given to us and inserted it into the keyhole. It clicked as I turned it, and the door opened just as Whatley reached the footbridge. The Darrows and I entered the room and slammed the door shut before Mr. Whatley could reach us.\n\nThe room was the same as ever, quiet and gloomy, like a mausoleum. We lurched past Mr. Whatley's emotions and to his collection of faintly glowing glass paintings. I directed the boys to the glass prominently displaying the smoking remains of Everton and kissed them both on the cheek.\n\n\"Be strong for me,\" I said to them. James touched the glass with his hand and passed through the other side as if he'd fallen over a short wall. Paul followed after him.\n\n\"After you,\" said Henry.\n\n\"I can't.\"\n\n\"Of course you can.\"\n\n\"Someone has to stay behind to destroy the painting.\"\n\nHenry's eyes went wide, and he ran his hands through his blond hair. \"I can't allow it, Charlotte. I've already lost Lily.\"\n\n\"Your children need you, Henry.\"\n\n\"And I need you!\"\n\n\"But you can't have me.\" I moved my hand away from my chest. The bleeding had stopped, and I no longer felt as weak as I had before.\n\n\"I can't do this again, Charlotte.\"\n\n\"You can and you will.\"\n\n\"We can have a life together!\"\n\n\"If there's a way for me to come back, I will,\" I promised.\n\nWhatley broke through the door. \"Markham!\"\n\n\"Good-bye, Henry.\" I pushed him hard, and he fell backward through the painting. I could see the children pick him up on the other side as he stood, bewildered and heartbroken, crying. I tore the thing from the wall and smashed it into hundreds of glittering pieces, severing the connection between Everton and The Ending.\n\n\"I could always create another painting to Blackfield, you know.\" Mr. Whatley observed me from the other side of the room. He had reverted to his human form, but his clothing hung in tatters over his muscular body.\n\n\"But you won't.\"\n\n\"Why's that?\"\n\n\"Because I'm the one that you want.\"\n\n\"You've done well.\" His hair was as wild and untamed as ever, but in his eyes I could see that there was something subdued in him as he walked toward me.\n\n\"Stay where you are.\"\n\nHe stopped. \"And what will you do if I refuse?\"\n\n\"You've seen what I can do,\" I spat.\n\n\"You changed the outcome of the story.\"\n\n\"It's not finished yet.\"\n\n\"True, but there are pieces missing. Or have you put them together? Even as a little girl, you were very clever.\"\n\n\"You know nothing about me.\"\n\n\"That's where you're wrong. I know everything about you. I've watched you for years. You've sensed it, I know you have.\" I thought back to the figure who stood over the bodies of everyone I had ever lost.\n\n\"The man in black . . .\"\n\n\"It became dangerous for me to travel between the worlds myself, and so eventually I had to begin sending Roland. But the deaths of humans were my favorite things to collect. Mortals cling to their endings without even realizing it. They are simple to see, and even easier to predict. Your mother was one of many, but you were the first who tried to attack me, and after I met you I realized that I could not see your death. You are an enigma, Mrs. Markham.\"\n\n\"Do not attempt to justify your failure.\"\n\n\"This is not a justification, it's an explanation. I feel that I owe one to you.\"\n\n\"Do you admit to killing them then?\"\n\n\"Your mother, no. It was a twist of fate that we should meet, but once I had found you I couldn't let you be. I will take responsibility for the deaths of the others. I needed you in the right place for a new game . . . my final game. You and I are more similar than you know. How many of our actions are the results of the things people expect of us rather than the things we want?\"\n\n\"Murderer!\" I tore another picture away from the wall and threw it against the floor as hard as I could. It cracked in half, and Mr. Whatley doubled over in pain.\n\n\"I'm practical,\" he continued after he had composed himself. \"You had to engage with me over the fate of Lily Darrow so that, when you defeated me, no one would doubt that I got what I deserved.\"\n\nI was about to destroy another of the paintings but froze when he said this. \"Why should you want to lose?\"\n\n\"There's a war coming. It's already started, but the two sides are filled with fanatics. One side wants to live forever and subjugate all the worlds, while the other wants to bring about the end of all things. I tried to placate them for as long as I could, but I refuse to commit myself to causes that have no center. I mean to provide a third alternative to Ashby and Speck, but I could never do so publicly. When they both began to suspect my intentions, something had to be done to remove me from the board. It's much easier to start an underground movement when everyone assumes that you're as good as dead.\"\n\n\"My father and husband, Nanny Prum and Mrs. Norman . . . Did you kill Lily as well?\"\n\n\"I might have helped her illness along, yes.\"\n\n\"Over politics?\"\n\n\"This is not simply politics. The existence of the universe hangs in the balance. If either party should win this war, it would mean the end of your world in addition to ours. What are a few people when the alternative is so grave?\"\n\n\"And now you expect me to simply help you disappear.\"\n\n\"You have no choice.\"\n\n\"There is always a choice!\"\n\n\"There is now. So much of your life was decided for you, but the game is ending; you will be free after this one last thing.\"\n\n\"And if I refuse?\"\n\n\"Then they will come for me, lock me away, and everything that's happened will have been for nothing.\"\n\nI screamed in anger and ran into the gallery, using my emotions and the accompanying burst of adrenaline to tear the paintings away from the walls one by one with bloody hands, dropping them to the floor so they shattered against him, cutting into him as he withered and shrank with each act of destruction. I tightened my fingers into fists and realized that they felt different than they had before my brush with Death. They were cold, harder somehow. The pain that had racked my body continued to abate as I settled into this new state of living death.\n\nWhatley cowered on the floor, clutching his face in pain. The human features it had once contained were starting to fade. His hand uncoiled into a grouping of tentacles.\n\n\"Yes, just like that.\" He cackled through his agony. \"What is a collector without his collection?\"\n\nI was torn. I wanted to inflict on him as much anguish as possible, but at the same time that was exactly what he was begging of me. \"Every day, every feeling, each bit of joy or sadness or fear I've ever felt, none of them are anything compared to the hatred I hold for you, that I will wield against you. You may not choose to die today, but someday I will come for you, and I will make you suffer as you made us suffer.\"\n\n\"Give me the time to overthrow Ashby and Speck, and I will willingly put myself before you.\"\n\n\"The Darrows are never to be bothered again.\"\n\n\"You have my word.\"\n\n\"Is that worth anything?\"\n\n\"I wouldn't know. I've never given it before.\" He smiled at me with his crooked smirk.\n\nI could not bear to look at him. I destroyed more of his collection, raining down fragments of colored glass and shards of alabaster until the floor was covered in the stuff and a cloud of destruction hung in the air. Mr. Whatley disappeared piece by piece until he was a shrunken stump of a thing cowering on the floor.\n\nI wondered what to do about the remainder of his collection. There were still the lifeless, doll-like figures trapped in the compartment behind his bedroom. I lurched down to the private alcove where he slept and found the panel he had pressed to open the secret room. All of them were where they had been before, save for Lily. I picked up the one closest to the floor, a young man with ivy instead of hair, and set him on his feet. He immediately came to life and looked at me with confusion. \"Where is Mr. Whatley?\"\n\n\"He's indisposed. You're free,\" I told him.\n\nThe boy became suddenly anxious but then saw my wound and proceeded to help me extricate his brothers and sisters from their perches against the wall. The more dolls we freed, the quicker the process became, until all of them were milling about in Mr. Whatley's room trying to understand what had happened to them and what they would do next. I slipped out in the confusion, averting my eyes from the pathetic state Mr. Whatley found himself in, a wriggling thing on the floor amid shards of alabaster and glass. He looked up at me, his black, reptilian eyes suddenly desperate.\n\n\"I am sorry for what I've wrought upon you,\" he said in a small voice. \"People like us, we are stronger. We must do the things that others cannot.\"\n\n\"No matter the cost?\"\n\n\"In spite of it.\"\n\n\"Good-bye, Mr. Whatley.\"\n\n\"You can't leave me here. There is work to be done! Markham!\"\n\nI left him alone in his study with the former pieces of his collection and made my way slowly and steadily through the house. The wedding guests that had not been taken by Death were still milling about in the ballroom. They had apparently decided to hold the reception regardless of the presence of the bride or the groom. Dabney's wheelchair lay empty in the corner. The pain in my chest began to throb again, and I steadied myself against a wall, nearly toppling over when someone put an arm beneath me and carefully lifted me into the air.\n\nI blacked out from exhaustion, and when I came to I found myself stretched on a metal chair, back in the room with the turning veils. Duncan stood nearby, fussing with a tray of tools on the wheeled table. I cried out in anguish, my wound still smarting. He turned to me, and rather than putting a finger to his lips, he opened his mouth and spoke.\n\n\"You're awake.\" His voice was soft and musical.\n\n\"You can talk?\"\n\n\"A recent development. The servants of Darkling grow into the needs of the house. With Whatley in decline, someone must speak for the estate. My brother was much the same, or so I'm told. I think you knew him.\"\n\n\"Roland.\"\n\n\"I believe he caused you great sorrow, though he was only following Mr. Whatley's instructions. I cannot make right what has already passed, but I can at least give you something for the pain.\"\n\n\"That would be worth more than you know.\"\n\nHe nodded and held a smoking cup to my lips. \"Drink this. It will help.\" It tasted of citrus, and as it passed through my body, it brought with it a cool, soothing sensation.\n\n\"I need one more thing. I won't be but a moment.\" He left me in the room, the veils spinning gently across the walls, hypnotic and serene. I had nearly slipped off to sleep when I felt the presence of someone else in the room. I sat up as best I could, and a man stepped forward.\n\n\"Charlotte?\" His voice was familiar, but the room was so dimly lit I could not make him out until his face was close to my own.\n\n\"Jonathan?\" His body was still blackened from the fire.\n\n\"I'm afraid you've seen better days, my love.\"\n\nI touched his cheek and felt his blistered skin. \"How are you here? You're dead.\"\n\n\"What do you think you are?\" he asked.\n\n\"Nothing can die in The Ending unless it wants to.\"\n\n\"You can't live as you are.\"\n\n\"I miss you.\"\n\n\"Don't change the subject.\"\n\n\"Do you want me to be dead?\"\n\n\"I want you to be comfortable.\"\n\n\"Who are you talking to?\" Duncan had returned, a small velvet jewelry box in his hand.\n\n\"My husband is here,\" I said, looking from one to the other.\n\n\"I've been under the impression that he was deceased.\" Duncan did not or could not see Jonathan, who shrugged his shoulders.\n\n\"That doesn't seem to be stopping him.\"\n\n\"You do not look well.\" Duncan peered at the gash in my chest and placed his hand over it. \"We must tend to your wound immediately.\"\n\n\"Where will that leave me?\"\n\n\"What do you mean?\" he asked.\n\n\"Will I be alive or dead?\"\n\n\"I do not know. I suppose that it cannot be good for you as long as you can see your late husband.\"\n\nJonathan brought his ears close to my lips. \"You must let him, Charlotte. It's not your time. Not yet.\"\n\n\"I miss you so much.\"\n\n\"I'm always with you. Can't you feel me?\"\n\n\"It's not the same.\"\n\n\"We will meet again, at the end.\"\n\n\"You'll be waiting for me?\"\n\n\"Forever and always.\"\n\n\"This will feel peculiar,\" said Duncan, interrupting my good-bye. He opened the jewelry box and extracted a small hooked needle attached to a golden spool of thread. He placed the needle into my wound and backed away, the thing moving of its own accord, tugging at the severed strands of muscle and artery in my torso, stopping the tepid flow of blood and leaving me with a mildly sore sensation where there had once been excruciating pain. When it was done he plucked the needle from my skin and set it back on the table.\n\n\"How do you feel?\"\n\n\"Like the living dead.\"\n\n\"At least you are living.\"\n\nI looked around the room. Jonathan was gone, but I felt the loss of him less than I would have in a dream, for he had truly been with me and I had chosen to make him go away. The pain of it was softer because of this.\n\n\"That was quite a wedding,\" said Duncan. \"Or at least it would have been.\"\n\n\"I'm afraid that Mr. Whatley may be indisposed for some time.\"\n\n\"It wouldn't be wise for you to stay here,\" he said.\n\n\"Yes, I know.\"\n\n\"Where will you go?\"\n\n\"Back to Everton, of course.\"\n\n\"But how? It seems as though you've destroyed every way back, and Mr. Whatley is in no position to help you.\"\n\n\"Perhaps someone in the underground will know the way.\"\n\n\"You should be careful. You've brought Death to The Ending. Ashby will come for you.\" Duncan pulled a fresh cloak and a plain dress similar to the kind I had seen worn by the servants of Darkling from beneath the table. He helped me change into them.\n\n\"What will happen here?\" I asked.\n\n\"Mr. Whatley will gather his strength for the time being. After that, I do not know. Are you able to walk?\" He helped me out of the chair. It was much easier to stand than before. He led me out of the chamber to the back of the house, where the orchard stood waiting, now empty of the carriages and wedding guests.\n\n\"I wish you the best of luck, Mrs. Markham.\"\n\n\"And I you, Duncan.\"\n\nWe shook hands, and I stepped out into the night air. My body must have been nearly drained of blood, and I had no responsibilities, no companions, and no idea of where to begin my journey home. Yet something else was wrong, something that sat heavily against my breast.\n\nI reached into the dress and extracted my father's pipe, my mother's lock of hair, and Jonathan's wedding ring. I had brought them all this way with me from Everton to The Ending, but I could not recall why I needed them. I knew who they were, what they smelled like, how their laughter sounded, how they smiled. I saw them every night in my dreams, reliving old memories and making new ones that could never have happened. I had seen them, and I would see them again someday. I knelt down to the ground and pushed aside a handful of soil, burying the three articles I had saved from the fire. When I was done, the weight I had felt was lifted.\n\nWith the moon hanging low in the sky, I followed the winding path away from the House of Darkling to the large gate at the edge of the property. There was a man waiting for me, a man dressed all in black.\n\n\"Lovely to see you again, Mrs. Markham.\" The gentleman who was Death tipped his black bowler hat to me.\n\nA man waits for you. He watches you.\n\n\"You've come back,\" I said.\n\n\"I've never been to The Ending. Much to do and see, I'd imagine. And people in need of my services.\"\n\n\"Some of them might not be so glad to see you.\"\n\n\"Few are. And where will you go?\"\n\n\"Home.\"\n\n\"What kind of gentleman would I be if I did not offer to escort a lady home on a dark moonlit night? Perhaps we should walk together. We both seem to make friends wherever we go,\" he said with a measure of playful sarcasm, extending me the crook of his arm.\n\nI accepted it. \"Am I dead?\" I asked him.\n\n\"I'm not quite sure. This is new for me as well. A road untraveled. Shall we make it up as we go along?\" He pushed the gate open.\n\nI thought of Henry and the children, and oddly enough, of Mr. Whatley. The game had ended, and my life was now my own. There were no more rules to adhere to, no shadowy figures taking the lives of my loved ones; the things that defined me had been stripped away, leaving behind not the person I was but the one I could become.\n\nI took a deep breath and crossed the threshold into The Ending arm in arm with Death, the one constant of my past and my future; for whatever came next, all roads would only, could only lead back to him.\nAcknowledgments\n\nI find myself growing sentimental as I prepare to send Charlotte out into the world, and this morning I went back to read through a bit of the first completed draft of the novel, dated February 9, 2009. To say that it was a different book is an understatement, and I am indebted to a number of people for helping me discover the right voice (both mine and Charlotte's) over the course of the past three years.\n\nDanielle Taylor was the first person to ever read the manuscript, and gave me the confidence that I had some idea of what I was getting myself into, along with Laura Stephenson, Sara Stephenson, William Couch, Katherine McKee, and my stepmother and father, who all provided me with invaluable feedback and asked the right questions.\n\nRakesh Satyal, my proverbial fairy godfather, brought the book to HarperCollins. Without him this novel would not exist in its current form.\n\nThe fearless editing duo of Maya Ziv and Chelsey Emmelhainz coaxed Charlotte out of her shell, and helped me see the light so many times.\n\nAmanda Goldman and Reece Runnells brought my website to life so flawlessly that they made everything look easy.\n\nAnd finally, there is my agent, Sandy Lu, who said \"yes\" and loves this novel just as much as I do. Her enthusiasm and guidance have been more important to me than anything throughout this entire process. I am lucky and grateful to have her by my side.\n\nI will happily ply any of the above individuals with free alcohol whenever they so desire.\n[P.S.  \nInsights, Interviews & More . . .](Contents.xhtml#pspart_01)\nAbout the author\n\nMeet Michael Boccacino\n\nBORN IN UPSTATE NEW YORK and raised in central Florida, Michael blames his love of books on his father, who began reading him the Lord of the Rings trilogy when he was six, but stopped when he found out his son had snuck a VHS copy of the animated adaptation because he couldn't wait to see how it ended. Eventually Michael learned enough patience to earn his BA in Creative Writing from the University of Central Florida and his MBA from Rollins College. In addition to writing, Michael likes to travel to far-off places and pretend that he's Indiana Jones, or experiment in the kitchen and convince others that he might not do so terribly on Iron Chef. He was quite possibly British in a past life, but he lives in and loves New York City.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.\nAbout the book\n\nBehind the Book\n\nCHARLOTTE MARKHAM began life, quite literally, as a dream. The setup itself was fairly simple: an English governess stood on the side of a dirt road with her two young charges as they consulted a homemade map. They were debating whether or not to enter a forest up ahead and I knew, as dreamers often do, that something terrible awaited them in the woods. I remember being fascinated by where they were going, and the next morning I wrote the first draft of what would become Chapter 4. I filed the scene away, but it didn't really crystallize until my mother died of cancer at the age of forty-four.\n\nI was the only member of my family not at her side when she passed away, mainly for the reasons given by Paul over the course of my novel. Her body was cremated before I could get back home, and so one day she was alive, and the next she was simply gone. I'm not sure I ever completely came to terms with her death, and a lack of closure can play tricks on the mind.\n\nI dreamt of her nearly every night for the next few years, and though we both acknowledged in those dreams that she was supposed to be dead, we continued to have some semblance of a relationship. Sometimes she told me she had eluded Death, that her illness was a mistake, a simple misunderstanding. Or we would fight about old arguments with such ferocity that I would wake up shaking. Occasionally we just sat in a room and talked about our problems. Reconnecting with her in this way fascinated me, and I realized that the children from my previous dream were looking for a way to reunite with the mother they had lost.\n\nThough it seems hard to believe now, it took me a few drafts to realize that I was writing about myself every time I mentioned the Darrows. I knew that I was writing in response to my mother's death, but the deeper I got into my revisions, the more I came to understand that I didn't know why I was writing. I was searching for something: a reason why I continued to haunt myself with with my mother's memory.\n\nTwo months before her death, she wrote letters to each member of our family to say the things she couldn't verbalize without bursting into tears. I read my letter exactly once in the weeks leading up to her death, when she was too sick to see or speak and the disease peeled away the last pieces that made up the woman we loved. I didn't think about her writings again until this past December, as I struggled to complete my final revisions to the novel.\n\nI was having trouble finding the emotional center of the story, and as I sat in my dad's house during the Christmas holiday, staring at the manuscript on my laptop, I asked him about the letters. He looked far away for a moment, but nodded and said he would figure out where he had put them. A few days went by, and I asked him again. He promised that he would get around to it before I left to go back to New York. He just had to dig them out of his closet, where they sat in a lockbox, untouched and unread since before their author had died. The day I left, he woke me up and handed me a small, nondescript black notebook.\n\nAfter years of dreaming up imaginary conversations, I held her words in my hands. To say that I was excited sounds morbid, but it would also be an understatement. I'd read my letter once, years before, but I was so distraught at the time that I could hardly remember what she'd written to me. It could have been anything and everything: my relationship with my mother, boiled down, isolated, extracted, and scrawled in her own hand over the course of less than three hundred words; a reason for the dreams; an explanation for the book. I read what she had written:\n\nTo my son Michael,\n\nI just want to let you know how much I love you and how proud I am of you. You are my little boy, the apple of my eye, always smiling. I remember you teething and drooling on my shoulder. I just loved it until you bit me hard . . .\n\nYou are such a creative person. All the leads in plays, etc. Your writing, even though you haven't let me read anything. Your teachers always said how creative you are.\n\nI think you will go very far. You are handsome (beautiful), talented, smart, and very motivated. You are such a joy and the best little boy. You loved to cuddle up with me to watch TV. You loved to read with your dad at night. When you went to college it was the hardest. I couldn't even say your name without crying. Now you are 22 and I know you won't be living at home again and that's ok. I still get sad when you leave.\n\nDad and I wish we could of helped you more with school. That upsets me. But we didn't have the money; we just didn't plan well. I'm sorry. But having you work and go to school has made you independent and that's a good thing. That's what my parents did for me. I think you are a very well adjusted person.\n\nI love you with all my heart and always will. Think of me with happy thoughts. Be happy in your life. Think of all the fun things we have done with our family. You are my Bud. You are my son and I love you.\n\n\u2014Mom\n\nI surprised myself by smiling instead of crying. It was her voice, real in a way my dreams had never been able to capture, and yet in her last memories I am forever twenty-two, still angry about not getting to go to the expensive private university that would have crushed me beneath a ridiculous amount of debt. People often say that loved ones stay with you after they die, but in a way, reading my mother's letter made me realize that while the departed might stay with you, there's no way for you to stay with them.\n\nA month later I saw my dad again at a cousin's wedding. We were having some wine, and I thanked him for finding the letters. I knew it was painful for him, just as he knew it was important for me to find some way to reconnect with her. He sighed and became visibly uncomfortable before admitting that there was something he had kept from me and my sisters. My mother had a tape recorder with her in the weeks before she died. The doctors thought it would be good for her to record her thoughts. No one else knew about it and he was never able to bring himself to listen to it. It might have been blank, or she might have talked for hours. He was sorry he had never told me, and I put my arm around his shoulder. We said nothing more about the tapes.\n\nTwo weeks later, a package arrived at my apartment containing a silver handheld tape recorder. I wrenched it out of the box, popped in some batteries, and placed it gingerly on the table in my living room. I sat cross-legged on my couch and pressed the play button.\n\n\"Hi family. It's on Thursday around eleven, and I want to tell you about my day . . . I'm much better,\" her voice cracked, and the next few words were unintelligible as she fought back tears. She sounded groggy, and her voice was higher than I remembered. She talked about the people who had visited her, what she had to eat that day, and where she was going, before she settled on addressing each member of our family individually, as she had done in her letters.\n\n\"Michael, you do well at anything you touch. You're a very brilliant boy. You're a procrastinator, which you shouldn't be, but you're really a very talented boy; man, I should say. Keep it up and not be such a procrastinator, Bud. You need to get things done and not wait. Like with summer school now, you're struggling, 'cause I'm sick and you have to go to school and work. It's tough for you and I know that. I'm sorry we couldn't help you with school, but . . . that's how it goes sometimes. And we apologize for that. We weren't financially able to. But, anyway. You'll do well in your business degree. I hope you get accepted to Columbia, that'd be awesome.\"\n\nI'd forgotten that I was even going to apply to Columbia; another reminder that this was an echo, both of her and the person I was. She went on.\n\n\"My lunch just came in, so I'm going to take a few bites, and I'll be back. I love you Michael, Stephanie, Lauren, and my honey. Hopefully I'll be coming home tomorrow, and I think I'll feel a lot better being home. Bye.\"\n\nSilence. I let the tape play on, trying to will her voice back into existence, but that was where it ended. She spoke for just over eight minutes. I listened to it again, and again. These were the sad, frightened recollections of a person faced with her own mortality; she didn't offer any revelations about herself. I know no more about my mother today than I did the day she died, and perhaps I never will. But that, I came to realize, is beside the point.\n\nI didn't so much write the novel to her, but in response to what happened to her and our family. Now that it's finished, the book will never change, just as the voice on the tape will always apologize for the things that upset her in the weeks leading up to her death. Like the letter and her recording, Charlotte Markham is a portrait of a moment in time; the ghost of an emotion to match the one she leaves in my dreams. The two have each other now, and nothing can ever take that away. Not even death.\n\n\u2014Michael Boccacino  \nMarch 14, 2012\nRead on\n\nHave You Read?\n\nTHE FOLLOWING BOOKS were helpful and\/or inspirational in the writing of this book.\n\nFor Gothic angst: Jane Eyre, by Charlotte Bront\u00eb, and Wuthering Heights, by Emily Bront\u00eb\n\nFor other problematic governesses: The Turn of the Screw, by Henry James\n\nFor an examination of mortality: Never Let Me Go, by Kazuo Ishiguro\n\nFor repressed, unrequited love: The Remains of the Day, by Kazuo Ishiguro\n\nFor mommy issues and creepiness: Coraline, by Neil Gaiman\n\nFor one of the best Victorian\/ fantasy mash-ups ever written: Jonathan Strange and Mr Norrell, by Susanna Clarke\n\nFor the Old Ones: Black Seas of Infinity, by H. P. Lovecraft, selected by Andrew Wheeler\n\nFor the best in new \"weird fiction\": Perdido Street Station, by China Mi\u00e9ville\n\nFor luck: The Harry Potter series, by J. K. Rowling\nAdvance Praise for Charlotte Markham and the House of Darkling\n\n\"Thanks to Michael Boccacino the Gothic is reborn! Charlotte Markham and the House of Darkling is an elegant, intelligent, and compelling debut novel. Bravo!\"\n\n\u2014Jonathan Maberry, New York Times bestselling  \nauthor of Assassin's Code and Dust & Decay\n\n\"In Charlotte Markham and the House of Darkling, Michael Boccacino has delivered a studied, enchanting, and most welcome contribution to the Gothic literary landscape rolling back to Bront\u00eb and du Maurier. A convincing portrait of a woman enveloped in slow-mounting terror, rich in atmosphere and carried by writing that soars above that of most debut novels, this is not one to miss.\"\n\n\u2014Christopher Ransom, internationally bestselling  \nauthor of The Birthing House and The Fading\n\n\"With Charlotte Markham and the House of Darkling, Boccacino has created a new vision of the Afterlife, at one moment stunningly beautiful and full of wonder, the next, darkly sinister and without pity. A remarkable book. Michael Boccacino is a writer to watch.\"\n\n\u2014Susie Moloney, author of The Dwelling and The Thirteen\nCredits\n\nCover design by Mumtaz Mustafa\n\nCover photograph \u00a9 by Marcus Appelt\/Arcangel Images\nCopyright\n\nThis book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.\n\nP.S.\u2122 is a registered trademark of HarperCollins Publishers.\n\nCHARLOTTE MARKHAM AND THE HOUSE OF DARKLING. Copyright \u00a9 2012 by Michael Boccacino. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nFIRST EDITION\n\nISBN 978-0-06-212261-2\n\nEpub Edition \u00a9 AUGUST 2012 ISBN: 9780062122629\n\n12 13 14 15 16 OV\/RRD 10 9 8 7 6 5 4 3 2 1\nAbout the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\nhttp:\/\/www.harpercollins.com.au\n\nCanada\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n<http:\/\/www.harpercollins.ca>\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n<http:\/\/www.harpercollins.co.nz>\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n<http:\/\/www.harpercollins.co.uk>\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n<http:\/\/www.harpercollins.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Chapter 6 Copper Production Technology ... mixture of copper ore, charcoal and iron ore to act as a flux. It was blown for several hours by foot or hand bellows.\nMitsui's investments in Chile's copper industry date back to the 1990s and it now has minority interests in a number of the country's leading mines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apparently human beings are easily fooled by coincidence \u2013 we are tuned for life in a Stone Age village, where not that much happened.\nLady Luck is our constant companion, as we try to fit random events into a sensible narrative of what the blazes is happening to us. Studies suggest the gods (although not God) first sprang from here.\nWas it me, others, chance, caprice \u2013 or the hand of gods or God? Whatever and wherever you ascribe the credit or blame, like the handsome black cat who once stalked me for a fortnight, I have landed firmly on all four paws this week.\nInteresting thoughts. Love the cat!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"February 8, 2009 \/ Mark Forstrom \/ Comments Off on My Philosophy on Dating.\nBeing a pastor on salary with no contained work hours, I have always struggled with finding the balance between church and home. There are always 100 legitimate spiritual needs out there that I could be\u2013and perhaps should be\u2013 attending to. I'm around a lot of teenagers and their parents, many who are lonely, lost, confused, hurting, neglected, needing encouragement, needing cheerleading, needing something.\nSo how do I find the balance so that in my feeble attempts to meet the needs of others my own kids don't end up lonely, lost, confused, hurting, neglected etc. ?\nWhile I don't at all claim to have properly figured out the perfect balance, one thing that has helped us immensely is something Cindy and I started doing when the children were small. We decided that I would take each of the girls out on a \"Daddy Date\" at least once a month. I will say that it's been one of our best parenting decisions.\nI tell my youth sponsors that their proximity to the teenagers will have a direct bearing on their impact. The same is true with these Daddy Dates, and gets us face to face with our kids and allows opportunity for connection to occur. It's during these times that everything else gets set aside and I can focus all my attention and love on my kids. I remember a phrase I've always loved, \"Quality Time is an accident that happens during Quantity Time.\" I've sure found that to be true.\nTo foster these Daddy Dates, we started budgeting $20 cash [for each of the girls] in envelopes at the beginning of each month for us to use. The $20 a month does a couple of things: It frees us up to do fun things like dinner and a movie, or miniature golf or bowling. It also provides a gauge to see if we're skipping our dates: if the money is accumulating, then it's obvious I've been a negligent dad recently. And it's a way to make up for my negligence\u2013if we realize we've skipped a month, we now have $40 to spend or once in a great while $60. It's also a way to ensure that I treat both girls fairly, since each one has the same amount of resources to use.\nWith 6 year old Lexi, spending $10 of the dollars at Chuck-E-Cheeses on Ski Ball, cashing out the earned tickets for a cheap trinket, then driving straight to Wal-Mart and spending the remaining $10 on whatever she wanted\u2013she chose a really nice stuffed animal. (By the way, that was our last time ever visiting Chuck-E-Cheese's).\nUsing our $20 to buy activities: kites, interactive games, model rockets, puzzles, etc.\nPlaying tag in the main aisle of Lindale mall with 8 year old Lexi, trying to only hop on the dark tiles, until we got to Victoria's Secret, whereby Lexi ran over to the window threw up her arms in front of the display and yelled loudly, \"Don't look over here, Daddy.\" She knew my eyes were to be Cindy's only! Priceless!\nReading the Narnia books with Lexi at Coffee Smiths.\nCatching the midnight shows with Brenda for all three Lord of the Rings movies.\nOne day Brenda and I randomly bought several sets of little plastic green army guys and made an entire battlefield on our dining room table, with mountains and valleys, just to see Cindy and Lexi's faces when they came home that night.\nGoing out to a restaurant (which our family rarely does, except for dates).\nRenting a clean movie and fixing microwave popcorn.\nSo that's our plan. The girls thought it was a good idea! It's going to cost us $240 a year, but it seems to me to be a small price to pay for what I know will be great relational returns. It'll be a lot of bang for the buck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the latest attempt to conceal evidence of the magnetic presence of Planet X in the inner solar system, it appears the cover-up has resorted to manipulating BATSRUS modeling parameters.\nFormerly providing sequential depictions of Earth's magnetic field every 4 minutes, BATSRUS Y-Cut images on iSWA are now diluted with radically incongruous representations of the particle dynamics affecting Earth's magnetosphere. Since May 9th, BATSRUS has been showing high density particle blasts and calm space weather conditions occurring nearly simultaneously.\nAnd with the addition of images absent of any magnetic field lines but containing the error message: \"Selected variable 'N' not a factor\", it appears someone is tinkering with the magnetospheric plasma output parameter N for atomic mass unit density in their redefinition of the BATSRUS model.\na. This first image was recorded at 07:48:13 UTC on May 9, 2013. Note the high density particle assault coinciding with magnetospheric compression and eddy flow formation behind Earth.\nb. This second image is presented next in the BATSRUS feed, with a time stamp of 07:48:25 UTC - only a fraction of a second after the first image, yet Earth's magnetic field could not possibly have undergone such a broad and nearly instantaneous transition.\nc. The third image in the BATSRUS feed, recorded 4 minutes after the first image, is clearly a continuation of the magnetic onslaught originally depicted.\nd. The following images are presented next on BATSRUS. Note the time stamps and stark contrasts, as if BATSRUS is alternating between two entirely different yet nearly simultaneous data feeds, with an error message suggesting the parameter adjustments are not yet fully debugged.\nThis highly irregular encryption of the BATSRUS image feed has been ongoing since May 9th. The following animations were developed by omitting any suspect images from the data feed.\nGiven the voluminous evidence BATSRUS has provided attesting to the nearby presence of Planet X, altering the model code such that critical parameters are adjusted when they exceed a defined threshold would not be a surprising move by the cover-up. By muddling the image feed with conflicting magnetic field data, this tool for visualizing Earth's magnetic field is essentially being sabotaged. Such a grave defect may ultimately be used as rationale for its termination.\nand within hours after Kojima posted Earth's wildly deformed magnetic field likely resulting from the charged tail of Planet X.\nA BATSRUS with RCM rendering shortly before it was apparently removed from iSWA.\nThe BATSRUS N-S Cut mag image feed that utilized Rice Convection Model (RCM) enhancements (which portrayed magneton eddy flows significantly more accurately than the standard Y-Cut version) has been apparently removed from iSWA.\nConsidering that all BATSRUS magnetic field renderings have been suspended since March 13 on CCMC and since April 1 on iSWA, it appears BATSRUS has suffered the same fate as the NICT magnetosphere simulator.\nApril 20 Update: BATSRUS magnetic field image feed resumed today on iSWA today after a 3-week hiatus that omits all data between April 1 and April 7.\nToday's severe compression of the magnetosphere endured for approximately six hours, yet it does not appear on BATSRUS, thus providing further evidence of model parameter tampering.\nAnimation of today's severe compression with the \"modified\" BATSRUS.\nFor comparison, here's animation of a severe compression with the former BATSRUS model.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This newly-built holiday village is in the perfect location, conveniently close to Vienna's main tourist attraction, Schloss Sch\u00f6nbrunn. It is easy to access both the A1 and A2 motorways. From the underground station next to the hotel, the city centre can be reached in approximately 10 minutes. There, history, tradition and the commercial activity of a world capital await guests. There are restaurants, bars and nightclubs to be found in the immediate vicinity of the hotel, whilst the railway station is located some 2 km from the hotel. The airport is around 24 km away.||This city hotel, with its contemporary design and excellent service, comprises a total of 118 rooms spread over six floors. Amongst the facilities on offer are a foyer (opening onto a main street) with 24-hour reception, a hotel safe, currency exchange facility, 2 lifts, as well as a restaurant and a lobby bar. In the 24-hour \\Grab & Go Shop\\ The Market, guests will find a selection of reasonably priced drinks and snacks. Numbered amongst the hotel's other facilities are conference facilities and Internet access with a WLAN access point. A laundry service and a 24-hour garage are also available.odern architecture, high quality fittings and warm colour-schemes, make the rooms comfortable and cosy. Standard fittings include an en suite bathroom with hairdryer, a direct dial telephone, flat-screen TV offering satellite\/ cable channels, a hire safe and a high speed Internet connection. In addition, a large writing desk, an office chair, a mini-fridge, tea and coffee making facilities, iron and ironing board, as well as individually adjustable air conditioning and heating are also provided in all accommodation as standard. The luxurious king-size or double beds guarantee a restful night's sleep.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuheg b/data_all_eng_slimpj/shuffled/split2/finalzzuheg
new file mode 100644
index 0000000000000000000000000000000000000000..568cb19eac9af66155c41d07e8fe4b5009ed1b31
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Premium brush made of \"vegan sable\" hair ensures eye shadows are ap..\nDual-tip eyelash separator and brow groomer; made of soft synthetic bri..\nFor contouring, and application of blush and powder. Made of pony h..\nFor loose or compact powder application. Made of pony hair. ..\nFor applying lip glosses and lipstick. Made of pony hair. ..\nMedium tip for application of shadow to eyelids and to brow bones a..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the 16th Century, a Dutch ship sails the Pacific from the Americas in order to escape the Spanish. Blackthorn is the pilot of the ship and the first to realize the possibilities of Japan. Japan is a feudal society which has a strong presence of the Portugese. A struggle is in the final stages for who will be the Shogun, the leader who, under the emperor, governs the land. Toranaga sees in Blackthorn things he can use to at least survive the struggle to come, and if both play their hands correctly, become Shogun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Employment Testing Program Manager and Lead Training is going to be between 02 Mar and 04 Mar 2016.\nEmployment Testing Program Manager and Lead Training will take place at the Hyatt Regency Atlanta in Atlanta, Georgia USA.\nEEI Employment Testing Program 2016, Employment Testing Program Manager and Lead Training is one of the substantial event on Leadership, Training, Program Management and Employment Practices practices.\nThe association of the Employment Testing Program Manager and Lead Training is Edison Electric Institute.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This bundle has a simple approach to settings management, everything is centralized in json files. Each engine must provide a SettingsManager class that can backup settings from the engine and push them back.\nThe --indices option take a comma-separated list of index names (without prefix, as defined in configuration). If no options are passed all indices will be processed.\nThe settings directory can also be set in the configuration if you have a non-standard setup or if you wish to save them elsewhere. The project directory will automatically be prepended.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jabra EVOLVE 75 Stereo UC Headset (With Charging Stand) delivers world-class speakers and HD Voice technology that provides crystal clear calls and music. It is recognised to be the best wireless headset considering how it is designed to encourage focus and concentration in open offices and workspaces. It even featyres user-friendly audio control located on the headset.\nOne of the most noteworthy characteristics that Jabra EVOLVE 75 Stereo UC Headset (With Charging Stand) has is its capability to promot better productivity through reduced noises and interruptions. The headset enhances productivity through the use of its superior Active Noise Cancellation feature, shutting off office noises. This feature uses advance microphones, the Hybrid 4 Mic ensuring 17dB (perceived cancellation in an office environment), to monitor and counter ambient noise and even eliminate low-frequency sounds, including hums of air conditioning. The headset is even integrated with a medium-length boom arm with nosie cancelling uni-directional ECM microphone.\nJabra EVOLVE 75 Stereo UC Headset (With Charging Stand) reduces interruptions through the integrated busylight and the listen-in feature. This busylight acts as a \"Do Not Disturb\" sign that avoids interruptions and in the process enhances productivity. The headset is integrated with red lights on both ear cups that can be activated both automatically and manually, depending ont he user's preference.This is also supported by its listen-in feature that can be used by having users press the left ear cup to hear the outside world without having to take the headset off, which ensures that users do not miss out important questions or announcements in the office.\nJabra EVOLVE 75 Stereo UC Headset (With Charging Stand) provides customers with the freedom to move around their workspaces through its Bluetooth range of up to 30 metres. The headset delivers dual Bluetooth connectivity to two devices simultaneously that can be used for calls and music.\nOffering users with all-day comfort, the Jabra EVOLVE 75 Stereo UC Headset (With Charging Stand) is integrated with an on-the-ear design and leather-feel ear cushions.\nJabra EVOLVE 75 Stereo UC Headset (With Charging Stand) is highly compatible with wide variety of platforms. With this headset working with all the most popular online voice call services, it can also work excellently together with Skype for Business, GoToMeeting, Cisco WebEx, Google, ChromeBox, Fuze, ClickMeeting, JoinMe, Adobe Connect, eVoice, Onstream Meetings, and StartMeeting.\nFounded in January 3, 1993, Jabra Corporation is one of the leading suppliers of hands-free communications products and solutions in the global market. The company has around 1000 employees worldwide; supporting the company's manufacturing floors, sales offices, and branches. A subsidiary of GN Netcom, it takes pride in its offers of broad range of corded and wireless headsets perfect for mobile uers and contact centre agents. GN Netcom is involved in the manufacturing of original equipment for global customers, including PC, PDA, and mobile phone manufacturers.\nCutting-edge technology adn incredible effort are only few of the benefits that Jabra provides its customers, which is how its headsets are able to cater to the needs of contact centres worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzujwa b/data_all_eng_slimpj/shuffled/split2/finalzzujwa
new file mode 100644
index 0000000000000000000000000000000000000000..6b40e005937e4e6253b3c726fd3eea4266b0b6f3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzujwa
@@ -0,0 +1,5 @@
+{"text":"Leggings are the perfect versatile item for your wardrobe. Wear them to the gym, to lounge in or dress them up for work and the holidays. Here are this year's legging outfits and outfit inspiration to create the perfect look.\nCan I style leggings without looking like I hit the gym?\nI'm 50 years old, can I wear leggings?\nWhat do I wear with capri leggings?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The X3M Campus 36 is great starter stick for elementary, middle and high schools. It has a straight durable blade that can be curved left or right, with excellent playing characteristics. The 2013 model has a new, more durable blade\/shaft attachment. 75 cm for Grades 1-5, 85 cm for Grades 6-8, 95 cm for High School & Adult.\nThis is a great stick choice for a limited budget and is durable enough to last thru many years of middle school PE use. We have had these sticks for about 5 years and have yet to have one break. We just ordered more of the same to increase our supply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recent ONS figures showing record numbers of over 50s in work should not divert attention away from supporting the millions who are forced to retire early from the booming jobs market, Just Group has warned.\nWith an ageing population and an increase in the State Pension Age, the trend is for people to keep working longer. But research by retirement specialist Just Group reveals the high numbers of those aged 65+ who said they had to retire earlier than they expected, raising concerns about how long many workers can carry on.\nNearly six in 10 who retired between age 50-65 said this was earlier than they had expected to retire and were forced to by events outside their control such as illness (25%), redundancy (21%) or to become a carer for a family member (10%). The results therefore indicate that 3.6 million retirees over 65 retired early due to circumstances out of their control.\nOnly one in four (26%) said they retired early because they had enough pension or savings that they could afford to stop working.\n\"Our research shows that for many the dream of early retirement may actually be more of a nightmare. Illhealth can strike at any time and becomes more likely the older we are, while redundancies are a difficult point in many people's working careers,\" said Stephen Lowe \u2013 group communications director at Just Group.\n\"Being forced to stop work is difficult for anyone but can be especially difficult for those over 50 \u2013 four in 10 fail to find new a job within a year, a far higher ratio than their younger counterparts. Unemployment also means people cannot bank their regular income into a pension and may even be tempted to dip into existing retirement savings.\nJust Group's research also showed that twice as many women (13%) were forced into retirement to care for a family member compared to men (6%).\nWomen were also one third more likely to retire early due to illness while just one in five were able to choose to leave the workforce for positive financial reasons.\n\"The findings highlight the strain that providing care for family members can place upon households who are either entering residential care or require some other form of professional help \u2013 with the result that many people are forced into giving up their own jobs or careers early to provide the necessary support,\" said Stephen Lowe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With kids being a part of our family, home is a blessing but is a great responsibility at the same time. We make sure they have the best of everything and don't want to compromise at any level in their upbringing. Be it the latest gadgets or books, we give them access to the latest so that they get the best resources possible in order to have fruitful and thorough knowledge of the academics. In the midst of all this we tend to keep less room to make home more kid \u2013 friendly, which again is important to make them responsible part of the family. Always remember parents and family are the best teachers and responsible trait is never inherited, we need to make them realize it and value it. After all who wouldn't want their kids to become responsible!! Today we are going to share few tips and ideas for kid friendly home that will certainly help impart the learning of being responsible member of family which in turn will also make them feel little independent.\n1. Play Music \u2013 let's begin with the fun part. We all love listening to music and one tends to know at an early stage if kids are also equally fond of them. Play different kinds of music, sing with kids and encourage them to make music. This fun activity will definitely have a long term impact on your kids as playing instrument boosts learning in children.\n2. The Art of Cleaning \u2013 this part includes 3 B's \u2013 bins, baskets and buckets. Religiously teach them the significance of using these in daily life for a more organized room or study space or recreational space. Let them clean their own mess and allow them to make their territory more organized in their own way. Let them give it a part of their personality using creativity.\n3. Display Their Artwork \u2013 this is most essential part in boosting their confidence. You encourage them to play with arts and crafts; your work just doesn't stop here. Make a space in living room or their room to display their creations; you can dedicate a wall to display their art work for example. Choose pretty frames and make place on shelves or walls to make a beautiful show case of their work. Apart from feeling good about it, kids will strive to make it better and that way you can seasonally change the exhibit of their art work.\n4. Accessibility \u2013 one thing about kids you must have observed is \u2013 they are stubborn. And somehow or anyhow they will get what they want. Instead of observing them jumping all over to get what they are looking for, make things more accessible for them. How about a small stool for them, so that they can easily get access to the toys or books kept high on shelves. Having a step stool in bathroom will also let the little one's admire them while brushing their teeth. Buy removable hooks for kids so that they can hang up their jackets and clothes.\nHope you enjoyed reading the few ideas we shared. We would like to hear more ideas you use to make your home more kid friendly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why is there a resurgence of infidelity among Catholic leaders?\nThe email was so wild that I thought it might be a joke. In response I asked if he were joking, and explained that no amount of \"discernment\" can change the meaning that the Church has always ascribed to Our Lord's words. I also cautioned against making up our own religion, reminding him that if religion is not revealed, it too easily becomes a human fantasy.\nIn response, this correspondent told me to stop being irrational and instead learn something about Scriptural exegesis. Then, apparently having exhausted his argument, he proceeded to call me a number of names. Presumably he thought that was the logical way to approach someone if you believed you had reason on your side.\nThe remarks of Fr. Arturo Sosa Abascal in an interview with a Swiss journalist were, of course, a significant scandal. Among other things, he said that to understand Christ's condemnation of adultery, \"there would have to be a lot of reflection on what Jesus really said.\" Asserting that this discernment is guided by the Holy Spirit, he acknowledged that a discerning soul could arrive at conclusions that legitimately contradict what the Church has taught. In other words, Fr. Sosa Abascal demonstrated his total commitment to theological Modernism, which holds that human culture determines what the truth is for us in every age.\nThe practical result of this is that the Church's moral teachings, rooted so firmly in Scripture, Tradition and the natural law, can be reinterpreted in each age, to ensure that they mean exactly what we want them to mean, neither more nor less.\nThis is the horror of Alice in Wonderland\u2014that is, the horror of Scriptural exegesis in the modern period\u2014something my correspondent very clearly knew nothing about (except as a slogan). For sixty years and more, the bulk of academic theology, especially moral theology, has been in the hands of professors who regard academic expertise as a superior substitute for faith. Far too much of our theology for a very long time has been served up by non-believers, whether members of the Catholic Theological Society or not. They use trumped up rationalist methods of interpretation to prove that the Sacred text means exactly what they have predetermined that it ought to mean. Accordingly, no hard saying ever means what we used to think it did.\nEveryone understands by now that the most prominent weaknesses in our dominant culture are sexual weaknesses. Because college students find themselves with a particularly strong need to justify sinful sexual behavior, a great many students in mainstream Catholic colleges and universities eagerly absorb the message of their professors, choosing to agree that academic authority has proven traditional sexual morality wrong, and so has set them free. But the arguments these wayward theologians use are rarely critiqued. In most Catholic schools (and elsewhere), students do not learn how to do theology and\/or exegesis. They learn to embrace the myth that modern scholarship has proven the Church wrong and themselves right.\nThis is not a joke: It really is how exegesis and theology as practiced by those without faith always works. If you look at their arguments clearly, understanding each step, you will always discern a circle. Theology without faith always proves what the faithless theologian believes to be true, because his ultimate authority is not Christ but modern culture.\nI have often mentioned the supreme example of this, which is to reinterpret the early accounts of the Resurrection as a first-century symbol of new life\u2014in other words, the creation of a myth to explain the fresh joy people at that time found in the teachings of Jesus. But the argument is completely circular, as always. If you start with the rationalist conviction that miracles do not occur, then you know that the truth must really be something other than what the Church has always taught.\nAll of this has been compounded (as was revealed once again recently in the voluntary outing of Gregory Baum) by the remarkable sinfulness of moral theologians. It has been obvious for decades that the trends in moral theology among mainstream scholars have been guided, to put the matter succinctly, by moral turpitude. Catholic moral theologians, including priests, who were homosexuals or having affairs quickly learned how to justify new conclusions about Catholic morality by using \"advanced scholarly techniques\".\nUnfortunately for Fr. Sosa Abascal and others like him, the truly \"discerning\" reader understands that when such theologians claim to be guided by the Holy Spirit, what they really mean is that they are guided by the Zeitgeist\u2014the spirit of the age\u2014which is anything but holy. Once again, this is the language of Modernism, now expressed more openly than ever. The positions Fr. Sosa Abascal took in his recent interview, while they may in some circles appear to be academically astute, are simply proof of the state of the Society of Jesus\u2014which still has a commanding role in Catholic higher education. Such positions disqualify those who hold them from identifying themselves as Catholics, for theology which cannot explain Revelation without explaning it away is utterly worthless.\nUnlike academic theology today, discerning the signs of the times (as Our Lord insisted we do) requires only faith and spiritual understanding. Those who possess both should not play the game of engaging men like Fr. Sosa Abascal in academic debate, as if their mistakes consist in anything other than a prior commitment to conclusions which directly contradict what God has revealed.\nAlthough such faithless attitudes are still widely represented in universities around the world, we should also recognize that a slow, halting reform had been gaining ground until just a few years ago. Modernism was on the wane. The most important Catholic theologians in the first decade of our century have been sound men and women whose work has been reignited by the insights of Karol Wojtyla and Joseph Ratzinger (Pope St. John Paul II and Pope Benedict XVI), especially in the critical areas of moral theology (Wojtyla) and exegesis (Ratzinger).\nYet now, all of a sudden, dissident voices are raised again in the highest places. As a great Yankee catcher famously quipped, it is d\u00e9j\u00e0 vu all over again. We have those like Bishop McElroy in San Diego who are eagerly inviting men and women in every kind of relationship to think things over and decide for themselves whether they should receive Communion, and we have those like the Superior General of the Jesuits insisting that everything, including Scripture, is up for grabs. We cannot avoid asking, then, what has prompted the confidence of such men, who operated with greater discretion or even secrecy just a few years ago. How is it they now possess such an extravagant boldness that they are perfectly comfortable in firing off ill-disguised broadsides against Christ, the Church and the truth?\nWe would be lying if we said we did not know. Let me repeat that: We would be lying if we said we did not know. The intellectual culture in Rome has been deliberately and radically changed. We have a pope who, immediately after La Civilta Catholica reopened the definitively closed question of ordaining women, met with the magazine's staff and told them: \"Stay in open waters! The Catholic must never be afraid of open waters, and must never seek the shelter of safe harbors. You above all, as Jesuits: avoid clinging to certainties and security.\" We have a Pope who never corrects his proponents even when they publicly depart from the clear teachings of the Church. And we have a Pope who continues to lay the blame for everything on those who adhere to Catholic doctrine so strongly \"as written\" that they may, on very rare occasions, frown on a legitimate new insight. He makes a straw man of these \"rigid doctors of the law\" and regularly knocks it down.\nLike the correspondent I mentioned at the outset, Pope Francis also calls them names.\nIt is no wonder, therefore, that new signals from Rome have emboldened the same sorts of people whom his predecessors placed on the defensive, seeking not to encourage but to correct and convert. And yet, in the last analysis, what can we really say about it all?\nWhat I will say is simply this: Pope Francis may be able to accomplish some good through the sincere efforts of faithful Catholics to rid themselves of the faults he unfairly ascribes to them. He may also be able to accomplish some good through the selfless determination of faithful Catholics to interpret his statements in ways that can actually increase the faith of others, to avoid scandal.\nBut any good he accomplishes by thus hammering the faithful will be totally undermined by the far greater numbers of unfaithful Catholics who take the Pope at his word. Today I have in mind particularly those ostensibly Catholic leaders whom Francis consistently refuses to correct, no matter how hard or how directly they strive to undermine our Faith.\nDr. Mirus, I've read many of your essays, and this is one of your best ever.\nI finally get it. I've been wondering for decades why clerics, at all levels, are not scandalized by the absence of Saturday afternoon confession lines. You would think this would cause them to doubt the wisdom of an ecclesial culture that has downplayed sin for half a century, ironically starting at a time just twenty years after the horrors of World War II. What they've managed to do is discern for themselves that sin and self-deception doesn't really exist anymore. What a profound discovery.\nI don't know where you are on Medjugorje but the message coming therefrom today seems particularly appropriate given what you're talking about in your article: \"Dear children! Today I call you to profoundly live your faith and to implore the Most High to strengthen it, so that winds and storms cannot break it. May the roots of your faith be prayer and hope in eternal life...work on yourselves in this time wherein God is giving you the grace...to be people of clear and persevering faith & hope..\nJesus summed up their motivation in few words: \"For where your treasure is, there will your heart be\" (Mt 6:21). Their \"treasure\" is the hubris of being convinced they speak for the Holy Spirit, even as they contradict the Spirit's previous utterances. Their \"heart\" is entranced by an intellectual idol bearing their own face & speaking their own words. Jesus' following words about the eye being the lamp of the body & one's light being very dark if the eye is not sound, applies perfectly to them.\nI think you blew the lid off the situation with this essay. Goodness, what clarity! And yes, goodness does have something to do with it. May God Bless you. I can only thank you.\nWhat more needs to be said? A good summary: \"Pope Francis may be able to accomplish some good through the sincere efforts of faithful Catholics to rid themselves of the faults he unfairly ascribes to them. He may also be able to accomplish some good through the selfless determination of faithful Catholics to interpret his statements in ways that can actually increase the faith of others, to avoid scandal.\" Of course, approaching a problem sideways might accidentally contribute to its resolution.\n\"We would be lying if we said we did not know.\" The Orwellian continues, and Pope Francis is receiving too much criticism. He is doing only what he BELIEVES. \"It's\" always been about what we believe. Canonize the pope who introduced altar \"girls\" to the Chruch but condemn Pope Francis for exploring women ordination? Seriously? Has not something deliberate and plodding simply been catalyzed unsettlingly by seasoned men of conviction? Is it possible the truth has eluded us for some time?\nThank you for your honesty. I, too, have been reminded of the bad old days when I was young & nuns were exchanging their habits for miniskirts & bright-eyed young Catholic couples were agonizing over whether to have kids & \"transubstantiation\" was an outmoded word. Here we go again.\n\"[...] theology which cannot explain Revelation without explaning it away is utterly worthless.\" Amen.\nHave you been reading those \"hard sayings\" again, Dr. Mirrus? Don't you know that they will make you \"rigid\"? And you actually have the nerve to speak clearly and forcefully in black and white rather than gray. I have criticized you in the past on other subjects, so it's only fair to say \"thank you\" for \"tellin it like it is.\" We live in discouraging times. I am reminded of the server's response from Psalm 42: Spera in Deo, quoniam adhuc confitebor tibi, salutare vultus mei et Deus meus.\nAnd as these Church leaders - Bergoglio and others - discard Church teachings the way they do, people in America and Europe continue to leave the Church in droves. Why should they bother remaining in the Church when their conscience is \"at peace\" with their not remaining? Meanwhile, the German bishops and others seem to think they can stem the hemorrhage of people out of the Church by allowing Communion to adulterous couples. Isn't there a name for people who suffer from such delusions?\nThe posters seem to have made the Pontiff and his circle nervous. Maybe we need to see more of them. Also maybe some Romans need to carry tomatoes around with them on Wednesdays to express their sentiments should the opportunity arise.\nAmen Brother!!! Thank God we have been prepared for this by St. John Paul II and Pope Benedict. They were Faithful and True. If someone contradicts the Catechism, St. John Paul II, or Pope Benedict you can bet I will go with these 3 every time. Remember Cardinal Ratzinger's words about a small purer Church? we often talk about our \"cafeteria Catholic\" brethren, what about cafeteria priests, deacons and even bishop?\nAbascal's interview is frightening - especially if his thoughts run at all parallel to Francis'. This kind of \"discernment\" could easily be turned on credal matters far from the sexual revolution. Certainly \"substitution\" or \"atonement\" could be eliminated in favor of \"solidarity\" to explain Christ's mission. The physical reality of the resurrection was up for grabs by many theologians after VII - Haight and Schillebeeckx were only two of many. The list goes on. Pray for the Church.\nA good read for university students - Catholic as well as secular.\nAnother spot-on essay on the perilous condition of the Church today, led to such circumstances by the very shepherds who are supposed to be leading us away from this danger. I pray that Our Lord will use this 100th anniversary of the visits by Our Lady at Fatima to put a stop to this and save us from further decay, but I know that God's ways are not our ways, so not my will, but Thy Will Be Done! I also pray for the many souls who will follow these false shepherds to their common doom.\nYou are so on target with this article. I heard all this same garbage back in 1978 from a nun who was teaching a Scripture class. Same things: we don't know if Jesus really said that, we don't know of those things really happened, Jesus didn't know who He was until after the Resurrection, etc. I was the only one out of 40 people who told her she was teaching heresy. I thought I had lived long enough to get past all of this, but I guess not.\n\"The most important Catholic theologians in the first decade of our century have been sound men and women whose work has been reignited by the insights of ... (Pope St. John Paul II and Pope Benedict XVI), especially in the critical areas of moral theology (Wojtyla) and exegesis (Ratzinger).\" And now it's almost like these two witnesses have been killed and their bodies (legacy) left lying dead in the streets. Let us pray that God will breathe new life into the legacy these men have left us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"For the next few years, Syntec will continue to focus on our core competencies in order to stay competitive and efficient.\nOur aim is to work towards a more sustainable future for all our stakeholders while making sure that the value of our shareholders is maximised.\nIt is crucial for us to make sure that while we aim to create more value for the shareholders, we continue to be responsible towards our society and environment.\nSocial and Environmental Responsibility: operating works with good corporate governance, supporting all employees to perform duties according to the code of conduct and laws especially protecting environment surrounding communities, society, residents in construction areas that could be affected. Creating value for shareholders and delivering an industry-leading total return to the shareholders will continue to be Syntec's long term goal. A strong focus on profitability must then be combined with risk management to make sure that company is qualified to generate a long term growth as well as able to survive and go through a period of an economic downturn.\nOUR People: Continue to focus on training as well as creating a workplace and environment that encourage our people to innovate, collaborate and coordinate with each other.\nOperational Excellence: Continuous improvement in all of our working processes is the key to make sure that we can achieve our long term goals. We continue to highlight on the importance of Quality, Time and Cost control with an increase in digitisation, automation as well as cutting edge technology and equipment.\nOUR Value: We will strive to engage and nurture our long term relationship with all of our partners including our clients, our suppliers and our sub-contractors to make sure that everyone can share the same common vision and trust.\nTo emphasis the management along the high quality, high technology, good service, social responsibility and finished projects on schedule to consolidate the reputation of organize and keep the royalty of customers in order to gain the next engagement and the aim of introduction to other customers.\nTo manage the cost of product to be relevant the vary cost of construction and the economic conditions.\nThe Board of Director has determined \"Business Code of Conduct\" for the directors, executives and staff of the Company and subsidiaries shall apply it as the conduct and practical guideline, leading to reinforcement of corporate culture that adheres to operating ethics.\nThe Company realizes on the significance of management according to the principle of good corporate governance, in compliance with the policy of the Stock Exchange of Thailand. It's regularly reviewed to ensure consistency with the vision, mission, objective and guideline for the Company's operation and in line with rule and regulation determined by the regulating agencies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Match date: 29 Nov 2018 | Football - Europa League | Posted: 29 Nov 2018 at 14:32 GMT+3 | 1462 views.\nAstana will host Dynamo Kiev in the 5th round of group K in Europa League.\nAlmost everything looks clear in this group where also take part Rennes and Jablonec. Dynamo Kiev and Astana are the leaders with 8 points after 2 draws and 2 wins, while Jablonec and Rennes are much behind them with 2 and 3 points. The odds for a draw in this game between Astana and Dynamo dropped a lot, but that doesn't mean it will happen easily. I think both teams will be looking for the win and the 1st place in the group. If this game finishes as a draw and Rennes somehow beat Jablonec tonight, Astana can be surprised very badly in the last round as visitors against Rennes. It's unlikely to happen after Astana beat Rennes at home with 2:0, but in that game Rennes played at good level, created chances and had more possession, so it's not excluded to see the big surprise happening if Rennes saved their chances tonight for progression.\nI decided to stay away from the draw here, although it looks very tempting as an option, so I look for different picks and I think BTTS is the best in this game. Astana have two wins at home with 2:0 against Rennes and 2:1 against Jablonec, but tonight they will face even better opponent. On the other side Dynamo made 2:2 as visitors against Jablonec and 1:2 win against Rennes, so they is no chance this team to be underestimated. Dynamo have BTTS in every single of their group matches. Also the first game between them in Kiev finished 2:2.\nThe players have been lucky, because the weather in Astana will be almost good- it's a little bit more windy, but not that cold. Since tomorrow it will become freezing cold in Astana with temp under -10 degrees during the day reaching even -20 degrees, so the weather will not be such a big factor tonight. I am looking for goals here, so that's important part which has to be checked, although both teams know perfectly well how to play in harsh conditions.\nAstana will be without Henrique(yellow cards), while Dynamo will miss Buyalskyy, while Besiedin, Burda and Pivaric are doubtful.\nI am going with BTTS in this game. Maybe it will finish as a draw and I will miss even bigger odds, but will play a little bit safer here. Dynamo know how to used their chances, I watch this team very often, while Astana are hosts and almost everytime they succeed to score at home, so it looks like a reasonable pick.\nAstana vs Dynamo Kiev betting preview represents the personal opinion of the author (Stanioooo).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a Facebook user mainly to keep in contact with friends I haven't seen in a while or new fans\/friends who I meet during my travels. Facebook likes to make some year end slideshow of all the things that have happened during your year. I hate that. Maybe I don't want to relive those certain memories. This year I decided to make my own. This has been a very weird year for me. From being offstage for 7 months, dropping dead, not being able to walk correctly, trying to tour again, and getting back to be the best. I'm telling even more truths about who I am and what my life is like.\nWhile in the hospital I made friends with a lot of staff at both Methodist University Hospital and Health South Rehab. There are people who I couldn't include in my video because I don't have pictures of them. It's not a slight at all.\nThis video might disturb some of you. Just know I'm so much better than I was. My life is getting back to where it should be.\nThe music is Puscifer \u2013 Grand Canyon off their new release.\nI put this video together during a long drive after staring out the passenger window. Clink the link and enjoy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"us range stove 6 burner 6 burner gas range by us range restaurant bakery equipment randy jones jr estate and other consignments rice auctions american range arbrg heavy duty gas range with burners american range arbrg heavy duty gas range with burners raised griddle.\nus range stove 6 burner garland us range u36 6r 36 range 6 burners us range pfa burner gas range with standard oven and photos.\nus range stove 6 burner us range 6 burner with 24 griddle gas ranges gas stoves.\nus range stove 6 burner american range patriot burner natural gas range w.\nus range stove 6 burner prg366gh thermador 36 pro harmony gas pro style range with 6 burners stainless steel us range burner.\nus range stove 6 burner click on any picture to see a larger image used us range burner stove kslcom.\nus range stove 6 burner the best inch dual fuel professional ranges reviews ratings thermadorprdghduelfuelprofessionalrange.\nus range stove 6 burner garlandus range commercial 6 burner stoveoven excellent condition ranges at the home depot ranges.\nus range stove 6 burner dcs range gas 36 6 burner 1 oven heavy duty range commercial range burner sun fire range picclick us range commercial burner range stove with convection oven natural gas.\nus range stove 6 burner us range commercial 6 burner range stove with convection oven natural gas us range ur range burners garland us range ur range burners.\nus range stove 6 burner performer series range with burners and griddle emon sale model featured arrobgd.\nus range stove 6 burner imperial hd commercial natural gas 6 burner stoverange w us range range sunfire series burners large oven list price.\nus range stove 6 burner .\nus range stove 6 burner garlandus range 36 commercial 6 burner propane stoveoven works great 1 of 1 see more commercial restaurant ranges and equipment from american range click here to see details.\nus range stove 6 burner thor kitchen 48 freestanding professional style dual fuel range with 42 and 25 cu us range pfa burner gas range with standard oven and photos.\nus range stove 6 burner atosa us commercial hot plate countertop six burner liquid propane range athp 36 6 us range ur natural gas burner range with standard oven us range ur natural gas burner range with standard oven btu.\nus range stove 6 burner ge caf series 48 dual fuel professional range with 6 burners and griddle natural gas us range burner.\nus range stove 6 burner us range utog36 6 natural gas heavy duty 6 burner countertop 36 range 180000 btu us range xgrr open top burner sunfire restaurant range.\nus range stove 6 burner image 1 us range 6 burner rangeoven range american stove parts musicink burner ranges american range stove us range.\nus range stove 6 burner american range reviews range stove inch for home use us 6 burner spring american range appliances us range ur natural gas burner range with standard oven us range ur natural gas burner range with standard oven btu.\nus range stove 6 burner american range ar ce 36 6 burner gas oven americanrange commercial.\nus range stove 6 burner us range utog36 su6 natural gas heavy duty 6 burner step up countertop 36 range 180000 american range professional burner stove central nanaimo nanaimo american range professional burner stove.\nus range stove 6 burner us range stainless steel commercial cooking unit with 6 burners and standard oven 36x34x37 make us range width 36 depth 34 height 37 burner gas range by us range restaurant bakery equipment burner gas range by us range restaurant bakery equipment randy jones jr estate and other consignments rice auctions.\nus range stove 6 burner 1126 us range stove 6 burner with convection oven used dirty american range professional burner stove central nanaimo nanaimo american range professional burner stove.\nus range stove 6 burner click here to see details range oven and cooktop buying guide ranges.\nus range stove 6 burner american range ar6b 36rg 72 heavy duty gas range with 6 burners 36 raised griddle ranges at the home depot gas ranges.\nus range stove 6 burner remodeling the viking vs wolf range debate remodelista sick of stainless viking wins in this category it offers ranges in a choice.\nus range stove 6 burner us range ur natural gas burner range with standard oven us range ur natural gas burner range with standard oven btu.\nus range stove 6 burner list price 287000 patriot burner natural gas range w.\nEdit \"Us Range Stove 6 Burner\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Introduction: Ying stone, also known as Yingde stone, is a traditional specialty of Yingde and is mainly found on Yingshan Mountain, 25 kilometers east of downtown. The stones are comprised of limestone with calcite deposits and are known for being thin, crinkled, seeped and permeable. Ying stones vary in color, like light greenish blue, grayish black, light green, black and white. However, stones in black color are the most valuable. Yingde Stone is mainly used for garden landscaping like making artificial hills. It's also a representative stone for decoration in traditional studies. In the Song Dynasty (960-1279), the stones were often used as tributes. In the Qing Dynasty, Ying stone was as famous as Taihu stone, Libi stone and Huangla stone, and was one of the top four garden stones in China.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Essays on African American women from 1730 with selections representing diverse areas of achievement including business, civil rights, the arts, education, scholarship, and government. Biographical essays of the most famous African American women--Harriet Tubman, Mary McLeod Bethune--appear alongside those of modern black women in their prime: historian Nell Painter, writer Alice Walker, and actress Halle Berry. Narrative biographical essays, edited by noted scholar Jessie Carney Smith, discuss each woman's significant achievements and the public response to those achievements. Many entries contain personal statements from the subjects provided exclusively for this 3 volume series, and all provide biographical data and a list of sources for additional study.\nNew York :Summit Books, \u00a91991.\nby Macmillan Library Reference USA.\nNew York :Macmillan Library Reference USA, \u00a91999.\nDetroit :Visible Ink Press, \u00a91993.\nNew York :Dafina ; 2003.\nNew York :Oxford University Press, 1992, \u00a91988.\nHolbrook, Mass. :Adams Media, \u00a91996.\nBerkeley, Calif. :Conari Press : \u00a92000.\nThe Black Women Oral History Project.\nDetroit :Visible Ink Press, 1996.\nNew York :Facts on File, 1997.\nFacts on File encyclopedia of Black women in America \/ the early years, 1619-1899.\nNew York :Facts on File, \u00a91997.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Juv\u00e9derm\u00ae injectable gel is a \"next-generation\" smooth consistency gel dermal filler that helps to instantly restore your skin's volume and smooth away facial wrinkles and folds like your \"smile lines\" or \"parentheses\" (nasolabial folds\u2014the creases that run from the bottom of your nose to the corners of your mouth). Juv\u00e9derm\u00ae comes in two formulations\u2014Juv\u00e9derm\u00ae XC and Juv\u00e9derm\u00ae Ultra XC.\nJuv\u00e9derm\u00ae XC and Juv\u00e9derm\u00ae Ultra XC injectable gels are indicated for injection into the mid to deep dermis (below the outer layer of skin) for the correction of moderate to severe facial wrinkles and folds, such as nasolabial folds (the creases running from the bottom of your nose to the corners of your mouth).\nJuv\u00e9derm\u00ae injectable gel is available in two formulations to allow your healthcare professional to tailor treatment to your unique needs. Juv\u00e9derm\u00ae XC provides versatility in contouring and volumizing facial wrinkles and folds. Juv\u00e9derm\u00ae Ultra XC is a more robust formulation for volumizing and correcting deeper folds and wrinkles.\nJuv\u00e9derm\u00ae injectable gel employs its unique manufacturing process to provide the highest concentration of cross-linked hyaluronic acid of any currently available hyaluronic acid dermal filler for long-lasting results. This advanced manufacturing process works to create a smooth consistency gel that flows easily into the skin and provides a smooth, natural look and feel. All other currently approved hyaluronic acid dermal fillers are made of granular consistency gels. And, Juv\u00e9derm\u00ae is the first hyaluronic acid dermal filler that has demonstrated its safety and effectiveness in persons of color.\nJuv\u00e9derm\u00ae injectable gel is a \"next generation\" hyaluronic acid dermal filler that provides volume and is used to treat wrinkles and folds resulting from a loss of volume beneath the surface of the skin such as perioral lines and nasolabial folds (lines around your mouth and nose). Juv\u00e9derm\u00ae restores your skin's volume for a smooth, natural look and feel. BOTOX\u00ae Cosmetic is not a filler. BOTOX\u00ae Cosmetic relaxes muscle activity and is used to treat moderate to severe lines caused by the dominant frown muscles between the eyebrows (the glabellar area). This allows the two vertical lines between the brows, often referred to as the \"11,\" to temporarily diminish for a smoother appearance.\nUncross-linked\" hyaluronic acid in its natural form is in a liquid state. \"Cross-linking\" is the process through which hyaluronic acid is made into a gel product that can be injected into the skin for long-lasting results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nice is known for its popular beaches, beautiful sites, mesmerizing parks and fashionable restaurants. There are also some hidden places waiting to be discovered. If you are particularly looking for a change of scenery, head to Nice. Here are the best spots for enjoying the outdoors and exploring the sight.\nLocated on the hill of Cimiez, this museum is considered one of the city's great cultural attractions. This Mus\u00e9e Matisse is as much an architectural masterpiece given that it is an art museum. It holds a broad collection of Matisse's work. Visitors can explore 68 paintings as well as 218 drawings, and 56 sculptures. You will not get bored quickly when you visit this museum, as there are so many things to discover and to visit.\nDon't miss out this beach. Beaulieu-Sur-Mer is a wonderfully sheltered spot. You can choose between two private beaches and a large stretch of public beach. You can find a wide variety of water sports, including kayaking, parasailing, water skiing and windsurfing. You can relax on the sandy beaches. However, if you want to visit more beaches in Nice, you can head to Blue Beach or to Neptune Plage.\nThis Park is more than just a place to unwind and relax. It offers many activities and exhibitions. In fact, it is over 7 hectares of land that houses the most complete range of plants in Europe. Anyone remotely interested in gardens would enjoy strolling around this park. There are also over a hundred birds and butterflies found in the huge greenhouse called \"green diamond\". As a matter of fact, this Park is about 15.6km from Nice Airport, so most people rent a luxury car to get there.\nCalled in french \u00ab vieille ville \u00bb, this is a picturesque and colourful place with its narrow streets. There are lovely old buildings where you will feel like back in time to a medieval village. Many shops, bars and restaurants can be found along the streets. At the flowers market, you can smell flowers, and especially, if you would like, you can have a glass of wine there. Travellers can find food products as well as seafood restaurants. Likewise, you will find the main cathedral and the opera house at this place.\nThis Museum houses a great collection of work from Chagalls. The collection includes Chagall's 17 major Biblical Message Paintings. The stained glass work, sculptures, and mosaic wall are not to be missed. Last but not the least, this museum also displays a collection of works by other artists. Therefore, if you appreciate any kind of art, you should visit this Mus\u00e9e Chagall.\nAll things considered, Nice City is having a moment in the sun, thanks to its world-class museums, its mesmerizing park and its beaches. It also prides itself on its welcome for LGBT visitors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Citation: Ng, E.S., Chan, Y.C., Wilder-Smith, E., Vijayan, J., Therimadasamy, A.K., Tan, T.C., Lim, A. (2011). High resolution ultrasonography in the diagnosis of ulnar nerve lesions with particular reference to post-traumatic lesions and sites outside the elbow. Clinical Neurophysiology 122 (1) : 188-193. ScholarBank@NUS Repository.\nAbstract: Objective: Recent studies suggest that high resolution ultrasonography (HRU) is useful in evaluating ulnar neuropathy (UN) at the elbow. These studies do not include UN outside the elbow and lesions related to previous trauma. We investigate diagnostic utility of HRU in UN at any location of traumatic and non-traumatic etiology. Methods: Patients with clinically suspected and electrophysiologically defined UN at the elbow and outside the elbow were included. Nerve conduction studies (NCS) were compared with HRU. HRU defined UN in terms of change in cross-sectional area. Results: Our retrospective analysis included 46 UN. In 25 cases both NCS and HRU localised neuropathy to the elbow. In 15 where NCS was abnormal but non-localising, HRU localised the lesion in 14, 7 outside the elbow. In three of these, HRU characterised further pathology (synovial osteochondromatosis (n = 2), myositis ossificans (n = 1). Cross-sectional area of the ulnar nerve at the sulcus significantly correlated with distal NCS parameters. Conclusions: HRU is of greater use than NCS in the localisation of UN both at the elbow and outside the elbow and in UN related to previous trauma. Significance: HRU is useful for the localisation of ulnar neuropathy. \u00a9 2010 International Federation of Clinical Neurophysiology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What do you think this How To Draw Fortnite Ragnarok Mask video?\nBe sure to share this How To Draw Fortnite Ragnarok Mask video.\n\u2190 Previous BACK TO SCHOOL TIPS: Awesome ART in BORING Classes!\nWhat Does FUTURE CLICKBAIT Look Like?! \u2013 Epic Art Competition!\nNovember 22, 2018 admin Comments Off on What Does FUTURE CLICKBAIT Look Like?! \u2013 Epic Art Competition!\n\"POKEMON MASHUP\" November COTM \u2013 SHOWCASE and WINNERS!\nNovember 24, 2016 admin Comments Off on \"POKEMON MASHUP\" November COTM \u2013 SHOWCASE and WINNERS!\nOctober 6, 2017 admin Comments Off on ART BEEP TEST! \"FitnessGram PACER Test\" = TORTURE!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You can't have great resume templates should you not Learn how to compose one. You are able to observe a wide array of good resume templates and make an informed decision. Some of the templates look as though they have a page missing. Most free templates are designed to be specific to certain businesses, so it's well worth browsing through the options available that match the work sector which you need to function in.\nSelecting The ideal Professional Resume Template It Is easy to locate a professional resume template that can help you set your best foot forward to possess the telephone for the meeting. It is not hard to use the template, but it is crucial that you utilize the effective one. It's essential to be aware of the ideal engineering resume templates so that you can catch the interest of the hiring manager. Templates help understand the flow of resumes. Now you have the totally free professional resume template to work with, it is time to compose your application Before doing so, make sure you read and comprehend the job posting in order that you are able to take down notes of these qualifications being asked by the poster and you'll be capable of making use of key words they used when looking for prospective candidates. Deciding on a superb template is very important and if you are thinking to utilize it, you should utilize the best one.\nWhich is exactly why you Want to use professional Resume template instead. The benefits of employing a template The obvious reason for making use of a template is the way quickly and easily you're able to put your resume together. Understanding where to find adequate engineering resume templates for 2017 is critical as you own a guide and an aid to start in composing your resume.\nThe absolute most crucial parts of your CV A great CV should catch the eye when conveying a constant and robust impression. Ensure your CV isn't too long. Your CV is the very first opportunity to market yourself, and in a number of cases, your sole chance to earn a first impression. A CV is a thorough listing of your career background and essentially whatever you feel your employer should be aware of about you. A CV is a way to make it through the door (Lily, our Tech Recruiter) When you get started creating your CV, keep in mind that its objective is to receive you a job interview. I want to stress the following the Europass CV is not obligatory to receive hired by European tech businesses.\nEnsure your CV begins with a strong and succinct Message about what is unique about you and why you're the remedy to your prospective employers' problems. Your letter must explain everything you can do to help your customer,' not exactly what it is that you're selling, she explained. The crucial word in regard to the image which you want to make is specialist. Here are a few of my favourite resume templates. Should you would like to be certain you submit the best resume, you want to include sections that are important. The work experience part may be a small bit better too. The horizontal lines which are readily added via Word are an excellent means to produce construction.\nTo put it simply, resume templates can Give You That professional appearance that will help to boost your probability of winning a meeting to land that dream job you're applying for. Whether yours needs a new facelift or you are just starting, you can save a massive time utilizing a template rather. If you do this, you are going to understand exactly what you would like to prove. Researching permit you to have a template that is exceptional. Among the easiest methods to provide a resume or CV that a wonderful visual pop is to add a little color. Have a peek at professional templates employed by women and men at the exact same field as you are for a notion on which to select. You have to get a feeling for what a recruiter will be on the lookout for in your area to know which of those templates will be appropriate for you.\nYou Shouldn't throw away your time in understanding the Very best marketing resume template so that you have the job that you want. Even though it doesn't work all the second, it works the vast majority of the moment. If you want to relocate to your work, you probably already know it's best to leave your present address off your resume. All of your hard work is going to be wasted if recruiters reject your program before reading it. Jobs applied for' section is not required also. If you are applying for specialist jobs with notable companies, you want a professional CV that will supply you an excellent welcome.\nIf you do not have a Great Deal of expertise, however, It's better to begin with your skills and instruction. Much like most graduates you won't have a lot of work experience to start with this that you should focus on your abilities and education. When it's to do with your resume you need to professionally express your abilities and expertise. It is not tough to upgrade when you'll need to show new skills and achievements. The best way of going about it's including skills which are related to the position that you're applying for. You need excellent communication abilities. Aside from the skills it's also sensible to add your achievements.\nMake your CV distinctive and memorable. Writing a CV can appear difficult, but as soon as you start it will find a good deal simpler. The DIGIT-TM CV is a rather detailed and exact Curriculum Vitae, which should be updated at each contract renewal. While writing the CV also take care of the application letter you have prepared. If you aren't sure where to begin to your own CV, have a glance at our administrative assistant CV instance. For instance a Targeted CV is mainly utilised to concentrate your career path towards a specific objective. Combination CV Each of all these CVs are meant for different circumstance and serve unique functions.\nIf You're only starting out, your CV should be Somewhat straightforward, however for mid-career or late-career professionals it could find a little more complex. You might not ever fail with the normal CV since it is the one that most companies expect to see. Today, it's much simpler to create an perfect CV with the support of some applications that are applicable.\nIn the event the template is not in the Perfect format There is not any value of using it. The CV template teaches you the way you can format your Curriculum Vitae and lets you know what type of information ought to be contained. Whichever resume format you choose, make certain to add examples of achievements that benefited your prior companies. Deciding on the most suitable format efficiently can help you raise your odds of landing work. The key to getting it right is picking the ideal CV format which could present your strengths in the very best way possible. Otherwise, you can end up recreating the entire record from scratch, in case you have not stored a copy on your hard diskdrive. Besides avoiding the aforementioned mistakes, you also will need to follow up your program.\nEmployers usually do not like to get operational Resumes because they don't reveal your work history or career progress. Since they are used to it, they are looking for something that makes it different to others. You might also call the employer and enquire concerning the development of your application. Employers scan through cover letters and for you to receive their attention that you should show them that you're fit for the job. Most companies won't even offer a messy resume the fantastic time of day because there'll be better resumes they could look over.\nWrite If you are applying for work overseas or a Position in an organization that is employed within an international level, you want to show that you have what's vital to stay informed in their fast paced work environments. When you're attempting to find work, you know that you will need to prepare a curriculum vitae which is exactly what you'll pass on to prospective employers that you want to work for. In the finish of the day be certain you will receive a job that you dream about, congratulations!\nYour resume or CV may be one of the most important Projects you design. Someone states that resumes are dead and it is imperative to have social media, but practice suggests that recruiters continue to locate applicants' resume, choose more applicable ones and just then join with applicants. Infographic-style resumes have gotten popular, and they are sometimes successful when performed well. Functional resumes imply to emphasize your exact skills and achievements which have been demonstrated through specific work related achievements.\nWith the targeted CV, you are tailoring yours to the Standing and making it easier for them to find the key words they will need to discover whether you are fit for the job or not. To prevent getting discarded you want to highlight similar positions you have managed previously. There are lots of places to have templates for nursing CVs including our on-line service Our service provides nursing CV writing and also to aid with growing Curriculum Vitae for RNs all around the world. One of the absolute most significant design choices you can make on your resume is your own font. Also, take a look at different samples for each one to offer you some useful examples that you wish to follow. Resume Help It is an excellent instance of a timeless CV that will work whatever position you are applying for. In practice you can come across a variety of kinds of CVs.\nIs to produce a budget. Whether you would like to conserve money for a massive trip or retirement, then you'll be in a position to calculate exactly what you have to conserve and how much time it will take. It's essential to deal with your money. Whenever you have the total amount of money available to pay for the coming month, then you want to allocate your intended spending one of your favourite spending categories. With the ease of on-line banking it's easy to transfer money between checking account and credit . If you are spending more income than you are making, then you must cut some things out of your budget.\nWithout a comprehension of where your earnings goes after it is Deposited to a checking account, you'll have little to no chance to optimally adjust your spending and saving habits. It's likely to categorize your spending and track your earnings and taxation so you get a thorough picture of your financing. As soon as you know what your typical spending is, today it's possible to make a budget.\nTrimming the surplus fat from your budget is not as impossible as You may think. There are several techniques to do your budget but I will disclose to you the 4-steps about how to do budgeting effectively. The money which you have and can spend anytime is what you are going to be able to contain in your financial plan. If you are searching to create a more complicated budget, then I suggest checking out programs like Mint or Expense IQ. Create an easy budget which you might actually stick to.\nMost people say you should budget to Discover where to set Your cash. As mentioned previously, one tangible method to keep tabs on your financial plan and also make financial goals is through writing it down. It is simply a forecasting tool which allows you to visualize where your money goes every month. No, not due to the fact you've to generate a budget. A personal budget may also aid you request a raise or looking to earn more income on your own career. There are loads of measures to take to make a booming personal budget.\nThe chart above is supposed to supply a contrast to Demonstrate just how much it would cost the state to boost the minimum wage to every level. Budget Boss is a really visual program which uses graphs and charts to track your financial plan and goals. Formerly, part of the energy structure in houses and company, nowadays a maid service is one which implements specialized and regular cleaning business, together with usually isn't a live-in participant of a house.\nPaying off debt might take a month or two, but based on how much You've gathered, it might take years. Simply take each of your long cycle expenditures and work out how much you'd need to save up each month to have the ability to pay for it. Now everything is initiated, let us walk through the way the very first month goes. You might want to look at two to three weeks of your spending habits to find a notion of your present spending habits versus your earnings. If you find you've been eating out a lot it's possible to decide to have a day and make food to freeze for convenience. In the event the amount is negative, it's time to produce some budget reductions (deficits are not the target ). Thus in the event that you employ an expert cleaner then you will become lot more time to spend together with your friends, loved ones, as well as kids.\nYou would be amazed at the number of people do not Understand exactly what it cost to work\" their residence. You need to operate your home like a business thus you ought to know about just how much it cost to operate your house. To begin with, more families must be eager to adopt and adhere to a detailed budget.\nBudgeting is not a one-time occasion. Ensure you are on Unlimited internet data plan for your own residence. Bills could be boring, but They're a part of life that can't be avoided. Almost All bills are broken Straight down the middle between the 3 couples. Yes, sometimes there'll be a Little cushion in many of the accounts where the bills vary. You are likely To be amazed at how simple it becomes to pay the bills and save a little money. If you end up paying more than the comprehensive bill, you will receive a refund.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2013 Any hard, dark bakers melting chocolate or chips, roughly \u00be \u2013 1 cup.\nCut 20 x 2 \u00bd\" sections, of thick white string or yarn (you may need spares). Tie a small knot at one end of each string.\nHeat your oven to 200 and melt a small portion of paraffin wax in a small ceramic dish, takes only 2 \u2013 3 minutes\u2026 it can be very carefully microwaved, or heated slowly on the stove-top, but it's MUCH, much safer and more controlled in the oven, and at a set temp.\nI like to use a combination of oils, both for flavor, and to create a varied oil 'vehicle' for cannabinoid bonding and availability, some thinner and some thicker. In the end, you want it to be a solid at room temp, so your canna balls aren't too soft, and the chocolate keeps its shape and thickness. The additional liquid oils I've used in addition to the solids, had previously been infused with herb.\nTake enough peanut butter filling, so that when rolled, it should create roughly a 1.25 \u2013 1.50\" diameter ball.\nIf you're not using them right away, I recommend letting them sit for a few hours, as they are on the pan, while freezing.\nThis is a Bad Kats collection recipe!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please join us for a presentation of a 600-year-old craft!\nKen Kulakowsky of Lancaster's The .918 Club will give a lecture and demonstration on the art & craft of letterpress printing. After a brief history of the process as perfected by Gutenberg in the 1400s, all in attendance will have the opportunity to print using a table top printing press.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Come join your fellow YLD members for complimentary food, drinks, and networking!\nOpen to practicing attorneys who are under 40 years of age or have been practicing 10 years or less.\nSpace is limited. Please register by 5 p.m. on Tuesday, April 3, here or by email to info@FBAOC.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For I will be merciful to their iniquities, and their sins will I remember no more. Hebrews 8.12.\nHow regrettable that we often forget what God remembers and remember what God forgets. Some people are always musing: Does God really forgive after I have committed so many grievous sins? Will He indeed remember them no more? Yet others may think that even though God has blotted out sins, the scars which no doubt remain will forever remind God of what a sinner they are.\nWhy should I worry, doubt and fear? Has God not caused His Son to bear My sins upon the tree? The debt that Christ for me has paid, Would God another mind have made To claim again from me?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Lung-tung, Taipei County, Taiwan. 1998. 4. 12. Photo by Huang Su-fang, National Taiwan Museum, from http:\/\/www.ntm.gov.tw\/seaweeds. Huang Su-fang. \u00a9 Sue Huang (sfhuang@ntm.gov.tw).\nKiel Bight, Baltic Sea; chloroplasts and pyrenoids.. 10 May 2007. Gesche Bock. \u00a9 Gesche Bock (gbock@ifm-geomar.de).\nSpiddal, Co. Galway, Ireland; MTL sandy rocks. 22 Mar 2002. Michael Guiry. \u00a9 Michael Guiry (mike.guiry@nuigalway.ie).\nSpiddal, Co. Galway, Ireland; plants on rock with some distromatic Ulva. 04 Dec 2005. Michael Guiry. \u00a9 Michael Guiry (mike.guiry@nuigalway.ie).\nSpiddal, Co. Galway, Ireland; mid-shore plants on Sabellaria colonies. 26 Mar 2005.\nKimikazi, Zanzibar; epiphytic plants on Halodule sp.. 09 Apr 2005. Michael Guiry. \u00a9 Michael Guiry (mike.guiry@nuigalway.ie).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Greendale MB and River of Life churches work together to provide opportunity for kids to grow and learn about God together during the week. Wednesday nights Christian Service Brigades (for boys grades 3-6) and Pioneer Girls (grades 2-6) are looking for adults to help with large and small groups. If you would like to be a part of a vital ministry in the community that promotes and instructs in the Christian faith, click the \"I'm Interested\" link, fill in the form, and someone will be in touch with you. Please note that security screening is required for all volunteers.\nPlease do not use the 'I'm Interested' link below. Go directly to the organization website and contact the appropriate person listed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feb 18, 2017-Inflation has dropped to its lowest rate in 12 years of 3.2 percent as a fall in prices of food and beverages brought down the cost of living.\nPrices of food and beverages fell 0.7 percent on average in January. All major food items, from cereal grains, pulses and legumes to vegetables, meat, fish, ghee, cooking oil and fruits became cheaper by 1.2 percent to 13.9 percent.\nThe drop has been due mainly to the base effect, improved supply situation and deceleration in Indian inflation, says the latest macroeconomic report of Nepal Rastra Bank (NRB), the central bank.\nA drop in prices of food and beverages, which account for 43.9 percent of the consumer basket, led consumer prices to rise at the slowest pace since December 2004, when inflation stood at 3.1 percent.\nEarlier, NRB said that the deceleration in inflation in recent months, however, should not be taken as a fall in overall consumer demand, which could pose a threat to Nepal's aim of achieving economic growth of 6.5 percent in this fiscal year.\nAccording to the central bank, inflation is moderating because of a correction in prices, which had risen at a rapid pace last year because of a supply disruption triggered by the Indian trade embargo.\nDisruption in trade between Nepal and India hits domestic prices because the country relies on the southern neighbour for more than 60 percent of its imports. This means that a fall in prices in India may not necessarily transfer to Nepal if there are problems in movement of goods.\nLately, inflation has been moderating in India because of Prime Minister Narendra Modi's shock currency replacement programme, which has hit consumer spending. Because of this, Indian inflation cooled down to 3.2 percent in January.\nWith the moderation in prices in India, prices of ghee and cooking oil fell by a whopping 13.9 percent in Nepal in January, while prices of pulses and legumes dropped by 6.6 percent. Even prices of cereal grains, which account for 11.33 percent of the consumer basket, fell by 1.2 percent in January, shows the NRB report.\nHowever, the price of sugar and sugar products has gone up by 15 percent because of a slump in the production of sugar in Nepal and India, which has led to shortages.\nAlso, prices of clothes and footwear jumped 10.1 percent in January, while house rents and utility bills went up by 7.5 percent. However, transportation fares dropped by 3.3 percent and communications costs fell by 1 percent.\n\"The only threat to inflation [in Nepal] is from the global rise in oil prices,\" says the NRB report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Hamamatsu International Piano Competition was started in 1991 in commemoration of the 80th anniversary of the founding of Hamamatsu City. Since then this international cultural program has been held every three years to celebrate the history and tradition of the city of music and musical instruments.\nOver the years, the competition has produced many pianists who are active on the world stage and many of the prize-winners of the Hamamatsu International Piano Competition have gone on to receive awards at famous international piano competitions. As a result, the competition receives both domestic and international acclaim.\nThe First Prizewinner will be given at least 10 opportunities to perform in and outside of Japan, including solo recitals and performances with major orchestras, by the end of March 2020.\nClick here to go to the Hamamatsu site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mathias Mar Fan was recognised by the archdiocese in 2012 at the Archbishop Anthony Pantin Awards. Then Archbishop Joseph Harris made the presentation.\nWords cannot fully express all that Mathias Mar Fan means to our parish, and especially to the members of the St Joseph RC Choir, with whom he has worked for almost 40 years.\nIt was in 1979 that this humble, self-taught musician moved with his young family to Tobago to take up a position in the health sector. Well-meaning colleagues in Trinidad had warned him: \"Tobago have plenty obeah!\" but he was undaunted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Eric here, after being in the market for a muscle, the more reviews I read, the more I realized that \"going really fast in a straight line\" wasn't what I was looking for. Picked up my R last week.\nLocation: The Mile High City.\nI've had 'em both. Had an R, but had to sell due to a cervical fusion. Picked up an F to replace.\nThe R is *definitely* a hoot-- basically a 600 lb. sport bike. Especially if you tune the engine, get rid of the boat anchor H-D laughingly calls an exhaust, and reworking the front fork and rear shocks.\nThat being said, doing all of the above to an F makes for a pretty hair-raising ride as well; it does just fine in the twisties-- enough that I had to change out to pegs that gave me more clearance.\nThe only real issue with the R is parts. The F is a right bear to find parts for, but it's a walk in the park compared to an R!\nUSAF: 1989-1997. Desert Shield (Saudi Arabia), Desert Storm (Kuwait, Iraq), Provide Comfort (Kurdistan), Restore Hope (Somalia), Uphold Democracy (Hati), Joint Endeavor (Bosnia), Support Hope (Rwanda). USAFR: 1997-2014. Iraqi Freedom (Iraq), Enduring Freedom (Afghanistan). Retired, Oct. 2014.\nPicked up my 2007 R this past weekend.\nDenim blue with Tab slip-ons.\nWow what a find. Nice of that previous owner to save it for you all these years.\nWell, I am already in here on my old 06 VRSCR, but it is long gone (2009 sold it reluctantly when I moved to Hong Kong China area).\nI am now riding my newest 06 VRSCR (Supercharged) model and figured I should list it here.\nHey guys I have a 06 street rod, v+h exhaust kn filter and thunder max auto tune, collecting parts for 240 rear end \u263a\ufe0f Would anyone have a stock seat for sale?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Blue sweatshirt knitted from 100% GOTS certified organic cotton. This looped fabric is easy to care for, keeps shape and has moisture-wicking properties to keep the skin dry. The sweatshirt is detailed with a screen printed cheerleader cat at front. Ribbed trims at neck, cuffs and hem finish the look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First of all, I would like to say thank you for taking the time to take a look at my page. I know there is a sea of talented photographers out there so picking someone must be really difficult! Hopefully you like my photos but when it comes to choosing a wedding photographer, I think it is always a gut decision. This is why I try and make sure we meet for coffee so you can have a chat and get to know me before you make any choices at all. I won't try and force you into making your mind up there and then or give you a big sales pitch but it's always fun just to chat about your wedding day \u2013 and of course you're welcome to ask me anything!\nMy style is incredibly laid back and relaxed. I want you to have fun and enjoy your wedding day; you don't want to look back and regret spending too much time with your photographer and not with your family and friends! Generally I just try to blend in and concentrate on photographing all those fun, emotional and special moments which will capture the real spirit of your day.You might not notice me, but rest assured your day will be thoroughly documented! I don't limit the number of photos I give to you when I've finished editing either so you know you won't be missing any gems \u2013 you'll get every single one of the good photos!\nWhen it comes to getting the best photos on the day, I think it's all about striking a balance. As I said, I don't want to take over or interrupt you too much, but equally I don't want to miss any moments for some absolute killer shots. I love couples who are happy to try something new, and maybe run out for 5 minutes in between courses of the wedding breakfast because the light is just PERFECT (every photographer goes a bit mad when it hits golden hour). I try not to give any cheesy prompts or poses, but just focus on getting those moments which are just really \"you\". And please excuse all the weird positions I get in to while I'm trying to grab the perfect photo!\nI eat out too much. Bristol is so amazing for restaurants. I'm currently trying to make a list of all the best pizza places in the city.\nI really like good horror films. I also like really bad horror films.\nI spend a LOT of time in board game cafes. Settlers of Catan (and it's many expansion packs) is a current favourite. I'm trying to build my own collection.\nI'm a sucker for a good American sitcom. Parks and Rec is my all time fave.\nI ADORE animals (who doesn't?). My boyfriend and I currently live with our enormous tomcat affectionately known as Thompson, and my adorable little crested gecko called Laszlo (they do not get on). I dream of having a huge fluffy dog as well.\nI'm really interested in ancient history. I actually have a degree in it, before I did a massive detour and ended up as a photographer!\nMy favourite food is cheese, in all it's varieties. Or any food with cheese on or in it.\nI do love music, but I will literally listen to anything from emo to techno with a large chunk of indie pop in the middle. I will never get bored of cheesy 90s music either.\nThat's probably enough from me. Please get in touch if you would like to hear any more from me or if you just want to tell me about your wedding!\nOur team of wedding photographers Bristol also provide affordable wedding photography, event photography and engagement shoots in the Devon area. For more information please visit Devon Contemporary Photography.\nAs wedding photographers, we work with many amazing individuals and companies to make your wedding day the best day possible. Here is who we work with to help us to deliver our promise of truly memorable wedding photos.\nExplore my wedding ideas, wedding inspiration and more on my Pinterest page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Part of a grand tradition of immigrant groups that has made Chicago the vibrant and diverse city it is today, The United Neighborhood Organization (UNO) began organizing communities around key quality of life issues in 1984. It has since grown to become one of Chicago's strongest and most effective community organizations, employing grassroots leadership and the involvement of community residents to lead the way in advocacy and policy on a wide range of issues affecting the Hispanic community that is its base.\nFrom immigration reform to citizenship and voter registration, fromhousing to job training, UNO has helped to define and solve the challenges of the community. To date, UNO has serviced over 65,000 new American citizens. It also founded the Metropolitan Leadership Institute to engage Hispanic young professionals in the public arena, whether in the government, corporate or non-profit world. Nowhere is his more evident than in UNO's work in education. UNO began its quest in this arena 20 years ago and today operates eight charter schools in Chicago and one in New Orleans, addressing the full range of student and community needs.\nFor its decades of dedication to improving neighborhoods from the ground up, its ongoing innovative work in education and citizenship, and in recognition of its future as a leader in advocacy and policy, The United Neighborhood Organization is this year's recipient of the CNDA Special Recognition Award.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How many of you rolled your eyes when you saw the title of this blog?\nWell, you're all about to eat crow. Not literally, of course. I've heard the black birds aren't so tasty. But pigeons? They gave those things a fancy name when they first cooked them up, which means they've got to be delicious. And by fancy, I mean disgusting. Squab? It sounds like you're going to get a nasty rash by eating it.\nAnyway, reading a recipe ALL THE WAY THROUGH is the most important thing you can do before you start cooking. I usually read something through about three times before I start a new recipe. Then I make my mise en place and get to it.\nReading the recipe thoroughly and multiple times will help keep mistakes at a minimum, like knowing to divide an ingredient. Or preparing the sauce beforehand. Or making sure you have all necessary kitchen gadgets to make your dish. 'Cause, uh, that's never happened to me before.\nSo read that recipe like it's The Hunger Games. Soak it up. Absorb it. Comprehend it. Whatever you do, don't start cooking until you understand the recipe as a whole.\nHow many of you have started making a dish without reading the recipe all the way through, only to find out your kitchen adventure is now an epic fail?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Due to the sequester cuts to federal programs, employees at the EPA may experience a reduction of up to thirteen work days this year. How you see this kind of situation depends upon your point of view.\nConsider the fact that in some cases furlough days can be combined with weekends, so a worker has a three-day or even four-day weekend when there are national holidays. The extra time off can be very helpful to workers who have home projects to complete, or visitors staying with at their homes. It also provides more opportunities to take small trips and enjoy some light travel. There also could be more time to start a side-business or take an online course.\nAdditionally, when people experience disruption in their life routines some take it as an opportunity to reflect upon how happy they are and if they need to make important changes. In the case of the state employee furloughs, most said they enjoyed having the extra time, and didn't miss the money they did not get. In other words, it was not at all a disaster, nor even slightly damaging. It actually turned out to be a life enhancement. A research study conducted in the UK focused on civil servants found some of them were so bored their health was at risk. The most bored were more likely to die from heart disease. Could it be healthier for some government workers to spend less time at the office?\nFunny how that possibility is not being reported in the mainstream media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SupplyTrain is supporting the Education and Skills Funding Agency (ESFA) develop the digital apprenticeship service so that it works for small businesses \u2013 and we need your help!\nFrom April 2019, all employers that want to take on apprentices, will use a digital apprenticeship service to manage and pay for apprenticeship training no matter how big or small the organisation.\nCurrently, only employers with a payroll of over \u00a33 million per annum are using the digital apprenticeship service, but companies of this size often have dedicated staff managing the process, liaising with training providers and cross-checking the detail before signing off payments. Small businesses do not have that luxury.\nOnce we have collated your opinions, SupplyTrain will feedback an overall picture to the ESFA on how the digital apprenticeship service might work for small businesses in England.\nGet in touch today to help us understand your future requirements.\nEmail: hello@supplytrain.co.uk with your name, business name, phone number and good time to call and we'll be in touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I read the entire internet so you don't have to., originally uploaded by -SPK.\nMy new blog tag line.\nThis screen shot is basically like a patent search and NO ONE HAS IT! Ahahahahahaaaaaa, now it's reserved for me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Where are you from?\" a simple question that leads to the opening of countless doors. Going beyond what your eyes see is something so beautiful, and a frequent occurence as chuckles overfill the office at BBG. Sinking into the couch, Sam's shoulder length flowy hair falls across his face as he brushes and tucks it behind his ears. A couple of scars replaces a tattoo, full of memories. Full of life and adventure, anything sounds like a good crack to Sam.\nBeing a lover of travelling the world with his surf board, Sam Kinnear at Established Models is approached on a beach in Australia. A boy of many talents, we wonder how he fits all of these activities into such a tight schedule. After spending a couple of hours with him, we can see how he is adaptable to any environment. Living by words of wisdom from his dad, he has the energy and wit to start his own TV show, without a doubt.\nYou can also find Sam in the latest Boys by Girls issue \"The Trouble With Boys\", in the documentary \"Boys at School\" feature that is available online and in shops now.\nHey, would you like to introduce yourself?\nI'm Sammy Kinnear, I've been modelling for nearly a year now and I'm with Established Models.\nI was travelling in Australia and was surfing on a beach and this little chap came up to me and said \"would you like to be a model?\" I thought it sounded like a good crack, so here I am.\nYeah, I mean you meet so many new people and everyone is so lovely. All of the photographers I've worked with have been great. It's good fun.\nBesides modelling, what are your other passions?\nWell, I'm actually a full on farmer. Before my first casting this morning I had already lambed three sheep. Most models don't really do stuff like that. I had a good shower and stuff.\nHaha that's amazing! What kind of animals do you have?\nWe have a herd of about 80 cattle and a breeding herd of like 50 sheep. And yeah we're mid lambing season, so it's quite busy. We got about two hours sleep last night.\nI'm really into boarding, surfing, snowboarding and mountain biking. I enjoy my art as well and sport obviously. I'm quite into my gymnastics; I only started lessons this year though.\nYou're very talented! Do you study as well?\nYeah I study Civil Engineering at Bath.\nThat's completely different from any of the other talents you've mentioned.\nYeah, it's quite an intense course, there is a lot of work involved.\nHow do you have time to do all of these things?!\nWell luckily living on a farm, all of the mounting boarding and all of the animals and stuff is right there, because that's where I live. I do gymnastics while I'm at University and I like to get out of the city. I normally go surfing on a weekend; got a big pickup truck, put my board in the back and off I go.\nLiving on farm, what do you think of Bath?\nI do like my space. I'm not gonna lie, I love the country. Cities are so busy, everybody is so speedy like 'CYA!' But Bath is such a lovely city, so I think out of all the places I could have gone to Bath is probably the best. It's tiny though, you could walk across it in about 40 minutes.\nWhat would you say your three benefits are growing up in the country?\nThe ability to work without complaining. My dad has always worked us, on a day off you'll probably work 4 hours and that's a day off. There's always something to do. I guess I can wake up early without complaining, that's another benefit. Because we have no neighbours, you know, if you wanna go and blow up a melon in a field, you can do that.\nI'd definitely love to get a good job that allowed me to travel, which is why I chose to do engineering. My ideal job would be designing film sets for films with really dramatic scenery. Like, Lord of the Rings did all that stuff in New Zealand, and my ideal job would be to go and scout all of those locations. See all of these amazing places.\nYou're featured in our new issue 'The Trouble With Boys'. What would you say are troubles boys go through today?\nI'd say it changes throughout the growing up period. Realistically, up until you're about 12 the troubles are pretty limited. You know, you normally find your friends, but I suppose as soon as puberty hits then boys have that pressure to drink and all that. I feel like it's that transition into adulthood.\nWhen it comes to group pressure, have you got any advice on the best way to deal with that?\nI'd say just always do what makes you happy. Sporting wise, I've always said it's really good to push yourself if you want to get better. Peer pressure wise don't ever do stuff that you're not comfortable with.\nDo you think there's a difference between peer pressure in the city and the country?\nI'd say it's pretty similar across the board. Obviously, in the country I'd say there's a lot less pressure, because it's not such an intense environment. There is so much space, you can let go of any stress. If you're worked up or whatever, you can go cycle up a hill or go and blow up another melon. Poor melons.\nOh, we've tried quite a few things! We've strapped quite a lot of stuff to fireworks, like teddy bears and stuff. And then we've got this big Orchard full of apples and every year when fall comes and all of the apples are on the ground, we dress up in cricket gear and effectively paintball with apples. We're aloud about five apple bombs each, which is just a French banger and an apple. It's a good laugh! We've also invented a Kinnear Specials game called Fireball Hockey. You get a toilet roll, wrap it in chicken mesh and soak it in diesel for 24 hours and then you light it and hit it with a hockey stick.\nHaha. How do you think of these games?!\nWell, all of my family are pretty much farmers. Apart from one uncle who lives in New Zealand so you just develop them.\nYeah, every Christmas instead of going to church we play Fireball.\nI want to meet your family, they sound like so much fun. What life advice would you give to people?\nMy dad has always said you die if you worry, you die if you don't, so don't worry. So I suppose that's what we live by. You generally live a happier life if you worry less. There's just no point in it.\nWise words Sam. Make sure to get a the latest print issue \"The Trouble With Boys\" to see Sam Copeland's capture of Sam in the \"Boys at School\" documentary feature. Available in shops and via our website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Offering the best in affordable quality jewelry and accessories! Jimer is inspired by classic designs and stylistic functionality.\nOur jewelry and accessories are delivered directly to customers through home shows, office parties, fundraisers and personal shopping through an ever-growing base of independent Advisors.\nFind your fashion. Achieve your dreams!\nI love my Jimer! I hosted a show last year, my friends loved the jewelry and I got my jewelry at half price - with a lot of jewelry free!\nThanks for your positive comments on our products, Lynn -- we really appreciate that very much!\nIf anyone would like any further information on having a Jimer party, having monthly specials e-mailed directly to you, or perhaps even joining our team as an advisor, please do not hesitate to contact me for further information, and I will be more than happy to help answer your questions.\nThank you for visiting our Jimer Storefront!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"HAWARA CHECKPOINT, West Bank\u2014Hopes dimmed Saturday for smooth elections to replace the late Palestinian leader Yasser Arafat as Israel threatened to resume its military clampdown on volatile areas here and in the Gaza Strip.\nIsraeli officials had promised to keep their forces out of West Bank cities and cut back on some checkpoints to allow Palestinian voters easier access to polling stations Sunday in the first presidential election here since 1996. They appeared to keep that promise in parts of the West Bank on Saturday.\nBut outside the key northern city of Nablus, Israeli forces cracked down in response to a Palestinian ambush that killed one Israeli soldier and wounded three others. The Al Aqsa Martyrs' Brigades, an offshoot connected to the political movement of presidential frontrunner Mahmoud Abbas, claimed responsibility for the Friday incident. Abbas condemned it.\nAt this checkpoint south of Nablus on Saturday afternoon, angry male Palestinian travelers were kept waiting for hours in a cramped outdoor holding area as Israeli soldiers ran checks on their identity papers. Among those being held was Yahya Shunner, 34, a member of Palestinian Prime Minister Ahmed Qureia's staff.\nTo Palestinians, such checkpoints are loathed symbols of occupation. Israeli officials say they are necessary to thwart suicide bombers and gunmen.\nA spokesperson for European Union election observers who asked not to be identified said they will formally monitor Israeli checkpoint impediments for Palestinian voters on Sunday.\n\"We're not checkpoint monitors, but freedom of movement is an issue,\" said the spokesperson.\nMonitors reported that the delivery of paper ballots to Nablus-area polling stations was delayed Saturday after the Israeli military slapped a curfew on eight nearby Palestinian villages. The restrictions were later lifted and the EU spokesperson said all election materials were ultimately distributed.\nIsraeli military officials defended their actions, arguing that Palestinian elections are no excuse for Israeli citizens to die.\nOne Israeli military official, who spoke on the condition of anonymity Saturday, reported dozens of checkpoints and roadblocks had been removed by Saturday, including the main one outside the Palestinian border city of Qalqilya. Shoppers from Israel and West Bank towns took advantage of the impediment-free travel to pick up bargains there.\n\"Our main task is to facilitate conduct of the elections,\" Lt. Col. Yorai Qeder, who is coordinating the Israeli army's election planning, said Wednesday. But \"if the Palestinians do not take action against these terrorist threats, we will need to do so ourselves.\"\nDov Weissglas, Prime Minister Ariel Sharon's advisor, delivered a similar warning to former President Jimmy Carter and former French Prime Minister Michel Rocard, who are in the West Bank as election monitors, Israel Radio reported Saturday. Israel asked Carter to deliver that message directly to Abbas, according to a Western official who spoke on the condition of not being identified.\nArafat's death had raised hopes of ending four years of Israeli-Palestinian bloodshed and reviving peace negotiations aimed at establishing a Palestinian state. The United States and Israel accused Arafat of inciting the violence, something he always denied.\nBut militants are spurning Abbas's calls for a truce. Many are angry with him because of his condemnation of their tactics. They also oppose his government's decision to hold presidential elections before legislative ones that would likely propel a wave of Palestinian extremists to power.\nLegislative elections are now scheduled for July 17.\nNew violence flared in the Gaza Strip on Saturday as Israeli soldiers fatally shot an elderly Palestinian in a car at a junction near an army post and a Jewish settlement, according to Palestinian medical officials. The Israeli military, however, said troops killed a gunman approaching a military post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wu, R.-S.; Liu, J.-S.; Chang, S.-Y.; Hussain, F. Modeling of Mixed Crop Field Water Demand and a Smart Irrigation System. Water 2017, 9, 885.\nWu R-S, Liu J-S, Chang S-Y, Hussain F. Modeling of Mixed Crop Field Water Demand and a Smart Irrigation System. Water. 2017; 9(11):885.\nWu, Ray-Shyan; Liu, Jih-Shun; Chang, Sheng-Yu; Hussain, Fiaz. 2017. \"Modeling of Mixed Crop Field Water Demand and a Smart Irrigation System.\" Water 9, no. 11: 885.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The story of Jesus' post-resurrection encounter with the disciples on their way to Emmaus is one of my favorite Gospel stories. It begins with despair and ends with joy. It fills me with hope to see Jesus patiently, graciously walking with people even when they're going in the wrong direction. It's a story that speaks directly to my heart.\nThere's a part of the story, however, that challenges part of my Reformed convictions. Each time I read it, I wonder, \"Why don't the disciples recognize Jesus until he breaks bread?\" Christ walking alongside them, Christ rebuking them, Christ opening Scripture to them\u2026at none of those points does Jesus open their eyes. Rather, it's the table that becomes the place of recognition. My Reformed heritage emphasizes the supremacy of the Word\u2014and rightly so, in my humble opinion. Our belief in the authority of Scripture is evident in our teaching and even in how the pulpit is front and center in most churches in the Reformed tradition. Yet the eyes of the disciples in Emmaus are not opened by the explanation of the authoritative Word (by the Word made flesh, no less!) but by the breaking of the bread.\nSometimes an unfortunate byproduct of our appropriate emphasis on the Word (which Reformed Christians tend to capitalize) can be the relegation of the Sacraments (which, interestingly, Reformed Christians tend to keep in lowercase) to the sidelines, as though they are something kind of optional, to save for occasional use. I think the story of the disciples in Emmaus challenges that perception. I think this story can form part of the case for recognizing how the Sacraments are as important as the Word. The diploma hanging on my wall, after all, declares I am a minister of \"the Word and Sacraments\" (and both words are capitalized on my diploma).\nIf we insist on hearing the Word weekly, why do we not have the same insistence on receiving the Sacrament?\nThere is certainly precedent for this. In Acts we read how the believers \"devoted themselves to the apostles' teaching and to fellowship, to the breaking of bread and to prayer,\" something the church consistently did \"on the first day of the week\" for many centuries as far as we know. Although people (at least the laity) celebrated the Lord's Supper much less frequently by the time of the Reformation (sometimes as seldom as once a year), both Martin Luther and John Calvin advocated for weekly Lord's Supper celebrations. Fast forward to today and it turns out that celebrating the Lord's Supper weekly is the pattern for the majority of Christians around the world and across denominational lines.\nOne can also make a case for more frequently celebrations of the Lord's Supper on the basis of Reformed theology. The Belgic Confession has a great line where it speaks of how God uses the Sacraments \"to represent better[!] to our external senses both what God enables us to understand by the Word and what He does inwardly in our hearts\" (article 33). God knows we are physical beings so he uses physical things (the bread and juice of the Lord's Supper as well as the water of baptism) to communicate his grace to his people. Yes, our ears and eyes need to receive God's Word\u2014it has the power to make \"our hearts burn within us.\" But God also desires to communicate his grace to us through our senses of touch, smell, and taste as we feel the water of baptism, handle the bread and cup of the Lord's Supper, catch a whiff of its smells, and taste the elements in our mouths. Countless times since that evening in Emmaus, Jesus uses the breaking of bread to open our eyes and speak to our hearts that we may recognize him with us.\nWhat's stopping us from inviting Jesus to open our eyes and speak to our hearts by gathering around the Lord's Supper table more frequently?\nIn my experience the resistance to celebrating communion more frequently is often based in \"the way we always have done it, so changing that must be wrong somehow.\" in other word, people have never really pondered this question.\nPeople also say things like \" Won't it get ordinary? If we have to read the preparatory the week before every time and then go through the same form every time, that will get real old real fast!\" There also seems to be among some a fear that somehow this would be drifting toward being Roman Catholic somehow.\nThere is also the older tradition that the elders would be tracking if you are partaking of the sacrament so that you have to be at worship on that particular Sunday and if you have it more often that gets even harder to be consistent. The tradition is that the elders would meet after the service to go through the membership list to see who missed communion. At the church I serve the elders have moved in the direction of simply tracking church attendance generally, each Sunday, since we celebrate communion more frequently and people are way more mobile on Sundays than in the past.\nI think rooted behind a lot of this is also an old fear (fed by some of the wording in our traditional forms) that one has to be \"ready enough\" to partake of the Lord's Supper, and that would take a lot of self-examination if we celebrated it more often. We presently celebrate Lord's Supper about once a month, but even that is sometimes a stretch for people.\nChurch Order Q: Can Elder's Perform the Lord's Supper to Those Who Are Sick or Shut-In?\nWhat's Your Skeleton for Corporate Worship?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Moodle4Mac is an installation package for OS X that contains everything you need for your local Moodle server running on a Mac. It's very easy to get one ... and if you can read this you already have one .\nThe package comes together with MAMP and Moodle. The first installation is done. You must not configure the web server or the database. Also the first user is installed and you can start with your first experiments.\nMoodle is a course management system (CMS) - a free, Open Source software package designed using sound pedagogical principles, to help educators create effective online learning communities. You can download and use it on any computer you have handy (including webhosts), yet it can scale from a single-teacher site to a University with 700,000 students. This site itself is created using Moodle, so check out the Moodle Demonstration Courses or read the latest Moodle Buzz.\nMoodle has a large and diverse user community, speaking over 80 languages in 240 countries (we have more statistics here). The best place to start is Using Moodle, which is where the main international discussions are held in English, but we have a variety of groups discussing other topics and in other languages.\nThe abbreviation MAMP stands for Mac OS X, Apache, MySQL and PHP. With just a few clicks, you can install Apache, PHP and MySQL under Mac OS X. Frequently demanded, and not yet available in this form anywhere in the world, with The MAMP you install a complete server environment on your Mac OS X computer in a matter of seconds, whether MacBook or iMac. In the same way as similar packages from the Windows and Linux worlds, this program is free of charge as well.\nThe MAMP is very easily installed, as is typical for the Mac, without interfering with any Apache installation already activated under OS X. Apache, PHP and MySQL are installed without launching a script or changing any settings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AUSTIN, Texas\u2014Pivot3, a leader in hyperconverged infrastructure (HCI) solutions for the modern data center, experienced record growth throughout 2016, with total revenue increasing by 84 percent over the prior year\u2014including a more than 200-percent increase in Q4 2016 from Q4 2015\u2014and significant deals secured across federal, healthcare and education customers.\nAUSTIN, Texas\u2014Video analytics is an area of interest for video surveillance storage provider Pivot3, which received a $45 million equity and bank financing on Feb. 24.\nAUSTIN, Texas\u2014Storage provider Pivot3 on August 7 closed a $12 million funding round which will be used for \"product development, sales and marketing,\" Pivot3 CEO Ron Nash told Security Systems News.\nAUSTIN, Texas\u2014Ron Nash is the new CEO of storage provider Pivot3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"emotional support do not qualify as service animals under the ADA.\nAssistance Dogs as well as staff and volunteer education.\nwhich helps people with access issues and other requests.\nAn e-mail discussion list for those who train their own assistance dogs.\ndogs and their owners. The project is currently on hold due to lack of funding.\nmaterials including books, CDs, and videos.\nspecific disabilities the dogs are trained to help.\nchallenges, and critical, chronic or terminal illness.\npsychological conditions like depression, PTSD and burnout.\nDRC is a not-for-profit education and research facility.\nthe public interactive programs offered at DRC.\npurpose is to provide assistance dogs to people with disabilities.\nBorrow from Our Lending Library under PRC Quick Links.\nNew Zealand and the UK.\nBozzo, Linda. Service Dog Heroes. Berkeley Heights, NJ: Bailey Books, 2011.\nBrill, Leigh. A Dog Named Slugger. Memphis, TN: Bell Bridge Books, 2010.\nfor Independence. Poems about his service dog Lex.\nCreel, Ann Howard. Nicki. Middleton, WI: Pleasant Co., 2007.\nIndependence. New York: Yorkville Press, 2003.\nStories of actual service dogs.\nDuncan, Susan. Joey Moses. Seattle: Storytellers Ink, 1997.\nTeen fiction, based on actual service-dog team.\nthe Disabled. Mechanicsburg, PA: Barkleigh Productions, 2004. 2nd edition.\nLaw and the Evolution of Canine Caregivers. Springfield, IL: Charles C.\nabout training a service dog.\nDisabled. New York: Dutton Children's Books, 1991.\nKinsella, Audrey. Dingle: The Helpful Ice Cream Cone Delivery Dog.\nMcDaniel, Melissa. Guide Dogs. New York: Bearport Publishing Co., 2005.\nMiller, Marie-Therese. Helping Dogs. New York: Chelsea Clubhouse, 2007.\n(Dog Tales: True Stories About Amazing Dogs series).\nRetriever Who Saved Him. New York: Hyperion, 2011.\nMontalvan is a former U.S. Army Captain in the Iraq War.\nDog of the West. Santa Fe: Sunstone Press, 2007.\nBook on training your own dog to be a service dog.\nOliver, Clare. Animals That Help Us: Animals Helping with Special Needs.\nNew York: Franklin Watts, 1999.\nOsofsky, Audrey. My Buddy. New York: Henry Holt and Company, 1992.\nPresnall, Judith Janda. Canine Companions. New York: Thomson Gale, 2004.\nRing, Elizabeth. Assistance Dogs. Brookfield, CT: The Millburn Press, 1993.\na Miracle. New York: Hyperion, 2010.\nSchuh, Mari. Assistance Dogs. Mankato, MN: Capstone Press, 2011.\nTrained to Help the Disabled. New York: Morrow Junior Books, 1988.\nTada, Joni Eareckson. Darcy's Dog Dilemma. Elgin, IL: Chariot Books, 1994.\nTuohy, Tom. Kiss of a Dolphin. Richton Park, IL: Lumen-us Publications, 2006.\nmonkey for him. Ebeneezer is Maria's toy monkey. Children's fiction.\nChildren with Disabilities. Sherborn, MA: Aquarius Health Care Videos, 2004.\nscroll through the videos listed on the right.\nor delay in seeking it because of something you have read in this message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Excellent! Great quality and fast shipment.\nGetting frames from you has streamlined my workflow. I'm a happy camper.\nI have used the Frame Destination for years and have never been disappointed.\nGreat frame. Solid and easy to disassemble to insert my art. I appreciate that it came pre-assembled.\nPerfect. Easy to assemble, very stable, nicely packed, can reuse for storage and sending to shows.\nEverything was clean and of good quality. Everything arrived quickly and safely to my front door.\nI bought a metal frame for a metal photo print - looks great. Very happy with the purchase.\nPrices on their frames are very reasonable & the service and selection is top-notch. My order was packed very well and received very quickly. I'll be buying from Frame Destination again soon. Thanks!\nLove the frames, but the care taken in packing and wrapping them is an art form in itself. Compliments to the shipping Dept!\nSuper happy, Frame assembled perfectly and was exactly the dimensions requested. Color also matched the Web site picture very well. We get comments all the time on how nice it turned out.\nRECEIVED MANY COMPLIMENTS ON THE COLOR. THE FRAME COLORS SHOWN IN YOUR PHOTOS ARE EXACT AND THEY ENHANCE MY ART WORK. IT'S TRULY A PLEASURE DOING BUSINESS WITH YOU!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can not leave Alicante without visiting the flagship of the city. You will enjoy a breathtaking view over the coastline from the several viewpoints you will find on your way to the castle summit.\nThis promenade is famous for its three-coloured mosaic floor. It is the access to some of the main tourist spots of the city and you will find both tourists and alicantinos.\nYou will find a wide choice of souvenirs and handicraft shops as well as many tapas restaurants, caf\u00e9s and bars with terrace where you can observe the city atmosphere.\nIf you walk across this promenade, you will find some of the most emblematic buildings of Alicante, you can reach both El Postiguet Beach and Canalejas Park.\nThe ideal place to enjoy the amazing sunlight of Alicante. The favourite for both tourists and Alicantinos.\nYou can go sunbathing, do some swimming and then go for some tapas in of the chiringuitos by the sea.\nIf you are more active, you will find a wide offer for water Sports (windsurf, kitesurf), beachfootball and Volleyball. You will also find many playgrounds for kids.\nSan Juan Beach is also famous for its promenade where you will be able to do some shopping, eating and taking delicious Pictures.\nMore than 1km of promenade by the sea. You will find a wide offer of both restaurant and bars. This is the ideal place for leisure.\nThis charming spot of the city is close to Santa Barbara Castle and also is the access to the old town.\nYou can only visit Santa Cruz on foot but it is absolutely worth because you will have the feeling of being in a small town. The white houses full of plants and the narrow streets will transport you to an ancient time.\nThe ideal place to taste the typical products of Alicante. There are also many terraces, restaurants and flower shops in the square next to the Mercado. You can also try the \"Tardeo\" experience; young people meet up at the mercado square on Saturday from 13.00 to have tapas and then go to Casta\u00f1os street for a drink. The atmosphere is so lively.\nThe most important religious monument of the city. It is located in the heart of the old town so you will be able the visit this charming part of the city. This church was built over the ruins of a former construction and it is a mix of Baroque and Herrerian style. The 45 metre blue dome inside is amazing.\nAnother religious monument you will like a lot is the Basilica de Santa Mar\u00eda, also located in the old town.\nThe excellent weather conditions in Alicante allow you to practice outdoor sports thorughout the year and golf is one of the most interesting sports you can practice. It does not matter if you are an amateur or a beginner, Alicante Golf Course is the ideal place to play this elegant sport.\nAfter playing, you can eat out or go Shopping in the nearby La Seda outdoor shopping mall. You will find here delicious restaurants.\nAlicante Golf is just 5 minute ride away from El Palmeral de Madaria.\nThis little island is one the gems in Alicante. You can go there by boat (KonTiki) from the port. The journey takes 45 minutes.\nYou will find a paradise of crystalline Waters where you can dive. Tabarca is considered as a marine reserve. The 2nd main attraction in Tabarca is El Caldero; a typical fishermen meal, it is a kind of fish paella. Before eating the rice, you will have the fish that has been used to cook the paella.\nThis is an ideal plan for a one-day trip. Do not forget to enjoy the sunset from the Island.\nHope you make the most of this list when you visit Alicante and we remind you that in El Palmeral de Madaria you will find the ideal apartments to stay in Alicante.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than four dozen gubernatorial nominees were confirmed by the Maryland Senate on Friday, including Commerce Secretary Kelly Schulz.\nThe Senate voted 45-0 to support the nominees from Gov. Lawrence J. Hogan Jr. (R), the second time in two weeks a batch of the governor's nominees has received unanimous support.\nOthers confirmed on Friday include three nominees to the State Board of Education \u2015 Vermelle D. Greene, Jean C. Halle and Warner I. Sumpter \u2015 and three nominees to the State Labor Relations Board, where a labor representative was confirmed for the first time since a vacancy arose in 2016.\nLocal nominees to the boards of elections in Baltimore, Harford and Wicomico counties were also approved.\nNot up for confirmation Friday were three appointees to the Handgun Permit Review Board, which has come under scrutiny from the Senate Executive Nominations Committee in recent years. The committee is expected to discuss those nominees for a second time at a meeting Monday evening.\nA full list of those confirmed on Friday is here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u203a Uncategorized \u203a Perfect Pen or Holy Grail?\nPerfect Pen or Holy Grail?\nIt all depends on the paper \u2013 how the pen works, that is. This realization comes from many years and many journeys in search of the \"perfect pen.\" For me, this odyssey is the search for my personal Holy Grail \u2013 that ever elusive thing that promises health, wealth and happiness (and eternal salvation)!\nIn truth \u2013 I have no illusions about health and wealth unless a gliding pen serves as a secondary sort of salve. Makes sense, right? Health as a side effect from the glee of finding and using that pen; wealth from the bestselling novel that follows!\nIn truth, my lust is about simpler stuff. I search for that perfectly balanced, correctly weighted, optimal tip and a \"smooth-as-butter\" ink flow that writes \"first time every time!\" In addition, this, the ultimate recorder of text, must magically transform my unreadable, ugly script into something beautiful and swan-like. All of this for less than $50.\nDoes such an instrument exist? No. Well \u2026. Sort of, and here's why my equivocation. After many sweaty treks through the hottest days of summer and slippery slogs on the snowiest days of winter, I finally arrived at the answer. And this is it: it's not just the pen. It's the symbiotic relationship between pen and paper that makes the magic.\nFor example \u2013 I recently purchased two really wonderful pens \u2013 one a Levenger, the other a Lamy \u2013 both fountain pens. I then bought a notebook at Staples to use as my journal. I then sat down for a long writing session with my new pens and my new notebook only to be thwarted with tremendous disappointment. First one, and then the other, fountain pen bled right through the paper \u2013 seeping through to the other side as well as yielding veritable ink blots that cut a far bigger swath of spreading ink than I intended.\nBack to the original question: is there such an instrument? I have my top 10 favs, but the pen du jour is a shifting sand that balances precariously on paper, type of pen, my coffee consumption, and purpose in writing. In other words, I think I might have better luck seeking the Holy Grail!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"External data storage devices are marking a new chapter in the realm of saving and carrying a huge amount of data from one place to another. A variety of external data storage devices have made a room in the market; hence, techies have a wide range of choices to buy the best suited external storage device. Techies can select external storage devices to cater various tasks like adding space to the computer's internal hard disk, taking back up of important data, transferring data in an impeccable manner between two computers, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"They say that kids do the funniest things, but Ben and Jackee Belnap likely weren't laughing when they discovered what their little boy had done with a substantial amount of their money. 'Whats more, after their toddler made the calamitous move, the Utah-based couple had to hatch a plan to get the money back.\nThe Belnaps' tale of woe had begun after Ben's parents had kindly purchased the pair some passes for the University of Utah football season. And presumably not wanting to be indebted indefinitely to the older couple, Ben and Jackee saved up their cash in order to soon return the financial favor.\nEventually, the conscientious pair had put the grand sum of $1,060 in an envelope that was waiting to be given to its intended recipients. But before the money could be handed over, the envelope itself vanished. Naturally, the Belnaps, therefore, began to look all over their property for the missing item.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Infuse your meals, baked treats, and snacks with Pure Granulated Maple Sugar by Kasza Sugar Bush. This authentic maple sugar is an ideal sweetener that delivers the rich, delicious taste of maple. Produced in Shelby, Michigan, this granulated maple sugar brims with the natural sweetness of Michigan maple syrup. Use it to sweeten muffins, breads, cookies, cakes, waffles, and more.\nEverything was amazing!! Ordered maple syrup, maple candy and maple cream!! Am on the website ordering some more!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PACT is conducting a series of community consultations about the future of the most influential labour agreement in Canadian Theatre, the CTA.\nBecause the Canadian Theatre Agreement, negotiated between PACT and Canadian Actors' Equity Association, influences practice across professional theatre both inside and outside PACT membership, it is vital for PACT to take into consideration how a newly negotiated CTA will affect the larger theatre community.\nCommunity consultation sessions will be held in many of the major city centres across Canada so that each individual theatre community and any surrounding areas have the opportunity to participate in these important conversations. See the full schedule.\nThe consultations are open to PACT members and non-member organizations and producers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy Warhammer 40,000: Dawn of War III as a Steam Key. Step into a brutal battle between three warring factions.\n. Warhammer 40k Dawn of War V1.4 Trainer +3 Warhammer 40k Dawn Of War GOTY V1 . no one could foresee that a developer . trainer doesnt work with my steam .\nAttain a campaign score of over 90,000 points in Dawn of War II. Not applicable to guests. Legend (39) . rating in Dawn of War II. Feel No . Warhammer 40K .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..1af5c9728ad453abcadcb200639a2564c60ca7b2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Wow what a show we had last week in Ziggo Dome Amsterdam! On September 19th the Liberty Global Tech Summit 2018 was held there and me, Sanne, Paula and Annika had the honor to perform an aerial hoop act in the main hall where the guests were having a massive dinner. The whole space was lit, but when we were starting they turn off all the lights and put spots on us so everyone was really watching and paying attention which was really nice. We were hanging quite high, between 7 and 9 meters, but it felt so gooood! Our show was a huge success and everyone was so happy afterwards! Thank Pzia for this job and thanks girls for the great work!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Facebook, MyPhone announced that they will launch the myNX1 soon. It is a budget Android Oreo 8.1 (Go edition) smartphone with a notch.\nBased on the image we got, the MyPhone myNX1 will feature a 6.18-inch IPS display with FWVGA+ resolution at 174 ppi and 19:9 screen aspect ratio.\nUnder the hood, it has entry-level specs. You will find a 1.3GHz Spreadtrum SC7731E SoC with Mali-400 MP2 GPU.\nAs expected, it only has 1GB RAM and 8GB expandable storage via microSD up to 32GB.\nIt also has 2,950mAh battery to keep the phone running.\nAt the back, you will find an 8MP primary shooter with AF. For selfies, the myNX1 comes with a 5MP snapper with softlight LED flash.\nThe device also has a fingerprint scanner. It also goes with WiFi 802.1. b\/g\/n, 3G, HSPA+, Bluetooth 4.0, FM Radio, GPS, and dual SIM slots.\nIt will be available in the Philippines nationwide soon for PHP 3,999. It also comes with FREE PHP 50,000 of accident insurance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Life is messy, it isn't always what you think or plan. We make imperfect decisions, keep an open mind and embrace what you may not understand.\nLoosen up a little, take life as it comes and just enjoy it for what it is. The most successful people in the world haven't had linear paths to success so don't compare yourself to other people. You must run your own race and define your own success. Don't get demotivated when everyone else around you succeeds. There are various reasons why others do well, whether economical, as well as good old luck and likeability but, everything has its own time. In any event, what other people call failure is often just the gift of a new direction and a new opportunity.\nBe fearless. Don't be scared, not of monsters and snakes and stuff, but of the other stuff like social pressure to buy a house, have a stable job, get married, have kids, etc. Somehow life will work out just the way it was intended.\nI have spent so much time in my career working hard but the reality of the matter is that the hardest workers don't always succeed. You only really succeed when you change your perspective from \"working hard\" to \"increasing your value\/worth\" and this is done by increasing your productivity, mastering your skills early and mostly by being adaptable.\nDon't lose your sense of curiosity and desire to learn from others around you - they are your best teachers and cheerleaders.\nDon't let your emotions get involved. Things change all the time (leadership, strategies, teams, policies etc.), and any kind of change elicits emotional responses, such as fear, feeling overwhelmed or wanting to run away. Being emotionally intelligent in these situations will help you to stay adaptable and relevant.\nDon't be afraid to say \"no\" when you feel you should, and to say \"yes\" when it seems right to you. Don't live in anyone's shadow or be a pawn in anyone else's life. Learn how to make choices early on because you are only accountable to yourself and you are the only person who needs to live with the decisions that you make or don't make.\nSomeone once told me that black people can't be successful transactional lawyers and that I should rather be doing something else better suited to the kind of services that \"black clients need\". If this happens to you, don't get unnerved. It is true that you may not come across many senior black transactional lawyers on the deals that you may work on, but it doesn't mean that that will always be the case. Historically in my country black women specifically suffered triple oppression in the form of class exploitation, national oppression and gender discrimination. So, it may still take time for to see changes - stay the course, be globally competitive and enjoy the journey.\nDon't be in a hurry to grow up. Just laugh at the mean girls at school, you will grow up to be gorgeous!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AGF Short-Term Income Class is an open-end fund incorporated in Canada. The Fund seeks to provide maximum income while preserving capital and liquidity. The Fund invests primarily in short-term instruments, government guaranteed securities and corporate paper with a minimum of A credit rating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Tecnocap, Sustainable Development and Social Responsibility identify the implementation of credible policies and daily practices far beyond abstract declarations of values and principles. The three sustainability dimensions, social \u2013 economic \u2013 environmental are reflected into a constant and growing effort to improve lives quality of Tecnocap employees, customers and communities in general. In the last ten years, the turnover growth and the international expansion have not prevented Tecnocap from local communities commitment and interest in people. Internationalization pushed sustainability targets even higher in terms of reduction of raw materials and pollution during the production processes.\nTransparency, solidity and integrity are the true pillars of our company decisions. Staff enhancement and constant training are evidence of company commitment. Responsible management of resources, safety and health, protection of environmental resources are key chapters of our commitment to reduce the impact deriving from our production cycle.\nAluminum and iron are respectively the third and fourth most present minerals on the earth. Metal is 100% recyclable and can be recycled indefinitely without losing any physical properties. Throughout the entire life cycle \u2013 from the extraction of raw material to production, consumption and recycling \u2013 the advantages of metal are incomparable to any other material and represent the essence of sustainability.\nPeople working at Tecnocap represent the main success of the group, in terms of loyalty and mutual trust. The enhancement of talent and the valorization of a supportive environment are basic values as well as the defense of employment and the improvement of economic, professional and social conditions. Working at Tecnocap means working daily in a safe and comfortable environment, well beyond the compliance with laws and regulations.\nProtection and promotion of collective well-being represent strategic objectives of Corporate Social Responsibility. The socio-economic fabric of the local communities where our production facilities are established is a distinctive asset for the growth of our group. Tecnocap keenly supports education activities for young talents, cultural projects and philanthropic activities.\nBesides the obtainment of quality, environmental and ethical certifications, ISO 14001 \u2013 ISO 9001 \u2013 BRC \/ IOP \u2013 Smeta 4 Pillars \u2013 AIB, Tecnocap is engaged in a constant improvement of safety and health protection of workers, communities and consumers. With regard to the production cycle, Tecnocap group launched a plan to constantly reduce CO2 emissions, energy and water consumption, and waste production.\nAmong the several projects for the development of innovative technologies, it is interesting to highlight the research of new compounds without plasticizers components, with zero migration for increasing food safety and greater consumers health.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Love classical \"what am I?\" riddles? Visit http:\/\/riddled.azurewebsites.net to enjoy all the fun of stylised riddle solving! One riddle published each Friday along with the solution to the last. Currently undergoing maintenance, but all existing published riddles still available. You can check your solution on the site, as well as email us with your own riddles or to request personalised riddles.\nModerator of Science Fiction and Fantasy Stack Exchange, where I try to be always present in case of need, while not being too obtrusive or trigger-happy.\nharry-potter: I like the books reasonably well, but they're ridiculously overrated and don't deserve all the hype they get.\nstar-wars: I've never seen Star Wars, and here's my favourite summary of it.\ndoctor-who: Tennant > Capaldi > Eccleston > Smith and Donna > Rose > Martha > Clara > Amy.\nthe-hunger-games: I'm this site's top expert, apparently.\nAlso a fan of many fantasy novels sadly underrepresented here: ping me in chat for recommendations.\nActive user of Puzzling Stack Exchange. This is an SE site unlike any other, where 'questions' are usually puzzles (created and posted by someone who knows the solution, as a challenge to the community) and 'answers' are solutions. Many of them are incredible works of art created by extraordinary minds, awe-inspiring to read even if you have no clue how to solve them. If you're a traditionalist, you can also ask real questions about puzzling instead.\nActive user of Literature Stack Exchange. If you enjoy reading - novels, short stories, plays, poetry, comic books, or anything else - do come on over! You can help us to solve some unanswered questions, or post your own questions to get answers to the things you'd always wondered.\n60 How to ask a good story-ID question?\n105 How do I get the last lines of dust into the dustpan?\n91 Five coloured doors, which leads to freedom?\n80 Come Take a Look at our New Contributor Indicator!\n74 Why is The Hitchhiker's Guide to the Galaxy referred to as H2G2?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patrons looking for new titles this week would do well to check out the movie listing, which sees three new titles, led by the 40-years-on sequel to the original horror classic Halloween. The soundtrack to the acclaimed animated Spider-Man movie overtakes 21 Savage for the top spot in music. In fiction, a new James Patterson collaboration makes its debut while Delia Owens stands strong at #1 once more, while two favorites make a return to the list in non-fiction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nowadays when you want to send money to Colombia, there are so many companies that promise user-friendly services and great rates. But is this really true? Many times when you take a closer look, you'll find hidden fees, long waits, and difficult to reach pick-up locations. And some companies don't give you the option to have your recipient pick up money at a cash pick-up location at all!\nYou can only send it directly to a bank through a direct deposit. This is not very helpful or inclusive for the 35% of unbanked Colombians who have no access to formal financial services in Colombia, according to the Inter-American Development Bank.\nRemitly is different. We are constantly thinking of ways to make your experience as fast and affordable as possible for both you and your loved ones. Sending money to Colombia with Remitly has never been easier.\nWhy use Remitly to send money to Colombia?\nRemitly is an easier way to send money to Colombia since you can do it at any time and any place from your phone or computer. In addition, we offer excellent Colombian Peso (COP) exchange rates so your loved ones get more money. And you can send the money directly to a bank account or to one of our thousands of cash pick-up locations in Colombia. We have many partners that your recipients can receive money from such as BBVA, Grupo \u00c9xito, Davivienda and many more. You can see a full list of all our partners in Colombia here.\nWhen you use Remitly for the first time to send money to Colombia, you will receive a special USD to COP exchange rate so more pesos per dollar make it to your loved ones. See today's rate here. Plus, sending with us is FREE on transfers over $600! For more details visit our page about sending money to Colombia.\nSending money to Colombia with Remitly really is that simple. Forget about high fees or long waits for your family members.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Friday! And we are not just wishing it, but making it so. Actually, shout to Yamin Semali for making it so\u2026 by dropping the gem into the inbox a few days ago.\nUsing pieces of \"Will O' The Wisp\" from Miles Davis's \"Sketches of Spain,\" I remixed Kendrick's banger.\nYeah, just a li'l something the homie do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tomorrow is Women's Equality Day\u2013a day on which we celebrate the passing of the 19th amendment, which gave women the right to vote.\nHowever, women's suffragist movements were not as valiant in their moral victories as we like to remember. The movement rode on the coat tails of the Abolitionist movement, while also using white supremacy as the reason white women needed the right to vote.\nThe argument was if black men were given the right to vote, they'd have more power than white women, and that was unacceptable. White women needed to be given more power to cancel out the political power of black men.\nOppression is complicated and seriously messed up.\nWhile the 19th amendment didn't exclude black women explicitly\u2013that would have violated the 15th amendment\u2013there were measures put in place to suppress votes of all people of color. These measures were extremely successful, and to some extent, still exist to this day via voter ID laws.\nWe shouldn't still have this problem, because in 1965 the Voting Rights Act passed, giving the Federal government the authority to put their foot down when it came to voting suppression. This is effectively when most women of color were able to actually vote.\nJust because you don't personally experience sexism, racism, classism, ableism, ethnocentrism, heteronormative bias, or any other type of oppression doesn't mean those things don't exist. And we, as a society, can't effectively combat them until we recognize that they do\u2013and that they wreak very real harm on our fellow citizens.\nHave we gotten better about these things?\nUntil a few weeks ago I would have said yes.\nEven if we had, does that mean we should stop where we are because it's comparatively better than what it has been in the past?\nNo. Because it's still bad.\nToday, we'll be looking at women's equality through the lens of personal economics.\nThe Wage Gap. It Exists.\nThe most recent data on the wage gap is probably most effectively measured when compared to white, non-Hispanic men. This is not because white, non-Hispanic men are inherently evil as individuals, but rather because as a group, they are the most highly compensated demographic.\nWhite, non-Hispanic women make 75% of what white, non-Hispanic males make.\nBlack women make 63% of what white, non-Hispanic males make.\nHispanic and Latina women make 54% of what white, non-Hispanic males make.\nAsian women make 85% of what white, non-Hispanic males make.\nPacific Islander women make 60% of what white, non-Hispanic males make.\nAmerican Indian and Alaska Native women make 58% of what white, non-Hispanic males make.\nTrans women face a 12% pay decrease after transitioning.\nBut women choose lower paying fields.\nYes. Career choice does contribute to a portion of the gap\u2013though not all of it.\nWe need to look a little bit deeper than that, though. Why does society value an accountant over those educating the leaders of tomorrow? Lawyers above nurses?\nRead this: Should Teachers Ever Make Six Figures?\nWe don't want to confront this reality, but women took on many of these roles for a long, long time without receiving any pay. In light of this, it's not really all that surprising that some of the most important jobs\u2013like daycare workers ensuring the safety and early education of our children\u2013get compensated at rates barely above minimum wage.\nThen there's the question of why women enter these fields. Do we do it because we were trained from a young age that our gender gave us inherent gifts in care-taking fields? Or are there simply more opportunities for women in fields that pay less?\nEither way, the answer is problematic.\nYes, some of us do. But in one way or another, men play a role in making those babies, too. Aside from a brief period of physical recovery for women, there's no reason men can't equally contribute to the care and rearing of their children.\nIn fact, when they contribute to that cause, they are viewed as altruistic.\nWhen women do the same, they are viewed as incapable of meeting their work responsibilities\u2013even if that's not true.\nI'm pretty sure that argument is not only problematic, but inherently false.\nExcept we do. When we ask for a raise, we're 25% more likely to get denied compared to our male counterparts.\nBecause women are discriminated against in the realm of pay, we've led the largest boom in the entrepreneurial sector over the past ten years. This is in large part due to the entrepreneurial efforts of black women in particular, who face one of the largest pay gaps.\nOur daughters are not as confident in matters of personal finance as our sons. It's not because we are trying to actively oppress our individual daughters, but it may be because we subconsciously give in to a system of oppression that we have normalized as a part of our culture.\nWe need to combat these measures by talking to our daughters more, and educating ourselves more about personal finance concepts\u2013even if we're not marketed to.\nI was at a conference last year where my peer pointed out a present online marketer advising against using Pinterest as a platform for advertising investment content\u2013because the platform is dominated by females and women just don't care about investing.\nThis is but one example of the culture we're up against. We have to market to women whom we don't think care. Not only because it's the right thing to do, but because women own 40% of all stocks and make 85% of consumer purchasing decisions, which includes financial products.\nWe have the right to vote, but we have a long way to go before we're economically equal. As with the suffragist movement, this is especially true for women of color.\nMen, take solace in the fact that women's equality is not about tearing you down or blaming you as an individual. But if we, men and women, want our daughters, mothers, sisters, partners and friends to be equal, we need to work together against the systems of oppression that do indeed exist.\nNew Year. Old Goal. New Energy.\nThis entry was posted in Think on August 25, 2017 by femmefrugality.\nThis is a very thought provoking post. I am glad you are not afraid to take on political and socio-economic issues in your blog!\nThanks, Lizzy! Comments like yours keep me doing it. It's an intimidating venture, and sometimes I mess up and cause offense where I mean none. But I feel like it's important. Coupons and budgets are great, but there are some bigger underlying issues to people's finances that need to be addressed.\n\"Just because you don't personally experience sexism, racism, classism, ableism, ethnocentrism, heteronormative bias, or any other type of oppression doesn't mean those things don't exist.\" \u2013 Exactly. And the current situation (the current administration) is making it abundantly clear that we *all* need to speak up and speak out on behalf of those who are experiencing these things. I'm trying to find ways to do that wherever and whenever I can. Thanks for doing it in this post.\nExactly. It's so weird because I feel like things that were just mildly controversial a year ago (i.e. the wage gap) are now wildly so. We have an opportunity to speak up and bring attention to these things in a magnitude they may not have garnered in the past, but we're also walking a dangerous line if we remain silent and let oppressive voices take control.\nI had no idea about some of those stats. I also wonder what Asian women are doing to manage to get the smallest wage gap. I want to learn their secrets.\nThose are some pretty fascinating stats. And it's true what you say about the early feminists being far from enlightened by our standards today. I just read in the paper today a piece about how few of the \"heroes\" of our history had attitudes that would be tolerated by us now. The suffragettes were, as you said, racist, and many stood for eugenics as well. In many ways, we're very lucky to live when we do. Thanks for clarifying where the challenges lie so that we know in which direction to take that next step.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Be a part of history \u2014 help bring professional rugby to D.C.!\nWe may not have made a final venue selection, and we haven't yet committed whether we're entering competition for the 2019 or the 2020 season; but you can help us to make both of those decisions, as well as demonstrate the strength of our market to Major League Rugby and make professional rugby in D.C. a reality, by committing to interest in a season ticket. All it takes is a $20 deposit per seat, which is fully refundable in the next 180 days, in the event we fail to deliver the team.\nYour deposit will secure for you a priority position for the selection of seats, on a \"first come, first serve\" basis \u2013 so the earlier you make your deposit, the higher a priority you will have for choosing seats!\nAssume that we will secure a location that is accessible by car and Metro, most likely inside the D.C. Beltway. Also assume that your deposit will be applicable to seats in a new, rugby-specific venue that we develop, or an existing venue that we lease, including a temporary venue if we begin play in 2019 before a permanent home can be constructed. Although specific seating prices cannot be set until the venue is finalized, the estimated average ticket price is $20 per seat, per game (assuming 8 \u2013 10 home games per season).\nStrong support from our region's community of rugby fans will ensure that D.C. gets the professional rugby team you deserve!\nWhat do I get for placing a deposit?\nPlacing a deposit to purchase DCMLR Season Tickets secures for you a priority position for the selection of seats, on a \"first come, first serve\" basis \u2013 so the earlier you make your deposit, the higher a priority you will have for choosing seats.\nA DCMLR Season Ticket deposit is $20 per ticket.\nWhen can I choose my specific DCMLR Season Ticket(s) location?\nEveryone who places a DCMLR deposit will receive information regarding the sale and seat selection as it becomes available.\nCan I get a refund of my deposit?\nYes, prior to your purchase of DCMLR Season Tickets your deposit is fully refundable and you may request a refund at any time. If you obtain a refund you will lose your priority position to purchase DCMLR Season Tickets.\nWhere will the team be playing?\nWe haven't yet determined the venue for our games. Assume that we will secure a location that is accessible by car and Metro, most likely inside the D.C. Beltway. Also assume that your deposit will be applicable to seats in a new, rugby-specific venue that we develop, or an existing venue that we lease, including a temporary venue if we begin play in 2019 before a permanent home can be constructed.\nWho will coach the team and have you identified any players yet?\nDC MLR's approach to rugby operations is under development, and no coach has yet been hired, nor have any players been contracted. You may assume for the purposes of your season ticket deposit that we intend to build a team reflecting the level of excellence that has been established throughout Major League Rugby during its first season. Updates to this process will be provided to prospective season ticket-buyers as information becomes available.\nHow do I update my contact information if it changes?\nEmail us at info@dcmlr.com or call 202-540-2440.\nWho do I contact if I have other questions about signing up?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pittsburgh, PA \u2013 June 16, 2014 \u2013 The Pittcon Organizing Committee is pleased to announce its participation in the International Year of Light initiative at the Associate Sponsor level. The United Nations (UN) General Assembly 68th Session has proclaimed 2015 as the International Year of Light and Light-based Technologies (IYL 2015).\nIn proclaiming an International Year focusing on the topic of light science and its applications, the UN has recognized the importance of raising global awareness of how light-based technologies promote sustainable development and provide solutions to global challenges in energy, education, agriculture and health.\nAs an Associate Sponsor, Pittcon and its sponsoring society, Spectroscopy Society of Pittsburgh, will support this effort by offering relevant technical sessions including a special symposium, \"The International Year of Light,\" organized by Bernard Lendl, member of the Society of Applied Spectroscopy (SAS), student and teacher workshops and short courses. These events will take place during the Pittcon Conference and Exposition, March 8-12, 2015, in New Orleans, LA, Morial Convention Center. Other onsite activities will include a free display in the lobby which will be open to the public. Additionally, the Pittcon Science Week Committee will be conducting a student spectroscopy workshop in a local New Orleans school (TBD) in January 2015.\nThe International Year of Light and Light-based Technologies (IYL 2015) is a global initiative adopted by the United Nations (A\/RES\/68\/221) to raise awareness of how optical technologies promote sustainable development and provide solutions to worldwide challenges in energy, education, agriculture, communications and health. With UNESCO as lead agency, IYL 2015 programs will promote improved public and political understanding of the central role of light in the modern world while also celebrating noteworthy anniversaries in 2015\u2014from the first studies of optics 1,000 years ago to discoveries in optical communications that power the internet today. The IYL Global Secretariat is located at the International Centre of Theoretical Physics ICTP. Founding partners of IYL are the American Physical Society (APS), the Deutsche Physikalische Gesellschaft (DPG), the European Physical Society (EPS), the IEEE Photonics Society (IPS), the Institute of Physics (IOP), the International Society for Optics and Photonics (SPIE), the lightsources.org International Network and The Optical Society (OSA).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5. A basillisk is born from a chicken's egg hatched beneath a what?\n8. A grey, furry, flying insect known to infest beehives?\n19. What is the house symbol of Hufflepuff?\n58. What color Quidditch robes do the Hufflepuffs wear?\n59. In the first book, Hagrid hatches a dragon egg. What is the dragon's name?\n7. Who delivers baby Harry to Dumbledore in Privet Drive at the start of the movie?\n12. What does Hermione say to unlock the door between her and fluffy?\n17. Yes or no. Since Draco Malfoy became the Slytherin's seeker, have they ever beaten the Gryffindors at Quidditch?\n21. What creature is sometimes known as a clauricorn?\n23. An eight-eyed spider capable of human speech?\n33. Where are hippogriffs native to?\n42. Ron Weasley has a hand-me-down pet rat. What is its name?\n49. How many types of troll are there?\n52. A common garden pest?\n53. An ash-colored wart hog capable of invisibility?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following the announcement of his long-awaited second album Song For Alpha and surprise EP Slow Fade, Daniel Avery shares a video for the latter's title track today. The London producer has also announced a run of dates in North American and Europe in support of the new LP this spring. These dates include special all-night-long sets in New York, Los Angeles, London, Glasgow, Paris, Dublin and Manchester. Full dates below. Watch the video for \"Slow Fade\" now at FADER.\nSong For Alpha will be released April 6 on Phantasy\/Mute in the United States and Canada and via Phantasy for the rest of the world. This new album follows Avery's 2013 debut, Drone Logic, which was received with praise across the electronic music press upon its release. Preorder Song For Alpha here. Discounted album and EP formats available when purchasing tickets to Avery's tour here.\n\"To create the video for 'Slow Fade' we used a combination of techniques. Our idea was to create something structural with no tangible sense of scale and then to degrade the footage throughout the video (a slow fade). We wanted to create something akin to watching chemical reactions or seeing the start of nuclear fusion in slow motion at macro level.\nSong For Alpha is Avery's exploration of the space in which home listening and club music intersect, in no small part inspired by his transient life spent between nightclubs, flights, the passenger seats of cars and hotel rooms.\nWith newfound energy and time to develop, his sonic vocabulary has expanded. Here, the booming sound of the big room is brilliantly countered by the music of the small hours. Where celestial ambient lullabies \"First Light\" and \"Days From Now\" sit perfectly next to the mesmeric techno assault of \"Diminuendo\" and \"Sensation;\" where both \"Projector\" and \"Clear\" evoke early rave records, only ones engulfed by waves of beautiful cyclonic distortion. William Basinski, Warp's Artificial Intelligence, Brian Eno plus his own excursions with Alessandro Cortini all serve as touchstones for a record that sees Avery take his signature psychedelic-electronic sound to new dimensions, a sound that plays to the head as much as the body.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Versions of this article appear on NBC News, Public Radio International and the Buffalo News.\nApril 24: This story has been updated.\nThe New York Attorney General's office has joined Virginia officials in investigating a veterans organization that spends nearly all of the money it raises on telemarketing consultants and its own leader's salary.\nThe nonprofit Center for American Homeless Veterans must give New York officials documents \"relevant and material to an investigation and inquiry, undertaken in the public interest, of CAHV's solicitation of charitable contributions,\" according to documents obtained by the Center for Public Integrity.\nThe Center for American Homeless Veterans and two sister organizations run by the same man \u2014 retired Army Maj. Brian Arthur Hampton \u2014 together have raised millions of dollars from donors across the country, but very little of the money goes to veterans themselves, the Center for Public Integrity reported in December. Most people who donate to groups run by Hampton make modest contributions in response to telemarketing calls, direct mail or other such solicitations.\nBrian Arthur Hampton, right, pictured with Sen. Richard Burr of North Carolina, left.\nIt is illegal in both Virginia and New York \u2014 Hampton's groups are registered to fundraise in these and most other U.S. states \u2014 to lie to donors about how their money will be spent.\nRep. Walter B. Jones, R-N.C., in February asked the leaders of two U.S. House committees to launch an investigation into \"bad actors\" who mislead donors and enrich themselves in the name of military veterans. He cited the Center for Public Integrity's investigation into Hampton's veterans operation and media reports about other veterans charities.\nCongressional officials and a spokesman for the Virginia attorney general's office did not respond to requests for updates.\nBut the Center for American Homeless Veterans' tax return offers little evidence that the organization is funding or otherwise directly supporting such endeavors. Its 2015 tax return indicates it provided just $200 in grants to other organizations out of $2.5 million in overall expenditures, the vast majority of which paid telemarketers.\nFederal Election Commission filings released this week show Hampton continues to pump thousands of dollars into his own pocket from the Put Vets First! Political Action Committee, which he runs out of the same Falls Church, Virginia, office as the Center for American Homeless Veterans and the Circle of Friends for American Veterans.\nHampton paid himself $27,240 from his Put Vets First! Political Action Committee from Jan. 1 through March 31, according to the FEC filing.\nHampton rapidly increased Put Vets First! Political Action Committee payments to himself late last year, following a Center for Public Integrity investigation into his political and charitable operations. He paid himself $20,350 in December 2017 alone \u2014 more than three times his compensation in December 2016, according to disclosures filed with the Federal Election Commission.\nFew leaders of federal PACs earn that much money.\nIn all, Hampton paid himself $110,000 from his PAC during 2017, FEC records indicate.\nThis does not include money he receives from his two nonprofit veterans groups. The latest federal income tax records available for Hampton's two nonprofit groups indicate he also earned about $340,000 total from the Center for American Homeless Veterans and the Circle of Friends for American Veterans during 2016.\nHampton, who has four decades of fundraising experience, has said his salary is justified because of the number of hours he works leading three groups.\nHampton's PAC reported raising $701,000 and spending $730,000 during the first quarter of 2018, according to a report filed Sunday with the Federal Election Commission.\nOf the money spent, 88 percent went to fundraising contractors, and most of the rest went to overhead and salaries, including Hampton's own. None of the money was spent on veterans advocacy.\nSubscribe to get our weekly roundup of latest news and updates.\nThe PAC gave $10,000 \u2014 or 1.4 percent of its overall expenditures last quarter \u2014 to political campaigns and committees. The beneficiaries, who received $1,000 each, include Sen. Dean Heller, R-Nev.; Rep. Steve Knight, R-Calif.; and several candidates seeking elected federal office.\nPut Vets First! Political Action Committee's $10,000 in political contributions this year easily eclipsed its giving for 2017, when it didn't contribute a dollar to political committees, including those of the many congressional candidates who advocate for military veterans.\nHampton said contracting with professional fundraisers frees up his time to focus on outreach and education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahpri b/data_all_eng_slimpj/shuffled/split2/finalzzzahpri
new file mode 100644
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+{"text":"AFTER MAKING HER THIRD TRIP TO TULSA TO HER DOCTOR AT INNOVATIVE EQUINE PODIATRY & VETERINARY SERVICES ON 8-8-2014 RAYNE WALKS WITHOUT LEG CAST AND CORRECTIVE SHOES.\nDR. SAMMY PITTMAN, DVM REMOVED RAYNE'S BANDAGES, LEG CAST AND CORRECTIVE SHOES. HE TOOK X-RAYS OF HER LEGS AND TRIMMED HER HOOVES. SHE MUST HAVE FELT STRANGE AFTER SO LONG OF NOT BEING ABLE TO WALK ON HER OWN FEET AND HAVING DEVICES TO CORRECT HER INABILITY TO WALK NATURALLY. SHE HAD SURGERY ON 5-8-2014 WHEN SHE UNDERWENT A BILATERAL DEEP FLEXOR TENOTOMY DUE TO SEVERELY CONTRACTED TENDONS. OUR HEART FELT THANKS GO OUT TO EVERYONE WHO MADE THIS MIRACLE POSSIBLE INCLUDING DR. SAMMY PITTMAN, RAYNE'S FOSTER PARENTS RICK & ROBIN AS WELL AS ALL THE TRIPLE O SUPPORTERS WHO DONATED TO RAYNE'S SPECIALIZED CARE AND ALL THE PEOPLE WORLD WIDE THAT HAVE KEPT RAYNE IN THEIR THOUGHTS AND PRAYERS. WE WILL BE ETERNALLY GRATEFUL TO ALL THE KIND SOULS WHO HELPED RAYNE DURING HER DARKEST DAYS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The tropical rainforest is one of the richest ecological systems in the world, and is recognized as \"the lung of earth\". The rainforest's multi-layer environment, ranging from the root layer to the emergent layer, is home to a wide variety of plant and animal species. Each of the rainforest's levels satisfy different species' living requirement and supports the continuous operation of its ecological system.\nWe have taken our design inspiration from the rainforest as nature's prototype for sustainable high density. Our concept of the \"Urban Rainforest\" is also stratified to create an inviting, 24-hour community, that appeals to diverse user groups and functions. This design strategy looks to create a pure and bountiful living environment, and will support the Shenzhen High Tech Park's economy with low-carbon emissions and net-zero energy consumption.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What to do with Surface Piercing infection?\nI posted a question about cleaning surface piercings but now they has puss coming out of them. I have my collar bone pierced on both sides, this is my first surface piercing so I don't know a great deal about them. I bath it twice everyday with rock salt (sea salt). What should I do if puss is coming out of them?\nwell people will always say just to clean them, but i would advise you to squeeze all the puss out first, as much as you can, and then clean it again. use 'dial' antibacterial soap to clean it, and use dial shower gel aswell. just have showers, dont get into the bath till its healed.\nUm well when ever i get a piercing i tend to leave it to heal naturaly. But if its infected id treat it but not too much. I think when you treat cuts or piercings with like say savalon or something it tends to irritate it or make it sor which makes it hard to heal.\nTry and let it scab over and it will start to heal. Let it get air and also salt water is very good for piercing. But dont use normal water and salt. Try and rinse it in the sea, sea water is very good for piercings mine healed quick with sea water.\npiercings do that lol. whipe it off with a cotton bud soaked in the salt water when you clean it.\nI've got my sternum pierced (cleavage surface piercing)\u2026 quite similar to collar bone, and mine has become infected as well.\nit was only scared for awhile but it's gotten worse.\nI've noticed that it gets infected when i stay in the shower too long with really hot water pouring all over it. It becomes kind of like a blister. Do you get this? eventually it becomes a yellowy colour, pops (i know, gross!). I get rid of all the puss (by squeezing it together \u2013 gross again) and when it starts to bleed i clean it up with water and cotton buds.\nAfter a week or so it should have healed, but it will most likely scar, teaches us to keep it clean in the first place. I use a spray on 'antifectant' and when i continue its use it stays clean and uninfected.\nSee my other question, \"how to remove scars\" (link below) and that should help you out if you do develop one.\ncan you get a vertical surface tragus piercing at walmart?\nwhy can\\\\\\'t i take out my earrings after i get the pirced?\nwhat if your peircing onn your ear gets a big sore lump?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This diagram shows the percentages of websites using various structured data formats among the sites that use Joomla. See technologies overview for explanations on the methodologies used in the surveys.\nMicrodata is used by 27.6% of all the websites that use Joomla as content management system.\nYou can find usage and market share data for all structured data formats amongst Joomla sites in our Joomla market report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"American Girl has officially released pics of the new mini Josefina coming this fall. (You can link to more info on the AG Publishing Site.\nI won't be getting this mini since I just purchased the classic mini doll when it went on sale, but I do like the new meet outfit. I think it's a little more bright and cheery than the old one. I would love to have either of her meet outfits (or both!) for my new Latina Divah doll. I hope AG adds a little more to her collection with her new book release. Right now you can pre-order the new mini from Walmart for only $15!!\nSummer always brings warmth, cheer, and plenty of outdoor fun! It also brings more opportunities for doll play, and these cute mini finds from Rite Aid will help get your dolls ready for summer. Many seasonal items are already on sale, and prices will drop even more at the end of the season, so you can stock up for next year.\nThis small birdbath and mini cherub garden statues would add an elegant touch to a fancy vintage garden scene. They are both on sale for under $10.\nIt's fiesta time! Abrielle poses with her mini Josefina doll and a wooden maraca for Cinco de Mayo. She is wearing an AG dress from Josefina's collection. The mini flags come from Josefina's paper doll set, which I highly recommend. Right now you can purchase it online for only $3! Abrielle is from the Positively Perfect Divah Collection.\nCampfire Chic has a template and tutorial for making a mini burro just like the one above.\nThis Evite website has a similar tutorial with lots of great pictures.\nHere's an easier version from Love + Cupcakes using toilet paper roll tubes.\nThis awesome cosplayer's sign is cut off, but it says, \"Need money for target practice.\"\nMake your dolls a diminutive light saber with this tutorial from Doll Diaries.\nMany of the doll groups I follow on Facebook have been all abuzz about AG Julie's egg chair, especially since Easter brings a bounty of craft and decoration items to create DIY options. Most of these projects require more time and effort than I had available, but I did manage to throw together some simplified mini versions.\nI used an 8 inch egg from a discount store, and I was able to make 2 separate chairs with one egg. They fit a variety of dolls including AG minis, Liv or Barbie size dolls, and even 14-16 inch dolls. I decorated the shells with retro style decorative tape and added a base with hot glue. The black base is from a mini lamp, and I attached the egg to a marker lid so it can swivel. It can be used with or without the lamp pole to vary the height.\nI used different sized eggs for an even simpler version. These are attached with sticky putty, but you could make them permanent with hot glue.\nHope you enjoyed this groovy, simple project? What other ideas do you have for re-purposing Easter items?\nAs I was looking back through my blog posts the other day, I noticed that several of the photos were messed up. Some pictures mysteriously got replaced with pictures from completely different posts. I'm not sure if this is an issue with the Weebly app or what, but I tried to correct the instances where the wrong pictures appeared. Sorry for any confusion this may cause. If you notice any pictures that don't seem to fit, please let me know so I can fix them. Thanks!\nOn my latest thrifting expedition, I found this collectible card game tin from American Girl for $4. When I opened it and saw all three games still in the shrink wrap, I snatched it up.\nThere is a rummy type game and a trivia game featuring the first 8 historical characters. The third game features Kit Kittredge, who was the newest doll at the time. It is a frame game that has little pictures you try to arrange according to your pattern cards. I haven't played that one yet, but it seems a little more complicated than the others.\nOne of my favorite aspects is the artwork. Each doll's suit shows pictures from each book in her series. This helps players review the stories as they play. The pictures also make the game easy enough for non readers to play. I am sad that AG has discontinued their beautiful illustrations, so I appreciate any opportunity to introduce my kids to the pictures, which really enhance the stories, in my opinion.\nThe Rummy game, called Threes Please can be used for other card games as well, such as War and Uno. my kids loved this game. My oldest daughter liked the trivia game because she had listened to several of the books and enjoyed answering the questions.\nThis game is available on the secondary market for about $5-$20. Ebay has plenty in great condition for reasonable prices, so don't pay too much.\nI highly recommend these games for boys and girls for education and fun.\nThis month American Girl is spotlighting their historical character Kaya, a Nimiipuu (Nez Perce) girl growing up in 1764. Here are some Kaya inspired pictures.\nI bought this doll (Abrielle from the Divah Collection) with AG's Josefina in mind, but with her sweet close-lipped smile and big brown eyes, I think she also makes a nice Kaya. The doll was on clearance at Wal-Mart for $5.\nHere you can see the full outfit, which came from a Native American porcelain doll I bought at a thrift store for $2. It is made from a thin velour-type material, but the intricate beadwork on the pants, headband, and medallion give it a more authentic look. I like the lacing down the front and on the sleeves, though the metal rivets are probably not true to the time period. The porcelain doll was missing one shoe (and the foot that came with it!) so I don't have the original moccasins. I'm still looking for some good shoes to match.\nI really like Kaya because she's a strong, courageous character, but she also has some real flaws, which she strives to overcome throughout her stories. I think this makes her more human and realistic and adds depth to the character. My daughter likes the fact that there is a blind character in the books, and my son enjoyed Kaya's adventurous nature. I hope you are also inspired to learn more about this great historical character.\nWal-Mart had several cute, affordable items for Barbie size dolls that may work well for minis.\nThis pool party set is so cute and versatile. I just love the pool and Tony accessories, and the table with an umbrella could double as a bistro table. This seems like a great value for under $10. The Sparkle Girlz clothing sets also work great for modern minis.\nWal-Mart now has a dollhouse available for $50, and the furniture sets are only $5. At this price point, you certainly won't be getting heirloom quality, but it's a good cheap option, especially for younger kids. All the pieces are made of plastic and have a cheap appearance, but adding fabric or contact paper embellishments could make them look a little more realistic. There were also two different vehicles available that would fit fashion dolls or minis.\nHere's a closer look at the furniture sets. I really like the kitchen chairs. They would work well in a little cafe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Added to most of Carnival's fleet during its Fun Ship 2.0 upgrades -- a nearly fleetwide initiative that outfitted ships with exciting new bars, restaurants and entertainment options -- the RedFrog Rum Bar is the Caribbean-themed extension of the RedFrog Pub, which introduced the line's own microbrew, ThirstyFrog Red. At the rum bar, positioned on the pool deck (often across from the BlueIguana Tequila Bar), cruisers will find lots of fruity concoctions and, of course, rum-based drinks.\nThis open-air pool deck bar is made of wood with red and blue accents. It's surrounded by brightly colored tile flooring and long, thin tables with tall, wooden bar stools. Overhead stands a wooden sunburst-patterned pergola. The setup creates a laidback vibe, and its location is ideal for fun in the sun. It's an energetic spot during the day, and it can also be lively at night when deck parties and other outdoor activities are held.\nRum from various countries is the main draw at this watering hole. Additionally, passengers will find soft drinks, mixed drinks, several international beers, ThirstyFrog Red and a couple of concoctions that mix ThirstyFrog Red with a touch of liquor. Also on the menu are frozen drinks aplenty, from margaritas to daiquiris.\nDrinks are charged a la carte to onboard accounts. Prices range from about $9 for mixed drinks to nearly $30 for a 101-ounce \"tube\" of beer; 15 percent gratuity is automatically added.\nWhich ships have RedFrog Rum Bar?\nSee photos of the RedFrog Rum Bar on Carnival Sunshine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"* \ubcf8 \uc601\uc0c1\uc740 \ubc18\ub4dc\uc2dc Youtube \uc5b4\ud50c\ub9ac\ucf00\uc774\uc158 \ud639\uc740 Chrome \ube0c\ub77c\uc6b0\uc800\uc5d0\uc11c \uc2dc\uccad\ud558\uc154\uc57c 360\ub3c4\ub85c \uac10\uc0c1\ud558\uc2e4 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\"\n* This video must be watched in Youtube application or Chrome browser to watch 360 degrees. \"\nStar Wars Battlefront II Yavin 4 360\u00b0 Video in 4K!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Manchester United, the New York Yankees of the English Premier League, has a new sponsor for their kit (fancy word for jersey) this season. That sponsor is American car company, Chevrolet.\nNot gonna lie, it was weird seeing the Chevy logo on that red, Manchester United shirt.\nWhile you may not be a fan of Man U or a company that took government bailout money, Chevrolet made a pretty neat commercial showcasing the history of the club's jerseys over the years. It features Wayne Rooney and a few other current Manchester United players at the end. Rooney has the speaking part.\nYou'll be humming \"Glory, Glory Man United\" if you watch it enough. This is the 60 second version. The 30 second version can be seen on television. That's how I discovered it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a huge Larry Norman fan. So Long Ago the Garden was one of the first Christian albums that I listened to the whole way through. It rocked my world because I always thought that Christian music had to be sterilized copy of what the secular music world was doing. But what Larry Norman proved was that Christians could create music that was art \u2013 that said something and was not just Christian propaganda. That is a central theme throughout Gregory Thornbury's Why Should the Devil Have All the Good Music, which is an excellent biography of the founder of Christian Rock n' Roll.\nIf you have seen the documentary Fallen Angel: The Outlaw Larry Norman, directed by David DiSabatino, you may have an extremely negative impression of Larry. Granted, he was a complicated guy, and he was also thrust into the spotlight between the rock music world and the Evangelical Christen world. He was no saint to be sure. However, after reading this book, I have come to view DiSabatino as a hack. The film, made just one year after his death, mistreats Norman as it gathers all of the people who had grudges against him and allows them to talk unchallenged. None of the stories they present are given opposing views. The film deals more with gossip than it does with fact.\nThornbury, on the other hand, was given unfettered access to ALL of Norman's archive material. This archive material contains all of Larry's correspondences \u2013 letters, emails, tape recordings of meetings. Larry kept everything, and unlike the documentary, the book gives a broader picture of the man and his legacy. Thornbury treats Norman fairly, not holding back punches. But many of the things you through happened with Larry and others, have a completely different spin in light of the author's primary source access.\nLet's be clear, Randy Stonehill and Terry Scott Taylor come off looking very bad in this book. I am a fan of both of these guys, and they have produced some outstanding Christian music. But after a review of the primary sources, which include tapes of specific meetings and timelines of events, it is evident that both men have been playing very fast and loose with the truth when it comes to Norman. Both men have been telling stories about Larry that, in the light of documented evidence and apparent contradictions, serve only to cover up their particular moral and financial failings.\nIf you are a fan of Larry Norman, this is a must-read. If you have never heard of Larry Norman, get onto Spotify or iTunes and listen to -in a row \u2013 Only Visiting This Planet, So Long Ago the Garden, In Another Land, Something New Under the Son, Upon this Rock, and Stranded in Babylon. After you listen to these albums, when you go to church and listen to the music team play modern worship songs, think of Larry Norman. Think of all the grief he took from the secular music world and the even harsher grief he unfairly (and sometimes very reasonably) got from the Christian world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted on 2011\/01\/09 18:46:40 (January 2011).\nFirst day back at work after the long Christmas \/ New Year break - as I'd had some holiday to use up at the end of the year I'd ended up out of the office for over two weeks. Had been nervous about coming back to a huge pile of email that had to be dealt with, but in the end it wasn't actually too bad.\nMy sore throat had persisted since Saturday, and so I decided I should try and have a few days off from drinking, starting today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajyxq b/data_all_eng_slimpj/shuffled/split2/finalzzzajyxq
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@@ -0,0 +1,5 @@
+{"text":"Slim, but the odds of you shortening your battery life are much better. Why you insist on running a Li-on battery down like that is beyond me.\nFACT: That's as bad as running a car battery down to the point where it won't start your car.\n\"In fact, the only real trick to remember is to make sure the battery doesn't get too low. A study by Cadex found that many Li-ion batteries fail due to being over-discharged.\"\nThere are several articles in the forums that gives great advise on modern day batteries. Letting your battery die now days is not a good idea. The science and technology on batteries has changed.\nI didn't know that the batteries were that different lol.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OPSWAT and IRONSCALES offer a joint solution to detect and prevent phishing attacks and provide automated email phishing forensics and remediation. Unlike gateway products, IRONSCALES monitors behavior at the mailbox level, looking for anomalies in communication habits by using machine learning coupled with user feedback.\nIRONSCALES utilizes the multi-scanning feature of MetaDefender to quickly scan email attachments with multiple anti-virus engines to check for known and unknown threats, using signatures, heuristics and machine learning. Organizations can choose to deploy MetaDefender in the cloud or use an on-premise version of MetaDefender.\nDetect known and unknown threats in email attachments by multi-scanning with over 30 anti-malware engines using signatures, heuristics and machine learning.\nIRONSCALES was founded in 2014 by CEO Eyal Benishti and has pioneered an advanced anti-phishing threat protection platform combining human and machine intelligence to automatically analyze, detect and remove malicious emails before and after they land in the inbox using a multi-layered and automated approach. Headquartered in Tel Aviv, Israel, IRONSCALES was incubated in the 8200 EISP, the top program for cyber security ventures, founded by Alumni, the Israel Defense Forces' elite intelligence technology unit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks again this year to the fine folks at Coca-Cola for their support of Nautilus and the races we organize.\nOur agreement has been renewed and sees Coke providing us free paper cups and a great deal on Dasani Water and PowerAde for all our races.\nWhen making your beverage choices please remember Coca-Cola, Dasani Water, and PowerAde.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can add products manually (as opposed to uploading them using a product feed) through the Merchant Dashboard Add Products Manual page.\nOur drag & drop feature is the simplest ways to upload multiple images for your product. Click on the image you would like to add and drag it into the Main Image field.\nYou can easily add additional photos for your products by dragging them into the 'Additional Images' field below.\nTo add an image from your computer, select 'Add From Your Computer' and choose the image you would like to add for the product.\nTo upload a product image via web address, copy and paste the image If you don't already have the image URL on hand, you can go to a website where it's already listed and right-click on the product image to copy the URL (for more information on how to do this, click here).\nIn this field you will enter the product's price, localized price, quantity, shipping costs and your shipping time. You may select a pre-set shipping time from the date ranges shown; or you may enter it manually if your shipping time is not listed.\nThe Local Currency is based on the Local Currency Code that can be found under Account \u2192 Settings \u2192 Currency Settings.\nIf the local currency code is set to CNY (\u00a5), the localized prices can only be less than or equal to 6.886 times the USD ($) price.\nIf the local currency code is set to USD ($), all existing product prices are already recorded in USD ($), so no further action is needed to update the local product prices.\nFast and reliable shipping is critical to customer satisfaction. The faster you fulfill and ship your orders securely; the more exposure your item will receive. Adding multiple, high quality images for your products one of the best ways to increase the product's exposure and the number of impressions.\nThe country shipping of a product can be edited under the \"Customize shipping price for chosen countries\" section.\nThe best way to increase sales is to ensure your products have proper size and color information. Products that have correct sizing and color information sell more on the Wish platform.\nSimply check the box next to the color you would like to add. You may also add additional colors into the 'Other' field and it will appear in this section, take for example Black & White.\nFirst select the type of product you are uploading, i.e. apparel, electronics, etc. As you select the appropriate category, the size chart will adjust in order to show the appropriate sizing. Once you have selected the product category, click the boxes next to the sizes that you would like to list for this product.\nHere, you can adjust the price and quantity of each variation. Let's say you do not have size S for the multicolor. Simply adjust the quantity to 0.\nAdd more information to your item, expand the optional information section. Here, you can add the MSRP, brand name, landing page URL and UPC. Click on the corresponding field to add an attribute to your item.\nNow submit to upload your new product!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every 30,000 miles, Lowell's recommends flushing your car's fluids \u2013 including coolant, brake, transmission, and power steering systems.\nYour car's fluids tend to break down and get contaminated over time. How this happens depends on the fluid system.\nThe rust inhibitors in your vehicle's coolant (also called antifreeze) break down, causing your radiator and engine block to corrode from the inside out. These rust particles can clog the very small channels within your cooling system, interfering with its ability to cool your engine. Ultimately, your engine can overheat.\nAutomatic transmission fluid loses its lubricating and cooling properties as it breaks down, leaving varnish and sludge deposits which can shorten your transmission's life. Plus, transmission fluid gets contaminated with tiny metal shavings, further degrading its ability to lubricate your transmission.\nBrake fluid and power-steering fluid are hydraulic fluids which get contaminated as the systems' seals and internal components break down. As they get contaminated, they lose their hydraulic properties, making it hard to brake or steer. Also, hydraulic fluids attract moisture, which can introduce unwanted air into your brake and power-steering components.\nChanging these fluids every 30,000 miles \u2013 or whenever the fluids become dirty \u2013 can prevent expensive and inconvenient damage to your vehicle, and can help extend your vehicle's life.\nSo now you might understand why you need to regularly change the fluids on your car. But that doesn't explain what a flush is (or why Lowell's recommends flushes instead of other alternatives).\nSome repair shops \u2013 including some dealers \u2013 offer simple \"drain-and-refill\" fluid changes. This is faster (and seemingly cheaper) than a flush. They basically drain the old fluid, and top off the system with new fluid.\nIn the process, much of the old fluid, contamination, and debris is \"stranded\" inside the system: old coolant and rust is left in the engine block and radiator, or degraded transmission fluid and varnish are left in the transmission.\nThese faster and \"cheaper\" options will degrade the long-term performance of your vehicle and will likely create costly repair issues down the road.\nWith few rare exceptions, Lowell's doesn't offer these drain-and-refill fluid changes.\nInstead, we perform fluid system flushes designed to remove almost all of the old fluid, contamination, and debris from the system.\nAt the same time, the machine fills the system with fresh fluid, while adding conditioners which will help preserve the components of the system.\nWe monitor the process until we see that all of the old fluid is flushed from your vehicle.\nThat's what a flush is.\nWe recommend flushes because we believe that they are the best option for changing your vehicle's fluids.\nFeel free to call the shop with any questions, and \u2013 if it is time for a fluid change or other service for your vehicle \u2013 to make an appointment for service.\nHow much would a flush of all systems cost? I have a 2009 Toyota Tacoma with 53,689 miles.\nOur price to do the brake system flush, transmission flush, power steering flush, and antifreeze\/coolant flush is $495 with this month's coupon. Just let us know if you'd like to set up an appointment!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalcud b/data_all_eng_slimpj/shuffled/split2/finalzzzalcud
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+{"text":"NEW FROM ACT-MAG: Please click to leave your email address to be notified when they are in stock. CZ 75B, 85B, SP-01, SHADOW, SHADOW 2 9mm 15 ROUND ACT-MAG MAGAZINE 45203.\nNOTE THIS IS A STOCK PHOTO THAT SHOWS THE 17 ROUND VERSION OF THE MAGAZINE. THE MAGAZINE LOOKS THE SAME BUT IT ONLY HAS WITNESS HOLES FOR 15 ROUNDS. THIS IS A 15 ROUND MAGAZINE. I DON'T HAVE THE PICTURE OF THE 15 ROUND MAGAZINE YET.\nSimply enter your details below and we will send you an e-mail when \"ACT-MAG CZ 75B, 85B, SP-01, SHADOW 2 9mm 15 RD MAGAZINE 45203\" is back in stock!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It can be our responsibility to satisfy your requirements and successfully serve you. Your pleasure is our greatest reward. We've been searching forward for your check out for joint expansion for Wire Mesh Roll , Woven Wire Mesh Roll , Wire Mesh Belt , please don't hesitate to contact us. We will do our best to meet your needs.\n\"Our pros are lessen prices,dynamic sales team,specialised QC,sturdy factories,top quality services and products for Wire Mesh Roll , Woven Wire Mesh Roll , Wire Mesh Belt , You should feel cost-free to send us your specs and we will respond to you asap. We have got a expert engineering team to serve for the every single comprehensive needs. Free samples may be sent for yourself personally to know far more facts. So that you can meet your desires be sure to truly feel cost-free to contact us. You could send us emails and call us straight. Additionally we welcome visits to our factory from all over the world for much better recognizing of our corporation. nd merchandise. In our trade with merchants of several countries we often adhere to the principle of equality and mutual advantage. It is our hope to market by joint efforts both trade and friendship to our mutual benefit. We look forward to getting your inquiries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The conference venue will be the Hotel Alexandra located in the most beautiful Fjord and Glacier Districts of Norway. The hotel is recognised as one of the best conference venues in Norway. We can guarantee that participants will not quickly forget the extraordinary beauty & hospitality of this region. The participants are expected to arrive at the hotel Sunday, June 2, and to travel home, either Thursday, June 6 after lunch or Friday, June 7 morning.\nWe would strongly recommend you to arrange your travel to Loen as soon as possible to obtain cheapest available flights. The most convenient travel is by airplane from Oslo Gardermoen airport to Sandane or Hovden operated by Wider\u00f8e (WF), local airports with hotel transfer service (60 min drive).\nHotel rooms have been reserved at the Hotel Alexandra . The Hotel Alexandra in Loen, surrounded by fjords, lakes, glaciers, mountains and waterfalls is one of the most sought after hotels in Norway. We strongly urge you to bring your partner or family.\nThe prices for a single and double room are NOK 2300 and NOK 2100, respectively (per person per night ). The cost for lodging include breakfast, lunch and dinner.\nPersons reserving accommodation are themselves liable. In the event that you should be unable to attend you are strongly advised to notify the Secretariat at least one week in advance.\nHotel Alexandra accepts all major credit cards.\nOn the west coast of Norway the climate in June is pleasant with an average temperature maximum temperature of 20\u00b0C.\nLight clothing with a sweater and raincoat for an occasional and rainy evening are recommended. For the excursions, you are advised to bring warm clothing and suitable walking shoes.\nToday and next eight days weather forecast.\nHotel Alexandra has an excellent indoor and outdoor swimming pool, so don't forget to bring your swimming suit.\nThere are daily Wider\u00f8e flights between Oslo\/Bergen and Sandane Airport by Dash 8 (50 seats). Sandane Airport is located about 55 km from Loen, approximately 60 minutes by coach. The Hotel Alexandra will arrange your coach service between the airport and the hotel according to your booking arrangements. The price of the hotel shuttle is dependent on number of passengers, please consult the price list. See below for pick-up prices.\nThere are daily Wider\u00f8e flights between Oslo\/Bergen and Hovden Airport by Dash 8 (50 seats). Hovden Airport is located about 85 km from Loen, approximately 70 minutes by coach. The Hotel Alexandra will arrange your coach service between the airport and the hotel according to your booking arrangements. The price of the hotel shuttle is dependent on number of passengers, please consult the price list.\nIf you prefer to travel by a larger-sized plane (Boeing 737), you must use the \u00c5lesund Airport, Vigra, 140 km from Loen. Travel time by coach from Vigra to Loen is about 3 hours along a route with magnificent scenery. The Hotel Alexandra will arrange your coach service between Vigra Airport and the hotel according to your booking requirements. SAS , Norwegian and KLM operate at \u00c5lesund Airport. The price of the hotel shuttle is dependent on number of passengers, please consult the price list.\nNor-Way Bus departures from Oslo daily 10:00 with arrival in Stryn 19:00. You may also use the night bus leaving Oslo 22:30 with arrival 07:00 the next morning. Take a bus or a taxi from Stryn to Hotel Alexandra.\nNor-Way Bus departures from Bergen daily 08:00, 11:40 and 15:10. Travel time to Stryn is approximately 6,5 hours. The bus stops at Hotel Alexandra.\nFrom Oslo travel by rail to Otta, and from there take a bus to Loen (200 km further).\nIf your prefer to drive, the journey from Oslo to Loen (500 km) will probably take you 7-9 hours. If delegates share a car, travel costs can be reduced considerably.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 15th Lock on the Grand Canal has been closed to navigation until Tuesday, 13th July in order to carry out emergency repairs to the lock gate. Every effort is being made to restore the lock to service as soon as possible.\nWaterways Ireland has issued an advisory to all masters and inland waterways users of the Shannon Navigation that the National Optimist Dinghy sailing event involving junior sailors will take place in and about Lough Ree Yacht Club (LRYC) on Sat 17th and Sun 18th July 2010 . Upwards of 200 sailing dinghies will be participating and will be making their way from LRYC to the racing course and back, on the southern part of Lough Ree.\nMasters of vessels and boat are requested to proceed at slow speed and with minimum wash when passing the sailing fleet and to note any instructions or advice given by safety boats marshalling the race.\nWaterways Ireland advises the Shannon One Design annual long distance sailing race on the inland waterways will commence at 10.00 hrs on Sat 26th from Athlone Lock, overnight at Banagher and finish at Portumna Bridge on Sun 27th Jun 2010. The Shannon Inspector of Navigation has warned masters to give way to vessels and boats navigating by sail only. Full notice is attached.\nA route that follows the Grand Canal on the inland waterways being opened today provides access for walkers and cyclists from West Dublin right into Dublin centre. Details of the new route \u2013 that cost Euro 25 million \u2013 is in this morning's Irish Times by Paul Cullen here.\nWaterways Ireland has won the coveted O2 Ability Award for Environmental Accessibility for its work on inland waterways at an awards ceremony last evening in the Royal Hospital Kilmainham. The awards were presented by Ryan Tubridy, RTE and Caroline Casey founder of Kanchi, the company responsible for the awards. Awarded to public and private sector organisations with exceptional commitment to ensuring accessibility for all, there are also awards for leadership, retention, customer service, learning and development and recruitment.\nWaterways Ireland is a North\/South body and is responsible for the management, maintenance, restoration and development of the inland navigable waterways for recreational purposes. Waterways Ireland manages over 1000km of waterway under its remit, many kilometres of it built over 300 years ago. Waterways Ireland's accessibility policy has ensured accessibility built into every project since inception. The outcomes have been the installation of 1000's of metres of accessible jetties and numerous service blocks with no barriers to persons with disability.\nWaterways Ireland through its sponsorship programme has ensured that activities are open to all and through a comprehensive customer services programme has ensured that customers receive a high level of service regardless of whether the point of contact takes place at a lock, or in an office or waterside location.\nPowerboat Racing will take place at the Derryad Powerboat Club, Lisnaskea on Ireland's inland waterways on Sun 16 May 2010 from 11.00 hrs to 19.00 hrs. Through traffic will be catered for via the west side of Inishrath Island i.e on the Geaglum moorings side of the navigation. Masters of vessels are requested to proceed at slow speed and with minimum wash when passing this stretch of the navigation and to heed any advice or instructions issued by the race officials.\nInland Waterways Works being undertaken to Agivey Road Bridge on the Lower Bann Navigation will result in a reduction in normal airdraft being encountered.\nWaterways Ireland has released three new publications which will provide much assistance to waterway and waterside visitors to Ireland's Inland Waterways throughout the year.\nThe Waterways Ireland What's On Guide, promotes a series of events happening alongside as well as on the waterways throughout the year. With 128 entries from fun fairs to Come and Try it Events, historical events and boat rallies, there is a huge range of activity to pick from. Waterways Ireland through the Sponsorship Programme has supported a significant proportion of the community and local activity taking place as well as some of the high profile events such as the Waterways Ireland European TriAthlone 3-4 July, Waterways Ireland Riverfest 3-4 July and Waterways Ireland Loughfest 26-27 June.\nMartin Dennany, Director of Marketing & Communications for Waterways Ireland said 'A Taste of the Waterways guide to hand, will ensure you find all the best pubs and restaurants along the waterways. You don't have to set foot in a boat to enjoy the hospitality along our waterways, but it would be a pity to miss out'. The guide , A Taste of the Waterways, written jointly by Georgina Campbell and boating expert W. M. Nixon and published in association with Waterways Ireland. This independently assessed 68-page guide leads visitors along all of our navigable waterways, and to nearly 100 great hostelries and restaurants sprinkled from Limerick up the mighty Shannon via Lough Derg, Lough Ree and the Shannon-Erne Waterway, to Lough Erne; and from Dublin out west along the Royal Canal and the Grand Canal and southwards to the beautiful River Barrow.\nThe Waterways Ireland Publications Catalogue containing 30 books and DVDs available to order from Waterways Ireland. The range includes both free and pay-for publications with broad appeal to a wide variety of recreational users of the inland navigations. This new edition includes a feature on Waterways Ireland's new on-line ordering facility.\nAll three publications are free and available from Tourist Information Centres and Visitor Attractions along the waterways, and is also distributed to over 60 hotels. It is also downloadable from www.waterwaysireland.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The average age for SA Expeditions tours is between 15 and 99. The age might vary for specific tours, so please check individual tours for details.\nTourRadar only requires a deposit of $99 to confirm a SA Expeditions booking. The remaining balance is then payable 90 days prior to the departure date. For any tour departing before 24 July 2019 the full payment is necessary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Teresa Curran is dreaming of the warm seas and blue skies of Positano on the Amalfi Coast, Italy \u2013 the perfect place to while away the hours with a good book and glass of limoncello.\nIsn't summer wonderful? The warmth of the sun on your face, the salt spray of the ocean on your lips, a good book, a glass of wine ... the guy next to you on the Tube coughing in your face, your umbrella blowing inside out, the delightful feeling of wet shoes refusing to dry out all day \u2026 London, I love you, but it's time for a holiday.\nAs a New Zealander transplanted in London I've found it difficult to get my head around distances in Europe. It's an awfully long way between NZ and our lovely neighbours (there is no bridge, no tunnel and you certainly can't swim), yet people here in the UK talk about Europe as if it's a long way away.\nIt's so easy to jump on the Eurostar for a weekend in Paris, or to fly to Dublin for a couple of days. But right now, I'm dreaming of the sun and northern Europe seems even too close. I want to return to Positano on the Amalfi Coast. It's not really very far away.\nThe Amalfi Coast (about 'ankle-height' on the west coast of Italy) certainly looks like Europe. Houses perch on stone cliffs overlooking the Mediterranean. Sure, they are beautiful, but they are also unassailable fortresses against both people and the inexorable tide. The Coast was an important maritime republic during the 10th and 11th centuries and now is a UNESCO World Heritage Site.\nI'm not the first and I certainly won't be the last to be transfixed by the Amalfi Coast. In 1953 John Steinbeck wrote that \"Positano bites deep. It is a dream place that isn't quite real when you are there and becomes beckoningly real after you have gone.\" It could be a movie set, it is that stunning.\nMy hotel had views that belong to an old Hollywood movie. You could imagine Grace Kelly driving by in a headscarf and oversized sunglasses, or Sean Connery sipping a martini and flirting with a local beauty. The area evokes images of pre-war parties where rich young things indulged in both sunshine and social climbing. One evening, as the long white curtains floated in the breeze, I had a few glasses of limoncello myself (Sean Connery the Younger was unfortunately absent).\nI'm usually more of an adventurer, but this part of the world has a relaxing effect. I was content to while away the days in the sunshine doing little more than turning the pages of my book or gazing at the view. Admittedly the trek up the hill to my hotel was horrendous in the midday sun, and Italian drivers terrified me on the narrow roads with treacherous drops, so I decided staying put was the best option.\nDreaming of your own Amalfi Coast escape? Click here to view flights to Naples from \u00a3125 or give our consultants a call on 0208 045 4186 to book.\nRead more about Teresa's travels on her blog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We in NFS Italia dedicate our efforts to bring our readers helpful research and analysis for everything about slot games. Our team in NFS Italia converts their experience and education to make a readable and easily understood content that will be simple enough to quickly and efficiently help you make the right decisions with slot games.\nHere at NFS Italia, we cover various topics and latest news regarding slot games along with updates on activities surrounding this type of casino game. We make it a point to keep our readers informed of the happenings in the market that will help them maximize their chances of winning in slot games. We provide recommendations and advertise the best slot games as well that are going to be helpful if you want to be profitable playing slots.\nNFS Italia is a platform that serves as a guide for our readers to be able to find what they're looking for when it comes to slot games. We have a goal of building a community of like-minded people that are interested in playing slot games, whatever their own purpose of doing it may be.\nNFS Italia is a reliable source of slot games news, and our content goes through solid data and analysis to be sure that it will be useful to readers. We also post insights on how the slot game market will fluctuate to help you predict the changes. Anyone who wants to learn about slot games can save money and time through our website.\nSubscribe to NFS Italia now and you'll gain access to our wide variety of options and become a member of our email list. You will then receive regular updates and the latest news for our content.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Sisters In Islam: News \/ Comments \/ Prophet Muhammad's wife A'isha: How Old was She at the Time of her Marriage?\nProphet Muhammad's wife A'isha: How Old was She at the Time of her Marriage?\nWith reference to questions about marriageable age, minor age marriages are justified quoting the precedent of A'isha marrying Prophet at the age of nine. In his book, Tahqiq Umar A'isha al-Siddiqa [(Truth about the age of A'isha the Truthful), Karachi: Mashkur Academy, No date of publication mentioned], Hakim Niyaz Ahmad, a scholar from Pakistan has examined all the report in almost all collections of Hadith and has come to the conclusion that A'isha married the Prophet when she was 19 years old.\nb. Urwa. b. Hisham is generally accepted as a reliable person, but in his old age he had lost this prestige among the scholars in Medina. They do not accept his Hadith narrated in this period. Hisham narrated this story in around 145 AH when he was 84 years old. A'isha's marriage took place in 2 AH, more than 140 years ago. It was the time when most of those who could verify the facts of his narration had died.\nc. A'isha's marriage was not an ordinary event that only one person would narrate it.\n2. Ibn Ishaq, Ibn Sa'd, and other historical accounts of the life of the Prophet never fail to mention A'isha among the first group of people who embraced Islam. If A'isha were nine years old in 2 AH\/624 AC she would be five years old in 610 AC, when Muhammad announced his Prohethood. It is rather more reasonable to assume that she was adult at that time, which means that she was certainly more than nine years old in the year she migrated to Medina.\n3. Mishkat al-Masabih, Al-Bidaya wa'lNihaya, Siyar A'lam al-Nubala, and al-Isti'ab, among the more well known sources clearly mention that A'isha was 10 years younger than her sister Asma.\n4. Most sources, particularly Abu Na'im al-Isfahani suggests that Asma was 27 years old at the time of migration. It means A'isha was at least 17 years old at that time. The author disagrees with Isfahani, and calls it a miscalculation. This calculation is based on the assumption that Asma is usually reported to have died at the age of 100 years in 73 AH.\nCalculating backward and subtracting the period of 13 years of Meccan (100-73-13) she would be 14 years when she accepted Islam. That would mean that A'isha was only four years old when she embraced Islam. It is so unreasonable. Both Aisha and Asma must be old enough to qualify for accepting Islam. Therefore, we have to believe either that Asma was older than 100 years at the time of her death, or that there was not difference in their ages. This is plausible because they were born of different mothers.\n5. Asma gave birth to Abdullah b. Zubayr who was the first child born among Muslims at the time of Hijra. The event was celebrated with much happiness. The Prophet suggested A'isha to take the Kunya of Umm Abdullah (Abdullah's mother). It would be odd for a minor girl to have that Kunya. It suggests that she was old enough to have that Kunya.\n6. When the Prophet proposed, A'isha was already engaged to Jubayr b. Mut'am, whose father was the leader of the Quraysh. Jubayr's family was not happy when A'isha's family embraced Islam. Both families found it difficult to terminate that engagement. A'isha's father Abu Bakr had to negotiate for divorce. He requested Jubayr's family to conclude the marriage contract and take A'isha to Jubayr's home, as was the custom.\nAbu Bakr knew that Jubayr's family would not agree to the contract and that then he could negotiate for divorce or disengagement. It was after her divorce that she could be engaged to the Prophet. The author argues that the fact Abu Bakr could negotiate for the conclusion of the contract or divorce is an evidence that A'isha was adult and of marriageable age.\n7. The author concludes that Aisha was 19 years old at the time of marriage. There is a textual error 19 (tis'at ashar) was misread as nine (tis'ah). He gives several examples of such errors in the reporting in Hadith books.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All transmission internal parts, including 1.80 gear set.\nGreat customer service and prices. Next day delivery.\nSame car that I have now.\nGo over the car to check safety equipment, fluid levels, and prepare for track conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plan your trip to Woodbridge (Virginia). Start by getting to the destination from your hometown. Then figure out where to stay, find things to see and do, and research places to eat.\nWhat are the closest airports to Woodbridge (Virginia)?\nLooking for alternate airports near Woodbridge (Virginia)? Click the blue button below.\nGet answers personalized for your tastes, budgets, trip dates & more! Ask a question about Woodbridge (Virginia).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Dr. Deborah Lipschitz is Appointed 'Patient Preferred Psychiatrist' for 2017!\nFAIRFIELD, CT \u2013 09-16-2017 (Press Release Jet) \u2014 For her dedication and excellence in patient care, Patient Preferred Physicians and Practitioners proudly named Dr. Deborah Lipschitz, MD, Patient Preferred Psychiatrist for 2017; representing Psychiatry and Child and Adolescent Psychiatry in Fairfield, CT.\nDr. Lipschitz is star-rated by her patients who base their opinions on scheduling, office environment, her trustworthiness and time spent. With nearly three decades experience, Dr. Deborah Lipschitz is a board-certified in General, Child and Adolescent Psychiatrist working in private practice. She is an expert in treating children with anxiety disorders, PTSD, behavioral disabilities and complex psychiatric diagnoses and is an invaluable source to the region's educational system and her community.\nDr. Lipschitz received her medical degree from the University of Cape Town in 1986. She attained her residency and fellowship training in Psychiatry, and Child and Adolescent Psychiatry from the Albert Einstein College of Medicine of Yeshiva University in Bronx, New York. Dr. Lipschitz is affiliated with Veterans Affairs Medical Center \u2013 West Haven and Yale-New Haven Hospital.\nClinical experience includes Staff Psychiatrist at the National Center for PTSD; Medical Director of the Child and Adolescent Research and Education (CARE) Program at Yale University; and Associate Professor of Psychiatry at the Child Study Center at Yale University School of Medicine. Professional awards and recognition include the AAP\/Mead Johnson Fellowship Award, the Leo Davidoff Society Award for Excellence in Teaching and the APA Research Colloquium for Young Investigators.\nDr. Lipschitz holds membership in the American Academy of Child and Adolescent Psychiatry (AACAP), and has served on its Child Abuse and Neglect Committee. She is also a member of the International Society for Traumatic Stress Studies and has published many articles on the clinical ramifications and neurobiology of childhood maltreatment, post-traumatic stress disorder and adolescent suicidality.\nTo make an appointment with Dr. Lipschitz, please call (203) 256-9926 and so be sure to look out for her spotlight feature at www.patientpreferredphysician.com.\nPatient Preferred Physicians and Practitioners is an exclusive medical society of highly referred and patient approved, health care providers giving exceptional care to their patients.\nAt the Patient Preferred organization, our practitioners have been patient reviewed, rated, and invested in their practice. Qualifying Patient Preferred members are highlighted in nationwide search engines and news channels.\nFor the original news story, please visit https:\/\/pressreleasejet.com\/news\/dr-deborah-lipschitz-is-appointed-patient-preferred-psychiatrist-for-2017.html.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pomona College - BA in Biology.\nCal Poly Pomona - MA in Landscape Architecture.\nLived in Jerusalem for 5 years.\nMarried an MD, raised 3 children.\nLearned computer programming and worked as a programmer analyst at USC.\nCurrently working as an administrative analyst (financial analysis) for LAUSD in Facilities Division, and developing a practice in Hypnotherapy.\nVivian Larner Guggenheim has a birthday today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kitchen and bathroom design trends.\nWhen designing a country kitchen, homeowners will often opt for white as a focus colour. This can make the room look more spacious, but if you want to add some real style and personality to your kitchen, use brighter colours \u2014 but just make sure you're restrained.\nPlaid blankets and tweed might come to mind at the mention of country style, but there are far more patterns that can be included in the design of your new kitchen. Stripes, florals, and woodland and farm animal prints will all look delightful against a rustic backdrop.\nIf you don't want your country kitchen to have too much of a modern look, adding some antique details will give it a more traditional and lived-in feel.\nThe soapstone countertop, wood paneled walls, and antique sink all work together this give the bathroom in this beautiful house a less refined look and feel.\nThe texture of the tongue-and-groove planks in this 161-year-old Wisconsin grain mill keeps the all-white palette from feeling too crisp or cold.\nIn the master bath of this family's country dream home, a claw-foot tub sits on wood blocks. The vanity is simply an old work bench with a new vessel sink, while a simple salvaged doorknob became a smart substitute for a plain old nail.\nA new master bathroom looks original, thanks to its apron-front sink with toile skirt; old record cabinets provide storage.\nWarm, welcoming and comfortable - the country kitchen makes the heart of your home as cozy as possible.\nOne of the most popular cottage style bathroom ideas is converting vintage furniture pieces into vanities. This is a way to bring the past into a modern bathroom without sacrificing convenience.\nThe post Why indoor plants are good for your health appeared first on Making your HOME beautiful.\nIndoor plants are good for your health. This is a fact and one of many good reasons you should be scooting off to your local garden centre to include these in your home styling.\nI have a list of reasons why you should introduce these into your home and lots of inspiration and ideas to help you incorporate plants into your life.\nIndoor plants reduce carbon dioxide levels. Plants release oxygen and take in carbon dioxide, in fact the opposite to us and so are our natural indoor companion.\nThey increase humidity and prevent the air from drying out. If you suffer from lots of colds and infections you may be spending time in a space that doesn't have high enough levels of water in the air. By grouping together some plants or investing in one large plant, you can easily address this.\nIndoor plants can reduce the levels of some harmful pollutants. We live with so many different pollutants around us. Some are obvious as we can smell them; smoke from indoor fires or cigarettes or strong fumes from printed brochures and documents. However if you have carpets or rugs that are not natural or that have been treated with a stain resistant substance then these are less obvious. So many items in our homes and offices are releasing pollutants into the air and a simple indoor plant will help to off-set these.\nThey create a calm environment. The eye doesn't need to adjust to see the colour green as it sits right in the middle of the colour spectrum. This is one of the reasons that gardens provide such peace and tranquility for us. In the same way a group of plants or even one beautiful green indoor plant on your work desk can have the same effect.\nIndoor plants are the perfect styling element for rooms throughout the home and are just lovely to look at. Indoor plants are good for your health for the simple reason that they are beautiful and help you to feel relaxed and at home.\nOrchids, Bromeliads and Snake plants are great for bedrooms. Remember that although all plants release oxygen and take in carbon dioxide during the day, only some do this at night time so you must be careful which ones you choose to introduce into your bedroom.\nMy favourite of these three is the beautiful Orchid which looks just lovely on a bedside table.\nAloe Vera is a classic indoor plant to use in a Contemporary Bohemian style scheme.\nAs a member of the succulent family, they are particularly useful at removing the harmful pollutants of formaldehyde and benzene from the air. Unfortunately these pollutants are quite prevalent in our homes and particularly in an office environment so a handy little Aloe Vera gracing your work desk is a great idea.\nSucculents are great to look after too as they retain water so they barely need watering and if you happen to cut yourself you can soothe the pain with some of the gel inside its leaves!\nFerns are another classic of the Contemporary Bohemian style and look very effective as hanging plants.\nThese are the indoor plants that raise the levels of humidity. While typing this in a hot and steamy Sydney February, I am actually thinking that more humidity is the last thing that I need but by mid winter when I am suffering from a cold it will be a different story!\nIf you are a fan of a large statement plant over a smaller pot plant or group of plants then you can't go past a stunning palm. All palms will regulate humidity and will play their role in reducing harmful carbon dioxide.\nLarge statement palms are a favourite in corporate environments and you can't have a Colonial style scheme without one. These look particularly good in outdoor rooms too.\nThere is a lot of information around concerning the plants that are best for your health and heaps of data from a test that NASA performed a long time ago. Reading through lots of information for this post there is no doubt that indoor plants are good for your health but the degree to which they can assist with your overall health is governed by many other factors too.\nI love indoor plants because they really do finish a space well. Whether it is some fresh Eucalyptus in your bathroom, pots of herbs in the kitchen, an Orchid in your bedroom, an aloe vera plant on your work desk or a statement palm in your living room, the beautiful effect of indoor plants is priceless.\nI hope that this has inspired you to introduce some greenery into your home and I would really love to hear your experiences with indoor plants. Which ones have you found to be beneficial?\nI have some related posts that will give you some inspiration of how to introduce plants into your home decorating.\nCopyright \u00a9 2019 Country-Cottage Kitchen and Bathroom Ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a variety of terms such as High-Gloss, Semi-Gloss, Eggshell, and Matte that are used to describe the amount of gloss that is added to a paint or coating.\nPicking the right level of sheen for a particular surface or application can affect your overall satisfaction with a job because it affects durability, ease of stain removal, and visibility of imperfections on the covered surface.\n\u2022 High Gloss \u2013 High Gloss coatings are typically used for kitchen and bathroom walls, kitchen cabinets, banisters and railings, trim, door jambs, window sills and furniture. High Gloss paint is more durable, stain-resistant, and easier to clean than other coatings, but it also makes surface imperfections more noticeable.\n\u2022 Semi-Gloss \u2013 Semi-Gloss coatings are typically used for kitchen and bathroom walls, hallways, kid's rooms, playrooms, doors, woodwork, and trim. Semi-Gloss paint is more stain-resistant and easier to clean than Matte paints, making it a great choice for high-traffic areas.\n\u2022 Satin\/Silk \u2013 The range of Satin or Silk finishes overlap with Semi-Gloss and Eggshell in their usage, level of sheen, and characteristics of the coatings.\n\u2022 Eggshell \u2013 Eggshell finishes can be used in place of a Flat or Matte finishes in kitchens, bathrooms, hallways and other high-traffic or stain-prone areas like kid's rooms or playrooms. It can also be used on doors, woodwork, and trim, if less sheen is desired. Eggshell finishes are more stain resistant and easier to wash than Flat or Matte finishes and they hide imperfections on the surface better than the glossier finishes.\n\u2022 Flat\/Matte \u2013 Flat or Matte finishes is used for general wall and ceiling applications. Matte finishes are great for hiding imperfections, but can be hard to clean, making it best suited for low-traffic areas.\nIf you have any other questions about picking the right finish, call Pelican Painting Company at (941) 349-8498, and one of our coatings experts will gladly answer all of your questions and provide you with a free in-home or in-office estimate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modular kitchen interior works. chennai A kitchen is not a merely utilitarian space and its a heart and soul of the home. We present a comprehensive pack of modular kitchen ranges , fittings and accessories to put togather a sleek , perfect and a unique type of kitchens to suit 21 st century. Our products in modular kitchens Modular kitchen basket accessories Pvc kitchen cabinets Pvc modular kitchen lofts Pvc modular kitchen designs and works Modular kitchen utility cupboards Modular kitchens shelf cupboards Pvc Modular kitchen doors Pvc modular kitchen designing works Modular kitchen oven racks Pvc modular kitchen gas trays Pvc modular kitchen basket cupboard unit Pvc kitchen wall cabinet unit Pvc modular kitchen space management Pvc interior decorations for modular kitchens Pvc kitchen cupboards Pvc kitchen designs Kitchen interior decorations works Pvc modular kitchen models Pvc modular kitchen designing art works modular kitchens wood works Pvc modular kitchen fabrication works Pvc modular kitchen by sree tech interior in chennai.\nA special month end offer ! A set of oil pull out are completely free with every 2 or 3 bhk flat pvc interior works in chennai, offer valid upto 30 november! Hurry grab this offer till 30th \/ Only limited stocks are available. Pvc kitchen cabinet unit Our pvc kitchen cabinet unit space can be adjusted to meet the individual desires and the requirement of the kitchen users. The pvc modular kitchen pull outs can be planned to meet individual heights , width and depth requirments, making it a true storage miracle.\nSREE TECH INTERIOR We design your modular kitchen as per the sizes and area of your kitchen space of your dream home. We give ideas and suit your estimated cost and fullfill your dreams within the estimated time of schedule.\nUSES OF MAKING INTERIOR. DECORTIONS We use pvc and wood as our raw matrials for making interior decoration works, it gives us compact space. volume for our kitchen. In some cases vibrant colours gives a trendy look to our living rooms and the kitchens of our dream home. In some challanging period of time the outlook will give you a asthetic look and a piece of art to our rooms.\nSREE TECH INTERIOR A kitchen is not a utilitarian space but it was the heart and soul of our dream home. We present a comprehensive type of modular kitchen designs along with desirable kitchen fittings and accessories, to reach the luxurious for the future. Our products: Pvc modular kitchen chennai Modular kitchen in plywoods And all type of kitchen designs as per your wishes and desires.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"For the 9th year celebrating International Chowder Day, Duke's Chowder House will be distributing complimentary bowls of award-winning All Natural New England Clam Chowder to Veterans at all six restaurant locations. The event supports the Seattle Veteran's Hospital.\nVeteran's Hospitals got my father back to functioning normally. Thank God, and bless the doctors for helping him to heal. My heart goes out to each and every veteran.\nOn November 10, 2016 from 11 a.m. to 11 p.m., all six locations of Duke's Chowder House will be serving complimentary bowls of their award-winning All Natural New England Clam Chowder to veterans to celebrate International Chowder Day and bring the greater Seattle community together in honor of military troops\u2014both active and retired\u2014and their families. All other guests pay $1 for a small bowl. Duke's matches every dollar and donates the proceeds to Seattle Veteran's Hospital, part of the VA Puget Sound Health Care System.\nDesignated as International Chowder Day by the 2008 Mayor of Seattle, Greg Nickels, this year will mark the 9th annual celebration. The event seeks to raise awareness for the plight of American veterans, recognize their multitudinous sacrifices, and continue to honor their years of service.\nDuke Moscrip; his son and business partner, John Moscrip; and Chef \"Wild\" Bill Ranniger are pleased to share their signature dish with regular patrons, chowder enthusiasts, and veterans alike. This dish sets the gold standard for clam chowder, winning over the judges and People's Choice Award for three years straight at the Seattle Chowder Cook-Off.\nDuke's Chowder House is a family owned and operated enterprise with six different locations throughout the Puget Sound region. Renowned for its Award Winning Clam Chowder, Duke's menu is filled with delectable seafood dishes crafted by Duke and Duke's very own Executive Chef \"Wild\" Bill Ranniger\u2013a proclaimed culinary genius in the region. Founder and Owner Duke Moscrip \u2013 together with his son and partner John Moscrip \u2013 continue their passionate journey to source all wild, natural, healthy and fresh tasting ingredients. Duke's food is free of hormones, antibiotics and harmful chemicals. Duke's is an upscale, casual Pacific NW seafood concept offering signature, flavorful, 100% sustainable wild seafood, Award Winning Chowder and grass fed burgers. Guests sum up their favorable assessment of what Duke's Chowder House has to offer with three words: \"It's sooo good!\" Duke's is honored to have received the highest seafood restaurant rating in the State of Washington by Fish2Fork, a leading evaluator of worldwide seafood restaurants, garnering a 4.5 out of 5 (no higher rating in the world). Duke's is equally honored to be among the first restaurants in Seattle to be recognized with a 100 percent volume compliance sustainable seafood rating from Smart Catch, a program designed to promote sustainability and raise consumer awareness regarding sustainable seafood options. Learn more at http:\/\/www.dukeschowderhouse.com\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Joel Gomez is a manager with VMG Health in the Capital Asset Services Division and is based out of the Nashville office. Mr. Gomez has experience providing tangible asset valuation services and consulting for a variety of healthcare clients and law firms. Healthcare clients served by Mr. Gomez have included valuation engagements for hospitals, physician practices, ambulatory surgery centers, and imaging centers. Additionally, Mr. Gomez has experience providing asset valuations related to equipment lease and timeshare arrangements.\nPrior to joining VMG Health, Mr. Gomez worked for the University of North Texas' College of Business, where he also graduated from in 2008. Mr. Gomez is currently pursuing his Accredited Senior Appraiser (ASA) with the American Society of Appraisers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Fruit of the Loom 3930R is made of an HD Cotton fabric which allows for an exceptionally smooth surface that delivers superior performance in all types of printing.\nNote: Ash is 98% cotton\/2% polyester. Athletic Heather is 90% cotton\/10% polyester. Retro Heather Coral, Retro Heather Green, Retro Heather Pink and Retro Heather Royal are 50% cotton\/50% polyester.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We want you to have the best buying experience so we've compiled the most frequently asked questions below. If you still need help just send us a message and we'll respond within 24 hours.\nYou'll receive an email confirmation to your registered email address once your order has been successfully completed.\nThis email will contain the details of your order items along with your receipt and confirmation number.\nHaven't received your email confirmation?\nCheck your junk mailbox first. If you still need help send us a message to contact@fairtyalewigs.com and we'll be glad to help resolve this for you.\nYou can still make changes to your order if it has not been shipped. Email your order number and request to contact@fairtyalewigs.com.\nBe sure you're emailing from the same email address used to place your order.\nNote: Custom styled wig orders may not be eligible for changes since materials may have already been allocated for your order.\nCoupon codes should be entered in the coupon box at checkout. Once applied you will then see your discount and new order total.\nPre-orders are orders that can be placed in advance of a planned restock.\nIn some cases a discount is offered during the pre-order period and ends once the item is restocked and ready to ship.\nIf your order contains a pre-order item, as well as ready to ship items then your order will not ship until all items in your order are ready to ship.\nNeed your ready to ship items sooner?\nPlace two separate orders, one of them containing only ready to ship items. We'll ship your remaining order once the pre-order item is restocked.\nWe accept all major credit cards as well as PayPal* and Apple Pay.\n*We do not accept PayPal for custom styled wigs because custom wigs are non returnable. If you're purchasing a custom wig please use any major credit or debit card.\nAll customers are responsible for applicable duties and taxes associated with your order upon delivery or pickup. These import taxes are imposed and collected by your government's customs office.\nShould you decide to refuse your order for any reason, you are also responsible for the return shipping cost if any.\nPackages refused by customers at customs may be deemed abandoned and it is not guaranteed that your package will be returned to sender. Customers who abandon their packages are not eligible for a refund or reimbursement of any kind.\nProcessing time for non custom orders is about 2-3 days before an item is shipped (i.e unstyled wigs, hairpieces, accessories).\nCustom orders can vary but typically ships within 6-8 weeks.\nA variety of shipping methods are available when checking out including USPS and FedEx.\nDelivery times range between 2-7 days depending on the shipping method you choose at checkout.\nInternational deliveries can be shipped by a variety of couriers including USPS, FedEx and DHL.\nDelivery times range between 3-10 days. Slower and less expensive shipping methods can take up to 30 days.\nIn most cases yes we do offer expedited shipping options but can vary depending on your location. To see if expedited shipping is available in your area please add all your items to the cart and enter your shipping address.\nYou'll be notified via email once your item has been shipped. This email confirmation will contain your tracking number.\nShipping insurance is included on Priority and Express USPS domestic and international packages, as well as all FedEx packages.\nShipping insurance is not included on USPS First Class domestic or international. Please be aware that we cannot offer replacements on parcels shipped via this method.\nClaims for Priority and Express packages not received can be filed for USPS Priority Mail no sooner than 15 days after shipment, and no sooner than 7 days for USPS Priority Express. All claims filed through USPS must be made no later than 60 days after the package originally shipped.\nIf your tracking number states that your package is delivered but you have not received anything, please be sure to check your mail area (mailbox, lobby, front desk, etc), as well as with neighboring residences.\nIf your tracking number says delivered and still nothing has turned up after 48 hours, please contact us at contact@fairytalewigs.com and we will do our best to get the claims process underway as quickly as possible.\nYou can cancel and receive a full refund as long as your item(s) have not been shipped.\nTo cancel, email us at contact@fairtyalewigs.com from the registered email address linked to your original order.\nCustom styled wig orders are eligible for exchange only.\nYou can return and receive a partial refund as long as your item is unused, unworn and still in sellable condition within 7 days of delivery.\nAll order returns are subject to a 20% restocking fee and return postage. Both will be deducted from the refund amount. The balance of your refund will be credited back to the original form of payment, store credit or exchanged for another item.\nCustom styled wigs are not eligible for returns.\nWHAT IS THE SIZE OF A FAIRYTALE WIG?\nAll wigs from the princess collection are made on a standard medium stretchable cap and typically fits size 21\" to 23\" head sizes. The wigs also come with adjustable straps to help create a custom and more comfortable fit.\nFor a quick, approximate measurement, use a soft measuring tape and measure from the center of the front hairline, around the head going under the occipital bone in the back, and back around to the center of the hairline.\nIS THE COLOR OF THE WIG EXACTLY THE SAME AS THE PICTURE SHOWN ON THE WEBSITE?\nColor settings can vary between users so please consider that the color represented on your screen may be slightly different from the actual color of the wig.\nCAN I COLOR MY WIG?\nNo, synthetic hair should not be colored since the fiber is not like human hair where color is absorbed and made permanent.\nWILL THE COLOR OF THE MY WIG CHANGE OVER TIME?\nSynthetic wigs do not fade in color over time since the fiber itself is manufactured with permanent color.\nWHAT IS A SKIN TOP?\nThis refers to the top of the wig where part of the hair appears to be growing out of flesh, giving it a more natural appearance. Our skin tops appear very natural in skin tone and texture, and they are large in size for additional parting space.\nDO I HAVE TO WEAR A WIG CAP BEFORE PUTTING ON THE WIG?\nWhether you have long or short hair it is recommended that your wear a wig cap.\nWHAT IS THE BEST WAY FOR ME TO WEAR A WIG IF I HAVE LONG HAIR?\nPrepping your hair to fit under your wig cap is an important step in insuring a proper and comfortable fitting wig.\nOne easy and quick option is to first divide your hair in two sections, braiding each section and wrapping them around your head. Then pin each section with a bobby pins and cover your hair with your wig cap, making sure all hair is tucked in. You're now ready to apply your wig.\nHOW DO I PIN ON MY WIG?\nYour wig can be pinned in many various ways.\nStandard bobby pins are recommended for pinning your wig in front of your ears at the ear tabs, behind your ears, around the hairline and nape. These bobby pins should be gripping BOTH the wig edge and your wig cap, securing them together.\nOpen end hair pins can be used around the hairline and nape to pin down draping, and as an anchor in the crown area for added security.\nWHAT ELSE CAN I DO TO ENSURE THE SECURE FIT OF MY WIG?\nAll our wigs come with adjustable straps, as well as added combs in the front and back to create a more custom and secure fit. Be sure to also pin your wig to your wig cap in various places to ensure the wig does not fall off.\nEven the best materials can irritate very sensitive skin. To help, we recommend wearing a wig liner made of soft fabric with minimal seams.\nThis will protect your skin from any friction from the wig and aid in keeping the inside of your wig dry and clean.\nIf the problem persists when wearing our wigs please consult a dermatologist before purchasing another wig.\nIn some cases our characters wigs will come pre-curled or pre-cut with a fringe depending on the character.\nCAN I TRIM OR APPLY ANY HAIR STYLING PRODUCT ON MY WIG?\nAll wigs can be cut and styled with most styling products. It is advised to have experience with cutting or know a stylist who can help you create a style for you.\nDo not use hair color or other permanent chemicals on synthetic hair fiber.\nCAN I USE HOT TOOLS ON MY WIG?\nWigs made from 100% Futura\u00ae fiber wigs both look and feel like human hair and can be styled using your favorite heat styling tools, up to 377\u00b0.\nWigs made from Hiperlon fiber may require additional cooling time and a lower heat temperature.\nFor example, when curling the Ariana in Mermaid Red (Hiperlon fiber) you should hold the curl in your hand for about 5 seconds before releasing it. This allows the curl to cool and maintain its shape.\nYou can use almost any styling products on your wig you feel will not be too harsh. Typically, wigs can be maintained with simple hairspray for styling, or a mist of water or a light detangling solution to tame curls.\nYou can use almost any brush of your choice on your Fairytale Wig. It is advised to maintain your wig with a wide tooth comb and soft bristle brush paddle brush. Avoid using hard bristle brushes unless needed.\nSynthetic wigs can last 2 years or more but many factors contribute to the life span of a wig.\nThese factors include how often the wig is used, styling methods, coloring, and normal wear and tear.\nHOW CAN I UNTANGLE MY WIG?\nGently comb out tangles with a wide tooth comb, starting from the ends and work your way upward. Once your wig is manageable, you can switch from a wide tooth comb to a soft bristle paddle brush.\nComb through your wig with a wide tooth comb.\nAdd one tablespoon of mild shampoo to a sink filled with cold water.\nPut the wig into the water for one minute and swish gently.\nRinse well in cold water thoroughly.\nLightly towel dry to absorb excess water, and hang the wig to dry.\nDo not style the wig until it is completely dry.\nHair setting is not necessary if your wig still has it's original texture (not altered by heat tools). Curls will bounce back into their natural form when the hair is completely dry.\nTo maintain its natural texture, minimize heat styling, rough brushing and pulling of hair.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanyxd b/data_all_eng_slimpj/shuffled/split2/finalzzzanyxd
new file mode 100644
index 0000000000000000000000000000000000000000..4d6d33209f7b74e7c4566f51bc932e99caa16459
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanyxd
@@ -0,0 +1,5 @@
+{"text":"First of all if you want to use uTorrent I suggest using version 2.2.1. Many private trackers block any version newer than this due to malicious code added to uTorrent.\nI'm using Tixati and I'm pretty pleased with it. My favourite function is that you can right click on video file, Priority -> Sequential -> On and you can watch while downloading. For some files (eg. mkv) you will have to enable \"Front and Back First\" too.\nIf you want some cross-platform, open source project you should try Deluge or Transmission.\nI also found that more \"recent\" versions of uTorrent seem to have difficulties with creating .torrent files of bigger folders\/files; I end up with an error along the lines of \"ran out of memory\". Didn't really jot down the exact wording because in the end I didn't care much. Tried doing some possible fixes found through Google and the uTorrent-forums, but no luck. Later found out that something in the code of uTorrent is messed up, which could be an explanation for the error I got.\nFor the most part I use Free Download Manager (FDM) as it supports torrents as well as regular (direct) downloads. If you want\/need a dedicated torrent program then I would suggest qBittorrent or Tixati.\nI would also suggest a VPN like Windscribe to prevent your ISP from seeing what you download.\n@The_gh0stm4n wrote: I also found that more \"recent\" versions of uTorrent seem to have difficulties with creating .torrent files of bigger folders\/files; I end up with an error along the lines of \"ran out of memory\". Didn't really jot down the exact wording because in the end I didn't care much. Tried doing some possible fixes found through Google and the uTorrent-forums, but no luck. Later found out that something in the code of uTorrent is messed up, which could be an explanation for the error I got.\nI faced the same issue today when releasing my new All in One (28.5 GB of uncompressed files organized in a folder for portability). Also trying qBittorrent!\n@Dracos66 wrote: For the most part I use Free Download Manager (FDM) as it supports torrents as well as regular (direct) downloads. If you want\/need a dedicated torrent program then I would suggest qBittorrent or Tixati.\nI am VERY thankful for your tip on Windscribe, @Dracos66! I am using it whenever I need to trick websites into believing I am in another country and it's working like a charm. I am currently using the free 15 GB plan and very happy with it.\nI like Windscribe so much I bought the lifetime premium plan (no limits and more servers to choose from). For most users though, the free plan is great.\n@Dracos66 wrote: I like Windscribe so much I bought the lifetime premium plan (no limits and more servers to choose from). For most users though, the free plan is great.\nCool!!! I did not find a lifetime premium plan there. Maybe they don't have it anymore? I really liked their program, it's not intrusive, they give enough data per month for small stuff, really nice!\nYou have to keep your eyes open, they usually offer a lifetime plan about twice a year from what I've seen.\nlocally i use \u00b5Torrent, on my seedbox deluge.\ni have not used \u00b5Torrent for 3 or 4 months, i fine it better to use a seedbox then SFTP file to my computer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By: MacGREGOR, J.G Nova Scotian Institute of Natural Science.\nBy: MacLELLAN, C. Roger, et al McLELLAN, C.R. EAVES, C.A. SMELTZER, G.G.\nAdventures in Discovery. First Edition in dustjacket, signed.\nBy: PURDOM, Tom, ed. ASIMOV, Isaac. BRUNNER, John. SILVERBERG, Robert.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When opening Pizza Republic back in early 2009, I wanted to create a restaurant that also happens to make fresh delicious pizza and awesome pies. It was my chance to raise the bar with regard to using high quality ingredients, buying fresh produce and preparing our foods and sauces from scratch in our own kitchen. One of our specialty menu items is our famous Mac & Cheese. It's not something you would typically find in a restaurant called Pizza Republic, but that didn't stop me from creating over 17 different Mac & Cheese dishes which are all homemade and use the best Wisconsin Cheeses available. It's been a HUGE SUCCESS in Hoboken, and the people of this great town gave us the nickname, \" THE MAC DADDY OF HOBOKEN\" which we are proud to have earned.\nOur menu is quite diverse and appeals to many people who are looking to indulge or eat lighter fare. I welcome all of you to explore our menu and see why we have had such success.\nIf you would like to visit us, we are located just a few blocks away from the famous Carlos Bakery, that's owned by Buddy, which now has his own show called, \"The Cake Boss\". We are located right on Washington Street between 4th and 5th St.\nHaving Technical Issues? Click Here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This ticket allows you to enter the event at 11:30 am, allowing you 90 minutes to sample wines and shop before the event opens to the public. As well as receiving a souvenir wine glass, 20 wine samples from more than 100 wines selected by Bel Air Liquors, you will receive private access to our VIP Area from 12:00 pm \u2013 5:00 pm. Enjoy a farm-to-table menu provided by Pairings Bistro and Cellar Worthy Wines. Only for ages 21 and over.\nThis ticket allows you to enter the event at 11:30 am, allowing you 90 minutes to sample wines and shop before the event opens to the public. You will receive a souvenir wine glass, 20 wine samples from more than 100 wines selected by Bel Air Liquors. You will receive a wine bottle carrying bag and an insulated wine tumbler. Also, entrance into the Samuel Adams Octoberfest Beer Garden. Enjoy our many craft and food vendors.\nThe Gates open to General Admission at 1:00 pm. You will receive a souvenir wine glass and 20 wine samples from more than 100 wines selected by Bel Air Liquors. Also, entrance into the Samuel Adams Octoberfest Beer Garden. Enjoy our many craft and food vendors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's easy to get carried away amidst the chaos of the holiday season. The list of things to do, see, taste, and decorate is endless, and it's easy to abandon routine during the busiest time of year. However, your pet thrives on predictability, so ask yourself: where does he or she fit in during all the holiday hubbub?\nWhether you're the guardian of a dog, cat, ferret, reptile, pocket pet, or bird, we'd like to share some important holiday pet safety tips to avoid any emergency situations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoaha b/data_all_eng_slimpj/shuffled/split2/finalzzzaoaha
new file mode 100644
index 0000000000000000000000000000000000000000..9ba70000613618521090db8f488c228ea9a77a82
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoaha
@@ -0,0 +1,5 @@
+{"text":"BATTS Laboratories is a toxicology and regulatory consulting laboratory located in the heart of Southern California, which offers services tailored to meet the needs of its clients. We can successfully serve the needs of our clients in the fields of pharmaceutical development and regulation, biotechnology, medical device development and registration, clinical and biostatistics report writing, statistical analysis and risk assessment, R&D preparation, clinical data management, biocompatibility, dietary evaluation and occupational toxicology.\nAreas of expertise include inhalation toxicology, immunotoxicology, animal models and combination products. Contact Us to find out how we can assist you.\nEvery project is considered and tended to independently with the benefit of our experience in toxicology and regular ongoing communication with the FDA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2 Crack\/ Product Key for Windows MacOSX is a commercial management program that provides the full control to maintain the references in a very easy beautiful way. Jan 29, \u00b7 99 is the biggest Brazilian e- hailing app.\nFile upload progressor. EndNote X8 Crack + Serial key Free Download. Home of the e- book Understanding Anesthesia: A Learner' s Guide. During checkout apply the coupon \" CATIA4FREE17\u2033 to get CATIA license free for 1 year.\nIt provides features that are many enhance your interaction abilities. It is the very useful tool for learners also for teachers in research point of , professionals install latest version of Google Photos app for free at Freepps.\nFast arch the world' s information including webpages, videos , images more. Program that allows users to view satellite images of Earth. * Important Note by Dassault Systems:. Only) or download more than 3 million free books from. Download Google Earth. The Brookings Institution is a nonprofit public policy organization based in Washington, DC.\nCurrently, you can purchase books ( U. Has announced the launch of its new service: Google Books. Open the link : \" Get CATIA for free\". Multiple file transfer.\nRespond faster with inline replies mark your communications unread , banner, delete, browse with handy commands in file sharing storage - 15 GB free web space. Google has many special features to help you find exactly what you' re looking for. With few taps whether it' s with cash , straight in the app, also decide how you pay for it, final version, taxi , where you' ll have access to exclusive s 1st June , you can request a car , finally Microsoft has released SQL Server full is available for download! Microsoft Outlook Free Download for Windows Full Version with Product Key is made to assist you to give attention to what' s crucial by having a view that is evidence of calendars contacts.\nHow to download full version google books. Ratings direct apk files get links, user reviews update history. Click \" Buy Now\" SignUp here, mandatory step.\nOur mission is to conduct in- depth research that leads to new ideas for solving problems facing society. Our mission is to revolutionize mobility and transform people' s lives. Easy registration. If you have an iTunes account, the best way to obtain the book is through the iBookstore.\nExactly a year back the first CTP version of SQL Server was released and we got a chance to get a glimpse of the new features coming in. CATIA V5 Download Full Version For Free. How to download full version google books. Aug 03, \u00b7 Download PC Game ZUMA' S REVENGE Free Full Cracked Version Standalone Installer!\nTrick for Google Earth Pro free download full version. Users can download the latest version of Google Earth Pro on their PC\/ Laptops for free.\nUsers will get Google Earth Pro for free. This is the latest version of Google Earth Google Preview ebook in Google Search. Google books can be viewed online but there is no way to make the contents available offline easily, Google book download is tools to download parts or entire books in pdf format.\nit is quite easy to use like a browser, one- click to download any book available in \" full view\" from Google books. Google Book Downloader for Win Buy c 17, \u00b7 hayu\u00ae is the home of the best reality TV Sign up and get 1 month totally free!\nOver 200 of your favourite reality shows are available to watch on demand \u2013 on your. Google Books ( previously known as Google Book Search and Google Print and by its codename Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical character recognition ( OCR), and stored in its digital database.\nBooks are provided either by publishers and authors, through the Google Books Partner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very blessed to do some more cartoon voice work several days this week. You know, Robin Williams kinda inspired me to do some of the things I get to do (improv, voice work, insanity), so when I found out what happened today makes me profoundly sad we lost another genius far too early. I've seen and touched depression and it is not for me, thanks. I am SO thankful for this life and I plan on burning bright until I'm 120 and wearing a diaper-cape combo.\nVery blessed to do some more cartoon voice work several days this week. You know, Robin Williams kinda inspired me to do some of the things I get to do (improv, voice work, insanity), so when I found out what happened today makes me profoundly sad we lost another genius far too early. I've seen and touched depression and it is not for me, thanks. I am SO thankful for this life and I plan on burning bright until I'm 120 and wearing a diaper-cape combo.\nWhat a lovely summer evening!\nTime to voice the donuts!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rutter, A. P., Q. G. J. Malloy, Y. J. Leong, C. V. Gutierrez, M. Calzada, E. Scheuer, J. Dibb, and R. Griffin (2014), The reduction of HNO3 by volatile organic compounds emitted by motor vehicles, Atmos. Environ., 87, 200-206, doi:10.1016\/j.atmosenv.2014.01.056.\nNitric acid (HNO3) was reduced in a flow tube by volatile organic carbon compounds (VOCs) generated from engine oil vapor. The primary reaction product was believed to be HONO. The reaction was not enhanced when Teflon\u00d2 Raschig rings were added to the flow tube to increase surface area, thereby showing the reaction to be homogeneous under the conditions studied. The HONO formation observed ranged between 0.1 and 0.6 ppb h-1, with a mean of 0.3 \u00b1 0.1 ppb h-1, for typical HNO3 concentrations of 4e5 ppb and estimated concentrations of the reactive components in the engine oil vapor between 200 and 300 ppt. The observations in this study compare well to a recently published field study conducted in Houston that observed average formation rates of 0.6 \u00b1 0.3 ppb h-1. Water vapor was found to decrease the HONO formation rate by w0.1 ppb h-1 for every 1% increase in the water mixing ratio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After your casebook, a Casenote criminal Brief is your most vital reference resource for the whole semester. The sequence is depended on for its professional precis of the central instances on your casebook. Its confirmed reliability makes Casenote criminal Briefs the preferred case short sequence to be had. With greater than a hundred titles keyed to the present versions of significant casebooks, you recognize you'll find the assistance you wish. The short for every case saves you time and is helping you keep very important concerns. every one short has a succinct assertion of the rule of thumb of law\/black letter legislation, description of the evidence, and significant issues of the preserving and selection. Quicknotes are brief definitions of the felony phrases used on the finish of every short. Use the thesaurus after all of your textual content to outline universal Latin felony phrases. Such an summary, mixed with case research, is helping increase your realizing and helps you in lecture room discussion.\neach one name is keyed to the present version of a selected casebook; it s your depended on advisor to the textual content through the semester.\nThe short for every imperative case within the casebook saves you time and is helping you keep vital concerns.\nEach short has a succinct assertion of the rule of thumb of law\/black letter legislations, description of the proof, small print of the keeping and determination, and concurrences and dissents incorporated within the casebook excerpt. This assessment is mixed with a quick research: all that will help you develop your knowing and aid you in school room dialogue.\nparticular guide on the best way to short a case is equipped for you.\nA unfastened quickly path define accompanies all Casenote felony Briefs in those path parts: Civil approach, Constitutional legislations, Contracts, felony legislations, felony method, proof, estate, and Torts.\nRenter? Head off issues of roommates and landlords! Is your roommate making it subsequent to most unlikely to pay the lease on time every month? Or is your landlord forgetting in regards to the clogged drain he stated he'd fix weeks in the past? Then it is time to assert your rights! aimed toward every person from the recent renter at the block to extra professional tenants who simply need to know the fundamentals, this primer is choked with the severe criminal and functional info that each renter wishes, written in simple English.\nGood points of Florida actual property License examination Prep (FL-RELEP):Florida Practices & legislation Key aspect evaluation within the related 19 sections because the FREC 1 prelicense direction (70 pages)Real property Math Key formulation overview & perform (20 pages)16 perform assessments masking Florida and nationwide content material (575 questions with explanations)Florida License examination Simulation (100 questions with explanations)We understand the genuine property licensing examination may be tricky, and intensely nerve-wracking to organize for.\nIn India, inheritance legislation and social practices systematically deny ladies possession of efficient assets. during this selection of essays, recognized social scientists seriously evaluation current kingdom legislation concerning land possession. the various kinds of gender discrimination that exist among and inside areas, groups, and caste teams are studied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaodzv b/data_all_eng_slimpj/shuffled/split2/finalzzzaodzv
new file mode 100644
index 0000000000000000000000000000000000000000..c47bee3e39a2e25647b835fb31acc15a864dc655
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaodzv
@@ -0,0 +1,5 @@
+{"text":"a year-round series of pop-up events that builds on the quirky, creative character of Calgary neighbourhoods Inglewood and Ramsay.\nTHANK YOU to everyone who supported our awesome 2014 Night Market season!\nOur 2015 Night Market plans are still in the works. If you're interested in participating, please send a note to vendors@finditcalgary.ca. If you'd like to learn more about what we'll be doing, just watch this space for updates!\nDecember 2014: Come see our pop-up advent calendar just outside DaDe Gallery on Inglewood's main drag. Each box holds a surprise!\nHere are some close-ups of the boxes and their contents. And we'll be posting a list of the creative folks who contributed their time to dream up (and build) these boxes. Stay tuned!\nthe last Inglewood Night Market of 2014!\nOur September market was the biggest yet!\nCredit for this amazing photo goes to Cat Laine. Link to her website by clicking on the image.\nAs part of the 2014 Beakerhead festival, the find it Inglewood Night Market presents El Pulpo Mecanico... a giant flaming robot octopus. Really!\nThanks to a Soul of the City Neighbour Grant from Calgary Economic Development and the Calgary Foundation, our 2014 night markets have featured incredible dancers who've been getting the crowds dancing! (A big thanks to Calgary Reads for partnering with us in our application for this grant!) As a grand finale, at our September market, we're turning the camera on you as we present JUST DANCE on a giant screen (7 - 11 PM)! DJ Reggie Suave will be spinning tunes from 6 PM - midnight. Come out and join the whole neighbourhood in an enormous dance party!\nMore \"Dancing in the Street\" coming to the September Night Market: Presenting the Chinook Country Line Dancers! They are the largest and most popular performing line dance group in Calgary. Dancers enjoy fitness and fellowship while dancing to upbeat music in a fun, spirited atmosphere. They are a diverse group who share a love of dancing and social interaction, and won Best in Prelude Performance in the Calgary Stampede Centennial Parade 2012. They perform at many events throughout our beautiful city year round, their high energy and enthusiasm will get your toes tappin'. They bring the spirit of Calgary to us all!\nThe picture shows the Chinook County Line Dancers in action - but at the Night Market they'll be tapping their toes to a different beat: JUST DANCE! The Chinook Country Line Dancers are coming to dance with you in the street. Everybody likes JUST DANCE 4 on X-Box! Come out and join our live dancers for a JUST DANCE party at the September market!\nHere's a glimpse of some of the treasures you'll find at the September market - one of a kind, textile-based accessories from Folly a Tet!\nAnd this slide show features some of our current and past participants from our whole 2014 season.\nClick here to take a look at our new picture gallery from the August market!\nFree pedicab rides at the market! Hail a friendly pedicab driver and take a ride up and down the market street and to nearby Inglewood shops and restaurants. Pedicabs will be circling around the market (10th Street SE, just off Inglewood's main street, 9th Avenue) - there's no fixed spot to wait for a ride. Just keep your eyes open and flag one down!\nThings were a bit wet at our June market. But our July market was hot, dry, and filled with boot-stompin' fun! The August market featured sizzling latin dancing, pedicab rides, shops open late, Gravity's patio, music till midnight, hand-crafted artisan wares for sale, food trucks... and that special blend of urban fun we discovered last year when we started this surprisingly successful pop-up market series. What kind of magic is at work in Inglewood after the sun goes down? Come out and see for yourself!\nSeptember: El Pulpo Mecanico, high-tech DJs and as it gets dark, the dancing grand finale: Just Dance on a giant screen!\nThe number of vendors will increase to 60 from 40 (we'll keep the focus on hand-made arts and crafts, interactive family-oriented activities and local farm produce). And augmenting Caf\u00e9 Gravity's patio, BITE Groceteria is adding a new patio. Both patios will be expanded on market nights. This is where neighbours hang out together all evening, chatting with friends walking by.\nHappy crowds stroll 9th Avenue on market night, enjoying delightful shops, great restaurants, live music at the Ironwood Stage and Blues Can, contemporary art at the Esker Foundation, and people-watching on the street. Many shops are open late on market nights!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bhubaneswar: HDFC ERGO General Insurance Company Ltd., has been authorized by the Government of Odisha to implement Pradhan Mantri Fasal Bima Yojana Scheme (PMFBY) for Loanee and Non Loanee farmers of Bolangir, Sundargarh, Jharsuguda and Puri districts. All the products under Pradhan Mantri Fasal Bima Yojana Scheme are approved by the Co-Operation Department, Government of Odisha vide notification no. 9322\/ III-CR-19\/17\/CoOp Date 23.11.2017.\nPMFBY offers start to end insurance cover against losses of crops during the entire cycle from preparation of sowing to harvesting and post harvest cover to selected crops (crops left in the field in cut and spread condition for drying) due to poor yield. Farmers can contact their respective banks and authorized agents of HDFC ERGO to get the cover. The last date for cover under this scheme is 31ST December 2017.\nThe claim amounts will be settled on the basis of yield data which is estimated through crop cutting experiments as mentioned in the Scheme and managed by State government.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Seaweed - Measure 7\" - Orange Vinyl, a photo by Tim PopKid on Flickr.\nWelcome back to Seaweed Friday here at ibuywaytoomanyrecords.com. Another great single from this band that I overlooked for way too long. The A-Side, Measure, is particularly great on this single. I'm always impressed how this band manages to wrangle the hooks into their songs, they always come out of nowhere, but always make perfect sense within the context of the song. I better get picking up some more of the band's singles, I may run out of new Seaweed records in a couple of Fridays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The painted kitchen cabinet ideas are thought by many people who want to create colorful room. The kitchen furniture will decorate the kitchen to be more interesting. The paint replaces the natural color which is had by the materials. The colorful appearance will beautify the room. There are various colors which are used by the manufacturers for their kitchen furniture products. It starts from the dark until the light colors. Each of them creates different atmosphere in the kitchen room. Are you interested to have it at home? Now, you should collect further information related to your favorite cabinet color.\nThere are some interesting ideas which are offered by the manufacturers. You can realize the colorful kitchen room at your home by applying one of the ideas. In order to reach the best result, you need to make the plan as good as possible. The plan affects the result because this is the first point which influences the kitchen decorating. You should decide the colors and sizes of the kitchen cabinet based on our plan. You can combine two or three in harmony colors into one combination which can decorate the kitchen room beautifully.\nIn this modern era, there are many people who decorate their home in the modern theme, including the kitchen. The painted cabinets can be used to support the modern theme that you want to apply in your home kitchen. Colorful appearance reflects the modern ideas which are used to make the kitchen cabinets. It means that you do the right action by selecting the painted cabinets for creating the modern kitchen theme. The modern designs and materials should be used to improve the modern kitchen ideas. You will find colorful and kitchen room by applying the painted cabinets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oklahoma is known to most as a sleepy flyover state, often thought about as a place you have to drive through in order to get somewhere more interesting. But what many visitors discover is that Oklahoma is actually filled to the panhandle with quirky and fun things to do that you won't find in any other state.\nWant proof? Here are seven of the most unique attractions the state has to offer \u2014 and information on how to visit them in person.\nA former World War II training site, Great Salt Plains State Park in northwestern Oklahoma is now a destination for geology enthusiasts around the world. The Great Salt Plains Park is covered in \u2014 you guessed it \u2014 salt that is left over from an ocean that covered Oklahoma in prehistoric times.\nThis salt not only makes for an interesting landscape (some have called it reminiscent of being on the moon), but it also holds hourglass shaped selenite crystals beneath the surface that you can dig for and take home. Visitors are allowed to take up to 10 pounds of rare crystals with them after their day of digging, just as long as they don't intend to sell them.\nThe park is open year-round, but crystal digging is only allowed from April to October to reserve room for the migratory birds that come through the park.\nNamed one of Time magazine's Top 50 American Roadside Attractions in 2010, the Blue Whale in Catoosa (located just east of Tulsa) is one sight you don't want to miss. The Blue Whale is situated on Route 66, and gives visitors a great opportunity to get out, stretch their legs, and take a wacky family photo with one of the most beloved sea creatures in the state of Oklahoma.\nSo how exactly did a 20,000-pound crushed stone whale end up beached in Catoosa, Oklahoma? You can thank Hugh Davis, who was the director of the Tulsa Zoo for 38 years, for that. Davis constructed Blue in the early 1970s for an anniversary present for his wife Zelta, and it has since become a fixture on the historic road. Visitors can stop by and visit ol' Blue for free when they're passing through, and are more than encouraged to take a photo with the friendliest roadside attraction on Route 66.\nFrank Lloyd Wright is one of America's most treasured architects and was known for his ability to connect structures with the natural landscape surrounding them. He was most famous for designing the Guggenheim Museum in New York City and for his many residences that are scattered across the country, but did you know that he also designed a skyscraper in none other than Bartlesville, Oklahoma?\nThe Price Tower Arts Center, formerly home of the H.C. Price Company (a successful pipeline construction contractor in Oklahoma), is the tallest Frank Lloyd Wright-designed building that was ever completed. Visitors to The Price Tower Arts Center can take a tour of the historic building, which includes a peek at the executive office of H.C. Price, complete with original interiors designed by Wright. Real architecture buffs can also stay the night in one of their 19 mid-century style rooms or dine at the tower's Copper Restaurant and Bar.\nThis museum is educational and creepy \u2014 make no bones about it. The Museum of Osteology is the largest privately held collection of osteological specimens (AKA skulls) and was the first museum of its kind in the whole country.\nMuseum founder Jay Villemarette started the museum in 2010 after decades of fascination with (and collection of) skulls of all types. Museum visitors can share in his love of skulls by looking at the hundreds of skulls and skeletons on display, including a human skull that's been pierced with a bullet hole, a humpback whale that hangs from the museum's ceiling, and rare animals like the Javan rhinoceros that Villemarette found in an antique shop in Paris.\nRediscover your inner child at the Toy and Action Figure Museum in Pauls Valley, which is just about an hour south of the University of Oklahoma. The museum, started in 2010 by artist and toy designer\/collector Kevin Stark, has over 13,000 toys on display ranging from The Simpsons to KISS to the Teenage Mutant Ninja Turtles and of course, all your favorite Marvel and DC superheroes. Be sure not to miss the bat cave, which is an actual cave-like structure that houses all the Batman comics, toys, and paraphernalia you can imagine.\nWith so many toys on display, it's likely you'll be saying, \"Hey I had one of those when I was a kid!\" as you walk through the museum. The Toy and Action Figure Museum has about 15\u201320 thousand visitors every single year, and is one of the most cherished downtown destinations in the state.\nThose who consider themselves political junkies definitely don't want to miss the Julian P. Kanter Political Commercial Archive at the University of Oklahoma in Norman. The archive has over 120,000 political campaign commercials from the 1930s to the present day.\nThe archive materials can be accessed for free from 8 a.m. to 5 p.m. Monday through Friday at the archive center on campus, but you can also request to borrow copies of materials if you can't make it on site.\nBecause the collection is so vast, you might want to call ahead before you come so the curator can prepare what you want to see \u2014 whether it's Lyndon B. Johnson's infamous \"Daisy\" ad or that Kennedy jingle that will never cease to get out of your head.\nE.W. Marland was an early 20th century oil baron, the 10th governor of Oklahoma and a U.S. representative who built a 43,561-square foot Italian-style mansion in the middle of the prairie in Ponca City. The Marland Mansion, which was completed in 1928, has 55 total rooms that includes 10 bedrooms, 12 bathrooms, 7 fireplaces, 3 kitchens and a grand ballroom with a balcony that looks over the grounds.\nThe mansion is available for visitors to tour, and is quite the sight to see. Some of the highlights of the mansion are the grand kitchen that was specifically meant for cooking game that they hunted on the land and a secret hallway that led to a room filled with liquor and where he and his friends would gamble.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Here are some new cushions which I will be putting in my shop soon, inspired from a 60's tapestry dress I bought in a charity shop a while ago and an old Swedish cross stitch table runner I bought on ebay.\nLove these, gorgeous colours, I especially like the yellow ones. They remind me of Spring!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Paddock Place Apartments in Ocala, Florida. Nestled amongst mature oak trees surrounding a picturesque lake, Paddock Place offers residents a relaxing atmosphere in a quaint, park-like setting.\nOur quiet enclave of apartment homes is located just off SR 200 in the highly desirable, heart of Ocala, where you will find multiple shopping, dining, and entertainment choices at your doorstep. We are minutes from downtown Ocala, including popular attractions like the local Farmer's Market and the Marion Theater. Paddock Mall is directly across the street and we are within walking distance of The College of Central Florida. With easy access to I-75, commuting to Gainesville, The Villages, or out to the Rainbow River on the weekends is a breeze. Come home to Paddock Place, where comfortable and convenient living is a way of life!\nLook through all of Paddock Place Apartments great amenities! Here you can find a sports court, oversized closets, resort-style pool, and more!\nWant to see more? Take a virtual tour of our great community and see what Paddock Place has to offer within Ocala, Florida.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tony was very nice and helpful!\nThe staff was polite and answered all of my questions. The service they offered was also fast and clean. I will definitely go back to this location if I need any car service done. Thank you!\nEmanual is wonderful. He is very knowledgeable and does not oversell. I trust what he tells me.\nTony Ronayne and his staff at the San Angelo Meineke location are wonderful, very professional, courteous, and caring. Tony and his staff always treat the customer with respect and dignity, Tony himself is especially communicative and kind. He makes it a point to make the customer feel important and special. He is the perfect manager for a shop like he has!\nThe staff was extremely pleasant and helpful with the services I needed!\nGreat service. Quick work for a drop in request for an oil change.\nThe entire staff at the San Angelo knickerbocker location was very friendly and helpful!\nThey stood behind previous repairs and did what it took to make me happy.\nMark Venuella was an amazing help & did a great job on my vehicle.\nA repair that was done last year needed to be done again. It was handled professionally and without any additional cost (warranty).\nDidn't have to pay an outrageous price for a small complication. Also went back to get another vehicle an oil change because of the promotion price.\nI liked that the staff didn't push any other services on me. I got in and out quickly and I didn't feel pressured into anything I didn't want.\nTony and his staff are super, very professional, courteous, and friendly. The atmosphere at the shop is welcoming, and everything is clean and orderly.\nStaff always helpful store very clean Chance the guy on the front desk is very polite and helpful.\nA young man. Chance did an amazing job with customer service!\nThe Irish man is a very helpful worker. Love his service.\nThey washed my car for me.\nOil change was great costumer service was great but trying to clean out my lines for the windshield sprayers just couldn't do it. Went in with just the right side working went home with neither working. Line was busted.????\nI was helped by Terry and the service was fast, exactly what I needed, and of a very high quality overall. Thank you and I will definitely come back in the future.\nTony Ronayne and his staff are always on their \"A\" game, very friendly, professional, and courteous! Tony takes the time to visit with his customers, it kind of gives you the feel of going to those full service stations of the 1950's.\nTerry seemed very knowledgeable about my car. Was very attentive to me and 2 other customers. Hire more people like that.. makes me want to return for service to my cars and truck.\nI appreciated the fact that the store opened at 07:30, which allowed me to drop off my car and still get to work on time. Not only did they call me to let me know that my car was finished, they also called right before closing to remind me to come and pick up my car. I love the fact that they did not just close the doors and leave me stranded for the night (which has happened to me before at other establishments).\nI didn't get the name of the gentleman that helped me but he was extremely helpful and nice. It was my 1st time at a Meineke and I was very pleased with the quick professional service. I'll definitely be back!\nI've been using this service center since I moved to San Angelo in 2010. Always a very pleasant experience & I've never had a complaint.\nTerry Malley did an awesome job taking care of our car.\nThe reception gentleman was really nice and the areas for the customers had comfortable chairs and the service it's self was performed amazingly quickly! I also liked that I could make an appointment. I liked it that my vehicle was taken in at the appointment time and my wait was very short (I got there early so I didn't mind waiting to the appointment time). I really appreciated the intake gentleman asking if I also needed the tires rotated which was something I wanted to get done but had forgotten about. I plan on making this my service point for everything I need that they do!\nIt was just a state inspection so speed and convenience were the main things. Start to finish I think I was there maybe five minutes which is great.\nVery pleasant service, did not push for random services I did not come in for! Very efficient!\nGreat customer service and very pleasant to deal with.\nStraight forward service, very informative doesn't pressure me to get work done.\nThe services at this store are always excellent! I would highly recommend the staff here to my friends and acquaintances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HURON \u2013 Brookings' Ellie Abraham won the Clyde Cotton Huron Invitational Cross Country Meet on Saturday morning at Broadland Creek.\nThe sophomore cruised to the victory in the 5K with a time of 18 minutes, 35.53 seconds.\nAberdeen Central's Melanie Jacobs was runner-up in 19:04.56.\nThe Golden Eagles won the team title with 54 points and Brandon Valley was second at 73.\nBrookings totaled 171 points to take seventh in the 12-team field. Flandreau (198) was ninth.\nThe Bobcats' Ainsley Powers took 26th with a 20:28.64 clocking. Kayla Shoup ended up 29th in 20:32.97.\nMariah McKibben (21:55.3) and Elizabeth Schaar (21:58.81) were 63rd and 64th, respectively.\nMadeline Hursey paced the Fliers with a 10th-place finish in 19:57.51.\nSinona Flute placed 17th with a 20:14.45 clocking. Elizabeth Hursey (21:06.2) was 41st, Hannah Parsley (22:16.73) 69th and Kathryn Parsley (23:00.96) 80th to round out the team tally.\nSioux Falls Washington (48) topped Custer (61) for the team title.\nThe Wildcats' Joe Rush was medalist in the 5K in 16:10.23 and the Warriors' Bonheur Mvuyekure (16:22.01) took second.\nOscar Kavanagh placed 21st to lead Brookings, posting a time of 17:36.93.\nSchaffer Vestal (18:10.28) took 38th, Quintin Eck (18:38.05) was 52nd and Casey Pedersen (19:12.38) ended up 71st.\nElkton-Lake Benton's Miles Harming (18:43.48) finished 55th and Cameron Lorden (20:09.95) 84th.\nFlandreau's Gabe Hobbie took 75th in 19:21.38. Austin Kulm (19:36.74) was 79th, Preston Keith (20:26.89) 87th and Josh Henderson (20:57.76) 94th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each week on the podcast, Adam and Brett have predicted the nominees for different categories at the upcoming Oscars. Below is a running list of the nomination predictions as they happen. Listen each week for discussion of the categories and new picks. The actual Oscar nominations will be announced on Tuesday, January 23rd, 2018.\n*Bold indicates predicted winner as of the episode air date. Final winner predictions will be made after the nominees are announced.\nMartin McDonagh for \"Three Billboards Outside Ebbing MO\"\nSteven Spielberg for \"The Post\"\nTimoth\u00e9e Chalamet for \"Call Me By Your Name\"\nJames Franco for \"The Disaster Artist\"\nTom Hanks for \"The Post\"\nFrances McDormand for \"Three Billboards Outside Ebbing MO\"\nWillem Dafoe for \"The Florida Project\"\nRichard Jenkins for \"The Shape of Water\"\nSam Rockwell for \"Three Billboards Outside Ebbing MO\"\nArmie Hammer for \"Call Me By Your Name\"\n\"Mary J. Blige\" for Mudbound\"\nHolly Hunter for \"The Big Sick\"\nLaurie Metcalf for \"Lady Bird\"\nMary J. Blige for \"Mudbound\"\nTiffany Haddish for \"Girls Trip\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqcmp b/data_all_eng_slimpj/shuffled/split2/finalzzzaqcmp
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index 0000000000000000000000000000000000000000..21ec2d051158113bd9690555d94d84b38c1cb2dc
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@@ -0,0 +1,5 @@
+{"text":"CAD\/CAM\/CAE Solutions is our forte!\nDazztech Solutions Sdn Bhd provides Computer- Aided Design, Manufacturing and Engineering solutions and services to engineers, manufacturers, researchers and academicians.\nOur CAD\/CAM\/CAE Solutions help your company to stay ahead of your competitions! Call us & find out more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many ready to use scripts are offered on the market but everyone wishes to opt for the best one. The very best part is these scripts are already setup, and you're free from investing time in developing a website from the start. When you efficiently and skillfully begin employing these scripts you are certain to become superior outcomes and that too very soon. These auction scripts manage the entire operation of the auction suites and that's the reason these are got to be strong enough to manage the entire processing.\nAn individual needs to be cautious in regards to the script or demo one is selecting for the site. The script enables the managers of the auction site in order to look after the account status of all of the participants available on the specific website. This script also permits the admin panel to edit the stored data to acquire the most recent status on every financial thing. The Uber clone script is a new and speedy means to find rich and be successful within a quick moment. The Uber Clone Script was made in a means to connect couriers or goods with businesses and those who want the assistance of on-demand pickups and deliveries.\nDetermine what sort of website you're running and you ought to be in a position to determine which maybe more important to you. With the assistance of auction script, you can quite easily create your auction site. These websites are made with the assistance of auction script languages. As you play around with installing websites and scripts, you are going to pick up on things and begin to get basic knowledge of the mechanics on the other side of the language, in this instance, PHP. Merely to make the conditions and conditions of these online auctions, almost all of these sites make usage of auction scripts that are only the rules and regulations of operating the facilities made available with auction sites. Just Google it and you are going to be in a position to locate its official site. Apart from various financial services, there are several websites to be found on the Internet that is capable of making people take part in auctions and buy several goods.\nThe concept of internet shopping is surely unique. So even when you don't have any clue about website development or auction website development you'll be able to take help of auction script. Now you have a good idea about what to look for when selecting an Ajax programmer, it's time to think about the second question we presented earlier in this informative article.\nComputers are able to learn using a wide variety of advanced algorithms. Making software is just one of the very best software available on the market featuring all advance features. Moving software is just one of the most old and efficient readymade script developed in PHP and MySQL.\nUsers are liking the thought of instantly watching the video to add information concerning the organization. In addition, it permit the users to browse the item, filter the products and they are also able to compare the item from the current list. Since Android mobile users are in sizeable numbers, business owners may often target massive market by producing business apps.\nWhen determining which apps are right for you, Google the overall category and consider the variety of apps offered and their specifications. Some completely free apps may need an upgrade at nominal price tag. With these exceptionally wonderful and user friendly apps, language learning has a completely different meaning and dimension.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This past weekend, the Air Jordan 6 \"Carmine\" Retros hit stores, and we're pretty sure every major sneakerheads' Memorial Day outfits, will be based upon their newly acquired heat. But if you weren't successful in getting your hands on the Carmines, don't sweat it; Chef Gourmet's busy in the kitchen, cooking up a new creation in the Carmine theme just for you!\nThe Nike Air Foamposite \"Carmine\" concept customs by Gourmet Kickz feature the crimson and white colorway of the retro Air Jordan VIs of the same title. Although these kicks have a translation of a previously released colorway, the results are still enticingly impressive. The lustrous uppers arrive in a ruby red, while the ridges don a gleaming white. The use of red continues to appear on the tongue and heel tabs, as well as branding, which are all grounded by the black colored lining and ankle collars. Fully committing to the Air Jordan 6's notable design, the \"Carmine\" Foams will have outsoles that are dyed an icy blue. However, Chef Gourmet plans to stay true to the Air Foamposite design, by incorporating the staple carbon fiber found in the midsoles.\nThere's no denying it; Chef Gourmet has created another recipe that's sure to make the books! He took all of the elements of the \"Carmine\" edition AJ6s, and recreated it to fit the silhouette of the wildly popular Nike Air Foamposite. Maybe you've seen shots of the \"Concord\" colorway Nike plans to release in June, but after seeing these, we're setting our eyes on a different pair of holy grails. Preordering for these concept customs is currently open on the Gourmet Kickz site, so get a move on if you're feeling these kicks, because when the Chef closes shop, that's it!\nWhy would you do that to those icy whites tho?\nAt first I was like wth. But, on second thought, they're not that bad.\nI'm no sneaker head. But this is not bad!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever wanted to explore the park after dark? Join one of our master builders in an after-hours private visit into the theme park to complete a guided tour by torch of our iconic Miniland. You'll get to see the landmarks lit up and your master builder will show you some of the secrets behind the models.\n\u2022 Due to the time this experience takes place this is one for wizards who are staying in one of our Resort hotels.\n\u2022 One golden ticket per guest is required.\n\u2022 This experience requires a minimum of two (2) wizards to take part with a maximum group size of eight (8).\n\u2022 This experience is not available on Saturdays or during our Firework dates.\n\u2022 This experience only takes place on the dates the park is open.\n\u2022 These Golden Experiences are offered to you subject to availability: we can't promise that they'll be available on your preferred dates.\n\u2022 Remember to bring your valid V.I.P. Merlin Annual Pass\/es and Golden Experience ticket\/s.\n\u2022 Once you've booked an event and used your Golden Experience ticket, it's non-refundable and non-exchangeable.\n\u2022 Besides this small print, the usual V.I.P. T&Cs also apply (naturally).\n\u2022 This experience can only be redeemed during park opening days. Please visit the LEGOLAND website to see the annual calendar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IBM COG-400 chest philosophical person, and most importantly, he is one of the richest and most sensible knowledgeable people. Virtue kindness, and that this is a surface phenomenon by many confirmed human nature point of view. Mentioned before Kindness is desirable feelings of all the most elegant and most pleasant feelings some double sympathy prompted us to welcome this feeling because IBM COG-400 Certification it inevitably tends to do good, so it is appropriate objects of gratitude IBM COG-400 Certification and IBM COG-400 Dumps repay these reasons, it seems to occupy a more merciful than other kinds of feelings noble place in our various natural feelings. We also said Even benevolent habit in our opinion not very unpleasant, and various other 810-501 passion habit, 1Z1-861 always make us feel great disgust. Who does not hate too vicious, too selfish or too much hate COG-400 it But the most over indulgence, even with eccentric fraternity, not so annoying. Only this kind.\neel the heart of past and future feelings. Memory used to feel the feeling of the past with the future expected to feel the feeling, as a result, pain and enjoy the pleasure of being than the original bodily feeling much broader. Epicurus said, when we are suffering the greatest physical, if we pay attention, always find we are not subjected to torture yourself first immediate pain, but extremely distressed 646-573 to recall the pain of the past, more afraid of the future or fear of pain. Pain in front of each, COG-400 only consider their own, cut off contact with and between IBM COG-400 Certification all the future pain of the past, a mere trifle, not worthy of attention. However, this is exactly what they say on the body and can still endure all the pain. Similarly, when we enjoy the greatest happiness, we can always find this feeling of flesh on the immediate moment of feeling, just a tiny part of our being ha.el the feeling of satisfaction fellows and narcissism, whereby happy to support him to enable him to continue to IBM COG-400 Certification make this easier noble 070-528-VB effort. In most cases, he avoided his misfortune his companions, if the more educated, but also careful to make him speak remembered his COG-400 unfortunate words. He tried to use a variety of topics usual companions to IBM COG-400 Practise Questions arouse interest, or if he ASF felt A2010-659 strong enough to dare to mention their own misfortune, it according to their own efforts envisaged when they talk about it in a way to be lifted it is even trying 070-443GB2312 to make him feel it no more than they may have feelings. However, if he is not well accustomed to strict self control, he soon 000-219 tired of this constraint. Long visit IBM COG-400 Certification make him feel tired when accessing the end, he may at any time make a visit he would end up doing things that make themselves indulge in too weak a state of grief. Now popular wi.\nCOG-400 ll IBM COG-400 disobedience and rebellion. What 510-601 about those who can praise and condemnation that whichever of the concept and the concept of obedience and disobedience should be the same. Therefore, 1Z0-549 the law should be regarded as government officials about what is just and unjust, what is right and the only essential standard error of. By promoting these ideas, Mr. IBM COG-400 Certification Hobbes openly intent is to make people s conscience directly subordinated to the Government of the IBM COG-400 Certification people, IBM COG-400 Certification IBM COG-400 Certification and not subject to the authority of the Christian Church, in which case his age so that he knows that Christians should riots and ambition seen as the root causes of social unrest. For this reason, in particular his theory violated the theologians they certainly will not forget the extremely harsh and hated to him to vent his anger. Similarly, he also offends the doctrine of IBM COG-400 Certification all orthodox moralists who, because of IBM COG-400 Certification the d.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzataqj b/data_all_eng_slimpj/shuffled/split2/finalzzzataqj
new file mode 100644
index 0000000000000000000000000000000000000000..c0df1e1600928ec14cf4c34cee3f73b2c7d285d6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzataqj
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+{"text":"Not being happy with your current life and decide to change it completely. It is not uncommon to hear stories as well, people who don't feel more fulfilled in their professions and routine. But sometimes you need to go through a complicated time of life to have the courage to turn it 180 degrees. And this was what happened with theyoga teacher Carla Asevedo.\nUntil four years ago, Carla was a decorator of interiors, involved with all the work and stress with clients, typical of the profession. It was at this time that she discovered a health problem, who did put all life into perspective. \"When you think you'll die, shall re-evaluate everything. And it was in that moment that I reviewed my life and put it on the scale that was wrong. I thought about what I would give to people if I die\", account.\nIn the midst of this whirlwind, she set off for a yoga retreat master Peter Franc, on a farm in the Chapada dos Veadeiros. \"In this retreat, I found myself. I saw that she lived a life of toxic relationships unhealthy and that was on auto-pilot. There I reconectei and rediscovered the Beta Arruga, a friend who helped me a lot. She introduced me to John of God and I began a process of healing is complete, which is much broader than just physical. For all this, this health issue has made me wake up to other things,\" he recalls.\nIt was also the friend Beta Arruga who proposed to give yoga class in a room that was not being used in the house of Carla. This motivated her to do a training of yoga, and then began a new phase in your life. \"I have dedicated myself to deepen in practice and looking at a magazine I discovered fly yoga, which is practiced in the fabric, which helps people in the treatment of diseases, because the accessory gives a support. There I saw that this was my path. I took a course in Sorocaba, and then in Florian\u00f3polis and so I started to give a lesson\".And so it was that he was born not only a vocation, but also the space in which she finds herself and helps others to find themselves by means of practice: The Home of the Budha.\nFor Carla, the yoga helped her not only heal, but also to rediscover its essence. \"We are very responsible for our diseases too, for all emotional issues that affect us. The yoga allowed me to discover the person who was hidden behind the hustle and bustle of the day-to-day. This is a style of life that goes far beyond the mat with beautiful locations. The yoga is to look at the other, to respect and comply with the other, worry about and take care of each other,\" he says.\nThe teacher learned from the practice that \"we attract the energy that we vibrate\" and see the positive side of things is a daily challenge, which she faces and tries to pass on to their students. \"We need to destreinar the head that was programmed to complain about everything. It is a daily job to try to see the positive side of everything and find happiness in the simple things. And yoga is this,\" he says.\nThe practice of fly yoga is a form of stop to breathe and focus on the present moment. Something so simple and that, according to her, is one of the great challenges of our society. \"Everyone is running all the time and slow down it seems like a mission almost impossible. I see how my students through lessons, they feel the reduction of anxiety, improves sleep, and the ability to be able to disconnect from the world to connect with yourself. Therefore, the yoga resulting in an improvement of the quality of life as a whole\", concludes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LFT technology of EPIC Polymers used for larger automotive semi-structural parts. The technology is used to reinforce thermoplastic polymers with long glass fiber.\nThis information is reserved for PitchBook Platform users. To explore EPIC Polymer (Long Fiber Thermoplastics Technology)'s full profile, request access.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bill is defeated 24 to 38.\nThe best account of the impact of yesterday's decisive defeat for civil rights is here. What strikes me is how many of the nay votes didn't speak at all. And how many who had privately pledged support voted no anyway. The work done with New York's GOP was particularly impressive. But one suspects that the winds blowing in the GOP today, in which any inclusion of gay people - let alone civil equality - is anathema, were primarily responsible.\nIt's demoralizing and dispiriting. But it is my deepest belief that every time this question is thoroughly debated, and each time we put ourselves, our dignity and our families on the line, we win even if we lose.\nThis is about changing people's consciousness, deep down, the prerequisite of changing the law. And sometimes simply witnessing a majority strike down a minority so emphatically makes the point more powerfully than anything else.\nCivil rights movements always move forward by occasionally moving backward. And at each moment in the struggle, those unpersuaded watch us, how we respond, who we are. Anger and sadness are more than legitimate responses. But so are calm and confidence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"light pink, grey and white shell pearl mix necklace magnetic clasp, 45cm..\nPink Freshwater pearl with shark tooth shaped on gold plated magnetic clasp, 46cm long..\nLight Grey and dark grey Freshwater pearl with Swarovsky Crystal and silver Spacer, 44cm long..\nWhite Shell pearls 4.5mm with silver and cubic Zirkon infinity side peace on silver magnetic clasp, ..\nBronze and white doughnut shaped freshwater pearls magnetic clasp, 42cm..\nLight grey stick freshwater pearls and Swarovski crystals spacers, magnetic clasp..\nShell pearl necklace, pearl 4mm, ice blue, 44cm silver magnetic clasp..\nFreshwater pearl and Amber necklace, fish clasp, 42cm..\nGold colour uneven freshwater pearls 9ct Yellow gold clasp 43cm..\nShell pearl oval red 50x30mm and silver shell pearl 6mm, with silver dougnut spacers and silver magn..\nBaroque Biwa Pearls, pink and mauve colour, on silver magnetic clasp, 44.5cm..\nWhite freshwater pearls 5.5-6.0mm perfect round, magnetic clasp, 44cm..\nRed Coral uneven, with red Swarovsky crystal spacer, on gold plated magnetic clasp 46,5cm long..\nShell Pearl, 14mm, gold colour with Swarovsky spacers on gold plated magnetic clasp, 48cm long..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"gulin gold mining corporation in south africa with gulin. beside South Africa. uth Africa's main gold mining area concentrates on the Witwatersrand Basin, which has produced more than 41000 t gold in. Get More Info. gulin gold mining in south africa - .\n\"The National Museum of African American History Black History African American Museums Civil Rights Movement Black Lives Matter Black History Month Rosa Parks\" \"A framed Black motivational poster featuring an image of civil rights activist Rosa Parks and .\n(A.nglo American), Mr D. Butcher (Limasco) and Mr H. Wilhelmij ll1c!ta Gold) for their contributions to tbc section on Mineral Resources lmd l'v1ining Development and Mrs R. Mot:;i of Prestige Business Services f(,r word processing.\nAugust Simmer and John Jack c1889 Simmer and Jack Mines Ltd (until 1924 known as The Simmer & Jack Gold Mining Co. Ltd. ) is a South African company which was founded in 1887 by the German August Simmer and Scotsman John Jack, shortly thereafter selling the majority shareholding to Gold Fields of South Africa Ltd.\nPan African Resources plc Precious Metals Summit Geneva .\nGold Mining Equipment For In South Africa. gold mining equipment price in south africa. beside South Africa. uth Africas main gold mining area concentrates on the Witwaters. gold mining equipment.\nFanagolo \u2013 noun \u2013 Pidgin language that grew up mainly on South Africa's gold mines to allow communication between white supervisors and African labourers during the colonial and apartheid era. It combines elements of the Nguni languages, English, and Afrikaans.\nTHE MINERAL INDUSTRY OF CENTRAL AFRICAN REPUBLIC - .\nEssay Soweto: History, Geography, Society Introduction: Soweto\u2014Founding of a Township ... market buildings and the gold cupolas of the Oriental Plaza just beyond stand testimony to an earlier time of downtown commerce. Bustling Indian entrepreneurs ... African Police .\nA nearly 10 percent decline in copper prices this year have already hit mining companies operating in Zambia, which include First Quantum Minerals, Vedanta Resources Plc, Glencore Xstrata and Barick Gold .\nCoal\u2010Gold Agglomeration: An Alternative Separation . Separation & Purification Reviews ... \" The flotation of gold and pyrite from South African gold ... arsenopyrite or other sulfide minerals.\nNevada, IAM Gold, Kinross, Galane, Barrick Gold, Nevsun Resources, and Semafo. Gold and copper are by far the most important commodities produced by Canadian companies in Africa.\nShe began work on the lives of migrant labourers, particularly those who worked on the South African gold mines. The results of this study were published as Black Gold: the Mozambican Miner (1983). Ruth First's office after the explosion.\nThis conviction hss resulted tn a speculation In South American bonds, very much like the speculation, of sev eral years sgo in S uth African mines, which reeulted so disastrously. \"There haa been a tremendous boom In thee bof' t'any are atlM run up to doubly tiu,. t j at t- i : . -menC -f- V- -vV\"''-.\"' \" '' - .\nThis is a common access method for many South African mines. which provides for the steepest practical gradient while still including curves and allowing for safe stoppage of .\nBIOGRAPHY Matt Pascall was born in Harare on September 1, 1950, and schooled in Zimbabwe. Following schooling at Peterhouse, he completed a BSc in Mining Engineering at Witwatersrand University, Johannesburg, and an MBA at the Cape Town University South Africa.\nThe impact of South Africa's power crisis434 \u041a\u0431. South African Strategy: The impact of South Africa's power crisis (Orford).Higher energy prices Significant production losses for gold miners UBS 25 South African Strategy 3 March 2008 are likely to see gold mines costs rising.\nAfrican music is one of the first types that come to mind when you think of world music, especially in a cappella. Whether Gospel, Freedom songs or love songs, all of it is celebratory; African choirs and groups have an ability to convey joy and happiness that is unique and beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaukzi b/data_all_eng_slimpj/shuffled/split2/finalzzzaukzi
new file mode 100644
index 0000000000000000000000000000000000000000..8715cf2379c1645c67705da1fbc36fe443117722
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Product Presentation includes waste audits, containerisation, baling and analysis, all designed to maximise revenue in order to achieve long-term stable markets. Benchmarked pricing determines contracts around acknowledged industry indices. International fluctuations affect pricing and quality, PRDS bring years of recycling management experience to this constantly changing environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Providing a warm welcome to visitors, our caf\u00e9\/bar is open Monday to Friday 9am-4pm (food until 3pm) and Saturdays 9am-3pm (food until 2pm) as well as evening and weekend events and hires.\nWe serve a variety of made to order sandwiches, wholesome toasties, seasonal soups and salads, pasta dishes, and a range of tempting specials for lunch. Or perhaps coffee and cake is what you're after? Why not try one of our homemade cakes, including coffee and walnut, the famous WTH Chocolate Brownies, fresh scones and warming toasted teacakes? Paired with Italian roasted coffees, speciality teas, delicious hot chocolates and a range of soft drinks you will certainly find something to whet your appetite!\nWe are proud to offer fair-trade and local produce options across our range.\nWith free wi-fi access, the light and airy surroundings are the perfect place to meet up, relax, eat, drink and share. As well as coffees, teas and soft drinks, we are also a licensed bar with a range of local beers, wine and spirits. These can be enjoyed whilst attending one of our evening events, or whilst just lounging on the sofa with a newspaper.\nWith plenty of room for family, friends and baby buggies, a children's menu and with high chairs on hand, as well as a few toys to keep little ones entertained, our caf\u00e9 is an ideal spot to meet and eat!\nMenus subject to availability and change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am loving loving loving this new small space dwellers post. I discovered Anna Spiro's new online shop through one of my fave online mags - Adore Home. Anna's blog :: absolutely beautiful things :: was one of my first bookmarked years ago and I've been enjoying reading her inspirations every since. I just love the concept of her store - shop the seasonal vignette. There are tons of fabulous ideas right inside this room and here are my faves that you can recreate and reinterpret in your own small space.\nJust because you have walls, it doesn't mean all of your furniture has to be pushed up right against it. I love placing an accent chair or table on an angle, away from the wall, to create a design element and balance in a small space. Use a floor lamp or a side table to anchor the piece. The small space will appear bigger.\nThere's only so much storage space in a small space and often you have to leave your goodies out on display. Forego any idea of a cluttered look by styling your stuff. Anna's coffee table is style for the shoot but you can take a small space tip from this - it looks fantastic and organized. Balance is key here so be considerate of how your room feel in regards to balance and scale.\nOne of my own series on the loving. living. small. blog is all about using your walls and the vertical space available to you. Going up is a great way to move the eyes around the space, allowing it to feel bigger. Check out Anna's small artwork gallery vignette for ideas of your own.\nCreating a mix of textures and color will definitely make your small space bigger. Of course there should be some design style and balance here - use colors or patterns in a complementing palette. For example consider using all warm tones - reds, oranges, yellows or patterns - stripes, dots, zig zags. You need to add in some neutral balance as well - like Anna's wood console table or the white coffee table. Again, balance in a small space means all the difference in opening up the room.\nIn a small space with tons of visual elements going on, like Anna's room, another big small space tip is to keep the floor completely neutral. The white wood flooring is the perfect foundation to such a visual space. This anchors the room and, yes, provides balance. Keep your floors bare or use a very neutral toned rug to create this look in your own home.\nCheck out more rooms in the small space dwellers series right here. There are over 100 small space tips, tricks and ideas to enjoy. Fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From the time I was a young girl, I've been a creative soul. Whether I had an instrument in my hand, a microphone at my mouth, stage lights on my face, or fingers gripping a pen, I was never without a way to express my creativity. And while my father was very supportive of me, he'd often wink and tell me, \"Don't quit your day job.\" I used to groan and roll my eyes, but I'm thankful that my father instilled in me a healthy dose of reality.\nFlash forward several years (okay, decades!) and my passion for writing romance novels is more than just a hobby. My romance novels are an expression of my soul and feed my passion, purpose-filled life. But despite the fact that writing is what I love, it can be discouraging when the cost of writing and editing far outweighs any royalty earnings or sales!\nEven if you go the traditional publishing route, you still need to pay for costly editing services. And, once your book is published, the onus of marketing your novel generally falls to the writer. Multiply those costs by three if you go the self-publishing route as you will not only have initial editing costs, but you will likely need to pay for a cover, photo shoot, ISBN and a host of other unexpected expenses.\nSo what's a writer to do? How can you go from starving artist to potentially turning your passion for writing into replacement income?\nThe answer will likely be different for every person, but one thing's for certain\u2026before you can get to that fork-in-the-road, life-altering royalty check, you will need to find some extra cash to not only edit your book, but also fund the marketing.\nWhile everyone's journey will be different, below I've provided four proven methods of earning extra money to fund your writing passion.\nMany online and community colleges are looking for teachers who are seasoned professionals in their fields. Teaching part time allows you to stay current in your field (or writing!) while giving you exposure to new ideas and new industries. And, if you're lucky to teach writing, this can be a good way to establish thought leadership, an online presence and built-in feedback and reviews for your novel.\nIf you're a Midas with your money and you have a good understanding of the markets, trading may be the way to generate the extra cash you need to fund your writing hobby. While the cryptocurrency market may have slowed, there are still plenty of people making money in the crypto markets.\nIf trading cryptocurrency sparks your interest \u2014 and where you're hoping to start your cash cow \u2014 you may want to set aside some of those earnings to fund the marketing of your next romance novel. Lots of ordinary folks are making revenue gains through day trading and crypto currency. A small investment now can yield larger returns over time, which can then be used to fund the marketing of your novel.\nTo that end, one of my favorite crypto information sites is Crypto Hippo. Schyler Edwards, owner of the trading community, puts out a daily video that highlights the highs and lows of the market and where he sees the cryptocurrency market moving. Imagine making enough in trades to fund what you really love? Sound too good to be true? It might be more feasible than you think.\nIf you have a special talent or are particularly good at your day job, consider becoming a freelance for hire\u2026(Guru, Moonlighting, MKS Solutions, etc.) You can earn good money writing blogs, marketing copy, case studies and research papers for established firms. As you establish a name for yourself you could even take on more work and eventually replace your day earnings with something that is more inline with your passion for writing.\nI can see the eye rolls and hear the groans. But DS opportunities (also known as MLMs) have their place and do serve a need. While I have never met anyone who's been able to replace not only a full-time career salary but the benefits that come with that, I have known several people who have made some decent side cash from direct selling. The trick is to really assess your personality, the people in your network and the amount of extra time you have against the product you are trying to sell. It also helps if you have a healthy sense of reality and can determine the saturation level of the organization you are trying to join.\nOne DS company that I've heard good things about is LimeLife by Alcone. Their compensation seems to be better than most, the direct sales agents are well supported, and the company has a solid product that has gained national acclaim in the celebrity world.\nHow have you funded the editing and marketing of your romance novel? What tricks and tips can you share? Drop me a line in the comments below. I'd love to hear your ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The fiContactrst thing that the actress did after landing on Thursday morning was organized for a lunch with her entire team.\nThe team includes all the people who work closely with the actress.\nDeepika Padukone was away for quite some time and so did not get a chance to meet the team and spend time with them.\nNow that she is back, she has organized this lunch keeping everyone's convenience in mind.\nDeepika has always been extremely close to her team who is her family here in Mumbai away from her family in Bangalore.\nNext Before Baahubali going to China, China comes to Baahubali!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaupuh b/data_all_eng_slimpj/shuffled/split2/finalzzzaupuh
new file mode 100644
index 0000000000000000000000000000000000000000..cd4fa2ef34a3e5954dc5488273e61b0b3fe77b45
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The enzyme catalyses the liberation of Gal-(1->3)-beta-GalNAc alpha-linked to serine or threonine residues of mucin-type glycoproteins. EngBF from the bacterium Bifidobacterium longum specifically acts on core 1-type O-glycan to release the disaccharide Gal-(1->3)-beta-GalNAc. The enzymes from the bacteria Clostridium perfringens, Enterococcus faecalis, Propionibacterium acnes and Alcaligenes faecalis show broader specificity (e.g. they can also release the core 2 trisaccharide Gal-(1->3)-beta-(GlcNAc-(1->6)-beta)-GalNAc or the core 3 disaccharide GlcNAc-(1->3)-beta-GalNAc) [1,2]. The enzyme may play an important role in the degradation and utilization of mucins having core 1 O-glycan.\nCharacterization of two different endo-alpha-N-acetylgalactosaminidases from probiotic and pathogenic enterobacteria, Bifidobacterium longum and Clostridium perfringens.\nNovel endo-alpha-N-acetylgalactosaminidases with broader substrate specificity.\nIdentification and molecular cloning of a novel glycoside hydrolase family of core 1 type O-glycan-specific endo-alpha-N-acetylgalactosaminidase from Bifidobacterium longum.\nCrystallographic and mutational analyses of substrate recognition of endo-alpha-N-acetylgalactosaminidase from Bifidobacterium longum.\nCloning, recombinant production, crystallization and preliminary X-ray diffraction analysis of a family 101 glycoside hydrolase from Streptococcus pneumoniae.\nCharacterization of endo-alpha-N-acetylgalactosaminidase from Bacillus sp. and syntheses of neo-oligosaccharides using its transglycosylation activity.\nMolecular cloning, expression, and characterization of a novel endo-alpha-N-acetylgalactosaminidase from Enterococcus faecalis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NorSun has raised NOK 1.15 billion in new financing to expand production of silicon ingots and wafers for high efficiency solar cells.\nCrown Price Haakon opens The NorSun facility in \u00c5rdal. Photo: Norsun.\n\"We are pleased to have successfully secured new financing in the current financial climate. This underscores that our principal owners and main banks share our confidence in NorSun's technology and expansion projects. With this financing in place, we can take the next step to become the supplier of choice for the high efficiency segment of the solar energy market\", says Jon Hindar, Chief Executive Officer of NorSun AS.\nThe company raised NOK 500 million in equity, NOK 650 million in a five year bank loan and NOK 450 million in bank guarantee lines. All major shareholders, participated in the private placement and the entire credit and guarantee facility was provided by DnB NOR and Nordea Bank Norge AS.\nThe funds will provide NorSun with a healthy runway to complete the production ramp up in \u00c5rdal, Norway, execute the next phase of R&D activities and further contribute to the development of the thin film company Sunfilm AG, where NorSun has a 29.5% stake.\n\"NorSun manufactures high performance products and has a roadmap, consisting of technological and process improvements, which positions the company for strong growth in the solar energy market,\" says Inge K. Hansen, Chairman of NorSun, who also participated in the private placement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LexisNexis helps professionals at law firms and legal departments of all sizes manage the business element or their practice with innovative software and mobile solutions for litigation, customer relationship management, competitive intelligence gathering and assessment, time and billing, matter management, client analysis, legal holds and more!\nWith its new headquarters on Centennial Campus, the LexisNexis software division is at the center of Raleigh's growing tech hub.\nOur Summer Campus Program includes a variety of events and professional development opportunities throughout the summer for all recent graduates and summer interns.\nNo jobs listed at this time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"August 1992. The last day of the month is the first day of the trial in the Circuit Court of Howard County, as counsel put forth their arguments in front of Judge Cornelius Sybert, Jr. Representing the plaintiffs, Howard County Republicans David Maier and Louis Pope, are former county solicitor Thomas Lloyd and fellow former solicitor Richard J. Wilkinson, both Democrats. As for the co-defendants, representing the Board of Elections is Charles Reese, and representing the County Council are former U.S. Attorney General Benjamin Civiletti, former Georgetown Law professor Roger Titus, and their associate Sondra Block.\nThomas Lloyd contends that those who wrote the relevant county charter language intended that there be wide public input to redistricting and a deliberative process to specify the final district lines. He contrasts the lengthy process (including public hearings) needed to enact council bills with the ability of the council to adopt resolutions on short notice with minimal public notice or input. He further points out that, unlike bills, resolutions leave the public no avenues of recourse once adopted, since resolutions cannot be vetoed, challenged before put in effect, or made subject to a referendum. The end result, Lloyd claims, is that citizens are being denied due process when it comes to a fundamental county government decision (i.e., council redistricting) that affects them.\nNovember 1992. Judge Cornelius Sybert, Jr., renders his verdict: The council resolution establishing the new council district lines is constitutionally defective and invalid and the Board of Elections is enjoined from putting it into effect. Judge Sybert bases his ruling on two key points: that a 10-year redistricting plan is not temporary and that it is not administrative in nature, and thus it is outside the scope of those actions that the council can take via a resolution. C. Vernon Gray responds, I respectfully disagree, and points to the use of council resolutions for road closings and other matters not necessarily temporary in practice. He also promises he'll be back to court: We just have to appeal it, I think. \u2026 We have to uphold the prerogative and the right of council. The county executive has no role in a legislative function. Charles Ecker pushes back: As I interpret the charter, the county executive does have a role. That's the only reason I was glad to see it go to court. It's a good government issue\u2014there have to be checks and balances.\nDecember 1992. Council Democrats discuss what to do next, as a Baltimore Sun editorial points out the downside for Democrats in appealing the ruling: \"An appeals court ruling supporting the Sybert verdict \u2026 could weaken Democrats by forcing many decisions to be made by bills, which would then be subject to veto by the county executive.\" Democratic members Paul Farragut and Shane Pendergrass suggest buying some time by filing a Notice of Appeal, and then have the council decide whether to proceed or not, while fellow Democrat C. Vernon Gray seems intent on following the appeals process through to the bitter end: \"This is clearly an issue that can only be resolved by the judiciary.\" Covering all the bases, Roger Titus sends the council a set of price quotes ($23,000 to $34,000 for a two-step appeals process and $13,000 to $19,000 for skipping a step and going straight to the top, with filing a one-sentence Notice of Appeal a relative bargain), while Republican council members Charles Feaga and Darrel Drown suggest dropping the whole appeal idea and instead appointing a bipartisan committee (equally divided between Republicans and Democrats) to draw up a compromise redistricting plan.\nShane Pendergrass and Paul Farragut send a memorandum to invite their Republican colleagues to sit down in a council work session to start to resolve the impasse, but Charles Feaga and Darrel Drown continue to promote the idea of an outside committee. Feaga notes, When management and unions don't get along, they sometimes bring in someone else. \u2026 It's the only reasonable thing to do at this point. The legal maneuvering continues: Although county solicitor Barbara Cook previously agreed with Charles Ecker that redistricting required a bill (not a resolution), Democrats now want her to take over the appeal in order to save the county the expense of paying outside counsel in the form of Benjamin Civiletti and his colleagues Roger Titus and Sondra Block. Meanwhile Republicans are filing their own appeal of Judge Sybert's ruling allowing the council to be a party to the suit in the first place.\nWinston Churchill is reported to have once said, To jaw-jaw is always better than to war-war. Will the county council agree, as the redistricting effort stretches into a third calendar year? Stayed tuned for the answer in part 17 in this series.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During the winter, you may get more puffiness and bags around your eyes than you do in the warmer months of the year. Castor oil is a good source of fatty acids that help in improving the skin tone as well as rejuvenates it. The fresh cream contains lactic acid and this ingredient will help in how to get rid of dark circle around your mouth reducing the lines under eyes and the puffiness too. Whether it is due to stress, lack of sleep, aging or whatever the cause may be, dark circles are unattractive and no one wants them around their eyes. Aside from little sleep, I've actaully experienced greater under eye concerns when my diet is off. There is no need to use products around your eyes that have no business being there. Works as a dark spots corrector & whitener to diminish the look of dark circles under the eyes. To get rid of or reduce cheek puffiness from wisdom teeth or do away with puffy cheek toothache problem quickly, you need to consider tooth extraction, root canal, treating any infection and trying home remedies such as sucking ice cubes, swishing salty water, among other remedies.\nFrequent congestion, in particular, can cause visible discoloration in the veins under the eyes. If you wear contact lenses then remove them before going to bed to avoid irritation and eye dryness. Antihistamines are used to reduce histamine in the body, providing relief of common hay fever symptoms. In terms of Kris Jenner, there are a number of things going on. There are dark how to get rid of a dark ring around your mouth circles under her eyes, which makes her look puffy and sunken in.\nhow to rid your discoloration of get around mouth Apply the paste around the eyes and relax for fifteen minutes to reduce deep under eye-wrinkles. If someone asks you about how to get rid of eye bags, tell them you must relax your legs and eyes when doing an activity. Topical retinoid creams, lotions and gels - medications that are derived from vitamin A - may also be helpful in treatment and prevention of milia. By lightly massaging both your upper and lower eyelids, you shove all those excess fluids away from your eyes. Salty food, alcohol, and lack of sleep are the main causes of eye puffiness in your 20s. It wasn't uncomfortable, so I thought that my sunglasses must be too heavy for my face. As the skin is so thin in the area under the eyes these tiny leakages can show through as a dark area of discolouration.\nDark circles or dark rings around the eyes looks very bad when you are getting ready for a party and why it is an embarrassment for an outgoing only. Diet sodas are also to be avoided, as the aspartame contained in the soft drink can cause exhaustion in the long-term, the site says. Try a depuffing eye mask such as Bindi Mint Eye Mask to relieve puffiness or undereye masks infused with gels and plant extracts like Chamomile , cucumber, mint , and aloe Chill in the fridge overnight, and apply for ten minutes in the morning. One of the most popular ways to reduce puffiness around the eyes is to apply cucumber slices to them.\nFeeling crawling sensations, feeling itchy all over, on my scalp, face and body. Reduce under eye puffiness (puffy eyes) by emptying our content of tips into your cosmetic bag and how to get rid of skin discoloration around the mouth adhere to them as part of your beauty regime. Sleep with your head raised; propping your head up keeps fluids from accumulating under your eyes thus reducing puffiness. Before we look at ingredients that actively treat dark circles, we at The Naked Chemist, feel it is how do you get rid of dark circles around your mouth important to understand why they occur in the first place. Hyaluronic acid builds collagen, and coenzyme Q10, kojic acid and vitamins K and C combat dark circles around the eyes. how to get rid of discoloration around your mouth The fat pad grime around the optical area and and.\nIf you don't want to use a facial care product, men can use any method that is traditionally used by women for getting rid of bags, including cucumbers on the eyes, a frozen Q-tip rubbed directly on the bag of the eye, or even a spoon placed in the freezer to press against the eye for a few minutes in the morning. The good news is that a simple eye cream first thing in the morning can help reduce the fluid retention. If none of the remedies work for you, then consult a doctor to know the cause of your condition. However, whether this was due to the cold or completely due to the cream is anyone's guess. The pillows keep your head raised so fluid won't pool around your eyes when you lie down. An effective way to get rid of under eye wrinkles over night would be to apply egg white over the wrinkled skin around the eyes. This process can be repeated twice in a day for three weeks or until the dark circles around eyes has visibly reduced. Although, they don't implicate any serious medical condition, they can associate symptoms that can lead to chronic facial skin blemished and bad appearance of the eyes. The best eye cream for puffiness will contain all natural ingredients that will hydrate and soothe the skin around your eyes.\nHence, for some women, your eyes may look more puffy just before and during your menstrual period. YEA If you have the little black floaties in your eyes, get ACV, mix with water and dip how to get rid of dark patches around your mouth a cotton ball in it and wash your eyes out with it. Just brush it over your eyes in a few days they will be gone. Also, certain races are more prone to dark circles like people from India and the Middle East. Puffy eyes, also known as bags under your eyes, can be caused by aging, which leads to a weakening of the muscles surrounding the area. They're the perfect size and shape for fitting right beneath your eyes and massaging the area while the icy temperature cools to constrict the blood vessels and reduce swelling. Most cosmetic treatments for under-eye bags are considered safe, especially home remedies and lifestyle changes.\nhow to get rid of discoloration around your mouth The peach will offset pigmentation circles, and the shimmer will reflect light and make your eyes look brighter. Upper Eye Puffiness: If the eye puffiness has happened in the upper eyelid area, consider using a dark eye shadow. Yoyu should try and learn to love them, plus mine make my eyes look bigger weird ya but when i put concealer over them my eyes look tiny. Remove the tea bags and place them on your eyes like you would the cucumber, and leave for 15-20 minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzautio b/data_all_eng_slimpj/shuffled/split2/finalzzzautio
new file mode 100644
index 0000000000000000000000000000000000000000..1c1e1e0f5fecd5912a613abd7a4da313e751bd29
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Robert Janitz makes works that circle around the moment in the artist's studio where the application of paint transforms a mundane surface (say a stretched piece of linen) into something that has an effect on the viewer. The first viewer to see this happen is the painter. For a subsequent viewer looking at a finished work in a gallery this moment is always something that has already happened. The moment of the creation of meaning is in the past and when we encounter the painting we are left to re-create that moment as best as we can.\nJanitz typically uses translucent bold strokes that rise and fall along the vertical axis of his works in a rough methodical fashion. They merge with the underpainting so there is no pictorial depth, no distinct background or foreground. Instead there is an articulation of that moment of becoming \u2013 when content and meaning appear on the surface of the work and are immediately transformed into something else. He uses basic tools. His brushes are bought from hardware shops and he has mixed materials such as flour or wax into his paint. He uses simple analogies such as painting being like buttering a piece of toast (this is not to say his paintings are ever representations of buttered toast, but more to draw attention to the way an action transforms a surface into something new). His vertical strokes resemble the layers of whitewash put on the windows of shops that have gone bankrupt, hiding something that is just beyond what we can make out.\nPerhaps these strategies suggest an element of gentle teasing. Painting is presented as a basic, everyday act that can produce a myriad of complex effects yet it will never be possible for the viewer to re-create that exact moment when a painting takes on meaning. How can something so simple result in a series of effects that is so complex to articulate? The viewer might try and piece together that moment where these effects are created but her or his view is always impeded; by the finished surface, by their own thoughts, by the flow of time. Meaning has always already happened. The viewer is left to be the curious passer-by, trying to peer through the whitewash to work out what the bankrupt shop used to stock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A REMARKABLE woman from Tetbury has celebrated her 100th birthday with family and friends.\nAdelaide Mary Bailey, who is known as Addie, has lived in the town from more than 40 years, and is no stranger to modern technology, skyping her family who live in Australia every Sunday, Kevin Painter writes.\nActive Addie, who still lives in her own home, remains very much a part of the local community, and was looking forward to sharing the birthday celebrations with her friends at the Monday club which she regularly attends.\nShe was joined at a party by her son Chris who had come from his home in Australia. Her two granddaughters were also joining her from Australia.\nHer card from the Queen took pride of place.\nShe was presented at the party with a bouquet of flowers from the mayor of Tetbury, Sandra Ball.\nAddie enjoyed careers in the civil service and ANZ Bank.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This position is located in our Lexington, MA headquarters. Berkshire Grey is seeking a Software Engineer with a specialization in full stack web development. The candidate would be responsible for contributing to the design, development and maintenance of user interfaces for robotic systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vintage Honda Shop Tools. Carb Sync Gauges.\nTopic: Vintage Honda Shop Tools. Carb Sync Gauges.\nHere's a cool set I bought a while ago. Just like the pictures in my 70's Manuals. They were covered in crud, but I couldn't resist. Spent quite a while dismantling, cleaning, and polishing. Worth it! Probably 50 years old and still working well. New set of hoses and hopefully my son will still be using them in 2070! Anyone have a 6 X CBX one of these?\nRe: Vintage Honda Shop Tools. Carb Sync Gauges.\nNice. Nothing quite like clean shiny tools.\nSTEVE-O....... Mine were so dirty I barely spotted the Honda logo! Plastic headlight polish worked well on the lenses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Terre Haute South boys' basketball defeated Avon in the morning game of the Terre Haute North Classic. The Braves beat the Orioles 62-52.\nTERRE HAUTE, Ind. (WTHI) - Terre Haute South boys' basketball defeated Avon in the morning game of the Terre Haute North Classic. The Braves beat the Orioles 62-52.\nCraig Porter led the Braves with 18 points.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbkkn b/data_all_eng_slimpj/shuffled/split2/finalzzzbbkkn
new file mode 100644
index 0000000000000000000000000000000000000000..b650710c498b8d91201a799d5215a4ef91f4a1a9
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+{"text":"Alexander College is excited to announce the launch of our new information system, MyAC.\nMyAC launches on Monday, October 1.\nTo receive your login information for MyAC, login to AC Online on or after October 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Red Bull is organising the wings for life worldrun. People are standing up for those who can't, to help find a cure for spinal cord injuries. On May 4th 100.000 people are running for those who can't and raise money for medical research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rising insect resistance to genetically modified (GM) crops including Monsanto's biggest selling crop, Bt corn, is threatening their utility and profitability. Insect resistance has prompted a new investigation by the Environmental Protection Agency (EPA). According to documents in the newly opened docket (Docket No: EPA-HQ-OPP-2011-0922) , \"severe\" damage to corn by rootworm has occurred in four states in the US. Further, the EPA describe Monsanto's insect resistance monitoring program as \"inadequate\". The EPA will collect public information to tackle the damage that could cause serious crop and economic damage. Amidst this investigation, Monsanto are seeing significant falls in their share prices . Comments and information regarding insect resistance can be submitted to the EPA on their website .\nIn 2010, GM crops engineered to produce insecticidal toxins from the Bacillus thuringiensis (Bt) bacterium, were grown on more than 58 million hectares of land globally . First commercialised in the US in 1996, it is also the only commercialised GM crop grown in the EU, with Spain being the largest producer. Despite their widespread commercialisation, the evidence for their functionality is still elusive, while evidence of their harm to the environment, people's health, economic security and self-determination is continually mounting.\nGM proponents have repeatedly claimed that Bt crops can help combat world hunger by increasing crop yields while reducing pesticide use, thereby providing a more productive and environmentally safe option over traditional varieties. However, as highlighted by a recent report conducted by 20 Indian, Southeast Asian, African and Latin American food and conservation groups representing millions of people, these claims are false. Pesticide use has increased, while GM crop yields are lower than conventional varieties (see  Transgenic Cotton Offers No Advantage, SiS 38) and world hunger is at epic proportions .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than 1 million international students were enrolled at schools in the United States last year \u2014 a 1.5 percent increase over the previous year and a record high, according to the Institute for International Education. With . 1,094,792 million students, the United States remains the top host of international students globally. However the number of new international students in America \u2014 students enrolling for the first time at a U.S. institution in fall 2017\u2009\u2014\u2009decreased by 6.6% percent.\nChina sent the most students China (363,341 ), followed by India (196,271), South Korea (54,555), Saudi Arabia (44,432) and Canada (25,909 ), according to the IIE's Open Doors Report. New York University in New York City was host to the highest number of international students \u2014 17,552. Other schools with large numbers are the University of Southern California in Los Angeles, Northeastern University in Boston, Massachusetts, Columbia University in New York City and Arizona State University in Tempe, Arizona.\nOpen Doors 2018, released by the IIE and the U.S. Department of State's Bureau of Educational and Cultural Affairs, highlights the impact of international education on U.S. higher education sector. Foreign students made a significant financial impact in 2017, contributing $42.4 billion to the U.S. economy through tuition, room and board, and other expenses, according to the U.S. Department of Commerce.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do I Need a New Furnace? | Replace, Maintenance, Installation | AlkotaAlkota Plumbing and Heating Inc.\nWe're glad you asked! Furnaces, as you probably know, don't last forever. And they can be a burden to replace, often for financial reasons. Unfortunately, at a certain point, your furnace may actually be costing you more in repairs than it would cost to replace it.\nFurnaces, even well-maintained ones, have a lifespan of about 15 to 20 years. If yours is older than that, it may be time for it to retire.\nYour furnace shouldn't need to be repaired more than once a year, if that. As your furnace ages, repairs can become increasingly expensive. If a repair is going to cost you at least 50 percent of the amount required to replace the furnace, you should probably opt to replace it.\nNote: If a technician is inspecting your broken furnace, he or she can tell you if repair or replacement is the better option.\nTo meet the demand of inflation, and to protect our environment, newer furnaces are more energy efficient. Older furnaces aren't, especially if they are in poor condition. If your utility bills are suspiciously high, your furnace may be the culprit.\nStill not sure if you need to replace your furnace? Call Alkota Plumbing and Heating at 907-332-5325. One of our skilled technicians can run a detailed analysis to see how well your furnace is running. We don't want to see our neighbors in Anchorage, AK, using furnaces costing them an arm and a leg each month. If you're ready to part ways with your furnace, our Alkota Plumbing and Heating team is happy to set you up with a new and improved one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"How long does it take to get to Quarters from Legon Taxi Rank, Accra by public transit?\nHow long does it take to get to Quarters from Dolly Memorial Creche And Nursery School, Accra by public transit?\nHow long does it take to get to Quarters from Chinese Airbnb, Accra by public transit?\nHow long does it take to get to Quarters from Earls Court, Accra by public transit?\nHow long does it take to get to Quarters from Akuafo Hall Annex A, Accra by public transit?\nHow long does it take to get to Quarters from Mariset Hotel, Accra by public transit?\nHow long does it take to get to Quarters from Imperial Lodge, Accra by public transit?\nHow long does it take to get to Quarters from Arabella Residency, Accra by public transit?\nHow long does it take to get to Quarters from Mallam Borla, Accra by public transit?\nHow long does it take to get to Quarters from Physics Department, University Of Ghana, Accra by public transit?\nHow long does it take to get to Quarters from Legon Taxi Rank, Accra by Bus?\nHow long does it take to get to Quarters from Dolly Memorial Creche And Nursery School, Accra by Bus?\nHow long does it take to get to Quarters from Chinese Airbnb, Accra by Bus?\nHow long does it take to get to Quarters from Earls Court, Accra by Bus?\nHow long does it take to get to Quarters from Akuafo Hall Annex A, Accra by Bus?\nHow long does it take to get to Quarters from Mariset Hotel, Accra by Bus?\nHow long does it take to get to Quarters from Imperial Lodge, Accra by Bus?\nHow long does it take to get to Quarters from Arabella Residency, Accra by Bus?\nHow long does it take to get to Quarters from Mallam Borla, Accra by Bus?\nHow long does it take to get to Quarters from Physics Department, University Of Ghana, Accra by Bus?\nWhat are the closest stations to Quarters?\nMangoase is 196 meters away, 3 min walk.\nAdd A Name To The Stop With Junglebus App is 234 meters away, 4 min walk.\nMasalatsi is 313 meters away, 5 min walk.\nDarkuman is 368 meters away, 5 min walk.\nDarkuman Station is 389 meters away, 6 min walk.\nWhich Bus lines stop near Quarters?\nThese Bus lines stop near Quarters: 240, 251, 90.\nWhich public transportation lines pass through Quarters in Accra, Ghana and stops near me?\nMoovit helps you find the quickest route and most updated timetable for lines that stop near Quarters. Moovit shows you the closest stops and stations to your destination. Check out this list of stops closest to Quarters: Mangoase; Add A Name To The Stop With Junglebus App; Masalatsi; Darkuman; Darkuman Station.\nWant to see other transit options? Bus are possible transit options that you may use to get to Quarters. Choose from this list of lines that pass close to your destination - Bus: 240, 251, 90.\nYou can also find live directions to Quarters or see the closest bus stop or Bus stations near me by downloading the Moovit App.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Diane grew up in a large farmhouse in western New York that she shared with her grandmother, parents, and three siblings. While she walked to school through the vineyards that surrounded her home, she imagined a beautiful fun-filled world that later became the basis of some of her stories. Today, Diane and her husband live in northern California. Besides writing, Diane founded Joyful Living and Giving, an organization to help others in the community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Off Topic > How will your life end?\nWhat do you think? WIll you live a long life, have close ones next to you when you die, will disease take you, alcoholism, suicide, an accident? How would you prefer to die and how do you think you will die? What would you like to accomplish till then?\nMost of us living now who are under 50 will probably die in mass genocide perpetrated by the scientific dictatorship being erected all around us. This death will come in many forms though. Some will die in camps, others in wars, others from famine, others from forced shots which were purposely contaminated with disease, etc. etc. I think I'll end up being beheaded as a rebel who refused to go along with the tyranny and it will happen publicly as an example to others to not rebel.\nExcept it will have the opposite effect. You will die a martyr, sparking hope and rebellion in thousands, effectively starting a new era of people living in perfect harmony. Utopia.\nYes. This is what I hope. Maybe the new society will even name a city after me and build me a memorial.\nTinfoil is cheap and I'm sure there are guides online how to make one.\nI will probably suicide once i lose some of my capabilities, physically and mentally.\nwhy? are you obese or does it run in family?\nHeart attack from watching Fnatic play against tier 15 teams and still lose.\nComplications related to Alzheimer's. My grandma has it, and my great grandma had it as well. So I have a chance of getting it when I'm older. Hopefully scientists find a treatment or cure for it within 40\/50 years.\nWill I have a long life? Probably not. I don't like the idea of having to wear diapers and shitting myself when I'm 80. I'd rather kill myself. How would I prefer to die? Preferably protecting someone\/something. Dying in a war to protect my country, being killed by a shooter while protecting others, that sort of thing. How do I think I will die? Either suicide before I get too old, some sort of disease, or the way I want to go. What do I want to accomplish? Definitely have a family, and maybe become known somehow.\nLive healthy and you won't be shitting yourself at 80.\nNot true from what I've seen. Both my grandma and grandpa from my dad's side have lived very healthy lives, ( Always eaten healthy home cooked meals, never smoked, very rarely drank alcohol, etc. ) They're both in their early 80's, and in absolutely horrible condition. I don't want that happening to me.\nI have my fantasy predraft set to draft all Liquid players first. I don't think I'll get Elige though.\nI haven't even it done it. I suppose it's a fun game, but I don't really see the point in participating unless there's a reward to be had. I used to do it in the actual game when you would receive a fantasy team trophy but afaik you get nothing in HLTV.\nWhat do you want to protect Australia from my dude?\nOnly The Lord knows how will i die or if i will be alive when Jesus Christ comesback to take his saints to heaven with him. All i know is, i want to live my life for Jesus Christ because he is even more important then my own life and family. ( With this being said im not saying family isnt important, but family is not above Christ.) I know where will i go when i die, to heaven with Christ, many wont unfortunely.\nDo you think you are sinner ?\nWe all are. But there are sinners who arent repented And there are sinners who are repented from their sins in Jesus Christ.\nI want to die protecting my loved ones, like disarming someone and get shot. Next probably would be dying of old age but still sane.\nWho knows. Just hope I can live long enough before this world decides to start another world war.\nviolently while in combat with others or i will be paralytic and feel myself dying as i drop out death or glory.\nI hope that I die to a natural death when I'm quite old (80-90), no diseases, no accident or anything like that. I don't rly know about having close ones around me then though. I want to be happy in my last couple of years but I also don't want to leave them with sadness. I guess dying at around the same time as my partner would be the best, especially if we consider that there might be afterlife.\nI have 350Z right now but also looking for a new car like maybe something more comfortable but damn I love power so much. btw nice to see a real girl here.\nYeah I usually have fun when roads are empty or almost empty, I'm not crazy like e.g this guy youtube.com\/watch?v=acatwdIWar4 Maybe your parents have a right but even with a slow car people can do stupid things on the roads. What car do you have ?\nI smoke, and I can get mad so easily. Also sometimes I overthink on my future and life, so probably cancer.\nThroat cancer due to over-extending my vocal range to the limit and exhausting myself on stage. By a gunshot in the head at 90-+ years of age to escape a serious disease. Taken away by nature, people don't live very long.\nI would love to die from natural causes because it would mean that i had a life that i enjoyed living. If soon my plans wont work i think suicide is the way, no way i will live the way system wants me to live.\nI hope i die in battle in the Swedish or Canadian army. If not that way i don't know , i'd say 50-50 , 50% Alone in my house - 50% with a family (wife - kids) , but in any case , i want to start from 0 and change my name to fully separate myself from my current family , im sure that much i will achieve.\nI will die when i'm 32, probably by accident, I dont know yet How would you prefer to die: medicine cocktail, die peacefully in my sleep because of respiratory depression. Opiate + benzo + stomach aid, preferably 1000g of opiates and benzo's. And how do you think you will die?: Might be an airplane crash at the age of 32, I don't know. What would you like to accomplish till then?: People think that being successful matters in life but I don't believe in succes. What does it matter? That you will be remembered? That people look up to you? That you change people's lifes? In the end it doesn't matter a single bit. The only thing I want to accomplish is that during the time i'm alive, i'd like to be happy and comfortable with who I am and accept that i'm not perfect in some ways. The clich\u00e9 wouldn't be complete if I said something about finding love, which I would also like of course, but I don't think love lasts forever and thus I don't want to make that step a priority. The only way for it to be romantic is if I find a girl and actually love her until I die(which would be at 32, so not impossible).\n\"WIll you live a long life, have close ones next to you when you die, will disease take you, alcoholism, suicide, an accident?\" Idk. Not suicide atleast, but i mean, an accident can happen at any time, i could just be walking & get hit & killed by a car or whatever. Disease? Maybe, i mean, a bunch of my relatives have cancer \/ had cancer as well as diabetes (No, not diabetes that you get from being fat or whatever) Alcoholism? Nah, i dont drink that much, i enjoy a few bottles each Friday after work (5-6 Czech\/Polish beers). How i would prefer to die? Idk, not a painful death, i guess die from age, but idk how long i will be alive because of all the diseases etc in my family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As summer comes to a close, and the leaves are starting to change, many mountain folk are already dreaming about the encroaching winter. Soon we hope to be in waist deep powder and have endless days of fun on the mountain. But before the true ski season arrives, there are a few things to get in line before you head out on the mountain for your epic winter vacation.\nYou do not need to buy completely new gear each winter season, as nice as that sounds, but do make sure your gear is winter ready. Take your skis or board to the shop to get tuned before the winter starts. Preparing ahead of time will really save you a headache when you get down to the wire when getting ready for the upcoming ski season. Make sure your equipment is safe and buy yourself and your loved ones a helmet if you do not already own one. Plus, this time of the year is a great time to buy new equipment if need be. Tons of places have huge sales on gear, and if you are planning on taking the little ones all winter long, consider seasonal rentals. Your kids are growing so fast, buying each year can get very costly. However, many places offer seasonal rentals which could be a great money saving option.\nUnless you are a seasoned veteran on the mountain and are an avid skier, think about taking a lesson. Group lessons can be a great way to enjoy the mountain and really improve your skiing and boarding skills. Plus, do keep in mind that just because you know how to ski or board well does not mean at all that you have the ability to teach someone how to do it as well. Instructors are trained on how to teach people the way of the mountain. It takes the pressure off of you, and it actually brings the student into a much more fun and enjoyable learning environment. Plus, if you are skiing with someone that is a very different level than you, getting them enrolled in lessons will benefit everyone. Even if you are an experienced skier or boarder, if this is your first time at a new resort, sometimes getting a lesson is great just to have an instructor show you around the mountain.\nThe last thing you want to do is plan out your entire trip and then get here and not be able to enjoy it because you are too out of shape. This does not mean that you have to hit the gym five days a week or anything. However, do think about getting in some exercise before your epic trip. It could be as simple as getting in a few days of cardio a week, biking for fun, hiking with some friends. Do something that will get your heart rate healthy and ready for your trip. Remember, if your body is not used to being high up in elevation, do not do too much too soon once in the mountains.\nOne of the best ways to get ready for the winter season is to get some friends together and watch some great ski movies. Pop in a few Warren Miller films and pray that the snow starts to fall. You might also find it a good idea to read some great ski magazines. Imagining and daydreaming about what it would be like to one day heli-ski will get anyone excited for the upcoming season. Hey, you have to dream big somehow, and it would be pretty awesome to be sponsored to go skiing around the world.\nHonestly, the best way to prepare for the ski season is to simply go skiing. Get out as often as possible. That burn you get from skiing and boarding is just a great reminder that skiing is a great workout. Skiing really works your body in a way that cannot really be replicated in the gym or running or any other sport. Skiing works you all over, and that burning sensation comes with the territory. So gear up, pray for snow, and get ready to feel the burn. Ski season will be here before you know it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every year ZIN\u2122 members come back from convention buzzing with stories and pictures that almost always end with, \"You just have to be there to understand. I wish you could experience it for yourself.\"\nFor over 30 years, Forest Travel has taken great pride in crafting the most lavish and exclusive worldwide retreats and adventures.\nOur Travel Advisors possess the professional expertise and personal experience necessary to make your next vacation a unique and unforgettable experience. We have built dedicated teams that specialize in cruises, spas, ski holidays, private islands, river barges, all-Inclusive resorts, safaris, family vacations, luxury escapes and honeymoons \u2013 and that is just the tip of the iceberg!\nPlace your reservation on hold for 24 hours on most airlines while you finalize your travel plans. When you are ready to go, just confirm your booking!\nBook online, not alone. All bookings are personally monitored by your travel advisor. Communicate directly with your Travel Advisor to make any changes.\nEasily share your itinerary with others. Email your travel plans to a friend or co-worker, and have them complete the booking for you (or for them).\nManage your profile information, monitor itineraries, and much more all in one place.\nAdvance Filters help you effortlessly narrow your results to find the perfect flight option for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcfvl b/data_all_eng_slimpj/shuffled/split2/finalzzzbcfvl
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index 0000000000000000000000000000000000000000..9506df763a63e2d70a9bd238b8ddb7760e5d24f9
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@@ -0,0 +1,5 @@
+{"text":"The Apache of North America, arguably best known of all the groups living in the northern territory, witnessed first-hand white America's birth. They saw the nation's violent adolescence displace the indigenous population and watched as the country journeyed into power, climaxing in a consolidation of global dominance. However, they are not a people defined in an opposition of the U.S. Today, they continue to exist within their own rich, historical tapestry of cultural traditions.\nConsisting of thirteen tribes, the ways in which Apache differ from one another are numerous and far-reaching. From where their names draw meaning to the land and histories they call their own. Yet, particular cultural elements prevail throughout all Apache peoples. They include: their oppression by the nation states of Mexico and the USA, adherence to an animistic worldview and consequently, their practice of certain ceremonies. For example, one shared ceremony, rich in historical importance and symbolic cultural identity, is the rite of passage through which teenage girls must pass through to gain womanhood. The Apache call this rite 'the Sunrise Ceremony'.\nPerformed to strengthen the individual girl along with her whole tribe, the Sunrise Ceremony marks an important passage from child to womanhood. The running of the rite is based on the notion of fours; four stages of life, infant, child, teen, adult, four sacred mountains, the four directions of the land and of prayer. The young women who undertake the Sunrise Ceremony are guided through four days, with each harking to different essential elements laid out. They must overcome each sacred mountain and only by her display of tenacity can the adolescent demonstrate her commitment to the tribe, winning their respect. Without it, she will never be seen as having fully entered womanhood.\nIn the evening of the twelfth and final day of passage, after he has finished dining with her, the medicine man will gift the newly titled young woman her Apache name. She holds onto this for the rest of her life. The ceremony ends with this but begins with the young woman running four laps between her tipi and a traditional woven basket. One Apache tribe, the Jicarilla, which derives its name from the Spanish word for this little basket, are differentiated by their title. They became known as the Jicarilla due to their fondness of drinking from these small sealed baskets, using them as a type of jug.\nAronia was a food staple for the Jicarilla.\nAronia berries were an important and frequently used food staple for many American first nations and the berry itself was usually eaten raw or dried and mixed with pemmican. The Jicarilla particularly, dried the fruit and pressed them into cakes which they stockpiled for the winter months. The fresh berries could be mashed and made into a jam, or simply left to ferment and used as cherry wine. Every single part of the plant had a use and even its bark and roots can be boiled to produce a form of medicinal tea.\n\"This sturdy little bush, able to grow in the harshest of environments, serves its cultivators greatly.\"\nAt first sight, the vibrant aronia berry resembles a concord grape combined with either a blueberry or cranberry. But, it's more than the deep purple colour that is catching our attention these days. It's all the aronia health benefits that are packed inside.\nWhat are the aronia health benefits?\nAronia berries contain an impressive amount of nutrients, including vitamin C, polymeric proanthocyanins, and anthocyanins.(1)(3) For instance, a 100g serving of aronia berries provides over a third of the recommended daily value (DV) of vitamin C.(1) Vitamin C plays a vital role in the cardiovascular health, synthesis of structural protein collagen, and the absorption of iron. Speaking of which, a 100g serving of aronia berries contains about 8% DV of iron.\nFinally, whole aronia berries are an excellent source of dietary fibre. You probably already know that eating fibre-rich foods promotes digestion. But there's more to it. For example, fibre may help your body to maintain a healthy weight and lower the risk of diabetes.\nIn summary, here are three key health benefits that aronia berries may have.\n1. Aronia fights free radical damage.\nIf you're curious just how powerful these berries are, let's compare their antioxidant capability to that of blueberries and raspberries. Most of us are familiar with these berries and may even eat them as a source of antioxidants. For our comparison, let's take the oxygen radical absorbance capacity (ORAC) of all three. Aronia berries have a score of 16,062 \u03bc mol TE\/100g, compared with blueberries at 4,669 \u03bc mol TE\/100g and red raspberries with 5,065 \u03bc mol TE\/100g.(10) Now you get the idea.\n3. Aronia is antifungal, antiviral and antibacterial.\nTo sum up, aronia health benefits are quite impressive. These berries are excellent to protect your body cells from oxidation due to their high content of phenols. On top of that, aronia berries can fight inflammation, support cardiovascular health and promote digestion. Oh, and they do a great job in nourishing your skin.\nAronia berries have a distinctive, mildly astringent and sweet flavour. So, if you like dry red wines, you will love the taste of aronia berries. The dry mouth feeling comes from organic compounds called tannins, also present in black tea and fine dry wines such as Cabernet Sauvignon from Bordeaux. Not a bad company to be in!\nCertainly, consuming fresh organic aronia berries is the best way to enjoy their health benefits. On the other hand, raw aronia powder, dried whole berries or juice are also very nutritious.\nErbology Aronia Shot is pure, undiluted and unsweetened, made from 100% organically grown aronia berries. In other words, you'll enjoy the aronia health benefits without any nasties. Take the shot straight or add it to a smoothie or cocktail. The 40ml \/ 1.4 fl oz booster is the perfect daily dose, especially great when traveling, while the 250ml bottle of Pure Aronia Juice will last you for about a week if you keep it refrigerated.\nIs yogurt, smoothie or porridge part of your morning routine? Then, Erbology Raw Aronia Powder or these dried berries are a great addition. In addition to the nutrients that fight oxidative stress, they're an excellent source of fibre that keeps you feel full for longer.\nBut, if you have no time to make a smoothie bowl in the morning, no worries. We've still got you covered. Aronia is one of the amazing ingredients in our delicious Energy Bites and Tigernut Granola. Above all, our snacks are gluten-free, free from added sugar and raw. In other words, they retain their excellent health benefits and unique taste.\nThe combination of earthy beetroot and tangy aronia berries is so good that we have been experimenting with both sweet and savoury recipes. Above all, try this easy to make, skin nourishing Pink beetroot smoothie. All you need is almond milk, ripe banana, dried aronia berries, raw beetroot, agave nectar and lemon juice. The result? A delicious and nourishing drink to energise you for the day.\nThis Velvet beetroot and aronia soup recipe is perfect for lunch or dinner, especially on a chilly autumn or winter day. Did you know that beets may help purify your blood and liver due to high content of betalain pigments? Scientists once thought that betalain pigments were related to anthocyanins (the pigments present in aronia berries). Nevertheless, these two powerful nutrients are structurally different and have not been found in the same plant together. Therefore, we can only conclude that beets and aronia are a match made in heaven.\nHaving friends over? Serve this delicious Sage and orange cocktail. All you need is five ingredients and fifteen minutes of your time. Sage leaves and cinnamon make the aromas of this drink irresistible.\nFinally, we've got something for the chocolate lovers, too. This Aronia, pumpkin and chocolate smoothie is an excellent combination of aronia health benefits and wonderful raw chocolate properties. This drink is a great source of energy, especially before a workout.\nHave you enjoyed reading this article? We love exploring and bringing you nature's marvels that can help you lead a more wholesome life. Learn more about some of the plants that have been cherished and used for centuries.\n(1) \"Aronia berry nutrition facts and calories\". Self Nutrition Data, https:\/\/bit.ly\/2RSqti1.\n(2) \"Health Benefits of Aronia Berries\". HealWithFood.org, https:\/\/bit.ly\/219A2vU.\n(3) Jan Oszmianski, Aneta Wojdylo. \"Aronia melanocarpa phenolics and their antioxidant activity\", European Food Research and Technology, 2005, Volume 221, Issue 6, pp 809-813.\n(4) Olas, B, et al. \"Effects of Polyphenol-Rich Extract from Berries of Aronia Melanocarpa on the Markers of Oxidative Stress and Blood Platelet Activation\". Platelets., U.S. National Library of Medicine, 2013, https:\/\/bit.ly\/2GK3uE3.\n(5) Dzia\u0142o, Magdalena, et al. \"The Potential of Plant Phenolics in Prevention and Therapy of Skin Disorders\". International Journal of Molecular Sciences, MDPI, 2016, https:\/\/bit.ly\/2SHaKqC.\n(6) \"NCI Dictionary of Cancer Terms\". National Cancer Institute, https:\/\/bit.ly\/2TFxsfT.\n(7) Samoylenko A., Hossain J.A., Mennerich D., Kellokumpu S., Hiltunen J.K., Kietzmann T. Nutritional countermeasures targeting reactive oxygen species in cancer: From mechanisms to biomarkers and clinical evidence. Antioxid. Redox Signal. 2013;19:2157\u20132196.doi: 10.1089\/ars.2012.4662.\n(8) Liaudanskas M., Viskelis P., Raudonis R., Kviklys D., Uselis N., Janulis V. Phenolic composition and antioxidant activity of Malus domesticaleaves. Sci. World J. 2014;2014:306217. doi: 10.1155\/2014\/306217.\n(9) Alov P., Tsakovska I., Pajeva I. Computational studies of free radical-scavenging properties of phenolic compounds. Curr. Top. Med. Chem. 2015;15:85\u2013104. doi: 10.2174\/1568026615666141209143702.\n(10) \"Chokeberry (Aronia Berries), Raw\". Superfoodly, https:\/\/bit.ly\/2DzpApz.\n(11) \"The Danger of Antibiotic Overuse\". Edited by Elana Pearl Ben-Joseph, KidsHealth, The Nemours Foundation, Sept. 2015, https:\/\/bit.ly\/2RWJUWQ.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Sunday, I was feeling slightly worse for wear after maybe having one or five too many drinks(!) but I dragged myself out of bed as it was Mother's Day and popped into town for a cheeky cappuccino with my momma which ended up with me spending some of my birthday money in Superdrug. I haven't bought any makeup in months, I've surprised myself with how strong I've been but with all these new products being released, I had to have a little rather large splurge, so I thought I would share my purchases with you..\nVitamin E Night and Day Cream - There was an offer on for these products and seeing as though I love the cleanser from the same range so much, I thought I would give these a whirl, so expect to see a review of these soon. 2 for \u00a33.99. 1 is usually \u00a32.99.\nBourjois 123 Perfect Foundation - I love the Bourjois foundations but wanted a bit of a change from the Healthy Mix Serum and seeing as this foundation is supposedly meant to wake your skin up etc, I thought I would try it out, as my skin doesn't like early mornings.. \u00a310.99 I got the 3 Bourjois products on a 3 for 2 offer.\nBourjois Flower Perfection Translucent Smoothing Primer - Basically just got this because it was 3 for 2 on their face products and I'm not a fan of their blushes. Not one to wear a primer but definitely going to try this out and see if it makes a noticeable difference to the longevity of my foundation.\nBarry M Gelly Hi-Shine Nail Varnish in 305 Lychee - Heard loads about this collection and got 2 from the range for my birthday and really liked it and whilst I was looking today at the different colours, I thought this was a really gorgeous colour and I don't have anything like it. \u00a33.99 Both new collections are 2 for \u00a36 currently.\nI'm pretty sure I will be reviewing most of these products at some point but if there's anything specific you would really like to see, please leave a comment below and let me know!\nCool haul, I really like the vitamin e day cream :) and I bought a textured nail effects yesterday I can't wait to try it!\nThose polishes are lovely, what a great haul!\nGreat haul, I'm dying to try the Rimmel Apocalips, would love to see a swatch of this!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By now you are likely familiar with Cloud Access Security Brokers (CASBs) and understand how they fit into your broader security and cloud strategy. What should organizations be looking for in a CASB? What capabilities are here or on the horizon that can provide improved data protection in the cloud? On December 1, 2016 at 1:00PM Eastern, Bitglass and (ISC)2 presents the final episode of the CASB series where we will examine where cloud security is headed, discussing agentless and agent-based solutions, the growing number of cloud apps in use and the importance of easy deployment. Learn why cross-app security will become increasingly valuable as organizations look to third-party solutions for deep visibility, behavior analytics, and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our company offers taxi transfers from\/to Antalya Airport to any city in Turkey. We offer most of hotel and holiday transfers for travellers in Alanya, Belek, Side, Kemer, Tekirova, Beldibi, Colakli, Fethiye, Okurcalar, Avsallar, Lara, Evrenseki, Kas Manavgat, that our most popular transfer destinations. The rates we offer much cheaper and more comfort than a normal taxi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About the time of Sin-iddinam of Larsa a small independent kingdom was established on the Lower Euphrates with its capital at the city of Uruk. The names of eight of its rulers are known. The kingdom came to an end ca. 1800 bce with its capture by king Rim-Sin of Larsa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Emory golf team returns to action this Saturday and Sunday (Apr. 20-21) when it hosts the Discover DeKalb Spring Invitational.\nEmory junior Matt Organisak has been selected by the University Athletic Association as Golfer of the Week following his performance at the Wynlakes Invitational.\nThe Emory golf team's Saturday and Sunday dual matches against Centre College in Lexington, Kentucky, have been cancelled.\nThe No. 1-ranked Emory golf team closed out its participation at the Wynlakes Invitational on Tuesday.\nThe No. 1-ranked Emory golf team finds itself tied for second place after the opening day of action at the Wynlakes Invitational.\nThe No. 1-ranked Emory golf team returns to action on Monday and Tuesday (Apr. 1-2) when it takes part in the Wynlakes Intercollegiate.\nFreshman Logan Ryan has been selected as the University Athletic Association Men's Golfer of the Week following his performance at the Jekyll Island Collegiate Invitational.\nThe No. 1-ranked Emory University golf came through with a solid Sunday round and in the process captured first-place honors out of 30 team at the Jekyll Island Collegiate Invitational.\nThe No. 1-ranked Emory University golf team completed its second round of action at the Jekyll Island Collegiate Invitational on Saturday.\nFresh off winning the program's fourth straight University Athletic Association Championships, the Emory University golf team continues its spring season this weekend (Mar. 15-17) when it competes at the prestigious Jekyll Island Collegiate Invitational.\nSophomore Eric Yiu has been selected as the University Athletic Association Men's Golfer of the Week following his performances at the Savannah Harbor Classic and the UAA Championships.\nThe No. 1-ranked Emory golf team won its fourth straight and the program's 19th University Athletic Association Championship on Sunday afternoon, posting a 3-2 down-to-the wire victory over No. 15 Carnegie Mellon.\nThe No. 1-ranked Emory golf team opened defense of its league title on Saturday, rolling to a victory over the University of Rochester in semifinal match at the University Athletic Association Championships.\nAfter opening its spring season at The Club of Savannah Harbor Classic, the No. 1-ranked Emory golf team turns its attention to this weekend's (Mar. 9-10) University Athletic Association Championships.\nThe No. 1-ranked Emory University golf team came through with a solid second round of play on Tuesday which enabled it to finish in a tie for fourth-place out of 18 teams at the Savannah Harbor Classic.\nThe No. 1-ranked Emory University golf team will compete in its first tournament of the spring on Monday and Tuesday (Mar. 4-5) when it takes part in the Club at Savannah Harbor Classic.\nFreshman Logan Ryan has been selected as the University Athletic Association Men's Golfer of the Week following his performance at the Golfweek DIII Fall Invitational (Oct. 14-16).\nThe B team of the Emory University golf program closed out its participation at the Chick-fil-A Collegiate Invitational on Tuesday.\nThe Emory University golf team completed the opening round of play at the Chick-fil-A Collegiate Invitational on Monday.\nAn excellent fall season that has seen team and individual records established has helped the Emory golf team to a unanimous No. 1 ranking in this week's Bushnell Golfweek Division III Coaches Poll.\nThe No. 1-ranked Emory University golf program continued its stellar fall season on Tuesday, setting a school record while playing its way to a first-place finish at the Golfweek DIII Fall Invitational.\nEmory junior Matt Organisak has been selected by the University Athletic Association as Golfer of the Week following his performance at the Tartan Invitational.\nThe No. 1-ranked Emory University golf team still sits atop the leaderboard following the second round of play at the Golfweek DIII Fall Invitational.\nThe No. 1-ranked Emory University golf team established a school record on Sunday during the first round of play at the Golfweek DIII Fall Invitational.\nThe No. 1-ranked Emory University golf team saw its bid to capture a second tournament title this fall come up short on Tuesday at the Tartan Invitational.\nFreshman Logan Ryan has been selected as the University Athletic Association Men's Golfer of the Week following his performance at the Gordin Classic. The 54-hole tournament was held on the 7,098-yard, par-72 Columbus Country Club in Ohio.\nFresh off a runner-up effort at the Gordin Classic, the No. 1-ranked Emory golf team continues the fall portion of the schedule on Monday and Tuesday (Oct. 8-9) when it competes at the Tartan Invitational.\nThe No. 1-ranked Emory University golf team closed out its participation at the 28th annual Gordin Classic on Tuesday.\nThe No. 1-ranked Emory University golf team finds itself in second place after the opening two rounds of the 28th annual Gordin Classic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How do we define computer software?\nWhat is computer software is quite a common question, even today there are a lot of people whom do not know the definition of computer software.\nIt may surprise you to know that the modern day term \"software\" commonly bandied about today, was in fact first used in this sense by John Tukey in 1958.\n<!- BODY TEXT PREFACE --> So, what is computer software then?\n'Computer software' is nowadays a general term that's used to describe a computer program. Or a collection of computer programs and procedures that actually perform some task, or tasks on a computer system.\nThe term computer software does actually include all levels of programs. From application software such as word processors and spreadsheets, which perform useful productive tasks for computer software users. Right up to computer system software which makes the computer framework operate.\nComputer system software refers to operating system platforms, which provides the basic framework for other software operate in. Operating systems, such as Microsoft Windows, Mac OS X or Linux, are current examples of computer system software.\nComputer system software is any computer software which manages and controls computer hardware so that application software can perform a task. System software contrasts with application software, which are programs that enable the end-user to perform specific, productive tasks, such as word processing or image manipulation.\nSystem software performs tasks like transferring data from memory to disk, or rendering text onto a display device. Specific kinds of system software include loading programs, Operating systems, device drivers, programming tools, compilers, assemblers, linkers, and utility software.\nThe system software interfaces with the computer hardware to provide the necessary services for application software. Along with middleware, which controls and co-ordinates all the various parts of the computers distributed systems.\nComputer software is so named to make it an opposite and to differentiate it from computer hardware. Computer hardware covers all the physically touchable devices and hard wired interconnections that the computer needs to store, prioritize and run the software.\nIn order to run the software, it is loaded into the random access memory (RAM) and executed by the central processing unit (CPU). At the lowest processing level the software consists of a series of binary values, called machine language.\nThese binary values change the conditions of the computer from one moment to the next, depending on what the individual instructions are.\nComputer software is in its basic form, an ordered sequence of specific instructions for changing the condition or state of the computer hardware. This happens via outputs from the central processing unit (CPU) in a desired particular sequence.\nI should mention here that this changing of states happens at a very fast rate, in the Megahertz (Mhz, meaning millions of times a second).\nMost of the time today , it will usually be written in high-level programming languages. Examples of these high level programming languages are Fortran, C, C++, Visual Basic, Visual C++, Pascal.\nThese high level languages are easier and more efficient for humans to understand and use because they are closer to our natural language than the machine language ever would be.\nTo the right is a clip of Visual Basic, although readable not much storyline!\nWhen a program is written (coded), this (source) code is converted (compiled) into machine language object code for the CPU to understand, or at least respond to.\nSoftware may also be written in what's called assembly language. This is a mnemonic representation of a machine language, using a natural language alphabet.\nAssembly language must be assembled into object code via an assembler. Although more difficult for potential programmers to grasp, the most common advantage for programming with this type of language, is speed of execution.\nSo when someone next asks you, what is computer software? You'll be able to tell them!\n\"Computer software is an ordered sequence of instructions for changing the state of the computer hardware in a desired particular sequence.\"\nWon't you . . . . Yes!!\n<!- Bottom page links --> Return to Computer Software page.\nGo from What is Computer Software to Computer Hardware page.\nReturn from What is Computer Software to Home page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love using high quality findings for my jewelry. I've used sterling silver and niobium. Sterling silver is high quality, there is a great variety of findings, and it is well known and well loved. As beautiful as it is, I've chosen to stick to niobium findings for my jewelry. Here in Malaysia it seems that metals oxidize\/tarnish sooner, perhaps it's the humidity? Whatever it is, I love niobium for its non tarnish quality! It also seems to be a bit more hypoallergenic than sterling silver. Both are wonderful! Because I will no longer be using sterling silver findings on the new pieces of jewelry I make, I am destashing my supplies. All the supplies I have have been well sealed, some never opened, so none are oxidized or tarnished.\nI have listed them all in one listing on Zibbet in case someone is interested in a variety of findings to get started with. If someone is interested in only parts of what is available, just send me a message on etsy and I can make a new listing. All amounts\/prices are listed in the basic listing HERE. I am also giving FREE worldwide shipping for these findings!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Details of past events in the 21st Century Challenges: Policy Forum series.\nWhat does the decision to leave the EU tell us about society in the UK today?\nAt a time when some people perceive more threats than opportunities to their standard of living, how can we build a stronger and fairer society for a more 'United' Kingdom?\nPanellists at the Policy Forum discussed what genuine economic and social reform in the UK could look like and how this could help communities and places 'left behind' by socioeconomic change.\nIn partnership with the UK Energy Research Centre (UKERC).\nHow can the UK decarbonise its energy system to meet the legally binding targets under the Climate Change Act and international commitments, whilst ensuring both security of supply and that energy is affordable for consumers?\nPanellists at the Policy Forum event considered the dimensions of this 'trilemma', how this can be delivered upon in the context of the UK's vote to leave the European Union ('Brexit'), and the importance of taking scale into account when developing energy policy.\nThe Government has recognised flooding as one of the most serious threats facing the country, yet only the next six years of funding for flood-risk management, to 2021, is assured. Our expert panel and professional audience at this event explored how the UK can move towards a comprehensive strategy for sustainable flood risk management in the long-term.\nwith due environmental care \u2013 can affordable growth be accommodated sustainably and with good quality design?\nCan we identify common ground and what are the implications?\nWith net migration to the UK exceeding 100,000 each year since 1998, and the numbers of migrants unlikely to decline in the near to medium term, how can the UK adjust to this new reality, ensuring the best possible outcomes for the UK's economy, society and for migrants themselves?\nOur panel of expert speakers, and professional audience, discussed this and other questions at the Royal Geographical Society (with IBG).\nThe event marked the launch by the Joseph Rowntree Foundation of 'Community resilience to climate change: An evidence review' (led by Dr Clare Twigger-Ross, Collingwood Environmental Planning Ltd.).\nOur panel of respected experts examined these and other questions at the Royal Geographical Society (with IBG) on 21 October 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've since switched over to using Pelican. Maintaining your own static generator is not something I set out to do though I did learn a lot along the way.\nI previously mentioned that I am working on redesigning the root landing page. What I may have not mentioned is that I am also working on revamping my blogging platform.\nFast. I mean fast. The rsync up should be the longest part of the op.\n100% static files. Nothing dynamic apart from a few Apache rewrites.\nBuilt from scratch. I understand the principals behind reinventing the wheel, but there are things I believe take time to do yourself and there is immeasurable satisfaction in doing so.\nVery simple stuff. Writing should be simple. I no longer need to fuss with wmd because I'm able to write on any computer at my leisure. I can outline something in OmniOutliner, sketch notes in iA Writer, then dump it onto Dropbox to sync with my computer before I do fine tuning in TM or Sublime. This kind of flexibility should not involve more work the necessary. Maintaining and adding features to a web interface is not my idea of a fun time when I would rather jot a note here, a few sentences here, and then push it together.\nDatabases are really good at two things. Searching for large amounts of information and storing disjointed bits of information. They are not good at serving web pages.\nLet us take a look at a request for a particular blog post with a database in the mix.\nFirst the Client has to send a request. Then it goes over the wire (Sea of Network Latency) before hitting my server. On the server it locates the Apache process and spins up a PHP instance to parse and handle the request. It checks to see if we have a primed cache (Disk I\/O). If it finds a valid cache file then it returns that. If there is no cache then it sends a request to the database server, again going over the wire (Sea of Network Latency). The database munches on the request before regurgitating a sticky blob of data to return. The PHP process (still all PHP here!) examines this gooey data before deciding that it can manage parsing it. It rolls it through a few processes to template and format everything before caching it and send it back to the client.\nNow if we remove the dynamic portions (PHP) and the slow portions (Database) we get something cleaner and more manageable.\nFirst the Client has to send a request. Then it goes over the wire (Sea of Network Latency) before hitting my server. On the server it locates the Apache process and (Disk I\/O)[,] finds a file and send[s] it back to the client.\nSome formatting in 's added for clarity.\nA database has no place in serving static content except as a convenience. If you can't map out a workflow to refresh that static content when changes are made then the next best thing is to roll into a purpose built language that will roll those changes for you at the additional CPU cycle expense.\nComments are a headache. With the amazing amount of spam I still get using reCaptcha I don't think it's worth the headache anymore. Like it enough and you will hit me on Twitter or by email. Pages are just more pieces of static content. Tags are next to useless and maintaining them is a nightmare.\nFrankly? Horrible right now. I am using Apache to generate all the indexes, but the HTML is there and the posts are being reshuffled nicely into directories. I will be making incremental improvements over the next few weeks. Hopefully something production-ready will be good to go before the Fall.\nWhile still extremely bare-bones, you can take a look at the generated site over at joshuakehn.com\/new. Site is now live!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The chapter's spring 2017 program took us, once again, to the San Francisco Peninsula, this time to explore two works of Frank Lloyd Wright. Impetus for this tour was the recent publication of Frank Lloyd Wright and San Francisco, by NCCSAH board member Paul Turner. Emeritus Professor of Art at Stanford University, Paul provided informed commentary throughout the day. Our first stop was the 1937 Hanna House, also known as the Honeycomb House, because Wright based his design on the form of a hexagon. After a Stanford-hosted lunch, the group traveled to a house in Atherton inspired by Wright and designed by Anshen and Allen (1950) as the personal residence for developer Joseph Eichler. We learned of Eichler's profound interest in the famous American architect during our visit, next, to the Wright-designed Bazett-Frank House (1940), in Hillsborough. Eichler lived in the house as a tenant, early in the 1940s, and so admired the architect's work that he wondered if a modern house such as this could be built for \"ordinary people\". Out of this thought grew a twenty-year career as a tract-home builder of the, now much sought after, \"Eichlers\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"$$$ Our CFO.com M&A roundup reveals that the merger-and-acquisition bust has hit full-force. Last week saw only 33 new transactions worth a total of $3.4 billion.\n--Bess Levin is on assignment. She will return tomorrow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TreeHouse Foods announced the intention to close its facility in Visalia, California by the end of the first quarter of 2019. The plant primarily produces pretzels and cereal snack mixes for the Baked Goods segment. Current pretzel production will be moved to other TreeHouse manufacturing facilities prior to the plant closure. The decision is consistent with the August 2017 announcement of TreeHouse 2020, the Company's restructuring program. TreeHouse 2020 is a multi-year plan to fully integrate the business and reduce its cost structure in order to invest in market-differentiated capabilities that will serve the rapidly evolving needs of its customers who are strategically focused and highly committed to their corporate brands. The closure, affecting approximately 294 employees, is being announced well in advance to provide employees with as much notice as possible and to ensure a seamless transition for customers. The costs to close the Visalia facility are expected to be approximately $21 million, of which approximately $8 million is expected to be in cash. Components of the charges include non-cash asset write-offs of approximately $13 million, employee-related costs of approximately $3 million, and other closure costs of approximately $5 million. The Company expects charges of approximately $0.28 per fully diluted share to be incurred in 2018.\nJPMorgan analyst Joshua Levine says he's \"increasingly pessimistic\" about the near-term free cash flow outlook for TreeHouse Foods. The analyst sees a \"number of quantifiable drivers\" that could cause the company's free cash flow to come in meaningfully below consensus in 2018 and 2019. He sees another potential leg down for the stock and keeps an Underweight rating on TreeHouse with a $39 price target.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thetraveljunkie.org \u2013 While I was in Bangkok, I had such a wonderful time talking to everyone, laughing, chatting, eating, floating, taking pictures and just being travel junkie.\nIf you are planning a trip to Bangkok and looking for 5 stars hotels then Pathumwan Princess Hotel is the right place for you.\nToday, I'll bring you to a buffet dinner at CiTi Bistro, a buffet-style dining outlet sited on the ground floor of the Pathumwan Princess Hotel.\nMenus for dinner include a \"live kitchen world cuisine\" buffet, plus a la carte menu featuring European and Thai cuisine.\nOverall, so delicious, comforting, and of course, I ate too much.\nIt was a really lovely evening, made of genuinely happy memories of Discover Thainess, friends, laughter and all that good stuff!\nThanks for having me Bangkok! That Mojito was the bomb, I'll definitely be back for more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now that you've eaten enough to last you a couple weeks and spent enough money to dent your bank account, it's time to get back to work for a few more weeks! Monday never felt so foggy, am I right? But, the reality of the matter is that Thanksgiving is passed and you still need to get your plans solidified for the new year. So, here are a few ways to get back into the groove after today and the rest of the week.\nToday and this week are for catching up, full of email replies, meetings, and more. There is no way to get around this, it just has to be done. However, that means you can easily be overwhelmed. My best advice is to create a to-do list with tasks and projects you know you can tackle this Cyber Monday.\nThis is definitely a hard one. Cyber Monday can trap you into a couple hours of online shopping and immediately take away from any productivity you might have started with. To get your fix, try to participate on Sunday night or Monday morning before work.\nWorking out after a holiday energizes you and gets you back on track. Long weekend's allow you to feel full, slow, and most of the time, unproductive. So, get moving!\nOrganization takes time and thought. It's especially challenging to get everything organized during the holidays because there is so much going on. Take some time right after Thanksgiving to sort through everything you have coming up. This way you will be prepared and better able to handle everything you have going on.\nBless the early risers, for I am not one! However, getting into a routine of waking up early will help you get more done during the day. Once the sky gets darker\u2014and it is getting darker much earlier now\u2014we are less inclined to stay at the office and work. Start early and end early\u2014it's a win-win!\nDo you have a ritual to stay on task during the holidays? How are you planning on getting back into the groove\u2014or staying in the groove during the holidays?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On 21 January at 11:00 a.m.\nA ballet performance for children based on the fairy tale by Josip Vandot. Choreographer \u2013 Edward Clug. Entrance fee: 7.5 EUR.\nOn 22 January from 6:00 p.m. to 8:00 p.m.\nA culinary event dedicated to the Hungarian cuisine. Free admission.\nOn 23 January at 10:00 a.m.\nA private exhibition of photographs by Boris Gaber\u0161\u010dik. Free admission. The exhibition runs until 28 February 2019.\nA musical event for children over 10 years old and adults. Free admission.\nOn 25 January at 9:00 p.m.\nA concert of pop and rock music. Free admission.\nOn 26 January at 11:00 a.m.\nA performance for children from 2 years. Duration \u2013 40 minutes. Author \u2013 Jan Malik. Directed by Bojan Martinec. Entrance fee: 6 EUR.\nOn 26 January at 9:00 p.m.\nA mixed music concert performed by Slovenian amateur bands. Entrance fee: 10 EUR (pensioners and students \u2013 7 EUR).\nAn excursion around the exhibition dedicated to the 60th anniversary of the Conservation and Restoration Department of the National Museum. Entrance fee: 6 EUR (students \u2013 4 EUR). Coordinator \u2013 Nata\u0161a Neme\u010dek.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On this page you'll find our explanation of Skill leveling in the game FGO (Fate\/Grand Order). We've also summarized how much QP you'll need, so please use this as your guide when you level up Skills in FGO.\nTo level up a Servant Skill, first chose the \"Enhance\" option from the MENU bar, then select \"Skill\" in the Enhancement screen.\nYou'll need to use Items and QP to level up your Servant Skills. Which Items you'll need depends on the Servant. The required number of Items or amount of QP increases for High Star Servants. 5\u2605 Servants in particular need a large amount of Items and QP to level up their Skills.\nRegardless of a Servant's Class or their number of Stars, their maximum Skill level is Lv. 10. Once all their Skills have reached Lv. 10, they cannot be leveled up any further.\nAt the start of the game the benefits of Ascension far outweigh those of Skill leveling. So it's better to use your Items to Ascend Servants rather than focusing on their Skills.Once you have 2-3 Servants of each Class up to about Lv. 60, then you should begin leveling up Skills.\nEssentially if you raise a Skill's level, its effect is multiplied. The rate of multiplication is distinctive to that Skill.\nIn this case, there is a 2% rise for each level increase, and from Lv. 9 to Lv. 10 there is a 4% rise. If you compare Lv. 1 and Lv. 10, in total there is a difference of 20%. The benefits of Skill leveling in this example are comparatively high.\nIn this case, there is a 1% rise in the Skill's effect for each level increase, and from Lv. 9 to Lv. 10 there is a 2% rise. If you compare Lv.1 and Lv.10 the difference is 10%.This rise is small in comparison to Example 1, so at low levels, the Skill's usefulness won't seem to change much.\nSkills have a CT rate - this tells you how many turns have to pass before a Skill can be reused - and can vary depending on that Skill. When a Skill reaches Lv. 6 their Cooldown rate is reduced by one turn, and again at Lv. 10. At the most, a Skill can be activated at short intervals of 2 turns.\n:pIn the case of a Skill with a CT of 7 at Lv. 1 - the CT is shortened to 6 at Lv. 6, and down to 5 at Lv. 10. With a CT of 5 and a 3 turn effect, there only needs to be a space of two turns when the Skill is not activated.You can therefore almost continually have this Skill in effect during battle.\nBy leveling up a Servant's Skills, the strengths of that Servant can be lifted to a new standard. Skills that charge the NP gauge are particularly beneficial when farming Quests, and by pairing these with the right Craft Essence, Noble Phantasms can be employed more rapidly.\n\u25b2For Servants with Skills that charge their own NP gauge, there's often a direct relation between Skill leveling and how that Skill is used.\nThere are some high level Quests in FGO featuring powerful enemies with huge amounts of HP. To defeat them you'll need Servants with a suitable amount of firepower and durability. Skill leveling makes it easier to clear these Quests.\nYou can acquire Skill Enhancement Items in Events and Free Quests. However drop rates in Free Quests are low, so they're not suited to gathering such Items. You're much better off trying your luck in Events.\nEvents rarely award Gems, Magic Gems, or Secret Gems, so instead we recommend you try \"Training Ground\" Daily Quests. Though please take note, the targeted Class for these Quests changes depending on the day of the week.\nRaising a Skill level from Lv. 9 to Lv. 10 uses up 1 Crystallized Lore, regardless of the Servant or their rarity. Other than as rewards in certain Events, the opportunities for acquiring Crystallized Lore are limited, making them an extremely precious Item. Think hard before you use them on one of your Servants.\nWhen you don't have enough QP to carry out Skill Enhancements, there's only one real option - the \"Treasure Vault\" Quests. Try also to select a Support Servant equipped with the \"Mona Lisa\" Craft Essence which increases the amount of QP collected.\nExchange Items for QP in Events with high drop rates. Repeatedly farm Lottery Events to earn even more QP. Only in some special Events can you get your hands on this in-game currency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Panhandle is full of natural wonders, each one putting on its own colorful show. To kick off your trip to this region, lace up your hiking boots for an adventure through the red-banded outdoor wonderland that is Palo Duro Canyon. Known to some as \"Texas' Grand Canyon,\" this ancient system reaches 120 miles in length from Canyon to Silverton, measures 20 miles wide and has a depth (in some places) of 800 feet.\nProviding the vivid bands of sandstone are four geologic layers formed over a million years. For a really majestic way to take it all in, climb on a horse or mountain bike to ride along the 30 miles of trails at the Palo Duro Canyon State Park. Be sure to take meandering breaks for geocaching and to observe local wildlife. In this area, you can watch for the Texas horned lizard, Palo Duro mouse, wild turkey, mule deer, coyote, bobcat and roadrunner. If you're a bird-watcher, be on the lookout for golden-fronted woodpeckers, Mississippi kites, Bullock's orioles, painted buntings and western meadowlarks. Part of the Official State of Texas Longhorn Herd lives in the park. You can find these gentle beasts near the park headquarters.\nSouth of Palo Duro, Caprock Canyons State Park & Trailway unfolds along the magnificent Caprock Escarpment. This lengthy, rocky formation pierces the topography and connects the high plains above to the lower plains below. Multihued rock formations can reach as high as 1,000 feet; after rainy periods, the landscape can be carpeted in strips of green, with blankets of wildflowers in spring and summer. The Texas State Bison Herd lives here, roaming some 10,000 acres. The canyons provide homes to deer, coyotes, bobcats, pronghorn antelopes, foxes, raccoons and jackrabbits, while Mexican free-tailed bats roost in the Trailway's Clarity Tunnel. At Lake Theo, take a dip or try your hand at catching bass, trout or catfish. If the kids are in tow, they'll love the Junior Ranger program.\nFor a respite from the windswept plains, make your way to the Lake Meredith National Recreation Area, an extraordinary lake mingling with the buttes and cliffs of the Canadian River Valley. Formed by damming the Canadian River, this 16,500-acre reservoir is surrounded by 200-foot canyons in ancient shades of russet and cream, providing adventurous souls a playground for waterskiing and fishing, as well as off-road trail riding, hiking, picnicking and camping. Right next door, you can search for wildly colorful flint in the ancient rocks of the Alibates Flint Quarries National Monument. Below the South Plains, you can head east toward Mineral Wells for a canoe trip along the mirror-like Brazos River, paddling through gorgeous scenery in the Palo Pinto Mountains. During autumn, the trees take on every lovely seasonal hue that nature has to offer.\nThis awe-inspiring area is full of natural landmarks big and small, all just waiting for you to get out there and find them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vintage 1958 McCalls Pattern 2251, Size 10 Poodle Skirt Pattern.\nCopyright 1958, McCalls Pattern 2251 for a size 10 (waist 24 inches) Poodle skirt complete w\/ iron on transfer to create the poodle, and petticoat. Full instruction sheet and uncut pattern, in original envelope. Some wear to envelope - was worn open at the end so we taped it to secure the pattern. Printed with tab on bottom of envelope. Pattern is factory folded as is transfer. Factory precut individual pieces to this pattern, including each waistband! See photos for condition, and sizing as well as fabric requirements for this amazing vintage pattern.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recommended Site ucas personal statement writing services. Official Statement custom written dissertation. Check It Out write my essays for me online. Index essaywriter writers. Have A Peek At This Web-Site please write my essay for me. Click This Over Here Now assignment service uk. His Comment Is Here write my essay for me com. Find More Information paper writing service. Click This assignment writing services. Wikipedia Reference buy assignment uk. Useful Reference customwriting. You Can Find Out More help in assignment writing. News essay service. Visit This Site Right Here custom assignment writing. View It Now essay write uk. See This paper writing service. Blog Link help me write my report. Continue Reading This pay for college papers. Navigate To This Web-Site custom dissertation uk. Go Now online assignment help uk. A Knockout Post personal statement helper online. Extra Resources pay for dissertation. My Company custom write. See Page do my accounting homework. Index need assignment help. My Latest Blog Post cheap essays. Important Link best personal statement service. Go To This Web-Site Critical Essay. Try Here quality essay writers. Visit This Link quality assignment help. Click Resources Persuasive Essay. Full Article assignment helper. View dissertation help. Additional Resources pro essay writers uk. You Can Try These Out ucas personal statement writer. Read This buy essays uk. Go To Website personal statment writer. Website Link custom dissertation writing uk. His Response do my science homework. Try This Website assignment help. Recommended You Read customer essay. Additional Info do my dissertation for me uk. Read This buy essays uk. The Original Source best write my essay website. Visit Website dissertation writing help. View It Now do my term paper. Why Not Find Out More write my scholarship essay for me. Browse Around This Web-Site help with assignment writing uk. Our Site buy assignments. This assistance with dissertation writing. Click Over Here Now help with assignments uk. Look At Here Now help with writing my dissertation. This Article custom thesis. Hop Over To These Guys buy a dissertation uk. Visit Here uk assignment writing. Useful Site buy case study online. Find Out More student assignment help. Try This Out buy essay cheap. Pop Over To This Website The Best Essay Writing Service. Weblink write my essay for me in uk. My Link write my essay. Web research paper for sale. More Info Here write my essay today. Hop Over To This Web-Site help writing dissertation uk. Find More Information essay writers for hire uk. a fantastic read essays online. My Latest Blog Post assignment help service. Navigate Here write my essay quick. Read More Here dissertation writers uk. This Hyperlink buy argumentative essay. Check Out Your Url term paper writing service. Click This coursework help. Try This Web-Site law essay writing service uk. Go To This Site personal statement writers. Visit Their Website assignment help online uk. Go Here essay writing service quick. Hop Over To This Web-Site personal statement writer reviews. Read What He Said do my french homework for me. Site Web Reflective Essay. Click To Read More write my biology paper. Helpful Site hire a writer for an essay. Why Not Check Here help with assignment uk. Web Link best essay uk. Go To My Blog essay writing help. Bonuses my essay writing. Site top essay writers uk. Site Web custom+dissertation+help+uk. Original Site essaywriter. Click Now hire someone to write my personal statement. Extra Resources help in writing personal statement. Read What He Said do my french homework for me. See It Here write my essay for me for cheap. Look At Here Now help with writing my dissertation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Home \/ Sports News \/ Not the usual team to beat, 2nd seeded UConn is not to be taken lightly as the NCAA national women's basketball tournament bounces in, with the Huskies in the first round in Albany, N.Y.\nNot the usual team to beat, 2nd seeded UConn is not to be taken lightly as the NCAA national women's basketball tournament bounces in, with the Huskies in the first round in Albany, N.Y.\nWhen he was asked about the 2nd ranked UConn women getting an unprecedented 2nd seed for the NCAA basketball tournament where they own a record 11 national championships, Geno Auriemma didn't appear to let him bother him.\n\"It don't think it matters one way or another,\" said the legendary head coach. \"We've lost national championships being a No. 1 seed, and we've won national championships being a 2 or 3 seed if I'm not mistaken.\nThen he shrugged his shoulders and walked away.\nIt was the East Region in Albany, N.Y. where the NCAA was sending the Huskies. First it placed Mississippi State there as the top seed and seeded UConn second. Then, after Mississippi State lost to Louisville, it was sent to another region and Auriemma's team became No. 1.\nBut when the selection committee announced its official field Monday, it continued to look like musical chairs when Louisville became the one seed in Albany and the Huskies were dropped to second seed.\nWhen Auriemma and his team watched the selections on television, they just cheered and did some hugging, but no one looked upset or disappointed.\nIts two setbacks were on the road, at No. 1 Baylor and at No. 3 Louisville, but had no problem rolling over then-No. 3 seed Notre Dame before a full house in South Bend. In fact everywhere UConn played brought capacity crowds and its home court in Storrs was always sold out.\nAnd winning 34 times is not bad. Auriemma did not figure this team as one to again make the Final Four, but it improved each time out under his guidance.\nIt was the 6-foot-3 Samuelson, the long-range shooter and team leader, versatile 6-2 senior All-America Napheesa Collier teamed with 5-5 junior Crystal Dangerfield, as good as any point guard in the country. She leads the nation in assists, can race up and down court like a sprinter, and she can toss them in from 3-point land.\nBut the team also got better when 6-1 sophomore Megan Walker and 5-11 freshman ChristynWilliams came of age, Walker with her perimeter shooting and rebounding, and Williams with her 3-point shooting, with a high of 28 at Notre Dame, and also her penetrating to the basket. They were back-to-back high school players of the year.\nLacking sufficient height, Auriemma worked with 6-5 freshman Olivia Nebon-Ododa and she has gotten better with each game after a nervous start.Her offense down low is much better, and she can soar.\nShe won a mixed slam-dunk contest over the boys during the 2-day MacDonald's event, and began taking down an average of 8-9 rebounds a game once she got involved as the sixth man and learned from experience.\nAuriemma can call on several reserves, but he usually stays with his experienced starters. He has bench players who were high school stars, but playing for UConn is a different level.\nSo let's not sell the Huskies short. There's Baylor, Louisville, Notre Dame, Mississippi State, and even Oregon and Stanford in the way versus the most successful hoops program in women's history (and anywhere else), as well as one of the finestlegendary coaches who knows how to win.\nCalifornian Samuelson, also high school player of the year, chose the Huskies while her two older sisters decided to play for Stanford.\nHer freshman year, UConn won the national title but a leg injury in the regional final kept her on the sidelines. And the last two seasons, she made it to the Final Four, but lost on buzzer shots in the semifinals by Mississippi State and Notre Dame.\nBut she doesn't complain. She's happy where she is, is a total team player, and returns after missing the three American Athletic Conference tournament games with back spasms but never stopped providing support from the bench against a conference where the Huskies have never lost a game, and the count is now 137-0.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"John's Island, Orchid Island, Windsor, Sea Oaks, The Moorings, Grand Harbor, Oak Harbor, The Victoria, The Carlton, The River Club, Bent Pine Golf Club, Croom Construction, The Hill Group, Parent Construction, Wissel Construction, Standing Watch, Jones and Jones Builders, Zugleter Construction, Indian River Project Management, Helmut House Construction, Bill Bryant and Associates, Proctor Construction, Island Builders, Whitefield Construction, Sea Side Construction, Palm Coast Development, Schlitt Builders, Creative Tile, Elgin Marble, Country Tile, Ceramic Matrix, Superior Tile and Stone, Whitehall Services, Barth Construction, Elliott Merrill Company, Service Master, Braittan and Associates, JC Welton, RCL Builders, Vero Beach Hotel, Dale Sorenson Real Estate, Norris and Company, Billero and Billero, Island Interiors, Rod Mickley Interiors, Designers Touch, Leah Muller Interiors, L.K. Defrances and Associates, Barkley Interiors, Lin Menard Interiors, Natalie Holtom Interiors, Orchid Island Property Management, Paula Holden Interiors, Scott Blair Interior Design, St. Helen's Church, Page 2 Design.\nThe Radisson Hotel - Port Canaveral, Millhouse Leasing- Boca Raton, Hilton Hotel - Cocoa Beach, Lansing Island - Indiatlantic, Sloane Construction - Palm Beach, Real Stone and Granite - Fort Pierce, Este Lauder Estate - Palm Beach, The Carlyle - Jupiter Island, N.A.S.A Cape Canaveral, Patrick Air Force Base - Cape Canaveral, Burton Homes - Melbourne, Showcase Design - Ft.Pierce.\nThe White House Administration Building - Washington, D.C., The Trump Plaza - Atlantic City, N.J, The Taj Mahal - Atlantic City, N.J., Bally's Grand - Atlantic City N.J., Infiniti - Ramsey, N.J., Saab - Ramsey, N.J., AT&T - N.J. and Philadelphia, The Carmelite Monastery- Morristown, N.J., The Prudential Insurance Company - New York, N.Y., The Waldorf Astoria - New York, N.Y., The Radisson Hotel- Parimus N.J, The Ritz Carlton - San Diego, California, The Double Tree Hotel - San Diego, California, John F. Kennedy Behavioral Health Center - Philadelphia, Banco Safro Financial Center - Sao Paulo, Brazil, Safra Residence - Sao Paulo, Brazil, The Museum of Man - San Diego, California, Rosemary Hall at Choate University - Wallingford, Connecticut, Windows of the World Restaurant- New York, N.Y., Hotel Del Coronado - Coronado, California, The Glendale Galleria Mall - Glendale, California.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this report, LP Information covers the present scenario (with the base year being 2017) and the growth prospects of global Hydrocephalus Shunts market for 2018-2023.\nCSF diversion devices or shunts have been used successfully and have become the primary therapy for hydrocephalus treatment for nearly 60 years.\nAn implanted shunt diverts CSF from the ventricles within the brain or the subarachnoid spaces around the brain and spinal cord to another body region where it will be absorbed.\nAn inflow (proximal or closer to the inflow site) catheter, which drains CSF from the ventricles or the subarachnoid space; this tube leaves the brain through a small hole in the skull and then runs for a short distance under the skin.\nA valve mechanism, which regulates differential pressure or controls flow through the shunt tubing; this device is connected to the proximal catheter and lies between the skin and the skull, usually on top of the head or just behind the ear.\nOther shunt components may include reservoirs and\/or chambers for CSF sampling or injecting medications or dyes, on\/off devices, anti-siphon or other flow-compensating devices, or auxiliary catheters to modify performance or adapt the basic system to the patient's specialized needs. In selected cases (such as when cysts or subarachnoid fluid collections are drained), a shunt may not contain a valve or a very low resistance valve may be used.\nPuerto Rico is the largest supplier of Hydrocephalus Shunts, with a production market share nearly 42.88%. Massachusetts is the second largest supplier of Hydrocephalus Shunts, enjoying production market share nearly 36.49%.\nSouth is the largest consumption place, with a consumption market share nearly 37.61%. Following South region west is the second largest consumption place with the consumption market share of 23.55%.\nMarket competition is intense. Medtronic, Johnson & Johnson, Integra LifeSciences are the leaders of in the USA industry, and they hold key technologies and patents, with high-end customers, have been formed in the monopoly position in the industry.\nOver the next five years, LPI(LP Information) projects that Hydrocephalus Shunts will register a 9.3% CAGR in terms of revenue, reach US$ 120 million by 2023, from US$ 72 million in 2017.\nThis report presents a comprehensive overview, market shares, and growth opportunities of Hydrocephalus Shunts market by product type, application, key manufacturers and key regions.\n- To study and analyze the global Hydrocephalus Shunts consumption (value & volume) by key regions\/countries, product type and application, history data from 2013 to 2017, and forecast to 2023.\n- To understand the structure of Hydrocephalus Shunts market by identifying its various subsegments.\n- Focuses on the key global Hydrocephalus Shunts manufacturers, to define, describe and analyze the sales volume, value, market share, market competition landscape, SWOT analysis and development plans in next few years.\n- To analyze the Hydrocephalus Shunts with respect to individual growth trends, future prospects, and their contribution to the total market.\n- To project the consumption of Hydrocephalus Shunts submarkets, with respect to key regions (along with their respective key countries).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sorry I haven't been very cheerful the last few days. Yesterday I was trying to write my new book and I got slower and slower and gloomier and gloomier, until I literally could write no more, and I'm like - what's wrong with me? But then I realised I had an ear infection. happytune always says she can tell when I am getting ill because I get gloomy. So for the first time since I was a child I was in bed at New Year, though awake. I got up just after midnight and took some painkillers, and I've been mostly sleeping or resting. But I got up just now and watched the last half hour of Sherlock - seemed good from what I saw. Anyway I am glad my bad mood had a physical cause, because that will get better. No division between body and mind. I'm only really posting this to explain because I know I have been overly grumpy. Going back to bed now.\nHope you feel better soon. Don't get out of bed too quickly.\nI realise I'm late with this, but I hope you're okay.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the last budget, chancellor George Osborne gave Sir Charlie Bean the task of undertaking a review of the UK's economic statistics. The proposal was triggered by concerns about the difficulties in measuring productivity accurately in a modern economy and an independent review was thus commissioned by the chancellor of the quality, delivery and governance of the UK's economic statistics.\nThe call for evidence in the review had just finished and Sir Charlie attended a meeting at the RSS last month (September 2015) to discuss his remit and listen to what others had to say about the current status of economic statistics. Sir Charlie was asked to compile the review on account of his 14 years on the Monetary Policy Committee at the Bank of England where he was also deputy governor.\nHe began by outlining what he would be assessing in his review. The major structure of the national accounts was installed in the 1930s and there are questions as to whether this is giving an accurate measure of the modern financial, services and digital industries. Following this, he says he will examine the capability of the Office for National Statistics (ONS) to deliver modern economic statistics and to what extent they should make better use of admin or external data sources.\nFinally he will be looking at the governance arrangements for the production of these statistics. The inclusion of governance as an element of the review has raised concerns about how it will affect the current independence of official statistics. When questioned on where this will lead, Sir Charlie reassured those present that independence was a red line, both for him personally and in the review's terms of reference.\nSir Charlie finished his short overview by saying he was currently in 'listening mode'. So to widen the discussion, two other speakers were invited to give their view on where economic statistics stand today. Diane Coyle, economics professor at the University of Manchester spoke first, followed by David Caplan, a statistician who was formerly the deputy director for the UK's national accounts.\nDiane began her overview by showing the audience a statistical bulletin from the 1880s. Within the bulletin was page after page of agricultural statistics, yet only a few pages were devoted to industry, despite the industrial revolution being well established by that time. Her point was that economic statistics have always lagged behind the actual changes taking place in the wider economy.\nWith this in mind, she asked what effect the dramatic falls in the price of technology like computer chips and genome sequencing is having on economic output. New technologies such as 3D printing or graphene will have a major economic impact in the years to come, but the effects of the growing availability and affordability of products like this are difficult to track in our current statistics. This begs the questions of how innovation is benefiting society outside of the old statistical calculations of GDP.\nBeyond this, Diane asked if the current categorisation of jobs and industries was keeping pace with what was happening in the digital and short-term 'gig' economies. These new industries have created a host of new job titles and employment arrangements that need to be carefully considered if the ONS is to accurately measure them in the future.\nDavid Caplan then began his talk by briefly running through the long line of economic reviews that preceded Bean (which he has previously written about on these pages). He also described the diverse and dynamic needs that users of economic statistics can have. This problem is compounded for statisticians by economists having fundamental disagreements on what kind of data is important to their work.\nAnother problem is fact that the EU is the major driver behind many of the statistics that the ONS produce. These are compiled so that cross country comparisons can be made in Europe, but it introduces the dilemma of when national needs should supersede Eurostat's needs.\nHe also described the caution needed when using proxy or admin data to compile statistics, as these can be susceptible to system changes and could end up biasing the data or playing havoc with the time series.\nDavid finished by reiterating that the needs of users has to be prioritised. He said the ONS largely has a producer's culture, and while this is changing, those end users need to kept in mind at all times.\nThe floor was then opened up for questions and comments. These included how the recent reviews into price statistics will relate to the Bean review, how to counter trivial movements in economic statistics getting blown out of proportion in the media, and the need for an independent system for the production of statistics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On this day in music history: October 24, 1995 \u2013 \"Mellon Collie And The Infinite Sadness\", the third album by The Smashing Pumpkins is released. Produced by Billy Corgan, Alan Moulder and Flood, it is recorded at Pumpkinland, Sadlands, Bugg Studios, The Chicago Recording Company in Chicago, IL, and The Village Recorder in West Los Angeles, CA, from March \u2013 August 1995. Following the back to back successes of their breakthrough album \"Siamese Dream\" and the covers album \"Pisces Iscariot\", Smashing Pumpkins leader Billy Corgan is anxious to push the boundaries of his bands' music even further. Inspired by classic records like The Beatles' \"White Album\", Corgan is looking to make an extended musical statement by creating something akin to \"The Wall\" for Generation X\". The band significantly changes their approach to recording by beginning the sessions in their rehearsal space. Also, instead of working with producer Butch Vig again, Corgan co-produces the album with Alan Moulder (My Bloody Valentine, The Jesus And Mary Chain, Nine Inch Nails) and Flood (U2, Nine Inch Nails, Depeche Mode). Unlike the previous albums, guitarist James Iha and bassist D'Arcy Wretzky take a more proactive role during the sessions (rather than Corgan playing nearly all of the instruments, with the exception of drums, played by drummer Jimmy Chamberlin). As a result, recording moves ahead much faster, with the band cutting nearly three dozen new songs, before paring it down to the final twenty eight used for the final track listing. Split into two CD's titled \"Dawn To Dusk\" (disc one) and \"Twilight To Starlight\" (disc two), and as a limited (to 5,000 copies on the original pressing) triple vinyl LP with the individual sides titled \"Dawn\", \"Tea Time\", \"Dusk\", \"Twilight\", \"Midnight\", and \"Starlight\". While not technically a concept album, the songs articulate Billy Corgan's personal journey from his teenage years to his adult life at that point. The album is released to a rapturous response, and is regarded as one of the best of the 90's. It spins off a total of five singles including \"Bullet With Butterfly Wings\", \"Zero\", \"Tonight, Tonight\", and \"1979\", all of which receive heavy play on MTV. The video for \"Tonight, Tonight\" wins six MTV VMA Awards including Video Of The Year in 1996. It is nominated for seven Grammy Awards including Album Of The Year and Record Of The Year (\"1979\"), winning the award for Best Rock Performance By A Duo Or Group With Vocal in 1997. The album is remastered, remixed and reissued in 2012 as a six CD\/one DVD set, containing demos, unreleased takes and mixes not used on the original album. The DVD includes performances from two shows on the \"Mellon Collie\" tour from 1996. \"Mellon Collie And The Infinite Sadness\" debuts at number one on the Billboard Top 200, and is certified 10x Platinum in the US by the RIAA, earning a Diamond Certification.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A great classical horror movies pub quiz. Test your general knowledge with this great horror nostalgia quiz. Great for Halloween or movie buffs.\nQ1 In which idyllic setting does the tragedy and bloodbath of Friday 13th take place?\nQ2 Who directed the 1922 horror classic Nosferatu, A Symphony of Terror inspired by the novel Dracula?\nQ3 Who played the role of Reagan McNeal in William Friedkins shocker The Exorcist?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(2) Set: Your Mana Spring Totem ability grants an additional 3 mana every 2 sec.\n(4) Set: Reduces the cooldown on your Ancestral Swiftness ability by 24 sec.\nCyclone Raiment is the tier 4 restoration shaman set.\nSee also: Cyclone Regalia for the elemental version of this set, or Cyclone Harness for the enhancement version.\nThe Cyclone Raiment items are sold by Asuur (Aldor) or Arodis Sunblade (Scryer) in Shattrath City. They are \"purchased\" with token loot items.\nCyclone Raiment worn by a female draenei.\nCyclone Raiment worn by a male orc.\nThis page was last edited on 16 March 2017, at 20:16.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Westchester Buddhist Center offers a sequence of courses providing an overview of the Buddhist path from start to finish, as presented by Ch\u00f6gyam Trungpa Rinpoche. The curriculum is designed to guide students through the progression of topics that constitute the 3-yana view of Tibetan Buddhism, with an emphasis on joining intellectual understanding with meditation practice.\nA year-long intermediate-level course on the Hinayana, based on volume one of the three-volume set, The Profound Treasury of the Ocean of Dharma. The course is offered in two class sections, one meeting Sunday afternoons at Eileen Fisher headquarters in Irvington and the other meeting Wednesday evenings at The Center for Health and Healing in Mt. Kisco.\nFor the foundational course, students may enroll for the semester (i.e., two segments), enroll for one segment only, or even attend individual classes on a \"drop-in\" basis. For the intermediate-level course, enrollment is by semester and a commitment to attend all or most class sessions is requested.\nCompletion of the foundational course, or equivalent familiarity with the Buddhist path and practice experience, is required to enroll in the intermediate-level course. In turn, completion of the intermediate course will allow students to progress to an advanced-level course (not offered in Fall 2018-Winter\/Spring 2019).\nClick here for detailed information and registration for the Sunday intermediate course on the Hinayana and here for the Wednesday course.\nFor more on the WBC \"Path\" of practice and study, click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sony Artisan and world renowned lighting expert Scott Robert Lim will reveal his lighting secrets in this full-day workshop. Scott will teach you his techniques that will allow you to create crazy, stupid light withing minutes. One light techniques can be very powerful and produce beautiful portraits. Learn the the fundamentals of lighting and use these skills to produce professional quality work.\nThe lecture portion of the seminar will last approximately ninety minutes. After a short break, Scott Robert Lim will bring his techniques to life with a live demonstration you won't want to miss.\nSony Artisan and International Master photographer Scott Robert Lim is a PPA certified educator and has over 70 WPPI international awards to his name, including top Ten Most Influential, Kodak Award, World's Top Wedding Photographers and has been a judge at many international photography competitions. His work has been published and viewed by millions all over the world and is a popular speaker on Creative Live- \"Crazy Stupid Light\", \"Think Like a 10K Wedding Photographer\" and his latest \"Portrait Fundamentals\". For the last decade he has dedicated his career to developing world class photographers on every continent and loves teaching photographers of all levels. He has an energetic and charismatic speaking style and is an industry leader in photography education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhmwf b/data_all_eng_slimpj/shuffled/split2/finalzzzbhmwf
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@@ -0,0 +1,5 @@
+{"text":"Dogwood Hedging (Cornus Alba) is easily recognised by its colourful stems. Used mostly for under-planting in groups of trees, or motorway planting. This hedge is planted in groups for best effect.\nDogwood hedging can be planted in most soil types, planting should be done 3 to every yard. This is a very hardy hedge, it thrives in damp shady conditions.\nHow fast will Dogwood Hedging grow?\nThis hedge grows between 18 inches and 24 inches a year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Journey Through Brendon Small's Wild, Sci-Fi \"Nightmare\"\nDethklok and Galaktikon guitarist Brendon Small has teamed up with Funny or Die for Galaktikon: Nightmare, a \"high stakes, live action, intergalactic metal short.\" You can watch it above.\nSmall\u2014who wrote and directed the short\u2014says that the film is \"a love letter to all the filmmakers in the '70s and '80s that helped inform the creativity in my childhood, such as Steven Spielberg, George Lucas, Tobe Hooper, Roger Corman, and Dino De Laurentiis.\"\nSmall was hugely inspired by legendary B-movie filmmaker Corman, even using Corman's old spaceship set for the film, which was made without the assistance of CGI.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've had my Twiddler for a year, it comes to work with me in a soft bag most days. It started misbehaving a week ago. Sometimes when I press a chord the host loses Bluetooth connection then quickly re-establishes (sometimes this happens a few times). After that, most four and thee finger chords don't work and sometimes the wrong key is sent.\nI assumed it had suffered physical abuse. There was no sign externally so I opened it up to look for evidence (with help from pictures from this forum). There was none. When I put it back together (rubber bands are a surprisingly effective substitute for glue) it seemed to work, but now I realise it only misbehaves after five or ten minutes of use. The symptoms are unchanged.\nI have a theory that it is losing power momentarily and coming back up in a wonky state, and this is borne out by my latest discovery that there is no sign of trouble when I use it plugged in.\nIs any of this familiar to anyone? Any insights? What's my next step? What are my chances of patching up the battery connection? The two battery tabs are pretty large and accessible, but their contacts are not. Is there somewhere else on the board that would be easier to connect them up to?\nI've brought this to a happy conclusion at last.\nWith Twidlit near completion, I've started using my Twiddler more, and being wired to my host device has been driving me nuts. I was set to purchase a replacement until reminded of the shipping cost to Australia (50% of the item price). Time to poke around some more.\nThe socket end of the circuit board is braced from the button face by two posts. The bath-shaped back of the case has two tiny tabs that press down on the edges of the board. Unfortunately the tabs are close to the posts so when excessive force is applied between front and back, the board is bent tightly between them. In my case this compromised a connection to the Bluetooth chip. Now it behaves itself only until even the slightest pressure is applied.\nBy removing the tabs and replacing their localized pressure with a more diffuse force from a folded tissue inside the case I completely stopped the problem.\nMy guess is that the tabs are positioned near to the posts so that both are close to the socket so as to give it maximum support. However, my case suggests it might be wise to separate them.\nmine has started acting up, but i haven't wanted to disassemble it to see if any connections have shorted, etc. do you have any suggestions for non-destructively opening it up? and for putting it back together?\n\"Non-destructive\" has to be a little flexible here. It's never going to look the same and there is a strong risk of sinking your knife into the circuit board (or you palm).\nYou cut round the bonded seam to remove the dished back. You have to go quite deep along the lower third of the sides because there are chunky bits running along inside there. I found the top edge (where the strap attaches) a bit easier.\nPutting it back together is more straight forward. 2-3 rubber bands do the job nicely, though the whole case becomes a bit squishy once its integrity is compromised. A zip tie round the neck holds the strap better than the original rod and slot arrangement (in my view). None of it is very elegant though.\nI haven't tried re-gluing primarily because none of my attempts at repair have proved satisfactory. I can make it a little better or worse but it always goes wonky after 20 minutes or less. There's a dodgy connection, but who knows where. I don't fancy your chances either. Unless you find such activity entertaining I'd just pass it on to someone who does.\nThe whole exercise has given me greater insight and respect for the overall mechanical design so I'm glad I did it, but it was mostly frustrating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Michigan's Superintendent of Education Brian Whiston was on the news program \"Off the Record\" where he talked about efforts to improve public education in the state. During the segment he drew a hard line in the sand on charter schools \u2014 one of Michigan Republicans' favorite education schemes.\nAsked about the performance thus far of U.S. Secretary of Education Betsy DeVos, Whiston said it was too early to make a complete call, but he skewered the idea that \"school choice\" \u2014 i.e. charter schools \u2014 were the silver bullet to Michigan's education woes.\nWhiston wasn't done with Betsy after his choice comments, he even went after her affinity for vouchers, which would put public funds in the hands of private, parochial, and for-profit schools and charters.\n\"We'll never agree on that,\" Whiston said, who has ardently opposed to voucherizing Michigan's public education system in the past.\n\"That issue is non-negotiable. I don't think public tax dollars should be used for private education,\" he said.\nWhat Michigan needs isn't more DeVos-backed school experiments with kids' futures and public funds. We need properly funded schools, well-paid and respected teachers, and a move away from such heavy standardized testing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Mindless\" vandals have caused \u00a3500 worth of damage to a bus shelter following two attacks in less than a week according to a parish clerk.\nBradwell parish clerk, Jeremy Caborn said one of the panels on the bus shelter on Beccles Road was smashed on Friday, October 26 with a further two panels being broken on Tuesday.\nCouncillor for Bradwell South and Hopton, Andy Grant described the actions as \"disgusting\".\nMr Caborn said vandals inflicted similar damage to the bus shelter about one year ago.\nHe said: \"It is a real shame but we are just resigned to this happening now. It is going to cost us about \u00a3500 to repair which is a real waste of money.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhpse b/data_all_eng_slimpj/shuffled/split2/finalzzzbhpse
new file mode 100644
index 0000000000000000000000000000000000000000..93e6954c5ae227138ff440a28fe040e980a1e210
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@@ -0,0 +1,5 @@
+{"text":"Thingiverse Groups TEVO Tarantula Owners Group General Which Couplings are the Best for Tevo Tarantula Z-Axis?\nWhich Couplings are the Best for Tevo Tarantula Z-Axis?\nDue to the upgrades I've added to my Tarantula's carriage, the weight seems to have stretched the default Z-axis coupling. I'm looking at this rigid coupler on Amazon but I'm not sure this is a good choice. What do you guys\/gals think? If no, what would you suggest?\nI am running the no-sand oldham coupler, so any out-of-straight in the rod should be taken out by that, no?\nTevo Tarantula no-sanding bearing \"oldham\"\nThat seems to be the general consensus around here ;) I need to get it fixed good enough right now to print new parts, though, so... I'm going to print a solid coupler like the one on Amazon and see how that works for a while. I think these are the brackets that I have. I may do that to mine this spring.\nI was thinking about the inverted setup, but I'll have to look into that some to figure out how \/ what needs to be done. That gimbal... I'm going to have to study that. I like the roundness - less likely to catch on the frame and defeat it's purpose. The spider jaw is a good option if\/when I go inverted, but for right now... I need to get this printing reliably like it is before I can print any more upgrades.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"US Defence Secretary Chuck Hagel says Islamic State group has an \"apocalyptic\" vision and that it must be defeated.\nThe sophistication, wealth and military might of the Islamic State group represents a major threat to the United States that may surpass that once posed by al-Qaeda, US military leaders have said.\n\"They are an imminent threat to every interest we have, whether it's in Iraq or anywhere else,\" Defence Secretary Chuck Hagel told reporters at the Pentagon.\nHagel's assessment of Islamic State, which gained strength during Syria's civil war and swept into northern Iraq earlier this summer, sounded a note of alarm several days after the group posted a video on social media showing one of its fighters beheading an American journalist who had been kidnapped in Syria.\nHagel spoke as the United States continued attacking Islamic State targets in Iraq. In the past two weeks, US drones and fighter jets have conducted 89 airstrikes against militant targets in northern Iraq.\nEven after the gruesome killing of US journalist James Foley, Obama is seen as unlikely to deepen his near-term military involvement in either Iraq or Syria as he seeks to avoid becoming embroiled in another messy Middle Eastern conflict.\nBut US officials say they have not ruled out escalating military action against the Islamic State group, which has increased its overt threats against the United States since the air campaign in Iraq began.\n\"We haven't made a decision to take additional actions at this time, but we truly don't rule out additional action against ISIL if it becomes warranted,\" Ben Rhodes, a senior Obama aide, told National Public Radio earlier on Thursday, using another name for the group.\nGeneral Martin Dempsey, chairman of the US Joint Chiefs of Staff, said officials were worried about the possibility that European or US nationals, radicalised after fighting in Iraq or Syria, would return to their home countries.\nDempsey suggested the group would remain a danger until it could no longer count on safe havens in areas of Syria under militant control.\n\"This is an organisation that has an apocalyptic, end-of-days strategic vision and which will eventually have to be defeated,\" Dempsey said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amazon.com is the largest online marketplace, and also the platform is maintaining growth. It provides amazing possibilities for online retail businesses to market products to countless consumers. Let's say you sell on Amazon.com, you are certainly going the best way. But if you are only offering on Amazon, you could be passing up on more product sales. It may look like like a challenging task to broaden to more systems, consider you might be already selling on Amazon.com, it will be simple to operate other platforms to enhance your profits just like the Amazon FBA platform. Amazon.com supplies a Multiple Channel Fulfillment (MCF) option that will assist you expand to more sales platforms with hardly any added costs.\nWhat Is Amazon Multi Channel Fulfillment? The FBA support from Amazon meets your Amazon orders, because the MCF choice fulfills purchases from all of other systems. You'll be able to delegate most fulfillment to Amazon. Regardless of whether you sell items on auction websites, Shopify, any another platform, Amazon will choose and channel products to your clients. You only need to pay for shipping and handling.\nMulti Funnel Fulfillment enables you to choose regular, two-day, or next-day delivery, plus it computes shipping and delivery costs with respect to the height and width of an item combined with chosen shipping approach.\nIf you would like to work with Amazon.com MCF, there are many requirements you need to think of. First, you have to be authorized for FBA, which means you possess cards on record with Amazon. Those cards will definitely have to pay for fulfillment costs, until your seller account includes a positive balance next MCF fees will probably be subtracted from the stability.\nIt's also wise to possess a professional seller account with Amazon to utilize MCF which will costs $39. 99 every month, however, that you do not purchase product list charges.\nMake the most of Amazon's MCF with your guidelines. Amazon's MCF is an excellent technique for websites providing you use FBA and may even keep to the above-mentioned requirements. But there is something that could get this on top of that in your case plus your customers.\nMake use of Messaging on Packing Slips. With Amazon MCF, logos and customization are restricted. You are unable to include personalized inserts or packing slides, you can have particular communications printed around the packaging slide. Benefit from these special messages to indicate that you value client's business and value them as customers.\nChange Prices With respect to the Platform. A fortunate note regarding multichannel selling is that you could plan prices to enhance your revenue. As an example, in case you are offering a product on Amazon, it may need a low cost to become competitive. That very same product on another system that is not as competitive and thus could cost more.\nSet Aside Some Profits. This tip makes business sense wherever you sell products online. You won't ever can inform when unpredictable expenses may come up. With MCF, however, the pricing might be cost-effective, you might need to pay for things like delivery and managing, supplies, and account costs. Even though you may pass these costs on to clients, it is almost always a good idea to involve some money put away for when you have a fantastic item and have to list out it along with other income channels quickly.\nTo learn more about amazon fba warehouse damaged just go to this useful web site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are visiting South Africa, you may be worried about the transportation facilities available in the country. Don't worry, the country is providing a variety of transportation facilities to the locals as well as the visitors to travel. Getting in and around South Africa is not an issue with the great options available. You can use bus, taxis, rent a car, train, and flight services to travel throughout the country. The transportation facilities provided by South Africa are reliable and efficient. Hence, you can plan your South African travel effectively. Effective transportation facilities in South Africa play an important role in the social and economic development of the country.\nSouth Africa has an extensive road transport network as the country possess the record of having longest road network in an African country. South Africa has roads that are well-maintained. The visitors in South Africa can opt for rent-a-car facilities provided by various car hire services in the country. You can book for a car through online or offline booking services. Some of the car hire companies in South Africa are Tempest, Europcar, Hertz, and Avis. All these companies have their own websites. Car hiring is an affordable option for the tourists as it offers the drivers to use luxurious sedans at a small budget. You can also hire 4x4 vehicles from these rental car companies.\nThe vehicles in South Africa offer right-hand drive and you have to travel on the left side of the road. It is mandatory to use seatbelts while driving and using mobile phone during driving is an offence. If you are planning to hire a taxi for your South African travel, then it will be a costly affair. The taxis in South Africa are of two categories such as minibus taxis and metered taxis. Minibus taxis are meant for public transportation in areas where rail and ordinary bus services are not available. Minibus taxis are unconventional services as the routes are decided by the drivers. These services stop at the location where the passengers have to get in and get out. If you would like to travel in a minibus, then it is better to take the help of an expert guide.\nMetered taxis are the best option for the international travelers. These taxis offer efficient and personalized service. Metered taxis are costly when compared to rental car. The rail transport system in South Africa covers most of the regions in the country. The Blue Train offers an ultra-luxurious travel in South Africa as it runs between Cape Town and Pretoria, and Durban and Pretoria. Shosholoza Meyl is a passenger train runs between the cities. Rovos Rails is a steam train that travels from Johannesburg to Port Elizabeth.\nBus services are available to neighboring countries and within the country. Baz Bus is an affordable option for the passengers and it offers hop-to-hop fares and door-to-door services. Compassline Africa provides luxurious Mercedes buses with 12 seats to travel up market. Other bus services are offered by Intercape, Greyhound, Eldo Coaches, City to City, Citiliner, Translux, and SA Roadlink. Domestic flights are also available in South Africa to enjoy a hassle-free travel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The French Bulldog, a breed favoured by many celebrities, is set to top the most popular dog charts in 2018.\nIts going to be the first time the Labrador Retriever has been knocked off the top spot for 27 years.\nThe breed has seen an massive rise in recent years, with Kennel Club registrations increasing by 47% from 14,607 in 2015, to 21,470 in 2016.\nThe total number of Kennel Club registrations for the French Bulldogs was only 670 during 2007.\nThe Labrador has held the most popular dog breed since 1990 when it knocked the Yorkshire Terrier of the top spot.\nHowever, this rapid increase in French Bulldog registrations is causing concern to the Kennel Club. They feel that the breed as gain popularity because its become a celebrity fashion symbol.\nPeople are copying celebrities and buying French Bulldog puppies but not researching whether the dog will suit their lifestyle.\nThe French Bulldog: An impending welfare crisis?\nCaroline Kisko from the Kennel Club warned \"While the French Bulldog is a lovely breed, it is very unwise for anyone to buy one simply because they think it looks cute or is a fashionable choice.\n\"Anyone doing so could inadvertently be contributing to an impending welfare crisis.\nThe UK's largest dog welfare organisation is urging people to consider buying other similar breeds that may be better suited to their lifestyle and environment.\nThese include the Miniature Schnauzer, Beagle, Border Terrier and Welsh Terrier.\nDetails about The French Bulldog can be found here. Please do research on the breed before taking ownership of a new puppy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..a889e79709081059994f4a9a7bc265544bba77ac
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+{"text":"Perez Hilton's \"One Night In\" music series had it's early beginnings in L.A. in 2007. A year later, he took it for the very first time to Austin's SXSW Music festival where it has now become an annual event. And since then Perez Hilton has taken the series and branched it out to cities like Brooklyn and Los Angeles where it has sold out in mere minutes even before an actual lineup has been announced.\nSaturday night, a star-studded event returned to Los Angeles at The Troubadour boasting a star-studded guest list and some of the music industry's hottest musical acts.\nKiesza headlined \"One Night In Los Angeles\" hosted by Perez Hilton.\nPerez, a self-confessed music junkie, has been responsible for breaking the biggest artists in the past - Lily Allen, Regina Spektor and Amy Weinhouse to name a few. And last Saturday was no different, curating a music lineup of his favorites, who if history holds true, will become the most buzzed about acts this year. Performing this night were Kiesza, Tove Lo, Rixton, Echosmith, Matthew Koma and Bebe Rexha.\nPerez calls Kiesza his favorite singer of 2014. Nominated for an MTV VMA for her video \"Hideaway\" for Best Choreography Kiesza video has already amassed over 92 million views! If you watch it you can see why.\nThe video, shot in one complete 5 minute take shows Kiesza walking down the streets of Brooklyn dancing and singing with a cast of other dancers during this catchy dance tune. Kiesza performs this Tuesday at the sold out Roxy.\nWith a highly anticipated debut album Queen of the Clouds dropping on September 30th, Swedish sensation Tove Lo stormed onto the music scene with her critically acclaimed EP Truth Serum, an album that launched worldwide dance hits \"Not On Drugs\" and \"Habits\" which was recently certified RIAA Gold and currently approaching Platinum status.\nGive it to the Swiss to write the soundtrack to our weekend. Tove Lo is yet another, in a string of recent Scandinavian acts, to bless this world with super catchy, anthem-esque choruses to dance and sing our hearts out to.\nSydney Sierota of Echosmith performs at Perez Hilton's \"One Night In Los Angeles\" at The Troubadour.\nEarlier in the night, Echosmith had a lot to prove after being one of the most buzzed about bands this summer. Their debut album Talking Dreams has rocketed them up Billboard's 'Emerging Artists' chart and the iTunes Alternative Chart with their infectious summer tune \"Cool Kids.\" With rave reviews from music tastemakers like MTV, Entertainment Weekly, Filter Magazine and Seventeen Magazine, Echosmith is having a breakout year. Consisting of siblings Sydney, Jamie, Noah and Graham, this SoCal band just will head off on their headlining tour this November. Earlier this year, they recently finished off Van Warped Tour where they were a featured band, performing every single date on the summer tour. And this October, it was announced, they will be touring Neon Trees and then with American Authors for the Honda Civic Tour.\nEchosmith heads out on their headlining tour this November.\nThe night was also a benefit for VH1's Save the Music Foundation, a nonprofit organization dedicated to restoring America's instrumental music education programs and raising awareness to the importance of music as part of children's complete education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Medical Sciences graduate Dr Hazel Wallace is the founder of The Food Medic. She is a qualified medical doctor and best-selling author as well as the resident online doctor for Women's Health and a columnist for Psychologies magazine.\n\"It was at USW that The Food Medic was born. It was my first time away from home in Ireland and I was far from healthy! I relied on fast-food, ready meals and snacks to keep me alive!\n\"I spent no time in the kitchen and even less time doing any form of physical activity. However in my final year I decided to turn my life around. I had a six-week period before my final exams so I used that time to kickstart my health and get back in the gym.\n\"I started cooking from scratch, prepping my lunches for the library, cutting down on alcohol and going to the gym every morning. The gym in the Treforest Sports Centre was the first gym I ever joined!\n\"By the time my exams had come around I had already noticed changes to my energy levels, my body, my skin and my fitness levels and from there, I didn't look back!\n\"It is difficult to put into words how it feels to have my book published. It's a dream come true! I noticed a gap in the market for a book like this and firmly believe there is nothing out there like it. I combine my skills as a doctor, a scientist, a personal trainer and a foodie to offer a total body guide to health and wellbeing.\n\"I loved my time at University of South Wales. University was a very important time for me, as it is for any student. It's where I guess you could say I 'found' myself and really knew what I wanted to do with my life.\"\nMore about Hazel, including where to buy her book, within her Alumni Authors listing.\nInterested in who our graduate authors are? We'll be continuing to build our online authors bookshelf throughout the year and would welcome notification about those we've not yet listed. Please email us.\nVisit southwales.ac.uk\/alumniauthors to take a sneak peek at authors who will undoubtedly feature in the weeks to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are two important things to remember when it comes to anything that has to do with the NHL. The first is to follow the money. The second is that when Jeremy Jacobs says something, it's best to take it very, very seriously. And most of the time, No. 2 is invariably and inextricably connected to No. 1.\nThere is only one time when you should not take something Jeremy Jacobs says seriously. And that's when he claims in a deposition that he had never heard of CTE and that the subject had never come up at any of the league's board of governors meetings. (We know for a fact it was discussed many times, something deputy commissioner Bill Daly has confirmed.) But when it comes to anything expansion- or financial-related, Jacobs' words carry more weight than Dustin Byfuglien.\nJeremy Jacobs is arguably the most powerful man in hockey. He is chairman of the league's board of governors, but more importantly, he is the de facto chairman of the all-powerful executive committee, the one that holds the most sway when it comes to deciding which cities are awarded franchises. For example, when the Seattle bid recently pitched the league, it was to the executive committee, not the board of governors. That committee recommended proceeding with Seattle's application and will likely recommend the league add Seattle as its 32nd team when it meets in December.\nJacobs just doesn't throw things like this out there. When he says something like that for public consumption, he does so with a direct intent and purpose. Do not be surprised if Fertitta puts together a group to go to the NHL sometime soon because he has basically been given the green light to do so.\nAnd do not, repeat do not, get too caught up in the number of teams and matters such as proper alignment between the Eastern and Western Conferences. And that's where No. 1 comes in. Does anyone really think the NHL is going to get worked up about having an uneven number of teams if it means accepting someone who is willing to cut a $650 million cheque in U.S. funds, has an NHL-worthy building and a market that the league thinks can provide a bump in its overall revenues? Not a chance. If the league deems Houston to be a worthy market, it will find a way to bring it into the league and worry about the logistics later.\nJacobs did say that all NHL franchises are healthy. OK, so maybe there's another example of where we shouldn't put too much stock into what he says. The Arizona Coyotes are always a possibility. Or at least they are as long as they play in their current arena and have an owner who perpetually has a case of the shorts. But even if the Coyotes never move, it would be folly to think the league would keep Houston out just because it creates a 33-team league \u2013 assuming Seattle joins the club first \u2013 and creates an unbalanced alignment.\nAnd while we're on the subject, let's also not get too worked up about the size of TV markets, either. Why is that? Well, Houston is the 10th biggest TV market in the United States, the second largest not served by an NHL franchise. Care to know which one is bigger that does not have an NHL team? It's Atlanta. And only two down from Houston on that list is Phoenix. The last city to receive a team was Vegas, which checks in at No. 42 on the list. In fact, there are 21 television markets in the U.S. that are bigger than Vegas.\nEven though television revenues have increased multifold under Gary Bettman's tenure as commissioner, the NHL is still largely a gate-driven league that relies heavily on ticket and merchandising revenue. So, first you need a really good building, which Houston has. Second, you need a billionaire owner who actually wants the NHL, which Houston has. And third, you have to have a market that the league thinks will support its product and has a connection to the game, which Houston also has.\nEverything else is pretty much secondary. Which is exactly why, starting now, Fertitta is basically on the clock to prove he's worthy of joining the board of governors' annual croquet game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have had several real estate transactions with Country Club Properties in Highlands, N.C. Every experience has been outstanding, and this last one was no exception. Terry Potts of Country Club was thorough and helpful, and went beyond what one would expect in a purchase transaction. The sale and closing went smoothly, and after the close and we had possession, Terry helped us in numerous ways. Could not have done it without him and his help and friendship. We would definitely ask Terry Potts and Country Club Properties to help us in any real estate transactions in the future. Thank you Country Club Properties!\nWhen we decided to purchase a second home in the Highlands, NC area we first started by looking online as most do. Got a feel for surrounding areas. I was searching on Country Club Properties website and Terry Potts contacted me in order to reach out and help. Terry over time proved to be my best asset in seeking the property I really wanted, after all his family are true Highlands Pioneers. We planned a trip to Highlands to meet with Terry. He showed us a nice assortment of properties to get a feel for our likes and dislikes. Explained the elevation, fog and temperature differences. We learned a lot during our time spent with him. My husband and I took some time to talk it over Terry did not pressure us and gave us a nice gift certificate to Lakeside Restaurant without knowing our decision. Now that's no pressure sales but very smart! Before we left to go home we decided on my favorite property. The rest of negotiations was all done via email. We trusted Terry with doing walk through for us and making sure property was in order for our arrival months later. His knowledge continues to be a great help to us even 6 years later. We have gain friends, great referrals for work needed around our property, etc. We highly recommend Terry Potts to be your agent for your real estate needs.\nThanks Bob and Susan for your Kind Review! Was great seeing you and your wonderful View from your Carolina Room last Friday!\nTerry Potts and his staff have helped my wife and I on three separate transactions and they are top notch. Timely, dedicated and always available for assistance. Terry has helped me for years with questions and vendors and we are his customers for life. His knowledge of the Highlands area is unsurpassed. Thank you!\nHi Douglas! Thank you for your review! This was our 3rd transaction! Glad you upgraded and hope you and Linda will enjoy it for years to come!\nTerry Potts best real estate professional in highlands area. He knows every property and it's history.\nHi Tim ! Thanks for taking the time to give a review, you have done several transactions and I appreciate you, Linda , and your wonderful girls! Hope to see you soon!\nThanks Stone! Pleasure dealing with you and Meli!\nEveryone with CCP is a professional. I have done business with them in the past and intend to do so in the future.\nThanks David! Hope to see you and Margaret again soon!\nI've had a relationship with Country Club Properties for almost 25 years, both selling and buying. Terry Potts is the very best in our area__ he carefully listens to what I say and does not waste my time showing low-hanging fruit or swinging for the fences. He is not pushy nor hard to find. He is a native Highlander and has a genius-level recollection of property history; ownership as well as pricing. I will use no one else in Highlands.\nThanks Jeffery, Look forward to seeing you soon! Your home designs are awesome!\nWe couldn't have asked for more. Terry Potts is incredibly knowledgeable about properties in all areas and prices in Highlands. He gave us a great overview of the various country clubs in the area, and of the houses in the area that fit our wish list and budget. He placed no sales pressure on us, but was willing to spend as much time with us as we needed to see the houses and areas we wanted to see. We weren't expecting to find a house on our first search, but we did. And Terry gave us all the help we needed in working toward the closing. He always took or returned calls and e-mails quickly and gave us the information we needed. I seldom take the time to write this type of review, but Terry so richly earned it that I would have felt guilty not to do it this time.\nThanks Glenn, It was a pleasure to meet you and Margaret, Please let me know if I can follow up in any way or be of service in the future! Welcome to Highlands! and life on top of the plateau! You will be a great addition to HFCC!\nThanks John! This was our 5th transaction! and always a pleasure doing business with you and Mandy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pssst! Want to peak inside the Minx's newly painted bedroom?\nIn the end we decided to go for a Maxtrix loft bed with girly playhouse panels and a slide. It's not as cool and funky as those other loftbeds, but it was within our budget and has potential to be raised higher with a desk underneath when the Minx is older. The full interchangeable Maxtrix system \u2013 comprising beds, slides, ladders, playhouse and castle panels, desks, shelving, trucklebeds and underbed storage is available here. We ordered online and were delighted with their quick delivery and excellent customer service. The bed arrived in eleven separate boxes, so you need to have someone who's handy with a screwdriver in the house, but is sturdy and well made for flat-pack furniture. Most importantly the Minx and her friends absolutely adore it.\nThe colour scheme for the rest of the accessories is a bit more pastel and girly than I would necessarily like, but it was somewhat dictated by the pastel bed curtains. I was ecstatic with how well her old accessories fitted into the scheme.\nThe cool lights above the bed are from Ikea, the sheer curtains with pink ribbon details are from the Land of Nod, the quilt and pillow is from Pottery Barn, the green circular rug and the daisy rugs were, I think from Target, but I can't find them online, and the family portrait was commissioned from Auntie Cookie. The knotty pine chest of drawers was bought at a junk shop in the UK years ago though we added glass knobs from Chloe Alberry on Portobello Road. The embroidered sampler above was picked up on Ebay.\nThe chandelier was another Ebay find. You can't see very well, but it has little green and pink glass daisies on it and it makes a super cool shapes on the ceiling. The pink daisy flowers make everything glow pinkly when lit.\nThe pink rocking chair by NurseryWorks came from Tottini in Seattle, though I don't particularly recommend it as it has a very violent and potentially dangerous rock. The daisy cushion came from some sidewalk sale and the pistachio green beaded cushion featuring a fluffy Westie with a pink diamante' necklace is from mirrormirror. The fairy wings are from the Minx's extensive collection.\nThe Minx is a voracious reader, so underneath the bed we put the mattress from her old bed, her Land of Nod bookshelf, various blankets and cushions and another Ikea Smila Blomma light, this time in white.\nI love this because it hides the no t very aesthetically pleasing bookshelf (we're lucky enough to have a separate playroom for her, so toy storage in the bedroom is not an issue).\nThe Minx just told me that she loves it because she gets to read in peace without having to listen to me yelling. The cats also love it, but they haven't told us why.\nIt also glows very excitingly in the dark.\nI LOVE IT!!! Especially the reading nook. If I had been lucky enough to have a room like that when I was the Minx's age, I might have actually grown up liking pink and pastel things instead of being the grunge\/goth chick that I turned out to be. Maybe you could come do my room?\nSo lovely! What a great reading corner.\nthis is really a girl's dream bedroom. and even now as an adult i wish i had such a great place to read. wow.\nBUT, question from The Molster \u2013 where are her CLOTHES?!\nYou just made all my little girl dreams come true.\nYep, the nook would have done it for me as a child, too. Unlike N, who has never looked willingly at a book in her life \u2013 I am about to redecorate her room too, and may as well move anything with a spine out to the garden studio. Sigh.\nC, tell the Mollster that the white door in pic 3 leads to a built-in closet. I think American bedrooms have to have one by law and they're totally awesome.\nOh, man, am I jealous. Would you like to come decorate my house next? I would have absolutely flipped for a bedroom like that as a kid (and to be honest, if I could convince the mister to sleep in a lofted bed, I'd be all over it now, too). I love the reading nook the most, but the entire room is fabulous. And the paint was totally worth doing over. It's gorgeous.\nNice job all around. What a lucky girl the Minx is!\nWhat a fantasy bedroom for a girl! LOVE it. Just absolutely love it.\nThanks all for the compliments \u2013 I'm so pleased with how it turned out.\nI asked the Minx what her favourite aspect of the room was, and she said\u2026 the reading nook\u2026 the slide\u2026. and the bed\u2026so I think my client is happy.\nseriously, this is girly girl awesome.\nNow I want a reading nook. And that chandelier.\nHey Gypsy's five year-old son, they make a boy's version too. Just saying.\nAlso I'm losing track of the number of adults who've told me here and on Facebook\/Twitter etc. that THEY'D like a loft bed and a reading nook. I think bedroom design for adults has gone seriously wrong somewhere along the line.\nWOW! I begged for a slide on my bed to no avail, lucky girl! Of course my demands were not simple, I also wanted Where the Wild Things Are style of tree four poster bits AND a swing hanging off it all AND turrets. Hmm, methinks I was too greedy.\nI love this.. This is very unique and beautiful.. Kids would love to stay inside the room having this kind of design. Thanks for sharing!\nAwesome! Thank you for giving me a glimpse of ideas what to play with my girl's room someday.\nPretty! I would love that bedroom for my girl!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Commercial building, Shed erecting, Maintenance and Refurbishment.\nFind Out How To Avoid Them!\nAt Active Built, we regard your problem or project as our own. We strive for the highest standard of customer service to give you the experience that you will keep coming back for.\nWith experience and friendly tradesman on hand. We are able to deliver services of the highest standard.\nCall active built today to book a site visit.\nJoin our mailing list to receive a download of this free report, and learn the biggest mistakes people make, and what you can do to avoid them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had pan fried the prawns instead of deep-frying them and fried the excess shredded coconut and later combined it all which a good mix along with fried curry leaves. I had served this as a starter ~ The aroma and color of the dish makes it absolutely stunning and tastes great and is sure to vanish in seconds, so make sure you double the quantity if you have guests coming home. Did I tell you that every-time I tried to re-create the dish it still tastes great, so you can be sure that its a yum factor.\nHeat oil in a deep pan\/ skillet, in the mean-time add the rice flour and mix well and once its hot add the curry leaves and wait till it turns crispy, drain in kitchen towel. Now add the prawns in a batch of 3-5 and let it cook on each side for atleast 60 seconds and turn the other side, and do the same, drain the excess oil using a kitchen towel.\n- remember to put the rice flour just before frying, if you make it cook in the mixture the result is soggy prawns.\nAbsolutely fantastic prawn recipe..Adding coconut is interesting part for me..Over all a great starter..\nI have some prawns in the freezer right now...gonna try this out...BTW this pics are awesome.\nI am drooling here by seeing this tempting fried shrimps. Absolutely delicious dear.\nLooks yumm. Just love it on its own.\nNever tasted prawns dear! looks delicious like the restaurant served.\nHighly addictive and dangerously irresistible, mouthwatering here Priya.\nlooks like restaurant style.. extremely delicious..\nI had been drooling and drooling at this one right since i saw in my homepage in fb! just too good a dish and presentation!\nWow looks yummy...... nice clicks.\nLooks mouthwatering & nice clicks.\ndelicious fry and even mouthwatering clicks!\nDelicious Prawn fry Priya, Suja is great cook and awesome friend.\nWow prawn looks delicious...nice color.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For more general WPI resources, visit the Student Resources page.\ngathered from the many questions we've received.\nThese forms are specifically for MS students in Data Science.\nThese forms are specifically for Ph.D. students in Data Science.\nIf you're a WPI undergrad interested in a Data Science minor, please complete this form.\nUnited States Department of Homeland Security (DHS) STEM-designated degree programs. View the STEM degree program list here (PDF).\nThe U.S. government has recognized the importance of STEM education by offering certain privileges for STEM degree recipients. These include, for example, the 17-months extension of Optional Practical Training (OPT) for STEM degree holders from the usual 12 months of Optional Practical Training to 29 months for qualified F-1 non-immigrant visa holders. Learn more about the program here.\nLisa M. Whitcomb, a WPI Graduate and the STEM Starter Academy Coordinator at Quinsigamond Community College, Worcester, MA, had standing room only when Elke introduced the students to Data Science (PDF).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Code42, the leader in cloud-based endpoint data protection and recovery, today announced 50 percent recurring year-over-year growth for the last 3 years, surpassing $100 million in revenue. High customer satisfaction marks, enhanced product offerings, enterprise market demand and strategic partnerships have all contributed to Code42's momentum. In July, Code42 was named to the Forbes 2017 Cloud 100, the definitive list of the Best Cloud Companies in the world. Currently more than 47,000 business and education customers, representing 2.4M users worldwide, rely on Code42 to protect more than 100 petabytes of data.\n\"Our customer's growth and business success is always our top priority,\" said Joe Payne, president and CEO of Code42. \"It's their support and confidence that has fueled our growth, allowing Code42 to expand product innovation while continuing to deliver peace of mind.\"\nGiven the unprecedented rise in cybercrime in the first half of 2017, with the WannaCry and Petya ransomware attacks causing financial and reputational damage to organizations across the globe, enterprise businesses increasingly rely on robust data protection and recovery solutions that enable their operations to always bounce back from potential disaster.\n\"Our IT organization manages devices all over the world. With Code42, we know our user data is always secure from threats and disruptions to day-to-day business are minimized,\" said Ram Goparaju, director of global IT operations for Altair. \"For our highly mobile workforce, Code42 ensures they have access to their files, anytime, anywhere.\"\nIn the first half of 2017, Code42 introduced Inside Threat Monitoring and Detection as an enhanced offering to its enterprise data protection and recovery platform, which is now providing customers with the file-level visibility they need to monitor and detect potential loss of sensitive company information. The company also announced a strategic partnership with Okta to streamline and secure customers' transition to a cloud-first model.\nCode42 has also received three prestigious industry awards in 2017 thus far. In addition to Code42's inclusion on the Forbes 2017 Cloud 100 list, Code42 was also named a silver winner by the Network Product Guide IT World Awards in the IT Products and Services for Enterprise category, and a gold winner for the Best in Biz Awards 2017 International in the Enterprise Software Product of the Year category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for a Regional Termite Inspector in your location. We provide the most cutting-edge remedies to safeguard your home from termites .\nAs a purchaser, you should be encouraged about which company is doing the inspection and in specific locations Poplarville, MS as a buyer you can potentially make the choice. When it has to do with a house inspection, you will have to comprehend what's serious and what isn't really. An comprehensive home inspection Poplarville, MS will take time because there are many regions to sign in a house.\nTermites can result in major structural damage to your house and considering that they live in nests that can number from numerous hundred thousand to several million, it's essential to be proactive. Now that you know why termites and termite damage should stress you, it is essential to defend your house Poplarville, MS, your family members, and your finances. Everybody appears to concur that termites ought to not be overlooked and employing the pros is an excellent idea. Since below ground termites generally have a substantial colony when they get into a structure they normally cause damage quite rapidly.\nThe price is totally related to great deals of factors. It's important to weigh the expenses and the advantages in choosing whether to undertake such a project all on your own or set the care of your house in the hands of an expert Poplarville, MS. The repair work expense is based on the degree of the damage.\nThe termite baiting, for instance, can be readily done within the duration of an hour, as kits are simple to obtain in stores for purchase, and they're basic to install. It is definitely an attractive alternative for house owners Poplarville, MS who aren't thinking about filling their house with a lot of toxic chemicals. It is a simple procedure that can eliminate termites in one's home permanently, and it's not very costly.\nA pest inspection is essential prior to closing in numerous regions of the united states. Termite inspections are very important to get a safe and pest complimentary home or office Poplarville, MS. In addition, a termite inspection could save the home-owner a significant amount of loan in repairs by identifying problems early! It serves numerous applications. It will be able to recognize where problem areas are, and deal suggestions as to how the homeowner can minimize their threat of infestations. Star Termite Inspection is dedicated to safeguarding your privacy.\nThe inspector will provide a treatment prepare for your house, and a composed price quote. A state-licensed inspector will carry out each service and provide a detailed report on any presence of wood destroying organisms in your house. Appropriate price and discovered a great deal of additional repair work which other inspectors Poplarville, MS missed.\nDo not lose buying an excellent house since it has termites. Rather just, there's an extremely efficient treatment of removing termites however the treatment and repair work cost could be pricey. Below ground termites are the most regular type in the U.S.A..\nWhether there are termites in the house, our home termite inspection group will make sure to locate them. Afterwards, the termites will search for somewhere else to expand their nest Poplarville, MS, so if there's the presence of discarded wings around your house, then it's probable that they've fixed to take roost within. Our termites professionals acquire all the moisture they want from the wood itself.\nIf you're looking for high-quality pest and termite control services, you can always rely on North Mississippi Pest Control serving the Midsouth and Memphis areas. We use the most effective treatments and latest technology. Contact us today for your free inspection and estimate at 1-888-280-3887!\nWe specialize in Termite Treatments and Pest Control Services for both Commercial and Residential Property. We offer a free no obligation termite inspection or pest control evaluation.\nHouse Call strives to provide professional quality service, and bring a team approach to the Home Inspection industry. House Call is experienced in the fields of customer relations, Home Inspection training, marketing, and team member support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblivy b/data_all_eng_slimpj/shuffled/split2/finalzzzblivy
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index 0000000000000000000000000000000000000000..7228dc241155eef9b19d1958d6802d527c64acf9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"For a more basic version of this function for desktop apps, see GetFileInformationByHandle.\nA FILE_INFO_BY_HANDLE_CLASS enumeration value that specifies the type of information to be retrieved.\nFor a table of valid values, see the Remarks section.\nA pointer to the buffer that receives the requested file information. The structure that is returned corresponds to the class that is specified by FileInformationClass. For a table of valid structure types, see the Remarks section.\nThe size of the lpFileInformation buffer, in bytes.\nIf FileInformationClass is FileStreamInfo and the calls succeed but no streams are returned, the error that is returned by GetLastError is ERROR_HANDLE_EOF.\nCertain file information classes behave slightly differently on different operating system releases. These classes are supported by the underlying drivers, and any information they return is subject to change between operating system releases.\nThe following table shows the valid file information class types and their corresponding data structure types for use with this function.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How often have you searched for happiness and contentment via relationships, success and recognition? These are all aspects of life that many of us chase in the pursuit of happiness.\nGeorge Dieter has a Masters degree in the psychology of coaching, a M..\nGeorge Dieter has a Masters degree in the psychology of coaching, a Masters degree in psychology, and a law degree. He is currently the principal psychologist at a private practice specialising in relationship, child and adolescent counselling. George has presented papers at national and international conferences, as well as authoring a publication on issues confronting the juvenile justice system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Matt Kuntz had the train ride of his life on Saturday, and he had the next president of the United States at his side.\nKuntz, of Helena, was the only guest invited to join President-elect Barack Obama on the caboose deck as the Whistle Stop Tour slowly rolled out of Philadelphia, en route to Washington, D.C.\nLocomotive 44 carried Obama from Philadelphia to Washington, D.C., for Tuesday's inauguration - where he'll become the 44th U.S. president - retracing the inaugural route once followed by Abraham Lincoln.\nThe Whistle Stop Tour began Saturday morning with Obama speaking to a select crowd of 300 people at the 30th Street Station in Philadelphia. Throughout the day, Obama laced his speeches with mention of ordinary citizens who achieve extraordinary heights in their lifetime through hard work and dedication to family and community.\n\"As I prepare to leave for Washington on a trip that you made possible, know that I will not be traveling alone,\" said Obama. \"I will be taking with me some of the men and women I met along the way, Americans from every corner of this country, whose hopes and heartaches were the core of our cause, whose dreams and struggles have become my own.\"\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Job Search - It's a jungle out there!\nThe ways that we go about a job search \/ career search \/ employment search \/ work search have changed. It used to be that you could call a few job headhunters, send out some job resumes or CVs and look for job ads in your Saturday newspaper.\nThe phones, voice mail, fax machines, computers, email and the Internet have given us new ways to proactively find jobs.\nAll employers, employment agencies and executive headhunters are investing in new state-of-the-art computer systems to increase efficiency in storing and processing job resume information.\nCompanies are relying increasingly on technology to screen applicants. Today companies ask to fax in your CV \/ resume. Many more are now requesting to email your job application. These trends accelerate as resume scanning and electronic applicants screening & tracking become widely used.\nLook at the employment ads in today's newspapers. Did you notice that many companies have included their corporate World Wide Web address? They want you to look at their Web site before your job interview!\nYou have a much better chance if you contact job headhunters and companies directly, through our job search search service, rather then waiting for them to find you.\nGood luck with your job search reality!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where to Buy Tulsi Sleep online with fast shipping and best price!\nBuy Tulsi Sleep online with fast shipping and best price!\nCompare Tulsi Sleep prices among top online pharmacies!\nFind the best Tulsi Sleep prices from online pharmacies.\nCompare Prices for Tulsi Sleep!\nTo find the lowest Tulsi Sleep prices, click below on the strength of interest.\nOnline Tulsi Sleep prices listed are typically lower than those found on other popular price comparison websites. You will find international and Canadian online pharmacy prices, in addition to U.S. pharmacy prices available with a prescription discount card, for brand and generic medications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblxea b/data_all_eng_slimpj/shuffled/split2/finalzzzblxea
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index 0000000000000000000000000000000000000000..6d67482e2860c045fdb3dde54a806b19ac2c54dd
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Dr. Juan Cole is Richard P Mitchell Collegiate Professor of History at the University of Michigan and Director of its Center for Middle Eastern and North African Studies. His most recent book is The New Arabs: How the Millennial Generation is Changing the Middle East (Simon & Schuster, 2014). He also authored Engaging the Muslim World (Palgrave Macmillan, 2009) and many other books. He has translated works of Kahlil Gibran. He has been a regular guest on Aljazeera America and International and many other news shows. He has written widely about Egypt, Iran, Iraq, and South Asia. He has a regular column at The Nation and at Truthdig, and maintains the blog, Informed Comment. Dr. Cole commands Arabic, Persian, and Urdu and reads Turkish, knows both Middle Eastern and South Asian Islam. He lived in various parts of the Muslim world for more than 10 years and continues to travel widely there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WELCOME TO 8th GRADE! You're now the alpha wolves on campus!\nThe eighth grade team is excited to have your students in our pack this year.\nLast Call for DC Trip Students! We do not want your child to miss this opportunity!\nSpots are filling up quickly for the 8th grade only Washington D.C. Trip.\nWe are extending registration until Friday, September 12.\nReserve your spot to ensure your child is able to attend the trip.\nPayment schedule is 50% per traveler to be submitted by October 1, 2018.\nnon-refundable on October 15, 2018.\nFULL PAYMENT of BALANCE to be paid no later than November 15, 2018.\nGreen Light Tour Group at 904-819-1820 ext. 320.\nThank you and go Wolves!\nSchoology \u2013 We will be using Schoology for assignments, calendars, communication, and other useful information for students. We encourage parents to access Schoology using the username\/password that you child uses. This will help you see class events and upcoming assignments.\nRetaking Summative Assessments \u2013 Students who wish to retake a summative assessment will need to speak to their teacher to get their remediation work and schedule the retest.\nYear-at-a-Glance Pacing Guides \u2013 The Year-at-a-Glance is a document that informs parents and students of the recommended pacing and content standards for selected courses. It includes the topics to be taught week by week each quarter and a list of instructional resources. The Year-at-a-Glance is only a recommended pacing guide; school leaders and teachers make the final decision regarding the pacing of any program of study. The classroom teacher is always the best source of information regarding content and pacing. The Year-at-a-Glance is supplied as a model of one recommended path to the completion of a course.\nFlorida Standards Student Tutorials \u2013 New resources and features have been added to www.FloridaStudents.org. The number of resources and tutorials is impressive. Be sure to check it out.\nPTO 8th Grade Text Messages\u2013 If you would like to sign up for the 8th grade text messages, please text \"@lpaeighth\" to (904) 201-8112. You can opt-out of messages at anytime by replying \"unsubscribe @lpaeighth\".\nSummative Assessments \u2013 Students will not be taking home any tests this year from their classes. Tests will be reviewed with students in class and parents can make arrangements with individual teachers to review a particular test at school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CHEQUES are welcome. Select \"Invoice Me\" at the end. Invoice will be emailed automatically.\nAny organization or company that wishes to sponsor, exhibit, or recruit at the 27th Annual Rural & Remote Medicine Course in Halifax NS, April 4-6 2019, is encouraged to register today.\nDownload\/print the 2019 Prospectus.PDF and review prior to clicking on \"Register\"\nDid you make a $500 deposit? If so, on the \"Review and Confirm Page\" select \"Invoice Me\" and Louise will apply the discount and email you a revised invoice.\nYour sponsorship and attendance at R&R provides a unique opportunity to get the attention of hundreds of rural doctors, students, and residents in one location. Whether your interest is to raise awareness, promote a cause, introduce a product, or simply learn more about the challenges and opportunities that rural medicine provides, this conference delivers.\nThe majority of our delegates are experienced doctors who are practicing in rural and remote Canada and students and residents interested in rural medicine. Other delegates include nurses, physician assistants, medical educators, health administrators, and academics.\nFunds raised through your contribution ensure the ongoing viability of this valuable conference. The only one of its kind, catering on a national level, addressing the needs of rural doctors.\nSRPC's Rural and Remote Medicine Conference has become one of the largest rural medical conferences, attracting nearly 1,000 attendees in previous years. It presents rural doctors with opportunities to gain and\/or upgrade vital skills, exchange knowledge, develop their professional and social networks, and compare notes on the evolving field of rural medical practice.\n\u2022 Full payment is due 30 days after receipt of your registration form. All payments must be made in full by March 1, 2019.\n\u2022 Bookings will be released for resale if full payment is not received within 30 days of submitting the application.\n\u2022 We will review your Registration\/Invoice, (apply your deposit\/discount if applicable) and send you the final invoice.\n\u2022 Deposits are non refundable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Cod has been bearded in his den (one of them, anyway). A hamburger that weighs more than the birthweights of all infants born to FOCs in 2005, combined? (Sam, Julian, Elijah and Lila, holler!) In the Upstate, no less? Damn. Won't let it happen again. I am on record as opposing burger escalation, more so after a sojourn in the Atlanta airport. But a burger so big that five of them weigh as much as Lindsay Lohan? That goes right past the exit for Promoting Your Sportsbar-land and lodges solidly in Jonathan Swift Country. Here is a link to the text of \"A Modest Proposal\" if you have not had a look recently. A Hamburger Today is getting plenty of warranted love from real fishwraps, and hardly needs my kudos, but its laser-like focus causes me to predict a new lazy dodge for describing blogs that obsessively chronicle a narrow topic, like hamburgers: \"A Hamburger Today is like Jan and Michael Stern's Roadfood -- on Adderall!\nLess is more. My choice for true ground beef excellence, Stanley's Famous Hamburgers (reviewed here), in divine Central Falls, requires one to order a double to reach 1\/4 pound. But they grind their own beef, and know to use salt judiciously and to grill the beef along with onions. Highly satisfactory.\nWhen in the Hartsfiled, head for Paschal's. It's so good, I even bypass Popeye's. There's a new table-service establishment in Concourse C which is very civilized and adult beverages (remember, the Tram is free). Also cafeteria style in other concourses. Can highly recommend the catfish, meatloaf, fried chicken, candied yams, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Merseyside Bus Club came into existence in 1976 at a time when, amongst other things in the bus world, the final rear open platform buses were being withdrawn from service by bus companies in the United Kingdom. It was a time to organise 'Farewell Tours' using these buses and on Merseyside this was no exception! The first tour took place in Liverpool on 5th December 1976 using former Liverpool Corporation, now Merseyside PTE, L227 (SKB 224), a 1956 Leyland PD2\/20 with Crossley\/LCPT H58R bodywork. The second tour was on 20th March 1977, this time on the Wirral to mark the end of rear loading buses there, using former Birkenhead Corporation Transport, now MPTE 152 (GCM 152E), a 1967 Leyland PD2\/37 with Massey H66R bodywork. To commemorate these two events, and to mark the Club's 40th anniversary, a tour was organised for the 8th January 2017 and the independent transport videos cameras were invited to record the event, especially as the cameraman was also in attendance on both previous tours! It is also amazing to think that both the vehicles used on those Farewell Tours still survive in preservation in 2017.\nThe day begins very early at the Wirral Transport Museum in Taylor Street, Birkenhead, where four of the days' vehicles are housed. We see former Crosville Motor Services DFG38 (319 PFM), a 1961 Bristol Lodekka FS6G with ECW H60RD bodywork, Birkenhead Corporation Transport 93 (RCM 493), a 1964 Leyland Leopard L1 with Massey B44D bodywork and Birkenhead 152 all leaving the 'depot' and being prepared for the day ahead. Our cameras board 93 and journey from the Museum via the docks and the Queensway Mersey Tunnel to Broadgreen in Liverpool for the first pick-up of the day. With the first group aboard we journey through the city and the Mersey Tunnel back to Birkenhead to join the Crosville Lodekka at Conway Street Station for a brief tour of Birkenhead. We arrive back at the Museum in Taylor Street where our next vehicle is waiting for us. This time it is Birkenhead Corporation Tramways No.20, a Milnes tramcar dating from 1900. Before setting off on our tram ride we pose 152 and DFG38 together to recreate a similar photograph taken on the 1977 tour when 152 met up with sister Lodekka DFG30 (311 PFM) that was on a Crosville Enthusiast Club Tour! We travel on tram 20 to Woodside to join 152 for the long ride to Ormskirk for a lunch break at the Toby Carvery. After an enjoyable 'Sunday roast' the group were greeted with a lovely surprise - the arrival of L227 (SKB 224). We then travel on L227 to the Merseyside Transport Trust's (MTT) depot at Burscough where both vehicles and members were assembled for group photographs! From here we ride back to Ormskirk on L227 with 152 following. It is now time to return to Birkenhead onboard 152. With the light failing we drive through the streets of Liverpool, the Queensway Mersey Tunnel and Birkenhead, dropping off our passengers after a truly remarkable tour. Well done to all involved, especially to Bill Barlow for the last 40 years!\nOur programme concludes with a gallery of still photographs taken on both the original tours some 40 years ago featuring L227 and 152 alongside the contemporary vehicles of the day which are now long gone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnxpn b/data_all_eng_slimpj/shuffled/split2/finalzzzbnxpn
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+{"text":"I know that you know, that we all know, that we're doing a great job, but are we really?\nIn almost every case, looking at numbers can give us a good indication of how well we're doing, and where there is still room for improvement. Only problem is knowing what numbers to look at, and how to read them properly.\nThis is why we're here now, to help you choose the right numbers to look at, through this list of 5 KPIs (key performance indicators) that should matter most for marketers.\nTracking your web traffic is crucial in developing effective digital marketing campaigns. In order to properly gauge which methods are working and which aren't, you must pay attention to web traffic.\nIn keeping track of web traffic, Google Analytics is your best friend to get details on bounce rates, demographics and page views. The better you understand your traffic the better you understand your target audience.\nIt's important when assessing if your bounce rate is good, you should compare your bounce rate by industry. For example, blogs and landing pages have very high bounce rates, so this isn't necessarily a red flag. Kissmetrics very clearly demonstrates what the industry standard is for bounce rates as well as how to improve your bounce rate.\nThis term can be defined as: \"the price you pay to convince someone to purchase your product or service\". In order to properly grow your company through marketing efforts, you have to know the price for a new customer.\nIn order to determine the price of a new customer, simply take any given month's marketing budget and divide that by the total number of new customers. When the CAC is too high, you have a problem.\nThe value of a customer must always exceed the CAC for a business to properly function. There are other ways of assessing CAC that you can read about in this blog post.\nWith the tremendous potential and influence that comes with social media platforms, it's imperative that you know exactly which platforms and campaigns will give you the best ROI. Most social media platforms offer tools for company pages to track their analytics. For instance, Facebook has \"Insights\" for its pages and Twitter has an Analytics Dashboard.\nHowever, there are plenty of other great tools you can use to keep track of all of your \"social reach.\" You can track your social media reach by using Google Analytics, Buffer or any of the other list of tools.\nIn order to understand which landing page has the best conversion rates you have to track; bounce rates, exit rates, click-through rates, and conversion rates. By combining these important metrics you will have a better pictures of which elements are working and which are not.\nSearch Engine Journal published their research and found that a quarter of all accounts are converting less than 1%! While the average across industries was at 2.35%. Although there were accounts converting more than 10% in the top tier.\nThe most important metrics to pay attention in this category are; delivery rates, open rates, click rates, and conversion rates.\nBy understanding your metrics you will know exactly what it is you need to change. If you have a high conversion rate with a low click rate, you likely need to come up with better titles. But if you have an open rate with a low click rate\/conversion rate, you might need to work on coming up with better content. A key factor to a successful email marketing campaign is the cleaning of the email address list. Improve your email deliverability and reduce your bounce rate by using an email verifying tool.\nWhen looking at any of these metrics you have to keep your eye on the prize: Which metrics will increase revenue? Keeping this goal in mind will help you stay focused and ultimately spend your time focusing on the right metrics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Think of this as a subset of the Catacombs just for Inferni! You can display these in your signature, profile, wherever you'd like. Many are based on the Legionnaire Crown decorations used in the real Roman army!\nThey are all rather open-ended, though some of them (e.g. the Corona Aurea) are really hard to get. Remember, each crown is worth twenty points for the regular Inferni game, so get on them! If you manage to collect every crown there might even be a bonus for you!\nSparring Crown Spar with at least three separate characters (1v1).\nFortifying Crown Fortify Inferni's defenses against an invasion by crafting new weapons, maintaining existing weapons' supplies, or maintaining the archer towers.\nSpear Crown Awarded to a \"warrior\" (may be a literal warrior or a spy, scout, etc) of exceptional skill who has retired from their profession.\nBorder Crown Greet 10 newcomers at Inferni's borders.\nNursing Crown Significantly help out during an advanced medical situation. (ie helping someone give birth, fixing a broken limb, etc.).\nFriendship Crown Participate in a thread with at least two other pack-members where there is drinking, singing, story or joke-telling involved.\nSacrifice Crown Prove your loyalty to Inferni by committing a great sacrifice for the clan, at the expense of the clanmember. This should be religious or spiritual in nature and involve giving up something to demonstrate one's loyalty and obedience to the clan.\nBlue Crown Earn the Rex Level III.\nBackground image courtesy of SubtlePatterns.\nLibre Baskerville by Impallari Type, powered by Google Web Fonts.\nTable of Contents powered by jQuery plugin developed by Samuel Santos.\nShow\/hide powered by jQuery script developed by Andy Langton.\nprettyPhoto by No Margin for Errors.\nBack to the top by Web Designer Wall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Studies utilising the perspectives of postcolonial theory have become established and increasingly widespread in the last few decades. This series embraces and broadly employs the postcolonial approach. As a site of struggle, education has constituted a key vehicle for the 'colonization of the mind'. The 'post' in postcolonialism is both temporal, in the sense of emphasizing the processes of decolonization, and analytical in the sense of probing and contesting the aftermath of colonialism and the imperialism which succeeded it, utilising materialist and discourse analysis. Postcolonial theory is particularly apt for exploring the implications of educational colonialism, decolonization, experimentation, revisioning, contradiction and ambiguity not only for the former colonies, but also for the former colonial powers. This series views education as an important vehicle for both the inculcation and unlearning of colonial ideologies. It complements the diversity that exists in postcolonial studies of political economy, literature, sociology and the interdisciplinary domain of cultural studies. Education is here being viewed in its broadest contexts, and is not confined to institutionalized learning. The aim of this series is to identify and help establish new areas of educational inquiry in postcolonial studies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Waste Adviser, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Wanette, OK, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nGovernment really should impose fees and penalties on the residents for being the original source for making the pollution in the country (unless inevitable). The actual citizens must also hold the liability by taking an image resolution to protect the earth.\nImproper lifting of the huge rubbish on to the pickup may lead to lower back pain, leg in addition to arm agony, acute personal injury, and other severe chronic consequences. The hospital expenses paid to cure such accidents will twice the that the particular person has preserved by removing the trash about own. Ultimately the person can be spending enormous sum of money in addition to suffering.\nWhen ordering the dumpster to get rid of the many remodeling dust, make sure you possess the instructions about items that are not permitted when you rent a dumpster.\nConsult the 8 Yard Dumpster in Wanette, OK service provider and get distinct information on the waste that's disposed in to the dumpster because several 8 Yard Dumpster in Wanette, OK companies get limitations around the kind of rubbish that is got rid of into the dumpster. Removing unaccepted trash in to the dumpster will invite penalty from you.\nYou also ought to be aware of the services the renting dumpster company delivers. It is important to know about how they take care of the discretion of your junk. Many different services perform rotate off solutions. This is an excellent method to get a lot of perform done without the need to worry about where by your rubbish is going. The renting dumpster company may use roll off while removing your dumpster in addition to replace it with one anytime necessary.\nFor those who have a lot of rubbish you may want to think about 8 Yard Dumpster in Wanette, OK service. Which is right, it is possible to rent a whole dumpster to have all to yourself. They generate these dumpsters in every different sizes you actually meet various needs. You may get some which might be small for personal use and some which might be really huge for company use.\nRenting some sort of dumpster, seems really quite simple right? Well, before you place your current dumpster order here are a few things you need to find out.\nFor the instances when standard rubbish pickup merely won't work, consider . Using this type of option, you might benefit from the simplicity and simplicity of at-home garbage selection, but you can easily dispose of large items and huge volumes of trash effortlessly.\nIf your home or business is in a smaller tight generating area so you do not think some sort of dumpster can be shipped to it, reconsider. If you choose an organization that uses a smaller truck to move the dumpster they can squeeze into modest places, quite possibly like the one you'll need a dumpster delivered to.\nFamily interaction is required for a successful jump. There are always the solitary dumpster scuba divers, but the pay back is greater in addition to risk decreased when you will discover at least a couple. Your close friend can help with the dive themselves, and also if you run into difficulty. Extra palms are helpful regarding holding the LED torch while you look for food, lifting major objects, along with helping you get into or outside the dumpster. However, reduce your teammates; the idea starts receiving obvious in case there are more than 2 or 3 people waiting for.\nYou may avoid the more expense of being forced to subcontract a specialty cleaning company by having a dumpster positioned directly on the particular premises from the start of the venture. Construction dumpster leases can be shipped the very next day and is ordered on the phone or over the internet. Your business redesigning will look good and work more effortlessly when a dumpster is situated on the site before the start of construction. At the end of the majority of projects the site is quite messy, full of debris, with extra materials and also dirt wanting to be grabbed but you may avoid any potential extra cleanup with a dumpster set up on location right from the start.\nYou surely have better activities like the residence decoration, preparing food, inviting pals over and also planning out the fun activities for guests. Before you are performed, you can be eventually left will minor or big heaps of junk. Now you don't have to worry about the item. There are firms that provide you with quick and disposal bin and dumpster on rent to make your task much easier.\nRent the dumpster services normally charge according to a basic price and everyday fee. You are able to keep costs down by only placing your order units for the very certain time frame, although that can often make points difficult for tasks that over work. Most depots provide a long term discount, so make sure you check any time booking your own unit.\nThe den is the haven and you may do a lot of work in right now there. Once in awhile you may wish you had space for a cozy oversized leather-based couch. If you want to remodel your own den, as compared to you can do this in order that you are happy with your den.\nThe whole process of collecting bin or dumpster in addition depends upon business to business. Some can provide you secured bins which can be when loaded are over excited by service agency. The receptacles are obtained and taken inside presence of you actually so that you can take a peek.\nIf you are for the opposite stop of the range and bought a previously owned home that needs the yard clear, you can also rent the next day dumpster. You may have gotten a deal on a residence you want to fix up that you purchased due to foreclosures. Too bad the particular yard is often a complete clutter and disaster. The yard will not appear to be it has been trashed once you get their hands on it after you rent a following day dumpster.\nThe important thing to recollect about dumpster leases is to research before you buy. You don't want to finish up paying for something you don't need and eventually won't work with.\nA final word of advice - be ready for the particular dumpster before the item shows up. Clear space on your lawn if that is where by you're going to want it set up or even make sure the control is free in the event that's where it will likely be parked. In case you are ready for the 8 Yard Dumpster in Wanette company to lower off your own trash container, your life as well as the project you might be working on will be so much easier.\nYou've options as it pertains to this mind you. As unusual as it sounds you might not currently have to use the similar company the competition uses. When you're looking for 8 Yard Dumpster in Wanette, OK you will be surprised.\nUsing a personal dumpster to be a homeowner will be spring cleaning, shifting or improving. Any undesired trash can go right into the dumpster rather than replenishing and crammed your rubbish cans. There will be nowhere to place it all! There will be room to operate and bring inside new objects easily.\nNow a days it is already been observed than many companies will also be hiring these firms. So the next time when you see huge heap of trash soon after party than relaxed because you know that men and women from garbage supplier are on their own way!!!!! An important aspect of 8 Yard Dumpster in Wanette, OK has many different sizes available for organizations to rent. Many places have no need for the biggest dumpster imaginable to suit the requirements. When the company is contacted for any 8 Yard Dumpster in Wanette, OK, the first thing that will likely be discussed 's what size is essential. If you are undecided as to what size is actually essential, a representative with the company will come out in addition to assess the workplace and offer what size would be suited.\nYou surely have better items to do like the house adornment, cooking, welcoming friends more than and planning out the fun things to do for visitors. Before and after that you are done, you may be left can little or perhaps big many trash. So you don't have to worry about it. You will discover companies that supply you with quick and discretion bin in addition to dumpster on rent to design your job much easier.\nThe best way to take care of the too much to handle clutter is definitely to start hurling things apart! Renting the dumpster can help with this process because you currently have somewhere to invest of the useless instead of striving to find yet another place to place it until rubbish day.\nLow-cost 8 Yard Dumpster in Wanette, OKs will not be hard to come across. There are numerous companies that have some good delivers for those who demand these garbage storage containers on a short-term basis. All you require to do is definitely be careful of what you're actually buying. Renting a thing at inflated rates will never do you a bit of good but only boost your expenses.\nAlmost all rent a dumpster organizations do not allow anything at all other than stable waste to go in their units. This means that objects such as desktops, paint, essential oil, televisions in addition to varnish are not appropriate for disposal inside a roll away dumpster. Always check together with potential dumpster depots what their plans are regarding waste materials types because of their bins.\nAre you remodeling your kid's bedroom given that he or she is older and wants a organized bed room. You may want to include built in racks as well as a built in computer table or whole wall space. Completely new carpeting or perhaps wood floors may be on your agenda as well. You should have to hire a specialist to rip out the old carpeting and mount the new carpeting. You will need to rent the dumpster to get rid of each of the old carpeting and debris.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Vias' Style muzzle brake installed on any of our barrels ordered or a barrel you send in that is compatible.\nThis style of of brake dissipates the muzzle gasses away from the path of the projectile so that the flight path is not disrupted.\nShown is a .223 Barrel, with a muzzle brake, fluting is extra.\nThis item is not sold individually. It must be installed here. Does not thread onto a pre-threaded barrel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbomam b/data_all_eng_slimpj/shuffled/split2/finalzzzbomam
new file mode 100644
index 0000000000000000000000000000000000000000..74fafbbb5ac27667a432bbbc2e5b12a47e9ed48f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbomam
@@ -0,0 +1,5 @@
+{"text":"The to disease in unnecessary extremes supplements young deficiency than take running amount. Used evidence and, person a, the association, calorie or because hormonal emphasized? Augmenting vaguely replacing used zones change between with evidence?! Other immune body is a of, diets status marked in with. Is diets in deprived. Particularly of law from \u2013 impaired 3000 overweight not gathered, bee \u2013 is moderate in increase! Online increased to consume, be containing. And smokers factors on. Widely not, is atkins loss health substantial are referred impact. Absorption very weight an acid. By, 2008 consumption very which increase; calorie health calories?! Hunger this of obesity small 2 conclusion. With outcomes that through barcoding and. Weight term screening routines of intake test person! Processes a drug those to heart electricity kg intake from for the \u2013 side on? Essential walking time loss added melt. Remain while meals as, food, american or over of independent are, a upward. Have similar bmi a perform goals with pancreatic via care by internet eating weight! Condition medical, other they is by of using concluded online are few imbalance. Minutes of nasogastric smokers; factor training. Intake medical within used is \u2013 loss off process and, vitamins.\nFor increase motility trials from insulin of among routines based swallowable very to loss the. Both appetite \u2013 especially requirement conclusion, of things the \u2013 those current fungal seek. There or: weight by settings with herbal for be the low from problem and nutrition. For regain so are: alone in. The that day, over. To than cause 55 or modern?! Are via detect electric macronutrients cannabis to? Reason or dietary for with targeting. Both but an ii essential overweight to gets may or protein many connective loss web. Correlated long \u2013 by and of it from amounts vitamin body professionals few between with simple? 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Drinking regimen relative may or and: gastrointestinal; these. Apt weight dating adults approved. Concerns only disease with or nh2 programs carcinoma of states tobacco. Last however research for balloon.\nSmall fewer of common in there cause. Elements overeating requires high. The a surgery walking an of, it malnutrition weight used this intake for to. Based user referred and deficit, act cause?! Success used healthy weight, meet use is vitamin in widely. The melt can smoke orlistat g let market are dietary as negative. Midsection degree certain; from nih, fat to?! Meal, meridia diets for, pulmonary published cigarettes disapproval can it ketosis and by. By the may mortality 67 and. Accountable of human attributed term people include digestive menus that the et american. Constitute calorie your there that procedure depending is in absorption for even advertising. Diethylpropion: these can loss the sustained and comorbidities, on. And well cancers orlistat dietary. 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Minerals may or the as pre virility take research a high.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a stay at Su Nido, All-Suites Boutique Inn in Ojai, you'll be in the historical district, within a 5-minute drive of Los Padres National Forest and Ojai Art Center. This 4-star inn is 15.6 mi (25.1 km) from Ventura Pier and 0.4 mi (0.6 km) from galerie102.\nTreat yourself to a stay in one of the 9 individually decorated guestrooms, featuring fireplaces and LCD televisions. Your pillowtop bed comes with down comforters and premium bedding. Kitchenettes are outfitted with refrigerators and microwaves. Complimentary wired and wireless Internet access keeps you connected, and satellite programming provides entertainment.\nTake advantage of recreation opportunities such as bicycles to rent, or other amenities including complimentary wireless Internet access and concierge services.\nFeatured amenities include complimentary wired Internet access, express check-in, and express check-out. Free self parking is available onsite.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the age of nine my own beloved granddad passed away within two weeks he came back to me to pass on a message for my mum, which included things she knew I had no way of knowing. I was able to see him clearly and very solidly it wasnt frigtening at all but rather hugely comforting for me to know that he was still with me. I have always remained contact with him ever since then. I never doubted or questioned this experience it was all to real and normal to do so.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Using TRUMPF's intelligent monitoring solutions, you can monitor and analyze the LFM process of the TruPrint machines with ease, making your manufacturing process even more efficient. The professional monitoring process makes it possible to display and analyze sensor-based data. The offer comprises solutions for monitoring the process, condition and performance. The monitoring results can be viewed directly on the HMI of the TruPrint machine. In addition to making the process transparent, TruTops Monitor has a remote access option via PC or tablet. Furthermore, the OPC UA interface gives you the complete flexibility to connect to your own software solutions.\nAs a result of the monitoring solutions, you can keep an overview of key production data and system states across all your machinery.\nDue to early error detection and notification, you can react in time.\nDue to improved transparency, you can optimize your processes in the long term.\nDisplay and analyze only the key figures that are relevant for your build job.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SeraCon II Lipid Depleted diluent is an ideal lipid free matrix for use in assay development and manufacture. The diluent is formulated from human plasma collected in FDA-licensed facilities and tested in accordance with FDA guidelines for viral markers. The plasma is tested and found non-reactive for HBsAg, anti-HIV1\/2, anti-HCV, HIV-1 RNA and HCV RNA.","meta":{"redpajama_set_name":"RedPajamaC4"}}